1,1,264,509,5.5194224,"\int \frac{\sqrt{d \cos (e+f x)} \sqrt{g \sin (e+f x)}}{a+b \cos (e+f x)} \, dx","Integrate[(Sqrt[d*Cos[e + f*x]]*Sqrt[g*Sin[e + f*x]])/(a + b*Cos[e + f*x]),x]","\frac{2 \sqrt{2} g \sqrt{\tan \left(\frac{1}{2} (e+f x)\right)} \sqrt{d \cos (e+f x)} \left(i \sqrt{-a-b} \sqrt{a-b} \Pi \left(-i;\left.\sin ^{-1}\left(\sqrt{\tan \left(\frac{1}{2} (e+f x)\right)}\right)\right|-1\right)-i \sqrt{-a-b} \sqrt{a-b} \Pi \left(i;\left.\sin ^{-1}\left(\sqrt{\tan \left(\frac{1}{2} (e+f x)\right)}\right)\right|-1\right)+a \left(\Pi \left(\frac{\sqrt{a-b}}{\sqrt{-a-b}};\left.\sin ^{-1}\left(\sqrt{\tan \left(\frac{1}{2} (e+f x)\right)}\right)\right|-1\right)-\Pi \left(-\frac{\sqrt{a-b}}{\sqrt{-a-b}};\left.\sin ^{-1}\left(\sqrt{\tan \left(\frac{1}{2} (e+f x)\right)}\right)\right|-1\right)\right)\right)}{b f \sqrt{-a-b} \sqrt{a-b} \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sqrt{g \sin (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{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,"(2*Sqrt[2]*g*Sqrt[d*Cos[e + f*x]]*(I*Sqrt[-a - b]*Sqrt[a - b]*EllipticPi[-I, ArcSin[Sqrt[Tan[(e + f*x)/2]]], -1] - I*Sqrt[-a - b]*Sqrt[a - b]*EllipticPi[I, ArcSin[Sqrt[Tan[(e + f*x)/2]]], -1] + a*(-EllipticPi[-(Sqrt[a - b]/Sqrt[-a - b]), ArcSin[Sqrt[Tan[(e + f*x)/2]]], -1] + EllipticPi[Sqrt[a - b]/Sqrt[-a - b], ArcSin[Sqrt[Tan[(e + f*x)/2]]], -1]))*Sqrt[Tan[(e + f*x)/2]])/(Sqrt[-a - b]*Sqrt[a - b]*b*f*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[g*Sin[e + f*x]])","C",1
2,1,594,209,9.6896697,"\int \frac{\sqrt{d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt{g \sin (e+f x)}} \, dx","Integrate[Sqrt[d*Cos[e + f*x]]/((a + b*Cos[e + f*x])*Sqrt[g*Sin[e + f*x]]),x]","\frac{2 \sqrt{\tan (e+f x)} \sec ^2(e+f x) \sqrt{d \cos (e+f x)} \left(a \sqrt{\tan ^2(e+f x)+1}+b\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\tan (e+f x)} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\tan ^2(e+f x),-\frac{a^2 \tan ^2(e+f x)}{a^2-b^2}\right)}{\sqrt{\tan ^2(e+f x)+1} \left(a^2 \left(\tan ^2(e+f x)+1\right)-b^2\right) \left(2 \tan ^2(e+f x) \left(2 a^2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\tan ^2(e+f x),-\frac{a^2 \tan ^2(e+f x)}{a^2-b^2}\right)+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\tan ^2(e+f x),-\frac{a^2 \tan ^2(e+f x)}{a^2-b^2}\right)\right)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\tan ^2(e+f x),-\frac{a^2 \tan ^2(e+f x)}{a^2-b^2}\right)\right)}+\frac{\sqrt{a} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\tan (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\tan (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(-\sqrt{2} \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}+\sqrt{a^2-b^2}+a \tan (e+f x)\right)+\log \left(\sqrt{2} \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}+\sqrt{a^2-b^2}+a \tan (e+f x)\right)\right)}{4 \sqrt{2} \left(a^2-b^2\right)^{3/4}}\right)}{f \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{g \sin (e+f x)} (a+b \cos (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[d*Cos[e + f*x]]*Sec[e + f*x]^2*Sqrt[Tan[e + f*x]]*(b + a*Sqrt[1 + Tan[e + f*x]^2])*((Sqrt[a]*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Tan[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Tan[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + a*Tan[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + a*Tan[e + f*x]]))/(4*Sqrt[2]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, -Tan[e + f*x]^2, -((a^2*Tan[e + f*x]^2)/(a^2 - b^2))]*Sqrt[Tan[e + f*x]])/(Sqrt[1 + Tan[e + f*x]^2]*(-5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, -Tan[e + f*x]^2, -((a^2*Tan[e + f*x]^2)/(a^2 - b^2))] + 2*(2*a^2*AppellF1[5/4, 1/2, 2, 9/4, -Tan[e + f*x]^2, -((a^2*Tan[e + f*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, -Tan[e + f*x]^2, -((a^2*Tan[e + f*x]^2)/(a^2 - b^2))])*Tan[e + f*x]^2)*(-b^2 + a^2*(1 + Tan[e + f*x]^2)))))/(f*(a + b*Cos[e + f*x])*Sqrt[g*Sin[e + f*x]]*(1 + Tan[e + f*x]^2)^(3/2))","C",0
