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

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Rubi [A]  time = 0.0527617, 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 \[ \frac{x^3}{3}+\frac{x}{2}+2 \sin (x)-2 x \cos (x)-\frac{1}{2} \sin (x) \cos (x) \]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (x + \sin{\left (x \right )}\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x+sin(x))**2,x)

[Out]

Integral((x + sin(x))**2, x)

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Mathematica [A]  time = 0.0629458, 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

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Maple [A]  time = 0.014, size = 25, normalized size = 0.8 \[{\frac{x}{2}}+{\frac{{x}^{3}}{3}}-2\,x\cos \left ( x \right ) +2\,\sin \left ( x \right ) -{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{2}} \]

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 = 1.44905, size = 32, normalized size = 1.07 \[ \frac{1}{3} \, x^{3} - 2 \, x \cos \left (x\right ) + \frac{1}{2} \, x - \frac{1}{4} \, \sin \left (2 \, x\right ) + 2 \, \sin \left (x\right ) \]

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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Fricas [A]  time = 0.220219, size = 30, normalized size = 1. \[ \frac{1}{3} \, x^{3} - 2 \, x \cos \left (x\right ) - \frac{1}{2} \,{\left (\cos \left (x\right ) - 4\right )} \sin \left (x\right ) + \frac{1}{2} \, x \]

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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Sympy [A]  time = 0.235873, size = 41, normalized size = 1.37 \[ \frac{x^{3}}{3} + \frac{x \sin ^{2}{\left (x \right )}}{2} + \frac{x \cos ^{2}{\left (x \right )}}{2} - 2 x \cos{\left (x \right )} - \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{2} + 2 \sin{\left (x \right )} \]

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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GIAC/XCAS [A]  time = 0.211972, size = 32, normalized size = 1.07 \[ \frac{1}{3} \, x^{3} - 2 \, x \cos \left (x\right ) + \frac{1}{2} \, x - \frac{1}{4} \, \sin \left (2 \, x\right ) + 2 \, \sin \left (x\right ) \]

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)