3.38 \(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^6 \, dx\)
Optimal. Leaf size=265 \[ \frac{11 a^3 c^6 (10 A-3 B) \cos ^7(e+f x)}{560 f}+\frac{a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac{11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}+\frac{11 a^3 c^6 (10 A-3 B) \sin (e+f x) \cos ^5(e+f x)}{480 f}+\frac{11 a^3 c^6 (10 A-3 B) \sin (e+f x) \cos ^3(e+f x)}{384 f}+\frac{11 a^3 c^6 (10 A-3 B) \sin (e+f x) \cos (e+f x)}{256 f}+\frac{11}{256} a^3 c^6 x (10 A-3 B)-\frac{a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f} \]
[Out]
(11*a^3*(10*A - 3*B)*c^6*x)/256 + (11*a^3*(10*A - 3*B)*c^6*Cos[e + f*x]^7)/(560*f) + (11*a^3*(10*A - 3*B)*c^6*
Cos[e + f*x]*Sin[e + f*x])/(256*f) + (11*a^3*(10*A - 3*B)*c^6*Cos[e + f*x]^3*Sin[e + f*x])/(384*f) + (11*a^3*(
10*A - 3*B)*c^6*Cos[e + f*x]^5*Sin[e + f*x])/(480*f) - (a^3*B*Cos[e + f*x]^7*(c^2 - c^2*Sin[e + f*x])^3)/(10*f
) + (a^3*(10*A - 3*B)*Cos[e + f*x]^7*(c^3 - c^3*Sin[e + f*x])^2)/(90*f) + (11*a^3*(10*A - 3*B)*Cos[e + f*x]^7*
(c^6 - c^6*Sin[e + f*x]))/(720*f)
________________________________________________________________________________________
Rubi [A] time = 0.390784, antiderivative size = 265, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used =
{2967, 2860, 2678, 2669, 2635, 8} \[ \frac{11 a^3 c^6 (10 A-3 B) \cos ^7(e+f x)}{560 f}+\frac{a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac{11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}+\frac{11 a^3 c^6 (10 A-3 B) \sin (e+f x) \cos ^5(e+f x)}{480 f}+\frac{11 a^3 c^6 (10 A-3 B) \sin (e+f x) \cos ^3(e+f x)}{384 f}+\frac{11 a^3 c^6 (10 A-3 B) \sin (e+f x) \cos (e+f x)}{256 f}+\frac{11}{256} a^3 c^6 x (10 A-3 B)-\frac{a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^6,x]
[Out]
(11*a^3*(10*A - 3*B)*c^6*x)/256 + (11*a^3*(10*A - 3*B)*c^6*Cos[e + f*x]^7)/(560*f) + (11*a^3*(10*A - 3*B)*c^6*
Cos[e + f*x]*Sin[e + f*x])/(256*f) + (11*a^3*(10*A - 3*B)*c^6*Cos[e + f*x]^3*Sin[e + f*x])/(384*f) + (11*a^3*(
10*A - 3*B)*c^6*Cos[e + f*x]^5*Sin[e + f*x])/(480*f) - (a^3*B*Cos[e + f*x]^7*(c^2 - c^2*Sin[e + f*x])^3)/(10*f
) + (a^3*(10*A - 3*B)*Cos[e + f*x]^7*(c^3 - c^3*Sin[e + f*x])^2)/(90*f) + (11*a^3*(10*A - 3*B)*Cos[e + f*x]^7*
(c^6 - c^6*Sin[e + f*x]))/(720*f)
Rule 2967
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I
ntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Rule 2860
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[(d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(f*g*(m + p + 1)), x]
+ Dist[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; Fre
eQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[m + p + 1, 0]
Rule 2678
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(b*(g
*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m + p)), x] + Dist[(a*(2*m + p - 1))/(m + p), Int[(
g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0]
