\(\int \frac {1}{(3+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx\) [763]

Optimal result

Integrand size = 27, antiderivative size = 468 \[ \int \frac {1}{(3+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx=\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 \left (9-b^2\right ) (b c-3 d) f (3+b \sin (e+f x))^2}+\frac {3 b^2 \left (6 b c-27 d+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 \left (9-b^2\right )^2 (b c-3 d)^2 f (3+b \sin (e+f x))}+\frac {3 b \left (6 b c-27 d+b^2 d\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{4 \left (9-b^2\right )^2 (b c-3 d)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (18 b c-63 d+b^2 d\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 \left (9-b^2\right )^2 (b c-3 d) f \sqrt {c+d \sin (e+f x)}}-\frac {\left (540 b c d+12 b^3 c d-1215 d^2-18 b^2 \left (4 c^2-3 d^2\right )-b^4 \left (4 c^2+3 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 (3-b)^2 (3+b)^3 (b c-3 d)^2 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

1/2*b^2*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/(a^2-b^2)/(-a*d+b*c)/f/(a+b*sin(f*x+e))^2+3/4*b^2*(-3*a^2*d+2*a*b*c+
b^2*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/(a^2-b^2)^2/(-a*d+b*c)^2/f/(a+b*sin(f*x+e))-3/4*b*(-3*a^2*d+2*a*b*c+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)*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)^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+1/4*(-7*a^
2*d+6*a*b*c+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)/f/(c+d*sin(f*x+e))^(1/2)
+1/4*(20*a^3*b*c*d+4*a*b^3*c*d-15*a^4*d^2-2*a^2*b^2*(4*c^2-3*d^2)-b^4*(4*c^2+3*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)^2/f/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.07, number of steps used = 10, 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} \[ \int \frac {1}{(3+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx=\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 (a+b \sin (e+f x))}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}-\frac {\left (-7 a^2 d+6 a b c+b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 (b c-a d) \sqrt {c+d \sin (e+f x)}}+\frac {3 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (-15 a^4 d^2+20 a^3 b c d-2 a^2 b^2 \left (4 c^2-3 d^2\right )+4 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{4 f (a-b)^2 (a+b)^3 (b c-a d)^2 \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

(b^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2) + (3*b^2*(2*a
*b*c - 3*a^2*d + b^2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e +
f*x])) + (3*b*(2*a*b*c - 3*a^2*d + b^2*d)*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)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((6*a*b*c - 7*a^2*d + b^2*d)*Ellipt
icF[(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)*f*Sqrt
[c + d*Sin[e + f*x]]) - ((20*a^3*b*c*d + 4*a*b^3*c*d - 15*a^4*d^2 - 2*a^2*b^2*(4*c^2 - 3*d^2) - b^4*(4*c^2 + 3
*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)^2*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) \sqrt {c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}-\frac {\int \frac {\frac {1}{2} \left (-4 a b c+4 a^2 d-3 b^2 d\right )+b (b c-2 a d) \sin (e+f x)+\frac {1}{2} b^2 d \sin ^2(e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)} \\ & = \frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac {3 b^2 \left (2 a b c-3 a^2 d+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}+\frac {\int \frac {\frac {1}{4} \left (-16 a^3 b c d-2 a b^3 c d+8 a^4 d^2+a^2 b^2 \left (8 c^2-5 d^2\right )+b^4 \left (4 c^2+3 d^2\right )\right )+\frac {1}{2} b d \left (5 a^2 b c+b^3 c-8 a^3 d+2 a b^2 d\right ) \sin (e+f x)+\frac {3}{4} b^2 d \left (2 a b c-3 a^2 d+b^2 d\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{2 \left (a^2-b^2\right )^2 (b c-a d)^2} \\ & = \frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac {3 b^2 \left (2 a b c-3 a^2 d+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}-\frac {\int \frac {\frac {1}{4} b d \left (7 a^3 b c d+5 a b^3 c d-8 a^4 d^2-a^2 b^2 \left (2 c^2-5 d^2\right )-b^4 \left (4 c^2+3 d^2\right )\right )+\frac {1}{4} b^2 d (b c-a d) \left (6 a b c-7 a^2 d+b^2 d\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{2 b \left (a^2-b^2\right )^2 d (b c-a d)^2}+\frac {\left (3 b \left (2 a b c-3 a^2 d+b^2 d\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)^2} \\ & = \frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac {3 b^2 \left (2 a b c-3 a^2 d+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}-\frac {\left (6 a b c-7 a^2 d+b^2 d\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)}-\frac {\left (20 a^3 b c d+4 a b^3 c d-15 a^4 d^2-2 a^2 b^2 \left (4 c^2-3 d^2\right )-b^4 \left (4 c^2+3 d^2\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)^2}+\frac {\left (3 b \left (2 a b c-3 a^2 d+b^2 d\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)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \\ & = \frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac {3 b^2 \left (2 a b c-3 a^2 d+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}+\frac {3 b \left (2 a b c-3 a^2 d+b^2 d\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)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (\left (6 a b c-7 a^2 d+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) \sqrt {c+d \sin (e+f x)}}-\frac {\left (\left (20 a^3 b c d+4 a b^3 c d-15 a^4 d^2-2 a^2 b^2 \left (4 c^2-3 d^2\right )-b^4 \left (4 c^2+3 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)^2 \sqrt {c+d \sin (e+f x)}} \\ & = \frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac {3 b^2 \left (2 a b c-3 a^2 d+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}+\frac {3 b \left (2 a b c-3 a^2 d+b^2 d\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)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (6 a b c-7 a^2 d+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) f \sqrt {c+d \sin (e+f x)}}-\frac {\left (20 a^3 b c d+4 a b^3 c d-15 a^4 d^2-2 a^2 b^2 \left (4 c^2-3 d^2\right )-b^4 \left (4 c^2+3 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)^2 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.81 (sec) , antiderivative size = 1016, normalized size of antiderivative = 2.17 \[ \int \frac {1}{(3+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx=\frac {\sqrt {c+d \sin (e+f x)} \left (-\frac {b^2 \cos (e+f x)}{2 \left (-9+b^2\right ) (b c-3 d) (3+b \sin (e+f x))^2}+\frac {3 \left (6 b^3 c \cos (e+f x)-27 b^2 d \cos (e+f x)+b^4 d \cos (e+f x)\right )}{4 \left (-9+b^2\right )^2 (b c-3 d)^2 (3+b \sin (e+f x))}\right )}{f}+\frac {-\frac {2 \left (144 b^2 c^2+8 b^4 c^2-864 b c d+6 b^3 c d+1296 d^2-171 b^2 d^2+9 b^4 d^2\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^2 c d+4 b^4 c d-864 b d^2+24 b^3 d^2\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^3 c d+81 b^2 d^2-3 b^4 d^2\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)^2 f} \]

