Optimal. Leaf size=183 \[ \frac{a^4 2^{\frac{n}{2}-2} n \left (n^2+8\right ) \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (1-\frac{n}{2},-\frac{n}{2},2-\frac{n}{2},\frac{a-\frac{1}{x}}{2 a}\right )}{3 (2-n)}+\frac{1}{24} a^3 \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (a \left (n^2+6\right )+\frac{2 n}{x}\right ) \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}}+\frac{a^2 \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}}}{4 x^2} \]
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Rubi [A] time = 0.123382, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6171, 100, 147, 69} \[ \frac{a^4 2^{\frac{n}{2}-2} n \left (n^2+8\right ) \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{2 a}\right )}{3 (2-n)}+\frac{1}{24} a^3 \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (a \left (n^2+6\right )+\frac{2 n}{x}\right ) \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}}+\frac{a^2 \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}}}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 100
Rule 147
Rule 69
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{x^5} \, dx &=-\operatorname{Subst}\left (\int x^3 \left (1-\frac{x}{a}\right )^{-n/2} \left (1+\frac{x}{a}\right )^{n/2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{a^2 \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{4 x^2}+\frac{1}{4} a^2 \operatorname{Subst}\left (\int x \left (1-\frac{x}{a}\right )^{-n/2} \left (1+\frac{x}{a}\right )^{n/2} \left (-2-\frac{n x}{a}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{24} a^3 \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} \left (a \left (6+n^2\right )+\frac{2 n}{x}\right )+\frac{a^2 \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{4 x^2}-\frac{1}{24} \left (a^3 n \left (8+n^2\right )\right ) \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}{24} a^3 \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} \left (a \left (6+n^2\right )+\frac{2 n}{x}\right )+\frac{a^2 \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{4 x^2}+\frac{2^{-2+\frac{n}{2}} a^4 n \left (8+n^2\right ) \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 )}{3 (2-n)}\\ \end{align*}
Mathematica [A] time = 0.484103, size = 148, normalized size = 0.81 \[ -\frac{1}{24} a^4 e^{n \coth ^{-1}(a x)} \left (-\frac{\left (n^2+8\right ) n^2 e^{2 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,-e^{2 \coth ^{-1}(a x)}\right )}{n+2}+\left (n^2+8\right ) n \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,-e^{2 \coth ^{-1}(a x)}\right )+\frac{n^2}{a^2 x^2}+\frac{2 n}{a^3 x^3}+\frac{6}{a^4 x^4}+\frac{n^3}{a x}+\frac{6 n}{a x}-n^2-6\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.078, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{{x}^{5}}}\, 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^{5}}\,{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^{5}}, 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^{5}}\, 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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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