Optimal. Leaf size=52 \[ \frac{x^{m+1} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{1}{2} (-m-1),\frac{1-m}{2},-\frac{1}{a^2 x^2}\right )}{m+1}+\frac{x^m}{a m} \]
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Rubi [A] time = 0.0398986, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6336, 30, 339, 364} \[ \frac{x^{m+1} \, _2F_1\left (-\frac{1}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};-\frac{1}{a^2 x^2}\right )}{m+1}+\frac{x^m}{a m} \]
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}(a x)} x^m \, dx &=\frac{\int x^{-1+m} \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^2}} x^m \, dx\\ &=\frac{x^m}{a m}-\left (\left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \sqrt{1+\frac{x^2}{a^2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{x^m}{a m}+\frac{x^{1+m} \, _2F_1\left (-\frac{1}{2},\frac{1}{2} (-1-m);\frac{1-m}{2};-\frac{1}{a^2 x^2}\right )}{1+m}\\ \end{align*}
Mathematica [A] time = 0.0454976, size = 54, normalized size = 1.04 \[ \frac{x^{m+1} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{1}{2} (-m-1),\frac{1}{2} (-m-1)+1,-\frac{1}{a^2 x^2}\right )}{m+1}+\frac{x^m}{a m} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.255, size = 0, normalized size = 0. \begin{align*} \int \left ({\frac{1}{ax}}+\sqrt{1+{\frac{1}{{a}^{2}{x}^{2}}}} \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 x^{m} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + x^{m}}{a x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.6337, size = 51, normalized size = 0.98 \begin{align*} - \frac{x^{m} \Gamma \left (- \frac{m}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} \\ \frac{m}{2} + 1 \end{matrix}\middle |{a^{2} x^{2} e^{i \pi }} \right )}}{2 a \Gamma \left (1 - \frac{m}{2}\right )} + \frac{\begin{cases} \frac{x^{m}}{m} & \text{for}\: m \neq 0 \\\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^{2}} + 1} + \frac{1}{a x}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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