\(\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx\) [252]

Optimal result

Integrand size = 35, antiderivative size = 336 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {1}{8} a^2 \left (12 A c^2+8 B c^2+16 A c d+14 B c d+7 A d^2+6 B d^2\right ) x+\frac {a^2 \left (5 A d \left (c^3-8 c^2 d-20 c d^2-8 d^3\right )-2 B \left (c^4-5 c^3 d+16 c^2 d^2+40 c d^3+18 d^4\right )\right ) \cos (e+f x)}{30 d^2 f}+\frac {a^2 \left (5 A d \left (2 c^2-16 c d-21 d^2\right )-B \left (4 c^3-20 c^2 d+66 c d^2+90 d^3\right )\right ) \cos (e+f x) \sin (e+f x)}{120 d f}+\frac {a^2 \left (5 A (c-8 d) d-2 B \left (c^2-5 c d+18 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{60 d^2 f}+\frac {a^2 (2 B (c-3 d)-5 A d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d^2 f}-\frac {B \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{5 d f} \] Output:

1/8*a^2*(12*A*c^2+16*A*c*d+7*A*d^2+8*B*c^2+14*B*c*d+6*B*d^2)*x+1/30*a^2*(5 
*A*d*(c^3-8*c^2*d-20*c*d^2-8*d^3)-2*B*(c^4-5*c^3*d+16*c^2*d^2+40*c*d^3+18* 
d^4))*cos(f*x+e)/d^2/f+1/120*a^2*(5*A*d*(2*c^2-16*c*d-21*d^2)-B*(4*c^3-20* 
c^2*d+66*c*d^2+90*d^3))*cos(f*x+e)*sin(f*x+e)/d/f+1/60*a^2*(5*A*(c-8*d)*d- 
2*B*(c^2-5*c*d+18*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^2/d^2/f+1/20*a^2*(2*B* 
(c-3*d)-5*A*d)*cos(f*x+e)*(c+d*sin(f*x+e))^3/d^2/f-1/5*B*cos(f*x+e)*(a^2+a 
^2*sin(f*x+e))*(c+d*sin(f*x+e))^3/d/f
 

Mathematica [A] (verified)

Time = 1.87 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.88 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {a^2 \cos (e+f x) \left (60 \left (2 B \left (4 c^2+7 c d+3 d^2\right )+A \left (12 c^2+16 c d+7 d^2\right )\right ) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (480 A c^2+440 B c^2+880 A c d+800 B c d+400 A d^2+378 B d^2-8 \left (10 A d (c+d)+B \left (5 c^2+20 c d+12 d^2\right )\right ) \cos (2 (e+f x))+6 B d^2 \cos (4 (e+f x))+120 A c^2 \sin (e+f x)+240 B c^2 \sin (e+f x)+480 A c d \sin (e+f x)+510 B c d \sin (e+f x)+255 A d^2 \sin (e+f x)+270 B d^2 \sin (e+f x)-30 B c d \sin (3 (e+f x))-15 A d^2 \sin (3 (e+f x))-30 B d^2 \sin (3 (e+f x))\right )\right )}{240 f \sqrt {\cos ^2(e+f x)}} \] Input:

Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]) 
^2,x]
 

Output:

-1/240*(a^2*Cos[e + f*x]*(60*(2*B*(4*c^2 + 7*c*d + 3*d^2) + A*(12*c^2 + 16 
*c*d + 7*d^2))*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^ 
2]*(480*A*c^2 + 440*B*c^2 + 880*A*c*d + 800*B*c*d + 400*A*d^2 + 378*B*d^2 
- 8*(10*A*d*(c + d) + B*(5*c^2 + 20*c*d + 12*d^2))*Cos[2*(e + f*x)] + 6*B* 
d^2*Cos[4*(e + f*x)] + 120*A*c^2*Sin[e + f*x] + 240*B*c^2*Sin[e + f*x] + 4 
80*A*c*d*Sin[e + f*x] + 510*B*c*d*Sin[e + f*x] + 255*A*d^2*Sin[e + f*x] + 
270*B*d^2*Sin[e + f*x] - 30*B*c*d*Sin[3*(e + f*x)] - 15*A*d^2*Sin[3*(e + f 
*x)] - 30*B*d^2*Sin[3*(e + f*x)])))/(f*Sqrt[Cos[e + f*x]^2])
 

Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.343, Rules used = {3042, 3455, 3042, 3447, 3042, 3502, 25, 3042, 3232, 25, 3042, 3213}

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.

