\(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx\) [259]

Optimal result

Integrand size = 35, antiderivative size = 463 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {1}{16} a^3 \left (B \left (30 c^2+52 c d+23 d^2\right )+A \left (40 c^2+60 c d+26 d^2\right )\right ) x-\frac {a^3 \left (2 A d \left (2 c^4-15 c^3 d+72 c^2 d^2+180 c d^3+76 d^4\right )-B \left (2 c^5-12 c^4 d+37 c^3 d^2-112 c^2 d^3-304 c d^4-136 d^5\right )\right ) \cos (e+f x)}{60 d^3 f}-\frac {a^3 \left (2 A d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )-B \left (4 c^4-24 c^3 d+76 c^2 d^2-236 c d^3-345 d^4\right )\right ) \cos (e+f x) \sin (e+f x)}{240 d^2 f}-\frac {a^3 \left (2 A d \left (2 c^2-15 c d+76 d^2\right )-B \left (2 c^3-12 c^2 d+41 c d^2-136 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^3 f}+\frac {a^3 \left (2 A (2 c-11 d) d-B \left (2 c^2-8 c d+21 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{40 d^3 f}-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3}{6 d f}+\frac {(3 B c-6 A d-8 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{30 d^2 f} \]

[Out]

1/16*a^3*(B*(30*c^2+52*c*d+23*d^2)+A*(40*c^2+60*c*d+26*d^2))*x-1/60*a^3*(2*A*d*(2*c^4-15*c^3*d+72*c^2*d^2+180*
c*d^3+76*d^4)-B*(2*c^5-12*c^4*d+37*c^3*d^2-112*c^2*d^3-304*c*d^4-136*d^5))*cos(f*x+e)/d^3/f-1/240*a^3*(2*A*d*(
4*c^3-30*c^2*d+146*c*d^2+195*d^3)-B*(4*c^4-24*c^3*d+76*c^2*d^2-236*c*d^3-345*d^4))*cos(f*x+e)*sin(f*x+e)/d^2/f
-1/120*a^3*(2*A*d*(2*c^2-15*c*d+76*d^2)-B*(2*c^3-12*c^2*d+41*c*d^2-136*d^3))*cos(f*x+e)*(c+d*sin(f*x+e))^2/d^3
/f+1/40*a^3*(2*A*(2*c-11*d)*d-B*(2*c^2-8*c*d+21*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^3/d^3/f-1/6*a*B*cos(f*x+e)*(
a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^3/d/f+1/30*(-6*A*d+3*B*c-8*B*d)*cos(f*x+e)*(a^3+a^3*sin(f*x+e))*(c+d*sin(f*
x+e))^3/d^2/f

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3055, 3047, 3102, 2832, 2813} \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {1}{16} a^3 x \left (A \left (40 c^2+60 c d+26 d^2\right )+B \left (30 c^2+52 c d+23 d^2\right )\right )+\frac {a^3 \left (2 A d (2 c-11 d)-B \left (2 c^2-8 c d+21 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{40 d^3 f}-\frac {a^3 \left (2 A d \left (2 c^2-15 c d+76 d^2\right )-B \left (2 c^3-12 c^2 d+41 c d^2-136 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^3 f}-\frac {a^3 \left (2 A d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )-B \left (4 c^4-24 c^3 d+76 c^2 d^2-236 c d^3-345 d^4\right )\right ) \sin (e+f x) \cos (e+f x)}{240 d^2 f}-\frac {a^3 \left (2 A d \left (2 c^4-15 c^3 d+72 c^2 d^2+180 c d^3+76 d^4\right )-B \left (2 c^5-12 c^4 d+37 c^3 d^2-112 c^2 d^3-304 c d^4-136 d^5\right )\right ) \cos (e+f x)}{60 d^3 f}+\frac {(-6 A d+3 B c-8 B d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^3}{30 d^2 f}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^3}{6 d f} \]

[In]

Int[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]

[Out]

(a^3*(B*(30*c^2 + 52*c*d + 23*d^2) + A*(40*c^2 + 60*c*d + 26*d^2))*x)/16 - (a^3*(2*A*d*(2*c^4 - 15*c^3*d + 72*
c^2*d^2 + 180*c*d^3 + 76*d^4) - B*(2*c^5 - 12*c^4*d + 37*c^3*d^2 - 112*c^2*d^3 - 304*c*d^4 - 136*d^5))*Cos[e +
f*x])/(60*d^3*f) - (a^3*(2*A*d*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^3) - B*(4*c^4 - 24*c^3*d + 76*c^2*d^2 -
236*c*d^3 - 345*d^4))*Cos[e + f*x]*Sin[e + f*x])/(240*d^2*f) - (a^3*(2*A*d*(2*c^2 - 15*c*d + 76*d^2) - B*(2*c^
3 - 12*c^2*d + 41*c*d^2 - 136*d^3))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(120*d^3*f) + (a^3*(2*A*(2*c - 11*d)*
d - B*(2*c^2 - 8*c*d + 21*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(40*d^3*f) - (a*B*Cos[e + f*x]*(a + a*Sin
[e + f*x])^2*(c + d*Sin[e + f*x])^3)/(6*d*f) + ((3*B*c - 6*A*d - 8*B*d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*
(c + d*Sin[e + f*x])^3)/(30*d^2*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 3055

