Integrand size = 10, antiderivative size = 19 \[ \int e^{2 i \arctan (a x)} \, dx=-x+\frac {2 i \log (i+a x)}{a} \]
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Time = 0.01 (sec) , antiderivative size = 19, 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 = 1.58 \[ \int e^{2 i \arctan (a x)} \, dx=-x+\frac {2 \arctan (a x)}{a}+\frac {i \log \left (1+a^2 x^2\right )}{a} \]
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Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05
method | result | size |
parallelrisch | \(\frac {2 i \ln \left (a x +i\right )-a x}{a}\) | \(20\) |
default | \(-x +\frac {i \ln \left (a^{2} x^{2}+1\right )}{a}+\frac {2 \arctan \left (a x \right )}{a}\) | \(30\) |
risch | \(-x +\frac {i \ln \left (a^{2} x^{2}+1\right )}{a}+\frac {2 \arctan \left (a x \right )}{a}\) | \(30\) |
meijerg | \(\frac {\arctan \left (a x \right )}{a}+\frac {i \ln \left (a^{2} x^{2}+1\right )}{a}-\frac {\frac {2 x \left (a^{2}\right )^{\frac {3}{2}}}{a^{2}}-\frac {2 \left (a^{2}\right )^{\frac {3}{2}} \arctan \left (a x \right )}{a^{3}}}{2 \sqrt {a^{2}}}\) | \(59\) |
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none
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int e^{2 i \arctan (a x)} \, dx=-\frac {a x - 2 i \, \log \left (\frac {a x + i}{a}\right )}{a} \]
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Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int e^{2 i \arctan (a x)} \, dx=- x + \frac {2 i \log {\left (a x + i \right )}}{a} \]
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none
Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int e^{2 i \arctan (a x)} \, dx=-x + \frac {2 \, \arctan \left (a x\right )}{a} + \frac {i \, \log \left (a^{2} x^{2} + 1\right )}{a} \]
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none
Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int e^{2 i \arctan (a x)} \, dx=-x + \frac {2 i \, \log \left (a x + i\right )}{a} \]
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Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int e^{2 i \arctan (a x)} \, dx=-x+\frac {\ln \left (x+\frac {1{}\mathrm {i}}{a}\right )\,2{}\mathrm {i}}{a} \]
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