\(\int \cos ^3(x) \sin ^4(x) \, dx\) [63]

Optimal result

Integrand size = 9, antiderivative size = 17 \[ \int \cos ^3(x) \sin ^4(x) \, dx=\frac {\sin ^5(x)}{5}-\frac {\sin ^7(x)}{7} \]

[Out]

1/5*sin(x)^5-1/7*sin(x)^7

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, 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 = {2644, 14} \[ \int \cos ^3(x) \sin ^4(x) \, dx=\frac {\sin ^5(x)}{5}-\frac {\sin ^7(x)}{7} \]

[In]

Int[Cos[x]^3*Sin[x]^4,x]

[Out]

Sin[x]^5/5 - Sin[x]^7/7

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\sin (x)\right ) \\ & = \text {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\sin (x)\right ) \\ & = \frac {\sin ^5(x)}{5}-\frac {\sin ^7(x)}{7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \cos ^3(x) \sin ^4(x) \, dx=\frac {3 \sin (x)}{64}-\frac {1}{64} \sin (3 x)-\frac {1}{320} \sin (5 x)+\frac {1}{448} \sin (7 x) \]

[In]

Integrate[Cos[x]^3*Sin[x]^4,x]

[Out]

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

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {\left (\sin ^{5}\left (x \right )\right )}{5}-\frac {\left (\sin ^{7}\left (x \right )\right )}{7}\)	\(14\)
default	\(\frac {\left (\sin ^{5}\left (x \right )\right )}{5}-\frac {\left (\sin ^{7}\left (x \right )\right )}{7}\)	\(14\)
risch	\(\frac {3 \sin \left (x \right )}{64}+\frac {\sin \left (7 x \right )}{448}-\frac {\sin \left (5 x \right )}{320}-\frac {\sin \left (3 x \right )}{64}\)	\(24\)
parallelrisch	\(\frac {3 \sin \left (x \right )}{64}+\frac {\sin \left (7 x \right )}{448}-\frac {\sin \left (5 x \right )}{320}-\frac {\sin \left (3 x \right )}{64}\)	\(24\)
norman	\(\frac {\frac {32 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{5}-\frac {192 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{35}+\frac {32 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{5}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{7}}\)	\(37\)

[In]

int(cos(x)^3*sin(x)^4,x,method=_RETURNVERBOSE)

[Out]

1/5*sin(x)^5-1/7*sin(x)^7

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int \cos ^3(x) \sin ^4(x) \, dx=\frac {1}{35} \, {\left (5 \, \cos \left (x\right )^{6} - 8 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 2\right )} \sin \left (x\right ) \]

[In]

integrate(cos(x)^3*sin(x)^4,x, algorithm="fricas")

[Out]

1/35*(5*cos(x)^6 - 8*cos(x)^4 + cos(x)^2 + 2)*sin(x)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \cos ^3(x) \sin ^4(x) \, dx=- \frac {\sin ^{7}{\left (x \right )}}{7} + \frac {\sin ^{5}{\left (x \right )}}{5} \]

[In]

integrate(cos(x)**3*sin(x)**4,x)

[Out]

-sin(x)**7/7 + sin(x)**5/5

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos ^3(x) \sin ^4(x) \, dx=-\frac {1}{7} \, \sin \left (x\right )^{7} + \frac {1}{5} \, \sin \left (x\right )^{5} \]

[In]

integrate(cos(x)^3*sin(x)^4,x, algorithm="maxima")

[Out]

-1/7*sin(x)^7 + 1/5*sin(x)^5

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos ^3(x) \sin ^4(x) \, dx=-\frac {1}{7} \, \sin \left (x\right )^{7} + \frac {1}{5} \, \sin \left (x\right )^{5} \]

[In]

integrate(cos(x)^3*sin(x)^4,x, algorithm="giac")

[Out]

-1/7*sin(x)^7 + 1/5*sin(x)^5

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \cos ^3(x) \sin ^4(x) \, dx=-\frac {{\sin \left (x\right )}^5\,\left (5\,{\sin \left (x\right )}^2-7\right )}{35} \]

[In]

int(cos(x)^3*sin(x)^4,x)

[Out]

-(sin(x)^5*(5*sin(x)^2 - 7))/35