Optimal. Leaf size=58 \[ -\frac{\sqrt{1-a x^2}}{6 a^3 \sqrt{\frac{1}{a x^2+1}}}+\frac{x^4}{12 a}+\frac{1}{6} x^6 e^{\text{sech}^{-1}\left (a x^2\right )} \]
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Rubi [A] time = 0.0412168, antiderivative size = 71, normalized size of antiderivative = 1.22, 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, 30, 259, 261} \[ -\frac{\sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{6 a^3}+\frac{x^4}{12 a}+\frac{1}{6} x^6 e^{\text{sech}^{-1}\left (a x^2\right )} \]
Warning: Unable to verify antiderivative.
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Rule 6335
Rule 30
Rule 259
Rule 261
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}\left (a x^2\right )} x^5 \, dx &=\frac{1}{6} e^{\text{sech}^{-1}\left (a x^2\right )} x^6+\frac{\int x^3 \, dx}{3 a}+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^3}{\sqrt{1-a x^2} \sqrt{1+a x^2}} \, dx}{3 a}\\ &=\frac{x^4}{12 a}+\frac{1}{6} e^{\text{sech}^{-1}\left (a x^2\right )} x^6+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^3}{\sqrt{1-a^2 x^4}} \, dx}{3 a}\\ &=\frac{x^4}{12 a}+\frac{1}{6} e^{\text{sech}^{-1}\left (a x^2\right )} x^6-\frac{\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{6 a^3}\\ \end{align*}
Mathematica [A] time = 0.0941505, size = 56, normalized size = 0.97 \[ \frac{\left (a x^2-1\right ) \sqrt{\frac{1-a x^2}{a x^2+1}} \left (a x^2+1\right )^2}{6 a^3}+\frac{x^4}{4 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.178, size = 60, normalized size = 1. \begin{align*}{\frac{{x}^{2} \left ({a}^{2}{x}^{4}-1 \right ) }{6\,{a}^{2}}\sqrt{-{\frac{a{x}^{2}-1}{a{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+1}{a{x}^{2}}}}}+{\frac{{x}^{4}}{4\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06829, size = 57, normalized size = 0.98 \begin{align*} \frac{x^{4}}{4 \, a} + \frac{{\left (a^{2} x^{4} - 1\right )} \sqrt{a x^{2} + 1} \sqrt{-a x^{2} + 1}}{6 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06129, size = 126, normalized size = 2.17 \begin{align*} \frac{3 \, a x^{4} + 2 \,{\left (a^{2} x^{6} - x^{2}\right )} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}}}{12 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18225, size = 240, normalized size = 4.14 \begin{align*} \frac{3 \,{\left (a^{2} x^{2} + a\right )}^{2} - 6 \,{\left (a^{2} x^{2} + a\right )} a - \frac{6 \, a^{3} \arcsin \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + a}}{2 \, \sqrt{a}}\right ) - \sqrt{a^{2} x^{2} + a} \sqrt{-a^{2} x^{2} + a}{\left ({\left (2 \, a^{2} x^{2} + 5 \, a\right )}{\left (a^{2} x^{2} - a\right )} + 3 \, a^{2}\right )} + 3 \,{\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}{a}}{12 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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