Optimal. Leaf size=93 \[ -\frac {4^p \left (1-\frac {1}{a x}\right )^{-p} \left (\frac {1}{a x}+1\right )^{1-p} F_1\left (1-p;-2 p,2;2-p;\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right ) \left (c-\frac {c}{a x}\right )^p}{a (1-p)} \]
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Rubi [A] time = 0.10, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6182, 6179, 136} \[ -\frac {4^p \left (1-\frac {1}{a x}\right )^{-p} \left (\frac {1}{a x}+1\right )^{1-p} F_1\left (1-p;-2 p,2;2-p;\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right ) \left (c-\frac {c}{a x}\right )^p}{a (1-p)} \]
Antiderivative was successfully verified.
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Rule 136
Rule 6179
Rule 6182
Rubi steps
\begin {align*} \int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx &=\left (\left (1-\frac {1}{a x}\right )^{-p} \left (c-\frac {c}{a x}\right )^p\right ) \int e^{-2 p \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^p \, dx\\ &=-\left (\left (\left (1-\frac {1}{a x}\right )^{-p} \left (c-\frac {c}{a x}\right )^p\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{2 p} \left (1+\frac {x}{a}\right )^{-p}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\frac {4^p \left (1-\frac {1}{a x}\right )^{-p} \left (1+\frac {1}{a x}\right )^{1-p} \left (c-\frac {c}{a x}\right )^p F_1\left (1-p;-2 p,2;2-p;\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (1-p)}\\ \end {align*}
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Mathematica [F] time = 0.66, size = 0, normalized size = 0.00 \[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a c x - c}{a x}\right )^{p}}{\left (\frac {a x - 1}{a x + 1}\right )^{p}}, x\right ) \]
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 \[ \int \frac {{\left (c - \frac {c}{a x}\right )}^{p}}{\left (\frac {a x - 1}{a x + 1}\right )^{p}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (c -\frac {c}{a x}\right )^{p} {\mathrm e}^{-2 p \,\mathrm {arccoth}\left (a x \right )}\, dx \]
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 \[ \int \frac {{\left (c - \frac {c}{a x}\right )}^{p}}{\left (\frac {a x - 1}{a x + 1}\right )^{p}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{-2\,p\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-\frac {c}{a\,x}\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- c \left (-1 + \frac {1}{a x}\right )\right )^{p} e^{- 2 p \operatorname {acoth}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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