Integrand size = 40, antiderivative size = 285 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {8 c^2 \left (C \left (19-8 m+4 m^2\right )+A \left (35+24 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{f (5+2 m) (7+2 m) \left (3+8 m+4 m^2\right ) \sqrt {c-c \sin (e+f x)}}+\frac {2 c \left (C \left (19-8 m+4 m^2\right )+A \left (35+24 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (3+2 m) (5+2 m) (7+2 m)}-\frac {4 C (1+2 m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m)}+\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{c f (7+2 m)} \]
-4*C*(1+2*m)*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(3/2)/f/(4*m^2 +24*m+35)+2*C*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2)/c/f/(7+ 2*m)+8*c^2*(C*(4*m^2-8*m+19)+A*(4*m^2+24*m+35))*cos(f*x+e)*(a+a*sin(f*x+e) )^m/f/(7+2*m)/(8*m^3+36*m^2+46*m+15)/(c-c*sin(f*x+e))^(1/2)+2*c*(C*(4*m^2- 8*m+19)+A*(4*m^2+24*m+35))*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^ (1/2)/f/(7+2*m)/(4*m^2+16*m+15)
Time = 5.87 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.93 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {c \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^m \sqrt {c-c \sin (e+f x)} \left (700 A+494 C+760 A m+284 C m+272 A m^2+136 C m^2+32 A m^3+16 C m^3-2 C \left (39+110 m+68 m^2+8 m^3\right ) \cos (2 (e+f x))-(1+2 m) \left (4 A \left (35+24 m+4 m^2\right )+C \left (253+80 m+12 m^2\right )\right ) \sin (e+f x)+15 C \sin (3 (e+f x))+46 C m \sin (3 (e+f x))+36 C m^2 \sin (3 (e+f x))+8 C m^3 \sin (3 (e+f x))\right )}{2 f (1+2 m) (3+2 m) (5+2 m) (7+2 m) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^m*Sqrt[c - c*Sin[e + f*x]]*(700*A + 494*C + 760*A*m + 284*C*m + 272*A*m^2 + 136*C*m^ 2 + 32*A*m^3 + 16*C*m^3 - 2*C*(39 + 110*m + 68*m^2 + 8*m^3)*Cos[2*(e + f*x )] - (1 + 2*m)*(4*A*(35 + 24*m + 4*m^2) + C*(253 + 80*m + 12*m^2))*Sin[e + f*x] + 15*C*Sin[3*(e + f*x)] + 46*C*m*Sin[3*(e + f*x)] + 36*C*m^2*Sin[3*( e + f*x)] + 8*C*m^3*Sin[3*(e + f*x)]))/(2*f*(1 + 2*m)*(3 + 2*m)*(5 + 2*m)* (7 + 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))
Time = 1.19 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.89, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {3042, 3519, 27, 3042, 3452, 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.
| \(\displaystyle \int (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m \left (A+C \sin ^2(e+f x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m \left (A+C \sin (e+f x)^2\right )dx\) |
| \(\Big \downarrow \) 3519 |
| \(\displaystyle \frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}-\frac {2 \int -\frac {1}{2} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2} (a c (C (5-2 m)+A (2 m+7))+2 a c C (2 m+1) \sin (e+f x))dx}{a c (2 m+7)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2} (a c (C (5-2 m)+A (2 m+7))+2 a c C (2 m+1) \sin (e+f x))dx}{a c (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2} (a c (C (5-2 m)+A (2 m+7))+2 a c C (2 m+1) \sin (e+f x))dx}{a c (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}\) |
| \(\Big \downarrow \) 3452 |
| \(\displaystyle \frac {\frac {a c \left (A \left (4 m^2+24 m+35\right )+C \left (4 m^2-8 m+19\right )\right ) \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2}dx}{2 m+5}-\frac {4 a c C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}}{a c (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a c \left (A \left (4 m^2+24 m+35\right )+C \left (4 m^2-8 m+19\right )\right ) \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2}dx}{2 m+5}-\frac {4 a c C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}}{a c (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle \frac {\frac {a c \left (A \left (4 m^2+24 m+35\right )+C \left (4 m^2-8 m+19\right )\right ) \left (\frac {4 c \int (\sin (e+f x) a+a)^m \sqrt {c-c \sin (e+f x)}dx}{2 m+3}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}\right )}{2 m+5}-\frac {4 a c C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}}{a c (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a c \left (A \left (4 m^2+24 m+35\right )+C \left (4 m^2-8 m+19\right )\right ) \left (\frac {4 c \int (\sin (e+f x) a+a)^m \sqrt {c-c \sin (e+f x)}dx}{2 m+3}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}\right )}{2 m+5}-\frac {4 a c C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}}{a c (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}\) |
