3.1 \(\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} (A+C \sin ^2(e+f x)) \, dx\)

Optimal. Leaf size=384 \[ \frac {64 c^3 \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9) \left (4 m^2+8 m+3\right ) \sqrt {c-c \sin (e+f x)}}+\frac {16 c^2 \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9) \left (4 m^2+16 m+15\right )}+\frac {2 c \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}-\frac {4 C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9)} \]

[Out]

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+39)+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)

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Rubi [A]  time = 0.93, antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3040, 2973, 2740, 2738} \[ \frac {16 c^2 \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9) \left (4 m^2+16 m+15\right )}+\frac {64 c^3 \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9) \left (4 m^2+8 m+3\right ) \sqrt {c-c \sin (e+f x)}}+\frac {2 c \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}-\frac {4 C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9)} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2)*(A + C*Sin[e + f*x]^2),x]

[Out]

(64*c^3*(C*(39 - 16*m + 4*m^2) + A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(7 +
2*m)*(9 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (16*c^2*(C*(39 - 16*m + 4*m^2) + A*(63 + 32*m +
4*m^2))*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)*(15 + 16*m + 4*m^
2)) + (2*c*(C*(39 - 16*m + 4*m^2) + A*(63 + 32*m + 4*m^2))*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)) - (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)) + (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))

Rule 2738

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)]

Rule 2740

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Sim
p[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n)/(f*(m + n)), x] + Dist[(a*(2*m - 1))/(m
+ n), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && E
qQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[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])

Rule 2973

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(B*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(f*(
m + n + 1)), x] - Dist[(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]

Rule 3040

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] + Dist[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]

Rubi steps

\begin {align*} \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 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)}-\frac {2 \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (-\frac {1}{2} a c (C (7-2 m)+A (9+2 m))-a c C (1+2 m) \sin (e+f x)\right ) \, dx}{a c (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)}+\frac {\left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx}{(7+2 m) (9+2 m)}\\ &=\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)}+\frac {\left (8 c \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right )\right ) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx}{(5+2 m) (7+2 m) (9+2 m)}\\ &=\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 (3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\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)}+\frac {\left (32 c^2 \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right )\right ) \int (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx}{(3+2 m) (5+2 m) (7+2 m) (9+2 m)}\\ &=\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 (1+2 m) (3+2 m) (5+2 m) (7+2 m) (9+2 m) \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 (3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\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)}\\ \end {align*}

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Mathematica [C]  time = 6.98, size = 899, normalized size = 2.34 \[ \frac {(a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^{5/2} \left (\frac {\left (64 A m^4+16 C m^4+896 A m^3+224 C m^3+5280 A m^2+1416 C m^2+15648 A m+648 C m+18900 A+12285 C\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 )}{(2 m+1) (2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac {\left (64 A m^4+16 C m^4+896 A m^3+224 C m^3+5280 A m^2+1416 C m^2+15648 A m+648 C m+18900 A+12285 C\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 )}{(2 m+1) (2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac {\left (24 A m^3+8 C m^3+292 A m^2+100 C m^2+1178 A m+414 C m+1575 A+1575 C\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 )}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac {\left (24 A m^3+8 C m^3+292 A m^2+100 C m^2+1178 A m+414 C m+1575 A+1575 C\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 )}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac {\left (4 A m^2+4 C m^2+32 A m+44 C m+63 A+189 C\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 )}{(2 m+5) (2 m+7) (2 m+9)}+\frac {\left (4 A m^2+4 C m^2+32 A m+44 C m+63 A+189 C\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 )}{(2 m+5) (2 m+7) (2 m+9)}+\frac {(2 m+15) \left (\left (\frac {3}{16}-\frac {3 i}{16}\right ) C \sin \left (\frac {7}{2} (e+f x)\right )-\left (\frac {3}{16}+\frac {3 i}{16}\right ) C \cos \left (\frac {7}{2} (e+f x)\right )\right )}{(2 m+7) (2 m+9)}+\frac {(2 m+15) \left (\left (\frac {3}{16}+\frac {3 i}{16}\right ) C \sin \left (\frac {7}{2} (e+f x)\right )-\left (\frac {3}{16}-\frac {3 i}{16}\right ) C \cos \left (\frac {7}{2} (e+f x)\right )\right )}{(2 m+7) (2 m+9)}+\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 )}{2 m+9}+\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 )}{2 m+9}\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2)*(A + C*Sin[e + f*x]^2),x]

[Out]

