Optimal. Leaf size=235 \[ \frac {12 i x^2 \text {Li}_2\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {12 i x^2 \text {Li}_2\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {48 x \text {Li}_3\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {48 x \text {Li}_3\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {96 i \text {Li}_4\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {96 i \text {Li}_4\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {4 x^3 \sin \left (\frac {x}{2}\right ) \tanh ^{-1}\left (e^{\frac {i x}{2}}\right )}{\sqrt {a-a \cos (x)}} \]
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Rubi [A] time = 0.17, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3319, 4183, 2531, 6609, 2282, 6589} \[ \frac {12 i x^2 \text {Li}_2\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {12 i x^2 \text {Li}_2\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {48 x \text {Li}_3\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {48 x \text {Li}_3\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {96 i \text {Li}_4\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {96 i \text {Li}_4\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {4 x^3 \sin \left (\frac {x}{2}\right ) \tanh ^{-1}\left (e^{\frac {i x}{2}}\right )}{\sqrt {a-a \cos (x)}} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 3319
Rule 4183
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {a-a \cos (x)}} \, dx &=\frac {\sin \left (\frac {x}{2}\right ) \int x^3 \csc \left (\frac {x}{2}\right ) \, dx}{\sqrt {a-a \cos (x)}}\\ &=-\frac {4 x^3 \tanh ^{-1}\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {\left (6 \sin \left (\frac {x}{2}\right )\right ) \int x^2 \log \left (1-e^{\frac {i x}{2}}\right ) \, dx}{\sqrt {a-a \cos (x)}}+\frac {\left (6 \sin \left (\frac {x}{2}\right )\right ) \int x^2 \log \left (1+e^{\frac {i x}{2}}\right ) \, dx}{\sqrt {a-a \cos (x)}}\\ &=-\frac {4 x^3 \tanh ^{-1}\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {12 i x^2 \text {Li}_2\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {12 i x^2 \text {Li}_2\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {\left (24 i \sin \left (\frac {x}{2}\right )\right ) \int x \text {Li}_2\left (-e^{\frac {i x}{2}}\right ) \, dx}{\sqrt {a-a \cos (x)}}+\frac {\left (24 i \sin \left (\frac {x}{2}\right )\right ) \int x \text {Li}_2\left (e^{\frac {i x}{2}}\right ) \, dx}{\sqrt {a-a \cos (x)}}\\ &=-\frac {4 x^3 \tanh ^{-1}\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {12 i x^2 \text {Li}_2\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {12 i x^2 \text {Li}_2\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {48 x \text {Li}_3\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {48 x \text {Li}_3\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {\left (48 \sin \left (\frac {x}{2}\right )\right ) \int \text {Li}_3\left (-e^{\frac {i x}{2}}\right ) \, dx}{\sqrt {a-a \cos (x)}}-\frac {\left (48 \sin \left (\frac {x}{2}\right )\right ) \int \text {Li}_3\left (e^{\frac {i x}{2}}\right ) \, dx}{\sqrt {a-a \cos (x)}}\\ &=-\frac {4 x^3 \tanh ^{-1}\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {12 i x^2 \text {Li}_2\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {12 i x^2 \text {Li}_2\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {48 x \text {Li}_3\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {48 x \text {Li}_3\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {\left (96 i \sin \left (\frac {x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{\frac {i x}{2}}\right )}{\sqrt {a-a \cos (x)}}+\frac {\left (96 i \sin \left (\frac {x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{\frac {i x}{2}}\right )}{\sqrt {a-a \cos (x)}}\\ &=-\frac {4 x^3 \tanh ^{-1}\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {12 i x^2 \text {Li}_2\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {12 i x^2 \text {Li}_2\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {48 x \text {Li}_3\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {48 x \text {Li}_3\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {96 i \text {Li}_4\left (-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {96 i \text {Li}_4\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 = 0.10, size = 170, normalized size = 0.72 \[ -\frac {i \sin \left (\frac {x}{2}\right ) \left (-48 x^2 \text {Li}_2\left (e^{-\frac {i x}{2}}\right )-48 x^2 \text {Li}_2\left (-e^{\frac {i x}{2}}\right )+192 i x \text {Li}_3\left (e^{-\frac {i x}{2}}\right )-192 i x \text {Li}_3\left (-e^{\frac {i x}{2}}\right )+384 \text {Li}_4\left (e^{-\frac {i x}{2}}\right )+384 \text {Li}_4\left (-e^{\frac {i x}{2}}\right )-x^4+8 i x^3 \log \left (1-e^{-\frac {i x}{2}}\right )-8 i x^3 \log \left (1+e^{\frac {i x}{2}}\right )+8 \pi ^4\right )}{4 \sqrt {a-a \cos (x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a \cos \relax (x) + a} x^{3}}{a \cos \relax (x) - a}, x\right ) \]
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 \[ \int \frac {x^{3}}{\sqrt {-a \cos \relax (x) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\sqrt {a -a \cos \relax (x )}}\, dx \]
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 \[ \int \frac {x^{3}}{\sqrt {-a \cos \relax (x) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3}{\sqrt {a-a\,\cos \relax (x)}} \,d x \]
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 \[ \int \frac {x^{3}}{\sqrt {- a \left (\cos {\relax (x )} - 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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