Optimal. Leaf size=30 \[ \frac {x}{2}-\frac {1}{2} \log \left (\tan \left (\frac {x}{2}\right )+1\right )-\frac {1}{2} \log (\sin (x)+\cos (x)+1) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3137, 3124, 31} \[ \frac {x}{2}-\frac {1}{2} \log \left (\tan \left (\frac {x}{2}\right )+1\right )-\frac {1}{2} \log (\sin (x)+\cos (x)+1) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 3124
Rule 3137
Rubi steps
\begin {align*} \int \frac {\sin (x)}{1+\cos (x)+\sin (x)} \, dx &=\frac {x}{2}-\frac {1}{2} \log (1+\cos (x)+\sin (x))-\frac {1}{2} \int \frac {1}{1+\cos (x)+\sin (x)} \, dx\\ &=\frac {x}{2}-\frac {1}{2} \log (1+\cos (x)+\sin (x))-\operatorname {Subst}\left (\int \frac {1}{2+2 x} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\frac {x}{2}-\frac {1}{2} \log (1+\cos (x)+\sin (x))-\frac {1}{2} \log \left (1+\tan \left (\frac {x}{2}\right )\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 22, normalized size = 0.73 \[ \frac {x}{2}-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 11, normalized size = 0.37 \[ \frac {1}{2} \, x - \frac {1}{2} \, \log \left (\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.02, size = 25, normalized size = 0.83 \[ \frac {1}{2} \, x + \frac {1}{2} \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) - \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 25, normalized size = 0.83 \[ \frac {x}{2}-\ln \left (\tan \left (\frac {x}{2}\right )+1\right )+\frac {\ln \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.15, size = 41, normalized size = 1.37 \[ \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) - \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right ) + \frac {1}{2} \, \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.31, size = 34, normalized size = 1.13 \[ -\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )+\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-\mathrm {i}\right )\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.32, size = 22, normalized size = 0.73 \[ \frac {x}{2} - \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} + \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________