Integrand size = 15, antiderivative size = 63 \[ \int x^3 \tan ^2(a+i \log (x)) \, dx=2 e^{2 i a} x^2-\frac {x^4}{4}-\frac {2 e^{6 i a}}{e^{2 i a}+x^2}-4 e^{4 i a} \log \left (e^{2 i a}+x^2\right ) \]
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Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4591, 456, 457, 78} \[ \int x^3 \tan ^2(a+i \log (x)) \, dx=2 e^{2 i a} x^2-\frac {2 e^{6 i a}}{x^2+e^{2 i a}}-4 e^{4 i a} \log \left (x^2+e^{2 i a}\right )-\frac {x^4}{4} \]
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Rule 78
Rule 456
Rule 457
Rule 4591
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (i-\frac {i e^{2 i a}}{x^2}\right )^2 x^3}{\left (1+\frac {e^{2 i a}}{x^2}\right )^2} \, dx \\ & = \int \frac {x^3 \left (-i e^{2 i a}+i x^2\right )^2}{\left (e^{2 i a}+x^2\right )^2} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {\left (-i e^{2 i a}+i x\right )^2 x}{\left (e^{2 i a}+x\right )^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (4 e^{2 i a}-x+\frac {4 e^{6 i a}}{\left (e^{2 i a}+x\right )^2}-\frac {8 e^{4 i a}}{e^{2 i a}+x}\right ) \, dx,x,x^2\right ) \\ & = 2 e^{2 i a} x^2-\frac {x^4}{4}-\frac {2 e^{6 i a}}{e^{2 i a}+x^2}-4 e^{4 i a} \log \left (e^{2 i a}+x^2\right ) \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(155\) vs. \(2(63)=126\).
Time = 0.19 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.46 \[ \int x^3 \tan ^2(a+i \log (x)) \, dx=-\frac {x^4}{4}+2 x^2 \cos (2 a)-4 i \arctan \left (\frac {\left (1+x^2\right ) \cot (a)}{-1+x^2}\right ) \cos (4 a)-2 \cos (4 a) \log \left (1+x^4+2 x^2 \cos (2 a)\right )+2 i x^2 \sin (2 a)+4 \arctan \left (\frac {\left (1+x^2\right ) \cot (a)}{-1+x^2}\right ) \sin (4 a)-2 i \log \left (1+x^4+2 x^2 \cos (2 a)\right ) \sin (4 a)-\frac {2 (\cos (5 a)+i \sin (5 a))}{\left (1+x^2\right ) \cos (a)-i \left (-1+x^2\right ) \sin (a)} \]
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Time = 5.16 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83
method | result | size |
risch | \(-\frac {9 x^{4}}{4}+\frac {2 x^{4}}{1+\frac {{\mathrm e}^{2 i a}}{x^{2}}}+4 \,{\mathrm e}^{2 i a} x^{2}-4 \,{\mathrm e}^{4 i a} \ln \left ({\mathrm e}^{2 i a}+x^{2}\right )\) | \(52\) |
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Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02 \[ \int x^3 \tan ^2(a+i \log (x)) \, dx=-\frac {x^{6} - 7 \, x^{4} e^{\left (2 i \, a\right )} - 8 \, x^{2} e^{\left (4 i \, a\right )} + 16 \, {\left (x^{2} e^{\left (4 i \, a\right )} + e^{\left (6 i \, a\right )}\right )} \log \left (x^{2} + e^{\left (2 i \, a\right )}\right ) + 8 \, e^{\left (6 i \, a\right )}}{4 \, {\left (x^{2} + e^{\left (2 i \, a\right )}\right )}} \]
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Time = 0.17 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.86 \[ \int x^3 \tan ^2(a+i \log (x)) \, dx=- \frac {x^{4}}{4} + 2 x^{2} e^{2 i a} - 4 e^{4 i a} \log {\left (x^{2} + e^{2 i a} \right )} - \frac {2 e^{6 i a}}{x^{2} + e^{2 i a}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (46) = 92\).
Time = 0.21 (sec) , antiderivative size = 217, normalized size of antiderivative = 3.44 \[ \int x^3 \tan ^2(a+i \log (x)) \, dx=-\frac {x^{6} - 7 \, x^{4} {\left (\cos \left (2 \, a\right ) + i \, \sin \left (2 \, a\right )\right )} - 8 \, {\left (2 \, {\left (-i \, \cos \left (4 \, a\right ) + \sin \left (4 \, a\right )\right )} \arctan \left (\sin \left (2 \, a\right ), x^{2} + \cos \left (2 \, a\right )\right ) + \cos \left (4 \, a\right ) + i \, \sin \left (4 \, a\right )\right )} x^{2} - 16 \, {\left ({\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \cos \left (4 \, a\right ) + {\left (\cos \left (2 \, a\right ) + i \, \sin \left (2 \, a\right )\right )} \sin \left (4 \, a\right )\right )} \arctan \left (\sin \left (2 \, a\right ), x^{2} + \cos \left (2 \, a\right )\right ) + 8 \, {\left (x^{2} {\left (\cos \left (4 \, a\right ) + i \, \sin \left (4 \, a\right )\right )} + {\left (\cos \left (2 \, a\right ) + i \, \sin \left (2 \, a\right )\right )} \cos \left (4 \, a\right ) - {\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \sin \left (4 \, a\right )\right )} \log \left (x^{4} + 2 \, x^{2} \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right ) + 8 \, \cos \left (6 \, a\right ) + 8 i \, \sin \left (6 \, a\right )}{4 \, {\left (x^{2} + \cos \left (2 \, a\right ) + i \, \sin \left (2 \, a\right )\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (46) = 92\).
Time = 0.46 (sec) , antiderivative size = 261, normalized size of antiderivative = 4.14 \[ \int x^3 \tan ^2(a+i \log (x)) \, dx=-\frac {x^{6}}{4 \, {\left (x^{2} + \frac {e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}\right )}} + \frac {3 \, x^{4} e^{\left (2 i \, a\right )}}{2 \, {\left (x^{2} + \frac {e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}\right )}} - \frac {4 \, x^{2} e^{\left (4 i \, a\right )} \log \left (-x^{2} - e^{\left (2 i \, a\right )}\right )}{x^{2} + \frac {e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}} + \frac {17 \, x^{2} e^{\left (4 i \, a\right )}}{4 \, {\left (x^{2} + \frac {e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}\right )}} - \frac {8 \, e^{\left (6 i \, a\right )} \log \left (-x^{2} - e^{\left (2 i \, a\right )}\right )}{x^{2} + \frac {e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}} + \frac {e^{\left (6 i \, a\right )}}{x^{2} + \frac {e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}} - \frac {4 \, e^{\left (8 i \, a\right )} \log \left (-x^{2} - e^{\left (2 i \, a\right )}\right )}{{\left (x^{2} + \frac {e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}\right )} x^{2}} - \frac {3 \, e^{\left (8 i \, a\right )}}{2 \, {\left (x^{2} + \frac {e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}\right )} x^{2}} \]
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Time = 27.49 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int x^3 \tan ^2(a+i \log (x)) \, dx=-\frac {2\,{\mathrm {e}}^{a\,6{}\mathrm {i}}}{x^2+{\mathrm {e}}^{a\,2{}\mathrm {i}}}-4\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,\ln \left (x^2+{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )+2\,x^2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}-\frac {x^4}{4} \]
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