\(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx\) [40]

Optimal result

Integrand size = 36, antiderivative size = 181 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=\frac {5}{128} a^3 (8 A-B) c^4 x+\frac {a^3 (8 A-B) c^4 \cos ^7(e+f x)}{56 f}+\frac {5 a^3 (8 A-B) c^4 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {5 a^3 (8 A-B) c^4 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac {a^3 (8 A-B) c^4 \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac {a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f} \]

[Out]

5/128*a^3*(8*A-B)*c^4*x+1/56*a^3*(8*A-B)*c^4*cos(f*x+e)^7/f+5/128*a^3*(8*A-B)*c^4*cos(f*x+e)*sin(f*x+e)/f+5/19
2*a^3*(8*A-B)*c^4*cos(f*x+e)^3*sin(f*x+e)/f+1/48*a^3*(8*A-B)*c^4*cos(f*x+e)^5*sin(f*x+e)/f-1/8*a^3*B*cos(f*x+e
)^7*(c^4-c^4*sin(f*x+e))/f

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {3046, 2939, 2748, 2715, 8} \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=\frac {a^3 c^4 (8 A-B) \cos ^7(e+f x)}{56 f}+\frac {a^3 c^4 (8 A-B) \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac {5 a^3 c^4 (8 A-B) \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac {5 a^3 c^4 (8 A-B) \sin (e+f x) \cos (e+f x)}{128 f}+\frac {5}{128} a^3 c^4 x (8 A-B)-\frac {a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f} \]

[In]

Int[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4,x]

[Out]

(5*a^3*(8*A - B)*c^4*x)/128 + (a^3*(8*A - B)*c^4*Cos[e + f*x]^7)/(56*f) + (5*a^3*(8*A - B)*c^4*Cos[e + f*x]*Si
n[e + f*x])/(128*f) + (5*a^3*(8*A - B)*c^4*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) + (a^3*(8*A - B)*c^4*Cos[e + f
*x]^5*Sin[e + f*x])/(48*f) - (a^3*B*Cos[e + f*x]^7*(c^4 - c^4*Sin[e + f*x]))/(8*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x
] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2939

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x
] + Dist[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; F
reeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[m + p + 1, 0]

Rule 3046

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I
ntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))

Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \cos ^6(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx \\ & = -\frac {a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f}+\frac {1}{8} \left (a^3 (8 A-B) c^3\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x)) \, dx \\ & = \frac {a^3 (8 A-B) c^4 \cos ^7(e+f x)}{56 f}-\frac {a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f}+\frac {1}{8} \left (a^3 (8 A-B) c^4\right ) \int \cos ^6(e+f x) \, dx \\ & = \frac {a^3 (8 A-B) c^4 \cos ^7(e+f x)}{56 f}+\frac {a^3 (8 A-B) c^4 \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac {a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f}+\frac {1}{48} \left (5 a^3 (8 A-B) c^4\right ) \int \cos ^4(e+f x) \, dx \\ & = \frac {a^3 (8 A-B) c^4 \cos ^7(e+f x)}{56 f}+\frac {5 a^3 (8 A-B) c^4 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac {a^3 (8 A-B) c^4 \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac {a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f}+\frac {1}{64} \left (5 a^3 (8 A-B) c^4\right ) \int \cos ^2(e+f x) \, dx \\ & = \frac {a^3 (8 A-B) c^4 \cos ^7(e+f x)}{56 f}+\frac {5 a^3 (8 A-B) c^4 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {5 a^3 (8 A-B) c^4 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac {a^3 (8 A-B) c^4 \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac {a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f}+\frac {1}{128} \left (5 a^3 (8 A-B) c^4\right ) \int 1 \, dx \\ & = \frac {5}{128} a^3 (8 A-B) c^4 x+\frac {a^3 (8 A-B) c^4 \cos ^7(e+f x)}{56 f}+\frac {5 a^3 (8 A-B) c^4 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {5 a^3 (8 A-B) c^4 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac {a^3 (8 A-B) c^4 \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac {a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.47 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.15 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=\frac {(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4 (840 (8 A-B) (e+f x)+1680 (A-B) \cos (e+f x)+1008 (A-B) \cos (3 (e+f x))+336 (A-B) \cos (5 (e+f x))+48 (A-B) \cos (7 (e+f x))+336 (15 A-B) \sin (2 (e+f x))+168 (6 A+B) \sin (4 (e+f x))+112 (A+B) \sin (6 (e+f x))+21 B \sin (8 (e+f x)))}{21504 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6} \]

