Optimal. Leaf size=19 \[ \sqrt {x}+\cos \left (\sqrt {x}\right ) \sin \left (\sqrt {x}\right ) \]
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Rubi [A]
time = 0.01, 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 = {3461, 2715, 8}
\begin {gather*} \sqrt {x}+\sin \left (\sqrt {x}\right ) \cos \left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 3461
Rubi steps
\begin {align*} \int \frac {\cos ^2\left (\sqrt {x}\right )}{\sqrt {x}} \, dx &=2 \text {Subst}\left (\int \cos ^2(x) \, dx,x,\sqrt {x}\right )\\ &=\cos \left (\sqrt {x}\right ) \sin \left (\sqrt {x}\right )+\text {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 = 0.02, size = 18, normalized size = 0.95 \begin {gather*} \sqrt {x}+\frac {1}{2} \sin \left (2 \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 14, normalized size = 0.74
method | result | size |
derivativedivides | \(\cos \left (\sqrt {x}\right ) \sin \left (\sqrt {x}\right )+\sqrt {x}\) | \(14\) |
default | \(\cos \left (\sqrt {x}\right ) \sin \left (\sqrt {x}\right )+\sqrt {x}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 12, normalized size = 0.63 \begin {gather*} \sqrt {x} + \frac {1}{2} \, \sin \left (2 \, \sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 13, normalized size = 0.68 \begin {gather*} \cos \left (\sqrt {x}\right ) \sin \left (\sqrt {x}\right ) + \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs.
\(2 (17) = 34\).
time = 0.08, size = 39, normalized size = 2.05 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 12, normalized size = 0.63 \begin {gather*} \sqrt {x} + \frac {1}{2} \, \sin \left (2 \, \sqrt {x}\right ) \end {gather*}
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 \begin {gather*} \frac {\sin \left (2\,\sqrt {x}\right )}{2}+\sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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