\(\int 35 \cos ^3(x) \sin ^4(x) \, dx\) [834]

Optimal result

Integrand size = 10, antiderivative size = 13 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=7 \sin ^5(x)-5 \sin ^7(x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {12, 2644, 14} \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=7 \sin ^5(x)-5 \sin ^7(x) \]

[In]

Int[35*Cos[x]^3*Sin[x]^4,x]

[Out]

7*Sin[x]^5 - 5*Sin[x]^7

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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}& = 35 \int \cos ^3(x) \sin ^4(x) \, dx \\ & = 35 \text {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\sin (x)\right ) \\ & = 35 \text {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\sin (x)\right ) \\ & = 7 \sin ^5(x)-5 \sin ^7(x) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(33\) vs. \(2(13)=26\).

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.54 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=35 \left (\frac {3 \sin (x)}{64}-\frac {1}{64} \sin (3 x)-\frac {1}{320} \sin (5 x)+\frac {1}{448} \sin (7 x)\right ) \]

[In]

Integrate[35*Cos[x]^3*Sin[x]^4,x]

[Out]

35*((3*Sin[x])/64 - Sin[3*x]/64 - Sin[5*x]/320 + Sin[7*x]/448)

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(7 \sin \left (x \right )^{5}-5 \sin \left (x \right )^{7}\)	\(14\)
default	\(7 \sin \left (x \right )^{5}-5 \sin \left (x \right )^{7}\)	\(14\)
risch	\(\frac {105 \sin \left (x \right )}{64}+\frac {5 \sin \left (7 x \right )}{64}-\frac {7 \sin \left (5 x \right )}{64}-\frac {35 \sin \left (3 x \right )}{64}\)	\(24\)
parallelrisch	\(\frac {105 \sin \left (x \right )}{64}+\frac {5 \sin \left (7 x \right )}{64}-\frac {7 \sin \left (5 x \right )}{64}-\frac {35 \sin \left (3 x \right )}{64}\)	\(24\)
norman	\(\frac {224 \tan \left (\frac {x}{2}\right )^{5}-192 \tan \left (\frac {x}{2}\right )^{7}+224 \tan \left (\frac {x}{2}\right )^{9}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{7}}\)	\(37\)

[In]

int(35*cos(x)^3*sin(x)^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.62 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx={\left (5 \, \cos \left (x\right )^{6} - 8 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 2\right )} \sin \left (x\right ) \]

[In]

integrate(35*cos(x)^3*sin(x)^4,x, algorithm="fricas")

[Out]

(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 = 0.92 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=- 5 \sin ^{7}{\left (x \right )} + 7 \sin ^{5}{\left (x \right )} \]

[In]

integrate(35*cos(x)**3*sin(x)**4,x)

[Out]

-5*sin(x)**7 + 7*sin(x)**5

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=-5 \, \sin \left (x\right )^{7} + 7 \, \sin \left (x\right )^{5} \]

[In]

integrate(35*cos(x)^3*sin(x)^4,x, algorithm="maxima")

[Out]

-5*sin(x)^7 + 7*sin(x)^5

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=-5 \, \sin \left (x\right )^{7} + 7 \, \sin \left (x\right )^{5} \]

[In]

integrate(35*cos(x)^3*sin(x)^4,x, algorithm="giac")

[Out]

-5*sin(x)^7 + 7*sin(x)^5

Mupad [B] (verification not implemented)

Time = 27.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=7\,{\sin \left (x\right )}^5-5\,{\sin \left (x\right )}^7 \]

[In]

int(35*cos(x)^3*sin(x)^4,x)

[Out]

7*sin(x)^5 - 5*sin(x)^7