Optimal. Leaf size=166 \[ -\frac{2 (n+2) \left (\frac{1}{a x}+1\right )^{n/2} \left (1-\frac{1}{a x}\right )^{-n/2} \text{Hypergeometric2F1}\left (1,-\frac{n}{2},1-\frac{n}{2},\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a c^2 n}-\frac{(n+3) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1}}{a c^2 (n+2)}+\frac{x \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1}}{c^2} \]
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Rubi [A] time = 0.115471, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6179, 129, 155, 12, 131} \[ -\frac{2 (n+2) \left (\frac{1}{a x}+1\right )^{n/2} \left (1-\frac{1}{a x}\right )^{-n/2} \, _2F_1\left (1,-\frac{n}{2};1-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a c^2 n}-\frac{(n+3) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1}}{a c^2 (n+2)}+\frac{x \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1}}{c^2} \]
Antiderivative was successfully verified.
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Rule 6179
Rule 129
Rule 155
Rule 12
Rule 131
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-2-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2}}{x^2} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x}{c^2}+\frac{\operatorname{Subst}\left (\int \frac{\left (-\frac{2+n}{a}-\frac{x}{a^2}\right ) \left (1-\frac{x}{a}\right )^{-2-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2}}{x} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=-\frac{(3+n) \left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a c^2 (2+n)}+\frac{\left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x}{c^2}-\frac{a \operatorname{Subst}\left (\int \frac{(2+n)^2 \left (1-\frac{x}{a}\right )^{-1-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2}}{a^2 x} \, dx,x,\frac{1}{x}\right )}{c^2 (2+n)}\\ &=-\frac{(3+n) \left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a c^2 (2+n)}+\frac{\left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x}{c^2}-\frac{(2+n) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-1-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2}}{x} \, dx,x,\frac{1}{x}\right )}{a c^2}\\ &=-\frac{(3+n) \left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a c^2 (2+n)}+\frac{\left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x}{c^2}-\frac{2 (2+n) \left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{n/2} \, _2F_1\left (1,-\frac{n}{2};1-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a c^2 n}\\ \end{align*}
Mathematica [A] time = 0.0725854, size = 113, normalized size = 0.68 \[ \frac{\left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{n/2} \left (n (a x+1) (n (a x-1)+2 a x-3)-2 (n+2)^2 (a x-1) \text{Hypergeometric2F1}\left (1,-\frac{n}{2},1-\frac{n}{2},\frac{a x-1}{a x+1}\right )\right )}{a c^2 n (n+2) (a x-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( c-{\frac{c}{ax}} \right ) ^{-2}}\, 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 \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (c - \frac{c}{a x}\right )}^{2}}\,{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 (\frac{a^{2} x^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}}, 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*} \frac{a^{2} \int \frac{x^{2} e^{n \operatorname{acoth}{\left (a x \right )}}}{a^{2} x^{2} - 2 a x + 1}\, dx}{c^{2}} \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 \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (c - \frac{c}{a x}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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