Optimal. Leaf size=52 \[ x \log \left (a \cos ^n(x)\right )+\frac {1}{2} i n \text {Li}_2\left (-e^{2 i x}\right )+\frac {1}{2} i n x^2-n x \log \left (1+e^{2 i x}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {2548, 12, 3719, 2190, 2279, 2391} \[ \frac {1}{2} i n \text {PolyLog}\left (2,-e^{2 i x}\right )+x \log \left (a \cos ^n(x)\right )+\frac {1}{2} i n x^2-n x \log \left (1+e^{2 i x}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 2190
Rule 2279
Rule 2391
Rule 2548
Rule 3719
Rubi steps
\begin {align*} \int \log \left (a \cos ^n(x)\right ) \, dx &=x \log \left (a \cos ^n(x)\right )+\int n x \tan (x) \, dx\\ &=x \log \left (a \cos ^n(x)\right )+n \int x \tan (x) \, dx\\ &=\frac {1}{2} i n x^2+x \log \left (a \cos ^n(x)\right )-(2 i n) \int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx\\ &=\frac {1}{2} i n x^2-n x \log \left (1+e^{2 i x}\right )+x \log \left (a \cos ^n(x)\right )+n \int \log \left (1+e^{2 i x}\right ) \, dx\\ &=\frac {1}{2} i n x^2-n x \log \left (1+e^{2 i x}\right )+x \log \left (a \cos ^n(x)\right )-\frac {1}{2} (i n) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i x}\right )\\ &=\frac {1}{2} i n x^2-n x \log \left (1+e^{2 i x}\right )+x \log \left (a \cos ^n(x)\right )+\frac {1}{2} i n \text {Li}_2\left (-e^{2 i x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 1.00 \[ x \log \left (a \cos ^n(x)\right )+\frac {1}{2} i n \text {Li}_2\left (-e^{2 i x}\right )+\frac {1}{2} i n x^2-n x \log \left (1+e^{2 i x}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 115, normalized size = 2.21 \[ -\frac {1}{2} \, n x \log \left (i \, \cos \relax (x) + \sin \relax (x) + 1\right ) - \frac {1}{2} \, n x \log \left (i \, \cos \relax (x) - \sin \relax (x) + 1\right ) - \frac {1}{2} \, n x \log \left (-i \, \cos \relax (x) + \sin \relax (x) + 1\right ) - \frac {1}{2} \, n x \log \left (-i \, \cos \relax (x) - \sin \relax (x) + 1\right ) + n x \log \left (\cos \relax (x)\right ) - \frac {1}{2} i \, n {\rm Li}_2\left (i \, \cos \relax (x) + \sin \relax (x)\right ) + \frac {1}{2} i \, n {\rm Li}_2\left (i \, \cos \relax (x) - \sin \relax (x)\right ) + \frac {1}{2} i \, n {\rm Li}_2\left (-i \, \cos \relax (x) + \sin \relax (x)\right ) - \frac {1}{2} i \, n {\rm Li}_2\left (-i \, \cos \relax (x) - \sin \relax (x)\right ) + x \log \relax (a) \]
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 \[ \int \log \left (a \cos \relax (x)^{n}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.61, size = 0, normalized size = 0.00 \[ \int \ln \left (a \left (\cos ^{n}\relax (x )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 65, normalized size = 1.25 \[ -\frac {1}{2} \, {\left (-i \, x^{2} + 2 i \, x \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) + x \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) - i \, {\rm Li}_2\left (-e^{\left (2 i \, x\right )}\right )\right )} n + x \log \left (a \cos \relax (x)^{n}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 41, normalized size = 0.79 \[ x\,\ln \left (a\,{\cos \relax (x)}^n\right )+\frac {n\,\mathrm {polylog}\left (2,-{\mathrm {e}}^{x\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}+\frac {n\,x\,\left (x+\ln \left ({\mathrm {e}}^{x\,2{}\mathrm {i}}+1\right )\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \]
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 \[ \int \log {\left (a \cos ^{n}{\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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