Optimal. Leaf size=109 \[ -\frac{3 \sqrt{\frac{1}{a x^3+1}} \sqrt{a x^3+1} x^{m-2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m-2}{6},\frac{m+4}{6},a^2 x^6\right )}{a \left (-m^2+m+2\right )}-\frac{3 x^{m-2}}{a \left (-m^2+m+2\right )}+\frac{x^{m+1} e^{\text{sech}^{-1}\left (a x^3\right )}}{m+1} \]
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Rubi [A] time = 0.0677183, antiderivative size = 109, 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, 30, 259, 364} \[ -\frac{3 \sqrt{\frac{1}{a x^3+1}} \sqrt{a x^3+1} x^{m-2} \, _2F_1\left (\frac{1}{2},\frac{m-2}{6};\frac{m+4}{6};a^2 x^6\right )}{a \left (-m^2+m+2\right )}-\frac{3 x^{m-2}}{a \left (-m^2+m+2\right )}+\frac{x^{m+1} e^{\text{sech}^{-1}\left (a x^3\right )}}{m+1} \]
Antiderivative was successfully verified.
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Rule 6335
Rule 30
Rule 259
Rule 364
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}\left (a x^3\right )} x^m \, dx &=\frac{e^{\text{sech}^{-1}\left (a x^3\right )} x^{1+m}}{1+m}+\frac{3 \int x^{-3+m} \, dx}{a (1+m)}+\frac{\left (3 \sqrt{\frac{1}{1+a x^3}} \sqrt{1+a x^3}\right ) \int \frac{x^{-3+m}}{\sqrt{1-a x^3} \sqrt{1+a x^3}} \, dx}{a (1+m)}\\ &=-\frac{3 x^{-2+m}}{a \left (2+m-m^2\right )}+\frac{e^{\text{sech}^{-1}\left (a x^3\right )} x^{1+m}}{1+m}+\frac{\left (3 \sqrt{\frac{1}{1+a x^3}} \sqrt{1+a x^3}\right ) \int \frac{x^{-3+m}}{\sqrt{1-a^2 x^6}} \, dx}{a (1+m)}\\ &=-\frac{3 x^{-2+m}}{a \left (2+m-m^2\right )}+\frac{e^{\text{sech}^{-1}\left (a x^3\right )} x^{1+m}}{1+m}-\frac{3 x^{-2+m} \sqrt{\frac{1}{1+a x^3}} \sqrt{1+a x^3} \, _2F_1\left (\frac{1}{2},\frac{1}{6} (-2+m);\frac{4+m}{6};a^2 x^6\right )}{a \left (2+m-m^2\right )}\\ \end{align*}
Mathematica [A] time = 2.49469, size = 159, normalized size = 1.46 \[ \frac{2^{\frac{m+1}{3}} x^{m+1} \left (a x^3\right )^{\frac{1}{3} (-m-1)} e^{\text{sech}^{-1}\left (a x^3\right )} \left (\frac{e^{\text{sech}^{-1}\left (a x^3\right )}}{e^{2 \text{sech}^{-1}\left (a x^3\right )}+1}\right )^{\frac{m+1}{3}} \left ((m+10) \text{Hypergeometric2F1}\left (1,\frac{2-m}{6},\frac{m+10}{6},-e^{2 \text{sech}^{-1}\left (a x^3\right )}\right )-(m+4) e^{2 \text{sech}^{-1}\left (a x^3\right )} \text{Hypergeometric2F1}\left (1,\frac{8-m}{6},\frac{m+16}{6},-e^{2 \text{sech}^{-1}\left (a x^3\right )}\right )\right )}{(m+4) (m+10)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.23, size = 0, normalized size = 0. \begin{align*} \int \left ({\frac{1}{{x}^{3}a}}+\sqrt{{\frac{1}{{x}^{3}a}}-1}\sqrt{{\frac{1}{{x}^{3}a}}+1} \right ){x}^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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^{3} x^{m} \sqrt{\frac{a x^{3} + 1}{a x^{3}}} \sqrt{-\frac{a x^{3} - 1}{a x^{3}}} + x^{m}}{a x^{3}}, 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^{m}{\left (\sqrt{\frac{1}{a x^{3}} + 1} \sqrt{\frac{1}{a x^{3}} - 1} + \frac{1}{a x^{3}}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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