Optimal. Leaf size=52 \[ \frac {1}{2} i n x^2-n x \log \left (1-e^{2 i x}\right )+x \log \left (a \sin ^n(x)\right )+\frac {1}{2} i n \text {Li}_2\left (e^{2 i x}\right ) \]
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Rubi [A]
time = 0.04, 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 = {2628, 12, 3798,
2221, 2317, 2438} \begin {gather*} \frac {1}{2} i n \text {PolyLog}\left (2,e^{2 i x}\right )+x \log \left (a \sin ^n(x)\right )+\frac {1}{2} i n x^2-n x \log \left (1-e^{2 i x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2221
Rule 2317
Rule 2438
Rule 2628
Rule 3798
Rubi steps
\begin {align*} \int \log \left (a \sin ^n(x)\right ) \, dx &=x \log \left (a \sin ^n(x)\right )-\int n x \cot (x) \, dx\\ &=x \log \left (a \sin ^n(x)\right )-n \int x \cot (x) \, dx\\ &=\frac {1}{2} i n x^2+x \log \left (a \sin ^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 \sin ^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 \sin ^n(x)\right )-\frac {1}{2} (i n) \text {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 \sin ^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 \begin {gather*} \frac {1}{2} i n x^2-n x \log \left (1-e^{2 i x}\right )+x \log \left (a \sin ^n(x)\right )+\frac {1}{2} i n \text {Li}_2\left (e^{2 i x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \ln \left (a \left (\sin ^{n}\left (x \right )\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 91 vs. \(2 (37) = 74\).
time = 0.65, size = 91, normalized size = 1.75 \begin {gather*} -\frac {1}{2} \, {\left (-i \, x^{2} + 2 i \, x \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - 2 i \, x \arctan \left (\sin \left (x\right ), -\cos \left (x\right ) + 1\right ) + x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - 2 i \, {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) - 2 i \, {\rm Li}_2\left (e^{\left (i \, x\right )}\right )\right )} n + x \log \left (a \sin \left (x\right )^{n}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 115 vs. \(2 (37) = 74\).
time = 0.39, size = 115, normalized size = 2.21 \begin {gather*} -\frac {1}{2} \, n x \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} \, n x \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} \, n x \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} \, n x \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + n x \log \left (\sin \left (x\right )\right ) + \frac {1}{2} i \, n {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, n {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, n {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac {1}{2} i \, n {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + x \log \left (a\right ) \end {gather*}
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 \begin {gather*} \int \log {\left (a \sin ^{n}{\left (x \right )} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \ln \left (a\,{\sin \left (x\right )}^n\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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