Integrand size = 40, antiderivative size = 384 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {64 c^3 \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{f (5+2 m) (7+2 m) (9+2 m) \left (3+8 m+4 m^2\right ) \sqrt {c-c \sin (e+f x)}}+\frac {16 c^2 \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (7+2 m) (9+2 m) \left (15+16 m+4 m^2\right )}+\frac {2 c \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m) (9+2 m)}-\frac {4 C (1+2 m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m) (9+2 m)}+\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{7/2}}{c f (9+2 m)} \]
2*c*(C*(4*m^2-16*m+39)+A*(4*m^2+32*m+63))*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c -c*sin(f*x+e))^(3/2)/f/(9+2*m)/(4*m^2+24*m+35)-4*C*(1+2*m)*cos(f*x+e)*(a+a *sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2)/f/(4*m^2+32*m+63)+2*C*cos(f*x+e)*(a+ a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(7/2)/c/f/(9+2*m)+64*c^3*(C*(4*m^2-16*m+3 9)+A*(4*m^2+32*m+63))*cos(f*x+e)*(a+a*sin(f*x+e))^m/f/(9+2*m)/(16*m^4+128* m^3+344*m^2+352*m+105)/(c-c*sin(f*x+e))^(1/2)+16*c^2*(C*(4*m^2-16*m+39)+A* (4*m^2+32*m+63))*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(1/2)/f/(9 +2*m)/(8*m^3+60*m^2+142*m+105)
Result contains complex when optimal does not.
Time = 13.45 (sec) , antiderivative size = 899, normalized size of antiderivative = 2.34 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {(a (1+\sin (e+f x)))^m (c-c \sin (e+f x))^{5/2} \left (\frac {\left (18900 A+12285 C+15648 A m+648 C m+5280 A m^2+1416 C m^2+896 A m^3+224 C m^3+64 A m^4+16 C m^4\right ) \left (\left (\frac {1}{8}+\frac {i}{8}\right ) \cos \left (\frac {1}{2} (e+f x)\right )+\left (\frac {1}{8}-\frac {i}{8}\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{(1+2 m) (3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {\left (18900 A+12285 C+15648 A m+648 C m+5280 A m^2+1416 C m^2+896 A m^3+224 C m^3+64 A m^4+16 C m^4\right ) \left (\left (\frac {1}{8}-\frac {i}{8}\right ) \cos \left (\frac {1}{2} (e+f x)\right )+\left (\frac {1}{8}+\frac {i}{8}\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{(1+2 m) (3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {\left (1575 A+1575 C+1178 A m+414 C m+292 A m^2+100 C m^2+24 A m^3+8 C m^3\right ) \left (\left (\frac {1}{4}-\frac {i}{4}\right ) \cos \left (\frac {3}{2} (e+f x)\right )-\left (\frac {1}{4}+\frac {i}{4}\right ) \sin \left (\frac {3}{2} (e+f x)\right )\right )}{(3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {\left (1575 A+1575 C+1178 A m+414 C m+292 A m^2+100 C m^2+24 A m^3+8 C m^3\right ) \left (\left (\frac {1}{4}+\frac {i}{4}\right ) \cos \left (\frac {3}{2} (e+f x)\right )-\left (\frac {1}{4}-\frac {i}{4}\right ) \sin \left (\frac {3}{2} (e+f x)\right )\right )}{(3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {\left (63 A+189 C+32 A m+44 C m+4 A m^2+4 C m^2\right ) \left (\left (-\frac {1}{4}+\frac {i}{4}\right ) \cos \left (\frac {5}{2} (e+f x)\right )-\left (\frac {1}{4}+\frac {i}{4}\right ) \sin \left (\frac {5}{2} (e+f x)\right )\right )}{(5+2 m) (7+2 m) (9+2 m)}+\frac {\left (63 A+189 C+32 A m+44 C m+4 A m^2+4 C m^2\right ) \left (\left (-\frac {1}{4}-\frac {i}{4}\right ) \cos \left (\frac {5}{2} (e+f x)\right )-\left (\frac {1}{4}-\frac {i}{4}\right ) \sin \left (\frac {5}{2} (e+f x)\right )\right )}{(5+2 m) (7+2 m) (9+2 m)}+\frac {(15+2 m) \left (\left (-\frac {3}{16}-\frac {3 i}{16}\right ) C \cos \left (\frac {7}{2} (e+f x)\right )+\left (\frac {3}{16}-\frac {3 i}{16}\right ) C \sin \left (\frac {7}{2} (e+f x)\right )\right )}{(7+2 m) (9+2 m)}+\frac {(15+2 m) \left (\left (-\frac {3}{16}+\frac {3 i}{16}\right ) C \cos \left (\frac {7}{2} (e+f x)\right )+\left (\frac {3}{16}+\frac {3 i}{16}\right ) C \sin \left (\frac {7}{2} (e+f x)\right )\right )}{(7+2 m) (9+2 m)}+\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) C \cos \left (\frac {9}{2} (e+f x)\right )+\left (\frac {1}{16}-\frac {i}{16}\right ) C \sin \left (\frac {9}{2} (e+f x)\right )}{9+2 m}+\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) C \cos \left (\frac {9}{2} (e+f x)\right )+\left (\frac {1}{16}+\frac {i}{16}\right ) C \sin \left (\frac {9}{2} (e+f x)\right )}{9+2 m}\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
((a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^(5/2)*(((18900*A + 12285*C + 15648*A*m + 648*C*m + 5280*A*m^2 + 1416*C*m^2 + 896*A*m^3 + 224*C*m^3 + 64*A*m^4 + 16*C*m^4)*((1/8 + I/8)*Cos[(e + f*x)/2] + (1/8 - I/8)*Sin[(e + f*x)/2]))/((1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((18900*A + 12285*C + 15648*A*m + 648*C*m + 5280*A*m^2 + 1416*C*m^2 + 896*A*m^3 + 22 4*C*m^3 + 64*A*m^4 + 16*C*m^4)*((1/8 - I/8)*Cos[(e + f*x)/2] + (1/8 + I/8) *Sin[(e + f*x)/2]))/((1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((1575*A + 1575*C + 1178*A*m + 414*C*m + 292*A*m^2 + 100*C*m^2 + 24*A*m^3 + 8*C*m^3)*((1/4 - I/4)*Cos[(3*(e + f*x))/2] - (1/4 + I/4)*Sin[(3*(e + f*x ))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((1575*A + 1575*C + 11 78*A*m + 414*C*m + 292*A*m^2 + 100*C*m^2 + 24*A*m^3 + 8*C*m^3)*((1/4 + I/4 )*Cos[(3*(e + f*x))/2] - (1/4 - I/4)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((63*A + 189*C + 32*A*m + 44*C*m + 4*A*m^2 + 4*C*m^2)*((-1/4 + I/4)*Cos[(5*(e + f*x))/2] - (1/4 + I/4)*Sin[(5*(e + f*x ))/2]))/((5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((63*A + 189*C + 32*A*m + 44*C*m + 4*A*m^2 + 4*C*m^2)*((-1/4 - I/4)*Cos[(5*(e + f*x))/2] - (1/4 - I/4)*Sin [(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((15 + 2*m)*((-3/16 - (3*I)/16)*C*Cos[(7*(e + f*x))/2] + (3/16 - (3*I)/16)*C*Sin[(7*(e + f*x)) /2]))/((7 + 2*m)*(9 + 2*m)) + ((15 + 2*m)*((-3/16 + (3*I)/16)*C*Cos[(7*(e + f*x))/2] + (3/16 + (3*I)/16)*C*Sin[(7*(e + f*x))/2]))/((7 + 2*m)*(9 +...
