\(\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx\) [244]

Optimal result

Integrand size = 33, antiderivative size = 327 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\frac {1}{8} a \left (B \left (4 c^3+12 c^2 d+9 c d^2+3 d^3\right )+A \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right )\right ) x-\frac {a \left (5 A d \left (3 c^3+16 c^2 d+12 c d^2+4 d^3\right )-B \left (3 c^4-15 c^3 d-52 c^2 d^2-60 c d^3-16 d^4\right )\right ) \cos (e+f x)}{30 d f}-\frac {a \left (5 A d \left (6 c^2+20 c d+9 d^2\right )-B \left (6 c^3-30 c^2 d-71 c d^2-45 d^3\right )\right ) \cos (e+f x) \sin (e+f x)}{120 f}-\frac {a \left (4 (5 A+4 B) d^2-3 c (B c-5 (A+B) d)\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{60 d f}+\frac {a (B c-5 (A+B) d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac {a B \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f} \]

[Out]

1/8*a*(B*(4*c^3+12*c^2*d+9*c*d^2+3*d^3)+A*(8*c^3+12*c^2*d+12*c*d^2+3*d^3))*x-1/30*a*(5*A*d*(3*c^3+16*c^2*d+12*
c*d^2+4*d^3)-B*(3*c^4-15*c^3*d-52*c^2*d^2-60*c*d^3-16*d^4))*cos(f*x+e)/d/f-1/120*a*(5*A*d*(6*c^2+20*c*d+9*d^2)
-B*(6*c^3-30*c^2*d-71*c*d^2-45*d^3))*cos(f*x+e)*sin(f*x+e)/f-1/60*a*(4*(5*A+4*B)*d^2-3*c*(B*c-5*(A+B)*d))*cos(
f*x+e)*(c+d*sin(f*x+e))^2/d/f+1/20*a*(B*c-5*(A+B)*d)*cos(f*x+e)*(c+d*sin(f*x+e))^3/d/f-1/5*a*B*cos(f*x+e)*(c+d
*sin(f*x+e))^4/d/f

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3047, 3102, 2832, 2813} \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=-\frac {a \left (5 A d \left (6 c^2+20 c d+9 d^2\right )-B \left (6 c^3-30 c^2 d-71 c d^2-45 d^3\right )\right ) \sin (e+f x) \cos (e+f x)}{120 f}+\frac {1}{8} a x \left (A \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right )+B \left (4 c^3+12 c^2 d+9 c d^2+3 d^3\right )\right )-\frac {a \left (5 A d \left (3 c^3+16 c^2 d+12 c d^2+4 d^3\right )-B \left (3 c^4-15 c^3 d-52 c^2 d^2-60 c d^3-16 d^4\right )\right ) \cos (e+f x)}{30 d f}-\frac {a \left (4 d^2 (5 A+4 B)-3 c (B c-5 d (A+B))\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{60 d f}+\frac {a (B c-5 d (A+B)) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac {a B \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f} \]

[In]

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

[Out]

(a*(B*(4*c^3 + 12*c^2*d + 9*c*d^2 + 3*d^3) + A*(8*c^3 + 12*c^2*d + 12*c*d^2 + 3*d^3))*x)/8 - (a*(5*A*d*(3*c^3
+ 16*c^2*d + 12*c*d^2 + 4*d^3) - B*(3*c^4 - 15*c^3*d - 52*c^2*d^2 - 60*c*d^3 - 16*d^4))*Cos[e + f*x])/(30*d*f)
- (a*(5*A*d*(6*c^2 + 20*c*d + 9*d^2) - B*(6*c^3 - 30*c^2*d - 71*c*d^2 - 45*d^3))*Cos[e + f*x]*Sin[e + f*x])/(
120*f) - (a*(4*(5*A + 4*B)*d^2 - 3*c*(B*c - 5*(A + B)*d))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(60*d*f) + (a*(
B*c - 5*(A + B)*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(20*d*f) - (a*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(
5*d*f)

Rule 2813

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)*(Cos[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 2832

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] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[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 3047

