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 = 0.0491912, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}
Mathematica [A] time = 0.108788, size = 72, normalized size = 1.2 \[ \frac{2 x (\cos (a)-i \sin (a))}{\left (x^2+1\right ) \cos (a)-i \left (x^2-1\right ) \sin (a)}+2 \cos (a) \tan ^{-1}(x (\cos (a)-i \sin (a)))-2 i \sin (a) \tan ^{-1}(x (\cos (a)-i \sin (a)))+\frac{1}{x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \tan \left ( a+i\ln \left ( x \right ) \right ) \right ) ^{2}}{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.6189, size = 312, normalized size = 5.2 \begin{align*} \frac{6 \, x^{2} -{\left (x^{3}{\left (2 \, \cos \left (a\right ) - 2 i \, \sin \left (a\right )\right )} +{\left ({\left (2 \, \cos \left (a\right ) - 2 i \, \sin \left (a\right )\right )} \cos \left (2 \, a\right ) + 2 \,{\left (i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} x\right )} \arctan \left (\frac{2 \, x \cos \left (a\right )}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}, \frac{x^{2} - \cos \left (a\right )^{2} - \sin \left (a\right )^{2}}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}\right ) +{\left (x^{3}{\left (-i \, \cos \left (a\right ) - \sin \left (a\right )\right )} +{\left ({\left (-i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \cos \left (2 \, a\right ) +{\left (\cos \left (a\right ) - i \, \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} x\right )} \log \left (\frac{x^{2} + \cos \left (a\right )^{2} + 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (x e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + x\right )}{\rm integral}\left (-\frac{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}{x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + x^{2}}, x\right ) + 2}{x e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.672558, size = 51, normalized size = 0.85 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21748, size = 99, normalized size = 1.65 \begin{align*} 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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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