Integrand size = 10, antiderivative size = 20 \[ \int e^{-2 i \arctan (a x)} \, dx=-x-\frac {2 i \log (i-a x)}{a} \]
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Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5169, 45} \[ \int e^{-2 i \arctan (a x)} \, dx=-x-\frac {2 i \log (-a x+i)}{a} \]
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Rule 45
Rule 5169
Rubi steps \begin{align*} \text {integral}& = \int \frac {1-i a x}{1+i a x} \, dx \\ & = \int \left (-1-\frac {2 i}{-i+a x}\right ) \, dx \\ & = -x-\frac {2 i \log (i-a x)}{a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.50 \[ \int e^{-2 i \arctan (a x)} \, dx=-x+\frac {2 \arctan (a x)}{a}-\frac {i \log \left (1+a^2 x^2\right )}{a} \]
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Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
default | \(-x -\frac {2 i \ln \left (-a x +i\right )}{a}\) | \(19\) |
risch | \(-x -\frac {i \ln \left (a^{2} x^{2}+1\right )}{a}+\frac {2 \arctan \left (a x \right )}{a}\) | \(30\) |
parallelrisch | \(\frac {2 i \ln \left (a x -i\right ) x a +a^{2} x^{2}+1+2 \ln \left (a x -i\right )}{a \left (-a x +i\right )}\) | \(44\) |
meijerg | \(\frac {i \left (\frac {i a x \left (3 i a x +6\right )}{3 i a x +3}-2 \ln \left (i a x +1\right )\right )}{a}+\frac {x}{i a x +1}\) | \(51\) |
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none
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int e^{-2 i \arctan (a x)} \, dx=-\frac {a x + 2 i \, \log \left (\frac {a x - i}{a}\right )}{a} \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int e^{-2 i \arctan (a x)} \, dx=- x - \frac {2 i \log {\left (a x - i \right )}}{a} \]
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none
Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int e^{-2 i \arctan (a x)} \, dx=-x - \frac {2 i \, \log \left (i \, a x + 1\right )}{a} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (16) = 32\).
Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 3.25 \[ \int e^{-2 i \arctan (a x)} \, dx=a^{2} {\left (\frac {i \, {\left (i \, a x + 1\right )}}{a^{3}} + \frac {2 i \, \log \left (\frac {1}{\sqrt {a^{2} x^{2} + 1} {\left | a \right |}}\right )}{a^{3}} - \frac {i}{{\left (i \, a x + 1\right )} a^{3}}\right )} + \frac {i}{{\left (i \, a x + 1\right )} a} \]
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Time = 0.45 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int e^{-2 i \arctan (a x)} \, dx=-x-\frac {\ln \left (x-\frac {1{}\mathrm {i}}{a}\right )\,2{}\mathrm {i}}{a} \]
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