3.18 \(\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} (A+B \sin (e+f x)+C \sin ^2(e+f x)) \, dx\)
Optimal. Leaf size=435 \[ -\frac{16 c^2 \left (-A \left (4 m^2+32 m+63\right )+B \left (-4 m^2-8 m+45\right )-C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9) \left (4 m^2+16 m+15\right )}-\frac{64 c^3 \left (-A \left (4 m^2+32 m+63\right )+B \left (-4 m^2-8 m+45\right )-C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9) \left (4 m^2+8 m+3\right ) \sqrt{c-c \sin (e+f x)}}-\frac{2 c \left (-A \left (4 m^2+32 m+63\right )+B \left (-4 m^2-8 m+45\right )-C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9)}-\frac{2 (2 B m+9 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9)}+\frac{2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)} \]
[Out]
(-64*c^3*(B*(45 - 8*m - 4*m^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)/(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*(B*(45 - 8*m - 4
*m^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*(B*(45 - 8*m - 4*m^2) - C*(39 - 16*m + 4*m^2) - A*(6
3 + 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)) - (2*(9*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])^(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))
________________________________________________________________________________________
Rubi [A] time = 0.891801, antiderivative size = 435, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used =
{3039, 2973, 2740, 2738} \[ -\frac{16 c^2 \left (-A \left (4 m^2+32 m+63\right )+B \left (-4 m^2-8 m+45\right )-C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9) \left (4 m^2+16 m+15\right )}-\frac{64 c^3 \left (-A \left (4 m^2+32 m+63\right )+B \left (-4 m^2-8 m+45\right )-C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9) \left (4 m^2+8 m+3\right ) \sqrt{c-c \sin (e+f x)}}-\frac{2 c \left (-A \left (4 m^2+32 m+63\right )+B \left (-4 m^2-8 m+45\right )-C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9)}-\frac{2 (2 B m+9 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9)}+\frac{2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]
[Out]
(-64*c^3*(B*(45 - 8*m - 4*m^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)/(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*(B*(45 - 8*m - 4
*m^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*(B*(45 - 8*m - 4*m^2) - C*(39 - 16*m + 4*m^2) - A*(6
3 + 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)) - (2*(9*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])^(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 3039
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] + 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*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]
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 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 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)]
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+B \sin (e+f x)+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))-\frac{1}{2} a c (9 B+2 C+2 B m+4 C m) \sin (e+f x)\right ) \, dx}{a c (9+2 m)}\\ &=-\frac{2 (9 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))^{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 (B \left (45-8 m-4 m^2\right )-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 (B \left (45-8 m-4 m^2\right )-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{2 (9 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))^{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 (B \left (45-8 m-4 m^2\right )-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 (B \left (45-8 m-4 m^2\right )-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 (B \left (45-8 m-4 m^2\right )-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{2 (9 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))^{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 (B \left (45-8 m-4 m^2\right )-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 (B \left (45-8 m-4 m^2\right )-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 (B \left (45-8 m-4 m^2\right )-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 (B \left (45-8 m-4 m^2\right )-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{2 (9 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))^{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*}
