Integrand size = 11, antiderivative size = 35 \[ \int x \cot (a+i \log (x)) \, dx=-\frac {i x^2}{2}-i e^{2 i a} \log \left (e^{2 i a}-x^2\right ) \]
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Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4592, 456, 455, 45} \[ \int x \cot (a+i \log (x)) \, dx=-i e^{2 i a} \log \left (-x^2+e^{2 i a}\right )-\frac {i x^2}{2} \]
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Rule 45
Rule 455
Rule 456
Rule 4592
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-i-\frac {i e^{2 i a}}{x^2}\right ) x}{1-\frac {e^{2 i a}}{x^2}} \, dx \\ & = \int \frac {x \left (-i e^{2 i a}-i x^2\right )}{-e^{2 i a}+x^2} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {-i e^{2 i a}-i x}{-e^{2 i a}+x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-i+\frac {2 i e^{2 i a}}{e^{2 i a}-x}\right ) \, dx,x,x^2\right ) \\ & = -\frac {i x^2}{2}-i e^{2 i a} \log \left (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. \(118\) vs. \(2(35)=70\).
Time = 0.02 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.37 \[ \int x \cot (a+i \log (x)) \, dx=-\frac {i x^2}{2}-\arctan \left (\frac {\left (-1+x^2\right ) \cos (a)}{-\sin (a)-x^2 \sin (a)}\right ) \cos (2 a)-\frac {1}{2} i \cos (2 a) \log \left (1+x^4-2 x^2 \cos (2 a)\right )-i \arctan \left (\frac {\left (-1+x^2\right ) \cos (a)}{-\sin (a)-x^2 \sin (a)}\right ) \sin (2 a)+\frac {1}{2} \log \left (1+x^4-2 x^2 \cos (2 a)\right ) \sin (2 a) \]
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Time = 0.46 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {i x^{2}}{2}-i {\mathrm e}^{2 i a} \ln \left ({\mathrm e}^{2 i a}-x^{2}\right )\) | \(28\) |
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Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.66 \[ \int x \cot (a+i \log (x)) \, dx=-\frac {1}{2} i \, x^{2} - i \, e^{\left (2 i \, a\right )} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int x \cot (a+i \log (x)) \, dx=- \frac {i x^{2}}{2} - i e^{2 i a} \log {\left (x^{2} - e^{2 i a} \right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (23) = 46\).
Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.11 \[ \int x \cot (a+i \log (x)) \, dx=-\frac {1}{2} i \, x^{2} + {\left (\cos \left (2 \, a\right ) + i \, \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) - {\left (\cos \left (2 \, a\right ) + i \, \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right ) + \frac {1}{2} \, {\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \log \left (x^{2} + 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) + \frac {1}{2} \, {\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \log \left (x^{2} - 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17 \[ \int x \cot (a+i \log (x)) \, dx=-\frac {1}{2} i \, x^{2} + \frac {1}{2} \, \pi e^{\left (2 i \, a\right )} - i \, e^{\left (2 i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) - i \, e^{\left (2 i \, a\right )} \log \left (-x + e^{\left (i \, a\right )}\right ) \]
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Time = 28.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int x \cot (a+i \log (x)) \, dx=-\ln \left (x^2-{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,1{}\mathrm {i}-\frac {x^2\,1{}\mathrm {i}}{2} \]
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