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 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = \int (c+d \sin (e+f x))^3 \left (a A+(a A+a B) \sin (e+f x)+a B \sin ^2(e+f x)\right ) \, dx \\ & = -\frac {a B \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac {\int (c+d \sin (e+f x))^3 (a (5 A+4 B) d-a (B c-5 (A+B) d) \sin (e+f x)) \, dx}{5 d} \\ & = \frac {a (B c-5 (A+B) d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac {a B \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac {\int (c+d \sin (e+f x))^2 \left (a d (20 A c+13 B c+15 A d+15 B d)+a \left (4 (5 A+4 B) d^2-3 c (B c-5 (A+B) d)\right ) \sin (e+f x)\right ) \, dx}{20 d} \\ & = -\frac {a \left (4 (5 A+4 B) d^2-3 c (B c-5 (A+B) d)\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{60 d f}+\frac {a (B c-5 (A+B) d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac {a B \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac {\int (c+d \sin (e+f x)) \left (a d \left (60 A c^2+33 B c^2+75 A c d+75 B c d+40 A d^2+32 B d^2\right )+a \left (5 A d \left (6 c^2+20 c d+9 d^2\right )-B \left (6 c^3-30 c^2 d-71 c d^2-45 d^3\right )\right ) \sin (e+f x)\right ) \, dx}{60 d} \\ & = \frac {1}{8} a \left (B \left (4 c^3+12 c^2 d+9 c d^2+3 d^3\right )+A \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right )\right ) x-\frac {a \left (5 A d \left (3 c^3+16 c^2 d+12 c d^2+4 d^3\right )-B \left (3 c^4-15 c^3 d-52 c^2 d^2-60 c d^3-16 d^4\right )\right ) \cos (e+f x)}{30 d f}-\frac {a \left (5 A d \left (6 c^2+20 c d+9 d^2\right )-B \left (6 c^3-30 c^2 d-71 c d^2-45 d^3\right )\right ) \cos (e+f x) \sin (e+f x)}{120 f}-\frac {a \left (4 (5 A+4 B) d^2-3 c (B c-5 (A+B) d)\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{60 d f}+\frac {a (B c-5 (A+B) d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac {a B \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.54 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.82 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\frac {a (1+\sin (e+f x)) \left (-60 \left (2 A \left (4 c^3+12 c^2 d+9 c d^2+3 d^3\right )+B \left (8 c^3+18 c^2 d+18 c d^2+5 d^3\right )\right ) \cos (e+f x)+10 d \left (4 A d (3 c+d)+B \left (12 c^2+12 c d+5 d^2\right )\right ) \cos (3 (e+f x))-6 B d^3 \cos (5 (e+f x))+15 \left (4 \left (B \left (4 c^3+12 c^2 d+9 c d^2+3 d^3\right )+A \left (8 c^3+12 c^2 d+12 c d^2+3 d^3\right )\right ) f x-8 \left (B (c+d)^3+A d \left (3 c^2+3 c d+d^2\right )\right ) \sin (2 (e+f x))+d^2 (A d+B (3 c+d)) \sin (4 (e+f x))\right )\right )}{480 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2} \]

[In]

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

[Out]

(a*(1 + Sin[e + f*x])*(-60*(2*A*(4*c^3 + 12*c^2*d + 9*c*d^2 + 3*d^3) + B*(8*c^3 + 18*c^2*d + 18*c*d^2 + 5*d^3)
)*Cos[e + f*x] + 10*d*(4*A*d*(3*c + d) + B*(12*c^2 + 12*c*d + 5*d^2))*Cos[3*(e + f*x)] - 6*B*d^3*Cos[5*(e + f*
x)] + 15*(4*(B*(4*c^3 + 12*c^2*d + 9*c*d^2 + 3*d^3) + A*(8*c^3 + 12*c^2*d + 12*c*d^2 + 3*d^3))*f*x - 8*(B*(c +
d)^3 + A*d*(3*c^2 + 3*c*d + d^2))*Sin[2*(e + f*x)] + d^2*(A*d + B*(3*c + d))*Sin[4*(e + f*x)])))/(480*f*(Cos[
(e + f*x)/2] + Sin[(e + f*x)/2])^2)

Maple [A] (verified)