&& GtQ[m, 0] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]
Rule 2669
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Rule 2635
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] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^6 \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx\\ &=-\frac{a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac{1}{10} \left (a^3 (10 A-3 B) c^3\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x))^3 \, dx\\ &=-\frac{a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac{a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac{1}{90} \left (11 a^3 (10 A-3 B) c^4\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x))^2 \, dx\\ &=-\frac{a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac{a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac{11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}+\frac{1}{80} \left (11 a^3 (10 A-3 B) c^5\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac{11 a^3 (10 A-3 B) c^6 \cos ^7(e+f x)}{560 f}-\frac{a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac{a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac{11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}+\frac{1}{80} \left (11 a^3 (10 A-3 B) c^6\right ) \int \cos ^6(e+f x) \, dx\\ &=\frac{11 a^3 (10 A-3 B) c^6 \cos ^7(e+f x)}{560 f}+\frac{11 a^3 (10 A-3 B) c^6 \cos ^5(e+f x) \sin (e+f x)}{480 f}-\frac{a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac{a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac{11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}+\frac{1}{96} \left (11 a^3 (10 A-3 B) c^6\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac{11 a^3 (10 A-3 B) c^6 \cos ^7(e+f x)}{560 f}+\frac{11 a^3 (10 A-3 B) c^6 \cos ^3(e+f x) \sin (e+f x)}{384 f}+\frac{11 a^3 (10 A-3 B) c^6 \cos ^5(e+f x) \sin (e+f x)}{480 f}-\frac{a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac{a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac{11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}+\frac{1}{128} \left (11 a^3 (10 A-3 B) c^6\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac{11 a^3 (10 A-3 B) c^6 \cos ^7(e+f x)}{560 f}+\frac{11 a^3 (10 A-3 B) c^6 \cos (e+f x) \sin (e+f x)}{256 f}+\frac{11 a^3 (10 A-3 B) c^6 \cos ^3(e+f x) \sin (e+f x)}{384 f}+\frac{11 a^3 (10 A-3 B) c^6 \cos ^5(e+f x) \sin (e+f x)}{480 f}-\frac{a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac{a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac{11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}+\frac{1}{256} \left (11 a^3 (10 A-3 B) c^6\right ) \int 1 \, dx\\ &=\frac{11}{256} a^3 (10 A-3 B) c^6 x+\frac{11 a^3 (10 A-3 B) c^6 \cos ^7(e+f x)}{560 f}+\frac{11 a^3 (10 A-3 B) c^6 \cos (e+f x) \sin (e+f x)}{256 f}+\frac{11 a^3 (10 A-3 B) c^6 \cos ^3(e+f x) \sin (e+f x)}{384 f}+\frac{11 a^3 (10 A-3 B) c^6 \cos ^5(e+f x) \sin (e+f x)}{480 f}-\frac{a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac{a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac{11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}\\ \end{align*}
