3.764 \(\int \frac{1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=682 \[ -\frac{d \left (a^2 b^2 d \left (13 c^2-29 d^2\right )+8 a^4 d^3-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{4 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right ) (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{\left (a^2 b^2 d \left (13 c^2-29 d^2\right )+8 a^4 d^3-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right ) (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{b \left (-2 a^2 b^2 \left (4 c^2-19 d^2\right )+28 a^3 b c d-35 a^4 d^2-4 a b^3 c d-b^4 \left (4 c^2+15 d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 f (a-b)^2 (a+b)^3 (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}-\frac{b \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}} \]

[Out]

-(d*(8*a^4*d^3 + a^2*b^2*d*(13*c^2 - 29*d^2) - b^4*d*(7*c^2 - 15*d^2) - 6*a*b^3*c*(c^2 - d^2))*Cos[e + f*x])/(
4*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c
- a*d)*f*(a + b*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]]) + (b^2*(6*a*b*c - 11*a^2*d + 5*b^2*d)*Cos[e + f*x])
/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]) - ((8*a^4*d^3 + a^2*b^2*d*(13
*c^2 - 29*d^2) - b^4*d*(7*c^2 - 15*d^2) - 6*a*b^3*c*(c^2 - d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*
Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (
b*(6*a*b*c - 11*a^2*d + 5*b^2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d
)])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - (b*(28*a^3*b*c*d - 4*a*b^3*c*d - 35*a^4*d^2 -
2*a^2*b^2*(4*c^2 - 19*d^2) - b^4*(4*c^2 + 15*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d
)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*(a + b)^3*(b*c - a*d)^3*f*Sqrt[c + d*Sin[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 2.76845, antiderivative size = 682, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.37, Rules used = {2802, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac{d \left (a^2 b^2 d \left (13 c^2-29 d^2\right )+8 a^4 d^3-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{4 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right ) (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{\left (a^2 b^2 d \left (13 c^2-29 d^2\right )+8 a^4 d^3-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right ) (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{b \left (-2 a^2 b^2 \left (4 c^2-19 d^2\right )+28 a^3 b c d-35 a^4 d^2-4 a b^3 c d-b^4 \left (4 c^2+15 d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 f (a-b)^2 (a+b)^3 (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}-\frac{b \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2)),x]

[Out]

-(d*(8*a^4*d^3 + a^2*b^2*d*(13*c^2 - 29*d^2) - b^4*d*(7*c^2 - 15*d^2) - 6*a*b^3*c*(c^2 - d^2))*Cos[e + f*x])/(
4*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c
- a*d)*f*(a + b*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]]) + (b^2*(6*a*b*c - 11*a^2*d + 5*b^2*d)*Cos[e + f*x])
/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]) - ((8*a^4*d^3 + a^2*b^2*d*(13
*c^2 - 29*d^2) - b^4*d*(7*c^2 - 15*d^2) - 6*a*b^3*c*(c^2 - d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*
Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (
b*(6*a*b*c - 11*a^2*d + 5*b^2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d
)])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - (b*(28*a^3*b*c*d - 4*a*b^3*c*d - 35*a^4*d^2 -
2*a^2*b^2*(4*c^2 - 19*d^2) - b^4*(4*c^2 + 15*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d
)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*(a + b)^3*(b*c - a*d)^3*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2802

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 -
b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n
*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n
+ 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !
(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3055

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3059

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) +
(f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[C/(b*d), Int[Sqrt[a + b*Sin[e + f*x]]
, x], x] - Dist[1/(b*d), Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e +
f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 3002

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[
(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[B/d, Int[(a + b*Sin[e + f*x])^m, x], x] - Dist[(B*c - A*d)/d, Int[(a +
b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2807

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d*
Sin[e + f*x])/(c + d)]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[c + d]), x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

