1,1,83,0,0.0614125,"\int \left(a+b \sec ^2(e+f x)\right) \sin ^7(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^7,x]","-\frac{(3 a-b) \cos ^5(e+f x)}{5 f}+\frac{(a-b) \cos ^3(e+f x)}{f}-\frac{(a-3 b) \cos (e+f x)}{f}+\frac{a \cos ^7(e+f x)}{7 f}+\frac{b \sec (e+f x)}{f}","-\frac{(3 a-b) \cos ^5(e+f x)}{5 f}+\frac{(a-b) \cos ^3(e+f x)}{f}-\frac{(a-3 b) \cos (e+f x)}{f}+\frac{a \cos ^7(e+f x)}{7 f}+\frac{b \sec (e+f x)}{f}",1,"-(((a - 3*b)*Cos[e + f*x])/f) + ((a - b)*Cos[e + f*x]^3)/f - ((3*a - b)*Cos[e + f*x]^5)/(5*f) + (a*Cos[e + f*x]^7)/(7*f) + (b*Sec[e + f*x])/f","A",3,2,21,0.09524,1,"{4133, 448}"
2,1,66,0,0.051205,"\int \left(a+b \sec ^2(e+f x)\right) \sin ^5(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^5,x]","\frac{(2 a-b) \cos ^3(e+f x)}{3 f}-\frac{(a-2 b) \cos (e+f x)}{f}-\frac{a \cos ^5(e+f x)}{5 f}+\frac{b \sec (e+f x)}{f}","\frac{(2 a-b) \cos ^3(e+f x)}{3 f}-\frac{(a-2 b) \cos (e+f x)}{f}-\frac{a \cos ^5(e+f x)}{5 f}+\frac{b \sec (e+f x)}{f}",1,"-(((a - 2*b)*Cos[e + f*x])/f) + ((2*a - b)*Cos[e + f*x]^3)/(3*f) - (a*Cos[e + f*x]^5)/(5*f) + (b*Sec[e + f*x])/f","A",3,2,21,0.09524,1,"{4133, 448}"
3,1,44,0,0.0389122,"\int \left(a+b \sec ^2(e+f x)\right) \sin ^3(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^3,x]","-\frac{(a-b) \cos (e+f x)}{f}+\frac{a \cos ^3(e+f x)}{3 f}+\frac{b \sec (e+f x)}{f}","-\frac{(a-b) \cos (e+f x)}{f}+\frac{a \cos ^3(e+f x)}{3 f}+\frac{b \sec (e+f x)}{f}",1,"-(((a - b)*Cos[e + f*x])/f) + (a*Cos[e + f*x]^3)/(3*f) + (b*Sec[e + f*x])/f","A",3,2,21,0.09524,1,"{4133, 448}"
4,1,24,0,0.0202495,"\int \left(a+b \sec ^2(e+f x)\right) \sin (e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)*Sin[e + f*x],x]","\frac{b \sec (e+f x)}{f}-\frac{a \cos (e+f x)}{f}","\frac{b \sec (e+f x)}{f}-\frac{a \cos (e+f x)}{f}",1,"-((a*Cos[e + f*x])/f) + (b*Sec[e + f*x])/f","A",3,2,19,0.1053,1,"{4133, 14}"
5,1,27,0,0.0308487,"\int \csc (e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Csc[e + f*x]*(a + b*Sec[e + f*x]^2),x]","\frac{b \sec (e+f x)}{f}-\frac{(a+b) \tanh ^{-1}(\cos (e+f x))}{f}","\frac{b \sec (e+f x)}{f}-\frac{(a+b) \tanh ^{-1}(\cos (e+f x))}{f}",1,"-(((a + b)*ArcTanh[Cos[e + f*x]])/f) + (b*Sec[e + f*x])/f","A",3,3,19,0.1579,1,"{4133, 453, 206}"
6,1,53,0,0.0522348,"\int \csc ^3(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2),x]","-\frac{(a+3 b) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{(a+b) \cot (e+f x) \csc (e+f x)}{2 f}+\frac{b \sec (e+f x)}{f}","-\frac{(a+3 b) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{(a+b) \cot (e+f x) \csc (e+f x)}{2 f}+\frac{b \sec (e+f x)}{f}",1,"-((a + 3*b)*ArcTanh[Cos[e + f*x]])/(2*f) - ((a + b)*Cot[e + f*x]*Csc[e + f*x])/(2*f) + (b*Sec[e + f*x])/f","A",4,4,21,0.1905,1,"{4133, 456, 453, 206}"
7,1,81,0,0.074872,"\int \csc ^5(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Csc[e + f*x]^5*(a + b*Sec[e + f*x]^2),x]","-\frac{3 (a+5 b) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{(a+b) \cot (e+f x) \csc ^3(e+f x)}{4 f}-\frac{(3 a+7 b) \cot (e+f x) \csc (e+f x)}{8 f}+\frac{b \sec (e+f x)}{f}","-\frac{3 (a+5 b) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{(a+b) \cot (e+f x) \csc ^3(e+f x)}{4 f}-\frac{(3 a+7 b) \cot (e+f x) \csc (e+f x)}{8 f}+\frac{b \sec (e+f x)}{f}",1,"(-3*(a + 5*b)*ArcTanh[Cos[e + f*x]])/(8*f) - ((3*a + 7*b)*Cot[e + f*x]*Csc[e + f*x])/(8*f) - ((a + b)*Cot[e + f*x]*Csc[e + f*x]^3)/(4*f) + (b*Sec[e + f*x])/f","A",5,4,21,0.1905,1,"{4133, 456, 453, 206}"
8,1,98,0,0.1041634,"\int \left(a+b \sec ^2(e+f x)\right) \sin ^6(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^6,x]","\frac{(13 a-6 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}-\frac{(11 a-18 b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} x (a-6 b)-\frac{a \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{b \tan (e+f x)}{f}","\frac{(13 a-6 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}-\frac{(11 a-18 b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} x (a-6 b)-\frac{a \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{b \tan (e+f x)}{f}",1,"(5*(a - 6*b)*x)/16 - ((11*a - 18*b)*Cos[e + f*x]*Sin[e + f*x])/(16*f) + ((13*a - 6*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) - (a*Cos[e + f*x]^5*Sin[e + f*x])/(6*f) + (b*Tan[e + f*x])/f","A",6,6,21,0.2857,1,"{4132, 455, 1814, 1157, 388, 203}"
9,1,70,0,0.0638165,"\int \left(a+b \sec ^2(e+f x)\right) \sin ^4(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^4,x]","-\frac{(5 a-4 b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} x (a-4 b)+\frac{a \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac{b \tan (e+f x)}{f}","-\frac{(5 a-4 b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} x (a-4 b)+\frac{a \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac{b \tan (e+f x)}{f}",1,"(3*(a - 4*b)*x)/8 - ((5*a - 4*b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a*Cos[e + f*x]^3*Sin[e + f*x])/(4*f) + (b*Tan[e + f*x])/f","A",5,5,21,0.2381,1,"{4132, 455, 1157, 388, 203}"
10,1,42,0,0.0438048,"\int \left(a+b \sec ^2(e+f x)\right) \sin ^2(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^2,x]","\frac{1}{2} x (a-2 b)-\frac{a \sin (e+f x) \cos (e+f x)}{2 f}+\frac{b \tan (e+f x)}{f}","\frac{1}{2} x (a-2 b)-\frac{a \sin (e+f x) \cos (e+f x)}{2 f}+\frac{b \tan (e+f x)}{f}",1,"((a - 2*b)*x)/2 - (a*Cos[e + f*x]*Sin[e + f*x])/(2*f) + (b*Tan[e + f*x])/f","A",4,4,21,0.1905,1,"{4132, 455, 388, 203}"
11,1,15,0,0.01241,"\int \left(a+b \sec ^2(e+f x)\right) \, dx","Int[a + b*Sec[e + f*x]^2,x]","a x+\frac{b \tan (e+f x)}{f}","a x+\frac{b \tan (e+f x)}{f}",1,"a*x + (b*Tan[e + f*x])/f","A",3,2,12,0.1667,1,"{3767, 8}"
12,1,26,0,0.0335316,"\int \csc ^2(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Csc[e + f*x]^2*(a + b*Sec[e + f*x]^2),x]","\frac{b \tan (e+f x)}{f}-\frac{(a+b) \cot (e+f x)}{f}","\frac{b \tan (e+f x)}{f}-\frac{(a+b) \cot (e+f x)}{f}",1,"-(((a + b)*Cot[e + f*x])/f) + (b*Tan[e + f*x])/f","A",3,2,21,0.09524,1,"{4132, 14}"
13,1,46,0,0.0461213,"\int \csc ^4(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Csc[e + f*x]^4*(a + b*Sec[e + f*x]^2),x]","-\frac{(a+b) \cot ^3(e+f x)}{3 f}-\frac{(a+2 b) \cot (e+f x)}{f}+\frac{b \tan (e+f x)}{f}","-\frac{(a+b) \cot ^3(e+f x)}{3 f}-\frac{(a+2 b) \cot (e+f x)}{f}+\frac{b \tan (e+f x)}{f}",1,"-(((a + 2*b)*Cot[e + f*x])/f) - ((a + b)*Cot[e + f*x]^3)/(3*f) + (b*Tan[e + f*x])/f","A",3,2,21,0.09524,1,"{4132, 448}"
14,1,68,0,0.0560181,"\int \csc ^6(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Csc[e + f*x]^6*(a + b*Sec[e + f*x]^2),x]","-\frac{(a+b) \cot ^5(e+f x)}{5 f}-\frac{(2 a+3 b) \cot ^3(e+f x)}{3 f}-\frac{(a+3 b) \cot (e+f x)}{f}+\frac{b \tan (e+f x)}{f}","-\frac{(a+b) \cot ^5(e+f x)}{5 f}-\frac{(2 a+3 b) \cot ^3(e+f x)}{3 f}-\frac{(a+3 b) \cot (e+f x)}{f}+\frac{b \tan (e+f x)}{f}",1,"-(((a + 3*b)*Cot[e + f*x])/f) - ((2*a + 3*b)*Cot[e + f*x]^3)/(3*f) - ((a + b)*Cot[e + f*x]^5)/(5*f) + (b*Tan[e + f*x])/f","A",3,2,21,0.09524,1,"{4132, 448}"
15,1,97,0,0.0897856,"\int \left(a+b \sec ^2(e+f x)\right)^2 \sin ^5(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^5,x]","-\frac{\left(a^2-4 a b+b^2\right) \cos (e+f x)}{f}-\frac{a^2 \cos ^5(e+f x)}{5 f}+\frac{2 a (a-b) \cos ^3(e+f x)}{3 f}+\frac{2 b (a-b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}","-\frac{\left(a^2-4 a b+b^2\right) \cos (e+f x)}{f}-\frac{a^2 \cos ^5(e+f x)}{5 f}+\frac{2 a (a-b) \cos ^3(e+f x)}{3 f}+\frac{2 b (a-b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"-(((a^2 - 4*a*b + b^2)*Cos[e + f*x])/f) + (2*a*(a - b)*Cos[e + f*x]^3)/(3*f) - (a^2*Cos[e + f*x]^5)/(5*f) + (2*(a - b)*b*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)","A",3,2,23,0.08696,1,"{4133, 448}"
16,1,72,0,0.072152,"\int \left(a+b \sec ^2(e+f x)\right)^2 \sin ^3(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^3,x]","\frac{a^2 \cos ^3(e+f x)}{3 f}-\frac{a (a-2 b) \cos (e+f x)}{f}+\frac{b (2 a-b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}","\frac{a^2 \cos ^3(e+f x)}{3 f}-\frac{a (a-2 b) \cos (e+f x)}{f}+\frac{b (2 a-b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"-((a*(a - 2*b)*Cos[e + f*x])/f) + (a^2*Cos[e + f*x]^3)/(3*f) + ((2*a - b)*b*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)","A",3,2,23,0.08696,1,"{4133, 448}"
17,1,46,0,0.0354167,"\int \left(a+b \sec ^2(e+f x)\right)^2 \sin (e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x],x]","-\frac{a^2 \cos (e+f x)}{f}+\frac{2 a b \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}","-\frac{a^2 \cos (e+f x)}{f}+\frac{2 a b \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"-((a^2*Cos[e + f*x])/f) + (2*a*b*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)","A",3,2,21,0.09524,1,"{4133, 270}"
18,1,52,0,0.0657563,"\int \csc (e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Csc[e + f*x]*(a + b*Sec[e + f*x]^2)^2,x]","\frac{b (2 a+b) \sec (e+f x)}{f}-\frac{(a+b)^2 \tanh ^{-1}(\cos (e+f x))}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}","\frac{b (2 a+b) \sec (e+f x)}{f}-\frac{(a+b)^2 \tanh ^{-1}(\cos (e+f x))}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"-(((a + b)^2*ArcTanh[Cos[e + f*x]])/f) + (b*(2*a + b)*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)","A",4,3,21,0.1429,1,"{4133, 461, 207}"
19,1,104,0,0.1101872,"\int \csc ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2,x]","-\frac{\left(3 a^2+6 a b+5 b^2\right) \cot (e+f x) \csc (e+f x)}{6 f}+\frac{b (6 a+5 b) \sec (e+f x)}{3 f}-\frac{(a+b) (a+5 b) \tanh ^{-1}(\cos (e+f x))}{2 f}+\frac{b^2 \csc ^2(e+f x) \sec ^3(e+f x)}{3 f}","-\frac{\left(3 a^2+6 a b+5 b^2\right) \cot (e+f x) \csc (e+f x)}{6 f}+\frac{b (6 a+5 b) \sec (e+f x)}{3 f}-\frac{(a+b) (a+5 b) \tanh ^{-1}(\cos (e+f x))}{2 f}+\frac{b^2 \csc ^2(e+f x) \sec ^3(e+f x)}{3 f}",1,"-((a + b)*(a + 5*b)*ArcTanh[Cos[e + f*x]])/(2*f) - ((3*a^2 + 6*a*b + 5*b^2)*Cot[e + f*x]*Csc[e + f*x])/(6*f) + (b*(6*a + 5*b)*Sec[e + f*x])/(3*f) + (b^2*Csc[e + f*x]^2*Sec[e + f*x]^3)/(3*f)","A",5,5,23,0.2174,1,"{4133, 462, 456, 453, 206}"
20,1,141,0,0.1384973,"\int \csc ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Csc[e + f*x]^5*(a + b*Sec[e + f*x]^2)^2,x]","-\frac{\left(3 a^2+30 a b+35 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{\left(3 a^2+6 a b+7 b^2\right) \cot (e+f x) \csc ^3(e+f x)}{12 f}+\frac{b (6 a+7 b) \sec (e+f x)}{3 f}-\frac{(3 a+7 b)^2 \cot (e+f x) \csc (e+f x)}{24 f}+\frac{b^2 \csc ^4(e+f x) \sec ^3(e+f x)}{3 f}","-\frac{\left(3 a^2+30 a b+35 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{\left(3 a^2+6 a b+7 b^2\right) \cot (e+f x) \csc ^3(e+f x)}{12 f}+\frac{b (6 a+7 b) \sec (e+f x)}{3 f}-\frac{(3 a+7 b)^2 \cot (e+f x) \csc (e+f x)}{24 f}+\frac{b^2 \csc ^4(e+f x) \sec ^3(e+f x)}{3 f}",1,"-((3*a^2 + 30*a*b + 35*b^2)*ArcTanh[Cos[e + f*x]])/(8*f) - ((3*a + 7*b)^2*Cot[e + f*x]*Csc[e + f*x])/(24*f) - ((3*a^2 + 6*a*b + 7*b^2)*Cot[e + f*x]*Csc[e + f*x]^3)/(12*f) + (b*(6*a + 7*b)*Sec[e + f*x])/(3*f) + (b^2*Csc[e + f*x]^4*Sec[e + f*x]^3)/(3*f)","A",6,5,23,0.2174,1,"{4133, 462, 456, 453, 206}"
21,1,148,0,0.1769639,"\int \left(a+b \sec ^2(e+f x)\right)^2 \sin ^6(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^6,x]","-\frac{\left(a^2-12 a b+12 b^2\right) \tan (e+f x)}{6 f}-\frac{\left(3 a^2-36 a b+8 b^2\right) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} x \left(a^2-12 a b+8 b^2\right)+\frac{a^2 \sin ^6(e+f x) \tan (e+f x)}{6 f}+\frac{a (a-12 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{b^2 \tan ^3(e+f x)}{3 f}","-\frac{\left(a^2-12 a b+12 b^2\right) \tan (e+f x)}{6 f}-\frac{\left(3 a^2-36 a b+8 b^2\right) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} x \left(a^2-12 a b+8 b^2\right)+\frac{a^2 \sin ^6(e+f x) \tan (e+f x)}{6 f}+\frac{a (a-12 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"(5*(a^2 - 12*a*b + 8*b^2)*x)/16 - ((3*a^2 - 36*a*b + 8*b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (a*(a - 12*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) - ((a^2 - 12*a*b + 12*b^2)*Tan[e + f*x])/(6*f) + (a^2*Sin[e + f*x]^6*Tan[e + f*x])/(6*f) + (b^2*Tan[e + f*x]^3)/(3*f)","A",7,6,23,0.2609,1,"{4132, 463, 455, 1814, 1153, 203}"
22,1,114,0,0.1207139,"\int \left(a+b \sec ^2(e+f x)\right)^2 \sin ^4(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^4,x]","-\frac{\left(a^2-8 a b+4 b^2\right) \tan (e+f x)}{4 f}+\frac{1}{8} x \left(3 a^2-24 a b+8 b^2\right)+\frac{a^2 \sin ^4(e+f x) \tan (e+f x)}{4 f}-\frac{a (a-8 b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{b^2 \tan ^3(e+f x)}{3 f}","-\frac{\left(a^2-8 a b+4 b^2\right) \tan (e+f x)}{4 f}+\frac{1}{8} x \left(3 a^2-24 a b+8 b^2\right)+\frac{a^2 \sin ^4(e+f x) \tan (e+f x)}{4 f}-\frac{a (a-8 b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"((3*a^2 - 24*a*b + 8*b^2)*x)/8 - (a*(a - 8*b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - ((a^2 - 8*a*b + 4*b^2)*Tan[e + f*x])/(4*f) + (a^2*Sin[e + f*x]^4*Tan[e + f*x])/(4*f) + (b^2*Tan[e + f*x]^3)/(3*f)","A",6,5,23,0.2174,1,"{4132, 463, 455, 1153, 203}"
23,1,73,0,0.0989284,"\int \left(a+b \sec ^2(e+f x)\right)^2 \sin ^2(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^2,x]","\frac{a^2 \sin ^2(e+f x) \tan (e+f x)}{2 f}-\frac{a (a-4 b) \tan (e+f x)}{2 f}+\frac{1}{2} a x (a-4 b)+\frac{b^2 \tan ^3(e+f x)}{3 f}","\frac{a^2 \sin ^2(e+f x) \tan (e+f x)}{2 f}-\frac{a (a-4 b) \tan (e+f x)}{2 f}+\frac{1}{2} a x (a-4 b)+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"(a*(a - 4*b)*x)/2 - (a*(a - 4*b)*Tan[e + f*x])/(2*f) + (a^2*Sin[e + f*x]^2*Tan[e + f*x])/(2*f) + (b^2*Tan[e + f*x]^3)/(3*f)","A",5,5,23,0.2174,1,"{4132, 463, 459, 321, 203}"
24,1,40,0,0.0286617,"\int \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[(a + b*Sec[e + f*x]^2)^2,x]","a^2 x+\frac{b (2 a+b) \tan (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}","a^2 x+\frac{b (2 a+b) \tan (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"a^2*x + (b*(2*a + b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)","A",4,3,14,0.2143,1,"{4128, 390, 203}"
25,1,50,0,0.0583803,"\int \csc ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Csc[e + f*x]^2*(a + b*Sec[e + f*x]^2)^2,x]","\frac{2 b (a+b) \tan (e+f x)}{f}-\frac{(a+b)^2 \cot (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}","\frac{2 b (a+b) \tan (e+f x)}{f}-\frac{(a+b)^2 \cot (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"-(((a + b)^2*Cot[e + f*x])/f) + (2*b*(a + b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)","A",3,2,23,0.08696,1,"{4132, 270}"
26,1,76,0,0.0756403,"\int \csc ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Csc[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2,x]","\frac{b (2 a+3 b) \tan (e+f x)}{f}-\frac{(a+b)^2 \cot ^3(e+f x)}{3 f}-\frac{(a+b) (a+3 b) \cot (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}","\frac{b (2 a+3 b) \tan (e+f x)}{f}-\frac{(a+b)^2 \cot ^3(e+f x)}{3 f}-\frac{(a+b) (a+3 b) \cot (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"-(((a + b)*(a + 3*b)*Cot[e + f*x])/f) - ((a + b)^2*Cot[e + f*x]^3)/(3*f) + (b*(2*a + 3*b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)","A",3,2,23,0.08696,1,"{4132, 448}"
27,1,103,0,0.0968982,"\int \csc ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Csc[e + f*x]^6*(a + b*Sec[e + f*x]^2)^2,x]","-\frac{\left(a^2+6 a b+6 b^2\right) \cot (e+f x)}{f}+\frac{2 b (a+2 b) \tan (e+f x)}{f}-\frac{(a+b)^2 \cot ^5(e+f x)}{5 f}-\frac{2 (a+b) (a+2 b) \cot ^3(e+f x)}{3 f}+\frac{b^2 \tan ^3(e+f x)}{3 f}","-\frac{\left(a^2+6 a b+6 b^2\right) \cot (e+f x)}{f}+\frac{2 b (a+2 b) \tan (e+f x)}{f}-\frac{(a+b)^2 \cot ^5(e+f x)}{5 f}-\frac{2 (a+b) (a+2 b) \cot ^3(e+f x)}{3 f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"-(((a^2 + 6*a*b + 6*b^2)*Cot[e + f*x])/f) - (2*(a + b)*(a + 2*b)*Cot[e + f*x]^3)/(3*f) - ((a + b)^2*Cot[e + f*x]^5)/(5*f) + (2*b*(a + 2*b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)","A",3,2,23,0.08696,1,"{4132, 448}"
28,1,98,0,0.1053064,"\int \frac{\sin ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Sin[e + f*x]^5/(a + b*Sec[e + f*x]^2),x]","\frac{(2 a+b) \cos ^3(e+f x)}{3 a^2 f}-\frac{(a+b)^2 \cos (e+f x)}{a^3 f}+\frac{\sqrt{b} (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{a^{7/2} f}-\frac{\cos ^5(e+f x)}{5 a f}","\frac{(2 a+b) \cos ^3(e+f x)}{3 a^2 f}-\frac{(a+b)^2 \cos (e+f x)}{a^3 f}+\frac{\sqrt{b} (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{a^{7/2} f}-\frac{\cos ^5(e+f x)}{5 a f}",1,"(Sqrt[b]*(a + b)^2*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(a^(7/2)*f) - ((a + b)^2*Cos[e + f*x])/(a^3*f) + ((2*a + b)*Cos[e + f*x]^3)/(3*a^2*f) - Cos[e + f*x]^5/(5*a*f)","A",4,3,23,0.1304,1,"{4133, 461, 205}"
29,1,71,0,0.0840075,"\int \frac{\sin ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2),x]","-\frac{(a+b) \cos (e+f x)}{a^2 f}+\frac{\sqrt{b} (a+b) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{a^{5/2} f}+\frac{\cos ^3(e+f x)}{3 a f}","-\frac{(a+b) \cos (e+f x)}{a^2 f}+\frac{\sqrt{b} (a+b) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{a^{5/2} f}+\frac{\cos ^3(e+f x)}{3 a f}",1,"(Sqrt[b]*(a + b)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(a^(5/2)*f) - ((a + b)*Cos[e + f*x])/(a^2*f) + Cos[e + f*x]^3/(3*a*f)","A",4,4,23,0.1739,1,"{4133, 459, 321, 205}"
30,1,47,0,0.0408325,"\int \frac{\sin (e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Sin[e + f*x]/(a + b*Sec[e + f*x]^2),x]","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{a^{3/2} f}-\frac{\cos (e+f x)}{a f}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{a^{3/2} f}-\frac{\cos (e+f x)}{a f}",1,"(Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(a^(3/2)*f) - Cos[e + f*x]/(a*f)","A",3,3,21,0.1429,1,"{4133, 321, 205}"
31,1,55,0,0.0705629,"\int \frac{\csc (e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Csc[e + f*x]/(a + b*Sec[e + f*x]^2),x]","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{\sqrt{a} f (a+b)}-\frac{\tanh ^{-1}(\cos (e+f x))}{f (a+b)}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{\sqrt{a} f (a+b)}-\frac{\tanh ^{-1}(\cos (e+f x))}{f (a+b)}",1,"(Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(Sqrt[a]*(a + b)*f) - ArcTanh[Cos[e + f*x]]/((a + b)*f)","A",4,4,21,0.1905,1,"{4133, 481, 206, 205}"
32,1,86,0,0.099954,"\int \frac{\csc ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Csc[e + f*x]^3/(a + b*Sec[e + f*x]^2),x]","\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{f (a+b)^2}-\frac{(a-b) \tanh ^{-1}(\cos (e+f x))}{2 f (a+b)^2}-\frac{\cot (e+f x) \csc (e+f x)}{2 f (a+b)}","\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{f (a+b)^2}-\frac{(a-b) \tanh ^{-1}(\cos (e+f x))}{2 f (a+b)^2}-\frac{\cot (e+f x) \csc (e+f x)}{2 f (a+b)}",1,"(Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/((a + b)^2*f) - ((a - b)*ArcTanh[Cos[e + f*x]])/(2*(a + b)^2*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*(a + b)*f)","A",5,5,23,0.2174,1,"{4133, 471, 522, 206, 205}"
33,1,129,0,0.1537261,"\int \frac{\csc ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2),x]","-\frac{\left(3 a^2-6 a b-b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f (a+b)^3}+\frac{a^{3/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{f (a+b)^3}-\frac{\cot (e+f x) \csc ^3(e+f x)}{4 f (a+b)}-\frac{(3 a-b) \cot (e+f x) \csc (e+f x)}{8 f (a+b)^2}","-\frac{\left(3 a^2-6 a b-b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f (a+b)^3}+\frac{a^{3/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{f (a+b)^3}-\frac{\cot (e+f x) \csc ^3(e+f x)}{4 f (a+b)}-\frac{(3 a-b) \cot (e+f x) \csc (e+f x)}{8 f (a+b)^2}",1,"(a^(3/2)*Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/((a + b)^3*f) - ((3*a^2 - 6*a*b - b^2)*ArcTanh[Cos[e + f*x]])/(8*(a + b)^3*f) - ((3*a - b)*Cot[e + f*x]*Csc[e + f*x])/(8*(a + b)^2*f) - (Cot[e + f*x]*Csc[e + f*x]^3)/(4*(a + b)*f)","A",6,6,23,0.2609,1,"{4133, 471, 527, 522, 206, 205}"
34,1,166,0,0.3351821,"\int \frac{\sin ^6(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Sin[e + f*x]^6/(a + b*Sec[e + f*x]^2),x]","-\frac{\left(11 a^2+18 a b+8 b^2\right) \sin (e+f x) \cos (e+f x)}{16 a^3 f}+\frac{x \left(30 a^2 b+5 a^3+40 a b^2+16 b^3\right)}{16 a^4}-\frac{\sqrt{b} (a+b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a^4 f}+\frac{(3 a+2 b) \sin (e+f x) \cos ^3(e+f x)}{8 a^2 f}+\frac{\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f}","-\frac{\left(11 a^2+18 a b+8 b^2\right) \sin (e+f x) \cos (e+f x)}{16 a^3 f}+\frac{x \left(30 a^2 b+5 a^3+40 a b^2+16 b^3\right)}{16 a^4}-\frac{\sqrt{b} (a+b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a^4 f}+\frac{(3 a+2 b) \sin (e+f x) \cos ^3(e+f x)}{8 a^2 f}+\frac{\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f}",1,"((5*a^3 + 30*a^2*b + 40*a*b^2 + 16*b^3)*x)/(16*a^4) - (Sqrt[b]*(a + b)^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a^4*f) - ((11*a^2 + 18*a*b + 8*b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*a^3*f) + ((3*a + 2*b)*Cos[e + f*x]^3*Sin[e + f*x])/(8*a^2*f) + (Cos[e + f*x]^3*Sin[e + f*x]^3)/(6*a*f)","A",7,7,23,0.3043,1,"{4132, 470, 578, 527, 522, 203, 205}"
35,1,117,0,0.1689493,"\int \frac{\sin ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Sin[e + f*x]^4/(a + b*Sec[e + f*x]^2),x]","\frac{x \left(3 a^2+12 a b+8 b^2\right)}{8 a^3}-\frac{\sqrt{b} (a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a^3 f}-\frac{(5 a+4 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f}","\frac{x \left(3 a^2+12 a b+8 b^2\right)}{8 a^3}-\frac{\sqrt{b} (a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a^3 f}-\frac{(5 a+4 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f}",1,"((3*a^2 + 12*a*b + 8*b^2)*x)/(8*a^3) - (Sqrt[b]*(a + b)^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a^3*f) - ((5*a + 4*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f)","A",6,6,23,0.2609,1,"{4132, 470, 527, 522, 203, 205}"
36,1,76,0,0.0983991,"\int \frac{\sin ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2),x]","-\frac{\sqrt{b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a^2 f}+\frac{x (a+2 b)}{2 a^2}-\frac{\sin (e+f x) \cos (e+f x)}{2 a f}","-\frac{\sqrt{b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a^2 f}+\frac{x (a+2 b)}{2 a^2}-\frac{\sin (e+f x) \cos (e+f x)}{2 a f}",1,"((a + 2*b)*x)/(2*a^2) - (Sqrt[b]*Sqrt[a + b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a^2*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*a*f)","A",5,5,23,0.2174,1,"{4132, 471, 522, 203, 205}"
37,1,45,0,0.0444918,"\int \frac{1}{a+b \sec ^2(e+f x)} \, dx","Int[(a + b*Sec[e + f*x]^2)^(-1),x]","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (e+f x)}{\sqrt{b}}\right)}{a f \sqrt{a+b}}+\frac{x}{a}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (e+f x)}{\sqrt{b}}\right)}{a f \sqrt{a+b}}+\frac{x}{a}",1,"x/a + (Sqrt[b]*ArcTan[(Sqrt[a + b]*Cot[e + f*x])/Sqrt[b]])/(a*Sqrt[a + b]*f)","A",3,3,14,0.2143,1,"{4127, 3181, 205}"
38,1,54,0,0.0729386,"\int \frac{\csc ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Csc[e + f*x]^2/(a + b*Sec[e + f*x]^2),x]","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{f (a+b)^{3/2}}-\frac{\cot (e+f x)}{f (a+b)}","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{f (a+b)^{3/2}}-\frac{\cot (e+f x)}{f (a+b)}",1,"-((Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/((a + b)^(3/2)*f)) - Cot[e + f*x]/((a + b)*f)","A",3,3,23,0.1304,1,"{4132, 325, 205}"
39,1,76,0,0.0944464,"\int \frac{\csc ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2),x]","-\frac{a \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{f (a+b)^{5/2}}-\frac{\cot ^3(e+f x)}{3 f (a+b)}-\frac{a \cot (e+f x)}{f (a+b)^2}","-\frac{a \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{f (a+b)^{5/2}}-\frac{\cot ^3(e+f x)}{3 f (a+b)}-\frac{a \cot (e+f x)}{f (a+b)^2}",1,"-((a*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/((a + b)^(5/2)*f)) - (a*Cot[e + f*x])/((a + b)^2*f) - Cot[e + f*x]^3/(3*(a + b)*f)","A",4,4,23,0.1739,1,"{4132, 453, 325, 205}"
40,1,105,0,0.1393968,"\int \frac{\csc ^6(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Csc[e + f*x]^6/(a + b*Sec[e + f*x]^2),x]","-\frac{a^2 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{f (a+b)^{7/2}}-\frac{a^2 \cot (e+f x)}{f (a+b)^3}-\frac{\cot ^5(e+f x)}{5 f (a+b)}-\frac{(2 a+b) \cot ^3(e+f x)}{3 f (a+b)^2}","-\frac{a^2 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{f (a+b)^{7/2}}-\frac{a^2 \cot (e+f x)}{f (a+b)^3}-\frac{\cot ^5(e+f x)}{5 f (a+b)}-\frac{(2 a+b) \cot ^3(e+f x)}{3 f (a+b)^2}",1,"-((a^2*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/((a + b)^(7/2)*f)) - (a^2*Cot[e + f*x])/((a + b)^3*f) - ((2*a + b)*Cot[e + f*x]^3)/(3*(a + b)^2*f) - Cot[e + f*x]^5/(5*(a + b)*f)","A",4,3,23,0.1304,1,"{4132, 461, 205}"
41,1,161,0,0.1779677,"\int \frac{\sin ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Sin[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{(a+b)^2 \cos ^5(e+f x)}{2 a^2 b f \left(a \cos ^2(e+f x)+b\right)}+\frac{(a+b) (3 a+7 b) \cos ^3(e+f x)}{6 a^3 b f}-\frac{(a+b) (3 a+7 b) \cos (e+f x)}{2 a^4 f}+\frac{\sqrt{b} (a+b) (3 a+7 b) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{2 a^{9/2} f}-\frac{\cos ^5(e+f x)}{5 a^2 f}","-\frac{(a+b)^2 \cos ^5(e+f x)}{2 a^2 b f \left(a \cos ^2(e+f x)+b\right)}+\frac{(a+b) (3 a+7 b) \cos ^3(e+f x)}{6 a^3 b f}-\frac{(a+b) (3 a+7 b) \cos (e+f x)}{2 a^4 f}+\frac{\sqrt{b} (a+b) (3 a+7 b) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{2 a^{9/2} f}-\frac{\cos ^5(e+f x)}{5 a^2 f}",1,"(Sqrt[b]*(a + b)*(3*a + 7*b)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(2*a^(9/2)*f) - ((a + b)*(3*a + 7*b)*Cos[e + f*x])/(2*a^4*f) + ((a + b)*(3*a + 7*b)*Cos[e + f*x]^3)/(6*a^3*b*f) - Cos[e + f*x]^5/(5*a^2*f) - ((a + b)^2*Cos[e + f*x]^5)/(2*a^2*b*f*(b + a*Cos[e + f*x]^2))","A",6,5,23,0.2174,1,"{4133, 463, 459, 302, 205}"
42,1,114,0,0.1121879,"\int \frac{\sin ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{b (a+b) \cos (e+f x)}{2 a^3 f \left(a \cos ^2(e+f x)+b\right)}-\frac{(a+2 b) \cos (e+f x)}{a^3 f}+\frac{\sqrt{b} (3 a+5 b) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{2 a^{7/2} f}+\frac{\cos ^3(e+f x)}{3 a^2 f}","-\frac{b (a+b) \cos (e+f x)}{2 a^3 f \left(a \cos ^2(e+f x)+b\right)}-\frac{(a+2 b) \cos (e+f x)}{a^3 f}+\frac{\sqrt{b} (3 a+5 b) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{2 a^{7/2} f}+\frac{\cos ^3(e+f x)}{3 a^2 f}",1,"(Sqrt[b]*(3*a + 5*b)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(2*a^(7/2)*f) - ((a + 2*b)*Cos[e + f*x])/(a^3*f) + Cos[e + f*x]^3/(3*a^2*f) - (b*(a + b)*Cos[e + f*x])/(2*a^3*f*(b + a*Cos[e + f*x]^2))","A",5,4,23,0.1739,1,"{4133, 455, 1153, 205}"
43,1,84,0,0.0505626,"\int \frac{\sin (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Sin[e + f*x]/(a + b*Sec[e + f*x]^2)^2,x]","\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{2 a^{5/2} f}-\frac{3 \cos (e+f x)}{2 a^2 f}+\frac{\cos ^3(e+f x)}{2 a f \left(a \cos ^2(e+f x)+b\right)}","\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{2 a^{5/2} f}-\frac{3 \cos (e+f x)}{2 a^2 f}+\frac{\cos ^3(e+f x)}{2 a f \left(a \cos ^2(e+f x)+b\right)}",1,"(3*Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(2*a^(5/2)*f) - (3*Cos[e + f*x])/(2*a^2*f) + Cos[e + f*x]^3/(2*a*f*(b + a*Cos[e + f*x]^2))","A",4,4,21,0.1905,1,"{4133, 288, 321, 205}"
44,1,99,0,0.1065689,"\int \frac{\csc (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Csc[e + f*x]/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{2 a^{3/2} f (a+b)^2}-\frac{b \cos (e+f x)}{2 a f (a+b) \left(a \cos ^2(e+f x)+b\right)}-\frac{\tanh ^{-1}(\cos (e+f x))}{f (a+b)^2}","\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{2 a^{3/2} f (a+b)^2}-\frac{b \cos (e+f x)}{2 a f (a+b) \left(a \cos ^2(e+f x)+b\right)}-\frac{\tanh ^{-1}(\cos (e+f x))}{f (a+b)^2}",1,"(Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(2*a^(3/2)*(a + b)^2*f) - ArcTanh[Cos[e + f*x]]/((a + b)^2*f) - (b*Cos[e + f*x])/(2*a*(a + b)*f*(b + a*Cos[e + f*x]^2))","A",5,5,21,0.2381,1,"{4133, 470, 522, 206, 205}"
45,1,147,0,0.1726848,"\int \frac{\csc ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Csc[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2,x]","\frac{(a-b) \cos (e+f x)}{2 f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)}+\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{2 \sqrt{a} f (a+b)^3}-\frac{(a-3 b) \tanh ^{-1}(\cos (e+f x))}{2 f (a+b)^3}-\frac{\cot (e+f x) \csc (e+f x)}{2 f (a+b) \left(a \cos ^2(e+f x)+b\right)}","\frac{(a-b) \cos (e+f x)}{2 f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)}+\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{2 \sqrt{a} f (a+b)^3}-\frac{(a-3 b) \tanh ^{-1}(\cos (e+f x))}{2 f (a+b)^3}-\frac{\cot (e+f x) \csc (e+f x)}{2 f (a+b) \left(a \cos ^2(e+f x)+b\right)}",1,"((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(2*Sqrt[a]*(a + b)^3*f) - ((a - 3*b)*ArcTanh[Cos[e + f*x]])/(2*(a + b)^3*f) + ((a - b)*Cos[e + f*x])/(2*(a + b)^2*f*(b + a*Cos[e + f*x]^2)) - (Cot[e + f*x]*Csc[e + f*x])/(2*(a + b)*f*(b + a*Cos[e + f*x]^2))","A",6,6,23,0.2609,1,"{4133, 470, 527, 522, 206, 205}"
46,1,197,0,0.2464369,"\int \frac{\csc ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{3 \left(a^2-6 a b+b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f (a+b)^4}+\frac{3 a (a-3 b) \cos (e+f x)}{8 f (a+b)^3 \left(a \cos ^2(e+f x)+b\right)}+\frac{3 \sqrt{a} \sqrt{b} (a-b) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{2 f (a+b)^4}-\frac{\cot (e+f x) \csc ^3(e+f x)}{4 f (a+b) \left(a \cos ^2(e+f x)+b\right)}-\frac{(a-5 b) \cot (e+f x) \csc (e+f x)}{8 f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)}","-\frac{3 \left(a^2-6 a b+b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f (a+b)^4}+\frac{3 a (a-3 b) \cos (e+f x)}{8 f (a+b)^3 \left(a \cos ^2(e+f x)+b\right)}+\frac{3 \sqrt{a} \sqrt{b} (a-b) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{2 f (a+b)^4}-\frac{\cot (e+f x) \csc ^3(e+f x)}{4 f (a+b) \left(a \cos ^2(e+f x)+b\right)}-\frac{(a-5 b) \cot (e+f x) \csc (e+f x)}{8 f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)}",1,"(3*Sqrt[a]*(a - b)*Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(2*(a + b)^4*f) - (3*(a^2 - 6*a*b + b^2)*ArcTanh[Cos[e + f*x]])/(8*(a + b)^4*f) + (3*a*(a - 3*b)*Cos[e + f*x])/(8*(a + b)^3*f*(b + a*Cos[e + f*x]^2)) - ((a - 5*b)*Cot[e + f*x]*Csc[e + f*x])/(8*(a + b)^2*f*(b + a*Cos[e + f*x]^2)) - (Cot[e + f*x]*Csc[e + f*x]^3)/(4*(a + b)*f*(b + a*Cos[e + f*x]^2))","A",7,6,23,0.2609,1,"{4133, 470, 527, 522, 206, 205}"
47,1,267,0,0.4264263,"\int \frac{\sin ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Sin[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{b \left(19 a^2+52 a b+32 b^2\right) \tan (e+f x)}{16 a^4 f \left(a+b \tan ^2(e+f x)+b\right)}-\frac{\left(33 a^2+82 a b+48 b^2\right) \sin (e+f x) \cos (e+f x)}{48 a^3 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x \left(60 a^2 b+5 a^3+120 a b^2+64 b^3\right)}{16 a^5}-\frac{\sqrt{b} (a+b)^{3/2} (3 a+8 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^5 f}+\frac{(9 a+8 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f \left(a+b \tan ^2(e+f x)+b\right)}","-\frac{b \left(19 a^2+52 a b+32 b^2\right) \tan (e+f x)}{16 a^4 f \left(a+b \tan ^2(e+f x)+b\right)}-\frac{\left(33 a^2+82 a b+48 b^2\right) \sin (e+f x) \cos (e+f x)}{48 a^3 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x \left(60 a^2 b+5 a^3+120 a b^2+64 b^3\right)}{16 a^5}-\frac{\sqrt{b} (a+b)^{3/2} (3 a+8 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^5 f}+\frac{(9 a+8 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f \left(a+b \tan ^2(e+f x)+b\right)}",1,"((5*a^3 + 60*a^2*b + 120*a*b^2 + 64*b^3)*x)/(16*a^5) - (Sqrt[b]*(a + b)^(3/2)*(3*a + 8*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^5*f) - ((33*a^2 + 82*a*b + 48*b^2)*Cos[e + f*x]*Sin[e + f*x])/(48*a^3*f*(a + b + b*Tan[e + f*x]^2)) + ((9*a + 8*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*(a + b + b*Tan[e + f*x]^2)) + (Cos[e + f*x]^3*Sin[e + f*x]^3)/(6*a*f*(a + b + b*Tan[e + f*x]^2)) - (b*(19*a^2 + 52*a*b + 32*b^2)*Tan[e + f*x])/(16*a^4*f*(a + b + b*Tan[e + f*x]^2))","A",8,7,23,0.3043,1,"{4132, 470, 578, 527, 522, 203, 205}"
48,1,191,0,0.255016,"\int \frac{\sin ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Sin[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]","\frac{3 x \left(a^2+8 a b+8 b^2\right)}{8 a^4}-\frac{3 \sqrt{b} \sqrt{a+b} (a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^4 f}-\frac{3 b (3 a+4 b) \tan (e+f x)}{8 a^3 f \left(a+b \tan ^2(e+f x)+b\right)}-\frac{(5 a+6 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)}","\frac{3 x \left(a^2+8 a b+8 b^2\right)}{8 a^4}-\frac{3 \sqrt{b} \sqrt{a+b} (a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^4 f}-\frac{3 b (3 a+4 b) \tan (e+f x)}{8 a^3 f \left(a+b \tan ^2(e+f x)+b\right)}-\frac{(5 a+6 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)}",1,"(3*(a^2 + 8*a*b + 8*b^2)*x)/(8*a^4) - (3*Sqrt[b]*Sqrt[a + b]*(a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^4*f) - ((5*a + 6*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^2)) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*(a + b + b*Tan[e + f*x]^2)) - (3*b*(3*a + 4*b)*Tan[e + f*x])/(8*a^3*f*(a + b + b*Tan[e + f*x]^2))","A",7,6,23,0.2609,1,"{4132, 470, 527, 522, 203, 205}"
49,1,130,0,0.1688727,"\int \frac{\sin ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{\sqrt{b} (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^3 f \sqrt{a+b}}-\frac{b \tan (e+f x)}{a^2 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x (a+4 b)}{2 a^3}-\frac{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)}","-\frac{\sqrt{b} (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^3 f \sqrt{a+b}}-\frac{b \tan (e+f x)}{a^2 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x (a+4 b)}{2 a^3}-\frac{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)}",1,"((a + 4*b)*x)/(2*a^3) - (Sqrt[b]*(3*a + 4*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^3*Sqrt[a + b]*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*(a + b + b*Tan[e + f*x]^2)) - (b*Tan[e + f*x])/(a^2*f*(a + b + b*Tan[e + f*x]^2))","A",6,6,23,0.2609,1,"{4132, 471, 527, 522, 203, 205}"
50,1,92,0,0.085927,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[(a + b*Sec[e + f*x]^2)^(-2),x]","-\frac{\sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 f (a+b)^{3/2}}+\frac{x}{a^2}-\frac{b \tan (e+f x)}{2 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}","-\frac{\sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 f (a+b)^{3/2}}+\frac{x}{a^2}-\frac{b \tan (e+f x)}{2 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}",1,"x/a^2 - (Sqrt[b]*(3*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(3/2)*f) - (b*Tan[e + f*x])/(2*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2))","A",5,5,14,0.3571,1,"{4128, 414, 522, 203, 205}"
51,1,91,0,0.0850206,"\int \frac{\csc ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Csc[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 f (a+b)^{5/2}}-\frac{3 \cot (e+f x)}{2 f (a+b)^2}+\frac{\cot (e+f x)}{2 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}","-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 f (a+b)^{5/2}}-\frac{3 \cot (e+f x)}{2 f (a+b)^2}+\frac{\cot (e+f x)}{2 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}",1,"(-3*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*(a + b)^(5/2)*f) - (3*Cot[e + f*x])/(2*(a + b)^2*f) + Cot[e + f*x]/(2*(a + b)*f*(a + b + b*Tan[e + f*x]^2))","A",4,4,23,0.1739,1,"{4132, 290, 325, 205}"
52,1,123,0,0.1746157,"\int \frac{\csc ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{\sqrt{b} (3 a-2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 f (a+b)^{7/2}}-\frac{a b \tan (e+f x)}{2 f (a+b)^3 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{\cot ^3(e+f x)}{3 f (a+b)^2}-\frac{(a-b) \cot (e+f x)}{f (a+b)^3}","-\frac{\sqrt{b} (3 a-2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 f (a+b)^{7/2}}-\frac{a b \tan (e+f x)}{2 f (a+b)^3 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{\cot ^3(e+f x)}{3 f (a+b)^2}-\frac{(a-b) \cot (e+f x)}{f (a+b)^3}",1,"-((3*a - 2*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*(a + b)^(7/2)*f) - ((a - b)*Cot[e + f*x])/((a + b)^3*f) - Cot[e + f*x]^3/(3*(a + b)^2*f) - (a*b*Tan[e + f*x])/(2*(a + b)^3*f*(a + b + b*Tan[e + f*x]^2))","A",5,4,23,0.1739,1,"{4132, 456, 1261, 205}"