\(\displaystyle \int (a \sin (e+f x)+a)^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sin (e+f x)+a)^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2dx\)

\(\Big \downarrow \) 3455

\(\displaystyle \frac {\int (\sin (e+f x) a+a) (c+d \sin (e+f x))^2 (a (5 A d+B (c+3 d))-a (2 B (c-3 d)-5 A d) \sin (e+f x))dx}{5 d}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (c+d \sin (e+f x))^3}{5 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (\sin (e+f x) a+a) (c+d \sin (e+f x))^2 (a (5 A d+B (c+3 d))-a (2 B (c-3 d)-5 A d) \sin (e+f x))dx}{5 d}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (c+d \sin (e+f x))^3}{5 d f}\)

\(\Big \downarrow \) 3447

\(\displaystyle \frac {\int (c+d \sin (e+f x))^2 \left (-\left ((2 B (c-3 d)-5 A d) \sin ^2(e+f x) a^2\right )+(5 A d+B (c+3 d)) a^2+\left (a^2 (5 A d+B (c+3 d))-a^2 (2 B (c-3 d)-5 A d)\right ) \sin (e+f x)\right )dx}{5 d}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (c+d \sin (e+f x))^3}{5 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (c+d \sin (e+f x))^2 \left (-\left ((2 B (c-3 d)-5 A d) \sin (e+f x)^2 a^2\right )+(5 A d+B (c+3 d)) a^2+\left (a^2 (5 A d+B (c+3 d))-a^2 (2 B (c-3 d)-5 A d)\right ) \sin (e+f x)\right )dx}{5 d}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (c+d \sin (e+f x))^3}{5 d f}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {\int -(c+d \sin (e+f x))^2 \left (d (2 B c-35 A d-30 B d) a^2+\left (5 A (c-8 d) d-2 B \left (c^2-5 d c+18 d^2\right )\right ) \sin (e+f x) a^2\right )dx}{4 d}+\frac {a^2 (2 B (c-3 d)-5 A d) \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f}}{5 d}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (c+d \sin (e+f x))^3}{5 d f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a^2 (2 B (c-3 d)-5 A d) \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f}-\frac {\int (c+d \sin (e+f x))^2 \left (d (2 B (c-15 d)-35 A d) a^2+\left (5 A (c-8 d) d-2 B \left (c^2-5 d c+18 d^2\right )\right ) \sin (e+f x) a^2\right )dx}{4 d}}{5 d}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (c+d \sin (e+f x))^3}{5 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {a^2 (2 B (c-3 d)-5 A d) \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f}-\frac {\int (c+d \sin (e+f x))^2 \left (d (2 B (c-15 d)-35 A d) a^2+\left (5 A (c-8 d) d-2 B \left (c^2-5 d c+18 d^2\right )\right ) \sin (e+f x) a^2\right )dx}{4 d}}{5 d}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (c+d \sin (e+f x))^3}{5 d f}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {a^2 (2 B (c-3 d)-5 A d) \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f}-\frac {\frac {1}{3} \int -\left ((c+d \sin (e+f x)) \left (a^2 d \left (5 A d (19 c+16 d)-B \left (2 c^2-70 d c-72 d^2\right )\right )-a^2 \left (5 A d \left (2 c^2-16 d c-21 d^2\right )-2 B \left (2 c^3-10 d c^2+33 d^2 c+45 d^3\right )\right ) \sin (e+f x)\right )\right )dx-\frac {a^2 \left (5 A d (c-8 d)-2 B \left (c^2-5 c d+18 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}}{4 d}}{5 d}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (c+d \sin (e+f x))^3}{5 d f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a^2 (2 B (c-3 d)-5 A d) \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f}-\frac {-\frac {1}{3} \int (c+d \sin (e+f x)) \left (a^2 d \left (5 A d (19 c+16 d)-B \left (2 c^2-70 d c-72 d^2\right )\right )-a^2 \left (5 A d \left (2 c^2-16 d c-21 d^2\right )-2 B \left (2 c^3-10 d c^2+33 d^2 c+45 d^3\right )\right ) \sin (e+f x)\right )dx-\frac {a^2 \left (5 A d (c-8 d)-2 B \left (c^2-5 c d+18 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}}{4 d}}{5 d}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (c+d \sin (e+f x))^3}{5 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {a^2 (2 B (c-3 d)-5 A d) \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f}-\frac {-\frac {1}{3} \int (c+d \sin (e+f x)) \left (a^2 d \left (5 A d (19 c+16 d)-B \left (2 c^2-70 d c-72 d^2\right )\right )-a^2 \left (5 A d \left (2 c^2-16 d c-21 d^2\right )-2 B \left (2 c^3-10 d c^2+33 d^2 c+45 d^3\right )\right ) \sin (e+f x)\right )dx-\frac {a^2 \left (5 A d (c-8 d)-2 B \left (c^2-5 c d+18 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}}{4 d}}{5 d}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (c+d \sin (e+f x))^3}{5 d f}\)

