Optimal. Leaf size=41 \[ \frac{(c-a c x)^{p+1}}{a c (p+1)}-\frac{2 (c-a c x)^p}{a p} \]
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Rubi [A] time = 0.0441538, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6130, 21, 43} \[ \frac{(c-a c x)^{p+1}}{a c (p+1)}-\frac{2 (c-a c x)^p}{a p} \]
Antiderivative was successfully verified.
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Rule 6130
Rule 21
Rule 43
Rubi steps
\begin{align*} \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\\ &=c \int (1+a x) (c-a c x)^{-1+p} \, dx\\ &=c \int \left (2 (c-a c x)^{-1+p}-\frac{(c-a c x)^p}{c}\right ) \, dx\\ &=-\frac{2 (c-a c x)^p}{a p}+\frac{(c-a c x)^{1+p}}{a c (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0214986, size = 29, normalized size = 0.71 \[ -\frac{(a p x+p+2) (c-a c x)^p}{a p (p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 30, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -acx+c \right ) ^{p} \left ( apx+p+2 \right ) }{ap \left ( 1+p \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13554, size = 47, normalized size = 1.15 \begin{align*} -\frac{{\left (a c^{p} p x + c^{p}{\left (p + 2\right )}\right )}{\left (-a x + 1\right )}^{p}}{{\left (p^{2} + p\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68245, size = 63, normalized size = 1.54 \begin{align*} -\frac{{\left (a p x + p + 2\right )}{\left (-a c x + c\right )}^{p}}{a p^{2} + a p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.13121, size = 126, normalized size = 3.07 \begin{align*} \begin{cases} c^{p} x & \text{for}\: a = 0 \\\frac{a x \log{\left (x - \frac{1}{a} \right )}}{a^{2} c x - a c} - \frac{\log{\left (x - \frac{1}{a} \right )}}{a^{2} c x - a c} - \frac{2}{a^{2} c x - a c} & \text{for}\: p = -1 \\- x - \frac{2 \log{\left (x - \frac{1}{a} \right )}}{a} & \text{for}\: p = 0 \\- \frac{a p x \left (- a c x + c\right )^{p}}{a p^{2} + a p} - \frac{p \left (- a c x + c\right )^{p}}{a p^{2} + a p} - \frac{2 \left (- a c x + c\right )^{p}}{a p^{2} + a p} & \text{otherwise} \end{cases} \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 )}^{2}{\left (-a c x + c\right )}^{p}}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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