Optimal. Leaf size=32 \[ 4 \sqrt {a \cos (x)+a}+2 x \tan \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \]
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Rubi [A] time = 0.05, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3319, 3296, 2638} \[ 4 \sqrt {a \cos (x)+a}+2 x \tan \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3319
Rubi steps
\begin {align*} \int x \sqrt {a+a \cos (x)} \, dx &=\left (\sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int x \cos \left (\frac {x}{2}\right ) \, dx\\ &=2 x \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )-\left (2 \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \sin \left (\frac {x}{2}\right ) \, dx\\ &=4 \sqrt {a+a \cos (x)}+2 x \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 0.69 \[ 2 \left (x \tan \left (\frac {x}{2}\right )+2\right ) \sqrt {a (\cos (x)+1)} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 31, normalized size = 0.97 \[ 2 \, \sqrt {2} {\left (x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right ) + 2 \, \cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 55, normalized size = 1.72 \[ -\frac {i \sqrt {2}\, \sqrt {a \left ({\mathrm e}^{i x}+1\right )^{2} {\mathrm e}^{-i x}}\, \left (2 i {\mathrm e}^{i x}+x \,{\mathrm e}^{i x}+2 i-x \right )}{{\mathrm e}^{i x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.50, size = 24, normalized size = 0.75 \[ 2 \, {\left (\sqrt {2} x \sin \left (\frac {1}{2} \, x\right ) + 2 \, \sqrt {2} \cos \left (\frac {1}{2} \, x\right )\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 50, normalized size = 1.56 \[ \frac {2\,\sqrt {a}\,\sqrt {\cos \relax (x)+1}\,\left (x\,\cos \relax (x)+\cos \relax (x)\,2{}\mathrm {i}-2\,\sin \relax (x)-x+x\,\sin \relax (x)\,1{}\mathrm {i}+2{}\mathrm {i}\right )}{\cos \relax (x)\,1{}\mathrm {i}-\sin \relax (x)+1{}\mathrm {i}} \]
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 \[ \int x \sqrt {a \left (\cos {\relax (x )} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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