Optimal. Leaf size=42 \[ \frac{2 (c-a c x)^p}{a p}-\frac{(c-a c x)^{p+1}}{a c (p+1)} \]
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Rubi [A] time = 0.0666535, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6167, 6130, 21, 43} \[ \frac{2 (c-a c x)^p}{a p}-\frac{(c-a c x)^{p+1}}{a c (p+1)} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 43
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\\ &=-\left (c \int (1+a x) (c-a c x)^{-1+p} \, dx\right )\\ &=-\left (c \int \left (2 (c-a c x)^{-1+p}-\frac{(c-a c x)^p}{c}\right ) \, dx\right )\\ &=\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 = 0.022598, size = 28, normalized size = 0.67 \[ \frac{(a p x+p+2) (c-a c x)^p}{a p (p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 29, normalized size = 0.7 \begin{align*}{\frac{ \left ( apx+p+2 \right ) \left ( -acx+c \right ) ^{p}}{ap \left ( 1+p \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06866, size = 66, normalized size = 1.57 \begin{align*} \frac{{\left (a c^{p} p x + c^{p}\right )}{\left (-a x + 1\right )}^{p}}{{\left (p^{2} + p\right )} a} + \frac{{\left (-a x + 1\right )}^{p} c^{p}}{a p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59576, size = 62, normalized size = 1.48 \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 = 0.83473, size = 124, normalized size = 2.95 \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 )}{\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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