Integrand size = 27, antiderivative size = 359 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {(2+n) \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (2+2 n+n^2\right ) \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^4}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {2 n \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1+n),\frac {1+n}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{a (1-n) \left (c-a^2 c x^2\right )^{3/2}} \]
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Time = 0.27 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6327, 6329, 105, 160, 12, 133} \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {2 n x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {n-1}{2},\frac {n+1}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{a (1-n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (n^2+2 n+2\right ) x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{a (1-n) (n+1) \left (c-a^2 c x^2\right )^{3/2}}+\frac {x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {(n+2) x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{a (n+1) \left (c-a^2 c x^2\right )^{3/2}} \]
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Rule 12
Rule 105
Rule 133
Rule 160
Rule 6327
Rule 6329
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}} \, 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 \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^2} \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}} \\ & = \frac {\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^4}{\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 \frac {\left (-\frac {n}{a}-\frac {2 x}{a^2}\right ) \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} \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}} \\ & = -\frac {(2+n) \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}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\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^4}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {\left (a \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\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 {3}{2}+\frac {n}{2}} \left (\frac {n (1+n)}{a^2}+\frac {(2+n) x}{a^3}\right )}{x} \, dx,x,\frac {1}{x}\right )}{(1+n) \left (c-a^2 c x^2\right )^{3/2}} \\ & = -\frac {(2+n) \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}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (2+2 n+n^2\right ) \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}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\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^4}{\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 \frac {n \left (1-n^2\right ) \left (1-\frac {x}{a}\right )^{\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}}}{a^3 x} \, dx,x,\frac {1}{x}\right )}{(1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}} \\ & = -\frac {(2+n) \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}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (2+2 n+n^2\right ) \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}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\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^4}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {\left (n \left (1-n^2\right ) \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\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 {3}{2}+\frac {n}{2}}}{x} \, dx,x,\frac {1}{x}\right )}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}} \\ & = -\frac {(2+n) \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}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (2+2 n+n^2\right ) \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}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\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^4}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {2 n \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 \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1+n),\frac {1+n}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{a (1-n) \left (c-a^2 c x^2\right )^{3/2}} \\ \end{align*}
Time = 0.92 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.37 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\frac {c e^{n \coth ^{-1}(a x)} (-1+a n x)}{-1+n^2}-\frac {c \left (-1+a^2 x^2\right ) \left (e^{n \coth ^{-1}(a x)} (1+n)+\frac {2 e^{(1+n) \coth ^{-1}(a x)} n \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},e^{2 \coth ^{-1}(a x)}\right )}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\right )}{1+n}}{a^4 c^2 \sqrt {c-a^2 c x^2}} \]
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\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} x^{3}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^{3} e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^3\,{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]
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