\(\int \cos ^6(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^2 \, dx\) [1235]

Optimal result

Integrand size = 29, antiderivative size = 238 \[ \int \cos ^6(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5 a b x}{512}-\frac {\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}+\frac {\left (2 a^2+3 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac {\left (a^2+3 b^2\right ) \cos ^{11}(c+d x)}{11 d}+\frac {b^2 \cos ^{13}(c+d x)}{13 d}+\frac {5 a b \cos (c+d x) \sin (c+d x)}{512 d}+\frac {5 a b \cos ^3(c+d x) \sin (c+d x)}{768 d}+\frac {a b \cos ^5(c+d x) \sin (c+d x)}{192 d}-\frac {a b \cos ^7(c+d x) \sin (c+d x)}{32 d}-\frac {a b \cos ^7(c+d x) \sin ^3(c+d x)}{12 d}-\frac {a b \cos ^7(c+d x) \sin ^5(c+d x)}{6 d} \] Output:

5/512*a*b*x-1/7*(a^2+b^2)*cos(d*x+c)^7/d+1/9*(2*a^2+3*b^2)*cos(d*x+c)^9/d- 
1/11*(a^2+3*b^2)*cos(d*x+c)^11/d+1/13*b^2*cos(d*x+c)^13/d+5/512*a*b*cos(d* 
x+c)*sin(d*x+c)/d+5/768*a*b*cos(d*x+c)^3*sin(d*x+c)/d+1/192*a*b*cos(d*x+c) 
^5*sin(d*x+c)/d-1/32*a*b*cos(d*x+c)^7*sin(d*x+c)/d-1/12*a*b*cos(d*x+c)^7*s 
in(d*x+c)^3/d-1/6*a*b*cos(d*x+c)^7*sin(d*x+c)^5/d
 

Mathematica [A] (verified)

Time = 2.03 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.88 \[ \int \cos ^6(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {360360 a b c+360360 a b d x-180180 \left (2 a^2+b^2\right ) \cos (c+d x)-15015 \left (8 a^2+3 b^2\right ) \cos (3 (c+d x))+36036 a^2 \cos (5 (c+d x))+27027 b^2 \cos (5 (c+d x))+25740 a^2 \cos (7 (c+d x))+7722 b^2 \cos (7 (c+d x))-4004 a^2 \cos (9 (c+d x))-6006 b^2 \cos (9 (c+d x))-3276 a^2 \cos (11 (c+d x))-819 b^2 \cos (11 (c+d x))+693 b^2 \cos (13 (c+d x))-135135 a b \sin (4 (c+d x))+27027 a b \sin (8 (c+d x))-3003 a b \sin (12 (c+d x))}{36900864 d} \] Input:

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]
 

Output:

(360360*a*b*c + 360360*a*b*d*x - 180180*(2*a^2 + b^2)*Cos[c + d*x] - 15015 
*(8*a^2 + 3*b^2)*Cos[3*(c + d*x)] + 36036*a^2*Cos[5*(c + d*x)] + 27027*b^2 
*Cos[5*(c + d*x)] + 25740*a^2*Cos[7*(c + d*x)] + 7722*b^2*Cos[7*(c + d*x)] 
 - 4004*a^2*Cos[9*(c + d*x)] - 6006*b^2*Cos[9*(c + d*x)] - 3276*a^2*Cos[11 
*(c + d*x)] - 819*b^2*Cos[11*(c + d*x)] + 693*b^2*Cos[13*(c + d*x)] - 1351 
35*a*b*Sin[4*(c + d*x)] + 27027*a*b*Sin[8*(c + d*x)] - 3003*a*b*Sin[12*(c 
+ d*x)])/(36900864*d)
 

Rubi [A] (verified)

Time = 1.49 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.26, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.655, Rules used = {3042, 3390, 3042, 3048, 3042, 3048, 3042, 3048, 3042, 3115, 3042, 3115, 3042, 3115, 24, 3680, 354, 86, 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.

