1,1,151,0,0.0722948,"\int (a+a \sec (c+d x)) \tan ^9(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Tan[c + d*x]^9,x]","\frac{a \sec ^9(c+d x)}{9 d}+\frac{a \sec ^8(c+d x)}{8 d}-\frac{4 a \sec ^7(c+d x)}{7 d}-\frac{2 a \sec ^6(c+d x)}{3 d}+\frac{6 a \sec ^5(c+d x)}{5 d}+\frac{3 a \sec ^4(c+d x)}{2 d}-\frac{4 a \sec ^3(c+d x)}{3 d}-\frac{2 a \sec ^2(c+d x)}{d}+\frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}","\frac{a \sec ^9(c+d x)}{9 d}+\frac{a \sec ^8(c+d x)}{8 d}-\frac{4 a \sec ^7(c+d x)}{7 d}-\frac{2 a \sec ^6(c+d x)}{3 d}+\frac{6 a \sec ^5(c+d x)}{5 d}+\frac{3 a \sec ^4(c+d x)}{2 d}-\frac{4 a \sec ^3(c+d x)}{3 d}-\frac{2 a \sec ^2(c+d x)}{d}+\frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Log[Cos[c + d*x]])/d) + (a*Sec[c + d*x])/d - (2*a*Sec[c + d*x]^2)/d - (4*a*Sec[c + d*x]^3)/(3*d) + (3*a*Sec[c + d*x]^4)/(2*d) + (6*a*Sec[c + d*x]^5)/(5*d) - (2*a*Sec[c + d*x]^6)/(3*d) - (4*a*Sec[c + d*x]^7)/(7*d) + (a*Sec[c + d*x]^8)/(8*d) + (a*Sec[c + d*x]^9)/(9*d)","A",3,2,19,0.1053,1,"{3879, 88}"
2,1,118,0,0.0619443,"\int (a+a \sec (c+d x)) \tan ^7(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Tan[c + d*x]^7,x]","\frac{a \sec ^7(c+d x)}{7 d}+\frac{a \sec ^6(c+d x)}{6 d}-\frac{3 a \sec ^5(c+d x)}{5 d}-\frac{3 a \sec ^4(c+d x)}{4 d}+\frac{a \sec ^3(c+d x)}{d}+\frac{3 a \sec ^2(c+d x)}{2 d}-\frac{a \sec (c+d x)}{d}+\frac{a \log (\cos (c+d x))}{d}","\frac{a \sec ^7(c+d x)}{7 d}+\frac{a \sec ^6(c+d x)}{6 d}-\frac{3 a \sec ^5(c+d x)}{5 d}-\frac{3 a \sec ^4(c+d x)}{4 d}+\frac{a \sec ^3(c+d x)}{d}+\frac{3 a \sec ^2(c+d x)}{2 d}-\frac{a \sec (c+d x)}{d}+\frac{a \log (\cos (c+d x))}{d}",1,"(a*Log[Cos[c + d*x]])/d - (a*Sec[c + d*x])/d + (3*a*Sec[c + d*x]^2)/(2*d) + (a*Sec[c + d*x]^3)/d - (3*a*Sec[c + d*x]^4)/(4*d) - (3*a*Sec[c + d*x]^5)/(5*d) + (a*Sec[c + d*x]^6)/(6*d) + (a*Sec[c + d*x]^7)/(7*d)","A",3,2,19,0.1053,1,"{3879, 88}"
3,1,87,0,0.0491738,"\int (a+a \sec (c+d x)) \tan ^5(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Tan[c + d*x]^5,x]","\frac{a \sec ^5(c+d x)}{5 d}+\frac{a \sec ^4(c+d x)}{4 d}-\frac{2 a \sec ^3(c+d x)}{3 d}-\frac{a \sec ^2(c+d x)}{d}+\frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}","\frac{a \sec ^5(c+d x)}{5 d}+\frac{a \sec ^4(c+d x)}{4 d}-\frac{2 a \sec ^3(c+d x)}{3 d}-\frac{a \sec ^2(c+d x)}{d}+\frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Log[Cos[c + d*x]])/d) + (a*Sec[c + d*x])/d - (a*Sec[c + d*x]^2)/d - (2*a*Sec[c + d*x]^3)/(3*d) + (a*Sec[c + d*x]^4)/(4*d) + (a*Sec[c + d*x]^5)/(5*d)","A",3,2,19,0.1053,1,"{3879, 88}"
4,1,57,0,0.0391315,"\int (a+a \sec (c+d x)) \tan ^3(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Tan[c + d*x]^3,x]","\frac{a \sec ^3(c+d x)}{3 d}+\frac{a \sec ^2(c+d x)}{2 d}-\frac{a \sec (c+d x)}{d}+\frac{a \log (\cos (c+d x))}{d}","\frac{a \sec ^3(c+d x)}{3 d}+\frac{a \sec ^2(c+d x)}{2 d}-\frac{a \sec (c+d x)}{d}+\frac{a \log (\cos (c+d x))}{d}",1,"(a*Log[Cos[c + d*x]])/d - (a*Sec[c + d*x])/d + (a*Sec[c + d*x]^2)/(2*d) + (a*Sec[c + d*x]^3)/(3*d)","A",3,2,19,0.1053,1,"{3879, 75}"
5,1,25,0,0.0192775,"\int (a+a \sec (c+d x)) \tan (c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Tan[c + d*x],x]","\frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}","\frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Log[Cos[c + d*x]])/d) + (a*Sec[c + d*x])/d","A",3,2,17,0.1176,1,"{3879, 43}"
6,1,16,0,0.0203078,"\int \cot (c+d x) (a+a \sec (c+d x)) \, dx","Int[Cot[c + d*x]*(a + a*Sec[c + d*x]),x]","\frac{a \log (1-\cos (c+d x))}{d}","\frac{a \log (1-\cos (c+d x))}{d}",1,"(a*Log[1 - Cos[c + d*x]])/d","A",2,2,17,0.1176,1,"{3879, 31}"
7,1,57,0,0.0439983,"\int \cot ^3(c+d x) (a+a \sec (c+d x)) \, dx","Int[Cot[c + d*x]^3*(a + a*Sec[c + d*x]),x]","-\frac{a}{2 d (1-\cos (c+d x))}-\frac{3 a \log (1-\cos (c+d x))}{4 d}-\frac{a \log (\cos (c+d x)+1)}{4 d}","-\frac{a}{2 d (1-\cos (c+d x))}-\frac{3 a \log (1-\cos (c+d x))}{4 d}-\frac{a \log (\cos (c+d x)+1)}{4 d}",1,"-a/(2*d*(1 - Cos[c + d*x])) - (3*a*Log[1 - Cos[c + d*x]])/(4*d) - (a*Log[1 + Cos[c + d*x]])/(4*d)","A",3,2,19,0.1053,1,"{3879, 88}"
8,1,95,0,0.0643238,"\int \cot ^5(c+d x) (a+a \sec (c+d x)) \, dx","Int[Cot[c + d*x]^5*(a + a*Sec[c + d*x]),x]","\frac{3 a}{4 d (1-\cos (c+d x))}+\frac{a}{8 d (\cos (c+d x)+1)}-\frac{a}{8 d (1-\cos (c+d x))^2}+\frac{11 a \log (1-\cos (c+d x))}{16 d}+\frac{5 a \log (\cos (c+d x)+1)}{16 d}","\frac{3 a}{4 d (1-\cos (c+d x))}+\frac{a}{8 d (\cos (c+d x)+1)}-\frac{a}{8 d (1-\cos (c+d x))^2}+\frac{11 a \log (1-\cos (c+d x))}{16 d}+\frac{5 a \log (\cos (c+d x)+1)}{16 d}",1,"-a/(8*d*(1 - Cos[c + d*x])^2) + (3*a)/(4*d*(1 - Cos[c + d*x])) + a/(8*d*(1 + Cos[c + d*x])) + (11*a*Log[1 - Cos[c + d*x]])/(16*d) + (5*a*Log[1 + Cos[c + d*x]])/(16*d)","A",3,2,19,0.1053,1,"{3879, 88}"
9,1,133,0,0.0823685,"\int \cot ^7(c+d x) (a+a \sec (c+d x)) \, dx","Int[Cot[c + d*x]^7*(a + a*Sec[c + d*x]),x]","-\frac{15 a}{16 d (1-\cos (c+d x))}-\frac{a}{4 d (\cos (c+d x)+1)}+\frac{9 a}{32 d (1-\cos (c+d x))^2}+\frac{a}{32 d (\cos (c+d x)+1)^2}-\frac{a}{24 d (1-\cos (c+d x))^3}-\frac{21 a \log (1-\cos (c+d x))}{32 d}-\frac{11 a \log (\cos (c+d x)+1)}{32 d}","-\frac{15 a}{16 d (1-\cos (c+d x))}-\frac{a}{4 d (\cos (c+d x)+1)}+\frac{9 a}{32 d (1-\cos (c+d x))^2}+\frac{a}{32 d (\cos (c+d x)+1)^2}-\frac{a}{24 d (1-\cos (c+d x))^3}-\frac{21 a \log (1-\cos (c+d x))}{32 d}-\frac{11 a \log (\cos (c+d x)+1)}{32 d}",1,"-a/(24*d*(1 - Cos[c + d*x])^3) + (9*a)/(32*d*(1 - Cos[c + d*x])^2) - (15*a)/(16*d*(1 - Cos[c + d*x])) + a/(32*d*(1 + Cos[c + d*x])^2) - a/(4*d*(1 + Cos[c + d*x])) - (21*a*Log[1 - Cos[c + d*x]])/(32*d) - (11*a*Log[1 + Cos[c + d*x]])/(32*d)","A",3,2,19,0.1053,1,"{3879, 88}"
10,1,129,0,0.128695,"\int (a+a \sec (c+d x)) \tan ^8(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Tan[c + d*x]^8,x]","\frac{35 a \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{\tan ^7(c+d x) (7 a \sec (c+d x)+8 a)}{56 d}-\frac{\tan ^5(c+d x) (35 a \sec (c+d x)+48 a)}{240 d}+\frac{\tan ^3(c+d x) (35 a \sec (c+d x)+64 a)}{192 d}-\frac{\tan (c+d x) (35 a \sec (c+d x)+128 a)}{128 d}+a x","\frac{35 a \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{\tan ^7(c+d x) (7 a \sec (c+d x)+8 a)}{56 d}-\frac{\tan ^5(c+d x) (35 a \sec (c+d x)+48 a)}{240 d}+\frac{\tan ^3(c+d x) (35 a \sec (c+d x)+64 a)}{192 d}-\frac{\tan (c+d x) (35 a \sec (c+d x)+128 a)}{128 d}+a x",1,"a*x + (35*a*ArcTanh[Sin[c + d*x]])/(128*d) - ((128*a + 35*a*Sec[c + d*x])*Tan[c + d*x])/(128*d) + ((64*a + 35*a*Sec[c + d*x])*Tan[c + d*x]^3)/(192*d) - ((48*a + 35*a*Sec[c + d*x])*Tan[c + d*x]^5)/(240*d) + ((8*a + 7*a*Sec[c + d*x])*Tan[c + d*x]^7)/(56*d)","A",6,2,19,0.1053,1,"{3881, 3770}"
11,1,102,0,0.0944652,"\int (a+a \sec (c+d x)) \tan ^6(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Tan[c + d*x]^6,x]","-\frac{5 a \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\tan ^5(c+d x) (5 a \sec (c+d x)+6 a)}{30 d}-\frac{\tan ^3(c+d x) (5 a \sec (c+d x)+8 a)}{24 d}+\frac{\tan (c+d x) (5 a \sec (c+d x)+16 a)}{16 d}-a x","-\frac{5 a \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\tan ^5(c+d x) (5 a \sec (c+d x)+6 a)}{30 d}-\frac{\tan ^3(c+d x) (5 a \sec (c+d x)+8 a)}{24 d}+\frac{\tan (c+d x) (5 a \sec (c+d x)+16 a)}{16 d}-a x",1,"-(a*x) - (5*a*ArcTanh[Sin[c + d*x]])/(16*d) + ((16*a + 5*a*Sec[c + d*x])*Tan[c + d*x])/(16*d) - ((8*a + 5*a*Sec[c + d*x])*Tan[c + d*x]^3)/(24*d) + ((6*a + 5*a*Sec[c + d*x])*Tan[c + d*x]^5)/(30*d)","A",5,2,19,0.1053,1,"{3881, 3770}"
12,1,73,0,0.0619919,"\int (a+a \sec (c+d x)) \tan ^4(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Tan[c + d*x]^4,x]","\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan ^3(c+d x) (3 a \sec (c+d x)+4 a)}{12 d}-\frac{\tan (c+d x) (3 a \sec (c+d x)+8 a)}{8 d}+a x","\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan ^3(c+d x) (3 a \sec (c+d x)+4 a)}{12 d}-\frac{\tan (c+d x) (3 a \sec (c+d x)+8 a)}{8 d}+a x",1,"a*x + (3*a*ArcTanh[Sin[c + d*x]])/(8*d) - ((8*a + 3*a*Sec[c + d*x])*Tan[c + d*x])/(8*d) + ((4*a + 3*a*Sec[c + d*x])*Tan[c + d*x]^3)/(12*d)","A",4,2,19,0.1053,1,"{3881, 3770}"
13,1,45,0,0.0339243,"\int (a+a \sec (c+d x)) \tan ^2(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Tan[c + d*x]^2,x]","-\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\tan (c+d x) (a \sec (c+d x)+2 a)}{2 d}-a x","-\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\tan (c+d x) (a \sec (c+d x)+2 a)}{2 d}-a x",1,"-(a*x) - (a*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*a + a*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",3,2,19,0.1053,1,"{3881, 3770}"
14,1,26,0,0.0239717,"\int \cot ^2(c+d x) (a+a \sec (c+d x)) \, dx","Int[Cot[c + d*x]^2*(a + a*Sec[c + d*x]),x]","-\frac{\cot (c+d x) (a \sec (c+d x)+a)}{d}-a x","-\frac{\cot (c+d x) (a \sec (c+d x)+a)}{d}-a x",1,"-(a*x) - (Cot[c + d*x]*(a + a*Sec[c + d*x]))/d","A",2,2,19,0.1053,1,"{3882, 8}"
15,1,55,0,0.0517449,"\int \cot ^4(c+d x) (a+a \sec (c+d x)) \, dx","Int[Cot[c + d*x]^4*(a + a*Sec[c + d*x]),x]","-\frac{\cot ^3(c+d x) (a \sec (c+d x)+a)}{3 d}+\frac{\cot (c+d x) (2 a \sec (c+d x)+3 a)}{3 d}+a x","-\frac{\cot ^3(c+d x) (a \sec (c+d x)+a)}{3 d}+\frac{\cot (c+d x) (2 a \sec (c+d x)+3 a)}{3 d}+a x",1,"a*x - (Cot[c + d*x]^3*(a + a*Sec[c + d*x]))/(3*d) + (Cot[c + d*x]*(3*a + 2*a*Sec[c + d*x]))/(3*d)","A",3,2,19,0.1053,1,"{3882, 8}"
16,1,84,0,0.0809355,"\int \cot ^6(c+d x) (a+a \sec (c+d x)) \, dx","Int[Cot[c + d*x]^6*(a + a*Sec[c + d*x]),x]","-\frac{\cot ^5(c+d x) (a \sec (c+d x)+a)}{5 d}+\frac{\cot ^3(c+d x) (4 a \sec (c+d x)+5 a)}{15 d}-\frac{\cot (c+d x) (8 a \sec (c+d x)+15 a)}{15 d}-a x","-\frac{\cot ^5(c+d x) (a \sec (c+d x)+a)}{5 d}+\frac{\cot ^3(c+d x) (4 a \sec (c+d x)+5 a)}{15 d}-\frac{\cot (c+d x) (8 a \sec (c+d x)+15 a)}{15 d}-a x",1,"-(a*x) - (Cot[c + d*x]^5*(a + a*Sec[c + d*x]))/(5*d) + (Cot[c + d*x]^3*(5*a + 4*a*Sec[c + d*x]))/(15*d) - (Cot[c + d*x]*(15*a + 8*a*Sec[c + d*x]))/(15*d)","A",4,2,19,0.1053,1,"{3882, 8}"
17,1,111,0,0.1135592,"\int \cot ^8(c+d x) (a+a \sec (c+d x)) \, dx","Int[Cot[c + d*x]^8*(a + a*Sec[c + d*x]),x]","-\frac{\cot ^7(c+d x) (a \sec (c+d x)+a)}{7 d}+\frac{\cot ^5(c+d x) (6 a \sec (c+d x)+7 a)}{35 d}-\frac{\cot ^3(c+d x) (24 a \sec (c+d x)+35 a)}{105 d}+\frac{\cot (c+d x) (16 a \sec (c+d x)+35 a)}{35 d}+a x","-\frac{\cot ^7(c+d x) (a \sec (c+d x)+a)}{7 d}+\frac{\cot ^5(c+d x) (6 a \sec (c+d x)+7 a)}{35 d}-\frac{\cot ^3(c+d x) (24 a \sec (c+d x)+35 a)}{105 d}+\frac{\cot (c+d x) (16 a \sec (c+d x)+35 a)}{35 d}+a x",1,"a*x - (Cot[c + d*x]^7*(a + a*Sec[c + d*x]))/(7*d) + (Cot[c + d*x]^5*(7*a + 6*a*Sec[c + d*x]))/(35*d) + (Cot[c + d*x]*(35*a + 16*a*Sec[c + d*x]))/(35*d) - (Cot[c + d*x]^3*(35*a + 24*a*Sec[c + d*x]))/(105*d)","A",5,2,19,0.1053,1,"{3882, 8}"
18,1,140,0,0.1447256,"\int \cot ^{10}(c+d x) (a+a \sec (c+d x)) \, dx","Int[Cot[c + d*x]^10*(a + a*Sec[c + d*x]),x]","-\frac{\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}+\frac{\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{63 d}-\frac{\cot ^5(c+d x) (16 a \sec (c+d x)+21 a)}{105 d}+\frac{\cot ^3(c+d x) (64 a \sec (c+d x)+105 a)}{315 d}-\frac{\cot (c+d x) (128 a \sec (c+d x)+315 a)}{315 d}-a x","-\frac{\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}+\frac{\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{63 d}-\frac{\cot ^5(c+d x) (16 a \sec (c+d x)+21 a)}{105 d}+\frac{\cot ^3(c+d x) (64 a \sec (c+d x)+105 a)}{315 d}-\frac{\cot (c+d x) (128 a \sec (c+d x)+315 a)}{315 d}-a x",1,"-(a*x) - (Cot[c + d*x]^9*(a + a*Sec[c + d*x]))/(9*d) + (Cot[c + d*x]^7*(9*a + 8*a*Sec[c + d*x]))/(63*d) - (Cot[c + d*x]^5*(21*a + 16*a*Sec[c + d*x]))/(105*d) + (Cot[c + d*x]^3*(105*a + 64*a*Sec[c + d*x]))/(315*d) - (Cot[c + d*x]*(315*a + 128*a*Sec[c + d*x]))/(315*d)","A",6,2,19,0.1053,1,"{3882, 8}"
19,1,192,0,0.0988049,"\int (a+a \sec (c+d x))^2 \tan ^9(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Tan[c + d*x]^9,x]","\frac{a^2 \sec ^{10}(c+d x)}{10 d}+\frac{2 a^2 \sec ^9(c+d x)}{9 d}-\frac{3 a^2 \sec ^8(c+d x)}{8 d}-\frac{8 a^2 \sec ^7(c+d x)}{7 d}+\frac{a^2 \sec ^6(c+d x)}{3 d}+\frac{12 a^2 \sec ^5(c+d x)}{5 d}+\frac{a^2 \sec ^4(c+d x)}{2 d}-\frac{8 a^2 \sec ^3(c+d x)}{3 d}-\frac{3 a^2 \sec ^2(c+d x)}{2 d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d}","\frac{a^2 \sec ^{10}(c+d x)}{10 d}+\frac{2 a^2 \sec ^9(c+d x)}{9 d}-\frac{3 a^2 \sec ^8(c+d x)}{8 d}-\frac{8 a^2 \sec ^7(c+d x)}{7 d}+\frac{a^2 \sec ^6(c+d x)}{3 d}+\frac{12 a^2 \sec ^5(c+d x)}{5 d}+\frac{a^2 \sec ^4(c+d x)}{2 d}-\frac{8 a^2 \sec ^3(c+d x)}{3 d}-\frac{3 a^2 \sec ^2(c+d x)}{2 d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d}",1,"-((a^2*Log[Cos[c + d*x]])/d) + (2*a^2*Sec[c + d*x])/d - (3*a^2*Sec[c + d*x]^2)/(2*d) - (8*a^2*Sec[c + d*x]^3)/(3*d) + (a^2*Sec[c + d*x]^4)/(2*d) + (12*a^2*Sec[c + d*x]^5)/(5*d) + (a^2*Sec[c + d*x]^6)/(3*d) - (8*a^2*Sec[c + d*x]^7)/(7*d) - (3*a^2*Sec[c + d*x]^8)/(8*d) + (2*a^2*Sec[c + d*x]^9)/(9*d) + (a^2*Sec[c + d*x]^10)/(10*d)","A",3,2,21,0.09524,1,"{3879, 88}"
20,1,132,0,0.0792304,"\int (a+a \sec (c+d x))^2 \tan ^7(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Tan[c + d*x]^7,x]","\frac{a^2 \sec ^8(c+d x)}{8 d}+\frac{2 a^2 \sec ^7(c+d x)}{7 d}-\frac{a^2 \sec ^6(c+d x)}{3 d}-\frac{6 a^2 \sec ^5(c+d x)}{5 d}+\frac{2 a^2 \sec ^3(c+d x)}{d}+\frac{a^2 \sec ^2(c+d x)}{d}-\frac{2 a^2 \sec (c+d x)}{d}+\frac{a^2 \log (\cos (c+d x))}{d}","\frac{a^2 \sec ^8(c+d x)}{8 d}+\frac{2 a^2 \sec ^7(c+d x)}{7 d}-\frac{a^2 \sec ^6(c+d x)}{3 d}-\frac{6 a^2 \sec ^5(c+d x)}{5 d}+\frac{2 a^2 \sec ^3(c+d x)}{d}+\frac{a^2 \sec ^2(c+d x)}{d}-\frac{2 a^2 \sec (c+d x)}{d}+\frac{a^2 \log (\cos (c+d x))}{d}",1,"(a^2*Log[Cos[c + d*x]])/d - (2*a^2*Sec[c + d*x])/d + (a^2*Sec[c + d*x]^2)/d + (2*a^2*Sec[c + d*x]^3)/d - (6*a^2*Sec[c + d*x]^5)/(5*d) - (a^2*Sec[c + d*x]^6)/(3*d) + (2*a^2*Sec[c + d*x]^7)/(7*d) + (a^2*Sec[c + d*x]^8)/(8*d)","A",3,2,21,0.09524,1,"{3879, 88}"
21,1,120,0,0.072974,"\int (a+a \sec (c+d x))^2 \tan ^5(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Tan[c + d*x]^5,x]","\frac{a^2 \sec ^6(c+d x)}{6 d}+\frac{2 a^2 \sec ^5(c+d x)}{5 d}-\frac{a^2 \sec ^4(c+d x)}{4 d}-\frac{4 a^2 \sec ^3(c+d x)}{3 d}-\frac{a^2 \sec ^2(c+d x)}{2 d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d}","\frac{a^2 \sec ^6(c+d x)}{6 d}+\frac{2 a^2 \sec ^5(c+d x)}{5 d}-\frac{a^2 \sec ^4(c+d x)}{4 d}-\frac{4 a^2 \sec ^3(c+d x)}{3 d}-\frac{a^2 \sec ^2(c+d x)}{2 d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d}",1,"-((a^2*Log[Cos[c + d*x]])/d) + (2*a^2*Sec[c + d*x])/d - (a^2*Sec[c + d*x]^2)/(2*d) - (4*a^2*Sec[c + d*x]^3)/(3*d) - (a^2*Sec[c + d*x]^4)/(4*d) + (2*a^2*Sec[c + d*x]^5)/(5*d) + (a^2*Sec[c + d*x]^6)/(6*d)","A",3,2,21,0.09524,1,"{3879, 88}"
22,1,65,0,0.0558754,"\int (a+a \sec (c+d x))^2 \tan ^3(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Tan[c + d*x]^3,x]","\frac{a^2 \sec ^4(c+d x)}{4 d}+\frac{2 a^2 \sec ^3(c+d x)}{3 d}-\frac{2 a^2 \sec (c+d x)}{d}+\frac{a^2 \log (\cos (c+d x))}{d}","\frac{a^2 \sec ^4(c+d x)}{4 d}+\frac{2 a^2 \sec ^3(c+d x)}{3 d}-\frac{2 a^2 \sec (c+d x)}{d}+\frac{a^2 \log (\cos (c+d x))}{d}",1,"(a^2*Log[Cos[c + d*x]])/d - (2*a^2*Sec[c + d*x])/d + (2*a^2*Sec[c + d*x]^3)/(3*d) + (a^2*Sec[c + d*x]^4)/(4*d)","A",3,2,21,0.09524,1,"{3879, 75}"
23,1,48,0,0.0356408,"\int (a+a \sec (c+d x))^2 \tan (c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Tan[c + d*x],x]","\frac{a^2 \sec ^2(c+d x)}{2 d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d}","\frac{a^2 \sec ^2(c+d x)}{2 d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d}",1,"-((a^2*Log[Cos[c + d*x]])/d) + (2*a^2*Sec[c + d*x])/d + (a^2*Sec[c + d*x]^2)/(2*d)","A",3,2,19,0.1053,1,"{3879, 43}"
24,1,35,0,0.0404139,"\int \cot (c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]*(a + a*Sec[c + d*x])^2,x]","\frac{2 a^2 \log (1-\cos (c+d x))}{d}-\frac{a^2 \log (\cos (c+d x))}{d}","\frac{2 a^2 \log (1-\cos (c+d x))}{d}-\frac{a^2 \log (\cos (c+d x))}{d}",1,"(2*a^2*Log[1 - Cos[c + d*x]])/d - (a^2*Log[Cos[c + d*x]])/d","A",3,2,19,0.1053,1,"{3879, 72}"
25,1,40,0,0.0496429,"\int \cot ^3(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]^3*(a + a*Sec[c + d*x])^2,x]","-\frac{a^2}{d (1-\cos (c+d x))}-\frac{a^2 \log (1-\cos (c+d x))}{d}","-\frac{a^2}{d (1-\cos (c+d x))}-\frac{a^2 \log (1-\cos (c+d x))}{d}",1,"-(a^2/(d*(1 - Cos[c + d*x]))) - (a^2*Log[1 - Cos[c + d*x]])/d","A",3,2,21,0.09524,1,"{3879, 43}"
26,1,85,0,0.0669472,"\int \cot ^5(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^2,x]","\frac{5 a^2}{4 d (1-\cos (c+d x))}-\frac{a^2}{4 d (1-\cos (c+d x))^2}+\frac{7 a^2 \log (1-\cos (c+d x))}{8 d}+\frac{a^2 \log (\cos (c+d x)+1)}{8 d}","\frac{5 a^2}{4 d (1-\cos (c+d x))}-\frac{a^2}{4 d (1-\cos (c+d x))^2}+\frac{7 a^2 \log (1-\cos (c+d x))}{8 d}+\frac{a^2 \log (\cos (c+d x)+1)}{8 d}",1,"-a^2/(4*d*(1 - Cos[c + d*x])^2) + (5*a^2)/(4*d*(1 - Cos[c + d*x])) + (7*a^2*Log[1 - Cos[c + d*x]])/(8*d) + (a^2*Log[1 + Cos[c + d*x]])/(8*d)","A",3,2,21,0.09524,1,"{3879, 88}"
27,1,127,0,0.0865423,"\int \cot ^7(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]^7*(a + a*Sec[c + d*x])^2,x]","-\frac{23 a^2}{16 d (1-\cos (c+d x))}-\frac{a^2}{16 d (\cos (c+d x)+1)}+\frac{a^2}{2 d (1-\cos (c+d x))^2}-\frac{a^2}{12 d (1-\cos (c+d x))^3}-\frac{13 a^2 \log (1-\cos (c+d x))}{16 d}-\frac{3 a^2 \log (\cos (c+d x)+1)}{16 d}","-\frac{23 a^2}{16 d (1-\cos (c+d x))}-\frac{a^2}{16 d (\cos (c+d x)+1)}+\frac{a^2}{2 d (1-\cos (c+d x))^2}-\frac{a^2}{12 d (1-\cos (c+d x))^3}-\frac{13 a^2 \log (1-\cos (c+d x))}{16 d}-\frac{3 a^2 \log (\cos (c+d x)+1)}{16 d}",1,"-a^2/(12*d*(1 - Cos[c + d*x])^3) + a^2/(2*d*(1 - Cos[c + d*x])^2) - (23*a^2)/(16*d*(1 - Cos[c + d*x])) - a^2/(16*d*(1 + Cos[c + d*x])) - (13*a^2*Log[1 - Cos[c + d*x]])/(16*d) - (3*a^2*Log[1 + Cos[c + d*x]])/(16*d)","A",3,2,21,0.09524,1,"{3879, 88}"
28,1,169,0,0.1110585,"\int \cot ^9(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]^9*(a + a*Sec[c + d*x])^2,x]","\frac{51 a^2}{32 d (1-\cos (c+d x))}+\frac{9 a^2}{64 d (\cos (c+d x)+1)}-\frac{3 a^2}{4 d (1-\cos (c+d x))^2}-\frac{a^2}{64 d (\cos (c+d x)+1)^2}+\frac{11 a^2}{48 d (1-\cos (c+d x))^3}-\frac{a^2}{32 d (1-\cos (c+d x))^4}+\frac{99 a^2 \log (1-\cos (c+d x))}{128 d}+\frac{29 a^2 \log (\cos (c+d x)+1)}{128 d}","\frac{51 a^2}{32 d (1-\cos (c+d x))}+\frac{9 a^2}{64 d (\cos (c+d x)+1)}-\frac{3 a^2}{4 d (1-\cos (c+d x))^2}-\frac{a^2}{64 d (\cos (c+d x)+1)^2}+\frac{11 a^2}{48 d (1-\cos (c+d x))^3}-\frac{a^2}{32 d (1-\cos (c+d x))^4}+\frac{99 a^2 \log (1-\cos (c+d x))}{128 d}+\frac{29 a^2 \log (\cos (c+d x)+1)}{128 d}",1,"-a^2/(32*d*(1 - Cos[c + d*x])^4) + (11*a^2)/(48*d*(1 - Cos[c + d*x])^3) - (3*a^2)/(4*d*(1 - Cos[c + d*x])^2) + (51*a^2)/(32*d*(1 - Cos[c + d*x])) - a^2/(64*d*(1 + Cos[c + d*x])^2) + (9*a^2)/(64*d*(1 + Cos[c + d*x])) + (99*a^2*Log[1 - Cos[c + d*x]])/(128*d) + (29*a^2*Log[1 + Cos[c + d*x]])/(128*d)","A",3,2,21,0.09524,1,"{3879, 88}"
29,1,161,0,0.1785656,"\int (a+a \sec (c+d x))^2 \tan ^6(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Tan[c + d*x]^6,x]","\frac{a^2 \tan ^7(c+d x)}{7 d}+\frac{a^2 \tan ^5(c+d x)}{5 d}-\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}-\frac{5 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \tan ^5(c+d x) \sec (c+d x)}{3 d}-\frac{5 a^2 \tan ^3(c+d x) \sec (c+d x)}{12 d}+\frac{5 a^2 \tan (c+d x) \sec (c+d x)}{8 d}-a^2 x","\frac{a^2 \tan ^7(c+d x)}{7 d}+\frac{a^2 \tan ^5(c+d x)}{5 d}-\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}-\frac{5 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \tan ^5(c+d x) \sec (c+d x)}{3 d}-\frac{5 a^2 \tan ^3(c+d x) \sec (c+d x)}{12 d}+\frac{5 a^2 \tan (c+d x) \sec (c+d x)}{8 d}-a^2 x",1,"-(a^2*x) - (5*a^2*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*Tan[c + d*x])/d + (5*a^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (a^2*Tan[c + d*x]^3)/(3*d) - (5*a^2*Sec[c + d*x]*Tan[c + d*x]^3)/(12*d) + (a^2*Tan[c + d*x]^5)/(5*d) + (a^2*Sec[c + d*x]*Tan[c + d*x]^5)/(3*d) + (a^2*Tan[c + d*x]^7)/(7*d)","A",12,7,21,0.3333,1,"{3886, 3473, 8, 2611, 3770, 2607, 30}"
30,1,119,0,0.139138,"\int (a+a \sec (c+d x))^2 \tan ^4(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Tan[c + d*x]^4,x]","\frac{a^2 \tan ^5(c+d x)}{5 d}+\frac{a^2 \tan ^3(c+d x)}{3 d}-\frac{a^2 \tan (c+d x)}{d}+\frac{3 a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^2 \tan ^3(c+d x) \sec (c+d x)}{2 d}-\frac{3 a^2 \tan (c+d x) \sec (c+d x)}{4 d}+a^2 x","\frac{a^2 \tan ^5(c+d x)}{5 d}+\frac{a^2 \tan ^3(c+d x)}{3 d}-\frac{a^2 \tan (c+d x)}{d}+\frac{3 a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^2 \tan ^3(c+d x) \sec (c+d x)}{2 d}-\frac{3 a^2 \tan (c+d x) \sec (c+d x)}{4 d}+a^2 x",1,"a^2*x + (3*a^2*ArcTanh[Sin[c + d*x]])/(4*d) - (a^2*Tan[c + d*x])/d - (3*a^2*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a^2*Tan[c + d*x]^3)/(3*d) + (a^2*Sec[c + d*x]*Tan[c + d*x]^3)/(2*d) + (a^2*Tan[c + d*x]^5)/(5*d)","A",10,7,21,0.3333,1,"{3886, 3473, 8, 2611, 3770, 2607, 30}"
31,1,72,0,0.1090607,"\int (a+a \sec (c+d x))^2 \tan ^2(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Tan[c + d*x]^2,x]","\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{d}-a^2 x","\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{d}-a^2 x",1,"-(a^2*x) - (a^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Tan[c + d*x])/d + (a^2*Sec[c + d*x]*Tan[c + d*x])/d + (a^2*Tan[c + d*x]^3)/(3*d)","A",8,7,21,0.3333,1,"{3886, 3473, 8, 2611, 3770, 2607, 30}"
32,1,35,0,0.0721307,"\int \cot ^2(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*(a + a*Sec[c + d*x])^2,x]","-\frac{2 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc (c+d x)}{d}+a^2 (-x)","-\frac{2 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc (c+d x)}{d}+a^2 (-x)",1,"-(a^2*x) - (2*a^2*Cot[c + d*x])/d - (2*a^2*Csc[c + d*x])/d","A",8,5,21,0.2381,1,"{3886, 3473, 8, 2606, 3767}"
33,1,69,0,0.1105134,"\int \cot ^4(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^2,x]","-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}+\frac{2 a^2 \csc (c+d x)}{d}+a^2 x","-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}+\frac{2 a^2 \csc (c+d x)}{d}+a^2 x",1,"a^2*x + (a^2*Cot[c + d*x])/d - (2*a^2*Cot[c + d*x]^3)/(3*d) + (2*a^2*Csc[c + d*x])/d - (2*a^2*Csc[c + d*x]^3)/(3*d)","A",9,6,21,0.2857,1,"{3886, 3473, 8, 2606, 2607, 30}"
34,1,107,0,0.1276374,"\int \cot ^6(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*(a + a*Sec[c + d*x])^2,x]","-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}+\frac{4 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}-a^2 x","-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}+\frac{4 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}-a^2 x",1,"-(a^2*x) - (a^2*Cot[c + d*x])/d + (a^2*Cot[c + d*x]^3)/(3*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (2*a^2*Csc[c + d*x])/d + (4*a^2*Csc[c + d*x]^3)/(3*d) - (2*a^2*Csc[c + d*x]^5)/(5*d)","A",11,7,21,0.3333,1,"{3886, 3473, 8, 2606, 194, 2607, 30}"
35,1,139,0,0.1405454,"\int \cot ^8(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]^8*(a + a*Sec[c + d*x])^2,x]","-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}+\frac{6 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{d}+\frac{2 a^2 \csc (c+d x)}{d}+a^2 x","-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}+\frac{6 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{d}+\frac{2 a^2 \csc (c+d x)}{d}+a^2 x",1,"a^2*x + (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) + (a^2*Cot[c + d*x]^5)/(5*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) + (2*a^2*Csc[c + d*x])/d - (2*a^2*Csc[c + d*x]^3)/d + (6*a^2*Csc[c + d*x]^5)/(5*d) - (2*a^2*Csc[c + d*x]^7)/(7*d)","A",12,7,21,0.3333,1,"{3886, 3473, 8, 2606, 194, 2607, 30}"
36,1,179,0,0.1549758,"\int \cot ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]^10*(a + a*Sec[c + d*x])^2,x]","-\frac{2 a^2 \cot ^9(c+d x)}{9 d}+\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^9(c+d x)}{9 d}+\frac{8 a^2 \csc ^7(c+d x)}{7 d}-\frac{12 a^2 \csc ^5(c+d x)}{5 d}+\frac{8 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}-a^2 x","-\frac{2 a^2 \cot ^9(c+d x)}{9 d}+\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^9(c+d x)}{9 d}+\frac{8 a^2 \csc ^7(c+d x)}{7 d}-\frac{12 a^2 \csc ^5(c+d x)}{5 d}+\frac{8 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}-a^2 x",1,"-(a^2*x) - (a^2*Cot[c + d*x])/d + (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]^5)/(5*d) + (a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Cot[c + d*x]^9)/(9*d) - (2*a^2*Csc[c + d*x])/d + (8*a^2*Csc[c + d*x]^3)/(3*d) - (12*a^2*Csc[c + d*x]^5)/(5*d) + (8*a^2*Csc[c + d*x]^7)/(7*d) - (2*a^2*Csc[c + d*x]^9)/(9*d)","A",13,7,21,0.3333,1,"{3886, 3473, 8, 2606, 194, 2607, 30}"
37,1,210,0,0.1026301,"\int (a+a \sec (c+d x))^3 \tan ^9(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^9,x]","\frac{a^3 \sec ^{11}(c+d x)}{11 d}+\frac{3 a^3 \sec ^{10}(c+d x)}{10 d}-\frac{a^3 \sec ^9(c+d x)}{9 d}-\frac{11 a^3 \sec ^8(c+d x)}{8 d}-\frac{6 a^3 \sec ^7(c+d x)}{7 d}+\frac{7 a^3 \sec ^6(c+d x)}{3 d}+\frac{14 a^3 \sec ^5(c+d x)}{5 d}-\frac{3 a^3 \sec ^4(c+d x)}{2 d}-\frac{11 a^3 \sec ^3(c+d x)}{3 d}-\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d}","\frac{a^3 \sec ^{11}(c+d x)}{11 d}+\frac{3 a^3 \sec ^{10}(c+d x)}{10 d}-\frac{a^3 \sec ^9(c+d x)}{9 d}-\frac{11 a^3 \sec ^8(c+d x)}{8 d}-\frac{6 a^3 \sec ^7(c+d x)}{7 d}+\frac{7 a^3 \sec ^6(c+d x)}{3 d}+\frac{14 a^3 \sec ^5(c+d x)}{5 d}-\frac{3 a^3 \sec ^4(c+d x)}{2 d}-\frac{11 a^3 \sec ^3(c+d x)}{3 d}-\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d}",1,"-((a^3*Log[Cos[c + d*x]])/d) + (3*a^3*Sec[c + d*x])/d - (a^3*Sec[c + d*x]^2)/(2*d) - (11*a^3*Sec[c + d*x]^3)/(3*d) - (3*a^3*Sec[c + d*x]^4)/(2*d) + (14*a^3*Sec[c + d*x]^5)/(5*d) + (7*a^3*Sec[c + d*x]^6)/(3*d) - (6*a^3*Sec[c + d*x]^7)/(7*d) - (11*a^3*Sec[c + d*x]^8)/(8*d) - (a^3*Sec[c + d*x]^9)/(9*d) + (3*a^3*Sec[c + d*x]^10)/(10*d) + (a^3*Sec[c + d*x]^11)/(11*d)","A",3,2,21,0.09524,1,"{3879, 88}"
38,1,137,0,0.0771652,"\int (a+a \sec (c+d x))^3 \tan ^7(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^7,x]","\frac{a^3 \sec ^9(c+d x)}{9 d}+\frac{3 a^3 \sec ^8(c+d x)}{8 d}-\frac{4 a^3 \sec ^6(c+d x)}{3 d}-\frac{6 a^3 \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \sec ^4(c+d x)}{2 d}+\frac{8 a^3 \sec ^3(c+d x)}{3 d}-\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d}","\frac{a^3 \sec ^9(c+d x)}{9 d}+\frac{3 a^3 \sec ^8(c+d x)}{8 d}-\frac{4 a^3 \sec ^6(c+d x)}{3 d}-\frac{6 a^3 \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \sec ^4(c+d x)}{2 d}+\frac{8 a^3 \sec ^3(c+d x)}{3 d}-\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d}",1,"(a^3*Log[Cos[c + d*x]])/d - (3*a^3*Sec[c + d*x])/d + (8*a^3*Sec[c + d*x]^3)/(3*d) + (3*a^3*Sec[c + d*x]^4)/(2*d) - (6*a^3*Sec[c + d*x]^5)/(5*d) - (4*a^3*Sec[c + d*x]^6)/(3*d) + (3*a^3*Sec[c + d*x]^8)/(8*d) + (a^3*Sec[c + d*x]^9)/(9*d)","A",3,2,21,0.09524,1,"{3879, 88}"
39,1,138,0,0.0772046,"\int (a+a \sec (c+d x))^3 \tan ^5(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^5,x]","\frac{a^3 \sec ^7(c+d x)}{7 d}+\frac{a^3 \sec ^6(c+d x)}{2 d}+\frac{a^3 \sec ^5(c+d x)}{5 d}-\frac{5 a^3 \sec ^4(c+d x)}{4 d}-\frac{5 a^3 \sec ^3(c+d x)}{3 d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d}","\frac{a^3 \sec ^7(c+d x)}{7 d}+\frac{a^3 \sec ^6(c+d x)}{2 d}+\frac{a^3 \sec ^5(c+d x)}{5 d}-\frac{5 a^3 \sec ^4(c+d x)}{4 d}-\frac{5 a^3 \sec ^3(c+d x)}{3 d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d}",1,"-((a^3*Log[Cos[c + d*x]])/d) + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d) - (5*a^3*Sec[c + d*x]^3)/(3*d) - (5*a^3*Sec[c + d*x]^4)/(4*d) + (a^3*Sec[c + d*x]^5)/(5*d) + (a^3*Sec[c + d*x]^6)/(2*d) + (a^3*Sec[c + d*x]^7)/(7*d)","A",3,2,21,0.09524,1,"{3879, 88}"
40,1,99,0,0.066601,"\int (a+a \sec (c+d x))^3 \tan ^3(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^3,x]","\frac{a^3 \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \sec ^4(c+d x)}{4 d}+\frac{2 a^3 \sec ^3(c+d x)}{3 d}-\frac{a^3 \sec ^2(c+d x)}{d}-\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d}","\frac{a^3 \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \sec ^4(c+d x)}{4 d}+\frac{2 a^3 \sec ^3(c+d x)}{3 d}-\frac{a^3 \sec ^2(c+d x)}{d}-\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d}",1,"(a^3*Log[Cos[c + d*x]])/d - (3*a^3*Sec[c + d*x])/d - (a^3*Sec[c + d*x]^2)/d + (2*a^3*Sec[c + d*x]^3)/(3*d) + (3*a^3*Sec[c + d*x]^4)/(4*d) + (a^3*Sec[c + d*x]^5)/(5*d)","A",3,2,21,0.09524,1,"{3879, 75}"
41,1,66,0,0.0388226,"\int (a+a \sec (c+d x))^3 \tan (c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x],x]","\frac{a^3 \sec ^3(c+d x)}{3 d}+\frac{3 a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d}","\frac{a^3 \sec ^3(c+d x)}{3 d}+\frac{3 a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d}",1,"-((a^3*Log[Cos[c + d*x]])/d) + (3*a^3*Sec[c + d*x])/d + (3*a^3*Sec[c + d*x]^2)/(2*d) + (a^3*Sec[c + d*x]^3)/(3*d)","A",3,2,19,0.1053,1,"{3879, 43}"
42,1,48,0,0.0460482,"\int \cot (c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cot[c + d*x]*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 \sec (c+d x)}{d}+\frac{4 a^3 \log (1-\cos (c+d x))}{d}-\frac{3 a^3 \log (\cos (c+d x))}{d}","\frac{a^3 \sec (c+d x)}{d}+\frac{4 a^3 \log (1-\cos (c+d x))}{d}-\frac{3 a^3 \log (\cos (c+d x))}{d}",1,"(4*a^3*Log[1 - Cos[c + d*x]])/d - (3*a^3*Log[Cos[c + d*x]])/d + (a^3*Sec[c + d*x])/d","A",3,2,19,0.1053,1,"{3879, 88}"
43,1,40,0,0.0503526,"\int \cot ^3(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cot[c + d*x]^3*(a + a*Sec[c + d*x])^3,x]","-\frac{2 a^3}{d (1-\cos (c+d x))}-\frac{a^3 \log (1-\cos (c+d x))}{d}","-\frac{2 a^3}{d (1-\cos (c+d x))}-\frac{a^3 \log (1-\cos (c+d x))}{d}",1,"(-2*a^3)/(d*(1 - Cos[c + d*x])) - (a^3*Log[1 - Cos[c + d*x]])/d","A",3,2,21,0.09524,1,"{3879, 43}"
44,1,61,0,0.058681,"\int \cot ^5(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^3,x]","\frac{2 a^3}{d (1-\cos (c+d x))}-\frac{a^3}{2 d (1-\cos (c+d x))^2}+\frac{a^3 \log (1-\cos (c+d x))}{d}","\frac{2 a^3}{d (1-\cos (c+d x))}-\frac{a^3}{2 d (1-\cos (c+d x))^2}+\frac{a^3 \log (1-\cos (c+d x))}{d}",1,"-a^3/(2*d*(1 - Cos[c + d*x])^2) + (2*a^3)/(d*(1 - Cos[c + d*x])) + (a^3*Log[1 - Cos[c + d*x]])/d","A",3,2,21,0.09524,1,"{3879, 43}"
45,1,107,0,0.0763585,"\int \cot ^7(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cot[c + d*x]^7*(a + a*Sec[c + d*x])^3,x]","-\frac{17 a^3}{8 d (1-\cos (c+d x))}+\frac{7 a^3}{8 d (1-\cos (c+d x))^2}-\frac{a^3}{6 d (1-\cos (c+d x))^3}-\frac{15 a^3 \log (1-\cos (c+d x))}{16 d}-\frac{a^3 \log (\cos (c+d x)+1)}{16 d}","-\frac{17 a^3}{8 d (1-\cos (c+d x))}+\frac{7 a^3}{8 d (1-\cos (c+d x))^2}-\frac{a^3}{6 d (1-\cos (c+d x))^3}-\frac{15 a^3 \log (1-\cos (c+d x))}{16 d}-\frac{a^3 \log (\cos (c+d x)+1)}{16 d}",1,"-a^3/(6*d*(1 - Cos[c + d*x])^3) + (7*a^3)/(8*d*(1 - Cos[c + d*x])^2) - (17*a^3)/(8*d*(1 - Cos[c + d*x])) - (15*a^3*Log[1 - Cos[c + d*x]])/(16*d) - (a^3*Log[1 + Cos[c + d*x]])/(16*d)","A",3,2,21,0.09524,1,"{3879, 88}"
46,1,149,0,0.097923,"\int \cot ^9(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cot[c + d*x]^9*(a + a*Sec[c + d*x])^3,x]","\frac{9 a^3}{4 d (1-\cos (c+d x))}+\frac{a^3}{32 d (\cos (c+d x)+1)}-\frac{39 a^3}{32 d (1-\cos (c+d x))^2}+\frac{5 a^3}{12 d (1-\cos (c+d x))^3}-\frac{a^3}{16 d (1-\cos (c+d x))^4}+\frac{57 a^3 \log (1-\cos (c+d x))}{64 d}+\frac{7 a^3 \log (\cos (c+d x)+1)}{64 d}","\frac{9 a^3}{4 d (1-\cos (c+d x))}+\frac{a^3}{32 d (\cos (c+d x)+1)}-\frac{39 a^3}{32 d (1-\cos (c+d x))^2}+\frac{5 a^3}{12 d (1-\cos (c+d x))^3}-\frac{a^3}{16 d (1-\cos (c+d x))^4}+\frac{57 a^3 \log (1-\cos (c+d x))}{64 d}+\frac{7 a^3 \log (\cos (c+d x)+1)}{64 d}",1,"-a^3/(16*d*(1 - Cos[c + d*x])^4) + (5*a^3)/(12*d*(1 - Cos[c + d*x])^3) - (39*a^3)/(32*d*(1 - Cos[c + d*x])^2) + (9*a^3)/(4*d*(1 - Cos[c + d*x])) + a^3/(32*d*(1 + Cos[c + d*x])) + (57*a^3*Log[1 - Cos[c + d*x]])/(64*d) + (7*a^3*Log[1 + Cos[c + d*x]])/(64*d)","A",3,2,21,0.09524,1,"{3879, 88}"
47,1,237,0,0.2993893,"\int (a+a \sec (c+d x))^3 \tan ^6(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^6,x]","\frac{3 a^3 \tan ^7(c+d x)}{7 d}+\frac{a^3 \tan ^5(c+d x)}{5 d}-\frac{a^3 \tan ^3(c+d x)}{3 d}+\frac{a^3 \tan (c+d x)}{d}-\frac{125 a^3 \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{a^3 \tan ^5(c+d x) \sec ^3(c+d x)}{8 d}-\frac{5 a^3 \tan ^3(c+d x) \sec ^3(c+d x)}{48 d}+\frac{5 a^3 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac{a^3 \tan ^5(c+d x) \sec (c+d x)}{2 d}-\frac{5 a^3 \tan ^3(c+d x) \sec (c+d x)}{8 d}+\frac{115 a^3 \tan (c+d x) \sec (c+d x)}{128 d}-a^3 x","\frac{3 a^3 \tan ^7(c+d x)}{7 d}+\frac{a^3 \tan ^5(c+d x)}{5 d}-\frac{a^3 \tan ^3(c+d x)}{3 d}+\frac{a^3 \tan (c+d x)}{d}-\frac{125 a^3 \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{a^3 \tan ^5(c+d x) \sec ^3(c+d x)}{8 d}-\frac{5 a^3 \tan ^3(c+d x) \sec ^3(c+d x)}{48 d}+\frac{5 a^3 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac{a^3 \tan ^5(c+d x) \sec (c+d x)}{2 d}-\frac{5 a^3 \tan ^3(c+d x) \sec (c+d x)}{8 d}+\frac{115 a^3 \tan (c+d x) \sec (c+d x)}{128 d}-a^3 x",1,"-(a^3*x) - (125*a^3*ArcTanh[Sin[c + d*x]])/(128*d) + (a^3*Tan[c + d*x])/d + (115*a^3*Sec[c + d*x]*Tan[c + d*x])/(128*d) + (5*a^3*Sec[c + d*x]^3*Tan[c + d*x])/(64*d) - (a^3*Tan[c + d*x]^3)/(3*d) - (5*a^3*Sec[c + d*x]*Tan[c + d*x]^3)/(8*d) - (5*a^3*Sec[c + d*x]^3*Tan[c + d*x]^3)/(48*d) + (a^3*Tan[c + d*x]^5)/(5*d) + (a^3*Sec[c + d*x]*Tan[c + d*x]^5)/(2*d) + (a^3*Sec[c + d*x]^3*Tan[c + d*x]^5)/(8*d) + (3*a^3*Tan[c + d*x]^7)/(7*d)","A",17,8,21,0.3810,1,"{3886, 3473, 8, 2611, 3770, 2607, 30, 3768}"
48,1,169,0,0.2243643,"\int (a+a \sec (c+d x))^3 \tan ^4(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^4,x]","\frac{3 a^3 \tan ^5(c+d x)}{5 d}+\frac{a^3 \tan ^3(c+d x)}{3 d}-\frac{a^3 \tan (c+d x)}{d}+\frac{19 a^3 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 \tan ^3(c+d x) \sec ^3(c+d x)}{6 d}-\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac{3 a^3 \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac{17 a^3 \tan (c+d x) \sec (c+d x)}{16 d}+a^3 x","\frac{3 a^3 \tan ^5(c+d x)}{5 d}+\frac{a^3 \tan ^3(c+d x)}{3 d}-\frac{a^3 \tan (c+d x)}{d}+\frac{19 a^3 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 \tan ^3(c+d x) \sec ^3(c+d x)}{6 d}-\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac{3 a^3 \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac{17 a^3 \tan (c+d x) \sec (c+d x)}{16 d}+a^3 x",1,"a^3*x + (19*a^3*ArcTanh[Sin[c + d*x]])/(16*d) - (a^3*Tan[c + d*x])/d - (17*a^3*Sec[c + d*x]*Tan[c + d*x])/(16*d) - (a^3*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (a^3*Tan[c + d*x]^3)/(3*d) + (3*a^3*Sec[c + d*x]*Tan[c + d*x]^3)/(4*d) + (a^3*Sec[c + d*x]^3*Tan[c + d*x]^3)/(6*d) + (3*a^3*Tan[c + d*x]^5)/(5*d)","A",14,8,21,0.3810,1,"{3886, 3473, 8, 2611, 3770, 2607, 30, 3768}"
49,1,98,0,0.1581167,"\int (a+a \sec (c+d x))^3 \tan ^2(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^2,x]","\frac{a^3 \tan ^3(c+d x)}{d}+\frac{a^3 \tan (c+d x)}{d}-\frac{13 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{11 a^3 \tan (c+d x) \sec (c+d x)}{8 d}-a^3 x","\frac{a^3 \tan ^3(c+d x)}{d}+\frac{a^3 \tan (c+d x)}{d}-\frac{13 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{11 a^3 \tan (c+d x) \sec (c+d x)}{8 d}-a^3 x",1,"-(a^3*x) - (13*a^3*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*Tan[c + d*x])/d + (11*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a^3*Tan[c + d*x]^3)/d","A",11,8,21,0.3810,1,"{3886, 3473, 8, 2611, 3770, 2607, 30, 3768}"
50,1,49,0,0.0970836,"\int \cot ^2(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cot[c + d*x]^2*(a + a*Sec[c + d*x])^3,x]","-\frac{4 a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+a^3 (-x)","-\frac{4 a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+a^3 (-x)",1,"-(a^3*x) + (a^3*ArcTanh[Sin[c + d*x]])/d - (4*a^3*Cot[c + d*x])/d - (4*a^3*Csc[c + d*x])/d","A",11,8,21,0.3810,1,"{3886, 3473, 8, 2606, 3767, 2621, 321, 207}"
51,1,69,0,0.132014,"\int \cot ^4(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^3,x]","-\frac{4 a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x","-\frac{4 a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x",1,"a^3*x + (a^3*Cot[c + d*x])/d - (4*a^3*Cot[c + d*x]^3)/(3*d) + (3*a^3*Csc[c + d*x])/d - (4*a^3*Csc[c + d*x]^3)/(3*d)","A",11,6,21,0.2857,1,"{3886, 3473, 8, 2606, 2607, 30}"
52,1,107,0,0.1640525,"\int \cot ^6(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cot[c + d*x]^6*(a + a*Sec[c + d*x])^3,x]","-\frac{4 a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^5(c+d x)}{5 d}+\frac{7 a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc (c+d x)}{d}-a^3 x","-\frac{4 a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^5(c+d x)}{5 d}+\frac{7 a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc (c+d x)}{d}-a^3 x",1,"-(a^3*x) - (a^3*Cot[c + d*x])/d + (a^3*Cot[c + d*x]^3)/(3*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (3*a^3*Csc[c + d*x])/d + (7*a^3*Csc[c + d*x]^3)/(3*d) - (4*a^3*Csc[c + d*x]^5)/(5*d)","A",14,8,21,0.3810,1,"{3886, 3473, 8, 2606, 194, 2607, 30, 14}"
53,1,141,0,0.1784991,"\int \cot ^8(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cot[c + d*x]^8*(a + a*Sec[c + d*x])^3,x]","-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^7(c+d x)}{7 d}+\frac{11 a^3 \csc ^5(c+d x)}{5 d}-\frac{10 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x","-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^7(c+d x)}{7 d}+\frac{11 a^3 \csc ^5(c+d x)}{5 d}-\frac{10 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x",1,"a^3*x + (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) + (a^3*Cot[c + d*x]^5)/(5*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) + (3*a^3*Csc[c + d*x])/d - (10*a^3*Csc[c + d*x]^3)/(3*d) + (11*a^3*Csc[c + d*x]^5)/(5*d) - (4*a^3*Csc[c + d*x]^7)/(7*d)","A",15,8,21,0.3810,1,"{3886, 3473, 8, 2606, 194, 2607, 30, 270}"
54,1,179,0,0.1993622,"\int \cot ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cot[c + d*x]^10*(a + a*Sec[c + d*x])^3,x]","-\frac{4 a^3 \cot ^9(c+d x)}{9 d}+\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^9(c+d x)}{9 d}+\frac{15 a^3 \csc ^7(c+d x)}{7 d}-\frac{21 a^3 \csc ^5(c+d x)}{5 d}+\frac{13 a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc (c+d x)}{d}-a^3 x","-\frac{4 a^3 \cot ^9(c+d x)}{9 d}+\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^9(c+d x)}{9 d}+\frac{15 a^3 \csc ^7(c+d x)}{7 d}-\frac{21 a^3 \csc ^5(c+d x)}{5 d}+\frac{13 a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc (c+d x)}{d}-a^3 x",1,"-(a^3*x) - (a^3*Cot[c + d*x])/d + (a^3*Cot[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x]^5)/(5*d) + (a^3*Cot[c + d*x]^7)/(7*d) - (4*a^3*Cot[c + d*x]^9)/(9*d) - (3*a^3*Csc[c + d*x])/d + (13*a^3*Csc[c + d*x]^3)/(3*d) - (21*a^3*Csc[c + d*x]^5)/(5*d) + (15*a^3*Csc[c + d*x]^7)/(7*d) - (4*a^3*Csc[c + d*x]^9)/(9*d)","A",16,8,21,0.3810,1,"{3886, 3473, 8, 2606, 194, 2607, 30, 270}"
55,1,213,0,0.2212976,"\int \cot ^{12}(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cot[c + d*x]^12*(a + a*Sec[c + d*x])^3,x]","-\frac{4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^{11}(c+d x)}{11 d}+\frac{19 a^3 \csc ^9(c+d x)}{9 d}-\frac{36 a^3 \csc ^7(c+d x)}{7 d}+\frac{34 a^3 \csc ^5(c+d x)}{5 d}-\frac{16 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x","-\frac{4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^{11}(c+d x)}{11 d}+\frac{19 a^3 \csc ^9(c+d x)}{9 d}-\frac{36 a^3 \csc ^7(c+d x)}{7 d}+\frac{34 a^3 \csc ^5(c+d x)}{5 d}-\frac{16 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x",1,"a^3*x + (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) + (a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/(7*d) + (a^3*Cot[c + d*x]^9)/(9*d) - (4*a^3*Cot[c + d*x]^11)/(11*d) + (3*a^3*Csc[c + d*x])/d - (16*a^3*Csc[c + d*x]^3)/(3*d) + (34*a^3*Csc[c + d*x]^5)/(5*d) - (36*a^3*Csc[c + d*x]^7)/(7*d) + (19*a^3*Csc[c + d*x]^9)/(9*d) - (4*a^3*Csc[c + d*x]^11)/(11*d)","A",17,8,21,0.3810,1,"{3886, 3473, 8, 2606, 194, 2607, 30, 270}"
56,1,135,0,0.0784222,"\int \frac{\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Tan[c + d*x]^9/(a + a*Sec[c + d*x]),x]","\frac{\sec ^7(c+d x)}{7 a d}-\frac{\sec ^6(c+d x)}{6 a d}-\frac{3 \sec ^5(c+d x)}{5 a d}+\frac{3 \sec ^4(c+d x)}{4 a d}+\frac{\sec ^3(c+d x)}{a d}-\frac{3 \sec ^2(c+d x)}{2 a d}-\frac{\sec (c+d x)}{a d}-\frac{\log (\cos (c+d x))}{a d}","\frac{\sec ^7(c+d x)}{7 a d}-\frac{\sec ^6(c+d x)}{6 a d}-\frac{3 \sec ^5(c+d x)}{5 a d}+\frac{3 \sec ^4(c+d x)}{4 a d}+\frac{\sec ^3(c+d x)}{a d}-\frac{3 \sec ^2(c+d x)}{2 a d}-\frac{\sec (c+d x)}{a d}-\frac{\log (\cos (c+d x))}{a d}",1,"-(Log[Cos[c + d*x]]/(a*d)) - Sec[c + d*x]/(a*d) - (3*Sec[c + d*x]^2)/(2*a*d) + Sec[c + d*x]^3/(a*d) + (3*Sec[c + d*x]^4)/(4*a*d) - (3*Sec[c + d*x]^5)/(5*a*d) - Sec[c + d*x]^6/(6*a*d) + Sec[c + d*x]^7/(7*a*d)","A",3,2,21,0.09524,1,"{3879, 88}"
57,1,97,0,0.0684093,"\int \frac{\tan ^7(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Tan[c + d*x]^7/(a + a*Sec[c + d*x]),x]","\frac{\sec ^5(c+d x)}{5 a d}-\frac{\sec ^4(c+d x)}{4 a d}-\frac{2 \sec ^3(c+d x)}{3 a d}+\frac{\sec ^2(c+d x)}{a d}+\frac{\sec (c+d x)}{a d}+\frac{\log (\cos (c+d x))}{a d}","\frac{\sec ^5(c+d x)}{5 a d}-\frac{\sec ^4(c+d x)}{4 a d}-\frac{2 \sec ^3(c+d x)}{3 a d}+\frac{\sec ^2(c+d x)}{a d}+\frac{\sec (c+d x)}{a d}+\frac{\log (\cos (c+d x))}{a d}",1,"Log[Cos[c + d*x]]/(a*d) + Sec[c + d*x]/(a*d) + Sec[c + d*x]^2/(a*d) - (2*Sec[c + d*x]^3)/(3*a*d) - Sec[c + d*x]^4/(4*a*d) + Sec[c + d*x]^5/(5*a*d)","A",3,2,21,0.09524,1,"{3879, 88}"
58,1,66,0,0.0572111,"\int \frac{\tan ^5(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Tan[c + d*x]^5/(a + a*Sec[c + d*x]),x]","\frac{\sec ^3(c+d x)}{3 a d}-\frac{\sec ^2(c+d x)}{2 a d}-\frac{\sec (c+d x)}{a d}-\frac{\log (\cos (c+d x))}{a d}","\frac{\sec ^3(c+d x)}{3 a d}-\frac{\sec ^2(c+d x)}{2 a d}-\frac{\sec (c+d x)}{a d}-\frac{\log (\cos (c+d x))}{a d}",1,"-(Log[Cos[c + d*x]]/(a*d)) - Sec[c + d*x]/(a*d) - Sec[c + d*x]^2/(2*a*d) + Sec[c + d*x]^3/(3*a*d)","A",3,2,21,0.09524,1,"{3879, 75}"
59,1,28,0,0.0481519,"\int \frac{\tan ^3(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Tan[c + d*x]^3/(a + a*Sec[c + d*x]),x]","\frac{\sec (c+d x)}{a d}+\frac{\log (\cos (c+d x))}{a d}","\frac{\sec (c+d x)}{a d}+\frac{\log (\cos (c+d x))}{a d}",1,"Log[Cos[c + d*x]]/(a*d) + Sec[c + d*x]/(a*d)","A",3,2,21,0.09524,1,"{3879, 43}"
60,1,17,0,0.0257337,"\int \frac{\tan (c+d x)}{a+a \sec (c+d x)} \, dx","Int[Tan[c + d*x]/(a + a*Sec[c + d*x]),x]","-\frac{\log (\cos (c+d x)+1)}{a d}","-\frac{\log (\cos (c+d x)+1)}{a d}",1,"-(Log[1 + Cos[c + d*x]]/(a*d))","A",2,2,19,0.1053,1,"{3879, 31}"
61,1,61,0,0.0556855,"\int \frac{\cot (c+d x)}{a+a \sec (c+d x)} \, dx","Int[Cot[c + d*x]/(a + a*Sec[c + d*x]),x]","\frac{1}{2 a d (\cos (c+d x)+1)}+\frac{\log (1-\cos (c+d x))}{4 a d}+\frac{3 \log (\cos (c+d x)+1)}{4 a d}","\frac{1}{2 a d (\cos (c+d x)+1)}+\frac{\log (1-\cos (c+d x))}{4 a d}+\frac{3 \log (\cos (c+d x)+1)}{4 a d}",1,"1/(2*a*d*(1 + Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(4*a*d) + (3*Log[1 + Cos[c + d*x]])/(4*a*d)","A",3,2,19,0.1053,1,"{3879, 88}"
62,1,103,0,0.0762475,"\int \frac{\cot ^3(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Cot[c + d*x]^3/(a + a*Sec[c + d*x]),x]","-\frac{1}{8 a d (1-\cos (c+d x))}-\frac{3}{4 a d (\cos (c+d x)+1)}+\frac{1}{8 a d (\cos (c+d x)+1)^2}-\frac{5 \log (1-\cos (c+d x))}{16 a d}-\frac{11 \log (\cos (c+d x)+1)}{16 a d}","-\frac{1}{8 a d (1-\cos (c+d x))}-\frac{3}{4 a d (\cos (c+d x)+1)}+\frac{1}{8 a d (\cos (c+d x)+1)^2}-\frac{5 \log (1-\cos (c+d x))}{16 a d}-\frac{11 \log (\cos (c+d x)+1)}{16 a d}",1,"-1/(8*a*d*(1 - Cos[c + d*x])) + 1/(8*a*d*(1 + Cos[c + d*x])^2) - 3/(4*a*d*(1 + Cos[c + d*x])) - (5*Log[1 - Cos[c + d*x]])/(16*a*d) - (11*Log[1 + Cos[c + d*x]])/(16*a*d)","A",3,2,21,0.09524,1,"{3879, 88}"
63,1,145,0,0.0980931,"\int \frac{\cot ^5(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Cot[c + d*x]^5/(a + a*Sec[c + d*x]),x]","\frac{1}{4 a d (1-\cos (c+d x))}+\frac{15}{16 a d (\cos (c+d x)+1)}-\frac{1}{32 a d (1-\cos (c+d x))^2}-\frac{9}{32 a d (\cos (c+d x)+1)^2}+\frac{1}{24 a d (\cos (c+d x)+1)^3}+\frac{11 \log (1-\cos (c+d x))}{32 a d}+\frac{21 \log (\cos (c+d x)+1)}{32 a d}","\frac{1}{4 a d (1-\cos (c+d x))}+\frac{15}{16 a d (\cos (c+d x)+1)}-\frac{1}{32 a d (1-\cos (c+d x))^2}-\frac{9}{32 a d (\cos (c+d x)+1)^2}+\frac{1}{24 a d (\cos (c+d x)+1)^3}+\frac{11 \log (1-\cos (c+d x))}{32 a d}+\frac{21 \log (\cos (c+d x)+1)}{32 a d}",1,"-1/(32*a*d*(1 - Cos[c + d*x])^2) + 1/(4*a*d*(1 - Cos[c + d*x])) + 1/(24*a*d*(1 + Cos[c + d*x])^3) - 9/(32*a*d*(1 + Cos[c + d*x])^2) + 15/(16*a*d*(1 + Cos[c + d*x])) + (11*Log[1 - Cos[c + d*x]])/(32*a*d) + (21*Log[1 + Cos[c + d*x]])/(32*a*d)","A",3,2,21,0.09524,1,"{3879, 88}"