Time = 1.83 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.75

method	result	size
parts	\(\frac {\left (A a \,d^{3}+3 B a c \,d^{2}+d^{3} B a \right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {\left (A \,c^{3} a +3 A a \,c^{2} d +B a \,c^{3}\right ) \cos \left (f x +e \right )}{f}-\frac {\left (3 A a c \,d^{2}+A a \,d^{3}+3 B a \,c^{2} d +3 B a c \,d^{2}\right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {\left (3 A a \,c^{2} d +3 A a c \,d^{2}+B a \,c^{3}+3 B a \,c^{2} d \right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+A a \,c^{3} x -\frac {d^{3} B a \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}\)	\(244\)
parallelrisch	\(\frac {\left (\left (\left (-A -B \right ) d^{3}-3 d^{2} c \left (A +B \right )-3 c^{2} d \left (A +B \right )-B \,c^{3}\right ) \sin \left (2 f x +2 e \right )+\left (\frac {\left (A +\frac {5 B}{4}\right ) d^{2}}{3}+d c \left (A +B \right )+B \,c^{2}\right ) d \cos \left (3 f x +3 e \right )+\frac {d^{2} \left (3 B c +\left (A +B \right ) d \right ) \sin \left (4 f x +4 e \right )}{8}-\frac {B \,d^{3} \cos \left (5 f x +5 e \right )}{20}+\left (\left (-\frac {5 B}{2}-3 A \right ) d^{3}-9 d^{2} c \left (A +B \right )-12 c^{2} \left (A +\frac {3 B}{4}\right ) d -4 c^{3} \left (A +B \right )\right ) \cos \left (f x +e \right )+\left (-\frac {32}{15} B +\frac {3}{2} f x A +\frac {3}{2} f x B -\frac {8}{3} A \right ) d^{3}+6 c \left (f x A +\frac {3}{4} f x B -\frac {4}{3} A -\frac {4}{3} B \right ) d^{2}+6 c^{2} \left (f x A +f x B -2 A -\frac {4}{3} B \right ) d +4 c^{3} \left (f x A +\frac {1}{2} f x B -A -B \right )\right ) a}{4 f}\)	\(258\)
derivativedivides	\(\frac {-A \cos \left (f x +e \right ) a \,c^{3}+3 A a \,c^{2} d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-A a c \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+A a \,d^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+B \,c^{3} a \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B a \,c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 B a c \,d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {d^{3} B a \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+A \,c^{3} a \left (f x +e \right )-3 A a \,c^{2} d \cos \left (f x +e \right )+3 A a c \,d^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {A a \,d^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-B \cos \left (f x +e \right ) a \,c^{3}+3 B a \,c^{2} d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B a c \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+d^{3} B a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\)	\(422\)
default	\(\frac {-A \cos \left (f x +e \right ) a \,c^{3}+3 A a \,c^{2} d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-A a c \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+A a \,d^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+B \,c^{3} a \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B a \,c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 B a c \,d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {d^{3} B a \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+A \,c^{3} a \left (f x +e \right )-3 A a \,c^{2} d \cos \left (f x +e \right )+3 A a c \,d^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {A a \,d^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-B \cos \left (f x +e \right ) a \,c^{3}+3 B a \,c^{2} d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B a c \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+d^{3} B a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\)	\(422\)
