Optimal. Leaf size=105 \[ -\frac{2 a^2 n (a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (2,1-\frac{n}{2},2-\frac{n}{2},\frac{1-a x}{a x+1}\right )}{2-n}-\frac{(a x+1)^{\frac{n+2}{2}} (1-a x)^{1-\frac{n}{2}}}{2 x^2} \]
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Rubi [A] time = 0.0489941, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6126, 96, 131} \[ -\frac{2 a^2 n (a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-a x}{a x+1}\right )}{2-n}-\frac{(a x+1)^{\frac{n+2}{2}} (1-a x)^{1-\frac{n}{2}}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 96
Rule 131
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{x^3} \, dx &=\int \frac{(1-a x)^{-n/2} (1+a x)^{n/2}}{x^3} \, dx\\ &=-\frac{(1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{2 x^2}+\frac{1}{2} (a n) \int \frac{(1-a x)^{-n/2} (1+a x)^{n/2}}{x^2} \, dx\\ &=-\frac{(1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{2 x^2}-\frac{2 a^2 n (1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-a x}{1+a x}\right )}{2-n}\\ \end{align*}
Mathematica [A] time = 0.029136, size = 91, normalized size = 0.87 \[ \frac{(1-a x)^{1-\frac{n}{2}} (a x+1)^{\frac{n}{2}-1} \left (4 a^2 n x^2 \text{Hypergeometric2F1}\left (2,1-\frac{n}{2},2-\frac{n}{2},\frac{1-a x}{a x+1}\right )-(n-2) (a x+1)^2\right )}{2 (n-2) x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{{x}^{3}}}\, 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}}{x^{3}}\,{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{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{x^{3}}, 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 \frac{e^{n \operatorname{atanh}{\left (a x \right )}}}{x^{3}}\, 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 \frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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