Optimal. Leaf size=59 \[ \frac{x^{m+1} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{1}{4} (-m-1),\frac{3-m}{4},-\frac{1}{a^2 x^4}\right )}{m+1}-\frac{x^{m-1}}{a (1-m)} \]
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Rubi [A] time = 0.0415475, antiderivative size = 59, 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 = {6336, 30, 339, 364} \[ \frac{x^{m+1} \, _2F_1\left (-\frac{1}{2},\frac{1}{4} (-m-1);\frac{3-m}{4};-\frac{1}{a^2 x^4}\right )}{m+1}-\frac{x^{m-1}}{a (1-m)} \]
Antiderivative was successfully verified.
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Rule 6336
Rule 30
Rule 339
Rule 364
Rubi steps
\begin{align*} \int e^{\text{csch}^{-1}\left (a x^2\right )} x^m \, dx &=\frac{\int x^{-2+m} \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^4}} x^m \, dx\\ &=-\frac{x^{-1+m}}{a (1-m)}-\left (\left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \sqrt{1+\frac{x^4}{a^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{x^{-1+m}}{a (1-m)}+\frac{x^{1+m} \, _2F_1\left (-\frac{1}{2},\frac{1}{4} (-1-m);\frac{3-m}{4};-\frac{1}{a^2 x^4}\right )}{1+m}\\ \end{align*}
Mathematica [A] time = 0.0612119, size = 55, normalized size = 0.93 \[ x^{m-1} \left (\frac{x^2 \text{Hypergeometric2F1}\left (-\frac{1}{2},-\frac{m}{4}-\frac{1}{4},\frac{3}{4}-\frac{m}{4},-\frac{1}{a^2 x^4}\right )}{m+1}+\frac{1}{a (m-1)}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.199, size = 0, normalized size = 0. \begin{align*} \int \left ({\frac{1}{a{x}^{2}}}+\sqrt{1+{\frac{1}{{a}^{2}{x}^{4}}}} \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^{2} x^{4} + 1}{a^{2} x^{4}}} + x^{m}}{a x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.5374, size = 66, normalized size = 1.12 \begin{align*} - \frac{x x^{m} \Gamma \left (- \frac{m}{4} - \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{m}{4} - \frac{1}{4} \\ \frac{3}{4} - \frac{m}{4} \end{matrix}\middle |{\frac{e^{i \pi }}{a^{2} x^{4}}} \right )}}{4 \Gamma \left (\frac{3}{4} - \frac{m}{4}\right )} + \frac{\begin{cases} \frac{x^{m}}{m x - x} & \text{for}\: m \neq 1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}}{a} \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^{2} x^{4}} + 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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