risch	\(-\frac {3 \sin \left (2 f x +2 e \right ) A a \,c^{2} d}{4 f}-\frac {3 \sin \left (2 f x +2 e \right ) A a c \,d^{2}}{4 f}-\frac {3 \sin \left (2 f x +2 e \right ) B a \,c^{2} d}{4 f}-\frac {3 \sin \left (2 f x +2 e \right ) B a c \,d^{2}}{4 f}+A a \,c^{3} x +\frac {3 A a \,d^{3} x}{8}+\frac {B a \,c^{3} x}{2}+\frac {3 B a \,d^{3} x}{8}-\frac {3 a \cos \left (f x +e \right ) c^{2} d A}{f}-\frac {9 a \cos \left (f x +e \right ) d^{2} c A}{4 f}-\frac {9 a \cos \left (f x +e \right ) c^{2} d B}{4 f}-\frac {9 a \cos \left (f x +e \right ) d^{2} c B}{4 f}+\frac {3 \sin \left (4 f x +4 e \right ) B a c \,d^{2}}{32 f}+\frac {a \,d^{2} \cos \left (3 f x +3 e \right ) A c}{4 f}+\frac {a d \cos \left (3 f x +3 e \right ) B \,c^{2}}{4 f}+\frac {a \,d^{2} \cos \left (3 f x +3 e \right ) c B}{4 f}+\frac {3 A a \,c^{2} d x}{2}+\frac {3 A a c \,d^{2} x}{2}+\frac {3 B a \,c^{2} d x}{2}+\frac {9 B a c \,d^{2} x}{8}-\frac {a \cos \left (f x +e \right ) A \,c^{3}}{f}-\frac {3 a \cos \left (f x +e \right ) A \,d^{3}}{4 f}-\frac {a \cos \left (f x +e \right ) B \,c^{3}}{f}-\frac {5 a \cos \left (f x +e \right ) d^{3} B}{8 f}+\frac {\sin \left (4 f x +4 e \right ) A a \,d^{3}}{32 f}+\frac {\sin \left (4 f x +4 e \right ) d^{3} B a}{32 f}+\frac {a \,d^{3} \cos \left (3 f x +3 e \right ) A}{12 f}+\frac {5 a \,d^{3} \cos \left (3 f x +3 e \right ) B}{48 f}-\frac {B a \,d^{3} \cos \left (5 f x +5 e \right )}{80 f}-\frac {\sin \left (2 f x +2 e \right ) A a \,d^{3}}{4 f}-\frac {\sin \left (2 f x +2 e \right ) B a \,c^{3}}{4 f}-\frac {\sin \left (2 f x +2 e \right ) d^{3} B a}{4 f}\)	\(513\)
norman	\(\text {Expression too large to display}\)	\(1032\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.73 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=-\frac {24 \, B a d^{3} \cos \left (f x + e\right )^{5} - 40 \, {\left (3 \, B a c^{2} d + 3 \, {\left (A + B\right )} a c d^{2} + {\left (A + 2 \, B\right )} a d^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (4 \, {\left (2 \, A + B\right )} a c^{3} + 12 \, {\left (A + B\right )} a c^{2} d + 3 \, {\left (4 \, A + 3 \, B\right )} a c d^{2} + 3 \, {\left (A + B\right )} a d^{3}\right )} f x + 120 \, {\left ({\left (A + B\right )} a c^{3} + 3 \, {\left (A + B\right )} a c^{2} d + 3 \, {\left (A + B\right )} a c d^{2} + {\left (A + B\right )} a d^{3}\right )} \cos \left (f x + e\right ) - 15 \, {\left (2 \, {\left (3 \, B a c d^{2} + {\left (A + B\right )} a d^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (4 \, B a c^{3} + 12 \, {\left (A + B\right )} a c^{2} d + 3 \, {\left (4 \, A + 5 \, B\right )} a c d^{2} + 5 \, {\left (A + B\right )} a d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{120 \, f} \]