53,1,188,0,0.2643397,"\int \frac{\csc ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Csc[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{b \left(5 a^2+2 b^2\right) \tan (e+f x)}{10 f (a+b)^4 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{\left(5 a^2-10 a b-b^2\right) \cot (e+f x)}{5 f (a+b)^4}-\frac{a \sqrt{b} (3 a-4 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 f (a+b)^{9/2}}-\frac{(10 a+3 b) \cot ^3(e+f x)}{15 f (a+b)^3}-\frac{\cot ^5(e+f x)}{5 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}","-\frac{b \left(5 a^2+2 b^2\right) \tan (e+f x)}{10 f (a+b)^4 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{\left(5 a^2-10 a b-b^2\right) \cot (e+f x)}{5 f (a+b)^4}-\frac{a \sqrt{b} (3 a-4 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 f (a+b)^{9/2}}-\frac{(10 a+3 b) \cot ^3(e+f x)}{15 f (a+b)^3}-\frac{\cot ^5(e+f x)}{5 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}",1,"-(a*(3*a - 4*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*(a + b)^(9/2)*f) - ((5*a^2 - 10*a*b - b^2)*Cot[e + f*x])/(5*(a + b)^4*f) - ((10*a + 3*b)*Cot[e + f*x]^3)/(15*(a + b)^3*f) - Cot[e + f*x]^5/(5*(a + b)*f*(a + b + b*Tan[e + f*x]^2)) - (b*(5*a^2 + 2*b^2)*Tan[e + f*x])/(10*(a + b)^4*f*(a + b + b*Tan[e + f*x]^2))","A",6,5,23,0.2174,1,"{4132, 462, 456, 1261, 205}"
54,1,214,0,0.25457,"\int \frac{\sin ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Sin[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{\left(3 a^2+14 a b+13 b^2\right) \cos (e+f x)}{2 a^5 f}+\frac{\sqrt{b} \left(15 a^2+70 a b+63 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{8 a^{11/2} f}-\frac{(a+b)^2 \cos ^7(e+f x)}{4 a^2 b f \left(a \cos ^2(e+f x)+b\right)^2}+\frac{(a+3 b) (3 a+5 b) \cos ^3(e+f x)}{12 a^4 b f}-\frac{b (a+b) (3 a+11 b) \cos (e+f x)}{8 a^5 f \left(a \cos ^2(e+f x)+b\right)}-\frac{\cos ^5(e+f x)}{5 a^3 f}","-\frac{\left(3 a^2+14 a b+13 b^2\right) \cos (e+f x)}{2 a^5 f}+\frac{\sqrt{b} \left(15 a^2+70 a b+63 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{8 a^{11/2} f}-\frac{(a+b)^2 \cos ^7(e+f x)}{4 a^2 b f \left(a \cos ^2(e+f x)+b\right)^2}+\frac{(a+3 b) (3 a+5 b) \cos ^3(e+f x)}{12 a^4 b f}-\frac{b (a+b) (3 a+11 b) \cos (e+f x)}{8 a^5 f \left(a \cos ^2(e+f x)+b\right)}-\frac{\cos ^5(e+f x)}{5 a^3 f}",1,"(Sqrt[b]*(15*a^2 + 70*a*b + 63*b^2)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(8*a^(11/2)*f) - ((3*a^2 + 14*a*b + 13*b^2)*Cos[e + f*x])/(2*a^5*f) + ((a + 3*b)*(3*a + 5*b)*Cos[e + f*x]^3)/(12*a^4*b*f) - Cos[e + f*x]^5/(5*a^3*f) - ((a + b)^2*Cos[e + f*x]^7)/(4*a^2*b*f*(b + a*Cos[e + f*x]^2)^2) - (b*(a + b)*(3*a + 11*b)*Cos[e + f*x])/(8*a^5*f*(b + a*Cos[e + f*x]^2))","A",6,5,23,0.2174,1,"{4133, 463, 455, 1810, 205}"
55,1,154,0,0.1883274,"\int \frac{\sin ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3,x]","\frac{b^2 (a+b) \cos (e+f x)}{4 a^4 f \left(a \cos ^2(e+f x)+b\right)^2}-\frac{b (9 a+13 b) \cos (e+f x)}{8 a^4 f \left(a \cos ^2(e+f x)+b\right)}-\frac{(a+3 b) \cos (e+f x)}{a^4 f}+\frac{5 \sqrt{b} (3 a+7 b) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{8 a^{9/2} f}+\frac{\cos ^3(e+f x)}{3 a^3 f}","\frac{b^2 (a+b) \cos (e+f x)}{4 a^4 f \left(a \cos ^2(e+f x)+b\right)^2}-\frac{b (9 a+13 b) \cos (e+f x)}{8 a^4 f \left(a \cos ^2(e+f x)+b\right)}-\frac{(a+3 b) \cos (e+f x)}{a^4 f}+\frac{5 \sqrt{b} (3 a+7 b) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{8 a^{9/2} f}+\frac{\cos ^3(e+f x)}{3 a^3 f}",1,"(5*Sqrt[b]*(3*a + 7*b)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(8*a^(9/2)*f) - ((a + 3*b)*Cos[e + f*x])/(a^4*f) + Cos[e + f*x]^3/(3*a^3*f) + (b^2*(a + b)*Cos[e + f*x])/(4*a^4*f*(b + a*Cos[e + f*x]^2)^2) - (b*(9*a + 13*b)*Cos[e + f*x])/(8*a^4*f*(b + a*Cos[e + f*x]^2))","A",6,5,23,0.2174,1,"{4133, 455, 1814, 1153, 205}"
56,1,116,0,0.0681461,"\int \frac{\sin (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Sin[e + f*x]/(a + b*Sec[e + f*x]^2)^3,x]","\frac{5 \cos ^3(e+f x)}{8 a^2 f \left(a \cos ^2(e+f x)+b\right)}+\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{8 a^{7/2} f}-\frac{15 \cos (e+f x)}{8 a^3 f}+\frac{\cos ^5(e+f x)}{4 a f \left(a \cos ^2(e+f x)+b\right)^2}","\frac{5 \cos ^3(e+f x)}{8 a^2 f \left(a \cos ^2(e+f x)+b\right)}+\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{8 a^{7/2} f}-\frac{15 \cos (e+f x)}{8 a^3 f}+\frac{\cos ^5(e+f x)}{4 a f \left(a \cos ^2(e+f x)+b\right)^2}",1,"(15*Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(8*a^(7/2)*f) - (15*Cos[e + f*x])/(8*a^3*f) + Cos[e + f*x]^5/(4*a*f*(b + a*Cos[e + f*x]^2)^2) + (5*Cos[e + f*x]^3)/(8*a^2*f*(b + a*Cos[e + f*x]^2))","A",5,4,21,0.1905,1,"{4133, 288, 321, 205}"
57,1,154,0,0.196444,"\int \frac{\csc (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Csc[e + f*x]/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\sqrt{b} \left(15 a^2+10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{8 a^{5/2} f (a+b)^3}-\frac{b (7 a+3 b) \cos (e+f x)}{8 a^2 f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)}-\frac{b \cos ^3(e+f x)}{4 a f (a+b) \left(a \cos ^2(e+f x)+b\right)^2}-\frac{\tanh ^{-1}(\cos (e+f x))}{f (a+b)^3}","\frac{\sqrt{b} \left(15 a^2+10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{8 a^{5/2} f (a+b)^3}-\frac{b (7 a+3 b) \cos (e+f x)}{8 a^2 f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)}-\frac{b \cos ^3(e+f x)}{4 a f (a+b) \left(a \cos ^2(e+f x)+b\right)^2}-\frac{\tanh ^{-1}(\cos (e+f x))}{f (a+b)^3}",1,"(Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(8*a^(5/2)*(a + b)^3*f) - ArcTanh[Cos[e + f*x]]/((a + b)^3*f) - (b*Cos[e + f*x]^3)/(4*a*(a + b)*f*(b + a*Cos[e + f*x]^2)^2) - (b*(7*a + 3*b)*Cos[e + f*x])/(8*a^2*(a + b)^2*f*(b + a*Cos[e + f*x]^2))","A",6,6,21,0.2857,1,"{4133, 470, 578, 522, 206, 205}"
58,1,213,0,0.3116597,"\int \frac{\csc ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Csc[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\left(4 a^2-9 a b-b^2\right) \cos (e+f x)}{8 a f (a+b)^3 \left(a \cos ^2(e+f x)+b\right)}+\frac{\sqrt{b} \left(15 a^2-10 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{8 a^{3/2} f (a+b)^4}-\frac{b (2 a-b) \cos (e+f x)}{4 a f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)^2}-\frac{\cos (e+f x) \cot ^2(e+f x)}{2 f (a+b) \left(a \cos ^2(e+f x)+b\right)^2}-\frac{(a-5 b) \tanh ^{-1}(\cos (e+f x))}{2 f (a+b)^4}","\frac{\left(4 a^2-9 a b-b^2\right) \cos (e+f x)}{8 a f (a+b)^3 \left(a \cos ^2(e+f x)+b\right)}+\frac{\sqrt{b} \left(15 a^2-10 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{8 a^{3/2} f (a+b)^4}-\frac{b (2 a-b) \cos (e+f x)}{4 a f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)^2}-\frac{\cos (e+f x) \cot ^2(e+f x)}{2 f (a+b) \left(a \cos ^2(e+f x)+b\right)^2}-\frac{(a-5 b) \tanh ^{-1}(\cos (e+f x))}{2 f (a+b)^4}",1,"(Sqrt[b]*(15*a^2 - 10*a*b - b^2)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(8*a^(3/2)*(a + b)^4*f) - ((a - 5*b)*ArcTanh[Cos[e + f*x]])/(2*(a + b)^4*f) - ((2*a - b)*b*Cos[e + f*x])/(4*a*(a + b)^2*f*(b + a*Cos[e + f*x]^2)^2) + ((4*a^2 - 9*a*b - b^2)*Cos[e + f*x])/(8*a*(a + b)^3*f*(b + a*Cos[e + f*x]^2)) - (Cos[e + f*x]*Cot[e + f*x]^2)/(2*(a + b)*f*(b + a*Cos[e + f*x]^2)^2)","A",7,7,23,0.3043,1,"{4133, 470, 578, 527, 522, 206, 205}"
59,1,257,0,0.366037,"\int \frac{\csc ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]","\frac{3 \left(a^2-6 a b+b^2\right) \cos (e+f x)}{8 f (a+b)^4 \left(a \cos ^2(e+f x)+b\right)}+\frac{\left(a^2-9 a b+2 b^2\right) \cos (e+f x)}{8 f (a+b)^3 \left(a \cos ^2(e+f x)+b\right)^2}+\frac{3 \sqrt{b} \left(5 a^2-10 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{8 \sqrt{a} f (a+b)^5}-\frac{3 \left(a^2-10 a b+5 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f (a+b)^5}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 f (a+b) \left(a \cos ^2(e+f x)+b\right)^2}-\frac{(a-7 b) \cot (e+f x) \csc (e+f x)}{8 f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)^2}","\frac{3 \left(a^2-6 a b+b^2\right) \cos (e+f x)}{8 f (a+b)^4 \left(a \cos ^2(e+f x)+b\right)}+\frac{\left(a^2-9 a b+2 b^2\right) \cos (e+f x)}{8 f (a+b)^3 \left(a \cos ^2(e+f x)+b\right)^2}+\frac{3 \sqrt{b} \left(5 a^2-10 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{8 \sqrt{a} f (a+b)^5}-\frac{3 \left(a^2-10 a b+5 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f (a+b)^5}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 f (a+b) \left(a \cos ^2(e+f x)+b\right)^2}-\frac{(a-7 b) \cot (e+f x) \csc (e+f x)}{8 f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)^2}",1,"(3*Sqrt[b]*(5*a^2 - 10*a*b + b^2)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(8*Sqrt[a]*(a + b)^5*f) - (3*(a^2 - 10*a*b + 5*b^2)*ArcTanh[Cos[e + f*x]])/(8*(a + b)^5*f) + ((a^2 - 9*a*b + 2*b^2)*Cos[e + f*x])/(8*(a + b)^3*f*(b + a*Cos[e + f*x]^2)^2) + (3*(a^2 - 6*a*b + b^2)*Cos[e + f*x])/(8*(a + b)^4*f*(b + a*Cos[e + f*x]^2)) - ((a - 7*b)*Cot[e + f*x]*Csc[e + f*x])/(8*(a + b)^2*f*(b + a*Cos[e + f*x]^2)^2) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*(a + b)*f*(b + a*Cos[e + f*x]^2)^2)","A",8,7,23,0.3043,1,"{4133, 470, 578, 527, 522, 206, 205}"
60,1,314,0,0.5049429,"\int \frac{\sin ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Sin[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{5 b \left(5 a^2+20 a b+16 b^2\right) \tan (e+f x)}{16 a^5 f \left(a+b \tan ^2(e+f x)+b\right)}-\frac{5 b \left(9 a^2+32 a b+24 b^2\right) \tan (e+f x)}{48 a^4 f \left(a+b \tan ^2(e+f x)+b\right)^2}-\frac{\left(33 a^2+110 a b+80 b^2\right) \sin (e+f x) \cos (e+f x)}{48 a^3 f \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{5 x (a+2 b) \left(a^2+16 a b+16 b^2\right)}{16 a^6}-\frac{5 \sqrt{b} \sqrt{a+b} (a+4 b) (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^6 f}+\frac{(9 a+10 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f \left(a+b \tan ^2(e+f x)+b\right)^2}","-\frac{5 b \left(5 a^2+20 a b+16 b^2\right) \tan (e+f x)}{16 a^5 f \left(a+b \tan ^2(e+f x)+b\right)}-\frac{5 b \left(9 a^2+32 a b+24 b^2\right) \tan (e+f x)}{48 a^4 f \left(a+b \tan ^2(e+f x)+b\right)^2}-\frac{\left(33 a^2+110 a b+80 b^2\right) \sin (e+f x) \cos (e+f x)}{48 a^3 f \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{5 x (a+2 b) \left(a^2+16 a b+16 b^2\right)}{16 a^6}-\frac{5 \sqrt{b} \sqrt{a+b} (a+4 b) (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^6 f}+\frac{(9 a+10 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"(5*(a + 2*b)*(a^2 + 16*a*b + 16*b^2)*x)/(16*a^6) - (5*Sqrt[b]*Sqrt[a + b]*(a + 4*b)*(3*a + 4*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^6*f) - ((33*a^2 + 110*a*b + 80*b^2)*Cos[e + f*x]*Sin[e + f*x])/(48*a^3*f*(a + b + b*Tan[e + f*x]^2)^2) + ((9*a + 10*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*(a + b + b*Tan[e + f*x]^2)^2) + (Cos[e + f*x]^3*Sin[e + f*x]^3)/(6*a*f*(a + b + b*Tan[e + f*x]^2)^2) - (5*b*(9*a^2 + 32*a*b + 24*b^2)*Tan[e + f*x])/(48*a^4*f*(a + b + b*Tan[e + f*x]^2)^2) - (5*b*(5*a^2 + 20*a*b + 16*b^2)*Tan[e + f*x])/(16*a^5*f*(a + b + b*Tan[e + f*x]^2))","A",9,7,23,0.3043,1,"{4132, 470, 578, 527, 522, 203, 205}"
61,1,238,0,0.3402877,"\int \frac{\sin ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Sin[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{3 \sqrt{b} \left(5 a^2+20 a b+16 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^5 f \sqrt{a+b}}+\frac{3 x \left(a^2+12 a b+16 b^2\right)}{8 a^5}-\frac{3 b (a+2 b) \tan (e+f x)}{2 a^4 f \left(a+b \tan ^2(e+f x)+b\right)}-\frac{b (7 a+12 b) \tan (e+f x)}{8 a^3 f \left(a+b \tan ^2(e+f x)+b\right)^2}-\frac{(5 a+8 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)^2}","-\frac{3 \sqrt{b} \left(5 a^2+20 a b+16 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^5 f \sqrt{a+b}}+\frac{3 x \left(a^2+12 a b+16 b^2\right)}{8 a^5}-\frac{3 b (a+2 b) \tan (e+f x)}{2 a^4 f \left(a+b \tan ^2(e+f x)+b\right)}-\frac{b (7 a+12 b) \tan (e+f x)}{8 a^3 f \left(a+b \tan ^2(e+f x)+b\right)^2}-\frac{(5 a+8 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"(3*(a^2 + 12*a*b + 16*b^2)*x)/(8*a^5) - (3*Sqrt[b]*(5*a^2 + 20*a*b + 16*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^5*Sqrt[a + b]*f) - ((5*a + 8*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^2)^2) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(7*a + 12*b)*Tan[e + f*x])/(8*a^3*f*(a + b + b*Tan[e + f*x]^2)^2) - (3*b*(a + 2*b)*Tan[e + f*x])/(2*a^4*f*(a + b + b*Tan[e + f*x]^2))","A",8,6,23,0.2609,1,"{4132, 470, 527, 522, 203, 205}"
62,1,184,0,0.2783058,"\int \frac{\sin ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{\sqrt{b} \left(15 a^2+40 a b+24 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^4 f (a+b)^{3/2}}-\frac{b (11 a+12 b) \tan (e+f x)}{8 a^3 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}-\frac{3 b \tan (e+f x)}{4 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{x (a+6 b)}{2 a^4}-\frac{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)^2}","-\frac{\sqrt{b} \left(15 a^2+40 a b+24 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^4 f (a+b)^{3/2}}-\frac{b (11 a+12 b) \tan (e+f x)}{8 a^3 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}-\frac{3 b \tan (e+f x)}{4 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{x (a+6 b)}{2 a^4}-\frac{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"((a + 6*b)*x)/(2*a^4) - (Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^4*(a + b)^(3/2)*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*(a + b + b*Tan[e + f*x]^2)^2) - (3*b*Tan[e + f*x])/(4*a^2*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(11*a + 12*b)*Tan[e + f*x])/(8*a^3*(a + b)*f*(a + b + b*Tan[e + f*x]^2))","A",7,6,23,0.2609,1,"{4132, 471, 527, 522, 203, 205}"
63,1,144,0,0.1884217,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[(a + b*Sec[e + f*x]^2)^(-3),x]","-\frac{\sqrt{b} \left(15 a^2+20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 f (a+b)^{5/2}}-\frac{b (7 a+4 b) \tan (e+f x)}{8 a^2 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x}{a^3}-\frac{b \tan (e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}","-\frac{\sqrt{b} \left(15 a^2+20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 f (a+b)^{5/2}}-\frac{b (7 a+4 b) \tan (e+f x)}{8 a^2 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x}{a^3}-\frac{b \tan (e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"x/a^3 - (Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(5/2)*f) - (b*Tan[e + f*x])/(4*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(7*a + 4*b)*Tan[e + f*x])/(8*a^2*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))","A",6,6,14,0.4286,1,"{4128, 414, 527, 522, 203, 205}"
64,1,124,0,0.1110207,"\int \frac{\csc ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Csc[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 f (a+b)^{7/2}}-\frac{15 \cot (e+f x)}{8 f (a+b)^3}+\frac{5 \cot (e+f x)}{8 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\cot (e+f x)}{4 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}","-\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 f (a+b)^{7/2}}-\frac{15 \cot (e+f x)}{8 f (a+b)^3}+\frac{5 \cot (e+f x)}{8 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\cot (e+f x)}{4 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"(-15*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*(a + b)^(7/2)*f) - (15*Cot[e + f*x])/(8*(a + b)^3*f) + Cot[e + f*x]/(4*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) + (5*Cot[e + f*x])/(8*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))","A",5,4,23,0.1739,1,"{4132, 290, 325, 205}"
65,1,164,0,0.2504495,"\int \frac{\csc ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{5 \sqrt{b} (3 a-4 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 f (a+b)^{9/2}}-\frac{b (7 a-4 b) \tan (e+f x)}{8 f (a+b)^4 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{a b \tan (e+f x)}{4 f (a+b)^3 \left(a+b \tan ^2(e+f x)+b\right)^2}-\frac{\cot ^3(e+f x)}{3 f (a+b)^3}-\frac{(a-2 b) \cot (e+f x)}{f (a+b)^4}","-\frac{5 \sqrt{b} (3 a-4 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 f (a+b)^{9/2}}-\frac{b (7 a-4 b) \tan (e+f x)}{8 f (a+b)^4 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{a b \tan (e+f x)}{4 f (a+b)^3 \left(a+b \tan ^2(e+f x)+b\right)^2}-\frac{\cot ^3(e+f x)}{3 f (a+b)^3}-\frac{(a-2 b) \cot (e+f x)}{f (a+b)^4}",1,"(-5*(3*a - 4*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*(a + b)^(9/2)*f) - ((a - 2*b)*Cot[e + f*x])/((a + b)^4*f) - Cot[e + f*x]^3/(3*(a + b)^3*f) - (a*b*Tan[e + f*x])/(4*(a + b)^3*f*(a + b + b*Tan[e + f*x]^2)^2) - ((7*a - 4*b)*b*Tan[e + f*x])/(8*(a + b)^4*f*(a + b + b*Tan[e + f*x]^2))","A",6,5,23,0.2174,1,"{4132, 456, 1259, 1261, 205}"
66,1,242,0,0.369588,"\int \frac{\csc ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Csc[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{\sqrt{b} \left(15 a^2-40 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 f (a+b)^{11/2}}-\frac{b \left(35 a^2-40 a b+24 b^2\right) \tan (e+f x)}{40 f (a+b)^5 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{b \left(5 a^2+4 b^2\right) \tan (e+f x)}{20 f (a+b)^4 \left(a+b \tan ^2(e+f x)+b\right)^2}-\frac{\left(5 a^2-20 a b+2 b^2\right) \cot (e+f x)}{5 f (a+b)^5}-\frac{(10 a+b) \cot ^3(e+f x)}{15 f (a+b)^4}-\frac{\cot ^5(e+f x)}{5 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}","-\frac{\sqrt{b} \left(15 a^2-40 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 f (a+b)^{11/2}}-\frac{b \left(35 a^2-40 a b+24 b^2\right) \tan (e+f x)}{40 f (a+b)^5 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{b \left(5 a^2+4 b^2\right) \tan (e+f x)}{20 f (a+b)^4 \left(a+b \tan ^2(e+f x)+b\right)^2}-\frac{\left(5 a^2-20 a b+2 b^2\right) \cot (e+f x)}{5 f (a+b)^5}-\frac{(10 a+b) \cot ^3(e+f x)}{15 f (a+b)^4}-\frac{\cot ^5(e+f x)}{5 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"-(Sqrt[b]*(15*a^2 - 40*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*(a + b)^(11/2)*f) - ((5*a^2 - 20*a*b + 2*b^2)*Cot[e + f*x])/(5*(a + b)^5*f) - ((10*a + b)*Cot[e + f*x]^3)/(15*(a + b)^4*f) - Cot[e + f*x]^5/(5*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(5*a^2 + 4*b^2)*Tan[e + f*x])/(20*(a + b)^4*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(35*a^2 - 40*a*b + 24*b^2)*Tan[e + f*x])/(40*(a + b)^5*f*(a + b + b*Tan[e + f*x]^2))","A",7,6,23,0.2609,1,"{4132, 462, 456, 1259, 1261, 205}"
67,1,139,0,0.1469463,"\int \sqrt{a+b \sec ^2(e+f x)} \sin ^5(e+f x) \, dx","Int[Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^5,x]","\frac{2 (5 a+b) \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{15 a^2 f}-\frac{\cos ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{5 a f}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}","\frac{2 (5 a+b) \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{15 a^2 f}-\frac{\cos ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{5 a f}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f - (Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/f + (2*(5*a + b)*Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2))/(15*a^2*f) - (Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2))/(5*a*f)","A",6,6,25,0.2400,1,"{4134, 462, 451, 277, 217, 206}"
68,1,100,0,0.0938918,"\int \sqrt{a+b \sec ^2(e+f x)} \sin ^3(e+f x) \, dx","Int[Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^3,x]","\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 a f}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}","\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 a f}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f - (Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/f + (Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2))/(3*a*f)","A",5,5,25,0.2000,1,"{4134, 451, 277, 217, 206}"
69,1,66,0,0.0533864,"\int \sqrt{a+b \sec ^2(e+f x)} \sin (e+f x) \, dx","Int[Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x],x]","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{f}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{f}",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f - (Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/f","A",4,4,23,0.1739,1,"{4134, 277, 217, 206}"
70,1,82,0,0.0899001,"\int \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Csc[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}-\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}-\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f - (Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f","A",6,6,23,0.2609,1,"{4134, 402, 217, 206, 377, 207}"
71,1,124,0,0.1357445,"\int \csc ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Csc[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}-\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f \sqrt{a+b}}-\frac{\cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 f}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}-\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f \sqrt{a+b}}-\frac{\cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 f}",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f - ((a + 2*b)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*Sqrt[a + b]*f) - (Cot[e + f*x]*Csc[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(2*f)","A",7,7,25,0.2800,1,"{4134, 467, 523, 217, 206, 377, 207}"
72,1,183,0,0.2201595,"\int \csc ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Csc[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\left(3 a^2+12 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{8 f (a+b)^{3/2}}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}-\frac{\cot (e+f x) \csc ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{4 f}-\frac{(3 a+4 b) \cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{8 f (a+b)}","-\frac{\left(3 a^2+12 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{8 f (a+b)^{3/2}}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}-\frac{\cot (e+f x) \csc ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{4 f}-\frac{(3 a+4 b) \cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{8 f (a+b)}",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f - ((3*a^2 + 12*a*b + 8*b^2)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(8*(a + b)^(3/2)*f) - ((3*a + 4*b)*Cot[e + f*x]*Csc[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(8*(a + b)*f) - (Cot[e + f*x]*Csc[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2])/(4*f)","A",8,8,25,0.3200,1,"{4134, 467, 578, 523, 217, 206, 377, 207}"
73,1,240,0,0.3844216,"\int \sqrt{a+b \sec ^2(e+f x)} \sin ^6(e+f x) \, dx","Int[Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^6,x]","\frac{\left(-15 a^2 b+5 a^3-5 a b^2-b^3\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{5/2} f}-\frac{(a-b) (5 a+b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{16 a^2 f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\sin ^5(e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{6 f}-\frac{(5 a-b) \sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{24 a f}","\frac{\left(-15 a^2 b+5 a^3-5 a b^2-b^3\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{5/2} f}-\frac{(a-b) (5 a+b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{16 a^2 f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\sin ^5(e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{6 f}-\frac{(5 a-b) \sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{24 a f}",1,"((5*a^3 - 15*a^2*b - 5*a*b^2 - b^3)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(5/2)*f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - ((a - b)*(5*a + b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(16*a^2*f) - ((5*a - b)*Cos[e + f*x]*Sin[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(24*a*f) - (Cos[e + f*x]*Sin[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(6*f)","A",9,8,25,0.3200,1,"{4132, 467, 578, 523, 217, 206, 377, 203}"
74,1,181,0,0.22117,"\int \sqrt{a+b \sec ^2(e+f x)} \sin ^4(e+f x) \, dx","Int[Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^4,x]","\frac{\left(3 a^2-6 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{3/2} f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 f}-\frac{(3 a-b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 a f}","\frac{\left(3 a^2-6 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{3/2} f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 f}-\frac{(3 a-b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 a f}",1,"((3*a^2 - 6*a*b - b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(3/2)*f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - ((3*a - b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*a*f) - (Cos[e + f*x]*Sin[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(4*f)","A",8,8,25,0.3200,1,"{4132, 467, 578, 523, 217, 206, 377, 203}"
75,1,123,0,0.1311256,"\int \sqrt{a+b \sec ^2(e+f x)} \sin ^2(e+f x) \, dx","Int[Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^2,x]","\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 \sqrt{a} f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}","\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 \sqrt{a} f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}",1,"((a - b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*Sqrt[a]*f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - (Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)","A",7,7,25,0.2800,1,"{4132, 467, 523, 217, 206, 377, 203}"
76,1,79,0,0.0508236,"\int \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}",1,"(Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f","A",6,6,16,0.3750,1,"{4128, 402, 217, 206, 377, 203}"
77,1,68,0,0.0808933,"\int \csc ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Csc[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f}",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - (Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/f","A",4,4,25,0.1600,1,"{4132, 277, 217, 206}"
78,1,105,0,0.0971405,"\int \csc ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Csc[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{3 f (a+b)}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{3 f (a+b)}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f}",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - (Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/f - (Cot[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^(3/2))/(3*(a + b)*f)","A",5,5,25,0.2000,1,"{4132, 451, 277, 217, 206}"
79,1,149,0,0.1401201,"\int \csc ^6(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Csc[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{5 f (a+b)}-\frac{2 (5 a+4 b) \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{15 f (a+b)^2}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{5 f (a+b)}-\frac{2 (5 a+4 b) \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{15 f (a+b)^2}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f}",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - (Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/f - (2*(5*a + 4*b)*Cot[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^(3/2))/(15*(a + b)^2*f) - (Cot[e + f*x]^5*(a + b + b*Tan[e + f*x]^2)^(3/2))/(5*(a + b)*f)","A",6,6,25,0.2400,1,"{4132, 462, 451, 277, 217, 206}"
80,1,196,0,0.1799434,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \sin ^5(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^5,x]","\frac{b (3 a-4 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 a f}-\frac{\cos ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^{5/2}}{5 a f}+\frac{2 \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{5/2}}{3 a f}-\frac{(3 a-4 b) \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 a f}+\frac{\sqrt{b} (3 a-4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f}","\frac{b (3 a-4 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 a f}-\frac{\cos ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^{5/2}}{5 a f}+\frac{2 \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{5/2}}{3 a f}-\frac{(3 a-4 b) \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 a f}+\frac{\sqrt{b} (3 a-4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f}",1,"((3*a - 4*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*f) + ((3*a - 4*b)*b*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(2*a*f) - ((3*a - 4*b)*Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2))/(3*a*f) + (2*Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(5/2))/(3*a*f) - (Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(5/2))/(5*a*f)","A",7,7,25,0.2800,1,"{4134, 462, 453, 277, 195, 217, 206}"
81,1,162,0,0.1424083,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \sin ^3(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^3,x]","\frac{b (3 a-2 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 a f}+\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{5/2}}{3 a f}-\frac{(3 a-2 b) \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 a f}+\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f}","\frac{b (3 a-2 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 a f}+\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{5/2}}{3 a f}-\frac{(3 a-2 b) \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 a f}+\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f}",1,"((3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*f) + ((3*a - 2*b)*b*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(2*a*f) - ((3*a - 2*b)*Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2))/(3*a*f) + (Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(5/2))/(3*a*f)","A",6,6,25,0.2400,1,"{4134, 453, 277, 195, 217, 206}"
82,1,100,0,0.0702479,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \sin (e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x],x]","\frac{3 b \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 f}-\frac{\cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{f}+\frac{3 a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f}","\frac{3 b \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 f}-\frac{\cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{f}+\frac{3 a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f}",1,"(3*a*Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*f) + (3*b*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(2*f) - (Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2))/f","A",5,5,23,0.2174,1,"{4134, 277, 195, 217, 206}"
83,1,122,0,0.1360503,"\int \csc (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{b \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 f}-\frac{(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}+\frac{\sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f}","\frac{b \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 f}-\frac{(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f}+\frac{\sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f}",1,"(Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*f) - ((a + b)^(3/2)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f + (b*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(2*f)","A",7,7,23,0.3043,1,"{4134, 416, 523, 217, 206, 377, 207}"
84,1,161,0,0.2031214,"\int \csc ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{b \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{f}+\frac{\sqrt{b} (3 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f}-\frac{\sqrt{a+b} (a+4 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f}-\frac{\cot (e+f x) \csc (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{2 f}","\frac{b \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{f}+\frac{\sqrt{b} (3 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f}-\frac{\sqrt{a+b} (a+4 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f}-\frac{\cot (e+f x) \csc (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{2 f}",1,"(Sqrt[b]*(3*a + 4*b)*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*f) - (Sqrt[a + b]*(a + 4*b)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*f) + (b*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/f - (Cot[e + f*x]*Csc[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2))/(2*f)","A",8,8,25,0.3200,1,"{4134, 467, 528, 523, 217, 206, 377, 207}"
85,1,218,0,0.3361474,"\int \csc ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{3 \left(a^2+8 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{8 f \sqrt{a+b}}+\frac{3 (a+4 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{8 f}-\frac{3 (a+2 b) \csc ^2(e+f x) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{8 f}+\frac{3 \sqrt{b} (a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f}-\frac{\cot (e+f x) \csc ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{4 f}","-\frac{3 \left(a^2+8 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{8 f \sqrt{a+b}}+\frac{3 (a+4 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{8 f}-\frac{3 (a+2 b) \csc ^2(e+f x) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{8 f}+\frac{3 \sqrt{b} (a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f}-\frac{\cot (e+f x) \csc ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{4 f}",1,"(3*Sqrt[b]*(a + 2*b)*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*f) - (3*(a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(8*Sqrt[a + b]*f) + (3*(a + 4*b)*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(8*f) - (3*(a + 2*b)*Csc[e + f*x]^2*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(8*f) - (Cot[e + f*x]*Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2))/(4*f)","A",9,9,25,0.3600,1,"{4134, 467, 577, 582, 523, 217, 206, 377, 207}"
86,1,298,0,0.473906,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \sin ^6(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^6,x]","\frac{\left(-45 a^2 b+5 a^3+15 a b^2+b^3\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{3/2} f}-\frac{\left(5 a^2-26 a b+b^2\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{16 a f}+\frac{\left(5 a^2-40 a b+3 b^2\right) \sin ^2(e+f x) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{48 a f}+\frac{(5 a-3 b) \sin ^4(e+f x) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{24 f}+\frac{\sqrt{b} (3 a-5 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}-\frac{\sin ^5(e+f x) \cos (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{6 f}","\frac{\left(-45 a^2 b+5 a^3+15 a b^2+b^3\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{3/2} f}-\frac{\left(5 a^2-26 a b+b^2\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{16 a f}+\frac{\left(5 a^2-40 a b+3 b^2\right) \sin ^2(e+f x) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{48 a f}+\frac{(5 a-3 b) \sin ^4(e+f x) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{24 f}+\frac{\sqrt{b} (3 a-5 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}-\frac{\sin ^5(e+f x) \cos (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{6 f}",1,"((5*a^3 - 45*a^2*b + 15*a*b^2 + b^3)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(3/2)*f) + ((3*a - 5*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) - ((5*a^2 - 26*a*b + b^2)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(16*a*f) + ((5*a^2 - 40*a*b + 3*b^2)*Sin[e + f*x]^2*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(48*a*f) + ((5*a - 3*b)*Sin[e + f*x]^4*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(24*f) - (Cos[e + f*x]*Sin[e + f*x]^5*(a + b + b*Tan[e + f*x]^2)^(3/2))/(6*f)","A",10,10,25,0.4000,1,"{4132, 467, 577, 578, 582, 523, 217, 206, 377, 203}"
