Integrand size = 29, antiderivative size = 621 \[ \int \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\frac {\sqrt {3+b} (c-d) \sqrt {c+d} E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{(b c-3 d) f}+\frac {\sqrt {c+d} (b c+3 d) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{b \sqrt {3+b} d f}-\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {3+b \sin (e+f x)}}+\frac {(3+b)^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{b \sqrt {c+d} f} \]
(a*d+b*c)*EllipticPi((a+b)^(1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*s in(f*x+e))^(1/2),b*(c+d)/(a+b)/d,((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*sec(f*x+ e)*(a+b*sin(f*x+e))*(c+d)^(1/2)*(-(-a*d+b*c)*(1-sin(f*x+e))/(c+d)/(a+b*sin (f*x+e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+e)))^(1/2)/b /d/f/(a+b)^(1/2)+(c-d)*EllipticE((a+b)^(1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^ (1/2)/(a+b*sin(f*x+e))^(1/2),((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*( a+b*sin(f*x+e))*(a+b)^(1/2)*(c+d)^(1/2)*(-(-a*d+b*c)*(1-sin(f*x+e))/(c+d)/ (a+b*sin(f*x+e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+e))) ^(1/2)/(-a*d+b*c)/f+(a+b)^(3/2)*EllipticF((c+d)^(1/2)*(a+b*sin(f*x+e))^(1/ 2)/(a+b)^(1/2)/(c+d*sin(f*x+e))^(1/2),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*sec (f*x+e)*(c+d*sin(f*x+e))*((-a*d+b*c)*(1-sin(f*x+e))/(a+b)/(c+d*sin(f*x+e)) )^(1/2)*(-(-a*d+b*c)*(1+sin(f*x+e))/(a-b)/(c+d*sin(f*x+e)))^(1/2)/b/f/(c+d )^(1/2)-b*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f/(a+b*sin(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 37.85 (sec) , antiderivative size = 196044, normalized size of antiderivative = 315.69 \[ \int \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\text {Result too large to show} \]
Time = 2.05 (sec) , antiderivative size = 681, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {3042, 3300, 27, 3042, 3532, 27, 3042, 3280, 3042, 3290, 3297, 3475}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}dx\) |
\(\Big \downarrow \) 3300 |
\(\displaystyle \frac {\int \frac {b d (b c+a d) \sin ^2(e+f x)+2 a d (b c+a d) \sin (e+f x)+d \left (2 c a^2+b d a-b^2 c\right )}{2 (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {b d (b c+a d) \sin ^2(e+f x)+2 a d (b c+a d) \sin (e+f x)+d \left (2 c a^2+b d a-b^2 c\right )}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {b d (b c+a d) \sin (e+f x)^2+2 a d (b c+a d) \sin (e+f x)+d \left (2 c a^2+b d a-b^2 c\right )}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}\) |
\(\Big \downarrow \) 3532 |
\(\displaystyle \frac {\frac {\int \frac {b \left (a^2-b^2\right ) d (b c-a d)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {d (a d+b c) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b}}{2 d}-\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {d \left (a^2-b^2\right ) (b c-a d) \int \frac {1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b}+\frac {d (a d+b c) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b}}{2 d}-\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {d \left (a^2-b^2\right ) (b c-a d) \int \frac {1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b}+\frac {d (a d+b c) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b}}{2 d}-\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}\) |
\(\Big \downarrow \) 3280 |
\(\displaystyle \frac {\frac {d \left (a^2-b^2\right ) (b c-a d) \left (\frac {\int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{a-b}-\frac {b \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{a-b}\right )}{b}+\frac {d (a d+b c) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b}}{2 d}-\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {d \left (a^2-b^2\right ) (b c-a d) \left (\frac {\int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{a-b}-\frac {b \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{a-b}\right )}{b}+\frac {d (a d+b c) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b}}{2 d}-\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}\) |
\(\Big \downarrow \) 3290 |
\(\displaystyle \frac {\frac {d \left (a^2-b^2\right ) (b c-a d) \left (\frac {\int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{a-b}-\frac {b \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{a-b}\right )}{b}+\frac {2 \sqrt {c+d} (a d+b c) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b f \sqrt {a+b}}}{2 d}-\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}\) |
\(\Big \downarrow \) 3297 |
\(\displaystyle \frac {\frac {d \left (a^2-b^2\right ) (b c-a d) \left (\frac {2 \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (a-b) \sqrt {c+d} (b c-a d)}-\frac {b \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{a-b}\right )}{b}+\frac {2 \sqrt {c+d} (a d+b c) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b f \sqrt {a+b}}}{2 d}-\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}\) |
