Integrand size = 27, antiderivative size = 277 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {2^{\frac {1+n}{2}} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} x^3 \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{(1-n) \left (c-a^2 c x^2\right )^{3/2}} \]
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Time = 0.26 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6327, 6330, 91, 80, 71} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {a^3 2^{\frac {n+1}{2}} x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{(1-n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^3 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {a^3 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{(n+1) \left (c-a^2 c x^2\right )^{3/2}} \]
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Rule 71
Rule 80
Rule 91
Rule 6327
Rule 6330
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4} \, dx}{\left (c-a^2 c x^2\right )^{3/2}} \\ & = -\frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \text {Subst}\left (\int x^2 \left (1-\frac {x}{a}\right )^{-\frac {3}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}} \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}} \\ & = -\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \text {Subst}\left (\int \left (1-\frac {x}{a}\right )^{-\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}} \left (\frac {n}{a}+\frac {(1+n) x}{a^2}\right ) \, dx,x,\frac {1}{x}\right )}{(1+n) \left (c-a^2 c x^2\right )^{3/2}} \\ & = -\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \text {Subst}\left (\int \left (1-\frac {x}{a}\right )^{-\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{\frac {1}{2} (-1+n)} \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}} \\ & = -\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {2^{\frac {1+n}{2}} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} x^3 \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{(1-n) \left (c-a^2 c x^2\right )^{3/2}} \\ \end{align*}
Time = 1.32 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.46 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a n x)-2 e^{\coth ^{-1}(a x)} (-1+n) \left (-1+a^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-e^{2 \coth ^{-1}(a x)}\right )\right )}{a c (-1+n) (1+n) \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a^2 c x^2}} \]
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\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{x \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x} \,d x } \]
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Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{x\,{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]
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