Integrand size = 18, antiderivative size = 104 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (n-2 p)} \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (n-2 p),-1-p,-p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+p} \]
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Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6311, 6316, 134} \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {x \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} (c-a c x)^p \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (n-2 p)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (n-2 p),-p-1,-p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{p+1} \]
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Rule 134
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \left (\left (1-\frac {1}{a x}\right )^{-p} x^{-p} (c-a c x)^p\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^p x^p \, dx \\ & = -\left (\left (\left (1-\frac {1}{a x}\right )^{-p} \left (\frac {1}{x}\right )^p (c-a c x)^p\right ) \text {Subst}\left (\int x^{-2-p} \left (1-\frac {x}{a}\right )^{-\frac {n}{2}+p} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (n-2 p)} \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (n-2 p),-1-p,-p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+p} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \left (\frac {-1+a x}{1+a x}\right )^{\frac {1}{2} (n-2 p)} (1+a x) (c-a c x)^p \operatorname {Hypergeometric2F1}\left (-1-p,\frac {n}{2}-p,-p,\frac {2}{1+a x}\right )}{a (1+p)} \]
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\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a c x +c \right )^{p}d x\]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^p \, dx=\int { {\left (-a c x + c\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^p \, dx=\int \left (- c \left (a x - 1\right )\right )^{p} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^p \, dx=\int { {\left (-a c x + c\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^p \, dx=\int { {\left (-a c x + c\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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Timed out. \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^p \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^p \,d x \]
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