Optimal. Leaf size=68 \[ -\frac {i \text {Li}_2\left (-e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{2 n}-\frac {i \cos ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\cos ^{-1}\left (a x^n\right ) \log \left (1+e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{n} \]
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Rubi [A] time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4831, 3719, 2190, 2279, 2391} \[ -\frac {i \text {PolyLog}\left (2,-e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{2 n}-\frac {i \cos ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\cos ^{-1}\left (a x^n\right ) \log \left (1+e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{n} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4831
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}\left (a x^n\right )}{x} \, dx &=-\frac {\operatorname {Subst}\left (\int x \tan (x) \, dx,x,\cos ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac {i \cos ^{-1}\left (a x^n\right )^2}{2 n}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\cos ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac {i \cos ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\cos ^{-1}\left (a x^n\right ) \log \left (1+e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{n}-\frac {\operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac {i \cos ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\cos ^{-1}\left (a x^n\right ) \log \left (1+e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{n}+\frac {i \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{2 n}\\ &=-\frac {i \cos ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\cos ^{-1}\left (a x^n\right ) \log \left (1+e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{n}-\frac {i \text {Li}_2\left (-e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{2 n}\\ \end {align*}
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Mathematica [B] time = 0.15, size = 141, normalized size = 2.07 \[ \frac {a \left (\text {Li}_2\left (e^{-2 \sinh ^{-1}\left (\sqrt {-a^2} x^n\right )}\right )-\sinh ^{-1}\left (\sqrt {-a^2} x^n\right )^2-2 \sinh ^{-1}\left (\sqrt {-a^2} x^n\right ) \log \left (1-e^{-2 \sinh ^{-1}\left (\sqrt {-a^2} x^n\right )}\right )+2 n \log (x) \log \left (\sqrt {1-a^2 x^{2 n}}+\sqrt {-a^2} x^n\right )\right )}{2 \sqrt {-a^2} n}+\log (x) \cos ^{-1}\left (a x^n\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {\arccos \left (a x^{n}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 89, normalized size = 1.31 \[ -\frac {i \arccos \left (a \,x^{n}\right )^{2}}{2 n}+\frac {\arccos \left (a \,x^{n}\right ) \ln \left (1+\left (a \,x^{n}+i \sqrt {1-a^{2} x^{2 n}}\right )^{2}\right )}{n}-\frac {i \polylog \left (2, -\left (a \,x^{n}+i \sqrt {1-a^{2} x^{2 n}}\right )^{2}\right )}{2 n} \]
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 \[ -a n \int \frac {\sqrt {a x^{n} + 1} \sqrt {-a x^{n} + 1} x^{n} \log \relax (x)}{a^{2} x x^{2 \, n} - x}\,{d x} + \arctan \left (\frac {\sqrt {a x^{n} + 1} \sqrt {-a x^{n} + 1}}{a x^{n}}\right ) \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acos}\left (a\,x^n\right )}{x} \,d x \]
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 \[ \int \frac {\operatorname {acos}{\left (a x^{n} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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