Integrand size = 39, antiderivative size = 250 \[ \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))} \, dx=-\frac {2 \sqrt {a+b} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{c f \sqrt {g}}+\frac {2 (b c-a d) \sqrt {-\cot ^2(e+f x)} \sqrt {\frac {b+a \csc (e+f x)}{a+b}} \operatorname {EllipticPi}\left (\frac {2 c}{c+d},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 a}{a+b}\right ) \sqrt {g \sin (e+f x)} \tan (e+f x)}{c (c+d) f g \sqrt {a+b \sin (e+f x)}} \]
-2*EllipticF(g^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(g*sin(f*x+e))^(1/ 2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*(a*(1-csc(f*x+e))/(a+b))^(1/2)*(a*(1+ csc(f*x+e))/(a-b))^(1/2)*tan(f*x+e)/c/f/g^(1/2)+2*(-a*d+b*c)*EllipticPi(1/ 2*(1-csc(f*x+e))^(1/2)*2^(1/2),2*c/(c+d),2^(1/2)*(a/(a+b))^(1/2))*(-cot(f* x+e)^2)^(1/2)*((b+a*csc(f*x+e))/(a+b))^(1/2)*(g*sin(f*x+e))^(1/2)*tan(f*x+ e)/c/(c+d)/f/g/(a+b*sin(f*x+e))^(1/2)
\[ \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))} \, dx=\int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))} \, dx \]
Time = 0.97 (sec) , antiderivative size = 250, 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, 3412, 3042, 3295, 3416}
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.
| \(\displaystyle \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))}dx\) |
| \(\Big \downarrow \) 3412 |
| \(\displaystyle \frac {(b c-a d) \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))}dx}{c g}+\frac {a \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx}{c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(b c-a d) \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))}dx}{c g}+\frac {a \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx}{c}\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \frac {(b c-a d) \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))}dx}{c g}-\frac {2 \sqrt {a+b} \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{c f \sqrt {g}}\) |
| \(\Big \downarrow \) 3416 |
| \(\displaystyle \frac {2 (b c-a d) \tan (e+f x) \sqrt {-\cot ^2(e+f x)} \sqrt {g \sin (e+f x)} \sqrt {\frac {a \csc (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (\frac {2 c}{c+d},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 a}{a+b}\right )}{c f g (c+d) \sqrt {a+b \sin (e+f x)}}-\frac {2 \sqrt {a+b} \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{c f \sqrt {g}}\) |
(-2*Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[ a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(c*f*Sqrt [g]) + (2*(b*c - a*d)*Sqrt[-Cot[e + f*x]^2]*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2* a)/(a + b)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/(c*(c + d)*f*g*Sqrt[a + b*S in[e + f*x]])
3.1.43.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] ], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(g_.)*sin[(e_.) + (f_. )*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[a/c I nt[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x], x] + Simp[(b*c - a*d)/(c*g) Int[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*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] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_. )*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[2*Sqrt[ -Cot[e + f*x]^2]*(Sqrt[g*Sin[e + f*x]]/(f*(c + d)*Cot[e + f*x]*Sqrt[a + b*S in[e + f*x]]))*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[2*(c/(c + d)), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], 2*(a/(a + b))], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2883\) vs. \(2(231)=462\).
Time = 2.50 (sec) , antiderivative size = 2884, normalized size of antiderivative = 11.54
1/f*(a*d-b*c)*(2*(-a^2+b^2)^(1/2)*(-c^2+d^2)^(1/2)*EllipticPi((1/(b+(-a^2+ b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2),(b+(-a^2 +b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1/2) *((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b+2*(-a^2+b^2)^(1/2)*(-c^2 +d^2)^(1/2)*EllipticPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+ (-a^2+b^2)^(1/2)+b))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)-a*(- c^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2) )^(1/2))*b-4*(-a^2+b^2)^(1/2)*(-c^2+d^2)^(1/2)*EllipticF((1/(b+(-a^2+b^2)^ (1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*( (b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b+(-a^2+b^2)^(1/2)*EllipticP i((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^ (1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-a*d+b* c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*c-2*(-a^2+ b^2)^(1/2)*EllipticPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+( -a^2+b^2)^(1/2)+b))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)+a*(-c ^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2)) ^(1/2))*b*d-(-a^2+b^2)^(1/2)*EllipticPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x +e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^ 2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2)) /(-a^2+b^2)^(1/2))^(1/2))*a*c+2*(-a^2+b^2)^(1/2)*EllipticPi((1/(b+(-a^2...
Timed out. \[ \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))} \, dx=\int \frac {\sqrt {a + b \sin {\left (e + f x \right )}}}{\sqrt {g \sin {\left (e + f x \right )}} \left (c + d \sin {\left (e + f x \right )}\right )}\, dx \]
\[ \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))} \, dx=\int { \frac {\sqrt {b \sin \left (f x + e\right ) + a}}{{\left (d \sin \left (f x + e\right ) + c\right )} \sqrt {g \sin \left (f x + e\right )}} \,d x } \]
\[ \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))} \, dx=\int { \frac {\sqrt {b \sin \left (f x + e\right ) + a}}{{\left (d \sin \left (f x + e\right ) + c\right )} \sqrt {g \sin \left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))} \, dx=\int \frac {\sqrt {a+b\,\sin \left (e+f\,x\right )}}{\sqrt {g\,\sin \left (e+f\,x\right )}\,\left (c+d\,\sin \left (e+f\,x\right )\right )} \,d x \]