1,1,196,0,0.3020054,"\int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx","Int[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5,x]","\frac{3 a^2 c^5 \tan ^5(e+f x)}{5 f}+\frac{a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac{a^2 c^5 \tan (e+f x)}{f}-\frac{19 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)}{6 f}+\frac{a^2 c^5 \tan (e+f x) \sec ^3(e+f x)}{8 f}-\frac{3 a^2 c^5 \tan ^3(e+f x) \sec (e+f x)}{4 f}+\frac{17 a^2 c^5 \tan (e+f x) \sec (e+f x)}{16 f}+a^2 c^5 x","\frac{3 a^2 c^5 \tan ^5(e+f x)}{5 f}+\frac{a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac{a^2 c^5 \tan (e+f x)}{f}-\frac{19 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)}{6 f}+\frac{a^2 c^5 \tan (e+f x) \sec ^3(e+f x)}{8 f}-\frac{3 a^2 c^5 \tan ^3(e+f x) \sec (e+f x)}{4 f}+\frac{17 a^2 c^5 \tan (e+f x) \sec (e+f x)}{16 f}+a^2 c^5 x",1,"a^2*c^5*x - (19*a^2*c^5*ArcTanh[Sin[e + f*x]])/(16*f) - (a^2*c^5*Tan[e + f*x])/f + (17*a^2*c^5*Sec[e + f*x]*Tan[e + f*x])/(16*f) + (a^2*c^5*Sec[e + f*x]^3*Tan[e + f*x])/(8*f) + (a^2*c^5*Tan[e + f*x]^3)/(3*f) - (3*a^2*c^5*Sec[e + f*x]*Tan[e + f*x]^3)/(4*f) - (a^2*c^5*Sec[e + f*x]^3*Tan[e + f*x]^3)/(6*f) + (3*a^2*c^5*Tan[e + f*x]^5)/(5*f)","A",15,9,26,0.3462,1,"{3904, 3886, 3473, 8, 2611, 3770, 2607, 30, 3768}"
2,1,140,0,0.1989879,"\int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4 \, dx","Int[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^4,x]","\frac{a^2 c^4 \tan ^5(e+f x)}{5 f}+\frac{a^2 c^4 \tan ^3(e+f x)}{3 f}-\frac{a^2 c^4 \tan (e+f x)}{f}-\frac{3 a^2 c^4 \tanh ^{-1}(\sin (e+f x))}{4 f}-\frac{a^2 c^4 \tan ^3(e+f x) \sec (e+f x)}{2 f}+\frac{3 a^2 c^4 \tan (e+f x) \sec (e+f x)}{4 f}+a^2 c^4 x","\frac{a^2 c^4 \tan ^5(e+f x)}{5 f}+\frac{a^2 c^4 \tan ^3(e+f x)}{3 f}-\frac{a^2 c^4 \tan (e+f x)}{f}-\frac{3 a^2 c^4 \tanh ^{-1}(\sin (e+f x))}{4 f}-\frac{a^2 c^4 \tan ^3(e+f x) \sec (e+f x)}{2 f}+\frac{3 a^2 c^4 \tan (e+f x) \sec (e+f x)}{4 f}+a^2 c^4 x",1,"a^2*c^4*x - (3*a^2*c^4*ArcTanh[Sin[e + f*x]])/(4*f) - (a^2*c^4*Tan[e + f*x])/f + (3*a^2*c^4*Sec[e + f*x]*Tan[e + f*x])/(4*f) + (a^2*c^4*Tan[e + f*x]^3)/(3*f) - (a^2*c^4*Sec[e + f*x]*Tan[e + f*x]^3)/(2*f) + (a^2*c^4*Tan[e + f*x]^5)/(5*f)","A",11,8,26,0.3077,1,"{3904, 3886, 3473, 8, 2611, 3770, 2607, 30}"
3,1,97,0,0.1179924,"\int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3 \, dx","Int[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^3,x]","-\frac{3 a^2 c^3 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a^2 \tan ^3(e+f x) \left(4 c^3-3 c^3 \sec (e+f x)\right)}{12 f}-\frac{a^2 \tan (e+f x) \left(8 c^3-3 c^3 \sec (e+f x)\right)}{8 f}+a^2 c^3 x","-\frac{3 a^2 c^3 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a^2 \tan ^3(e+f x) \left(4 c^3-3 c^3 \sec (e+f x)\right)}{12 f}-\frac{a^2 \tan (e+f x) \left(8 c^3-3 c^3 \sec (e+f x)\right)}{8 f}+a^2 c^3 x",1,"a^2*c^3*x - (3*a^2*c^3*ArcTanh[Sin[e + f*x]])/(8*f) - (a^2*(8*c^3 - 3*c^3*Sec[e + f*x])*Tan[e + f*x])/(8*f) + (a^2*(4*c^3 - 3*c^3*Sec[e + f*x])*Tan[e + f*x]^3)/(12*f)","A",5,3,26,0.1154,1,"{3904, 3881, 3770}"
4,1,47,0,0.0646025,"\int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^2 \, dx","Int[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^2,x]","\frac{a^2 c^2 \tan ^3(e+f x)}{3 f}-\frac{a^2 c^2 \tan (e+f x)}{f}+a^2 c^2 x","\frac{a^2 c^2 \tan ^3(e+f x)}{3 f}-\frac{a^2 c^2 \tan (e+f x)}{f}+a^2 c^2 x",1,"a^2*c^2*x - (a^2*c^2*Tan[e + f*x])/f + (a^2*c^2*Tan[e + f*x]^3)/(3*f)","A",4,3,26,0.1154,1,"{3904, 3473, 8}"
5,1,55,0,0.0626252,"\int (a+a \sec (e+f x))^2 (c-c \sec (e+f x)) \, dx","Int[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x]),x]","\frac{a^2 c \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac{c \tan (e+f x) \left(a^2 \sec (e+f x)+2 a^2\right)}{2 f}+a^2 c x","\frac{a^2 c \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac{c \tan (e+f x) \left(a^2 \sec (e+f x)+2 a^2\right)}{2 f}+a^2 c x",1,"a^2*c*x + (a^2*c*ArcTanh[Sin[e + f*x]])/(2*f) - (c*(2*a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(2*f)","A",4,3,24,0.1250,1,"{3904, 3881, 3770}"
6,1,56,0,0.1639214,"\int \frac{(a+a \sec (e+f x))^2}{c-c \sec (e+f x)} \, dx","Int[(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x]),x]","-\frac{a^2 \tanh ^{-1}(\sin (e+f x))}{c f}-\frac{4 a^2 \tan (e+f x)}{c f (1-\sec (e+f x))}+\frac{a^2 x}{c}","-\frac{a^2 \tanh ^{-1}(\sin (e+f x))}{c f}-\frac{4 a^2 \tan (e+f x)}{c f (1-\sec (e+f x))}+\frac{a^2 x}{c}",1,"(a^2*x)/c - (a^2*ArcTanh[Sin[e + f*x]])/(c*f) - (4*a^2*Tan[e + f*x])/(c*f*(1 - Sec[e + f*x]))","A",8,6,26,0.2308,1,"{3903, 3777, 8, 3794, 3789, 3770}"
7,1,71,0,0.2363148,"\int \frac{(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^2} \, dx","Int[(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^2,x]","-\frac{4 a^2 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))}-\frac{4 a^2 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))^2}+\frac{a^2 x}{c^2}","-\frac{4 a^2 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))}-\frac{4 a^2 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))^2}+\frac{a^2 x}{c^2}",1,"(a^2*x)/c^2 - (4*a^2*Tan[e + f*x])/(3*c^2*f*(1 - Sec[e + f*x])^2) - (4*a^2*Tan[e + f*x])/(3*c^2*f*(1 - Sec[e + f*x]))","A",9,6,26,0.2308,1,"{3903, 3777, 3919, 3794, 3796, 3797}"
8,1,102,0,0.3290806,"\int \frac{(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^3} \, dx","Int[(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^3,x]","-\frac{23 a^2 \tan (e+f x)}{15 c^3 f (1-\sec (e+f x))}-\frac{8 a^2 \tan (e+f x)}{15 c^3 f (1-\sec (e+f x))^2}-\frac{4 a^2 \tan (e+f x)}{5 c^3 f (1-\sec (e+f x))^3}+\frac{a^2 x}{c^3}","-\frac{23 a^2 \tan (e+f x)}{15 c^3 f (1-\sec (e+f x))}-\frac{8 a^2 \tan (e+f x)}{15 c^3 f (1-\sec (e+f x))^2}-\frac{4 a^2 \tan (e+f x)}{5 c^3 f (1-\sec (e+f x))^3}+\frac{a^2 x}{c^3}",1,"(a^2*x)/c^3 - (4*a^2*Tan[e + f*x])/(5*c^3*f*(1 - Sec[e + f*x])^3) - (8*a^2*Tan[e + f*x])/(15*c^3*f*(1 - Sec[e + f*x])^2) - (23*a^2*Tan[e + f*x])/(15*c^3*f*(1 - Sec[e + f*x]))","A",12,7,26,0.2692,1,"{3903, 3777, 3922, 3919, 3794, 3796, 3797}"
9,1,133,0,0.4272204,"\int \frac{(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^4} \, dx","Int[(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^4,x]","-\frac{164 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))}-\frac{59 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}-\frac{12 a^2 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac{4 a^2 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}+\frac{a^2 x}{c^4}","-\frac{164 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))}-\frac{59 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}-\frac{12 a^2 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac{4 a^2 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}+\frac{a^2 x}{c^4}",1,"(a^2*x)/c^4 - (4*a^2*Tan[e + f*x])/(7*c^4*f*(1 - Sec[e + f*x])^4) - (12*a^2*Tan[e + f*x])/(35*c^4*f*(1 - Sec[e + f*x])^3) - (59*a^2*Tan[e + f*x])/(105*c^4*f*(1 - Sec[e + f*x])^2) - (164*a^2*Tan[e + f*x])/(105*c^4*f*(1 - Sec[e + f*x]))","A",15,7,26,0.2692,1,"{3903, 3777, 3922, 3919, 3794, 3796, 3797}"
10,1,164,0,0.5427921,"\int \frac{(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^5} \, dx","Int[(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^5,x]","-\frac{494 a^2 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))}-\frac{179 a^2 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}-\frac{37 a^2 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}-\frac{16 a^2 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac{4 a^2 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}+\frac{a^2 x}{c^5}","-\frac{494 a^2 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))}-\frac{179 a^2 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}-\frac{37 a^2 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}-\frac{16 a^2 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac{4 a^2 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}+\frac{a^2 x}{c^5}",1,"(a^2*x)/c^5 - (4*a^2*Tan[e + f*x])/(9*c^5*f*(1 - Sec[e + f*x])^5) - (16*a^2*Tan[e + f*x])/(63*c^5*f*(1 - Sec[e + f*x])^4) - (37*a^2*Tan[e + f*x])/(105*c^5*f*(1 - Sec[e + f*x])^3) - (179*a^2*Tan[e + f*x])/(315*c^5*f*(1 - Sec[e + f*x])^2) - (494*a^2*Tan[e + f*x])/(315*c^5*f*(1 - Sec[e + f*x]))","A",18,7,26,0.2692,1,"{3903, 3777, 3922, 3919, 3794, 3796, 3797}"
11,1,188,0,0.2372103,"\int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx","Int[(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5,x]","-\frac{a^3 c^5 \tan ^7(e+f x)}{7 f}-\frac{a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac{a^3 c^5 \tan ^3(e+f x)}{3 f}-\frac{a^3 c^5 \tan (e+f x)}{f}-\frac{5 a^3 c^5 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a^3 c^5 \tan ^5(e+f x) \sec (e+f x)}{3 f}-\frac{5 a^3 c^5 \tan ^3(e+f x) \sec (e+f x)}{12 f}+\frac{5 a^3 c^5 \tan (e+f x) \sec (e+f x)}{8 f}+a^3 c^5 x","-\frac{a^3 c^5 \tan ^7(e+f x)}{7 f}-\frac{a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac{a^3 c^5 \tan ^3(e+f x)}{3 f}-\frac{a^3 c^5 \tan (e+f x)}{f}-\frac{5 a^3 c^5 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a^3 c^5 \tan ^5(e+f x) \sec (e+f x)}{3 f}-\frac{5 a^3 c^5 \tan ^3(e+f x) \sec (e+f x)}{12 f}+\frac{5 a^3 c^5 \tan (e+f x) \sec (e+f x)}{8 f}+a^3 c^5 x",1,"a^3*c^5*x - (5*a^3*c^5*ArcTanh[Sin[e + f*x]])/(8*f) - (a^3*c^5*Tan[e + f*x])/f + (5*a^3*c^5*Sec[e + f*x]*Tan[e + f*x])/(8*f) + (a^3*c^5*Tan[e + f*x]^3)/(3*f) - (5*a^3*c^5*Sec[e + f*x]*Tan[e + f*x]^3)/(12*f) - (a^3*c^5*Tan[e + f*x]^5)/(5*f) + (a^3*c^5*Sec[e + f*x]*Tan[e + f*x]^5)/(3*f) - (a^3*c^5*Tan[e + f*x]^7)/(7*f)","A",13,8,26,0.3077,1,"{3904, 3886, 3473, 8, 2611, 3770, 2607, 30}"
12,1,132,0,0.1510083,"\int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4 \, dx","Int[(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^4,x]","-\frac{5 a^3 c^4 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac{a^3 \tan ^5(e+f x) \left(6 c^4-5 c^4 \sec (e+f x)\right)}{30 f}+\frac{a^3 \tan ^3(e+f x) \left(8 c^4-5 c^4 \sec (e+f x)\right)}{24 f}-\frac{a^3 \tan (e+f x) \left(16 c^4-5 c^4 \sec (e+f x)\right)}{16 f}+a^3 c^4 x","-\frac{5 a^3 c^4 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac{a^3 \tan ^5(e+f x) \left(6 c^4-5 c^4 \sec (e+f x)\right)}{30 f}+\frac{a^3 \tan ^3(e+f x) \left(8 c^4-5 c^4 \sec (e+f x)\right)}{24 f}-\frac{a^3 \tan (e+f x) \left(16 c^4-5 c^4 \sec (e+f x)\right)}{16 f}+a^3 c^4 x",1,"a^3*c^4*x - (5*a^3*c^4*ArcTanh[Sin[e + f*x]])/(16*f) - (a^3*(16*c^4 - 5*c^4*Sec[e + f*x])*Tan[e + f*x])/(16*f) + (a^3*(8*c^4 - 5*c^4*Sec[e + f*x])*Tan[e + f*x]^3)/(24*f) - (a^3*(6*c^4 - 5*c^4*Sec[e + f*x])*Tan[e + f*x]^5)/(30*f)","A",6,3,26,0.1154,1,"{3904, 3881, 3770}"
13,1,68,0,0.07435,"\int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx","Int[(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^3,x]","-\frac{a^3 c^3 \tan ^5(e+f x)}{5 f}+\frac{a^3 c^3 \tan ^3(e+f x)}{3 f}-\frac{a^3 c^3 \tan (e+f x)}{f}+a^3 c^3 x","-\frac{a^3 c^3 \tan ^5(e+f x)}{5 f}+\frac{a^3 c^3 \tan ^3(e+f x)}{3 f}-\frac{a^3 c^3 \tan (e+f x)}{f}+a^3 c^3 x",1,"a^3*c^3*x - (a^3*c^3*Tan[e + f*x])/f + (a^3*c^3*Tan[e + f*x]^3)/(3*f) - (a^3*c^3*Tan[e + f*x]^5)/(5*f)","A",5,3,26,0.1154,1,"{3904, 3473, 8}"
14,1,97,0,0.1143574,"\int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2 \, dx","Int[(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^2,x]","\frac{3 a^3 c^2 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{c^2 \tan ^3(e+f x) \left(3 a^3 \sec (e+f x)+4 a^3\right)}{12 f}-\frac{c^2 \tan (e+f x) \left(3 a^3 \sec (e+f x)+8 a^3\right)}{8 f}+a^3 c^2 x","\frac{3 a^3 c^2 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{c^2 \tan ^3(e+f x) \left(3 a^3 \sec (e+f x)+4 a^3\right)}{12 f}-\frac{c^2 \tan (e+f x) \left(3 a^3 \sec (e+f x)+8 a^3\right)}{8 f}+a^3 c^2 x",1,"a^3*c^2*x + (3*a^3*c^2*ArcTanh[Sin[e + f*x]])/(8*f) - (c^2*(8*a^3 + 3*a^3*Sec[e + f*x])*Tan[e + f*x])/(8*f) + (c^2*(4*a^3 + 3*a^3*Sec[e + f*x])*Tan[e + f*x]^3)/(12*f)","A",5,3,26,0.1154,1,"{3904, 3881, 3770}"
15,1,77,0,0.1473245,"\int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx","Int[(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x]),x]","-\frac{a^3 c \tan ^3(e+f x)}{3 f}-\frac{a^3 c \tan (e+f x)}{f}+\frac{a^3 c \tanh ^{-1}(\sin (e+f x))}{f}-\frac{a^3 c \tan (e+f x) \sec (e+f x)}{f}+a^3 c x","-\frac{a^3 c \tan ^3(e+f x)}{3 f}-\frac{a^3 c \tan (e+f x)}{f}+\frac{a^3 c \tanh ^{-1}(\sin (e+f x))}{f}-\frac{a^3 c \tan (e+f x) \sec (e+f x)}{f}+a^3 c x",1,"a^3*c*x + (a^3*c*ArcTanh[Sin[e + f*x]])/f - (a^3*c*Tan[e + f*x])/f - (a^3*c*Sec[e + f*x]*Tan[e + f*x])/f - (a^3*c*Tan[e + f*x]^3)/(3*f)","A",9,8,24,0.3333,1,"{3904, 3886, 3473, 8, 2611, 3770, 2607, 30}"
16,1,78,0,0.2087924,"\int \frac{(a+a \sec (e+f x))^3}{c-c \sec (e+f x)} \, dx","Int[(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x]),x]","-\frac{a^3 \tan (e+f x)}{c f}+\frac{8 a^3 \cot (e+f x)}{c f}+\frac{8 a^3 \csc (e+f x)}{c f}-\frac{4 a^3 \tanh ^{-1}(\sin (e+f x))}{c f}+\frac{a^3 x}{c}","-\frac{a^3 \tan (e+f x)}{c f}+\frac{8 a^3 \cot (e+f x)}{c f}+\frac{8 a^3 \csc (e+f x)}{c f}-\frac{4 a^3 \tanh ^{-1}(\sin (e+f x))}{c f}+\frac{a^3 x}{c}",1,"(a^3*x)/c - (4*a^3*ArcTanh[Sin[e + f*x]])/(c*f) + (8*a^3*Cot[e + f*x])/(c*f) + (8*a^3*Csc[e + f*x])/(c*f) - (a^3*Tan[e + f*x])/(c*f)","A",15,11,26,0.4231,1,"{3904, 3886, 3473, 8, 2606, 3767, 2621, 321, 207, 2620, 14}"
17,1,88,0,0.3610249,"\int \frac{(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^2} \, dx","Int[(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^2,x]","\frac{a^3 \tanh ^{-1}(\sin (e+f x))}{c^2 f}+\frac{4 a^3 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))}-\frac{8 a^3 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))^2}+\frac{a^3 x}{c^2}","\frac{a^3 \tanh ^{-1}(\sin (e+f x))}{c^2 f}+\frac{4 a^3 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))}-\frac{8 a^3 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))^2}+\frac{a^3 x}{c^2}",1,"(a^3*x)/c^2 + (a^3*ArcTanh[Sin[e + f*x]])/(c^2*f) - (8*a^3*Tan[e + f*x])/(3*c^2*f*(1 - Sec[e + f*x])^2) + (4*a^3*Tan[e + f*x])/(3*c^2*f*(1 - Sec[e + f*x]))","A",13,9,26,0.3462,1,"{3903, 3777, 3919, 3794, 3796, 3797, 3799, 3998, 3770}"
18,1,102,0,0.4516165,"\int \frac{(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^3} \, dx","Int[(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^3,x]","-\frac{26 a^3 \tan (e+f x)}{15 c^3 f (1-\sec (e+f x))}+\frac{4 a^3 \tan (e+f x)}{15 c^3 f (1-\sec (e+f x))^2}-\frac{8 a^3 \tan (e+f x)}{5 c^3 f (1-\sec (e+f x))^3}+\frac{a^3 x}{c^3}","-\frac{26 a^3 \tan (e+f x)}{15 c^3 f (1-\sec (e+f x))}+\frac{4 a^3 \tan (e+f x)}{15 c^3 f (1-\sec (e+f x))^2}-\frac{8 a^3 \tan (e+f x)}{5 c^3 f (1-\sec (e+f x))^3}+\frac{a^3 x}{c^3}",1,"(a^3*x)/c^3 - (8*a^3*Tan[e + f*x])/(5*c^3*f*(1 - Sec[e + f*x])^3) + (4*a^3*Tan[e + f*x])/(15*c^3*f*(1 - Sec[e + f*x])^2) - (26*a^3*Tan[e + f*x])/(15*c^3*f*(1 - Sec[e + f*x]))","A",15,9,26,0.3462,1,"{3903, 3777, 3922, 3919, 3794, 3796, 3797, 3799, 4000}"
19,1,133,0,0.5792502,"\int \frac{(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx","Int[(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^4,x]","-\frac{167 a^3 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))}-\frac{62 a^3 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}+\frac{4 a^3 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac{8 a^3 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}+\frac{a^3 x}{c^4}","-\frac{167 a^3 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))}-\frac{62 a^3 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}+\frac{4 a^3 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac{8 a^3 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}+\frac{a^3 x}{c^4}",1,"(a^3*x)/c^4 - (8*a^3*Tan[e + f*x])/(7*c^4*f*(1 - Sec[e + f*x])^4) + (4*a^3*Tan[e + f*x])/(35*c^4*f*(1 - Sec[e + f*x])^3) - (62*a^3*Tan[e + f*x])/(105*c^4*f*(1 - Sec[e + f*x])^2) - (167*a^3*Tan[e + f*x])/(105*c^4*f*(1 - Sec[e + f*x]))","A",19,9,26,0.3462,1,"{3903, 3777, 3922, 3919, 3794, 3796, 3797, 3799, 4000}"
20,1,164,0,0.7333999,"\int \frac{(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^5} \, dx","Int[(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^5,x]","-\frac{496 a^3 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))}-\frac{181 a^3 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}-\frac{38 a^3 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}+\frac{4 a^3 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac{8 a^3 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}+\frac{a^3 x}{c^5}","-\frac{496 a^3 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))}-\frac{181 a^3 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}-\frac{38 a^3 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}+\frac{4 a^3 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac{8 a^3 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}+\frac{a^3 x}{c^5}",1,"(a^3*x)/c^5 - (8*a^3*Tan[e + f*x])/(9*c^5*f*(1 - Sec[e + f*x])^5) + (4*a^3*Tan[e + f*x])/(63*c^5*f*(1 - Sec[e + f*x])^4) - (38*a^3*Tan[e + f*x])/(105*c^5*f*(1 - Sec[e + f*x])^3) - (181*a^3*Tan[e + f*x])/(315*c^5*f*(1 - Sec[e + f*x])^2) - (496*a^3*Tan[e + f*x])/(315*c^5*f*(1 - Sec[e + f*x]))","A",23,9,26,0.3462,1,"{3903, 3777, 3922, 3919, 3794, 3796, 3797, 3799, 4000}"
21,1,153,0,0.4021106,"\int \frac{(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx","Int[(c - c*Sec[e + f*x])^5/(a + a*Sec[e + f*x])^2,x]","\frac{7 c^5 \tan (e+f x)}{a^2 f}-\frac{64 c^5 \cot ^3(e+f x)}{3 a^2 f}-\frac{48 c^5 \cot (e+f x)}{a^2 f}+\frac{131 c^5 \csc ^3(e+f x)}{6 a^2 f}+\frac{33 c^5 \csc (e+f x)}{2 a^2 f}-\frac{47 c^5 \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}-\frac{c^5 \csc ^3(e+f x) \sec ^2(e+f x)}{2 a^2 f}+\frac{c^5 x}{a^2}","\frac{13 c^5 \tan (e+f x)}{2 a^2 f}-\frac{47 c^5 \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}+\frac{112 c^5 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}+\frac{\tan (e+f x) \left(c^5-c^5 \sec (e+f x)\right)}{2 a^2 f}+\frac{c^5 x}{a^2}-\frac{32 c^5 \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2}",1,"(c^5*x)/a^2 - (47*c^5*ArcTanh[Sin[e + f*x]])/(2*a^2*f) - (48*c^5*Cot[e + f*x])/(a^2*f) - (64*c^5*Cot[e + f*x]^3)/(3*a^2*f) + (33*c^5*Csc[e + f*x])/(2*a^2*f) + (131*c^5*Csc[e + f*x]^3)/(6*a^2*f) - (c^5*Csc[e + f*x]^3*Sec[e + f*x]^2)/(2*a^2*f) + (7*c^5*Tan[e + f*x])/(a^2*f)","A",26,14,26,0.5385,1,"{3904, 3886, 3473, 8, 2606, 2607, 30, 3767, 2621, 302, 207, 2620, 270, 288}"
22,1,102,0,0.3089674,"\int \frac{(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx","Int[(c - c*Sec[e + f*x])^4/(a + a*Sec[e + f*x])^2,x]","\frac{c^4 \tan (e+f x)}{a^2 f}-\frac{32 c^4 \cot ^3(e+f x)}{3 a^2 f}-\frac{16 c^4 \cot (e+f x)}{a^2 f}+\frac{32 c^4 \csc ^3(e+f x)}{3 a^2 f}-\frac{6 c^4 \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac{c^4 x}{a^2}","\frac{c^4 \tan (e+f x)}{a^2 f}-\frac{32 c^4 \cot ^3(e+f x)}{3 a^2 f}-\frac{16 c^4 \cot (e+f x)}{a^2 f}+\frac{32 c^4 \csc ^3(e+f x)}{3 a^2 f}-\frac{6 c^4 \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac{c^4 x}{a^2}",1,"(c^4*x)/a^2 - (6*c^4*ArcTanh[Sin[e + f*x]])/(a^2*f) - (16*c^4*Cot[e + f*x])/(a^2*f) - (32*c^4*Cot[e + f*x]^3)/(3*a^2*f) + (32*c^4*Csc[e + f*x]^3)/(3*a^2*f) + (c^4*Tan[e + f*x])/(a^2*f)","A",21,13,26,0.5000,1,"{3904, 3886, 3473, 8, 2606, 2607, 30, 3767, 2621, 302, 207, 2620, 270}"
23,1,85,0,0.3309863,"\int \frac{(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^2} \, dx","Int[(c - c*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^2,x]","-\frac{c^3 \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac{4 c^3 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{8 c^3 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)^2}+\frac{c^3 x}{a^2}","-\frac{c^3 \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac{4 c^3 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{8 c^3 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)^2}+\frac{c^3 x}{a^2}",1,"(c^3*x)/a^2 - (c^3*ArcTanh[Sin[e + f*x]])/(a^2*f) - (8*c^3*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x])^2) + (4*c^3*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x]))","A",13,9,26,0.3462,1,"{3903, 3777, 3919, 3794, 3796, 3797, 3799, 3998, 3770}"
24,1,67,0,0.2284893,"\int \frac{(c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^2} \, dx","Int[(c - c*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^2,x]","-\frac{4 c^2 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{4 c^2 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)^2}+\frac{c^2 x}{a^2}","-\frac{4 c^2 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{4 c^2 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)^2}+\frac{c^2 x}{a^2}",1,"(c^2*x)/a^2 - (4*c^2*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x])^2) - (4*c^2*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x]))","A",9,6,26,0.2308,1,"{3903, 3777, 3919, 3794, 3796, 3797}"
25,1,61,0,0.1451324,"\int \frac{c-c \sec (e+f x)}{(a+a \sec (e+f x))^2} \, dx","Int[(c - c*Sec[e + f*x])/(a + a*Sec[e + f*x])^2,x]","-\frac{5 c \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{2 c \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)^2}+\frac{c x}{a^2}","-\frac{5 c \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{2 c \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)^2}+\frac{c x}{a^2}",1,"(c*x)/a^2 - (2*c*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x])^2) - (5*c*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x]))","A",7,5,24,0.2083,1,"{3903, 3777, 3919, 3794, 3796}"
26,1,69,0,0.1137391,"\int \frac{1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))} \, dx","Int[1/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])),x]","-\frac{\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}+\frac{\cot (e+f x) (3-2 \sec (e+f x))}{3 a^2 c f}+\frac{x}{a^2 c}","-\frac{\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}+\frac{\cot (e+f x) (3-2 \sec (e+f x))}{3 a^2 c f}+\frac{x}{a^2 c}",1,"x/(a^2*c) + (Cot[e + f*x]*(3 - 2*Sec[e + f*x]))/(3*a^2*c*f) - (Cot[e + f*x]^3*(1 - Sec[e + f*x]))/(3*a^2*c*f)","A",4,3,26,0.1154,1,"{3904, 3882, 8}"
27,1,46,0,0.0713255,"\int \frac{1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^2} \, dx","Int[1/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^2),x]","-\frac{\cot ^3(e+f x)}{3 a^2 c^2 f}+\frac{\cot (e+f x)}{a^2 c^2 f}+\frac{x}{a^2 c^2}","-\frac{\cot ^3(e+f x)}{3 a^2 c^2 f}+\frac{\cot (e+f x)}{a^2 c^2 f}+\frac{x}{a^2 c^2}",1,"x/(a^2*c^2) + Cot[e + f*x]/(a^2*c^2*f) - Cot[e + f*x]^3/(3*a^2*c^2*f)","A",4,3,26,0.1154,1,"{3904, 3473, 8}"
28,1,98,0,0.1447861,"\int \frac{1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx","Int[1/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^3),x]","\frac{\cot ^5(e+f x) (\sec (e+f x)+1)}{5 a^2 c^3 f}-\frac{\cot ^3(e+f x) (4 \sec (e+f x)+5)}{15 a^2 c^3 f}+\frac{\cot (e+f x) (8 \sec (e+f x)+15)}{15 a^2 c^3 f}+\frac{x}{a^2 c^3}","\frac{\cot ^5(e+f x) (\sec (e+f x)+1)}{5 a^2 c^3 f}-\frac{\cot ^3(e+f x) (4 \sec (e+f x)+5)}{15 a^2 c^3 f}+\frac{\cot (e+f x) (8 \sec (e+f x)+15)}{15 a^2 c^3 f}+\frac{x}{a^2 c^3}",1,"x/(a^2*c^3) + (Cot[e + f*x]^5*(1 + Sec[e + f*x]))/(5*a^2*c^3*f) - (Cot[e + f*x]^3*(5 + 4*Sec[e + f*x]))/(15*a^2*c^3*f) + (Cot[e + f*x]*(15 + 8*Sec[e + f*x]))/(15*a^2*c^3*f)","A",5,3,26,0.1154,1,"{3904, 3882, 8}"
29,1,166,0,0.2112474,"\int \frac{1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4} \, dx","Int[1/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^4),x]","-\frac{2 \cot ^7(e+f x)}{7 a^2 c^4 f}+\frac{\cot ^5(e+f x)}{5 a^2 c^4 f}-\frac{\cot ^3(e+f x)}{3 a^2 c^4 f}+\frac{\cot (e+f x)}{a^2 c^4 f}-\frac{2 \csc ^7(e+f x)}{7 a^2 c^4 f}+\frac{6 \csc ^5(e+f x)}{5 a^2 c^4 f}-\frac{2 \csc ^3(e+f x)}{a^2 c^4 f}+\frac{2 \csc (e+f x)}{a^2 c^4 f}+\frac{x}{a^2 c^4}","-\frac{2 \cot ^7(e+f x)}{7 a^2 c^4 f}+\frac{\cot ^5(e+f x)}{5 a^2 c^4 f}-\frac{\cot ^3(e+f x)}{3 a^2 c^4 f}+\frac{\cot (e+f x)}{a^2 c^4 f}-\frac{2 \csc ^7(e+f x)}{7 a^2 c^4 f}+\frac{6 \csc ^5(e+f x)}{5 a^2 c^4 f}-\frac{2 \csc ^3(e+f x)}{a^2 c^4 f}+\frac{2 \csc (e+f x)}{a^2 c^4 f}+\frac{x}{a^2 c^4}",1,"x/(a^2*c^4) + Cot[e + f*x]/(a^2*c^4*f) - Cot[e + f*x]^3/(3*a^2*c^4*f) + Cot[e + f*x]^5/(5*a^2*c^4*f) - (2*Cot[e + f*x]^7)/(7*a^2*c^4*f) + (2*Csc[e + f*x])/(a^2*c^4*f) - (2*Csc[e + f*x]^3)/(a^2*c^4*f) + (6*Csc[e + f*x]^5)/(5*a^2*c^4*f) - (2*Csc[e + f*x]^7)/(7*a^2*c^4*f)","A",13,8,26,0.3077,1,"{3904, 3886, 3473, 8, 2606, 194, 2607, 30}"
30,1,210,0,0.2861628,"\int \frac{1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5} \, dx","Int[1/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5),x]","\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{\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac{\cot ^3(e+f x)}{3 a^2 c^5 f}+\frac{\cot (e+f x)}{a^2 c^5 f}+\frac{4 \csc ^9(e+f x)}{9 a^2 c^5 f}-\frac{15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac{21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac{13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac{3 \csc (e+f x)}{a^2 c^5 f}+\frac{x}{a^2 c^5}","\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{\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac{\cot ^3(e+f x)}{3 a^2 c^5 f}+\frac{\cot (e+f x)}{a^2 c^5 f}+\frac{4 \csc ^9(e+f x)}{9 a^2 c^5 f}-\frac{15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac{21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac{13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac{3 \csc (e+f x)}{a^2 c^5 f}+\frac{x}{a^2 c^5}",1,"x/(a^2*c^5) + Cot[e + f*x]/(a^2*c^5*f) - Cot[e + f*x]^3/(3*a^2*c^5*f) + Cot[e + f*x]^5/(5*a^2*c^5*f) - Cot[e + f*x]^7/(7*a^2*c^5*f) + (4*Cot[e + f*x]^9)/(9*a^2*c^5*f) + (3*Csc[e + f*x])/(a^2*c^5*f) - (13*Csc[e + f*x]^3)/(3*a^2*c^5*f) + (21*Csc[e + f*x]^5)/(5*a^2*c^5*f) - (15*Csc[e + f*x]^7)/(7*a^2*c^5*f) + (4*Csc[e + f*x]^9)/(9*a^2*c^5*f)","A",17,9,26,0.3462,1,"{3904, 3886, 3473, 8, 2606, 194, 2607, 30, 270}"
31,1,162,0,0.4434826,"\int \frac{(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx","Int[(c - c*Sec[e + f*x])^5/(a + a*Sec[e + f*x])^3,x]","-\frac{c^5 \tan (e+f x)}{a^3 f}+\frac{128 c^5 \cot ^5(e+f x)}{5 a^3 f}+\frac{128 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac{32 c^5 \cot (e+f x)}{a^3 f}-\frac{128 c^5 \csc ^5(e+f x)}{5 a^3 f}+\frac{64 c^5 \csc ^3(e+f x)}{3 a^3 f}-\frac{16 c^5 \csc (e+f x)}{a^3 f}+\frac{8 c^5 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac{c^5 x}{a^3}","-\frac{c^5 \tan (e+f x)}{a^3 f}+\frac{128 c^5 \cot ^5(e+f x)}{5 a^3 f}+\frac{128 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac{32 c^5 \cot (e+f x)}{a^3 f}-\frac{128 c^5 \csc ^5(e+f x)}{5 a^3 f}+\frac{64 c^5 \csc ^3(e+f x)}{3 a^3 f}-\frac{16 c^5 \csc (e+f x)}{a^3 f}+\frac{8 c^5 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac{c^5 x}{a^3}",1,"(c^5*x)/a^3 + (8*c^5*ArcTanh[Sin[e + f*x]])/(a^3*f) + (32*c^5*Cot[e + f*x])/(a^3*f) + (128*c^5*Cot[e + f*x]^3)/(3*a^3*f) + (128*c^5*Cot[e + f*x]^5)/(5*a^3*f) - (16*c^5*Csc[e + f*x])/(a^3*f) + (64*c^5*Csc[e + f*x]^3)/(3*a^3*f) - (128*c^5*Csc[e + f*x]^5)/(5*a^3*f) - (c^5*Tan[e + f*x])/(a^3*f)","A",29,15,26,0.5769,1,"{3904, 3886, 3473, 8, 2606, 194, 2607, 30, 14, 3767, 2621, 302, 207, 2620, 270}"
32,1,148,0,0.6060572,"\int \frac{(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx","Int[(c - c*Sec[e + f*x])^4/(a + a*Sec[e + f*x])^3,x]","\frac{c^4 \tanh ^{-1}(\sin (e+f x))}{a^3 f}-\frac{c^4 \tan (e+f x) \sec ^2(e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}-\frac{23 c^4 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)}+\frac{14 c^4 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^2}-\frac{3 c^4 \tan (e+f x)}{a^3 f (\sec (e+f x)+1)^3}+\frac{c^4 x}{a^3}","\frac{c^4 \tanh ^{-1}(\sin (e+f x))}{a^3 f}-\frac{c^4 \tan (e+f x) \sec ^2(e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}-\frac{23 c^4 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)}+\frac{14 c^4 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^2}-\frac{3 c^4 \tan (e+f x)}{a^3 f (\sec (e+f x)+1)^3}+\frac{c^4 x}{a^3}",1,"(c^4*x)/a^3 + (c^4*ArcTanh[Sin[e + f*x]])/(a^3*f) - (3*c^4*Tan[e + f*x])/(a^3*f*(1 + Sec[e + f*x])^3) - (c^4*Sec[e + f*x]^2*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x])^3) + (14*c^4*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x])^2) - (23*c^4*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x]))","A",20,13,26,0.5000,1,"{3903, 3777, 3922, 3919, 3794, 3796, 3797, 3799, 4000, 3816, 4008, 3998, 3770}"
33,1,96,0,0.4205622,"\int \frac{(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^3} \, dx","Int[(c - c*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^3,x]","-\frac{26 c^3 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)}+\frac{4 c^3 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)^2}-\frac{8 c^3 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}+\frac{c^3 x}{a^3}","-\frac{26 c^3 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)}+\frac{4 c^3 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)^2}-\frac{8 c^3 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}+\frac{c^3 x}{a^3}",1,"(c^3*x)/a^3 - (8*c^3*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x])^3) + (4*c^3*Tan[e + f*x])/(15*a^3*f*(1 + Sec[e + f*x])^2) - (26*c^3*Tan[e + f*x])/(15*a^3*f*(1 + Sec[e + f*x]))","A",15,9,26,0.3462,1,"{3903, 3777, 3922, 3919, 3794, 3796, 3797, 3799, 4000}"
34,1,96,0,0.3038655,"\int \frac{(c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^3} \, dx","Int[(c - c*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^3,x]","-\frac{23 c^2 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)}-\frac{8 c^2 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)^2}-\frac{4 c^2 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}+\frac{c^2 x}{a^3}","-\frac{23 c^2 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)}-\frac{8 c^2 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)^2}-\frac{4 c^2 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}+\frac{c^2 x}{a^3}",1,"(c^2*x)/a^3 - (4*c^2*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x])^3) - (8*c^2*Tan[e + f*x])/(15*a^3*f*(1 + Sec[e + f*x])^2) - (23*c^2*Tan[e + f*x])/(15*a^3*f*(1 + Sec[e + f*x]))","A",12,7,26,0.2692,1,"{3903, 3777, 3922, 3919, 3794, 3796, 3797}"
35,1,88,0,0.2021232,"\int \frac{c-c \sec (e+f x)}{(a+a \sec (e+f x))^3} \, dx","Int[(c - c*Sec[e + f*x])/(a + a*Sec[e + f*x])^3,x]","-\frac{8 c \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)}-\frac{3 c \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^2}-\frac{2 c \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}+\frac{c x}{a^3}","-\frac{8 c \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)}-\frac{3 c \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^2}-\frac{2 c \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}+\frac{c x}{a^3}",1,"(c*x)/a^3 - (2*c*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x])^3) - (3*c*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x])^2) - (8*c*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x]))","A",9,6,24,0.2500,1,"{3903, 3777, 3922, 3919, 3794, 3796}"
36,1,126,0,0.1932623,"\int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))} \, dx","Int[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])),x]","\frac{2 \cot ^5(e+f x)}{5 a^3 c f}-\frac{\cot ^3(e+f x)}{3 a^3 c f}+\frac{\cot (e+f x)}{a^3 c f}-\frac{2 \csc ^5(e+f x)}{5 a^3 c f}+\frac{4 \csc ^3(e+f x)}{3 a^3 c f}-\frac{2 \csc (e+f x)}{a^3 c f}+\frac{x}{a^3 c}","\frac{2 \cot ^5(e+f x)}{5 a^3 c f}-\frac{\cot ^3(e+f x)}{3 a^3 c f}+\frac{\cot (e+f x)}{a^3 c f}-\frac{2 \csc ^5(e+f x)}{5 a^3 c f}+\frac{4 \csc ^3(e+f x)}{3 a^3 c f}-\frac{2 \csc (e+f x)}{a^3 c f}+\frac{x}{a^3 c}",1,"x/(a^3*c) + Cot[e + f*x]/(a^3*c*f) - Cot[e + f*x]^3/(3*a^3*c*f) + (2*Cot[e + f*x]^5)/(5*a^3*c*f) - (2*Csc[e + f*x])/(a^3*c*f) + (4*Csc[e + f*x]^3)/(3*a^3*c*f) - (2*Csc[e + f*x]^5)/(5*a^3*c*f)","A",12,8,26,0.3077,1,"{3904, 3886, 3473, 8, 2606, 194, 2607, 30}"
37,1,100,0,0.1428633,"\int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2} \, dx","Int[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^2),x]","\frac{\cot ^5(e+f x) (1-\sec (e+f x))}{5 a^3 c^2 f}-\frac{\cot ^3(e+f x) (5-4 \sec (e+f x))}{15 a^3 c^2 f}+\frac{\cot (e+f x) (15-8 \sec (e+f x))}{15 a^3 c^2 f}+\frac{x}{a^3 c^2}","\frac{\cot ^5(e+f x) (1-\sec (e+f x))}{5 a^3 c^2 f}-\frac{\cot ^3(e+f x) (5-4 \sec (e+f x))}{15 a^3 c^2 f}+\frac{\cot (e+f x) (15-8 \sec (e+f x))}{15 a^3 c^2 f}+\frac{x}{a^3 c^2}",1,"x/(a^3*c^2) + (Cot[e + f*x]*(15 - 8*Sec[e + f*x]))/(15*a^3*c^2*f) - (Cot[e + f*x]^3*(5 - 4*Sec[e + f*x]))/(15*a^3*c^2*f) + (Cot[e + f*x]^5*(1 - Sec[e + f*x]))/(5*a^3*c^2*f)","A",5,3,26,0.1154,1,"{3904, 3882, 8}"
38,1,67,0,0.0805624,"\int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3} \, dx","Int[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^3),x]","\frac{\cot ^5(e+f x)}{5 a^3 c^3 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^3 f}+\frac{\cot (e+f x)}{a^3 c^3 f}+\frac{x}{a^3 c^3}","\frac{\cot ^5(e+f x)}{5 a^3 c^3 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^3 f}+\frac{\cot (e+f x)}{a^3 c^3 f}+\frac{x}{a^3 c^3}",1,"x/(a^3*c^3) + Cot[e + f*x]/(a^3*c^3*f) - Cot[e + f*x]^3/(3*a^3*c^3*f) + Cot[e + f*x]^5/(5*a^3*c^3*f)","A",5,3,26,0.1154,1,"{3904, 3473, 8}"
