\(\int \frac {x \cos (\sqrt {3} \sqrt {2+x^2})}{\sqrt {2+x^2}} \, dx\) [49]

Optimal result

Integrand size = 27, antiderivative size = 22 \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {\sin \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {3}} \]

[Out]

1/3*sin(3^(1/2)*(x^2+2)^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6847, 3513, 15, 2717} \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {\sin \left (\sqrt {3} \sqrt {x^2+2}\right )}{\sqrt {3}} \]

[In]

Int[(x*Cos[Sqrt[3]*Sqrt[2 + x^2]])/Sqrt[2 + x^2],x]

[Out]

Sin[Sqrt[3]*Sqrt[2 + x^2]]/Sqrt[3]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 2717

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

Rule 3513

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_.)*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :
> Dist[1/(n*f), Subst[Int[ExpandIntegrand[(a + b*Cos[c + d*x])^p, x^(1/n - 1)*(g - e*(h/f) + h*(x^(1/n)/f))^m,
x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\cos \left (\sqrt {3} \sqrt {2+x}\right )}{\sqrt {2+x}} \, dx,x,x^2\right ) \\ & = \text {Subst}\left (\int \frac {x \cos \left (\sqrt {3} x\right )}{\sqrt {x^2}} \, dx,x,\sqrt {2+x^2}\right ) \\ & = 1 \text {Subst}\left (\int \cos \left (\sqrt {3} x\right ) \, dx,x,\sqrt {2+x^2}\right ) \\ & = \frac {\sin \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {\sin \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {3}} \]

[In]

Integrate[(x*Cos[Sqrt[3]*Sqrt[2 + x^2]])/Sqrt[2 + x^2],x]

[Out]

Sin[Sqrt[3]*Sqrt[2 + x^2]]/Sqrt[3]

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {\sin \left (\sqrt {3}\, \sqrt {x^{2}+2}\right ) \sqrt {3}}{3}\)	\(18\)
default	\(\frac {\sin \left (\sqrt {3}\, \sqrt {x^{2}+2}\right ) \sqrt {3}}{3}\)	\(18\)

[In]

int(x*cos(3^(1/2)*(x^2+2)^(1/2))/(x^2+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*sin(3^(1/2)*(x^2+2)^(1/2))*3^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).

Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {2 \, \sqrt {3} \tan \left (\frac {1}{2} \, \sqrt {3} \sqrt {x^{2} + 2}\right )}{3 \, {\left (\tan \left (\frac {1}{2} \, \sqrt {3} \sqrt {x^{2} + 2}\right )^{2} + 1\right )}} \]

[In]

integrate(x*cos(3^(1/2)*(x^2+2)^(1/2))/(x^2+2)^(1/2),x, algorithm="fricas")

[Out]

2/3*sqrt(3)*tan(1/2*sqrt(3)*sqrt(x^2 + 2))/(tan(1/2*sqrt(3)*sqrt(x^2 + 2))^2 + 1)

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {\sqrt {3} \sin {\left (\sqrt {3} \sqrt {x^{2} + 2} \right )}}{3} \]

[In]

integrate(x*cos(3**(1/2)*(x**2+2)**(1/2))/(x**2+2)**(1/2),x)

[Out]

sqrt(3)*sin(sqrt(3)*sqrt(x**2 + 2))/3

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {1}{3} \, \sqrt {3} \sin \left (\sqrt {3} \sqrt {x^{2} + 2}\right ) \]

[In]

integrate(x*cos(3^(1/2)*(x^2+2)^(1/2))/(x^2+2)^(1/2),x, algorithm="maxima")

[Out]

1/3*sqrt(3)*sin(sqrt(3)*sqrt(x^2 + 2))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {1}{3} \, \sqrt {3} \sin \left (\sqrt {3} \sqrt {x^{2} + 2}\right ) \]

[In]

integrate(x*cos(3^(1/2)*(x^2+2)^(1/2))/(x^2+2)^(1/2),x, algorithm="giac")

[Out]

1/3*sqrt(3)*sin(sqrt(3)*sqrt(x^2 + 2))

Mupad [B] (verification not implemented)

Time = 26.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {\sqrt {3}\,\sin \left (\sqrt {3\,x^2+6}\right )}{3} \]

[In]

int((x*cos(3^(1/2)*(x^2 + 2)^(1/2)))/(x^2 + 2)^(1/2),x)

[Out]

(3^(1/2)*sin((3*x^2 + 6)^(1/2)))/3