Mathematica [C] time = 7.2327, size = 1029, normalized size = 2.37 \[ \frac{(a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^{5/2} \left (\frac{\left (64 A m^4-16 B m^4+16 C m^4+896 A m^3-208 B m^3+224 C m^3+5280 A m^2-832 B m^2+1416 C m^2+15648 A m-4140 B m+648 C m+18900 A-14175 B+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 B m^4+16 C m^4+896 A m^3-208 B m^3+224 C m^3+5280 A m^2-832 B m^2+1416 C m^2+15648 A m-4140 B m+648 C m+18900 A-14175 B+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 (48 A m^3-24 B m^3+16 C m^3+584 A m^2-316 B m^2+200 C m^2+2356 A m-1706 B m+828 C m+3150 A-3465 B+3150 C\right ) \left (\left (\frac{1}{8}-\frac{i}{8}\right ) \cos \left (\frac{3}{2} (e+f x)\right )-\left (\frac{1}{8}+\frac{i}{8}\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 (48 A m^3-24 B m^3+16 C m^3+584 A m^2-316 B m^2+200 C m^2+2356 A m-1706 B m+828 C m+3150 A-3465 B+3150 C\right ) \left (\left (\frac{1}{8}+\frac{i}{8}\right ) \cos \left (\frac{3}{2} (e+f x)\right )-\left (\frac{1}{8}-\frac{i}{8}\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 (8 A m^2-12 B m^2+8 C m^2+64 A m-124 B m+88 C m+126 A-315 B+378 C\right ) \left (\left (-\frac{1}{8}+\frac{i}{8}\right ) \cos \left (\frac{5}{2} (e+f x)\right )-\left (\frac{1}{8}+\frac{i}{8}\right ) \sin \left (\frac{5}{2} (e+f x)\right )\right )}{(2 m+5) (2 m+7) (2 m+9)}+\frac{\left (8 A m^2-12 B m^2+8 C m^2+64 A m-124 B m+88 C m+126 A-315 B+378 C\right ) \left (\left (-\frac{1}{8}-\frac{i}{8}\right ) \cos \left (\frac{5}{2} (e+f x)\right )-\left (\frac{1}{8}-\frac{i}{8}\right ) \sin \left (\frac{5}{2} (e+f x)\right )\right )}{(2 m+5) (2 m+7) (2 m+9)}+\frac{(4 m B+18 B-45 C-6 C m) \left (\left (\frac{1}{16}-\frac{i}{16}\right ) \cos \left (\frac{7}{2} (e+f x)\right )-\left (\frac{1}{16}+\frac{i}{16}\right ) \sin \left (\frac{7}{2} (e+f x)\right )\right )}{(2 m+7) (2 m+9)}+\frac{(4 m B+18 B-45 C-6 C m) \left (\left (\frac{1}{16}+\frac{i}{16}\right ) \cos \left (\frac{7}{2} (e+f x)\right )-\left (\frac{1}{16}-\frac{i}{16}\right ) \sin \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 + B*Sin[e + f*x] + 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 - 14175*B + 12285*C + 15648*A*m - 4140*B*m + 6
48*C*m + 5280*A*m^2 - 832*B*m^2 + 1416*C*m^2 + 896*A*m^3 - 208*B*m^3 + 224*C*m^3 + 64*A*m^4 - 16*B*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 - 14175*B + 12285*C + 15648*A*m - 4140*B*m + 648*C*m + 5280*A*m^2 - 832*B*m^2 + 1416*C*m
^2 + 896*A*m^3 - 208*B*m^3 + 224*C*m^3 + 64*A*m^4 - 16*B*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)) + ((3150*A - 3465*B + 3150*C + 2
356*A*m - 1706*B*m + 828*C*m + 584*A*m^2 - 316*B*m^2 + 200*C*m^2 + 48*A*m^3 - 24*B*m^3 + 16*C*m^3)*((1/8 - I/8
)*Cos[(3*(e + f*x))/2] - (1/8 + I/8)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((3150
*A - 3465*B + 3150*C + 2356*A*m - 1706*B*m + 828*C*m + 584*A*m^2 - 316*B*m^2 + 200*C*m^2 + 48*A*m^3 - 24*B*m^3
+ 16*C*m^3)*((1/8 + I/8)*Cos[(3*(e + f*x))/2] - (1/8 - I/8)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 +
2*m)*(9 + 2*m)) + ((126*A - 315*B + 378*C + 64*A*m - 124*B*m + 88*C*m + 8*A*m^2 - 12*B*m^2 + 8*C*m^2)*((-1/8 +
I/8)*Cos[(5*(e + f*x))/2] - (1/8 + I/8)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((126*A - 31
5*B + 378*C + 64*A*m - 124*B*m + 88*C*m + 8*A*m^2 - 12*B*m^2 + 8*C*m^2)*((-1/8 - I/8)*Cos[(5*(e + f*x))/2] - (
1/8 - I/8)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((18*B - 45*C + 4*B*m - 6*C*m)*((1/16 - I/
16)*Cos[(7*(e + f*x))/2] - (1/16 + I/16)*Sin[(7*(e + f*x))/2]))/((7 + 2*m)*(9 + 2*m)) + ((18*B - 45*C + 4*B*m
