3.315 \(\int (x+\sin (x))^2 \, dx\)

Optimal. Leaf size=30 \[ \frac{x^3}{3}+\frac{x}{2}+2 \sin (x)-2 x \cos (x)-\frac{1}{2} \sin (x) \cos (x) \]

[Out]

x/2 + x^3/3 - 2*x*Cos[x] + 2*Sin[x] - (Cos[x]*Sin[x])/2

________________________________________________________________________________________

Rubi [A]  time = 0.0340623, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6742, 3296, 2637, 2635, 8} \[ \frac{x^3}{3}+\frac{x}{2}+2 \sin (x)-2 x \cos (x)-\frac{1}{2} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[(x + Sin[x])^2,x]

[Out]

x/2 + x^3/3 - 2*x*Cos[x] + 2*Sin[x] - (Cos[x]*Sin[x])/2

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0578726, size = 30, normalized size = 1. \[ \frac{1}{6} x \left (2 x^2+3\right )+2 \sin (x)-\frac{1}{4} \sin (2 x)-2 x \cos (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(x + Sin[x])^2,x]

[Out]

(x*(3 + 2*x^2))/6 - 2*x*Cos[x] + 2*Sin[x] - Sin[2*x]/4

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 25, normalized size = 0.8 \begin{align*}{\frac{x}{2}}+{\frac{{x}^{3}}{3}}-2\,x\cos \left ( x \right ) +2\,\sin \left ( x \right ) -{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+sin(x))^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.935999, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{3} \, x^{3} - 2 \, x \cos \left (x\right ) + \frac{1}{2} \, x - \frac{1}{4} \, \sin \left (2 \, x\right ) + 2 \, \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.17985, size = 76, normalized size = 2.53 \begin{align*} \frac{1}{3} \, x^{3} - 2 \, x \cos \left (x\right ) - \frac{1}{2} \,{\left (\cos \left (x\right ) - 4\right )} \sin \left (x\right ) + \frac{1}{2} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.192206, size = 41, normalized size = 1.37 \begin{align*} \frac{x^{3}}{3} + \frac{x \sin ^{2}{\left (x \right )}}{2} + \frac{x \cos ^{2}{\left (x \right )}}{2} - 2 x \cos{\left (x \right )} - \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{2} + 2 \sin{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3/3 + x*sin(x)**2/2 + x*cos(x)**2/2 - 2*x*cos(x) - sin(x)*cos(x)/2 + 2*sin(x)

________________________________________________________________________________________

Giac [A]  time = 1.05383, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{3} \, x^{3} - 2 \, x \cos \left (x\right ) + \frac{1}{2} \, x - \frac{1}{4} \, \sin \left (2 \, x\right ) + 2 \, \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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