Integrand size = 15, antiderivative size = 57 \[ \int \frac {\cot ^2(a+i \log (x))}{x^3} \, dx=\frac {2 e^{-2 i a}}{1-\frac {e^{2 i a}}{x^2}}+\frac {1}{2 x^2}+2 e^{-2 i a} \log \left (1-\frac {e^{2 i a}}{x^2}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4592, 455, 45} \[ \int \frac {\cot ^2(a+i \log (x))}{x^3} \, dx=\frac {2 e^{-2 i a}}{1-\frac {e^{2 i a}}{x^2}}+2 e^{-2 i a} \log \left (1-\frac {e^{2 i a}}{x^2}\right )+\frac {1}{2 x^2} \]
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Rule 45
Rule 455
Rule 4592
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-i-\frac {i e^{2 i a}}{x^2}\right )^2}{\left (1-\frac {e^{2 i a}}{x^2}\right )^2 x^3} \, dx \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {\left (-i-i e^{2 i a} x\right )^2}{\left (1-e^{2 i a} x\right )^2} \, dx,x,\frac {1}{x^2}\right )\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \left (-1-\frac {4}{\left (-1+e^{2 i a} x\right )^2}-\frac {4}{-1+e^{2 i a} x}\right ) \, dx,x,\frac {1}{x^2}\right )\right ) \\ & = \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 ) \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(153\) vs. \(2(57)=114\).
Time = 0.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.68 \[ \int \frac {\cot ^2(a+i \log (x))}{x^3} \, dx=\frac {1}{2 x^2}+\cos (2 a) \left (-4 \log (x)+\log \left (1+x^4-2 x^2 \cos (2 a)\right )\right )+\frac {2 \cos (a)}{\left (-1+x^2\right ) \cos (a)-i \left (1+x^2\right ) \sin (a)}+\frac {2 \sin (a)}{i \left (-1+x^2\right ) \cos (a)+\left (1+x^2\right ) \sin (a)}+\arctan \left (\frac {\cot (a)-x^2 \cot (a)}{1+x^2}\right ) (-2 i \cos (2 a)-4 \cos (a) \sin (a))+4 i \log (x) \sin (2 a)-i \log \left (1+x^4-2 x^2 \cos (2 a)\right ) \sin (2 a) \]
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Time = 3.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93
method | result | size |
risch | \(\frac {1}{2 x^{2}}-\frac {2}{x^{2} \left (\frac {{\mathrm e}^{2 i a}}{x^{2}}-1\right )}+2 \,{\mathrm e}^{-2 i a} \ln \left ({\mathrm e}^{2 i a}-x^{2}\right )-4 \,{\mathrm e}^{-2 i a} \ln \left (x \right )\) | \(53\) |
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none
Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.42 \[ \int \frac {\cot ^2(a+i \log (x))}{x^3} \, dx=\frac {5 \, x^{2} e^{\left (2 i \, a\right )} + 4 \, {\left (x^{4} - x^{2} e^{\left (2 i \, a\right )}\right )} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right ) - 8 \, {\left (x^{4} - x^{2} e^{\left (2 i \, a\right )}\right )} \log \left (x\right ) - e^{\left (4 i \, a\right )}}{2 \, {\left (x^{4} e^{\left (2 i \, a\right )} - x^{2} e^{\left (4 i \, a\right )}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.05 \[ \int \frac {\cot ^2(a+i \log (x))}{x^3} \, dx=- \frac {- 5 x^{2} + e^{2 i a}}{2 x^{4} - 2 x^{2} e^{2 i a}} - 4 e^{- 2 i a} \log {\left (x \right )} + 2 e^{- 2 i a} \log {\left (x^{2} - e^{2 i a} \right )} \]
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Exception generated. \[ \int \frac {\cot ^2(a+i \log (x))}{x^3} \, dx=\text {Exception raised: RuntimeError} \]
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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 = 0.32 (sec) , antiderivative size = 190, normalized size of antiderivative = 3.33 \[ \int \frac {\cot ^2(a+i \log (x))}{x^3} \, dx=\frac {2 \, x^{4} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right )}{x^{4} e^{\left (2 i \, a\right )} - x^{2} e^{\left (4 i \, a\right )}} - \frac {4 \, x^{4} \log \left (x\right )}{x^{4} e^{\left (2 i \, a\right )} - x^{2} e^{\left (4 i \, a\right )}} - \frac {2 \, x^{2} e^{\left (2 i \, a\right )} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right )}{x^{4} e^{\left (2 i \, a\right )} - x^{2} e^{\left (4 i \, a\right )}} + \frac {4 \, x^{2} e^{\left (2 i \, a\right )} \log \left (x\right )}{x^{4} e^{\left (2 i \, a\right )} - x^{2} e^{\left (4 i \, a\right )}} + \frac {5 \, x^{2} e^{\left (2 i \, a\right )}}{2 \, {\left (x^{4} e^{\left (2 i \, a\right )} - x^{2} e^{\left (4 i \, a\right )}\right )}} - \frac {e^{\left (4 i \, a\right )}}{2 \, {\left (x^{4} e^{\left (2 i \, a\right )} - x^{2} e^{\left (4 i \, a\right )}\right )}} \]
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Time = 27.57 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.05 \[ \int \frac {\cot ^2(a+i \log (x))}{x^3} \, dx=-4\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\ln \left (x\right )+2\,\ln \left (x^2-{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}+\frac {\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}}{2}-\frac {5\,x^2}{2}}{x^2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}-x^4} \]
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