Integrand size = 33, antiderivative size = 213 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {1}{8} a \left (4 A \left (2 c^2+2 c d+d^2\right )+B \left (4 c^2+8 c d+3 d^2\right )\right ) x-\frac {a \left (4 A d \left (c^2+3 c d+d^2\right )-B \left (c^3-4 c^2 d-8 c d^2-4 d^3\right )\right ) \cos (e+f x)}{6 d f}-\frac {a \left (3 (4 A+3 B) d^2-2 c (B c-4 (A+B) d)\right ) \cos (e+f x) \sin (e+f x)}{24 f}+\frac {a (B c-4 (A+B) d) \cos (e+f x) (c+d \sin (e+f x))^2}{12 d f}-\frac {a B \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f} \]
1/8*a*(4*A*(2*c^2+2*c*d+d^2)+B*(4*c^2+8*c*d+3*d^2))*x-1/6*a*(4*A*d*(c^2+3* c*d+d^2)-B*(c^3-4*c^2*d-8*c*d^2-4*d^3))*cos(f*x+e)/d/f-1/24*a*(3*(4*A+3*B) *d^2-2*c*(B*c-4*(A+B)*d))*cos(f*x+e)*sin(f*x+e)/f+1/12*a*(B*c-4*(A+B)*d)*c os(f*x+e)*(c+d*sin(f*x+e))^2/d/f-1/4*a*B*cos(f*x+e)*(c+d*sin(f*x+e))^3/d/f
Time = 1.64 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.87 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {a (1+\sin (e+f x)) \left (-24 \left (B \left (4 c^2+6 c d+3 d^2\right )+A \left (4 c^2+8 c d+3 d^2\right )\right ) \cos (e+f x)+8 d (A d+B (2 c+d)) \cos (3 (e+f x))+3 \left (4 \left (4 A \left (2 c^2+2 c d+d^2\right )+B \left (4 c^2+8 c d+3 d^2\right )\right ) f x-8 \left (B (c+d)^2+A d (2 c+d)\right ) \sin (2 (e+f x))+B d^2 \sin (4 (e+f x))\right )\right )}{96 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2} \]
(a*(1 + Sin[e + f*x])*(-24*(B*(4*c^2 + 6*c*d + 3*d^2) + A*(4*c^2 + 8*c*d + 3*d^2))*Cos[e + f*x] + 8*d*(A*d + B*(2*c + d))*Cos[3*(e + f*x)] + 3*(4*(4 *A*(2*c^2 + 2*c*d + d^2) + B*(4*c^2 + 8*c*d + 3*d^2))*f*x - 8*(B*(c + d)^2 + A*d*(2*c + d))*Sin[2*(e + f*x)] + B*d^2*Sin[4*(e + f*x)])))/(96*f*(Cos[ (e + f*x)/2] + Sin[(e + f*x)/2])^2)
Time = 0.75 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3042, 3447, 3042, 3502, 3042, 3232, 3042, 3213}
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.
