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

Optimal. Leaf size=285 \[ \frac{8 c^2 \left (A \left (4 m^2+24 m+35\right )+C \left (4 m^2-8 m+19\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) \left (4 m^2+8 m+3\right ) \sqrt{c-c \sin (e+f x)}}+\frac{2 c \left (A \left (4 m^2+24 m+35\right )+C \left (4 m^2-8 m+19\right )\right ) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3) (2 m+5) (2 m+7)}+\frac{2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}-\frac{4 C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7)} \]

[Out]

(8*c^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*(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)) - (4*C*(1 + 2*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))

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Rubi [A]  time = 0.719355, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3040, 2973, 2740, 2738} \[ \frac{8 c^2 \left (A \left (4 m^2+24 m+35\right )+C \left (4 m^2-8 m+19\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) \left (4 m^2+8 m+3\right ) \sqrt{c-c \sin (e+f x)}}+\frac{2 c \left (A \left (4 m^2+24 m+35\right )+C \left (4 m^2-8 m+19\right )\right ) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3) (2 m+5) (2 m+7)}+\frac{2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}-\frac{4 C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7)} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2)*(A + C*Sin[e + f*x]^2),x]

[Out]

(8*c^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*(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)) - (4*C*(1 + 2*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 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]

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+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))-a c C (1+2 m) \sin (e+f x)\right ) \, dx}{a c (7+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))^{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 (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 (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{4 C (1+2 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 (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 (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 (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{4 C (1+2 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 = 3.61078, size = 264, normalized size = 0.93 \[ \frac{c \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) (a (\sin (e+f x)+1))^m \left (-(2 m+1) \left (4 A \left (4 m^2+24 m+35\right )+C \left (12 m^2+80 m+253\right )\right ) \sin (e+f x)+32 A m^3+272 A m^2+760 A m+700 A+8 C m^3 \sin (3 (e+f x))+36 C m^2 \sin (3 (e+f x))-2 C \left (8 m^3+68 m^2+110 m+39\right ) \cos (2 (e+f x))+46 C m \sin (3 (e+f x))+15 C \sin (3 (e+f x))+16 C m^3+136 C m^2+284 C m+494 C\right )}{2 f (2 m+1) (2 m+3) (2 m+5) (2 m+7) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2)*(A + 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 + 494*C + 76
0*A*m + 284*C*m + 272*A*m^2 + 136*C*m^2 + 32*A*m^3 + 16*C*m^3 - 2*C*(39 + 110*m + 68*m^2 + 8*m^3)*Cos[2*(e + f
*x)] - (1 + 2*m)*(4*A*(35 + 24*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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Maple [F]  time = 0.688, 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+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+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+C*sin(f*x+e)^2),x)

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Maxima [B]  time = 1.72613, size = 875, normalized size = 3.07 \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+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)) + 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 = 2.00517, size = 1199, normalized size = 4.21 \begin{align*} -\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 + C\right )} c m^{2} +{\left (8 \, C c m^{3} + 68 \, C c m^{2} + 110 \, C c m + 39 \, C c\right )} \cos \left (f x + e\right )^{3} - 32 \,{\left (3 \, A - C\right )} c m -{\left (8 \,{\left (A + C\right )} c m^{3} + 4 \,{\left (13 \, A + 5 \, C\right )} c m^{2} + 94 \,{\left (A + C\right )} c m +{\left (35 \, A + 43 \, C\right )} c\right )} \cos \left (f x + e\right )^{2} - 4 \,{\left (35 \, A + 19 \, C\right )} c -{\left (8 \,{\left (A + C\right )} c m^{3} + 68 \,{\left (A + C\right )} c m^{2} + 2 \,{\left (95 \, A + 63 \, C\right )} c m +{\left (175 \, A + 143 \, C\right )} c\right )} \cos \left (f x + e\right ) -{\left (16 \,{\left (A + 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 - C\right )} c m - 8 \,{\left (4 \, C c m^{2} + 8 \, C c m + 3 \, C c\right )} \cos \left (f x + e\right )^{2} + 4 \,{\left (35 \, A + 19 \, C\right )} c -{\left (8 \,{\left (A + C\right )} c m^{3} + 52 \,{\left (A + C\right )} c m^{2} + 2 \,{\left (47 \, A + 79 \, C\right )} c m +{\left (35 \, A + 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} \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+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 + C)*c*m^2 + (8*C*c*m^3 + 68*C*c*m^2 +
110*C*c*m + 39*C*c)*cos(f*x + e)^3 - 32*(3*A - C)*c*m - (8*(A + C)*c*m^3 + 4*(13*A + 5*C)*c*m^2 + 94*(A + C)*
c*m + (35*A + 43*C)*c)*cos(f*x + e)^2 - 4*(35*A + 19*C)*c - (8*(A + C)*c*m^3 + 68*(A + C)*c*m^2 + 2*(95*A + 63
*C)*c*m + (175*A + 143*C)*c)*cos(f*x + e) - (16*(A + C)*c*m^2 + (8*C*c*m^3 + 36*C*c*m^2 + 46*C*c*m + 15*C*c)*c
os(f*x + e)^3 + 32*(3*A - C)*c*m - 8*(4*C*c*m^2 + 8*C*c*m + 3*C*c)*cos(f*x + e)^2 + 4*(35*A + 19*C)*c - (8*(A
+ C)*c*m^3 + 52*(A + C)*c*m^2 + 2*(47*A + 79*C)*c*m + (35*A + 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 + 34
4*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+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+C*sin(f*x+e)^2),x, algorithm="giac")

[Out]

Timed out