[In]

Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4,x]

[Out]

((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4*(840*(8*A - B)*(e + f*x) + 1680*(A - B)*Cos[e + f*x] + 1008*(A
- B)*Cos[3*(e + f*x)] + 336*(A - B)*Cos[5*(e + f*x)] + 48*(A - B)*Cos[7*(e + f*x)] + 336*(15*A - B)*Sin[2*(e +
f*x)] + 168*(6*A + B)*Sin[4*(e + f*x)] + 112*(A + B)*Sin[6*(e + f*x)] + 21*B*Sin[8*(e + f*x)]))/(21504*f*(Cos
[(e + f*x)/2] - Sin[(e + f*x)/2])^8*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)

Maple [A] (verified)

Time = 2.94 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.82

method	result	size
parallelrisch	\(\frac {c^{4} \left (3 \left (A -B \right ) \cos \left (3 f x +3 e \right )+\left (A -B \right ) \cos \left (5 f x +5 e \right )+\frac {\left (A -B \right ) \cos \left (7 f x +7 e \right )}{7}+\left (15 A -B \right ) \sin \left (2 f x +2 e \right )+\left (\frac {B}{2}+3 A \right ) \sin \left (4 f x +4 e \right )+\frac {\left (A +B \right ) \sin \left (6 f x +6 e \right )}{3}+\frac {B \sin \left (8 f x +8 e \right )}{16}+5 \left (A -B \right ) \cos \left (f x +e \right )+20 f x A -\frac {5 f x B}{2}+\frac {64 A}{7}-\frac {64 B}{7}\right ) a^{3}}{64 f}\)	\(149\)
risch	\(\frac {5 a^{3} c^{4} x A}{16}-\frac {5 a^{3} c^{4} x B}{128}+\frac {5 c^{4} a^{3} \cos \left (f x +e \right ) A}{64 f}-\frac {5 c^{4} a^{3} \cos \left (f x +e \right ) B}{64 f}+\frac {B \,a^{3} c^{4} \sin \left (8 f x +8 e \right )}{1024 f}+\frac {c^{4} a^{3} \cos \left (7 f x +7 e \right ) A}{448 f}-\frac {c^{4} a^{3} \cos \left (7 f x +7 e \right ) B}{448 f}+\frac {\sin \left (6 f x +6 e \right ) A \,a^{3} c^{4}}{192 f}+\frac {\sin \left (6 f x +6 e \right ) B \,a^{3} c^{4}}{192 f}+\frac {c^{4} a^{3} \cos \left (5 f x +5 e \right ) A}{64 f}-\frac {c^{4} a^{3} \cos \left (5 f x +5 e \right ) B}{64 f}+\frac {3 \sin \left (4 f x +4 e \right ) A \,a^{3} c^{4}}{64 f}+\frac {\sin \left (4 f x +4 e \right ) B \,a^{3} c^{4}}{128 f}+\frac {3 c^{4} a^{3} \cos \left (3 f x +3 e \right ) A}{64 f}-\frac {3 c^{4} a^{3} \cos \left (3 f x +3 e \right ) B}{64 f}+\frac {15 \sin \left (2 f x +2 e \right ) A \,a^{3} c^{4}}{64 f}-\frac {\sin \left (2 f x +2 e \right ) B \,a^{3} c^{4}}{64 f}\)	\(331\)
parts	\(\frac {\left (-3 A \,a^{3} c^{4}-B \,a^{3} c^{4}\right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {\left (-3 A \,a^{3} c^{4}+3 B \,a^{3} c^{4}\right ) \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}+\frac {\left (-A \,a^{3} c^{4}-3 B \,a^{3} c^{4}\right ) \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}-\frac {\left (-A \,a^{3} c^{4}+B \,a^{3} c^{4}\right ) \cos \left (f x +e \right )}{f}-\frac {\left (A \,a^{3} c^{4}-B \,a^{3} c^{4}\right ) \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7 f}-\frac {\left (3 A \,a^{3} c^{4}-3 B \,a^{3} c^{4}\right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {\left (3 A \,a^{3} c^{4}+3 B \,a^{3} c^{4}\right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}+a^{3} c^{4} x A +\frac {B \,a^{3} c^{4} \left (-\frac {\left (\sin ^{7}\left (f x +e \right )+\frac {7 \left (\sin ^{5}\left (f x +e \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (f x +e \right )\right )}{24}+\frac {35 \sin \left (f x +e \right )}{16}\right ) \cos \left (f x +e \right )}{8}+\frac {35 f x}{128}+\frac {35 e}{128}\right )}{f}\)	\(420\)
derivativedivides	\(\frac {A \,a^{3} c^{4} \left (f x +e \right )+3 A \,a^{3} c^{4} \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^{4} \left (-\frac {\left (\sin ^{7}\left (f x +e \right )+\frac {7 \left (\sin ^{5}\left (f x +e \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (f x +e \right )\right )}{24}+\frac {35 \sin \left (f x +e \right )}{16}\right ) \cos \left (f x +e \right )}{8}+\frac {35 f x}{128}+\frac {35 e}{128}\right )+\frac {B \,a^{3} c^{4} \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}-3 B \,a^{3} c^{4} \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 )-\frac {3 B \,a^{3} c^{4} \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^{4} \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^{3} c^{4} \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}-A \,a^{3} c^{4} \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 )+\frac {3 A \,a^{3} c^{4} \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 \,a^{3} c^{4} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 B \,a^{3} c^{4} \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 )-3 A \,a^{3} c^{4} \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^{4} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+A \,a^{3} c^{4} \cos \left (f x +e \right )-B \,a^{3} c^{4} \cos \left (f x +e \right )}{f}\)	\(568\)