64,1,105,0,0.1440013,"\int \frac{\tan ^8(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Tan[c + d*x]^8/(a + a*Sec[c + d*x]),x]","-\frac{5 \tanh ^{-1}(\sin (c+d x))}{16 a d}-\frac{\tan ^5(c+d x) (6-5 \sec (c+d x))}{30 a d}+\frac{\tan ^3(c+d x) (8-5 \sec (c+d x))}{24 a d}-\frac{\tan (c+d x) (16-5 \sec (c+d x))}{16 a d}+\frac{x}{a}","-\frac{5 \tanh ^{-1}(\sin (c+d x))}{16 a d}-\frac{\tan ^5(c+d x) (6-5 \sec (c+d x))}{30 a d}+\frac{\tan ^3(c+d x) (8-5 \sec (c+d x))}{24 a d}-\frac{\tan (c+d x) (16-5 \sec (c+d x))}{16 a d}+\frac{x}{a}",1,"x/a - (5*ArcTanh[Sin[c + d*x]])/(16*a*d) - ((16 - 5*Sec[c + d*x])*Tan[c + d*x])/(16*a*d) + ((8 - 5*Sec[c + d*x])*Tan[c + d*x]^3)/(24*a*d) - ((6 - 5*Sec[c + d*x])*Tan[c + d*x]^5)/(30*a*d)","A",6,3,21,0.1429,1,"{3888, 3881, 3770}"
65,1,78,0,0.1086351,"\int \frac{\tan ^6(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Tan[c + d*x]^6/(a + a*Sec[c + d*x]),x]","\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac{\tan ^3(c+d x) (4-3 \sec (c+d x))}{12 a d}+\frac{\tan (c+d x) (8-3 \sec (c+d x))}{8 a d}-\frac{x}{a}","\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac{\tan ^3(c+d x) (4-3 \sec (c+d x))}{12 a d}+\frac{\tan (c+d x) (8-3 \sec (c+d x))}{8 a d}-\frac{x}{a}",1,"-(x/a) + (3*ArcTanh[Sin[c + d*x]])/(8*a*d) + ((8 - 3*Sec[c + d*x])*Tan[c + d*x])/(8*a*d) - ((4 - 3*Sec[c + d*x])*Tan[c + d*x]^3)/(12*a*d)","A",5,3,21,0.1429,1,"{3888, 3881, 3770}"
66,1,49,0,0.0771262,"\int \frac{\tan ^4(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Tan[c + d*x]^4/(a + a*Sec[c + d*x]),x]","-\frac{\tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{\tan (c+d x) (2-\sec (c+d x))}{2 a d}+\frac{x}{a}","-\frac{\tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{\tan (c+d x) (2-\sec (c+d x))}{2 a d}+\frac{x}{a}",1,"x/a - ArcTanh[Sin[c + d*x]]/(2*a*d) - ((2 - Sec[c + d*x])*Tan[c + d*x])/(2*a*d)","A",4,3,21,0.1429,1,"{3888, 3881, 3770}"
67,1,21,0,0.0503932,"\int \frac{\tan ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Tan[c + d*x]^2/(a + a*Sec[c + d*x]),x]","\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{x}{a}","\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{x}{a}",1,"-(x/a) + ArcTanh[Sin[c + d*x]]/(a*d)","A",3,2,21,0.09524,1,"{3888, 3770}"
68,1,61,0,0.0973967,"\int \frac{\cot ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Cot[c + d*x]^2/(a + a*Sec[c + d*x]),x]","\frac{\cot ^3(c+d x) (1-\sec (c+d x))}{3 a d}-\frac{\cot (c+d x) (3-2 \sec (c+d x))}{3 a d}-\frac{x}{a}","\frac{\cot ^3(c+d x) (1-\sec (c+d x))}{3 a d}-\frac{\cot (c+d x) (3-2 \sec (c+d x))}{3 a d}-\frac{x}{a}",1,"-(x/a) - (Cot[c + d*x]*(3 - 2*Sec[c + d*x]))/(3*a*d) + (Cot[c + d*x]^3*(1 - Sec[c + d*x]))/(3*a*d)","A",4,3,21,0.1429,1,"{3888, 3882, 8}"
69,1,88,0,0.1273823,"\int \frac{\cot ^4(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Cot[c + d*x]^4/(a + a*Sec[c + d*x]),x]","\frac{\cot ^5(c+d x) (1-\sec (c+d x))}{5 a d}-\frac{\cot ^3(c+d x) (5-4 \sec (c+d x))}{15 a d}+\frac{\cot (c+d x) (15-8 \sec (c+d x))}{15 a d}+\frac{x}{a}","\frac{\cot ^5(c+d x) (1-\sec (c+d x))}{5 a d}-\frac{\cot ^3(c+d x) (5-4 \sec (c+d x))}{15 a d}+\frac{\cot (c+d x) (15-8 \sec (c+d x))}{15 a d}+\frac{x}{a}",1,"x/a + (Cot[c + d*x]*(15 - 8*Sec[c + d*x]))/(15*a*d) - (Cot[c + d*x]^3*(5 - 4*Sec[c + d*x]))/(15*a*d) + (Cot[c + d*x]^5*(1 - Sec[c + d*x]))/(5*a*d)","A",5,3,21,0.1429,1,"{3888, 3882, 8}"
70,1,117,0,0.1619613,"\int \frac{\cot ^6(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Cot[c + d*x]^6/(a + a*Sec[c + d*x]),x]","\frac{\cot ^7(c+d x) (1-\sec (c+d x))}{7 a d}-\frac{\cot ^5(c+d x) (7-6 \sec (c+d x))}{35 a d}+\frac{\cot ^3(c+d x) (35-24 \sec (c+d x))}{105 a d}-\frac{\cot (c+d x) (35-16 \sec (c+d x))}{35 a d}-\frac{x}{a}","\frac{\cot ^7(c+d x) (1-\sec (c+d x))}{7 a d}-\frac{\cot ^5(c+d x) (7-6 \sec (c+d x))}{35 a d}+\frac{\cot ^3(c+d x) (35-24 \sec (c+d x))}{105 a d}-\frac{\cot (c+d x) (35-16 \sec (c+d x))}{35 a d}-\frac{x}{a}",1,"-(x/a) + (Cot[c + d*x]^3*(35 - 24*Sec[c + d*x]))/(105*a*d) - (Cot[c + d*x]*(35 - 16*Sec[c + d*x]))/(35*a*d) - (Cot[c + d*x]^5*(7 - 6*Sec[c + d*x]))/(35*a*d) + (Cot[c + d*x]^7*(1 - Sec[c + d*x]))/(7*a*d)","A",6,3,21,0.1429,1,"{3888, 3882, 8}"
71,1,120,0,0.0736004,"\int \frac{\tan ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]^9/(a + a*Sec[c + d*x])^2,x]","\frac{\sec ^6(c+d x)}{6 a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{\sec ^4(c+d x)}{4 a^2 d}+\frac{4 \sec ^3(c+d x)}{3 a^2 d}-\frac{\sec ^2(c+d x)}{2 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d}","\frac{\sec ^6(c+d x)}{6 a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{\sec ^4(c+d x)}{4 a^2 d}+\frac{4 \sec ^3(c+d x)}{3 a^2 d}-\frac{\sec ^2(c+d x)}{2 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d}",1,"-(Log[Cos[c + d*x]]/(a^2*d)) - (2*Sec[c + d*x])/(a^2*d) - Sec[c + d*x]^2/(2*a^2*d) + (4*Sec[c + d*x]^3)/(3*a^2*d) - Sec[c + d*x]^4/(4*a^2*d) - (2*Sec[c + d*x]^5)/(5*a^2*d) + Sec[c + d*x]^6/(6*a^2*d)","A",3,2,21,0.09524,1,"{3879, 88}"
72,1,65,0,0.0578504,"\int \frac{\tan ^7(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]^7/(a + a*Sec[c + d*x])^2,x]","\frac{\sec ^4(c+d x)}{4 a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}+\frac{2 \sec (c+d x)}{a^2 d}+\frac{\log (\cos (c+d x))}{a^2 d}","\frac{\sec ^4(c+d x)}{4 a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}+\frac{2 \sec (c+d x)}{a^2 d}+\frac{\log (\cos (c+d x))}{a^2 d}",1,"Log[Cos[c + d*x]]/(a^2*d) + (2*Sec[c + d*x])/(a^2*d) - (2*Sec[c + d*x]^3)/(3*a^2*d) + Sec[c + d*x]^4/(4*a^2*d)","A",3,2,21,0.09524,1,"{3879, 75}"
73,1,48,0,0.0497635,"\int \frac{\tan ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]^5/(a + a*Sec[c + d*x])^2,x]","\frac{\sec ^2(c+d x)}{2 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d}","\frac{\sec ^2(c+d x)}{2 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d}",1,"-(Log[Cos[c + d*x]]/(a^2*d)) - (2*Sec[c + d*x])/(a^2*d) + Sec[c + d*x]^2/(2*a^2*d)","A",3,2,21,0.09524,1,"{3879, 43}"
74,1,33,0,0.0448387,"\int \frac{\tan ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]^3/(a + a*Sec[c + d*x])^2,x]","\frac{2 \log (\cos (c+d x)+1)}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d}","\frac{2 \log (\cos (c+d x)+1)}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d}",1,"-(Log[Cos[c + d*x]]/(a^2*d)) + (2*Log[1 + Cos[c + d*x]])/(a^2*d)","A",3,2,21,0.09524,1,"{3879, 72}"
75,1,36,0,0.0347561,"\int \frac{\tan (c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]/(a + a*Sec[c + d*x])^2,x]","-\frac{1}{a^2 d (\cos (c+d x)+1)}-\frac{\log (\cos (c+d x)+1)}{a^2 d}","-\frac{1}{a^2 d (\cos (c+d x)+1)}-\frac{\log (\cos (c+d x)+1)}{a^2 d}",1,"-(1/(a^2*d*(1 + Cos[c + d*x]))) - Log[1 + Cos[c + d*x]]/(a^2*d)","A",3,2,19,0.1053,1,"{3879, 43}"
76,1,81,0,0.060841,"\int \frac{\cot (c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Cot[c + d*x]/(a + a*Sec[c + d*x])^2,x]","\frac{5}{4 a^2 d (\cos (c+d x)+1)}-\frac{1}{4 a^2 d (\cos (c+d x)+1)^2}+\frac{\log (1-\cos (c+d x))}{8 a^2 d}+\frac{7 \log (\cos (c+d x)+1)}{8 a^2 d}","\frac{5}{4 a^2 d (\cos (c+d x)+1)}-\frac{1}{4 a^2 d (\cos (c+d x)+1)^2}+\frac{\log (1-\cos (c+d x))}{8 a^2 d}+\frac{7 \log (\cos (c+d x)+1)}{8 a^2 d}",1,"-1/(4*a^2*d*(1 + Cos[c + d*x])^2) + 5/(4*a^2*d*(1 + Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(8*a^2*d) + (7*Log[1 + Cos[c + d*x]])/(8*a^2*d)","A",3,2,19,0.1053,1,"{3879, 88}"
77,1,123,0,0.086958,"\int \frac{\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Cot[c + d*x]^3/(a + a*Sec[c + d*x])^2,x]","-\frac{1}{16 a^2 d (1-\cos (c+d x))}-\frac{23}{16 a^2 d (\cos (c+d x)+1)}+\frac{1}{2 a^2 d (\cos (c+d x)+1)^2}-\frac{1}{12 a^2 d (\cos (c+d x)+1)^3}-\frac{3 \log (1-\cos (c+d x))}{16 a^2 d}-\frac{13 \log (\cos (c+d x)+1)}{16 a^2 d}","-\frac{1}{16 a^2 d (1-\cos (c+d x))}-\frac{23}{16 a^2 d (\cos (c+d x)+1)}+\frac{1}{2 a^2 d (\cos (c+d x)+1)^2}-\frac{1}{12 a^2 d (\cos (c+d x)+1)^3}-\frac{3 \log (1-\cos (c+d x))}{16 a^2 d}-\frac{13 \log (\cos (c+d x)+1)}{16 a^2 d}",1,"-1/(16*a^2*d*(1 - Cos[c + d*x])) - 1/(12*a^2*d*(1 + Cos[c + d*x])^3) + 1/(2*a^2*d*(1 + Cos[c + d*x])^2) - 23/(16*a^2*d*(1 + Cos[c + d*x])) - (3*Log[1 - Cos[c + d*x]])/(16*a^2*d) - (13*Log[1 + Cos[c + d*x]])/(16*a^2*d)","A",3,2,21,0.09524,1,"{3879, 88}"
78,1,165,0,0.1106809,"\int \frac{\cot ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^2,x]","\frac{9}{64 a^2 d (1-\cos (c+d x))}+\frac{51}{32 a^2 d (\cos (c+d x)+1)}-\frac{1}{64 a^2 d (1-\cos (c+d x))^2}-\frac{3}{4 a^2 d (\cos (c+d x)+1)^2}+\frac{11}{48 a^2 d (\cos (c+d x)+1)^3}-\frac{1}{32 a^2 d (\cos (c+d x)+1)^4}+\frac{29 \log (1-\cos (c+d x))}{128 a^2 d}+\frac{99 \log (\cos (c+d x)+1)}{128 a^2 d}","\frac{9}{64 a^2 d (1-\cos (c+d x))}+\frac{51}{32 a^2 d (\cos (c+d x)+1)}-\frac{1}{64 a^2 d (1-\cos (c+d x))^2}-\frac{3}{4 a^2 d (\cos (c+d x)+1)^2}+\frac{11}{48 a^2 d (\cos (c+d x)+1)^3}-\frac{1}{32 a^2 d (\cos (c+d x)+1)^4}+\frac{29 \log (1-\cos (c+d x))}{128 a^2 d}+\frac{99 \log (\cos (c+d x)+1)}{128 a^2 d}",1,"-1/(64*a^2*d*(1 - Cos[c + d*x])^2) + 9/(64*a^2*d*(1 - Cos[c + d*x])) - 1/(32*a^2*d*(1 + Cos[c + d*x])^4) + 11/(48*a^2*d*(1 + Cos[c + d*x])^3) - 3/(4*a^2*d*(1 + Cos[c + d*x])^2) + 51/(32*a^2*d*(1 + Cos[c + d*x])) + (29*Log[1 - Cos[c + d*x]])/(128*a^2*d) + (99*Log[1 + Cos[c + d*x]])/(128*a^2*d)","A",3,2,21,0.09524,1,"{3879, 88}"
79,1,119,0,0.1902589,"\int \frac{\tan ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]^8/(a + a*Sec[c + d*x])^2,x]","\frac{\tan ^5(c+d x)}{5 a^2 d}+\frac{\tan ^3(c+d x)}{3 a^2 d}-\frac{\tan (c+d x)}{a^2 d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac{\tan ^3(c+d x) \sec (c+d x)}{2 a^2 d}+\frac{3 \tan (c+d x) \sec (c+d x)}{4 a^2 d}+\frac{x}{a^2}","\frac{\tan ^5(c+d x)}{5 a^2 d}+\frac{\tan ^3(c+d x)}{3 a^2 d}-\frac{\tan (c+d x)}{a^2 d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac{\tan ^3(c+d x) \sec (c+d x)}{2 a^2 d}+\frac{3 \tan (c+d x) \sec (c+d x)}{4 a^2 d}+\frac{x}{a^2}",1,"x/a^2 - (3*ArcTanh[Sin[c + d*x]])/(4*a^2*d) - Tan[c + d*x]/(a^2*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(4*a^2*d) + Tan[c + d*x]^3/(3*a^2*d) - (Sec[c + d*x]*Tan[c + d*x]^3)/(2*a^2*d) + Tan[c + d*x]^5/(5*a^2*d)","A",11,8,21,0.3810,1,"{3888, 3886, 3473, 8, 2611, 3770, 2607, 30}"
80,1,72,0,0.1491595,"\int \frac{\tan ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]^6/(a + a*Sec[c + d*x])^2,x]","\frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{\tan (c+d x)}{a^2 d}+\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{\tan (c+d x) \sec (c+d x)}{a^2 d}-\frac{x}{a^2}","\frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{\tan (c+d x)}{a^2 d}+\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{\tan (c+d x) \sec (c+d x)}{a^2 d}-\frac{x}{a^2}",1,"-(x/a^2) + ArcTanh[Sin[c + d*x]]/(a^2*d) + Tan[c + d*x]/(a^2*d) - (Sec[c + d*x]*Tan[c + d*x])/(a^2*d) + Tan[c + d*x]^3/(3*a^2*d)","A",9,8,21,0.3810,1,"{3888, 3886, 3473, 8, 2611, 3770, 2607, 30}"
81,1,34,0,0.0653077,"\int \frac{\tan ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]^4/(a + a*Sec[c + d*x])^2,x]","\frac{\tan (c+d x)}{a^2 d}-\frac{2 \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{x}{a^2}","\frac{\tan (c+d x)}{a^2 d}-\frac{2 \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{x}{a^2}",1,"x/a^2 - (2*ArcTanh[Sin[c + d*x]])/(a^2*d) + Tan[c + d*x]/(a^2*d)","A",5,5,21,0.2381,1,"{3888, 3773, 3770, 3767, 8}"
82,1,35,0,0.1117425,"\int \frac{\tan ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]^2/(a + a*Sec[c + d*x])^2,x]","-\frac{2 \cot (c+d x)}{a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}-\frac{x}{a^2}","\frac{2 \tan (c+d x)}{a d (a \sec (c+d x)+a)}-\frac{x}{a^2}",1,"-(x/a^2) - (2*Cot[c + d*x])/(a^2*d) + (2*Csc[c + d*x])/(a^2*d)","A",9,6,21,0.2857,1,"{3888, 3886, 3473, 8, 2606, 3767}"
83,1,107,0,0.1737476,"\int \frac{\cot ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Cot[c + d*x]^2/(a + a*Sec[c + d*x])^2,x]","-\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{\cot (c+d x)}{a^2 d}+\frac{2 \csc ^5(c+d x)}{5 a^2 d}-\frac{4 \csc ^3(c+d x)}{3 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}-\frac{x}{a^2}","-\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{\cot (c+d x)}{a^2 d}+\frac{2 \csc ^5(c+d x)}{5 a^2 d}-\frac{4 \csc ^3(c+d x)}{3 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}-\frac{x}{a^2}",1,"-(x/a^2) - Cot[c + d*x]/(a^2*d) + Cot[c + d*x]^3/(3*a^2*d) - (2*Cot[c + d*x]^5)/(5*a^2*d) + (2*Csc[c + d*x])/(a^2*d) - (4*Csc[c + d*x]^3)/(3*a^2*d) + (2*Csc[c + d*x]^5)/(5*a^2*d)","A",12,8,21,0.3810,1,"{3888, 3886, 3473, 8, 2606, 194, 2607, 30}"
84,1,139,0,0.190764,"\int \frac{\cot ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Cot[c + d*x]^4/(a + a*Sec[c + d*x])^2,x]","-\frac{2 \cot ^7(c+d x)}{7 a^2 d}+\frac{\cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{\cot (c+d x)}{a^2 d}+\frac{2 \csc ^7(c+d x)}{7 a^2 d}-\frac{6 \csc ^5(c+d x)}{5 a^2 d}+\frac{2 \csc ^3(c+d x)}{a^2 d}-\frac{2 \csc (c+d x)}{a^2 d}+\frac{x}{a^2}","-\frac{2 \cot ^7(c+d x)}{7 a^2 d}+\frac{\cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{\cot (c+d x)}{a^2 d}+\frac{2 \csc ^7(c+d x)}{7 a^2 d}-\frac{6 \csc ^5(c+d x)}{5 a^2 d}+\frac{2 \csc ^3(c+d x)}{a^2 d}-\frac{2 \csc (c+d x)}{a^2 d}+\frac{x}{a^2}",1,"x/a^2 + Cot[c + d*x]/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d) + Cot[c + d*x]^5/(5*a^2*d) - (2*Cot[c + d*x]^7)/(7*a^2*d) - (2*Csc[c + d*x])/(a^2*d) + (2*Csc[c + d*x]^3)/(a^2*d) - (6*Csc[c + d*x]^5)/(5*a^2*d) + (2*Csc[c + d*x]^7)/(7*a^2*d)","A",13,8,21,0.3810,1,"{3888, 3886, 3473, 8, 2606, 194, 2607, 30}"
85,1,179,0,0.2078809,"\int \frac{\cot ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^2,x]","-\frac{2 \cot ^9(c+d x)}{9 a^2 d}+\frac{\cot ^7(c+d x)}{7 a^2 d}-\frac{\cot ^5(c+d x)}{5 a^2 d}+\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{\cot (c+d x)}{a^2 d}+\frac{2 \csc ^9(c+d x)}{9 a^2 d}-\frac{8 \csc ^7(c+d x)}{7 a^2 d}+\frac{12 \csc ^5(c+d x)}{5 a^2 d}-\frac{8 \csc ^3(c+d x)}{3 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}-\frac{x}{a^2}","-\frac{2 \cot ^9(c+d x)}{9 a^2 d}+\frac{\cot ^7(c+d x)}{7 a^2 d}-\frac{\cot ^5(c+d x)}{5 a^2 d}+\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{\cot (c+d x)}{a^2 d}+\frac{2 \csc ^9(c+d x)}{9 a^2 d}-\frac{8 \csc ^7(c+d x)}{7 a^2 d}+\frac{12 \csc ^5(c+d x)}{5 a^2 d}-\frac{8 \csc ^3(c+d x)}{3 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}-\frac{x}{a^2}",1,"-(x/a^2) - Cot[c + d*x]/(a^2*d) + Cot[c + d*x]^3/(3*a^2*d) - Cot[c + d*x]^5/(5*a^2*d) + Cot[c + d*x]^7/(7*a^2*d) - (2*Cot[c + d*x]^9)/(9*a^2*d) + (2*Csc[c + d*x])/(a^2*d) - (8*Csc[c + d*x]^3)/(3*a^2*d) + (12*Csc[c + d*x]^5)/(5*a^2*d) - (8*Csc[c + d*x]^7)/(7*a^2*d) + (2*Csc[c + d*x]^9)/(9*a^2*d)","A",14,8,21,0.3810,1,"{3888, 3886, 3473, 8, 2606, 194, 2607, 30}"
86,1,137,0,0.0768255,"\int \frac{\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Tan[c + d*x]^11/(a + a*Sec[c + d*x])^3,x]","\frac{\sec ^7(c+d x)}{7 a^3 d}-\frac{\sec ^6(c+d x)}{2 a^3 d}+\frac{\sec ^5(c+d x)}{5 a^3 d}+\frac{5 \sec ^4(c+d x)}{4 a^3 d}-\frac{5 \sec ^3(c+d x)}{3 a^3 d}-\frac{\sec ^2(c+d x)}{2 a^3 d}+\frac{3 \sec (c+d x)}{a^3 d}+\frac{\log (\cos (c+d x))}{a^3 d}","\frac{\sec ^7(c+d x)}{7 a^3 d}-\frac{\sec ^6(c+d x)}{2 a^3 d}+\frac{\sec ^5(c+d x)}{5 a^3 d}+\frac{5 \sec ^4(c+d x)}{4 a^3 d}-\frac{5 \sec ^3(c+d x)}{3 a^3 d}-\frac{\sec ^2(c+d x)}{2 a^3 d}+\frac{3 \sec (c+d x)}{a^3 d}+\frac{\log (\cos (c+d x))}{a^3 d}",1,"Log[Cos[c + d*x]]/(a^3*d) + (3*Sec[c + d*x])/(a^3*d) - Sec[c + d*x]^2/(2*a^3*d) - (5*Sec[c + d*x]^3)/(3*a^3*d) + (5*Sec[c + d*x]^4)/(4*a^3*d) + Sec[c + d*x]^5/(5*a^3*d) - Sec[c + d*x]^6/(2*a^3*d) + Sec[c + d*x]^7/(7*a^3*d)","A",3,2,21,0.09524,1,"{3879, 88}"
87,1,99,0,0.0670373,"\int \frac{\tan ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Tan[c + d*x]^9/(a + a*Sec[c + d*x])^3,x]","\frac{\sec ^5(c+d x)}{5 a^3 d}-\frac{3 \sec ^4(c+d x)}{4 a^3 d}+\frac{2 \sec ^3(c+d x)}{3 a^3 d}+\frac{\sec ^2(c+d x)}{a^3 d}-\frac{3 \sec (c+d x)}{a^3 d}-\frac{\log (\cos (c+d x))}{a^3 d}","\frac{\sec ^5(c+d x)}{5 a^3 d}-\frac{3 \sec ^4(c+d x)}{4 a^3 d}+\frac{2 \sec ^3(c+d x)}{3 a^3 d}+\frac{\sec ^2(c+d x)}{a^3 d}-\frac{3 \sec (c+d x)}{a^3 d}-\frac{\log (\cos (c+d x))}{a^3 d}",1,"-(Log[Cos[c + d*x]]/(a^3*d)) - (3*Sec[c + d*x])/(a^3*d) + Sec[c + d*x]^2/(a^3*d) + (2*Sec[c + d*x]^3)/(3*a^3*d) - (3*Sec[c + d*x]^4)/(4*a^3*d) + Sec[c + d*x]^5/(5*a^3*d)","A",3,2,21,0.09524,1,"{3879, 75}"
88,1,65,0,0.0568776,"\int \frac{\tan ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Tan[c + d*x]^7/(a + a*Sec[c + d*x])^3,x]","\frac{\sec ^3(c+d x)}{3 a^3 d}-\frac{3 \sec ^2(c+d x)}{2 a^3 d}+\frac{3 \sec (c+d x)}{a^3 d}+\frac{\log (\cos (c+d x))}{a^3 d}","\frac{\sec ^3(c+d x)}{3 a^3 d}-\frac{3 \sec ^2(c+d x)}{2 a^3 d}+\frac{3 \sec (c+d x)}{a^3 d}+\frac{\log (\cos (c+d x))}{a^3 d}",1,"Log[Cos[c + d*x]]/(a^3*d) + (3*Sec[c + d*x])/(a^3*d) - (3*Sec[c + d*x]^2)/(2*a^3*d) + Sec[c + d*x]^3/(3*a^3*d)","A",3,2,21,0.09524,1,"{3879, 43}"
89,1,46,0,0.0541435,"\int \frac{\tan ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Tan[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]","\frac{\sec (c+d x)}{a^3 d}+\frac{3 \log (\cos (c+d x))}{a^3 d}-\frac{4 \log (\cos (c+d x)+1)}{a^3 d}","\frac{\sec (c+d x)}{a^3 d}+\frac{3 \log (\cos (c+d x))}{a^3 d}-\frac{4 \log (\cos (c+d x)+1)}{a^3 d}",1,"(3*Log[Cos[c + d*x]])/(a^3*d) - (4*Log[1 + Cos[c + d*x]])/(a^3*d) + Sec[c + d*x]/(a^3*d)","A",3,2,21,0.09524,1,"{3879, 88}"
90,1,35,0,0.0514823,"\int \frac{\tan ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Tan[c + d*x]^3/(a + a*Sec[c + d*x])^3,x]","\frac{2}{a^3 d (\cos (c+d x)+1)}+\frac{\log (\cos (c+d x)+1)}{a^3 d}","\frac{2}{a^3 d (\cos (c+d x)+1)}+\frac{\log (\cos (c+d x)+1)}{a^3 d}",1,"2/(a^3*d*(1 + Cos[c + d*x])) + Log[1 + Cos[c + d*x]]/(a^3*d)","A",3,2,21,0.09524,1,"{3879, 43}"
91,1,56,0,0.0413221,"\int \frac{\tan (c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Tan[c + d*x]/(a + a*Sec[c + d*x])^3,x]","-\frac{2}{a^3 d (\cos (c+d x)+1)}+\frac{1}{2 a^3 d (\cos (c+d x)+1)^2}-\frac{\log (\cos (c+d x)+1)}{a^3 d}","-\frac{2}{a^3 d (\cos (c+d x)+1)}+\frac{1}{2 a^3 d (\cos (c+d x)+1)^2}-\frac{\log (\cos (c+d x)+1)}{a^3 d}",1,"1/(2*a^3*d*(1 + Cos[c + d*x])^2) - 2/(a^3*d*(1 + Cos[c + d*x])) - Log[1 + Cos[c + d*x]]/(a^3*d)","A",3,2,19,0.1053,1,"{3879, 43}"
92,1,101,0,0.0686666,"\int \frac{\cot (c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Cot[c + d*x]/(a + a*Sec[c + d*x])^3,x]","\frac{17}{8 a^3 d (\cos (c+d x)+1)}-\frac{7}{8 a^3 d (\cos (c+d x)+1)^2}+\frac{1}{6 a^3 d (\cos (c+d x)+1)^3}+\frac{\log (1-\cos (c+d x))}{16 a^3 d}+\frac{15 \log (\cos (c+d x)+1)}{16 a^3 d}","\frac{17}{8 a^3 d (\cos (c+d x)+1)}-\frac{7}{8 a^3 d (\cos (c+d x)+1)^2}+\frac{1}{6 a^3 d (\cos (c+d x)+1)^3}+\frac{\log (1-\cos (c+d x))}{16 a^3 d}+\frac{15 \log (\cos (c+d x)+1)}{16 a^3 d}",1,"1/(6*a^3*d*(1 + Cos[c + d*x])^3) - 7/(8*a^3*d*(1 + Cos[c + d*x])^2) + 17/(8*a^3*d*(1 + Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(16*a^3*d) + (15*Log[1 + Cos[c + d*x]])/(16*a^3*d)","A",3,2,19,0.1053,1,"{3879, 88}"
93,1,143,0,0.0966191,"\int \frac{\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Cot[c + d*x]^3/(a + a*Sec[c + d*x])^3,x]","-\frac{1}{32 a^3 d (1-\cos (c+d x))}-\frac{9}{4 a^3 d (\cos (c+d x)+1)}+\frac{39}{32 a^3 d (\cos (c+d x)+1)^2}-\frac{5}{12 a^3 d (\cos (c+d x)+1)^3}+\frac{1}{16 a^3 d (\cos (c+d x)+1)^4}-\frac{7 \log (1-\cos (c+d x))}{64 a^3 d}-\frac{57 \log (\cos (c+d x)+1)}{64 a^3 d}","-\frac{1}{32 a^3 d (1-\cos (c+d x))}-\frac{9}{4 a^3 d (\cos (c+d x)+1)}+\frac{39}{32 a^3 d (\cos (c+d x)+1)^2}-\frac{5}{12 a^3 d (\cos (c+d x)+1)^3}+\frac{1}{16 a^3 d (\cos (c+d x)+1)^4}-\frac{7 \log (1-\cos (c+d x))}{64 a^3 d}-\frac{57 \log (\cos (c+d x)+1)}{64 a^3 d}",1,"-1/(32*a^3*d*(1 - Cos[c + d*x])) + 1/(16*a^3*d*(1 + Cos[c + d*x])^4) - 5/(12*a^3*d*(1 + Cos[c + d*x])^3) + 39/(32*a^3*d*(1 + Cos[c + d*x])^2) - 9/(4*a^3*d*(1 + Cos[c + d*x])) - (7*Log[1 - Cos[c + d*x]])/(64*a^3*d) - (57*Log[1 + Cos[c + d*x]])/(64*a^3*d)","A",3,2,21,0.09524,1,"{3879, 88}"
94,1,185,0,0.1235723,"\int \frac{\cot ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]","\frac{5}{64 a^3 d (1-\cos (c+d x))}+\frac{303}{128 a^3 d (\cos (c+d x)+1)}-\frac{1}{128 a^3 d (1-\cos (c+d x))^2}-\frac{99}{64 a^3 d (\cos (c+d x)+1)^2}+\frac{35}{48 a^3 d (\cos (c+d x)+1)^3}-\frac{13}{64 a^3 d (\cos (c+d x)+1)^4}+\frac{1}{40 a^3 d (\cos (c+d x)+1)^5}+\frac{37 \log (1-\cos (c+d x))}{256 a^3 d}+\frac{219 \log (\cos (c+d x)+1)}{256 a^3 d}","\frac{5}{64 a^3 d (1-\cos (c+d x))}+\frac{303}{128 a^3 d (\cos (c+d x)+1)}-\frac{1}{128 a^3 d (1-\cos (c+d x))^2}-\frac{99}{64 a^3 d (\cos (c+d x)+1)^2}+\frac{35}{48 a^3 d (\cos (c+d x)+1)^3}-\frac{13}{64 a^3 d (\cos (c+d x)+1)^4}+\frac{1}{40 a^3 d (\cos (c+d x)+1)^5}+\frac{37 \log (1-\cos (c+d x))}{256 a^3 d}+\frac{219 \log (\cos (c+d x)+1)}{256 a^3 d}",1,"-1/(128*a^3*d*(1 - Cos[c + d*x])^2) + 5/(64*a^3*d*(1 - Cos[c + d*x])) + 1/(40*a^3*d*(1 + Cos[c + d*x])^5) - 13/(64*a^3*d*(1 + Cos[c + d*x])^4) + 35/(48*a^3*d*(1 + Cos[c + d*x])^3) - 99/(64*a^3*d*(1 + Cos[c + d*x])^2) + 303/(128*a^3*d*(1 + Cos[c + d*x])) + (37*Log[1 - Cos[c + d*x]])/(256*a^3*d) + (219*Log[1 + Cos[c + d*x]])/(256*a^3*d)","A",3,2,21,0.09524,1,"{3879, 88}"
95,1,237,0,0.3629984,"\int \frac{\tan ^{12}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Tan[c + d*x]^12/(a + a*Sec[c + d*x])^3,x]","-\frac{3 \tan ^7(c+d x)}{7 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{\tan (c+d x)}{a^3 d}-\frac{125 \tanh ^{-1}(\sin (c+d x))}{128 a^3 d}+\frac{\tan ^5(c+d x) \sec ^3(c+d x)}{8 a^3 d}-\frac{5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a^3 d}+\frac{5 \tan (c+d x) \sec ^3(c+d x)}{64 a^3 d}+\frac{\tan ^5(c+d x) \sec (c+d x)}{2 a^3 d}-\frac{5 \tan ^3(c+d x) \sec (c+d x)}{8 a^3 d}+\frac{115 \tan (c+d x) \sec (c+d x)}{128 a^3 d}+\frac{x}{a^3}","-\frac{3 \tan ^7(c+d x)}{7 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{\tan (c+d x)}{a^3 d}-\frac{125 \tanh ^{-1}(\sin (c+d x))}{128 a^3 d}+\frac{\tan ^5(c+d x) \sec ^3(c+d x)}{8 a^3 d}-\frac{5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a^3 d}+\frac{5 \tan (c+d x) \sec ^3(c+d x)}{64 a^3 d}+\frac{\tan ^5(c+d x) \sec (c+d x)}{2 a^3 d}-\frac{5 \tan ^3(c+d x) \sec (c+d x)}{8 a^3 d}+\frac{115 \tan (c+d x) \sec (c+d x)}{128 a^3 d}+\frac{x}{a^3}",1,"x/a^3 - (125*ArcTanh[Sin[c + d*x]])/(128*a^3*d) - Tan[c + d*x]/(a^3*d) + (115*Sec[c + d*x]*Tan[c + d*x])/(128*a^3*d) + (5*Sec[c + d*x]^3*Tan[c + d*x])/(64*a^3*d) + Tan[c + d*x]^3/(3*a^3*d) - (5*Sec[c + d*x]*Tan[c + d*x]^3)/(8*a^3*d) - (5*Sec[c + d*x]^3*Tan[c + d*x]^3)/(48*a^3*d) - Tan[c + d*x]^5/(5*a^3*d) + (Sec[c + d*x]*Tan[c + d*x]^5)/(2*a^3*d) + (Sec[c + d*x]^3*Tan[c + d*x]^5)/(8*a^3*d) - (3*Tan[c + d*x]^7)/(7*a^3*d)","A",18,9,21,0.4286,1,"{3888, 3886, 3473, 8, 2611, 3770, 2607, 30, 3768}"
96,1,169,0,0.2680652,"\int \frac{\tan ^{10}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Tan[c + d*x]^10/(a + a*Sec[c + d*x])^3,x]","-\frac{3 \tan ^5(c+d x)}{5 a^3 d}-\frac{\tan ^3(c+d x)}{3 a^3 d}+\frac{\tan (c+d x)}{a^3 d}+\frac{19 \tanh ^{-1}(\sin (c+d x))}{16 a^3 d}+\frac{\tan ^3(c+d x) \sec ^3(c+d x)}{6 a^3 d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{8 a^3 d}+\frac{3 \tan ^3(c+d x) \sec (c+d x)}{4 a^3 d}-\frac{17 \tan (c+d x) \sec (c+d x)}{16 a^3 d}-\frac{x}{a^3}","-\frac{3 \tan ^5(c+d x)}{5 a^3 d}-\frac{\tan ^3(c+d x)}{3 a^3 d}+\frac{\tan (c+d x)}{a^3 d}+\frac{19 \tanh ^{-1}(\sin (c+d x))}{16 a^3 d}+\frac{\tan ^3(c+d x) \sec ^3(c+d x)}{6 a^3 d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{8 a^3 d}+\frac{3 \tan ^3(c+d x) \sec (c+d x)}{4 a^3 d}-\frac{17 \tan (c+d x) \sec (c+d x)}{16 a^3 d}-\frac{x}{a^3}",1,"-(x/a^3) + (19*ArcTanh[Sin[c + d*x]])/(16*a^3*d) + Tan[c + d*x]/(a^3*d) - (17*Sec[c + d*x]*Tan[c + d*x])/(16*a^3*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(8*a^3*d) - Tan[c + d*x]^3/(3*a^3*d) + (3*Sec[c + d*x]*Tan[c + d*x]^3)/(4*a^3*d) + (Sec[c + d*x]^3*Tan[c + d*x]^3)/(6*a^3*d) - (3*Tan[c + d*x]^5)/(5*a^3*d)","A",15,9,21,0.4286,1,"{3888, 3886, 3473, 8, 2611, 3770, 2607, 30, 3768}"
97,1,99,0,0.2042994,"\int \frac{\tan ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Tan[c + d*x]^8/(a + a*Sec[c + d*x])^3,x]","-\frac{\tan ^3(c+d x)}{a^3 d}-\frac{\tan (c+d x)}{a^3 d}-\frac{13 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{4 a^3 d}+\frac{11 \tan (c+d x) \sec (c+d x)}{8 a^3 d}+\frac{x}{a^3}","-\frac{\tan ^3(c+d x)}{a^3 d}-\frac{\tan (c+d x)}{a^3 d}-\frac{13 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{4 a^3 d}+\frac{11 \tan (c+d x) \sec (c+d x)}{8 a^3 d}+\frac{x}{a^3}",1,"x/a^3 - (13*ArcTanh[Sin[c + d*x]])/(8*a^3*d) - Tan[c + d*x]/(a^3*d) + (11*Sec[c + d*x]*Tan[c + d*x])/(8*a^3*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(4*a^3*d) - Tan[c + d*x]^3/(a^3*d)","A",12,9,21,0.4286,1,"{3888, 3886, 3473, 8, 2611, 3770, 2607, 30, 3768}"
98,1,66,0,0.0908163,"\int \frac{\tan ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Tan[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]","-\frac{5 \tan (c+d x)}{2 a^3 d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{\tan (c+d x) (1-\sec (c+d x))}{2 a^3 d}-\frac{x}{a^3}","-\frac{5 \tan (c+d x)}{2 a^3 d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{\tan (c+d x) (1-\sec (c+d x))}{2 a^3 d}-\frac{x}{a^3}",1,"-(x/a^3) + (7*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (5*Tan[c + d*x])/(2*a^3*d) - ((1 - Sec[c + d*x])*Tan[c + d*x])/(2*a^3*d)","A",6,6,21,0.2857,1,"{3888, 3775, 3914, 3767, 8, 3770}"
99,1,48,0,0.1409028,"\int \frac{\tan ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Tan[c + d*x]^4/(a + a*Sec[c + d*x])^3,x]","\frac{4 \cot (c+d x)}{a^3 d}-\frac{4 \csc (c+d x)}{a^3 d}+\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{x}{a^3}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{4 \tan (c+d x)}{a^2 d (a \sec (c+d x)+a)}+\frac{x}{a^3}",1,"x/a^3 + ArcTanh[Sin[c + d*x]]/(a^3*d) + (4*Cot[c + d*x])/(a^3*d) - (4*Csc[c + d*x])/(a^3*d)","A",12,9,21,0.4286,1,"{3888, 3886, 3473, 8, 2606, 3767, 2621, 321, 207}"
100,1,71,0,0.1733232,"\int \frac{\tan ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Tan[c + d*x]^2/(a + a*Sec[c + d*x])^3,x]","\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{x}{a^3}","\frac{2 \tan (c+d x)}{a^2 d (a \sec (c+d x)+a)}-\frac{x}{a^3}-\frac{\tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^3}",1,"-(x/a^3) - Cot[c + d*x]/(a^3*d) + (4*Cot[c + d*x]^3)/(3*a^3*d) + (3*Csc[c + d*x])/(a^3*d) - (4*Csc[c + d*x]^3)/(3*a^3*d)","A",12,7,21,0.3333,1,"{3888, 3886, 3473, 8, 2606, 2607, 30}"
101,1,143,0,0.2362582,"\int \frac{\cot ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Cot[c + d*x]^2/(a + a*Sec[c + d*x])^3,x]","\frac{4 \cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \csc ^7(c+d x)}{7 a^3 d}+\frac{11 \csc ^5(c+d x)}{5 a^3 d}-\frac{10 \csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{x}{a^3}","\frac{4 \cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \csc ^7(c+d x)}{7 a^3 d}+\frac{11 \csc ^5(c+d x)}{5 a^3 d}-\frac{10 \csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{x}{a^3}",1,"-(x/a^3) - Cot[c + d*x]/(a^3*d) + Cot[c + d*x]^3/(3*a^3*d) - Cot[c + d*x]^5/(5*a^3*d) + (4*Cot[c + d*x]^7)/(7*a^3*d) + (3*Csc[c + d*x])/(a^3*d) - (10*Csc[c + d*x]^3)/(3*a^3*d) + (11*Csc[c + d*x]^5)/(5*a^3*d) - (4*Csc[c + d*x]^7)/(7*a^3*d)","A",16,9,21,0.4286,1,"{3888, 3886, 3473, 8, 2606, 194, 2607, 30, 270}"
102,1,177,0,0.2534954,"\int \frac{\cot ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Cot[c + d*x]^4/(a + a*Sec[c + d*x])^3,x]","\frac{4 \cot ^9(c+d x)}{9 a^3 d}-\frac{\cot ^7(c+d x)}{7 a^3 d}+\frac{\cot ^5(c+d x)}{5 a^3 d}-\frac{\cot ^3(c+d x)}{3 a^3 d}+\frac{\cot (c+d x)}{a^3 d}-\frac{4 \csc ^9(c+d x)}{9 a^3 d}+\frac{15 \csc ^7(c+d x)}{7 a^3 d}-\frac{21 \csc ^5(c+d x)}{5 a^3 d}+\frac{13 \csc ^3(c+d x)}{3 a^3 d}-\frac{3 \csc (c+d x)}{a^3 d}+\frac{x}{a^3}","\frac{4 \cot ^9(c+d x)}{9 a^3 d}-\frac{\cot ^7(c+d x)}{7 a^3 d}+\frac{\cot ^5(c+d x)}{5 a^3 d}-\frac{\cot ^3(c+d x)}{3 a^3 d}+\frac{\cot (c+d x)}{a^3 d}-\frac{4 \csc ^9(c+d x)}{9 a^3 d}+\frac{15 \csc ^7(c+d x)}{7 a^3 d}-\frac{21 \csc ^5(c+d x)}{5 a^3 d}+\frac{13 \csc ^3(c+d x)}{3 a^3 d}-\frac{3 \csc (c+d x)}{a^3 d}+\frac{x}{a^3}",1,"x/a^3 + Cot[c + d*x]/(a^3*d) - Cot[c + d*x]^3/(3*a^3*d) + Cot[c + d*x]^5/(5*a^3*d) - Cot[c + d*x]^7/(7*a^3*d) + (4*Cot[c + d*x]^9)/(9*a^3*d) - (3*Csc[c + d*x])/(a^3*d) + (13*Csc[c + d*x]^3)/(3*a^3*d) - (21*Csc[c + d*x]^5)/(5*a^3*d) + (15*Csc[c + d*x]^7)/(7*a^3*d) - (4*Csc[c + d*x]^9)/(9*a^3*d)","A",17,9,21,0.4286,1,"{3888, 3886, 3473, 8, 2606, 194, 2607, 30, 270}"
103,1,215,0,0.278093,"\int \frac{\cot ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]","\frac{4 \cot ^{11}(c+d x)}{11 a^3 d}-\frac{\cot ^9(c+d x)}{9 a^3 d}+\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac{19 \csc ^9(c+d x)}{9 a^3 d}-\frac{36 \csc ^7(c+d x)}{7 a^3 d}+\frac{34 \csc ^5(c+d x)}{5 a^3 d}-\frac{16 \csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{x}{a^3}","\frac{4 \cot ^{11}(c+d x)}{11 a^3 d}-\frac{\cot ^9(c+d x)}{9 a^3 d}+\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac{19 \csc ^9(c+d x)}{9 a^3 d}-\frac{36 \csc ^7(c+d x)}{7 a^3 d}+\frac{34 \csc ^5(c+d x)}{5 a^3 d}-\frac{16 \csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{x}{a^3}",1,"-(x/a^3) - Cot[c + d*x]/(a^3*d) + Cot[c + d*x]^3/(3*a^3*d) - Cot[c + d*x]^5/(5*a^3*d) + Cot[c + d*x]^7/(7*a^3*d) - Cot[c + d*x]^9/(9*a^3*d) + (4*Cot[c + d*x]^11)/(11*a^3*d) + (3*Csc[c + d*x])/(a^3*d) - (16*Csc[c + d*x]^3)/(3*a^3*d) + (34*Csc[c + d*x]^5)/(5*a^3*d) - (36*Csc[c + d*x]^7)/(7*a^3*d) + (19*Csc[c + d*x]^9)/(9*a^3*d) - (4*Csc[c + d*x]^11)/(11*a^3*d)","A",18,9,21,0.4286,1,"{3888, 3886, 3473, 8, 2606, 194, 2607, 30, 270}"
104,1,310,0,0.320431,"\int (a+a \sec (c+d x)) (e \tan (c+d x))^{5/2} \, dx","Int[(a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2),x]","\frac{a e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}-\frac{a e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}-\frac{a e^{5/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{a e^{5/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{6 a e^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 d \sqrt{\sin (2 c+2 d x)}}-\frac{6 a e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac{2 e (3 a \sec (c+d x)+5 a) (e \tan (c+d x))^{3/2}}{15 d}","\frac{a e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}-\frac{a e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}-\frac{a e^{5/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{a e^{5/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{6 a e^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 d \sqrt{\sin (2 c+2 d x)}}-\frac{6 a e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac{2 e (3 a \sec (c+d x)+5 a) (e \tan (c+d x))^{3/2}}{15 d}",1,"(a*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (a*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (a*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) + (a*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) + (6*a*e^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*d*Sqrt[Sin[2*c + 2*d*x]]) - (6*a*e*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*d) + (2*e*(5*a + 3*a*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2))/(15*d)","A",17,14,23,0.6087,1,"{3881, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639}"
105,1,282,0,0.2740868,"\int (a+a \sec (c+d x)) (e \tan (c+d x))^{3/2} \, dx","Int[(a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2),x]","\frac{a e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}-\frac{a e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}+\frac{a e^{3/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{a e^{3/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{a e^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d \sqrt{e \tan (c+d x)}}+\frac{2 e (a \sec (c+d x)+3 a) \sqrt{e \tan (c+d x)}}{3 d}","\frac{a e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}-\frac{a e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}+\frac{a e^{3/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{a e^{3/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{a e^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d \sqrt{e \tan (c+d x)}}+\frac{2 e (a \sec (c+d x)+3 a) \sqrt{e \tan (c+d x)}}{3 d}",1,"(a*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (a*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) + (a*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (a*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (a*e^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*d*Sqrt[e*Tan[c + d*x]]) + (2*e*(3*a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]])/(3*d)","A",16,13,23,0.5652,1,"{3881, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641}"
106,1,272,0,0.2439566,"\int (a+a \sec (c+d x)) \sqrt{e \tan (c+d x)} \, dx","Int[(a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]],x]","-\frac{a \sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}+\frac{a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}+\frac{a \sqrt{e} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{a \sqrt{e} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{2 a \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac{2 a \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{d \sqrt{\sin (2 c+2 d x)}}","-\frac{a \sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}+\frac{a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}+\frac{a \sqrt{e} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{a \sqrt{e} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{2 a \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac{2 a \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{d \sqrt{\sin (2 c+2 d x)}}",1,"-((a*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d)) + (a*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) + (a*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (a*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (2*a*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(d*Sqrt[Sin[2*c + 2*d*x]]) + (2*a*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(d*e)","A",16,13,23,0.5652,1,"{3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639}"
107,1,244,0,0.2128785,"\int \frac{a+a \sec (c+d x)}{\sqrt{e \tan (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])/Sqrt[e*Tan[c + d*x]],x]","-\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \sqrt{e}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \sqrt{e}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{a \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{d \sqrt{e \tan (c+d x)}}","-\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \sqrt{e}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \sqrt{e}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{a \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{d \sqrt{e \tan (c+d x)}}",1,"-((a*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e])) + (a*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) + (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) + (a*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(d*Sqrt[e*Tan[c + d*x]])","A",15,12,23,0.5217,1,"{3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641}"
108,1,305,0,0.3033285,"\int \frac{a+a \sec (c+d x)}{(e \tan (c+d x))^{3/2}} \, dx","Int[(a + a*Sec[c + d*x])/(e*Tan[c + d*x])^(3/2),x]","\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{3/2}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{3/2}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}+\frac{2 a \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e^3}-\frac{2 a \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{d e^2 \sqrt{\sin (2 c+2 d x)}}-\frac{2 (a \sec (c+d x)+a)}{d e \sqrt{e \tan (c+d x)}}","\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{3/2}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{3/2}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}+\frac{2 a \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e^3}-\frac{2 a \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{d e^2 \sqrt{\sin (2 c+2 d x)}}-\frac{2 (a \sec (c+d x)+a)}{d e \sqrt{e \tan (c+d x)}}",1,"(a*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2)) - (a*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2)) - (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2)) + (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2)) - (2*(a + a*Sec[c + d*x]))/(d*e*Sqrt[e*Tan[c + d*x]]) - (2*a*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(d*e^2*Sqrt[Sin[2*c + 2*d*x]]) + (2*a*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(d*e^3)","A",17,14,23,0.6087,1,"{3882, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639}"
109,1,282,0,0.275203,"\int \frac{a+a \sec (c+d x)}{(e \tan (c+d x))^{5/2}} \, dx","Int[(a + a*Sec[c + d*x])/(e*Tan[c + d*x])^(5/2),x]","\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{5/2}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{5/2}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}-\frac{a \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d e^2 \sqrt{e \tan (c+d x)}}-\frac{2 (a \sec (c+d x)+a)}{3 d e (e \tan (c+d x))^{3/2}}","\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{5/2}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{5/2}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}-\frac{a \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d e^2 \sqrt{e \tan (c+d x)}}-\frac{2 (a \sec (c+d x)+a)}{3 d e (e \tan (c+d x))^{3/2}}",1,"(a*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2)) - (a*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2)) + (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2)) - (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2)) - (2*(a + a*Sec[c + d*x]))/(3*d*e*(e*Tan[c + d*x])^(3/2)) - (a*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*d*e^2*Sqrt[e*Tan[c + d*x]])","A",16,13,23,0.5652,1,"{3882, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641}"
110,1,346,0,0.3654971,"\int \frac{a+a \sec (c+d x)}{(e \tan (c+d x))^{7/2}} \, dx","Int[(a + a*Sec[c + d*x])/(e*Tan[c + d*x])^(7/2),x]","-\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{7/2}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{7/2}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}-\frac{6 a \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d e^5}+\frac{2 (3 a \sec (c+d x)+5 a)}{5 d e^3 \sqrt{e \tan (c+d x)}}+\frac{6 a \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 d e^4 \sqrt{\sin (2 c+2 d x)}}-\frac{2 (a \sec (c+d x)+a)}{5 d e (e \tan (c+d x))^{5/2}}","-\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{7/2}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{7/2}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}-\frac{6 a \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d e^5}+\frac{2 (3 a \sec (c+d x)+5 a)}{5 d e^3 \sqrt{e \tan (c+d x)}}+\frac{6 a \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 d e^4 \sqrt{\sin (2 c+2 d x)}}-\frac{2 (a \sec (c+d x)+a)}{5 d e (e \tan (c+d x))^{5/2}}",1,"-((a*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2))) + (a*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) + (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2)) - (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2)) - (2*(a + a*Sec[c + d*x]))/(5*d*e*(e*Tan[c + d*x])^(5/2)) + (2*(5*a + 3*a*Sec[c + d*x]))/(5*d*e^3*Sqrt[e*Tan[c + d*x]]) + (6*a*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*d*e^4*Sqrt[Sin[2*c + 2*d*x]]) - (6*a*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*d*e^5)","A",18,14,23,0.6087,1,"{3882, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639}"
111,1,366,0,0.4328959,"\int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx","Int[(a + a*Sec[c + d*x])^2*(e*Tan[c + d*x])^(5/2),x]","\frac{a^2 e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}-\frac{a^2 e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}-\frac{a^2 e^{5/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{a^2 e^{5/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{12 a^2 e^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 d \sqrt{\sin (2 c+2 d x)}}+\frac{2 a^2 (e \tan (c+d x))^{7/2}}{7 d e}+\frac{2 a^2 e (e \tan (c+d x))^{3/2}}{3 d}-\frac{12 a^2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac{4 a^2 e \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 d}","\frac{a^2 e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}-\frac{a^2 e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}-\frac{a^2 e^{5/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{a^2 e^{5/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{12 a^2 e^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 d \sqrt{\sin (2 c+2 d x)}}+\frac{2 a^2 (e \tan (c+d x))^{7/2}}{7 d e}+\frac{2 a^2 e (e \tan (c+d x))^{3/2}}{3 d}-\frac{12 a^2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac{4 a^2 e \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 d}",1,"(a^2*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (a^2*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (a^2*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) + (a^2*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) + (12*a^2*e^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*d*Sqrt[Sin[2*c + 2*d*x]]) + (2*a^2*e*(e*Tan[c + d*x])^(3/2))/(3*d) - (12*a^2*e*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*d) + (4*a^2*e*Sec[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*d) + (2*a^2*(e*Tan[c + d*x])^(7/2))/(7*d*e)","A",21,17,25,0.6800,1,"{3886, 3473, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2611, 2613, 2615, 2572, 2639, 2607, 32}"
112,1,335,0,0.3853153,"\int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{3/2} \, dx","Int[(a + a*Sec[c + d*x])^2*(e*Tan[c + d*x])^(3/2),x]","\frac{a^2 e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}-\frac{a^2 e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}+\frac{a^2 e^{3/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{a^2 e^{3/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{2 a^2 e^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d \sqrt{e \tan (c+d x)}}+\frac{2 a^2 (e \tan (c+d x))^{5/2}}{5 d e}+\frac{2 a^2 e \sqrt{e \tan (c+d x)}}{d}+\frac{4 a^2 e \sec (c+d x) \sqrt{e \tan (c+d x)}}{3 d}","\frac{a^2 e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}-\frac{a^2 e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}+\frac{a^2 e^{3/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{a^2 e^{3/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{2 a^2 e^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d \sqrt{e \tan (c+d x)}}+\frac{2 a^2 (e \tan (c+d x))^{5/2}}{5 d e}+\frac{2 a^2 e \sqrt{e \tan (c+d x)}}{d}+\frac{4 a^2 e \sec (c+d x) \sqrt{e \tan (c+d x)}}{3 d}",1,"(a^2*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (a^2*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) + (a^2*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (a^2*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (2*a^2*e^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*d*Sqrt[e*Tan[c + d*x]]) + (2*a^2*e*Sqrt[e*Tan[c + d*x]])/d + (4*a^2*e*Sec[c + d*x]*Sqrt[e*Tan[c + d*x]])/(3*d) + (2*a^2*(e*Tan[c + d*x])^(5/2))/(5*d*e)","A",20,16,25,0.6400,1,"{3886, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2611, 2614, 2573, 2641, 2607, 32}"