Mathematica [A] time = 4.2867, size = 255, normalized size = 0.96 \[ \frac{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^6 (27720 (10 A-3 B) (e+f x)+1260 (144 A-25 B) \sin (2 (e+f x))+2520 (6 A+7 B) \sin (4 (e+f x))-210 (32 A-51 B) \sin (6 (e+f x))-315 (6 A-5 B) \sin (8 (e+f x))+5040 (33 A-19 B) \cos (e+f x)+3360 (29 A-15 B) \cos (3 (e+f x))+10080 (3 A-B) \cos (5 (e+f x))+360 (9 A+5 B) \cos (7 (e+f x))-280 (A-3 B) \cos (9 (e+f x))-126 B \sin (10 (e+f x)))}{645120 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{12} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^6,x]
[Out]
((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6*(27720*(10*A - 3*B)*(e + f*x) + 5040*(33*A - 19*B)*Cos[e + f*x]
+ 3360*(29*A - 15*B)*Cos[3*(e + f*x)] + 10080*(3*A - B)*Cos[5*(e + f*x)] + 360*(9*A + 5*B)*Cos[7*(e + f*x)] -
280*(A - 3*B)*Cos[9*(e + f*x)] + 1260*(144*A - 25*B)*Sin[2*(e + f*x)] + 2520*(6*A + 7*B)*Sin[4*(e + f*x)] - 2
10*(32*A - 51*B)*Sin[6*(e + f*x)] - 315*(6*A - 5*B)*Sin[8*(e + f*x)] - 126*B*Sin[10*(e + f*x)]))/(645120*f*(Co
s[(e + f*x)/2] - Sin[(e + f*x)/2])^12*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)
________________________________________________________________________________________
Maple [B] time = 0.148, size = 651, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^6,x)
[Out]
1/f*(A*a^3*c^6*(f*x+e)-8/3*A*a^3*c^6*(2+sin(f*x+e)^2)*cos(f*x+e)+8*B*a^3*c^6*(-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*a^3*c^6*cos(f*x+e)-3*B*a^3*c^6*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-B*a
^3*c^6*cos(f*x+e)-1/9*A*a^3*c^6*(128/35+sin(f*x+e)^8+8/7*sin(f*x+e)^6+48/35*sin(f*x+e)^4+64/35*sin(f*x+e)^2)*c
os(f*x+e)-3*A*a^3*c^6*(-1/8*(sin(f*x+e)^7+7/6*sin(f*x+e)^5+35/24*sin(f*x+e)^3+35/16*sin(f*x+e))*cos(f*x+e)+35/
128*f*x+35/128*e)-8/7*B*a^3*c^6*(16/5+sin(f*x+e)^6+6/5*sin(f*x+e)^4+8/5*sin(f*x+e)^2)*cos(f*x+e)+1/3*B*a^3*c^6
*(128/35+sin(f*x+e)^8+8/7*sin(f*x+e)^6+48/35*sin(f*x+e)^4+64/35*sin(f*x+e)^2)*cos(f*x+e)+6/5*A*a^3*c^6*(8/3+si
n(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)-6*A*a^3*c^6*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8
*e)+B*a^3*c^6*(-1/10*(sin(f*x+e)^9+9/8*sin(f*x+e)^7+21/16*sin(f*x+e)^5+105/64*sin(f*x+e)^3+315/128*sin(f*x+e))
*cos(f*x+e)+63/256*f*x+63/256*e)+8*A*a^3*c^6*(-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)+6/5*B*a^3*c^6*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)-6*B*a^3*c^6*(-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))
________________________________________________________________________________________
Maxima [B] time = 1.04242, size = 892, normalized size = 3.37 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^6,x, algorithm="maxima")
