Optimal. Leaf size=19 \[ \sqrt{x}+\sin \left (\sqrt{x}\right ) \cos \left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.019536, antiderivative size = 19, normalized size of antiderivative = 1., 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 3380
Rule 2635
Rule 8
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*}
Mathematica [A] time = 0.026684, 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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Maple [A] time = 0.009, size = 14, normalized size = 0.7 \begin{align*} \cos \left ( \sqrt{x} \right ) \sin \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 = 0.92742, size = 16, normalized size = 0.84 \begin{align*} \sqrt{x} + \frac{1}{2} \, \sin \left (2 \, \sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87875, size = 50, normalized size = 2.63 \begin{align*} \cos \left (\sqrt{x}\right ) \sin \left (\sqrt{x}\right ) + \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.320803, size = 39, normalized size = 2.05 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04795, size = 16, normalized size = 0.84 \begin{align*} \sqrt{x} + \frac{1}{2} \, \sin \left (2 \, \sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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