| \(\displaystyle \int (a \sin (e+f x)+a) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2dx\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \int (c+d \sin (e+f x))^2 \left ((a A+a B) \sin (e+f x)+a A+a B \sin ^2(e+f x)\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d \sin (e+f x))^2 \left ((a A+a B) \sin (e+f x)+a A+a B \sin (e+f x)^2\right )dx\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\int (c+d \sin (e+f x))^2 (a (4 A+3 B) d-a (B c-4 (A+B) d) \sin (e+f x))dx}{4 d}-\frac {a B \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (c+d \sin (e+f x))^2 (a (4 A+3 B) d-a (B c-4 (A+B) d) \sin (e+f x))dx}{4 d}-\frac {a B \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{3} \int (c+d \sin (e+f x)) \left (a d (12 A c+7 B c+8 A d+8 B d)-a \left (2 B c^2-8 (A+B) d c-3 (4 A+3 B) d^2\right ) \sin (e+f x)\right )dx+\frac {a (B c-4 d (A+B)) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}}{4 d}-\frac {a B \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \int (c+d \sin (e+f x)) \left (a d (12 A c+7 B c+8 A d+8 B d)-a \left (2 B c^2-8 (A+B) d c-3 (4 A+3 B) d^2\right ) \sin (e+f x)\right )dx+\frac {a (B c-4 d (A+B)) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}}{4 d}-\frac {a B \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {a d \left (-8 c d (A+B)-3 d^2 (4 A+3 B)+2 B c^2\right ) \sin (e+f x) \cos (e+f x)}{2 f}+\frac {3}{2} a d x \left (4 A \left (2 c^2+2 c d+d^2\right )+B \left (4 c^2+8 c d+3 d^2\right )\right )-\frac {2 a \left (4 A d \left (c^2+3 c d+d^2\right )-B \left (c^3-4 c^2 d-8 c d^2-4 d^3\right )\right ) \cos (e+f x)}{f}\right )+\frac {a (B c-4 d (A+B)) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}}{4 d}-\frac {a B \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f}\) |
-1/4*(a*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(d*f) + ((a*(B*c - 4*(A + B )*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*f) + ((3*a*d*(4*A*(2*c^2 + 2* c*d + d^2) + B*(4*c^2 + 8*c*d + 3*d^2))*x)/2 - (2*a*(4*A*d*(c^2 + 3*c*d + d^2) - B*(c^3 - 4*c^2*d - 8*c*d^2 - 4*d^3))*Cos[e + f*x])/f + (a*d*(2*B*c^ 2 - 8*(A + B)*c*d - 3*(4*A + 3*B)*d^2)*Cos[e + f*x]*Sin[e + f*x])/(2*f))/3 )/(4*d)
3.3.45.3.1 Defintions of rubi rules used
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)*(Co s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
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] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[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]
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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[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]
Time = 1.12 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.81
| method | result | size |
| parts | \(-\frac {\left (A a \,d^{2}+2 B a c d +d^{2} B a \right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}-\frac {\left (a \,c^{2} A +2 A a c d +B a \,c^{2}\right ) \cos \left (f x +e \right )}{f}+\frac {\left (2 A a c d +A a \,d^{2}+B a \,c^{2}+2 B a c d \right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+A a \,c^{2} x +\frac {d^{2} B a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\) | \(172\) |