Rubi steps

\begin{align*} \int \frac{1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx &=\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}-\frac{\int \frac{\frac{1}{2} \left (-4 a b c+4 a^2 d-5 b^2 d\right )+b (b c-2 a d) \sin (e+f x)+\frac{3}{2} b^2 d \sin ^2(e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)}\\ &=\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \left (6 a b c-11 a^2 d+5 b^2 d\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}+\frac{\int \frac{\frac{1}{4} \left (-16 a^3 b c d+10 a b^3 c d+8 a^4 d^2+a^2 b^2 \left (8 c^2-29 d^2\right )+b^4 \left (4 c^2+15 d^2\right )\right )+\frac{1}{2} b d \left (3 a^2 b c+3 b^3 c-8 a^3 d+2 a b^2 d\right ) \sin (e+f x)-\frac{1}{4} b^2 d \left (6 a b c-11 a^2 d+5 b^2 d\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx}{2 \left (a^2-b^2\right )^2 (b c-a d)^2}\\ &=-\frac{d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \left (6 a b c-11 a^2 d+5 b^2 d\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}+\frac{\int \frac{\frac{1}{8} \left (-8 a^5 c d^3+2 a b^4 c d \left (3 c^2-7 d^2\right )-8 a^3 b^2 c d \left (3 c^2-5 d^2\right )+24 a^4 b d^2 \left (c^2-d^2\right )+b^5 \left (4 c^4+11 c^2 d^2-15 d^4\right )+a^2 b^3 \left (8 c^4-41 c^2 d^2+33 d^4\right )\right )-\frac{1}{4} d \left (4 a^4 b c d^2+4 a^5 d^3-b^5 c \left (c^2-5 d^2\right )+4 a^3 b^2 d \left (3 c^2-5 d^2\right )-2 a b^4 d \left (3 c^2-5 d^2\right )-a^2 b^3 c \left (5 c^2+3 d^2\right )\right ) \sin (e+f x)-\frac{1}{8} b d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{\left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right )}\\ &=-\frac{d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \left (6 a b c-11 a^2 d+5 b^2 d\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}-\frac{\int \frac{\frac{1}{8} b^2 d \left (c^2-d^2\right ) \left (11 a^3 b c d+a b^3 c d-24 a^4 d^2-a^2 b^2 \left (2 c^2-33 d^2\right )-b^4 \left (4 c^2+15 d^2\right )\right )+\frac{1}{8} b^3 d (b c-a d) \left (6 a b c-11 a^2 d+5 b^2 d\right ) \left (c^2-d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{b \left (a^2-b^2\right )^2 d (b c-a d)^3 \left (c^2-d^2\right )}-\frac{\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right )}\\ &=-\frac{d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \left (6 a b c-11 a^2 d+5 b^2 d\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}-\frac{\left (b \left (6 a b c-11 a^2 d+5 b^2 d\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)^2}-\frac{\left (b \left (28 a^3 b c d-4 a b^3 c d-35 a^4 d^2-2 a^2 b^2 \left (4 c^2-19 d^2\right )-b^4 \left (4 c^2+15 d^2\right )\right )\right ) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)^3}-\frac{\left (\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}\\ &=-\frac{d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \left (6 a b c-11 a^2 d+5 b^2 d\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}-\frac{\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left (b \left (6 a b c-11 a^2 d+5 b^2 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left (b \left (28 a^3 b c d-4 a b^3 c d-35 a^4 d^2-2 a^2 b^2 \left (4 c^2-19 d^2\right )-b^4 \left (4 c^2+15 d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \left (6 a b c-11 a^2 d+5 b^2 d\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}-\frac{\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{b \left (6 a b c-11 a^2 d+5 b^2 d\right ) F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f \sqrt{c+d \sin (e+f x)}}-\frac{b \left (28 a^3 b c d-4 a b^3 c d-35 a^4 d^2-2 a^2 b^2 \left (4 c^2-19 d^2\right )-b^4 \left (4 c^2+15 d^2\right )\right ) \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{4 (a-b)^2 (a+b)^3 (b c-a d)^3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 8.70336, size = 1318, normalized size = 1.93 \[ \frac{\sqrt{c+d \sin (e+f x)} \left (-\frac{2 \cos (e+f x) d^4}{(b c-a d)^3 \left (c^2-d^2\right ) (c+d \sin (e+f x))}-\frac{7 d \cos (e+f x) b^5+6 a c \cos (e+f x) b^4-13 a^2 d \cos (e+f x) b^3}{4 \left (a^2-b^2\right )^2 (a d-b c)^3 (a+b \sin (e+f x))}+\frac{b^3 \cos (e+f x)}{2 \left (a^2-b^2\right ) (a d-b c)^2 (a+b \sin (e+f x))^2}\right )}{f}+\frac{-\frac{2 \left (16 c d^3 a^5+56 b d^4 a^4-48 b c^2 d^2 a^4-80 b^2 c d^3 a^3+48 b^2 c^3 d a^3-16 b^3 c^4 a^2-95 b^3 d^4 a^2+95 b^3 c^2 d^2 a^2+34 b^4 c d^3 a-18 b^4 c^3 d a-8 b^5 c^4+45 b^5 d^4-29 b^5 c^2 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left (16 d^4 a^5+16 b c d^3 a^4-80 b^2 d^4 a^3+48 b^2 c^2 d^2 a^3-12 b^3 c d^3 a^2-20 b^3 c^3 d a^2+40 b^4 d^4 a-24 b^4 c^2 d^2 a+20 b^5 c d^3-4 b^5 c^3 d\right ) \cos (e+f x) \left ((b c-a d) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )+a d \Pi \left (\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )\right ) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left (-15 d^4 b^5+7 c^2 d^2 b^5-6 a c d^3 b^4+6 a c^3 d b^4+29 a^2 d^4 b^3-13 a^2 c^2 d^2 b^3-8 a^4 d^4 b\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (c-d) (b c-a d) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )+d \left (\left (2 a^2-b^2\right ) d \Pi \left (\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )-2 (a+b) (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )\right )\right ) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left (-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right ) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{16 (a-b)^2 (a+b)^2 (c-d) (c+d) (a d-b c)^3 f} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2)),x]

[Out]

