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

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

[Out]

1/4*(x+sin(x))^4

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Rubi [A]  time = 0.04, antiderivative size = 10, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, size = 10, normalized size = 1.00 \[ \frac {1}{4} (x+\sin (x))^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x + Sin[x])^4/4

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fricas [B]  time = 0.62, size = 45, normalized size = 4.50 \[ \frac {1}{4} \, x^{4} + \frac {1}{4} \, \cos \relax (x)^{4} - \frac {1}{2} \, {\left (3 \, x^{2} + 1\right )} \cos \relax (x)^{2} + \frac {3}{2} \, x^{2} + {\left (x^{3} - x \cos \relax (x)^{2} + x\right )} \sin \relax (x) \]

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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giac [B]  time = 0.15, size = 61, normalized size = 6.10 \[ \frac {1}{4} \, x^{4} + \frac {3}{4} \, x^{2} - \frac {1}{4} \, {\left (3 \, x^{2} - 1\right )} \cos \left (2 \, x\right ) - \frac {1}{4} \, x \sin \left (3 \, x\right ) + \frac {1}{4} \, {\left (4 \, x^{3} - 21 \, x\right )} \sin \relax (x) + 6 \, x \sin \relax (x) + \frac {1}{32} \, \cos \left (4 \, x\right ) - \frac {3}{8} \, \cos \left (2 \, x\right ) \]

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)

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maple [B]  time = 0.09, size = 65, normalized size = 6.50 \[ x^{3} \sin \relax (x )-\frac {3 \left (\cos ^{2}\relax (x )\right ) x^{2}}{2}+3 x \left (\frac {\cos \relax (x ) \sin \relax (x )}{2}+\frac {x}{2}\right )-\frac {3 x^{2}}{2}+x \left (\sin ^{3}\relax (x )\right )+\frac {\left (\sin ^{4}\relax (x )\right )}{4}+\frac {x^{4}}{4}+3 x \left (-\frac {\cos \relax (x ) \sin \relax (x )}{2}+\frac {x}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*sin(x)-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 = 0.33, size = 8, normalized size = 0.80 \[ \frac {1}{4} \, {\left (x + \sin \relax (x)\right )}^{4} \]

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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mupad [B]  time = 3.15, size = 8, normalized size = 0.80 \[ \frac {{\left (x+\sin \relax (x)\right )}^4}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(x) + 1)*(x + sin(x))^3,x)

[Out]

(x + sin(x))^4/4

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sympy [B]  time = 0.52, size = 36, normalized size = 3.60 \[ \frac {x^{4}}{4} + x^{3} \sin {\relax (x )} + \frac {3 x^{2} \sin ^{2}{\relax (x )}}{2} + x \sin ^{3}{\relax (x )} + \frac {\sin ^{4}{\relax (x )}}{4} \]

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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