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-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] + Dist[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 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}& = -\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3}{6 d f}+\frac {\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2 (a (2 B c+6 A d+3 B d)-a (3 B c-6 A d-8 B d) \sin (e+f x)) \, dx}{6 d} \\ & = -\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3}{6 d f}+\frac {(3 B c-6 A d-8 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{30 d^2 f}+\frac {\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^2 \left (3 a^2 \left (2 A d (c+8 d)-B \left (c^2-3 c d-13 d^2\right )\right )-3 a^2 \left (2 A (2 c-11 d) d-B \left (2 c^2-8 c d+21 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{30 d^2} \\ & = -\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3}{6 d f}+\frac {(3 B c-6 A d-8 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{30 d^2 f}+\frac {\int (c+d \sin (e+f x))^2 \left (3 a^3 \left (2 A d (c+8 d)-B \left (c^2-3 c d-13 d^2\right )\right )+\left (3 a^3 \left (2 A d (c+8 d)-B \left (c^2-3 c d-13 d^2\right )\right )-3 a^3 \left (2 A (2 c-11 d) d-B \left (2 c^2-8 c d+21 d^2\right )\right )\right ) \sin (e+f x)-3 a^3 \left (2 A (2 c-11 d) d-B \left (2 c^2-8 c d+21 d^2\right )\right ) \sin ^2(e+f x)\right ) \, dx}{30 d^2} \\ & = \frac {a^3 \left (2 A (2 c-11 d) d-B \left (2 c^2-8 c d+21 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{40 d^3 f}-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3}{6 d f}+\frac {(3 B c-6 A d-8 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{30 d^2 f}+\frac {\int (c+d \sin (e+f x))^2 \left (-3 a^3 d \left (2 A (2 c-65 d) d-B \left (2 c^2-12 c d+115 d^2\right )\right )+3 a^3 \left (2 A d \left (2 c^2-15 c d+76 d^2\right )-B \left (2 c^3-12 c^2 d+41 c d^2-136 d^3\right )\right ) \sin (e+f x)\right ) \, dx}{120 d^3} \\ & = -\frac {a^3 \left (2 A d \left (2 c^2-15 c d+76 d^2\right )-B \left (2 c^3-12 c^2 d+41 c d^2-136 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^3 f}+\frac {a^3 \left (2 A (2 c-11 d) d-B \left (2 c^2-8 c d+21 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{40 d^3 f}-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3}{6 d f}+\frac {(3 B c-6 A d-8 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{30 d^2 f}+\frac {\int (c+d \sin (e+f x)) \left (-3 a^3 d \left (2 A d \left (2 c^2-165 c d-152 d^2\right )-B \left (2 c^3-12 c^2 d+263 c d^2+272 d^3\right )\right )+3 a^3 \left (2 A d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )-B \left (4 c^4-24 c^3 d+76 c^2 d^2-236 c d^3-345 d^4\right )\right ) \sin (e+f x)\right ) \, dx}{360 d^3} \\ & = \frac {1}{16} a^3 \left (B \left (30 c^2+52 c d+23 d^2\right )+A \left (40 c^2+60 c d+26 d^2\right )\right ) x-\frac {a^3 \left (2 A d \left (2 c^4-15 c^3 d+72 c^2 d^2+180 c d^3+76 d^4\right )-B \left (2 c^5-12 c^4 d+37 c^3 d^2-112 c^2 d^3-304 c d^4-136 d^5\right )\right ) \cos (e+f x)}{60 d^3 f}-\frac {a^3 \left (2 A d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )-B \left (4 c^4-24 c^3 d+76 c^2 d^2-236 c d^3-345 d^4\right )\right ) \cos (e+f x) \sin (e+f x)}{240 d^2 f}-\frac {a^3 \left (2 A d \left (2 c^2-15 c d+76 d^2\right )-B \left (2 c^3-12 c^2 d+41 c d^2-136 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^3 f}+\frac {a^3 \left (2 A (2 c-11 d) d-B \left (2 c^2-8 c d+21 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{40 d^3 f}-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3}{6 d f}+\frac {(3 B c-6 A d-8 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{30 d^2 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.59 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.77 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {a^3 \cos (e+f x) \left (60 \left (B \left (30 c^2+52 c d+23 d^2\right )+A \left (40 c^2+60 c d+26 d^2\right )\right ) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (1840 A c^2+1680 B c^2+3360 A c d+3112 B c d+1556 A d^2+1468 B d^2-16 \left (A \left (5 c^2+30 c d+22 d^2\right )+B \left (15 c^2+44 c d+26 d^2\right )\right ) \cos (2 (e+f x))+12 d (2 B c+A d+3 B d) \cos (4 (e+f x))+720 A c^2 \sin (e+f x)+990 B c^2 \sin (e+f x)+1980 A c d \sin (e+f x)+2100 B c d \sin (e+f x)+1050 A d^2 \sin (e+f x)+1085 B d^2 \sin (e+f x)-30 B c^2 \sin (3 (e+f x))-60 A c d \sin (3 (e+f x))-180 B c d \sin (3 (e+f x))-90 A d^2 \sin (3 (e+f x))-140 B d^2 \sin (3 (e+f x))+5 B d^2 \sin (5 (e+f x))\right )\right )}{480 f \sqrt {\cos ^2(e+f x)}} \]

