Optimal. Leaf size=48 \[ -\frac{2 e^{2 i a} x}{-x^2+e^{2 i a}}+2 e^{i a} \tanh ^{-1}\left (e^{-i a} x\right )-x \]
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Rubi [F] time = 0.0105502, 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 = {} \[ \int \cot ^2(a+i \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \cot ^2(a+i \log (x)) \, dx &=\int \cot ^2(a+i \log (x)) \, dx\\ \end{align*}
Mathematica [A] time = 0.0845853, size = 70, normalized size = 1.46 \[ \frac{-x \left (x^2-3\right ) \cos (a)+i x \left (x^2+3\right ) \sin (a)}{\left (x^2-1\right ) \cos (a)-i \left (x^2+1\right ) \sin (a)}+2 (\cos (a)+i \sin (a)) \tanh ^{-1}(x (\cos (a)-i \sin (a))) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 56, normalized size = 1.2 \begin{align*} -3\,x-2\,{\frac{x}{ \left ({{\rm e}^{i \left ( a+i\ln \left ( x \right ) \right ) }} \right ) ^{2}-1}}-{{\rm e}^{ia}}\ln \left ({{\rm e}^{ia}}-x \right ) +{{\rm e}^{ia}}\ln \left ({{\rm e}^{ia}}+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.2429, size = 375, normalized size = 7.81 \begin{align*} -\frac{2 \,{\left ({\left (-i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) +{\left (-i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right )\right )} x^{2} + 2 \, x^{3} - x{\left (6 \, \cos \left (2 \, a\right ) + 6 i \, \sin \left (2 \, a\right )\right )} +{\left (2 \,{\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \cos \left (2 \, a\right ) -{\left (2 \, \cos \left (a\right ) + 2 i \, \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) +{\left (2 \,{\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \cos \left (2 \, a\right ) -{\left (2 \, \cos \left (a\right ) + 2 i \, \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right ) -{\left (x^{2}{\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} -{\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} \cos \left (2 \, a\right ) +{\left (-i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} \log \left (x^{2} + 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) +{\left (x^{2}{\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} -{\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} \cos \left (2 \, a\right ) -{\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} \log \left (x^{2} - 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \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 = 0.480368, size = 42, normalized size = 0.88 \begin{align*} - x + \frac{2 x e^{2 i a}}{x^{2} - e^{2 i a}} - \left (\log{\left (x - e^{i a} \right )} - \log{\left (x + 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 = 1.33802, size = 107, normalized size = 2.23 \begin{align*} -\frac{x^{3}}{x^{2} - e^{\left (2 i \, a\right )}} - 2 \,{\left (\frac{\arctan \left (\frac{x}{\sqrt{-e^{\left (2 i \, a\right )}}}\right )}{\sqrt{-e^{\left (2 i \, a\right )}}} - \frac{x}{x^{2} - e^{\left (2 i \, a\right )}}\right )} e^{\left (2 i \, a\right )} + \frac{5 \, x e^{\left (2 i \, a\right )}}{x^{2} - e^{\left (2 i \, a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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