\(\displaystyle \int \sin ^5(c+d x) \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin (c+d x)^5 \cos (c+d x)^6 (a+b \sin (c+d x))^2dx\)

\(\Big \downarrow \) 3390

\(\displaystyle \int \cos ^6(c+d x) \sin ^5(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right )dx+2 a b \int \cos ^6(c+d x) \sin ^6(c+d x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^5 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \int \cos (c+d x)^6 \sin (c+d x)^6dx\)

\(\Big \downarrow \) 3048

\(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^5 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {5}{12} \int \cos ^6(c+d x) \sin ^4(c+d x)dx-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^5 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {5}{12} \int \cos (c+d x)^6 \sin (c+d x)^4dx-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

\(\Big \downarrow \) 3048

\(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^5 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {5}{12} \left (\frac {3}{10} \int \cos ^6(c+d x) \sin ^2(c+d x)dx-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^5 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {5}{12} \left (\frac {3}{10} \int \cos (c+d x)^6 \sin (c+d x)^2dx-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

\(\Big \downarrow \) 3048

\(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^5 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \int \cos ^6(c+d x)dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^5 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^5 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^5 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^5 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^5 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^5 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^5 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

\(\Big \downarrow \) 3680

\(\displaystyle \frac {\sqrt {\cos ^2(c+d x)} \sec (c+d x) \int \sin ^5(c+d x) \left (1-\sin ^2(c+d x)\right )^{5/2} \left (a^2+b^2 \sin ^2(c+d x)\right )d\sin (c+d x)}{d}+2 a b \left (\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {\sqrt {\cos ^2(c+d x)} \sec (c+d x) \int \sin ^4(c+d x) \left (1-\sin ^2(c+d x)\right )^{5/2} \left (a^2+b^2 \sin ^2(c+d x)\right )d\sin ^2(c+d x)}{2 d}+2 a b \left (\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {\sqrt {\cos ^2(c+d x)} \sec (c+d x) \int \left (-b^2 \left (1-\sin ^2(c+d x)\right )^{11/2}+\left (a^2+3 b^2\right ) \left (1-\sin ^2(c+d x)\right )^{9/2}+\left (-2 a^2-3 b^2\right ) \left (1-\sin ^2(c+d x)\right )^{7/2}+\left (a^2+b^2\right ) \left (1-\sin ^2(c+d x)\right )^{5/2}\right )d\sin ^2(c+d x)}{2 d}+2 a b \left (\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {\cos ^2(c+d x)} \sec (c+d x) \left (-\frac {2}{11} \left (a^2+3 b^2\right ) \left (1-\sin ^2(c+d x)\right )^{11/2}+\frac {2}{9} \left (2 a^2+3 b^2\right ) \left (1-\sin ^2(c+d x)\right )^{9/2}-\frac {2}{7} \left (a^2+b^2\right ) \left (1-\sin ^2(c+d x)\right )^{7/2}+\frac {2}{13} b^2 \left (1-\sin ^2(c+d x)\right )^{13/2}\right )}{2 d}+2 a b \left (\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 d}\right )\)

Input:

Int[Cos[c + d*x]^6*Sin[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]
 

Output:

(Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*((-2*(a^2 + b^2)*(1 - Sin[c + d*x]^2)^( 
7/2))/7 + (2*(2*a^2 + 3*b^2)*(1 - Sin[c + d*x]^2)^(9/2))/9 - (2*(a^2 + 3*b 
^2)*(1 - Sin[c + d*x]^2)^(11/2))/11 + (2*b^2*(1 - Sin[c + d*x]^2)^(13/2))/ 
13))/(2*d) + 2*a*b*(-1/12*(Cos[c + d*x]^7*Sin[c + d*x]^5)/d + (5*(-1/10*(C 
os[c + d*x]^7*Sin[c + d*x]^3)/d + (3*(-1/8*(Cos[c + d*x]^7*Sin[c + d*x])/d 
 + ((Cos[c + d*x]^5*Sin[c + d*x])/(6*d) + (5*((Cos[c + d*x]^3*Sin[c + d*x] 
)/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4))/6)/8))/10))/12 
)
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

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 3048

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 
1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n))   Int[(b*Cos[e + f*x])^n 
*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] 
 && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
 

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3390

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[2*a*(b/d) 
Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x], x] + Int[(g*Cos[e + f* 
x])^p*(d*Sin[e + f*x])^n*(a^2 + b^2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, 
e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0]
 

rule 3680

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_ 
) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeFac 
tors[Sin[e + f*x], x]}, Simp[ff*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x]))   S 
ubst[Int[(d*ff*x)^n*(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, S 
in[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[m/2]
 

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.95

\[\frac {a^{2} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{7}}{11}-\frac {4 \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}}{99}-\frac {8 \cos \left (d x +c \right )^{7}}{693}\right )+2 a b \left (-\frac {\sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{7}}{12}-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{7}}{24}-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{64}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )+b^{2} \left (-\frac {\sin \left (d x +c \right )^{6} \cos \left (d x +c \right )^{7}}{13}-\frac {6 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{7}}{143}-\frac {8 \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}}{429}-\frac {16 \cos \left (d x +c \right )^{7}}{3003}\right )}{d}\]

Input:

int(cos(d*x+c)^6*sin(d*x+c)^5*(a+b*sin(d*x+c))^2,x)
 

