Optimal. Leaf size=14 \[ \frac {x}{2}+\frac {1}{2} \sin (x) \cos (x) \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2635, 8} \[ \frac {x}{2}+\frac {1}{2} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2635
Rubi steps
\begin {align*} \int \cos ^2(x) \, dx &=\frac {1}{2} \cos (x) \sin (x)+\frac {\int 1 \, dx}{2}\\ &=\frac {x}{2}+\frac {1}{2} \cos (x) \sin (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \[ \frac {x}{2}+\frac {1}{4} \sin (2 x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^2(x) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.14, size = 10, normalized size = 0.71 \[ \frac {1}{2} \, \cos \relax (x) \sin \relax (x) + \frac {1}{2} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.83, size = 10, normalized size = 0.71 \[ \frac {1}{2} \, x + \frac {1}{4} \, \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 11, normalized size = 0.79
method | result | size |
default | \(\frac {x}{2}+\frac {\cos \relax (x ) \sin \relax (x )}{2}\) | \(11\) |
risch | \(\frac {x}{2}+\frac {\sin \left (2 x \right )}{4}\) | \(11\) |
norman | \(\frac {x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\frac {x}{2}-\left (\tan ^{3}\left (\frac {x}{2}\right )\right )+\frac {x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{2}+\tan \left (\frac {x}{2}\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2}}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 10, normalized size = 0.71 \[ \frac {1}{2} \, x + \frac {1}{4} \, \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 10, normalized size = 0.71 \[ \frac {x}{2}+\frac {\sin \left (2\,x\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.06, size = 10, normalized size = 0.71 \[ \frac {x}{2} + \frac {\sin {\relax (x )} \cos {\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________