\(\Big \downarrow \) 3213

\(\displaystyle \frac {\frac {a^2 (2 B (c-3 d)-5 A d) \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f}-\frac {\frac {1}{3} \left (-\frac {15}{2} a^2 d^2 x \left (12 A c^2+16 A c d+7 A d^2+8 B c^2+14 B c d+6 B d^2\right )-\frac {a^2 d \left (5 A d \left (2 c^2-16 c d-21 d^2\right )-B \left (4 c^3-20 c^2 d+66 c d^2+90 d^3\right )\right ) \sin (e+f x) \cos (e+f x)}{2 f}-\frac {2 a^2 \left (5 A d \left (c^3-8 c^2 d-20 c d^2-8 d^3\right )-2 B \left (c^4-5 c^3 d+16 c^2 d^2+40 c d^3+18 d^4\right )\right ) \cos (e+f x)}{f}\right )-\frac {a^2 \left (5 A d (c-8 d)-2 B \left (c^2-5 c d+18 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}}{4 d}}{5 d}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (c+d \sin (e+f x))^3}{5 d f}\)

Input:

Int[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]
 

Output:

-1/5*(B*Cos[e + f*x]*(a^2 + a^2*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(d*f 
) + ((a^2*(2*B*(c - 3*d) - 5*A*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(4* 
d*f) - (-1/3*(a^2*(5*A*(c - 8*d)*d - 2*B*(c^2 - 5*c*d + 18*d^2))*Cos[e + f 
*x]*(c + d*Sin[e + f*x])^2)/f + ((-15*a^2*d^2*(12*A*c^2 + 8*B*c^2 + 16*A*c 
*d + 14*B*c*d + 7*A*d^2 + 6*B*d^2)*x)/2 - (2*a^2*(5*A*d*(c^3 - 8*c^2*d - 2 
0*c*d^2 - 8*d^3) - 2*B*(c^4 - 5*c^3*d + 16*c^2*d^2 + 40*c*d^3 + 18*d^4))*C 
os[e + f*x])/f - (a^2*d*(5*A*d*(2*c^2 - 16*c*d - 21*d^2) - B*(4*c^3 - 20*c 
^2*d + 66*c*d^2 + 90*d^3))*Cos[e + f*x]*Sin[e + f*x])/(2*f))/3)/(4*d))/(5* 
d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

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

rule 3213

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) 
*(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co 
s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free 
Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
 

rule 3232

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( 
f*(m + 1))), x] + Simp[1/(m + 1)   Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* 
d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ 
[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 
 0] && IntegerQ[2*m]
 

rule 3447

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a 
 + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), 
 x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
 

rule 3455

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n 
 + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1))   Int[(a + b*Sin[e + 
f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 
) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + 
f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 
 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1 
] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
 

rule 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m 
+ 2))   Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

Maple [A] (verified)

