\(\int \frac {1}{(3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx\) [756]

Optimal result

Integrand size = 27, antiderivative size = 426 \[ \int \frac {1}{(3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx=\frac {d \left (18 d^2+b^2 \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{\left (9-b^2\right ) (b c-3 d)^2 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \cos (e+f x)}{\left (9-b^2\right ) (b c-3 d) f (3+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}+\frac {\left (18 d^2+b^2 \left (c^2-3 d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{\left (9-b^2\right ) (b c-3 d)^2 \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {b \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{\left (9-b^2\right ) (b c-3 d) f \sqrt {c+d \sin (e+f x)}}+\frac {b \left (6 b c-45 d+3 b^2 d\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(3-b) (3+b)^2 (b c-3 d)^2 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

d*(2*a^2*d^2+b^2*(c^2-3*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))^(1/2)+b^2*cos(f*x
+e)/(a^2-b^2)/(-a*d+b*c)/f/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2)-(2*a^2*d^2+b^2*(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)*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)^2/(c^2-d^2)/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+b*(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*si
n(f*x+e))/(c+d))^(1/2)/(a^2-b^2)/(-a*d+b*c)/f/(c+d*sin(f*x+e))^(1/2)-b*(-5*a^2*d+2*a*b*c+3*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)*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)/(a+b)^2/(-a*d+b*c)^2/f/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.05, 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} \[ \int \frac {1}{(3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx=\frac {d \left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \left (c^2-d^2\right ) (b c-a d)^2 \sqrt {c+d \sin (e+f x)}}+\frac {\left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \left (a^2-b^2\right ) \left (c^2-d^2\right ) (b c-a d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {b \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}+\frac {b \left (-5 a^2 d+2 a b c+3 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f (a-b) (a+b)^2 (b c-a d)^2 \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

(d*(2*a^2*d^2 + b^2*(c^2 - 3*d^2))*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f
*x]]) + (b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]) + ((2*a^2
*d^2 + b^2*(c^2 - 3*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*
(b*c - a*d)^2*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (b*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c +
d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a^2 - b^2)*(b*c - a*d)*f*Sqrt[c + d*Sin[e + f*x]]) + (b*(2*a*b*c - 5
*a^2*d + 3*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)^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)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {\frac {1}{2} \left (-2 a b c+2 a^2 d-3 b^2 d\right )-a b d \sin (e+f x)+\frac {1}{2} b^2 d \sin ^2(e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx}{\left (a^2-b^2\right ) (b c-a d)} \\ & = \frac {d \left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{\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 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {2 \int \frac {\frac {1}{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 )-3 b^3 d \left (c^2-d^2\right )\right )-\frac {1}{2} d \left (a^2 b c d-b^3 c d+a^3 d^2+a b^2 \left (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 (c^2-3 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 ) (b c-a d)^2 \left (c^2-d^2\right )} \\ & = \frac {d \left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{\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 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}+\frac {2 \int \frac {\frac {1}{4} b^2 d \left (a b c-4 a^2 d+3 b^2 d\right ) \left (c^2-d^2\right )-\frac {1}{4} b^3 d (b c-a d) \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 ) d (b c-a d)^2 \left (c^2-d^2\right )}+\frac {\left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{2 \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 (c^2-3 d^2\right )\right ) \cos (e+f x)}{\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 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {b \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {\left (b \left (2 a b c-5 a^2 d+3 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)^2}+\frac {\left (\left (2 a^2 d^2+b^2 \left (c^2-3 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}{2 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \\ & = \frac {d \left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{\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 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}+\frac {\left (2 a^2 d^2+b^2 \left (c^2-3 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)}}{\left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (b \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}{2 \left (a^2-b^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}+\frac {\left (b \left (2 a b c-5 a^2 d+3 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)^2 \sqrt {c+d \sin (e+f x)}} \\ & = \frac {d \left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{\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 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}+\frac {\left (2 a^2 d^2+b^2 \left (c^2-3 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)}}{\left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {b \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}}}{\left (a^2-b^2\right ) (b c-a d) f \sqrt {c+d \sin (e+f x)}}+\frac {b \left (2 a b c-5 a^2 d+3 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)^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 = 6.73 (sec) , antiderivative size = 1014, normalized size of antiderivative = 2.38 \[ \int \frac {1}{(3+b \sin (e+f x))^2 (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)}{\left (-9+b^2\right ) (b c-3 d)^2 (3+b \sin (e+f x))}+\frac {2 d^3 \cos (e+f x)}{(b c-3 d)^2 \left (c^2-d^2\right ) (c+d \sin (e+f x))}\right )}{f}+\frac {-\frac {2 \left (-12 b^2 c^3+72 b c^2 d-7 b^3 c^2 d-108 c d^2+24 b^2 c d^2-90 b d^3+9 b^3 d^3\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 (-12 b^2 c^2 d-36 b c d^2+4 b^3 c d^2-108 d^3+24 b^2 d^3\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 (b^3 c^2 d+18 b d^3-3 b^3 d^3\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}}}}{4 (-3+b) (3+b) (b c-3 d)^2 (c-d) (c+d) f} \]

[In]

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

[Out]

(Sqrt[c + d*Sin[e + f*x]]*(-((b^3*Cos[e + f*x])/((-9 + b^2)*(b*c - 3*d)^2*(3 + b*Sin[e + f*x]))) + (2*d^3*Cos[
e + f*x])/((b*c - 3*d)^2*(c^2 - d^2)*(c + d*Sin[e + f*x]))))/f + ((-2*(-12*b^2*c^3 + 72*b*c^2*d - 7*b^3*c^2*d
- 108*c*d^2 + 24*b^2*c*d^2 - 90*b*d^3 + 9*b^3*d^3)*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)*(-12*b^2*c^2*d - 36*b*c*d^2
+ 4*b^3*c*d^2 - 108*d^3 + 24*b^2*d^3)*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
)*(b^3*c^2*d + 18*b*d^3 - 3*b^3*d^3)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(b*c - 3*d)*(c - d)*EllipticE[I*ArcSin
h[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*ArcSi
nh[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*Sq
rt[-(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)]))/(4*(-3 +
b)*(3 + b)*(b*c - 3*d)^2*(c - d)*(c + d)*f)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1265\) vs. \(2(533)=1066\).

Time = 8.85 (sec) , antiderivative size = 1266, normalized size of antiderivative = 2.97

method	result	size
default	\(\text {Expression too large to display}\)	\(1266\)

[In]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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