Optimal. Leaf size=52 \[ \frac{1}{2} i n \text{PolyLog}\left (2,-e^{2 i x}\right )+x \log \left (a \cos ^n(x)\right )+\frac{1}{2} i n x^2-n x \log \left (1+e^{2 i x}\right ) \]
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Rubi [A] time = 0.0581344, antiderivative size = 52, normalized size of antiderivative = 1., 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 2548
Rule 12
Rule 3719
Rule 2190
Rule 2279
Rule 2391
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*}
Mathematica [A] time = 0.0231103, size = 52, normalized size = 1. \[ \frac{1}{2} i n \text{PolyLog}\left (2,-e^{2 i x}\right )+x \log \left (a \cos ^n(x)\right )+\frac{1}{2} i n x^2-n x \log \left (1+e^{2 i x}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.108, size = 0, normalized size = 0. \begin{align*} \int \ln \left ( a \left ( \cos \left ( x \right ) \right ) ^{n} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.45841, size = 88, normalized size = 1.69 \begin{align*} -\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 \left (x\right )^{n}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.25467, size = 433, normalized size = 8.33 \begin{align*} -\frac{1}{2} \, n x \log \left (i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) - \frac{1}{2} \, n x \log \left (i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) - \frac{1}{2} \, n x \log \left (-i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) - \frac{1}{2} \, n x \log \left (-i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) + n x \log \left (\cos \left (x\right )\right ) - \frac{1}{2} i \, n{\rm Li}_2\left (i \, \cos \left (x\right ) + \sin \left (x\right )\right ) + \frac{1}{2} i \, n{\rm Li}_2\left (i \, \cos \left (x\right ) - \sin \left (x\right )\right ) + \frac{1}{2} i \, n{\rm Li}_2\left (-i \, \cos \left (x\right ) + \sin \left (x\right )\right ) - \frac{1}{2} i \, n{\rm Li}_2\left (-i \, \cos \left (x\right ) - \sin \left (x\right )\right ) + x \log \left (a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (a \cos ^{n}{\left (x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left (a \cos \left (x\right )^{n}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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