1,1,499,105,1.6970838,"\int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^4 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^4,x]","-\frac{a c^4 \sec (e) \sec ^5(e+f x) \left(-1920 \sin (2 e+f x)+780 \sin (e+2 f x)+780 \sin (3 e+2 f x)+640 \sin (2 e+3 f x)-720 \sin (4 e+3 f x)+30 \sin (3 e+4 f x)+30 \sin (5 e+4 f x)+272 \sin (4 e+5 f x)+525 \cos (2 e+3 f x) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+525 \cos (4 e+3 f x) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+105 \cos (4 e+5 f x) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+105 \cos (6 e+5 f x) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+1050 \cos (f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+1050 \cos (2 e+f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-525 \cos (2 e+3 f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-525 \cos (4 e+3 f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-105 \cos (4 e+5 f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-105 \cos (6 e+5 f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+800 \sin (f x)\right)}{3840 f}","\frac{a c^4 \tan ^5(e+f x)}{5 f}+\frac{4 a c^4 \tan ^3(e+f x)}{3 f}+\frac{7 a c^4 \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac{3 a c^4 \tan (e+f x) \sec ^3(e+f x)}{4 f}-\frac{a c^4 \tan (e+f x) \sec (e+f x)}{8 f}",1,"-1/3840*(a*c^4*Sec[e]*Sec[e + f*x]^5*(525*Cos[2*e + 3*f*x]*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 525*Cos[4*e + 3*f*x]*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 105*Cos[4*e + 5*f*x]*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 105*Cos[6*e + 5*f*x]*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 1050*Cos[f*x]*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + 1050*Cos[2*e + f*x]*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - 525*Cos[2*e + 3*f*x]*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - 525*Cos[4*e + 3*f*x]*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - 105*Cos[4*e + 5*f*x]*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - 105*Cos[6*e + 5*f*x]*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 800*Sin[f*x] - 1920*Sin[2*e + f*x] + 780*Sin[e + 2*f*x] + 780*Sin[3*e + 2*f*x] + 640*Sin[2*e + 3*f*x] - 720*Sin[4*e + 3*f*x] + 30*Sin[3*e + 4*f*x] + 30*Sin[5*e + 4*f*x] + 272*Sin[4*e + 5*f*x]))/f","B",1
2,1,887,86,6.4825105,"\int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^3 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^3,x]","a \left(\frac{5 \cos ^3(e+f x) \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right) (c-c \sec (e+f x))^3 \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{64 f}-\frac{5 \cos ^3(e+f x) \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right) (c-c \sec (e+f x))^3 \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{64 f}+\frac{\cos ^3(e+f x) (c-c \sec (e+f x))^3 \sin \left(\frac{f x}{2}\right) \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{12 f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}+\frac{\cos ^3(e+f x) (c-c \sec (e+f x))^3 \sin \left(\frac{f x}{2}\right) \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{12 f \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}+\frac{\cos ^3(e+f x) (c-c \sec (e+f x))^3 \left(\cos \left(\frac{e}{2}\right)-17 \sin \left(\frac{e}{2}\right)\right) \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{384 f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{\cos ^3(e+f x) (c-c \sec (e+f x))^3 \left(-\cos \left(\frac{e}{2}\right)-17 \sin \left(\frac{e}{2}\right)\right) \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{384 f \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}-\frac{\cos ^3(e+f x) (c-c \sec (e+f x))^3 \sin \left(\frac{f x}{2}\right) \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{24 f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^3}-\frac{\cos ^3(e+f x) (c-c \sec (e+f x))^3 \sin \left(\frac{f x}{2}\right) \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{24 f \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^3}+\frac{\cos ^3(e+f x) (c-c \sec (e+f x))^3 \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{128 f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^4}-\frac{\cos ^3(e+f x) (c-c \sec (e+f x))^3 \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{128 f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^4}\right)","\frac{2 a c^3 \tan ^3(e+f x)}{3 f}+\frac{5 a c^3 \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac{a c^3 \tan (e+f x) \sec ^3(e+f x)}{4 f}-\frac{3 a c^3 \tan (e+f x) \sec (e+f x)}{8 f}",1,"a*((5*Cos[e + f*x]^3*Csc[e/2 + (f*x)/2]^6*Log[Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2]]*(c - c*Sec[e + f*x])^3)/(64*f) - (5*Cos[e + f*x]^3*Csc[e/2 + (f*x)/2]^6*Log[Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2]]*(c - c*Sec[e + f*x])^3)/(64*f) + (Cos[e + f*x]^3*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^3)/(128*f*(Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2])^4) - (Cos[e + f*x]^3*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^3*Sin[(f*x)/2])/(24*f*(Cos[e/2] - Sin[e/2])*(Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2])^3) + (Cos[e + f*x]^3*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^3*(Cos[e/2] - 17*Sin[e/2]))/(384*f*(Cos[e/2] - Sin[e/2])*(Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2])^2) + (Cos[e + f*x]^3*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^3*Sin[(f*x)/2])/(12*f*(Cos[e/2] - Sin[e/2])*(Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2])) - (Cos[e + f*x]^3*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^3)/(128*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^4) - (Cos[e + f*x]^3*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^3*Sin[(f*x)/2])/(24*f*(Cos[e/2] + Sin[e/2])*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^3) + (Cos[e + f*x]^3*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^3*(-Cos[e/2] - 17*Sin[e/2]))/(384*f*(Cos[e/2] + Sin[e/2])*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (Cos[e + f*x]^3*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^3*Sin[(f*x)/2])/(12*f*(Cos[e/2] + Sin[e/2])*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])))","B",1
3,1,313,61,0.7218371,"\int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^2 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^2,x]","-\frac{a c^2 \sec (e) \sec ^3(e+f x) \left(-12 \sin (2 e+f x)+6 \sin (e+2 f x)+6 \sin (3 e+2 f x)+4 \sin (2 e+3 f x)+3 \cos (2 e+3 f x) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+3 \cos (4 e+3 f x) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+9 \cos (f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+9 \cos (2 e+f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-3 \cos (2 e+3 f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-3 \cos (4 e+3 f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{48 f}","\frac{a c^2 \tan ^3(e+f x)}{3 f}+\frac{a c^2 \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac{a c^2 \tan (e+f x) \sec (e+f x)}{2 f}",1,"-1/48*(a*c^2*Sec[e]*Sec[e + f*x]^3*(3*Cos[2*e + 3*f*x]*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 3*Cos[4*e + 3*f*x]*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 9*Cos[f*x]*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + 9*Cos[2*e + f*x]*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - 3*Cos[2*e + 3*f*x]*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - 3*Cos[4*e + 3*f*x]*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - 12*Sin[2*e + f*x] + 6*Sin[e + 2*f*x] + 6*Sin[3*e + 2*f*x] + 4*Sin[2*e + 3*f*x]))/f","B",1
4,1,38,38,0.0280384,"\int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x)) \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x]),x]","-a c \left(\frac{\tan (e+f x) \sec (e+f x)}{2 f}-\frac{\tanh ^{-1}(\sin (e+f x))}{2 f}\right)","\frac{a c \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac{a c \tan (e+f x) \sec (e+f x)}{2 f}",1,"-(a*c*(-1/2*ArcTanh[Sin[e + f*x]]/f + (Sec[e + f*x]*Tan[e + f*x])/(2*f)))","A",1
5,1,77,42,0.0715368,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))}{c-c \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c - c*Sec[e + f*x]),x]","-\frac{a \left(-\frac{2 \cot \left(\frac{1}{2} (e+f x)\right)}{f}-\frac{\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f}+\frac{\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f}\right)}{c}","-\frac{a \tanh ^{-1}(\sin (e+f x))}{c f}-\frac{2 a \tan (e+f x)}{f (c-c \sec (e+f x))}",1,"-((a*((-2*Cot[(e + f*x)/2])/f - Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]/f + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]/f))/c)","A",1
6,1,50,36,0.2515942,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c - c*Sec[e + f*x])^2,x]","\frac{a \csc \left(\frac{e}{2}\right) \left(\sin \left(e+\frac{3 f x}{2}\right)-3 \sin \left(e+\frac{f x}{2}\right)\right) \csc ^3\left(\frac{1}{2} (e+f x)\right)}{12 c^2 f}","-\frac{\tan (e+f x) (a \sec (e+f x)+a)}{3 f (c-c \sec (e+f x))^2}",1,"(a*Csc[e/2]*Csc[(e + f*x)/2]^3*(-3*Sin[e + (f*x)/2] + Sin[e + (3*f*x)/2]))/(12*c^2*f)","A",1
7,1,87,76,0.4069195,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c - c*Sec[e + f*x])^3,x]","-\frac{a \csc \left(\frac{e}{2}\right) \left(15 \sin \left(e+\frac{f x}{2}\right)-5 \sin \left(e+\frac{3 f x}{2}\right)-15 \sin \left(2 e+\frac{3 f x}{2}\right)+4 \sin \left(2 e+\frac{5 f x}{2}\right)+25 \sin \left(\frac{f x}{2}\right)\right) \csc ^5\left(\frac{1}{2} (e+f x)\right)}{240 c^3 f}","-\frac{\tan (e+f x) (a \sec (e+f x)+a)}{15 c f (c-c \sec (e+f x))^2}-\frac{\tan (e+f x) (a \sec (e+f x)+a)}{5 f (c-c \sec (e+f x))^3}",1,"-1/240*(a*Csc[e/2]*Csc[(e + f*x)/2]^5*(25*Sin[(f*x)/2] + 15*Sin[e + (f*x)/2] - 5*Sin[e + (3*f*x)/2] - 15*Sin[2*e + (3*f*x)/2] + 4*Sin[2*e + (5*f*x)/2]))/(c^3*f)","A",1
8,1,113,116,0.4640667,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^4} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c - c*Sec[e + f*x])^4,x]","-\frac{a \csc \left(\frac{e}{2}\right) \left(455 \sin \left(e+\frac{f x}{2}\right)-273 \sin \left(e+\frac{3 f x}{2}\right)-210 \sin \left(2 e+\frac{3 f x}{2}\right)+56 \sin \left(2 e+\frac{5 f x}{2}\right)+105 \sin \left(3 e+\frac{5 f x}{2}\right)-23 \sin \left(3 e+\frac{7 f x}{2}\right)+350 \sin \left(\frac{f x}{2}\right)\right) \csc ^7\left(\frac{1}{2} (e+f x)\right)}{6720 c^4 f}","-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)}{105 f \left(c^2-c^2 \sec (e+f x)\right)^2}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)}{35 c f (c-c \sec (e+f x))^3}-\frac{\tan (e+f x) (a \sec (e+f x)+a)}{7 f (c-c \sec (e+f x))^4}",1,"-1/6720*(a*Csc[e/2]*Csc[(e + f*x)/2]^7*(350*Sin[(f*x)/2] + 455*Sin[e + (f*x)/2] - 273*Sin[e + (3*f*x)/2] - 210*Sin[2*e + (3*f*x)/2] + 56*Sin[2*e + (5*f*x)/2] + 105*Sin[3*e + (5*f*x)/2] - 23*Sin[3*e + (7*f*x)/2]))/(c^4*f)","A",1
9,1,139,158,0.3649305,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^5} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c - c*Sec[e + f*x])^5,x]","-\frac{a \csc \left(\frac{e}{2}\right) \left(3465 \sin \left(e+\frac{f x}{2}\right)-2247 \sin \left(e+\frac{3 f x}{2}\right)-2625 \sin \left(2 e+\frac{3 f x}{2}\right)+1143 \sin \left(2 e+\frac{5 f x}{2}\right)+945 \sin \left(3 e+\frac{5 f x}{2}\right)-207 \sin \left(3 e+\frac{7 f x}{2}\right)-315 \sin \left(4 e+\frac{7 f x}{2}\right)+58 \sin \left(4 e+\frac{9 f x}{2}\right)+3843 \sin \left(\frac{f x}{2}\right)\right) \csc ^9\left(\frac{1}{2} (e+f x)\right)}{80640 c^5 f}","-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)}{315 c f \left(c^2-c^2 \sec (e+f x)\right)^2}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)}{105 c^2 f (c-c \sec (e+f x))^3}-\frac{\tan (e+f x) (a \sec (e+f x)+a)}{21 c f (c-c \sec (e+f x))^4}-\frac{\tan (e+f x) (a \sec (e+f x)+a)}{9 f (c-c \sec (e+f x))^5}",1,"-1/80640*(a*Csc[e/2]*Csc[(e + f*x)/2]^9*(3843*Sin[(f*x)/2] + 3465*Sin[e + (f*x)/2] - 2247*Sin[e + (3*f*x)/2] - 2625*Sin[2*e + (3*f*x)/2] + 1143*Sin[2*e + (5*f*x)/2] + 945*Sin[3*e + (5*f*x)/2] - 207*Sin[3*e + (7*f*x)/2] - 315*Sin[4*e + (7*f*x)/2] + 58*Sin[4*e + (9*f*x)/2]))/(c^5*f)","A",1
10,1,102,171,1.8520158,"\int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5,x]","\frac{a^2 c^5 \left(10080 \tanh ^{-1}(\sin (e+f x))-(2520 \sin (e+f x)-455 \sin (2 (e+f x))-616 \sin (3 (e+f x))+2380 \sin (4 (e+f x))-392 \sin (5 (e+f x))+245 \sin (6 (e+f x))+184 \sin (7 (e+f x))) \sec ^7(e+f x)\right)}{17920 f}","-\frac{a^2 c^5 \tan ^7(e+f x)}{7 f}-\frac{4 a^2 c^5 \tan ^5(e+f x)}{5 f}+\frac{9 a^2 c^5 \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac{a^2 c^5 \tan ^3(e+f x) \sec ^3(e+f x)}{2 f}-\frac{3 a^2 c^5 \tan (e+f x) \sec ^3(e+f x)}{8 f}+\frac{a^2 c^5 \tan ^3(e+f x) \sec (e+f x)}{4 f}-\frac{3 a^2 c^5 \tan (e+f x) \sec (e+f x)}{16 f}",1,"(a^2*c^5*(10080*ArcTanh[Sin[e + f*x]] - Sec[e + f*x]^7*(2520*Sin[e + f*x] - 455*Sin[2*(e + f*x)] - 616*Sin[3*(e + f*x)] + 2380*Sin[4*(e + f*x)] - 392*Sin[5*(e + f*x)] + 245*Sin[6*(e + f*x)] + 184*Sin[7*(e + f*x)])))/(17920*f)","A",1
11,1,91,150,1.3962189,"\int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^4,x]","\frac{a^2 c^4 \left(1680 \tanh ^{-1}(\sin (e+f x))+(330 \sin (e+f x)-240 \sin (2 (e+f x))-445 \sin (3 (e+f x))+192 \sin (4 (e+f x))-135 \sin (5 (e+f x))-48 \sin (6 (e+f x))) \sec ^6(e+f x)\right)}{3840 f}","-\frac{2 a^2 c^4 \tan ^5(e+f x)}{5 f}+\frac{7 a^2 c^4 \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac{a^2 c^4 \tan ^3(e+f x) \sec ^3(e+f x)}{6 f}-\frac{a^2 c^4 \tan (e+f x) \sec ^3(e+f x)}{8 f}+\frac{a^2 c^4 \tan ^3(e+f x) \sec (e+f x)}{4 f}-\frac{5 a^2 c^4 \tan (e+f x) \sec (e+f x)}{16 f}",1,"(a^2*c^4*(1680*ArcTanh[Sin[e + f*x]] + Sec[e + f*x]^6*(330*Sin[e + f*x] - 240*Sin[2*(e + f*x)] - 445*Sin[3*(e + f*x)] + 192*Sin[4*(e + f*x)] - 135*Sin[5*(e + f*x)] - 48*Sin[6*(e + f*x)])))/(3840*f)","A",1
12,1,82,94,0.7498377,"\int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^3,x]","\frac{a^2 c^3 \left(120 \tanh ^{-1}(\sin (e+f x))-(40 \sin (e+f x)+10 \sin (2 (e+f x))-20 \sin (3 (e+f x))+25 \sin (4 (e+f x))+4 \sin (5 (e+f x))) \sec ^5(e+f x)\right)}{320 f}","-\frac{a^2 c^3 \tan ^5(e+f x)}{5 f}+\frac{3 a^2 c^3 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a^2 c^3 \tan ^3(e+f x) \sec (e+f x)}{4 f}-\frac{3 a^2 c^3 \tan (e+f x) \sec (e+f x)}{8 f}",1,"(a^2*c^3*(120*ArcTanh[Sin[e + f*x]] - Sec[e + f*x]^5*(40*Sin[e + f*x] + 10*Sin[2*(e + f*x)] - 20*Sin[3*(e + f*x)] + 25*Sin[4*(e + f*x)] + 4*Sin[5*(e + f*x)])))/(320*f)","A",1
13,1,51,73,0.1520939,"\int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^2 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^2,x]","\frac{a^2 c^2 \left(6 \tanh ^{-1}(\sin (e+f x))-(5 \cos (2 (e+f x))+1) \tan (e+f x) \sec ^3(e+f x)\right)}{16 f}","\frac{3 a^2 c^2 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a^2 c^2 \tan ^3(e+f x) \sec (e+f x)}{4 f}-\frac{3 a^2 c^2 \tan (e+f x) \sec (e+f x)}{8 f}",1,"(a^2*c^2*(6*ArcTanh[Sin[e + f*x]] - (1 + 5*Cos[2*(e + f*x)])*Sec[e + f*x]^3*Tan[e + f*x]))/(16*f)","A",1
14,1,45,61,0.1066235,"\int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x)) \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x]),x]","\frac{a^2 c \left(-2 \tan ^3(e+f x)+3 \tanh ^{-1}(\sin (e+f x))-3 \tan (e+f x) \sec (e+f x)\right)}{6 f}","-\frac{a^2 c \tan ^3(e+f x)}{3 f}+\frac{a^2 c \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac{a^2 c \tan (e+f x) \sec (e+f x)}{2 f}",1,"(a^2*c*(3*ArcTanh[Sin[e + f*x]] - 3*Sec[e + f*x]*Tan[e + f*x] - 2*Tan[e + f*x]^3))/(6*f)","A",1
15,1,220,74,1.8653538,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{c-c \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c - c*Sec[e + f*x]),x]","\frac{2 a^2 \sin \left(\frac{1}{2} (e+f x)\right) \cos \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \left(4 \csc \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sec \left(\frac{1}{2} (e+f x)\right)+\tan \left(\frac{1}{2} (e+f x)\right) \left(\frac{\sin (f x)}{\left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}-3 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+3 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}{f (c-c \sec (e+f x))}","-\frac{3 a^2 \tan (e+f x)}{c f}-\frac{3 a^2 \tanh ^{-1}(\sin (e+f x))}{c f}-\frac{2 \tan (e+f x) \left(a^2 \sec (e+f x)+a^2\right)}{f (c-c \sec (e+f x))}",1,"(2*a^2*Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]*(4*Csc[e/2]*Sec[(e + f*x)/2]*Sin[(f*x)/2] + (-3*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 3*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + Sin[f*x]/((Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))*Tan[(e + f*x)/2]))/(f*(c - c*Sec[e + f*x]))","B",1
16,1,109,89,0.0875401,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c - c*Sec[e + f*x])^2,x]","\frac{a^2 \left(-\frac{4 \cot \left(\frac{1}{2} (e+f x)\right)}{3 f}-\frac{2 \cot \left(\frac{1}{2} (e+f x)\right) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{3 f}-\frac{\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f}+\frac{\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f}\right)}{c^2}","\frac{a^2 \tanh ^{-1}(\sin (e+f x))}{c^2 f}+\frac{2 a^2 \tan (e+f x)}{f \left(c^2-c^2 \sec (e+f x)\right)}-\frac{2 \tan (e+f x) \left(a^2 \sec (e+f x)+a^2\right)}{3 f (c-c \sec (e+f x))^2}",1,"(a^2*((-4*Cot[(e + f*x)/2])/(3*f) - (2*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2)/(3*f) - Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]/f + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]/f))/c^2","A",1
17,1,25,38,0.1231901,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c - c*Sec[e + f*x])^3,x]","\frac{a^2 \cot ^5\left(\frac{1}{2} (e+f x)\right)}{5 c^3 f}","-\frac{\tan (e+f x) (a \sec (e+f x)+a)^2}{5 f (c-c \sec (e+f x))^3}",1,"(a^2*Cot[(e + f*x)/2]^5)/(5*c^3*f)","A",1
18,1,115,80,0.4398023,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^4} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c - c*Sec[e + f*x])^4,x]","-\frac{a^2 \csc \left(\frac{e}{2}\right) \left(140 \sin \left(e+\frac{f x}{2}\right)-91 \sin \left(e+\frac{3 f x}{2}\right)-35 \sin \left(2 e+\frac{3 f x}{2}\right)+7 \sin \left(2 e+\frac{5 f x}{2}\right)+35 \sin \left(3 e+\frac{5 f x}{2}\right)-6 \sin \left(3 e+\frac{7 f x}{2}\right)+70 \sin \left(\frac{f x}{2}\right)\right) \csc ^7\left(\frac{1}{2} (e+f x)\right)}{2240 c^4 f}","-\frac{\tan (e+f x) (a \sec (e+f x)+a)^2}{35 c f (c-c \sec (e+f x))^3}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^2}{7 f (c-c \sec (e+f x))^4}",1,"-1/2240*(a^2*Csc[e/2]*Csc[(e + f*x)/2]^7*(70*Sin[(f*x)/2] + 140*Sin[e + (f*x)/2] - 91*Sin[e + (3*f*x)/2] - 35*Sin[2*e + (3*f*x)/2] + 7*Sin[2*e + (5*f*x)/2] + 35*Sin[3*e + (5*f*x)/2] - 6*Sin[3*e + (7*f*x)/2]))/(c^4*f)","A",1
19,1,141,121,0.4352264,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^5} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c - c*Sec[e + f*x])^5,x]","-\frac{a^2 \csc \left(\frac{e}{2}\right) \left(2520 \sin \left(e+\frac{f x}{2}\right)-1638 \sin \left(e+\frac{3 f x}{2}\right)-2310 \sin \left(2 e+\frac{3 f x}{2}\right)+1062 \sin \left(2 e+\frac{5 f x}{2}\right)+630 \sin \left(3 e+\frac{5 f x}{2}\right)-108 \sin \left(3 e+\frac{7 f x}{2}\right)-315 \sin \left(4 e+\frac{7 f x}{2}\right)+47 \sin \left(4 e+\frac{9 f x}{2}\right)+3402 \sin \left(\frac{f x}{2}\right)\right) \csc ^9\left(\frac{1}{2} (e+f x)\right)}{80640 c^5 f}","-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^2}{315 c^2 f (c-c \sec (e+f x))^3}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^2}{63 c f (c-c \sec (e+f x))^4}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^2}{9 f (c-c \sec (e+f x))^5}",1,"-1/80640*(a^2*Csc[e/2]*Csc[(e + f*x)/2]^9*(3402*Sin[(f*x)/2] + 2520*Sin[e + (f*x)/2] - 1638*Sin[e + (3*f*x)/2] - 2310*Sin[2*e + (3*f*x)/2] + 1062*Sin[2*e + (5*f*x)/2] + 630*Sin[3*e + (5*f*x)/2] - 108*Sin[3*e + (7*f*x)/2] - 315*Sin[4*e + (7*f*x)/2] + 47*Sin[4*e + (9*f*x)/2]))/(c^5*f)","A",1
20,1,167,163,0.6748266,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^6} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c - c*Sec[e + f*x])^6,x]","-\frac{a^2 \csc \left(\frac{e}{2}\right) \left(37422 \sin \left(e+\frac{f x}{2}\right)-27060 \sin \left(e+\frac{3 f x}{2}\right)-23100 \sin \left(2 e+\frac{3 f x}{2}\right)+11220 \sin \left(2 e+\frac{5 f x}{2}\right)+13860 \sin \left(3 e+\frac{5 f x}{2}\right)-4895 \sin \left(3 e+\frac{7 f x}{2}\right)-3465 \sin \left(4 e+\frac{7 f x}{2}\right)+517 \sin \left(4 e+\frac{9 f x}{2}\right)+1155 \sin \left(5 e+\frac{9 f x}{2}\right)-152 \sin \left(5 e+\frac{11 f x}{2}\right)+32802 \sin \left(\frac{f x}{2}\right)\right) \csc ^{11}\left(\frac{1}{2} (e+f x)\right)}{1182720 c^6 f}","-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^2}{1155 f \left(c^2-c^2 \sec (e+f x)\right)^3}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^2}{231 c^2 f (c-c \sec (e+f x))^4}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^2}{33 c f (c-c \sec (e+f x))^5}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^2}{11 f (c-c \sec (e+f x))^6}",1,"-1/1182720*(a^2*Csc[e/2]*Csc[(e + f*x)/2]^11*(32802*Sin[(f*x)/2] + 37422*Sin[e + (f*x)/2] - 27060*Sin[e + (3*f*x)/2] - 23100*Sin[2*e + (3*f*x)/2] + 11220*Sin[2*e + (5*f*x)/2] + 13860*Sin[3*e + (5*f*x)/2] - 4895*Sin[3*e + (7*f*x)/2] - 3465*Sin[4*e + (7*f*x)/2] + 517*Sin[4*e + (9*f*x)/2] + 1155*Sin[5*e + (9*f*x)/2] - 152*Sin[5*e + (11*f*x)/2]))/(c^6*f)","A",1
21,1,122,227,3.3937709,"\int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^6,x]","\frac{a^3 c^6 \left(443520 \tanh ^{-1}(\sin (e+f x))-(-88704 \sin (e+f x)+88074 \sin (2 (e+f x))+37632 \sin (3 (e+f x))-2142 \sin (4 (e+f x))+2304 \sin (5 (e+f x))+39858 \sin (6 (e+f x))-7488 \sin (7 (e+f x))+4599 \sin (8 (e+f x))+1856 \sin (9 (e+f x))) \sec ^9(e+f x)\right)}{1032192 f}","\frac{a^3 c^6 \tan ^9(e+f x)}{9 f}+\frac{4 a^3 c^6 \tan ^7(e+f x)}{7 f}+\frac{55 a^3 c^6 \tanh ^{-1}(\sin (e+f x))}{128 f}-\frac{3 a^3 c^6 \tan ^5(e+f x) \sec ^3(e+f x)}{8 f}+\frac{5 a^3 c^6 \tan ^3(e+f x) \sec ^3(e+f x)}{16 f}-\frac{15 a^3 c^6 \tan (e+f x) \sec ^3(e+f x)}{64 f}-\frac{a^3 c^6 \tan ^5(e+f x) \sec (e+f x)}{6 f}+\frac{5 a^3 c^6 \tan ^3(e+f x) \sec (e+f x)}{24 f}-\frac{25 a^3 c^6 \tan (e+f x) \sec (e+f x)}{128 f}",1,"(a^3*c^6*(443520*ArcTanh[Sin[e + f*x]] - Sec[e + f*x]^9*(-88704*Sin[e + f*x] + 88074*Sin[2*(e + f*x)] + 37632*Sin[3*(e + f*x)] - 2142*Sin[4*(e + f*x)] + 2304*Sin[5*(e + f*x)] + 39858*Sin[6*(e + f*x)] - 7488*Sin[7*(e + f*x)] + 4599*Sin[8*(e + f*x)] + 1856*Sin[9*(e + f*x)])))/(1032192*f)","A",1
22,1,111,206,2.5399548,"\int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5,x]","-\frac{a^3 c^5 \left((5705 \sin (e+f x)-1792 \sin (2 (e+f x))+21 \sin (3 (e+f x))+1792 \sin (4 (e+f x))+2065 \sin (5 (e+f x))-768 \sin (6 (e+f x))+581 \sin (7 (e+f x))+128 \sin (8 (e+f x))) \sec ^8(e+f x)-20160 \tanh ^{-1}(\sin (e+f x))\right)}{57344 f}","\frac{2 a^3 c^5 \tan ^7(e+f x)}{7 f}+\frac{45 a^3 c^5 \tanh ^{-1}(\sin (e+f x))}{128 f}-\frac{a^3 c^5 \tan ^5(e+f x) \sec ^3(e+f x)}{8 f}+\frac{5 a^3 c^5 \tan ^3(e+f x) \sec ^3(e+f x)}{48 f}-\frac{5 a^3 c^5 \tan (e+f x) \sec ^3(e+f x)}{64 f}-\frac{a^3 c^5 \tan ^5(e+f x) \sec (e+f x)}{6 f}+\frac{5 a^3 c^5 \tan ^3(e+f x) \sec (e+f x)}{24 f}-\frac{35 a^3 c^5 \tan (e+f x) \sec (e+f x)}{128 f}",1,"-1/57344*(a^3*c^5*(-20160*ArcTanh[Sin[e + f*x]] + Sec[e + f*x]^8*(5705*Sin[e + f*x] - 1792*Sin[2*(e + f*x)] + 21*Sin[3*(e + f*x)] + 1792*Sin[4*(e + f*x)] + 2065*Sin[5*(e + f*x)] - 768*Sin[6*(e + f*x)] + 581*Sin[7*(e + f*x)] + 128*Sin[8*(e + f*x)])))/f","A",1
23,1,102,121,1.6384567,"\int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^4,x]","\frac{a^3 c^4 \left(3360 \tanh ^{-1}(\sin (e+f x))-(-840 \sin (e+f x)+595 \sin (2 (e+f x))+504 \sin (3 (e+f x))+196 \sin (4 (e+f x))-168 \sin (5 (e+f x))+231 \sin (6 (e+f x))+24 \sin (7 (e+f x))) \sec ^7(e+f x)\right)}{10752 f}","\frac{a^3 c^4 \tan ^7(e+f x)}{7 f}+\frac{5 a^3 c^4 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac{a^3 c^4 \tan ^5(e+f x) \sec (e+f x)}{6 f}+\frac{5 a^3 c^4 \tan ^3(e+f x) \sec (e+f x)}{24 f}-\frac{5 a^3 c^4 \tan (e+f x) \sec (e+f x)}{16 f}",1,"(a^3*c^4*(3360*ArcTanh[Sin[e + f*x]] - Sec[e + f*x]^7*(-840*Sin[e + f*x] + 595*Sin[2*(e + f*x)] + 504*Sin[3*(e + f*x)] + 196*Sin[4*(e + f*x)] - 168*Sin[5*(e + f*x)] + 231*Sin[6*(e + f*x)] + 24*Sin[7*(e + f*x)])))/(10752*f)","A",1
24,1,60,100,0.2499265,"\int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^3,x]","-\frac{a^3 c^3 \left((28 \cos (2 (e+f x))+33 \cos (4 (e+f x))+59) \tan (e+f x) \sec ^5(e+f x)-120 \tanh ^{-1}(\sin (e+f x))\right)}{384 f}","\frac{5 a^3 c^3 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac{a^3 c^3 \tan ^5(e+f x) \sec (e+f x)}{6 f}+\frac{5 a^3 c^3 \tan ^3(e+f x) \sec (e+f x)}{24 f}-\frac{5 a^3 c^3 \tan (e+f x) \sec (e+f x)}{16 f}",1,"-1/384*(a^3*c^3*(-120*ArcTanh[Sin[e + f*x]] + (59 + 28*Cos[2*(e + f*x)] + 33*Cos[4*(e + f*x)])*Sec[e + f*x]^5*Tan[e + f*x]))/f","A",1
25,1,81,94,0.806821,"\int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^2,x]","\frac{a^3 c^2 \left(120 \tanh ^{-1}(\sin (e+f x))+(40 \sin (e+f x)-10 \sin (2 (e+f x))-20 \sin (3 (e+f x))-25 \sin (4 (e+f x))+4 \sin (5 (e+f x))) \sec ^5(e+f x)\right)}{320 f}","\frac{a^3 c^2 \tan ^5(e+f x)}{5 f}+\frac{3 a^3 c^2 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a^3 c^2 \tan ^3(e+f x) \sec (e+f x)}{4 f}-\frac{3 a^3 c^2 \tan (e+f x) \sec (e+f x)}{8 f}",1,"(a^3*c^2*(120*ArcTanh[Sin[e + f*x]] + Sec[e + f*x]^5*(40*Sin[e + f*x] - 10*Sin[2*(e + f*x)] - 20*Sin[3*(e + f*x)] - 25*Sin[4*(e + f*x)] + 4*Sin[5*(e + f*x)])))/(320*f)","A",1
26,1,70,86,0.6004538,"\int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x]),x]","\frac{a^3 c \left(60 \tanh ^{-1}(\sin (e+f x))-(33 \sin (e+f x)+16 \sin (2 (e+f x))+9 \sin (3 (e+f x))-8 \sin (4 (e+f x))) \sec ^4(e+f x)\right)}{96 f}","-\frac{2 a^3 c \tan ^3(e+f x)}{3 f}+\frac{5 a^3 c \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac{a^3 c \tan (e+f x) \sec ^3(e+f x)}{4 f}-\frac{3 a^3 c \tan (e+f x) \sec (e+f x)}{8 f}",1,"(a^3*c*(60*ArcTanh[Sin[e + f*x]] - Sec[e + f*x]^4*(33*Sin[e + f*x] + 16*Sin[2*(e + f*x)] + 9*Sin[3*(e + f*x)] - 8*Sin[4*(e + f*x)])))/(96*f)","A",1
27,1,287,100,2.6431553,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{c-c \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c - c*Sec[e + f*x]),x]","\frac{a^3 \cos ^2(e+f x) \tan \left(\frac{1}{2} (e+f x)\right) \sec ^4\left(\frac{1}{2} (e+f x)\right) (\sec (e+f x)+1)^3 \left(32 \csc \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sec \left(\frac{1}{2} (e+f x)\right)+\tan \left(\frac{1}{2} (e+f x)\right) \left(\frac{16 \sin (f x)}{\left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{1}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{1}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}-30 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+30 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}{16 f (c-c \sec (e+f x))}","-\frac{10 a^3 \tan (e+f x)}{c f}-\frac{15 a^3 \tanh ^{-1}(\sin (e+f x))}{2 c f}-\frac{5 a^3 \tan (e+f x) \sec (e+f x)}{2 c f}-\frac{2 a \tan (e+f x) (a \sec (e+f x)+a)^2}{f (c-c \sec (e+f x))}",1,"(a^3*Cos[e + f*x]^2*Sec[(e + f*x)/2]^4*(1 + Sec[e + f*x])^3*Tan[(e + f*x)/2]*(32*Csc[e/2]*Sec[(e + f*x)/2]*Sin[(f*x)/2] + (-30*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 30*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(-2) - (Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^(-2) + (16*Sin[f*x])/((Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))*Tan[(e + f*x)/2]))/(16*f*(c - c*Sec[e + f*x]))","B",1
28,1,402,119,3.4525008,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c - c*Sec[e + f*x])^2,x]","\frac{a^3 (\cos (e+f x)+1)^3 \tan \left(\frac{1}{2} (e+f x)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right) \left(-48 \sin (e) \csc ^3\left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sin ^7\left(\frac{1}{2} (e+f x)\right) \csc ^4(e+f x)+\frac{1}{16} \csc ^3\left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) (120 \cos (e+f x)-46 \cos (2 (e+f x))-76 \cos (2 e+f x)+23 \cos (e+2 f x)+23 \cos (3 e+2 f x)+42 \cos (e)-76 \cos (f x)-74) \sec ^5\left(\frac{1}{2} (e+f x)\right)-4 \cot ^2\left(\frac{e}{2}\right) \csc \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) (5 \cos (e+f x)-1) \tan ^2\left(\frac{1}{2} (e+f x)\right) \sec ^3\left(\frac{1}{2} (e+f x)\right)+\cos (e) \csc ^2\left(\frac{e}{2}\right) \cos (e+f x) \tan \left(\frac{1}{2} (e+f x)\right) \sec ^4\left(\frac{1}{2} (e+f x)\right) \left(15 \sin ^2\left(\frac{1}{2} (e+f x)\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+4 \cot \left(\frac{e}{2}\right)\right)\right)}{6 c^2 f \left(\cot \left(\frac{e}{2}\right)-1\right) \left(\cot \left(\frac{e}{2}\right)+1\right) (\cos (e+f x)-1)^2 \left(\tan \left(\frac{1}{2} (e+f x)\right)-1\right) \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)}","\frac{5 a^3 \tan (e+f x)}{c^2 f}+\frac{5 a^3 \tanh ^{-1}(\sin (e+f x))}{c^2 f}+\frac{10 \tan (e+f x) \left(a^3 \sec (e+f x)+a^3\right)}{3 f \left(c^2-c^2 \sec (e+f x)\right)}-\frac{2 a \tan (e+f x) (a \sec (e+f x)+a)^2}{3 f (c-c \sec (e+f x))^2}",1,"(a^3*(1 + Cos[e + f*x])^3*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]*(((-74 + 42*Cos[e] - 76*Cos[f*x] + 120*Cos[e + f*x] - 46*Cos[2*(e + f*x)] - 76*Cos[2*e + f*x] + 23*Cos[e + 2*f*x] + 23*Cos[3*e + 2*f*x])*Csc[e/2]^3*Sec[(e + f*x)/2]^5*Sin[(f*x)/2])/16 - 48*Csc[e/2]^3*Csc[e + f*x]^4*Sin[e]*Sin[(f*x)/2]*Sin[(e + f*x)/2]^7 + Cos[e]*Cos[e + f*x]*Csc[e/2]^2*Sec[(e + f*x)/2]^4*(4*Cot[e/2] + 15*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[(e + f*x)/2]^2)*Tan[(e + f*x)/2] - 4*(-1 + 5*Cos[e + f*x])*Cot[e/2]^2*Csc[e/2]*Sec[(e + f*x)/2]^3*Sin[(f*x)/2]*Tan[(e + f*x)/2]^2))/(6*c^2*f*(-1 + Cos[e + f*x])^2*(-1 + Cot[e/2])*(1 + Cot[e/2])*(-1 + Tan[(e + f*x)/2])*(1 + Tan[(e + f*x)/2]))","B",1
29,1,139,132,0.1146219,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c - c*Sec[e + f*x])^3,x]","-\frac{a^3 \left(-\frac{26 \cot \left(\frac{1}{2} (e+f x)\right)}{15 f}-\frac{2 \cot \left(\frac{1}{2} (e+f x)\right) \csc ^4\left(\frac{1}{2} (e+f x)\right)}{5 f}+\frac{2 \cot \left(\frac{1}{2} (e+f x)\right) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{15 f}-\frac{\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f}+\frac{\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f}\right)}{c^3}","-\frac{a^3 \tanh ^{-1}(\sin (e+f x))}{c^3 f}-\frac{2 a^3 \tan (e+f x)}{f \left(c^3-c^3 \sec (e+f x)\right)}+\frac{2 \tan (e+f x) \left(a^3 \sec (e+f x)+a^3\right)}{3 c f (c-c \sec (e+f x))^2}-\frac{2 a \tan (e+f x) (a \sec (e+f x)+a)^2}{5 f (c-c \sec (e+f x))^3}",1,"-((a^3*((-26*Cot[(e + f*x)/2])/(15*f) + (2*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2)/(15*f) - (2*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^4)/(5*f) - Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]/f + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]/f))/c^3)","A",1
30,1,25,38,0.1550115,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c - c*Sec[e + f*x])^4,x]","-\frac{a^3 \cot ^7\left(\frac{1}{2} (e+f x)\right)}{7 c^4 f}","-\frac{\tan (e+f x) (a \sec (e+f x)+a)^3}{7 f (c-c \sec (e+f x))^4}",1,"-1/7*(a^3*Cot[(e + f*x)/2]^7)/(c^4*f)","A",1
31,1,141,80,0.4030181,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^5} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c - c*Sec[e + f*x])^5,x]","-\frac{a^3 \csc \left(\frac{e}{2}\right) \left(315 \sin \left(e+\frac{f x}{2}\right)-189 \sin \left(e+\frac{3 f x}{2}\right)-483 \sin \left(2 e+\frac{3 f x}{2}\right)+225 \sin \left(2 e+\frac{5 f x}{2}\right)+63 \sin \left(3 e+\frac{5 f x}{2}\right)-9 \sin \left(3 e+\frac{7 f x}{2}\right)-63 \sin \left(4 e+\frac{7 f x}{2}\right)+8 \sin \left(4 e+\frac{9 f x}{2}\right)+693 \sin \left(\frac{f x}{2}\right)\right) \csc ^9\left(\frac{1}{2} (e+f x)\right)}{16128 c^5 f}","-\frac{\tan (e+f x) (a \sec (e+f x)+a)^3}{63 c f (c-c \sec (e+f x))^4}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^3}{9 f (c-c \sec (e+f x))^5}",1,"-1/16128*(a^3*Csc[e/2]*Csc[(e + f*x)/2]^9*(693*Sin[(f*x)/2] + 315*Sin[e + (f*x)/2] - 189*Sin[e + (3*f*x)/2] - 483*Sin[2*e + (3*f*x)/2] + 225*Sin[2*e + (5*f*x)/2] + 63*Sin[3*e + (5*f*x)/2] - 9*Sin[3*e + (7*f*x)/2] - 63*Sin[4*e + (7*f*x)/2] + 8*Sin[4*e + (9*f*x)/2]))/(c^5*f)","A",1
32,1,167,121,0.6998878,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^6} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c - c*Sec[e + f*x])^6,x]","-\frac{a^3 \csc \left(\frac{e}{2}\right) \left(21252 \sin \left(e+\frac{f x}{2}\right)-15444 \sin \left(e+\frac{3 f x}{2}\right)-10626 \sin \left(2 e+\frac{3 f x}{2}\right)+4950 \sin \left(2 e+\frac{5 f x}{2}\right)+8085 \sin \left(3 e+\frac{5 f x}{2}\right)-2959 \sin \left(3 e+\frac{7 f x}{2}\right)-1386 \sin \left(4 e+\frac{7 f x}{2}\right)+176 \sin \left(4 e+\frac{9 f x}{2}\right)+693 \sin \left(5 e+\frac{9 f x}{2}\right)-79 \sin \left(5 e+\frac{11 f x}{2}\right)+15246 \sin \left(\frac{f x}{2}\right)\right) \csc ^{11}\left(\frac{1}{2} (e+f x)\right)}{709632 c^6 f}","-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{693 c^2 f (c-c \sec (e+f x))^4}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{99 c f (c-c \sec (e+f x))^5}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^3}{11 f (c-c \sec (e+f x))^6}",1,"-1/709632*(a^3*Csc[e/2]*Csc[(e + f*x)/2]^11*(15246*Sin[(f*x)/2] + 21252*Sin[e + (f*x)/2] - 15444*Sin[e + (3*f*x)/2] - 10626*Sin[2*e + (3*f*x)/2] + 4950*Sin[2*e + (5*f*x)/2] + 8085*Sin[3*e + (5*f*x)/2] - 2959*Sin[3*e + (7*f*x)/2] - 1386*Sin[4*e + (7*f*x)/2] + 176*Sin[4*e + (9*f*x)/2] + 693*Sin[5*e + (9*f*x)/2] - 79*Sin[5*e + (11*f*x)/2]))/(c^6*f)","A",1
33,1,193,162,0.5921685,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^7} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c - c*Sec[e + f*x])^7,x]","-\frac{a^3 \csc \left(\frac{e}{2}\right) \left(246246 \sin \left(e+\frac{f x}{2}\right)-182754 \sin \left(e+\frac{3 f x}{2}\right)-216216 \sin \left(2 e+\frac{3 f x}{2}\right)+122551 \sin \left(2 e+\frac{5 f x}{2}\right)+99099 \sin \left(3 e+\frac{5 f x}{2}\right)-37609 \sin \left(3 e+\frac{7 f x}{2}\right)-51051 \sin \left(4 e+\frac{7 f x}{2}\right)+15171 \sin \left(4 e+\frac{9 f x}{2}\right)+9009 \sin \left(5 e+\frac{9 f x}{2}\right)-1027 \sin \left(5 e+\frac{11 f x}{2}\right)-3003 \sin \left(6 e+\frac{11 f x}{2}\right)+310 \sin \left(6 e+\frac{13 f x}{2}\right)+285714 \sin \left(\frac{f x}{2}\right)\right) \csc ^{13}\left(\frac{1}{2} (e+f x)\right)}{12300288 c^7 f}","-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{3003 c^3 f (c-c \sec (e+f x))^4}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{429 c^2 f (c-c \sec (e+f x))^5}-\frac{3 \tan (e+f x) (a \sec (e+f x)+a)^3}{143 c f (c-c \sec (e+f x))^6}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^3}{13 f (c-c \sec (e+f x))^7}",1,"-1/12300288*(a^3*Csc[e/2]*Csc[(e + f*x)/2]^13*(285714*Sin[(f*x)/2] + 246246*Sin[e + (f*x)/2] - 182754*Sin[e + (3*f*x)/2] - 216216*Sin[2*e + (3*f*x)/2] + 122551*Sin[2*e + (5*f*x)/2] + 99099*Sin[3*e + (5*f*x)/2] - 37609*Sin[3*e + (7*f*x)/2] - 51051*Sin[4*e + (7*f*x)/2] + 15171*Sin[4*e + (9*f*x)/2] + 9009*Sin[5*e + (9*f*x)/2] - 1027*Sin[5*e + (11*f*x)/2] - 3003*Sin[6*e + (11*f*x)/2] + 310*Sin[6*e + (13*f*x)/2]))/(c^7*f)","A",1
34,1,1036,121,6.4229406,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^4)/(a + a*Sec[e + f*x]),x]","\frac{2 \cos ^3(e+f x) \cot \left(\frac{e}{2}+\frac{f x}{2}\right) \sec \left(\frac{e}{2}\right) (c-c \sec (e+f x))^4 \sin \left(\frac{f x}{2}\right) \csc ^7\left(\frac{e}{2}+\frac{f x}{2}\right)}{f (\sec (e+f x) a+a)}+\frac{35 \cos ^3(e+f x) \cot ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right) (c-c \sec (e+f x))^4 \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{16 f (\sec (e+f x) a+a)}-\frac{35 \cos ^3(e+f x) \cot ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right) (c-c \sec (e+f x))^4 \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{16 f (\sec (e+f x) a+a)}+\frac{35 \cos ^3(e+f x) \cot ^2\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4 \sin \left(\frac{f x}{2}\right) \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{24 f (\sec (e+f x) a+a) \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}+\frac{35 \cos ^3(e+f x) \cot ^2\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4 \sin \left(\frac{f x}{2}\right) \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{24 f (\sec (e+f x) a+a) \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}+\frac{\cos ^3(e+f x) \cot ^2\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4 \left(8 \sin \left(\frac{e}{2}\right)-7 \cos \left(\frac{e}{2}\right)\right) \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{48 f (\sec (e+f x) a+a) \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{\cos ^3(e+f x) \cot ^2\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4 \left(7 \cos \left(\frac{e}{2}\right)+8 \sin \left(\frac{e}{2}\right)\right) \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{48 f (\sec (e+f x) a+a) \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{\cos ^3(e+f x) \cot ^2\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4 \sin \left(\frac{f x}{2}\right) \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{48 f (\sec (e+f x) a+a) \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^3}+\frac{\cos ^3(e+f x) \cot ^2\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4 \sin \left(\frac{f x}{2}\right) \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{48 f (\sec (e+f x) a+a) \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^3}","\frac{7 c^4 \tan ^3(e+f x)}{3 a f}+\frac{28 c^4 \tan (e+f x)}{a f}-\frac{35 c^4 \tanh ^{-1}(\sin (e+f x))}{2 a f}-\frac{21 c^4 \tan (e+f x) \sec (e+f x)}{2 a f}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^3}{f (a \sec (e+f x)+a)}",1,"(35*Cos[e + f*x]^3*Cot[e/2 + (f*x)/2]^2*Csc[e/2 + (f*x)/2]^6*Log[Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2]]*(c - c*Sec[e + f*x])^4)/(16*f*(a + a*Sec[e + f*x])) - (35*Cos[e + f*x]^3*Cot[e/2 + (f*x)/2]^2*Csc[e/2 + (f*x)/2]^6*Log[Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2]]*(c - c*Sec[e + f*x])^4)/(16*f*(a + a*Sec[e + f*x])) + (2*Cos[e + f*x]^3*Cot[e/2 + (f*x)/2]*Csc[e/2 + (f*x)/2]^7*Sec[e/2]*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(f*(a + a*Sec[e + f*x])) + (Cos[e + f*x]^3*Cot[e/2 + (f*x)/2]^2*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(48*f*(a + a*Sec[e + f*x])*(Cos[e/2] - Sin[e/2])*(Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2])^3) + (Cos[e + f*x]^3*Cot[e/2 + (f*x)/2]^2*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^4*(-7*Cos[e/2] + 8*Sin[e/2]))/(48*f*(a + a*Sec[e + f*x])*(Cos[e/2] - Sin[e/2])*(Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2])^2) + (35*Cos[e + f*x]^3*Cot[e/2 + (f*x)/2]^2*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(24*f*(a + a*Sec[e + f*x])*(Cos[e/2] - Sin[e/2])*(Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2])) + (Cos[e + f*x]^3*Cot[e/2 + (f*x)/2]^2*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(48*f*(a + a*Sec[e + f*x])*(Cos[e/2] + Sin[e/2])*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^3) + (Cos[e + f*x]^3*Cot[e/2 + (f*x)/2]^2*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^4*(7*Cos[e/2] + 8*Sin[e/2]))/(48*f*(a + a*Sec[e + f*x])*(Cos[e/2] + Sin[e/2])*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (35*Cos[e + f*x]^3*Cot[e/2 + (f*x)/2]^2*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(24*f*(a + a*Sec[e + f*x])*(Cos[e/2] + Sin[e/2])*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2]))","B",1
35,1,287,100,2.6515094,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^3}{a+a \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^3)/(a + a*Sec[e + f*x]),x]","\frac{\cos ^2(e+f x) \cot \left(\frac{1}{2} (e+f x)\right) \csc ^4\left(\frac{1}{2} (e+f x)\right) (c-c \sec (e+f x))^3 \left(\cot \left(\frac{1}{2} (e+f x)\right) \left(-\frac{16 \sin (f x)}{\left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{1}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{1}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}-30 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+30 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-32 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \csc \left(\frac{1}{2} (e+f x)\right)\right)}{16 a f (\sec (e+f x)+1)}","\frac{10 c^3 \tan (e+f x)}{a f}-\frac{15 c^3 \tanh ^{-1}(\sin (e+f x))}{2 a f}-\frac{5 c^3 \tan (e+f x) \sec (e+f x)}{2 a f}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}",1,"(Cos[e + f*x]^2*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^4*(c - c*Sec[e + f*x])^3*(-32*Csc[(e + f*x)/2]*Sec[e/2]*Sin[(f*x)/2] + Cot[(e + f*x)/2]*(-30*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 30*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(-2) - (Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^(-2) - (16*Sin[f*x])/((Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))))/(16*a*f*(1 + Sec[e + f*x]))","B",1
36,1,220,74,1.6569799,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^2}{a+a \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^2)/(a + a*Sec[e + f*x]),x]","\frac{2 c^2 \sin \left(\frac{1}{2} (e+f x)\right) \cos \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \left(4 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \csc \left(\frac{1}{2} (e+f x)\right)+\cot \left(\frac{1}{2} (e+f x)\right) \left(\frac{\sin (f x)}{\left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}+3 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-3 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}{a f (\sec (e+f x)+1)}","\frac{3 c^2 \tan (e+f x)}{a f}-\frac{3 c^2 \tanh ^{-1}(\sin (e+f x))}{a f}+\frac{2 \tan (e+f x) \left(c^2-c^2 \sec (e+f x)\right)}{f (a \sec (e+f x)+a)}",1,"(2*c^2*Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]*(4*Csc[(e + f*x)/2]*Sec[e/2]*Sin[(f*x)/2] + Cot[(e + f*x)/2]*(3*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 3*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + Sin[f*x]/((Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))))/(a*f*(1 + Sec[e + f*x]))","B",1
37,1,77,41,0.0643115,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))}{a+a \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x]))/(a + a*Sec[e + f*x]),x]","-\frac{c \left(-\frac{2 \tan \left(\frac{1}{2} (e+f x)\right)}{f}-\frac{\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f}+\frac{\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f}\right)}{a}","\frac{2 c \tan (e+f x)}{f (a \sec (e+f x)+a)}-\frac{c \tanh ^{-1}(\sin (e+f x))}{a f}",1,"-((c*(-(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]/f) + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]/f - (2*Tan[(e + f*x)/2])/f))/a)","A",1
38,1,16,16,0.0273304,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) (c-c \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])),x]","\frac{\csc (e+f x)}{a c f}","\frac{\csc (e+f x)}{a c f}",1,"Csc[e + f*x]/(a*c*f)","A",1
39,1,81,59,0.4818384,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) (c-c \sec (e+f x))^2} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^2),x]","\frac{\csc (e) (-10 \sin (e+f x)+5 \sin (2 (e+f x))-6 \sin (2 e+f x)+2 \sin (e+2 f x)+6 \sin (e)+2 \sin (f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right) \csc (e+f x)}{24 a c^2 f}","-\frac{\cot ^3(e+f x)}{3 a c^2 f}-\frac{\csc ^3(e+f x)}{3 a c^2 f}+\frac{\csc (e+f x)}{a c^2 f}",1,"(Csc[e]*Csc[(e + f*x)/2]^2*Csc[e + f*x]*(6*Sin[e] + 2*Sin[f*x] - 10*Sin[e + f*x] + 5*Sin[2*(e + f*x)] - 6*Sin[2*e + f*x] + 2*Sin[e + 2*f*x]))/(24*a*c^2*f)","A",1
40,1,107,78,0.8498646,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) (c-c \sec (e+f x))^3} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^3),x]","-\frac{\csc (e) (65 \sin (e+f x)-52 \sin (2 (e+f x))+13 \sin (3 (e+f x))+40 \sin (2 e+f x)-12 \sin (e+2 f x)-20 \sin (3 e+2 f x)+8 \sin (2 e+3 f x)-40 \sin (e)) \csc ^4\left(\frac{1}{2} (e+f x)\right) \csc (e+f x)}{320 a c^3 f}","\frac{2 \cot ^5(e+f x)}{5 a c^3 f}+\frac{2 \csc ^5(e+f x)}{5 a c^3 f}-\frac{\csc ^3(e+f x)}{a c^3 f}+\frac{\csc (e+f x)}{a c^3 f}",1,"-1/320*(Csc[e]*Csc[(e + f*x)/2]^4*Csc[e + f*x]*(-40*Sin[e] + 65*Sin[e + f*x] - 52*Sin[2*(e + f*x)] + 13*Sin[3*(e + f*x)] + 40*Sin[2*e + f*x] - 12*Sin[e + 2*f*x] - 20*Sin[3*e + 2*f*x] + 8*Sin[2*e + 3*f*x]))/(a*c^3*f)","A",1
41,1,145,120,0.9019436,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) (c-c \sec (e+f x))^4} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^4),x]","\frac{\csc (e) (-1946 \sin (e+f x)+1946 \sin (2 (e+f x))-834 \sin (3 (e+f x))+139 \sin (4 (e+f x))-1400 \sin (2 e+f x)+616 \sin (e+2 f x)+840 \sin (3 e+2 f x)-344 \sin (2 e+3 f x)-280 \sin (4 e+3 f x)+104 \sin (3 e+4 f x)+840 \sin (e)-56 \sin (f x)) \csc ^6\left(\frac{1}{2} (e+f x)\right) \csc (e+f x)}{17920 a c^4 f}","-\frac{4 \cot ^7(e+f x)}{7 a c^4 f}-\frac{\cot ^5(e+f x)}{5 a c^4 f}-\frac{4 \csc ^7(e+f x)}{7 a c^4 f}+\frac{9 \csc ^5(e+f x)}{5 a c^4 f}-\frac{2 \csc ^3(e+f x)}{a c^4 f}+\frac{\csc (e+f x)}{a c^4 f}",1,"(Csc[e]*Csc[(e + f*x)/2]^6*Csc[e + f*x]*(840*Sin[e] - 56*Sin[f*x] - 1946*Sin[e + f*x] + 1946*Sin[2*(e + f*x)] - 834*Sin[3*(e + f*x)] + 139*Sin[4*(e + f*x)] - 1400*Sin[2*e + f*x] + 616*Sin[e + 2*f*x] + 840*Sin[3*e + 2*f*x] - 344*Sin[2*e + 3*f*x] - 280*Sin[4*e + 3*f*x] + 104*Sin[3*e + 4*f*x]))/(17920*a*c^4*f)","A",1
