Optimal. Leaf size=47 \[ \frac {i \text {Li}_2\left (\frac {i x^{-n}}{a}\right )}{2 n}-\frac {i \text {Li}_2\left (-\frac {i x^{-n}}{a}\right )}{2 n} \]
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Rubi [A] time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5032, 4849, 2391} \[ \frac {i \text {PolyLog}\left (2,\frac {i x^{-n}}{a}\right )}{2 n}-\frac {i \text {PolyLog}\left (2,-\frac {i x^{-n}}{a}\right )}{2 n} \]
Antiderivative was successfully verified.
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Rule 2391
Rule 4849
Rule 5032
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}\left (a x^n\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cot ^{-1}(a x)}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i}{a x}\right )}{x} \, dx,x,x^n\right )}{2 n}-\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {i}{a x}\right )}{x} \, dx,x,x^n\right )}{2 n}\\ &=-\frac {i \text {Li}_2\left (-\frac {i x^{-n}}{a}\right )}{2 n}+\frac {i \text {Li}_2\left (\frac {i x^{-n}}{a}\right )}{2 n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 0.85 \[ -\frac {i \left (\text {Li}_2\left (-\frac {i x^{-n}}{a}\right )-\text {Li}_2\left (\frac {i x^{-n}}{a}\right )\right )}{2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.58, size = 63, normalized size = 1.34 \[ \frac {2 \, n \operatorname {arccot}\left (a x^{n}\right ) \log \relax (x) - i \, n \log \left (i \, a x^{n} + 1\right ) \log \relax (x) + i \, n \log \left (-i \, a x^{n} + 1\right ) \log \relax (x) + i \, {\rm Li}_2\left (i \, a x^{n}\right ) - i \, {\rm Li}_2\left (-i \, a x^{n}\right )}{2 \, n} \]
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 \[ \int \frac {\operatorname {arccot}\left (a x^{n}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 94, normalized size = 2.00 \[ \frac {\ln \left (a \,x^{n}\right ) \mathrm {arccot}\left (a \,x^{n}\right )}{n}-\frac {i \ln \left (a \,x^{n}\right ) \ln \left (1+i a \,x^{n}\right )}{2 n}+\frac {i \ln \left (a \,x^{n}\right ) \ln \left (1-i a \,x^{n}\right )}{2 n}-\frac {i \dilog \left (1+i a \,x^{n}\right )}{2 n}+\frac {i \dilog \left (1-i a \,x^{n}\right )}{2 n} \]
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 \[ a n \int \frac {x^{n} \log \relax (x)}{a^{2} x x^{2 \, n} + x}\,{d x} + \arctan \left (\frac {1}{a x^{n}}\right ) \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {acot}\left (a\,x^n\right )}{x} \,d x \]
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 \[ \int \frac {\operatorname {acot}{\left (a x^{n} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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