3.19 \(\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} (A+B \sin (e+f x)+C \sin ^2(e+f x)) \, dx\)

Optimal. Leaf size=322 \[ -\frac {8 c^2 \left (-A \left (4 m^2+24 m+35\right )+B \left (-4 m^2-8 m+21\right )-C \left (4 m^2-8 m+19\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) \left (4 m^2+8 m+3\right ) \sqrt {c-c \sin (e+f x)}}-\frac {2 c \left (-A \left (4 m^2+24 m+35\right )+B \left (-4 m^2-8 m+21\right )-C \left (4 m^2-8 m+19\right )\right ) \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3) (2 m+5) (2 m+7)}-\frac {2 (2 B m+7 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)} \]

[Out]

-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
+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*(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)

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Rubi [A]  time = 0.71, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3039, 2973, 2740, 2738} \[ -\frac {8 c^2 \left (-A \left (4 m^2+24 m+35\right )+B \left (-4 m^2-8 m+21\right )-C \left (4 m^2-8 m+19\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) \left (4 m^2+8 m+3\right ) \sqrt {c-c \sin (e+f x)}}-\frac {2 c \left (-A \left (4 m^2+24 m+35\right )+B \left (-4 m^2-8 m+21\right )-C \left (4 m^2-8 m+19\right )\right ) \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3) (2 m+5) (2 m+7)}-\frac {2 (2 B m+7 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)} \]

Antiderivative was successfully verified.

[In]

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]

[Out]

(-8*c^2*(B*(21 - 8*m - 4*m^2) - C*(19 - 8*m + 4*m^2) - A*(35 + 24*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x]
)^m)/(f*(5 + 2*m)*(7 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) - (2*c*(B*(21 - 8*m - 4*m^2) - C*(19 -
8*m + 4*m^2) - A*(35 + 24*m + 4*m^2))*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)) - (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)) + (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))

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 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]

Rubi steps

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

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Mathematica [A]  time = 5.11, size = 306, normalized size = 0.95 \[ \frac {c \sqrt {c-c \sin (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) (a (\sin (e+f x)+1))^m \left (-(2 m+1) \left (4 A \left (4 m^2+24 m+35\right )-4 B \left (4 m^2+32 m+63\right )+C \left (12 m^2+80 m+253\right )\right ) \sin (e+f x)+32 A m^3+272 A m^2+760 A m+700 A+2 \left (4 m^2+8 m+3\right ) (B (2 m+7)-C (2 m+13)) \cos (2 (e+f x))-16 B m^3-120 B m^2-380 B m-546 B+8 C m^3 \sin (3 (e+f x))+36 C m^2 \sin (3 (e+f x))+46 C m \sin (3 (e+f x))+15 C \sin (3 (e+f x))+16 C m^3+136 C m^2+284 C m+494 C\right )}{2 f (2 m+1) (2 m+3) (2 m+5) (2 m+7) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

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]

[Out]

(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 + 49
4*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*(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]))

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fricas [A]  time = 0.54, size = 563, normalized size = 1.75 \[ -\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} \]

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))^(3/2)*(A+B*sin(f*x+e)+C*sin(f*x+e)^2),x, algorithm="fricas")

[Out]

-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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]

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))^(3/2)*(A+B*sin(f*x+e)+C*sin(f*x+e)^2),x, algorithm="giac")

[Out]

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)

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maple [F]  time = 6.07, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +B \sin \left (f x +e \right )+C \left (\sin ^{2}\left (f x +e \right )\right )\right )\, dx \]

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))^(3/2)*(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))^(3/2)*(A+B*sin(f*x+e)+C*sin(f*x+e)^2),x)

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maxima [B]  time = 0.59, size = 950, normalized size = 2.95 \[ \text {result too large to display} \]

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))^(3/2)*(A+B*sin(f*x+e)+C*sin(f*x+e)^2),x, algorithm="maxima")

[Out]

-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
)/(cos(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*lo
g(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)/(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)*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/(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

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mupad [B]  time = 23.16, size = 790, normalized size = 2.45 \[ \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}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

((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 + 344*m^2 + 128*m^3 + 16*m^4 + 105))

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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))**(3/2)*(A+B*sin(f*x+e)+C*sin(f*x+e)**2),x)

[Out]

Timed out

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