Optimal. Leaf size=18 \[ \frac {1}{2} x \sin (a)-\frac {1}{4} \cos (a+2 x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4574, 2638} \[ \frac {1}{2} x \sin (a)-\frac {1}{4} \cos (a+2 x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2638
Rule 4574
Rubi steps
\begin {align*} \int \cos (x) \sin (a+x) \, dx &=\int \left (\frac {\sin (a)}{2}+\frac {1}{2} \sin (a+2 x)\right ) \, dx\\ &=\frac {1}{2} x \sin (a)+\frac {1}{2} \int \sin (a+2 x) \, dx\\ &=-\frac {1}{4} \cos (a+2 x)+\frac {1}{2} x \sin (a)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 18, normalized size = 1.00 \[ \frac {1}{4} (2 x \sin (a)-\cos (a+2 x)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 28, normalized size = 1.56 \[ -\frac {1}{2} \, \cos \left (a + x\right )^{2} \cos \relax (a) - \frac {1}{2} \, \cos \left (a + x\right ) \sin \left (a + x\right ) \sin \relax (a) + \frac {1}{2} \, x \sin \relax (a) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.04, size = 14, normalized size = 0.78 \[ \frac {1}{2} \, x \sin \relax (a) - \frac {1}{4} \, \cos \left (a + 2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 15, normalized size = 0.83 \[ \frac {x \sin \relax (a )}{2}-\frac {\cos \left (a +2 x \right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 14, normalized size = 0.78 \[ \frac {1}{2} \, x \sin \relax (a) - \frac {1}{4} \, \cos \left (a + 2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 14, normalized size = 0.78 \[ \frac {x\,\sin \relax (a)}{2}-\frac {\cos \left (a+2\,x\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.54, size = 32, normalized size = 1.78 \[ - \frac {x \sin {\relax (x )} \cos {\left (a + x \right )}}{2} + \frac {x \sin {\left (a + x \right )} \cos {\relax (x )}}{2} + \frac {\sin {\relax (x )} \sin {\left (a + x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________