\(\int \cos ^2(x) \sin ^5(x) \, dx\) [342]

Optimal result

Integrand size = 9, antiderivative size = 25 \[ \int \cos ^2(x) \sin ^5(x) \, dx=-\frac {1}{3} \cos ^3(x)+\frac {2 \cos ^5(x)}{5}-\frac {\cos ^7(x)}{7} \]

[Out]

-1/3*cos(x)^3+2/5*cos(x)^5-1/7*cos(x)^7

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2645, 276} \[ \int \cos ^2(x) \sin ^5(x) \, dx=-\frac {1}{7} \cos ^7(x)+\frac {2 \cos ^5(x)}{5}-\frac {\cos ^3(x)}{3} \]

[In]

Int[Cos[x]^2*Sin[x]^5,x]

[Out]

-1/3*Cos[x]^3 + (2*Cos[x]^5)/5 - Cos[x]^7/7

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

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^2 \left (1-x^2\right )^2 \, dx,x,\cos (x)\right ) \\ & = -\text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (x)\right ) \\ & = -\frac {1}{3} \cos ^3(x)+\frac {2 \cos ^5(x)}{5}-\frac {\cos ^7(x)}{7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \cos ^2(x) \sin ^5(x) \, dx=-\frac {5 \cos (x)}{64}-\frac {1}{192} \cos (3 x)+\frac {3}{320} \cos (5 x)-\frac {1}{448} \cos (7 x) \]

[In]

Integrate[Cos[x]^2*Sin[x]^5,x]

[Out]

(-5*Cos[x])/64 - Cos[3*x]/192 + (3*Cos[5*x])/320 - Cos[7*x]/448

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(-\frac {\left (\cos ^{3}\left (x \right )\right )}{3}+\frac {2 \left (\cos ^{5}\left (x \right )\right )}{5}-\frac {\left (\cos ^{7}\left (x \right )\right )}{7}\)	\(20\)
default	\(-\frac {\left (\cos ^{3}\left (x \right )\right )}{3}+\frac {2 \left (\cos ^{5}\left (x \right )\right )}{5}-\frac {\left (\cos ^{7}\left (x \right )\right )}{7}\)	\(20\)
risch	\(-\frac {5 \cos \left (x \right )}{64}-\frac {\cos \left (7 x \right )}{448}+\frac {3 \cos \left (5 x \right )}{320}-\frac {\cos \left (3 x \right )}{192}\)	\(24\)
parallelrisch	\(\frac {8}{35}-\frac {5 \cos \left (x \right )}{64}-\frac {\cos \left (7 x \right )}{448}+\frac {3 \cos \left (5 x \right )}{320}-\frac {\cos \left (3 x \right )}{192}\)	\(25\)
norman	\(\frac {-\frac {32 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{3}-\frac {16 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{5}-\frac {16 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{15}+\frac {16 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{3}-\frac {16}{105}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{7}}\)	\(46\)

[In]

int(cos(x)^2*sin(x)^5,x,method=_RETURNVERBOSE)

[Out]

-1/3*cos(x)^3+2/5*cos(x)^5-1/7*cos(x)^7

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \cos ^2(x) \sin ^5(x) \, dx=-\frac {1}{7} \, \cos \left (x\right )^{7} + \frac {2}{5} \, \cos \left (x\right )^{5} - \frac {1}{3} \, \cos \left (x\right )^{3} \]

[In]

integrate(cos(x)^2*sin(x)^5,x, algorithm="fricas")

[Out]

-1/7*cos(x)^7 + 2/5*cos(x)^5 - 1/3*cos(x)^3

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \cos ^2(x) \sin ^5(x) \, dx=- \frac {\cos ^{7}{\left (x \right )}}{7} + \frac {2 \cos ^{5}{\left (x \right )}}{5} - \frac {\cos ^{3}{\left (x \right )}}{3} \]

[In]

integrate(cos(x)**2*sin(x)**5,x)

[Out]

-cos(x)**7/7 + 2*cos(x)**5/5 - cos(x)**3/3

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \cos ^2(x) \sin ^5(x) \, dx=-\frac {1}{7} \, \cos \left (x\right )^{7} + \frac {2}{5} \, \cos \left (x\right )^{5} - \frac {1}{3} \, \cos \left (x\right )^{3} \]

[In]

integrate(cos(x)^2*sin(x)^5,x, algorithm="maxima")

[Out]

-1/7*cos(x)^7 + 2/5*cos(x)^5 - 1/3*cos(x)^3

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \cos ^2(x) \sin ^5(x) \, dx=-\frac {1}{7} \, \cos \left (x\right )^{7} + \frac {2}{5} \, \cos \left (x\right )^{5} - \frac {1}{3} \, \cos \left (x\right )^{3} \]

[In]

integrate(cos(x)^2*sin(x)^5,x, algorithm="giac")

[Out]

-1/7*cos(x)^7 + 2/5*cos(x)^5 - 1/3*cos(x)^3

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \cos ^2(x) \sin ^5(x) \, dx=-\frac {{\cos \left (x\right )}^7}{7}+\frac {2\,{\cos \left (x\right )}^5}{5}-\frac {{\cos \left (x\right )}^3}{3} \]

[In]

int(cos(x)^2*sin(x)^5,x)

[Out]

(2*cos(x)^5)/5 - cos(x)^3/3 - cos(x)^7/7