### 3.110 $$\int \cos ^5(x) \sin (x) \, dx$$

Optimal. Leaf size=8 $-\frac{1}{6} \cos ^6(x)$

[Out]

-Cos[x]^6/6

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Rubi [A]  time = 0.0125415, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {2565, 30} $-\frac{1}{6} \cos ^6(x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Cos[x]^6/6

Rule 2565

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0012081, size = 8, normalized size = 1. $-\frac{1}{6} \cos ^6(x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Cos[x]^6/6

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Maple [A]  time = 0.001, size = 7, normalized size = 0.9 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{6}}{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*cos(x)^6

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Maxima [A]  time = 0.927721, size = 8, normalized size = 1. \begin{align*} -\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),x, algorithm="maxima")

[Out]

-1/6*cos(x)^6

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Fricas [A]  time = 1.95143, size = 20, normalized size = 2.5 \begin{align*} -\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),x, algorithm="fricas")

[Out]

-1/6*cos(x)^6

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Sympy [A]  time = 0.055917, size = 7, normalized size = 0.88 \begin{align*} - \frac{\cos ^{6}{\left (x \right )}}{6} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)**6/6

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Giac [A]  time = 1.05263, size = 8, normalized size = 1. \begin{align*} -\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),x, algorithm="giac")

[Out]

-1/6*cos(x)^6