Integrand size = 9, antiderivative size = 27 \[ \int \tan (a+i \log (x)) \, dx=i x-2 i e^{i a} \arctan \left (e^{-i a} x\right ) \]
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Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4587, 381, 396, 209} \[ \int \tan (a+i \log (x)) \, dx=i x-2 i e^{i a} \arctan \left (e^{-i a} x\right ) \]
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Rule 209
Rule 381
Rule 396
Rule 4587
Rubi steps \begin{align*} \text {integral}& = \int \frac {i-\frac {i e^{2 i a}}{x^2}}{1+\frac {e^{2 i a}}{x^2}} \, dx \\ & = \int \frac {-i e^{2 i a}+i x^2}{e^{2 i a}+x^2} \, dx \\ & = i x-\left (2 i e^{2 i a}\right ) \int \frac {1}{e^{2 i a}+x^2} \, dx \\ & = i x-2 i e^{i a} \arctan \left (e^{-i a} x\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \tan (a+i \log (x)) \, dx=i x-2 i \arctan (x \cos (a)-i x \sin (a)) \cos (a)+2 \arctan (x \cos (a)-i x \sin (a)) \sin (a) \]
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Time = 1.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81
method | result | size |
risch | \(i x -2 i \arctan \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{i a}\) | \(22\) |
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Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \tan (a+i \log (x)) \, dx=e^{\left (i \, a\right )} \log \left (x + i \, e^{\left (i \, a\right )}\right ) - e^{\left (i \, a\right )} \log \left (x - i \, e^{\left (i \, a\right )}\right ) + i \, x \]
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Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \tan (a+i \log (x)) \, dx=i x + \left (- \log {\left (x - i e^{i a} \right )} + \log {\left (x + i e^{i a} \right )}\right ) e^{i a} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (17) = 34\).
Time = 0.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.52 \[ \int \tan (a+i \log (x)) \, dx={\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \arctan \left (\frac {2 \, x \cos \left (a\right )}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}, \frac {x^{2} - \cos \left (a\right )^{2} - \sin \left (a\right )^{2}}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}\right ) - \frac {1}{2} \, {\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} \log \left (\frac {x^{2} + \cos \left (a\right )^{2} + 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}\right ) + i \, x \]
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Time = 0.32 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \tan (a+i \log (x)) \, dx=-2 i \, \arctan \left (x e^{\left (-i \, a\right )}\right ) e^{\left (i \, a\right )} + i \, x \]
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Time = 27.55 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \tan (a+i \log (x)) \, dx=x\,1{}\mathrm {i}-\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}\,\mathrm {atan}\left (\frac {x}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,2{}\mathrm {i} \]
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