- 6*C*m)*((1/16 + I/16)*Cos[(7*(e + f*x))/2] - (1/16 - I/16)*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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Maple [F] time = 0.725, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sin \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}} \left ( A+B\sin \left ( fx+e \right ) +C \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \, dx \end{align*}
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+B*sin(f*x+e)+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+B*sin(f*x+e)+C*sin(f*x+e)^2),x)
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Maxima [B] time = 2.02756, size = 1787, normalized size = 4.11 \begin{align*} \text{result too large to display} \end{align*}
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+B*sin(f*x+e)+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)) - 2*((4*m^2 + 40*m + 115)*a^m*c^(5/2) - 2*(4*m^3 + 40*m^2 + 115*m)*a^m*c^(5/2)*sin(f*x + e)/(
cos(f*x + e) + 1) + 2*(12*m^3 + 76*m^2 + 97*m + 175)*a^m*c^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - (16*m^3
+ 76*m^2 + 260*m - 175)*a^m*c^(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - (16*m^3 + 76*m^2 + 260*m - 175)*a^m
*c^(5/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 2*(12*m^3 + 76*m^2 + 97*m + 175)*a^m*c^(5/2)*sin(f*x + e)^5/(co
s(f*x + e) + 1)^5 - 2*(4*m^3 + 40*m^2 + 115*m)*a^m*c^(5/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + (4*m^2 + 40*m
+ 115)*a^m*c^(5/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7)*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))/((16*m^4 + 128*m^3 + 344*m^2 + 352*m + (16*m^4 + 128*m^3 + 344*
m^2 + 352*m + 105)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 105)*(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 + 56
7)*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(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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Fricas [B] time = 2.35935, size = 2410, normalized size = 5.54 \begin{align*} \text{result too large to display} \end{align*}
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+B*sin(f*x+e)+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 + B + C)*c
^2*m^2 + (16*(B - C)*c^2*m^4 + 16*(9*B - 14*C)*c^2*m^3 + 8*(52*B - 97*C)*c^2*m^2 + 4*(111*B - 226*C)*c^2*m + 1
5*(9*B - 19*C)*c^2)*cos(f*x + e)^4 + 256*(4*A + B - 2*C)*c^2*m - (16*(A - 2*B + 3*C)*c^2*m^4 + 16*(10*A - 23*B
+ 32*C)*c^2*m^3 + 8*(65*A - 169*B + 253*C)*c^2*m^2 + 4*(150*A - 417*B + 656*C)*c^2*m + 3*(63*A - 180*B + 289*
C)*c^2)*cos(f*x + e)^3 + 96*(21*A - 15*B + 13*C)*c^2 + (16*(A - B + C)*c^2*m^4 + 32*(7*A - 5*B + 7*C)*c^2*m^3
+ 8*(133*A - 97*B + 85*C)*c^2*m^2 + 8*(233*A - 235*B + 233*C)*c^2*m + 3*(231*A - 255*B + 263*C)*c^2)*cos(f*x +
e)^2 + 2*(16*(A - B + C)*c^2*m^4 + 192*(A - B + C)*c^2*m^3 + 8*(107*A - 99*B + 107*C)*c^2*m^2 + 16*(109*A - 8
9*B + 85*C)*c^2*m + 3*(483*A - 435*B + 419*C)*c^2)*cos(f*x + e) + (128*(A + B + C)*c^2*m^2 + (16*C*c^2*m^4 + 1
28*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 + 256*(4*A + B - 2*C)*c^2*m - (16*(B -
2*C)*c^2*m^4 + 16*(9*B - 22*C)*c^2*m^3 + 32*(13*B - 35*C)*c^2*m^2 + 4*(111*B - 314*C)*c^2*m + 15*(9*B - 26*C)*
c^2)*cos(f*x + e)^3 + 96*(21*A - 15*B + 13*C)*c^2 - (16*(A - B + C)*c^2*m^4 + 32*(5*A - 7*B + 5*C)*c^2*m^3 + 8
*(65*A - 117*B + 113*C)*c^2*m^2 + 24*(25*A - 51*B + 57*C)*c^2*m + 9*(21*A - 45*B + 53*C)*c^2)*cos(f*x + e)^2 -
2*(16*(A - B + C)*c^2*m^4 + 192*(A - B + C)*c^2*m^3 + 8*(99*A - 107*B + 99*C)*c^2*m^2 + 16*(77*A - 97*B + 101
*C)*c^2*m + 3*(147*A - 195*B + 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 + 1840*f*m^3 + 38
00*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)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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+B*sin(f*x+e)+C*sin(f*x+e)**2),x)
[Out]
Timed out
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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+B*sin(f*x+e)+C*sin(f*x+e)^2),x, algorithm="giac")
[Out]
Timed out