3.491 \(\int x \tan ^3(x) \, dx\)

Optimal. Leaf size=59 \[ -\frac{1}{2} i \text{PolyLog}\left (2,-e^{2 i x}\right )-\frac{i x^2}{2}+\frac{x}{2}+x \log \left (1+e^{2 i x}\right )+\frac{1}{2} x \tan ^2(x)-\frac{\tan (x)}{2} \]

[Out]

x/2 - (I/2)*x^2 + x*Log[1 + E^((2*I)*x)] - (I/2)*PolyLog[2, -E^((2*I)*x)] - Tan[
x]/2 + (x*Tan[x]^2)/2

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Rubi [A]  time = 0.0987397, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.167 \[ -\frac{1}{2} i \text{PolyLog}\left (2,-e^{2 i x}\right )-\frac{i x^2}{2}+\frac{x}{2}+x \log \left (1+e^{2 i x}\right )+\frac{1}{2} x \tan ^2(x)-\frac{\tan (x)}{2} \]

Antiderivative was successfully verified.

[In]  Int[x*Tan[x]^3,x]

[Out]

x/2 - (I/2)*x^2 + x*Log[1 + E^((2*I)*x)] - (I/2)*PolyLog[2, -E^((2*I)*x)] - Tan[
x]/2 + (x*Tan[x]^2)/2

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Rubi in Sympy [A]  time = 13.2233, size = 48, normalized size = 0.81 \[ - \frac{i x^{2}}{2} + x \log{\left (e^{2 i x} + 1 \right )} + \frac{x \tan ^{2}{\left (x \right )}}{2} + \frac{x}{2} - \frac{\tan{\left (x \right )}}{2} - \frac{i \operatorname{Li}_{2}\left (- e^{2 i x}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*sin(x)**3/cos(x)**3,x)

[Out]

-I*x**2/2 + x*log(exp(2*I*x) + 1) + x*tan(x)**2/2 + x/2 - tan(x)/2 - I*polylog(2
, -exp(2*I*x))/2

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Mathematica [A]  time = 0.0125968, size = 54, normalized size = 0.92 \[ -\frac{1}{2} i \text{PolyLog}\left (2,-e^{2 i x}\right )-\frac{i x^2}{2}+x \log \left (1+e^{2 i x}\right )-\frac{\tan (x)}{2}+\frac{1}{2} x \sec ^2(x) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Tan[x]^3,x]

[Out]

(-I/2)*x^2 + x*Log[1 + E^((2*I)*x)] - (I/2)*PolyLog[2, -E^((2*I)*x)] + (x*Sec[x]
^2)/2 - Tan[x]/2

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Maple [A]  time = 0.116, size = 59, normalized size = 1. \[ -{\frac{i}{2}}{x}^{2}+{\frac{-i{{\rm e}^{2\,ix}}+2\,x{{\rm e}^{2\,ix}}-i}{ \left ( 1+{{\rm e}^{2\,ix}} \right ) ^{2}}}+x\ln \left ( 1+{{\rm e}^{2\,ix}} \right ) -{\frac{i}{2}}{\it polylog} \left ( 2,-{{\rm e}^{2\,ix}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*sin(x)^3/cos(x)^3,x)

[Out]

-1/2*I*x^2+(-I*exp(2*I*x)+2*x*exp(2*I*x)-I)/(1+exp(2*I*x))^2+x*ln(1+exp(2*I*x))-
1/2*I*polylog(2,-exp(2*I*x))

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Maxima [A]  time = 1.89677, size = 288, normalized size = 4.88 \[ -\frac{x^{2} \cos \left (4 \, x\right ) + i \, x^{2} \sin \left (4 \, x\right ) + x^{2} -{\left (2 \, x \cos \left (4 \, x\right ) + 4 \, x \cos \left (2 \, x\right ) + 2 i \, x \sin \left (4 \, x\right ) + 4 i \, x \sin \left (2 \, x\right ) + 2 \, x\right )} \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) + 2 \,{\left (x^{2} + 2 i \, x + 1\right )} \cos \left (2 \, x\right ) +{\left (\cos \left (4 \, x\right ) + 2 \, \cos \left (2 \, x\right ) + i \, \sin \left (4 \, x\right ) + 2 i \, \sin \left (2 \, x\right ) + 1\right )}{\rm Li}_2\left (-e^{\left (2 i \, x\right )}\right ) -{\left (-i \, x \cos \left (4 \, x\right ) - 2 i \, x \cos \left (2 \, x\right ) + x \sin \left (4 \, x\right ) + 2 \, x \sin \left (2 \, x\right ) - i \, x\right )} \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) -{\left (-2 i \, x^{2} + 4 \, x - 2 i\right )} \sin \left (2 \, x\right ) + 2}{-2 i \, \cos \left (4 \, x\right ) - 4 i \, \cos \left (2 \, x\right ) + 2 \, \sin \left (4 \, x\right ) + 4 \, \sin \left (2 \, x\right ) - 2 i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(x)^3/cos(x)^3,x, algorithm="maxima")

[Out]

