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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3055, 3047, 3102, 2832, 2813} \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=-\frac {a^3 \left (-14 A c d+91 A d^2+6 B c^2-27 B c d+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{210 d^3 f}-\frac {a^3 \left (7 A d \left (2 c^2-18 c d+115 d^2\right )-B \left (6 c^3-42 c^2 d+177 c d^2-735 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{840 d^3 f}+\frac {1}{16} a^3 x \left (A \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right )+3 B \left (10 c^3+26 c^2 d+23 c d^2+7 d^3\right )\right )-\frac {a^3 \left (7 A d \left (2 c^3-18 c^2 d+111 c d^2+136 d^3\right )-B \left (6 c^4-42 c^3 d+165 c^2 d^2-651 c d^3-864 d^4\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{840 d^3 f}-\frac {a^3 \left (7 A d \left (4 c^4-36 c^3 d+216 c^2 d^2+626 c d^3+345 d^4\right )-3 B \left (4 c^5-28 c^4 d+104 c^3 d^2-392 c^2 d^3-1263 c d^4-735 d^5\right )\right ) \sin (e+f x) \cos (e+f x)}{1680 d^2 f}-\frac {a^3 \left (7 A d \left (2 c^5-18 c^4 d+107 c^3 d^2+472 c^2 d^3+456 c d^4+136 d^5\right )-3 B \left (2 c^6-14 c^5 d+51 c^4 d^2-189 c^3 d^3-920 c^2 d^4-952 c d^5-288 d^6\right )\right ) \cos (e+f x)}{420 d^3 f}+\frac {(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{42 d^2 f}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f} \]

[In]

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

[Out]

(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)/16 - (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[e + f*x])/(420*d^3*f) - (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[e + f*x]*Sin[e + f*x])/(1680*d^2*f) - (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)/(840*d^3*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
)/(840*d^3*f) - (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
)/(210*d^3*f) - (a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^4)/(7*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)/(42*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))^4}{7 d f}+\frac {\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3 (a (7 A d+2 B (c+2 d))-a (3 B c-7 A d-9 B d) \sin (e+f x)) \, dx}{7 d} \\ & = -\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}+\frac {\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^3 \left (a^2 \left (7 A d (c+10 d)-B \left (3 c^2-9 c d-60 d^2\right )\right )+a^2 \left (6 B c^2-14 A c d-27 B c d+91 A d^2+87 B d^2\right ) \sin (e+f x)\right ) \, dx}{42 d^2} \\ & = -\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}+\frac {\int (c+d \sin (e+f x))^3 \left (a^3 \left (7 A d (c+10 d)-B \left (3 c^2-9 c d-60 d^2\right )\right )+\left (a^3 \left (6 B c^2-14 A c d-27 B c d+91 A d^2+87 B d^2\right )+a^3 \left (7 A d (c+10 d)-B \left (3 c^2-9 c d-60 d^2\right )\right )\right ) \sin (e+f x)+a^3 \left (6 B c^2-14 A c d-27 B c d+91 A d^2+87 B d^2\right ) \sin ^2(e+f x)\right ) \, dx}{42 d^2} \\ & = -\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}+\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 c d+72 d^2\right )\right )+a^3 \left (7 A d \left (2 c^2-18 c d+115 d^2\right )-3 B \left (2 c^3-14 c^2 d+59 c d^2-245 d^3\right )\right ) \sin (e+f x)\right ) \, dx}{210 d^3} \\ & = -\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}+\frac {\int (c+d \sin (e+f x))^2 \left (-3 a^3 d \left (7 A d \left (2 c^2-118 c d-115 d^2\right )-B \left (6 c^3-42 c^2 d+687 c d^2+735 d^3\right )\right )+3 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 ) \sin (e+f x)\right ) \, dx}{840 d^3} \\ & = -\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}+\frac {\int (c+d \sin (e+f x)) \left (-3 a^3 d \left (7 A d \left (2 c^3-318 c^2 d-567 c d^2-272 d^3\right )-3 B \left (2 c^4-14 c^3 d+577 c^2 d^2+1169 c d^3+576 d^4\right )\right )+3 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 ) \sin (e+f x)\right ) \, dx}{2520 d^3} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.05 (sec) , antiderivative size = 528, normalized size of antiderivative = 0.87 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=-\frac {a^3 \cos (e+f x) \left (420 \left (3 B \left (10 c^3+26 c^2 d+23 c d^2+7 d^3\right )+A \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right )\right ) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (12880 A c^3+11760 B c^3+35280 A c^2 d+32676 B c^2 d+32676 A c d^2+30828 B c d^2+10276 A d^3+9762 B d^3-\left (112 A \left (5 c^3+45 c^2 d+66 c d^2+26 d^3\right )+3 B \left (560 c^3+2464 c^2 d+2912 c d^2+1083 d^3\right )\right ) \cos (2 (e+f x))+18 d \left (14 A d (c+d)+B \left (14 c^2+42 c d+23 d^2\right )\right ) \cos (4 (e+f x))-15 B d^3 \cos (6 (e+f x))+5040 A c^3 \sin (e+f x)+6930 B c^3 \sin (e+f x)+20790 A c^2 d \sin (e+f x)+22050 B c^2 d \sin (e+f x)+22050 A c d^2 \sin (e+f x)+22785 B c d^2 \sin (e+f x)+7595 A d^3 \sin (e+f x)+7665 B d^3 \sin (e+f x)-210 B c^3 \sin (3 (e+f x))-630 A c^2 d \sin (3 (e+f x))-1890 B c^2 d \sin (3 (e+f x))-1890 A c d^2 \sin (3 (e+f x))-2940 B c d^2 \sin (3 (e+f x))-980 A d^3 \sin (3 (e+f x))-1260 B d^3 \sin (3 (e+f x))+105 B c d^2 \sin (5 (e+f x))+35 A d^3 \sin (5 (e+f x))+105 B d^3 \sin (5 (e+f x))\right )\right )}{3360 f \sqrt {\cos ^2(e+f x)}} \]

