Definite Integral

Author

Eric W. Weisstein
May 20, 2011

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/DefiniteIntegral.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DefiniteIntegral.html.

©2011 Wolfram Research, Inc. except for portions noted otherwise

Coxeter's Integrals

(Sqrt[a-b]) (∫_0^b)(((tan^(-1))((Sqrt[x])))/((a-x) (Sqrt[b-x])))dx==π[(tan^(-1))((Sqrt[((b+1)/(a-b))]))-(tan^(-1))((Sqrt[(1/a)]))]

Numerical Check

Table[Subtract@@{(Sqrt[a-b]) NIntegrate[((ArcTan[(Sqrt[x])])/((a-x) (Sqrt[b-x]))),{x,0,b},WorkingPrecision->20,MaxRecursion->20],π(ArcTan[(Sqrt[((b+1)/(a-b))])]-ArcTan[(Sqrt[(1/a)])])}/.Thread[{a,b}->Reverse[Sort[Table[Random[Real,{0,5},20],{2}]]]],{20}]//Chop

v5.1

(Sqrt[a-b])Integrate[((ArcTan[(Sqrt[x])])/((a-x) (Sqrt[b-x]))),{x,0,b}]//Timing

(Sqrt[a-b])Integrate[((ArcTan[(Sqrt[x])])/((a-x) (Sqrt[b-x]))),{x,0,b},Assumptions->b<a]//Timing

v6

(Sqrt[a-b])Integrate[((ArcTan[(Sqrt[x])])/((a-x) (Sqrt[b-x]))),{x,0,b}]//Timing

(Sqrt[a-b])Integrate[((ArcTan[(Sqrt[x])])/((a-x) (Sqrt[b-x]))),{x,0,b},Assumptions->b<a]//Timing

(∫_a^b)(((2+(1/(Sqrt[x+3]))) (sec^(-1))(x))/((Sqrt[x+1]) (x+2)))dx

<<Graphics`Colors`

Plot[(1/Sqrt[x+3]+2)ArcSec[x]/Sqrt[x+1]/(x+2),{x,1,10},PlotStyle->Red]

4.2

{#,N[#]}&[Integrate[(1/Sqrt[x+3]+2)ArcSec[x]/Sqrt[x+1]/(x+2),{x,1,6}]]//Timing

{#,N[#]}&[Integrate[(1/Sqrt[x+3]+2)ArcSec[x]/Sqrt[x+1]/(x+2),{x,1,Infinity}]]//Timing

v5.0.0

Integrate[(1/Sqrt[x+3]+2)ArcSec[x]/Sqrt[x+1]/(x+2),{x,1,6}]//Timing

Integrate[(1/Sqrt[x+3]+2)ArcSec[x]/Sqrt[x+1]/(x+2),{x,1,Infinity}]

5.0.1

Integrate[ArcSec[x]/Sqrt[(x+1)(x+3)]/(x+2),{x,1,6}]+Integrate[2ArcSec[x]/Sqrt[x+1]/(x+2),{x,1,6}]//Timing

Integrate[(1/Sqrt[x+3])ArcSec[x]/Sqrt[x+1]/(x+2),{x,1,∞}]+Integrate[(2)ArcSec[x]/Sqrt[x+1]/(x+2),{x,1,∞}]//Timing

Integrate[(1/Sqrt[x+3]+2)ArcSec[x]/Sqrt[x+1]/(x+2),{x,1,6}]//Timing

Integrate[(1/Sqrt[x+3]+2)ArcSec[x]/Sqrt[x+1]/(x+2),{x,1,∞}]//Timing

v6

Integrate[(1/Sqrt[x+3]+2)ArcSec[x]/Sqrt[x+1]/(x+2),{x,1,6}]//Timing

Integrate[(1/Sqrt[x+3]+2)ArcSec[x]/Sqrt[x+1]/(x+2),{x,1,∞}]//Timing

Definite Integral Not Obviously Equal to 0

Plot[Log[2 Cos[x/2]],{x,-Pi,Pi}]

4.2

$Version

Integrate[Log[2 Cos[x/2]],{x,-π,π}]

5.0

$Version

Integrate[Log[2 Cos[x/2]],{x,-π,π}]

Woods' Integral (∫_0^π)log((α^2)-2 cos(x) α+1)dx

v4.2

Timing[i=Integrate[Log[1-2α Cos[x]+α^2],{x,0,Pi}]]

Table[NIntegrate[Log[1-2α Cos[x]+α^2],{x,0,Pi},WorkingPrecision->16]-2Pi Log[Abs[α]]/.α->Random[Real,{1,10}](-1)^Random[Integer],{10}]//Chop

Wrong

FullSimplify[i,α>1]

Wrong

FullSimplify[i,α<-1]

Integrate[Log[1-2α Cos[x]+α^2],{x,0,Pi},Assumptions->α>1]

FullSimplify[%,α>1]

Integrate[Log[1-2α Cos[x]+α^2],{x,0,Pi},Assumptions->α<-1]

FullSimplify[%,α<-1]

i=Integrate[TrigToExp[Log[1-2a Cos[x]+a^2]],{x,0,Pi}]

FullSimplify[i,a>1]

FullSimplify[i,a<-1]

v4.2 Dumping of internal variables

{#,FullSimplify[#]}&[Integrate[Log[1-2α Cos[x]+α^2],{x,0,Pi}]]

v5.0

$Version

Timing[i=Integrate[Log[1-2a Cos[x]+a^2],{x,0,Pi}]]

FullSimplify[i,a∈Reals]

FullSimplify[i,a>1]

FullSimplify[i,a<-1]

v5.1

(i=Integrate[Log[1-2α Cos[x]+α^2],{x,0,Pi}])//Timing

(i=Integrate[Log[1-2α Cos[x]+α^2],{x,0,Pi},Assumptions->α<-1||α>1])//Timing

FullSimplify[i,α>0]

v6

$Version

(i=Integrate[Log[1-2α Cos[x]+α^2],{x,0,Pi}])//Timing

(i=Integrate[Log[1-2α Cos[x]+α^2],{x,0,Pi},Assumptions->α<-1||α>1])//Timing

FullSimplify[i,α<-1||α>1]

(∫_0^∞)(((Sqrt[x]) (log^n)(x))/(((1-x))^n))dx

v4.2

i[n_]:=Integrate[Sqrt[x]Log[x]^n/(1-x)^n,{x,0,Infinity}]

i[2]//Timing

i[3]//Timing

i[4]//Timing

v5.0

i[n_]:=Integrate[Sqrt[x]Log[x]^n/(1-x)^n,{x,0,Infinity}]

i[2]//Timing

i[3]//Timing

i[4]//Timing

(2/(Sqrt[3])) (∫_0^1)(((log^n)(x) (tan^(-1))(((x (Sqrt[3]))/(x-2))))/(x+1))dx

(2/(Sqrt[3])) (∫_0^1)(((log^6)(x) (tan^(-1))(((x (Sqrt[3]))/(x-2))))/(x+1))dx

Bailey et al. 2006, p. 61

(2/(Sqrt[3]))Integrate[((((Log[x])^6)ArcTan[((x(Sqrt[3]))/(x-2))])/(x+1)),{x,0,1}]

(2/(Sqrt[3]))NIntegrate[((((Log[x])^6)ArcTan[((x(Sqrt[3]))/(x-2))])/(x+1)),{x,0,1},WorkingPrecision->40]

L[3,s_]:=Sum[(1/(((3n-2))^s))-(1/(((3n-1))^s)),{n,∞}]

L[3,s]

N[(1/81648)(-229635L[3,8]+29852550L[3,7]Log[3]-1632960L[3,6](π^2)+27760320L[3,5]Zeta[3]-275184L[3,4](π^4)+36288000L[3,3]Zeta[5]-30008L[3,2](π^6)-57030120L[3,1]Zeta[7]),30]

(2/(Sqrt[3])) (∫_0^1)(((log^8)(x) (tan^(-1))(((x (Sqrt[3]))/(x-2))))/(x+1))dx

p. 226

(2/(Sqrt[3]))NIntegrate[((((Log[x])^8)ArcTan[((x(Sqrt[3]))/(x-2))])/(x+1)),{x,0,1},WorkingPrecision->40]

(2/(Sqrt[3]))Integrate[((((Log[x])^8)ArcTan[((x(Sqrt[3]))/(x-2))])/(x+1)),{x,0,1}]

Glasser (∫_0^(π/2))((x^2)/((x^2)+(log^2)(2 cos(x))))dx==(1/8) π (log(2 π)-ℽ+1)

(∫_0^(π/2))((x^2)/((x^2)+(log^2)(2 cos(x))))\[DifferentialD]x==(1/8) π (log(2 π)-ℽ+1)

(π/8)(1-EulerGamma+Log[2π])//N

N[(π/8)(1-EulerGamma+Log[2π]),20]

Definite

NIntegrate[((x^2)/((x^2)+((Log[2Cos[x]])^2))),{x,0,π/2}]

Integrate[((x^2)/((x^2)+((Log[2Cos[x]])^2))),{x,0,π/2}]

Integrate[(x/(x+Log[2Cos[x]])),{x,0,π/2}]

Indefinite

Integrate[((x^2)/((x^2)+((Log[2Cos[x]])^2))),x]

y->(x^2)

(1/2)NIntegrate[((Sqrt[y])/(y+((Log[2Cos[(Sqrt[y])]])^2))),{y,0,((π^2)/4)}]

Integrate[((Sqrt[y])/(y+((Log[2Cos[(Sqrt[y])]])^2))),{y,0,((π^2)/4)}]

Integrate[((Sqrt[y])/(y+((Log[2Cos[(Sqrt[y])]])^2))),y]

y=2Cos[x]

NIntegrate[(1/2)(((ArcCos[(y/2)])^2)/(((ArcCos[(y/2)])^2)+((Log[y])^2)))(1/(Sqrt[1-((y^2)/4)])),{y,0,2}]

FullSimplify[(1/2)(((ArcCos[(y/2)])^2)/(((ArcCos[(y/2)])^2)+((Log[y])^2)))(1/(Sqrt[1-((y^2)/4)])),0<y<2]

Integrate[(((ArcCos[(y/2)])^2)/((Sqrt[4-(y^2)]) (((ArcCos[(y/2)])^2)+((Log[y])^2)))),{y,0,2}]

Integrate[(((ArcCos[(y/2)])^2)/((Sqrt[4-(y^2)]) (((ArcCos[(y/2)])^2)+((Log[y])^2)))),y]

