1,1,509,0,0.8153704,"\int \frac{\sqrt{d \cos (e+f x)} \sqrt{g \sin (e+f x)}}{a+b \cos (e+f x)} \, dx","Int[(Sqrt[d*Cos[e + f*x]]*Sqrt[g*Sin[e + f*x]])/(a + b*Cos[e + f*x]),x]","\frac{2 \sqrt{2} a d \sqrt{g} \sqrt{\cos (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b f \sqrt{b-a} \sqrt{a+b} \sqrt{d \cos (e+f x)}}-\frac{2 \sqrt{2} a d \sqrt{g} \sqrt{\cos (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b f \sqrt{b-a} \sqrt{a+b} \sqrt{d \cos (e+f x)}}-\frac{\sqrt{d} \sqrt{g} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{d \cos (e+f x)}}\right)}{\sqrt{2} b f}+\frac{\sqrt{d} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{d \cos (e+f x)}}+1\right)}{\sqrt{2} b f}+\frac{\sqrt{d} \sqrt{g} \log \left(-\frac{\sqrt{2} \sqrt{d} \sqrt{g \sin (e+f x)}}{\sqrt{d \cos (e+f x)}}+\sqrt{g} \tan (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b f}-\frac{\sqrt{d} \sqrt{g} \log \left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \sin (e+f x)}}{\sqrt{d \cos (e+f x)}}+\sqrt{g} \tan (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b f}","\frac{2 \sqrt{2} a d \sqrt{g} \sqrt{\cos (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b f \sqrt{b-a} \sqrt{a+b} \sqrt{d \cos (e+f x)}}-\frac{2 \sqrt{2} a d \sqrt{g} \sqrt{\cos (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b f \sqrt{b-a} \sqrt{a+b} \sqrt{d \cos (e+f x)}}-\frac{\sqrt{d} \sqrt{g} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{d \cos (e+f x)}}\right)}{\sqrt{2} b f}+\frac{\sqrt{d} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{d \cos (e+f x)}}+1\right)}{\sqrt{2} b f}+\frac{\sqrt{d} \sqrt{g} \log \left(-\frac{\sqrt{2} \sqrt{d} \sqrt{g \sin (e+f x)}}{\sqrt{d \cos (e+f x)}}+\sqrt{g} \tan (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b f}-\frac{\sqrt{d} \sqrt{g} \log \left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \sin (e+f x)}}{\sqrt{d \cos (e+f x)}}+\sqrt{g} \tan (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b f}",1,"-((Sqrt[d]*Sqrt[g]*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Sin[e + f*x]])/(Sqrt[g]*Sqrt[d*Cos[e + f*x]])])/(Sqrt[2]*b*f)) + (Sqrt[d]*Sqrt[g]*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Sin[e + f*x]])/(Sqrt[g]*Sqrt[d*Cos[e + f*x]])])/(Sqrt[2]*b*f) + (2*Sqrt[2]*a*d*Sqrt[g]*Sqrt[Cos[e + f*x]]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Sin[e + f*x]]/(Sqrt[g]*Sqrt[1 + Cos[e + f*x]])], -1])/(b*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Cos[e + f*x]]) - (2*Sqrt[2]*a*d*Sqrt[g]*Sqrt[Cos[e + f*x]]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Sin[e + f*x]]/(Sqrt[g]*Sqrt[1 + Cos[e + f*x]])], -1])/(b*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Cos[e + f*x]]) + (Sqrt[d]*Sqrt[g]*Log[Sqrt[g] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Sin[e + f*x]])/Sqrt[d*Cos[e + f*x]] + Sqrt[g]*Tan[e + f*x]])/(2*Sqrt[2]*b*f) - (Sqrt[d]*Sqrt[g]*Log[Sqrt[g] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Sin[e + f*x]])/Sqrt[d*Cos[e + f*x]] + Sqrt[g]*Tan[e + f*x]])/(2*Sqrt[2]*b*f)","A",16,12,37,0.3243,1,"{2909, 2574, 297, 1162, 617, 204, 1165, 628, 2906, 2905, 490, 1218}"
2,1,209,0,0.4098493,"\int \frac{\sqrt{d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt{g \sin (e+f x)}} \, dx","Int[Sqrt[d*Cos[e + f*x]]/((a + b*Cos[e + f*x])*Sqrt[g*Sin[e + f*x]]),x]","\frac{2 \sqrt{2} \sqrt{d} \sqrt{\sin (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \cos (e+f x)}}{\sqrt{d} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b^2-a^2} \sqrt{g \sin (e+f x)}}-\frac{2 \sqrt{2} \sqrt{d} \sqrt{\sin (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \cos (e+f x)}}{\sqrt{d} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b^2-a^2} \sqrt{g \sin (e+f x)}}","\frac{2 \sqrt{2} \sqrt{d} \sqrt{\sin (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \cos (e+f x)}}{\sqrt{d} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b^2-a^2} \sqrt{g \sin (e+f x)}}-\frac{2 \sqrt{2} \sqrt{d} \sqrt{\sin (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \cos (e+f x)}}{\sqrt{d} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b^2-a^2} \sqrt{g \sin (e+f x)}}",1,"(2*Sqrt[2]*Sqrt[d]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Cos[e + f*x]]/(Sqrt[d]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(Sqrt[-a^2 + b^2]*f*Sqrt[g*Sin[e + f*x]]) - (2*Sqrt[2]*Sqrt[d]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Cos[e + f*x]]/(Sqrt[d]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(Sqrt[-a^2 + b^2]*f*Sqrt[g*Sin[e + f*x]])","A",4,3,37,0.08108,1,"{2908, 2907, 1218}"
3,1,208,0,0.3901428,"\int \frac{\sqrt{g \sin (e+f x)}}{\sqrt{d \cos (e+f x)} (a+b \cos (e+f x))} \, dx","Int[Sqrt[g*Sin[e + f*x]]/(Sqrt[d*Cos[e + f*x]]*(a + b*Cos[e + f*x])),x]","\frac{2 \sqrt{2} \sqrt{g} \sqrt{\cos (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b-a} \sqrt{a+b} \sqrt{d \cos (e+f x)}}-\frac{2 \sqrt{2} \sqrt{g} \sqrt{\cos (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b-a} \sqrt{a+b} \sqrt{d \cos (e+f x)}}","\frac{2 \sqrt{2} \sqrt{g} \sqrt{\cos (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b-a} \sqrt{a+b} \sqrt{d \cos (e+f x)}}-\frac{2 \sqrt{2} \sqrt{g} \sqrt{\cos (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b-a} \sqrt{a+b} \sqrt{d \cos (e+f x)}}",1,"(-2*Sqrt[2]*Sqrt[g]*Sqrt[Cos[e + f*x]]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Sin[e + f*x]]/(Sqrt[g]*Sqrt[1 + Cos[e + f*x]])], -1])/(Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Cos[e + f*x]]) + (2*Sqrt[2]*Sqrt[g]*Sqrt[Cos[e + f*x]]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Sin[e + f*x]]/(Sqrt[g]*Sqrt[1 + Cos[e + f*x]])], -1])/(Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Cos[e + f*x]])","A",5,4,37,0.1081,1,"{2906, 2905, 490, 1218}"
4,1,273,0,0.6181506,"\int \frac{1}{\sqrt{d \cos (e+f x)} (a+b \cos (e+f x)) \sqrt{g \sin (e+f x)}} \, dx","Int[1/(Sqrt[d*Cos[e + f*x]]*(a + b*Cos[e + f*x])*Sqrt[g*Sin[e + f*x]]),x]","-\frac{2 \sqrt{2} b \sqrt{\sin (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \cos (e+f x)}}{\sqrt{d} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a \sqrt{d} f \sqrt{b^2-a^2} \sqrt{g \sin (e+f x)}}+\frac{2 \sqrt{2} b \sqrt{\sin (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \cos (e+f x)}}{\sqrt{d} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a \sqrt{d} f \sqrt{b^2-a^2} \sqrt{g \sin (e+f x)}}+\frac{\sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a f \sqrt{d \cos (e+f x)} \sqrt{g \sin (e+f x)}}","-\frac{2 \sqrt{2} b \sqrt{\sin (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \cos (e+f x)}}{\sqrt{d} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a \sqrt{d} f \sqrt{b^2-a^2} \sqrt{g \sin (e+f x)}}+\frac{2 \sqrt{2} b \sqrt{\sin (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \cos (e+f x)}}{\sqrt{d} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a \sqrt{d} f \sqrt{b^2-a^2} \sqrt{g \sin (e+f x)}}+\frac{\sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a f \sqrt{d \cos (e+f x)} \sqrt{g \sin (e+f x)}}",1,"(-2*Sqrt[2]*b*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Cos[e + f*x]]/(Sqrt[d]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*Sqrt[-a^2 + b^2]*Sqrt[d]*f*Sqrt[g*Sin[e + f*x]]) + (2*Sqrt[2]*b*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Cos[e + f*x]]/(Sqrt[d]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*Sqrt[-a^2 + b^2]*Sqrt[d]*f*Sqrt[g*Sin[e + f*x]]) + (EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a*f*Sqrt[d*Cos[e + f*x]]*Sqrt[g*Sin[e + f*x]])","A",7,6,37,0.1622,1,"{2910, 2573, 2641, 2908, 2907, 1218}"