Optimal. Leaf size=107 \[ -\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} x^{m-1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m-1}{4},\frac{m+3}{4},a^2 x^4\right )}{a \left (1-m^2\right )}-\frac{2 x^{m-1}}{a \left (1-m^2\right )}+\frac{x^{m+1} e^{\text{sech}^{-1}\left (a x^2\right )}}{m+1} \]
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Rubi [A] time = 0.0557968, antiderivative size = 107, 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} \[ -\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} x^{m-1} \, _2F_1\left (\frac{1}{2},\frac{m-1}{4};\frac{m+3}{4};a^2 x^4\right )}{a \left (1-m^2\right )}-\frac{2 x^{m-1}}{a \left (1-m^2\right )}+\frac{x^{m+1} e^{\text{sech}^{-1}\left (a x^2\right )}}{m+1} \]
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^2\right )} x^m \, dx &=\frac{e^{\text{sech}^{-1}\left (a x^2\right )} x^{1+m}}{1+m}+\frac{2 \int x^{-2+m} \, dx}{a (1+m)}+\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^{-2+m}}{\sqrt{1-a x^2} \sqrt{1+a x^2}} \, dx}{a (1+m)}\\ &=-\frac{2 x^{-1+m}}{a \left (1-m^2\right )}+\frac{e^{\text{sech}^{-1}\left (a x^2\right )} x^{1+m}}{1+m}+\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^{-2+m}}{\sqrt{1-a^2 x^4}} \, dx}{a (1+m)}\\ &=-\frac{2 x^{-1+m}}{a \left (1-m^2\right )}+\frac{e^{\text{sech}^{-1}\left (a x^2\right )} x^{1+m}}{1+m}-\frac{2 x^{-1+m} \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \, _2F_1\left (\frac{1}{2},\frac{1}{4} (-1+m);\frac{3+m}{4};a^2 x^4\right )}{a \left (1-m^2\right )}\\ \end{align*}
Mathematica [A] time = 2.30329, size = 159, normalized size = 1.49 \[ \frac{2^{\frac{m+1}{2}} x^{m+1} \left (a x^2\right )^{\frac{1}{2} (-m-1)} e^{\text{sech}^{-1}\left (a x^2\right )} \left (\frac{e^{\text{sech}^{-1}\left (a x^2\right )}}{e^{2 \text{sech}^{-1}\left (a x^2\right )}+1}\right )^{\frac{m+1}{2}} \left ((m+7) \text{Hypergeometric2F1}\left (1,\frac{1-m}{4},\frac{m+7}{4},-e^{2 \text{sech}^{-1}\left (a x^2\right )}\right )-(m+3) e^{2 \text{sech}^{-1}\left (a x^2\right )} \text{Hypergeometric2F1}\left (1,\frac{5-m}{4},\frac{m+11}{4},-e^{2 \text{sech}^{-1}\left (a x^2\right )}\right )\right )}{(m+3) (m+7)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.194, size = 0, normalized size = 0. \begin{align*} \int \left ({\frac{1}{a{x}^{2}}}+\sqrt{{\frac{1}{a{x}^{2}}}-1}\sqrt{{\frac{1}{a{x}^{2}}}+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^{2} x^{m} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} + x^{m}}{a x^{2}}, 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^{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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