Optimal. Leaf size=54 \[ \frac{x (1-a x)^{-p} F_1(1-p;-2 p,p;2-p;a x,-a x) \left (c-\frac{c}{a x}\right )^p}{1-p} \]
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Rubi [A] time = 0.10718, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {6134, 6129, 133} \[ \frac{x (1-a x)^{-p} F_1(1-p;-2 p,p;2-p;a x,-a x) \left (c-\frac{c}{a x}\right )^p}{1-p} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6129
Rule 133
Rubi steps
\begin{align*} \int e^{-2 p \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^p \, dx &=\left (\left (c-\frac{c}{a x}\right )^p x^p (1-a x)^{-p}\right ) \int e^{-2 p \tanh ^{-1}(a x)} x^{-p} (1-a x)^p \, dx\\ &=\left (\left (c-\frac{c}{a x}\right )^p x^p (1-a x)^{-p}\right ) \int x^{-p} (1-a x)^{2 p} (1+a x)^{-p} \, dx\\ &=\frac{\left (c-\frac{c}{a x}\right )^p x (1-a x)^{-p} F_1(1-p;-2 p,p;2-p;a x,-a x)}{1-p}\\ \end{align*}
Mathematica [F] time = 0.425185, size = 0, normalized size = 0. \[ \int e^{-2 p \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^p \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.147, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{{\rm e}^{2\,p{\it Artanh} \left ( ax \right ) }}} \left ( c-{\frac{c}{ax}} \right ) ^{p}}\, 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 (c - \frac{c}{a x}\right )}^{p}}{\left (\frac{a x + 1}{a x - 1}\right )^{p}}\,{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 (\frac{a c x - c}{a x}\right )^{p}}{\left (\frac{a x + 1}{a x - 1}\right )^{p}}, 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 \frac{{\left (c - \frac{c}{a x}\right )}^{p}}{\left (\frac{a x + 1}{a x - 1}\right )^{p}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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