\(\int 35 \cos ^3(x) \sin ^4(x) \, dx\) [777]

Optimal result

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

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

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 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=35 \left (\frac {3 \sin (x)}{64}-\frac {1}{64} \sin (3 x)-\frac {1}{320} \sin (5 x)+\frac {1}{448} \sin (7 x)\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.46, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {27, 3042, 3044, 244, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int 35 \sin ^4(x) \cos ^3(x) \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle 35 \int \cos ^3(x) \sin ^4(x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 35 \int \cos (x)^3 \sin (x)^4dx\)

\(\Big \downarrow \) 3044

\(\displaystyle 35 \int \sin ^4(x) \left (1-\sin ^2(x)\right )d\sin (x)\)

\(\Big \downarrow \) 244

\(\displaystyle 35 \int \left (\sin ^4(x)-\sin ^6(x)\right )d\sin (x)\)

\(\Big \downarrow \) 2009

\(\displaystyle 35 \left (\frac {\sin ^5(x)}{5}-\frac {\sin ^7(x)}{7}\right )\)

Input:

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

Output:

35*(Sin[x]^5/5 - Sin[x]^7/7)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3044

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ 
Symbol] :> Simp[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] &&  !(I 
ntegerQ[(m - 1)/2] && LtQ[0, m, n])
 

Maple [A] (verified)

Time = 1.54 (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	\(7 \sin \left (x \right )^{5}-5 \sin \left (x \right )^{7}\)	\(14\)
orering	\(7 \cos \left (x \right )^{2} \sin \left (x \right )^{5}+2 \sin \left (x \right )^{7}\)	\(18\)
risch	\(\frac {105 \sin \left (x \right )}{64}+\frac {5 \sin \left (7 x \right )}{64}-\frac {7 \sin \left (5 x \right )}{64}-\frac {35 \sin \left (3 x \right )}{64}\)	\(24\)
norman	\(\frac {224 \tan \left (\frac {x}{2}\right )^{5}-192 \tan \left (\frac {x}{2}\right )^{7}+224 \tan \left (\frac {x}{2}\right )^{9}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{7}}\)	\(37\)
parallelrisch	\(\frac {\left (\sin \left (\frac {5 x}{2}\right )-5 \sin \left (\frac {3 x}{2}\right )+10 \sin \left (\frac {x}{2}\right )\right ) \left (9+5 \cos \left (2 x \right )\right ) \left (\cos \left (\frac {5 x}{2}\right )+5 \cos \left (\frac {3 x}{2}\right )+10 \cos \left (\frac {x}{2}\right )\right )}{16}\)	\(45\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.62 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx={\left (5 \, \cos \left (x\right )^{6} - 8 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 2\right )} \sin \left (x\right ) \] Input:

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

Output:

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=-5 \, \sin \left (x\right )^{7} + 7 \, \sin \left (x\right )^{5} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=-5 \, \sin \left (x\right )^{7} + 7 \, \sin \left (x\right )^{5} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=7\,{\sin \left (x\right )}^5-5\,{\sin \left (x\right )}^7 \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int 35 \cos ^3(x) \sin ^4(x) \, dx=\sin \left (x \right )^{5} \left (-5 \sin \left (x \right )^{2}+7\right ) \] Input:

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

Output:

sin(x)**5*( - 5*sin(x)**2 + 7)