[Out]
-1/645120*(2048*(35*cos(f*x + e)^9 - 180*cos(f*x + e)^7 + 378*cos(f*x + e)^5 - 420*cos(f*x + e)^3 + 315*cos(f*
x + e))*A*a^3*c^6 - 258048*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*A*a^3*c^6 - 1720320*(cos(f
*x + e)^3 - 3*cos(f*x + e))*A*a^3*c^6 + 630*(128*sin(2*f*x + 2*e)^3 + 840*f*x + 840*e + 3*sin(8*f*x + 8*e) + 1
68*sin(4*f*x + 4*e) - 768*sin(2*f*x + 2*e))*A*a^3*c^6 - 26880*(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))*A*a^3*c^6 + 120960*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A
*a^3*c^6 - 645120*(f*x + e)*A*a^3*c^6 - 6144*(35*cos(f*x + e)^9 - 180*cos(f*x + e)^7 + 378*cos(f*x + e)^5 - 42
0*cos(f*x + e)^3 + 315*cos(f*x + e))*B*a^3*c^6 - 147456*(5*cos(f*x + e)^7 - 21*cos(f*x + e)^5 + 35*cos(f*x + e
)^3 - 35*cos(f*x + e))*B*a^3*c^6 - 258048*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a^3*c^6 +
63*(32*sin(2*f*x + 2*e)^5 - 640*sin(2*f*x + 2*e)^3 - 2520*f*x - 2520*e - 25*sin(8*f*x + 8*e) - 600*sin(4*f*x
+ 4*e) + 2560*sin(2*f*x + 2*e))*B*a^3*c^6 + 20160*(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^6 - 161280*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a^3*c^6 + 4
83840*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^3*c^6 - 1935360*A*a^3*c^6*cos(f*x + e) + 645120*B*a^3*c^6*cos(f*x +
e))/f
________________________________________________________________________________________
Fricas [A] time = 1.75442, size = 455, normalized size = 1.72 \begin{align*} -\frac{8960 \,{\left (A - 3 \, B\right )} a^{3} c^{6} \cos \left (f x + e\right )^{9} - 46080 \,{\left (A - B\right )} a^{3} c^{6} \cos \left (f x + e\right )^{7} - 3465 \,{\left (10 \, A - 3 \, B\right )} a^{3} c^{6} f x + 21 \,{\left (384 \, B a^{3} c^{6} \cos \left (f x + e\right )^{9} + 48 \,{\left (30 \, A - 41 \, B\right )} a^{3} c^{6} \cos \left (f x + e\right )^{7} - 88 \,{\left (10 \, A - 3 \, B\right )} a^{3} c^{6} \cos \left (f x + e\right )^{5} - 110 \,{\left (10 \, A - 3 \, B\right )} a^{3} c^{6} \cos \left (f x + e\right )^{3} - 165 \,{\left (10 \, A - 3 \, B\right )} a^{3} c^{6} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{80640 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^6,x, algorithm="fricas")
[Out]
-1/80640*(8960*(A - 3*B)*a^3*c^6*cos(f*x + e)^9 - 46080*(A - B)*a^3*c^6*cos(f*x + e)^7 - 3465*(10*A - 3*B)*a^3
*c^6*f*x + 21*(384*B*a^3*c^6*cos(f*x + e)^9 + 48*(30*A - 41*B)*a^3*c^6*cos(f*x + e)^7 - 88*(10*A - 3*B)*a^3*c^
6*cos(f*x + e)^5 - 110*(10*A - 3*B)*a^3*c^6*cos(f*x + e)^3 - 165*(10*A - 3*B)*a^3*c^6*cos(f*x + e))*sin(f*x +
e))/f
________________________________________________________________________________________
Sympy [A] time = 83.288, size = 1948, normalized size = 7.35 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**6,x)
[Out]