FullSimplify[TrigToExp[(((ArcCos[(y/2)])^2)/((Sqrt[4-(y^2)]) (((ArcCos[(y/2)])^2)+((Log[y])^2))))],0<y<2]

NIntegrate[((((π+2 I Log[(1/2) (I y+(Sqrt[4-(y^2)]))]))^2)/((Sqrt[4-(y^2)]) (4 ((Log[y])^2)+(((π+2 I Log[(1/2) (I y+(Sqrt[4-(y^2)]))]))^2)))),{y,0,2}]

Integrate[((((π+2 I Log[(1/2) (I y+(Sqrt[4-(y^2)]))]))^2)/((Sqrt[4-(y^2)]) (4 ((Log[y])^2)+(((π+2 I Log[(1/2) (I y+(Sqrt[4-(y^2)]))]))^2)))),{y,0,2}]

Integrate[((((π+2 I Log[(1/2) (I y+(Sqrt[4-(y^2)]))]))^2)/((Sqrt[4-(y^2)]) (4 ((Log[y])^2)+(((π+2 I Log[(1/2) (I y+(Sqrt[4-(y^2)]))]))^2)))),y]

Factor

TrigFactor[((x^2)/((x^2)+((Log[2Cos[x]])^2)))]

NIntegrate[((x^2)/((x+I Log[2Cos[x]])(x-I Log[2Cos[x]]))),{x,0,π/2}]

Apart[((x^2)/((x+I Log[2Cos[x]])(x-I Log[2Cos[x]])))]

NIntegrate[-((I x)/(2 (-I x+Log[2 Cos[x]])))+((I x)/(2 (I x+Log[2 Cos[x]]))),{x,0,π/2}]

Integrate[((I x)/(2 (I x+Log[2 Cos[x]]))),{x,0,π/2}]

Integrate[((I x)/(2 (I x+Log[2 Cos[x]]))),x]

Integrate[((I )/(2 (I +((Log[2 Cos[x]])/x)))),{x,0,π/2}]

Series[((I )/(2 (I +((Log[2 Cos[x]])/x)))),{x,0,5}]

Plot[Im[-((I x)/(2 (-I x+Log[2 Cos[x]])))+((I x)/(2 (I x+Log[2 Cos[x]])))],{x,0,π/2}]

ComplexExpand[Re[-((I x)/(2 (-I x+Log[2 Cos[x]])))],0<x<π/2]

Plot[Im[((I )/(2 (I +((Log[2 Cos[x]])/x))))],{x,0,π/2}]

Integrate[((I )/(2 (I +((Log[2 Cos[x]])/x)))),x]

TrigToExp[(x/(2 (x-I Log[2 Cos[x]])))]//FullSimplify

Log[2]+Log[3]-Log[6]//Simplify

FullSimplify[Log[Cos[x]]]

NIntegrate[(-((I ArcCos[y/2])/(2 (-I ArcCos[y/2]+Log[y])))+((I ArcCos[y/2])/(2 (I ArcCos[y/2]+Log[y]))))(1/(2(Sqrt[1-(y^2)/4]))),{y,0,2}]

FullSimplify[(-((I ArcCos[y/2])/(2 (-I ArcCos[y/2]+Log[y]))))(1/(2(Sqrt[1-(y^2)/4])))]

Integrate[

Series[Log[2Cos[x]]^2,{x,0,8}]

SeriesTerm[Log[2Cos[x]]^2,{x,0,n}]

Borwein and Bailey "Complicated and Not Useful"

Mathematics by Experiment: Plausible Reasoning in the 21st Century, p. 61

Experimental Mathematicss in Action, Ch. 3: p. 62: "are complicated"

(∫_0^1)(((t^2) log(t))/(((t^2)-1) ((t^4)+1)))dt

v4.1

Timing[i1=Integrate[(((t^2)Log[t])/(((t^2)-1)((t^4)+1))),{t,0,1}]]

v4.2.1

Timing[i1=Integrate[(((t^2)Log[t])/(((t^2)-1)((t^4)+1))),{t,0,1}]]

FullSimplify[i1/.Zeta[n_,r_,o_]:>Zeta[n,r]]

v5.0.1

Timing[i1=Integrate[(((t^2)Log[t])/(((t^2)-1)((t^4)+1))),{t,0,1}]]

v5.2

Timing[i1=Integrate[(((t^2)Log[t])/(((t^2)-1)((t^4)+1))),{t,0,1}]]

v6

Timing[i1=Integrate[(((t^2)Log[t])/(((t^2)-1)((t^4)+1))),{t,0,1}]]

V8.0

$Version

Timing[i1=Integrate[(((t^2)Log[t])/(((t^2)-1)((t^4)+1))),{t,0,1}]]

V8.1

$Version

Timing[i1=Integrate[(((t^2)Log[t])/(((t^2)-1)((t^4)+1))),{t,0,1}]]

(∫_0^(π/4))((t^2)/((sin^2)(t)))dt

v4.1

Timing[Integrate[((t^2)/((Sin[t])^2)),{t,0,π/4}]]

v4.2.1

Timing[i2=Integrate[((t^2)/((Sin[t])^2)),{t,0,π/4}]]

FullSimplify[i2]

v5.0.1

Timing[i2=Integrate[((t^2)/((Sin[t])^2)),{t,0,π/4}]]

v6

Timing[i2=Integrate[((t^2)/((Sin[t])^2)),{t,0,π/4}]]

FullSimplify[i2]

V8.0

Timing[i2=Integrate[((t^2)/((Sin[t])^2)),{t,0,π/4}]]

V8.1

$Version

Timing[i2=Integrate[((t^2)/((Sin[t])^2)),{t,0,π/4}]]

(∫_0^π)((x sin(x))/((cos^2)(x)+1))dx

This gives machine-dependent results

v4.1

Timing[Integrate[((x Sin[x])/(1+((Cos[x])^2))),{x,0,π}]]

v4.2.1

Timing[i3=Integrate[((x Sin[x])/(1+((Cos[x])^2))),{x,0,π}]]

FullSimplify in verison 5!

