3.34 \(\int \cos ^3(x) \sin (x) \, dx\)

Optimal. Leaf size=8 \[ -\frac {1}{4} \cos ^4(x) \]

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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} \[ -\frac {1}{4} \cos ^4(x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^3*Sin[x],x]

[Out]

-Cos[x]^4/4

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 ^3(x) \sin (x) \, dx &=-\operatorname {Subst}\left (\int x^3 \, dx,x,\cos (x)\right )\\ &=-\frac {1}{4} \cos ^4(x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 1.00 \[ -\frac {1}{4} \cos ^4(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^3*Sin[x],x]

[Out]

-1/4*Cos[x]^4

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^3(x) \sin (x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Cos[x]^3*Sin[x],x]

[Out]

Could not integrate

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fricas [A]  time = 1.33, size = 6, normalized size = 0.75 \[ -\frac {1}{4} \, \cos \relax (x)^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x),x, algorithm="fricas")

[Out]

-1/4*cos(x)^4

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giac [A]  time = 0.94, size = 6, normalized size = 0.75 \[ -\frac {1}{4} \, \cos \relax (x)^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x),x, algorithm="giac")

[Out]

-1/4*cos(x)^4

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maple [A]  time = 0.04, size = 7, normalized size = 0.88

method	result	size
derivativedivides	\(-\frac {\left (\cos ^{4}\relax (x )\right )}{4}\)	\(7\)
default	\(-\frac {\left (\cos ^{4}\relax (x )\right )}{4}\)	\(7\)
risch	\(-\frac {\cos \left (4 x \right )}{32}-\frac {\cos \left (2 x \right )}{8}\)	\(14\)
norman	\(\frac {2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{4}}\)	\(29\)
meijerg	\(\frac {\sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (2 x \right )}{\sqrt {\pi }}\right )}{8}+\frac {\sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (4 x \right )}{\sqrt {\pi }}\right )}{32}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3*sin(x),x,method=_RETURNVERBOSE)

[Out]

-1/4*cos(x)^4

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maxima [A]  time = 0.43, size = 6, normalized size = 0.75 \[ -\frac {1}{4} \, \cos \relax (x)^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x),x, algorithm="maxima")

[Out]

-1/4*cos(x)^4

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mupad [B]  time = 0.03, size = 12, normalized size = 1.50 \[ -\frac {{\sin \relax (x)}^2\,\left ({\sin \relax (x)}^2-2\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3*sin(x),x)

[Out]

-(sin(x)^2*(sin(x)^2 - 2))/4

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sympy [A]  time = 0.06, size = 7, normalized size = 0.88 \[ - \frac {\cos ^{4}{\relax (x )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**3*sin(x),x)

[Out]

-cos(x)**4/4

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