Optimal. Leaf size=150 \[ \frac{x^m \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{m}{2},1-\frac{m}{2},\frac{1}{a^2 x^2}\right )}{a m}-\frac{4 x^m \text{Hypergeometric2F1}\left (\frac{3}{2},-\frac{m}{2},1-\frac{m}{2},\frac{1}{a^2 x^2}\right )}{a m}-\frac{3 x^{m+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (-m-1),\frac{1-m}{2},\frac{1}{a^2 x^2}\right )}{m+1}+\frac{4 x^{m+1} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{1}{2} (-m-1),\frac{1-m}{2},\frac{1}{a^2 x^2}\right )}{m+1} \]
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Rubi [A] time = 1.05072, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6172, 6742, 364, 850, 808} \[ \frac{x^m \, _2F_1\left (\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right )}{a m}-\frac{4 x^m \, _2F_1\left (\frac{3}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right )}{a m}-\frac{3 x^{m+1} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};\frac{1}{a^2 x^2}\right )}{m+1}+\frac{4 x^{m+1} \, _2F_1\left (\frac{3}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};\frac{1}{a^2 x^2}\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 6172
Rule 6742
Rule 364
Rule 850
Rule 808
Rubi steps
\begin{align*} \int e^{-3 \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 )^2}{\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 \left (-\frac{3 x^{-2-m}}{\sqrt{1-\frac{x^2}{a^2}}}+\frac{x^{-1-m}}{a \sqrt{1-\frac{x^2}{a^2}}}+\frac{4 x^{-2-m}}{\left (1+\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}}\right ) \, dx,x,\frac{1}{x}\right )\right )\\ &=\left (3 \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 )-\left (4 \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 )-\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{3 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}-\left (4 \left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-2-m} \left (1-\frac{x}{a}\right )}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3 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}-\left (4 \left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-2-m}}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )+\frac{\left (4 \left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-1-m}}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{3 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}+\frac{4 x^{1+m} \, _2F_1\left (\frac{3}{2},\frac{1}{2} (-1-m);\frac{1-m}{2};\frac{1}{a^2 x^2}\right )}{1+m}-\frac{4 x^m \, _2F_1\left (\frac{3}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right )}{a m}\\ \end{align*}
Mathematica [C] time = 0.240829, size = 192, normalized size = 1.28 \[ \frac{x^{m+1} \left (m \sqrt{1-a x} \sqrt{x^2-\frac{1}{a^2}} \text{Hypergeometric2F1}\left (-\frac{1}{2},-\frac{m}{2}-\frac{1}{2},\frac{1}{2}-\frac{m}{2},\frac{1}{a^2 x^2}\right )-3 (m+1) \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{\frac{a x-1}{a^2}} F_1\left (m;-\frac{1}{2},\frac{1}{2};m+1;a x,-a x\right )+2 (m+1) \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{\frac{a x-1}{a^2}} F_1\left (m;-\frac{1}{2},\frac{3}{2};m+1;a x,-a x\right )\right )}{m (m+1) \sqrt{1-a x} \sqrt{x^2-\frac{1}{a^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.178, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\, 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} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{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 (\frac{{\left (a x - 1\right )} x^{m} \sqrt{\frac{a x - 1}{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} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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