Time = 1.47 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.81, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {3042, 3519, 27, 3042, 3452, 3042, 3219, 3042, 3219, 3042, 3217}
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.
| \(\displaystyle \int (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m \left (A+C \sin ^2(e+f x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m \left (A+C \sin (e+f x)^2\right )dx\) |
| \(\Big \downarrow \) 3519 |
| \(\displaystyle \frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}-\frac {2 \int -\frac {1}{2} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{5/2} (a c (C (7-2 m)+A (2 m+9))+2 a c C (2 m+1) \sin (e+f x))dx}{a c (2 m+9)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{5/2} (a c (C (7-2 m)+A (2 m+9))+2 a c C (2 m+1) \sin (e+f x))dx}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{5/2} (a c (C (7-2 m)+A (2 m+9))+2 a c C (2 m+1) \sin (e+f x))dx}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3452 |
| \(\displaystyle \frac {\frac {a c \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{5/2}dx}{2 m+7}-\frac {4 a c C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a c \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{5/2}dx}{2 m+7}-\frac {4 a c C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle \frac {\frac {a c \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \left (\frac {8 c \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2}dx}{2 m+5}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}\right )}{2 m+7}-\frac {4 a c C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a c \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \left (\frac {8 c \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2}dx}{2 m+5}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}\right )}{2 m+7}-\frac {4 a c C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle \frac {\frac {a c \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \left (\frac {8 c \left (\frac {4 c \int (\sin (e+f x) a+a)^m \sqrt {c-c \sin (e+f x)}dx}{2 m+3}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}\right )}{2 m+5}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}\right )}{2 m+7}-\frac {4 a c C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a c \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \left (\frac {8 c \left (\frac {4 c \int (\sin (e+f x) a+a)^m \sqrt {c-c \sin (e+f x)}dx}{2 m+3}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}\right )}{2 m+5}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}\right )}{2 m+7}-\frac {4 a c C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3217 |
| \(\displaystyle \frac {\frac {a c \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \left (\frac {8 c \left (\frac {8 c^2 \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) (2 m+3) \sqrt {c-c \sin (e+f x)}}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}\right )}{2 m+5}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}\right )}{2 m+7}-\frac {4 a c C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
(2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(7/2))/(c*f* (9 + 2*m)) + ((-4*a*c*C*(1 + 2*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(f*(7 + 2*m)) + (a*c*(C*(39 - 16*m + 4*m^2) + A*(6 3 + 32*m + 4*m^2))*((2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)) + (8*c*((8*c^2*Cos[e + f*x]*(a + a*Sin[e + f* x])^m)/(f*(1 + 2*m)*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + (2*c*Cos[e + f*x ]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(3 + 2*m))))/(5 + 2* m)))/(7 + 2*m))/(a*c*(9 + 2*m))
3.1.1.3.1 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[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f _.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e, f, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x])^ (m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Simp[a*((2*m - 1)/(m + n )) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; Fre eQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I GtQ[m - 1/2, 0] && !LtQ[n, -1] && !(IGtQ[n - 1/2, 0] && LtQ[n, m]) && !( ILtQ[m + n, 0] && GtQ[2*m + n + 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] :> Sim p[(-B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(f*(m + n + 1))), x] - Simp[(B*c*(m - n) - A*d*(m + n + 1))/(d*(m + n + 1)) Int[( a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 1, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1 )/(d*f*(m + n + 2))), x] + Simp[1/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f* x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1 )) - b*c*C*(2*m + 1)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A , C, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1 )] && NeQ[m + n + 2, 0]
\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +C \left (\sin ^{2}\left (f x +e \right )\right )\right )d x\]
Leaf count of result is larger than twice the leaf count of optimal. 777 vs. \(2 (364) = 728\).
