Optimal. Leaf size=257 \[ \frac {2 i x \text {Li}_2\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {2 i x \text {Li}_2\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {4 \text {Li}_3\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}+\frac {4 \text {Li}_3\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {i x^2 \cos \left (\frac {x}{2}\right ) \tan ^{-1}\left (e^{\frac {i x}{2}}\right )}{a \sqrt {a \cos (x)+a}}+\frac {x^2 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a \cos (x)+a}}-\frac {2 x}{a \sqrt {a \cos (x)+a}}+\frac {4 \cos \left (\frac {x}{2}\right ) \tanh ^{-1}\left (\sin \left (\frac {x}{2}\right )\right )}{a \sqrt {a \cos (x)+a}} \]
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Rubi [A] time = 0.19, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3319, 4186, 3770, 4181, 2531, 2282, 6589} \[ \frac {2 i x \text {Li}_2\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {2 i x \text {Li}_2\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {4 \text {Li}_3\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}+\frac {4 \text {Li}_3\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {i x^2 \cos \left (\frac {x}{2}\right ) \tan ^{-1}\left (e^{\frac {i x}{2}}\right )}{a \sqrt {a \cos (x)+a}}+\frac {x^2 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a \cos (x)+a}}-\frac {2 x}{a \sqrt {a \cos (x)+a}}+\frac {4 \cos \left (\frac {x}{2}\right ) \tanh ^{-1}\left (\sin \left (\frac {x}{2}\right )\right )}{a \sqrt {a \cos (x)+a}} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 3319
Rule 3770
Rule 4181
Rule 4186
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^2}{(a+a \cos (x))^{3/2}} \, dx &=\frac {\cos \left (\frac {x}{2}\right ) \int x^2 \sec ^3\left (\frac {x}{2}\right ) \, dx}{2 a \sqrt {a+a \cos (x)}}\\ &=-\frac {2 x}{a \sqrt {a+a \cos (x)}}+\frac {x^2 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}+\frac {\cos \left (\frac {x}{2}\right ) \int x^2 \sec \left (\frac {x}{2}\right ) \, dx}{4 a \sqrt {a+a \cos (x)}}+\frac {\left (2 \cos \left (\frac {x}{2}\right )\right ) \int \sec \left (\frac {x}{2}\right ) \, dx}{a \sqrt {a+a \cos (x)}}\\ &=-\frac {2 x}{a \sqrt {a+a \cos (x)}}-\frac {i x^2 \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {4 \tanh ^{-1}\left (\sin \left (\frac {x}{2}\right )\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x^2 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}-\frac {\cos \left (\frac {x}{2}\right ) \int x \log \left (1-i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}+\frac {\cos \left (\frac {x}{2}\right ) \int x \log \left (1+i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}\\ &=-\frac {2 x}{a \sqrt {a+a \cos (x)}}-\frac {i x^2 \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {4 \tanh ^{-1}\left (\sin \left (\frac {x}{2}\right )\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {2 i x \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {2 i x \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^2 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}-\frac {\left (2 i \cos \left (\frac {x}{2}\right )\right ) \int \text {Li}_2\left (-i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}+\frac {\left (2 i \cos \left (\frac {x}{2}\right )\right ) \int \text {Li}_2\left (i e^{\frac {i x}{2}}\right ) \, dx}{a \sqrt {a+a \cos (x)}}\\ &=-\frac {2 x}{a \sqrt {a+a \cos (x)}}-\frac {i x^2 \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {4 \tanh ^{-1}\left (\sin \left (\frac {x}{2}\right )\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {2 i x \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {2 i x \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^2 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}-\frac {\left (4 \cos \left (\frac {x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {\left (4 \cos \left (\frac {x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}\\ &=-\frac {2 x}{a \sqrt {a+a \cos (x)}}-\frac {i x^2 \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {4 \tanh ^{-1}\left (\sin \left (\frac {x}{2}\right )\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {2 i x \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {2 i x \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {4 \cos \left (\frac {x}{2}\right ) \text {Li}_3\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}+\frac {4 \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^2 \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 185, normalized size = 0.72 \[ \frac {\cos \left (\frac {x}{2}\right ) \left (4 i x \text {Li}_2\left (-i e^{\frac {i x}{2}}\right ) \cos ^2\left (\frac {x}{2}\right )-4 i x \text {Li}_2\left (i e^{\frac {i x}{2}}\right ) \cos ^2\left (\frac {x}{2}\right )-8 \text {Li}_3\left (-i e^{\frac {i x}{2}}\right ) \cos ^2\left (\frac {x}{2}\right )+8 \text {Li}_3\left (i e^{\frac {i x}{2}}\right ) \cos ^2\left (\frac {x}{2}\right )+x^2 \sin \left (\frac {x}{2}\right )-2 i x^2 \cos ^2\left (\frac {x}{2}\right ) \tan ^{-1}\left (e^{\frac {i x}{2}}\right )-4 x \cos \left (\frac {x}{2}\right )+8 \cos ^2\left (\frac {x}{2}\right ) \tanh ^{-1}\left (\sin \left (\frac {x}{2}\right )\right )\right )}{(a (\cos (x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cos \relax (x) + a} x^{2}}{a^{2} \cos \relax (x)^{2} + 2 \, a^{2} \cos \relax (x) + a^{2}}, x\right ) \]
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 \[ \int \frac {x^{2}}{{\left (a \cos \relax (x) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a +a \cos \relax (x )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {x^2}{{\left (a+a\,\cos \relax (x)\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2}}{\left (a \left (\cos {\relax (x )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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