3.1.76 \(\int \frac {\cos ^2(\sqrt {x})}{\sqrt {x}} \, dx\) [76]

Optimal. Leaf size=19 \[ \sqrt {x}+\cos \left (\sqrt {x}\right ) \sin \left (\sqrt {x}\right ) \]

[Out]

cos(x^(1/2))*sin(x^(1/2))+x^(1/2)

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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.

[In]

Int[Cos[Sqrt[x]]^2/Sqrt[x],x]

[Out]

Sqrt[x] + Cos[Sqrt[x]]*Sin[Sqrt[x]]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 3461

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

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.

[In]

Integrate[Cos[Sqrt[x]]^2/Sqrt[x],x]

[Out]

Sqrt[x] + Sin[2*Sqrt[x]]/2

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Mathics [A]
time = 1.99, size = 12, normalized size = 0.63 \begin {gather*} \sqrt {x}+\frac {\text {Sin}\left [2 \sqrt {x}\right ]}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Cos[Sqrt[x]]^2/Sqrt[x],x]')

[Out]

Sqrt[x] + Sin[2 Sqrt[x]] / 2

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Maple [A]
time = 0.03, 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.

[In]

int(cos(x^(1/2))^2/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

cos(x^(1/2))*sin(x^(1/2))+x^(1/2)

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Maxima [A]
time = 0.26, 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.

[In]

integrate(cos(x^(1/2))^2/x^(1/2),x, algorithm="maxima")

[Out]

sqrt(x) + 1/2*sin(2*sqrt(x))

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Fricas [A]
time = 0.34, 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.

[In]

integrate(cos(x^(1/2))^2/x^(1/2),x, algorithm="fricas")

[Out]

cos(sqrt(x))*sin(sqrt(x)) + sqrt(x)

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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 = 0.11, 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.

[In]

integrate(cos(x**(1/2))**2/x**(1/2),x)

[Out]

sqrt(x)*sin(sqrt(x))**2 + sqrt(x)*cos(sqrt(x))**2 + sin(sqrt(x))*cos(sqrt(x))

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Giac [A]
time = 0.00, size = 22, normalized size = 1.16 \begin {gather*} 2 \left (\frac {\sqrt {x}}{2}+\frac {\sin \left (2 \sqrt {x}\right )}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x^(1/2))^2/x^(1/2),x)

[Out]

sqrt(x) + 1/2*sin(2*sqrt(x))

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Mupad [B]
time = 0.27, 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.

[In]

int(cos(x^(1/2))^2/x^(1/2),x)

[Out]

sin(2*x^(1/2))/2 + x^(1/2)

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