Time = 296.27 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.66

method	result	size
parallelrisch	\(\frac {a^{2} \left (\left (\left (-3 A -3 B \right ) d^{2}-6 c \left (A +B \right ) d -\frac {3 c^{2} \left (A +2 B \right )}{2}\right ) \sin \left (2 f x +2 e \right )+\left (\left (A +\frac {9 B}{8}\right ) d^{2}+c \left (A +2 B \right ) d +\frac {B \,c^{2}}{2}\right ) \cos \left (3 f x +3 e \right )+\frac {3 \left (\left (A +2 B \right ) d +2 B c \right ) d \sin \left (4 f x +4 e \right )}{16}-\frac {3 B \,d^{2} \cos \left (5 f x +5 e \right )}{40}+\left (\left (-\frac {33 B}{4}-9 A \right ) d^{2}-21 \left (A +\frac {6 B}{7}\right ) c d -12 c^{2} \left (A +\frac {7 B}{8}\right )\right ) \cos \left (f x +e \right )+\left (-\frac {36}{5} B +\frac {21}{4} f x A +\frac {9}{2} f x B -8 A \right ) d^{2}+12 \left (f x A +\frac {7}{8} f x B -\frac {5}{3} A -\frac {4}{3} B \right ) c d +9 c^{2} \left (f x A +\frac {2}{3} f x B -\frac {4}{3} A -\frac {10}{9} B \right )\right )}{6 f}\)	\(222\)
parts	\(\frac {\left (a^{2} A \,d^{2}+2 a^{2} B c d +2 a^{2} B \,d^{2}\right ) \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {\left (2 a^{2} A \,c^{2}+2 a^{2} A c d +a^{2} B \,c^{2}\right ) \cos \left (f x +e \right )}{f}-\frac {\left (2 a^{2} A c d +2 a^{2} A \,d^{2}+a^{2} B \,c^{2}+4 a^{2} B c d +a^{2} B \,d^{2}\right ) \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3 f}+\frac {\left (a^{2} A \,c^{2}+4 a^{2} A c d +a^{2} A \,d^{2}+2 a^{2} B \,c^{2}+2 a^{2} B c d \right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+a^{2} A \,c^{2} x -\frac {a^{2} B \,d^{2} \left (\frac {8}{3}+\sin \left (f x +e \right )^{4}+\frac {4 \sin \left (f x +e \right )^{2}}{3}\right ) \cos \left (f x +e \right )}{5 f}\)	\(280\)
risch	\(-\frac {B \,a^{2} d^{2} \cos \left (5 f x +5 e \right )}{80 f}+\frac {a^{2} \cos \left (3 f x +3 e \right ) B c d}{3 f}+\frac {3 a^{2} A \,c^{2} x}{2}-\frac {7 a^{2} \cos \left (f x +e \right ) A c d}{2 f}+B \,a^{2} c^{2} x +\frac {3 B \,a^{2} d^{2} x}{4}-\frac {\sin \left (2 f x +2 e \right ) a^{2} A c d}{f}-\frac {\sin \left (2 f x +2 e \right ) a^{2} B c d}{f}+2 A \,a^{2} c d x +\frac {7 B \,a^{2} c d x}{4}-\frac {2 a^{2} \cos \left (f x +e \right ) A \,c^{2}}{f}-\frac {3 a^{2} \cos \left (f x +e \right ) A \,d^{2}}{2 f}-\frac {7 a^{2} \cos \left (f x +e \right ) B \,c^{2}}{4 f}-\frac {11 a^{2} \cos \left (f x +e \right ) B \,d^{2}}{8 f}+\frac {\sin \left (4 f x +4 e \right ) a^{2} A \,d^{2}}{32 f}+\frac {\sin \left (4 f x +4 e \right ) a^{2} B \,d^{2}}{16 f}-\frac {3 a^{2} \cos \left (f x +e \right ) B c d}{f}+\frac {\sin \left (4 f x +4 e \right ) a^{2} B c d}{16 f}+\frac {a^{2} \cos \left (3 f x +3 e \right ) A c d}{6 f}+\frac {7 A \,a^{2} d^{2} x}{8}+\frac {a^{2} \cos \left (3 f x +3 e \right ) A \,d^{2}}{6 f}+\frac {a^{2} \cos \left (3 f x +3 e \right ) B \,c^{2}}{12 f}+\frac {3 a^{2} \cos \left (3 f x +3 e \right ) B \,d^{2}}{16 f}-\frac {\sin \left (2 f x +2 e \right ) a^{2} A \,c^{2}}{4 f}-\frac {\sin \left (2 f x +2 e \right ) a^{2} A \,d^{2}}{2 f}-\frac {\sin \left (2 f x +2 e \right ) a^{2} B \,c^{2}}{2 f}-\frac {\sin \left (2 f x +2 e \right ) a^{2} B \,d^{2}}{2 f}\)	\(475\)
derivativedivides	\(\frac {a^{2} A \,c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 a^{2} A c d \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}+a^{2} A \,d^{2} \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{2} B \,c^{2} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}+2 a^{2} B c d \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{2} B \,d^{2} \left (\frac {8}{3}+\sin \left (f x +e \right )^{4}+\frac {4 \sin \left (f x +e \right )^{2}}{3}\right ) \cos \left (f x +e \right )}{5}-2 a^{2} A \,c^{2} \cos \left (f x +e \right )+4 a^{2} A c d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 a^{2} A \,d^{2} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}+2 a^{2} B \,c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {4 a^{2} B c d \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}+2 a^{2} B \,d^{2} \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+a^{2} A \,c^{2} \left (f x +e \right )-2 a^{2} A c d \cos \left (f x +e \right )+a^{2} A \,d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{2} B \,c^{2} \cos \left (f x +e \right )+2 a^{2} B c d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {a^{2} B \,d^{2} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}}{f}\)	\(496\)
default	\(\frac {a^{2} A \,c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 a^{2} A c d \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}+a^{2} A \,d^{2} \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{2} B \,c^{2} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}+2 a^{2} B c d \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{2} B \,d^{2} \left (\frac {8}{3}+\sin \left (f x +e \right )^{4}+\frac {4 \sin \left (f x +e \right )^{2}}{3}\right ) \cos \left (f x +e \right )}{5}-2 a^{2} A \,c^{2} \cos \left (f x +e \right )+4 a^{2} A c d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 a^{2} A \,d^{2} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}+2 a^{2} B \,c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {4 a^{2} B c d \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}+2 a^{2} B \,d^{2} \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+a^{2} A \,c^{2} \left (f x +e \right )-2 a^{2} A c d \cos \left (f x +e \right )+a^{2} A \,d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{2} B \,c^{2} \cos \left (f x +e \right )+2 a^{2} B c d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {a^{2} B \,d^{2} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}}{f}\)	\(496\)
norman	\(\frac {\left (\frac {3}{2} a^{2} A \,c^{2}+2 a^{2} A c d +\frac {7}{8} a^{2} A \,d^{2}+a^{2} B \,c^{2}+\frac {7}{4} a^{2} B c d +\frac {3}{4} a^{2} B \,d^{2}\right ) x +\left (15 a^{2} A \,c^{2}+20 a^{2} A c d +\frac {35}{4} a^{2} A \,d^{2}+10 a^{2} B \,c^{2}+\frac {35}{2} a^{2} B c d +\frac {15}{2} a^{2} B \,d^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (15 a^{2} A \,c^{2}+20 a^{2} A c d +\frac {35}{4} a^{2} A \,d^{2}+10 a^{2} B \,c^{2}+\frac {35}{2} a^{2} B c d +\frac {15}{2} a^{2} B \,d^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+\left (\frac {3}{2} a^{2} A \,c^{2}+2 a^{2} A c d +\frac {7}{8} a^{2} A \,d^{2}+a^{2} B \,c^{2}+\frac {7}{4} a^{2} B c d +\frac {3}{4} a^{2} B \,d^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}+\left (\frac {15}{2} a^{2} A \,c^{2}+10 a^{2} A c d +\frac {35}{8} a^{2} A \,d^{2}+5 a^{2} B \,c^{2}+\frac {35}{4} a^{2} B c d +\frac {15}{4} a^{2} B \,d^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (\frac {15}{2} a^{2} A \,c^{2}+10 a^{2} A c d +\frac {35}{8} a^{2} A \,d^{2}+5 a^{2} B \,c^{2}+\frac {35}{4} a^{2} B c d +\frac {15}{4} a^{2} B \,d^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-\frac {60 a^{2} A \,c^{2}+100 a^{2} A c d +40 a^{2} A \,d^{2}+50 a^{2} B \,c^{2}+80 a^{2} B c d +36 a^{2} B \,d^{2}}{15 f}-\frac {\left (4 a^{2} A \,c^{2}+4 a^{2} A c d +2 a^{2} B \,c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{f}-\frac {2 \left (8 a^{2} A \,c^{2}+12 a^{2} A c d +4 a^{2} A \,d^{2}+6 a^{2} B \,c^{2}+8 a^{2} B c d +2 a^{2} B \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{f}-\frac {2 \left (36 a^{2} A \,c^{2}+64 a^{2} A c d +28 a^{2} A \,d^{2}+32 a^{2} B \,c^{2}+56 a^{2} B c d +30 a^{2} B \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{3 f}-\frac {\left (48 a^{2} A \,c^{2}+88 a^{2} A c d +40 a^{2} A \,d^{2}+44 a^{2} B \,c^{2}+80 a^{2} B c d +36 a^{2} B \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{3 f}-\frac {a^{2} \left (4 A \,c^{2}+16 A c d +7 A \,d^{2}+8 B \,c^{2}+14 B c d +6 B \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {a^{2} \left (4 A \,c^{2}+16 A c d +7 A \,d^{2}+8 B \,c^{2}+14 B c d +6 B \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{4 f}-\frac {a^{2} \left (4 A \,c^{2}+16 A c d +11 A \,d^{2}+8 B \,c^{2}+22 B c d +14 B \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{2 f}+\frac {a^{2} \left (4 A \,c^{2}+16 A c d +11 A \,d^{2}+8 B \,c^{2}+22 B c d +14 B \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{2 f}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{5}}\)	\(923\)
orering	\(\text {Expression too large to display}\)	\(8158\)

