Integrand size = 24, antiderivative size = 15 \[ \int \frac {e^{i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {i \log (i+a x)}{a} \]
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Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5181, 31} \[ \int \frac {e^{i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {i \log (a x+i)}{a} \]
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Rule 31
Rule 5181
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{1-i a x} \, dx \\ & = \frac {i \log (i+a x)}{a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {e^{i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {i \log (i+a x)}{a} \]
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Time = 0.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93
method | result | size |
parallelrisch | \(\frac {i \ln \left (a x +i\right )}{a}\) | \(14\) |
default | \(\frac {i \ln \left (a^{2} x^{2}+1\right )}{2 a}+\frac {\arctan \left (a x \right )}{a}\) | \(26\) |
meijerg | \(\frac {i \ln \left (a^{2} x^{2}+1\right )}{2 a}+\frac {\arctan \left (a x \right )}{a}\) | \(26\) |
risch | \(\frac {i \ln \left (a^{2} x^{2}+1\right )}{2 a}+\frac {\arctan \left (a x \right )}{a}\) | \(26\) |
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none
Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {e^{i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {i \, \log \left (\frac {a x + i}{a}\right )}{a} \]
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Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.53 \[ \int \frac {e^{i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {i \log {\left (a x + i \right )}}{a} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (11) = 22\).
Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.60 \[ \int \frac {e^{i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {\arctan \left (a x\right )}{a} + \frac {i \, \log \left (a^{2} x^{2} + 1\right )}{2 \, a} \]
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none
Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {e^{i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {i \, \log \left (a x + i\right )}{a} \]
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Time = 0.55 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {e^{i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {\ln \left (x+\frac {1{}\mathrm {i}}{a}\right )\,1{}\mathrm {i}}{a} \]
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