FullSimplify[(1/4) (-(π^2)+2 π ArcCos[-(Sqrt[2])]+2 I π Log[1-I]-2 I π Log[1+I]-2 I ArcCos[-(Sqrt[2])] Log[-(((1/2)-(I/2))/(2^(1/4)))]+2 I ArcCos[(Sqrt[2])] Log[(((1/2)-(I/2))/(2^(1/4)))]-2 I ArcCos[-(Sqrt[2])] Log[(((1/2)+(I/2))/(2^(1/4)))]+2 I ArcCos[(Sqrt[2])] Log[(((1/2)+(I/2))/(2^(1/4)))]+2 I π Log[(1+I)-I (Sqrt[2])]-2 I ArcCos[-(Sqrt[2])] Log[(1+I)-I (Sqrt[2])]+2 Log[-(((1/2)-(I/2))/(2^(1/4)))] Log[(1+I)-I (Sqrt[2])]+2 Log[(((1/2)-(I/2))/(2^(1/4)))] Log[(1+I)-I (Sqrt[2])]-4 Log[(((1/2)+(I/2))/(2^(1/4)))] Log[(1+I)-I (Sqrt[2])]+2 I π Log[(1+I)+I (Sqrt[2])]-2 I ArcCos[(Sqrt[2])] Log[(1+I)+I (Sqrt[2])]-2 Log[-(((1/2)-(I/2))/(2^(1/4)))] Log[(1+I)+I (Sqrt[2])]-2 Log[(((1/2)-(I/2))/(2^(1/4)))] Log[(1+I)+I (Sqrt[2])]+4 Log[(((1/2)+(I/2))/(2^(1/4)))] Log[(1+I)+I (Sqrt[2])]+2 I ArcCos[(Sqrt[2])] Log[((Sqrt[2])/(-1+(Sqrt[2])))]+2 I ArcCos[(Sqrt[2])] Log[((2-(Sqrt[2]))/(-1+(Sqrt[2])))]-2 I ArcCos[(Sqrt[2])] Log[((((1/2)-(I/2)) (((-1))^(1/4)))/(Sqrt[-1+(Sqrt[2])]))]-2 I ArcCos[(Sqrt[2])] Log[-((((1/2)+(I/2)) (((-1))^(3/4)))/(Sqrt[-1+(Sqrt[2])]))]-2 Log[((((1/2)-(I/2)) (((-1))^(1/4)))/(Sqrt[-1+(Sqrt[2])]))] Log[-2 I (-1+(Sqrt[2]))]+2 Log[-((((1/2)+(I/2)) (((-1))^(3/4)))/(Sqrt[-1+(Sqrt[2])]))] Log[-2 I (-1+(Sqrt[2]))]+2 Log[((((1/2)-(I/2)) (((-1))^(1/4)))/(Sqrt[-1+(Sqrt[2])]))] Log[2 I (-1+(Sqrt[2]))]-2 Log[-((((1/2)+(I/2)) (((-1))^(3/4)))/(Sqrt[-1+(Sqrt[2])]))] Log[2 I (-1+(Sqrt[2]))]-2 I π Log[I ((-1-I)+(Sqrt[2]))]+2 I ArcCos[-(Sqrt[2])] Log[I ((-1-I)+(Sqrt[2]))]-2 Log[-(((1/2)-(I/2))/(2^(1/4)))] Log[I ((-1-I)+(Sqrt[2]))]-2 Log[(((1/2)-(I/2))/(2^(1/4)))] Log[I ((-1-I)+(Sqrt[2]))]+4 Log[(((1/2)+(I/2))/(2^(1/4)))] Log[I ((-1-I)+(Sqrt[2]))]-2 I ArcCos[(Sqrt[2])] Log[(((1-I) (Sqrt[2]))/((-1+I)+(Sqrt[2])))]+2 Log[(1+I)+I (Sqrt[2])] Log[(((1-I) (Sqrt[2]))/((-1+I)+(Sqrt[2])))]-2 I ArcCos[(Sqrt[2])] Log[-(((1+I) (-2+(Sqrt[2])))/((-1+I)+(Sqrt[2])))]-2 Log[(1+I)+I (Sqrt[2])] Log[-(((1+I) (-2+(Sqrt[2])))/((-1+I)+(Sqrt[2])))]-2 I ArcCos[-(Sqrt[2])] Log[((Sqrt[2])/(1+(Sqrt[2])))]+2 I ArcCos[-(Sqrt[2])] Log[((((1/2)+(I/2)) (((-1))^(1/4)))/(Sqrt[1+(Sqrt[2])]))]+2 I ArcCos[-(Sqrt[2])] Log[((((1/2)-(I/2)) (((-1))^(3/4)))/(Sqrt[1+(Sqrt[2])]))]+2 Log[((((1/2)+(I/2)) (((-1))^(1/4)))/(Sqrt[1+(Sqrt[2])]))] Log[-2 I (1+(Sqrt[2]))]-2 Log[((((1/2)-(I/2)) (((-1))^(3/4)))/(Sqrt[1+(Sqrt[2])]))] Log[-2 I (1+(Sqrt[2]))]-2 Log[((((1/2)+(I/2)) (((-1))^(1/4)))/(Sqrt[1+(Sqrt[2])]))] Log[2 I (1+(Sqrt[2]))]+2 Log[((((1/2)-(I/2)) (((-1))^(3/4)))/(Sqrt[1+(Sqrt[2])]))] Log[2 I (1+(Sqrt[2]))]+2 I π Log[1+(Sqrt[2])]+2 I ArcCos[-(Sqrt[2])] Log[(((1+I) (Sqrt[2]))/((1+I)+(Sqrt[2])))]-2 Log[(1+I)-I (Sqrt[2])] Log[(((1+I) (Sqrt[2]))/((1+I)+(Sqrt[2])))]+2 Log[I ((-1-I)+(Sqrt[2]))] Log[(((1+I) (Sqrt[2]))/((1+I)+(Sqrt[2])))]-2 I π Log[-I ((1+I)+(Sqrt[2]))]+2 I ArcCos[(Sqrt[2])] Log[-I ((1+I)+(Sqrt[2]))]+2 Log[-(((1/2)-(I/2))/(2^(1/4)))] Log[-I ((1+I)+(Sqrt[2]))]+2 Log[(((1/2)-(I/2))/(2^(1/4)))] Log[-I ((1+I)+(Sqrt[2]))]-4 Log[(((1/2)+(I/2))/(2^(1/4)))] Log[-I ((1+I)+(Sqrt[2]))]-2 Log[(((1-I) (Sqrt[2]))/((-1+I)+(Sqrt[2])))] Log[-I ((1+I)+(Sqrt[2]))]+2 Log[-(((1+I) (-2+(Sqrt[2])))/((-1+I)+(Sqrt[2])))] Log[-I ((1+I)+(Sqrt[2]))]+((Log[-((I ((1+I)+(Sqrt[2])))/((-1+I)+(Sqrt[2])))])^2)-2 I ArcCos[-(Sqrt[2])] Log[((2+(Sqrt[2]))/(1+(Sqrt[2])))]+2 I ArcCos[-(Sqrt[2])] Log[-(((1-I) (2+(Sqrt[2])))/((1+I)+(Sqrt[2])))]+2 Log[(1+I)-I (Sqrt[2])] Log[-(((1-I) (2+(Sqrt[2])))/((1+I)+(Sqrt[2])))]-2 Log[I ((-1-I)+(Sqrt[2]))] Log[-(((1-I) (2+(Sqrt[2])))/((1+I)+(Sqrt[2])))]-((Log[((I ((1+3 I)+(1+2 I) (Sqrt[2])))/((1+I)+(Sqrt[2])))])^2)-2 PolyLog[2,(1/(-1-(Sqrt[2])))]-2 PolyLog[2,1-(Sqrt[2])]+2 PolyLog[2,-1+(Sqrt[2])]+2 PolyLog[2,(1/(1+(Sqrt[2])))]-2 PolyLog[2,(((1+I)-I (Sqrt[2]))/((1+I)+(Sqrt[2])))]+2 PolyLog[2,-((I ((-1+I)+(Sqrt[2])))/((1+I)+(Sqrt[2])))]-2 PolyLog[2,((I ((1+I)+(Sqrt[2])))/((1+3 I)+(1+2 I) (Sqrt[2])))]+2 PolyLog[2,(((-3-I)+(2+I) (Sqrt[2]))/((-1+I)+(Sqrt[2])))])+(1/4) ((π^2)+2 π ArcCos[-(Sqrt[2])]+2 I ArcCos[-(Sqrt[2])] Log[-(((1/2)-(I/2))/(2^(1/4)))]-2 I ArcCos[(Sqrt[2])] Log[(((1/2)-(I/2))/(2^(1/4)))]+2 I ArcCos[-(Sqrt[2])] Log[(((1/2)+(I/2))/(2^(1/4)))]-2 I ArcCos[(Sqrt[2])] Log[(((1/2)+(I/2))/(2^(1/4)))]-2 I π Log[(1+I)-I (Sqrt[2])]-2 I ArcCos[-(Sqrt[2])] Log[(1+I)-I (Sqrt[2])]-2 Log[-(((1/2)-(I/2))/(2^(1/4)))] Log[(1+I)-I (Sqrt[2])]-2 Log[(((1/2)-(I/2))/(2^(1/4)))] Log[(1+I)-I (Sqrt[2])]+4 Log[(((1/2)+(I/2))/(2^(1/4)))] Log[(1+I)-I (Sqrt[2])]-2 I π Log[(1+I)+I (Sqrt[2])]-2 I ArcCos[(Sqrt[2])] Log[(1+I)+I (Sqrt[2])]+2 Log[-(((1/2)-(I/2))/(2^(1/4)))] Log[(1+I)+I (Sqrt[2])]+2 Log[(((1/2)-(I/2))/(2^(1/4)))] Log[(1+I)+I (Sqrt[2])]-4 Log[(((1/2)+(I/2))/(2^(1/4)))] Log[(1+I)+I (Sqrt[2])]-2 I ArcCos[(Sqrt[2])] Log[((Sqrt[2])/(-1+(Sqrt[2])))]-2 I ArcCos[(Sqrt[2])] Log[((2-(Sqrt[2]))/(-1+(Sqrt[2])))]+2 I ArcCos[(Sqrt[2])] Log[((((1/2)-(I/2)) (((-1))^(1/4)))/(Sqrt[-1+(Sqrt[2])]))]+2 I ArcCos[(Sqrt[2])] Log[-((((1/2)+(I/2)) (((-1))^(3/4)))/(Sqrt[-1+(Sqrt[2])]))]-2 Log[((((1/2)-(I/2)) (((-1))^(1/4)))/(Sqrt[-1+(Sqrt[2])]))] Log[-2 I (-1+(Sqrt[2]))]+2 Log[-((((1/2)+(I/2)) (((-1))^(3/4)))/(Sqrt[-1+(Sqrt[2])]))] Log[-2 I (-1+(Sqrt[2]))]+2 Log[((((1/2)-(I/2)) (((-1))^(1/4)))/(Sqrt[-1+(Sqrt[2])]))] Log[2 I (-1+(Sqrt[2]))]-2 Log[-((((1/2)+(I/2)) (((-1))^(3/4)))/(Sqrt[-1+(Sqrt[2])]))] Log[2 I (-1+(Sqrt[2]))]+2 I ArcCos[(Sqrt[2])] Log[(((1+I) (Sqrt[2]))/((-1-I)+(Sqrt[2])))]+2 Log[(1+I)+I (Sqrt[2])] Log[(((1+I) (Sqrt[2]))/((-1-I)+(Sqrt[2])))]+2 I ArcCos[(Sqrt[2])] Log[-(((1-I) (-2+(Sqrt[2])))/((-1-I)+(Sqrt[2])))]-2 Log[(1+I)+I (Sqrt[2])] Log[-(((1-I) (-2+(Sqrt[2])))/((-1-I)+(Sqrt[2])))]+2 I π Log[I ((-1-I)+(Sqrt[2]))]+2 I ArcCos[-(Sqrt[2])] Log[I ((-1-I)+(Sqrt[2]))]+2 Log[-(((1/2)-(I/2))/(2^(1/4)))] Log[I ((-1-I)+(Sqrt[2]))]+2 Log[(((1/2)-(I/2))/(2^(1/4)))] Log[I ((-1-I)+(Sqrt[2]))]-4 Log[(((1/2)+(I/2))/(2^(1/4)))] Log[I ((-1-I)+(Sqrt[2]))]+2 I ArcCos[-(Sqrt[2])] Log[((Sqrt[2])/(1+(Sqrt[2])))]-2 I ArcCos[-(Sqrt[2])] Log[((((1/2)+(I/2)) (((-1))^(1/4)))/(Sqrt[1+(Sqrt[2])]))]-2 I ArcCos[-(Sqrt[2])] Log[((((1/2)-(I/2)) (((-1))^(3/4)))/(Sqrt[1+(Sqrt[2])]))]+2 Log[((((1/2)+(I/2)) (((-1))^(1/4)))/(Sqrt[1+(Sqrt[2])]))] Log[-2 I (1+(Sqrt[2]))]-2 Log[((((1/2)-(I/2)) (((-1))^(3/4)))/(Sqrt[1+(Sqrt[2])]))] Log[-2 I (1+(Sqrt[2]))]-2 Log[((((1/2)+(I/2)) (((-1))^(1/4)))/(Sqrt[1+(Sqrt[2])]))] Log[2 I (1+(Sqrt[2]))]+2 Log[((((1/2)-(I/2)) (((-1))^(3/4)))/(Sqrt[1+(Sqrt[2])]))] Log[2 I (1+(Sqrt[2]))]+2 I π Log[1+(Sqrt[2])]-2 I ArcCos[-(Sqrt[2])] Log[(((1-I) (Sqrt[2]))/((1-I)+(Sqrt[2])))]-2 Log[(1+I)-I (Sqrt[2])] Log[(((1-I) (Sqrt[2]))/((1-I)+(Sqrt[2])))]+2 Log[I ((-1-I)+(Sqrt[2]))] Log[(((1-I) (Sqrt[2]))/((1-I)+(Sqrt[2])))]-((Log[((I ((1-I)+(Sqrt[2])))/((-1-I)+(Sqrt[2])))])^2)+2 I π Log[-I ((1+I)+(Sqrt[2]))]+2 I ArcCos[(Sqrt[2])] Log[-I ((1+I)+(Sqrt[2]))]-2 Log[-(((1/2)-(I/2))/(2^(1/4)))] Log[-I ((1+I)+(Sqrt[2]))]-2 Log[(((1/2)-(I/2))/(2^(1/4)))] Log[-I ((1+I)+(Sqrt[2]))]+4 Log[(((1/2)+(I/2))/(2^(1/4)))] Log[-I ((1+I)+(Sqrt[2]))]-2 Log[(((1+I) (Sqrt[2]))/((-1-I)+(Sqrt[2])))] Log[-I ((1+I)+(Sqrt[2]))]+2 Log[-(((1-I) (-2+(Sqrt[2])))/((-1-I)+(Sqrt[2])))] Log[-I ((1+I)+(Sqrt[2]))]+2 I ArcCos[-(Sqrt[2])] Log[((2+(Sqrt[2]))/(1+(Sqrt[2])))]-2 I ArcCos[-(Sqrt[2])] Log[-(((1+I) (2+(Sqrt[2])))/((1-I)+(Sqrt[2])))]+2 Log[(1+I)-I (Sqrt[2])] Log[-(((1+I) (2+(Sqrt[2])))/((1-I)+(Sqrt[2])))]-2 Log[I ((-1-I)+(Sqrt[2]))] Log[-(((1+I) (2+(Sqrt[2])))/((1-I)+(Sqrt[2])))]+((Log[-(((3+I)+(2+I) (Sqrt[2]))/((1-I)+(Sqrt[2])))])^2)+2 PolyLog[2,(1/(-1-(Sqrt[2])))]+2 PolyLog[2,1-(Sqrt[2])]-2 PolyLog[2,-1+(Sqrt[2])]-2 PolyLog[2,(1/(1+(Sqrt[2])))]-2 PolyLog[2,-((I ((-1-3 I)+(1+2 I) (Sqrt[2])))/((-1-I)+(Sqrt[2])))]+2 PolyLog[2,(((1-I)+(Sqrt[2]))/((3+I)+(2+I) (Sqrt[2])))])]

