Integrand size = 48, antiderivative size = 322 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=-\frac {8 c^2 \left (B \left (21-8 m-4 m^2\right )-C \left (19-8 m+4 m^2\right )-A \left (35+24 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{f (5+2 m) (7+2 m) \left (3+8 m+4 m^2\right ) \sqrt {c-c \sin (e+f x)}}-\frac {2 c \left (B \left (21-8 m-4 m^2\right )-C \left (19-8 m+4 m^2\right )-A \left (35+24 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (3+2 m) (5+2 m) (7+2 m)}-\frac {2 (7 B+2 C+2 B m+4 C m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m)}+\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{c f (7+2 m)} \]
-2*(2*B*m+4*C*m+7*B+2*C)*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*(B*(-4*m^2-8*m+21)-C*(4*m^2-8*m+19)-A*(4*m^2+24*m+3 5))*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*si n(f*x+e))^(1/2)-2*c*(B*(-4*m^2-8*m+21)-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/(8*m^3+60*m^2+142* m+105)
Time = 6.44 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.95 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\frac {c \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^m \sqrt {c-c \sin (e+f x)} \left (700 A-546 B+494 C+760 A m-380 B m+284 C m+272 A m^2-120 B m^2+136 C m^2+32 A m^3-16 B m^3+16 C m^3+2 \left (3+8 m+4 m^2\right ) (B (7+2 m)-C (13+2 m)) \cos (2 (e+f x))-(1+2 m) \left (4 A \left (35+24 m+4 m^2\right )-4 B \left (63+32 m+4 m^2\right )+C \left (253+80 m+12 m^2\right )\right ) \sin (e+f x)+15 C \sin (3 (e+f x))+46 C m \sin (3 (e+f x))+36 C m^2 \sin (3 (e+f x))+8 C m^3 \sin (3 (e+f x))\right )}{2 f (1+2 m) (3+2 m) (5+2 m) (7+2 m) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]
(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 - 546*B + 494*C + 760*A*m - 380*B*m + 284*C*m + 27 2*A*m^2 - 120*B*m^2 + 136*C*m^2 + 32*A*m^3 - 16*B*m^3 + 16*C*m^3 + 2*(3 + 8*m + 4*m^2)*(B*(7 + 2*m) - C*(13 + 2*m))*Cos[2*(e + f*x)] - (1 + 2*m)*(4* A*(35 + 24*m + 4*m^2) - 4*B*(63 + 32*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 = 1.22 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.87, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3042, 3518, 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.
| \(\displaystyle \int (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m \left (A+B \sin (e+f x)+C \sin (e+f x)^2\right )dx\) |
| \(\Big \downarrow \) 3518 |
| \(\displaystyle \frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}-\frac {2 \int -\frac {1}{2} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2} (a c (C (5-2 m)+A (2 m+7))+a c (2 m B+7 B+2 C+4 C m) \sin (e+f x))dx}{a c (2 m+7)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2} (a c (C (5-2 m)+A (2 m+7))+a c (2 m B+7 B+2 C+4 C m) \sin (e+f x))dx}{a c (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2} (a c (C (5-2 m)+A (2 m+7))+a c (2 m B+7 B+2 C+4 C m) \sin (e+f x))dx}{a c (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}\) |
| \(\Big \downarrow \) 3452 |
| \(\displaystyle \frac {-\frac {a c \left (-A \left (4 m^2+24 m+35\right )+B \left (-4 m^2-8 m+21\right )-C \left (4 m^2-8 m+19\right )\right ) \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2}dx}{2 m+5}-\frac {2 a c (2 B m+7 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}}{a c (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {a c \left (-A \left (4 m^2+24 m+35\right )+B \left (-4 m^2-8 m+21\right )-C \left (4 m^2-8 m+19\right )\right ) \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2}dx}{2 m+5}-\frac {2 a c (2 B m+7 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}}{a c (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle \frac {-\frac {a c \left (-A \left (4 m^2+24 m+35\right )+B \left (-4 m^2-8 m+21\right )-C \left (4 m^2-8 m+19\right )\right ) \left (\frac {4 c \int (\sin (e+f x) a+a)^m \sqrt {c-c \sin (e+f x)}dx}{2 m+3}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}\right )}{2 m+5}-\frac {2 a c (2 B m+7 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}}{a c (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {a c \left (-A \left (4 m^2+24 m+35\right )+B \left (-4 m^2-8 m+21\right )-C \left (4 m^2-8 m+19\right )\right ) \left (\frac {4 c \int (\sin (e+f x) a+a)^m \sqrt {c-c \sin (e+f x)}dx}{2 m+3}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}\right )}{2 m+5}-\frac {2 a c (2 B m+7 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}}{a c (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}\) |