87,1,217,0,0.3408621,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \sin ^4(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^4,x]","\frac{3 \left(a^2-6 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 \sqrt{a} f}-\frac{3 (a-3 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 f}+\frac{3 (a-b) \sin ^2(e+f x) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 f}+\frac{3 \sqrt{b} (a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}-\frac{\sin ^3(e+f x) \cos (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{4 f}","\frac{3 \left(a^2-6 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 \sqrt{a} f}-\frac{3 (a-3 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 f}+\frac{3 (a-b) \sin ^2(e+f x) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 f}+\frac{3 \sqrt{b} (a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}-\frac{\sin ^3(e+f x) \cos (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{4 f}",1,"(3*(a^2 - 6*a*b + b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*Sqrt[a]*f) + (3*(a - b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) - (3*(a - 3*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*f) + (3*(a - b)*Sin[e + f*x]^2*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*f) - (Cos[e + f*x]*Sin[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^(3/2))/(4*f)","A",9,9,25,0.3600,1,"{4132, 467, 577, 582, 523, 217, 206, 377, 203}"
88,1,161,0,0.1977325,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \sin ^2(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^2,x]","\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f}+\frac{\sqrt{a} (a-3 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}+\frac{\sqrt{b} (3 a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}-\frac{\sin (e+f x) \cos (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{2 f}","\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f}+\frac{\sqrt{a} (a-3 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}+\frac{\sqrt{b} (3 a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}-\frac{\sin (e+f x) \cos (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{2 f}",1,"(Sqrt[a]*(a - 3*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + ((3*a - b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (b*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/f - (Cos[e + f*x]*Sin[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(2*f)","A",8,8,25,0.3200,1,"{4132, 467, 528, 523, 217, 206, 377, 203}"
89,1,118,0,0.0961235,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}+\frac{\sqrt{b} (3 a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}+\frac{\sqrt{b} (3 a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}",1,"(a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*(3*a + b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (b*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)","A",7,7,16,0.4375,1,"{4128, 416, 523, 217, 206, 377, 203}"
90,1,105,0,0.1049941,"\int \csc ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]^2*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{3 b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}+\frac{3 \sqrt{b} (a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{f}","\frac{3 b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}+\frac{3 \sqrt{b} (a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{f}",1,"(3*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (3*b*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f) - (Cot[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/f","A",5,5,25,0.2000,1,"{4132, 277, 195, 217, 206}"
91,1,172,0,0.1527811,"\int \csc ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]^4*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{b (3 a+5 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f (a+b)}+\frac{\sqrt{b} (3 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{5/2}}{3 f (a+b)}-\frac{(3 a+5 b) \cot (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{3 f (a+b)}","\frac{b (3 a+5 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f (a+b)}+\frac{\sqrt{b} (3 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{5/2}}{3 f (a+b)}-\frac{(3 a+5 b) \cot (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{3 f (a+b)}",1,"(Sqrt[b]*(3*a + 5*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (b*(3*a + 5*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*(a + b)*f) - ((3*a + 5*b)*Cot[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(3*(a + b)*f) - (Cot[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^(5/2))/(3*(a + b)*f)","A",6,6,25,0.2400,1,"{4132, 453, 277, 195, 217, 206}"
92,1,209,0,0.2045399,"\int \csc ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]^6*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{b (3 a+7 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f (a+b)}+\frac{\sqrt{b} (3 a+7 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{5/2}}{5 f (a+b)}-\frac{2 \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{5/2}}{3 f (a+b)}-\frac{(3 a+7 b) \cot (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{3 f (a+b)}","\frac{b (3 a+7 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f (a+b)}+\frac{\sqrt{b} (3 a+7 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{5/2}}{5 f (a+b)}-\frac{2 \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{5/2}}{3 f (a+b)}-\frac{(3 a+7 b) \cot (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{3 f (a+b)}",1,"(Sqrt[b]*(3*a + 7*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (b*(3*a + 7*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*(a + b)*f) - ((3*a + 7*b)*Cot[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(3*(a + b)*f) - (2*Cot[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^(5/2))/(3*(a + b)*f) - (Cot[e + f*x]^5*(a + b + b*Tan[e + f*x]^2)^(5/2))/(5*(a + b)*f)","A",7,7,25,0.2800,1,"{4132, 462, 453, 277, 195, 217, 206}"
93,1,123,0,0.1387164,"\int \frac{\sin ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Sin[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\left(15 a^2+20 a b+8 b^2\right) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{15 a^3 f}+\frac{2 (5 a+2 b) \cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{15 a^2 f}-\frac{\cos ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{5 a f}","-\frac{\left(15 a^2+20 a b+8 b^2\right) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{15 a^3 f}+\frac{2 (5 a+2 b) \cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{15 a^2 f}-\frac{\cos ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{5 a f}",1,"-((15*a^2 + 20*a*b + 8*b^2)*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(15*a^3*f) + (2*(5*a + 2*b)*Cos[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2])/(15*a^2*f) - (Cos[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2])/(5*a*f)","A",4,4,25,0.1600,1,"{4134, 462, 453, 264}"
94,1,74,0,0.0910306,"\int \frac{\sin ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Sin[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{3 a f}-\frac{(3 a+2 b) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{3 a^2 f}","\frac{\cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{3 a f}-\frac{(3 a+2 b) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{3 a^2 f}",1,"-((3*a + 2*b)*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(3*a^2*f) + (Cos[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2])/(3*a*f)","A",3,3,25,0.1200,1,"{4134, 453, 264}"
95,1,30,0,0.0420239,"\int \frac{\sin (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Sin[e + f*x]/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{a f}","-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{a f}",1,"-((Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(a*f))","A",2,2,23,0.08696,1,"{4134, 264}"
96,1,43,0,0.0677984,"\int \frac{\csc (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Csc[e + f*x]/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f \sqrt{a+b}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f \sqrt{a+b}}",1,"-(ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]]/(Sqrt[a + b]*f))","A",3,3,23,0.1304,1,"{4134, 377, 207}"
97,1,87,0,0.1095788,"\int \frac{\csc ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Csc[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{a \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f (a+b)^{3/2}}-\frac{\cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 f (a+b)}","-\frac{a \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f (a+b)^{3/2}}-\frac{\cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 f (a+b)}",1,"-(a*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*(a + b)^(3/2)*f) - (Cot[e + f*x]*Csc[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(2*(a + b)*f)","A",5,5,25,0.2000,1,"{4134, 471, 12, 377, 207}"
98,1,138,0,0.1650517,"\int \frac{\csc ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Csc[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{8 f (a+b)^{5/2}}-\frac{\cot ^3(e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{4 f (a+b)}-\frac{(5 a+2 b) \cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{8 f (a+b)^2}","-\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{8 f (a+b)^{5/2}}-\frac{\cot ^3(e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{4 f (a+b)}-\frac{(5 a+2 b) \cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{8 f (a+b)^2}",1,"(-3*a^2*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(8*(a + b)^(5/2)*f) - ((5*a + 2*b)*Cot[e + f*x]*Csc[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(8*(a + b)^2*f) - (Cot[e + f*x]^3*Csc[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(4*(a + b)*f)","A",6,6,25,0.2400,1,"{4134, 470, 527, 12, 377, 207}"
99,1,193,0,0.2783491,"\int \frac{\sin ^6(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Sin[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\left(33 a^2+40 a b+15 b^2\right) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{48 a^3 f}+\frac{5 (a+b)^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{7/2} f}+\frac{(9 a+5 b) \sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{24 a^2 f}+\frac{\sin ^3(e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{6 a f}","-\frac{\left(33 a^2+40 a b+15 b^2\right) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{48 a^3 f}+\frac{5 (a+b)^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{7/2} f}+\frac{(9 a+5 b) \sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{24 a^2 f}+\frac{\sin ^3(e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{6 a f}",1,"(5*(a + b)^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(7/2)*f) - ((33*a^2 + 40*a*b + 15*b^2)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(48*a^3*f) + ((9*a + 5*b)*Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(24*a^2*f) + (Cos[e + f*x]^3*Sin[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(6*a*f)","A",7,7,25,0.2800,1,"{4132, 470, 578, 527, 12, 377, 203}"
100,1,135,0,0.1522403,"\int \frac{\sin ^4(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Sin[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{3 (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{5/2} f}-\frac{(5 a+3 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 a^2 f}+\frac{\sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 a f}","\frac{3 (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{5/2} f}-\frac{(5 a+3 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 a^2 f}+\frac{\sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 a f}",1,"(3*(a + b)^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(5/2)*f) - ((5*a + 3*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*a^2*f) + (Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(4*a*f)","A",6,6,25,0.2400,1,"{4132, 470, 527, 12, 377, 203}"
101,1,85,0,0.1093128,"\int \frac{\sin ^2(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Sin[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 a^{3/2} f}-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 a f}","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 a^{3/2} f}-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 a f}",1,"((a + b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*a^(3/2)*f) - (Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*a*f)","A",5,5,25,0.2000,1,"{4132, 471, 12, 377, 203}"
102,1,39,0,0.0288085,"\int \frac{1}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[1/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}",1,"ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f)","A",3,3,16,0.1875,1,"{4128, 377, 203}"
103,1,33,0,0.0710493,"\int \frac{\csc ^2(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Csc[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f (a+b)}","-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f (a+b)}",1,"-((Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/((a + b)*f))","A",2,2,25,0.08000,1,"{4132, 264}"
104,1,78,0,0.0970371,"\int \frac{\csc ^4(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Csc[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f (a+b)}-\frac{(3 a+b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f (a+b)^2}","-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f (a+b)}-\frac{(3 a+b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f (a+b)^2}",1,"-((3*a + b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*(a + b)^2*f) - (Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*(a + b)*f)","A",3,3,25,0.1200,1,"{4132, 453, 264}"
105,1,132,0,0.1399567,"\int \frac{\csc ^6(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Csc[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\left(15 a^2+10 a b+3 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^3}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 f (a+b)}-\frac{2 (5 a+3 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^2}","-\frac{\left(15 a^2+10 a b+3 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^3}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 f (a+b)}-\frac{2 (5 a+3 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^2}",1,"-((15*a^2 + 10*a*b + 3*b^2)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*(a + b)^3*f) - (2*(5*a + 3*b)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*(a + b)^2*f) - (Cot[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(5*(a + b)*f)","A",4,4,25,0.1600,1,"{4132, 462, 453, 264}"
106,1,171,0,0.1870013,"\int \frac{\sin ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{2 b \left(15 a^2+40 a b+24 b^2\right) \sec (e+f x)}{15 a^4 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{2 (5 a+3 b) \cos ^3(e+f x)}{15 a^2 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\left(\frac{8 b (5 a+3 b)}{a^2}+15\right) \cos (e+f x)}{15 a f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\cos ^5(e+f x)}{5 a f \sqrt{a+b \sec ^2(e+f x)}}","-\frac{2 b \left(15 a^2+40 a b+24 b^2\right) \sec (e+f x)}{15 a^4 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\left(15 a^2+40 a b+24 b^2\right) \cos (e+f x)}{15 a^3 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{2 (5 a+3 b) \cos ^3(e+f x)}{15 a^2 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\cos ^5(e+f x)}{5 a f \sqrt{a+b \sec ^2(e+f x)}}",1,"-((15 + (8*b*(5*a + 3*b))/a^2)*Cos[e + f*x])/(15*a*f*Sqrt[a + b*Sec[e + f*x]^2]) + (2*(5*a + 3*b)*Cos[e + f*x]^3)/(15*a^2*f*Sqrt[a + b*Sec[e + f*x]^2]) - Cos[e + f*x]^5/(5*a*f*Sqrt[a + b*Sec[e + f*x]^2]) - (2*b*(15*a^2 + 40*a*b + 24*b^2)*Sec[e + f*x])/(15*a^4*f*Sqrt[a + b*Sec[e + f*x]^2])","A",5,5,25,0.2000,1,"{4134, 462, 453, 271, 191}"
107,1,114,0,0.1169662,"\int \frac{\sin ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{2 b (3 a+4 b) \sec (e+f x)}{3 a^3 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{(3 a+4 b) \cos (e+f x)}{3 a^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\cos ^3(e+f x)}{3 a f \sqrt{a+b \sec ^2(e+f x)}}","-\frac{2 b (3 a+4 b) \sec (e+f x)}{3 a^3 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{(3 a+4 b) \cos (e+f x)}{3 a^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\cos ^3(e+f x)}{3 a f \sqrt{a+b \sec ^2(e+f x)}}",1,"-((3*a + 4*b)*Cos[e + f*x])/(3*a^2*f*Sqrt[a + b*Sec[e + f*x]^2]) + Cos[e + f*x]^3/(3*a*f*Sqrt[a + b*Sec[e + f*x]^2]) - (2*b*(3*a + 4*b)*Sec[e + f*x])/(3*a^3*f*Sqrt[a + b*Sec[e + f*x]^2])","A",4,4,25,0.1600,1,"{4134, 453, 271, 191}"
108,1,62,0,0.0561524,"\int \frac{\sin (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{2 b \sec (e+f x)}{a^2 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\cos (e+f x)}{a f \sqrt{a+b \sec ^2(e+f x)}}","-\frac{2 b \sec (e+f x)}{a^2 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\cos (e+f x)}{a f \sqrt{a+b \sec ^2(e+f x)}}",1,"-(Cos[e + f*x]/(a*f*Sqrt[a + b*Sec[e + f*x]^2])) - (2*b*Sec[e + f*x])/(a^2*f*Sqrt[a + b*Sec[e + f*x]^2])","A",3,3,23,0.1304,1,"{4134, 271, 191}"
109,1,80,0,0.0921376,"\int \frac{\csc (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Csc[e + f*x]/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{b \sec (e+f x)}{a f (a+b) \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f (a+b)^{3/2}}","-\frac{b \sec (e+f x)}{a f (a+b) \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f (a+b)^{3/2}}",1,"-(ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]]/((a + b)^(3/2)*f)) - (b*Sec[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2])","A",4,4,23,0.1739,1,"{4134, 382, 377, 207}"
110,1,126,0,0.1531051,"\int \frac{\csc ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Csc[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{3 b \sec (e+f x)}{2 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}-\frac{(a-2 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f (a+b)^{5/2}}-\frac{\cot (e+f x) \csc (e+f x)}{2 f (a+b) \sqrt{a+b \sec ^2(e+f x)}}","-\frac{3 b \sec (e+f x)}{2 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}-\frac{(a-2 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f (a+b)^{5/2}}-\frac{\cot (e+f x) \csc (e+f x)}{2 f (a+b) \sqrt{a+b \sec ^2(e+f x)}}",1,"-((a - 2*b)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*(a + b)^(5/2)*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]) - (3*b*Sec[e + f*x])/(2*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2])","A",6,6,25,0.2400,1,"{4134, 471, 527, 12, 377, 207}"
111,1,177,0,0.2164741,"\int \frac{\csc ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{b (13 a-2 b) \sec (e+f x)}{8 f (a+b)^3 \sqrt{a+b \sec ^2(e+f x)}}-\frac{3 a (a-4 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{8 f (a+b)^{7/2}}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 f (a+b) \sqrt{a+b \sec ^2(e+f x)}}-\frac{5 a \cot (e+f x) \csc (e+f x)}{8 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}","-\frac{b (13 a-2 b) \sec (e+f x)}{8 f (a+b)^3 \sqrt{a+b \sec ^2(e+f x)}}-\frac{3 a (a-4 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{8 f (a+b)^{7/2}}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 f (a+b) \sqrt{a+b \sec ^2(e+f x)}}-\frac{5 a \cot (e+f x) \csc (e+f x)}{8 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}",1,"(-3*a*(a - 4*b)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(8*(a + b)^(7/2)*f) - (5*a*Cot[e + f*x]*Csc[e + f*x])/(8*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]) - ((13*a - 2*b)*b*Sec[e + f*x])/(8*(a + b)^3*f*Sqrt[a + b*Sec[e + f*x]^2])","A",7,6,25,0.2400,1,"{4134, 470, 527, 12, 377, 207}"
112,1,242,0,0.3736285,"\int \frac{\sin ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{b \left(81 a^2+190 a b+105 b^2\right) \tan (e+f x)}{48 a^4 f \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{5 (a+b)^2 (a+7 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{9/2} f}+\frac{(9 a+7 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{(a+b) (33 a+35 b) \sin (e+f x) \cos (e+f x)}{48 a^3 f \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f \sqrt{a+b \tan ^2(e+f x)+b}}","-\frac{b \left(81 a^2+190 a b+105 b^2\right) \tan (e+f x)}{48 a^4 f \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{5 (a+b)^2 (a+7 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{9/2} f}+\frac{(9 a+7 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{(a+b) (33 a+35 b) \sin (e+f x) \cos (e+f x)}{48 a^3 f \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f \sqrt{a+b \tan ^2(e+f x)+b}}",1,"(5*(a + b)^2*(a + 7*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(9/2)*f) - ((a + b)*(33*a + 35*b)*Cos[e + f*x]*Sin[e + f*x])/(48*a^3*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + ((9*a + 7*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + (Cos[e + f*x]^3*Sin[e + f*x]^3)/(6*a*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - (b*(81*a^2 + 190*a*b + 105*b^2)*Tan[e + f*x])/(48*a^4*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",8,7,25,0.2800,1,"{4132, 470, 578, 527, 12, 377, 203}"
113,1,175,0,0.2171532,"\int \frac{\sin ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{3 (a+b) (a+5 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{7/2} f}-\frac{b (13 a+15 b) \tan (e+f x)}{8 a^3 f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{5 (a+b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \sqrt{a+b \tan ^2(e+f x)+b}}","\frac{3 (a+b) (a+5 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{7/2} f}-\frac{b (13 a+15 b) \tan (e+f x)}{8 a^3 f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{5 (a+b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \sqrt{a+b \tan ^2(e+f x)+b}}",1,"(3*(a + b)*(a + 5*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(7/2)*f) - (5*(a + b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - (b*(13*a + 15*b)*Tan[e + f*x])/(8*a^3*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",7,6,25,0.2400,1,"{4132, 470, 527, 12, 377, 203}"
114,1,121,0,0.1495282,"\int \frac{\sin ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{(a+3 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 a^{5/2} f}-\frac{3 b \tan (e+f x)}{2 a^2 f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\sin (e+f x) \cos (e+f x)}{2 a f \sqrt{a+b \tan ^2(e+f x)+b}}","\frac{(a+3 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 a^{5/2} f}-\frac{3 b \tan (e+f x)}{2 a^2 f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\sin (e+f x) \cos (e+f x)}{2 a f \sqrt{a+b \tan ^2(e+f x)+b}}",1,"((a + 3*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*a^(5/2)*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - (3*b*Tan[e + f*x])/(2*a^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",6,6,25,0.2400,1,"{4132, 471, 527, 12, 377, 203}"
115,1,77,0,0.0501041,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[(a + b*Sec[e + f*x]^2)^(-3/2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}-\frac{b \tan (e+f x)}{a f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}-\frac{b \tan (e+f x)}{a f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}",1,"ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f) - (b*Tan[e + f*x])/(a*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",4,4,16,0.2500,1,"{4128, 382, 377, 203}"
116,1,68,0,0.0914931,"\int \frac{\csc ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Csc[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{2 b \tan (e+f x)}{f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\cot (e+f x)}{f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}","-\frac{2 b \tan (e+f x)}{f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\cot (e+f x)}{f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}",1,"-(Cot[e + f*x]/((a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])) - (2*b*Tan[e + f*x])/((a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",3,3,25,0.1200,1,"{4132, 271, 191}"
117,1,123,0,0.1278236,"\int \frac{\csc ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{2 b (3 a-b) \tan (e+f x)}{3 f (a+b)^3 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\cot ^3(e+f x)}{3 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{(3 a-b) \cot (e+f x)}{3 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}","-\frac{2 b (3 a-b) \tan (e+f x)}{3 f (a+b)^3 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\cot ^3(e+f x)}{3 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{(3 a-b) \cot (e+f x)}{3 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}",1,"-((3*a - b)*Cot[e + f*x])/(3*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - Cot[e + f*x]^3/(3*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - (2*(3*a - b)*b*Tan[e + f*x])/(3*(a + b)^3*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",4,4,25,0.1600,1,"{4132, 453, 271, 191}"
118,1,183,0,0.1815746,"\int \frac{\csc ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Csc[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{2 b \left(15 a^2-10 a b-b^2\right) \tan (e+f x)}{15 f (a+b)^4 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\left(15 a^2-10 a b-b^2\right) \cot (e+f x)}{15 f (a+b)^3 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\cot ^5(e+f x)}{5 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{2 (5 a+2 b) \cot ^3(e+f x)}{15 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}","-\frac{2 b \left(15 a^2-10 a b-b^2\right) \tan (e+f x)}{15 f (a+b)^4 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\left(15 a^2-10 a b-b^2\right) \cot (e+f x)}{15 f (a+b)^3 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\cot ^5(e+f x)}{5 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{2 (5 a+2 b) \cot ^3(e+f x)}{15 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}",1,"-((15*a^2 - 10*a*b - b^2)*Cot[e + f*x])/(15*(a + b)^3*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - (2*(5*a + 2*b)*Cot[e + f*x]^3)/(15*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - Cot[e + f*x]^5/(5*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - (2*b*(15*a^2 - 10*a*b - b^2)*Tan[e + f*x])/(15*(a + b)^4*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",5,5,25,0.2000,1,"{4132, 462, 453, 271, 191}"
119,1,219,0,0.2152924,"\int \frac{\sin ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{8 b \left(5 a^2+20 a b+16 b^2\right) \sec (e+f x)}{15 a^5 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{4 b \left(5 a^2+20 a b+16 b^2\right) \sec (e+f x)}{15 a^4 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}+\frac{2 (5 a+4 b) \cos ^3(e+f x)}{15 a^2 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\left(\frac{4 b (5 a+4 b)}{a^2}+5\right) \cos (e+f x)}{5 a f \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\cos ^5(e+f x)}{5 a f \left(a+b \sec ^2(e+f x)\right)^{3/2}}","-\frac{8 b \left(5 a^2+20 a b+16 b^2\right) \sec (e+f x)}{15 a^5 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{4 b \left(5 a^2+20 a b+16 b^2\right) \sec (e+f x)}{15 a^4 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\left(5 a^2+20 a b+16 b^2\right) \cos (e+f x)}{5 a^3 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}+\frac{2 (5 a+4 b) \cos ^3(e+f x)}{15 a^2 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\cos ^5(e+f x)}{5 a f \left(a+b \sec ^2(e+f x)\right)^{3/2}}",1,"-((5 + (4*b*(5*a + 4*b))/a^2)*Cos[e + f*x])/(5*a*f*(a + b*Sec[e + f*x]^2)^(3/2)) + (2*(5*a + 4*b)*Cos[e + f*x]^3)/(15*a^2*f*(a + b*Sec[e + f*x]^2)^(3/2)) - Cos[e + f*x]^5/(5*a*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (4*b*(5*a^2 + 20*a*b + 16*b^2)*Sec[e + f*x])/(15*a^4*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (8*b*(5*a^2 + 20*a*b + 16*b^2)*Sec[e + f*x])/(15*a^5*f*Sqrt[a + b*Sec[e + f*x]^2])","A",6,6,25,0.2400,1,"{4134, 462, 453, 271, 192, 191}"
120,1,146,0,0.1380493,"\int \frac{\sin ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{8 b (a+2 b) \sec (e+f x)}{3 a^4 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{4 b (a+2 b) \sec (e+f x)}{3 a^3 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{(a+2 b) \cos (e+f x)}{a^2 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}+\frac{\cos ^3(e+f x)}{3 a f \left(a+b \sec ^2(e+f x)\right)^{3/2}}","-\frac{8 b (a+2 b) \sec (e+f x)}{3 a^4 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{4 b (a+2 b) \sec (e+f x)}{3 a^3 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{(a+2 b) \cos (e+f x)}{a^2 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}+\frac{\cos ^3(e+f x)}{3 a f \left(a+b \sec ^2(e+f x)\right)^{3/2}}",1,"-(((a + 2*b)*Cos[e + f*x])/(a^2*f*(a + b*Sec[e + f*x]^2)^(3/2))) + Cos[e + f*x]^3/(3*a*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (4*b*(a + 2*b)*Sec[e + f*x])/(3*a^3*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (8*b*(a + 2*b)*Sec[e + f*x])/(3*a^4*f*Sqrt[a + b*Sec[e + f*x]^2])","A",5,5,25,0.2000,1,"{4134, 453, 271, 192, 191}"
121,1,97,0,0.0684935,"\int \frac{\sin (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{8 b \sec (e+f x)}{3 a^3 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{4 b \sec (e+f x)}{3 a^2 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\cos (e+f x)}{a f \left(a+b \sec ^2(e+f x)\right)^{3/2}}","-\frac{8 b \sec (e+f x)}{3 a^3 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{4 b \sec (e+f x)}{3 a^2 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\cos (e+f x)}{a f \left(a+b \sec ^2(e+f x)\right)^{3/2}}",1,"-(Cos[e + f*x]/(a*f*(a + b*Sec[e + f*x]^2)^(3/2))) - (4*b*Sec[e + f*x])/(3*a^2*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (8*b*Sec[e + f*x])/(3*a^3*f*Sqrt[a + b*Sec[e + f*x]^2])","A",4,4,23,0.1739,1,"{4134, 271, 192, 191}"
122,1,127,0,0.1417609,"\int \frac{\csc (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Csc[e + f*x]/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{b (5 a+2 b) \sec (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}-\frac{b \sec (e+f x)}{3 a f (a+b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f (a+b)^{5/2}}","-\frac{b (5 a+2 b) \sec (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}-\frac{b \sec (e+f x)}{3 a f (a+b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{f (a+b)^{5/2}}",1,"-(ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]]/((a + b)^(5/2)*f)) - (b*Sec[e + f*x])/(3*a*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (b*(5*a + 2*b)*Sec[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2])","A",6,6,23,0.2609,1,"{4134, 414, 527, 12, 377, 207}"
123,1,171,0,0.2064488,"\int \frac{\csc ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Csc[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{b (13 a-2 b) \sec (e+f x)}{6 a f (a+b)^3 \sqrt{a+b \sec ^2(e+f x)}}-\frac{5 b \sec (e+f x)}{6 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{(a-4 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f (a+b)^{7/2}}-\frac{\cot (e+f x) \csc (e+f x)}{2 f (a+b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}","-\frac{b (13 a-2 b) \sec (e+f x)}{6 a f (a+b)^3 \sqrt{a+b \sec ^2(e+f x)}}-\frac{5 b \sec (e+f x)}{6 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{(a-4 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{2 f (a+b)^{7/2}}-\frac{\cot (e+f x) \csc (e+f x)}{2 f (a+b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}",1,"-((a - 4*b)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*(a + b)^(7/2)*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (5*b*Sec[e + f*x])/(6*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^(3/2)) - ((13*a - 2*b)*b*Sec[e + f*x])/(6*a*(a + b)^3*f*Sqrt[a + b*Sec[e + f*x]^2])","A",7,6,25,0.2400,1,"{4134, 471, 527, 12, 377, 207}"
124,1,234,0,0.3239453,"\int \frac{\csc ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{\left(3 a^2-24 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{8 f (a+b)^{9/2}}-\frac{5 b (11 a-10 b) \sec (e+f x)}{24 f (a+b)^4 \sqrt{a+b \sec ^2(e+f x)}}-\frac{b (23 a-12 b) \sec (e+f x)}{24 f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 f (a+b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{(5 a-2 b) \cot (e+f x) \csc (e+f x)}{8 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^{3/2}}","-\frac{\left(3 a^2-24 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right)}{8 f (a+b)^{9/2}}-\frac{5 b (11 a-10 b) \sec (e+f x)}{24 f (a+b)^4 \sqrt{a+b \sec ^2(e+f x)}}-\frac{b (23 a-12 b) \sec (e+f x)}{24 f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 f (a+b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{(5 a-2 b) \cot (e+f x) \csc (e+f x)}{8 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^{3/2}}",1,"-((3*a^2 - 24*a*b + 8*b^2)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(8*(a + b)^(9/2)*f) - ((5*a - 2*b)*Cot[e + f*x]*Csc[e + f*x])/(8*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)) - ((23*a - 12*b)*b*Sec[e + f*x])/(24*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (5*(11*a - 10*b)*b*Sec[e + f*x])/(24*(a + b)^4*f*Sqrt[a + b*Sec[e + f*x]^2])","A",8,6,25,0.2400,1,"{4134, 470, 527, 12, 377, 207}"
125,1,288,0,0.4418568,"\int \frac{\sin ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{5 (a+b) \left(a^2+14 a b+21 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{11/2} f}-\frac{b \left(113 a^2+420 a b+315 b^2\right) \tan (e+f x)}{48 a^5 f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{7 b (a+b) (7 a+15 b) \tan (e+f x)}{48 a^4 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{3 (a+b) \sin (e+f x) \cos ^3(e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{(a+b) (11 a+21 b) \sin (e+f x) \cos (e+f x)}{16 a^3 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","\frac{5 (a+b) \left(a^2+14 a b+21 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{11/2} f}-\frac{b \left(113 a^2+420 a b+315 b^2\right) \tan (e+f x)}{48 a^5 f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{7 b (a+b) (7 a+15 b) \tan (e+f x)}{48 a^4 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{3 (a+b) \sin (e+f x) \cos ^3(e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{(a+b) (11 a+21 b) \sin (e+f x) \cos (e+f x)}{16 a^3 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"(5*(a + b)*(a^2 + 14*a*b + 21*b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(11/2)*f) - ((a + b)*(11*a + 21*b)*Cos[e + f*x]*Sin[e + f*x])/(16*a^3*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (3*(a + b)*Cos[e + f*x]^3*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (Cos[e + f*x]^3*Sin[e + f*x]^3)/(6*a*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (7*b*(a + b)*(7*a + 15*b)*Tan[e + f*x])/(48*a^4*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(113*a^2 + 420*a*b + 315*b^2)*Tan[e + f*x])/(48*a^5*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",9,7,25,0.2800,1,"{4132, 470, 578, 527, 12, 377, 203}"
126,1,227,0,0.302317,"\int \frac{\sin ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\left(3 a^2+30 a b+35 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{9/2} f}-\frac{5 b (11 a+21 b) \tan (e+f x)}{24 a^4 f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{b (23 a+35 b) \tan (e+f x)}{24 a^3 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{(5 a+7 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","\frac{\left(3 a^2+30 a b+35 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{9/2} f}-\frac{5 b (11 a+21 b) \tan (e+f x)}{24 a^4 f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{b (23 a+35 b) \tan (e+f x)}{24 a^3 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{(5 a+7 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"((3*a^2 + 30*a*b + 35*b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(9/2)*f) - ((5*a + 7*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(23*a + 35*b)*Tan[e + f*x])/(24*a^3*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (5*b*(11*a + 21*b)*Tan[e + f*x])/(24*a^4*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",8,6,25,0.2400,1,"{4132, 470, 527, 12, 377, 203}"
127,1,167,0,0.201752,"\int \frac{\sin ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{(a+5 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 a^{7/2} f}-\frac{b (13 a+15 b) \tan (e+f x)}{6 a^3 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{5 b \tan (e+f x)}{6 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","\frac{(a+5 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 a^{7/2} f}-\frac{b (13 a+15 b) \tan (e+f x)}{6 a^3 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{5 b \tan (e+f x)}{6 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"((a + 5*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*a^(7/2)*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (5*b*Tan[e + f*x])/(6*a^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(13*a + 15*b)*Tan[e + f*x])/(6*a^3*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",7,6,25,0.2400,1,"{4132, 471, 527, 12, 377, 203}"
