4.26

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
4.26.E1	${\displaystyle{\displaystyle\int\sin x\mathrm{d}x=-\cos x}}$ \int\sin@@{x}\diff{x} = -\cos@@{x}		int(sin(x), x) = - cos(x)	Integrate[Sin[x], x, GenerateConditions->None] == - Cos[x]	Successful	Successful	-	Successful [Tested: 3]
4.26.E2	${\displaystyle{\displaystyle\int\cos x\mathrm{d}x=\sin x}}$ \int\cos@@{x}\diff{x} = \sin@@{x}		int(cos(x), x) = sin(x)	Integrate[Cos[x], x, GenerateConditions->None] == Sin[x]	Successful	Successful	-	Successful [Tested: 3]
4.26.E3	${\displaystyle{\displaystyle\int\tan x\mathrm{d}x=-\ln\left(\cos x\right)}}$ \int\tan@@{x}\diff{x} = -\ln@{\cos@@{x}}	${\displaystyle{\displaystyle-\tfrac{1}{2}\pi<x,x<\tfrac{1}{2}\pi}}$	int(tan(x), x) = - ln(cos(x))	Integrate[Tan[x], x, GenerateConditions->None] == - Log[Cos[x]]	Successful	Successful	-	Successful [Tested: 2]
4.26.E4	${\displaystyle{\displaystyle\int\csc x\mathrm{d}x=\ln\left(\tan\tfrac{1}{2}x\right)}}$ \int\csc@@{x}\diff{x} = \ln@{\tan@@{\tfrac{1}{2}x}}	${\displaystyle{\displaystyle 0<x,x<\pi}}$	int(csc(x), x) = ln(tan((1)/(2)*x))	Integrate[Csc[x], x, GenerateConditions->None] == Log[Tan[Divide[1,2]*x]]	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
4.26.E5	${\displaystyle{\displaystyle\int\sec x\mathrm{d}x={\operatorname{gd}^{-1}}\left(x\right)}}$ \int\sec@@{x}\diff{x} = \aGudermannian@{x}	${\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}}$	int(sec(x), x) = arctanh(sin(x))	Integrate[Sec[x], x, GenerateConditions->None] == InverseGudermannian[x]	Failure	Failure	Successful [Tested: 2]	Successful [Tested: 2]
4.26.E6	${\displaystyle{\displaystyle\int\cot x\mathrm{d}x=\ln\left(\sin x\right)}}$ \int\cot@@{x}\diff{x} = \ln@{\sin@@{x}}	${\displaystyle{\displaystyle 0<x,x<\pi}}$	int(cot(x), x) = ln(sin(x))	Integrate[Cot[x], x, GenerateConditions->None] == Log[Sin[x]]	Successful	Successful	-	Successful [Tested: 3]
4.26.E7	${\displaystyle{\displaystyle\int e^{ax}\sin\left(bx\right)\mathrm{d}x=\frac{e^{ax}}{a^{2}+b^{2}}(a\sin\left(bx\right)-b\cos\left(bx\right))}}$ \int e^{ax}\sin@{bx}\diff{x} = \frac{e^{ax}}{a^{2}+b^{2}}(a\sin@{bx}-b\cos@{bx})		int(exp(a*x)*sin(b*x), x) = (exp(a*x))/((a)^(2)+ (b)^(2))*(a*sin(b*x)- b*cos(b*x))	Integrate[Exp[a*x]*Sin[b*x], x, GenerateConditions->None] == Divide[Exp[a*x],(a)^(2)+ (b)^(2)]*(a*Sin[b*x]- b*Cos[b*x])	Successful	Successful	-	Successful [Tested: 108]
4.26.E8	${\displaystyle{\displaystyle\int e^{ax}\cos\left(bx\right)\mathrm{d}x=\frac{e^{ax}}{a^{2}+b^{2}}(a\cos\left(bx\right)+b\sin\left(bx\right))}}$ \int e^{ax}\cos@{bx}\diff{x} = \frac{e^{ax}}{a^{2}+b^{2}}(a\cos@{bx}+b\sin@{bx})		int(exp(a*x)*cos(b*x), x) = (exp(a*x))/((a)^(2)+ (b)^(2))*(a*cos(b*x)+ b*sin(b*x))	Integrate[Exp[a*x]*Cos[b*x], x, GenerateConditions->None] == Divide[Exp[a*x],(a)^(2)+ (b)^(2)]*(a*Cos[b*x]+ b*Sin[b*x])	Successful	Successful	-	Successful [Tested: 108]
4.26.E9	${\displaystyle{\displaystyle\int_{0}^{\pi}\sin\left(mt\right)\sin\left(nt\right)\mathrm{d}t=0}}$ \int_{0}^{\pi}\sin@{mt}\sin@{nt}\diff{t} = 0	${\displaystyle{\displaystyle m\neq n}}$	int(sin(m*t)*sin(n*t), t = 0..Pi) = 0	Integrate[Sin[m*t]*Sin[n*t], {t, 0, Pi}, GenerateConditions->None] == 0	Successful	Failure	-	Successful [Tested: 6]