v5.0.1 on slow machine

Timing[i3=Integrate[((x Sin[x])/(1+((Cos[x])^2))),{x,0,π}]]

v5.2 on slow machine

Timing[i3=Integrate[((x Sin[x])/(1+((Cos[x])^2))),{x,0,π}]]

v6 on hammer

(i3=Integrate[((x Sin[x])/(1+((Cos[x])^2))),{x,0,π}])//Timing

(i3=Integrate[((x Sin[x])/(1+((Cos[x])^2))),{x,0,π}])//Timing

N[i3]//Chop

v6 on hammer again

N[i3]//Chop

NIntegrate[((x Sin[x])/(1+((Cos[x])^2))),{x,0,π}]

(π^2)/4.

V8.0

$Version

Timing[i=Integrate[((x Sin[x])/(1+((Cos[x])^2))),{x,0,π}]]

V8.1

$Version

Timing[i=Integrate[((x Sin[x])/(1+((Cos[x])^2))),{x,0,π}]]

(∫_0^(π/4))((t^3)/((sin^2)(t)))dt

v4.1

Timing[i4=Integrate[((t^3)/((Sin[t])^2)),{t,0,Pi/4}]]

v4.2.1

Timing[i4=Integrate[((t^3)/((Sin[t])^2)),{t,0,Pi/4}]]

FullSimplify[i4]

v5.0.1

Timing[i4=Integrate[((t^3)/((Sin[t])^2)),{t,0,Pi/4}]]

v5.2

<<Utilities`Simplify`

Timing[i4=Integrate[((t^3)/((Sin[t])^2)),{t,0,Pi/4}]]

i4//LogExpand//Expand

v6

<<Utilities`Simplify`

Timing[i4=Integrate[((t^3)/((Sin[t])^2)),{t,0,Pi/4}]]

i4//LogExpand//Expand

V8

$Version

Timing[i4=Integrate[((t^3)/((Sin[t])^2)),{t,0,Pi/4}]]

i4//LogExpand//Expand

V8.1

$Version

Timing[i4=Integrate[((t^3)/((Sin[t])^2)),{t,0,Pi/4}]]

Putnam Problems

1985-B5

B&B1, p. 75

Integrate[(t^(-1/2))Exp[-1985(t+1/t)],{t,0,∞}]

1987-B1

B&B1, p. 75

Integrate[((Sqrt[Log[9-x]])/(Sqrt[Log[9-x]]+Sqrt[Log[3+x]])),{x,2,4}]

NIntegrate[((Sqrt[Log[9-x]])/(Sqrt[Log[9-x]]+Sqrt[Log[3+x]])),{x,2,4}]

(∫_0^(π/2))(1/((tan^a)(x)+1))dx

v4.2

$Version

i[a_]:=Integrate[(1/(1+Tan[x]^a)),{x,0,Pi/2}]

Arbitrary a

i[a]

Integers

Table[i[a],{a,-5,5}]//Timing

Half-integers

{#,FullSimplify[#]}&[Table[i[n/2],{n,-5,5,2}]]//Timing

a=(Sqrt[n])

Table[i[Sqrt[n]],{n,5}]//Timing

Rational a

{#,FullSimplify[#]}&[i[2/3]]//Timing

{#,FullSimplify[#]}&[i[1/4]]//Timing

{#,FullSimplify[#]}&[i[6/5]]//Timing

Numerical a

i[1.2]

v5.0.1

$Version

i[a_]:=Integrate[(1/(1+Tan[x]^a)),{x,0,Pi/2}]

Arbitrary a

i[a]//Timing

Integer a

Table[i[a],{a,-5,5}]//Timing

Half-integer a

Table[i[n/2],{n,-5,5,2}]//Timing

a=(Sqrt[n])

Table[i[Sqrt[n]],{n,5}]//Timing

Rational a

{#,FullSimplify[#]}&[i[2/3]]//Timing

{#,FullSimplify[#]}&[i[1/4]]//Timing

{#,FullSimplify[#]}&[i[6/5]]//Timing

Numerical a

i[1.2]//Timing

v6

$Version

i[a_]:=Integrate[(1/(1+Tan[x]^a)),{x,0,Pi/2}]

Arbitrary a

i[a]//Timing

Integer a

Table[i[a],{a,-5,5}]//Timing

Half-integer a

Table[i[n/2],{n,-5,5,2}]//Timing

a=(Sqrt[n])

Table[i[Sqrt[n]],{n,5}]//Timing

Rational a

{#,FullSimplify[#]}&[i[2/3]]//Timing

{#,FullSimplify[#]}&[i[1/4]]//Timing

{#,FullSimplify[#]}&[i[6/5]]//Timing

Numerical a

i[1.2]//Timing

Math by Experiment, Vol. 1, p. 208

a=Integrate[Exp[-t]Sin[x t]/Sqrt[t],{t,0,∞},Assumptions->x>0]

b=Integrate[Exp[-t]Cos[x t]/Sqrt[t],{t,0,∞},Assumptions->x>0]

FullSimplify[#,x>0]&/@FunctionExpand/@{a,b}

many other PFQ identities could simplify

Experimental Math in Action, Ch. 1

<<Utilities`Simplify`

(∫_0^1)((x^2)/(((x^4)+1) (Sqrt[1-(x^4)])))dx==(π/8)

Integrate[((x^2)/((1+(x^4))(Sqrt[1-(x^4)]))),{x,0,1}]

(∫_0^∞)x (e^(-x)) (Sqrt[1-(e^(-2 x))])dx==(1/8) π (2 log(2)+1)

Integrate[x Exp[-x](Sqrt[1-Exp[-2x]]),{x,0,∞}]//LogExpand

(∫_0^∞)((x^2)/(Sqrt[(e^x)-1]))dx==4 π (log^2)(2)+((π^3)/3)

Integrate[((x^2)/(Sqrt[Exp[x]-1])),{x,0,∞}]//LogExpand

(∫_0^(π/4))x tan(x)dx==(1/8) (4 C-π log(2))

Integrate[x Tan[x],{x,0,(π/4)}]

(∫_0^(π/2))((x^2)/(1-cos(x)))dx==π log(2)-((π^2)/4)+4 C

Integrate[((x^2)/(1-Cos[x])),{x,0,(π/2)}]

(∫_0^(π/4))((π/4)-x tan(x)) tan(x)dx==(1/32) (4 π (log(2)-2)+(π^2)+16 log(2))

Integrate[((π/4)-x Tan[x])Tan[x],{x,0,(π/4)}]

(∫_0^(π/2))((x^2)/((sin^2)(x)))dx==π log(2)

Integrate[((x^2)/((Sin[x])^2)),{x,0,(π/2)}]

(∫_0^(π/2))(log^2)(cos(x))dx==(1/24) π (12 (log^2)(2)+(π^2))

Integrate[((Log[Cos[x]])^2),{x,0,(π/2)}]//LogExpand

(∫_0^1)(((log^2)(x))/((x^2)+x+1))dx==((8 (π^3))/(81 (Sqrt[3])))

Integrate[(((Log[x])^2)/((x^2)+x+1)),{x,0,1}]

(∫_0^1)((log((x^2)+1))/(x^2))dx==(π/2)-log(2)

Integrate[((Log[1+(x^2)])/(x^2)),{x,0,1}]

(∫_0^∞)((log((x^3)+1))/((x^2)-x+1))dx==((2 π log(3))/(Sqrt[3]))

Integrate[((Log[1+(x^3)])/(1-x+(x^2))),{x,0,∞}]//FullSimplify//LogExpand//Timing

(∫_0^∞)((log(x))/((cosh^2)(x)))dx==-2 log(2)+log(π)-ℽ

Integrate[((Log[x])/((Cosh[x])^2)),{x,0,∞}]//LogExpand//LogExpand

(∫_0^1)(((tan^(-1))(x))/(x (Sqrt[1-(x^2)])))dx==(1/2) π (sinh^(-1))(1)

Integrate[((ArcTan[x])/(x(Sqrt[1-(x^2)]))),{x,0,1}]

(∫_0^(π/2))(tan^(1/2))(t)dt==(π/(Sqrt[2]))

Integrate[(Sqrt[Tan[t]]),{t,0,(π/2)}]

Experimental Math in Action, Ch. 3

V?

