Integrand size = 6, antiderivative size = 8 \[ \int x \cos \left (x^2\right ) \, dx=\frac {\sin \left (x^2\right )}{2} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3461, 2717} \[ \int x \cos \left (x^2\right ) \, dx=\frac {\sin \left (x^2\right )}{2} \]
[In]
[Out]
Rule 2717
Rule 3461
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \cos (x) \, dx,x,x^2\right ) \\ & = \frac {\sin \left (x^2\right )}{2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int x \cos \left (x^2\right ) \, dx=\frac {\sin \left (x^2\right )}{2} \]
[In]
[Out]
Time = 0.62 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\sin \left (x^{2}\right )}{2}\) | \(7\) |
default | \(\frac {\sin \left (x^{2}\right )}{2}\) | \(7\) |
meijerg | \(\frac {\sin \left (x^{2}\right )}{2}\) | \(7\) |
risch | \(\frac {\sin \left (x^{2}\right )}{2}\) | \(7\) |
parallelrisch | \(\frac {\sin \left (x^{2}\right )}{2}\) | \(7\) |
norman | \(\frac {\tan \left (\frac {x^{2}}{2}\right )}{1+\tan \left (\frac {x^{2}}{2}\right )^{2}}\) | \(20\) |
parts | \(\frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, x}{\sqrt {\pi }}\right ) x}{2}-\frac {\pi \left (\frac {\operatorname {FresnelC}\left (\frac {\sqrt {2}\, x}{\sqrt {\pi }}\right ) \sqrt {2}\, x}{\sqrt {\pi }}-\frac {\sin \left (x^{2}\right )}{\pi }\right )}{2}\) | \(50\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int x \cos \left (x^2\right ) \, dx=\frac {1}{2} \, \sin \left (x^{2}\right ) \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.62 \[ \int x \cos \left (x^2\right ) \, dx=\frac {\sin {\left (x^{2} \right )}}{2} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int x \cos \left (x^2\right ) \, dx=\frac {1}{2} \, \sin \left (x^{2}\right ) \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int x \cos \left (x^2\right ) \, dx=\frac {1}{2} \, \sin \left (x^{2}\right ) \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int x \cos \left (x^2\right ) \, dx=\frac {\sin \left (x^2\right )}{2} \]
[In]
[Out]