42,1,380,164,1.1981939,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^5)/(a + a*Sec[e + f*x])^2,x]","\frac{\cot \left(\frac{1}{2} (e+f x)\right) \csc ^6\left(\frac{1}{2} (e+f x)\right) (c-c \sec (e+f x))^5 \left(\sec \left(\frac{e}{2}\right) \sec (e) \left(-2901 \sin \left(e-\frac{f x}{2}\right)+1197 \sin \left(e+\frac{f x}{2}\right)-3027 \sin \left(2 e+\frac{f x}{2}\right)-273 \sin \left(e+\frac{3 f x}{2}\right)+1827 \sin \left(2 e+\frac{3 f x}{2}\right)-1693 \sin \left(3 e+\frac{3 f x}{2}\right)+1995 \sin \left(e+\frac{5 f x}{2}\right)-117 \sin \left(2 e+\frac{5 f x}{2}\right)+1143 \sin \left(3 e+\frac{5 f x}{2}\right)-969 \sin \left(4 e+\frac{5 f x}{2}\right)+1173 \sin \left(2 e+\frac{7 f x}{2}\right)+117 \sin \left(3 e+\frac{7 f x}{2}\right)+747 \sin \left(4 e+\frac{7 f x}{2}\right)-309 \sin \left(5 e+\frac{7 f x}{2}\right)+494 \sin \left(3 e+\frac{9 f x}{2}\right)+142 \sin \left(4 e+\frac{9 f x}{2}\right)+352 \sin \left(5 e+\frac{9 f x}{2}\right)-1323 \sin \left(\frac{f x}{2}\right)+3247 \sin \left(\frac{3 f x}{2}\right)\right) \csc ^3\left(\frac{1}{2} (e+f x)\right)+20160 \cos ^3(e+f x) \cot ^3\left(\frac{1}{2} (e+f x)\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}{3072 a^2 f (\sec (e+f x)+1)^2}","-\frac{7 c^5 \tan ^3(e+f x)}{a^2 f}-\frac{84 c^5 \tan (e+f x)}{a^2 f}+\frac{105 c^5 \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}+\frac{63 c^5 \tan (e+f x) \sec (e+f x)}{2 a^2 f}-\frac{6 c^2 \tan (e+f x) (c-c \sec (e+f x))^3}{f \left(a^2 \sec (e+f x)+a^2\right)}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^4}{3 f (a \sec (e+f x)+a)^2}",1,"(Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^6*(c - c*Sec[e + f*x])^5*(20160*Cos[e + f*x]^3*Cot[(e + f*x)/2]^3*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + Csc[(e + f*x)/2]^3*Sec[e/2]*Sec[e]*(-1323*Sin[(f*x)/2] + 3247*Sin[(3*f*x)/2] - 2901*Sin[e - (f*x)/2] + 1197*Sin[e + (f*x)/2] - 3027*Sin[2*e + (f*x)/2] - 273*Sin[e + (3*f*x)/2] + 1827*Sin[2*e + (3*f*x)/2] - 1693*Sin[3*e + (3*f*x)/2] + 1995*Sin[e + (5*f*x)/2] - 117*Sin[2*e + (5*f*x)/2] + 1143*Sin[3*e + (5*f*x)/2] - 969*Sin[4*e + (5*f*x)/2] + 1173*Sin[2*e + (7*f*x)/2] + 117*Sin[3*e + (7*f*x)/2] + 747*Sin[4*e + (7*f*x)/2] - 309*Sin[5*e + (7*f*x)/2] + 494*Sin[3*e + (9*f*x)/2] + 142*Sin[4*e + (9*f*x)/2] + 352*Sin[5*e + (9*f*x)/2])))/(3072*a^2*f*(1 + Sec[e + f*x])^2)","B",1
43,1,349,150,1.9702154,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^4)/(a + a*Sec[e + f*x])^2,x]","\frac{c^4 \sin ^3\left(\frac{1}{2} (e+f x)\right) \cos \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \left(-32 \tan \left(\frac{e}{2}\right) \cot \left(\frac{1}{2} (e+f x)\right) \csc ^2\left(\frac{1}{2} (e+f x)\right)-32 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \csc ^3\left(\frac{1}{2} (e+f x)\right)+3 \cot ^3\left(\frac{1}{2} (e+f x)\right) \left(-\frac{24 \sin (f x)}{\left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{1}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{1}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}-70 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+70 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-256 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \cot ^2\left(\frac{1}{2} (e+f x)\right) \csc \left(\frac{1}{2} (e+f x)\right)\right)}{3 a^2 f (\sec (e+f x)+1)^2}","-\frac{70 c^4 \tan (e+f x)}{3 a^2 f}+\frac{35 c^4 \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}+\frac{35 c^4 \tan (e+f x) \sec (e+f x)}{6 a^2 f}-\frac{14 \tan (e+f x) \left(c^2-c^2 \sec (e+f x)\right)^2}{3 f \left(a^2 \sec (e+f x)+a^2\right)}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}",1,"(c^4*Cos[(e + f*x)/2]*Sec[e + f*x]^2*Sin[(e + f*x)/2]^3*(-256*Cot[(e + f*x)/2]^2*Csc[(e + f*x)/2]*Sec[e/2]*Sin[(f*x)/2] - 32*Csc[(e + f*x)/2]^3*Sec[e/2]*Sin[(f*x)/2] + 3*Cot[(e + f*x)/2]^3*(-70*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 70*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(-2) - (Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^(-2) - (24*Sin[f*x])/((Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))) - 32*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2*Tan[e/2]))/(3*a^2*f*(1 + Sec[e + f*x])^2)","B",1
44,1,485,119,4.3223347,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^3)/(a + a*Sec[e + f*x])^2,x]","\frac{c^3 (\cos (e+f x)-1)^3 \cot \left(\frac{1}{2} (e+f x)\right) \csc ^2\left(\frac{1}{2} (e+f x)\right) \left(-\frac{1}{16} \sec ^3\left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) (-80 \cos (e+f x)-40 \cos (2 (e+f x))+66 \cos (2 e+f x)+23 \cos (e+2 f x)+17 \cos (3 e+2 f x)+40 \cos (e)+78 \cos (f x)-40) \csc ^5\left(\frac{1}{2} (e+f x)\right)-26 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \cot ^4\left(\frac{1}{2} (e+f x)\right) \csc \left(\frac{1}{2} (e+f x)\right)+20 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \cot ^2\left(\frac{1}{2} (e+f x)\right) \csc \left(\frac{1}{2} (e+f x)\right)+2 \left(\sin \left(\frac{3 e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \sec ^3\left(\frac{e}{2}\right) \cot \left(\frac{1}{2} (e+f x)\right) \csc ^2\left(\frac{1}{2} (e+f x)\right)-15 \cos (e) \sec ^2\left(\frac{e}{2}\right) \cot ^5\left(\frac{1}{2} (e+f x)\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-\left(\tan ^2\left(\frac{e}{2}\right)-1\right) \cot ^3\left(\frac{1}{2} (e+f x)\right) \left(15 \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-4 \tan \left(\frac{e}{2}\right) \csc ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}{6 a^2 f \left(\tan \left(\frac{e}{2}\right)-1\right) \left(\tan \left(\frac{e}{2}\right)+1\right) (\cos (e+f x)+1)^2 \left(\cot \left(\frac{1}{2} (e+f x)\right)-1\right) \left(\cot \left(\frac{1}{2} (e+f x)\right)+1\right)}","-\frac{5 c^3 \tan (e+f x)}{a^2 f}+\frac{5 c^3 \tanh ^{-1}(\sin (e+f x))}{a^2 f}-\frac{10 \tan (e+f x) \left(c^3-c^3 \sec (e+f x)\right)}{3 f \left(a^2 \sec (e+f x)+a^2\right)}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^2}{3 f (a \sec (e+f x)+a)^2}",1,"(c^3*(-1 + Cos[e + f*x])^3*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2*(-15*Cos[e]*Cot[(e + f*x)/2]^5*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sec[e/2]^2 + 2*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2*Sec[e/2]^3*(-Sin[e/2] + Sin[(3*e)/2]) + 20*Cot[(e + f*x)/2]^2*Csc[(e + f*x)/2]*Sec[e/2]*Sin[(f*x)/2] - 26*Cot[(e + f*x)/2]^4*Csc[(e + f*x)/2]*Sec[e/2]*Sin[(f*x)/2] - ((-40 + 40*Cos[e] + 78*Cos[f*x] - 80*Cos[e + f*x] - 40*Cos[2*(e + f*x)] + 66*Cos[2*e + f*x] + 23*Cos[e + 2*f*x] + 17*Cos[3*e + 2*f*x])*Csc[(e + f*x)/2]^5*Sec[e/2]^3*Sin[(f*x)/2])/16 - Cot[(e + f*x)/2]^3*(15*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - 4*Csc[(e + f*x)/2]^2*Tan[e/2])*(-1 + Tan[e/2]^2)))/(6*a^2*f*(1 + Cos[e + f*x])^2*(-1 + Cot[(e + f*x)/2])*(1 + Cot[(e + f*x)/2])*(-1 + Tan[e/2])*(1 + Tan[e/2]))","B",1
45,1,109,88,0.1105897,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^2)/(a + a*Sec[e + f*x])^2,x]","\frac{c^2 \left(-\frac{4 \tan \left(\frac{1}{2} (e+f x)\right)}{3 f}-\frac{2 \tan \left(\frac{1}{2} (e+f x)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{3 f}-\frac{\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f}+\frac{\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f}\right)}{a^2}","\frac{c^2 \tanh ^{-1}(\sin (e+f x))}{a^2 f}-\frac{2 c^2 \tan (e+f x)}{f \left(a^2 \sec (e+f x)+a^2\right)}+\frac{2 \tan (e+f x) \left(c^2-c^2 \sec (e+f x)\right)}{3 f (a \sec (e+f x)+a)^2}",1,"(c^2*(-(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]/f) + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]/f - (4*Tan[(e + f*x)/2])/(3*f) - (2*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3*f)))/a^2","A",1
46,1,23,36,0.0880673,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x]))/(a + a*Sec[e + f*x])^2,x]","-\frac{c \tan ^3\left(\frac{1}{2} (e+f x)\right)}{3 a^2 f}","\frac{\tan (e+f x) (c-c \sec (e+f x))}{3 f (a \sec (e+f x)+a)^2}",1,"-1/3*(c*Tan[(e + f*x)/2]^3)/(a^2*f)","A",1
47,1,83,59,0.5734384,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])),x]","-\frac{\csc (e) \sin ^2\left(\frac{1}{2} (e+f x)\right) (10 \sin (e+f x)+5 \sin (2 (e+f x))-6 \sin (2 e+f x)-2 \sin (e+2 f x)-6 \sin (e)+2 \sin (f x)) \csc ^3(e+f x)}{6 a^2 c f}","\frac{\cot ^3(e+f x)}{3 a^2 c f}-\frac{\csc ^3(e+f x)}{3 a^2 c f}+\frac{\csc (e+f x)}{a^2 c f}",1,"-1/6*(Csc[e]*Csc[e + f*x]^3*Sin[(e + f*x)/2]^2*(-6*Sin[e] + 2*Sin[f*x] + 10*Sin[e + f*x] + 5*Sin[2*(e + f*x)] - 6*Sin[2*e + f*x] - 2*Sin[e + 2*f*x]))/(a^2*c*f)","A",1
48,1,33,38,0.0518964,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^2} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^2),x]","\frac{\frac{\csc (e+f x)}{f}-\frac{\csc ^3(e+f x)}{3 f}}{a^2 c^2}","\frac{\csc (e+f x)}{a^2 c^2 f}-\frac{\csc ^3(e+f x)}{3 a^2 c^2 f}",1,"(Csc[e + f*x]/f - Csc[e + f*x]^3/(3*f))/(a^2*c^2)","A",1
49,1,147,80,0.9638891,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^3),x]","-\frac{\csc (e) (534 \sin (e+f x)-178 \sin (2 (e+f x))-178 \sin (3 (e+f x))+89 \sin (4 (e+f x))+40 \sin (2 e+f x)-168 \sin (e+2 f x)+120 \sin (3 e+2 f x)+72 \sin (2 e+3 f x)-120 \sin (4 e+3 f x)+24 \sin (3 e+4 f x)-200 \sin (e)+104 \sin (f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right) \csc ^3(e+f x)}{1920 a^2 c^3 f}","\frac{\cot ^5(e+f x)}{5 a^2 c^3 f}+\frac{\csc ^5(e+f x)}{5 a^2 c^3 f}-\frac{2 \csc ^3(e+f x)}{3 a^2 c^3 f}+\frac{\csc (e+f x)}{a^2 c^3 f}",1,"-1/1920*(Csc[e]*Csc[(e + f*x)/2]^2*Csc[e + f*x]^3*(-200*Sin[e] + 104*Sin[f*x] + 534*Sin[e + f*x] - 178*Sin[2*(e + f*x)] - 178*Sin[3*(e + f*x)] + 89*Sin[4*(e + f*x)] + 40*Sin[2*e + f*x] - 168*Sin[e + 2*f*x] + 120*Sin[3*e + 2*f*x] + 72*Sin[2*e + 3*f*x] - 120*Sin[4*e + 3*f*x] + 24*Sin[3*e + 4*f*x]))/(a^2*c^3*f)","A",1
50,1,179,98,0.9304286,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^4),x]","\frac{\csc (e) (-182 \sin (e+f x)+104 \sin (2 (e+f x))+39 \sin (3 (e+f x))-52 \sin (4 (e+f x))+13 \sin (5 (e+f x))-56 \sin (2 e+f x)+76 \sin (e+2 f x)-28 \sin (3 e+2 f x)-24 \sin (2 e+3 f x)+42 \sin (4 e+3 f x)-3 \sin (3 e+4 f x)-21 \sin (5 e+4 f x)+6 \sin (4 e+5 f x)+42 \sin (e)-28 \sin (f x)) \csc ^4\left(\frac{1}{2} (e+f x)\right) \csc ^3(e+f x)}{1344 a^2 c^4 f}","-\frac{2 \cot ^7(e+f x)}{7 a^2 c^4 f}-\frac{2 \csc ^7(e+f x)}{7 a^2 c^4 f}+\frac{\csc ^5(e+f x)}{a^2 c^4 f}-\frac{4 \csc ^3(e+f x)}{3 a^2 c^4 f}+\frac{\csc (e+f x)}{a^2 c^4 f}",1,"(Csc[e]*Csc[(e + f*x)/2]^4*Csc[e + f*x]^3*(42*Sin[e] - 28*Sin[f*x] - 182*Sin[e + f*x] + 104*Sin[2*(e + f*x)] + 39*Sin[3*(e + f*x)] - 52*Sin[4*(e + f*x)] + 13*Sin[5*(e + f*x)] - 56*Sin[2*e + f*x] + 76*Sin[e + 2*f*x] - 28*Sin[3*e + 2*f*x] - 24*Sin[2*e + 3*f*x] + 42*Sin[4*e + 3*f*x] - 3*Sin[3*e + 4*f*x] - 21*Sin[5*e + 4*f*x] + 6*Sin[4*e + 5*f*x]))/(1344*a^2*c^4*f)","A",1
51,1,211,141,1.432165,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5),x]","-\frac{\csc (e) (36252 \sin (e+f x)-27189 \sin (2 (e+f x))-2014 \sin (3 (e+f x))+12084 \sin (4 (e+f x))-6042 \sin (5 (e+f x))+1007 \sin (6 (e+f x))+12096 \sin (2 e+f x)-14400 \sin (e+2 f x)-2016 \sin (3 e+2 f x)+7520 \sin (2 e+3 f x)-8736 \sin (4 e+3 f x)+1248 \sin (3 e+4 f x)+6048 \sin (5 e+4 f x)-1632 \sin (4 e+5 f x)-2016 \sin (6 e+5 f x)+608 \sin (5 e+6 f x)-9408 \sin (e)+9792 \sin (f x)) \csc ^6\left(\frac{1}{2} (e+f x)\right) \csc ^3(e+f x)}{516096 a^2 c^5 f}","\frac{4 \cot ^9(e+f x)}{9 a^2 c^5 f}+\frac{\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac{4 \csc ^9(e+f x)}{9 a^2 c^5 f}-\frac{13 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac{3 \csc ^5(e+f x)}{a^2 c^5 f}-\frac{7 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac{\csc (e+f x)}{a^2 c^5 f}",1,"-1/516096*(Csc[e]*Csc[(e + f*x)/2]^6*Csc[e + f*x]^3*(-9408*Sin[e] + 9792*Sin[f*x] + 36252*Sin[e + f*x] - 27189*Sin[2*(e + f*x)] - 2014*Sin[3*(e + f*x)] + 12084*Sin[4*(e + f*x)] - 6042*Sin[5*(e + f*x)] + 1007*Sin[6*(e + f*x)] + 12096*Sin[2*e + f*x] - 14400*Sin[e + 2*f*x] - 2016*Sin[3*e + 2*f*x] + 7520*Sin[2*e + 3*f*x] - 8736*Sin[4*e + 3*f*x] + 1248*Sin[3*e + 4*f*x] + 6048*Sin[5*e + 4*f*x] - 1632*Sin[4*e + 5*f*x] - 2016*Sin[6*e + 5*f*x] + 608*Sin[5*e + 6*f*x]))/(a^2*c^5*f)","A",1
52,1,406,215,2.1658539,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^6}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^6)/(a + a*Sec[e + f*x])^3,x]","\frac{c^6 \cos \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) \left(\sec \left(\frac{e}{2}\right) \sec (e) \left(-130340 \sin \left(e-\frac{f x}{2}\right)+75600 \sin \left(e+\frac{f x}{2}\right)-120176 \sin \left(2 e+\frac{f x}{2}\right)-34230 \sin \left(e+\frac{3 f x}{2}\right)+82278 \sin \left(2 e+\frac{3 f x}{2}\right)-79450 \sin \left(3 e+\frac{3 f x}{2}\right)+91670 \sin \left(e+\frac{5 f x}{2}\right)-14730 \sin \left(2 e+\frac{5 f x}{2}\right)+61920 \sin \left(3 e+\frac{5 f x}{2}\right)-44480 \sin \left(4 e+\frac{5 f x}{2}\right)+53593 \sin \left(2 e+\frac{7 f x}{2}\right)-1735 \sin \left(3 e+\frac{7 f x}{2}\right)+38123 \sin \left(4 e+\frac{7 f x}{2}\right)-17205 \sin \left(5 e+\frac{7 f x}{2}\right)+23735 \sin \left(3 e+\frac{9 f x}{2}\right)+2455 \sin \left(4 e+\frac{9 f x}{2}\right)+17785 \sin \left(5 e+\frac{9 f x}{2}\right)-3495 \sin \left(6 e+\frac{9 f x}{2}\right)+5446 \sin \left(4 e+\frac{11 f x}{2}\right)+1190 \sin \left(5 e+\frac{11 f x}{2}\right)+4256 \sin \left(6 e+\frac{11 f x}{2}\right)-65436 \sin \left(\frac{f x}{2}\right)+127498 \sin \left(\frac{3 f x}{2}\right)\right) \sec ^3(e+f x)+887040 \cos ^5\left(\frac{1}{2} (e+f x)\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}{960 a^3 f (\sec (e+f x)+1)^3}","\frac{77 c^6 \tan ^3(e+f x)}{5 a^3 f}+\frac{924 c^6 \tan (e+f x)}{5 a^3 f}-\frac{231 c^6 \tanh ^{-1}(\sin (e+f x))}{2 a^3 f}-\frac{693 c^6 \tan (e+f x) \sec (e+f x)}{10 a^3 f}+\frac{66 \tan (e+f x) \left(c^2-c^2 \sec (e+f x)\right)^3}{5 f \left(a^3 \sec (e+f x)+a^3\right)}-\frac{22 c^2 \tan (e+f x) (c-c \sec (e+f x))^4}{15 a f (a \sec (e+f x)+a)^2}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^5}{5 f (a \sec (e+f x)+a)^3}",1,"(c^6*Cos[(e + f*x)/2]*Sec[e + f*x]^3*(887040*Cos[(e + f*x)/2]^5*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + Sec[e/2]*Sec[e]*Sec[e + f*x]^3*(-65436*Sin[(f*x)/2] + 127498*Sin[(3*f*x)/2] - 130340*Sin[e - (f*x)/2] + 75600*Sin[e + (f*x)/2] - 120176*Sin[2*e + (f*x)/2] - 34230*Sin[e + (3*f*x)/2] + 82278*Sin[2*e + (3*f*x)/2] - 79450*Sin[3*e + (3*f*x)/2] + 91670*Sin[e + (5*f*x)/2] - 14730*Sin[2*e + (5*f*x)/2] + 61920*Sin[3*e + (5*f*x)/2] - 44480*Sin[4*e + (5*f*x)/2] + 53593*Sin[2*e + (7*f*x)/2] - 1735*Sin[3*e + (7*f*x)/2] + 38123*Sin[4*e + (7*f*x)/2] - 17205*Sin[5*e + (7*f*x)/2] + 23735*Sin[3*e + (9*f*x)/2] + 2455*Sin[4*e + (9*f*x)/2] + 17785*Sin[5*e + (9*f*x)/2] - 3495*Sin[6*e + (9*f*x)/2] + 5446*Sin[4*e + (11*f*x)/2] + 1190*Sin[5*e + (11*f*x)/2] + 4256*Sin[6*e + (11*f*x)/2])))/(960*a^3*f*(1 + Sec[e + f*x])^3)","A",1
53,1,380,193,1.4077975,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^5)/(a + a*Sec[e + f*x])^3,x]","\frac{\cot \left(\frac{1}{2} (e+f x)\right) \csc ^4\left(\frac{1}{2} (e+f x)\right) (c-c \sec (e+f x))^5 \left(\sec \left(\frac{e}{2}\right) \sec (e) \left(7351 \sin \left(e-\frac{f x}{2}\right)-5271 \sin \left(e+\frac{f x}{2}\right)+5545 \sin \left(2 e+\frac{f x}{2}\right)+2205 \sin \left(e+\frac{3 f x}{2}\right)-4515 \sin \left(2 e+\frac{3 f x}{2}\right)+3805 \sin \left(3 e+\frac{3 f x}{2}\right)-4407 \sin \left(e+\frac{5 f x}{2}\right)+585 \sin \left(2 e+\frac{5 f x}{2}\right)-3447 \sin \left(3 e+\frac{5 f x}{2}\right)+1545 \sin \left(4 e+\frac{5 f x}{2}\right)-2155 \sin \left(2 e+\frac{7 f x}{2}\right)-75 \sin \left(3 e+\frac{7 f x}{2}\right)-1755 \sin \left(4 e+\frac{7 f x}{2}\right)+325 \sin \left(5 e+\frac{7 f x}{2}\right)-496 \sin \left(3 e+\frac{9 f x}{2}\right)-80 \sin \left(4 e+\frac{9 f x}{2}\right)-416 \sin \left(5 e+\frac{9 f x}{2}\right)+3465 \sin \left(\frac{f x}{2}\right)-6115 \sin \left(\frac{3 f x}{2}\right)\right) \csc ^5\left(\frac{1}{2} (e+f x)\right)-40320 \cos ^2(e+f x) \cot ^5\left(\frac{1}{2} (e+f x)\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}{5120 a^3 f (\sec (e+f x)+1)^3}","\frac{42 c^5 \tan (e+f x)}{a^3 f}-\frac{63 c^5 \tanh ^{-1}(\sin (e+f x))}{2 a^3 f}-\frac{21 c^5 \tan (e+f x) \sec (e+f x)}{2 a^3 f}+\frac{42 c \tan (e+f x) \left(c^2-c^2 \sec (e+f x)\right)^2}{5 f \left(a^3 \sec (e+f x)+a^3\right)}-\frac{6 c^2 \tan (e+f x) (c-c \sec (e+f x))^3}{5 a f (a \sec (e+f x)+a)^2}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}",1,"(Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^4*(c - c*Sec[e + f*x])^5*(-40320*Cos[e + f*x]^2*Cot[(e + f*x)/2]^5*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + Csc[(e + f*x)/2]^5*Sec[e/2]*Sec[e]*(3465*Sin[(f*x)/2] - 6115*Sin[(3*f*x)/2] + 7351*Sin[e - (f*x)/2] - 5271*Sin[e + (f*x)/2] + 5545*Sin[2*e + (f*x)/2] + 2205*Sin[e + (3*f*x)/2] - 4515*Sin[2*e + (3*f*x)/2] + 3805*Sin[3*e + (3*f*x)/2] - 4407*Sin[e + (5*f*x)/2] + 585*Sin[2*e + (5*f*x)/2] - 3447*Sin[3*e + (5*f*x)/2] + 1545*Sin[4*e + (5*f*x)/2] - 2155*Sin[2*e + (7*f*x)/2] - 75*Sin[3*e + (7*f*x)/2] - 1755*Sin[4*e + (7*f*x)/2] + 325*Sin[5*e + (7*f*x)/2] - 496*Sin[3*e + (9*f*x)/2] - 80*Sin[4*e + (9*f*x)/2] - 416*Sin[5*e + (9*f*x)/2])))/(5120*a^3*f*(1 + Sec[e + f*x])^3)","A",1
54,1,826,164,6.3216083,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^4)/(a + a*Sec[e + f*x])^3,x]","\frac{2 \cos (e+f x) \cot \left(\frac{e}{2}+\frac{f x}{2}\right) \sec \left(\frac{e}{2}\right) (c-c \sec (e+f x))^4 \sin \left(\frac{f x}{2}\right) \csc ^7\left(\frac{e}{2}+\frac{f x}{2}\right)}{5 f (\sec (e+f x) a+a)^3}+\frac{2 \cos (e+f x) \cot ^2\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4 \tan \left(\frac{e}{2}\right) \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{5 f (\sec (e+f x) a+a)^3}+\frac{8 \cos (e+f x) \cot ^3\left(\frac{e}{2}+\frac{f x}{2}\right) \sec \left(\frac{e}{2}\right) (c-c \sec (e+f x))^4 \sin \left(\frac{f x}{2}\right) \csc ^5\left(\frac{e}{2}+\frac{f x}{2}\right)}{15 f (\sec (e+f x) a+a)^3}+\frac{8 \cos (e+f x) \cot ^4\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4 \tan \left(\frac{e}{2}\right) \csc ^4\left(\frac{e}{2}+\frac{f x}{2}\right)}{15 f (\sec (e+f x) a+a)^3}+\frac{76 \cos (e+f x) \cot ^5\left(\frac{e}{2}+\frac{f x}{2}\right) \sec \left(\frac{e}{2}\right) (c-c \sec (e+f x))^4 \sin \left(\frac{f x}{2}\right) \csc ^3\left(\frac{e}{2}+\frac{f x}{2}\right)}{15 f (\sec (e+f x) a+a)^3}+\frac{7 \cos (e+f x) \cot ^6\left(\frac{e}{2}+\frac{f x}{2}\right) \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right) (c-c \sec (e+f x))^4 \csc ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{2 f (\sec (e+f x) a+a)^3}-\frac{7 \cos (e+f x) \cot ^6\left(\frac{e}{2}+\frac{f x}{2}\right) \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right) (c-c \sec (e+f x))^4 \csc ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{2 f (\sec (e+f x) a+a)^3}+\frac{\cos (e+f x) \cot ^6\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4 \sin \left(\frac{f x}{2}\right) \csc ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{2 f (\sec (e+f x) a+a)^3 \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}+\frac{\cos (e+f x) \cot ^6\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4 \sin \left(\frac{f x}{2}\right) \csc ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{2 f (\sec (e+f x) a+a)^3 \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}","\frac{7 c^4 \tan (e+f x)}{a^3 f}-\frac{7 c^4 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac{14 \tan (e+f x) \left(c^4-c^4 \sec (e+f x)\right)}{3 f \left(a^3 \sec (e+f x)+a^3\right)}-\frac{14 \tan (e+f x) \left(c^2-c^2 \sec (e+f x)\right)^2}{15 a f (a \sec (e+f x)+a)^2}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3}",1,"(7*Cos[e + f*x]*Cot[e/2 + (f*x)/2]^6*Csc[e/2 + (f*x)/2]^2*Log[Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2]]*(c - c*Sec[e + f*x])^4)/(2*f*(a + a*Sec[e + f*x])^3) - (7*Cos[e + f*x]*Cot[e/2 + (f*x)/2]^6*Csc[e/2 + (f*x)/2]^2*Log[Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2]]*(c - c*Sec[e + f*x])^4)/(2*f*(a + a*Sec[e + f*x])^3) + (76*Cos[e + f*x]*Cot[e/2 + (f*x)/2]^5*Csc[e/2 + (f*x)/2]^3*Sec[e/2]*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(15*f*(a + a*Sec[e + f*x])^3) + (8*Cos[e + f*x]*Cot[e/2 + (f*x)/2]^3*Csc[e/2 + (f*x)/2]^5*Sec[e/2]*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(15*f*(a + a*Sec[e + f*x])^3) + (2*Cos[e + f*x]*Cot[e/2 + (f*x)/2]*Csc[e/2 + (f*x)/2]^7*Sec[e/2]*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(5*f*(a + a*Sec[e + f*x])^3) + (Cos[e + f*x]*Cot[e/2 + (f*x)/2]^6*Csc[e/2 + (f*x)/2]^2*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(2*f*(a + a*Sec[e + f*x])^3*(Cos[e/2] - Sin[e/2])*(Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2])) + (Cos[e + f*x]*Cot[e/2 + (f*x)/2]^6*Csc[e/2 + (f*x)/2]^2*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(2*f*(a + a*Sec[e + f*x])^3*(Cos[e/2] + Sin[e/2])*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])) + (8*Cos[e + f*x]*Cot[e/2 + (f*x)/2]^4*Csc[e/2 + (f*x)/2]^4*(c - c*Sec[e + f*x])^4*Tan[e/2])/(15*f*(a + a*Sec[e + f*x])^3) + (2*Cos[e + f*x]*Cot[e/2 + (f*x)/2]^2*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^4*Tan[e/2])/(5*f*(a + a*Sec[e + f*x])^3)","B",1
55,1,139,131,0.1222889,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^3)/(a + a*Sec[e + f*x])^3,x]","-\frac{c^3 \left(-\frac{26 \tan \left(\frac{1}{2} (e+f x)\right)}{15 f}-\frac{2 \tan \left(\frac{1}{2} (e+f x)\right) \sec ^4\left(\frac{1}{2} (e+f x)\right)}{5 f}+\frac{2 \tan \left(\frac{1}{2} (e+f x)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{15 f}-\frac{\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f}+\frac{\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f}\right)}{a^3}","-\frac{c^3 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac{2 c^3 \tan (e+f x)}{f \left(a^3 \sec (e+f x)+a^3\right)}-\frac{2 \tan (e+f x) \left(c^3-c^3 \sec (e+f x)\right)}{3 a f (a \sec (e+f x)+a)^2}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^2}{5 f (a \sec (e+f x)+a)^3}",1,"-((c^3*(-(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]/f) + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]/f - (26*Tan[(e + f*x)/2])/(15*f) + (2*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(15*f) - (2*Sec[(e + f*x)/2]^4*Tan[(e + f*x)/2])/(5*f)))/a^3)","A",1
56,1,25,38,0.1261693,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^2)/(a + a*Sec[e + f*x])^3,x]","\frac{c^2 \tan ^5\left(\frac{1}{2} (e+f x)\right)}{5 a^3 f}","\frac{\tan (e+f x) (c-c \sec (e+f x))^2}{5 f (a \sec (e+f x)+a)^3}",1,"(c^2*Tan[(e + f*x)/2]^5)/(5*a^3*f)","A",1
57,1,87,76,0.3454209,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x]))/(a + a*Sec[e + f*x])^3,x]","\frac{c \sec \left(\frac{e}{2}\right) \left(-15 \sin \left(e+\frac{f x}{2}\right)+5 \sin \left(e+\frac{3 f x}{2}\right)-15 \sin \left(2 e+\frac{3 f x}{2}\right)+4 \sin \left(2 e+\frac{5 f x}{2}\right)+25 \sin \left(\frac{f x}{2}\right)\right) \sec ^5\left(\frac{1}{2} (e+f x)\right)}{240 a^3 f}","\frac{\tan (e+f x) (c-c \sec (e+f x))}{15 a f (a \sec (e+f x)+a)^2}+\frac{\tan (e+f x) (c-c \sec (e+f x))}{5 f (a \sec (e+f x)+a)^3}",1,"(c*Sec[e/2]*Sec[(e + f*x)/2]^5*(25*Sin[(f*x)/2] - 15*Sin[e + (f*x)/2] + 5*Sin[e + (3*f*x)/2] - 15*Sin[2*e + (3*f*x)/2] + 4*Sin[2*e + (5*f*x)/2]))/(240*a^3*f)","A",1
58,1,109,78,0.8211893,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])),x]","-\frac{\csc (e) \sin ^4\left(\frac{1}{2} (e+f x)\right) (65 \sin (e+f x)+52 \sin (2 (e+f x))+13 \sin (3 (e+f x))-40 \sin (2 e+f x)-12 \sin (e+2 f x)-20 \sin (3 e+2 f x)-8 \sin (2 e+3 f x)-40 \sin (e)) \csc ^5(e+f x)}{20 a^3 c f}","-\frac{2 \cot ^5(e+f x)}{5 a^3 c f}+\frac{2 \csc ^5(e+f x)}{5 a^3 c f}-\frac{\csc ^3(e+f x)}{a^3 c f}+\frac{\csc (e+f x)}{a^3 c f}",1,"-1/20*(Csc[e]*Csc[e + f*x]^5*Sin[(e + f*x)/2]^4*(-40*Sin[e] + 65*Sin[e + f*x] + 52*Sin[2*(e + f*x)] + 13*Sin[3*(e + f*x)] - 40*Sin[2*e + f*x] - 12*Sin[e + 2*f*x] - 20*Sin[3*e + 2*f*x] - 8*Sin[2*e + 3*f*x]))/(a^3*c*f)","A",1
59,1,147,80,0.8245514,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^2),x]","\frac{\csc (e) \sin ^2\left(\frac{1}{2} (e+f x)\right) (-534 \sin (e+f x)-178 \sin (2 (e+f x))+178 \sin (3 (e+f x))+89 \sin (4 (e+f x))+40 \sin (2 e+f x)+168 \sin (e+2 f x)-120 \sin (3 e+2 f x)+72 \sin (2 e+3 f x)-120 \sin (4 e+3 f x)-24 \sin (3 e+4 f x)+200 \sin (e)+104 \sin (f x)) \csc ^5(e+f x)}{480 a^3 c^2 f}","-\frac{\cot ^5(e+f x)}{5 a^3 c^2 f}+\frac{\csc ^5(e+f x)}{5 a^3 c^2 f}-\frac{2 \csc ^3(e+f x)}{3 a^3 c^2 f}+\frac{\csc (e+f x)}{a^3 c^2 f}",1,"(Csc[e]*Csc[e + f*x]^5*Sin[(e + f*x)/2]^2*(200*Sin[e] + 104*Sin[f*x] - 534*Sin[e + f*x] - 178*Sin[2*(e + f*x)] + 178*Sin[3*(e + f*x)] + 89*Sin[4*(e + f*x)] + 40*Sin[2*e + f*x] + 168*Sin[e + 2*f*x] - 120*Sin[3*e + 2*f*x] + 72*Sin[2*e + 3*f*x] - 120*Sin[4*e + 3*f*x] - 24*Sin[3*e + 4*f*x]))/(480*a^3*c^2*f)","A",1
60,1,50,59,0.0745994,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^3),x]","-\frac{-\frac{\csc ^5(e+f x)}{5 f}+\frac{2 \csc ^3(e+f x)}{3 f}-\frac{\csc (e+f x)}{f}}{a^3 c^3}","\frac{\csc ^5(e+f x)}{5 a^3 c^3 f}-\frac{2 \csc ^3(e+f x)}{3 a^3 c^3 f}+\frac{\csc (e+f x)}{a^3 c^3 f}",1,"-((-(Csc[e + f*x]/f) + (2*Csc[e + f*x]^3)/(3*f) - Csc[e + f*x]^5/(5*f))/(a^3*c^3))","A",1
61,1,211,99,1.4064344,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^4),x]","\frac{\csc (e) (-7620 \sin (e+f x)+1905 \sin (2 (e+f x))+3810 \sin (3 (e+f x))-1524 \sin (4 (e+f x))-762 \sin (5 (e+f x))+381 \sin (6 (e+f x))-2016 \sin (2 e+f x)+2080 \sin (e+2 f x)-1680 \sin (3 e+2 f x)+240 \sin (2 e+3 f x)+560 \sin (4 e+3 f x)-880 \sin (3 e+4 f x)+560 \sin (5 e+4 f x)+400 \sin (4 e+5 f x)-560 \sin (6 e+5 f x)+80 \sin (5 e+6 f x)+2912 \sin (e)+416 \sin (f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right) \csc ^5(e+f x)}{35840 a^3 c^4 f}","-\frac{\cot ^7(e+f x)}{7 a^3 c^4 f}-\frac{\csc ^7(e+f x)}{7 a^3 c^4 f}+\frac{3 \csc ^5(e+f x)}{5 a^3 c^4 f}-\frac{\csc ^3(e+f x)}{a^3 c^4 f}+\frac{\csc (e+f x)}{a^3 c^4 f}",1,"(Csc[e]*Csc[(e + f*x)/2]^2*Csc[e + f*x]^5*(2912*Sin[e] + 416*Sin[f*x] - 7620*Sin[e + f*x] + 1905*Sin[2*(e + f*x)] + 3810*Sin[3*(e + f*x)] - 1524*Sin[4*(e + f*x)] - 762*Sin[5*(e + f*x)] + 381*Sin[6*(e + f*x)] - 2016*Sin[2*e + f*x] + 2080*Sin[e + 2*f*x] - 1680*Sin[3*e + 2*f*x] + 240*Sin[2*e + 3*f*x] + 560*Sin[4*e + 3*f*x] - 880*Sin[3*e + 4*f*x] + 560*Sin[5*e + 4*f*x] + 400*Sin[4*e + 5*f*x] - 560*Sin[6*e + 5*f*x] + 80*Sin[5*e + 6*f*x]))/(35840*a^3*c^4*f)","B",1
62,1,257,120,1.6074662,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5),x]","-\frac{\csc (e) (76455 \sin (e+f x)-33980 \sin (2 (e+f x))-32281 \sin (3 (e+f x))+27184 \sin (4 (e+f x))+1699 \sin (5 (e+f x))-6796 \sin (6 (e+f x))+1699 \sin (7 (e+f x))+22656 \sin (2 e+f x)-17216 \sin (e+2 f x)+4416 \sin (3 e+2 f x)+3200 \sin (2 e+3 f x)-15360 \sin (4 e+3 f x)+12160 \sin (3 e+4 f x)-1920 \sin (5 e+4 f x)-5120 \sin (4 e+5 f x)+5760 \sin (6 e+5 f x)+320 \sin (5 e+6 f x)-2880 \sin (7 e+6 f x)+640 \sin (6 e+7 f x)-33024 \sin (e)+6144 \sin (f x)) \tan (e+f x) \sec ^7(e+f x)}{184320 a^3 c^5 f (\sec (e+f x)-1)^5 (\sec (e+f x)+1)^3}","\frac{2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac{2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac{\csc ^7(e+f x)}{a^3 c^5 f}+\frac{9 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac{5 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac{\csc (e+f x)}{a^3 c^5 f}",1,"-1/184320*(Csc[e]*Sec[e + f*x]^7*(-33024*Sin[e] + 6144*Sin[f*x] + 76455*Sin[e + f*x] - 33980*Sin[2*(e + f*x)] - 32281*Sin[3*(e + f*x)] + 27184*Sin[4*(e + f*x)] + 1699*Sin[5*(e + f*x)] - 6796*Sin[6*(e + f*x)] + 1699*Sin[7*(e + f*x)] + 22656*Sin[2*e + f*x] - 17216*Sin[e + 2*f*x] + 4416*Sin[3*e + 2*f*x] + 3200*Sin[2*e + 3*f*x] - 15360*Sin[4*e + 3*f*x] + 12160*Sin[3*e + 4*f*x] - 1920*Sin[5*e + 4*f*x] - 5120*Sin[4*e + 5*f*x] + 5760*Sin[6*e + 5*f*x] + 320*Sin[5*e + 6*f*x] - 2880*Sin[7*e + 6*f*x] + 640*Sin[6*e + 7*f*x])*Tan[e + f*x])/(a^3*c^5*f*(-1 + Sec[e + f*x])^5*(1 + Sec[e + f*x])^3)","B",1
63,1,289,162,2.3203045,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^6),x]","\frac{\csc (e) (-3440690 \sin (e+f x)+2064414 \sin (2 (e+f x))+1063486 \sin (3 (e+f x))-1563950 \sin (4 (e+f x))+312790 \sin (5 (e+f x))+312790 \sin (6 (e+f x))-187674 \sin (7 (e+f x))+31279 \sin (8 (e+f x))-1499520 \sin (2 e+f x)+1051776 \sin (e+2 f x)+4224 \sin (3 e+2 f x)-85376 \sin (2 e+3 f x)+629376 \sin (4 e+3 f x)-483200 \sin (3 e+4 f x)-316800 \sin (5 e+4 f x)+392320 \sin (4 e+5 f x)-232320 \sin (6 e+5 f x)-30080 \sin (5 e+6 f x)+190080 \sin (7 e+6 f x)-32640 \sin (6 e+7 f x)-63360 \sin (8 e+7 f x)+16000 \sin (7 e+8 f x)+1119360 \sin (e)-260480 \sin (f x)) \tan (e+f x) \sec ^8(e+f x)}{8110080 a^3 c^6 f (\sec (e+f x)-1)^6 (\sec (e+f x)+1)^3}","-\frac{4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}-\frac{\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac{4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac{17 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac{4 \csc ^7(e+f x)}{a^3 c^6 f}+\frac{22 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac{8 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac{\csc (e+f x)}{a^3 c^6 f}",1,"(Csc[e]*Sec[e + f*x]^8*(1119360*Sin[e] - 260480*Sin[f*x] - 3440690*Sin[e + f*x] + 2064414*Sin[2*(e + f*x)] + 1063486*Sin[3*(e + f*x)] - 1563950*Sin[4*(e + f*x)] + 312790*Sin[5*(e + f*x)] + 312790*Sin[6*(e + f*x)] - 187674*Sin[7*(e + f*x)] + 31279*Sin[8*(e + f*x)] - 1499520*Sin[2*e + f*x] + 1051776*Sin[e + 2*f*x] + 4224*Sin[3*e + 2*f*x] - 85376*Sin[2*e + 3*f*x] + 629376*Sin[4*e + 3*f*x] - 483200*Sin[3*e + 4*f*x] - 316800*Sin[5*e + 4*f*x] + 392320*Sin[4*e + 5*f*x] - 232320*Sin[6*e + 5*f*x] - 30080*Sin[5*e + 6*f*x] + 190080*Sin[7*e + 6*f*x] - 32640*Sin[6*e + 7*f*x] - 63360*Sin[8*e + 7*f*x] + 16000*Sin[7*e + 8*f*x])*Tan[e + f*x])/(8110080*a^3*c^6*f*(-1 + Sec[e + f*x])^6*(1 + Sec[e + f*x])^3)","A",1
64,1,86,163,0.9100913,"\int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{7/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(7/2),x]","\frac{a c^3 \cos ^2\left(\frac{1}{2} (e+f x)\right) (1617 \cos (e+f x)-642 \cos (2 (e+f x))+319 \cos (3 (e+f x))-782) \cot \left(\frac{1}{2} (e+f x)\right) \sec ^4(e+f x) \sqrt{c-c \sec (e+f x)}}{315 f}","-\frac{256 c^4 \tan (e+f x) (a \sec (e+f x)+a)}{315 f \sqrt{c-c \sec (e+f x)}}-\frac{64 c^3 \tan (e+f x) (a \sec (e+f x)+a) \sqrt{c-c \sec (e+f x)}}{105 f}-\frac{8 c^2 \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{3/2}}{21 f}-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{5/2}}{9 f}",1,"(a*c^3*Cos[(e + f*x)/2]^2*(-782 + 1617*Cos[e + f*x] - 642*Cos[2*(e + f*x)] + 319*Cos[3*(e + f*x)])*Cot[(e + f*x)/2]*Sec[e + f*x]^4*Sqrt[c - c*Sec[e + f*x]])/(315*f)","A",1
65,1,76,122,0.4857959,"\int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(5/2),x]","\frac{2 a c^2 \cos ^2\left(\frac{1}{2} (e+f x)\right) (-108 \cos (e+f x)+71 \cos (2 (e+f x))+101) \cot \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) \sqrt{c-c \sec (e+f x)}}{105 f}","-\frac{64 c^3 \tan (e+f x) (a \sec (e+f x)+a)}{105 f \sqrt{c-c \sec (e+f x)}}-\frac{16 c^2 \tan (e+f x) (a \sec (e+f x)+a) \sqrt{c-c \sec (e+f x)}}{35 f}-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{3/2}}{7 f}",1,"(2*a*c^2*Cos[(e + f*x)/2]^2*(101 - 108*Cos[e + f*x] + 71*Cos[2*(e + f*x)])*Cot[(e + f*x)/2]*Sec[e + f*x]^3*Sqrt[c - c*Sec[e + f*x]])/(105*f)","A",1
66,1,64,81,0.2754751,"\int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(3/2),x]","\frac{4 a c \cos ^2\left(\frac{1}{2} (e+f x)\right) (7 \cos (e+f x)-3) \cot \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{c-c \sec (e+f x)}}{15 f}","-\frac{8 c^2 \tan (e+f x) (a \sec (e+f x)+a)}{15 f \sqrt{c-c \sec (e+f x)}}-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a) \sqrt{c-c \sec (e+f x)}}{5 f}",1,"(4*a*c*Cos[(e + f*x)/2]^2*(-3 + 7*Cos[e + f*x])*Cot[(e + f*x)/2]*Sec[e + f*x]^2*Sqrt[c - c*Sec[e + f*x]])/(15*f)","A",1
67,1,51,39,0.1838534,"\int \sec (e+f x) (a+a \sec (e+f x)) \sqrt{c-c \sec (e+f x)} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]],x]","\frac{4 a \cos ^2\left(\frac{1}{2} (e+f x)\right) \cot \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{c-c \sec (e+f x)}}{3 f}","-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)}{3 f \sqrt{c-c \sec (e+f x)}}",1,"(4*a*Cos[(e + f*x)/2]^2*Cot[(e + f*x)/2]*Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(3*f)","A",1
68,1,132,77,0.643998,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))}{\sqrt{c-c \sec (e+f x)}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x]))/Sqrt[c - c*Sec[e + f*x]],x]","-\frac{i \sqrt{2} a \left(-1+e^{i (e+f x)}\right) \left(\sqrt{2} \left(1+e^{i (e+f x)}\right)-2 \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)\right)}{f \left(1+e^{2 i (e+f x)}\right) \sqrt{c-c \sec (e+f x)}}","\frac{2 a \tan (e+f x)}{f \sqrt{c-c \sec (e+f x)}}-\frac{2 \sqrt{2} a \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{\sqrt{c} f}",1,"((-I)*Sqrt[2]*a*(-1 + E^(I*(e + f*x)))*(Sqrt[2]*(1 + E^(I*(e + f*x))) - 2*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])]))/((1 + E^((2*I)*(e + f*x)))*f*Sqrt[c - c*Sec[e + f*x]])","C",1
69,1,246,76,1.1459414,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^{3/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c - c*Sec[e + f*x])^(3/2),x]","\frac{a \left(\frac{i \sqrt{2} \left(-1+e^{i (e+f x)}\right)^3 \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)}{\left(1+e^{2 i (e+f x)}\right)^{3/2}}+8 \sin \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sin ^3\left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x)-8 \cos \left(\frac{e}{2}\right) \cos \left(\frac{f x}{2}\right) \sin ^3\left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x)+4 \cot \left(\frac{e}{2}\right) \sin ^2\left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x)-4 \csc \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sin \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x)\right)}{2 c f (\sec (e+f x)-1) \sqrt{c-c \sec (e+f x)}}","\frac{a \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{\sqrt{2} c^{3/2} f}-\frac{a \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}",1,"(a*((I*Sqrt[2]*(-1 + E^(I*(e + f*x)))^3*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])])/(1 + E^((2*I)*(e + f*x)))^(3/2) - 4*Csc[e/2]*Sec[e + f*x]^2*Sin[(f*x)/2]*Sin[(e + f*x)/2] + 4*Cot[e/2]*Sec[e + f*x]^2*Sin[(e + f*x)/2]^2 - 8*Cos[e/2]*Cos[(f*x)/2]*Sec[e + f*x]^2*Sin[(e + f*x)/2]^3 + 8*Sec[e + f*x]^2*Sin[e/2]*Sin[(f*x)/2]*Sin[(e + f*x)/2]^3))/(2*c*f*(-1 + Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]])","C",1
70,1,309,113,1.1248402,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^{5/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c - c*Sec[e + f*x])^(5/2),x]","\frac{a \left(-\frac{i \sqrt{2} \left(-1+e^{i (e+f x)}\right)^5 \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)}{\left(1+e^{2 i (e+f x)}\right)^{5/2}}+48 \sin \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sin ^5\left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x)-48 \cos \left(\frac{e}{2}\right) \cos \left(\frac{f x}{2}\right) \sin ^5\left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x)+56 \cot \left(\frac{e}{2}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x)-16 \cot \left(\frac{e}{2}\right) \sin ^2\left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x)-56 \csc \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sin ^3\left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x)+16 \csc \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sin \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x)\right)}{16 c^2 f (\sec (e+f x)-1)^2 \sqrt{c-c \sec (e+f x)}}","\frac{a \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{8 \sqrt{2} c^{5/2} f}+\frac{a \tan (e+f x)}{8 c f (c-c \sec (e+f x))^{3/2}}-\frac{a \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}",1,"(a*(((-I)*Sqrt[2]*(-1 + E^(I*(e + f*x)))^5*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])])/(1 + E^((2*I)*(e + f*x)))^(5/2) + 16*Csc[e/2]*Sec[e + f*x]^3*Sin[(f*x)/2]*Sin[(e + f*x)/2] - 16*Cot[e/2]*Sec[e + f*x]^3*Sin[(e + f*x)/2]^2 - 56*Csc[e/2]*Sec[e + f*x]^3*Sin[(f*x)/2]*Sin[(e + f*x)/2]^3 + 56*Cot[e/2]*Sec[e + f*x]^3*Sin[(e + f*x)/2]^4 - 48*Cos[e/2]*Cos[(f*x)/2]*Sec[e + f*x]^3*Sin[(e + f*x)/2]^5 + 48*Sec[e + f*x]^3*Sin[e/2]*Sin[(f*x)/2]*Sin[(e + f*x)/2]^5))/(16*c^2*f*(-1 + Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]])","C",1
71,1,88,171,1.6407566,"\int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{7/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(7/2),x]","\frac{2 a^2 c^3 \cos ^4\left(\frac{1}{2} (e+f x)\right) (3419 \cos (e+f x)-1510 \cos (2 (e+f x))+533 \cos (3 (e+f x))-1930) \cot \left(\frac{1}{2} (e+f x)\right) \sec ^5(e+f x) \sqrt{c-c \sec (e+f x)}}{1155 f}","-\frac{256 c^4 \tan (e+f x) (a \sec (e+f x)+a)^2}{1155 f \sqrt{c-c \sec (e+f x)}}-\frac{64 c^3 \tan (e+f x) (a \sec (e+f x)+a)^2 \sqrt{c-c \sec (e+f x)}}{231 f}-\frac{8 c^2 \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{3/2}}{33 f}-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}{11 f}",1,"(2*a^2*c^3*Cos[(e + f*x)/2]^4*(-1930 + 3419*Cos[e + f*x] - 1510*Cos[2*(e + f*x)] + 533*Cos[3*(e + f*x)])*Cot[(e + f*x)/2]*Sec[e + f*x]^5*Sqrt[c - c*Sec[e + f*x]])/(1155*f)","A",1
72,1,78,128,1.2416902,"\int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(5/2),x]","\frac{4 a^2 c^2 \cos ^4\left(\frac{1}{2} (e+f x)\right) (-220 \cos (e+f x)+107 \cos (2 (e+f x))+177) \cot \left(\frac{1}{2} (e+f x)\right) \sec ^4(e+f x) \sqrt{c-c \sec (e+f x)}}{315 f}","-\frac{64 c^3 \tan (e+f x) (a \sec (e+f x)+a)^2}{315 f \sqrt{c-c \sec (e+f x)}}-\frac{16 c^2 \tan (e+f x) (a \sec (e+f x)+a)^2 \sqrt{c-c \sec (e+f x)}}{63 f}-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{3/2}}{9 f}",1,"(4*a^2*c^2*Cos[(e + f*x)/2]^4*(177 - 220*Cos[e + f*x] + 107*Cos[2*(e + f*x)])*Cot[(e + f*x)/2]*Sec[e + f*x]^4*Sqrt[c - c*Sec[e + f*x]])/(315*f)","A",1
73,1,66,85,0.7819235,"\int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{3/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(3/2),x]","\frac{8 a^2 c \cos ^4\left(\frac{1}{2} (e+f x)\right) (9 \cos (e+f x)-5) \cot \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) \sqrt{c-c \sec (e+f x)}}{35 f}","-\frac{8 c^2 \tan (e+f x) (a \sec (e+f x)+a)^2}{35 f \sqrt{c-c \sec (e+f x)}}-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^2 \sqrt{c-c \sec (e+f x)}}{7 f}",1,"(8*a^2*c*Cos[(e + f*x)/2]^4*(-5 + 9*Cos[e + f*x])*Cot[(e + f*x)/2]*Sec[e + f*x]^3*Sqrt[c - c*Sec[e + f*x]])/(35*f)","A",1
74,1,55,41,0.441399,"\int \sec (e+f x) (a+a \sec (e+f x))^2 \sqrt{c-c \sec (e+f x)} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]],x]","\frac{8 a^2 \cos ^4\left(\frac{1}{2} (e+f x)\right) \cot \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{c-c \sec (e+f x)}}{5 f}","-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^2}{5 f \sqrt{c-c \sec (e+f x)}}",1,"(8*a^2*Cos[(e + f*x)/2]^4*Cot[(e + f*x)/2]*Sec[e + f*x]^2*Sqrt[c - c*Sec[e + f*x]])/(5*f)","A",1
75,1,173,117,0.9289882,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{\sqrt{c-c \sec (e+f x)}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/Sqrt[c - c*Sec[e + f*x]],x]","\frac{4 a^2 e^{-\frac{1}{2} i (e+f x)} \sin \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)+i \sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right) (\sec (e+f x)+7)-3 \sqrt{2} e^{-\frac{1}{2} i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)\right)}{3 f \sqrt{c-c \sec (e+f x)}}","-\frac{4 \sqrt{2} a^2 \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{\sqrt{c} f}-\frac{2 a^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{3 c f}+\frac{16 a^2 \tan (e+f x)}{3 f \sqrt{c-c \sec (e+f x)}}",1,"(4*a^2*Sec[e + f*x]*((-3*Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])])/E^((I/2)*(e + f*x)) + Cos[(e + f*x)/2]*(7 + Sec[e + f*x]))*(Cos[(e + f*x)/2] + I*Sin[(e + f*x)/2])*Sin[(e + f*x)/2])/(3*E^((I/2)*(e + f*x))*f*Sqrt[c - c*Sec[e + f*x]])","C",1
76,1,184,113,1.5298247,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{3/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c - c*Sec[e + f*x])^(3/2),x]","\frac{a^2 e^{-2 i (e+f x)} \sin \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \left(4 \left(1+e^{3 i (e+f x)}\right)-3 \sqrt{2} \left(-1+e^{i (e+f x)}\right)^2 \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+i \sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 c f (\sec (e+f x)-1) \sqrt{c-c \sec (e+f x)}}","\frac{3 \sqrt{2} a^2 \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{c^{3/2} f}-\frac{2 a^2 \tan (e+f x)}{c f \sqrt{c-c \sec (e+f x)}}-\frac{2 a^2 \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}",1,"(a^2*(4*(1 + E^((3*I)*(e + f*x))) - 3*Sqrt[2]*(-1 + E^(I*(e + f*x)))^2*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])])*Sec[e + f*x]^2*(Cos[(e + f*x)/2] + I*Sin[(e + f*x)/2])*Sin[(e + f*x)/2])/(2*c*E^((2*I)*(e + f*x))*f*(-1 + Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]])","C",1
77,1,359,117,2.6297667,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{5/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c - c*Sec[e + f*x])^(5/2),x]","-\frac{a^2 \csc \left(\frac{e}{2}\right) e^{-\frac{1}{2} i (e+f x)} \tan \left(\frac{1}{2} (e+f x)\right) \sec ^3\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)} (\sec (e+f x)+1)^2 \left(3 \sin \left(\frac{e}{2}\right) \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \sin ^4\left(\frac{1}{2} (e+f x)\right) \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)+\frac{e^{-\frac{3 i e}{2}} \left(-1+e^{i e}\right) \left(\cos \left(\frac{f x}{2}\right)+i \sin \left(\frac{f x}{2}\right)\right) \left(-9 e^{i e} \sin \left(\frac{f x}{2}\right)+9 e^{2 i e} \sin \left(\frac{f x}{2}\right)-e^{3 i e} \sin \left(\frac{3 f x}{2}\right)-9 i e^{i e} \left(1+e^{i e}\right) \cos \left(\frac{f x}{2}\right)+i \left(1+e^{3 i e}\right) \cos \left(\frac{3 f x}{2}\right)+\sin \left(\frac{3 f x}{2}\right)\right)}{16 \sqrt{\sec (e+f x)}}\right)}{4 c^2 f (\sec (e+f x)-1)^2 \sqrt{c-c \sec (e+f x)}}","-\frac{3 a^2 \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{4 \sqrt{2} c^{5/2} f}+\frac{5 a^2 \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{3/2}}-\frac{a^2 \tan (e+f x)}{f (c-c \sec (e+f x))^{5/2}}",1,"-1/4*(a^2*Csc[e/2]*Sec[(e + f*x)/2]^3*Sqrt[Sec[e + f*x]]*(1 + Sec[e + f*x])^2*(((-1 + E^(I*e))*(Cos[(f*x)/2] + I*Sin[(f*x)/2])*((-9*I)*E^(I*e)*(1 + E^(I*e))*Cos[(f*x)/2] + I*(1 + E^((3*I)*e))*Cos[(3*f*x)/2] - 9*E^(I*e)*Sin[(f*x)/2] + 9*E^((2*I)*e)*Sin[(f*x)/2] + Sin[(3*f*x)/2] - E^((3*I)*e)*Sin[(3*f*x)/2]))/(16*E^(((3*I)/2)*e)*Sqrt[Sec[e + f*x]]) + 3*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])]*Sin[e/2]*Sin[(e + f*x)/2]^4)*Tan[(e + f*x)/2])/(c^2*E^((I/2)*(e + f*x))*f*(-1 + Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]])","C",0
78,1,398,164,5.896973,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{7/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c - c*Sec[e + f*x])^(7/2),x]","\frac{a^2 \csc \left(\frac{e}{2}\right) e^{-\frac{1}{2} i (e+f x)} \tan \left(\frac{1}{2} (e+f x)\right) \sec ^3\left(\frac{1}{2} (e+f x)\right) \sec ^{\frac{3}{2}}(e+f x) (\sec (e+f x)+1)^2 \left(e^{\frac{i e}{2}} \sqrt{\sec (e+f x)} \left(e^{\frac{i f x}{2}} \sin \left(\frac{f x}{2}\right) \left(-43 \sin ^2\left(\frac{1}{2} (e+f x)\right)+14 \sin ^2\left(\frac{e}{2}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)+34\right) \sin ^2\left(\frac{1}{2} (e+f x)\right)-\frac{1}{8} \cos \left(\frac{e}{2}\right) e^{\frac{i f x}{2}} \sin \left(\frac{1}{2} (e+f x)\right) (36 \cos (e+f x)-43 \cos (2 (e+f x))-57)-\frac{7}{2} \sin (e) e^{\frac{i f x}{2}} \sin (f x) \csc \left(\frac{f x}{2}\right) \sin ^6\left(\frac{1}{2} (e+f x)\right)+4 i e^{i f x}-4 i\right)-3 \sin \left(\frac{e}{2}\right) \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \sin ^6\left(\frac{1}{2} (e+f x)\right) \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)\right)}{24 c^3 f (\sec (e+f x)-1)^3 \sqrt{c-c \sec (e+f x)}}","-\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{16 \sqrt{2} c^{7/2} f}-\frac{a^2 \tan (e+f x)}{16 c^2 f (c-c \sec (e+f x))^{3/2}}+\frac{a^2 \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{5/2}}-\frac{\tan (e+f x) \left(a^2 \sec (e+f x)+a^2\right)}{3 f (c-c \sec (e+f x))^{7/2}}",1,"(a^2*Csc[e/2]*Sec[(e + f*x)/2]^3*Sec[e + f*x]^(3/2)*(1 + Sec[e + f*x])^2*(-3*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])]*Sin[e/2]*Sin[(e + f*x)/2]^6 + E^((I/2)*e)*Sqrt[Sec[e + f*x]]*(-4*I + (4*I)*E^(I*f*x) - (E^((I/2)*f*x)*Cos[e/2]*(-57 + 36*Cos[e + f*x] - 43*Cos[2*(e + f*x)])*Sin[(e + f*x)/2])/8 - (7*E^((I/2)*f*x)*Csc[(f*x)/2]*Sin[e]*Sin[f*x]*Sin[(e + f*x)/2]^6)/2 + E^((I/2)*f*x)*Sin[(f*x)/2]*Sin[(e + f*x)/2]^2*(34 - 43*Sin[(e + f*x)/2]^2 + 14*Sin[e/2]^2*Sin[(e + f*x)/2]^4)))*Tan[(e + f*x)/2])/(24*c^3*E^((I/2)*(e + f*x))*f*(-1 + Sec[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]])","C",1
79,1,88,171,2.5404291,"\int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{7/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(7/2),x]","\frac{4 a^3 c^3 \cos ^6\left(\frac{1}{2} (e+f x)\right) (6285 \cos (e+f x)-2842 \cos (2 (e+f x))+835 \cos (3 (e+f x))-3766) \cot \left(\frac{1}{2} (e+f x)\right) \sec ^6(e+f x) \sqrt{c-c \sec (e+f x)}}{3003 f}","-\frac{256 c^4 \tan (e+f x) (a \sec (e+f x)+a)^3}{3003 f \sqrt{c-c \sec (e+f x)}}-\frac{64 c^3 \tan (e+f x) (a \sec (e+f x)+a)^3 \sqrt{c-c \sec (e+f x)}}{429 f}-\frac{24 c^2 \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}{143 f}-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}{13 f}",1,"(4*a^3*c^3*Cos[(e + f*x)/2]^6*(-3766 + 6285*Cos[e + f*x] - 2842*Cos[2*(e + f*x)] + 835*Cos[3*(e + f*x)])*Cot[(e + f*x)/2]*Sec[e + f*x]^6*Sqrt[c - c*Sec[e + f*x]])/(3003*f)","A",1
80,1,78,128,1.5616636,"\int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2),x]","\frac{8 a^3 c^2 \cos ^6\left(\frac{1}{2} (e+f x)\right) (-364 \cos (e+f x)+151 \cos (2 (e+f x))+277) \cot \left(\frac{1}{2} (e+f x)\right) \sec ^5(e+f x) \sqrt{c-c \sec (e+f x)}}{693 f}","-\frac{64 c^3 \tan (e+f x) (a \sec (e+f x)+a)^3}{693 f \sqrt{c-c \sec (e+f x)}}-\frac{16 c^2 \tan (e+f x) (a \sec (e+f x)+a)^3 \sqrt{c-c \sec (e+f x)}}{99 f}-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}{11 f}",1,"(8*a^3*c^2*Cos[(e + f*x)/2]^6*(277 - 364*Cos[e + f*x] + 151*Cos[2*(e + f*x)])*Cot[(e + f*x)/2]*Sec[e + f*x]^5*Sqrt[c - c*Sec[e + f*x]])/(693*f)","A",1