default	\(\frac {A \,a^{3} c^{4} \left (f x +e \right )+3 A \,a^{3} c^{4} \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^{4} \left (-\frac {\left (\sin ^{7}\left (f x +e \right )+\frac {7 \left (\sin ^{5}\left (f x +e \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (f x +e \right )\right )}{24}+\frac {35 \sin \left (f x +e \right )}{16}\right ) \cos \left (f x +e \right )}{8}+\frac {35 f x}{128}+\frac {35 e}{128}\right )+\frac {B \,a^{3} c^{4} \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}-3 B \,a^{3} c^{4} \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 )-\frac {3 B \,a^{3} c^{4} \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^{4} \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^{3} c^{4} \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}-A \,a^{3} c^{4} \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 )+\frac {3 A \,a^{3} c^{4} \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 \,a^{3} c^{4} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 B \,a^{3} c^{4} \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 )-3 A \,a^{3} c^{4} \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^{4} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+A \,a^{3} c^{4} \cos \left (f x +e \right )-B \,a^{3} c^{4} \cos \left (f x +e \right )}{f}\)	\(568\)
norman	\(\frac {\frac {c^{4} a^{3} \left (488 A -397 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{192 f}+\left (\frac {5}{16} A \,a^{3} c^{4}-\frac {5}{128} B \,a^{3} c^{4}\right ) x +\frac {c^{4} a^{3} \left (88 A +5 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{64 f}-\frac {5 c^{4} a^{3} \left (136 A -353 B \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{192 f}-\frac {c^{4} a^{3} \left (904 A +895 B \right ) \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{192 f}+\frac {c^{4} a^{3} \left (904 A +895 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{192 f}+\frac {10 \left (A \,a^{3} c^{4}-B \,a^{3} c^{4}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {5 c^{4} a^{3} \left (136 A -353 B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{192 f}-\frac {c^{4} a^{3} \left (488 A -397 B \right ) \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{192 f}-\frac {c^{4} a^{3} \left (88 A +5 B \right ) \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}+\frac {2 \left (5 A \,a^{3} c^{4}-5 B \,a^{3} c^{4}\right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\left (\frac {5}{16} A \,a^{3} c^{4}-\frac {5}{128} B \,a^{3} c^{4}\right ) x \left (\tan ^{16}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {2 A \,a^{3} c^{4}-2 B \,a^{3} c^{4}}{7 f}+\frac {2 \left (3 A \,a^{3} c^{4}-3 B \,a^{3} c^{4}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {2 \left (A \,a^{3} c^{4}-B \,a^{3} c^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 f}+\frac {2 \left (A \,a^{3} c^{4}-B \,a^{3} c^{4}\right ) \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {2 \left (A \,a^{3} c^{4}-B \,a^{3} c^{4}\right ) \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {2 \left (3 A \,a^{3} c^{4}-3 B \,a^{3} c^{4}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\left (\frac {175}{8} A \,a^{3} c^{4}-\frac {175}{64} B \,a^{3} c^{4}\right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {35}{4} A \,a^{3} c^{4}-\frac {35}{32} B \,a^{3} c^{4}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {35}{4} A \,a^{3} c^{4}-\frac {35}{32} B \,a^{3} c^{4}\right ) x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {5}{2} A \,a^{3} c^{4}-\frac {5}{16} B \,a^{3} c^{4}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {35}{2} A \,a^{3} c^{4}-\frac {35}{16} B \,a^{3} c^{4}\right ) x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {35}{2} A \,a^{3} c^{4}-\frac {35}{16} B \,a^{3} c^{4}\right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {5}{2} A \,a^{3} c^{4}-\frac {5}{16} B \,a^{3} c^{4}\right ) x \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{8}}\)	\(790\)