| parallelrisch | \(\frac {\left (\left (\left (-3 B -3 A \right ) d^{2}-6 d c \left (A +B \right )-3 B \,c^{2}\right ) \sin \left (2 f x +2 e \right )+\left (2 B c +\left (A +B \right ) d \right ) d \cos \left (3 f x +3 e \right )+\frac {3 d^{2} B \sin \left (4 f x +4 e \right )}{8}+\left (\left (-9 A -9 B \right ) d^{2}-24 c \left (A +\frac {3 B}{4}\right ) d -12 c^{2} \left (A +B \right )\right ) \cos \left (f x +e \right )+\left (-8 B +6 f x A +\frac {9}{2} f x B -8 A \right ) d^{2}+12 c \left (f x A +f x B -2 A -\frac {4}{3} B \right ) d +12 c^{2} \left (f x A +\frac {1}{2} f x B -A -B \right )\right ) a}{12 f}\) | \(176\) |
| derivativedivides | \(\frac {-a \,c^{2} A \cos \left (f x +e \right )+2 A a c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {A a \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+B a \,c^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 B a c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+d^{2} B a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+a \,c^{2} A \left (f x +e \right )-2 A a c d \cos \left (f x +e \right )+A a \,d^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B a \,c^{2} \cos \left (f x +e \right )+2 B a c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {d^{2} B a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\) | \(274\) |
| default | \(\frac {-a \,c^{2} A \cos \left (f x +e \right )+2 A a c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {A a \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+B a \,c^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 B a c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+d^{2} B a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+a \,c^{2} A \left (f x +e \right )-2 A a c d \cos \left (f x +e \right )+A a \,d^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B a \,c^{2} \cos \left (f x +e \right )+2 B a c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {d^{2} B a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\) | \(274\) |
| risch | \(A a \,c^{2} x +A a c d x +\frac {A a \,d^{2} x}{2}+\frac {B a \,c^{2} x}{2}+B a c d x +\frac {3 B a \,d^{2} x}{8}-\frac {a \cos \left (f x +e \right ) A \,c^{2}}{f}-\frac {2 a \cos \left (f x +e \right ) A c d}{f}-\frac {3 a \cos \left (f x +e \right ) A \,d^{2}}{4 f}-\frac {a \cos \left (f x +e \right ) B \,c^{2}}{f}-\frac {3 a \cos \left (f x +e \right ) c d B}{2 f}-\frac {3 a \cos \left (f x +e \right ) d^{2} B}{4 f}+\frac {d^{2} B a \sin \left (4 f x +4 e \right )}{32 f}+\frac {d^{2} a \cos \left (3 f x +3 e \right ) A}{12 f}+\frac {d a \cos \left (3 f x +3 e \right ) B c}{6 f}+\frac {d^{2} a \cos \left (3 f x +3 e \right ) B}{12 f}-\frac {\sin \left (2 f x +2 e \right ) A a c d}{2 f}-\frac {\sin \left (2 f x +2 e \right ) A a \,d^{2}}{4 f}-\frac {\sin \left (2 f x +2 e \right ) B a \,c^{2}}{4 f}-\frac {\sin \left (2 f x +2 e \right ) B a c d}{2 f}-\frac {\sin \left (2 f x +2 e \right ) d^{2} B a}{4 f}\) | \(307\) |
| norman | \(\frac {\left (a \,c^{2} A +A a c d +\frac {1}{2} A a \,d^{2}+\frac {1}{2} B a \,c^{2}+B a c d +\frac {3}{8} d^{2} B a \right ) x +\left (a \,c^{2} A +A a c d +\frac {1}{2} A a \,d^{2}+\frac {1}{2} B a \,c^{2}+B a c d +\frac {3}{8} d^{2} B a \right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (4 a \,c^{2} A +4 A a c d +2 A a \,d^{2}+2 B a \,c^{2}+4 B a c d +\frac {3}{2} d^{2} B a \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (4 a \,c^{2} A +4 A a c d +2 A a \,d^{2}+2 B a \,c^{2}+4 B a c d +\frac {3}{2} d^{2} B a \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (6 a \,c^{2} A +6 A a c d +3 A a \,d^{2}+3 B a \,c^{2}+6 B a