### 3.52 $$\int \sin (\sqrt{x}) \, dx$$

Optimal. Leaf size=22 $2 \sin \left (\sqrt{x}\right )-2 \sqrt{x} \cos \left (\sqrt{x}\right )$

[Out]

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

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Rubi [A]  time = 0.0108838, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {3361, 3296, 2637} $2 \sin \left (\sqrt{x}\right )-2 \sqrt{x} \cos \left (\sqrt{x}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 3361

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]

Rule 3296

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 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 \sin \left (\sqrt{x}\right ) \, dx &=2 \operatorname{Subst}\left (\int x \sin (x) \, dx,x,\sqrt{x}\right )\\ &=-2 \sqrt{x} \cos \left (\sqrt{x}\right )+2 \operatorname{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]  time = 0.0122321, size = 22, normalized size = 1. $2 \sin \left (\sqrt{x}\right )-2 \sqrt{x} \cos \left (\sqrt{x}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 17, normalized size = 0.8 \begin{align*} 2\,\sin \left ( \sqrt{x} \right ) -2\,\cos \left ( \sqrt{x} \right ) \sqrt{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.934698, size = 22, normalized size = 1. \begin{align*} -2 \, \sqrt{x} \cos \left (\sqrt{x}\right ) + 2 \, \sin \left (\sqrt{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.25847, size = 57, normalized size = 2.59 \begin{align*} -2 \, \sqrt{x} \cos \left (\sqrt{x}\right ) + 2 \, \sin \left (\sqrt{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.293694, size = 20, normalized size = 0.91 \begin{align*} - 2 \sqrt{x} \cos{\left (\sqrt{x} \right )} + 2 \sin{\left (\sqrt{x} \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.04925, size = 22, normalized size = 1. \begin{align*} -2 \, \sqrt{x} \cos \left (\sqrt{x}\right ) + 2 \, \sin \left (\sqrt{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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