3.348 \(\int \cos ^2(x) \sin ^2(x) \, dx\)

Optimal. Leaf size=24 \[ \frac {x}{8}-\frac {1}{4} \sin (x) \cos ^3(x)+\frac {1}{8} \sin (x) \cos (x) \]

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Rubi [A]  time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2568, 2635, 8} \[ \frac {x}{8}-\frac {1}{4} \sin (x) \cos ^3(x)+\frac {1}{8} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/8 + (Cos[x]*Sin[x])/8 - (Cos[x]^3*Sin[x])/4

Rule 8

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

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 14, normalized size = 0.58 \[ \frac {x}{8}-\frac {1}{32} \sin (4 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/8 - Sin[4*x]/32

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^2(x) \sin ^2(x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Cos[x]^2*Sin[x]^2,x]

[Out]

Could not integrate

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fricas [A]  time = 0.70, size = 19, normalized size = 0.79 \[ -\frac {1}{8} \, {\left (2 \, \cos \relax (x)^{3} - \cos \relax (x)\right )} \sin \relax (x) + \frac {1}{8} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.00, size = 10, normalized size = 0.42 \[ \frac {1}{8} \, x - \frac {1}{32} \, \sin \left (4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x - 1/32*sin(4*x)

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maple [A]  time = 0.05, size = 11, normalized size = 0.46

method	result	size
risch	\(\frac {x}{8}-\frac {\sin \left (4 x \right )}{32}\)	\(11\)
default	\(\frac {x}{8}+\frac {\cos \relax (x ) \sin \relax (x )}{8}-\frac {\left (\cos ^{3}\relax (x )\right ) \sin \relax (x )}{4}\)	\(19\)
norman	\(\frac {\frac {x}{8}+\frac {7 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{4}-\frac {7 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4}+\frac {\left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{4}+\frac {x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}+\frac {3 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{4}+\frac {x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{2}+\frac {x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{8}-\frac {\tan \left (\frac {x}{2}\right )}{4}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{4}}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^2*sin(x)^2,x,method=_RETURNVERBOSE)

[Out]

1/8*x-1/32*sin(4*x)

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maxima [A]  time = 0.57, size = 10, normalized size = 0.42 \[ \frac {1}{8} \, x - \frac {1}{32} \, \sin \left (4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x - 1/32*sin(4*x)

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mupad [B]  time = 0.04, size = 18, normalized size = 0.75 \[ \frac {\cos \relax (x)\,{\sin \relax (x)}^3}{4}-\frac {\cos \relax (x)\,\sin \relax (x)}{8}+\frac {x}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/8 - (cos(x)*sin(x))/8 + (cos(x)*sin(x)^3)/4

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sympy [A]  time = 0.07, size = 14, normalized size = 0.58 \[ \frac {x}{8} - \frac {\sin {\left (2 x \right )} \cos {\left (2 x \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/8 - sin(2*x)*cos(2*x)/16

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