| \(\Big \downarrow \) 3217 |
| \(\displaystyle \frac {\frac {a c \left (A \left (4 m^2+24 m+35\right )+C \left (4 m^2-8 m+19\right )\right ) \left (\frac {8 c^2 \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) (2 m+3) \sqrt {c-c \sin (e+f x)}}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}\right )}{2 m+5}-\frac {4 a c C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}}{a c (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}\) |
(2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(c*f* (7 + 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])^(3/2))/(f*(5 + 2*m)) + (a*c*(C*(19 - 8*m + 4*m^2) + A*(35 + 24*m + 4*m^2))*((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))/(a*c*(7 + 2*m) )
3.1.2.3.1 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]
\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +C \left (\sin ^{2}\left (f x +e \right )\right )\right )d x\]
Time = 0.31 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.62 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=-\frac {2 \, {\left ({\left (8 \, C c m^{3} + 36 \, C c m^{2} + 46 \, C c m + 15 \, C c\right )} \cos \left (f x + e\right )^{4} - 16 \, {\left (A + C\right )} c m^{2} + {\left (8 \, C c m^{3} + 68 \, C c m^{2} + 110 \, C c m + 39 \, C c\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (3 \, A - C\right )} c m - {\left (8 \, {\left (A + C\right )} c m^{3} + 4 \, {\left (13 \, A + 5 \, C\right )} c m^{2} + 94 \, {\left (A + C\right )} c m + {\left (35 \, A + 43 \, C\right )} c\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left (35 \, A + 19 \, C\right )} c - {\left (8 \, {\left (A + C\right )} c m^{3} + 68 \, {\left (A + C\right )} c m^{2} + 2 \, {\left (95 \, A + 63 \, C\right )} c m + {\left (175 \, A + 143 \, C\right )} c\right )} \cos \left (f x + e\right ) - {\left (16 \, {\left (A + C\right )} c m^{2} + {\left (8 \, C c m^{3} + 36 \, C c m^{2} + 46 \, C c m + 15 \, C c\right )} \cos \left (f x + e\right )^{3} + 32 \, {\left (3 \, A - C\right )} c m - 8 \, {\left (4 \, C c m^{2} + 8 \, C c m + 3 \, C c\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left (35 \, A + 19 \, C\right )} c - {\left (8 \, {\left (A + C\right )} c m^{3} + 52 \, {\left (A + C\right )} c m^{2} + 2 \, {\left (47 \, A + 79 \, C\right )} c m + {\left (35 \, A + 67 \, C\right )} c\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + {\left (16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + 105 \, f\right )} \cos \left (f x + e\right ) - {\left (16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + 105 \, f\right )} \sin \left (f x + e\right ) + 105 \, f} \]
-2*((8*C*c*m^3 + 36*C*c*m^2 + 46*C*c*m + 15*C*c)*cos(f*x + e)^4 - 16*(A + C)*c*m^2 + (8*C*c*m^3 + 68*C*c*m^2 + 110*C*c*m + 39*C*c)*cos(f*x + e)^3 - 32*(3*A - C)*c*m - (8*(A + C)*c*m^3 + 4*(13*A + 5*C)*c*m^2 + 94*(A + C)*c* m + (35*A + 43*C)*c)*cos(f*x + e)^2 - 4*(35*A + 19*C)*c - (8*(A + C)*c*m^3 + 68*(A + C)*c*m^2 + 2*(95*A + 63*C)*c*m + (175*A + 143*C)*c)*cos(f*x + e ) - (16*(A + C)*c*m^2 + (8*C*c*m^3 + 36*C*c*m^2 + 46*C*c*m + 15*C*c)*cos(f *x + e)^3 + 32*(3*A - C)*c*m - 8*(4*C*c*m^2 + 8*C*c*m + 3*C*c)*cos(f*x + e )^2 + 4*(35*A + 19*C)*c - (8*(A + C)*c*m^3 + 52*(A + C)*c*m^2 + 2*(47*A + 79*C)*c*m + (35*A + 67*C)*c)*cos(f*x + e))*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m/(16*f*m^4 + 128*f*m^3 + 344*f*m^2 + 352*f* m + (16*f*m^4 + 128*f*m^3 + 344*f*m^2 + 352*f*m + 105*f)*cos(f*x + e) - (1 6*f*m^4 + 128*f*m^3 + 344*f*m^2 + 352*f*m + 105*f)*sin(f*x + e) + 105*f)
Timed out. \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 648 vs. \(2 (271) = 542\).
