Optimal. Leaf size=75 \[ \frac{x^{m+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (-m-1),\frac{1-m}{2},\frac{1}{a^2 x^2}\right )}{m+1}-\frac{x^m \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{m}{2},1-\frac{m}{2},\frac{1}{a^2 x^2}\right )}{a m} \]
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Rubi [A] time = 0.0601647, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6172, 808, 364} \[ \frac{x^{m+1} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};\frac{1}{a^2 x^2}\right )}{m+1}-\frac{x^m \, _2F_1\left (\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right )}{a m} \]
Antiderivative was successfully verified.
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Rule 6172
Rule 808
Rule 364
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} x^m \, dx &=-\left (\left (\left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-2-m} \left (1-\frac{x}{a}\right )}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\left (\left (\left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-2-m}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\right )+\frac{\left (\left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-1-m}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\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}-\frac{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 [C] time = 0.242454, size = 115, normalized size = 1.53 \[ x^{m+1} \left (\frac{\text{Hypergeometric2F1}\left (-\frac{1}{2},-\frac{m}{2}-\frac{1}{2},\frac{1}{2}-\frac{m}{2},\frac{1}{a^2 x^2}\right )}{m+1}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} \sqrt{\frac{a x-1}{a^2}} F_1\left (m;-\frac{1}{2},\frac{1}{2};m+1;a x,-a x\right )}{m \sqrt{1-a x} \sqrt{x^2-\frac{1}{a^2}}}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.188, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt{{\frac{ax-1}{ax+1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \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 (x^{m} \sqrt{\frac{a x - 1}{a x + 1}}, 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} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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