Integrand size = 34, antiderivative size = 140 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\frac {1}{8} a^3 (5 A+2 B) c x-\frac {a^3 (5 A+2 B) c \cos ^3(e+f x)}{12 f}+\frac {a^3 (5 A+2 B) c \cos (e+f x) \sin (e+f x)}{8 f}-\frac {a B c \cos ^3(e+f x) (a+a \sin (e+f x))^2}{5 f}-\frac {(5 A+2 B) c \cos ^3(e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{20 f} \]
1/8*a^3*(5*A+2*B)*c*x-1/12*a^3*(5*A+2*B)*c*cos(f*x+e)^3/f+1/8*a^3*(5*A+2*B )*c*cos(f*x+e)*sin(f*x+e)/f-1/5*a*B*c*cos(f*x+e)^3*(a+a*sin(f*x+e))^2/f-1/ 20*(5*A+2*B)*c*cos(f*x+e)^3*(a^3+a^3*sin(f*x+e))/f
Time = 0.89 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.01 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\frac {a^3 c \cos (e+f x) \left (-30 (5 A+2 B) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (-8 (10 A+7 B)+15 (3 A-2 B) \sin (e+f x)+16 (5 A+2 B) \sin ^2(e+f x)+30 (A+2 B) \sin ^3(e+f x)+24 B \sin ^4(e+f x)\right )\right )}{120 f \sqrt {\cos ^2(e+f x)}} \]
(a^3*c*Cos[e + f*x]*(-30*(5*A + 2*B)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2] ] + Sqrt[Cos[e + f*x]^2]*(-8*(10*A + 7*B) + 15*(3*A - 2*B)*Sin[e + f*x] + 16*(5*A + 2*B)*Sin[e + f*x]^2 + 30*(A + 2*B)*Sin[e + f*x]^3 + 24*B*Sin[e + f*x]^4)))/(120*f*Sqrt[Cos[e + f*x]^2])
Time = 0.64 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.87, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {3042, 3446, 3042, 3339, 3042, 3157, 3042, 3148, 3042, 3115, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c-c \sin (e+f x)) (A+B \sin (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c-c \sin (e+f x)) (A+B \sin (e+f x))dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a c \int \cos ^2(e+f x) (\sin (e+f x) a+a)^2 (A+B \sin (e+f x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \int \cos (e+f x)^2 (\sin (e+f x) a+a)^2 (A+B \sin (e+f x))dx\) |
| \(\Big \downarrow \) 3339 |
| \(\displaystyle a c \left (\frac {1}{5} (5 A+2 B) \int \cos ^2(e+f x) (\sin (e+f x) a+a)^2dx-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)^2}{5 f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {1}{5} (5 A+2 B) \int \cos (e+f x)^2 (\sin (e+f x) a+a)^2dx-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)^2}{5 f}\right )\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle a c \left (\frac {1}{5} (5 A+2 B) \left (\frac {5}{4} a \int \cos ^2(e+f x) (\sin (e+f x) a+a)dx-\frac {\cos ^3(e+f x) \left (a^2 \sin (e+f x)+a^2\right )}{4 f}\right )-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)^2}{5 f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {1}{5} (5 A+2 B) \left (\frac {5}{4} a \int \cos (e+f x)^2 (\sin (e+f x) a+a)dx-\frac {\cos ^3(e+f x) \left (a^2 \sin (e+f x)+a^2\right )}{4 f}\right )-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)^2}{5 f}\right )\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle a c \left (\frac {1}{5} (5 A+2 B) \left (\frac {5}{4} a \left (a \int \cos ^2(e+f x)dx-\frac {a \cos ^3(e+f x)}{3 f}\right )-\frac {\cos ^3(e+f x) \left (a^2 \sin (e+f x)+a^2\right )}{4 f}\right )-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)^2}{5 f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {1}{5} (5 A+2 B) \left (\frac {5}{4} a \left (a \int \sin \left (e+f x+\frac {\pi }{2}\right )^2dx-\frac {a \cos ^3(e+f x)}{3 f}\right )-\frac {\cos ^3(e+f x) \left (a^2 \sin (e+f x)+a^2\right )}{4 f}\right )-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)^2}{5 f}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a c \left (\frac {1}{5} (5 A+2 B) \left (\frac {5}{4} a \left (a \left (\frac {\int 1dx}{2}+\frac {\sin (e+f x) \cos (e+f x)}{2 f}\right )-\frac {a \cos ^3(e+f x)}{3 f}\right )-\frac {\cos ^3(e+f x) \left (a^2 \sin (e+f x)+a^2\right )}{4 f}\right )-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)^2}{5 f}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a c \left (\frac {1}{5} (5 A+2 B) \left (\frac {5}{4} a \left (a \left (\frac {\sin (e+f x) \cos (e+f x)}{2 f}+\frac {x}{2}\right )-\frac {a \cos ^3(e+f x)}{3 f}\right )-\frac {\cos ^3(e+f x) \left (a^2 \sin (e+f x)+a^2\right )}{4 f}\right )-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)^2}{5 f}\right )\) |
