Optimal. Leaf size=114 \[ \frac{a^2 2^{n/2} n \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (1-\frac{n}{2},-\frac{n}{2},2-\frac{n}{2},\frac{a-\frac{1}{x}}{2 a}\right )}{2-n}+\frac{1}{2} a^2 \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \]
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Rubi [A] time = 0.0560491, antiderivative size = 114, 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 = {6171, 80, 69} \[ \frac{a^2 2^{n/2} n \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{2 a}\right )}{2-n}+\frac{1}{2} a^2 \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 80
Rule 69
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{x^3} \, dx &=-\operatorname{Subst}\left (\int x \left (1-\frac{x}{a}\right )^{-n/2} \left (1+\frac{x}{a}\right )^{n/2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} a^2 \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}-\frac{1}{2} (a n) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{-n/2} \left (1+\frac{x}{a}\right )^{n/2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} a^2 \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}+\frac{2^{n/2} a^2 n \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{2 a}\right )}{2-n}\\ \end{align*}
Mathematica [A] time = 0.33554, size = 107, normalized size = 0.94 \[ -\frac{a^2 e^{n \coth ^{-1}(a x)} \left ((n+2) \left (n \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,-e^{2 \coth ^{-1}(a x)}\right )+\frac{1}{a^2 x^2}+\frac{n}{a x}-1\right )-n^2 e^{2 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,-e^{2 \coth ^{-1}(a x)}\right )\right )}{2 (n+2)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.075, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\rm arccoth} \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{acoth}{\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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