3.2.24 \(\int \frac {\sin (\sqrt {x})}{\sqrt {x}} \, dx\) [124]

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

[Out]

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

________________________________________________________________________________________

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 = {3460, 2718} \begin {gather*} -2 \cos \left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Cos[Sqrt[x]]

Rule 2718

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

Rule 3460

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

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Cos[Sqrt[x]]

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 7, normalized size = 0.88

method	result	size
derivativedivides	\(-2 \cos \left (\sqrt {x}\right )\)	\(7\)
default	\(-2 \cos \left (\sqrt {x}\right )\)	\(7\)
meijerg	\(2 \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (\sqrt {x}\right )}{\sqrt {\pi }}\right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 6, normalized size = 0.75 \begin {gather*} -2 \, \cos \left (\sqrt {x}\right ) \end {gather*}

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))

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 6, normalized size = 0.75 \begin {gather*} -2 \, \cos \left (\sqrt {x}\right ) \end {gather*}

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))

________________________________________________________________________________________

Sympy [A]
time = 0.10, size = 8, normalized size = 1.00 \begin {gather*} - 2 \cos {\left (\sqrt {x} \right )} \end {gather*}

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))

________________________________________________________________________________________

Giac [A]
time = 4.49, size = 6, normalized size = 0.75 \begin {gather*} -2 \, \cos \left (\sqrt {x}\right ) \end {gather*}

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))

________________________________________________________________________________________

Mupad [B]
time = 4.57, size = 6, normalized size = 0.75 \begin {gather*} -2\,\cos \left (\sqrt {x}\right ) \end {gather*}

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))

________________________________________________________________________________________