Input:

int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x,method=_RETUR 
NVERBOSE)
 

Output:

1/6*a^2*(((-3*A-3*B)*d^2-6*c*(A+B)*d-3/2*c^2*(A+2*B))*sin(2*f*x+2*e)+((A+9 
/8*B)*d^2+c*(A+2*B)*d+1/2*B*c^2)*cos(3*f*x+3*e)+3/16*((A+2*B)*d+2*B*c)*d*s 
in(4*f*x+4*e)-3/40*B*d^2*cos(5*f*x+5*e)+((-33/4*B-9*A)*d^2-21*(A+6/7*B)*c* 
d-12*c^2*(A+7/8*B))*cos(f*x+e)+(-36/5*B+21/4*f*x*A+9/2*f*x*B-8*A)*d^2+12*( 
f*x*A+7/8*f*x*B-5/3*A-4/3*B)*c*d+9*c^2*(f*x*A+2/3*f*x*B-4/3*A-10/9*B))/f
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.73 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {24 \, B a^{2} d^{2} \cos \left (f x + e\right )^{5} - 40 \, {\left (B a^{2} c^{2} + 2 \, {\left (A + 2 \, B\right )} a^{2} c d + {\left (2 \, A + 3 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (4 \, {\left (3 \, A + 2 \, B\right )} a^{2} c^{2} + 2 \, {\left (8 \, A + 7 \, B\right )} a^{2} c d + {\left (7 \, A + 6 \, B\right )} a^{2} d^{2}\right )} f x + 240 \, {\left ({\left (A + B\right )} a^{2} c^{2} + 2 \, {\left (A + B\right )} a^{2} c d + {\left (A + B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right ) - 15 \, {\left (2 \, {\left (2 \, B a^{2} c d + {\left (A + 2 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (4 \, {\left (A + 2 \, B\right )} a^{2} c^{2} + 2 \, {\left (8 \, A + 9 \, B\right )} a^{2} c d + {\left (9 \, A + 10 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{120 \, f} \] Input:

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, algori 
thm="fricas")
 

Output:

-1/120*(24*B*a^2*d^2*cos(f*x + e)^5 - 40*(B*a^2*c^2 + 2*(A + 2*B)*a^2*c*d 
+ (2*A + 3*B)*a^2*d^2)*cos(f*x + e)^3 - 15*(4*(3*A + 2*B)*a^2*c^2 + 2*(8*A 
 + 7*B)*a^2*c*d + (7*A + 6*B)*a^2*d^2)*f*x + 240*((A + B)*a^2*c^2 + 2*(A + 
 B)*a^2*c*d + (A + B)*a^2*d^2)*cos(f*x + e) - 15*(2*(2*B*a^2*c*d + (A + 2* 
B)*a^2*d^2)*cos(f*x + e)^3 - (4*(A + 2*B)*a^2*c^2 + 2*(8*A + 9*B)*a^2*c*d 
+ (9*A + 10*B)*a^2*d^2)*cos(f*x + e))*sin(f*x + e))/f
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1129 vs. \(2 (330) = 660\).

Time = 0.37 (sec) , antiderivative size = 1129, normalized size of antiderivative = 3.36 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))**2*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))**2,x)
 

Output:

Piecewise((A*a**2*c**2*x*sin(e + f*x)**2/2 + A*a**2*c**2*x*cos(e + f*x)**2 
/2 + A*a**2*c**2*x - A*a**2*c**2*sin(e + f*x)*cos(e + f*x)/(2*f) - 2*A*a** 
2*c**2*cos(e + f*x)/f + 2*A*a**2*c*d*x*sin(e + f*x)**2 + 2*A*a**2*c*d*x*co 
s(e + f*x)**2 - 2*A*a**2*c*d*sin(e + f*x)**2*cos(e + f*x)/f - 2*A*a**2*c*d 
*sin(e + f*x)*cos(e + f*x)/f - 4*A*a**2*c*d*cos(e + f*x)**3/(3*f) - 2*A*a* 
*2*c*d*cos(e + f*x)/f + 3*A*a**2*d**2*x*sin(e + f*x)**4/8 + 3*A*a**2*d**2* 
x*sin(e + f*x)**2*cos(e + f*x)**2/4 + A*a**2*d**2*x*sin(e + f*x)**2/2 + 3* 
A*a**2*d**2*x*cos(e + f*x)**4/8 + A*a**2*d**2*x*cos(e + f*x)**2/2 - 5*A*a* 
*2*d**2*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 2*A*a**2*d**2*sin(e + f*x)**2 
*cos(e + f*x)/f - 3*A*a**2*d**2*sin(e + f*x)*cos(e + f*x)**3/(8*f) - A*a** 
2*d**2*sin(e + f*x)*cos(e + f*x)/(2*f) - 4*A*a**2*d**2*cos(e + f*x)**3/(3* 
f) + B*a**2*c**2*x*sin(e + f*x)**2 + B*a**2*c**2*x*cos(e + f*x)**2 - B*a** 
2*c**2*sin(e + f*x)**2*cos(e + f*x)/f - B*a**2*c**2*sin(e + f*x)*cos(e + f 
*x)/f - 2*B*a**2*c**2*cos(e + f*x)**3/(3*f) - B*a**2*c**2*cos(e + f*x)/f + 
 3*B*a**2*c*d*x*sin(e + f*x)**4/4 + 3*B*a**2*c*d*x*sin(e + f*x)**2*cos(e + 
 f*x)**2/2 + B*a**2*c*d*x*sin(e + f*x)**2 + 3*B*a**2*c*d*x*cos(e + f*x)**4 
/4 + B*a**2*c*d*x*cos(e + f*x)**2 - 5*B*a**2*c*d*sin(e + f*x)**3*cos(e + f 
*x)/(4*f) - 4*B*a**2*c*d*sin(e + f*x)**2*cos(e + f*x)/f - 3*B*a**2*c*d*sin 
(e + f*x)*cos(e + f*x)**3/(4*f) - B*a**2*c*d*sin(e + f*x)*cos(e + f*x)/f - 
 8*B*a**2*c*d*cos(e + f*x)**3/(3*f) + 3*B*a**2*d**2*x*sin(e + f*x)**4/4...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.42 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {120 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c^{2} + 480 \, {\left (f x + e\right )} A a^{2} c^{2} + 160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c^{2} + 240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{2} + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} c d + 480 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c d + 640 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c d + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c d + 240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c d + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} d^{2} + 15 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} d^{2} + 120 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} d^{2} - 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a^{2} d^{2} + 160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} d^{2} + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} d^{2} - 960 \, A a^{2} c^{2} \cos \left (f x + e\right ) - 480 \, B a^{2} c^{2} \cos \left (f x + e\right ) - 960 \, A a^{2} c d \cos \left (f x + e\right )}{480 \, f} \] Input:

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, algori 
thm="maxima")
 

Output:

