Optimal. Leaf size=22 \[ 2 \sqrt {x} \sin \left (\sqrt {x}\right )+2 \cos \left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3362, 3296, 2638} \[ 2 \sqrt {x} \sin \left (\sqrt {x}\right )+2 \cos \left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3362
Rubi steps
\begin {align*} \int \cos \left (\sqrt {x}\right ) \, dx &=2 \operatorname {Subst}\left (\int x \cos (x) \, dx,x,\sqrt {x}\right )\\ &=2 \sqrt {x} \sin \left (\sqrt {x}\right )-2 \operatorname {Subst}\left (\int \sin (x) \, dx,x,\sqrt {x}\right )\\ &=2 \cos \left (\sqrt {x}\right )+2 \sqrt {x} \sin \left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ 2 \sqrt {x} \sin \left (\sqrt {x}\right )+2 \cos \left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 16, normalized size = 0.73 \[ 2 \, \sqrt {x} \sin \left (\sqrt {x}\right ) + 2 \, \cos \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 16, normalized size = 0.73 \[ 2 \, \sqrt {x} \sin \left (\sqrt {x}\right ) + 2 \, \cos \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 17, normalized size = 0.77 \[ 2 \cos \left (\sqrt {x}\right )+2 \sin \left (\sqrt {x}\right ) \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 16, normalized size = 0.73 \[ 2 \, \sqrt {x} \sin \left (\sqrt {x}\right ) + 2 \, \cos \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 16, normalized size = 0.73 \[ 2\,\cos \left (\sqrt {x}\right )+2\,\sqrt {x}\,\sin \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 20, normalized size = 0.91 \[ 2 \sqrt {x} \sin {\left (\sqrt {x} \right )} + 2 \cos {\left (\sqrt {x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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