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]
(-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))
________________________________________________________________________________________
Rubi [A] time = 0.711224, antiderivative size = 322, normalized size of antiderivative = 1.,
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 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))^{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*}
Mathematica [A] time = 5.13681, 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]))
________________________________________________________________________________________
Maple [F] time = 0.715, 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{3}{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))^(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 = 1.86125, size = 1283, normalized size = 3.98 \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))^(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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Fricas [A] time = 2.06133, size = 1463, normalized size = 4.54 \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))^(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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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))**(3/2)*(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))^(3/2)*(A+B*sin(f*x+e)+C*sin(f*x+e)^2),x, algorithm="giac")
[Out]
Timed out