Integrand size = 39, antiderivative size = 252 \[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\frac {g E\left (\arcsin \left (\frac {\cos (e+f x)}{1+\sin (e+f x)}\right )|-\frac {a-b}{a+b}\right ) \sqrt {\frac {\sin (e+f x)}{1+\sin (e+f x)}} \sqrt {a+b \sin (e+f x)}}{(a-b) c f \sqrt {g \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a+b) (1+\sin (e+f x))}}}-\frac {2 \sqrt {a+b} \sqrt {g} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{(a-b) c f} \]
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)/(a-b)/c/f/(g*sin(f*x+e))^(1/2)/ ((a+b*sin(f*x+e))/(a+b)/(1+sin(f*x+e)))^(1/2)-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)*g^(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)/(a-b)/c/f
Leaf count is larger than twice the leaf count of optimal. \(4464\) vs. \(2(252)=504\).
Time = 28.85 (sec) , antiderivative size = 4464, normalized size of antiderivative = 17.71 \[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\text {Result too large to show} \]
(2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[g*Sin[e + f *x]]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*f*(c + c*Sin[e + f*x])) + (Cot[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sqrt[g*Sin[e + f*x]]*Sqr t[a + b*Sin[e + f*x]]*(-1/2*(a*Sqrt[Sin[e + f*x]])/((a - b)*Sqrt[a + b*Sin [e + f*x]]) - (b*Sqrt[Sin[e + f*x]])/(2*(a - b)*Sqrt[a + b*Sin[e + f*x]]) - (b*Cos[(3*(e + f*x))/2]*Sec[(e + f*x)/2]*Sqrt[Sin[e + f*x]])/(2*(a - b)* Sqrt[a + b*Sin[e + f*x]]) - (b*Sec[(e + f*x)/2]*Sqrt[Sin[e + f*x]]*Sin[(3* (e + f*x))/2])/(2*(a - b)*Sqrt[a + b*Sin[e + f*x]]) + (a*Sqrt[Sin[e + f*x] ]*Tan[(e + f*x)/2])/(2*(a - b)*Sqrt[a + b*Sin[e + f*x]]) - (b*Sqrt[Sin[e + f*x]]*Tan[(e + f*x)/2])/(2*(a - b)*Sqrt[a + b*Sin[e + f*x]]))*(-2*Tan[(e + f*x)/2]*(1 + Tan[(e + f*x)/2]) + (2*Sqrt[-a^2 + b^2]*Sqrt[(a*Sec[(e + f* x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-(EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt [-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + EllipticF[ArcSi n[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[ 2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2] )/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^ 2]))]))/((a + b*Sin[e + f*x])*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])))/(2*(a - b)*f*(c + c*Sin[e + f*x])*((b*Cos[e + f*x]*Cot[(e + f*x) /2]*Sqrt[Sin[e + f*x]]*(-2*Tan[(e + f*x)/2]*(1 + Tan[(e + f*x)/2]) + (2...
Time = 0.95 (sec) , antiderivative size = 252, 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, 3415, 3042, 3295, 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.
| \(\displaystyle \int \frac {\sqrt {g \sin (e+f x)}}{(c \sin (e+f x)+c) \sqrt {a+b \sin (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {g \sin (e+f x)}}{(c \sin (e+f x)+c) \sqrt {a+b \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3415 |
| \(\displaystyle \frac {a g \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx}{c (a-b)}-\frac {g \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (\sin (e+f x) c+c)}dx}{a-b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a g \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx}{c (a-b)}-\frac {g \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (\sin (e+f x) c+c)}dx}{a-b}\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle -\frac {g \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (\sin (e+f x) c+c)}dx}{a-b}-\frac {2 \sqrt {g} \sqrt {a+b} \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{c f (a-b)}\) |
| \(\Big \downarrow \) 3411 |
| \(\displaystyle \frac {g \sqrt {\frac {\sin (e+f x)}{\sin (e+f x)+1}} \sqrt {a+b \sin (e+f x)} E\left (\arcsin \left (\frac {\cos (e+f x)}{\sin (e+f x)+1}\right )|-\frac {a-b}{a+b}\right )}{c f (a-b) \sqrt {g \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}-\frac {2 \sqrt {g} \sqrt {a+b} \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{c f (a-b)}\) |
(g*EllipticE[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]])/((a - b)*c *f*Sqrt[g*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f* x]))]) - (2*Sqrt[a + b]*Sqrt[g]*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])/((a - b)*c*f)
3.1.33.3.1 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[(-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]
Int[Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_. )*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(-a)*(g /(b*c - a*d)) Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x], x] + Simp[c*(g/(b*c - a*d)) Int[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] && N eQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(3056\) vs. \(2(232)=464\).
Time = 2.48 (sec) , antiderivative size = 3057, normalized size of antiderivative = 12.13
1/2/c/f*(g/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)*(csc(f*x+e)-cot(f*x+e)))^(1/2 )*((1-cos(f*x+e))^2*csc(f*x+e)^2+1)*((a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b* (csc(f*x+e)-cot(f*x+e))+a)/((1-cos(f*x+e))^2*csc(f*x+e)^2+1))^(1/2)*((1/(b +(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2)*( 1/(-a^2+b^2)^(1/2)*(-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))^(1/2)* (-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+(- a^2+b^2)^(1/2))*(a*(csc(f*x+e)-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))*(-a^2+b^2)^(1/2)*2^ (1/2)*a*(csc(f*x+e)-cot(f*x+e))-2*(1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-c ot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2)*(1/(-a^2+b^2)^(1/2)*(-a*(csc(f*x+e)- cot(f*x+e))+(-a^2+b^2)^(1/2)-b))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e )-cot(f*x+e)))^(1/2)*EllipticE((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-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))*(-a^2+b^2)^(1/2)*2^(1/2)*b*(csc(f*x+e)-cot(f*x+e))-(1/ (b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2) *(1/(-a^2+b^2)^(1/2)*(-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))^(1/2 )*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+ (-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-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))*2^(1/2)*a^2*(csc( f*x+e)-cot(f*x+e))+(1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(...
\[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\int { \frac {\sqrt {g \sin \left (f x + e\right )}}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )}} \,d x } \]
integral(-sqrt(b*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))/(b*c*cos(f*x + e)^ 2 - (a + b)*c*sin(f*x + e) - (a + b)*c), x)
\[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\frac {\int \frac {\sqrt {g \sin {\left (e + f x \right )}}}{\sqrt {a + b \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + \sqrt {a + b \sin {\left (e + f x \right )}}}\, dx}{c} \]
Integral(sqrt(g*sin(e + f*x))/(sqrt(a + b*sin(e + f*x))*sin(e + f*x) + sqr t(a + b*sin(e + f*x))), x)/c
\[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\int { \frac {\sqrt {g \sin \left (f x + e\right )}}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )}} \,d x } \]
\[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\int { \frac {\sqrt {g \sin \left (f x + e\right )}}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\int \frac {\sqrt {g\,\sin \left (e+f\,x\right )}}{\sqrt {a+b\,\sin \left (e+f\,x\right )}\,\left (c+c\,\sin \left (e+f\,x\right )\right )} \,d x \]