Optimal. Leaf size=75 \[ -\frac{i \left (a^2 x^2+1\right )^{3/2}}{3 a^3}+\frac{x \sqrt{a^2 x^2+1}}{2 a^2}+\frac{i \sqrt{a^2 x^2+1}}{a^3}-\frac{\sinh ^{-1}(a x)}{2 a^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0473174, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5060, 797, 641, 195, 215} \[ -\frac{i \left (a^2 x^2+1\right )^{3/2}}{3 a^3}+\frac{x \sqrt{a^2 x^2+1}}{2 a^2}+\frac{i \sqrt{a^2 x^2+1}}{a^3}-\frac{\sinh ^{-1}(a x)}{2 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5060
Rule 797
Rule 641
Rule 195
Rule 215
Rubi steps
\begin{align*} \int e^{-i \tan ^{-1}(a x)} x^2 \, dx &=\int \frac{x^2 (1-i a x)}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\int \frac{1-i a x}{\sqrt{1+a^2 x^2}} \, dx}{a^2}+\frac{\int (1-i a x) \sqrt{1+a^2 x^2} \, dx}{a^2}\\ &=\frac{i \sqrt{1+a^2 x^2}}{a^3}-\frac{i \left (1+a^2 x^2\right )^{3/2}}{3 a^3}-\frac{\int \frac{1}{\sqrt{1+a^2 x^2}} \, dx}{a^2}+\frac{\int \sqrt{1+a^2 x^2} \, dx}{a^2}\\ &=\frac{i \sqrt{1+a^2 x^2}}{a^3}+\frac{x \sqrt{1+a^2 x^2}}{2 a^2}-\frac{i \left (1+a^2 x^2\right )^{3/2}}{3 a^3}-\frac{\sinh ^{-1}(a x)}{a^3}+\frac{\int \frac{1}{\sqrt{1+a^2 x^2}} \, dx}{2 a^2}\\ &=\frac{i \sqrt{1+a^2 x^2}}{a^3}+\frac{x \sqrt{1+a^2 x^2}}{2 a^2}-\frac{i \left (1+a^2 x^2\right )^{3/2}}{3 a^3}-\frac{\sinh ^{-1}(a x)}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.0311688, size = 46, normalized size = 0.61 \[ \frac{-3 \sinh ^{-1}(a x)+\left (-2 i a^2 x^2+3 a x+4 i\right ) \sqrt{a^2 x^2+1}}{6 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.077, size = 168, normalized size = 2.2 \begin{align*}{\frac{-{\frac{i}{3}}}{{a}^{3}} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{x}{2\,{a}^{2}}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{1}{2\,{a}^{2}}\ln \left ({{a}^{2}x{\frac{1}{\sqrt{{a}^{2}}}}}+\sqrt{{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{i}{{a}^{3}}\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) }}-{\frac{1}{{a}^{2}}\ln \left ({ \left ( ia+{a}^{2} \left ( x-{\frac{i}{a}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}}}}}+\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.54917, size = 80, normalized size = 1.07 \begin{align*} \frac{\sqrt{a^{2} x^{2} + 1} x}{2 \, a^{2}} - \frac{i \,{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{3 \, a^{3}} - \frac{\operatorname{arsinh}\left (a x\right )}{2 \, a^{3}} + \frac{i \, \sqrt{a^{2} x^{2} + 1}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.67148, size = 124, normalized size = 1.65 \begin{align*} \frac{\sqrt{a^{2} x^{2} + 1}{\left (-2 i \, a^{2} x^{2} + 3 \, a x + 4 i\right )} + 3 \, \log \left (-a x + \sqrt{a^{2} x^{2} + 1}\right )}{6 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \sqrt{a^{2} x^{2} + 1}}{i a x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10836, size = 85, normalized size = 1.13 \begin{align*} -\frac{1}{6} \, \sqrt{a^{2} x^{2} + 1}{\left ({\left (\frac{2 \, i x}{a} - \frac{3}{a^{2}}\right )} x - \frac{4 \, i}{a^{3}}\right )} + \frac{\log \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1}\right )}{2 \, a^{2}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]