Integrand size = 25, antiderivative size = 215 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx=\frac {27}{8} (2 c+d) \left (2 c^2+3 c d+2 d^2\right ) x+\frac {9 \left (c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right ) \cos (e+f x)}{10 d f}+\frac {9 \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right ) \cos (e+f x) \sin (e+f x)}{40 f}+\frac {9 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac {9 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac {9 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f} \]
3/8*a^2*(2*c+d)*(2*c^2+3*c*d+2*d^2)*x+1/10*a^2*(c^4-10*c^3*d-44*c^2*d^2-40 *c*d^3-12*d^4)*cos(f*x+e)/d/f+1/40*a^2*(2*c^3-20*c^2*d-57*c*d^2-30*d^3)*co s(f*x+e)*sin(f*x+e)/f+1/20*a^2*(c^2-10*c*d-12*d^2)*cos(f*x+e)*(c+d*sin(f*x +e))^2/d/f+1/20*a^2*(c-10*d)*cos(f*x+e)*(c+d*sin(f*x+e))^3/d/f-1/5*a^2*cos (f*x+e)*(c+d*sin(f*x+e))^4/d/f
Time = 0.73 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx=\frac {9 \cos (e+f x) \left (-8 \left (10 c^3+25 c^2 d+20 c d^2+6 d^3\right )-\frac {30 \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right ) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(e+f x)}}-5 \left (4 c^3+24 c^2 d+21 c d^2+6 d^3\right ) \sin (e+f x)-8 d \left (5 c^2+10 c d+3 d^2\right ) \sin ^2(e+f x)-10 d^2 (3 c+2 d) \sin ^3(e+f x)-8 d^3 \sin ^4(e+f x)\right )}{40 f} \]
(9*Cos[e + f*x]*(-8*(10*c^3 + 25*c^2*d + 20*c*d^2 + 6*d^3) - (30*(4*c^3 + 8*c^2*d + 7*c*d^2 + 2*d^3)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]])/Sqrt[Co s[e + f*x]^2] - 5*(4*c^3 + 24*c^2*d + 21*c*d^2 + 6*d^3)*Sin[e + f*x] - 8*d *(5*c^2 + 10*c*d + 3*d^2)*Sin[e + f*x]^2 - 10*d^2*(3*c + 2*d)*Sin[e + f*x] ^3 - 8*d^3*Sin[e + f*x]^4))/(40*f)
Time = 0.76 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 3242, 3042, 3232, 27, 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.
| \(\displaystyle \int (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^3dx\) |
| \(\Big \downarrow \) 3242 |
| \(\displaystyle \frac {\int \left (9 a^2 d-a^2 (c-10 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^3dx}{5 d}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (9 a^2 d-a^2 (c-10 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^3dx}{5 d}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{4} \int 3 (c+d \sin (e+f x))^2 \left (a^2 d (11 c+10 d)-a^2 \left (c^2-10 d c-12 d^2\right ) \sin (e+f x)\right )dx+\frac {a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3}{4} \int (c+d \sin (e+f x))^2 \left (a^2 d (11 c+10 d)-a^2 \left (c^2-10 d c-12 d^2\right ) \sin (e+f x)\right )dx+\frac {a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{4} \int (c+d \sin (e+f x))^2 \left (a^2 d (11 c+10 d)-a^2 \left (c^2-10 d c-12 d^2\right ) \sin (e+f x)\right )dx+\frac {a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{3} \int (c+d \sin (e+f x)) \left (a^2 d \left (31 c^2+50 d c+24 d^2\right )-a^2 \left (2 c^3-20 d c^2-57 d^2 c-30 d^3\right ) \sin (e+f x)\right )dx+\frac {a^2 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}\right )+\frac {a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{3} \int (c+d \sin (e+f x)) \left (a^2 d \left (31 c^2+50 d c+24 d^2\right )-a^2 \left (2 c^3-20 d c^2-57 d^2 c-30 d^3\right ) \sin (e+f x)\right )dx+\frac {a^2 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}\right )+\frac {a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {a^2 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}+\frac {1}{3} \left (\frac {15}{2} a^2 d x (2 c+d) \left (2 c^2+3 c d+2 d^2\right )+\frac {a^2 d \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right ) \sin (e+f x) \cos (e+f x)}{2 f}+\frac {2 a^2 \left (c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right ) \cos (e+f x)}{f}\right )\right )+\frac {a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}\) |