[In]

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

[Out]

-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 + 6
6*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] + 693
0*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*Si
n[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])

Maple [A] (verified)

Time = 2.72 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.64

method	result	size
parallelrisch	\(\frac {\left (\left (\frac {\left (19 A +\frac {81 B}{4}\right ) d^{3}}{4}+\frac {51 c \left (A +\frac {19 B}{17}\right ) d^{2}}{4}+9 \left (\frac {17 B}{12}+A \right ) c^{2} d +c^{3} \left (A +3 B \right )\right ) \cos \left (3 f x +3 e \right )+3 \left (\frac {\left (-63 A -61 B \right ) d^{3}}{16}-12 c \left (A +\frac {63 B}{64}\right ) d^{2}-12 c^{2} d \left (A +B \right )-3 \left (A +\frac {4 B}{3}\right ) c^{3}\right ) \sin \left (2 f x +2 e \right )+\frac {3 \left (\frac {\left (11 B +9 A \right ) d^{3}}{2}+9 \left (\frac {3 B}{2}+A \right ) c \,d^{2}+3 c^{2} \left (A +3 B \right ) d +B \,c^{3}\right ) \sin \left (4 f x +4 e \right )}{8}-\frac {9 d \left (\left (A +\frac {19 B}{12}\right ) d^{2}+c \left (A +3 B \right ) d +B \,c^{2}\right ) \cos \left (5 f x +5 e \right )}{20}-\frac {\left (\left (A +3 B \right ) d +3 B c \right ) d^{2} \sin \left (6 f x +6 e \right )}{16}+\frac {3 B \,d^{3} \cos \left (7 f x +7 e \right )}{112}+3 \left (\frac {\left (-\frac {155 B}{8}-21 A \right ) d^{3}}{2}-\frac {69 \left (\frac {21 B}{23}+A \right ) c \,d^{2}}{2}-39 c^{2} \left (\frac {23 B}{26}+A \right ) d -15 c^{3} \left (A +\frac {13 B}{15}\right )\right ) \cos \left (f x +e \right )+\left (-\frac {864}{35} B +\frac {69}{4} f x A +\frac {63}{4} f x B -\frac {136}{5} A \right ) d^{3}+\frac {117 c \left (f x A +\frac {23}{26} f x B -\frac {304}{195} A -\frac {272}{195} B \right ) d^{2}}{2}+\frac {135 c^{2} \left (f x A +\frac {13}{15} f x B -\frac {8}{5} A -\frac {304}{225} B \right ) d}{2}+30 c^{3} \left (f x A +\frac {3}{4} f x B -\frac {22}{15} A -\frac {6}{5} B \right )\right ) a^{3}}{12 f}\)	\(384\)
parts	\(\frac {\left (A \,a^{3} d^{3}+3 B \,a^{3} d^{2} c +3 B \,a^{3} d^{3}\right ) \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}-\frac {\left (3 A \,a^{3} c^{3}+3 A \,a^{3} c^{2} d +B \,a^{3} c^{3}\right ) \cos \left (f x +e \right )}{f}-\frac {\left (3 A \,a^{3} d^{2} c +3 A \,a^{3} d^{3}+3 B \,a^{3} c^{2} d +9 B \,a^{3} d^{2} c +3 B \,a^{3} d^{3}\right ) \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}+\frac {\left (3 A \,a^{3} c^{3}+9 A \,a^{3} c^{2} d +3 A \,a^{3} d^{2} c +3 B \,a^{3} c^{3}+3 B \,a^{3} c^{2} d \right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {\left (3 A \,a^{3} c^{2} d +9 A \,a^{3} d^{2} c +3 A \,a^{3} d^{3}+B \,a^{3} c^{3}+9 B \,a^{3} c^{2} d +9 B \,a^{3} d^{2} c +B \,a^{3} d^{3}\right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {\left (A \,a^{3} c^{3}+9 A \,a^{3} c^{2} d +9 A \,a^{3} d^{2} c +A \,a^{3} d^{3}+3 B \,a^{3} c^{3}+9 B \,a^{3} c^{2} d +3 B \,a^{3} d^{2} c \right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+a^{3} A \,c^{3} x -\frac {B \,a^{3} d^{3} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7 f}\)	\(518\)
