3.2.10 \(\int \cos ^5(x) \sin (x) \, dx\) [110]

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 = {2645, 30} \begin {gather*} -\frac {1}{6} \cos ^6(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*Cos[x]^6

Rule 30

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

Rule 2645

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 &=-\text {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 \begin {gather*} -\frac {1}{6} \cos ^6(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*Cos[x]^6

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Maple [A]
time = 0.03, size = 7, normalized size = 0.88

method	result	size
derivativedivides	\(-\frac {\left (\cos ^{6}\left (x \right )\right )}{6}\)	\(7\)
default	\(-\frac {\left (\cos ^{6}\left (x \right )\right )}{6}\)	\(7\)
risch	\(-\frac {\cos \left (6 x \right )}{192}-\frac {\cos \left (4 x \right )}{32}-\frac {5 \cos \left (2 x \right )}{64}\)	\(20\)
norman	\(\frac {-5 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )-5 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )-\frac {\left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{3}-\frac {1}{3}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{6}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^5*sin(x),x,method=_RETURNVERBOSE)

[Out]

-1/6*cos(x)^6

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Maxima [A]
time = 0.94, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{6} \, \cos \left (x\right )^{6} \end {gather*}

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 = 1.87, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{6} \, \cos \left (x\right )^{6} \end {gather*}

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 = 0.01, size = 7, normalized size = 0.88 \begin {gather*} - \frac {\cos ^{6}{\left (x \right )}}{6} \end {gather*}

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 = 0.69, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{6} \, \cos \left (x\right )^{6} \end {gather*}

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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Mupad [B]
time = 0.03, size = 19, normalized size = 2.38 \begin {gather*} \frac {{\sin \left (x\right )}^6}{6}-\frac {{\sin \left (x\right )}^4}{2}+\frac {{\sin \left (x\right )}^2}{2} \end {gather*}

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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