Optimal. Leaf size=57 \[ \frac {(2-p) \left (c-\frac {c}{a x}\right )^p \, _2F_1\left (1,p;p+1;1-\frac {1}{a x}\right )}{a p}+x \left (c-\frac {c}{a x}\right )^p \]
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Rubi [A] time = 0.11, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 25
Rule 65
Rule 78
Rule 375
Rule 514
Rule 6133
Rule 6167
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*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.81 \[ \frac {\left (c-\frac {c}{a x}\right )^p \left (a p x-(p-2) \, _2F_1\left (1,p;p+1;1-\frac {1}{a x}\right )\right )}{a p} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x + 1\right )} \left (\frac {a c x - c}{a x}\right )^{p}}{a x - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{p}}{a x - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right ) \left (c -\frac {c}{a x}\right )^{p}}{a x -1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{p}}{a x - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^p\,\left (a\,x+1\right )}{a\,x-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.83, size = 272, normalized size = 4.77 \[ a \left (\begin {cases} \frac {0^{p} x}{a} + \frac {0^{p} \log {\left (a x - 1 \right )}}{a^{2}} - \frac {a^{- p} c^{p} p x^{2} x^{- p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (2 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 2 - p \\ 3 - p \end {matrix}\middle | {a x} \right )}}{\Gamma \left (3 - p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{a x}\right | > 1 \\\frac {0^{p} x}{a} + \frac {0^{p} \log {\left (- a x + 1 \right )}}{a^{2}} - \frac {a^{- p} c^{p} p x^{2} x^{- p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (2 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 2 - p \\ 3 - p \end {matrix}\middle | {a x} \right )}}{\Gamma \left (3 - p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases}\right ) + \begin {cases} \frac {0^{p} \log {\left (a x - 1 \right )}}{a} - \frac {a^{- p} c^{p} p x x^{- p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 1 - p \\ 2 - p \end {matrix}\middle | {a x} \right )}}{\Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{a x}\right | > 1 \\\frac {0^{p} \log {\left (- a x + 1 \right )}}{a} - \frac {a^{- p} c^{p} p x x^{- p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 1 - p \\ 2 - p \end {matrix}\middle | {a x} \right )}}{\Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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