### 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]

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

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Rubi [A]  time = 0.0299897, antiderivative size = 25, normalized size of antiderivative = 1., 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 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])

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 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])

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*}

Mathematica [A]  time = 0.0102496, size = 25, normalized size = 1. $-\frac{5}{512} \cos (2 x)+\frac{5 \cos (6 x)}{3072}-\frac{\cos (10 x)}{5120}$

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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Maple [A]  time = 0.007, size = 28, normalized size = 1.1 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{6} \left ( \sin \left ( x \right ) \right ) ^{4}}{10}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{2} \left ( \cos \left ( x \right ) \right ) ^{6}}{20}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{6}}{60}} \end{align*}

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 = 0.927547, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{10} \, \sin \left (x\right )^{10} - \frac{1}{4} \, \sin \left (x\right )^{8} + \frac{1}{6} \, \sin \left (x\right )^{6} \end{align*}

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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Fricas [A]  time = 2.00251, size = 63, normalized size = 2.52 \begin{align*} -\frac{1}{10} \, \cos \left (x\right )^{10} + \frac{1}{4} \, \cos \left (x\right )^{8} - \frac{1}{6} \, \cos \left (x\right )^{6} \end{align*}

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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Sympy [A]  time = 0.067664, size = 19, normalized size = 0.76 \begin{align*} \frac{\sin ^{10}{\left (x \right )}}{10} - \frac{\sin ^{8}{\left (x \right )}}{4} + \frac{\sin ^{6}{\left (x \right )}}{6} \end{align*}

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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Giac [A]  time = 1.05139, size = 26, normalized size = 1.04 \begin{align*} -\frac{1}{10} \, \cos \left (x\right )^{10} + \frac{1}{4} \, \cos \left (x\right )^{8} - \frac{1}{6} \, \cos \left (x\right )^{6} \end{align*}

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