Integrand size = 24, antiderivative size = 80 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-1+\frac {n}{2}} \, dx=\frac {2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{n/2} x (c-a c x)^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{n} \]
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Time = 0.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6311, 6316, 133} \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-1+\frac {n}{2}} \, dx=\frac {2 x \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{n/2} (c-a c x)^{\frac {n-2}{2}} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{n} \]
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Rule 133
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \left (\left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} x^{1-\frac {n}{2}} (c-a c x)^{-1+\frac {n}{2}}\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{-1+\frac {n}{2}} x^{-1+\frac {n}{2}} \, dx \\ & = -\left (\left (\left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{x}\right )^{-1+\frac {n}{2}} (c-a c x)^{-1+\frac {n}{2}}\right ) \text {Subst}\left (\int \frac {x^{-1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{1-\frac {x}{a}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{n/2} x (c-a c x)^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{n} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-1+\frac {n}{2}} \, dx=-\frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} (c-a c x)^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {2}{1+a x}\right )}{a c n} \]
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\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a c x +c \right )^{-1+\frac {n}{2}}d x\]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-1+\frac {n}{2}} \, dx=\int { {\left (-a c x + c\right )}^{\frac {1}{2} \, n - 1} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-1+\frac {n}{2}} \, dx=\int \left (- c \left (a x - 1\right )\right )^{\frac {n}{2} - 1} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-1+\frac {n}{2}} \, dx=\int { {\left (-a c x + c\right )}^{\frac {1}{2} \, n - 1} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-1+\frac {n}{2}} \, dx=\int { {\left (-a c x + c\right )}^{\frac {1}{2} \, n - 1} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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Timed out. \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-1+\frac {n}{2}} \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}-1} \,d x \]
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