\(\Big \downarrow \) 3475 |
\(\displaystyle \frac {\frac {d \left (a^2-b^2\right ) (b c-a d) \left (\frac {2 \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (a-b) \sqrt {c+d} (b c-a d)}+\frac {2 b (c-d) \sqrt {c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{f (a-b) \sqrt {a+b} (b c-a d)^2}\right )}{b}+\frac {2 \sqrt {c+d} (a d+b c) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b f \sqrt {a+b}}}{2 d}-\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}\) |
-((b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + b*Sin[e + f*x]])) + ((2*Sqrt[c + d]*(b*c + a*d)*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[( Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x] ])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d) *(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b *Sqrt[a + b]*f) + ((a^2 - b^2)*d*(b*c - a*d)*((2*b*(c - d)*Sqrt[c + d]*Ell ipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt [-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[ ((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Si n[e + f*x]))/((a - b)*Sqrt[a + b]*(b*c - a*d)^2*f) + (2*Sqrt[a + b]*Ellipt icF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d* Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[(( b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b* c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((a - b)*Sqrt[c + d]*(b*c - a*d)*f)))/b)/(2*d)
3.8.67.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin [(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[1/(a - b) Int[1/(Sqrt[a + b*Sin [e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] - Simp[b/(a - b) Int[(1 + Si n[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[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] && NeQ[c^2 - d^2, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[2*((a + b*Sin[e + f*x])/(d*f*Rt[(a + b)/ (c + d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticPi[b*((c + d)/(d*(a + b))), ArcSin[Rt[(a + b)/( c + d), 2]*(Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])], (a - b)*(( c + d)/((a + b)*(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] && PosQ[(a + b)/(c + d)]
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_ .) + (f_.)*(x_)]]), x_Symbol] :> Simp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d )*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e + f*x] )/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/ ((a - b)*(c + d*Sin[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(S qrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c - d)/((a - b)*(c + d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && N eQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/(a + b)]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x] )^(m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Simp[1/(d*(m + n)) I nt[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n - 1)*Simp[a^2*c*d*( m + n) + b*d*(b*c*(m - 1) + a*d*n) + (a*d*(2*b*c + a*d)*(m + n) - b*d*(a*c - b*d*(m + n - 1)))*Sin[e + f*x] + b*d*(b*c*n + a*d*(2*m + n - 1))*Sin[e + f*x]^2, x], x], 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] && LtQ[0, m, 2] && LtQ[-1, n, 2] && NeQ[m + n, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.) *(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Sim p[-2*A*(c - d)*((a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2 ]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticE[ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin[e + f*x]] /Sqrt[a + b*Sin[e + f*x]])], (a - b)*((c + d)/((a + b)*(c - d)))], x] /; Fr eeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[C/b^2 Int[Sqrt[a + b*Sin[e + f*x]] /Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[1/b^2 Int[(A*b^2 - a^2*C + b*(b* B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[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]
Result contains complex when optimal does not.
Time = 12.13 (sec) , antiderivative size = 132368, normalized size of antiderivative = 213.15
\[ \int \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\int { \sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]
\[ \int \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\int \sqrt {a + b \sin {\left (e + f x \right )}} \sqrt {c + d \sin {\left (e + f x \right )}}\, dx \]
\[ \int \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\int { \sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]
\[ \int \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\int { \sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]
Timed out. \[ \int \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\int \sqrt {a+b\,\sin \left (e+f\,x\right )}\,\sqrt {c+d\,\sin \left (e+f\,x\right )} \,d x \]