Optimal. Leaf size=81 \[ -\frac{256 c^3 \left (1-\frac{1}{a x}\right )^{4-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n-8}{2}} \text{Hypergeometric2F1}\left (8,4-\frac{n}{2},5-\frac{n}{2},\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (8-n)} \]
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Rubi [A] time = 0.135166, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6191, 6195, 131} \[ -\frac{256 c^3 \left (1-\frac{1}{a x}\right )^{4-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n-8}{2}} \, _2F_1\left (8,4-\frac{n}{2};5-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (8-n)} \]
Antiderivative was successfully verified.
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Rule 6191
Rule 6195
Rule 131
Rubi steps
\begin{align*} \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=-\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 ) \operatorname{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)} \, _2F_1\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*}
Mathematica [B] time = 2.28098, size = 267, normalized size = 3.3 \[ -\frac{c^3 e^{n \coth ^{-1}(a x)} \left (n \left (n^5-2 n^4-52 n^3+104 n^2+576 n-1152\right ) e^{2 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,e^{2 \coth ^{-1}(a x)}\right )+\left (n^6-56 n^4+784 n^2-2304\right ) \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,e^{2 \coth ^{-1}(a x)}\right )+24 a^5 n^2 x^5+6 a^4 n^3 x^4+2 a^3 n^4 x^3-152 a^3 n^2 x^3+a^2 n^5 x^2-64 a^2 n^3 x^2+120 a^6 n x^6-576 a^4 n x^4+1368 a^2 n x^2+720 a^7 x^7-3024 a^5 x^5+5040 a^3 x^3+a n^6 x-58 a n^4 x+912 a n^2 x-5040 a x-n^5+58 n^3-912 n\right )}{5040 a} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.22, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - c^{3} \left (\int 3 a^{2} x^{2} e^{n \operatorname{acoth}{\left (a x \right )}}\, dx + \int - 3 a^{4} x^{4} e^{n \operatorname{acoth}{\left (a x \right )}}\, dx + \int a^{6} x^{6} e^{n \operatorname{acoth}{\left (a x \right )}}\, dx + \int - e^{n \operatorname{acoth}{\left (a x \right )}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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