113,1,309,0,0.3355173,"\int (a+a \sec (c+d x))^2 \sqrt{e \tan (c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]],x]","-\frac{a^2 \sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}+\frac{a^2 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}+\frac{2 a^2 (e \tan (c+d x))^{3/2}}{3 d e}+\frac{a^2 \sqrt{e} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{a^2 \sqrt{e} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{4 a^2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac{4 a^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{d \sqrt{\sin (2 c+2 d x)}}","-\frac{a^2 \sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}+\frac{a^2 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}+\frac{2 a^2 (e \tan (c+d x))^{3/2}}{3 d e}+\frac{a^2 \sqrt{e} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{a^2 \sqrt{e} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{4 a^2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac{4 a^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{d \sqrt{\sin (2 c+2 d x)}}",1,"-((a^2*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d)) + (a^2*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) + (a^2*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (a^2*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (4*a^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(d*Sqrt[Sin[2*c + 2*d*x]]) + (2*a^2*(e*Tan[c + d*x])^(3/2))/(3*d*e) + (4*a^2*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(d*e)","A",19,15,25,0.6000,1,"{3886, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639, 2607, 32}"
114,1,278,0,0.3031583,"\int \frac{(a+a \sec (c+d x))^2}{\sqrt{e \tan (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^2/Sqrt[e*Tan[c + d*x]],x]","-\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \sqrt{e}}+\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \sqrt{e}}+\frac{2 a^2 \sqrt{e \tan (c+d x)}}{d e}-\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{d \sqrt{e \tan (c+d x)}}","-\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \sqrt{e}}+\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \sqrt{e}}+\frac{2 a^2 \sqrt{e \tan (c+d x)}}{d e}-\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{d \sqrt{e \tan (c+d x)}}",1,"-((a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e])) + (a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) + (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) + (2*a^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(d*Sqrt[e*Tan[c + d*x]]) + (2*a^2*Sqrt[e*Tan[c + d*x]])/(d*e)","A",18,14,25,0.5600,1,"{3886, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641, 2607, 32}"
115,1,310,0,0.3878745,"\int \frac{(a+a \sec (c+d x))^2}{(e \tan (c+d x))^{3/2}} \, dx","Int[(a + a*Sec[c + d*x])^2/(e*Tan[c + d*x])^(3/2),x]","\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{3/2}}-\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{3/2}}-\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}+\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}-\frac{4 a^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{d e^2 \sqrt{\sin (2 c+2 d x)}}-\frac{4 a^2}{d e \sqrt{e \tan (c+d x)}}-\frac{4 a^2 \cos (c+d x)}{d e \sqrt{e \tan (c+d x)}}","\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{3/2}}-\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{3/2}}-\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}+\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}-\frac{4 a^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{d e^2 \sqrt{\sin (2 c+2 d x)}}-\frac{4 a^2}{d e \sqrt{e \tan (c+d x)}}-\frac{4 a^2 \cos (c+d x)}{d e \sqrt{e \tan (c+d x)}}",1,"(a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2)) - (a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2)) - (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2)) + (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2)) - (4*a^2)/(d*e*Sqrt[e*Tan[c + d*x]]) - (4*a^2*Cos[c + d*x])/(d*e*Sqrt[e*Tan[c + d*x]]) - (4*a^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(d*e^2*Sqrt[Sin[2*c + 2*d*x]])","A",20,16,25,0.6400,1,"{3886, 3474, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2608, 2615, 2572, 2639, 2607, 32}"
116,1,316,0,0.3862157,"\int \frac{(a+a \sec (c+d x))^2}{(e \tan (c+d x))^{5/2}} \, dx","Int[(a + a*Sec[c + d*x])^2/(e*Tan[c + d*x])^(5/2),x]","\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{5/2}}-\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{5/2}}+\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}-\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}-\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d e^2 \sqrt{e \tan (c+d x)}}-\frac{4 a^2}{3 d e (e \tan (c+d x))^{3/2}}-\frac{4 a^2 \sec (c+d x)}{3 d e (e \tan (c+d x))^{3/2}}","\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{5/2}}-\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{5/2}}+\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}-\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}-\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d e^2 \sqrt{e \tan (c+d x)}}-\frac{4 a^2}{3 d e (e \tan (c+d x))^{3/2}}-\frac{4 a^2 \sec (c+d x)}{3 d e (e \tan (c+d x))^{3/2}}",1,"(a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2)) - (a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2)) + (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2)) - (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2)) - (4*a^2)/(3*d*e*(e*Tan[c + d*x])^(3/2)) - (4*a^2*Sec[c + d*x])/(3*d*e*(e*Tan[c + d*x])^(3/2)) - (2*a^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*d*e^2*Sqrt[e*Tan[c + d*x]])","A",20,16,25,0.6400,1,"{3886, 3474, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2609, 2614, 2573, 2641, 2607, 32}"
117,1,370,0,0.4558751,"\int \frac{(a+a \sec (c+d x))^2}{(e \tan (c+d x))^{7/2}} \, dx","Int[(a + a*Sec[c + d*x])^2/(e*Tan[c + d*x])^(7/2),x]","-\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{7/2}}+\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{7/2}}+\frac{2 a^2}{d e^3 \sqrt{e \tan (c+d x)}}+\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}-\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}+\frac{12 a^2 \cos (c+d x)}{5 d e^3 \sqrt{e \tan (c+d x)}}+\frac{12 a^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 d e^4 \sqrt{\sin (2 c+2 d x)}}-\frac{4 a^2}{5 d e (e \tan (c+d x))^{5/2}}-\frac{4 a^2 \sec (c+d x)}{5 d e (e \tan (c+d x))^{5/2}}","-\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{7/2}}+\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{7/2}}+\frac{2 a^2}{d e^3 \sqrt{e \tan (c+d x)}}+\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}-\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}+\frac{12 a^2 \cos (c+d x)}{5 d e^3 \sqrt{e \tan (c+d x)}}+\frac{12 a^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 d e^4 \sqrt{\sin (2 c+2 d x)}}-\frac{4 a^2}{5 d e (e \tan (c+d x))^{5/2}}-\frac{4 a^2 \sec (c+d x)}{5 d e (e \tan (c+d x))^{5/2}}",1,"-((a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2))) + (a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) + (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2)) - (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2)) - (4*a^2)/(5*d*e*(e*Tan[c + d*x])^(5/2)) - (4*a^2*Sec[c + d*x])/(5*d*e*(e*Tan[c + d*x])^(5/2)) + (2*a^2)/(d*e^3*Sqrt[e*Tan[c + d*x]]) + (12*a^2*Cos[c + d*x])/(5*d*e^3*Sqrt[e*Tan[c + d*x]]) + (12*a^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*d*e^4*Sqrt[Sin[2*c + 2*d*x]])","A",22,17,25,0.6800,1,"{3886, 3474, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2609, 2608, 2615, 2572, 2639, 2607, 32}"
118,1,330,0,0.4190454,"\int \frac{(e \tan (c+d x))^{11/2}}{a+a \sec (c+d x)} \, dx","Int[(e*Tan[c + d*x])^(11/2)/(a + a*Sec[c + d*x]),x]","\frac{e^{11/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}-\frac{e^{11/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}+\frac{e^{11/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{e^{11/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{2 e^5 (21-5 \sec (c+d x)) \sqrt{e \tan (c+d x)}}{21 a d}-\frac{2 e^3 (7-5 \sec (c+d x)) (e \tan (c+d x))^{5/2}}{35 a d}+\frac{5 e^6 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{21 a d \sqrt{e \tan (c+d x)}}","\frac{e^{11/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}-\frac{e^{11/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}+\frac{e^{11/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{e^{11/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{2 e^5 (21-5 \sec (c+d x)) \sqrt{e \tan (c+d x)}}{21 a d}-\frac{2 e^3 (7-5 \sec (c+d x)) (e \tan (c+d x))^{5/2}}{35 a d}+\frac{5 e^6 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{21 a d \sqrt{e \tan (c+d x)}}",1,"(e^(11/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) - (e^(11/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) + (e^(11/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) - (e^(11/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) + (5*e^6*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(21*a*d*Sqrt[e*Tan[c + d*x]]) + (2*e^5*(21 - 5*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]])/(21*a*d) - (2*e^3*(7 - 5*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2))/(35*a*d)","A",18,14,25,0.5600,1,"{3888, 3881, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641}"
119,1,326,0,0.3936702,"\int \frac{(e \tan (c+d x))^{9/2}}{a+a \sec (c+d x)} \, dx","Int[(e*Tan[c + d*x])^(9/2)/(a + a*Sec[c + d*x]),x]","-\frac{e^{9/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}+\frac{e^{9/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}+\frac{e^{9/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{e^{9/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{6 e^3 \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 a d}-\frac{2 e^3 (5-3 \sec (c+d x)) (e \tan (c+d x))^{3/2}}{15 a d}+\frac{6 e^4 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 a d \sqrt{\sin (2 c+2 d x)}}","-\frac{e^{9/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}+\frac{e^{9/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}+\frac{e^{9/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{e^{9/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{6 e^3 \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 a d}-\frac{2 e^3 (5-3 \sec (c+d x)) (e \tan (c+d x))^{3/2}}{15 a d}+\frac{6 e^4 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 a d \sqrt{\sin (2 c+2 d x)}}",1,"-((e^(9/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d)) + (e^(9/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) + (e^(9/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) - (e^(9/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) + (6*e^4*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*a*d*Sqrt[Sin[2*c + 2*d*x]]) - (6*e^3*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*a*d) - (2*e^3*(5 - 3*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2))/(15*a*d)","A",18,15,25,0.6000,1,"{3888, 3881, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639}"
120,1,295,0,0.3525028,"\int \frac{(e \tan (c+d x))^{7/2}}{a+a \sec (c+d x)} \, dx","Int[(e*Tan[c + d*x])^(7/2)/(a + a*Sec[c + d*x]),x]","-\frac{e^{7/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}+\frac{e^{7/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}-\frac{e^{7/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{e^{7/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{2 e^3 (3-\sec (c+d x)) \sqrt{e \tan (c+d x)}}{3 a d}-\frac{e^4 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a d \sqrt{e \tan (c+d x)}}","-\frac{e^{7/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}+\frac{e^{7/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}-\frac{e^{7/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{e^{7/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{2 e^3 (3-\sec (c+d x)) \sqrt{e \tan (c+d x)}}{3 a d}-\frac{e^4 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a d \sqrt{e \tan (c+d x)}}",1,"-((e^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d)) + (e^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) - (e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) + (e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) - (e^4*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*a*d*Sqrt[e*Tan[c + d*x]]) - (2*e^3*(3 - Sec[c + d*x])*Sqrt[e*Tan[c + d*x]])/(3*a*d)","A",17,14,25,0.5600,1,"{3888, 3881, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641}"
121,1,285,0,0.3256083,"\int \frac{(e \tan (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx","Int[(e*Tan[c + d*x])^(5/2)/(a + a*Sec[c + d*x]),x]","\frac{e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}-\frac{e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}-\frac{e^{5/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{e^{5/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{2 e^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{a d \sqrt{\sin (2 c+2 d x)}}+\frac{2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{a d}","\frac{e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}-\frac{e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}-\frac{e^{5/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{e^{5/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{2 e^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{a d \sqrt{\sin (2 c+2 d x)}}+\frac{2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{a d}",1,"(e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) - (e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) - (e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) + (e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) - (2*e^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(a*d*Sqrt[Sin[2*c + 2*d*x]]) + (2*e*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(a*d)","A",17,14,25,0.5600,1,"{3888, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639}"
122,1,257,0,0.2919121,"\int \frac{(e \tan (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx","Int[(e*Tan[c + d*x])^(3/2)/(a + a*Sec[c + d*x]),x]","\frac{e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}-\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}+\frac{e^{3/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{e^{3/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{e^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a d \sqrt{e \tan (c+d x)}}","\frac{e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}-\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}+\frac{e^{3/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{e^{3/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{e^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a d \sqrt{e \tan (c+d x)}}",1,"(e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) - (e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) + (e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) - (e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) + (e^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(a*d*Sqrt[e*Tan[c + d*x]])","A",16,13,25,0.5200,1,"{3888, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641}"
123,1,315,0,0.3754576,"\int \frac{\sqrt{e \tan (c+d x)}}{a+a \sec (c+d x)} \, dx","Int[Sqrt[e*Tan[c + d*x]]/(a + a*Sec[c + d*x]),x]","-\frac{\sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}+\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{a d e}+\frac{2 e (1-\sec (c+d x))}{a d \sqrt{e \tan (c+d x)}}-\frac{2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{a d \sqrt{\sin (2 c+2 d x)}}","-\frac{\sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}+\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{a d e}+\frac{2 e (1-\sec (c+d x))}{a d \sqrt{e \tan (c+d x)}}-\frac{2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{a d \sqrt{\sin (2 c+2 d x)}}",1,"-((Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d)) + (Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) + (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) - (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) + (2*e*(1 - Sec[c + d*x]))/(a*d*Sqrt[e*Tan[c + d*x]]) - (2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(a*d*Sqrt[Sin[2*c + 2*d*x]]) + (2*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(a*d*e)","A",18,15,25,0.6000,1,"{3888, 3882, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639}"
124,1,290,0,0.345878,"\int \frac{1}{(a+a \sec (c+d x)) \sqrt{e \tan (c+d x)}} \, dx","Int[1/((a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]),x]","-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d \sqrt{e}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d \sqrt{e}}-\frac{\log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d \sqrt{e}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d \sqrt{e}}+\frac{2 e (1-\sec (c+d x))}{3 a d (e \tan (c+d x))^{3/2}}-\frac{\sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a d \sqrt{e \tan (c+d x)}}","-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d \sqrt{e}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d \sqrt{e}}-\frac{\log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d \sqrt{e}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d \sqrt{e}}+\frac{2 e (1-\sec (c+d x))}{3 a d (e \tan (c+d x))^{3/2}}-\frac{\sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a d \sqrt{e \tan (c+d x)}}",1,"-(ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*Sqrt[e]) - Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*Sqrt[e]) + Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*Sqrt[e]) + (2*e*(1 - Sec[c + d*x]))/(3*a*d*(e*Tan[c + d*x])^(3/2)) - (EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*a*d*Sqrt[e*Tan[c + d*x]])","A",17,14,25,0.5600,1,"{3888, 3882, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641}"
125,1,359,0,0.4520683,"\int \frac{1}{(a+a \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx","Int[1/((a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2)),x]","\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d e^{3/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d e^{3/2}}-\frac{\log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d e^{3/2}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d e^{3/2}}-\frac{6 \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 a d e^3}+\frac{6 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 a d e^2 \sqrt{\sin (2 c+2 d x)}}-\frac{2 (5-3 \sec (c+d x))}{5 a d e \sqrt{e \tan (c+d x)}}+\frac{2 e (1-\sec (c+d x))}{5 a d (e \tan (c+d x))^{5/2}}","\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d e^{3/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d e^{3/2}}-\frac{\log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d e^{3/2}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d e^{3/2}}-\frac{6 \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 a d e^3}+\frac{6 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 a d e^2 \sqrt{\sin (2 c+2 d x)}}-\frac{2 (5-3 \sec (c+d x))}{5 a d e \sqrt{e \tan (c+d x)}}+\frac{2 e (1-\sec (c+d x))}{5 a d (e \tan (c+d x))^{5/2}}",1,"ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*e^(3/2)) - ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*e^(3/2)) - Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*e^(3/2)) + Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*e^(3/2)) + (2*e*(1 - Sec[c + d*x]))/(5*a*d*(e*Tan[c + d*x])^(5/2)) - (2*(5 - 3*Sec[c + d*x]))/(5*a*d*e*Sqrt[e*Tan[c + d*x]]) + (6*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*a*d*e^2*Sqrt[Sin[2*c + 2*d*x]]) - (6*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*a*d*e^3)","A",19,15,25,0.6000,1,"{3888, 3882, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639}"
126,1,328,0,0.4165428,"\int \frac{1}{(a+a \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx","Int[1/((a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2)),x]","\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d e^{5/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d e^{5/2}}+\frac{\log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d e^{5/2}}-\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d e^{5/2}}+\frac{5 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{21 a d e^2 \sqrt{e \tan (c+d x)}}-\frac{2 (7-5 \sec (c+d x))}{21 a d e (e \tan (c+d x))^{3/2}}+\frac{2 e (1-\sec (c+d x))}{7 a d (e \tan (c+d x))^{7/2}}","\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d e^{5/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d e^{5/2}}+\frac{\log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d e^{5/2}}-\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d e^{5/2}}+\frac{5 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{21 a d e^2 \sqrt{e \tan (c+d x)}}-\frac{2 (7-5 \sec (c+d x))}{21 a d e (e \tan (c+d x))^{3/2}}+\frac{2 e (1-\sec (c+d x))}{7 a d (e \tan (c+d x))^{7/2}}",1,"ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*e^(5/2)) - ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*e^(5/2)) + Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*e^(5/2)) - Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*e^(5/2)) + (2*e*(1 - Sec[c + d*x]))/(7*a*d*(e*Tan[c + d*x])^(7/2)) - (2*(7 - 5*Sec[c + d*x]))/(21*a*d*e*(e*Tan[c + d*x])^(3/2)) + (5*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(21*a*d*e^2*Sqrt[e*Tan[c + d*x]])","A",18,14,25,0.5600,1,"{3888, 3882, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641}"
127,1,372,0,0.4916119,"\int \frac{(e \tan (c+d x))^{13/2}}{(a+a \sec (c+d x))^2} \, dx","Int[(e*Tan[c + d*x])^(13/2)/(a + a*Sec[c + d*x])^2,x]","\frac{e^{13/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}-\frac{e^{13/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}+\frac{2 e^5 (e \tan (c+d x))^{3/2}}{3 a^2 d}+\frac{2 e^3 (e \tan (c+d x))^{7/2}}{7 a^2 d}-\frac{e^{13/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{e^{13/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{12 e^5 \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 a^2 d}-\frac{4 e^5 \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 a^2 d}-\frac{12 e^6 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 a^2 d \sqrt{\sin (2 c+2 d x)}}","\frac{e^{13/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}-\frac{e^{13/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}+\frac{2 e^5 (e \tan (c+d x))^{3/2}}{3 a^2 d}+\frac{2 e^3 (e \tan (c+d x))^{7/2}}{7 a^2 d}-\frac{e^{13/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{e^{13/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{12 e^5 \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 a^2 d}-\frac{4 e^5 \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 a^2 d}-\frac{12 e^6 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 a^2 d \sqrt{\sin (2 c+2 d x)}}",1,"(e^(13/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) - (e^(13/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) - (e^(13/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) + (e^(13/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (12*e^6*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*a^2*d*Sqrt[Sin[2*c + 2*d*x]]) + (2*e^5*(e*Tan[c + d*x])^(3/2))/(3*a^2*d) + (12*e^5*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*a^2*d) - (4*e^5*Sec[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*a^2*d) + (2*e^3*(e*Tan[c + d*x])^(7/2))/(7*a^2*d)","A",22,18,25,0.7200,1,"{3888, 3886, 3473, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2611, 2613, 2615, 2572, 2639, 2607, 32}"
128,1,339,0,0.4583372,"\int \frac{(e \tan (c+d x))^{11/2}}{(a+a \sec (c+d x))^2} \, dx","Int[(e*Tan[c + d*x])^(11/2)/(a + a*Sec[c + d*x])^2,x]","\frac{e^{11/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}-\frac{e^{11/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}+\frac{2 e^5 \sqrt{e \tan (c+d x)}}{a^2 d}+\frac{2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d}+\frac{e^{11/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{e^{11/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{4 e^5 \sec (c+d x) \sqrt{e \tan (c+d x)}}{3 a^2 d}+\frac{2 e^6 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a^2 d \sqrt{e \tan (c+d x)}}","\frac{e^{11/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}-\frac{e^{11/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}+\frac{2 e^5 \sqrt{e \tan (c+d x)}}{a^2 d}+\frac{2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d}+\frac{e^{11/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{e^{11/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{4 e^5 \sec (c+d x) \sqrt{e \tan (c+d x)}}{3 a^2 d}+\frac{2 e^6 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a^2 d \sqrt{e \tan (c+d x)}}",1,"(e^(11/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) - (e^(11/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) + (e^(11/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (e^(11/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) + (2*e^6*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*a^2*d*Sqrt[e*Tan[c + d*x]]) + (2*e^5*Sqrt[e*Tan[c + d*x]])/(a^2*d) - (4*e^5*Sec[c + d*x]*Sqrt[e*Tan[c + d*x]])/(3*a^2*d) + (2*e^3*(e*Tan[c + d*x])^(5/2))/(5*a^2*d)","A",21,17,25,0.6800,1,"{3888, 3886, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2611, 2614, 2573, 2641, 2607, 32}"
129,1,312,0,0.4106492,"\int \frac{(e \tan (c+d x))^{9/2}}{(a+a \sec (c+d x))^2} \, dx","Int[(e*Tan[c + d*x])^(9/2)/(a + a*Sec[c + d*x])^2,x]","-\frac{e^{9/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}+\frac{e^{9/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}+\frac{2 e^3 (e \tan (c+d x))^{3/2}}{3 a^2 d}+\frac{e^{9/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{e^{9/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{4 e^3 \cos (c+d x) (e \tan (c+d x))^{3/2}}{a^2 d}+\frac{4 e^4 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{a^2 d \sqrt{\sin (2 c+2 d x)}}","-\frac{e^{9/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}+\frac{e^{9/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}+\frac{2 e^3 (e \tan (c+d x))^{3/2}}{3 a^2 d}+\frac{e^{9/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{e^{9/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{4 e^3 \cos (c+d x) (e \tan (c+d x))^{3/2}}{a^2 d}+\frac{4 e^4 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{a^2 d \sqrt{\sin (2 c+2 d x)}}",1,"-((e^(9/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d)) + (e^(9/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) + (e^(9/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (e^(9/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) + (4*e^4*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(a^2*d*Sqrt[Sin[2*c + 2*d*x]]) + (2*e^3*(e*Tan[c + d*x])^(3/2))/(3*a^2*d) - (4*e^3*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(a^2*d)","A",20,16,25,0.6400,1,"{3888, 3886, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639, 2607, 32}"
130,1,281,0,0.3781747,"\int \frac{(e \tan (c+d x))^{7/2}}{(a+a \sec (c+d x))^2} \, dx","Int[(e*Tan[c + d*x])^(7/2)/(a + a*Sec[c + d*x])^2,x]","-\frac{e^{7/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}+\frac{e^{7/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}+\frac{2 e^3 \sqrt{e \tan (c+d x)}}{a^2 d}-\frac{e^{7/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{e^{7/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{2 e^4 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a^2 d \sqrt{e \tan (c+d x)}}","-\frac{e^{7/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}+\frac{e^{7/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}+\frac{2 e^3 \sqrt{e \tan (c+d x)}}{a^2 d}-\frac{e^{7/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{e^{7/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{2 e^4 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a^2 d \sqrt{e \tan (c+d x)}}",1,"-((e^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d)) + (e^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) - (e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) + (e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (2*e^4*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(a^2*d*Sqrt[e*Tan[c + d*x]]) + (2*e^3*Sqrt[e*Tan[c + d*x]])/(a^2*d)","A",19,15,25,0.6000,1,"{3888, 3886, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641, 2607, 32}"
131,1,310,0,0.4651324,"\int \frac{(e \tan (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx","Int[(e*Tan[c + d*x])^(5/2)/(a + a*Sec[c + d*x])^2,x]","\frac{e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}-\frac{e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}-\frac{4 e^3}{a^2 d \sqrt{e \tan (c+d x)}}-\frac{e^{5/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{e^{5/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{4 e^3 \cos (c+d x)}{a^2 d \sqrt{e \tan (c+d x)}}+\frac{4 e^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{a^2 d \sqrt{\sin (2 c+2 d x)}}","\frac{e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}-\frac{e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}-\frac{4 e^3}{a^2 d \sqrt{e \tan (c+d x)}}-\frac{e^{5/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{e^{5/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{4 e^3 \cos (c+d x)}{a^2 d \sqrt{e \tan (c+d x)}}+\frac{4 e^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{a^2 d \sqrt{\sin (2 c+2 d x)}}",1,"(e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) - (e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) - (e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) + (e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (4*e^3)/(a^2*d*Sqrt[e*Tan[c + d*x]]) + (4*e^3*Cos[c + d*x])/(a^2*d*Sqrt[e*Tan[c + d*x]]) + (4*e^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(a^2*d*Sqrt[Sin[2*c + 2*d*x]])","A",21,17,25,0.6800,1,"{3888, 3886, 3474, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2608, 2615, 2572, 2639, 2607, 32}"
132,1,316,0,0.4587536,"\int \frac{(e \tan (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx","Int[(e*Tan[c + d*x])^(3/2)/(a + a*Sec[c + d*x])^2,x]","\frac{e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}-\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}-\frac{4 e^3}{3 a^2 d (e \tan (c+d x))^{3/2}}+\frac{e^{3/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{e^{3/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{4 e^3 \sec (c+d x)}{3 a^2 d (e \tan (c+d x))^{3/2}}+\frac{2 e^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a^2 d \sqrt{e \tan (c+d x)}}","\frac{e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}-\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}-\frac{4 e^3}{3 a^2 d (e \tan (c+d x))^{3/2}}+\frac{e^{3/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{e^{3/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{4 e^3 \sec (c+d x)}{3 a^2 d (e \tan (c+d x))^{3/2}}+\frac{2 e^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a^2 d \sqrt{e \tan (c+d x)}}",1,"(e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) - (e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) + (e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (4*e^3)/(3*a^2*d*(e*Tan[c + d*x])^(3/2)) + (4*e^3*Sec[c + d*x])/(3*a^2*d*(e*Tan[c + d*x])^(3/2)) + (2*e^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*a^2*d*Sqrt[e*Tan[c + d*x]])","A",21,17,25,0.6800,1,"{3888, 3886, 3474, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2609, 2614, 2573, 2641, 2607, 32}"
133,1,363,0,0.5248642,"\int \frac{\sqrt{e \tan (c+d x)}}{(a+a \sec (c+d x))^2} \, dx","Int[Sqrt[e*Tan[c + d*x]]/(a + a*Sec[c + d*x])^2,x]","-\frac{4 e^3}{5 a^2 d (e \tan (c+d x))^{5/2}}+\frac{4 e^3 \sec (c+d x)}{5 a^2 d (e \tan (c+d x))^{5/2}}-\frac{\sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}+\frac{2 e}{a^2 d \sqrt{e \tan (c+d x)}}+\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{12 e \cos (c+d x)}{5 a^2 d \sqrt{e \tan (c+d x)}}-\frac{12 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 a^2 d \sqrt{\sin (2 c+2 d x)}}","-\frac{4 e^3}{5 a^2 d (e \tan (c+d x))^{5/2}}+\frac{4 e^3 \sec (c+d x)}{5 a^2 d (e \tan (c+d x))^{5/2}}-\frac{\sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}+\frac{2 e}{a^2 d \sqrt{e \tan (c+d x)}}+\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{12 e \cos (c+d x)}{5 a^2 d \sqrt{e \tan (c+d x)}}-\frac{12 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 a^2 d \sqrt{\sin (2 c+2 d x)}}",1,"-((Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d)) + (Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) + (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (4*e^3)/(5*a^2*d*(e*Tan[c + d*x])^(5/2)) + (4*e^3*Sec[c + d*x])/(5*a^2*d*(e*Tan[c + d*x])^(5/2)) + (2*e)/(a^2*d*Sqrt[e*Tan[c + d*x]]) - (12*e*Cos[c + d*x])/(5*a^2*d*Sqrt[e*Tan[c + d*x]]) - (12*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*a^2*d*Sqrt[Sin[2*c + 2*d*x]])","A",23,18,25,0.7200,1,"{3888, 3886, 3474, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2609, 2608, 2615, 2572, 2639, 2607, 32}"
134,1,365,0,0.5308671,"\int \frac{1}{(a+a \sec (c+d x))^2 \sqrt{e \tan (c+d x)}} \, dx","Int[1/((a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]),x]","-\frac{4 e^3}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac{4 e^3 \sec (c+d x)}{7 a^2 d (e \tan (c+d x))^{7/2}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d \sqrt{e}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d \sqrt{e}}+\frac{2 e}{3 a^2 d (e \tan (c+d x))^{3/2}}-\frac{\log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d \sqrt{e}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d \sqrt{e}}-\frac{20 e \sec (c+d x)}{21 a^2 d (e \tan (c+d x))^{3/2}}-\frac{10 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{21 a^2 d \sqrt{e \tan (c+d x)}}","-\frac{4 e^3}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac{4 e^3 \sec (c+d x)}{7 a^2 d (e \tan (c+d x))^{7/2}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d \sqrt{e}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d \sqrt{e}}+\frac{2 e}{3 a^2 d (e \tan (c+d x))^{3/2}}-\frac{\log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d \sqrt{e}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d \sqrt{e}}-\frac{20 e \sec (c+d x)}{21 a^2 d (e \tan (c+d x))^{3/2}}-\frac{10 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{21 a^2 d \sqrt{e \tan (c+d x)}}",1,"-(ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a^2*d*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a^2*d*Sqrt[e]) - Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a^2*d*Sqrt[e]) + Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a^2*d*Sqrt[e]) - (4*e^3)/(7*a^2*d*(e*Tan[c + d*x])^(7/2)) + (4*e^3*Sec[c + d*x])/(7*a^2*d*(e*Tan[c + d*x])^(7/2)) + (2*e)/(3*a^2*d*(e*Tan[c + d*x])^(3/2)) - (20*e*Sec[c + d*x])/(21*a^2*d*(e*Tan[c + d*x])^(3/2)) - (10*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(21*a^2*d*Sqrt[e*Tan[c + d*x]])","A",23,17,25,0.6800,1,"{3888, 3886, 3474, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2609, 2614, 2573, 2641, 2607, 32}"
135,1,147,0,0.1149708,"\int \sqrt{a+a \sec (c+d x)} \tan ^5(c+d x) \, dx","Int[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^5,x]","\frac{2 (a \sec (c+d x)+a)^{9/2}}{9 a^4 d}-\frac{6 (a \sec (c+d x)+a)^{7/2}}{7 a^3 d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a^2 d}+\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a d}+\frac{2 \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}","\frac{2 (a \sec (c+d x)+a)^{9/2}}{9 a^4 d}-\frac{6 (a \sec (c+d x)+a)^{7/2}}{7 a^3 d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a^2 d}+\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a d}+\frac{2 \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}",1,"(-2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (2*Sqrt[a + a*Sec[c + d*x]])/d + (2*(a + a*Sec[c + d*x])^(3/2))/(3*a*d) + (2*(a + a*Sec[c + d*x])^(5/2))/(5*a^2*d) - (6*(a + a*Sec[c + d*x])^(7/2))/(7*a^3*d) + (2*(a + a*Sec[c + d*x])^(9/2))/(9*a^4*d)","A",8,5,23,0.2174,1,"{3880, 88, 50, 63, 207}"
136,1,99,0,0.0837909,"\int \sqrt{a+a \sec (c+d x)} \tan ^3(c+d x) \, dx","Int[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^3,x]","\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a^2 d}-\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a d}-\frac{2 \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}","\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a^2 d}-\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a d}-\frac{2 \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}",1,"(2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (2*Sqrt[a + a*Sec[c + d*x]])/d - (2*(a + a*Sec[c + d*x])^(3/2))/(3*a*d) + (2*(a + a*Sec[c + d*x])^(5/2))/(5*a^2*d)","A",6,5,23,0.2174,1,"{3880, 80, 50, 63, 207}"
137,1,51,0,0.0445641,"\int \sqrt{a+a \sec (c+d x)} \tan (c+d x) \, dx","Int[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x],x]","\frac{2 \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}","\frac{2 \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}",1,"(-2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (2*Sqrt[a + a*Sec[c + d*x]])/d","A",4,4,21,0.1905,1,"{3880, 50, 63, 207}"
138,1,73,0,0.0713238,"\int \cot (c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{\sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}","\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{\sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}",1,"(2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d","A",6,4,21,0.1905,1,"{3880, 86, 63, 207}"
139,1,131,0,0.1184591,"\int \cot ^3(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Cot[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]],x]","\frac{a}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a}{2 d (1-\sec (c+d x)) \sqrt{a \sec (c+d x)+a}}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{7 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} d}","\frac{a}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a}{2 d (1-\sec (c+d x)) \sqrt{a \sec (c+d x)+a}}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{7 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} d}",1,"(-2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (7*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*d) + a/(4*d*Sqrt[a + a*Sec[c + d*x]]) + a/(2*d*(1 - Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]])","A",8,6,23,0.2609,1,"{3880, 103, 152, 156, 63, 207}"
140,1,193,0,0.1578033,"\int \cot ^5(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Cot[c + d*x]^5*Sqrt[a + a*Sec[c + d*x]],x]","\frac{43 a^2}{96 d (a \sec (c+d x)+a)^{3/2}}-\frac{15 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}-\frac{21 a}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{107 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{64 \sqrt{2} d}","\frac{43 a^2}{96 d (a \sec (c+d x)+a)^{3/2}}-\frac{15 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}-\frac{21 a}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{107 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{64 \sqrt{2} d}",1,"(2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (107*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(64*Sqrt[2]*d) + (43*a^2)/(96*d*(a + a*Sec[c + d*x])^(3/2)) - a^2/(4*d*(1 - Sec[c + d*x])^2*(a + a*Sec[c + d*x])^(3/2)) - (15*a^2)/(16*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(3/2)) - (21*a)/(64*d*Sqrt[a + a*Sec[c + d*x]])","A",10,7,23,0.3043,1,"{3880, 103, 151, 152, 156, 63, 207}"
141,1,222,0,0.1060123,"\int \sqrt{a+a \sec (c+d x)} \tan ^6(c+d x) \, dx","Int[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^6,x]","\frac{2 a^6 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{10 a^5 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{2 a^4 \tan ^7(c+d x)}{d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^3 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a^2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\frac{2 a^6 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{10 a^5 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{2 a^4 \tan ^7(c+d x)}{d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^3 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a^2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(-2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) - (2*a^2*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^3*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2)) + (2*a^4*Tan[c + d*x]^7)/(d*(a + a*Sec[c + d*x])^(7/2)) + (10*a^5*Tan[c + d*x]^9)/(9*d*(a + a*Sec[c + d*x])^(9/2)) + (2*a^6*Tan[c + d*x]^11)/(11*d*(a + a*Sec[c + d*x])^(11/2))","A",4,3,23,0.1304,1,"{3887, 461, 203}"
142,1,160,0,0.0947099,"\int \sqrt{a+a \sec (c+d x)} \tan ^4(c+d x) \, dx","Int[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^4,x]","\frac{2 a^4 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{6 a^3 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a^2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\frac{2 a^4 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{6 a^3 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a^2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (2*a*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (6*a^3*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2)) + (2*a^4*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2))","A",4,3,23,0.1304,1,"{3887, 461, 203}"
143,1,96,0,0.0769082,"\int \sqrt{a+a \sec (c+d x)} \tan ^2(c+d x) \, dx","Int[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^2,x]","\frac{2 a^2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\frac{2 a^2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(-2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2))","A",4,4,23,0.1739,1,"{3887, 459, 321, 203}"
144,1,109,0,0.0990509,"\int \cot ^2(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Cot[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]],x]","-\frac{\cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{2} d}","-\frac{\cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{2} d}",1,"(-2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[2]*d) - (Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/d","A",5,4,23,0.1739,1,"{3887, 480, 522, 203}"
145,1,196,0,0.2033869,"\int \cot ^4(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Cot[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]],x]","\frac{\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 a d}+\frac{7 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{8 d}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{9 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{8 \sqrt{2} d}-\frac{\cos (c+d x) \cot ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{4 a d}","\frac{\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 a d}+\frac{7 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{8 d}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{9 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{8 \sqrt{2} d}-\frac{\cos (c+d x) \cot ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{4 a d}",1,"(2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (9*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(8*Sqrt[2]*d) + (7*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(8*d) + (Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(12*a*d) - (Cos[c + d*x]*Cot[c + d*x]^3*Sec[(c + d*x)/2]^2*(a + a*Sec[c + d*x])^(3/2))/(4*a*d)","A",7,5,23,0.2174,1,"{3887, 472, 583, 522, 203}"
146,1,280,0,0.2713052,"\int \cot ^6(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Cot[c + d*x]^6*Sqrt[a + a*Sec[c + d*x]],x]","\frac{87 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{160 a^2 d}-\frac{\cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{16 a^2 d}-\frac{17 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{32 a^2 d}-\frac{23 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{192 a d}-\frac{105 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{128 d}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{151 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{128 \sqrt{2} d}","\frac{87 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{160 a^2 d}-\frac{\cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{16 a^2 d}-\frac{17 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{32 a^2 d}-\frac{23 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{192 a d}-\frac{105 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{128 d}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{151 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{128 \sqrt{2} d}",1,"(-2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (151*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(128*Sqrt[2]*d) - (105*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(128*d) - (23*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(192*a*d) + (87*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(160*a^2*d) - (17*Cos[c + d*x]*Cot[c + d*x]^5*Sec[(c + d*x)/2]^2*(a + a*Sec[c + d*x])^(5/2))/(32*a^2*d) - (Cos[c + d*x]^2*Cot[c + d*x]^5*Sec[(c + d*x)/2]^4*(a + a*Sec[c + d*x])^(5/2))/(16*a^2*d)","A",9,6,23,0.2609,1,"{3887, 472, 579, 583, 522, 203}"
