\(\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx\) [28]

Optimal result

Integrand size = 36, antiderivative size = 147 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\frac {1}{16} a^2 (6 A-B) c^3 x+\frac {a^2 (6 A-B) c^3 \cos ^5(e+f x)}{30 f}+\frac {a^2 (6 A-B) c^3 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {a^2 (6 A-B) c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac {a^2 B \cos ^5(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{6 f} \]

[Out]

1/16*a^2*(6*A-B)*c^3*x+1/30*a^2*(6*A-B)*c^3*cos(f*x+e)^5/f+1/16*a^2*(6*A-B)*c^3*cos(f*x+e)*sin(f*x+e)/f+1/24*a
^2*(6*A-B)*c^3*cos(f*x+e)^3*sin(f*x+e)/f-1/6*a^2*B*cos(f*x+e)^5*(c^3-c^3*sin(f*x+e))/f

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {3046, 2939, 2748, 2715, 8} \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\frac {a^2 c^3 (6 A-B) \cos ^5(e+f x)}{30 f}+\frac {a^2 c^3 (6 A-B) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {a^2 c^3 (6 A-B) \sin (e+f x) \cos (e+f x)}{16 f}+\frac {1}{16} a^2 c^3 x (6 A-B)-\frac {a^2 B \cos ^5(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{6 f} \]

[In]

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

[Out]

(a^2*(6*A - B)*c^3*x)/16 + (a^2*(6*A - B)*c^3*Cos[e + f*x]^5)/(30*f) + (a^2*(6*A - B)*c^3*Cos[e + f*x]*Sin[e +
f*x])/(16*f) + (a^2*(6*A - B)*c^3*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) - (a^2*B*Cos[e + f*x]^5*(c^3 - c^3*Sin[
e + f*x]))/(6*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^2 c^2\right ) \int \cos ^4(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx \\ & = -\frac {a^2 B \cos ^5(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{6 f}+\frac {1}{6} \left (a^2 (6 A-B) c^2\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x)) \, dx \\ & = \frac {a^2 (6 A-B) c^3 \cos ^5(e+f x)}{30 f}-\frac {a^2 B \cos ^5(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{6 f}+\frac {1}{6} \left (a^2 (6 A-B) c^3\right ) \int \cos ^4(e+f x) \, dx \\ & = \frac {a^2 (6 A-B) c^3 \cos ^5(e+f x)}{30 f}+\frac {a^2 (6 A-B) c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac {a^2 B \cos ^5(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{6 f}+\frac {1}{8} \left (a^2 (6 A-B) c^3\right ) \int \cos ^2(e+f x) \, dx \\ & = \frac {a^2 (6 A-B) c^3 \cos ^5(e+f x)}{30 f}+\frac {a^2 (6 A-B) c^3 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {a^2 (6 A-B) c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac {a^2 B \cos ^5(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{6 f}+\frac {1}{16} \left (a^2 (6 A-B) c^3\right ) \int 1 \, dx \\ & = \frac {1}{16} a^2 (6 A-B) c^3 x+\frac {a^2 (6 A-B) c^3 \cos ^5(e+f x)}{30 f}+\frac {a^2 (6 A-B) c^3 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {a^2 (6 A-B) c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac {a^2 B \cos ^5(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{6 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.03 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.93 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\frac {a^2 c^3 (360 A e-60 B e+360 A f x-60 B f x+120 (A-B) \cos (e+f x)+60 (A-B) \cos (3 (e+f x))+12 A \cos (5 (e+f x))-12 B \cos (5 (e+f x))+240 A \sin (2 (e+f x))-15 B \sin (2 (e+f x))+30 A \sin (4 (e+f x))+15 B \sin (4 (e+f x))+5 B \sin (6 (e+f x)))}{960 f} \]

[In]

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

[Out]

