Optimal. Leaf size=55 \[ -\frac{2 e^{-2 i a}}{1+\frac{e^{2 i a}}{x^2}}-2 e^{-2 i a} \log \left (1+\frac{e^{2 i a}}{x^2}\right )+\frac{1}{2 x^2} \]
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Rubi [F] time = 0.052695, 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 = {} \[ \int \frac{\tan ^2(a+i \log (x))}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\tan ^2(a+i \log (x))}{x^3} \, dx &=\int \frac{\tan ^2(a+i \log (x))}{x^3} \, dx\\ \end{align*}
Mathematica [B] time = 0.170479, size = 150, normalized size = 2.73 \[ -\cos (2 a) \log \left (2 x^2 \cos (2 a)+x^4+1\right )+\frac{2 \cos (a)-2 i \sin (a)}{\left (x^2+1\right ) \cos (a)-i \left (x^2-1\right ) \sin (a)}+i \sin (2 a) \log \left (2 x^2 \cos (2 a)+x^4+1\right )-2 i \cos (2 a) \tan ^{-1}\left (\frac{\left (x^2+1\right ) \cot (a)}{x^2-1}\right )-2 \sin (2 a) \tan ^{-1}\left (\frac{\left (x^2+1\right ) \cot (a)}{x^2-1}\right )-4 i \sin (2 a) \log (x)+4 \cos (2 a) \log (x)+\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 61, normalized size = 1.1 \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 ( \left ({{\rm e}^{ia}} \right ) ^{2}+{x}^{2} \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 = 1.0463, size = 60, normalized size = 1.09 \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 = 1.21621, size = 240, normalized size = 4.36 \begin{align*} -\frac{2 \, \log \left (-x^{2} - e^{\left (2 i \, a\right )}\right )}{\frac{e^{\left (4 i \, a\right )}}{x^{2}} + e^{\left (2 i \, a\right )}} + \frac{4 \, \log \left (x\right )}{\frac{e^{\left (4 i \, a\right )}}{x^{2}} + e^{\left (2 i \, a\right )}} - \frac{2}{\frac{e^{\left (4 i \, a\right )}}{x^{2}} + e^{\left (2 i \, a\right )}} - \frac{2 \, e^{\left (2 i \, a\right )} \log \left (-x^{2} - e^{\left (2 i \, a\right )}\right )}{x^{2}{\left (\frac{e^{\left (4 i \, a\right )}}{x^{2}} + e^{\left (2 i \, a\right )}\right )}} + \frac{4 \, e^{\left (2 i \, a\right )} \log \left (x\right )}{x^{2}{\left (\frac{e^{\left (4 i \, a\right )}}{x^{2}} + e^{\left (2 i \, a\right )}\right )}} + \frac{e^{\left (2 i \, a\right )}}{2 \, x^{2}{\left (\frac{e^{\left (4 i \, a\right )}}{x^{2}} + e^{\left (2 i \, a\right )}\right )}} + \frac{e^{\left (4 i \, a\right )}}{2 \, x^{4}{\left (\frac{e^{\left (4 i \, a\right )}}{x^{2}} + e^{\left (2 i \, a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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