147,1,169,0,0.1361473,"\int (a+a \sec (c+d x))^{3/2} \tan ^5(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^5,x]","\frac{2 (a \sec (c+d x)+a)^{11/2}}{11 a^4 d}-\frac{2 (a \sec (c+d x)+a)^{9/2}}{3 a^3 d}+\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a^2 d}-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a d}+\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac{2 a \sqrt{a \sec (c+d x)+a}}{d}","\frac{2 (a \sec (c+d x)+a)^{11/2}}{11 a^4 d}-\frac{2 (a \sec (c+d x)+a)^{9/2}}{3 a^3 d}+\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a^2 d}-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a d}+\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac{2 a \sqrt{a \sec (c+d x)+a}}{d}",1,"(-2*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (2*a*Sqrt[a + a*Sec[c + d*x]])/d + (2*(a + a*Sec[c + d*x])^(3/2))/(3*d) + (2*(a + a*Sec[c + d*x])^(5/2))/(5*a*d) + (2*(a + a*Sec[c + d*x])^(7/2))/(7*a^2*d) - (2*(a + a*Sec[c + d*x])^(9/2))/(3*a^3*d) + (2*(a + a*Sec[c + d*x])^(11/2))/(11*a^4*d)","A",9,5,23,0.2174,1,"{3880, 88, 50, 63, 207}"
148,1,121,0,0.1035799,"\int (a+a \sec (c+d x))^{3/2} \tan ^3(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^3,x]","\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a^2 d}+\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a d}-\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 d}-\frac{2 a \sqrt{a \sec (c+d x)+a}}{d}","\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a^2 d}+\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a d}-\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 d}-\frac{2 a \sqrt{a \sec (c+d x)+a}}{d}",1,"(2*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (2*a*Sqrt[a + a*Sec[c + d*x]])/d - (2*(a + a*Sec[c + d*x])^(3/2))/(3*d) - (2*(a + a*Sec[c + d*x])^(5/2))/(5*a*d) + (2*(a + a*Sec[c + d*x])^(7/2))/(7*a^2*d)","A",7,5,23,0.2174,1,"{3880, 80, 50, 63, 207}"
149,1,73,0,0.0552833,"\int (a+a \sec (c+d x))^{3/2} \tan (c+d x) \, dx","Int[(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x],x]","-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{2 a \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 d}","-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{2 a \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(-2*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (2*a*Sqrt[a + a*Sec[c + d*x]])/d + (2*(a + a*Sec[c + d*x])^(3/2))/(3*d)","A",5,4,21,0.1905,1,"{3880, 50, 63, 207}"
150,1,73,0,0.0788902,"\int \cot (c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]*(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{2 \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}","\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{2 \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}",1,"(2*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (2*Sqrt[2]*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d","A",6,4,21,0.1905,1,"{3880, 83, 63, 207}"
151,1,109,0,0.107463,"\int \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2),x]","-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{5 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} d}+\frac{a \sqrt{a \sec (c+d x)+a}}{2 d (1-\sec (c+d x))}","-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{5 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} d}+\frac{a \sqrt{a \sec (c+d x)+a}}{2 d (1-\sec (c+d x))}",1,"(-2*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (5*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*d) + (a*Sqrt[a + a*Sec[c + d*x]])/(2*d*(1 - Sec[c + d*x]))","A",7,5,23,0.2174,1,"{3880, 103, 156, 63, 207}"
152,1,171,0,0.1430663,"\int \cot ^5(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2),x]","\frac{7 a^2}{32 d \sqrt{a \sec (c+d x)+a}}-\frac{13 a^2}{16 d (1-\sec (c+d x)) \sqrt{a \sec (c+d x)+a}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 \sqrt{a \sec (c+d x)+a}}+\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{71 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{32 \sqrt{2} d}","\frac{7 a^2}{32 d \sqrt{a \sec (c+d x)+a}}-\frac{13 a^2}{16 d (1-\sec (c+d x)) \sqrt{a \sec (c+d x)+a}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 \sqrt{a \sec (c+d x)+a}}+\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{71 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{32 \sqrt{2} d}",1,"(2*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (71*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(32*Sqrt[2]*d) + (7*a^2)/(32*d*Sqrt[a + a*Sec[c + d*x]]) - a^2/(4*d*(1 - Sec[c + d*x])^2*Sqrt[a + a*Sec[c + d*x]]) - (13*a^2)/(16*d*(1 - Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]])","A",9,7,23,0.3043,1,"{3880, 103, 151, 152, 156, 63, 207}"
153,1,258,0,0.1217986,"\int (a+a \sec (c+d x))^{3/2} \tan ^6(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^6,x]","\frac{2 a^8 \tan ^{13}(c+d x)}{13 d (a \sec (c+d x)+a)^{13/2}}+\frac{14 a^7 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{34 a^6 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{30 a^5 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^4 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a^3 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\frac{2 a^8 \tan ^{13}(c+d x)}{13 d (a \sec (c+d x)+a)^{13/2}}+\frac{14 a^7 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{34 a^6 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{30 a^5 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^4 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a^3 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(-2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) - (2*a^3*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^4*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2)) + (30*a^5*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2)) + (34*a^6*Tan[c + d*x]^9)/(9*d*(a + a*Sec[c + d*x])^(9/2)) + (14*a^7*Tan[c + d*x]^11)/(11*d*(a + a*Sec[c + d*x])^(11/2)) + (2*a^8*Tan[c + d*x]^13)/(13*d*(a + a*Sec[c + d*x])^(13/2))","A",4,3,23,0.1304,1,"{3887, 461, 203}"
154,1,194,0,0.1099261,"\int (a+a \sec (c+d x))^{3/2} \tan ^4(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^4,x]","\frac{2 a^6 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{10 a^5 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{14 a^4 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a^3 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{2 a^2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\frac{2 a^6 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{10 a^5 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{14 a^4 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a^3 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{2 a^2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (2*a^2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (14*a^4*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2)) + (10*a^5*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2)) + (2*a^6*Tan[c + d*x]^9)/(9*d*(a + a*Sec[c + d*x])^(9/2))","A",4,3,23,0.1304,1,"{3887, 461, 203}"
155,1,128,0,0.0927509,"\int (a+a \sec (c+d x))^{3/2} \tan ^2(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^2,x]","\frac{2 a^4 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a^3 \tan ^3(c+d x)}{d (a \sec (c+d x)+a)^{3/2}}-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\frac{2 a^4 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a^3 \tan ^3(c+d x)}{d (a \sec (c+d x)+a)^{3/2}}-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(-2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*Tan[c + d*x]^3)/(d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^4*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2))","A",4,3,23,0.1304,1,"{3887, 461, 203}"
156,1,64,0,0.0709612,"\int \cot ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2),x]","-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{2 a \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d}","-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{2 a \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d}",1,"(-2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (2*a*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/d","A",3,3,23,0.1304,1,"{3887, 325, 203}"
157,1,144,0,0.1484644,"\int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} d}-\frac{\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac{3 a \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} d}-\frac{\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac{3 a \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}",1,"(2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*d) + (3*a*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(2*d) - (Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(3*d)","A",6,5,23,0.2174,1,"{3887, 480, 583, 522, 203}"
158,1,226,0,0.2309549,"\int \cot ^6(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^6*(a + a*Sec[c + d*x])^(3/2),x]","-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{11 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} d}+\frac{3 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 a d}+\frac{5 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{24 d}-\frac{21 a \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{16 d}-\frac{\cos (c+d x) \cot ^5(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{4 a d}","-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{11 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} d}+\frac{3 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 a d}+\frac{5 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{24 d}-\frac{21 a \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{16 d}-\frac{\cos (c+d x) \cot ^5(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{4 a d}",1,"(-2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (11*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*d) - (21*a*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(16*d) + (5*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(24*d) + (3*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(20*a*d) - (Cos[c + d*x]*Cot[c + d*x]^5*Sec[(c + d*x)/2]^2*(a + a*Sec[c + d*x])^(5/2))/(4*a*d)","A",8,5,23,0.2174,1,"{3887, 472, 583, 522, 203}"
159,1,193,0,0.1458929,"\int (a+a \sec (c+d x))^{5/2} \tan ^5(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^5,x]","\frac{2 (a \sec (c+d x)+a)^{13/2}}{13 a^4 d}-\frac{6 (a \sec (c+d x)+a)^{11/2}}{11 a^3 d}+\frac{2 (a \sec (c+d x)+a)^{9/2}}{9 a^2 d}+\frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 d}+\frac{2 a (a \sec (c+d x)+a)^{3/2}}{3 d}","\frac{2 (a \sec (c+d x)+a)^{13/2}}{13 a^4 d}-\frac{6 (a \sec (c+d x)+a)^{11/2}}{11 a^3 d}+\frac{2 (a \sec (c+d x)+a)^{9/2}}{9 a^2 d}+\frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 d}+\frac{2 a (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(-2*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (2*a^2*Sqrt[a + a*Sec[c + d*x]])/d + (2*a*(a + a*Sec[c + d*x])^(3/2))/(3*d) + (2*(a + a*Sec[c + d*x])^(5/2))/(5*d) + (2*(a + a*Sec[c + d*x])^(7/2))/(7*a*d) + (2*(a + a*Sec[c + d*x])^(9/2))/(9*a^2*d) - (6*(a + a*Sec[c + d*x])^(11/2))/(11*a^3*d) + (2*(a + a*Sec[c + d*x])^(13/2))/(13*a^4*d)","A",10,5,23,0.2174,1,"{3880, 88, 50, 63, 207}"
160,1,145,0,0.1183746,"\int (a+a \sec (c+d x))^{5/2} \tan ^3(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^3,x]","\frac{2 (a \sec (c+d x)+a)^{9/2}}{9 a^2 d}-\frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a d}-\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 d}-\frac{2 a (a \sec (c+d x)+a)^{3/2}}{3 d}","\frac{2 (a \sec (c+d x)+a)^{9/2}}{9 a^2 d}-\frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a d}-\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 d}-\frac{2 a (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(2*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (2*a^2*Sqrt[a + a*Sec[c + d*x]])/d - (2*a*(a + a*Sec[c + d*x])^(3/2))/(3*d) - (2*(a + a*Sec[c + d*x])^(5/2))/(5*d) - (2*(a + a*Sec[c + d*x])^(7/2))/(7*a*d) + (2*(a + a*Sec[c + d*x])^(9/2))/(9*a^2*d)","A",8,5,23,0.2174,1,"{3880, 80, 50, 63, 207}"
161,1,97,0,0.0650882,"\int (a+a \sec (c+d x))^{5/2} \tan (c+d x) \, dx","Int[(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x],x]","\frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{2 a (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 d}","\frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{2 a (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 d}",1,"(-2*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (2*a^2*Sqrt[a + a*Sec[c + d*x]])/d + (2*a*(a + a*Sec[c + d*x])^(3/2))/(3*d) + (2*(a + a*Sec[c + d*x])^(5/2))/(5*d)","A",6,4,21,0.1905,1,"{3880, 50, 63, 207}"
162,1,95,0,0.0891302,"\int \cot (c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Cot[c + d*x]*(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{4 \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}","\frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{4 \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}",1,"(2*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (4*Sqrt[2]*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (2*a^2*Sqrt[a + a*Sec[c + d*x]])/d","A",7,5,21,0.2381,1,"{3880, 84, 156, 63, 207}"
163,1,106,0,0.1028849,"\int \cot ^3(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2),x]","\frac{a^2 \sqrt{a \sec (c+d x)+a}}{d (1-\sec (c+d x))}-\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{3 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} d}","\frac{a^2 \sqrt{a \sec (c+d x)+a}}{d (1-\sec (c+d x))}-\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{3 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} d}",1,"(-2*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (3*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*d) + (a^2*Sqrt[a + a*Sec[c + d*x]])/(d*(1 - Sec[c + d*x]))","A",7,5,23,0.2174,1,"{3880, 99, 156, 63, 207}"
164,1,147,0,0.1268553,"\int \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2),x]","-\frac{11 a^2 \sqrt{a \sec (c+d x)+a}}{16 d (1-\sec (c+d x))}-\frac{a^2 \sqrt{a \sec (c+d x)+a}}{4 d (1-\sec (c+d x))^2}+\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{43 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} d}","-\frac{11 a^2 \sqrt{a \sec (c+d x)+a}}{16 d (1-\sec (c+d x))}-\frac{a^2 \sqrt{a \sec (c+d x)+a}}{4 d (1-\sec (c+d x))^2}+\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{43 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} d}",1,"(2*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (43*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(16*Sqrt[2]*d) - (a^2*Sqrt[a + a*Sec[c + d*x]])/(4*d*(1 - Sec[c + d*x])^2) - (11*a^2*Sqrt[a + a*Sec[c + d*x]])/(16*d*(1 - Sec[c + d*x]))","A",8,6,23,0.2609,1,"{3880, 103, 151, 156, 63, 207}"
165,1,290,0,0.1279094,"\int (a+a \sec (c+d x))^{5/2} \tan ^6(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^6,x]","\frac{2 a^{10} \tan ^{15}(c+d x)}{15 d (a \sec (c+d x)+a)^{15/2}}+\frac{18 a^9 \tan ^{13}(c+d x)}{13 d (a \sec (c+d x)+a)^{13/2}}+\frac{62 a^8 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{98 a^7 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{62 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^5 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^3 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\frac{2 a^{10} \tan ^{15}(c+d x)}{15 d (a \sec (c+d x)+a)^{15/2}}+\frac{18 a^9 \tan ^{13}(c+d x)}{13 d (a \sec (c+d x)+a)^{13/2}}+\frac{62 a^8 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{98 a^7 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{62 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^5 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^3 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(-2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) - (2*a^4*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^5*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2)) + (62*a^6*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2)) + (98*a^7*Tan[c + d*x]^9)/(9*d*(a + a*Sec[c + d*x])^(9/2)) + (62*a^8*Tan[c + d*x]^11)/(11*d*(a + a*Sec[c + d*x])^(11/2)) + (18*a^9*Tan[c + d*x]^13)/(13*d*(a + a*Sec[c + d*x])^(13/2)) + (2*a^10*Tan[c + d*x]^15)/(15*d*(a + a*Sec[c + d*x])^(15/2))","A",4,3,23,0.1304,1,"{3887, 461, 203}"
166,1,224,0,0.1110373,"\int (a+a \sec (c+d x))^{5/2} \tan ^4(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^4,x]","\frac{2 a^8 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{14 a^7 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{34 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{6 a^5 \tan ^5(c+d x)}{d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{2 a^3 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\frac{2 a^8 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{14 a^7 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{34 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{6 a^5 \tan ^5(c+d x)}{d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{2 a^3 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (2*a^3*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^4*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (6*a^5*Tan[c + d*x]^5)/(d*(a + a*Sec[c + d*x])^(5/2)) + (34*a^6*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2)) + (14*a^7*Tan[c + d*x]^9)/(9*d*(a + a*Sec[c + d*x])^(9/2)) + (2*a^8*Tan[c + d*x]^11)/(11*d*(a + a*Sec[c + d*x])^(11/2))","A",4,3,23,0.1304,1,"{3887, 461, 203}"
167,1,160,0,0.0987673,"\int (a+a \sec (c+d x))^{5/2} \tan ^2(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^2,x]","\frac{2 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^5 \tan ^5(c+d x)}{d (a \sec (c+d x)+a)^{5/2}}+\frac{14 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^3 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\frac{2 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^5 \tan ^5(c+d x)}{d (a \sec (c+d x)+a)^{5/2}}+\frac{14 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^3 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(-2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (14*a^4*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^5*Tan[c + d*x]^5)/(d*(a + a*Sec[c + d*x])^(5/2)) + (2*a^6*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2))","A",4,3,23,0.1304,1,"{3887, 461, 203}"
168,1,66,0,0.0710611,"\int \cot ^2(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Cot[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2),x]","-\frac{4 a^2 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","-\frac{4 a^2 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(-2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (4*a^2*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/d","A",3,3,23,0.1304,1,"{3887, 453, 203}"
169,1,96,0,0.0760531,"\int \cot ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 a^2 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{2 a \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}","\frac{2 a^2 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{2 a \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/d - (2*a*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(3*d)","A",4,3,23,0.1304,1,"{3887, 325, 203}"
170,1,176,0,0.1837072,"\int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Cot[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2),x]","-\frac{7 a^2 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{2} d}-\frac{\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}+\frac{a \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d}","-\frac{7 a^2 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{2} d}-\frac{\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}+\frac{a \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d}",1,"(-2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(4*Sqrt[2]*d) - (7*a^2*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(4*d) + (a*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(2*d) - (Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(5*d)","A",7,5,23,0.2174,1,"{3887, 480, 583, 522, 203}"
171,1,126,0,0.101379,"\int \frac{\tan ^5(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Tan[c + d*x]^5/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a^4 d}-\frac{6 (a \sec (c+d x)+a)^{5/2}}{5 a^3 d}+\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a^2 d}+\frac{2 \sqrt{a \sec (c+d x)+a}}{a d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}","\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a^4 d}-\frac{6 (a \sec (c+d x)+a)^{5/2}}{5 a^3 d}+\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a^2 d}+\frac{2 \sqrt{a \sec (c+d x)+a}}{a d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"(-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + (2*Sqrt[a + a*Sec[c + d*x]])/(a*d) + (2*(a + a*Sec[c + d*x])^(3/2))/(3*a^2*d) - (6*(a + a*Sec[c + d*x])^(5/2))/(5*a^3*d) + (2*(a + a*Sec[c + d*x])^(7/2))/(7*a^4*d)","A",7,5,23,0.2174,1,"{3880, 88, 50, 63, 207}"
172,1,78,0,0.0740956,"\int \frac{\tan ^3(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Tan[c + d*x]^3/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a^2 d}-\frac{2 \sqrt{a \sec (c+d x)+a}}{a d}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}","\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a^2 d}-\frac{2 \sqrt{a \sec (c+d x)+a}}{a d}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"(2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - (2*Sqrt[a + a*Sec[c + d*x]])/(a*d) + (2*(a + a*Sec[c + d*x])^(3/2))/(3*a^2*d)","A",5,5,23,0.2174,1,"{3880, 80, 50, 63, 207}"
173,1,31,0,0.0392376,"\int \frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Tan[c + d*x]/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"(-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d)","A",3,3,21,0.1429,1,"{3880, 63, 207}"
174,1,92,0,0.0864493,"\int \frac{\cot (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Cot[c + d*x]/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{1}{d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}","-\frac{1}{d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}",1,"(2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(Sqrt[2]*Sqrt[a]*d) - 1/(d*Sqrt[a + a*Sec[c + d*x]])","A",7,5,21,0.2381,1,"{3880, 85, 156, 63, 207}"
175,1,152,0,0.1352953,"\int \frac{\cot ^3(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Cot[c + d*x]^3/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{a}{12 d (a \sec (c+d x)+a)^{3/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}+\frac{7}{8 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}+\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{8 \sqrt{2} \sqrt{a} d}","-\frac{a}{12 d (a \sec (c+d x)+a)^{3/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}+\frac{7}{8 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}+\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{8 \sqrt{2} \sqrt{a} d}",1,"(-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + (9*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(8*Sqrt[2]*Sqrt[a]*d) - a/(12*d*(a + a*Sec[c + d*x])^(3/2)) + a/(2*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(3/2)) + 7/(8*d*Sqrt[a + a*Sec[c + d*x]])","A",9,6,23,0.2609,1,"{3880, 103, 152, 156, 63, 207}"
176,1,214,0,0.1778528,"\int \frac{\cot ^5(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Cot[c + d*x]^5/Sqrt[a + a*Sec[c + d*x]],x]","\frac{87 a^2}{160 d (a \sec (c+d x)+a)^{5/2}}-\frac{17 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}+\frac{23 a}{192 d (a \sec (c+d x)+a)^{3/2}}-\frac{105}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}-\frac{151 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{128 \sqrt{2} \sqrt{a} d}","\frac{87 a^2}{160 d (a \sec (c+d x)+a)^{5/2}}-\frac{17 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}+\frac{23 a}{192 d (a \sec (c+d x)+a)^{3/2}}-\frac{105}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}-\frac{151 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{128 \sqrt{2} \sqrt{a} d}",1,"(2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - (151*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(128*Sqrt[2]*Sqrt[a]*d) + (87*a^2)/(160*d*(a + a*Sec[c + d*x])^(5/2)) - a^2/(4*d*(1 - Sec[c + d*x])^2*(a + a*Sec[c + d*x])^(5/2)) - (17*a^2)/(16*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(5/2)) + (23*a)/(192*d*(a + a*Sec[c + d*x])^(3/2)) - 105/(128*d*Sqrt[a + a*Sec[c + d*x]])","A",11,7,23,0.3043,1,"{3880, 103, 151, 152, 156, 63, 207}"
177,1,189,0,0.1005384,"\int \frac{\tan ^6(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Tan[c + d*x]^6/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 a^4 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{6 a^3 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^2 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\frac{2 a^4 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{6 a^3 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^2 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) + (2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) - (2*a*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^2*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2)) + (6*a^3*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2)) + (2*a^4*Tan[c + d*x]^9)/(9*d*(a + a*Sec[c + d*x])^(9/2))","A",4,3,23,0.1304,1,"{3887, 461, 203}"
178,1,125,0,0.0838666,"\int \frac{\tan ^4(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Tan[c + d*x]^4/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 a^2 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\frac{2 a^2 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^2*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2))","A",5,4,23,0.1739,1,"{3887, 459, 302, 203}"
179,1,63,0,0.0626098,"\int \frac{\tan ^2(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Tan[c + d*x]^2/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) + (2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",3,3,23,0.1304,1,"{3887, 321, 203}"
180,1,165,0,0.1414877,"\int \frac{\cot ^2(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Cot[c + d*x]^2/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{\cot (c+d x) \sqrt{a \sec (c+d x)+a}}{4 a d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{7 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{2} \sqrt{a} d}-\frac{\cos (c+d x) \cot (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{4 a d}","-\frac{\cot (c+d x) \sqrt{a \sec (c+d x)+a}}{4 a d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{7 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{2} \sqrt{a} d}-\frac{\cos (c+d x) \cot (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{4 a d}",1,"(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) + (7*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(4*Sqrt[2]*Sqrt[a]*d) - (Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(4*a*d) - (Cos[c + d*x]*Cot[c + d*x]*Sec[(c + d*x)/2]^2*Sqrt[a + a*Sec[c + d*x]])/(4*a*d)","A",6,5,23,0.2174,1,"{3887, 472, 583, 522, 203}"
181,1,251,0,0.2259165,"\int \frac{\cot ^4(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Cot[c + d*x]^4/Sqrt[a + a*Sec[c + d*x]],x]","\frac{43 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{96 a^2 d}-\frac{\cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{16 a^2 d}-\frac{15 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{32 a^2 d}+\frac{21 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{64 a d}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{107 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{64 \sqrt{2} \sqrt{a} d}","\frac{43 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{96 a^2 d}-\frac{\cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{16 a^2 d}-\frac{15 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{32 a^2 d}+\frac{21 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{64 a d}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{107 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{64 \sqrt{2} \sqrt{a} d}",1,"(2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (107*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(64*Sqrt[2]*Sqrt[a]*d) + (21*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(64*a*d) + (43*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(96*a^2*d) - (15*Cos[c + d*x]*Cot[c + d*x]^3*Sec[(c + d*x)/2]^2*(a + a*Sec[c + d*x])^(3/2))/(32*a^2*d) - (Cos[c + d*x]^2*Cot[c + d*x]^3*Sec[(c + d*x)/2]^4*(a + a*Sec[c + d*x])^(3/2))/(16*a^2*d)","A",8,6,23,0.2609,1,"{3887, 472, 579, 583, 522, 203}"
182,1,335,0,0.3215123,"\int \frac{\cot ^6(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Cot[c + d*x]^6/Sqrt[a + a*Sec[c + d*x]],x]","\frac{579 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{640 a^3 d}-\frac{323 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{768 a^2 d}-\frac{\cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{48 a^3 d}-\frac{23 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{192 a^3 d}-\frac{101 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{128 a^3 d}-\frac{189 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{512 a d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{835 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{512 \sqrt{2} \sqrt{a} d}","\frac{579 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{640 a^3 d}-\frac{323 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{768 a^2 d}-\frac{\cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{48 a^3 d}-\frac{23 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{192 a^3 d}-\frac{101 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{128 a^3 d}-\frac{189 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{512 a d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{835 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{512 \sqrt{2} \sqrt{a} d}",1,"(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) + (835*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(512*Sqrt[2]*Sqrt[a]*d) - (189*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(512*a*d) - (323*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(768*a^2*d) + (579*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(640*a^3*d) - (101*Cos[c + d*x]*Cot[c + d*x]^5*Sec[(c + d*x)/2]^2*(a + a*Sec[c + d*x])^(5/2))/(128*a^3*d) - (23*Cos[c + d*x]^2*Cot[c + d*x]^5*Sec[(c + d*x)/2]^4*(a + a*Sec[c + d*x])^(5/2))/(192*a^3*d) - (Cos[c + d*x]^3*Cot[c + d*x]^5*Sec[(c + d*x)/2]^6*(a + a*Sec[c + d*x])^(5/2))/(48*a^3*d)","A",10,6,23,0.2609,1,"{3887, 472, 579, 583, 522, 203}"
183,1,100,0,0.0972202,"\int \frac{\tan ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Tan[c + d*x]^5/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a^4 d}-\frac{2 (a \sec (c+d x)+a)^{3/2}}{a^3 d}+\frac{2 \sqrt{a \sec (c+d x)+a}}{a^2 d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}","\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a^4 d}-\frac{2 (a \sec (c+d x)+a)^{3/2}}{a^3 d}+\frac{2 \sqrt{a \sec (c+d x)+a}}{a^2 d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}",1,"(-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + (2*Sqrt[a + a*Sec[c + d*x]])/(a^2*d) - (2*(a + a*Sec[c + d*x])^(3/2))/(a^3*d) + (2*(a + a*Sec[c + d*x])^(5/2))/(5*a^4*d)","A",6,5,23,0.2174,1,"{3880, 88, 50, 63, 207}"
184,1,54,0,0.0749947,"\int \frac{\tan ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Tan[c + d*x]^3/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 \sqrt{a \sec (c+d x)+a}}{a^2 d}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}","\frac{2 \sqrt{a \sec (c+d x)+a}}{a^2 d}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}",1,"(2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + (2*Sqrt[a + a*Sec[c + d*x]])/(a^2*d)","A",4,4,23,0.1739,1,"{3880, 80, 63, 207}"
185,1,54,0,0.0492996,"\int \frac{\tan (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Tan[c + d*x]/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2}{a d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}","\frac{2}{a d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}",1,"(-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + 2/(a*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,21,0.1905,1,"{3880, 51, 63, 207}"
186,1,120,0,0.1066952,"\int \frac{\cot (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Cot[c + d*x]/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{1}{3 d (a \sec (c+d x)+a)^{3/2}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{1}{3 d (a \sec (c+d x)+a)^{3/2}}",1,"(2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(2*Sqrt[2]*a^(3/2)*d) - 1/(3*d*(a + a*Sec[c + d*x])^(3/2)) - 3/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",8,6,21,0.2857,1,"{3880, 85, 152, 156, 63, 207}"
187,1,176,0,0.1621974,"\int \frac{\cot ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Cot[c + d*x]^3/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} d}-\frac{3 a}{20 d (a \sec (c+d x)+a)^{5/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}+\frac{5}{24 d (a \sec (c+d x)+a)^{3/2}}+\frac{21}{16 a d \sqrt{a \sec (c+d x)+a}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} d}-\frac{3 a}{20 d (a \sec (c+d x)+a)^{5/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}+\frac{5}{24 d (a \sec (c+d x)+a)^{3/2}}+\frac{21}{16 a d \sqrt{a \sec (c+d x)+a}}",1,"(-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + (11*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(16*Sqrt[2]*a^(3/2)*d) - (3*a)/(20*d*(a + a*Sec[c + d*x])^(5/2)) + a/(2*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(5/2)) + 5/(24*d*(a + a*Sec[c + d*x])^(3/2)) + 21/(16*a*d*Sqrt[a + a*Sec[c + d*x]])","A",10,6,23,0.2609,1,"{3880, 103, 152, 156, 63, 207}"
188,1,238,0,0.2047774,"\int \frac{\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^(3/2),x]","\frac{139 a^2}{224 d (a \sec (c+d x)+a)^{7/2}}-\frac{19 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}-\frac{203 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{256 \sqrt{2} a^{3/2} d}+\frac{15 a}{64 d (a \sec (c+d x)+a)^{5/2}}-\frac{53}{384 d (a \sec (c+d x)+a)^{3/2}}-\frac{309}{256 a d \sqrt{a \sec (c+d x)+a}}","\frac{139 a^2}{224 d (a \sec (c+d x)+a)^{7/2}}-\frac{19 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}-\frac{203 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{256 \sqrt{2} a^{3/2} d}+\frac{15 a}{64 d (a \sec (c+d x)+a)^{5/2}}-\frac{53}{384 d (a \sec (c+d x)+a)^{3/2}}-\frac{309}{256 a d \sqrt{a \sec (c+d x)+a}}",1,"(2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - (203*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(256*Sqrt[2]*a^(3/2)*d) + (139*a^2)/(224*d*(a + a*Sec[c + d*x])^(7/2)) - a^2/(4*d*(1 - Sec[c + d*x])^2*(a + a*Sec[c + d*x])^(7/2)) - (19*a^2)/(16*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(7/2)) + (15*a)/(64*d*(a + a*Sec[c + d*x])^(5/2)) - 53/(384*d*(a + a*Sec[c + d*x])^(3/2)) - 309/(256*a*d*Sqrt[a + a*Sec[c + d*x]])","A",12,7,23,0.3043,1,"{3880, 103, 151, 152, 156, 63, 207}"
189,1,157,0,0.0970172,"\int \frac{\tan ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Tan[c + d*x]^6/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 a^2 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{2 a \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 \tan (c+d x)}{a d \sqrt{a \sec (c+d x)+a}}","\frac{2 a^2 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{2 a \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 \tan (c+d x)}{a d \sqrt{a \sec (c+d x)+a}}",1,"(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + (2*Tan[c + d*x])/(a*d*Sqrt[a + a*Sec[c + d*x]]) - (2*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2)) + (2*a^2*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2))","A",5,4,23,0.1739,1,"{3887, 459, 302, 203}"
190,1,95,0,0.0795231,"\int \frac{\tan ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Tan[c + d*x]^4/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \tan (c+d x)}{a d \sqrt{a \sec (c+d x)+a}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \tan (c+d x)}{a d \sqrt{a \sec (c+d x)+a}}",1,"(2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - (2*Tan[c + d*x])/(a*d*Sqrt[a + a*Sec[c + d*x]]) + (2*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2))","A",4,3,23,0.1304,1,"{3887, 302, 203}"
191,1,85,0,0.0861983,"\int \frac{\tan ^2(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Tan[c + d*x]^2/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}","\frac{2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}",1,"(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + (2*Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(a^(3/2)*d)","A",4,3,23,0.1304,1,"{3887, 481, 203}"
192,1,215,0,0.1951167,"\int \frac{\cot ^2(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Cot[c + d*x]^2/(a + a*Sec[c + d*x])^(3/2),x]","\frac{7 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{32 a^2 d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{71 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{32 \sqrt{2} a^{3/2} d}-\frac{\cos ^2(c+d x) \cot (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{16 a^2 d}-\frac{13 \cos (c+d x) \cot (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{32 a^2 d}","\frac{7 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{32 a^2 d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{71 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{32 \sqrt{2} a^{3/2} d}-\frac{\cos ^2(c+d x) \cot (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{16 a^2 d}-\frac{13 \cos (c+d x) \cot (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{32 a^2 d}",1,"(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + (71*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(32*Sqrt[2]*a^(3/2)*d) + (7*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(32*a^2*d) - (13*Cos[c + d*x]*Cot[c + d*x]*Sec[(c + d*x)/2]^2*Sqrt[a + a*Sec[c + d*x]])/(32*a^2*d) - (Cos[c + d*x]^2*Cot[c + d*x]*Sec[(c + d*x)/2]^4*Sqrt[a + a*Sec[c + d*x]])/(16*a^2*d)","A",7,6,23,0.2609,1,"{3887, 472, 579, 583, 522, 203}"
193,1,303,0,0.2815499,"\int \frac{\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Cot[c + d*x]^4/(a + a*Sec[c + d*x])^(3/2),x]","\frac{277 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{384 a^3 d}-\frac{21 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{256 a^2 d}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{533 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{256 \sqrt{2} a^{3/2} d}-\frac{\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{48 a^3 d}-\frac{7 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{64 a^3 d}-\frac{81 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{128 a^3 d}","\frac{277 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{384 a^3 d}-\frac{21 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{256 a^2 d}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{533 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{256 \sqrt{2} a^{3/2} d}-\frac{\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{48 a^3 d}-\frac{7 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{64 a^3 d}-\frac{81 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{128 a^3 d}",1,"(2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - (533*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(256*Sqrt[2]*a^(3/2)*d) - (21*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(256*a^2*d) + (277*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(384*a^3*d) - (81*Cos[c + d*x]*Cot[c + d*x]^3*Sec[(c + d*x)/2]^2*(a + a*Sec[c + d*x])^(3/2))/(128*a^3*d) - (7*Cos[c + d*x]^2*Cot[c + d*x]^3*Sec[(c + d*x)/2]^4*(a + a*Sec[c + d*x])^(3/2))/(64*a^3*d) - (Cos[c + d*x]^3*Cot[c + d*x]^3*Sec[(c + d*x)/2]^6*(a + a*Sec[c + d*x])^(3/2))/(48*a^3*d)","A",9,6,23,0.2609,1,"{3887, 472, 579, 583, 522, 203}"
194,1,387,0,0.3677996,"\int \frac{\cot ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^(3/2),x]","\frac{12267 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{10240 a^4 d}-\frac{8171 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12288 a^3 d}-\frac{21 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{8192 a^2 d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{16363 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{8192 \sqrt{2} a^{3/2} d}-\frac{\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{128 a^4 d}-\frac{29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{768 a^4 d}-\frac{511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{3072 a^4 d}-\frac{2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{2048 a^4 d}","\frac{12267 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{10240 a^4 d}-\frac{8171 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12288 a^3 d}-\frac{21 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{8192 a^2 d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{16363 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{8192 \sqrt{2} a^{3/2} d}-\frac{\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{128 a^4 d}-\frac{29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{768 a^4 d}-\frac{511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{3072 a^4 d}-\frac{2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{2048 a^4 d}",1,"(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + (16363*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(8192*Sqrt[2]*a^(3/2)*d) - (21*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(8192*a^2*d) - (8171*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(12288*a^3*d) + (12267*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(10240*a^4*d) - (2045*Cos[c + d*x]*Cot[c + d*x]^5*Sec[(c + d*x)/2]^2*(a + a*Sec[c + d*x])^(5/2))/(2048*a^4*d) - (511*Cos[c + d*x]^2*Cot[c + d*x]^5*Sec[(c + d*x)/2]^4*(a + a*Sec[c + d*x])^(5/2))/(3072*a^4*d) - (29*Cos[c + d*x]^3*Cot[c + d*x]^5*Sec[(c + d*x)/2]^6*(a + a*Sec[c + d*x])^(5/2))/(768*a^4*d) - (Cos[c + d*x]^4*Cot[c + d*x]^5*Sec[(c + d*x)/2]^8*(a + a*Sec[c + d*x])^(5/2))/(128*a^4*d)","A",11,6,23,0.2609,1,"{3887, 472, 579, 583, 522, 203}"
195,1,78,0,0.090066,"\int \frac{\tan ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Tan[c + d*x]^5/(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a^4 d}-\frac{6 \sqrt{a \sec (c+d x)+a}}{a^3 d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}","\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a^4 d}-\frac{6 \sqrt{a \sec (c+d x)+a}}{a^3 d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}",1,"(-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) - (6*Sqrt[a + a*Sec[c + d*x]])/(a^3*d) + (2*(a + a*Sec[c + d*x])^(3/2))/(3*a^4*d)","A",5,4,23,0.1739,1,"{3880, 88, 63, 207}"
196,1,54,0,0.072335,"\int \frac{\tan ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Tan[c + d*x]^3/(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}-\frac{4}{a^2 d \sqrt{a \sec (c+d x)+a}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}-\frac{4}{a^2 d \sqrt{a \sec (c+d x)+a}}",1,"(2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) - 4/(a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,23,0.1739,1,"{3880, 78, 63, 207}"
197,1,78,0,0.0582467,"\int \frac{\tan (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Tan[c + d*x]/(a + a*Sec[c + d*x])^(5/2),x]","\frac{2}{a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{2}{3 a d (a \sec (c+d x)+a)^{3/2}}","\frac{2}{a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{2}{3 a d (a \sec (c+d x)+a)^{3/2}}",1,"(-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) + 2/(3*a*d*(a + a*Sec[c + d*x])^(3/2)) + 2/(a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",5,4,21,0.1905,1,"{3880, 51, 63, 207}"