(a^2*c^3*(360*A*e - 60*B*e + 360*A*f*x - 60*B*f*x + 120*(A - B)*Cos[e + f*x] + 60*(A - B)*Cos[3*(e + f*x)] + 1
2*A*Cos[5*(e + f*x)] - 12*B*Cos[5*(e + f*x)] + 240*A*Sin[2*(e + f*x)] - 15*B*Sin[2*(e + f*x)] + 30*A*Sin[4*(e
+ f*x)] + 15*B*Sin[4*(e + f*x)] + 5*B*Sin[6*(e + f*x)]))/(960*f)

Maple [A] (verified)

Time = 2.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.81

method	result	size
parallelrisch	\(\frac {c^{3} \left (5 \left (A -B \right ) \cos \left (3 f x +3 e \right )+\left (A -B \right ) \cos \left (5 f x +5 e \right )+5 \left (4 A -\frac {B}{4}\right ) \sin \left (2 f x +2 e \right )+\frac {5 \left (A +\frac {B}{2}\right ) \sin \left (4 f x +4 e \right )}{2}+\frac {5 B \sin \left (6 f x +6 e \right )}{12}+10 \left (A -B \right ) \cos \left (f x +e \right )+30 f x A -5 f x B +16 A -16 B \right ) a^{2}}{80 f}\)	\(119\)
risch	\(\frac {3 a^{2} c^{3} x A}{8}-\frac {a^{2} c^{3} x B}{16}+\frac {c^{3} a^{2} \cos \left (f x +e \right ) A}{8 f}-\frac {c^{3} a^{2} \cos \left (f x +e \right ) B}{8 f}+\frac {B \,a^{2} c^{3} \sin \left (6 f x +6 e \right )}{192 f}+\frac {c^{3} a^{2} \cos \left (5 f x +5 e \right ) A}{80 f}-\frac {c^{3} a^{2} \cos \left (5 f x +5 e \right ) B}{80 f}+\frac {\sin \left (4 f x +4 e \right ) A \,a^{2} c^{3}}{32 f}+\frac {\sin \left (4 f x +4 e \right ) B \,a^{2} c^{3}}{64 f}+\frac {c^{3} a^{2} \cos \left (3 f x +3 e \right ) A}{16 f}-\frac {c^{3} a^{2} \cos \left (3 f x +3 e \right ) B}{16 f}+\frac {\sin \left (2 f x +2 e \right ) A \,a^{2} c^{3}}{4 f}-\frac {\sin \left (2 f x +2 e \right ) B \,a^{2} c^{3}}{64 f}\)	\(247\)
parts	\(\frac {\left (-2 A \,a^{2} c^{3}-B \,a^{2} c^{3}\right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {\left (-A \,a^{2} c^{3}+B \,a^{2} c^{3}\right ) \cos \left (f x +e \right )}{f}-\frac {\left (-A \,a^{2} c^{3}+B \,a^{2} c^{3}\right ) \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}+\frac {\left (A \,a^{2} c^{3}+2 B \,a^{2} c^{3}\right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {\left (2 A \,a^{2} c^{3}-2 B \,a^{2} c^{3}\right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+a^{2} c^{3} x A -\frac {B \,a^{2} c^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}\)	\(284\)
derivativedivides	\(\frac {\frac {A \,a^{2} c^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+A \,a^{2} c^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {2 A \,a^{2} c^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-2 A \,a^{2} c^{3} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{2} c^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-\frac {B \,a^{2} c^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+2 B \,a^{2} c^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {2 B \,a^{2} c^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+A \,a^{2} c^{3} \cos \left (f x +e \right )-B \,a^{2} c^{3} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,a^{2} c^{3} \left (f x +e \right )-B \,a^{2} c^{3} \cos \left (f x +e \right )}{f}\)	\(365\)
default	\(\frac {\frac {A \,a^{2} c^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+A \,a^{2} c^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {2 A \,a^{2} c^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-2 A \,a^{2} c^{3} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{2} c^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-\frac {B \,a^{2} c^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+2 B \,a^{2} c^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {2 B \,a^{2} c^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+A \,a^{2} c^{3} \cos \left (f x +e \right )-B \,a^{2} c^{3} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,a^{2} c^{3} \left (f x +e \right )-B \,a^{2} c^{3} \cos \left (f x +e \right )}{f}\)	\(365\)
norman	\(\frac {\left (\frac {3}{8} A \,a^{2} c^{3}-\frac {1}{16} B \,a^{2} c^{3}\right ) x +\left (\frac {3}{8} A \,a^{2} c^{3}-\frac {1}{16} B \,a^{2} c^{3}\right ) x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {9}{4} A \,a^{2} c^{3}-\frac {3}{8} B \,a^{2} c^{3}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {9}{4} A \,a^{2} c^{3}-\frac {3}{8} B \,a^{2} c^{3}\right ) x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {15}{2} A \,a^{2} c^{3}-\frac {5}{4} B \,a^{2} c^{3}\right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {45}{8} A \,a^{2} c^{3}-\frac {15}{16} B \,a^{2} c^{3}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {45}{8} A \,a^{2} c^{3}-\frac {15}{16} B \,a^{2} c^{3}\right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {\left (2 A \,a^{2} c^{3}-2 B \,a^{2} c^{3}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {\left (4 A \,a^{2} c^{3}-4 B \,a^{2} c^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {2 A \,a^{2} c^{3}-2 B \,a^{2} c^{3}}{5 f}+\frac {2 \left (A \,a^{2} c^{3}-B \,a^{2} c^{3}\right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {4 \left (A \,a^{2} c^{3}-B \,a^{2} c^{3}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {2 \left (A \,a^{2} c^{3}-B \,a^{2} c^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f}+\frac {c^{3} a^{2} \left (2 A +13 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {c^{3} a^{2} \left (2 A +13 B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {c^{3} a^{2} \left (10 A +B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}-\frac {c^{3} a^{2} \left (10 A +B \right ) \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}+\frac {c^{3} a^{2} \left (42 A -47 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}-\frac {c^{3} a^{2} \left (42 A -47 B \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{6}}\)	\(593\)

