Optimal. Leaf size=111 \[ -\frac{x^2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{16 a^3}+\frac{\sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sin ^{-1}\left (a x^2\right )}{16 a^4}+\frac{x^6}{24 a}+\frac{1}{8} x^8 e^{\text{sech}^{-1}\left (a x^2\right )} \]
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Rubi [A] time = 0.0616554, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6335, 30, 259, 275, 321, 216} \[ -\frac{x^2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{16 a^3}+\frac{\sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sin ^{-1}\left (a x^2\right )}{16 a^4}+\frac{x^6}{24 a}+\frac{1}{8} x^8 e^{\text{sech}^{-1}\left (a x^2\right )} \]
Antiderivative was successfully verified.
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Rule 6335
Rule 30
Rule 259
Rule 275
Rule 321
Rule 216
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}\left (a x^2\right )} x^7 \, dx &=\frac{1}{8} e^{\text{sech}^{-1}\left (a x^2\right )} x^8+\frac{\int x^5 \, dx}{4 a}+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^5}{\sqrt{1-a x^2} \sqrt{1+a x^2}} \, dx}{4 a}\\ &=\frac{x^6}{24 a}+\frac{1}{8} e^{\text{sech}^{-1}\left (a x^2\right )} x^8+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^5}{\sqrt{1-a^2 x^4}} \, dx}{4 a}\\ &=\frac{x^6}{24 a}+\frac{1}{8} e^{\text{sech}^{-1}\left (a x^2\right )} x^8+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-a^2 x^2}} \, dx,x,x^2\right )}{8 a}\\ &=\frac{x^6}{24 a}+\frac{1}{8} e^{\text{sech}^{-1}\left (a x^2\right )} x^8-\frac{x^2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{16 a^3}+\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 )}{16 a^3}\\ &=\frac{x^6}{24 a}+\frac{1}{8} e^{\text{sech}^{-1}\left (a x^2\right )} x^8-\frac{x^2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{16 a^3}+\frac{\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sin ^{-1}\left (a x^2\right )}{16 a^4}\\ \end{align*}
Mathematica [C] time = 0.181418, size = 111, normalized size = 1. \[ \frac{8 a^3 x^6-3 a \sqrt{\frac{1-a x^2}{a x^2+1}} \left (-2 a^3 x^8-2 a^2 x^6+a x^4+x^2\right )+3 i \log \left (2 \sqrt{\frac{1-a x^2}{a x^2+1}} \left (a x^2+1\right )-2 i a x^2\right )}{48 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.322, size = 137, normalized size = 1.2 \begin{align*}{\frac{{x}^{2}}{16\,{a}^{4}}\sqrt{-{\frac{a{x}^{2}-1}{a{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+1}{a{x}^{2}}}} \left ( 2\,{x}^{6}\sqrt{-{\frac{{a}^{2}{x}^{4}-1}{{a}^{2}}}}{a}^{4}-{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}^{6}}{6\,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^{6}}{6 \, a} + \frac{-\frac{{\left (-a^{2} x^{4} + 1\right )}^{\frac{3}{2}} x^{2}}{8 \, a^{2}} + \frac{\sqrt{-a^{2} x^{4} + 1} x^{2}}{16 \, a^{2}} + \frac{\arcsin \left (\frac{a^{2} x^{2}}{\sqrt{a^{2}}}\right )}{16 \, \sqrt{a^{2}} a^{2}}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12503, size = 251, normalized size = 2.26 \begin{align*} \frac{8 \, a^{3} x^{6} + 3 \,{\left (2 \, a^{4} x^{8} - a^{2} x^{4}\right )} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} - 6 \, \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 )}{48 \, a^{4}} \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 = 1.19495, size = 360, normalized size = 3.24 \begin{align*} \frac{8 \, a^{4} x^{6} - \frac{30 \, a^{4} \arcsin \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + a}}{2 \, \sqrt{a}}\right ) - 12 \,{\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^{2} - \sqrt{a^{2} x^{2} + a} \sqrt{-a^{2} x^{2} + a}{\left (15 \, a^{3} +{\left (a^{2} x^{2} - a\right )}{\left (2 \,{\left (3 \, a^{2} x^{2} + 14 \, a\right )}{\left (a^{2} x^{2} - a\right )} + 59 \, a^{2}\right )}\right )} - 8 \,{\left (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 )}\right )} a}{a^{3}}}{48 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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