[In]

Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]

[Out]

-1/480*(a^3*Cos[e + f*x]*(60*(B*(30*c^2 + 52*c*d + 23*d^2) + A*(40*c^2 + 60*c*d + 26*d^2))*ArcSin[Sqrt[1 - Sin
[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(1840*A*c^2 + 1680*B*c^2 + 3360*A*c*d + 3112*B*c*d + 1556*A*d^2 + 1
468*B*d^2 - 16*(A*(5*c^2 + 30*c*d + 22*d^2) + B*(15*c^2 + 44*c*d + 26*d^2))*Cos[2*(e + f*x)] + 12*d*(2*B*c + A
*d + 3*B*d)*Cos[4*(e + f*x)] + 720*A*c^2*Sin[e + f*x] + 990*B*c^2*Sin[e + f*x] + 1980*A*c*d*Sin[e + f*x] + 210
0*B*c*d*Sin[e + f*x] + 1050*A*d^2*Sin[e + f*x] + 1085*B*d^2*Sin[e + f*x] - 30*B*c^2*Sin[3*(e + f*x)] - 60*A*c*
d*Sin[3*(e + f*x)] - 180*B*c*d*Sin[3*(e + f*x)] - 90*A*d^2*Sin[3*(e + f*x)] - 140*B*d^2*Sin[3*(e + f*x)] + 5*B
*d^2*Sin[5*(e + f*x)])))/(f*Sqrt[Cos[e + f*x]^2])

Maple [A] (verified)

