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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 2638

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

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*}

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Mathematica [A]  time = 0.01, size = 8, normalized size = 1.00 \[ -2 \cos \left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Cos[Sqrt[x]]

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fricas [A]  time = 0.70, size = 6, normalized size = 0.75 \[ -2 \, \cos \left (\sqrt {x}\right ) \]

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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giac [A]  time = 0.51, size = 6, normalized size = 0.75 \[ -2 \, \cos \left (\sqrt {x}\right ) \]

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))

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maple [A]  time = 0.01, size = 7, normalized size = 0.88 \[ -2 \cos \left (\sqrt {x}\right ) \]

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 = 0.31, size = 6, normalized size = 0.75 \[ -2 \, \cos \left (\sqrt {x}\right ) \]

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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mupad [B]  time = 4.57, size = 6, normalized size = 0.75 \[ -2\,\cos \left (\sqrt {x}\right ) \]

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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sympy [A]  time = 0.32, size = 8, normalized size = 1.00 \[ - 2 \cos {\left (\sqrt {x} \right )} \]

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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