\(\int \sin (\sqrt {x}) \, dx\) [192]

Optimal result

Integrand size = 6, antiderivative size = 22 \[ \int \sin \left (\sqrt {x}\right ) \, dx=-2 \sqrt {x} \cos \left (\sqrt {x}\right )+2 \sin \left (\sqrt {x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3442, 3377, 2717} \[ \int \sin \left (\sqrt {x}\right ) \, dx=2 \sin \left (\sqrt {x}\right )-2 \sqrt {x} \cos \left (\sqrt {x}\right ) \]

[In]

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

[Out]

-2*Sqrt[x]*Cos[Sqrt[x]] + 2*Sin[Sqrt[x]]

Rule 2717

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

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3442

Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :> Dist[1/(n*f), Subst[Int[x
^(1/n - 1)*(a + b*Sin[c + d*x])^p, x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && In
tegerQ[1/n]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \sin \left (\sqrt {x}\right ) \, dx=-2 \sqrt {x} \cos \left (\sqrt {x}\right )+2 \sin \left (\sqrt {x}\right ) \]

[In]

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

[Out]

-2*Sqrt[x]*Cos[Sqrt[x]] + 2*Sin[Sqrt[x]]

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(2 \sin \left (\sqrt {x}\right )-2 \cos \left (\sqrt {x}\right ) \sqrt {x}\)	\(17\)
default	\(2 \sin \left (\sqrt {x}\right )-2 \cos \left (\sqrt {x}\right ) \sqrt {x}\)	\(17\)
meijerg	\(4 \sqrt {\pi }\, \left (-\frac {\sqrt {x}\, \cos \left (\sqrt {x}\right )}{2 \sqrt {\pi }}+\frac {\sin \left (\sqrt {x}\right )}{2 \sqrt {\pi }}\right )\)	\(28\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \sin \left (\sqrt {x}\right ) \, dx=-2 \, \sqrt {x} \cos \left (\sqrt {x}\right ) + 2 \, \sin \left (\sqrt {x}\right ) \]

[In]

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

[Out]

-2*sqrt(x)*cos(sqrt(x)) + 2*sin(sqrt(x))

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \sin \left (\sqrt {x}\right ) \, dx=- 2 \sqrt {x} \cos {\left (\sqrt {x} \right )} + 2 \sin {\left (\sqrt {x} \right )} \]

[In]

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

[Out]

-2*sqrt(x)*cos(sqrt(x)) + 2*sin(sqrt(x))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \sin \left (\sqrt {x}\right ) \, dx=-2 \, \sqrt {x} \cos \left (\sqrt {x}\right ) + 2 \, \sin \left (\sqrt {x}\right ) \]

[In]

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

[Out]

-2*sqrt(x)*cos(sqrt(x)) + 2*sin(sqrt(x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \sin \left (\sqrt {x}\right ) \, dx=-2 \, \sqrt {x} \cos \left (\sqrt {x}\right ) + 2 \, \sin \left (\sqrt {x}\right ) \]

[In]

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

[Out]

-2*sqrt(x)*cos(sqrt(x)) + 2*sin(sqrt(x))

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \sin \left (\sqrt {x}\right ) \, dx=2\,\sin \left (\sqrt {x}\right )-2\,\sqrt {x}\,\cos \left (\sqrt {x}\right ) \]

[In]

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

[Out]

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