Integrand size = 11, antiderivative size = 33 \[ \int \cos (x) \sqrt {\cos (2 x)} \, dx=\frac {\arcsin \left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}}+\frac {1}{2} \sqrt {\cos (2 x)} \sin (x) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4441, 201, 222} \[ \int \cos (x) \sqrt {\cos (2 x)} \, dx=\frac {\arcsin \left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}}+\frac {1}{2} \sin (x) \sqrt {\cos (2 x)} \]
[In]
[Out]
Rule 201
Rule 222
Rule 4441
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \sqrt {1-2 x^2} \, dx,x,\sin (x)\right ) \\ & = \frac {1}{2} \sqrt {\cos (2 x)} \sin (x)+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-2 x^2}} \, dx,x,\sin (x)\right ) \\ & = \frac {\arcsin \left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}}+\frac {1}{2} \sqrt {\cos (2 x)} \sin (x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \cos (x) \sqrt {\cos (2 x)} \, dx=\frac {1}{4} \left (\sqrt {2} \arcsin \left (\sqrt {2} \sin (x)\right )+2 \sqrt {\cos (2 x)} \sin (x)\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(23)=46\).
Time = 0.47 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88
method | result | size |
default | \(-\frac {\sqrt {\left (2 \left (\cos ^{2}\left (x \right )\right )-1\right ) \left (\sin ^{2}\left (x \right )\right )}\, \left (-\sqrt {2}\, \arcsin \left (4 \left (\sin ^{2}\left (x \right )\right )-1\right )-4 \sqrt {-2 \left (\sin ^{4}\left (x \right )\right )+\sin ^{2}\left (x \right )}\right )}{8 \sin \left (x \right ) \sqrt {2 \left (\cos ^{2}\left (x \right )\right )-1}}\) | \(62\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.33 \[ \int \cos (x) \sqrt {\cos (2 x)} \, dx=-\frac {1}{16} \, \sqrt {2} \arctan \left (\frac {{\left (32 \, \sqrt {2} \cos \left (x\right )^{4} - 48 \, \sqrt {2} \cos \left (x\right )^{2} + 17 \, \sqrt {2}\right )} \sqrt {2 \, \cos \left (x\right )^{2} - 1}}{8 \, {\left (8 \, \cos \left (x\right )^{4} - 10 \, \cos \left (x\right )^{2} + 3\right )} \sin \left (x\right )}\right ) + \frac {1}{2} \, \sqrt {2 \, \cos \left (x\right )^{2} - 1} \sin \left (x\right ) \]
[In]
[Out]
\[ \int \cos (x) \sqrt {\cos (2 x)} \, dx=\int \cos {\left (x \right )} \sqrt {\cos {\left (2 x \right )}}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (23) = 46\).
Time = 0.32 (sec) , antiderivative size = 488, normalized size of antiderivative = 14.79 \[ \int \cos (x) \sqrt {\cos (2 x)} \, dx=\text {Too large to display} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \cos (x) \sqrt {\cos (2 x)} \, dx=\frac {1}{4} \, \sqrt {2} \arcsin \left (\sqrt {2} \sin \left (x\right )\right ) + \frac {1}{2} \, \sqrt {-2 \, \sin \left (x\right )^{2} + 1} \sin \left (x\right ) \]
[In]
[Out]
Timed out. \[ \int \cos (x) \sqrt {\cos (2 x)} \, dx=\int \sqrt {\cos \left (2\,x\right )}\,\cos \left (x\right ) \,d x \]
[In]
[Out]