Output:

1/d*(a^2*(-1/11*sin(d*x+c)^4*cos(d*x+c)^7-4/99*cos(d*x+c)^7*sin(d*x+c)^2-8 
/693*cos(d*x+c)^7)+2*a*b*(-1/12*sin(d*x+c)^5*cos(d*x+c)^7-1/24*sin(d*x+c)^ 
3*cos(d*x+c)^7-1/64*cos(d*x+c)^7*sin(d*x+c)+1/384*(cos(d*x+c)^5+5/4*cos(d* 
x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/1024*d*x+5/1024*c)+b^2*(-1/13*sin(d*x 
+c)^6*cos(d*x+c)^7-6/143*sin(d*x+c)^4*cos(d*x+c)^7-8/429*cos(d*x+c)^7*sin( 
d*x+c)^2-16/3003*cos(d*x+c)^7))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.68 \[ \int \cos ^6(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {354816 \, b^{2} \cos \left (d x + c\right )^{13} - 419328 \, {\left (a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{11} + 512512 \, {\left (2 \, a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{9} - 658944 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{7} + 45045 \, a b d x - 3003 \, {\left (256 \, a b \cos \left (d x + c\right )^{11} - 640 \, a b \cos \left (d x + c\right )^{9} + 432 \, a b \cos \left (d x + c\right )^{7} - 8 \, a b \cos \left (d x + c\right )^{5} - 10 \, a b \cos \left (d x + c\right )^{3} - 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4612608 \, d} \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^5*(a+b*sin(d*x+c))^2,x, algorithm="frica 
s")
 

Output:

1/4612608*(354816*b^2*cos(d*x + c)^13 - 419328*(a^2 + 3*b^2)*cos(d*x + c)^ 
11 + 512512*(2*a^2 + 3*b^2)*cos(d*x + c)^9 - 658944*(a^2 + b^2)*cos(d*x + 
c)^7 + 45045*a*b*d*x - 3003*(256*a*b*cos(d*x + c)^11 - 640*a*b*cos(d*x + c 
)^9 + 432*a*b*cos(d*x + c)^7 - 8*a*b*cos(d*x + c)^5 - 10*a*b*cos(d*x + c)^ 
3 - 15*a*b*cos(d*x + c))*sin(d*x + c))/d
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (224) = 448\).

Time = 3.37 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.05 \[ \int \cos ^6(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\begin {cases} - \frac {a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {4 a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {8 a^{2} \cos ^{11}{\left (c + d x \right )}}{693 d} + \frac {5 a b x \sin ^{12}{\left (c + d x \right )}}{512} + \frac {15 a b x \sin ^{10}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {75 a b x \sin ^{8}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{512} + \frac {25 a b x \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {75 a b x \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{512} + \frac {15 a b x \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{256} + \frac {5 a b x \cos ^{12}{\left (c + d x \right )}}{512} + \frac {5 a b \sin ^{11}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{512 d} + \frac {85 a b \sin ^{9}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{1536 d} + \frac {33 a b \sin ^{7}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{256 d} - \frac {33 a b \sin ^{5}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{256 d} - \frac {85 a b \sin ^{3}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{1536 d} - \frac {5 a b \sin {\left (c + d x \right )} \cos ^{11}{\left (c + d x \right )}}{512 d} - \frac {b^{2} \sin ^{6}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {2 b^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{21 d} - \frac {8 b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{11}{\left (c + d x \right )}}{231 d} - \frac {16 b^{2} \cos ^{13}{\left (c + d x \right )}}{3003 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{2} \sin ^{5}{\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:

integrate(cos(d*x+c)**6*sin(d*x+c)**5*(a+b*sin(d*x+c))**2,x)
 

Output:

