Optimal. Leaf size=39 \[ \frac {2 x}{a^2}+\frac {i x^2}{a}-\frac {x^3}{3}-\frac {2 i \log (i+a x)}{a^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 = {5170, 78}
\begin {gather*} -\frac {2 i \log (a x+i)}{a^3}+\frac {2 x}{a^2}+\frac {i x^2}{a}-\frac {x^3}{3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 5170
Rubi steps
\begin {align*} \int e^{2 i \tan ^{-1}(a x)} x^2 \, dx &=\int \frac {x^2 (1+i a x)}{1-i a x} \, dx\\ &=\int \left (\frac {2}{a^2}+\frac {2 i x}{a}-x^2-\frac {2 i}{a^2 (i+a x)}\right ) \, dx\\ &=\frac {2 x}{a^2}+\frac {i x^2}{a}-\frac {x^3}{3}-\frac {2 i \log (i+a x)}{a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 39, normalized size = 1.00 \begin {gather*} \frac {2 x}{a^2}+\frac {i x^2}{a}-\frac {x^3}{3}-\frac {2 i \log (i+a x)}{a^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 55, normalized size = 1.41
method | result | size |
risch | \(\frac {2 x}{a^{2}}-\frac {x^{3}}{3}+\frac {i x^{2}}{a}-\frac {i \ln \left (a^{2} x^{2}+1\right )}{a^{3}}-\frac {2 \arctan \left (a x \right )}{a^{3}}\) | \(47\) |
default | \(\frac {2 x -\frac {1}{3} a^{2} x^{3}+i a \,x^{2}}{a^{2}}+\frac {-\frac {i \ln \left (a^{2} x^{2}+1\right )}{a}-\frac {2 \arctan \left (a x \right )}{a}}{a^{2}}\) | \(55\) |
meijerg | \(\frac {\frac {2 x \left (a^{2}\right )^{\frac {3}{2}}}{a^{2}}-\frac {2 \left (a^{2}\right )^{\frac {3}{2}} \arctan \left (a x \right )}{a^{3}}}{2 a^{2} \sqrt {a^{2}}}+\frac {i \left (a^{2} x^{2}-\ln \left (a^{2} x^{2}+1\right )\right )}{a^{3}}-\frac {-\frac {2 x \left (a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{15 a^{4}}+\frac {2 \left (a^{2}\right )^{\frac {5}{2}} \arctan \left (a x \right )}{a^{5}}}{2 a^{2} \sqrt {a^{2}}}\) | \(110\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.46, size = 47, normalized size = 1.21 \begin {gather*} -\frac {a^{2} x^{3} - 3 i \, a x^{2} - 6 \, x}{3 \, a^{2}} - \frac {2 \, \arctan \left (a x\right )}{a^{3}} - \frac {i \, \log \left (a^{2} x^{2} + 1\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 4.11, size = 37, normalized size = 0.95 \begin {gather*} -\frac {a^{3} x^{3} - 3 i \, a^{2} x^{2} - 6 \, a x + 6 i \, \log \left (\frac {a x + i}{a}\right )}{3 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.06, size = 31, normalized size = 0.79 \begin {gather*} - \frac {x^{3}}{3} + \frac {i x^{2}}{a} + \frac {2 x}{a^{2}} - \frac {2 i \log {\left (a x + i \right )}}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.39, size = 37, normalized size = 0.95 \begin {gather*} -\frac {a^{3} x^{3} - 3 i \, a^{2} x^{2} - 6 \, a x}{3 \, a^{3}} - \frac {2 i \, \log \left (a x + i\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.42, size = 36, normalized size = 0.92 \begin {gather*} \frac {2\,x}{a^2}-\frac {\ln \left (x+\frac {1{}\mathrm {i}}{a}\right )\,2{}\mathrm {i}}{a^3}-\frac {x^3}{3}+\frac {x^2\,1{}\mathrm {i}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________