\(\int \frac {1}{(3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx\) [764]

Optimal result

Integrand size = 27, antiderivative size = 636 \[ \int \frac {1}{(3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=-\frac {d \left (648 d^3+9 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-18 b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{4 \left (9-b^2\right )^2 (b c-3 d)^3 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \cos (e+f x)}{2 \left (9-b^2\right ) (b c-3 d) f (3+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \left (18 b c-99 d+5 b^2 d\right ) \cos (e+f x)}{4 \left (9-b^2\right )^2 (b c-3 d)^2 f (3+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\left (648 d^3+9 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-18 b^3 c \left (c^2-d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{4 \left (9-b^2\right )^2 (b c-3 d)^3 \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {b \left (18 b c-99 d+5 b^2 d\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 \left (9-b^2\right )^2 (b c-3 d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {b \left (756 b c d-12 b^3 c d-2835 d^2-18 b^2 \left (4 c^2-19 d^2\right )-b^4 \left (4 c^2+15 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 (3-b)^2 (3+b)^3 (b c-3 d)^3 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

-1/4*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(f*x+e)/(a^2-b^2)^2/(
-a*d+b*c)^3/(c^2-d^2)/f/(c+d*sin(f*x+e))^(1/2)+1/2*b^2*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)/f/(a+b*sin(f*x+e))^2/(c
+d*sin(f*x+e))^(1/2)+1/4*b^2*(-11*a^2*d+6*a*b*c+5*b^2*d)*cos(f*x+e)/(a^2-b^2)^2/(-a*d+b*c)^2/f/(a+b*sin(f*x+e)
)/(c+d*sin(f*x+e))^(1/2)+1/4*(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))*(s
in(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+
d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/(a^2-b^2)^2/(-a*d+b*c)^3/(c^2-d^2)/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+1/4*b*(-
11*a^2*d+6*a*b*c+5*b^2*d)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/
4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/(a^2-b^2)^2/(-a*d+b*c)^2/f/(c+d*sin(f*x+
e))^(1/2)+1/4*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))*(sin(1/2*e+1
/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticPi(cos(1/2*e+1/4*Pi+1/2*f*x),2*b/(a+b),2^(1/2)*(d/(c
+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/(a-b)^2/(a+b)^3/(-a*d+b*c)^3/f/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.79 (sec) , antiderivative size = 682, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {2881, 3134, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \frac {1}{(3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}-\frac {b \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 \sqrt {c+d \sin (e+f x)}}-\frac {d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{4 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right ) (b c-a d)^3 \sqrt {c+d \sin (e+f x)}}-\frac {\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right ) (b c-a d)^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {b \left (-35 a^4 d^2+28 a^3 b c d-2 a^2 b^2 \left (4 c^2-19 d^2\right )-4 a b^3 c d-b^4 \left (4 c^2+15 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{4 f (a-b)^2 (a+b)^3 (b c-a d)^3 \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

-1/4*(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
])/((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 2732

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

Rule 2734

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

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

Rule 2742

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2881

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-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 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(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]

Rule 2886

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/
(c + d))*Sin[e + f*x]]), 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 3081

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 3134

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] + D
ist[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] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3138

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \operatorname {EllipticF}\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 ) \operatorname {EllipticPi}\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] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.51 (sec) , antiderivative size = 1231, normalized size of antiderivative = 1.94 \[ \int \frac {1}{(3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\frac {\sqrt {c+d \sin (e+f x)} \left (-\frac {b^3 \cos (e+f x)}{2 \left (-9+b^2\right ) (b c-3 d)^2 (3+b \sin (e+f x))^2}+\frac {18 b^4 c \cos (e+f x)-117 b^3 d \cos (e+f x)+7 b^5 d \cos (e+f x)}{4 \left (-9+b^2\right )^2 (b c-3 d)^3 (3+b \sin (e+f x))}-\frac {2 d^4 \cos (e+f x)}{(b c-3 d)^3 \left (c^2-d^2\right ) (c+d \sin (e+f x))}\right )}{f}+\frac {-\frac {2 \left (144 b^3 c^4+8 b^5 c^4-1296 b^2 c^3 d+54 b^4 c^3 d+3888 b c^2 d^2-855 b^3 c^2 d^2+29 b^5 c^2 d^2-3888 c d^3+2160 b^2 c d^3-102 b^4 c d^3-4536 b d^4+855 b^3 d^4-45 b^5 d^4\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+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}}}{(3+b) \sqrt {c+d \sin (e+f x)}}-\frac {2 i \left (180 b^3 c^3 d+4 b^5 c^3 d-1296 b^2 c^2 d^2+72 b^4 c^2 d^2-1296 b c d^3+108 b^3 c d^3-20 b^5 c d^3-3888 d^4+2160 b^2 d^4-120 b^4 d^4\right ) \cos (e+f x) \left ((b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )+3 d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\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 {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b (b c-3 d) d^2 \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}-\frac {2 i \left (-18 b^4 c^3 d+117 b^3 c^2 d^2-7 b^5 c^2 d^2+18 b^4 c d^3+648 b d^4-261 b^3 d^4+15 b^5 d^4\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (b c-3 d) (c-d) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+d \left (2 (3+b) (b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )-\left (-18+b^2\right ) d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\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 {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b^2 (b c-3 d) d \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \left (-2 c^2+d^2+4 c (c+d \sin (e+f x))-2 (c+d \sin (e+f x))^2\right ) \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}}{16 (-3+b)^2 (3+b)^2 (b c-3 d)^3 (c-d) (c+d) f} \]

