Integrand size = 14, antiderivative size = 27 \[ \int \frac {e^{-2 i \arctan (a x)}}{x^2} \, dx=-\frac {1}{x}-2 i a \log (x)+2 i a \log (i-a x) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 27, 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} \[ \int \frac {e^{-2 i \arctan (a x)}}{x^2} \, dx=-2 i a \log (x)+2 i a \log (-a x+i)-\frac {1}{x} \]
[In]
[Out]
Rule 78
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {1-i a x}{x^2 (1+i a x)} \, dx \\ & = \int \left (\frac {1}{x^2}-\frac {2 i a}{x}+\frac {2 i a^2}{-i+a x}\right ) \, dx \\ & = -\frac {1}{x}-2 i a \log (x)+2 i a \log (i-a x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 i \arctan (a x)}}{x^2} \, dx=-\frac {1}{x}-2 i a \log (x)+2 i a \log (i-a x) \]
[In]
[Out]
Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {1}{x}-2 i a \ln \left (x \right )+2 i a \ln \left (-a x +i\right )\) | \(25\) |
risch | \(-\frac {1}{x}-2 i a \ln \left (x \right )-2 a \arctan \left (a x \right )+i a \ln \left (a^{2} x^{2}+1\right )\) | \(34\) |
parallelrisch | \(-\frac {2 i a^{2} \ln \left (x \right ) x -2 i a^{2} \ln \left (a x -i\right ) x +a}{a x}\) | \(34\) |
meijerg | \(\frac {a^{2} x}{i a x +1}+i a \left (\frac {3 i a x}{3 i a x +3}+2 \ln \left (i a x +1\right )-1-2 \ln \left (x \right )-2 \ln \left (i a \right )+\frac {i}{x a}\right )\) | \(66\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-2 i \arctan (a x)}}{x^2} \, dx=\frac {-2 i \, a x \log \left (x\right ) + 2 i \, a x \log \left (\frac {a x - i}{a}\right ) - 1}{x} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-2 i \arctan (a x)}}{x^2} \, dx=- 2 a \left (i \log {\left (4 a^{2} x \right )} - i \log {\left (4 a^{2} x - 4 i a \right )}\right ) - \frac {1}{x} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {e^{-2 i \arctan (a x)}}{x^2} \, dx=2 i \, a \log \left (i \, a x + 1\right ) - 2 i \, a \log \left (x\right ) - \frac {a x - i}{a x^{2} - i \, x} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {e^{-2 i \arctan (a x)}}{x^2} \, dx=-2 i \, a \log \left (\frac {i}{i \, a x + 1} - i\right ) - \frac {a}{\frac {i}{i \, a x + 1} - i} \]
[In]
[Out]
Time = 0.49 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {e^{-2 i \arctan (a x)}}{x^2} \, dx=-4\,a\,\mathrm {atan}\left (2\,a\,x-\mathrm {i}\right )-\frac {1}{x} \]
[In]
[Out]