Integrand size = 7, antiderivative size = 15 \[ \int \cos (x) \cos (2 x) \, dx=\frac {\sin (x)}{2}+\frac {1}{6} \sin (3 x) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4368} \[ \int \cos (x) \cos (2 x) \, dx=\frac {\sin (x)}{2}+\frac {1}{6} \sin (3 x) \]
[In]
[Out]
Rule 4368
Rubi steps \begin{align*} \text {integral}& = \frac {\sin (x)}{2}+\frac {1}{6} \sin (3 x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (2 x) \, dx=\frac {\sin (x)}{2}+\frac {1}{6} \sin (3 x) \]
[In]
[Out]
Time = 0.37 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {\sin \left (x \right )}{2}+\frac {\sin \left (3 x \right )}{6}\) | \(12\) |
risch | \(\frac {\sin \left (x \right )}{2}+\frac {\sin \left (3 x \right )}{6}\) | \(12\) |
parallelrisch | \(\frac {\sin \left (x \right )}{2}+\frac {\sin \left (3 x \right )}{6}\) | \(12\) |
norman | \(\frac {-\frac {4 \tan \left (x \right ) \tan \left (\frac {x}{2}\right )^{2}}{3}+\frac {2 \tan \left (x \right )^{2} \tan \left (\frac {x}{2}\right )}{3}+\frac {4 \tan \left (x \right )}{3}-\frac {2 \tan \left (\frac {x}{2}\right )}{3}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right ) \left (1+\tan \left (x \right )^{2}\right )}\) | \(51\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \cos (x) \cos (2 x) \, dx=\frac {1}{3} \, {\left (2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int \cos (x) \cos (2 x) \, dx=- \frac {\sin {\left (x \right )} \cos {\left (2 x \right )}}{3} + \frac {2 \sin {\left (2 x \right )} \cos {\left (x \right )}}{3} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \cos (x) \cos (2 x) \, dx=\frac {1}{6} \, \sin \left (3 \, x\right ) + \frac {1}{2} \, \sin \left (x\right ) \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \cos (x) \cos (2 x) \, dx=\frac {1}{6} \, \sin \left (3 \, x\right ) + \frac {1}{2} \, \sin \left (x\right ) \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \cos (x) \cos (2 x) \, dx=\sin \left (x\right )-\frac {2\,{\sin \left (x\right )}^3}{3} \]
[In]
[Out]