Optimal. Leaf size=33 \[ \frac {\sin ^{-1}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}}+\frac {1}{2} \sin (x) \sqrt {\cos (2 x)} \]
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Rubi [A] time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4356, 195, 216} \[ \frac {\sin ^{-1}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}}+\frac {1}{2} \sin (x) \sqrt {\cos (2 x)} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 4356
Rubi steps
\begin {align*} \int \cos (x) \sqrt {\cos (2 x)} \, dx &=\operatorname {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} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^2}} \, dx,x,\sin (x)\right )\\ &=\frac {\sin ^{-1}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}}+\frac {1}{2} \sqrt {\cos (2 x)} \sin (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 0.97 \[ \frac {1}{4} \left (\sqrt {2} \sin ^{-1}\left (\sqrt {2} \sin (x)\right )+2 \sin (x) \sqrt {\cos (2 x)}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos (x) \sqrt {\cos (2 x)} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.32, size = 77, normalized size = 2.33 \[ -\frac {1}{16} \, \sqrt {2} \arctan \left (\frac {{\left (32 \, \sqrt {2} \cos \relax (x)^{4} - 48 \, \sqrt {2} \cos \relax (x)^{2} + 17 \, \sqrt {2}\right )} \sqrt {2 \, \cos \relax (x)^{2} - 1}}{8 \, {\left (8 \, \cos \relax (x)^{4} - 10 \, \cos \relax (x)^{2} + 3\right )} \sin \relax (x)}\right ) + \frac {1}{2} \, \sqrt {2 \, \cos \relax (x)^{2} - 1} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.69, size = 27, normalized size = 0.82 \[ \frac {1}{4} \, \sqrt {2} \arcsin \left (\sqrt {2} \sin \relax (x)\right ) + \frac {1}{2} \, \sqrt {-2 \, \sin \relax (x)^{2} + 1} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 62, normalized size = 1.88
method | result | size |
default | \(-\frac {\sqrt {\left (2 \left (\cos ^{2}\relax (x )\right )-1\right ) \left (\sin ^{2}\relax (x )\right )}\, \left (-\sqrt {2}\, \arcsin \left (4 \left (\sin ^{2}\relax (x )\right )-1\right )-4 \sqrt {-2 \left (\sin ^{4}\relax (x )\right )+\sin ^{2}\relax (x )}\right )}{8 \sin \relax (x ) \sqrt {2 \left (\cos ^{2}\relax (x )\right )-1}}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.15, size = 488, normalized size = 14.79 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \sqrt {\cos \left (2\,x\right )}\,\cos \relax (x) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\relax (x )} \sqrt {\cos {\left (2 x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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