Optimal. Leaf size=18 \[ -\log (x)+i \cot (a+i \log (x)) \]
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Rubi [A] time = 0.0235847, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3473, 8} \[ -\log (x)+i \cot (a+i \log (x)) \]
Antiderivative was successfully verified.
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Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \frac{\cot ^2(a+i \log (x))}{x} \, dx &=\operatorname{Subst}\left (\int \cot ^2(a+i x) \, dx,x,\log (x)\right )\\ &=i \cot (a+i \log (x))-\operatorname{Subst}(\int 1 \, dx,x,\log (x))\\ &=i \cot (a+i \log (x))-\log (x)\\ \end{align*}
Mathematica [C] time = 0.0498985, size = 34, normalized size = 1.89 \[ i \cot (a+i \log (x)) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2(a+i \log (x))\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 27, normalized size = 1.5 \begin{align*} i\cot \left ( a+i\ln \left ( x \right ) \right ) -{\frac{i}{2}}\pi +i \left ( a+i\ln \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.72516, size = 26, normalized size = 1.44 \begin{align*} i \, a + \frac{i}{\tan \left (a + i \, \log \left (x\right )\right )} - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.476125, size = 97, normalized size = 5.39 \begin{align*} -\frac{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} \log \left (x\right ) - \log \left (x\right ) + 2}{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 = 0.538921, size = 20, normalized size = 1.11 \begin{align*} - \log{\left (x \right )} + \frac{2 e^{2 i a}}{x^{2} - e^{2 i a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.31752, size = 43, normalized size = 2.39 \begin{align*} i \, a + \frac{i}{2 \, \tan \left (\frac{1}{2} \, a + \frac{1}{2} i \, \log \left (x\right )\right )} - \log \left (x\right ) - \frac{1}{2} i \, \tan \left (\frac{1}{2} \, a + \frac{1}{2} i \, \log \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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