3,1,375,208,3.0935658,"\int \frac{\sqrt{g \sin (e+f x)}}{\sqrt{d \cos (e+f x)} (a+b \cos (e+f x))} \, dx","Integrate[Sqrt[g*Sin[e + f*x]]/(Sqrt[d*Cos[e + f*x]]*(a + b*Cos[e + f*x])),x]","\frac{2 \sqrt{g \sin (e+f x)} \left(a \sqrt{\sec ^2(e+f x)}+b\right) \left(\frac{b \tan ^{\frac{3}{2}}(e+f x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),-\frac{a^2 \tan ^2(e+f x)}{a^2-b^2}\right)}{3 \left(b^2-a^2\right)}+\frac{-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\tan (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\tan (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)+\log \left(-\sqrt{2} \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}+\sqrt{a^2-b^2}+a \tan (e+f x)\right)-\log \left(\sqrt{2} \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}+\sqrt{a^2-b^2}+a \tan (e+f x)\right)}{4 \sqrt{2} \sqrt{a} \sqrt[4]{a^2-b^2}}\right)}{f \sqrt{\tan (e+f x)} \sqrt{\sec ^2(e+f x)} \sqrt{d \cos (e+f x)} (a+b \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*(b + a*Sqrt[Sec[e + f*x]^2])*Sqrt[g*Sin[e + f*x]]*((-2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Tan[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Tan[e + f*x]])/(a^2 - b^2)^(1/4)] + Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + a*Tan[e + f*x]] - Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + a*Tan[e + f*x]])/(4*Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)) + (b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, -((a^2*Tan[e + f*x]^2)/(a^2 - b^2))]*Tan[e + f*x]^(3/2))/(3*(-a^2 + b^2))))/(f*Sqrt[d*Cos[e + f*x]]*(a + b*Cos[e + f*x])*Sqrt[Sec[e + f*x]^2]*Sqrt[Tan[e + f*x]])","C",0
4,1,496,273,9.2312831,"\int \frac{1}{\sqrt{d \cos (e+f x)} (a+b \cos (e+f x)) \sqrt{g \sin (e+f x)}} \, dx","Integrate[1/(Sqrt[d*Cos[e + f*x]]*(a + b*Cos[e + f*x])*Sqrt[g*Sin[e + f*x]]),x]","\frac{18 (a+b) \sqrt{g \sin (e+f x)} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\tan ^2\left(\frac{1}{2} (e+f x)\right),\frac{(b-a) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}\right)+5 F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\tan ^2\left(\frac{1}{2} (e+f x)\right),\frac{(b-a) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}\right)\right)}{f g \sqrt{d \cos (e+f x)} (a+b \cos (e+f x)) \left(\tan ^2\left(\frac{1}{2} (e+f x)\right) \left(10 \tan ^2\left(\frac{1}{2} (e+f x)\right) \left((a+b) F_1\left(\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\tan ^2\left(\frac{1}{2} (e+f x)\right),\frac{(b-a) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}\right)-2 (a-b) F_1\left(\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\tan ^2\left(\frac{1}{2} (e+f x)\right),\frac{(b-a) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}\right)\right)+45 (a+b) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\tan ^2\left(\frac{1}{2} (e+f x)\right),\frac{(b-a) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}\right)-36 (a-b) F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\tan ^2\left(\frac{1}{2} (e+f x)\right),\frac{(b-a) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}\right)+18 (a+b) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\tan ^2\left(\frac{1}{2} (e+f x)\right),\frac{(b-a) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}\right)\right)+45 (a+b) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\tan ^2\left(\frac{1}{2} (e+f x)\right),\frac{(b-a) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}\right)\right)}","-\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,"(18*(a + b)*Sqrt[g*Sin[e + f*x]]*(5*AppellF1[1/4, 1/2, 1, 5/4, Tan[(e + f*x)/2]^2, ((-a + b)*Tan[(e + f*x)/2]^2)/(a + b)] + AppellF1[5/4, 1/2, 1, 9/4, Tan[(e + f*x)/2]^2, ((-a + b)*Tan[(e + f*x)/2]^2)/(a + b)]*Tan[(e + f*x)/2]^2))/(f*g*Sqrt[d*Cos[e + f*x]]*(a + b*Cos[e + f*x])*(45*(a + b)*AppellF1[1/4, 1/2, 1, 5/4, Tan[(e + f*x)/2]^2, ((-a + b)*Tan[(e + f*x)/2]^2)/(a + b)] + Tan[(e + f*x)/2]^2*(45*(a + b)*AppellF1[5/4, 1/2, 1, 9/4, Tan[(e + f*x)/2]^2, ((-a + b)*Tan[(e + f*x)/2]^2)/(a + b)] - 36*(a - b)*AppellF1[5/4, 1/2, 2, 9/4, Tan[(e + f*x)/2]^2, ((-a + b)*Tan[(e + f*x)/2]^2)/(a + b)] + 18*(a + b)*AppellF1[5/4, 3/2, 1, 9/4, Tan[(e + f*x)/2]^2, ((-a + b)*Tan[(e + f*x)/2]^2)/(a + b)] + 10*(-2*(a - b)*AppellF1[9/4, 1/2, 2, 13/4, Tan[(e + f*x)/2]^2, ((-a + b)*Tan[(e + f*x)/2]^2)/(a + b)] + (a + b)*AppellF1[9/4, 3/2, 1, 13/4, Tan[(e + f*x)/2]^2, ((-a + b)*Tan[(e + f*x)/2]^2)/(a + b)])*Tan[(e + f*x)/2]^2)))","C",0