\(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx\) [258]

Optimal result

Integrand size = 35, antiderivative size = 604 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\frac {1}{16} a^3 \left (3 B \left (10 c^3+26 c^2 d+23 c d^2+7 d^3\right )+A \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right )\right ) x-\frac {a^3 \left (7 A d \left (2 c^5-18 c^4 d+107 c^3 d^2+472 c^2 d^3+456 c d^4+136 d^5\right )-3 B \left (2 c^6-14 c^5 d+51 c^4 d^2-189 c^3 d^3-920 c^2 d^4-952 c d^5-288 d^6\right )\right ) \cos (e+f x)}{420 d^3 f}-\frac {a^3 \left (7 A d \left (4 c^4-36 c^3 d+216 c^2 d^2+626 c d^3+345 d^4\right )-3 B \left (4 c^5-28 c^4 d+104 c^3 d^2-392 c^2 d^3-1263 c d^4-735 d^5\right )\right ) \cos (e+f x) \sin (e+f x)}{1680 d^2 f}-\frac {a^3 \left (7 A d \left (2 c^3-18 c^2 d+111 c d^2+136 d^3\right )-B \left (6 c^4-42 c^3 d+165 c^2 d^2-651 c d^3-864 d^4\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{840 d^3 f}-\frac {a^3 \left (7 A d \left (2 c^2-18 c d+115 d^2\right )-B \left (6 c^3-42 c^2 d+177 c d^2-735 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{840 d^3 f}-\frac {a^3 \left (6 B c^2-14 A c d-27 B c d+91 A d^2+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{210 d^3 f}-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^4}{7 d f}+\frac {(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{42 d^2 f} \] Output:

1/16*a^3*(3*B*(10*c^3+26*c^2*d+23*c*d^2+7*d^3)+A*(40*c^3+90*c^2*d+78*c*d^2 
+23*d^3))*x-1/420*a^3*(7*A*d*(2*c^5-18*c^4*d+107*c^3*d^2+472*c^2*d^3+456*c 
*d^4+136*d^5)-3*B*(2*c^6-14*c^5*d+51*c^4*d^2-189*c^3*d^3-920*c^2*d^4-952*c 
*d^5-288*d^6))*cos(f*x+e)/d^3/f-1/1680*a^3*(7*A*d*(4*c^4-36*c^3*d+216*c^2* 
d^2+626*c*d^3+345*d^4)-3*B*(4*c^5-28*c^4*d+104*c^3*d^2-392*c^2*d^3-1263*c* 
d^4-735*d^5))*cos(f*x+e)*sin(f*x+e)/d^2/f-1/840*a^3*(7*A*d*(2*c^3-18*c^2*d 
+111*c*d^2+136*d^3)-B*(6*c^4-42*c^3*d+165*c^2*d^2-651*c*d^3-864*d^4))*cos( 
f*x+e)*(c+d*sin(f*x+e))^2/d^3/f-1/840*a^3*(7*A*d*(2*c^2-18*c*d+115*d^2)-B* 
(6*c^3-42*c^2*d+177*c*d^2-735*d^3))*cos(f*x+e)*(c+d*sin(f*x+e))^3/d^3/f-1/ 
210*a^3*(-14*A*c*d+91*A*d^2+6*B*c^2-27*B*c*d+87*B*d^2)*cos(f*x+e)*(c+d*sin 
(f*x+e))^4/d^3/f-1/7*a*B*cos(f*x+e)*(a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^4/ 
d/f+1/42*(3*B*(c-3*d)-7*A*d)*cos(f*x+e)*(a^3+a^3*sin(f*x+e))*(c+d*sin(f*x+ 
e))^4/d^2/f
 

Mathematica [A] (warning: unable to verify)

Time = 5.65 (sec) , antiderivative size = 528, normalized size of antiderivative = 0.87 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=-\frac {a^3 \cos (e+f x) \left (420 \left (3 B \left (10 c^3+26 c^2 d+23 c d^2+7 d^3\right )+A \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right )\right ) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (12880 A c^3+11760 B c^3+35280 A c^2 d+32676 B c^2 d+32676 A c d^2+30828 B c d^2+10276 A d^3+9762 B d^3-\left (112 A \left (5 c^3+45 c^2 d+66 c d^2+26 d^3\right )+3 B \left (560 c^3+2464 c^2 d+2912 c d^2+1083 d^3\right )\right ) \cos (2 (e+f x))+18 d \left (14 A d (c+d)+B \left (14 c^2+42 c d+23 d^2\right )\right ) \cos (4 (e+f x))-15 B d^3 \cos (6 (e+f x))+5040 A c^3 \sin (e+f x)+6930 B c^3 \sin (e+f x)+20790 A c^2 d \sin (e+f x)+22050 B c^2 d \sin (e+f x)+22050 A c d^2 \sin (e+f x)+22785 B c d^2 \sin (e+f x)+7595 A d^3 \sin (e+f x)+7665 B d^3 \sin (e+f x)-210 B c^3 \sin (3 (e+f x))-630 A c^2 d \sin (3 (e+f x))-1890 B c^2 d \sin (3 (e+f x))-1890 A c d^2 \sin (3 (e+f x))-2940 B c d^2 \sin (3 (e+f x))-980 A d^3 \sin (3 (e+f x))-1260 B d^3 \sin (3 (e+f x))+105 B c d^2 \sin (5 (e+f x))+35 A d^3 \sin (5 (e+f x))+105 B d^3 \sin (5 (e+f x))\right )\right )}{3360 f \sqrt {\cos ^2(e+f x)}} \] Input:

Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]) 
^3,x]
 

Output:

-1/3360*(a^3*Cos[e + f*x]*(420*(3*B*(10*c^3 + 26*c^2*d + 23*c*d^2 + 7*d^3) 
 + A*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3))*ArcSin[Sqrt[1 - Sin[e + f*x] 
]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(12880*A*c^3 + 11760*B*c^3 + 35280*A*c^2 
*d + 32676*B*c^2*d + 32676*A*c*d^2 + 30828*B*c*d^2 + 10276*A*d^3 + 9762*B* 
d^3 - (112*A*(5*c^3 + 45*c^2*d + 66*c*d^2 + 26*d^3) + 3*B*(560*c^3 + 2464* 
c^2*d + 2912*c*d^2 + 1083*d^3))*Cos[2*(e + f*x)] + 18*d*(14*A*d*(c + d) + 
B*(14*c^2 + 42*c*d + 23*d^2))*Cos[4*(e + f*x)] - 15*B*d^3*Cos[6*(e + f*x)] 
 + 5040*A*c^3*Sin[e + f*x] + 6930*B*c^3*Sin[e + f*x] + 20790*A*c^2*d*Sin[e 
 + f*x] + 22050*B*c^2*d*Sin[e + f*x] + 22050*A*c*d^2*Sin[e + f*x] + 22785* 
B*c*d^2*Sin[e + f*x] + 7595*A*d^3*Sin[e + f*x] + 7665*B*d^3*Sin[e + f*x] - 
 210*B*c^3*Sin[3*(e + f*x)] - 630*A*c^2*d*Sin[3*(e + f*x)] - 1890*B*c^2*d* 
Sin[3*(e + f*x)] - 1890*A*c*d^2*Sin[3*(e + f*x)] - 2940*B*c*d^2*Sin[3*(e + 
 f*x)] - 980*A*d^3*Sin[3*(e + f*x)] - 1260*B*d^3*Sin[3*(e + f*x)] + 105*B* 
c*d^2*Sin[5*(e + f*x)] + 35*A*d^3*Sin[5*(e + f*x)] + 105*B*d^3*Sin[5*(e + 
f*x)])))/(f*Sqrt[Cos[e + f*x]^2])
 

Rubi [A] (verified)

Time = 2.63 (sec) , antiderivative size = 628, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.457, Rules used = {3042, 3455, 3042, 3455, 3042, 3447, 3042, 3502, 25, 3042, 3232, 27, 3042, 3232, 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.

\(\displaystyle \int (a \sin (e+f x)+a)^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sin (e+f x)+a)^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3dx\)

\(\Big \downarrow \) 3455

\(\displaystyle \frac {\int (\sin (e+f x) a+a)^2 (c+d \sin (e+f x))^3 (a (7 A d+2 B (c+2 d))-a (3 B (c-3 d)-7 A d) \sin (e+f x))dx}{7 d}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (\sin (e+f x) a+a)^2 (c+d \sin (e+f x))^3 (a (7 A d+2 B (c+2 d))-a (3 B (c-3 d)-7 A d) \sin (e+f x))dx}{7 d}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}\)

\(\Big \downarrow \) 3455

\(\displaystyle \frac {\frac {\int (\sin (e+f x) a+a) (c+d \sin (e+f x))^3 \left (\left (7 A d (c+10 d)-B \left (3 c^2-9 d c-60 d^2\right )\right ) a^2+\left (6 B c^2-14 A d c-27 B d c+91 A d^2+87 B d^2\right ) \sin (e+f x) a^2\right )dx}{6 d}+\frac {(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}}{7 d}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int (\sin (e+f x) a+a) (c+d \sin (e+f x))^3 \left (\left (7 A d (c+10 d)-B \left (3 c^2-9 d c-60 d^2\right )\right ) a^2+\left (6 B c^2-14 A d c-27 B d c+91 A d^2+87 B d^2\right ) \sin (e+f x) a^2\right )dx}{6 d}+\frac {(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}}{7 d}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}\)