[In]

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4,x,method=_RETURNVERBOSE)

[Out]

1/64*c^4*(3*(A-B)*cos(3*f*x+3*e)+(A-B)*cos(5*f*x+5*e)+1/7*(A-B)*cos(7*f*x+7*e)+(15*A-B)*sin(2*f*x+2*e)+(1/2*B+
3*A)*sin(4*f*x+4*e)+1/3*(A+B)*sin(6*f*x+6*e)+1/16*B*sin(8*f*x+8*e)+5*(A-B)*cos(f*x+e)+20*f*x*A-5/2*f*x*B+64/7*
A-64/7*B)*a^3/f

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.76 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=\frac {384 \, {\left (A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )^{7} + 105 \, {\left (8 \, A - B\right )} a^{3} c^{4} f x + 7 \, {\left (48 \, B a^{3} c^{4} \cos \left (f x + e\right )^{7} + 8 \, {\left (8 \, A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )^{5} + 10 \, {\left (8 \, A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )^{3} + 15 \, {\left (8 \, A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2688 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4,x, algorithm="fricas")

[Out]

1/2688*(384*(A - B)*a^3*c^4*cos(f*x + e)^7 + 105*(8*A - B)*a^3*c^4*f*x + 7*(48*B*a^3*c^4*cos(f*x + e)^7 + 8*(8
*A - B)*a^3*c^4*cos(f*x + e)^5 + 10*(8*A - B)*a^3*c^4*cos(f*x + e)^3 + 15*(8*A - B)*a^3*c^4*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. 1579 vs. \(2 (163) = 326\).

Time = 0.87 (sec) , antiderivative size = 1579, normalized size of antiderivative = 8.72 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**4,x)

[Out]

