Optimal. Leaf size=60 \[ \frac {3 x}{x^2+e^{2 i a}}+\frac {e^{2 i a}}{x \left (x^2+e^{2 i a}\right )}+2 e^{-i a} \tan ^{-1}\left (e^{-i a} x\right ) \]
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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 = {} \[ \int \frac {\tan ^2(a+i \log (x))}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\tan ^2(a+i \log (x))}{x^2} \, dx &=\int \frac {\tan ^2(a+i \log (x))}{x^2} \, dx\\ \end {align*}
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Mathematica [A] time = 0.11, size = 72, normalized size = 1.20 \[ \frac {2 x (\cos (a)-i \sin (a))}{\left (x^2+1\right ) \cos (a)-i \left (x^2-1\right ) \sin (a)}+2 \cos (a) \tan ^{-1}(x (\cos (a)-i \sin (a)))-2 i \sin (a) \tan ^{-1}(x (\cos (a)-i \sin (a)))+\frac {1}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 78, normalized size = 1.30 \[ \frac {3 \, x^{2} e^{\left (i \, a\right )} + {\left (i \, x^{3} + i \, x e^{\left (2 i \, a\right )}\right )} \log \left (x + i \, e^{\left (i \, a\right )}\right ) + {\left (-i \, x^{3} - i \, x e^{\left (2 i \, a\right )}\right )} \log \left (x - i \, e^{\left (i \, a\right )}\right ) + e^{\left (3 i \, a\right )}}{x^{3} e^{\left (i \, a\right )} + x e^{\left (3 i \, a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.56, size = 73, normalized size = 1.22 \[ 2 \, {\left (\arctan \left (x e^{\left (-i \, a\right )}\right ) e^{\left (-3 i \, a\right )} + \frac {x e^{\left (-2 i \, a\right )}}{x^{2} + e^{\left (2 i \, a\right )}}\right )} e^{\left (2 i \, a\right )} + \frac {5}{x {\left (\frac {e^{\left (2 i \, a\right )}}{x^{2}} + 1\right )}} + \frac {e^{\left (2 i \, a\right )}}{x^{3} {\left (\frac {e^{\left (2 i \, a\right )}}{x^{2}} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 38, normalized size = 0.63 \[ \frac {1}{x}+\frac {2}{x \left (1+\frac {{\mathrm e}^{2 i a}}{x^{2}}\right )}+2 \arctan \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{-i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 231, normalized size = 3.85 \[ \frac {6 \, x^{2} - {\left (x^{3} {\left (2 \, \cos \relax (a) - 2 i \, \sin \relax (a)\right )} + {\left ({\left (2 \, \cos \relax (a) - 2 i \, \sin \relax (a)\right )} \cos \left (2 \, a\right ) + 2 \, {\left (i \, \cos \relax (a) + \sin \relax (a)\right )} \sin \left (2 \, a\right )\right )} x\right )} \arctan \left (\frac {2 \, x \cos \relax (a)}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}, \frac {x^{2} - \cos \relax (a)^{2} - \sin \relax (a)^{2}}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}\right ) + {\left (x^{3} {\left (-i \, \cos \relax (a) - \sin \relax (a)\right )} + {\left ({\left (-i \, \cos \relax (a) - \sin \relax (a)\right )} \cos \left (2 \, a\right ) + {\left (\cos \relax (a) - i \, \sin \relax (a)\right )} \sin \left (2 \, a\right )\right )} x\right )} \log \left (\frac {x^{2} + \cos \relax (a)^{2} + 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}\right ) + 2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )}{2 \, x^{3} + x {\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.20, size = 45, normalized size = 0.75 \[ \frac {2\,\mathrm {atan}\left (\frac {x}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}+\frac {3\,x^2+{\mathrm {e}}^{a\,2{}\mathrm {i}}}{x^3+{\mathrm {e}}^{a\,2{}\mathrm {i}}\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 54, normalized size = 0.90 \[ - \frac {- 3 x^{2} - e^{2 i a}}{x^{3} + x e^{2 i a}} - \left (i \log {\left (x - i e^{i a} \right )} - i \log {\left (x + i e^{i a} \right )}\right ) e^{- i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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