[In]

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

[Out]

-1/120*(24*B*a*d^3*cos(f*x + e)^5 - 40*(3*B*a*c^2*d + 3*(A + B)*a*c*d^2 + (A + 2*B)*a*d^3)*cos(f*x + e)^3 - 15
*(4*(2*A + B)*a*c^3 + 12*(A + B)*a*c^2*d + 3*(4*A + 3*B)*a*c*d^2 + 3*(A + B)*a*d^3)*f*x + 120*((A + B)*a*c^3 +
3*(A + B)*a*c^2*d + 3*(A + B)*a*c*d^2 + (A + B)*a*d^3)*cos(f*x + e) - 15*(2*(3*B*a*c*d^2 + (A + B)*a*d^3)*cos
(f*x + e)^3 - (4*B*a*c^3 + 12*(A + B)*a*c^2*d + 3*(4*A + 5*B)*a*c*d^2 + 5*(A + B)*a*d^3)*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. 996 vs. \(2 (311) = 622\).

Time = 0.34 (sec) , antiderivative size = 996, normalized size of antiderivative = 3.05 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((A*a*c**3*x - A*a*c**3*cos(e + f*x)/f + 3*A*a*c**2*d*x*sin(e + f*x)**2/2 + 3*A*a*c**2*d*x*cos(e + f*
x)**2/2 - 3*A*a*c**2*d*sin(e + f*x)*cos(e + f*x)/(2*f) - 3*A*a*c**2*d*cos(e + f*x)/f + 3*A*a*c*d**2*x*sin(e +
f*x)**2/2 + 3*A*a*c*d**2*x*cos(e + f*x)**2/2 - 3*A*a*c*d**2*sin(e + f*x)**2*cos(e + f*x)/f - 3*A*a*c*d**2*sin(
e + f*x)*cos(e + f*x)/(2*f) - 2*A*a*c*d**2*cos(e + f*x)**3/f + 3*A*a*d**3*x*sin(e + f*x)**4/8 + 3*A*a*d**3*x*s
in(e + f*x)**2*cos(e + f*x)**2/4 + 3*A*a*d**3*x*cos(e + f*x)**4/8 - 5*A*a*d**3*sin(e + f*x)**3*cos(e + f*x)/(8
*f) - A*a*d**3*sin(e + f*x)**2*cos(e + f*x)/f - 3*A*a*d**3*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 2*A*a*d**3*cos
(e + f*x)**3/(3*f) + B*a*c**3*x*sin(e + f*x)**2/2 + B*a*c**3*x*cos(e + f*x)**2/2 - B*a*c**3*sin(e + f*x)*cos(e
+ f*x)/(2*f) - B*a*c**3*cos(e + f*x)/f + 3*B*a*c**2*d*x*sin(e + f*x)**2/2 + 3*B*a*c**2*d*x*cos(e + f*x)**2/2
- 3*B*a*c**2*d*sin(e + f*x)**2*cos(e + f*x)/f - 3*B*a*c**2*d*sin(e + f*x)*cos(e + f*x)/(2*f) - 2*B*a*c**2*d*co
s(e + f*x)**3/f + 9*B*a*c*d**2*x*sin(e + f*x)**4/8 + 9*B*a*c*d**2*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 9*B*a*
c*d**2*x*cos(e + f*x)**4/8 - 15*B*a*c*d**2*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 3*B*a*c*d**2*sin(e + f*x)**2*c
os(e + f*x)/f - 9*B*a*c*d**2*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 2*B*a*c*d**2*cos(e + f*x)**3/f + 3*B*a*d**3*
x*sin(e + f*x)**4/8 + 3*B*a*d**3*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 3*B*a*d**3*x*cos(e + f*x)**4/8 - B*a*d*
*3*sin(e + f*x)**4*cos(e + f*x)/f - 5*B*a*d**3*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 4*B*a*d**3*sin(e + f*x)**2
*cos(e + f*x)**3/(3*f) - 3*B*a*d**3*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 8*B*a*d**3*cos(e + f*x)**5/(15*f), Ne
(f, 0)), (x*(A + B*sin(e))*(c + d*sin(e))**3*(a*sin(e) + a), True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.24 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\frac {480 \, {\left (f x + e\right )} A a c^{3} + 120 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a c^{3} + 360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a c^{2} d + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a c^{2} d + 360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a c^{2} d + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a c d^{2} + 360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a c d^{2} + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a c d^{2} + 45 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a c d^{2} + 160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a d^{3} + 15 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a d^{3} - 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a d^{3} + 15 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a d^{3} - 480 \, A a c^{3} \cos \left (f x + e\right ) - 480 \, B a c^{3} \cos \left (f x + e\right ) - 1440 \, A a c^{2} d \cos \left (f x + e\right )}{480 \, f} \]

[In]

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

[Out]

1/480*(480*(f*x + e)*A*a*c^3 + 120*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a*c^3 + 360*(2*f*x + 2*e - sin(2*f*x + 2
*e))*A*a*c^2*d + 480*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a*c^2*d + 360*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a*c^
2*d + 480*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a*c*d^2 + 360*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a*c*d^2 + 480*(
cos(f*x + e)^3 - 3*cos(f*x + e))*B*a*c*d^2 + 45*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a*c*
d^2 + 160*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a*d^3 + 15*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e
))*A*a*d^3 - 32*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a*d^3 + 15*(12*f*x + 12*e + sin(4*f
*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a*d^3 - 480*A*a*c^3*cos(f*x + e) - 480*B*a*c^3*cos(f*x + e) - 1440*A*a*c^2*d
*cos(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.94 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=-\frac {B a d^{3} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {1}{8} \, {\left (8 \, A a c^{3} + 4 \, B a c^{3} + 12 \, A a c^{2} d + 12 \, B a c^{2} d + 12 \, A a c d^{2} + 9 \, B a c d^{2} + 3 \, A a d^{3} + 3 \, B a d^{3}\right )} x + \frac {{\left (12 \, B a c^{2} d + 12 \, A a c d^{2} + 12 \, B a c d^{2} + 4 \, A a d^{3} + 5 \, B a d^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (8 \, A a c^{3} + 8 \, B a c^{3} + 24 \, A a c^{2} d + 18 \, B a c^{2} d + 18 \, A a c d^{2} + 18 \, B a c d^{2} + 6 \, A a d^{3} + 5 \, B a d^{3}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {{\left (3 \, B a c d^{2} + A a d^{3} + B a d^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac {{\left (B a c^{3} + 3 \, A a c^{2} d + 3 \, B a c^{2} d + 3 \, A a c d^{2} + 3 \, B a c d^{2} + A a d^{3} + B a d^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]

