Optimal. Leaf size=61 \[ -\frac{2^{-p} (1-a x)^p (c-a c x)^{p+1} \text{Hypergeometric2F1}\left (p,2 p+1,2 (p+1),\frac{1}{2} (1-a x)\right )}{a c (2 p+1)} \]
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Rubi [A] time = 0.0587585, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {6130, 23, 69} \[ -\frac{2^{-p} (1-a x)^p (c-a c x)^{p+1} \, _2F_1\left (p,2 p+1;2 (p+1);\frac{1}{2} (1-a x)\right )}{a c (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 6130
Rule 23
Rule 69
Rubi steps
\begin{align*} \int e^{-2 p \tanh ^{-1}(a x)} (c-a c x)^p \, dx &=\int (1-a x)^p (1+a x)^{-p} (c-a c x)^p \, dx\\ &=\left ((1-a x)^p (c-a c x)^{-p}\right ) \int (1+a x)^{-p} (c-a c x)^{2 p} \, dx\\ &=-\frac{2^{-p} (1-a x)^p (c-a c x)^{1+p} \, _2F_1\left (p,1+2 p;2 (1+p);\frac{1}{2} (1-a x)\right )}{a c (1+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0258814, size = 56, normalized size = 0.92 \[ -\frac{2^{-p} (1-a x)^{p+1} (c-a c x)^p \text{Hypergeometric2F1}\left (p,2 p+1,2 p+2,\frac{1}{2}-\frac{a x}{2}\right )}{2 a p+a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.343, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -acx+c \right ) ^{p}}{{{\rm e}^{2\,p{\it Artanh} \left ( ax \right ) }}}}\, 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 (-a c x + c\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 (-a c x + c\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 (-a c x + c\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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