Optimal. Leaf size=9 \[ \frac {\sin (x)}{\cos (x)+1} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2648} \[ \frac {\sin (x)}{\cos (x)+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2648
Rubi steps
\begin {align*} \int \frac {1}{1+\cos (x)} \, dx &=\frac {\sin (x)}{1+\cos (x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 6, normalized size = 0.67 \[ \tan \left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{1+\cos (x)} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 9, normalized size = 1.00 \[ \frac {\sin \relax (x)}{\cos \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.85, size = 30, normalized size = 3.33 \[ -\frac {2 \, \tan \left (\frac {1}{2} \, x\right )}{{\left (x^{2} + 1\right )} {\left (\frac {x^{2} - 1}{x^{2} + 1} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 5, normalized size = 0.56
method | result | size |
default | \(\tan \left (\frac {x}{2}\right )\) | \(5\) |
norman | \(\tan \left (\frac {x}{2}\right )\) | \(5\) |
risch | \(\frac {2 i}{{\mathrm e}^{i x}+1}\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 9, normalized size = 1.00 \[ \frac {\sin \relax (x)}{\cos \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.19, size = 4, normalized size = 0.44 \[ \mathrm {tan}\left (\frac {x}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.20, size = 3, normalized size = 0.33 \[ \tan {\left (\frac {x}{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________