3.793 \(\int (1+\cos (x)) (x+\sin (x))^3 \, dx\)

Optimal. Leaf size=10 \[ \frac{1}{4} (x+\sin (x))^4 \]

[Out]

(x + Sin[x])^4/4

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Rubi [A]  time = 0.0378183, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6686} \[ \frac{1}{4} (x+\sin (x))^4 \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cos[x])*(x + Sin[x])^3,x]

[Out]

(x + Sin[x])^4/4

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int (1+\cos (x)) (x+\sin (x))^3 \, dx &=\frac{1}{4} (x+\sin (x))^4\\ \end{align*}

Mathematica [A]  time = 0.0184468, size = 10, normalized size = 1. \[ \frac{1}{4} (x+\sin (x))^4 \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cos[x])*(x + Sin[x])^3,x]

[Out]

(x + Sin[x])^4/4

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Maple [B]  time = 0.036, size = 65, normalized size = 6.5 \begin{align*} \sin \left ( x \right ){x}^{3}-{\frac{3\, \left ( \cos \left ( x \right ) \right ) ^{2}{x}^{2}}{2}}+3\,x \left ( 1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) -{\frac{3\,{x}^{2}}{2}}+x \left ( \sin \left ( x \right ) \right ) ^{3}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}}{4}}+{\frac{{x}^{4}}{4}}+3\,x \left ( -1/2\,\cos \left ( x \right ) \sin \left ( x \right ) +x/2 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+cos(x))*(x+sin(x))^3,x)

[Out]

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

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Maxima [A]  time = 0.954098, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{4} \,{\left (x + \sin \left (x\right )\right )}^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cos(x))*(x+sin(x))^3,x, algorithm="maxima")

[Out]

1/4*(x + sin(x))^4

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Fricas [B]  time = 2.03644, size = 126, normalized size = 12.6 \begin{align*} \frac{1}{4} \, x^{4} + \frac{1}{4} \, \cos \left (x\right )^{4} - \frac{1}{2} \,{\left (3 \, x^{2} + 1\right )} \cos \left (x\right )^{2} + \frac{3}{2} \, x^{2} +{\left (x^{3} - x \cos \left (x\right )^{2} + x\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cos(x))*(x+sin(x))^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.61257, size = 36, normalized size = 3.6 \begin{align*} \frac{x^{4}}{4} + x^{3} \sin{\left (x \right )} + \frac{3 x^{2} \sin ^{2}{\left (x \right )}}{2} + x \sin ^{3}{\left (x \right )} + \frac{\sin ^{4}{\left (x \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cos(x))*(x+sin(x))**3,x)

[Out]

x**4/4 + x**3*sin(x) + 3*x**2*sin(x)**2/2 + x*sin(x)**3 + sin(x)**4/4

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Giac [B]  time = 1.08073, size = 82, normalized size = 8.2 \begin{align*} \frac{1}{4} \, x^{4} + \frac{3}{4} \, x^{2} - \frac{1}{4} \,{\left (3 \, x^{2} - 1\right )} \cos \left (2 \, x\right ) - \frac{1}{4} \, x \sin \left (3 \, x\right ) + \frac{1}{4} \,{\left (4 \, x^{3} - 21 \, x\right )} \sin \left (x\right ) + 6 \, x \sin \left (x\right ) + \frac{1}{32} \, \cos \left (4 \, x\right ) - \frac{3}{8} \, \cos \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cos(x))*(x+sin(x))^3,x, algorithm="giac")

[Out]

1/4*x^4 + 3/4*x^2 - 1/4*(3*x^2 - 1)*cos(2*x) - 1/4*x*sin(3*x) + 1/4*(4*x^3 - 21*x)*sin(x) + 6*x*sin(x) + 1/32*
cos(4*x) - 3/8*cos(2*x)