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]

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

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Rubi [A]  time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, 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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 2637

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

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 6742

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

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*}

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Mathematica [A]  time = 0.06, size = 30, normalized size = 1.00 \[ \frac {1}{6} x \left (2 x^2+3\right )+2 \sin (x)-\frac {1}{4} \sin (2 x)-2 x \cos (x) \]

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

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fricas [A]  time = 0.43, size = 22, normalized size = 0.73 \[ \frac {1}{3} \, x^{3} - 2 \, x \cos \relax (x) - \frac {1}{2} \, {\left (\cos \relax (x) - 4\right )} \sin \relax (x) + \frac {1}{2} \, x \]

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

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giac [A]  time = 0.94, size = 24, normalized size = 0.80 \[ \frac {1}{3} \, x^{3} - 2 \, x \cos \relax (x) + \frac {1}{2} \, x - \frac {1}{4} \, \sin \left (2 \, x\right ) + 2 \, \sin \relax (x) \]

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)

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maple [A]  time = 0.02, size = 25, normalized size = 0.83 \[ \frac {x^{3}}{3}-2 x \cos \relax (x )-\frac {\cos \relax (x ) \sin \relax (x )}{2}+\frac {x}{2}+2 \sin \relax (x ) \]

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)

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maxima [A]  time = 0.58, size = 24, normalized size = 0.80 \[ \frac {1}{3} \, x^{3} - 2 \, x \cos \relax (x) + \frac {1}{2} \, x - \frac {1}{4} \, \sin \left (2 \, x\right ) + 2 \, \sin \relax (x) \]

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)

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mupad [B]  time = 0.31, size = 24, normalized size = 0.80 \[ \frac {x}{2}+2\,\sin \relax (x)-\frac {\cos \relax (x)\,\sin \relax (x)}{2}-2\,x\,\cos \relax (x)+\frac {x^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + sin(x))^2,x)

[Out]

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

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sympy [A]  time = 0.22, size = 41, normalized size = 1.37 \[ \frac {x^{3}}{3} + \frac {x \sin ^{2}{\relax (x )}}{2} + \frac {x \cos ^{2}{\relax (x )}}{2} - 2 x \cos {\relax (x )} - \frac {\sin {\relax (x )} \cos {\relax (x )}}{2} + 2 \sin {\relax (x )} \]

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)

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