39,1,129,0,0.1729451,"\int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx","Int[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^4),x]","-\frac{\cot ^7(e+f x) (\sec (e+f x)+1)}{7 a^3 c^4 f}+\frac{\cot ^5(e+f x) (6 \sec (e+f x)+7)}{35 a^3 c^4 f}-\frac{\cot ^3(e+f x) (24 \sec (e+f x)+35)}{105 a^3 c^4 f}+\frac{\cot (e+f x) (16 \sec (e+f x)+35)}{35 a^3 c^4 f}+\frac{x}{a^3 c^4}","-\frac{\cot ^7(e+f x) (\sec (e+f x)+1)}{7 a^3 c^4 f}+\frac{\cot ^5(e+f x) (6 \sec (e+f x)+7)}{35 a^3 c^4 f}-\frac{\cot ^3(e+f x) (24 \sec (e+f x)+35)}{105 a^3 c^4 f}+\frac{\cot (e+f x) (16 \sec (e+f x)+35)}{35 a^3 c^4 f}+\frac{x}{a^3 c^4}",1,"x/(a^3*c^4) - (Cot[e + f*x]^7*(1 + Sec[e + f*x]))/(7*a^3*c^4*f) + (Cot[e + f*x]^5*(7 + 6*Sec[e + f*x]))/(35*a^3*c^4*f) + (Cot[e + f*x]*(35 + 16*Sec[e + f*x]))/(35*a^3*c^4*f) - (Cot[e + f*x]^3*(35 + 24*Sec[e + f*x]))/(105*a^3*c^4*f)","A",6,3,26,0.1154,1,"{3904, 3882, 8}"
40,1,210,0,0.2364064,"\int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx","Int[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5),x]","\frac{2 \cot ^9(e+f x)}{9 a^3 c^5 f}-\frac{\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac{\cot (e+f x)}{a^3 c^5 f}+\frac{2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac{8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac{12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac{8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac{2 \csc (e+f x)}{a^3 c^5 f}+\frac{x}{a^3 c^5}","\frac{2 \cot ^9(e+f x)}{9 a^3 c^5 f}-\frac{\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac{\cot (e+f x)}{a^3 c^5 f}+\frac{2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac{8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac{12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac{8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac{2 \csc (e+f x)}{a^3 c^5 f}+\frac{x}{a^3 c^5}",1,"x/(a^3*c^5) + Cot[e + f*x]/(a^3*c^5*f) - Cot[e + f*x]^3/(3*a^3*c^5*f) + Cot[e + f*x]^5/(5*a^3*c^5*f) - Cot[e + f*x]^7/(7*a^3*c^5*f) + (2*Cot[e + f*x]^9)/(9*a^3*c^5*f) + (2*Csc[e + f*x])/(a^3*c^5*f) - (8*Csc[e + f*x]^3)/(3*a^3*c^5*f) + (12*Csc[e + f*x]^5)/(5*a^3*c^5*f) - (8*Csc[e + f*x]^7)/(7*a^3*c^5*f) + (2*Csc[e + f*x]^9)/(9*a^3*c^5*f)","A",14,8,26,0.3077,1,"{3904, 3886, 3473, 8, 2606, 194, 2607, 30}"
41,1,252,0,0.2995342,"\int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx","Int[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^6),x]","-\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{\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac{\cot (e+f x)}{a^3 c^6 f}-\frac{4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac{19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac{36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac{34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac{16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac{3 \csc (e+f x)}{a^3 c^6 f}+\frac{x}{a^3 c^6}","-\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{\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac{\cot (e+f x)}{a^3 c^6 f}-\frac{4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac{19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac{36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac{34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac{16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac{3 \csc (e+f x)}{a^3 c^6 f}+\frac{x}{a^3 c^6}",1,"x/(a^3*c^6) + Cot[e + f*x]/(a^3*c^6*f) - Cot[e + f*x]^3/(3*a^3*c^6*f) + Cot[e + f*x]^5/(5*a^3*c^6*f) - Cot[e + f*x]^7/(7*a^3*c^6*f) + Cot[e + f*x]^9/(9*a^3*c^6*f) - (4*Cot[e + f*x]^11)/(11*a^3*c^6*f) + (3*Csc[e + f*x])/(a^3*c^6*f) - (16*Csc[e + f*x]^3)/(3*a^3*c^6*f) + (34*Csc[e + f*x]^5)/(5*a^3*c^6*f) - (36*Csc[e + f*x]^7)/(7*a^3*c^6*f) + (19*Csc[e + f*x]^9)/(9*a^3*c^6*f) - (4*Csc[e + f*x]^11)/(11*a^3*c^6*f)","A",18,9,26,0.3462,1,"{3904, 3886, 3473, 8, 2606, 194, 2607, 30, 270}"
42,1,175,0,0.1761845,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^4 \, dx","Int[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^4,x]","\frac{2 a^4 c^4 \tan ^7(e+f x)}{7 f (a \sec (e+f x)+a)^{7/2}}-\frac{2 a^3 c^4 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^2 c^4 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 \sqrt{a} c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a c^4 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","\frac{2 a^4 c^4 \tan ^7(e+f x)}{7 f (a \sec (e+f x)+a)^{7/2}}-\frac{2 a^3 c^4 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^2 c^4 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 \sqrt{a} c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a c^4 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*Sqrt[a]*c^4*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a*c^4*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*c^4*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2)) - (2*a^3*c^4*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2)) + (2*a^4*c^4*Tan[e + f*x]^7)/(7*f*(a + a*Sec[e + f*x])^(7/2))","A",5,4,28,0.1429,1,"{3904, 3887, 302, 203}"
43,1,140,0,0.1676138,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^3 \, dx","Int[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^3,x]","-\frac{2 a^3 c^3 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^2 c^3 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 \sqrt{a} c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a c^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","-\frac{2 a^3 c^3 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^2 c^3 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 \sqrt{a} c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a c^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*Sqrt[a]*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a*c^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*c^3*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2)) - (2*a^3*c^3*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2))","A",5,4,28,0.1429,1,"{3904, 3887, 302, 203}"
44,1,105,0,0.1584859,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^2 \, dx","Int[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^2,x]","\frac{2 a^2 c^2 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 \sqrt{a} c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a c^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","\frac{2 a^2 c^2 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 \sqrt{a} c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a c^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*Sqrt[a]*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a*c^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*c^2*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2))","A",5,4,28,0.1429,1,"{3904, 3887, 302, 203}"
45,1,66,0,0.109402,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x)) \, dx","Int[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x]),x]","\frac{2 \sqrt{a} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a c \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","\frac{2 \sqrt{a} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a c \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*Sqrt[a]*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a*c*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])","A",4,4,26,0.1538,1,"{3904, 3887, 321, 203}"
46,1,69,0,0.147882,"\int \frac{\sqrt{a+a \sec (e+f x)}}{c-c \sec (e+f x)} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x]),x]","\frac{2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c f}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}","\frac{2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c f}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}",1,"(2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c*f) + (2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c*f)","A",4,4,28,0.1429,1,"{3904, 3887, 325, 203}"
47,1,104,0,0.1573382,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^2} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^2,x]","-\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a c^2 f}+\frac{2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^2 f}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^2 f}","-\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a c^2 f}+\frac{2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^2 f}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^2 f}",1,"(2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^2*f) + (2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^2*f) - (2*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*a*c^2*f)","A",5,4,28,0.1429,1,"{3904, 3887, 325, 203}"
48,1,139,0,0.1674371,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^3} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^3,x]","\frac{2 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a^2 c^3 f}-\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a c^3 f}+\frac{2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^3 f}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^3 f}","\frac{2 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a^2 c^3 f}-\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a c^3 f}+\frac{2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^3 f}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^3 f}",1,"(2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^3*f) + (2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^3*f) - (2*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*a*c^3*f) + (2*Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(5*a^2*c^3*f)","A",6,4,28,0.1429,1,"{3904, 3887, 325, 203}"
49,1,174,0,0.1765756,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^4} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^4,x]","-\frac{2 \cot ^7(e+f x) (a \sec (e+f x)+a)^{7/2}}{7 a^3 c^4 f}+\frac{2 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a^2 c^4 f}-\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a c^4 f}+\frac{2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^4 f}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^4 f}","-\frac{2 \cot ^7(e+f x) (a \sec (e+f x)+a)^{7/2}}{7 a^3 c^4 f}+\frac{2 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a^2 c^4 f}-\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a c^4 f}+\frac{2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^4 f}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^4 f}",1,"(2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^4*f) + (2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^4*f) - (2*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*a*c^4*f) + (2*Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(5*a^2*c^4*f) - (2*Cot[e + f*x]^7*(a + a*Sec[e + f*x])^(7/2))/(7*a^3*c^4*f)","A",7,4,28,0.1429,1,"{3904, 3887, 325, 203}"
50,1,177,0,0.1838372,"\int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^3 \, dx","Int[(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^3,x]","-\frac{2 a^5 c^3 \tan ^7(e+f x)}{7 f (a \sec (e+f x)+a)^{7/2}}-\frac{2 a^4 c^3 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^3 c^3 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{3/2} c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^2 c^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","-\frac{2 a^5 c^3 \tan ^7(e+f x)}{7 f (a \sec (e+f x)+a)^{7/2}}-\frac{2 a^4 c^3 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^3 c^3 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{3/2} c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^2 c^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*a^(3/2)*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a^2*c^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^3*c^3*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2)) - (2*a^4*c^3*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2)) - (2*a^5*c^3*Tan[e + f*x]^7)/(7*f*(a + a*Sec[e + f*x])^(7/2))","A",6,5,28,0.1786,1,"{3904, 3887, 459, 302, 203}"
51,1,142,0,0.1700425,"\int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^2 \, dx","Int[(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^2,x]","\frac{2 a^4 c^2 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^3 c^2 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{3/2} c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^2 c^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","\frac{2 a^4 c^2 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^3 c^2 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{3/2} c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^2 c^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*a^(3/2)*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a^2*c^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^3*c^2*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2)) + (2*a^4*c^2*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2))","A",6,5,28,0.1786,1,"{3904, 3887, 459, 302, 203}"
52,1,101,0,0.1234694,"\int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x)) \, dx","Int[(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x]),x]","-\frac{2 a^3 c \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{3/2} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^2 c \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","-\frac{2 a^3 c \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{3/2} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^2 c \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*a^(3/2)*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a^2*c*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (2*a^3*c*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2))","A",5,5,26,0.1923,1,"{3904, 3887, 459, 321, 203}"
53,1,70,0,0.1561487,"\int \frac{(a+a \sec (e+f x))^{3/2}}{c-c \sec (e+f x)} \, dx","Int[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x]),x]","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}+\frac{4 a \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c f}","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}+\frac{4 a \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c f}",1,"(2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c*f) + (4*a*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c*f)","A",4,4,28,0.1429,1,"{3904, 3887, 453, 203}"
54,1,102,0,0.1621585,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^2} \, dx","Int[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^2,x]","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^2 f}-\frac{4 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^2 f}+\frac{2 a \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^2 f}","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^2 f}-\frac{4 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^2 f}+\frac{2 a \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^2 f}",1,"(2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^2*f) + (2*a*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^2*f) - (4*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*c^2*f)","A",5,5,28,0.1786,1,"{3904, 3887, 453, 325, 203}"
55,1,137,0,0.1777963,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^3} \, dx","Int[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^3,x]","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^3 f}+\frac{4 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a c^3 f}-\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^3 f}+\frac{2 a \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^3 f}","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^3 f}+\frac{4 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a c^3 f}-\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^3 f}+\frac{2 a \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^3 f}",1,"(2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^3*f) + (2*a*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^3*f) - (2*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*c^3*f) + (4*Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(5*a*c^3*f)","A",6,5,28,0.1786,1,"{3904, 3887, 453, 325, 203}"
56,1,172,0,0.1866271,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^4} \, dx","Int[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^4,x]","-\frac{4 \cot ^7(e+f x) (a \sec (e+f x)+a)^{7/2}}{7 a^2 c^4 f}+\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^4 f}+\frac{2 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a c^4 f}-\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^4 f}+\frac{2 a \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^4 f}","-\frac{4 \cot ^7(e+f x) (a \sec (e+f x)+a)^{7/2}}{7 a^2 c^4 f}+\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^4 f}+\frac{2 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a c^4 f}-\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^4 f}+\frac{2 a \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^4 f}",1,"(2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^4*f) + (2*a*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^4*f) - (2*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*c^4*f) + (2*Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(5*a*c^4*f) - (4*Cot[e + f*x]^7*(a + a*Sec[e + f*x])^(7/2))/(7*a^2*c^4*f)","A",7,5,28,0.1786,1,"{3904, 3887, 453, 325, 203}"
57,1,212,0,0.1909082,"\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^3 \, dx","Int[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^3,x]","-\frac{2 a^7 c^3 \tan ^9(e+f x)}{9 f (a \sec (e+f x)+a)^{9/2}}-\frac{6 a^6 c^3 \tan ^7(e+f x)}{7 f (a \sec (e+f x)+a)^{7/2}}-\frac{2 a^5 c^3 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^4 c^3 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{5/2} c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^3 c^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","-\frac{2 a^7 c^3 \tan ^9(e+f x)}{9 f (a \sec (e+f x)+a)^{9/2}}-\frac{6 a^6 c^3 \tan ^7(e+f x)}{7 f (a \sec (e+f x)+a)^{7/2}}-\frac{2 a^5 c^3 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^4 c^3 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{5/2} c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^3 c^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*a^(5/2)*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a^3*c^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^4*c^3*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2)) - (2*a^5*c^3*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2)) - (6*a^6*c^3*Tan[e + f*x]^7)/(7*f*(a + a*Sec[e + f*x])^(7/2)) - (2*a^7*c^3*Tan[e + f*x]^9)/(9*f*(a + a*Sec[e + f*x])^(9/2))","A",5,4,28,0.1429,1,"{3904, 3887, 461, 203}"
58,1,177,0,0.1743062,"\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^2 \, dx","Int[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^2,x]","\frac{2 a^6 c^2 \tan ^7(e+f x)}{7 f (a \sec (e+f x)+a)^{7/2}}+\frac{6 a^5 c^2 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^4 c^2 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{5/2} c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^3 c^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","\frac{2 a^6 c^2 \tan ^7(e+f x)}{7 f (a \sec (e+f x)+a)^{7/2}}+\frac{6 a^5 c^2 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^4 c^2 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{5/2} c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^3 c^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*a^(5/2)*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a^3*c^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^4*c^2*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2)) + (6*a^5*c^2*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2)) + (2*a^6*c^2*Tan[e + f*x]^7)/(7*f*(a + a*Sec[e + f*x])^(7/2))","A",5,4,28,0.1429,1,"{3904, 3887, 461, 203}"
59,1,132,0,0.1406824,"\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x)) \, dx","Int[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x]),x]","-\frac{2 a^5 c \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}-\frac{2 a^4 c \tan ^3(e+f x)}{f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{5/2} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^3 c \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","-\frac{2 a^5 c \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}-\frac{2 a^4 c \tan ^3(e+f x)}{f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{5/2} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^3 c \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*a^(5/2)*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a^3*c*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (2*a^4*c*Tan[e + f*x]^3)/(f*(a + a*Sec[e + f*x])^(3/2)) - (2*a^5*c*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2))","A",5,4,26,0.1538,1,"{3904, 3887, 461, 203}"
60,1,103,0,0.1719659,"\int \frac{(a+a \sec (e+f x))^{5/2}}{c-c \sec (e+f x)} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x]),x]","\frac{8 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c f}+\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}-\frac{2 a^3 \tan (e+f x)}{c f \sqrt{a \sec (e+f x)+a}}","\frac{8 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c f}+\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}-\frac{2 a^3 \tan (e+f x)}{c f \sqrt{a \sec (e+f x)+a}}",1,"(2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c*f) + (8*a^2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c*f) - (2*a^3*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]])","A",5,4,28,0.1429,1,"{3904, 3887, 461, 203}"
61,1,74,0,0.1679219,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^2} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^2,x]","\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^2 f}-\frac{8 a \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^2 f}","\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^2 f}-\frac{8 a \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^2 f}",1,"(2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^2*f) - (8*a*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*c^2*f)","A",5,4,28,0.1429,1,"{3904, 3887, 461, 203}"
62,1,104,0,0.1770502,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^3} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^3,x]","\frac{2 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^3 f}+\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^3 f}+\frac{8 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 c^3 f}","\frac{2 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^3 f}+\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^3 f}+\frac{8 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 c^3 f}",1,"(2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^3*f) + (2*a^2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^3*f) + (8*Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(5*c^3*f)","A",5,4,28,0.1429,1,"{3904, 3887, 461, 203}"
63,1,140,0,0.1840249,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^4} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^4,x]","\frac{2 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^4 f}+\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^4 f}-\frac{8 \cot ^7(e+f x) (a \sec (e+f x)+a)^{7/2}}{7 a c^4 f}-\frac{2 a \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^4 f}","\frac{2 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^4 f}+\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^4 f}-\frac{8 \cot ^7(e+f x) (a \sec (e+f x)+a)^{7/2}}{7 a c^4 f}-\frac{2 a \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^4 f}",1,"(2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^4*f) + (2*a^2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^4*f) - (2*a*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*c^4*f) - (8*Cot[e + f*x]^7*(a + a*Sec[e + f*x])^(7/2))/(7*a*c^4*f)","A",5,4,28,0.1429,1,"{3904, 3887, 461, 203}"
64,1,172,0,0.1982717,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^5} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^5,x]","\frac{8 \cot ^9(e+f x) (a \sec (e+f x)+a)^{9/2}}{9 a^2 c^5 f}+\frac{2 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^5 f}+\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^5 f}+\frac{2 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 c^5 f}-\frac{2 a \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^5 f}","\frac{8 \cot ^9(e+f x) (a \sec (e+f x)+a)^{9/2}}{9 a^2 c^5 f}+\frac{2 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^5 f}+\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^5 f}+\frac{2 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 c^5 f}-\frac{2 a \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^5 f}",1,"(2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^5*f) + (2*a^2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^5*f) - (2*a*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*c^5*f) + (2*Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(5*c^5*f) + (8*Cot[e + f*x]^9*(a + a*Sec[e + f*x])^(9/2))/(9*a^2*c^5*f)","A",5,4,28,0.1429,1,"{3904, 3887, 461, 203}"
65,1,185,0,0.2761443,"\int \frac{(c-c \sec (e+f x))^4}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[(c - c*Sec[e + f*x])^4/Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 a^2 c^4 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}-\frac{2 a c^4 \tan ^3(e+f x)}{f (a \sec (e+f x)+a)^{3/2}}+\frac{2 c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{16 \sqrt{2} c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}+\frac{14 c^4 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","\frac{2 a^2 c^4 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}-\frac{2 a c^4 \tan ^3(e+f x)}{f (a \sec (e+f x)+a)^{3/2}}+\frac{2 c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{16 \sqrt{2} c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}+\frac{14 c^4 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*c^4*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*f) - (16*Sqrt[2]*c^4*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*f) + (14*c^4*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (2*a*c^4*Tan[e + f*x]^3)/(f*(a + a*Sec[e + f*x])^(3/2)) + (2*a^2*c^4*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2))","A",8,6,28,0.2143,1,"{3904, 3887, 479, 582, 522, 203}"
66,1,152,0,0.2284973,"\int \frac{(c-c \sec (e+f x))^3}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[(c - c*Sec[e + f*x])^3/Sqrt[a + a*Sec[e + f*x]],x]","-\frac{2 a c^3 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{8 \sqrt{2} c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}+\frac{6 c^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","-\frac{2 a c^3 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{8 \sqrt{2} c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}+\frac{6 c^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*f) - (8*Sqrt[2]*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*f) + (6*c^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (2*a*c^3*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2))","A",7,6,28,0.2143,1,"{3904, 3887, 479, 582, 522, 203}"
67,1,119,0,0.18764,"\int \frac{(c-c \sec (e+f x))^2}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[(c - c*Sec[e + f*x])^2/Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{4 \sqrt{2} c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}+\frac{2 c^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","\frac{2 c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{4 \sqrt{2} c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}+\frac{2 c^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*f) - (4*Sqrt[2]*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*f) + (2*c^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])","A",6,5,28,0.1786,1,"{3904, 3887, 479, 522, 203}"
68,1,87,0,0.1377229,"\int \frac{c-c \sec (e+f x)}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[(c - c*Sec[e + f*x])/Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{2 \sqrt{2} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}","\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{2 \sqrt{2} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}",1,"(2*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*f) - (2*Sqrt[2]*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*f)","A",5,4,26,0.1538,1,"{3904, 3887, 481, 203}"
69,1,121,0,0.1923333,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx","Int[1/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])),x]","\frac{\cot (e+f x) \sqrt{a \sec (e+f x)+a}}{a c f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{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}","\frac{\cot (e+f x) \sqrt{a \sec (e+f x)+a}}{a c f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{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,"(2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*c*f) - ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])]/(Sqrt[2]*Sqrt[a]*c*f) + (Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(a*c*f)","A",6,5,28,0.1786,1,"{3904, 3887, 480, 522, 203}"
70,1,161,0,0.2345235,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^2} \, dx","Int[1/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^2),x]","-\frac{\cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a^2 c^2 f}+\frac{3 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{2 a c^2 f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} c^2 f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{2 \sqrt{2} \sqrt{a} c^2 f}","-\frac{\cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a^2 c^2 f}+\frac{3 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{2 a c^2 f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} c^2 f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{2 \sqrt{2} \sqrt{a} c^2 f}",1,"(2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*c^2*f) - ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])]/(2*Sqrt[2]*Sqrt[a]*c^2*f) + (3*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(2*a*c^2*f) - (Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*a^2*c^2*f)","A",7,6,28,0.2143,1,"{3904, 3887, 480, 583, 522, 203}"
71,1,196,0,0.2842378,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^3} \, dx","Int[1/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^3),x]","\frac{\cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a^3 c^3 f}-\frac{\cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{2 a^2 c^3 f}+\frac{7 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{4 a c^3 f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} c^3 f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{4 \sqrt{2} \sqrt{a} c^3 f}","\frac{\cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a^3 c^3 f}-\frac{\cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{2 a^2 c^3 f}+\frac{7 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{4 a c^3 f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} c^3 f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{4 \sqrt{2} \sqrt{a} c^3 f}",1,"(2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*c^3*f) - ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])]/(4*Sqrt[2]*Sqrt[a]*c^3*f) + (7*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(4*a*c^3*f) - (Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(2*a^2*c^3*f) + (Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(5*a^3*c^3*f)","A",8,6,28,0.2143,1,"{3904, 3887, 480, 583, 522, 203}"
72,1,203,0,0.2862326,"\int \frac{(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^{3/2}} \, dx","Int[(c - c*Sec[e + f*x])^4/(a + a*Sec[e + f*x])^(3/2),x]","\frac{2 c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}+\frac{12 \sqrt{2} c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}+\frac{8 c^4 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}-\frac{14 c^4 \tan (e+f x)}{a f \sqrt{a \sec (e+f x)+a}}-\frac{a c^4 \sin (e+f x) \tan ^4(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{f (a \sec (e+f x)+a)^{5/2}}","\frac{2 c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}+\frac{12 \sqrt{2} c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}+\frac{8 c^4 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}-\frac{14 c^4 \tan (e+f x)}{a f \sqrt{a \sec (e+f x)+a}}-\frac{a c^4 \sin (e+f x) \tan ^4(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{f (a \sec (e+f x)+a)^{5/2}}",1,"(2*c^4*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*f) + (12*Sqrt[2]*c^4*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(a^(3/2)*f) - (14*c^4*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]) + (8*c^4*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2)) - (a*c^4*Sec[(e + f*x)/2]^2*Sin[e + f*x]*Tan[e + f*x]^4)/(f*(a + a*Sec[e + f*x])^(5/2))","A",8,6,28,0.2143,1,"{3904, 3887, 470, 582, 522, 203}"
73,1,169,0,0.2403926,"\int \frac{(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^{3/2}} \, dx","Int[(c - c*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^(3/2),x]","\frac{2 c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}+\frac{2 \sqrt{2} c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}-\frac{4 c^3 \tan (e+f x)}{a f \sqrt{a \sec (e+f x)+a}}+\frac{c^3 \sin (e+f x) \tan ^2(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{f (a \sec (e+f x)+a)^{3/2}}","\frac{2 c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}+\frac{2 \sqrt{2} c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}-\frac{4 c^3 \tan (e+f x)}{a f \sqrt{a \sec (e+f x)+a}}+\frac{c^3 \sin (e+f x) \tan ^2(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{f (a \sec (e+f x)+a)^{3/2}}",1,"(2*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*f) + (2*Sqrt[2]*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(a^(3/2)*f) - (4*c^3*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]) + (c^3*Sec[(e + f*x)/2]^2*Sin[e + f*x]*Tan[e + f*x]^2)/(f*(a + a*Sec[e + f*x])^(3/2))","A",7,6,28,0.2143,1,"{3904, 3887, 470, 582, 522, 203}"
