Optimal. Leaf size=22 \[ 2 \cos \left (\sqrt {x}\right )+2 \sqrt {x} \sin \left (\sqrt {x}\right ) \]
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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 = {3443, 3377,
2718} \begin {gather*} 2 \sqrt {x} \sin \left (\sqrt {x}\right )+2 \cos \left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 3443
Rubi steps
\begin {align*} \int \cos \left (\sqrt {x}\right ) \, dx &=2 \text {Subst}\left (\int x \cos (x) \, dx,x,\sqrt {x}\right )\\ &=2 \sqrt {x} \sin \left (\sqrt {x}\right )-2 \text {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 \begin {gather*} 2 \cos \left (\sqrt {x}\right )+2 \sqrt {x} \sin \left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 17, normalized size = 0.77
method | result | size |
derivativedivides | \(2 \cos \left (\sqrt {x}\right )+2 \sin \left (\sqrt {x}\right ) \sqrt {x}\) | \(17\) |
default | \(2 \cos \left (\sqrt {x}\right )+2 \sin \left (\sqrt {x}\right ) \sqrt {x}\) | \(17\) |
meijerg | \(4 \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (\sqrt {x}\right )}{2 \sqrt {\pi }}+\frac {\sqrt {x}\, \sin \left (\sqrt {x}\right )}{2 \sqrt {\pi }}\right )\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.70, size = 16, normalized size = 0.73 \begin {gather*} 2 \, \sqrt {x} \sin \left (\sqrt {x}\right ) + 2 \, \cos \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.62, size = 16, normalized size = 0.73 \begin {gather*} 2 \, \sqrt {x} \sin \left (\sqrt {x}\right ) + 2 \, \cos \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 20, normalized size = 0.91 \begin {gather*} 2 \sqrt {x} \sin {\left (\sqrt {x} \right )} + 2 \cos {\left (\sqrt {x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 16, normalized size = 0.73 \begin {gather*} 2 \, \sqrt {x} \sin \left (\sqrt {x}\right ) + 2 \, \cos \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 16, normalized size = 0.73 \begin {gather*} 2\,\cos \left (\sqrt {x}\right )+2\,\sqrt {x}\,\sin \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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