81,1,66,85,1.1105458,"\int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(3/2),x]","\frac{16 a^3 c \cos ^6\left(\frac{1}{2} (e+f x)\right) (11 \cos (e+f x)-7) \cot \left(\frac{1}{2} (e+f x)\right) \sec ^4(e+f x) \sqrt{c-c \sec (e+f x)}}{63 f}","-\frac{8 c^2 \tan (e+f x) (a \sec (e+f x)+a)^3}{63 f \sqrt{c-c \sec (e+f x)}}-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^3 \sqrt{c-c \sec (e+f x)}}{9 f}",1,"(16*a^3*c*Cos[(e + f*x)/2]^6*(-7 + 11*Cos[e + f*x])*Cot[(e + f*x)/2]*Sec[e + f*x]^4*Sqrt[c - c*Sec[e + f*x]])/(63*f)","A",1
82,1,55,41,0.7086303,"\int \sec (e+f x) (a+a \sec (e+f x))^3 \sqrt{c-c \sec (e+f x)} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]],x]","\frac{16 a^3 \cos ^6\left(\frac{1}{2} (e+f x)\right) \cot \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) \sqrt{c-c \sec (e+f x)}}{7 f}","-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^3}{7 f \sqrt{c-c \sec (e+f x)}}",1,"(16*a^3*Cos[(e + f*x)/2]^6*Cot[(e + f*x)/2]*Sec[e + f*x]^3*Sqrt[c - c*Sec[e + f*x]])/(7*f)","A",1
83,1,185,164,1.5349348,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{\sqrt{c-c \sec (e+f x)}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/Sqrt[c - c*Sec[e + f*x]],x]","\frac{4 a^3 e^{-\frac{1}{2} i (e+f x)} \sin \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)+i \sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right) \left(3 \sec ^2(e+f x)+16 \sec (e+f x)+73\right)-30 \sqrt{2} e^{-\frac{1}{2} i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)\right)}{15 f \sqrt{c-c \sec (e+f x)}}","-\frac{8 \sqrt{2} a^3 \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{\sqrt{c} f}+\frac{8 a^3 \tan (e+f x)}{f \sqrt{c-c \sec (e+f x)}}+\frac{4 \tan (e+f x) \left(a^3 \sec (e+f x)+a^3\right)}{3 f \sqrt{c-c \sec (e+f x)}}+\frac{2 a \tan (e+f x) (a \sec (e+f x)+a)^2}{5 f \sqrt{c-c \sec (e+f x)}}",1,"(4*a^3*Sec[e + f*x]*((-30*Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])])/E^((I/2)*(e + f*x)) + Cos[(e + f*x)/2]*(73 + 16*Sec[e + f*x] + 3*Sec[e + f*x]^2))*(Cos[(e + f*x)/2] + I*Sin[(e + f*x)/2])*Sin[(e + f*x)/2])/(15*E^((I/2)*(e + f*x))*f*Sqrt[c - c*Sec[e + f*x]])","C",1
84,1,324,168,3.0993872,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^{3/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c - c*Sec[e + f*x])^(3/2),x]","-\frac{a^3 \csc \left(\frac{e}{2}\right) e^{-\frac{1}{2} i (e+f x)} \tan ^3\left(\frac{1}{2} (e+f x)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right) (\sec (e+f x)+1)^3 \left(-\frac{i \left(-1+e^{i e}\right) e^{\frac{i f x}{2}} \left(-24 e^{i (e+f x)}+34 e^{2 i (e+f x)}-24 e^{3 i (e+f x)}+19 e^{4 i (e+f x)}+19\right) \sqrt{\sec (e+f x)}}{2 \left(-1+e^{i (e+f x)}\right)^2 \left(1+e^{2 i (e+f x)}\right)}-15 \sin \left(\frac{e}{2}\right) \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \sec \left(\frac{1}{2} (e+f x)\right) \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)\right)}{3 c f (\sec (e+f x)-1) \sec ^{\frac{3}{2}}(e+f x) \sqrt{c-c \sec (e+f x)}}","\frac{10 \sqrt{2} a^3 \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{c^{3/2} f}-\frac{10 a^3 \tan (e+f x)}{c f \sqrt{c-c \sec (e+f x)}}-\frac{5 \tan (e+f x) \left(a^3 \sec (e+f x)+a^3\right)}{3 c f \sqrt{c-c \sec (e+f x)}}-\frac{a \tan (e+f x) (a \sec (e+f x)+a)^2}{f (c-c \sec (e+f x))^{3/2}}",1,"-1/3*(a^3*Csc[e/2]*Sec[(e + f*x)/2]^2*(1 + Sec[e + f*x])^3*(((-1/2*I)*E^((I/2)*f*x)*(-1 + E^(I*e))*(19 - 24*E^(I*(e + f*x)) + 34*E^((2*I)*(e + f*x)) - 24*E^((3*I)*(e + f*x)) + 19*E^((4*I)*(e + f*x)))*Sqrt[Sec[e + f*x]])/((-1 + E^(I*(e + f*x)))^2*(1 + E^((2*I)*(e + f*x)))) - 15*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])]*Sec[(e + f*x)/2]*Sin[e/2])*Tan[(e + f*x)/2]^3)/(c*E^((I/2)*(e + f*x))*f*(-1 + Sec[e + f*x])*Sec[e + f*x]^(3/2)*Sqrt[c - c*Sec[e + f*x]])","C",1
85,1,263,174,4.5738248,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^{5/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c - c*Sec[e + f*x])^(5/2),x]","-\frac{a^3 e^{-\frac{1}{2} i (2 e+f x)} \tan ^5\left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) (\sec (e+f x)+1)^3 \left(120 e^{\frac{i e}{2}} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)+\left(25 \cos \left(\frac{3}{2} (e+f x)\right)-9 \cos \left(\frac{5}{2} (e+f x)\right)\right) \csc ^4\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)} \left(\cos \left(e+\frac{f x}{2}\right)+i \sin \left(e+\frac{f x}{2}\right)\right)\right)}{32 c^2 f (\sec (e+f x)-1)^2 \sqrt{\sec (e+f x)} \sqrt{c-c \sec (e+f x)}}","-\frac{15 a^3 \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{2 \sqrt{2} c^{5/2} f}+\frac{15 a^3 \tan (e+f x)}{4 c^2 f \sqrt{c-c \sec (e+f x)}}+\frac{5 \tan (e+f x) \left(a^3 \sec (e+f x)+a^3\right)}{4 c f (c-c \sec (e+f x))^{3/2}}-\frac{a \tan (e+f x) (a \sec (e+f x)+a)^2}{2 f (c-c \sec (e+f x))^{5/2}}",1,"-1/32*(a^3*Sec[(e + f*x)/2]*(1 + Sec[e + f*x])^3*(120*E^((I/2)*e)*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])] + (25*Cos[(3*(e + f*x))/2] - 9*Cos[(5*(e + f*x))/2])*Csc[(e + f*x)/2]^4*Sqrt[Sec[e + f*x]]*(Cos[e + (f*x)/2] + I*Sin[e + (f*x)/2]))*Tan[(e + f*x)/2]^5)/(c^2*E^((I/2)*(2*e + f*x))*f*(-1 + Sec[e + f*x])^2*Sqrt[Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","C",1
86,1,86,142,0.7379151,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^{7/2}}{a+a \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^(7/2))/(a + a*Sec[e + f*x]),x]","-\frac{c^3 (245 \cos (e+f x)+86 \cos (2 (e+f x))+91 \cos (3 (e+f x))+90) \cot \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{c-c \sec (e+f x)}}{10 a f (\cos (e+f x)+1)}","\frac{128 c^4 \tan (e+f x)}{5 a f \sqrt{c-c \sec (e+f x)}}+\frac{32 c^3 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{5 a f}+\frac{12 c^2 \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{5 a f}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{f (a \sec (e+f x)+a)}",1,"-1/10*(c^3*(90 + 245*Cos[e + f*x] + 86*Cos[2*(e + f*x)] + 91*Cos[3*(e + f*x)])*Cot[(e + f*x)/2]*Sec[e + f*x]^2*Sqrt[c - c*Sec[e + f*x]])/(a*f*(1 + Cos[e + f*x]))","A",1
87,1,74,108,0.435407,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{a+a \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^(5/2))/(a + a*Sec[e + f*x]),x]","-\frac{c^2 (20 \cos (e+f x)+23 \cos (2 (e+f x))+21) \cot \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{c-c \sec (e+f x)}}{3 a f (\cos (e+f x)+1)}","\frac{32 c^3 \tan (e+f x)}{3 a f \sqrt{c-c \sec (e+f x)}}+\frac{8 c^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{3 a f}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{f (a \sec (e+f x)+a)}",1,"-1/3*(c^2*(21 + 20*Cos[e + f*x] + 23*Cos[2*(e + f*x)])*Cot[(e + f*x)/2]*Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a*f*(1 + Cos[e + f*x]))","A",1
88,1,54,72,0.2449466,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^{3/2}}{a+a \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^(3/2))/(a + a*Sec[e + f*x]),x]","-\frac{2 c (3 \cos (e+f x)+1) \cot \left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)}}{a f (\cos (e+f x)+1)}","\frac{4 c^2 \tan (e+f x)}{a f \sqrt{c-c \sec (e+f x)}}+\frac{2 c \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f (a \sec (e+f x)+a)}",1,"(-2*c*(1 + 3*Cos[e + f*x])*Cot[(e + f*x)/2]*Sqrt[c - c*Sec[e + f*x]])/(a*f*(1 + Cos[e + f*x]))","A",1
89,1,29,39,0.1368406,"\int \frac{\sec (e+f x) \sqrt{c-c \sec (e+f x)}}{a+a \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a + a*Sec[e + f*x]),x]","-\frac{2 \cot (e+f x) \sqrt{c-c \sec (e+f x)}}{a f}","\frac{2 c \tan (e+f x)}{f (a \sec (e+f x)+a) \sqrt{c-c \sec (e+f x)}}",1,"(-2*Cot[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a*f)","A",1
90,1,155,89,0.4794408,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) \sqrt{c-c \sec (e+f x)}} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]]),x]","-\frac{i \left(-1+e^{2 i (e+f x)}\right) \left(2 \left(1+e^{2 i (e+f x)}\right)-\sqrt{2} \left(1+e^{i (e+f x)}\right) \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)\right)}{2 a f \left(1+e^{2 i (e+f x)}\right)^2 (\sec (e+f x)+1) \sqrt{c-c \sec (e+f x)}}","\frac{\tan (e+f x)}{f (a \sec (e+f x)+a) \sqrt{c-c \sec (e+f x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{\sqrt{2} a \sqrt{c} f}",1,"((-1/2*I)*(-1 + E^((2*I)*(e + f*x)))*(2*(1 + E^((2*I)*(e + f*x))) - Sqrt[2]*(1 + E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])]))/(a*(1 + E^((2*I)*(e + f*x)))^2*f*(1 + Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]])","C",1
91,1,183,122,1.4170102,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2}} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(3/2)),x]","\frac{e^{-\frac{1}{2} i (e+f x)} \csc (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)+i \sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-8 (\cos (e+f x)-3)+\frac{6 \sqrt{2} e^{-i (e+f x)} \left(-1+e^{i (e+f x)}\right)^2 \left(1+e^{i (e+f x)}\right) \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)}{\sqrt{1+e^{2 i (e+f x)}}}\right)}{32 a c f \sqrt{c-c \sec (e+f x)}}","-\frac{3 \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{4 \sqrt{2} a c^{3/2} f}-\frac{3 \tan (e+f x)}{4 a f (c-c \sec (e+f x))^{3/2}}+\frac{\tan (e+f x)}{f (a \sec (e+f x)+a) (c-c \sec (e+f x))^{3/2}}",1,"(((6*Sqrt[2]*(-1 + E^(I*(e + f*x)))^2*(1 + E^(I*(e + f*x)))*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])])/(E^(I*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]) - 8*(-3 + Cos[e + f*x]))*Csc[e + f*x]*(Cos[(e + f*x)/2] + I*Sin[(e + f*x)/2]))/(32*a*c*E^((I/2)*(e + f*x))*f*Sqrt[c - c*Sec[e + f*x]])","C",1
92,1,306,156,2.9328857,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2}} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(5/2)),x]","\frac{e^{-\frac{1}{2} i (e+f x)} \sin \left(\frac{1}{2} (e+f x)\right) \cos \left(\frac{1}{2} (e+f x)\right) \sec ^{\frac{7}{2}}(e+f x) \left(-\frac{1}{32} e^{-\frac{5}{2} i (e+f x)} \left(40 e^{i (e+f x)}-51 e^{2 i (e+f x)}+80 e^{3 i (e+f x)}-51 e^{4 i (e+f x)}+40 e^{5 i (e+f x)}+3 e^{6 i (e+f x)}+3\right) \sqrt{\sec (e+f x)}-\frac{15}{2} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \sin (e+f x) \sin ^3\left(\frac{1}{2} (e+f x)\right) \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)\right)}{4 a c^2 f (\sec (e+f x)-1)^2 (\sec (e+f x)+1) \sqrt{c-c \sec (e+f x)}}","-\frac{15 \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{32 \sqrt{2} a c^{5/2} f}-\frac{15 \tan (e+f x)}{32 a c f (c-c \sec (e+f x))^{3/2}}-\frac{5 \tan (e+f x)}{8 a f (c-c \sec (e+f x))^{5/2}}+\frac{\tan (e+f x)}{f (a \sec (e+f x)+a) (c-c \sec (e+f x))^{5/2}}",1,"(Cos[(e + f*x)/2]*Sec[e + f*x]^(7/2)*Sin[(e + f*x)/2]*(-1/32*((3 + 40*E^(I*(e + f*x)) - 51*E^((2*I)*(e + f*x)) + 80*E^((3*I)*(e + f*x)) - 51*E^((4*I)*(e + f*x)) + 40*E^((5*I)*(e + f*x)) + 3*E^((6*I)*(e + f*x)))*Sqrt[Sec[e + f*x]])/E^(((5*I)/2)*(e + f*x)) - (15*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])]*Sin[(e + f*x)/2]^3*Sin[e + f*x])/2))/(4*a*c^2*E^((I/2)*(e + f*x))*f*(-1 + Sec[e + f*x])^2*(1 + Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]])","C",1
93,1,84,155,0.9398448,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^{7/2}}{(a+a \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^(7/2))/(a + a*Sec[e + f*x])^2,x]","\frac{c^3 (195 \cos (e+f x)+138 \cos (2 (e+f x))+45 \cos (3 (e+f x))+134) \cot \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{c-c \sec (e+f x)}}{6 a^2 f (\cos (e+f x)+1)^2}","-\frac{64 c^4 \tan (e+f x)}{3 a^2 f \sqrt{c-c \sec (e+f x)}}-\frac{16 c^3 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{3 a^2 f}-\frac{4 c^2 \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{f \left(a^2 \sec (e+f x)+a^2\right)}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f (a \sec (e+f x)+a)^2}",1,"(c^3*(134 + 195*Cos[e + f*x] + 138*Cos[2*(e + f*x)] + 45*Cos[3*(e + f*x)])*Cot[(e + f*x)/2]*Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(6*a^2*f*(1 + Cos[e + f*x])^2)","A",1
94,1,68,123,0.3938185,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^(5/2))/(a + a*Sec[e + f*x])^2,x]","\frac{c^2 (36 \cos (e+f x)+11 \cos (2 (e+f x))+17) \cot \left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)}}{3 a^2 f (\cos (e+f x)+1)^2}","-\frac{16 c^3 \tan (e+f x)}{3 a^2 f \sqrt{c-c \sec (e+f x)}}-\frac{8 c^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{3 f \left(a^2 \sec (e+f x)+a^2\right)}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{3 f (a \sec (e+f x)+a)^2}",1,"(c^2*(17 + 36*Cos[e + f*x] + 11*Cos[2*(e + f*x)])*Cot[(e + f*x)/2]*Sqrt[c - c*Sec[e + f*x]])/(3*a^2*f*(1 + Cos[e + f*x])^2)","A",1
95,1,60,89,0.2603263,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^{3/2}}{(a+a \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^(3/2))/(a + a*Sec[e + f*x])^2,x]","\frac{2 c \cos (e+f x) (\cos (e+f x)+3) \cot \left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)}}{3 a^2 f (\cos (e+f x)+1)^2}","\frac{2 c \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{3 f (a \sec (e+f x)+a)^2}-\frac{4 c^2 \tan (e+f x)}{3 f \left(a^2 \sec (e+f x)+a^2\right) \sqrt{c-c \sec (e+f x)}}",1,"(2*c*Cos[e + f*x]*(3 + Cos[e + f*x])*Cot[(e + f*x)/2]*Sqrt[c - c*Sec[e + f*x]])/(3*a^2*f*(1 + Cos[e + f*x])^2)","A",1
96,1,55,41,0.1527039,"\int \frac{\sec (e+f x) \sqrt{c-c \sec (e+f x)}}{(a+a \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a + a*Sec[e + f*x])^2,x]","-\frac{\cos ^2(e+f x) \csc \left(\frac{1}{2} (e+f x)\right) \sec ^3\left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)}}{6 a^2 f}","\frac{2 c \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 \sqrt{c-c \sec (e+f x)}}",1,"-1/6*(Cos[e + f*x]^2*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]^3*Sqrt[c - c*Sec[e + f*x]])/(a^2*f)","A",1
97,1,259,138,2.0797763,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^2 \sqrt{c-c \sec (e+f x)}} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]]),x]","\frac{2 e^{-\frac{1}{2} i (e+f x)} \sin \left(\frac{1}{2} (e+f x)\right) \cos \left(\frac{1}{2} (e+f x)\right) \sec ^{\frac{5}{2}}(e+f x) \left(\frac{1}{8} e^{-\frac{3}{2} i (e+f x)} \left(6 e^{i (e+f x)}+10 e^{2 i (e+f x)}+6 e^{3 i (e+f x)}+5 e^{4 i (e+f x)}+5\right) \sqrt{\sec (e+f x)}-3 \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \cos ^3\left(\frac{1}{2} (e+f x)\right) \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)\right)}{3 a^2 f (\sec (e+f x)+1)^2 \sqrt{c-c \sec (e+f x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{2 \sqrt{2} a^2 \sqrt{c} f}+\frac{\tan (e+f x)}{2 f \left(a^2 \sec (e+f x)+a^2\right) \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 \sqrt{c-c \sec (e+f x)}}",1,"(2*Cos[(e + f*x)/2]*(-3*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])]*Cos[(e + f*x)/2]^3 + ((5 + 6*E^(I*(e + f*x)) + 10*E^((2*I)*(e + f*x)) + 6*E^((3*I)*(e + f*x)) + 5*E^((4*I)*(e + f*x)))*Sqrt[Sec[e + f*x]])/(8*E^(((3*I)/2)*(e + f*x))))*Sec[e + f*x]^(5/2)*Sin[(e + f*x)/2])/(3*a^2*E^((I/2)*(e + f*x))*f*(1 + Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]])","C",1
98,1,365,169,1.3924099,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{3/2}} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(3/2)),x]","\frac{-\frac{15 i \sqrt{2} \left(-1+e^{i (e+f x)}\right)^3 \left(1+e^{i (e+f x)}\right)^4 \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)}{\left(1+e^{2 i (e+f x)}\right)^{7/2}}-416 \sin \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sin ^8(e+f x) \csc \left(\frac{1}{2} (e+f x)\right) \csc ^4(2 (e+f x))-40 \tan ^3(e+f x) \sec (e+f x)+416 \cos \left(\frac{e}{2}\right) \cos \left(\frac{f x}{2}\right) \sin ^3\left(\frac{1}{2} (e+f x)\right) \cos ^4\left(\frac{1}{2} (e+f x)\right) \sec ^4(e+f x)+32 \sin ^3\left(\frac{1}{2} (e+f x)\right) \cos \left(\frac{1}{2} (e+f x)\right) \sec ^4(e+f x)+48 \cot \left(\frac{e}{2}\right) \sin ^2\left(\frac{1}{2} (e+f x)\right) \cos ^4\left(\frac{1}{2} (e+f x)\right) \sec ^4(e+f x)-48 \csc \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sin \left(\frac{1}{2} (e+f x)\right) \cos ^4\left(\frac{1}{2} (e+f x)\right) \sec ^4(e+f x)}{48 a^2 c f (\sec (e+f x)-1) (\sec (e+f x)+1)^2 \sqrt{c-c \sec (e+f x)}}","-\frac{5 \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{8 \sqrt{2} a^2 c^{3/2} f}-\frac{5 \tan (e+f x)}{8 a^2 f (c-c \sec (e+f x))^{3/2}}+\frac{5 \tan (e+f x)}{6 f \left(a^2 \sec (e+f x)+a^2\right) (c-c \sec (e+f x))^{3/2}}+\frac{\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{3/2}}",1,"(((-15*I)*Sqrt[2]*(-1 + E^(I*(e + f*x)))^3*(1 + E^(I*(e + f*x)))^4*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])])/(1 + E^((2*I)*(e + f*x)))^(7/2) - 48*Cos[(e + f*x)/2]^4*Csc[e/2]*Sec[e + f*x]^4*Sin[(f*x)/2]*Sin[(e + f*x)/2] + 48*Cos[(e + f*x)/2]^4*Cot[e/2]*Sec[e + f*x]^4*Sin[(e + f*x)/2]^2 + 32*Cos[(e + f*x)/2]*Sec[e + f*x]^4*Sin[(e + f*x)/2]^3 + 416*Cos[e/2]*Cos[(f*x)/2]*Cos[(e + f*x)/2]^4*Sec[e + f*x]^4*Sin[(e + f*x)/2]^3 - 416*Csc[(e + f*x)/2]*Csc[2*(e + f*x)]^4*Sin[e/2]*Sin[(f*x)/2]*Sin[e + f*x]^8 - 40*Sec[e + f*x]*Tan[e + f*x]^3)/(48*a^2*c*f*(-1 + Sec[e + f*x])*(1 + Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]])","C",1
99,1,434,203,2.3340718,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(5/2)),x]","\frac{\cot ^4(e+f x) \left(\frac{105 i \sqrt{2} \left(-1+e^{i (e+f x)}\right)^5 \left(1+e^{i (e+f x)}\right)^4 \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)}{\left(1+e^{2 i (e+f x)}\right)^{9/2}}-3648 \csc \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sin ^9(e+f x) \csc \left(\frac{1}{2} (e+f x)\right) \csc ^5(2 (e+f x))-5504 \sin \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sin \left(\frac{1}{2} (e+f x)\right) \sin ^9(e+f x) \csc ^5(2 (e+f x))-13312 \sin ^2\left(\frac{1}{2} (e+f x)\right) \sin ^8(e+f x) \csc ^5(2 (e+f x))+2752 \cos \left(\frac{e}{2}\right) \cos \left(\frac{f x}{2}\right) \sin ^5\left(\frac{1}{2} (e+f x)\right) \cos ^4\left(\frac{1}{2} (e+f x)\right) \sec ^5(e+f x)+256 \sin ^5\left(\frac{1}{2} (e+f x)\right) \cos \left(\frac{1}{2} (e+f x)\right) \sec ^5(e+f x)+114 \cot \left(\frac{e}{2}\right) \tan ^4(e+f x) \sec (e+f x)-192 \cot \left(\frac{e}{2}\right) \sin ^2\left(\frac{1}{2} (e+f x)\right) \cos ^4\left(\frac{1}{2} (e+f x)\right) \sec ^5(e+f x)+192 \csc \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sin \left(\frac{1}{2} (e+f x)\right) \cos ^4\left(\frac{1}{2} (e+f x)\right) \sec ^5(e+f x)\right)}{384 a^2 c^2 f \sqrt{c-c \sec (e+f x)}}","-\frac{35 \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{64 \sqrt{2} a^2 c^{5/2} f}-\frac{35 \tan (e+f x)}{64 a^2 c f (c-c \sec (e+f x))^{3/2}}-\frac{35 \tan (e+f x)}{48 a^2 f (c-c \sec (e+f x))^{5/2}}+\frac{7 \tan (e+f x)}{6 f \left(a^2 \sec (e+f x)+a^2\right) (c-c \sec (e+f x))^{5/2}}+\frac{\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}",1,"(Cot[e + f*x]^4*(((105*I)*Sqrt[2]*(-1 + E^(I*(e + f*x)))^5*(1 + E^(I*(e + f*x)))^4*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])])/(1 + E^((2*I)*(e + f*x)))^(9/2) + 192*Cos[(e + f*x)/2]^4*Csc[e/2]*Sec[e + f*x]^5*Sin[(f*x)/2]*Sin[(e + f*x)/2] - 192*Cos[(e + f*x)/2]^4*Cot[e/2]*Sec[e + f*x]^5*Sin[(e + f*x)/2]^2 + 256*Cos[(e + f*x)/2]*Sec[e + f*x]^5*Sin[(e + f*x)/2]^5 + 2752*Cos[e/2]*Cos[(f*x)/2]*Cos[(e + f*x)/2]^4*Sec[e + f*x]^5*Sin[(e + f*x)/2]^5 - 13312*Csc[2*(e + f*x)]^5*Sin[(e + f*x)/2]^2*Sin[e + f*x]^8 - 3648*Csc[e/2]*Csc[(e + f*x)/2]*Csc[2*(e + f*x)]^5*Sin[(f*x)/2]*Sin[e + f*x]^9 - 5504*Csc[2*(e + f*x)]^5*Sin[e/2]*Sin[(f*x)/2]*Sin[(e + f*x)/2]*Sin[e + f*x]^9 + 114*Cot[e/2]*Sec[e + f*x]*Tan[e + f*x]^4))/(384*a^2*c^2*f*Sqrt[c - c*Sec[e + f*x]])","C",1
100,1,78,169,0.6331873,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^{7/2}}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^(7/2))/(a + a*Sec[e + f*x])^3,x]","-\frac{c^3 (249 \cos (e+f x)+110 \cos (2 (e+f x))+23 \cos (3 (e+f x))+130) \cot \left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)}}{10 a^3 f (\cos (e+f x)+1)^3}","\frac{32 c^4 \tan (e+f x)}{5 a^3 f \sqrt{c-c \sec (e+f x)}}+\frac{16 c^3 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{5 f \left(a^3 \sec (e+f x)+a^3\right)}-\frac{4 c^2 \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{5 a f (a \sec (e+f x)+a)^2}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{5 f (a \sec (e+f x)+a)^3}",1,"-1/10*(c^3*(130 + 249*Cos[e + f*x] + 110*Cos[2*(e + f*x)] + 23*Cos[3*(e + f*x)])*Cot[(e + f*x)/2]*Sqrt[c - c*Sec[e + f*x]])/(a^3*f*(1 + Cos[e + f*x])^3)","A",1
101,1,74,135,0.3459479,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^(5/2))/(a + a*Sec[e + f*x])^3,x]","-\frac{c^2 \cos (e+f x) (20 \cos (e+f x)+7 \cos (2 (e+f x))+37) \cot \left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)}}{15 a^3 f (\cos (e+f x)+1)^3}","\frac{16 c^3 \tan (e+f x)}{15 f \left(a^3 \sec (e+f x)+a^3\right) \sqrt{c-c \sec (e+f x)}}-\frac{8 c^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{15 a f (a \sec (e+f x)+a)^2}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{5 f (a \sec (e+f x)+a)^3}",1,"-1/15*(c^2*Cos[e + f*x]*(37 + 20*Cos[e + f*x] + 7*Cos[2*(e + f*x)])*Cot[(e + f*x)/2]*Sqrt[c - c*Sec[e + f*x]])/(a^3*f*(1 + Cos[e + f*x])^3)","A",1
102,1,60,88,0.3187471,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^{3/2}}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^(3/2))/(a + a*Sec[e + f*x])^3,x]","-\frac{2 c (\cos (e+f x)-5) \cot \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{c-c \sec (e+f x)}}{15 a^3 f (\sec (e+f x)+1)^3}","\frac{2 c \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{5 f (a \sec (e+f x)+a)^3}-\frac{4 c^2 \tan (e+f x)}{15 a f (a \sec (e+f x)+a)^2 \sqrt{c-c \sec (e+f x)}}",1,"(-2*c*(-5 + Cos[e + f*x])*Cot[(e + f*x)/2]*Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(15*a^3*f*(1 + Sec[e + f*x])^3)","A",1
103,1,55,41,0.1535307,"\int \frac{\sec (e+f x) \sqrt{c-c \sec (e+f x)}}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a + a*Sec[e + f*x])^3,x]","-\frac{\cos ^3(e+f x) \csc \left(\frac{1}{2} (e+f x)\right) \sec ^5\left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)}}{20 a^3 f}","\frac{2 c \tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 \sqrt{c-c \sec (e+f x)}}",1,"-1/20*(Cos[e + f*x]^3*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]^5*Sqrt[c - c*Sec[e + f*x]])/(a^3*f)","A",1
104,1,225,181,1.532929,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 \sqrt{c-c \sec (e+f x)}} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]]),x]","\frac{2 e^{-\frac{1}{2} i (e+f x)} \sin \left(\frac{1}{2} (e+f x)\right) \cos \left(\frac{1}{2} (e+f x)\right) \sec ^{\frac{7}{2}}(e+f x) \left(\frac{e^{\frac{1}{2} i (e+f x)} (80 \cos (e+f x)+37 \cos (2 (e+f x))+67)}{8 \sqrt{\sec (e+f x)}}-15 \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \cos ^5\left(\frac{1}{2} (e+f x)\right) \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)\right)}{15 a^3 f (\sec (e+f x)+1)^3 \sqrt{c-c \sec (e+f x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{4 \sqrt{2} a^3 \sqrt{c} f}+\frac{\tan (e+f x)}{4 f \left(a^3 \sec (e+f x)+a^3\right) \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{6 a f (a \sec (e+f x)+a)^2 \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 \sqrt{c-c \sec (e+f x)}}",1,"(2*Cos[(e + f*x)/2]*(-15*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])]*Cos[(e + f*x)/2]^5 + (E^((I/2)*(e + f*x))*(67 + 80*Cos[e + f*x] + 37*Cos[2*(e + f*x)]))/(8*Sqrt[Sec[e + f*x]]))*Sec[e + f*x]^(7/2)*Sin[(e + f*x)/2])/(15*a^3*E^((I/2)*(e + f*x))*f*(1 + Sec[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]])","C",1
105,1,398,212,6.4667944,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2}} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(3/2)),x]","\frac{\sin ^3\left(\frac{e}{2}+\frac{f x}{2}\right) \cos ^6\left(\frac{e}{2}+\frac{f x}{2}\right) \sec ^5(e+f x) \left(\frac{278 \sin \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right)}{15 f}-\frac{278 \cos \left(\frac{e}{2}\right) \cos \left(\frac{f x}{2}\right)}{15 f}+\frac{2 \sec ^5\left(\frac{e}{2}+\frac{f x}{2}\right)}{5 f}-\frac{56 \sec ^3\left(\frac{e}{2}+\frac{f x}{2}\right)}{15 f}+\frac{242 \sec \left(\frac{e}{2}+\frac{f x}{2}\right)}{15 f}-\frac{\cot \left(\frac{e}{2}\right) \csc \left(\frac{e}{2}+\frac{f x}{2}\right)}{f}+\frac{\csc \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \csc ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{f}\right)}{(a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}+\frac{7 e^{-\frac{1}{2} i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \sin ^3\left(\frac{e}{2}+\frac{f x}{2}\right) \cos ^6\left(\frac{e}{2}+\frac{f x}{2}\right) \sec ^{\frac{9}{2}}(e+f x) \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)}{f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}","-\frac{7 \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{16 \sqrt{2} a^3 c^{3/2} f}-\frac{7 \tan (e+f x)}{16 a^3 f (c-c \sec (e+f x))^{3/2}}+\frac{7 \tan (e+f x)}{12 f \left(a^3 \sec (e+f x)+a^3\right) (c-c \sec (e+f x))^{3/2}}+\frac{7 \tan (e+f x)}{30 a f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{3/2}}+\frac{\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}",1,"(7*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])]*Cos[e/2 + (f*x)/2]^6*Sec[e + f*x]^(9/2)*Sin[e/2 + (f*x)/2]^3)/(E^((I/2)*(e + f*x))*f*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(3/2)) + (Cos[e/2 + (f*x)/2]^6*Sec[e + f*x]^5*((-278*Cos[e/2]*Cos[(f*x)/2])/(15*f) - (Cot[e/2]*Csc[e/2 + (f*x)/2])/f + (242*Sec[e/2 + (f*x)/2])/(15*f) - (56*Sec[e/2 + (f*x)/2]^3)/(15*f) + (2*Sec[e/2 + (f*x)/2]^5)/(5*f) + (Csc[e/2]*Csc[e/2 + (f*x)/2]^2*Sin[(f*x)/2])/f + (278*Sin[e/2]*Sin[(f*x)/2])/(15*f))*Sin[e/2 + (f*x)/2]^3)/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(3/2))","C",1
106,1,468,246,6.6631151,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2)),x]","\frac{\sin ^5\left(\frac{e}{2}+\frac{f x}{2}\right) \cos ^6\left(\frac{e}{2}+\frac{f x}{2}\right) \sec ^6(e+f x) \left(-\frac{257 \sin \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right)}{10 f}+\frac{257 \cos \left(\frac{e}{2}\right) \cos \left(\frac{f x}{2}\right)}{10 f}-\frac{2 \sec ^5\left(\frac{e}{2}+\frac{f x}{2}\right)}{5 f}+\frac{22 \sec ^3\left(\frac{e}{2}+\frac{f x}{2}\right)}{5 f}-\frac{124 \sec \left(\frac{e}{2}+\frac{f x}{2}\right)}{5 f}-\frac{\cot \left(\frac{e}{2}\right) \csc ^3\left(\frac{e}{2}+\frac{f x}{2}\right)}{2 f}+\frac{23 \cot \left(\frac{e}{2}\right) \csc \left(\frac{e}{2}+\frac{f x}{2}\right)}{4 f}+\frac{\csc \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \csc ^4\left(\frac{e}{2}+\frac{f x}{2}\right)}{2 f}-\frac{23 \csc \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \csc ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{4 f}\right)}{(a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}-\frac{63 e^{-\frac{1}{2} i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \sin ^5\left(\frac{e}{2}+\frac{f x}{2}\right) \cos ^6\left(\frac{e}{2}+\frac{f x}{2}\right) \sec ^{\frac{11}{2}}(e+f x) \tanh ^{-1}\left(\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)}{4 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}","-\frac{63 \tan ^{-1}\left(\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right)}{128 \sqrt{2} a^3 c^{5/2} f}-\frac{63 \tan (e+f x)}{128 a^3 c f (c-c \sec (e+f x))^{3/2}}-\frac{21 \tan (e+f x)}{32 a^3 f (c-c \sec (e+f x))^{5/2}}+\frac{21 \tan (e+f x)}{20 f \left(a^3 \sec (e+f x)+a^3\right) (c-c \sec (e+f x))^{5/2}}+\frac{3 \tan (e+f x)}{10 a f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}+\frac{\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}",1,"(-63*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])]*Cos[e/2 + (f*x)/2]^6*Sec[e + f*x]^(11/2)*Sin[e/2 + (f*x)/2]^5)/(4*E^((I/2)*(e + f*x))*f*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2)) + (Cos[e/2 + (f*x)/2]^6*Sec[e + f*x]^6*((257*Cos[e/2]*Cos[(f*x)/2])/(10*f) + (23*Cot[e/2]*Csc[e/2 + (f*x)/2])/(4*f) - (Cot[e/2]*Csc[e/2 + (f*x)/2]^3)/(2*f) - (124*Sec[e/2 + (f*x)/2])/(5*f) + (22*Sec[e/2 + (f*x)/2]^3)/(5*f) - (2*Sec[e/2 + (f*x)/2]^5)/(5*f) - (23*Csc[e/2]*Csc[e/2 + (f*x)/2]^2*Sin[(f*x)/2])/(4*f) + (Csc[e/2]*Csc[e/2 + (f*x)/2]^4*Sin[(f*x)/2])/(2*f) - (257*Sin[e/2]*Sin[(f*x)/2])/(10*f))*Sin[e/2 + (f*x)/2]^5)/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2))","C",1
107,1,87,43,0.4657334,"\int \sec (e+f x) \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2} \, dx","Integrate[Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2),x]","\frac{c^2 (-6 \cos (e+f x)+3 \cos (2 (e+f x))+5) \csc \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{12 f}","\frac{a \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt{a \sec (e+f x)+a}}",1,"(c^2*(5 - 6*Cos[e + f*x] + 3*Cos[2*(e + f*x)])*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*Sec[e + f*x]^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(12*f)","B",1
108,1,73,43,0.3055064,"\int \sec (e+f x) \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2} \, dx","Integrate[Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2),x]","\frac{c (2 \cos (e+f x)-1) \csc \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{4 f}","\frac{a \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}}",1,"(c*(-1 + 2*Cos[e + f*x])*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*Sec[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(4*f)","A",1
109,1,56,41,0.1695918,"\int \sec (e+f x) \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)} \, dx","Integrate[Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]],x]","\frac{\csc \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{2 f}","-\frac{c \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f \sqrt{c-c \sec (e+f x)}}",1,"(Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(2*f)","A",1
110,1,99,51,0.8386673,"\int \frac{\sec (e+f x) \sqrt{a+a \sec (e+f x)}}{\sqrt{c-c \sec (e+f x)}} \, dx","Integrate[(Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/Sqrt[c - c*Sec[e + f*x]],x]","-\frac{i \left(-1+e^{i (e+f x)}\right) \left(2 \log \left(1-e^{i (e+f x)}\right)-\log \left(1+e^{2 i (e+f x)}\right)\right) \sqrt{a (\sec (e+f x)+1)}}{f \left(1+e^{i (e+f x)}\right) \sqrt{c-c \sec (e+f x)}}","\frac{a \tan (e+f x) \log (1-\sec (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"((-I)*(-1 + E^(I*(e + f*x)))*(2*Log[1 - E^(I*(e + f*x))] - Log[1 + E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sec[e + f*x])])/((1 + E^(I*(e + f*x)))*f*Sqrt[c - c*Sec[e + f*x]])","C",1
111,1,62,42,0.2472124,"\int \frac{\sec (e+f x) \sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx","Integrate[(Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c - c*Sec[e + f*x])^(3/2),x]","\frac{\tan \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{a (\sec (e+f x)+1)}}{c f (\sec (e+f x)-1) \sqrt{c-c \sec (e+f x)}}","-\frac{\tan (e+f x) \sqrt{a \sec (e+f x)+a}}{2 f (c-c \sec (e+f x))^{3/2}}",1,"(Sec[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(c*f*(-1 + Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]])","A",1
112,1,69,43,0.3897945,"\int \frac{\sec (e+f x) \sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx","Integrate[(Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c - c*Sec[e + f*x])^(5/2),x]","-\frac{(2 \cos (e+f x)-1) \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)}}{2 c^2 f (\cos (e+f x)-1)^2 \sqrt{c-c \sec (e+f x)}}","-\frac{a \tan (e+f x)}{2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}",1,"-1/2*((-1 + 2*Cos[e + f*x])*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(c^2*f*(-1 + Cos[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]])","A",1
113,1,108,89,1.0591431,"\int \sec (e+f x) (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{7/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(7/2),x]","\frac{a c^3 (-10 \cos (e+f x)+20 \cos (2 (e+f x))-10 \cos (3 (e+f x))+5 \cos (4 (e+f x))+7) \csc \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sec ^4(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{80 f}","\frac{a^2 \tan (e+f x) (c-c \sec (e+f x))^{7/2}}{10 f \sqrt{a \sec (e+f x)+a}}+\frac{a \tan (e+f x) \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}{5 f}",1,"(a*c^3*(7 - 10*Cos[e + f*x] + 20*Cos[2*(e + f*x)] - 10*Cos[3*(e + f*x)] + 5*Cos[4*(e + f*x)])*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*Sec[e + f*x]^4*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(80*f)","A",1
114,1,97,89,0.5859927,"\int \sec (e+f x) (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{5/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(5/2),x]","\frac{a c^2 (5 \cos (e+f x)-3 \cos (2 (e+f x))+3 \cos (3 (e+f x))) \csc \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{24 f}","\frac{a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{6 f \sqrt{a \sec (e+f x)+a}}+\frac{a \tan (e+f x) \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}{4 f}",1,"(a*c^2*(5*Cos[e + f*x] - 3*Cos[2*(e + f*x)] + 3*Cos[3*(e + f*x)])*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*Sec[e + f*x]^3*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(24*f)","A",1
115,1,78,89,0.4269229,"\int \sec (e+f x) (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(3/2),x]","\frac{a c (3 \cos (2 (e+f x))+1) \csc \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{12 f}","-\frac{c^2 \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{3 f \sqrt{c-c \sec (e+f x)}}-\frac{c \tan (e+f x) (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}{3 f}",1,"(a*c*(1 + 3*Cos[2*(e + f*x)])*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*Sec[e + f*x]^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(12*f)","A",1
116,1,73,43,0.2769288,"\int \sec (e+f x) (a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]],x]","\frac{a (2 \cos (e+f x)+1) \csc \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{4 f}","-\frac{c \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sec (e+f x)}}",1,"(a*(1 + 2*Cos[e + f*x])*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*Sec[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(4*f)","A",1
117,1,174,95,1.4158591,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{\sqrt{c-c \sec (e+f x)}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/Sqrt[c - c*Sec[e + f*x]],x]","\frac{\sqrt{2} a \sin \left(\frac{1}{2} (e+f x)\right) \sec ^{\frac{3}{2}}(e+f x) \sqrt{a (\sec (e+f x)+1)} \left(1+\left(4 \log \left(1-e^{i (e+f x)}\right)-2 \log \left(1+e^{2 i (e+f x)}\right)\right) \cos (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+i \sin \left(\frac{1}{2} (e+f x)\right)\right)}{f \left(1+e^{i (e+f x)}\right) \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{c-c \sec (e+f x)}}","\frac{2 a^2 \tan (e+f x) \log (1-\sec (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{a \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f \sqrt{c-c \sec (e+f x)}}",1,"(Sqrt[2]*a*(1 + Cos[e + f*x]*(4*Log[1 - E^(I*(e + f*x))] - 2*Log[1 + E^((2*I)*(e + f*x))]))*Sec[e + f*x]^(3/2)*Sqrt[a*(1 + Sec[e + f*x])]*(Cos[(e + f*x)/2] + I*Sin[(e + f*x)/2])*Sin[(e + f*x)/2])/((1 + E^(I*(e + f*x)))*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*f*Sqrt[c - c*Sec[e + f*x]])","C",1
118,1,134,99,0.695225,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{3/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(3/2),x]","-\frac{a \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(-2 \log \left(1-e^{i (e+f x)}\right)+\log \left(1+e^{2 i (e+f x)}\right)+\left(2 \log \left(1-e^{i (e+f x)}\right)-\log \left(1+e^{2 i (e+f x)}\right)\right) \cos (e+f x)+2\right)}{c f (\cos (e+f x)-1) \sqrt{c-c \sec (e+f x)}}","-\frac{a^2 \tan (e+f x) \log (1-\sec (e+f x))}{c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f (c-c \sec (e+f x))^{3/2}}",1,"-((a*(2 - 2*Log[1 - E^(I*(e + f*x))] + Cos[e + f*x]*(2*Log[1 - E^(I*(e + f*x))] - Log[1 + E^((2*I)*(e + f*x))]) + Log[1 + E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(c*f*(-1 + Cos[e + f*x])*Sqrt[c - c*Sec[e + f*x]]))","C",1
119,1,63,42,0.48225,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(5/2),x]","\frac{a \tan \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{c^3 f (\sec (e+f x)-1)^3}","-\frac{\tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{4 f (c-c \sec (e+f x))^{5/2}}",1,"(a*Sec[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]]*Tan[(e + f*x)/2])/(c^3*f*(-1 + Sec[e + f*x])^3)","A",1
120,1,80,88,0.5556957,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{7/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(7/2),x]","\frac{a (3 \cos (e+f x)-3 \cos (2 (e+f x))-4) \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)}}{6 c^3 f (\cos (e+f x)-1)^3 \sqrt{c-c \sec (e+f x)}}","-\frac{\tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{24 c f (c-c \sec (e+f x))^{5/2}}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{6 f (c-c \sec (e+f x))^{7/2}}",1,"(a*(-4 + 3*Cos[e + f*x] - 3*Cos[2*(e + f*x)])*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(6*c^3*f*(-1 + Cos[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]])","A",1
121,1,90,92,0.8275815,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{9/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(9/2),x]","-\frac{a (17 \cos (e+f x)-6 \cos (2 (e+f x))+3 \cos (3 (e+f x))-8) \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)}}{12 c^4 f (\cos (e+f x)-1)^4 \sqrt{c-c \sec (e+f x)}}","\frac{a^2 \tan (e+f x)}{12 c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}-\frac{a \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{4 f (c-c \sec (e+f x))^{9/2}}",1,"-1/12*(a*(-8 + 17*Cos[e + f*x] - 6*Cos[2*(e + f*x)] + 3*Cos[3*(e + f*x)])*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(c^4*f*(-1 + Cos[e + f*x])^4*Sqrt[c - c*Sec[e + f*x]])","A",1
122,1,100,92,1.2191657,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{11/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(11/2),x]","\frac{a (75 \cos (e+f x)-50 \cos (2 (e+f x))+15 \cos (3 (e+f x))-5 \cos (4 (e+f x))-51) \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)}}{40 c^5 f (\cos (e+f x)-1)^5 \sqrt{c-c \sec (e+f x)}}","\frac{a^2 \tan (e+f x)}{20 c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{9/2}}-\frac{a \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{5 f (c-c \sec (e+f x))^{11/2}}",1,"(a*(-51 + 75*Cos[e + f*x] - 50*Cos[2*(e + f*x)] + 15*Cos[3*(e + f*x)] - 5*Cos[4*(e + f*x)])*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(40*c^5*f*(-1 + Cos[e + f*x])^5*Sqrt[c - c*Sec[e + f*x]])","A",1
123,1,113,134,1.297506,"\int \sec (e+f x) (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{7/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(7/2),x]","\frac{a^2 c^3 (78 \cos (e+f x)+5 (7 \cos (3 (e+f x))-3 \cos (4 (e+f x))+3 \cos (5 (e+f x))-5)) \csc \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sec ^5(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{480 f}","\frac{a^3 \tan (e+f x) (c-c \sec (e+f x))^{7/2}}{15 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}{15 f}+\frac{a \tan (e+f x) (a \sec (e+f x)+a)^{3/2} (c-c \sec (e+f x))^{7/2}}{6 f}",1,"(a^2*c^3*(78*Cos[e + f*x] + 5*(-5 + 7*Cos[3*(e + f*x)] - 3*Cos[4*(e + f*x)] + 3*Cos[5*(e + f*x)]))*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*Sec[e + f*x]^5*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(480*f)","A",1
124,1,92,134,0.7786723,"\int \sec (e+f x) (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{5/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(5/2),x]","\frac{a^2 c^2 (20 \cos (2 (e+f x))+15 \cos (4 (e+f x))+29) \csc \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sec ^4(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{240 f}","-\frac{2 c^3 \tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{15 f \sqrt{c-c \sec (e+f x)}}-\frac{c^2 \tan (e+f x) (a \sec (e+f x)+a)^{5/2} \sqrt{c-c \sec (e+f x)}}{5 f}-\frac{c \tan (e+f x) (a \sec (e+f x)+a)^{5/2} (c-c \sec (e+f x))^{3/2}}{5 f}",1,"(a^2*c^2*(29 + 20*Cos[2*(e + f*x)] + 15*Cos[4*(e + f*x)])*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*Sec[e + f*x]^4*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(240*f)","A",1
125,1,96,89,0.5938566,"\int \sec (e+f x) (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2),x]","\frac{a^2 c (5 \cos (e+f x)+3 (\cos (2 (e+f x))+\cos (3 (e+f x)))) \csc \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{24 f}","-\frac{c^2 \tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{6 f \sqrt{c-c \sec (e+f x)}}-\frac{c \tan (e+f x) (a \sec (e+f x)+a)^{5/2} \sqrt{c-c \sec (e+f x)}}{4 f}",1,"(a^2*c*(5*Cos[e + f*x] + 3*(Cos[2*(e + f*x)] + Cos[3*(e + f*x)]))*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*Sec[e + f*x]^3*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(24*f)","A",1
126,1,88,43,0.4680721,"\int \sec (e+f x) (a+a \sec (e+f x))^{5/2} \sqrt{c-c \sec (e+f x)} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]],x]","\frac{a^2 \cot \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)} \left(4 \cos (e+f x)+\cos ^2(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)+2\right)}{6 f}","-\frac{c \tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sec (e+f x)}}",1,"(a^2*Cot[(e + f*x)/2]*(2 + 4*Cos[e + f*x] + Cos[e + f*x]^2*Sec[(e + f*x)/2]^2)*Sec[e + f*x]^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(6*f)","B",1
127,1,328,141,6.6038952,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{\sqrt{c-c \sec (e+f x)}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2))/Sqrt[c - c*Sec[e + f*x]],x]","\frac{\sin \left(\frac{e}{2}+\frac{f x}{2}\right) \sec (e+f x) (a (\sec (e+f x)+1))^{5/2} \sqrt{(\cos (e+f x)+1) \sec (e+f x)} \left(\frac{5 \sec \left(\frac{e}{2}+\frac{f x}{2}\right)}{2 f}+\frac{\cos \left(\frac{e}{2}+\frac{f x}{2}\right) \sec (e+f x)}{f}\right)}{(\sec (e+f x)+1)^{5/2} \sqrt{c-c \sec (e+f x)}}+\frac{4 \sqrt{2} e^{\frac{1}{2} i (e+f x)} \sqrt{\frac{\left(1+e^{i (e+f x)}\right)^2}{1+e^{2 i (e+f x)}}} \left(2 \log \left(1-e^{i (e+f x)}\right)-\log \left(1+e^{2 i (e+f x)}\right)\right) \sin \left(\frac{e}{2}+\frac{f x}{2}\right) \sqrt{\sec (e+f x)} (a (\sec (e+f x)+1))^{5/2}}{f \left(1+e^{i (e+f x)}\right) \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} (\sec (e+f x)+1)^{5/2} \sqrt{c-c \sec (e+f x)}}","\frac{4 a^3 \tan (e+f x) \log (1-\sec (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{2 a^2 \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f \sqrt{c-c \sec (e+f x)}}+\frac{a \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sec (e+f x)}}",1,"(4*Sqrt[2]*E^((I/2)*(e + f*x))*Sqrt[(1 + E^(I*(e + f*x)))^2/(1 + E^((2*I)*(e + f*x)))]*(2*Log[1 - E^(I*(e + f*x))] - Log[1 + E^((2*I)*(e + f*x))])*Sqrt[Sec[e + f*x]]*(a*(1 + Sec[e + f*x]))^(5/2)*Sin[e/2 + (f*x)/2])/((1 + E^(I*(e + f*x)))*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*f*(1 + Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]]) + (Sec[e + f*x]*Sqrt[(1 + Cos[e + f*x])*Sec[e + f*x]]*(a*(1 + Sec[e + f*x]))^(5/2)*((5*Sec[e/2 + (f*x)/2])/(2*f) + (Cos[e/2 + (f*x)/2]*Sec[e + f*x])/f)*Sin[e/2 + (f*x)/2])/((1 + Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]])","C",1
128,1,188,145,1.4962267,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{3/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2))/(c - c*Sec[e + f*x])^(3/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{a (\sec (e+f x)+1)} \left(-4 \log \left(1-e^{i (e+f x)}\right)+2 \log \left(1+e^{2 i (e+f x)}\right)+\left(8 \log \left(1-e^{i (e+f x)}\right)-4 \log \left(1+e^{2 i (e+f x)}\right)-5\right) \cos (e+f x)+\left(2 \log \left(1+e^{2 i (e+f x)}\right)-4 \log \left(1-e^{i (e+f x)}\right)\right) \cos (2 (e+f x))+1\right)}{c f (\cos (e+f x)-1) \sqrt{c-c \sec (e+f x)}}","-\frac{4 a^3 \tan (e+f x) \log (1-\sec (e+f x))}{c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{2 a^2 \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{c f \sqrt{c-c \sec (e+f x)}}-\frac{a \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{f (c-c \sec (e+f x))^{3/2}}",1,"(a^2*(1 - 4*Log[1 - E^(I*(e + f*x))] + Cos[e + f*x]*(-5 + 8*Log[1 - E^(I*(e + f*x))] - 4*Log[1 + E^((2*I)*(e + f*x))]) + 2*Log[1 + E^((2*I)*(e + f*x))] + Cos[2*(e + f*x)]*(-4*Log[1 - E^(I*(e + f*x))] + 2*Log[1 + E^((2*I)*(e + f*x))]))*Sec[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(c*f*(-1 + Cos[e + f*x])*Sqrt[c - c*Sec[e + f*x]])","C",1
129,1,182,145,1.3433345,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{5/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2))/(c - c*Sec[e + f*x])^(5/2),x]","-\frac{a^2 \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(-6 \log \left(1-e^{i (e+f x)}\right)+3 \log \left(1+e^{2 i (e+f x)}\right)+\left(8 \log \left(1-e^{i (e+f x)}\right)-4 \log \left(1+e^{2 i (e+f x)}\right)\right) \cos (e+f x)+\left(\log \left(1+e^{2 i (e+f x)}\right)-2 \log \left(1-e^{i (e+f x)}\right)\right) \cos (2 (e+f x))+4\right)}{2 c^2 f (\cos (e+f x)-1)^2 \sqrt{c-c \sec (e+f x)}}","\frac{a^3 \tan (e+f x) \log (1-\sec (e+f x))}{c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{a^2 \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{c f (c-c \sec (e+f x))^{3/2}}-\frac{a \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{2 f (c-c \sec (e+f x))^{5/2}}",1,"-1/2*(a^2*(4 - 6*Log[1 - E^(I*(e + f*x))] + Cos[e + f*x]*(8*Log[1 - E^(I*(e + f*x))] - 4*Log[1 + E^((2*I)*(e + f*x))]) + 3*Log[1 + E^((2*I)*(e + f*x))] + Cos[2*(e + f*x)]*(-2*Log[1 - E^(I*(e + f*x))] + Log[1 + E^((2*I)*(e + f*x))]))*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(c^2*f*(-1 + Cos[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]])","C",1
130,1,76,42,0.5598627,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{7/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2))/(c - c*Sec[e + f*x])^(7/2),x]","\frac{a^2 (3 \cos (2 (e+f x))+5) \csc ^5\left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)}}{48 c^3 f \sqrt{c-c \sec (e+f x)}}","-\frac{\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{6 f (c-c \sec (e+f x))^{7/2}}",1,"(a^2*(5 + 3*Cos[2*(e + f*x)])*Csc[(e + f*x)/2]^5*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])])/(48*c^3*f*Sqrt[c - c*Sec[e + f*x]])","A",1
