Optimal. Leaf size=28 \[ -\frac {x}{2}+\tan \left (\frac {x}{2}\right )+\frac {\sin (x) \log \left (\cos \left (\frac {x}{2}\right )\right )}{\cos (x)+1} \]
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Rubi [A] time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2648, 2554, 12, 3473, 8} \[ -\frac {x}{2}+\tan \left (\frac {x}{2}\right )+\frac {\sin (x) \log \left (\cos \left (\frac {x}{2}\right )\right )}{\cos (x)+1} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 2554
Rule 2648
Rule 3473
Rubi steps
\begin {align*} \int \frac {\log \left (\cos \left (\frac {x}{2}\right )\right )}{1+\cos (x)} \, dx &=\frac {\log \left (\cos \left (\frac {x}{2}\right )\right ) \sin (x)}{1+\cos (x)}-\int -\frac {1}{2} \tan ^2\left (\frac {x}{2}\right ) \, dx\\ &=\frac {\log \left (\cos \left (\frac {x}{2}\right )\right ) \sin (x)}{1+\cos (x)}+\frac {1}{2} \int \tan ^2\left (\frac {x}{2}\right ) \, dx\\ &=\frac {\log \left (\cos \left (\frac {x}{2}\right )\right ) \sin (x)}{1+\cos (x)}+\tan \left (\frac {x}{2}\right )-\frac {\int 1 \, dx}{2}\\ &=-\frac {x}{2}+\frac {\log \left (\cos \left (\frac {x}{2}\right )\right ) \sin (x)}{1+\cos (x)}+\tan \left (\frac {x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 32, normalized size = 1.14 \[ -\frac {\sin (x) \left (x \cot \left (\frac {x}{2}\right )-2 \left (\log \left (\cos \left (\frac {x}{2}\right )\right )+1\right )\right )}{2 (\cos (x)+1)} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\cos \left (\frac {x}{2}\right )\right )}{1+\cos (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.90, size = 32, normalized size = 1.14 \[ -\frac {x \cos \left (\frac {1}{2} \, x\right ) - 2 \, \log \left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right ) - 2 \, \sin \left (\frac {1}{2} \, x\right )}{2 \, \cos \left (\frac {1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 43, normalized size = 1.54 \[ -\frac {1}{2} \, x - \frac {2 \, \log \left (\cos \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )}{{\left (x^{2} + 1\right )} {\left (\frac {x^{2} - 1}{x^{2} + 1} - 1\right )}} + \tan \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.25, size = 164, normalized size = 5.86
method | result | size |
risch | \(-\frac {2 i \ln \left ({\mathrm e}^{\frac {i x}{2}}\right )}{{\mathrm e}^{i x}+1}+\frac {-i \ln \left ({\mathrm e}^{i x}+1\right ) {\mathrm e}^{i x}+\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{i x}+1\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-\frac {i x}{2}}\right ) \mathrm {csgn}\left (i \cos \left (\frac {x}{2}\right )\right )-\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{i x}+1\right )\right ) \mathrm {csgn}\left (i \cos \left (\frac {x}{2}\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-\frac {i x}{2}}\right ) \mathrm {csgn}\left (i \cos \left (\frac {x}{2}\right )\right )^{2}+\pi \mathrm {csgn}\left (i \cos \left (\frac {x}{2}\right )\right )^{3}-x \,{\mathrm e}^{i x}+i \ln \left ({\mathrm e}^{i x}+1\right )-2 i \ln \relax (2)+2 i-x}{{\mathrm e}^{i x}+1}\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 56, normalized size = 2.00 \[ \frac {\log \left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \relax (x)}{\cos \relax (x) + 1} - \frac {x \cos \relax (x)^{2} + x \sin \relax (x)^{2} + 2 \, x \cos \relax (x) + x - 4 \, \sin \relax (x)}{2 \, {\left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 39, normalized size = 1.39 \[ \mathrm {tan}\left (\frac {x}{2}\right )-x+\mathrm {tan}\left (\frac {x}{2}\right )\,\ln \left (\cos \left (\frac {x}{2}\right )\right )+\ln \left (\cos \left (\frac {x}{2}\right )\right )\,1{}\mathrm {i}-\ln \left (\cos \relax (x)+1+\sin \relax (x)\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \]
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 \[ \int \frac {\log {\left (\cos {\left (\frac {x}{2} \right )} \right )}}{\cos {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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