Optimal. Leaf size=57 \[ \frac {2 e^{-2 i a}}{1-\frac {e^{2 i a}}{x^2}}+\frac {1}{2 x^2}+2 e^{-2 i a} \log \left (1-\frac {e^{2 i a}}{x^2}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4592, 455, 45}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 455
Rule 4592
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*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(153\) vs. \(2(57)=114\).
time = 0.17, size = 153, normalized size = 2.68 \begin {gather*} \frac {1}{2 x^2}+\cos (2 a) \left (-4 \log (x)+\log \left (1+x^4-2 x^2 \cos (2 a)\right )\right )+\frac {2 \cos (a)}{\left (-1+x^2\right ) \cos (a)-i \left (1+x^2\right ) \sin (a)}+\frac {2 \sin (a)}{i \left (-1+x^2\right ) \cos (a)+\left (1+x^2\right ) \sin (a)}+\text {ArcTan}\left (\frac {\cot (a)-x^2 \cot (a)}{1+x^2}\right ) (-2 i \cos (2 a)-4 \cos (a) \sin (a))+4 i \log (x) \sin (2 a)-i \log \left (1+x^4-2 x^2 \cos (2 a)\right ) \sin (2 a) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 53, normalized size = 0.93
method | result | size |
risch | \(\frac {1}{2 x^{2}}-\frac {2}{x^{2} \left (\frac {{\mathrm e}^{2 i a}}{x^{2}}-1\right )}+2 \,{\mathrm e}^{-2 i a} \ln \left ({\mathrm e}^{2 i a}-x^{2}\right )-4 \,{\mathrm e}^{-2 i a} \ln \left (x \right )\) | \(53\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.87, size = 81, normalized size = 1.42 \begin {gather*} \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 \left (x\right ) - 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 60, normalized size = 1.05 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 190 vs. \(2 (42) = 84\).
time = 0.45, size = 190, normalized size = 3.33 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.23, size = 60, normalized size = 1.05 \begin {gather*} -4\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\ln \left (x\right )+2\,\ln \left (x^2-{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}+\frac {\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}}{2}-\frac {5\,x^2}{2}}{x^2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}-x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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