131,1,92,88,0.8269731,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{9/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2))/(c - c*Sec[e + f*x])^(9/2),x]","-\frac{a^2 (17 \cos (e+f x)-3 \cos (2 (e+f x))+3 \cos (3 (e+f x))-5) \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)}}{12 c^4 f (\cos (e+f x)-1)^4 \sqrt{c-c \sec (e+f x)}}","-\frac{\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{48 c f (c-c \sec (e+f x))^{7/2}}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{8 f (c-c \sec (e+f x))^{9/2}}",1,"-1/12*(a^2*(-5 + 17*Cos[e + f*x] - 3*Cos[2*(e + f*x)] + 3*Cos[3*(e + f*x)])*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(c^4*f*(-1 + Cos[e + f*x])^4*Sqrt[c - c*Sec[e + f*x]])","A",1
132,1,102,133,1.237561,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2))/(c - c*Sec[e + f*x])^(11/2),x]","\frac{a^2 (170 \cos (e+f x)-140 \cos (2 (e+f x))+30 \cos (3 (e+f x))-15 \cos (4 (e+f x))-141) \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)}}{120 c^5 f (\cos (e+f x)-1)^5 \sqrt{c-c \sec (e+f x)}}","-\frac{\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{240 c^2 f (c-c \sec (e+f x))^{7/2}}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{40 c f (c-c \sec (e+f x))^{9/2}}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{10 f (c-c \sec (e+f x))^{11/2}}",1,"(a^2*(-141 + 170*Cos[e + f*x] - 140*Cos[2*(e + f*x)] + 30*Cos[3*(e + f*x)] - 15*Cos[4*(e + f*x)])*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(120*c^5*f*(-1 + Cos[e + f*x])^5*Sqrt[c - c*Sec[e + f*x]])","A",1
133,1,141,139,1.4837468,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^(5/2))/Sqrt[a + a*Sec[e + f*x]],x]","\frac{c^2 \cot \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{c-c \sec (e+f x)} \left(8 \log \left(1+e^{i (e+f x)}\right)-4 \log \left(1+e^{2 i (e+f x)}\right)-6 \cos (e+f x)+\left(8 \log \left(1+e^{i (e+f x)}\right)-4 \log \left(1+e^{2 i (e+f x)}\right)\right) \cos (2 (e+f x))+1\right)}{2 f \sqrt{a (\sec (e+f x)+1)}}","-\frac{4 c^3 \tan (e+f x) \log (\sec (e+f x)+1)}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{2 c^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}-\frac{c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}}",1,"(c^2*Cot[(e + f*x)/2]*(1 - 6*Cos[e + f*x] + 8*Log[1 + E^(I*(e + f*x))] + Cos[2*(e + f*x)]*(8*Log[1 + E^(I*(e + f*x))] - 4*Log[1 + E^((2*I)*(e + f*x))]) - 4*Log[1 + E^((2*I)*(e + f*x))])*Sec[e + f*x]^2*Sqrt[c - c*Sec[e + f*x]])/(2*f*Sqrt[a*(1 + Sec[e + f*x])])","C",1
134,1,173,94,1.8337882,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^{3/2}}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^(3/2))/Sqrt[a + a*Sec[e + f*x]],x]","\frac{c e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2 \cos \left(\frac{1}{2} (e+f x)\right) \cot \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) \sqrt{c-c \sec (e+f x)} \left(-1+\left(4 \log \left(1+e^{i (e+f x)}\right)-2 \log \left(1+e^{2 i (e+f x)}\right)\right) \cos (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+i \sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 f \left(1+e^{i (e+f x)}\right) \sqrt{a (\sec (e+f x)+1)}}","-\frac{2 c^2 \tan (e+f x) \log (\sec (e+f x)+1)}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{c \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}",1,"(c*(1 + E^((2*I)*(e + f*x)))^2*Cos[(e + f*x)/2]*Cot[(e + f*x)/2]*(-1 + Cos[e + f*x]*(4*Log[1 + E^(I*(e + f*x))] - 2*Log[1 + E^((2*I)*(e + f*x))]))*Sec[e + f*x]^3*Sqrt[c - c*Sec[e + f*x]]*(Cos[(e + f*x)/2] + I*Sin[(e + f*x)/2]))/(2*E^((2*I)*(e + f*x))*(1 + E^(I*(e + f*x)))*f*Sqrt[a*(1 + Sec[e + f*x])])","C",1
135,1,140,50,0.4276886,"\int \frac{\sec (e+f x) \sqrt{c-c \sec (e+f x)}}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/Sqrt[a + a*Sec[e + f*x]],x]","\frac{i \left(1+e^{i (e+f x)}\right) \sqrt{\frac{c \left(-1+e^{i (e+f x)}\right)^2}{1+e^{2 i (e+f x)}}} \left(2 \log \left(1+e^{i (e+f x)}\right)-\log \left(1+e^{2 i (e+f x)}\right)\right)}{f \left(-1+e^{i (e+f x)}\right) \sqrt{\frac{a \left(1+e^{i (e+f x)}\right)^2}{1+e^{2 i (e+f x)}}}}","-\frac{c \tan (e+f x) \log (\sec (e+f x)+1)}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(I*(1 + E^(I*(e + f*x)))*Sqrt[(c*(-1 + E^(I*(e + f*x)))^2)/(1 + E^((2*I)*(e + f*x)))]*(2*Log[1 + E^(I*(e + f*x))] - Log[1 + E^((2*I)*(e + f*x))]))/((-1 + E^(I*(e + f*x)))*Sqrt[(a*(1 + E^(I*(e + f*x)))^2)/(1 + E^((2*I)*(e + f*x)))]*f)","C",1
136,1,94,47,0.7884779,"\int \frac{\sec (e+f x)}{\sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}} \, dx","Integrate[Sec[e + f*x]/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]),x]","\frac{4 i \left(-1+e^{i (e+f x)}\right) \cos ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \tanh ^{-1}\left(e^{i (e+f x)}\right)}{f \left(1+e^{i (e+f x)}\right) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","-\frac{\tan (e+f x) \tanh ^{-1}(\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"((4*I)*(-1 + E^(I*(e + f*x)))*ArcTanh[E^(I*(e + f*x))]*Cos[(e + f*x)/2]^2*Sec[e + f*x])/((1 + E^(I*(e + f*x)))*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
137,1,79,95,0.8968743,"\int \frac{\sec (e+f x)}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}} \, dx","Integrate[Sec[e + f*x]/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)),x]","-\frac{\tan (e+f x) \left(1+2 (\cos (e+f x)-1) \tanh ^{-1}\left(e^{i (e+f x)}\right)\right)}{2 c f (\cos (e+f x)-1) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","-\frac{\tan (e+f x)}{2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{\tan (e+f x) \tanh ^{-1}(\cos (e+f x))}{2 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"-1/2*((1 + 2*ArcTanh[E^(I*(e + f*x))]*(-1 + Cos[e + f*x]))*Tan[e + f*x])/(c*f*(-1 + Cos[e + f*x])*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
138,1,91,140,0.8288488,"\int \frac{\sec (e+f x)}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}} \, dx","Integrate[Sec[e + f*x]/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)),x]","-\frac{\tan (e+f x) \left(3 \cos (e+f x)+8 \sin ^4\left(\frac{1}{2} (e+f x)\right) \tanh ^{-1}\left(e^{i (e+f x)}\right)-2\right)}{4 c^2 f (\cos (e+f x)-1)^2 \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","-\frac{\tan (e+f x) \tanh ^{-1}(\cos (e+f x))}{4 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{4 c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{\tan (e+f x)}{4 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}",1,"-1/4*((-2 + 3*Cos[e + f*x] + 8*ArcTanh[E^(I*(e + f*x))]*Sin[(e + f*x)/2]^4)*Tan[e + f*x])/(c^2*f*(-1 + Cos[e + f*x])^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
139,1,183,142,1.047399,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^(5/2))/(a + a*Sec[e + f*x])^(3/2),x]","-\frac{c^2 \cot \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{c-c \sec (e+f x)} \left(4 \log \left(1+e^{i (e+f x)}\right)-2 \log \left(1+e^{2 i (e+f x)}\right)+\left(8 \log \left(1+e^{i (e+f x)}\right)-4 \log \left(1+e^{2 i (e+f x)}\right)-5\right) \cos (e+f x)+\left(4 \log \left(1+e^{i (e+f x)}\right)-2 \log \left(1+e^{2 i (e+f x)}\right)\right) \cos (2 (e+f x))-1\right)}{a f (\cos (e+f x)+1) \sqrt{a (\sec (e+f x)+1)}}","\frac{4 c^3 \tan (e+f x) \log (\sec (e+f x)+1)}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{2 c^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{a f \sqrt{a \sec (e+f x)+a}}+\frac{c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{f (a \sec (e+f x)+a)^{3/2}}",1,"-((c^2*Cot[(e + f*x)/2]*(-1 + 4*Log[1 + E^(I*(e + f*x))] + Cos[e + f*x]*(-5 + 8*Log[1 + E^(I*(e + f*x))] - 4*Log[1 + E^((2*I)*(e + f*x))]) + Cos[2*(e + f*x)]*(4*Log[1 + E^(I*(e + f*x))] - 2*Log[1 + E^((2*I)*(e + f*x))]) - 2*Log[1 + E^((2*I)*(e + f*x))])*Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a*f*(1 + Cos[e + f*x])*Sqrt[a*(1 + Sec[e + f*x])]))","C",1
140,1,132,95,1.0639485,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^{3/2}}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^(3/2))/(a + a*Sec[e + f*x])^(3/2),x]","-\frac{c \cot \left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)} \left(2 \log \left(1+e^{i (e+f x)}\right)-\log \left(1+e^{2 i (e+f x)}\right)+\left(2 \log \left(1+e^{i (e+f x)}\right)-\log \left(1+e^{2 i (e+f x)}\right)\right) \cos (e+f x)-2\right)}{a f (\cos (e+f x)+1) \sqrt{a (\sec (e+f x)+1)}}","\frac{c^2 \tan (e+f x) \log (\sec (e+f x)+1)}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{c \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f (a \sec (e+f x)+a)^{3/2}}",1,"-((c*Cot[(e + f*x)/2]*(-2 + 2*Log[1 + E^(I*(e + f*x))] + Cos[e + f*x]*(2*Log[1 + E^(I*(e + f*x))] - Log[1 + E^((2*I)*(e + f*x))]) - Log[1 + E^((2*I)*(e + f*x))])*Sqrt[c - c*Sec[e + f*x]])/(a*f*(1 + Cos[e + f*x])*Sqrt[a*(1 + Sec[e + f*x])]))","C",1
141,1,42,42,0.193622,"\int \frac{\sec (e+f x) \sqrt{c-c \sec (e+f x)}}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a + a*Sec[e + f*x])^(3/2),x]","\frac{\csc (e+f x) \sqrt{c-c \sec (e+f x)}}{a f \sqrt{a (\sec (e+f x)+1)}}","\frac{\tan (e+f x) \sqrt{c-c \sec (e+f x)}}{2 f (a \sec (e+f x)+a)^{3/2}}",1,"(Csc[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
142,1,157,95,1.4012501,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)}} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]]),x]","-\frac{\sin \left(\frac{1}{2} (e+f x)\right) \sec ^{\frac{3}{2}}(e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)+i \sin \left(\frac{1}{2} (e+f x)\right)\right) \left(1+2 (\cos (e+f x)+1) \tanh ^{-1}\left(e^{i (e+f x)}\right)\right)}{\sqrt{2} a f \left(1+e^{i (e+f x)}\right) \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","\frac{\tan (e+f x)}{2 f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x) \tanh ^{-1}(\cos (e+f x))}{2 a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"-(((1 + 2*ArcTanh[E^(I*(e + f*x))]*(1 + Cos[e + f*x]))*Sec[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + I*Sin[(e + f*x)/2])*Sin[(e + f*x)/2])/(Sqrt[2]*a*(1 + E^(I*(e + f*x)))*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]]))","C",1
143,1,69,104,0.7958744,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2}} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(3/2)),x]","\frac{\csc (e+f x)-2 \tan (e+f x) \tanh ^{-1}\left(e^{i (e+f x)}\right)}{2 a c f \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","\frac{\csc (e+f x)}{2 a c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x) \tanh ^{-1}(\cos (e+f x))}{2 a c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(Csc[e + f*x] - 2*ArcTanh[E^(I*(e + f*x))]*Tan[e + f*x])/(2*a*c*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
144,1,122,146,1.583916,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{5/2}} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(5/2)),x]","-\frac{\tan (e+f x) \left(-2 \cos (e+f x)+5 \cos (2 (e+f x))+24 \sin ^2\left(\frac{1}{2} (e+f x)\right) \sin ^2(e+f x) \tanh ^{-1}\left(e^{i (e+f x)}\right)+1\right)}{16 a c^2 f (\cos (e+f x)-1)^2 (\cos (e+f x)+1) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","\frac{3 \csc (e+f x)}{8 a c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{3 \tan (e+f x) \tanh ^{-1}(\cos (e+f x))}{8 a c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{4 f (a \sec (e+f x)+a)^{3/2} (c-c \sec (e+f x))^{5/2}}",1,"-1/16*((1 - 2*Cos[e + f*x] + 5*Cos[2*(e + f*x)] + 24*ArcTanh[E^(I*(e + f*x))]*Sin[(e + f*x)/2]^2*Sin[e + f*x]^2)*Tan[e + f*x])/(a*c^2*f*(-1 + Cos[e + f*x])^2*(1 + Cos[e + f*x])*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
145,1,178,145,1.3949855,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^{5/2}} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^(5/2))/(a + a*Sec[e + f*x])^(5/2),x]","\frac{c^2 \cot \left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)} \left(6 \log \left(1+e^{i (e+f x)}\right)-3 \log \left(1+e^{2 i (e+f x)}\right)+\left(8 \log \left(1+e^{i (e+f x)}\right)-4 \log \left(1+e^{2 i (e+f x)}\right)\right) \cos (e+f x)+\left(2 \log \left(1+e^{i (e+f x)}\right)-\log \left(1+e^{2 i (e+f x)}\right)\right) \cos (2 (e+f x))-4\right)}{2 a^2 f (\cos (e+f x)+1)^2 \sqrt{a (\sec (e+f x)+1)}}","-\frac{c^3 \tan (e+f x) \log (\sec (e+f x)+1)}{a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{c^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{a f (a \sec (e+f x)+a)^{3/2}}+\frac{c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f (a \sec (e+f x)+a)^{5/2}}",1,"(c^2*Cot[(e + f*x)/2]*(-4 + 6*Log[1 + E^(I*(e + f*x))] + Cos[e + f*x]*(8*Log[1 + E^(I*(e + f*x))] - 4*Log[1 + E^((2*I)*(e + f*x))]) + Cos[2*(e + f*x)]*(2*Log[1 + E^(I*(e + f*x))] - Log[1 + E^((2*I)*(e + f*x))]) - 3*Log[1 + E^((2*I)*(e + f*x))])*Sqrt[c - c*Sec[e + f*x]])/(2*a^2*f*(1 + Cos[e + f*x])^2*Sqrt[a*(1 + Sec[e + f*x])])","C",1
146,1,68,42,0.3127891,"\int \frac{\sec (e+f x) (c-c \sec (e+f x))^{3/2}}{(a+a \sec (e+f x))^{5/2}} \, dx","Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^(3/2))/(a + a*Sec[e + f*x])^(5/2),x]","\frac{c \cos (e+f x) \csc \left(\frac{1}{2} (e+f x)\right) \sec ^3\left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)}}{4 a^2 f \sqrt{a (\sec (e+f x)+1)}}","\frac{\tan (e+f x) (c-c \sec (e+f x))^{3/2}}{4 f (a \sec (e+f x)+a)^{5/2}}",1,"(c*Cos[e + f*x]*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]^3*Sqrt[c - c*Sec[e + f*x]])/(4*a^2*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
147,1,71,43,0.2565293,"\int \frac{\sec (e+f x) \sqrt{c-c \sec (e+f x)}}{(a+a \sec (e+f x))^{5/2}} \, dx","Integrate[(Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a + a*Sec[e + f*x])^(5/2),x]","\frac{(2 \cos (e+f x)+1) \csc \left(\frac{1}{2} (e+f x)\right) \sec ^3\left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)}}{8 a^2 f \sqrt{a (\sec (e+f x)+1)}}","\frac{c \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{5/2} \sqrt{c-c \sec (e+f x)}}",1,"((1 + 2*Cos[e + f*x])*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]^3*Sqrt[c - c*Sec[e + f*x]])/(8*a^2*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
148,1,91,140,1.0398942,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} \sqrt{c-c \sec (e+f x)}} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]]),x]","-\frac{\tan (e+f x) \left(3 \cos (e+f x)+8 \cos ^4\left(\frac{1}{2} (e+f x)\right) \tanh ^{-1}\left(e^{i (e+f x)}\right)+2\right)}{4 a^2 f (\cos (e+f x)+1)^2 \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","-\frac{\tan (e+f x) \tanh ^{-1}(\cos (e+f x))}{4 a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{4 a f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2} \sqrt{c-c \sec (e+f x)}}",1,"-1/4*((2 + 8*ArcTanh[E^(I*(e + f*x))]*Cos[(e + f*x)/2]^4 + 3*Cos[e + f*x])*Tan[e + f*x])/(a^2*f*(1 + Cos[e + f*x])^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
149,1,130,146,1.3924798,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2}} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2)),x]","-\frac{\tan (e+f x) \left(2 \cos (e+f x)+5 \cos (2 (e+f x))-3 (\cos (e+f x)-2 \cos (2 (e+f x))-\cos (3 (e+f x))+2) \tanh ^{-1}\left(e^{i (e+f x)}\right)+1\right)}{16 a^2 c f (\cos (e+f x)-1) (\cos (e+f x)+1)^2 \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","\frac{3 \csc (e+f x)}{8 a^2 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{3 \tan (e+f x) \tanh ^{-1}(\cos (e+f x))}{8 a^2 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2} (c-c \sec (e+f x))^{3/2}}",1,"-1/16*((1 + 2*Cos[e + f*x] + 5*Cos[2*(e + f*x)] - 3*ArcTanh[E^(I*(e + f*x))]*(2 + Cos[e + f*x] - 2*Cos[2*(e + f*x)] - Cos[3*(e + f*x)]))*Tan[e + f*x])/(a^2*c*f*(-1 + Cos[e + f*x])*(1 + Cos[e + f*x])^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
150,1,84,160,1.3789111,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{5/2}} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(5/2)),x]","\frac{(1-5 \cos (2 (e+f x))) \csc ^3(e+f x)-12 \tan (e+f x) \tanh ^{-1}\left(e^{i (e+f x)}\right)}{16 a^2 c^2 f \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","\frac{3 \csc (e+f x)}{8 a^2 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\cot ^2(e+f x) \csc (e+f x)}{4 a^2 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{3 \tan (e+f x) \tanh ^{-1}(\cos (e+f x))}{8 a^2 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"((1 - 5*Cos[2*(e + f*x)])*Csc[e + f*x]^3 - 12*ArcTanh[E^(I*(e + f*x))]*Tan[e + f*x])/(16*a^2*c^2*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
151,0,0,101,1.3467182,"\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^n \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^n,x]","\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^n \, dx","-\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (1-\sec (e+f x))^{\frac{1}{2}-n} (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{n-1} \, _2F_1\left(m+\frac{1}{2},\frac{1}{2}-n;m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1)\right)}{f (2 m+1)}",1,"Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^n, x]","F",-1
152,1,89,92,0.2483727,"\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^2 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^2,x]","\frac{c^2 2^{m+\frac{1}{2}} \tan (e+f x) (\sec (e+f x)-1)^2 (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a (\sec (e+f x)+1))^m \, _2F_1\left(\frac{5}{2},\frac{1}{2}-m;\frac{7}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{5 f}","\frac{a 2^{m+\frac{1}{2}} \tan (e+f x) (c-c \sec (e+f x))^2 (\sec (e+f x)+1)^{\frac{1}{2}-m} (a \sec (e+f x)+a)^{m-1} \, _2F_1\left(\frac{5}{2},\frac{1}{2}-m;\frac{7}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{5 f}",1,"(2^(1/2 + m)*c^2*Hypergeometric2F1[5/2, 1/2 - m, 7/2, (1 - Sec[e + f*x])/2]*(-1 + Sec[e + f*x])^2*(1 + Sec[e + f*x])^(-1/2 - m)*(a*(1 + Sec[e + f*x]))^m*Tan[e + f*x])/(5*f)","A",1
153,1,85,90,0.1583216,"\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x)) \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x]),x]","-\frac{c 2^{m+\frac{1}{2}} \tan (e+f x) (\sec (e+f x)-1) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a (\sec (e+f x)+1))^m \, _2F_1\left(\frac{3}{2},\frac{1}{2}-m;\frac{5}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{3 f}","\frac{a 2^{m+\frac{1}{2}} \tan (e+f x) (c-c \sec (e+f x)) (\sec (e+f x)+1)^{\frac{1}{2}-m} (a \sec (e+f x)+a)^{m-1} \, _2F_1\left(\frac{3}{2},\frac{1}{2}-m;\frac{5}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{3 f}",1,"-1/3*(2^(1/2 + m)*c*Hypergeometric2F1[3/2, 1/2 - m, 5/2, (1 - Sec[e + f*x])/2]*(-1 + Sec[e + f*x])*(1 + Sec[e + f*x])^(-1/2 - m)*(a*(1 + Sec[e + f*x]))^m*Tan[e + f*x])/f","A",1
154,0,0,90,0.5692029,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^m}{c-c \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^m)/(c - c*Sec[e + f*x]),x]","\int \frac{\sec (e+f x) (a+a \sec (e+f x))^m}{c-c \sec (e+f x)} \, dx","-\frac{a 2^{m+\frac{1}{2}} \tan (e+f x) (\sec (e+f x)+1)^{\frac{1}{2}-m} (a \sec (e+f x)+a)^{m-1} \, _2F_1\left(-\frac{1}{2},\frac{1}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f (c-c \sec (e+f x))}",1,"Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^m)/(c - c*Sec[e + f*x]), x]","F",-1
155,0,0,92,2.5907042,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^m}{(c-c \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^m)/(c - c*Sec[e + f*x])^2,x]","\int \frac{\sec (e+f x) (a+a \sec (e+f x))^m}{(c-c \sec (e+f x))^2} \, dx","-\frac{a 2^{m+\frac{1}{2}} \tan (e+f x) (\sec (e+f x)+1)^{\frac{1}{2}-m} (a \sec (e+f x)+a)^{m-1} \, _2F_1\left(-\frac{3}{2},\frac{1}{2}-m;-\frac{1}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{3 f (c-c \sec (e+f x))^2}",1,"Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^m)/(c - c*Sec[e + f*x])^2, x]","F",-1
156,0,0,160,15.6464076,"\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{5/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(5/2),x]","\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{5/2} \, dx","-\frac{64 c^3 \tan (e+f x) (a \sec (e+f x)+a)^m}{f (2 m+5) \left(4 m^2+8 m+3\right) \sqrt{c-c \sec (e+f x)}}-\frac{16 c^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)} (a \sec (e+f x)+a)^m}{f \left(4 m^2+16 m+15\right)}-\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2} (a \sec (e+f x)+a)^m}{f (2 m+5)}",1,"Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(5/2), x]","F",-1
157,0,0,100,38.4363636,"\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{3/2} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(3/2),x]","\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{3/2} \, dx","-\frac{8 c^2 \tan (e+f x) (a \sec (e+f x)+a)^m}{f \left(4 m^2+8 m+3\right) \sqrt{c-c \sec (e+f x)}}-\frac{2 c \tan (e+f x) \sqrt{c-c \sec (e+f x)} (a \sec (e+f x)+a)^m}{f (2 m+3)}",1,"Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(3/2), x]","F",-1
158,1,163,46,18.8951003,"\int \sec (e+f x) (a+a \sec (e+f x))^m \sqrt{c-c \sec (e+f x)} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*Sqrt[c - c*Sec[e + f*x]],x]","\frac{\sqrt{2} e^{-\frac{1}{2} i (e+f x)} \left(1+e^{i (e+f x)}\right) \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \left(\frac{\left(1+e^{i (e+f x)}\right)^2}{1+e^{2 i (e+f x)}}\right)^m \csc \left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)} (\sec (e+f x)+1)^{-m} (a (\sec (e+f x)+1))^m}{(2 f m+f) \sqrt{\sec (e+f x)}}","-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^m}{f (2 m+1) \sqrt{c-c \sec (e+f x)}}",1,"(Sqrt[2]*(1 + E^(I*(e + f*x)))*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*((1 + E^(I*(e + f*x)))^2/(1 + E^((2*I)*(e + f*x))))^m*Csc[(e + f*x)/2]*(a*(1 + Sec[e + f*x]))^m*Sqrt[c - c*Sec[e + f*x]])/(E^((I/2)*(e + f*x))*(f + 2*f*m)*Sqrt[Sec[e + f*x]]*(1 + Sec[e + f*x])^m)","C",1
159,0,0,69,0.6532839,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^m}{\sqrt{c-c \sec (e+f x)}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^m)/Sqrt[c - c*Sec[e + f*x]],x]","\int \frac{\sec (e+f x) (a+a \sec (e+f x))^m}{\sqrt{c-c \sec (e+f x)}} \, dx","-\frac{\tan (e+f x) (a \sec (e+f x)+a)^m \, _2F_1\left(1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1)\right)}{f (2 m+1) \sqrt{c-c \sec (e+f x)}}",1,"Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^m)/Sqrt[c - c*Sec[e + f*x]], x]","F",-1
160,0,0,74,0.7134887,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^m}{(c-c \sec (e+f x))^{3/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^m)/(c - c*Sec[e + f*x])^(3/2),x]","\int \frac{\sec (e+f x) (a+a \sec (e+f x))^m}{(c-c \sec (e+f x))^{3/2}} \, dx","-\frac{\tan (e+f x) (a \sec (e+f x)+a)^m \, _2F_1\left(2,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1)\right)}{2 c f (2 m+1) \sqrt{c-c \sec (e+f x)}}",1,"Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^m)/(c - c*Sec[e + f*x])^(3/2), x]","F",-1
161,0,0,74,2.1423832,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^m}{(c-c \sec (e+f x))^{5/2}} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^m)/(c - c*Sec[e + f*x])^(5/2),x]","\int \frac{\sec (e+f x) (a+a \sec (e+f x))^m}{(c-c \sec (e+f x))^{5/2}} \, dx","-\frac{\tan (e+f x) (a \sec (e+f x)+a)^m \, _2F_1\left(3,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1)\right)}{4 c^2 f (2 m+1) \sqrt{c-c \sec (e+f x)}}",1,"Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^m)/(c - c*Sec[e + f*x])^(5/2), x]","F",-1
162,1,321,169,9.0848175,"\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-3-m} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-3 - m),x]","-\frac{i 2^{m+3} \left(1+e^{i (e+f x)}\right) \left(-i e^{-\frac{1}{2} i (e+f x)} \left(-1+e^{i (e+f x)}\right)\right)^{-2 m} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{-m} \left(\frac{\left(1+e^{i (e+f x)}\right)^2}{1+e^{2 i (e+f x)}}\right)^m \left(\left(4 m^2+12 m+7\right) e^{4 i (e+f x)}+\left(8 m^2+24 m+22\right) e^{2 i (e+f x)}-4 (2 m+3) e^{i (e+f x)}-4 (2 m+3) e^{3 i (e+f x)}+4 m^2+12 m+7\right) \sin ^{-2 (-m-3)}\left(\frac{e}{2}+\frac{f x}{2}\right) \sec ^{m+3}(e+f x) (\sec (e+f x)+1)^{-m} (a (\sec (e+f x)+1))^m (c-c \sec (e+f x))^{-m-3}}{f (2 m+1) (2 m+3) (2 m+5) \left(-1+e^{i (e+f x)}\right)^5}","-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^{m+2} (c-c \sec (e+f x))^{-m-3}}{a^2 f (2 m+1) \left(4 m^2+16 m+15\right)}+\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^{m+1} (c-c \sec (e+f x))^{-m-3}}{a f \left(4 m^2+8 m+3\right)}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-3}}{f (2 m+1)}",1,"((-I)*2^(3 + m)*(1 + E^(I*(e + f*x)))*((1 + E^(I*(e + f*x)))^2/(1 + E^((2*I)*(e + f*x))))^m*(7 + 12*m + 4*m^2 - 4*E^(I*(e + f*x))*(3 + 2*m) - 4*E^((3*I)*(e + f*x))*(3 + 2*m) + E^((4*I)*(e + f*x))*(7 + 12*m + 4*m^2) + E^((2*I)*(e + f*x))*(22 + 24*m + 8*m^2))*Sec[e + f*x]^(3 + m)*(a*(1 + Sec[e + f*x]))^m*(c - c*Sec[e + f*x])^(-3 - m))/((-1 + E^(I*(e + f*x)))^5*(((-I)*(-1 + E^(I*(e + f*x))))/E^((I/2)*(e + f*x)))^(2*m)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*f*(1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(1 + Sec[e + f*x])^m*Sin[e/2 + (f*x)/2]^(2*(-3 - m)))","C",0
163,1,250,104,3.0499337,"\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-2-m} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-2 - m),x]","\frac{i 2^{m+3} \left(1+e^{i (e+f x)}\right) \left(-i e^{-\frac{1}{2} i (e+f x)} \left(-1+e^{i (e+f x)}\right)\right)^{-2 m} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{-m} \left(\frac{\left(1+e^{i (e+f x)}\right)^2}{1+e^{2 i (e+f x)}}\right)^m \left((m+1) e^{2 i (e+f x)}-e^{i (e+f x)}+m+1\right) \sin ^{2 (m+2)}\left(\frac{1}{2} (e+f x)\right) \sec ^{m+2}(e+f x) (\sec (e+f x)+1)^{-m} (a (\sec (e+f x)+1))^m (c-c \sec (e+f x))^{-m-2}}{f (2 m+1) (2 m+3) \left(-1+e^{i (e+f x)}\right)^3}","\frac{\tan (e+f x) (a \sec (e+f x)+a)^{m+1} (c-c \sec (e+f x))^{-m-2}}{a f \left(4 m^2+8 m+3\right)}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-2}}{f (2 m+1)}",1,"(I*2^(3 + m)*(1 + E^(I*(e + f*x)))*((1 + E^(I*(e + f*x)))^2/(1 + E^((2*I)*(e + f*x))))^m*(1 - E^(I*(e + f*x)) + m + E^((2*I)*(e + f*x))*(1 + m))*Sec[e + f*x]^(2 + m)*(a*(1 + Sec[e + f*x]))^m*(c - c*Sec[e + f*x])^(-2 - m)*Sin[(e + f*x)/2]^(2*(2 + m)))/((-1 + E^(I*(e + f*x)))^3*(((-I)*(-1 + E^(I*(e + f*x))))/E^((I/2)*(e + f*x)))^(2*m)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*f*(1 + 2*m)*(3 + 2*m)*(1 + Sec[e + f*x])^m)","C",0
164,1,208,47,1.1784831,"\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-1-m} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-1 - m),x]","-\frac{2^{m+1} e^{-\frac{1}{2} i (e+f x)} \left(1+e^{i (e+f x)}\right) \left(-i e^{-\frac{1}{2} i (e+f x)} \left(-1+e^{i (e+f x)}\right)\right)^{-2 m-1} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{-m} \left(\frac{\left(1+e^{i (e+f x)}\right)^2}{1+e^{2 i (e+f x)}}\right)^m \sin ^{2 (m+1)}\left(\frac{1}{2} (e+f x)\right) \sec ^{m+1}(e+f x) (\sec (e+f x)+1)^{-m} (a (\sec (e+f x)+1))^m (c-c \sec (e+f x))^{-m-1}}{2 f m+f}","-\frac{\tan (e+f x) (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-1}}{f (2 m+1)}",1,"-((2^(1 + m)*(((-I)*(-1 + E^(I*(e + f*x))))/E^((I/2)*(e + f*x)))^(-1 - 2*m)*(1 + E^(I*(e + f*x)))*((1 + E^(I*(e + f*x)))^2/(1 + E^((2*I)*(e + f*x))))^m*Sec[e + f*x]^(1 + m)*(a*(1 + Sec[e + f*x]))^m*(c - c*Sec[e + f*x])^(-1 - m)*Sin[(e + f*x)/2]^(2*(1 + m)))/(E^((I/2)*(e + f*x))*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*(f + 2*f*m)*(1 + Sec[e + f*x])^m))","C",0
165,1,257,101,1.4197458,"\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-m} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^m)/(c - c*Sec[e + f*x])^m,x]","\frac{2^{m-1} \left(-i e^{-\frac{1}{2} i (e+f x)} \left(-1+e^{i (e+f x)}\right)\right)^{-2 m} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{-m} \left(\frac{\left(1+e^{i (e+f x)}\right)^2}{1+e^{2 i (e+f x)}}\right)^m \left(\, _2F_1\left(1,-2 m;1-2 m;\frac{i \left(-1+e^{i (e+f x)}\right)}{1+e^{i (e+f x)}}\right)-\, _2F_1\left(1,-2 m;1-2 m;-\frac{i \left(-1+e^{i (e+f x)}\right)}{1+e^{i (e+f x)}}\right)\right) \sin ^{2 m}\left(\frac{1}{2} (e+f x)\right) \left(\frac{\sec (e+f x)}{\sec (e+f x)+1}\right)^m (a (\sec (e+f x)+1))^m (c-c \sec (e+f x))^{-m}}{f m}","-\frac{c 2^{\frac{1}{2}-m} \tan (e+f x) (1-\sec (e+f x))^{m+\frac{1}{2}} (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-1} \, _2F_1\left(m+\frac{1}{2},m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1)\right)}{f (2 m+1)}",1,"(2^(-1 + m)*((1 + E^(I*(e + f*x)))^2/(1 + E^((2*I)*(e + f*x))))^m*(-Hypergeometric2F1[1, -2*m, 1 - 2*m, ((-I)*(-1 + E^(I*(e + f*x))))/(1 + E^(I*(e + f*x)))] + Hypergeometric2F1[1, -2*m, 1 - 2*m, (I*(-1 + E^(I*(e + f*x))))/(1 + E^(I*(e + f*x)))])*(Sec[e + f*x]/(1 + Sec[e + f*x]))^m*(a*(1 + Sec[e + f*x]))^m*Sin[(e + f*x)/2]^(2*m))/((((-I)*(-1 + E^(I*(e + f*x))))/E^((I/2)*(e + f*x)))^(2*m)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*f*m*(c - c*Sec[e + f*x])^m)","C",1
166,0,0,99,2.2624121,"\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{1-m} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(1 - m),x]","\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{1-m} \, dx","-\frac{c 2^{\frac{3}{2}-m} \tan (e+f x) (1-\sec (e+f x))^{m-\frac{1}{2}} (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m} \, _2F_1\left(m-\frac{1}{2},m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1)\right)}{f (2 m+1)}",1,"Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(1 - m), x]","F",-1
167,0,0,101,2.7080952,"\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{2-m} \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(2 - m),x]","\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{2-m} \, dx","-\frac{c^2 2^{\frac{5}{2}-m} \tan (e+f x) (1-\sec (e+f x))^{m-\frac{1}{2}} (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m} \, _2F_1\left(m-\frac{3}{2},m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1)\right)}{f (2 m+1)}",1,"Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(2 - m), x]","F",-1
168,1,68,105,0.2990905,"\int \sec ^2(e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx","Integrate[Sec[e + f*x]^2*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x]),x]","\frac{a^3 c \left(15 \tanh ^{-1}(\sin (e+f x))-\tan (e+f x) \left(12 \tan ^4(e+f x)+40 \tan ^2(e+f x)+30 \sec ^3(e+f x)-15 \sec (e+f x)\right)\right)}{60 f}","-\frac{a^3 c \tan ^5(e+f x)}{5 f}-\frac{2 a^3 c \tan ^3(e+f x)}{3 f}+\frac{a^3 c \tanh ^{-1}(\sin (e+f x))}{4 f}-\frac{a^3 c \tan (e+f x) \sec ^3(e+f x)}{2 f}+\frac{a^3 c \tan (e+f x) \sec (e+f x)}{4 f}",1,"(a^3*c*(15*ArcTanh[Sin[e + f*x]] - Tan[e + f*x]*(-15*Sec[e + f*x] + 30*Sec[e + f*x]^3 + 40*Tan[e + f*x]^2 + 12*Tan[e + f*x]^4)))/(60*f)","A",1
169,1,57,86,0.1789163,"\int \sec ^2(e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x)) \, dx","Integrate[Sec[e + f*x]^2*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x]),x]","\frac{a^2 c \left(3 \tanh ^{-1}(\sin (e+f x))+\tan (e+f x) \left(-8 \tan ^2(e+f x)-6 \sec ^3(e+f x)+3 \sec (e+f x)\right)\right)}{24 f}","-\frac{a^2 c \tan ^3(e+f x)}{3 f}+\frac{a^2 c \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac{a^2 c \tan (e+f x) \sec ^3(e+f x)}{4 f}+\frac{a^2 c \tan (e+f x) \sec (e+f x)}{8 f}",1,"(a^2*c*(3*ArcTanh[Sin[e + f*x]] + Tan[e + f*x]*(3*Sec[e + f*x] - 6*Sec[e + f*x]^3 - 8*Tan[e + f*x]^2)))/(24*f)","A",1
170,1,17,17,0.0175412,"\int \sec ^2(e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x)) \, dx","Integrate[Sec[e + f*x]^2*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x]),x]","-\frac{a c \tan ^3(e+f x)}{3 f}","-\frac{a c \tan ^3(e+f x)}{3 f}",1,"-1/3*(a*c*Tan[e + f*x]^3)/f","A",1
171,1,154,56,0.6191747,"\int \frac{\sec ^2(e+f x) (c-c \sec (e+f x))}{a+a \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]^2*(c - c*Sec[e + f*x]))/(a + a*Sec[e + f*x]),x]","-\frac{c \left(\frac{2 \tan \left(\frac{1}{2} (e+f x)\right)}{f}+\frac{\sin \left(\frac{1}{2} (e+f x)\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{\sin \left(\frac{1}{2} (e+f x)\right)}{f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f}-\frac{2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f}\right)}{a}","-\frac{c \tan (e+f x)}{a f}+\frac{2 c \tanh ^{-1}(\sin (e+f x))}{a f}-\frac{2 c \tan (e+f x)}{f (a \sec (e+f x)+a)}",1,"-((c*((2*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]])/f - (2*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])/f + Sin[(e + f*x)/2]/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])) + Sin[(e + f*x)/2]/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) + (2*Tan[(e + f*x)/2])/f))/a)","B",1
172,1,335,70,0.4556563,"\int \frac{\sec ^2(e+f x) (c-c \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]^2*(c - c*Sec[e + f*x]))/(a + a*Sec[e + f*x])^2,x]","\frac{c \sec \left(\frac{e}{2}\right) \cos \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \left(-6 \sin \left(e+\frac{f x}{2}\right)+10 \sin \left(e+\frac{3 f x}{2}\right)+3 \cos \left(e+\frac{3 f x}{2}\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+3 \cos \left(2 e+\frac{3 f x}{2}\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+9 \cos \left(\frac{f x}{2}\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+9 \cos \left(e+\frac{f x}{2}\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-3 \cos \left(e+\frac{3 f x}{2}\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-3 \cos \left(2 e+\frac{3 f x}{2}\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+24 \sin \left(\frac{f x}{2}\right)\right)}{6 a^2 f (\sec (e+f x)+1)^2}","-\frac{c \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac{7 c \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{2 c \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2}",1,"(c*Cos[(e + f*x)/2]*Sec[e/2]*Sec[e + f*x]^2*(3*Cos[e + (3*f*x)/2]*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 3*Cos[2*e + (3*f*x)/2]*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 9*Cos[(f*x)/2]*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + 9*Cos[e + (f*x)/2]*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - 3*Cos[e + (3*f*x)/2]*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - 3*Cos[2*e + (3*f*x)/2]*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 24*Sin[(f*x)/2] - 6*Sin[e + (f*x)/2] + 10*Sin[e + (3*f*x)/2]))/(6*a^2*f*(1 + Sec[e + f*x])^2)","B",1
173,1,43,86,0.173128,"\int \frac{\sec ^2(e+f x) (c-c \sec (e+f x))}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]^2*(c - c*Sec[e + f*x]))/(a + a*Sec[e + f*x])^3,x]","-\frac{c (\cos (e+f x)+4) \tan ^3\left(\frac{1}{2} (e+f x)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{30 a^3 f}","-\frac{4 c \tan (e+f x)}{15 f \left(a^3 \sec (e+f x)+a^3\right)}+\frac{11 c \tan (e+f x)}{15 a f (a \sec (e+f x)+a)^2}-\frac{2 c \tan (e+f x)}{5 f (a \sec (e+f x)+a)^3}",1,"-1/30*(c*(4 + Cos[e + f*x])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]^3)/(a^3*f)","A",1
174,1,7087,140,30.6140601,"\int (g \sec (e+f x))^p (a+a \sec (e+f x))^2 (c-c \sec (e+f x)) \, dx","Integrate[(g*Sec[e + f*x])^p*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x]),x]","\text{Result too large to show}","-\frac{a^2 c \tan ^3(e+f x) \cos ^2(e+f x)^{\frac{p+3}{2}} (g \sec (e+f x))^p \, _2F_1\left(\frac{3}{2},\frac{p+3}{2};\frac{5}{2};\sin ^2(e+f x)\right)}{3 f}-\frac{a^2 c \tan ^3(e+f x) \cos ^2(e+f x)^{\frac{p+4}{2}} (g \sec (e+f x))^{p+1} \, _2F_1\left(\frac{3}{2},\frac{p+4}{2};\frac{5}{2};\sin ^2(e+f x)\right)}{3 f g}",1,"Result too large to show","C",0
175,1,72,65,0.3344937,"\int (g \sec (e+f x))^p (a+a \sec (e+f x)) (c-c \sec (e+f x)) \, dx","Integrate[(g*Sec[e + f*x])^p*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x]),x]","-\frac{a c \tan (e+f x) (g \sec (e+f x))^p \left(\frac{\, _2F_1\left(\frac{1}{2},\frac{p}{2};\frac{p+2}{2};\sec ^2(e+f x)\right)}{\sqrt{-\tan ^2(e+f x)}}+p\right)}{f p (p+1)}","-\frac{a c \tan ^3(e+f x) \cos ^2(e+f x)^{\frac{p+3}{2}} (g \sec (e+f x))^p \, _2F_1\left(\frac{3}{2},\frac{p+3}{2};\frac{5}{2};\sin ^2(e+f x)\right)}{3 f}",1,"-((a*c*(g*Sec[e + f*x])^p*Tan[e + f*x]*(p + Hypergeometric2F1[1/2, p/2, (2 + p)/2, Sec[e + f*x]^2]/Sqrt[-Tan[e + f*x]^2]))/(f*p*(1 + p)))","A",1
176,1,3396,180,16.8076296,"\int \frac{(g \sec (e+f x))^p (c-c \sec (e+f x))}{a+a \sec (e+f x)} \, dx","Integrate[((g*Sec[e + f*x])^p*(c - c*Sec[e + f*x]))/(a + a*Sec[e + f*x]),x]","\text{Result too large to show}","-\frac{c g (1-2 p) \sin (e+f x) (g \sec (e+f x))^{p-1} \, _2F_1\left(\frac{1}{2},\frac{1-p}{2};\frac{3-p}{2};\cos ^2(e+f x)\right)}{a f (1-p) \sqrt{\sin ^2(e+f x)}}+\frac{2 c \sin (e+f x) (g \sec (e+f x))^p \, _2F_1\left(\frac{1}{2},-\frac{p}{2};\frac{2-p}{2};\cos ^2(e+f x)\right)}{a f \sqrt{\sin ^2(e+f x)}}-\frac{2 c \tan (e+f x) (g \sec (e+f x))^p}{f (a \sec (e+f x)+a)}",1,"(-6*c*Sec[e + f*x]^p*(g*Sec[e + f*x])^p*Tan[(e + f*x)/2]^3*(-((AppellF1[1/2, p, 1 - p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2)/(3*AppellF1[1/2, p, 1 - p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + p)*AppellF1[3/2, p, 2 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + p*AppellF1[3/2, 1 + p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + AppellF1[1/2, p, -p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]/(3*AppellF1[1/2, p, -p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*p*(AppellF1[3/2, p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + p, -p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))/(a*f*(3*Sec[(e + f*x)/2]^2*Sec[e + f*x]^p*(-((AppellF1[1/2, p, 1 - p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2)/(3*AppellF1[1/2, p, 1 - p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + p)*AppellF1[3/2, p, 2 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + p*AppellF1[3/2, 1 + p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + AppellF1[1/2, p, -p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]/(3*AppellF1[1/2, p, -p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*p*(AppellF1[3/2, p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + p, -p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + 6*p*Sec[e + f*x]^(1 + p)*Sin[e + f*x]*Tan[(e + f*x)/2]*(-((AppellF1[1/2, p, 1 - p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2)/(3*AppellF1[1/2, p, 1 - p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + p)*AppellF1[3/2, p, 2 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + p*AppellF1[3/2, 1 + p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + AppellF1[1/2, p, -p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]/(3*AppellF1[1/2, p, -p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*p*(AppellF1[3/2, p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + p, -p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + 6*Sec[e + f*x]^p*Tan[(e + f*x)/2]*((AppellF1[1/2, p, 1 - p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]*Sin[(e + f*x)/2])/(3*AppellF1[1/2, p, 1 - p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + p)*AppellF1[3/2, p, 2 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + p*AppellF1[3/2, 1 + p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (Cos[(e + f*x)/2]^2*(-1/3*((1 - p)*AppellF1[3/2, p, 2 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (p*AppellF1[3/2, 1 + p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, p, 1 - p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + p)*AppellF1[3/2, p, 2 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + p*AppellF1[3/2, 1 + p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + ((p*AppellF1[3/2, p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + (p*AppellF1[3/2, 1 + p, -p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3)/(3*AppellF1[1/2, p, -p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*p*(AppellF1[3/2, p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + p, -p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (AppellF1[1/2, p, -p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(2*p*(AppellF1[3/2, p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + p, -p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*((p*AppellF1[3/2, p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + (p*AppellF1[3/2, 1 + p, -p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*p*Tan[(e + f*x)/2]^2*((-3*(1 - p)*AppellF1[5/2, p, 2 - p, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (6*p*AppellF1[5/2, 1 + p, 1 - p, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + p)*AppellF1[5/2, 2 + p, -p, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5)))/(3*AppellF1[1/2, p, -p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*p*(AppellF1[3/2, p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + p, -p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (AppellF1[1/2, p, 1 - p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2*(2*((-1 + p)*AppellF1[3/2, p, 2 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + p*AppellF1[3/2, 1 + p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - p)*AppellF1[3/2, p, 2 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (p*AppellF1[3/2, 1 + p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*Tan[(e + f*x)/2]^2*((-1 + p)*((-3*(2 - p)*AppellF1[5/2, p, 3 - p, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*p*AppellF1[5/2, 1 + p, 2 - p, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + p*((-3*(1 - p)*AppellF1[5/2, 1 + p, 2 - p, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + p)*AppellF1[5/2, 2 + p, 1 - p, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, p, 1 - p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + p)*AppellF1[3/2, p, 2 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + p*AppellF1[3/2, 1 + p, 1 - p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2)))","C",0
177,0,0,226,13.3287943,"\int \frac{(g \sec (e+f x))^p (c-c \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx","Integrate[((g*Sec[e + f*x])^p*(c - c*Sec[e + f*x]))/(a + a*Sec[e + f*x])^2,x]","\int \frac{(g \sec (e+f x))^p (c-c \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx","-\frac{c g (3-4 p) \sin (e+f x) (g \sec (e+f x))^{p-1} \, _2F_1\left(\frac{1}{2},\frac{1-p}{2};\frac{3-p}{2};\cos ^2(e+f x)\right)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}+\frac{c (5-4 p) \sin (e+f x) (g \sec (e+f x))^p \, _2F_1\left(\frac{1}{2},-\frac{p}{2};\frac{2-p}{2};\cos ^2(e+f x)\right)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{c (5-4 p) \tan (e+f x) (g \sec (e+f x))^p}{3 a^2 f (\sec (e+f x)+1)}-\frac{2 c \tan (e+f x) (g \sec (e+f x))^p}{3 f (a \sec (e+f x)+a)^2}",1,"Integrate[((g*Sec[e + f*x])^p*(c - c*Sec[e + f*x]))/(a + a*Sec[e + f*x])^2, x]","F",-1
178,1,162,104,1.5342173,"\int \frac{(g \sec (e+f x))^{3/2} \sqrt{a+a \sec (e+f x)}}{c-c \sec (e+f x)} \, dx","Integrate[((g*Sec[e + f*x])^(3/2)*Sqrt[a + a*Sec[e + f*x]])/(c - c*Sec[e + f*x]),x]","\frac{2 \cot \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} (g \sec (e+f x))^{3/2} \left(\sqrt{\sec (e+f x)} \sqrt{\sec (e+f x)+1}+\sqrt{\tan ^2(e+f x)} \left(\log (\sec (e+f x)+1)-\log \left(\sec ^{\frac{3}{2}}(e+f x)+\sqrt{\sec (e+f x)}+\sqrt{\tan ^2(e+f x)} \sqrt{\sec (e+f x)+1}\right)\right)\right)}{c f \sec ^{\frac{3}{2}}(e+f x) (\sec (e+f x)+1)^{3/2}}","\frac{2 g \cot (e+f x) \sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}{c f}-\frac{2 \sqrt{a} g^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{g} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}\right)}{c f}",1,"(2*Cot[(e + f*x)/2]*(g*Sec[e + f*x])^(3/2)*Sqrt[a*(1 + Sec[e + f*x])]*(Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]] + (Log[1 + Sec[e + f*x]] - Log[Sqrt[Sec[e + f*x]] + Sec[e + f*x]^(3/2) + Sqrt[1 + Sec[e + f*x]]*Sqrt[Tan[e + f*x]^2]])*Sqrt[Tan[e + f*x]^2]))/(c*f*Sec[e + f*x]^(3/2)*(1 + Sec[e + f*x])^(3/2))","A",0
179,1,73,81,0.4714365,"\int \frac{\sec ^2(e+f x)}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]^2/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])),x]","-\frac{\cot \left(\frac{1}{2} (e+f x)\right) \left(\sqrt{2} \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)-2\right)}{2 c f \sqrt{a (\sec (e+f x)+1)}}","\frac{\cot (e+f x) \sqrt{a \sec (e+f x)+a}}{a c f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{2} \sqrt{a} c f}",1,"-1/2*(Cot[(e + f*x)/2]*(-2 + Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Sqrt[-1 + Sec[e + f*x]]))/(c*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
