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]
________________________________________________________________________________________
Rubi [A] time = 0.0347725, 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, 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.
[In]
[Out]
Rule 2648
Rule 2554
Rule 12
Rule 3473
Rule 8
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*}
Mathematica [A] time = 0.0845528, 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]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.163, size = 164, normalized size = 5.9 \begin{align*}{\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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.955665, size = 76, normalized size = 2.71 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.61238, size = 105, normalized size = 3.75 \begin{align*} -\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (\cos{\left (\frac{x}{2} \right )} \right )}}{\cos{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.08722, size = 58, normalized size = 2.07 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]