Optimal. Leaf size=64 \[ \frac{2 x^m \text{Hypergeometric2F1}\left (-\frac{1}{2},-\frac{m}{2},1-\frac{m}{2},-\frac{1}{a^2 x^2}\right )}{a m}-\frac{2 x^{m-1}}{a^2 (1-m)}+\frac{x^{m+1}}{m+1} \]
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Rubi [A] time = 0.321878, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6338, 6742, 339, 364} \[ \frac{2 x^m \, _2F_1\left (-\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};-\frac{1}{a^2 x^2}\right )}{a m}-\frac{2 x^{m-1}}{a^2 (1-m)}+\frac{x^{m+1}}{m+1} \]
Antiderivative was successfully verified.
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Rule 6338
Rule 6742
Rule 339
Rule 364
Rubi steps
\begin{align*} \int e^{2 \text{csch}^{-1}(a x)} x^m \, dx &=\int \left (\sqrt{1+\frac{1}{a^2 x^2}}+\frac{1}{a x}\right )^2 x^m \, dx\\ &=\int \left (\frac{2 x^{-2+m}}{a^2}+\frac{2 \sqrt{1+\frac{1}{a^2 x^2}} x^{-1+m}}{a}+x^m\right ) \, dx\\ &=-\frac{2 x^{-1+m}}{a^2 (1-m)}+\frac{x^{1+m}}{1+m}+\frac{2 \int \sqrt{1+\frac{1}{a^2 x^2}} x^{-1+m} \, dx}{a}\\ &=-\frac{2 x^{-1+m}}{a^2 (1-m)}+\frac{x^{1+m}}{1+m}-\frac{\left (2 \left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int x^{-1-m} \sqrt{1+\frac{x^2}{a^2}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{2 x^{-1+m}}{a^2 (1-m)}+\frac{x^{1+m}}{1+m}+\frac{2 x^m \, _2F_1\left (-\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};-\frac{1}{a^2 x^2}\right )}{a m}\\ \end{align*}
Mathematica [A] time = 0.0648334, size = 57, normalized size = 0.89 \[ x^m \left (\frac{2 \text{Hypergeometric2F1}\left (-\frac{1}{2},-\frac{m}{2},1-\frac{m}{2},-\frac{1}{a^2 x^2}\right )}{a m}+\frac{2}{a^2 (m-1) x}+\frac{x}{m+1}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.185, size = 0, normalized size = 0. \begin{align*} \int \left ({\frac{1}{ax}}+\sqrt{1+{\frac{1}{{a}^{2}{x}^{2}}}} \right ) ^{2}{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{2 \, a x x^{m} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} +{\left (a^{2} x^{2} + 2\right )} x^{m}}{a^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.1205, size = 71, normalized size = 1.11 \begin{align*} \begin{cases} \frac{x^{m + 1}}{m + 1} & \text{for}\: m \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases} - \frac{x^{m} \Gamma \left (- \frac{m}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{m}{2} \\ 1 - \frac{m}{2} \end{matrix}\middle |{\frac{e^{i \pi }}{a^{2} x^{2}}} \right )}}{a \Gamma \left (1 - \frac{m}{2}\right )} + \frac{2 \left (\begin{cases} \frac{x^{m}}{m x - x} & \text{for}\: m \neq 1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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