Optimal. Leaf size=111 \[ -\frac{2^{\frac{1}{2}-\frac{n}{2}} \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} F_1\left (\frac{n+2}{2};\frac{n+1}{2},2;\frac{n+4}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right )}{a (n+2) \sqrt{c-\frac{c}{a x}}} \]
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Rubi [A] time = 0.149866, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6182, 6179, 136} \[ -\frac{2^{\frac{1}{2}-\frac{n}{2}} \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} F_1\left (\frac{n+2}{2};\frac{n+1}{2},2;\frac{n+4}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right )}{a (n+2) \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6179
Rule 136
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx &=\frac{\sqrt{1-\frac{1}{a x}} \int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{1-\frac{1}{a x}}} \, dx}{\sqrt{c-\frac{c}{a x}}}\\ &=-\frac{\sqrt{1-\frac{1}{a x}} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2}}{x^2} \, dx,x,\frac{1}{x}\right )}{\sqrt{c-\frac{c}{a x}}}\\ &=-\frac{2^{\frac{1}{2}-\frac{n}{2}} \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} F_1\left (\frac{2+n}{2};\frac{1+n}{2},2;\frac{4+n}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right )}{a (2+n) \sqrt{c-\frac{c}{a x}}}\\ \end{align*}
Mathematica [F] time = 180.006, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [F] time = 0.172, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}{\frac{1}{\sqrt{c-{\frac{c}{ax}}}}}}\, 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 \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{\sqrt{c - \frac{c}{a x}}}\,{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{a x \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n} \sqrt{\frac{a c x - c}{a x}}}{a c x - c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{n \operatorname{acoth}{\left (a x \right )}}}{\sqrt{- c \left (-1 + \frac{1}{a x}\right )}}\, dx \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 \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{\sqrt{c - \frac{c}{a x}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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