Optimal. Leaf size=46 \[ -\frac{2 e^{2 i a} x}{x^2+e^{2 i a}}+2 e^{i a} \tan ^{-1}\left (e^{-i a} x\right )-x \]
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Rubi [F] time = 0.0104115, 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 = {} \[ \int \tan ^2(a+i \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \tan ^2(a+i \log (x)) \, dx &=\int \tan ^2(a+i \log (x)) \, dx\\ \end{align*}
Mathematica [A] time = 0.0887918, size = 70, normalized size = 1.52 \[ \frac{-x \left (x^2+3\right ) \cos (a)+i x \left (x^2-3\right ) \sin (a)}{\left (x^2+1\right ) \cos (a)-i \left (x^2-1\right ) \sin (a)}+2 (\cos (a)+i \sin (a)) \tan ^{-1}(x (\cos (a)-i \sin (a))) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.061, size = 0, normalized size = 0. \begin{align*} \int \left ( \tan \left ( a+i\ln \left ( x \right ) \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.59874, size = 305, normalized size = 6.63 \begin{align*} -\frac{2 \, x^{3} + x{\left (6 \, \cos \left (2 \, a\right ) + 6 i \, \sin \left (2 \, a\right )\right )} +{\left (x^{2}{\left (2 \, \cos \left (a\right ) + 2 i \, \sin \left (a\right )\right )} +{\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 )} \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^{2}{\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} +{\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 )} \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 \, x^{2} + 2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\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 (e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1\right )}{\rm integral}\left (-\frac{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 3}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1}, x\right ) + 2 \, x}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.60995, size = 51, normalized size = 1.11 \begin{align*} - x - \frac{2 x e^{2 i a}}{x^{2} + 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 [B] time = 1.18675, size = 154, normalized size = 3.35 \begin{align*} -\frac{x^{3}}{x^{2} + \frac{e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}} + 2 \,{\left (\arctan \left (x e^{\left (-i \, a\right )}\right ) e^{\left (-i \, a\right )} - \frac{x}{x^{2} + e^{\left (2 i \, a\right )}}\right )} e^{\left (2 i \, a\right )} - \frac{6 \, x e^{\left (2 i \, a\right )}}{x^{2} + \frac{e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}} - \frac{5 \, e^{\left (4 i \, a\right )}}{{\left (x^{2} + \frac{e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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