180,1,724,140,6.4647238,"\int \frac{\sec ^{\frac{5}{2}}(e+f x)}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]^(5/2)/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])),x]","\frac{\sin (e+f x) \sin ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \cos (e+f x) (\sec (e+f x)+1)^{3/2} \sqrt{\sec ^2(e+f x)-1} \left(\log \left(-3 \sec ^2(e+f x)-2 \sqrt{2} \sqrt{\sec (e+f x)+1} \sqrt{\sec ^2(e+f x)-1} \sqrt{\sec (e+f x)}-2 \sec (e+f x)+1\right)-\log \left(-3 \sec ^2(e+f x)+2 \sqrt{2} \sqrt{\sec (e+f x)+1} \sqrt{\sec ^2(e+f x)-1} \sqrt{\sec (e+f x)}-2 \sec (e+f x)+1\right)\right)}{2 f (\cos (e+f x)+1) \sqrt{2-2 \cos ^2(e+f x)} \sqrt{1-\cos ^2(e+f x)} \sqrt{a (\sec (e+f x)+1)} (c-c \sec (e+f x))}+\frac{\sin (e+f x) \sin ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \cos (e+f x) (\sec (e+f x)+1)^{3/2} \sqrt{\sec ^2(e+f x)-1} \left(\sqrt{2} \left(\log \left(-3 \sec ^2(e+f x)+2 \sqrt{2} \sqrt{\sec (e+f x)+1} \sqrt{\sec ^2(e+f x)-1} \sqrt{\sec (e+f x)}-2 \sec (e+f x)+1\right)-\log \left(-3 \sec ^2(e+f x)-2 \sqrt{2} \sqrt{\sec (e+f x)+1} \sqrt{\sec ^2(e+f x)-1} \sqrt{\sec (e+f x)}-2 \sec (e+f x)+1\right)\right)+8 \log \left(\sec ^{\frac{3}{2}}(e+f x)+\sqrt{\sec (e+f x)+1} \sqrt{\sec ^2(e+f x)-1}+\sqrt{\sec (e+f x)}\right)-8 \log (\sec (e+f x)+1)\right)}{2 f (\cos (e+f x)+1) \left(1-\cos ^2(e+f x)\right) \sqrt{a (\sec (e+f x)+1)} (c-c \sec (e+f x))}+\frac{\sin ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \sec ^{\frac{3}{2}}(e+f x) \sqrt{\sec (e+f x)+1} \sqrt{(\cos (e+f x)+1) \sec (e+f x)} \left(\frac{\csc \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \csc \left(\frac{e}{2}+\frac{f x}{2}\right)}{f}+\frac{\sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sec \left(\frac{e}{2}+\frac{f x}{2}\right)}{f}-\frac{2 \cot (e)}{f}\right)}{\sqrt{a (\sec (e+f x)+1)} (c-c \sec (e+f x))}","\frac{\csc (e+f x) \sqrt{a \sec (e+f x)+a}}{a c f \sqrt{\sec (e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x) \sqrt{\sec (e+f x)}}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{2} \sqrt{a} c f}-\frac{2 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} c f}",1,"(Sec[e + f*x]^(3/2)*Sqrt[(1 + Cos[e + f*x])*Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*((-2*Cot[e])/f + (Csc[e/2]*Csc[e/2 + (f*x)/2]*Sin[(f*x)/2])/f + (Sec[e/2]*Sec[e/2 + (f*x)/2]*Sin[(f*x)/2])/f)*Sin[e/2 + (f*x)/2]^2)/(Sqrt[a*(1 + Sec[e + f*x])]*(c - c*Sec[e + f*x])) + (Cos[e + f*x]*(Log[1 - 2*Sec[e + f*x] - 3*Sec[e + f*x]^2 - 2*Sqrt[2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*Sqrt[-1 + Sec[e + f*x]^2]] - Log[1 - 2*Sec[e + f*x] - 3*Sec[e + f*x]^2 + 2*Sqrt[2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*Sqrt[-1 + Sec[e + f*x]^2]])*(1 + Sec[e + f*x])^(3/2)*Sqrt[-1 + Sec[e + f*x]^2]*Sin[e/2 + (f*x)/2]^2*Sin[e + f*x])/(2*f*(1 + Cos[e + f*x])*Sqrt[2 - 2*Cos[e + f*x]^2]*Sqrt[1 - Cos[e + f*x]^2]*Sqrt[a*(1 + Sec[e + f*x])]*(c - c*Sec[e + f*x])) + (Cos[e + f*x]*(-8*Log[1 + Sec[e + f*x]] + 8*Log[Sqrt[Sec[e + f*x]] + Sec[e + f*x]^(3/2) + Sqrt[1 + Sec[e + f*x]]*Sqrt[-1 + Sec[e + f*x]^2]] + Sqrt[2]*(-Log[1 - 2*Sec[e + f*x] - 3*Sec[e + f*x]^2 - 2*Sqrt[2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*Sqrt[-1 + Sec[e + f*x]^2]] + Log[1 - 2*Sec[e + f*x] - 3*Sec[e + f*x]^2 + 2*Sqrt[2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*Sqrt[-1 + Sec[e + f*x]^2]]))*(1 + Sec[e + f*x])^(3/2)*Sqrt[-1 + Sec[e + f*x]^2]*Sin[e/2 + (f*x)/2]^2*Sin[e + f*x])/(2*f*(1 + Cos[e + f*x])*(1 - Cos[e + f*x]^2)*Sqrt[a*(1 + Sec[e + f*x])]*(c - c*Sec[e + f*x]))","B",0
181,1,236,116,2.9821404,"\int \frac{(g \sec (e+f x))^{3/2}}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx","Integrate[(g*Sec[e + f*x])^(3/2)/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])),x]","-\frac{a \sin ^3\left(\frac{1}{2} (e+f x)\right) \cos \left(\frac{1}{2} (e+f x)\right) (g \sec (e+f x))^{5/2} \left(-4 \sec (e+f x)+\frac{\sqrt{\tan ^2(e+f x)} \left(\log \left(-3 \sec ^2(e+f x)-2 \sec (e+f x)-2 \sqrt{2} \sqrt{\tan ^2(e+f x)} \sqrt{\sec (e+f x)+1} \sqrt{\sec (e+f x)}+1\right)-\log \left(-3 \sec ^2(e+f x)-2 \sec (e+f x)+2 \sqrt{2} \sqrt{\tan ^2(e+f x)} \sqrt{\sec (e+f x)+1} \sqrt{\sec (e+f x)}+1\right)\right)}{\sqrt{\sec ^2\left(\frac{1}{2} (e+f x)\right)}}-4\right)}{c f g (\sec (e+f x)-1)^2 (a (\sec (e+f x)+1))^{3/2}}","\frac{g \cot (e+f x) \sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}{a c f}-\frac{g^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{g} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}\right)}{\sqrt{2} \sqrt{a} c f}",1,"-((a*Cos[(e + f*x)/2]*(g*Sec[e + f*x])^(5/2)*Sin[(e + f*x)/2]^3*(-4 - 4*Sec[e + f*x] + ((Log[1 - 2*Sec[e + f*x] - 3*Sec[e + f*x]^2 - 2*Sqrt[2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*Sqrt[Tan[e + f*x]^2]] - Log[1 - 2*Sec[e + f*x] - 3*Sec[e + f*x]^2 + 2*Sqrt[2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*Sqrt[Tan[e + f*x]^2]])*Sqrt[Tan[e + f*x]^2])/Sqrt[Sec[(e + f*x)/2]^2]))/(c*f*g*(-1 + Sec[e + f*x])^2*(a*(1 + Sec[e + f*x]))^(3/2)))","B",1
182,1,328,179,2.285308,"\int \frac{(g \sec (e+f x))^{5/2}}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx","Integrate[(g*Sec[e + f*x])^(5/2)/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])),x]","-\frac{\sin ^3(e+f x) \sqrt{\sec (e+f x)+1} (g \sec (e+f x))^{5/2} \left(8 \sqrt{\sec (e+f x)} \sqrt{\sec (e+f x)+1}+\sqrt{\tan ^2(e+f x)} \left(16 \log (\sec (e+f x)+1)-16 \log \left(\sec ^{\frac{3}{2}}(e+f x)+\sqrt{\sec (e+f x)}+\sqrt{\tan ^2(e+f x)} \sqrt{\sec (e+f x)+1}\right)+\sqrt{2} \left(\log \left(-3 \sec ^2(e+f x)-2 \sec (e+f x)-2 \sqrt{2} \sqrt{\tan ^2(e+f x)} \sqrt{\sec (e+f x)+1} \sqrt{\sec (e+f x)}+1\right)-\log \left(-3 \sec ^2(e+f x)-2 \sec (e+f x)+2 \sqrt{2} \sqrt{\tan ^2(e+f x)} \sqrt{\sec (e+f x)+1} \sqrt{\sec (e+f x)}+1\right)\right)\right)\right)}{8 c f (\cos (e+f x)-1) (\cos (e+f x)+1)^2 (\sec (e+f x)-1) \sec ^{\frac{5}{2}}(e+f x) \sqrt{a (\sec (e+f x)+1)}}","-\frac{2 g^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{g} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}\right)}{\sqrt{a} c f}+\frac{g^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{g} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}\right)}{\sqrt{2} \sqrt{a} c f}+\frac{g^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}{a c f}",1,"-1/8*((g*Sec[e + f*x])^(5/2)*Sqrt[1 + Sec[e + f*x]]*Sin[e + f*x]^3*(8*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]] + (16*Log[1 + Sec[e + f*x]] - 16*Log[Sqrt[Sec[e + f*x]] + Sec[e + f*x]^(3/2) + Sqrt[1 + Sec[e + f*x]]*Sqrt[Tan[e + f*x]^2]] + Sqrt[2]*(Log[1 - 2*Sec[e + f*x] - 3*Sec[e + f*x]^2 - 2*Sqrt[2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*Sqrt[Tan[e + f*x]^2]] - Log[1 - 2*Sec[e + f*x] - 3*Sec[e + f*x]^2 + 2*Sqrt[2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*Sqrt[Tan[e + f*x]^2]]))*Sqrt[Tan[e + f*x]^2]))/(c*f*(-1 + Cos[e + f*x])*(1 + Cos[e + f*x])^2*(-1 + Sec[e + f*x])*Sec[e + f*x]^(5/2)*Sqrt[a*(1 + Sec[e + f*x])])","A",0
183,1,94,46,0.776714,"\int \frac{\sec ^2(e+f x)}{\sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}} \, dx","Integrate[Sec[e + f*x]^2/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]),x]","\frac{4 i \left(-1+e^{i (e+f x)}\right) \cos ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \tanh ^{-1}\left(e^{2 i (e+f x)}\right)}{f \left(1+e^{i (e+f x)}\right) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","\frac{\tan (e+f x) \log (\tan (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"((4*I)*(-1 + E^(I*(e + f*x)))*ArcTanh[E^((2*I)*(e + f*x))]*Cos[(e + f*x)/2]^2*Sec[e + f*x])/((1 + E^(I*(e + f*x)))*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
184,1,98,65,0.2641553,"\int \frac{\sec (e+f x) \sqrt{a+a \sec (e+f x)}}{c-d \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c - d*Sec[e + f*x]),x]","\frac{\sqrt{2} \sqrt{\cos (e+f x)} \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c-d} \sqrt{\cos (e+f x)}}\right)}{\sqrt{d} f \sqrt{c-d}}","\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{c-d} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{d} f \sqrt{c-d}}",1,"(Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[d]*Sin[(e + f*x)/2])/(Sqrt[c - d]*Sqrt[Cos[e + f*x]])]*Sqrt[Cos[e + f*x]]*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])])/(Sqrt[c - d]*Sqrt[d]*f)","A",1
185,1,153,236,1.7421601,"\int \sec (e+f x) (a+a \sec (e+f x)) (c+d \sec (e+f x))^4 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^4,x]","\frac{a \left(\tan (e+f x) \left(80 d^2 \left(3 c^2+2 c d+d^2\right) \tan ^2(e+f x)+15 d \left(16 c^3+24 c^2 d+12 c d^2+3 d^3\right) \sec (e+f x)+30 d^3 (4 c+d) \sec ^3(e+f x)+120 (c+d)^4+24 d^4 \tan ^4(e+f x)\right)+15 \left(8 c^4+16 c^3 d+24 c^2 d^2+12 c d^3+3 d^4\right) \tanh ^{-1}(\sin (e+f x))\right)}{120 f}","\frac{a \left(12 c^2+35 c d+16 d^2\right) \tan (e+f x) (c+d \sec (e+f x))^2}{60 f}+\frac{a d \left(24 c^3+130 c^2 d+116 c d^2+45 d^3\right) \tan (e+f x) \sec (e+f x)}{120 f}+\frac{a \left(12 c^4+95 c^3 d+112 c^2 d^2+80 c d^3+16 d^4\right) \tan (e+f x)}{30 f}+\frac{a \left(8 c^4+16 c^3 d+24 c^2 d^2+12 c d^3+3 d^4\right) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a \tan (e+f x) (c+d \sec (e+f x))^4}{5 f}+\frac{a (4 c+5 d) \tan (e+f x) (c+d \sec (e+f x))^3}{20 f}",1,"(a*(15*(8*c^4 + 16*c^3*d + 24*c^2*d^2 + 12*c*d^3 + 3*d^4)*ArcTanh[Sin[e + f*x]] + Tan[e + f*x]*(120*(c + d)^4 + 15*d*(16*c^3 + 24*c^2*d + 12*c*d^2 + 3*d^3)*Sec[e + f*x] + 30*d^3*(4*c + d)*Sec[e + f*x]^3 + 80*d^2*(3*c^2 + 2*c*d + d^2)*Tan[e + f*x]^2 + 24*d^4*Tan[e + f*x]^4)))/(120*f)","A",1
186,1,103,171,0.6813725,"\int \sec (e+f x) (a+a \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^3,x]","\frac{a \left(3 \left(8 c^3+12 c^2 d+12 c d^2+3 d^3\right) \tanh ^{-1}(\sin (e+f x))+\tan (e+f x) \left(8 d^2 (3 c+d) \tan ^2(e+f x)+9 d (2 c+d)^2 \sec (e+f x)+24 (c+d)^3+6 d^3 \sec ^3(e+f x)\right)\right)}{24 f}","\frac{a d \left(6 c^2+20 c d+9 d^2\right) \tan (e+f x) \sec (e+f x)}{24 f}+\frac{a \left(3 c^3+16 c^2 d+12 c d^2+4 d^3\right) \tan (e+f x)}{6 f}+\frac{a \left(8 c^3+12 c^2 d+12 c d^2+3 d^3\right) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}+\frac{a (3 c+4 d) \tan (e+f x) (c+d \sec (e+f x))^2}{12 f}",1,"(a*(3*(8*c^3 + 12*c^2*d + 12*c*d^2 + 3*d^3)*ArcTanh[Sin[e + f*x]] + Tan[e + f*x]*(24*(c + d)^3 + 9*d*(2*c + d)^2*Sec[e + f*x] + 6*d^3*Sec[e + f*x]^3 + 8*d^2*(3*c + d)*Tan[e + f*x]^2)))/(24*f)","A",1
187,1,75,108,0.4315223,"\int \sec (e+f x) (a+a \sec (e+f x)) (c+d \sec (e+f x))^2 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^2,x]","\frac{a \left(3 \left(2 c^2+2 c d+d^2\right) \tanh ^{-1}(\sin (e+f x))+\tan (e+f x) \left(2 \left(3 (c+d)^2+d^2 \tan ^2(e+f x)\right)+3 d (2 c+d) \sec (e+f x)\right)\right)}{6 f}","\frac{2 a \left(c^2+3 c d+d^2\right) \tan (e+f x)}{3 f}+\frac{a \left(2 c^2+2 c d+d^2\right) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{a \tan (e+f x) (c+d \sec (e+f x))^2}{3 f}+\frac{a d (2 c+3 d) \tan (e+f x) \sec (e+f x)}{6 f}",1,"(a*(3*(2*c^2 + 2*c*d + d^2)*ArcTanh[Sin[e + f*x]] + Tan[e + f*x]*(3*d*(2*c + d)*Sec[e + f*x] + 2*(3*(c + d)^2 + d^2*Tan[e + f*x]^2))))/(6*f)","A",1
188,1,75,56,0.0300678,"\int \sec (e+f x) (a+a \sec (e+f x)) (c+d \sec (e+f x)) \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x]),x]","\frac{a c \tan (e+f x)}{f}+\frac{a c \tanh ^{-1}(\sin (e+f x))}{f}+\frac{a d \tan (e+f x)}{f}+\frac{a d \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{a d \tan (e+f x) \sec (e+f x)}{2 f}","\frac{a (c+d) \tan (e+f x)}{f}+\frac{a (2 c+d) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{a d \tan (e+f x) \sec (e+f x)}{2 f}",1,"(a*c*ArcTanh[Sin[e + f*x]])/f + (a*d*ArcTanh[Sin[e + f*x]])/(2*f) + (a*c*Tan[e + f*x])/f + (a*d*Tan[e + f*x])/f + (a*d*Sec[e + f*x]*Tan[e + f*x])/(2*f)","A",1
189,1,107,69,0.1909161,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))}{c+d \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c + d*Sec[e + f*x]),x]","\frac{a \left(\frac{2 (c-d) \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{d f}","\frac{a \tanh ^{-1}(\sin (e+f x))}{d f}-\frac{2 a \sqrt{c-d} \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{d f \sqrt{c+d}}",1,"(a*((2*(c - d)*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] - Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]))/(d*f)","A",1
190,1,75,79,0.2288581,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))}{(c+d \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c + d*Sec[e + f*x])^2,x]","\frac{a \left(\frac{\sin (e+f x)}{c \cos (e+f x)+d}-\frac{2 \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}\right)}{f (c+d)}","\frac{2 a \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f \sqrt{c-d} (c+d)^{3/2}}+\frac{a \tan (e+f x)}{f (c+d) (c+d \sec (e+f x))}",1,"(a*((-2*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] + Sin[e + f*x]/(d + c*Cos[e + f*x])))/((c + d)*f)","A",1
191,1,167,131,1.2433785,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))}{(c+d \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c + d*Sec[e + f*x])^3,x]","\frac{a (\cos (e+f x)+1) \sec ^2\left(\frac{1}{2} (e+f x)\right) \left(\sqrt{c^2-d^2} \sin (e+f x) \left(\left(2 c^2-2 c d-d^2\right) \cos (e+f x)+d (c-2 d)\right)-2 (2 c-d) (c \cos (e+f x)+d)^2 \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)\right)}{4 f (c-d) (c+d)^2 \sqrt{c^2-d^2} (c \cos (e+f x)+d)^2}","\frac{a (2 c-d) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f (c-d)^{3/2} (c+d)^{5/2}}+\frac{a (c-2 d) \tan (e+f x)}{2 f (c-d) (c+d)^2 (c+d \sec (e+f x))}+\frac{a \tan (e+f x)}{2 f (c+d) (c+d \sec (e+f x))^2}",1,"(a*(1 + Cos[e + f*x])*Sec[(e + f*x)/2]^2*(-2*(2*c - d)*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x])^2 + Sqrt[c^2 - d^2]*((c - 2*d)*d + (2*c^2 - 2*c*d - d^2)*Cos[e + f*x])*Sin[e + f*x]))/(4*(c - d)*(c + d)^2*Sqrt[c^2 - d^2]*f*(d + c*Cos[e + f*x])^2)","A",1
192,1,247,189,3.3534703,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))}{(c+d \sec (e+f x))^4} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c + d*Sec[e + f*x])^4,x]","-\frac{a (\cos (e+f x)+1) \sec ^2\left(\frac{1}{2} (e+f x)\right) \left(6 \left(2 c^2-2 c d+d^2\right) (c \cos (e+f x)+d)^3 \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)-\frac{1}{2} \sqrt{c^2-d^2} \sin (e+f x) \left(6 c^4-12 c^3 d+2 c^2 d^2+6 d \left(2 c^3-7 c^2 d+2 c d^2+d^3\right) \cos (e+f x)+\left(6 c^4-12 c^3 d-2 c^2 d^2+3 c d^3+2 d^4\right) \cos (2 (e+f x))-15 c d^3+10 d^4\right)\right)}{12 f (c-d)^2 (c+d)^3 \sqrt{c^2-d^2} (c \cos (e+f x)+d)^3}","\frac{a \left(2 c^2-2 c d+d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f (c-d)^{5/2} (c+d)^{7/2}}+\frac{a (c-4 d) (2 c-d) \tan (e+f x)}{6 f (c-d)^2 (c+d)^3 (c+d \sec (e+f x))}+\frac{a (2 c-3 d) \tan (e+f x)}{6 f (c-d) (c+d)^2 (c+d \sec (e+f x))^2}+\frac{a \tan (e+f x)}{3 f (c+d) (c+d \sec (e+f x))^3}",1,"-1/12*(a*(1 + Cos[e + f*x])*Sec[(e + f*x)/2]^2*(6*(2*c^2 - 2*c*d + d^2)*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x])^3 - (Sqrt[c^2 - d^2]*(6*c^4 - 12*c^3*d + 2*c^2*d^2 - 15*c*d^3 + 10*d^4 + 6*d*(2*c^3 - 7*c^2*d + 2*c*d^2 + d^3)*Cos[e + f*x] + (6*c^4 - 12*c^3*d - 2*c^2*d^2 + 3*c*d^3 + 2*d^4)*Cos[2*(e + f*x)])*Sin[e + f*x])/2))/((c - d)^2*(c + d)^3*Sqrt[c^2 - d^2]*f*(d + c*Cos[e + f*x])^3)","A",1
193,1,460,327,2.0343052,"\int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^4 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^4,x]","-\frac{a^2 (\cos (e+f x)+1)^2 \sec ^4\left(\frac{1}{2} (e+f x)\right) \sec ^6(e+f x) \left(240 \left(24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right) \cos ^6(e+f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-2 \sin (e+f x) \left(1200 c^4 \cos (3 (e+f x))+120 c^4 \cos (4 (e+f x))+240 c^4 \cos (5 (e+f x))+360 c^4+4640 c^3 d \cos (3 (e+f x))+960 c^3 d \cos (4 (e+f x))+800 c^3 d \cos (5 (e+f x))+2880 c^3 d+6720 c^2 d^2 \cos (3 (e+f x))+1260 c^2 d^2 \cos (4 (e+f x))+960 c^2 d^2 \cos (5 (e+f x))+5220 c^2 d^2+32 \left(75 c^4+310 c^3 d+480 c^2 d^2+336 c d^3+88 d^4\right) \cos (e+f x)+20 \left(24 c^4+192 c^3 d+324 c^2 d^2+240 c d^3+55 d^4\right) \cos (2 (e+f x))+4032 c d^3 \cos (3 (e+f x))+720 c d^3 \cos (4 (e+f x))+576 c d^3 \cos (5 (e+f x))+4080 c d^3+896 d^4 \cos (3 (e+f x))+165 d^4 \cos (4 (e+f x))+128 d^4 \cos (5 (e+f x))+1255 d^4\right)\right)}{15360 f}","-\frac{a^2 \left(4 c^2-48 c d-55 d^2\right) \tan (e+f x) (c+d \sec (e+f x))^3}{120 d f}-\frac{a^2 \left(4 c^3-48 c^2 d-123 c d^2-64 d^3\right) \tan (e+f x) (c+d \sec (e+f x))^2}{120 d f}+\frac{a^2 \left(24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right) \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac{a^2 \left(8 c^4-96 c^3 d-438 c^2 d^2-464 c d^3-165 d^4\right) \tan (e+f x) \sec (e+f x)}{240 f}-\frac{a^2 \left(4 c^5-48 c^4 d-311 c^3 d^2-448 c^2 d^3-288 c d^4-64 d^5\right) \tan (e+f x)}{60 d f}+\frac{a^2 \tan (e+f x) (c+d \sec (e+f x))^5}{6 d f}-\frac{a^2 (c-12 d) \tan (e+f x) (c+d \sec (e+f x))^4}{30 d f}",1,"-1/15360*(a^2*(1 + Cos[e + f*x])^2*Sec[(e + f*x)/2]^4*Sec[e + f*x]^6*(240*(24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*Cos[e + f*x]^6*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - 2*(360*c^4 + 2880*c^3*d + 5220*c^2*d^2 + 4080*c*d^3 + 1255*d^4 + 32*(75*c^4 + 310*c^3*d + 480*c^2*d^2 + 336*c*d^3 + 88*d^4)*Cos[e + f*x] + 20*(24*c^4 + 192*c^3*d + 324*c^2*d^2 + 240*c*d^3 + 55*d^4)*Cos[2*(e + f*x)] + 1200*c^4*Cos[3*(e + f*x)] + 4640*c^3*d*Cos[3*(e + f*x)] + 6720*c^2*d^2*Cos[3*(e + f*x)] + 4032*c*d^3*Cos[3*(e + f*x)] + 896*d^4*Cos[3*(e + f*x)] + 120*c^4*Cos[4*(e + f*x)] + 960*c^3*d*Cos[4*(e + f*x)] + 1260*c^2*d^2*Cos[4*(e + f*x)] + 720*c*d^3*Cos[4*(e + f*x)] + 165*d^4*Cos[4*(e + f*x)] + 240*c^4*Cos[5*(e + f*x)] + 800*c^3*d*Cos[5*(e + f*x)] + 960*c^2*d^2*Cos[5*(e + f*x)] + 576*c*d^3*Cos[5*(e + f*x)] + 128*d^4*Cos[5*(e + f*x)])*Sin[e + f*x]))/f","A",1
194,1,326,242,1.3949817,"\int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^3 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^3,x]","-\frac{a^2 (\cos (e+f x)+1)^2 \sec ^4\left(\frac{1}{2} (e+f x)\right) \sec ^5(e+f x) \left(120 \left(4 c^3+8 c^2 d+7 c d^2+2 d^3\right) \cos ^5(e+f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-2 \sin (e+f x) \left(20 c^3 \cos (3 (e+f x))+40 c^3 \cos (4 (e+f x))+120 c^3+120 c^2 d \cos (3 (e+f x))+100 c^2 d \cos (4 (e+f x))+380 c^2 d+5 \left(12 c^3+72 c^2 d+87 c d^2+34 d^3\right) \cos (e+f x)+16 \left(10 c^3+30 c^2 d+30 c d^2+9 d^3\right) \cos (2 (e+f x))+105 c d^2 \cos (3 (e+f x))+80 c d^2 \cos (4 (e+f x))+400 c d^2+30 d^3 \cos (3 (e+f x))+24 d^3 \cos (4 (e+f x))+152 d^3\right)\right)}{1280 f}","\frac{3 a^2 (2 c+d) \left(2 c^2+3 c d+2 d^2\right) \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac{a^2 \left(c^2-10 c d-12 d^2\right) \tan (e+f x) (c+d \sec (e+f x))^2}{20 d f}-\frac{a^2 \left(2 c^3-20 c^2 d-57 c d^2-30 d^3\right) \tan (e+f x) \sec (e+f x)}{40 f}-\frac{a^2 \left(c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right) \tan (e+f x)}{10 d f}+\frac{a^2 \tan (e+f x) (c+d \sec (e+f x))^4}{5 d f}-\frac{a^2 (c-10 d) \tan (e+f x) (c+d \sec (e+f x))^3}{20 d f}",1,"-1/1280*(a^2*(1 + Cos[e + f*x])^2*Sec[(e + f*x)/2]^4*Sec[e + f*x]^5*(120*(4*c^3 + 8*c^2*d + 7*c*d^2 + 2*d^3)*Cos[e + f*x]^5*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - 2*(120*c^3 + 380*c^2*d + 400*c*d^2 + 152*d^3 + 5*(12*c^3 + 72*c^2*d + 87*c*d^2 + 34*d^3)*Cos[e + f*x] + 16*(10*c^3 + 30*c^2*d + 30*c*d^2 + 9*d^3)*Cos[2*(e + f*x)] + 20*c^3*Cos[3*(e + f*x)] + 120*c^2*d*Cos[3*(e + f*x)] + 105*c*d^2*Cos[3*(e + f*x)] + 30*d^3*Cos[3*(e + f*x)] + 40*c^3*Cos[4*(e + f*x)] + 100*c^2*d*Cos[4*(e + f*x)] + 80*c*d^2*Cos[4*(e + f*x)] + 24*d^3*Cos[4*(e + f*x)])*Sin[e + f*x]))/f","A",1
195,1,479,176,0.9981806,"\int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2,x]","-\frac{a^2 \sec ^4(e+f x) \left(12 \left(12 c^2+16 c d+7 d^2\right) \cos (2 (e+f x)) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+3 \left(12 c^2+16 c d+7 d^2\right) \cos (4 (e+f x)) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-24 c^2 \sin (e+f x)-96 c^2 \sin (2 (e+f x))-24 c^2 \sin (3 (e+f x))-48 c^2 \sin (4 (e+f x))+108 c^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-108 c^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-96 c d \sin (e+f x)-224 c d \sin (2 (e+f x))-96 c d \sin (3 (e+f x))-80 c d \sin (4 (e+f x))+144 c d \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-144 c d \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-90 d^2 \sin (e+f x)-128 d^2 \sin (2 (e+f x))-42 d^2 \sin (3 (e+f x))-32 d^2 \sin (4 (e+f x))+63 d^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-63 d^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{192 f}","\frac{a^2 \left(12 c^2+16 c d+7 d^2\right) \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac{a^2 \left(c^3-8 c^2 d-20 c d^2-8 d^3\right) \tan (e+f x)}{6 d f}-\frac{a^2 \left(2 c (c-8 d)-21 d^2\right) \tan (e+f x) \sec (e+f x)}{24 f}+\frac{a^2 \tan (e+f x) (c+d \sec (e+f x))^3}{4 d f}-\frac{a^2 (c-8 d) \tan (e+f x) (c+d \sec (e+f x))^2}{12 d f}",1,"-1/192*(a^2*Sec[e + f*x]^4*(108*c^2*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 144*c*d*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 63*d^2*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 12*(12*c^2 + 16*c*d + 7*d^2)*Cos[2*(e + f*x)]*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + 3*(12*c^2 + 16*c*d + 7*d^2)*Cos[4*(e + f*x)]*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - 108*c^2*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - 144*c*d*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - 63*d^2*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - 24*c^2*Sin[e + f*x] - 96*c*d*Sin[e + f*x] - 90*d^2*Sin[e + f*x] - 96*c^2*Sin[2*(e + f*x)] - 224*c*d*Sin[2*(e + f*x)] - 128*d^2*Sin[2*(e + f*x)] - 24*c^2*Sin[3*(e + f*x)] - 96*c*d*Sin[3*(e + f*x)] - 42*d^2*Sin[3*(e + f*x)] - 48*c^2*Sin[4*(e + f*x)] - 80*c*d*Sin[4*(e + f*x)] - 32*d^2*Sin[4*(e + f*x)]))/f","B",1
196,1,481,103,6.3619633,"\int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x)) \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x]),x]","\frac{a^2 \cos ^3(e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right) (\sec (e+f x)+1)^2 (c+d \sec (e+f x)) \left(\frac{4 (6 c+5 d) \sin \left(\frac{f x}{2}\right)}{\left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{4 (6 c+5 d) \sin \left(\frac{f x}{2}\right)}{\left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{(3 c+7 d) \cos \left(\frac{e}{2}\right)-(3 c+5 d) \sin \left(\frac{e}{2}\right)}{\left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{(3 c+5 d) \sin \left(\frac{e}{2}\right)+(3 c+7 d) \cos \left(\frac{e}{2}\right)}{\left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}-6 (3 c+2 d) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+6 (3 c+2 d) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\frac{2 d \sin \left(\frac{f x}{2}\right)}{\left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{2 d \sin \left(\frac{f x}{2}\right)}{\left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}\right)}{48 f (c \cos (e+f x)+d)}","\frac{2 a^2 (3 c+2 d) \tan (e+f x)}{3 f}+\frac{a^2 (3 c+2 d) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{a^2 (3 c+2 d) \tan (e+f x) \sec (e+f x)}{6 f}+\frac{d \tan (e+f x) (a \sec (e+f x)+a)^2}{3 f}",1,"(a^2*Cos[e + f*x]^3*Sec[(e + f*x)/2]^4*(1 + Sec[e + f*x])^2*(c + d*Sec[e + f*x])*(-6*(3*c + 2*d)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 6*(3*c + 2*d)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (2*d*Sin[(f*x)/2])/((Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) + ((3*c + 7*d)*Cos[e/2] - (3*c + 5*d)*Sin[e/2])/((Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (4*(6*c + 5*d)*Sin[(f*x)/2])/((Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])) + (2*d*Sin[(f*x)/2])/((Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3) - ((3*c + 7*d)*Cos[e/2] + (3*c + 5*d)*Sin[e/2])/((Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) + (4*(6*c + 5*d)*Sin[(f*x)/2])/((Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))))/(48*f*(d + c*Cos[e + f*x]))","B",1
197,1,329,95,2.019105,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{c+d \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c + d*Sec[e + f*x]),x]","\frac{a^2 \cos (e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right) (\sec (e+f x)+1)^2 (c \cos (e+f x)+d) \left(-\frac{2 i (c-d)^2 (\cos (e)-i \sin (e)) \tan ^{-1}\left(\frac{(\sin (e)+i \cos (e)) \left(\tan \left(\frac{f x}{2}\right) (c \cos (e)-d)+c \sin (e)\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}+(c-2 d) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-(c-2 d) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\frac{d \sin \left(\frac{f x}{2}\right)}{\left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{d \sin \left(\frac{f x}{2}\right)}{\left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}\right)}{4 d^2 f (c+d \sec (e+f x))}","-\frac{a^2 (c-2 d) \tanh ^{-1}(\sin (e+f x))}{d^2 f}+\frac{2 a^2 (c-d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{d^2 f \sqrt{c+d}}+\frac{a^2 \tan (e+f x)}{d f}",1,"(a^2*Cos[e + f*x]*(d + c*Cos[e + f*x])*Sec[(e + f*x)/2]^4*(1 + Sec[e + f*x])^2*((c - 2*d)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - (c - 2*d)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - ((2*I)*(c - d)^2*ArcTan[((I*Cos[e] + Sin[e])*(c*Sin[e] + (-d + c*Cos[e])*Tan[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*(Cos[e] - I*Sin[e]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + (d*Sin[(f*x)/2])/((Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])) + (d*Sin[(f*x)/2])/((Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))))/(4*d^2*f*(c + d*Sec[e + f*x]))","C",1
198,1,312,117,1.5053843,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c+d \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^2,x]","\frac{a^2 \sec ^4\left(\frac{1}{2} (e+f x)\right) (\sec (e+f x)+1)^2 (c \cos (e+f x)+d) \left(\frac{2 \left(c^2+c d-2 d^2\right) (\sin (e)+i \cos (e)) (c \cos (e+f x)+d) \tan ^{-1}\left(\frac{(\sin (e)+i \cos (e)) \left(\tan \left(\frac{f x}{2}\right) (c \cos (e)-d)+c \sin (e)\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{(c+d) \sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}+\frac{d (c-d) (d \sin (e)-c \sin (f x))}{c (c+d) \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right)}-\left((c \cos (e+f x)+d) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)+(c \cos (e+f x)+d) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{4 d^2 f (c+d \sec (e+f x))^2}","-\frac{2 a^2 \sqrt{c-d} (c+2 d) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{d^2 f (c+d)^{3/2}}-\frac{a^2 (c-d) \tan (e+f x)}{d f (c+d) (c+d \sec (e+f x))}+\frac{a^2 \tanh ^{-1}(\sin (e+f x))}{d^2 f}",1,"(a^2*(d + c*Cos[e + f*x])*Sec[(e + f*x)/2]^4*(1 + Sec[e + f*x])^2*(-((d + c*Cos[e + f*x])*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]) + (d + c*Cos[e + f*x])*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (2*(c^2 + c*d - 2*d^2)*ArcTan[((I*Cos[e] + Sin[e])*(c*Sin[e] + (-d + c*Cos[e])*Tan[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*(d + c*Cos[e + f*x])*(I*Cos[e] + Sin[e]))/((c + d)*Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + ((c - d)*d*(d*Sin[e] - c*Sin[f*x]))/(c*(c + d)*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2]))))/(4*d^2*f*(c + d*Sec[e + f*x])^2)","C",1
199,1,249,130,1.2451706,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c+d \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^3,x]","\frac{a^2 \sec ^4\left(\frac{1}{2} (e+f x)\right) \sec (e+f x) (\sec (e+f x)+1)^2 (c \cos (e+f x)+d) \left(-\frac{6 i (\cos (e)-i \sin (e)) (c \cos (e+f x)+d)^2 \tan ^{-1}\left(\frac{(\sin (e)+i \cos (e)) \left(\tan \left(\frac{f x}{2}\right) (c \cos (e)-d)+c \sin (e)\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}+\frac{\sec (e) \left(\left(c^2-4 c d-2 d^2\right) \sin (e)+c (4 c+d) \sin (f x)\right) (c \cos (e+f x)+d)}{c^2}+\frac{(c-d) (c+d) \sec (e) (c \sin (f x)-d \sin (e))}{c^2}\right)}{8 f (c+d)^2 (c+d \sec (e+f x))^3}","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f \sqrt{c-d} (c+d)^{5/2}}+\frac{3 a^2 \tan (e+f x)}{2 f (c+d)^2 (c+d \sec (e+f x))}+\frac{\tan (e+f x) \left(a^2 \sec (e+f x)+a^2\right)}{2 f (c+d) (c+d \sec (e+f x))^2}",1,"(a^2*(d + c*Cos[e + f*x])*Sec[(e + f*x)/2]^4*Sec[e + f*x]*(1 + Sec[e + f*x])^2*(((-6*I)*ArcTan[((I*Cos[e] + Sin[e])*(c*Sin[e] + (-d + c*Cos[e])*Tan[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*(d + c*Cos[e + f*x])^2*(Cos[e] - I*Sin[e]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + ((c - d)*(c + d)*Sec[e]*(-(d*Sin[e]) + c*Sin[f*x]))/c^2 + ((d + c*Cos[e + f*x])*Sec[e]*((c^2 - 4*c*d - 2*d^2)*Sin[e] + c*(4*c + d)*Sin[f*x]))/c^2))/(8*(c + d)^2*f*(c + d*Sec[e + f*x])^3)","C",1
200,1,211,213,4.7798747,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c+d \sec (e+f x))^4} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^4,x]","\frac{a^2 (c-d)^2 \left(24 (3 c-2 d) (c \cos (e+f x)+d)^3 \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)-2 \sqrt{c^2-d^2} \sin (e+f x) \left(12 c^3-5 c^2 d+6 \left(c^3+6 c^2 d-7 c d^2-2 d^3\right) \cos (e+f x)+\left(12 c^3-7 c^2 d-6 c d^2-2 d^3\right) \cos (2 (e+f x))+6 c d^2-22 d^3\right)\right)}{24 f (d-c)^3 (c+d)^3 \sqrt{c^2-d^2} (c \cos (e+f x)+d)^3}","\frac{a^2 (3 c-2 d) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f (c-d)^{3/2} (c+d)^{7/2}}+\frac{a^2 (3 c-2 d) \tan (e+f x)}{2 f (c-d) (c+d)^3 (c+d \sec (e+f x))}+\frac{(3 c-2 d) \tan (e+f x) \left(a^2 \sec (e+f x)+a^2\right)}{6 f (c-d) (c+d)^2 (c+d \sec (e+f x))^2}-\frac{d \tan (e+f x) (a \sec (e+f x)+a)^2}{3 f \left(c^2-d^2\right) (c+d \sec (e+f x))^3}",1,"(a^2*(c - d)^2*(24*(3*c - 2*d)*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x])^3 - 2*Sqrt[c^2 - d^2]*(12*c^3 - 5*c^2*d + 6*c*d^2 - 22*d^3 + 6*(c^3 + 6*c^2*d - 7*c*d^2 - 2*d^3)*Cos[e + f*x] + (12*c^3 - 7*c^2*d - 6*c*d^2 - 2*d^3)*Cos[2*(e + f*x)])*Sin[e + f*x]))/(24*(-c + d)^3*(c + d)^3*Sqrt[c^2 - d^2]*f*(d + c*Cos[e + f*x])^3)","A",1
201,1,322,276,9.0588393,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c+d \sec (e+f x))^5} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^5,x]","\frac{a^2 \left(\frac{\sin (e+f x) \left(48 c^5 \cos (3 (e+f x))+24 c^5-68 c^4 d \cos (3 (e+f x))+192 c^4 d-16 c^3 d^2 \cos (3 (e+f x))-446 c^3 d^2+5 c^2 d^3 \cos (3 (e+f x))+128 c^2 d^3+\left(144 c^5-172 c^4 d+208 c^3 d^2-785 c^2 d^3+368 c d^4+102 d^5\right) \cos (e+f x)+2 \left(12 c^5+96 c^4 d-227 c^3 d^2+32 c^2 d^3+44 c d^4+16 d^5\right) \cos (2 (e+f x))+16 c d^4 \cos (3 (e+f x))-148 c d^4+6 d^5 \cos (3 (e+f x))+160 d^5\right)}{(c \cos (e+f x)+d)^4}-\frac{24 \left(12 c^2-16 c d+7 d^2\right) \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}\right)}{96 f (c-d)^2 (c+d)^4}","\frac{a^2 \left(12 c^2-16 c d+7 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{4 f (c-d)^{5/2} (c+d)^{9/2}}+\frac{a^2 \left(2 c^2+16 c d-21 d^2\right) \tan (e+f x)}{24 d f (c-d) (c+d)^3 (c+d \sec (e+f x))^2}+\frac{a^2 \left(2 c^3+16 c^2 d-59 c d^2+32 d^3\right) \tan (e+f x)}{24 d f (c-d)^2 (c+d)^4 (c+d \sec (e+f x))}+\frac{a^2 (c+8 d) \tan (e+f x)}{12 d f (c+d)^2 (c+d \sec (e+f x))^3}-\frac{a^2 (c-d) \tan (e+f x)}{4 d f (c+d) (c+d \sec (e+f x))^4}",1,"(a^2*((-24*(12*c^2 - 16*c*d + 7*d^2)*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] + ((24*c^5 + 192*c^4*d - 446*c^3*d^2 + 128*c^2*d^3 - 148*c*d^4 + 160*d^5 + (144*c^5 - 172*c^4*d + 208*c^3*d^2 - 785*c^2*d^3 + 368*c*d^4 + 102*d^5)*Cos[e + f*x] + 2*(12*c^5 + 96*c^4*d - 227*c^3*d^2 + 32*c^2*d^3 + 44*c*d^4 + 16*d^5)*Cos[2*(e + f*x)] + 48*c^5*Cos[3*(e + f*x)] - 68*c^4*d*Cos[3*(e + f*x)] - 16*c^3*d^2*Cos[3*(e + f*x)] + 5*c^2*d^3*Cos[3*(e + f*x)] + 16*c*d^4*Cos[3*(e + f*x)] + 6*d^5*Cos[3*(e + f*x)])*Sin[e + f*x])/(d + c*Cos[e + f*x])^4))/(96*(c - d)^2*(c + d)^4*f)","A",1
202,1,380,288,2.6939509,"\int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3,x]","-\frac{a^3 (\cos (e+f x)+1)^3 \sec ^6\left(\frac{1}{2} (e+f x)\right) \sec ^6(e+f x) \left(240 \left(40 c^3+90 c^2 d+78 c d^2+23 d^3\right) \cos ^6(e+f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-2 \sin (e+f x) \left(2360 c^3 \cos (3 (e+f x))+360 c^3 \cos (4 (e+f x))+440 c^3 \cos (5 (e+f x))+1080 c^3+6840 c^2 d \cos (3 (e+f x))+1350 c^2 d \cos (4 (e+f x))+1080 c^2 d \cos (5 (e+f x))+4770 c^2 d+16 \left(305 c^3+945 c^2 d+984 c d^2+344 d^3\right) \cos (e+f x)+20 \left(72 c^3+306 c^2 d+342 c d^2+115 d^3\right) \cos (2 (e+f x))+6384 c d^2 \cos (3 (e+f x))+1170 c d^2 \cos (4 (e+f x))+912 c d^2 \cos (5 (e+f x))+5670 c d^2+1904 d^3 \cos (3 (e+f x))+345 d^3 \cos (4 (e+f x))+272 d^3 \cos (5 (e+f x))+2275 d^3\right)\right)}{30720 f}","\frac{a^3 \left(40 c^3+90 c^2 d+78 c d^2+23 d^3\right) \tan (e+f x)}{16 f}+\frac{a^3 \left(40 c^3+90 c^2 d+78 c d^2+23 d^3\right) \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac{\left(40 c^3+90 c^2 d+78 c d^2+23 d^3\right) \tan (e+f x) \left(a^3 \sec (e+f x)+a^3\right)}{48 f}+\frac{a \tan (e+f x) (a \sec (e+f x)+a)^2 \left(d \left(6 c^2+62 c d+31 d^2\right) \sec (e+f x)+2 \left(4 c^3+74 c^2 d+66 c d^2+21 d^3\right)\right)}{120 f}+\frac{a \tan (e+f x) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))^3}{6 f}+\frac{a (3 c+8 d) \tan (e+f x) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))^2}{30 f}",1,"-1/30720*(a^3*(1 + Cos[e + f*x])^3*Sec[(e + f*x)/2]^6*Sec[e + f*x]^6*(240*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*Cos[e + f*x]^6*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - 2*(1080*c^3 + 4770*c^2*d + 5670*c*d^2 + 2275*d^3 + 16*(305*c^3 + 945*c^2*d + 984*c*d^2 + 344*d^3)*Cos[e + f*x] + 20*(72*c^3 + 306*c^2*d + 342*c*d^2 + 115*d^3)*Cos[2*(e + f*x)] + 2360*c^3*Cos[3*(e + f*x)] + 6840*c^2*d*Cos[3*(e + f*x)] + 6384*c*d^2*Cos[3*(e + f*x)] + 1904*d^3*Cos[3*(e + f*x)] + 360*c^3*Cos[4*(e + f*x)] + 1350*c^2*d*Cos[4*(e + f*x)] + 1170*c*d^2*Cos[4*(e + f*x)] + 345*d^3*Cos[4*(e + f*x)] + 440*c^3*Cos[5*(e + f*x)] + 1080*c^2*d*Cos[5*(e + f*x)] + 912*c*d^2*Cos[5*(e + f*x)] + 272*d^3*Cos[5*(e + f*x)])*Sin[e + f*x]))/f","A",1
203,1,433,257,2.6918815,"\int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2,x]","-\frac{a^3 (\cos (e+f x)+1)^3 \sec ^6\left(\frac{1}{2} (e+f x)\right) \sec ^5(e+f x) \left(240 \left(20 c^2+30 c d+13 d^2\right) \cos ^5(e+f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-\sec (e) \left(-240 \left(7 c^2+10 c d+3 d^2\right) \sin (2 e+f x)+80 \left(34 c^2+60 c d+29 d^2\right) \sin (f x)+360 c^2 \sin (e+2 f x)+360 c^2 \sin (3 e+2 f x)+1840 c^2 \sin (2 e+3 f x)-360 c^2 \sin (4 e+3 f x)+180 c^2 \sin (3 e+4 f x)+180 c^2 \sin (5 e+4 f x)+440 c^2 \sin (4 e+5 f x)+1140 c d \sin (e+2 f x)+1140 c d \sin (3 e+2 f x)+3360 c d \sin (2 e+3 f x)-240 c d \sin (4 e+3 f x)+450 c d \sin (3 e+4 f x)+450 c d \sin (5 e+4 f x)+720 c d \sin (4 e+5 f x)+750 d^2 \sin (e+2 f x)+750 d^2 \sin (3 e+2 f x)+1520 d^2 \sin (2 e+3 f x)+195 d^2 \sin (3 e+4 f x)+195 d^2 \sin (5 e+4 f x)+304 d^2 \sin (4 e+5 f x)\right)\right)}{15360 f}","\frac{a^3 \left(20 c^2+30 c d+13 d^2\right) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a^3 \left(2 c^2-15 c d+76 d^2\right) \tan (e+f x) (c+d \sec (e+f x))^2}{60 d^2 f}+\frac{a^3 \left(4 c^3-30 c^2 d+146 c d^2+195 d^3\right) \tan (e+f x) \sec (e+f x)}{120 d f}+\frac{a^3 \left(2 c^4-15 c^3 d+72 c^2 d^2+180 c d^3+76 d^4\right) \tan (e+f x)}{30 d^2 f}-\frac{a^3 (2 c-11 d) \tan (e+f x) (c+d \sec (e+f x))^3}{20 d^2 f}+\frac{\tan (e+f x) \left(a^3 \sec (e+f x)+a^3\right) (c+d \sec (e+f x))^3}{5 d f}",1,"-1/15360*(a^3*(1 + Cos[e + f*x])^3*Sec[(e + f*x)/2]^6*Sec[e + f*x]^5*(240*(20*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x]^5*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - Sec[e]*(80*(34*c^2 + 60*c*d + 29*d^2)*Sin[f*x] - 240*(7*c^2 + 10*c*d + 3*d^2)*Sin[2*e + f*x] + 360*c^2*Sin[e + 2*f*x] + 1140*c*d*Sin[e + 2*f*x] + 750*d^2*Sin[e + 2*f*x] + 360*c^2*Sin[3*e + 2*f*x] + 1140*c*d*Sin[3*e + 2*f*x] + 750*d^2*Sin[3*e + 2*f*x] + 1840*c^2*Sin[2*e + 3*f*x] + 3360*c*d*Sin[2*e + 3*f*x] + 1520*d^2*Sin[2*e + 3*f*x] - 360*c^2*Sin[4*e + 3*f*x] - 240*c*d*Sin[4*e + 3*f*x] + 180*c^2*Sin[3*e + 4*f*x] + 450*c*d*Sin[3*e + 4*f*x] + 195*d^2*Sin[3*e + 4*f*x] + 180*c^2*Sin[5*e + 4*f*x] + 450*c*d*Sin[5*e + 4*f*x] + 195*d^2*Sin[5*e + 4*f*x] + 440*c^2*Sin[4*e + 5*f*x] + 720*c*d*Sin[4*e + 5*f*x] + 304*d^2*Sin[4*e + 5*f*x])))/f","A",1
204,1,273,125,1.3678129,"\int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x]),x]","-\frac{a^3 (\cos (e+f x)+1)^3 \sec ^6\left(\frac{1}{2} (e+f x)\right) \sec ^4(e+f x) \left(120 (4 c+3 d) \cos ^4(e+f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-\sec (e) (-24 (11 c+9 d) \sin (e)+(36 c+69 d) \sin (f x)+36 c \sin (2 e+f x)+280 c \sin (e+2 f x)-72 c \sin (3 e+2 f x)+36 c \sin (2 e+3 f x)+36 c \sin (4 e+3 f x)+88 c \sin (3 e+4 f x)+69 d \sin (2 e+f x)+264 d \sin (e+2 f x)-24 d \sin (3 e+2 f x)+45 d \sin (2 e+3 f x)+45 d \sin (4 e+3 f x)+72 d \sin (3 e+4 f x))\right)}{1536 f}","\frac{a^3 (4 c+3 d) \tan ^3(e+f x)}{12 f}+\frac{a^3 (4 c+3 d) \tan (e+f x)}{f}+\frac{5 a^3 (4 c+3 d) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{3 a^3 (4 c+3 d) \tan (e+f x) \sec (e+f x)}{8 f}+\frac{d \tan (e+f x) (a \sec (e+f x)+a)^3}{4 f}",1,"-1/1536*(a^3*(1 + Cos[e + f*x])^3*Sec[(e + f*x)/2]^6*Sec[e + f*x]^4*(120*(4*c + 3*d)*Cos[e + f*x]^4*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - Sec[e]*(-24*(11*c + 9*d)*Sin[e] + (36*c + 69*d)*Sin[f*x] + 36*c*Sin[2*e + f*x] + 69*d*Sin[2*e + f*x] + 280*c*Sin[e + 2*f*x] + 264*d*Sin[e + 2*f*x] - 72*c*Sin[3*e + 2*f*x] - 24*d*Sin[3*e + 2*f*x] + 36*c*Sin[2*e + 3*f*x] + 45*d*Sin[2*e + 3*f*x] + 36*c*Sin[4*e + 3*f*x] + 45*d*Sin[4*e + 3*f*x] + 88*c*Sin[3*e + 4*f*x] + 72*d*Sin[3*e + 4*f*x])))/f","B",1
205,1,419,153,2.4136799,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{c+d \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c + d*Sec[e + f*x]),x]","\frac{a^3 \cos ^2(e+f x) \sec ^6\left(\frac{1}{2} (e+f x)\right) (\sec (e+f x)+1)^3 (c \cos (e+f x)+d) \left(-2 \left(2 c^2-6 c d+7 d^2\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+2 \left(2 c^2-6 c d+7 d^2\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\frac{8 (c-d)^3 (\sin (e)+i \cos (e)) \tan ^{-1}\left(\frac{(\sin (e)+i \cos (e)) \left(\tan \left(\frac{f x}{2}\right) (c \cos (e)-d)+c \sin (e)\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}-\frac{4 d (c-3 d) \sin \left(\frac{f x}{2}\right)}{\left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}-\frac{4 d (c-3 d) \sin \left(\frac{f x}{2}\right)}{\left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{d^2}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{d^2}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}\right)}{32 d^3 f (c+d \sec (e+f x))}","\frac{a^3 \left(c^2-3 c d+3 d^2\right) \tanh ^{-1}(\sin (e+f x))}{d^3 f}-\frac{2 a^3 (c-d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{d^3 f \sqrt{c+d}}-\frac{a^3 (c-3 d) \tan (e+f x)}{d^2 f}+\frac{a^3 \tanh ^{-1}(\sin (e+f x))}{2 d f}+\frac{a^3 \tan (e+f x) \sec (e+f x)}{2 d f}",1,"(a^3*Cos[e + f*x]^2*(d + c*Cos[e + f*x])*Sec[(e + f*x)/2]^6*(1 + Sec[e + f*x])^3*(-2*(2*c^2 - 6*c*d + 7*d^2)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 2*(2*c^2 - 6*c*d + 7*d^2)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (8*(c - d)^3*ArcTan[((I*Cos[e] + Sin[e])*(c*Sin[e] + (-d + c*Cos[e])*Tan[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*(I*Cos[e] + Sin[e]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + d^2/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 - (4*(c - 3*d)*d*Sin[(f*x)/2])/((Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])) - d^2/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - (4*(c - 3*d)*d*Sin[(f*x)/2])/((Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))))/(32*d^3*f*(c + d*Sec[e + f*x]))","C",1
206,1,455,161,4.2136092,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^2,x]","\frac{a^3 \cos (e+f x) \sec ^6\left(\frac{1}{2} (e+f x)\right) (\sec (e+f x)+1)^3 (c \cos (e+f x)+d) \left(-\frac{2 i (2 c+3 d) (c-d)^2 (\cos (e)-i \sin (e)) (c \cos (e+f x)+d) \tan ^{-1}\left(\frac{(\sin (e)+i \cos (e)) \left(\tan \left(\frac{f x}{2}\right) (c \cos (e)-d)+c \sin (e)\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{(c+d) \sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}+\frac{d (c-d)^2 (c \sin (f x)-d \sin (e))}{c (c+d) \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right)}+\frac{d \sin \left(\frac{f x}{2}\right) (c \cos (e+f x)+d)}{\left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{d \sin \left(\frac{f x}{2}\right) (c \cos (e+f x)+d)}{\left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}+(2 c-3 d) (c \cos (e+f x)+d) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+(3 d-2 c) (c \cos (e+f x)+d) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{8 d^3 f (c+d \sec (e+f x))^2}","-\frac{a^3 (2 c-3 d) \tanh ^{-1}(\sin (e+f x))}{d^3 f}+\frac{2 a^3 (c-d)^{3/2} (2 c+3 d) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{d^3 f (c+d)^{3/2}}+\frac{2 a^3 c \tan (e+f x)}{d^2 f (c+d)}-\frac{(c-d) \tan (e+f x) \left(a^3 \sec (e+f x)+a^3\right)}{d f (c+d) (c+d \sec (e+f x))}",1,"(a^3*Cos[e + f*x]*(d + c*Cos[e + f*x])*Sec[(e + f*x)/2]^6*(1 + Sec[e + f*x])^3*((2*c - 3*d)*(d + c*Cos[e + f*x])*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + (-2*c + 3*d)*(d + c*Cos[e + f*x])*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - ((2*I)*(c - d)^2*(2*c + 3*d)*ArcTan[((I*Cos[e] + Sin[e])*(c*Sin[e] + (-d + c*Cos[e])*Tan[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*(d + c*Cos[e + f*x])*(Cos[e] - I*Sin[e]))/((c + d)*Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + ((c - d)^2*d*(-(d*Sin[e]) + c*Sin[f*x]))/(c*(c + d)*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])) + (d*(d + c*Cos[e + f*x])*Sin[(f*x)/2])/((Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])) + (d*(d + c*Cos[e + f*x])*Sin[(f*x)/2])/((Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))))/(8*d^3*f*(c + d*Sec[e + f*x])^2)","C",1
207,1,393,188,3.6671595,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^3,x]","\frac{a^3 \sec ^6\left(\frac{1}{2} (e+f x)\right) (\sec (e+f x)+1)^3 (c \cos (e+f x)+d) \left(\frac{4 \left(2 c^3+4 c^2 d+c d^2-7 d^3\right) (\sin (e)+i \cos (e)) (c \cos (e+f x)+d)^2 \tan ^{-1}\left(\frac{(\sin (e)+i \cos (e)) \left(\tan \left(\frac{f x}{2}\right) (c \cos (e)-d)+c \sin (e)\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{(c+d)^2 \sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}+\frac{d (c-d) \sec (e) \left(\left(2 c^4+6 c^3 d+5 c^2 d^2+12 c d^3+2 d^4\right) \sin (e)-c \left(-d \left(c^2+6 c d+2 d^2\right) \sin (2 e+f x)+c \left(2 c^2+6 c d+d^2\right) \sin (e+2 f x)+d \left(7 c^2+18 c d+2 d^2\right) \sin (f x)\right)\right)}{c^2 (c+d)^2}-4 (c \cos (e+f x)+d)^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+4 (c \cos (e+f x)+d)^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{32 d^3 f (c+d \sec (e+f x))^3}","-\frac{a^3 \sqrt{c-d} \left(2 c^2+6 c d+7 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{d^3 f (c+d)^{5/2}}-\frac{a^3 (c-d) (2 c+5 d) \tan (e+f x)}{2 d^2 f (c+d)^2 (c+d \sec (e+f x))}-\frac{(c-d) \tan (e+f x) \left(a^3 \sec (e+f x)+a^3\right)}{2 d f (c+d) (c+d \sec (e+f x))^2}+\frac{a^3 \tanh ^{-1}(\sin (e+f x))}{d^3 f}",1,"(a^3*(d + c*Cos[e + f*x])*Sec[(e + f*x)/2]^6*(1 + Sec[e + f*x])^3*(-4*(d + c*Cos[e + f*x])^2*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 4*(d + c*Cos[e + f*x])^2*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (4*(2*c^3 + 4*c^2*d + c*d^2 - 7*d^3)*ArcTan[((I*Cos[e] + Sin[e])*(c*Sin[e] + (-d + c*Cos[e])*Tan[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*(d + c*Cos[e + f*x])^2*(I*Cos[e] + Sin[e]))/((c + d)^2*Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + ((c - d)*d*Sec[e]*((2*c^4 + 6*c^3*d + 5*c^2*d^2 + 12*c*d^3 + 2*d^4)*Sin[e] - c*(d*(7*c^2 + 18*c*d + 2*d^2)*Sin[f*x] - d*(c^2 + 6*c*d + 2*d^2)*Sin[2*e + f*x] + c*(2*c^2 + 6*c*d + d^2)*Sin[e + 2*f*x])))/(c^2*(c + d)^2)))/(32*d^3*f*(c + d*Sec[e + f*x])^3)","C",1
208,1,398,178,3.5767816,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^4} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^4,x]","\frac{a^3 \sec ^6\left(\frac{1}{2} (e+f x)\right) \sec (e+f x) (\sec (e+f x)+1)^3 (c \cos (e+f x)+d) \left(\frac{c \sec (e) \left(c \left(c \left(22 c^2+9 c d+2 d^2\right) \sin (2 e+3 f x)+3 \left(3 c^3+38 c^2 d+12 c d^2+2 d^3\right) \sin (e+2 f x)+3 \left(3 c^3-6 c^2 d-6 c d^2-2 d^3\right) \sin (3 e+2 f x)\right)-3 \left(6 c^4-3 c^3 d+30 c^2 d^2+18 c d^3+4 d^4\right) \sin (2 e+f x)+6 \left(8 c^4+6 c^3 d+30 c^2 d^2+9 c d^3+2 d^4\right) \sin (f x)\right)-2 d \left(66 c^4+27 c^3 d+50 c^2 d^2+18 c d^3+4 d^4\right) \tan (e)}{c^3}-\frac{120 i (\cos (e)-i \sin (e)) (c \cos (e+f x)+d)^3 \tan ^{-1}\left(\frac{(\sin (e)+i \cos (e)) \left(\tan \left(\frac{f x}{2}\right) (c \cos (e)-d)+c \sin (e)\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{192 f (c+d)^3 (c+d \sec (e+f x))^4}","\frac{5 a^3 \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f \sqrt{c-d} (c+d)^{7/2}}+\frac{5 a^3 (c+4 d) \tan (e+f x)}{6 d f (c+d)^3 (c+d \sec (e+f x))}-\frac{5 a^3 (c-d) \tan (e+f x)}{6 d f (c+d)^2 (c+d \sec (e+f x))^2}+\frac{a \tan (e+f x) (a \sec (e+f x)+a)^2}{3 f (c+d) (c+d \sec (e+f x))^3}",1,"(a^3*(d + c*Cos[e + f*x])*Sec[(e + f*x)/2]^6*Sec[e + f*x]*(1 + Sec[e + f*x])^3*(((-120*I)*ArcTan[((I*Cos[e] + Sin[e])*(c*Sin[e] + (-d + c*Cos[e])*Tan[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*(d + c*Cos[e + f*x])^3*(Cos[e] - I*Sin[e]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + (c*Sec[e]*(6*(8*c^4 + 6*c^3*d + 30*c^2*d^2 + 9*c*d^3 + 2*d^4)*Sin[f*x] - 3*(6*c^4 - 3*c^3*d + 30*c^2*d^2 + 18*c*d^3 + 4*d^4)*Sin[2*e + f*x] + c*(3*(3*c^3 + 38*c^2*d + 12*c*d^2 + 2*d^3)*Sin[e + 2*f*x] + 3*(3*c^3 - 6*c^2*d - 6*c*d^2 - 2*d^3)*Sin[3*e + 2*f*x] + c*(22*c^2 + 9*c*d + 2*d^2)*Sin[2*e + 3*f*x])) - 2*d*(66*c^4 + 27*c^3*d + 50*c^2*d^2 + 18*c*d^3 + 4*d^4)*Tan[e])/c^3))/(192*(c + d)^3*f*(c + d*Sec[e + f*x])^4)","C",1
209,1,274,266,9.1908741,"\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^5} \, dx","Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^5,x]","\frac{a^3 \left(-\frac{120 (4 c-3 d) \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}-\frac{\sin (e+f x) \left(-88 c^4 \cos (3 (e+f x))-72 c^4+36 c^3 d \cos (3 (e+f x))-478 c^3 d+37 c^2 d^2 \cos (3 (e+f x))+336 c^2 d^2+\left(-296 c^4-84 c^3 d-577 c^2 d^2+984 c d^3+198 d^4\right) \cos (e+f x)+\left(-72 c^4-470 c^3 d+384 c^2 d^2+200 c d^3+48 d^4\right) \cos (2 (e+f x))+24 c d^3 \cos (3 (e+f x))+28 c d^3+6 d^4 \cos (3 (e+f x))+336 d^4\right)}{(c \cos (e+f x)+d)^4}\right)}{96 f (c-d) (c+d)^4}","\frac{5 a^3 (4 c-3 d) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{4 f (c-d)^{3/2} (c+d)^{9/2}}+\frac{5 a^3 (4 c-3 d) (c+4 d) \tan (e+f x)}{24 d f (c-d) (c+d)^4 (c+d \sec (e+f x))}-\frac{5 a^3 (4 c-3 d) \tan (e+f x)}{24 d f (c+d)^3 (c+d \sec (e+f x))^2}-\frac{d \tan (e+f x) (a \sec (e+f x)+a)^3}{4 f \left(c^2-d^2\right) (c+d \sec (e+f x))^4}+\frac{a (4 c-3 d) \tan (e+f x) (a \sec (e+f x)+a)^2}{12 f (c-d) (c+d)^2 (c+d \sec (e+f x))^3}",1,"(a^3*((-120*(4*c - 3*d)*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] - ((-72*c^4 - 478*c^3*d + 336*c^2*d^2 + 28*c*d^3 + 336*d^4 + (-296*c^4 - 84*c^3*d - 577*c^2*d^2 + 984*c*d^3 + 198*d^4)*Cos[e + f*x] + (-72*c^4 - 470*c^3*d + 384*c^2*d^2 + 200*c*d^3 + 48*d^4)*Cos[2*(e + f*x)] - 88*c^4*Cos[3*(e + f*x)] + 36*c^3*d*Cos[3*(e + f*x)] + 37*c^2*d^2*Cos[3*(e + f*x)] + 24*c*d^3*Cos[3*(e + f*x)] + 6*d^4*Cos[3*(e + f*x)])*Sin[e + f*x])/(d + c*Cos[e + f*x])^4))/(96*(c - d)*(c + d)^4*f)","A",1