Piecewise((-5*A*a**3*c**4*x*sin(e + f*x)**6/16 - 15*A*a**3*c**4*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 9*A*a**
3*c**4*x*sin(e + f*x)**4/8 - 15*A*a**3*c**4*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 9*A*a**3*c**4*x*sin(e + f*x
)**2*cos(e + f*x)**2/4 - 3*A*a**3*c**4*x*sin(e + f*x)**2/2 - 5*A*a**3*c**4*x*cos(e + f*x)**6/16 + 9*A*a**3*c**
4*x*cos(e + f*x)**4/8 - 3*A*a**3*c**4*x*cos(e + f*x)**2/2 + A*a**3*c**4*x - A*a**3*c**4*sin(e + f*x)**6*cos(e
+ f*x)/f + 11*A*a**3*c**4*sin(e + f*x)**5*cos(e + f*x)/(16*f) - 2*A*a**3*c**4*sin(e + f*x)**4*cos(e + f*x)**3/
f + 3*A*a**3*c**4*sin(e + f*x)**4*cos(e + f*x)/f + 5*A*a**3*c**4*sin(e + f*x)**3*cos(e + f*x)**3/(6*f) - 15*A*
a**3*c**4*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 8*A*a**3*c**4*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) + 4*A*a**3*
c**4*sin(e + f*x)**2*cos(e + f*x)**3/f - 3*A*a**3*c**4*sin(e + f*x)**2*cos(e + f*x)/f + 5*A*a**3*c**4*sin(e +
f*x)*cos(e + f*x)**5/(16*f) - 9*A*a**3*c**4*sin(e + f*x)*cos(e + f*x)**3/(8*f) + 3*A*a**3*c**4*sin(e + f*x)*co
s(e + f*x)/(2*f) - 16*A*a**3*c**4*cos(e + f*x)**7/(35*f) + 8*A*a**3*c**4*cos(e + f*x)**5/(5*f) - 2*A*a**3*c**4
*cos(e + f*x)**3/f + A*a**3*c**4*cos(e + f*x)/f + 35*B*a**3*c**4*x*sin(e + f*x)**8/128 + 35*B*a**3*c**4*x*sin(
e + f*x)**6*cos(e + f*x)**2/32 - 15*B*a**3*c**4*x*sin(e + f*x)**6/16 + 105*B*a**3*c**4*x*sin(e + f*x)**4*cos(e
+ f*x)**4/64 - 45*B*a**3*c**4*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 9*B*a**3*c**4*x*sin(e + f*x)**4/8 + 35*B
*a**3*c**4*x*sin(e + f*x)**2*cos(e + f*x)**6/32 - 45*B*a**3*c**4*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 9*B*a*
*3*c**4*x*sin(e + f*x)**2*cos(e + f*x)**2/4 - B*a**3*c**4*x*sin(e + f*x)**2/2 + 35*B*a**3*c**4*x*cos(e + f*x)*
*8/128 - 15*B*a**3*c**4*x*cos(e + f*x)**6/16 + 9*B*a**3*c**4*x*cos(e + f*x)**4/8 - B*a**3*c**4*x*cos(e + f*x)*
*2/2 - 93*B*a**3*c**4*sin(e + f*x)**7*cos(e + f*x)/(128*f) + B*a**3*c**4*sin(e + f*x)**6*cos(e + f*x)/f - 511*
B*a**3*c**4*sin(e + f*x)**5*cos(e + f*x)**3/(384*f) + 33*B*a**3*c**4*sin(e + f*x)**5*cos(e + f*x)/(16*f) + 2*B
*a**3*c**4*sin(e + f*x)**4*cos(e + f*x)**3/f - 3*B*a**3*c**4*sin(e + f*x)**4*cos(e + f*x)/f - 385*B*a**3*c**4*
sin(e + f*x)**3*cos(e + f*x)**5/(384*f) + 5*B*a**3*c**4*sin(e + f*x)**3*cos(e + f*x)**3/(2*f) - 15*B*a**3*c**4
*sin(e + f*x)**3*cos(e + f*x)/(8*f) + 8*B*a**3*c**4*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) - 4*B*a**3*c**4*sin(
e + f*x)**2*cos(e + f*x)**3/f + 3*B*a**3*c**4*sin(e + f*x)**2*cos(e + f*x)/f - 35*B*a**3*c**4*sin(e + f*x)*cos
(e + f*x)**7/(128*f) + 15*B*a**3*c**4*sin(e + f*x)*cos(e + f*x)**5/(16*f) - 9*B*a**3*c**4*sin(e + f*x)*cos(e +
f*x)**3/(8*f) + B*a**3*c**4*sin(e + f*x)*cos(e + f*x)/(2*f) + 16*B*a**3*c**4*cos(e + f*x)**7/(35*f) - 8*B*a**
3*c**4*cos(e + f*x)**5/(5*f) + 2*B*a**3*c**4*cos(e + f*x)**3/f - B*a**3*c**4*cos(e + f*x)/f, Ne(f, 0)), (x*(A
+ B*sin(e))*(a*sin(e) + a)**3*(-c*sin(e) + c)**4, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (170) = 340\).

Time = 0.24 (sec) , antiderivative size = 571, normalized size of antiderivative = 3.15 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=\frac {3072 \, {\left (5 \, \cos \left (f x + e\right )^{7} - 21 \, \cos \left (f x + e\right )^{5} + 35 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )\right )} A a^{3} c^{4} + 21504 \, {\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} c^{4} + 107520 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} c^{4} - 560 \, {\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 )} A a^{3} c^{4} + 10080 \, {\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^{4} - 80640 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c^{4} + 107520 \, {\left (f x + e\right )} A a^{3} c^{4} - 3072 \, {\left (5 \, \cos \left (f x + e\right )^{7} - 21 \, \cos \left (f x + e\right )^{5} + 35 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )\right )} B a^{3} c^{4} - 21504 \, {\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^{4} - 107520 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} c^{4} + 35 \, {\left (128 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 840 \, f x + 840 \, e + 3 \, \sin \left (8 \, f x + 8 \, e\right ) + 168 \, \sin \left (4 \, f x + 4 \, e\right ) - 768 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c^{4} - 1680 \, {\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} c^{4} + 10080 \, {\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^{4} - 26880 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c^{4} + 107520 \, A a^{3} c^{4} \cos \left (f x + e\right ) - 107520 \, B a^{3} c^{4} \cos \left (f x + e\right )}{107520 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4,x, algorithm="maxima")