| \(\Big \downarrow \) 3217 |
| \(\displaystyle \frac {-\frac {a c \left (-A \left (4 m^2+24 m+35\right )+B \left (-4 m^2-8 m+21\right )-C \left (4 m^2-8 m+19\right )\right ) \left (\frac {8 c^2 \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) (2 m+3) \sqrt {c-c \sin (e+f x)}}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}\right )}{2 m+5}-\frac {2 a c (2 B m+7 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}}{a c (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}\) |
Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]
(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)) + ((-2*a*c*(7*B + 2*C + 2*B*m + 4*C*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*(B*(21 - 8*m - 4*m^2) - 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))
3.1.19.3.1 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_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (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*B*d*(m + n + 2) - b*c*C*(2*m + 1))* Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, 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]
\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +B \sin \left (f x +e \right )+C \left (\sin ^{2}\left (f x +e \right )\right )\right )d x\]
Time = 0.34 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.75 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (A+B \sin (e+f x)+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 + B + C\right )} c m^{2} - {\left (8 \, {\left (B - C\right )} c m^{3} + 4 \, {\left (11 \, B - 17 \, C\right )} c m^{2} + 2 \, {\left (31 \, B - 55 \, C\right )} c m + 3 \, {\left (7 \, B - 13 \, C\right )} c\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (3 \, A + B - C\right )} c m - {\left (8 \, {\left (A + C\right )} c m^{3} + 4 \, {\left (13 \, A - 6 \, B + 5 \, C\right )} c m^{2} + 2 \, {\left (47 \, A - 48 \, B + 47 \, C\right )} c m + {\left (35 \, A - 42 \, B + 43 \, C\right )} c\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left (35 \, A - 21 \, B + 19 \, C\right )} c - {\left (8 \, {\left (A - B + C\right )} c m^{3} + 4 \, {\left (17 \, A - 13 \, B + 17 \, C\right )} c m^{2} + 2 \, {\left (95 \, A - 63 \, B + 63 \, C\right )} c m + {\left (175 \, A - 147 \, B + 143 \, C\right )} c\right )} \cos \left (f x + e\right ) - {\left (16 \, {\left (A + B + 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 + B - C\right )} c m + {\left (8 \, B c m^{3} + 4 \, {\left (11 \, B - 8 \, C\right )} c m^{2} + 2 \, {\left (31 \, B - 32 \, C\right )} c m + 3 \, {\left (7 \, B - 8 \, C\right )} c\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left (35 \, A - 21 \, B + 19 \, C\right )} c - {\left (8 \, {\left (A - B + C\right )} c m^{3} + 4 \, {\left (13 \, A - 17 \, B + 13 \, C\right )} c m^{2} + 2 \, {\left (47 \, A - 79 \, B + 79 \, C\right )} c m + {\left (35 \, A - 63 \, B + 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} \]
integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(3/2)*(A+B*sin(f*x+e)+C*sin( f*x+e)^2),x, algorithm="fricas")
-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 + B + C)*c*m^2 - (8*(B - C)*c*m^3 + 4*(11*B - 17*C)*c*m^2 + 2*(31*B - 55*C)* c*m + 3*(7*B - 13*C)*c)*cos(f*x + e)^3 - 32*(3*A + B - C)*c*m - (8*(A + C) *c*m^3 + 4*(13*A - 6*B + 5*C)*c*m^2 + 2*(47*A - 48*B + 47*C)*c*m + (35*A - 42*B + 43*C)*c)*cos(f*x + e)^2 - 4*(35*A - 21*B + 19*C)*c - (8*(A - B + C )*c*m^3 + 4*(17*A - 13*B + 17*C)*c*m^2 + 2*(95*A - 63*B + 63*C)*c*m + (175 *A - 147*B + 143*C)*c)*cos(f*x + e) - (16*(A + B + 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 + B - C)*c*m + ( 8*B*c*m^3 + 4*(11*B - 8*C)*c*m^2 + 2*(31*B - 32*C)*c*m + 3*(7*B - 8*C)*c)* cos(f*x + e)^2 + 4*(35*A - 21*B + 19*C)*c - (8*(A - B + C)*c*m^3 + 4*(13*A - 17*B + 13*C)*c*m^2 + 2*(47*A - 79*B + 79*C)*c*m + (35*A - 63*B + 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) - (16*f*m^4 + 128*f*m^3 + 344* f*m^2 + 352*f*m + 105*f)*sin(f*x + e) + 105*f)
Timed out. \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 950 vs. \(2 (302) = 604\).