Time = 0.32 (sec) , antiderivative size = 777, normalized size of antiderivative = 2.02 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {2 \, {\left ({\left (16 \, C c^{2} m^{4} + 128 \, C c^{2} m^{3} + 344 \, C c^{2} m^{2} + 352 \, C c^{2} m + 105 \, C c^{2}\right )} \cos \left (f x + e\right )^{5} + 128 \, {\left (A + C\right )} c^{2} m^{2} - {\left (16 \, C c^{2} m^{4} + 224 \, C c^{2} m^{3} + 776 \, C c^{2} m^{2} + 904 \, C c^{2} m + 285 \, C c^{2}\right )} \cos \left (f x + e\right )^{4} + 512 \, {\left (2 \, A - C\right )} c^{2} m - {\left (16 \, {\left (A + 3 \, C\right )} c^{2} m^{4} + 32 \, {\left (5 \, A + 16 \, C\right )} c^{2} m^{3} + 8 \, {\left (65 \, A + 253 \, C\right )} c^{2} m^{2} + 8 \, {\left (75 \, A + 328 \, C\right )} c^{2} m + 3 \, {\left (63 \, A + 289 \, C\right )} c^{2}\right )} \cos \left (f x + e\right )^{3} + 96 \, {\left (21 \, A + 13 \, C\right )} c^{2} + {\left (16 \, {\left (A + C\right )} c^{2} m^{4} + 224 \, {\left (A + C\right )} c^{2} m^{3} + 8 \, {\left (133 \, A + 85 \, C\right )} c^{2} m^{2} + 1864 \, {\left (A + C\right )} c^{2} m + 3 \, {\left (231 \, A + 263 \, C\right )} c^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (16 \, {\left (A + C\right )} c^{2} m^{4} + 192 \, {\left (A + C\right )} c^{2} m^{3} + 856 \, {\left (A + C\right )} c^{2} m^{2} + 16 \, {\left (109 \, A + 85 \, C\right )} c^{2} m + 3 \, {\left (483 \, A + 419 \, C\right )} c^{2}\right )} \cos \left (f x + e\right ) + {\left (128 \, {\left (A + C\right )} c^{2} m^{2} + {\left (16 \, C c^{2} m^{4} + 128 \, C c^{2} m^{3} + 344 \, C c^{2} m^{2} + 352 \, C c^{2} m + 105 \, C c^{2}\right )} \cos \left (f x + e\right )^{4} + 512 \, {\left (2 \, A - C\right )} c^{2} m + 2 \, {\left (16 \, C c^{2} m^{4} + 176 \, C c^{2} m^{3} + 560 \, C c^{2} m^{2} + 628 \, C c^{2} m + 195 \, C c^{2}\right )} \cos \left (f x + e\right )^{3} + 96 \, {\left (21 \, A + 13 \, C\right )} c^{2} - {\left (16 \, {\left (A + C\right )} c^{2} m^{4} + 160 \, {\left (A + C\right )} c^{2} m^{3} + 8 \, {\left (65 \, A + 113 \, C\right )} c^{2} m^{2} + 24 \, {\left (25 \, A + 57 \, C\right )} c^{2} m + 9 \, {\left (21 \, A + 53 \, C\right )} c^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (16 \, {\left (A + C\right )} c^{2} m^{4} + 192 \, {\left (A + C\right )} c^{2} m^{3} + 792 \, {\left (A + C\right )} c^{2} m^{2} + 16 \, {\left (77 \, A + 101 \, C\right )} c^{2} m + 3 \, {\left (147 \, A + 211 \, C\right )} c^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{32 \, f m^{5} + 400 \, f m^{4} + 1840 \, f m^{3} + 3800 \, f m^{2} + 3378 \, f m + {\left (32 \, f m^{5} + 400 \, f m^{4} + 1840 \, f m^{3} + 3800 \, f m^{2} + 3378 \, f m + 945 \, f\right )} \cos \left (f x + e\right ) - {\left (32 \, f m^{5} + 400 \, f m^{4} + 1840 \, f m^{3} + 3800 \, f m^{2} + 3378 \, f m + 945 \, f\right )} \sin \left (f x + e\right ) + 945 \, f} \]
