Optimal. Leaf size=31 \[ \frac{1}{3} x^3 \left (\frac{1}{a^2 x^2}+1\right )^{3/2}+\frac{x^2}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0219103, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6336, 30, 264} \[ \frac{1}{3} x^3 \left (\frac{1}{a^2 x^2}+1\right )^{3/2}+\frac{x^2}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6336
Rule 30
Rule 264
Rubi steps
\begin{align*} \int e^{\text{csch}^{-1}(a x)} x^2 \, dx &=\frac{\int x \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^2}} x^2 \, dx\\ &=\frac{x^2}{2 a}+\frac{1}{3} \left (1+\frac{1}{a^2 x^2}\right )^{3/2} x^3\\ \end{align*}
Mathematica [A] time = 0.0379749, size = 38, normalized size = 1.23 \[ \frac{2 \sqrt{\frac{1}{a^2 x^2}+1} \left (a^2 x^3+x\right )+3 a x^2}{6 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.171, size = 43, normalized size = 1.4 \begin{align*}{\frac{x \left ({a}^{2}{x}^{2}+1 \right ) }{3\,{a}^{2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}}}+{\frac{{x}^{2}}{2\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02121, size = 34, normalized size = 1.1 \begin{align*} \frac{1}{3} \, x^{3}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + \frac{x^{2}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.53478, size = 90, normalized size = 2.9 \begin{align*} \frac{3 \, a x^{2} + 2 \,{\left (a^{2} x^{3} + x\right )} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}}}{6 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.45725, size = 41, normalized size = 1.32 \begin{align*} \frac{x^{2} \sqrt{a^{2} x^{2} + 1}}{3 a} + \frac{x^{2}}{2 a} + \frac{\sqrt{a^{2} x^{2} + 1}}{3 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1483, size = 61, normalized size = 1.97 \begin{align*} -\frac{{\left | a \right |} \mathrm{sgn}\left (x\right )}{3 \, a^{4}} + \frac{2 \,{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right ) + 3 \,{\left (a^{2} x^{2} + 1\right )} a}{6 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]