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[itex]\int\sqrt{E-Bk^{2}\frac{cos^{2}(kr)}{sin^{2}(kr)}-k\frac{cos(kr)}{sin(kr)}\sqrt{D+Fk^{2}\frac{cos^{2}(kr)}{sin^{2}(kr)}}-\frac{Ak^{2}}{sin^{2}(kr)}}dr[/itex]

where [itex]A,B,C,D,E,F,k[/itex] are constants.

Substitution [itex]t=sin(kr)[/itex] leads to

[itex]\int\sqrt{\frac{E}{k^{2}(1-t^{2})}-\frac{B}{t^{2}}-\frac{1}{kt\sqrt{1-t^{2}}}\sqrt{D+Fk^{2}\frac{1-t^{2}}{t^{2}}}-\frac{A}{t^{2}(1-t^{2})}}dt[/itex]

The last integral is equal to

[itex]\int\frac{1}{t\sqrt{1-t^{2}}}\sqrt{(\frac{E}{k^{2}}+B)t^{2}-(A+B)-\sqrt{(F-\frac{D}{k^{2}})t^{4}+(\frac{D}{k^{2}}-2F)t^{2}+F}}dt[/itex]

I don't have any idea what to do now. Maybe the substitution [itex]t=sin(kr)[/itex] is not appropriate here?

Any help is appreciated.