Optimal. Leaf size=12 \[ \frac {2 \sin (x)}{\sqrt {\cos (x)+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2646} \[ \frac {2 \sin (x)}{\sqrt {\cos (x)+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2646
Rubi steps
\begin {align*} \int \sqrt {1+\cos (x)} \, dx &=\frac {2 \sin (x)}{\sqrt {1+\cos (x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 16, normalized size = 1.33 \[ 2 \sqrt {\cos (x)+1} \tan \left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 10, normalized size = 0.83 \[ \frac {2 \, \sin \relax (x)}{\sqrt {\cos \relax (x) + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.04, size = 14, normalized size = 1.17 \[ 2 \, \sqrt {2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 22, normalized size = 1.83 \[ \frac {4 \sqrt {2}\, \cos \left (\frac {x}{2}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {2 \cos \relax (x )+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.16, size = 9, normalized size = 0.75 \[ 2 \, \sqrt {2} \sin \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.15, size = 10, normalized size = 0.83 \[ \frac {2\,\sin \relax (x)}{\sqrt {\cos \relax (x)+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cos {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________