[Out]

1/107520*(3072*(5*cos(f*x + e)^7 - 21*cos(f*x + e)^5 + 35*cos(f*x + e)^3 - 35*cos(f*x + e))*A*a^3*c^4 + 21504*
(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*A*a^3*c^4 + 107520*(cos(f*x + e)^3 - 3*cos(f*x + e))*
A*a^3*c^4 - 560*(4*sin(2*f*x + 2*e)^3 + 60*f*x + 60*e + 9*sin(4*f*x + 4*e) - 48*sin(2*f*x + 2*e))*A*a^3*c^4 +
10080*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^3*c^4 - 80640*(2*f*x + 2*e - sin(2*f*x + 2*e
))*A*a^3*c^4 + 107520*(f*x + e)*A*a^3*c^4 - 3072*(5*cos(f*x + e)^7 - 21*cos(f*x + e)^5 + 35*cos(f*x + e)^3 - 3
5*cos(f*x + e))*B*a^3*c^4 - 21504*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a^3*c^4 - 107520*
(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^3*c^4 + 35*(128*sin(2*f*x + 2*e)^3 + 840*f*x + 840*e + 3*sin(8*f*x + 8*e
) + 168*sin(4*f*x + 4*e) - 768*sin(2*f*x + 2*e))*B*a^3*c^4 - 1680*(4*sin(2*f*x + 2*e)^3 + 60*f*x + 60*e + 9*si
n(4*f*x + 4*e) - 48*sin(2*f*x + 2*e))*B*a^3*c^4 + 10080*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e)
)*B*a^3*c^4 - 26880*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^3*c^4 + 107520*A*a^3*c^4*cos(f*x + e) - 107520*B*a^3*
c^4*cos(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.46 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=\frac {B a^{3} c^{4} \sin \left (8 \, f x + 8 \, e\right )}{1024 \, f} + \frac {5}{128} \, {\left (8 \, A a^{3} c^{4} - B a^{3} c^{4}\right )} x + \frac {{\left (A a^{3} c^{4} - B a^{3} c^{4}\right )} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} + \frac {{\left (A a^{3} c^{4} - B a^{3} c^{4}\right )} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} + \frac {3 \, {\left (A a^{3} c^{4} - B a^{3} c^{4}\right )} \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} + \frac {5 \, {\left (A a^{3} c^{4} - B a^{3} c^{4}\right )} \cos \left (f x + e\right )}{64 \, f} + \frac {{\left (A a^{3} c^{4} + B a^{3} c^{4}\right )} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {{\left (6 \, A a^{3} c^{4} + B a^{3} c^{4}\right )} \sin \left (4 \, f x + 4 \, e\right )}{128 \, f} + \frac {{\left (15 \, A a^{3} c^{4} - B a^{3} c^{4}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4,x, algorithm="giac")

[Out]

1/1024*B*a^3*c^4*sin(8*f*x + 8*e)/f + 5/128*(8*A*a^3*c^4 - B*a^3*c^4)*x + 1/448*(A*a^3*c^4 - B*a^3*c^4)*cos(7*
f*x + 7*e)/f + 1/64*(A*a^3*c^4 - B*a^3*c^4)*cos(5*f*x + 5*e)/f + 3/64*(A*a^3*c^4 - B*a^3*c^4)*cos(3*f*x + 3*e)
/f + 5/64*(A*a^3*c^4 - B*a^3*c^4)*cos(f*x + e)/f + 1/192*(A*a^3*c^4 + B*a^3*c^4)*sin(6*f*x + 6*e)/f + 1/128*(6
*A*a^3*c^4 + B*a^3*c^4)*sin(4*f*x + 4*e)/f + 1/64*(15*A*a^3*c^4 - B*a^3*c^4)*sin(2*f*x + 2*e)/f

Mupad [B] (verification not implemented)

