Integrand size = 25, antiderivative size = 144 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^2 \, dx=\frac {27}{8} \left (20 c^2+30 c d+13 d^2\right ) x-\frac {108 (c+d)^2 \cos (e+f x)}{f}+\frac {9 \left (c^2+6 c d+5 d^2\right ) \cos ^3(e+f x)}{f}-\frac {27 d^2 \cos ^5(e+f x)}{5 f}-\frac {27 \left (12 c^2+30 c d+13 d^2\right ) \cos (e+f x) \sin (e+f x)}{8 f}-\frac {27 d (2 c+3 d) \cos (e+f x) \sin ^3(e+f x)}{4 f} \]
1/8*a^3*(20*c^2+30*c*d+13*d^2)*x-4*a^3*(c+d)^2*cos(f*x+e)/f+1/3*a^3*(c^2+6 *c*d+5*d^2)*cos(f*x+e)^3/f-1/5*a^3*d^2*cos(f*x+e)^5/f-1/8*a^3*(12*c^2+30*c *d+13*d^2)*cos(f*x+e)*sin(f*x+e)/f-1/4*a^3*d*(2*c+3*d)*cos(f*x+e)*sin(f*x+ e)^3/f
Time = 0.57 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.11 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^2 \, dx=\frac {9 \cos (e+f x) \left (-8 \left (55 c^2+90 c d+38 d^2\right )-\frac {30 \left (20 c^2+30 c d+13 d^2\right ) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(e+f x)}}-15 \left (12 c^2+30 c d+13 d^2\right ) \sin (e+f x)-8 \left (5 c^2+30 c d+19 d^2\right ) \sin ^2(e+f x)-30 d (2 c+3 d) \sin ^3(e+f x)-24 d^2 \sin ^4(e+f x)\right )}{40 f} \]
(9*Cos[e + f*x]*(-8*(55*c^2 + 90*c*d + 38*d^2) - (30*(20*c^2 + 30*c*d + 13 *d^2)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]])/Sqrt[Cos[e + f*x]^2] - 15*(1 2*c^2 + 30*c*d + 13*d^2)*Sin[e + f*x] - 8*(5*c^2 + 30*c*d + 19*d^2)*Sin[e + f*x]^2 - 30*d*(2*c + 3*d)*Sin[e + f*x]^3 - 24*d^2*Sin[e + f*x]^4))/(40*f )
Time = 0.56 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 3240, 3042, 3230, 3042, 3124, 2009}
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.
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2dx\) |
| \(\Big \downarrow \) 3240 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^3 \left (a \left (5 c^2+4 d^2\right )+a (10 c-d) d \sin (e+f x)\right )dx}{5 a}-\frac {d^2 \cos (e+f x) (a \sin (e+f x)+a)^4}{5 a f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^3 \left (a \left (5 c^2+4 d^2\right )+a (10 c-d) d \sin (e+f x)\right )dx}{5 a}-\frac {d^2 \cos (e+f x) (a \sin (e+f x)+a)^4}{5 a f}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {1}{4} a \left (20 c^2+30 c d+13 d^2\right ) \int (\sin (e+f x) a+a)^3dx-\frac {a d (10 c-d) \cos (e+f x) (a \sin (e+f x)+a)^3}{4 f}}{5 a}-\frac {d^2 \cos (e+f x) (a \sin (e+f x)+a)^4}{5 a f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} a \left (20 c^2+30 c d+13 d^2\right ) \int (\sin (e+f x) a+a)^3dx-\frac {a d (10 c-d) \cos (e+f x) (a \sin (e+f x)+a)^3}{4 f}}{5 a}-\frac {d^2 \cos (e+f x) (a \sin (e+f x)+a)^4}{5 a f}\) |
| \(\Big \downarrow \) 3124 |
| \(\displaystyle \frac {\frac {1}{4} a \left (20 c^2+30 c d+13 d^2\right ) \int \left (\sin ^3(e+f x) a^3+3 \sin ^2(e+f x) a^3+3 \sin (e+f x) a^3+a^3\right )dx-\frac {a d (10 c-d) \cos (e+f x) (a \sin (e+f x)+a)^3}{4 f}}{5 a}-\frac {d^2 \cos (e+f x) (a \sin (e+f x)+a)^4}{5 a f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{4} a \left (20 c^2+30 c d+13 d^2\right ) \left (\frac {a^3 \cos ^3(e+f x)}{3 f}-\frac {4 a^3 \cos (e+f x)}{f}-\frac {3 a^3 \sin (e+f x) \cos (e+f x)}{2 f}+\frac {5 a^3 x}{2}\right )-\frac {a d (10 c-d) \cos (e+f x) (a \sin (e+f x)+a)^3}{4 f}}{5 a}-\frac {d^2 \cos (e+f x) (a \sin (e+f x)+a)^4}{5 a f}\) |
-1/5*(d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^4)/(a*f) + (-1/4*(a*(10*c - d) *d*Cos[e + f*x]*(a + a*Sin[e + f*x])^3)/f + (a*(20*c^2 + 30*c*d + 13*d^2)* ((5*a^3*x)/2 - (4*a^3*Cos[e + f*x])/f + (a^3*Cos[e + f*x]^3)/(3*f) - (3*a^ 3*Cos[e + f*x]*Sin[e + f*x])/(2*f)))/4)/(5*a)
