3.492 \(\int \frac{2 x+\sin (2 x)}{(\cos (x)+x \sin (x))^2} \, dx\)

Optimal. Leaf size=12 \[ \frac{2}{\frac{\cot (x)}{x}+1} \]

[Out]

2/(1 + Cot[x]/x)

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Rubi [A]  time = 0.0988092, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6711, 32} \[ \frac{2}{\frac{\cot (x)}{x}+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/(1 + Cot[x]/x)

Rule 6711

Int[(u_)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w
, x])]}, Dist[c*p, Subst[Int[(b + a*x^p)^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}
, x] && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.213904, size = 14, normalized size = 1.17 \[ \frac{2 x \sin (x)}{x \sin (x)+\cos (x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sin[x])/(Cos[x] + x*Sin[x])

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Maple [C]  time = 0.711, size = 44, normalized size = 3.7 \begin{align*}{\frac{-2\,i}{x+i}}-{\frac{4\,ix}{ \left ( x+i \right ) \left ( x{{\rm e}^{2\,ix}}-x+i{{\rm e}^{2\,ix}}+i \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x+sin(2*x))/(cos(x)+x*sin(x))^2,x)

[Out]

-2*I/(x+I)-4*I*x/(x+I)/(x*exp(2*I*x)-x+I*exp(2*I*x)+I)

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Maxima [B]  time = 1.35291, size = 105, normalized size = 8.75 \begin{align*} -\frac{2 \,{\left (\cos \left (2 \, x\right )^{2} + 2 \, x \sin \left (2 \, x\right ) + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )}}{{\left (x^{2} + 1\right )} \cos \left (2 \, x\right )^{2} +{\left (x^{2} + 1\right )} \sin \left (2 \, x\right )^{2} + x^{2} - 2 \,{\left (x^{2} - 1\right )} \cos \left (2 \, x\right ) + 4 \, x \sin \left (2 \, x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(cos(2*x)^2 + 2*x*sin(2*x) + sin(2*x)^2 + 2*cos(2*x) + 1)/((x^2 + 1)*cos(2*x)^2 + (x^2 + 1)*sin(2*x)^2 + x^
2 - 2*(x^2 - 1)*cos(2*x) + 4*x*sin(2*x) + 1)

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Fricas [A]  time = 1.95124, size = 42, normalized size = 3.5 \begin{align*} -\frac{2 \, \cos \left (x\right )}{x \sin \left (x\right ) + \cos \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(x)/(x*sin(x) + cos(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 x + \sin{\left (2 x \right )}}{\left (x \sin{\left (x \right )} + \cos{\left (x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x+sin(2*x))/(cos(x)+x*sin(x))**2,x)

[Out]

Integral((2*x + sin(2*x))/(x*sin(x) + cos(x))**2, x)

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Giac [A]  time = 1.06736, size = 14, normalized size = 1.17 \begin{align*} -\frac{2}{x \tan \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(x*tan(x) + 1)