\(\int \frac {1}{(3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx\) [757]

Optimal result

Integrand size = 27, antiderivative size = 621 \[ \int \frac {1}{(3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\frac {d \left (18 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 \left (9-b^2\right ) (b c-3 d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {b^2 \cos (e+f x)}{\left (9-b^2\right ) (b c-3 d) f (3+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {\left (216 c d^4-24 b^2 c d^4-36 b d^3 \left (5 c^2-3 d^2\right )-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{3 \left (9-b^2\right ) (b c-3 d)^3 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (216 c d^3-24 b^2 c d^3-36 b d^2 \left (5 c^2-3 d^2\right )-b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 \left (9-b^2\right ) (b c-3 d)^3 \left (c^2-d^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (18 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 \left (9-b^2\right ) (b c-3 d)^2 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \left (6 b c-63 d+5 b^2 d\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(3-b) (3+b)^2 (b c-3 d)^3 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

1/3*d*(2*a^2*d^2+b^2*(3*c^2-5*d^2))*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)^2/(c^2-d^2)/f/(c+d*sin(f*x+e))^(3/2)+b^2*c
os(f*x+e)/(a^2-b^2)/(-a*d+b*c)/f/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(3/2)-1/3*(8*a^3*c*d^4-8*a*b^2*c*d^4-4*a^2*
b*d^3*(5*c^2-3*d^2)-b^3*(3*c^4*d-26*c^2*d^3+15*d^5))*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)^3/(c^2-d^2)^2/f/(c+d*sin(
f*x+e))^(1/2)+1/3*(8*a^3*c*d^3-8*a*b^2*c*d^3-4*a^2*b*d^2*(5*c^2-3*d^2)-b^3*(3*c^4-26*c^2*d^2+15*d^4))*(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)*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)/(-a*d+b*c)^3/(c^2-d^2)^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+1/3*(2*a^2*d^2
+b^2*(3*c^2-5*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)*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)/(-a*d+b*c)^2/(c^2-d^2)/f/(c+d*sin(f*
x+e))^(1/2)-b^2*(-7*a^2*d+2*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)*Ellip
ticPi(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)/(a+b)^
2/(-a*d+b*c)^3/f/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.79 (sec) , antiderivative size = 661, normalized size of antiderivative = 1.06, 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} \[ \int \frac {1}{(3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\frac {d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (a^2-b^2\right ) \left (c^2-d^2\right ) (b c-a d)^2 (c+d \sin (e+f x))^{3/2}}-\frac {\left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{3 f \left (a^2-b^2\right ) \left (c^2-d^2\right ) (b c-a d)^2 \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}+\frac {b^2 \left (-7 a^2 d+2 a b c+5 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f (a-b) (a+b)^2 (b c-a d)^3 \sqrt {c+d \sin (e+f x)}}-\frac {\left (8 a^3 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-8 a b^2 c d^3-\left (b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 f \left (a^2-b^2\right ) \left (c^2-d^2\right )^2 (b c-a d)^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{3 f \left (a^2-b^2\right ) \left (c^2-d^2\right )^2 (b c-a d)^3 \sqrt {c+d \sin (e+f x)}} \]

[In]

Int[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2)),x]

[Out]