-1/5*(a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(d*f) + ((a^2*(c - 10*d)*Co s[e + f*x]*(c + d*Sin[e + f*x])^3)/(4*f) + (3*((a^2*(c^2 - 10*c*d - 12*d^2 )*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*f) + ((15*a^2*d*(2*c + d)*(2*c^2 + 3*c*d + 2*d^2)*x)/2 + (2*a^2*(c^4 - 10*c^3*d - 44*c^2*d^2 - 40*c*d^3 - 12*d^4)*Cos[e + f*x])/f + (a^2*d*(2*c^3 - 20*c^2*d - 57*c*d^2 - 30*d^3)*Co s[e + f*x]*Sin[e + f*x])/(2*f))/3))/4)/(5*d)
3.5.35.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c* (m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !LtQ[ n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[ c, 0]))
Time = 2.19 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.78
| method | result | size |
| parallelrisch | \(-\frac {a^{2} \left (\left (c^{3}+6 c^{2} d +6 c \,d^{2}+2 d^{3}\right ) \sin \left (2 f x +2 e \right )-\left (c +\frac {3 d}{2}\right ) d \left (c +\frac {d}{2}\right ) \cos \left (3 f x +3 e \right )-\frac {3 d^{2} \left (c +\frac {2 d}{3}\right ) \sin \left (4 f x +4 e \right )}{8}+\frac {d^{3} \cos \left (5 f x +5 e \right )}{20}+\left (8 c^{3}+21 c^{2} d +18 c \,d^{2}+\frac {11}{2} d^{3}\right ) \cos \left (f x +e \right )+\left (-3 f x +\frac {24}{5}\right ) d^{3}+c \left (-\frac {21 f x}{2}+16\right ) d^{2}+\left (-12 f x +20\right ) c^{2} d +\left (-6 f x +8\right ) c^{3}\right )}{4 f}\) | \(168\) |
| parts | \(a^{2} c^{3} x +\frac {\left (3 a^{2} c \,d^{2}+2 a^{2} d^{3}\right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {\left (2 a^{2} c^{3}+3 a^{2} c^{2} d \right ) \cos \left (f x +e \right )}{f}-\frac {\left (3 a^{2} c^{2} d +6 a^{2} c \,d^{2}+a^{2} d^{3}\right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {\left (a^{2} c^{3}+6 a^{2} c^{2} d +3 a^{2} c \,d^{2}\right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {a^{2} d^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}\) | \(230\) |
| risch | \(\frac {3 a^{2} c^{3} x}{2}+3 a^{2} c^{2} d x +\frac {21 a^{2} c \,d^{2} x}{8}+\frac {3 a^{2} d^{3} x}{4}-\frac {2 a^{2} \cos \left (f x +e \right ) c^{3}}{f}-\frac {21 a^{2} \cos \left (f x +e \right ) c^{2} d}{4 f}-\frac {9 a^{2} \cos \left (f x +e \right ) c \,d^{2}}{2 f}-\frac {11 a^{2} \cos \left (f x +e \right ) d^{3}}{8 f}-\frac {a^{2} d^{3} \cos \left (5 f x +5 e \right )}{80 f}+\frac {3 \sin \left (4 f x +4 e \right ) a^{2} c \,d^{2}}{32 f}+\frac {\sin \left (4 f x +4 e \right ) a^{2} d^{3}}{16 f}+\frac {a^{2} d \cos \left (3 f x +3 e \right ) c^{2}}{4 f}+\frac {a^{2} d^{2} \cos \left (3 f x +3 e \right ) c}{2 f}+\frac {3 a^{2} d^{3} \cos \left (3 f x +3 e \right )}{16 f}-\frac {\sin \left (2 f x +2 e \right ) a^{2} c^{3}}{4 f}-\frac {3 \sin \left (2 f x +2 e \right ) a^{2} c^{2} d}{2 f}-\frac {3 \sin \left (2 f x +2 e \right ) a^{2} c \,d^{2}}{2 f}-\frac {\sin \left (2 f x +2 e \right ) a^{2} d^{3}}{2 f}\) | \(315\) |
| derivativedivides | \(\frac {a^{2} c^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{2} c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a^{2} c \,d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{2} d^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-2 a^{2} c^{3} \cos \left (f x +e \right )+6 a^{2} c^{2} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 a^{2} c \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+2 a^{2} d^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+a^{2} c^{3} \left (f x +e \right )-3 a^{2} c^{2} d \cos \left (f x +e \right )+3 a^{2} c \,d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {a^{2} d^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\) | \(329\) |
