Optimal. Leaf size=79 \[ -\frac{8 c \left (1-\frac{1}{a x}\right )^{2-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n-4}{2}} \text{Hypergeometric2F1}\left (3,2-\frac{n}{2},3-\frac{n}{2},\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (4-n)} \]
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Rubi [A] time = 0.075156, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {6175, 6180, 131} \[ -\frac{8 c \left (1-\frac{1}{a x}\right )^{2-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n-4}{2}} \, _2F_1\left (3,2-\frac{n}{2};3-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (4-n)} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6180
Rule 131
Rubi steps
\begin{align*} \int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx &=-\left ((a c) \int e^{n \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right ) x \, dx\right )\\ &=(a c) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{1-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2}}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{8 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 (3,2-\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 = 0.494428, size = 104, normalized size = 1.32 \[ -\frac{c e^{n \coth ^{-1}(a x)} \left ((n+2) \left ((n-2) \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,e^{2 \coth ^{-1}(a x)}\right )+a^2 x^2+a (n-2) x-1\right )+(n-2) n e^{2 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,e^{2 \coth ^{-1}(a x)}\right )\right )}{2 a (n+2)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.064, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( -acx+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 c x - 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 c x - 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 x 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 c x - 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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