c d +\frac {9}{4} d^{2} B a \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {6 a \,c^{2} A +12 A a c d +4 A a \,d^{2}+6 B a \,c^{2}+8 B a c d +4 d^{2} B a}{3 f}-\frac {2 \left (a \,c^{2} A +2 A a c d +B a \,c^{2}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (3 a \,c^{2} A +6 A a c d +2 A a \,d^{2}+3 B a \,c^{2}+4 B a c d +2 d^{2} B a \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (9 a \,c^{2} A +18 A a c d +8 A a \,d^{2}+9 B a \,c^{2}+16 B a c d +8 d^{2} B a \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {a \left (8 A c d +4 A \,d^{2}+4 B \,c^{2}+8 c d B +3 d^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {a \left (8 A c d +4 A \,d^{2}+4 B \,c^{2}+8 c d B +3 d^{2} B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {a \left (8 A c d +4 A \,d^{2}+4 B \,c^{2}+8 c d B +11 d^{2} B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {a \left (8 A c d +4 A \,d^{2}+4 B \,c^{2}+8 c d B +11 d^{2} B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4}}\) | \(648\) |
-1/3*(A*a*d^2+2*B*a*c*d+B*a*d^2)/f*(2+sin(f*x+e)^2)*cos(f*x+e)-(A*a*c^2+2* A*a*c*d+B*a*c^2)/f*cos(f*x+e)+(2*A*a*c*d+A*a*d^2+B*a*c^2+2*B*a*c*d)/f*(-1/ 2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)+A*a*c^2*x+d^2*B*a/f*(-1/4*(sin(f*x+ e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)
Time = 0.27 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.75 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {8 \, {\left (2 \, B a c d + {\left (A + B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (4 \, {\left (2 \, A + B\right )} a c^{2} + 8 \, {\left (A + B\right )} a c d + {\left (4 \, A + 3 \, B\right )} a d^{2}\right )} f x - 24 \, {\left ({\left (A + B\right )} a c^{2} + 2 \, {\left (A + B\right )} a c d + {\left (A + B\right )} a d^{2}\right )} \cos \left (f x + e\right ) + 3 \, {\left (2 \, B a d^{2} \cos \left (f x + e\right )^{3} - {\left (4 \, B a c^{2} + 8 \, {\left (A + B\right )} a c d + {\left (4 \, A + 5 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \]
1/24*(8*(2*B*a*c*d + (A + B)*a*d^2)*cos(f*x + e)^3 + 3*(4*(2*A + B)*a*c^2 + 8*(A + B)*a*c*d + (4*A + 3*B)*a*d^2)*f*x - 24*((A + B)*a*c^2 + 2*(A + B) *a*c*d + (A + B)*a*d^2)*cos(f*x + e) + 3*(2*B*a*d^2*cos(f*x + e)^3 - (4*B* a*c^2 + 8*(A + B)*a*c*d + (4*A + 5*B)*a*d^2)*cos(f*x + e))*sin(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (201) = 402\).
Time = 0.23 (sec) , antiderivative size = 571, normalized size of antiderivative = 2.68 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\begin {cases} A a c^{2} x - \frac {A a c^{2} \cos {\left (e + f x \right )}}{f} + A a c d x \sin ^{2}{\left (e + f x \right )} + A a c d x \cos ^{2}{\left (e + f x \right )} - \frac {A a c d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 A a c d \cos {\left (e + f x \right )}}{f} + \frac {A a d^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {A a d^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {A a d^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {A a d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 A a d^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {B a c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {B a c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {B a c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {B a c^{2} \cos {\left (e + f x \right )}}{f} + B a c d x \sin ^{2}{\left (e + f x \right )} + B a c d x \cos ^{2}{\left (e + f x \right )} - \frac {2 B a c d \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {B a c d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {4 B a c d \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {3 B a d^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {3 B a d^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {3 B a d^{2} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {5 B a d^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {B a d^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 B a d^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {2 B a d^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\left (e \right )}\right ) \left (c + d \sin {\left (e \right )}\right )^{2} \left (a \sin {\left (e \right )} + a\right ) & \text {otherwise} \end {cases} \]
Piecewise((A*a*c**2*x - A*a*c**2*cos(e + f*x)/f + A*a*c*d*x*sin(e + f*x)** 2 + A*a*c*d*x*cos(e + f*x)**2 - A*a*c*d*sin(e + f*x)*cos(e + f*x)/f - 2*A* a*c*d*cos(e + f*x)/f + A*a*d**2*x*sin(e + f*x)**2/2 + A*a*d**2*x*cos(e + f *x)**2/2 - A*a*d**2*sin(e + f*x)**2*cos(e + f*x)/f - A*a*d**2*sin(e + f*x) *cos(e + f*x)/(2*f) - 2*A*a*d**2*cos(e + f*x)**3/(3*f) + B*a*c**2*x*sin(e + f*x)**2/2 + B*a*c**2*x*cos(e + f*x)**2/2 - B*a*c**2*sin(e + f*x)*cos(e + f*x)/(2*f) - B*a*c**2*cos(e + f*x)/f + B*a*c*d*x*sin(e + f*x)**2 + B*a*c* d*x*cos(e + f*x)**2 - 2*B*a*c*d*sin(e + f*x)**2*cos(e + f*x)/f - B*a*c*d*s in(e + f*x)*cos(e + f*x)/f - 4*B*a*c*d*cos(e + f*x)**3/(3*f) + 3*B*a*d**2* x*sin(e + f*x)**4/8 + 3*B*a*d**2*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 3*B *a*d**2*x*cos(e + f*x)**4/8 - 5*B*a*d**2*sin(e + f*x)**3*cos(e + f*x)/(8*f ) - B*a*d**2*sin(e + f*x)**2*cos(e + f*x)/f - 3*B*a*d**2*sin(e + f*x)*cos( e + f*x)**3/(8*f) - 2*B*a*d**2*cos(e + f*x)**3/(3*f), Ne(f, 0)), (x*(A + B *sin(e))*(c + d*sin(e))**2*(a*sin(e) + a), True))
Time = 0.23 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.24 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {96 \, {\left (f x + e\right )} A a c^{2} + 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a c^{2} + 48 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a c d + 64 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a c d + 48 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a c d + 32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a d^{2} + 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a d^{2} + 32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a d^{2} + 3 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a d^{2} - 96 \, A a c^{2} \cos \left (f x + e\right ) - 96 \, B a c^{2} \cos \left (f x + e\right ) - 192 \, A a c d \cos \left (f x + e\right )}{96 \, f} \]
1/96*(96*(f*x + e)*A*a*c^2 + 24*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a*c^2 + 48*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a*c*d + 64*(cos(f*x + e)^3 - 3*cos( f*x + e))*B*a*c*d + 48*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a*c*d + 32*(cos( f*x + e)^3 - 3*cos(f*x + e))*A*a*d^2 + 24*(2*f*x + 2*e - sin(2*f*x + 2*e)) *A*a*d^2 + 32*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a*d^2 + 3*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a*d^2 - 96*A*a*c^2*cos(f*x + e ) - 96*B*a*c^2*cos(f*x + e) - 192*A*a*c*d*cos(f*x + e))/f
