Optimal. Leaf size=94 \[ \frac{2^{\frac{n}{2}+1} (1-a x)^{-n/2} \text{Hypergeometric2F1}\left (-\frac{n}{2},-\frac{n}{2},1-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a^2 c n}-\frac{(1-a x)^{-n/2} (a x+1)^{n/2}}{a^2 c n} \]
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Rubi [A] time = 0.0828106, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {6150, 79, 69} \[ \frac{2^{\frac{n}{2}+1} (1-a x)^{-n/2} \, _2F_1\left (-\frac{n}{2},-\frac{n}{2};1-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a^2 c n}-\frac{(1-a x)^{-n/2} (a x+1)^{n/2}}{a^2 c n} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 79
Rule 69
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)} x}{c-a^2 c x^2} \, dx &=\frac{\int x (1-a x)^{-1-\frac{n}{2}} (1+a x)^{-1+\frac{n}{2}} \, dx}{c}\\ &=-\frac{(1-a x)^{-n/2} (1+a x)^{n/2}}{a^2 c n}+\frac{\int (1-a x)^{-1-\frac{n}{2}} (1+a x)^{n/2} \, dx}{a c}\\ &=-\frac{(1-a x)^{-n/2} (1+a x)^{n/2}}{a^2 c n}+\frac{2^{1+\frac{n}{2}} (1-a x)^{-n/2} \, _2F_1\left (-\frac{n}{2},-\frac{n}{2};1-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a^2 c n}\\ \end{align*}
Mathematica [A] time = 0.055578, size = 74, normalized size = 0.79 \[ \frac{(1-a x)^{-n/2} \left (2^{\frac{n}{2}+1} \text{Hypergeometric2F1}\left (-\frac{n}{2},-\frac{n}{2},1-\frac{n}{2},\frac{1}{2} (1-a x)\right )-(a x+1)^{n/2}\right )}{a^2 c n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.184, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}x}{-{a}^{2}c{x}^{2}+c}}\, 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{x \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}\,{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{x \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}, 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{\int \frac{x e^{n \operatorname{atanh}{\left (a x \right )}}}{a^{2} x^{2} - 1}\, dx}{c} \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{x \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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