Time = 2.10 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.58

method	result	size
parallelrisch	\(\frac {a^{3} \left (\left (\frac {\left (17 A +19 B \right ) d^{2}}{4}+6 \left (\frac {17 B}{12}+A \right ) c d +c^{2} \left (A +3 B \right )\right ) \cos \left (3 f x +3 e \right )+3 \left (\left (-\frac {63 B}{16}-4 A \right ) d^{2}-8 d c \left (A +B \right )-3 \left (A +\frac {4 B}{3}\right ) c^{2}\right ) \sin \left (2 f x +2 e \right )+\frac {3 \left (\frac {3 \left (\frac {3 B}{2}+A \right ) d^{2}}{2}+c \left (A +3 B \right ) d +\frac {B \,c^{2}}{2}\right ) \sin \left (4 f x +4 e \right )}{4}-\frac {3 \left (\left (A +3 B \right ) d +2 B c \right ) d \cos \left (5 f x +5 e \right )}{20}-\frac {B \,d^{2} \sin \left (6 f x +6 e \right )}{16}+3 \left (\frac {\left (-23 A -21 B \right ) d^{2}}{2}-26 c \left (\frac {23 B}{26}+A \right ) d -15 c^{2} \left (A +\frac {13 B}{15}\right )\right ) \cos \left (f x +e \right )+\left (-\frac {136}{5} B +\frac {39}{2} f x A +\frac {69}{4} f x B -\frac {152}{5} A \right ) d^{2}+45 c \left (f x A +\frac {13}{15} f x B -\frac {8}{5} A -\frac {304}{225} B \right ) d +30 c^{2} \left (f x A +\frac {3}{4} f x B -\frac {22}{15} A -\frac {6}{5} B \right )\right )}{12 f}\)	\(268\)
parts	\(-\frac {\left (A \,a^{3} d^{2}+2 B \,a^{3} c d +3 B \,a^{3} d^{2}\right ) \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}-\frac {\left (3 A \,a^{3} c^{2}+2 A \,a^{3} c d +B \,a^{3} c^{2}\right ) \cos \left (f x +e \right )}{f}+\frac {\left (2 A \,a^{3} c d +3 A \,a^{3} d^{2}+B \,a^{3} c^{2}+6 B \,a^{3} c d +3 B \,a^{3} d^{2}\right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}+\frac {\left (3 A \,a^{3} c^{2}+6 A \,a^{3} c d +A \,a^{3} d^{2}+3 B \,a^{3} c^{2}+2 B \,a^{3} c d \right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {\left (A \,a^{3} c^{2}+6 A \,a^{3} c d +3 A \,a^{3} d^{2}+3 B \,a^{3} c^{2}+6 B \,a^{3} c d +B \,a^{3} d^{2}\right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+a^{3} c^{2} x A +\frac {B \,a^{3} d^{2} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}\)	\(380\)
risch	\(\frac {13 B \,a^{3} c d x}{4}-\frac {15 a^{3} \cos \left (f x +e \right ) A \,c^{2}}{4 f}-\frac {23 a^{3} \cos \left (f x +e \right ) A \,d^{2}}{8 f}-\frac {13 a^{3} \cos \left (f x +e \right ) B \,c^{2}}{4 f}-\frac {21 a^{3} \cos \left (f x +e \right ) d^{2} B}{8 f}-\frac {B \,a^{3} d^{2} \sin \left (6 f x +6 e \right )}{192 f}+\frac {15 B \,a^{3} c^{2} x}{8}+\frac {23 B \,a^{3} d^{2} x}{16}+\frac {13 A \,a^{3} d^{2} x}{8}-\frac {2 \sin \left (2 f x +2 e \right ) A \,a^{3} c d}{f}-\frac {2 \sin \left (2 f x +2 e \right ) B \,a^{3} c d}{f}-\frac {13 a^{3} \cos \left (f x +e \right ) A c d}{2 f}-\frac {23 a^{3} \cos \left (f x +e \right ) c d B}{4 f}-\frac {a^{3} d \cos \left (5 f x +5 e \right ) B c}{40 f}+\frac {\sin \left (4 f x +4 e \right ) A \,a^{3} c d}{16 f}+\frac {3 \sin \left (4 f x +4 e \right ) B \,a^{3} c d}{16 f}+\frac {a^{3} \cos \left (3 f x +3 e \right ) A c d}{2 f}+\frac {17 a^{3} \cos \left (3 f x +3 e \right ) c d B}{24 f}+\frac {15 A \,a^{3} c d x}{4}-\frac {a^{3} d^{2} \cos \left (5 f x +5 e \right ) A}{80 f}-\frac {3 a^{3} d^{2} \cos \left (5 f x +5 e \right ) B}{80 f}+\frac {3 \sin \left (4 f x +4 e \right ) A \,a^{3} d^{2}}{32 f}+\frac {\sin \left (4 f x +4 e \right ) B \,a^{3} c^{2}}{32 f}+\frac {9 \sin \left (4 f x +4 e \right ) B \,a^{3} d^{2}}{64 f}+\frac {a^{3} \cos \left (3 f x +3 e \right ) A \,c^{2}}{12 f}+\frac {17 a^{3} \cos \left (3 f x +3 e \right ) A \,d^{2}}{48 f}+\frac {a^{3} \cos \left (3 f x +3 e \right ) B \,c^{2}}{4 f}+\frac {19 a^{3} \cos \left (3 f x +3 e \right ) d^{2} B}{48 f}-\frac {3 \sin \left (2 f x +2 e \right ) A \,a^{3} c^{2}}{4 f}-\frac {\sin \left (2 f x +2 e \right ) A \,a^{3} d^{2}}{f}-\frac {\sin \left (2 f x +2 e \right ) B \,a^{3} c^{2}}{f}-\frac {63 \sin \left (2 f x +2 e \right ) B \,a^{3} d^{2}}{64 f}+\frac {5 a^{3} c^{2} x A}{2}\)	\(600\)
derivativedivides	\(\frac {-\frac {A \,a^{3} c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 A \,a^{3} c d \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 {A \,a^{3} d^{2} \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}+B \,a^{3} c^{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 {2 B \,a^{3} c d \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}+B \,a^{3} d^{2} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+3 A \,a^{3} c^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 A \,a^{3} c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 A \,a^{3} 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 )-B \,a^{3} c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+6 B \,a^{3} c d \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 {3 B \,a^{3} d^{2} \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}-3 A \,a^{3} c^{2} \cos \left (f x +e \right )+6 A \,a^{3} c 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^{3} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 B \,a^{3} c^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 B \,a^{3} c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 B \,a^{3} 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 )+A \,a^{3} c^{2} \left (f x +e \right )-2 A \,a^{3} c d \cos \left (f x +e \right )+A \,a^{3} 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 )-B \,a^{3} c^{2} \cos \left (f x +e \right )+2 B \,a^{3} c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {B \,a^{3} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\)	\(725\)
default	\(\frac {-\frac {A \,a^{3} c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 A \,a^{3} c d \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 {A \,a^{3} d^{2} \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}+B \,a^{3} c^{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 {2 B \,a^{3} c d \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}+B \,a^{3} d^{2} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+3 A \,a^{3} c^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 A \,a^{3} c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 A \,a^{3} 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 )-B \,a^{3} c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+6 B \,a^{3} c d \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 {3 B \,a^{3} d^{2} \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}-3 A \,a^{3} c^{2} \cos \left (f x +e \right )+6 A \,a^{3} c 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^{3} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 B \,a^{3} c^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 B \,a^{3} c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 B \,a^{3} 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 )+A \,a^{3} c^{2} \left (f x +e \right )-2 A \,a^{3} c d \cos \left (f x +e \right )+A \,a^{3} 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 )-B \,a^{3} c^{2} \cos \left (f x +e \right )+2 B \,a^{3} c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {B \,a^{3} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\)	\(725\)
norman	\(\text {Expression too large to display}\)	\(1167\)