198,1,144,0,0.125346,"\int \frac{\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Cot[c + d*x]/(a + a*Sec[c + d*x])^(5/2),x]","-\frac{7}{4 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}-\frac{1}{2 a d (a \sec (c+d x)+a)^{3/2}}-\frac{1}{5 d (a \sec (c+d x)+a)^{5/2}}","-\frac{7}{4 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}-\frac{1}{2 a d (a \sec (c+d x)+a)^{3/2}}-\frac{1}{5 d (a \sec (c+d x)+a)^{5/2}}",1,"(2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) - ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(4*Sqrt[2]*a^(5/2)*d) - 1/(5*d*(a + a*Sec[c + d*x])^(5/2)) - 1/(2*a*d*(a + a*Sec[c + d*x])^(3/2)) - 7/(4*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",9,6,21,0.2857,1,"{3880, 85, 152, 156, 63, 207}"
199,1,200,0,0.1748873,"\int \frac{\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Cot[c + d*x]^3/(a + a*Sec[c + d*x])^(5/2),x]","\frac{51}{32 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{13 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{32 \sqrt{2} a^{5/2} d}-\frac{5 a}{28 d (a \sec (c+d x)+a)^{7/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}+\frac{3}{40 d (a \sec (c+d x)+a)^{5/2}}+\frac{19}{48 a d (a \sec (c+d x)+a)^{3/2}}","\frac{51}{32 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{13 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{32 \sqrt{2} a^{5/2} d}-\frac{5 a}{28 d (a \sec (c+d x)+a)^{7/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}+\frac{3}{40 d (a \sec (c+d x)+a)^{5/2}}+\frac{19}{48 a d (a \sec (c+d x)+a)^{3/2}}",1,"(-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) + (13*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(32*Sqrt[2]*a^(5/2)*d) - (5*a)/(28*d*(a + a*Sec[c + d*x])^(7/2)) + a/(2*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(7/2)) + 3/(40*d*(a + a*Sec[c + d*x])^(5/2)) + 19/(48*a*d*(a + a*Sec[c + d*x])^(3/2)) + 51/(32*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",11,6,23,0.2609,1,"{3880, 103, 152, 156, 63, 207}"
200,1,262,0,0.2265663,"\int \frac{\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^(5/2),x]","\frac{199 a^2}{288 d (a \sec (c+d x)+a)^{9/2}}-\frac{21 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}-\frac{761}{512 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}-\frac{263 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{512 \sqrt{2} a^{5/2} d}+\frac{135 a}{448 d (a \sec (c+d x)+a)^{7/2}}+\frac{7}{640 d (a \sec (c+d x)+a)^{5/2}}-\frac{83}{256 a d (a \sec (c+d x)+a)^{3/2}}","\frac{199 a^2}{288 d (a \sec (c+d x)+a)^{9/2}}-\frac{21 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}-\frac{761}{512 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}-\frac{263 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{512 \sqrt{2} a^{5/2} d}+\frac{135 a}{448 d (a \sec (c+d x)+a)^{7/2}}+\frac{7}{640 d (a \sec (c+d x)+a)^{5/2}}-\frac{83}{256 a d (a \sec (c+d x)+a)^{3/2}}",1,"(2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) - (263*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(512*Sqrt[2]*a^(5/2)*d) + (199*a^2)/(288*d*(a + a*Sec[c + d*x])^(9/2)) - a^2/(4*d*(1 - Sec[c + d*x])^2*(a + a*Sec[c + d*x])^(9/2)) - (21*a^2)/(16*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(9/2)) + (135*a)/(448*d*(a + a*Sec[c + d*x])^(7/2)) + 7/(640*d*(a + a*Sec[c + d*x])^(5/2)) - 83/(256*a*d*(a + a*Sec[c + d*x])^(3/2)) - 761/(512*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",13,7,23,0.3043,1,"{3880, 103, 151, 152, 156, 63, 207}"
201,1,127,0,0.0865185,"\int \frac{\tan ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Tan[c + d*x]^6/(a + a*Sec[c + d*x])^(5/2),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{2 \tan (c+d x)}{a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 \tan ^3(c+d x)}{3 a d (a \sec (c+d x)+a)^{3/2}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{2 \tan (c+d x)}{a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 \tan ^3(c+d x)}{3 a d (a \sec (c+d x)+a)^{3/2}}",1,"(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + (2*Tan[c + d*x])/(a^2*d*Sqrt[a + a*Sec[c + d*x]]) - (2*Tan[c + d*x]^3)/(3*a*d*(a + a*Sec[c + d*x])^(3/2)) + (2*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2))","A",4,3,23,0.1304,1,"{3887, 302, 203}"
202,1,113,0,0.1068454,"\int \frac{\tan ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Tan[c + d*x]^4/(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{2 \tan (c+d x)}{a^2 d \sqrt{a \sec (c+d x)+a}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{2 \tan (c+d x)}{a^2 d \sqrt{a \sec (c+d x)+a}}",1,"(2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - (4*Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(a^(5/2)*d) + (2*Tan[c + d*x])/(a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",5,4,23,0.1739,1,"{3887, 479, 522, 203}"
203,1,127,0,0.1061086,"\int \frac{\tan ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Tan[c + d*x]^2/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\sin (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{2 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{2} a^{5/2} d}","\frac{\sin (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{2 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{2} a^{5/2} d}",1,"(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + (3*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[2]*a^(5/2)*d) + (Sec[(c + d*x)/2]^2*Sin[c + d*x])/(2*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",5,4,23,0.1739,1,"{3887, 471, 522, 203}"
204,1,265,0,0.2331896,"\int \frac{\cot ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Cot[c + d*x]^2/(a + a*Sec[c + d*x])^(5/2),x]","\frac{63 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{128 a^3 d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{319 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{128 \sqrt{2} a^{5/2} d}-\frac{\cos ^3(c+d x) \cot (c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{48 a^3 d}-\frac{19 \cos ^2(c+d x) \cot (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{192 a^3 d}-\frac{191 \cos (c+d x) \cot (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{384 a^3 d}","\frac{63 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{128 a^3 d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{319 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{128 \sqrt{2} a^{5/2} d}-\frac{\cos ^3(c+d x) \cot (c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{48 a^3 d}-\frac{19 \cos ^2(c+d x) \cot (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{192 a^3 d}-\frac{191 \cos (c+d x) \cot (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{384 a^3 d}",1,"(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + (319*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(128*Sqrt[2]*a^(5/2)*d) + (63*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(128*a^3*d) - (191*Cos[c + d*x]*Cot[c + d*x]*Sec[(c + d*x)/2]^2*Sqrt[a + a*Sec[c + d*x]])/(384*a^3*d) - (19*Cos[c + d*x]^2*Cot[c + d*x]*Sec[(c + d*x)/2]^4*Sqrt[a + a*Sec[c + d*x]])/(192*a^3*d) - (Cos[c + d*x]^3*Cot[c + d*x]*Sec[(c + d*x)/2]^6*Sqrt[a + a*Sec[c + d*x]])/(48*a^3*d)","A",8,6,23,0.2609,1,"{3887, 472, 579, 583, 522, 203}"
205,1,355,0,0.3294245,"\int \frac{\cot ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Cot[c + d*x]^4/(a + a*Sec[c + d*x])^(5/2),x]","\frac{5587 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{6144 a^4 d}-\frac{1491 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{4096 a^3 d}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{9683 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{4096 \sqrt{2} a^{5/2} d}-\frac{\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{128 a^4 d}-\frac{9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{256 a^4 d}-\frac{145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{1024 a^4 d}-\frac{1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{2048 a^4 d}","\frac{5587 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{6144 a^4 d}-\frac{1491 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{4096 a^3 d}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{9683 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{4096 \sqrt{2} a^{5/2} d}-\frac{\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{128 a^4 d}-\frac{9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{256 a^4 d}-\frac{145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{1024 a^4 d}-\frac{1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{2048 a^4 d}",1,"(2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - (9683*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(4096*Sqrt[2]*a^(5/2)*d) - (1491*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(4096*a^3*d) + (5587*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(6144*a^4*d) - (1527*Cos[c + d*x]*Cot[c + d*x]^3*Sec[(c + d*x)/2]^2*(a + a*Sec[c + d*x])^(3/2))/(2048*a^4*d) - (145*Cos[c + d*x]^2*Cot[c + d*x]^3*Sec[(c + d*x)/2]^4*(a + a*Sec[c + d*x])^(3/2))/(1024*a^4*d) - (9*Cos[c + d*x]^3*Cot[c + d*x]^3*Sec[(c + d*x)/2]^6*(a + a*Sec[c + d*x])^(3/2))/(256*a^4*d) - (Cos[c + d*x]^4*Cot[c + d*x]^3*Sec[(c + d*x)/2]^8*(a + a*Sec[c + d*x])^(3/2))/(128*a^4*d)","A",10,6,23,0.2609,1,"{3887, 472, 579, 583, 522, 203}"
206,1,439,0,0.4201032,"\int \frac{\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^(5/2),x]","\frac{58077 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{40960 a^5 d}-\frac{41693 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{49152 a^4 d}+\frac{8925 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{32768 a^3 d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{74461 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{32768 \sqrt{2} a^{5/2} d}-\frac{\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{320 a^5 d}-\frac{7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{512 a^5 d}-\frac{155 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{3072 a^5 d}-\frac{2473 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{12288 a^5 d}-\frac{9467 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{8192 a^5 d}","\frac{58077 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{40960 a^5 d}-\frac{41693 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{49152 a^4 d}+\frac{8925 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{32768 a^3 d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{74461 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{32768 \sqrt{2} a^{5/2} d}-\frac{\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{320 a^5 d}-\frac{7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{512 a^5 d}-\frac{155 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{3072 a^5 d}-\frac{2473 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{12288 a^5 d}-\frac{9467 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{8192 a^5 d}",1,"(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + (74461*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(32768*Sqrt[2]*a^(5/2)*d) + (8925*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(32768*a^3*d) - (41693*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(49152*a^4*d) + (58077*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(40960*a^5*d) - (9467*Cos[c + d*x]*Cot[c + d*x]^5*Sec[(c + d*x)/2]^2*(a + a*Sec[c + d*x])^(5/2))/(8192*a^5*d) - (2473*Cos[c + d*x]^2*Cot[c + d*x]^5*Sec[(c + d*x)/2]^4*(a + a*Sec[c + d*x])^(5/2))/(12288*a^5*d) - (155*Cos[c + d*x]^3*Cot[c + d*x]^5*Sec[(c + d*x)/2]^6*(a + a*Sec[c + d*x])^(5/2))/(3072*a^5*d) - (7*Cos[c + d*x]^4*Cot[c + d*x]^5*Sec[(c + d*x)/2]^8*(a + a*Sec[c + d*x])^(5/2))/(512*a^5*d) - (Cos[c + d*x]^5*Cot[c + d*x]^5*Sec[(c + d*x)/2]^10*(a + a*Sec[c + d*x])^(5/2))/(320*a^5*d)","A",12,6,23,0.2609,1,"{3887, 472, 579, 583, 522, 203}"
207,1,227,0,0.1795001,"\int \frac{\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx","Int[Tan[e + f*x]^2/(a + a*Sec[e + f*x])^(9/2),x]","\frac{27 \sin (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{64 a^4 f \sqrt{a \sec (e+f x)+a}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{9/2} f}+\frac{91 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{32 \sqrt{2} a^{9/2} f}+\frac{\sin (e+f x) \cos ^2(e+f x) \sec ^6\left(\frac{1}{2} (e+f x)\right)}{24 a^4 f \sqrt{a \sec (e+f x)+a}}+\frac{11 \sin (e+f x) \cos (e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)}{96 a^4 f \sqrt{a \sec (e+f x)+a}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{9/2} f}+\frac{91 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{32 \sqrt{2} a^{9/2} f}+\frac{27 \tan (e+f x)}{32 a^3 f (a \sec (e+f x)+a)^{3/2}}+\frac{11 \tan (e+f x)}{24 a^2 f (a \sec (e+f x)+a)^{5/2}}+\frac{\tan (e+f x)}{3 a f (a \sec (e+f x)+a)^{7/2}}",1,"(-2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(9/2)*f) + (91*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(32*Sqrt[2]*a^(9/2)*f) + (27*Sec[(e + f*x)/2]^2*Sin[e + f*x])/(64*a^4*f*Sqrt[a + a*Sec[e + f*x]]) + (11*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Sin[e + f*x])/(96*a^4*f*Sqrt[a + a*Sec[e + f*x]]) + (Cos[e + f*x]^2*Sec[(e + f*x)/2]^6*Sin[e + f*x])/(24*a^4*f*Sqrt[a + a*Sec[e + f*x]])","A",7,5,23,0.2174,1,"{3887, 471, 527, 522, 203}"
208,1,125,0,0.0884521,"\int (a+a \sec (c+d x))^n (e \tan (c+d x))^m \, dx","Int[(a + a*Sec[c + d*x])^n*(e*Tan[c + d*x])^m,x]","\frac{2^{m+n+1} (a \sec (c+d x)+a)^n (e \tan (c+d x))^{m+1} \left(\frac{1}{\sec (c+d x)+1}\right)^{m+n+1} F_1\left(\frac{m+1}{2};m+n,1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d e (m+1)}","\frac{2^{m+n+1} (a \sec (c+d x)+a)^n (e \tan (c+d x))^{m+1} \left(\frac{1}{\sec (c+d x)+1}\right)^{m+n+1} F_1\left(\frac{m+1}{2};m+n,1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d e (m+1)}",1,"(2^(1 + m + n)*AppellF1[(1 + m)/2, m + n, 1, (3 + m)/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*((1 + Sec[c + d*x])^(-1))^(1 + m + n)*(a + a*Sec[c + d*x])^n*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))","A",1,1,23,0.04348,1,"{3889}"
209,1,243,0,0.2291599,"\int (a+a \sec (c+d x))^3 (e \tan (c+d x))^m \, dx","Int[(a + a*Sec[c + d*x])^3*(e*Tan[c + d*x])^m,x]","\frac{a^3 (e \tan (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d e (m+1)}+\frac{a^3 \sec ^3(c+d x) \cos ^2(c+d x)^{\frac{m+4}{2}} (e \tan (c+d x))^{m+1} \, _2F_1\left(\frac{m+1}{2},\frac{m+4}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{3 a^3 \sec (c+d x) \cos ^2(c+d x)^{\frac{m+2}{2}} (e \tan (c+d x))^{m+1} \, _2F_1\left(\frac{m+1}{2},\frac{m+2}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{3 a^3 (e \tan (c+d x))^{m+1}}{d e (m+1)}","\frac{a^3 (e \tan (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d e (m+1)}+\frac{a^3 \sec ^3(c+d x) \cos ^2(c+d x)^{\frac{m+4}{2}} (e \tan (c+d x))^{m+1} \, _2F_1\left(\frac{m+1}{2},\frac{m+4}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{3 a^3 \sec (c+d x) \cos ^2(c+d x)^{\frac{m+2}{2}} (e \tan (c+d x))^{m+1} \, _2F_1\left(\frac{m+1}{2},\frac{m+2}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{3 a^3 (e \tan (c+d x))^{m+1}}{d e (m+1)}",1,"(3*a^3*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)) + (a^3*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)) + (3*a^3*(Cos[c + d*x]^2)^((2 + m)/2)*Hypergeometric2F1[(1 + m)/2, (2 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)) + (a^3*(Cos[c + d*x]^2)^((4 + m)/2)*Hypergeometric2F1[(1 + m)/2, (4 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]^3*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))","A",8,6,23,0.2609,1,"{3886, 3476, 364, 2617, 2607, 32}"
210,1,161,0,0.1694255,"\int (a+a \sec (c+d x))^2 (e \tan (c+d x))^m \, dx","Int[(a + a*Sec[c + d*x])^2*(e*Tan[c + d*x])^m,x]","\frac{a^2 (e \tan (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d e (m+1)}+\frac{2 a^2 \sec (c+d x) \cos ^2(c+d x)^{\frac{m+2}{2}} (e \tan (c+d x))^{m+1} \, _2F_1\left(\frac{m+1}{2},\frac{m+2}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a^2 (e \tan (c+d x))^{m+1}}{d e (m+1)}","\frac{a^2 (e \tan (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d e (m+1)}+\frac{2 a^2 \sec (c+d x) \cos ^2(c+d x)^{\frac{m+2}{2}} (e \tan (c+d x))^{m+1} \, _2F_1\left(\frac{m+1}{2},\frac{m+2}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a^2 (e \tan (c+d x))^{m+1}}{d e (m+1)}",1,"(a^2*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)) + (a^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)) + (2*a^2*(Cos[c + d*x]^2)^((2 + m)/2)*Hypergeometric2F1[(1 + m)/2, (2 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))","A",7,6,23,0.2609,1,"{3886, 3476, 364, 2617, 2607, 32}"
211,1,129,0,0.0824419,"\int (a+a \sec (c+d x)) (e \tan (c+d x))^m \, dx","Int[(a + a*Sec[c + d*x])*(e*Tan[c + d*x])^m,x]","\frac{a (e \tan (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d e (m+1)}+\frac{a \sec (c+d x) \cos ^2(c+d x)^{\frac{m+2}{2}} (e \tan (c+d x))^{m+1} \, _2F_1\left(\frac{m+1}{2},\frac{m+2}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}","\frac{a (e \tan (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d e (m+1)}+\frac{a \sec (c+d x) \cos ^2(c+d x)^{\frac{m+2}{2}} (e \tan (c+d x))^{m+1} \, _2F_1\left(\frac{m+1}{2},\frac{m+2}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}",1,"(a*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)) + (a*(Cos[c + d*x]^2)^((2 + m)/2)*Hypergeometric2F1[(1 + m)/2, (2 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))","A",4,4,21,0.1905,1,"{3884, 3476, 364, 2617}"
212,1,130,0,0.1592158,"\int \frac{(e \tan (c+d x))^m}{a+a \sec (c+d x)} \, dx","Int[(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x]),x]","\frac{e (e \tan (c+d x))^{m-1} \, _2F_1\left(1,\frac{m-1}{2};\frac{m+1}{2};-\tan ^2(c+d x)\right)}{a d (1-m)}-\frac{e \sec (c+d x) \cos ^2(c+d x)^{m/2} (e \tan (c+d x))^{m-1} \, _2F_1\left(\frac{m-1}{2},\frac{m}{2};\frac{m+1}{2};\sin ^2(c+d x)\right)}{a d (1-m)}","\frac{e (e \tan (c+d x))^{m-1} \, _2F_1\left(1,\frac{m-1}{2};\frac{m+1}{2};-\tan ^2(c+d x)\right)}{a d (1-m)}-\frac{e \sec (c+d x) \cos ^2(c+d x)^{m/2} (e \tan (c+d x))^{m-1} \, _2F_1\left(\frac{m-1}{2},\frac{m}{2};\frac{m+1}{2};\sin ^2(c+d x)\right)}{a d (1-m)}",1,"(e*Hypergeometric2F1[1, (-1 + m)/2, (1 + m)/2, -Tan[c + d*x]^2]*(e*Tan[c + d*x])^(-1 + m))/(a*d*(1 - m)) - (e*(Cos[c + d*x]^2)^(m/2)*Hypergeometric2F1[(-1 + m)/2, m/2, (1 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Tan[c + d*x])^(-1 + m))/(a*d*(1 - m))","A",5,5,23,0.2174,1,"{3888, 3884, 3476, 364, 2617}"
213,1,169,0,0.2723738,"\int \frac{(e \tan (c+d x))^m}{(a+a \sec (c+d x))^2} \, dx","Int[(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x])^2,x]","-\frac{e^3 (e \tan (c+d x))^{m-3} \, _2F_1\left(1,\frac{m-3}{2};\frac{m-1}{2};-\tan ^2(c+d x)\right)}{a^2 d (3-m)}+\frac{2 e^3 \sec (c+d x) \cos ^2(c+d x)^{\frac{m-2}{2}} (e \tan (c+d x))^{m-3} \, _2F_1\left(\frac{m-3}{2},\frac{m-2}{2};\frac{m-1}{2};\sin ^2(c+d x)\right)}{a^2 d (3-m)}-\frac{e^3 (e \tan (c+d x))^{m-3}}{a^2 d (3-m)}","-\frac{e^3 (e \tan (c+d x))^{m-3} \, _2F_1\left(1,\frac{m-3}{2};\frac{m-1}{2};-\tan ^2(c+d x)\right)}{a^2 d (3-m)}+\frac{2 e^3 \sec (c+d x) \cos ^2(c+d x)^{\frac{m-2}{2}} (e \tan (c+d x))^{m-3} \, _2F_1\left(\frac{m-3}{2},\frac{m-2}{2};\frac{m-1}{2};\sin ^2(c+d x)\right)}{a^2 d (3-m)}-\frac{e^3 (e \tan (c+d x))^{m-3}}{a^2 d (3-m)}",1,"-((e^3*(e*Tan[c + d*x])^(-3 + m))/(a^2*d*(3 - m))) - (e^3*Hypergeometric2F1[1, (-3 + m)/2, (-1 + m)/2, -Tan[c + d*x]^2]*(e*Tan[c + d*x])^(-3 + m))/(a^2*d*(3 - m)) + (2*e^3*(Cos[c + d*x]^2)^((-2 + m)/2)*Hypergeometric2F1[(-3 + m)/2, (-2 + m)/2, (-1 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Tan[c + d*x])^(-3 + m))/(a^2*d*(3 - m))","A",8,7,23,0.3043,1,"{3888, 3886, 3476, 364, 2617, 2607, 32}"
214,1,252,0,0.3413054,"\int \frac{(e \tan (c+d x))^m}{(a+a \sec (c+d x))^3} \, dx","Int[(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x])^3,x]","\frac{e^5 (e \tan (c+d x))^{m-5} \, _2F_1\left(1,\frac{m-5}{2};\frac{m-3}{2};-\tan ^2(c+d x)\right)}{a^3 d (5-m)}-\frac{e^5 \sec ^3(c+d x) \cos ^2(c+d x)^{\frac{m-2}{2}} (e \tan (c+d x))^{m-5} \, _2F_1\left(\frac{m-5}{2},\frac{m-2}{2};\frac{m-3}{2};\sin ^2(c+d x)\right)}{a^3 d (5-m)}-\frac{3 e^5 \sec (c+d x) \cos ^2(c+d x)^{\frac{m-4}{2}} (e \tan (c+d x))^{m-5} \, _2F_1\left(\frac{m-5}{2},\frac{m-4}{2};\frac{m-3}{2};\sin ^2(c+d x)\right)}{a^3 d (5-m)}+\frac{3 e^5 (e \tan (c+d x))^{m-5}}{a^3 d (5-m)}","\frac{e^5 (e \tan (c+d x))^{m-5} \, _2F_1\left(1,\frac{m-5}{2};\frac{m-3}{2};-\tan ^2(c+d x)\right)}{a^3 d (5-m)}-\frac{e^5 \sec ^3(c+d x) \cos ^2(c+d x)^{\frac{m-2}{2}} (e \tan (c+d x))^{m-5} \, _2F_1\left(\frac{m-5}{2},\frac{m-2}{2};\frac{m-3}{2};\sin ^2(c+d x)\right)}{a^3 d (5-m)}-\frac{3 e^5 \sec (c+d x) \cos ^2(c+d x)^{\frac{m-4}{2}} (e \tan (c+d x))^{m-5} \, _2F_1\left(\frac{m-5}{2},\frac{m-4}{2};\frac{m-3}{2};\sin ^2(c+d x)\right)}{a^3 d (5-m)}+\frac{3 e^5 (e \tan (c+d x))^{m-5}}{a^3 d (5-m)}",1,"(3*e^5*(e*Tan[c + d*x])^(-5 + m))/(a^3*d*(5 - m)) + (e^5*Hypergeometric2F1[1, (-5 + m)/2, (-3 + m)/2, -Tan[c + d*x]^2]*(e*Tan[c + d*x])^(-5 + m))/(a^3*d*(5 - m)) - (3*e^5*(Cos[c + d*x]^2)^((-4 + m)/2)*Hypergeometric2F1[(-5 + m)/2, (-4 + m)/2, (-3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Tan[c + d*x])^(-5 + m))/(a^3*d*(5 - m)) - (e^5*(Cos[c + d*x]^2)^((-2 + m)/2)*Hypergeometric2F1[(-5 + m)/2, (-2 + m)/2, (-3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]^3*(e*Tan[c + d*x])^(-5 + m))/(a^3*d*(5 - m))","A",9,7,23,0.3043,1,"{3888, 3886, 3476, 364, 2617, 2607, 32}"
215,1,131,0,0.0973926,"\int (a+a \sec (c+d x))^{3/2} (e \tan (c+d x))^m \, dx","Int[(a + a*Sec[c + d*x])^(3/2)*(e*Tan[c + d*x])^m,x]","\frac{2^{m+\frac{5}{2}} (a \sec (c+d x)+a)^{3/2} \left(\frac{1}{\sec (c+d x)+1}\right)^{m+\frac{5}{2}} (e \tan (c+d x))^{m+1} F_1\left(\frac{m+1}{2};m+\frac{3}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d e (m+1)}","\frac{2^{m+\frac{5}{2}} (a \sec (c+d x)+a)^{3/2} \left(\frac{1}{\sec (c+d x)+1}\right)^{m+\frac{5}{2}} (e \tan (c+d x))^{m+1} F_1\left(\frac{m+1}{2};m+\frac{3}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d e (m+1)}",1,"(2^(5/2 + m)*AppellF1[(1 + m)/2, 3/2 + m, 1, (3 + m)/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*((1 + Sec[c + d*x])^(-1))^(5/2 + m)*(a + a*Sec[c + d*x])^(3/2)*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))","A",1,1,25,0.04000,1,"{3889}"
216,1,131,0,0.0855779,"\int \sqrt{a+a \sec (c+d x)} (e \tan (c+d x))^m \, dx","Int[Sqrt[a + a*Sec[c + d*x]]*(e*Tan[c + d*x])^m,x]","\frac{2^{m+\frac{3}{2}} \sqrt{a \sec (c+d x)+a} \left(\frac{1}{\sec (c+d x)+1}\right)^{m+\frac{3}{2}} (e \tan (c+d x))^{m+1} F_1\left(\frac{m+1}{2};m+\frac{1}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d e (m+1)}","\frac{2^{m+\frac{3}{2}} \sqrt{a \sec (c+d x)+a} \left(\frac{1}{\sec (c+d x)+1}\right)^{m+\frac{3}{2}} (e \tan (c+d x))^{m+1} F_1\left(\frac{m+1}{2};m+\frac{1}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d e (m+1)}",1,"(2^(3/2 + m)*AppellF1[(1 + m)/2, 1/2 + m, 1, (3 + m)/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*((1 + Sec[c + d*x])^(-1))^(3/2 + m)*Sqrt[a + a*Sec[c + d*x]]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))","A",1,1,25,0.04000,1,"{3889}"
217,1,131,0,0.0884013,"\int \frac{(e \tan (c+d x))^m}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(e*Tan[c + d*x])^m/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2^{m+\frac{1}{2}} \left(\frac{1}{\sec (c+d x)+1}\right)^{m+\frac{1}{2}} (e \tan (c+d x))^{m+1} F_1\left(\frac{m+1}{2};m-\frac{1}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d e (m+1) \sqrt{a \sec (c+d x)+a}}","\frac{2^{m+\frac{1}{2}} \left(\frac{1}{\sec (c+d x)+1}\right)^{m+\frac{1}{2}} (e \tan (c+d x))^{m+1} F_1\left(\frac{m+1}{2};m-\frac{1}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d e (m+1) \sqrt{a \sec (c+d x)+a}}",1,"(2^(1/2 + m)*AppellF1[(1 + m)/2, -1/2 + m, 1, (3 + m)/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*((1 + Sec[c + d*x])^(-1))^(1/2 + m)*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)*Sqrt[a + a*Sec[c + d*x]])","A",1,1,25,0.04000,1,"{3889}"
218,1,131,0,0.1048771,"\int \frac{(e \tan (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2^{m-\frac{1}{2}} \left(\frac{1}{\sec (c+d x)+1}\right)^{m-\frac{1}{2}} (e \tan (c+d x))^{m+1} F_1\left(\frac{m+1}{2};m-\frac{3}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d e (m+1) (a \sec (c+d x)+a)^{3/2}}","\frac{2^{m-\frac{1}{2}} \left(\frac{1}{\sec (c+d x)+1}\right)^{m-\frac{1}{2}} (e \tan (c+d x))^{m+1} F_1\left(\frac{m+1}{2};m-\frac{3}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d e (m+1) (a \sec (c+d x)+a)^{3/2}}",1,"(2^(-1/2 + m)*AppellF1[(1 + m)/2, -3/2 + m, 1, (3 + m)/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*((1 + Sec[c + d*x])^(-1))^(-1/2 + m)*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)*(a + a*Sec[c + d*x])^(3/2))","A",1,1,25,0.04000,1,"{3889}"
219,1,123,0,0.1004693,"\int (a+a \sec (c+d x))^n \tan ^7(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^7,x]","\frac{(a \sec (c+d x)+a)^{n+4} \, _2F_1(1,n+4;n+5;\sec (c+d x)+1)}{a^4 d (n+4)}+\frac{7 (a \sec (c+d x)+a)^{n+4}}{a^4 d (n+4)}-\frac{5 (a \sec (c+d x)+a)^{n+5}}{a^5 d (n+5)}+\frac{(a \sec (c+d x)+a)^{n+6}}{a^6 d (n+6)}","\frac{(a \sec (c+d x)+a)^{n+4} \, _2F_1(1,n+4;n+5;\sec (c+d x)+1)}{a^4 d (n+4)}+\frac{7 (a \sec (c+d x)+a)^{n+4}}{a^4 d (n+4)}-\frac{5 (a \sec (c+d x)+a)^{n+5}}{a^5 d (n+5)}+\frac{(a \sec (c+d x)+a)^{n+6}}{a^6 d (n+6)}",1,"(7*(a + a*Sec[c + d*x])^(4 + n))/(a^4*d*(4 + n)) + (Hypergeometric2F1[1, 4 + n, 5 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(4 + n))/(a^4*d*(4 + n)) - (5*(a + a*Sec[c + d*x])^(5 + n))/(a^5*d*(5 + n)) + (a + a*Sec[c + d*x])^(6 + n)/(a^6*d*(6 + n))","A",4,3,21,0.1429,1,"{3880, 88, 65}"
220,1,97,0,0.0817055,"\int (a+a \sec (c+d x))^n \tan ^5(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^5,x]","-\frac{(a \sec (c+d x)+a)^{n+3} \, _2F_1(1,n+3;n+4;\sec (c+d x)+1)}{a^3 d (n+3)}-\frac{3 (a \sec (c+d x)+a)^{n+3}}{a^3 d (n+3)}+\frac{(a \sec (c+d x)+a)^{n+4}}{a^4 d (n+4)}","-\frac{(a \sec (c+d x)+a)^{n+3} \, _2F_1(1,n+3;n+4;\sec (c+d x)+1)}{a^3 d (n+3)}-\frac{3 (a \sec (c+d x)+a)^{n+3}}{a^3 d (n+3)}+\frac{(a \sec (c+d x)+a)^{n+4}}{a^4 d (n+4)}",1,"(-3*(a + a*Sec[c + d*x])^(3 + n))/(a^3*d*(3 + n)) - (Hypergeometric2F1[1, 3 + n, 4 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3 + n))/(a^3*d*(3 + n)) + (a + a*Sec[c + d*x])^(4 + n)/(a^4*d*(4 + n))","A",4,3,21,0.1429,1,"{3880, 88, 65}"
221,1,69,0,0.0645631,"\int (a+a \sec (c+d x))^n \tan ^3(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^3,x]","\frac{(a \sec (c+d x)+a)^{n+2} \, _2F_1(1,n+2;n+3;\sec (c+d x)+1)}{a^2 d (n+2)}+\frac{(a \sec (c+d x)+a)^{n+2}}{a^2 d (n+2)}","\frac{(a \sec (c+d x)+a)^{n+2} \, _2F_1(1,n+2;n+3;\sec (c+d x)+1)}{a^2 d (n+2)}+\frac{(a \sec (c+d x)+a)^{n+2}}{a^2 d (n+2)}",1,"(a + a*Sec[c + d*x])^(2 + n)/(a^2*d*(2 + n)) + (Hypergeometric2F1[1, 2 + n, 3 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2 + n))/(a^2*d*(2 + n))","A",3,3,21,0.1429,1,"{3880, 80, 65}"
222,1,43,0,0.0413094,"\int (a+a \sec (c+d x))^n \tan (c+d x) \, dx","Int[(a + a*Sec[c + d*x])^n*Tan[c + d*x],x]","-\frac{(a \sec (c+d x)+a)^{n+1} \, _2F_1(1,n+1;n+2;\sec (c+d x)+1)}{a d (n+1)}","-\frac{(a \sec (c+d x)+a)^{n+1} \, _2F_1(1,n+1;n+2;\sec (c+d x)+1)}{a d (n+1)}",1,"-((Hypergeometric2F1[1, 1 + n, 2 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1 + n))/(a*d*(1 + n)))","A",2,2,19,0.1053,1,"{3880, 65}"
223,1,74,0,0.0608846,"\int \cot (c+d x) (a+a \sec (c+d x))^n \, dx","Int[Cot[c + d*x]*(a + a*Sec[c + d*x])^n,x]","\frac{(a \sec (c+d x)+a)^n \, _2F_1(1,n;n+1;\sec (c+d x)+1)}{d n}-\frac{(a \sec (c+d x)+a)^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (\sec (c+d x)+1)\right)}{2 d n}","\frac{(a \sec (c+d x)+a)^n \, _2F_1(1,n;n+1;\sec (c+d x)+1)}{d n}-\frac{(a \sec (c+d x)+a)^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (\sec (c+d x)+1)\right)}{2 d n}",1,"-(Hypergeometric2F1[1, n, 1 + n, (1 + Sec[c + d*x])/2]*(a + a*Sec[c + d*x])^n)/(2*d*n) + (Hypergeometric2F1[1, n, 1 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^n)/(d*n)","A",4,4,19,0.2105,1,"{3880, 86, 65, 68}"
224,1,127,0,0.109211,"\int \cot ^3(c+d x) (a+a \sec (c+d x))^n \, dx","Int[Cot[c + d*x]^3*(a + a*Sec[c + d*x])^n,x]","-\frac{a (4-n) (a \sec (c+d x)+a)^{n-1} \, _2F_1\left(1,n-1;n;\frac{1}{2} (\sec (c+d x)+1)\right)}{4 d (1-n)}+\frac{a (a \sec (c+d x)+a)^{n-1} \, _2F_1(1,n-1;n;\sec (c+d x)+1)}{d (1-n)}+\frac{a (a \sec (c+d x)+a)^{n-1}}{2 d (1-\sec (c+d x))}","-\frac{a (4-n) (a \sec (c+d x)+a)^{n-1} \, _2F_1\left(1,n-1;n;\frac{1}{2} (\sec (c+d x)+1)\right)}{4 d (1-n)}+\frac{a (a \sec (c+d x)+a)^{n-1} \, _2F_1(1,n-1;n;\sec (c+d x)+1)}{d (1-n)}+\frac{a (a \sec (c+d x)+a)^{n-1}}{2 d (1-\sec (c+d x))}",1,"-(a*(4 - n)*Hypergeometric2F1[1, -1 + n, n, (1 + Sec[c + d*x])/2]*(a + a*Sec[c + d*x])^(-1 + n))/(4*d*(1 - n)) + (a*Hypergeometric2F1[1, -1 + n, n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(-1 + n))/(d*(1 - n)) + (a*(a + a*Sec[c + d*x])^(-1 + n))/(2*d*(1 - Sec[c + d*x]))","A",5,5,21,0.2381,1,"{3880, 103, 156, 65, 68}"
225,1,106,0,0.0553183,"\int (a+a \sec (c+d x))^n \tan ^4(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^4,x]","\frac{2^{n+5} \tan ^5(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n+5} (a \sec (c+d x)+a)^n F_1\left(\frac{5}{2};n+4,1;\frac{7}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{5 d}","\frac{2^{n+5} \tan ^5(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n+5} (a \sec (c+d x)+a)^n F_1\left(\frac{5}{2};n+4,1;\frac{7}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{5 d}",1,"(2^(5 + n)*AppellF1[5/2, 4 + n, 1, 7/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*((1 + Sec[c + d*x])^(-1))^(5 + n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x]^5)/(5*d)","A",1,1,21,0.04762,1,"{3889}"
226,1,106,0,0.0553637,"\int (a+a \sec (c+d x))^n \tan ^2(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^2,x]","\frac{2^{n+3} \tan ^3(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n+3} (a \sec (c+d x)+a)^n F_1\left(\frac{3}{2};n+2,1;\frac{5}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{3 d}","\frac{2^{n+3} \tan ^3(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n+3} (a \sec (c+d x)+a)^n F_1\left(\frac{3}{2};n+2,1;\frac{5}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{3 d}",1,"(2^(3 + n)*AppellF1[3/2, 2 + n, 1, 5/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*((1 + Sec[c + d*x])^(-1))^(3 + n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x]^3)/(3*d)","A",1,1,21,0.04762,1,"{3889}"
227,1,102,0,0.0553295,"\int \cot ^2(c+d x) (a+a \sec (c+d x))^n \, dx","Int[Cot[c + d*x]^2*(a + a*Sec[c + d*x])^n,x]","-\frac{2^{n-1} \cot (c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n-1} (a \sec (c+d x)+a)^n F_1\left(-\frac{1}{2};n-2,1;\frac{1}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d}","-\frac{2^{n-1} \cot (c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n-1} (a \sec (c+d x)+a)^n F_1\left(-\frac{1}{2};n-2,1;\frac{1}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d}",1,"-((2^(-1 + n)*AppellF1[-1/2, -2 + n, 1, 1/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*Cot[c + d*x]*((1 + Sec[c + d*x])^(-1))^(-1 + n)*(a + a*Sec[c + d*x])^n)/d)","A",1,1,21,0.04762,1,"{3889}"
228,1,106,0,0.0555987,"\int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx","Int[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^n,x]","-\frac{2^{n-3} \cot ^3(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n-3} (a \sec (c+d x)+a)^n F_1\left(-\frac{3}{2};n-4,1;-\frac{1}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{3 d}","-\frac{2^{n-3} \cot ^3(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n-3} (a \sec (c+d x)+a)^n F_1\left(-\frac{3}{2};n-4,1;-\frac{1}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{3 d}",1,"-(2^(-3 + n)*AppellF1[-3/2, -4 + n, 1, -1/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*Cot[c + d*x]^3*((1 + Sec[c + d*x])^(-1))^(-3 + n)*(a + a*Sec[c + d*x])^n)/(3*d)","A",1,1,21,0.04762,1,"{3889}"
229,1,114,0,0.0640971,"\int (a+a \sec (c+d x))^n \tan ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^(3/2),x]","\frac{2^{n+\frac{7}{2}} \tan ^{\frac{5}{2}}(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n+\frac{5}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{5}{4};n+\frac{3}{2},1;\frac{9}{4};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{5 d}","\frac{2^{n+\frac{7}{2}} \tan ^{\frac{5}{2}}(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n+\frac{5}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{5}{4};n+\frac{3}{2},1;\frac{9}{4};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{5 d}",1,"(2^(7/2 + n)*AppellF1[5/4, 3/2 + n, 1, 9/4, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*((1 + Sec[c + d*x])^(-1))^(5/2 + n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x]^(5/2))/(5*d)","A",1,1,23,0.04348,1,"{3889}"
230,1,114,0,0.0583088,"\int (a+a \sec (c+d x))^n \sqrt{\tan (c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^n*Sqrt[Tan[c + d*x]],x]","\frac{2^{n+\frac{5}{2}} \tan ^{\frac{3}{2}}(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n+\frac{3}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{3}{4};n+\frac{1}{2},1;\frac{7}{4};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{3 d}","\frac{2^{n+\frac{5}{2}} \tan ^{\frac{3}{2}}(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n+\frac{3}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{3}{4};n+\frac{1}{2},1;\frac{7}{4};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{3 d}",1,"(2^(5/2 + n)*AppellF1[3/4, 1/2 + n, 1, 7/4, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*((1 + Sec[c + d*x])^(-1))^(3/2 + n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x]^(3/2))/(3*d)","A",1,1,23,0.04348,1,"{3889}"
231,1,111,0,0.06267,"\int \frac{(a+a \sec (c+d x))^n}{\sqrt{\tan (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^n/Sqrt[Tan[c + d*x]],x]","\frac{2^{n+\frac{3}{2}} \sqrt{\tan (c+d x)} \left(\frac{1}{\sec (c+d x)+1}\right)^{n+\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{1}{4};n-\frac{1}{2},1;\frac{5}{4};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d}","\frac{2^{n+\frac{3}{2}} \sqrt{\tan (c+d x)} \left(\frac{1}{\sec (c+d x)+1}\right)^{n+\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{1}{4};n-\frac{1}{2},1;\frac{5}{4};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d}",1,"(2^(3/2 + n)*AppellF1[1/4, -1/2 + n, 1, 5/4, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*((1 + Sec[c + d*x])^(-1))^(1/2 + n)*(a + a*Sec[c + d*x])^n*Sqrt[Tan[c + d*x]])/d","A",1,1,23,0.04348,1,"{3889}"
232,1,112,0,0.0673982,"\int \frac{(a+a \sec (c+d x))^n}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^n/Tan[c + d*x]^(3/2),x]","-\frac{2^{n+\frac{1}{2}} \left(\frac{1}{\sec (c+d x)+1}\right)^{n-\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left(-\frac{1}{4};n-\frac{3}{2},1;\frac{3}{4};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d \sqrt{\tan (c+d x)}}","-\frac{2^{n+\frac{1}{2}} \left(\frac{1}{\sec (c+d x)+1}\right)^{n-\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left(-\frac{1}{4};n-\frac{3}{2},1;\frac{3}{4};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d \sqrt{\tan (c+d x)}}",1,"-((2^(1/2 + n)*AppellF1[-1/4, -3/2 + n, 1, 3/4, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*((1 + Sec[c + d*x])^(-1))^(-1/2 + n)*(a + a*Sec[c + d*x])^n)/(d*Sqrt[Tan[c + d*x]]))","A",1,1,23,0.04348,1,"{3889}"
233,1,320,0,0.2635232,"\int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx","Int[(e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x]),x]","\frac{a \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}{\sqrt{2} d}-\frac{a \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}{\sqrt{2} d}+\frac{a \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{a \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 \tan (c+d x) (a \sec (c+d x)+a) (e \cot (c+d x))^{5/2}}{3 d}-\frac{a \sqrt{\sin (2 c+2 d x)} \tan ^2(c+d x) \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) (e \cot (c+d x))^{5/2}}{3 d}","\frac{a \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}{\sqrt{2} d}-\frac{a \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}{\sqrt{2} d}+\frac{a \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{a \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 \tan (c+d x) (a \sec (c+d x)+a) (e \cot (c+d x))^{5/2}}{3 d}-\frac{a \sqrt{\sin (2 c+2 d x)} \tan ^2(c+d x) \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) (e \cot (c+d x))^{5/2}}{3 d}",1,"(-2*(e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])*Tan[c + d*x])/(3*d) - (a*(e*Cot[c + d*x])^(5/2)*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]]*Tan[c + d*x]^2)/(3*d) + (a*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))/(Sqrt[2]*d) - (a*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))/(Sqrt[2]*d) + (a*(e*Cot[c + d*x])^(5/2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(5/2))/(2*Sqrt[2]*d) - (a*(e*Cot[c + d*x])^(5/2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(5/2))/(2*Sqrt[2]*d)","A",17,14,23,0.6087,1,"{3900, 3882, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641}"
234,1,346,0,0.2794046,"\int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx","Int[(e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x]),x]","\frac{a \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} d}-\frac{a \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} d}-\frac{a \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{a \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 a \sin (c+d x) \tan ^2(c+d x) (e \cot (c+d x))^{3/2}}{d}-\frac{2 \tan (c+d x) (a \sec (c+d x)+a) (e \cot (c+d x))^{3/2}}{d}-\frac{2 a \sin (c+d x) \tan (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) (e \cot (c+d x))^{3/2}}{d \sqrt{\sin (2 c+2 d x)}}","\frac{a \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} d}-\frac{a \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} d}-\frac{a \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{a \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 a \sin (c+d x) \tan ^2(c+d x) (e \cot (c+d x))^{3/2}}{d}-\frac{2 \tan (c+d x) (a \sec (c+d x)+a) (e \cot (c+d x))^{3/2}}{d}-\frac{2 a \sin (c+d x) \tan (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) (e \cot (c+d x))^{3/2}}{d \sqrt{\sin (2 c+2 d x)}}",1,"(-2*(e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x])*Tan[c + d*x])/d - (2*a*(e*Cot[c + d*x])^(3/2)*EllipticE[c - Pi/4 + d*x, 2]*Sin[c + d*x]*Tan[c + d*x])/(d*Sqrt[Sin[2*c + 2*d*x]]) + (a*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*d) - (a*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*d) - (a*(e*Cot[c + d*x])^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*d) + (a*(e*Cot[c + d*x])^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*d) + (2*a*(e*Cot[c + d*x])^(3/2)*Sin[c + d*x]*Tan[c + d*x]^2)/d","A",18,15,23,0.6522,1,"{3900, 3882, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639}"
235,1,274,0,0.2012906,"\int \sqrt{e \cot (c+d x)} (a+a \sec (c+d x)) \, dx","Int[Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x]),x]","-\frac{a \sqrt{\tan (c+d x)} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \sqrt{e \cot (c+d x)}}{\sqrt{2} d}+\frac{a \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}{\sqrt{2} d}-\frac{a \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{a \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{a \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \cot (c+d x)}}{d}","-\frac{a \sqrt{\tan (c+d x)} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \sqrt{e \cot (c+d x)}}{\sqrt{2} d}+\frac{a \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}{\sqrt{2} d}-\frac{a \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{a \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{a \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \cot (c+d x)}}{d}",1,"(a*Sqrt[e*Cot[c + d*x]]*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/d - (a*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[2]*d) + (a*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[2]*d) - (a*Sqrt[e*Cot[c + d*x]]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[Tan[c + d*x]])/(2*Sqrt[2]*d) + (a*Sqrt[e*Cot[c + d*x]]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[Tan[c + d*x]])/(2*Sqrt[2]*d)","A",16,13,23,0.5652,1,"{3900, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641}"
236,1,299,0,0.2333194,"\int \frac{a+a \sec (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])/Sqrt[e*Cot[c + d*x]],x]","\frac{2 a \sin (c+d x)}{d \sqrt{e \cot (c+d x)}}-\frac{a \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{a \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{a \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{a \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{2 a \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{d \sqrt{\sin (2 c+2 d x)} \sqrt{e \cot (c+d x)}}","\frac{2 a \sin (c+d x)}{d \sqrt{e \cot (c+d x)}}-\frac{a \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{a \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{a \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{a \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{2 a \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{d \sqrt{\sin (2 c+2 d x)} \sqrt{e \cot (c+d x)}}",1,"(2*a*Sin[c + d*x])/(d*Sqrt[e*Cot[c + d*x]]) - (2*a*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2])/(d*Sqrt[e*Cot[c + d*x]]*Sqrt[Sin[2*c + 2*d*x]]) - (a*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + (a*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + (a*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) - (a*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])","A",17,14,23,0.6087,1,"{3900, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639}"
237,1,320,0,0.251212,"\int \frac{a+a \sec (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx","Int[(a + a*Sec[c + d*x])/(e*Cot[c + d*x])^(3/2),x]","\frac{a \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{2 \cot (c+d x) (a \sec (c+d x)+3 a)}{3 d (e \cot (c+d x))^{3/2}}+\frac{a \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a \sqrt{\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d (e \cot (c+d x))^{3/2}}","\frac{a \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{2 \cot (c+d x) (a \sec (c+d x)+3 a)}{3 d (e \cot (c+d x))^{3/2}}+\frac{a \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a \sqrt{\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d (e \cot (c+d x))^{3/2}}",1,"(2*Cot[c + d*x]*(3*a + a*Sec[c + d*x]))/(3*d*(e*Cot[c + d*x])^(3/2)) - (a*Cot[c + d*x]*Csc[c + d*x]*EllipticF[c - Pi/4 + d*x, 2]*Sqrt[Sin[2*c + 2*d*x]])/(3*d*(e*Cot[c + d*x])^(3/2)) + (a*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - (a*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) + (a*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - (a*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))","A",17,14,23,0.6087,1,"{3900, 3881, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641}"
238,1,357,0,0.3359219,"\int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx","Int[(e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}{\sqrt{2} d}-\frac{a^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}{\sqrt{2} d}-\frac{4 a^2 \tan (c+d x) (e \cot (c+d x))^{5/2}}{3 d}+\frac{a^2 \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{a^2 \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{4 a^2 \tan (c+d x) \sec (c+d x) (e \cot (c+d x))^{5/2}}{3 d}-\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \tan ^2(c+d x) \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) (e \cot (c+d x))^{5/2}}{3 d}","\frac{a^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}{\sqrt{2} d}-\frac{a^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}{\sqrt{2} d}-\frac{4 a^2 \tan (c+d x) (e \cot (c+d x))^{5/2}}{3 d}+\frac{a^2 \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{a^2 \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{4 a^2 \tan (c+d x) \sec (c+d x) (e \cot (c+d x))^{5/2}}{3 d}-\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \tan ^2(c+d x) \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) (e \cot (c+d x))^{5/2}}{3 d}",1,"(-4*a^2*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x])/(3*d) - (4*a^2*(e*Cot[c + d*x])^(5/2)*Sec[c + d*x]*Tan[c + d*x])/(3*d) - (2*a^2*(e*Cot[c + d*x])^(5/2)*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]]*Tan[c + d*x]^2)/(3*d) + (a^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))/(Sqrt[2]*d) - (a^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))/(Sqrt[2]*d) + (a^2*(e*Cot[c + d*x])^(5/2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(5/2))/(2*Sqrt[2]*d) - (a^2*(e*Cot[c + d*x])^(5/2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(5/2))/(2*Sqrt[2]*d)","A",21,17,25,0.6800,1,"{3900, 3886, 3474, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2609, 2614, 2573, 2641, 2607, 30}"