128,1,125,0,0.0990468,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[(a + b*Sec[e + f*x]^2)^(-5/2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}-\frac{b (5 a+3 b) \tan (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{b \tan (e+f x)}{3 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}-\frac{b (5 a+3 b) \tan (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{b \tan (e+f x)}{3 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f) - (b*Tan[e + f*x])/(3*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(5*a + 3*b)*Tan[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",6,6,16,0.3750,1,"{4128, 414, 527, 12, 377, 203}"
129,1,106,0,0.1082491,"\int \frac{\csc ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Csc[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{8 b \tan (e+f x)}{3 f (a+b)^3 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{4 b \tan (e+f x)}{3 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{\cot (e+f x)}{f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","-\frac{8 b \tan (e+f x)}{3 f (a+b)^3 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{4 b \tan (e+f x)}{3 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{\cot (e+f x)}{f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"-(Cot[e + f*x]/((a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2))) - (4*b*Tan[e + f*x])/(3*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (8*b*Tan[e + f*x])/(3*(a + b)^3*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",4,4,25,0.1600,1,"{4132, 271, 192, 191}"
130,1,158,0,0.1607173,"\int \frac{\csc ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{8 b (a-b) \tan (e+f x)}{3 f (a+b)^4 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{4 b (a-b) \tan (e+f x)}{3 f (a+b)^3 \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{\cot ^3(e+f x)}{3 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{(a-b) \cot (e+f x)}{f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","-\frac{8 b (a-b) \tan (e+f x)}{3 f (a+b)^4 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{4 b (a-b) \tan (e+f x)}{3 f (a+b)^3 \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{\cot ^3(e+f x)}{3 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{(a-b) \cot (e+f x)}{f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"-(((a - b)*Cot[e + f*x])/((a + b)^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2))) - Cot[e + f*x]^3/(3*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (4*(a - b)*b*Tan[e + f*x])/(3*(a + b)^3*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (8*(a - b)*b*Tan[e + f*x])/(3*(a + b)^4*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",5,5,25,0.2000,1,"{4132, 453, 271, 192, 191}"
131,1,226,0,0.2397596,"\int \frac{\csc ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Csc[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{8 b \left(5 a^2-10 a b+b^2\right) \tan (e+f x)}{15 f (a+b)^5 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{4 b \left(5 a^2-10 a b+b^2\right) \tan (e+f x)}{15 f (a+b)^4 \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{\left(5 a^2-10 a b+b^2\right) \cot (e+f x)}{5 f (a+b)^3 \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{\cot ^5(e+f x)}{5 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{2 (5 a+b) \cot ^3(e+f x)}{15 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","-\frac{8 b \left(5 a^2-10 a b+b^2\right) \tan (e+f x)}{15 f (a+b)^5 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{4 b \left(5 a^2-10 a b+b^2\right) \tan (e+f x)}{15 f (a+b)^4 \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{\left(5 a^2-10 a b+b^2\right) \cot (e+f x)}{5 f (a+b)^3 \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{\cot ^5(e+f x)}{5 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}-\frac{2 (5 a+b) \cot ^3(e+f x)}{15 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"-((5*a^2 - 10*a*b + b^2)*Cot[e + f*x])/(5*(a + b)^3*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (2*(5*a + b)*Cot[e + f*x]^3)/(15*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - Cot[e + f*x]^5/(5*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (4*b*(5*a^2 - 10*a*b + b^2)*Tan[e + f*x])/(15*(a + b)^4*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (8*b*(5*a^2 - 10*a*b + b^2)*Tan[e + f*x])/(15*(a + b)^5*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",6,6,25,0.2400,1,"{4132, 462, 453, 271, 192, 191}"
132,0,0,0,0.0454144,"\int \left(a+b \sec ^2(e+f x)\right)^p (d \sin (e+f x))^m \, dx","Int[(a + b*Sec[e + f*x]^2)^p*(d*Sin[e + f*x])^m,x]","\int \left(a+b \sec ^2(e+f x)\right)^p (d \sin (e+f x))^m \, dx","\frac{\tan (e+f x) \cos ^2(e+f x)^{p+\frac{1}{2}} (d \sin (e+f x))^m \left(\frac{-a \sin ^2(e+f x)+a+b}{a+b}\right)^{-p} \left(a+b \sec ^2(e+f x)\right)^p F_1\left(\frac{m+1}{2};p+\frac{1}{2},-p;\frac{m+3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)}{f (m+1)}",1,"Defer[Int][(a + b*Sec[e + f*x]^2)^p*(d*Sin[e + f*x])^m, x]","F",0,0,0,0,-1,"{}"
133,1,182,0,0.1921251,"\int \left(a+b \sec ^2(e+f x)\right)^p \sin ^5(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^5,x]","-\frac{\left(15 a^2+10 a b (1-2 p)+b^2 \left(4 p^2-8 p+3\right)\right) \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \left(\frac{b \sec ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a}\right)}{15 a^2 f}+\frac{(10 a+b (3-2 p)) \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{p+1}}{15 a^2 f}-\frac{\cos ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^{p+1}}{5 a f}","-\frac{\left(15 a^2+10 a b (1-2 p)+b^2 \left(4 p^2-8 p+3\right)\right) \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \left(\frac{b \sec ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a}\right)}{15 a^2 f}+\frac{(10 a+b (3-2 p)) \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{p+1}}{15 a^2 f}-\frac{\cos ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^{p+1}}{5 a f}",1,"((10*a + b*(3 - 2*p))*Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(1 + p))/(15*a^2*f) - (Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(1 + p))/(5*a*f) - ((15*a^2 + 10*a*b*(1 - 2*p) + b^2*(3 - 8*p + 4*p^2))*Cos[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/a)]*(a + b*Sec[e + f*x]^2)^p)/(15*a^2*f*(1 + (b*Sec[e + f*x]^2)/a)^p)","A",5,5,23,0.2174,1,"{4134, 462, 453, 365, 364}"
134,1,117,0,0.0953044,"\int \left(a+b \sec ^2(e+f x)\right)^p \sin ^3(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^3,x]","\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{p+1}}{3 a f}-\frac{(3 a-2 b p+b) \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \left(\frac{b \sec ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a}\right)}{3 a f}","\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{p+1}}{3 a f}-\frac{(3 a-2 b p+b) \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \left(\frac{b \sec ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a}\right)}{3 a f}",1,"(Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(1 + p))/(3*a*f) - ((3*a + b - 2*b*p)*Cos[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/a)]*(a + b*Sec[e + f*x]^2)^p)/(3*a*f*(1 + (b*Sec[e + f*x]^2)/a)^p)","A",4,4,23,0.1739,1,"{4134, 453, 365, 364}"
135,1,68,0,0.0486125,"\int \left(a+b \sec ^2(e+f x)\right)^p \sin (e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x],x]","-\frac{\cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \left(\frac{b \sec ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a}\right)}{f}","-\frac{\cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \left(\frac{b \sec ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a}\right)}{f}",1,"-((Cos[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/a)]*(a + b*Sec[e + f*x]^2)^p)/(f*(1 + (b*Sec[e + f*x]^2)/a)^p))","A",3,3,21,0.1429,1,"{4134, 365, 364}"
136,1,77,0,0.077101,"\int \csc (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Csc[e + f*x]*(a + b*Sec[e + f*x]^2)^p,x]","-\frac{\sec (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \left(\frac{b \sec ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};\sec ^2(e+f x),-\frac{b \sec ^2(e+f x)}{a}\right)}{f}","-\frac{\sec (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \left(\frac{b \sec ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};\sec ^2(e+f x),-\frac{b \sec ^2(e+f x)}{a}\right)}{f}",1,"-((AppellF1[1/2, 1, -p, 3/2, Sec[e + f*x]^2, -((b*Sec[e + f*x]^2)/a)]*Sec[e + f*x]*(a + b*Sec[e + f*x]^2)^p)/(f*(1 + (b*Sec[e + f*x]^2)/a)^p))","A",3,3,21,0.1429,1,"{4134, 430, 429}"
137,1,81,0,0.0914773,"\int \csc ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\sec ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \left(\frac{b \sec ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{3}{2};2,-p;\frac{5}{2};\sec ^2(e+f x),-\frac{b \sec ^2(e+f x)}{a}\right)}{3 f}","\frac{\sec ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \left(\frac{b \sec ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{3}{2};2,-p;\frac{5}{2};\sec ^2(e+f x),-\frac{b \sec ^2(e+f x)}{a}\right)}{3 f}",1,"(AppellF1[3/2, 2, -p, 5/2, Sec[e + f*x]^2, -((b*Sec[e + f*x]^2)/a)]*Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^p)/(3*f*(1 + (b*Sec[e + f*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{4134, 511, 510}"
138,1,88,0,0.1218439,"\int \left(a+b \sec ^2(e+f x)\right)^p \sin ^4(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^4,x]","\frac{\tan ^5(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{5}{2};3,-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{5 f}","\frac{\tan ^5(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{5}{2};3,-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{5 f}",1,"(AppellF1[5/2, 3, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^5*(a + b + b*Tan[e + f*x]^2)^p)/(5*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",3,3,23,0.1304,1,"{4132, 511, 510}"
139,1,88,0,0.0985019,"\int \left(a+b \sec ^2(e+f x)\right)^p \sin ^2(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^2,x]","\frac{\tan ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{3}{2};2,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{3 f}","\frac{\tan ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{3}{2};2,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{3 f}",1,"(AppellF1[3/2, 2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^p)/(3*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",3,3,23,0.1304,1,"{4132, 511, 510}"
140,1,83,0,0.0502151,"\int \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[(a + b*Sec[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}",1,"(AppellF1[1/2, 1, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",3,3,14,0.2143,1,"{4128, 430, 429}"
141,1,73,0,0.0666715,"\int \csc ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Csc[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p,x]","-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}","-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}",1,"-((Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p))","A",3,3,23,0.1304,1,"{4132, 365, 364}"
142,1,128,0,0.106278,"\int \csc ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Csc[e + f*x]^4*(a + b*Sec[e + f*x]^2)^p,x]","-\frac{(3 a+2 b (p+1)) \cot (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)}{3 f (a+b)}-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{p+1}}{3 f (a+b)}","-\frac{(3 a+2 b (p+1)) \cot (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)}{3 f (a+b)}-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{p+1}}{3 f (a+b)}",1,"-(Cot[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^(1 + p))/(3*(a + b)*f) - ((3*a + 2*b*(1 + p))*Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + b + b*Tan[e + f*x]^2)^p)/(3*(a + b)*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",4,4,23,0.1739,1,"{4132, 453, 365, 364}"
143,1,192,0,0.1735664,"\int \csc ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Csc[e + f*x]^6*(a + b*Sec[e + f*x]^2)^p,x]","-\frac{\left(15 a^2+20 a b (p+1)+4 b^2 \left(p^2+3 p+2\right)\right) \cot (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)}{15 f (a+b)^2}-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{p+1}}{5 f (a+b)}-\frac{(10 a+b (2 p+7)) \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{p+1}}{15 f (a+b)^2}","-\frac{\left(15 a^2+20 a b (p+1)+4 b^2 \left(p^2+3 p+2\right)\right) \cot (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)}{15 f (a+b)^2}-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{p+1}}{5 f (a+b)}-\frac{(10 a+b (2 p+7)) \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{p+1}}{15 f (a+b)^2}",1,"-((10*a + b*(7 + 2*p))*Cot[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^(1 + p))/(15*(a + b)^2*f) - (Cot[e + f*x]^5*(a + b + b*Tan[e + f*x]^2)^(1 + p))/(5*(a + b)*f) - ((15*a^2 + 20*a*b*(1 + p) + 4*b^2*(2 + 3*p + p^2))*Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + b + b*Tan[e + f*x]^2)^p)/(15*(a + b)^2*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",5,5,23,0.2174,1,"{4132, 462, 453, 365, 364}"
144,1,74,0,0.047288,"\int \left(a-a \sec ^2(c+d x)\right)^4 \, dx","Int[(a - a*Sec[c + d*x]^2)^4,x]","\frac{a^4 \tan ^7(c+d x)}{7 d}-\frac{a^4 \tan ^5(c+d x)}{5 d}+\frac{a^4 \tan ^3(c+d x)}{3 d}-\frac{a^4 \tan (c+d x)}{d}+a^4 x","\frac{a^4 \tan ^7(c+d x)}{7 d}-\frac{a^4 \tan ^5(c+d x)}{5 d}+\frac{a^4 \tan ^3(c+d x)}{3 d}-\frac{a^4 \tan (c+d x)}{d}+a^4 x",1,"a^4*x - (a^4*Tan[c + d*x])/d + (a^4*Tan[c + d*x]^3)/(3*d) - (a^4*Tan[c + d*x]^5)/(5*d) + (a^4*Tan[c + d*x]^7)/(7*d)","A",6,3,15,0.2000,1,"{4120, 3473, 8}"
145,1,56,0,0.0385128,"\int \left(a-a \sec ^2(c+d x)\right)^3 \, dx","Int[(a - a*Sec[c + d*x]^2)^3,x]","-\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}+a^3 x","-\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}+a^3 x",1,"a^3*x - (a^3*Tan[c + d*x])/d + (a^3*Tan[c + d*x]^3)/(3*d) - (a^3*Tan[c + d*x]^5)/(5*d)","A",5,3,15,0.2000,1,"{4120, 3473, 8}"
146,1,38,0,0.0300588,"\int \left(a-a \sec ^2(c+d x)\right)^2 \, dx","Int[(a - a*Sec[c + d*x]^2)^2,x]","\frac{a^2 \tan ^3(c+d x)}{3 d}-\frac{a^2 \tan (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}+a^2 x",1,"a^2*x - (a^2*Tan[c + d*x])/d + (a^2*Tan[c + d*x]^3)/(3*d)","A",4,3,15,0.2000,1,"{4120, 3473, 8}"
147,1,16,0,0.0126255,"\int \left(a-a \sec ^2(c+d x)\right) \, dx","Int[a - a*Sec[c + d*x]^2,x]","a x-\frac{a \tan (c+d x)}{d}","a x-\frac{a \tan (c+d x)}{d}",1,"a*x - (a*Tan[c + d*x])/d","A",3,2,13,0.1538,1,"{3767, 8}"
148,1,19,0,0.0214506,"\int \frac{1}{a-a \sec ^2(c+d x)} \, dx","Int[(a - a*Sec[c + d*x]^2)^(-1),x]","\frac{\cot (c+d x)}{a d}+\frac{x}{a}","\frac{\cot (c+d x)}{a d}+\frac{x}{a}",1,"x/a + Cot[c + d*x]/(a*d)","A",3,3,15,0.2000,1,"{4120, 3473, 8}"
149,1,37,0,0.0299385,"\int \frac{1}{\left(a-a \sec ^2(c+d x)\right)^2} \, dx","Int[(a - a*Sec[c + d*x]^2)^(-2),x]","-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{\cot (c+d x)}{a^2 d}+\frac{x}{a^2}","-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{\cot (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)","A",4,3,15,0.2000,1,"{4120, 3473, 8}"
150,1,55,0,0.0387751,"\int \frac{1}{\left(a-a \sec ^2(c+d x)\right)^3} \, dx","Int[(a - a*Sec[c + d*x]^2)^(-3),x]","\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{x}{a^3}","\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{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)","A",5,3,15,0.2000,1,"{4120, 3473, 8}"
151,1,73,0,0.0453561,"\int \frac{1}{\left(a-a \sec ^2(c+d x)\right)^4} \, dx","Int[(a - a*Sec[c + d*x]^2)^(-4),x]","-\frac{\cot ^7(c+d x)}{7 a^4 d}+\frac{\cot ^5(c+d x)}{5 a^4 d}-\frac{\cot ^3(c+d x)}{3 a^4 d}+\frac{\cot (c+d x)}{a^4 d}+\frac{x}{a^4}","-\frac{\cot ^7(c+d x)}{7 a^4 d}+\frac{\cot ^5(c+d x)}{5 a^4 d}-\frac{\cot ^3(c+d x)}{3 a^4 d}+\frac{\cot (c+d x)}{a^4 d}+\frac{x}{a^4}",1,"x/a^4 + Cot[c + d*x]/(a^4*d) - Cot[c + d*x]^3/(3*a^4*d) + Cot[c + d*x]^5/(5*a^4*d) - Cot[c + d*x]^7/(7*a^4*d)","A",6,3,15,0.2000,1,"{4120, 3473, 8}"
152,1,98,0,0.0593756,"\int \sec ^5(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Sec[e + f*x]^5*(a + b*Sec[e + f*x]^2),x]","\frac{(6 a+5 b) \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac{(6 a+5 b) \tan (e+f x) \sec ^3(e+f x)}{24 f}+\frac{(6 a+5 b) \tan (e+f x) \sec (e+f x)}{16 f}+\frac{b \tan (e+f x) \sec ^5(e+f x)}{6 f}","\frac{(6 a+5 b) \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac{(6 a+5 b) \tan (e+f x) \sec ^3(e+f x)}{24 f}+\frac{(6 a+5 b) \tan (e+f x) \sec (e+f x)}{16 f}+\frac{b \tan (e+f x) \sec ^5(e+f x)}{6 f}",1,"((6*a + 5*b)*ArcTanh[Sin[e + f*x]])/(16*f) + ((6*a + 5*b)*Sec[e + f*x]*Tan[e + f*x])/(16*f) + ((6*a + 5*b)*Sec[e + f*x]^3*Tan[e + f*x])/(24*f) + (b*Sec[e + f*x]^5*Tan[e + f*x])/(6*f)","A",4,3,21,0.1429,1,"{4046, 3768, 3770}"
153,1,70,0,0.0457003,"\int \sec ^3(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2),x]","\frac{(4 a+3 b) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{(4 a+3 b) \tan (e+f x) \sec (e+f x)}{8 f}+\frac{b \tan (e+f x) \sec ^3(e+f x)}{4 f}","\frac{(4 a+3 b) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{(4 a+3 b) \tan (e+f x) \sec (e+f x)}{8 f}+\frac{b \tan (e+f x) \sec ^3(e+f x)}{4 f}",1,"((4*a + 3*b)*ArcTanh[Sin[e + f*x]])/(8*f) + ((4*a + 3*b)*Sec[e + f*x]*Tan[e + f*x])/(8*f) + (b*Sec[e + f*x]^3*Tan[e + f*x])/(4*f)","A",3,3,21,0.1429,1,"{4046, 3768, 3770}"
154,1,40,0,0.0248988,"\int \sec (e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Sec[e + f*x]*(a + b*Sec[e + f*x]^2),x]","\frac{(2 a+b) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{b \tan (e+f x) \sec (e+f x)}{2 f}","\frac{(2 a+b) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{b \tan (e+f x) \sec (e+f x)}{2 f}",1,"((2*a + b)*ArcTanh[Sin[e + f*x]])/(2*f) + (b*Sec[e + f*x]*Tan[e + f*x])/(2*f)","A",2,2,19,0.1053,1,"{4046, 3770}"
155,1,24,0,0.0275183,"\int \cos (e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Cos[e + f*x]*(a + b*Sec[e + f*x]^2),x]","\frac{a \sin (e+f x)}{f}+\frac{b \tanh ^{-1}(\sin (e+f x))}{f}","\frac{a \sin (e+f x)}{f}+\frac{b \tanh ^{-1}(\sin (e+f x))}{f}",1,"(b*ArcTanh[Sin[e + f*x]])/f + (a*Sin[e + f*x])/f","A",2,2,19,0.1053,1,"{4045, 3770}"
156,1,30,0,0.045325,"\int \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2),x]","\frac{(a+b) \sin (e+f x)}{f}-\frac{a \sin ^3(e+f x)}{3 f}","\frac{(a+b) \sin (e+f x)}{f}-\frac{a \sin ^3(e+f x)}{3 f}",1,"((a + b)*Sin[e + f*x])/f - (a*Sin[e + f*x]^3)/(3*f)","A",3,2,21,0.09524,1,"{4044, 3013}"
157,1,50,0,0.0662125,"\int \cos ^5(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2),x]","-\frac{(2 a+b) \sin ^3(e+f x)}{3 f}+\frac{(a+b) \sin (e+f x)}{f}+\frac{a \sin ^5(e+f x)}{5 f}","-\frac{(2 a+b) \sin ^3(e+f x)}{3 f}+\frac{(a+b) \sin (e+f x)}{f}+\frac{a \sin ^5(e+f x)}{5 f}",1,"((a + b)*Sin[e + f*x])/f - ((2*a + b)*Sin[e + f*x]^3)/(3*f) + (a*Sin[e + f*x]^5)/(5*f)","A",4,3,21,0.1429,1,"{4044, 3013, 373}"
158,1,87,0,0.049371,"\int \sec ^6(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Sec[e + f*x]^6*(a + b*Sec[e + f*x]^2),x]","\frac{(7 a+6 b) \tan ^5(e+f x)}{35 f}+\frac{2 (7 a+6 b) \tan ^3(e+f x)}{21 f}+\frac{(7 a+6 b) \tan (e+f x)}{7 f}+\frac{b \tan (e+f x) \sec ^6(e+f x)}{7 f}","\frac{(7 a+6 b) \tan ^5(e+f x)}{35 f}+\frac{2 (7 a+6 b) \tan ^3(e+f x)}{21 f}+\frac{(7 a+6 b) \tan (e+f x)}{7 f}+\frac{b \tan (e+f x) \sec ^6(e+f x)}{7 f}",1,"((7*a + 6*b)*Tan[e + f*x])/(7*f) + (b*Sec[e + f*x]^6*Tan[e + f*x])/(7*f) + (2*(7*a + 6*b)*Tan[e + f*x]^3)/(21*f) + ((7*a + 6*b)*Tan[e + f*x]^5)/(35*f)","A",3,2,21,0.09524,1,"{4046, 3767}"
159,1,65,0,0.0432743,"\int \sec ^4(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Sec[e + f*x]^4*(a + b*Sec[e + f*x]^2),x]","\frac{(5 a+4 b) \tan ^3(e+f x)}{15 f}+\frac{(5 a+4 b) \tan (e+f x)}{5 f}+\frac{b \tan (e+f x) \sec ^4(e+f x)}{5 f}","\frac{(5 a+4 b) \tan ^3(e+f x)}{15 f}+\frac{(5 a+4 b) \tan (e+f x)}{5 f}+\frac{b \tan (e+f x) \sec ^4(e+f x)}{5 f}",1,"((5*a + 4*b)*Tan[e + f*x])/(5*f) + (b*Sec[e + f*x]^4*Tan[e + f*x])/(5*f) + ((5*a + 4*b)*Tan[e + f*x]^3)/(15*f)","A",3,2,21,0.09524,1,"{4046, 3767}"
160,1,43,0,0.0383772,"\int \sec ^2(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Sec[e + f*x]^2*(a + b*Sec[e + f*x]^2),x]","\frac{(3 a+2 b) \tan (e+f x)}{3 f}+\frac{b \tan (e+f x) \sec ^2(e+f x)}{3 f}","\frac{(3 a+2 b) \tan (e+f x)}{3 f}+\frac{b \tan (e+f x) \sec ^2(e+f x)}{3 f}",1,"((3*a + 2*b)*Tan[e + f*x])/(3*f) + (b*Sec[e + f*x]^2*Tan[e + f*x])/(3*f)","A",3,3,21,0.1429,1,"{4046, 3767, 8}"
161,1,15,0,0.0125057,"\int \left(a+b \sec ^2(e+f x)\right) \, dx","Int[a + b*Sec[e + f*x]^2,x]","a x+\frac{b \tan (e+f x)}{f}","a x+\frac{b \tan (e+f x)}{f}",1,"a*x + (b*Tan[e + f*x])/f","A",3,2,12,0.1667,1,"{3767, 8}"
162,1,31,0,0.0273794,"\int \cos ^2(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Cos[e + f*x]^2*(a + b*Sec[e + f*x]^2),x]","\frac{1}{2} x (a+2 b)+\frac{a \sin (e+f x) \cos (e+f x)}{2 f}","\frac{1}{2} x (a+2 b)+\frac{a \sin (e+f x) \cos (e+f x)}{2 f}",1,"((a + 2*b)*x)/2 + (a*Cos[e + f*x]*Sin[e + f*x])/(2*f)","A",2,2,21,0.09524,1,"{4045, 8}"
163,1,61,0,0.041084,"\int \cos ^4(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2),x]","\frac{(3 a+4 b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} x (3 a+4 b)+\frac{a \sin (e+f x) \cos ^3(e+f x)}{4 f}","\frac{(3 a+4 b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} x (3 a+4 b)+\frac{a \sin (e+f x) \cos ^3(e+f x)}{4 f}",1,"((3*a + 4*b)*x)/8 + ((3*a + 4*b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a*Cos[e + f*x]^3*Sin[e + f*x])/(4*f)","A",3,3,21,0.1429,1,"{4045, 2635, 8}"
164,1,89,0,0.052592,"\int \cos ^6(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Cos[e + f*x]^6*(a + b*Sec[e + f*x]^2),x]","\frac{(5 a+6 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{(5 a+6 b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} x (5 a+6 b)+\frac{a \sin (e+f x) \cos ^5(e+f x)}{6 f}","\frac{(5 a+6 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{(5 a+6 b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} x (5 a+6 b)+\frac{a \sin (e+f x) \cos ^5(e+f x)}{6 f}",1,"((5*a + 6*b)*x)/16 + ((5*a + 6*b)*Cos[e + f*x]*Sin[e + f*x])/(16*f) + ((5*a + 6*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) + (a*Cos[e + f*x]^5*Sin[e + f*x])/(6*f)","A",4,3,21,0.1429,1,"{4045, 2635, 8}"
165,1,165,0,0.1410822,"\int \sec ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Sec[e + f*x]^5*(a + b*Sec[e + f*x]^2)^2,x]","\frac{\left(48 a^2+80 a b+35 b^2\right) \tanh ^{-1}(\sin (e+f x))}{128 f}+\frac{\left(48 a^2+80 a b+35 b^2\right) \tan (e+f x) \sec ^3(e+f x)}{192 f}+\frac{\left(48 a^2+80 a b+35 b^2\right) \tan (e+f x) \sec (e+f x)}{128 f}+\frac{b (10 a+7 b) \tan (e+f x) \sec ^5(e+f x)}{48 f}+\frac{b \tan (e+f x) \sec ^7(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{8 f}","\frac{\left(48 a^2+80 a b+35 b^2\right) \tanh ^{-1}(\sin (e+f x))}{128 f}+\frac{\left(48 a^2+80 a b+35 b^2\right) \tan (e+f x) \sec ^3(e+f x)}{192 f}+\frac{\left(48 a^2+80 a b+35 b^2\right) \tan (e+f x) \sec (e+f x)}{128 f}+\frac{b (10 a+7 b) \tan (e+f x) \sec ^5(e+f x)}{48 f}+\frac{b \tan (e+f x) \sec ^7(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{8 f}",1,"((48*a^2 + 80*a*b + 35*b^2)*ArcTanh[Sin[e + f*x]])/(128*f) + ((48*a^2 + 80*a*b + 35*b^2)*Sec[e + f*x]*Tan[e + f*x])/(128*f) + ((48*a^2 + 80*a*b + 35*b^2)*Sec[e + f*x]^3*Tan[e + f*x])/(192*f) + (b*(10*a + 7*b)*Sec[e + f*x]^5*Tan[e + f*x])/(48*f) + (b*Sec[e + f*x]^7*(a + b - a*Sin[e + f*x]^2)*Tan[e + f*x])/(8*f)","A",6,5,23,0.2174,1,"{4147, 413, 385, 199, 206}"
166,1,129,0,0.1340583,"\int \sec ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2,x]","\frac{\left(8 a^2+12 a b+5 b^2\right) \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac{\left(8 a^2+12 a b+5 b^2\right) \tan (e+f x) \sec (e+f x)}{16 f}+\frac{b (8 a+5 b) \tan (e+f x) \sec ^3(e+f x)}{24 f}+\frac{b \tan (e+f x) \sec ^5(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{6 f}","\frac{\left(8 a^2+12 a b+5 b^2\right) \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac{\left(8 a^2+12 a b+5 b^2\right) \tan (e+f x) \sec (e+f x)}{16 f}+\frac{b (8 a+5 b) \tan (e+f x) \sec ^3(e+f x)}{24 f}+\frac{b \tan (e+f x) \sec ^5(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{6 f}",1,"((8*a^2 + 12*a*b + 5*b^2)*ArcTanh[Sin[e + f*x]])/(16*f) + ((8*a^2 + 12*a*b + 5*b^2)*Sec[e + f*x]*Tan[e + f*x])/(16*f) + (b*(8*a + 5*b)*Sec[e + f*x]^3*Tan[e + f*x])/(24*f) + (b*Sec[e + f*x]^5*(a + b - a*Sin[e + f*x]^2)*Tan[e + f*x])/(6*f)","A",5,5,23,0.2174,1,"{4147, 413, 385, 199, 206}"
167,1,91,0,0.0739133,"\int \sec (e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Sec[e + f*x]*(a + b*Sec[e + f*x]^2)^2,x]","\frac{\left(8 a^2+8 a b+3 b^2\right) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{3 b (2 a+b) \tan (e+f x) \sec (e+f x)}{8 f}+\frac{b \tan (e+f x) \sec ^3(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{4 f}","\frac{\left(8 a^2+8 a b+3 b^2\right) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{3 b (2 a+b) \tan (e+f x) \sec (e+f x)}{8 f}+\frac{b \tan (e+f x) \sec ^3(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{4 f}",1,"((8*a^2 + 8*a*b + 3*b^2)*ArcTanh[Sin[e + f*x]])/(8*f) + (3*b*(2*a + b)*Sec[e + f*x]*Tan[e + f*x])/(8*f) + (b*Sec[e + f*x]^3*(a + b - a*Sin[e + f*x]^2)*Tan[e + f*x])/(4*f)","A",4,4,21,0.1905,1,"{4147, 413, 385, 206}"
168,1,56,0,0.0673528,"\int \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^2,x]","\frac{a^2 \sin (e+f x)}{f}+\frac{b (4 a+b) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{b^2 \tan (e+f x) \sec (e+f x)}{2 f}","\frac{a^2 \sin (e+f x)}{f}+\frac{b (4 a+b) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{b^2 \tan (e+f x) \sec (e+f x)}{2 f}",1,"(b*(4*a + b)*ArcTanh[Sin[e + f*x]])/(2*f) + (a^2*Sin[e + f*x])/f + (b^2*Sec[e + f*x]*Tan[e + f*x])/(2*f)","A",5,4,21,0.1905,1,"{4147, 390, 385, 206}"
169,1,49,0,0.0608709,"\int \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2,x]","-\frac{a^2 \sin ^3(e+f x)}{3 f}+\frac{a (a+2 b) \sin (e+f x)}{f}+\frac{b^2 \tanh ^{-1}(\sin (e+f x))}{f}","-\frac{a^2 \sin ^3(e+f x)}{3 f}+\frac{a (a+2 b) \sin (e+f x)}{f}+\frac{b^2 \tanh ^{-1}(\sin (e+f x))}{f}",1,"(b^2*ArcTanh[Sin[e + f*x]])/f + (a*(a + 2*b)*Sin[e + f*x])/f - (a^2*Sin[e + f*x]^3)/(3*f)","A",4,3,23,0.1304,1,"{4147, 390, 206}"
170,1,53,0,0.0662634,"\int \cos ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^2,x]","\frac{a^2 \sin ^5(e+f x)}{5 f}-\frac{2 a (a+b) \sin ^3(e+f x)}{3 f}+\frac{(a+b)^2 \sin (e+f x)}{f}","\frac{a^2 \sin ^5(e+f x)}{5 f}-\frac{2 a (a+b) \sin ^3(e+f x)}{3 f}+\frac{(a+b)^2 \sin (e+f x)}{f}",1,"((a + b)^2*Sin[e + f*x])/f - (2*a*(a + b)*Sin[e + f*x]^3)/(3*f) + (a^2*Sin[e + f*x]^5)/(5*f)","A",3,2,23,0.08696,1,"{4147, 194}"
171,1,106,0,0.089263,"\int \sec ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Sec[e + f*x]^6*(a + b*Sec[e + f*x]^2)^2,x]","\frac{\left(a^2+6 a b+6 b^2\right) \tan ^5(e+f x)}{5 f}+\frac{2 b (a+2 b) \tan ^7(e+f x)}{7 f}+\frac{2 (a+b) (a+2 b) \tan ^3(e+f x)}{3 f}+\frac{(a+b)^2 \tan (e+f x)}{f}+\frac{b^2 \tan ^9(e+f x)}{9 f}","\frac{\left(a^2+6 a b+6 b^2\right) \tan ^5(e+f x)}{5 f}+\frac{2 b (a+2 b) \tan ^7(e+f x)}{7 f}+\frac{2 (a+b) (a+2 b) \tan ^3(e+f x)}{3 f}+\frac{(a+b)^2 \tan (e+f x)}{f}+\frac{b^2 \tan ^9(e+f x)}{9 f}",1,"((a + b)^2*Tan[e + f*x])/f + (2*(a + b)*(a + 2*b)*Tan[e + f*x]^3)/(3*f) + ((a^2 + 6*a*b + 6*b^2)*Tan[e + f*x]^5)/(5*f) + (2*b*(a + 2*b)*Tan[e + f*x]^7)/(7*f) + (b^2*Tan[e + f*x]^9)/(9*f)","A",3,2,23,0.08696,1,"{4146, 373}"
172,1,80,0,0.0759502,"\int \sec ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Sec[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2,x]","\frac{b (2 a+3 b) \tan ^5(e+f x)}{5 f}+\frac{(a+b) (a+3 b) \tan ^3(e+f x)}{3 f}+\frac{(a+b)^2 \tan (e+f x)}{f}+\frac{b^2 \tan ^7(e+f x)}{7 f}","\frac{b (2 a+3 b) \tan ^5(e+f x)}{5 f}+\frac{(a+b) (a+3 b) \tan ^3(e+f x)}{3 f}+\frac{(a+b)^2 \tan (e+f x)}{f}+\frac{b^2 \tan ^7(e+f x)}{7 f}",1,"((a + b)^2*Tan[e + f*x])/f + ((a + b)*(a + 3*b)*Tan[e + f*x]^3)/(3*f) + (b*(2*a + 3*b)*Tan[e + f*x]^5)/(5*f) + (b^2*Tan[e + f*x]^7)/(7*f)","A",3,2,23,0.08696,1,"{4146, 373}"
173,1,53,0,0.06289,"\int \sec ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Sec[e + f*x]^2*(a + b*Sec[e + f*x]^2)^2,x]","\frac{2 b (a+b) \tan ^3(e+f x)}{3 f}+\frac{(a+b)^2 \tan (e+f x)}{f}+\frac{b^2 \tan ^5(e+f x)}{5 f}","\frac{2 b (a+b) \tan ^3(e+f x)}{3 f}+\frac{(a+b)^2 \tan (e+f x)}{f}+\frac{b^2 \tan ^5(e+f x)}{5 f}",1,"((a + b)^2*Tan[e + f*x])/f + (2*b*(a + b)*Tan[e + f*x]^3)/(3*f) + (b^2*Tan[e + f*x]^5)/(5*f)","A",3,2,23,0.08696,1,"{4146, 194}"
174,1,40,0,0.0294881,"\int \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[(a + b*Sec[e + f*x]^2)^2,x]","a^2 x+\frac{b (2 a+b) \tan (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}","a^2 x+\frac{b (2 a+b) \tan (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"a^2*x + (b*(2*a + b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)","A",4,3,14,0.2143,1,"{4128, 390, 203}"
175,1,47,0,0.0722403,"\int \cos ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Cos[e + f*x]^2*(a + b*Sec[e + f*x]^2)^2,x]","\frac{a^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{1}{2} a x (a+4 b)+\frac{b^2 \tan (e+f x)}{f}","\frac{a^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{1}{2} a x (a+4 b)+\frac{b^2 \tan (e+f x)}{f}",1,"(a*(a + 4*b)*x)/2 + (a^2*Cos[e + f*x]*Sin[e + f*x])/(2*f) + (b^2*Tan[e + f*x])/f","A",5,4,23,0.1739,1,"{4146, 390, 385, 203}"
176,1,81,0,0.0863722,"\int \cos ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2,x]","\frac{1}{8} x \left(3 a^2+8 a b+8 b^2\right)+\frac{3 a (a+2 b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{a \sin (e+f x) \cos ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)}{4 f}","\frac{1}{8} x \left(3 a^2+8 a b+8 b^2\right)+\frac{3 a (a+2 b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{a \sin (e+f x) \cos ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)}{4 f}",1,"((3*a^2 + 8*a*b + 8*b^2)*x)/8 + (3*a*(a + 2*b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a*Cos[e + f*x]^3*Sin[e + f*x]*(a + b + b*Tan[e + f*x]^2))/(4*f)","A",4,4,23,0.1739,1,"{4146, 413, 385, 203}"
177,1,119,0,0.1456375,"\int \cos ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Cos[e + f*x]^6*(a + b*Sec[e + f*x]^2)^2,x]","\frac{\left(5 a^2+12 a b+8 b^2\right) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} x \left(5 a^2+12 a b+8 b^2\right)+\frac{a (5 a+8 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{a \sin (e+f x) \cos ^5(e+f x) \left(a+b \tan ^2(e+f x)+b\right)}{6 f}","\frac{\left(5 a^2+12 a b+8 b^2\right) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} x \left(5 a^2+12 a b+8 b^2\right)+\frac{a (5 a+8 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{a \sin (e+f x) \cos ^5(e+f x) \left(a+b \tan ^2(e+f x)+b\right)}{6 f}",1,"((5*a^2 + 12*a*b + 8*b^2)*x)/16 + ((5*a^2 + 12*a*b + 8*b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (a*(5*a + 8*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) + (a*Cos[e + f*x]^5*Sin[e + f*x]*(a + b + b*Tan[e + f*x]^2))/(6*f)","A",5,5,23,0.2174,1,"{4146, 413, 385, 199, 203}"
178,1,73,0,0.0444889,"\int \left(a+b \sec ^2(c+d x)\right)^3 \, dx","Int[(a + b*Sec[c + d*x]^2)^3,x]","\frac{b \left(3 a^2+3 a b+b^2\right) \tan (c+d x)}{d}+a^3 x+\frac{b^2 (3 a+2 b) \tan ^3(c+d x)}{3 d}+\frac{b^3 \tan ^5(c+d x)}{5 d}","\frac{b \left(3 a^2+3 a b+b^2\right) \tan (c+d x)}{d}+a^3 x+\frac{b^2 (3 a+2 b) \tan ^3(c+d x)}{3 d}+\frac{b^3 \tan ^5(c+d x)}{5 d}",1,"a^3*x + (b*(3*a^2 + 3*a*b + b^2)*Tan[c + d*x])/d + (b^2*(3*a + 2*b)*Tan[c + d*x]^3)/(3*d) + (b^3*Tan[c + d*x]^5)/(5*d)","A",4,3,14,0.2143,1,"{4128, 390, 203}"
179,1,111,0,0.0648342,"\int \left(a+b \sec ^2(c+d x)\right)^4 \, dx","Int[(a + b*Sec[c + d*x]^2)^4,x]","\frac{b^2 \left(6 a^2+8 a b+3 b^2\right) \tan ^3(c+d x)}{3 d}+\frac{b (2 a+b) \left(2 a^2+2 a b+b^2\right) \tan (c+d x)}{d}+a^4 x+\frac{b^3 (4 a+3 b) \tan ^5(c+d x)}{5 d}+\frac{b^4 \tan ^7(c+d x)}{7 d}","\frac{b^2 \left(6 a^2+8 a b+3 b^2\right) \tan ^3(c+d x)}{3 d}+\frac{b (2 a+b) \left(2 a^2+2 a b+b^2\right) \tan (c+d x)}{d}+a^4 x+\frac{b^3 (4 a+3 b) \tan ^5(c+d x)}{5 d}+\frac{b^4 \tan ^7(c+d x)}{7 d}",1,"a^4*x + (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Tan[c + d*x])/d + (b^2*(6*a^2 + 8*a*b + 3*b^2)*Tan[c + d*x]^3)/(3*d) + (b^3*(4*a + 3*b)*Tan[c + d*x]^5)/(5*d) + (b^4*Tan[c + d*x]^7)/(7*d)","A",4,3,14,0.2143,1,"{4128, 390, 203}"
180,1,86,0,0.1205963,"\int \frac{\sec ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2),x]","\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{b^2 f \sqrt{a+b}}-\frac{(2 a-b) \tanh ^{-1}(\sin (e+f x))}{2 b^2 f}+\frac{\tan (e+f x) \sec (e+f x)}{2 b f}","\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{b^2 f \sqrt{a+b}}-\frac{(2 a-b) \tanh ^{-1}(\sin (e+f x))}{2 b^2 f}+\frac{\tan (e+f x) \sec (e+f x)}{2 b f}",1,"-((2*a - b)*ArcTanh[Sin[e + f*x]])/(2*b^2*f) + (a^(3/2)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(b^2*Sqrt[a + b]*f) + (Sec[e + f*x]*Tan[e + f*x])/(2*b*f)","A",5,5,23,0.2174,1,"{4147, 414, 522, 206, 208}"
181,1,55,0,0.0717949,"\int \frac{\sec ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Sec[e + f*x]^3/(a + b*Sec[e + f*x]^2),x]","\frac{\tanh ^{-1}(\sin (e+f x))}{b f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{b f \sqrt{a+b}}","\frac{\tanh ^{-1}(\sin (e+f x))}{b f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{b f \sqrt{a+b}}",1,"ArcTanh[Sin[e + f*x]]/(b*f) - (Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(b*Sqrt[a + b]*f)","A",4,4,23,0.1739,1,"{4147, 391, 206, 208}"
182,1,36,0,0.042072,"\int \frac{\sec (e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Sec[e + f*x]/(a + b*Sec[e + f*x]^2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{a} f \sqrt{a+b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{a} f \sqrt{a+b}}",1,"ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]/(Sqrt[a]*Sqrt[a + b]*f)","A",2,2,21,0.09524,1,"{4147, 208}"
183,1,52,0,0.0632139,"\int \frac{\cos (e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Cos[e + f*x]/(a + b*Sec[e + f*x]^2),x]","\frac{\sin (e+f x)}{a f}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{a^{3/2} f \sqrt{a+b}}","\frac{\sin (e+f x)}{a f}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{a^{3/2} f \sqrt{a+b}}",1,"-((b*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(a^(3/2)*Sqrt[a + b]*f)) + Sin[e + f*x]/(a*f)","A",3,3,21,0.1429,1,"{4147, 388, 208}"
184,1,76,0,0.0890151,"\int \frac{\cos ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2),x]","\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{a^{5/2} f \sqrt{a+b}}+\frac{(a-b) \sin (e+f x)}{a^2 f}-\frac{\sin ^3(e+f x)}{3 a f}","\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{a^{5/2} f \sqrt{a+b}}+\frac{(a-b) \sin (e+f x)}{a^2 f}-\frac{\sin ^3(e+f x)}{3 a f}",1,"(b^2*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(a^(5/2)*Sqrt[a + b]*f) + ((a - b)*Sin[e + f*x])/(a^2*f) - Sin[e + f*x]^3/(3*a*f)","A",4,3,23,0.1304,1,"{4147, 390, 208}"
185,1,108,0,0.102014,"\int \frac{\cos ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2),x]","\frac{\left(a^2-a b+b^2\right) \sin (e+f x)}{a^3 f}-\frac{b^3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{a^{7/2} f \sqrt{a+b}}-\frac{(2 a-b) \sin ^3(e+f x)}{3 a^2 f}+\frac{\sin ^5(e+f x)}{5 a f}","\frac{\left(a^2-a b+b^2\right) \sin (e+f x)}{a^3 f}-\frac{b^3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{a^{7/2} f \sqrt{a+b}}-\frac{(2 a-b) \sin ^3(e+f x)}{3 a^2 f}+\frac{\sin ^5(e+f x)}{5 a f}",1,"-((b^3*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(a^(7/2)*Sqrt[a + b]*f)) + ((a^2 - a*b + b^2)*Sin[e + f*x])/(a^3*f) - ((2*a - b)*Sin[e + f*x]^3)/(3*a^2*f) + Sin[e + f*x]^5/(5*a*f)","A",4,3,23,0.1304,1,"{4147, 390, 208}"