[In]

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

[Out]

-1/80*B*a*d^3*cos(5*f*x + 5*e)/f + 1/8*(8*A*a*c^3 + 4*B*a*c^3 + 12*A*a*c^2*d + 12*B*a*c^2*d + 12*A*a*c*d^2 + 9
*B*a*c*d^2 + 3*A*a*d^3 + 3*B*a*d^3)*x + 1/48*(12*B*a*c^2*d + 12*A*a*c*d^2 + 12*B*a*c*d^2 + 4*A*a*d^3 + 5*B*a*d
^3)*cos(3*f*x + 3*e)/f - 1/8*(8*A*a*c^3 + 8*B*a*c^3 + 24*A*a*c^2*d + 18*B*a*c^2*d + 18*A*a*c*d^2 + 18*B*a*c*d^
2 + 6*A*a*d^3 + 5*B*a*d^3)*cos(f*x + e)/f + 1/32*(3*B*a*c*d^2 + A*a*d^3 + B*a*d^3)*sin(4*f*x + 4*e)/f - 1/4*(B
*a*c^3 + 3*A*a*c^2*d + 3*B*a*c^2*d + 3*A*a*c*d^2 + 3*B*a*c*d^2 + A*a*d^3 + B*a*d^3)*sin(2*f*x + 2*e)/f

Mupad [B] (verification not implemented)

Time = 16.27 (sec) , antiderivative size = 830, normalized size of antiderivative = 2.54 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (8\,A\,c^3+3\,A\,d^3+4\,B\,c^3+3\,B\,d^3+12\,A\,c\,d^2+12\,A\,c^2\,d+9\,B\,c\,d^2+12\,B\,c^2\,d\right )}{4\,\left (2\,A\,a\,c^3+\frac {3\,A\,a\,d^3}{4}+B\,a\,c^3+\frac {3\,B\,a\,d^3}{4}+3\,A\,a\,c\,d^2+3\,A\,a\,c^2\,d+\frac {9\,B\,a\,c\,d^2}{4}+3\,B\,a\,c^2\,d\right )}\right )\,\left (8\,A\,c^3+3\,A\,d^3+4\,B\,c^3+3\,B\,d^3+12\,A\,c\,d^2+12\,A\,c^2\,d+9\,B\,c\,d^2+12\,B\,c^2\,d\right )}{4\,f}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3\,A\,a\,d^3}{4}+B\,a\,c^3+\frac {3\,B\,a\,d^3}{4}+3\,A\,a\,c\,d^2+3\,A\,a\,c^2\,d+\frac {9\,B\,a\,c\,d^2}{4}+3\,B\,a\,c^2\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (2\,A\,a\,c^3+2\,B\,a\,c^3+6\,A\,a\,c^2\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (8\,A\,a\,c^3+\frac {20\,A\,a\,d^3}{3}+8\,B\,a\,c^3+\frac {16\,B\,a\,d^3}{3}+20\,A\,a\,c\,d^2+24\,A\,a\,c^2\,d+20\,B\,a\,c\,d^2+20\,B\,a\,c^2\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (12\,A\,a\,c^3+\frac {28\,A\,a\,d^3}{3}+12\,B\,a\,c^3+\frac {32\,B\,a\,d^3}{3}+28\,A\,a\,c\,d^2+36\,A\,a\,c^2\,d+28\,B\,a\,c\,d^2+28\,B\,a\,c^2\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (8\,A\,a\,c^3+4\,A\,a\,d^3+8\,B\,a\,c^3+12\,A\,a\,c\,d^2+24\,A\,a\,c^2\,d+12\,B\,a\,c\,d^2+12\,B\,a\,c^2\,d\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (\frac {3\,A\,a\,d^3}{4}+B\,a\,c^3+\frac {3\,B\,a\,d^3}{4}+3\,A\,a\,c\,d^2+3\,A\,a\,c^2\,d+\frac {9\,B\,a\,c\,d^2}{4}+3\,B\,a\,c^2\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {7\,A\,a\,d^3}{2}+2\,B\,a\,c^3+\frac {7\,B\,a\,d^3}{2}+6\,A\,a\,c\,d^2+6\,A\,a\,c^2\,d+\frac {21\,B\,a\,c\,d^2}{2}+6\,B\,a\,c^2\,d\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (\frac {7\,A\,a\,d^3}{2}+2\,B\,a\,c^3+\frac {7\,B\,a\,d^3}{2}+6\,A\,a\,c\,d^2+6\,A\,a\,c^2\,d+\frac {21\,B\,a\,c\,d^2}{2}+6\,B\,a\,c^2\,d\right )+2\,A\,a\,c^3+\frac {4\,A\,a\,d^3}{3}+2\,B\,a\,c^3+\frac {16\,B\,a\,d^3}{15}+4\,A\,a\,c\,d^2+6\,A\,a\,c^2\,d+4\,B\,a\,c\,d^2+4\,B\,a\,c^2\,d}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]