\(\Big \downarrow \) 3447

\(\displaystyle \frac {\frac {\int (c+d \sin (e+f x))^3 \left (\left (6 B c^2-14 A d c-27 B d c+91 A d^2+87 B d^2\right ) \sin ^2(e+f x) a^3+\left (7 A d (c+10 d)-B \left (3 c^2-9 d c-60 d^2\right )\right ) a^3+\left (\left (6 B c^2-14 A d c-27 B d c+91 A d^2+87 B d^2\right ) a^3+\left (7 A d (c+10 d)-B \left (3 c^2-9 d c-60 d^2\right )\right ) a^3\right ) \sin (e+f x)\right )dx}{6 d}+\frac {(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}}{7 d}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int (c+d \sin (e+f x))^3 \left (\left (6 B c^2-14 A d c-27 B d c+91 A d^2+87 B d^2\right ) \sin (e+f x)^2 a^3+\left (7 A d (c+10 d)-B \left (3 c^2-9 d c-60 d^2\right )\right ) a^3+\left (\left (6 B c^2-14 A d c-27 B d c+91 A d^2+87 B d^2\right ) a^3+\left (7 A d (c+10 d)-B \left (3 c^2-9 d c-60 d^2\right )\right ) a^3\right ) \sin (e+f x)\right )dx}{6 d}+\frac {(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}}{7 d}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {\frac {\int -(c+d \sin (e+f x))^3 \left (3 a^3 d \left (7 A (c-34 d) d-3 B \left (c^2-7 d c+72 d^2\right )\right )-a^3 \left (7 A d \left (2 c^2-18 d c+115 d^2\right )-B \left (6 c^3-42 d c^2+177 d^2 c-735 d^3\right )\right ) \sin (e+f x)\right )dx}{5 d}-\frac {a^3 \left (-14 A c d+91 A d^2+6 B c^2-27 B c d+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}}{6 d}+\frac {(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}}{7 d}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {-\frac {\int (c+d \sin (e+f x))^3 \left (3 a^3 d \left (7 A (c-34 d) d-3 B \left (c^2-7 d c+72 d^2\right )\right )-a^3 \left (7 A d \left (2 c^2-18 d c+115 d^2\right )-B \left (6 c^3-42 d c^2+177 d^2 c-735 d^3\right )\right ) \sin (e+f x)\right )dx}{5 d}-\frac {a^3 \left (-14 A c d+91 A d^2+6 B c^2-27 B c d+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}}{6 d}+\frac {(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}}{7 d}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\int (c+d \sin (e+f x))^3 \left (3 a^3 d \left (7 A (c-34 d) d-3 B \left (c^2-7 d c+72 d^2\right )\right )-a^3 \left (7 A d \left (2 c^2-18 d c+115 d^2\right )-B \left (6 c^3-42 d c^2+177 d^2 c-735 d^3\right )\right ) \sin (e+f x)\right )dx}{5 d}-\frac {a^3 \left (-14 A c d+91 A d^2+6 B c^2-27 B c d+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}}{6 d}+\frac {(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}}{7 d}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {-\frac {\frac {1}{4} \int 3 (c+d \sin (e+f x))^2 \left (a^3 d \left (7 A d \left (2 c^2-118 d c-115 d^2\right )-3 B \left (2 c^3-14 d c^2+229 d^2 c+245 d^3\right )\right )-a^3 \left (7 A d \left (2 c^3-18 d c^2+111 d^2 c+136 d^3\right )-B \left (6 c^4-42 d c^3+165 d^2 c^2-651 d^3 c-864 d^4\right )\right ) \sin (e+f x)\right )dx+\frac {a^3 \left (7 A d \left (2 c^2-18 c d+115 d^2\right )-B \left (6 c^3-42 c^2 d+177 c d^2-735 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}-\frac {a^3 \left (-14 A c d+91 A d^2+6 B c^2-27 B c d+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}}{6 d}+\frac {(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}}{7 d}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {\frac {3}{4} \int (c+d \sin (e+f x))^2 \left (a^3 d \left (7 A d \left (2 c^2-118 d c-115 d^2\right )-3 B \left (2 c^3-14 d c^2+229 d^2 c+245 d^3\right )\right )-a^3 \left (7 A d \left (2 c^3-18 d c^2+111 d^2 c+136 d^3\right )-B \left (6 c^4-42 d c^3+165 d^2 c^2-651 d^3 c-864 d^4\right )\right ) \sin (e+f x)\right )dx+\frac {a^3 \left (7 A d \left (2 c^2-18 c d+115 d^2\right )-B \left (6 c^3-42 c^2 d+177 c d^2-735 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}-\frac {a^3 \left (-14 A c d+91 A d^2+6 B c^2-27 B c d+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}}{6 d}+\frac {(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}}{7 d}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\frac {3}{4} \int (c+d \sin (e+f x))^2 \left (a^3 d \left (7 A d \left (2 c^2-118 d c-115 d^2\right )-3 B \left (2 c^3-14 d c^2+229 d^2 c+245 d^3\right )\right )-a^3 \left (7 A d \left (2 c^3-18 d c^2+111 d^2 c+136 d^3\right )-B \left (6 c^4-42 d c^3+165 d^2 c^2-651 d^3 c-864 d^4\right )\right ) \sin (e+f x)\right )dx+\frac {a^3 \left (7 A d \left (2 c^2-18 c d+115 d^2\right )-B \left (6 c^3-42 c^2 d+177 c d^2-735 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}-\frac {a^3 \left (-14 A c d+91 A d^2+6 B c^2-27 B c d+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}}{6 d}+\frac {(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}}{7 d}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {-\frac {\frac {3}{4} \left (\frac {1}{3} \int (c+d \sin (e+f x)) \left (a^3 d \left (7 A d \left (2 c^3-318 d c^2-567 d^2 c-272 d^3\right )-3 B \left (2 c^4-14 d c^3+577 d^2 c^2+1169 d^3 c+576 d^4\right )\right )-a^3 \left (7 A d \left (4 c^4-36 d c^3+216 d^2 c^2+626 d^3 c+345 d^4\right )-3 B \left (4 c^5-28 d c^4+104 d^2 c^3-392 d^3 c^2-1263 d^4 c-735 d^5\right )\right ) \sin (e+f x)\right )dx+\frac {a^3 \left (7 A d \left (2 c^3-18 c^2 d+111 c d^2+136 d^3\right )-B \left (6 c^4-42 c^3 d+165 c^2 d^2-651 c d^3-864 d^4\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}\right )+\frac {a^3 \left (7 A d \left (2 c^2-18 c d+115 d^2\right )-B \left (6 c^3-42 c^2 d+177 c d^2-735 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}-\frac {a^3 \left (-14 A c d+91 A d^2+6 B c^2-27 B c d+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}}{6 d}+\frac {(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}}{7 d}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\frac {3}{4} \left (\frac {1}{3} \int (c+d \sin (e+f x)) \left (a^3 d \left (7 A d \left (2 c^3-318 d c^2-567 d^2 c-272 d^3\right )-3 B \left (2 c^4-14 d c^3+577 d^2 c^2+1169 d^3 c+576 d^4\right )\right )-a^3 \left (7 A d \left (4 c^4-36 d c^3+216 d^2 c^2+626 d^3 c+345 d^4\right )-3 B \left (4 c^5-28 d c^4+104 d^2 c^3-392 d^3 c^2-1263 d^4 c-735 d^5\right )\right ) \sin (e+f x)\right )dx+\frac {a^3 \left (7 A d \left (2 c^3-18 c^2 d+111 c d^2+136 d^3\right )-B \left (6 c^4-42 c^3 d+165 c^2 d^2-651 c d^3-864 d^4\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}\right )+\frac {a^3 \left (7 A d \left (2 c^2-18 c d+115 d^2\right )-B \left (6 c^3-42 c^2 d+177 c d^2-735 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}-\frac {a^3 \left (-14 A c d+91 A d^2+6 B c^2-27 B c d+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}}{6 d}+\frac {(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}}{7 d}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}\)