Piecewise((-105*A*a**3*c**6*x*sin(e + f*x)**8/128 - 105*A*a**3*c**6*x*sin(e + f*x)**6*cos(e + f*x)**2/32 + 5*A
*a**3*c**6*x*sin(e + f*x)**6/2 - 315*A*a**3*c**6*x*sin(e + f*x)**4*cos(e + f*x)**4/64 + 15*A*a**3*c**6*x*sin(e
+ f*x)**4*cos(e + f*x)**2/2 - 9*A*a**3*c**6*x*sin(e + f*x)**4/4 - 105*A*a**3*c**6*x*sin(e + f*x)**2*cos(e + f
*x)**6/32 + 15*A*a**3*c**6*x*sin(e + f*x)**2*cos(e + f*x)**4/2 - 9*A*a**3*c**6*x*sin(e + f*x)**2*cos(e + f*x)*
*2/2 - 105*A*a**3*c**6*x*cos(e + f*x)**8/128 + 5*A*a**3*c**6*x*cos(e + f*x)**6/2 - 9*A*a**3*c**6*x*cos(e + f*x
)**4/4 + A*a**3*c**6*x - A*a**3*c**6*sin(e + f*x)**8*cos(e + f*x)/f + 279*A*a**3*c**6*sin(e + f*x)**7*cos(e +
f*x)/(128*f) - 8*A*a**3*c**6*sin(e + f*x)**6*cos(e + f*x)**3/(3*f) + 511*A*a**3*c**6*sin(e + f*x)**5*cos(e + f
*x)**3/(128*f) - 11*A*a**3*c**6*sin(e + f*x)**5*cos(e + f*x)/(2*f) - 16*A*a**3*c**6*sin(e + f*x)**4*cos(e + f*
x)**5/(5*f) + 6*A*a**3*c**6*sin(e + f*x)**4*cos(e + f*x)/f + 385*A*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)**5/(
128*f) - 20*A*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)**3/(3*f) + 15*A*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)/(4
*f) - 64*A*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)**7/(35*f) + 8*A*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)**3/f
- 8*A*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)/f + 105*A*a**3*c**6*sin(e + f*x)*cos(e + f*x)**7/(128*f) - 5*A*a*
*3*c**6*sin(e + f*x)*cos(e + f*x)**5/(2*f) + 9*A*a**3*c**6*sin(e + f*x)*cos(e + f*x)**3/(4*f) - 128*A*a**3*c**
6*cos(e + f*x)**9/(315*f) + 16*A*a**3*c**6*cos(e + f*x)**5/(5*f) - 16*A*a**3*c**6*cos(e + f*x)**3/(3*f) + 3*A*
a**3*c**6*cos(e + f*x)/f + 63*B*a**3*c**6*x*sin(e + f*x)**10/256 + 315*B*a**3*c**6*x*sin(e + f*x)**8*cos(e + f
*x)**2/256 + 315*B*a**3*c**6*x*sin(e + f*x)**6*cos(e + f*x)**4/128 - 15*B*a**3*c**6*x*sin(e + f*x)**6/8 + 315*
B*a**3*c**6*x*sin(e + f*x)**4*cos(e + f*x)**6/128 - 45*B*a**3*c**6*x*sin(e + f*x)**4*cos(e + f*x)**2/8 + 3*B*a
**3*c**6*x*sin(e + f*x)**4 + 315*B*a**3*c**6*x*sin(e + f*x)**2*cos(e + f*x)**8/256 - 45*B*a**3*c**6*x*sin(e +
f*x)**2*cos(e + f*x)**4/8 + 6*B*a**3*c**6*x*sin(e + f*x)**2*cos(e + f*x)**2 - 3*B*a**3*c**6*x*sin(e + f*x)**2/
2 + 63*B*a**3*c**6*x*cos(e + f*x)**10/256 - 15*B*a**3*c**6*x*cos(e + f*x)**6/8 + 3*B*a**3*c**6*x*cos(e + f*x)*
*4 - 3*B*a**3*c**6*x*cos(e + f*x)**2/2 - 193*B*a**3*c**6*sin(e + f*x)**9*cos(e + f*x)/(256*f) + 3*B*a**3*c**6*
sin(e + f*x)**8*cos(e + f*x)/f - 237*B*a**3*c**6*sin(e + f*x)**7*cos(e + f*x)**3/(128*f) + 8*B*a**3*c**6*sin(e
+ f*x)**6*cos(e + f*x)**3/f - 8*B*a**3*c**6*sin(e + f*x)**6*cos(e + f*x)/f - 21*B*a**3*c**6*sin(e + f*x)**5*c
os(e + f*x)**5/(10*f) + 33*B*a**3*c**6*sin(e + f*x)**5*cos(e + f*x)/(8*f) + 48*B*a**3*c**6*sin(e + f*x)**4*cos
(e + f*x)**5/(5*f) - 16*B*a**3*c**6*sin(e + f*x)**4*cos(e + f*x)**3/f + 6*B*a**3*c**6*sin(e + f*x)**4*cos(e +
