Optimal. Leaf size=163 \[ -\frac {4 x^2 \tanh ^{-1}\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {8 i x \text {PolyLog}\left (2,-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {8 i x \text {PolyLog}\left (2,e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {16 \text {PolyLog}\left (3,-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {16 \text {PolyLog}\left (3,e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}} \]
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Rubi [A]
time = 0.10, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3400, 4268,
2611, 2320, 6724} \begin {gather*} \frac {8 i x \text {Li}_2\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {8 i x \text {Li}_2\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {16 \text {Li}_3\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {16 \text {Li}_3\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {4 x^2 \sin \left (\frac {x}{2}\right ) \tanh ^{-1}\left (e^{\frac {i x}{2}}\right )}{\sqrt {a-a \cos (x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 3400
Rule 4268
Rule 6724
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a-a \cos (x)}} \, dx &=\frac {\sin \left (\frac {x}{2}\right ) \int x^2 \csc \left (\frac {x}{2}\right ) \, dx}{\sqrt {a-a \cos (x)}}\\ &=-\frac {4 x^2 \tanh ^{-1}\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {\left (4 \sin \left (\frac {x}{2}\right )\right ) \int x \log \left (1-e^{\frac {i x}{2}}\right ) \, dx}{\sqrt {a-a \cos (x)}}+\frac {\left (4 \sin \left (\frac {x}{2}\right )\right ) \int x \log \left (1+e^{\frac {i x}{2}}\right ) \, dx}{\sqrt {a-a \cos (x)}}\\ &=-\frac {4 x^2 \tanh ^{-1}\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {8 i x \text {Li}_2\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {8 i x \text {Li}_2\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {\left (8 i \sin \left (\frac {x}{2}\right )\right ) \int \text {Li}_2\left (-e^{\frac {i x}{2}}\right ) \, dx}{\sqrt {a-a \cos (x)}}+\frac {\left (8 i \sin \left (\frac {x}{2}\right )\right ) \int \text {Li}_2\left (e^{\frac {i x}{2}}\right ) \, dx}{\sqrt {a-a \cos (x)}}\\ &=-\frac {4 x^2 \tanh ^{-1}\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {8 i x \text {Li}_2\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {8 i x \text {Li}_2\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {\left (16 \sin \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\frac {i x}{2}}\right )}{\sqrt {a-a \cos (x)}}+\frac {\left (16 \sin \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\frac {i x}{2}}\right )}{\sqrt {a-a \cos (x)}}\\ &=-\frac {4 x^2 \tanh ^{-1}\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {8 i x \text {Li}_2\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {8 i x \text {Li}_2\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {16 \text {Li}_3\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {16 \text {Li}_3\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 117, normalized size = 0.72 \begin {gather*} \frac {2 \left (x^2 \log \left (1-e^{\frac {i x}{2}}\right )-x^2 \log \left (1+e^{\frac {i x}{2}}\right )+4 i x \text {PolyLog}\left (2,-e^{\frac {i x}{2}}\right )-4 i x \text {PolyLog}\left (2,e^{\frac {i x}{2}}\right )-8 \text {PolyLog}\left (3,-e^{\frac {i x}{2}}\right )+8 \text {PolyLog}\left (3,e^{\frac {i x}{2}}\right )\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\sqrt {a -a \cos \left (x \right )}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {- a \left (\cos {\left (x \right )} - 1\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 = -0.01 \begin {gather*} \int \frac {x^2}{\sqrt {a-a\,\cos \left (x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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