Optimal. Leaf size=28 \[ -\frac{x}{2}+\tan \left (\frac{x}{2}\right )+\frac{\sin (x) \log \left (\cos \left (\frac{x}{2}\right )\right )}{\cos (x)+1} \]
[Out]
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Rubi [A] time = 0.0565151, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ -\frac{x}{2}+\tan \left (\frac{x}{2}\right )+\frac{\sin (x) \log \left (\cos \left (\frac{x}{2}\right )\right )}{\cos (x)+1} \]
Antiderivative was successfully verified.
[In] Int[Log[Cos[x/2]]/(1 + Cos[x]),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sin{\left (x \right )} \tan{\left (\frac{x}{2} \right )}}{2 \cos{\left (x \right )} + 2}\, dx + \frac{\log{\left (\cos{\left (\frac{x}{2} \right )} \right )} \sin{\left (x \right )}}{\cos{\left (x \right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(ln(cos(1/2*x))/(1+cos(x)),x)
[Out]
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Mathematica [A] time = 0.0842314, 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.
[In] Integrate[Log[Cos[x/2]]/(1 + Cos[x]),x]
[Out]
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Maple [C] time = 0.217, size = 164, normalized size = 5.9 \[{\frac{-2\,i\ln \left ({{\rm e}^{{\frac{i}{2}}x}} \right ) }{{{\rm e}^{ix}}+1}}+{\frac{1}{{{\rm e}^{ix}}+1} \left ( -i\ln \left ({{\rm e}^{ix}}+1 \right ){{\rm e}^{ix}}+\pi \,{\it csgn} \left ( i \left ({{\rm e}^{ix}}+1 \right ) \right ){\it csgn} \left ( i{{\rm e}^{-{\frac{i}{2}}x}} \right ){\it csgn} \left ( i\cos \left ({\frac{x}{2}} \right ) \right ) -\pi \,{\it csgn} \left ( i \left ({{\rm e}^{ix}}+1 \right ) \right ) \left ({\it csgn} \left ( i\cos \left ({\frac{x}{2}} \right ) \right ) \right ) ^{2}-\pi \,{\it csgn} \left ( i{{\rm e}^{-{\frac{i}{2}}x}} \right ) \left ({\it csgn} \left ( i\cos \left ({\frac{x}{2}} \right ) \right ) \right ) ^{2}+\pi \, \left ({\it csgn} \left ( i\cos \left ({\frac{x}{2}} \right ) \right ) \right ) ^{3}-x{{\rm e}^{ix}}+i\ln \left ({{\rm e}^{ix}}+1 \right ) -2\,i\ln \left ( 2 \right ) +2\,i-x \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(ln(cos(1/2*x))/(1+cos(x)),x)
[Out]
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Maxima [A] time = 1.37029, size = 76, normalized size = 2.71 \[ \frac{\log \left (\cos \left (\frac{1}{2} \, x\right )\right ) \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{x \cos \left (x\right )^{2} + x \sin \left (x\right )^{2} + 2 \, x \cos \left (x\right ) + x - 4 \, \sin \left (x\right )}{2 \,{\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(log(cos(1/2*x))/(cos(x) + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.252857, size = 43, normalized size = 1.54 \[ -\frac{x \cos \left (\frac{1}{2} \, x\right ) - 2 \, \log \left (\cos \left (\frac{1}{2} \, x\right )\right ) \sin \left (\frac{1}{2} \, x\right ) - 2 \, \sin \left (\frac{1}{2} \, x\right )}{2 \, \cos \left (\frac{1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(log(cos(1/2*x))/(cos(x) + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\log{\left (\cos{\left (\frac{x}{2} \right )} \right )}}{\cos{\left (x \right )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(ln(cos(1/2*x))/(1+cos(x)),x)
[Out]
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GIAC/XCAS [A] time = 0.218164, size = 58, normalized size = 2.07 \[ -\frac{1}{2} \, x - \frac{2 \,{\rm ln}\left (\cos \left (\frac{1}{2} \, x\right )\right ) \tan \left (\frac{1}{2} \, x\right )}{{\left (x^{2} + 1\right )}{\left (\frac{x^{2} - 1}{x^{2} + 1} - 1\right )}} + \tan \left (\frac{1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(log(cos(1/2*x))/(cos(x) + 1),x, algorithm="giac")
[Out]