Integrand size = 14, antiderivative size = 16 \[ \int \frac {e^{4 i \arctan (a x)}}{x} \, dx=\frac {4 i}{i+a x}+\log (x) \]
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Time = 0.02 (sec) , antiderivative size = 16, 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, 90} \[ \int \frac {e^{4 i \arctan (a x)}}{x} \, dx=\log (x)+\frac {4 i}{a x+i} \]
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Rule 90
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1+i a x)^2}{x (1-i a x)^2} \, dx \\ & = \int \left (\frac {1}{x}-\frac {4 i a}{(i+a x)^2}\right ) \, dx \\ & = \frac {4 i}{i+a x}+\log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {e^{4 i \arctan (a x)}}{x} \, dx=\frac {4 i}{i+a x}+\log (x) \]
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Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {4 i}{a x +i}+\ln \left (x \right )\) | \(15\) |
risch | \(\frac {4 i}{a x +i}+\ln \left (-x \right )\) | \(17\) |
norman | \(\frac {-4 a^{2} x^{2}+4 i a x}{a^{2} x^{2}+1}+\ln \left (x \right )\) | \(30\) |
parallelrisch | \(\frac {a^{2} \ln \left (x \right ) x^{2}-4 a^{2} x^{2}+4 i a x +\ln \left (x \right )}{a^{2} x^{2}+1}\) | \(38\) |
meijerg | \(-\frac {a^{2} x^{2}}{2 a^{2} x^{2}+2}+\frac {1}{2}+\ln \left (x \right )+\frac {\ln \left (a^{2}\right )}{2}+\frac {2 i a \left (\frac {2 x \sqrt {a^{2}}}{2 a^{2} x^{2}+2}+\frac {\sqrt {a^{2}}\, \arctan \left (a x \right )}{a}\right )}{\sqrt {a^{2}}}-\frac {7 a^{2} x^{2}}{2 \left (a^{2} x^{2}+1\right )}-\frac {2 i a \left (-\frac {x \left (a^{2}\right )^{\frac {3}{2}}}{a^{2} \left (a^{2} x^{2}+1\right )}+\frac {\left (a^{2}\right )^{\frac {3}{2}} \arctan \left (a x \right )}{a^{3}}\right )}{\sqrt {a^{2}}}\) | \(138\) |
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none
Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {e^{4 i \arctan (a x)}}{x} \, dx=\frac {{\left (a x + i\right )} \log \left (x\right ) + 4 i}{a x + i} \]
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Time = 0.10 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \frac {e^{4 i \arctan (a x)}}{x} \, dx=\log {\left (x \right )} + \frac {4 i}{a x + i} \]
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none
Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {e^{4 i \arctan (a x)}}{x} \, dx=-\frac {4 \, {\left (-i \, a x - 1\right )}}{a^{2} x^{2} + 1} + \log \left (x\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {e^{4 i \arctan (a x)}}{x} \, dx=\frac {4 i}{a x + i} + \log \left ({\left | x \right |}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {e^{4 i \arctan (a x)}}{x} \, dx=\ln \left (x\right )+\frac {4{}\mathrm {i}}{a\,x+1{}\mathrm {i}} \]
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