Time = 15.20 (sec) , antiderivative size = 661, normalized size of antiderivative = 3.65 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (6\,A\,a^3\,c^4-6\,B\,a^3\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (2\,A\,a^3\,c^4-2\,B\,a^3\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (6\,A\,a^3\,c^4-6\,B\,a^3\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}\,\left (2\,A\,a^3\,c^4-2\,B\,a^3\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {2\,A\,a^3\,c^4}{7}-\frac {2\,B\,a^3\,c^4}{7}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (10\,A\,a^3\,c^4-10\,B\,a^3\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (10\,A\,a^3\,c^4-10\,B\,a^3\,c^4\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{15}\,\left (\frac {11\,A\,a^3\,c^4}{8}+\frac {5\,B\,a^3\,c^4}{64}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {61\,A\,a^3\,c^4}{24}-\frac {397\,B\,a^3\,c^4}{192}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}\,\left (\frac {61\,A\,a^3\,c^4}{24}-\frac {397\,B\,a^3\,c^4}{192}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (\frac {113\,A\,a^3\,c^4}{24}+\frac {895\,B\,a^3\,c^4}{192}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\left (\frac {113\,A\,a^3\,c^4}{24}+\frac {895\,B\,a^3\,c^4}{192}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (\frac {85\,A\,a^3\,c^4}{24}-\frac {1765\,B\,a^3\,c^4}{192}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (\frac {85\,A\,a^3\,c^4}{24}-\frac {1765\,B\,a^3\,c^4}{192}\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {11\,A\,a^3\,c^4}{8}+\frac {5\,B\,a^3\,c^4}{64}\right )+\frac {2\,A\,a^3\,c^4}{7}-\frac {2\,B\,a^3\,c^4}{7}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}+8\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+\frac {5\,a^3\,c^4\,\mathrm {atan}\left (\frac {5\,a^3\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (8\,A-B\right )}{64\,\left (\frac {5\,A\,a^3\,c^4}{8}-\frac {5\,B\,a^3\,c^4}{64}\right )}\right )\,\left (8\,A-B\right )}{64\,f} \]

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^4,x)

[Out]

(tan(e/2 + (f*x)/2)^4*(6*A*a^3*c^4 - 6*B*a^3*c^4) + tan(e/2 + (f*x)/2)^12*(2*A*a^3*c^4 - 2*B*a^3*c^4) + tan(e/
2 + (f*x)/2)^6*(6*A*a^3*c^4 - 6*B*a^3*c^4) + tan(e/2 + (f*x)/2)^14*(2*A*a^3*c^4 - 2*B*a^3*c^4) + tan(e/2 + (f*
x)/2)^2*((2*A*a^3*c^4)/7 - (2*B*a^3*c^4)/7) + tan(e/2 + (f*x)/2)^8*(10*A*a^3*c^4 - 10*B*a^3*c^4) + tan(e/2 + (
f*x)/2)^10*(10*A*a^3*c^4 - 10*B*a^3*c^4) - tan(e/2 + (f*x)/2)^15*((11*A*a^3*c^4)/8 + (5*B*a^3*c^4)/64) + tan(e
/2 + (f*x)/2)^3*((61*A*a^3*c^4)/24 - (397*B*a^3*c^4)/192) - tan(e/2 + (f*x)/2)^13*((61*A*a^3*c^4)/24 - (397*B*
a^3*c^4)/192) + tan(e/2 + (f*x)/2)^5*((113*A*a^3*c^4)/24 + (895*B*a^3*c^4)/192) - tan(e/2 + (f*x)/2)^11*((113*
A*a^3*c^4)/24 + (895*B*a^3*c^4)/192) + tan(e/2 + (f*x)/2)^7*((85*A*a^3*c^4)/24 - (1765*B*a^3*c^4)/192) - tan(e
/2 + (f*x)/2)^9*((85*A*a^3*c^4)/24 - (1765*B*a^3*c^4)/192) + tan(e/2 + (f*x)/2)*((11*A*a^3*c^4)/8 + (5*B*a^3*c
^4)/64) + (2*A*a^3*c^4)/7 - (2*B*a^3*c^4)/7)/(f*(8*tan(e/2 + (f*x)/2)^2 + 28*tan(e/2 + (f*x)/2)^4 + 56*tan(e/2
+ (f*x)/2)^6 + 70*tan(e/2 + (f*x)/2)^8 + 56*tan(e/2 + (f*x)/2)^10 + 28*tan(e/2 + (f*x)/2)^12 + 8*tan(e/2 + (f
*x)/2)^14 + tan(e/2 + (f*x)/2)^16 + 1)) + (5*a^3*c^4*atan((5*a^3*c^4*tan(e/2 + (f*x)/2)*(8*A - B))/(64*((5*A*a
^3*c^4)/8 - (5*B*a^3*c^4)/64)))*(8*A - B))/(64*f)