\(\Big \downarrow \) 3213

\(\displaystyle \frac {\frac {-\frac {a^3 \left (-14 A c d+91 A d^2+6 B c^2-27 B c d+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {\frac {a^3 \left (7 A d \left (2 c^2-18 c d+115 d^2\right )-B \left (6 c^3-42 c^2 d+177 c d^2-735 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}+\frac {3}{4} \left (\frac {a^3 \left (7 A d \left (2 c^3-18 c^2 d+111 c d^2+136 d^3\right )-B \left (6 c^4-42 c^3 d+165 c^2 d^2-651 c d^3-864 d^4\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}+\frac {1}{3} \left (-\frac {105}{2} a^3 d^3 x \left (A \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right )+3 B \left (10 c^3+26 c^2 d+23 c d^2+7 d^3\right )\right )+\frac {a^3 d \left (7 A d \left (4 c^4-36 c^3 d+216 c^2 d^2+626 c d^3+345 d^4\right )-3 B \left (4 c^5-28 c^4 d+104 c^3 d^2-392 c^2 d^3-1263 c d^4-735 d^5\right )\right ) \sin (e+f x) \cos (e+f x)}{2 f}+\frac {2 a^3 \left (7 A d \left (2 c^5-18 c^4 d+107 c^3 d^2+472 c^2 d^3+456 c d^4+136 d^5\right )-3 B \left (2 c^6-14 c^5 d+51 c^4 d^2-189 c^3 d^3-920 c^2 d^4-952 c d^5-288 d^6\right )\right ) \cos (e+f x)}{f}\right )\right )}{5 d}}{6 d}+\frac {(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}}{7 d}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}\)

Input:

Int[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3,x]
 

Output:

-1/7*(a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^4)/(d*f 
) + (((3*B*(c - 3*d) - 7*A*d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d 
*Sin[e + f*x])^4)/(6*d*f) + (-1/5*(a^3*(6*B*c^2 - 14*A*c*d - 27*B*c*d + 91 
*A*d^2 + 87*B*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(d*f) - ((a^3*(7*A 
*d*(2*c^2 - 18*c*d + 115*d^2) - B*(6*c^3 - 42*c^2*d + 177*c*d^2 - 735*d^3) 
)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(4*f) + (3*((a^3*(7*A*d*(2*c^3 - 18 
*c^2*d + 111*c*d^2 + 136*d^3) - B*(6*c^4 - 42*c^3*d + 165*c^2*d^2 - 651*c* 
d^3 - 864*d^4))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*f) + ((-105*a^3*d^ 
3*(3*B*(10*c^3 + 26*c^2*d + 23*c*d^2 + 7*d^3) + A*(40*c^3 + 90*c^2*d + 78* 
c*d^2 + 23*d^3))*x)/2 + (2*a^3*(7*A*d*(2*c^5 - 18*c^4*d + 107*c^3*d^2 + 47 
2*c^2*d^3 + 456*c*d^4 + 136*d^5) - 3*B*(2*c^6 - 14*c^5*d + 51*c^4*d^2 - 18 
9*c^3*d^3 - 920*c^2*d^4 - 952*c*d^5 - 288*d^6))*Cos[e + f*x])/f + (a^3*d*( 
7*A*d*(4*c^4 - 36*c^3*d + 216*c^2*d^2 + 626*c*d^3 + 345*d^4) - 3*B*(4*c^5 
- 28*c^4*d + 104*c^3*d^2 - 392*c^2*d^3 - 1263*c*d^4 - 735*d^5))*Cos[e + f* 
x]*Sin[e + f*x])/(2*f))/3))/4)/(5*d))/(6*d))/(7*d)
 

Defintions of rubi rules used

rule 25

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

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 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 = 1.07 (sec) , antiderivative size = 1077, normalized size of antiderivative = 1.78

\[\text {Expression too large to display}\]

Input:

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3,x)
 

Output:

1/f*(-3/5*a^3*A*c*d^2*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)-3/5*a 
^3*B*c^2*d*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)-3*a^3*A*c^2*d*(2 
+sin(f*x+e)^2)*cos(f*x+e)-9/5*a^3*B*c*d^2*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e) 
^2)*cos(f*x+e)-3*a^3*A*c*d^2*(2+sin(f*x+e)^2)*cos(f*x+e)-3*a^3*B*c^2*d*(2+ 
sin(f*x+e)^2)*cos(f*x+e)-3/5*a^3*B*d^3*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2) 
*cos(f*x+e)-1/3*a^3*A*d^3*(2+sin(f*x+e)^2)*cos(f*x+e)+3*a^3*A*c^2*d*(-1/4* 
(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)+3*a^3*B*c*d^2*(-1/ 
6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e)+5/16*f*x+5/16 
*e)-1/3*a^3*A*c^3*(2+sin(f*x+e)^2)*cos(f*x+e)+a^3*A*c^3*(f*x+e)-a^3*B*c^3* 
cos(f*x+e)+a^3*B*d^3*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f* 
x+3/8*e)+3*a^3*A*c^3*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)+a^3*A*d^3* 
(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e)+5/16*f*x+ 
5/16*e)+3*a^3*B*d^3*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))* 
cos(f*x+e)+5/16*f*x+5/16*e)-3*a^3*A*c^3*cos(f*x+e)+3*a^3*A*d^3*(-1/4*(sin( 
f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)+3*a^3*B*c^3*(-1/2*sin(f 
*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)+a^3*B*c^3*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+ 
e))*cos(f*x+e)+3/8*f*x+3/8*e)+9*a^3*A*c*d^2*(-1/4*(sin(f*x+e)^3+3/2*sin(f* 
x+e))*cos(f*x+e)+3/8*f*x+3/8*e)+9*a^3*B*c^2*d*(-1/4*(sin(f*x+e)^3+3/2*sin( 
f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)+9*a^3*A*c^2*d*(-1/2*sin(f*x+e)*cos(f*x+e 
)+1/2*f*x+1/2*e)+3*a^3*A*c*d^2*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*...
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 432, normalized size of antiderivative = 0.72 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\frac {240 \, B a^{3} d^{3} \cos \left (f x + e\right )^{7} - 1008 \, {\left (B a^{3} c^{2} d + {\left (A + 3 \, B\right )} a^{3} c d^{2} + {\left (A + 2 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )^{5} + 560 \, {\left ({\left (A + 3 \, B\right )} a^{3} c^{3} + 3 \, {\left (3 \, A + 5 \, B\right )} a^{3} c^{2} d + 3 \, {\left (5 \, A + 7 \, B\right )} a^{3} c d^{2} + {\left (7 \, A + 9 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )^{3} + 105 \, {\left (10 \, {\left (4 \, A + 3 \, B\right )} a^{3} c^{3} + 6 \, {\left (15 \, A + 13 \, B\right )} a^{3} c^{2} d + 3 \, {\left (26 \, A + 23 \, B\right )} a^{3} c d^{2} + {\left (23 \, A + 21 \, B\right )} a^{3} d^{3}\right )} f x - 6720 \, {\left ({\left (A + B\right )} a^{3} c^{3} + 3 \, {\left (A + B\right )} a^{3} c^{2} d + 3 \, {\left (A + B\right )} a^{3} c d^{2} + {\left (A + B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right ) - 35 \, {\left (8 \, {\left (3 \, B a^{3} c d^{2} + {\left (A + 3 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )^{5} - 2 \, {\left (6 \, B a^{3} c^{3} + 18 \, {\left (A + 3 \, B\right )} a^{3} c^{2} d + 3 \, {\left (18 \, A + 31 \, B\right )} a^{3} c d^{2} + {\left (31 \, A + 45 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (2 \, {\left (12 \, A + 17 \, B\right )} a^{3} c^{3} + 6 \, {\left (17 \, A + 19 \, B\right )} a^{3} c^{2} d + 3 \, {\left (38 \, A + 41 \, B\right )} a^{3} c d^{2} + {\left (41 \, A + 43 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{1680 \, f} \] Input:

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3,x, algori 
thm="fricas")
 

Output:

1/1680*(240*B*a^3*d^3*cos(f*x + e)^7 - 1008*(B*a^3*c^2*d + (A + 3*B)*a^3*c 
*d^2 + (A + 2*B)*a^3*d^3)*cos(f*x + e)^5 + 560*((A + 3*B)*a^3*c^3 + 3*(3*A 
 + 5*B)*a^3*c^2*d + 3*(5*A + 7*B)*a^3*c*d^2 + (7*A + 9*B)*a^3*d^3)*cos(f*x 
 + e)^3 + 105*(10*(4*A + 3*B)*a^3*c^3 + 6*(15*A + 13*B)*a^3*c^2*d + 3*(26* 
A + 23*B)*a^3*c*d^2 + (23*A + 21*B)*a^3*d^3)*f*x - 6720*((A + B)*a^3*c^3 + 
 3*(A + B)*a^3*c^2*d + 3*(A + B)*a^3*c*d^2 + (A + B)*a^3*d^3)*cos(f*x + e) 
 - 35*(8*(3*B*a^3*c*d^2 + (A + 3*B)*a^3*d^3)*cos(f*x + e)^5 - 2*(6*B*a^3*c 
^3 + 18*(A + 3*B)*a^3*c^2*d + 3*(18*A + 31*B)*a^3*c*d^2 + (31*A + 45*B)*a^ 
3*d^3)*cos(f*x + e)^3 + 3*(2*(12*A + 17*B)*a^3*c^3 + 6*(17*A + 19*B)*a^3*c 
^2*d + 3*(38*A + 41*B)*a^3*c*d^2 + (41*A + 43*B)*a^3*d^3)*cos(f*x + e))*si 
n(f*x + e))/f
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2878 vs. \(2 (598) = 1196\).

