3.833 \(\int 8 \cos ^2(x) \sin ^4(x) \, dx\)

Optimal. Leaf size=34 \[ \frac{x}{2}-\frac{4}{3} \sin ^3(x) \cos ^3(x)-\sin (x) \cos ^3(x)+\frac{1}{2} \sin (x) \cos (x) \]

[Out]

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

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Rubi [A]  time = 0.0502599, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {12, 2568, 2635, 8} \[ \frac{x}{2}-\frac{4}{3} \sin ^3(x) \cos ^3(x)-\sin (x) \cos ^3(x)+\frac{1}{2} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[8*Cos[x]^2*Sin[x]^4,x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

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 8 \cos ^2(x) \sin ^4(x) \, dx &=8 \int \cos ^2(x) \sin ^4(x) \, dx\\ &=-\frac{4}{3} \cos ^3(x) \sin ^3(x)+4 \int \cos ^2(x) \sin ^2(x) \, dx\\ &=-\cos ^3(x) \sin (x)-\frac{4}{3} \cos ^3(x) \sin ^3(x)+\int \cos ^2(x) \, dx\\ &=\frac{1}{2} \cos (x) \sin (x)-\cos ^3(x) \sin (x)-\frac{4}{3} \cos ^3(x) \sin ^3(x)+\frac{\int 1 \, dx}{2}\\ &=\frac{x}{2}+\frac{1}{2} \cos (x) \sin (x)-\cos ^3(x) \sin (x)-\frac{4}{3} \cos ^3(x) \sin ^3(x)\\ \end{align*}

Mathematica [A]  time = 0.0100087, size = 32, normalized size = 0.94 \[ 8 \left (\frac{x}{16}-\frac{1}{64} \sin (2 x)-\frac{1}{64} \sin (4 x)+\frac{1}{192} \sin (6 x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[8*Cos[x]^2*Sin[x]^4,x]

[Out]

8*(x/16 - Sin[2*x]/64 - Sin[4*x]/64 + Sin[6*x]/192)

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Maple [A]  time = 0.005, size = 29, normalized size = 0.9 \begin{align*}{\frac{x}{2}}+{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{2}}- \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) -{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{3} \left ( \sin \left ( x \right ) \right ) ^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(8*cos(x)^2*sin(x)^4,x)

[Out]

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

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Maxima [A]  time = 0.960433, size = 24, normalized size = 0.71 \begin{align*} -\frac{1}{6} \, \sin \left (2 \, x\right )^{3} + \frac{1}{2} \, x - \frac{1}{8} \, \sin \left (4 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8*cos(x)^2*sin(x)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.39833, size = 78, normalized size = 2.29 \begin{align*} \frac{1}{6} \,{\left (8 \, \cos \left (x\right )^{5} - 14 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{1}{2} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8*cos(x)^2*sin(x)^4,x, algorithm="fricas")

[Out]

1/6*(8*cos(x)^5 - 14*cos(x)^3 + 3*cos(x))*sin(x) + 1/2*x

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Sympy [A]  time = 0.061079, size = 32, normalized size = 0.94 \begin{align*} \frac{x}{2} + \frac{4 \sin ^{5}{\left (x \right )} \cos{\left (x \right )}}{3} - \frac{\sin ^{3}{\left (x \right )} \cos{\left (x \right )}}{3} - \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8*cos(x)**2*sin(x)**4,x)

[Out]

x/2 + 4*sin(x)**5*cos(x)/3 - sin(x)**3*cos(x)/3 - sin(x)*cos(x)/2

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Giac [A]  time = 1.08186, size = 30, normalized size = 0.88 \begin{align*} \frac{1}{2} \, x + \frac{1}{24} \, \sin \left (6 \, x\right ) - \frac{1}{8} \, \sin \left (4 \, x\right ) - \frac{1}{8} \, \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8*cos(x)^2*sin(x)^4,x, algorithm="giac")

[Out]

1/2*x + 1/24*sin(6*x) - 1/8*sin(4*x) - 1/8*sin(2*x)