Optimal. Leaf size=79 \[ -\frac{16 c \left (1-\frac{1}{a x}\right )^{2-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n-4}{2}} \text{Hypergeometric2F1}\left (4,2-\frac{n}{2},3-\frac{n}{2},\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (4-n)} \]
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Rubi [A] time = 0.0925007, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6191, 6195, 131} \[ -\frac{16 c \left (1-\frac{1}{a x}\right )^{2-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n-4}{2}} \, _2F_1\left (4,2-\frac{n}{2};3-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (4-n)} \]
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 ) \, dx &=-\left (\left (a^2 c\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right ) x^2 \, dx\right )\\ &=\left (a^2 c\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{1-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{1+\frac{n}{2}}}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{16 c \left (1-\frac{1}{a x}\right )^{2-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-4+n)} \, _2F_1\left (4,2-\frac{n}{2};3-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (4-n)}\\ \end{align*}
Mathematica [A] time = 0.704379, size = 111, normalized size = 1.41 \[ -\frac{c e^{n \coth ^{-1}(a x)} \left (\left (n^2-4\right ) \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,e^{2 \coth ^{-1}(a x)}\right )+(n-2) n e^{2 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,e^{2 \coth ^{-1}(a x)}\right )+a^2 n x^2+2 a^3 x^3+a n^2 x-6 a x-n\right )}{6 a} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.06, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( -{a}^{2}c{x}^{2}+c \right ) \, 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 )} \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^{2} c x^{2} - c\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 \left (\int a^{2} x^{2} 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 )} \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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