3.46 \(\int \frac{\cos (\sqrt{x})}{\sqrt{x}} \, dx\)

Optimal. Leaf size=8 \[ 2 \sin \left (\sqrt{x}\right ) \]

[Out]

2*Sin[Sqrt[x]]

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Rubi [A]  time = 0.0088485, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3380, 2637} \[ 2 \sin \left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\cos \left (\sqrt{x}\right )}{\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \cos (x) \, dx,x,\sqrt{x}\right )\\ &=2 \sin \left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0019517, size = 8, normalized size = 1. \[ 2 \sin \left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sin[Sqrt[x]]

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Maple [A]  time = 0.023, size = 7, normalized size = 0.9 \begin{align*} 2\,\sin \left ( \sqrt{x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14184, size = 8, normalized size = 1. \begin{align*} 2 \, \sin \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sin(sqrt(x))

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Fricas [A]  time = 1.62154, size = 22, normalized size = 2.75 \begin{align*} 2 \, \sin \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sin(sqrt(x))

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Sympy [A]  time = 0.419689, size = 7, normalized size = 0.88 \begin{align*} 2 \sin{\left (\sqrt{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sin(sqrt(x))

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Giac [A]  time = 1.14308, size = 8, normalized size = 1. \begin{align*} 2 \, \sin \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sin(sqrt(x))