<<Utilities`Simplify`

Integrate[((x^2)/((1+(x^4))(Sqrt[1-(x^4)]))),{x,0,1}]//Timing

Integrate[x Exp[-x](Sqrt[1-Exp[-2x]]),{x,0,∞}]//LogExpand//Expand//Timing

Integrate[((x^2)/(Sqrt[Exp[x]-1])),{x,0,∞}]//LogExpand//Expand//Timing

Integrate[x Tan[x],{x,0,π/4}]//Expand//Timing

Integrate[((x^2)/(1-Cos[x])),{x,0,π/2}]//Timing

Integrate[(π/4-x Tan[x])Tan[x],{x,0,π/4}]//Expand//Timing

Integrate[((x^2)/((Sin[x])^2)),{x,0,π/2}]//Timing

Integrate[((Log[Cos[x]])^2),{x,0,π/2}]//LogExpand//Expand//Timing

Integrate[(((Log[x])^2)/((x^2)+x+1)),{x,0,1}]//Timing

Integrate[((Log[1+(x^2)])/(x^2)),{x,0,1}]//Timing

Integrate[((Log[1+(x^3)])/(1-x+(x^2))),{x,0,∞}]//FullSimplify//LogExpand//Timing

Integrate[((Log[x])/((Cosh[x])^2)),{x,0,∞}]//LogExpand//Timing

Integrate[((ArcTan[x])/(x(Sqrt[1-(x^2)]))),{x,0,1}]//TrigToExp//Timing

Integrate[(Sqrt[Tan[t]]),{t,0,π/2}]//Timing

Experimental Math in Action, Ch. 7 (Victor Moll)

(p. 162) (∫_0^1)(1/(Sqrt[2-(x^2)]))dx==(π/4)

Integrate[(1/(Sqrt[2-(x^2)])),{x,0,1}]

(p. 162) (∫_0^1)(1/(Sqrt[9-(x^2)]))dx==(sin^(-1))((1/3))

Integrate[(1/(Sqrt[9-(x^2)])),{x,0,1}]

(∫_0^1)((log(x))/(Sqrt[x (1-(x^2))]))dx==-(1/4) (Sqrt[(π/2)]) ((Γ((1/4)))^2)

<<Utilities`Simplify`

Integrate[((Log[x])/(Sqrt[x(1-(x^2))])),{x,0,1}]

GammaSimplify[%]

(∫_0^x)(1/((t^2)+1))dt==(tan^(-1))(x)

Integrate[(1/(1+(t^2))),{t,0,x}]

(∫_0^∞)((t^(x-1))/((e^t)+1))dt==(2^(-x)) (-2+(2^x)) Γ(x) ζ(x)

Integrate[((t^(x-1))/(Exp[t]+1)),{t,0,∞}]

(∫_0^(π/2))((((x/(sin(x)))))^4)dx==(1/3) (π^3) log(2)+2 π log(2)-(3/2) π ζ(3)-((π^3)/12)

Integrate[((((x/(Sin[x]))))^4),{x,0,π/2}]

Expand[%//LogExpand]

(∫_0^(π/2))(log^n)(cos(x))dx

Integrate[Log[Cos[x]],{x,0,π/2}]

Integrate[Log[Cos[x]]^2,{x,0,π/2}]//LogExpand//Expand

Integrate[Log[Cos[x]]^3,{x,0,π/2}]//LogExpand//Expand

Integrate[Log[Cos[x]]^4,{x,0,π/2}]//LogExpand//Expand

Integrate[Log[Cos[x]]^5,{x,0,π/2}]//LogExpand//Expand

(p. 166) (∫_(π/4)^(π/2))log(log(tan(x)))dx==(π/2)log(((Γ((3/4)) (Sqrt[2 π]))/(Γ((1/4)))))

NIntegrate[Log[Log[Tan[x]]],{x,π/4,π/2}]//Chop

(π/2)Log[((Gamma[3/4])/(Gamma[1/4]))(Sqrt[2π])]//N

v6.1

Integrate[Log[Log[Tan[x]]],{x,π/2,π/4}]//Timing

(p. 166) (∫_0^∞)((log(u))/(cosh(u)))du==(1/2) π log(((4 (π^3))/((Γ((1/4)))^4)))

NIntegrate[((Log[u])/(Cosh[u])),{u,0,∞}]//Chop

v6.1

(i=Integrate[((Log[u])/(Cosh[u])),{u,0,∞}]//FullSimplify)//Timing

N[i]

(∫_0^∞)((log(x))/((cosh^2)(x)))dx==-2 log(2)+log(π)-ℽ

Integrate[((Log[x])/((Cosh[x])^2)),{x,0,∞}]//LogExpand

(∫_0^∞)((log(cosh((x/2))))/(cosh(x)))dx==C-(1/4) π log(2)

Integrate[((Log[Cosh[(x/2)]])/(Cosh[x])),{x,0,∞}]

Integrate[((Log[(E^(x/2))-(E^(-x/2))])/((E^x)+(E^(-x)))),{x,0,∞}]//Expand

(∫_0^∞)((log(cosh((x/3))))/(cosh(x)))dx==very complicated polylog expresion

NIntegrate[((Log[Cosh[(x/3)]])/(Cosh[x])),{x,0,∞}]

v5.2

Integrate[((Log[Cosh[(x/3)]])/(Cosh[x])),{x,0,∞}]//Timing

v5.2.1

(i=Integrate[((Log[Cosh[(x/3)]])/(Cosh[x])),{x,0,∞}]//FullSimplify)//Timing

N[i]//Chop

v6.0.1

(i=Integrate[((Log[Cosh[(x/3)]])/(Cosh[x])),{x,0,∞}])//FullSimplify//Timing

N[i]//Chop

v6.1

(i=Integrate[((Log[Cosh[(x/3)]])/(Cosh[x])),{x,0,∞}])//FullSimplify//Timing

N[i]//Chop

(∫_0^∞)((log(cosh(x)))/(cosh(3 x)))dx==complicated polylog expresion

NIntegrate[((Log[Cosh[x]])/(Cosh[3x])),{x,0,∞}]

v6.0.1

(i=Integrate[((Log[Cosh[x]])/(Cosh[3x])),{x,0,∞}]//FullSimplify)//Timing

v6.1

(i=Integrate[((Log[Cosh[x]])/(Cosh[3x])),{x,0,∞}]//FullSimplify)//Timing

N[i]//Chop

Simplification of polylogs

{N[#],#,#[[2]]//N,1/#[[2]]//N}&/@(expr=List@@(PolyLog[2,((1/6)+(I/6)) (3-I (Sqrt[3]))]+PolyLog[2,-((1+I)/(-I+(Sqrt[3])))]-PolyLog[2,((1-I)/(-I+(Sqrt[3])))]-PolyLog[2,-((1-I)/(I+(Sqrt[3])))]+PolyLog[2,((1+I)/(I+(Sqrt[3])))]-PolyLog[2,((1/6)+(I/6)) (-3 I+(Sqrt[3]))]+PolyLog[2,-((2-2 I)/(3 I+(Sqrt[3])))]-PolyLog[2,(-(1/6)-(I/6)) (3 I+(Sqrt[3]))]))//Chop//TableForm

Total[expr]//N

ComplexExpand[Total[expr/.PolyLog[2,x_]:>x^k],k∈Integers&&k>0]//FullSimplify

TrigReduce[I (2^(2-(k/2))) (3^(-k/2)) ((2^k) Cos[((k π)/6)]-(3^(k/2)) Cos[((k π)/3)]) Sin[((k π)/4)]]

Table[Cos[((k π)/6)] Sin[((k π)/4)],{k,12}]

Table[Cos[((k π)/3)] Sin[((k π)/4)],{k,12}]

Sum[((I (2^(2+(k/2))) (3^(-k/2)) Cos[((k π)/6)] Sin[((k π)/4)])/(k^2))-((I (2^(2-(k/2))) Cos[((k π)/3)] Sin[((k π)/4)])/(k^2)),{k,∞}]//FullSimplify

N[%//Chop

(∫_0^∞)((sinh(a x))/(sinh(b x)))dx==(π/(2 b)) tan(((a π)/(2 b)))

Integrate[((Sinh[a x])/(Sinh[b x])),{x,0,∞}]//Timing

(∫_0^∞)((sinh(a x))/(cosh(b x)))dx==(((Subscript[H, (1/4) ((a/b)-3)])-(Subscript[H, (1/4) ((a/b)-1)])+(Subscript[H, -((a+b)/(4 b))])-(Subscript[H, -((a+3 b)/(4 b))]))/(4 b))

Integrate[((Sinh[a x])/(Cosh[b x])),{x,0,∞}]//Timing

FullSimplify[((PolyGamma[0,(3/4)-(a/(4 b))]-PolyGamma[0,(1/4) (3+(a/b))]-PolyGamma[0,((-a+b)/(4 b))]+PolyGamma[0,((a+b)/(4 b))])/(4 b))]

(p. 184; G&R wrong) (∫_0^∞)(1/(((((x^2)+1))^(3/2)) (Sqrt[ϕ(x)+(Sqrt[ϕ(x)])])))dx

v6

ϕ[x_]:=1+((4(x^2))/(3(1+(x^2))))

NIntegrate[(1/((((1+(x^2)))^(3/2))(((ϕ[x]+(Sqrt[ϕ[x]])))^(1/2)))),{x,0,∞}]

Integrate[(1/((((1+(x^2)))^(3/2))(((ϕ[x]+(Sqrt[ϕ[x]])))^(1/2)))),{x,0,∞}]

FullSimplify[(1/((((1+(x^2)))^(3/2))(((ϕ[x]+(Sqrt[ϕ[x]])))^(1/2)))),x>0]

Integrate[((Sqrt[3])/((1+(x^2)) (Sqrt[3+7 (x^2)+(Sqrt[9+30 (x^2)+21 (x^4)])]))),{x,0,∞}]

(π/(2(Sqrt[6])))//N

...wrong...

(1/π)(∫_0^∞)(1/((((x^2)+1))^(m+1)))dx

(1/π)Integrate[(1/((((x^2)+1))^(m+1))),{x,0,∞}]

(1/π)Integrate[(1/((((x^2)+1))^(m+1))),{x,0,∞},Assumptions->Re[m]>-(1/2)]//FullSimplify

Subtract@@{%,(2^(-2m-1))Binomial[2m,m]}//FullSimplify

(∫_(-∞)^∞)(1/((((1-x+(e^x)))^2)+(π^2)))dx

Plot[(1/((((Exp[x]-x+1))^2)+(π^2))),{x,-20,20},PlotStyle->Red,PlotRange->All]

NIntegrate[(1/((((Exp[x]-x+1))^2)+(π^2))),{x,-∞,∞}]

NIntegrate[(1/((((Exp[x]-x+1))^2)+(π^2))),{x,0,∞},WorkingPrecision->50]

NIntegrate[(1/((((Exp[x]-x+1))^2)+(π^2))),{x,-∞,0},WorkingPrecision->50]

v5.2

Integrate[(1/((((Exp[x]-x+1))^2)+(π^2))),{x,-∞,∞}]//Timing

v6

Integrate[(1/((((Exp[x]-x+1))^2)+(π^2))),{x,-∞,∞}]//Timing

(∫_(-∞)^∞)(1/((((1-x+(e^x)))^2)+(π^2)))dx

Plot[(1/((((Exp[x]-x+1))^2)+(π^2))),{x,-20,20},PlotStyle->Red,PlotRange->All]

NIntegrate[(1/((((Exp[x]-x))^2)+(π^2))),{x,-∞,∞}]

N[(1/(1+ProductLog[1])),20]

v5.2

Integrate[(1/((((Exp[x]-x))^2)+(π^2))),{x,-∞,∞}]//Timing

v6

Integrate[(1/((((Exp[x]-x))^2)+(π^2))),{x,-∞,∞}]//Timing

(p. 185) ∫ζ(z,q)dq==((ζ(z-1,q))/(1-z))

Integrate[Zeta[z,q],q]

(p. 185) (∫_0^∞)(q^(α-1)) ζ(z,a+b q)dq==(b^(-α)) Β(α,z-α) ζ(z-α,a)

Integrate[(q^(α-1))Zeta[z,a+b q],{q,0,∞}]

(p. 185) (∫_0^1)sin(2 π q) ζ(z,q)dq==((((2 π))^z)/(4 Γ(z)))csc(((z π)/2))

Integrate[Sin[2π q]Zeta[z,q],{q,0,1}]

(p. 186) (∫_0^1)log(Γ(q))dq==(1/2) log(2 π)

Integrate[Log[Gamma[q]],{q,0,1}]

(p. 186) (∫_0^1)(log^2)(Γ(q))dq==...