(d*(2*a^2*d^2 + b^2*(3*c^2 - 5*d^2))*Cos[e + f*x])/(3*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*Sin[e + f
*x])^(3/2)) + (b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) -
((8*a^3*c*d^4 - 8*a*b^2*c*d^4 - 4*a^2*b*d^3*(5*c^2 - 3*d^2) - b^3*(3*c^4*d - 26*c^2*d^3 + 15*d^5))*Cos[e + f*
x])/(3*(a^2 - b^2)*(b*c - a*d)^3*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((8*a^3*c*d^3 - 8*a*b^2*c*d^3 - 4
*a^2*b*d^2*(5*c^2 - 3*d^2) - b^3*(3*c^4 - 26*c^2*d^2 + 15*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*S
qrt[c + d*Sin[e + f*x]])/(3*(a^2 - b^2)*(b*c - a*d)^3*(c^2 - d^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((
2*a^2*d^2 + b^2*(3*c^2 - 5*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d
)])/(3*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (b^2*(2*a*b*c - 7*a^2*d + 5*b^2*d)*
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)*(a +
b)^2*(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)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {\int \frac {\frac {1}{2} \left (-2 a b c+2 a^2 d-5 b^2 d\right )-a b d \sin (e+f x)+\frac {3}{2} b^2 d \sin ^2(e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx}{\left (a^2-b^2\right ) (b c-a d)} \\ & = \frac {d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {2 \int \frac {-\frac {3}{4} \left (2 a^3 c d^2+2 a b^2 c \left (c^2-2 d^2\right )-4 a^2 b d \left (c^2-d^2\right )+5 b^3 d \left (c^2-d^2\right )\right )-\frac {1}{2} d \left (3 a^2 b c d-3 b^3 c d-a^3 d^2+a b^2 \left (3 c^2-2 d^2\right )\right ) \sin (e+f x)+\frac {1}{4} b d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx}{3 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right )} \\ & = \frac {d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {\left (8 a^3 c d^4-8 a b^2 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}-\frac {4 \int \frac {\frac {1}{8} \left (-2 a^3 b c d^2 \left (9 c^2-5 d^2\right )-15 b^4 d \left (c^2-d^2\right )^2+2 a^2 b^2 d \left (9 c^4-21 c^2 d^2+8 d^4\right )-2 a b^3 c \left (3 c^4-15 c^2 d^2+8 d^4\right )+2 a^4 \left (3 c^2 d^3+d^5\right )\right )+\frac {1}{4} d \left (4 a^4 c d^3+b^4 c d \left (9 c^2-5 d^2\right )-7 a^3 b d^2 \left (c^2-d^2\right )-a^2 b^2 c d \left (9 c^2-d^2\right )-a b^3 \left (3 c^4-13 c^2 d^2+10 d^4\right )\right ) \sin (e+f x)+\frac {1}{8} b \left (8 a^3 c d^4-8 a b^2 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{3 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2} \\ & = \frac {d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {\left (8 a^3 c d^4-8 a b^2 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {4 \int \frac {-\frac {1}{8} b d \left (c^2-d^2\right ) \left (2 a^3 b c d^2-2 a^4 d^3+2 a^2 b^2 d \left (9 c^2-8 d^2\right )-15 b^4 d \left (c^2-d^2\right )-a b^3 c \left (3 c^2-d^2\right )\right )-\frac {1}{8} b^2 d (b c-a d) \left (c^2-d^2\right ) \left (3 b^2 c^2+2 a^2 d^2-5 b^2 d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{3 b \left (a^2-b^2\right ) d (b c-a d)^3 \left (c^2-d^2\right )^2}-\frac {\left (8 a^3 c d^3-8 a b^2 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{6 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2} \\ & = \frac {d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {\left (8 a^3 c d^4-8 a b^2 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {\left (b^2 \left (2 a b c-7 a^2 d+5 b^2 d\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)^3}-\frac {\left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{6 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right )}-\frac {\left (\left (8 a^3 c d^3-8 a b^2 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-b^3 \left (3 c^4-26 c^2 d^2+15 d^4\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}{6 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \\ & = \frac {d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {\left (8 a^3 c d^4-8 a b^2 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (8 a^3 c d^3-8 a b^2 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-b^3 \left (3 c^4-26 c^2 d^2+15 d^4\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)}}{3 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (b^2 \left (2 a b c-7 a^2 d+5 b^2 d\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}{2 \left (a^2-b^2\right ) (b c-a d)^3 \sqrt {c+d \sin (e+f x)}}-\frac {\left (\left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\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}{6 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}} \\ & = \frac {d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {\left (8 a^3 c d^4-8 a b^2 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (8 a^3 c d^3-8 a b^2 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-b^3 \left (3 c^4-26 c^2 d^2+15 d^4\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)}}{3 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\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}}}{3 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \left (2 a b c-7 a^2 d+5 b^2 d\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}}}{(a-b) (a+b)^2 (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 = 7.52 (sec) , antiderivative size = 1244, normalized size of antiderivative = 2.00 \[ \int \frac {1}{(3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\frac {\sqrt {c+d \sin (e+f x)} \left (-\frac {b^4 \cos (e+f x)}{\left (-9+b^2\right ) (b c-3 d)^3 (3+b \sin (e+f x))}+\frac {2 d^3 \cos (e+f x)}{3 (b c-3 d)^2 \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}+\frac {4 \left (5 b c^2 d^3 \cos (e+f x)-6 c d^4 \cos (e+f x)-3 b d^5 \cos (e+f x)\right )}{3 (b c-3 d)^3 \left (c^2-d^2\right )^2 (c+d \sin (e+f x))}\right )}{f}+\frac {-\frac {2 \left (-36 b^3 c^5+324 b^2 c^4 d-33 b^4 c^4 d-972 b c^3 d^2+180 b^3 c^3 d^2+972 c^2 d^3-936 b^2 c^2 d^3+86 b^4 c^2 d^3+756 b c d^4-120 b^3 c d^4+324 d^5+396 b^2 d^5-45 b^4 d^5\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 (-36 b^3 c^4 d-324 b^2 c^3 d^2+36 b^4 c^3 d^2-756 b c^2 d^3+156 b^3 c^2 d^3+1296 c d^4+36 b^2 c d^4-20 b^4 c d^4+756 b d^5-120 b^3 d^5\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 (3 b^4 c^4 d+180 b^2 c^2 d^3-26 b^4 c^2 d^3-216 b c d^4+24 b^3 c d^4-108 b^2 d^5+15 b^4 d^5\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}}}}{12 (-3+b) (3+b) (b c-3 d)^3 (c-d)^2 (c+d)^2 f} \]