2*((16*C*c^2*m^4 + 128*C*c^2*m^3 + 344*C*c^2*m^2 + 352*C*c^2*m + 105*C*c^2 )*cos(f*x + e)^5 + 128*(A + C)*c^2*m^2 - (16*C*c^2*m^4 + 224*C*c^2*m^3 + 7 76*C*c^2*m^2 + 904*C*c^2*m + 285*C*c^2)*cos(f*x + e)^4 + 512*(2*A - C)*c^2 *m - (16*(A + 3*C)*c^2*m^4 + 32*(5*A + 16*C)*c^2*m^3 + 8*(65*A + 253*C)*c^ 2*m^2 + 8*(75*A + 328*C)*c^2*m + 3*(63*A + 289*C)*c^2)*cos(f*x + e)^3 + 96 *(21*A + 13*C)*c^2 + (16*(A + C)*c^2*m^4 + 224*(A + C)*c^2*m^3 + 8*(133*A + 85*C)*c^2*m^2 + 1864*(A + C)*c^2*m + 3*(231*A + 263*C)*c^2)*cos(f*x + e) ^2 + 2*(16*(A + C)*c^2*m^4 + 192*(A + C)*c^2*m^3 + 856*(A + C)*c^2*m^2 + 1 6*(109*A + 85*C)*c^2*m + 3*(483*A + 419*C)*c^2)*cos(f*x + e) + (128*(A + C )*c^2*m^2 + (16*C*c^2*m^4 + 128*C*c^2*m^3 + 344*C*c^2*m^2 + 352*C*c^2*m + 105*C*c^2)*cos(f*x + e)^4 + 512*(2*A - C)*c^2*m + 2*(16*C*c^2*m^4 + 176*C* c^2*m^3 + 560*C*c^2*m^2 + 628*C*c^2*m + 195*C*c^2)*cos(f*x + e)^3 + 96*(21 *A + 13*C)*c^2 - (16*(A + C)*c^2*m^4 + 160*(A + C)*c^2*m^3 + 8*(65*A + 113 *C)*c^2*m^2 + 24*(25*A + 57*C)*c^2*m + 9*(21*A + 53*C)*c^2)*cos(f*x + e)^2 - 2*(16*(A + C)*c^2*m^4 + 192*(A + C)*c^2*m^3 + 792*(A + C)*c^2*m^2 + 16* (77*A + 101*C)*c^2*m + 3*(147*A + 211*C)*c^2)*cos(f*x + e))*sin(f*x + e))* sqrt(-c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m/(32*f*m^5 + 400*f*m^4 + 1 840*f*m^3 + 3800*f*m^2 + 3378*f*m + (32*f*m^5 + 400*f*m^4 + 1840*f*m^3 + 3 800*f*m^2 + 3378*f*m + 945*f)*cos(f*x + e) - (32*f*m^5 + 400*f*m^4 + 1840* f*m^3 + 3800*f*m^2 + 3378*f*m + 945*f)*sin(f*x + e) + 945*f)
Timed out. \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 892 vs. \(2 (364) = 728\).
Time = 0.39 (sec) , antiderivative size = 892, normalized size of antiderivative = 2.32 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\text {Too large to display} \]
-2*(((4*m^2 + 24*m + 43)*a^m*c^(5/2) - (12*m^2 + 40*m - 15)*a^m*c^(5/2)*si n(f*x + e)/(cos(f*x + e) + 1) + 2*(4*m^2 + 8*m + 35)*a^m*c^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2*(4*m^2 + 8*m + 35)*a^m*c^(5/2)*sin(f*x + e) ^3/(cos(f*x + e) + 1)^3 - (12*m^2 + 40*m - 15)*a^m*c^(5/2)*sin(f*x + e)^4/ (cos(f*x + e) + 1)^4 + (4*m^2 + 24*m + 43)*a^m*c^(5/2)*sin(f*x + e)^5/(cos (f*x + e) + 1)^5)*A*e^(2*m*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1) - m*lo g(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1))/((8*m^3 + 36*m^2 + 46*m + 15)* (sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(5/2)) + 4*(2*(4*m^2 + 56*m + 21 9)*a^m*c^(5/2) - 4*(4*m^3 + 56*m^2 + 219*m)*a^m*c^(5/2)*sin(f*x + e)/(cos( f*x + e) + 1) + (16*m^4 + 240*m^3 + 1136*m^2 + 1380*m + 1971)*a^m*c^(5/2)* sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - (48*m^4 + 496*m^3 + 1568*m^2 + 3108* m - 315)*a^m*c^(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 4*(8*m^4 + 68*m ^3 + 290*m^2 + 111*m + 567)*a^m*c^(5/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^ 4 + 4*(8*m^4 + 68*m^3 + 290*m^2 + 111*m + 567)*a^m*c^(5/2)*sin(f*x + e)^5/ (cos(f*x + e) + 1)^5 - (48*m^4 + 496*m^3 + 1568*m^2 + 3108*m - 315)*a^m*c^ (5/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + (16*m^4 + 240*m^3 + 1136*m^2 + 1380*m + 1971)*a^m*c^(5/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 4*(4*m^3 + 56*m^2 + 219*m)*a^m*c^(5/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 2*(4* m^2 + 56*m + 219)*a^m*c^(5/2)*sin(f*x + e)^9/(cos(f*x + e) + 1)^9)*C*e^(2* m*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1) - m*log(sin(f*x + e)^2/(cos(...