239,1,343,0,0.3292551,"\int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx","Int[(e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2,x]","-\frac{4 a^2 \sin (c+d x) (e \cot (c+d x))^{3/2}}{d}+\frac{a^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} d}-\frac{a^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} d}-\frac{4 a^2 \tan (c+d x) (e \cot (c+d x))^{3/2}}{d}-\frac{a^2 \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{a^2 \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{4 a^2 \sin (c+d x) \tan (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) (e \cot (c+d x))^{3/2}}{d \sqrt{\sin (2 c+2 d x)}}","-\frac{4 a^2 \sin (c+d x) (e \cot (c+d x))^{3/2}}{d}+\frac{a^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} d}-\frac{a^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} d}-\frac{4 a^2 \tan (c+d x) (e \cot (c+d x))^{3/2}}{d}-\frac{a^2 \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{a^2 \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{4 a^2 \sin (c+d x) \tan (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) (e \cot (c+d x))^{3/2}}{d \sqrt{\sin (2 c+2 d x)}}",1,"(-4*a^2*(e*Cot[c + d*x])^(3/2)*Sin[c + d*x])/d - (4*a^2*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x])/d - (4*a^2*(e*Cot[c + d*x])^(3/2)*EllipticE[c - Pi/4 + d*x, 2]*Sin[c + d*x]*Tan[c + d*x])/(d*Sqrt[Sin[2*c + 2*d*x]]) + (a^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*d) - (a^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*d) - (a^2*(e*Cot[c + d*x])^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*d) + (a^2*(e*Cot[c + d*x])^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*d)","A",21,17,25,0.6800,1,"{3900, 3886, 3474, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2608, 2615, 2572, 2639, 2607, 30}"
240,1,311,0,0.2872159,"\int \sqrt{e \cot (c+d x)} (a+a \sec (c+d x))^2 \, dx","Int[Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])^2,x]","-\frac{a^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}{\sqrt{2} d}+\frac{a^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}{\sqrt{2} d}+\frac{2 a^2 \tan (c+d x) \sqrt{e \cot (c+d x)}}{d}-\frac{a^2 \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{a^2 \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \cot (c+d x)}}{d}","-\frac{a^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}{\sqrt{2} d}+\frac{a^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}{\sqrt{2} d}+\frac{2 a^2 \tan (c+d x) \sqrt{e \cot (c+d x)}}{d}-\frac{a^2 \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{a^2 \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \cot (c+d x)}}{d}",1,"(2*a^2*Sqrt[e*Cot[c + d*x]]*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/d - (a^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[2]*d) + (a^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[2]*d) - (a^2*Sqrt[e*Cot[c + d*x]]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[Tan[c + d*x]])/(2*Sqrt[2]*d) + (a^2*Sqrt[e*Cot[c + d*x]]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*a^2*Sqrt[e*Cot[c + d*x]]*Tan[c + d*x])/d","A",19,15,25,0.6000,1,"{3900, 3886, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641, 2607, 30}"
241,1,339,0,0.3183665,"\int \frac{(a+a \sec (c+d x))^2}{\sqrt{e \cot (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^2/Sqrt[e*Cot[c + d*x]],x]","\frac{4 a^2 \sin (c+d x)}{d \sqrt{e \cot (c+d x)}}-\frac{a^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{a^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{2 a^2 \tan (c+d x)}{3 d \sqrt{e \cot (c+d x)}}+\frac{a^2 \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{a^2 \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{4 a^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{d \sqrt{\sin (2 c+2 d x)} \sqrt{e \cot (c+d x)}}","\frac{4 a^2 \sin (c+d x)}{d \sqrt{e \cot (c+d x)}}-\frac{a^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{a^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{2 a^2 \tan (c+d x)}{3 d \sqrt{e \cot (c+d x)}}+\frac{a^2 \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{a^2 \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{4 a^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{d \sqrt{\sin (2 c+2 d x)} \sqrt{e \cot (c+d x)}}",1,"(4*a^2*Sin[c + d*x])/(d*Sqrt[e*Cot[c + d*x]]) - (4*a^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2])/(d*Sqrt[e*Cot[c + d*x]]*Sqrt[Sin[2*c + 2*d*x]]) - (a^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + (a^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + (a^2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) - (a^2*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + (2*a^2*Tan[c + d*x])/(3*d*Sqrt[e*Cot[c + d*x]])","A",20,16,25,0.6400,1,"{3900, 3886, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639, 2607, 30}"
242,1,375,0,0.3284235,"\int \frac{(a+a \sec (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx","Int[(a + a*Sec[c + d*x])^2/(e*Cot[c + d*x])^(3/2),x]","\frac{2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac{a^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}+\frac{4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}+\frac{a^2 \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a^2 \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d (e \cot (c+d x))^{3/2}}","\frac{2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac{a^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}+\frac{4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}+\frac{a^2 \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a^2 \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d (e \cot (c+d x))^{3/2}}",1,"(2*a^2*Cot[c + d*x])/(d*(e*Cot[c + d*x])^(3/2)) + (4*a^2*Csc[c + d*x])/(3*d*(e*Cot[c + d*x])^(3/2)) - (2*a^2*Cot[c + d*x]*Csc[c + d*x]*EllipticF[c - Pi/4 + d*x, 2]*Sqrt[Sin[2*c + 2*d*x]])/(3*d*(e*Cot[c + d*x])^(3/2)) + (a^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - (a^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) + (a^2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - (a^2*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) + (2*a^2*Tan[c + d*x])/(5*d*(e*Cot[c + d*x])^(3/2))","A",21,17,25,0.6800,1,"{3900, 3886, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2611, 2614, 2573, 2641, 2607, 30}"
243,1,405,0,0.4267192,"\int \frac{(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx","Int[(e*Cot[c + d*x])^(3/2)/(a + a*Sec[c + d*x]),x]","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} a d}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} a d}+\frac{2 \cot (c+d x) (1-\sec (c+d x)) (e \cot (c+d x))^{3/2}}{5 a d}-\frac{\tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d}+\frac{\tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d}-\frac{6 \sin (c+d x) \tan ^2(c+d x) (e \cot (c+d x))^{3/2}}{5 a d}-\frac{2 \tan (c+d x) (5-3 \sec (c+d x)) (e \cot (c+d x))^{3/2}}{5 a d}+\frac{6 \sin (c+d x) \tan (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) (e \cot (c+d x))^{3/2}}{5 a d \sqrt{\sin (2 c+2 d x)}}","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} a d}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} a d}+\frac{2 \cot (c+d x) (1-\sec (c+d x)) (e \cot (c+d x))^{3/2}}{5 a d}-\frac{\tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d}+\frac{\tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d}-\frac{6 \sin (c+d x) \tan ^2(c+d x) (e \cot (c+d x))^{3/2}}{5 a d}-\frac{2 \tan (c+d x) (5-3 \sec (c+d x)) (e \cot (c+d x))^{3/2}}{5 a d}+\frac{6 \sin (c+d x) \tan (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) (e \cot (c+d x))^{3/2}}{5 a d \sqrt{\sin (2 c+2 d x)}}",1,"(2*Cot[c + d*x]*(e*Cot[c + d*x])^(3/2)*(1 - Sec[c + d*x]))/(5*a*d) - (2*(e*Cot[c + d*x])^(3/2)*(5 - 3*Sec[c + d*x])*Tan[c + d*x])/(5*a*d) + (6*(e*Cot[c + d*x])^(3/2)*EllipticE[c - Pi/4 + d*x, 2]*Sin[c + d*x]*Tan[c + d*x])/(5*a*d*Sqrt[Sin[2*c + 2*d*x]]) + (ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*a*d) - (ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*a*d) - ((e*Cot[c + d*x])^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*a*d) + ((e*Cot[c + d*x])^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*a*d) - (6*(e*Cot[c + d*x])^(3/2)*Sin[c + d*x]*Tan[c + d*x]^2)/(5*a*d)","A",20,16,25,0.6400,1,"{3900, 3888, 3882, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639}"
244,1,325,0,0.3246594,"\int \frac{\sqrt{e \cot (c+d x)}}{a+a \sec (c+d x)} \, dx","Int[Sqrt[e*Cot[c + d*x]]/(a + a*Sec[c + d*x]),x]","-\frac{\sqrt{\tan (c+d x)} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \sqrt{e \cot (c+d x)}}{\sqrt{2} a d}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}{\sqrt{2} a d}+\frac{2 \cot (c+d x) (1-\sec (c+d x)) \sqrt{e \cot (c+d x)}}{3 a d}-\frac{\sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d}+\frac{\sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d}-\frac{\sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \cot (c+d x)}}{3 a d}","-\frac{\sqrt{\tan (c+d x)} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \sqrt{e \cot (c+d x)}}{\sqrt{2} a d}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}{\sqrt{2} a d}+\frac{2 \cot (c+d x) (1-\sec (c+d x)) \sqrt{e \cot (c+d x)}}{3 a d}-\frac{\sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d}+\frac{\sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d}-\frac{\sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \cot (c+d x)}}{3 a d}",1,"(2*Cot[c + d*x]*Sqrt[e*Cot[c + d*x]]*(1 - Sec[c + d*x]))/(3*a*d) - (Sqrt[e*Cot[c + d*x]]*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*a*d) - (ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[2]*a*d) + (ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[2]*a*d) - (Sqrt[e*Cot[c + d*x]]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[Tan[c + d*x]])/(2*Sqrt[2]*a*d) + (Sqrt[e*Cot[c + d*x]]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[Tan[c + d*x]])/(2*Sqrt[2]*a*d)","A",18,15,25,0.6000,1,"{3900, 3888, 3882, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641}"
245,1,347,0,0.3565337,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+a \sec (c+d x))} \, dx","Int[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])),x]","\frac{2 \sin (c+d x)}{a d \sqrt{e \cot (c+d x)}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{2 \cot (c+d x) (1-\sec (c+d x))}{a d \sqrt{e \cot (c+d x)}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a d \sqrt{\sin (2 c+2 d x)} \sqrt{e \cot (c+d x)}}","\frac{2 \sin (c+d x)}{a d \sqrt{e \cot (c+d x)}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{2 \cot (c+d x) (1-\sec (c+d x))}{a d \sqrt{e \cot (c+d x)}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a d \sqrt{\sin (2 c+2 d x)} \sqrt{e \cot (c+d x)}}",1,"(2*Cot[c + d*x]*(1 - Sec[c + d*x]))/(a*d*Sqrt[e*Cot[c + d*x]]) + (2*Sin[c + d*x])/(a*d*Sqrt[e*Cot[c + d*x]]) - (2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2])/(a*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Sin[2*c + 2*d*x]]) - ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])","A",19,16,25,0.6400,1,"{3900, 3888, 3882, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639}"
246,1,290,0,0.2890105,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))} \, dx","Int[1/((e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x])),x]","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{\sqrt{\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a d (e \cot (c+d x))^{3/2}}","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{\sqrt{\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a d (e \cot (c+d x))^{3/2}}",1,"(Cot[c + d*x]*Csc[c + d*x]*EllipticF[c - Pi/4 + d*x, 2]*Sqrt[Sin[2*c + 2*d*x]])/(a*d*(e*Cot[c + d*x])^(3/2)) + ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))","A",17,14,25,0.5600,1,"{3900, 3888, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641}"
247,1,325,0,0.3230623,"\int \frac{1}{(e \cot (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx","Int[1/((e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])),x]","\frac{2 \cos (c+d x) \cot (c+d x)}{a d (e \cot (c+d x))^{5/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}+\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac{2 \cos (c+d x) \cot ^2(c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a d \sqrt{\sin (2 c+2 d x)} (e \cot (c+d x))^{5/2}}","\frac{2 \cos (c+d x) \cot (c+d x)}{a d (e \cot (c+d x))^{5/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}+\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac{2 \cos (c+d x) \cot ^2(c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a d \sqrt{\sin (2 c+2 d x)} (e \cot (c+d x))^{5/2}}",1,"(2*Cos[c + d*x]*Cot[c + d*x])/(a*d*(e*Cot[c + d*x])^(5/2)) - (2*Cos[c + d*x]*Cot[c + d*x]^2*EllipticE[c - Pi/4 + d*x, 2])/(a*d*(e*Cot[c + d*x])^(5/2)*Sqrt[Sin[2*c + 2*d*x]]) + ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2)) - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2)) - Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2)) + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))","A",18,15,25,0.6000,1,"{3900, 3888, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639}"
248,1,335,0,0.3373748,"\int \frac{1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx","Int[1/((e*Cot[c + d*x])^(7/2)*(a + a*Sec[c + d*x])),x]","-\frac{2 \cot ^3(c+d x) (3-\sec (c+d x))}{3 a d (e \cot (c+d x))^{7/2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}-\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}+\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}-\frac{\sqrt{\sin (2 c+2 d x)} \cot ^3(c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a d (e \cot (c+d x))^{7/2}}","-\frac{2 \cot ^3(c+d x) (3-\sec (c+d x))}{3 a d (e \cot (c+d x))^{7/2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}-\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}+\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}-\frac{\sqrt{\sin (2 c+2 d x)} \cot ^3(c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a d (e \cot (c+d x))^{7/2}}",1,"(-2*Cot[c + d*x]^3*(3 - Sec[c + d*x]))/(3*a*d*(e*Cot[c + d*x])^(7/2)) - (Cot[c + d*x]^3*Csc[c + d*x]*EllipticF[c - Pi/4 + d*x, 2]*Sqrt[Sin[2*c + 2*d*x]])/(3*a*d*(e*Cot[c + d*x])^(7/2)) - ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2)) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2)) - Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2)) + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2))","A",18,15,25,0.6000,1,"{3900, 3888, 3881, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641}"
249,1,371,0,0.3675969,"\int \frac{1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx","Int[1/((e*Cot[c + d*x])^(9/2)*(a + a*Sec[c + d*x])),x]","-\frac{6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac{2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{6 \cos (c+d x) \cot ^4(c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{5 a d \sqrt{\sin (2 c+2 d x)} (e \cot (c+d x))^{9/2}}","-\frac{6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac{2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{6 \cos (c+d x) \cot ^4(c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{5 a d \sqrt{\sin (2 c+2 d x)} (e \cot (c+d x))^{9/2}}",1,"(-6*Cos[c + d*x]*Cot[c + d*x]^3)/(5*a*d*(e*Cot[c + d*x])^(9/2)) - (2*Cot[c + d*x]^3*(5 - 3*Sec[c + d*x]))/(15*a*d*(e*Cot[c + d*x])^(9/2)) + (6*Cos[c + d*x]*Cot[c + d*x]^4*EllipticE[c - Pi/4 + d*x, 2])/(5*a*d*(e*Cot[c + d*x])^(9/2)*Sqrt[Sin[2*c + 2*d*x]]) - ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2)) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2)) + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2)) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2))","A",19,16,25,0.6400,1,"{3900, 3888, 3881, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639}"
250,1,413,0,0.4270953,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx","Int[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","-\frac{4 \cot ^3(c+d x)}{5 a^2 d \sqrt{e \cot (c+d x)}}+\frac{2 \cot (c+d x)}{a^2 d \sqrt{e \cot (c+d x)}}-\frac{12 \cos (c+d x) \cot (c+d x)}{5 a^2 d \sqrt{e \cot (c+d x)}}+\frac{4 \cot ^2(c+d x) \csc (c+d x)}{5 a^2 d \sqrt{e \cot (c+d x)}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{12 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{5 a^2 d \sqrt{\sin (2 c+2 d x)} \sqrt{e \cot (c+d x)}}","-\frac{4 \cot ^3(c+d x)}{5 a^2 d \sqrt{e \cot (c+d x)}}+\frac{2 \cot (c+d x)}{a^2 d \sqrt{e \cot (c+d x)}}-\frac{12 \cos (c+d x) \cot (c+d x)}{5 a^2 d \sqrt{e \cot (c+d x)}}+\frac{4 \cot ^2(c+d x) \csc (c+d x)}{5 a^2 d \sqrt{e \cot (c+d x)}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{12 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{5 a^2 d \sqrt{\sin (2 c+2 d x)} \sqrt{e \cot (c+d x)}}",1,"(2*Cot[c + d*x])/(a^2*d*Sqrt[e*Cot[c + d*x]]) - (12*Cos[c + d*x]*Cot[c + d*x])/(5*a^2*d*Sqrt[e*Cot[c + d*x]]) - (4*Cot[c + d*x]^3)/(5*a^2*d*Sqrt[e*Cot[c + d*x]]) + (4*Cot[c + d*x]^2*Csc[c + d*x])/(5*a^2*d*Sqrt[e*Cot[c + d*x]]) - (12*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2])/(5*a^2*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Sin[2*c + 2*d*x]]) - ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])","A",24,19,25,0.7600,1,"{3900, 3888, 3886, 3474, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2609, 2608, 2615, 2572, 2639, 2607, 30}"
251,1,359,0,0.3886786,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx","Int[1/((e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2),x]","-\frac{4 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac{4 \cot ^2(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{2 \sqrt{\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a^2 d (e \cot (c+d x))^{3/2}}","-\frac{4 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac{4 \cot ^2(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{2 \sqrt{\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a^2 d (e \cot (c+d x))^{3/2}}",1,"(-4*Cot[c + d*x]^3)/(3*a^2*d*(e*Cot[c + d*x])^(3/2)) + (4*Cot[c + d*x]^2*Csc[c + d*x])/(3*a^2*d*(e*Cot[c + d*x])^(3/2)) + (2*Cot[c + d*x]*Csc[c + d*x]*EllipticF[c - Pi/4 + d*x, 2]*Sqrt[Sin[2*c + 2*d*x]])/(3*a^2*d*(e*Cot[c + d*x])^(3/2)) + ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))","A",22,18,25,0.7200,1,"{3900, 3888, 3886, 3474, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2609, 2614, 2573, 2641, 2607, 30}"
252,1,355,0,0.417798,"\int \frac{1}{(e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2} \, dx","Int[1/((e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2),x]","-\frac{4 \cot ^3(c+d x)}{a^2 d (e \cot (c+d x))^{5/2}}+\frac{4 \cos (c+d x) \cot ^3(c+d x)}{a^2 d (e \cot (c+d x))^{5/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}+\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}+\frac{4 \cos (c+d x) \cot ^2(c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a^2 d \sqrt{\sin (2 c+2 d x)} (e \cot (c+d x))^{5/2}}","-\frac{4 \cot ^3(c+d x)}{a^2 d (e \cot (c+d x))^{5/2}}+\frac{4 \cos (c+d x) \cot ^3(c+d x)}{a^2 d (e \cot (c+d x))^{5/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}+\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}+\frac{4 \cos (c+d x) \cot ^2(c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a^2 d \sqrt{\sin (2 c+2 d x)} (e \cot (c+d x))^{5/2}}",1,"(-4*Cot[c + d*x]^3)/(a^2*d*(e*Cot[c + d*x])^(5/2)) + (4*Cos[c + d*x]*Cot[c + d*x]^3)/(a^2*d*(e*Cot[c + d*x])^(5/2)) + (4*Cos[c + d*x]*Cot[c + d*x]^2*EllipticE[c - Pi/4 + d*x, 2])/(a^2*d*(e*Cot[c + d*x])^(5/2)*Sqrt[Sin[2*c + 2*d*x]]) + ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2)) - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2)) - Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2)) + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))","A",22,18,25,0.7200,1,"{3900, 3888, 3886, 3474, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2608, 2615, 2572, 2639, 2607, 30}"
253,1,321,0,0.3723849,"\int \frac{1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx","Int[1/((e*Cot[c + d*x])^(7/2)*(a + a*Sec[c + d*x])^2),x]","\frac{2 \cot ^3(c+d x)}{a^2 d (e \cot (c+d x))^{7/2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}-\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}+\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}-\frac{2 \sqrt{\sin (2 c+2 d x)} \cot ^3(c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a^2 d (e \cot (c+d x))^{7/2}}","\frac{2 \cot ^3(c+d x)}{a^2 d (e \cot (c+d x))^{7/2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}-\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}+\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}-\frac{2 \sqrt{\sin (2 c+2 d x)} \cot ^3(c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a^2 d (e \cot (c+d x))^{7/2}}",1,"(2*Cot[c + d*x]^3)/(a^2*d*(e*Cot[c + d*x])^(7/2)) - (2*Cot[c + d*x]^3*Csc[c + d*x]*EllipticF[c - Pi/4 + d*x, 2]*Sqrt[Sin[2*c + 2*d*x]])/(a^2*d*(e*Cot[c + d*x])^(7/2)) - ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2)) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2)) - Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2)) + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2))","A",20,16,25,0.6400,1,"{3900, 3888, 3886, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641, 2607, 30}"
254,1,357,0,0.3939554,"\int \frac{1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))^2} \, dx","Int[1/((e*Cot[c + d*x])^(9/2)*(a + a*Sec[c + d*x])^2),x]","\frac{2 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{9/2}}-\frac{4 \cos (c+d x) \cot ^3(c+d x)}{a^2 d (e \cot (c+d x))^{9/2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{4 \cos (c+d x) \cot ^4(c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a^2 d \sqrt{\sin (2 c+2 d x)} (e \cot (c+d x))^{9/2}}","\frac{2 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{9/2}}-\frac{4 \cos (c+d x) \cot ^3(c+d x)}{a^2 d (e \cot (c+d x))^{9/2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{4 \cos (c+d x) \cot ^4(c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a^2 d \sqrt{\sin (2 c+2 d x)} (e \cot (c+d x))^{9/2}}",1,"(2*Cot[c + d*x]^3)/(3*a^2*d*(e*Cot[c + d*x])^(9/2)) - (4*Cos[c + d*x]*Cot[c + d*x]^3)/(a^2*d*(e*Cot[c + d*x])^(9/2)) + (4*Cos[c + d*x]*Cot[c + d*x]^4*EllipticE[c - Pi/4 + d*x, 2])/(a^2*d*(e*Cot[c + d*x])^(9/2)*Sqrt[Sin[2*c + 2*d*x]]) - ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2)) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2)) + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2)) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2))","A",21,17,25,0.6800,1,"{3900, 3888, 3886, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639, 2607, 30}"
255,1,389,0,0.4804198,"\int \frac{1}{(e \cot (c+d x))^{11/2} (a+a \sec (c+d x))^2} \, dx","Int[1/((e*Cot[c + d*x])^(11/2)*(a + a*Sec[c + d*x])^2),x]","\frac{2 \cot ^5(c+d x)}{a^2 d (e \cot (c+d x))^{11/2}}+\frac{2 \cot ^3(c+d x)}{5 a^2 d (e \cot (c+d x))^{11/2}}-\frac{4 \cot ^4(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{11/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \tan ^{\frac{11}{2}}(c+d x) (e \cot (c+d x))^{11/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \tan ^{\frac{11}{2}}(c+d x) (e \cot (c+d x))^{11/2}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{11}{2}}(c+d x) (e \cot (c+d x))^{11/2}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{11}{2}}(c+d x) (e \cot (c+d x))^{11/2}}+\frac{2 \sqrt{\sin (2 c+2 d x)} \cot ^5(c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a^2 d (e \cot (c+d x))^{11/2}}","\frac{2 \cot ^5(c+d x)}{a^2 d (e \cot (c+d x))^{11/2}}+\frac{2 \cot ^3(c+d x)}{5 a^2 d (e \cot (c+d x))^{11/2}}-\frac{4 \cot ^4(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{11/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \tan ^{\frac{11}{2}}(c+d x) (e \cot (c+d x))^{11/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \tan ^{\frac{11}{2}}(c+d x) (e \cot (c+d x))^{11/2}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{11}{2}}(c+d x) (e \cot (c+d x))^{11/2}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{11}{2}}(c+d x) (e \cot (c+d x))^{11/2}}+\frac{2 \sqrt{\sin (2 c+2 d x)} \cot ^5(c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a^2 d (e \cot (c+d x))^{11/2}}",1,"(2*Cot[c + d*x]^3)/(5*a^2*d*(e*Cot[c + d*x])^(11/2)) + (2*Cot[c + d*x]^5)/(a^2*d*(e*Cot[c + d*x])^(11/2)) - (4*Cot[c + d*x]^4*Csc[c + d*x])/(3*a^2*d*(e*Cot[c + d*x])^(11/2)) + (2*Cot[c + d*x]^5*Csc[c + d*x]*EllipticF[c - Pi/4 + d*x, 2]*Sqrt[Sin[2*c + 2*d*x]])/(3*a^2*d*(e*Cot[c + d*x])^(11/2)) + ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(11/2)*Tan[c + d*x]^(11/2)) - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(11/2)*Tan[c + d*x]^(11/2)) + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(11/2)*Tan[c + d*x]^(11/2)) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(11/2)*Tan[c + d*x]^(11/2))","A",22,18,25,0.7200,1,"{3900, 3888, 3886, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2611, 2614, 2573, 2641, 2607, 30}"
256,1,111,0,0.1560444,"\int (a+b \sec (c+d x)) \tan ^7(c+d x) \, dx","Int[(a + b*Sec[c + d*x])*Tan[c + d*x]^7,x]","\frac{\tan ^6(c+d x) (7 a+6 b \sec (c+d x))}{42 d}-\frac{\tan ^4(c+d x) (35 a+24 b \sec (c+d x))}{140 d}+\frac{\tan ^2(c+d x) (35 a+16 b \sec (c+d x))}{70 d}+\frac{a \log (\cos (c+d x))}{d}-\frac{16 b \sec (c+d x)}{35 d}","\frac{\tan ^6(c+d x) (7 a+6 b \sec (c+d x))}{42 d}-\frac{\tan ^4(c+d x) (35 a+24 b \sec (c+d x))}{140 d}+\frac{\tan ^2(c+d x) (35 a+16 b \sec (c+d x))}{70 d}+\frac{a \log (\cos (c+d x))}{d}-\frac{16 b \sec (c+d x)}{35 d}",1,"(a*Log[Cos[c + d*x]])/d - (16*b*Sec[c + d*x])/(35*d) + ((35*a + 16*b*Sec[c + d*x])*Tan[c + d*x]^2)/(70*d) - ((35*a + 24*b*Sec[c + d*x])*Tan[c + d*x]^4)/(140*d) + ((7*a + 6*b*Sec[c + d*x])*Tan[c + d*x]^6)/(42*d)","A",7,5,19,0.2632,1,"{3881, 3884, 3475, 2606, 8}"
257,1,84,0,0.0935648,"\int (a+b \sec (c+d x)) \tan ^5(c+d x) \, dx","Int[(a + b*Sec[c + d*x])*Tan[c + d*x]^5,x]","\frac{\tan ^4(c+d x) (5 a+4 b \sec (c+d x))}{20 d}-\frac{\tan ^2(c+d x) (15 a+8 b \sec (c+d x))}{30 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{8 b \sec (c+d x)}{15 d}","\frac{\tan ^4(c+d x) (5 a+4 b \sec (c+d x))}{20 d}-\frac{\tan ^2(c+d x) (15 a+8 b \sec (c+d x))}{30 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{8 b \sec (c+d x)}{15 d}",1,"-((a*Log[Cos[c + d*x]])/d) + (8*b*Sec[c + d*x])/(15*d) - ((15*a + 8*b*Sec[c + d*x])*Tan[c + d*x]^2)/(30*d) + ((5*a + 4*b*Sec[c + d*x])*Tan[c + d*x]^4)/(20*d)","A",6,5,19,0.2632,1,"{3881, 3884, 3475, 2606, 8}"
258,1,55,0,0.0661531,"\int (a+b \sec (c+d x)) \tan ^3(c+d x) \, dx","Int[(a + b*Sec[c + d*x])*Tan[c + d*x]^3,x]","\frac{\tan ^2(c+d x) (3 a+2 b \sec (c+d x))}{6 d}+\frac{a \log (\cos (c+d x))}{d}-\frac{2 b \sec (c+d x)}{3 d}","\frac{\tan ^2(c+d x) (3 a+2 b \sec (c+d x))}{6 d}+\frac{a \log (\cos (c+d x))}{d}-\frac{2 b \sec (c+d x)}{3 d}",1,"(a*Log[Cos[c + d*x]])/d - (2*b*Sec[c + d*x])/(3*d) + ((3*a + 2*b*Sec[c + d*x])*Tan[c + d*x]^2)/(6*d)","A",5,5,19,0.2632,1,"{3881, 3884, 3475, 2606, 8}"
259,1,25,0,0.0367905,"\int (a+b \sec (c+d x)) \tan (c+d x) \, dx","Int[(a + b*Sec[c + d*x])*Tan[c + d*x],x]","\frac{b \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}","\frac{b \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Log[Cos[c + d*x]])/d) + (b*Sec[c + d*x])/d","A",4,4,17,0.2353,1,"{3884, 3475, 2606, 8}"
260,1,43,0,0.080858,"\int \cot (c+d x) (a+b \sec (c+d x)) \, dx","Int[Cot[c + d*x]*(a + b*Sec[c + d*x]),x]","\frac{(a+b) \log (1-\cos (c+d x))}{2 d}+\frac{(a-b) \log (\cos (c+d x)+1)}{2 d}","\frac{(a+b) \log (1-\cos (c+d x))}{2 d}+\frac{(a-b) \log (\cos (c+d x)+1)}{2 d}",1,"((a + b)*Log[1 - Cos[c + d*x]])/(2*d) + ((a - b)*Log[1 + Cos[c + d*x]])/(2*d)","A",5,4,17,0.2353,1,"{3883, 2668, 633, 31}"
261,1,72,0,0.1084308,"\int \cot ^3(c+d x) (a+b \sec (c+d x)) \, dx","Int[Cot[c + d*x]^3*(a + b*Sec[c + d*x]),x]","-\frac{(2 a+b) \log (1-\cos (c+d x))}{4 d}-\frac{(2 a-b) \log (\cos (c+d x)+1)}{4 d}-\frac{\cot ^2(c+d x) (a+b \sec (c+d x))}{2 d}","-\frac{(2 a+b) \log (1-\cos (c+d x))}{4 d}-\frac{(2 a-b) \log (\cos (c+d x)+1)}{4 d}-\frac{\cot ^2(c+d x) (a+b \sec (c+d x))}{2 d}",1,"-((2*a + b)*Log[1 - Cos[c + d*x]])/(4*d) - ((2*a - b)*Log[1 + Cos[c + d*x]])/(4*d) - (Cot[c + d*x]^2*(a + b*Sec[c + d*x]))/(2*d)","A",6,5,19,0.2632,1,"{3882, 3883, 2668, 633, 31}"
262,1,102,0,0.1314367,"\int \cot ^5(c+d x) (a+b \sec (c+d x)) \, dx","Int[Cot[c + d*x]^5*(a + b*Sec[c + d*x]),x]","\frac{(8 a+3 b) \log (1-\cos (c+d x))}{16 d}+\frac{(8 a-3 b) \log (\cos (c+d x)+1)}{16 d}-\frac{\cot ^4(c+d x) (a+b \sec (c+d x))}{4 d}+\frac{\cot ^2(c+d x) (4 a+3 b \sec (c+d x))}{8 d}","\frac{(8 a+3 b) \log (1-\cos (c+d x))}{16 d}+\frac{(8 a-3 b) \log (\cos (c+d x)+1)}{16 d}-\frac{\cot ^4(c+d x) (a+b \sec (c+d x))}{4 d}+\frac{\cot ^2(c+d x) (4 a+3 b \sec (c+d x))}{8 d}",1,"((8*a + 3*b)*Log[1 - Cos[c + d*x]])/(16*d) + ((8*a - 3*b)*Log[1 + Cos[c + d*x]])/(16*d) - (Cot[c + d*x]^4*(a + b*Sec[c + d*x]))/(4*d) + (Cot[c + d*x]^2*(4*a + 3*b*Sec[c + d*x]))/(8*d)","A",7,5,19,0.2632,1,"{3882, 3883, 2668, 633, 31}"
263,1,130,0,0.1759708,"\int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx","Int[Cot[c + d*x]^7*(a + b*Sec[c + d*x]),x]","-\frac{(16 a+5 b) \log (1-\cos (c+d x))}{32 d}-\frac{(16 a-5 b) \log (\cos (c+d x)+1)}{32 d}-\frac{\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac{\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac{\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d}","-\frac{(16 a+5 b) \log (1-\cos (c+d x))}{32 d}-\frac{(16 a-5 b) \log (\cos (c+d x)+1)}{32 d}-\frac{\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac{\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac{\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d}",1,"-((16*a + 5*b)*Log[1 - Cos[c + d*x]])/(32*d) - ((16*a - 5*b)*Log[1 + Cos[c + d*x]])/(32*d) - (Cot[c + d*x]^6*(a + b*Sec[c + d*x]))/(6*d) + (Cot[c + d*x]^4*(6*a + 5*b*Sec[c + d*x]))/(24*d) - (Cot[c + d*x]^2*(8*a + 5*b*Sec[c + d*x]))/(16*d)","A",8,5,19,0.2632,1,"{3882, 3883, 2668, 633, 31}"
264,1,102,0,0.0953595,"\int (a+b \sec (c+d x)) \tan ^6(c+d x) \, dx","Int[(a + b*Sec[c + d*x])*Tan[c + d*x]^6,x]","\frac{\tan ^5(c+d x) (6 a+5 b \sec (c+d x))}{30 d}-\frac{\tan ^3(c+d x) (8 a+5 b \sec (c+d x))}{24 d}+\frac{\tan (c+d x) (16 a+5 b \sec (c+d x))}{16 d}-a x-\frac{5 b \tanh ^{-1}(\sin (c+d x))}{16 d}","\frac{\tan ^5(c+d x) (6 a+5 b \sec (c+d x))}{30 d}-\frac{\tan ^3(c+d x) (8 a+5 b \sec (c+d x))}{24 d}+\frac{\tan (c+d x) (16 a+5 b \sec (c+d x))}{16 d}-a x-\frac{5 b \tanh ^{-1}(\sin (c+d x))}{16 d}",1,"-(a*x) - (5*b*ArcTanh[Sin[c + d*x]])/(16*d) + ((16*a + 5*b*Sec[c + d*x])*Tan[c + d*x])/(16*d) - ((8*a + 5*b*Sec[c + d*x])*Tan[c + d*x]^3)/(24*d) + ((6*a + 5*b*Sec[c + d*x])*Tan[c + d*x]^5)/(30*d)","A",5,2,19,0.1053,1,"{3881, 3770}"
265,1,73,0,0.065274,"\int (a+b \sec (c+d x)) \tan ^4(c+d x) \, dx","Int[(a + b*Sec[c + d*x])*Tan[c + d*x]^4,x]","\frac{\tan ^3(c+d x) (4 a+3 b \sec (c+d x))}{12 d}-\frac{\tan (c+d x) (8 a+3 b \sec (c+d x))}{8 d}+a x+\frac{3 b \tanh ^{-1}(\sin (c+d x))}{8 d}","\frac{\tan ^3(c+d x) (4 a+3 b \sec (c+d x))}{12 d}-\frac{\tan (c+d x) (8 a+3 b \sec (c+d x))}{8 d}+a x+\frac{3 b \tanh ^{-1}(\sin (c+d x))}{8 d}",1,"a*x + (3*b*ArcTanh[Sin[c + d*x]])/(8*d) - ((8*a + 3*b*Sec[c + d*x])*Tan[c + d*x])/(8*d) + ((4*a + 3*b*Sec[c + d*x])*Tan[c + d*x]^3)/(12*d)","A",4,2,19,0.1053,1,"{3881, 3770}"
266,1,45,0,0.0354112,"\int (a+b \sec (c+d x)) \tan ^2(c+d x) \, dx","Int[(a + b*Sec[c + d*x])*Tan[c + d*x]^2,x]","\frac{\tan (c+d x) (2 a+b \sec (c+d x))}{2 d}-a x-\frac{b \tanh ^{-1}(\sin (c+d x))}{2 d}","\frac{\tan (c+d x) (2 a+b \sec (c+d x))}{2 d}-a x-\frac{b \tanh ^{-1}(\sin (c+d x))}{2 d}",1,"-(a*x) - (b*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*a + b*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",3,2,19,0.1053,1,"{3881, 3770}"
267,1,26,0,0.0264657,"\int \cot ^2(c+d x) (a+b \sec (c+d x)) \, dx","Int[Cot[c + d*x]^2*(a + b*Sec[c + d*x]),x]","-\frac{\cot (c+d x) (a+b \sec (c+d x))}{d}-a x","-\frac{\cot (c+d x) (a+b \sec (c+d x))}{d}-a x",1,"-(a*x) - (Cot[c + d*x]*(a + b*Sec[c + d*x]))/d","A",2,2,19,0.1053,1,"{3882, 8}"
268,1,55,0,0.0517472,"\int \cot ^4(c+d x) (a+b \sec (c+d x)) \, dx","Int[Cot[c + d*x]^4*(a + b*Sec[c + d*x]),x]","-\frac{\cot ^3(c+d x) (a+b \sec (c+d x))}{3 d}+\frac{\cot (c+d x) (3 a+2 b \sec (c+d x))}{3 d}+a x","-\frac{\cot ^3(c+d x) (a+b \sec (c+d x))}{3 d}+\frac{\cot (c+d x) (3 a+2 b \sec (c+d x))}{3 d}+a x",1,"a*x - (Cot[c + d*x]^3*(a + b*Sec[c + d*x]))/(3*d) + (Cot[c + d*x]*(3*a + 2*b*Sec[c + d*x]))/(3*d)","A",3,2,19,0.1053,1,"{3882, 8}"
269,1,84,0,0.0811312,"\int \cot ^6(c+d x) (a+b \sec (c+d x)) \, dx","Int[Cot[c + d*x]^6*(a + b*Sec[c + d*x]),x]","-\frac{\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac{\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{15 d}-\frac{\cot (c+d x) (15 a+8 b \sec (c+d x))}{15 d}-a x","-\frac{\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac{\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{15 d}-\frac{\cot (c+d x) (15 a+8 b \sec (c+d x))}{15 d}-a x",1,"-(a*x) - (Cot[c + d*x]^5*(a + b*Sec[c + d*x]))/(5*d) + (Cot[c + d*x]^3*(5*a + 4*b*Sec[c + d*x]))/(15*d) - (Cot[c + d*x]*(15*a + 8*b*Sec[c + d*x]))/(15*d)","A",4,2,19,0.1053,1,"{3882, 8}"
270,1,111,0,0.1114273,"\int \cot ^8(c+d x) (a+b \sec (c+d x)) \, dx","Int[Cot[c + d*x]^8*(a + b*Sec[c + d*x]),x]","-\frac{\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac{\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}-\frac{\cot ^3(c+d x) (35 a+24 b \sec (c+d x))}{105 d}+\frac{\cot (c+d x) (35 a+16 b \sec (c+d x))}{35 d}+a x","-\frac{\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac{\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}-\frac{\cot ^3(c+d x) (35 a+24 b \sec (c+d x))}{105 d}+\frac{\cot (c+d x) (35 a+16 b \sec (c+d x))}{35 d}+a x",1,"a*x - (Cot[c + d*x]^7*(a + b*Sec[c + d*x]))/(7*d) + (Cot[c + d*x]^5*(7*a + 6*b*Sec[c + d*x]))/(35*d) + (Cot[c + d*x]*(35*a + 16*b*Sec[c + d*x]))/(35*d) - (Cot[c + d*x]^3*(35*a + 24*b*Sec[c + d*x]))/(105*d)","A",5,2,19,0.1053,1,"{3882, 8}"
271,1,217,0,0.1311869,"\int (a+b \sec (c+d x))^2 \tan ^9(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^9,x]","\frac{\left(a^2-4 b^2\right) \sec ^8(c+d x)}{8 d}-\frac{\left(2 a^2-3 b^2\right) \sec ^6(c+d x)}{3 d}+\frac{\left(3 a^2-2 b^2\right) \sec ^4(c+d x)}{2 d}-\frac{\left(4 a^2-b^2\right) \sec ^2(c+d x)}{2 d}-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^9(c+d x)}{9 d}-\frac{8 a b \sec ^7(c+d x)}{7 d}+\frac{12 a b \sec ^5(c+d x)}{5 d}-\frac{8 a b \sec ^3(c+d x)}{3 d}+\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \sec ^{10}(c+d x)}{10 d}","\frac{a^2 \sec ^8(c+d x)}{8 d}-\frac{2 a^2 \sec ^6(c+d x)}{3 d}+\frac{3 a^2 \sec ^4(c+d x)}{2 d}-\frac{2 a^2 \sec ^2(c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^9(c+d x)}{9 d}-\frac{8 a b \sec ^7(c+d x)}{7 d}+\frac{12 a b \sec ^5(c+d x)}{5 d}-\frac{8 a b \sec ^3(c+d x)}{3 d}+\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \tan ^{10}(c+d x)}{10 d}",1,"-((a^2*Log[Cos[c + d*x]])/d) + (2*a*b*Sec[c + d*x])/d - ((4*a^2 - b^2)*Sec[c + d*x]^2)/(2*d) - (8*a*b*Sec[c + d*x]^3)/(3*d) + ((3*a^2 - 2*b^2)*Sec[c + d*x]^4)/(2*d) + (12*a*b*Sec[c + d*x]^5)/(5*d) - ((2*a^2 - 3*b^2)*Sec[c + d*x]^6)/(3*d) - (8*a*b*Sec[c + d*x]^7)/(7*d) + ((a^2 - 4*b^2)*Sec[c + d*x]^8)/(8*d) + (2*a*b*Sec[c + d*x]^9)/(9*d) + (b^2*Sec[c + d*x]^10)/(10*d)","A",3,2,21,0.09524,1,"{3885, 948}"
272,1,169,0,0.1107689,"\int (a+b \sec (c+d x))^2 \tan ^7(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^7,x]","\frac{\left(a^2-3 b^2\right) \sec ^6(c+d x)}{6 d}-\frac{3 \left(a^2-b^2\right) \sec ^4(c+d x)}{4 d}+\frac{\left(3 a^2-b^2\right) \sec ^2(c+d x)}{2 d}+\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^7(c+d x)}{7 d}-\frac{6 a b \sec ^5(c+d x)}{5 d}+\frac{2 a b \sec ^3(c+d x)}{d}-\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \sec ^8(c+d x)}{8 d}","\frac{a^2 \sec ^6(c+d x)}{6 d}-\frac{3 a^2 \sec ^4(c+d x)}{4 d}+\frac{3 a^2 \sec ^2(c+d x)}{2 d}+\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^7(c+d x)}{7 d}-\frac{6 a b \sec ^5(c+d x)}{5 d}+\frac{2 a b \sec ^3(c+d x)}{d}-\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \tan ^8(c+d x)}{8 d}",1,"(a^2*Log[Cos[c + d*x]])/d - (2*a*b*Sec[c + d*x])/d + ((3*a^2 - b^2)*Sec[c + d*x]^2)/(2*d) + (2*a*b*Sec[c + d*x]^3)/d - (3*(a^2 - b^2)*Sec[c + d*x]^4)/(4*d) - (6*a*b*Sec[c + d*x]^5)/(5*d) + ((a^2 - 3*b^2)*Sec[c + d*x]^6)/(6*d) + (2*a*b*Sec[c + d*x]^7)/(7*d) + (b^2*Sec[c + d*x]^8)/(8*d)","A",3,2,21,0.09524,1,"{3885, 948}"
273,1,131,0,0.091351,"\int (a+b \sec (c+d x))^2 \tan ^5(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^5,x]","\frac{\left(a^2-2 b^2\right) \sec ^4(c+d x)}{4 d}-\frac{\left(2 a^2-b^2\right) \sec ^2(c+d x)}{2 d}-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^5(c+d x)}{5 d}-\frac{4 a b \sec ^3(c+d x)}{3 d}+\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \sec ^6(c+d x)}{6 d}","\frac{a^2 \sec ^4(c+d x)}{4 d}-\frac{a^2 \sec ^2(c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^5(c+d x)}{5 d}-\frac{4 a b \sec ^3(c+d x)}{3 d}+\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \tan ^6(c+d x)}{6 d}",1,"-((a^2*Log[Cos[c + d*x]])/d) + (2*a*b*Sec[c + d*x])/d - ((2*a^2 - b^2)*Sec[c + d*x]^2)/(2*d) - (4*a*b*Sec[c + d*x]^3)/(3*d) + ((a^2 - 2*b^2)*Sec[c + d*x]^4)/(4*d) + (2*a*b*Sec[c + d*x]^5)/(5*d) + (b^2*Sec[c + d*x]^6)/(6*d)","A",3,2,21,0.09524,1,"{3885, 948}"
274,1,87,0,0.069152,"\int (a+b \sec (c+d x))^2 \tan ^3(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^3,x]","\frac{\left(a^2-b^2\right) \sec ^2(c+d x)}{2 d}+\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^3(c+d x)}{3 d}-\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \sec ^4(c+d x)}{4 d}","\frac{\left(a^2-b^2\right) \sec ^2(c+d x)}{2 d}+\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^3(c+d x)}{3 d}-\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \sec ^4(c+d x)}{4 d}",1,"(a^2*Log[Cos[c + d*x]])/d - (2*a*b*Sec[c + d*x])/d + ((a^2 - b^2)*Sec[c + d*x]^2)/(2*d) + (2*a*b*Sec[c + d*x]^3)/(3*d) + (b^2*Sec[c + d*x]^4)/(4*d)","A",3,2,21,0.09524,1,"{3885, 894}"
275,1,47,0,0.0336619,"\int (a+b \sec (c+d x))^2 \tan (c+d x) \, dx","Int[(a + b*Sec[c + d*x])^2*Tan[c + d*x],x]","-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \sec ^2(c+d x)}{2 d}","-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \sec ^2(c+d x)}{2 d}",1,"-((a^2*Log[Cos[c + d*x]])/d) + (2*a*b*Sec[c + d*x])/d + (b^2*Sec[c + d*x]^2)/(2*d)","A",3,2,19,0.1053,1,"{3885, 43}"
276,1,61,0,0.0996327,"\int \cot (c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]*(a + b*Sec[c + d*x])^2,x]","\frac{a^2 \log (\cos (c+d x))}{d}+\frac{(a+b)^2 \log (1-\sec (c+d x))}{2 d}+\frac{(a-b)^2 \log (\sec (c+d x)+1)}{2 d}","\frac{a^2 \log (\cos (c+d x))}{d}+\frac{(a+b)^2 \log (1-\sec (c+d x))}{2 d}+\frac{(a-b)^2 \log (\sec (c+d x)+1)}{2 d}",1,"(a^2*Log[Cos[c + d*x]])/d + ((a + b)^2*Log[1 - Sec[c + d*x]])/(2*d) + ((a - b)^2*Log[1 + Sec[c + d*x]])/(2*d)","A",3,2,19,0.1053,1,"{3885, 1802}"
277,1,92,0,0.129499,"\int \cot ^3(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]^3*(a + b*Sec[c + d*x])^2,x]","-\frac{\cot ^2(c+d x) \left(a^2+2 a b \sec (c+d x)+b^2\right)}{2 d}-\frac{a^2 \log (\cos (c+d x))}{d}-\frac{a (a+b) \log (1-\sec (c+d x))}{2 d}-\frac{a (a-b) \log (\sec (c+d x)+1)}{2 d}","-\frac{\cot ^2(c+d x) \left(a^2+2 a b \sec (c+d x)+b^2\right)}{2 d}-\frac{a^2 \log (\cos (c+d x))}{d}-\frac{a (a+b) \log (1-\sec (c+d x))}{2 d}-\frac{a (a-b) \log (\sec (c+d x)+1)}{2 d}",1,"-((a^2*Log[Cos[c + d*x]])/d) - (a*(a + b)*Log[1 - Sec[c + d*x]])/(2*d) - (a*(a - b)*Log[1 + Sec[c + d*x]])/(2*d) - (Cot[c + d*x]^2*(a^2 + b^2 + 2*a*b*Sec[c + d*x]))/(2*d)","A",4,3,21,0.1429,1,"{3885, 1805, 801}"
278,1,126,0,0.1594819,"\int \cot ^5(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]^5*(a + b*Sec[c + d*x])^2,x]","-\frac{\cot ^4(c+d x) \left(a^2+2 a b \sec (c+d x)+b^2\right)}{4 d}+\frac{a^2 \log (\cos (c+d x))}{d}+\frac{a (4 a+3 b) \log (1-\sec (c+d x))}{8 d}+\frac{a (4 a-3 b) \log (\sec (c+d x)+1)}{8 d}+\frac{a \cot ^2(c+d x) (2 a+3 b \sec (c+d x))}{4 d}","-\frac{\cot ^4(c+d x) \left(a^2+2 a b \sec (c+d x)+b^2\right)}{4 d}+\frac{a^2 \log (\cos (c+d x))}{d}+\frac{a (4 a+3 b) \log (1-\sec (c+d x))}{8 d}+\frac{a (4 a-3 b) \log (\sec (c+d x)+1)}{8 d}+\frac{a \cot ^2(c+d x) (2 a+3 b \sec (c+d x))}{4 d}",1,"(a^2*Log[Cos[c + d*x]])/d + (a*(4*a + 3*b)*Log[1 - Sec[c + d*x]])/(8*d) + (a*(4*a - 3*b)*Log[1 + Sec[c + d*x]])/(8*d) + (a*Cot[c + d*x]^2*(2*a + 3*b*Sec[c + d*x]))/(4*d) - (Cot[c + d*x]^4*(a^2 + b^2 + 2*a*b*Sec[c + d*x]))/(4*d)","A",5,4,21,0.1905,1,"{3885, 1805, 823, 801}"
279,1,157,0,0.1977914,"\int (a+b \sec (c+d x))^2 \tan ^6(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^6,x]","\frac{a^2 \tan ^5(c+d x)}{5 d}-\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}-a^2 x-\frac{5 a b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a b \tan ^5(c+d x) \sec (c+d x)}{3 d}-\frac{5 a b \tan ^3(c+d x) \sec (c+d x)}{12 d}+\frac{5 a b \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b^2 \tan ^7(c+d x)}{7 d}","\frac{a^2 \tan ^5(c+d x)}{5 d}-\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}-a^2 x-\frac{5 a b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a b \tan ^5(c+d x) \sec (c+d x)}{3 d}-\frac{5 a b \tan ^3(c+d x) \sec (c+d x)}{12 d}+\frac{5 a b \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b^2 \tan ^7(c+d x)}{7 d}",1,"-(a^2*x) - (5*a*b*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*Tan[c + d*x])/d + (5*a*b*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (a^2*Tan[c + d*x]^3)/(3*d) - (5*a*b*Sec[c + d*x]*Tan[c + d*x]^3)/(12*d) + (a^2*Tan[c + d*x]^5)/(5*d) + (a*b*Sec[c + d*x]*Tan[c + d*x]^5)/(3*d) + (b^2*Tan[c + d*x]^7)/(7*d)","A",12,7,21,0.3333,1,"{3886, 3473, 8, 2611, 3770, 2607, 30}"
280,1,116,0,0.1521661,"\int (a+b \sec (c+d x))^2 \tan ^4(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^4,x]","\frac{a^2 \tan ^3(c+d x)}{3 d}-\frac{a^2 \tan (c+d x)}{d}+a^2 x+\frac{3 a b \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b \tan ^3(c+d x) \sec (c+d x)}{2 d}-\frac{3 a b \tan (c+d x) \sec (c+d x)}{4 d}+\frac{b^2 \tan ^5(c+d x)}{5 d}","\frac{a^2 \tan ^3(c+d x)}{3 d}-\frac{a^2 \tan (c+d x)}{d}+a^2 x+\frac{3 a b \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b \tan ^3(c+d x) \sec (c+d x)}{2 d}-\frac{3 a b \tan (c+d x) \sec (c+d x)}{4 d}+\frac{b^2 \tan ^5(c+d x)}{5 d}",1,"a^2*x + (3*a*b*ArcTanh[Sin[c + d*x]])/(4*d) - (a^2*Tan[c + d*x])/d - (3*a*b*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a^2*Tan[c + d*x]^3)/(3*d) + (a*b*Sec[c + d*x]*Tan[c + d*x]^3)/(2*d) + (b^2*Tan[c + d*x]^5)/(5*d)","A",10,7,21,0.3333,1,"{3886, 3473, 8, 2611, 3770, 2607, 30}"
281,1,70,0,0.1144411,"\int (a+b \sec (c+d x))^2 \tan ^2(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^2,x]","\frac{a^2 \tan (c+d x)}{d}-a^2 x-\frac{a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b \tan (c+d x) \sec (c+d x)}{d}+\frac{b^2 \tan ^3(c+d x)}{3 d}","\frac{a^2 \tan (c+d x)}{d}-a^2 x-\frac{a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b \tan (c+d x) \sec (c+d x)}{d}+\frac{b^2 \tan ^3(c+d x)}{3 d}",1,"-(a^2*x) - (a*b*ArcTanh[Sin[c + d*x]])/d + (a^2*Tan[c + d*x])/d + (a*b*Sec[c + d*x]*Tan[c + d*x])/d + (b^2*Tan[c + d*x]^3)/(3*d)","A",8,7,21,0.3333,1,"{3886, 3473, 8, 2611, 3770, 2607, 30}"