3.5.45.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTri g[(a + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n, 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[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^2, x_Symbol] :> Simp[(-d^2)*Cos[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[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ [a^2 - b^2, 0] && !LtQ[m, -1]
Time = 2.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.95
| method | result | size |
| parallelrisch | \(-\frac {3 \left (\left (-\frac {1}{9} c^{2}-\frac {2}{3} c d -\frac {17}{36} d^{2}\right ) \cos \left (3 f x +3 e \right )+\left (c +\frac {2 d}{3}\right ) \left (c +2 d \right ) \sin \left (2 f x +2 e \right )-\frac {\left (c +\frac {3 d}{2}\right ) d \sin \left (4 f x +4 e \right )}{12}+\frac {d^{2} \cos \left (5 f x +5 e \right )}{60}+\left (5 c^{2}+\frac {26}{3} c d +\frac {23}{6} d^{2}\right ) \cos \left (f x +e \right )+\left (-\frac {13 f x}{6}+\frac {152}{45}\right ) d^{2}+\left (-5 f x +8\right ) c d -\frac {10 \left (f x -\frac {22}{15}\right ) c^{2}}{3}\right ) a^{3}}{4 f}\) | \(137\) |
| parts | \(a^{3} c^{2} x +\frac {\left (2 a^{3} c d +3 a^{3} d^{2}\right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {\left (3 a^{3} c^{2}+2 a^{3} c d \right ) \cos \left (f x +e \right )}{f}-\frac {\left (a^{3} c^{2}+6 a^{3} c d +3 a^{3} d^{2}\right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {\left (3 a^{3} c^{2}+6 a^{3} c d +a^{3} d^{2}\right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {a^{3} d^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}\) | \(220\) |
| risch | \(\frac {5 a^{3} c^{2} x}{2}+\frac {15 a^{3} c d x}{4}+\frac {13 a^{3} d^{2} x}{8}-\frac {15 c^{2} a^{3} \cos \left (f x +e \right )}{4 f}-\frac {13 a^{3} \cos \left (f x +e \right ) c d}{2 f}-\frac {23 a^{3} \cos \left (f x +e \right ) d^{2}}{8 f}-\frac {a^{3} d^{2} \cos \left (5 f x +5 e \right )}{80 f}+\frac {\sin \left (4 f x +4 e \right ) a^{3} c d}{16 f}+\frac {3 \sin \left (4 f x +4 e \right ) a^{3} d^{2}}{32 f}+\frac {a^{3} \cos \left (3 f x +3 e \right ) c^{2}}{12 f}+\frac {a^{3} \cos \left (3 f x +3 e \right ) c d}{2 f}+\frac {17 a^{3} \cos \left (3 f x +3 e \right ) d^{2}}{48 f}-\frac {3 \sin \left (2 f x +2 e \right ) a^{3} c^{2}}{4 f}-\frac {2 \sin \left (2 f x +2 e \right ) a^{3} c d}{f}-\frac {\sin \left (2 f x +2 e \right ) a^{3} d^{2}}{f}\) | \(255\) |
| derivativedivides | \(\frac {-\frac {a^{3} c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 a^{3} c d \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{3} d^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+3 a^{3} c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 a^{3} c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a^{3} d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-3 a^{3} c^{2} \cos \left (f x +e \right )+6 a^{3} c d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{3} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+a^{3} c^{2} \left (f x +e \right )-2 a^{3} c d \cos \left (f x +e \right )+a^{3} d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\) | \(319\) |