Experimental Math in Action, Ch. 8

pp. 208–209

Integrate[Log[Gamma[q]]^2,{q,0,1}]

Integrate[(((Log[1-x])^2)/x),{x,0,1}]

Integrate[((PolyLog[2,x]^2)/x),{x,0,1}]

Integrate[((PolyLog[2,x]Log[1-x]Log[x])/x),{x,0,1}]

Integrate[((((Log[x])^2)((Log[1-x])^2))/(2x)),{x,0,1}]

Integrate[((PolyLog[3,x]Log[1-x])/x),{x,0,1}]

Integrate[((PolyLog[3,x]PolyLog[2,x])/x),{x,0,1}]

(∫_0^1)((log(cos(((π x)/2))))/(x (x+1)))dx==(((log^2)(2))/2)-log(2) log(π)

AMM Problem 11152, May 2005. Bailey etal 2006 p,p.259–260

NIntegrate[((Log[Cos[((π x)/2)]])/(x(1+x))),{x,0,1},WorkingPrecision->20]//Timing

N[(1/2)((Log[2])^2)-Log[2]Log[π],20]

Integrate[((Log[Cos[((π x)/2)]])/(x(1+x))),{x,0,1}]//Timing

TrigToExp[((Log[Cos[((π x)/2)]])/(x(1+x)))]

Integrate[((Log[(1/2) ((E^(-(1/2) I π x))+(E^((I π x)/2)))])/(x (1+x))),{x,0,1}]//Timing

Integrate[((Log[Cos[((π x)/2)]])/(x(1+x))),x]

(∫_0^∞)(((x^8)-4 (x^6)+9 (x^4)-5 (x^2)+1)/((x^12)-10 (x^10)+37 (x^8)-42 (x^6)+26 (x^4)-8 (x^2)+1))dx==(π/2)

AMM Problem 11148, Apr. 2005.  Bailey etal 2006, p. 258

NIntegrate[(((x^8)-4(x^6)+9(x^4)-5(x^2)+1)/((x^12)-10(x^10)+37(x^8)-42(x^6)+26(x^4)-8(x^2)+1)),{x,0,∞}]

Factor[(((x^8)-4(x^6)+9(x^4)-5(x^2)+1)/((x^12)-10(x^10)+37(x^8)-42(x^6)+26(x^4)-8(x^2)+1))]

v5

(i=Integrate[(((x^8)-4(x^6)+9(x^4)-5(x^2)+1)/((x^12)-10(x^10)+37(x^8)-42(x^6)+26(x^4)-8(x^2)+1)),{x,0,∞}])//Timing

RootReduce[i]

v6

(i=Integrate[(((x^8)-4(x^6)+9(x^4)-5(x^2)+1)/((x^12)-10(x^10)+37(x^8)-42(x^6)+26(x^4)-8(x^2)+1)),{x,0,∞}])//Timing

RootReduce[i]//Timing

(∫_0^1)(((tan^(-1))((Sqrt[(a^2)+(x^2)])))/((Sqrt[(a^2)+(x^2)]) ((x^2)+1)))dx

Borwein et al. (Experimental Math in Action, Ch. 1) p. 42

(∫_0^1)(((tan^(-1))((Sqrt[(a^2)+(x^2)])))/((Sqrt[(a^2)+(x^2)]) ((x^2)+1)))dx

Integrate[((ArcTan[(Sqrt[(x^2)+(a^2)])])/((Sqrt[(x^2)+(a^2)])((x^2)+1))),{x,0,1}]//Timing

a==0

With[{a=0},Integrate[((ArcTan[(Sqrt[(x^2)+(a^2)])])/((Sqrt[(x^2)+(a^2)])((x^2)+1))),{x,0,1}]]

a==1

With[{a=1},Integrate[((ArcTan[(Sqrt[(x^2)+(a^2)])])/((Sqrt[(x^2)+(a^2)])((x^2)+1))),{x,0,1}]]//FullSimplify

%19 /. ArcCot[x_]:>Pi/2-ArcTan[(Sqrt[2])]//FullSimplify

a==(Sqrt[2])

With[{a=(Sqrt[2])},Integrate[((ArcTan[(Sqrt[(x^2)+(a^2)])])/((Sqrt[(x^2)+(a^2)])((x^2)+1))),{x,0,1}]]

{With[{a=(Sqrt[2])},NIntegrate[((ArcTan[(Sqrt[(x^2)+(a^2)])])/((Sqrt[(x^2)+(a^2)])((x^2)+1))),{x,0,1}]],5(π^2)/96}//N

a==(Sqrt[3])

With[{a=(Sqrt[3])},Integrate[((ArcTan[(Sqrt[(x^2)+(a^2)])])/((Sqrt[(x^2)+(a^2)])((x^2)+1))),{x,0,1}]]//Timing

a==2

With[{a=2},Integrate[((ArcTan[(Sqrt[(x^2)+(a^2)])])/((Sqrt[(x^2)+(a^2)])((x^2)+1))),{x,0,1}]]//Timing

(∫_0^∞)(((tan^(-1))((Sqrt[(a^2)+(x^2)])))/((Sqrt[(a^2)+(x^2)]) ((x^2)+1)))dx

Integrate[((ArcTan[(Sqrt[(x^2)+(a^2)])])/((Sqrt[(x^2)+(a^2)])((x^2)+1))),{x,0,∞}]

(π/(2 (Sqrt[(a^2)-1]))) (2 (tan^(-1))((Sqrt[(a^2)-1]))-(tan^(-1))((Sqrt[(a^4)-1])))

(∫_0^∞)(t^(7/4)) (e^(-t))dt==((21 π (Sqrt[2]))/(16 Γ((1/4))))

Bailey and Plouffe

Integrate[(t^(7/4))Exp[-t],{t,0,∞}]

FullSimplify[Subtract@@{Gamma[11/4],21Pi Sqrt[2]/16/Gamma[1/4]}]

Whittaker and Watson Examples

<<Utilities`Typesetting`

Whittaker and Watson pp. 112–122

(∫_0^(2 π))((dθ)/((p^2)-2p  cos θ p+1))

Integrate[(1/(1-2p Cos[θ]+(p^2))),{θ,0,2π},Assumptions->0<p<1]//Timing

Integrate[(1/(1-2p Cos[θ]+(p^2))),{θ,0,2π}]//Timing

(∫_0^(2 π))(((cos^2)(3 θ))/((p^2)-2p cos(2 θ)+1))dθ

Integrate[(((Cos[3θ])^2)/(1-2p Cos[2θ]+(p^2))),{θ,0,2π},Assumptions->0<p<1]//Timing

Integrate[(((Cos[3θ])^2)/(1-2p Cos[2θ]+(p^2))),{θ,0,2π}]//Timing

(∫_0^(2 π))(e^(cosθ)) cos(n θ-sinθ)dθ

Integrate[Exp[Cos[θ]]Cos[n θ-Sin[θ]],{θ,0,2π},Assumptions->n>0&&n∈Integers]//Timing

Should be 2 π/n!

Table[Integrate[Exp[Cos[θ]]Cos[n θ-Sin[θ]],{θ,0,2π}],{n,5}]

NIntegrate[Exp[Cos[θ]]Cos[θ-Sin[θ]],{θ,0,2π}]

Integrate[Exp[Cos[θ]]Cos[θ-Sin[θ]],{θ,0,2π}]

Integrate[Exp[Cos[θ]]Cos[2θ-Sin[θ]],{θ,0,2π}]

(∫_0^(2 π))(e^(cosθ)) sin(n θ-sinθ)dθ

Integrate[Exp[Cos[θ]]Sin[n θ-Sin[θ]],{θ,0,2π},Assumptions->n>0&&n∈Integers]//Timing

Should be 0

Table[Integrate[Exp[Cos[θ]]Sin[n θ-Sin[θ]],{θ,0,2π}],{n,2}]//Timing

(∫_0^(2 π))((dθ)/(((a+b (cosθ)))^2))

Integrate[(1/(((a+b Cos[θ]))^2)),{θ,0,2π},Assumptions->0<b<a]//Timing

Integrate[(1/(((a+b Cos[θ]))^2)),{θ,0,2π}]//Timing

(∫_0^(2 π))((dθ)/(((b (cos^2)(θ)+a))^2))

Integrate[(1/(((a+b ((Cos[θ])^2)))^2)),{θ,0,2π},Assumptions->0<b<a]//Timing

Integrate[(1/(((a+b ((Cos[θ])^2)))^2)),{θ,0,2π}]//Timing

(∫_(-∞)^∞)((dx)/((((x^2)+1))^3))

Integrate[(1/((((x^2)+1))^3)),{x,-∞,∞}]

(∫_(-∞)^∞)(((x^4)dx)/(((b (x^2)+a))^4))

Integrate[((x^4)/(((a+b (x^2)))^4)),{x,-∞,∞},Assumptions->a>0&&b>0]

Integrate[((x^4)/(((a+b (x^2)))^4)),{x,-∞,∞}]//Timing

(∫_0^∞)((cos x dx)/((a^2)+(x^2)))

Integrate[((Cos[x])/((x^2)+(a^2))),{x,0,∞},Assumptions->a>0]

Integrate[((Cos[x])/((x^2)+(a^2))),{x,0,∞}]//Timing

(∫_0^∞)((cos(2 a x)-cos(2 b x))/(x^2))dx

Integrate[((Cos[2a x]-Cos[2b x])/(x^2)),{x,0,∞},Assumptions->a≥0&&b≥0]