(Sqrt[c + d*Sin[e + f*x]]*((b^3*Cos[e + f*x])/(2*(a^2 - b^2)*(-(b*c) + a*d)^2*(a + b*Sin[e + f*x])^2) - (6*a*b
^4*c*Cos[e + f*x] - 13*a^2*b^3*d*Cos[e + f*x] + 7*b^5*d*Cos[e + f*x])/(4*(a^2 - b^2)^2*(-(b*c) + a*d)^3*(a + b
*Sin[e + f*x])) - (2*d^4*Cos[e + f*x])/((b*c - a*d)^3*(c^2 - d^2)*(c + d*Sin[e + f*x]))))/f + ((-2*(-16*a^2*b^
3*c^4 - 8*b^5*c^4 + 48*a^3*b^2*c^3*d - 18*a*b^4*c^3*d - 48*a^4*b*c^2*d^2 + 95*a^2*b^3*c^2*d^2 - 29*b^5*c^2*d^2
+ 16*a^5*c*d^3 - 80*a^3*b^2*c*d^3 + 34*a*b^4*c*d^3 + 56*a^4*b*d^4 - 95*a^2*b^3*d^4 + 45*b^5*d^4)*EllipticPi[(
2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[
e + f*x]]) - ((2*I)*(-20*a^2*b^3*c^3*d - 4*b^5*c^3*d + 48*a^3*b^2*c^2*d^2 - 24*a*b^4*c^2*d^2 + 16*a^4*b*c*d^3
- 12*a^2*b^3*c*d^3 + 20*b^5*c*d^3 + 16*a^5*d^4 - 80*a^3*b^2*d^4 + 40*a*b^4*d^4)*Cos[e + f*x]*((b*c - a*d)*Elli
pticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(
b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x
])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d
)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x])
+ (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(6*a*b^4*c^3*d - 13*a^2*b^3*c^2*d^2 + 7*b^5*c^2*d^2 - 6*a*b^4*c*d^3
- 8*a^4*b*d^4 + 29*a^2*b^3*d^4 - 15*b^5*d^4)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE
[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*Elli
pticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(
b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d -
d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d
*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[
e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d
^2)]))/(16*(a - b)^2*(a + b)^2*(c - d)*(c + d)*(-(b*c) + a*d)^3*f)

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Maple [B]  time = 11.483, size = 2099, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(-2*d^2/(a*d-b*c)^3*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f
*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(-c/d+a/b)*Ellipti
cPi(((c+d*sin(f*x+e))/(c-d))^(1/2),(-c/d+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2))-b*d/(a*d-b*c)^2*(-b^2/(a^3*d-a^2*b
*c-a*b^2*d+b^3*c)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(a+b*sin(f*x+e))-a*d/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(
c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin
(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-b*d/(a^3*d-a^2*b*
c-a*b^2*d+b^3*c)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d
))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+
d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))+(3*a^2*d-2*a*b*c-b^2*d)/(a^3*d-a^2*b
*c-a*b^2*d+b^3*c)/b*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(
c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(-c/d+a/b)*EllipticPi(((c+d*sin(f*x+e))/(c-d))^(1/2),(-c/d
+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2)))-b/(a*d-b*c)*(-1/2*b^2/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(-(-d*sin(f*x+e)-c)*c
os(f*x+e)^2)^(1/2)/(a+b*sin(f*x+e))^2-3/4*b^2*(3*a^2*d-2*a*b*c-b^2*d)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)^2*(-(-d*si
n(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(a+b*sin(f*x+e))-1/4*d*(7*a^3*d-4*a^2*b*c-a*b^2*d-2*b^3*c)/(a^3*d-a^2*b*c-a*b^
2*d+b^3*c)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(
1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-3/4
*b*d*(3*a^2*d-2*a*b*c-b^2*d)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(
f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*Ellipti
cE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^
(1/2)))+1/4*(15*a^4*d^2-20*a^3*b*c*d+8*a^2*b^2*c^2-6*a^2*b^2*d^2-4*a*b^3*c*d+4*b^4*c^2+3*b^4*d^2)/(a^3*d-a^2*b
*c-a*b^2*d+b^3*c)^2/b*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d
/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(-c/d+a/b)*EllipticPi(((c+d*sin(f*x+e))/(c-d))^(1/2),(-c
/d+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2)))+d^3/(a*d-b*c)^3*(2*d*cos(f*x+e)^2/(c^2-d^2)/(-(-d*sin(f*x+e)-c)*cos(f*x
+e)^2)^(1/2)+2*c/(c^2-d^2)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)
-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+
d))^(1/2))+2/(c^2-d^2)*d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1
)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((
c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))))/cos(f*x+e)/(c+d*sin(f*x+e)
)^(1/2)/f

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Maxima [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(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [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(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

Timed out

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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(1/(a+b*sin(f*x+e))**3/(c+d*sin(f*x+e))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate(1/((b*sin(f*x + e) + a)^3*(d*sin(f*x + e) + c)^(3/2)), x)