Integrand size = 14, antiderivative size = 14 \[ \int \frac {e^{-2 i \arctan (a x)}}{x} \, dx=\log (x)-2 \log (i-a x) \]
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Time = 0.02 (sec) , antiderivative size = 14, 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 = 14, 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 = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00
method | result | size |
default | \(\ln \left (x \right )-2 \ln \left (-a x +i\right )\) | \(14\) |
parallelrisch | \(\frac {\ln \left (x \right ) a -2 \ln \left (a x -i\right ) a}{a}\) | \(20\) |
risch | \(\ln \left (x \right )-\ln \left (a^{2} x^{2}+1\right )-2 i \arctan \left (a x \right )\) | \(23\) |
meijerg | \(\frac {i a x}{i a x +1}-2 \ln \left (i a x +1\right )-\frac {2 i a x}{2 i a x +2}+1+\ln \left (x \right )+\ln \left (i a \right )\) | \(48\) |
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none
Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-2 i \arctan (a x)}}{x} \, dx=\log \left (x\right ) - 2 \, \log \left (\frac {a x - i}{a}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.21 \[ \int \frac {e^{-2 i \arctan (a x)}}{x} \, dx=\log {\left (3 a x \right )} - 2 \log {\left (3 a x - 3 i \right )} \]
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none
Time = 0.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {e^{-2 i \arctan (a x)}}{x} \, dx=-2 \, \log \left (i \, a x + 1\right ) + \log \left (x\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (12) = 24\).
Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 3.14 \[ \int \frac {e^{-2 i \arctan (a x)}}{x} \, dx=i \, a {\left (-\frac {i \, \log \left (\frac {i}{i \, a x + 1} - i\right )}{a} - \frac {i \, \log \left (\frac {1}{\sqrt {a^{2} x^{2} + 1} {\left | a \right |}}\right )}{a}\right )} \]
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Time = 0.50 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 i \arctan (a x)}}{x} \, dx=\ln \left (x\right )-2\,\ln \left (x-\frac {1{}\mathrm {i}}{a}\right ) \]
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