Integrate[((Cos[2a x]-Cos[2b x])/(x^2)),{x,0,∞}]//Timing

(∫_0^∞)(((3 (x^2)-(a^2)) cos(m x))/((((b^2)+(x^2)))^2))dx

Integrate[((3(x^2)-(a^2))/((((x^2)+(b^2)))^2))Cos[m x],{x,0,∞},Assumptions->b>0&&m≥0]//Factor//Timing

Integrate[((3(x^2)-(a^2))/((((x^2)+(b^2)))^2))Cos[m x],{x,0,∞}]//Timing

(∫_0^∞)((x sin(a x))/((k^2)+(x^2)))dx

Integrate[((x Sin[a x])/((x^2)+(k^2))),{x,0,∞},Assumptions->k>0&&a>0]//Timing

Integrate[((x Sin[a x])/((x^2)+(k^2))),{x,0,∞}]//Timing

(∫_0^∞)((sin(m x))/(x ((((a^2)+(x^2)))^2)))dx

Integrate[((Sin[m x])/(x((((x^2)+(a^2)))^2))),{x,0,∞},Assumptions->m≥0&&a>0]//Apart//Timing

Integrate[((Sin[m x])/(x((((x^2)+(a^2)))^2))),{x,0,∞}]//Timing

(∫_0^∞)(((e^(-t))-(e^(-t x)))/t)dt

v5

Integrate[((Exp[-t]-Exp[-t z])/t),{t,0,∞},Assumptions->Re[z]>0]//Timing

v6

Integrate[((Exp[-t]-Exp[-t z])/t),{t,0,∞},Assumptions->Re[z]>0]//Timing

f[z_?NumericQ]:= N[Floor[((π+I π (Sqrt[-(1/z)]) (Sqrt[z])-Arg[(1/z)])/(2 π))]-Floor[(1/2) (1+((I (Sqrt[-z]))/(Sqrt[z]))-((Arg[z])/π))]]

f[z_?NumericQ,prec_]:= N[Floor[((π+I π (Sqrt[-(1/z)]) (Sqrt[z])-Arg[(1/z)])/(2 π))]-Floor[(1/2) (1+((I (Sqrt[-z]))/(Sqrt[z]))-((Arg[z])/π))],prec]

Plot3D[Evaluate[Re[I π Floor[((π+I π (Sqrt[-(1/z)]) (Sqrt[z])-Arg[(1/z)])/(2 π))]-I π Floor[(1/2) (1+((I (Sqrt[-z]))/(Sqrt[z]))-((Arg[z])/π))]/.{z->x+I y}]],{x,-5,5},{y,-5,5},PlotPoints->100,Mesh->False]

Integrate[((Exp[-t]-Exp[-t z])/t),{t,0,∞}]//Timing

(∫_0^∞)((x^(a-1))/(x+1))dx

Integrate[((x^(a-1))/(1+x)),{x,0,∞},Assumptions->0<a<1]//Timing

Integrate[((x^(a-1))/(1+x)),{x,0,∞}]//Timing

(∫_0^∞)((x^(a-1))/(1-x))dx

Integrate[((x^(a-1))/(1-x)),{x,0,∞},Assumptions->0<a<1,PrincipalValue->True]//Timing

Integrate[((x^(a-1))/(1-x)),{x,0,∞},PrincipalValue->True]//Timing

(∫_0^∞)((t^(z-1))/(t+(e^(i a))))dt

Integrate[((t^(z-1))/(t+Exp[I a])),{t,0,∞},Assumptions->0<z<1&&-Pi<a<Pi]//Timing

f[z_?NumericQ,a_]:=NIntegrate[((t^(z-1))/(t+Exp[I a])),{t,0,∞}]-(E^(I a (-1+z))) π Csc[π z]

Plot3D[Evaluate[Re[f[x,a]]],{x,0,1},{a,-Pi,Pi},PlotRange->All,PlotPoints->50,Mesh->False]//Timing

Integrate[((t^(z-1))/(t+Exp[I a])),{t,0,∞}]//Timing

(∫_0^∞)((x^z)/((((x^2)+1))^2))dx

Integrate[((x^z)/(((1+(x^2)))^2)),{x,0,∞},Assumptions->-1<z<3]

Integrate[((x^z)/(((1+(x^2)))^2)),{x,0,∞}]

(∫_0^∞)((x^(-p))/((x^2)+2 cos(λ) x+1))dx

Integrate[((x^(-p))/(1+2x Cos[λ]+(x^2))),{x,0,∞},Assumptions->-1<p<1&&-π<λ<π]//Timing

Integrate[((x^(-p))/(1+2x Cos[λ]+(x^2))),{x,0,∞},Assumptions->-1<p<1&&-π/2<λ<π/2]//Timing

Integrate[((x^(-p))/(1+2x Cos[λ]+(x^2))),{x,0,∞}]//Timing

(∫_0^∞)((sin(a x))/((e^(2 π x))-1))dx

Integrate[((Sin[a x])/(Exp[2π x]-1)),{x,0,∞}]//Timing

Simplify[Subtract@@{((-2+a Coth[(a/2)])/(4 a)),(1/4)((Exp[a]+1)/(Exp[a]-1))-(1/(2a))}]

(∫_0^∞)(((e^(a cos(b x))) sin(a sin(b x)))/x)dx

Integrate[Exp[a Cos[b x]]Sin[a Sin[b x]]/x,{x,0,∞},Assumptions->a>0&&b>0]//Timing

Integrate[Exp[a Cos[b x]]Sin[a Sin[b x]]/x,{x,0,∞}]//Timing

Actual answer is π((e^a)-1)/2

(∫_0^(π/2))((a sin(2 x))/((a^2)-2 cos(2 x) a+1))dx

(int=Integrate[((a Sin[2x]x)/(1-2a Cos[2x]+(a^2))),{x,0,π/2},Assumptions->a∈Reals])//Timing

FullSimplify[int,0<a<1]

FullSimplify[int,-1<a<0]

FullSimplify[int,a==0]

Integrate[((a Sin[2x]x)/(1-2a Cos[2x]+(a^2))),{x,0,π/2},Assumptions->Abs[a]>1]//Timing

Integrate[((a Sin[2x]x)/(1-2a Cos[2x]+(a^2))),{x,0,π/2},Assumptions->a^2>1]//Timing

FullSimplify[-(1/16) I ((π^2)+2 I π Log[1+(1/a)]+2 I π Log[1+a]+2 PolyLog[2,-(1/a)]-2 PolyLog[2,(1/a)]+2 PolyLog[2,-a]-2 PolyLog[2,a]),a^2>1]

Integrate[((a Sin[2x]x)/(1-2a Cos[2x]+(a^2))),{x,0,π/2},Assumptions->a<-1]//Timing

Integrate[((a Sin[2x]x)/(1-2a Cos[2x]+(a^2))),{x,0,π/2},Assumptions->a>1]//Timing

...wrong...

Integrate[((a Sin[2x]x)/(1-2a Cos[2x]+(a^2))),{x,0,π/2}]//Timing

(∫_(-∞)^∞)(1/(((b^2)+(x^2)) ((((a^2)+(x^2)))^2)))dx

Integrate[(1/(((x^2)+(b^2))((((x^2)+(a^2)))^2))),{x,-∞,∞}]//Timing

(∫_0^∞)(1/(((b (x^2)+a))^n))dx

Integrate[(1/(((a+b (x^2)))^n)),{x,0,∞},Assumptions->(a|b)∈Reals]//Timing

Integrate[(1/(((a+b (x^2)))^n)),{x,0,∞}]//Timing

(∫_0^∞)(((sin^n)(m t))/(t^n))dt

Integrate[(((Sin[m t])^n)/(t^n)),{t,0,∞},Assumptions->m>0]//Timing

Integrate[(((Sin[m t])^n)/(t^n)),{t,0,∞}]//Timing

((π (m^(n-1)))/((2^n)(n-1)!)) Sum[((Pochhammer[-n,k])/(k!))(((n-2k))^(n-1)),{k,0,∞}]

Table[((π (m^(n-1)))/((2^n)(n-1)!)) Sum[((Pochhammer[-n,k])/(k!))(((n-2k))^(n-1)),{k,0,∞}],{m,3},{n,3}]

(∫_0^∞)(((x^(a-1)) sin(((a π)/2)-b x) r)/((r^2)+(x^2)))dx

Integrate[(x^(a-1))Sin[(1/2)a π-b x] (r/((x^2)+(r^2))),{x,0,∞},Assumptions->0<a<2&&b>0&&r>0]//Timing

Integrate[(x^(a-1))Sin[(1/2)a π-b x] (r/((x^2)+(r^2))),{x,0,∞}]//Timing

FullSimplify[r(1/2) ((b^2) ((Abs[b])^(-a)) Gamma[-2+a] HypergeometricPFQ[{1},{(3/2)-(a/2),2-(a/2)},(((b^2) (r^2))/4)] (-1+Sign[b]) Sin[a π]+π ((((1/(r^2))))^(1-(a/2))) (Cosh[(Sqrt[(r^2)]) Abs[b]]-Sign[b] Sinh[(Sqrt[(r^2)]) Abs[b]])),0<a<2&&b>0&&r>0]

Arfken pp. 416-421

(∫_0^(2 π))(1/(a±b cos(θ)))dθ

v6

Integrate[(1/(a+b Cos[θ])),{θ,0,2π}]//Timing

Reduce[b≠a&&a+b≠0&&(Re[(a/b)]≥1||1+Re[(a/b)]≤0||(Im[(a/b)]≠0&&Im[((a+b)/b)]≠0))&&a>b]

FullSimplify[((2 (-Log[((a-b)/(Sqrt[-(a^2)+(b^2)]))]+Log[((Sqrt[-(a^2)+(b^2)])/(a+b))]))/(Sqrt[-(a^2)+(b^2)])),a>0&&(-a<b<0||0<b<a)]

Simplify[%]

Integrate[(1/(a+b Cos[θ])),{θ,0,2π},Assumptions->(a|b)∈Reals]

FullSimplify[((2 I (Log[-2 I (a-b)]-Log[2 I (a-b)]))/(Sqrt[(a-b) (a+b)])),a>b]

Integrate[(1/(a+b Cos[θ])),{θ,0,2π},Assumptions->a>b||a>-b]//Timing

Integrate[(1/(a-b Cos[θ])),{θ,0,2π},Assumptions->a>b]//Timing

Integrate[(1/(a+b Sin[θ])),{θ,0,2π},Assumptions->a>b]//Timing

Integrate[(1/(a-b Sin[θ])),{θ,0,2π},Assumptions->a>b]//Timing

(∫_0^π)(1/(((a+cos(θ)))^2))dθ

Integrate[(1/(((a+Cos[θ]))^2)),{θ,0,π}]//Timing

Integrate[(1/(((a+Cos[θ]))^2)),{θ,0,π},Assumptions->a>1]//Timing

(∫_0^(2 π))(1/((t^2)-2 cos(θ) t+1))dθ

Integrate[(1/(1-2t Cos[θ]+(t^2))),{θ,0,2π}]