210,1,1243,183,6.4586622,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^4)/(a + a*Sec[e + f*x]),x]","\frac{\cos \left(\frac{e}{2}+\frac{f x}{2}\right) \sec \left(\frac{e}{2}\right) \sec (e) \left(-18 \sin \left(\frac{f x}{2}\right) c^4+18 \sin \left(\frac{3 f x}{2}\right) c^4-18 \sin \left(2 e+\frac{f x}{2}\right) c^4+18 \sin \left(2 e+\frac{3 f x}{2}\right) c^4-6 \sin \left(2 e+\frac{5 f x}{2}\right) c^4-6 \sin \left(4 e+\frac{5 f x}{2}\right) c^4+6 \sin \left(2 e+\frac{7 f x}{2}\right) c^4+6 \sin \left(4 e+\frac{7 f x}{2}\right) c^4+72 d \sin \left(\frac{f x}{2}\right) c^3-72 d \sin \left(\frac{3 f x}{2}\right) c^3+72 d \sin \left(2 e+\frac{f x}{2}\right) c^3-72 d \sin \left(2 e+\frac{3 f x}{2}\right) c^3+24 d \sin \left(2 e+\frac{5 f x}{2}\right) c^3+24 d \sin \left(4 e+\frac{5 f x}{2}\right) c^3-24 d \sin \left(2 e+\frac{7 f x}{2}\right) c^3-24 d \sin \left(4 e+\frac{7 f x}{2}\right) c^3-36 d^2 \sin \left(\frac{f x}{2}\right) c^2+180 d^2 \sin \left(\frac{3 f x}{2}\right) c^2-72 d^2 \sin \left(e-\frac{f x}{2}\right) c^2-36 d^2 \sin \left(e+\frac{f x}{2}\right) c^2-144 d^2 \sin \left(2 e+\frac{f x}{2}\right) c^2+72 d^2 \sin \left(e+\frac{3 f x}{2}\right) c^2+72 d^2 \sin \left(2 e+\frac{3 f x}{2}\right) c^2-36 d^2 \sin \left(3 e+\frac{3 f x}{2}\right) c^2+36 d^2 \sin \left(e+\frac{5 f x}{2}\right) c^2-36 d^2 \sin \left(4 e+\frac{5 f x}{2}\right) c^2+72 d^2 \sin \left(2 e+\frac{7 f x}{2}\right) c^2+36 d^2 \sin \left(3 e+\frac{7 f x}{2}\right) c^2+36 d^2 \sin \left(4 e+\frac{7 f x}{2}\right) c^2+24 d^3 \sin \left(\frac{f x}{2}\right) c-108 d^3 \sin \left(\frac{3 f x}{2}\right) c+48 d^3 \sin \left(e-\frac{f x}{2}\right) c+24 d^3 \sin \left(e+\frac{f x}{2}\right) c+96 d^3 \sin \left(2 e+\frac{f x}{2}\right) c-36 d^3 \sin \left(e+\frac{3 f x}{2}\right) c-36 d^3 \sin \left(2 e+\frac{3 f x}{2}\right) c+36 d^3 \sin \left(3 e+\frac{3 f x}{2}\right) c-12 d^3 \sin \left(e+\frac{5 f x}{2}\right) c+12 d^3 \sin \left(2 e+\frac{5 f x}{2}\right) c+12 d^3 \sin \left(3 e+\frac{5 f x}{2}\right) c+36 d^3 \sin \left(4 e+\frac{5 f x}{2}\right) c-48 d^3 \sin \left(2 e+\frac{7 f x}{2}\right) c-24 d^3 \sin \left(3 e+\frac{7 f x}{2}\right) c-24 d^3 \sin \left(4 e+\frac{7 f x}{2}\right) c+6 d^4 \sin \left(\frac{f x}{2}\right)+39 d^4 \sin \left(\frac{3 f x}{2}\right)-24 d^4 \sin \left(e-\frac{f x}{2}\right)-6 d^4 \sin \left(e+\frac{f x}{2}\right)-24 d^4 \sin \left(2 e+\frac{f x}{2}\right)+21 d^4 \sin \left(e+\frac{3 f x}{2}\right)+9 d^4 \sin \left(2 e+\frac{3 f x}{2}\right)-9 d^4 \sin \left(3 e+\frac{3 f x}{2}\right)+7 d^4 \sin \left(e+\frac{5 f x}{2}\right)+d^4 \sin \left(2 e+\frac{5 f x}{2}\right)-3 d^4 \sin \left(3 e+\frac{5 f x}{2}\right)-9 d^4 \sin \left(4 e+\frac{5 f x}{2}\right)+16 d^4 \sin \left(2 e+\frac{7 f x}{2}\right)+10 d^4 \sin \left(3 e+\frac{7 f x}{2}\right)+6 d^4 \sin \left(4 e+\frac{7 f x}{2}\right)\right) (c+d \sec (e+f x))^4}{48 f (d+c \cos (e+f x))^4 (\sec (e+f x) a+a)}+\frac{\left(3 d^4-12 c d^3+12 c^2 d^2-8 c^3 d\right) \cos ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \cos ^3(e+f x) \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right) (c+d \sec (e+f x))^4}{f (d+c \cos (e+f x))^4 (\sec (e+f x) a+a)}+\frac{\left(-3 d^4+12 c d^3-12 c^2 d^2+8 c^3 d\right) \cos ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \cos ^3(e+f x) \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right) (c+d \sec (e+f x))^4}{f (d+c \cos (e+f x))^4 (\sec (e+f x) a+a)}","\frac{d \left(8 c^3-12 c^2 d+12 c d^2-3 d^3\right) \tanh ^{-1}(\sin (e+f x))}{2 a f}-\frac{d \tan (e+f x) \left(d \left(6 c^2-20 c d+9 d^2\right) \sec (e+f x)+4 \left(3 c^3-16 c^2 d+12 c d^2-4 d^3\right)\right)}{6 a f}+\frac{(c-d) \tan (e+f x) (c+d \sec (e+f x))^3}{f (a \sec (e+f x)+a)}-\frac{d (3 c-4 d) \tan (e+f x) (c+d \sec (e+f x))^2}{3 a f}",1,"((-8*c^3*d + 12*c^2*d^2 - 12*c*d^3 + 3*d^4)*Cos[e/2 + (f*x)/2]^2*Cos[e + f*x]^3*Log[Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2]]*(c + d*Sec[e + f*x])^4)/(f*(d + c*Cos[e + f*x])^4*(a + a*Sec[e + f*x])) + ((8*c^3*d - 12*c^2*d^2 + 12*c*d^3 - 3*d^4)*Cos[e/2 + (f*x)/2]^2*Cos[e + f*x]^3*Log[Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2]]*(c + d*Sec[e + f*x])^4)/(f*(d + c*Cos[e + f*x])^4*(a + a*Sec[e + f*x])) + (Cos[e/2 + (f*x)/2]*Sec[e/2]*Sec[e]*(c + d*Sec[e + f*x])^4*(-18*c^4*Sin[(f*x)/2] + 72*c^3*d*Sin[(f*x)/2] - 36*c^2*d^2*Sin[(f*x)/2] + 24*c*d^3*Sin[(f*x)/2] + 6*d^4*Sin[(f*x)/2] + 18*c^4*Sin[(3*f*x)/2] - 72*c^3*d*Sin[(3*f*x)/2] + 180*c^2*d^2*Sin[(3*f*x)/2] - 108*c*d^3*Sin[(3*f*x)/2] + 39*d^4*Sin[(3*f*x)/2] - 72*c^2*d^2*Sin[e - (f*x)/2] + 48*c*d^3*Sin[e - (f*x)/2] - 24*d^4*Sin[e - (f*x)/2] - 36*c^2*d^2*Sin[e + (f*x)/2] + 24*c*d^3*Sin[e + (f*x)/2] - 6*d^4*Sin[e + (f*x)/2] - 18*c^4*Sin[2*e + (f*x)/2] + 72*c^3*d*Sin[2*e + (f*x)/2] - 144*c^2*d^2*Sin[2*e + (f*x)/2] + 96*c*d^3*Sin[2*e + (f*x)/2] - 24*d^4*Sin[2*e + (f*x)/2] + 72*c^2*d^2*Sin[e + (3*f*x)/2] - 36*c*d^3*Sin[e + (3*f*x)/2] + 21*d^4*Sin[e + (3*f*x)/2] + 18*c^4*Sin[2*e + (3*f*x)/2] - 72*c^3*d*Sin[2*e + (3*f*x)/2] + 72*c^2*d^2*Sin[2*e + (3*f*x)/2] - 36*c*d^3*Sin[2*e + (3*f*x)/2] + 9*d^4*Sin[2*e + (3*f*x)/2] - 36*c^2*d^2*Sin[3*e + (3*f*x)/2] + 36*c*d^3*Sin[3*e + (3*f*x)/2] - 9*d^4*Sin[3*e + (3*f*x)/2] + 36*c^2*d^2*Sin[e + (5*f*x)/2] - 12*c*d^3*Sin[e + (5*f*x)/2] + 7*d^4*Sin[e + (5*f*x)/2] - 6*c^4*Sin[2*e + (5*f*x)/2] + 24*c^3*d*Sin[2*e + (5*f*x)/2] + 12*c*d^3*Sin[2*e + (5*f*x)/2] + d^4*Sin[2*e + (5*f*x)/2] + 12*c*d^3*Sin[3*e + (5*f*x)/2] - 3*d^4*Sin[3*e + (5*f*x)/2] - 6*c^4*Sin[4*e + (5*f*x)/2] + 24*c^3*d*Sin[4*e + (5*f*x)/2] - 36*c^2*d^2*Sin[4*e + (5*f*x)/2] + 36*c*d^3*Sin[4*e + (5*f*x)/2] - 9*d^4*Sin[4*e + (5*f*x)/2] + 6*c^4*Sin[2*e + (7*f*x)/2] - 24*c^3*d*Sin[2*e + (7*f*x)/2] + 72*c^2*d^2*Sin[2*e + (7*f*x)/2] - 48*c*d^3*Sin[2*e + (7*f*x)/2] + 16*d^4*Sin[2*e + (7*f*x)/2] + 36*c^2*d^2*Sin[3*e + (7*f*x)/2] - 24*c*d^3*Sin[3*e + (7*f*x)/2] + 10*d^4*Sin[3*e + (7*f*x)/2] + 6*c^4*Sin[4*e + (7*f*x)/2] - 24*c^3*d*Sin[4*e + (7*f*x)/2] + 36*c^2*d^2*Sin[4*e + (7*f*x)/2] - 24*c*d^3*Sin[4*e + (7*f*x)/2] + 6*d^4*Sin[4*e + (7*f*x)/2]))/(48*f*(d + c*Cos[e + f*x])^4*(a + a*Sec[e + f*x]))","B",1
211,1,275,117,2.6577292,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^3}{a+a \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^3)/(a + a*Sec[e + f*x]),x]","\frac{\cos ^6\left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \left(\left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) \left(3 d \left(2 c^2-2 c d+d^2\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-3 d \left(2 c^2-2 c d+d^2\right) \tan ^2\left(\frac{1}{2} (e+f x)\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-2 \left(c^3-3 c^2 d+9 c d^2-3 d^3\right) \tan \left(\frac{1}{2} (e+f x)\right)+2 (c-d)^3 \tan ^3\left(\frac{1}{2} (e+f x)\right)\right)+16 d^3 \sin ^4\left(\frac{1}{2} (e+f x)\right) \csc ^3(e+f x)\right)}{a f (\cos (e+f x)+1)}","\frac{3 d \left(2 c^2-2 c d+d^2\right) \tanh ^{-1}(\sin (e+f x))}{2 a f}-\frac{d \tan (e+f x) \left(4 \left(c^2-3 c d+d^2\right)+d (2 c-3 d) \sec (e+f x)\right)}{2 a f}+\frac{(c-d) \tan (e+f x) (c+d \sec (e+f x))^2}{f (a \sec (e+f x)+a)}",1,"(Cos[(e + f*x)/2]^6*Sec[e + f*x]^2*(16*d^3*Csc[e + f*x]^3*Sin[(e + f*x)/2]^4 + (-1 + Tan[(e + f*x)/2]^2)*(3*d*(2*c^2 - 2*c*d + d^2)*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - 2*(c^3 - 3*c^2*d + 9*c*d^2 - 3*d^3)*Tan[(e + f*x)/2] - 3*d*(2*c^2 - 2*c*d + d^2)*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Tan[(e + f*x)/2]^2 + 2*(c - d)^3*Tan[(e + f*x)/2]^3)))/(a*f*(1 + Cos[e + f*x]))","B",1
212,1,237,68,1.734811,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^2}{a+a \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^2)/(a + a*Sec[e + f*x]),x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \cos (e+f x) (c+d \sec (e+f x))^2 \left((c-d)^2 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right)+d \cos \left(\frac{1}{2} (e+f x)\right) \left(\frac{d \sin (f x)}{\left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}-(2 c-d) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)\right)}{a f (\sec (e+f x)+1) (c \cos (e+f x)+d)^2}","\frac{d (2 c-d) \tanh ^{-1}(\sin (e+f x))}{a f}+\frac{(c-d)^2 \tan (e+f x)}{f (a \sec (e+f x)+a)}+\frac{d^2 \tan (e+f x)}{a f}",1,"(2*Cos[(e + f*x)/2]*Cos[e + f*x]*(c + d*Sec[e + f*x])^2*((c - d)^2*Sec[e/2]*Sin[(f*x)/2] + d*Cos[(e + f*x)/2]*(-((2*c - d)*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])) + (d*Sin[f*x])/((Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))))/(a*f*(d + c*Cos[e + f*x])^2*(1 + Sec[e + f*x]))","B",1
213,1,109,43,0.2655419,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))}{a+a \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x]))/(a + a*Sec[e + f*x]),x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \left((c-d) \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right)+d \cos \left(\frac{1}{2} (e+f x)\right) \left(\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}{a f (\cos (e+f x)+1)}","\frac{(c-d) \tan (e+f x)}{f (a \sec (e+f x)+a)}+\frac{d \tanh ^{-1}(\sin (e+f x))}{a f}",1,"(2*Cos[(e + f*x)/2]*(d*Cos[(e + f*x)/2]*(-Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + (c - d)*Sec[e/2]*Sin[(f*x)/2]))/(a*f*(1 + Cos[e + f*x]))","B",1
214,1,160,83,0.6684587,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) (c+d \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])),x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \left(\sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right)+\frac{2 d (\sin (e)+i \cos (e)) \cos \left(\frac{1}{2} (e+f x)\right) \tan ^{-1}\left(\frac{(\sin (e)+i \cos (e)) \left(\tan \left(\frac{f x}{2}\right) (c \cos (e)-d)+c \sin (e)\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{a f (c-d) (\cos (e+f x)+1)}","\frac{\tan (e+f x)}{f (c-d) (a \sec (e+f x)+a)}-\frac{2 d \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{a f (c-d)^{3/2} \sqrt{c+d}}",1,"(2*Cos[(e + f*x)/2]*((2*d*ArcTan[((I*Cos[e] + Sin[e])*(c*Sin[e] + (-d + c*Cos[e])*Tan[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*Cos[(e + f*x)/2]*(I*Cos[e] + Sin[e]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + Sec[e/2]*Sin[(f*x)/2]))/(a*(c - d)*f*(1 + Cos[e + f*x]))","C",1
215,1,286,145,3.2484808,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) (c+d \sec (e+f x))^2} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^2),x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) (c \cos (e+f x)+d) \left(\frac{2 d (2 c+d) (\sin (e)+i \cos (e)) \cos \left(\frac{1}{2} (e+f x)\right) (c \cos (e+f x)+d) \tan ^{-1}\left(\frac{(\sin (e)+i \cos (e)) \left(\tan \left(\frac{f x}{2}\right) (c \cos (e)-d)+c \sin (e)\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{(c+d) \sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}+\frac{d^2 \cos \left(\frac{1}{2} (e+f x)\right) (c \sin (f x)-d \sin (e))}{c (c+d) \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right)}+\sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) (c \cos (e+f x)+d)\right)}{a f (c-d)^2 (\sec (e+f x)+1) (c+d \sec (e+f x))^2}","-\frac{d \tan (e+f x)}{f \left(c^2-d^2\right) (a \sec (e+f x)+a) (c+d \sec (e+f x))}-\frac{2 d (2 c+d) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{a f (c-d)^{5/2} (c+d)^{3/2}}+\frac{(c+2 d) \tan (e+f x)}{f (c-d)^2 (c+d) (a \sec (e+f x)+a)}",1,"(2*Cos[(e + f*x)/2]*(d + c*Cos[e + f*x])*Sec[e + f*x]^3*((2*d*(2*c + d)*ArcTan[((I*Cos[e] + Sin[e])*(c*Sin[e] + (-d + c*Cos[e])*Tan[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*Cos[(e + f*x)/2]*(d + c*Cos[e + f*x])*(I*Cos[e] + Sin[e]))/((c + d)*Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + (d + c*Cos[e + f*x])*Sec[e/2]*Sin[(f*x)/2] + (d^2*Cos[(e + f*x)/2]*(-(d*Sin[e]) + c*Sin[f*x]))/(c*(c + d)*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2]))))/(a*(c - d)^2*f*(1 + Sec[e + f*x])*(c + d*Sec[e + f*x])^2)","C",1
216,1,1422,207,6.8232887,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) (c+d \sec (e+f x))^3} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^3),x]","\frac{\left(2 c^2+2 d c+d^2\right) \cos ^2\left(\frac{e}{2}+\frac{f x}{2}\right) (d+c \cos (e+f x))^3 \left(-\frac{6 i d \tan ^{-1}\left(\sec \left(\frac{f x}{2}\right) \left(\frac{\cos (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}-\frac{i \sin (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}\right) \left(i c \sin \left(e+\frac{f x}{2}\right)-i d \sin \left(\frac{f x}{2}\right)\right)\right) \cos (e)}{\sqrt{c^2-d^2} f \sqrt{\cos (2 e)-i \sin (2 e)}}-\frac{6 d \tan ^{-1}\left(\sec \left(\frac{f x}{2}\right) \left(\frac{\cos (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}-\frac{i \sin (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}\right) \left(i c \sin \left(e+\frac{f x}{2}\right)-i d \sin \left(\frac{f x}{2}\right)\right)\right) \sin (e)}{\sqrt{c^2-d^2} f \sqrt{\cos (2 e)-i \sin (2 e)}}\right) \sec ^4(e+f x)}{(d-c)^3 (c+d)^2 (\sec (e+f x) a+a) (c+d \sec (e+f x))^3}+\frac{\cos \left(\frac{e}{2}+\frac{f x}{2}\right) (d+c \cos (e+f x)) \sec \left(\frac{e}{2}\right) \sec (e) \left(4 \sin \left(e-\frac{f x}{2}\right) c^6-4 \sin \left(e+\frac{f x}{2}\right) c^6+2 \sin \left(e+\frac{3 f x}{2}\right) c^6+2 \sin \left(3 e+\frac{3 f x}{2}\right) c^6-2 \sin \left(e+\frac{5 f x}{2}\right) c^6-2 \sin \left(3 e+\frac{5 f x}{2}\right) c^6+8 d \sin \left(\frac{f x}{2}\right) c^5-8 d \sin \left(\frac{3 f x}{2}\right) c^5+8 d \sin \left(e-\frac{f x}{2}\right) c^5-8 d \sin \left(e+\frac{f x}{2}\right) c^5+8 d \sin \left(2 e+\frac{f x}{2}\right) c^5+4 d \sin \left(e+\frac{3 f x}{2}\right) c^5-8 d \sin \left(2 e+\frac{3 f x}{2}\right) c^5+4 d \sin \left(3 e+\frac{3 f x}{2}\right) c^5-4 d \sin \left(e+\frac{5 f x}{2}\right) c^5-4 d \sin \left(3 e+\frac{5 f x}{2}\right) c^5+10 d^2 \sin \left(\frac{f x}{2}\right) c^4-22 d^2 \sin \left(\frac{3 f x}{2}\right) c^4+18 d^2 \sin \left(e-\frac{f x}{2}\right) c^4-6 d^2 \sin \left(e+\frac{f x}{2}\right) c^4+22 d^2 \sin \left(2 e+\frac{f x}{2}\right) c^4-4 d^2 \sin \left(e+\frac{3 f x}{2}\right) c^4-16 d^2 \sin \left(2 e+\frac{3 f x}{2}\right) c^4+2 d^2 \sin \left(3 e+\frac{3 f x}{2}\right) c^4-8 d^2 \sin \left(e+\frac{5 f x}{2}\right) c^4-6 d^2 \sin \left(2 e+\frac{5 f x}{2}\right) c^4-2 d^2 \sin \left(3 e+\frac{5 f x}{2}\right) c^4-11 d^3 \sin \left(\frac{f x}{2}\right) c^3-27 d^3 \sin \left(\frac{3 f x}{2}\right) c^3+35 d^3 \sin \left(e-\frac{f x}{2}\right) c^3-7 d^3 \sin \left(e+\frac{f x}{2}\right) c^3+17 d^3 \sin \left(2 e+\frac{f x}{2}\right) c^3-19 d^3 \sin \left(e+\frac{3 f x}{2}\right) c^3-d^3 \sin \left(2 e+\frac{3 f x}{2}\right) c^3+7 d^3 \sin \left(3 e+\frac{3 f x}{2}\right) c^3-2 d^3 \sin \left(e+\frac{5 f x}{2}\right) c^3-2 d^3 \sin \left(2 e+\frac{5 f x}{2}\right) c^3-17 d^4 \sin \left(\frac{f x}{2}\right) c^2-5 d^4 \sin \left(\frac{3 f x}{2}\right) c^2+25 d^4 \sin \left(e-\frac{f x}{2}\right) c^2+5 d^4 \sin \left(e+\frac{f x}{2}\right) c^2+13 d^4 \sin \left(2 e+\frac{f x}{2}\right) c^2-5 d^4 \sin \left(e+\frac{3 f x}{2}\right) c^2+2 d^4 \sin \left(2 e+\frac{3 f x}{2}\right) c^2+2 d^4 \sin \left(3 e+\frac{3 f x}{2}\right) c^2+d^4 \sin \left(e+\frac{5 f x}{2}\right) c^2+d^4 \sin \left(2 e+\frac{5 f x}{2}\right) c^2-2 d^5 \sin \left(\frac{f x}{2}\right) c+2 d^5 \sin \left(\frac{3 f x}{2}\right) c+2 d^5 \sin \left(e-\frac{f x}{2}\right) c+2 d^5 \sin \left(e+\frac{f x}{2}\right) c+2 d^5 \sin \left(2 e+\frac{f x}{2}\right) c+2 d^5 \sin \left(e+\frac{3 f x}{2}\right) c-2 d^5 \sin \left(2 e+\frac{3 f x}{2}\right) c-2 d^5 \sin \left(3 e+\frac{3 f x}{2}\right) c+2 d^6 \sin \left(\frac{f x}{2}\right)-2 d^6 \sin \left(e-\frac{f x}{2}\right)-2 d^6 \sin \left(e+\frac{f x}{2}\right)-2 d^6 \sin \left(2 e+\frac{f x}{2}\right)\right) \sec ^4(e+f x)}{8 c^2 (d-c)^3 (c+d)^2 f (\sec (e+f x) a+a) (c+d \sec (e+f x))^3}","-\frac{3 d \left(2 c^2+2 c d+d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{a f (c-d)^{7/2} (c+d)^{5/2}}+\frac{d (2 c+d) (c+4 d) \tan (e+f x)}{2 a f (c-d)^3 (c+d)^2 (c+d \sec (e+f x))}+\frac{d (2 c+3 d) \tan (e+f x)}{2 a f (c-d)^2 (c+d) (c+d \sec (e+f x))^2}+\frac{\tan (e+f x)}{f (c-d) (a \sec (e+f x)+a) (c+d \sec (e+f x))^2}",1,"((2*c^2 + 2*c*d + d^2)*Cos[e/2 + (f*x)/2]^2*(d + c*Cos[e + f*x])^3*Sec[e + f*x]^4*(((-6*I)*d*ArcTan[Sec[(f*x)/2]*(Cos[e]/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (I*Sin[e])/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]))*((-I)*d*Sin[(f*x)/2] + I*c*Sin[e + (f*x)/2])]*Cos[e])/(Sqrt[c^2 - d^2]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (6*d*ArcTan[Sec[(f*x)/2]*(Cos[e]/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (I*Sin[e])/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]))*((-I)*d*Sin[(f*x)/2] + I*c*Sin[e + (f*x)/2])]*Sin[e])/(Sqrt[c^2 - d^2]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((-c + d)^3*(c + d)^2*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^3) + (Cos[e/2 + (f*x)/2]*(d + c*Cos[e + f*x])*Sec[e/2]*Sec[e]*Sec[e + f*x]^4*(8*c^5*d*Sin[(f*x)/2] + 10*c^4*d^2*Sin[(f*x)/2] - 11*c^3*d^3*Sin[(f*x)/2] - 17*c^2*d^4*Sin[(f*x)/2] - 2*c*d^5*Sin[(f*x)/2] + 2*d^6*Sin[(f*x)/2] - 8*c^5*d*Sin[(3*f*x)/2] - 22*c^4*d^2*Sin[(3*f*x)/2] - 27*c^3*d^3*Sin[(3*f*x)/2] - 5*c^2*d^4*Sin[(3*f*x)/2] + 2*c*d^5*Sin[(3*f*x)/2] + 4*c^6*Sin[e - (f*x)/2] + 8*c^5*d*Sin[e - (f*x)/2] + 18*c^4*d^2*Sin[e - (f*x)/2] + 35*c^3*d^3*Sin[e - (f*x)/2] + 25*c^2*d^4*Sin[e - (f*x)/2] + 2*c*d^5*Sin[e - (f*x)/2] - 2*d^6*Sin[e - (f*x)/2] - 4*c^6*Sin[e + (f*x)/2] - 8*c^5*d*Sin[e + (f*x)/2] - 6*c^4*d^2*Sin[e + (f*x)/2] - 7*c^3*d^3*Sin[e + (f*x)/2] + 5*c^2*d^4*Sin[e + (f*x)/2] + 2*c*d^5*Sin[e + (f*x)/2] - 2*d^6*Sin[e + (f*x)/2] + 8*c^5*d*Sin[2*e + (f*x)/2] + 22*c^4*d^2*Sin[2*e + (f*x)/2] + 17*c^3*d^3*Sin[2*e + (f*x)/2] + 13*c^2*d^4*Sin[2*e + (f*x)/2] + 2*c*d^5*Sin[2*e + (f*x)/2] - 2*d^6*Sin[2*e + (f*x)/2] + 2*c^6*Sin[e + (3*f*x)/2] + 4*c^5*d*Sin[e + (3*f*x)/2] - 4*c^4*d^2*Sin[e + (3*f*x)/2] - 19*c^3*d^3*Sin[e + (3*f*x)/2] - 5*c^2*d^4*Sin[e + (3*f*x)/2] + 2*c*d^5*Sin[e + (3*f*x)/2] - 8*c^5*d*Sin[2*e + (3*f*x)/2] - 16*c^4*d^2*Sin[2*e + (3*f*x)/2] - c^3*d^3*Sin[2*e + (3*f*x)/2] + 2*c^2*d^4*Sin[2*e + (3*f*x)/2] - 2*c*d^5*Sin[2*e + (3*f*x)/2] + 2*c^6*Sin[3*e + (3*f*x)/2] + 4*c^5*d*Sin[3*e + (3*f*x)/2] + 2*c^4*d^2*Sin[3*e + (3*f*x)/2] + 7*c^3*d^3*Sin[3*e + (3*f*x)/2] + 2*c^2*d^4*Sin[3*e + (3*f*x)/2] - 2*c*d^5*Sin[3*e + (3*f*x)/2] - 2*c^6*Sin[e + (5*f*x)/2] - 4*c^5*d*Sin[e + (5*f*x)/2] - 8*c^4*d^2*Sin[e + (5*f*x)/2] - 2*c^3*d^3*Sin[e + (5*f*x)/2] + c^2*d^4*Sin[e + (5*f*x)/2] - 6*c^4*d^2*Sin[2*e + (5*f*x)/2] - 2*c^3*d^3*Sin[2*e + (5*f*x)/2] + c^2*d^4*Sin[2*e + (5*f*x)/2] - 2*c^6*Sin[3*e + (5*f*x)/2] - 4*c^5*d*Sin[3*e + (5*f*x)/2] - 2*c^4*d^2*Sin[3*e + (5*f*x)/2]))/(8*c^2*(-c + d)^3*(c + d)^2*f*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^3)","C",0
217,1,446,258,4.0279236,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^5)/(a + a*Sec[e + f*x])^2,x]","\frac{240 d^2 \left(-4 c^3+8 c^2 d-7 c d^2+2 d^3\right) \cos ^4\left(\frac{1}{2} (e+f x)\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+2 \sin \left(\frac{1}{2} (e+f x)\right) \cos \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) \left(2 c^5 \cos (3 (e+f x))+2 c^5 \cos (4 (e+f x))+6 c^5+20 c^4 d \cos (3 (e+f x))+5 c^4 d \cos (4 (e+f x))+15 c^4 d-100 c^3 d^2 \cos (3 (e+f x))-40 c^3 d^2 \cos (4 (e+f x))-120 c^3 d^2+280 c^2 d^3 \cos (3 (e+f x))+100 c^2 d^3 \cos (4 (e+f x))+420 c^2 d^3+\left(6 c^5+60 c^4 d-300 c^3 d^2+840 c^2 d^3-585 c d^4+190 d^5\right) \cos (e+f x)+4 \left(2 c^5+5 c^4 d-40 c^3 d^2+130 c^2 d^3-95 c d^4+30 d^5\right) \cos (2 (e+f x))-215 c d^4 \cos (3 (e+f x))-80 c d^4 \cos (4 (e+f x))-300 c d^4+66 d^5 \cos (3 (e+f x))+24 d^5 \cos (4 (e+f x))+104 d^5\right)}{24 a^2 f (\cos (e+f x)+1)^2}","\frac{5 d^2 (2 c-d) \left(2 c^2-3 c d+2 d^2\right) \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}-\frac{d \left(c^2+10 c d-12 d^2\right) \tan (e+f x) (c+d \sec (e+f x))^2}{3 a^2 f}-\frac{d \tan (e+f x) \left(d \left(2 c^3+20 c^2 d-57 c d^2+30 d^3\right) \sec (e+f x)+4 \left(c^4+10 c^3 d-44 c^2 d^2+40 c d^3-12 d^4\right)\right)}{6 a^2 f}+\frac{(c-d) (c+10 d) \tan (e+f x) (c+d \sec (e+f x))^3}{3 f \left(a^2 \sec (e+f x)+a^2\right)}+\frac{(c-d) \tan (e+f x) (c+d \sec (e+f x))^4}{3 f (a \sec (e+f x)+a)^2}",1,"(240*d^2*(-4*c^3 + 8*c^2*d - 7*c*d^2 + 2*d^3)*Cos[(e + f*x)/2]^4*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + 2*Cos[(e + f*x)/2]*(6*c^5 + 15*c^4*d - 120*c^3*d^2 + 420*c^2*d^3 - 300*c*d^4 + 104*d^5 + (6*c^5 + 60*c^4*d - 300*c^3*d^2 + 840*c^2*d^3 - 585*c*d^4 + 190*d^5)*Cos[e + f*x] + 4*(2*c^5 + 5*c^4*d - 40*c^3*d^2 + 130*c^2*d^3 - 95*c*d^4 + 30*d^5)*Cos[2*(e + f*x)] + 2*c^5*Cos[3*(e + f*x)] + 20*c^4*d*Cos[3*(e + f*x)] - 100*c^3*d^2*Cos[3*(e + f*x)] + 280*c^2*d^3*Cos[3*(e + f*x)] - 215*c*d^4*Cos[3*(e + f*x)] + 66*d^5*Cos[3*(e + f*x)] + 2*c^5*Cos[4*(e + f*x)] + 5*c^4*d*Cos[4*(e + f*x)] - 40*c^3*d^2*Cos[4*(e + f*x)] + 100*c^2*d^3*Cos[4*(e + f*x)] - 80*c*d^4*Cos[4*(e + f*x)] + 24*d^5*Cos[4*(e + f*x)])*Sec[e + f*x]^3*Sin[(e + f*x)/2])/(24*a^2*f*(1 + Cos[e + f*x])^2)","A",1
218,1,310,193,2.8266949,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^4)/(a + a*Sec[e + f*x])^2,x]","\frac{2 \sin \left(\frac{1}{2} (e+f x)\right) \cos \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \left(2 c^4 \cos (3 (e+f x))+2 c^4+4 c^3 d \cos (3 (e+f x))+16 c^3 d-24 c^2 d^2 \cos (3 (e+f x))-60 c^2 d^2+6 \left(c^4+2 c^3 d-12 c^2 d^2+28 c d^3-10 d^4\right) \cos (e+f x)+\left(2 c^4+16 c^3 d-60 c^2 d^2+112 c d^3-43 d^4\right) \cos (2 (e+f x))+40 c d^3 \cos (3 (e+f x))+112 c d^3-16 d^4 \cos (3 (e+f x))-37 d^4\right)-24 d^2 \left(12 c^2-16 c d+7 d^2\right) \cos ^4\left(\frac{1}{2} (e+f x)\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{12 a^2 f (\cos (e+f x)+1)^2}","\frac{d^2 \left(12 c^2-16 c d+7 d^2\right) \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}-\frac{d \tan (e+f x) \left(d \left(2 c^2+16 c d-21 d^2\right) \sec (e+f x)+4 \left(c^3+8 c^2 d-20 c d^2+8 d^3\right)\right)}{6 a^2 f}+\frac{(c-d) (c+8 d) \tan (e+f x) (c+d \sec (e+f x))^2}{3 f \left(a^2 \sec (e+f x)+a^2\right)}+\frac{(c-d) \tan (e+f x) (c+d \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}",1,"(-24*d^2*(12*c^2 - 16*c*d + 7*d^2)*Cos[(e + f*x)/2]^4*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + 2*Cos[(e + f*x)/2]*(2*c^4 + 16*c^3*d - 60*c^2*d^2 + 112*c*d^3 - 37*d^4 + 6*(c^4 + 2*c^3*d - 12*c^2*d^2 + 28*c*d^3 - 10*d^4)*Cos[e + f*x] + (2*c^4 + 16*c^3*d - 60*c^2*d^2 + 112*c*d^3 - 43*d^4)*Cos[2*(e + f*x)] + 2*c^4*Cos[3*(e + f*x)] + 4*c^3*d*Cos[3*(e + f*x)] - 24*c^2*d^2*Cos[3*(e + f*x)] + 40*c*d^3*Cos[3*(e + f*x)] - 16*d^4*Cos[3*(e + f*x)])*Sec[e + f*x]^2*Sin[(e + f*x)/2])/(12*a^2*f*(1 + Cos[e + f*x])^2)","A",1
219,1,294,133,1.7032943,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^3)/(a + a*Sec[e + f*x])^2,x]","\frac{2 \cos ^6\left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \left(2 \left(2 c^3+3 c^2 d-12 c d^2+13 d^3\right) \tan \left(\frac{1}{2} (e+f x)\right)+6 d^2 (2 d-3 c) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+6 d^2 (3 c-2 d) \tan ^2\left(\frac{1}{2} (e+f x)\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-2 (c-d)^2 (2 c+7 d) \tan ^3\left(\frac{1}{2} (e+f x)\right)+32 (c-d)^3 \sin ^8\left(\frac{1}{2} (e+f x)\right) \csc ^5(e+f x)-8 (c-d)^3 \sin ^4\left(\frac{1}{2} (e+f x)\right) \csc ^3(e+f x)\right)}{3 a^2 f (\cos (e+f x)+1)^2}","\frac{\tan (e+f x) \left(c^3+4 c^2 d-d^2 (c-4 d) \sec (e+f x)-12 c d^2+10 d^3\right)}{3 f \left(a^2 \sec (e+f x)+a^2\right)}+\frac{d^2 (3 c-2 d) \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac{(c-d) \tan (e+f x) (c+d \sec (e+f x))^2}{3 f (a \sec (e+f x)+a)^2}",1,"(2*Cos[(e + f*x)/2]^6*Sec[e + f*x]*(6*d^2*(-3*c + 2*d)*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - 8*(c - d)^3*Csc[e + f*x]^3*Sin[(e + f*x)/2]^4 + 32*(c - d)^3*Csc[e + f*x]^5*Sin[(e + f*x)/2]^8 + 2*(2*c^3 + 3*c^2*d - 12*c*d^2 + 13*d^3)*Tan[(e + f*x)/2] + 6*(3*c - 2*d)*d^2*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Tan[(e + f*x)/2]^2 - 2*(c - d)^2*(2*c + 7*d)*Tan[(e + f*x)/2]^3))/(3*a^2*f*(1 + Cos[e + f*x])^2)","B",1
220,1,181,89,0.7849912,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^2)/(a + a*Sec[e + f*x])^2,x]","-\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \left(-4 \left(c^2+c d-2 d^2\right) \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \cos ^2\left(\frac{1}{2} (e+f x)\right)+(c-d)^2 \tan \left(\frac{e}{2}\right) \cos \left(\frac{1}{2} (e+f x)\right)+(c-d)^2 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right)+6 d^2 \cos ^3\left(\frac{1}{2} (e+f x)\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}{3 a^2 f (\cos (e+f x)+1)^2}","\frac{(c+5 d) (c-d) \tan (e+f x)}{3 f \left(a^2 \sec (e+f x)+a^2\right)}+\frac{d^2 \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac{(c-d)^2 \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2}",1,"(-2*Cos[(e + f*x)/2]*(6*d^2*Cos[(e + f*x)/2]^3*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + (c - d)^2*Sec[e/2]*Sin[(f*x)/2] - 4*(c^2 + c*d - 2*d^2)*Cos[(e + f*x)/2]^2*Sec[e/2]*Sin[(f*x)/2] + (c - d)^2*Cos[(e + f*x)/2]*Tan[e/2]))/(3*a^2*f*(1 + Cos[e + f*x])^2)","B",1
221,1,76,65,0.2217389,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x]))/(a + a*Sec[e + f*x])^2,x]","\frac{\sec \left(\frac{e}{2}\right) \cos \left(\frac{1}{2} (e+f x)\right) \left((2 c+d) \sin \left(e+\frac{3 f x}{2}\right)+3 (c+d) \sin \left(\frac{f x}{2}\right)-3 c \sin \left(e+\frac{f x}{2}\right)\right)}{3 a^2 f (\cos (e+f x)+1)^2}","\frac{(c+2 d) \tan (e+f x)}{3 f \left(a^2 \sec (e+f x)+a^2\right)}+\frac{(c-d) \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2}",1,"(Cos[(e + f*x)/2]*Sec[e/2]*(3*(c + d)*Sin[(f*x)/2] - 3*c*Sin[e + (f*x)/2] + (2*c + d)*Sin[e + (3*f*x)/2]))/(3*a^2*f*(1 + Cos[e + f*x])^2)","A",1
222,1,209,129,1.6491958,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^2 (c+d \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])),x]","\frac{\cos \left(\frac{1}{2} (e+f x)\right) \left(\sec \left(\frac{e}{2}\right) \left(-3 (c-2 d) \sin \left(e+\frac{f x}{2}\right)+(2 c-5 d) \sin \left(e+\frac{3 f x}{2}\right)+3 (c-3 d) \sin \left(\frac{f x}{2}\right)\right)-\frac{24 i d^2 (\cos (e)-i \sin (e)) \cos ^3\left(\frac{1}{2} (e+f x)\right) \tan ^{-1}\left(\frac{(\sin (e)+i \cos (e)) \left(\tan \left(\frac{f x}{2}\right) (c \cos (e)-d)+c \sin (e)\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{3 a^2 f (c-d)^2 (\cos (e+f x)+1)^2}","\frac{2 d^2 \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{a^2 f (c-d)^{5/2} \sqrt{c+d}}+\frac{(c-4 d) \tan (e+f x)}{3 f (c-d)^2 \left(a^2 \sec (e+f x)+a^2\right)}+\frac{\tan (e+f x)}{3 f (c-d) (a \sec (e+f x)+a)^2}",1,"(Cos[(e + f*x)/2]*(((-24*I)*d^2*ArcTan[((I*Cos[e] + Sin[e])*(c*Sin[e] + (-d + c*Cos[e])*Tan[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*Cos[(e + f*x)/2]^3*(Cos[e] - I*Sin[e]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + Sec[e/2]*(3*(c - 3*d)*Sin[(f*x)/2] - 3*(c - 2*d)*Sin[e + (f*x)/2] + (2*c - 5*d)*Sin[e + (3*f*x)/2])))/(3*a^2*(c - d)^2*f*(1 + Cos[e + f*x])^2)","C",1
223,1,376,211,3.9621042,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2),x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \sec ^4(e+f x) (c \cos (e+f x)+d) \left(\frac{12 d^2 (3 c+2 d) (\sin (e)+i \cos (e)) \cos ^3\left(\frac{1}{2} (e+f x)\right) (c \cos (e+f x)+d) \tan ^{-1}\left(\frac{(\sin (e)+i \cos (e)) \left(\tan \left(\frac{f x}{2}\right) (c \cos (e)-d)+c \sin (e)\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{(c+d) \sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}+\frac{6 d^3 \cos ^3\left(\frac{1}{2} (e+f x)\right) (c \sin (f x)-d \sin (e))}{c (c+d) \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right)}+(c-d) \tan \left(\frac{e}{2}\right) \cos \left(\frac{1}{2} (e+f x)\right) (c \cos (e+f x)+d)-4 (c-4 d) \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \cos ^2\left(\frac{1}{2} (e+f x)\right) (c \cos (e+f x)+d)+(c-d) \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) (c \cos (e+f x)+d)\right)}{3 a^2 f (d-c)^3 (\sec (e+f x)+1)^2 (c+d \sec (e+f x))^2}","\frac{d \left(c^2-6 c d-10 d^2\right) \tan (e+f x)}{3 a^2 f (c-d)^3 (c+d) (c+d \sec (e+f x))}+\frac{2 d^2 (3 c+2 d) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{a^2 f (c-d)^{7/2} (c+d)^{3/2}}+\frac{(c-6 d) \tan (e+f x)}{3 a^2 f (c-d)^2 (\sec (e+f x)+1) (c+d \sec (e+f x))}+\frac{\tan (e+f x)}{3 f (c-d) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))}",1,"(2*Cos[(e + f*x)/2]*(d + c*Cos[e + f*x])*Sec[e + f*x]^4*((12*d^2*(3*c + 2*d)*ArcTan[((I*Cos[e] + Sin[e])*(c*Sin[e] + (-d + c*Cos[e])*Tan[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*Cos[(e + f*x)/2]^3*(d + c*Cos[e + f*x])*(I*Cos[e] + Sin[e]))/((c + d)*Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + (c - d)*(d + c*Cos[e + f*x])*Sec[e/2]*Sin[(f*x)/2] - 4*(c - 4*d)*Cos[(e + f*x)/2]^2*(d + c*Cos[e + f*x])*Sec[e/2]*Sin[(f*x)/2] + (6*d^3*Cos[(e + f*x)/2]^3*(-(d*Sin[e]) + c*Sin[f*x]))/(c*(c + d)*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])) + (c - d)*Cos[(e + f*x)/2]*(d + c*Cos[e + f*x])*Tan[e/2]))/(3*a^2*(-c + d)^3*f*(1 + Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2)","C",1
224,1,2220,284,7.3472713,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^2 (c+d \sec (e+f x))^3} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^3),x]","\text{Result too large to show}","\frac{d^2 \left(12 c^2+16 c d+7 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{a^2 f (c-d)^{9/2} (c+d)^{5/2}}+\frac{d \left(2 c^2-16 c d-21 d^2\right) \tan (e+f x)}{6 a^2 f (c-d)^3 (c+d) (c+d \sec (e+f x))^2}+\frac{d \left(2 c^3-16 c^2 d-59 c d^2-32 d^3\right) \tan (e+f x)}{6 a^2 f (c-d)^4 (c+d)^2 (c+d \sec (e+f x))}+\frac{(c-8 d) \tan (e+f x)}{3 a^2 f (c-d)^2 (\sec (e+f x)+1) (c+d \sec (e+f x))^2}+\frac{\tan (e+f x)}{3 f (c-d) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))^2}",1,"((12*c^2 + 16*c*d + 7*d^2)*Cos[e/2 + (f*x)/2]^4*(d + c*Cos[e + f*x])^3*Sec[e + f*x]^5*(((-4*I)*d^2*ArcTan[Sec[(f*x)/2]*(Cos[e]/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (I*Sin[e])/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]))*((-I)*d*Sin[(f*x)/2] + I*c*Sin[e + (f*x)/2])]*Cos[e])/(Sqrt[c^2 - d^2]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (4*d^2*ArcTan[Sec[(f*x)/2]*(Cos[e]/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (I*Sin[e])/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]))*((-I)*d*Sin[(f*x)/2] + I*c*Sin[e + (f*x)/2])]*Sin[e])/(Sqrt[c^2 - d^2]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((-c + d)^4*(c + d)^2*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^3) + (Cos[e/2 + (f*x)/2]*(d + c*Cos[e + f*x])*Sec[e/2]*Sec[e]*Sec[e + f*x]^5*(-16*c^7*Sin[(f*x)/2] + 14*c^6*d*Sin[(f*x)/2] + 220*c^5*d^2*Sin[(f*x)/2] + 334*c^4*d^3*Sin[(f*x)/2] + 54*c^3*d^4*Sin[(f*x)/2] - 156*c^2*d^5*Sin[(f*x)/2] - 48*c*d^6*Sin[(f*x)/2] + 18*d^7*Sin[(f*x)/2] + 14*c^7*Sin[(3*f*x)/2] - 16*c^6*d*Sin[(3*f*x)/2] - 226*c^5*d^2*Sin[(3*f*x)/2] - 532*c^4*d^3*Sin[(3*f*x)/2] - 583*c^3*d^4*Sin[(3*f*x)/2] - 232*c^2*d^5*Sin[(3*f*x)/2] - 6*c*d^6*Sin[(3*f*x)/2] + 6*d^7*Sin[(3*f*x)/2] - 12*c^7*Sin[e - (f*x)/2] + 20*c^6*d*Sin[e - (f*x)/2] + 236*c^5*d^2*Sin[e - (f*x)/2] + 628*c^4*d^3*Sin[e - (f*x)/2] + 778*c^3*d^4*Sin[e - (f*x)/2] + 420*c^2*d^5*Sin[e - (f*x)/2] + 48*c*d^6*Sin[e - (f*x)/2] - 18*d^7*Sin[e - (f*x)/2] + 12*c^7*Sin[e + (f*x)/2] - 20*c^6*d*Sin[e + (f*x)/2] - 236*c^5*d^2*Sin[e + (f*x)/2] - 460*c^4*d^3*Sin[e + (f*x)/2] - 310*c^3*d^4*Sin[e + (f*x)/2] + 39*c^2*d^5*Sin[e + (f*x)/2] + 48*c*d^6*Sin[e + (f*x)/2] - 18*d^7*Sin[e + (f*x)/2] - 16*c^7*Sin[2*e + (f*x)/2] + 14*c^6*d*Sin[2*e + (f*x)/2] + 220*c^5*d^2*Sin[2*e + (f*x)/2] + 502*c^4*d^3*Sin[2*e + (f*x)/2] + 522*c^3*d^4*Sin[2*e + (f*x)/2] + 303*c^2*d^5*Sin[2*e + (f*x)/2] + 48*c*d^6*Sin[2*e + (f*x)/2] - 18*d^7*Sin[2*e + (f*x)/2] - 6*c^7*Sin[e + (3*f*x)/2] + 6*c^6*d*Sin[e + (3*f*x)/2] + 126*c^5*d^2*Sin[e + (3*f*x)/2] + 114*c^4*d^3*Sin[e + (3*f*x)/2] - 159*c^3*d^4*Sin[e + (3*f*x)/2] - 144*c^2*d^5*Sin[e + (3*f*x)/2] - 6*c*d^6*Sin[e + (3*f*x)/2] + 6*d^7*Sin[e + (3*f*x)/2] + 14*c^7*Sin[2*e + (3*f*x)/2] - 16*c^6*d*Sin[2*e + (3*f*x)/2] - 226*c^5*d^2*Sin[2*e + (3*f*x)/2] - 412*c^4*d^3*Sin[2*e + (3*f*x)/2] - 235*c^3*d^4*Sin[2*e + (3*f*x)/2] - 7*c^2*d^5*Sin[2*e + (3*f*x)/2] + 6*c*d^6*Sin[2*e + (3*f*x)/2] - 6*d^7*Sin[2*e + (3*f*x)/2] - 6*c^7*Sin[3*e + (3*f*x)/2] + 6*c^6*d*Sin[3*e + (3*f*x)/2] + 126*c^5*d^2*Sin[3*e + (3*f*x)/2] + 234*c^4*d^3*Sin[3*e + (3*f*x)/2] + 189*c^3*d^4*Sin[3*e + (3*f*x)/2] + 81*c^2*d^5*Sin[3*e + (3*f*x)/2] + 6*c*d^6*Sin[3*e + (3*f*x)/2] - 6*d^7*Sin[3*e + (3*f*x)/2] + 6*c^7*Sin[e + (5*f*x)/2] - 14*c^6*d*Sin[e + (5*f*x)/2] - 134*c^5*d^2*Sin[e + (5*f*x)/2] - 274*c^4*d^3*Sin[e + (5*f*x)/2] - 193*c^3*d^4*Sin[e + (5*f*x)/2] - 27*c^2*d^5*Sin[e + (5*f*x)/2] + 6*c*d^6*Sin[e + (5*f*x)/2] - 6*c^7*Sin[2*e + (5*f*x)/2] + 12*c^6*d*Sin[2*e + (5*f*x)/2] + 42*c^5*d^2*Sin[2*e + (5*f*x)/2] - 48*c^4*d^3*Sin[2*e + (5*f*x)/2] - 105*c^3*d^4*Sin[2*e + (5*f*x)/2] - 27*c^2*d^5*Sin[2*e + (5*f*x)/2] + 6*c*d^6*Sin[2*e + (5*f*x)/2] + 6*c^7*Sin[3*e + (5*f*x)/2] - 14*c^6*d*Sin[3*e + (5*f*x)/2] - 134*c^5*d^2*Sin[3*e + (5*f*x)/2] - 202*c^4*d^3*Sin[3*e + (5*f*x)/2] - 61*c^3*d^4*Sin[3*e + (5*f*x)/2] + 12*c^2*d^5*Sin[3*e + (5*f*x)/2] - 6*c*d^6*Sin[3*e + (5*f*x)/2] - 6*c^7*Sin[4*e + (5*f*x)/2] + 12*c^6*d*Sin[4*e + (5*f*x)/2] + 42*c^5*d^2*Sin[4*e + (5*f*x)/2] + 24*c^4*d^3*Sin[4*e + (5*f*x)/2] + 27*c^3*d^4*Sin[4*e + (5*f*x)/2] + 12*c^2*d^5*Sin[4*e + (5*f*x)/2] - 6*c*d^6*Sin[4*e + (5*f*x)/2] + 4*c^7*Sin[2*e + (7*f*x)/2] - 14*c^6*d*Sin[2*e + (7*f*x)/2] - 40*c^5*d^2*Sin[2*e + (7*f*x)/2] - 46*c^4*d^3*Sin[2*e + (7*f*x)/2] - 12*c^3*d^4*Sin[2*e + (7*f*x)/2] + 3*c^2*d^5*Sin[2*e + (7*f*x)/2] - 24*c^4*d^3*Sin[3*e + (7*f*x)/2] - 12*c^3*d^4*Sin[3*e + (7*f*x)/2] + 3*c^2*d^5*Sin[3*e + (7*f*x)/2] + 4*c^7*Sin[4*e + (7*f*x)/2] - 14*c^6*d*Sin[4*e + (7*f*x)/2] - 40*c^5*d^2*Sin[4*e + (7*f*x)/2] - 22*c^4*d^3*Sin[4*e + (7*f*x)/2]))/(48*c^2*(-c + d)^4*(c + d)^2*f*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^3)","C",0
225,1,1338,363,6.719061,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^6}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^6)/(a + a*Sec[e + f*x])^3,x]","\frac{2 \cos ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \cos ^3(e+f x) \sec \left(\frac{e}{2}\right) \left(\sin \left(\frac{e}{2}\right) c^6-6 d \sin \left(\frac{e}{2}\right) c^5+15 d^2 \sin \left(\frac{e}{2}\right) c^4-20 d^3 \sin \left(\frac{e}{2}\right) c^3+15 d^4 \sin \left(\frac{e}{2}\right) c^2-6 d^5 \sin \left(\frac{e}{2}\right) c+d^6 \sin \left(\frac{e}{2}\right)\right) (c+d \sec (e+f x))^6}{5 f (d+c \cos (e+f x))^6 (\sec (e+f x) a+a)^3}+\frac{8 \cos ^4\left(\frac{e}{2}+\frac{f x}{2}\right) \cos ^3(e+f x) \sec \left(\frac{e}{2}\right) \left(-4 \sin \left(\frac{e}{2}\right) c^6+9 d \sin \left(\frac{e}{2}\right) c^5+15 d^2 \sin \left(\frac{e}{2}\right) c^4-70 d^3 \sin \left(\frac{e}{2}\right) c^3+90 d^4 \sin \left(\frac{e}{2}\right) c^2-51 d^5 \sin \left(\frac{e}{2}\right) c+11 d^6 \sin \left(\frac{e}{2}\right)\right) (c+d \sec (e+f x))^6}{15 f (d+c \cos (e+f x))^6 (\sec (e+f x) a+a)^3}+\frac{2 \cos \left(\frac{e}{2}+\frac{f x}{2}\right) \cos ^3(e+f x) \sec \left(\frac{e}{2}\right) \left(\sin \left(\frac{f x}{2}\right) c^6-6 d \sin \left(\frac{f x}{2}\right) c^5+15 d^2 \sin \left(\frac{f x}{2}\right) c^4-20 d^3 \sin \left(\frac{f x}{2}\right) c^3+15 d^4 \sin \left(\frac{f x}{2}\right) c^2-6 d^5 \sin \left(\frac{f x}{2}\right) c+d^6 \sin \left(\frac{f x}{2}\right)\right) (c+d \sec (e+f x))^6}{5 f (d+c \cos (e+f x))^6 (\sec (e+f x) a+a)^3}+\frac{8 \cos ^3\left(\frac{e}{2}+\frac{f x}{2}\right) \cos ^3(e+f x) \sec \left(\frac{e}{2}\right) \left(-4 \sin \left(\frac{f x}{2}\right) c^6+9 d \sin \left(\frac{f x}{2}\right) c^5+15 d^2 \sin \left(\frac{f x}{2}\right) c^4-70 d^3 \sin \left(\frac{f x}{2}\right) c^3+90 d^4 \sin \left(\frac{f x}{2}\right) c^2-51 d^5 \sin \left(\frac{f x}{2}\right) c+11 d^6 \sin \left(\frac{f x}{2}\right)\right) (c+d \sec (e+f x))^6}{15 f (d+c \cos (e+f x))^6 (\sec (e+f x) a+a)^3}+\frac{8 \cos ^5\left(\frac{e}{2}+\frac{f x}{2}\right) \cos ^3(e+f x) \sec \left(\frac{e}{2}\right) \left(7 \sin \left(\frac{f x}{2}\right) c^6+18 d \sin \left(\frac{f x}{2}\right) c^5+30 d^2 \sin \left(\frac{f x}{2}\right) c^4-440 d^3 \sin \left(\frac{f x}{2}\right) c^3+855 d^4 \sin \left(\frac{f x}{2}\right) c^2-642 d^5 \sin \left(\frac{f x}{2}\right) c+172 d^6 \sin \left(\frac{f x}{2}\right)\right) (c+d \sec (e+f x))^6}{15 f (d+c \cos (e+f x))^6 (\sec (e+f x) a+a)^3}+\frac{8 d^6 \cos ^6\left(\frac{e}{2}+\frac{f x}{2}\right) \sec (e) \sin (f x) (c+d \sec (e+f x))^6}{3 f (d+c \cos (e+f x))^6 (\sec (e+f x) a+a)^3}-\frac{4 \cos ^6\left(\frac{e}{2}+\frac{f x}{2}\right) \cos ^2(e+f x) \sec (e) \left(9 \sin (e) d^6-40 \sin (f x) d^6-18 c \sin (e) d^5+108 c \sin (f x) d^5-90 c^2 \sin (f x) d^4\right) (c+d \sec (e+f x))^6}{3 f (d+c \cos (e+f x))^6 (\sec (e+f x) a+a)^3}+\frac{4 \cos ^6\left(\frac{e}{2}+\frac{f x}{2}\right) \cos (e+f x) \sec (e) \left(2 \sin (e) d^6-9 \sin (f x) d^6+18 c \sin (f x) d^5\right) (c+d \sec (e+f x))^6}{3 f (d+c \cos (e+f x))^6 (\sec (e+f x) a+a)^3}+\frac{4 \left(23 d^6-78 c d^5+90 c^2 d^4-40 c^3 d^3\right) \cos ^6\left(\frac{e}{2}+\frac{f x}{2}\right) \cos ^3(e+f x) \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right) (c+d \sec (e+f x))^6}{f (d+c \cos (e+f x))^6 (\sec (e+f x) a+a)^3}-\frac{4 \left(23 d^6-78 c d^5+90 c^2 d^4-40 c^3 d^3\right) \cos ^6\left(\frac{e}{2}+\frac{f x}{2}\right) \cos ^3(e+f x) \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right) (c+d \sec (e+f x))^6}{f (d+c \cos (e+f x))^6 (\sec (e+f x) a+a)^3}","\frac{(c-d) \left(2 c^2+18 c d+115 d^2\right) \tan (e+f x) (c+d \sec (e+f x))^3}{15 f \left(a^3 \sec (e+f x)+a^3\right)}+\frac{d^3 \left(40 c^3-90 c^2 d+78 c d^2-23 d^3\right) \tanh ^{-1}(\sin (e+f x))}{2 a^3 f}-\frac{d \left(2 c^3+18 c^2 d+111 c d^2-136 d^3\right) \tan (e+f x) (c+d \sec (e+f x))^2}{15 a^3 f}-\frac{d^2 \left(4 c^4+36 c^3 d+216 c^2 d^2-626 c d^3+345 d^4\right) \tan (e+f x) \sec (e+f x)}{30 a^3 f}-\frac{2 d \left(2 c^5+18 c^4 d+107 c^3 d^2-472 c^2 d^3+456 c d^4-136 d^5\right) \tan (e+f x)}{15 a^3 f}+\frac{(c-d) \tan (e+f x) (c+d \sec (e+f x))^5}{5 f (a \sec (e+f x)+a)^3}+\frac{(c-d) (2 c+13 d) \tan (e+f x) (c+d \sec (e+f x))^4}{15 a f (a \sec (e+f x)+a)^2}",1,"(4*(-40*c^3*d^3 + 90*c^2*d^4 - 78*c*d^5 + 23*d^6)*Cos[e/2 + (f*x)/2]^6*Cos[e + f*x]^3*Log[Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2]]*(c + d*Sec[e + f*x])^6)/(f*(d + c*Cos[e + f*x])^6*(a + a*Sec[e + f*x])^3) - (4*(-40*c^3*d^3 + 90*c^2*d^4 - 78*c*d^5 + 23*d^6)*Cos[e/2 + (f*x)/2]^6*Cos[e + f*x]^3*Log[Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2]]*(c + d*Sec[e + f*x])^6)/(f*(d + c*Cos[e + f*x])^6*(a + a*Sec[e + f*x])^3) + (2*Cos[e/2 + (f*x)/2]^2*Cos[e + f*x]^3*Sec[e/2]*(c + d*Sec[e + f*x])^6*(c^6*Sin[e/2] - 6*c^5*d*Sin[e/2] + 15*c^4*d^2*Sin[e/2] - 20*c^3*d^3*Sin[e/2] + 15*c^2*d^4*Sin[e/2] - 6*c*d^5*Sin[e/2] + d^6*Sin[e/2]))/(5*f*(d + c*Cos[e + f*x])^6*(a + a*Sec[e + f*x])^3) + (8*Cos[e/2 + (f*x)/2]^4*Cos[e + f*x]^3*Sec[e/2]*(c + d*Sec[e + f*x])^6*(-4*c^6*Sin[e/2] + 9*c^5*d*Sin[e/2] + 15*c^4*d^2*Sin[e/2] - 70*c^3*d^3*Sin[e/2] + 90*c^2*d^4*Sin[e/2] - 51*c*d^5*Sin[e/2] + 11*d^6*Sin[e/2]))/(15*f*(d + c*Cos[e + f*x])^6*(a + a*Sec[e + f*x])^3) + (2*Cos[e/2 + (f*x)/2]*Cos[e + f*x]^3*Sec[e/2]*(c + d*Sec[e + f*x])^6*(c^6*Sin[(f*x)/2] - 6*c^5*d*Sin[(f*x)/2] + 15*c^4*d^2*Sin[(f*x)/2] - 20*c^3*d^3*Sin[(f*x)/2] + 15*c^2*d^4*Sin[(f*x)/2] - 6*c*d^5*Sin[(f*x)/2] + d^6*Sin[(f*x)/2]))/(5*f*(d + c*Cos[e + f*x])^6*(a + a*Sec[e + f*x])^3) + (8*Cos[e/2 + (f*x)/2]^3*Cos[e + f*x]^3*Sec[e/2]*(c + d*Sec[e + f*x])^6*(-4*c^6*Sin[(f*x)/2] + 9*c^5*d*Sin[(f*x)/2] + 15*c^4*d^2*Sin[(f*x)/2] - 70*c^3*d^3*Sin[(f*x)/2] + 90*c^2*d^4*Sin[(f*x)/2] - 51*c*d^5*Sin[(f*x)/2] + 11*d^6*Sin[(f*x)/2]))/(15*f*(d + c*Cos[e + f*x])^6*(a + a*Sec[e + f*x])^3) + (8*Cos[e/2 + (f*x)/2]^5*Cos[e + f*x]^3*Sec[e/2]*(c + d*Sec[e + f*x])^6*(7*c^6*Sin[(f*x)/2] + 18*c^5*d*Sin[(f*x)/2] + 30*c^4*d^2*Sin[(f*x)/2] - 440*c^3*d^3*Sin[(f*x)/2] + 855*c^2*d^4*Sin[(f*x)/2] - 642*c*d^5*Sin[(f*x)/2] + 172*d^6*Sin[(f*x)/2]))/(15*f*(d + c*Cos[e + f*x])^6*(a + a*Sec[e + f*x])^3) + (8*d^6*Cos[e/2 + (f*x)/2]^6*Sec[e]*(c + d*Sec[e + f*x])^6*Sin[f*x])/(3*f*(d + c*Cos[e + f*x])^6*(a + a*Sec[e + f*x])^3) - (4*Cos[e/2 + (f*x)/2]^6*Cos[e + f*x]^2*Sec[e]*(c + d*Sec[e + f*x])^6*(-18*c*d^5*Sin[e] + 9*d^6*Sin[e] - 90*c^2*d^4*Sin[f*x] + 108*c*d^5*Sin[f*x] - 40*d^6*Sin[f*x]))/(3*f*(d + c*Cos[e + f*x])^6*(a + a*Sec[e + f*x])^3) + (4*Cos[e/2 + (f*x)/2]^6*Cos[e + f*x]*Sec[e]*(c + d*Sec[e + f*x])^6*(2*d^6*Sin[e] + 18*c*d^5*Sin[f*x] - 9*d^6*Sin[f*x]))/(3*f*(d + c*Cos[e + f*x])^6*(a + a*Sec[e + f*x])^3)","B",1
226,1,439,287,2.2309682,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^5)/(a + a*Sec[e + f*x])^3,x]","\frac{2 \sin \left(\frac{1}{2} (e+f x)\right) \cos \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \left(12 c^5 \cos (3 (e+f x))+7 c^5 \cos (4 (e+f x))+29 c^5+90 c^4 d \cos (3 (e+f x))+15 c^4 d \cos (4 (e+f x))+105 c^4 d+120 c^3 d^2 \cos (3 (e+f x))+20 c^3 d^2 \cos (4 (e+f x))+340 c^3 d^2-1020 c^2 d^3 \cos (3 (e+f x))-220 c^2 d^3 \cos (4 (e+f x))-1940 c^2 d^3+3 \left(12 c^5+90 c^4 d+120 c^3 d^2-1020 c^2 d^3+1910 c d^4-777 d^5\right) \cos (e+f x)+6 \left(6 c^5+20 c^4 d+60 c^3 d^2-360 c^2 d^3+630 c d^4-261 d^5\right) \cos (2 (e+f x))+1710 c d^4 \cos (3 (e+f x))+360 c d^4 \cos (4 (e+f x))+3420 c d^4-717 d^5 \cos (3 (e+f x))-152 d^5 \cos (4 (e+f x))-1354 d^5\right)-480 d^3 \left(20 c^2-30 c d+13 d^2\right) \cos ^6\left(\frac{1}{2} (e+f x)\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{120 a^3 f (\cos (e+f x)+1)^3}","\frac{(c-d) \left(2 c^2+15 c d+76 d^2\right) \tan (e+f x) (c+d \sec (e+f x))^2}{15 f \left(a^3 \sec (e+f x)+a^3\right)}+\frac{d^3 \left(20 c^2-30 c d+13 d^2\right) \tanh ^{-1}(\sin (e+f x))}{2 a^3 f}-\frac{d^2 \left(4 c^3+30 c^2 d+146 c d^2-195 d^3\right) \tan (e+f x) \sec (e+f x)}{30 a^3 f}-\frac{2 d \left(2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right) \tan (e+f x)}{15 a^3 f}+\frac{(c-d) \tan (e+f x) (c+d \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}+\frac{(c-d) (2 c+11 d) \tan (e+f x) (c+d \sec (e+f x))^3}{15 a f (a \sec (e+f x)+a)^2}",1,"(-480*d^3*(20*c^2 - 30*c*d + 13*d^2)*Cos[(e + f*x)/2]^6*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + 2*Cos[(e + f*x)/2]*(29*c^5 + 105*c^4*d + 340*c^3*d^2 - 1940*c^2*d^3 + 3420*c*d^4 - 1354*d^5 + 3*(12*c^5 + 90*c^4*d + 120*c^3*d^2 - 1020*c^2*d^3 + 1910*c*d^4 - 777*d^5)*Cos[e + f*x] + 6*(6*c^5 + 20*c^4*d + 60*c^3*d^2 - 360*c^2*d^3 + 630*c*d^4 - 261*d^5)*Cos[2*(e + f*x)] + 12*c^5*Cos[3*(e + f*x)] + 90*c^4*d*Cos[3*(e + f*x)] + 120*c^3*d^2*Cos[3*(e + f*x)] - 1020*c^2*d^3*Cos[3*(e + f*x)] + 1710*c*d^4*Cos[3*(e + f*x)] - 717*d^5*Cos[3*(e + f*x)] + 7*c^5*Cos[4*(e + f*x)] + 15*c^4*d*Cos[4*(e + f*x)] + 20*c^3*d^2*Cos[4*(e + f*x)] - 220*c^2*d^3*Cos[4*(e + f*x)] + 360*c*d^4*Cos[4*(e + f*x)] - 152*d^5*Cos[4*(e + f*x)])*Sec[e + f*x]^2*Sin[(e + f*x)/2])/(120*a^3*f*(1 + Cos[e + f*x])^3)","A",1
227,1,292,205,2.3929328,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^4)/(a + a*Sec[e + f*x])^3,x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \left(4 (c-d)^2 \left(7 c^2+26 c d+57 d^2\right) \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \cos ^4\left(\frac{1}{2} (e+f x)\right)-60 d^3 \cos ^5\left(\frac{1}{2} (e+f x)\right) \left((4 c-3 d) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-d \sec (e) \sin (f x) \sec (e+f x)\right)-8 (c-d)^3 (2 c+3 d) \tan \left(\frac{e}{2}\right) \cos ^3\left(\frac{1}{2} (e+f x)\right)+3 (c-d)^4 \tan \left(\frac{e}{2}\right) \cos \left(\frac{1}{2} (e+f x)\right)+3 (c-d)^4 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right)-8 (c-d)^3 (2 c+3 d) \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \cos ^2\left(\frac{1}{2} (e+f x)\right)\right)}{15 a^3 f (\cos (e+f x)+1)^3}","\frac{\tan (e+f x) \left(2 c^4+8 c^3 d-d^2 \left(2 c^2+10 c d-27 d^2\right) \sec (e+f x)+21 c^2 d^2-88 c d^3+72 d^4\right)}{15 f \left(a^3 \sec (e+f x)+a^3\right)}+\frac{d^3 (4 c-3 d) \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac{(c-d) \tan (e+f x) (c+d \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3}+\frac{(c-d) (2 c+9 d) \tan (e+f x) (c+d \sec (e+f x))^2}{15 a f (a \sec (e+f x)+a)^2}",1,"(2*Cos[(e + f*x)/2]*(3*(c - d)^4*Sec[e/2]*Sin[(f*x)/2] - 8*(c - d)^3*(2*c + 3*d)*Cos[(e + f*x)/2]^2*Sec[e/2]*Sin[(f*x)/2] + 4*(c - d)^2*(7*c^2 + 26*c*d + 57*d^2)*Cos[(e + f*x)/2]^4*Sec[e/2]*Sin[(f*x)/2] - 60*d^3*Cos[(e + f*x)/2]^5*((4*c - 3*d)*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - d*Sec[e]*Sec[e + f*x]*Sin[f*x]) + 3*(c - d)^4*Cos[(e + f*x)/2]*Tan[e/2] - 8*(c - d)^3*(2*c + 3*d)*Cos[(e + f*x)/2]^3*Tan[e/2]))/(15*a^3*f*(1 + Cos[e + f*x])^3)","A",1
228,1,295,133,1.480122,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^3)/(a + a*Sec[e + f*x])^3,x]","\frac{(c-d) \sec \left(\frac{e}{2}\right) \cos \left(\frac{1}{2} (e+f x)\right) \left(-15 \left(2 c^2+5 c d+5 d^2\right) \sin \left(e+\frac{f x}{2}\right)+5 \left(8 c^2+17 c d+29 d^2\right) \sin \left(\frac{f x}{2}\right)+20 c^2 \sin \left(e+\frac{3 f x}{2}\right)-15 c^2 \sin \left(2 e+\frac{3 f x}{2}\right)+7 c^2 \sin \left(2 e+\frac{5 f x}{2}\right)+65 c d \sin \left(e+\frac{3 f x}{2}\right)-15 c d \sin \left(2 e+\frac{3 f x}{2}\right)+16 c d \sin \left(2 e+\frac{5 f x}{2}\right)+95 d^2 \sin \left(e+\frac{3 f x}{2}\right)-15 d^2 \sin \left(2 e+\frac{3 f x}{2}\right)+22 d^2 \sin \left(2 e+\frac{5 f x}{2}\right)\right)-240 d^3 \cos ^6\left(\frac{1}{2} (e+f x)\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{30 a^3 f (\cos (e+f x)+1)^3}","\frac{d^3 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac{(c-d) \tan (e+f x) \left(\left(2 c^2+11 c d+29 d^2\right) \sec (e+f x)+2 \left(2 c^2+8 c d+11 d^2\right)\right)}{15 a f (a \sec (e+f x)+a)^2}+\frac{(c-d) \tan (e+f x) (c+d \sec (e+f x))^2}{5 f (a \sec (e+f x)+a)^3}",1,"(-240*d^3*Cos[(e + f*x)/2]^6*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + (c - d)*Cos[(e + f*x)/2]*Sec[e/2]*(5*(8*c^2 + 17*c*d + 29*d^2)*Sin[(f*x)/2] - 15*(2*c^2 + 5*c*d + 5*d^2)*Sin[e + (f*x)/2] + 20*c^2*Sin[e + (3*f*x)/2] + 65*c*d*Sin[e + (3*f*x)/2] + 95*d^2*Sin[e + (3*f*x)/2] - 15*c^2*Sin[2*e + (3*f*x)/2] - 15*c*d*Sin[2*e + (3*f*x)/2] - 15*d^2*Sin[2*e + (3*f*x)/2] + 7*c^2*Sin[2*e + (5*f*x)/2] + 16*c*d*Sin[2*e + (5*f*x)/2] + 22*d^2*Sin[2*e + (5*f*x)/2]))/(30*a^3*f*(1 + Cos[e + f*x])^3)","B",1
