### 3.76 $$\int \frac{\cos ^2(\sqrt{x})}{\sqrt{x}} \, dx$$

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

[Out]

Sqrt[x] + Cos[Sqrt[x]]*Sin[Sqrt[x]]

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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.

[In]

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

[Out]

Sqrt[x] + Cos[Sqrt[x]]*Sin[Sqrt[x]]

Rule 3380

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]))

Rule 2635

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] && Integer
Q[2*n]

Rule 8

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

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.

[In]

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

[Out]

Sqrt[x] + Sin[2*Sqrt[x]]/2

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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.

[In]

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

[Out]

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

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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.

[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 = 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.

[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]  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.

[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 = 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.

[In]

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

[Out]

sqrt(x) + 1/2*sin(2*sqrt(x))