[In]

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

[Out]

1/80*c^3*(5*(A-B)*cos(3*f*x+3*e)+(A-B)*cos(5*f*x+5*e)+5*(4*A-1/4*B)*sin(2*f*x+2*e)+5/2*(A+1/2*B)*sin(4*f*x+4*e
)+5/12*B*sin(6*f*x+6*e)+10*(A-B)*cos(f*x+e)+30*f*x*A-5*f*x*B+16*A-16*B)*a^2/f

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.78 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\frac {48 \, {\left (A - B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{5} + 15 \, {\left (6 \, A - B\right )} a^{2} c^{3} f x + 5 \, {\left (8 \, B a^{2} c^{3} \cos \left (f x + e\right )^{5} + 2 \, {\left (6 \, A - B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{3} + 3 \, {\left (6 \, A - B\right )} a^{2} c^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \]

[In]

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

[Out]

1/240*(48*(A - B)*a^2*c^3*cos(f*x + e)^5 + 15*(6*A - B)*a^2*c^3*f*x + 5*(8*B*a^2*c^3*cos(f*x + e)^5 + 2*(6*A -
B)*a^2*c^3*cos(f*x + e)^3 + 3*(6*A - B)*a^2*c^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. 910 vs. \(2 (128) = 256\).

Time = 0.44 (sec) , antiderivative size = 910, normalized size of antiderivative = 6.19 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\begin {cases} \frac {3 A a^{2} c^{3} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {3 A a^{2} c^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - A a^{2} c^{3} x \sin ^{2}{\left (e + f x \right )} + \frac {3 A a^{2} c^{3} x \cos ^{4}{\left (e + f x \right )}}{8} - A a^{2} c^{3} x \cos ^{2}{\left (e + f x \right )} + A a^{2} c^{3} x + \frac {A a^{2} c^{3} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 A a^{2} c^{3} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} + \frac {4 A a^{2} c^{3} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {2 A a^{2} c^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 A a^{2} c^{3} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac {A a^{2} c^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {8 A a^{2} c^{3} \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {4 A a^{2} c^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {A a^{2} c^{3} \cos {\left (e + f x \right )}}{f} - \frac {5 B a^{2} c^{3} x \sin ^{6}{\left (e + f x \right )}}{16} - \frac {15 B a^{2} c^{3} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} + \frac {3 B a^{2} c^{3} x \sin ^{4}{\left (e + f x \right )}}{4} - \frac {15 B a^{2} c^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} + \frac {3 B a^{2} c^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} - \frac {B a^{2} c^{3} x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {5 B a^{2} c^{3} x \cos ^{6}{\left (e + f x \right )}}{16} + \frac {3 B a^{2} c^{3} x \cos ^{4}{\left (e + f x \right )}}{4} - \frac {B a^{2} c^{3} x \cos ^{2}{\left (e + f x \right )}}{2} + \frac {11 B a^{2} c^{3} \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{16 f} - \frac {B a^{2} c^{3} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {5 B a^{2} c^{3} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{6 f} - \frac {5 B a^{2} c^{3} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f} - \frac {4 B a^{2} c^{3} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {2 B a^{2} c^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {5 B a^{2} c^{3} \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} - \frac {3 B a^{2} c^{3} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{4 f} + \frac {B a^{2} c^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {8 B a^{2} c^{3} \cos ^{5}{\left (e + f x \right )}}{15 f} + \frac {4 B a^{2} c^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {B a^{2} c^{3} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\left (e \right )}\right ) \left (a \sin {\left (e \right )} + a\right )^{2} \left (- c \sin {\left (e \right )} + c\right )^{3} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (138) = 276\).

Time = 0.23 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.45 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\frac {64 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} A a^{2} c^{3} + 640 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} c^{3} + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c^{3} - 480 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c^{3} + 960 \, {\left (f x + e\right )} A a^{2} c^{3} - 64 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a^{2} c^{3} - 640 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c^{3} - 5 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{3} + 60 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{3} - 240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{3} + 960 \, A a^{2} c^{3} \cos \left (f x + e\right ) - 960 \, B a^{2} c^{3} \cos \left (f x + e\right )}{960 \, f} \]

[In]

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

[Out]

1/960*(64*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*A*a^2*c^3 + 640*(cos(f*x + e)^3 - 3*cos(f*x
+ e))*A*a^2*c^3 + 30*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^2*c^3 - 480*(2*f*x + 2*e - s
in(2*f*x + 2*e))*A*a^2*c^3 + 960*(f*x + e)*A*a^2*c^3 - 64*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x +
e))*B*a^2*c^3 - 640*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^2*c^3 - 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^2*c^3 + 60*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e)
)*B*a^2*c^3 - 240*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^2*c^3 + 960*A*a^2*c^3*cos(f*x + e) - 960*B*a^2*c^3*cos(
f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.37 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\frac {B a^{2} c^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {1}{16} \, {\left (6 \, A a^{2} c^{3} - B a^{2} c^{3}\right )} x + \frac {{\left (A a^{2} c^{3} - B a^{2} c^{3}\right )} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {{\left (A a^{2} c^{3} - B a^{2} c^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{16 \, f} + \frac {{\left (A a^{2} c^{3} - B a^{2} c^{3}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {{\left (2 \, A a^{2} c^{3} + B a^{2} c^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} + \frac {{\left (16 \, A a^{2} c^{3} - B a^{2} c^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]

[In]

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

[Out]

1/192*B*a^2*c^3*sin(6*f*x + 6*e)/f + 1/16*(6*A*a^2*c^3 - B*a^2*c^3)*x + 1/80*(A*a^2*c^3 - B*a^2*c^3)*cos(5*f*x
+ 5*e)/f + 1/16*(A*a^2*c^3 - B*a^2*c^3)*cos(3*f*x + 3*e)/f + 1/8*(A*a^2*c^3 - B*a^2*c^3)*cos(f*x + e)/f + 1/6
4*(2*A*a^2*c^3 + B*a^2*c^3)*sin(4*f*x + 4*e)/f + 1/64*(16*A*a^2*c^3 - B*a^2*c^3)*sin(2*f*x + 2*e)/f