Time = 0.36 (sec) , antiderivative size = 648, normalized size of antiderivative = 2.27 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=-\frac {2 \, {\left (\frac {{\left (a^{m} c^{\frac {3}{2}} {\left (2 \, m + 5\right )} - \frac {a^{m} c^{\frac {3}{2}} {\left (2 \, m - 3\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {a^{m} c^{\frac {3}{2}} {\left (2 \, m - 3\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{m} c^{\frac {3}{2}} {\left (2 \, m + 5\right )} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )} A e^{\left (2 \, m \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) - m \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )\right )}}{{\left (4 \, m^{2} + 8 \, m + 3\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {3}{2}}} + \frac {4 \, {\left (2 \, a^{m} c^{\frac {3}{2}} {\left (2 \, m + 13\right )} - \frac {4 \, {\left (2 \, m^{2} + 13 \, m\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {{\left (8 \, m^{3} + 60 \, m^{2} + 66 \, m + 91\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {{\left (8 \, m^{3} + 20 \, m^{2} + 82 \, m - 35\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {{\left (8 \, m^{3} + 20 \, m^{2} + 82 \, m - 35\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {{\left (8 \, m^{3} + 60 \, m^{2} + 66 \, m + 91\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {4 \, {\left (2 \, m^{2} + 13 \, m\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {2 \, a^{m} c^{\frac {3}{2}} {\left (2 \, m + 13\right )} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}\right )} C e^{\left (2 \, m \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) - m \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )\right )}}{{\left (16 \, m^{4} + 128 \, m^{3} + 344 \, m^{2} + 352 \, m + \frac {2 \, {\left (16 \, m^{4} + 128 \, m^{3} + 344 \, m^{2} + 352 \, m + 105\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {{\left (16 \, m^{4} + 128 \, m^{3} + 344 \, m^{2} + 352 \, m + 105\right )} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 105\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {3}{2}}}\right )}}{f} \]
-2*((a^m*c^(3/2)*(2*m + 5) - a^m*c^(3/2)*(2*m - 3)*sin(f*x + e)/(cos(f*x + e) + 1) - a^m*c^(3/2)*(2*m - 3)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^m *c^(3/2)*(2*m + 5)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3)*A*e^(2*m*log(sin(f *x + e)/(cos(f*x + e) + 1) + 1) - m*log(sin(f*x + e)^2/(cos(f*x + e) + 1)^ 2 + 1))/((4*m^2 + 8*m + 3)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(3/2) ) + 4*(2*a^m*c^(3/2)*(2*m + 13) - 4*(2*m^2 + 13*m)*a^m*c^(3/2)*sin(f*x + e )/(cos(f*x + e) + 1) + (8*m^3 + 60*m^2 + 66*m + 91)*a^m*c^(3/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - (8*m^3 + 20*m^2 + 82*m - 35)*a^m*c^(3/2)*sin(f *x + e)^3/(cos(f*x + e) + 1)^3 - (8*m^3 + 20*m^2 + 82*m - 35)*a^m*c^(3/2)* sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + (8*m^3 + 60*m^2 + 66*m + 91)*a^m*c^( 3/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 4*(2*m^2 + 13*m)*a^m*c^(3/2)*si n(f*x + e)^6/(cos(f*x + e) + 1)^6 + 2*a^m*c^(3/2)*(2*m + 13)*sin(f*x + e)^ 7/(cos(f*x + e) + 1)^7)*C*e^(2*m*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1) - m*log(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1))/((16*m^4 + 128*m^3 + 344 *m^2 + 352*m + 2*(16*m^4 + 128*m^3 + 344*m^2 + 352*m + 105)*sin(f*x + e)^2 /(cos(f*x + e) + 1)^2 + (16*m^4 + 128*m^3 + 344*m^2 + 352*m + 105)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 105)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(3/2)))/f