74,1,134,0,0.1917339,"\int \frac{(c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^{3/2}} \, dx","Int[(c - c*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^(3/2),x]","\frac{2 c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}-\frac{\sqrt{2} c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}-\frac{c^2 \sin (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{a f \sqrt{a \sec (e+f x)+a}}","\frac{2 c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}-\frac{\sqrt{2} c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}-\frac{2 c^2 \tan (e+f x)}{f (a \sec (e+f x)+a)^{3/2}}",1,"(2*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*f) - (Sqrt[2]*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(a^(3/2)*f) - (c^2*Sec[(e + f*x)/2]^2*Sin[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]])","A",7,6,28,0.2143,1,"{3904, 3887, 470, 12, 391, 203}"
75,1,130,0,0.1663902,"\int \frac{c-c \sec (e+f x)}{(a+a \sec (e+f x))^{3/2}} \, dx","Int[(c - c*Sec[e + f*x])/(a + a*Sec[e + f*x])^(3/2),x]","\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}-\frac{3 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{2} a^{3/2} f}-\frac{c \sin (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{2 a f \sqrt{a \sec (e+f x)+a}}","\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}-\frac{3 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{2} a^{3/2} f}-\frac{c \tan (e+f x)}{f (a \sec (e+f x)+a)^{3/2}}",1,"(2*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*f) - (3*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[2]*a^(3/2)*f) - (c*Sec[(e + f*x)/2]^2*Sin[e + f*x])/(2*a*f*Sqrt[a + a*Sec[e + f*x]])","A",6,5,26,0.1923,1,"{3904, 3887, 471, 522, 203}"
76,1,177,0,0.2444412,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))} \, dx","Int[1/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])),x]","\frac{\cot (e+f x) \sqrt{a \sec (e+f x)+a}}{4 a^2 c f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} c f}-\frac{7 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{4 \sqrt{2} a^{3/2} c f}+\frac{\cos (e+f x) \cot (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{a \sec (e+f x)+a}}{4 a^2 c f}","\frac{\cot (e+f x) \sqrt{a \sec (e+f x)+a}}{4 a^2 c f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} c f}-\frac{7 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{4 \sqrt{2} a^{3/2} c f}+\frac{\cos (e+f x) \cot (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{a \sec (e+f x)+a}}{4 a^2 c f}",1,"(2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*c*f) - (7*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(4*Sqrt[2]*a^(3/2)*c*f) + (Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(4*a^2*c*f) + (Cos[e + f*x]*Cot[e + f*x]*Sec[(e + f*x)/2]^2*Sqrt[a + a*Sec[e + f*x]])/(4*a^2*c*f)","A",7,6,28,0.2143,1,"{3904, 3887, 472, 583, 522, 203}"
77,1,214,0,0.2784581,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^2} \, dx","Int[1/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^2),x]","\frac{\cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{12 a^3 c^2 f}+\frac{7 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{8 a^2 c^2 f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} c^2 f}-\frac{9 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{8 \sqrt{2} a^{3/2} c^2 f}-\frac{\cos (e+f x) \cot ^3(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) (a \sec (e+f x)+a)^{3/2}}{4 a^3 c^2 f}","\frac{\cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{12 a^3 c^2 f}+\frac{7 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{8 a^2 c^2 f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} c^2 f}-\frac{9 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{8 \sqrt{2} a^{3/2} c^2 f}-\frac{\cos (e+f x) \cot ^3(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) (a \sec (e+f x)+a)^{3/2}}{4 a^3 c^2 f}",1,"(2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*c^2*f) - (9*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(8*Sqrt[2]*a^(3/2)*c^2*f) + (7*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(8*a^2*c^2*f) + (Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(12*a^3*c^2*f) - (Cos[e + f*x]*Cot[e + f*x]^3*Sec[(e + f*x)/2]^2*(a + a*Sec[e + f*x])^(3/2))/(4*a^3*c^2*f)","A",8,6,28,0.2143,1,"{3904, 3887, 472, 583, 522, 203}"
78,1,249,0,0.3399608,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^3} \, dx","Int[1/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^3),x]","-\frac{3 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{20 a^4 c^3 f}-\frac{5 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{24 a^3 c^3 f}+\frac{21 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{16 a^2 c^3 f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} c^3 f}-\frac{11 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{16 \sqrt{2} a^{3/2} c^3 f}+\frac{\cos (e+f x) \cot ^5(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) (a \sec (e+f x)+a)^{5/2}}{4 a^4 c^3 f}","-\frac{3 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{20 a^4 c^3 f}-\frac{5 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{24 a^3 c^3 f}+\frac{21 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{16 a^2 c^3 f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} c^3 f}-\frac{11 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{16 \sqrt{2} a^{3/2} c^3 f}+\frac{\cos (e+f x) \cot ^5(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) (a \sec (e+f x)+a)^{5/2}}{4 a^4 c^3 f}",1,"(2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*c^3*f) - (11*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(16*Sqrt[2]*a^(3/2)*c^3*f) + (21*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(16*a^2*c^3*f) - (5*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(24*a^3*c^3*f) - (3*Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(20*a^4*c^3*f) + (Cos[e + f*x]*Cot[e + f*x]^5*Sec[(e + f*x)/2]^2*(a + a*Sec[e + f*x])^(5/2))/(4*a^4*c^3*f)","A",9,6,28,0.2143,1,"{3904, 3887, 472, 583, 522, 203}"
79,1,260,0,0.3414866,"\int \frac{(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^{5/2}} \, dx","Int[(c - c*Sec[e + f*x])^5/(a + a*Sec[e + f*x])^(5/2),x]","\frac{2 c^5 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{23 \sqrt{2} c^5 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}+\frac{21 c^5 \tan (e+f x)}{a^2 f \sqrt{a \sec (e+f x)+a}}-\frac{19 c^5 \tan ^3(e+f x)}{6 a f (a \sec (e+f x)+a)^{3/2}}+\frac{a c^5 \sin ^2(e+f x) \tan ^5(e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)}{4 f (a \sec (e+f x)+a)^{7/2}}+\frac{3 c^5 \sin (e+f x) \tan ^4(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{4 f (a \sec (e+f x)+a)^{5/2}}","\frac{2 c^5 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{23 \sqrt{2} c^5 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}+\frac{21 c^5 \tan (e+f x)}{a^2 f \sqrt{a \sec (e+f x)+a}}-\frac{19 c^5 \tan ^3(e+f x)}{6 a f (a \sec (e+f x)+a)^{3/2}}+\frac{a c^5 \sin ^2(e+f x) \tan ^5(e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)}{4 f (a \sec (e+f x)+a)^{7/2}}+\frac{3 c^5 \sin (e+f x) \tan ^4(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{4 f (a \sec (e+f x)+a)^{5/2}}",1,"(2*c^5*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*f) - (23*Sqrt[2]*c^5*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(a^(5/2)*f) + (21*c^5*Tan[e + f*x])/(a^2*f*Sqrt[a + a*Sec[e + f*x]]) - (19*c^5*Tan[e + f*x]^3)/(6*a*f*(a + a*Sec[e + f*x])^(3/2)) + (3*c^5*Sec[(e + f*x)/2]^2*Sin[e + f*x]*Tan[e + f*x]^4)/(4*f*(a + a*Sec[e + f*x])^(5/2)) + (a*c^5*Sec[(e + f*x)/2]^4*Sin[e + f*x]^2*Tan[e + f*x]^5)/(4*f*(a + a*Sec[e + f*x])^(7/2))","A",9,7,28,0.2500,1,"{3904, 3887, 470, 578, 582, 522, 203}"
80,1,229,0,0.2959974,"\int \frac{(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^{5/2}} \, dx","Int[(c - c*Sec[e + f*x])^4/(a + a*Sec[e + f*x])^(5/2),x]","\frac{2 c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{11 c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{2} a^{5/2} f}+\frac{7 c^4 \tan (e+f x)}{2 a^2 f \sqrt{a \sec (e+f x)+a}}-\frac{c^4 \sin ^2(e+f x) \tan ^3(e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)}{4 f (a \sec (e+f x)+a)^{5/2}}-\frac{c^4 \sin (e+f x) \tan ^2(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{4 a f (a \sec (e+f x)+a)^{3/2}}","\frac{2 c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{11 c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{2} a^{5/2} f}+\frac{7 c^4 \tan (e+f x)}{2 a^2 f \sqrt{a \sec (e+f x)+a}}-\frac{c^4 \sin ^2(e+f x) \tan ^3(e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)}{4 f (a \sec (e+f x)+a)^{5/2}}-\frac{c^4 \sin (e+f x) \tan ^2(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{4 a f (a \sec (e+f x)+a)^{3/2}}",1,"(2*c^4*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*f) - (11*c^4*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[2]*a^(5/2)*f) + (7*c^4*Tan[e + f*x])/(2*a^2*f*Sqrt[a + a*Sec[e + f*x]]) - (c^4*Sec[(e + f*x)/2]^2*Sin[e + f*x]*Tan[e + f*x]^2)/(4*a*f*(a + a*Sec[e + f*x])^(3/2)) - (c^4*Sec[(e + f*x)/2]^4*Sin[e + f*x]^2*Tan[e + f*x]^3)/(4*f*(a + a*Sec[e + f*x])^(5/2))","A",8,7,28,0.2500,1,"{3904, 3887, 470, 578, 582, 522, 203}"
81,1,191,0,0.2520546,"\int \frac{(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^{5/2}} \, dx","Int[(c - c*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^(5/2),x]","-\frac{c^3 \sin (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{4 a^2 f \sqrt{a \sec (e+f x)+a}}+\frac{2 c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{7 c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{2 \sqrt{2} a^{5/2} f}+\frac{c^3 \sin ^2(e+f x) \tan (e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)}{4 a f (a \sec (e+f x)+a)^{3/2}}","-\frac{c^3 \sin (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{4 a^2 f \sqrt{a \sec (e+f x)+a}}+\frac{2 c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{7 c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{2 \sqrt{2} a^{5/2} f}+\frac{c^3 \sin ^2(e+f x) \tan (e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)}{4 a f (a \sec (e+f x)+a)^{3/2}}",1,"(2*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*f) - (7*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(2*Sqrt[2]*a^(5/2)*f) - (c^3*Sec[(e + f*x)/2]^2*Sin[e + f*x])/(4*a^2*f*Sqrt[a + a*Sec[e + f*x]]) + (c^3*Sec[(e + f*x)/2]^4*Sin[e + f*x]^2*Tan[e + f*x])/(4*a*f*(a + a*Sec[e + f*x])^(3/2))","A",7,6,28,0.2143,1,"{3904, 3887, 470, 578, 522, 203}"
82,1,189,0,0.2295775,"\int \frac{(c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx","Int[(c - c*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^(5/2),x]","-\frac{3 c^2 \sin (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 a^2 f \sqrt{a \sec (e+f x)+a}}+\frac{2 c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{11 c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{4 \sqrt{2} a^{5/2} f}-\frac{c^2 \sin (e+f x) \cos (e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)}{4 a^2 f \sqrt{a \sec (e+f x)+a}}","-\frac{3 c^2 \sin (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 a^2 f \sqrt{a \sec (e+f x)+a}}+\frac{2 c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{11 c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{4 \sqrt{2} a^{5/2} f}-\frac{c^2 \sin (e+f x) \cos (e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)}{4 a^2 f \sqrt{a \sec (e+f x)+a}}",1,"(2*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*f) - (11*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(4*Sqrt[2]*a^(5/2)*f) - (3*c^2*Sec[(e + f*x)/2]^2*Sin[e + f*x])/(8*a^2*f*Sqrt[a + a*Sec[e + f*x]]) - (c^2*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Sin[e + f*x])/(4*a^2*f*Sqrt[a + a*Sec[e + f*x]])","A",7,6,28,0.2143,1,"{3904, 3887, 470, 527, 522, 203}"
83,1,181,0,0.1915759,"\int \frac{c-c \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx","Int[(c - c*Sec[e + f*x])/(a + a*Sec[e + f*x])^(5/2),x]","-\frac{7 c \sin (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{16 a^2 f \sqrt{a \sec (e+f x)+a}}+\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{23 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{8 \sqrt{2} a^{5/2} f}-\frac{c \sin (e+f x) \cos (e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)}{8 a^2 f \sqrt{a \sec (e+f x)+a}}","\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{23 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{8 \sqrt{2} a^{5/2} f}-\frac{7 c \tan (e+f x)}{8 a f (a \sec (e+f x)+a)^{3/2}}-\frac{c \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{5/2}}",1,"(2*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*f) - (23*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(8*Sqrt[2]*a^(5/2)*f) - (7*c*Sec[(e + f*x)/2]^2*Sin[e + f*x])/(16*a^2*f*Sqrt[a + a*Sec[e + f*x]]) - (c*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Sin[e + f*x])/(8*a^2*f*Sqrt[a + a*Sec[e + f*x]])","A",7,6,26,0.2308,1,"{3904, 3887, 471, 527, 522, 203}"
84,1,230,0,0.3064513,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))} \, dx","Int[1/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])),x]","-\frac{7 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{32 a^3 c f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} c f}-\frac{71 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{32 \sqrt{2} a^{5/2} c f}+\frac{\cos ^2(e+f x) \cot (e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right) \sqrt{a \sec (e+f x)+a}}{16 a^3 c f}+\frac{13 \cos (e+f x) \cot (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{a \sec (e+f x)+a}}{32 a^3 c f}","-\frac{7 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{32 a^3 c f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} c f}-\frac{71 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{32 \sqrt{2} a^{5/2} c f}+\frac{\cos ^2(e+f x) \cot (e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right) \sqrt{a \sec (e+f x)+a}}{16 a^3 c f}+\frac{13 \cos (e+f x) \cot (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{a \sec (e+f x)+a}}{32 a^3 c f}",1,"(2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*c*f) - (71*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(32*Sqrt[2]*a^(5/2)*c*f) - (7*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(32*a^3*c*f) + (13*Cos[e + f*x]*Cot[e + f*x]*Sec[(e + f*x)/2]^2*Sqrt[a + a*Sec[e + f*x]])/(32*a^3*c*f) + (Cos[e + f*x]^2*Cot[e + f*x]*Sec[(e + f*x)/2]^4*Sqrt[a + a*Sec[e + f*x]])/(16*a^3*c*f)","A",8,7,28,0.2500,1,"{3904, 3887, 472, 579, 583, 522, 203}"
85,1,269,0,0.3352346,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^2} \, dx","Int[1/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^2),x]","\frac{43 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{96 a^4 c^2 f}+\frac{21 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{64 a^3 c^2 f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} c^2 f}-\frac{107 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{64 \sqrt{2} a^{5/2} c^2 f}-\frac{\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right) (a \sec (e+f x)+a)^{3/2}}{16 a^4 c^2 f}-\frac{15 \cos (e+f x) \cot ^3(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) (a \sec (e+f x)+a)^{3/2}}{32 a^4 c^2 f}","\frac{43 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{96 a^4 c^2 f}+\frac{21 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{64 a^3 c^2 f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} c^2 f}-\frac{107 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{64 \sqrt{2} a^{5/2} c^2 f}-\frac{\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right) (a \sec (e+f x)+a)^{3/2}}{16 a^4 c^2 f}-\frac{15 \cos (e+f x) \cot ^3(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) (a \sec (e+f x)+a)^{3/2}}{32 a^4 c^2 f}",1,"(2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*c^2*f) - (107*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(64*Sqrt[2]*a^(5/2)*c^2*f) + (21*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(64*a^3*c^2*f) + (43*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(96*a^4*c^2*f) - (15*Cos[e + f*x]*Cot[e + f*x]^3*Sec[(e + f*x)/2]^2*(a + a*Sec[e + f*x])^(3/2))/(32*a^4*c^2*f) - (Cos[e + f*x]^2*Cot[e + f*x]^3*Sec[(e + f*x)/2]^4*(a + a*Sec[e + f*x])^(3/2))/(16*a^4*c^2*f)","A",9,7,28,0.2500,1,"{3904, 3887, 472, 579, 583, 522, 203}"
86,1,185,0,0.3723234,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2),x]","-\frac{a c^3 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}-\frac{a c^2 \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}}+\frac{a c^4 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt{a \sec (e+f x)+a}}","-\frac{a c^3 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}-\frac{a c^2 \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}}+\frac{a c^4 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt{a \sec (e+f x)+a}}",1,"(a*c^4*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a*c^3*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (a*c^2*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*Sqrt[a + a*Sec[e + f*x]]) - (a*c*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]])","A",5,3,30,0.1000,1,"{3906, 3905, 3475}"
87,1,139,0,0.2728153,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2),x]","-\frac{a c^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}+\frac{a c^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}}","-\frac{a c^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}+\frac{a c^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}}",1,"(a*c^3*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a*c^2*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (a*c*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*Sqrt[a + a*Sec[e + f*x]])","A",4,3,30,0.1000,1,"{3906, 3905, 3475}"
88,1,93,0,0.1783255,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2),x]","\frac{a c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}","\frac{a c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}",1,"(a*c^2*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a*c*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])","A",3,3,30,0.1000,1,"{3906, 3905, 3475}"
89,1,48,0,0.0838858,"\int \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]],x]","\frac{a c \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","\frac{a c \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(a*c*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",2,2,30,0.06667,1,"{3905, 3475}"
90,1,51,0,0.0863869,"\int \frac{\sqrt{a+a \sec (e+f x)}}{\sqrt{c-c \sec (e+f x)}} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]/Sqrt[c - c*Sec[e + f*x]],x]","\frac{a \tan (e+f x) \log (1-\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","\frac{a \tan (e+f x) \log (1-\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(a*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",2,2,30,0.06667,1,"{3911, 31}"
91,1,96,0,0.1786922,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^(3/2),x]","\frac{a \tan (e+f x) \log (1-\cos (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)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}","\frac{a \tan (e+f x) \log (1-\cos (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)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}",1,"-((a*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2))) + (a*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,3,30,0.1000,1,"{3907, 3911, 31}"
92,1,142,0,0.2711433,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^(5/2),x]","\frac{a \tan (e+f x) \log (1-\cos (e+f x))}{c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a \tan (e+f x)}{c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{a \tan (e+f x)}{2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}","\frac{a \tan (e+f x) \log (1-\cos (e+f x))}{c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a \tan (e+f x)}{c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{a \tan (e+f x)}{2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}",1,"-(a*Tan[e + f*x])/(2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)) - (a*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",4,3,30,0.1000,1,"{3907, 3911, 31}"
93,1,188,0,0.366465,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{7/2}} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^(7/2),x]","-\frac{a \tan (e+f x)}{c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}+\frac{a \tan (e+f x) \log (1-\cos (e+f x))}{c^3 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a \tan (e+f x)}{2 c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac{a \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}","-\frac{a \tan (e+f x)}{c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}+\frac{a \tan (e+f x) \log (1-\cos (e+f x))}{c^3 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a \tan (e+f x)}{2 c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac{a \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}",1,"-(a*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2)) - (a*Tan[e + f*x])/(2*c*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)) - (a*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^3*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",5,3,30,0.1000,1,"{3907, 3911, 31}"
94,1,190,0,0.3639961,"\int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{5/2} \, dx","Int[(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(5/2),x]","-\frac{a^2 c^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}+\frac{a^2 c^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}}+\frac{a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt{a \sec (e+f x)+a}}","-\frac{a^2 c^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}+\frac{a^2 c^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}}+\frac{a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt{a \sec (e+f x)+a}}",1,"(a^2*c^3*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a^2*c^2*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (a^2*c*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*Sqrt[a + a*Sec[e + f*x]]) + (a^2*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]])","A",5,4,30,0.1333,1,"{3909, 3906, 3905, 3475}"
95,1,103,0,0.1095071,"\int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx","Int[(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(3/2),x]","\frac{a^2 c^2 \tan ^3(e+f x)}{2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{a^2 c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","\frac{a^2 c^2 \tan ^3(e+f x)}{2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{a^2 c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(a^2*c^2*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (a^2*c^2*Tan[e + f*x]^3)/(2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,3,30,0.1000,1,"{3905, 3473, 3475}"
96,1,93,0,0.169297,"\int (a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)} \, dx","Int[(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]],x]","\frac{a^2 c \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f \sqrt{c-c \sec (e+f x)}}","\frac{a^2 c \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f \sqrt{c-c \sec (e+f x)}}",1,"(a^2*c*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a*c*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]])","A",3,3,30,0.1000,1,"{3906, 3905, 3475}"
97,1,104,0,0.1033202,"\int \frac{(a+a \sec (e+f x))^{3/2}}{\sqrt{c-c \sec (e+f x)}} \, dx","Int[(a + a*Sec[e + f*x])^(3/2)/Sqrt[c - c*Sec[e + f*x]],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^2 \tan (e+f x) \log (\cos (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) \log (1-\sec (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{a^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(a^2*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (2*a^2*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,2,30,0.06667,1,"{3912, 72}"
98,1,100,0,0.1855994,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{3/2}} \, dx","Int[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^(3/2),x]","\frac{a^2 \tan (e+f x) \log (1-\cos (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)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}","\frac{a^2 \tan (e+f x) \log (1-\cos (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)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}",1,"(-2*a^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a^2*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,3,30,0.1000,1,"{3908, 3911, 31}"
99,1,146,0,0.2828222,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx","Int[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^(5/2),x]","\frac{a^2 \tan (e+f x) \log (1-\cos (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)}{c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{a^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}","\frac{a^2 \tan (e+f x) \log (1-\cos (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)}{c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{a^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}",1,"-((a^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2))) - (a^2*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a^2*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",4,4,30,0.1333,1,"{3908, 3907, 3911, 31}"
100,1,196,0,0.3824365,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{7/2}} \, dx","Int[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^(7/2),x]","-\frac{a^2 \tan (e+f x)}{c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}+\frac{a^2 \tan (e+f x) \log (1-\cos (e+f x))}{c^3 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^2 \tan (e+f x)}{2 c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac{2 a^2 \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}","-\frac{a^2 \tan (e+f x)}{c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}+\frac{a^2 \tan (e+f x) \log (1-\cos (e+f x))}{c^3 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^2 \tan (e+f x)}{2 c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac{2 a^2 \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}",1,"(-2*a^2*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2)) - (a^2*Tan[e + f*x])/(2*c*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)) - (a^2*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a^2*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^3*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",5,4,30,0.1333,1,"{3908, 3907, 3911, 31}"
101,1,153,0,0.1201687,"\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{5/2} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(5/2),x]","-\frac{a^3 c^3 \tan ^5(e+f x)}{4 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{a^3 c^3 \tan ^3(e+f x)}{2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{a^3 c^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","-\frac{a^3 c^3 \tan ^5(e+f x)}{4 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{a^3 c^3 \tan ^3(e+f x)}{2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{a^3 c^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(a^3*c^3*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (a^3*c^3*Tan[e + f*x]^3)/(2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a^3*c^3*Tan[e + f*x]^5)/(4*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",4,3,30,0.1000,1,"{3905, 3473, 3475}"
102,1,190,0,0.3628384,"\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2),x]","-\frac{a^2 c^2 \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f \sqrt{c-c \sec (e+f x)}}+\frac{a^3 c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c^2 \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sec (e+f x)}}+\frac{c^2 \tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sec (e+f x)}}","-\frac{a^2 c^2 \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f \sqrt{c-c \sec (e+f x)}}+\frac{a^3 c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c^2 \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sec (e+f x)}}+\frac{c^2 \tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sec (e+f x)}}",1,"(a^3*c^2*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a^2*c^2*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]]) - (a*c^2*(a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*Sqrt[c - c*Sec[e + f*x]]) + (c^2*(a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*f*Sqrt[c - c*Sec[e + f*x]])","A",5,4,30,0.1333,1,"{3909, 3906, 3905, 3475}"
103,1,139,0,0.2628026,"\int (a+a \sec (e+f x))^{5/2} \sqrt{c-c \sec (e+f x)} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]],x]","-\frac{a^2 c \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f \sqrt{c-c \sec (e+f x)}}+\frac{a^3 c \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sec (e+f x)}}","-\frac{a^2 c \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f \sqrt{c-c \sec (e+f x)}}+\frac{a^3 c \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sec (e+f x)}}",1,"(a^3*c*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a^2*c*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]]) - (a*c*(a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*Sqrt[c - c*Sec[e + f*x]])","A",4,3,30,0.1000,1,"{3906, 3905, 3475}"
104,1,152,0,0.1137347,"\int \frac{(a+a \sec (e+f x))^{5/2}}{\sqrt{c-c \sec (e+f x)}} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)/Sqrt[c - c*Sec[e + f*x]],x]","\frac{a^3 \tan (e+f x) \sec (e+f x)}{f \sqrt{a \sec (e+f x)+a} \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{a^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","\frac{a^3 \tan (e+f x) \sec (e+f x)}{f \sqrt{a \sec (e+f x)+a} \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{a^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(a^3*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (4*a^3*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (a^3*Sec[e + f*x]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,2,30,0.06667,1,"{3912, 72}"
105,1,96,0,0.1792632,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{3/2}} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(3/2),x]","\frac{a^3 \tan (e+f x) \log (\cos (e+f x))}{c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 a^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}","\frac{a^3 \tan (e+f x) \log (\cos (e+f x))}{c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 a^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}",1,"(-4*a^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a^3*Log[Cos[e + f*x]]*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,3,30,0.1000,1,"{3910, 3905, 3475}"
106,1,100,0,0.1807794,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{5/2}} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(5/2),x]","\frac{a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{2 a^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}","\frac{a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{2 a^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}",1,"(-2*a^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)) + (a^3*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,3,30,0.1000,1,"{3910, 3911, 31}"
107,1,148,0,0.2781287,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{7/2}} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(7/2),x]","-\frac{a^3 \tan (e+f x)}{c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}+\frac{a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^3 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 a^3 \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}","-\frac{a^3 \tan (e+f x)}{c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}+\frac{a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^3 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 a^3 \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}",1,"(-4*a^3*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2)) - (a^3*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a^3*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^3*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",4,4,30,0.1333,1,"{3910, 3907, 3911, 31}"
108,1,194,0,0.3755362,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{9/2}} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(9/2),x]","-\frac{a^3 \tan (e+f x)}{c^3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{a^3 \tan (e+f x)}{2 c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}+\frac{a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^4 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{9/2}}","-\frac{a^3 \tan (e+f x)}{c^3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{a^3 \tan (e+f x)}{2 c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}+\frac{a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^4 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{9/2}}",1,"-((a^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(9/2))) - (a^3*Tan[e + f*x])/(2*c^2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)) - (a^3*Tan[e + f*x])/(c^3*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a^3*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^4*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",5,4,30,0.1333,1,"{3910, 3907, 3911, 31}"
