Optimal. Leaf size=423 \[ \frac{3 i x^2 \text{Li}_2\left (-i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}-\frac{3 i x^2 \text{Li}_2\left (i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}-\frac{12 x \text{Li}_3\left (-i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}+\frac{12 x \text{Li}_3\left (i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}+\frac{24 i \text{Li}_2\left (-i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}-\frac{24 i \text{Li}_2\left (i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}-\frac{24 i \text{Li}_4\left (-i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}+\frac{24 i \text{Li}_4\left (i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}-\frac{3 x^2}{a \sqrt{a \cos (x)+a}}-\frac{i x^3 \cos \left (\frac{x}{2}\right ) \tan ^{-1}\left (e^{\frac{i x}{2}}\right )}{a \sqrt{a \cos (x)+a}}+\frac{x^3 \tan \left (\frac{x}{2}\right )}{2 a \sqrt{a \cos (x)+a}}-\frac{24 i x \cos \left (\frac{x}{2}\right ) \tan ^{-1}\left (e^{\frac{i x}{2}}\right )}{a \sqrt{a \cos (x)+a}} \]
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Rubi [A] time = 0.259535, antiderivative size = 423, normalized size of antiderivative = 1., 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 = {3319, 4186, 4181, 2279, 2391, 2531, 6609, 2282, 6589} \[ \frac{3 i x^2 \text{Li}_2\left (-i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}-\frac{3 i x^2 \text{Li}_2\left (i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}-\frac{12 x \text{Li}_3\left (-i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}+\frac{12 x \text{Li}_3\left (i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}+\frac{24 i \text{Li}_2\left (-i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}-\frac{24 i \text{Li}_2\left (i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}-\frac{24 i \text{Li}_4\left (-i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}+\frac{24 i \text{Li}_4\left (i e^{\frac{i x}{2}}\right ) \cos \left (\frac{x}{2}\right )}{a \sqrt{a \cos (x)+a}}-\frac{3 x^2}{a \sqrt{a \cos (x)+a}}-\frac{i x^3 \cos \left (\frac{x}{2}\right ) \tan ^{-1}\left (e^{\frac{i x}{2}}\right )}{a \sqrt{a \cos (x)+a}}+\frac{x^3 \tan \left (\frac{x}{2}\right )}{2 a \sqrt{a \cos (x)+a}}-\frac{24 i x \cos \left (\frac{x}{2}\right ) \tan ^{-1}\left (e^{\frac{i x}{2}}\right )}{a \sqrt{a \cos (x)+a}} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 4186
Rule 4181
Rule 2279
Rule 2391
Rule 2531
Rule 6609
Rule 2282
Rule 6589
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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.459208, size = 257, normalized size = 0.61 \[ -\frac{i \cos \left (\frac{x}{2}\right ) \left (-6 \left (x^2+8\right ) \text{Li}_2\left (-i e^{\frac{i x}{2}}\right ) \cos ^2\left (\frac{x}{2}\right )+6 \left (x^2+8\right ) \text{Li}_2\left (i e^{\frac{i x}{2}}\right ) \cos ^2\left (\frac{x}{2}\right )-24 i x \text{Li}_3\left (-i e^{\frac{i x}{2}}\right ) \cos ^2\left (\frac{x}{2}\right )+24 i x \text{Li}_3\left (i e^{\frac{i x}{2}}\right ) \cos ^2\left (\frac{x}{2}\right )+48 \text{Li}_4\left (-i e^{\frac{i x}{2}}\right ) \cos ^2\left (\frac{x}{2}\right )-48 \text{Li}_4\left (i e^{\frac{i x}{2}}\right ) \cos ^2\left (\frac{x}{2}\right )+i x^3 \sin \left (\frac{x}{2}\right )-6 i x^2 \cos \left (\frac{x}{2}\right )+2 x^3 \cos ^2\left (\frac{x}{2}\right ) \tan ^{-1}\left (e^{\frac{i x}{2}}\right )+48 x \cos ^2\left (\frac{x}{2}\right ) \tan ^{-1}\left (e^{\frac{i x}{2}}\right )\right )}{(a (\cos (x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.112, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+a\cos \left ( x \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{2} \cos \left (x\right )^{2} + 2 \, a^{2} \cos \left (x\right ) + a^{2}}, 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}}{\left (a \left (\cos{\left (x \right )} + 1\right )\right )^{\frac{3}{2}}}\, 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}}{{\left (a \cos \left (x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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