Optimal. Leaf size=29 \[ \frac{2 \log (a x+i)}{a^2}+\frac{2 i x}{a}-\frac{x^2}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0194471, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5062, 77} \[ \frac{2 \log (a x+i)}{a^2}+\frac{2 i x}{a}-\frac{x^2}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5062
Rule 77
Rubi steps
\begin{align*} \int e^{2 i \tan ^{-1}(a x)} x \, dx &=\int \frac{x (1+i a x)}{1-i a x} \, dx\\ &=\int \left (\frac{2 i}{a}-x+\frac{2}{a (i+a x)}\right ) \, dx\\ &=\frac{2 i x}{a}-\frac{x^2}{2}+\frac{2 \log (i+a x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.010063, size = 29, normalized size = 1. \[ \frac{2 \log (a x+i)}{a^2}+\frac{2 i x}{a}-\frac{x^2}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.036, size = 38, normalized size = 1.3 \begin{align*} -{\frac{{x}^{2}}{2}}+{\frac{2\,ix}{a}}+{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{2}}}-{\frac{2\,i\arctan \left ( ax \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.47589, size = 51, normalized size = 1.76 \begin{align*} -\frac{a x^{2} - 4 i \, x}{2 \, a} - \frac{2 i \, \arctan \left (a x\right )}{a^{2}} + \frac{\log \left (a^{2} x^{2} + 1\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.59842, size = 69, normalized size = 2.38 \begin{align*} -\frac{a^{2} x^{2} - 4 i \, a x - 4 \, \log \left (\frac{a x + i}{a}\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.165334, size = 22, normalized size = 0.76 \begin{align*} - \frac{x^{2}}{2} + \frac{2 i x}{a} + \frac{2 \log{\left (a x + i \right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.08694, size = 41, normalized size = 1.41 \begin{align*} -\frac{a^{2} x^{2} - 4 \, a i x}{2 \, a^{2}} + \frac{2 \, \log \left (a x + i\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]