| default | \(\frac {-\frac {a^{3} c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 a^{3} c d \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{3} d^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+3 a^{3} c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 a^{3} c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a^{3} d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-3 a^{3} c^{2} \cos \left (f x +e \right )+6 a^{3} c d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{3} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+a^{3} c^{2} \left (f x +e \right )-2 a^{3} c d \cos \left (f x +e \right )+a^{3} d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\) | \(319\) |
| norman | \(\frac {\left (\frac {5}{2} a^{3} c^{2}+\frac {15}{4} a^{3} c d +\frac {13}{8} a^{3} d^{2}\right ) x +\left (25 a^{3} c^{2}+\frac {75}{2} a^{3} c d +\frac {65}{4} a^{3} d^{2}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (25 a^{3} c^{2}+\frac {75}{2} a^{3} c d +\frac {65}{4} a^{3} d^{2}\right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {5}{2} a^{3} c^{2}+\frac {15}{4} a^{3} c d +\frac {13}{8} a^{3} d^{2}\right ) x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {25}{2} a^{3} c^{2}+\frac {75}{4} a^{3} c d +\frac {65}{8} a^{3} d^{2}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {25}{2} a^{3} c^{2}+\frac {75}{4} a^{3} c d +\frac {65}{8} a^{3} d^{2}\right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {110 a^{3} c^{2}+180 a^{3} c d +76 a^{3} d^{2}}{15 f}-\frac {\left (6 a^{3} c^{2}+4 a^{3} c d \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (14 a^{3} c^{2}+20 a^{3} c d +6 a^{3} d^{2}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (68 a^{3} c^{2}+120 a^{3} c d +58 a^{3} d^{2}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {\left (92 a^{3} c^{2}+168 a^{3} c d +76 a^{3} d^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {a^{3} \left (12 c^{2}+30 c d +13 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {a^{3} \left (12 c^{2}+30 c d +13 d^{2}\right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {a^{3} \left (12 c^{2}+38 c d +25 d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}+\frac {a^{3} \left (12 c^{2}+38 c d +25 d^{2}\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}}\) | \(544\) |
-3/4*((-1/9*c^2-2/3*c*d-17/36*d^2)*cos(3*f*x+3*e)+(c+2/3*d)*(c+2*d)*sin(2* f*x+2*e)-1/12*(c+3/2*d)*d*sin(4*f*x+4*e)+1/60*d^2*cos(5*f*x+5*e)+(5*c^2+26 /3*c*d+23/6*d^2)*cos(f*x+e)+(-13/6*f*x+152/45)*d^2+(-5*f*x+8)*c*d-10/3*(f* x-22/15)*c^2)*a^3/f
Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.25 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^2 \, dx=-\frac {24 \, a^{3} d^{2} \cos \left (f x + e\right )^{5} - 40 \, {\left (a^{3} c^{2} + 6 \, a^{3} c d + 5 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (20 \, a^{3} c^{2} + 30 \, a^{3} c d + 13 \, a^{3} d^{2}\right )} f x + 480 \, {\left (a^{3} c^{2} + 2 \, a^{3} c d + a^{3} d^{2}\right )} \cos \left (f x + e\right ) - 15 \, {\left (2 \, {\left (2 \, a^{3} c d + 3 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (12 \, a^{3} c^{2} + 34 \, a^{3} c d + 19 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{120 \, f} \]
-1/120*(24*a^3*d^2*cos(f*x + e)^5 - 40*(a^3*c^2 + 6*a^3*c*d + 5*a^3*d^2)*c os(f*x + e)^3 - 15*(20*a^3*c^2 + 30*a^3*c*d + 13*a^3*d^2)*f*x + 480*(a^3*c ^2 + 2*a^3*c*d + a^3*d^2)*cos(f*x + e) - 15*(2*(2*a^3*c*d + 3*a^3*d^2)*cos (f*x + e)^3 - (12*a^3*c^2 + 34*a^3*c*d + 19*a^3*d^2)*cos(f*x + e))*sin(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 702 vs. \(2 (153) = 306\).
