Integrand size = 24, antiderivative size = 29 \[ \int \frac {e^{-i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=-\frac {1}{2 a (i-a x)}+\frac {\arctan (a x)}{2 a} \]
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Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5181, 46, 209} \[ \int \frac {e^{-i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=\frac {\arctan (a x)}{2 a}-\frac {1}{2 a (-a x+i)} \]
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Rule 46
Rule 209
Rule 5181
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(1-i a x) (1+i a x)^2} \, dx \\ & = \int \left (-\frac {1}{2 (-i+a x)^2}+\frac {1}{2 \left (1+a^2 x^2\right )}\right ) \, dx \\ & = -\frac {1}{2 a (i-a x)}+\frac {1}{2} \int \frac {1}{1+a^2 x^2} \, dx \\ & = -\frac {1}{2 a (i-a x)}+\frac {\arctan (a x)}{2 a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {e^{-i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=\frac {\frac {1}{-i+a x}+\arctan (a x)}{2 a} \]
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Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {1}{2 a \left (a x -i\right )}+\frac {\arctan \left (a x \right )}{2 a}\) | \(24\) |
default | \(-\frac {i \ln \left (-a x +i\right )}{4 a}-\frac {1}{2 a \left (-a x +i\right )}+\frac {i \ln \left (a x +i\right )}{4 a}\) | \(43\) |
parallelrisch | \(\frac {i \ln \left (a x -i\right ) x a -i \ln \left (a x +i\right ) a x +2 i a x +\ln \left (a x -i\right )-\ln \left (a x +i\right )}{4 \left (-a x +i\right ) a}\) | \(61\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {e^{-i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=\frac {{\left (i \, a x + 1\right )} \log \left (\frac {a x + i}{a}\right ) + {\left (-i \, a x - 1\right )} \log \left (\frac {a x - i}{a}\right ) + 2}{4 \, {\left (a^{2} x - i \, a\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=- i \left (\frac {i}{2 a^{2} x - 2 i a} + \frac {\frac {\log {\left (x - \frac {i}{a} \right )}}{4} - \frac {\log {\left (x + \frac {i}{a} \right )}}{4}}{a}\right ) \]
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Exception generated. \[ \int \frac {e^{-i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {e^{-i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=\frac {i \, \log \left (a x + i\right )}{4 \, a} - \frac {i \, \log \left (a x - i\right )}{4 \, a} + \frac {1}{2 \, {\left (a x - i\right )} a} \]
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Time = 0.60 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {e^{-i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx=\frac {\mathrm {atan}\left (a\,x\right )}{2\,a}-\frac {1}{2\,\left (-a^2\,x+a\,1{}\mathrm {i}\right )} \]
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