Optimal. Leaf size=19 \[ \sqrt {x}+\sin \left (\sqrt {x}\right ) \cos \left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3380, 2635, 8} \[ \sqrt {x}+\sin \left (\sqrt {x}\right ) \cos \left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3380
Rubi steps
\begin {align*} \int \frac {\cos ^2\left (\sqrt {x}\right )}{\sqrt {x}} \, dx &=2 \operatorname {Subst}\left (\int \cos ^2(x) \, dx,x,\sqrt {x}\right )\\ &=\cos \left (\sqrt {x}\right ) \sin \left (\sqrt {x}\right )+\operatorname {Subst}\left (\int 1 \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x}+\cos \left (\sqrt {x}\right ) \sin \left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 18, normalized size = 0.95 \[ \sqrt {x}+\frac {1}{2} \sin \left (2 \sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 13, normalized size = 0.68 \[ \cos \left (\sqrt {x}\right ) \sin \left (\sqrt {x}\right ) + \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 12, normalized size = 0.63 \[ \sqrt {x} + \frac {1}{2} \, \sin \left (2 \, \sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 14, normalized size = 0.74 \[ \cos \left (\sqrt {x}\right ) \sin \left (\sqrt {x}\right )+\sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 12, normalized size = 0.63 \[ \sqrt {x} + \frac {1}{2} \, \sin \left (2 \, \sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 12, normalized size = 0.63 \[ \frac {\sin \left (2\,\sqrt {x}\right )}{2}+\sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.26, size = 39, normalized size = 2.05 \[ \sqrt {x} \sin ^{2}{\left (\sqrt {x} \right )} + \sqrt {x} \cos ^{2}{\left (\sqrt {x} \right )} + \sin {\left (\sqrt {x} \right )} \cos {\left (\sqrt {x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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