[In]

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

[Out]

(Sqrt[c + d*Sin[e + f*x]]*(-1/2*(b^3*Cos[e + f*x])/((-9 + b^2)*(b*c - 3*d)^2*(3 + b*Sin[e + f*x])^2) + (18*b^4
*c*Cos[e + f*x] - 117*b^3*d*Cos[e + f*x] + 7*b^5*d*Cos[e + f*x])/(4*(-9 + b^2)^2*(b*c - 3*d)^3*(3 + b*Sin[e +
f*x])) - (2*d^4*Cos[e + f*x])/((b*c - 3*d)^3*(c^2 - d^2)*(c + d*Sin[e + f*x]))))/f + ((-2*(144*b^3*c^4 + 8*b^5
*c^4 - 1296*b^2*c^3*d + 54*b^4*c^3*d + 3888*b*c^2*d^2 - 855*b^3*c^2*d^2 + 29*b^5*c^2*d^2 - 3888*c*d^3 + 2160*b
^2*c*d^3 - 102*b^4*c*d^3 - 4536*b*d^4 + 855*b^3*d^4 - 45*b^5*d^4)*EllipticPi[(2*b)/(3 + b), (-e + Pi/2 - f*x)/
2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((3 + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(180*b^3*c^3
*d + 4*b^5*c^3*d - 1296*b^2*c^2*d^2 + 72*b^4*c^2*d^2 - 1296*b*c*d^3 + 108*b^3*c*d^3 - 20*b^5*c*d^3 - 3888*d^4
+ 2160*b^2*d^4 - 120*b^4*d^4)*Cos[e + f*x]*((b*c - 3*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin
[e + f*x]]], (c + d)/(c - d)] + 3*d*EllipticPi[(b*(c + d))/(b*c - 3*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) + 3*d + b*(c + d*Sin[e + f*x])))/(b*(b*c - 3*d)*d^2*Sqrt[-(c + d)^(-1)]*(3 + 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)*(-18*b^
4*c^3*d + 117*b^3*c^2*d^2 - 7*b^5*c^2*d^2 + 18*b^4*c*d^3 + 648*b*d^4 - 261*b^3*d^4 + 15*b^5*d^4)*Cos[e + f*x]*
Cos[2*(e + f*x)]*(2*b*(b*c - 3*d)*(c - d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (
c + d)/(c - d)] + d*(2*(3 + b)*(b*c - 3*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]],
(c + d)/(c - d)] - (-18 + b^2)*d*EllipticPi[(b*(c + d))/(b*c - 3*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) + 3*d + b*(c + d*Sin[e + f*x])))/(b^2*(b*c - 3*d)*d*Sqrt[-(c + d)^(-1)]*(3 + b*Sin[e + f*x])*Sqrt[1 - Si
n[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*(-3 + b)^2*(3 + b)^2*(b*c - 3*d)^3*(c - d)*(c + d)*f)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2098\) vs. \(2(751)=1502\).

Time = 22.00 (sec) , antiderivative size = 2099, normalized size of antiderivative = 3.30

method	result	size
default	\(\text {Expression too large to display}\)	\(2099\)

[In]

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

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(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)*(1/(c-d)*(-
sin(f*x+e)-1)*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)*(1/(c-d)*(-
sin(f*x+e)-1)*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))))-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)*cos(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*sin(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)*(1/(c-d)*(-sin(f*x+e)-1)*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)*(1/(c-d)*(-sin(f*x+e)-1)*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)))+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)*(1/(c-d)*(-sin(f*x+e)-1)*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)))-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)*(1/(c-d)*(-sin(f*x+e)-1)*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*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)*(1/(c-d)*(-sin(f*x+e)-1)*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)*(1/(c-d)*(-sin(f*x+e)-1)*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)*(1/(c-d)*(-sin(f*x+e)-1)*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))))/cos(f*x+e
)/(c+d*sin(f*x+e))^(1/2)/f

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \frac {1}{(3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Giac [F]

\[ \int \frac {1}{(3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(3+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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