Integrand size = 33, antiderivative size = 154 \[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\frac {2 c \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{a f \sqrt {c+d \sin (e+f x)}}-\frac {2 (b c-a d) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{a (a+b) f \sqrt {c+d \sin (e+f x)}} \]
-2*c*(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)*Ellipti cPi(cos(1/2*e+1/4*Pi+1/2*f*x),2,2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e)) /(c+d))^(1/2)/a/f/(c+d*sin(f*x+e))^(1/2)+2*(-a*d+b*c)*(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/( a+b)/f/(c+d*sin(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 30.02 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.16 \[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\frac {2 i \left (\operatorname {EllipticPi}\left (\frac {c+d}{c},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )-\operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-a d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right ) \sec (e+f x) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {\frac {d (1+\sin (e+f x))}{-c+d}}}{a \sqrt {-\frac {1}{c+d}} f} \]
((2*I)*(EllipticPi[(c + d)/c, I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin [e + f*x]]], (c + d)/(c - d)] - EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcS inh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sec[e + f*x]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[(d*(1 + Sin[e + f*x] ))/(-c + d)])/(a*Sqrt[-(c + d)^(-1)]*f)
Time = 0.83 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3042, 3414, 3042, 3286, 3042, 3284}
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.
| \(\displaystyle \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {c+d \sin (e+f x)}}{\sin (e+f x) (a+b \sin (e+f x))}dx\) |
| \(\Big \downarrow \) 3414 |
| \(\displaystyle \frac {c \int \frac {\csc (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{a}-\frac {(b c-a d) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {c \int \frac {1}{\sin (e+f x) \sqrt {c+d \sin (e+f x)}}dx}{a}-\frac {(b c-a d) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{a}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {c \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {\csc (e+f x)}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{a \sqrt {c+d \sin (e+f x)}}-\frac {(b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{a \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {c \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sin (e+f x) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{a \sqrt {c+d \sin (e+f x)}}-\frac {(b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{a \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {2 c \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{a f \sqrt {c+d \sin (e+f x)}}-\frac {2 (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{a f (a+b) \sqrt {c+d \sin (e+f x)}}\) |
(2*c*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[e + f*x]]) - (2*(b*c - a*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*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])
3.1.40.3.1 Defintions of rubi rules used
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]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[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] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(sin[(e_.) + (f_.)*(x_)]*((c _) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[a/c Int[1/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x], x] + Simp[(b*c - a*d)/c Int[1/(Sqrt [a + b*Sin[e + f*x]]*(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]
Time = 1.29 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.23
| method | result | size |
| default | \(-\frac {2 \left (\Pi \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \frac {c -d}{c}, \sqrt {\frac {c -d}{c +d}}\right )-\Pi \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, -\frac {\left (c -d \right ) b}{a d -b c}, \sqrt {\frac {c -d}{c +d}}\right )\right ) \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \left (c -d \right )}{a \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(190\) |
-2*(EllipticPi(((c+d*sin(f*x+e))/(c-d))^(1/2),(c-d)/c,((c-d)/(c+d))^(1/2)) -EllipticPi(((c+d*sin(f*x+e))/(c-d))^(1/2),-(c-d)*b/(a*d-b*c),((c-d)/(c+d) )^(1/2)))*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)* ((c+d*sin(f*x+e))/(c-d))^(1/2)*(c-d)/a/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f
Timed out. \[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\text {Timed out} \]
\[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int \frac {\sqrt {c + d \sin {\left (e + f x \right )}}}{\left (a + b \sin {\left (e + f x \right )}\right ) \sin {\left (e + f x \right )}}\, dx \]
\[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (b \sin \left (f x + e\right ) + a\right )} \sin \left (f x + e\right )} \,d x } \]
\[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (b \sin \left (f x + e\right ) + a\right )} \sin \left (f x + e\right )} \,d x } \]
Timed out. \[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]