risch	\(-\frac {189 \sin \left (2 f x +2 e \right ) B \,a^{3} d^{2} c}{64 f}-\frac {15 a^{3} \cos \left (f x +e \right ) A \,c^{3}}{4 f}-\frac {21 a^{3} \cos \left (f x +e \right ) A \,d^{3}}{8 f}-\frac {13 a^{3} \cos \left (f x +e \right ) B \,c^{3}}{4 f}-\frac {155 a^{3} \cos \left (f x +e \right ) d^{3} B}{64 f}-\frac {\sin \left (6 f x +6 e \right ) A \,a^{3} d^{3}}{192 f}-\frac {\sin \left (6 f x +6 e \right ) B \,a^{3} d^{3}}{64 f}-\frac {3 a^{3} d^{3} \cos \left (5 f x +5 e \right ) A}{80 f}-\frac {19 a^{3} d^{3} \cos \left (5 f x +5 e \right ) B}{320 f}+\frac {9 \sin \left (4 f x +4 e \right ) A \,a^{3} d^{3}}{64 f}+\frac {\sin \left (4 f x +4 e \right ) B \,a^{3} c^{3}}{32 f}+\frac {11 \sin \left (4 f x +4 e \right ) B \,a^{3} d^{3}}{64 f}+\frac {a^{3} \cos \left (3 f x +3 e \right ) A \,c^{3}}{12 f}+\frac {19 a^{3} \cos \left (3 f x +3 e \right ) A \,d^{3}}{48 f}+\frac {a^{3} \cos \left (3 f x +3 e \right ) B \,c^{3}}{4 f}+\frac {27 a^{3} \cos \left (3 f x +3 e \right ) d^{3} B}{64 f}+\frac {15 B \,a^{3} c^{3} x}{8}+\frac {21 B \,a^{3} d^{3} x}{16}-\frac {3 \sin \left (2 f x +2 e \right ) A \,a^{3} c^{3}}{4 f}-\frac {63 \sin \left (2 f x +2 e \right ) A \,a^{3} d^{3}}{64 f}-\frac {\sin \left (2 f x +2 e \right ) B \,a^{3} c^{3}}{f}-\frac {61 \sin \left (2 f x +2 e \right ) B \,a^{3} d^{3}}{64 f}+\frac {B \,a^{3} d^{3} \cos \left (7 f x +7 e \right )}{448 f}+\frac {45 A \,a^{3} c^{2} d x}{8}+\frac {39 A \,a^{3} c \,d^{2} x}{8}+\frac {39 B \,a^{3} c^{2} d x}{8}+\frac {69 B \,a^{3} c \,d^{2} x}{16}+\frac {23 A \,a^{3} d^{3} x}{16}+\frac {5 a^{3} A \,c^{3} x}{2}-\frac {39 a^{3} \cos \left (f x +e \right ) c^{2} d A}{4 f}-\frac {69 a^{3} \cos \left (f x +e \right ) d^{2} c A}{8 f}-\frac {69 a^{3} \cos \left (f x +e \right ) c^{2} d B}{8 f}-\frac {63 a^{3} \cos \left (f x +e \right ) d^{2} c B}{8 f}-\frac {\sin \left (6 f x +6 e \right ) B \,a^{3} d^{2} c}{64 f}-\frac {3 a^{3} d^{2} \cos \left (5 f x +5 e \right ) A c}{80 f}-\frac {3 a^{3} d \cos \left (5 f x +5 e \right ) B \,c^{2}}{80 f}-\frac {9 a^{3} d^{2} \cos \left (5 f x +5 e \right ) c B}{80 f}+\frac {3 \sin \left (4 f x +4 e \right ) A \,a^{3} c^{2} d}{32 f}+\frac {9 \sin \left (4 f x +4 e \right ) A \,a^{3} d^{2} c}{32 f}+\frac {9 \sin \left (4 f x +4 e \right ) B \,a^{3} c^{2} d}{32 f}+\frac {27 \sin \left (4 f x +4 e \right ) B \,a^{3} d^{2} c}{64 f}+\frac {3 a^{3} \cos \left (3 f x +3 e \right ) c^{2} d A}{4 f}+\frac {17 a^{3} \cos \left (3 f x +3 e \right ) d^{2} c A}{16 f}+\frac {17 a^{3} \cos \left (3 f x +3 e \right ) c^{2} d B}{16 f}+\frac {19 a^{3} \cos \left (3 f x +3 e \right ) d^{2} c B}{16 f}-\frac {3 \sin \left (2 f x +2 e \right ) A \,a^{3} c^{2} d}{f}-\frac {3 \sin \left (2 f x +2 e \right ) A \,a^{3} d^{2} c}{f}-\frac {3 \sin \left (2 f x +2 e \right ) B \,a^{3} c^{2} d}{f}\)	\(922\)
derivativedivides	\(\text {Expression too large to display}\)	\(1077\)
default	\(\text {Expression too large to display}\)	\(1077\)
norman	\(\text {Expression too large to display}\)	\(1740\)