186,1,77,0,0.0880261,"\int \frac{\sec ^6(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Sec[e + f*x]^6/(a + b*Sec[e + f*x]^2),x]","\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{b^{5/2} f \sqrt{a+b}}-\frac{(a-b) \tan (e+f x)}{b^2 f}+\frac{\tan ^3(e+f x)}{3 b f}","\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{b^{5/2} f \sqrt{a+b}}-\frac{(a-b) \tan (e+f x)}{b^2 f}+\frac{\tan ^3(e+f x)}{3 b f}",1,"(a^2*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(b^(5/2)*Sqrt[a + b]*f) - ((a - b)*Tan[e + f*x])/(b^2*f) + Tan[e + f*x]^3/(3*b*f)","A",4,3,23,0.1304,1,"{4146, 390, 205}"
187,1,52,0,0.0676325,"\int \frac{\sec ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Sec[e + f*x]^4/(a + b*Sec[e + f*x]^2),x]","\frac{\tan (e+f x)}{b f}-\frac{a \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{b^{3/2} f \sqrt{a+b}}","\frac{\tan (e+f x)}{b f}-\frac{a \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{b^{3/2} f \sqrt{a+b}}",1,"-((a*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(b^(3/2)*Sqrt[a + b]*f)) + Tan[e + f*x]/(b*f)","A",3,3,23,0.1304,1,"{4146, 388, 205}"
188,1,36,0,0.0565247,"\int \frac{\sec ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{b} f \sqrt{a+b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{b} f \sqrt{a+b}}",1,"ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b]*f)","A",2,2,23,0.08696,1,"{4146, 205}"
189,1,45,0,0.0420105,"\int \frac{1}{a+b \sec ^2(e+f x)} \, dx","Int[(a + b*Sec[e + f*x]^2)^(-1),x]","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (e+f x)}{\sqrt{b}}\right)}{a f \sqrt{a+b}}+\frac{x}{a}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (e+f x)}{\sqrt{b}}\right)}{a f \sqrt{a+b}}+\frac{x}{a}",1,"x/a + (Sqrt[b]*ArcTan[(Sqrt[a + b]*Cot[e + f*x])/Sqrt[b]])/(a*Sqrt[a + b]*f)","A",3,3,14,0.2143,1,"{4127, 3181, 205}"
190,1,75,0,0.1033761,"\int \frac{\cos ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2),x]","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a^2 f \sqrt{a+b}}+\frac{x (a-2 b)}{2 a^2}+\frac{\sin (e+f x) \cos (e+f x)}{2 a f}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a^2 f \sqrt{a+b}}+\frac{x (a-2 b)}{2 a^2}+\frac{\sin (e+f x) \cos (e+f x)}{2 a f}",1,"((a - 2*b)*x)/(2*a^2) + (b^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a^2*Sqrt[a + b]*f) + (Cos[e + f*x]*Sin[e + f*x])/(2*a*f)","A",5,5,23,0.2174,1,"{4146, 414, 522, 203, 205}"
191,1,117,0,0.1566642,"\int \frac{\cos ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Cos[e + f*x]^4/(a + b*Sec[e + f*x]^2),x]","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a^3 f \sqrt{a+b}}+\frac{x \left(3 a^2-4 a b+8 b^2\right)}{8 a^3}+\frac{(3 a-4 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f}","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a^3 f \sqrt{a+b}}+\frac{x \left(3 a^2-4 a b+8 b^2\right)}{8 a^3}+\frac{(3 a-4 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f}",1,"((3*a^2 - 4*a*b + 8*b^2)*x)/(8*a^3) - (b^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a^3*Sqrt[a + b]*f) + ((3*a - 4*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f)","A",6,6,23,0.2609,1,"{4146, 414, 527, 522, 203, 205}"
192,1,163,0,0.2448247,"\int \frac{\cos ^6(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2),x]","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a^4 f \sqrt{a+b}}+\frac{\left(5 a^2-6 a b+8 b^2\right) \sin (e+f x) \cos (e+f x)}{16 a^3 f}+\frac{x \left(-6 a^2 b+5 a^3+8 a b^2-16 b^3\right)}{16 a^4}+\frac{(5 a-6 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f}+\frac{\sin (e+f x) \cos ^5(e+f x)}{6 a f}","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a^4 f \sqrt{a+b}}+\frac{\left(5 a^2-6 a b+8 b^2\right) \sin (e+f x) \cos (e+f x)}{16 a^3 f}+\frac{x \left(-6 a^2 b+5 a^3+8 a b^2-16 b^3\right)}{16 a^4}+\frac{(5 a-6 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f}+\frac{\sin (e+f x) \cos ^5(e+f x)}{6 a f}",1,"((5*a^3 - 6*a^2*b + 8*a*b^2 - 16*b^3)*x)/(16*a^4) + (b^(7/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a^4*Sqrt[a + b]*f) + ((5*a^2 - 6*a*b + 8*b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*a^3*f) + ((5*a - 6*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f) + (Cos[e + f*x]^5*Sin[e + f*x])/(6*a*f)","A",7,6,23,0.2609,1,"{4146, 414, 527, 522, 203, 205}"
193,1,102,0,0.1402507,"\int \frac{\sec ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{\sqrt{a} (2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{2 b^2 f (a+b)^{3/2}}-\frac{a \sin (e+f x)}{2 b f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{\tanh ^{-1}(\sin (e+f x))}{b^2 f}","-\frac{\sqrt{a} (2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{2 b^2 f (a+b)^{3/2}}-\frac{a \sin (e+f x)}{2 b f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{\tanh ^{-1}(\sin (e+f x))}{b^2 f}",1,"ArcTanh[Sin[e + f*x]]/(b^2*f) - (Sqrt[a]*(2*a + 3*b)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(2*b^2*(a + b)^(3/2)*f) - (a*Sin[e + f*x])/(2*b*(a + b)*f*(a + b - a*Sin[e + f*x]^2))","A",5,5,23,0.2174,1,"{4147, 414, 522, 206, 208}"
194,1,74,0,0.0703302,"\int \frac{\sec ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Sec[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sin (e+f x)}{2 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{2 \sqrt{a} f (a+b)^{3/2}}","\frac{\sin (e+f x)}{2 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{2 \sqrt{a} f (a+b)^{3/2}}",1,"ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]/(2*Sqrt[a]*(a + b)^(3/2)*f) + Sin[e + f*x]/(2*(a + b)*f*(a + b - a*Sin[e + f*x]^2))","A",3,3,23,0.1304,1,"{4147, 199, 208}"
195,1,83,0,0.0683364,"\int \frac{\sec (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Sec[e + f*x]/(a + b*Sec[e + f*x]^2)^2,x]","\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{2 a^{3/2} f (a+b)^{3/2}}-\frac{b \sin (e+f x)}{2 a f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)}","\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{2 a^{3/2} f (a+b)^{3/2}}-\frac{b \sin (e+f x)}{2 a f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)}",1,"((2*a + b)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(2*a^(3/2)*(a + b)^(3/2)*f) - (b*Sin[e + f*x])/(2*a*(a + b)*f*(a + b - a*Sin[e + f*x]^2))","A",3,3,21,0.1429,1,"{4147, 385, 208}"
196,1,101,0,0.1321115,"\int \frac{\cos (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Cos[e + f*x]/(a + b*Sec[e + f*x]^2)^2,x]","\frac{b^2 \sin (e+f x)}{2 a^2 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)}-\frac{b (4 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{2 a^{5/2} f (a+b)^{3/2}}+\frac{\sin (e+f x)}{a^2 f}","\frac{b^2 \sin (e+f x)}{2 a^2 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)}-\frac{b (4 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{2 a^{5/2} f (a+b)^{3/2}}+\frac{\sin (e+f x)}{a^2 f}",1,"-(b*(4*a + 3*b)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(2*a^(5/2)*(a + b)^(3/2)*f) + Sin[e + f*x]/(a^2*f) + (b^2*Sin[e + f*x])/(2*a^2*(a + b)*f*(a + b - a*Sin[e + f*x]^2))","A",5,4,21,0.1905,1,"{4147, 390, 385, 208}"
197,1,126,0,0.1551939,"\int \frac{\cos ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{b^3 \sin (e+f x)}{2 a^3 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{b^2 (6 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{2 a^{7/2} f (a+b)^{3/2}}+\frac{(a-2 b) \sin (e+f x)}{a^3 f}-\frac{\sin ^3(e+f x)}{3 a^2 f}","-\frac{b^3 \sin (e+f x)}{2 a^3 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{b^2 (6 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{2 a^{7/2} f (a+b)^{3/2}}+\frac{(a-2 b) \sin (e+f x)}{a^3 f}-\frac{\sin ^3(e+f x)}{3 a^2 f}",1,"(b^2*(6*a + 5*b)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(2*a^(7/2)*(a + b)^(3/2)*f) + ((a - 2*b)*Sin[e + f*x])/(a^3*f) - Sin[e + f*x]^3/(3*a^2*f) - (b^3*Sin[e + f*x])/(2*a^3*(a + b)*f*(a + b - a*Sin[e + f*x]^2))","A",5,4,23,0.1739,1,"{4147, 390, 385, 208}"
198,1,157,0,0.1684728,"\int \frac{\cos ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2,x]","\frac{b^4 \sin (e+f x)}{2 a^4 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{\left(a^2-2 a b+3 b^2\right) \sin (e+f x)}{a^4 f}-\frac{b^3 (8 a+7 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{2 a^{9/2} f (a+b)^{3/2}}-\frac{2 (a-b) \sin ^3(e+f x)}{3 a^3 f}+\frac{\sin ^5(e+f x)}{5 a^2 f}","\frac{b^4 \sin (e+f x)}{2 a^4 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{\left(a^2-2 a b+3 b^2\right) \sin (e+f x)}{a^4 f}-\frac{b^3 (8 a+7 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{2 a^{9/2} f (a+b)^{3/2}}-\frac{2 (a-b) \sin ^3(e+f x)}{3 a^3 f}+\frac{\sin ^5(e+f x)}{5 a^2 f}",1,"-(b^3*(8*a + 7*b)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(2*a^(9/2)*(a + b)^(3/2)*f) + ((a^2 - 2*a*b + 3*b^2)*Sin[e + f*x])/(a^4*f) - (2*(a - b)*Sin[e + f*x]^3)/(3*a^3*f) + Sin[e + f*x]^5/(5*a^2*f) + (b^4*Sin[e + f*x])/(2*a^4*(a + b)*f*(a + b - a*Sin[e + f*x]^2))","A",5,4,23,0.1739,1,"{4147, 390, 385, 208}"
199,1,100,0,0.1351157,"\int \frac{\sec ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Sec[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2,x]","\frac{a^2 \tan (e+f x)}{2 b^2 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}-\frac{a (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 b^{5/2} f (a+b)^{3/2}}+\frac{\tan (e+f x)}{b^2 f}","\frac{a^2 \tan (e+f x)}{2 b^2 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}-\frac{a (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 b^{5/2} f (a+b)^{3/2}}+\frac{\tan (e+f x)}{b^2 f}",1,"-(a*(3*a + 4*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*b^(5/2)*(a + b)^(3/2)*f) + Tan[e + f*x]/(b^2*f) + (a^2*Tan[e + f*x])/(2*b^2*(a + b)*f*(a + b + b*Tan[e + f*x]^2))","A",5,4,23,0.1739,1,"{4146, 390, 385, 205}"
200,1,82,0,0.0823868,"\int \frac{\sec ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Sec[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]","\frac{(a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 b^{3/2} f (a+b)^{3/2}}-\frac{a \tan (e+f x)}{2 b f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}","\frac{(a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 b^{3/2} f (a+b)^{3/2}}-\frac{a \tan (e+f x)}{2 b f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}",1,"((a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*b^(3/2)*(a + b)^(3/2)*f) - (a*Tan[e + f*x])/(2*b*(a + b)*f*(a + b + b*Tan[e + f*x]^2))","A",3,3,23,0.1304,1,"{4146, 385, 205}"
201,1,73,0,0.0735063,"\int \frac{\sec ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 \sqrt{b} f (a+b)^{3/2}}+\frac{\tan (e+f x)}{2 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 \sqrt{b} f (a+b)^{3/2}}+\frac{\tan (e+f x)}{2 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}",1,"ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]]/(2*Sqrt[b]*(a + b)^(3/2)*f) + Tan[e + f*x]/(2*(a + b)*f*(a + b + b*Tan[e + f*x]^2))","A",3,3,23,0.1304,1,"{4146, 199, 205}"
202,1,92,0,0.0815188,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[(a + b*Sec[e + f*x]^2)^(-2),x]","-\frac{\sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 f (a+b)^{3/2}}+\frac{x}{a^2}-\frac{b \tan (e+f x)}{2 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}","-\frac{\sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 f (a+b)^{3/2}}+\frac{x}{a^2}-\frac{b \tan (e+f x)}{2 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}",1,"x/a^2 - (Sqrt[b]*(3*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(3/2)*f) - (b*Tan[e + f*x])/(2*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2))","A",5,5,14,0.3571,1,"{4128, 414, 522, 203, 205}"
203,1,142,0,0.1937845,"\int \frac{\cos ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2,x]","\frac{b^{3/2} (5 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^3 f (a+b)^{3/2}}+\frac{b (a+2 b) \tan (e+f x)}{2 a^2 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x (a-4 b)}{2 a^3}+\frac{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)}","\frac{b^{3/2} (5 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^3 f (a+b)^{3/2}}+\frac{b (a+2 b) \tan (e+f x)}{2 a^2 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x (a-4 b)}{2 a^3}+\frac{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)}",1,"((a - 4*b)*x)/(2*a^3) + (b^(3/2)*(5*a + 4*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^3*(a + b)^(3/2)*f) + (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*(a + b + b*Tan[e + f*x]^2)) + (b*(a + 2*b)*Tan[e + f*x])/(2*a^2*(a + b)*f*(a + b + b*Tan[e + f*x]^2))","A",6,6,23,0.2609,1,"{4146, 414, 527, 522, 203, 205}"
204,1,203,0,0.2899477,"\int \frac{\cos ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Cos[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{b^{5/2} (7 a+6 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^4 f (a+b)^{3/2}}+\frac{x \left(3 a^2-8 a b+24 b^2\right)}{8 a^4}+\frac{b (a-3 b) (3 a+4 b) \tan (e+f x)}{8 a^3 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}+\frac{3 (a-2 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)}","-\frac{b^{5/2} (7 a+6 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^4 f (a+b)^{3/2}}+\frac{x \left(3 a^2-8 a b+24 b^2\right)}{8 a^4}+\frac{b (a-3 b) (3 a+4 b) \tan (e+f x)}{8 a^3 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}+\frac{3 (a-2 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)}",1,"((3*a^2 - 8*a*b + 24*b^2)*x)/(8*a^4) - (b^(5/2)*(7*a + 6*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^4*(a + b)^(3/2)*f) + (3*(a - 2*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^2)) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*(a + b + b*Tan[e + f*x]^2)) + ((a - 3*b)*b*(3*a + 4*b)*Tan[e + f*x])/(8*a^3*(a + b)*f*(a + b + b*Tan[e + f*x]^2))","A",7,6,23,0.2609,1,"{4146, 414, 527, 522, 203, 205}"
205,1,278,0,0.3467156,"\int \frac{\cos ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2,x]","\frac{b^{7/2} (9 a+8 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^5 f (a+b)^{3/2}}+\frac{b \left(-7 a^2 b+5 a^3+12 a b^2+32 b^3\right) \tan (e+f x)}{16 a^4 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\left(15 a^2-26 a b+48 b^2\right) \sin (e+f x) \cos (e+f x)}{48 a^3 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x \left(-12 a^2 b+5 a^3+24 a b^2-64 b^3\right)}{16 a^5}+\frac{(5 a-8 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\sin (e+f x) \cos ^5(e+f x)}{6 a f \left(a+b \tan ^2(e+f x)+b\right)}","\frac{b^{7/2} (9 a+8 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^5 f (a+b)^{3/2}}+\frac{b \left(-7 a^2 b+5 a^3+12 a b^2+32 b^3\right) \tan (e+f x)}{16 a^4 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\left(15 a^2-26 a b+48 b^2\right) \sin (e+f x) \cos (e+f x)}{48 a^3 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x \left(-12 a^2 b+5 a^3+24 a b^2-64 b^3\right)}{16 a^5}+\frac{(5 a-8 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\sin (e+f x) \cos ^5(e+f x)}{6 a f \left(a+b \tan ^2(e+f x)+b\right)}",1,"((5*a^3 - 12*a^2*b + 24*a*b^2 - 64*b^3)*x)/(16*a^5) + (b^(7/2)*(9*a + 8*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^5*(a + b)^(3/2)*f) + ((15*a^2 - 26*a*b + 48*b^2)*Cos[e + f*x]*Sin[e + f*x])/(48*a^3*f*(a + b + b*Tan[e + f*x]^2)) + ((5*a - 8*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*(a + b + b*Tan[e + f*x]^2)) + (Cos[e + f*x]^5*Sin[e + f*x])/(6*a*f*(a + b + b*Tan[e + f*x]^2)) + (b*(5*a^3 - 7*a^2*b + 12*a*b^2 + 32*b^3)*Tan[e + f*x])/(16*a^4*(a + b)*f*(a + b + b*Tan[e + f*x]^2))","A",8,6,23,0.2609,1,"{4146, 414, 527, 522, 203, 205}"
206,1,108,0,0.0993723,"\int \frac{\sec ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]","\frac{3 \sin (e+f x)}{8 f (a+b)^2 \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{\sin (e+f x)}{4 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^2}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{8 \sqrt{a} f (a+b)^{5/2}}","\frac{3 \sin (e+f x)}{8 f (a+b)^2 \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{\sin (e+f x)}{4 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^2}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{8 \sqrt{a} f (a+b)^{5/2}}",1,"(3*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(8*Sqrt[a]*(a + b)^(5/2)*f) + Sin[e + f*x]/(4*(a + b)*f*(a + b - a*Sin[e + f*x]^2)^2) + (3*Sin[e + f*x])/(8*(a + b)^2*f*(a + b - a*Sin[e + f*x]^2))","A",4,3,23,0.1304,1,"{4147, 199, 208}"
207,1,125,0,0.106025,"\int \frac{\sec ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Sec[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3,x]","\frac{(4 a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{8 a^{3/2} f (a+b)^{5/2}}+\frac{(4 a+b) \sin (e+f x)}{8 a f (a+b)^2 \left(-a \sin ^2(e+f x)+a+b\right)}-\frac{b \sin (e+f x)}{4 a f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^2}","\frac{(4 a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{8 a^{3/2} f (a+b)^{5/2}}+\frac{(4 a+b) \sin (e+f x)}{8 a f (a+b)^2 \left(-a \sin ^2(e+f x)+a+b\right)}-\frac{b \sin (e+f x)}{4 a f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^2}",1,"((4*a + b)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(8*a^(3/2)*(a + b)^(5/2)*f) - (b*Sin[e + f*x])/(4*a*(a + b)*f*(a + b - a*Sin[e + f*x]^2)^2) + ((4*a + b)*Sin[e + f*x])/(8*a*(a + b)^2*f*(a + b - a*Sin[e + f*x]^2))","A",4,4,23,0.1739,1,"{4147, 385, 199, 208}"
208,1,144,0,0.131073,"\int \frac{\sec (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Sec[e + f*x]/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\left(8 a^2+8 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{8 a^{5/2} f (a+b)^{5/2}}-\frac{3 b (2 a+b) \sin (e+f x)}{8 a^2 f (a+b)^2 \left(-a \sin ^2(e+f x)+a+b\right)}-\frac{b \sin (e+f x) \cos ^2(e+f x)}{4 a f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^2}","\frac{\left(8 a^2+8 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{8 a^{5/2} f (a+b)^{5/2}}-\frac{3 b (2 a+b) \sin (e+f x)}{8 a^2 f (a+b)^2 \left(-a \sin ^2(e+f x)+a+b\right)}-\frac{b \sin (e+f x) \cos ^2(e+f x)}{4 a f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^2}",1,"((8*a^2 + 8*a*b + 3*b^2)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(8*a^(5/2)*(a + b)^(5/2)*f) - (b*Cos[e + f*x]^2*Sin[e + f*x])/(4*a*(a + b)*f*(a + b - a*Sin[e + f*x]^2)^2) - (3*b*(2*a + b)*Sin[e + f*x])/(8*a^2*(a + b)^2*f*(a + b - a*Sin[e + f*x]^2))","A",4,4,21,0.1905,1,"{4147, 413, 385, 208}"
209,1,156,0,0.1935031,"\int \frac{\cos (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Cos[e + f*x]/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{b^3 \sin (e+f x)}{4 a^3 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^2}+\frac{3 b^2 (4 a+3 b) \sin (e+f x)}{8 a^3 f (a+b)^2 \left(-a \sin ^2(e+f x)+a+b\right)}-\frac{3 b \left(4 (a+b)^2+(2 a+b)^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{8 a^{7/2} f (a+b)^{5/2}}+\frac{\sin (e+f x)}{a^3 f}","-\frac{b^3 \sin (e+f x)}{4 a^3 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^2}+\frac{3 b^2 (4 a+3 b) \sin (e+f x)}{8 a^3 f (a+b)^2 \left(-a \sin ^2(e+f x)+a+b\right)}-\frac{3 b \left(4 (a+b)^2+(2 a+b)^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{8 a^{7/2} f (a+b)^{5/2}}+\frac{\sin (e+f x)}{a^3 f}",1,"(-3*b*(4*(a + b)^2 + (2*a + b)^2)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(8*a^(7/2)*(a + b)^(5/2)*f) + Sin[e + f*x]/(a^3*f) - (b^3*Sin[e + f*x])/(4*a^3*(a + b)*f*(a + b - a*Sin[e + f*x]^2)^2) + (3*b^2*(4*a + 3*b)*Sin[e + f*x])/(8*a^3*(a + b)^2*f*(a + b - a*Sin[e + f*x]^2))","A",6,5,21,0.2381,1,"{4147, 390, 1157, 385, 208}"
210,1,181,0,0.2427734,"\int \frac{\cos ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3,x]","\frac{b^4 \sin (e+f x)}{4 a^4 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^2}-\frac{b^3 (16 a+13 b) \sin (e+f x)}{8 a^4 f (a+b)^2 \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{b^2 \left(48 a^2+80 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{8 a^{9/2} f (a+b)^{5/2}}+\frac{(a-3 b) \sin (e+f x)}{a^4 f}-\frac{\sin ^3(e+f x)}{3 a^3 f}","\frac{b^4 \sin (e+f x)}{4 a^4 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^2}-\frac{b^3 (16 a+13 b) \sin (e+f x)}{8 a^4 f (a+b)^2 \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{b^2 \left(48 a^2+80 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{8 a^{9/2} f (a+b)^{5/2}}+\frac{(a-3 b) \sin (e+f x)}{a^4 f}-\frac{\sin ^3(e+f x)}{3 a^3 f}",1,"(b^2*(48*a^2 + 80*a*b + 35*b^2)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(8*a^(9/2)*(a + b)^(5/2)*f) + ((a - 3*b)*Sin[e + f*x])/(a^4*f) - Sin[e + f*x]^3/(3*a^3*f) + (b^4*Sin[e + f*x])/(4*a^4*(a + b)*f*(a + b - a*Sin[e + f*x]^2)^2) - (b^3*(16*a + 13*b)*Sin[e + f*x])/(8*a^4*(a + b)^2*f*(a + b - a*Sin[e + f*x]^2))","A",6,5,23,0.2174,1,"{4147, 390, 1157, 385, 208}"
211,1,214,0,0.2583655,"\int \frac{\cos ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{b^5 \sin (e+f x)}{4 a^5 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^2}+\frac{b^4 (20 a+17 b) \sin (e+f x)}{8 a^5 f (a+b)^2 \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{\left(a^2-3 a b+6 b^2\right) \sin (e+f x)}{a^5 f}-\frac{b^3 \left(80 a^2+140 a b+63 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{8 a^{11/2} f (a+b)^{5/2}}-\frac{(2 a-3 b) \sin ^3(e+f x)}{3 a^4 f}+\frac{\sin ^5(e+f x)}{5 a^3 f}","-\frac{b^5 \sin (e+f x)}{4 a^5 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^2}+\frac{b^4 (20 a+17 b) \sin (e+f x)}{8 a^5 f (a+b)^2 \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{\left(a^2-3 a b+6 b^2\right) \sin (e+f x)}{a^5 f}-\frac{b^3 \left(80 a^2+140 a b+63 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{8 a^{11/2} f (a+b)^{5/2}}-\frac{(2 a-3 b) \sin ^3(e+f x)}{3 a^4 f}+\frac{\sin ^5(e+f x)}{5 a^3 f}",1,"-(b^3*(80*a^2 + 140*a*b + 63*b^2)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(8*a^(11/2)*(a + b)^(5/2)*f) + ((a^2 - 3*a*b + 6*b^2)*Sin[e + f*x])/(a^5*f) - ((2*a - 3*b)*Sin[e + f*x]^3)/(3*a^4*f) + Sin[e + f*x]^5/(5*a^3*f) - (b^5*Sin[e + f*x])/(4*a^5*(a + b)*f*(a + b - a*Sin[e + f*x]^2)^2) + (b^4*(20*a + 17*b)*Sin[e + f*x])/(8*a^5*(a + b)^2*f*(a + b - a*Sin[e + f*x]^2))","A",6,5,23,0.2174,1,"{4147, 390, 1157, 385, 208}"
212,1,142,0,0.1566465,"\int \frac{\sec ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Sec[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\left(3 a^2+8 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 b^{5/2} f (a+b)^{5/2}}-\frac{3 a (a+2 b) \tan (e+f x)}{8 b^2 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{a \tan (e+f x) \sec ^2(e+f x)}{4 b f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}","\frac{\left(3 a^2+8 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 b^{5/2} f (a+b)^{5/2}}-\frac{3 a (a+2 b) \tan (e+f x)}{8 b^2 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{a \tan (e+f x) \sec ^2(e+f x)}{4 b f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"((3*a^2 + 8*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*b^(5/2)*(a + b)^(5/2)*f) - (a*Sec[e + f*x]^2*Tan[e + f*x])/(4*b*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (3*a*(a + 2*b)*Tan[e + f*x])/(8*b^2*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))","A",4,4,23,0.1739,1,"{4146, 413, 385, 205}"
213,1,123,0,0.1016825,"\int \frac{\sec ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Sec[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]","\frac{(a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 b^{3/2} f (a+b)^{5/2}}+\frac{(a+4 b) \tan (e+f x)}{8 b f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{a \tan (e+f x)}{4 b f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}","\frac{(a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 b^{3/2} f (a+b)^{5/2}}+\frac{(a+4 b) \tan (e+f x)}{8 b f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{a \tan (e+f x)}{4 b f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"((a + 4*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*b^(3/2)*(a + b)^(5/2)*f) - (a*Tan[e + f*x])/(4*b*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) + ((a + 4*b)*Tan[e + f*x])/(8*b*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))","A",4,4,23,0.1739,1,"{4146, 385, 199, 205}"
214,1,106,0,0.0818153,"\int \frac{\sec ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3,x]","\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 \sqrt{b} f (a+b)^{5/2}}+\frac{3 \tan (e+f x)}{8 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\tan (e+f x)}{4 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 \sqrt{b} f (a+b)^{5/2}}+\frac{3 \tan (e+f x)}{8 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\tan (e+f x)}{4 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"(3*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*Sqrt[b]*(a + b)^(5/2)*f) + Tan[e + f*x]/(4*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) + (3*Tan[e + f*x])/(8*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))","A",4,3,23,0.1304,1,"{4146, 199, 205}"
215,1,144,0,0.1748351,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[(a + b*Sec[e + f*x]^2)^(-3),x]","-\frac{\sqrt{b} \left(15 a^2+20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 f (a+b)^{5/2}}-\frac{b (7 a+4 b) \tan (e+f x)}{8 a^2 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x}{a^3}-\frac{b \tan (e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}","-\frac{\sqrt{b} \left(15 a^2+20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 f (a+b)^{5/2}}-\frac{b (7 a+4 b) \tan (e+f x)}{8 a^2 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x}{a^3}-\frac{b \tan (e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"x/a^3 - (Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(5/2)*f) - (b*Tan[e + f*x])/(4*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(7*a + 4*b)*Tan[e + f*x])/(8*a^2*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))","A",6,6,14,0.4286,1,"{4128, 414, 527, 522, 203, 205}"
216,1,201,0,0.3357448,"\int \frac{\cos ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3,x]","\frac{b^{3/2} \left(35 a^2+56 a b+24 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^4 f (a+b)^{5/2}}+\frac{b (4 a+3 b) (a+4 b) \tan (e+f x)}{8 a^3 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{b (2 a+3 b) \tan (e+f x)}{4 a^2 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{x (a-6 b)}{2 a^4}+\frac{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)^2}","\frac{b^{3/2} \left(35 a^2+56 a b+24 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^4 f (a+b)^{5/2}}+\frac{b (4 a+3 b) (a+4 b) \tan (e+f x)}{8 a^3 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{b (2 a+3 b) \tan (e+f x)}{4 a^2 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{x (a-6 b)}{2 a^4}+\frac{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"((a - 6*b)*x)/(2*a^4) + (b^(3/2)*(35*a^2 + 56*a*b + 24*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^4*(a + b)^(5/2)*f) + (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*(a + b + b*Tan[e + f*x]^2)^2) + (b*(2*a + 3*b)*Tan[e + f*x])/(4*a^2*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) + (b*(4*a + 3*b)*(a + 4*b)*Tan[e + f*x])/(8*a^3*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))","A",7,6,23,0.2609,1,"{4146, 414, 527, 522, 203, 205}"
217,1,269,0,0.3767861,"\int \frac{\cos ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Cos[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{3 b^{5/2} \left(21 a^2+36 a b+16 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^5 f (a+b)^{5/2}}+\frac{3 b (a+2 b) \left(a^2-4 a b-4 b^2\right) \tan (e+f x)}{8 a^4 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{b \left(3 a^2-7 a b-12 b^2\right) \tan (e+f x)}{8 a^3 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{3 x \left(a^2-4 a b+16 b^2\right)}{8 a^5}+\frac{(3 a-8 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)^2}","-\frac{3 b^{5/2} \left(21 a^2+36 a b+16 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^5 f (a+b)^{5/2}}+\frac{3 b (a+2 b) \left(a^2-4 a b-4 b^2\right) \tan (e+f x)}{8 a^4 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{b \left(3 a^2-7 a b-12 b^2\right) \tan (e+f x)}{8 a^3 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{3 x \left(a^2-4 a b+16 b^2\right)}{8 a^5}+\frac{(3 a-8 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"(3*(a^2 - 4*a*b + 16*b^2)*x)/(8*a^5) - (3*b^(5/2)*(21*a^2 + 36*a*b + 16*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^5*(a + b)^(5/2)*f) + ((3*a - 8*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^2)^2) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*(a + b + b*Tan[e + f*x]^2)^2) + (b*(3*a^2 - 7*a*b - 12*b^2)*Tan[e + f*x])/(8*a^3*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) + (3*b*(a + 2*b)*(a^2 - 4*a*b - 4*b^2)*Tan[e + f*x])/(8*a^4*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))","A",8,6,23,0.2609,1,"{4146, 414, 527, 522, 203, 205}"
218,1,352,0,0.4752892,"\int \frac{\cos ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]","\frac{b^{7/2} \left(99 a^2+176 a b+80 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^6 f (a+b)^{5/2}}+\frac{b \left(17 a^2 b^2-8 a^3 b+5 a^4+116 a b^3+80 b^4\right) \tan (e+f x)}{16 a^5 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{b \left(-29 a^2 b+15 a^3+64 a b^2+120 b^3\right) \tan (e+f x)}{48 a^4 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{\left(15 a^2-34 a b+80 b^2\right) \sin (e+f x) \cos (e+f x)}{48 a^3 f \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{x \left(-18 a^2 b+5 a^3+48 a b^2-160 b^3\right)}{16 a^6}+\frac{5 (a-2 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{\sin (e+f x) \cos ^5(e+f x)}{6 a f \left(a+b \tan ^2(e+f x)+b\right)^2}","\frac{b^{7/2} \left(99 a^2+176 a b+80 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^6 f (a+b)^{5/2}}+\frac{b \left(17 a^2 b^2-8 a^3 b+5 a^4+116 a b^3+80 b^4\right) \tan (e+f x)}{16 a^5 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{b \left(-29 a^2 b+15 a^3+64 a b^2+120 b^3\right) \tan (e+f x)}{48 a^4 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{\left(15 a^2-34 a b+80 b^2\right) \sin (e+f x) \cos (e+f x)}{48 a^3 f \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{x \left(-18 a^2 b+5 a^3+48 a b^2-160 b^3\right)}{16 a^6}+\frac{5 (a-2 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^2}+\frac{\sin (e+f x) \cos ^5(e+f x)}{6 a f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"((5*a^3 - 18*a^2*b + 48*a*b^2 - 160*b^3)*x)/(16*a^6) + (b^(7/2)*(99*a^2 + 176*a*b + 80*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^6*(a + b)^(5/2)*f) + ((15*a^2 - 34*a*b + 80*b^2)*Cos[e + f*x]*Sin[e + f*x])/(48*a^3*f*(a + b + b*Tan[e + f*x]^2)^2) + (5*(a - 2*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*(a + b + b*Tan[e + f*x]^2)^2) + (Cos[e + f*x]^5*Sin[e + f*x])/(6*a*f*(a + b + b*Tan[e + f*x]^2)^2) + (b*(15*a^3 - 29*a^2*b + 64*a*b^2 + 120*b^3)*Tan[e + f*x])/(48*a^4*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) + (b*(5*a^4 - 8*a^3*b + 17*a^2*b^2 + 116*a*b^3 + 80*b^4)*Tan[e + f*x])/(16*a^5*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))","A",9,6,23,0.2609,1,"{4146, 414, 527, 522, 203, 205}"
219,1,204,0,0.3348448,"\int \frac{1}{\left(a+b \sec ^2(c+d x)\right)^4} \, dx","Int[(a + b*Sec[c + d*x]^2)^(-4),x]","-\frac{\sqrt{b} \left(70 a^2 b+35 a^3+56 a b^2+16 b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a+b}}\right)}{16 a^4 d (a+b)^{7/2}}-\frac{b \left(19 a^2+22 a b+8 b^2\right) \tan (c+d x)}{16 a^3 d (a+b)^3 \left(a+b \tan ^2(c+d x)+b\right)}-\frac{b (11 a+6 b) \tan (c+d x)}{24 a^2 d (a+b)^2 \left(a+b \tan ^2(c+d x)+b\right)^2}+\frac{x}{a^4}-\frac{b \tan (c+d x)}{6 a d (a+b) \left(a+b \tan ^2(c+d x)+b\right)^3}","-\frac{\sqrt{b} \left(70 a^2 b+35 a^3+56 a b^2+16 b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a+b}}\right)}{16 a^4 d (a+b)^{7/2}}-\frac{b \left(19 a^2+22 a b+8 b^2\right) \tan (c+d x)}{16 a^3 d (a+b)^3 \left(a+b \tan ^2(c+d x)+b\right)}-\frac{b (11 a+6 b) \tan (c+d x)}{24 a^2 d (a+b)^2 \left(a+b \tan ^2(c+d x)+b\right)^2}+\frac{x}{a^4}-\frac{b \tan (c+d x)}{6 a d (a+b) \left(a+b \tan ^2(c+d x)+b\right)^3}",1,"x/a^4 - (Sqrt[b]*(35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a + b]])/(16*a^4*(a + b)^(7/2)*d) - (b*Tan[c + d*x])/(6*a*(a + b)*d*(a + b + b*Tan[c + d*x]^2)^3) - (b*(11*a + 6*b)*Tan[c + d*x])/(24*a^2*(a + b)^2*d*(a + b + b*Tan[c + d*x]^2)^2) - (b*(19*a^2 + 22*a*b + 8*b^2)*Tan[c + d*x])/(16*a^3*(a + b)^3*d*(a + b + b*Tan[c + d*x]^2))","A",7,6,14,0.4286,1,"{4128, 414, 527, 522, 203, 205}"
220,1,134,0,0.0623407,"\int \left(a-a \sec ^2(c+d x)\right)^{7/2} \, dx","Int[(a - a*Sec[c + d*x]^2)^(7/2),x]","-\frac{a^3 \tan ^5(c+d x) \sqrt{-a \tan ^2(c+d x)}}{6 d}+\frac{a^3 \tan ^3(c+d x) \sqrt{-a \tan ^2(c+d x)}}{4 d}-\frac{a^3 \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}-\frac{a^3 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d}","-\frac{a^3 \tan ^5(c+d x) \sqrt{-a \tan ^2(c+d x)}}{6 d}+\frac{a^3 \tan ^3(c+d x) \sqrt{-a \tan ^2(c+d x)}}{4 d}-\frac{a^3 \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}-\frac{a^3 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d}",1,"-((a^3*Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[-(a*Tan[c + d*x]^2)])/d) - (a^3*Tan[c + d*x]*Sqrt[-(a*Tan[c + d*x]^2)])/(2*d) + (a^3*Tan[c + d*x]^3*Sqrt[-(a*Tan[c + d*x]^2)])/(4*d) - (a^3*Tan[c + d*x]^5*Sqrt[-(a*Tan[c + d*x]^2)])/(6*d)","A",6,4,17,0.2353,1,"{4121, 3658, 3473, 3475}"
221,1,101,0,0.0513671,"\int \left(a-a \sec ^2(c+d x)\right)^{5/2} \, dx","Int[(a - a*Sec[c + d*x]^2)^(5/2),x]","\frac{a^2 \tan ^3(c+d x) \sqrt{-a \tan ^2(c+d x)}}{4 d}-\frac{a^2 \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}-\frac{a^2 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d}","\frac{a^2 \tan ^3(c+d x) \sqrt{-a \tan ^2(c+d x)}}{4 d}-\frac{a^2 \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}-\frac{a^2 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d}",1,"-((a^2*Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[-(a*Tan[c + d*x]^2)])/d) - (a^2*Tan[c + d*x]*Sqrt[-(a*Tan[c + d*x]^2)])/(2*d) + (a^2*Tan[c + d*x]^3*Sqrt[-(a*Tan[c + d*x]^2)])/(4*d)","A",5,4,17,0.2353,1,"{4121, 3658, 3473, 3475}"
222,1,64,0,0.0407729,"\int \left(a-a \sec ^2(c+d x)\right)^{3/2} \, dx","Int[(a - a*Sec[c + d*x]^2)^(3/2),x]","-\frac{a \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}-\frac{a \cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d}","-\frac{a \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}-\frac{a \cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d}",1,"-((a*Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[-(a*Tan[c + d*x]^2)])/d) - (a*Tan[c + d*x]*Sqrt[-(a*Tan[c + d*x]^2)])/(2*d)","A",4,4,17,0.2353,1,"{4121, 3658, 3473, 3475}"
223,1,33,0,0.0300599,"\int \sqrt{a-a \sec ^2(c+d x)} \, dx","Int[Sqrt[a - a*Sec[c + d*x]^2],x]","-\frac{\cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d}","-\frac{\cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d}",1,"-((Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[-(a*Tan[c + d*x]^2)])/d)","A",3,3,17,0.1765,1,"{4121, 3658, 3475}"
224,1,32,0,0.0349387,"\int \frac{1}{\sqrt{a-a \sec ^2(c+d x)}} \, dx","Int[1/Sqrt[a - a*Sec[c + d*x]^2],x]","\frac{\tan (c+d x) \log (\sin (c+d x))}{d \sqrt{-a \tan ^2(c+d x)}}","\frac{\tan (c+d x) \log (\sin (c+d x))}{d \sqrt{-a \tan ^2(c+d x)}}",1,"(Log[Sin[c + d*x]]*Tan[c + d*x])/(d*Sqrt[-(a*Tan[c + d*x]^2)])","A",3,3,17,0.1765,1,"{4121, 3658, 3475}"
225,1,67,0,0.0415855,"\int \frac{1}{\left(a-a \sec ^2(c+d x)\right)^{3/2}} \, dx","Int[(a - a*Sec[c + d*x]^2)^(-3/2),x]","\frac{\cot (c+d x)}{2 a d \sqrt{-a \tan ^2(c+d x)}}+\frac{\tan (c+d x) \log (\sin (c+d x))}{a d \sqrt{-a \tan ^2(c+d x)}}","\frac{\cot (c+d x)}{2 a d \sqrt{-a \tan ^2(c+d x)}}+\frac{\tan (c+d x) \log (\sin (c+d x))}{a d \sqrt{-a \tan ^2(c+d x)}}",1,"Cot[c + d*x]/(2*a*d*Sqrt[-(a*Tan[c + d*x]^2)]) + (Log[Sin[c + d*x]]*Tan[c + d*x])/(a*d*Sqrt[-(a*Tan[c + d*x]^2)])","A",4,4,17,0.2353,1,"{4121, 3658, 3473, 3475}"