229,1,180,115,0.4803986,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^2)/(a + a*Sec[e + f*x])^3,x]","\frac{\sec \left(\frac{e}{2}\right) \cos \left(\frac{1}{2} (e+f x)\right) \left(10 \left(4 c^2+3 c d+2 d^2\right) \sin \left(\frac{f x}{2}\right)+20 c^2 \sin \left(e+\frac{3 f x}{2}\right)-15 c^2 \sin \left(2 e+\frac{3 f x}{2}\right)+7 c^2 \sin \left(2 e+\frac{5 f x}{2}\right)-30 c (c+d) \sin \left(e+\frac{f x}{2}\right)+30 c d \sin \left(e+\frac{3 f x}{2}\right)+6 c d \sin \left(2 e+\frac{5 f x}{2}\right)+10 d^2 \sin \left(e+\frac{3 f x}{2}\right)+2 d^2 \sin \left(2 e+\frac{5 f x}{2}\right)\right)}{30 a^3 f (\cos (e+f x)+1)^3}","\frac{\left(2 c^2+6 c d+7 d^2\right) \tan (e+f x)}{15 f \left(a^3 \sec (e+f x)+a^3\right)}+\frac{(c-d)^2 \tan (e+f x)}{5 f (a \sec (e+f x)+a)^3}+\frac{2 (c+4 d) (c-d) \tan (e+f x)}{15 a f (a \sec (e+f x)+a)^2}",1,"(Cos[(e + f*x)/2]*Sec[e/2]*(10*(4*c^2 + 3*c*d + 2*d^2)*Sin[(f*x)/2] - 30*c*(c + d)*Sin[e + (f*x)/2] + 20*c^2*Sin[e + (3*f*x)/2] + 30*c*d*Sin[e + (3*f*x)/2] + 10*d^2*Sin[e + (3*f*x)/2] - 15*c^2*Sin[2*e + (3*f*x)/2] + 7*c^2*Sin[2*e + (5*f*x)/2] + 6*c*d*Sin[2*e + (5*f*x)/2] + 2*d^2*Sin[2*e + (5*f*x)/2]))/(30*a^3*f*(1 + Cos[e + f*x])^3)","A",1
230,1,135,102,0.3399798,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))}{(a+a \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x]))/(a + a*Sec[e + f*x])^3,x]","\frac{\sec \left(\frac{e}{2}\right) \cos \left(\frac{1}{2} (e+f x)\right) \left(-15 (2 c+d) \sin \left(e+\frac{f x}{2}\right)+5 (8 c+3 d) \sin \left(\frac{f x}{2}\right)+20 c \sin \left(e+\frac{3 f x}{2}\right)-15 c \sin \left(2 e+\frac{3 f x}{2}\right)+7 c \sin \left(2 e+\frac{5 f x}{2}\right)+15 d \sin \left(e+\frac{3 f x}{2}\right)+3 d \sin \left(2 e+\frac{5 f x}{2}\right)\right)}{30 a^3 f (\cos (e+f x)+1)^3}","\frac{(2 c+3 d) \tan (e+f x)}{15 f \left(a^3 \sec (e+f x)+a^3\right)}+\frac{(2 c+3 d) \tan (e+f x)}{15 a f (a \sec (e+f x)+a)^2}+\frac{(c-d) \tan (e+f x)}{5 f (a \sec (e+f x)+a)^3}",1,"(Cos[(e + f*x)/2]*Sec[e/2]*(5*(8*c + 3*d)*Sin[(f*x)/2] - 15*(2*c + d)*Sin[e + (f*x)/2] + 20*c*Sin[e + (3*f*x)/2] + 15*d*Sin[e + (3*f*x)/2] - 15*c*Sin[2*e + (3*f*x)/2] + 7*c*Sin[2*e + (5*f*x)/2] + 3*d*Sin[2*e + (5*f*x)/2]))/(30*a^3*f*(1 + Cos[e + f*x])^3)","A",1
231,1,345,181,3.0254778,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])),x]","\frac{\cos \left(\frac{1}{2} (e+f x)\right) \left(\sec \left(\frac{e}{2}\right) \left(-15 \left(2 c^2-7 c d+9 d^2\right) \sin \left(e+\frac{f x}{2}\right)+5 \left(8 c^2-27 c d+37 d^2\right) \sin \left(\frac{f x}{2}\right)+20 c^2 \sin \left(e+\frac{3 f x}{2}\right)-15 c^2 \sin \left(2 e+\frac{3 f x}{2}\right)+7 c^2 \sin \left(2 e+\frac{5 f x}{2}\right)-75 c d \sin \left(e+\frac{3 f x}{2}\right)+45 c d \sin \left(2 e+\frac{3 f x}{2}\right)-24 c d \sin \left(2 e+\frac{5 f x}{2}\right)+115 d^2 \sin \left(e+\frac{3 f x}{2}\right)-45 d^2 \sin \left(2 e+\frac{3 f x}{2}\right)+32 d^2 \sin \left(2 e+\frac{5 f x}{2}\right)\right)+\frac{480 d^3 (\sin (e)+i \cos (e)) \cos ^5\left(\frac{1}{2} (e+f x)\right) \tan ^{-1}\left(\frac{(\sin (e)+i \cos (e)) \left(\tan \left(\frac{f x}{2}\right) (c \cos (e)-d)+c \sin (e)\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{30 a^3 f (c-d)^3 (\cos (e+f x)+1)^3}","\frac{\left(2 c^2-9 c d+22 d^2\right) \tan (e+f x)}{15 f (c-d)^3 \left(a^3 \sec (e+f x)+a^3\right)}-\frac{2 d^3 \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{a^3 f (c-d)^{7/2} \sqrt{c+d}}+\frac{(2 c-7 d) \tan (e+f x)}{15 a f (c-d)^2 (a \sec (e+f x)+a)^2}+\frac{\tan (e+f x)}{5 f (c-d) (a \sec (e+f x)+a)^3}",1,"(Cos[(e + f*x)/2]*((480*d^3*ArcTan[((I*Cos[e] + Sin[e])*(c*Sin[e] + (-d + c*Cos[e])*Tan[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*Cos[(e + f*x)/2]^5*(I*Cos[e] + Sin[e]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + Sec[e/2]*(5*(8*c^2 - 27*c*d + 37*d^2)*Sin[(f*x)/2] - 15*(2*c^2 - 7*c*d + 9*d^2)*Sin[e + (f*x)/2] + 20*c^2*Sin[e + (3*f*x)/2] - 75*c*d*Sin[e + (3*f*x)/2] + 115*d^2*Sin[e + (3*f*x)/2] - 15*c^2*Sin[2*e + (3*f*x)/2] + 45*c*d*Sin[2*e + (3*f*x)/2] - 45*d^2*Sin[2*e + (3*f*x)/2] + 7*c^2*Sin[2*e + (5*f*x)/2] - 24*c*d*Sin[2*e + (5*f*x)/2] + 32*d^2*Sin[2*e + (5*f*x)/2])))/(30*a^3*(c - d)^3*f*(1 + Cos[e + f*x])^3)","C",1
232,1,1772,288,7.0938508,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2),x]","\frac{(4 c+3 d) (d+c \cos (e+f x))^2 \sec ^5(e+f x) \left(\frac{16 i \tan ^{-1}\left(\sec \left(\frac{f x}{2}\right) \left(\frac{\cos (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}-\frac{i \sin (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}\right) \left(i c \sin \left(e+\frac{f x}{2}\right)-i d \sin \left(\frac{f x}{2}\right)\right)\right) \cos (e) d^3}{\sqrt{c^2-d^2} f \sqrt{\cos (2 e)-i \sin (2 e)}}+\frac{16 \tan ^{-1}\left(\sec \left(\frac{f x}{2}\right) \left(\frac{\cos (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}-\frac{i \sin (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}\right) \left(i c \sin \left(e+\frac{f x}{2}\right)-i d \sin \left(\frac{f x}{2}\right)\right)\right) \sin (e) d^3}{\sqrt{c^2-d^2} f \sqrt{\cos (2 e)-i \sin (2 e)}}\right) \cos ^6\left(\frac{e}{2}+\frac{f x}{2}\right)}{(d-c)^4 (c+d) (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^2}+\frac{(d+c \cos (e+f x)) \sec \left(\frac{e}{2}\right) \sec (e) \sec ^5(e+f x) \left(-55 \sin \left(\frac{f x}{2}\right) c^5+47 \sin \left(\frac{3 f x}{2}\right) c^5-50 \sin \left(e-\frac{f x}{2}\right) c^5+50 \sin \left(e+\frac{f x}{2}\right) c^5-55 \sin \left(2 e+\frac{f x}{2}\right) c^5-30 \sin \left(e+\frac{3 f x}{2}\right) c^5+47 \sin \left(2 e+\frac{3 f x}{2}\right) c^5-30 \sin \left(3 e+\frac{3 f x}{2}\right) c^5+20 \sin \left(e+\frac{5 f x}{2}\right) c^5-15 \sin \left(2 e+\frac{5 f x}{2}\right) c^5+20 \sin \left(3 e+\frac{5 f x}{2}\right) c^5-15 \sin \left(4 e+\frac{5 f x}{2}\right) c^5+7 \sin \left(2 e+\frac{7 f x}{2}\right) c^5+7 \sin \left(4 e+\frac{7 f x}{2}\right) c^5+135 d \sin \left(\frac{f x}{2}\right) c^4-137 d \sin \left(\frac{3 f x}{2}\right) c^4+130 d \sin \left(e-\frac{f x}{2}\right) c^4-130 d \sin \left(e+\frac{f x}{2}\right) c^4+135 d \sin \left(2 e+\frac{f x}{2}\right) c^4+90 d \sin \left(e+\frac{3 f x}{2}\right) c^4-137 d \sin \left(2 e+\frac{3 f x}{2}\right) c^4+90 d \sin \left(3 e+\frac{3 f x}{2}\right) c^4-76 d \sin \left(e+\frac{5 f x}{2}\right) c^4+45 d \sin \left(2 e+\frac{5 f x}{2}\right) c^4-76 d \sin \left(3 e+\frac{5 f x}{2}\right) c^4+45 d \sin \left(4 e+\frac{5 f x}{2}\right) c^4-27 d \sin \left(2 e+\frac{7 f x}{2}\right) c^4-27 d \sin \left(4 e+\frac{7 f x}{2}\right) c^4-20 d^2 \sin \left(\frac{f x}{2}\right) c^3+88 d^2 \sin \left(\frac{3 f x}{2}\right) c^3-10 d^2 \sin \left(e-\frac{f x}{2}\right) c^3+10 d^2 \sin \left(e+\frac{f x}{2}\right) c^3-20 d^2 \sin \left(2 e+\frac{f x}{2}\right) c^3-60 d^2 \sin \left(e+\frac{3 f x}{2}\right) c^3+88 d^2 \sin \left(2 e+\frac{3 f x}{2}\right) c^3-60 d^2 \sin \left(3 e+\frac{3 f x}{2}\right) c^3+106 d^2 \sin \left(e+\frac{5 f x}{2}\right) c^3-30 d^2 \sin \left(2 e+\frac{5 f x}{2}\right) c^3+106 d^2 \sin \left(3 e+\frac{5 f x}{2}\right) c^3-30 d^2 \sin \left(4 e+\frac{5 f x}{2}\right) c^3+38 d^2 \sin \left(2 e+\frac{7 f x}{2}\right) c^3+38 d^2 \sin \left(4 e+\frac{7 f x}{2}\right) c^3-810 d^3 \sin \left(\frac{f x}{2}\right) c^2+812 d^3 \sin \left(\frac{3 f x}{2}\right) c^2-1030 d^3 \sin \left(e-\frac{f x}{2}\right) c^2+1030 d^3 \sin \left(e+\frac{f x}{2}\right) c^2-810 d^3 \sin \left(2 e+\frac{f x}{2}\right) c^2-360 d^3 \sin \left(e+\frac{3 f x}{2}\right) c^2+812 d^3 \sin \left(2 e+\frac{3 f x}{2}\right) c^2-360 d^3 \sin \left(3 e+\frac{3 f x}{2}\right) c^2+346 d^3 \sin \left(e+\frac{5 f x}{2}\right) c^2-90 d^3 \sin \left(2 e+\frac{5 f x}{2}\right) c^2+346 d^3 \sin \left(3 e+\frac{5 f x}{2}\right) c^2-90 d^3 \sin \left(4 e+\frac{5 f x}{2}\right) c^2+72 d^3 \sin \left(2 e+\frac{7 f x}{2}\right) c^2+72 d^3 \sin \left(4 e+\frac{7 f x}{2}\right) c^2-450 d^4 \sin \left(\frac{f x}{2}\right) c+690 d^4 \sin \left(\frac{3 f x}{2}\right) c-990 d^4 \sin \left(e-\frac{f x}{2}\right) c+765 d^4 \sin \left(e+\frac{f x}{2}\right) c-675 d^4 \sin \left(2 e+\frac{f x}{2}\right) c-30 d^4 \sin \left(e+\frac{3 f x}{2}\right) c+525 d^4 \sin \left(2 e+\frac{3 f x}{2}\right) c-195 d^4 \sin \left(3 e+\frac{3 f x}{2}\right) c+219 d^4 \sin \left(e+\frac{5 f x}{2}\right) c+75 d^4 \sin \left(2 e+\frac{5 f x}{2}\right) c+144 d^4 \sin \left(3 e+\frac{5 f x}{2}\right) c+15 d^4 \sin \left(2 e+\frac{7 f x}{2}\right) c+15 d^4 \sin \left(3 e+\frac{7 f x}{2}\right) c+150 d^5 \sin \left(\frac{f x}{2}\right)+75 d^5 \sin \left(\frac{3 f x}{2}\right)-150 d^5 \sin \left(e-\frac{f x}{2}\right)-150 d^5 \sin \left(e+\frac{f x}{2}\right)-150 d^5 \sin \left(2 e+\frac{f x}{2}\right)+75 d^5 \sin \left(e+\frac{3 f x}{2}\right)-75 d^5 \sin \left(2 e+\frac{3 f x}{2}\right)-75 d^5 \sin \left(3 e+\frac{3 f x}{2}\right)+15 d^5 \sin \left(e+\frac{5 f x}{2}\right)+15 d^5 \sin \left(2 e+\frac{5 f x}{2}\right)-15 d^5 \sin \left(3 e+\frac{5 f x}{2}\right)-15 d^5 \sin \left(4 e+\frac{5 f x}{2}\right)\right) \cos \left(\frac{e}{2}+\frac{f x}{2}\right)}{120 c (d-c)^4 (c+d) f (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^2}","\frac{\left(2 c^2-12 c d+45 d^2\right) \tan (e+f x)}{15 f (c-d)^3 \left(a^3 \sec (e+f x)+a^3\right) (c+d \sec (e+f x))}+\frac{d \left(2 c^3-12 c^2 d+43 c d^2+72 d^3\right) \tan (e+f x)}{15 a^3 f (c-d)^4 (c+d) (c+d \sec (e+f x))}-\frac{2 d^3 (4 c+3 d) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{a^3 f (c-d)^{9/2} (c+d)^{3/2}}+\frac{(2 c-9 d) \tan (e+f x)}{15 a f (c-d)^2 (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))}+\frac{\tan (e+f x)}{5 f (c-d) (a \sec (e+f x)+a)^3 (c+d \sec (e+f x))}",1,"((4*c + 3*d)*Cos[e/2 + (f*x)/2]^6*(d + c*Cos[e + f*x])^2*Sec[e + f*x]^5*(((16*I)*d^3*ArcTan[Sec[(f*x)/2]*(Cos[e]/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (I*Sin[e])/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]))*((-I)*d*Sin[(f*x)/2] + I*c*Sin[e + (f*x)/2])]*Cos[e])/(Sqrt[c^2 - d^2]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) + (16*d^3*ArcTan[Sec[(f*x)/2]*(Cos[e]/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (I*Sin[e])/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]))*((-I)*d*Sin[(f*x)/2] + I*c*Sin[e + (f*x)/2])]*Sin[e])/(Sqrt[c^2 - d^2]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((-c + d)^4*(c + d)*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2) + (Cos[e/2 + (f*x)/2]*(d + c*Cos[e + f*x])*Sec[e/2]*Sec[e]*Sec[e + f*x]^5*(-55*c^5*Sin[(f*x)/2] + 135*c^4*d*Sin[(f*x)/2] - 20*c^3*d^2*Sin[(f*x)/2] - 810*c^2*d^3*Sin[(f*x)/2] - 450*c*d^4*Sin[(f*x)/2] + 150*d^5*Sin[(f*x)/2] + 47*c^5*Sin[(3*f*x)/2] - 137*c^4*d*Sin[(3*f*x)/2] + 88*c^3*d^2*Sin[(3*f*x)/2] + 812*c^2*d^3*Sin[(3*f*x)/2] + 690*c*d^4*Sin[(3*f*x)/2] + 75*d^5*Sin[(3*f*x)/2] - 50*c^5*Sin[e - (f*x)/2] + 130*c^4*d*Sin[e - (f*x)/2] - 10*c^3*d^2*Sin[e - (f*x)/2] - 1030*c^2*d^3*Sin[e - (f*x)/2] - 990*c*d^4*Sin[e - (f*x)/2] - 150*d^5*Sin[e - (f*x)/2] + 50*c^5*Sin[e + (f*x)/2] - 130*c^4*d*Sin[e + (f*x)/2] + 10*c^3*d^2*Sin[e + (f*x)/2] + 1030*c^2*d^3*Sin[e + (f*x)/2] + 765*c*d^4*Sin[e + (f*x)/2] - 150*d^5*Sin[e + (f*x)/2] - 55*c^5*Sin[2*e + (f*x)/2] + 135*c^4*d*Sin[2*e + (f*x)/2] - 20*c^3*d^2*Sin[2*e + (f*x)/2] - 810*c^2*d^3*Sin[2*e + (f*x)/2] - 675*c*d^4*Sin[2*e + (f*x)/2] - 150*d^5*Sin[2*e + (f*x)/2] - 30*c^5*Sin[e + (3*f*x)/2] + 90*c^4*d*Sin[e + (3*f*x)/2] - 60*c^3*d^2*Sin[e + (3*f*x)/2] - 360*c^2*d^3*Sin[e + (3*f*x)/2] - 30*c*d^4*Sin[e + (3*f*x)/2] + 75*d^5*Sin[e + (3*f*x)/2] + 47*c^5*Sin[2*e + (3*f*x)/2] - 137*c^4*d*Sin[2*e + (3*f*x)/2] + 88*c^3*d^2*Sin[2*e + (3*f*x)/2] + 812*c^2*d^3*Sin[2*e + (3*f*x)/2] + 525*c*d^4*Sin[2*e + (3*f*x)/2] - 75*d^5*Sin[2*e + (3*f*x)/2] - 30*c^5*Sin[3*e + (3*f*x)/2] + 90*c^4*d*Sin[3*e + (3*f*x)/2] - 60*c^3*d^2*Sin[3*e + (3*f*x)/2] - 360*c^2*d^3*Sin[3*e + (3*f*x)/2] - 195*c*d^4*Sin[3*e + (3*f*x)/2] - 75*d^5*Sin[3*e + (3*f*x)/2] + 20*c^5*Sin[e + (5*f*x)/2] - 76*c^4*d*Sin[e + (5*f*x)/2] + 106*c^3*d^2*Sin[e + (5*f*x)/2] + 346*c^2*d^3*Sin[e + (5*f*x)/2] + 219*c*d^4*Sin[e + (5*f*x)/2] + 15*d^5*Sin[e + (5*f*x)/2] - 15*c^5*Sin[2*e + (5*f*x)/2] + 45*c^4*d*Sin[2*e + (5*f*x)/2] - 30*c^3*d^2*Sin[2*e + (5*f*x)/2] - 90*c^2*d^3*Sin[2*e + (5*f*x)/2] + 75*c*d^4*Sin[2*e + (5*f*x)/2] + 15*d^5*Sin[2*e + (5*f*x)/2] + 20*c^5*Sin[3*e + (5*f*x)/2] - 76*c^4*d*Sin[3*e + (5*f*x)/2] + 106*c^3*d^2*Sin[3*e + (5*f*x)/2] + 346*c^2*d^3*Sin[3*e + (5*f*x)/2] + 144*c*d^4*Sin[3*e + (5*f*x)/2] - 15*d^5*Sin[3*e + (5*f*x)/2] - 15*c^5*Sin[4*e + (5*f*x)/2] + 45*c^4*d*Sin[4*e + (5*f*x)/2] - 30*c^3*d^2*Sin[4*e + (5*f*x)/2] - 90*c^2*d^3*Sin[4*e + (5*f*x)/2] - 15*d^5*Sin[4*e + (5*f*x)/2] + 7*c^5*Sin[2*e + (7*f*x)/2] - 27*c^4*d*Sin[2*e + (7*f*x)/2] + 38*c^3*d^2*Sin[2*e + (7*f*x)/2] + 72*c^2*d^3*Sin[2*e + (7*f*x)/2] + 15*c*d^4*Sin[2*e + (7*f*x)/2] + 15*c*d^4*Sin[3*e + (7*f*x)/2] + 7*c^5*Sin[4*e + (7*f*x)/2] - 27*c^4*d*Sin[4*e + (7*f*x)/2] + 38*c^3*d^2*Sin[4*e + (7*f*x)/2] + 72*c^2*d^3*Sin[4*e + (7*f*x)/2]))/(120*c*(-c + d)^4*(c + d)*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2)","C",0
233,1,1096,368,7.828289,"\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx","Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3),x]","\frac{4 \cos ^4\left(\frac{e}{2}+\frac{f x}{2}\right) (d+c \cos (e+f x))^3 \sec \left(\frac{e}{2}\right) \left(23 d \sin \left(\frac{e}{2}\right)-8 c \sin \left(\frac{e}{2}\right)\right) \sec ^6(e+f x)}{15 (d-c)^4 f (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3}+\frac{\left(20 c^2+30 d c+13 d^2\right) \cos ^6\left(\frac{e}{2}+\frac{f x}{2}\right) (d+c \cos (e+f x))^3 \left(-\frac{8 i \tan ^{-1}\left(\sec \left(\frac{f x}{2}\right) \left(\frac{\cos (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}-\frac{i \sin (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}\right) \left(i c \sin \left(e+\frac{f x}{2}\right)-i d \sin \left(\frac{f x}{2}\right)\right)\right) \cos (e) d^3}{\sqrt{c^2-d^2} f \sqrt{\cos (2 e)-i \sin (2 e)}}-\frac{8 \tan ^{-1}\left(\sec \left(\frac{f x}{2}\right) \left(\frac{\cos (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}-\frac{i \sin (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}\right) \left(i c \sin \left(e+\frac{f x}{2}\right)-i d \sin \left(\frac{f x}{2}\right)\right)\right) \sin (e) d^3}{\sqrt{c^2-d^2} f \sqrt{\cos (2 e)-i \sin (2 e)}}\right) \sec ^6(e+f x)}{(d-c)^5 (c+d)^2 (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3}-\frac{2 \cos \left(\frac{e}{2}+\frac{f x}{2}\right) (d+c \cos (e+f x))^3 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sec ^6(e+f x)}{5 (d-c)^3 f (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3}+\frac{4 \cos ^3\left(\frac{e}{2}+\frac{f x}{2}\right) (d+c \cos (e+f x))^3 \sec \left(\frac{e}{2}\right) \left(23 d \sin \left(\frac{f x}{2}\right)-8 c \sin \left(\frac{f x}{2}\right)\right) \sec ^6(e+f x)}{15 (d-c)^4 f (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3}-\frac{8 \cos ^5\left(\frac{e}{2}+\frac{f x}{2}\right) (d+c \cos (e+f x))^3 \sec \left(\frac{e}{2}\right) \left(7 \sin \left(\frac{f x}{2}\right) c^2-44 d \sin \left(\frac{f x}{2}\right) c+127 d^2 \sin \left(\frac{f x}{2}\right)\right) \sec ^6(e+f x)}{15 (d-c)^5 f (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3}+\frac{4 \cos ^6\left(\frac{e}{2}+\frac{f x}{2}\right) (d+c \cos (e+f x)) \sec (e) \left(d^6 \sin (e)-c d^5 \sin (f x)\right) \sec ^6(e+f x)}{c^2 (d-c)^4 (c+d) f (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3}-\frac{4 \cos ^6\left(\frac{e}{2}+\frac{f x}{2}\right) (d+c \cos (e+f x))^2 \sec (e) \left(2 \sin (e) d^7-6 c \sin (e) d^6-c \sin (f x) d^6-11 c^2 \sin (e) d^5+6 c^2 \sin (f x) d^5+10 c^3 \sin (f x) d^4\right) \sec ^6(e+f x)}{c^2 (d-c)^5 (c+d)^2 f (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3}-\frac{2 \cos ^2\left(\frac{e}{2}+\frac{f x}{2}\right) (d+c \cos (e+f x))^3 \tan \left(\frac{e}{2}\right) \sec ^6(e+f x)}{5 (d-c)^3 f (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3}","\frac{\left(2 c^2-15 c d+76 d^2\right) \tan (e+f x)}{15 f (c-d)^3 \left(a^3 \sec (e+f x)+a^3\right) (c+d \sec (e+f x))^2}-\frac{d^3 \left(20 c^2+30 c d+13 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{a^3 f (c-d)^{11/2} (c+d)^{5/2}}+\frac{d \left(4 c^3-30 c^2 d+146 c d^2+195 d^3\right) \tan (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sec (e+f x))^2}+\frac{d \left(4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right) \tan (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 (c+d \sec (e+f x))}+\frac{(2 c-11 d) \tan (e+f x)}{15 a f (c-d)^2 (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))^2}+\frac{\tan (e+f x)}{5 f (c-d) (a \sec (e+f x)+a)^3 (c+d \sec (e+f x))^2}",1,"(4*Cos[e/2 + (f*x)/2]^4*(d + c*Cos[e + f*x])^3*Sec[e/2]*Sec[e + f*x]^6*(-8*c*Sin[e/2] + 23*d*Sin[e/2]))/(15*(-c + d)^4*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3) + ((20*c^2 + 30*c*d + 13*d^2)*Cos[e/2 + (f*x)/2]^6*(d + c*Cos[e + f*x])^3*Sec[e + f*x]^6*(((-8*I)*d^3*ArcTan[Sec[(f*x)/2]*(Cos[e]/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (I*Sin[e])/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]))*((-I)*d*Sin[(f*x)/2] + I*c*Sin[e + (f*x)/2])]*Cos[e])/(Sqrt[c^2 - d^2]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (8*d^3*ArcTan[Sec[(f*x)/2]*(Cos[e]/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (I*Sin[e])/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]))*((-I)*d*Sin[(f*x)/2] + I*c*Sin[e + (f*x)/2])]*Sin[e])/(Sqrt[c^2 - d^2]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((-c + d)^5*(c + d)^2*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3) - (2*Cos[e/2 + (f*x)/2]*(d + c*Cos[e + f*x])^3*Sec[e/2]*Sec[e + f*x]^6*Sin[(f*x)/2])/(5*(-c + d)^3*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3) + (4*Cos[e/2 + (f*x)/2]^3*(d + c*Cos[e + f*x])^3*Sec[e/2]*Sec[e + f*x]^6*(-8*c*Sin[(f*x)/2] + 23*d*Sin[(f*x)/2]))/(15*(-c + d)^4*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3) - (8*Cos[e/2 + (f*x)/2]^5*(d + c*Cos[e + f*x])^3*Sec[e/2]*Sec[e + f*x]^6*(7*c^2*Sin[(f*x)/2] - 44*c*d*Sin[(f*x)/2] + 127*d^2*Sin[(f*x)/2]))/(15*(-c + d)^5*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3) + (4*Cos[e/2 + (f*x)/2]^6*(d + c*Cos[e + f*x])*Sec[e]*Sec[e + f*x]^6*(d^6*Sin[e] - c*d^5*Sin[f*x]))/(c^2*(-c + d)^4*(c + d)*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3) - (4*Cos[e/2 + (f*x)/2]^6*(d + c*Cos[e + f*x])^2*Sec[e]*Sec[e + f*x]^6*(-11*c^2*d^5*Sin[e] - 6*c*d^6*Sin[e] + 2*d^7*Sin[e] + 10*c^3*d^4*Sin[f*x] + 6*c^2*d^5*Sin[f*x] - c*d^6*Sin[f*x]))/(c^2*(-c + d)^5*(c + d)^2*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3) - (2*Cos[e/2 + (f*x)/2]^2*(d + c*Cos[e + f*x])^3*Sec[e + f*x]^6*Tan[e/2])/(5*(-c + d)^3*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3)","C",1
234,1,102,61,0.2339972,"\int \frac{\sec (e+f x) \sqrt{a+a \sec (e+f x)}}{\sqrt{c+d \sec (e+f x)}} \, dx","Integrate[(Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/Sqrt[c + d*Sec[e + f*x]],x]","\frac{\sqrt{2} \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \sqrt{c \cos (e+f x)+d} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c \cos (e+f x)+d}}\right)}{\sqrt{d} f \sqrt{c+d \sec (e+f x)}}","\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{d} f}",1,"(Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[d]*Sin[(e + f*x)/2])/Sqrt[d + c*Cos[e + f*x]]]*Sqrt[d + c*Cos[e + f*x]]*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])])/(Sqrt[d]*f*Sqrt[c + d*Sec[e + f*x]])","A",1
235,1,187,140,17.0200046,"\int \frac{\sec (e+f x) \sqrt{c+d \sec (e+f x)}}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(Sec[e + f*x]*Sqrt[c + d*Sec[e + f*x]])/Sqrt[a + a*Sec[e + f*x]],x]","\frac{\sqrt{c} \sin (e+f x) \sqrt{c+d \sec (e+f x)} \left(2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{c \cos (e+f x)+d}}{\sqrt{d} \sqrt{c-c \cos (e+f x)}}\right)-\sqrt{2} \sqrt{c-d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{c \cos (e+f x)+d}}{\sqrt{c-d} \sqrt{c-c \cos (e+f x)}}\right)\right)}{f \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \cos (e+f x)} \sqrt{c \cos (e+f x)+d}}","\frac{\sqrt{2} \sqrt{c-d} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{a} f}+\frac{2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{a} f}",1,"(Sqrt[c]*(-(Sqrt[2]*Sqrt[c - d]*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + c*Cos[e + f*x]])/(Sqrt[c - d]*Sqrt[c - c*Cos[e + f*x]])]) + 2*Sqrt[d]*ArcTanh[(Sqrt[c]*Sqrt[d + c*Cos[e + f*x]])/(Sqrt[d]*Sqrt[c - c*Cos[e + f*x]])])*Sqrt[c + d*Sec[e + f*x]]*Sin[e + f*x])/(f*Sqrt[c - c*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]*Sqrt[a*(1 + Sec[e + f*x])])","A",1
236,1,107,78,0.2213435,"\int \frac{\sec (e+f x)}{\sqrt{a+a \sec (e+f x)} \sqrt{c+d \sec (e+f x)}} \, dx","Integrate[Sec[e + f*x]/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]),x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{c \cos (e+f x)+d} \tan ^{-1}\left(\frac{\sqrt{c-d} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c \cos (e+f x)+d}}\right)}{f \sqrt{c-d} \sqrt{a (\sec (e+f x)+1)} \sqrt{c+d \sec (e+f x)}}","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{a} f \sqrt{c-d}}",1,"(2*ArcTan[(Sqrt[c - d]*Sin[(e + f*x)/2])/Sqrt[d + c*Cos[e + f*x]]]*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*Sec[e + f*x])/(Sqrt[c - d]*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c + d*Sec[e + f*x]])","A",1
237,1,171,141,0.2776274,"\int \frac{\sec ^2(e+f x)}{\sqrt{a+a \sec (e+f x)} \sqrt{c+d \sec (e+f x)}} \, dx","Integrate[Sec[e + f*x]^2/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]),x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{c \cos (e+f x)+d} \left(\sqrt{2} \sqrt{c-d} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c \cos (e+f x)+d}}\right)-\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{c-d} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c \cos (e+f x)+d}}\right)\right)}{\sqrt{d} f \sqrt{c-d} \sqrt{a (\sec (e+f x)+1)} \sqrt{c+d \sec (e+f x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{a} \sqrt{d} f}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{a} f \sqrt{c-d}}",1,"(2*(-(Sqrt[d]*ArcTan[(Sqrt[c - d]*Sin[(e + f*x)/2])/Sqrt[d + c*Cos[e + f*x]]]) + Sqrt[2]*Sqrt[c - d]*ArcTanh[(Sqrt[2]*Sqrt[d]*Sin[(e + f*x)/2])/Sqrt[d + c*Cos[e + f*x]]])*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*Sec[e + f*x])/(Sqrt[c - d]*Sqrt[d]*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c + d*Sec[e + f*x]])","A",1
238,1,94,61,0.2263877,"\int \frac{\sec (e+f x) \sqrt{a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c + d*Sec[e + f*x]),x]","\frac{\sqrt{2} \sqrt{\cos (e+f x)} \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d} \sqrt{\cos (e+f x)}}\right)}{\sqrt{d} f \sqrt{c+d}}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{c+d} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{d} f \sqrt{c+d}}",1,"(Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(e + f*x)/2])/(Sqrt[c + d]*Sqrt[Cos[e + f*x]])]*Sqrt[Cos[e + f*x]]*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])])/(Sqrt[d]*Sqrt[c + d]*f)","A",1
239,1,427,149,1.369452,"\int \frac{(g \sec (e+f x))^{3/2} \sqrt{a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx","Integrate[((g*Sec[e + f*x])^(3/2)*Sqrt[a + a*Sec[e + f*x]])/(c + d*Sec[e + f*x]),x]","\frac{\left(\sqrt{2}-2 i\right) g^2 \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(i \left(2 \sqrt{c+d} \log \left(2 \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{2}\right)+2 \sqrt{c} \log \left(\sqrt{2} \sqrt{c+d}-2 \sqrt{c} \sin \left(\frac{1}{2} (e+f x)\right)\right)-2 \sqrt{c} \log \left(\sqrt{2} \sqrt{c+d}+2 \sqrt{c} \sin \left(\frac{1}{2} (e+f x)\right)\right)-\sqrt{c+d} \log \left(-\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)-\sqrt{2} \cos \left(\frac{1}{2} (e+f x)\right)+2\right)-\sqrt{c+d} \log \left(-\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{2} \cos \left(\frac{1}{2} (e+f x)\right)+2\right)\right)+2 \sqrt{c+d} \tan ^{-1}\left(\frac{\cos \left(\frac{1}{4} (e+f x)\right)-\left(\sqrt{2}-1\right) \sin \left(\frac{1}{4} (e+f x)\right)}{\left(1+\sqrt{2}\right) \cos \left(\frac{1}{4} (e+f x)\right)-\sin \left(\frac{1}{4} (e+f x)\right)}\right)+2 \sqrt{c+d} \tan ^{-1}\left(\frac{\cos \left(\frac{1}{4} (e+f x)\right)-\left(1+\sqrt{2}\right) \sin \left(\frac{1}{4} (e+f x)\right)}{\left(\sqrt{2}-1\right) \cos \left(\frac{1}{4} (e+f x)\right)-\sin \left(\frac{1}{4} (e+f x)\right)}\right)\right)}{4 \left(\sqrt{2}+i\right) d f \sqrt{c+d} \sqrt{g \sec (e+f x)}}","\frac{2 \sqrt{a} g^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{g} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}\right)}{d f}-\frac{2 \sqrt{a} \sqrt{c} g^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \sqrt{g} \tan (e+f x)}{\sqrt{c+d} \sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}\right)}{d f \sqrt{c+d}}",1,"((-2*I + Sqrt[2])*g^2*(2*Sqrt[c + d]*ArcTan[(Cos[(e + f*x)/4] - (-1 + Sqrt[2])*Sin[(e + f*x)/4])/((1 + Sqrt[2])*Cos[(e + f*x)/4] - Sin[(e + f*x)/4])] + 2*Sqrt[c + d]*ArcTan[(Cos[(e + f*x)/4] - (1 + Sqrt[2])*Sin[(e + f*x)/4])/((-1 + Sqrt[2])*Cos[(e + f*x)/4] - Sin[(e + f*x)/4])] + I*(2*Sqrt[c + d]*Log[Sqrt[2] + 2*Sin[(e + f*x)/2]] - Sqrt[c + d]*Log[2 - Sqrt[2]*Cos[(e + f*x)/2] - Sqrt[2]*Sin[(e + f*x)/2]] - Sqrt[c + d]*Log[2 + Sqrt[2]*Cos[(e + f*x)/2] - Sqrt[2]*Sin[(e + f*x)/2]] + 2*Sqrt[c]*Log[Sqrt[2]*Sqrt[c + d] - 2*Sqrt[c]*Sin[(e + f*x)/2]] - 2*Sqrt[c]*Log[Sqrt[2]*Sqrt[c + d] + 2*Sqrt[c]*Sin[(e + f*x)/2]]))*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])])/(4*(I + Sqrt[2])*d*Sqrt[c + d]*f*Sqrt[g*Sec[e + f*x]])","C",1
240,1,229015,122,33.7118766,"\int \frac{\sec (e+f x)}{\sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]","\text{Result too large to show}","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f (c-d)}-\frac{2 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{c+d} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f (c-d) \sqrt{c+d}}",1,"Result too large to show","C",0
241,1,141,124,0.4081889,"\int \frac{\sec ^2(e+f x)}{\sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]^2/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]","-\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \left(\sqrt{d} \sqrt{c+d} \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{\cos (e+f x)}}\right)-\sqrt{2} c \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d} \sqrt{\cos (e+f x)}}\right)\right)}{\sqrt{d} f (c-d) \sqrt{c+d} \sqrt{\cos (e+f x)} \sqrt{a (\sec (e+f x)+1)}}","\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{c+d} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} \sqrt{d} f (c-d) \sqrt{c+d}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f (c-d)}",1,"(-2*(Sqrt[d]*Sqrt[c + d]*ArcTan[Sin[(e + f*x)/2]/Sqrt[Cos[e + f*x]]] - Sqrt[2]*c*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(e + f*x)/2])/(Sqrt[c + d]*Sqrt[Cos[e + f*x]])])*Cos[(e + f*x)/2])/((c - d)*Sqrt[d]*Sqrt[c + d]*f*Sqrt[Cos[e + f*x]]*Sqrt[a*(1 + Sec[e + f*x])])","A",1
242,1,198,167,0.3945492,"\int \frac{(g \sec (e+f x))^{3/2}}{\sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx","Integrate[(g*Sec[e + f*x])^(3/2)/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]","\frac{g \cos \left(\frac{1}{2} (e+f x)\right) \sqrt{g \sec (e+f x)} \left(\sqrt{2} \sqrt{c} \left(\log \left(\sqrt{2} \sqrt{c+d}+2 \sqrt{c} \sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sqrt{2} \sqrt{c+d}-2 \sqrt{c} \sin \left(\frac{1}{2} (e+f x)\right)\right)\right)+2 \sqrt{c+d} \log \left(\cos \left(\frac{1}{4} (e+f x)\right)-\sin \left(\frac{1}{4} (e+f x)\right)\right)-2 \sqrt{c+d} \log \left(\sin \left(\frac{1}{4} (e+f x)\right)+\cos \left(\frac{1}{4} (e+f x)\right)\right)\right)}{f (c-d) \sqrt{c+d} \sqrt{a (\sec (e+f x)+1)}}","\frac{2 \sqrt{c} g^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \sqrt{g} \tan (e+f x)}{\sqrt{c+d} \sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}\right)}{\sqrt{a} f (c-d) \sqrt{c+d}}-\frac{\sqrt{2} g^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{g} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}\right)}{\sqrt{a} f (c-d)}",1,"(g*Cos[(e + f*x)/2]*(2*Sqrt[c + d]*Log[Cos[(e + f*x)/4] - Sin[(e + f*x)/4]] - 2*Sqrt[c + d]*Log[Cos[(e + f*x)/4] + Sin[(e + f*x)/4]] + Sqrt[2]*Sqrt[c]*(-Log[Sqrt[2]*Sqrt[c + d] - 2*Sqrt[c]*Sin[(e + f*x)/2]] + Log[Sqrt[2]*Sqrt[c + d] + 2*Sqrt[c]*Sin[(e + f*x)/2]]))*Sqrt[g*Sec[e + f*x]])/((c - d)*Sqrt[c + d]*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
243,1,155,231,0.3818266,"\int \frac{(g \sec (e+f x))^{5/2}}{\sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx","Integrate[(g*Sec[e + f*x])^(5/2)/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]","\frac{2 g^2 \cos \left(\frac{1}{2} (e+f x)\right) \sqrt{g \sec (e+f x)} \left(\sqrt{2} \left((c-d) \sqrt{c+d} \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right)-c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)\right)+d \sqrt{c+d} \tanh ^{-1}\left(\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)}{d f (c-d) \sqrt{c+d} \sqrt{a (\sec (e+f x)+1)}}","-\frac{2 c^{3/2} g^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \sqrt{g} \tan (e+f x)}{\sqrt{c+d} \sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}\right)}{\sqrt{a} d f (c-d) \sqrt{c+d}}+\frac{\sqrt{2} g^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{g} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}\right)}{\sqrt{a} f (c-d)}+\frac{2 g^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{g} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}\right)}{\sqrt{a} d f}",1,"(2*g^2*(d*Sqrt[c + d]*ArcTanh[Sin[(e + f*x)/2]] + Sqrt[2]*((c - d)*Sqrt[c + d]*ArcTanh[Sqrt[2]*Sin[(e + f*x)/2]] - c^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sin[(e + f*x)/2])/Sqrt[c + d]]))*Cos[(e + f*x)/2]*Sqrt[g*Sec[e + f*x]])/((c - d)*d*Sqrt[c + d]*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
244,1,201,250,4.4788507,"\int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^4 \, dx","Integrate[Sec[e + f*x]*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^4,x]","\frac{15 \left(a \left(8 c^4+24 c^2 d^2+3 d^4\right)+4 b c d \left(4 c^2+3 d^2\right)\right) \tanh ^{-1}(\sin (e+f x))+\tan (e+f x) \left(8 \left(10 d^2 \left(2 a c d+b \left(3 c^2+d^2\right)\right) \tan ^2(e+f x)+15 \left(4 a c d \left(c^2+d^2\right)+b \left(c^4+6 c^2 d^2+d^4\right)\right)+3 b d^4 \tan ^4(e+f x)\right)+15 d \left(3 a d \left(8 c^2+d^2\right)+4 b \left(4 c^3+3 c d^2\right)\right) \sec (e+f x)+30 d^3 (a d+4 b c) \sec ^3(e+f x)\right)}{120 f}","\frac{\left(35 a c d+12 b c^2+16 b d^2\right) \tan (e+f x) (c+d \sec (e+f x))^2}{60 f}+\frac{d \left(130 a c^2 d+45 a d^3+24 b c^3+116 b c d^2\right) \tan (e+f x) \sec (e+f x)}{120 f}+\frac{\left(95 a c^3 d+80 a c d^3+12 b c^4+112 b c^2 d^2+16 b d^4\right) \tan (e+f x)}{30 f}+\frac{\left(8 a c^4+24 a c^2 d^2+3 a d^4+16 b c^3 d+12 b c d^3\right) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{(5 a d+4 b c) \tan (e+f x) (c+d \sec (e+f x))^3}{20 f}+\frac{b \tan (e+f x) (c+d \sec (e+f x))^4}{5 f}",1,"(15*(4*b*c*d*(4*c^2 + 3*d^2) + a*(8*c^4 + 24*c^2*d^2 + 3*d^4))*ArcTanh[Sin[e + f*x]] + Tan[e + f*x]*(15*d*(3*a*d*(8*c^2 + d^2) + 4*b*(4*c^3 + 3*c*d^2))*Sec[e + f*x] + 30*d^3*(4*b*c + a*d)*Sec[e + f*x]^3 + 8*(15*(4*a*c*d*(c^2 + d^2) + b*(c^4 + 6*c^2*d^2 + d^4)) + 10*d^2*(2*a*c*d + b*(3*c^2 + d^2))*Tan[e + f*x]^2 + 3*b*d^4*Tan[e + f*x]^4)))/(120*f)","A",1
245,1,143,180,1.1101721,"\int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx","Integrate[Sec[e + f*x]*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^3,x]","\frac{3 \left(4 a \left(2 c^3+3 c d^2\right)+3 b d \left(4 c^2+d^2\right)\right) \tanh ^{-1}(\sin (e+f x))+\tan (e+f x) \left(9 d \left(4 a c d+b \left(4 c^2+d^2\right)\right) \sec (e+f x)+8 \left(d^2 (a d+3 b c) \tan ^2(e+f x)+3 a d \left(3 c^2+d^2\right)+3 b \left(c^3+3 c d^2\right)\right)+6 b d^3 \sec ^3(e+f x)\right)}{24 f}","\frac{d \left(20 a c d+6 b c^2+9 b d^2\right) \tan (e+f x) \sec (e+f x)}{24 f}+\frac{\left(4 a d \left(4 c^2+d^2\right)+3 b \left(c^3+4 c d^2\right)\right) \tan (e+f x)}{6 f}+\frac{\left(8 a c^3+12 a c d^2+12 b c^2 d+3 b d^3\right) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{(4 a d+3 b c) \tan (e+f x) (c+d \sec (e+f x))^2}{12 f}+\frac{b \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}",1,"(3*(3*b*d*(4*c^2 + d^2) + 4*a*(2*c^3 + 3*c*d^2))*ArcTanh[Sin[e + f*x]] + Tan[e + f*x]*(9*d*(4*a*c*d + b*(4*c^2 + d^2))*Sec[e + f*x] + 6*b*d^3*Sec[e + f*x]^3 + 8*(3*a*d*(3*c^2 + d^2) + 3*b*(c^3 + 3*c*d^2) + d^2*(3*b*c + a*d)*Tan[e + f*x]^2)))/(24*f)","A",1
246,1,88,115,0.6077787,"\int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^2 \, dx","Integrate[Sec[e + f*x]*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^2,x]","\frac{3 \left(a \left(2 c^2+d^2\right)+2 b c d\right) \tanh ^{-1}(\sin (e+f x))+\tan (e+f x) \left(3 d (a d+2 b c) \sec (e+f x)+12 a c d+6 b \left(c^2+d^2\right)+2 b d^2 \tan ^2(e+f x)\right)}{6 f}","\frac{2 \left(3 a c d+b \left(c^2+d^2\right)\right) \tan (e+f x)}{3 f}+\frac{\left(a \left(2 c^2+d^2\right)+2 b c d\right) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{d (3 a d+2 b c) \tan (e+f x) \sec (e+f x)}{6 f}+\frac{b \tan (e+f x) (c+d \sec (e+f x))^2}{3 f}",1,"(3*(2*b*c*d + a*(2*c^2 + d^2))*ArcTanh[Sin[e + f*x]] + Tan[e + f*x]*(12*a*c*d + 6*b*(c^2 + d^2) + 3*d*(2*b*c + a*d)*Sec[e + f*x] + 2*b*d^2*Tan[e + f*x]^2))/(6*f)","A",1
247,1,75,61,0.0248105,"\int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x)) \, dx","Integrate[Sec[e + f*x]*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x]),x]","\frac{a c \tanh ^{-1}(\sin (e+f x))}{f}+\frac{a d \tan (e+f x)}{f}+\frac{b c \tan (e+f x)}{f}+\frac{b d \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{b d \tan (e+f x) \sec (e+f x)}{2 f}","\frac{(a d+b c) \tan (e+f x)}{f}+\frac{(2 a c+b d) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{b d \tan (e+f x) \sec (e+f x)}{2 f}",1,"(a*c*ArcTanh[Sin[e + f*x]])/f + (b*d*ArcTanh[Sin[e + f*x]])/(2*f) + (b*c*Tan[e + f*x])/f + (a*d*Tan[e + f*x])/f + (b*d*Sec[e + f*x]*Tan[e + f*x])/(2*f)","A",1
248,1,112,76,0.2121285,"\int \frac{\sec (e+f x) (a+b \sec (e+f x))}{c+d \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(a + b*Sec[e + f*x]))/(c + d*Sec[e + f*x]),x]","\frac{\frac{2 (b c-a d) \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}+b \left(\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)}{d f}","\frac{b \tanh ^{-1}(\sin (e+f x))}{d f}-\frac{2 (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{d f \sqrt{c-d} \sqrt{c+d}}",1,"((2*(b*c - a*d)*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] + b*(-Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]))/(d*f)","A",1
249,1,97,99,0.3867388,"\int \frac{\sec (e+f x) (a+b \sec (e+f x))}{(c+d \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(a + b*Sec[e + f*x]))/(c + d*Sec[e + f*x])^2,x]","\frac{\frac{(b c-a d) \sin (e+f x)}{(c-d) (c+d) (c \cos (e+f x)+d)}-\frac{2 (a c-b d) \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\left(c^2-d^2\right)^{3/2}}}{f}","\frac{(b c-a d) \tan (e+f x)}{f \left(c^2-d^2\right) (c+d \sec (e+f x))}+\frac{2 (a c-b d) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f (c-d)^{3/2} (c+d)^{3/2}}",1,"((-2*(a*c - b*d)*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(3/2) + ((b*c - a*d)*Sin[e + f*x])/((c - d)*(c + d)*(d + c*Cos[e + f*x])))/f","A",1
250,1,172,166,0.9809308,"\int \frac{\sec (e+f x) (a+b \sec (e+f x))}{(c+d \sec (e+f x))^3} \, dx","Integrate[(Sec[e + f*x]*(a + b*Sec[e + f*x]))/(c + d*Sec[e + f*x])^3,x]","\frac{\frac{\left(a d \left(d^2-4 c^2\right)+b c \left(2 c^2+d^2\right)\right) \sin (e+f x)}{c (c-d)^2 (c+d)^2 (c \cos (e+f x)+d)}-\frac{2 \left(a \left(2 c^2+d^2\right)-3 b c d\right) \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\left(c^2-d^2\right)^{5/2}}+\frac{d (a d-b c) \sin (e+f x)}{c (c-d) (c+d) (c \cos (e+f x)+d)^2}}{2 f}","-\frac{\left(3 b c d-a \left(2 c^2+d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f (c-d)^{5/2} (c+d)^{5/2}}-\frac{\left(3 a c d-b \left(c^2+2 d^2\right)\right) \tan (e+f x)}{2 f \left(c^2-d^2\right)^2 (c+d \sec (e+f x))}+\frac{(b c-a d) \tan (e+f x)}{2 f \left(c^2-d^2\right) (c+d \sec (e+f x))^2}",1,"((-2*(-3*b*c*d + a*(2*c^2 + d^2))*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(5/2) + (d*(-(b*c) + a*d)*Sin[e + f*x])/(c*(c - d)*(c + d)*(d + c*Cos[e + f*x])^2) + ((a*d*(-4*c^2 + d^2) + b*c*(2*c^2 + d^2))*Sin[e + f*x])/(c*(c - d)^2*(c + d)^2*(d + c*Cos[e + f*x])))/(2*f)","A",1
251,1,405,237,1.0821637,"\int \frac{\sec (e+f x) (a+b \sec (e+f x))}{(c+d \sec (e+f x))^4} \, dx","Integrate[(Sec[e + f*x]*(a + b*Sec[e + f*x]))/(c + d*Sec[e + f*x])^4,x]","\frac{\sec ^3(e+f x) (a+b \sec (e+f x)) (c \cos (e+f x)+d) \left(\frac{24 \left(a \left(2 c^3+3 c d^2\right)-b d \left(4 c^2+d^2\right)\right) (c \cos (e+f x)+d)^3 \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}+18 a c^4 d \sin (e+f x)+18 a c^4 d \sin (3 (e+f x))+54 a c^3 d^2 \sin (2 (e+f x))+39 a c^2 d^3 \sin (e+f x)-5 a c^2 d^3 \sin (3 (e+f x))+6 a c d^4 \sin (2 (e+f x))+18 a d^5 \sin (e+f x)+2 a d^5 \sin (3 (e+f x))-6 b c^5 \sin (e+f x)-6 b c^5 \sin (3 (e+f x))-12 b c^4 d \sin (2 (e+f x))-18 b c^3 d^2 \sin (e+f x)-10 b c^3 d^2 \sin (3 (e+f x))-54 b c^2 d^3 \sin (2 (e+f x))-51 b c d^4 \sin (e+f x)+b c d^4 \sin (3 (e+f x))+6 b d^5 \sin (2 (e+f x))\right)}{24 f \left(d^2-c^2\right)^3 (a \cos (e+f x)+b) (c+d \sec (e+f x))^4}","\frac{\left(-5 a c d+2 b c^2+3 b d^2\right) \tan (e+f x)}{6 f \left(c^2-d^2\right)^2 (c+d \sec (e+f x))^2}+\frac{(b c-a d) \tan (e+f x)}{3 f \left(c^2-d^2\right) (c+d \sec (e+f x))^3}+\frac{\left(2 a c^3+3 a c d^2-4 b c^2 d-b d^3\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f (c-d)^{7/2} (c+d)^{7/2}}+\frac{\left(-11 a c^2 d-4 a d^3+2 b c^3+13 b c d^2\right) \tan (e+f x)}{6 f \left(c^2-d^2\right)^3 (c+d \sec (e+f x))}",1,"((d + c*Cos[e + f*x])*Sec[e + f*x]^3*(a + b*Sec[e + f*x])*((24*(-(b*d*(4*c^2 + d^2)) + a*(2*c^3 + 3*c*d^2))*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x])^3)/Sqrt[c^2 - d^2] - 6*b*c^5*Sin[e + f*x] + 18*a*c^4*d*Sin[e + f*x] - 18*b*c^3*d^2*Sin[e + f*x] + 39*a*c^2*d^3*Sin[e + f*x] - 51*b*c*d^4*Sin[e + f*x] + 18*a*d^5*Sin[e + f*x] - 12*b*c^4*d*Sin[2*(e + f*x)] + 54*a*c^3*d^2*Sin[2*(e + f*x)] - 54*b*c^2*d^3*Sin[2*(e + f*x)] + 6*a*c*d^4*Sin[2*(e + f*x)] + 6*b*d^5*Sin[2*(e + f*x)] - 6*b*c^5*Sin[3*(e + f*x)] + 18*a*c^4*d*Sin[3*(e + f*x)] - 10*b*c^3*d^2*Sin[3*(e + f*x)] - 5*a*c^2*d^3*Sin[3*(e + f*x)] + b*c*d^4*Sin[3*(e + f*x)] + 2*a*d^5*Sin[3*(e + f*x)]))/(24*(-c^2 + d^2)^3*f*(b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^4)","A",1
252,1,580,247,4.6949695,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^4}{a+b \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^4)/(a + b*Sec[e + f*x]),x]","\frac{\cos ^3(e+f x) (a \cos (e+f x)+b) (c+d \sec (e+f x))^4 \left(\frac{4 b d^2 \left(3 a^2 d^2-12 a b c d+2 b^2 \left(9 c^2+d^2\right)\right) \sin \left(\frac{1}{2} (e+f x)\right)}{\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)}+\frac{4 b d^2 \left(3 a^2 d^2-12 a b c d+2 b^2 \left(9 c^2+d^2\right)\right) \sin \left(\frac{1}{2} (e+f x)\right)}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}-\frac{24 (b c-a d)^4 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-6 d \left(-2 a^3 d^3+8 a^2 b c d^2-a b^2 d \left(12 c^2+d^2\right)+4 b^3 c \left(2 c^2+d^2\right)\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-6 d \left(2 a^3 d^3-8 a^2 b c d^2+a b^2 d \left(12 c^2+d^2\right)-4 b^3 c \left(2 c^2+d^2\right)\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\frac{b^2 d^3 (b (12 c+d)-3 a d)}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{b^2 d^3 (b (12 c+d)-3 a d)}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{2 b^3 d^4 \sin \left(\frac{1}{2} (e+f x)\right)}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{2 b^3 d^4 \sin \left(\frac{1}{2} (e+f x)\right)}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}\right)}{12 b^4 f (a+b \sec (e+f x)) (c \cos (e+f x)+d)^4}","\frac{d (2 b c-a d) \left(a^2 d^2-2 a b c d+2 b^2 c^2\right) \tanh ^{-1}(\sin (e+f x))}{b^4 f}+\frac{d^2 \left(a^2 d^2-4 a b c d+6 b^2 c^2\right) \tan (e+f x)}{b^3 f}+\frac{2 (b c-a d)^4 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{b^4 f \sqrt{a-b} \sqrt{a+b}}+\frac{d^3 (4 b c-a d) \tanh ^{-1}(\sin (e+f x))}{2 b^2 f}+\frac{d^3 (4 b c-a d) \tan (e+f x) \sec (e+f x)}{2 b^2 f}+\frac{d^4 \tan ^3(e+f x)}{3 b f}+\frac{d^4 \tan (e+f x)}{b f}",1,"(Cos[e + f*x]^3*(b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^4*((-24*(b*c - a*d)^4*ArcTanh[((-a + b)*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 6*d*(8*a^2*b*c*d^2 - 2*a^3*d^3 + 4*b^3*c*(2*c^2 + d^2) - a*b^2*d*(12*c^2 + d^2))*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 6*d*(-8*a^2*b*c*d^2 + 2*a^3*d^3 - 4*b^3*c*(2*c^2 + d^2) + a*b^2*d*(12*c^2 + d^2))*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (b^2*d^3*(-3*a*d + b*(12*c + d)))/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + (2*b^3*d^4*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + (4*b*d^2*(-12*a*b*c*d + 3*a^2*d^2 + 2*b^2*(9*c^2 + d^2))*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) + (2*b^3*d^4*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - (b^2*d^3*(-3*a*d + b*(12*c + d)))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (4*b*d^2*(-12*a*b*c*d + 3*a^2*d^2 + 2*b^2*(9*c^2 + d^2))*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))/(12*b^4*f*(d + c*Cos[e + f*x])^4*(a + b*Sec[e + f*x]))","B",1
253,1,389,170,1.4735124,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^3}{a+b \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^3)/(a + b*Sec[e + f*x]),x]","\frac{\cos ^2(e+f x) (a \cos (e+f x)+b) (c+d \sec (e+f x))^3 \left(-2 d \left(2 a^2 d^2-6 a b c d+b^2 \left(6 c^2+d^2\right)\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+2 d \left(2 a^2 d^2-6 a b c d+b^2 \left(6 c^2+d^2\right)\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\frac{8 (a d-b c)^3 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{4 b d^2 (3 b c-a d) \sin \left(\frac{1}{2} (e+f x)\right)}{\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)}+\frac{4 b d^2 (3 b c-a d) \sin \left(\frac{1}{2} (e+f x)\right)}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}+\frac{b^2 d^3}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{b^2 d^3}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}\right)}{4 b^3 f (a+b \sec (e+f x)) (c \cos (e+f x)+d)^3}","\frac{d \left(a^2 d^2-3 a b c d+3 b^2 c^2\right) \tanh ^{-1}(\sin (e+f x))}{b^3 f}+\frac{2 (b c-a d)^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{b^3 f \sqrt{a-b} \sqrt{a+b}}+\frac{d^2 (3 b c-a d) \tan (e+f x)}{b^2 f}+\frac{d^3 \tanh ^{-1}(\sin (e+f x))}{2 b f}+\frac{d^3 \tan (e+f x) \sec (e+f x)}{2 b f}",1,"(Cos[e + f*x]^2*(b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^3*((8*(-(b*c) + a*d)^3*ArcTanh[((-a + b)*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 2*d*(-6*a*b*c*d + 2*a^2*d^2 + b^2*(6*c^2 + d^2))*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 2*d*(-6*a*b*c*d + 2*a^2*d^2 + b^2*(6*c^2 + d^2))*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (b^2*d^3)/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + (4*b*d^2*(3*b*c - a*d)*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - (b^2*d^3)/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (4*b*d^2*(3*b*c - a*d)*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))/(4*b^3*f*(d + c*Cos[e + f*x])^3*(a + b*Sec[e + f*x]))","B",1
254,1,135,103,0.8268846,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^2}{a+b \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^2)/(a + b*Sec[e + f*x]),x]","\frac{d \left(b d \tan (e+f x)-(2 b c-a d) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)-\frac{2 (b c-a d)^2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}}{b^2 f}","\frac{d (2 b c-a d) \tanh ^{-1}(\sin (e+f x))}{b^2 f}+\frac{2 (b c-a d)^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{b^2 f \sqrt{a-b} \sqrt{a+b}}+\frac{d^2 \tan (e+f x)}{b f}",1,"((-2*(b*c - a*d)^2*ArcTanh[((-a + b)*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + d*(-((2*b*c - a*d)*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])) + b*d*Tan[e + f*x]))/(b^2*f)","A",1
255,1,112,76,0.1828397,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))}{a+b \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x]))/(a + b*Sec[e + f*x]),x]","\frac{\frac{2 (a d-b c) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+d \left(\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)}{b f}","\frac{2 (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{b f \sqrt{a-b} \sqrt{a+b}}+\frac{d \tanh ^{-1}(\sin (e+f x))}{b f}",1,"((2*(-(b*c) + a*d)*ArcTanh[((-a + b)*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + d*(-Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]))/(b*f)","A",1
