Optimal. Leaf size=118 \[ \frac{x \left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{-p-\frac{1}{2}} \left (1-\frac{1}{a x}\right )^{p+\frac{1}{2}} \left (\frac{1}{a x}+1\right )^{p+\frac{1}{2}} \left (c-a^2 c x^2\right )^p \text{Hypergeometric2F1}\left (-2 p-1,-p-\frac{1}{2},-2 p,\frac{2}{x \left (a+\frac{1}{x}\right )}\right )}{2 p+1} \]
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Rubi [A] time = 0.138725, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6192, 6196, 132} \[ \frac{x \left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{-p-\frac{1}{2}} \left (1-\frac{1}{a x}\right )^{p+\frac{1}{2}} \left (\frac{1}{a x}+1\right )^{p+\frac{1}{2}} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-2 p-1,-p-\frac{1}{2};-2 p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{2 p+1} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6196
Rule 132
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-\frac{1}{a^2 x^2}\right )^{-p} x^{-2 p} \left (c-a^2 c x^2\right )^p\right ) \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^p x^{2 p} \, dx\\ &=-\left (\left (\left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (\frac{1}{x}\right )^{2 p} \left (c-a^2 c x^2\right )^p\right ) \operatorname{Subst}\left (\int x^{-2-2 p} \left (1-\frac{x}{a}\right )^{\frac{1}{2}+p} \left (1+\frac{x}{a}\right )^{-\frac{1}{2}+p} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{\left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{-\frac{1}{2}-p} \left (1-\frac{1}{a x}\right )^{\frac{1}{2}+p} \left (1+\frac{1}{a x}\right )^{\frac{1}{2}+p} x \left (c-a^2 c x^2\right )^p \, _2F_1\left (-1-2 p,-\frac{1}{2}-p;-2 p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{1+2 p}\\ \end{align*}
Mathematica [A] time = 0.220012, size = 118, normalized size = 1. \[ -\frac{4^{p+1} \left (a x \sqrt{1-\frac{1}{a^2 x^2}}\right )^{-2 p} e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \left (1-e^{2 \coth ^{-1}(a x)}\right )^{2 p} \left (\frac{e^{\coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}\right )^{2 p} \text{Hypergeometric2F1}\left (p+\frac{1}{2},2 p+2,p+\frac{3}{2},e^{2 \coth ^{-1}(a x)}\right )}{2 a p+a} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.359, size = 0, normalized size = 0. \begin{align*} \int \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}\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{\left (-a^{2} c x^{2} + c\right )}^{p} \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 ({\left (-a^{2} c x^{2} + c\right )}^{p} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{a x - 1}{a x + 1}} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p}\, 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{\left (-a^{2} c x^{2} + c\right )}^{p} \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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