Integrand size = 13, antiderivative size = 47 \[ \int x^3 \tan (a+i \log (x)) \, dx=-i e^{2 i a} x^2+\frac {i x^4}{4}+i e^{4 i a} \log \left (e^{2 i a}+x^2\right ) \]
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Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4591, 456, 457, 78} \[ \int x^3 \tan (a+i \log (x)) \, dx=-i e^{2 i a} x^2+i e^{4 i a} \log \left (x^2+e^{2 i a}\right )+\frac {i x^4}{4} \]
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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 ) x^3}{1+\frac {e^{2 i a}}{x^2}} \, dx \\ & = \int \frac {x^3 \left (-i e^{2 i a}+i x^2\right )}{e^{2 i a}+x^2} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {\left (-i e^{2 i a}+i x\right ) x}{e^{2 i a}+x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-2 i e^{2 i a}+i x+\frac {2 i e^{4 i a}}{e^{2 i a}+x}\right ) \, dx,x,x^2\right ) \\ & = -i e^{2 i a} x^2+\frac {i x^4}{4}+i 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. \(132\) vs. \(2(47)=94\).
Time = 0.03 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.81 \[ \int x^3 \tan (a+i \log (x)) \, dx=\frac {i x^4}{4}-i x^2 \cos (2 a)+\arctan \left (\frac {\left (1+x^2\right ) \cos (a)}{\sin (a)-x^2 \sin (a)}\right ) \cos (4 a)+\frac {1}{2} i \cos (4 a) \log \left (1+x^4+2 x^2 \cos (2 a)\right )+x^2 \sin (2 a)+i \arctan \left (\frac {\left (1+x^2\right ) \cos (a)}{\sin (a)-x^2 \sin (a)}\right ) \sin (4 a)-\frac {1}{2} \log \left (1+x^4+2 x^2 \cos (2 a)\right ) \sin (4 a) \]
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Time = 4.36 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.79
method | result | size |
risch | \(-i {\mathrm e}^{2 i a} x^{2}+\frac {i x^{4}}{4}+i {\mathrm e}^{4 i a} \ln \left ({\mathrm e}^{2 i a}+x^{2}\right )\) | \(37\) |
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Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.64 \[ \int x^3 \tan (a+i \log (x)) \, dx=\frac {1}{4} i \, x^{4} - i \, x^{2} e^{\left (2 i \, a\right )} + i \, e^{\left (4 i \, a\right )} \log \left (x^{2} + e^{\left (2 i \, a\right )}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.79 \[ \int x^3 \tan (a+i \log (x)) \, dx=\frac {i x^{4}}{4} - i x^{2} e^{2 i a} + i e^{4 i a} \log {\left (x^{2} + e^{2 i a} \right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (30) = 60\).
Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.87 \[ \int x^3 \tan (a+i \log (x)) \, dx=\frac {1}{4} i \, x^{4} + x^{2} {\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} - {\left (\cos \left (4 \, a\right ) + i \, \sin \left (4 \, a\right )\right )} \arctan \left (\sin \left (2 \, a\right ), x^{2} + \cos \left (2 \, a\right )\right ) + \frac {1}{2} \, {\left (i \, \cos \left (4 \, a\right ) - \sin \left (4 \, a\right )\right )} \log \left (x^{4} + 2 \, x^{2} \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right ) \]
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Time = 0.35 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.79 \[ \int x^3 \tan (a+i \log (x)) \, dx=\frac {1}{4} i \, x^{4} - i \, x^{2} e^{\left (2 i \, a\right )} - \frac {1}{2} \, \pi e^{\left (4 i \, a\right )} + i \, e^{\left (4 i \, a\right )} \log \left (x^{2} + e^{\left (2 i \, a\right )}\right ) \]
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Time = 25.98 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77 \[ \int x^3 \tan (a+i \log (x)) \, dx={\mathrm {e}}^{a\,4{}\mathrm {i}}\,\ln \left (x^2+{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}-x^2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,1{}\mathrm {i}+\frac {x^4\,1{}\mathrm {i}}{4} \]
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