109,1,244,0,0.4741198,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(11/2),x]","-\frac{a^3 \tan (e+f x)}{c^4 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{a^3 \tan (e+f x)}{2 c^3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac{a^3 \tan (e+f x)}{3 c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}+\frac{a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^5 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 a^3 \tan (e+f x)}{5 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{11/2}}","-\frac{a^3 \tan (e+f x)}{c^4 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{a^3 \tan (e+f x)}{2 c^3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac{a^3 \tan (e+f x)}{3 c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}+\frac{a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^5 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 a^3 \tan (e+f x)}{5 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{11/2}}",1,"(-4*a^3*Tan[e + f*x])/(5*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(11/2)) - (a^3*Tan[e + f*x])/(3*c^2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2)) - (a^3*Tan[e + f*x])/(2*c^3*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)) - (a^3*Tan[e + f*x])/(c^4*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a^3*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^5*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",6,4,30,0.1333,1,"{3910, 3907, 3911, 31}"
110,1,204,0,0.1235924,"\int \frac{(c-c \sec (e+f x))^{7/2}}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[(c - c*Sec[e + f*x])^(7/2)/Sqrt[a + a*Sec[e + f*x]],x]","\frac{c^4 \tan (e+f x) \sec ^2(e+f x)}{2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 c^4 \tan (e+f x) \sec (e+f x)}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{8 c^4 \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^4 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","\frac{c^4 \tan (e+f x) \sec ^2(e+f x)}{2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 c^4 \tan (e+f x) \sec (e+f x)}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{8 c^4 \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^4 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(c^4*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (8*c^4*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (4*c^4*Sec[e + f*x]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (c^4*Sec[e + f*x]^2*Tan[e + f*x])/(2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,2,30,0.06667,1,"{3912, 72}"
111,1,151,0,0.1134596,"\int \frac{(c-c \sec (e+f x))^{5/2}}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[(c - c*Sec[e + f*x])^(5/2)/Sqrt[a + a*Sec[e + f*x]],x]","-\frac{c^3 \tan (e+f x) \sec (e+f x)}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\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{c^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","-\frac{c^3 \tan (e+f x) \sec (e+f x)}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\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{c^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(c^3*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (4*c^3*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (c^3*Sec[e + f*x]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,2,30,0.06667,1,"{3912, 72}"
112,1,102,0,0.1068631,"\int \frac{(c-c \sec (e+f x))^{3/2}}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[(c - c*Sec[e + f*x])^(3/2)/Sqrt[a + a*Sec[e + f*x]],x]","\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^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","\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^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(c^2*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (2*c^2*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,2,30,0.06667,1,"{3912, 72}"
113,1,49,0,0.084185,"\int \frac{\sqrt{c-c \sec (e+f x)}}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[Sqrt[c - c*Sec[e + f*x]]/Sqrt[a + a*Sec[e + f*x]],x]","\frac{c \tan (e+f x) \log (\cos (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) \log (\cos (e+f x)+1)}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(c*Log[1 + Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",2,2,30,0.06667,1,"{3911, 31}"
114,1,46,0,0.0896478,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}} \, dx","Int[1/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]),x]","\frac{\tan (e+f x) \log (\sin (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","\frac{\tan (e+f x) \log (\sin (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(Log[Sin[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",2,2,30,0.06667,1,"{3905, 3475}"
115,1,217,0,0.136506,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}} \, dx","Int[1/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)),x]","-\frac{\tan (e+f x)}{2 c f (1-\sec (e+f x)) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{3 \tan (e+f x) \log (1-\sec (e+f x))}{4 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\sec (e+f x)+1)}{4 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","\frac{\tan (e+f x)}{2 c f (1-\cos (e+f x)) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{3 \tan (e+f x) \log (1-\cos (e+f x))}{4 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x)+1)}{4 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(Log[Cos[e + f*x]]*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (3*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(4*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(4*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(2*c*f*(1 - Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,2,30,0.06667,1,"{3912, 72}"
116,1,274,0,0.1609438,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}} \, dx","Int[1/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)),x]","-\frac{3 \tan (e+f x)}{4 c^2 f (1-\sec (e+f x)) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{4 c^2 f (1-\sec (e+f x))^2 \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{7 \tan (e+f x) \log (1-\sec (e+f x))}{8 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\sec (e+f x)+1)}{8 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","-\frac{3 \tan (e+f x)}{4 c^2 f (1-\sec (e+f x)) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{4 c^2 f (1-\sec (e+f x))^2 \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{7 \tan (e+f x) \log (1-\sec (e+f x))}{8 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\sec (e+f x)+1)}{8 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(Log[Cos[e + f*x]]*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (7*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(8*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(8*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(4*c^2*f*(1 - Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (3*Tan[e + f*x])/(4*c^2*f*(1 - Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,2,30,0.06667,1,"{3912, 72}"
117,1,215,0,0.1449476,"\int \frac{(c-c \sec (e+f x))^{7/2}}{(a+a \sec (e+f x))^{3/2}} \, dx","Int[(c - c*Sec[e + f*x])^(7/2)/(a + a*Sec[e + f*x])^(3/2),x]","\frac{c^4 \tan (e+f x) \sec (e+f x)}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{8 c^4 \tan (e+f x)}{a f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 c^4 \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^4 \tan (e+f x) \log (\cos (e+f x))}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","\frac{c^4 \tan (e+f x) \sec (e+f x)}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{8 c^4 \tan (e+f x)}{a f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 c^4 \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^4 \tan (e+f x) \log (\cos (e+f x))}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(c^4*Log[Cos[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (4*c^4*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (c^4*Sec[e + f*x]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (8*c^4*Tan[e + f*x])/(a*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,2,30,0.06667,1,"{3912, 88}"
118,1,96,0,0.193182,"\int \frac{(c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^{3/2}} \, dx","Int[(c - c*Sec[e + f*x])^(5/2)/(a + a*Sec[e + f*x])^(3/2),x]","\frac{c^3 \tan (e+f x) \log (\cos (e+f x))}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 c^3 \tan (e+f x)}{f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}","\frac{c^3 \tan (e+f x) \log (\cos (e+f x))}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 c^3 \tan (e+f x)}{f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}",1,"(-4*c^3*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]]) + (c^3*Log[Cos[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,3,30,0.1000,1,"{3910, 3905, 3475}"
119,1,98,0,0.1957619,"\int \frac{(c-c \sec (e+f x))^{3/2}}{(a+a \sec (e+f x))^{3/2}} \, dx","Int[(c - c*Sec[e + f*x])^(3/2)/(a + a*Sec[e + f*x])^(3/2),x]","\frac{c^2 \tan (e+f x) \log (\cos (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)}{f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}","\frac{c^2 \tan (e+f x) \log (\cos (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)}{f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}",1,"(-2*c^2*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]]) + (c^2*Log[1 + Cos[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,3,30,0.1000,1,"{3908, 3911, 31}"
120,1,94,0,0.1894194,"\int \frac{\sqrt{c-c \sec (e+f x)}}{(a+a \sec (e+f x))^{3/2}} \, dx","Int[Sqrt[c - c*Sec[e + f*x]]/(a + a*Sec[e + f*x])^(3/2),x]","\frac{c \tan (e+f x) \log (\cos (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)}{f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}","\frac{c \tan (e+f x) \log (\cos (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)}{f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}",1,"-((c*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]])) + (c*Log[1 + Cos[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,3,30,0.1000,1,"{3907, 3911, 31}"
121,1,215,0,0.1461462,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)}} \, dx","Int[1/((a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]]),x]","-\frac{\tan (e+f x)}{2 a f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (1-\sec (e+f x))}{4 a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{3 \tan (e+f x) \log (\sec (e+f x)+1)}{4 a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","-\frac{\tan (e+f x)}{2 a f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (1-\sec (e+f x))}{4 a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{3 \tan (e+f x) \log (\sec (e+f x)+1)}{4 a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(Log[Cos[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(4*a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (3*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(4*a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(2*a*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,2,30,0.06667,1,"{3912, 72}"
122,1,101,0,0.1214614,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2}} \, dx","Int[1/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(3/2)),x]","\frac{\cot (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) \log (\sin (e+f x))}{a c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","\frac{\cot (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) \log (\sin (e+f x))}{a c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"Cot[e + f*x]/(2*a*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (Log[Sin[e + f*x]]*Tan[e + f*x])/(a*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,3,30,0.1000,1,"{3905, 3473, 3475}"
123,1,347,0,0.1839146,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{5/2}} \, dx","Int[1/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(5/2)),x]","-\frac{\tan (e+f x)}{2 a c^2 f (1-\sec (e+f x)) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{8 a c^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{8 a c^2 f (1-\sec (e+f x))^2 \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{11 \tan (e+f x) \log (1-\sec (e+f x))}{16 a c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{5 \tan (e+f x) \log (\sec (e+f x)+1)}{16 a c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{a c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","-\frac{\tan (e+f x)}{2 a c^2 f (1-\sec (e+f x)) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{8 a c^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{8 a c^2 f (1-\sec (e+f x))^2 \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{11 \tan (e+f x) \log (1-\sec (e+f x))}{16 a c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{5 \tan (e+f x) \log (\sec (e+f x)+1)}{16 a c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{a c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(Log[Cos[e + f*x]]*Tan[e + f*x])/(a*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (11*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(16*a*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (5*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(16*a*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(8*a*c^2*f*(1 - Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(2*a*c^2*f*(1 - Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(8*a*c^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,2,30,0.06667,1,"{3912, 88}"
124,1,220,0,0.1438065,"\int \frac{(c-c \sec (e+f x))^{7/2}}{(a+a \sec (e+f x))^{5/2}} \, dx","Int[(c - c*Sec[e + f*x])^(7/2)/(a + a*Sec[e + f*x])^(5/2),x]","\frac{4 c^4 \tan (e+f x)}{a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 c^4 \tan (e+f x)}{a^2 f (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{2 c^4 \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^4 \tan (e+f x) \log (\cos (e+f x))}{a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","\frac{4 c^4 \tan (e+f x)}{a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 c^4 \tan (e+f x)}{a^2 f (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{2 c^4 \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^4 \tan (e+f x) \log (\cos (e+f x))}{a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(c^4*Log[Cos[e + f*x]]*Tan[e + f*x])/(a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (2*c^4*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (4*c^4*Tan[e + f*x])/(a^2*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (4*c^4*Tan[e + f*x])/(a^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,2,30,0.06667,1,"{3912, 88}"
125,1,98,0,0.1871877,"\int \frac{(c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^{5/2}} \, dx","Int[(c - c*Sec[e + f*x])^(5/2)/(a + a*Sec[e + f*x])^(5/2),x]","\frac{c^3 \tan (e+f x) \log (\cos (e+f x)+1)}{a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{2 c^3 \tan (e+f x)}{f (a \sec (e+f x)+a)^{5/2} \sqrt{c-c \sec (e+f x)}}","\frac{c^3 \tan (e+f x) \log (\cos (e+f x)+1)}{a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{2 c^3 \tan (e+f x)}{f (a \sec (e+f x)+a)^{5/2} \sqrt{c-c \sec (e+f x)}}",1,"(-2*c^3*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]]) + (c^3*Log[1 + Cos[e + f*x]]*Tan[e + f*x])/(a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,3,30,0.1000,1,"{3910, 3911, 31}"
126,1,144,0,0.2877506,"\int \frac{(c-c \sec (e+f x))^{3/2}}{(a+a \sec (e+f x))^{5/2}} \, dx","Int[(c - c*Sec[e + f*x])^(3/2)/(a + a*Sec[e + f*x])^(5/2),x]","\frac{c^2 \tan (e+f x) \log (\cos (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)}{a f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}-\frac{c^2 \tan (e+f x)}{f (a \sec (e+f x)+a)^{5/2} \sqrt{c-c \sec (e+f x)}}","\frac{c^2 \tan (e+f x) \log (\cos (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)}{a f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}-\frac{c^2 \tan (e+f x)}{f (a \sec (e+f x)+a)^{5/2} \sqrt{c-c \sec (e+f x)}}",1,"-((c^2*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]])) - (c^2*Tan[e + f*x])/(a*f*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]]) + (c^2*Log[1 + Cos[e + f*x]]*Tan[e + f*x])/(a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",4,4,30,0.1333,1,"{3908, 3907, 3911, 31}"
127,1,140,0,0.282233,"\int \frac{\sqrt{c-c \sec (e+f x)}}{(a+a \sec (e+f x))^{5/2}} \, dx","Int[Sqrt[c - c*Sec[e + f*x]]/(a + a*Sec[e + f*x])^(5/2),x]","\frac{c \tan (e+f x) \log (\cos (e+f x)+1)}{a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{c \tan (e+f x)}{a f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}-\frac{c \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{5/2} \sqrt{c-c \sec (e+f x)}}","\frac{c \tan (e+f x) \log (\cos (e+f x)+1)}{a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{c \tan (e+f x)}{a f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}-\frac{c \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{5/2} \sqrt{c-c \sec (e+f x)}}",1,"-(c*Tan[e + f*x])/(2*f*(a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]]) - (c*Tan[e + f*x])/(a*f*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]]) + (c*Log[1 + Cos[e + f*x]]*Tan[e + f*x])/(a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",4,3,30,0.1000,1,"{3907, 3911, 31}"
128,1,270,0,0.1535839,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} \sqrt{c-c \sec (e+f x)}} \, dx","Int[1/((a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]]),x]","-\frac{3 \tan (e+f x)}{4 a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{4 a^2 f (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (1-\sec (e+f x))}{8 a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{7 \tan (e+f x) \log (\sec (e+f x)+1)}{8 a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","-\frac{3 \tan (e+f x)}{4 a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{4 a^2 f (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (1-\sec (e+f x))}{8 a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{7 \tan (e+f x) \log (\sec (e+f x)+1)}{8 a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(Log[Cos[e + f*x]]*Tan[e + f*x])/(a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(8*a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (7*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(8*a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(4*a^2*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (3*Tan[e + f*x])/(4*a^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,2,30,0.06667,1,"{3912, 72}"
129,1,345,0,0.1808218,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2}} \, dx","Int[1/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2)),x]","-\frac{\tan (e+f x)}{8 a^2 c f (1-\sec (e+f x)) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{2 a^2 c f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{8 a^2 c f (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{5 \tan (e+f x) \log (1-\sec (e+f x))}{16 a^2 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{11 \tan (e+f x) \log (\sec (e+f x)+1)}{16 a^2 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{a^2 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","-\frac{\tan (e+f x)}{8 a^2 c f (1-\sec (e+f x)) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{2 a^2 c f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{8 a^2 c f (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{5 \tan (e+f x) \log (1-\sec (e+f x))}{16 a^2 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{11 \tan (e+f x) \log (\sec (e+f x)+1)}{16 a^2 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{a^2 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(Log[Cos[e + f*x]]*Tan[e + f*x])/(a^2*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (5*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(16*a^2*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (11*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(16*a^2*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(8*a^2*c*f*(1 - Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(8*a^2*c*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(2*a^2*c*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",3,2,30,0.06667,1,"{3912, 88}"
130,1,151,0,0.1311104,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{5/2}} \, dx","Int[1/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(5/2)),x]","-\frac{\cot ^3(e+f x)}{4 a^2 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\cot (e+f x)}{2 a^2 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\sin (e+f x))}{a^2 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}","-\frac{\cot ^3(e+f x)}{4 a^2 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\cot (e+f x)}{2 a^2 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\sin (e+f x))}{a^2 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"Cot[e + f*x]/(2*a^2*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Cot[e + f*x]^3/(4*a^2*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (Log[Sin[e + f*x]]*Tan[e + f*x])/(a^2*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])","A",4,3,30,0.1000,1,"{3905, 3473, 3475}"
131,1,92,0,0.0936537,"\int (1+\sec (e+f x))^m (c-c \sec (e+f x))^n \, dx","Int[(1 + Sec[e + f*x])^m*(c - c*Sec[e + f*x])^n,x]","\frac{2^{m+\frac{1}{2}} \tan (e+f x) (c-c \sec (e+f x))^n F_1\left(n+\frac{1}{2};\frac{1}{2}-m,1;n+\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{\sec (e+f x)+1}}","\frac{2^{m+\frac{1}{2}} \tan (e+f x) (c-c \sec (e+f x))^n F_1\left(n+\frac{1}{2};\frac{1}{2}-m,1;n+\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{\sec (e+f x)+1}}",1,"(2^(1/2 + m)*AppellF1[1/2 + n, 1/2 - m, 1, 3/2 + n, (1 - Sec[e + f*x])/2, 1 - Sec[e + f*x]]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[1 + Sec[e + f*x]])","A",2,2,24,0.08333,1,"{3912, 136}"
132,1,109,0,0.1216913,"\int (a+a \sec (e+f x))^m (c-c \sec (e+f x))^n \, dx","Int[(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^n,x]","\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} F_1\left(m+\frac{1}{2};\frac{1}{2}-n,1;m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right)}{f (2 m+1)}","\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} F_1\left(m+\frac{1}{2};\frac{1}{2}-n,1;m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right)}{f (2 m+1)}",1,"(2^(1/2 + n)*c*AppellF1[1/2 + m, 1/2 - n, 1, 3/2 + m, (1 + Sec[e + f*x])/2, 1 + Sec[e + f*x]]*(1 - Sec[e + f*x])^(1/2 - n)*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-1 + n)*Tan[e + f*x])/(f*(1 + 2*m))","A",3,3,26,0.1154,1,"{3912, 137, 136}"
133,1,101,0,0.1001258,"\int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^n \, dx","Int[(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^n,x]","\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (a \sec (e+f x)+a)^3 (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left(\frac{7}{2};\frac{1}{2}-n,1;\frac{9}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right) (c-c \sec (e+f x))^{n-1}}{7 f}","\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (a \sec (e+f x)+a)^3 (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left(\frac{7}{2};\frac{1}{2}-n,1;\frac{9}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right) (c-c \sec (e+f x))^{n-1}}{7 f}",1,"(2^(1/2 + n)*c*AppellF1[7/2, 1/2 - n, 1, 9/2, (1 + Sec[e + f*x])/2, 1 + Sec[e + f*x]]*(1 - Sec[e + f*x])^(1/2 - n)*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(-1 + n)*Tan[e + f*x])/(7*f)","A",3,3,26,0.1154,1,"{3912, 137, 136}"
134,1,101,0,0.0990131,"\int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^n \, dx","Int[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^n,x]","\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (a \sec (e+f x)+a)^2 (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left(\frac{5}{2};\frac{1}{2}-n,1;\frac{7}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right) (c-c \sec (e+f x))^{n-1}}{5 f}","\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (a \sec (e+f x)+a)^2 (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left(\frac{5}{2};\frac{1}{2}-n,1;\frac{7}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right) (c-c \sec (e+f x))^{n-1}}{5 f}",1,"(2^(1/2 + n)*c*AppellF1[5/2, 1/2 - n, 1, 7/2, (1 + Sec[e + f*x])/2, 1 + Sec[e + f*x]]*(1 - Sec[e + f*x])^(1/2 - n)*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(-1 + n)*Tan[e + f*x])/(5*f)","A",3,3,26,0.1154,1,"{3912, 137, 136}"
135,1,99,0,0.0754091,"\int (a+a \sec (e+f x)) (c-c \sec (e+f x))^n \, dx","Int[(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^n,x]","\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (a \sec (e+f x)+a) (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left(\frac{3}{2};\frac{1}{2}-n,1;\frac{5}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right) (c-c \sec (e+f x))^{n-1}}{3 f}","\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (a \sec (e+f x)+a) (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left(\frac{3}{2};\frac{1}{2}-n,1;\frac{5}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right) (c-c \sec (e+f x))^{n-1}}{3 f}",1,"(2^(1/2 + n)*c*AppellF1[3/2, 1/2 - n, 1, 5/2, (1 + Sec[e + f*x])/2, 1 + Sec[e + f*x]]*(1 - Sec[e + f*x])^(1/2 - n)*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(-1 + n)*Tan[e + f*x])/(3*f)","A",3,3,24,0.1250,1,"{3912, 137, 136}"
136,1,99,0,0.1003691,"\int \frac{(c-c \sec (e+f x))^n}{a+a \sec (e+f x)} \, dx","Int[(c - c*Sec[e + f*x])^n/(a + a*Sec[e + f*x]),x]","-\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left(-\frac{1}{2};\frac{1}{2}-n,1;\frac{1}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right) (c-c \sec (e+f x))^{n-1}}{f (a \sec (e+f x)+a)}","-\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left(-\frac{1}{2};\frac{1}{2}-n,1;\frac{1}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right) (c-c \sec (e+f x))^{n-1}}{f (a \sec (e+f x)+a)}",1,"-((2^(1/2 + n)*c*AppellF1[-1/2, 1/2 - n, 1, 1/2, (1 + Sec[e + f*x])/2, 1 + Sec[e + f*x]]*(1 - Sec[e + f*x])^(1/2 - n)*(c - c*Sec[e + f*x])^(-1 + n)*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])))","A",3,3,26,0.1154,1,"{3912, 137, 136}"
137,1,101,0,0.1002201,"\int \frac{(c-c \sec (e+f x))^n}{(a+a \sec (e+f x))^2} \, dx","Int[(c - c*Sec[e + f*x])^n/(a + a*Sec[e + f*x])^2,x]","-\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left(-\frac{3}{2};\frac{1}{2}-n,1;-\frac{1}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right) (c-c \sec (e+f x))^{n-1}}{3 f (a \sec (e+f x)+a)^2}","-\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left(-\frac{3}{2};\frac{1}{2}-n,1;-\frac{1}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right) (c-c \sec (e+f x))^{n-1}}{3 f (a \sec (e+f x)+a)^2}",1,"-(2^(1/2 + n)*c*AppellF1[-3/2, 1/2 - n, 1, -1/2, (1 + Sec[e + f*x])/2, 1 + Sec[e + f*x]]*(1 - Sec[e + f*x])^(1/2 - n)*(c - c*Sec[e + f*x])^(-1 + n)*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2)","A",3,3,26,0.1154,1,"{3912, 137, 136}"
138,1,172,0,0.1453384,"\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^n \, dx","Int[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^n,x]","\frac{2 a^3 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{6 a^3 \tan (e+f x) (c-c \sec (e+f x))^n}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}-\frac{2 a^3 \tan (e+f x) (c-c \sec (e+f x))^{n+1}}{c f (2 n+3) \sqrt{a \sec (e+f x)+a}}","\frac{2 a^3 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{6 a^3 \tan (e+f x) (c-c \sec (e+f x))^n}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}-\frac{2 a^3 \tan (e+f x) (c-c \sec (e+f x))^{n+1}}{c f (2 n+3) \sqrt{a \sec (e+f x)+a}}",1,"(6*a^3*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) + (2*a^3*Hypergeometric2F1[1, 1/2 + n, 3/2 + n, 1 - Sec[e + f*x]]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) - (2*a^3*(c - c*Sec[e + f*x])^(1 + n)*Tan[e + f*x])/(c*f*(3 + 2*n)*Sqrt[a + a*Sec[e + f*x]])","A",4,3,28,0.1071,1,"{3912, 88, 65}"
139,1,119,0,0.1620677,"\int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^n \, dx","Int[(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^n,x]","\frac{2 a^2 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (c-c \sec (e+f x))^n}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}","\frac{2 a^2 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (c-c \sec (e+f x))^n}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}",1,"(2*a^2*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*Hypergeometric2F1[1, 1/2 + n, 3/2 + n, 1 - Sec[e + f*x]]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]])","A",3,3,28,0.1071,1,"{3909, 3912, 65}"
140,1,68,0,0.0827766,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^n \, dx","Int[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^n,x]","\frac{2 a \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}","\frac{2 a \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}",1,"(2*a*Hypergeometric2F1[1, 1/2 + n, 3/2 + n, 1 - Sec[e + f*x]]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]])","A",2,2,28,0.07143,1,"{3912, 65}"
141,1,139,0,0.1120584,"\int \frac{(c-c \sec (e+f x))^n}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[(c - c*Sec[e + f*x])^n/Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}","\frac{2 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}",1,"-((Hypergeometric2F1[1, 1/2 + n, 3/2 + n, (1 - Sec[e + f*x])/2]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]])) + (2*Hypergeometric2F1[1, 1/2 + n, 3/2 + n, 1 - Sec[e + f*x]]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]])","A",4,4,28,0.1429,1,"{3912, 86, 65, 68}"
142,1,205,0,0.1678741,"\int \frac{(c-c \sec (e+f x))^n}{(a+a \sec (e+f x))^{3/2}} \, dx","Int[(c - c*Sec[e + f*x])^n/(a + a*Sec[e + f*x])^(3/2),x]","-\frac{(5-2 n) \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{4 a f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right)}{a f (2 n+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x) (c-c \sec (e+f x))^n}{2 a f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}","-\frac{(5-2 n) \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{4 a f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right)}{a f (2 n+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x) (c-c \sec (e+f x))^n}{2 a f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}",1,"-((5 - 2*n)*Hypergeometric2F1[1, 1/2 + n, 3/2 + n, (1 - Sec[e + f*x])/2]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(4*a*f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) + (2*Hypergeometric2F1[1, 1/2 + n, 3/2 + n, 1 - Sec[e + f*x]]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(a*f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) - ((c - c*Sec[e + f*x])^n*Tan[e + f*x])/(2*a*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]])","A",5,5,28,0.1786,1,"{3912, 103, 156, 65, 68}"
143,1,91,0,0.0824256,"\int \frac{\sqrt{a+a \sec (e+f x)}}{c+c \sec (e+f x)} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]/(c + c*Sec[e + f*x]),x]","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}-\frac{\sqrt{2} \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{c f}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}-\frac{\sqrt{2} \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{c f}",1,"(2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c*f) - (Sqrt[2]*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(c*f)","A",6,5,27,0.1852,1,"{21, 3776, 3774, 203, 3795}"
144,1,231,0,0.2573235,"\int \frac{(c+d \sec (e+f x))^{3/2}}{a+a \sec (e+f x)} \, dx","Int[(c + d*Sec[e + f*x])^(3/2)/(a + a*Sec[e + f*x]),x]","-\frac{2 c \cot (e+f x) \sqrt{-\frac{d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt{\frac{d (\sec (e+f x)+1)}{c+d \sec (e+f x)}} (c+d \sec (e+f x)) \Pi \left(\frac{c}{c+d};\sin ^{-1}\left(\frac{\sqrt{c+d}}{\sqrt{c+d \sec (e+f x)}}\right)|\frac{c-d}{c+d}\right)}{a f \sqrt{c+d}}-\frac{(c-d) \sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{c+d \sec (e+f x)} E\left(\sin ^{-1}\left(\frac{\tan (e+f x)}{\sec (e+f x)+1}\right)|\frac{c-d}{c+d}\right)}{a f \sqrt{\frac{c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}}","-\frac{2 c \cot (e+f x) \sqrt{-\frac{d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt{\frac{d (\sec (e+f x)+1)}{c+d \sec (e+f x)}} (c+d \sec (e+f x)) \Pi \left(\frac{c}{c+d};\sin ^{-1}\left(\frac{\sqrt{c+d}}{\sqrt{c+d \sec (e+f x)}}\right)|\frac{c-d}{c+d}\right)}{a f \sqrt{c+d}}-\frac{(c-d) \sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{c+d \sec (e+f x)} E\left(\sin ^{-1}\left(\frac{\tan (e+f x)}{\sec (e+f x)+1}\right)|\frac{c-d}{c+d}\right)}{a f \sqrt{\frac{c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}}",1,"(-2*c*Cot[e + f*x]*EllipticPi[c/(c + d), ArcSin[Sqrt[c + d]/Sqrt[c + d*Sec[e + f*x]]], (c - d)/(c + d)]*Sqrt[-((d*(1 - Sec[e + f*x]))/(c + d*Sec[e + f*x]))]*Sqrt[(d*(1 + Sec[e + f*x]))/(c + d*Sec[e + f*x])]*(c + d*Sec[e + f*x]))/(a*Sqrt[c + d]*f) - ((c - d)*EllipticE[ArcSin[Tan[e + f*x]/(1 + Sec[e + f*x])], (c - d)/(c + d)]*Sqrt[(1 + Sec[e + f*x])^(-1)]*Sqrt[c + d*Sec[e + f*x]])/(a*f*Sqrt[(c + d*Sec[e + f*x])/((c + d)*(1 + Sec[e + f*x]))])","A",3,3,27,0.1111,1,"{3927, 3780, 3968}"
145,1,225,0,0.2107301,"\int \frac{\sqrt{c+d \sec (e+f x)}}{a+a \sec (e+f x)} \, dx","Int[Sqrt[c + d*Sec[e + f*x]]/(a + a*Sec[e + f*x]),x]","-\frac{2 \cot (e+f x) \sqrt{-\frac{d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt{\frac{d (\sec (e+f x)+1)}{c+d \sec (e+f x)}} (c+d \sec (e+f x)) \Pi \left(\frac{c}{c+d};\sin ^{-1}\left(\frac{\sqrt{c+d}}{\sqrt{c+d \sec (e+f x)}}\right)|\frac{c-d}{c+d}\right)}{a f \sqrt{c+d}}-\frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{c+d \sec (e+f x)} E\left(\sin ^{-1}\left(\frac{\tan (e+f x)}{\sec (e+f x)+1}\right)|\frac{c-d}{c+d}\right)}{a f \sqrt{\frac{c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}}","-\frac{2 \cot (e+f x) \sqrt{-\frac{d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt{\frac{d (\sec (e+f x)+1)}{c+d \sec (e+f x)}} (c+d \sec (e+f x)) \Pi \left(\frac{c}{c+d};\sin ^{-1}\left(\frac{\sqrt{c+d}}{\sqrt{c+d \sec (e+f x)}}\right)|\frac{c-d}{c+d}\right)}{a f \sqrt{c+d}}-\frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{c+d \sec (e+f x)} E\left(\sin ^{-1}\left(\frac{\tan (e+f x)}{\sec (e+f x)+1}\right)|\frac{c-d}{c+d}\right)}{a f \sqrt{\frac{c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}}",1,"(-2*Cot[e + f*x]*EllipticPi[c/(c + d), ArcSin[Sqrt[c + d]/Sqrt[c + d*Sec[e + f*x]]], (c - d)/(c + d)]*Sqrt[-((d*(1 - Sec[e + f*x]))/(c + d*Sec[e + f*x]))]*Sqrt[(d*(1 + Sec[e + f*x]))/(c + d*Sec[e + f*x])]*(c + d*Sec[e + f*x]))/(a*Sqrt[c + d]*f) - (EllipticE[ArcSin[Tan[e + f*x]/(1 + Sec[e + f*x])], (c - d)/(c + d)]*Sqrt[(1 + Sec[e + f*x])^(-1)]*Sqrt[c + d*Sec[e + f*x]])/(a*f*Sqrt[(c + d*Sec[e + f*x])/((c + d)*(1 + Sec[e + f*x]))])","A",3,3,27,0.1111,1,"{3925, 3780, 3968}"