-(x^2*cos(4*x) + I*x^2*sin(4*x) + x^2 - (2*x*cos(4*x) + 4*x*cos(2*x) + 2*I*x*sin
(4*x) + 4*I*x*sin(2*x) + 2*x)*arctan2(sin(2*x), cos(2*x) + 1) + 2*(x^2 + 2*I*x +
1)*cos(2*x) + (cos(4*x) + 2*cos(2*x) + I*sin(4*x) + 2*I*sin(2*x) + 1)*dilog(-e^
(2*I*x)) - (-I*x*cos(4*x) - 2*I*x*cos(2*x) + x*sin(4*x) + 2*x*sin(2*x) - I*x)*lo
g(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1) - (-2*I*x^2 + 4*x - 2*I)*sin(2*x) +
2)/(-2*I*cos(4*x) - 4*I*cos(2*x) + 2*sin(4*x) + 4*sin(2*x) - 2*I)

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Fricas [A]  time = 0.277717, size = 338, normalized size = 5.73 \[ \frac{x \cos \left (x\right )^{2} \log \left (\frac{\left (i + 1\right ) \, \cos \left (x\right ) + \left (i + 1\right ) \, \sin \left (x\right ) + i + 1}{i \, \cos \left (x\right ) + \sin \left (x\right ) + i}\right ) + x \cos \left (x\right )^{2} \log \left (\frac{-\left (i - 1\right ) \, \cos \left (x\right ) - \left (i - 1\right ) \, \sin \left (x\right ) - i + 1}{-i \, \cos \left (x\right ) + \sin \left (x\right ) - i}\right ) + x \cos \left (x\right )^{2} \log \left (\frac{\left (i - 1\right ) \, \cos \left (x\right ) - \left (i - 1\right ) \, \sin \left (x\right ) + i - 1}{i \, \cos \left (x\right ) + \sin \left (x\right ) + i}\right ) + x \cos \left (x\right )^{2} \log \left (\frac{-\left (i + 1\right ) \, \cos \left (x\right ) + \left (i + 1\right ) \, \sin \left (x\right ) - i - 1}{-i \, \cos \left (x\right ) + \sin \left (x\right ) - i}\right ) - i \, \cos \left (x\right )^{2}{\rm Li}_2\left (-\frac{\left (i + 1\right ) \, \cos \left (x\right ) + \left (i + 1\right ) \, \sin \left (x\right ) + i + 1}{i \, \cos \left (x\right ) + \sin \left (x\right ) + i} + 1\right ) + i \, \cos \left (x\right )^{2}{\rm Li}_2\left (-\frac{-\left (i - 1\right ) \, \cos \left (x\right ) - \left (i - 1\right ) \, \sin \left (x\right ) - i + 1}{-i \, \cos \left (x\right ) + \sin \left (x\right ) - i} + 1\right ) - i \, \cos \left (x\right )^{2}{\rm Li}_2\left (-\frac{\left (i - 1\right ) \, \cos \left (x\right ) - \left (i - 1\right ) \, \sin \left (x\right ) + i - 1}{i \, \cos \left (x\right ) + \sin \left (x\right ) + i} + 1\right ) + i \, \cos \left (x\right )^{2}{\rm Li}_2\left (-\frac{-\left (i + 1\right ) \, \cos \left (x\right ) + \left (i + 1\right ) \, \sin \left (x\right ) - i - 1}{-i \, \cos \left (x\right ) + \sin \left (x\right ) - i} + 1\right ) - \cos \left (x\right ) \sin \left (x\right ) + x}{2 \, \cos \left (x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(x)^3/cos(x)^3,x, algorithm="fricas")

[Out]

1/2*(x*cos(x)^2*log(((I + 1)*cos(x) + (I + 1)*sin(x) + I + 1)/(I*cos(x) + sin(x)
+ I)) + x*cos(x)^2*log((-(I - 1)*cos(x) - (I - 1)*sin(x) - I + 1)/(-I*cos(x) +
sin(x) - I)) + x*cos(x)^2*log(((I - 1)*cos(x) - (I - 1)*sin(x) + I - 1)/(I*cos(x
) + sin(x) + I)) + x*cos(x)^2*log((-(I + 1)*cos(x) + (I + 1)*sin(x) - I - 1)/(-I
*cos(x) + sin(x) - I)) - I*cos(x)^2*dilog(-((I + 1)*cos(x) + (I + 1)*sin(x) + I
+ 1)/(I*cos(x) + sin(x) + I) + 1) + I*cos(x)^2*dilog(-(-(I - 1)*cos(x) - (I - 1)
*sin(x) - I + 1)/(-I*cos(x) + sin(x) - I) + 1) - I*cos(x)^2*dilog(-((I - 1)*cos(
x) - (I - 1)*sin(x) + I - 1)/(I*cos(x) + sin(x) + I) + 1) + I*cos(x)^2*dilog(-(-
(I + 1)*cos(x) + (I + 1)*sin(x) - I - 1)/(-I*cos(x) + sin(x) - I) + 1) - cos(x)*
sin(x) + x)/cos(x)^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \sin ^{3}{\left (x \right )}}{\cos ^{3}{\left (x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(x)**3/cos(x)**3,x)

[Out]

Integral(x*sin(x)**3/cos(x)**3, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \sin \left (x\right )^{3}}{\cos \left (x\right )^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(x)^3/cos(x)^3,x, algorithm="giac")

[Out]

integrate(x*sin(x)^3/cos(x)^3, x)