Optimal. Leaf size=96 \[ \frac{2 x \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{n+1}{2}} \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{n+1}{2},\frac{1}{2},\frac{2}{x \left (a+\frac{1}{x}\right )}\right )}{\sqrt{c-a c x}} \]
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Rubi [A] time = 0.179679, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6176, 6181, 132} \[ \frac{2 x \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{n+1}{2}} \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \, _2F_1\left (-\frac{1}{2},\frac{n+1}{2};\frac{1}{2};\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{\sqrt{c-a c x}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 132
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx &=\frac{\left (\sqrt{1-\frac{1}{a x}} \sqrt{x}\right ) \int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{1-\frac{1}{a x}} \sqrt{x}} \, dx}{\sqrt{c-a c 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^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ &=\frac{2 \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1+n}{2}} \left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x \, _2F_1\left (-\frac{1}{2},\frac{1+n}{2};\frac{1}{2};\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{\sqrt{c-a c x}}\\ \end{align*}
Mathematica [A] time = 0.0534178, size = 96, normalized size = 1. \[ \frac{2 (a x+1) \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{n/2} \left (\frac{a x-1}{a x+1}\right )^{\frac{n+1}{2}} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{n+1}{2},\frac{1}{2},\frac{2}{a x+1}\right )}{a \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.341, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}{\frac{1}{\sqrt{-acx+c}}}}\, 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{-a c x + c}}\,{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{\sqrt{-a c x + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{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 (a x - 1\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{-a c x + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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