a*c*(-1/5*(B*Cos[e + f*x]^3*(a + a*Sin[e + f*x])^2)/f + ((5*A + 2*B)*(-1/4 *(Cos[e + f*x]^3*(a^2 + a^2*Sin[e + f*x]))/f + (5*a*(-1/3*(a*Cos[e + f*x]^ 3)/f + a*(x/2 + (Cos[e + f*x]*Sin[e + f*x])/(2*f))))/4))/5)
3.1.43.3.1 Defintions of rubi rules used
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] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[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])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && Integers Q[2*m, 2*p]
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] + S imp[(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] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[ a^2 - b^2, 0] && NeQ[m + p + 1, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si mp[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] && IntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Time = 1.40 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.69
| method | result | size |
| parallelrisch | \(\frac {\left (\left (-\frac {2 A}{3}-\frac {5 B}{12}\right ) \cos \left (3 f x +3 e \right )+\left (-\frac {A}{8}-\frac {B}{4}\right ) \sin \left (4 f x +4 e \right )+\frac {\cos \left (5 f x +5 e \right ) B}{20}+A \sin \left (2 f x +2 e \right )+\left (-2 A -\frac {3 B}{2}\right ) \cos \left (f x +e \right )+\frac {5 f x A}{2}+f x B -\frac {8 A}{3}-\frac {28 B}{15}\right ) c \,a^{3}}{4 f}\) | \(97\) |
| risch | \(\frac {5 a^{3} c x A}{8}+\frac {a^{3} c x B}{4}-\frac {a^{3} c \cos \left (f x +e \right ) A}{2 f}-\frac {3 a^{3} c \cos \left (f x +e \right ) B}{8 f}+\frac {B \,a^{3} c \cos \left (5 f x +5 e \right )}{80 f}-\frac {\sin \left (4 f x +4 e \right ) A \,a^{3} c}{32 f}-\frac {\sin \left (4 f x +4 e \right ) B \,a^{3} c}{16 f}-\frac {a^{3} c \cos \left (3 f x +3 e \right ) A}{6 f}-\frac {5 a^{3} c \cos \left (3 f x +3 e \right ) B}{48 f}+\frac {A \,a^{3} c \sin \left (2 f x +2 e \right )}{4 f}\) | \(164\) |
| parts | \(\frac {\left (-A \,a^{3} c -2 B \,a^{3} c \right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {\left (2 A \,a^{3} c +B \,a^{3} c \right ) \cos \left (f x +e \right )}{f}+a^{3} c x A +\frac {2 A \,a^{3} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {2 B \,a^{3} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {B \,a^{3} c \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}\) | \(180\) |
| derivativedivides | \(\frac {-A \,a^{3} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {2 A \,a^{3} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+\frac {B \,a^{3} c \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-2 B \,a^{3} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-2 A \cos \left (f x +e \right ) a^{3} c +2 B \,a^{3} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,a^{3} c \left (f x +e \right )-B \cos \left (f x +e \right ) a^{3} c}{f}\) | \(208\) |
| default | \(\frac {-A \,a^{3} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {2 A \,a^{3} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+\frac {B \,a^{3} c \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-2 B \,a^{3} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-2 A \cos \left (f x +e \right ) a^{3} c +2 B \,a^{3} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,a^{3} c \left (f x +e \right )-B \cos \left (f x +e \right ) a^{3} c}{f}\) | \(208\) |
