Optimal. Leaf size=35 \[ -i e^{2 i a} \log \left (-x^2+e^{2 i a}\right )-\frac{i x^2}{2} \]
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Rubi [F] time = 0.014523, 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 = {} \[ \int x \cot (a+i \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x \cot (a+i \log (x)) \, dx &=\int x \cot (a+i \log (x)) \, dx\\ \end{align*}
Mathematica [B] time = 0.0233421, size = 118, normalized size = 3.37 \[ -\frac{1}{2} i \cos (2 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )+\frac{1}{2} \sin (2 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )-\cos (2 a) \tan ^{-1}\left (\frac{\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )-i \sin (2 a) \tan ^{-1}\left (\frac{\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )-\frac{i x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 53, normalized size = 1.5 \begin{align*}{\frac{i}{2}}{x}^{2}+i \left ( -{x}^{2}- \left ({{\rm e}^{ia}} \right ) ^{2}\ln \left ({{\rm e}^{ia}}-x \right ) - \left ({{\rm e}^{ia}} \right ) ^{2}\ln \left ({{\rm e}^{ia}}+x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09912, size = 154, normalized size = 4.4 \begin{align*} -\frac{1}{2} i \, x^{2} + \frac{1}{2} \,{\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) - \frac{1}{2} \,{\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right ) + \frac{1}{2} \,{\left (-i \, \cos \left (2 \, a\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 ) + \frac{1}{2} \,{\left (-i \, \cos \left (2 \, a\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 ) \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*}{\rm integral}\left (\frac{i \, x e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i \, x}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.505025, size = 27, normalized size = 0.77 \begin{align*} - \frac{i x^{2}}{2} - i e^{2 i a} \log{\left (x^{2} - e^{2 i a} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37946, size = 55, normalized size = 1.57 \begin{align*} -\frac{1}{2} i \, x^{2} + \frac{1}{2} \, \pi e^{\left (2 i \, a\right )} - i \, e^{\left (2 i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) - i \, e^{\left (2 i \, a\right )} \log \left (-x + e^{\left (i \, a\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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