Piecewise((-a**2*sin(c + d*x)**4*cos(c + d*x)**7/(7*d) - 4*a**2*sin(c + d* 
x)**2*cos(c + d*x)**9/(63*d) - 8*a**2*cos(c + d*x)**11/(693*d) + 5*a*b*x*s 
in(c + d*x)**12/512 + 15*a*b*x*sin(c + d*x)**10*cos(c + d*x)**2/256 + 75*a 
*b*x*sin(c + d*x)**8*cos(c + d*x)**4/512 + 25*a*b*x*sin(c + d*x)**6*cos(c 
+ d*x)**6/128 + 75*a*b*x*sin(c + d*x)**4*cos(c + d*x)**8/512 + 15*a*b*x*si 
n(c + d*x)**2*cos(c + d*x)**10/256 + 5*a*b*x*cos(c + d*x)**12/512 + 5*a*b* 
sin(c + d*x)**11*cos(c + d*x)/(512*d) + 85*a*b*sin(c + d*x)**9*cos(c + d*x 
)**3/(1536*d) + 33*a*b*sin(c + d*x)**7*cos(c + d*x)**5/(256*d) - 33*a*b*si 
n(c + d*x)**5*cos(c + d*x)**7/(256*d) - 85*a*b*sin(c + d*x)**3*cos(c + d*x 
)**9/(1536*d) - 5*a*b*sin(c + d*x)*cos(c + d*x)**11/(512*d) - b**2*sin(c + 
 d*x)**6*cos(c + d*x)**7/(7*d) - 2*b**2*sin(c + d*x)**4*cos(c + d*x)**9/(2 
1*d) - 8*b**2*sin(c + d*x)**2*cos(c + d*x)**11/(231*d) - 16*b**2*cos(c + d 
*x)**13/(3003*d), Ne(d, 0)), (x*(a + b*sin(c))**2*sin(c)**5*cos(c)**6, Tru 
e))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.57 \[ \int \cos ^6(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {53248 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{2} - 3003 \, {\left (4 \, \sin \left (4 \, d x + 4 \, c\right )^{3} + 120 \, d x + 120 \, c + 9 \, \sin \left (8 \, d x + 8 \, c\right ) - 48 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b - 12288 \, {\left (231 \, \cos \left (d x + c\right )^{13} - 819 \, \cos \left (d x + c\right )^{11} + 1001 \, \cos \left (d x + c\right )^{9} - 429 \, \cos \left (d x + c\right )^{7}\right )} b^{2}}{36900864 \, d} \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^5*(a+b*sin(d*x+c))^2,x, algorithm="maxim 
a")
 

Output:

-1/36900864*(53248*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + 
 c)^7)*a^2 - 3003*(4*sin(4*d*x + 4*c)^3 + 120*d*x + 120*c + 9*sin(8*d*x + 
8*c) - 48*sin(4*d*x + 4*c))*a*b - 12288*(231*cos(d*x + c)^13 - 819*cos(d*x 
 + c)^11 + 1001*cos(d*x + c)^9 - 429*cos(d*x + c)^7)*b^2)/d
 

Giac [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.90 \[ \int \cos ^6(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5}{512} \, a b x + \frac {b^{2} \cos \left (13 \, d x + 13 \, c\right )}{53248 \, d} - \frac {a b \sin \left (12 \, d x + 12 \, c\right )}{12288 \, d} + \frac {3 \, a b \sin \left (8 \, d x + 8 \, c\right )}{4096 \, d} - \frac {15 \, a b \sin \left (4 \, d x + 4 \, c\right )}{4096 \, d} - \frac {{\left (4 \, a^{2} + b^{2}\right )} \cos \left (11 \, d x + 11 \, c\right )}{45056 \, d} - \frac {{\left (2 \, a^{2} + 3 \, b^{2}\right )} \cos \left (9 \, d x + 9 \, c\right )}{18432 \, d} + \frac {{\left (10 \, a^{2} + 3 \, b^{2}\right )} \cos \left (7 \, d x + 7 \, c\right )}{14336 \, d} + \frac {{\left (4 \, a^{2} + 3 \, b^{2}\right )} \cos \left (5 \, d x + 5 \, c\right )}{4096 \, d} - \frac {5 \, {\left (8 \, a^{2} + 3 \, b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{12288 \, d} - \frac {5 \, {\left (2 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )}{1024 \, d} \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^5*(a+b*sin(d*x+c))^2,x, algorithm="giac" 
)
 

Output:

5/512*a*b*x + 1/53248*b^2*cos(13*d*x + 13*c)/d - 1/12288*a*b*sin(12*d*x + 
12*c)/d + 3/4096*a*b*sin(8*d*x + 8*c)/d - 15/4096*a*b*sin(4*d*x + 4*c)/d - 
 1/45056*(4*a^2 + b^2)*cos(11*d*x + 11*c)/d - 1/18432*(2*a^2 + 3*b^2)*cos( 
9*d*x + 9*c)/d + 1/14336*(10*a^2 + 3*b^2)*cos(7*d*x + 7*c)/d + 1/4096*(4*a 
^2 + 3*b^2)*cos(5*d*x + 5*c)/d - 5/12288*(8*a^2 + 3*b^2)*cos(3*d*x + 3*c)/ 
d - 5/1024*(2*a^2 + b^2)*cos(d*x + c)/d
 

Mupad [B] (verification not implemented)

