Integrand size = 24, antiderivative size = 116 \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {32 \left (1-\frac {1}{a x}\right )^{\frac {5-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-5+n)} \left (c-a^2 c x^2\right )^{3/2} \operatorname {Hypergeometric2F1}\left (5,\frac {5-n}{2},\frac {7-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^4 (5-n) \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \]
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Time = 0.16 (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} \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {32 \left (c-a^2 c x^2\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {5-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-5}{2}} \operatorname {Hypergeometric2F1}\left (5,\frac {5-n}{2},\frac {7-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^4 (5-n) x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}} \]
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Rule 133
Rule 6327
Rule 6330
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c-a^2 c x^2\right )^{3/2} \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \, dx}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \\ & = -\frac {\left (c-a^2 c x^2\right )^{3/2} \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{\frac {3}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{\frac {3}{2}+\frac {n}{2}}}{x^5} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \\ & = \frac {32 \left (1-\frac {1}{a x}\right )^{\frac {5-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-5+n)} \left (c-a^2 c x^2\right )^{3/2} \operatorname {Hypergeometric2F1}\left (5,\frac {5-n}{2},\frac {7-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^4 (5-n) \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(280\) vs. \(2(116)=232\).
Time = 2.93 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.41 \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {c^2 \left (96 a^3 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \left (a e^{n \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x (n+a x)+2 e^{(1+n) \coth ^{-1}(a x)} (-1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )-c \left (-1+a^2 x^2\right ) \left (2 e^{n \coth ^{-1}(a x)} \left (-1+a^2 x^2\right )^2 \left (-a \left (-21+n^2\right ) x+2 n \left (1-n^2+\left (3+n^2\right ) \cosh \left (2 \coth ^{-1}(a x)\right )\right )+a \left (3+n^2\right ) \sqrt {1-\frac {1}{a^2 x^2}} x \cosh \left (3 \coth ^{-1}(a x)\right )\right )+16 a e^{(1+n) \coth ^{-1}(a x)} \left (-3+3 n-n^2+n^3\right ) \sqrt {1-\frac {1}{a^2 x^2}} x \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{192 a \left (c-a^2 c x^2\right )^{3/2}} \]
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\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}d x\]
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\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
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\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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Exception generated. \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \]
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