Integrand size = 24, antiderivative size = 116 \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {8 \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} \sqrt {c-a^2 c x^2} \operatorname {Hypergeometric2F1}\left (3,\frac {3-n}{2},\frac {5-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 (3-n) \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.12 (sec) , antiderivative size = 116, 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 = {6327, 6330, 133} \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {8 \sqrt {c-a^2 c x^2} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \operatorname {Hypergeometric2F1}\left (3,\frac {3-n}{2},\frac {5-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 (3-n) x \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 133
Rule 6327
Rule 6330
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a^2 c x^2} \int e^{n \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = -\frac {\sqrt {c-a^2 c x^2} \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{\frac {1}{2}+\frac {n}{2}}}{x^3} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {8 \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} \sqrt {c-a^2 c x^2} \operatorname {Hypergeometric2F1}\left (3,\frac {3-n}{2},\frac {5-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 (3-n) \sqrt {1-\frac {1}{a^2 x^2}} x} \\ \end{align*}
Time = 0.85 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.87 \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=-\frac {c e^{n \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x \left (a \sqrt {1-\frac {1}{a^2 x^2}} x (n+a x)+2 e^{\coth ^{-1}(a x)} (-1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )}{2 \sqrt {c-a^2 c x^2}} \]
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\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \sqrt {-a^{2} c \,x^{2}+c}d x\]
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\[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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\[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
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\[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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Exception generated. \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,\sqrt {c-a^2\,c\,x^2} \,d x \]
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