| norman | \(\frac {\left (\frac {5}{8} A \,a^{3} c +\frac {1}{4} B \,a^{3} c \right ) x +\left (\frac {5}{8} A \,a^{3} c +\frac {1}{4} B \,a^{3} c \right ) x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {25}{4} A \,a^{3} c +\frac {5}{2} B \,a^{3} c \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {25}{4} A \,a^{3} c +\frac {5}{2} B \,a^{3} c \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {25}{8} A \,a^{3} c +\frac {5}{4} B \,a^{3} c \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {25}{8} A \,a^{3} c +\frac {5}{4} B \,a^{3} c \right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {20 A \,a^{3} c +14 B \,a^{3} c}{15 f}-\frac {\left (4 A \,a^{3} c +2 B \,a^{3} c \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (4 A \,a^{3} c +4 B \,a^{3} c \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (8 A \,a^{3} c +2 B \,a^{3} c \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {\left (8 A \,a^{3} c +8 B \,a^{3} c \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}+\frac {a^{3} c \left (3 A -2 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}-\frac {a^{3} c \left (3 A -2 B \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {a^{3} c \left (6 B +7 A \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}-\frac {a^{3} c \left (6 B +7 A \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{5}}\) | \(425\) |
1/4*((-2/3*A-5/12*B)*cos(3*f*x+3*e)+(-1/8*A-1/4*B)*sin(4*f*x+4*e)+1/20*cos (5*f*x+5*e)*B+A*sin(2*f*x+2*e)+(-2*A-3/2*B)*cos(f*x+e)+5/2*f*x*A+f*x*B-8/3 *A-28/15*B)*c*a^3/f
Time = 0.26 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.71 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\frac {24 \, B a^{3} c \cos \left (f x + e\right )^{5} - 80 \, {\left (A + B\right )} a^{3} c \cos \left (f x + e\right )^{3} + 15 \, {\left (5 \, A + 2 \, B\right )} a^{3} c f x - 15 \, {\left (2 \, {\left (A + 2 \, B\right )} a^{3} c \cos \left (f x + e\right )^{3} - {\left (5 \, A + 2 \, B\right )} a^{3} c \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{120 \, f} \]
1/120*(24*B*a^3*c*cos(f*x + e)^5 - 80*(A + B)*a^3*c*cos(f*x + e)^3 + 15*(5 *A + 2*B)*a^3*c*f*x - 15*(2*(A + 2*B)*a^3*c*cos(f*x + e)^3 - (5*A + 2*B)*a ^3*c*cos(f*x + e))*sin(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (128) = 256\).
Time = 0.30 (sec) , antiderivative size = 486, normalized size of antiderivative = 3.47 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\begin {cases} - \frac {3 A a^{3} c x \sin ^{4}{\left (e + f x \right )}}{8} - \frac {3 A a^{3} c x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - \frac {3 A a^{3} c x \cos ^{4}{\left (e + f x \right )}}{8} + A a^{3} c x + \frac {5 A a^{3} c \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} + \frac {2 A a^{3} c \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {3 A a^{3} c \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac {4 A a^{3} c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {2 A a^{3} c \cos {\left (e + f x \right )}}{f} - \frac {3 B a^{3} c x \sin ^{4}{\left (e + f x \right )}}{4} - \frac {3 B a^{3} c x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} + B a^{3} c x \sin ^{2}{\left (e + f x \right )} - \frac {3 B a^{3} c x \cos ^{4}{\left (e + f x \right )}}{4} + B a^{3} c x \cos ^{2}{\left (e + f x \right )} + \frac {B a^{3} c \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {5 B a^{3} c \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f} + \frac {4 B a^{3} c \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {3 B a^{3} c \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{4 f} - \frac {B a^{3} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {8 B a^{3} c \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {B a^{3} c \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\left (e \right )}\right ) \left (a \sin {\left (e \right )} + a\right )^{3} \left (- c \sin {\left (e \right )} + c\right ) & \text {otherwise} \end {cases} \]
Piecewise((-3*A*a**3*c*x*sin(e + f*x)**4/8 - 3*A*a**3*c*x*sin(e + f*x)**2* cos(e + f*x)**2/4 - 3*A*a**3*c*x*cos(e + f*x)**4/8 + A*a**3*c*x + 5*A*a**3 *c*sin(e + f*x)**3*cos(e + f*x)/(8*f) + 2*A*a**3*c*sin(e + f*x)**2*cos(e + f*x)/f + 3*A*a**3*c*sin(e + f*x)*cos(e + f*x)**3/(8*f) + 4*A*a**3*c*cos(e + f*x)**3/(3*f) - 2*A*a**3*c*cos(e + f*x)/f - 3*B*a**3*c*x*sin(e + f*x)** 4/4 - 3*B*a**3*c*x*sin(e + f*x)**2*cos(e + f*x)**2/2 + B*a**3*c*x*sin(e + f*x)**2 - 3*B*a**3*c*x*cos(e + f*x)**4/4 + B*a**3*c*x*cos(e + f*x)**2 + B* a**3*c*sin(e + f*x)**4*cos(e + f*x)/f + 5*B*a**3*c*sin(e + f*x)**3*cos(e + f*x)/(4*f) + 4*B*a**3*c*sin(e + f*x)**2*cos(e + f*x)**3/(3*f) + 3*B*a**3* c*sin(e + f*x)*cos(e + f*x)**3/(4*f) - B*a**3*c*sin(e + f*x)*cos(e + f*x)/ f + 8*B*a**3*c*cos(e + f*x)**5/(15*f) - B*a**3*c*cos(e + f*x)/f, Ne(f, 0)) , (x*(A + B*sin(e))*(a*sin(e) + a)**3*(-c*sin(e) + c), True))