Time = 0.76 (sec) , antiderivative size = 2878, normalized size of antiderivative = 4.76 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))**3,x)
 

Output:

Piecewise((3*A*a**3*c**3*x*sin(e + f*x)**2/2 + 3*A*a**3*c**3*x*cos(e + f*x 
)**2/2 + A*a**3*c**3*x - A*a**3*c**3*sin(e + f*x)**2*cos(e + f*x)/f - 3*A* 
a**3*c**3*sin(e + f*x)*cos(e + f*x)/(2*f) - 2*A*a**3*c**3*cos(e + f*x)**3/ 
(3*f) - 3*A*a**3*c**3*cos(e + f*x)/f + 9*A*a**3*c**2*d*x*sin(e + f*x)**4/8 
 + 9*A*a**3*c**2*d*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 9*A*a**3*c**2*d*x 
*sin(e + f*x)**2/2 + 9*A*a**3*c**2*d*x*cos(e + f*x)**4/8 + 9*A*a**3*c**2*d 
*x*cos(e + f*x)**2/2 - 15*A*a**3*c**2*d*sin(e + f*x)**3*cos(e + f*x)/(8*f) 
 - 9*A*a**3*c**2*d*sin(e + f*x)**2*cos(e + f*x)/f - 9*A*a**3*c**2*d*sin(e 
+ f*x)*cos(e + f*x)**3/(8*f) - 9*A*a**3*c**2*d*sin(e + f*x)*cos(e + f*x)/( 
2*f) - 6*A*a**3*c**2*d*cos(e + f*x)**3/f - 3*A*a**3*c**2*d*cos(e + f*x)/f 
+ 27*A*a**3*c*d**2*x*sin(e + f*x)**4/8 + 27*A*a**3*c*d**2*x*sin(e + f*x)** 
2*cos(e + f*x)**2/4 + 3*A*a**3*c*d**2*x*sin(e + f*x)**2/2 + 27*A*a**3*c*d* 
*2*x*cos(e + f*x)**4/8 + 3*A*a**3*c*d**2*x*cos(e + f*x)**2/2 - 3*A*a**3*c* 
d**2*sin(e + f*x)**4*cos(e + f*x)/f - 45*A*a**3*c*d**2*sin(e + f*x)**3*cos 
(e + f*x)/(8*f) - 4*A*a**3*c*d**2*sin(e + f*x)**2*cos(e + f*x)**3/f - 9*A* 
a**3*c*d**2*sin(e + f*x)**2*cos(e + f*x)/f - 27*A*a**3*c*d**2*sin(e + f*x) 
*cos(e + f*x)**3/(8*f) - 3*A*a**3*c*d**2*sin(e + f*x)*cos(e + f*x)/(2*f) - 
 8*A*a**3*c*d**2*cos(e + f*x)**5/(5*f) - 6*A*a**3*c*d**2*cos(e + f*x)**3/f 
 + 5*A*a**3*d**3*x*sin(e + f*x)**6/16 + 15*A*a**3*d**3*x*sin(e + f*x)**4*c 
os(e + f*x)**2/16 + 9*A*a**3*d**3*x*sin(e + f*x)**4/8 + 15*A*a**3*d**3*...
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 1056, normalized size of antiderivative = 1.75 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3,x, algori 
thm="maxima")
 

Output:

1/6720*(2240*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a^3*c^3 + 5040*(2*f*x + 2 
*e - sin(2*f*x + 2*e))*A*a^3*c^3 + 6720*(f*x + e)*A*a^3*c^3 + 6720*(cos(f* 
x + e)^3 - 3*cos(f*x + e))*B*a^3*c^3 + 210*(12*f*x + 12*e + sin(4*f*x + 4* 
e) - 8*sin(2*f*x + 2*e))*B*a^3*c^3 + 5040*(2*f*x + 2*e - sin(2*f*x + 2*e)) 
*B*a^3*c^3 + 20160*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a^3*c^2*d + 630*(12 
*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^3*c^2*d + 15120*( 
2*f*x + 2*e - sin(2*f*x + 2*e))*A*a^3*c^2*d - 1344*(3*cos(f*x + e)^5 - 10* 
cos(f*x + e)^3 + 15*cos(f*x + e))*B*a^3*c^2*d + 20160*(cos(f*x + e)^3 - 3* 
cos(f*x + e))*B*a^3*c^2*d + 1890*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin 
(2*f*x + 2*e))*B*a^3*c^2*d + 5040*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^3*c 
^2*d - 1344*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*A*a^3 
*c*d^2 + 20160*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a^3*c*d^2 + 1890*(12*f* 
x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^3*c*d^2 + 5040*(2*f* 
x + 2*e - sin(2*f*x + 2*e))*A*a^3*c*d^2 - 4032*(3*cos(f*x + e)^5 - 10*cos( 
f*x + e)^3 + 15*cos(f*x + e))*B*a^3*c*d^2 + 6720*(cos(f*x + e)^3 - 3*cos(f 
*x + e))*B*a^3*c*d^2 + 105*(4*sin(2*f*x + 2*e)^3 + 60*f*x + 60*e + 9*sin(4 
*f*x + 4*e) - 48*sin(2*f*x + 2*e))*B*a^3*c*d^2 + 1890*(12*f*x + 12*e + sin 
(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a^3*c*d^2 - 1344*(3*cos(f*x + e)^5 - 
 10*cos(f*x + e)^3 + 15*cos(f*x + e))*A*a^3*d^3 + 2240*(cos(f*x + e)^3 - 3 
*cos(f*x + e))*A*a^3*d^3 + 35*(4*sin(2*f*x + 2*e)^3 + 60*f*x + 60*e + 9...
 

Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 559, normalized size of antiderivative = 0.93 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\frac {B a^{3} d^{3} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} + \frac {1}{16} \, {\left (40 \, A a^{3} c^{3} + 30 \, B a^{3} c^{3} + 90 \, A a^{3} c^{2} d + 78 \, B a^{3} c^{2} d + 78 \, A a^{3} c d^{2} + 69 \, B a^{3} c d^{2} + 23 \, A a^{3} d^{3} + 21 \, B a^{3} d^{3}\right )} x - \frac {{\left (12 \, B a^{3} c^{2} d + 12 \, A a^{3} c d^{2} + 36 \, B a^{3} c d^{2} + 12 \, A a^{3} d^{3} + 19 \, B a^{3} d^{3}\right )} \cos \left (5 \, f x + 5 \, e\right )}{320 \, f} + \frac {{\left (16 \, A a^{3} c^{3} + 48 \, B a^{3} c^{3} + 144 \, A a^{3} c^{2} d + 204 \, B a^{3} c^{2} d + 204 \, A a^{3} c d^{2} + 228 \, B a^{3} c d^{2} + 76 \, A a^{3} d^{3} + 81 \, B a^{3} d^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{192 \, f} - \frac {{\left (240 \, A a^{3} c^{3} + 208 \, B a^{3} c^{3} + 624 \, A a^{3} c^{2} d + 552 \, B a^{3} c^{2} d + 552 \, A a^{3} c d^{2} + 504 \, B a^{3} c d^{2} + 168 \, A a^{3} d^{3} + 155 \, B a^{3} d^{3}\right )} \cos \left (f x + e\right )}{64 \, f} - \frac {{\left (3 \, B a^{3} c d^{2} + A a^{3} d^{3} + 3 \, B a^{3} d^{3}\right )} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {{\left (2 \, B a^{3} c^{3} + 6 \, A a^{3} c^{2} d + 18 \, B a^{3} c^{2} d + 18 \, A a^{3} c d^{2} + 27 \, B a^{3} c d^{2} + 9 \, A a^{3} d^{3} + 11 \, B a^{3} d^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {{\left (48 \, A a^{3} c^{3} + 64 \, B a^{3} c^{3} + 192 \, A a^{3} c^{2} d + 192 \, B a^{3} c^{2} d + 192 \, A a^{3} c d^{2} + 189 \, B a^{3} c d^{2} + 63 \, A a^{3} d^{3} + 61 \, B a^{3} d^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \] Input:

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3,x, algori 
thm="giac")
 

Output:

1/448*B*a^3*d^3*cos(7*f*x + 7*e)/f + 1/16*(40*A*a^3*c^3 + 30*B*a^3*c^3 + 9 
0*A*a^3*c^2*d + 78*B*a^3*c^2*d + 78*A*a^3*c*d^2 + 69*B*a^3*c*d^2 + 23*A*a^ 
3*d^3 + 21*B*a^3*d^3)*x - 1/320*(12*B*a^3*c^2*d + 12*A*a^3*c*d^2 + 36*B*a^ 
3*c*d^2 + 12*A*a^3*d^3 + 19*B*a^3*d^3)*cos(5*f*x + 5*e)/f + 1/192*(16*A*a^ 
3*c^3 + 48*B*a^3*c^3 + 144*A*a^3*c^2*d + 204*B*a^3*c^2*d + 204*A*a^3*c*d^2 
 + 228*B*a^3*c*d^2 + 76*A*a^3*d^3 + 81*B*a^3*d^3)*cos(3*f*x + 3*e)/f - 1/6 
4*(240*A*a^3*c^3 + 208*B*a^3*c^3 + 624*A*a^3*c^2*d + 552*B*a^3*c^2*d + 552 
*A*a^3*c*d^2 + 504*B*a^3*c*d^2 + 168*A*a^3*d^3 + 155*B*a^3*d^3)*cos(f*x + 
e)/f - 1/192*(3*B*a^3*c*d^2 + A*a^3*d^3 + 3*B*a^3*d^3)*sin(6*f*x + 6*e)/f 
+ 1/64*(2*B*a^3*c^3 + 6*A*a^3*c^2*d + 18*B*a^3*c^2*d + 18*A*a^3*c*d^2 + 27 
*B*a^3*c*d^2 + 9*A*a^3*d^3 + 11*B*a^3*d^3)*sin(4*f*x + 4*e)/f - 1/64*(48*A 
*a^3*c^3 + 64*B*a^3*c^3 + 192*A*a^3*c^2*d + 192*B*a^3*c^2*d + 192*A*a^3*c* 
d^2 + 189*B*a^3*c*d^2 + 63*A*a^3*d^3 + 61*B*a^3*d^3)*sin(2*f*x + 2*e)/f
 

Mupad [B] (verification not implemented)

Time = 38.13 (sec) , antiderivative size = 1395, normalized size of antiderivative = 2.31 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \] Input:

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^3,x)
                                                                                    
                                                                                    
 

Output:

