3.4.49 \(\int \cos ^4(x) \sin ^4(x) \, dx\) [349]

Optimal. Leaf size=46 \[ \frac {3 x}{128}+\frac {3}{128} \cos (x) \sin (x)+\frac {1}{64} \cos ^3(x) \sin (x)-\frac {1}{16} \cos ^5(x) \sin (x)-\frac {1}{8} \cos ^5(x) \sin ^3(x) \]

[Out]

3/128*x+3/128*cos(x)*sin(x)+1/64*cos(x)^3*sin(x)-1/16*cos(x)^5*sin(x)-1/8*cos(x)^5*sin(x)^3

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Rubi [A]
time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2648, 2715, 8} \begin {gather*} \frac {3 x}{128}-\frac {1}{8} \sin ^3(x) \cos ^5(x)-\frac {1}{16} \sin (x) \cos ^5(x)+\frac {1}{64} \sin (x) \cos ^3(x)+\frac {3}{128} \sin (x) \cos (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x)/128 + (3*Cos[x]*Sin[x])/128 + (Cos[x]^3*Sin[x])/64 - (Cos[x]^5*Sin[x])/16 - (Cos[x]^5*Sin[x]^3)/8

Rule 8

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

Rule 2648

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 2715

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] && Integ
erQ[2*n]

Rubi steps

\begin {align*} \int \cos ^4(x) \sin ^4(x) \, dx &=-\frac {1}{8} \cos ^5(x) \sin ^3(x)+\frac {3}{8} \int \cos ^4(x) \sin ^2(x) \, dx\\ &=-\frac {1}{16} \cos ^5(x) \sin (x)-\frac {1}{8} \cos ^5(x) \sin ^3(x)+\frac {1}{16} \int \cos ^4(x) \, dx\\ &=\frac {1}{64} \cos ^3(x) \sin (x)-\frac {1}{16} \cos ^5(x) \sin (x)-\frac {1}{8} \cos ^5(x) \sin ^3(x)+\frac {3}{64} \int \cos ^2(x) \, dx\\ &=\frac {3}{128} \cos (x) \sin (x)+\frac {1}{64} \cos ^3(x) \sin (x)-\frac {1}{16} \cos ^5(x) \sin (x)-\frac {1}{8} \cos ^5(x) \sin ^3(x)+\frac {3 \int 1 \, dx}{128}\\ &=\frac {3 x}{128}+\frac {3}{128} \cos (x) \sin (x)+\frac {1}{64} \cos ^3(x) \sin (x)-\frac {1}{16} \cos ^5(x) \sin (x)-\frac {1}{8} \cos ^5(x) \sin ^3(x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.48 \begin {gather*} \frac {3 x}{128}-\frac {1}{128} \sin (4 x)+\frac {\sin (8 x)}{1024} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x)/128 - Sin[4*x]/128 + Sin[8*x]/1024

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Maple [A]
time = 0.08, size = 36, normalized size = 0.78

method	result	size
risch	\(\frac {3 x}{128}+\frac {\sin \left (8 x \right )}{1024}-\frac {\sin \left (4 x \right )}{128}\)	\(17\)
default	\(-\frac {\left (\cos ^{5}\left (x \right )\right ) \left (\sin ^{3}\left (x \right )\right )}{8}-\frac {\left (\cos ^{5}\left (x \right )\right ) \sin \left (x \right )}{16}+\frac {\left (\cos ^{3}\left (x \right )+\frac {3 \cos \left (x \right )}{2}\right ) \sin \left (x \right )}{64}+\frac {3 x}{128}\)	\(36\)
norman	\(\frac {\frac {3 x}{128}-\frac {3 \tan \left (\frac {x}{2}\right )}{64}+\frac {21 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{32}+\frac {3 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{16}-\frac {23 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{64}+\frac {3 x \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{16}+\frac {3 x \left (\tan ^{16}\left (\frac {x}{2}\right )\right )}{128}+\frac {21 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{16}+\frac {105 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{64}+\frac {333 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{64}-\frac {671 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{64}+\frac {671 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{64}-\frac {333 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{64}+\frac {21 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{16}+\frac {21 x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{32}+\frac {23 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{64}+\frac {3 \left (\tan ^{15}\left (\frac {x}{2}\right )\right )}{64}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{8}}\)	\(150\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^4*sin(x)^4,x,method=_RETURNVERBOSE)

[Out]

-1/8*cos(x)^5*sin(x)^3-1/16*cos(x)^5*sin(x)+1/64*(cos(x)^3+3/2*cos(x))*sin(x)+3/128*x

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Maxima [A]
time = 1.81, size = 16, normalized size = 0.35 \begin {gather*} \frac {3}{128} \, x + \frac {1}{1024} \, \sin \left (8 \, x\right ) - \frac {1}{128} \, \sin \left (4 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/128*x + 1/1024*sin(8*x) - 1/128*sin(4*x)

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Fricas [A]
time = 0.91, size = 31, normalized size = 0.67 \begin {gather*} \frac {1}{128} \, {\left (16 \, \cos \left (x\right )^{7} - 24 \, \cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {3}{128} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(16*cos(x)^7 - 24*cos(x)^5 + 2*cos(x)^3 + 3*cos(x))*sin(x) + 3/128*x

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Sympy [A]
time = 0.01, size = 31, normalized size = 0.67 \begin {gather*} \frac {3 x}{128} - \frac {\sin ^{3}{\left (2 x \right )} \cos {\left (2 x \right )}}{128} - \frac {3 \sin {\left (2 x \right )} \cos {\left (2 x \right )}}{256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x/128 - sin(2*x)**3*cos(2*x)/128 - 3*sin(2*x)*cos(2*x)/256

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Giac [A]
time = 0.87, size = 16, normalized size = 0.35 \begin {gather*} \frac {3}{128} \, x + \frac {1}{1024} \, \sin \left (8 \, x\right ) - \frac {1}{128} \, \sin \left (4 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/128*x + 1/1024*sin(8*x) - 1/128*sin(4*x)

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Mupad [B]
time = 0.04, size = 32, normalized size = 0.70 \begin {gather*} \left (\frac {{\cos \left (x\right )}^3}{8}+\frac {\cos \left (x\right )}{16}\right )\,{\sin \left (x\right )}^5+\frac {3\,x}{128}-\frac {\sin \left (2\,x\right )}{64}+\frac {\sin \left (4\,x\right )}{512} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*x)/128 - sin(2*x)/64 + sin(4*x)/512 + sin(x)^5*(cos(x)/16 + cos(x)^3/8)

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