Mupad [B] (verification not implemented)

Time = 14.57 (sec) , antiderivative size = 542, normalized size of antiderivative = 3.69 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (4\,A\,a^2\,c^3-4\,B\,a^2\,c^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (2\,A\,a^2\,c^3-2\,B\,a^2\,c^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (4\,A\,a^2\,c^3-4\,B\,a^2\,c^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (2\,A\,a^2\,c^3-2\,B\,a^2\,c^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {2\,A\,a^2\,c^3}{5}-\frac {2\,B\,a^2\,c^3}{5}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (\frac {A\,a^2\,c^3}{2}+\frac {13\,B\,a^2\,c^3}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (\frac {A\,a^2\,c^3}{2}+\frac {13\,B\,a^2\,c^3}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\left (\frac {5\,A\,a^2\,c^3}{4}+\frac {B\,a^2\,c^3}{8}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {7\,A\,a^2\,c^3}{4}-\frac {47\,B\,a^2\,c^3}{24}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (\frac {7\,A\,a^2\,c^3}{4}-\frac {47\,B\,a^2\,c^3}{24}\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {5\,A\,a^2\,c^3}{4}+\frac {B\,a^2\,c^3}{8}\right )+\frac {2\,A\,a^2\,c^3}{5}-\frac {2\,B\,a^2\,c^3}{5}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+\frac {a^2\,c^3\,\mathrm {atan}\left (\frac {a^2\,c^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (6\,A-B\right )}{8\,\left (\frac {3\,A\,a^2\,c^3}{4}-\frac {B\,a^2\,c^3}{8}\right )}\right )\,\left (6\,A-B\right )}{8\,f}-\frac {a^2\,c^3\,\left (6\,A-B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )}{8\,f} \]

[In]

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

[Out]

(tan(e/2 + (f*x)/2)^4*(4*A*a^2*c^3 - 4*B*a^2*c^3) + tan(e/2 + (f*x)/2)^8*(2*A*a^2*c^3 - 2*B*a^2*c^3) + tan(e/2
+ (f*x)/2)^6*(4*A*a^2*c^3 - 4*B*a^2*c^3) + tan(e/2 + (f*x)/2)^10*(2*A*a^2*c^3 - 2*B*a^2*c^3) + tan(e/2 + (f*x
)/2)^2*((2*A*a^2*c^3)/5 - (2*B*a^2*c^3)/5) + tan(e/2 + (f*x)/2)^5*((A*a^2*c^3)/2 + (13*B*a^2*c^3)/4) - tan(e/2
+ (f*x)/2)^7*((A*a^2*c^3)/2 + (13*B*a^2*c^3)/4) - tan(e/2 + (f*x)/2)^11*((5*A*a^2*c^3)/4 + (B*a^2*c^3)/8) + t
an(e/2 + (f*x)/2)^3*((7*A*a^2*c^3)/4 - (47*B*a^2*c^3)/24) - tan(e/2 + (f*x)/2)^9*((7*A*a^2*c^3)/4 - (47*B*a^2*
c^3)/24) + tan(e/2 + (f*x)/2)*((5*A*a^2*c^3)/4 + (B*a^2*c^3)/8) + (2*A*a^2*c^3)/5 - (2*B*a^2*c^3)/5)/(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)) + (a^2*c^3*atan((a^2*c^3*tan(e/2 + (f*x)/2)*(6*A - B))/(8*((3*A*a^2*
c^3)/4 - (B*a^2*c^3)/8)))*(6*A - B))/(8*f) - (a^2*c^3*(6*A - B)*(atan(tan(e/2 + (f*x)/2)) - (f*x)/2))/(8*f)