4.26.E10	${\displaystyle{\displaystyle\int_{0}^{\pi}\cos\left(mt\right)\cos\left(nt\right)\mathrm{d}t=0}}$ \int_{0}^{\pi}\cos@{mt}\cos@{nt}\diff{t} = 0	${\displaystyle{\displaystyle m\neq n}}$	int(cos(m*t)*cos(n*t), t = 0..Pi) = 0	Integrate[Cos[m*t]*Cos[n*t], {t, 0, Pi}, GenerateConditions->None] == 0	Successful	Failure	-	Successful [Tested: 6]
4.26.E11	${\displaystyle{\displaystyle\int_{0}^{\pi}{\sin^{2}}\left(nt\right)\mathrm{d}t=\int_{0}^{\pi}{\cos^{2}}\left(nt\right)\mathrm{d}t}}$ \int_{0}^{\pi}\sin^{2}@{nt}\diff{t} = \int_{0}^{\pi}\cos^{2}@{nt}\diff{t}	${\displaystyle{\displaystyle n\neq 0}}$	int((sin(n*t))^(2), t = 0..Pi) = int((cos(n*t))^(2), t = 0..Pi)	Integrate[(Sin[n*t])^(2), {t, 0, Pi}, GenerateConditions->None] == Integrate[(Cos[n*t])^(2), {t, 0, Pi}, GenerateConditions->None]	Successful	Failure	-	Successful [Tested: 3]
4.26.E11	${\displaystyle{\displaystyle\int_{0}^{\pi}{\cos^{2}}\left(nt\right)\mathrm{d}t=\tfrac{1}{2}\pi}}$ \int_{0}^{\pi}\cos^{2}@{nt}\diff{t} = \tfrac{1}{2}\pi	${\displaystyle{\displaystyle n\neq 0}}$	int((cos(n*t))^(2), t = 0..Pi) = (1)/(2)*Pi	Integrate[(Cos[n*t])^(2), {t, 0, Pi}, GenerateConditions->None] == Divide[1,2]*Pi	Successful	Failure	-	Successful [Tested: 3]
4.26.E13	${\displaystyle{\displaystyle\int_{0}^{\infty}\sin\left(t^{2}\right)\mathrm{d}t=\int_{0}^{\infty}\cos\left(t^{2}\right)\mathrm{d}t}}$ \int_{0}^{\infty}\sin@{t^{2}}\diff{t} = \int_{0}^{\infty}\cos@{t^{2}}\diff{t}		int(sin((t)^(2)), t = 0..infinity) = int(cos((t)^(2)), t = 0..infinity)	Integrate[Sin[(t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Cos[(t)^(2)], {t, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 1]
4.26.E13	${\displaystyle{\displaystyle\int_{0}^{\infty}\cos\left(t^{2}\right)\mathrm{d}t=\frac{1}{2}\sqrt{\frac{\pi}{2}}}}$ \int_{0}^{\infty}\cos@{t^{2}}\diff{t} = \frac{1}{2}\sqrt{\frac{\pi}{2}}		int(cos((t)^(2)), t = 0..infinity) = (1)/(2)*sqrt((Pi)/(2))	Integrate[Cos[(t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2]*Sqrt[Divide[Pi,2]]	Successful	Successful	-	Successful [Tested: 1]
4.26.E14	${\displaystyle{\displaystyle\int\operatorname{arcsin}x\mathrm{d}x=x\operatorname{arcsin}x+(1-x^{2})^{1/2}}}$ \int\asin@@{x}\diff{x} = x\asin@@{x}+(1-x^{2})^{1/2}	${\displaystyle{\displaystyle-1<x,x<1}}$	int(arcsin(x), x) = x*arcsin(x)+(1 - (x)^(2))^(1/2)	Integrate[ArcSin[x], x, GenerateConditions->None] == x*ArcSin[x]+(1 - (x)^(2))^(1/2)	Successful	Successful	-	Successful [Tested: 1]
4.26.E15	${\displaystyle{\displaystyle\int\operatorname{arccos}x\mathrm{d}x=x\operatorname{arccos}x-(1-x^{2})^{1/2}}}$ \int\acos@@{x}\diff{x} = x\acos@@{x}-(1-x^{2})^{1/2}	${\displaystyle{\displaystyle-1<x,x<1}}$	int(arccos(x), x) = x*arccos(x)-(1 - (x)^(2))^(1/2)	Integrate[ArcCos[x], x, GenerateConditions->None] == x*ArcCos[x]-(1 - (x)^(2))^(1/2)	Successful	Successful	-	Successful [Tested: 1]
