Optimal. Leaf size=57 \[ \frac{2 e^{-2 i a}}{1-\frac{e^{2 i a}}{x^2}}+2 e^{-2 i a} \log \left (1-\frac{e^{2 i a}}{x^2}\right )+\frac{1}{2 x^2} \]
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Rubi [F] time = 0.050647, 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 = {} \[ \int \frac{\cot ^2(a+i \log (x))}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\cot ^2(a+i \log (x))}{x^3} \, dx &=\int \frac{\cot ^2(a+i \log (x))}{x^3} \, dx\\ \end{align*}
Mathematica [B] time = 0.221049, size = 153, normalized size = 2.68 \[ \cos (2 a) \left (\log \left (-2 x^2 \cos (2 a)+x^4+1\right )-4 \log (x)\right )+\frac{2 \cos (a)}{\left (x^2-1\right ) \cos (a)-i \left (x^2+1\right ) \sin (a)}+\frac{2 \sin (a)}{\left (x^2+1\right ) \sin (a)+i \left (x^2-1\right ) \cos (a)}-i \sin (2 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )+(-4 \sin (a) \cos (a)-2 i \cos (2 a)) \tan ^{-1}\left (\frac{\cot (a)-x^2 \cot (a)}{x^2+1}\right )+4 i \sin (2 a) \log (x)+\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 76, normalized size = 1.3 \begin{align*}{\frac{1}{2\,{x}^{2}}}-2\,{\frac{1}{{x}^{2} \left ( \left ({{\rm e}^{i \left ( a+i\ln \left ( x \right ) \right ) }} \right ) ^{2}-1 \right ) }}-4\,{\frac{\ln \left ( x \right ) }{ \left ({{\rm e}^{ia}} \right ) ^{2}}}+2\,{\frac{\ln \left ({{\rm e}^{ia}}-x \right ) }{ \left ({{\rm e}^{ia}} \right ) ^{2}}}+2\,{\frac{\ln \left ({{\rm e}^{ia}}+x \right ) }{ \left ({{\rm e}^{ia}} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{2}\right )}{\rm integral}\left (-\frac{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 3}{x^{3} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{3}}, x\right ) - 2}{x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.17795, size = 60, normalized size = 1.05 \begin{align*} \frac{5 x^{2} - e^{2 i a}}{2 x^{4} - 2 x^{2} e^{2 i a}} - 4 e^{- 2 i a} \log{\left (x \right )} + 2 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 [B] time = 1.34441, size = 257, normalized size = 4.51 \begin{align*} \frac{2 \, x^{4} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right )}{x^{4} e^{\left (2 i \, a\right )} - x^{2} e^{\left (4 i \, a\right )}} - \frac{4 \, x^{4} \log \left (x\right )}{x^{4} e^{\left (2 i \, a\right )} - x^{2} e^{\left (4 i \, a\right )}} - \frac{2 \, x^{2} e^{\left (2 i \, a\right )} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right )}{x^{4} e^{\left (2 i \, a\right )} - x^{2} e^{\left (4 i \, a\right )}} + \frac{4 \, x^{2} e^{\left (2 i \, a\right )} \log \left (x\right )}{x^{4} e^{\left (2 i \, a\right )} - x^{2} e^{\left (4 i \, a\right )}} + \frac{5 \, x^{2} e^{\left (2 i \, a\right )}}{2 \,{\left (x^{4} e^{\left (2 i \, a\right )} - x^{2} e^{\left (4 i \, a\right )}\right )}} - \frac{e^{\left (4 i \, a\right )}}{2 \,{\left (x^{4} e^{\left (2 i \, a\right )} - x^{2} e^{\left (4 i \, a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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