[In]

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

[Out]

1/12*((1/4*(19*A+81/4*B)*d^3+51/4*c*(A+19/17*B)*d^2+9*(17/12*B+A)*c^2*d+c^3*(A+3*B))*cos(3*f*x+3*e)+3*(1/16*(-
63*A-61*B)*d^3-12*c*(A+63/64*B)*d^2-12*c^2*d*(A+B)-3*(A+4/3*B)*c^3)*sin(2*f*x+2*e)+3/8*(1/2*(11*B+9*A)*d^3+9*(
3/2*B+A)*c*d^2+3*c^2*(A+3*B)*d+B*c^3)*sin(4*f*x+4*e)-9/20*d*((A+19/12*B)*d^2+c*(A+3*B)*d+B*c^2)*cos(5*f*x+5*e)
-1/16*((A+3*B)*d+3*B*c)*d^2*sin(6*f*x+6*e)+3/112*B*d^3*cos(7*f*x+7*e)+3*(1/2*(-155/8*B-21*A)*d^3-69/2*(21/23*B
+A)*c*d^2-39*c^2*(23/26*B+A)*d-15*c^3*(A+13/15*B))*cos(f*x+e)+(-864/35*B+69/4*f*x*A+63/4*f*x*B-136/5*A)*d^3+11
7/2*c*(f*x*A+23/26*f*x*B-304/195*A-272/195*B)*d^2+135/2*c^2*(f*x*A+13/15*f*x*B-8/5*A-304/225*B)*d+30*c^3*(f*x*
A+3/4*f*x*B-22/15*A-6/5*B))*a^3/f

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 432, normalized size of antiderivative = 0.72 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\frac {240 \, B a^{3} d^{3} \cos \left (f x + e\right )^{7} - 1008 \, {\left (B a^{3} c^{2} d + {\left (A + 3 \, B\right )} a^{3} c d^{2} + {\left (A + 2 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )^{5} + 560 \, {\left ({\left (A + 3 \, B\right )} a^{3} c^{3} + 3 \, {\left (3 \, A + 5 \, B\right )} a^{3} c^{2} d + 3 \, {\left (5 \, A + 7 \, B\right )} a^{3} c d^{2} + {\left (7 \, A + 9 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )^{3} + 105 \, {\left (10 \, {\left (4 \, A + 3 \, B\right )} a^{3} c^{3} + 6 \, {\left (15 \, A + 13 \, B\right )} a^{3} c^{2} d + 3 \, {\left (26 \, A + 23 \, B\right )} a^{3} c d^{2} + {\left (23 \, A + 21 \, B\right )} a^{3} d^{3}\right )} f x - 6720 \, {\left ({\left (A + B\right )} a^{3} c^{3} + 3 \, {\left (A + B\right )} a^{3} c^{2} d + 3 \, {\left (A + B\right )} a^{3} c d^{2} + {\left (A + B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right ) - 35 \, {\left (8 \, {\left (3 \, B a^{3} c d^{2} + {\left (A + 3 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )^{5} - 2 \, {\left (6 \, B a^{3} c^{3} + 18 \, {\left (A + 3 \, B\right )} a^{3} c^{2} d + 3 \, {\left (18 \, A + 31 \, B\right )} a^{3} c d^{2} + {\left (31 \, A + 45 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (2 \, {\left (12 \, A + 17 \, B\right )} a^{3} c^{3} + 6 \, {\left (17 \, A + 19 \, B\right )} a^{3} c^{2} d + 3 \, {\left (38 \, A + 41 \, B\right )} a^{3} c d^{2} + {\left (41 \, A + 43 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{1680 \, f} \]

[In]

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

[Out]