| default | \(\frac {a^{2} c^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{2} c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a^{2} c \,d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{2} d^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-2 a^{2} c^{3} \cos \left (f x +e \right )+6 a^{2} c^{2} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 a^{2} c \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+2 a^{2} d^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+a^{2} c^{3} \left (f x +e \right )-3 a^{2} c^{2} d \cos \left (f x +e \right )+3 a^{2} c \,d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {a^{2} d^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\) | \(329\) |
| norman | \(\frac {\left (\frac {3}{2} a^{2} c^{3}+3 a^{2} c^{2} d +\frac {21}{8} a^{2} c \,d^{2}+\frac {3}{4} a^{2} d^{3}\right ) x +\left (15 a^{2} c^{3}+30 a^{2} c^{2} d +\frac {105}{4} a^{2} c \,d^{2}+\frac {15}{2} a^{2} d^{3}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (15 a^{2} c^{3}+30 a^{2} c^{2} d +\frac {105}{4} a^{2} c \,d^{2}+\frac {15}{2} a^{2} d^{3}\right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {3}{2} a^{2} c^{3}+3 a^{2} c^{2} d +\frac {21}{8} a^{2} c \,d^{2}+\frac {3}{4} a^{2} d^{3}\right ) x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {15}{2} a^{2} c^{3}+15 a^{2} c^{2} d +\frac {105}{8} a^{2} c \,d^{2}+\frac {15}{4} a^{2} d^{3}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {15}{2} a^{2} c^{3}+15 a^{2} c^{2} d +\frac {105}{8} a^{2} c \,d^{2}+\frac {15}{4} a^{2} d^{3}\right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {20 a^{2} c^{3}+50 a^{2} c^{2} d +40 a^{2} c \,d^{2}+12 a^{2} d^{3}}{5 f}-\frac {\left (4 a^{2} c^{3}+6 a^{2} c^{2} d \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (8 a^{2} c^{3}+18 a^{2} c^{2} d +12 a^{2} c \,d^{2}+2 a^{2} d^{3}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (12 a^{2} c^{3}+32 a^{2} c^{2} d +28 a^{2} c \,d^{2}+10 a^{2} d^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {\left (16 a^{2} c^{3}+44 a^{2} c^{2} d +40 a^{2} c \,d^{2}+12 a^{2} d^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {a^{2} \left (4 c^{3}+24 c^{2} d +21 c \,d^{2}+6 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {a^{2} \left (4 c^{3}+24 c^{2} d +21 c \,d^{2}+6 d^{3}\right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {a^{2} \left (4 c^{3}+24 c^{2} d +33 c \,d^{2}+14 d^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}+\frac {a^{2} \left (4 c^{3}+24 c^{2} d +33 c \,d^{2}+14 d^{3}\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}}\) | \(688\) |
-1/4*a^2*((c^3+6*c^2*d+6*c*d^2+2*d^3)*sin(2*f*x+2*e)-(c+3/2*d)*d*(c+1/2*d) *cos(3*f*x+3*e)-3/8*d^2*(c+2/3*d)*sin(4*f*x+4*e)+1/20*d^3*cos(5*f*x+5*e)+( 8*c^3+21*c^2*d+18*c*d^2+11/2*d^3)*cos(f*x+e)+(-3*f*x+24/5)*d^3+c*(-21/2*f* x+16)*d^2+(-12*f*x+20)*c^2*d+(-6*f*x+8)*c^3)/f
Time = 0.30 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.01 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx=-\frac {8 \, a^{2} d^{3} \cos \left (f x + e\right )^{5} - 40 \, {\left (a^{2} c^{2} d + 2 \, a^{2} c d^{2} + a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (4 \, a^{2} c^{3} + 8 \, a^{2} c^{2} d + 7 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} f x + 80 \, {\left (a^{2} c^{3} + 3 \, a^{2} c^{2} d + 3 \, a^{2} c d^{2} + a^{2} d^{3}\right )} \cos \left (f x + e\right ) - 5 \, {\left (2 \, {\left (3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (4 \, a^{2} c^{3} + 24 \, a^{2} c^{2} d + 27 \, a^{2} c d^{2} + 10 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{40 \, f} \]
-1/40*(8*a^2*d^3*cos(f*x + e)^5 - 40*(a^2*c^2*d + 2*a^2*c*d^2 + a^2*d^3)*c os(f*x + e)^3 - 15*(4*a^2*c^3 + 8*a^2*c^2*d + 7*a^2*c*d^2 + 2*a^2*d^3)*f*x + 80*(a^2*c^3 + 3*a^2*c^2*d + 3*a^2*c*d^2 + a^2*d^3)*cos(f*x + e) - 5*(2* (3*a^2*c*d^2 + 2*a^2*d^3)*cos(f*x + e)^3 - (4*a^2*c^3 + 24*a^2*c^2*d + 27* a^2*c*d^2 + 10*a^2*d^3)*cos(f*x + e))*sin(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 729 vs. \(2 (218) = 436\).
