Optimal. Leaf size=67 \[ \frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{a} x\right ),-1\right )}{3 a^{3/2}}+\frac{1}{3} x^3 e^{\text{sech}^{-1}\left (a x^2\right )}+\frac{2 x}{3 a} \]
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Rubi [A] time = 0.0279784, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6335, 8, 248, 221} \[ \frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{3 a^{3/2}}+\frac{1}{3} x^3 e^{\text{sech}^{-1}\left (a x^2\right )}+\frac{2 x}{3 a} \]
Antiderivative was successfully verified.
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Rule 6335
Rule 8
Rule 248
Rule 221
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}\left (a x^2\right )} x^2 \, dx &=\frac{1}{3} e^{\text{sech}^{-1}\left (a x^2\right )} x^3+\frac{2 \int 1 \, dx}{3 a}+\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1}{\sqrt{1-a x^2} \sqrt{1+a x^2}} \, dx}{3 a}\\ &=\frac{2 x}{3 a}+\frac{1}{3} e^{\text{sech}^{-1}\left (a x^2\right )} x^3+\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}{3 a}\\ &=\frac{2 x}{3 a}+\frac{1}{3} e^{\text{sech}^{-1}\left (a x^2\right )} x^3+\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 )}{3 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.189033, size = 116, normalized size = 1.73 \[ -\frac{2 i \sqrt{\frac{1-a x^2}{a x^2+1}} \sqrt{1-a^2 x^4} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-a} x\right ),-1\right )}{3 (-a)^{3/2} \left (a x^2-1\right )}+\frac{\sqrt{\frac{1-a x^2}{a x^2+1}} \left (a x^3+x\right )}{3 a}+\frac{x}{a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.204, size = 102, normalized size = 1.5 \begin{align*}{\frac{{x}^{2}}{3\,{a}^{2}{x}^{4}-3}\sqrt{-{\frac{a{x}^{2}-1}{a{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+1}{a{x}^{2}}}} \left ({x}^{5}{a}^{{\frac{5}{2}}}-2\,{\it EllipticF} \left ( x\sqrt{a},i \right ) \sqrt{-a{x}^{2}+1}\sqrt{a{x}^{2}+1}-x\sqrt{a} \right ){\frac{1}{\sqrt{a}}}}+{\frac{x}{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}{a} + \frac{\int \sqrt{a x^{2} + 1} \sqrt{-a x^{2} + 1}\,{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^{2} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} + 1}{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 1\, dx + \int a x^{2} \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^{2}{\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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