1/480*(120*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a^2*c^2 + 480*(f*x + e)*A*a^ 
2*c^2 + 160*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^2*c^2 + 240*(2*f*x + 2*e 
 - sin(2*f*x + 2*e))*B*a^2*c^2 + 320*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a 
^2*c*d + 480*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a^2*c*d + 640*(cos(f*x + e 
)^3 - 3*cos(f*x + e))*B*a^2*c*d + 30*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8 
*sin(2*f*x + 2*e))*B*a^2*c*d + 240*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^2* 
c*d + 320*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a^2*d^2 + 15*(12*f*x + 12*e 
+ sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^2*d^2 + 120*(2*f*x + 2*e - si 
n(2*f*x + 2*e))*A*a^2*d^2 - 32*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15* 
cos(f*x + e))*B*a^2*d^2 + 160*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^2*d^2 
+ 30*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a^2*d^2 - 9 
60*A*a^2*c^2*cos(f*x + e) - 480*B*a^2*c^2*cos(f*x + e) - 960*A*a^2*c*d*cos 
(f*x + e))/f
 

Giac [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.91 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {B a^{2} d^{2} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {1}{8} \, {\left (12 \, A a^{2} c^{2} + 8 \, B a^{2} c^{2} + 16 \, A a^{2} c d + 14 \, B a^{2} c d + 7 \, A a^{2} d^{2} + 6 \, B a^{2} d^{2}\right )} x + \frac {{\left (4 \, B a^{2} c^{2} + 8 \, A a^{2} c d + 16 \, B a^{2} c d + 8 \, A a^{2} d^{2} + 9 \, B a^{2} d^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (16 \, A a^{2} c^{2} + 14 \, B a^{2} c^{2} + 28 \, A a^{2} c d + 24 \, B a^{2} c d + 12 \, A a^{2} d^{2} + 11 \, B a^{2} d^{2}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {{\left (2 \, B a^{2} c d + A a^{2} d^{2} + 2 \, B a^{2} d^{2}\right )} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac {{\left (A a^{2} c^{2} + 2 \, B a^{2} c^{2} + 4 \, A a^{2} c d + 4 \, B a^{2} c d + 2 \, A a^{2} d^{2} + 2 \, B a^{2} d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \] Input:

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, algori 
thm="giac")
 

Output:

-1/80*B*a^2*d^2*cos(5*f*x + 5*e)/f + 1/8*(12*A*a^2*c^2 + 8*B*a^2*c^2 + 16* 
A*a^2*c*d + 14*B*a^2*c*d + 7*A*a^2*d^2 + 6*B*a^2*d^2)*x + 1/48*(4*B*a^2*c^ 
2 + 8*A*a^2*c*d + 16*B*a^2*c*d + 8*A*a^2*d^2 + 9*B*a^2*d^2)*cos(3*f*x + 3* 
e)/f - 1/8*(16*A*a^2*c^2 + 14*B*a^2*c^2 + 28*A*a^2*c*d + 24*B*a^2*c*d + 12 
*A*a^2*d^2 + 11*B*a^2*d^2)*cos(f*x + e)/f + 1/32*(2*B*a^2*c*d + A*a^2*d^2 
+ 2*B*a^2*d^2)*sin(4*f*x + 4*e)/f - 1/4*(A*a^2*c^2 + 2*B*a^2*c^2 + 4*A*a^2 
*c*d + 4*B*a^2*c*d + 2*A*a^2*d^2 + 2*B*a^2*d^2)*sin(2*f*x + 2*e)/f
 

Mupad [B] (verification not implemented)

Time = 37.56 (sec) , antiderivative size = 765, normalized size of antiderivative = 2.28 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx =\text {Too large to display} \] Input:

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^2,x)
 

Output:

(a^2*atan((a^2*tan(e/2 + (f*x)/2)*(12*A*c^2 + 7*A*d^2 + 8*B*c^2 + 6*B*d^2 
+ 16*A*c*d + 14*B*c*d))/(4*(3*A*a^2*c^2 + (7*A*a^2*d^2)/4 + 2*B*a^2*c^2 + 
(3*B*a^2*d^2)/2 + 4*A*a^2*c*d + (7*B*a^2*c*d)/2)))*(12*A*c^2 + 7*A*d^2 + 8 
*B*c^2 + 6*B*d^2 + 16*A*c*d + 14*B*c*d))/(4*f) - (tan(e/2 + (f*x)/2)^8*(4* 
A*a^2*c^2 + 2*B*a^2*c^2 + 4*A*a^2*c*d) + tan(e/2 + (f*x)/2)*(A*a^2*c^2 + ( 
7*A*a^2*d^2)/4 + 2*B*a^2*c^2 + (3*B*a^2*d^2)/2 + 4*A*a^2*c*d + (7*B*a^2*c* 
d)/2) - tan(e/2 + (f*x)/2)^9*(A*a^2*c^2 + (7*A*a^2*d^2)/4 + 2*B*a^2*c^2 + 
(3*B*a^2*d^2)/2 + 4*A*a^2*c*d + (7*B*a^2*c*d)/2) + tan(e/2 + (f*x)/2)^3*(2 
*A*a^2*c^2 + (11*A*a^2*d^2)/2 + 4*B*a^2*c^2 + 7*B*a^2*d^2 + 8*A*a^2*c*d + 
11*B*a^2*c*d) - tan(e/2 + (f*x)/2)^7*(2*A*a^2*c^2 + (11*A*a^2*d^2)/2 + 4*B 
*a^2*c^2 + 7*B*a^2*d^2 + 8*A*a^2*c*d + 11*B*a^2*c*d) + tan(e/2 + (f*x)/2)^ 
6*(16*A*a^2*c^2 + 8*A*a^2*d^2 + 12*B*a^2*c^2 + 4*B*a^2*d^2 + 24*A*a^2*c*d 
+ 16*B*a^2*c*d) + tan(e/2 + (f*x)/2)^2*(16*A*a^2*c^2 + (40*A*a^2*d^2)/3 + 
(44*B*a^2*c^2)/3 + 12*B*a^2*d^2 + (88*A*a^2*c*d)/3 + (80*B*a^2*c*d)/3) + t 
an(e/2 + (f*x)/2)^4*(24*A*a^2*c^2 + (56*A*a^2*d^2)/3 + (64*B*a^2*c^2)/3 + 
20*B*a^2*d^2 + (128*A*a^2*c*d)/3 + (112*B*a^2*c*d)/3) + 4*A*a^2*c^2 + (8*A 
*a^2*d^2)/3 + (10*B*a^2*c^2)/3 + (12*B*a^2*d^2)/5 + (20*A*a^2*c*d)/3 + (16 
*B*a^2*c*d)/3)/(f*(5*tan(e/2 + (f*x)/2)^2 + 10*tan(e/2 + (f*x)/2)^4 + 10*t 
an(e/2 + (f*x)/2)^6 + 5*tan(e/2 + (f*x)/2)^8 + tan(e/2 + (f*x)/2)^10 + 1))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.32 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {a^{2} \left (-60 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b \,d^{2}-80 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a \,d^{2}-72 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b \,d^{2}-105 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a \,d^{2}-120 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b \,c^{2}-90 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b \,d^{2}-400 \cos \left (f x +e \right ) a c d -320 \cos \left (f x +e \right ) b c d +180 a \,c^{2} f x +105 a \,d^{2} f x +120 b \,c^{2} f x +90 b \,d^{2} f x -160 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b c d -240 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a c d -210 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b c d +240 a c d f x +210 b c d f x +160 a \,d^{2}+200 b \,c^{2}+144 b \,d^{2}+240 a \,c^{2}-40 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b \,c^{2}-60 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a \,c^{2}-60 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b c d -80 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a c d -24 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} b \,d^{2}-30 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a \,d^{2}-240 \cos \left (f x +e \right ) a \,c^{2}-160 \cos \left (f x +e \right ) a \,d^{2}-200 \cos \left (f x +e \right ) b \,c^{2}-144 \cos \left (f x +e \right ) b \,d^{2}+400 a c d +320 b c d \right )}{120 f} \] Input:

int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x)
 

Output:

(a**2*( - 24*cos(e + f*x)*sin(e + f*x)**4*b*d**2 - 30*cos(e + f*x)*sin(e + 
 f*x)**3*a*d**2 - 60*cos(e + f*x)*sin(e + f*x)**3*b*c*d - 60*cos(e + f*x)* 
sin(e + f*x)**3*b*d**2 - 80*cos(e + f*x)*sin(e + f*x)**2*a*c*d - 80*cos(e 
+ f*x)*sin(e + f*x)**2*a*d**2 - 40*cos(e + f*x)*sin(e + f*x)**2*b*c**2 - 1 
60*cos(e + f*x)*sin(e + f*x)**2*b*c*d - 72*cos(e + f*x)*sin(e + f*x)**2*b* 
d**2 - 60*cos(e + f*x)*sin(e + f*x)*a*c**2 - 240*cos(e + f*x)*sin(e + f*x) 
*a*c*d - 105*cos(e + f*x)*sin(e + f*x)*a*d**2 - 120*cos(e + f*x)*sin(e + f 
*x)*b*c**2 - 210*cos(e + f*x)*sin(e + f*x)*b*c*d - 90*cos(e + f*x)*sin(e + 
 f*x)*b*d**2 - 240*cos(e + f*x)*a*c**2 - 400*cos(e + f*x)*a*c*d - 160*cos( 
e + f*x)*a*d**2 - 200*cos(e + f*x)*b*c**2 - 320*cos(e + f*x)*b*c*d - 144*c 
os(e + f*x)*b*d**2 + 180*a*c**2*f*x + 240*a*c**2 + 240*a*c*d*f*x + 400*a*c 
*d + 105*a*d**2*f*x + 160*a*d**2 + 120*b*c**2*f*x + 200*b*c**2 + 210*b*c*d 
*f*x + 320*b*c*d + 90*b*d**2*f*x + 144*b*d**2))/(120*f)