Time = 0.36 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.39 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx=\begin {cases} \frac {a^{2} c^{3} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c^{3} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c^{3} x - \frac {a^{2} c^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a^{2} c^{3} \cos {\left (e + f x \right )}}{f} + 3 a^{2} c^{2} d x \sin ^{2}{\left (e + f x \right )} + 3 a^{2} c^{2} d x \cos ^{2}{\left (e + f x \right )} - \frac {3 a^{2} c^{2} d \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a^{2} c^{2} d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 a^{2} c^{2} d \cos ^{3}{\left (e + f x \right )}}{f} - \frac {3 a^{2} c^{2} d \cos {\left (e + f x \right )}}{f} + \frac {9 a^{2} c d^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {9 a^{2} c d^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {3 a^{2} c d^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {9 a^{2} c d^{2} x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {3 a^{2} c d^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {15 a^{2} c d^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {6 a^{2} c d^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {9 a^{2} c d^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {3 a^{2} c d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {4 a^{2} c d^{2} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {3 a^{2} d^{3} x \sin ^{4}{\left (e + f x \right )}}{4} + \frac {3 a^{2} d^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} + \frac {3 a^{2} d^{3} x \cos ^{4}{\left (e + f x \right )}}{4} - \frac {a^{2} d^{3} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 a^{2} d^{3} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f} - \frac {4 a^{2} d^{3} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {a^{2} d^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a^{2} d^{3} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{4 f} - \frac {8 a^{2} d^{3} \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {2 a^{2} d^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (c + d \sin {\left (e \right )}\right )^{3} \left (a \sin {\left (e \right )} + a\right )^{2} & \text {otherwise} \end {cases} \]
Piecewise((a**2*c**3*x*sin(e + f*x)**2/2 + a**2*c**3*x*cos(e + f*x)**2/2 + a**2*c**3*x - a**2*c**3*sin(e + f*x)*cos(e + f*x)/(2*f) - 2*a**2*c**3*cos (e + f*x)/f + 3*a**2*c**2*d*x*sin(e + f*x)**2 + 3*a**2*c**2*d*x*cos(e + f* x)**2 - 3*a**2*c**2*d*sin(e + f*x)**2*cos(e + f*x)/f - 3*a**2*c**2*d*sin(e + f*x)*cos(e + f*x)/f - 2*a**2*c**2*d*cos(e + f*x)**3/f - 3*a**2*c**2*d*c os(e + f*x)/f + 9*a**2*c*d**2*x*sin(e + f*x)**4/8 + 9*a**2*c*d**2*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 3*a**2*c*d**2*x*sin(e + f*x)**2/2 + 9*a**2*c *d**2*x*cos(e + f*x)**4/8 + 3*a**2*c*d**2*x*cos(e + f*x)**2/2 - 15*a**2*c* d**2*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 6*a**2*c*d**2*sin(e + f*x)**2*co s(e + f*x)/f - 9*a**2*c*d**2*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 3*a**2*c *d**2*sin(e + f*x)*cos(e + f*x)/(2*f) - 4*a**2*c*d**2*cos(e + f*x)**3/f + 3*a**2*d**3*x*sin(e + f*x)**4/4 + 3*a**2*d**3*x*sin(e + f*x)**2*cos(e + f* x)**2/2 + 3*a**2*d**3*x*cos(e + f*x)**4/4 - a**2*d**3*sin(e + f*x)**4*cos( e + f*x)/f - 5*a**2*d**3*sin(e + f*x)**3*cos(e + f*x)/(4*f) - 4*a**2*d**3* sin(e + f*x)**2*cos(e + f*x)**3/(3*f) - a**2*d**3*sin(e + f*x)**2*cos(e + f*x)/f - 3*a**2*d**3*sin(e + f*x)*cos(e + f*x)**3/(4*f) - 8*a**2*d**3*cos( e + f*x)**5/(15*f) - 2*a**2*d**3*cos(e + f*x)**3/(3*f), Ne(f, 0)), (x*(c + d*sin(e))**3*(a*sin(e) + a)**2, True))
Time = 0.24 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.48 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx=\frac {120 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{3} + 480 \, {\left (f x + e\right )} a^{2} c^{3} + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c^{2} d + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} d + 960 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c d^{2} + 45 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d^{2} + 360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d^{2} - 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} d^{3} + 160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} d^{3} + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{3} - 960 \, a^{2} c^{3} \cos \left (f x + e\right ) - 1440 \, a^{2} c^{2} d \cos \left (f x + e\right )}{480 \, f} \]
1/480*(120*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^2*c^3 + 480*(f*x + e)*a^2*c^ 3 + 480*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^2*c^2*d + 720*(2*f*x + 2*e - s in(2*f*x + 2*e))*a^2*c^2*d + 960*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^2*c*d ^2 + 45*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^2*c*d^2 + 360*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^2*c*d^2 - 32*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^2*d^3 + 160*(cos(f*x + e)^3 - 3*cos (f*x + e))*a^2*d^3 + 30*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^2*d^3 - 960*a^2*c^3*cos(f*x + e) - 1440*a^2*c^2*d*cos(f*x + e))/f
Time = 0.39 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.47 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx=-\frac {a^{2} d^{3} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {a^{2} d^{3} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {3 \, a^{2} c d^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{8} \, {\left (4 \, a^{2} c^{3} + 24 \, a^{2} c^{2} d + 9 \, a^{2} c d^{2} + 6 \, a^{2} d^{3}\right )} x + \frac {1}{2} \, {\left (2 \, a^{2} c^{3} + 3 \, a^{2} c d^{2}\right )} x + \frac {{\left (12 \, a^{2} c^{2} d + 24 \, a^{2} c d^{2} + 5 \, a^{2} d^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (16 \, a^{2} c^{3} + 18 \, a^{2} c^{2} d + 36 \, a^{2} c d^{2} + 5 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )}{8 \, f} - \frac {3 \, {\left (4 \, a^{2} c^{2} d + a^{2} d^{3}\right )} \cos \left (f x + e\right )}{4 \, f} + \frac {{\left (3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac {{\left (a^{2} c^{3} + 6 \, a^{2} c^{2} d + 3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
