Integrand size = 8, antiderivative size = 78 \[ \int e^{n \coth ^{-1}(a x)} \, dx=\frac {4 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (2-n)} \]
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Time = 0.02 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6305, 133} \[ \int e^{n \coth ^{-1}(a x)} \, dx=\frac {4 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (2-n)} \]
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Rule 133
Rule 6305
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {4 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (2-n)} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.05 \[ \int e^{n \coth ^{-1}(a x)} \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} n \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (a x+\operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{a (2+n)} \]
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\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}d x\]
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\[ \int e^{n \coth ^{-1}(a x)} \, dx=\int { \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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\[ \int e^{n \coth ^{-1}(a x)} \, dx=\int e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
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\[ \int e^{n \coth ^{-1}(a x)} \, dx=\int { \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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\[ \int e^{n \coth ^{-1}(a x)} \, dx=\int { \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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Timed out. \[ \int e^{n \coth ^{-1}(a x)} \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )} \,d x \]
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