146,1,319,0,0.3699605,"\int \frac{1}{(a+a \sec (e+f x)) \sqrt{c+d \sec (e+f x)}} \, dx","Int[1/((a + a*Sec[e + f*x])*Sqrt[c + d*Sec[e + f*x]]),x]","\frac{2 \sqrt{c+d} \cot (e+f x) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (\sec (e+f x)+1)}{c-d}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right)|\frac{c+d}{c-d}\right)}{a f (c-d)}-\frac{2 \sqrt{c+d} \cot (e+f x) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (\sec (e+f x)+1)}{c-d}} \Pi \left(\frac{c+d}{c};\sin ^{-1}\left(\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right)|\frac{c+d}{c-d}\right)}{a c f}-\frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{c+d \sec (e+f x)} E\left(\sin ^{-1}\left(\frac{\tan (e+f x)}{\sec (e+f x)+1}\right)|\frac{c-d}{c+d}\right)}{a f (c-d) \sqrt{\frac{c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}}","\frac{2 \sqrt{c+d} \cot (e+f x) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (\sec (e+f x)+1)}{c-d}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right)|\frac{c+d}{c-d}\right)}{a f (c-d)}-\frac{2 \sqrt{c+d} \cot (e+f x) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (\sec (e+f x)+1)}{c-d}} \Pi \left(\frac{c+d}{c};\sin ^{-1}\left(\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right)|\frac{c+d}{c-d}\right)}{a c f}-\frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{c+d \sec (e+f x)} E\left(\sin ^{-1}\left(\frac{\tan (e+f x)}{\sec (e+f x)+1}\right)|\frac{c-d}{c+d}\right)}{a f (c-d) \sqrt{\frac{c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}}",1,"(2*Sqrt[c + d]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[c + d*Sec[e + f*x]]/Sqrt[c + d]], (c + d)/(c - d)]*Sqrt[(d*(1 - Sec[e + f*x]))/(c + d)]*Sqrt[-((d*(1 + Sec[e + f*x]))/(c - d))])/(a*(c - d)*f) - (2*Sqrt[c + d]*Cot[e + f*x]*EllipticPi[(c + d)/c, ArcSin[Sqrt[c + d*Sec[e + f*x]]/Sqrt[c + d]], (c + d)/(c - d)]*Sqrt[(d*(1 - Sec[e + f*x]))/(c + d)]*Sqrt[-((d*(1 + Sec[e + f*x]))/(c - d))])/(a*c*f) - (EllipticE[ArcSin[Tan[e + f*x]/(1 + Sec[e + f*x])], (c - d)/(c + d)]*Sqrt[(1 + Sec[e + f*x])^(-1)]*Sqrt[c + d*Sec[e + f*x]])/(a*(c - d)*f*Sqrt[(c + d*Sec[e + f*x])/((c + d)*(1 + Sec[e + f*x]))])","A",5,5,27,0.1852,1,"{3929, 3921, 3784, 3832, 3968}"
147,1,271,0,0.1728941,"\int \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^4 \, dx","Int[Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^4,x]","\frac{2 a^{3/2} c^4 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{2 d^4 \tan (e+f x) (a-a \sec (e+f x))^3}{7 a^2 f \sqrt{a \sec (e+f x)+a}}-\frac{2 d^2 \left(6 c^2+8 c d+3 d^2\right) \tan (e+f x) (a-a \sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a d (2 c+d) \left(2 c^2+2 c d+d^2\right) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 (4 c+3 d) \tan (e+f x) (a-a \sec (e+f x))^2}{5 a f \sqrt{a \sec (e+f x)+a}}","\frac{2 a^{3/2} c^4 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{2 d^4 \tan (e+f x) (a-a \sec (e+f x))^3}{7 a^2 f \sqrt{a \sec (e+f x)+a}}-\frac{2 d^2 \left(6 c^2+8 c d+3 d^2\right) \tan (e+f x) (a-a \sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a d (2 c+d) \left(2 c^2+2 c d+d^2\right) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 (4 c+3 d) \tan (e+f x) (a-a \sec (e+f x))^2}{5 a f \sqrt{a \sec (e+f x)+a}}",1,"(2*a*d*(2*c + d)*(2*c^2 + 2*c*d + d^2)*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(3/2)*c^4*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*d^2*(6*c^2 + 8*c*d + 3*d^2)*(a - a*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]) + (2*d^3*(4*c + 3*d)*(a - a*Sec[e + f*x])^2*Tan[e + f*x])/(5*a*f*Sqrt[a + a*Sec[e + f*x]]) - (2*d^4*(a - a*Sec[e + f*x])^3*Tan[e + f*x])/(7*a^2*f*Sqrt[a + a*Sec[e + f*x]])","A",5,4,27,0.1481,1,"{3940, 88, 63, 206}"
148,1,205,0,0.1409357,"\int \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^3 \, dx","Int[Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^3,x]","\frac{2 a^{3/2} c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a d \left(3 c^2+3 c d+d^2\right) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}-\frac{2 d^2 (3 c+2 d) \tan (e+f x) (a-a \sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 \tan (e+f x) (a-a \sec (e+f x))^2}{5 a f \sqrt{a \sec (e+f x)+a}}","\frac{2 a^{3/2} c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a d \left(3 c^2+3 c d+d^2\right) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}-\frac{2 d^2 (3 c+2 d) \tan (e+f x) (a-a \sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 \tan (e+f x) (a-a \sec (e+f x))^2}{5 a f \sqrt{a \sec (e+f x)+a}}",1,"(2*a*d*(3*c^2 + 3*c*d + d^2)*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(3/2)*c^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*d^2*(3*c + 2*d)*(a - a*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]) + (2*d^3*(a - a*Sec[e + f*x])^2*Tan[e + f*x])/(5*a*f*Sqrt[a + a*Sec[e + f*x]])","A",5,4,27,0.1481,1,"{3940, 88, 63, 206}"
149,1,144,0,0.1160504,"\int \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^2 \, dx","Int[Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2,x]","\frac{2 a^{3/2} c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a d (2 c+d) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}-\frac{2 d^2 \tan (e+f x) (a-a \sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}","\frac{2 a^{3/2} c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a d (2 c+d) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}-\frac{2 d^2 \tan (e+f x) (a-a \sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}",1,"(2*a*d*(2*c + d)*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(3/2)*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*d^2*(a - a*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]])","A",5,4,27,0.1481,1,"{3940, 88, 63, 206}"
150,1,66,0,0.0868663,"\int \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x)) \, dx","Int[Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]),x]","\frac{2 \sqrt{a} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}+\frac{2 a d \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","\frac{2 \sqrt{a} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}+\frac{2 a d \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*Sqrt[a]*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f + (2*a*d*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])","A",4,4,25,0.1600,1,"{3915, 3774, 203, 3792}"
151,1,105,0,0.2296103,"\int \frac{\sqrt{a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x]),x]","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}-\frac{2 \sqrt{a} \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)}{c f \sqrt{c+d}}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}-\frac{2 \sqrt{a} \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)}{c f \sqrt{c+d}}",1,"(2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c*f) - (2*Sqrt[a]*Sqrt[d]*ArcTan[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])])/(c*Sqrt[c + d]*f)","A",5,5,27,0.1852,1,"{3925, 3774, 203, 3967, 205}"
152,1,219,0,0.2221723,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c+d \sec (e+f x))^2} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x])^2,x]","-\frac{a^{3/2} \sqrt{d} (3 c+2 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 f (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{3/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{a d \tan (e+f x)}{c f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}","-\frac{a^{3/2} \sqrt{d} (3 c+2 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 f (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{3/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{a d \tan (e+f x)}{c f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}",1,"(2*a^(3/2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (a^(3/2)*Sqrt[d]*(3*c + 2*d)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^2*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (a*d*Tan[e + f*x])/(c*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))","A",7,6,27,0.2222,1,"{3940, 103, 156, 63, 206, 208}"
153,1,287,0,0.3068893,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c+d \sec (e+f x))^3} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x])^3,x]","-\frac{a^{3/2} \sqrt{d} \left(15 c^2+20 c d+8 d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{4 c^3 f (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{3/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{a d (7 c+4 d) \tan (e+f x)}{4 c^2 f (c+d)^2 \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}-\frac{a d \tan (e+f x)}{2 c f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))^2}","-\frac{a^{3/2} \sqrt{d} \left(15 c^2+20 c d+8 d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{4 c^3 f (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{3/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{a d (7 c+4 d) \tan (e+f x)}{4 c^2 f (c+d)^2 \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}-\frac{a d \tan (e+f x)}{2 c f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))^2}",1,"(2*a^(3/2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (a^(3/2)*Sqrt[d]*(15*c^2 + 20*c*d + 8*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(4*c^3*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (a*d*Tan[e + f*x])/(2*c*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2) - (a*d*(7*c + 4*d)*Tan[e + f*x])/(4*c^2*(c + d)^2*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))","A",8,7,27,0.2593,1,"{3940, 103, 151, 156, 63, 206, 208}"
154,1,241,0,0.1997159,"\int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3 \, dx","Int[(a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^3,x]","\frac{2 a^2 \tan (e+f x) \left(d \left(24 c^2+111 c d+52 d^2\right) \sec (e+f x)+2 \left(243 c^2 d+36 c^3+189 c d^2+52 d^3\right)\right)}{105 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{5/2} c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (c+d \sec (e+f x))^3}{7 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 (6 c+13 d) \tan (e+f x) (c+d \sec (e+f x))^2}{35 f \sqrt{a \sec (e+f x)+a}}","\frac{2 a^2 \tan (e+f x) \left(d \left(24 c^2+111 c d+52 d^2\right) \sec (e+f x)+2 \left(243 c^2 d+36 c^3+189 c d^2+52 d^3\right)\right)}{105 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{5/2} c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (c+d \sec (e+f x))^3}{7 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 (6 c+13 d) \tan (e+f x) (c+d \sec (e+f x))^2}{35 f \sqrt{a \sec (e+f x)+a}}",1,"(2*a^(5/2)*c^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(6*c + 13*d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(35*f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(7*f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(2*(36*c^3 + 243*c^2*d + 189*c*d^2 + 52*d^3) + d*(24*c^2 + 111*c*d + 52*d^2)*Sec[e + f*x])*Tan[e + f*x])/(105*f*Sqrt[a + a*Sec[e + f*x]])","A",6,5,27,0.1852,1,"{3940, 153, 147, 63, 206}"
155,1,176,0,0.1435388,"\int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^2 \, dx","Int[(a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^2,x]","\frac{2 a^2 \tan (e+f x) \left(2 \left(6 c^2+25 c d+9 d^2\right)+d (4 c+9 d) \sec (e+f x)\right)}{15 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{5/2} c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (c+d \sec (e+f x))^2}{5 f \sqrt{a \sec (e+f x)+a}}","\frac{2 a^2 \tan (e+f x) \left(2 \left(6 c^2+25 c d+9 d^2\right)+d (4 c+9 d) \sec (e+f x)\right)}{15 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{5/2} c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (c+d \sec (e+f x))^2}{5 f \sqrt{a \sec (e+f x)+a}}",1,"(2*a^(5/2)*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(2*(6*c^2 + 25*c*d + 9*d^2) + d*(4*c + 9*d)*Sec[e + f*x])*Tan[e + f*x])/(15*f*Sqrt[a + a*Sec[e + f*x]])","A",5,5,27,0.1852,1,"{3940, 153, 147, 63, 206}"
156,1,105,0,0.1500981,"\int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x)) \, dx","Int[(a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x]),x]","\frac{2 a^2 (3 c+4 d) \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{3/2} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}+\frac{2 a d \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{3 f}","\frac{2 a^2 (3 c+4 d) \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{3/2} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}+\frac{2 a d \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{3 f}",1,"(2*a^(3/2)*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f + (2*a^2*(3*c + 4*d)*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]) + (2*a*d*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(3*f)","A",5,5,25,0.2000,1,"{3917, 3915, 3774, 203, 3792}"
157,1,110,0,0.2405826,"\int \frac{(a+a \sec (e+f x))^{3/2}}{c+d \sec (e+f x)} \, dx","Int[(a + a*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x]),x]","\frac{2 a^{3/2} (c-d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{c+d} \sqrt{a \sec (e+f x)+a}}\right)}{c \sqrt{d} f \sqrt{c+d}}+\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}","\frac{2 a^{3/2} (c-d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{c+d} \sqrt{a \sec (e+f x)+a}}\right)}{c \sqrt{d} f \sqrt{c+d}}+\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}",1,"(2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c*f) + (2*a^(3/2)*(c - d)*ArcTan[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])])/(c*Sqrt[d]*Sqrt[c + d]*f)","A",5,5,27,0.1852,1,"{3927, 3774, 203, 3967, 205}"
158,1,229,0,0.2489399,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^2} \, dx","Int[(a + a*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^2,x]","\frac{a^{5/2} \left(c^2-3 c d-2 d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 \sqrt{d} f (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{5/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{a^2 (c-d) \tan (e+f x)}{c f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}","\frac{a^{5/2} \left(c^2-3 c d-2 d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 \sqrt{d} f (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{5/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{a^2 (c-d) \tan (e+f x)}{c f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}",1,"(2*a^(5/2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (a^(5/2)*(c^2 - 3*c*d - 2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^2*Sqrt[d]*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (a^2*(c - d)*Tan[e + f*x])/(c*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))","A",7,6,27,0.2222,1,"{3940, 151, 156, 63, 206, 208}"
159,1,310,0,0.3417037,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^3} \, dx","Int[(a + a*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^3,x]","\frac{a^2 \left(3 c^2-7 c d-4 d^2\right) \tan (e+f x)}{4 c^2 f (c+d)^2 \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{a^{5/2} \left(-15 c^2 d+3 c^3-20 c d^2-8 d^3\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{4 c^3 \sqrt{d} f (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{5/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{a^2 (c-d) \tan (e+f x)}{2 c f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))^2}","\frac{a^2 \left(3 c^2-7 c d-4 d^2\right) \tan (e+f x)}{4 c^2 f (c+d)^2 \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{a^{5/2} \left(-15 c^2 d+3 c^3-20 c d^2-8 d^3\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{4 c^3 \sqrt{d} f (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{5/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{a^2 (c-d) \tan (e+f x)}{2 c f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))^2}",1,"(2*a^(5/2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (a^(5/2)*(3*c^3 - 15*c^2*d - 20*c*d^2 - 8*d^3)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(4*c^3*Sqrt[d]*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (a^2*(c - d)*Tan[e + f*x])/(2*c*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2) + (a^2*(3*c^2 - 7*c*d - 4*d^2)*Tan[e + f*x])/(4*c^2*(c + d)^2*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))","A",8,6,27,0.2222,1,"{3940, 151, 156, 63, 206, 208}"
160,1,336,0,0.2088888,"\int (a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))^3 \, dx","Int[(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^3,x]","-\frac{2 \left(12 c^2 d+c^3+24 c d^2+12 d^3\right) \tan (e+f x) \left(a^3-a^3 \sec (e+f x)\right)}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^3 \left(12 c^2 d+3 c^3+12 c d^2+4 d^3\right) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{7/2} c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a \sec (e+f x)+a} \sqrt{a-a \sec (e+f x)}}+\frac{2 a d \left(3 c^2+15 c d+13 d^2\right) \tan (e+f x) (a-a \sec (e+f x))^2}{5 f \sqrt{a \sec (e+f x)+a}}-\frac{6 d^2 (c+2 d) \tan (e+f x) (a-a \sec (e+f x))^3}{7 f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 \tan (e+f x) (a-a \sec (e+f x))^4}{9 a f \sqrt{a \sec (e+f x)+a}}","-\frac{2 \left(12 c^2 d+c^3+24 c d^2+12 d^3\right) \tan (e+f x) \left(a^3-a^3 \sec (e+f x)\right)}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^3 \left(12 c^2 d+3 c^3+12 c d^2+4 d^3\right) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{7/2} c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a \sec (e+f x)+a} \sqrt{a-a \sec (e+f x)}}+\frac{2 a d \left(3 c^2+15 c d+13 d^2\right) \tan (e+f x) (a-a \sec (e+f x))^2}{5 f \sqrt{a \sec (e+f x)+a}}-\frac{6 d^2 (c+2 d) \tan (e+f x) (a-a \sec (e+f x))^3}{7 f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 \tan (e+f x) (a-a \sec (e+f x))^4}{9 a f \sqrt{a \sec (e+f x)+a}}",1,"(2*a^3*(3*c^3 + 12*c^2*d + 12*c*d^2 + 4*d^3)*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(7/2)*c^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*a*d*(3*c^2 + 15*c*d + 13*d^2)*(a - a*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*Sqrt[a + a*Sec[e + f*x]]) - (6*d^2*(c + 2*d)*(a - a*Sec[e + f*x])^3*Tan[e + f*x])/(7*f*Sqrt[a + a*Sec[e + f*x]]) + (2*d^3*(a - a*Sec[e + f*x])^4*Tan[e + f*x])/(9*a*f*Sqrt[a + a*Sec[e + f*x]]) - (2*(c^3 + 12*c^2*d + 24*c*d^2 + 12*d^3)*(a^3 - a^3*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]])","A",5,4,27,0.1481,1,"{3940, 180, 63, 206}"
161,1,258,0,0.1773101,"\int (a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))^2 \, dx","Int[(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^2,x]","-\frac{2 \left(c^2+8 c d+8 d^2\right) \tan (e+f x) \left(a^3-a^3 \sec (e+f x)\right)}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{7/2} c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^3 (c+2 d) (3 c+2 d) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}+\frac{2 a d (2 c+5 d) \tan (e+f x) (a-a \sec (e+f x))^2}{5 f \sqrt{a \sec (e+f x)+a}}-\frac{2 d^2 \tan (e+f x) (a-a \sec (e+f x))^3}{7 f \sqrt{a \sec (e+f x)+a}}","-\frac{2 \left(c^2+8 c d+8 d^2\right) \tan (e+f x) \left(a^3-a^3 \sec (e+f x)\right)}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{7/2} c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^3 (c+2 d) (3 c+2 d) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}+\frac{2 a d (2 c+5 d) \tan (e+f x) (a-a \sec (e+f x))^2}{5 f \sqrt{a \sec (e+f x)+a}}-\frac{2 d^2 \tan (e+f x) (a-a \sec (e+f x))^3}{7 f \sqrt{a \sec (e+f x)+a}}",1,"(2*a^3*(c + 2*d)*(3*c + 2*d)*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(7/2)*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*a*d*(2*c + 5*d)*(a - a*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*Sqrt[a + a*Sec[e + f*x]]) - (2*d^2*(a - a*Sec[e + f*x])^3*Tan[e + f*x])/(7*f*Sqrt[a + a*Sec[e + f*x]]) - (2*(c^2 + 8*c*d + 8*d^2)*(a^3 - a^3*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]])","A",5,4,27,0.1481,1,"{3940, 180, 63, 206}"
162,1,142,0,0.2319337,"\int (a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x)) \, dx","Int[(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x]),x]","\frac{2 a^3 (35 c+32 d) \tan (e+f x)}{15 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 (5 c+8 d) \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{15 f}+\frac{2 a^{5/2} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}+\frac{2 a d \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{5 f}","\frac{2 a^3 (35 c+32 d) \tan (e+f x)}{15 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 (5 c+8 d) \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{15 f}+\frac{2 a^{5/2} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}+\frac{2 a d \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{5 f}",1,"(2*a^(5/2)*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f + (2*a^3*(35*c + 32*d)*Tan[e + f*x])/(15*f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(5*c + 8*d)*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(15*f) + (2*a*d*(a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(5*f)","A",6,5,25,0.2000,1,"{3917, 3915, 3774, 203, 3792}"
163,1,203,0,0.2318011,"\int \frac{(a+a \sec (e+f x))^{5/2}}{c+d \sec (e+f x)} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x]),x]","-\frac{2 a^{7/2} (c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c d^{3/2} f \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{7/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^3 \tan (e+f x)}{d f \sqrt{a \sec (e+f x)+a}}","-\frac{2 a^{7/2} (c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c d^{3/2} f \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{7/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^3 \tan (e+f x)}{d f \sqrt{a \sec (e+f x)+a}}",1,"(2*a^3*Tan[e + f*x])/(d*f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(7/2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*a^(7/2)*(c - d)^2*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c*d^(3/2)*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])","A",7,5,27,0.1852,1,"{3940, 180, 63, 206, 208}"
164,1,329,0,0.3372777,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^2} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^2,x]","\frac{2 a^{7/2} (c-d) \sqrt{c+d} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 d^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{7/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{a^{7/2} (c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c d^{3/2} f (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{a^3 (c-d)^2 \tan (e+f x)}{c d f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}","\frac{2 a^{7/2} (c-d) \sqrt{c+d} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 d^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{7/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{a^{7/2} (c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c d^{3/2} f (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{a^3 (c-d)^2 \tan (e+f x)}{c d f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}",1,"(2*a^(7/2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (a^(7/2)*(c - d)^2*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c*d^(3/2)*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(7/2)*(c - d)*Sqrt[c + d]*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^2*d^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (a^3*(c - d)^2*Tan[e + f*x])/(c*d*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))","A",10,6,27,0.2222,1,"{3940, 180, 63, 206, 51, 208}"
165,1,536,0,0.4920226,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^3} \, dx","Int[(a + a*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^3,x]","\frac{a^{7/2} (c-d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 d^{3/2} f \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{a^3 (c-d) \tan (e+f x)}{c^2 d f \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}-\frac{2 a^{7/2} \sqrt{d} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^3 f \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{7/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 a^{7/2} (c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{4 c d^{3/2} f (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 a^3 (c-d)^2 \tan (e+f x)}{4 c d f (c+d)^2 \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}-\frac{a^3 (c-d)^2 \tan (e+f x)}{2 c d f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))^2}","\frac{a^{7/2} (c-d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 d^{3/2} f \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{a^3 (c-d) \tan (e+f x)}{c^2 d f \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}-\frac{2 a^{7/2} \sqrt{d} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^3 f \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{7/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 a^{7/2} (c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{4 c d^{3/2} f (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 a^3 (c-d)^2 \tan (e+f x)}{4 c d f (c+d)^2 \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}-\frac{a^3 (c-d)^2 \tan (e+f x)}{2 c d f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))^2}",1,"(2*a^(7/2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*a^(7/2)*(c - d)^2*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(4*c*d^(3/2)*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (a^(7/2)*(c - d)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^2*d^(3/2)*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*a^(7/2)*Sqrt[d]*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^3*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (a^3*(c - d)^2*Tan[e + f*x])/(2*c*d*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2) + (a^3*(c - d)*Tan[e + f*x])/(c^2*d*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])) - (3*a^3*(c - d)^2*Tan[e + f*x])/(4*c*d*(c + d)^2*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))","A",14,6,27,0.2222,1,"{3940, 180, 63, 206, 51, 208}"
166,1,258,0,0.2073055,"\int \frac{(c+d \sec (e+f x))^3}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[(c + d*Sec[e + f*x])^3/Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 \sqrt{a} c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^2 (3 c-d) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \sqrt{a} (c-d)^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{2 d^3 \tan (e+f x) (1-\sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","\frac{2 \sqrt{a} c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^2 (3 c-d) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \sqrt{a} (c-d)^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{2 d^3 \tan (e+f x) (1-\sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*(3*c - d)*d^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*d^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (2*d^3*(1 - Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]) + (2*Sqrt[a]*c^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*Sqrt[a]*(c - d)^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])","A",9,5,27,0.1852,1,"{3940, 180, 63, 206, 43}"
167,1,183,0,0.1580576,"\int \frac{(c+d \sec (e+f x))^2}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[(c + d*Sec[e + f*x])^2/Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 \sqrt{a} c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \sqrt{a} (c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}","\frac{2 \sqrt{a} c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \sqrt{a} (c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*d^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*Sqrt[a]*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*Sqrt[a]*(c - d)^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])","A",7,4,27,0.1481,1,"{3940, 180, 63, 206}"
168,1,91,0,0.1095333,"\int \frac{c+d \sec (e+f x)}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[(c + d*Sec[e + f*x])/Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{\sqrt{2} (c-d) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}","\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{\sqrt{2} (c-d) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}",1,"(2*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*f) - (Sqrt[2]*(c - d)*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*f)","A",5,4,25,0.1600,1,"{3920, 3774, 203, 3795}"
169,1,166,0,0.3724477,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx","Int[1/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]","\frac{2 d^{3/2} \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} c 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)}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} c f}","\frac{2 d^{3/2} \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} c 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)}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} c f}",1,"(2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*c*f) - (Sqrt[2]*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*(c - d)*f) + (2*d^(3/2)*ArcTan[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*c*(c - d)*Sqrt[c + d]*f)","A",8,7,27,0.2593,1,"{3929, 3920, 3774, 203, 3795, 3967, 205}"
170,1,416,0,0.422065,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^2} \, dx","Int[1/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2),x]","\frac{d^2 \tan (e+f x)}{c f \left(c^2-d^2\right) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{2 \sqrt{a} d^{3/2} (2 c-d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 f (c-d)^2 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \sqrt{a} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{\sqrt{a} d^{3/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c f (c-d) (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \sqrt{a} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}","\frac{d^2 \tan (e+f x)}{c f \left(c^2-d^2\right) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{2 \sqrt{a} d^{3/2} (2 c-d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 f (c-d)^2 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \sqrt{a} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{\sqrt{a} d^{3/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c f (c-d) (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \sqrt{a} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"(2*Sqrt[a]*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/((c - d)^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (Sqrt[a]*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c*(c - d)*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*Sqrt[a]*(2*c - d)*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^2*(c - d)^2*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (d^2*Tan[e + f*x])/(c*(c^2 - d^2)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))","A",12,6,27,0.2222,1,"{3940, 180, 63, 206, 51, 208}"
171,1,653,0,0.6463448,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^3} \, dx","Int[1/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^3),x]","\frac{d^2 (2 c-d) \tan (e+f x)}{c^2 f (c-d)^2 (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{d^2 \tan (e+f x)}{2 c f \left(c^2-d^2\right) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))^2}+\frac{2 \sqrt{a} d^{3/2} \left(3 c^2-3 c d+d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^3 f (c-d)^3 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{\sqrt{a} d^{3/2} (2 c-d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 f (c-d)^2 (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \sqrt{a} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{3 d^2 \tan (e+f x)}{4 c f (c-d) (c+d)^2 \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{3 \sqrt{a} d^{3/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{4 c f (c-d) (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \sqrt{a} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}","\frac{d^2 (2 c-d) \tan (e+f x)}{c^2 f (c-d)^2 (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{d^2 \tan (e+f x)}{2 c f \left(c^2-d^2\right) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))^2}+\frac{2 \sqrt{a} d^{3/2} \left(3 c^2-3 c d+d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^3 f (c-d)^3 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{\sqrt{a} d^{3/2} (2 c-d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 f (c-d)^2 (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \sqrt{a} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{3 d^2 \tan (e+f x)}{4 c f (c-d) (c+d)^2 \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{3 \sqrt{a} d^{3/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{4 c f (c-d) (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \sqrt{a} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"(2*Sqrt[a]*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/((c - d)^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (3*Sqrt[a]*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(4*c*(c - d)*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (Sqrt[a]*(2*c - d)*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^2*(c - d)^2*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*Sqrt[a]*d^(3/2)*(3*c^2 - 3*c*d + d^2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^3*(c - d)^3*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (d^2*Tan[e + f*x])/(2*c*(c^2 - d^2)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2) + (3*d^2*Tan[e + f*x])/(4*c*(c - d)*(c + d)^2*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])) + ((2*c - d)*d^2*Tan[e + f*x])/(c^2*(c - d)^2*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))","A",16,6,27,0.2222,1,"{3940, 180, 63, 206, 51, 208}"