[In]

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

1/12*a^3*((1/4*(17*A+19*B)*d^2+6*(17/12*B+A)*c*d+c^2*(A+3*B))*cos(3*f*x+3*e)+3*((-63/16*B-4*A)*d^2-8*d*c*(A+B)
-3*(A+4/3*B)*c^2)*sin(2*f*x+2*e)+3/4*(3/2*(3/2*B+A)*d^2+c*(A+3*B)*d+1/2*B*c^2)*sin(4*f*x+4*e)-3/20*((A+3*B)*d+
2*B*c)*d*cos(5*f*x+5*e)-1/16*B*d^2*sin(6*f*x+6*e)+3*(1/2*(-23*A-21*B)*d^2-26*c*(23/26*B+A)*d-15*c^2*(A+13/15*B
))*cos(f*x+e)+(-136/5*B+39/2*f*x*A+69/4*f*x*B-152/5*A)*d^2+45*c*(f*x*A+13/15*f*x*B-8/5*A-304/225*B)*d+30*c^2*(
f*x*A+3/4*f*x*B-22/15*A-6/5*B))/f

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.65 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {48 \, {\left (2 \, B a^{3} c d + {\left (A + 3 \, B\right )} a^{3} d^{2}\right )} \cos \left (f x + e\right )^{5} - 80 \, {\left ({\left (A + 3 \, B\right )} a^{3} c^{2} + 2 \, {\left (3 \, A + 5 \, B\right )} a^{3} c d + {\left (5 \, A + 7 \, B\right )} a^{3} d^{2}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (10 \, {\left (4 \, A + 3 \, B\right )} a^{3} c^{2} + 4 \, {\left (15 \, A + 13 \, B\right )} a^{3} c d + {\left (26 \, A + 23 \, B\right )} a^{3} d^{2}\right )} f x + 960 \, {\left ({\left (A + B\right )} a^{3} c^{2} + 2 \, {\left (A + B\right )} a^{3} c d + {\left (A + B\right )} a^{3} d^{2}\right )} \cos \left (f x + e\right ) + 5 \, {\left (8 \, B a^{3} d^{2} \cos \left (f x + e\right )^{5} - 2 \, {\left (6 \, B a^{3} c^{2} + 12 \, {\left (A + 3 \, B\right )} a^{3} c d + {\left (18 \, A + 31 \, B\right )} a^{3} d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (2 \, {\left (12 \, A + 17 \, B\right )} a^{3} c^{2} + 4 \, {\left (17 \, A + 19 \, B\right )} a^{3} c d + {\left (38 \, A + 41 \, B\right )} a^{3} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

-1/240*(48*(2*B*a^3*c*d + (A + 3*B)*a^3*d^2)*cos(f*x + e)^5 - 80*((A + 3*B)*a^3*c^2 + 2*(3*A + 5*B)*a^3*c*d +
(5*A + 7*B)*a^3*d^2)*cos(f*x + e)^3 - 15*(10*(4*A + 3*B)*a^3*c^2 + 4*(15*A + 13*B)*a^3*c*d + (26*A + 23*B)*a^3
*d^2)*f*x + 960*((A + B)*a^3*c^2 + 2*(A + B)*a^3*c*d + (A + B)*a^3*d^2)*cos(f*x + e) + 5*(8*B*a^3*d^2*cos(f*x
+ e)^5 - 2*(6*B*a^3*c^2 + 12*(A + 3*B)*a^3*c*d + (18*A + 31*B)*a^3*d^2)*cos(f*x + e)^3 + 3*(2*(12*A + 17*B)*a^
3*c^2 + 4*(17*A + 19*B)*a^3*c*d + (38*A + 41*B)*a^3*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. 1804 vs. \(2 (449) = 898\).

Time = 0.50 (sec) , antiderivative size = 1804, normalized size of antiderivative = 3.90 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))**2,x)

[Out]