f*x)/f - 147*B*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)**7/(128*f) + 5*B*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)*
*3/f - 5*B*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)/f + 192*B*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)**7/(35*f) -
64*B*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) + 8*B*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)**3/f - 63*B
*a**3*c**6*sin(e + f*x)*cos(e + f*x)**9/(256*f) + 15*B*a**3*c**6*sin(e + f*x)*cos(e + f*x)**5/(8*f) - 3*B*a**3
*c**6*sin(e + f*x)*cos(e + f*x)**3/f + 3*B*a**3*c**6*sin(e + f*x)*cos(e + f*x)/(2*f) + 128*B*a**3*c**6*cos(e +
f*x)**9/(105*f) - 128*B*a**3*c**6*cos(e + f*x)**7/(35*f) + 16*B*a**3*c**6*cos(e + f*x)**5/(5*f) - B*a**3*c**6
*cos(e + f*x)/f, Ne(f, 0)), (x*(A + B*sin(e))*(a*sin(e) + a)**3*(-c*sin(e) + c)**6, True))
________________________________________________________________________________________
Giac [A] time = 1.48789, size = 468, normalized size = 1.77 \begin{align*} -\frac{B a^{3} c^{6} \sin \left (10 \, f x + 10 \, e\right )}{5120 \, f} + \frac{11}{256} \,{\left (10 \, A a^{3} c^{6} - 3 \, B a^{3} c^{6}\right )} x - \frac{{\left (A a^{3} c^{6} - 3 \, B a^{3} c^{6}\right )} \cos \left (9 \, f x + 9 \, e\right )}{2304 \, f} + \frac{{\left (9 \, A a^{3} c^{6} + 5 \, B a^{3} c^{6}\right )} \cos \left (7 \, f x + 7 \, e\right )}{1792 \, f} + \frac{{\left (3 \, A a^{3} c^{6} - B a^{3} c^{6}\right )} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} + \frac{{\left (29 \, A a^{3} c^{6} - 15 \, B a^{3} c^{6}\right )} \cos \left (3 \, f x + 3 \, e\right )}{192 \, f} + \frac{{\left (33 \, A a^{3} c^{6} - 19 \, B a^{3} c^{6}\right )} \cos \left (f x + e\right )}{128 \, f} - \frac{{\left (6 \, A a^{3} c^{6} - 5 \, B a^{3} c^{6}\right )} \sin \left (8 \, f x + 8 \, e\right )}{2048 \, f} - \frac{{\left (32 \, A a^{3} c^{6} - 51 \, B a^{3} c^{6}\right )} \sin \left (6 \, f x + 6 \, e\right )}{3072 \, f} + \frac{{\left (6 \, A a^{3} c^{6} + 7 \, B a^{3} c^{6}\right )} \sin \left (4 \, f x + 4 \, e\right )}{256 \, f} + \frac{{\left (144 \, A a^{3} c^{6} - 25 \, B a^{3} c^{6}\right )} \sin \left (2 \, f x + 2 \, e\right )}{512 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^6,x, algorithm="giac")
[Out]
-1/5120*B*a^3*c^6*sin(10*f*x + 10*e)/f + 11/256*(10*A*a^3*c^6 - 3*B*a^3*c^6)*x - 1/2304*(A*a^3*c^6 - 3*B*a^3*c
^6)*cos(9*f*x + 9*e)/f + 1/1792*(9*A*a^3*c^6 + 5*B*a^3*c^6)*cos(7*f*x + 7*e)/f + 1/64*(3*A*a^3*c^6 - B*a^3*c^6
)*cos(5*f*x + 5*e)/f + 1/192*(29*A*a^3*c^6 - 15*B*a^3*c^6)*cos(3*f*x + 3*e)/f + 1/128*(33*A*a^3*c^6 - 19*B*a^3
*c^6)*cos(f*x + e)/f - 1/2048*(6*A*a^3*c^6 - 5*B*a^3*c^6)*sin(8*f*x + 8*e)/f - 1/3072*(32*A*a^3*c^6 - 51*B*a^3
*c^6)*sin(6*f*x + 6*e)/f + 1/256*(6*A*a^3*c^6 + 7*B*a^3*c^6)*sin(4*f*x + 4*e)/f + 1/512*(144*A*a^3*c^6 - 25*B*
a^3*c^6)*sin(2*f*x + 2*e)/f