Integrand size = 14, antiderivative size = 36 \[ \int \frac {e^{2 i \arctan (a x)}}{x^3} \, dx=-\frac {1}{2 x^2}-\frac {2 i a}{x}-2 a^2 \log (x)+2 a^2 \log (i+a x) \]
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Time = 0.02 (sec) , antiderivative size = 36, 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, 78} \[ \int \frac {e^{2 i \arctan (a x)}}{x^3} \, dx=-2 a^2 \log (x)+2 a^2 \log (a x+i)-\frac {2 i a}{x}-\frac {1}{2 x^2} \]
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Rule 78
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+i a x}{x^3 (1-i a x)} \, dx \\ & = \int \left (\frac {1}{x^3}+\frac {2 i a}{x^2}-\frac {2 a^2}{x}+\frac {2 a^3}{i+a x}\right ) \, dx \\ & = -\frac {1}{2 x^2}-\frac {2 i a}{x}-2 a^2 \log (x)+2 a^2 \log (i+a x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 i \arctan (a x)}}{x^3} \, dx=-\frac {1}{2 x^2}-\frac {2 i a}{x}-2 a^2 \log (x)+2 a^2 \log (i+a x) \]
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Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06
method | result | size |
parallelrisch | \(-\frac {4 a^{2} \ln \left (x \right ) x^{2}-4 a^{2} \ln \left (a x +i\right ) x^{2}+4 i a x +1}{2 x^{2}}\) | \(38\) |
risch | \(\frac {-2 i a x -\frac {1}{2}}{x^{2}}-2 a^{2} \ln \left (x \right )-2 i a^{2} \arctan \left (a x \right )+a^{2} \ln \left (a^{2} x^{2}+1\right )\) | \(44\) |
default | \(-\frac {1}{2 x^{2}}-\frac {2 i a}{x}-2 a^{2} \ln \left (x \right )-2 a^{3} \left (-\frac {\ln \left (a^{2} x^{2}+1\right )}{2 a}+\frac {i \arctan \left (a x \right )}{a}\right )\) | \(52\) |
meijerg | \(\frac {a^{2} \left (\ln \left (a^{2} x^{2}+1\right )-2 \ln \left (x \right )-\ln \left (a^{2}\right )-\frac {1}{a^{2} x^{2}}\right )}{2}+\frac {i a^{3} \left (-\frac {2}{x \sqrt {a^{2}}}-\frac {2 a \arctan \left (a x \right )}{\sqrt {a^{2}}}\right )}{\sqrt {a^{2}}}-\frac {a^{2} \left (-\ln \left (a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (a^{2}\right )\right )}{2}\) | \(96\) |
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Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08 \[ \int \frac {e^{2 i \arctan (a x)}}{x^3} \, dx=-\frac {4 \, a^{2} x^{2} \log \left (x\right ) - 4 \, a^{2} x^{2} \log \left (\frac {a x + i}{a}\right ) + 4 i \, a x + 1}{2 \, x^{2}} \]
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Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \[ \int \frac {e^{2 i \arctan (a x)}}{x^3} \, dx=- 2 a^{2} \left (\log {\left (4 a^{3} x \right )} - \log {\left (4 a^{3} x + 4 i a^{2} \right )}\right ) - \frac {4 i a x + 1}{2 x^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \[ \int \frac {e^{2 i \arctan (a x)}}{x^3} \, dx=-2 i \, a^{2} \arctan \left (a x\right ) + a^{2} \log \left (a^{2} x^{2} + 1\right ) - 2 \, a^{2} \log \left (x\right ) - \frac {4 i \, a x + 1}{2 \, x^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {e^{2 i \arctan (a x)}}{x^3} \, dx=2 \, a^{2} \log \left (a x + i\right ) - 2 \, a^{2} \log \left ({\left | x \right |}\right ) - \frac {4 i \, a x + 1}{2 \, x^{2}} \]
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Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {e^{2 i \arctan (a x)}}{x^3} \, dx=-a^2\,\mathrm {atan}\left (2\,a\,x+1{}\mathrm {i}\right )\,4{}\mathrm {i}-\frac {\frac {1}{2}+a\,x\,2{}\mathrm {i}}{x^2} \]
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