Piecewise((3*A*a**3*c**2*x*sin(e + f*x)**2/2 + 3*A*a**3*c**2*x*cos(e + f*x)**2/2 + A*a**3*c**2*x - A*a**3*c**2
*sin(e + f*x)**2*cos(e + f*x)/f - 3*A*a**3*c**2*sin(e + f*x)*cos(e + f*x)/(2*f) - 2*A*a**3*c**2*cos(e + f*x)**
3/(3*f) - 3*A*a**3*c**2*cos(e + f*x)/f + 3*A*a**3*c*d*x*sin(e + f*x)**4/4 + 3*A*a**3*c*d*x*sin(e + f*x)**2*cos
(e + f*x)**2/2 + 3*A*a**3*c*d*x*sin(e + f*x)**2 + 3*A*a**3*c*d*x*cos(e + f*x)**4/4 + 3*A*a**3*c*d*x*cos(e + f*
x)**2 - 5*A*a**3*c*d*sin(e + f*x)**3*cos(e + f*x)/(4*f) - 6*A*a**3*c*d*sin(e + f*x)**2*cos(e + f*x)/f - 3*A*a*
*3*c*d*sin(e + f*x)*cos(e + f*x)**3/(4*f) - 3*A*a**3*c*d*sin(e + f*x)*cos(e + f*x)/f - 4*A*a**3*c*d*cos(e + f*
x)**3/f - 2*A*a**3*c*d*cos(e + f*x)/f + 9*A*a**3*d**2*x*sin(e + f*x)**4/8 + 9*A*a**3*d**2*x*sin(e + f*x)**2*co
s(e + f*x)**2/4 + A*a**3*d**2*x*sin(e + f*x)**2/2 + 9*A*a**3*d**2*x*cos(e + f*x)**4/8 + A*a**3*d**2*x*cos(e +
f*x)**2/2 - A*a**3*d**2*sin(e + f*x)**4*cos(e + f*x)/f - 15*A*a**3*d**2*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 4
*A*a**3*d**2*sin(e + f*x)**2*cos(e + f*x)**3/(3*f) - 3*A*a**3*d**2*sin(e + f*x)**2*cos(e + f*x)/f - 9*A*a**3*d
**2*sin(e + f*x)*cos(e + f*x)**3/(8*f) - A*a**3*d**2*sin(e + f*x)*cos(e + f*x)/(2*f) - 8*A*a**3*d**2*cos(e + f
*x)**5/(15*f) - 2*A*a**3*d**2*cos(e + f*x)**3/f + 3*B*a**3*c**2*x*sin(e + f*x)**4/8 + 3*B*a**3*c**2*x*sin(e +
f*x)**2*cos(e + f*x)**2/4 + 3*B*a**3*c**2*x*sin(e + f*x)**2/2 + 3*B*a**3*c**2*x*cos(e + f*x)**4/8 + 3*B*a**3*c
**2*x*cos(e + f*x)**2/2 - 5*B*a**3*c**2*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 3*B*a**3*c**2*sin(e + f*x)**2*cos
(e + f*x)/f - 3*B*a**3*c**2*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 3*B*a**3*c**2*sin(e + f*x)*cos(e + f*x)/(2*f)
- 2*B*a**3*c**2*cos(e + f*x)**3/f - B*a**3*c**2*cos(e + f*x)/f + 9*B*a**3*c*d*x*sin(e + f*x)**4/4 + 9*B*a**3*
c*d*x*sin(e + f*x)**2*cos(e + f*x)**2/2 + B*a**3*c*d*x*sin(e + f*x)**2 + 9*B*a**3*c*d*x*cos(e + f*x)**4/4 + B*
a**3*c*d*x*cos(e + f*x)**2 - 2*B*a**3*c*d*sin(e + f*x)**4*cos(e + f*x)/f - 15*B*a**3*c*d*sin(e + f*x)**3*cos(e
+ f*x)/(4*f) - 8*B*a**3*c*d*sin(e + f*x)**2*cos(e + f*x)**3/(3*f) - 6*B*a**3*c*d*sin(e + f*x)**2*cos(e + f*x)
/f - 9*B*a**3*c*d*sin(e + f*x)*cos(e + f*x)**3/(4*f) - B*a**3*c*d*sin(e + f*x)*cos(e + f*x)/f - 16*B*a**3*c*d*
cos(e + f*x)**5/(15*f) - 4*B*a**3*c*d*cos(e + f*x)**3/f + 5*B*a**3*d**2*x*sin(e + f*x)**6/16 + 15*B*a**3*d**2*
x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 9*B*a**3*d**2*x*sin(e + f*x)**4/8 + 15*B*a**3*d**2*x*sin(e + f*x)**2*co
s(e + f*x)**4/16 + 9*B*a**3*d**2*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 5*B*a**3*d**2*x*cos(e + f*x)**6/16 + 9*
B*a**3*d**2*x*cos(e + f*x)**4/8 - 11*B*a**3*d**2*sin(e + f*x)**5*cos(e + f*x)/(16*f) - 3*B*a**3*d**2*sin(e + f
*x)**4*cos(e + f*x)/f - 5*B*a**3*d**2*sin(e + f*x)**3*cos(e + f*x)**3/(6*f) - 15*B*a**3*d**2*sin(e + f*x)**3*c
os(e + f*x)/(8*f) - 4*B*a**3*d**2*sin(e + f*x)**2*cos(e + f*x)**3/f - B*a**3*d**2*sin(e + f*x)**2*cos(e + f*x)
/f - 5*B*a**3*d**2*sin(e + f*x)*cos(e + f*x)**5/(16*f) - 9*B*a**3*d**2*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 8*
B*a**3*d**2*cos(e + f*x)**5/(5*f) - 2*B*a**3*d**2*cos(e + f*x)**3/(3*f), Ne(f, 0)), (x*(A + B*sin(e))*(c + d*s
in(e))**2*(a*sin(e) + a)**3, True))