4.26.E16	${\displaystyle{\displaystyle\int\operatorname{arctan}x\mathrm{d}x=x\operatorname{arctan}x-\tfrac{1}{2}\ln\left(1+x^{2}\right)}}$ \int\atan@@{x}\diff{x} = x\atan@@{x}-\tfrac{1}{2}\ln@{1+x^{2}}	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	int(arctan(x), x) = x*arctan(x)-(1)/(2)*ln(1 + (x)^(2))	Integrate[ArcTan[x], x, GenerateConditions->None] == x*ArcTan[x]-Divide[1,2]*Log[1 + (x)^(2)]	Successful	Successful	-	Successful [Tested: 3]
4.26.E17	${\displaystyle{\displaystyle\int\operatorname{arccsc}x\mathrm{d}x=x\operatorname{arccsc}x+\ln\left(x+(x^{2}-1)^{1/2}\right)}}$ \int\acsc@@{x}\diff{x} = x\acsc@@{x}+\ln@{x+(x^{2}-1)^{1/2}}	${\displaystyle{\displaystyle 1<x,x<\infty}}$	int(arccsc(x), x) = x*arccsc(x)+ ln(x +((x)^(2)- 1)^(1/2))	Integrate[ArcCsc[x], x, GenerateConditions->None] == x*ArcCsc[x]+ Log[x +((x)^(2)- 1)^(1/2)]	Successful	Failure	-	Failed [2 / 2] Result: Complex[-1.1102230246251565*^-16, -1.5707963267948966] Test Values: {Rule[x, 1.5]} Result: Complex[-4.440892098500626*^-16, -1.5707963267948966] Test Values: {Rule[x, 2]}
4.26.E18	${\displaystyle{\displaystyle\int\operatorname{arcsec}x\mathrm{d}x=x\operatorname{arcsec}x-\ln\left(x+(x^{2}-1)^{1/2}\right)}}$ \int\asec@@{x}\diff{x} = x\asec@@{x}-\ln@{x+(x^{2}-1)^{1/2}}	${\displaystyle{\displaystyle 1<x,x<\infty}}$	int(arcsec(x), x) = x*arcsec(x)- ln(x +((x)^(2)- 1)^(1/2))	Integrate[ArcSec[x], x, GenerateConditions->None] == x*ArcSec[x]- Log[x +((x)^(2)- 1)^(1/2)]	Successful	Failure	-	Failed [2 / 2] Result: Complex[1.1102230246251565*^-16, 1.5707963267948966] Test Values: {Rule[x, 1.5]} Result: Complex[4.440892098500626*^-16, 1.5707963267948966] Test Values: {Rule[x, 2]}
4.26.E19	${\displaystyle{\displaystyle\int\operatorname{arccot}x\mathrm{d}x=x\operatorname{arccot}x+\tfrac{1}{2}\ln\left(1+x^{2}\right)}}$ \int\acot@@{x}\diff{x} = x\acot@@{x}+\tfrac{1}{2}\ln@{1+x^{2}}	${\displaystyle{\displaystyle 0<x,x<\infty}}$	int(arccot(x), x) = x*arccot(x)+(1)/(2)*ln(1 + (x)^(2))	Integrate[ArcCot[x], x, GenerateConditions->None] == x*ArcCot[x]+Divide[1,2]*Log[1 + (x)^(2)]	Successful	Successful	-	Successful [Tested: 3]
4.26.E20	${\displaystyle{\displaystyle\int x\operatorname{arcsin}x\mathrm{d}x=\left(\frac{x^{2}}{2}-\frac{1}{4}\right)\operatorname{arcsin}x+\frac{x}{4}(1-x^{2})^{1/2}}}$ \int x\asin@@{x}\diff{x} = \left(\frac{x^{2}}{2}-\frac{1}{4}\right)\asin@@{x}+\frac{x}{4}(1-x^{2})^{1/2}	${\displaystyle{\displaystyle-1<x,x<1}}$	int(x*arcsin(x), x) = (((x)^(2))/(2)-(1)/(4))*arcsin(x)+(x)/(4)*(1 - (x)^(2))^(1/2)	Integrate[x*ArcSin[x], x, GenerateConditions->None] == (Divide[(x)^(2),2]-Divide[1,4])*ArcSin[x]+Divide[x,4]*(1 - (x)^(2))^(1/2)	Successful	Successful	-	Successful [Tested: 1]
4.26.E21	${\displaystyle{\displaystyle\int x\operatorname{arccos}x\mathrm{d}x=\left(\frac{x^{2}}{2}-\frac{1}{4}\right)\operatorname{arccos}x-\frac{x}{4}(1-x^{2})^{1/2}}}$ \int x\acos@@{x}\diff{x} = \left(\frac{x^{2}}{2}-\frac{1}{4}\right)\acos@@{x}-\frac{x}{4}(1-x^{2})^{1/2}	${\displaystyle{\displaystyle-1<x,x<1}}$	int(x*arccos(x), x) = (((x)^(2))/(2)-(1)/(4))*arccos(x)-(x)/(4)*(1 - (x)^(2))^(1/2)	Integrate[x*ArcCos[x], x, GenerateConditions->None] == (Divide[(x)^(2),2]-Divide[1,4])*ArcCos[x]-Divide[x,4]*(1 - (x)^(2))^(1/2)	Failure	Failure	Failed [1 / 1] Result: .3926990817 Test Values: {x = .5}	Failed [1 / 1] Result: 0.3926990816987242 Test Values: {Rule[x, 0.5]}