Integrand size = 22, antiderivative size = 88 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=-\frac {2^{\frac {3}{2}+p} \left (1-\frac {1}{a x}\right )^{-p} \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^p \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2}-p,2,\frac {3}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a} \]
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Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6317, 6314, 141} \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=-\frac {2^{p+\frac {3}{2}} \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p-\frac {1}{2},2,\frac {3}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right ) \left (c-\frac {c}{a x}\right )^p}{a} \]
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Rule 141
Rule 6314
Rule 6317
Rubi steps \begin{align*} \text {integral}& = \left (\left (1-\frac {1}{a x}\right )^{-p} \left (c-\frac {c}{a x}\right )^p\right ) \int e^{-\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 ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{\frac {1}{2}+p}}{x^2 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\frac {2^{\frac {3}{2}+p} \left (1-\frac {1}{a x}\right )^{-p} \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^p \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2}-p,2,\frac {3}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a} \\ \end{align*}
\[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx \]
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\[\int \left (c -\frac {c}{a x}\right )^{p} \sqrt {\frac {a x -1}{a x +1}}d x\]
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\[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int { {\left (c - \frac {c}{a x}\right )}^{p} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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\[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int \sqrt {\frac {a x - 1}{a x + 1}} \left (- c \left (-1 + \frac {1}{a x}\right )\right )^{p}\, dx \]
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\[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int { {\left (c - \frac {c}{a x}\right )}^{p} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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\[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int { {\left (c - \frac {c}{a x}\right )}^{p} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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Timed out. \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int {\left (c-\frac {c}{a\,x}\right )}^p\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]
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