Optimal. Leaf size=115 \[ \frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{a} x\right ),-1\right )}{21 a^{7/2}}-\frac{2 x \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{21 a^3}+\frac{2 x^5}{35 a}+\frac{1}{7} x^7 e^{\text{sech}^{-1}\left (a x^2\right )} \]
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Rubi [A] time = 0.051338, antiderivative size = 115, 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, 321, 221} \[ -\frac{2 x \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{21 a^3}+\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{21 a^{7/2}}+\frac{2 x^5}{35 a}+\frac{1}{7} x^7 e^{\text{sech}^{-1}\left (a x^2\right )} \]
Antiderivative was successfully verified.
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Rule 6335
Rule 30
Rule 259
Rule 321
Rule 221
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}\left (a x^2\right )} x^6 \, dx &=\frac{1}{7} e^{\text{sech}^{-1}\left (a x^2\right )} x^7+\frac{2 \int x^4 \, dx}{7 a}+\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^4}{\sqrt{1-a x^2} \sqrt{1+a x^2}} \, dx}{7 a}\\ &=\frac{2 x^5}{35 a}+\frac{1}{7} e^{\text{sech}^{-1}\left (a x^2\right )} x^7+\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^4}{\sqrt{1-a^2 x^4}} \, dx}{7 a}\\ &=\frac{2 x^5}{35 a}+\frac{1}{7} e^{\text{sech}^{-1}\left (a x^2\right )} x^7-\frac{2 x \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{21 a^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}{21 a^3}\\ &=\frac{2 x^5}{35 a}+\frac{1}{7} e^{\text{sech}^{-1}\left (a x^2\right )} x^7-\frac{2 x \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{21 a^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 )}{21 a^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.332462, size = 139, normalized size = 1.21 \[ -\frac{2 i \sqrt{\frac{1-a x^2}{a x^2+1}} \sqrt{1-a^2 x^4} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-a} x\right ),-1\right )}{21 (-a)^{7/2} \left (a x^2-1\right )}+\frac{x \sqrt{\frac{1-a x^2}{a x^2+1}} \left (3 a^3 x^6+3 a^2 x^4-2 a x^2-2\right )}{21 a^3}+\frac{x^5}{5 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.19, size = 114, normalized size = 1. \begin{align*}{\frac{{x}^{2}}{21\,{a}^{2}{x}^{4}-21}\sqrt{-{\frac{a{x}^{2}-1}{a{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+1}{a{x}^{2}}}} \left ( 3\,{x}^{9}{a}^{9/2}-5\,{x}^{5}{a}^{5/2}-2\,{\it EllipticF} \left ( x\sqrt{a},i \right ) \sqrt{-a{x}^{2}+1}\sqrt{a{x}^{2}+1}+2\,x\sqrt{a} \right ){a}^{-{\frac{5}{2}}}}+{\frac{{x}^{5}}{5\,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^{5}}{5 \, a} + \frac{\int \sqrt{a x^{2} + 1} \sqrt{-a x^{2} + 1} x^{4}\,{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^{6} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} + x^{4}}{a}, x\right ) \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{6}{\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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