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 704, normalized size of antiderivative = 1.52 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} c^{2} + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c^{2} + 960 \, {\left (f x + e\right )} A a^{3} c^{2} + 960 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} c^{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^{3} c^{2} + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c^{2} + 1920 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} c d + 60 \, {\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^{3} c d + 1440 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c d - 128 \, {\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^{3} c d + 1920 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} c d + 180 \, {\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^{3} c d + 480 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c d - 64 \, {\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 )} A a^{3} d^{2} + 960 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} d^{2} + 90 \, {\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^{3} d^{2} + 240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} d^{2} - 192 \, {\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^{3} d^{2} + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} d^{2} + 5 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} d^{2} + 90 \, {\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^{3} d^{2} - 2880 \, A a^{3} c^{2} \cos \left (f x + e\right ) - 960 \, B a^{3} c^{2} \cos \left (f x + e\right ) - 1920 \, A a^{3} c d \cos \left (f x + e\right )}{960 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

1/960*(320*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a^3*c^2 + 720*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a^3*c^2 + 960*
(f*x + e)*A*a^3*c^2 + 960*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^3*c^2 + 30*(12*f*x + 12*e + sin(4*f*x + 4*e) -
8*sin(2*f*x + 2*e))*B*a^3*c^2 + 720*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^3*c^2 + 1920*(cos(f*x + e)^3 - 3*cos
(f*x + e))*A*a^3*c*d + 60*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^3*c*d + 1440*(2*f*x + 2*
e - sin(2*f*x + 2*e))*A*a^3*c*d - 128*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a^3*c*d + 192
0*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^3*c*d + 180*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*
a^3*c*d + 480*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^3*c*d - 64*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f
*x + e))*A*a^3*d^2 + 960*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a^3*d^2 + 90*(12*f*x + 12*e + sin(4*f*x + 4*e) -
8*sin(2*f*x + 2*e))*A*a^3*d^2 + 240*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a^3*d^2 - 192*(3*cos(f*x + e)^5 - 10*co
s(f*x + e)^3 + 15*cos(f*x + e))*B*a^3*d^2 + 320*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^3*d^2 + 5*(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*d^2 + 90*(12*f*x + 12*e + sin(4*f*x
+ 4*e) - 8*sin(2*f*x + 2*e))*B*a^3*d^2 - 2880*A*a^3*c^2*cos(f*x + e) - 960*B*a^3*c^2*cos(f*x + e) - 1920*A*a^3
*c*d*cos(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.81 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {B a^{3} d^{2} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {1}{16} \, {\left (40 \, A a^{3} c^{2} + 30 \, B a^{3} c^{2} + 60 \, A a^{3} c d + 52 \, B a^{3} c d + 26 \, A a^{3} d^{2} + 23 \, B a^{3} d^{2}\right )} x - \frac {{\left (2 \, B a^{3} c d + A a^{3} d^{2} + 3 \, B a^{3} d^{2}\right )} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {{\left (4 \, A a^{3} c^{2} + 12 \, B a^{3} c^{2} + 24 \, A a^{3} c d + 34 \, B a^{3} c d + 17 \, A a^{3} d^{2} + 19 \, B a^{3} d^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (30 \, A a^{3} c^{2} + 26 \, B a^{3} c^{2} + 52 \, A a^{3} c d + 46 \, B a^{3} c d + 23 \, A a^{3} d^{2} + 21 \, B a^{3} d^{2}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {{\left (2 \, B a^{3} c^{2} + 4 \, A a^{3} c d + 12 \, B a^{3} c d + 6 \, A a^{3} d^{2} + 9 \, B a^{3} d^{2}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {{\left (48 \, A a^{3} c^{2} + 64 \, B a^{3} c^{2} + 128 \, A a^{3} c d + 128 \, B a^{3} c d + 64 \, A a^{3} d^{2} + 63 \, B a^{3} d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, algorithm="giac")

[Out]

-1/192*B*a^3*d^2*sin(6*f*x + 6*e)/f + 1/16*(40*A*a^3*c^2 + 30*B*a^3*c^2 + 60*A*a^3*c*d + 52*B*a^3*c*d + 26*A*a
^3*d^2 + 23*B*a^3*d^2)*x - 1/80*(2*B*a^3*c*d + A*a^3*d^2 + 3*B*a^3*d^2)*cos(5*f*x + 5*e)/f + 1/48*(4*A*a^3*c^2
+ 12*B*a^3*c^2 + 24*A*a^3*c*d + 34*B*a^3*c*d + 17*A*a^3*d^2 + 19*B*a^3*d^2)*cos(3*f*x + 3*e)/f - 1/8*(30*A*a^
3*c^2 + 26*B*a^3*c^2 + 52*A*a^3*c*d + 46*B*a^3*c*d + 23*A*a^3*d^2 + 21*B*a^3*d^2)*cos(f*x + e)/f + 1/64*(2*B*a
^3*c^2 + 4*A*a^3*c*d + 12*B*a^3*c*d + 6*A*a^3*d^2 + 9*B*a^3*d^2)*sin(4*f*x + 4*e)/f - 1/64*(48*A*a^3*c^2 + 64*
B*a^3*c^2 + 128*A*a^3*c*d + 128*B*a^3*c*d + 64*A*a^3*d^2 + 63*B*a^3*d^2)*sin(2*f*x + 2*e)/f