256,1,119,121,0.2537639,"\int \frac{\sec (e+f x)}{(a+b \sec (e+f x)) (c+d \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]/((a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])),x]","-\frac{2 b \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2} (b c-a d)}-\frac{2 d \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{f \sqrt{c^2-d^2} (a d-b c)}","\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{f \sqrt{a-b} \sqrt{a+b} (b c-a d)}-\frac{2 d \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f \sqrt{c-d} \sqrt{c+d} (b c-a d)}",1,"(-2*b*ArcTanh[((-a + b)*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*(b*c - a*d)*f) - (2*d*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((-(b*c) + a*d)*Sqrt[c^2 - d^2]*f)","A",1
257,1,229,187,0.7195289,"\int \frac{\sec (e+f x)}{(a+b \sec (e+f x)) (c+d \sec (e+f x))^2} \, dx","Integrate[Sec[e + f*x]/((a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^2),x]","\frac{-2 b^2 \left(c^2-d^2\right)^{3/2} (c \cos (e+f x)+d) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a^2-b^2}}\right)-d \sqrt{a^2-b^2} \left(d \sqrt{c^2-d^2} (a d-b c) \sin (e+f x)-2 \left(-a c d+2 b c^2-b d^2\right) (c \cos (e+f x)+d) \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)\right)}{f \sqrt{a^2-b^2} (c-d) (c+d) \sqrt{c^2-d^2} (b c-a d)^2 (c \cos (e+f x)+d)}","\frac{2 b^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{f \sqrt{a-b} \sqrt{a+b} (b c-a d)^2}+\frac{d^2 \sin (e+f x)}{f \left(c^2-d^2\right) (b c-a d) (c \cos (e+f x)+d)}-\frac{2 d \left(-a c d+2 b c^2-b d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f (c-d)^{3/2} (c+d)^{3/2} (b c-a d)^2}",1,"(-2*b^2*(c^2 - d^2)^(3/2)*ArcTanh[((-a + b)*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]]*(d + c*Cos[e + f*x]) - Sqrt[a^2 - b^2]*d*(-2*(2*b*c^2 - a*c*d - b*d^2)*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x]) + d*(-(b*c) + a*d)*Sqrt[c^2 - d^2]*Sin[e + f*x]))/(Sqrt[a^2 - b^2]*(c - d)*(c + d)*(b*c - a*d)^2*Sqrt[c^2 - d^2]*f*(d + c*Cos[e + f*x]))","A",1
258,1,1137,379,6.5471463,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^5}{(a+b \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^5)/(a + b*Sec[e + f*x])^2,x]","\frac{(b+a \cos (e+f x)) \left(12 d^5 \sin (e+f x) b^5+60 c^2 d^3 \sin (e+f x) b^5+6 c^5 \sin (2 (e+f x)) b^5+30 c d^4 \sin (2 (e+f x)) b^5+4 d^5 \sin (3 (e+f x)) b^5+60 c^2 d^3 \sin (3 (e+f x)) b^5+3 c^5 \sin (4 (e+f x)) b^5-45 a c d^4 \sin (e+f x) b^4-4 a d^5 \sin (2 (e+f x)) b^4+60 a c^2 d^3 \sin (2 (e+f x)) b^4-30 a c^4 d \sin (2 (e+f x)) b^4-45 a c d^4 \sin (3 (e+f x)) b^4+2 a d^5 \sin (4 (e+f x)) b^4+30 a c^2 d^3 \sin (4 (e+f x)) b^4-15 a c^4 d \sin (4 (e+f x)) b^4-60 a^2 c^2 d^3 \sin (e+f x) b^3-90 a^2 c d^4 \sin (2 (e+f x)) b^3+60 a^2 c^3 d^2 \sin (2 (e+f x)) b^3+8 a^2 d^5 \sin (3 (e+f x)) b^3-60 a^2 c^2 d^3 \sin (3 (e+f x)) b^3-30 a^2 c d^4 \sin (4 (e+f x)) b^3+30 a^2 c^3 d^2 \sin (4 (e+f x)) b^3+45 a^3 c d^4 \sin (e+f x) b^2+22 a^3 d^5 \sin (2 (e+f x)) b^2-120 a^3 c^2 d^3 \sin (2 (e+f x)) b^2+45 a^3 c d^4 \sin (3 (e+f x)) b^2+7 a^3 d^5 \sin (4 (e+f x)) b^2-60 a^3 c^2 d^3 \sin (4 (e+f x)) b^2-12 a^4 d^5 \sin (e+f x) b+90 a^4 c d^4 \sin (2 (e+f x)) b-12 a^4 d^5 \sin (3 (e+f x)) b+45 a^4 c d^4 \sin (4 (e+f x)) b-24 a^5 d^5 \sin (2 (e+f x))-12 a^5 d^5 \sin (4 (e+f x))\right) (c+d \sec (e+f x))^5}{24 b^4 \left(b^2-a^2\right) f (d+c \cos (e+f x))^5 (a+b \sec (e+f x))^2}+\frac{\left(8 a^3 d^5+2 a b^2 d^5-5 b^3 c d^4-30 a^2 b c d^4+40 a b^2 c^2 d^3-20 b^3 c^3 d^2\right) \cos ^3(e+f x) (b+a \cos (e+f x))^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sec (e+f x))^5}{2 b^5 f (d+c \cos (e+f x))^5 (a+b \sec (e+f x))^2}+\frac{\left(-8 a^3 d^5-2 a b^2 d^5+5 b^3 c d^4+30 a^2 b c d^4-40 a b^2 c^2 d^3+20 b^3 c^3 d^2\right) \cos ^3(e+f x) (b+a \cos (e+f x))^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sec (e+f x))^5}{2 b^5 f (d+c \cos (e+f x))^5 (a+b \sec (e+f x))^2}-\frac{2 (b c-a d)^4 \left(-4 d a^2-b c a+5 b^2 d\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a^2-b^2}}\right) \cos ^3(e+f x) (b+a \cos (e+f x))^2 (c+d \sec (e+f x))^5}{b^5 \sqrt{a^2-b^2} \left(b^2-a^2\right) f (d+c \cos (e+f x))^5 (a+b \sec (e+f x))^2}","\frac{d^3 \left(3 a^2 d^2-10 a b c d+10 b^2 c^2\right) \tan (e+f x)}{b^4 f}-\frac{(b c-a d)^5 \sin (e+f x)}{b^4 f \left(a^2-b^2\right) (a \cos (e+f x)+b)}+\frac{d^2 \left(-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right) \tanh ^{-1}(\sin (e+f x))}{b^5 f}+\frac{2 (b c-a d)^4 (4 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a b^5 f \sqrt{a-b} \sqrt{a+b}}+\frac{d^4 (5 b c-2 a d) \tanh ^{-1}(\sin (e+f x))}{2 b^3 f}+\frac{d^4 (5 b c-2 a d) \tan (e+f x) \sec (e+f x)}{2 b^3 f}+\frac{2 (b c-a d)^5 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a b^3 f (a-b)^{3/2} (a+b)^{3/2}}+\frac{d^5 \tan ^3(e+f x)}{3 b^2 f}+\frac{d^5 \tan (e+f x)}{b^2 f}",1,"(-2*(b*c - a*d)^4*(-(a*b*c) - 4*a^2*d + 5*b^2*d)*ArcTanh[((-a + b)*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]]*Cos[e + f*x]^3*(b + a*Cos[e + f*x])^2*(c + d*Sec[e + f*x])^5)/(b^5*Sqrt[a^2 - b^2]*(-a^2 + b^2)*f*(d + c*Cos[e + f*x])^5*(a + b*Sec[e + f*x])^2) + ((-20*b^3*c^3*d^2 + 40*a*b^2*c^2*d^3 - 30*a^2*b*c*d^4 - 5*b^3*c*d^4 + 8*a^3*d^5 + 2*a*b^2*d^5)*Cos[e + f*x]^3*(b + a*Cos[e + f*x])^2*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(c + d*Sec[e + f*x])^5)/(2*b^5*f*(d + c*Cos[e + f*x])^5*(a + b*Sec[e + f*x])^2) + ((20*b^3*c^3*d^2 - 40*a*b^2*c^2*d^3 + 30*a^2*b*c*d^4 + 5*b^3*c*d^4 - 8*a^3*d^5 - 2*a*b^2*d^5)*Cos[e + f*x]^3*(b + a*Cos[e + f*x])^2*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(c + d*Sec[e + f*x])^5)/(2*b^5*f*(d + c*Cos[e + f*x])^5*(a + b*Sec[e + f*x])^2) + ((b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^5*(-60*a^2*b^3*c^2*d^3*Sin[e + f*x] + 60*b^5*c^2*d^3*Sin[e + f*x] + 45*a^3*b^2*c*d^4*Sin[e + f*x] - 45*a*b^4*c*d^4*Sin[e + f*x] - 12*a^4*b*d^5*Sin[e + f*x] + 12*b^5*d^5*Sin[e + f*x] + 6*b^5*c^5*Sin[2*(e + f*x)] - 30*a*b^4*c^4*d*Sin[2*(e + f*x)] + 60*a^2*b^3*c^3*d^2*Sin[2*(e + f*x)] - 120*a^3*b^2*c^2*d^3*Sin[2*(e + f*x)] + 60*a*b^4*c^2*d^3*Sin[2*(e + f*x)] + 90*a^4*b*c*d^4*Sin[2*(e + f*x)] - 90*a^2*b^3*c*d^4*Sin[2*(e + f*x)] + 30*b^5*c*d^4*Sin[2*(e + f*x)] - 24*a^5*d^5*Sin[2*(e + f*x)] + 22*a^3*b^2*d^5*Sin[2*(e + f*x)] - 4*a*b^4*d^5*Sin[2*(e + f*x)] - 60*a^2*b^3*c^2*d^3*Sin[3*(e + f*x)] + 60*b^5*c^2*d^3*Sin[3*(e + f*x)] + 45*a^3*b^2*c*d^4*Sin[3*(e + f*x)] - 45*a*b^4*c*d^4*Sin[3*(e + f*x)] - 12*a^4*b*d^5*Sin[3*(e + f*x)] + 8*a^2*b^3*d^5*Sin[3*(e + f*x)] + 4*b^5*d^5*Sin[3*(e + f*x)] + 3*b^5*c^5*Sin[4*(e + f*x)] - 15*a*b^4*c^4*d*Sin[4*(e + f*x)] + 30*a^2*b^3*c^3*d^2*Sin[4*(e + f*x)] - 60*a^3*b^2*c^2*d^3*Sin[4*(e + f*x)] + 30*a*b^4*c^2*d^3*Sin[4*(e + f*x)] + 45*a^4*b*c*d^4*Sin[4*(e + f*x)] - 30*a^2*b^3*c*d^4*Sin[4*(e + f*x)] - 12*a^5*d^5*Sin[4*(e + f*x)] + 7*a^3*b^2*d^5*Sin[4*(e + f*x)] + 2*a*b^4*d^5*Sin[4*(e + f*x)]))/(24*b^4*(-a^2 + b^2)*f*(d + c*Cos[e + f*x])^5*(a + b*Sec[e + f*x])^2)","B",1
259,1,511,297,4.1195206,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^4}{(a+b \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^4)/(a + b*Sec[e + f*x])^2,x]","\frac{\cos ^2(e+f x) (a \cos (e+f x)+b) (c+d \sec (e+f x))^4 \left(-2 d^2 \left(6 a^2 d^2-16 a b c d+b^2 \left(12 c^2+d^2\right)\right) (a \cos (e+f x)+b) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+2 d^2 \left(6 a^2 d^2-16 a b c d+b^2 \left(12 c^2+d^2\right)\right) (a \cos (e+f x)+b) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\frac{8 (a d-b c)^3 \left(3 a^2 d+a b c-4 b^2 d\right) (a \cos (e+f x)+b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{b^2 d^4 (a \cos (e+f x)+b)}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{b^2 d^4 (a \cos (e+f x)+b)}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{8 b d^3 (2 b c-a d) \sin \left(\frac{1}{2} (e+f x)\right) (a \cos (e+f x)+b)}{\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)}+\frac{8 b d^3 (2 b c-a d) \sin \left(\frac{1}{2} (e+f x)\right) (a \cos (e+f x)+b)}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}+\frac{4 b (b c-a d)^4 \sin (e+f x)}{(b-a) (a+b)}\right)}{4 b^4 f (a+b \sec (e+f x))^2 (c \cos (e+f x)+d)^4}","\frac{d^2 \left(3 a^2 d^2-8 a b c d+6 b^2 c^2\right) \tanh ^{-1}(\sin (e+f x))}{b^4 f}-\frac{(b c-a d)^4 \sin (e+f x)}{b^3 f \left(a^2-b^2\right) (a \cos (e+f x)+b)}+\frac{2 (b c-a d)^3 (3 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a b^4 f \sqrt{a-b} \sqrt{a+b}}+\frac{2 d^3 (2 b c-a d) \tan (e+f x)}{b^3 f}+\frac{2 (b c-a d)^4 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a b^2 f (a-b)^{3/2} (a+b)^{3/2}}+\frac{d^4 \tanh ^{-1}(\sin (e+f x))}{2 b^2 f}+\frac{d^4 \tan (e+f x) \sec (e+f x)}{2 b^2 f}",1,"(Cos[e + f*x]^2*(b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^4*((8*(-(b*c) + a*d)^3*(a*b*c + 3*a^2*d - 4*b^2*d)*ArcTanh[((-a + b)*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[e + f*x]))/(a^2 - b^2)^(3/2) - 2*d^2*(-16*a*b*c*d + 6*a^2*d^2 + b^2*(12*c^2 + d^2))*(b + a*Cos[e + f*x])*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 2*d^2*(-16*a*b*c*d + 6*a^2*d^2 + b^2*(12*c^2 + d^2))*(b + a*Cos[e + f*x])*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (b^2*d^4*(b + a*Cos[e + f*x]))/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + (8*b*d^3*(2*b*c - a*d)*(b + a*Cos[e + f*x])*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - (b^2*d^4*(b + a*Cos[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (8*b*d^3*(2*b*c - a*d)*(b + a*Cos[e + f*x])*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (4*b*(b*c - a*d)^4*Sin[e + f*x])/((-a + b)*(a + b))))/(4*b^4*f*(d + c*Cos[e + f*x])^4*(a + b*Sec[e + f*x])^2)","A",1
260,1,362,228,1.804316,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^3}{(a+b \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^3)/(a + b*Sec[e + f*x])^2,x]","\frac{\cos (e+f x) (a \cos (e+f x)+b) (c+d \sec (e+f x))^3 \left(-\frac{2 (b c-a d)^2 \left(2 a^2 d+a b c-3 b^2 d\right) (a \cos (e+f x)+b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+d^2 (2 a d-3 b c) (a \cos (e+f x)+b) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+d^2 (3 b c-2 a d) (a \cos (e+f x)+b) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\frac{b (b c-a d)^3 \sin (e+f x)}{(b-a) (a+b)}+\frac{b d^3 \sin \left(\frac{1}{2} (e+f x)\right) (a \cos (e+f x)+b)}{\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)}+\frac{b d^3 \sin \left(\frac{1}{2} (e+f x)\right) (a \cos (e+f x)+b)}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}\right)}{b^3 f (a+b \sec (e+f x))^2 (c \cos (e+f x)+d)^3}","-\frac{(b c-a d)^3 \sin (e+f x)}{b^2 f \left(a^2-b^2\right) (a \cos (e+f x)+b)}+\frac{d^2 (3 b c-2 a d) \tanh ^{-1}(\sin (e+f x))}{b^3 f}+\frac{2 (b c-a d)^2 (2 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a b^3 f \sqrt{a-b} \sqrt{a+b}}+\frac{2 (b c-a d)^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a b f (a-b)^{3/2} (a+b)^{3/2}}+\frac{d^3 \tan (e+f x)}{b^2 f}",1,"(Cos[e + f*x]*(b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^3*((-2*(b*c - a*d)^2*(a*b*c + 2*a^2*d - 3*b^2*d)*ArcTanh[((-a + b)*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[e + f*x]))/(a^2 - b^2)^(3/2) + d^2*(-3*b*c + 2*a*d)*(b + a*Cos[e + f*x])*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + d^2*(3*b*c - 2*a*d)*(b + a*Cos[e + f*x])*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (b*d^3*(b + a*Cos[e + f*x])*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) + (b*d^3*(b + a*Cos[e + f*x])*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (b*(b*c - a*d)^3*Sin[e + f*x])/((-a + b)*(a + b))))/(b^3*f*(d + c*Cos[e + f*x])^3*(a + b*Sec[e + f*x])^2)","A",1
261,1,180,198,0.7205705,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))^2}{(a+b \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^2)/(a + b*Sec[e + f*x])^2,x]","\frac{\frac{2 \left(a^3 d^2-a b^2 \left(c^2+2 d^2\right)+2 b^3 c d\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{b (b c-a d)^2 \sin (e+f x)}{(b-a) (a+b) (a \cos (e+f x)+b)}+d^2 \left(-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)+d^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{b^2 f}","\frac{2 \left(b^2 c^2-a^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a b^2 f \sqrt{a-b} \sqrt{a+b}}-\frac{(b c-a d)^2 \sin (e+f x)}{b f \left(a^2-b^2\right) (a \cos (e+f x)+b)}+\frac{2 (b c-a d)^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a f (a-b)^{3/2} (a+b)^{3/2}}+\frac{d^2 \tanh ^{-1}(\sin (e+f x))}{b^2 f}",1,"((2*(2*b^3*c*d + a^3*d^2 - a*b^2*(c^2 + 2*d^2))*ArcTanh[((-a + b)*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - d^2*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + d^2*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (b*(b*c - a*d)^2*Sin[e + f*x])/((-a + b)*(a + b)*(b + a*Cos[e + f*x])))/(b^2*f)","A",1
262,1,97,100,0.3978625,"\int \frac{\sec (e+f x) (c+d \sec (e+f x))}{(a+b \sec (e+f x))^2} \, dx","Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x]))/(a + b*Sec[e + f*x])^2,x]","\frac{\frac{(a d-b c) \sin (e+f x)}{(a-b) (a+b) (a \cos (e+f x)+b)}-\frac{2 (a c-b d) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}}{f}","\frac{2 (a c-b d) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{f (a-b)^{3/2} (a+b)^{3/2}}-\frac{(b c-a d) \tan (e+f x)}{f \left(a^2-b^2\right) (a+b \sec (e+f x))}",1,"((-2*(a*c - b*d)*ArcTanh[((-a + b)*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + ((-(b*c) + a*d)*Sin[e + f*x])/((a - b)*(a + b)*(b + a*Cos[e + f*x])))/f","A",1
263,1,176,186,1.0578716,"\int \frac{\sec (e+f x)}{(a+b \sec (e+f x))^2 (c+d \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]/((a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])),x]","\frac{\frac{2 d^2 \left(a^2-b^2\right) \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}+\frac{2 b \left(-2 a^2 d+a b c+b^2 d\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{b^2 (b c-a d) \sin (e+f x)}{a \cos (e+f x)+b}}{f (b-a) (a+b) (b c-a d)^2}","-\frac{b^2 \sin (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a \cos (e+f x)+b)}+\frac{2 b \left(-2 a^2 d+a b c+b^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{f (a-b)^{3/2} (a+b)^{3/2} (b c-a d)^2}+\frac{2 d^2 \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f \sqrt{c-d} \sqrt{c+d} (b c-a d)^2}",1,"((2*b*(a*b*c - 2*a^2*d + b^2*d)*ArcTanh[((-a + b)*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (2*(a^2 - b^2)*d^2*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] + (b^2*(b*c - a*d)*Sin[e + f*x])/(b + a*Cos[e + f*x]))/((-a + b)*(a + b)*(b*c - a*d)^2*f)","A",1
264,1,183,213,3.8478122,"\int \frac{\sec (e+f x) \sqrt{a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]])/(c + d*Sec[e + f*x]),x]","\frac{4 \cos ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \sqrt{a+b \sec (e+f x)} \left((a-b) (c+d) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)+2 (b c-a d) \Pi \left(\frac{c-d}{c+d};\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{f (c-d) (c+d) (a \cos (e+f x)+b)}","\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d f}-\frac{2 (b c-a d) \tan (e+f x) \sqrt{\frac{a+b \sec (e+f x)}{a+b}} \Pi \left(\frac{2 d}{c+d};\sin ^{-1}\left(\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right)|\frac{2 b}{a+b}\right)}{d f (c+d) \sqrt{-\tan ^2(e+f x)} \sqrt{a+b \sec (e+f x)}}",1,"(4*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*((a - b)*(c + d)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] + 2*(b*c - a*d)*EllipticPi[(c - d)/(c + d), ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)])*Sqrt[a + b*Sec[e + f*x]])/((c - d)*(c + d)*f*(b + a*Cos[e + f*x]))","A",1
265,1,44216,196,32.5908287,"\int \frac{\sec (e+f x) \sqrt{a+b \sec (e+f x)}}{\sqrt{c+d \sec (e+f x)}} \, dx","Integrate[(Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]])/Sqrt[c + d*Sec[e + f*x]],x]","\text{Result too large to show}","\frac{2 \cot (e+f x) (a+b \sec (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{\frac{a+b}{c+d}} \sqrt{c+d \sec (e+f x)}}{\sqrt{a+b \sec (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{d f \sqrt{\frac{a+b}{c+d}}}",1,"Result too large to show","C",0
266,1,233,192,3.5971226,"\int \frac{\sec (e+f x)}{\sqrt{a+b \sec (e+f x)} \sqrt{c+d \sec (e+f x)}} \, dx","Integrate[Sec[e + f*x]/(Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]),x]","\frac{4 \sin ^2\left(\frac{1}{2} (e+f x)\right) \csc (e+f x) \sqrt{a+b \sec (e+f x)} \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(a+b) \csc ^2\left(\frac{1}{2} (e+f x)\right) (c \cos (e+f x)+d)}{a d-b c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{a d-b c}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right)}{f (a+b) \sqrt{c+d \sec (e+f x)} \sqrt{\frac{(c+d) \csc ^2\left(\frac{1}{2} (e+f x)\right) (a \cos (e+f x)+b)}{b c-a d}}}","\frac{2 \sqrt{a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f \sqrt{c+d} (b c-a d)}",1,"(4*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(-(b*c) + a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(-(b*c) + a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sqrt[a + b*Sec[e + f*x]]*Sin[(e + f*x)/2]^2)/((a + b)*f*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[c + d*Sec[e + f*x]])","A",1
267,1,176,110,1.7912658,"\int \frac{\sec (e+f x)}{\sqrt{2+3 \sec (e+f x)} \sqrt{-4+5 \sec (e+f x)}} \, dx","Integrate[Sec[e + f*x]/(Sqrt[2 + 3*Sec[e + f*x]]*Sqrt[-4 + 5*Sec[e + f*x]]),x]","-\frac{4 \sin ^4\left(\frac{1}{2} (e+f x)\right) \sqrt{-\cot ^2\left(\frac{1}{2} (e+f x)\right)} \csc (e+f x) \sec (e+f x) \sqrt{-\left((2 \cos (e+f x)+3) \csc ^2\left(\frac{1}{2} (e+f x)\right)\right)} \sqrt{-\left((4 \cos (e+f x)-5) \csc ^2\left(\frac{1}{2} (e+f x)\right)\right)} F\left(\sin ^{-1}\left(\sqrt{\frac{5}{22}} \sqrt{\frac{4 \cos (e+f x)-5}{\cos (e+f x)-1}}\right)|\frac{44}{45}\right)}{3 \sqrt{5} f \sqrt{3 \sec (e+f x)+2} \sqrt{5 \sec (e+f x)-4}}","\frac{2 \cot (e+f x) (4-5 \sec (e+f x)) \sqrt{\frac{1-\sec (e+f x)}{4-5 \sec (e+f x)}} \sqrt{\frac{\sec (e+f x)+1}{4-5 \sec (e+f x)}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{3 \sec (e+f x)+2}}{\sqrt{5} \sqrt{5 \sec (e+f x)-4}}\right)\right|45\right)}{f}",1,"(-4*Sqrt[-Cot[(e + f*x)/2]^2]*Sqrt[-((3 + 2*Cos[e + f*x])*Csc[(e + f*x)/2]^2)]*Sqrt[-((-5 + 4*Cos[e + f*x])*Csc[(e + f*x)/2]^2)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[5/22]*Sqrt[(-5 + 4*Cos[e + f*x])/(-1 + Cos[e + f*x])]], 44/45]*Sec[e + f*x]*Sin[(e + f*x)/2]^4)/(3*Sqrt[5]*f*Sqrt[2 + 3*Sec[e + f*x]]*Sqrt[-4 + 5*Sec[e + f*x]])","A",1
268,1,176,125,0.4660301,"\int \frac{\sec (e+f x)}{\sqrt{4-5 \sec (e+f x)} \sqrt{2+3 \sec (e+f x)}} \, dx","Integrate[Sec[e + f*x]/(Sqrt[4 - 5*Sec[e + f*x]]*Sqrt[2 + 3*Sec[e + f*x]]),x]","-\frac{4 \sin ^4\left(\frac{1}{2} (e+f x)\right) \sqrt{-\cot ^2\left(\frac{1}{2} (e+f x)\right)} \csc (e+f x) \sec (e+f x) \sqrt{-\left((2 \cos (e+f x)+3) \csc ^2\left(\frac{1}{2} (e+f x)\right)\right)} \sqrt{-\left((4 \cos (e+f x)-5) \csc ^2\left(\frac{1}{2} (e+f x)\right)\right)} F\left(\sin ^{-1}\left(\sqrt{\frac{5}{22}} \sqrt{\frac{4 \cos (e+f x)-5}{\cos (e+f x)-1}}\right)|\frac{44}{45}\right)}{3 \sqrt{5} f \sqrt{4-5 \sec (e+f x)} \sqrt{3 \sec (e+f x)+2}}","\frac{2 i \cot (e+f x) \sqrt{\frac{1-\sec (e+f x)}{3 \sec (e+f x)+2}} \sqrt{\frac{\sec (e+f x)+1}{3 \sec (e+f x)+2}} (3 \sec (e+f x)+2) F\left(i \sinh ^{-1}\left(\frac{\sqrt{5} \sqrt{4-5 \sec (e+f x)}}{\sqrt{3 \sec (e+f x)+2}}\right)|\frac{1}{45}\right)}{3 \sqrt{5} f}",1,"(-4*Sqrt[-Cot[(e + f*x)/2]^2]*Sqrt[-((3 + 2*Cos[e + f*x])*Csc[(e + f*x)/2]^2)]*Sqrt[-((-5 + 4*Cos[e + f*x])*Csc[(e + f*x)/2]^2)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[5/22]*Sqrt[(-5 + 4*Cos[e + f*x])/(-1 + Cos[e + f*x])]], 44/45]*Sec[e + f*x]*Sin[(e + f*x)/2]^4)/(3*Sqrt[5]*f*Sqrt[4 - 5*Sec[e + f*x]]*Sqrt[2 + 3*Sec[e + f*x]])","A",1
269,1,39039,396,32.4769171,"\int \frac{\sec ^2(e+f x)}{\sqrt{a+b \sec (e+f x)} \sqrt{c+d \sec (e+f x)}} \, dx","Integrate[Sec[e + f*x]^2/(Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]),x]","\text{Result too large to show}","\frac{2 \cot (e+f x) (a+b \sec (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{\frac{a+b}{c+d}} \sqrt{c+d \sec (e+f x)}}{\sqrt{a+b \sec (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b d f \sqrt{\frac{a+b}{c+d}}}-\frac{2 a \sqrt{a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{b f \sqrt{c+d} (b c-a d)}",1,"Result too large to show","C",0
270,1,223,170,4.016322,"\int \frac{(g \sec (e+f x))^{3/2} \sqrt{c+d \sec (e+f x)}}{a+b \sec (e+f x)} \, dx","Integrate[((g*Sec[e + f*x])^(3/2)*Sqrt[c + d*Sec[e + f*x]])/(a + b*Sec[e + f*x]),x]","-\frac{2 i g \cot (e+f x) \sqrt{g \sec (e+f x)} \sqrt{-\frac{c (\cos (e+f x)-1)}{c+d}} \sqrt{\frac{c (\cos (e+f x)+1)}{c-d}} \sqrt{c+d \sec (e+f x)} \left(\Pi \left(1-\frac{c}{d};i \sinh ^{-1}\left(\sqrt{\frac{1}{c-d}} \sqrt{d+c \cos (e+f x)}\right)|\frac{d-c}{c+d}\right)-\Pi \left(\frac{a (d-c)}{a d-b c};i \sinh ^{-1}\left(\sqrt{\frac{1}{c-d}} \sqrt{d+c \cos (e+f x)}\right)|\frac{d-c}{c+d}\right)\right)}{b f \sqrt{\frac{1}{c-d}} \sqrt{c \cos (e+f x)+d}}","\frac{2 g (b c-a d) \sqrt{g \sec (e+f x)} \sqrt{\frac{c \cos (e+f x)+d}{c+d}} \Pi \left(\frac{2 a}{a+b};\frac{1}{2} (e+f x)|\frac{2 c}{c+d}\right)}{b f (a+b) \sqrt{c+d \sec (e+f x)}}+\frac{2 d g \sqrt{g \sec (e+f x)} \sqrt{\frac{c \cos (e+f x)+d}{c+d}} \Pi \left(2;\frac{1}{2} (e+f x)|\frac{2 c}{c+d}\right)}{b f \sqrt{c+d \sec (e+f x)}}",1,"((-2*I)*g*Sqrt[-((c*(-1 + Cos[e + f*x]))/(c + d))]*Sqrt[(c*(1 + Cos[e + f*x]))/(c - d)]*Cot[e + f*x]*(EllipticPi[1 - c/d, I*ArcSinh[Sqrt[(c - d)^(-1)]*Sqrt[d + c*Cos[e + f*x]]], (-c + d)/(c + d)] - EllipticPi[(a*(-c + d))/(-(b*c) + a*d), I*ArcSinh[Sqrt[(c - d)^(-1)]*Sqrt[d + c*Cos[e + f*x]]], (-c + d)/(c + d)])*Sqrt[g*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])/(b*Sqrt[(c - d)^(-1)]*f*Sqrt[d + c*Cos[e + f*x]])","C",1
271,1,83,83,0.233074,"\int \frac{(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt{c+d \sec (e+f x)}} \, dx","Integrate[(g*Sec[e + f*x])^(3/2)/((a + b*Sec[e + f*x])*Sqrt[c + d*Sec[e + f*x]]),x]","\frac{2 g \sqrt{g \sec (e+f x)} \sqrt{\frac{c \cos (e+f x)+d}{c+d}} \Pi \left(\frac{2 a}{a+b};\frac{1}{2} (e+f x)|\frac{2 c}{c+d}\right)}{f (a+b) \sqrt{c+d \sec (e+f x)}}","\frac{2 g \sqrt{g \sec (e+f x)} \sqrt{\frac{c \cos (e+f x)+d}{c+d}} \Pi \left(\frac{2 a}{a+b};\frac{1}{2} (e+f x)|\frac{2 c}{c+d}\right)}{f (a+b) \sqrt{c+d \sec (e+f x)}}",1,"(2*g*Sqrt[(d + c*Cos[e + f*x])/(c + d)]*EllipticPi[(2*a)/(a + b), (e + f*x)/2, (2*c)/(c + d)]*Sqrt[g*Sec[e + f*x]])/((a + b)*f*Sqrt[c + d*Sec[e + f*x]])","A",1
272,1,222,168,3.8924857,"\int \frac{\sqrt{g \sec (e+f x)} \sqrt{c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx","Integrate[(Sqrt[g*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])/(a + b*Cos[e + f*x]),x]","-\frac{2 i \cot (e+f x) \sqrt{g \sec (e+f x)} \sqrt{-\frac{c (\cos (e+f x)-1)}{c+d}} \sqrt{\frac{c (\cos (e+f x)+1)}{c-d}} \sqrt{c+d \sec (e+f x)} \left(\Pi \left(1-\frac{c}{d};i \sinh ^{-1}\left(\sqrt{\frac{1}{c-d}} \sqrt{d+c \cos (e+f x)}\right)|\frac{d-c}{c+d}\right)-\Pi \left(\frac{b (d-c)}{b d-a c};i \sinh ^{-1}\left(\sqrt{\frac{1}{c-d}} \sqrt{d+c \cos (e+f x)}\right)|\frac{d-c}{c+d}\right)\right)}{a f \sqrt{\frac{1}{c-d}} \sqrt{c \cos (e+f x)+d}}","\frac{2 (a c-b d) \sqrt{g \sec (e+f x)} \sqrt{\frac{c \cos (e+f x)+d}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} (e+f x)|\frac{2 c}{c+d}\right)}{a f (a+b) \sqrt{c+d \sec (e+f x)}}+\frac{2 d \sqrt{g \sec (e+f x)} \sqrt{\frac{c \cos (e+f x)+d}{c+d}} \Pi \left(2;\frac{1}{2} (e+f x)|\frac{2 c}{c+d}\right)}{a f \sqrt{c+d \sec (e+f x)}}",1,"((-2*I)*Sqrt[-((c*(-1 + Cos[e + f*x]))/(c + d))]*Sqrt[(c*(1 + Cos[e + f*x]))/(c - d)]*Cot[e + f*x]*(EllipticPi[1 - c/d, I*ArcSinh[Sqrt[(c - d)^(-1)]*Sqrt[d + c*Cos[e + f*x]]], (-c + d)/(c + d)] - EllipticPi[(b*(-c + d))/(-(a*c) + b*d), I*ArcSinh[Sqrt[(c - d)^(-1)]*Sqrt[d + c*Cos[e + f*x]]], (-c + d)/(c + d)])*Sqrt[g*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])/(a*Sqrt[(c - d)^(-1)]*f*Sqrt[d + c*Cos[e + f*x]])","C",1
273,1,264,95,5.752444,"\int \frac{\sec (e+f x) \sqrt{a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]])/(c + c*Sec[e + f*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)} \sqrt{a+b \sec (e+f x)} \left(\frac{2 \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sqrt{\sec (e+f x)+1} \sec ^4\left(\frac{1}{2} (e+f x)\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)}{\left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}}}+\frac{\left(\sin \left(\frac{3}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{\sec (e+f x)+1} \sec ^5\left(\frac{1}{2} (e+f x)\right)}{\left(\frac{1}{\cos (e+f x)+1}\right)^{3/2}}-8 \sqrt{\sec (e+f x)} \left(\sin (e+f x)-\tan \left(\frac{1}{2} (e+f x)\right)\right)\right)}{4 c f (\sec (e+f x)+1)}","\frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{a+b \sec (e+f x)} E\left(\sin ^{-1}\left(\frac{\tan (e+f x)}{\sec (e+f x)+1}\right)|\frac{a-b}{a+b}\right)}{c f \sqrt{\frac{a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}}}",1,"(Cos[(e + f*x)/2]^2*Sqrt[Sec[e + f*x]]*Sqrt[a + b*Sec[e + f*x]]*((2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*EllipticE[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sec[(e + f*x)/2]^4*Sqrt[1 + Sec[e + f*x]])/(((1 + Cos[e + f*x])^(-1))^(3/2)*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]) + (Sec[(e + f*x)/2]^5*Sqrt[1 + Sec[e + f*x]]*(-Sin[(e + f*x)/2] + Sin[(3*(e + f*x))/2]))/((1 + Cos[e + f*x])^(-1))^(3/2) - 8*Sqrt[Sec[e + f*x]]*(Sin[e + f*x] - Tan[(e + f*x)/2])))/(4*c*f*(1 + Sec[e + f*x]))","B",1
274,0,0,295,20.0517621,"\int \frac{(g \sec (e+f x))^{3/2} \sqrt{a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx","Integrate[((g*Sec[e + f*x])^(3/2)*Sqrt[a + b*Sec[e + f*x]])/(c + c*Sec[e + f*x]),x]","\int \frac{(g \sec (e+f x))^{3/2} \sqrt{a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx","-\frac{g \sin (e+f x) \sqrt{g \sec (e+f x)} (a \cos (e+f x)+b)}{f (c \cos (e+f x)+c) \sqrt{a+b \sec (e+f x)}}+\frac{g (a-b) \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} F\left(\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right)}{c f \sqrt{a+b \sec (e+f x)}}+\frac{g \sqrt{g \sec (e+f x)} (a \cos (e+f x)+b) E\left(\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right)}{c f \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \sqrt{a+b \sec (e+f x)}}+\frac{2 b g \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right)}{c f \sqrt{a+b \sec (e+f x)}}",1,"Integrate[((g*Sec[e + f*x])^(3/2)*Sqrt[a + b*Sec[e + f*x]])/(c + c*Sec[e + f*x]), x]","F",-1
275,1,375,209,14.7021193,"\int \frac{\sec (e+f x)}{\sqrt{a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]/(Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])),x]","\frac{\cos ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \sec ^2(e+f x) (a \cos (e+f x)+b) \left(\frac{2 \sin (e+f x)}{b-a}-\frac{2 \tan \left(\frac{1}{2} (e+f x)\right)}{b-a}\right)}{f (c \sec (e+f x)+c) \sqrt{a+b \sec (e+f x)}}-\frac{2 \cos ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) \sec ^{\frac{3}{2}}(e+f x) \sqrt{\cos ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x)} \left(\sqrt{2} \sqrt{\frac{a-b}{a+b}} \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \tan \left(\frac{1}{2} (e+f x)\right) (a \cos (e+f x)+b)+(a-b) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right) (a \cos (e+f x)+b)}{a+b}} E\left(\sin ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a+b}{a-b}\right)\right)}{f \left(\frac{a-b}{a+b}\right)^{3/2} (a+b) (c \sec (e+f x)+c) \sqrt{\cos (e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)} \sqrt{a+b \sec (e+f x)}}","\frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{a+b \sec (e+f x)} E\left(\sin ^{-1}\left(\frac{\tan (e+f x)}{\sec (e+f x)+1}\right)|\frac{a-b}{a+b}\right)}{c f (a-b) \sqrt{\frac{a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}}}-\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{c f (a-b)}",1,"(Cos[e/2 + (f*x)/2]^2*(b + a*Cos[e + f*x])*Sec[e + f*x]^2*((2*Sin[e + f*x])/(-a + b) - (2*Tan[(e + f*x)/2])/(-a + b)))/(f*Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])) - (2*Cos[e/2 + (f*x)/2]^2*Sec[e + f*x]^(3/2)*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*((a - b)*EllipticE[ArcSin[Sqrt[(a - b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[e + f*x])*Sec[(e + f*x)/2]^2)/(a + b)] + Sqrt[2]*Sqrt[(a - b)/(a + b)]*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*(b + a*Cos[e + f*x])*Tan[(e + f*x)/2])*(-1 + Tan[(e + f*x)/2]^2))/(((a - b)/(a + b))^(3/2)*(a + b)*f*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x]))","A",0
276,1,156,214,5.3146654,"\int \frac{\sec ^2(e+f x)}{\sqrt{a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]^2/(Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])),x]","\frac{4 \cos ^4\left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \left((a+b) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)-2 a F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{c f (b-a) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} (\cos (e+f x)+1)^2 \sqrt{a+b \sec (e+f x)}}","\frac{2 a \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b c f (a-b)}-\frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{a+b \sec (e+f x)} E\left(\sin ^{-1}\left(\frac{\tan (e+f x)}{\sec (e+f x)+1}\right)|\frac{a-b}{a+b}\right)}{c f (a-b) \sqrt{\frac{a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}}}",1,"(4*Cos[(e + f*x)/2]^4*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*((a + b)*EllipticE[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] - 2*a*EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]))/((-a + b)*c*f*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*(1 + Cos[e + f*x])^2*Sqrt[a + b*Sec[e + f*x]])","A",1
277,1,1019,229,8.4579784,"\int \frac{(g \sec (e+f x))^{3/2}}{\sqrt{a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx","Integrate[(g*Sec[e + f*x])^(3/2)/(Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])),x]","\frac{(b+a \cos (e+f x)) (g \sec (e+f x))^{3/2} \left(\frac{2 \csc (e)}{(b-a) f}+\frac{2 \sec \left(\frac{e}{2}\right) \sec \left(\frac{e}{2}+\frac{f x}{2}\right) \sin \left(\frac{f x}{2}\right)}{(b-a) f}\right) \cos ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{\sqrt{a+b \sec (e+f x)} (\sec (e+f x) c+c)}+\frac{a \sqrt{b+a \cos (e+f x)} \csc \left(\frac{e}{2}\right) \sec \left(\frac{e}{2}\right) (g \sec (e+f x))^{3/2} \left(\frac{F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{\sec (e) \left(b+a \cos (e) \cos \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{a \sqrt{\tan ^2(e)+1} \left(1-\frac{b \sec (e)}{a \sqrt{\tan ^2(e)+1}}\right)},-\frac{\sec (e) \left(b+a \cos (e) \cos \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{a \sqrt{\tan ^2(e)+1} \left(-\frac{b \sec (e)}{a \sqrt{\tan ^2(e)+1}}-1\right)}\right) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \tan (e)}{\sqrt{\tan ^2(e)+1} \sqrt{\frac{a \sqrt{\tan ^2(e)+1}-a \cos \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} a+b \sec (e)}} \sqrt{\frac{\cos \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1} a+\sqrt{\tan ^2(e)+1} a}{a \sqrt{\tan ^2(e)+1}-b \sec (e)}} \sqrt{b+a \cos (e) \cos \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}}}-\frac{\frac{\sin \left(f x+\tan ^{-1}(\tan (e))\right) \tan (e)}{\sqrt{\tan ^2(e)+1}}+\frac{2 a \cos (e) \left(b+a \cos (e) \cos \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{a^2 \cos ^2(e)+a^2 \sin ^2(e)}}{\sqrt{b+a \cos (e) \cos \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}}}\right) \cos ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{2 (b-a) f \sqrt{a+b \sec (e+f x)} (\sec (e+f x) c+c)}+\frac{F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{\csc (e) \left(b-a \sqrt{\cot ^2(e)+1} \sin (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)\right)}{a \sqrt{\cot ^2(e)+1} \left(\frac{b \csc (e)}{a \sqrt{\cot ^2(e)+1}}+1\right)},\frac{\csc (e) \left(b-a \sqrt{\cot ^2(e)+1} \sin (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)\right)}{a \sqrt{\cot ^2(e)+1} \left(\frac{b \csc (e)}{a \sqrt{\cot ^2(e)+1}}-1\right)}\right) \sqrt{b+a \cos (e+f x)} \csc \left(\frac{e}{2}\right) \sec \left(\frac{e}{2}\right) (g \sec (e+f x))^{3/2} \sec \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\frac{a \sqrt{\cot ^2(e)+1}-a \sqrt{\cot ^2(e)+1} \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{a \sqrt{\cot ^2(e)+1}-b \csc (e)}} \sqrt{\frac{\sqrt{\cot ^2(e)+1} \sin \left(f x-\tan ^{-1}(\cot (e))\right) a+\sqrt{\cot ^2(e)+1} a}{\sqrt{\cot ^2(e)+1} a+b \csc (e)}} \sqrt{b-a \sqrt{\cot ^2(e)+1} \sin (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)} \cos ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{(b-a) f \sqrt{\cot ^2(e)+1} \sqrt{a+b \sec (e+f x)} (\sec (e+f x) c+c)}","-\frac{g \sin (e+f x) \sqrt{g \sec (e+f x)} (a \cos (e+f x)+b)}{f (a-b) (c \cos (e+f x)+c) \sqrt{a+b \sec (e+f x)}}+\frac{g \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} F\left(\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right)}{c f \sqrt{a+b \sec (e+f x)}}+\frac{g \sqrt{g \sec (e+f x)} (a \cos (e+f x)+b) E\left(\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right)}{c f (a-b) \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \sqrt{a+b \sec (e+f x)}}",1,"(Cos[e/2 + (f*x)/2]^2*(b + a*Cos[e + f*x])*(g*Sec[e + f*x])^(3/2)*((2*Csc[e])/((-a + b)*f) + (2*Sec[e/2]*Sec[e/2 + (f*x)/2]*Sin[(f*x)/2])/((-a + b)*f)))/(Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])) + (AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[e]*(b - a*Sqrt[1 + Cot[e]^2]*Sin[e]*Sin[f*x - ArcTan[Cot[e]]]))/(a*Sqrt[1 + Cot[e]^2]*(1 + (b*Csc[e])/(a*Sqrt[1 + Cot[e]^2]))), (Csc[e]*(b - a*Sqrt[1 + Cot[e]^2]*Sin[e]*Sin[f*x - ArcTan[Cot[e]]]))/(a*Sqrt[1 + Cot[e]^2]*(-1 + (b*Csc[e])/(a*Sqrt[1 + Cot[e]^2])))]*Cos[e/2 + (f*x)/2]^2*Sqrt[b + a*Cos[e + f*x]]*Csc[e/2]*Sec[e/2]*(g*Sec[e + f*x])^(3/2)*Sec[f*x - ArcTan[Cot[e]]]*Sqrt[(a*Sqrt[1 + Cot[e]^2] - a*Sqrt[1 + Cot[e]^2]*Sin[f*x - ArcTan[Cot[e]]])/(a*Sqrt[1 + Cot[e]^2] - b*Csc[e])]*Sqrt[(a*Sqrt[1 + Cot[e]^2] + a*Sqrt[1 + Cot[e]^2]*Sin[f*x - ArcTan[Cot[e]]])/(a*Sqrt[1 + Cot[e]^2] + b*Csc[e])]*Sqrt[b - a*Sqrt[1 + Cot[e]^2]*Sin[e]*Sin[f*x - ArcTan[Cot[e]]]])/((-a + b)*f*Sqrt[1 + Cot[e]^2]*Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])) + (a*Cos[e/2 + (f*x)/2]^2*Sqrt[b + a*Cos[e + f*x]]*Csc[e/2]*Sec[e/2]*(g*Sec[e + f*x])^(3/2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[e]*(b + a*Cos[e]*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(a*Sqrt[1 + Tan[e]^2]*(1 - (b*Sec[e])/(a*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(b + a*Cos[e]*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(a*Sqrt[1 + Tan[e]^2]*(-1 - (b*Sec[e])/(a*Sqrt[1 + Tan[e]^2]))))]*Sin[f*x + ArcTan[Tan[e]]]*Tan[e])/(Sqrt[1 + Tan[e]^2]*Sqrt[(a*Sqrt[1 + Tan[e]^2] - a*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(b*Sec[e] + a*Sqrt[1 + Tan[e]^2])]*Sqrt[(a*Sqrt[1 + Tan[e]^2] + a*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(b*Sec[e]) + a*Sqrt[1 + Tan[e]^2])]*Sqrt[b + a*Cos[e]*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]]) - ((Sin[f*x + ArcTan[Tan[e]]]*Tan[e])/Sqrt[1 + Tan[e]^2] + (2*a*Cos[e]*(b + a*Cos[e]*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(a^2*Cos[e]^2 + a^2*Sin[e]^2))/Sqrt[b + a*Cos[e]*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]]))/(2*(-a + b)*f*Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x]))","C",0
278,0,0,312,15.5513528,"\int \frac{(g \sec (e+f x))^{5/2}}{\sqrt{a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx","Integrate[(g*Sec[e + f*x])^(5/2)/(Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])),x]","\int \frac{(g \sec (e+f x))^{5/2}}{\sqrt{a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx","\frac{g^2 \sin (e+f x) \sqrt{g \sec (e+f x)} (a \cos (e+f x)+b)}{f (a-b) (c \cos (e+f x)+c) \sqrt{a+b \sec (e+f x)}}-\frac{g^2 \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} F\left(\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right)}{c f \sqrt{a+b \sec (e+f x)}}-\frac{g^2 \sqrt{g \sec (e+f x)} (a \cos (e+f x)+b) E\left(\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right)}{c f (a-b) \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \sqrt{a+b \sec (e+f x)}}+\frac{2 g^2 \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right)}{c f \sqrt{a+b \sec (e+f x)}}",1,"Integrate[(g*Sec[e + f*x])^(5/2)/(Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])), x]","F",-1
279,1,183,213,0.3958422,"\int \frac{\sec (e+f x) \sqrt{a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx","Integrate[(Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]])/(c + d*Sec[e + f*x]),x]","\frac{4 \cos ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \sqrt{a+b \sec (e+f x)} \left((a-b) (c+d) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)+2 (b c-a d) \Pi \left(\frac{c-d}{c+d};\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{f (c-d) (c+d) (a \cos (e+f x)+b)}","\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d f}-\frac{2 (b c-a d) \tan (e+f x) \sqrt{\frac{a+b \sec (e+f x)}{a+b}} \Pi \left(\frac{2 d}{c+d};\sin ^{-1}\left(\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right)|\frac{2 b}{a+b}\right)}{d f (c+d) \sqrt{-\tan ^2(e+f x)} \sqrt{a+b \sec (e+f x)}}",1,"(4*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*((a - b)*(c + d)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] + 2*(b*c - a*d)*EllipticPi[(c - d)/(c + d), ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)])*Sqrt[a + b*Sec[e + f*x]])/((c - d)*(c + d)*f*(b + a*Cos[e + f*x]))","A",1
280,1,223,170,3.9847619,"\int \frac{(g \sec (e+f x))^{3/2} \sqrt{a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx","Integrate[((g*Sec[e + f*x])^(3/2)*Sqrt[a + b*Sec[e + f*x]])/(c + d*Sec[e + f*x]),x]","-\frac{2 i g \cot (e+f x) \sqrt{g \sec (e+f x)} \sqrt{-\frac{a (\cos (e+f x)-1)}{a+b}} \sqrt{\frac{a (\cos (e+f x)+1)}{a-b}} \sqrt{a+b \sec (e+f x)} \left(\Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (e+f x)}\right)|\frac{b-a}{a+b}\right)-\Pi \left(\frac{(a-b) c}{a d-b c};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (e+f x)}\right)|\frac{b-a}{a+b}\right)\right)}{d f \sqrt{\frac{1}{a-b}} \sqrt{a \cos (e+f x)+b}}","\frac{2 b g \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right)}{d f \sqrt{a+b \sec (e+f x)}}-\frac{2 g (b c-a d) \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \Pi \left(\frac{2 c}{c+d};\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right)}{d f (c+d) \sqrt{a+b \sec (e+f x)}}",1,"((-2*I)*g*Sqrt[-((a*(-1 + Cos[e + f*x]))/(a + b))]*Sqrt[(a*(1 + Cos[e + f*x]))/(a - b)]*Cot[e + f*x]*(EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[e + f*x]]], (-a + b)/(a + b)] - EllipticPi[((a - b)*c)/(-(b*c) + a*d), I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[e + f*x]]], (-a + b)/(a + b)])*Sqrt[g*Sec[e + f*x]]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[(a - b)^(-1)]*d*f*Sqrt[b + a*Cos[e + f*x]])","C",1
281,1,187,102,4.2100335,"\int \frac{\sec (e+f x)}{\sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]","\frac{2 \sqrt{\sec (e+f x)} \sqrt{\sec (e+f x)+1} \sqrt{\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \left((c+d) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)-2 d \Pi \left(\frac{c-d}{c+d};\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{f (c-d) (c+d) \sqrt{\sec ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{a+b \sec (e+f x)}}","\frac{2 \tan (e+f x) \sqrt{\frac{a+b \sec (e+f x)}{a+b}} \Pi \left(\frac{2 d}{c+d};\sin ^{-1}\left(\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right)|\frac{2 b}{a+b}\right)}{f (c+d) \sqrt{-\tan ^2(e+f x)} \sqrt{a+b \sec (e+f x)}}",1,"(2*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*((c + d)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] - 2*d*EllipticPi[(c - d)/(c + d), ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)])*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]])/((c - d)*(c + d)*f*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[a + b*Sec[e + f*x]])","A",1
282,1,165,209,3.0612038,"\int \frac{\sec ^2(e+f x)}{\sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx","Integrate[Sec[e + f*x]^2/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]","-\frac{4 \cos ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sec (e+f x) \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \left((c+d) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)-2 c \Pi \left(\frac{c-d}{c+d};\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{f (c-d) (c+d) \sqrt{a+b \sec (e+f x)}}","\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d f}-\frac{2 c \tan (e+f x) \sqrt{\frac{a+b \sec (e+f x)}{a+b}} \Pi \left(\frac{2 d}{c+d};\sin ^{-1}\left(\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right)|\frac{2 b}{a+b}\right)}{d f (c+d) \sqrt{-\tan ^2(e+f x)} \sqrt{a+b \sec (e+f x)}}",1,"(-4*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*((c + d)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] - 2*c*EllipticPi[(c - d)/(c + d), ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)])*Sec[e + f*x])/((c - d)*(c + d)*f*Sqrt[a + b*Sec[e + f*x]])","A",1
283,1,83,83,0.2857661,"\int \frac{(g \sec (e+f x))^{3/2}}{\sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx","Integrate[(g*Sec[e + f*x])^(3/2)/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]","\frac{2 g \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \Pi \left(\frac{2 c}{c+d};\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right)}{f (c+d) \sqrt{a+b \sec (e+f x)}}","\frac{2 g \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \Pi \left(\frac{2 c}{c+d};\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right)}{f (c+d) \sqrt{a+b \sec (e+f x)}}",1,"(2*g*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), (e + f*x)/2, (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/((c + d)*f*Sqrt[a + b*Sec[e + f*x]])","A",1
284,1,246,166,4.1219509,"\int \frac{(g \sec (e+f x))^{5/2}}{\sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx","Integrate[(g*Sec[e + f*x])^(5/2)/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]","-\frac{2 i g \cot (e+f x) (g \sec (e+f x))^{3/2} \sqrt{-\frac{a (\cos (e+f x)-1)}{a+b}} \sqrt{\frac{a (\cos (e+f x)+1)}{a-b}} \sqrt{a \cos (e+f x)+b} \left((a d-b c) \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (e+f x)}\right)|\frac{b-a}{a+b}\right)+b c \Pi \left(\frac{(a-b) c}{a d-b c};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (e+f x)}\right)|\frac{b-a}{a+b}\right)\right)}{b d f \sqrt{\frac{1}{a-b}} (a d-b c) \sqrt{a+b \sec (e+f x)}}","\frac{2 g^2 \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right)}{d f \sqrt{a+b \sec (e+f x)}}-\frac{2 c g^2 \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \Pi \left(\frac{2 c}{c+d};\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right)}{d f (c+d) \sqrt{a+b \sec (e+f x)}}",1,"((-2*I)*g*Sqrt[-((a*(-1 + Cos[e + f*x]))/(a + b))]*Sqrt[(a*(1 + Cos[e + f*x]))/(a - b)]*Sqrt[b + a*Cos[e + f*x]]*Cot[e + f*x]*((-(b*c) + a*d)*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[e + f*x]]], (-a + b)/(a + b)] + b*c*EllipticPi[((a - b)*c)/(-(b*c) + a*d), I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[e + f*x]]], (-a + b)/(a + b)])*(g*Sec[e + f*x])^(3/2))/(Sqrt[(a - b)^(-1)]*b*d*(-(b*c) + a*d)*f*Sqrt[a + b*Sec[e + f*x]])","C",1
285,1,151,67,0.9645845,"\int \frac{\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^7} \, dx","Integrate[(Sec[e + f*x]*Tan[e + f*x]^4)/(c - c*Sec[e + f*x])^7,x]","\frac{\csc \left(\frac{e}{2}\right) \left(-718830 \sin \left(e+\frac{f x}{2}\right)+467208 \sin \left(e+\frac{3 f x}{2}\right)+659400 \sin \left(2 e+\frac{3 f x}{2}\right)-303192 \sin \left(2 e+\frac{5 f x}{2}\right)-179640 \sin \left(3 e+\frac{5 f x}{2}\right)+30753 \sin \left(3 e+\frac{7 f x}{2}\right)+89955 \sin \left(4 e+\frac{7 f x}{2}\right)-13427 \sin \left(4 e+\frac{9 f x}{2}\right)+15 \sin \left(5 e+\frac{9 f x}{2}\right)-971082 \sin \left(\frac{f x}{2}\right)\right) \csc ^9\left(\frac{1}{2} (e+f x)\right)}{23063040 c^7 f}","\frac{\cot ^9\left(\frac{1}{2} (e+f x)\right)}{36 c^7 f}-\frac{\cot ^7\left(\frac{1}{2} (e+f x)\right)}{14 c^7 f}+\frac{\cot ^5\left(\frac{1}{2} (e+f x)\right)}{20 c^7 f}",1,"(Csc[e/2]*Csc[(e + f*x)/2]^9*(-971082*Sin[(f*x)/2] - 718830*Sin[e + (f*x)/2] + 467208*Sin[e + (3*f*x)/2] + 659400*Sin[2*e + (3*f*x)/2] - 303192*Sin[2*e + (5*f*x)/2] - 179640*Sin[3*e + (5*f*x)/2] + 30753*Sin[3*e + (7*f*x)/2] + 89955*Sin[4*e + (7*f*x)/2] - 13427*Sin[4*e + (9*f*x)/2] + 15*Sin[5*e + (9*f*x)/2]))/(23063040*c^7*f)","B",1
286,1,175,89,1.1189886,"\int \frac{\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^8} \, dx","Integrate[(Sec[e + f*x]*Tan[e + f*x]^4)/(c - c*Sec[e + f*x])^8,x]","-\frac{\csc \left(\frac{e}{2}\right) \left(486024 \sin \left(e+\frac{f x}{2}\right)-351450 \sin \left(e+\frac{3 f x}{2}\right)-299970 \sin \left(2 e+\frac{3 f x}{2}\right)+145695 \sin \left(2 e+\frac{5 f x}{2}\right)+180015 \sin \left(3 e+\frac{5 f x}{2}\right)-63580 \sin \left(3 e+\frac{7 f x}{2}\right)-44990 \sin \left(4 e+\frac{7 f x}{2}\right)+6710 \sin \left(4 e+\frac{9 f x}{2}\right)+15004 \sin \left(5 e+\frac{9 f x}{2}\right)-1975 \sin \left(5 e+\frac{11 f x}{2}\right)+\sin \left(6 e+\frac{11 f x}{2}\right)+425964 \sin \left(\frac{f x}{2}\right)\right) \csc ^{11}\left(\frac{1}{2} (e+f x)\right)}{15375360 c^8 f}","-\frac{\cot ^{11}\left(\frac{1}{2} (e+f x)\right)}{88 c^8 f}+\frac{\cot ^9\left(\frac{1}{2} (e+f x)\right)}{24 c^8 f}-\frac{3 \cot ^7\left(\frac{1}{2} (e+f x)\right)}{56 c^8 f}+\frac{\cot ^5\left(\frac{1}{2} (e+f x)\right)}{40 c^8 f}",1,"-1/15375360*(Csc[e/2]*Csc[(e + f*x)/2]^11*(425964*Sin[(f*x)/2] + 486024*Sin[e + (f*x)/2] - 351450*Sin[e + (3*f*x)/2] - 299970*Sin[2*e + (3*f*x)/2] + 145695*Sin[2*e + (5*f*x)/2] + 180015*Sin[3*e + (5*f*x)/2] - 63580*Sin[3*e + (7*f*x)/2] - 44990*Sin[4*e + (7*f*x)/2] + 6710*Sin[4*e + (9*f*x)/2] + 15004*Sin[5*e + (9*f*x)/2] - 1975*Sin[5*e + (11*f*x)/2] + Sin[6*e + (11*f*x)/2]))/(c^8*f)","A",1