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))*sin(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 = 0.72 (sec) , antiderivative size = 2878, normalized size of antiderivative = 4.76 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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*cos(e + f*x)**2/16 + 9*A*
a**3*d**3*x*sin(e + f*x)**4/8 + 15*A*a**3*d**3*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 9*A*a**3*d**3*x*sin(e +
f*x)**2*cos(e + f*x)**2/4 + 5*A*a**3*d**3*x*cos(e + f*x)**6/16 + 9*A*a**3*d**3*x*cos(e + f*x)**4/8 - 11*A*a**3
*d**3*sin(e + f*x)**5*cos(e + f*x)/(16*f) - 3*A*a**3*d**3*sin(e + f*x)**4*cos(e + f*x)/f - 5*A*a**3*d**3*sin(e
+ f*x)**3*cos(e + f*x)**3/(6*f) - 15*A*a**3*d**3*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 4*A*a**3*d**3*sin(e + f
*x)**2*cos(e + f*x)**3/f - A*a**3*d**3*sin(e + f*x)**2*cos(e + f*x)/f - 5*A*a**3*d**3*sin(e + f*x)*cos(e + f*x
)**5/(16*f) - 9*A*a**3*d**3*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 8*A*a**3*d**3*cos(e + f*x)**5/(5*f) - 2*A*a**
3*d**3*cos(e + f*x)**3/(3*f) + 3*B*a**3*c**3*x*sin(e + f*x)**4/8 + 3*B*a**3*c**3*x*sin(e + f*x)**2*cos(e + f*x
)**2/4 + 3*B*a**3*c**3*x*sin(e + f*x)**2/2 + 3*B*a**3*c**3*x*cos(e + f*x)**4/8 + 3*B*a**3*c**3*x*cos(e + f*x)*
*2/2 - 5*B*a**3*c**3*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 3*B*a**3*c**3*sin(e + f*x)**2*cos(e + f*x)/f - 3*B*a
**3*c**3*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 3*B*a**3*c**3*sin(e + f*x)*cos(e + f*x)/(2*f) - 2*B*a**3*c**3*co
s(e + f*x)**3/f - B*a**3*c**3*cos(e + f*x)/f + 27*B*a**3*c**2*d*x*sin(e + f*x)**4/8 + 27*B*a**3*c**2*d*x*sin(e
+ f*x)**2*cos(e + f*x)**2/4 + 3*B*a**3*c**2*d*x*sin(e + f*x)**2/2 + 27*B*a**3*c**2*d*x*cos(e + f*x)**4/8 + 3*
B*a**3*c**2*d*x*cos(e + f*x)**2/2 - 3*B*a**3*c**2*d*sin(e + f*x)**4*cos(e + f*x)/f - 45*B*a**3*c**2*d*sin(e +
f*x)**3*cos(e + f*x)/(8*f) - 4*B*a**3*c**2*d*sin(e + f*x)**2*cos(e + f*x)**3/f - 9*B*a**3*c**2*d*sin(e + f*x)*
*2*cos(e + f*x)/f - 27*B*a**3*c**2*d*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 3*B*a**3*c**2*d*sin(e + f*x)*cos(e +
f*x)/(2*f) - 8*B*a**3*c**2*d*cos(e + f*x)**5/(5*f) - 6*B*a**3*c**2*d*cos(e + f*x)**3/f + 15*B*a**3*c*d**2*x*s
in(e + f*x)**6/16 + 45*B*a**3*c*d**2*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 27*B*a**3*c*d**2*x*sin(e + f*x)**4
/8 + 45*B*a**3*c*d**2*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 27*B*a**3*c*d**2*x*sin(e + f*x)**2*cos(e + f*x)**
2/4 + 15*B*a**3*c*d**2*x*cos(e + f*x)**6/16 + 27*B*a**3*c*d**2*x*cos(e + f*x)**4/8 - 33*B*a**3*c*d**2*sin(e +
f*x)**5*cos(e + f*x)/(16*f) - 9*B*a**3*c*d**2*sin(e + f*x)**4*cos(e + f*x)/f - 5*B*a**3*c*d**2*sin(e + f*x)**3
*cos(e + f*x)**3/(2*f) - 45*B*a**3*c*d**2*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 12*B*a**3*c*d**2*sin(e + f*x)**
2*cos(e + f*x)**3/f - 3*B*a**3*c*d**2*sin(e + f*x)**2*cos(e + f*x)/f - 15*B*a**3*c*d**2*sin(e + f*x)*cos(e + f
*x)**5/(16*f) - 27*B*a**3*c*d**2*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 24*B*a**3*c*d**2*cos(e + f*x)**5/(5*f) -
2*B*a**3*c*d**2*cos(e + f*x)**3/f + 15*B*a**3*d**3*x*sin(e + f*x)**6/16 + 45*B*a**3*d**3*x*sin(e + f*x)**4*co
s(e + f*x)**2/16 + 3*B*a**3*d**3*x*sin(e + f*x)**4/8 + 45*B*a**3*d**3*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 3
*B*a**3*d**3*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 15*B*a**3*d**3*x*cos(e + f*x)**6/16 + 3*B*a**3*d**3*x*cos(e
