Integrand size = 14, antiderivative size = 13 \[ \int \frac {e^{2 i \arctan (a x)}}{x} \, dx=\log (x)-2 \log (i+a x) \]
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Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5170, 78} \[ \int \frac {e^{2 i \arctan (a x)}}{x} \, dx=\log (x)-2 \log (a x+i) \]
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Rule 78
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+i a x}{x (1-i a x)} \, dx \\ & = \int \left (\frac {1}{x}-\frac {2 a}{i+a x}\right ) \, dx \\ & = \log (x)-2 \log (i+a x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 i \arctan (a x)}}{x} \, dx=\log (x)-2 \log (i+a x) \]
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Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(\ln \left (x \right )-2 \ln \left (a x +i\right )\) | \(13\) |
risch | \(\ln \left (-x \right )-\ln \left (a^{2} x^{2}+1\right )+2 i \arctan \left (a x \right )\) | \(25\) |
meijerg | \(-\ln \left (a^{2} x^{2}+1\right )+\ln \left (x \right )+\frac {\ln \left (a^{2}\right )}{2}+2 i \arctan \left (a x \right )\) | \(29\) |
default | \(\ln \left (x \right )+2 a \left (-\frac {\ln \left (a^{2} x^{2}+1\right )}{2 a}+\frac {i \arctan \left (a x \right )}{a}\right )\) | \(33\) |
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Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {e^{2 i \arctan (a x)}}{x} \, dx=\log \left (x\right ) - 2 \, \log \left (\frac {a x + i}{a}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.31 \[ \int \frac {e^{2 i \arctan (a x)}}{x} \, dx=\log {\left (3 a x \right )} - 2 \log {\left (3 a x + 3 i \right )} \]
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Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.62 \[ \int \frac {e^{2 i \arctan (a x)}}{x} \, dx=2 i \, \arctan \left (a x\right ) - \log \left (a^{2} x^{2} + 1\right ) + \log \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {e^{2 i \arctan (a x)}}{x} \, dx=-2 \, \log \left (a x + i\right ) + \log \left ({\left | x \right |}\right ) \]
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Time = 0.50 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {e^{2 i \arctan (a x)}}{x} \, dx=\ln \left (x\right )-2\,\ln \left (x+\frac {1{}\mathrm {i}}{a}\right ) \]
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