Optimal. Leaf size=88 \[ -\frac{2^{p+\frac{3}{2}} \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{-p} F_1\left (\frac{1}{2};-p-\frac{1}{2},2;\frac{3}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right ) \left (c-\frac{c}{a x}\right )^p}{a} \]
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Rubi [A] time = 0.0853336, antiderivative size = 88, 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 = {6182, 6179, 136} \[ -\frac{2^{p+\frac{3}{2}} \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{-p} F_1\left (\frac{1}{2};-p-\frac{1}{2},2;\frac{3}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right ) \left (c-\frac{c}{a x}\right )^p}{a} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6179
Rule 136
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^p \, dx &=\left (\left (1-\frac{1}{a x}\right )^{-p} \left (c-\frac{c}{a x}\right )^p\right ) \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^p \, dx\\ &=-\left (\left (\left (1-\frac{1}{a x}\right )^{-p} \left (c-\frac{c}{a x}\right )^p\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{\frac{1}{2}+p}}{x^2 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{2^{\frac{3}{2}+p} \left (1-\frac{1}{a x}\right )^{-p} \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^p F_1\left (\frac{1}{2};-\frac{1}{2}-p,2;\frac{3}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right )}{a}\\ \end{align*}
Mathematica [F] time = 1.08491, size = 0, normalized size = 0. \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^p \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.174, size = 0, normalized size = 0. \begin{align*} \int \left ( c-{\frac{c}{ax}} \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 (c - \frac{c}{a x}\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 (\frac{a c x - c}{a x}\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(-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{\left (c - \frac{c}{a x}\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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