Integrate[(1/(1-2t Cos[θ]+(t^2))),{θ,0,2π},Assumptions->-1<t<1]

Integrate[(1/(1-2t Cos[θ]+(t^2))),{θ,0,2π},Assumptions->t==0]

(∫_0^π)(cos^(2 n))(θ)dθ

Integrate[((Cos[θ])^(2n)),{θ,0,π}]//Timing

Integrate[((Cos[θ])^(2n)),{θ,0,π},Assumptions->n∈Integers]//FullSimplify//Timing

Table[(((4^n) (π^2))/(((Gamma[(1/2)-n])^2) Gamma[1+2 n])),{n,4}]

Table[Integrate[((Cos[θ])^(2n)),{θ,0,π}],{n,4}]

Table[π(((2n)!)/((2^(2n))(((n!))^2))),{n,4}]

Table[π(((2n-1)!!)/((2n)!!)),{n,4}]

FullSimplify[FunctionExpand[Subtract@@{(((4^n) (π^2))/(((Gamma[(1/2)-n])^2) Gamma[1+2 n])),π(((2n-1)!!)/((2n)!!))}],n∈Integers]

Table[%,{n,4}]

(∫_(-∞)^∞)((cos(b x)-cos(a x))/(x^2))dx

Integrate[((Cos[b x]-Cos[a x])/(x^2)),{x,-∞,∞}]//Timing

Integrate[((Cos[b x]-Cos[a x])/(x^2)),{x,-∞,∞},Assumptions->a>b>0]//Timing

(∫_0^∞)(((sin^2)(x))/(x^2))dx

Integrate[(((Sin[x])^2)/(x^2)),{x,0,∞}]

(∫_(-∞)^∞)((cos(k x))/((a^2)+(x^2)))dx

Integrate[((Cos[k x])/((x^2)+(a^2))),{x,-∞,∞}]//Timing

Integrate[((Cos[k x])/((x^2)+(a^2))),{x,-∞,∞},Assumptions->a>0&&k>0]//Timing

(∫_(-∞)^∞)((x sin(k x))/((a^2)+(x^2)))dx

Integrate[((x Sin[k x])/((x^2)+(a^2))),{x,-∞,∞}]//Timing

Integrate[((x Sin[k x])/((x^2)+(a^2))),{x,-∞,∞},Assumptions->a>0&&k>0]//Timing

(∫_(-∞)^∞)((sin(x))/x)dx

Integrate[((Sin[x])/x),{x,-∞,∞}]//Timing

(∫_(-∞)^∞)((sin(t) (e^(i p t)))/t)dt

Integrate[((Sin[t])/t)Exp[I p t],{t,-∞,∞}]//Timing

Integrate[(((E^(I p t)) Sin[t])/t),{t,-∞,∞},Assumptions->-1≤Re[p]≤1||Im[p]≠0]

Integrate[(((E^(I p t)) Sin[t])/t),{t,-∞,∞},Assumptions->Im[p]==0&&(Re[p]==-1||Re[p]==1)]

(∫_0^∞)(((log^2)(x))/((x^2)+1))dx

Integrate[(((Log[x])^2)/(1+(x^2))),{x,0,∞}]//Timing

(∫_0^∞)((x^a)/(((x+1))^2))dx

Integrate[((x^a)/(((x+1))^2)),{x,0,∞}]//Timing

Integrate[((x^a)/(((x+1))^2)),{x,0,∞},Assumptions->-1<a<1]

(∫_0^∞)((x^(-a))/(x+1))dx

Integrate[((x^(-a))/(x+1)),{x,0,∞}]//Timing

(∫_0^∞)(1/((((a^2)+(x^2)))^2))dx

Integrate[(1/((((x^2)+(a^2)))^2)),{x,0,∞}]//Timing

(∫_(-∞)^∞)((x^2)/((x^4)+1))dx

Integrate[((x^2)/(1+(x^4))),{x,-∞,∞}]//Timing

(∫_0^∞)cos((t^2))dt

Integrate[Cos[(t^2)],{t,0,∞}]//Timing

Integrate[Sin[(t^2)],{t,0,∞}]//Timing

(∫_0^∞)(1/((x^n)+1))dx

Integrate[(1/(1+(x^n))),{x,0,∞}]//Timing

(∫_(-∞)^∞)(1/((x^4)-2 cos(2 θ) (x^2)+1))dx

v4.2

(expr=FullSimplify[Integrate[(1/((x^4)-2Cos[2θ](x^2)+1)),{x,-∞,∞},Assumptions->0<θ<2π],0<θ<2π])//Timing

Table[Subtract@@{(π/(2Sin[θ])),expr}/.θ->Random[Real,{1,0},20]π,{20}]//Chop//Union

v5.0

(expr=FullSimplify[Integrate[(1/((x^4)-2Cos[2θ](x^2)+1)),{x,-∞,∞},Assumptions->0<θ<2π],0<θ<2π])//Timing

Table[Subtract@@{(π/(2Abs[Sin[θ]])),expr}/.θ->Random[Real,{1,0},20]2π,{20}]//Chop//Union

FullSimplify[(π (-(Sqrt[-I Abs[Sin[2 θ]]-Cos[2 θ]]) (Sqrt[(1/(Sign[I Abs[Sin[2 θ]]-Cos[2 θ]]))]) (Sqrt[Sign[I Abs[Sin[2 θ]]-Cos[2 θ]]])+(Sqrt[I Abs[Sin[2 θ]]-Cos[2 θ]]) (Sqrt[-(1/(Sign[I Abs[Sin[2 θ]]+Cos[2 θ]]))]) (Sqrt[-Sign[I Abs[Sin[2 θ]]+Cos[2 θ]]])))/((Sqrt[2]) (Sqrt[-I Abs[Sin[2 θ]]-Cos[2 θ]]) (Sqrt[I Abs[Sin[2 θ]]-Cos[2 θ]]) (Sqrt[-1+Cos[4 θ]])),0<θ<2π]

v6

Integrate[(1/((x^4)-2Cos[2θ](x^2)+1)),{x,-∞,∞}]//Timing

Integrate[(1/((x^4)-2Cos[2θ](x^2)+1)),{x,-∞,∞},Assumptions->0<θ<2π]//Timing

Subtract@@@Table[
With[{θ=π Random[Real,{0,1},20]},{NIntegrate[(1/((x^4)-2Cos[2θ](x^2)+1)),{x,-∞,∞},MaxRecursion->10],(π/(2Sin[θ]))}],
{10}]//Chop//Union

Subtract@@@Table[
With[{θ=2π Random[Real,{0,1},20]},{NIntegrate[(1/((x^4)-2Cos[2θ](x^2)+1)),{x,-∞,∞},MaxRecursion->10],(π/(2Abs[Sin[θ]]))}],
{10}]//Chop//Union

(∫_(-∞)^∞)((x^2)/((x^4)-2 cos(2 θ) (x^2)+1))dx

v4.2

(expr=FullSimplify[Integrate[((x^2)/((x^4)-2Cos[2θ](x^2)+1)),{x,-∞,∞},Assumptions->0<θ<2π],0<θ<2π])//Timing

FullSimplify[expr,0<θ<2π]

Subtract@@@Table[
With[{θ=2π Random[Real,{0,1},20]},{NIntegrate[((x^2)/((x^4)-2Cos[2θ](x^2)+1)),{x,-∞,∞},MaxRecursion->10],(π/(2Abs[Sin[θ]]))}],
{10}]//Chop//Union

v5.0

(expr=FullSimplify[Integrate[((x^2)/((x^4)-2Cos[2θ](x^2)+1)),{x,-∞,∞},Assumptions->0<θ<2π],0<θ<2π])//Timing

FullSimplify[(π ((Sqrt[-I Abs[Sin[2 θ]]-Cos[2 θ]]) (Sqrt[I Abs[Sin[2 θ]]-Cos[2 θ]]) (Sqrt[(1/(Sign[I Abs[Sin[2 θ]]-Cos[2 θ]]))]) (Sqrt[Sign[I Abs[Sin[2 θ]]-Cos[2 θ]]])+(I Abs[Sin[2 θ]]+Cos[2 θ]) (Sqrt[-(1/(Sign[I Abs[Sin[2 θ]]+Cos[2 θ]]))]) (Sqrt[-Sign[I Abs[Sin[2 θ]]+Cos[2 θ]]])))/((Sqrt[2]) (Sqrt[-I Abs[Sin[2 θ]]-Cos[2 θ]]) (Sqrt[-1+Cos[4 θ]])),0<θ<2π]

Subtract@@@Table[
With[{θ=2π Random[Real,{0,1},20]},{NIntegrate[((x^2)/((x^4)-2Cos[2θ](x^2)+1)),{x,-∞,∞},MaxRecursion->10],(π/(2Abs[Sin[θ]]))}],
{10}]//Chop//Union

v5.1

(expr=FullSimplify[Integrate[((x^2)/((x^4)-2Cos[2θ](x^2)+1)),{x,-∞,∞},Assumptions->0<θ<2π],0<θ<2π])//Timing

(∫_(-∞)^∞)(1/((x^2)-(σ^2)))dx

Integrate[(1/((x^2)-(σ^2))),{x,-∞,∞},Assumptions->σ>0]//Timing

Integrate[(1/((x^2)-(σ^2))),{x,-∞,∞},PrincipalValue->True]//Timing

Integrate[(1/((x^2)-(σ^2))),{x,-∞,∞},Assumptions->Im[σ]≤0,PrincipalValue->True]

Integrate[(1/((x^2)-(σ^2))),{x,-∞,∞},Assumptions->Im[σ]==0,PrincipalValue->True]

(∫_0^∞)(((e^(-y)) (y^2))/((e^(-2 y))+1))dy

Integrate[Exp[-y]((y^2)/(1+Exp[-2y])),{y,0,∞}]//Timing

FoxTrot Integral

(∫_0^(2 π))(∫_0^(π/4))(∫_0^4)ρ cos(ϕ) (ρ^2) sin(ϕ)\[DifferentialD]ρ\[DifferentialD]ϕ\[DifferentialD]θ

Integrate[ρ Cos[ϕ](ρ^2)Sin[ϕ],{θ,0,2π},{ϕ,0,π/4},{ρ,0,4}]