Optimal. Leaf size=423 \[ -\frac {3 x^2}{a \sqrt {a+a \cos (x)}}-\frac {24 i x \text {ArcTan}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}-\frac {i x^3 \text {ArcTan}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {24 i \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {24 i \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {12 x \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (3,-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {12 x \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (3,i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {24 i \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (4,-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {24 i \cos \left (\frac {x}{2}\right ) \text {PolyLog}\left (4,i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}} \]
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Rubi [A]
time = 0.18, antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3400, 4271,
4266, 2317, 2438, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {i x^3 \text {ArcTan}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {24 i x \text {ArcTan}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}+\frac {3 i x^2 \text {Li}_2\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {3 i x^2 \text {Li}_2\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {12 x \text {Li}_3\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}+\frac {12 x \text {Li}_3\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}+\frac {24 i \text {Li}_2\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {24 i \text {Li}_2\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {24 i \text {Li}_4\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}+\frac {24 i \text {Li}_4\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a \cos (x)+a}}-\frac {3 x^2}{a \sqrt {a \cos (x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3400
Rule 4266
Rule 4271
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {x^3}{(a+a \cos (x))^{3/2}} \, dx &=\frac {\cos \left (\frac {x}{2}\right ) \int x^3 \sec ^3\left (\frac {x}{2}\right ) \, dx}{2 a \sqrt {a+a \cos (x)}}\\ &=-\frac {3 x^2}{a \sqrt {a+a \cos (x)}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}+\frac {\cos \left (\frac {x}{2}\right ) \int x^3 \sec \left (\frac {x}{2}\right ) \, dx}{4 a \sqrt {a+a \cos (x)}}+\frac {\left (6 \cos \left (\frac {x}{2}\right )\right ) \int x \sec \left (\frac {x}{2}\right ) \, dx}{a \sqrt {a+a \cos (x)}}\\ &=-\frac {3 x^2}{a \sqrt {a+a \cos (x)}}-\frac {24 i x \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}-\frac {i x^3 \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}-\frac {\left (3 \cos \left (\frac {x}{2}\right )\right ) \int x^2 \log \left (1-i e^{\frac {i x}{2}}\right ) \, dx}{2 a \sqrt {a+a \cos (x)}}+\frac {\left (3 \cos \left (\frac {x}{2}\right )\right ) \int x^2 \log \left (1+i e^{\frac {i x}{2}}\right ) \, dx}{2 a \sqrt {a+a \cos (x)}}-\frac {\left (12 \cos \left (\frac {x}{2}\right )\right ) \int \log \left (1-i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}+\frac {\left (12 \cos \left (\frac {x}{2}\right )\right ) \int \log \left (1+i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}\\ &=-\frac {3 x^2}{a \sqrt {a+a \cos (x)}}-\frac {24 i x \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}-\frac {i x^3 \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}-\frac {\left (6 i \cos \left (\frac {x}{2}\right )\right ) \int x \text {Li}_2\left (-i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}+\frac {\left (6 i \cos \left (\frac {x}{2}\right )\right ) \int x \text {Li}_2\left (i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}+\frac {\left (24 i \cos \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {\left (24 i \cos \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}\\ &=-\frac {3 x^2}{a \sqrt {a+a \cos (x)}}-\frac {24 i x \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}-\frac {i x^3 \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {12 x \cos \left (\frac {x}{2}\right ) \text {Li}_3\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {12 x \cos \left (\frac {x}{2}\right ) \text {Li}_3\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}+\frac {\left (12 \cos \left (\frac {x}{2}\right )\right ) \int \text {Li}_3\left (-i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}-\frac {\left (12 \cos \left (\frac {x}{2}\right )\right ) \int \text {Li}_3\left (i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}\\ &=-\frac {3 x^2}{a \sqrt {a+a \cos (x)}}-\frac {24 i x \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}-\frac {i x^3 \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {12 x \cos \left (\frac {x}{2}\right ) \text {Li}_3\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {12 x \cos \left (\frac {x}{2}\right ) \text {Li}_3\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}-\frac {\left (24 i \cos \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {\left (24 i \cos \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}\\ &=-\frac {3 x^2}{a \sqrt {a+a \cos (x)}}-\frac {24 i x \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}-\frac {i x^3 \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {3 i x^2 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {12 x \cos \left (\frac {x}{2}\right ) \text {Li}_3\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {12 x \cos \left (\frac {x}{2}\right ) \text {Li}_3\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_4\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {24 i \cos \left (\frac {x}{2}\right ) \text {Li}_4\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x^3 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 257, normalized size = 0.61 \begin {gather*} -\frac {i \cos \left (\frac {x}{2}\right ) \left (-6 i x^2 \cos \left (\frac {x}{2}\right )+48 x \text {ArcTan}\left (e^{\frac {i x}{2}}\right ) \cos ^2\left (\frac {x}{2}\right )+2 x^3 \text {ArcTan}\left (e^{\frac {i x}{2}}\right ) \cos ^2\left (\frac {x}{2}\right )-6 \left (8+x^2\right ) \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (2,-i e^{\frac {i x}{2}}\right )+6 \left (8+x^2\right ) \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (2,i e^{\frac {i x}{2}}\right )-24 i x \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (3,-i e^{\frac {i x}{2}}\right )+24 i x \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (3,i e^{\frac {i x}{2}}\right )+48 \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (4,-i e^{\frac {i x}{2}}\right )-48 \cos ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (4,i e^{\frac {i x}{2}}\right )+i x^3 \sin \left (\frac {x}{2}\right )\right )}{(a (1+\cos (x)))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (a +a \cos \left (x \right )\right )^{\frac {3}{2}}}\, dx\]
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^{3}}{\left (a \left (\cos {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}\, 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 = -0.00 \begin {gather*} \int \frac {x^3}{{\left (a+a\,\cos \left (x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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