+ f*x)**4/8 - B*a**3*d**3*sin(e + f*x)**6*cos(e + f*x)/f - 33*B*a**3*d**3*sin(e + f*x)**5*cos(e + f*x)/(16*f)
- 2*B*a**3*d**3*sin(e + f*x)**4*cos(e + f*x)**3/f - 3*B*a**3*d**3*sin(e + f*x)**4*cos(e + f*x)/f - 5*B*a**3*d
**3*sin(e + f*x)**3*cos(e + f*x)**3/(2*f) - 5*B*a**3*d**3*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 8*B*a**3*d**3*s
in(e + f*x)**2*cos(e + f*x)**5/(5*f) - 4*B*a**3*d**3*sin(e + f*x)**2*cos(e + f*x)**3/f - 15*B*a**3*d**3*sin(e
+ f*x)*cos(e + f*x)**5/(16*f) - 3*B*a**3*d**3*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 16*B*a**3*d**3*cos(e + f*x)
**7/(35*f) - 8*B*a**3*d**3*cos(e + f*x)**5/(5*f), Ne(f, 0)), (x*(A + B*sin(e))*(c + d*sin(e))**3*(a*sin(e) + a
)**3, True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 1056, normalized size of antiderivative = 1.75 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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 + 6
720*(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 + 151
20*(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 + 189
0*(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*sin(4*f*x + 4*e) - 48*sin(2*f*x + 2*e))*A*a^3*d^3 + 630*(12*f*
x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^3*d^3 + 192*(5*cos(f*x + e)^7 - 21*cos(f*x + e)^5 + 35*c
os(f*x + e)^3 - 35*cos(f*x + e))*B*a^3*d^3 - 1344*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a
^3*d^3 + 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*d^3 + 210
*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a^3*d^3 - 20160*A*a^3*c^3*cos(f*x + e) - 6720*B*a^3
*c^3*cos(f*x + e) - 20160*A*a^3*c^2*d*cos(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 559, normalized size of antiderivative = 0.93 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\frac {B a^{3} d^{3} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} + \frac {1}{16} \, {\left (40 \, A a^{3} c^{3} + 30 \, B a^{3} c^{3} + 90 \, A a^{3} c^{2} d + 78 \, B a^{3} c^{2} d + 78 \, A a^{3} c d^{2} + 69 \, B a^{3} c d^{2} + 23 \, A a^{3} d^{3} + 21 \, B a^{3} d^{3}\right )} x - \frac {{\left (12 \, B a^{3} c^{2} d + 12 \, A a^{3} c d^{2} + 36 \, B a^{3} c d^{2} + 12 \, A a^{3} d^{3} + 19 \, B a^{3} d^{3}\right )} \cos \left (5 \, f x + 5 \, e\right )}{320 \, f} + \frac {{\left (16 \, A a^{3} c^{3} + 48 \, B a^{3} c^{3} + 144 \, A a^{3} c^{2} d + 204 \, B a^{3} c^{2} d + 204 \, A a^{3} c d^{2} + 228 \, B a^{3} c d^{2} + 76 \, A a^{3} d^{3} + 81 \, B a^{3} d^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{192 \, f} - \frac {{\left (240 \, A a^{3} c^{3} + 208 \, B a^{3} c^{3} + 624 \, A a^{3} c^{2} d + 552 \, B a^{3} c^{2} d + 552 \, A a^{3} c d^{2} + 504 \, B a^{3} c d^{2} + 168 \, A a^{3} d^{3} + 155 \, B a^{3} d^{3}\right )} \cos \left (f x + e\right )}{64 \, f} - \frac {{\left (3 \, B a^{3} c d^{2} + A a^{3} d^{3} + 3 \, B a^{3} d^{3}\right )} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {{\left (2 \, B a^{3} c^{3} + 6 \, A a^{3} c^{2} d + 18 \, B a^{3} c^{2} d + 18 \, A a^{3} c d^{2} + 27 \, B a^{3} c d^{2} + 9 \, A a^{3} d^{3} + 11 \, B a^{3} d^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {{\left (48 \, A a^{3} c^{3} + 64 \, B a^{3} c^{3} + 192 \, A a^{3} c^{2} d + 192 \, B a^{3} c^{2} d + 192 \, A a^{3} c d^{2} + 189 \, B a^{3} c d^{2} + 63 \, A a^{3} d^{3} + 61 \, B a^{3} d^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]

