3.68 \(\int \cos ^5(x) \sin ^5(x) \, dx\)

Optimal. Leaf size=25 \[ \frac {\sin ^{10}(x)}{10}-\frac {\sin ^8(x)}{4}+\frac {\sin ^6(x)}{6} \]

[Out]

1/6*sin(x)^6-1/4*sin(x)^8+1/10*sin(x)^10

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Rubi [A]  time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2564, 266, 43} \[ \frac {\sin ^{10}(x)}{10}-\frac {\sin ^8(x)}{4}+\frac {\sin ^6(x)}{6} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^5*Sin[x]^5,x]

[Out]

Sin[x]^6/6 - Sin[x]^8/4 + Sin[x]^10/10

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rubi steps

\begin {align*} \int \cos ^5(x) \sin ^5(x) \, dx &=\operatorname {Subst}\left (\int x^5 \left (1-x^2\right )^2 \, dx,x,\sin (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int (1-x)^2 x^2 \, dx,x,\sin ^2(x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (x^2-2 x^3+x^4\right ) \, dx,x,\sin ^2(x)\right )\\ &=\frac {\sin ^6(x)}{6}-\frac {\sin ^8(x)}{4}+\frac {\sin ^{10}(x)}{10}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ -\frac {5}{512} \cos (2 x)+\frac {5 \cos (6 x)}{3072}-\frac {\cos (10 x)}{5120} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^5*Sin[x]^5,x]

[Out]

(-5*Cos[2*x])/512 + (5*Cos[6*x])/3072 - Cos[10*x]/5120

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fricas [A]  time = 0.41, size = 19, normalized size = 0.76 \[ -\frac {1}{10} \, \cos \relax (x)^{10} + \frac {1}{4} \, \cos \relax (x)^{8} - \frac {1}{6} \, \cos \relax (x)^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^5*sin(x)^5,x, algorithm="fricas")

[Out]

-1/10*cos(x)^10 + 1/4*cos(x)^8 - 1/6*cos(x)^6

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giac [A]  time = 0.92, size = 19, normalized size = 0.76 \[ -\frac {1}{10} \, \cos \relax (x)^{10} + \frac {1}{4} \, \cos \relax (x)^{8} - \frac {1}{6} \, \cos \relax (x)^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^5*sin(x)^5,x, algorithm="giac")

[Out]

-1/10*cos(x)^10 + 1/4*cos(x)^8 - 1/6*cos(x)^6

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maple [A]  time = 0.01, size = 28, normalized size = 1.12 \[ -\frac {\left (\cos ^{6}\relax (x )\right ) \left (\sin ^{4}\relax (x )\right )}{10}-\frac {\left (\cos ^{6}\relax (x )\right ) \left (\sin ^{2}\relax (x )\right )}{20}-\frac {\left (\cos ^{6}\relax (x )\right )}{60} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^5*sin(x)^5,x)

[Out]

-1/10*cos(x)^6*sin(x)^4-1/20*sin(x)^2*cos(x)^6-1/60*cos(x)^6

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maxima [A]  time = 0.45, size = 19, normalized size = 0.76 \[ \frac {1}{10} \, \sin \relax (x)^{10} - \frac {1}{4} \, \sin \relax (x)^{8} + \frac {1}{6} \, \sin \relax (x)^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^5*sin(x)^5,x, algorithm="maxima")

[Out]

1/10*sin(x)^10 - 1/4*sin(x)^8 + 1/6*sin(x)^6

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mupad [B]  time = 0.05, size = 19, normalized size = 0.76 \[ \frac {{\sin \relax (x)}^{10}}{10}-\frac {{\sin \relax (x)}^8}{4}+\frac {{\sin \relax (x)}^6}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^5*sin(x)^5,x)

[Out]

sin(x)^6/6 - sin(x)^8/4 + sin(x)^10/10

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sympy [A]  time = 0.07, size = 19, normalized size = 0.76 \[ \frac {\sin ^{10}{\relax (x )}}{10} - \frac {\sin ^{8}{\relax (x )}}{4} + \frac {\sin ^{6}{\relax (x )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**5*sin(x)**5,x)

[Out]

sin(x)**10/10 - sin(x)**8/4 + sin(x)**6/6

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