Time = 0.30 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.91 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {B a d^{2} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {1}{8} \, {\left (8 \, A a c^{2} + 4 \, B a c^{2} + 8 \, A a c d + 8 \, B a c d + 4 \, A a d^{2} + 3 \, B a d^{2}\right )} x + \frac {{\left (2 \, B a c d + A a d^{2} + B a d^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {{\left (4 \, A a c^{2} + 4 \, B a c^{2} + 8 \, A a c d + 6 \, B a c d + 3 \, A a d^{2} + 3 \, B a d^{2}\right )} \cos \left (f x + e\right )}{4 \, f} - \frac {{\left (B a c^{2} + 2 \, A a c d + 2 \, B a c d + A a d^{2} + B a d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
1/32*B*a*d^2*sin(4*f*x + 4*e)/f + 1/8*(8*A*a*c^2 + 4*B*a*c^2 + 8*A*a*c*d + 8*B*a*c*d + 4*A*a*d^2 + 3*B*a*d^2)*x + 1/12*(2*B*a*c*d + A*a*d^2 + B*a*d^ 2)*cos(3*f*x + 3*e)/f - 1/4*(4*A*a*c^2 + 4*B*a*c^2 + 8*A*a*c*d + 6*B*a*c*d + 3*A*a*d^2 + 3*B*a*d^2)*cos(f*x + e)/f - 1/4*(B*a*c^2 + 2*A*a*c*d + 2*B* a*c*d + A*a*d^2 + B*a*d^2)*sin(2*f*x + 2*e)/f
Time = 15.70 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.57 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (8\,A\,c^2+4\,A\,d^2+4\,B\,c^2+3\,B\,d^2+8\,A\,c\,d+8\,B\,c\,d\right )}{4\,\left (2\,A\,a\,c^2+A\,a\,d^2+B\,a\,c^2+\frac {3\,B\,a\,d^2}{4}+2\,A\,a\,c\,d+2\,B\,a\,c\,d\right )}\right )\,\left (8\,A\,c^2+4\,A\,d^2+4\,B\,c^2+3\,B\,d^2+8\,A\,c\,d+8\,B\,c\,d\right )}{4\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (2\,A\,a\,c^2+2\,B\,a\,c^2+4\,A\,a\,c\,d\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (A\,a\,d^2+B\,a\,c^2+\frac {3\,B\,a\,d^2}{4}+2\,A\,a\,c\,d+2\,B\,a\,c\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (6\,A\,a\,c^2+4\,A\,a\,d^2+6\,B\,a\,c^2+4\,B\,a\,d^2+12\,A\,a\,c\,d+8\,B\,a\,c\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (6\,A\,a\,c^2+\frac {16\,A\,a\,d^2}{3}+6\,B\,a\,c^2+\frac {16\,B\,a\,d^2}{3}+12\,A\,a\,c\,d+\frac {32\,B\,a\,c\,d}{3}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (A\,a\,d^2+B\,a\,c^2+\frac {3\,B\,a\,d^2}{4}+2\,A\,a\,c\,d+2\,B\,a\,c\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (A\,a\,d^2+B\,a\,c^2+\frac {11\,B\,a\,d^2}{4}+2\,A\,a\,c\,d+2\,B\,a\,c\,d\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (A\,a\,d^2+B\,a\,c^2+\frac {11\,B\,a\,d^2}{4}+2\,A\,a\,c\,d+2\,B\,a\,c\,d\right )+2\,A\,a\,c^2+\frac {4\,A\,a\,d^2}{3}+2\,B\,a\,c^2+\frac {4\,B\,a\,d^2}{3}+4\,A\,a\,c\,d+\frac {8\,B\,a\,c\,d}{3}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
(a*atan((a*tan(e/2 + (f*x)/2)*(8*A*c^2 + 4*A*d^2 + 4*B*c^2 + 3*B*d^2 + 8*A *c*d + 8*B*c*d))/(4*(2*A*a*c^2 + A*a*d^2 + B*a*c^2 + (3*B*a*d^2)/4 + 2*A*a *c*d + 2*B*a*c*d)))*(8*A*c^2 + 4*A*d^2 + 4*B*c^2 + 3*B*d^2 + 8*A*c*d + 8*B *c*d))/(4*f) - (tan(e/2 + (f*x)/2)^6*(2*A*a*c^2 + 2*B*a*c^2 + 4*A*a*c*d) + tan(e/2 + (f*x)/2)*(A*a*d^2 + B*a*c^2 + (3*B*a*d^2)/4 + 2*A*a*c*d + 2*B*a *c*d) + tan(e/2 + (f*x)/2)^4*(6*A*a*c^2 + 4*A*a*d^2 + 6*B*a*c^2 + 4*B*a*d^ 2 + 12*A*a*c*d + 8*B*a*c*d) + tan(e/2 + (f*x)/2)^2*(6*A*a*c^2 + (16*A*a*d^ 2)/3 + 6*B*a*c^2 + (16*B*a*d^2)/3 + 12*A*a*c*d + (32*B*a*c*d)/3) - tan(e/2 + (f*x)/2)^7*(A*a*d^2 + B*a*c^2 + (3*B*a*d^2)/4 + 2*A*a*c*d + 2*B*a*c*d) + tan(e/2 + (f*x)/2)^3*(A*a*d^2 + B*a*c^2 + (11*B*a*d^2)/4 + 2*A*a*c*d + 2 *B*a*c*d) - tan(e/2 + (f*x)/2)^5*(A*a*d^2 + B*a*c^2 + (11*B*a*d^2)/4 + 2*A *a*c*d + 2*B*a*c*d) + 2*A*a*c^2 + (4*A*a*d^2)/3 + 2*B*a*c^2 + (4*B*a*d^2)/ 3 + 4*A*a*c*d + (8*B*a*c*d)/3)/(f*(4*tan(e/2 + (f*x)/2)^2 + 6*tan(e/2 + (f *x)/2)^4 + 4*tan(e/2 + (f*x)/2)^6 + tan(e/2 + (f*x)/2)^8 + 1))