3.124 \(\int \frac{\sin (\sqrt{x})}{\sqrt{x}} \, dx\)

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

[Out]

-2*Cos[Sqrt[x]]

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Rubi [A]  time = 0.009238, 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 = {3379, 2638} \[ -2 \cos \left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sin[Sqrt[x]]/Sqrt[x],x]

[Out]

-2*Cos[Sqrt[x]]

Rule 3379

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[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 2638

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

Rubi steps

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

Mathematica [A]  time = 0.0102346, size = 8, normalized size = 1. \[ -2 \cos \left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[Sqrt[x]]/Sqrt[x],x]

[Out]

-2*Cos[Sqrt[x]]

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Maple [A]  time = 0.005, size = 7, normalized size = 0.9 \begin{align*} -2\,\cos \left ( \sqrt{x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(x^(1/2))

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Maxima [A]  time = 0.938496, size = 8, normalized size = 1. \begin{align*} -2 \, \cos \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(sqrt(x))

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Fricas [A]  time = 1.62489, size = 23, normalized size = 2.88 \begin{align*} -2 \, \cos \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(sqrt(x))

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Sympy [A]  time = 0.306216, size = 8, normalized size = 1. \begin{align*} - 2 \cos{\left (\sqrt{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(sqrt(x))

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Giac [A]  time = 1.1002, size = 8, normalized size = 1. \begin{align*} -2 \, \cos \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(sqrt(x))