Integrand size = 4, antiderivative size = 11 \[ \int \cos ^3(x) \, dx=\sin (x)-\frac {\sin ^3(x)}{3} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2713} \[ \int \cos ^3(x) \, dx=\sin (x)-\frac {\sin ^3(x)}{3} \]
[In]
[Out]
Rule 2713
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (x)\right ) \\ & = \sin (x)-\frac {\sin ^3(x)}{3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \cos ^3(x) \, dx=\sin (x)-\frac {\sin ^3(x)}{3} \]
[In]
[Out]
Time = 0.41 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {\left (2+\cos ^{2}\left (x \right )\right ) \sin \left (x \right )}{3}\) | \(11\) |
risch | \(\frac {3 \sin \left (x \right )}{4}+\frac {\sin \left (3 x \right )}{12}\) | \(12\) |
parallelrisch | \(\frac {3 \sin \left (x \right )}{4}+\frac {\sin \left (3 x \right )}{12}\) | \(12\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \cos ^3(x) \, dx=\frac {1}{3} \, {\left (\cos \left (x\right )^{2} + 2\right )} \sin \left (x\right ) \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \cos ^3(x) \, dx=- \frac {\sin ^{3}{\left (x \right )}}{3} + \sin {\left (x \right )} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \cos ^3(x) \, dx=-\frac {1}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \cos ^3(x) \, dx=-\frac {1}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \cos ^3(x) \, dx=\sin \left (x\right )-\frac {{\sin \left (x\right )}^3}{3} \]
[In]
[Out]