172,1,324,0,0.2365155,"\int \frac{(c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^{3/2}} \, dx","Int[(c + d*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^(3/2),x]","\frac{2 c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^3 \tan (e+f x)}{2 a f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} \sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} (c-d)^2 (c+2 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 \tan (e+f x)}{a f \sqrt{a \sec (e+f x)+a}}","\frac{2 c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^3 \tan (e+f x)}{2 a f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} \sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} (c-d)^2 (c+2 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 \tan (e+f x)}{a f \sqrt{a \sec (e+f x)+a}}",1,"(2*d^3*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]) - ((c - d)^3*Tan[e + f*x])/(2*a*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*c^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(Sqrt[a]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - d)^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*Sqrt[a]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c - d)^2*(c + 2*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(Sqrt[a]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])","A",10,5,27,0.1852,1,"{3940, 180, 63, 206, 51}"
173,1,290,0,0.2169368,"\int \frac{(c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^{3/2}} \, dx","Int[(c + d*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^(3/2),x]","-\frac{\sqrt{2} \left(c^2-d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^2 \tan (e+f x)}{2 a f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} \sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}","-\frac{\sqrt{2} \left(c^2-d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^2 \tan (e+f x)}{2 a f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} \sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"-((c - d)^2*Tan[e + f*x])/(2*a*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(Sqrt[a]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - d)^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*Sqrt[a]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c^2 - d^2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(Sqrt[a]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])","A",10,5,27,0.1852,1,"{3940, 180, 63, 206, 51}"
174,1,127,0,0.1832728,"\int \frac{c+d \sec (e+f x)}{(a+a \sec (e+f x))^{3/2}} \, dx","Int[(c + d*Sec[e + f*x])/(a + a*Sec[e + f*x])^(3/2),x]","-\frac{(5 c-d) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f}+\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}-\frac{(c-d) \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{3/2}}","-\frac{(5 c-d) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f}+\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}-\frac{(c-d) \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{3/2}}",1,"(2*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*f) - ((5*c - d)*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f) - ((c - d)*Tan[e + f*x])/(2*f*(a + a*Sec[e + f*x])^(3/2))","A",6,5,25,0.2000,1,"{3922, 3920, 3774, 203, 3795}"
175,1,394,0,0.3349933,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))} \, dx","Int[1/((a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])),x]","-\frac{2 d^{5/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{\sqrt{a} c f (c-d)^2 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{2 a f (c-d) (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} \sqrt{a} f (c-d) \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} (c-2 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{a} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} c f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}","-\frac{2 d^{5/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{\sqrt{a} c f (c-d)^2 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{2 a f (c-d) (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} \sqrt{a} f (c-d) \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} (c-2 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{a} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} c f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"-Tan[e + f*x]/(2*a*(c - d)*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(Sqrt[a]*c*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c - 2*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(Sqrt[a]*(c - d)^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*Sqrt[a]*(c - d)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(Sqrt[a]*c*(c - d)^2*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])","A",12,6,27,0.2222,1,"{3940, 180, 63, 206, 51, 208}"
176,1,560,0,0.5106821,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^2} \, dx","Int[1/((a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^2),x]","-\frac{2 d^{5/2} (3 c-d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{\sqrt{a} c^2 f (c-d)^3 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{d^3 \tan (e+f x)}{a c f (c-d)^2 (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}-\frac{d^{5/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{\sqrt{a} c f (c-d)^2 (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{2 a f (c-d)^2 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} \sqrt{a} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} (c-3 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{a} f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}","-\frac{2 d^{5/2} (3 c-d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{\sqrt{a} c^2 f (c-d)^3 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{d^3 \tan (e+f x)}{a c f (c-d)^2 (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}-\frac{d^{5/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{\sqrt{a} c f (c-d)^2 (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{2 a f (c-d)^2 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} \sqrt{a} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} (c-3 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{a} f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"-Tan[e + f*x]/(2*a*(c - d)^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(Sqrt[a]*c^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c - 3*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(Sqrt[a]*(c - d)^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*Sqrt[a]*(c - d)^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(Sqrt[a]*c*(c - d)^2*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*(3*c - d)*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(Sqrt[a]*c^2*(c - d)^3*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (d^3*Tan[e + f*x])/(a*c*(c - d)^2*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))","A",15,6,27,0.2222,1,"{3940, 180, 63, 206, 51, 208}"
177,1,802,0,0.7965664,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3} \, dx","Int[1/((a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^3),x]","-\frac{(3 c-d) \tan (e+f x) d^3}{a c^2 (c-d)^3 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}-\frac{3 \tan (e+f x) d^3}{4 a c \left(c^2-d^2\right)^2 f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}-\frac{\tan (e+f x) d^3}{2 a c (c-d)^2 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))^2}-\frac{2 \left(6 c^2-4 d c+d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{5/2}}{\sqrt{a} c^3 (c-d)^4 \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{(3 c-d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{5/2}}{\sqrt{a} c^2 (c-d)^3 (c+d)^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{5/2}}{4 \sqrt{a} c (c-d)^2 (c+d)^{5/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right) \tan (e+f x)}{\sqrt{a} c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right) \tan (e+f x)}{2 \sqrt{2} \sqrt{a} (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\sqrt{2} (c-4 d) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right) \tan (e+f x)}{\sqrt{a} (c-d)^4 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\tan (e+f x)}{2 a (c-d)^3 f (\sec (e+f x)+1) \sqrt{\sec (e+f x) a+a}}","-\frac{(3 c-d) \tan (e+f x) d^3}{a c^2 (c-d)^3 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}-\frac{3 \tan (e+f x) d^3}{4 a c \left(c^2-d^2\right)^2 f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}-\frac{\tan (e+f x) d^3}{2 a c (c-d)^2 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))^2}-\frac{2 \left(6 c^2-4 d c+d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{5/2}}{\sqrt{a} c^3 (c-d)^4 \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{(3 c-d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{5/2}}{\sqrt{a} c^2 (c-d)^3 (c+d)^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{5/2}}{4 \sqrt{a} c (c-d)^2 (c+d)^{5/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right) \tan (e+f x)}{\sqrt{a} c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right) \tan (e+f x)}{2 \sqrt{2} \sqrt{a} (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\sqrt{2} (c-4 d) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right) \tan (e+f x)}{\sqrt{a} (c-d)^4 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\tan (e+f x)}{2 a (c-d)^3 f (\sec (e+f x)+1) \sqrt{\sec (e+f x) a+a}}",1,"-Tan[e + f*x]/(2*a*(c - d)^3*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(Sqrt[a]*c^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c - 4*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(Sqrt[a]*(c - d)^4*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*Sqrt[a]*(c - d)^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(4*Sqrt[a]*c*(c - d)^2*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((3*c - d)*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(Sqrt[a]*c^2*(c - d)^3*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*d^(5/2)*(6*c^2 - 4*c*d + d^2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(Sqrt[a]*c^3*(c - d)^4*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (d^3*Tan[e + f*x])/(2*a*c*(c - d)^2*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2) - ((3*c - d)*d^3*Tan[e + f*x])/(a*c^2*(c - d)^3*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])) - (3*d^3*Tan[e + f*x])/(4*a*c*(c^2 - d^2)^2*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))","A",19,6,27,0.2222,1,"{3940, 180, 63, 206, 51, 208}"
178,1,480,0,0.3156437,"\int \frac{(c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^{5/2}} \, dx","Int[(c + d*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^(5/2),x]","-\frac{\sqrt{2} \left(c^3-d^3\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 (c-d)^3 \tan (e+f x)}{16 a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^2 (c+2 d) \tan (e+f x)}{2 a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^3 \tan (e+f x)}{4 a^2 f (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}-\frac{3 (c-d)^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^2 (c+2 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}","-\frac{\sqrt{2} \left(c^3-d^3\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 (c-d)^3 \tan (e+f x)}{16 a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^2 (c+2 d) \tan (e+f x)}{2 a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^3 \tan (e+f x)}{4 a^2 f (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}-\frac{3 (c-d)^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^2 (c+2 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"-((c - d)^3*Tan[e + f*x])/(4*a^2*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]) - (3*(c - d)^3*Tan[e + f*x])/(16*a^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) - ((c - d)^2*(c + 2*d)*Tan[e + f*x])/(2*a^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*c^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*(c - d)^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(16*Sqrt[2]*a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - d)^2*(c + 2*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c^3 - d^3)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])","A",14,5,27,0.1852,1,"{3940, 180, 63, 206, 51}"
179,1,468,0,0.2948679,"\int \frac{(c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx","Int[(c + d*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^(5/2),x]","-\frac{\left(c^2-d^2\right) \tan (e+f x)}{2 a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\left(c^2-d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 (c-d)^2 \tan (e+f x)}{16 a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^2 \tan (e+f x)}{4 a^2 f (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}-\frac{3 (c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}","-\frac{\left(c^2-d^2\right) \tan (e+f x)}{2 a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\left(c^2-d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 (c-d)^2 \tan (e+f x)}{16 a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^2 \tan (e+f x)}{4 a^2 f (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}-\frac{3 (c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"-((c - d)^2*Tan[e + f*x])/(4*a^2*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]) - (3*(c - d)^2*Tan[e + f*x])/(16*a^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) - ((c^2 - d^2)*Tan[e + f*x])/(2*a^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*(c - d)^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(16*Sqrt[2]*a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c^2 - d^2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])","A",14,5,27,0.1852,1,"{3940, 180, 63, 206, 51}"
180,1,164,0,0.2649299,"\int \frac{c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx","Int[(c + d*Sec[e + f*x])/(a + a*Sec[e + f*x])^(5/2),x]","-\frac{(43 c-3 d) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f}+\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{(11 c-3 d) \tan (e+f x)}{16 a f (a \sec (e+f x)+a)^{3/2}}-\frac{(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}}","-\frac{(43 c-3 d) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f}+\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{(11 c-3 d) \tan (e+f x)}{16 a f (a \sec (e+f x)+a)^{3/2}}-\frac{(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}}",1,"(2*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*f) - ((43*c - 3*d)*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f) - ((c - d)*Tan[e + f*x])/(4*f*(a + a*Sec[e + f*x])^(5/2)) - ((11*c - 3*d)*Tan[e + f*x])/(16*a*f*(a + a*Sec[e + f*x])^(3/2))","A",7,5,25,0.2000,1,"{3922, 3920, 3774, 203, 3795}"
181,1,592,0,0.4567265,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))} \, dx","Int[1/((a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])),x]","-\frac{\sqrt{2} \left(c^2-3 c d+3 d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{a^{3/2} f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^{7/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{a^{3/2} c f (c-d)^3 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x)}{16 a^2 f (c-d) (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-2 d) \tan (e+f x)}{2 a^2 f (c-d)^2 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{4 a^2 f (c-d) (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} f (c-d) \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-2 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{a^{3/2} c f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}","-\frac{\sqrt{2} \left(c^2-3 c d+3 d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{a^{3/2} f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^{7/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{a^{3/2} c f (c-d)^3 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x)}{16 a^2 f (c-d) (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-2 d) \tan (e+f x)}{2 a^2 f (c-d)^2 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{4 a^2 f (c-d) (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} f (c-d) \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-2 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{a^{3/2} c f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"-Tan[e + f*x]/(4*a^2*(c - d)*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]) - ((c - 2*d)*Tan[e + f*x])/(2*a^2*(c - d)^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) - (3*Tan[e + f*x])/(16*a^2*(c - d)*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(a^(3/2)*c*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - 2*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*a^(3/2)*(c - d)^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(16*Sqrt[2]*a^(3/2)*(c - d)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c^2 - 3*c*d + 3*d^2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(a^(3/2)*(c - d)^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*d^(7/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(a^(3/2)*c*(c - d)^3*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])","A",16,6,27,0.2222,1,"{3940, 180, 63, 206, 51, 208}"
182,1,756,0,0.633782,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))^2} \, dx","Int[1/((a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^2),x]","\frac{2 d^{7/2} (4 c-d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{a^{3/2} c^2 f (c-d)^4 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \left(c^2-4 c d+6 d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{a^{3/2} f (c-d)^4 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{a^{3/2} c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{d^4 \tan (e+f x)}{a^2 c f (c-d)^3 (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{d^{7/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{a^{3/2} c f (c-d)^3 (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x)}{16 a^2 f (c-d)^2 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-3 d) \tan (e+f x)}{2 a^2 f (c-d)^3 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{4 a^2 f (c-d)^2 (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-3 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}","\frac{2 d^{7/2} (4 c-d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{a^{3/2} c^2 f (c-d)^4 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \left(c^2-4 c d+6 d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{a^{3/2} f (c-d)^4 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{a^{3/2} c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{d^4 \tan (e+f x)}{a^2 c f (c-d)^3 (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{d^{7/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{a^{3/2} c f (c-d)^3 (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x)}{16 a^2 f (c-d)^2 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-3 d) \tan (e+f x)}{2 a^2 f (c-d)^3 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{4 a^2 f (c-d)^2 (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-3 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"-Tan[e + f*x]/(4*a^2*(c - d)^2*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]) - ((c - 3*d)*Tan[e + f*x])/(2*a^2*(c - d)^3*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) - (3*Tan[e + f*x])/(16*a^2*(c - d)^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(a^(3/2)*c^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - 3*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*a^(3/2)*(c - d)^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(16*Sqrt[2]*a^(3/2)*(c - d)^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c^2 - 4*c*d + 6*d^2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(a^(3/2)*(c - d)^4*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (d^(7/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(a^(3/2)*c*(c - d)^3*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*(4*c - d)*d^(7/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(a^(3/2)*c^2*(c - d)^4*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (d^4*Tan[e + f*x])/(a^2*c*(c - d)^3*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))","A",19,6,27,0.2222,1,"{3940, 180, 63, 206, 51, 208}"
183,1,999,0,0.9141618,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))^3} \, dx","Int[1/((a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^3),x]","\frac{(4 c-d) \tan (e+f x) d^4}{a^2 c^2 (c-d)^4 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}+\frac{3 \tan (e+f x) d^4}{4 a^2 c (c-d)^3 (c+d)^2 f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}+\frac{\tan (e+f x) d^4}{2 a^2 c (c-d)^3 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))^2}+\frac{2 \left(10 c^2-5 d c+d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{7/2}}{a^{3/2} c^3 (c-d)^5 \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}+\frac{(4 c-d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{7/2}}{a^{3/2} c^2 (c-d)^4 (c+d)^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{7/2}}{4 a^{3/2} c (c-d)^3 (c+d)^{5/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right) \tan (e+f x)}{a^{3/2} c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\sqrt{2} \left(c^2-5 d c+10 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right) \tan (e+f x)}{a^{3/2} (c-d)^5 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right) \tan (e+f x)}{16 \sqrt{2} a^{3/2} (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{(c-4 d) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right) \tan (e+f x)}{2 \sqrt{2} a^{3/2} (c-d)^4 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{3 \tan (e+f x)}{16 a^2 (c-d)^3 f (\sec (e+f x)+1) \sqrt{\sec (e+f x) a+a}}-\frac{(c-4 d) \tan (e+f x)}{2 a^2 (c-d)^4 f (\sec (e+f x)+1) \sqrt{\sec (e+f x) a+a}}-\frac{\tan (e+f x)}{4 a^2 (c-d)^3 f (\sec (e+f x)+1)^2 \sqrt{\sec (e+f x) a+a}}","\frac{(4 c-d) \tan (e+f x) d^4}{a^2 c^2 (c-d)^4 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}+\frac{3 \tan (e+f x) d^4}{4 a^2 c (c-d)^3 (c+d)^2 f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}+\frac{\tan (e+f x) d^4}{2 a^2 c (c-d)^3 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))^2}+\frac{2 \left(10 c^2-5 d c+d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{7/2}}{a^{3/2} c^3 (c-d)^5 \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}+\frac{(4 c-d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{7/2}}{a^{3/2} c^2 (c-d)^4 (c+d)^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{7/2}}{4 a^{3/2} c (c-d)^3 (c+d)^{5/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right) \tan (e+f x)}{a^{3/2} c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\sqrt{2} \left(c^2-5 d c+10 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right) \tan (e+f x)}{a^{3/2} (c-d)^5 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right) \tan (e+f x)}{16 \sqrt{2} a^{3/2} (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{(c-4 d) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right) \tan (e+f x)}{2 \sqrt{2} a^{3/2} (c-d)^4 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{3 \tan (e+f x)}{16 a^2 (c-d)^3 f (\sec (e+f x)+1) \sqrt{\sec (e+f x) a+a}}-\frac{(c-4 d) \tan (e+f x)}{2 a^2 (c-d)^4 f (\sec (e+f x)+1) \sqrt{\sec (e+f x) a+a}}-\frac{\tan (e+f x)}{4 a^2 (c-d)^3 f (\sec (e+f x)+1)^2 \sqrt{\sec (e+f x) a+a}}",1,"-Tan[e + f*x]/(4*a^2*(c - d)^3*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]) - ((c - 4*d)*Tan[e + f*x])/(2*a^2*(c - d)^4*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) - (3*Tan[e + f*x])/(16*a^2*(c - d)^3*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(a^(3/2)*c^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - 4*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*a^(3/2)*(c - d)^4*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(16*Sqrt[2]*a^(3/2)*(c - d)^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c^2 - 5*c*d + 10*d^2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(a^(3/2)*(c - d)^5*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (3*d^(7/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(4*a^(3/2)*c*(c - d)^3*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((4*c - d)*d^(7/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(a^(3/2)*c^2*(c - d)^4*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*d^(7/2)*(10*c^2 - 5*c*d + d^2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(a^(3/2)*c^3*(c - d)^5*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (d^4*Tan[e + f*x])/(2*a^2*c*(c - d)^3*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2) + (3*d^4*Tan[e + f*x])/(4*a^2*c*(c - d)^3*(c + d)^2*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])) + ((4*c - d)*d^4*Tan[e + f*x])/(a^2*c^2*(c - d)^4*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))","A",23,6,27,0.2222,1,"{3940, 180, 63, 206, 51, 208}"
184,1,123,0,0.3386344,"\int \sqrt{a+a \sec (e+f x)} \sqrt{c+d \sec (e+f x)} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]],x]","\frac{2 \sqrt{a} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{f}+\frac{2 \sqrt{a} \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)}{f}","\frac{2 \sqrt{a} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{f}+\frac{2 \sqrt{a} \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)}{f}",1,"(2*Sqrt[a]*Sqrt[c]*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/f + (2*Sqrt[a]*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/f","A",5,5,29,0.1724,1,"{3932, 3934, 203, 3980, 206}"
185,1,61,0,0.0966942,"\int \frac{\sqrt{a+a \sec (e+f x)}}{\sqrt{c+d \sec (e+f x)}} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]/Sqrt[c + d*Sec[e + f*x]],x]","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{c} f}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{c} f}",1,"(2*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[c]*f)","A",2,2,29,0.06897,1,"{3934, 203}"
186,1,111,0,0.3571179,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c+d \sec (e+f x))^{3/2}} \, dx","Int[Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x])^(3/2),x]","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{c^{3/2} f}-\frac{2 a d \tan (e+f x)}{c f (c+d) \sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{c^{3/2} f}-\frac{2 a d \tan (e+f x)}{c f (c+d) \sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}",1,"(2*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(c^(3/2)*f) - (2*a*d*Tan[e + f*x])/(c*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])","A",5,5,29,0.1724,1,"{3939, 3934, 203, 3987, 37}"
187,1,141,0,0.368627,"\int \frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[Sqrt[c + d*Sec[e + f*x]]/Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{a} f}-\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{c} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{a} f}-\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}",1,"(2*Sqrt[c]*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[a]*f) - (Sqrt[2]*Sqrt[c - d]*ArcTan[(Sqrt[a]*Sqrt[c - d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[a]*f)","A",5,4,29,0.1379,1,"{3935, 3934, 203, 3983}"
188,1,141,0,0.3386353,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} \sqrt{c+d \sec (e+f x)}} \, dx","Int[1/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{a} \sqrt{c} 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}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{a} \sqrt{c} 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*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[a]*Sqrt[c]*f) - (Sqrt[2]*ArcTan[(Sqrt[a]*Sqrt[c - d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[a]*Sqrt[c - d]*f)","A",5,4,29,0.1379,1,"{3938, 3934, 203, 3983}"
189,1,67,0,0.1255558,"\int \frac{a+b \sec (e+f x)}{c+d \sec (e+f x)} \, dx","Int[(a + b*Sec[e + f*x])/(c + d*Sec[e + f*x]),x]","\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)}{c f \sqrt{c-d} \sqrt{c+d}}+\frac{a x}{c}","\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)}{c f \sqrt{c-d} \sqrt{c+d}}+\frac{a x}{c}",1,"(a*x)/c + (2*(b*c - a*d)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(c*Sqrt[c - d]*Sqrt[c + d]*f)","A",4,4,23,0.1739,1,"{3919, 3831, 2659, 208}"
190,1,123,0,0.2467995,"\int \frac{a+b \sec (e+f x)}{(c+d \sec (e+f x))^2} \, dx","Int[(a + b*Sec[e + f*x])/(c + d*Sec[e + f*x])^2,x]","\frac{2 \left(-2 a c^2 d+a d^3+b c^3\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^2 f (c-d)^{3/2} (c+d)^{3/2}}-\frac{d (b c-a d) \tan (e+f x)}{c f \left(c^2-d^2\right) (c+d \sec (e+f x))}+\frac{a x}{c^2}","\frac{2 \left(-2 a c^2 d+a d^3+b c^3\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^2 f (c-d)^{3/2} (c+d)^{3/2}}-\frac{d (b c-a d) \tan (e+f x)}{c f \left(c^2-d^2\right) (c+d \sec (e+f x))}+\frac{a x}{c^2}",1,"(a*x)/c^2 + (2*(b*c^3 - 2*a*c^2*d + a*d^3)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(c^2*(c - d)^(3/2)*(c + d)^(3/2)*f) - (d*(b*c - a*d)*Tan[e + f*x])/(c*(c^2 - d^2)*f*(c + d*Sec[e + f*x]))","A",5,5,23,0.2174,1,"{3923, 3919, 3831, 2659, 208}"
191,1,204,0,0.5096319,"\int \frac{a+b \sec (e+f x)}{(c+d \sec (e+f x))^3} \, dx","Int[(a + b*Sec[e + f*x])/(c + d*Sec[e + f*x])^3,x]","\frac{\left(b c^3 \left(2 c^2+d^2\right)-a d \left(-5 c^2 d^2+6 c^4+2 d^4\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^3 f (c-d)^{5/2} (c+d)^{5/2}}-\frac{d \left(-5 a c^2 d+2 a d^3+3 b c^3\right) \tan (e+f x)}{2 c^2 f \left(c^2-d^2\right)^2 (c+d \sec (e+f x))}-\frac{d (b c-a d) \tan (e+f x)}{2 c f \left(c^2-d^2\right) (c+d \sec (e+f x))^2}+\frac{a x}{c^3}","\frac{\left(b c^3 \left(2 c^2+d^2\right)-a d \left(-5 c^2 d^2+6 c^4+2 d^4\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^3 f (c-d)^{5/2} (c+d)^{5/2}}-\frac{d \left(-5 a c^2 d+2 a d^3+3 b c^3\right) \tan (e+f x)}{2 c^2 f \left(c^2-d^2\right)^2 (c+d \sec (e+f x))}-\frac{d (b c-a d) \tan (e+f x)}{2 c f \left(c^2-d^2\right) (c+d \sec (e+f x))^2}+\frac{a x}{c^3}",1,"(a*x)/c^3 + ((b*c^3*(2*c^2 + d^2) - a*d*(6*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(c^3*(c - d)^(5/2)*(c + d)^(5/2)*f) - (d*(b*c - a*d)*Tan[e + f*x])/(2*c*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^2) - (d*(3*b*c^3 - 5*a*c^2*d + 2*a*d^3)*Tan[e + f*x])/(2*c^2*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x]))","A",6,6,23,0.2609,1,"{3923, 4060, 3919, 3831, 2659, 208}"
192,1,133,0,0.2849518,"\int \frac{(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^2} \, dx","Int[(a + b*Sec[e + f*x])^2/(c + d*Sec[e + f*x])^2,x]","\frac{a^2 x}{c^2}+\frac{(b c-a d)^2 \sin (e+f x)}{c f \left(c^2-d^2\right) (c \cos (e+f x)+d)}+\frac{2 (b c-a d) \left(2 a c^2-a d^2-b c d\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^2 f (c-d)^{3/2} (c+d)^{3/2}}","\frac{a^2 x}{c^2}+\frac{(b c-a d)^2 \sin (e+f x)}{c f \left(c^2-d^2\right) (c \cos (e+f x)+d)}+\frac{2 (b c-a d) \left(2 a c^2-a d^2-b c d\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^2 f (c-d)^{3/2} (c+d)^{3/2}}",1,"(a^2*x)/c^2 + (2*(b*c - a*d)*(2*a*c^2 - b*c*d - a*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(c^2*(c - d)^(3/2)*(c + d)^(3/2)*f) + ((b*c - a*d)^2*Sin[e + f*x])/(c*(c^2 - d^2)*f*(d + c*Cos[e + f*x]))","A",5,5,25,0.2000,1,"{3941, 2790, 2735, 2659, 208}"
193,1,237,0,0.7956084,"\int \frac{(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^3} \, dx","Int[(a + b*Sec[e + f*x])^2/(c + d*Sec[e + f*x])^3,x]","-\frac{\left(a^2 \left(-5 c^2 d^3+6 c^4 d+2 d^5\right)-2 a b c^3 \left(2 c^2+d^2\right)+3 b^2 c^4 d\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^3 f (c-d)^{5/2} (c+d)^{5/2}}+\frac{a^2 x}{c^3}-\frac{(b c-a d) \left(3 a d \left(2 c^2-d^2\right)-b c \left(2 c^2+d^2\right)\right) \sin (e+f x)}{2 c^2 f \left(c^2-d^2\right)^2 (c \cos (e+f x)+d)}-\frac{d (b c-a d)^2 \sin (e+f x)}{2 c^2 f \left(c^2-d^2\right) (c \cos (e+f x)+d)^2}","-\frac{\left(a^2 \left(-5 c^2 d^3+6 c^4 d+2 d^5\right)-2 a b c^3 \left(2 c^2+d^2\right)+3 b^2 c^4 d\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^3 f (c-d)^{5/2} (c+d)^{5/2}}+\frac{a^2 x}{c^3}-\frac{(b c-a d) \left(3 a d \left(2 c^2-d^2\right)-b c \left(2 c^2+d^2\right)\right) \sin (e+f x)}{2 c^2 f \left(c^2-d^2\right)^2 (c \cos (e+f x)+d)}-\frac{d (b c-a d)^2 \sin (e+f x)}{2 c^2 f \left(c^2-d^2\right) (c \cos (e+f x)+d)^2}",1,"(a^2*x)/c^3 - ((3*b^2*c^4*d - 2*a*b*c^3*(2*c^2 + d^2) + a^2*(6*c^4*d - 5*c^2*d^3 + 2*d^5))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(c^3*(c - d)^(5/2)*(c + d)^(5/2)*f) - (d*(b*c - a*d)^2*Sin[e + f*x])/(2*c^2*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^2) - ((b*c - a*d)*(3*a*d*(2*c^2 - d^2) - b*c*(2*c^2 + d^2))*Sin[e + f*x])/(2*c^2*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x]))","A",6,6,25,0.2400,1,"{3941, 2988, 3021, 2735, 2659, 208}"
