Optimal. Leaf size=235 \[ -\frac {4 x^3 \tanh ^{-1}\left (e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {12 i x^2 \text {PolyLog}\left (2,-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {12 i x^2 \text {PolyLog}\left (2,e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {48 x \text {PolyLog}\left (3,-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {48 x \text {PolyLog}\left (3,e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}-\frac {96 i \text {PolyLog}\left (4,-e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}}+\frac {96 i \text {PolyLog}\left (4,e^{\frac {i x}{2}}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, 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 = {3400, 4268,
2611, 6744, 2320, 6724} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2320
Rule 2611
Rule 3400
Rule 4268
Rule 6724
Rule 6744
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 ) \text {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 ) \text {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 = 0.07, size = 170, normalized size = 0.72 \begin {gather*} -\frac {i \left (8 \pi ^4-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 )-48 x^2 \text {PolyLog}\left (2,e^{-\frac {i x}{2}}\right )-48 x^2 \text {PolyLog}\left (2,-e^{\frac {i x}{2}}\right )+192 i x \text {PolyLog}\left (3,e^{-\frac {i x}{2}}\right )-192 i x \text {PolyLog}\left (3,-e^{\frac {i x}{2}}\right )+384 \text {PolyLog}\left (4,e^{-\frac {i x}{2}}\right )+384 \text {PolyLog}\left (4,-e^{\frac {i x}{2}}\right )\right ) \sin \left (\frac {x}{2}\right )}{4 \sqrt {a-a \cos (x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\sqrt {a -a \cos \left (x \right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\sqrt {- a \left (\cos {\left (x \right )} - 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3}{\sqrt {a-a\,\cos \left (x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________