226,1,100,0,0.0505803,"\int \frac{1}{\left(a-a \sec ^2(c+d x)\right)^{5/2}} \, dx","Int[(a - a*Sec[c + d*x]^2)^(-5/2),x]","-\frac{\cot ^3(c+d x)}{4 a^2 d \sqrt{-a \tan ^2(c+d x)}}+\frac{\cot (c+d x)}{2 a^2 d \sqrt{-a \tan ^2(c+d x)}}+\frac{\tan (c+d x) \log (\sin (c+d x))}{a^2 d \sqrt{-a \tan ^2(c+d x)}}","-\frac{\cot ^3(c+d x)}{4 a^2 d \sqrt{-a \tan ^2(c+d x)}}+\frac{\cot (c+d x)}{2 a^2 d \sqrt{-a \tan ^2(c+d x)}}+\frac{\tan (c+d x) \log (\sin (c+d x))}{a^2 d \sqrt{-a \tan ^2(c+d x)}}",1,"Cot[c + d*x]/(2*a^2*d*Sqrt[-(a*Tan[c + d*x]^2)]) - Cot[c + d*x]^3/(4*a^2*d*Sqrt[-(a*Tan[c + d*x]^2)]) + (Log[Sin[c + d*x]]*Tan[c + d*x])/(a^2*d*Sqrt[-(a*Tan[c + d*x]^2)])","A",5,4,17,0.2353,1,"{4121, 3658, 3473, 3475}"
227,1,133,0,0.0612302,"\int \frac{1}{\left(a-a \sec ^2(c+d x)\right)^{7/2}} \, dx","Int[(a - a*Sec[c + d*x]^2)^(-7/2),x]","\frac{\cot ^5(c+d x)}{6 a^3 d \sqrt{-a \tan ^2(c+d x)}}-\frac{\cot ^3(c+d x)}{4 a^3 d \sqrt{-a \tan ^2(c+d x)}}+\frac{\cot (c+d x)}{2 a^3 d \sqrt{-a \tan ^2(c+d x)}}+\frac{\tan (c+d x) \log (\sin (c+d x))}{a^3 d \sqrt{-a \tan ^2(c+d x)}}","\frac{\cot ^5(c+d x)}{6 a^3 d \sqrt{-a \tan ^2(c+d x)}}-\frac{\cot ^3(c+d x)}{4 a^3 d \sqrt{-a \tan ^2(c+d x)}}+\frac{\cot (c+d x)}{2 a^3 d \sqrt{-a \tan ^2(c+d x)}}+\frac{\tan (c+d x) \log (\sin (c+d x))}{a^3 d \sqrt{-a \tan ^2(c+d x)}}",1,"Cot[c + d*x]/(2*a^3*d*Sqrt[-(a*Tan[c + d*x]^2)]) - Cot[c + d*x]^3/(4*a^3*d*Sqrt[-(a*Tan[c + d*x]^2)]) + Cot[c + d*x]^5/(6*a^3*d*Sqrt[-(a*Tan[c + d*x]^2)]) + (Log[Sin[c + d*x]]*Tan[c + d*x])/(a^3*d*Sqrt[-(a*Tan[c + d*x]^2)])","A",6,4,17,0.2353,1,"{4121, 3658, 3473, 3475}"
228,1,471,0,0.6884791,"\int \sec ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Sec[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\left(2 a^2-3 a b-8 b^2\right) \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{15 b^2 f \sqrt{a \cos ^2(e+f x)+b}}+\frac{\left(2 a^2-3 a b-8 b^2\right) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 b^2 f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b}}+\frac{\tan (e+f x) \sec ^3(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{5 f \sqrt{a \cos ^2(e+f x)+b}}+\frac{(a+4 b) \tan (e+f x) \sec (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{15 b f \sqrt{a \cos ^2(e+f x)+b}}-\frac{(a-8 b) (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 b f \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}","-\frac{\left(2 a^2-3 a b-8 b^2\right) \sin (e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{15 b^2 f}+\frac{\left(2 a^2-3 a b-8 b^2\right) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 b^2 f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}+\frac{\tan (e+f x) \sec ^3(e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{5 f}+\frac{(a+4 b) \tan (e+f x) \sec (e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{15 b f}-\frac{(a-8 b) (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 b f \left(-a \sin ^2(e+f x)+a+b\right)}",1,"-((2*a^2 - 3*a*b - 8*b^2)*Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(15*b^2*f*Sqrt[b + a*Cos[e + f*x]^2]) + ((2*a^2 - 3*a*b - 8*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])/(15*b^2*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - ((a - 8*b)*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(15*b*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]) + ((a + 4*b)*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]*Tan[e + f*x])/(15*b*f*Sqrt[b + a*Cos[e + f*x]^2]) + (Sec[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]*Tan[e + f*x])/(5*f*Sqrt[b + a*Cos[e + f*x]^2])","A",11,10,25,0.4000,1,"{4148, 6722, 1974, 412, 527, 524, 426, 424, 421, 419}"
229,1,364,0,0.5096752,"\int \sec ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Sec[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{(a+2 b) \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{3 b f \sqrt{a \cos ^2(e+f x)+b}}+\frac{\tan (e+f x) \sec (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{3 f \sqrt{a \cos ^2(e+f x)+b}}+\frac{2 (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 f \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}-\frac{(a+2 b) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 b f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b}}","\frac{(a+2 b) \sin (e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{3 b f}+\frac{\tan (e+f x) \sec (e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{3 f}+\frac{2 (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 f \left(-a \sin ^2(e+f x)+a+b\right)}-\frac{(a+2 b) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 b f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}",1,"((a + 2*b)*Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*b*f*Sqrt[b + a*Cos[e + f*x]^2]) - ((a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*b*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + (2*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]) + (Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]*Tan[e + f*x])/(3*f*Sqrt[b + a*Cos[e + f*x]^2])","A",10,10,25,0.4000,1,"{4148, 6722, 1974, 412, 527, 524, 426, 424, 421, 419}"
230,1,271,0,0.3980855,"\int \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{f \sqrt{a \cos ^2(e+f x)+b}}+\frac{(a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{f \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}-\frac{\sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b}}","\frac{\sin (e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{f}+\frac{(a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{f \left(-a \sin ^2(e+f x)+a+b\right)}-\frac{\sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}",1,"(Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(f*Sqrt[b + a*Cos[e + f*x]^2]) - (Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])/(f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + ((a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",10,10,23,0.4348,1,"{4148, 6722, 1974, 412, 12, 493, 426, 424, 421, 419}"
231,1,103,0,0.1501853,"\int \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b}}","\frac{\sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}",1,"(Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])/(f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])","A",5,5,23,0.2174,1,"{4148, 6722, 1974, 426, 424}"
232,1,299,0,0.389358,"\int \cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Cos[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sin (e+f x) \cos ^2(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{3 f \sqrt{a \cos ^2(e+f x)+b}}-\frac{b (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a f \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}+\frac{(2 a+b) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b}}","\frac{\sin (e+f x) \cos ^2(e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{3 f}-\frac{b (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a f \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{(2 a+b) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}",1,"(Cos[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*f*Sqrt[b + a*Cos[e + f*x]^2]) + ((2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*a*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*a*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",9,9,25,0.3600,1,"{4148, 6722, 1974, 417, 524, 426, 424, 421, 419}"
233,1,400,0,0.5712856,"\int \cos ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Cos[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\left(8 a^2+3 a b-2 b^2\right) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^2 f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b}}-\frac{2 b (2 a-b) (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^2 f \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}+\frac{\sin (e+f x) \cos ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)^{3/2} \sqrt{a+b \sec ^2(e+f x)}}{5 a f \sqrt{a \cos ^2(e+f x)+b}}+\frac{2 (2 a-b) \sin (e+f x) \cos ^2(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{15 a f \sqrt{a \cos ^2(e+f x)+b}}","\frac{\left(8 a^2+3 a b-2 b^2\right) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^2 f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}-\frac{2 b (2 a-b) (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^2 f \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{2 (2 a-b) \sin (e+f x) \cos ^2(e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{15 a f}+\frac{\sin (e+f x) \cos ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{5 a f}",1,"(2*(2*a - b)*Cos[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(15*a*f*Sqrt[b + a*Cos[e + f*x]^2]) + (Cos[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2)^(3/2))/(5*a*f*Sqrt[b + a*Cos[e + f*x]^2]) + ((8*a^2 + 3*a*b - 2*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])/(15*a^2*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (2*(2*a - b)*b*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(15*a^2*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",10,10,25,0.4000,1,"{4148, 6722, 1974, 416, 528, 524, 426, 424, 421, 419}"
234,1,186,0,0.1701199,"\int \sec ^6(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Sec[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\left(a^2-2 a b+5 b^2\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{16 b^2 f}+\frac{(a+b) \left(a^2-2 a b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 b^{5/2} f}-\frac{(3 a-5 b) \tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{24 b^2 f}+\frac{\tan (e+f x) \sec ^2(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{6 b f}","\frac{\left(a^2-2 a b+5 b^2\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{16 b^2 f}+\frac{(a+b) \left(a^2-2 a b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 b^{5/2} f}-\frac{(3 a-5 b) \tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{24 b^2 f}+\frac{\tan (e+f x) \sec ^2(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{6 b f}",1,"((a + b)*(a^2 - 2*a*b + 5*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*b^(5/2)*f) + ((a^2 - 2*a*b + 5*b^2)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(16*b^2*f) - ((3*a - 5*b)*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(24*b^2*f) + (Sec[e + f*x]^2*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(6*b*f)","A",6,6,25,0.2400,1,"{4146, 416, 388, 195, 217, 206}"
235,1,122,0,0.1061591,"\int \sec ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Sec[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{(a-3 b) (a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 b^{3/2} f}+\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{4 b f}-\frac{(a-3 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 b f}","-\frac{(a-3 b) (a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 b^{3/2} f}+\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{4 b f}-\frac{(a-3 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 b f}",1,"-((a - 3*b)*(a + b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*b^(3/2)*f) - ((a - 3*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*b*f) + (Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(4*b*f)","A",5,5,25,0.2000,1,"{4146, 388, 195, 217, 206}"
236,1,76,0,0.0814039,"\int \sec ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Sec[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}+\frac{(a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 \sqrt{b} f}","\frac{\tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}+\frac{(a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 \sqrt{b} f}",1,"((a + b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*Sqrt[b]*f) + (Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)","A",4,4,25,0.1600,1,"{4146, 195, 217, 206}"
237,1,79,0,0.0493114,"\int \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}",1,"(Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f","A",6,6,16,0.3750,1,"{4128, 402, 217, 206, 377, 203}"
238,1,82,0,0.0919363,"\int \cos ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Cos[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 \sqrt{a} f}+\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 \sqrt{a} f}+\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}",1,"((a + b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*Sqrt[a]*f) + (Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)","A",4,4,25,0.1600,1,"{4146, 378, 377, 203}"
239,1,140,0,0.1225732,"\int \cos ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Cos[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{(3 a-b) (a+b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{3/2} f}+\frac{\sin (e+f x) \cos ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{4 a f}+\frac{(3 a-b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 a f}","\frac{(3 a-b) (a+b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{3/2} f}+\frac{\sin (e+f x) \cos ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{4 a f}+\frac{(3 a-b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 a f}",1,"((3*a - b)*(a + b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(3/2)*f) + ((3*a - b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*a*f) + (Cos[e + f*x]^3*Sin[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(4*a*f)","A",5,5,25,0.2000,1,"{4146, 382, 378, 377, 203}"
240,1,196,0,0.2058604,"\int \cos ^6(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Cos[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{(a+b) \left(5 a^2-2 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{5/2} f}+\frac{(3 a-b) (5 a+3 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{48 a^2 f}+\frac{\sin (e+f x) \cos ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{6 f}+\frac{(5 a+b) \sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{24 a f}","\frac{(a+b) \left(5 a^2-2 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{5/2} f}+\frac{(3 a-b) (5 a+3 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{48 a^2 f}+\frac{\sin (e+f x) \cos ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{6 f}+\frac{(5 a+b) \sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{24 a f}",1,"((a + b)*(5*a^2 - 2*a*b + b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(5/2)*f) + ((3*a - b)*(5*a + 3*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(48*a^2*f) + ((5*a + b)*Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(24*a*f) + (Cos[e + f*x]^5*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(6*f)","A",7,6,25,0.2400,1,"{4146, 412, 527, 12, 377, 203}"
241,1,572,0,0.8936313,"\int \sec ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Sec[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{2 (a+2 b) \left(a^2-4 a b-4 b^2\right) \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{35 b^2 f \sqrt{a \cos ^2(e+f x)+b}}+\frac{\left(a^2+11 a b+8 b^2\right) \tan (e+f x) \sec (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{35 b f \sqrt{a \cos ^2(e+f x)+b}}-\frac{(a+b) \left(a^2-16 a b-16 b^2\right) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{35 b f \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}+\frac{2 (a+2 b) \left(a^2-4 a b-4 b^2\right) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{35 b^2 f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b}}+\frac{b \tan (e+f x) \sec ^5(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{7 f \sqrt{a \cos ^2(e+f x)+b}}+\frac{2 (4 a+3 b) \tan (e+f x) \sec ^3(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{35 f \sqrt{a \cos ^2(e+f x)+b}}","-\frac{2 (a+2 b) \left(a^2-4 a b-4 b^2\right) \sin (e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{35 b^2 f}+\frac{\left(a^2+11 a b+8 b^2\right) \tan (e+f x) \sec (e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{35 b f}-\frac{(a+b) \left(a^2-16 a b-16 b^2\right) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{35 b f \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{2 (a+2 b) \left(a^2-4 a b-4 b^2\right) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{35 b^2 f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}+\frac{b \tan (e+f x) \sec ^5(e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{7 f}+\frac{2 (4 a+3 b) \tan (e+f x) \sec ^3(e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{35 f}",1,"(-2*(a + 2*b)*(a^2 - 4*a*b - 4*b^2)*Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(35*b^2*f*Sqrt[b + a*Cos[e + f*x]^2]) + (2*(a + 2*b)*(a^2 - 4*a*b - 4*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])/(35*b^2*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - ((a + b)*(a^2 - 16*a*b - 16*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(35*b*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]) + ((a^2 + 11*a*b + 8*b^2)*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]*Tan[e + f*x])/(35*b*f*Sqrt[b + a*Cos[e + f*x]^2]) + (2*(4*a + 3*b)*Sec[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]*Tan[e + f*x])/(35*f*Sqrt[b + a*Cos[e + f*x]^2]) + (b*Sec[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]*Tan[e + f*x])/(7*f*Sqrt[b + a*Cos[e + f*x]^2])","A",12,10,25,0.4000,1,"{4148, 6722, 1974, 413, 527, 524, 426, 424, 421, 419}"
242,1,470,0,0.7003384,"\int \sec ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\left(3 a^2+13 a b+8 b^2\right) \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{15 b f \sqrt{a \cos ^2(e+f x)+b}}-\frac{\left(3 a^2+13 a b+8 b^2\right) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 b f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b}}+\frac{b \tan (e+f x) \sec ^3(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{5 f \sqrt{a \cos ^2(e+f x)+b}}+\frac{2 (3 a+2 b) \tan (e+f x) \sec (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{15 f \sqrt{a \cos ^2(e+f x)+b}}+\frac{(a+b) (9 a+8 b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 f \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}","\frac{\left(3 a^2+13 a b+8 b^2\right) \sin (e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{15 b f}-\frac{\left(3 a^2+13 a b+8 b^2\right) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 b f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}+\frac{b \tan (e+f x) \sec ^3(e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{5 f}+\frac{2 (3 a+2 b) \tan (e+f x) \sec (e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{15 f}+\frac{(a+b) (9 a+8 b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 f \left(-a \sin ^2(e+f x)+a+b\right)}",1,"((3*a^2 + 13*a*b + 8*b^2)*Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(15*b*f*Sqrt[b + a*Cos[e + f*x]^2]) - ((3*a^2 + 13*a*b + 8*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])/(15*b*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + ((a + b)*(9*a + 8*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(15*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]) + (2*(3*a + 2*b)*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]*Tan[e + f*x])/(15*f*Sqrt[b + a*Cos[e + f*x]^2]) + (b*Sec[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]*Tan[e + f*x])/(5*f*Sqrt[b + a*Cos[e + f*x]^2])","A",11,10,25,0.4000,1,"{4148, 6722, 1974, 413, 527, 524, 426, 424, 421, 419}"
243,1,366,0,0.5291562,"\int \sec (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Sec[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{2 (2 a+b) \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{3 f \sqrt{a \cos ^2(e+f x)+b}}+\frac{b \tan (e+f x) \sec (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{3 f \sqrt{a \cos ^2(e+f x)+b}}+\frac{(a+b) (3 a+2 b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 f \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}-\frac{2 (2 a+b) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b}}","\frac{2 (2 a+b) \sin (e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{3 f}+\frac{b \tan (e+f x) \sec (e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{3 f}+\frac{(a+b) (3 a+2 b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 f \left(-a \sin ^2(e+f x)+a+b\right)}-\frac{2 (2 a+b) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}",1,"(2*(2*a + b)*Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*f*Sqrt[b + a*Cos[e + f*x]^2]) - (2*(2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + ((a + b)*(3*a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]) + (b*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]*Tan[e + f*x])/(3*f*Sqrt[b + a*Cos[e + f*x]^2])","A",10,10,23,0.4348,1,"{4148, 6722, 1974, 413, 527, 524, 426, 424, 421, 419}"
244,1,277,0,0.3000542,"\int \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{b \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{f \sqrt{a \cos ^2(e+f x)+b}}+\frac{b (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{f \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}+\frac{(a-b) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b}}","\frac{b \sin (e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{f}+\frac{b (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{f \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{(a-b) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}",1,"(b*Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(f*Sqrt[b + a*Cos[e + f*x]^2]) + ((a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])/(f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + (b*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",9,9,23,0.3913,1,"{4148, 6722, 1974, 413, 524, 426, 424, 421, 419}"
245,1,294,0,0.4210353,"\int \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{a \sin (e+f x) \cos ^2(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{3 f \sqrt{a \cos ^2(e+f x)+b}}-\frac{b (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 f \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}+\frac{2 (a+2 b) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b}}","\frac{a \sin (e+f x) \cos ^2(e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{3 f}-\frac{b (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 f \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{2 (a+2 b) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}",1,"(a*Cos[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*f*Sqrt[b + a*Cos[e + f*x]^2]) + (2*(a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",9,9,25,0.3600,1,"{4148, 6722, 1974, 416, 524, 426, 424, 421, 419}"
246,1,395,0,0.641289,"\int \cos ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\left(8 a^2+13 a b+3 b^2\right) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b}}+\frac{a \sin (e+f x) \cos ^4(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{5 f \sqrt{a \cos ^2(e+f x)+b}}-\frac{2 (a-3 (a+b)) \sin (e+f x) \cos ^2(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}{15 f \sqrt{a \cos ^2(e+f x)+b}}-\frac{b (a+b) (4 a+3 b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a f \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}","\frac{\left(8 a^2+13 a b+3 b^2\right) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a f \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}+\frac{a \sin (e+f x) \cos ^4(e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{5 f}-\frac{2 (a-3 (a+b)) \sin (e+f x) \cos ^2(e+f x) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}{15 f}-\frac{b (a+b) (4 a+3 b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a f \left(-a \sin ^2(e+f x)+a+b\right)}",1,"(-2*(a - 3*(a + b))*Cos[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(15*f*Sqrt[b + a*Cos[e + f*x]^2]) + (a*Cos[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(5*f*Sqrt[b + a*Cos[e + f*x]^2]) + ((8*a^2 + 13*a*b + 3*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])/(15*a*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(a + b)*(4*a + 3*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(15*a*f*Sqrt[b + a*Cos[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",10,10,25,0.4000,1,"{4148, 6722, 1974, 416, 528, 524, 426, 424, 421, 419}"
247,1,243,0,0.2228903,"\int \sec ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Sec[e + f*x]^6*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\left(3 a^2-10 a b+35 b^2\right) \tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{192 b^2 f}+\frac{(a+b) \left(3 a^2-10 a b+35 b^2\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{128 b^2 f}+\frac{(a+b)^2 \left(3 a^2-10 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{128 b^{5/2} f}-\frac{(3 a-7 b) \tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{5/2}}{48 b^2 f}+\frac{\tan (e+f x) \sec ^2(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{5/2}}{8 b f}","\frac{\left(3 a^2-10 a b+35 b^2\right) \tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{192 b^2 f}+\frac{(a+b) \left(3 a^2-10 a b+35 b^2\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{128 b^2 f}+\frac{(a+b)^2 \left(3 a^2-10 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{128 b^{5/2} f}-\frac{(3 a-7 b) \tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{5/2}}{48 b^2 f}+\frac{\tan (e+f x) \sec ^2(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{5/2}}{8 b f}",1,"((a + b)^2*(3*a^2 - 10*a*b + 35*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(128*b^(5/2)*f) + ((a + b)*(3*a^2 - 10*a*b + 35*b^2)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(128*b^2*f) + ((3*a^2 - 10*a*b + 35*b^2)*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(192*b^2*f) - ((3*a - 7*b)*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(5/2))/(48*b^2*f) + (Sec[e + f*x]^2*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(5/2))/(8*b*f)","A",7,6,25,0.2400,1,"{4146, 416, 388, 195, 217, 206}"
248,1,165,0,0.1306783,"\int \sec ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Sec[e + f*x]^4*(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{(a-5 b) (a+b)^2 \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 b^{3/2} f}+\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{5/2}}{6 b f}-\frac{(a-5 b) \tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{24 b f}-\frac{(a-5 b) (a+b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{16 b f}","-\frac{(a-5 b) (a+b)^2 \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 b^{3/2} f}+\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{5/2}}{6 b f}-\frac{(a-5 b) \tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{24 b f}-\frac{(a-5 b) (a+b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{16 b f}",1,"-((a - 5*b)*(a + b)^2*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*b^(3/2)*f) - ((a - 5*b)*(a + b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(16*b*f) - ((a - 5*b)*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(24*b*f) + (Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(5/2))/(6*b*f)","A",6,5,25,0.2000,1,"{4146, 388, 195, 217, 206}"
249,1,111,0,0.1024511,"\int \sec ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Sec[e + f*x]^2*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{3 (a+b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 f}+\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{4 f}+\frac{3 (a+b)^2 \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 \sqrt{b} f}","\frac{3 (a+b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 f}+\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{4 f}+\frac{3 (a+b)^2 \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 \sqrt{b} f}",1,"(3*(a + b)^2*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*Sqrt[b]*f) + (3*(a + b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*f) + (Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(4*f)","A",5,4,25,0.1600,1,"{4146, 195, 217, 206}"
250,1,118,0,0.0920687,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}+\frac{\sqrt{b} (3 a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}+\frac{\sqrt{b} (3 a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}",1,"(a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*(3*a + b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (b*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)","A",7,7,16,0.4375,1,"{4128, 416, 523, 217, 206, 377, 203}"
251,1,124,0,0.1363192,"\int \cos ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Cos[e + f*x]^2*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{\sqrt{a} (a+3 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}+\frac{a \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}","\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{\sqrt{a} (a+3 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}+\frac{a \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}",1,"(Sqrt[a]*(a + 3*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (b^(3/2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + (a*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)","A",7,7,25,0.2800,1,"{4146, 413, 523, 217, 206, 377, 203}"
252,1,125,0,0.1200047,"\int \cos ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{3 (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 \sqrt{a} f}+\frac{\sin (e+f x) \cos ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{4 f}+\frac{3 (a+b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 f}","\frac{3 (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 \sqrt{a} f}+\frac{\sin (e+f x) \cos ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{4 f}+\frac{3 (a+b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 f}",1,"(3*(a + b)^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*Sqrt[a]*f) + (3*(a + b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*f) + (Cos[e + f*x]^3*Sin[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(4*f)","A",5,4,25,0.1600,1,"{4146, 378, 377, 203}"
253,1,193,0,0.1623243,"\int \cos ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Cos[e + f*x]^6*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{(5 a-b) (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{3/2} f}+\frac{\sin (e+f x) \cos ^5(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{5/2}}{6 a f}+\frac{(5 a-b) \sin (e+f x) \cos ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{24 a f}+\frac{(5 a-b) (a+b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{16 a f}","\frac{(5 a-b) (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{3/2} f}+\frac{\sin (e+f x) \cos ^5(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{5/2}}{6 a f}+\frac{(5 a-b) \sin (e+f x) \cos ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}{24 a f}+\frac{(5 a-b) (a+b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{16 a f}",1,"((5*a - b)*(a + b)^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(3/2)*f) + ((5*a - b)*(a + b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(16*a*f) + ((5*a - b)*Cos[e + f*x]^3*Sin[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(24*a*f) + (Cos[e + f*x]^5*Sin[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(5/2))/(6*a*f)","A",6,5,25,0.2000,1,"{4146, 382, 378, 377, 203}"
254,1,166,0,0.1718853,"\int \left(a+b \sec ^2(c+d x)\right)^{5/2} \, dx","Int[(a + b*Sec[c + d*x]^2)^(5/2),x]","\frac{\sqrt{b} \left(15 a^2+10 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a+b \tan ^2(c+d x)+b}}\right)}{8 d}+\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+b \tan ^2(c+d x)+b}}\right)}{d}+\frac{b \tan (c+d x) \left(a+b \tan ^2(c+d x)+b\right)^{3/2}}{4 d}+\frac{b (7 a+3 b) \tan (c+d x) \sqrt{a+b \tan ^2(c+d x)+b}}{8 d}","\frac{\sqrt{b} \left(15 a^2+10 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a+b \tan ^2(c+d x)+b}}\right)}{8 d}+\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+b \tan ^2(c+d x)+b}}\right)}{d}+\frac{b \tan (c+d x) \left(a+b \tan ^2(c+d x)+b\right)^{3/2}}{4 d}+\frac{b (7 a+3 b) \tan (c+d x) \sqrt{a+b \tan ^2(c+d x)+b}}{8 d}",1,"(a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + b + b*Tan[c + d*x]^2]])/d + (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTanh[(Sqrt[b]*Tan[c + d*x])/Sqrt[a + b + b*Tan[c + d*x]^2]])/(8*d) + (b*(7*a + 3*b)*Tan[c + d*x]*Sqrt[a + b + b*Tan[c + d*x]^2])/(8*d) + (b*Tan[c + d*x]*(a + b + b*Tan[c + d*x]^2)^(3/2))/(4*d)","A",8,8,16,0.5000,1,"{4128, 416, 528, 523, 217, 206, 377, 203}"
255,1,42,0,0.0368925,"\int \left(1+\sec ^2(x)\right)^{3/2} \, dx","Int[(1 + Sec[x]^2)^(3/2),x]","\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{\tan ^2(x)+2}}\right)+\frac{1}{2} \tan (x) \sqrt{\tan ^2(x)+2}+2 \sinh ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)","\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{\tan ^2(x)+2}}\right)+\frac{1}{2} \tan (x) \sqrt{\tan ^2(x)+2}+2 \sinh ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)",1,"2*ArcSinh[Tan[x]/Sqrt[2]] + ArcTan[Tan[x]/Sqrt[2 + Tan[x]^2]] + (Tan[x]*Sqrt[2 + Tan[x]^2])/2","A",6,6,10,0.6000,1,"{4128, 416, 523, 215, 377, 203}"
256,1,24,0,0.0185345,"\int \sqrt{1+\sec ^2(x)} \, dx","Int[Sqrt[1 + Sec[x]^2],x]","\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{\tan ^2(x)+2}}\right)+\sinh ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)","\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{\tan ^2(x)+2}}\right)+\sinh ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)",1,"ArcSinh[Tan[x]/Sqrt[2]] + ArcTan[Tan[x]/Sqrt[2 + Tan[x]^2]]","A",5,5,10,0.5000,1,"{4128, 402, 215, 377, 203}"
257,1,380,0,0.5717596,"\int \frac{\sec ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Sec[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{2 (a-b) \tan (e+f x) \sec (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{3 b^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{2 (a-b) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 b^2 f \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}+\frac{\tan (e+f x) \sec ^3(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{3 b f \sqrt{a+b \sec ^2(e+f x)}}-\frac{(a-2 b) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 b f \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}","-\frac{2 (a-b) \tan (e+f x) \sec (e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{3 b^2 f \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{2 (a-b) \left(-a \sin ^2(e+f x)+a+b\right) E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 b^2 f \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{\tan (e+f x) \sec ^3(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{3 b f \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{(a-2 b) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 b f \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"(2*(a - b)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*b^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - ((a - 2*b)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*b*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]) - (2*(a - b)*Sqrt[b + a*Cos[e + f*x]^2]*Sec[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2]*Tan[e + f*x])/(3*b^2*f*Sqrt[a + b*Sec[e + f*x]^2]) + (Sqrt[b + a*Cos[e + f*x]^2]*Sec[e + f*x]^3*Sqrt[a + b - a*Sin[e + f*x]^2]*Tan[e + f*x])/(3*b*f*Sqrt[a + b*Sec[e + f*x]^2])","A",10,10,25,0.4000,1,"{4148, 6722, 1974, 414, 527, 524, 426, 424, 421, 419}"
258,1,202,0,0.370607,"\int \frac{\sec ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Sec[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\tan (e+f x) \sec (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{b f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\sqrt{a} \sqrt{a+b} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)|\frac{a+b}{a}\right)}{b f \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}","\frac{\tan (e+f x) \sec (e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{b f \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{\sqrt{a} \sqrt{a+b} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)|\frac{a+b}{a}\right)}{b f \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"-((Sqrt[a]*Sqrt[a + b]*Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]], (a + b)/a]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(b*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])) + (Sqrt[b + a*Cos[e + f*x]^2]*Sec[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2]*Tan[e + f*x])/(b*f*Sqrt[a + b*Sec[e + f*x]^2])","A",7,7,25,0.2800,1,"{4148, 6722, 1974, 414, 21, 427, 424}"
