Optimal. Leaf size=57 \[ \frac{(2-p) \left (c-\frac{c}{a x}\right )^p \text{Hypergeometric2F1}\left (1,p,p+1,1-\frac{1}{a x}\right )}{a p}+x \left (c-\frac{c}{a x}\right )^p \]
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Rubi [A] time = 0.106547, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6167, 6133, 25, 514, 375, 78, 65} \[ \frac{(2-p) \left (c-\frac{c}{a x}\right )^p \, _2F_1\left (1,p;p+1;1-\frac{1}{a x}\right )}{a p}+x \left (c-\frac{c}{a x}\right )^p \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 375
Rule 78
Rule 65
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^p \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^p \, dx\\ &=-\int \frac{\left (c-\frac{c}{a x}\right )^p (1+a x)}{1-a x} \, dx\\ &=\frac{c \int \frac{\left (c-\frac{c}{a x}\right )^{-1+p} (1+a x)}{x} \, dx}{a}\\ &=\frac{c \int \left (a+\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{-1+p} \, dx}{a}\\ &=-\frac{c \operatorname{Subst}\left (\int \frac{(a+x) \left (c-\frac{c x}{a}\right )^{-1+p}}{x^2} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\left (c-\frac{c}{a x}\right )^p x-\frac{(c (2-p)) \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{-1+p}}{x} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\left (c-\frac{c}{a x}\right )^p x+\frac{(2-p) \left (c-\frac{c}{a x}\right )^p \, _2F_1\left (1,p;1+p;1-\frac{1}{a x}\right )}{a p}\\ \end{align*}
Mathematica [A] time = 0.0235685, size = 46, normalized size = 0.81 \[ \frac{\left (c-\frac{c}{a x}\right )^p \left (a p x-(p-2) \text{Hypergeometric2F1}\left (1,p,p+1,1-\frac{1}{a x}\right )\right )}{a p} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.345, size = 0, normalized size = 0. \begin{align*} \int{\frac{ax+1}{ax-1} \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 (a x + 1\right )}{\left (c - \frac{c}{a x}\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 (\frac{a c x - c}{a x}\right )^{p}}{a x - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 7.64406, size = 272, normalized size = 4.77 \begin{align*} a \left (\begin{cases} \frac{0^{p} x}{a} + \frac{0^{p} \log{\left (a x - 1 \right )}}{a^{2}} - \frac{a^{- p} c^{p} p x^{2} x^{- p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (2 - p\right ){{}_{2}F_{1}\left (\begin{matrix} 1 - p, 2 - p \\ 3 - p \end{matrix}\middle |{a x} \right )}}{\Gamma \left (3 - p\right ) \Gamma \left (p + 1\right )} & \text{for}\: \left |{a x}\right | > 1 \\\frac{0^{p} x}{a} + \frac{0^{p} \log{\left (- a x + 1 \right )}}{a^{2}} - \frac{a^{- p} c^{p} p x^{2} x^{- p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (2 - p\right ){{}_{2}F_{1}\left (\begin{matrix} 1 - p, 2 - p \\ 3 - p \end{matrix}\middle |{a x} \right )}}{\Gamma \left (3 - p\right ) \Gamma \left (p + 1\right )} & \text{otherwise} \end{cases}\right ) + \begin{cases} \frac{0^{p} \log{\left (a x - 1 \right )}}{a} - \frac{a^{- p} c^{p} p x x^{- p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (1 - p\right ){{}_{2}F_{1}\left (\begin{matrix} 1 - p, 1 - p \\ 2 - p \end{matrix}\middle |{a x} \right )}}{\Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} & \text{for}\: \left |{a x}\right | > 1 \\\frac{0^{p} \log{\left (- a x + 1 \right )}}{a} - \frac{a^{- p} c^{p} p x x^{- p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (1 - p\right ){{}_{2}F_{1}\left (\begin{matrix} 1 - p, 1 - p \\ 2 - p \end{matrix}\middle |{a x} \right )}}{\Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} & \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 (c - \frac{c}{a x}\right )}^{p}}{a x - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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