Optimal. Leaf size=62 \[ -\frac {2 e^{2 i a} x^3}{x^2+e^{2 i a}}+6 e^{2 i a} x-6 e^{3 i a} \tan ^{-1}\left (e^{-i a} x\right )-\frac {x^3}{3} \]
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Rubi [F] time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^2 \tan ^2(a+i \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^2 \tan ^2(a+i \log (x)) \, dx &=\int x^2 \tan ^2(a+i \log (x)) \, dx\\ \end {align*}
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Mathematica [A] time = 0.13, size = 100, normalized size = 1.61 \[ \frac {2 x (\cos (3 a)+i \sin (3 a))}{\left (x^2+1\right ) \cos (a)-i \left (x^2-1\right ) \sin (a)}+4 i x \sin (2 a)+4 x \cos (2 a)-6 \cos (3 a) \tan ^{-1}(x (\cos (a)-i \sin (a)))-6 i \sin (3 a) \tan ^{-1}(x (\cos (a)-i \sin (a)))-\frac {x^3}{3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 86, normalized size = 1.39 \[ -\frac {x^{5} - 11 \, x^{3} e^{\left (2 i \, a\right )} - 18 \, x e^{\left (4 i \, a\right )} - {\left (-9 i \, x^{2} e^{\left (3 i \, a\right )} - 9 i \, e^{\left (5 i \, a\right )}\right )} \log \left (x + i \, e^{\left (i \, a\right )}\right ) - {\left (9 i \, x^{2} e^{\left (3 i \, a\right )} + 9 i \, e^{\left (5 i \, a\right )}\right )} \log \left (x - i \, e^{\left (i \, a\right )}\right )}{3 \, {\left (x^{2} + e^{\left (2 i \, a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.56, size = 141, normalized size = 2.27 \[ -\frac {x^{5}}{3 \, {\left (x^{2} + \frac {e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}\right )}} + \frac {10 \, x^{3} e^{\left (2 i \, a\right )}}{3 \, {\left (x^{2} + \frac {e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}\right )}} - 6 \, \arctan \left (x e^{\left (-i \, a\right )}\right ) e^{\left (3 i \, a\right )} + \frac {35 \, x e^{\left (4 i \, a\right )}}{3 \, {\left (x^{2} + \frac {e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}\right )}} + \frac {2 \, x e^{\left (4 i \, a\right )}}{x^{2} + e^{\left (2 i \, a\right )}} + \frac {8 \, e^{\left (6 i \, a\right )}}{{\left (x^{2} + \frac {e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 48, normalized size = 0.77 \[ -\frac {7 x^{3}}{3}+\frac {2 x^{3}}{1+\frac {{\mathrm e}^{2 i a}}{x^{2}}}+6 \,{\mathrm e}^{2 i a} x -6 \arctan \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{3 i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 269, normalized size = 4.34 \[ -\frac {2 \, x^{5} - x^{3} {\left (22 \, \cos \left (2 \, a\right ) + 22 i \, \sin \left (2 \, a\right )\right )} - x {\left (36 \, \cos \left (4 \, a\right ) + 36 i \, \sin \left (4 \, a\right )\right )} - {\left (x^{2} {\left (18 \, \cos \left (3 \, a\right ) + 18 i \, \sin \left (3 \, a\right )\right )} + {\left (18 \, \cos \left (2 \, a\right ) + 18 i \, \sin \left (2 \, a\right )\right )} \cos \left (3 \, a\right ) - 18 \, {\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \sin \left (3 \, a\right )\right )} \arctan \left (\frac {2 \, x \cos \relax (a)}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}, \frac {x^{2} - \cos \relax (a)^{2} - \sin \relax (a)^{2}}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}\right ) + {\left (9 \, x^{2} {\left (-i \, \cos \left (3 \, a\right ) + \sin \left (3 \, a\right )\right )} + 9 \, {\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \cos \left (3 \, a\right ) + {\left (9 \, \cos \left (2 \, a\right ) + 9 i \, \sin \left (2 \, a\right )\right )} \sin \left (3 \, a\right )\right )} \log \left (\frac {x^{2} + \cos \relax (a)^{2} + 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}\right )}{6 \, x^{2} + 6 \, \cos \left (2 \, a\right ) + 6 i \, \sin \left (2 \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.23, size = 52, normalized size = 0.84 \[ -6\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\right )}^{3/2}\,\mathrm {atan}\left (\frac {x}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )-\frac {x^3}{3}+4\,x\,{\mathrm {e}}^{a\,2{}\mathrm {i}}+\frac {2\,x\,{\mathrm {e}}^{a\,4{}\mathrm {i}}}{x^2+{\mathrm {e}}^{a\,2{}\mathrm {i}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 66, normalized size = 1.06 \[ - \frac {x^{3}}{3} + 4 x e^{2 i a} + \frac {2 x e^{4 i a}}{x^{2} + e^{2 i a}} - 3 \left (- i \log {\left (x - i e^{i a} \right )} + i \log {\left (x + i e^{i a} \right )}\right ) e^{3 i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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