(a^3*atan((a^3*tan(e/2 + (f*x)/2)*(40*A*c^3 + 23*A*d^3 + 30*B*c^3 + 21*B*d 
^3 + 78*A*c*d^2 + 90*A*c^2*d + 69*B*c*d^2 + 78*B*c^2*d))/(8*(5*A*a^3*c^3 + 
 (23*A*a^3*d^3)/8 + (15*B*a^3*c^3)/4 + (21*B*a^3*d^3)/8 + (39*A*a^3*c*d^2) 
/4 + (45*A*a^3*c^2*d)/4 + (69*B*a^3*c*d^2)/8 + (39*B*a^3*c^2*d)/4)))*(40*A 
*c^3 + 23*A*d^3 + 30*B*c^3 + 21*B*d^3 + 78*A*c*d^2 + 90*A*c^2*d + 69*B*c*d 
^2 + 78*B*c^2*d))/(8*f) - (tan(e/2 + (f*x)/2)*(3*A*a^3*c^3 + (23*A*a^3*d^3 
)/8 + (15*B*a^3*c^3)/4 + (21*B*a^3*d^3)/8 + (39*A*a^3*c*d^2)/4 + (45*A*a^3 
*c^2*d)/4 + (69*B*a^3*c*d^2)/8 + (39*B*a^3*c^2*d)/4) + tan(e/2 + (f*x)/2)^ 
10*(40*A*a^3*c^3 + 4*A*a^3*d^3 + 24*B*a^3*c^3 + 36*A*a^3*c*d^2 + 72*A*a^3* 
c^2*d + 12*B*a^3*c*d^2 + 36*B*a^3*c^2*d) - tan(e/2 + (f*x)/2)^13*(3*A*a^3* 
c^3 + (23*A*a^3*d^3)/8 + (15*B*a^3*c^3)/4 + (21*B*a^3*d^3)/8 + (39*A*a^3*c 
*d^2)/4 + (45*A*a^3*c^2*d)/4 + (69*B*a^3*c*d^2)/8 + (39*B*a^3*c^2*d)/4) + 
tan(e/2 + (f*x)/2)^3*(12*A*a^3*c^3 + (115*A*a^3*d^3)/6 + 17*B*a^3*c^3 + (3 
5*B*a^3*d^3)/2 + 57*A*a^3*c*d^2 + 51*A*a^3*c^2*d + (115*B*a^3*c*d^2)/2 + 5 
7*B*a^3*c^2*d) - tan(e/2 + (f*x)/2)^11*(12*A*a^3*c^3 + (115*A*a^3*d^3)/6 + 
 17*B*a^3*c^3 + (35*B*a^3*d^3)/2 + 57*A*a^3*c*d^2 + 51*A*a^3*c^2*d + (115* 
B*a^3*c*d^2)/2 + 57*B*a^3*c^2*d) + tan(e/2 + (f*x)/2)^8*((322*A*a^3*c^3)/3 
 + (148*A*a^3*d^3)/3 + 82*B*a^3*c^3 + 32*B*a^3*d^3 + 188*A*a^3*c*d^2 + 246 
*A*a^3*c^2*d + 148*B*a^3*c*d^2 + 188*B*a^3*c^2*d) + tan(e/2 + (f*x)/2)^6*( 
(448*A*a^3*c^3)/3 + (328*A*a^3*d^3)/3 + 128*B*a^3*c^3 + 112*B*a^3*d^3 +...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 869, normalized size of antiderivative = 1.44 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx =\text {Too large to display} \] Input:

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3,x)
 

Output:

(a**3*( - 240*cos(e + f*x)*sin(e + f*x)**6*b*d**3 - 280*cos(e + f*x)*sin(e 
 + f*x)**5*a*d**3 - 840*cos(e + f*x)*sin(e + f*x)**5*b*c*d**2 - 840*cos(e 
+ f*x)*sin(e + f*x)**5*b*d**3 - 1008*cos(e + f*x)*sin(e + f*x)**4*a*c*d**2 
 - 1008*cos(e + f*x)*sin(e + f*x)**4*a*d**3 - 1008*cos(e + f*x)*sin(e + f* 
x)**4*b*c**2*d - 3024*cos(e + f*x)*sin(e + f*x)**4*b*c*d**2 - 1296*cos(e + 
 f*x)*sin(e + f*x)**4*b*d**3 - 1260*cos(e + f*x)*sin(e + f*x)**3*a*c**2*d 
- 3780*cos(e + f*x)*sin(e + f*x)**3*a*c*d**2 - 1610*cos(e + f*x)*sin(e + f 
*x)**3*a*d**3 - 420*cos(e + f*x)*sin(e + f*x)**3*b*c**3 - 3780*cos(e + f*x 
)*sin(e + f*x)**3*b*c**2*d - 4830*cos(e + f*x)*sin(e + f*x)**3*b*c*d**2 - 
1470*cos(e + f*x)*sin(e + f*x)**3*b*d**3 - 560*cos(e + f*x)*sin(e + f*x)** 
2*a*c**3 - 5040*cos(e + f*x)*sin(e + f*x)**2*a*c**2*d - 6384*cos(e + f*x)* 
sin(e + f*x)**2*a*c*d**2 - 1904*cos(e + f*x)*sin(e + f*x)**2*a*d**3 - 1680 
*cos(e + f*x)*sin(e + f*x)**2*b*c**3 - 6384*cos(e + f*x)*sin(e + f*x)**2*b 
*c**2*d - 5712*cos(e + f*x)*sin(e + f*x)**2*b*c*d**2 - 1728*cos(e + f*x)*s 
in(e + f*x)**2*b*d**3 - 2520*cos(e + f*x)*sin(e + f*x)*a*c**3 - 9450*cos(e 
 + f*x)*sin(e + f*x)*a*c**2*d - 8190*cos(e + f*x)*sin(e + f*x)*a*c*d**2 - 
2415*cos(e + f*x)*sin(e + f*x)*a*d**3 - 3150*cos(e + f*x)*sin(e + f*x)*b*c 
**3 - 8190*cos(e + f*x)*sin(e + f*x)*b*c**2*d - 7245*cos(e + f*x)*sin(e + 
f*x)*b*c*d**2 - 2205*cos(e + f*x)*sin(e + f*x)*b*d**3 - 6160*cos(e + f*x)* 
a*c**3 - 15120*cos(e + f*x)*a*c**2*d - 12768*cos(e + f*x)*a*c*d**2 - 38...