Optimal. Leaf size=183 \[ \frac {1}{24} a^3 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} \left (a \left (6+n^2\right )+\frac {2 n}{x}\right )+\frac {a^2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}}{4 x^2}+\frac {2^{-2+\frac {n}{2}} a^4 n \left (8+n^2\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {a-\frac {1}{x}}{2 a}\right )}{3 (2-n)} \]
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Rubi [A]
time = 0.09, antiderivative size = 183, normalized size of antiderivative = 1.00, 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 = {6306, 102, 152,
71} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 102
Rule 152
Rule 6306
Rubi steps
\begin {align*} \int \frac {e^{n \coth ^{-1}(a x)}}{x^5} \, dx &=-\text {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 \text {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 ) \text {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*}
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Mathematica [A]
time = 0.35, size = 148, normalized size = 0.81 \begin {gather*} -\frac {1}{24} a^4 e^{n \coth ^{-1}(a x)} \left (-6-n^2+\frac {6}{a^4 x^4}+\frac {2 n}{a^3 x^3}+\frac {n^2}{a^2 x^2}+\frac {6 n}{a x}+\frac {n^3}{a x}-\frac {e^{2 \coth ^{-1}(a x)} n^2 \left (8+n^2\right ) \, _2F_1\left (1,1+\frac {n}{2};2+\frac {n}{2};-e^{2 \coth ^{-1}(a x)}\right )}{2+n}+n \left (8+n^2\right ) \, _2F_1\left (1,\frac {n}{2};1+\frac {n}{2};-e^{2 \coth ^{-1}(a x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{x^{5}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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