Integrand size = 22, antiderivative size = 81 \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {256 c^3 \left (1-\frac {1}{a x}\right )^{4-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-8+n)} \operatorname {Hypergeometric2F1}\left (8,4-\frac {n}{2},5-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (8-n)} \]
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Time = 0.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6326, 6330, 133} \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {256 c^3 \left (1-\frac {1}{a x}\right )^{4-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-8}{2}} \operatorname {Hypergeometric2F1}\left (8,4-\frac {n}{2},5-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (8-n)} \]
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Rule 133
Rule 6326
Rule 6330
Rubi steps \begin{align*} \text {integral}& = -\left (\left (a^6 c^3\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^3 x^6 \, dx\right ) \\ & = \left (a^6 c^3\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{3+\frac {n}{2}}}{x^8} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {256 c^3 \left (1-\frac {1}{a x}\right )^{4-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-8+n)} \operatorname {Hypergeometric2F1}\left (8,4-\frac {n}{2},5-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (8-n)} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(267\) vs. \(2(81)=162\).
Time = 2.39 (sec) , antiderivative size = 267, normalized size of antiderivative = 3.30 \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {c^3 e^{n \coth ^{-1}(a x)} \left (-912 n+58 n^3-n^5-5040 a x+912 a n^2 x-58 a n^4 x+a n^6 x+1368 a^2 n x^2-64 a^2 n^3 x^2+a^2 n^5 x^2+5040 a^3 x^3-152 a^3 n^2 x^3+2 a^3 n^4 x^3-576 a^4 n x^4+6 a^4 n^3 x^4-3024 a^5 x^5+24 a^5 n^2 x^5+120 a^6 n x^6+720 a^7 x^7+e^{2 \coth ^{-1}(a x)} n \left (-1152+576 n+104 n^2-52 n^3-2 n^4+n^5\right ) \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+\left (-2304+784 n^2-56 n^4+n^6\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )}{5040 a} \]
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\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{3}d x\]
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\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\int { -{\left (a^{2} c x^{2} - c\right )}^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=- c^{3} \left (\int 3 a^{2} x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx + \int \left (- 3 a^{4} x^{4} e^{n \operatorname {acoth}{\left (a x \right )}}\right )\, dx + \int a^{6} x^{6} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx + \int \left (- e^{n \operatorname {acoth}{\left (a x \right )}}\right )\, dx\right ) \]
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\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\int { -{\left (a^{2} c x^{2} - c\right )}^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\int { -{\left (a^{2} c x^{2} - c\right )}^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
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Timed out. \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^3 \,d x \]
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