Optimal. Leaf size=150 \[ \frac {i \text {Li}_2\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {i \text {Li}_2\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {1}{a \sqrt {a \cos (x)+a}}-\frac {i x \cos \left (\frac {x}{2}\right ) \tan ^{-1}\left (e^{\frac {i x}{2}}\right )}{a \sqrt {a \cos (x)+a}}+\frac {x \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a \cos (x)+a}} \]
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Rubi [A] time = 0.12, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3319, 4185, 4181, 2279, 2391} \[ \frac {i \text {Li}_2\left (-i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {i \text {Li}_2\left (i e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a \cos (x)+a}}-\frac {1}{a \sqrt {a \cos (x)+a}}-\frac {i x \cos \left (\frac {x}{2}\right ) \tan ^{-1}\left (e^{\frac {i x}{2}}\right )}{a \sqrt {a \cos (x)+a}}+\frac {x \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a \cos (x)+a}} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 3319
Rule 4181
Rule 4185
Rubi steps
\begin {align*} \int \frac {x}{(a+a \cos (x))^{3/2}} \, dx &=\frac {\cos \left (\frac {x}{2}\right ) \int x \sec ^3\left (\frac {x}{2}\right ) \, dx}{2 a \sqrt {a+a \cos (x)}}\\ &=-\frac {1}{a \sqrt {a+a \cos (x)}}+\frac {x \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}+\frac {\cos \left (\frac {x}{2}\right ) \int x \sec \left (\frac {x}{2}\right ) \, dx}{4 a \sqrt {a+a \cos (x)}}\\ &=-\frac {1}{a \sqrt {a+a \cos (x)}}-\frac {i x \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}-\frac {\cos \left (\frac {x}{2}\right ) \int \log \left (1-i e^{\frac {i x}{2}}\right ) \, dx}{2 a \sqrt {a+a \cos (x)}}+\frac {\cos \left (\frac {x}{2}\right ) \int \log \left (1+i e^{\frac {i x}{2}}\right ) \, dx}{2 a \sqrt {a+a \cos (x)}}\\ &=-\frac {1}{a \sqrt {a+a \cos (x)}}-\frac {i x \tan ^{-1}\left (e^{\frac {i x}{2}}\right ) \cos \left (\frac {x}{2}\right )}{a \sqrt {a+a \cos (x)}}+\frac {x \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}+\frac {\left (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 (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 {1}{a \sqrt {a+a \cos (x)}}-\frac {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 \cos \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{\frac {i x}{2}}\right )}{a \sqrt {a+a \cos (x)}}-\frac {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 {x \tan \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cos (x)}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 165, normalized size = 1.10 \[ \frac {\sec \left (\frac {x}{2}\right ) \left (2 i \text {Li}_2\left (-i e^{\frac {i x}{2}}\right ) (\cos (x)+1)-2 i \text {Li}_2\left (i e^{\frac {i x}{2}}\right ) (\cos (x)+1)+x \log \left (1-i e^{\frac {i x}{2}}\right )-x \log \left (1+i e^{\frac {i x}{2}}\right )+2 x \sin \left (\frac {x}{2}\right )-4 \cos \left (\frac {x}{2}\right )+x \log \left (1-i e^{\frac {i x}{2}}\right ) \cos (x)-x \log \left (1+i e^{\frac {i x}{2}}\right ) \cos (x)\right )}{4 a \sqrt {a (\cos (x)+1)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cos \relax (x) + a} x}{a^{2} \cos \relax (x)^{2} + 2 \, a^{2} \cos \relax (x) + a^{2}}, x\right ) \]
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 \[ \int \frac {x}{{\left (a \cos \relax (x) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x}{\left (a +a \cos \relax (x )\right )^{\frac {3}{2}}}\, dx \]
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 = -0.01 \[ \int \frac {x}{{\left (a+a\,\cos \relax (x)\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x}{\left (a \left (\cos {\relax (x )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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