Optimal. Leaf size=183 \[ \frac {a^4 2^{\frac {n}{2}-2} n \left (n^2+8\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {a-\frac {1}{x}}{2 a}\right )}{3 (2-n)}+\frac {1}{24} a^3 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (a \left (n^2+6\right )+\frac {2 n}{x}\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}}+\frac {a^2 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}}}{4 x^2} \]
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Rubi [A] time = 0.12, 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 = {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 69
Rule 100
Rule 147
Rule 6171
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*}
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Mathematica [A] time = 0.55, size = 148, normalized size = 0.81 \[ -\frac {1}{24} a^4 e^{n \coth ^{-1}(a x)} \left (\frac {6}{a^4 x^4}+\frac {2 n}{a^3 x^3}+\frac {n^2}{a^2 x^2}-\frac {\left (n^2+8\right ) n^2 e^{2 \coth ^{-1}(a x)} \, _2F_1\left (1,\frac {n}{2}+1;\frac {n}{2}+2;-e^{2 \coth ^{-1}(a x)}\right )}{n+2}+\left (n^2+8\right ) n \, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;-e^{2 \coth ^{-1}(a x)}\right )+\frac {n^3}{a x}+\frac {6 n}{a x}-n^2-6\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{x^{5}}, x\right ) \]
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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, 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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{x^{5}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{x^5} \,d x \]
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 \[ \int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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