Optimal. Leaf size=44 \[ \frac{(c-a c x)^{p+2} \text{Hypergeometric2F1}\left (1,p+2,p+3,\frac{1}{2} (1-a x)\right )}{2 a c^2 (p+2)} \]
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Rubi [A] time = 0.0617144, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6167, 6130, 21, 68} \[ \frac{(c-a c x)^{p+2} \, _2F_1\left (1,p+2;p+3;\frac{1}{2} (1-a x)\right )}{2 a c^2 (p+2)} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 68
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^p \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^p \, dx\\ &=-\int \frac{(1-a x) (c-a c x)^p}{1+a x} \, dx\\ &=-\frac{\int \frac{(c-a c x)^{1+p}}{1+a x} \, dx}{c}\\ &=\frac{(c-a c x)^{2+p} \, _2F_1\left (1,2+p;3+p;\frac{1}{2} (1-a x)\right )}{2 a c^2 (2+p)}\\ \end{align*}
Mathematica [A] time = 0.0152882, size = 44, normalized size = 1. \[ -\frac{(a x-1) (c-a c x)^p \left (\text{Hypergeometric2F1}\left (1,p+1,p+2,\frac{1}{2} (1-a x)\right )-1\right )}{a (p+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.52, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -acx+c \right ) ^{p} \left ( ax-1 \right ) }{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 \frac{{\left (a x - 1\right )}{\left (-a c x + c\right )}^{p}}{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 (\frac{{\left (a x - 1\right )}{\left (-a c x + c\right )}^{p}}{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 \frac{\left (- c \left (a x - 1\right )\right )^{p} \left (a x - 1\right )}{a x + 1}\, 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 (a x - 1\right )}{\left (-a c x + c\right )}^{p}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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