[In]

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

[Out]

(Sqrt[c + d*Sin[e + f*x]]*(-1/2*(b^2*Cos[e + f*x])/((-9 + b^2)*(b*c - 3*d)*(3 + b*Sin[e + f*x])^2) + (3*(6*b^3
*c*Cos[e + f*x] - 27*b^2*d*Cos[e + f*x] + b^4*d*Cos[e + f*x]))/(4*(-9 + b^2)^2*(b*c - 3*d)^2*(3 + b*Sin[e + f*
x]))))/f + ((-2*(144*b^2*c^2 + 8*b^4*c^2 - 864*b*c*d + 6*b^3*c*d + 1296*d^2 - 171*b^2*d^2 + 9*b^4*d^2)*Ellipti
cPi[(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^2*c*d + 4*b^4*c*d - 864*b*d^2 + 24*b^3*d^2)*Cos[e + f*x]*((b*c - 3*d)*Elliptic
F[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^3*c*d + 81*b^2*d^2 - 3*b^4*d^2)*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 - 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*(-3 + b)^2*(3 + b)^2*(b*c - 3*d)^2*f)

Maple [A] (verified)

Time = 9.16 (sec) , antiderivative size = 867, normalized size of antiderivative = 1.85

method	result	size
default	\(\frac {\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {b^{2} \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}{2 \left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right ) \left (a +b \sin \left (f x +e \right )\right )^{2}}-\frac {3 b^{2} \left (3 a^{2} d -2 a b c -b^{2} d \right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}{4 \left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right )^{2} \left (a +b \sin \left (f x +e \right )\right )}-\frac {d \left (7 a^{3} d -4 a^{2} b c -a \,b^{2} d -2 b^{3} c \right ) \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{4 \left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right )^{2} \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}-\frac {3 b d \left (3 a^{2} d -2 a b c -b^{2} d \right ) \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{4 \left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right )^{2} \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {\left (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}\right ) \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \Pi \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \frac {-\frac {c}{d}+1}{-\frac {c}{d}+\frac {a}{b}}, \sqrt {\frac {c -d}{c +d}}\right )}{4 \left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right )^{2} b \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {c}{d}+\frac {a}{b}\right )}\right )}{\cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(867\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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