Time = 25.00 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.85 \[ \int \cos ^6(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^2 \, dx =\text {Too large to display} \] Input:

int(cos(c + d*x)^6*sin(c + d*x)^5*(a + b*sin(c + d*x))^2,x)
 

Output:

(5*a*b*x)/512 - (tan(c/2 + (d*x)/2)^16*(16*a^2 - 96*b^2) - tan(c/2 + (d*x) 
/2)^18*((16*a^2)/3 - 32*b^2) + tan(c/2 + (d*x)/2)^14*((128*a^2)/3 + 192*b^ 
2) - tan(c/2 + (d*x)/2)^6*((256*a^2)/63 - (64*b^2)/21) + tan(c/2 + (d*x)/2 
)^4*((416*a^2)/231 + (64*b^2)/77) + tan(c/2 + (d*x)/2)^10*((96*a^2)/7 + (7 
68*b^2)/7) + tan(c/2 + (d*x)/2)^2*((208*a^2)/693 + (32*b^2)/231) - tan(c/2 
 + (d*x)/2)^12*((64*a^2)/21 + (1216*b^2)/7) + tan(c/2 + (d*x)/2)^8*((1376* 
a^2)/63 - (512*b^2)/21) + (32*a^2*tan(c/2 + (d*x)/2)^20)/3 + (16*a^2)/693 
+ (32*b^2)/3003 + (95*a*b*tan(c/2 + (d*x)/2)^3)/384 + (277*a*b*tan(c/2 + ( 
d*x)/2)^5)/192 - (4025*a*b*tan(c/2 + (d*x)/2)^7)/128 + (59435*a*b*tan(c/2 
+ (d*x)/2)^9)/768 - (16813*a*b*tan(c/2 + (d*x)/2)^11)/192 + (16813*a*b*tan 
(c/2 + (d*x)/2)^15)/192 - (59435*a*b*tan(c/2 + (d*x)/2)^17)/768 + (4025*a* 
b*tan(c/2 + (d*x)/2)^19)/128 - (277*a*b*tan(c/2 + (d*x)/2)^21)/192 - (95*a 
*b*tan(c/2 + (d*x)/2)^23)/384 - (5*a*b*tan(c/2 + (d*x)/2)^25)/256 + (5*a*b 
*tan(c/2 + (d*x)/2))/256)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^13)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.51 \[ \int \cos ^6(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {354816 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{12} b^{2}+768768 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{11} a b +419328 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{10} a^{2}-870912 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{10} b^{2}-1921920 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9} a b -1071616 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} a^{2}+569856 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} b^{2}+1297296 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} a b +752128 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a^{2}-7680 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} b^{2}-24024 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a b -19968 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{2}-9216 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b^{2}-30030 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a b -26624 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2}-12288 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b^{2}-45045 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a b -53248 \cos \left (d x +c \right ) a^{2}-24576 \cos \left (d x +c \right ) b^{2}+53248 a^{2}+45045 a b d x +24576 b^{2}}{4612608 d} \] Input:

int(cos(d*x+c)^6*sin(d*x+c)^5*(a+b*sin(d*x+c))^2,x)
 

Output:

(354816*cos(c + d*x)*sin(c + d*x)**12*b**2 + 768768*cos(c + d*x)*sin(c + d 
*x)**11*a*b + 419328*cos(c + d*x)*sin(c + d*x)**10*a**2 - 870912*cos(c + d 
*x)*sin(c + d*x)**10*b**2 - 1921920*cos(c + d*x)*sin(c + d*x)**9*a*b - 107 
1616*cos(c + d*x)*sin(c + d*x)**8*a**2 + 569856*cos(c + d*x)*sin(c + d*x)* 
*8*b**2 + 1297296*cos(c + d*x)*sin(c + d*x)**7*a*b + 752128*cos(c + d*x)*s 
in(c + d*x)**6*a**2 - 7680*cos(c + d*x)*sin(c + d*x)**6*b**2 - 24024*cos(c 
 + d*x)*sin(c + d*x)**5*a*b - 19968*cos(c + d*x)*sin(c + d*x)**4*a**2 - 92 
16*cos(c + d*x)*sin(c + d*x)**4*b**2 - 30030*cos(c + d*x)*sin(c + d*x)**3* 
a*b - 26624*cos(c + d*x)*sin(c + d*x)**2*a**2 - 12288*cos(c + d*x)*sin(c + 
 d*x)**2*b**2 - 45045*cos(c + d*x)*sin(c + d*x)*a*b - 53248*cos(c + d*x)*a 
**2 - 24576*cos(c + d*x)*b**2 + 53248*a**2 + 45045*a*b*d*x + 24576*b**2)/( 
4612608*d)