Optimal. Leaf size=36 \[ -2 a^2 \log (x)+2 a^2 \log (a x+i)-\frac{2 i a}{x}-\frac{1}{2 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0268669, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5062, 77} \[ -2 a^2 \log (x)+2 a^2 \log (a x+i)-\frac{2 i a}{x}-\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5062
Rule 77
Rubi steps
\begin{align*} \int \frac{e^{2 i \tan ^{-1}(a x)}}{x^3} \, dx &=\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*}
Mathematica [A] time = 0.0098901, size = 36, normalized size = 1. \[ -2 a^2 \log (x)+2 a^2 \log (a x+i)-\frac{2 i a}{x}-\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.041, size = 45, normalized size = 1.3 \begin{align*} -2\,i{a}^{2}\arctan \left ( ax \right ) +{a}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) -{\frac{1}{2\,{x}^{2}}}-{\frac{2\,ia}{x}}-2\,{a}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.51449, size = 57, normalized size = 1.58 \begin{align*} -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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.59904, size = 97, normalized size = 2.69 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.459619, size = 29, normalized size = 0.81 \begin{align*} - 2 a^{2} \left (\log{\left (x \right )} - \log{\left (x + \frac{i}{a} \right )}\right ) - \frac{4 i a x + 1}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11688, size = 43, normalized size = 1.19 \begin{align*} 2 \, a^{2} \log \left (a x + i\right ) - 2 \, a^{2} \log \left ({\left | x \right |}\right ) - \frac{4 \, a i x + 1}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]