[In]

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

[Out]

(a*atan((a*tan(e/2 + (f*x)/2)*(8*A*c^3 + 3*A*d^3 + 4*B*c^3 + 3*B*d^3 + 12*A*c*d^2 + 12*A*c^2*d + 9*B*c*d^2 + 1
2*B*c^2*d))/(4*(2*A*a*c^3 + (3*A*a*d^3)/4 + B*a*c^3 + (3*B*a*d^3)/4 + 3*A*a*c*d^2 + 3*A*a*c^2*d + (9*B*a*c*d^2
)/4 + 3*B*a*c^2*d)))*(8*A*c^3 + 3*A*d^3 + 4*B*c^3 + 3*B*d^3 + 12*A*c*d^2 + 12*A*c^2*d + 9*B*c*d^2 + 12*B*c^2*d
))/(4*f) - (tan(e/2 + (f*x)/2)*((3*A*a*d^3)/4 + B*a*c^3 + (3*B*a*d^3)/4 + 3*A*a*c*d^2 + 3*A*a*c^2*d + (9*B*a*c
*d^2)/4 + 3*B*a*c^2*d) + tan(e/2 + (f*x)/2)^8*(2*A*a*c^3 + 2*B*a*c^3 + 6*A*a*c^2*d) + tan(e/2 + (f*x)/2)^2*(8*
A*a*c^3 + (20*A*a*d^3)/3 + 8*B*a*c^3 + (16*B*a*d^3)/3 + 20*A*a*c*d^2 + 24*A*a*c^2*d + 20*B*a*c*d^2 + 20*B*a*c^
2*d) + tan(e/2 + (f*x)/2)^4*(12*A*a*c^3 + (28*A*a*d^3)/3 + 12*B*a*c^3 + (32*B*a*d^3)/3 + 28*A*a*c*d^2 + 36*A*a
*c^2*d + 28*B*a*c*d^2 + 28*B*a*c^2*d) + tan(e/2 + (f*x)/2)^6*(8*A*a*c^3 + 4*A*a*d^3 + 8*B*a*c^3 + 12*A*a*c*d^2
+ 24*A*a*c^2*d + 12*B*a*c*d^2 + 12*B*a*c^2*d) - tan(e/2 + (f*x)/2)^9*((3*A*a*d^3)/4 + B*a*c^3 + (3*B*a*d^3)/4
+ 3*A*a*c*d^2 + 3*A*a*c^2*d + (9*B*a*c*d^2)/4 + 3*B*a*c^2*d) + tan(e/2 + (f*x)/2)^3*((7*A*a*d^3)/2 + 2*B*a*c^
3 + (7*B*a*d^3)/2 + 6*A*a*c*d^2 + 6*A*a*c^2*d + (21*B*a*c*d^2)/2 + 6*B*a*c^2*d) - tan(e/2 + (f*x)/2)^7*((7*A*a
*d^3)/2 + 2*B*a*c^3 + (7*B*a*d^3)/2 + 6*A*a*c*d^2 + 6*A*a*c^2*d + (21*B*a*c*d^2)/2 + 6*B*a*c^2*d) + 2*A*a*c^3
+ (4*A*a*d^3)/3 + 2*B*a*c^3 + (16*B*a*d^3)/15 + 4*A*a*c*d^2 + 6*A*a*c^2*d + 4*B*a*c*d^2 + 4*B*a*c^2*d)/(f*(5*t
an(e/2 + (f*x)/2)^2 + 10*tan(e/2 + (f*x)/2)^4 + 10*tan(e/2 + (f*x)/2)^6 + 5*tan(e/2 + (f*x)/2)^8 + tan(e/2 + (
f*x)/2)^10 + 1))