Integrand size = 27, antiderivative size = 164 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {e^{n \coth ^{-1}(a x)} (n-a x)}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 c (1-n) \sqrt {c-a^2 c x^2}} \]
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Time = 0.23 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6324, 6327, 6330, 133} \[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {2 x \sqrt {1-\frac {1}{a^2 x^2}} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 c (1-n) \sqrt {c-a^2 c x^2}}-\frac {(n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \]
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Rule 133
Rule 6324
Rule 6327
Rule 6330
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{n \coth ^{-1}(a x)} (n-a x)}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {\int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx}{a^2 c} \\ & = -\frac {e^{n \coth ^{-1}(a x)} (n-a x)}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {\left (\sqrt {1-\frac {1}{a^2 x^2}} x\right ) \int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a^2 x^2}} x} \, dx}{a^2 c \sqrt {c-a^2 c x^2}} \\ & = -\frac {e^{n \coth ^{-1}(a x)} (n-a x)}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {\left (\sqrt {1-\frac {1}{a^2 x^2}} x\right ) \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} \, dx,x,\frac {1}{x}\right )}{a^2 c \sqrt {c-a^2 c x^2}} \\ & = -\frac {e^{n \coth ^{-1}(a x)} (n-a x)}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-\frac {1}{a^2 x^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 \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 c (1-n) \sqrt {c-a^2 c x^2}} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.77 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {e^{n \coth ^{-1}(a x)} \left (a \sqrt {1-\frac {1}{a^2 x^2}} x (-n+a x)+2 e^{\coth ^{-1}(a x)} (-1+n) \left (-1+a^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )}{a^4 c (-1+n) (1+n) \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a^2 c x^2}} \]
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\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} x^{2}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^2\,{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]
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