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

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

[Out]

-1/6*cos(x)^6

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 30

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

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])

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*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 1.00 \[ -\frac {1}{6} \cos ^6(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*Cos[x]^6

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fricas [A]  time = 0.42, size = 6, normalized size = 0.75 \[ -\frac {1}{6} \, \cos \relax (x)^{6} \]

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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giac [A]  time = 0.80, size = 6, normalized size = 0.75 \[ -\frac {1}{6} \, \cos \relax (x)^{6} \]

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

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maple [A]  time = 0.00, size = 7, normalized size = 0.88 \[ -\frac {\left (\cos ^{6}\relax (x )\right )}{6} \]

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 = 0.43, size = 6, normalized size = 0.75 \[ -\frac {1}{6} \, \cos \relax (x)^{6} \]

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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mupad [B]  time = 0.03, size = 19, normalized size = 2.38 \[ \frac {{\sin \relax (x)}^6}{6}-\frac {{\sin \relax (x)}^4}{2}+\frac {{\sin \relax (x)}^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)^2/2 - sin(x)^4/2 + sin(x)^6/6

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sympy [A]  time = 0.07, size = 7, normalized size = 0.88 \[ - \frac {\cos ^{6}{\relax (x )}}{6} \]

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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