\[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\int { {\left (C \sin \left (f x + e\right )^{2} + A\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
Time = 20.95 (sec) , antiderivative size = 714, normalized size of antiderivative = 2.51 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {C\,c\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (m^3\,8{}\mathrm {i}+m^2\,36{}\mathrm {i}+m\,46{}\mathrm {i}+15{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}+\frac {c\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (1260\,A+735\,C+1144\,A\,m-18\,C\,m+336\,A\,m^2+32\,A\,m^3+100\,C\,m^2+8\,C\,m^3\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {c\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (A\,1260{}\mathrm {i}+C\,735{}\mathrm {i}+A\,m\,1144{}\mathrm {i}-C\,m\,18{}\mathrm {i}+A\,m^2\,336{}\mathrm {i}+A\,m^3\,32{}\mathrm {i}+C\,m^2\,100{}\mathrm {i}+C\,m^3\,8{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {C\,c\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (8\,m^3+36\,m^2+46\,m+15\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {C\,c\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (8\,m^3+100\,m^2+174\,m+63\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}+\frac {C\,c\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (m^3\,8{}\mathrm {i}+m^2\,100{}\mathrm {i}+m\,174{}\mathrm {i}+63{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}+\frac {c\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (2\,m+1\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (140\,A+175\,C+96\,A\,m+16\,C\,m+16\,A\,m^2+4\,C\,m^2\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {c\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (2\,m+1\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (A\,140{}\mathrm {i}+C\,175{}\mathrm {i}+A\,m\,96{}\mathrm {i}+C\,m\,16{}\mathrm {i}+A\,m^2\,16{}\mathrm {i}+C\,m^2\,4{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}\right )}{{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (m^4\,16{}\mathrm {i}+m^3\,128{}\mathrm {i}+m^2\,344{}\mathrm {i}+m\,352{}\mathrm {i}+105{}\mathrm {i}\right )}{16\,m^4+128\,m^3+344\,m^2+352\,m+105}} \]
((c - c*sin(e + f*x))^(1/2)*((C*c*(a + a*sin(e + f*x))^m*(m*46i + m^2*36i + m^3*8i + 15i))/(4*f*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105)) + (c*exp (e*3i + f*x*3i)*(a + a*sin(e + f*x))^m*(1260*A + 735*C + 1144*A*m - 18*C*m + 336*A*m^2 + 32*A*m^3 + 100*C*m^2 + 8*C*m^3))/(4*f*(352*m + 344*m^2 + 12 8*m^3 + 16*m^4 + 105)) - (c*exp(e*4i + f*x*4i)*(a + a*sin(e + f*x))^m*(A*1 260i + C*735i + A*m*1144i - C*m*18i + A*m^2*336i + A*m^3*32i + C*m^2*100i + C*m^3*8i))/(4*f*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105)) - (C*c*exp(e *7i + f*x*7i)*(a + a*sin(e + f*x))^m*(46*m + 36*m^2 + 8*m^3 + 15))/(4*f*(3 52*m + 344*m^2 + 128*m^3 + 16*m^4 + 105)) - (C*c*exp(e*1i + f*x*1i)*(a + a *sin(e + f*x))^m*(174*m + 100*m^2 + 8*m^3 + 63))/(4*f*(352*m + 344*m^2 + 1 28*m^3 + 16*m^4 + 105)) + (C*c*exp(e*6i + f*x*6i)*(a + a*sin(e + f*x))^m*( m*174i + m^2*100i + m^3*8i + 63i))/(4*f*(352*m + 344*m^2 + 128*m^3 + 16*m^ 4 + 105)) + (c*exp(e*5i + f*x*5i)*(2*m + 1)*(a + a*sin(e + f*x))^m*(140*A + 175*C + 96*A*m + 16*C*m + 16*A*m^2 + 4*C*m^2))/(4*f*(352*m + 344*m^2 + 1 28*m^3 + 16*m^4 + 105)) - (c*exp(e*2i + f*x*2i)*(2*m + 1)*(a + a*sin(e + f *x))^m*(A*140i + C*175i + A*m*96i + C*m*16i + A*m^2*16i + C*m^2*4i))/(4*f* (352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105))))/(exp(e*4i + f*x*4i) - (exp(e *3i + f*x*3i)*(m*352i + m^2*344i + m^3*128i + m^4*16i + 105i))/(352*m + 34 4*m^2 + 128*m^3 + 16*m^4 + 105))