((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 + 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)) + ((1575*A + 1575*C + 1178*A*m + 4
14*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 + 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)) + ((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 + 2*m)) + ((1/16 + I/16)*C*Cos[(9*(e + f*x))/2] + (1/16 - I/16)*
C*Sin[(9*(e + f*x))/2])/(9 + 2*m) + ((1/16 - I/16)*C*Cos[(9*(e + f*x))/2] + (1/16 + I/16)*C*Sin[(9*(e + f*x))/
2])/(9 + 2*m)))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5)

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fricas [B]  time = 0.56, size = 777, normalized size = 2.02 \[ \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} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2)*(A+C*sin(f*x+e)^2),x, algorithm="fricas")

[Out]

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 + 776*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*(13
3*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 + 16*(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 + 51
2*(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 + 1840*f*m^3 + 3800*f*m^2 + 3378*f*m + (32*f*m^5 + 400*f*m^4 + 18
40*f*m^3 + 3800*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 + 337
8*f*m + 945*f)*sin(f*x + e) + 945*f)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sin \left (f x + e\right )^{2} + A\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2)*(A+C*sin(f*x+e)^2),x, algorithm="giac")

[Out]

integrate((C*sin(f*x + e)^2 + A)*(-c*sin(f*x + e) + c)^(5/2)*(a*sin(f*x + e) + a)^m, x)

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maple [F]  time = 3.66, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +C \left (\sin ^{2}\left (f x +e \right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2)*(A+C*sin(f*x+e)^2),x)

[Out]

int((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2)*(A+C*sin(f*x+e)^2),x)

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maxima [B]  time = 0.88, size = 892, normalized size = 2.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2)*(A+C*sin(f*x+e)^2),x, algorithm="maxima")

[Out]

-2*(((4*m^2 + 24*m + 43)*a^m*c^(5/2) - (12*m^2 + 40*m - 15)*a^m*c^(5/2)*sin(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*log(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 + 219)*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 - 3
15)*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(f*x + e) + 1)^2 + 1))/((32*m^5 + 400*m^4 + 1840*m^3 + 3800*m^2
+ 3378*m + 2*(32*m^5 + 400*m^4 + 1840*m^3 + 3800*m^2 + 3378*m + 945)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + (32
*m^5 + 400*m^4 + 1840*m^3 + 3800*m^2 + 3378*m + 945)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 945)*(sin(f*x + e)^
2/(cos(f*x + e) + 1)^2 + 1)^(5/2)))/f

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mupad [B]  time = 23.16, size = 1110, normalized size = 2.89 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + C*sin(e + f*x)^2)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(5/2),x)

[Out]

((c - c*sin(e + f*x))^(1/2)*((C*c^2*(a + a*sin(e + f*x))^m*(m*352i + m^2*344i + 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 + 1
6*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*1
416i + C*m^3*224i + C*m^4*16i))/(4*f*(m*3378i + 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 + 414*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*3800i + 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*1
840i + m^4*400i + m^5*32i + 945i)) - (3*C*c^2*exp(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 + 945i)) - (3*C*c^2*exp(e*8i
+ f*x*8i)*(a + a*sin(e + f*x))^m*(m*720i + m^2*632i + m^3*192i + m^4*16i + 225i))/(8*f*(m*3378i + m^2*3800i +
m^3*1840i + m^4*400i + m^5*32i + 945i)) - (c^2*exp(e*7i + f*x*7i)*(a + a*sin(e + f*x))^m*(8*m + 4*m^2 + 3)*(63
*A + 189*C + 32*A*m + 44*C*m + 4*A*m^2 + 4*C*m^2))/(2*f*(m*3378i + m^2*3800i + m^3*1840i + m^4*400i + m^5*32i
+ 945i)) - (c^2*exp(e*2i + f*x*2i)*(a + a*sin(e + f*x))^m*(8*m + 4*m^2 + 3)*(A*63i + C*189i + A*m*32i + C*m*44
i + A*m^2*4i + C*m^2*4i))/(2*f*(m*3378i + m^2*3800i + m^3*1840i + m^4*400i + m^5*32i + 945i))))/(exp(e*5i + f*
x*5i) + (exp(e*4i + f*x*4i)*(3378*m + 3800*m^2 + 1840*m^3 + 400*m^4 + 32*m^5 + 945))/(m*3378i + m^2*3800i + m^
3*1840i + m^4*400i + m^5*32i + 945i))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**(5/2)*(A+C*sin(f*x+e)**2),x)

[Out]

Timed out

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