Optimal. Leaf size=20 \[ \frac{x}{4}-\frac{1}{4} \cos \left (2 x+\frac{\pi }{6}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0144671, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4574, 2638} \[ \frac{x}{4}-\frac{1}{4} \cos \left (2 x+\frac{\pi }{6}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4574
Rule 2638
Rubi steps
\begin{align*} \int \cos (x) \sin \left (\frac{\pi }{6}+x\right ) \, dx &=\int \left (\frac{1}{4}+\frac{1}{2} \sin \left (\frac{\pi }{6}+2 x\right )\right ) \, dx\\ &=\frac{x}{4}+\frac{1}{2} \int \sin \left (\frac{\pi }{6}+2 x\right ) \, dx\\ &=\frac{x}{4}-\frac{1}{4} \cos \left (\frac{\pi }{6}+2 x\right )\\ \end{align*}
Mathematica [A] time = 0.0108803, size = 20, normalized size = 1. \[ \frac{x}{4}-\frac{1}{4} \cos \left (2 x+\frac{\pi }{6}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.029, size = 15, normalized size = 0.8 \begin{align*}{\frac{x}{4}}-{\frac{1}{4}\cos \left ({\frac{\pi }{6}}+2\,x \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.932546, size = 19, normalized size = 0.95 \begin{align*} \frac{1}{4} \, x - \frac{1}{4} \, \cos \left (\frac{1}{6} \, \pi + 2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.98278, size = 105, normalized size = 5.25 \begin{align*} -\frac{1}{4} \, \sqrt{3} \cos \left (\frac{1}{6} \, \pi + x\right )^{2} - \frac{1}{4} \, \cos \left (\frac{1}{6} \, \pi + x\right ) \sin \left (\frac{1}{6} \, \pi + x\right ) + \frac{1}{4} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.544632, size = 37, normalized size = 1.85 \begin{align*} - \frac{x \sin{\left (x \right )} \cos{\left (x + \frac{\pi }{6} \right )}}{2} + \frac{x \sin{\left (x + \frac{\pi }{6} \right )} \cos{\left (x \right )}}{2} - \frac{\cos{\left (x \right )} \cos{\left (x + \frac{\pi }{6} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.06048, size = 19, normalized size = 0.95 \begin{align*} \frac{1}{4} \, x - \frac{1}{4} \, \cos \left (\frac{1}{6} \, \pi + 2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]