\(\int \cos ^3(x) \sin (x) \, dx\) [34]

Optimal result

Integrand size = 7, antiderivative size = 8 \[ \int \cos ^3(x) \sin (x) \, dx=-\frac {1}{4} \cos ^4(x) \]

[Out]

-1/4*cos(x)^4

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2645, 30} \[ \int \cos ^3(x) \sin (x) \, dx=-\frac {1}{4} \cos ^4(x) \]

[In]

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

[Out]

-1/4*Cos[x]^4

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \cos ^3(x) \sin (x) \, dx=-\frac {1}{4} \cos ^4(x) \]

[In]

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

[Out]

-1/4*Cos[x]^4

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(-\frac {\left (\cos ^{4}\left (x \right )\right )}{4}\)	\(7\)
default	\(-\frac {\left (\cos ^{4}\left (x \right )\right )}{4}\)	\(7\)
risch	\(-\frac {\cos \left (4 x \right )}{32}-\frac {\cos \left (2 x \right )}{8}\)	\(14\)
parallelrisch	\(-\frac {\cos \left (4 x \right )}{32}+\frac {5}{32}-\frac {\cos \left (2 x \right )}{8}\)	\(15\)
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	\(\frac {\sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (2 x \right )}{\sqrt {\pi }}\right )}{8}+\frac {\sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (4 x \right )}{\sqrt {\pi }}\right )}{32}\)	\(38\)

[In]

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

[Out]

-1/4*cos(x)^4

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \cos ^3(x) \sin (x) \, dx=-\frac {1}{4} \, \cos \left (x\right )^{4} \]

[In]

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

[Out]

-1/4*cos(x)^4

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88 \[ \int \cos ^3(x) \sin (x) \, dx=- \frac {\cos ^{4}{\left (x \right )}}{4} \]

[In]

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

[Out]

-cos(x)**4/4

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \cos ^3(x) \sin (x) \, dx=-\frac {1}{4} \, \cos \left (x\right )^{4} \]

[In]

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

[Out]

-1/4*cos(x)^4

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \cos ^3(x) \sin (x) \, dx=-\frac {1}{4} \, \cos \left (x\right )^{4} \]

[In]

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

[Out]

-1/4*cos(x)^4

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.50 \[ \int \cos ^3(x) \sin (x) \, dx=-\frac {{\sin \left (x\right )}^2\,\left ({\sin \left (x\right )}^2-2\right )}{4} \]

[In]

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

[Out]

-(sin(x)^2*(sin(x)^2 - 2))/4