Mupad [B] (verification not implemented)

Time = 16.12 (sec) , antiderivative size = 976, normalized size of antiderivative = 2.11 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\text {Too large to display} \]

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^2,x)

[Out]

(a^3*atan((a^3*tan(e/2 + (f*x)/2)*(40*A*c^2 + 26*A*d^2 + 30*B*c^2 + 23*B*d^2 + 60*A*c*d + 52*B*c*d))/(8*(5*A*a
^3*c^2 + (13*A*a^3*d^2)/4 + (15*B*a^3*c^2)/4 + (23*B*a^3*d^2)/8 + (15*A*a^3*c*d)/2 + (13*B*a^3*c*d)/2)))*(40*A
*c^2 + 26*A*d^2 + 30*B*c^2 + 23*B*d^2 + 60*A*c*d + 52*B*c*d))/(8*f) - (tan(e/2 + (f*x)/2)^10*(6*A*a^3*c^2 + 2*
B*a^3*c^2 + 4*A*a^3*c*d) + tan(e/2 + (f*x)/2)*(3*A*a^3*c^2 + (13*A*a^3*d^2)/4 + (15*B*a^3*c^2)/4 + (23*B*a^3*d
^2)/8 + (15*A*a^3*c*d)/2 + (13*B*a^3*c*d)/2) - tan(e/2 + (f*x)/2)^11*(3*A*a^3*c^2 + (13*A*a^3*d^2)/4 + (15*B*a
^3*c^2)/4 + (23*B*a^3*d^2)/8 + (15*A*a^3*c*d)/2 + (13*B*a^3*c*d)/2) + tan(e/2 + (f*x)/2)^8*(34*A*a^3*c^2 + 12*
A*a^3*d^2 + 22*B*a^3*c^2 + 4*B*a^3*d^2 + 44*A*a^3*c*d + 24*B*a^3*c*d) + tan(e/2 + (f*x)/2)^5*(6*A*a^3*c^2 + (2
5*A*a^3*d^2)/2 + (19*B*a^3*c^2)/2 + (75*B*a^3*d^2)/4 + 19*A*a^3*c*d + 25*B*a^3*c*d) - tan(e/2 + (f*x)/2)^7*(6*
A*a^3*c^2 + (25*A*a^3*d^2)/2 + (19*B*a^3*c^2)/2 + (75*B*a^3*d^2)/4 + 19*A*a^3*c*d + 25*B*a^3*c*d) + tan(e/2 +
(f*x)/2)^4*(76*A*a^3*c^2 + 64*A*a^3*d^2 + 68*B*a^3*c^2 + 64*B*a^3*d^2 + 136*A*a^3*c*d + 128*B*a^3*c*d) + tan(e
/2 + (f*x)/2)^3*(9*A*a^3*c^2 + (63*A*a^3*d^2)/4 + (53*B*a^3*c^2)/4 + (391*B*a^3*d^2)/24 + (53*A*a^3*c*d)/2 + (
63*B*a^3*c*d)/2) - tan(e/2 + (f*x)/2)^9*(9*A*a^3*c^2 + (63*A*a^3*d^2)/4 + (53*B*a^3*c^2)/4 + (391*B*a^3*d^2)/2
4 + (53*A*a^3*c*d)/2 + (63*B*a^3*c*d)/2) + tan(e/2 + (f*x)/2)^2*(38*A*a^3*c^2 + (152*A*a^3*d^2)/5 + 34*B*a^3*c
^2 + (136*B*a^3*d^2)/5 + 68*A*a^3*c*d + (304*B*a^3*c*d)/5) + tan(e/2 + (f*x)/2)^6*((220*A*a^3*c^2)/3 + (152*A*
a^3*d^2)/3 + 60*B*a^3*c^2 + (136*B*a^3*d^2)/3 + 120*A*a^3*c*d + (304*B*a^3*c*d)/3) + (22*A*a^3*c^2)/3 + (76*A*
a^3*d^2)/15 + 6*B*a^3*c^2 + (68*B*a^3*d^2)/15 + 12*A*a^3*c*d + (152*B*a^3*c*d)/15)/(f*(6*tan(e/2 + (f*x)/2)^2
+ 15*tan(e/2 + (f*x)/2)^4 + 20*tan(e/2 + (f*x)/2)^6 + 15*tan(e/2 + (f*x)/2)^8 + 6*tan(e/2 + (f*x)/2)^10 + tan(
e/2 + (f*x)/2)^12 + 1))