Optimal. Leaf size=22 \[ 2 \sin \left (\sqrt{x}\right )-2 \sqrt{x} \cos \left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0108838, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3361, 3296, 2637} \[ 2 \sin \left (\sqrt{x}\right )-2 \sqrt{x} \cos \left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 3361
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int \sin \left (\sqrt{x}\right ) \, dx &=2 \operatorname{Subst}\left (\int x \sin (x) \, dx,x,\sqrt{x}\right )\\ &=-2 \sqrt{x} \cos \left (\sqrt{x}\right )+2 \operatorname{Subst}\left (\int \cos (x) \, dx,x,\sqrt{x}\right )\\ &=-2 \sqrt{x} \cos \left (\sqrt{x}\right )+2 \sin \left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0122321, size = 22, normalized size = 1. \[ 2 \sin \left (\sqrt{x}\right )-2 \sqrt{x} \cos \left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 17, normalized size = 0.8 \begin{align*} 2\,\sin \left ( \sqrt{x} \right ) -2\,\cos \left ( \sqrt{x} \right ) \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.934698, size = 22, normalized size = 1. \begin{align*} -2 \, \sqrt{x} \cos \left (\sqrt{x}\right ) + 2 \, \sin \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25847, size = 57, normalized size = 2.59 \begin{align*} -2 \, \sqrt{x} \cos \left (\sqrt{x}\right ) + 2 \, \sin \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.293694, size = 20, normalized size = 0.91 \begin{align*} - 2 \sqrt{x} \cos{\left (\sqrt{x} \right )} + 2 \sin{\left (\sqrt{x} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04925, size = 22, normalized size = 1. \begin{align*} -2 \, \sqrt{x} \cos \left (\sqrt{x}\right ) + 2 \, \sin \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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