3.50 \(\int \frac{(-1+2 x) \cos (\sqrt{6+3 (-1+2 x)^2})}{\sqrt{6+3 (-1+2 x)^2}} \, dx\)

Optimal. Leaf size=24 \[ \frac{1}{6} \sin \left (\sqrt{3} \sqrt{(2 x-1)^2+2}\right ) \]

[Out]

Sin[Sqrt[3]*Sqrt[2 + (-1 + 2*x)^2]]/6

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Rubi [A]  time = 0.493079, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {6715, 3432, 15, 2637} \[ \frac{1}{6} \sin \left (\sqrt{3} \sqrt{(2 x-1)^2+2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[((-1 + 2*x)*Cos[Sqrt[6 + 3*(-1 + 2*x)^2]])/Sqrt[6 + 3*(-1 + 2*x)^2],x]

[Out]

Sin[Sqrt[3]*Sqrt[2 + (-1 + 2*x)^2]]/6

Rule 6715

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]

Rule 3432

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 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 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 \frac{(-1+2 x) \cos \left (\sqrt{6+3 (-1+2 x)^2}\right )}{\sqrt{6+3 (-1+2 x)^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x \cos \left (\sqrt{6+3 x^2}\right )}{\sqrt{6+3 x^2}} \, dx,x,-1+2 x\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{\cos \left (\sqrt{6+3 x}\right )}{\sqrt{6+3 x}} \, dx,x,(-1+2 x)^2\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{x \cos (x)}{\sqrt{x^2}} \, dx,x,\sqrt{3} \sqrt{2+(-1+2 x)^2}\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \cos (x) \, dx,x,\sqrt{3} \sqrt{2+(-1+2 x)^2}\right )\\ &=\frac{1}{6} \sin \left (\sqrt{3} \sqrt{2+(-1+2 x)^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.15551, size = 20, normalized size = 0.83 \[ \frac{1}{6} \sin \left (\sqrt{3 (1-2 x)^2+6}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((-1 + 2*x)*Cos[Sqrt[6 + 3*(-1 + 2*x)^2]])/Sqrt[6 + 3*(-1 + 2*x)^2],x]

[Out]

Sin[Sqrt[6 + 3*(1 - 2*x)^2]]/6

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Maple [A]  time = 0.025, size = 16, normalized size = 0.7 \begin{align*}{\frac{1}{6}\sin \left ( \sqrt{12\,{x}^{2}-12\,x+9} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+2*x)*cos((6+3*(-1+2*x)^2)^(1/2))/(6+3*(-1+2*x)^2)^(1/2),x)

[Out]

1/6*sin((12*x^2-12*x+9)^(1/2))

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Maxima [A]  time = 1.13861, size = 22, normalized size = 0.92 \begin{align*} \frac{1}{6} \, \sin \left (\sqrt{3 \,{\left (2 \, x - 1\right )}^{2} + 6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)*cos((6+3*(-1+2*x)^2)^(1/2))/(6+3*(-1+2*x)^2)^(1/2),x, algorithm="maxima")

[Out]

1/6*sin(sqrt(3*(2*x - 1)^2 + 6))

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Fricas [A]  time = 2.1526, size = 46, normalized size = 1.92 \begin{align*} \frac{1}{6} \, \sin \left (\sqrt{12 \, x^{2} - 12 \, x + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)*cos((6+3*(-1+2*x)^2)^(1/2))/(6+3*(-1+2*x)^2)^(1/2),x, algorithm="fricas")

[Out]

1/6*sin(sqrt(12*x^2 - 12*x + 9))

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Sympy [A]  time = 7.19953, size = 15, normalized size = 0.62 \begin{align*} \frac{\sin{\left (\sqrt{3 \left (2 x - 1\right )^{2} + 6} \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)*cos((6+3*(-1+2*x)**2)**(1/2))/(6+3*(-1+2*x)**2)**(1/2),x)

[Out]

sin(sqrt(3*(2*x - 1)**2 + 6))/6

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Giac [A]  time = 1.09801, size = 26, normalized size = 1.08 \begin{align*} \frac{1}{6} \, \sin \left (\sqrt{3} \sqrt{4 \, x^{2} - 4 \, x + 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)*cos((6+3*(-1+2*x)^2)^(1/2))/(6+3*(-1+2*x)^2)^(1/2),x, algorithm="giac")

[Out]

1/6*sin(sqrt(3)*sqrt(4*x^2 - 4*x + 3))