[In]

Integrate[1/((3 + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2)),x]

[Out]

(Sqrt[c + d*Sin[e + f*x]]*(-((b^4*Cos[e + f*x])/((-9 + b^2)*(b*c - 3*d)^3*(3 + b*Sin[e + f*x]))) + (2*d^3*Cos[
e + f*x])/(3*(b*c - 3*d)^2*(c^2 - d^2)*(c + d*Sin[e + f*x])^2) + (4*(5*b*c^2*d^3*Cos[e + f*x] - 6*c*d^4*Cos[e
+ f*x] - 3*b*d^5*Cos[e + f*x]))/(3*(b*c - 3*d)^3*(c^2 - d^2)^2*(c + d*Sin[e + f*x]))))/f + ((-2*(-36*b^3*c^5 +
324*b^2*c^4*d - 33*b^4*c^4*d - 972*b*c^3*d^2 + 180*b^3*c^3*d^2 + 972*c^2*d^3 - 936*b^2*c^2*d^3 + 86*b^4*c^2*d
^3 + 756*b*c*d^4 - 120*b^3*c*d^4 + 324*d^5 + 396*b^2*d^5 - 45*b^4*d^5)*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)*(-36*b^
3*c^4*d - 324*b^2*c^3*d^2 + 36*b^4*c^3*d^2 - 756*b*c^2*d^3 + 156*b^3*c^2*d^3 + 1296*c*d^4 + 36*b^2*c*d^4 - 20*
b^4*c*d^4 + 756*b*d^5 - 120*b^3*d^5)*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)]*S
qrt[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])*Sq
rt[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)*
(3*b^4*c^4*d + 180*b^2*c^2*d^3 - 26*b^4*c^2*d^3 - 216*b*c*d^4 + 24*b^3*c*d^4 - 108*b^2*d^5 + 15*b^4*d^5)*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)]*Sqr
t[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])*Sqr
t[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)]))/(12*(-3 + b)*(3 + b)*(b*c - 3*d)^3*(c - d)^2*(c +
d)^2*f)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1730\) vs. \(2(735)=1470\).

Time = 22.78 (sec) , antiderivative size = 1731, normalized size of antiderivative = 2.79

method	result	size
default	\(\text {Expression too large to display}\)	\(1731\)

[In]

int(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(d^2/(a*d-b*c)^2*(2/3/(c^2-d^2)/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1
/2)/(sin(f*x+e)+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d^2)^2*c/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*(3*c^2+d^2)/
(3*c^4-6*c^2*d^2+3*d^4)*(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))+8/3*c*d/(c^2-d^2)^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))))+b^2/(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)))-2*d^2/(a*d-b*c)^3*b*(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*s
in(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))))+4
*b/(a*d-b*c)^3*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+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. \[ \int \frac {1}{(3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(a+b*sin(f*x+e))**2/(c+d*sin(f*x+e))**(5/2),x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \frac {1}{(3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \frac {1}{(3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((b*sin(f*x + e) + a)^2*(d*sin(f*x + e) + c)^(5/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]

[In]

int(1/((a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(5/2)),x)

[Out]

int(1/((a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(5/2)), x)