282,1,48,0,0.0746202,"\int \cot ^2(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*(a + b*Sec[c + d*x])^2,x]","-\frac{a^2 \cot (c+d x)}{d}+a^2 (-x)-\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot (c+d x)}{d}","-\frac{a^2 \cot (c+d x)}{d}+a^2 (-x)-\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot (c+d x)}{d}",1,"-(a^2*x) - (a^2*Cot[c + d*x])/d - (b^2*Cot[c + d*x])/d - (2*a*b*Csc[c + d*x])/d","A",8,5,21,0.2381,1,"{3886, 3473, 8, 2606, 3767}"
283,1,85,0,0.1153511,"\int \cot ^4(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*(a + b*Sec[c + d*x])^2,x]","-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+a^2 x-\frac{2 a b \csc ^3(c+d x)}{3 d}+\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot ^3(c+d x)}{3 d}","-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+a^2 x-\frac{2 a b \csc ^3(c+d x)}{3 d}+\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot ^3(c+d x)}{3 d}",1,"a^2*x + (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) - (b^2*Cot[c + d*x]^3)/(3*d) + (2*a*b*Csc[c + d*x])/d - (2*a*b*Csc[c + d*x]^3)/(3*d)","A",9,6,21,0.2857,1,"{3886, 3473, 8, 2606, 2607, 30}"
284,1,122,0,0.1347438,"\int \cot ^6(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*(a + b*Sec[c + d*x])^2,x]","-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-a^2 x-\frac{2 a b \csc ^5(c+d x)}{5 d}+\frac{4 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot ^5(c+d x)}{5 d}","-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-a^2 x-\frac{2 a b \csc ^5(c+d x)}{5 d}+\frac{4 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot ^5(c+d x)}{5 d}",1,"-(a^2*x) - (a^2*Cot[c + d*x])/d + (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]^5)/(5*d) - (b^2*Cot[c + d*x]^5)/(5*d) - (2*a*b*Csc[c + d*x])/d + (4*a*b*Csc[c + d*x]^3)/(3*d) - (2*a*b*Csc[c + d*x]^5)/(5*d)","A",11,7,21,0.3333,1,"{3886, 3473, 8, 2606, 194, 2607, 30}"
285,1,153,0,0.147713,"\int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cot[c + d*x]^8*(a + b*Sec[c + d*x])^2,x]","-\frac{a^2 \cot ^7(c+d x)}{7 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+a^2 x-\frac{2 a b \csc ^7(c+d x)}{7 d}+\frac{6 a b \csc ^5(c+d x)}{5 d}-\frac{2 a b \csc ^3(c+d x)}{d}+\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot ^7(c+d x)}{7 d}","-\frac{a^2 \cot ^7(c+d x)}{7 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+a^2 x-\frac{2 a b \csc ^7(c+d x)}{7 d}+\frac{6 a b \csc ^5(c+d x)}{5 d}-\frac{2 a b \csc ^3(c+d x)}{d}+\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot ^7(c+d x)}{7 d}",1,"a^2*x + (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) + (a^2*Cot[c + d*x]^5)/(5*d) - (a^2*Cot[c + d*x]^7)/(7*d) - (b^2*Cot[c + d*x]^7)/(7*d) + (2*a*b*Csc[c + d*x])/d - (2*a*b*Csc[c + d*x]^3)/d + (6*a*b*Csc[c + d*x]^5)/(5*d) - (2*a*b*Csc[c + d*x]^7)/(7*d)","A",12,7,21,0.3333,1,"{3886, 3473, 8, 2606, 194, 2607, 30}"
286,1,250,0,0.1968661,"\int \frac{\tan ^9(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Tan[c + d*x]^9/(a + b*Sec[c + d*x]),x]","\frac{\left(a^2-4 b^2\right) \sec ^5(c+d x)}{5 b^3 d}-\frac{a \left(a^2-4 b^2\right) \sec ^4(c+d x)}{4 b^4 d}+\frac{\left(-4 a^2 b^2+a^4+6 b^4\right) \sec ^3(c+d x)}{3 b^5 d}-\frac{a \left(-4 a^2 b^2+a^4+6 b^4\right) \sec ^2(c+d x)}{2 b^6 d}+\frac{\left(-4 a^4 b^2+6 a^2 b^4+a^6-4 b^6\right) \sec (c+d x)}{b^7 d}-\frac{\left(a^2-b^2\right)^4 \log (a+b \sec (c+d x))}{a b^8 d}-\frac{a \sec ^6(c+d x)}{6 b^2 d}-\frac{\log (\cos (c+d x))}{a d}+\frac{\sec ^7(c+d x)}{7 b d}","\frac{\left(a^2-4 b^2\right) \sec ^5(c+d x)}{5 b^3 d}-\frac{a \left(a^2-4 b^2\right) \sec ^4(c+d x)}{4 b^4 d}+\frac{\left(-4 a^2 b^2+a^4+6 b^4\right) \sec ^3(c+d x)}{3 b^5 d}-\frac{a \left(-4 a^2 b^2+a^4+6 b^4\right) \sec ^2(c+d x)}{2 b^6 d}+\frac{\left(-4 a^4 b^2+6 a^2 b^4+a^6-4 b^6\right) \sec (c+d x)}{b^7 d}-\frac{\left(a^2-b^2\right)^4 \log (a+b \sec (c+d x))}{a b^8 d}-\frac{a \sec ^6(c+d x)}{6 b^2 d}-\frac{\log (\cos (c+d x))}{a d}+\frac{\sec ^7(c+d x)}{7 b d}",1,"-(Log[Cos[c + d*x]]/(a*d)) - ((a^2 - b^2)^4*Log[a + b*Sec[c + d*x]])/(a*b^8*d) + ((a^6 - 4*a^4*b^2 + 6*a^2*b^4 - 4*b^6)*Sec[c + d*x])/(b^7*d) - (a*(a^4 - 4*a^2*b^2 + 6*b^4)*Sec[c + d*x]^2)/(2*b^6*d) + ((a^4 - 4*a^2*b^2 + 6*b^4)*Sec[c + d*x]^3)/(3*b^5*d) - (a*(a^2 - 4*b^2)*Sec[c + d*x]^4)/(4*b^4*d) + ((a^2 - 4*b^2)*Sec[c + d*x]^5)/(5*b^3*d) - (a*Sec[c + d*x]^6)/(6*b^2*d) + Sec[c + d*x]^7/(7*b*d)","A",3,2,21,0.09524,1,"{3885, 894}"
287,1,170,0,0.1392068,"\int \frac{\tan ^7(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Tan[c + d*x]^7/(a + b*Sec[c + d*x]),x]","\frac{\left(a^2-3 b^2\right) \sec ^3(c+d x)}{3 b^3 d}-\frac{a \left(a^2-3 b^2\right) \sec ^2(c+d x)}{2 b^4 d}+\frac{\left(-3 a^2 b^2+a^4+3 b^4\right) \sec (c+d x)}{b^5 d}-\frac{\left(a^2-b^2\right)^3 \log (a+b \sec (c+d x))}{a b^6 d}-\frac{a \sec ^4(c+d x)}{4 b^2 d}+\frac{\log (\cos (c+d x))}{a d}+\frac{\sec ^5(c+d x)}{5 b d}","\frac{\left(a^2-3 b^2\right) \sec ^3(c+d x)}{3 b^3 d}-\frac{a \left(a^2-3 b^2\right) \sec ^2(c+d x)}{2 b^4 d}+\frac{\left(-3 a^2 b^2+a^4+3 b^4\right) \sec (c+d x)}{b^5 d}-\frac{\left(a^2-b^2\right)^3 \log (a+b \sec (c+d x))}{a b^6 d}-\frac{a \sec ^4(c+d x)}{4 b^2 d}+\frac{\log (\cos (c+d x))}{a d}+\frac{\sec ^5(c+d x)}{5 b d}",1,"Log[Cos[c + d*x]]/(a*d) - ((a^2 - b^2)^3*Log[a + b*Sec[c + d*x]])/(a*b^6*d) + ((a^4 - 3*a^2*b^2 + 3*b^4)*Sec[c + d*x])/(b^5*d) - (a*(a^2 - 3*b^2)*Sec[c + d*x]^2)/(2*b^4*d) + ((a^2 - 3*b^2)*Sec[c + d*x]^3)/(3*b^3*d) - (a*Sec[c + d*x]^4)/(4*b^2*d) + Sec[c + d*x]^5/(5*b*d)","A",3,2,21,0.09524,1,"{3885, 894}"
288,1,108,0,0.0956663,"\int \frac{\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Tan[c + d*x]^5/(a + b*Sec[c + d*x]),x]","\frac{\left(a^2-2 b^2\right) \sec (c+d x)}{b^3 d}-\frac{\left(a^2-b^2\right)^2 \log (a+b \sec (c+d x))}{a b^4 d}-\frac{a \sec ^2(c+d x)}{2 b^2 d}-\frac{\log (\cos (c+d x))}{a d}+\frac{\sec ^3(c+d x)}{3 b d}","\frac{\left(a^2-2 b^2\right) \sec (c+d x)}{b^3 d}-\frac{\left(a^2-b^2\right)^2 \log (a+b \sec (c+d x))}{a b^4 d}-\frac{a \sec ^2(c+d x)}{2 b^2 d}-\frac{\log (\cos (c+d x))}{a d}+\frac{\sec ^3(c+d x)}{3 b d}",1,"-(Log[Cos[c + d*x]]/(a*d)) - ((a^2 - b^2)^2*Log[a + b*Sec[c + d*x]])/(a*b^4*d) + ((a^2 - 2*b^2)*Sec[c + d*x])/(b^3*d) - (a*Sec[c + d*x]^2)/(2*b^2*d) + Sec[c + d*x]^3/(3*b*d)","A",3,2,21,0.09524,1,"{3885, 894}"
289,1,59,0,0.0710039,"\int \frac{\tan ^3(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Tan[c + d*x]^3/(a + b*Sec[c + d*x]),x]","-\frac{\left(a^2-b^2\right) \log (a+b \sec (c+d x))}{a b^2 d}+\frac{\log (\cos (c+d x))}{a d}+\frac{\sec (c+d x)}{b d}","-\frac{\left(a^2-b^2\right) \log (a+b \sec (c+d x))}{a b^2 d}+\frac{\log (\cos (c+d x))}{a d}+\frac{\sec (c+d x)}{b d}",1,"Log[Cos[c + d*x]]/(a*d) - ((a^2 - b^2)*Log[a + b*Sec[c + d*x]])/(a*b^2*d) + Sec[c + d*x]/(b*d)","A",3,2,21,0.09524,1,"{3885, 894}"
290,1,35,0,0.0315418,"\int \frac{\tan (c+d x)}{a+b \sec (c+d x)} \, dx","Int[Tan[c + d*x]/(a + b*Sec[c + d*x]),x]","-\frac{\log (a+b \sec (c+d x))}{a d}-\frac{\log (\cos (c+d x))}{a d}","-\frac{\log (a+b \sec (c+d x))}{a d}-\frac{\log (\cos (c+d x))}{a d}",1,"-(Log[Cos[c + d*x]]/(a*d)) - Log[a + b*Sec[c + d*x]]/(a*d)","A",4,4,19,0.2105,1,"{3885, 36, 29, 31}"
291,1,94,0,0.1022865,"\int \frac{\cot (c+d x)}{a+b \sec (c+d x)} \, dx","Int[Cot[c + d*x]/(a + b*Sec[c + d*x]),x]","-\frac{b^2 \log (a+b \sec (c+d x))}{a d \left(a^2-b^2\right)}+\frac{\log (1-\sec (c+d x))}{2 d (a+b)}+\frac{\log (\sec (c+d x)+1)}{2 d (a-b)}+\frac{\log (\cos (c+d x))}{a d}","-\frac{b^2 \log (a+b \sec (c+d x))}{a d \left(a^2-b^2\right)}+\frac{\log (1-\sec (c+d x))}{2 d (a+b)}+\frac{\log (\sec (c+d x)+1)}{2 d (a-b)}+\frac{\log (\cos (c+d x))}{a d}",1,"Log[Cos[c + d*x]]/(a*d) + Log[1 - Sec[c + d*x]]/(2*(a + b)*d) + Log[1 + Sec[c + d*x]]/(2*(a - b)*d) - (b^2*Log[a + b*Sec[c + d*x]])/(a*(a^2 - b^2)*d)","A",3,2,19,0.1053,1,"{3885, 894}"
292,1,157,0,0.1810462,"\int \frac{\cot ^3(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Cot[c + d*x]^3/(a + b*Sec[c + d*x]),x]","-\frac{b^4 \log (a+b \sec (c+d x))}{a d \left(a^2-b^2\right)^2}+\frac{1}{4 d (a+b) (1-\sec (c+d x))}+\frac{1}{4 d (a-b) (\sec (c+d x)+1)}-\frac{(2 a+3 b) \log (1-\sec (c+d x))}{4 d (a+b)^2}-\frac{(2 a-3 b) \log (\sec (c+d x)+1)}{4 d (a-b)^2}-\frac{\log (\cos (c+d x))}{a d}","-\frac{b^4 \log (a+b \sec (c+d x))}{a d \left(a^2-b^2\right)^2}+\frac{1}{4 d (a+b) (1-\sec (c+d x))}+\frac{1}{4 d (a-b) (\sec (c+d x)+1)}-\frac{(2 a+3 b) \log (1-\sec (c+d x))}{4 d (a+b)^2}-\frac{(2 a-3 b) \log (\sec (c+d x)+1)}{4 d (a-b)^2}-\frac{\log (\cos (c+d x))}{a d}",1,"-(Log[Cos[c + d*x]]/(a*d)) - ((2*a + 3*b)*Log[1 - Sec[c + d*x]])/(4*(a + b)^2*d) - ((2*a - 3*b)*Log[1 + Sec[c + d*x]])/(4*(a - b)^2*d) - (b^4*Log[a + b*Sec[c + d*x]])/(a*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - Sec[c + d*x])) + 1/(4*(a - b)*d*(1 + Sec[c + d*x]))","A",3,2,21,0.09524,1,"{3885, 894}"
293,1,234,0,0.2946189,"\int \frac{\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Cot[c + d*x]^5/(a + b*Sec[c + d*x]),x]","-\frac{b^6 \log (a+b \sec (c+d x))}{a d \left(a^2-b^2\right)^3}+\frac{\left(8 a^2+21 a b+15 b^2\right) \log (1-\sec (c+d x))}{16 d (a+b)^3}+\frac{\left(8 a^2-21 a b+15 b^2\right) \log (\sec (c+d x)+1)}{16 d (a-b)^3}-\frac{5 a+7 b}{16 d (a+b)^2 (1-\sec (c+d x))}-\frac{5 a-7 b}{16 d (a-b)^2 (\sec (c+d x)+1)}-\frac{1}{16 d (a+b) (1-\sec (c+d x))^2}-\frac{1}{16 d (a-b) (\sec (c+d x)+1)^2}+\frac{\log (\cos (c+d x))}{a d}","-\frac{b^6 \log (a+b \sec (c+d x))}{a d \left(a^2-b^2\right)^3}+\frac{\left(8 a^2+21 a b+15 b^2\right) \log (1-\sec (c+d x))}{16 d (a+b)^3}+\frac{\left(8 a^2-21 a b+15 b^2\right) \log (\sec (c+d x)+1)}{16 d (a-b)^3}-\frac{5 a+7 b}{16 d (a+b)^2 (1-\sec (c+d x))}-\frac{5 a-7 b}{16 d (a-b)^2 (\sec (c+d x)+1)}-\frac{1}{16 d (a+b) (1-\sec (c+d x))^2}-\frac{1}{16 d (a-b) (\sec (c+d x)+1)^2}+\frac{\log (\cos (c+d x))}{a d}",1,"Log[Cos[c + d*x]]/(a*d) + ((8*a^2 + 21*a*b + 15*b^2)*Log[1 - Sec[c + d*x]])/(16*(a + b)^3*d) + ((8*a^2 - 21*a*b + 15*b^2)*Log[1 + Sec[c + d*x]])/(16*(a - b)^3*d) - (b^6*Log[a + b*Sec[c + d*x]])/(a*(a^2 - b^2)^3*d) - 1/(16*(a + b)*d*(1 - Sec[c + d*x])^2) - (5*a + 7*b)/(16*(a + b)^2*d*(1 - Sec[c + d*x])) - 1/(16*(a - b)*d*(1 + Sec[c + d*x])^2) - (5*a - 7*b)/(16*(a - b)^2*d*(1 + Sec[c + d*x]))","A",3,2,21,0.09524,1,"{3885, 894}"
294,1,271,0,0.3727169,"\int \frac{\tan ^6(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Tan[c + d*x]^6/(a + b*Sec[c + d*x]),x]","-\frac{a \left(a^2-3 b^2\right) \tan (c+d x)}{b^4 d}+\frac{\left(a^2-3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}+\frac{\left(-3 a^2 b^2+a^4+3 b^4\right) \tanh ^{-1}(\sin (c+d x))}{b^5 d}+\frac{\left(a^2-3 b^2\right) \tan (c+d x) \sec (c+d x)}{2 b^3 d}-\frac{a \tan ^3(c+d x)}{3 b^2 d}-\frac{a \tan (c+d x)}{b^2 d}-\frac{2 (a-b)^{5/2} (a+b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b^5 d}-\frac{x}{a}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 b d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{4 b d}+\frac{3 \tan (c+d x) \sec (c+d x)}{8 b d}","-\frac{a \left(a^2-2 b^2\right) \tan (c+d x)}{b^4 d}+\frac{\left(-20 a^2 b^2+8 a^4+15 b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 b^5 d}+\frac{\left(4 a^2-7 b^2\right) \tan (c+d x) \sec (c+d x)}{8 b^3 d}-\frac{a \tan ^3(c+d x)}{3 b^2 d}-\frac{2 (a-b)^{5/2} (a+b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b^5 d}-\frac{x}{a}+\frac{\tan ^3(c+d x) \sec (c+d x)}{4 b d}",1,"-(x/a) + (3*ArcTanh[Sin[c + d*x]])/(8*b*d) + ((a^2 - 3*b^2)*ArcTanh[Sin[c + d*x]])/(2*b^3*d) + ((a^4 - 3*a^2*b^2 + 3*b^4)*ArcTanh[Sin[c + d*x]])/(b^5*d) - (2*(a - b)^(5/2)*(a + b)^(5/2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*b^5*d) - (a*Tan[c + d*x])/(b^2*d) - (a*(a^2 - 3*b^2)*Tan[c + d*x])/(b^4*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(8*b*d) + ((a^2 - 3*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*b^3*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(4*b*d) - (a*Tan[c + d*x]^3)/(3*b^2*d)","A",15,8,21,0.3810,1,"{3898, 2897, 2659, 208, 3770, 3767, 8, 3768}"
295,1,126,0,0.3427259,"\int \frac{\tan ^4(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Tan[c + d*x]^4/(a + b*Sec[c + d*x]),x]","\frac{\left(2 a^2-3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}-\frac{a \tan (c+d x)}{b^2 d}-\frac{2 (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b^3 d}+\frac{x}{a}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d}","\frac{\left(2 a^2-3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}-\frac{a \tan (c+d x)}{b^2 d}-\frac{2 (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b^3 d}+\frac{x}{a}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d}",1,"x/a + ((2*a^2 - 3*b^2)*ArcTanh[Sin[c + d*x]])/(2*b^3*d) - (2*(a - b)^(3/2)*(a + b)^(3/2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*b^3*d) - (a*Tan[c + d*x])/(b^2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*b*d)","A",6,6,21,0.2857,1,"{3898, 2893, 3057, 2659, 208, 3770}"
296,1,76,0,0.1845864,"\int \frac{\tan ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Tan[c + d*x]^2/(a + b*Sec[c + d*x]),x]","-\frac{2 \sqrt{a-b} \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b d}-\frac{x}{a}+\frac{\tanh ^{-1}(\sin (c+d x))}{b d}","-\frac{2 \sqrt{a-b} \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b d}-\frac{x}{a}+\frac{\tanh ^{-1}(\sin (c+d x))}{b d}",1,"-(x/a) + ArcTanh[Sin[c + d*x]]/(b*d) - (2*Sqrt[a - b]*Sqrt[a + b]*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*b*d)","A",7,7,21,0.3333,1,"{3894, 4051, 3770, 3919, 3831, 2659, 208}"
297,1,135,0,0.2430038,"\int \frac{\cot ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Cot[c + d*x]^2/(a + b*Sec[c + d*x]),x]","-\frac{a \cot (c+d x)}{d \left(a^2-b^2\right)}+\frac{b \csc (c+d x)}{d \left(a^2-b^2\right)}+\frac{b^2 x}{a \left(a^2-b^2\right)}-\frac{a x}{a^2-b^2}-\frac{2 b^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a d (a-b)^{3/2} (a+b)^{3/2}}","-\frac{a \cot (c+d x)}{d \left(a^2-b^2\right)}+\frac{b \csc (c+d x)}{d \left(a^2-b^2\right)}-\frac{2 b^3 \tanh ^{-1}\left(\frac{\sqrt{a^2-b^2} \tan \left(\frac{1}{2} (c+d x)\right)}{a+b}\right)}{a d \left(a^2-b^2\right)^{3/2}}-\frac{x}{a}",1,"-((a*x)/(a^2 - b^2)) + (b^2*x)/(a*(a^2 - b^2)) - (2*b^3*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*(a - b)^(3/2)*(a + b)^(3/2)*d) - (a*Cot[c + d*x])/((a^2 - b^2)*d) + (b*Csc[c + d*x])/((a^2 - b^2)*d)","A",9,8,21,0.3810,1,"{3898, 2902, 2606, 8, 3473, 2735, 2659, 208}"
298,1,256,0,0.3869047,"\int \frac{\cot ^4(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Cot[c + d*x]^4/(a + b*Sec[c + d*x]),x]","-\frac{a \cot ^3(c+d x)}{3 d \left(a^2-b^2\right)}-\frac{a b^2 \cot (c+d x)}{d \left(a^2-b^2\right)^2}+\frac{a \cot (c+d x)}{d \left(a^2-b^2\right)}+\frac{b \csc ^3(c+d x)}{3 d \left(a^2-b^2\right)}+\frac{b^3 \csc (c+d x)}{d \left(a^2-b^2\right)^2}-\frac{b \csc (c+d x)}{d \left(a^2-b^2\right)}+\frac{b^4 x}{a \left(a^2-b^2\right)^2}-\frac{a b^2 x}{\left(a^2-b^2\right)^2}+\frac{a x}{a^2-b^2}-\frac{2 b^5 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a d (a-b)^{5/2} (a+b)^{5/2}}","-\frac{a \cot ^3(c+d x)}{3 d \left(a^2-b^2\right)}+\frac{a \left(a^2-2 b^2\right) \cot (c+d x)}{d \left(a^2-b^2\right)^2}+\frac{b \csc ^3(c+d x)}{3 d \left(a^2-b^2\right)}-\frac{b \left(a^2-2 b^2\right) \csc (c+d x)}{d \left(a^2-b^2\right)^2}-\frac{2 b^5 \tanh ^{-1}\left(\frac{\sqrt{a^2-b^2} \tan \left(\frac{1}{2} (c+d x)\right)}{a+b}\right)}{a d \left(a^2-b^2\right)^{5/2}}+\frac{x}{a}",1,"-((a*b^2*x)/(a^2 - b^2)^2) + (b^4*x)/(a*(a^2 - b^2)^2) + (a*x)/(a^2 - b^2) - (2*b^5*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*(a - b)^(5/2)*(a + b)^(5/2)*d) - (a*b^2*Cot[c + d*x])/((a^2 - b^2)^2*d) + (a*Cot[c + d*x])/((a^2 - b^2)*d) - (a*Cot[c + d*x]^3)/(3*(a^2 - b^2)*d) + (b^3*Csc[c + d*x])/((a^2 - b^2)^2*d) - (b*Csc[c + d*x])/((a^2 - b^2)*d) + (b*Csc[c + d*x]^3)/(3*(a^2 - b^2)*d)","A",15,8,21,0.3810,1,"{3898, 2902, 2606, 3473, 8, 2735, 2659, 208}"
299,1,255,0,0.2051093,"\int \frac{\tan ^9(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]^9/(a + b*Sec[c + d*x])^2,x]","\frac{\left(3 a^2-4 b^2\right) \sec ^4(c+d x)}{4 b^4 d}-\frac{4 a \left(a^2-2 b^2\right) \sec ^3(c+d x)}{3 b^5 d}+\frac{\left(-12 a^2 b^2+5 a^4+6 b^4\right) \sec ^2(c+d x)}{2 b^6 d}-\frac{2 a \left(-8 a^2 b^2+3 a^4+6 b^4\right) \sec (c+d x)}{b^7 d}+\frac{\left(a^2-b^2\right)^4}{a b^8 d (a+b \sec (c+d x))}+\frac{\left(a^2-b^2\right)^3 \left(7 a^2+b^2\right) \log (a+b \sec (c+d x))}{a^2 b^8 d}-\frac{\log (\cos (c+d x))}{a^2 d}-\frac{2 a \sec ^5(c+d x)}{5 b^3 d}+\frac{\sec ^6(c+d x)}{6 b^2 d}","\frac{\left(3 a^2-4 b^2\right) \sec ^4(c+d x)}{4 b^4 d}-\frac{4 a \left(a^2-2 b^2\right) \sec ^3(c+d x)}{3 b^5 d}+\frac{\left(-12 a^2 b^2+5 a^4+6 b^4\right) \sec ^2(c+d x)}{2 b^6 d}-\frac{2 a \left(-8 a^2 b^2+3 a^4+6 b^4\right) \sec (c+d x)}{b^7 d}+\frac{\left(a^2-b^2\right)^4}{a b^8 d (a+b \sec (c+d x))}+\frac{\left(a^2-b^2\right)^3 \left(7 a^2+b^2\right) \log (a+b \sec (c+d x))}{a^2 b^8 d}-\frac{\log (\cos (c+d x))}{a^2 d}-\frac{2 a \sec ^5(c+d x)}{5 b^3 d}+\frac{\sec ^6(c+d x)}{6 b^2 d}",1,"-(Log[Cos[c + d*x]]/(a^2*d)) + ((a^2 - b^2)^3*(7*a^2 + b^2)*Log[a + b*Sec[c + d*x]])/(a^2*b^8*d) - (2*a*(3*a^4 - 8*a^2*b^2 + 6*b^4)*Sec[c + d*x])/(b^7*d) + ((5*a^4 - 12*a^2*b^2 + 6*b^4)*Sec[c + d*x]^2)/(2*b^6*d) - (4*a*(a^2 - 2*b^2)*Sec[c + d*x]^3)/(3*b^5*d) + ((3*a^2 - 4*b^2)*Sec[c + d*x]^4)/(4*b^4*d) - (2*a*Sec[c + d*x]^5)/(5*b^3*d) + Sec[c + d*x]^6/(6*b^2*d) + (a^2 - b^2)^4/(a*b^8*d*(a + b*Sec[c + d*x]))","A",3,2,21,0.09524,1,"{3885, 894}"
300,1,179,0,0.1456368,"\int \frac{\tan ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]^7/(a + b*Sec[c + d*x])^2,x]","\frac{3 \left(a^2-b^2\right) \sec ^2(c+d x)}{2 b^4 d}-\frac{2 a \left(2 a^2-3 b^2\right) \sec (c+d x)}{b^5 d}+\frac{\left(a^2-b^2\right)^3}{a b^6 d (a+b \sec (c+d x))}+\frac{\left(a^2-b^2\right)^2 \left(5 a^2+b^2\right) \log (a+b \sec (c+d x))}{a^2 b^6 d}+\frac{\log (\cos (c+d x))}{a^2 d}-\frac{2 a \sec ^3(c+d x)}{3 b^3 d}+\frac{\sec ^4(c+d x)}{4 b^2 d}","\frac{3 \left(a^2-b^2\right) \sec ^2(c+d x)}{2 b^4 d}-\frac{2 a \left(2 a^2-3 b^2\right) \sec (c+d x)}{b^5 d}+\frac{\left(a^2-b^2\right)^3}{a b^6 d (a+b \sec (c+d x))}+\frac{\left(a^2-b^2\right)^2 \left(5 a^2+b^2\right) \log (a+b \sec (c+d x))}{a^2 b^6 d}+\frac{\log (\cos (c+d x))}{a^2 d}-\frac{2 a \sec ^3(c+d x)}{3 b^3 d}+\frac{\sec ^4(c+d x)}{4 b^2 d}",1,"Log[Cos[c + d*x]]/(a^2*d) + ((a^2 - b^2)^2*(5*a^2 + b^2)*Log[a + b*Sec[c + d*x]])/(a^2*b^6*d) - (2*a*(2*a^2 - 3*b^2)*Sec[c + d*x])/(b^5*d) + (3*(a^2 - b^2)*Sec[c + d*x]^2)/(2*b^4*d) - (2*a*Sec[c + d*x]^3)/(3*b^3*d) + Sec[c + d*x]^4/(4*b^2*d) + (a^2 - b^2)^3/(a*b^6*d*(a + b*Sec[c + d*x]))","A",3,2,21,0.09524,1,"{3885, 894}"
301,1,121,0,0.1043156,"\int \frac{\tan ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]","\frac{\left(a^2-b^2\right)^2}{a b^4 d (a+b \sec (c+d x))}+\frac{\left(3 a^2+b^2\right) \left(a^2-b^2\right) \log (a+b \sec (c+d x))}{a^2 b^4 d}-\frac{\log (\cos (c+d x))}{a^2 d}-\frac{2 a \sec (c+d x)}{b^3 d}+\frac{\sec ^2(c+d x)}{2 b^2 d}","\frac{\left(a^2-b^2\right)^2}{a b^4 d (a+b \sec (c+d x))}+\frac{\left(3 a^2+b^2\right) \left(a^2-b^2\right) \log (a+b \sec (c+d x))}{a^2 b^4 d}-\frac{\log (\cos (c+d x))}{a^2 d}-\frac{2 a \sec (c+d x)}{b^3 d}+\frac{\sec ^2(c+d x)}{2 b^2 d}",1,"-(Log[Cos[c + d*x]]/(a^2*d)) + ((a^2 - b^2)*(3*a^2 + b^2)*Log[a + b*Sec[c + d*x]])/(a^2*b^4*d) - (2*a*Sec[c + d*x])/(b^3*d) + Sec[c + d*x]^2/(2*b^2*d) + (a^2 - b^2)^2/(a*b^4*d*(a + b*Sec[c + d*x]))","A",3,2,21,0.09524,1,"{3885, 894}"
302,1,74,0,0.0836568,"\int \frac{\tan ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]","\frac{a^2-b^2}{a b^2 d (a+b \sec (c+d x))}+\frac{\left(a^2+b^2\right) \log (a+b \sec (c+d x))}{a^2 b^2 d}+\frac{\log (\cos (c+d x))}{a^2 d}","\frac{a^2-b^2}{a b^2 d (a+b \sec (c+d x))}+\frac{\left(a^2+b^2\right) \log (a+b \sec (c+d x))}{a^2 b^2 d}+\frac{\log (\cos (c+d x))}{a^2 d}",1,"Log[Cos[c + d*x]]/(a^2*d) + ((a^2 + b^2)*Log[a + b*Sec[c + d*x]])/(a^2*b^2*d) + (a^2 - b^2)/(a*b^2*d*(a + b*Sec[c + d*x]))","A",3,2,21,0.09524,1,"{3885, 894}"
303,1,54,0,0.0428775,"\int \frac{\tan (c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]/(a + b*Sec[c + d*x])^2,x]","-\frac{\log (a+b \sec (c+d x))}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d}+\frac{1}{a d (a+b \sec (c+d x))}","-\frac{\log (a+b \sec (c+d x))}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d}+\frac{1}{a d (a+b \sec (c+d x))}",1,"-(Log[Cos[c + d*x]]/(a^2*d)) - Log[a + b*Sec[c + d*x]]/(a^2*d) + 1/(a*d*(a + b*Sec[c + d*x]))","A",3,2,19,0.1053,1,"{3885, 44}"
304,1,138,0,0.1423802,"\int \frac{\cot (c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Cot[c + d*x]/(a + b*Sec[c + d*x])^2,x]","\frac{b^2}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{b^2 \left(3 a^2-b^2\right) \log (a+b \sec (c+d x))}{a^2 d \left(a^2-b^2\right)^2}+\frac{\log (\cos (c+d x))}{a^2 d}+\frac{\log (1-\sec (c+d x))}{2 d (a+b)^2}+\frac{\log (\sec (c+d x)+1)}{2 d (a-b)^2}","\frac{b^2}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{b^2 \left(3 a^2-b^2\right) \log (a+b \sec (c+d x))}{a^2 d \left(a^2-b^2\right)^2}+\frac{\log (\cos (c+d x))}{a^2 d}+\frac{\log (1-\sec (c+d x))}{2 d (a+b)^2}+\frac{\log (\sec (c+d x)+1)}{2 d (a-b)^2}",1,"Log[Cos[c + d*x]]/(a^2*d) + Log[1 - Sec[c + d*x]]/(2*(a + b)^2*d) + Log[1 + Sec[c + d*x]]/(2*(a - b)^2*d) - (b^2*(3*a^2 - b^2)*Log[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)^2*d) + b^2/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",3,2,19,0.1053,1,"{3885, 894}"
305,1,197,0,0.2287707,"\int \frac{\cot ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Cot[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]","\frac{b^4}{a d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{b^4 \left(5 a^2-b^2\right) \log (a+b \sec (c+d x))}{a^2 d \left(a^2-b^2\right)^3}-\frac{\log (\cos (c+d x))}{a^2 d}+\frac{1}{4 d (a+b)^2 (1-\sec (c+d x))}+\frac{1}{4 d (a-b)^2 (\sec (c+d x)+1)}-\frac{(a+2 b) \log (1-\sec (c+d x))}{2 d (a+b)^3}-\frac{(a-2 b) \log (\sec (c+d x)+1)}{2 d (a-b)^3}","\frac{b^4}{a d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{b^4 \left(5 a^2-b^2\right) \log (a+b \sec (c+d x))}{a^2 d \left(a^2-b^2\right)^3}-\frac{\log (\cos (c+d x))}{a^2 d}+\frac{1}{4 d (a+b)^2 (1-\sec (c+d x))}+\frac{1}{4 d (a-b)^2 (\sec (c+d x)+1)}-\frac{(a+2 b) \log (1-\sec (c+d x))}{2 d (a+b)^3}-\frac{(a-2 b) \log (\sec (c+d x)+1)}{2 d (a-b)^3}",1,"-(Log[Cos[c + d*x]]/(a^2*d)) - ((a + 2*b)*Log[1 - Sec[c + d*x]])/(2*(a + b)^3*d) - ((a - 2*b)*Log[1 + Sec[c + d*x]])/(2*(a - b)^3*d) - (b^4*(5*a^2 - b^2)*Log[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)^3*d) + 1/(4*(a + b)^2*d*(1 - Sec[c + d*x])) + 1/(4*(a - b)^2*d*(1 + Sec[c + d*x])) + b^4/(a*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",3,2,21,0.09524,1,"{3885, 894}"
306,1,278,0,0.3713013,"\int \frac{\cot ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Cot[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]","\frac{b^6}{a d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{b^6 \left(7 a^2-b^2\right) \log (a+b \sec (c+d x))}{a^2 d \left(a^2-b^2\right)^4}+\frac{\left(4 a^2+13 a b+12 b^2\right) \log (1-\sec (c+d x))}{8 d (a+b)^4}+\frac{\left(4 a^2-13 a b+12 b^2\right) \log (\sec (c+d x)+1)}{8 d (a-b)^4}+\frac{\log (\cos (c+d x))}{a^2 d}-\frac{5 a+9 b}{16 d (a+b)^3 (1-\sec (c+d x))}-\frac{5 a-9 b}{16 d (a-b)^3 (\sec (c+d x)+1)}-\frac{1}{16 d (a+b)^2 (1-\sec (c+d x))^2}-\frac{1}{16 d (a-b)^2 (\sec (c+d x)+1)^2}","\frac{b^6}{a d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{b^6 \left(7 a^2-b^2\right) \log (a+b \sec (c+d x))}{a^2 d \left(a^2-b^2\right)^4}+\frac{\left(4 a^2+13 a b+12 b^2\right) \log (1-\sec (c+d x))}{8 d (a+b)^4}+\frac{\left(4 a^2-13 a b+12 b^2\right) \log (\sec (c+d x)+1)}{8 d (a-b)^4}+\frac{\log (\cos (c+d x))}{a^2 d}-\frac{5 a+9 b}{16 d (a+b)^3 (1-\sec (c+d x))}-\frac{5 a-9 b}{16 d (a-b)^3 (\sec (c+d x)+1)}-\frac{1}{16 d (a+b)^2 (1-\sec (c+d x))^2}-\frac{1}{16 d (a-b)^2 (\sec (c+d x)+1)^2}",1,"Log[Cos[c + d*x]]/(a^2*d) + ((4*a^2 + 13*a*b + 12*b^2)*Log[1 - Sec[c + d*x]])/(8*(a + b)^4*d) + ((4*a^2 - 13*a*b + 12*b^2)*Log[1 + Sec[c + d*x]])/(8*(a - b)^4*d) - (b^6*(7*a^2 - b^2)*Log[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)^4*d) - 1/(16*(a + b)^2*d*(1 - Sec[c + d*x])^2) - (5*a + 9*b)/(16*(a + b)^3*d*(1 - Sec[c + d*x])) - 1/(16*(a - b)^2*d*(1 + Sec[c + d*x])^2) - (5*a - 9*b)/(16*(a - b)^3*d*(1 + Sec[c + d*x])) + b^6/(a*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",3,2,21,0.09524,1,"{3885, 894}"
307,1,283,0,0.4309172,"\int \frac{\tan ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]^6/(a + b*Sec[c + d*x])^2,x]","\frac{3 \left(a^2-b^2\right) \tan (c+d x)}{b^4 d}-\frac{2 a \left(2 a^2-3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{b^5 d}+\frac{\left(a^2-b^2\right)^2 \sin (c+d x)}{a b^4 d (a \cos (c+d x)+b)}-\frac{2 (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 b^3 d}+\frac{4 (a-b)^{3/2} (a+b)^{3/2} \left(2 a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 b^5 d}-\frac{x}{a^2}-\frac{a \tanh ^{-1}(\sin (c+d x))}{b^3 d}-\frac{a \tan (c+d x) \sec (c+d x)}{b^3 d}+\frac{\tan ^3(c+d x)}{3 b^2 d}+\frac{\tan (c+d x)}{b^2 d}","\frac{\left(3 a^2-2 b^2\right) \tan (c+d x)}{b^4 d}-\frac{a \left(4 a^2-5 b^2\right) \tanh ^{-1}(\sin (c+d x))}{b^5 d}+\frac{\left(a^2-b^2\right)^2 \sin (c+d x)}{a b^4 d (a \cos (c+d x)+b)}+\frac{2 (a-b)^{3/2} (a+b)^{3/2} \left(4 a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 b^5 d}-\frac{x}{a^2}-\frac{a \tan (c+d x) \sec (c+d x)}{b^3 d}+\frac{\tan ^3(c+d x)}{3 b^2 d}",1,"-(x/a^2) - (a*ArcTanh[Sin[c + d*x]])/(b^3*d) - (2*a*(2*a^2 - 3*b^2)*ArcTanh[Sin[c + d*x]])/(b^5*d) - (2*(a - b)^(3/2)*(a + b)^(3/2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*b^3*d) + (4*(a - b)^(3/2)*(a + b)^(3/2)*(2*a^2 + b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*b^5*d) + ((a^2 - b^2)^2*Sin[c + d*x])/(a*b^4*d*(b + a*Cos[c + d*x])) + Tan[c + d*x]/(b^2*d) + (3*(a^2 - b^2)*Tan[c + d*x])/(b^4*d) - (a*Sec[c + d*x]*Tan[c + d*x])/(b^3*d) + Tan[c + d*x]^3/(3*b^2*d)","A",16,10,21,0.4762,1,"{3898, 2897, 2664, 12, 2659, 208, 3770, 3767, 8, 3768}"
308,1,150,0,0.3257313,"\int \frac{\tan ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]^4/(a + b*Sec[c + d*x])^2,x]","\frac{\left(2 a^2-b^2\right) \sin (c+d x)}{a b^2 d (a \cos (c+d x)+b)}+\frac{2 \sqrt{a-b} \sqrt{a+b} \left(2 a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 b^3 d}+\frac{x}{a^2}-\frac{2 a \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac{\tan (c+d x)}{b d (a \cos (c+d x)+b)}","\frac{\left(2 a^2-b^2\right) \sin (c+d x)}{a b^2 d (a \cos (c+d x)+b)}+\frac{2 \sqrt{a-b} \sqrt{a+b} \left(2 a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 b^3 d}+\frac{x}{a^2}-\frac{2 a \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac{\tan (c+d x)}{b d (a \cos (c+d x)+b)}",1,"x/a^2 - (2*a*ArcTanh[Sin[c + d*x]])/(b^3*d) + (2*Sqrt[a - b]*Sqrt[a + b]*(2*a^2 + b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*b^3*d) + ((2*a^2 - b^2)*Sin[c + d*x])/(a*b^2*d*(b + a*Cos[c + d*x])) + Tan[c + d*x]/(b*d*(b + a*Cos[c + d*x]))","A",6,6,21,0.2857,1,"{3898, 2890, 3057, 2659, 208, 3770}"
309,1,85,0,0.1458792,"\int \frac{\tan ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Tan[c + d*x]^2/(a + b*Sec[c + d*x])^2,x]","\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{x}{a^2}+\frac{\tan (c+d x)}{a d (a+b \sec (c+d x))}","\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{x}{a^2}+\frac{\tan (c+d x)}{a d (a+b \sec (c+d x))}",1,"-(x/a^2) + (2*b*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) + Tan[c + d*x]/(a*d*(a + b*Sec[c + d*x]))","A",6,6,21,0.2857,1,"{3894, 4061, 12, 3783, 2659, 208}"
310,1,227,0,0.4112946,"\int \frac{\cot ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Cot[c + d*x]^2/(a + b*Sec[c + d*x])^2,x]","\frac{b^4 \sin (c+d x)}{a d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{2 b^5 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{4 b^3 \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{x}{a^2}-\frac{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}","\frac{b^4 \sin (c+d x)}{a d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{2 b^5 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{4 b^3 \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{x}{a^2}-\frac{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}",1,"-(x/a^2) - (2*b^5*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(5/2)*(a + b)^(5/2)*d) - (4*b^3*(2*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(5/2)*(a + b)^(5/2)*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(2*(a - b)^2*d*(1 + Cos[c + d*x])) + (b^4*Sin[c + d*x])/(a*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",11,7,21,0.3333,1,"{3898, 2897, 2648, 2664, 12, 2659, 208}"
311,1,360,0,0.5670127,"\int \frac{\cot ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Cot[c + d*x]^4/(a + b*Sec[c + d*x])^2,x]","\frac{b^6 \sin (c+d x)}{a d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{2 b^7 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{4 b^5 \left(3 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{x}{a^2}+\frac{(3 a+5 b) \sin (c+d x)}{4 d (a+b)^3 (1-\cos (c+d x))}-\frac{\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)}-\frac{(3 a-5 b) \sin (c+d x)}{4 d (a-b)^3 (\cos (c+d x)+1)}-\frac{\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))^2}+\frac{\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)^2}","\frac{b^6 \sin (c+d x)}{a d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{2 b^7 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{4 b^5 \left(3 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{x}{a^2}+\frac{(3 a+5 b) \sin (c+d x)}{4 d (a+b)^3 (1-\cos (c+d x))}-\frac{\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)}-\frac{(3 a-5 b) \sin (c+d x)}{4 d (a-b)^3 (\cos (c+d x)+1)}-\frac{\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))^2}+\frac{\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)^2}",1,"x/a^2 - (2*b^7*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(7/2)*(a + b)^(7/2)*d) - (4*b^5*(3*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(7/2)*(a + b)^(7/2)*d) - Sin[c + d*x]/(12*(a + b)^2*d*(1 - Cos[c + d*x])^2) - Sin[c + d*x]/(12*(a + b)^2*d*(1 - Cos[c + d*x])) + ((3*a + 5*b)*Sin[c + d*x])/(4*(a + b)^3*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(12*(a - b)^2*d*(1 + Cos[c + d*x])^2) - ((3*a - 5*b)*Sin[c + d*x])/(4*(a - b)^3*d*(1 + Cos[c + d*x])) + Sin[c + d*x]/(12*(a - b)^2*d*(1 + Cos[c + d*x])) + (b^6*Sin[c + d*x])/(a*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x]))","A",15,8,21,0.3810,1,"{3898, 2897, 2650, 2648, 2664, 12, 2659, 208}"
312,1,761,0,1.1747554,"\int \frac{(e \tan (c+d x))^{5/2}}{a+b \sec (c+d x)} \, dx","Int[(e*Tan[c + d*x])^(5/2)/(a + b*Sec[c + d*x]),x]","-\frac{e^{5/2} \left(a^2-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a b^2 d}+\frac{e^{5/2} \left(a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a b^2 d}+\frac{e^{5/2} \left(a^2-b^2\right) \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a b^2 d}-\frac{e^{5/2} \left(a^2-b^2\right) \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a b^2 d}+\frac{a e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} b^2 d}-\frac{a e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} b^2 d}-\frac{a e^{5/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} b^2 d}+\frac{a e^{5/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} b^2 d}+\frac{2 \sqrt{2} e^2 \sqrt{a-b} \sqrt{a+b} \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{a b d \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} e^2 \sqrt{a-b} \sqrt{a+b} \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{a b d \sqrt{\sin (c+d x)}}-\frac{2 e^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{b d \sqrt{\sin (2 c+2 d x)}}+\frac{2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{b d}","-\frac{e^{5/2} \left(a^2-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a b^2 d}+\frac{e^{5/2} \left(a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a b^2 d}+\frac{e^{5/2} \left(a^2-b^2\right) \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a b^2 d}-\frac{e^{5/2} \left(a^2-b^2\right) \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a b^2 d}+\frac{a e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} b^2 d}-\frac{a e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} b^2 d}-\frac{a e^{5/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} b^2 d}+\frac{a e^{5/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} b^2 d}+\frac{2 \sqrt{2} e^2 \sqrt{a-b} \sqrt{a+b} \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{a b d \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} e^2 \sqrt{a-b} \sqrt{a+b} \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{a b d \sqrt{\sin (c+d x)}}-\frac{2 e^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{b d \sqrt{\sin (2 c+2 d x)}}+\frac{2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{b d}",1,"(a*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*b^2*d) - ((a^2 - b^2)*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*b^2*d) - (a*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*b^2*d) + ((a^2 - b^2)*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*b^2*d) - (a*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*b^2*d) + ((a^2 - b^2)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*b^2*d) + (a*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*b^2*d) - ((a^2 - b^2)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*b^2*d) + (2*Sqrt[2]*Sqrt[a - b]*Sqrt[a + b]*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[-(Sqrt[a - b]/Sqrt[a + b]), ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(a*b*d*Sqrt[Sin[c + d*x]]) - (2*Sqrt[2]*Sqrt[a - b]*Sqrt[a + b]*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[Sqrt[a - b]/Sqrt[a + b], ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(a*b*d*Sqrt[Sin[c + d*x]]) - (2*e^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(b*d*Sqrt[Sin[2*c + 2*d*x]]) + (2*e*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(b*d)","A",38,22,25,0.8800,1,"{3891, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639, 3890, 2733, 2730, 2906, 2905, 490, 1213, 537}"
313,1,740,0,1.014978,"\int \frac{(e \tan (c+d x))^{3/2}}{a+b \sec (c+d x)} \, dx","Int[(e*Tan[c + d*x])^(3/2)/(a + b*Sec[c + d*x]),x]","-\frac{e^{3/2} \left(a^2-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a b^2 d}+\frac{e^{3/2} \left(a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a b^2 d}-\frac{e^{3/2} \left(a^2-b^2\right) \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a b^2 d}+\frac{e^{3/2} \left(a^2-b^2\right) \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a b^2 d}-\frac{2 \sqrt{2} e^2 \sqrt{a^2-b^2} \sqrt{\sin (c+d x)} \Pi \left(\frac{b}{a-\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right)}{a b d \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}+\frac{2 \sqrt{2} e^2 \sqrt{a^2-b^2} \sqrt{\sin (c+d x)} \Pi \left(\frac{b}{a+\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right)}{a b d \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}+\frac{a e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} b^2 d}-\frac{a e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} b^2 d}+\frac{a e^{3/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} b^2 d}-\frac{a e^{3/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} b^2 d}+\frac{e^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{b d \sqrt{e \tan (c+d x)}}","-\frac{e^{3/2} \left(a^2-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a b^2 d}+\frac{e^{3/2} \left(a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a b^2 d}-\frac{e^{3/2} \left(a^2-b^2\right) \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a b^2 d}+\frac{e^{3/2} \left(a^2-b^2\right) \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a b^2 d}-\frac{2 \sqrt{2} e^2 \sqrt{a^2-b^2} \sqrt{\sin (c+d x)} \Pi \left(\frac{b}{a-\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right)}{a b d \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}+\frac{2 \sqrt{2} e^2 \sqrt{a^2-b^2} \sqrt{\sin (c+d x)} \Pi \left(\frac{b}{a+\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right)}{a b d \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}+\frac{a e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} b^2 d}-\frac{a e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} b^2 d}+\frac{a e^{3/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} b^2 d}-\frac{a e^{3/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} b^2 d}+\frac{e^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{b d \sqrt{e \tan (c+d x)}}",1,"(a*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*b^2*d) - ((a^2 - b^2)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*b^2*d) - (a*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*b^2*d) + ((a^2 - b^2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*b^2*d) + (a*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*b^2*d) - ((a^2 - b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*b^2*d) - (a*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*b^2*d) + ((a^2 - b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*b^2*d) - (2*Sqrt[2]*Sqrt[a^2 - b^2]*e^2*EllipticPi[b/(a - Sqrt[a^2 - b^2]), ArcSin[Sqrt[-Cos[c + d*x]]/Sqrt[1 + Sin[c + d*x]]], -1]*Sqrt[Sin[c + d*x]])/(a*b*d*Sqrt[-Cos[c + d*x]]*Sqrt[e*Tan[c + d*x]]) + (2*Sqrt[2]*Sqrt[a^2 - b^2]*e^2*EllipticPi[b/(a + Sqrt[a^2 - b^2]), ArcSin[Sqrt[-Cos[c + d*x]]/Sqrt[1 + Sin[c + d*x]]], -1]*Sqrt[Sin[c + d*x]])/(a*b*d*Sqrt[-Cos[c + d*x]]*Sqrt[e*Tan[c + d*x]]) + (e^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(b*d*Sqrt[e*Tan[c + d*x]])","A",35,19,25,0.7600,1,"{3891, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641, 3892, 2733, 2729, 2907, 1213, 537}"
314,1,415,0,0.7049306,"\int \frac{\sqrt{e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx","Int[Sqrt[e*Tan[c + d*x]]/(a + b*Sec[c + d*x]),x]","\frac{2 \sqrt{2} b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{a d \sqrt{a-b} \sqrt{a+b} \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{a d \sqrt{a-b} \sqrt{a+b} \sqrt{\sin (c+d x)}}-\frac{\sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}+\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}","\frac{2 \sqrt{2} b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{a d \sqrt{a-b} \sqrt{a+b} \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{a d \sqrt{a-b} \sqrt{a+b} \sqrt{\sin (c+d x)}}-\frac{\sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}+\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}",1,"-((Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d)) + (Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) + (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) - (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) + (2*Sqrt[2]*b*Sqrt[Cos[c + d*x]]*EllipticPi[-(Sqrt[a - b]/Sqrt[a + b]), ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(a*Sqrt[a - b]*Sqrt[a + b]*d*Sqrt[Sin[c + d*x]]) - (2*Sqrt[2]*b*Sqrt[Cos[c + d*x]]*EllipticPi[Sqrt[a - b]/Sqrt[a + b], ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(a*Sqrt[a - b]*Sqrt[a + b]*d*Sqrt[Sin[c + d*x]])","A",21,16,25,0.6400,1,"{3890, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2733, 2730, 2906, 2905, 490, 1213, 537}"
315,1,422,0,0.5642164,"\int \frac{1}{(a+b \sec (c+d x)) \sqrt{e \tan (c+d x)}} \, dx","Int[1/((a + b*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]),x]","-\frac{2 \sqrt{2} b \sqrt{\sin (c+d x)} \Pi \left(\frac{b}{a-\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right)}{a d \sqrt{a^2-b^2} \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}+\frac{2 \sqrt{2} b \sqrt{\sin (c+d x)} \Pi \left(\frac{b}{a+\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right)}{a d \sqrt{a^2-b^2} \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d \sqrt{e}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d \sqrt{e}}-\frac{\log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d \sqrt{e}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d \sqrt{e}}","-\frac{2 \sqrt{2} b \sqrt{\sin (c+d x)} \Pi \left(\frac{b}{a-\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right)}{a d \sqrt{a^2-b^2} \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}+\frac{2 \sqrt{2} b \sqrt{\sin (c+d x)} \Pi \left(\frac{b}{a+\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right)}{a d \sqrt{a^2-b^2} \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d \sqrt{e}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d \sqrt{e}}-\frac{\log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d \sqrt{e}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d \sqrt{e}}",1,"-(ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*Sqrt[e]) - Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*Sqrt[e]) + Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*Sqrt[e]) - (2*Sqrt[2]*b*EllipticPi[b/(a - Sqrt[a^2 - b^2]), ArcSin[Sqrt[-Cos[c + d*x]]/Sqrt[1 + Sin[c + d*x]]], -1]*Sqrt[Sin[c + d*x]])/(a*Sqrt[a^2 - b^2]*d*Sqrt[-Cos[c + d*x]]*Sqrt[e*Tan[c + d*x]]) + (2*Sqrt[2]*b*EllipticPi[b/(a + Sqrt[a^2 - b^2]), ArcSin[Sqrt[-Cos[c + d*x]]/Sqrt[1 + Sin[c + d*x]]], -1]*Sqrt[Sin[c + d*x]])/(a*Sqrt[a^2 - b^2]*d*Sqrt[-Cos[c + d*x]]*Sqrt[e*Tan[c + d*x]])","A",19,14,25,0.5600,1,"{3892, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2733, 2729, 2907, 1213, 537}"
316,1,863,0,1.2499451,"\int \frac{1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx","Int[1/((a + b*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2)),x]","\frac{2 \sqrt{2} \sqrt{\cos (c+d x)} \Pi \left(-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right) \sqrt{e \tan (c+d x)} b^3}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} \sqrt{\cos (c+d x)} \Pi \left(\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right) \sqrt{e \tan (c+d x)} b^3}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt{\sin (c+d x)}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right) b^2}{\sqrt{2} a \left(a^2-b^2\right) d e^{3/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right) b^2}{\sqrt{2} a \left(a^2-b^2\right) d e^{3/2}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right) b^2}{2 \sqrt{2} a \left(a^2-b^2\right) d e^{3/2}}-\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right) b^2}{2 \sqrt{2} a \left(a^2-b^2\right) d e^{3/2}}-\frac{2 \cos (c+d x) (e \tan (c+d x))^{3/2} b}{\left(a^2-b^2\right) d e^3}+\frac{2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)} b}{\left(a^2-b^2\right) d e^2 \sqrt{\sin (2 c+2 d x)}}+\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} \left(a^2-b^2\right) d e^{3/2}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} \left(a^2-b^2\right) d e^{3/2}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right)}{2 \sqrt{2} \left(a^2-b^2\right) d e^{3/2}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right)}{2 \sqrt{2} \left(a^2-b^2\right) d e^{3/2}}-\frac{2 (a-b \sec (c+d x))}{\left(a^2-b^2\right) d e \sqrt{e \tan (c+d x)}}","\frac{2 \sqrt{2} \sqrt{\cos (c+d x)} \Pi \left(-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right) \sqrt{e \tan (c+d x)} b^3}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} \sqrt{\cos (c+d x)} \Pi \left(\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right) \sqrt{e \tan (c+d x)} b^3}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt{\sin (c+d x)}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right) b^2}{\sqrt{2} a \left(a^2-b^2\right) d e^{3/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right) b^2}{\sqrt{2} a \left(a^2-b^2\right) d e^{3/2}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right) b^2}{2 \sqrt{2} a \left(a^2-b^2\right) d e^{3/2}}-\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right) b^2}{2 \sqrt{2} a \left(a^2-b^2\right) d e^{3/2}}-\frac{2 \cos (c+d x) (e \tan (c+d x))^{3/2} b}{\left(a^2-b^2\right) d e^3}+\frac{2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)} b}{\left(a^2-b^2\right) d e^2 \sqrt{\sin (2 c+2 d x)}}+\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} \left(a^2-b^2\right) d e^{3/2}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} \left(a^2-b^2\right) d e^{3/2}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right)}{2 \sqrt{2} \left(a^2-b^2\right) d e^{3/2}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right)}{2 \sqrt{2} \left(a^2-b^2\right) d e^{3/2}}-\frac{2 (a-b \sec (c+d x))}{\left(a^2-b^2\right) d e \sqrt{e \tan (c+d x)}}",1,"(a*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 - b^2)*d*e^(3/2)) - (b^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*(a^2 - b^2)*d*e^(3/2)) - (a*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 - b^2)*d*e^(3/2)) + (b^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*(a^2 - b^2)*d*e^(3/2)) - (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*(a^2 - b^2)*d*e^(3/2)) + (b^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*(a^2 - b^2)*d*e^(3/2)) + (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*(a^2 - b^2)*d*e^(3/2)) - (b^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*(a^2 - b^2)*d*e^(3/2)) - (2*(a - b*Sec[c + d*x]))/((a^2 - b^2)*d*e*Sqrt[e*Tan[c + d*x]]) + (2*Sqrt[2]*b^3*Sqrt[Cos[c + d*x]]*EllipticPi[-(Sqrt[a - b]/Sqrt[a + b]), ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(a*(a - b)^(3/2)*(a + b)^(3/2)*d*e^2*Sqrt[Sin[c + d*x]]) - (2*Sqrt[2]*b^3*Sqrt[Cos[c + d*x]]*EllipticPi[Sqrt[a - b]/Sqrt[a + b], ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(a*(a - b)^(3/2)*(a + b)^(3/2)*d*e^2*Sqrt[Sin[c + d*x]]) + (2*b*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/((a^2 - b^2)*d*e^2*Sqrt[Sin[2*c + 2*d*x]]) - (2*b*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/((a^2 - b^2)*d*e^3)","A",39,23,25,0.9200,1,"{3893, 3882, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639, 3890, 2733, 2730, 2906, 2905, 490, 1213, 537}"