[In]

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

[Out]

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 + 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)*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 -
1/64*(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 + 1
1*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 = 16.31 (sec) , antiderivative size = 1395, normalized size of antiderivative = 2.31 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

(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 + (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)^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 + 344*A*a^3*c*d^2 + 384*A*a^3*c^2*d + 328*B*a^3*c*d^2 + 344*B*a^
3*c^2*d) + tan(e/2 + (f*x)/2)^2*((136*A*a^3*c^3)/3 + (476*A*a^3*d^3)/15 + 40*B*a^3*c^3 + (144*B*a^3*d^3)/5 + (
532*A*a^3*c*d^2)/5 + 120*A*a^3*c^2*d + (476*B*a^3*c*d^2)/5 + (532*B*a^3*c^2*d)/5) + tan(e/2 + (f*x)/2)^5*(15*A
*a^3*c^3 + (841*A*a^3*d^3)/24 + (91*B*a^3*c^3)/4 + (345*B*a^3*d^3)/8 + (339*A*a^3*c*d^2)/4 + (273*A*a^3*c^2*d)
/4 + (841*B*a^3*c*d^2)/8 + (339*B*a^3*c^2*d)/4) - tan(e/2 + (f*x)/2)^9*(15*A*a^3*c^3 + (841*A*a^3*d^3)/24 + (9
1*B*a^3*c^3)/4 + (345*B*a^3*d^3)/8 + (339*A*a^3*c*d^2)/4 + (273*A*a^3*c^2*d)/4 + (841*B*a^3*c*d^2)/8 + (339*B*
a^3*c^2*d)/4) + tan(e/2 + (f*x)/2)^4*(114*A*a^3*c^3 + (456*A*a^3*d^3)/5 + 102*B*a^3*c^3 + (432*B*a^3*d^3)/5 +
(1416*A*a^3*c*d^2)/5 + 306*A*a^3*c^2*d + (1368*B*a^3*c*d^2)/5 + (1416*B*a^3*c^2*d)/5) + tan(e/2 + (f*x)/2)^12*
(6*A*a^3*c^3 + 2*B*a^3*c^3 + 6*A*a^3*c^2*d) + (22*A*a^3*c^3)/3 + (68*A*a^3*d^3)/15 + 6*B*a^3*c^3 + (144*B*a^3*
d^3)/35 + (76*A*a^3*c*d^2)/5 + 18*A*a^3*c^2*d + (68*B*a^3*c*d^2)/5 + (76*B*a^3*c^2*d)/5)/(f*(7*tan(e/2 + (f*x)
/2)^2 + 21*tan(e/2 + (f*x)/2)^4 + 35*tan(e/2 + (f*x)/2)^6 + 35*tan(e/2 + (f*x)/2)^8 + 21*tan(e/2 + (f*x)/2)^10
+ 7*tan(e/2 + (f*x)/2)^12 + tan(e/2 + (f*x)/2)^14 + 1))