Integrand size = 13, antiderivative size = 29 \[ \int \frac {\tan (a+i \log (x))}{x^2} \, dx=\frac {i}{x}+2 i e^{-i a} \arctan \left (e^{-i a} x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4591, 456, 464, 209} \[ \int \frac {\tan (a+i \log (x))}{x^2} \, dx=2 i e^{-i a} \arctan \left (e^{-i a} x\right )+\frac {i}{x} \]
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Rule 209
Rule 456
Rule 464
Rule 4591
Rubi steps \begin{align*} \text {integral}& = \int \frac {i-\frac {i e^{2 i a}}{x^2}}{\left (1+\frac {e^{2 i a}}{x^2}\right ) x^2} \, dx \\ & = \int \frac {-i e^{2 i a}+i x^2}{x^2 \left (e^{2 i a}+x^2\right )} \, dx \\ & = \frac {i}{x}+2 i \int \frac {1}{e^{2 i a}+x^2} \, dx \\ & = \frac {i}{x}+2 i e^{-i a} \arctan \left (e^{-i a} x\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {\tan (a+i \log (x))}{x^2} \, dx=\frac {i}{x}+2 i \arctan (x \cos (a)-i x \sin (a)) \cos (a)+2 \arctan (x \cos (a)-i x \sin (a)) \sin (a) \]
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Time = 1.68 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {i}{x}+2 i \arctan \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{-i a}\) | \(24\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {\tan (a+i \log (x))}{x^2} \, dx=-\frac {{\left (x \log \left (x + i \, e^{\left (i \, a\right )}\right ) - x \log \left (x - i \, e^{\left (i \, a\right )}\right ) - i \, e^{\left (i \, a\right )}\right )} e^{\left (-i \, a\right )}}{x} \]
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Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {\tan (a+i \log (x))}{x^2} \, dx=\left (\log {\left (x - i e^{i a} \right )} - \log {\left (x + i e^{i a} \right )}\right ) e^{- i a} + \frac {i}{x} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (19) = 38\).
Time = 0.30 (sec) , antiderivative size = 127, normalized size of antiderivative = 4.38 \[ \int \frac {\tan (a+i \log (x))}{x^2} \, dx=\frac {2 \, x {\left (-i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \arctan \left (\frac {2 \, x \cos \left (a\right )}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}, \frac {x^{2} - \cos \left (a\right )^{2} - \sin \left (a\right )^{2}}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}\right ) + x {\left (\cos \left (a\right ) - i \, \sin \left (a\right )\right )} \log \left (\frac {x^{2} + \cos \left (a\right )^{2} + 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}\right ) + 2 i}{2 \, x} \]
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none
Time = 0.36 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {\tan (a+i \log (x))}{x^2} \, dx=2 i \, \arctan \left (x e^{\left (-i \, a\right )}\right ) e^{\left (-i \, a\right )} + \frac {i}{x} \]
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Time = 27.70 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {\tan (a+i \log (x))}{x^2} \, dx=\frac {\mathrm {atan}\left (\frac {x}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,2{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}+\frac {1{}\mathrm {i}}{x} \]
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