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 = 0.173309, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, 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 3319
Rule 4183
Rule 2531
Rule 6609
Rule 2282
Rule 6589
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*}
Mathematica [A] time = 0.0978472, 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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Maple [F] time = 0.125, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3}{\frac{1}{\sqrt{a-a\cos \left ( x \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{-a \cos \left (x\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a \cos \left (x\right ) + a} x^{3}}{a \cos \left (x\right ) - a}, x\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 \frac{x^{3}}{\sqrt{- a \left (\cos{\left (x \right )} - 1\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 \frac{x^{3}}{\sqrt{-a \cos \left (x\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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