259,1,103,0,0.2717982,"\int \frac{\sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Sec[e + f*x]/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{f \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}","\frac{\sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{f \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"(Sqrt[b + a*Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",5,5,23,0.2174,1,"{4148, 6722, 1974, 421, 419}"
260,1,128,0,0.1662385,"\int \frac{\cos (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Cos[e + f*x]/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{a+b} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)|\frac{a+b}{a}\right)}{\sqrt{a} f \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}","\frac{\sqrt{a+b} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)|\frac{a+b}{a}\right)}{\sqrt{a} f \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"(Sqrt[a + b]*Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]], (a + b)/a]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(Sqrt[a]*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",5,5,23,0.2174,1,"{4148, 6722, 1974, 427, 424}"
261,1,296,0,0.4067972,"\int \frac{\cos ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Cos[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{b (a-2 b) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^2 f \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}+\frac{2 (a-b) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^2 f \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{3 a f \sqrt{a+b \sec ^2(e+f x)}}","-\frac{b (a-2 b) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^2 f \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{2 (a-b) \left(-a \sin ^2(e+f x)+a+b\right) E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^2 f \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{\sin (e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{3 a f \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"(Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*a*f*Sqrt[a + b*Sec[e + f*x]^2]) + (2*(a - b)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*a^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - ((a - 2*b)*b*Sqrt[b + a*Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*a^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",9,9,25,0.3600,1,"{4148, 6722, 1974, 416, 524, 426, 424, 421, 419}"
262,1,395,0,0.5736368,"\int \frac{\cos ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Cos[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{b \left(4 a^2-3 a b+8 b^2\right) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}+\frac{\left(8 a^2-7 a b+8 b^2\right) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}+\frac{4 (a-b) \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{15 a^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sin (e+f x) \cos ^2(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{5 a f \sqrt{a+b \sec ^2(e+f x)}}","-\frac{b \left(4 a^2-3 a b+8 b^2\right) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{\left(8 a^2-7 a b+8 b^2\right) \left(-a \sin ^2(e+f x)+a+b\right) E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{4 (a-b) \sin (e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{15 a^2 f \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{\sin (e+f x) \cos ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{5 a f \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"(4*(a - b)*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(15*a^2*f*Sqrt[a + b*Sec[e + f*x]^2]) + (Cos[e + f*x]^2*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(5*a*f*Sqrt[a + b*Sec[e + f*x]^2]) + ((8*a^2 - 7*a*b + 8*b^2)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(15*a^3*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(4*a^2 - 3*a*b + 8*b^2)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(15*a^3*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",10,10,25,0.4000,1,"{4148, 6722, 1974, 416, 528, 524, 426, 424, 421, 419}"
263,1,137,0,0.1345422,"\int \frac{\sec ^6(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Sec[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\left(3 a^2-2 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 b^{5/2} f}-\frac{3 (a-b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 b^2 f}+\frac{\tan (e+f x) \sec ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 b f}","\frac{\left(3 a^2-2 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 b^{5/2} f}-\frac{3 (a-b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 b^2 f}+\frac{\tan (e+f x) \sec ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 b f}",1,"((3*a^2 - 2*a*b + 3*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*b^(5/2)*f) - (3*(a - b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*b^2*f) + (Sec[e + f*x]^2*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(4*b*f)","A",5,5,25,0.2000,1,"{4146, 416, 388, 217, 206}"
264,1,81,0,0.0912598,"\int \frac{\sec ^4(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Sec[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 b f}-\frac{(a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 b^{3/2} f}","\frac{\tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 b f}-\frac{(a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 b^{3/2} f}",1,"-((a - b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*b^(3/2)*f) + (Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*b*f)","A",4,4,25,0.1600,1,"{4146, 388, 217, 206}"
265,1,39,0,0.0714095,"\int \frac{\sec ^2(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Sec[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{b} f}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{b} f}",1,"ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[b]*f)","A",3,3,25,0.1200,1,"{4146, 217, 206}"
266,1,39,0,0.0299862,"\int \frac{1}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[1/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}",1,"ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f)","A",3,3,16,0.1875,1,"{4128, 377, 203}"
267,1,87,0,0.0992395,"\int \frac{\cos ^2(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Cos[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 a^{3/2} f}+\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 a f}","\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 a^{3/2} f}+\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 a f}",1,"((a - b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*a^(3/2)*f) + (Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*a*f)","A",4,4,25,0.1600,1,"{4146, 382, 377, 203}"
268,1,143,0,0.1415026,"\int \frac{\cos ^4(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Cos[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\left(3 a^2-2 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{5/2} f}+\frac{3 (a-b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 a^2 f}+\frac{\sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 a f}","\frac{\left(3 a^2-2 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{5/2} f}+\frac{3 (a-b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 a^2 f}+\frac{\sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 a f}",1,"((3*a^2 - 2*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(5/2)*f) + (3*(a - b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*a^2*f) + (Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(4*a*f)","A",6,6,25,0.2400,1,"{4146, 414, 527, 12, 377, 203}"
269,1,204,0,0.2068833,"\int \frac{\cos ^6(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Cos[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{(a-b) \left(5 a^2+2 a b+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{7/2} f}+\frac{\left(15 a^2-14 a b+15 b^2\right) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{48 a^3 f}+\frac{5 (a-b) \sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{24 a^2 f}+\frac{\sin (e+f x) \cos ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{6 a f}","\frac{(a-b) \left(5 a^2+2 a b+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{7/2} f}+\frac{\left(15 a^2-14 a b+15 b^2\right) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{48 a^3 f}+\frac{5 (a-b) \sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{24 a^2 f}+\frac{\sin (e+f x) \cos ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{6 a f}",1,"((a - b)*(5*a^2 + 2*a*b + 5*b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(7/2)*f) + ((15*a^2 - 14*a*b + 15*b^2)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(48*a^3*f) + (5*(a - b)*Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(24*a^2*f) + (Cos[e + f*x]^5*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(6*a*f)","A",7,6,25,0.2400,1,"{4146, 414, 527, 12, 377, 203}"
270,1,367,0,0.590233,"\int \frac{\sec ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{a (2 a+b) \sin (e+f x) \sqrt{a \cos ^2(e+f x)+b}}{b^2 f (a+b) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}-\frac{(2 a+b) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{b^2 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}+\frac{\tan (e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)+b}}{b f \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{b f \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}","\frac{a (2 a+b) \sin (e+f x)}{b^2 f (a+b) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{(2 a+b) \left(-a \sin ^2(e+f x)+a+b\right) E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{b^2 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{\tan (e+f x) \sec (e+f x)}{b f \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{\sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{b f \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"(a*(2*a + b)*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(b^2*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]) - ((2*a + b)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(b^2*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + (Sqrt[b + a*Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(b*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]) + (Sqrt[b + a*Cos[e + f*x]^2]*Sec[e + f*x]*Tan[e + f*x])/(b*f*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",10,10,25,0.4000,1,"{4148, 6722, 1974, 414, 527, 524, 426, 424, 421, 419}"
271,1,182,0,0.3839543,"\int \frac{\sec ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Sec[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{b f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}-\frac{a \sin (e+f x) \sqrt{a \cos ^2(e+f x)+b}}{b f (a+b) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}","\frac{\left(-a \sin ^2(e+f x)+a+b\right) E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{b f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{a \sin (e+f x)}{b f (a+b) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"-((a*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(b*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])) + (Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(b*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])","A",7,7,25,0.2800,1,"{4148, 6722, 1974, 414, 21, 426, 424}"
272,1,284,0,0.4478856,"\int \frac{\sec (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Sec[e + f*x]/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sin (e+f x) \sqrt{a \cos ^2(e+f x)+b}}{f (a+b) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{a f \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}-\frac{\sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{a f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}","\frac{\sin (e+f x)}{f (a+b) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{\sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{a f \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{\left(-a \sin ^2(e+f x)+a+b\right) E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{a f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"(Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/((a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]) - (Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(a*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + (Sqrt[b + a*Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(a*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",9,9,23,0.3913,1,"{4148, 6722, 1974, 412, 493, 426, 424, 421, 419}"
273,1,295,0,0.3371208,"\int \frac{\cos (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Cos[e + f*x]/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{2 b \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{a^2 f \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}+\frac{(a+2 b) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{a^2 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}-\frac{b \sin (e+f x) \sqrt{a \cos ^2(e+f x)+b}}{a f (a+b) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}","-\frac{2 b \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{a^2 f \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{(a+2 b) \left(-a \sin ^2(e+f x)+a+b\right) E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{a^2 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{b \sin (e+f x)}{a f (a+b) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"-((b*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])) + ((a + 2*b)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(a^2*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (2*b*Sqrt[b + a*Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(a^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",9,9,23,0.3913,1,"{4148, 6722, 1974, 413, 524, 426, 424, 421, 419}"
274,1,399,0,0.5608398,"\int \frac{\cos ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\left(2 a^2-3 a b-8 b^2\right) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^3 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}+\frac{(a+4 b) \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{3 a^2 f (a+b) \sqrt{a+b \sec ^2(e+f x)}}-\frac{b (a-8 b) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}-\frac{b \sin (e+f x) \cos ^2(e+f x) \sqrt{a \cos ^2(e+f x)+b}}{a f (a+b) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}","\frac{\left(2 a^2-3 a b-8 b^2\right) \left(-a \sin ^2(e+f x)+a+b\right) E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^3 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{(a+4 b) \sin (e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{3 a^2 f (a+b) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{b (a-8 b) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^3 f \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{b \sin (e+f x) \cos ^2(e+f x)}{a f (a+b) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"-((b*Cos[e + f*x]^2*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])) + ((a + 4*b)*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*a^2*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]) + ((2*a^2 - 3*a*b - 8*b^2)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*a^3*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - ((a - 8*b)*b*Sqrt[b + a*Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*a^3*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",10,10,25,0.4000,1,"{4148, 6722, 1974, 413, 528, 524, 426, 424, 421, 419}"
275,1,509,0,0.7510633,"\int \frac{\cos ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\left(4 a^2-5 a b-24 b^2\right) \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{15 a^3 f (a+b) \sqrt{a+b \sec ^2(e+f x)}}-\frac{4 b \left(a^2-2 a b+12 b^2\right) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^4 f \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}+\frac{\left(-9 a^2 b+8 a^3+16 a b^2+48 b^3\right) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^4 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}+\frac{(a+6 b) \sin (e+f x) \cos ^2(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{5 a^2 f (a+b) \sqrt{a+b \sec ^2(e+f x)}}-\frac{b \sin (e+f x) \cos ^4(e+f x) \sqrt{a \cos ^2(e+f x)+b}}{a f (a+b) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}","\frac{\left(4 a^2-5 a b-24 b^2\right) \sin (e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{15 a^3 f (a+b) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{4 b \left(a^2-2 a b+12 b^2\right) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^4 f \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{\left(-9 a^2 b+8 a^3+16 a b^2+48 b^3\right) \left(-a \sin ^2(e+f x)+a+b\right) E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^4 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{(a+6 b) \sin (e+f x) \cos ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{5 a^2 f (a+b) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{b \sin (e+f x) \cos ^4(e+f x)}{a f (a+b) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"-((b*Cos[e + f*x]^4*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])) + ((4*a^2 - 5*a*b - 24*b^2)*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(15*a^3*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]) + ((a + 6*b)*Cos[e + f*x]^2*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(5*a^2*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]) + ((8*a^3 - 9*a^2*b + 16*a*b^2 + 48*b^3)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(15*a^4*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (4*b*(a^2 - 2*a*b + 12*b^2)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(15*a^4*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",11,10,25,0.4000,1,"{4148, 6722, 1974, 413, 528, 524, 426, 424, 421, 419}"
276,1,138,0,0.14593,"\int \frac{\sec ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Sec[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{(3 a+b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 b^2 f (a+b)}-\frac{(3 a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 b^{5/2} f}-\frac{a \tan (e+f x) \sec ^2(e+f x)}{b f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}","\frac{(3 a+b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 b^2 f (a+b)}-\frac{(3 a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 b^{5/2} f}-\frac{a \tan (e+f x) \sec ^2(e+f x)}{b f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}",1,"-((3*a - b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*b^(5/2)*f) - (a*Sec[e + f*x]^2*Tan[e + f*x])/(b*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + ((3*a + b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*b^2*(a + b)*f)","A",5,5,25,0.2000,1,"{4146, 413, 388, 217, 206}"
277,1,77,0,0.0969246,"\int \frac{\sec ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Sec[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{b^{3/2} f}-\frac{a \tan (e+f x)}{b f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{b^{3/2} f}-\frac{a \tan (e+f x)}{b f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}",1,"ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(b^(3/2)*f) - (a*Tan[e + f*x])/(b*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",4,4,25,0.1600,1,"{4146, 385, 217, 206}"
278,1,32,0,0.0745116,"\int \frac{\sec ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\tan (e+f x)}{f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}","\frac{\tan (e+f x)}{f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}",1,"Tan[e + f*x]/((a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",2,2,25,0.08000,1,"{4146, 191}"
279,1,77,0,0.0486906,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[(a + b*Sec[e + f*x]^2)^(-3/2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}-\frac{b \tan (e+f x)}{a f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}-\frac{b \tan (e+f x)}{a f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}",1,"ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f) - (b*Tan[e + f*x])/(a*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",4,4,16,0.2500,1,"{4128, 382, 377, 203}"
280,1,131,0,0.1550554,"\int \frac{\cos ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{(a-3 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 a^{5/2} f}+\frac{b (a+3 b) \tan (e+f x)}{2 a^2 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\sin (e+f x) \cos (e+f x)}{2 a f \sqrt{a+b \tan ^2(e+f x)+b}}","\frac{(a-3 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 a^{5/2} f}+\frac{b (a+3 b) \tan (e+f x)}{2 a^2 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\sin (e+f x) \cos (e+f x)}{2 a f \sqrt{a+b \tan ^2(e+f x)+b}}",1,"((a - 3*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*a^(5/2)*f) + (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + (b*(a + 3*b)*Tan[e + f*x])/(2*a^2*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",6,6,25,0.2400,1,"{4146, 414, 527, 12, 377, 203}"
281,1,194,0,0.225588,"\int \frac{\cos ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Cos[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{3 \left(a^2-2 a b+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{7/2} f}+\frac{b (a-3 b) (3 a+5 b) \tan (e+f x)}{8 a^3 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{(3 a-5 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \sqrt{a+b \tan ^2(e+f x)+b}}","\frac{3 \left(a^2-2 a b+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{7/2} f}+\frac{b (a-3 b) (3 a+5 b) \tan (e+f x)}{8 a^3 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{(3 a-5 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \sqrt{a+b \tan ^2(e+f x)+b}}",1,"(3*(a^2 - 2*a*b + 5*b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(7/2)*f) + ((3*a - 5*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + ((a - 3*b)*b*(3*a + 5*b)*Tan[e + f*x])/(8*a^3*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",7,6,25,0.2400,1,"{4146, 414, 527, 12, 377, 203}"
282,1,271,0,0.3153951,"\int \frac{\cos ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\left(-9 a^2 b+5 a^3+15 a b^2-35 b^3\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{9/2} f}+\frac{b \left(-17 a^2 b+15 a^3+25 a b^2+105 b^3\right) \tan (e+f x)}{48 a^4 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\left(15 a^2-22 a b+35 b^2\right) \sin (e+f x) \cos (e+f x)}{48 a^3 f \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{(5 a-7 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\sin (e+f x) \cos ^5(e+f x)}{6 a f \sqrt{a+b \tan ^2(e+f x)+b}}","\frac{\left(-9 a^2 b+5 a^3+15 a b^2-35 b^3\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{9/2} f}+\frac{b \left(-17 a^2 b+15 a^3+25 a b^2+105 b^3\right) \tan (e+f x)}{48 a^4 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\left(15 a^2-22 a b+35 b^2\right) \sin (e+f x) \cos (e+f x)}{48 a^3 f \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{(5 a-7 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\sin (e+f x) \cos ^5(e+f x)}{6 a f \sqrt{a+b \tan ^2(e+f x)+b}}",1,"((5*a^3 - 9*a^2*b + 15*a*b^2 - 35*b^3)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(9/2)*f) + ((15*a^2 - 22*a*b + 35*b^2)*Cos[e + f*x]*Sin[e + f*x])/(48*a^3*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + ((5*a - 7*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + (Cos[e + f*x]^5*Sin[e + f*x])/(6*a*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + (b*(15*a^3 - 17*a^2*b + 25*a*b^2 + 105*b^3)*Tan[e + f*x])/(48*a^4*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",8,6,25,0.2400,1,"{4146, 414, 527, 12, 377, 203}"
283,1,383,0,0.6369586,"\int \frac{\sec ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{2 a (a+2 b) \sin (e+f x) \sqrt{a \cos ^2(e+f x)+b}}{3 b^2 f (a+b)^2 \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}+\frac{2 (a+2 b) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 b^2 f (a+b)^2 \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}-\frac{a \sin (e+f x) \sqrt{a \cos ^2(e+f x)+b}}{3 b f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^{3/2} \sqrt{a+b \sec ^2(e+f x)}}-\frac{\sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 b f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}","-\frac{2 a (a+2 b) \sin (e+f x)}{3 b^2 f (a+b)^2 \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{2 (a+2 b) \left(-a \sin ^2(e+f x)+a+b\right) E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 b^2 f (a+b)^2 \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{a \sin (e+f x)}{3 b f (a+b) \left(-a \sin ^2(e+f x)+a+b\right) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{\sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 b f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"-(a*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(3*b*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]*(a + b - a*Sin[e + f*x]^2)^(3/2)) - (2*a*(a + 2*b)*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(3*b^2*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]) + (2*(a + 2*b)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*b^2*(a + b)^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (Sqrt[b + a*Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*b*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",10,10,25,0.4000,1,"{4148, 6722, 1974, 414, 527, 524, 426, 424, 421, 419}"
284,1,381,0,0.5647334,"\int \frac{\sec ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Sec[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{(a-b) \sin (e+f x) \sqrt{a \cos ^2(e+f x)+b}}{3 b f (a+b)^2 \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sin (e+f x) \sqrt{a \cos ^2(e+f x)+b}}{3 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^{3/2} \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}+\frac{(a-b) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a b f (a+b)^2 \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}","-\frac{(a-b) \sin (e+f x)}{3 b f (a+b)^2 \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{\sin (e+f x)}{3 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{\sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{(a-b) \left(-a \sin ^2(e+f x)+a+b\right) E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a b f (a+b)^2 \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"(Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(3*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]*(a + b - a*Sin[e + f*x]^2)^(3/2)) - ((a - b)*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(3*b*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]) + ((a - b)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*a*b*(a + b)^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + (Sqrt[b + a*Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*a*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",10,10,25,0.4000,1,"{4148, 6722, 1974, 412, 527, 524, 426, 424, 421, 419}"
285,1,389,0,0.5794034,"\int \frac{\sec (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Sec[e + f*x]/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{(3 a+2 b) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^2 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}-\frac{2 (2 a+b) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^2 f (a+b)^2 \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}+\frac{2 (2 a+b) \sin (e+f x) \sqrt{a \cos ^2(e+f x)+b}}{3 a f (a+b)^2 \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}-\frac{b \sin (e+f x) \sqrt{a \cos ^2(e+f x)+b}}{3 a f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^{3/2} \sqrt{a+b \sec ^2(e+f x)}}","\frac{(3 a+2 b) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^2 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{2 (2 a+b) \left(-a \sin ^2(e+f x)+a+b\right) E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^2 f (a+b)^2 \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{2 (2 a+b) \sin (e+f x)}{3 a f (a+b)^2 \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{b \sin (e+f x)}{3 a f (a+b) \left(-a \sin ^2(e+f x)+a+b\right) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"-(b*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(3*a*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]*(a + b - a*Sin[e + f*x]^2)^(3/2)) + (2*(2*a + b)*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(3*a*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]) - (2*(2*a + b)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*a^2*(a + b)^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + ((3*a + 2*b)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*a^2*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",10,10,23,0.4348,1,"{4148, 6722, 1974, 413, 527, 524, 426, 424, 421, 419}"
286,1,411,0,0.4875393,"\int \frac{\cos (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Cos[e + f*x]/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\left(3 a^2+13 a b+8 b^2\right) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^3 f (a+b)^2 \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}-\frac{2 b (3 a+2 b) \sin (e+f x) \sqrt{a \cos ^2(e+f x)+b}}{3 a^2 f (a+b)^2 \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}-\frac{b (9 a+8 b) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^3 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}-\frac{b \sin (e+f x) \cos ^2(e+f x) \sqrt{a \cos ^2(e+f x)+b}}{3 a f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^{3/2} \sqrt{a+b \sec ^2(e+f x)}}","\frac{\left(3 a^2+13 a b+8 b^2\right) \left(-a \sin ^2(e+f x)+a+b\right) E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^3 f (a+b)^2 \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{2 b (3 a+2 b) \sin (e+f x)}{3 a^2 f (a+b)^2 \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{b (9 a+8 b) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^3 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{b \sin (e+f x) \cos ^2(e+f x)}{3 a f (a+b) \left(-a \sin ^2(e+f x)+a+b\right) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"-(b*Cos[e + f*x]^2*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(3*a*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]*(a + b - a*Sin[e + f*x]^2)^(3/2)) - (2*b*(3*a + 2*b)*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]) + ((3*a^2 + 13*a*b + 8*b^2)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*a^3*(a + b)^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(9*a + 8*b)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*a^3*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",10,10,23,0.4348,1,"{4148, 6722, 1974, 413, 526, 524, 426, 424, 421, 419}"
287,1,512,0,0.7305066,"\int \frac{\cos ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\left(a^2+11 a b+8 b^2\right) \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{3 a^3 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}-\frac{b \left(a^2-16 a b-16 b^2\right) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^4 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}+\frac{2 (a+2 b) \left(a^2-4 a b-4 b^2\right) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^4 f (a+b)^2 \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}-\frac{2 b (4 a+3 b) \sin (e+f x) \cos ^2(e+f x) \sqrt{a \cos ^2(e+f x)+b}}{3 a^2 f (a+b)^2 \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}-\frac{b \sin (e+f x) \cos ^4(e+f x) \sqrt{a \cos ^2(e+f x)+b}}{3 a f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^{3/2} \sqrt{a+b \sec ^2(e+f x)}}","\frac{\left(a^2+11 a b+8 b^2\right) \sin (e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{3 a^3 f (a+b)^2 \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{b \left(a^2-16 a b-16 b^2\right) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^4 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{2 (a+2 b) \left(a^2-4 a b-4 b^2\right) \left(-a \sin ^2(e+f x)+a+b\right) E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{3 a^4 f (a+b)^2 \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{2 b (4 a+3 b) \sin (e+f x) \cos ^2(e+f x)}{3 a^2 f (a+b)^2 \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{b \sin (e+f x) \cos ^4(e+f x)}{3 a f (a+b) \left(-a \sin ^2(e+f x)+a+b\right) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"-(b*Cos[e + f*x]^4*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(3*a*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]*(a + b - a*Sin[e + f*x]^2)^(3/2)) - (2*b*(4*a + 3*b)*Cos[e + f*x]^2*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]) + ((a^2 + 11*a*b + 8*b^2)*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*a^3*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]) + (2*(a + 2*b)*(a^2 - 4*a*b - 4*b^2)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*a^4*(a + b)^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(a^2 - 16*a*b - 16*b^2)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*a^4*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",11,11,25,0.4400,1,"{4148, 6722, 1974, 413, 526, 528, 524, 426, 424, 421, 419}"
288,1,639,0,0.9058562,"\int \frac{\cos ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\left(3 a^2+61 a b+48 b^2\right) \sin (e+f x) \cos ^2(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{15 a^3 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}+\frac{2 \left(-3 a^2 b+2 a^3-42 a b^2-32 b^3\right) \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b}}{15 a^4 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}-\frac{b \left(-9 a^2 b+4 a^3+120 a b^2+128 b^3\right) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a \cos ^2(e+f x)+b} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^5 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}+\frac{\left(27 a^2 b^2-11 a^3 b+8 a^4+184 a b^3+128 b^4\right) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos ^2(e+f x)+b} E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^5 f (a+b)^2 \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{a+b \sec ^2(e+f x)}}-\frac{2 b (5 a+4 b) \sin (e+f x) \cos ^4(e+f x) \sqrt{a \cos ^2(e+f x)+b}}{3 a^2 f (a+b)^2 \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)}}-\frac{b \sin (e+f x) \cos ^6(e+f x) \sqrt{a \cos ^2(e+f x)+b}}{3 a f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)^{3/2} \sqrt{a+b \sec ^2(e+f x)}}","\frac{2 \left(-3 a^2 b+2 a^3-42 a b^2-32 b^3\right) \sin (e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{15 a^4 f (a+b)^2 \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{\left(3 a^2+61 a b+48 b^2\right) \sin (e+f x) \cos ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{15 a^3 f (a+b)^2 \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{b \left(-9 a^2 b+4 a^3+120 a b^2+128 b^3\right) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} F\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^5 f (a+b) \sqrt{\cos ^2(e+f x)} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}+\frac{\left(27 a^2 b^2-11 a^3 b+8 a^4+184 a b^3+128 b^4\right) \left(-a \sin ^2(e+f x)+a+b\right) E\left(\sin ^{-1}(\sin (e+f x))|\frac{a}{a+b}\right)}{15 a^5 f (a+b)^2 \sqrt{\cos ^2(e+f x)} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{2 b (5 a+4 b) \sin (e+f x) \cos ^4(e+f x)}{3 a^2 f (a+b)^2 \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}-\frac{b \sin (e+f x) \cos ^6(e+f x)}{3 a f (a+b) \left(-a \sin ^2(e+f x)+a+b\right) \sqrt{\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}}",1,"-(b*Cos[e + f*x]^6*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(3*a*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]*(a + b - a*Sin[e + f*x]^2)^(3/2)) - (2*b*(5*a + 4*b)*Cos[e + f*x]^4*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]) + (2*(2*a^3 - 3*a^2*b - 42*a*b^2 - 32*b^3)*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(15*a^4*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]) + ((3*a^2 + 61*a*b + 48*b^2)*Cos[e + f*x]^2*Sqrt[b + a*Cos[e + f*x]^2]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(15*a^3*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]) + ((8*a^4 - 11*a^3*b + 27*a^2*b^2 + 184*a*b^3 + 128*b^4)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(15*a^5*(a + b)^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(4*a^3 - 9*a^2*b + 120*a*b^2 + 128*b^3)*Sqrt[b + a*Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(15*a^5*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])","A",12,11,25,0.4400,1,"{4148, 6722, 1974, 413, 526, 528, 524, 426, 424, 421, 419}"
