Integrand size = 39, antiderivative size = 267 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx=\frac {2 \sqrt {g} \operatorname {EllipticPi}\left (\frac {b}{a+b},\arcsin \left (\frac {\sqrt {a+b} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {a+b \sin (e+f x)}}\right ),-\frac {a-b}{a+b}\right ) \sec (e+f x) \sqrt {\frac {a (1-\sin (e+f x))}{a+b \sin (e+f x)}} \sqrt {\frac {a (1+\sin (e+f x))}{a+b \sin (e+f x)}} (a+b \sin (e+f x))}{\sqrt {a+b} c f}+\frac {g E\left (\arcsin \left (\frac {\cos (e+f x)}{1+\sin (e+f x)}\right )|-\frac {a-b}{a+b}\right ) \sqrt {\frac {\sin (e+f x)}{1+\sin (e+f x)}} \sqrt {a+b \sin (e+f x)}}{c f \sqrt {g \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a+b) (1+\sin (e+f x))}}} \]
2*EllipticPi((a+b)^(1/2)*(g*sin(f*x+e))^(1/2)/g^(1/2)/(a+b*sin(f*x+e))^(1/ 2),b/(a+b),((-a+b)/(a+b))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*g^(1/2)*(a*(1 -sin(f*x+e))/(a+b*sin(f*x+e)))^(1/2)*(a*(1+sin(f*x+e))/(a+b*sin(f*x+e)))^( 1/2)/c/f/(a+b)^(1/2)+g*EllipticE(cos(f*x+e)/(1+sin(f*x+e)),((-a+b)/(a+b))^ (1/2))*(sin(f*x+e)/(1+sin(f*x+e)))^(1/2)*(a+b*sin(f*x+e))^(1/2)/c/f/(g*sin (f*x+e))^(1/2)/((a+b*sin(f*x+e))/(a+b)/(1+sin(f*x+e)))^(1/2)
Result contains complex when optimal does not.
Time = 32.78 (sec) , antiderivative size = 13199, normalized size of antiderivative = 49.43 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx=\text {Result too large to show} \]
Time = 0.96 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {3042, 3407, 3042, 3290, 3411}
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.
\(\displaystyle \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{c \sin (e+f x)+c} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{c \sin (e+f x)+c}dx\) |
\(\Big \downarrow \) 3407 |
\(\displaystyle \frac {g \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)}}dx}{c}-g \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (\sin (e+f x) c+c)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {g \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)}}dx}{c}-g \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (\sin (e+f x) c+c)}dx\) |
\(\Big \downarrow \) 3290 |
\(\displaystyle \frac {2 \sqrt {g} \sec (e+f x) \sqrt {\frac {a (1-\sin (e+f x))}{a+b \sin (e+f x)}} \sqrt {\frac {a (\sin (e+f x)+1)}{a+b \sin (e+f x)}} (a+b \sin (e+f x)) \operatorname {EllipticPi}\left (\frac {b}{a+b},\arcsin \left (\frac {\sqrt {a+b} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {a+b \sin (e+f x)}}\right ),-\frac {a-b}{a+b}\right )}{c f \sqrt {a+b}}-g \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (\sin (e+f x) c+c)}dx\) |
\(\Big \downarrow \) 3411 |
\(\displaystyle \frac {g \sqrt {\frac {\sin (e+f x)}{\sin (e+f x)+1}} \sqrt {a+b \sin (e+f x)} E\left (\arcsin \left (\frac {\cos (e+f x)}{\sin (e+f x)+1}\right )|-\frac {a-b}{a+b}\right )}{c f \sqrt {g \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}+\frac {2 \sqrt {g} \sec (e+f x) \sqrt {\frac {a (1-\sin (e+f x))}{a+b \sin (e+f x)}} \sqrt {\frac {a (\sin (e+f x)+1)}{a+b \sin (e+f x)}} (a+b \sin (e+f x)) \operatorname {EllipticPi}\left (\frac {b}{a+b},\arcsin \left (\frac {\sqrt {a+b} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {a+b \sin (e+f x)}}\right ),-\frac {a-b}{a+b}\right )}{c f \sqrt {a+b}}\) |
(2*Sqrt[g]*EllipticPi[b/(a + b), ArcSin[(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]]) /(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])], -((a - b)/(a + b))]*Sec[e + f*x]*Sqr t[(a*(1 - Sin[e + f*x]))/(a + b*Sin[e + f*x])]*Sqrt[(a*(1 + Sin[e + f*x])) /(a + b*Sin[e + f*x])]*(a + b*Sin[e + f*x]))/(Sqrt[a + b]*c*f) + (g*Ellipt icE[ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -((a - b)/(a + b))]*Sqrt[Sin[ e + f*x]/(1 + Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]])/(c*f*Sqrt[g*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x]))])
3.1.31.3.1 Defintions of rubi rules used
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[(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_. )*(x_)]])/((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g/d I nt[Sqrt[a + b*Sin[e + f*x]]/Sqrt[g*Sin[e + f*x]], x], x] - Simp[c*(g/d) I nt[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x] , x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(g_.)*sin[(e_.) + (f_. )*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(-Sqrt[ a + b*Sin[e + f*x]])*(Sqrt[d*(Sin[e + f*x]/(c + d*Sin[e + f*x]))]/(d*f*Sqrt [g*Sin[e + f*x]]*Sqrt[c^2*((a + b*Sin[e + f*x])/((a*c + b*d)*(c + d*Sin[e + f*x])))]))*EllipticE[ArcSin[c*(Cos[e + f*x]/(c + d*Sin[e + f*x]))], (b*c - a*d)/(b*c + a*d)], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
Result contains complex when optimal does not.
Time = 2.60 (sec) , antiderivative size = 9956, normalized size of antiderivative = 37.29
\[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx=\int { \frac {\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{c \sin \left (f x + e\right ) + c} \,d x } \]
\[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx=\frac {\int \frac {\sqrt {g \sin {\left (e + f x \right )}} \sqrt {a + b \sin {\left (e + f x \right )}}}{\sin {\left (e + f x \right )} + 1}\, dx}{c} \]
\[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx=\int { \frac {\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{c \sin \left (f x + e\right ) + c} \,d x } \]
\[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx=\int { \frac {\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{c \sin \left (f x + e\right ) + c} \,d x } \]
Timed out. \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx=\int \frac {\sqrt {g\,\sin \left (e+f\,x\right )}\,\sqrt {a+b\,\sin \left (e+f\,x\right )}}{c+c\,\sin \left (e+f\,x\right )} \,d x \]