194,1,377,0,1.9989589,"\int \frac{(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^4} \, dx","Int[(a + b*Sec[e + f*x])^2/(c + d*Sec[e + f*x])^4,x]","-\frac{\left(-a^2 d^2 \left(-28 c^2 d^2+34 c^4+9 d^4\right)+2 a b c d \left(-5 c^2 d^2+18 c^4+2 d^4\right)+b^2 \left(-\left(10 c^4 d^2-c^2 d^4+6 c^6\right)\right)\right) \sin (e+f x)}{6 c^3 f \left(c^2-d^2\right)^3 (c \cos (e+f x)+d)}-\frac{\left(a^2 \left(7 c^2 d^5-8 c^4 d^3+8 c^6 d-2 d^7\right)-a b \left(6 c^5 d^2+4 c^7\right)+b^2 c^4 d \left(4 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)}{c^4 f (c-d)^{7/2} (c+d)^{7/2}}+\frac{a^2 x}{c^4}-\frac{d (b c-a d) \left(-8 a c^2 d+3 a d^3+6 b c^3-b c d^2\right) \sin (e+f x)}{6 c^3 f \left(c^2-d^2\right)^2 (c \cos (e+f x)+d)^2}+\frac{d^2 \sin (e+f x) (a \cos (e+f x)+b)^2}{3 c f \left(c^2-d^2\right) (c \cos (e+f x)+d)^3}","-\frac{\left(-a^2 d^2 \left(-28 c^2 d^2+34 c^4+9 d^4\right)+2 a b c d \left(-5 c^2 d^2+18 c^4+2 d^4\right)+b^2 \left(-\left(10 c^4 d^2-c^2 d^4+6 c^6\right)\right)\right) \sin (e+f x)}{6 c^3 f \left(c^2-d^2\right)^3 (c \cos (e+f x)+d)}-\frac{\left(a^2 \left(7 c^2 d^5-8 c^4 d^3+8 c^6 d-2 d^7\right)-a b \left(6 c^5 d^2+4 c^7\right)+b^2 c^4 d \left(4 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)}{c^4 f (c-d)^{7/2} (c+d)^{7/2}}+\frac{a^2 x}{c^4}-\frac{d (b c-a d) \left(-8 a c^2 d+3 a d^3+6 b c^3-b c d^2\right) \sin (e+f x)}{6 c^3 f \left(c^2-d^2\right)^2 (c \cos (e+f x)+d)^2}+\frac{d^2 \sin (e+f x) (a \cos (e+f x)+b)^2}{3 c f \left(c^2-d^2\right) (c \cos (e+f x)+d)^3}",1,"(a^2*x)/c^4 - ((b^2*c^4*d*(4*c^2 + d^2) - a*b*(4*c^7 + 6*c^5*d^2) + a^2*(8*c^6*d - 8*c^4*d^3 + 7*c^2*d^5 - 2*d^7))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(c^4*(c - d)^(7/2)*(c + d)^(7/2)*f) + (d^2*(b + a*Cos[e + f*x])^2*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^3) - (d*(b*c - a*d)*(6*b*c^3 - 8*a*c^2*d - b*c*d^2 + 3*a*d^3)*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])^2) - ((2*a*b*c*d*(18*c^4 - 5*c^2*d^2 + 2*d^4) - a^2*d^2*(34*c^4 - 28*c^2*d^2 + 9*d^4) - b^2*(6*c^6 + 10*c^4*d^2 - c^2*d^4))*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^3*f*(d + c*Cos[e + f*x]))","A",7,7,25,0.2800,1,"{3941, 3048, 3031, 3021, 2735, 2659, 208}"
195,1,254,0,1.1324412,"\int \frac{(a+b \sec (e+f x))^3}{(c+d \sec (e+f x))^3} \, dx","Int[(a + b*Sec[e + f*x])^3/(c + d*Sec[e + f*x])^3,x]","-\frac{(b c-a d) \left(a^2 \left(-\left(-5 c^2 d^2+6 c^4+2 d^4\right)\right)+2 a b c d \left(4 c^2-d^2\right)-b^2 c^2 \left(c^2+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)}{c^3 f (c-d)^{5/2} (c+d)^{5/2}}+\frac{a^3 x}{c^3}+\frac{(b c-a d)^2 \left(5 a c^2-2 a d^2-3 b c d\right) \sin (e+f x)}{2 c^2 f \left(c^2-d^2\right)^2 (c \cos (e+f x)+d)}+\frac{(b c-a d)^2 \sin (e+f x) (a \cos (e+f x)+b)}{2 c f \left(c^2-d^2\right) (c \cos (e+f x)+d)^2}","-\frac{(b c-a d) \left(a^2 \left(-\left(-5 c^2 d^2+6 c^4+2 d^4\right)\right)+2 a b c d \left(4 c^2-d^2\right)-b^2 c^2 \left(c^2+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)}{c^3 f (c-d)^{5/2} (c+d)^{5/2}}+\frac{a^3 x}{c^3}+\frac{(b c-a d)^2 \left(5 a c^2-2 a d^2-3 b c d\right) \sin (e+f x)}{2 c^2 f \left(c^2-d^2\right)^2 (c \cos (e+f x)+d)}+\frac{(b c-a d)^2 \sin (e+f x) (a \cos (e+f x)+b)}{2 c f \left(c^2-d^2\right) (c \cos (e+f x)+d)^2}",1,"(a^3*x)/c^3 - ((b*c - a*d)*(2*a*b*c*d*(4*c^2 - d^2) - b^2*c^2*(c^2 + 2*d^2) - a^2*(6*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(c^3*(c - d)^(5/2)*(c + d)^(5/2)*f) + ((b*c - a*d)^2*(b + a*Cos[e + f*x])*Sin[e + f*x])/(2*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^2) + ((b*c - a*d)^2*(5*a*c^2 - 3*b*c*d - 2*a*d^2)*Sin[e + f*x])/(2*c^2*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x]))","A",6,6,25,0.2400,1,"{3941, 2792, 3021, 2735, 2659, 208}"
196,1,412,0,1.0622997,"\int \frac{(a+b \sec (e+f x))^3}{(c+d \sec (e+f x))^4} \, dx","Int[(a + b*Sec[e + f*x])^3/(c + d*Sec[e + f*x])^4,x]","-\frac{(b c-a d) \left(a^2 \left(-28 c^2 d^3+34 c^4 d+9 d^5\right)-a b c \left(17 c^2 d^2+18 c^4-5 d^4\right)+b^2 c^2 d \left(13 c^2+2 d^2\right)\right) \sin (e+f x)}{6 c^3 f \left(c^2-d^2\right)^3 (c \cos (e+f x)+d)}-\frac{\left(-a^2 b \left(9 c^5 d^2+6 c^7\right)+a^3 \left(7 c^2 d^5-8 c^4 d^3+8 c^6 d-2 d^7\right)+3 a b^2 c^4 d \left(4 c^2+d^2\right)-b^3 c^5 \left(c^2+4 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^4 f \sqrt{c-d} \sqrt{c+d} \left(c^2-d^2\right)^3}+\frac{a^3 x}{c^4}+\frac{(b c-a d)^2 \left(-8 a c^2 d+3 a d^3+3 b c^3+2 b c d^2\right) \sin (e+f x)}{6 c^3 f \left(c^2-d^2\right)^2 (c \cos (e+f x)+d)^2}-\frac{d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^2}{3 c f \left(c^2-d^2\right) (c \cos (e+f x)+d)^3}","-\frac{(b c-a d) \left(a^2 \left(-28 c^2 d^3+34 c^4 d+9 d^5\right)-a b c \left(17 c^2 d^2+18 c^4-5 d^4\right)+b^2 c^2 d \left(13 c^2+2 d^2\right)\right) \sin (e+f x)}{6 c^3 f \left(c^2-d^2\right)^3 (c \cos (e+f x)+d)}-\frac{\left(-a^2 b \left(9 c^5 d^2+6 c^7\right)+a^3 \left(7 c^2 d^5-8 c^4 d^3+8 c^6 d-2 d^7\right)+3 a b^2 c^4 d \left(4 c^2+d^2\right)-b^3 c^5 \left(c^2+4 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^4 f \sqrt{c-d} \sqrt{c+d} \left(c^2-d^2\right)^3}+\frac{a^3 x}{c^4}+\frac{(b c-a d)^2 \left(-8 a c^2 d+3 a d^3+3 b c^3+2 b c d^2\right) \sin (e+f x)}{6 c^3 f \left(c^2-d^2\right)^2 (c \cos (e+f x)+d)^2}-\frac{d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^2}{3 c f \left(c^2-d^2\right) (c \cos (e+f x)+d)^3}",1,"(a^3*x)/c^4 - ((3*a*b^2*c^4*d*(4*c^2 + d^2) - b^3*c^5*(c^2 + 4*d^2) - a^2*b*(6*c^7 + 9*c^5*d^2) + a^3*(8*c^6*d - 8*c^4*d^3 + 7*c^2*d^5 - 2*d^7))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(c^4*Sqrt[c - d]*Sqrt[c + d]*(c^2 - d^2)^3*f) - (d*(b*c - a*d)*(b + a*Cos[e + f*x])^2*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^3) + ((b*c - a*d)^2*(3*b*c^3 - 8*a*c^2*d + 2*b*c*d^2 + 3*a*d^3)*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])^2) - ((b*c - a*d)*(b^2*c^2*d*(13*c^2 + 2*d^2) - a*b*c*(18*c^4 + 17*c^2*d^2 - 5*d^4) + a^2*(34*c^4*d - 28*c^2*d^3 + 9*d^5))*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^3*f*(d + c*Cos[e + f*x]))","A",7,7,25,0.2800,1,"{3941, 2989, 3031, 3021, 2735, 2659, 208}"
197,1,622,0,1.7689017,"\int \frac{(a+b \sec (e+f x))^3}{(c+d \sec (e+f x))^5} \, dx","Int[(a + b*Sec[e + f*x])^3/(c + d*Sec[e + f*x])^5,x]","-\frac{\left(a^2 b c d \left(10 c^4 d^2+49 c^2 d^4+272 c^6-16 d^6\right)+a^3 \left(-\left(139 c^2 d^6-210 c^4 d^4+212 c^6 d^2-36 d^8\right)\right)-3 a b^2 c^2 \left(84 c^4 d^2-5 c^2 d^4+24 c^6+2 d^6\right)+b^3 c^3 d \left(39 c^2 d^2+68 c^4-2 d^4\right)\right) \sin (e+f x)}{24 c^4 f \left(c^2-d^2\right)^4 (c \cos (e+f x)+d)}-\frac{(b c-a d) \left(-a^2 d^2 \left(-35 c^2 d^2+58 c^4+12 d^4\right)+2 a b c d \left(c^2 d^2+32 c^4+2 d^4\right)+b^2 \left(-\left(25 c^4 d^2-2 c^2 d^4+12 c^6\right)\right)\right) \sin (e+f x)}{24 c^4 f \left(c^2-d^2\right)^3 (c \cos (e+f x)+d)^2}-\frac{\left(-3 a^2 b c^5 \left(24 c^2 d^2+8 c^4+3 d^4\right)+a^3 \left(-36 c^2 d^7+63 c^4 d^5-40 c^6 d^3+40 c^8 d+8 d^9\right)+15 a b^2 c^6 d \left(4 c^2+3 d^2\right)-b^3 c^5 \left(27 c^2 d^2+4 c^4+4 d^4\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{4 c^5 f \sqrt{c-d} \sqrt{c+d} \left(c^2-d^2\right)^4}+\frac{a^3 x}{c^5}-\frac{d \left(-11 a c^2 d+4 a d^3+8 b c^3-b c d^2\right) \sin (e+f x) (a \cos (e+f x)+b)^2}{12 c^2 f \left(c^2-d^2\right)^2 (c \cos (e+f x)+d)^3}+\frac{d^2 \sin (e+f x) (a \cos (e+f x)+b)^3}{4 c f \left(c^2-d^2\right) (c \cos (e+f x)+d)^4}","-\frac{\left(a^2 b c d \left(10 c^4 d^2+49 c^2 d^4+272 c^6-16 d^6\right)+a^3 \left(-\left(139 c^2 d^6-210 c^4 d^4+212 c^6 d^2-36 d^8\right)\right)-3 a b^2 c^2 \left(84 c^4 d^2-5 c^2 d^4+24 c^6+2 d^6\right)+b^3 c^3 d \left(39 c^2 d^2+68 c^4-2 d^4\right)\right) \sin (e+f x)}{24 c^4 f \left(c^2-d^2\right)^4 (c \cos (e+f x)+d)}-\frac{(b c-a d) \left(-a^2 d^2 \left(-35 c^2 d^2+58 c^4+12 d^4\right)+2 a b c d \left(c^2 d^2+32 c^4+2 d^4\right)+b^2 \left(-\left(25 c^4 d^2-2 c^2 d^4+12 c^6\right)\right)\right) \sin (e+f x)}{24 c^4 f \left(c^2-d^2\right)^3 (c \cos (e+f x)+d)^2}-\frac{\left(-3 a^2 b c^5 \left(24 c^2 d^2+8 c^4+3 d^4\right)+a^3 \left(-36 c^2 d^7+63 c^4 d^5-40 c^6 d^3+40 c^8 d+8 d^9\right)+15 a b^2 c^6 d \left(4 c^2+3 d^2\right)-b^3 c^5 \left(27 c^2 d^2+4 c^4+4 d^4\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{4 c^5 f \sqrt{c-d} \sqrt{c+d} \left(c^2-d^2\right)^4}+\frac{a^3 x}{c^5}-\frac{d \left(-11 a c^2 d+4 a d^3+8 b c^3-b c d^2\right) \sin (e+f x) (a \cos (e+f x)+b)^2}{12 c^2 f \left(c^2-d^2\right)^2 (c \cos (e+f x)+d)^3}+\frac{d^2 \sin (e+f x) (a \cos (e+f x)+b)^3}{4 c f \left(c^2-d^2\right) (c \cos (e+f x)+d)^4}",1,"(a^3*x)/c^5 - ((15*a*b^2*c^6*d*(4*c^2 + 3*d^2) - 3*a^2*b*c^5*(8*c^4 + 24*c^2*d^2 + 3*d^4) - b^3*c^5*(4*c^4 + 27*c^2*d^2 + 4*d^4) + a^3*(40*c^8*d - 40*c^6*d^3 + 63*c^4*d^5 - 36*c^2*d^7 + 8*d^9))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(4*c^5*Sqrt[c - d]*Sqrt[c + d]*(c^2 - d^2)^4*f) + (d^2*(b + a*Cos[e + f*x])^3*Sin[e + f*x])/(4*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^4) - (d*(8*b*c^3 - 11*a*c^2*d - b*c*d^2 + 4*a*d^3)*(b + a*Cos[e + f*x])^2*Sin[e + f*x])/(12*c^2*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])^3) - ((b*c - a*d)*(2*a*b*c*d*(32*c^4 + c^2*d^2 + 2*d^4) - a^2*d^2*(58*c^4 - 35*c^2*d^2 + 12*d^4) - b^2*(12*c^6 + 25*c^4*d^2 - 2*c^2*d^4))*Sin[e + f*x])/(24*c^4*(c^2 - d^2)^3*f*(d + c*Cos[e + f*x])^2) - ((b^3*c^3*d*(68*c^4 + 39*c^2*d^2 - 2*d^4) + a^2*b*c*d*(272*c^6 + 10*c^4*d^2 + 49*c^2*d^4 - 16*d^6) - 3*a*b^2*c^2*(24*c^6 + 84*c^4*d^2 - 5*c^2*d^4 + 2*d^6) - a^3*(212*c^6*d^2 - 210*c^4*d^4 + 139*c^2*d^6 - 36*d^8))*Sin[e + f*x])/(24*c^4*(c^2 - d^2)^4*f*(d + c*Cos[e + f*x]))","A",8,8,25,0.3200,1,"{3941, 3048, 3047, 3031, 3021, 2735, 2659, 208}"
198,1,320,0,0.2829334,"\int \sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x)) \, dx","Int[Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x]),x]","\frac{2 \sqrt{a+b} (a d+b (c-d)) \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 f}-\frac{2 c \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}-\frac{2 d (a-b) \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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b f}","\frac{2 \sqrt{a+b} (a d+b (c-d)) \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 f}-\frac{2 c \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}-\frac{2 d (a-b) \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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b f}",1,"(-2*(a - b)*Sqrt[a + b]*d*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(b*f) + (2*Sqrt[a + b]*(b*(c - d) + a*d)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(b*f) - (2*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f","A",5,5,25,0.2000,1,"{3916, 3784, 4005, 3832, 4004}"
199,1,220,0,0.2438521,"\int \frac{\sqrt{a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx","Int[Sqrt[a + b*Sec[e + f*x]]/(c + d*Sec[e + f*x]),x]","\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)}{c f (c+d) \sqrt{-\tan ^2(e+f x)} \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{c 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)}{c f (c+d) \sqrt{-\tan ^2(e+f x)} \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{c f}",1,"(-2*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(c*f) + (2*(b*c - a*d)*EllipticPi[(2*d)/(c + d), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*b)/(a + b)]*Sqrt[(a + b*Sec[e + f*x])/(a + b)]*Tan[e + f*x])/(c*(c + d)*f*Sqrt[a + b*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])","A",3,3,27,0.1111,1,"{3926, 3784, 3973}"
200,1,380,0,0.4330035,"\int (a+b \sec (e+f x))^{3/2} (c+d \sec (e+f x)) \, dx","Int[(a + b*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x]),x]","\frac{2 \sqrt{a+b} \left(3 a^2 d+a b (6 c-4 d)-b^2 (3 c-d)\right) \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)}{3 b f}-\frac{2 (a-b) \sqrt{a+b} (4 a d+3 b c) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b f}-\frac{2 a c \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}+\frac{2 b d \tan (e+f x) \sqrt{a+b \sec (e+f x)}}{3 f}","\frac{2 \sqrt{a+b} \left(3 a^2 d+a b (6 c-4 d)-b^2 (3 c-d)\right) \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)}{3 b f}-\frac{2 (a-b) \sqrt{a+b} (4 a d+3 b c) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b f}-\frac{2 a c \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}+\frac{2 b d \tan (e+f x) \sqrt{a+b \sec (e+f x)}}{3 f}",1,"(-2*(a - b)*Sqrt[a + b]*(3*b*c + 4*a*d)*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(3*b*f) + (2*Sqrt[a + b]*(a*b*(6*c - 4*d) - b^2*(3*c - d) + 3*a^2*d)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(3*b*f) - (2*a*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f + (2*b*d*Sqrt[a + b*Sec[e + f*x]]*Tan[e + f*x])/(3*f)","A",6,6,25,0.2400,1,"{3918, 4058, 3921, 3784, 3832, 4004}"
201,1,326,0,0.3614916,"\int \frac{(a+b \sec (e+f x))^{3/2}}{c+d \sec (e+f x)} \, dx","Int[(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x]),x]","-\frac{2 (b c-a d)^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)}{c d f (c+d) \sqrt{-\tan ^2(e+f x)} \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{c f}+\frac{2 b \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)^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)}{c d f (c+d) \sqrt{-\tan ^2(e+f x)} \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{c f}+\frac{2 b \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}",1,"(2*b*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(d*f) - (2*a*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(c*f) - (2*(b*c - a*d)^2*EllipticPi[(2*d)/(c + d), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*b)/(a + b)]*Sqrt[(a + b*Sec[e + f*x])/(a + b)]*Tan[e + f*x])/(c*d*(c + d)*f*Sqrt[a + b*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])","A",5,5,27,0.1852,1,"{3928, 3921, 3784, 3832, 3973}"
202,1,442,0,0.6292782,"\int (a+b \sec (e+f x))^{5/2} (c+d \sec (e+f x)) \, dx","Int[(a + b*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x]),x]","\frac{2 \sqrt{a+b} \left(a^2 b (45 c-23 d)+15 a^3 d-a b^2 (35 c-17 d)+b^3 (5 c-9 d)\right) \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)}{15 b f}-\frac{2 (a-b) \sqrt{a+b} \left(23 a^2 d+35 a b c+9 b^2 d\right) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b f}-\frac{2 a^2 c \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}+\frac{2 b (8 a d+5 b c) \tan (e+f x) \sqrt{a+b \sec (e+f x)}}{15 f}+\frac{2 b d \tan (e+f x) (a+b \sec (e+f x))^{3/2}}{5 f}","\frac{2 \sqrt{a+b} \left(a^2 b (45 c-23 d)+15 a^3 d-a b^2 (35 c-17 d)+b^3 (5 c-9 d)\right) \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)}{15 b f}-\frac{2 (a-b) \sqrt{a+b} \left(23 a^2 d+35 a b c+9 b^2 d\right) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b f}-\frac{2 a^2 c \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}+\frac{2 b (8 a d+5 b c) \tan (e+f x) \sqrt{a+b \sec (e+f x)}}{15 f}+\frac{2 b d \tan (e+f x) (a+b \sec (e+f x))^{3/2}}{5 f}",1,"(-2*(a - b)*Sqrt[a + b]*(35*a*b*c + 23*a^2*d + 9*b^2*d)*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(15*b*f) + (2*Sqrt[a + b]*(a^2*b*(45*c - 23*d) - a*b^2*(35*c - 17*d) + b^3*(5*c - 9*d) + 15*a^3*d)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(15*b*f) - (2*a^2*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f + (2*b*(5*b*c + 8*a*d)*Sqrt[a + b*Sec[e + f*x]]*Tan[e + f*x])/(15*f) + (2*b*d*(a + b*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(5*f)","A",7,7,25,0.2800,1,"{3918, 4056, 4058, 3921, 3784, 3832, 4004}"
203,1,208,0,0.116644,"\int \frac{c+d \sec (e+f x)}{\sqrt{a+b \sec (e+f x)}} \, dx","Int[(c + d*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]],x]","\frac{2 d \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 f}-\frac{2 c \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a f}","\frac{2 d \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 f}-\frac{2 c \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a f}",1,"(2*Sqrt[a + b]*d*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(b*f) - (2*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*f)","A",3,3,25,0.1200,1,"{3921, 3784, 3832}"
204,1,216,0,0.2382349,"\int \frac{1}{\sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx","Int[1/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]","-\frac{2 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)}{c f (c+d) \sqrt{-\tan ^2(e+f x)} \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a c f}","-\frac{2 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)}{c f (c+d) \sqrt{-\tan ^2(e+f x)} \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a c f}",1,"(-2*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*c*f) - (2*d*EllipticPi[(2*d)/(c + d), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*b)/(a + b)]*Sqrt[(a + b*Sec[e + f*x])/(a + b)]*Tan[e + f*x])/(c*(c + d)*f*Sqrt[a + b*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])","A",3,3,27,0.1111,1,"{3930, 3784, 3973}"
205,1,376,0,0.4275586,"\int \frac{c+d \sec (e+f x)}{(a+b \sec (e+f x))^{3/2}} \, dx","Int[(c + d*Sec[e + f*x])/(a + b*Sec[e + f*x])^(3/2),x]","\frac{2 b (b c-a d) \tan (e+f x)}{a f \left(a^2-b^2\right) \sqrt{a+b \sec (e+f x)}}-\frac{2 c \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 f}-\frac{2 (b c-a d) \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)}{a b f \sqrt{a+b}}+\frac{2 (b c-a d) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b f \sqrt{a+b}}","\frac{2 b (b c-a d) \tan (e+f x)}{a f \left(a^2-b^2\right) \sqrt{a+b \sec (e+f x)}}-\frac{2 c \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 f}-\frac{2 (b c-a d) \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)}{a b f \sqrt{a+b}}+\frac{2 (b c-a d) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b f \sqrt{a+b}}",1,"(2*(b*c - a*d)*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*b*Sqrt[a + b]*f) - (2*(b*c - a*d)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*b*Sqrt[a + b]*f) - (2*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a^2*f) + (2*b*(b*c - a*d)*Tan[e + f*x])/(a*(a^2 - b^2)*f*Sqrt[a + b*Sec[e + f*x]])","A",6,6,25,0.2400,1,"{3923, 4058, 3921, 3784, 3832, 4004}"
206,1,495,0,0.7775033,"\int \frac{c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx","Int[(c + d*Sec[e + f*x])/(a + b*Sec[e + f*x])^(5/2),x]","\frac{2 b \left(7 a^2 b c-4 a^3 d-3 b^3 c\right) \tan (e+f x)}{3 a^2 f \left(a^2-b^2\right)^2 \sqrt{a+b \sec (e+f x)}}+\frac{2 b (b c-a d) \tan (e+f x)}{3 a f \left(a^2-b^2\right) (a+b \sec (e+f x))^{3/2}}-\frac{2 \left(6 a^2 b c+a^2 b d-3 a^3 d-a b^2 c-3 b^3 c\right) \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)}{3 a^2 b f (a-b) (a+b)^{3/2}}+\frac{2 \left(7 a^2 b c-4 a^3 d-3 b^3 c\right) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 b f (a-b) (a+b)^{3/2}}-\frac{2 c \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 f}","\frac{2 b \left(7 a^2 b c-4 a^3 d-3 b^3 c\right) \tan (e+f x)}{3 a^2 f \left(a^2-b^2\right)^2 \sqrt{a+b \sec (e+f x)}}+\frac{2 b (b c-a d) \tan (e+f x)}{3 a f \left(a^2-b^2\right) (a+b \sec (e+f x))^{3/2}}-\frac{2 \left(6 a^2 b c+a^2 b d-3 a^3 d-a b^2 c-3 b^3 c\right) \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)}{3 a^2 b f (a-b) (a+b)^{3/2}}+\frac{2 \left(7 a^2 b c-4 a^3 d-3 b^3 c\right) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 b f (a-b) (a+b)^{3/2}}-\frac{2 c \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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 f}",1,"(2*(7*a^2*b*c - 3*b^3*c - 4*a^3*d)*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(3*a^2*(a - b)*b*(a + b)^(3/2)*f) - (2*(6*a^2*b*c - a*b^2*c - 3*b^3*c - 3*a^3*d + a^2*b*d)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(3*a^2*(a - b)*b*(a + b)^(3/2)*f) - (2*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a^3*f) + (2*b*(b*c - a*d)*Tan[e + f*x])/(3*a*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^(3/2)) + (2*b*(7*a^2*b*c - 3*b^3*c - 4*a^3*d)*Tan[e + f*x])/(3*a^2*(a^2 - b^2)^2*f*Sqrt[a + b*Sec[e + f*x]])","A",7,7,25,0.2800,1,"{3923, 4060, 4058, 3921, 3784, 3832, 4004}"
207,1,389,0,0.4466058,"\int \sqrt{a+b \sec (e+f x)} \sqrt{c+d \sec (e+f x)} \, dx","Int[Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]],x]","\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)}{f \sqrt{\frac{a+b}{c+d}}}-\frac{2 \sqrt{c+d} \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{a (c+d)}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{f \sqrt{a+b}}","\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)}{f \sqrt{\frac{a+b}{c+d}}}-\frac{2 \sqrt{c+d} \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{a (c+d)}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{f \sqrt{a+b}}",1,"(-2*Sqrt[c + d]*Cot[e + f*x]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e + f*x]))/(Sqrt[a + b]*f) + (2*Cot[e + f*x]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[(a + b)/(c + d)]*Sqrt[c + d*Sec[e + f*x]])/Sqrt[a + b*Sec[e + f*x]]], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e + f*x]))/(Sqrt[(a + b)/(c + d)]*f)","A",3,3,29,0.1034,1,"{3932, 3936, 3982}"
208,1,198,0,0.1061486,"\int \frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{c+d \sec (e+f x)}} \, dx","Int[Sqrt[a + b*Sec[e + f*x]]/Sqrt[c + d*Sec[e + f*x]],x]","-\frac{2 \sqrt{c+d} \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{a (c+d)}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{c f \sqrt{a+b}}","-\frac{2 \sqrt{c+d} \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{a (c+d)}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{c f \sqrt{a+b}}",1,"(-2*Sqrt[c + d]*Cot[e + f*x]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e + f*x]))/(Sqrt[a + b]*c*f)","A",1,1,29,0.03448,1,"{3936}"
209,1,598,0,0.9020301,"\int \frac{\sqrt{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{3/2}} \, dx","Int[Sqrt[a + b*Sec[e + f*x]]/(c + d*Sec[e + f*x])^(3/2),x]","-\frac{2 \sqrt{c+d} \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{a (c+d)}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{c^2 f \sqrt{a+b}}-\frac{2 d (a-b) \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)}{c f (c-d) \sqrt{c+d} (b c-a d)}-\frac{2 d \sqrt{a+b} \cot (e+f x) (\sec (e+f x)+1) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} E\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)}{c f (c-d) \sqrt{c+d} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}}}","-\frac{2 \sqrt{c+d} \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{a (c+d)}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{c^2 f \sqrt{a+b}}-\frac{2 d (a-b) \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)}{c f (c-d) \sqrt{c+d} (b c-a d)}-\frac{2 d \sqrt{a+b} \cot (e+f x) (\sec (e+f x)+1) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} E\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)}{c f (c-d) \sqrt{c+d} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}}}",1,"(-2*Sqrt[c + d]*Cot[e + f*x]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e + f*x]))/(Sqrt[a + b]*c^2*f) - (2*Sqrt[a + b]*d*Cot[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*(1 + Sec[e + f*x])*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))])/(c*(c - d)*Sqrt[c + d]*f*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]) - (2*(a - b)*Sqrt[a + b]*d*Cot[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d*Sec[e + f*x]))/(c*(c - d)*Sqrt[c + d]*(b*c - a*d)*f)","A",5,5,29,0.1724,1,"{3939, 3936, 3986, 3984, 3994}"
210,1,899,0,2.2575608,"\int \frac{\sqrt{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx","Int[Sqrt[a + b*Sec[e + f*x]]/(c + d*Sec[e + f*x])^(5/2),x]","\frac{2 \sqrt{a+b \sec (e+f x)} \sin (e+f x) d^2}{3 c \left(c^2-d^2\right) f (d+c \cos (e+f x)) \sqrt{c+d \sec (e+f x)}}+\frac{2 (a-b) \sqrt{a+b} \left(6 b c^3-7 a d c^2-2 b d^2 c+3 a d^3\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} d}{3 c^2 (c-d)^2 (c+d)^{3/2} (b c-a d)^2 f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{a+b} \left(b c^2 \left(3 c^2+3 d c-2 d^2\right)-a d \left(9 c^3-2 d c^2-6 d^2 c+3 d^3\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{3 c^3 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{c^3 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}","\frac{2 \sqrt{a+b \sec (e+f x)} \sin (e+f x) d^2}{3 c \left(c^2-d^2\right) f (d+c \cos (e+f x)) \sqrt{c+d \sec (e+f x)}}+\frac{2 (a-b) \sqrt{a+b} \left(6 b c^3-7 a d c^2-2 b d^2 c+3 a d^3\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} d}{3 c^2 (c-d)^2 (c+d)^{3/2} (b c-a d)^2 f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{a+b} \left(b c^2 \left(3 c^2+3 d c-2 d^2\right)-a d \left(9 c^3-2 d c^2-6 d^2 c+3 d^3\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{3 c^3 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{c^3 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}",1,"(2*(a - b)*Sqrt[a + b]*d*(6*b*c^3 - 7*a*c^2*d - 2*b*c*d^2 + 3*a*d^3)*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(3*c^2*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^2*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*Sqrt[a + b]*(b*c^2*(3*c^2 + 3*c*d - 2*d^2) - a*d*(9*c^3 - 2*c^2*d - 6*c*d^2 + 3*d^3))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(3*c^3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^3*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*d^2*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]])","A",7,7,29,0.2414,1,"{3942, 3048, 3053, 2811, 2998, 2818, 2996}"
211,1,744,0,1.1062871,"\int \frac{(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx","Int[(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^(3/2),x]","-\frac{2 \sqrt{a+b} (b c-a (2 c-d)) \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{c^2 f (c-d) \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}-\frac{2 a \sqrt{a+b} \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{c^2 f \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}-\frac{2 (a-b) \sqrt{a+b} \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{c f (c-d) \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}","-\frac{2 \sqrt{a+b} (b c-a (2 c-d)) \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{c^2 f (c-d) \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}-\frac{2 a \sqrt{a+b} \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{c^2 f \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}-\frac{2 (a-b) \sqrt{a+b} \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{c f (c-d) \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}",1,"(-2*(a - b)*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c*(c - d)*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*Sqrt[a + b]*(b*c - a*(2*c - d))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^2*(c - d)*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*a*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^2*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])","A",6,6,29,0.2069,1,"{3942, 2798, 2811, 2998, 2818, 2996}"
212,1,919,0,2.1324439,"\int \frac{(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{5/2}} \, dx","Int[(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^(5/2),x]","-\frac{2 (a-b) \sqrt{a+b} \left(3 b c^3-7 a d c^2+b d^2 c+3 a d^3\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} (d+c \cos (e+f x))^{3/2}}{3 c^2 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 \sqrt{a+b} \left(b^2 (3 c+d) c^3-2 a b \left(3 c^2+2 d c-d^2\right) c^2+a^2 d \left(9 c^3-2 d c^2-6 d^2 c+3 d^3\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} (d+c \cos (e+f x))^{3/2}}{3 c^3 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 a \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} (d+c \cos (e+f x))^{3/2}}{c^3 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 d (b c-a d) \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{3 c \left(c^2-d^2\right) f \sqrt{c+d \sec (e+f x)} (d+c \cos (e+f x))}","-\frac{2 (a-b) \sqrt{a+b} \left(3 b c^3-7 a d c^2+b d^2 c+3 a d^3\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} (d+c \cos (e+f x))^{3/2}}{3 c^2 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 \sqrt{a+b} \left(b^2 (3 c+d) c^3-2 a b \left(3 c^2+2 d c-d^2\right) c^2+a^2 d \left(9 c^3-2 d c^2-6 d^2 c+3 d^3\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} (d+c \cos (e+f x))^{3/2}}{3 c^3 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 a \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} (d+c \cos (e+f x))^{3/2}}{c^3 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 d (b c-a d) \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{3 c \left(c^2-d^2\right) f \sqrt{c+d \sec (e+f x)} (d+c \cos (e+f x))}",1,"(-2*(a - b)*Sqrt[a + b]*(3*b*c^3 - 7*a*c^2*d + b*c*d^2 + 3*a*d^3)*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(3*c^2*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*Sqrt[a + b]*(b^2*c^3*(3*c + d) - 2*a*b*c^2*(3*c^2 + 2*c*d - d^2) + a^2*d*(9*c^3 - 2*c^2*d - 6*c*d^2 + 3*d^3))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(3*c^3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*a*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^3*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*d*(b*c - a*d)*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]])","A",7,7,29,0.2414,1,"{3942, 2989, 3053, 2811, 2998, 2818, 2996}"
213,1,1122,0,3.1627908,"\int \frac{(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{7/2}} \, dx","Int[(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^(7/2),x]","\frac{2 (b+a \cos (e+f x)) \sqrt{a+b \sec (e+f x)} \sin (e+f x) d^2}{5 c \left(c^2-d^2\right) f (d+c \cos (e+f x))^2 \sqrt{c+d \sec (e+f x)}}-\frac{2 \left(10 b c^3-13 a d c^2-2 b d^2 c+5 a d^3\right) \sqrt{a+b \sec (e+f x)} \sin (e+f x) d}{15 c^2 \left(c^2-d^2\right)^2 f (d+c \cos (e+f x)) \sqrt{c+d \sec (e+f x)}}+\frac{2 (a-b) \sqrt{a+b} \left(-\left(15 c^6+19 d^2 c^4-2 d^4 c^2\right) b^2+2 a c d \left(35 c^4-8 d^2 c^2+5 d^4\right) b-a^2 d^2 \left(58 c^4-41 d^2 c^2+15 d^4\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{15 c^3 (c-d)^3 (c+d)^{5/2} (b c-a d)^2 f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 \sqrt{a+b} \left(b^2 \left(15 c^3+10 d c^2+9 d^2 c-2 d^3\right) c^3-2 a b \left(15 c^4+20 d c^3-4 d^2 c^2-4 d^3 c+5 d^4\right) c^2+a^2 d \left(60 c^5-2 d c^4-66 d^2 c^3+25 d^3 c^2+30 d^4 c-15 d^5\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{15 c^4 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 a \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{c^4 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}","\frac{2 (b+a \cos (e+f x)) \sqrt{a+b \sec (e+f x)} \sin (e+f x) d^2}{5 c \left(c^2-d^2\right) f (d+c \cos (e+f x))^2 \sqrt{c+d \sec (e+f x)}}-\frac{2 \left(10 b c^3-13 a d c^2-2 b d^2 c+5 a d^3\right) \sqrt{a+b \sec (e+f x)} \sin (e+f x) d}{15 c^2 \left(c^2-d^2\right)^2 f (d+c \cos (e+f x)) \sqrt{c+d \sec (e+f x)}}+\frac{2 (a-b) \sqrt{a+b} \left(-\left(15 c^6+19 d^2 c^4-2 d^4 c^2\right) b^2+2 a c d \left(35 c^4-8 d^2 c^2+5 d^4\right) b-a^2 d^2 \left(58 c^4-41 d^2 c^2+15 d^4\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{15 c^3 (c-d)^3 (c+d)^{5/2} (b c-a d)^2 f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 \sqrt{a+b} \left(b^2 \left(15 c^3+10 d c^2+9 d^2 c-2 d^3\right) c^3-2 a b \left(15 c^4+20 d c^3-4 d^2 c^2-4 d^3 c+5 d^4\right) c^2+a^2 d \left(60 c^5-2 d c^4-66 d^2 c^3+25 d^3 c^2+30 d^4 c-15 d^5\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{15 c^4 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 a \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{c^4 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}",1,"(2*(a - b)*Sqrt[a + b]*(2*a*b*c*d*(35*c^4 - 8*c^2*d^2 + 5*d^4) - a^2*d^2*(58*c^4 - 41*c^2*d^2 + 15*d^4) - b^2*(15*c^6 + 19*c^4*d^2 - 2*c^2*d^4))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(15*c^3*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)^2*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*Sqrt[a + b]*(b^2*c^3*(15*c^3 + 10*c^2*d + 9*c*d^2 - 2*d^3) - 2*a*b*c^2*(15*c^4 + 20*c^3*d - 4*c^2*d^2 - 4*c*d^3 + 5*d^4) + a^2*d*(60*c^5 - 2*c^4*d - 66*c^3*d^2 + 25*c^2*d^3 + 30*c*d^4 - 15*d^5))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(15*c^4*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*a*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^4*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*d^2*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(5*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^2*Sqrt[c + d*Sec[e + f*x]]) - (2*d*(10*b*c^3 - 13*a*c^2*d - 2*b*c*d^2 + 5*a*d^3)*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(15*c^2*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]])","A",8,8,29,0.2759,1,"{3942, 3048, 3047, 3053, 2811, 2998, 2818, 2996}"
214,1,891,0,2.008923,"\int \frac{(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{5/2}} \, dx","Int[(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^(5/2),x]","-\frac{2 \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} a^2}{c^3 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 (b c-a d)^2 \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{3 c \left(c^2-d^2\right) f (d+c \cos (e+f x)) \sqrt{c+d \sec (e+f x)}}-\frac{2 (a-b) \sqrt{a+b} \left(7 a c^2-4 b d c-3 a d^2\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{3 c^2 (c-d)^2 (c+d)^{3/2} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{a+b} \left(\left(9 c^3-2 d c^2-6 d^2 c+3 d^3\right) a^2-b c \left(7 c^2+4 d c-3 d^2\right) a+b^2 c^2 (c+3 d)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{3 c^3 (c-d)^2 (c+d)^{3/2} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}","-\frac{2 \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} a^2}{c^3 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 (b c-a d)^2 \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{3 c \left(c^2-d^2\right) f (d+c \cos (e+f x)) \sqrt{c+d \sec (e+f x)}}-\frac{2 (a-b) \sqrt{a+b} \left(7 a c^2-4 b d c-3 a d^2\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{3 c^2 (c-d)^2 (c+d)^{3/2} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{a+b} \left(\left(9 c^3-2 d c^2-6 d^2 c+3 d^3\right) a^2-b c \left(7 c^2+4 d c-3 d^2\right) a+b^2 c^2 (c+3 d)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{3 c^3 (c-d)^2 (c+d)^{3/2} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}",1,"(-2*(a - b)*Sqrt[a + b]*(7*a*c^2 - 4*b*c*d - 3*a*d^2)*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(3*c^2*(c - d)^2*(c + d)^(3/2)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*Sqrt[a + b]*(b^2*c^2*(c + 3*d) - a*b*c*(7*c^2 + 4*c*d - 3*d^2) + a^2*(9*c^3 - 2*c^2*d - 6*c*d^2 + 3*d^3))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(3*c^3*(c - d)^2*(c + d)^(3/2)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*a^2*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^3*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*(b*c - a*d)^2*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]])","A",7,7,29,0.2414,1,"{3942, 2792, 3053, 2811, 2998, 2818, 2996}"