317,1,836,0,1.0634665,"\int \frac{1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx","Int[1/((a + b*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2)),x]","-\frac{2 \sqrt{2} \Pi \left(\frac{b}{a-\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right) \sqrt{\sin (c+d x)} b^3}{a \left(a^2-b^2\right)^{3/2} d e^2 \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}+\frac{2 \sqrt{2} \Pi \left(\frac{b}{a+\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right) \sqrt{\sin (c+d x)} b^3}{a \left(a^2-b^2\right)^{3/2} d e^2 \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right) b^2}{\sqrt{2} a \left(a^2-b^2\right) d e^{5/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right) b^2}{\sqrt{2} a \left(a^2-b^2\right) d e^{5/2}}-\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right) b^2}{2 \sqrt{2} a \left(a^2-b^2\right) d e^{5/2}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right) b^2}{2 \sqrt{2} a \left(a^2-b^2\right) d e^{5/2}}+\frac{F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sec (c+d x) \sqrt{\sin (2 c+2 d x)} b}{3 \left(a^2-b^2\right) d e^2 \sqrt{e \tan (c+d x)}}+\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} \left(a^2-b^2\right) d e^{5/2}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} \left(a^2-b^2\right) d e^{5/2}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right)}{2 \sqrt{2} \left(a^2-b^2\right) d e^{5/2}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right)}{2 \sqrt{2} \left(a^2-b^2\right) d e^{5/2}}-\frac{2 (a-b \sec (c+d x))}{3 \left(a^2-b^2\right) d e (e \tan (c+d x))^{3/2}}","-\frac{2 \sqrt{2} \Pi \left(\frac{b}{a-\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right) \sqrt{\sin (c+d x)} b^3}{a \left(a^2-b^2\right)^{3/2} d e^2 \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}+\frac{2 \sqrt{2} \Pi \left(\frac{b}{a+\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right) \sqrt{\sin (c+d x)} b^3}{a \left(a^2-b^2\right)^{3/2} d e^2 \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right) b^2}{\sqrt{2} a \left(a^2-b^2\right) d e^{5/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right) b^2}{\sqrt{2} a \left(a^2-b^2\right) d e^{5/2}}-\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right) b^2}{2 \sqrt{2} a \left(a^2-b^2\right) d e^{5/2}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right) b^2}{2 \sqrt{2} a \left(a^2-b^2\right) d e^{5/2}}+\frac{F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sec (c+d x) \sqrt{\sin (2 c+2 d x)} b}{3 \left(a^2-b^2\right) d e^2 \sqrt{e \tan (c+d x)}}+\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} \left(a^2-b^2\right) d e^{5/2}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} \left(a^2-b^2\right) d e^{5/2}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right)}{2 \sqrt{2} \left(a^2-b^2\right) d e^{5/2}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right)}{2 \sqrt{2} \left(a^2-b^2\right) d e^{5/2}}-\frac{2 (a-b \sec (c+d x))}{3 \left(a^2-b^2\right) d e (e \tan (c+d x))^{3/2}}",1,"(a*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 - b^2)*d*e^(5/2)) - (b^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*(a^2 - b^2)*d*e^(5/2)) - (a*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 - b^2)*d*e^(5/2)) + (b^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*(a^2 - b^2)*d*e^(5/2)) + (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*(a^2 - b^2)*d*e^(5/2)) - (b^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*(a^2 - b^2)*d*e^(5/2)) - (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*(a^2 - b^2)*d*e^(5/2)) + (b^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*(a^2 - b^2)*d*e^(5/2)) - (2*(a - b*Sec[c + d*x]))/(3*(a^2 - b^2)*d*e*(e*Tan[c + d*x])^(3/2)) - (2*Sqrt[2]*b^3*EllipticPi[b/(a - Sqrt[a^2 - b^2]), ArcSin[Sqrt[-Cos[c + d*x]]/Sqrt[1 + Sin[c + d*x]]], -1]*Sqrt[Sin[c + d*x]])/(a*(a^2 - b^2)^(3/2)*d*e^2*Sqrt[-Cos[c + d*x]]*Sqrt[e*Tan[c + d*x]]) + (2*Sqrt[2]*b^3*EllipticPi[b/(a + Sqrt[a^2 - b^2]), ArcSin[Sqrt[-Cos[c + d*x]]/Sqrt[1 + Sin[c + d*x]]], -1]*Sqrt[Sin[c + d*x]])/(a*(a^2 - b^2)^(3/2)*d*e^2*Sqrt[-Cos[c + d*x]]*Sqrt[e*Tan[c + d*x]]) + (b*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*(a^2 - b^2)*d*e^2*Sqrt[e*Tan[c + d*x]])","A",36,20,25,0.8000,1,"{3893, 3882, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641, 3892, 2733, 2729, 2907, 1213, 537}"
318,1,169,0,0.1702901,"\int \sqrt{a+b \sec (c+d x)} \tan ^5(c+d x) \, dx","Int[Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x]^5,x]","\frac{2 \left(3 a^2-2 b^2\right) (a+b \sec (c+d x))^{5/2}}{5 b^4 d}-\frac{2 a \left(a^2-2 b^2\right) (a+b \sec (c+d x))^{3/2}}{3 b^4 d}+\frac{2 (a+b \sec (c+d x))^{9/2}}{9 b^4 d}-\frac{6 a (a+b \sec (c+d x))^{7/2}}{7 b^4 d}+\frac{2 \sqrt{a+b \sec (c+d x)}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{d}","\frac{2 \left(3 a^2-2 b^2\right) (a+b \sec (c+d x))^{5/2}}{5 b^4 d}-\frac{2 a \left(a^2-2 b^2\right) (a+b \sec (c+d x))^{3/2}}{3 b^4 d}+\frac{2 (a+b \sec (c+d x))^{9/2}}{9 b^4 d}-\frac{6 a (a+b \sec (c+d x))^{7/2}}{7 b^4 d}+\frac{2 \sqrt{a+b \sec (c+d x)}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{d}",1,"(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/d + (2*Sqrt[a + b*Sec[c + d*x]])/d - (2*a*(a^2 - 2*b^2)*(a + b*Sec[c + d*x])^(3/2))/(3*b^4*d) + (2*(3*a^2 - 2*b^2)*(a + b*Sec[c + d*x])^(5/2))/(5*b^4*d) - (6*a*(a + b*Sec[c + d*x])^(7/2))/(7*b^4*d) + (2*(a + b*Sec[c + d*x])^(9/2))/(9*b^4*d)","A",5,4,23,0.1739,1,"{3885, 898, 1261, 207}"
319,1,100,0,0.1127876,"\int \sqrt{a+b \sec (c+d x)} \tan ^3(c+d x) \, dx","Int[Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x]^3,x]","\frac{2 (a+b \sec (c+d x))^{5/2}}{5 b^2 d}-\frac{2 a (a+b \sec (c+d x))^{3/2}}{3 b^2 d}-\frac{2 \sqrt{a+b \sec (c+d x)}}{d}+\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{d}","\frac{2 (a+b \sec (c+d x))^{5/2}}{5 b^2 d}-\frac{2 a (a+b \sec (c+d x))^{3/2}}{3 b^2 d}-\frac{2 \sqrt{a+b \sec (c+d x)}}{d}+\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{d}",1,"(2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/d - (2*Sqrt[a + b*Sec[c + d*x]])/d - (2*a*(a + b*Sec[c + d*x])^(3/2))/(3*b^2*d) + (2*(a + b*Sec[c + d*x])^(5/2))/(5*b^2*d)","A",5,4,23,0.1739,1,"{3885, 898, 1261, 207}"
320,1,51,0,0.0476281,"\int \sqrt{a+b \sec (c+d x)} \tan (c+d x) \, dx","Int[Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x],x]","\frac{2 \sqrt{a+b \sec (c+d x)}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{d}","\frac{2 \sqrt{a+b \sec (c+d x)}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{d}",1,"(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/d + (2*Sqrt[a + b*Sec[c + d*x]])/d","A",4,4,21,0.1905,1,"{3885, 50, 63, 207}"
321,1,106,0,0.1588673,"\int \cot (c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Int[Cot[c + d*x]*Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{d}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{d}-\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{d}","\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{d}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{d}-\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{d}",1,"(2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/d - (Sqrt[a - b]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/d - (Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/d","A",7,5,21,0.2381,1,"{3885, 898, 1287, 206, 207}"
322,1,215,0,0.2949871,"\int \cot ^3(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Int[Cot[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]],x]","-\frac{\cot ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{d}-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{4 d \sqrt{a-b}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{d \sqrt{a-b}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{d \sqrt{a+b}}+\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{4 d \sqrt{a+b}}","-\frac{\cot ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{d}-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{4 d \sqrt{a-b}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{d \sqrt{a-b}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{d \sqrt{a+b}}+\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{4 d \sqrt{a+b}}",1,"(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/d + (a*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(Sqrt[a - b]*d) - (3*b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(4*Sqrt[a - b]*d) + (a*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/(Sqrt[a + b]*d) + (3*b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/(4*Sqrt[a + b]*d) - (Cot[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]])/(2*d)","A",13,9,23,0.3913,1,"{3885, 898, 1315, 1178, 12, 1093, 206, 1170, 207}"
323,1,344,0,0.3850663,"\int \sqrt{a+b \sec (c+d x)} \tan ^2(c+d x) \, dx","Int[Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x]^2,x]","-\frac{2 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d}+\frac{2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}-\frac{2 \sqrt{a+b} (a+2 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}","-\frac{2 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d}+\frac{2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}-\frac{2 \sqrt{a+b} (a+2 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}",1,"(-2*a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) - (2*Sqrt[a + b]*(a + 2*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",7,7,23,0.3043,1,"{3894, 4057, 4058, 3921, 3784, 3832, 4004}"
324,1,125,0,0.0244018,"\int \sqrt{a+b \sec (c+d x)} \, dx","Int[Sqrt[a + b*Sec[c + d*x]],x]","-\frac{2 \cot (c+d x) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (\sec (c+d x)+1)}{a+b \sec (c+d x)}} (a+b \sec (c+d x)) \Pi \left(\frac{a}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right)|\frac{a-b}{a+b}\right)}{d \sqrt{a+b}}","-\frac{2 \cot (c+d x) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (\sec (c+d x)+1)}{a+b \sec (c+d x)}} (a+b \sec (c+d x)) \Pi \left(\frac{a}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right)|\frac{a-b}{a+b}\right)}{d \sqrt{a+b}}",1,"(-2*Cot[c + d*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sec[c + d*x]]], (a - b)/(a + b)]*Sqrt[-((b*(1 - Sec[c + d*x]))/(a + b*Sec[c + d*x]))]*Sqrt[(b*(1 + Sec[c + d*x]))/(a + b*Sec[c + d*x])]*(a + b*Sec[c + d*x]))/(Sqrt[a + b]*d)","A",1,1,14,0.07143,1,"{3780}"
325,1,246,0,0.2119572,"\int \cot ^2(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Int[Cot[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]],x]","-\frac{\cot (c+d x) \sqrt{a+b \sec (c+d x)}}{d}+\frac{\sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{2 \cot (c+d x) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (\sec (c+d x)+1)}{a+b \sec (c+d x)}} (a+b \sec (c+d x)) \Pi \left(\frac{a}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right)|\frac{a-b}{a+b}\right)}{d \sqrt{a+b}}","-\frac{\cot (c+d x) \sqrt{a+b \sec (c+d x)}}{d}+\frac{\sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{2 \cot (c+d x) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (\sec (c+d x)+1)}{a+b \sec (c+d x)}} (a+b \sec (c+d x)) \Pi \left(\frac{a}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right)|\frac{a-b}{a+b}\right)}{d \sqrt{a+b}}",1,"(Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (Cot[c + d*x]*Sqrt[a + b*Sec[c + d*x]])/d + (2*Cot[c + d*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sec[c + d*x]]], (a - b)/(a + b)]*Sqrt[-((b*(1 - Sec[c + d*x]))/(a + b*Sec[c + d*x]))]*Sqrt[(b*(1 + Sec[c + d*x]))/(a + b*Sec[c + d*x])]*(a + b*Sec[c + d*x]))/(Sqrt[a + b]*d)","A",5,4,23,0.1739,1,"{3896, 3780, 3875, 3832}"
326,1,148,0,0.1433415,"\int \frac{\tan ^5(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Tan[c + d*x]^5/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \left(3 a^2-2 b^2\right) (a+b \sec (c+d x))^{3/2}}{3 b^4 d}-\frac{2 a \left(a^2-2 b^2\right) \sqrt{a+b \sec (c+d x)}}{b^4 d}+\frac{2 (a+b \sec (c+d x))^{7/2}}{7 b^4 d}-\frac{6 a (a+b \sec (c+d x))^{5/2}}{5 b^4 d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}","\frac{2 \left(3 a^2-2 b^2\right) (a+b \sec (c+d x))^{3/2}}{3 b^4 d}-\frac{2 a \left(a^2-2 b^2\right) \sqrt{a+b \sec (c+d x)}}{b^4 d}+\frac{2 (a+b \sec (c+d x))^{7/2}}{7 b^4 d}-\frac{6 a (a+b \sec (c+d x))^{5/2}}{5 b^4 d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"(-2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - (2*a*(a^2 - 2*b^2)*Sqrt[a + b*Sec[c + d*x]])/(b^4*d) + (2*(3*a^2 - 2*b^2)*(a + b*Sec[c + d*x])^(3/2))/(3*b^4*d) - (6*a*(a + b*Sec[c + d*x])^(5/2))/(5*b^4*d) + (2*(a + b*Sec[c + d*x])^(7/2))/(7*b^4*d)","A",5,4,23,0.1739,1,"{3885, 898, 1153, 207}"
327,1,79,0,0.0984735,"\int \frac{\tan ^3(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Tan[c + d*x]^3/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 (a+b \sec (c+d x))^{3/2}}{3 b^2 d}-\frac{2 a \sqrt{a+b \sec (c+d x)}}{b^2 d}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}","\frac{2 (a+b \sec (c+d x))^{3/2}}{3 b^2 d}-\frac{2 a \sqrt{a+b \sec (c+d x)}}{b^2 d}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"(2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - (2*a*Sqrt[a + b*Sec[c + d*x]])/(b^2*d) + (2*(a + b*Sec[c + d*x])^(3/2))/(3*b^2*d)","A",5,4,23,0.1739,1,"{3885, 898, 1153, 207}"
328,1,31,0,0.0426453,"\int \frac{\tan (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Tan[c + d*x]/Sqrt[a + b*Sec[c + d*x]],x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"(-2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d)","A",3,3,21,0.1429,1,"{3885, 63, 207}"
329,1,106,0,0.1340643,"\int \frac{\cot (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Cot[c + d*x]/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{d \sqrt{a-b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{d \sqrt{a+b}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{d \sqrt{a-b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{d \sqrt{a+b}}",1,"(2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]]/(Sqrt[a - b]*d) - ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b]*d)","A",7,5,21,0.2381,1,"{3885, 898, 1170, 206, 207}"
330,1,260,0,0.2633818,"\int \frac{\cot ^3(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Cot[c + d*x]^3/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{a+b \sec (c+d x)}}{4 d (a+b) (1-\sec (c+d x))}+\frac{\sqrt{a+b \sec (c+d x)}}{4 d (a-b) (\sec (c+d x)+1)}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{4 d (a-b)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{d \sqrt{a-b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{d \sqrt{a+b}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{4 d (a+b)^{3/2}}","\frac{\sqrt{a+b \sec (c+d x)}}{4 d (a+b) (1-\sec (c+d x))}+\frac{\sqrt{a+b \sec (c+d x)}}{4 d (a-b) (\sec (c+d x)+1)}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{4 d (a-b)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{d \sqrt{a-b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{d \sqrt{a+b}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{4 d (a+b)^{3/2}}",1,"(-2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]]/(Sqrt[a - b]*d) - (b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(3/2)*d) + (b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(3/2)*d) + ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b]*d) + Sqrt[a + b*Sec[c + d*x]]/(4*(a + b)*d*(1 - Sec[c + d*x])) + Sqrt[a + b*Sec[c + d*x]]/(4*(a - b)*d*(1 + Sec[c + d*x]))","A",11,6,23,0.2609,1,"{3885, 898, 1238, 206, 199, 207}"
331,1,610,0,0.7634767,"\int \frac{\tan ^4(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Tan[c + d*x]^4/Sqrt[a + b*Sec[c + d*x]],x]","-\frac{2 \sqrt{a+b} \left(8 a^2-2 a b+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left(8 a^2+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^4 d}-\frac{8 a \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b^2 d}+\frac{4 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d}+\frac{2 \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b d}+\frac{4 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}","\frac{2 \sqrt{a+b} \left(-8 a^2+2 a b+21 b^2\right) \cot (c+d x) \sqrt{-\frac{b (\sec (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sec (c+d x)+1)}{b-a}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left(8 a^2-21 b^2\right) \cot (c+d x) \sqrt{-\frac{b (\sec (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^4 d}-\frac{8 a \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b^2 d}+\frac{2 \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b d}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"(4*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) - (2*(a - b)*Sqrt[a + b]*(8*a^2 + 9*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*d) + (4*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*Sqrt[a + b]*(8*a^2 - 2*a*b + 9*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) - (8*a*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^2*d) + (2*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b*d)","A",11,8,23,0.3478,1,"{3895, 3784, 3837, 3832, 4004, 3860, 4082, 4005}"
332,1,310,0,0.2476799,"\int \frac{\tan ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Tan[c + d*x]^2/Sqrt[a + b*Sec[c + d*x]],x]","-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}","-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"(-2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)","A",6,6,23,0.2609,1,"{3894, 4059, 3921, 3784, 3832, 4004}"
333,1,106,0,0.0208312,"\int \frac{1}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[1/Sqrt[a + b*Sec[c + d*x]],x]","-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}","-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"(-2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)","A",1,1,14,0.07143,1,"{3784}"
334,1,361,0,0.4204322,"\int \frac{\cot ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Cot[c + d*x]^2/Sqrt[a + b*Sec[c + d*x]],x]","\frac{b^2 \tan (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{\cot (c+d x)}{d \sqrt{a+b \sec (c+d x)}}-\frac{\cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d \sqrt{a+b}}+\frac{\cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d \sqrt{a+b}}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}","\frac{b^2 \tan (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{\cot (c+d x)}{d \sqrt{a+b \sec (c+d x)}}-\frac{\cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d \sqrt{a+b}}+\frac{\cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d \sqrt{a+b}}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"(Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(Sqrt[a + b]*d) - (Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(Sqrt[a + b]*d) + (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) - Cot[c + d*x]/(d*Sqrt[a + b*Sec[c + d*x]]) + (b^2*Tan[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",9,8,23,0.3478,1,"{3896, 3784, 3875, 3833, 21, 3829, 3832, 4004}"
335,1,148,0,0.1715337,"\int \frac{\tan ^5(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Tan[c + d*x]^5/(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 \left(3 a^2-2 b^2\right) \sqrt{a+b \sec (c+d x)}}{b^4 d}+\frac{2 \left(a^2-b^2\right)^2}{a b^4 d \sqrt{a+b \sec (c+d x)}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{2 (a+b \sec (c+d x))^{5/2}}{5 b^4 d}-\frac{2 a (a+b \sec (c+d x))^{3/2}}{b^4 d}","\frac{2 \left(3 a^2-2 b^2\right) \sqrt{a+b \sec (c+d x)}}{b^4 d}+\frac{2 \left(a^2-b^2\right)^2}{a b^4 d \sqrt{a+b \sec (c+d x)}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{2 (a+b \sec (c+d x))^{5/2}}{5 b^4 d}-\frac{2 a (a+b \sec (c+d x))^{3/2}}{b^4 d}",1,"(-2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + (2*(a^2 - b^2)^2)/(a*b^4*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^2 - 2*b^2)*Sqrt[a + b*Sec[c + d*x]])/(b^4*d) - (2*a*(a + b*Sec[c + d*x])^(3/2))/(b^4*d) + (2*(a + b*Sec[c + d*x])^(5/2))/(5*b^4*d)","A",5,4,23,0.1739,1,"{3885, 898, 1261, 206}"
336,1,88,0,0.1238033,"\int \frac{\tan ^3(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Tan[c + d*x]^3/(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 \left(a^2-b^2\right)}{a b^2 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{2 \sqrt{a+b \sec (c+d x)}}{b^2 d}","\frac{2 \left(a^2-b^2\right)}{a b^2 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{2 \sqrt{a+b \sec (c+d x)}}{b^2 d}",1,"(2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + (2*(a^2 - b^2))/(a*b^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*Sqrt[a + b*Sec[c + d*x]])/(b^2*d)","A",5,4,23,0.1739,1,"{3885, 898, 1261, 206}"
337,1,54,0,0.0528239,"\int \frac{\tan (c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Tan[c + d*x]/(a + b*Sec[c + d*x])^(3/2),x]","\frac{2}{a d \sqrt{a+b \sec (c+d x)}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}","\frac{2}{a d \sqrt{a+b \sec (c+d x)}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}",1,"(-2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + 2/(a*d*Sqrt[a + b*Sec[c + d*x]])","A",4,4,21,0.1905,1,"{3885, 51, 63, 207}"
338,1,142,0,0.1969042,"\int \frac{\cot (c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Cot[c + d*x]/(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 b^2}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{d (a-b)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{d (a+b)^{3/2}}","\frac{2 b^2}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{d (a-b)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{d (a+b)^{3/2}}",1,"(2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]]/((a - b)^(3/2)*d) - ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]]/((a + b)^(3/2)*d) + (2*b^2)/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",7,4,21,0.1905,1,"{3885, 898, 1287, 206}"
339,1,316,0,0.3848169,"\int \frac{\cot ^3(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Cot[c + d*x]^3/(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 b^4}{a d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{\sqrt{a+b \sec (c+d x)}}{4 d (a+b)^2 (1-\sec (c+d x))}+\frac{\sqrt{a+b \sec (c+d x)}}{4 d (a-b)^2 (\sec (c+d x)+1)}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{4 d (a-b)^{5/2}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{4 d (a+b)^{5/2}}+\frac{(2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{2 d (a-b)^{5/2}}+\frac{(2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{2 d (a+b)^{5/2}}","\frac{2 b^4}{a d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{\sqrt{a+b \sec (c+d x)}}{4 d (a+b)^2 (1-\sec (c+d x))}+\frac{\sqrt{a+b \sec (c+d x)}}{4 d (a-b)^2 (\sec (c+d x)+1)}+\frac{(4 a-7 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{4 d (a-b)^{5/2}}+\frac{(4 a+7 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{4 d (a+b)^{5/2}}",1,"(-2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + ((2*a - 3*b)*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(2*(a - b)^(5/2)*d) - (b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(5/2)*d) + (b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(5/2)*d) + ((2*a + 3*b)*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/(2*(a + b)^(5/2)*d) + (2*b^4)/(a*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + Sqrt[a + b*Sec[c + d*x]]/(4*(a + b)^2*d*(1 - Sec[c + d*x])) + Sqrt[a + b*Sec[c + d*x]]/(4*(a - b)^2*d*(1 + Sec[c + d*x]))","A",11,5,23,0.2174,1,"{3885, 898, 1335, 206, 199}"
340,1,907,0,1.2975862,"\int \frac{\tan ^4(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Tan[c + d*x]^4/(a + b*Sec[c + d*x])^(3/2),x]","-\frac{2 \sec (c+d x) \tan (c+d x) a^2}{b \left(a^2-b^2\right) d \sqrt{a+b \sec (c+d x)}}-\frac{4 \tan (c+d x) a}{\left(a^2-b^2\right) d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(8 a^2-5 b^2\right) \cot (c+d x) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} a}{3 b^4 \sqrt{a+b} d}-\frac{4 \cot (c+d x) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} a}{b^2 \sqrt{a+b} d}+\frac{2 \left(4 a^2-b^2\right) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{3 b^2 \left(a^2-b^2\right) d}+\frac{2 (2 a+b) (4 a+b) \cot (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}}}{3 b^3 \sqrt{a+b} d}-\frac{4 \cot (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}}}{b \sqrt{a+b} d}+\frac{2 b^2 \tan (c+d x)}{\left(a^2-b^2\right) d \sqrt{a+b \sec (c+d x)} a}+\frac{2 \cot (c+d x) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}}}{\sqrt{a+b} d a}-\frac{2 \cot (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}}}{\sqrt{a+b} d a}-\frac{2 \sqrt{a+b} \cot (c+d x) \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}}}{d a^2}","-\frac{2 a^2 \tan (c+d x) \sec (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{4 a \tan (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(4 a^2-b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}+\frac{2 b^2 \tan (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 (2 a+b) \left(4 a^2+a b-3 b^2\right) \cot (c+d x) \sqrt{-\frac{b (\sec (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a b^3 d \sqrt{a+b}}+\frac{2 \left(-11 a^2 b^2+8 a^4+3 b^4\right) \cot (c+d x) \sqrt{-\frac{b (\sec (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a b^4 d \sqrt{a+b}}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}",1,"(2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (4*a*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) + (2*a*(8*a^2 - 5*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*d) - (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (4*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*Sqrt[a + b]*d) + (2*(2*a + b)*(4*a + b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) - (4*a*Tan[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*a^2*Sec[c + d*x]*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(4*a^2 - b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d)","A",17,11,23,0.4783,1,"{3895, 3785, 4058, 3921, 3784, 3832, 4004, 3836, 4005, 3845, 4082}"
341,1,344,0,0.4037093,"\int \frac{\tan ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Tan[c + d*x]^2/(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b^2 d}+\frac{2 \tan (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d}","\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b^2 d}+\frac{2 \tan (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d}",1,"(2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b^2*d) + (2*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*Tan[c + d*x])/(a*d*Sqrt[a + b*Sec[c + d*x]])","A",7,7,23,0.3043,1,"{3894, 4061, 4059, 3921, 3784, 3832, 4004}"
342,1,347,0,0.3192091,"\int \frac{1}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(a + b*Sec[c + d*x])^(-3/2),x]","\frac{2 b^2 \tan (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}+\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}","\frac{2 b^2 \tan (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}+\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}",1,"(2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*b^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",6,6,14,0.4286,1,"{3785, 4058, 3921, 3784, 3832, 4004}"
343,1,664,0,0.9533056,"\int \frac{\cot ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Cot[c + d*x]^2/(a + b*Sec[c + d*x])^(3/2),x]","-\frac{2 b^2 \tan (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{4 a b^2 \tan (c+d x)}{d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \tan (c+d x)}{d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}-\frac{\cot (c+d x)}{d (a+b \sec (c+d x))^{3/2}}+\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}-\frac{(3 a-b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d (a-b) (a+b)^{3/2}}-\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}+\frac{4 a \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d (a-b) (a+b)^{3/2}}","\frac{2 b^2 \left(a^2+b^2\right) \tan (c+d x)}{a d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \tan (c+d x)}{d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{\left(a^2-a b+2 b^2\right) \cot (c+d x) \sqrt{-\frac{b (\sec (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d (a-b) (a+b)^{3/2}}+\frac{2 \left(a^2+b^2\right) \cot (c+d x) \sqrt{-\frac{b (\sec (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d (a-b) (a+b)^{3/2}}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}-\frac{\cot (c+d x)}{d (a+b \sec (c+d x))^{3/2}}",1,"(4*a*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/((a - b)*(a + b)^(3/2)*d) - (2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - ((3*a - b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/((a - b)*(a + b)^(3/2)*d) + (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) + (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) - Cot[c + d*x]/(d*(a + b*Sec[c + d*x])^(3/2)) + (b^2*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (4*a*b^2*Tan[c + d*x])/((a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*b^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",14,11,23,0.4783,1,"{3896, 3785, 4058, 3921, 3784, 3832, 4004, 3875, 3833, 4003, 4005}"
344,1,245,0,0.2717271,"\int (a+b \sec (e+f x))^3 (d \tan (e+f x))^n \, dx","Int[(a + b*Sec[e + f*x])^3*(d*Tan[e + f*x])^n,x]","\frac{3 a^2 b \sec (e+f x) \cos ^2(e+f x)^{\frac{n+2}{2}} (d \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{d f (n+1)}+\frac{a^3 (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{d f (n+1)}+\frac{3 a b^2 (d \tan (e+f x))^{n+1}}{d f (n+1)}+\frac{b^3 \sec ^3(e+f x) \cos ^2(e+f x)^{\frac{n+4}{2}} (d \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+4}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{d f (n+1)}","\frac{3 a^2 b \sec (e+f x) \cos ^2(e+f x)^{\frac{n+2}{2}} (d \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{d f (n+1)}+\frac{a^3 (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{d f (n+1)}+\frac{3 a b^2 (d \tan (e+f x))^{n+1}}{d f (n+1)}+\frac{b^3 \sec ^3(e+f x) \cos ^2(e+f x)^{\frac{n+4}{2}} (d \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+4}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{d f (n+1)}",1,"(3*a*b^2*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (a^3*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (3*a^2*b*(Cos[e + f*x]^2)^((2 + n)/2)*Hypergeometric2F1[(1 + n)/2, (2 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (b^3*(Cos[e + f*x]^2)^((4 + n)/2)*Hypergeometric2F1[(1 + n)/2, (4 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*Sec[e + f*x]^3*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n))","A",8,6,23,0.2609,1,"{3886, 3476, 364, 2617, 2607, 32}"
345,1,160,0,0.1820922,"\int (a+b \sec (e+f x))^2 (d \tan (e+f x))^n \, dx","Int[(a + b*Sec[e + f*x])^2*(d*Tan[e + f*x])^n,x]","\frac{a^2 (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{d f (n+1)}+\frac{2 a b \sec (e+f x) \cos ^2(e+f x)^{\frac{n+2}{2}} (d \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{d f (n+1)}+\frac{b^2 (d \tan (e+f x))^{n+1}}{d f (n+1)}","\frac{a^2 (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{d f (n+1)}+\frac{2 a b \sec (e+f x) \cos ^2(e+f x)^{\frac{n+2}{2}} (d \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{d f (n+1)}+\frac{b^2 (d \tan (e+f x))^{n+1}}{d f (n+1)}",1,"(b^2*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (a^2*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (2*a*b*(Cos[e + f*x]^2)^((2 + n)/2)*Hypergeometric2F1[(1 + n)/2, (2 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n))","A",7,6,23,0.2609,1,"{3886, 3476, 364, 2617, 2607, 32}"
346,1,129,0,0.0852123,"\int (a+b \sec (e+f x)) (d \tan (e+f x))^n \, dx","Int[(a + b*Sec[e + f*x])*(d*Tan[e + f*x])^n,x]","\frac{a (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{d f (n+1)}+\frac{b \sec (e+f x) \cos ^2(e+f x)^{\frac{n+2}{2}} (d \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{d f (n+1)}","\frac{a (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{d f (n+1)}+\frac{b \sec (e+f x) \cos ^2(e+f x)^{\frac{n+2}{2}} (d \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{d f (n+1)}",1,"(a*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (b*(Cos[e + f*x]^2)^((2 + n)/2)*Hypergeometric2F1[(1 + n)/2, (2 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n))","A",4,4,21,0.1905,1,"{3884, 3476, 364, 2617}"
347,0,0,0,0.0498594,"\int \frac{(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx","Int[(d*Tan[e + f*x])^n/(a + b*Sec[e + f*x]),x]","\int \frac{(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx","\frac{d \left(-\tan ^2(e+f x)\right)^{\frac{1-n}{2}+\frac{n-1}{2}} (d \tan (e+f x))^{n-1} \left(-\frac{b (1-\sec (e+f x))}{a+b \sec (e+f x)}\right)^{\frac{1-n}{2}} \left(\frac{b (\sec (e+f x)+1)}{a+b \sec (e+f x)}\right)^{\frac{1-n}{2}} F_1\left(1-n;\frac{1-n}{2},\frac{1-n}{2};2-n;\frac{a+b}{a+b \sec (e+f x)},\frac{a-b}{a+b \sec (e+f x)}\right)}{a f (1-n)}-\frac{d \left(-\tan ^2(e+f x)\right)^{\frac{1-n}{2}+\frac{n+1}{2}} (d \tan (e+f x))^{n-1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{a f (n+1)}",1,"Defer[Int][(d*Tan[e + f*x])^n/(a + b*Sec[e + f*x]), x]","F",0,0,0,0,-1,"{}"
348,0,0,0,0.068786,"\int (a+b \sec (c+d x))^{3/2} (e \tan (c+d x))^m \, dx","Int[(a + b*Sec[c + d*x])^(3/2)*(e*Tan[c + d*x])^m,x]","\int (a+b \sec (c+d x))^{3/2} (e \tan (c+d x))^m \, dx","\text{Int}\left((a+b \sec (c+d x))^{3/2} (e \tan (c+d x))^m,x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(3/2)*(e*Tan[c + d*x])^m, x]","A",0,0,0,0,-1,"{}"
349,0,0,0,0.0610669,"\int \sqrt{a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx","Int[Sqrt[a + b*Sec[c + d*x]]*(e*Tan[c + d*x])^m,x]","\int \sqrt{a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx","\text{Int}\left(\sqrt{a+b \sec (c+d x)} (e \tan (c+d x))^m,x\right)",0,"Defer[Int][Sqrt[a + b*Sec[c + d*x]]*(e*Tan[c + d*x])^m, x]","A",0,0,0,0,-1,"{}"
350,0,0,0,0.0624351,"\int \frac{(e \tan (c+d x))^m}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(e*Tan[c + d*x])^m/Sqrt[a + b*Sec[c + d*x]],x]","\int \frac{(e \tan (c+d x))^m}{\sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{(e \tan (c+d x))^m}{\sqrt{a+b \sec (c+d x)}},x\right)",0,"Defer[Int][(e*Tan[c + d*x])^m/Sqrt[a + b*Sec[c + d*x]], x]","A",0,0,0,0,-1,"{}"
351,0,0,0,0.0700814,"\int \frac{(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(e*Tan[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2),x]","\int \frac{(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}},x\right)",0,"Defer[Int][(e*Tan[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
352,0,0,0,0.0450369,"\int (a+b \sec (c+d x))^n (e \tan (c+d x))^m \, dx","Int[(a + b*Sec[c + d*x])^n*(e*Tan[c + d*x])^m,x]","\int (a+b \sec (c+d x))^n (e \tan (c+d x))^m \, dx","\text{Int}\left((e \tan (c+d x))^m (a+b \sec (c+d x))^n,x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^n*(e*Tan[c + d*x])^m, x]","A",0,0,0,0,-1,"{}"
353,1,177,0,0.2014188,"\int (a+b \sec (c+d x))^n \tan ^5(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^5,x]","-\frac{a \left(a^2-2 b^2\right) (a+b \sec (c+d x))^{n+1}}{b^4 d (n+1)}+\frac{\left(3 a^2-2 b^2\right) (a+b \sec (c+d x))^{n+2}}{b^4 d (n+2)}-\frac{3 a (a+b \sec (c+d x))^{n+3}}{b^4 d (n+3)}+\frac{(a+b \sec (c+d x))^{n+4}}{b^4 d (n+4)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a d (n+1)}","-\frac{a \left(a^2-2 b^2\right) (a+b \sec (c+d x))^{n+1}}{b^4 d (n+1)}+\frac{\left(3 a^2-2 b^2\right) (a+b \sec (c+d x))^{n+2}}{b^4 d (n+2)}-\frac{3 a (a+b \sec (c+d x))^{n+3}}{b^4 d (n+3)}+\frac{(a+b \sec (c+d x))^{n+4}}{b^4 d (n+4)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a d (n+1)}",1,"-((a*(a^2 - 2*b^2)*(a + b*Sec[c + d*x])^(1 + n))/(b^4*d*(1 + n))) - (Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a*d*(1 + n)) + ((3*a^2 - 2*b^2)*(a + b*Sec[c + d*x])^(2 + n))/(b^4*d*(2 + n)) - (3*a*(a + b*Sec[c + d*x])^(3 + n))/(b^4*d*(3 + n)) + (a + b*Sec[c + d*x])^(4 + n)/(b^4*d*(4 + n))","A",5,4,21,0.1905,1,"{3885, 952, 1620, 65}"
354,1,102,0,0.0923938,"\int (a+b \sec (c+d x))^n \tan ^3(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^3,x]","-\frac{a (a+b \sec (c+d x))^{n+1}}{b^2 d (n+1)}+\frac{(a+b \sec (c+d x))^{n+2}}{b^2 d (n+2)}+\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a d (n+1)}","-\frac{a (a+b \sec (c+d x))^{n+1}}{b^2 d (n+1)}+\frac{(a+b \sec (c+d x))^{n+2}}{b^2 d (n+2)}+\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a d (n+1)}",1,"-((a*(a + b*Sec[c + d*x])^(1 + n))/(b^2*d*(1 + n))) + (Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a*d*(1 + n)) + (a + b*Sec[c + d*x])^(2 + n)/(b^2*d*(2 + n))","A",4,4,21,0.1905,1,"{3885, 952, 80, 65}"
355,1,48,0,0.0371419,"\int (a+b \sec (c+d x))^n \tan (c+d x) \, dx","Int[(a + b*Sec[c + d*x])^n*Tan[c + d*x],x]","-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a d (n+1)}","-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a d (n+1)}",1,"-((Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a*d*(1 + n)))","A",2,2,19,0.1053,1,"{3885, 65}"
356,1,162,0,0.1771707,"\int \cot (c+d x) (a+b \sec (c+d x))^n \, dx","Int[Cot[c + d*x]*(a + b*Sec[c + d*x])^n,x]","-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{2 d (n+1) (a-b)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{2 d (n+1) (a+b)}+\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a d (n+1)}","-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{2 d (n+1) (a-b)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{2 d (n+1) (a+b)}+\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a d (n+1)}",1,"-(Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a - b)]*(a + b*Sec[c + d*x])^(1 + n))/(2*(a - b)*d*(1 + n)) - (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x])^(1 + n))/(2*(a + b)*d*(1 + n)) + (Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a*d*(1 + n))","A",8,5,19,0.2632,1,"{3885, 961, 65, 831, 68}"
357,1,279,0,0.2307588,"\int \cot ^3(c+d x) (a+b \sec (c+d x))^n \, dx","Int[Cot[c + d*x]^3*(a + b*Sec[c + d*x])^n,x]","\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{2 d (n+1) (a-b)}+\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{2 d (n+1) (a+b)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a d (n+1)}-\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{4 d (n+1) (a-b)^2}+\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{4 d (n+1) (a+b)^2}","\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{2 d (n+1) (a-b)}+\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{2 d (n+1) (a+b)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a d (n+1)}-\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{4 d (n+1) (a-b)^2}+\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{4 d (n+1) (a+b)^2}",1,"(Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a - b)]*(a + b*Sec[c + d*x])^(1 + n))/(2*(a - b)*d*(1 + n)) + (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x])^(1 + n))/(2*(a + b)*d*(1 + n)) - (Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a*d*(1 + n)) - (b*Hypergeometric2F1[2, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a - b)]*(a + b*Sec[c + d*x])^(1 + n))/(4*(a - b)^2*d*(1 + n)) + (b*Hypergeometric2F1[2, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x])^(1 + n))/(4*(a + b)^2*d*(1 + n))","A",10,5,21,0.2381,1,"{3885, 961, 68, 65, 831}"
358,0,0,0,0.0392137,"\int (a+b \sec (c+d x))^n \tan ^4(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^4,x]","\int (a+b \sec (c+d x))^n \tan ^4(c+d x) \, dx","\text{Int}\left(\tan ^4(c+d x) (a+b \sec (c+d x))^n,x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^n*Tan[c + d*x]^4, x]","A",0,0,0,0,-1,"{}"
359,0,0,0,0.3296078,"\int (a+b \sec (c+d x))^n \tan ^2(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^2,x]","\int (a+b \sec (c+d x))^n \tan ^2(c+d x) \, dx","-\text{Int}\left((a+b \sec (c+d x))^n,x\right)+\frac{\sqrt{2} (a+b) \tan (c+d x) (a+b \sec (c+d x))^n \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n-1;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1}}-\frac{\sqrt{2} a \tan (c+d x) (a+b \sec (c+d x))^n \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1}}",0,"(Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -1 - n, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^n*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^n) - (Sqrt[2]*a*AppellF1[1/2, 1/2, -n, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^n*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^n) - Defer[Int][(a + b*Sec[c + d*x])^n, x]","A",0,0,0,0,-1,"{}"
360,0,0,0,0.0405327,"\int \cot ^2(c+d x) (a+b \sec (c+d x))^n \, dx","Int[Cot[c + d*x]^2*(a + b*Sec[c + d*x])^n,x]","\int \cot ^2(c+d x) (a+b \sec (c+d x))^n \, dx","\text{Int}\left(\cot ^2(c+d x) (a+b \sec (c+d x))^n,x\right)",0,"Defer[Int][Cot[c + d*x]^2*(a + b*Sec[c + d*x])^n, x]","A",0,0,0,0,-1,"{}"
361,0,0,0,0.0407908,"\int \cot ^4(c+d x) (a+b \sec (c+d x))^n \, dx","Int[Cot[c + d*x]^4*(a + b*Sec[c + d*x])^n,x]","\int \cot ^4(c+d x) (a+b \sec (c+d x))^n \, dx","\text{Int}\left(\cot ^4(c+d x) (a+b \sec (c+d x))^n,x\right)",0,"Defer[Int][Cot[c + d*x]^4*(a + b*Sec[c + d*x])^n, x]","A",0,0,0,0,-1,"{}"
362,0,0,0,0.0459208,"\int (a+b \sec (c+d x))^n \tan ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^(3/2),x]","\int (a+b \sec (c+d x))^n \tan ^{\frac{3}{2}}(c+d x) \, dx","\text{Int}\left(\tan ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^n,x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^n*Tan[c + d*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
363,0,0,0,0.0428503,"\int (a+b \sec (c+d x))^n \sqrt{\tan (c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^n*Sqrt[Tan[c + d*x]],x]","\int (a+b \sec (c+d x))^n \sqrt{\tan (c+d x)} \, dx","\text{Int}\left(\sqrt{\tan (c+d x)} (a+b \sec (c+d x))^n,x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^n*Sqrt[Tan[c + d*x]], x]","A",0,0,0,0,-1,"{}"
364,0,0,0,0.0440214,"\int \frac{(a+b \sec (c+d x))^n}{\sqrt{\tan (c+d x)}} \, dx","Int[(a + b*Sec[c + d*x])^n/Sqrt[Tan[c + d*x]],x]","\int \frac{(a+b \sec (c+d x))^n}{\sqrt{\tan (c+d x)}} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^n}{\sqrt{\tan (c+d x)}},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^n/Sqrt[Tan[c + d*x]], x]","A",0,0,0,0,-1,"{}"
365,0,0,0,0.0463841,"\int \frac{(a+b \sec (c+d x))^n}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^n/Tan[c + d*x]^(3/2),x]","\int \frac{(a+b \sec (c+d x))^n}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^n}{\tan ^{\frac{3}{2}}(c+d x)},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^n/Tan[c + d*x]^(3/2), x]","A",0,0,0,0,-1,"{}"