Time = 0.21 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.43 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=-\frac {320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} c + 15 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c - 480 \, {\left (f x + e\right )} A a^{3} c - 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a^{3} c + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c - 240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c + 960 \, A a^{3} c \cos \left (f x + e\right ) + 480 \, B a^{3} c \cos \left (f x + e\right )}{480 \, f} \]
-1/480*(320*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a^3*c + 15*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^3*c - 480*(f*x + e)*A*a^3*c - 32*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a^3*c + 30* (12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a^3*c - 240*(2*f *x + 2*e - sin(2*f*x + 2*e))*B*a^3*c + 960*A*a^3*c*cos(f*x + e) + 480*B*a^ 3*c*cos(f*x + e))/f
Time = 0.44 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\frac {B a^{3} c \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {A a^{3} c \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{8} \, {\left (5 \, A a^{3} c + 2 \, B a^{3} c\right )} x - \frac {{\left (8 \, A a^{3} c + 5 \, B a^{3} c\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (4 \, A a^{3} c + 3 \, B a^{3} c\right )} \cos \left (f x + e\right )}{8 \, f} - \frac {{\left (A a^{3} c + 2 \, B a^{3} c\right )} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} \]
1/80*B*a^3*c*cos(5*f*x + 5*e)/f + 1/4*A*a^3*c*sin(2*f*x + 2*e)/f + 1/8*(5* A*a^3*c + 2*B*a^3*c)*x - 1/48*(8*A*a^3*c + 5*B*a^3*c)*cos(3*f*x + 3*e)/f - 1/8*(4*A*a^3*c + 3*B*a^3*c)*cos(f*x + e)/f - 1/32*(A*a^3*c + 2*B*a^3*c)*s in(4*f*x + 4*e)/f
Time = 14.34 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.79 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\frac {a^3\,c\,\mathrm {atan}\left (\frac {a^3\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (5\,A+2\,B\right )}{4\,\left (\frac {5\,A\,a^3\,c}{4}+\frac {B\,a^3\,c}{2}\right )}\right )\,\left (5\,A+2\,B\right )}{4\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (4\,A\,a^3\,c+2\,B\,a^3\,c\right )-\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3\,A\,a^3\,c}{4}-\frac {B\,a^3\,c}{2}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {7\,A\,a^3\,c}{2}+3\,B\,a^3\,c\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (\frac {7\,A\,a^3\,c}{2}+3\,B\,a^3\,c\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (\frac {3\,A\,a^3\,c}{4}-\frac {B\,a^3\,c}{2}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (8\,A\,a^3\,c+8\,B\,a^3\,c\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {8\,A\,a^3\,c}{3}+\frac {8\,B\,a^3\,c}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {16\,A\,a^3\,c}{3}+\frac {4\,B\,a^3\,c}{3}\right )+\frac {4\,A\,a^3\,c}{3}+\frac {14\,B\,a^3\,c}{15}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {a^3\,c\,\left (5\,A+2\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )}{4\,f} \]
(a^3*c*atan((a^3*c*tan(e/2 + (f*x)/2)*(5*A + 2*B))/(4*((5*A*a^3*c)/4 + (B* a^3*c)/2)))*(5*A + 2*B))/(4*f) - (tan(e/2 + (f*x)/2)^8*(4*A*a^3*c + 2*B*a^ 3*c) - tan(e/2 + (f*x)/2)*((3*A*a^3*c)/4 - (B*a^3*c)/2) - tan(e/2 + (f*x)/ 2)^3*((7*A*a^3*c)/2 + 3*B*a^3*c) + tan(e/2 + (f*x)/2)^7*((7*A*a^3*c)/2 + 3 *B*a^3*c) + tan(e/2 + (f*x)/2)^9*((3*A*a^3*c)/4 - (B*a^3*c)/2) + tan(e/2 + (f*x)/2)^6*(8*A*a^3*c + 8*B*a^3*c) + tan(e/2 + (f*x)/2)^2*((8*A*a^3*c)/3 + (8*B*a^3*c)/3) + tan(e/2 + (f*x)/2)^4*((16*A*a^3*c)/3 + (4*B*a^3*c)/3) + (4*A*a^3*c)/3 + (14*B*a^3*c)/15)/(f*(5*tan(e/2 + (f*x)/2)^2 + 10*tan(e/2 + (f*x)/2)^4 + 10*tan(e/2 + (f*x)/2)^6 + 5*tan(e/2 + (f*x)/2)^8 + tan(e/2 + (f*x)/2)^10 + 1)) - (a^3*c*(5*A + 2*B)*(atan(tan(e/2 + (f*x)/2)) - (f*x) /2))/(4*f)