Optimal. Leaf size=99 \[ -\frac{2^{n/2} n (1-a x)^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (1-\frac{n}{2},-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a^2 (2-n)}-\frac{(a x+1)^{\frac{n+2}{2}} (1-a x)^{1-\frac{n}{2}}}{2 a^2} \]
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Rubi [A] time = 0.0414173, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6126, 80, 69} \[ -\frac{2^{n/2} n (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a^2 (2-n)}-\frac{(a x+1)^{\frac{n+2}{2}} (1-a x)^{1-\frac{n}{2}}}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 80
Rule 69
Rubi steps
\begin{align*} \int e^{n \tanh ^{-1}(a x)} x \, dx &=\int x (1-a x)^{-n/2} (1+a x)^{n/2} \, dx\\ &=-\frac{(1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{2 a^2}+\frac{n \int (1-a x)^{-n/2} (1+a x)^{n/2} \, dx}{2 a}\\ &=-\frac{(1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{2 a^2}-\frac{2^{n/2} n (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a^2 (2-n)}\\ \end{align*}
Mathematica [A] time = 0.0242738, size = 86, normalized size = 0.87 \[ -\frac{(1-a x)^{1-\frac{n}{2}} \left ((n-2) (a x+1)^{\frac{n}{2}+1}-2^{\frac{n}{2}+1} n \text{Hypergeometric2F1}\left (1-\frac{n}{2},-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (1-a x)\right )\right )}{2 a^2 (n-2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}x\, 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 x \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 (x \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*} \int x e^{n \operatorname{atanh}{\left (a x \right )}}\, dx \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 x \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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