Optimal. Leaf size=36 \[ \frac{x}{2}+\frac{1}{2} \cos ^2\left (\sqrt{x}\right )+\sqrt{x} \sin \left (\sqrt{x}\right ) \cos \left (\sqrt{x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0213152, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3362, 3310, 30} \[ \frac{x}{2}+\frac{1}{2} \cos ^2\left (\sqrt{x}\right )+\sqrt{x} \sin \left (\sqrt{x}\right ) \cos \left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3362
Rule 3310
Rule 30
Rubi steps
\begin{align*} \int \cos ^2\left (\sqrt{x}\right ) \, dx &=2 \operatorname{Subst}\left (\int x \cos ^2(x) \, dx,x,\sqrt{x}\right )\\ &=\frac{1}{2} \cos ^2\left (\sqrt{x}\right )+\sqrt{x} \cos \left (\sqrt{x}\right ) \sin \left (\sqrt{x}\right )+\operatorname{Subst}\left (\int x \, dx,x,\sqrt{x}\right )\\ &=\frac{x}{2}+\frac{1}{2} \cos ^2\left (\sqrt{x}\right )+\sqrt{x} \cos \left (\sqrt{x}\right ) \sin \left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0278696, size = 31, normalized size = 0.86 \[ \frac{1}{4} \left (2 \left (x+\sqrt{x} \sin \left (2 \sqrt{x}\right )\right )+\cos \left (2 \sqrt{x}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.026, size = 34, normalized size = 0.9 \begin{align*} 2\,\sqrt{x} \left ( 1/2\,\cos \left ( \sqrt{x} \right ) \sin \left ( \sqrt{x} \right ) +1/2\,\sqrt{x} \right ) -{\frac{x}{2}}-{\frac{1}{2} \left ( \sin \left ( \sqrt{x} \right ) \right ) ^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.07688, size = 31, normalized size = 0.86 \begin{align*} \frac{1}{2} \, \sqrt{x} \sin \left (2 \, \sqrt{x}\right ) + \frac{1}{2} \, x + \frac{1}{4} \, \cos \left (2 \, \sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.60506, size = 86, normalized size = 2.39 \begin{align*} \sqrt{x} \cos \left (\sqrt{x}\right ) \sin \left (\sqrt{x}\right ) + \frac{1}{2} \, \cos \left (\sqrt{x}\right )^{2} + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.386695, size = 51, normalized size = 1.42 \begin{align*} \sqrt{x} \sin{\left (\sqrt{x} \right )} \cos{\left (\sqrt{x} \right )} + \frac{x \sin ^{2}{\left (\sqrt{x} \right )}}{2} + \frac{x \cos ^{2}{\left (\sqrt{x} \right )}}{2} + \frac{\cos ^{2}{\left (\sqrt{x} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11741, size = 31, normalized size = 0.86 \begin{align*} \frac{1}{2} \, \sqrt{x} \sin \left (2 \, \sqrt{x}\right ) + \frac{1}{2} \, x + \frac{1}{4} \, \cos \left (2 \, \sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]