-1/80*a^2*d^3*cos(5*f*x + 5*e)/f + 1/12*a^2*d^3*cos(3*f*x + 3*e)/f - 3/4*a ^2*c*d^2*sin(2*f*x + 2*e)/f + 1/8*(4*a^2*c^3 + 24*a^2*c^2*d + 9*a^2*c*d^2 + 6*a^2*d^3)*x + 1/2*(2*a^2*c^3 + 3*a^2*c*d^2)*x + 1/48*(12*a^2*c^2*d + 24 *a^2*c*d^2 + 5*a^2*d^3)*cos(3*f*x + 3*e)/f - 1/8*(16*a^2*c^3 + 18*a^2*c^2* d + 36*a^2*c*d^2 + 5*a^2*d^3)*cos(f*x + e)/f - 3/4*(4*a^2*c^2*d + a^2*d^3) *cos(f*x + e)/f + 1/32*(3*a^2*c*d^2 + 2*a^2*d^3)*sin(4*f*x + 4*e)/f - 1/4* (a^2*c^3 + 6*a^2*c^2*d + 3*a^2*c*d^2 + 2*a^2*d^3)*sin(2*f*x + 2*e)/f
Time = 9.76 (sec) , antiderivative size = 611, normalized size of antiderivative = 2.84 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx=\frac {3\,a^2\,\mathrm {atan}\left (\frac {3\,a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c+d\right )\,\left (2\,c^2+3\,c\,d+2\,d^2\right )}{4\,\left (3\,a^2\,c^3+6\,a^2\,c^2\,d+\frac {21\,a^2\,c\,d^2}{4}+\frac {3\,a^2\,d^3}{2}\right )}\right )\,\left (2\,c+d\right )\,\left (2\,c^2+3\,c\,d+2\,d^2\right )}{4\,f}-\frac {3\,a^2\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )\,\left (4\,c^3+8\,c^2\,d+7\,c\,d^2+2\,d^3\right )}{4\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (4\,a^2\,c^3+6\,d\,a^2\,c^2\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (a^2\,c^3+6\,a^2\,c^2\,d+\frac {21\,a^2\,c\,d^2}{4}+\frac {3\,a^2\,d^3}{2}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,a^2\,c^3+12\,a^2\,c^2\,d+\frac {33\,a^2\,c\,d^2}{2}+7\,a^2\,d^3\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (2\,a^2\,c^3+12\,a^2\,c^2\,d+\frac {33\,a^2\,c\,d^2}{2}+7\,a^2\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (16\,a^2\,c^3+36\,a^2\,c^2\,d+24\,a^2\,c\,d^2+4\,a^2\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (16\,a^2\,c^3+44\,a^2\,c^2\,d+40\,a^2\,c\,d^2+12\,a^2\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (24\,a^2\,c^3+64\,a^2\,c^2\,d+56\,a^2\,c\,d^2+20\,a^2\,d^3\right )+4\,a^2\,c^3+\frac {12\,a^2\,d^3}{5}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a^2\,c^3+6\,a^2\,c^2\,d+\frac {21\,a^2\,c\,d^2}{4}+\frac {3\,a^2\,d^3}{2}\right )+8\,a^2\,c\,d^2+10\,a^2\,c^2\,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 )} \]
(3*a^2*atan((3*a^2*tan(e/2 + (f*x)/2)*(2*c + d)*(3*c*d + 2*c^2 + 2*d^2))/( 4*(3*a^2*c^3 + (3*a^2*d^3)/2 + (21*a^2*c*d^2)/4 + 6*a^2*c^2*d)))*(2*c + d) *(3*c*d + 2*c^2 + 2*d^2))/(4*f) - (3*a^2*(atan(tan(e/2 + (f*x)/2)) - (f*x) /2)*(7*c*d^2 + 8*c^2*d + 4*c^3 + 2*d^3))/(4*f) - (tan(e/2 + (f*x)/2)^8*(4* a^2*c^3 + 6*a^2*c^2*d) - tan(e/2 + (f*x)/2)^9*(a^2*c^3 + (3*a^2*d^3)/2 + ( 21*a^2*c*d^2)/4 + 6*a^2*c^2*d) + tan(e/2 + (f*x)/2)^3*(2*a^2*c^3 + 7*a^2*d ^3 + (33*a^2*c*d^2)/2 + 12*a^2*c^2*d) - tan(e/2 + (f*x)/2)^7*(2*a^2*c^3 + 7*a^2*d^3 + (33*a^2*c*d^2)/2 + 12*a^2*c^2*d) + tan(e/2 + (f*x)/2)^6*(16*a^ 2*c^3 + 4*a^2*d^3 + 24*a^2*c*d^2 + 36*a^2*c^2*d) + tan(e/2 + (f*x)/2)^2*(1 6*a^2*c^3 + 12*a^2*d^3 + 40*a^2*c*d^2 + 44*a^2*c^2*d) + tan(e/2 + (f*x)/2) ^4*(24*a^2*c^3 + 20*a^2*d^3 + 56*a^2*c*d^2 + 64*a^2*c^2*d) + 4*a^2*c^3 + ( 12*a^2*d^3)/5 + tan(e/2 + (f*x)/2)*(a^2*c^3 + (3*a^2*d^3)/2 + (21*a^2*c*d^ 2)/4 + 6*a^2*c^2*d) + 8*a^2*c*d^2 + 10*a^2*c^2*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))