Integrand size = 6, antiderivative size = 59 \[ \int x \tan ^3(x) \, dx=\frac {x}{2}-\frac {i x^2}{2}+x \log \left (1+e^{2 i x}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {\tan (x)}{2}+\frac {1}{2} x \tan ^2(x) \]
[Out]
Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {3801, 3554, 8, 3800, 2221, 2317, 2438} \[ \int x \tan ^3(x) \, dx=-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {i x^2}{2}+\frac {x}{2}+x \log \left (1+e^{2 i x}\right )+\frac {1}{2} x \tan ^2(x)-\frac {\tan (x)}{2} \]
[In]
[Out]
Rule 8
Rule 2221
Rule 2317
Rule 2438
Rule 3554
Rule 3800
Rule 3801
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x \tan ^2(x)-\frac {1}{2} \int \tan ^2(x) \, dx-\int x \tan (x) \, dx \\ & = -\frac {i x^2}{2}-\frac {\tan (x)}{2}+\frac {1}{2} x \tan ^2(x)+2 i \int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx+\frac {\int 1 \, dx}{2} \\ & = \frac {x}{2}-\frac {i x^2}{2}+x \log \left (1+e^{2 i x}\right )-\frac {\tan (x)}{2}+\frac {1}{2} x \tan ^2(x)-\int \log \left (1+e^{2 i x}\right ) \, dx \\ & = \frac {x}{2}-\frac {i x^2}{2}+x \log \left (1+e^{2 i x}\right )-\frac {\tan (x)}{2}+\frac {1}{2} x \tan ^2(x)+\frac {1}{2} i \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i x}\right ) \\ & = \frac {x}{2}-\frac {i x^2}{2}+x \log \left (1+e^{2 i x}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {\tan (x)}{2}+\frac {1}{2} x \tan ^2(x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int x \tan ^3(x) \, dx=-\frac {i x^2}{2}+x \log \left (1+e^{2 i x}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )+\frac {1}{2} x \sec ^2(x)-\frac {\tan (x)}{2} \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {i x^{2}}{2}+\frac {2 x \,{\mathrm e}^{2 i x}-i {\mathrm e}^{2 i x}-i}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}+x \ln \left ({\mathrm e}^{2 i x}+1\right )-\frac {i \operatorname {Li}_{2}\left (-{\mathrm e}^{2 i x}\right )}{2}\) | \(59\) |
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (38) = 76\).
Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.34 \[ \int x \tan ^3(x) \, dx=\frac {x \cos \left (x\right )^{2} \log \left (i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) + x \cos \left (x\right )^{2} \log \left (i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) + x \cos \left (x\right )^{2} \log \left (-i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) + x \cos \left (x\right )^{2} \log \left (-i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) + i \, \cos \left (x\right )^{2} {\rm Li}_2\left (i \, \cos \left (x\right ) + \sin \left (x\right )\right ) - i \, \cos \left (x\right )^{2} {\rm Li}_2\left (i \, \cos \left (x\right ) - \sin \left (x\right )\right ) - i \, \cos \left (x\right )^{2} {\rm Li}_2\left (-i \, \cos \left (x\right ) + \sin \left (x\right )\right ) + i \, \cos \left (x\right )^{2} {\rm Li}_2\left (-i \, \cos \left (x\right ) - \sin \left (x\right )\right ) - \cos \left (x\right ) \sin \left (x\right ) + x}{2 \, \cos \left (x\right )^{2}} \]
[In]
[Out]
\[ \int x \tan ^3(x) \, dx=\int \frac {x \sin ^{3}{\left (x \right )}}{\cos ^{3}{\left (x \right )}}\, dx \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (38) = 76\).
Time = 0.29 (sec) , antiderivative size = 210, normalized size of antiderivative = 3.56 \[ \int x \tan ^3(x) \, dx=-\frac {x^{2} \cos \left (4 \, x\right ) + i \, x^{2} \sin \left (4 \, x\right ) + x^{2} - 2 \, {\left (x \cos \left (4 \, x\right ) + 2 \, x \cos \left (2 \, x\right ) + i \, x \sin \left (4 \, x\right ) + 2 i \, x \sin \left (2 \, x\right ) + x\right )} \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) + 2 \, {\left (x^{2} + 2 i \, x + 1\right )} \cos \left (2 \, x\right ) + {\left (\cos \left (4 \, x\right ) + 2 \, \cos \left (2 \, x\right ) + i \, \sin \left (4 \, x\right ) + 2 i \, \sin \left (2 \, x\right ) + 1\right )} {\rm Li}_2\left (-e^{\left (2 i \, x\right )}\right ) - {\left (-i \, x \cos \left (4 \, x\right ) - 2 i \, x \cos \left (2 \, x\right ) + x \sin \left (4 \, x\right ) + 2 \, x \sin \left (2 \, x\right ) - i \, x\right )} \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) + 2 \, {\left (i \, x^{2} - 2 \, x + i\right )} \sin \left (2 \, x\right ) + 2}{-2 i \, \cos \left (4 \, x\right ) - 4 i \, \cos \left (2 \, x\right ) + 2 \, \sin \left (4 \, x\right ) + 4 \, \sin \left (2 \, x\right ) - 2 i} \]
[In]
[Out]
\[ \int x \tan ^3(x) \, dx=\int { \frac {x \sin \left (x\right )^{3}}{\cos \left (x\right )^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int x \tan ^3(x) \, dx=\int \frac {x\,{\sin \left (x\right )}^3}{{\cos \left (x\right )}^3} \,d x \]
[In]
[Out]