215,1,1150,0,3.4351435,"\int \frac{(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{7/2}} \, dx","Int[(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^(7/2),x]","-\frac{2 \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} a^2}{c^4 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 (b c-a d) \left(5 b c^3-13 a d c^2+3 b d^2 c+5 a d^3\right) \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{15 c^2 \left(c^2-d^2\right)^2 f (d+c \cos (e+f x)) \sqrt{c+d \sec (e+f x)}}-\frac{2 d (b c-a d) (b+a \cos (e+f x)) \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{5 c \left(c^2-d^2\right) f (d+c \cos (e+f x))^2 \sqrt{c+d \sec (e+f x)}}+\frac{2 (a-b) \sqrt{a+b} \left(\left(15 d^5-41 c^2 d^3+58 c^4 d\right) a^2-b c \left(35 c^4+34 d^2 c^2-5 d^4\right) a+b^2 c^2 d \left(29 c^2+3 d^2\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{15 c^3 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{a+b} \left(b^3 \left(5 c^2+24 d c+3 d^2\right) c^4-a b^2 \left(35 c^3+42 d c^2+21 d^2 c-2 d^3\right) c^3+a^2 b \left(45 c^4+48 d c^3+d^2 c^2-8 d^3 c+10 d^4\right) c^2-a^3 d \left(60 c^5-2 d c^4-66 d^2 c^3+25 d^3 c^2+30 d^4 c-15 d^5\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{15 c^4 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}","-\frac{2 \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} a^2}{c^4 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 (b c-a d) \left(5 b c^3-13 a d c^2+3 b d^2 c+5 a d^3\right) \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{15 c^2 \left(c^2-d^2\right)^2 f (d+c \cos (e+f x)) \sqrt{c+d \sec (e+f x)}}-\frac{2 d (b c-a d) (b+a \cos (e+f x)) \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{5 c \left(c^2-d^2\right) f (d+c \cos (e+f x))^2 \sqrt{c+d \sec (e+f x)}}+\frac{2 (a-b) \sqrt{a+b} \left(\left(15 d^5-41 c^2 d^3+58 c^4 d\right) a^2-b c \left(35 c^4+34 d^2 c^2-5 d^4\right) a+b^2 c^2 d \left(29 c^2+3 d^2\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{15 c^3 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{a+b} \left(b^3 \left(5 c^2+24 d c+3 d^2\right) c^4-a b^2 \left(35 c^3+42 d c^2+21 d^2 c-2 d^3\right) c^3+a^2 b \left(45 c^4+48 d c^3+d^2 c^2-8 d^3 c+10 d^4\right) c^2-a^3 d \left(60 c^5-2 d c^4-66 d^2 c^3+25 d^3 c^2+30 d^4 c-15 d^5\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{15 c^4 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}",1,"(2*(a - b)*Sqrt[a + b]*(b^2*c^2*d*(29*c^2 + 3*d^2) - a*b*c*(35*c^4 + 34*c^2*d^2 - 5*d^4) + a^2*(58*c^4*d - 41*c^2*d^3 + 15*d^5))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(15*c^3*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*Sqrt[a + b]*(b^3*c^4*(5*c^2 + 24*c*d + 3*d^2) - a*b^2*c^3*(35*c^3 + 42*c^2*d + 21*c*d^2 - 2*d^3) + a^2*b*c^2*(45*c^4 + 48*c^3*d + c^2*d^2 - 8*c*d^3 + 10*d^4) - a^3*d*(60*c^5 - 2*c^4*d - 66*c^3*d^2 + 25*c^2*d^3 + 30*c*d^4 - 15*d^5))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(15*c^4*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*a^2*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^4*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*d*(b*c - a*d)*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(5*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^2*Sqrt[c + d*Sec[e + f*x]]) + (2*(b*c - a*d)*(5*b*c^3 - 13*a*c^2*d + 3*b*c*d^2 + 5*a*d^3)*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(15*c^2*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]])","A",8,8,29,0.2759,1,"{3942, 2989, 3047, 3053, 2811, 2998, 2818, 2996}"
216,1,1428,0,5.4378885,"\int \frac{(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{9/2}} \, dx","Int[(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^(9/2),x]","-\frac{2 \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} a^2}{c^5 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 \left(-\left(35 c^6+67 d^2 c^4-6 d^4 c^2\right) b^2+2 a c d \left(91 c^4-2 d^2 c^2+7 d^4\right) b-a^2 d^2 \left(162 c^4-101 d^2 c^2+35 d^4\right)\right) \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{105 c^3 \left(c^2-d^2\right)^3 f (d+c \cos (e+f x)) \sqrt{c+d \sec (e+f x)}}-\frac{2 d \left(14 b c^3-19 a d c^2-2 b d^2 c+7 a d^3\right) (b+a \cos (e+f x)) \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{35 c^2 \left(c^2-d^2\right)^2 f (d+c \cos (e+f x))^2 \sqrt{c+d \sec (e+f x)}}+\frac{2 d^2 (b+a \cos (e+f x))^2 \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{7 c \left(c^2-d^2\right) f (d+c \cos (e+f x))^3 \sqrt{c+d \sec (e+f x)}}+\frac{2 (a-b) \sqrt{a+b} \left(-\left(-105 d^8+392 c^2 d^6-485 c^4 d^4+582 c^6 d^2\right) a^3+2 b c d \left(406 c^6+73 d^2 c^4+132 d^4 c^2-35 d^6\right) a^2-b^2 c^2 \left(245 c^6+852 d^2 c^4+41 d^4 c^2+14 d^6\right) a+2 b^3 c^3 d \left(133 c^4+62 d^2 c^2-3 d^4\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{105 c^4 (c-d)^4 (c+d)^{7/2} (b c-a d)^2 f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{a+b} \left(b^3 \left(35 c^4+231 d c^3+67 d^2 c^2+57 d^3 c-6 d^4\right) c^4-a b^2 \left(245 c^5+413 d c^4+439 d^2 c^3+53 d^3 c^2-12 d^4 c+14 d^5\right) c^3+a^2 b \left(315 c^6+497 d c^5+219 d^2 c^4-73 d^3 c^3+208 d^4 c^2+56 d^5 c-70 d^6\right) c^2-a^3 d \left(525 c^7+57 d c^6-699 d^2 c^5+214 d^3 c^4+672 d^4 c^3-280 d^5 c^2-210 d^6 c+105 d^7\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{105 c^5 (c-d)^4 (c+d)^{7/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}","-\frac{2 \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} a^2}{c^5 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 \left(-\left(35 c^6+67 d^2 c^4-6 d^4 c^2\right) b^2+2 a c d \left(91 c^4-2 d^2 c^2+7 d^4\right) b-a^2 d^2 \left(162 c^4-101 d^2 c^2+35 d^4\right)\right) \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{105 c^3 \left(c^2-d^2\right)^3 f (d+c \cos (e+f x)) \sqrt{c+d \sec (e+f x)}}-\frac{2 d \left(14 b c^3-19 a d c^2-2 b d^2 c+7 a d^3\right) (b+a \cos (e+f x)) \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{35 c^2 \left(c^2-d^2\right)^2 f (d+c \cos (e+f x))^2 \sqrt{c+d \sec (e+f x)}}+\frac{2 d^2 (b+a \cos (e+f x))^2 \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{7 c \left(c^2-d^2\right) f (d+c \cos (e+f x))^3 \sqrt{c+d \sec (e+f x)}}+\frac{2 (a-b) \sqrt{a+b} \left(-\left(-105 d^8+392 c^2 d^6-485 c^4 d^4+582 c^6 d^2\right) a^3+2 b c d \left(406 c^6+73 d^2 c^4+132 d^4 c^2-35 d^6\right) a^2-b^2 c^2 \left(245 c^6+852 d^2 c^4+41 d^4 c^2+14 d^6\right) a+2 b^3 c^3 d \left(133 c^4+62 d^2 c^2-3 d^4\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{105 c^4 (c-d)^4 (c+d)^{7/2} (b c-a d)^2 f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{a+b} \left(b^3 \left(35 c^4+231 d c^3+67 d^2 c^2+57 d^3 c-6 d^4\right) c^4-a b^2 \left(245 c^5+413 d c^4+439 d^2 c^3+53 d^3 c^2-12 d^4 c+14 d^5\right) c^3+a^2 b \left(315 c^6+497 d c^5+219 d^2 c^4-73 d^3 c^3+208 d^4 c^2+56 d^5 c-70 d^6\right) c^2-a^3 d \left(525 c^7+57 d c^6-699 d^2 c^5+214 d^3 c^4+672 d^4 c^3-280 d^5 c^2-210 d^6 c+105 d^7\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{105 c^5 (c-d)^4 (c+d)^{7/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}",1,"(2*(a - b)*Sqrt[a + b]*(2*b^3*c^3*d*(133*c^4 + 62*c^2*d^2 - 3*d^4) + 2*a^2*b*c*d*(406*c^6 + 73*c^4*d^2 + 132*c^2*d^4 - 35*d^6) - a*b^2*c^2*(245*c^6 + 852*c^4*d^2 + 41*c^2*d^4 + 14*d^6) - a^3*(582*c^6*d^2 - 485*c^4*d^4 + 392*c^2*d^6 - 105*d^8))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(105*c^4*(c - d)^4*(c + d)^(7/2)*(b*c - a*d)^2*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*Sqrt[a + b]*(b^3*c^4*(35*c^4 + 231*c^3*d + 67*c^2*d^2 + 57*c*d^3 - 6*d^4) - a*b^2*c^3*(245*c^5 + 413*c^4*d + 439*c^3*d^2 + 53*c^2*d^3 - 12*c*d^4 + 14*d^5) + a^2*b*c^2*(315*c^6 + 497*c^5*d + 219*c^4*d^2 - 73*c^3*d^3 + 208*c^2*d^4 + 56*c*d^5 - 70*d^6) - a^3*d*(525*c^7 + 57*c^6*d - 699*c^5*d^2 + 214*c^4*d^3 + 672*c^3*d^4 - 280*c^2*d^5 - 210*c*d^6 + 105*d^7))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(105*c^5*(c - d)^4*(c + d)^(7/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*a^2*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^5*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*d^2*(b + a*Cos[e + f*x])^2*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(7*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^3*Sqrt[c + d*Sec[e + f*x]]) - (2*d*(14*b*c^3 - 19*a*c^2*d - 2*b*c*d^2 + 7*a*d^3)*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(35*c^2*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])^2*Sqrt[c + d*Sec[e + f*x]]) - (2*(2*a*b*c*d*(91*c^4 - 2*c^2*d^2 + 7*d^4) - a^2*d^2*(162*c^4 - 101*c^2*d^2 + 35*d^4) - b^2*(35*c^6 + 67*c^4*d^2 - 6*c^2*d^4))*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(105*c^3*(c^2 - d^2)^3*f*(d + c*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]])","A",9,8,29,0.2759,1,"{3942, 3048, 3047, 3053, 2811, 2998, 2818, 2996}"
217,0,0,0,0.092558,"\int \frac{(c+d \sec (e+f x))^{3/2}}{\sqrt{a+b \sec (e+f x)}} \, dx","Int[(c + d*Sec[e + f*x])^(3/2)/Sqrt[a + b*Sec[e + f*x]],x]","\int \frac{(c+d \sec (e+f x))^{3/2}}{\sqrt{a+b \sec (e+f x)}} \, dx","\frac{2 (b c-a d) \cot (e+f x) \sqrt{a+b \sec (e+f x)} \sqrt{c+d \sec (e+f x)} \sqrt{\frac{(b c-a d) (\sec (e+f x)-1)}{(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))}} F\left(\sin ^{-1}\left(\sqrt{\frac{(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{a b f \sqrt{\frac{(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}}-\frac{2 c (c+d) \cot (e+f x) (a+b \sec (e+f x))^{3/2} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \sqrt{\frac{(a+b) (b c-a d) (\sec (e+f x)-1) (c+d \sec (e+f x))}{(c+d)^2 (a+b \sec (e+f x))^2}} \Pi \left(\frac{a (c+d)}{(a+b) c};\sin ^{-1}\left(\sqrt{\frac{(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{a f (a+b) \sqrt{c+d \sec (e+f x)}}+\frac{2 d (c+d) \cot (e+f x) (a+b \sec (e+f x))^{3/2} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \sqrt{-\frac{(a+b) (a d-b c) (\sec (e+f x)-1) (c+d \sec (e+f x))}{(c+d)^2 (a+b \sec (e+f x))^2}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\sqrt{\frac{(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b f (a+b) \sqrt{c+d \sec (e+f x)}}",1,"Defer[Int][(c + d*Sec[e + f*x])^(3/2)/Sqrt[a + b*Sec[e + f*x]], x]","F",0,0,0,0,-1,"{}"
218,1,198,0,0.1109883,"\int \frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{a+b \sec (e+f x)}} \, dx","Int[Sqrt[c + d*Sec[e + f*x]]/Sqrt[a + b*Sec[e + f*x]],x]","-\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))}} \Pi \left(\frac{(a+b) c}{a (c+d)};\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)}{a f \sqrt{c+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))}} \Pi \left(\frac{(a+b) c}{a (c+d)};\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)}{a f \sqrt{c+d}}",1,"(-2*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d*Sec[e + f*x]))/(a*Sqrt[c + d]*f)","A",1,1,29,0.03448,1,"{3936}"
219,1,398,0,0.436726,"\int \frac{1}{\sqrt{a+b \sec (e+f x)} \sqrt{c+d \sec (e+f x)}} \, dx","Int[1/(Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]),x]","-\frac{2 b \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)}{a f \sqrt{c+d} (b c-a d)}-\frac{2 \sqrt{c+d} \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{a (c+d)}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{a c f \sqrt{a+b}}","-\frac{2 b \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)}{a f \sqrt{c+d} (b c-a d)}-\frac{2 \sqrt{c+d} \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{a (c+d)}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{a c f \sqrt{a+b}}",1,"(-2*Sqrt[c + d]*Cot[e + f*x]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e + f*x]))/(a*Sqrt[a + b]*c*f) - (2*b*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d*Sec[e + f*x]))/(a*Sqrt[c + d]*(b*c - a*d)*f)","A",3,3,29,0.1034,1,"{3938, 3936, 3984}"
220,1,763,0,1.315971,"\int \frac{1}{\sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx","Int[1/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^(3/2)),x]","-\frac{2 d \sqrt{a+b} (2 c-d) \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{c^2 f (c-d) \sqrt{c+d} (b c-a d) \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}-\frac{2 \sqrt{a+b} \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{a c^2 f \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}-\frac{2 d^2 (a-b) \sqrt{a+b} \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{c f (c-d) \sqrt{c+d} (b c-a d)^2 \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}","-\frac{2 d \sqrt{a+b} (2 c-d) \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)}{c^2 f (c-d) \sqrt{c+d} (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))}} \Pi \left(\frac{(a+b) c}{a (c+d)};\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)}{a c^2 f \sqrt{c+d}}-\frac{2 d^2 (a-b) \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))}} E\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)}{c f (c-d) \sqrt{c+d} (b c-a d)^2}",1,"(-2*(a - b)*Sqrt[a + b]*d^2*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c*(c - d)*Sqrt[c + d]*(b*c - a*d)^2*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*Sqrt[a + b]*(2*c - d)*d*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^2*(c - d)*Sqrt[c + d]*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(a*c^2*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])","A",6,6,29,0.2069,1,"{3942, 3054, 2811, 2998, 2818, 2996}"
221,0,0,0,0.1908791,"\int \frac{\sqrt[3]{a+b \sec (e+f x)}}{\sqrt[3]{c+d \sec (e+f x)}} \, dx","Int[(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(1/3),x]","\int \frac{\sqrt[3]{a+b \sec (e+f x)}}{\sqrt[3]{c+d \sec (e+f x)}} \, dx","\frac{\sqrt[3]{a+b \sec (e+f x)} \sqrt[3]{c \cos (e+f x)+d} \text{Int}\left(\frac{\sqrt[3]{a \cos (e+f x)+b}}{\sqrt[3]{c \cos (e+f x)+d}},x\right)}{\sqrt[3]{a \cos (e+f x)+b} \sqrt[3]{c+d \sec (e+f x)}}",0,"((d + c*Cos[e + f*x])^(1/3)*(a + b*Sec[e + f*x])^(1/3)*Defer[Int][(b + a*Cos[e + f*x])^(1/3)/(d + c*Cos[e + f*x])^(1/3), x])/((b + a*Cos[e + f*x])^(1/3)*(c + d*Sec[e + f*x])^(1/3))","A",0,0,0,0,-1,"{}"
222,0,0,0,0.0920152,"\int \frac{\sqrt[3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{4/3}} \, dx","Int[(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(4/3),x]","\int \frac{\sqrt[3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{4/3}} \, dx","\text{Int}\left(\frac{\sqrt[3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{4/3}},x\right)",0,"Defer[Int][(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(4/3), x]","A",0,0,0,0,-1,"{}"
223,0,0,0,0.091233,"\int \frac{\sqrt[3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{7/3}} \, dx","Int[(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(7/3),x]","\int \frac{\sqrt[3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{7/3}} \, dx","\text{Int}\left(\frac{\sqrt[3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{7/3}},x\right)",0,"Defer[Int][(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(7/3), x]","A",0,0,0,0,-1,"{}"
224,0,0,0,0.2202014,"\int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{2/3}} \, dx","Int[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(2/3),x]","\int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{2/3}} \, dx","\frac{(a+b \sec (e+f x))^{2/3} (c \cos (e+f x)+d)^{2/3} \text{Int}\left(\frac{(a \cos (e+f x)+b)^{2/3}}{(c \cos (e+f x)+d)^{2/3}},x\right)}{(a \cos (e+f x)+b)^{2/3} (c+d \sec (e+f x))^{2/3}}",0,"((d + c*Cos[e + f*x])^(2/3)*(a + b*Sec[e + f*x])^(2/3)*Defer[Int][(b + a*Cos[e + f*x])^(2/3)/(d + c*Cos[e + f*x])^(2/3), x])/((b + a*Cos[e + f*x])^(2/3)*(c + d*Sec[e + f*x])^(2/3))","A",0,0,0,0,-1,"{}"
225,0,0,0,0.0967324,"\int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{5/3}} \, dx","Int[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(5/3),x]","\int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{5/3}} \, dx","\text{Int}\left(\frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{5/3}},x\right)",0,"Defer[Int][(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(5/3), x]","A",0,0,0,0,-1,"{}"
226,0,0,0,0.0962087,"\int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx","Int[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(8/3),x]","\int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx","\text{Int}\left(\frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}},x\right)",0,"Defer[Int][(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(8/3), x]","A",0,0,0,0,-1,"{}"
227,0,0,0,0.2181742,"\int \frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{4/3}} \, dx","Int[(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(4/3),x]","\int \frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{4/3}} \, dx","\frac{(a+b \sec (e+f x))^{4/3} (c \cos (e+f x)+d)^{4/3} \text{Int}\left(\frac{(a \cos (e+f x)+b)^{4/3}}{(c \cos (e+f x)+d)^{4/3}},x\right)}{(a \cos (e+f x)+b)^{4/3} (c+d \sec (e+f x))^{4/3}}",0,"((d + c*Cos[e + f*x])^(4/3)*(a + b*Sec[e + f*x])^(4/3)*Defer[Int][(b + a*Cos[e + f*x])^(4/3)/(d + c*Cos[e + f*x])^(4/3), x])/((b + a*Cos[e + f*x])^(4/3)*(c + d*Sec[e + f*x])^(4/3))","A",0,0,0,0,-1,"{}"
228,0,0,0,0.0974406,"\int \frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{7/3}} \, dx","Int[(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(7/3),x]","\int \frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{7/3}} \, dx","\text{Int}\left(\frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{7/3}},x\right)",0,"Defer[Int][(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(7/3), x]","A",0,0,0,0,-1,"{}"
229,0,0,0,0.096693,"\int \frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx","Int[(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(10/3),x]","\int \frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx","\text{Int}\left(\frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}},x\right)",0,"Defer[Int][(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(10/3), x]","A",0,0,0,0,-1,"{}"
230,1,106,0,0.1893581,"\int \left(c (d \sec (e+f x))^p\right)^n (a+a \sec (e+f x))^m \, dx","Int[(c*(d*Sec[e + f*x])^p)^n*(a + a*Sec[e + f*x])^m,x]","-\frac{\tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(n p;\frac{1}{2},\frac{1}{2}-m;n p+1;\sec (e+f x),-\sec (e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p \sqrt{1-\sec (e+f x)}}","-\frac{\tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(n p;\frac{1}{2},\frac{1}{2}-m;n p+1;\sec (e+f x),-\sec (e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p \sqrt{1-\sec (e+f x)}}",1,"-((AppellF1[n*p, 1/2, 1/2 - m, 1 + n*p, Sec[e + f*x], -Sec[e + f*x]]*(c*(d*Sec[e + f*x])^p)^n*(1 + Sec[e + f*x])^(-1/2 - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*n*p*Sqrt[1 - Sec[e + f*x]]))","A",4,4,27,0.1481,1,"{3948, 3828, 3827, 133}"
231,1,275,0,0.4366033,"\int \left(c (d \sec (e+f x))^p\right)^n (a+a \sec (e+f x))^3 \, dx","Int[(c*(d*Sec[e + f*x])^p)^n*(a + a*Sec[e + f*x])^3,x]","-\frac{a^3 (4 n p+1) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f \left(1-n^2 p^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (4 n p+7) \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p (n p+2) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (2 n p+5) \tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+1) (n p+2)}+\frac{\tan (e+f x) \left(a^3 \sec (e+f x)+a^3\right) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+2)}","-\frac{a^3 (4 n p+1) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f \left(1-n^2 p^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (4 n p+7) \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p (n p+2) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (2 n p+5) \tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+1) (n p+2)}+\frac{\tan (e+f x) \left(a^3 \sec (e+f x)+a^3\right) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+2)}",1,"(a^3*(7 + 4*n*p)*Hypergeometric2F1[1/2, -(n*p)/2, (2 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*n*p*(2 + n*p)*Sqrt[Sin[e + f*x]^2]) - (a^3*(1 + 4*n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 - n*p)/2, (3 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*(1 - n^2*p^2)*Sqrt[Sin[e + f*x]^2]) + (a^3*(5 + 2*n*p)*(c*(d*Sec[e + f*x])^p)^n*Tan[e + f*x])/(f*(1 + n*p)*(2 + n*p)) + ((c*(d*Sec[e + f*x])^p)^n*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(f*(2 + n*p))","A",8,6,27,0.2222,1,"{3948, 3814, 3997, 3787, 3772, 2643}"
232,1,205,0,0.2487291,"\int \left(c (d \sec (e+f x))^p\right)^n (a+a \sec (e+f x))^2 \, dx","Int[(c*(d*Sec[e + f*x])^p)^n*(a + a*Sec[e + f*x])^2,x]","-\frac{a^2 (2 n p+1) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f \left(1-n^2 p^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{2 a^2 \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p \sqrt{\sin ^2(e+f x)}}+\frac{a^2 \tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+1)}","-\frac{a^2 (2 n p+1) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f \left(1-n^2 p^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{2 a^2 \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p \sqrt{\sin ^2(e+f x)}}+\frac{a^2 \tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+1)}",1,"(2*a^2*Hypergeometric2F1[1/2, -(n*p)/2, (2 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*n*p*Sqrt[Sin[e + f*x]^2]) - (a^2*(1 + 2*n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 - n*p)/2, (3 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*(1 - n^2*p^2)*Sqrt[Sin[e + f*x]^2]) + (a^2*(c*(d*Sec[e + f*x])^p)^n*Tan[e + f*x])/(f*(1 + n*p))","A",7,5,27,0.1852,1,"{3948, 3788, 3772, 2643, 4046}"
233,1,156,0,0.1460116,"\int \left(c (d \sec (e+f x))^p\right)^n (a+a \sec (e+f x)) \, dx","Int[(c*(d*Sec[e + f*x])^p)^n*(a + a*Sec[e + f*x]),x]","\frac{a \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p \sqrt{\sin ^2(e+f x)}}-\frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f (1-n p) \sqrt{\sin ^2(e+f x)}}","\frac{a \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p \sqrt{\sin ^2(e+f x)}}-\frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f (1-n p) \sqrt{\sin ^2(e+f x)}}",1,"(a*Hypergeometric2F1[1/2, -(n*p)/2, (2 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*n*p*Sqrt[Sin[e + f*x]^2]) - (a*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 - n*p)/2, (3 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*(1 - n*p)*Sqrt[Sin[e + f*x]^2])","A",6,4,25,0.1600,1,"{3948, 3787, 3772, 2643}"
234,1,208,0,0.2710225,"\int \frac{\left(c (d \sec (e+f x))^p\right)^n}{a+a \sec (e+f x)} \, dx","Int[(c*(d*Sec[e + f*x])^p)^n/(a + a*Sec[e + f*x]),x]","-\frac{\sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{a f \sqrt{\sin ^2(e+f x)}}+\frac{(1-n p) \sin (e+f x) \cos ^2(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (2-n p);\frac{1}{2} (4-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{a f (2-n p) \sqrt{\sin ^2(e+f x)}}+\frac{\sin (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f (a \sec (e+f x)+a)}","-\frac{\sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{a f \sqrt{\sin ^2(e+f x)}}+\frac{(1-n p) \sin (e+f x) \cos ^2(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (2-n p);\frac{1}{2} (4-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{a f (2-n p) \sqrt{\sin ^2(e+f x)}}+\frac{\sin (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f (a \sec (e+f x)+a)}",1,"((c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*(a + a*Sec[e + f*x])) - (Cos[e + f*x]*Hypergeometric2F1[1/2, (1 - n*p)/2, (3 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(a*f*Sqrt[Sin[e + f*x]^2]) + ((1 - n*p)*Cos[e + f*x]^2*Hypergeometric2F1[1/2, (2 - n*p)/2, (4 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(a*f*(2 - n*p)*Sqrt[Sin[e + f*x]^2])","A",7,5,27,0.1852,1,"{3948, 3820, 3787, 3772, 2643}"
235,1,248,0,0.4514743,"\int \frac{\left(c (d \sec (e+f x))^p\right)^n}{(a+a \sec (e+f x))^2} \, dx","Int[(c*(d*Sec[e + f*x])^p)^n/(a + a*Sec[e + f*x])^2,x]","\frac{2 (2-n p) \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{(3-2 n p) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{2 (2-n p) \tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{3 a^2 f (\sec (e+f x)+1)}-\frac{\tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{3 f (a \sec (e+f x)+a)^2}","\frac{2 (2-n p) \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{(3-2 n p) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{2 (2-n p) \tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{3 a^2 f (\sec (e+f x)+1)}-\frac{\tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{3 f (a \sec (e+f x)+a)^2}",1,"(2*(2 - n*p)*Hypergeometric2F1[1/2, -(n*p)/2, (2 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(3*a^2*f*Sqrt[Sin[e + f*x]^2]) - ((3 - 2*n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 - n*p)/2, (3 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(3*a^2*f*Sqrt[Sin[e + f*x]^2]) - (2*(2 - n*p)*(c*(d*Sec[e + f*x])^p)^n*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x])) - ((c*(d*Sec[e + f*x])^p)^n*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2)","A",8,6,27,0.2222,1,"{3948, 3817, 4020, 3787, 3772, 2643}"
236,0,0,0,0.1162309,"\int \left(c (d \sec (e+f x))^p\right)^n (a+b \sec (e+f x))^m \, dx","Int[(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x])^m,x]","\int \left(c (d \sec (e+f x))^p\right)^n (a+b \sec (e+f x))^m \, dx","(d \sec (e+f x))^{-n p} \left(c (d \sec (e+f x))^p\right)^n \text{Int}\left((a+b \sec (e+f x))^m (d \sec (e+f x))^{n p},x\right)",0,"((c*(d*Sec[e + f*x])^p)^n*Defer[Int][(d*Sec[e + f*x])^(n*p)*(a + b*Sec[e + f*x])^m, x])/(d*Sec[e + f*x])^(n*p)","A",0,0,0,0,-1,"{}"
237,1,296,0,0.5147257,"\int \left(c (d \sec (e+f x))^p\right)^n (a+b \sec (e+f x))^3 \, dx","Int[(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x])^3,x]","-\frac{a \left(a^2 (n p+1)+3 b^2 n p\right) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f \left(1-n^2 p^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{b \left(3 a^2 (n p+2)+b^2 (n p+1)\right) \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p (n p+2) \sqrt{\sin ^2(e+f x)}}+\frac{a b^2 (2 n p+5) \tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+1) (n p+2)}+\frac{b^2 \tan (e+f x) (a+b \sec (e+f x)) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+2)}","-\frac{a \left(a^2 (n p+1)+3 b^2 n p\right) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f \left(1-n^2 p^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{b \left(3 a^2 (n p+2)+b^2 (n p+1)\right) \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p (n p+2) \sqrt{\sin ^2(e+f x)}}+\frac{a b^2 (2 n p+5) \tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+1) (n p+2)}+\frac{b^2 \tan (e+f x) (a+b \sec (e+f x)) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+2)}",1,"(b*(b^2*(1 + n*p) + 3*a^2*(2 + n*p))*Hypergeometric2F1[1/2, -(n*p)/2, (2 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*n*p*(2 + n*p)*Sqrt[Sin[e + f*x]^2]) - (a*(3*b^2*n*p + a^2*(1 + n*p))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 - n*p)/2, (3 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*(1 - n^2*p^2)*Sqrt[Sin[e + f*x]^2]) + (a*b^2*(5 + 2*n*p)*(c*(d*Sec[e + f*x])^p)^n*Tan[e + f*x])/(f*(1 + n*p)*(2 + n*p)) + (b^2*(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x])*Tan[e + f*x])/(f*(2 + n*p))","A",8,6,27,0.2222,1,"{3948, 3842, 4047, 3772, 2643, 4046}"
238,1,211,0,0.2423887,"\int \left(c (d \sec (e+f x))^p\right)^n (a+b \sec (e+f x))^2 \, dx","Int[(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x])^2,x]","-\frac{\left(a^2 (n p+1)+b^2 n p\right) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f \left(1-n^2 p^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{2 a b \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p \sqrt{\sin ^2(e+f x)}}+\frac{b^2 \tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+1)}","-\frac{\left(a^2 (n p+1)+b^2 n p\right) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f \left(1-n^2 p^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{2 a b \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p \sqrt{\sin ^2(e+f x)}}+\frac{b^2 \tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+1)}",1,"(2*a*b*Hypergeometric2F1[1/2, -(n*p)/2, (2 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*n*p*Sqrt[Sin[e + f*x]^2]) - ((b^2*n*p + a^2*(1 + n*p))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 - n*p)/2, (3 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*(1 - n^2*p^2)*Sqrt[Sin[e + f*x]^2]) + (b^2*(c*(d*Sec[e + f*x])^p)^n*Tan[e + f*x])/(f*(1 + n*p))","A",7,5,27,0.1852,1,"{3948, 3788, 3772, 2643, 4046}"
239,1,156,0,0.1435643,"\int \left(c (d \sec (e+f x))^p\right)^n (a+b \sec (e+f x)) \, dx","Int[(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x]),x]","\frac{b \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p \sqrt{\sin ^2(e+f x)}}-\frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f (1-n p) \sqrt{\sin ^2(e+f x)}}","\frac{b \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p \sqrt{\sin ^2(e+f x)}}-\frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f (1-n p) \sqrt{\sin ^2(e+f x)}}",1,"(b*Hypergeometric2F1[1/2, -(n*p)/2, (2 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*n*p*Sqrt[Sin[e + f*x]^2]) - (a*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 - n*p)/2, (3 - n*p)/2, Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*(1 - n*p)*Sqrt[Sin[e + f*x]^2])","A",6,4,25,0.1600,1,"{3948, 3787, 3772, 2643}"
240,1,206,0,0.4023043,"\int \frac{\left(c (d \sec (e+f x))^p\right)^n}{a+b \sec (e+f x)} \, dx","Int[(c*(d*Sec[e + f*x])^p)^n/(a + b*Sec[e + f*x]),x]","\frac{a \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p-1)} \left(c (d \sec (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (n p-1),1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{b \sin (e+f x) \cos ^2(e+f x)^{\frac{n p}{2}} \left(c (d \sec (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{n p}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}","\frac{a \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p-1)} \left(c (d \sec (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (n p-1),1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{b \sin (e+f x) \cos ^2(e+f x)^{\frac{n p}{2}} \left(c (d \sec (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{n p}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}",1,"-((b*AppellF1[1/2, (n*p)/2, 1, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*(Cos[e + f*x]^2)^((n*p)/2)*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/((a^2 - b^2)*f)) + (a*AppellF1[1/2, (-1 + n*p)/2, 1, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(Cos[e + f*x]^2)^((-1 + n*p)/2)*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/((a^2 - b^2)*f)","A",7,5,27,0.1852,1,"{3948, 3869, 2823, 3189, 429}"
241,1,322,0,0.5593358,"\int \frac{\left(c (d \sec (e+f x))^p\right)^n}{(a+b \sec (e+f x))^2} \, dx","Int[(c*(d*Sec[e + f*x])^p)^n/(a + b*Sec[e + f*x])^2,x]","-\frac{2 a b \sin (e+f x) \cos ^2(e+f x)^{\frac{n p}{2}} \left(c (d \sec (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (n p-2),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}+\frac{a^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p-1)} \left(c (d \sec (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (n p-3),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}+\frac{b^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p-1)} \left(c (d \sec (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (n p-1),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}","-\frac{2 a b \sin (e+f x) \cos ^2(e+f x)^{\frac{n p}{2}} \left(c (d \sec (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (n p-2),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}+\frac{a^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p-1)} \left(c (d \sec (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (n p-3),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}+\frac{b^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p-1)} \left(c (d \sec (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (n p-1),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}",1,"(-2*a*b*AppellF1[1/2, (-2 + n*p)/2, 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*(Cos[e + f*x]^2)^((n*p)/2)*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/((a^2 - b^2)^2*f) + (a^2*AppellF1[1/2, (-3 + n*p)/2, 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(Cos[e + f*x]^2)^((-1 + n*p)/2)*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/((a^2 - b^2)^2*f) + (b^2*AppellF1[1/2, (-1 + n*p)/2, 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(Cos[e + f*x]^2)^((-1 + n*p)/2)*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/((a^2 - b^2)^2*f)","A",10,5,27,0.1852,1,"{3948, 3869, 2824, 3189, 429}"