Optimal. Leaf size=112 \[ -\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{a} x\right ),-1\right )}{5 a^{5/2}}+\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} E\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{5 a^{5/2}}+\frac{2 x^3}{15 a}+\frac{1}{5} x^5 e^{\text{sech}^{-1}\left (a x^2\right )} \]
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Rubi [A] time = 0.0665785, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {6335, 30, 259, 307, 221, 1199, 424} \[ -\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{5 a^{5/2}}+\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} E\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{5 a^{5/2}}+\frac{2 x^3}{15 a}+\frac{1}{5} x^5 e^{\text{sech}^{-1}\left (a x^2\right )} \]
Antiderivative was successfully verified.
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Rule 6335
Rule 30
Rule 259
Rule 307
Rule 221
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}\left (a x^2\right )} x^4 \, dx &=\frac{1}{5} e^{\text{sech}^{-1}\left (a x^2\right )} x^5+\frac{2 \int x^2 \, dx}{5 a}+\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^2}{\sqrt{1-a x^2} \sqrt{1+a x^2}} \, dx}{5 a}\\ &=\frac{2 x^3}{15 a}+\frac{1}{5} e^{\text{sech}^{-1}\left (a x^2\right )} x^5+\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^2}{\sqrt{1-a^2 x^4}} \, dx}{5 a}\\ &=\frac{2 x^3}{15 a}+\frac{1}{5} e^{\text{sech}^{-1}\left (a x^2\right )} x^5-\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1}{\sqrt{1-a^2 x^4}} \, dx}{5 a^2}+\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1+a x^2}{\sqrt{1-a^2 x^4}} \, dx}{5 a^2}\\ &=\frac{2 x^3}{15 a}+\frac{1}{5} e^{\text{sech}^{-1}\left (a x^2\right )} x^5-\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} F\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{5 a^{5/2}}+\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{\sqrt{1+a x^2}}{\sqrt{1-a x^2}} \, dx}{5 a^2}\\ &=\frac{2 x^3}{15 a}+\frac{1}{5} e^{\text{sech}^{-1}\left (a x^2\right )} x^5+\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} E\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{5 a^{5/2}}-\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} F\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{5 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.435428, size = 140, normalized size = 1.25 \[ \frac{1}{15} \left (\frac{6 i \sqrt{\frac{1-a x^2}{a x^2+1}} \sqrt{1-a^2 x^4} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-a} x\right )\right |-1\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-a} x\right ),-1\right )\right )}{(-a)^{5/2} \left (a x^2-1\right )}+\frac{5 x^3}{a}+\frac{3 \sqrt{\frac{1-a x^2}{a x^2+1}} \left (a x^5+x^3\right )}{a}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.181, size = 136, normalized size = 1.2 \begin{align*}{\frac{{x}^{2}}{5\,{a}^{2}{x}^{4}-5}\sqrt{-{\frac{a{x}^{2}-1}{a{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+1}{a{x}^{2}}}} \left ({a}^{{\frac{7}{2}}}{x}^{7}-{x}^{3}{a}^{{\frac{3}{2}}}+2\,{\it EllipticF} \left ( x\sqrt{a},i \right ) \sqrt{-a{x}^{2}+1}\sqrt{a{x}^{2}+1}-2\,\sqrt{-a{x}^{2}+1}\sqrt{a{x}^{2}+1}{\it EllipticE} \left ( x\sqrt{a},i \right ) \right ){a}^{-{\frac{3}{2}}}}+{\frac{{x}^{3}}{3\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{x^{3}}{3 \, a} + \frac{\int \sqrt{a x^{2} + 1} \sqrt{-a x^{2} + 1} x^{2}\,{d x}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a x^{4} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} + x^{2}}{a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int x^{2}\, dx + \int a x^{4} \sqrt{-1 + \frac{1}{a x^{2}}} \sqrt{1 + \frac{1}{a x^{2}}}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4}{\left (\sqrt{\frac{1}{a x^{2}} + 1} \sqrt{\frac{1}{a x^{2}} - 1} + \frac{1}{a x^{2}}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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