Time = 0.38 (sec) , antiderivative size = 950, normalized size of antiderivative = 2.95 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\text {Too large to display} \]
integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(3/2)*(A+B*sin(f*x+e)+C*sin( f*x+e)^2),x, algorithm="maxima")
-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) ) - 2*(a^m*c^(3/2)*(2*m + 9) - 2*(2*m^2 + 9*m)*a^m*c^(3/2)*sin(f*x + e)/(c os(f*x + e) + 1) + (4*m^2 + 15)*a^m*c^(3/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + (4*m^2 + 15)*a^m*c^(3/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 2*( 2*m^2 + 9*m)*a^m*c^(3/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^m*c^(3/2) *(2*m + 9)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)*B*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))/ ((8*m^3 + 36*m^2 + 46*m + (8*m^3 + 36*m^2 + 46*m + 15)*sin(f*x + e)^2/(cos (f*x + e) + 1)^2 + 15)*(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)/(c os(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)*sin(f* x + e)^6/(cos(f*x + e) + 1)^6 + 2*a^m*c^(3/2)*(2*m + 13)*sin(f*x + e)^7...
\[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\int { {\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(3/2)*(A+B*sin(f*x+e)+C*sin( f*x+e)^2),x, algorithm="giac")
integrate((C*sin(f*x + e)^2 + B*sin(f*x + e) + A)*(-c*sin(f*x + e) + c)^(3 /2)*(a*sin(f*x + e) + a)^m, x)
Time = 24.21 (sec) , antiderivative size = 790, normalized size of antiderivative = 2.45 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (A+B \sin (e+f x)+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-840\,B+735\,C+1144\,A\,m-128\,B\,m-18\,C\,m+336\,A\,m^2+32\,A\,m^3+32\,B\,m^2+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}-B\,840{}\mathrm {i}+C\,735{}\mathrm {i}+A\,m\,1144{}\mathrm {i}-B\,m\,128{}\mathrm {i}-C\,m\,18{}\mathrm {i}+A\,m^2\,336{}\mathrm {i}+A\,m^3\,32{}\mathrm {i}+B\,m^2\,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\,{\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-210\,B+175\,C+96\,A\,m-88\,B\,m+16\,C\,m+16\,A\,m^2-8\,B\,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}-B\,210{}\mathrm {i}+C\,175{}\mathrm {i}+A\,m\,96{}\mathrm {i}-B\,m\,88{}\mathrm {i}+C\,m\,16{}\mathrm {i}+A\,m^2\,16{}\mathrm {i}-B\,m^2\,8{}\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 )}+\frac {c\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (4\,m^2+8\,m+3\right )\,\left (14\,B-21\,C+4\,B\,m-2\,C\,m\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {c\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (4\,m^2+8\,m+3\right )\,\left (B\,14{}\mathrm {i}-C\,21{}\mathrm {i}+B\,m\,4{}\mathrm {i}-C\,m\,2{}\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 )}\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}} \]
int((a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(3/2)*(A + B*sin(e + f*x) + C*sin(e + f*x)^2),x)
((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 - 840*B + 735*C + 1144*A*m - 128*B*m - 18*C*m + 336*A*m^2 + 32*A*m^3 + 32*B*m^2 + 100*C*m^2 + 8*C*m^3 ))/(4*f*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105)) - (c*exp(e*4i + f*x*4i )*(a + a*sin(e + f*x))^m*(A*1260i - B*840i + C*735i + A*m*1144i - B*m*128i - C*m*18i + A*m^2*336i + A*m^3*32i + B*m^2*32i + C*m^2*100i + C*m^3*8i))/ (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 - 210*B + 175*C + 96*A*m - 88*B*m + 16*C*m + 16*A*m^2 - 8*B*m^2 + 4*C*m^2))/(4*f*(352*m + 344*m^2 + 128*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 - B*210i + C*175i + A*m*96i - B*m*88i + C*m*16i + A*m^2*16i - B*m^2 *8i + C*m^2*4i))/(4*f*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105)) + (c*exp (e*1i + f*x*1i)*(a + a*sin(e + f*x))^m*(8*m + 4*m^2 + 3)*(14*B - 21*C + 4* B*m - 2*C*m))/(4*f*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105)) - (c*exp(e* 6i + f*x*6i)*(a + a*sin(e + f*x))^m*(8*m + 4*m^2 + 3)*(B*14i - C*21i + B*m *4i - C*m*2i))/(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* (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 +...