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 = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}
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Mathematica [B] time = 0.19, size = 150, normalized size = 2.73 \[ \frac {2 \cos (a)-2 i \sin (a)}{\left (x^2+1\right ) \cos (a)-i \left (x^2-1\right ) \sin (a)}-2 i \cos (2 a) \tan ^{-1}\left (\frac {\left (x^2+1\right ) \cot (a)}{x^2-1}\right )-2 \sin (2 a) \tan ^{-1}\left (\frac {\left (x^2+1\right ) \cot (a)}{x^2-1}\right )-\cos (2 a) \log \left (2 x^2 \cos (2 a)+x^4+1\right )+i \sin (2 a) \log \left (2 x^2 \cos (2 a)+x^4+1\right )-4 i \sin (2 a) \log (x)+4 \cos (2 a) \log (x)+\frac {1}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 74, normalized size = 1.35 \[ \frac {5 \, x^{2} e^{\left (2 i \, a\right )} - 4 \, {\left (x^{4} + x^{2} e^{\left (2 i \, a\right )}\right )} \log \left (x^{2} + e^{\left (2 i \, a\right )}\right ) + 8 \, {\left (x^{4} + x^{2} e^{\left (2 i \, a\right )}\right )} \log \relax (x) + e^{\left (4 i \, a\right )}}{2 \, {\left (x^{4} e^{\left (2 i \, a\right )} + x^{2} e^{\left (4 i \, a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.77, size = 178, normalized size = 3.24 \[ -\frac {2 \, \log \left (-x^{2} - e^{\left (2 i \, a\right )}\right )}{\frac {e^{\left (4 i \, a\right )}}{x^{2}} + e^{\left (2 i \, a\right )}} + \frac {4 \, \log \relax (x)}{\frac {e^{\left (4 i \, a\right )}}{x^{2}} + e^{\left (2 i \, a\right )}} - \frac {2}{\frac {e^{\left (4 i \, a\right )}}{x^{2}} + e^{\left (2 i \, a\right )}} - \frac {2 \, e^{\left (2 i \, a\right )} \log \left (-x^{2} - e^{\left (2 i \, a\right )}\right )}{x^{2} {\left (\frac {e^{\left (4 i \, a\right )}}{x^{2}} + e^{\left (2 i \, a\right )}\right )}} + \frac {4 \, e^{\left (2 i \, a\right )} \log \relax (x)}{x^{2} {\left (\frac {e^{\left (4 i \, a\right )}}{x^{2}} + e^{\left (2 i \, a\right )}\right )}} + \frac {e^{\left (2 i \, a\right )}}{2 \, x^{2} {\left (\frac {e^{\left (4 i \, a\right )}}{x^{2}} + e^{\left (2 i \, a\right )}\right )}} + \frac {e^{\left (4 i \, a\right )}}{2 \, x^{4} {\left (\frac {e^{\left (4 i \, a\right )}}{x^{2}} + e^{\left (2 i \, a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 51, normalized size = 0.93 \[ \frac {1}{2 x^{2}}+\frac {2}{x^{2} \left (1+\frac {{\mathrm e}^{2 i a}}{x^{2}}\right )}+4 \,{\mathrm e}^{-2 i a} \ln \relax (x )-2 \,{\mathrm e}^{-2 i a} \ln \left ({\mathrm e}^{2 i a}+x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.21, size = 56, normalized size = 1.02 \[ -2\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\ln \left (x^2+{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )+4\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\ln \relax (x)+\frac {\frac {5\,x^2}{2}+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}}{2}}{x^4+{\mathrm {e}}^{a\,2{}\mathrm {i}}\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 61, normalized size = 1.11 \[ - \frac {- 5 x^{2} - e^{2 i a}}{2 x^{4} + 2 x^{2} e^{2 i a}} + 4 e^{- 2 i a} \log {\relax (x )} - 2 e^{- 2 i a} \log {\left (x^{2} + e^{2 i a} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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