Time = 0.34 (sec) , antiderivative size = 702, normalized size of antiderivative = 4.88 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^2 \, dx=\begin {cases} \frac {3 a^{3} c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 a^{3} c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{3} c^{2} x - \frac {a^{3} c^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a^{3} c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a^{3} c^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {3 a^{3} c^{2} \cos {\left (e + f x \right )}}{f} + \frac {3 a^{3} c d x \sin ^{4}{\left (e + f x \right )}}{4} + \frac {3 a^{3} c d x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} + 3 a^{3} c d x \sin ^{2}{\left (e + f x \right )} + \frac {3 a^{3} c d x \cos ^{4}{\left (e + f x \right )}}{4} + 3 a^{3} c d x \cos ^{2}{\left (e + f x \right )} - \frac {5 a^{3} c d \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f} - \frac {6 a^{3} c d \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a^{3} c d \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{4 f} - \frac {3 a^{3} c d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {4 a^{3} c d \cos ^{3}{\left (e + f x \right )}}{f} - \frac {2 a^{3} c d \cos {\left (e + f x \right )}}{f} + \frac {9 a^{3} d^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {9 a^{3} d^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {a^{3} d^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {9 a^{3} d^{2} x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {a^{3} d^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {a^{3} d^{2} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {15 a^{3} d^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {4 a^{3} d^{2} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {3 a^{3} d^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {9 a^{3} d^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {a^{3} d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {8 a^{3} d^{2} \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {2 a^{3} d^{2} \cos ^{3}{\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (c + d \sin {\left (e \right )}\right )^{2} \left (a \sin {\left (e \right )} + a\right )^{3} & \text {otherwise} \end {cases} \]
Piecewise((3*a**3*c**2*x*sin(e + f*x)**2/2 + 3*a**3*c**2*x*cos(e + f*x)**2 /2 + a**3*c**2*x - a**3*c**2*sin(e + f*x)**2*cos(e + f*x)/f - 3*a**3*c**2* sin(e + f*x)*cos(e + f*x)/(2*f) - 2*a**3*c**2*cos(e + f*x)**3/(3*f) - 3*a* *3*c**2*cos(e + f*x)/f + 3*a**3*c*d*x*sin(e + f*x)**4/4 + 3*a**3*c*d*x*sin (e + f*x)**2*cos(e + f*x)**2/2 + 3*a**3*c*d*x*sin(e + f*x)**2 + 3*a**3*c*d *x*cos(e + f*x)**4/4 + 3*a**3*c*d*x*cos(e + f*x)**2 - 5*a**3*c*d*sin(e + f *x)**3*cos(e + f*x)/(4*f) - 6*a**3*c*d*sin(e + f*x)**2*cos(e + f*x)/f - 3* a**3*c*d*sin(e + f*x)*cos(e + f*x)**3/(4*f) - 3*a**3*c*d*sin(e + f*x)*cos( e + f*x)/f - 4*a**3*c*d*cos(e + f*x)**3/f - 2*a**3*c*d*cos(e + f*x)/f + 9* a**3*d**2*x*sin(e + f*x)**4/8 + 9*a**3*d**2*x*sin(e + f*x)**2*cos(e + f*x) **2/4 + a**3*d**2*x*sin(e + f*x)**2/2 + 9*a**3*d**2*x*cos(e + f*x)**4/8 + a**3*d**2*x*cos(e + f*x)**2/2 - a**3*d**2*sin(e + f*x)**4*cos(e + f*x)/f - 15*a**3*d**2*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 4*a**3*d**2*sin(e + f*x )**2*cos(e + f*x)**3/(3*f) - 3*a**3*d**2*sin(e + f*x)**2*cos(e + f*x)/f - 9*a**3*d**2*sin(e + f*x)*cos(e + f*x)**3/(8*f) - a**3*d**2*sin(e + f*x)*co s(e + f*x)/(2*f) - 8*a**3*d**2*cos(e + f*x)**5/(15*f) - 2*a**3*d**2*cos(e + f*x)**3/f, Ne(f, 0)), (x*(c + d*sin(e))**2*(a*sin(e) + a)**3, True))
Time = 0.21 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.14 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^2 \, dx=\frac {160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c^{2} + 360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{2} + 480 \, {\left (f x + e\right )} a^{3} c^{2} + 960 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c d + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c d + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c d - 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{3} d^{2} + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} d^{2} + 45 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} d^{2} + 120 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} d^{2} - 1440 \, a^{3} c^{2} \cos \left (f x + e\right ) - 960 \, a^{3} c d \cos \left (f x + e\right )}{480 \, f} \]