\[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\int { {\left (C \sin \left (f x + e\right )^{2} + A\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
Time = 22.87 (sec) , antiderivative size = 1110, normalized size of antiderivative = 2.89 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\text {Too large to display} \]
((c - c*sin(e + f*x))^(1/2)*((C*c^2*(a + a*sin(e + f*x))^m*(m*352i + m^2*3 44i + m^3*128i + m^4*16i + 105i))/(8*f*(m*3378i + m^2*3800i + m^3*1840i + m^4*400i + m^5*32i + 945i)) + (c^2*exp(e*5i + f*x*5i)*(a + a*sin(e + f*x)) ^m*(18900*A + 12285*C + 15648*A*m + 648*C*m + 5280*A*m^2 + 896*A*m^3 + 64* A*m^4 + 1416*C*m^2 + 224*C*m^3 + 16*C*m^4))/(4*f*(m*3378i + m^2*3800i + m^ 3*1840i + m^4*400i + m^5*32i + 945i)) + (c^2*exp(e*4i + f*x*4i)*(a + a*sin (e + f*x))^m*(A*18900i + C*12285i + A*m*15648i + C*m*648i + A*m^2*5280i + A*m^3*896i + A*m^4*64i + C*m^2*1416i + C*m^3*224i + C*m^4*16i))/(4*f*(m*33 78i + m^2*3800i + m^3*1840i + m^4*400i + m^5*32i + 945i)) + (c^2*exp(e*3i + f*x*3i)*(2*m + 1)*(a + a*sin(e + f*x))^m*(1575*A + 1575*C + 1178*A*m + 4 14*C*m + 292*A*m^2 + 24*A*m^3 + 100*C*m^2 + 8*C*m^3))/(2*f*(m*3378i + m^2* 3800i + m^3*1840i + m^4*400i + m^5*32i + 945i)) + (c^2*exp(e*6i + f*x*6i)* (2*m + 1)*(a + a*sin(e + f*x))^m*(A*1575i + C*1575i + A*m*1178i + C*m*414i + A*m^2*292i + A*m^3*24i + C*m^2*100i + C*m^3*8i))/(2*f*(m*3378i + m^2*38 00i + m^3*1840i + m^4*400i + m^5*32i + 945i)) + (C*c^2*exp(e*9i + f*x*9i)* (a + a*sin(e + f*x))^m*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105))/(8*f*(m *3378i + m^2*3800i + m^3*1840i + m^4*400i + m^5*32i + 945i)) - (3*C*c^2*ex p(e*1i + f*x*1i)*(a + a*sin(e + f*x))^m*(720*m + 632*m^2 + 192*m^3 + 16*m^ 4 + 225))/(8*f*(m*3378i + m^2*3800i + m^3*1840i + m^4*400i + m^5*32i + 945 i)) - (3*C*c^2*exp(e*8i + f*x*8i)*(a + a*sin(e + f*x))^m*(m*720i + m^2*...