Optimal. Leaf size=63 \[ \frac{\sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sin ^{-1}\left (a x^2\right )}{4 a^2}+\frac{x^2}{4 a}+\frac{1}{4} x^4 e^{\text{sech}^{-1}\left (a x^2\right )} \]
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Rubi [A] time = 0.0368681, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6335, 30, 259, 275, 216} \[ \frac{\sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sin ^{-1}\left (a x^2\right )}{4 a^2}+\frac{x^2}{4 a}+\frac{1}{4} x^4 e^{\text{sech}^{-1}\left (a x^2\right )} \]
Antiderivative was successfully verified.
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Rule 6335
Rule 30
Rule 259
Rule 275
Rule 216
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}\left (a x^2\right )} x^3 \, dx &=\frac{1}{4} e^{\text{sech}^{-1}\left (a x^2\right )} x^4+\frac{\int x \, dx}{2 a}+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x}{\sqrt{1-a x^2} \sqrt{1+a x^2}} \, dx}{2 a}\\ &=\frac{x^2}{4 a}+\frac{1}{4} e^{\text{sech}^{-1}\left (a x^2\right )} x^4+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x}{\sqrt{1-a^2 x^4}} \, dx}{2 a}\\ &=\frac{x^2}{4 a}+\frac{1}{4} e^{\text{sech}^{-1}\left (a x^2\right )} x^4+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx,x,x^2\right )}{4 a}\\ &=\frac{x^2}{4 a}+\frac{1}{4} e^{\text{sech}^{-1}\left (a x^2\right )} x^4+\frac{\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sin ^{-1}\left (a x^2\right )}{4 a^2}\\ \end{align*}
Mathematica [C] time = 0.103778, size = 92, normalized size = 1.46 \[ \frac{2 a x^2+a \sqrt{\frac{1-a x^2}{a x^2+1}} \left (a x^4+x^2\right )+i \log \left (2 \sqrt{\frac{1-a x^2}{a x^2+1}} \left (a x^2+1\right )-2 i a x^2\right )}{4 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.33, size = 112, normalized size = 1.8 \begin{align*}{\frac{{x}^{2}}{4\,{a}^{2}}\sqrt{-{\frac{a{x}^{2}-1}{a{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+1}{a{x}^{2}}}} \left ({x}^{2}\sqrt{-{\frac{{a}^{2}{x}^{4}-1}{{a}^{2}}}}{a}^{2}+\arctan \left ({{x}^{2}{\frac{1}{\sqrt{-{\frac{{a}^{2}{x}^{4}-1}{{a}^{2}}}}}}} \right ) \right ){\frac{1}{\sqrt{-{\frac{{a}^{2}{x}^{4}-1}{{a}^{2}}}}}}}+{\frac{{x}^{2}}{2\,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^{2}}{2 \, a} + \frac{\frac{1}{4} \, \sqrt{-a^{2} x^{4} + 1} x^{2} + \frac{\arcsin \left (\frac{a^{2} x^{2}}{\sqrt{a^{2}}}\right )}{4 \, \sqrt{a^{2}}}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05342, size = 225, normalized size = 3.57 \begin{align*} \frac{a^{2} x^{4} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} + 2 \, a x^{2} - 2 \, \arctan \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^{2}}\right )}{4 \, a^{2}} \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\, dx + \int a x^{3} \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 [A] time = 1.16296, size = 107, normalized size = 1.7 \begin{align*} \frac{2 \,{\left (a^{2} x^{2} + a\right )} a^{2} +{\left (\sqrt{a^{2} x^{2} + a} \sqrt{-a^{2} x^{2} + a} a^{2} x^{2} - 2 \, a^{2} \arcsin \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + a}}{2 \, \sqrt{a}}\right )\right )} a}{4 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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