Optimal. Leaf size=38 \[ \frac{2 x^3 \left (\frac{1}{a^2 x^2}+1\right )^{3/2}}{3 a}+\frac{x^2}{a^2}+\frac{x^4}{4} \]
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Rubi [A] time = 0.224786, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6338, 6742, 264} \[ \frac{2 x^3 \left (\frac{1}{a^2 x^2}+1\right )^{3/2}}{3 a}+\frac{x^2}{a^2}+\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Rule 6338
Rule 6742
Rule 264
Rubi steps
\begin{align*} \int e^{2 \text{csch}^{-1}(a x)} x^3 \, dx &=\int \left (\sqrt{1+\frac{1}{a^2 x^2}}+\frac{1}{a x}\right )^2 x^3 \, dx\\ &=\int \left (\frac{2 x}{a^2}+\frac{2 \sqrt{1+\frac{1}{a^2 x^2}} x^2}{a}+x^3\right ) \, dx\\ &=\frac{x^2}{a^2}+\frac{x^4}{4}+\frac{2 \int \sqrt{1+\frac{1}{a^2 x^2}} x^2 \, dx}{a}\\ &=\frac{x^2}{a^2}+\frac{2 \left (1+\frac{1}{a^2 x^2}\right )^{3/2} x^3}{3 a}+\frac{x^4}{4}\\ \end{align*}
Mathematica [A] time = 0.0467991, size = 44, normalized size = 1.16 \[ \frac{x^2}{a^2}+\frac{2 \sqrt{\frac{1}{a^2 x^2}+1} \left (a^2 x^3+x\right )}{3 a^3}+\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.183, size = 61, normalized size = 1.6 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{{a}^{2}{x}^{4}}{4}}+{\frac{{x}^{2}}{2}} \right ) }+{\frac{2\,x \left ({a}^{2}{x}^{2}+1 \right ) }{3\,{a}^{3}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}}}+{\frac{{x}^{2}}{2\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991612, size = 43, normalized size = 1.13 \begin{align*} \frac{1}{4} \, x^{4} + \frac{2 \, x^{3}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{3}{2}}}{3 \, a} + \frac{x^{2}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.5358, size = 109, normalized size = 2.87 \begin{align*} \frac{3 \, a^{3} x^{4} + 12 \, a x^{2} + 8 \,{\left (a^{2} x^{3} + x\right )} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}}}{12 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.48901, size = 51, normalized size = 1.34 \begin{align*} \frac{x^{4}}{4} + \frac{2 x^{2} \sqrt{a^{2} x^{2} + 1}}{3 a^{2}} + \frac{x^{2}}{a^{2}} + \frac{2 \sqrt{a^{2} x^{2} + 1}}{3 a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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