1/480*(160*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^3*c^2 + 360*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^3*c^2 + 480*(f*x + e)*a^3*c^2 + 960*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^3*c*d + 30*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f *x + 2*e))*a^3*c*d + 720*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^3*c*d - 32*(3* cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^3*d^2 + 480*(cos(f *x + e)^3 - 3*cos(f*x + e))*a^3*d^2 + 45*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^3*d^2 + 120*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^3* d^2 - 1440*a^3*c^2*cos(f*x + e) - 960*a^3*c*d*cos(f*x + e))/f
Time = 0.48 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.69 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^2 \, dx=-\frac {a^{3} d^{2} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} - \frac {2 \, a^{3} c d \cos \left (f x + e\right )}{f} - \frac {a^{3} d^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {3}{8} \, {\left (4 \, a^{3} c^{2} + 10 \, a^{3} c d + 3 \, a^{3} d^{2}\right )} x + \frac {1}{2} \, {\left (2 \, a^{3} c^{2} + a^{3} d^{2}\right )} x + \frac {{\left (4 \, a^{3} c^{2} + 24 \, a^{3} c d + 17 \, a^{3} d^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (30 \, a^{3} c^{2} + 36 \, a^{3} c d + 23 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {{\left (2 \, a^{3} c d + 3 \, a^{3} d^{2}\right )} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac {{\left (3 \, a^{3} c^{2} + 8 \, a^{3} c d + 3 \, a^{3} d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
-1/80*a^3*d^2*cos(5*f*x + 5*e)/f - 2*a^3*c*d*cos(f*x + e)/f - 1/4*a^3*d^2* sin(2*f*x + 2*e)/f + 3/8*(4*a^3*c^2 + 10*a^3*c*d + 3*a^3*d^2)*x + 1/2*(2*a ^3*c^2 + a^3*d^2)*x + 1/48*(4*a^3*c^2 + 24*a^3*c*d + 17*a^3*d^2)*cos(3*f*x + 3*e)/f - 1/8*(30*a^3*c^2 + 36*a^3*c*d + 23*a^3*d^2)*cos(f*x + e)/f + 1/ 32*(2*a^3*c*d + 3*a^3*d^2)*sin(4*f*x + 4*e)/f - 1/4*(3*a^3*c^2 + 8*a^3*c*d + 3*a^3*d^2)*sin(2*f*x + 2*e)/f
Time = 8.51 (sec) , antiderivative size = 493, normalized size of antiderivative = 3.42 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^2 \, dx=\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (20\,c^2+30\,c\,d+13\,d^2\right )}{4\,\left (5\,a^3\,c^2+\frac {15\,a^3\,c\,d}{2}+\frac {13\,a^3\,d^2}{4}\right )}\right )\,\left (20\,c^2+30\,c\,d+13\,d^2\right )}{4\,f}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (3\,a^3\,c^2+\frac {15\,a^3\,c\,d}{2}+\frac {13\,a^3\,d^2}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (3\,a^3\,c^2+\frac {15\,a^3\,c\,d}{2}+\frac {13\,a^3\,d^2}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (6\,a^3\,c^2+19\,a^3\,c\,d+\frac {25\,a^3\,d^2}{2}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (6\,a^3\,c^2+19\,a^3\,c\,d+\frac {25\,a^3\,d^2}{2}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (28\,a^3\,c^2+40\,a^3\,c\,d+12\,a^3\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {92\,a^3\,c^2}{3}+56\,a^3\,c\,d+\frac {76\,a^3\,d^2}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {136\,a^3\,c^2}{3}+80\,a^3\,c\,d+\frac {116\,a^3\,d^2}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (6\,a^3\,c^2+4\,d\,a^3\,c\right )+\frac {22\,a^3\,c^2}{3}+\frac {76\,a^3\,d^2}{15}+12\,a^3\,c\,d}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {a^3\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )\,\left (20\,c^2+30\,c\,d+13\,d^2\right )}{4\,f} \]
(a^3*atan((a^3*tan(e/2 + (f*x)/2)*(30*c*d + 20*c^2 + 13*d^2))/(4*(5*a^3*c^ 2 + (13*a^3*d^2)/4 + (15*a^3*c*d)/2)))*(30*c*d + 20*c^2 + 13*d^2))/(4*f) - (tan(e/2 + (f*x)/2)*(3*a^3*c^2 + (13*a^3*d^2)/4 + (15*a^3*c*d)/2) - tan(e /2 + (f*x)/2)^9*(3*a^3*c^2 + (13*a^3*d^2)/4 + (15*a^3*c*d)/2) + tan(e/2 + (f*x)/2)^3*(6*a^3*c^2 + (25*a^3*d^2)/2 + 19*a^3*c*d) - tan(e/2 + (f*x)/2)^ 7*(6*a^3*c^2 + (25*a^3*d^2)/2 + 19*a^3*c*d) + tan(e/2 + (f*x)/2)^6*(28*a^3 *c^2 + 12*a^3*d^2 + 40*a^3*c*d) + tan(e/2 + (f*x)/2)^2*((92*a^3*c^2)/3 + ( 76*a^3*d^2)/3 + 56*a^3*c*d) + tan(e/2 + (f*x)/2)^4*((136*a^3*c^2)/3 + (116 *a^3*d^2)/3 + 80*a^3*c*d) + tan(e/2 + (f*x)/2)^8*(6*a^3*c^2 + 4*a^3*c*d) + (22*a^3*c^2)/3 + (76*a^3*d^2)/15 + 12*a^3*c*d)/(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*(atan(tan(e/2 + (f*x)/2)) - (f*x) /2)*(30*c*d + 20*c^2 + 13*d^2))/(4*f)