289,1,133,0,0.1394649,"\int \frac{\sec ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Sec[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{a (3 a+5 b) \tan (e+f x)}{3 b^2 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{b^{5/2} f}-\frac{a \tan (e+f x) \sec ^2(e+f x)}{3 b f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","-\frac{a (3 a+5 b) \tan (e+f x)}{3 b^2 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{b^{5/2} f}-\frac{a \tan (e+f x) \sec ^2(e+f x)}{3 b f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(b^(5/2)*f) - (a*Sec[e + f*x]^2*Tan[e + f*x])/(3*b*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (a*(3*a + 5*b)*Tan[e + f*x])/(3*b^2*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",5,5,25,0.2000,1,"{4146, 413, 385, 217, 206}"
290,1,79,0,0.0928341,"\int \frac{\sec ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Sec[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{2 \tan (e+f x)}{3 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\tan (e+f x) \sec ^2(e+f x)}{3 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","\frac{2 \tan (e+f x)}{3 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\tan (e+f x) \sec ^2(e+f x)}{3 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"(Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (2*Tan[e + f*x])/(3*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",3,3,25,0.1200,1,"{4146, 378, 191}"
291,1,71,0,0.0895528,"\int \frac{\sec ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{2 \tan (e+f x)}{3 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\tan (e+f x)}{3 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","\frac{2 \tan (e+f x)}{3 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\tan (e+f x)}{3 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"Tan[e + f*x]/(3*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (2*Tan[e + f*x])/(3*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",3,3,25,0.1200,1,"{4146, 192, 191}"
292,1,125,0,0.1023122,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[(a + b*Sec[e + f*x]^2)^(-5/2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}-\frac{b (5 a+3 b) \tan (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{b \tan (e+f x)}{3 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}-\frac{b (5 a+3 b) \tan (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{b \tan (e+f x)}{3 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f) - (b*Tan[e + f*x])/(3*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(5*a + 3*b)*Tan[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",6,6,16,0.3750,1,"{4128, 414, 527, 12, 377, 203}"
293,1,187,0,0.2425836,"\int \frac{\cos ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{b \left(3 a^2+22 a b+15 b^2\right) \tan (e+f x)}{6 a^3 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{(a-5 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 a^{7/2} f}+\frac{b (3 a+5 b) \tan (e+f x)}{6 a^2 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","\frac{b \left(3 a^2+22 a b+15 b^2\right) \tan (e+f x)}{6 a^3 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{(a-5 b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 a^{7/2} f}+\frac{b (3 a+5 b) \tan (e+f x)}{6 a^2 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"((a - 5*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*a^(7/2)*f) + (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (b*(3*a + 5*b)*Tan[e + f*x])/(6*a^2*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (b*(3*a^2 + 22*a*b + 15*b^2)*Tan[e + f*x])/(6*a^3*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",7,6,25,0.2400,1,"{4146, 414, 527, 12, 377, 203}"
294,1,261,0,0.3383825,"\int \frac{\cos ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Cos[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\left(3 a^2-10 a b+35 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{9/2} f}+\frac{b \left(-15 a^2 b+9 a^3-145 a b^2-105 b^3\right) \tan (e+f x)}{24 a^4 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{b \left(9 a^2-18 a b-35 b^2\right) \tan (e+f x)}{24 a^3 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{(3 a-7 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","\frac{\left(3 a^2-10 a b+35 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 a^{9/2} f}+\frac{b \left(-15 a^2 b+9 a^3-145 a b^2-105 b^3\right) \tan (e+f x)}{24 a^4 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{b \left(9 a^2-18 a b-35 b^2\right) \tan (e+f x)}{24 a^3 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{(3 a-7 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"((3*a^2 - 10*a*b + 35*b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(9/2)*f) + ((3*a - 7*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (b*(9*a^2 - 18*a*b - 35*b^2)*Tan[e + f*x])/(24*a^3*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (b*(9*a^3 - 15*a^2*b - 145*a*b^2 - 105*b^3)*Tan[e + f*x])/(24*a^4*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",8,6,25,0.2400,1,"{4146, 414, 527, 12, 377, 203}"
295,1,332,0,0.4287343,"\int \frac{\cos ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{5 (a-3 b) \left(a^2+7 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{11/2} f}+\frac{b \left(38 a^2 b^2-20 a^3 b+15 a^4+420 a b^3+315 b^4\right) \tan (e+f x)}{48 a^5 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{b \left(-25 a^2 b+15 a^3+49 a b^2+105 b^3\right) \tan (e+f x)}{48 a^4 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{\left(5 a^2-10 a b+21 b^2\right) \sin (e+f x) \cos (e+f x)}{16 a^3 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{(5 a-9 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{\sin (e+f x) \cos ^5(e+f x)}{6 a f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","\frac{5 (a-3 b) \left(a^2+7 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 a^{11/2} f}+\frac{b \left(38 a^2 b^2-20 a^3 b+15 a^4+420 a b^3+315 b^4\right) \tan (e+f x)}{48 a^5 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{b \left(-25 a^2 b+15 a^3+49 a b^2+105 b^3\right) \tan (e+f x)}{48 a^4 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{\left(5 a^2-10 a b+21 b^2\right) \sin (e+f x) \cos (e+f x)}{16 a^3 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{(5 a-9 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}+\frac{\sin (e+f x) \cos ^5(e+f x)}{6 a f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"(5*(a - 3*b)*(a^2 + 7*b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(11/2)*f) + ((5*a^2 - 10*a*b + 21*b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*a^3*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + ((5*a - 9*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (Cos[e + f*x]^5*Sin[e + f*x])/(6*a*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (b*(15*a^3 - 25*a^2*b + 49*a*b^2 + 105*b^3)*Tan[e + f*x])/(48*a^4*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (b*(15*a^4 - 20*a^3*b + 38*a^2*b^2 + 420*a*b^3 + 315*b^4)*Tan[e + f*x])/(48*a^5*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",9,6,25,0.2400,1,"{4146, 414, 527, 12, 377, 203}"
296,1,179,0,0.1938071,"\int \frac{1}{\left(a+b \sec ^2(c+d x)\right)^{7/2}} \, dx","Int[(a + b*Sec[c + d*x]^2)^(-7/2),x]","-\frac{b \left(33 a^2+40 a b+15 b^2\right) \tan (c+d x)}{15 a^3 d (a+b)^3 \sqrt{a+b \tan ^2(c+d x)+b}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+b \tan ^2(c+d x)+b}}\right)}{a^{7/2} d}-\frac{b (9 a+5 b) \tan (c+d x)}{15 a^2 d (a+b)^2 \left(a+b \tan ^2(c+d x)+b\right)^{3/2}}-\frac{b \tan (c+d x)}{5 a d (a+b) \left(a+b \tan ^2(c+d x)+b\right)^{5/2}}","-\frac{b \left(33 a^2+40 a b+15 b^2\right) \tan (c+d x)}{15 a^3 d (a+b)^3 \sqrt{a+b \tan ^2(c+d x)+b}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+b \tan ^2(c+d x)+b}}\right)}{a^{7/2} d}-\frac{b (9 a+5 b) \tan (c+d x)}{15 a^2 d (a+b)^2 \left(a+b \tan ^2(c+d x)+b\right)^{3/2}}-\frac{b \tan (c+d x)}{5 a d (a+b) \left(a+b \tan ^2(c+d x)+b\right)^{5/2}}",1,"ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + b + b*Tan[c + d*x]^2]]/(a^(7/2)*d) - (b*Tan[c + d*x])/(5*a*(a + b)*d*(a + b + b*Tan[c + d*x]^2)^(5/2)) - (b*(9*a + 5*b)*Tan[c + d*x])/(15*a^2*(a + b)^2*d*(a + b + b*Tan[c + d*x]^2)^(3/2)) - (b*(33*a^2 + 40*a*b + 15*b^2)*Tan[c + d*x])/(15*a^3*(a + b)^3*d*Sqrt[a + b + b*Tan[c + d*x]^2])","A",7,6,16,0.3750,1,"{4128, 414, 527, 12, 377, 203}"
297,1,14,0,0.0197271,"\int \frac{1}{\sqrt{1+\sec ^2(x)}} \, dx","Int[1/Sqrt[1 + Sec[x]^2],x]","\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{\tan ^2(x)+2}}\right)","\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{\tan ^2(x)+2}}\right)",1,"ArcTan[Tan[x]/Sqrt[2 + Tan[x]^2]]","A",3,3,10,0.3000,1,"{4128, 377, 203}"
298,0,0,0,0.0467519,"\int (d \sec (e+f x))^m \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[(d*Sec[e + f*x])^m*(a + b*Sec[e + f*x]^2)^p,x]","\int (d \sec (e+f x))^m \left(a+b \sec ^2(e+f x)\right)^p \, dx","\frac{\sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^m \left(a+b \sec ^2(e+f x)\right)^p \left(\frac{b \sec ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{m}{2};\frac{1}{2},-p;\frac{m+2}{2};\sec ^2(e+f x),-\frac{b \sec ^2(e+f x)}{a}\right)}{f m}",1,"Defer[Int][(d*Sec[e + f*x])^m*(a + b*Sec[e + f*x]^2)^p, x]","F",0,0,0,0,-1,"{}"
299,1,124,0,0.2234198,"\int \sec ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\sin (e+f x) \cos ^2(e+f x)^p \left(-a \sin ^2(e+f x)+a+b\right)^p \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)^{-p} \left(a \cos ^2(e+f x)+b\right)^{-p} \left(a+b \sec ^2(e+f x)\right)^p F_1\left(\frac{1}{2};p+2,-p;\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)}{f}","\frac{\sin (e+f x) \cos ^2(e+f x)^p \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)^{-p} F_1\left(\frac{1}{2};p+2,-p;\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)\right)^p}{f}",1,"(AppellF1[1/2, 2 + p, -p, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*(Cos[e + f*x]^2)^p*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2)^p)/(f*(b + a*Cos[e + f*x]^2)^p*(1 - (a*Sin[e + f*x]^2)/(a + b))^p)","A",5,5,23,0.2174,1,"{4148, 6722, 1974, 430, 429}"
300,1,124,0,0.1742497,"\int \sec (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Sec[e + f*x]*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\sin (e+f x) \cos ^2(e+f x)^p \left(-a \sin ^2(e+f x)+a+b\right)^p \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)^{-p} \left(a \cos ^2(e+f x)+b\right)^{-p} \left(a+b \sec ^2(e+f x)\right)^p F_1\left(\frac{1}{2};p+1,-p;\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)}{f}","\frac{\sin (e+f x) \cos ^2(e+f x)^p \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)^{-p} F_1\left(\frac{1}{2};p+1,-p;\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)\right)^p}{f}",1,"(AppellF1[1/2, 1 + p, -p, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*(Cos[e + f*x]^2)^p*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2)^p)/(f*(b + a*Cos[e + f*x]^2)^p*(1 - (a*Sin[e + f*x]^2)/(a + b))^p)","A",5,5,21,0.2381,1,"{4148, 6722, 1974, 430, 429}"
301,1,122,0,0.1207284,"\int \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\sin (e+f x) \cos ^2(e+f x)^p \left(-a \sin ^2(e+f x)+a+b\right)^p \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)^{-p} \left(a \cos ^2(e+f x)+b\right)^{-p} \left(a+b \sec ^2(e+f x)\right)^p F_1\left(\frac{1}{2};p,-p;\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)}{f}","\frac{\sin (e+f x) \cos ^2(e+f x)^p \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)^{-p} F_1\left(\frac{1}{2};p,-p;\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)\right)^p}{f}",1,"(AppellF1[1/2, p, -p, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*(Cos[e + f*x]^2)^p*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2)^p)/(f*(b + a*Cos[e + f*x]^2)^p*(1 - (a*Sin[e + f*x]^2)/(a + b))^p)","A",5,5,21,0.2381,1,"{4148, 6722, 1974, 430, 429}"
302,1,124,0,0.1610563,"\int \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\sin (e+f x) \cos ^2(e+f x)^p \left(-a \sin ^2(e+f x)+a+b\right)^p \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)^{-p} \left(a \cos ^2(e+f x)+b\right)^{-p} \left(a+b \sec ^2(e+f x)\right)^p F_1\left(\frac{1}{2};p-1,-p;\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)}{f}","\frac{\sin (e+f x) \cos ^2(e+f x)^p \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)^{-p} F_1\left(\frac{1}{2};p-1,-p;\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)\right)^p}{f}",1,"(AppellF1[1/2, -1 + p, -p, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*(Cos[e + f*x]^2)^p*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2)^p)/(f*(b + a*Cos[e + f*x]^2)^p*(1 - (a*Sin[e + f*x]^2)/(a + b))^p)","A",5,5,23,0.2174,1,"{4148, 6722, 1974, 430, 429}"
303,1,124,0,0.1805778,"\int \cos ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\sin (e+f x) \cos ^2(e+f x)^p \left(-a \sin ^2(e+f x)+a+b\right)^p \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)^{-p} \left(a \cos ^2(e+f x)+b\right)^{-p} \left(a+b \sec ^2(e+f x)\right)^p F_1\left(\frac{1}{2};p-2,-p;\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)}{f}","\frac{\sin (e+f x) \cos ^2(e+f x)^p \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)^{-p} F_1\left(\frac{1}{2};p-2,-p;\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)\right)^p}{f}",1,"(AppellF1[1/2, -2 + p, -p, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*(Cos[e + f*x]^2)^p*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2)^p)/(f*(b + a*Cos[e + f*x]^2)^p*(1 - (a*Sin[e + f*x]^2)/(a + b))^p)","A",5,5,23,0.2174,1,"{4148, 6722, 1974, 430, 429}"
304,1,216,0,0.2260094,"\int \sec ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Sec[e + f*x]^6*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\left(3 a^2-4 a b (p+1)+4 b^2 \left(p^2+3 p+2\right)\right) \tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)}{b^2 f (2 p+3) (2 p+5)}-\frac{(3 a-2 b (p+2)) \tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{p+1}}{b^2 f (2 p+3) (2 p+5)}+\frac{\tan (e+f x) \sec ^2(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{p+1}}{b f (2 p+5)}","\frac{\left(3 a^2-4 a b (p+1)+4 b^2 \left(p^2+3 p+2\right)\right) \tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)}{b^2 f (2 p+3) (2 p+5)}-\frac{(3 a-2 b (p+2)) \tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{p+1}}{b^2 f (2 p+3) (2 p+5)}+\frac{\tan (e+f x) \sec ^2(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{p+1}}{b f (2 p+5)}",1,"-(((3*a - 2*b*(2 + p))*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(1 + p))/(b^2*f*(3 + 2*p)*(5 + 2*p))) + (Sec[e + f*x]^2*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(1 + p))/(b*f*(5 + 2*p)) + ((3*a^2 - 4*a*b*(1 + p) + 4*b^2*(2 + 3*p + p^2))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(b^2*f*(3 + 2*p)*(5 + 2*p)*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",5,5,23,0.2174,1,"{4146, 416, 388, 246, 245}"
305,1,129,0,0.0992134,"\int \sec ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Sec[e + f*x]^4*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{p+1}}{b f (2 p+3)}-\frac{(a-2 b (p+1)) \tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)}{b f (2 p+3)}","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^{p+1}}{b f (2 p+3)}-\frac{(a-2 b (p+1)) \tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)}{b f (2 p+3)}",1,"(Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(1 + p))/(b*f*(3 + 2*p)) - ((a - 2*b*(1 + p))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(b*f*(3 + 2*p)*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",4,4,23,0.1739,1,"{4146, 388, 246, 245}"
306,1,72,0,0.0641613,"\int \sec ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Sec[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}",1,"(Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",3,3,23,0.1304,1,"{4146, 246, 245}"
307,1,83,0,0.0621683,"\int \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[(a + b*Sec[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}",1,"(AppellF1[1/2, 1, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",3,3,14,0.2143,1,"{4128, 430, 429}"
308,1,83,0,0.0820397,"\int \cos ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{1}{2};2,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{1}{2};2,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}",1,"(AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",3,3,23,0.1304,1,"{4146, 430, 429}"
309,1,83,0,0.0808565,"\int \cos ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{1}{2};3,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{1}{2};3,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}",1,"(AppellF1[1/2, 3, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",3,3,23,0.1304,1,"{4146, 430, 429}"
310,1,83,0,0.0800884,"\int \cos ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^6*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{1}{2};4,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{1}{2};4,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}",1,"(AppellF1[1/2, 4, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",3,3,23,0.1304,1,"{4146, 430, 429}"
311,1,72,0,0.0619932,"\int \left(a+b \sec ^2(e+f x)\right) \tan ^5(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^5,x]","\frac{(a-2 b) \sec ^4(e+f x)}{4 f}-\frac{(2 a-b) \sec ^2(e+f x)}{2 f}-\frac{a \log (\cos (e+f x))}{f}+\frac{b \sec ^6(e+f x)}{6 f}","\frac{(a-2 b) \sec ^4(e+f x)}{4 f}-\frac{(2 a-b) \sec ^2(e+f x)}{2 f}-\frac{a \log (\cos (e+f x))}{f}+\frac{b \sec ^6(e+f x)}{6 f}",1,"-((a*Log[Cos[e + f*x]])/f) - ((2*a - b)*Sec[e + f*x]^2)/(2*f) + ((a - 2*b)*Sec[e + f*x]^4)/(4*f) + (b*Sec[e + f*x]^6)/(6*f)","A",4,3,21,0.1429,1,"{4138, 446, 76}"
312,1,49,0,0.0488434,"\int \left(a+b \sec ^2(e+f x)\right) \tan ^3(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^3,x]","\frac{(a-b) \sec ^2(e+f x)}{2 f}+\frac{a \log (\cos (e+f x))}{f}+\frac{b \sec ^4(e+f x)}{4 f}","\frac{(a-b) \sec ^2(e+f x)}{2 f}+\frac{a \log (\cos (e+f x))}{f}+\frac{b \sec ^4(e+f x)}{4 f}",1,"(a*Log[Cos[e + f*x]])/f + ((a - b)*Sec[e + f*x]^2)/(2*f) + (b*Sec[e + f*x]^4)/(4*f)","A",4,3,21,0.1429,1,"{4138, 446, 76}"
313,1,30,0,0.0236449,"\int \left(a+b \sec ^2(e+f x)\right) \tan (e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)*Tan[e + f*x],x]","\frac{b \sec ^2(e+f x)}{2 f}-\frac{a \log (\cos (e+f x))}{f}","\frac{b \sec ^2(e+f x)}{2 f}-\frac{a \log (\cos (e+f x))}{f}",1,"-((a*Log[Cos[e + f*x]])/f) + (b*Sec[e + f*x]^2)/(2*f)","A",3,2,19,0.1053,1,"{4138, 14}"
314,1,28,0,0.0467835,"\int \cot (e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Cot[e + f*x]*(a + b*Sec[e + f*x]^2),x]","\frac{(a+b) \log (\sin (e+f x))}{f}-\frac{b \log (\cos (e+f x))}{f}","\frac{(a+b) \log (\sin (e+f x))}{f}-\frac{b \log (\cos (e+f x))}{f}",1,"-((b*Log[Cos[e + f*x]])/f) + ((a + b)*Log[Sin[e + f*x]])/f","A",4,3,19,0.1579,1,"{4138, 446, 72}"
315,1,32,0,0.0505279,"\int \cot ^3(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Cot[e + f*x]^3*(a + b*Sec[e + f*x]^2),x]","-\frac{(a+b) \csc ^2(e+f x)}{2 f}-\frac{a \log (\sin (e+f x))}{f}","-\frac{(a+b) \csc ^2(e+f x)}{2 f}-\frac{a \log (\sin (e+f x))}{f}",1,"-((a + b)*Csc[e + f*x]^2)/(2*f) - (a*Log[Sin[e + f*x]])/f","A",4,3,21,0.1429,1,"{4138, 444, 43}"
316,1,51,0,0.0714023,"\int \cot ^5(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Cot[e + f*x]^5*(a + b*Sec[e + f*x]^2),x]","-\frac{(a+b) \csc ^4(e+f x)}{4 f}+\frac{(2 a+b) \csc ^2(e+f x)}{2 f}+\frac{a \log (\sin (e+f x))}{f}","-\frac{(a+b) \csc ^4(e+f x)}{4 f}+\frac{(2 a+b) \csc ^2(e+f x)}{2 f}+\frac{a \log (\sin (e+f x))}{f}",1,"((2*a + b)*Csc[e + f*x]^2)/(2*f) - ((a + b)*Csc[e + f*x]^4)/(4*f) + (a*Log[Sin[e + f*x]])/f","A",4,3,21,0.1429,1,"{4138, 446, 77}"
317,1,64,0,0.0615938,"\int \left(a+b \sec ^2(e+f x)\right) \tan ^6(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^6,x]","\frac{a \tan ^5(e+f x)}{5 f}-\frac{a \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f}-a x+\frac{b \tan ^7(e+f x)}{7 f}","\frac{a \tan ^5(e+f x)}{5 f}-\frac{a \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f}-a x+\frac{b \tan ^7(e+f x)}{7 f}",1,"-(a*x) + (a*Tan[e + f*x])/f - (a*Tan[e + f*x]^3)/(3*f) + (a*Tan[e + f*x]^5)/(5*f) + (b*Tan[e + f*x]^7)/(7*f)","A",4,3,21,0.1429,1,"{4141, 1802, 203}"
318,1,48,0,0.0565195,"\int \left(a+b \sec ^2(e+f x)\right) \tan ^4(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^4,x]","\frac{a \tan ^3(e+f x)}{3 f}-\frac{a \tan (e+f x)}{f}+a x+\frac{b \tan ^5(e+f x)}{5 f}","\frac{a \tan ^3(e+f x)}{3 f}-\frac{a \tan (e+f x)}{f}+a x+\frac{b \tan ^5(e+f x)}{5 f}",1,"a*x - (a*Tan[e + f*x])/f + (a*Tan[e + f*x]^3)/(3*f) + (b*Tan[e + f*x]^5)/(5*f)","A",4,3,21,0.1429,1,"{4141, 1802, 203}"
319,1,32,0,0.0520089,"\int \left(a+b \sec ^2(e+f x)\right) \tan ^2(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^2,x]","\frac{a \tan (e+f x)}{f}-a x+\frac{b \tan ^3(e+f x)}{3 f}","\frac{a \tan (e+f x)}{f}-a x+\frac{b \tan ^3(e+f x)}{3 f}",1,"-(a*x) + (a*Tan[e + f*x])/f + (b*Tan[e + f*x]^3)/(3*f)","A",4,3,21,0.1429,1,"{4141, 1802, 203}"
320,1,15,0,0.0126401,"\int \left(a+b \sec ^2(e+f x)\right) \, dx","Int[a + b*Sec[e + f*x]^2,x]","a x+\frac{b \tan (e+f x)}{f}","a x+\frac{b \tan (e+f x)}{f}",1,"a*x + (b*Tan[e + f*x])/f","A",3,2,12,0.1667,1,"{3767, 8}"
321,1,19,0,0.0537068,"\int \cot ^2(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Cot[e + f*x]^2*(a + b*Sec[e + f*x]^2),x]","-\frac{(a+b) \cot (e+f x)}{f}-a x","-\frac{(a+b) \cot (e+f x)}{f}-a x",1,"-(a*x) - ((a + b)*Cot[e + f*x])/f","A",4,3,21,0.1429,1,"{4141, 1802, 203}"
322,1,33,0,0.0585496,"\int \cot ^4(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Cot[e + f*x]^4*(a + b*Sec[e + f*x]^2),x]","-\frac{(a+b) \cot ^3(e+f x)}{3 f}+\frac{a \cot (e+f x)}{f}+a x","-\frac{(a+b) \cot ^3(e+f x)}{3 f}+\frac{a \cot (e+f x)}{f}+a x",1,"a*x + (a*Cot[e + f*x])/f - ((a + b)*Cot[e + f*x]^3)/(3*f)","A",4,3,21,0.1429,1,"{4141, 1802, 203}"
323,1,51,0,0.0628051,"\int \cot ^6(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Int[Cot[e + f*x]^6*(a + b*Sec[e + f*x]^2),x]","-\frac{(a+b) \cot ^5(e+f x)}{5 f}+\frac{a \cot ^3(e+f x)}{3 f}-\frac{a \cot (e+f x)}{f}-a x","-\frac{(a+b) \cot ^5(e+f x)}{5 f}+\frac{a \cot ^3(e+f x)}{3 f}-\frac{a \cot (e+f x)}{f}-a x",1,"-(a*x) - (a*Cot[e + f*x])/f + (a*Cot[e + f*x]^3)/(3*f) - ((a + b)*Cot[e + f*x]^5)/(5*f)","A",4,3,21,0.1429,1,"{4141, 1802, 203}"
324,1,100,0,0.1006604,"\int \left(a+b \sec ^2(e+f x)\right)^2 \tan ^5(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^5,x]","\frac{\left(a^2-4 a b+b^2\right) \sec ^4(e+f x)}{4 f}-\frac{a^2 \log (\cos (e+f x))}{f}+\frac{b (a-b) \sec ^6(e+f x)}{3 f}-\frac{a (a-b) \sec ^2(e+f x)}{f}+\frac{b^2 \sec ^8(e+f x)}{8 f}","\frac{\left(a^2-4 a b+b^2\right) \sec ^4(e+f x)}{4 f}-\frac{a^2 \log (\cos (e+f x))}{f}+\frac{b (a-b) \sec ^6(e+f x)}{3 f}-\frac{a (a-b) \sec ^2(e+f x)}{f}+\frac{b^2 \sec ^8(e+f x)}{8 f}",1,"-((a^2*Log[Cos[e + f*x]])/f) - (a*(a - b)*Sec[e + f*x]^2)/f + ((a^2 - 4*a*b + b^2)*Sec[e + f*x]^4)/(4*f) + ((a - b)*b*Sec[e + f*x]^6)/(3*f) + (b^2*Sec[e + f*x]^8)/(8*f)","A",4,3,23,0.1304,1,"{4138, 446, 88}"
325,1,77,0,0.0840397,"\int \left(a+b \sec ^2(e+f x)\right)^2 \tan ^3(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^3,x]","\frac{a^2 \log (\cos (e+f x))}{f}+\frac{b (2 a-b) \sec ^4(e+f x)}{4 f}+\frac{a (a-2 b) \sec ^2(e+f x)}{2 f}+\frac{b^2 \sec ^6(e+f x)}{6 f}","\frac{a^2 \log (\cos (e+f x))}{f}+\frac{b (2 a-b) \sec ^4(e+f x)}{4 f}+\frac{a (a-2 b) \sec ^2(e+f x)}{2 f}+\frac{b^2 \sec ^6(e+f x)}{6 f}",1,"(a^2*Log[Cos[e + f*x]])/f + (a*(a - 2*b)*Sec[e + f*x]^2)/(2*f) + ((2*a - b)*b*Sec[e + f*x]^4)/(4*f) + (b^2*Sec[e + f*x]^6)/(6*f)","A",4,3,23,0.1304,1,"{4138, 446, 76}"
326,1,48,0,0.0419613,"\int \left(a+b \sec ^2(e+f x)\right)^2 \tan (e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x],x]","-\frac{a^2 \log (\cos (e+f x))}{f}+\frac{a b \sec ^2(e+f x)}{f}+\frac{b^2 \sec ^4(e+f x)}{4 f}","-\frac{a^2 \log (\cos (e+f x))}{f}+\frac{a b \sec ^2(e+f x)}{f}+\frac{b^2 \sec ^4(e+f x)}{4 f}",1,"-((a^2*Log[Cos[e + f*x]])/f) + (a*b*Sec[e + f*x]^2)/f + (b^2*Sec[e + f*x]^4)/(4*f)","A",4,3,21,0.1429,1,"{4138, 266, 43}"
327,1,53,0,0.0738816,"\int \cot (e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Cot[e + f*x]*(a + b*Sec[e + f*x]^2)^2,x]","\frac{(a+b)^2 \log (\sin (e+f x))}{f}-\frac{b (2 a+b) \log (\cos (e+f x))}{f}+\frac{b^2 \sec ^2(e+f x)}{2 f}","\frac{(a+b)^2 \log (\sin (e+f x))}{f}-\frac{b (2 a+b) \log (\cos (e+f x))}{f}+\frac{b^2 \sec ^2(e+f x)}{2 f}",1,"-((b*(2*a + b)*Log[Cos[e + f*x]])/f) + ((a + b)^2*Log[Sin[e + f*x]])/f + (b^2*Sec[e + f*x]^2)/(2*f)","A",4,3,21,0.1429,1,"{4138, 446, 88}"
328,1,57,0,0.0805385,"\int \cot ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Cot[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2,x]","-\frac{\left(a^2-b^2\right) \log (\sin (e+f x))}{f}-\frac{(a+b)^2 \csc ^2(e+f x)}{2 f}-\frac{b^2 \log (\cos (e+f x))}{f}","-\frac{\left(a^2-b^2\right) \log (\sin (e+f x))}{f}-\frac{(a+b)^2 \csc ^2(e+f x)}{2 f}-\frac{b^2 \log (\cos (e+f x))}{f}",1,"-((a + b)^2*Csc[e + f*x]^2)/(2*f) - (b^2*Log[Cos[e + f*x]])/f - ((a^2 - b^2)*Log[Sin[e + f*x]])/f","A",4,3,23,0.1304,1,"{4138, 446, 88}"
329,1,51,0,0.0865154,"\int \cot ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Cot[e + f*x]^5*(a + b*Sec[e + f*x]^2)^2,x]","\frac{a^2 \log (\sin (e+f x))}{f}-\frac{(a+b)^2 \csc ^4(e+f x)}{4 f}+\frac{a (a+b) \csc ^2(e+f x)}{f}","\frac{a^2 \log (\sin (e+f x))}{f}-\frac{(a+b)^2 \csc ^4(e+f x)}{4 f}+\frac{a (a+b) \csc ^2(e+f x)}{f}",1,"(a*(a + b)*Csc[e + f*x]^2)/f - ((a + b)^2*Csc[e + f*x]^4)/(4*f) + (a^2*Log[Sin[e + f*x]])/f","A",4,3,23,0.1304,1,"{4138, 444, 43}"
330,1,95,0,0.1075155,"\int \left(a+b \sec ^2(e+f x)\right)^2 \tan ^6(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^6,x]","\frac{a^2 \tan ^5(e+f x)}{5 f}-\frac{a^2 \tan ^3(e+f x)}{3 f}+\frac{a^2 \tan (e+f x)}{f}-a^2 x+\frac{b (2 a+b) \tan ^7(e+f x)}{7 f}+\frac{b^2 \tan ^9(e+f x)}{9 f}","\frac{a^2 \tan ^5(e+f x)}{5 f}-\frac{a^2 \tan ^3(e+f x)}{3 f}+\frac{a^2 \tan (e+f x)}{f}-a^2 x+\frac{b (2 a+b) \tan ^7(e+f x)}{7 f}+\frac{b^2 \tan ^9(e+f x)}{9 f}",1,"-(a^2*x) + (a^2*Tan[e + f*x])/f - (a^2*Tan[e + f*x]^3)/(3*f) + (a^2*Tan[e + f*x]^5)/(5*f) + (b*(2*a + b)*Tan[e + f*x]^7)/(7*f) + (b^2*Tan[e + f*x]^9)/(9*f)","A",4,3,23,0.1304,1,"{4141, 1802, 203}"
331,1,77,0,0.097092,"\int \left(a+b \sec ^2(e+f x)\right)^2 \tan ^4(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^4,x]","\frac{a^2 \tan ^3(e+f x)}{3 f}-\frac{a^2 \tan (e+f x)}{f}+a^2 x+\frac{b (2 a+b) \tan ^5(e+f x)}{5 f}+\frac{b^2 \tan ^7(e+f x)}{7 f}","\frac{a^2 \tan ^3(e+f x)}{3 f}-\frac{a^2 \tan (e+f x)}{f}+a^2 x+\frac{b (2 a+b) \tan ^5(e+f x)}{5 f}+\frac{b^2 \tan ^7(e+f x)}{7 f}",1,"a^2*x - (a^2*Tan[e + f*x])/f + (a^2*Tan[e + f*x]^3)/(3*f) + (b*(2*a + b)*Tan[e + f*x]^5)/(5*f) + (b^2*Tan[e + f*x]^7)/(7*f)","A",4,3,23,0.1304,1,"{4141, 1802, 203}"
332,1,59,0,0.0926242,"\int \left(a+b \sec ^2(e+f x)\right)^2 \tan ^2(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^2,x]","\frac{a^2 \tan (e+f x)}{f}-a^2 x+\frac{b (2 a+b) \tan ^3(e+f x)}{3 f}+\frac{b^2 \tan ^5(e+f x)}{5 f}","\frac{a^2 \tan (e+f x)}{f}-a^2 x+\frac{b (2 a+b) \tan ^3(e+f x)}{3 f}+\frac{b^2 \tan ^5(e+f x)}{5 f}",1,"-(a^2*x) + (a^2*Tan[e + f*x])/f + (b*(2*a + b)*Tan[e + f*x]^3)/(3*f) + (b^2*Tan[e + f*x]^5)/(5*f)","A",4,3,23,0.1304,1,"{4141, 1802, 203}"
333,1,40,0,0.0289237,"\int \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[(a + b*Sec[e + f*x]^2)^2,x]","a^2 x+\frac{b (2 a+b) \tan (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}","a^2 x+\frac{b (2 a+b) \tan (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"a^2*x + (b*(2*a + b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)","A",4,3,14,0.2143,1,"{4128, 390, 203}"
334,1,36,0,0.0803038,"\int \cot ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Cot[e + f*x]^2*(a + b*Sec[e + f*x]^2)^2,x]","a^2 (-x)-\frac{(a+b)^2 \cot (e+f x)}{f}+\frac{b^2 \tan (e+f x)}{f}","a^2 (-x)-\frac{(a+b)^2 \cot (e+f x)}{f}+\frac{b^2 \tan (e+f x)}{f}",1,"-(a^2*x) - ((a + b)^2*Cot[e + f*x])/f + (b^2*Tan[e + f*x])/f","A",4,3,23,0.1304,1,"{4141, 1802, 203}"
335,1,45,0,0.0868653,"\int \cot ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Cot[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2,x]","\frac{\left(a^2-b^2\right) \cot (e+f x)}{f}+a^2 x-\frac{(a+b)^2 \cot ^3(e+f x)}{3 f}","\frac{\left(a^2-b^2\right) \cot (e+f x)}{f}+a^2 x-\frac{(a+b)^2 \cot ^3(e+f x)}{3 f}",1,"a^2*x + ((a^2 - b^2)*Cot[e + f*x])/f - ((a + b)^2*Cot[e + f*x]^3)/(3*f)","A",4,3,23,0.1304,1,"{4141, 1802, 203}"
336,1,65,0,0.0933106,"\int \cot ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Int[Cot[e + f*x]^6*(a + b*Sec[e + f*x]^2)^2,x]","\frac{\left(a^2-b^2\right) \cot ^3(e+f x)}{3 f}-\frac{a^2 \cot (e+f x)}{f}-a^2 x-\frac{(a+b)^2 \cot ^5(e+f x)}{5 f}","\frac{\left(a^2-b^2\right) \cot ^3(e+f x)}{3 f}-\frac{a^2 \cot (e+f x)}{f}-a^2 x-\frac{(a+b)^2 \cot ^5(e+f x)}{5 f}",1,"-(a^2*x) - (a^2*Cot[e + f*x])/f + ((a^2 - b^2)*Cot[e + f*x]^3)/(3*f) - ((a + b)^2*Cot[e + f*x]^5)/(5*f)","A",4,3,23,0.1304,1,"{4141, 1802, 203}"
337,1,69,0,0.099578,"\int \frac{\tan ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2),x]","-\frac{(a+b)^2 \log \left(a \cos ^2(e+f x)+b\right)}{2 a b^2 f}+\frac{(a+2 b) \log (\cos (e+f x))}{b^2 f}+\frac{\sec ^2(e+f x)}{2 b f}","-\frac{(a+b)^2 \log \left(a \cos ^2(e+f x)+b\right)}{2 a b^2 f}+\frac{(a+2 b) \log (\cos (e+f x))}{b^2 f}+\frac{\sec ^2(e+f x)}{2 b f}",1,"((a + 2*b)*Log[Cos[e + f*x]])/(b^2*f) - ((a + b)^2*Log[b + a*Cos[e + f*x]^2])/(2*a*b^2*f) + Sec[e + f*x]^2/(2*b*f)","A",4,3,23,0.1304,1,"{4138, 446, 88}"
338,1,45,0,0.0772854,"\int \frac{\tan ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Tan[e + f*x]^3/(a + b*Sec[e + f*x]^2),x]","\frac{(a+b) \log \left(a \cos ^2(e+f x)+b\right)}{2 a b f}-\frac{\log (\cos (e+f x))}{b f}","\frac{(a+b) \log \left(a \cos ^2(e+f x)+b\right)}{2 a b f}-\frac{\log (\cos (e+f x))}{b f}",1,"-(Log[Cos[e + f*x]]/(b*f)) + ((a + b)*Log[b + a*Cos[e + f*x]^2])/(2*a*b*f)","A",4,3,23,0.1304,1,"{4138, 446, 72}"
339,1,23,0,0.0308976,"\int \frac{\tan (e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Tan[e + f*x]/(a + b*Sec[e + f*x]^2),x]","-\frac{\log \left(a \cos ^2(e+f x)+b\right)}{2 a f}","-\frac{\log \left(a \cos ^2(e+f x)+b\right)}{2 a f}",1,"-Log[b + a*Cos[e + f*x]^2]/(2*a*f)","A",2,2,21,0.09524,1,"{4138, 260}"
340,1,46,0,0.0803005,"\int \frac{\cot (e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Cot[e + f*x]/(a + b*Sec[e + f*x]^2),x]","\frac{\log (\sin (e+f x))}{f (a+b)}+\frac{b \log \left(a \cos ^2(e+f x)+b\right)}{2 a f (a+b)}","\frac{\log (\sin (e+f x))}{f (a+b)}+\frac{b \log \left(a \cos ^2(e+f x)+b\right)}{2 a f (a+b)}",1,"(b*Log[b + a*Cos[e + f*x]^2])/(2*a*(a + b)*f) + Log[Sin[e + f*x]]/((a + b)*f)","A",4,3,21,0.1429,1,"{4138, 446, 72}"
341,1,74,0,0.1100462,"\int \frac{\cot ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2),x]","-\frac{b^2 \log \left(a \cos ^2(e+f x)+b\right)}{2 a f (a+b)^2}-\frac{\csc ^2(e+f x)}{2 f (a+b)}-\frac{(a+2 b) \log (\sin (e+f x))}{f (a+b)^2}","-\frac{b^2 \log \left(a \cos ^2(e+f x)+b\right)}{2 a f (a+b)^2}-\frac{\csc ^2(e+f x)}{2 f (a+b)}-\frac{(a+2 b) \log (\sin (e+f x))}{f (a+b)^2}",1,"-Csc[e + f*x]^2/(2*(a + b)*f) - (b^2*Log[b + a*Cos[e + f*x]^2])/(2*a*(a + b)^2*f) - ((a + 2*b)*Log[Sin[e + f*x]])/((a + b)^2*f)","A",4,3,23,0.1304,1,"{4138, 446, 88}"
342,1,108,0,0.1484553,"\int \frac{\cot ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2),x]","\frac{\left(a^2+3 a b+3 b^2\right) \log (\sin (e+f x))}{f (a+b)^3}+\frac{b^3 \log \left(a \cos ^2(e+f x)+b\right)}{2 a f (a+b)^3}-\frac{\csc ^4(e+f x)}{4 f (a+b)}+\frac{(2 a+3 b) \csc ^2(e+f x)}{2 f (a+b)^2}","\frac{\left(a^2+3 a b+3 b^2\right) \log (\sin (e+f x))}{f (a+b)^3}+\frac{b^3 \log \left(a \cos ^2(e+f x)+b\right)}{2 a f (a+b)^3}-\frac{\csc ^4(e+f x)}{4 f (a+b)}+\frac{(2 a+3 b) \csc ^2(e+f x)}{2 f (a+b)^2}",1,"((2*a + 3*b)*Csc[e + f*x]^2)/(2*(a + b)^2*f) - Csc[e + f*x]^4/(4*(a + b)*f) + (b^3*Log[b + a*Cos[e + f*x]^2])/(2*a*(a + b)^3*f) + ((a^2 + 3*a*b + 3*b^2)*Log[Sin[e + f*x]])/((a + b)^3*f)","A",4,3,23,0.1304,1,"{4138, 446, 88}"
343,1,83,0,0.2726734,"\int \frac{\tan ^6(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Tan[e + f*x]^6/(a + b*Sec[e + f*x]^2),x]","\frac{(a+b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a b^{5/2} f}-\frac{(a+2 b) \tan (e+f x)}{b^2 f}-\frac{x}{a}+\frac{\tan ^3(e+f x)}{3 b f}","\frac{(a+b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a b^{5/2} f}-\frac{(a+2 b) \tan (e+f x)}{b^2 f}-\frac{x}{a}+\frac{\tan ^3(e+f x)}{3 b f}",1,"-(x/a) + ((a + b)^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a*b^(5/2)*f) - ((a + 2*b)*Tan[e + f*x])/(b^2*f) + Tan[e + f*x]^3/(3*b*f)","A",7,7,23,0.3043,1,"{4141, 1975, 479, 582, 522, 203, 205}"
344,1,59,0,0.1674379,"\int \frac{\tan ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Tan[e + f*x]^4/(a + b*Sec[e + f*x]^2),x]","-\frac{(a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a b^{3/2} f}+\frac{x}{a}+\frac{\tan (e+f x)}{b f}","-\frac{(a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a b^{3/2} f}+\frac{x}{a}+\frac{\tan (e+f x)}{b f}",1,"x/a - ((a + b)^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a*b^(3/2)*f) + Tan[e + f*x]/(b*f)","A",6,6,23,0.2609,1,"{4141, 1975, 479, 522, 203, 205}"
345,1,46,0,0.1262753,"\int \frac{\tan ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Tan[e + f*x]^2/(a + b*Sec[e + f*x]^2),x]","\frac{\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a \sqrt{b} f}-\frac{x}{a}","\frac{\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a \sqrt{b} f}-\frac{x}{a}",1,"-(x/a) + (Sqrt[a + b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a*Sqrt[b]*f)","A",5,5,23,0.2174,1,"{4141, 1975, 481, 203, 205}"
346,1,45,0,0.0454022,"\int \frac{1}{a+b \sec ^2(e+f x)} \, dx","Int[(a + b*Sec[e + f*x]^2)^(-1),x]","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (e+f x)}{\sqrt{b}}\right)}{a f \sqrt{a+b}}+\frac{x}{a}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (e+f x)}{\sqrt{b}}\right)}{a f \sqrt{a+b}}+\frac{x}{a}",1,"x/a + (Sqrt[b]*ArcTan[(Sqrt[a + b]*Cot[e + f*x])/Sqrt[b]])/(a*Sqrt[a + b]*f)","A",3,3,14,0.2143,1,"{4127, 3181, 205}"
347,1,62,0,0.1739566,"\int \frac{\cot ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Cot[e + f*x]^2/(a + b*Sec[e + f*x]^2),x]","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a f (a+b)^{3/2}}-\frac{\cot (e+f x)}{f (a+b)}-\frac{x}{a}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a f (a+b)^{3/2}}-\frac{\cot (e+f x)}{f (a+b)}-\frac{x}{a}",1,"-(x/a) + (b^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a*(a + b)^(3/2)*f) - Cot[e + f*x]/((a + b)*f)","A",6,6,23,0.2609,1,"{4141, 1975, 480, 522, 203, 205}"
348,1,86,0,0.2481739,"\int \frac{\cot ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2),x]","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a f (a+b)^{5/2}}-\frac{\cot ^3(e+f x)}{3 f (a+b)}+\frac{(a+2 b) \cot (e+f x)}{f (a+b)^2}+\frac{x}{a}","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a f (a+b)^{5/2}}-\frac{\cot ^3(e+f x)}{3 f (a+b)}+\frac{(a+2 b) \cot (e+f x)}{f (a+b)^2}+\frac{x}{a}",1,"x/a - (b^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a*(a + b)^(5/2)*f) + ((a + 2*b)*Cot[e + f*x])/((a + b)^2*f) - Cot[e + f*x]^3/(3*(a + b)*f)","A",7,7,23,0.3043,1,"{4141, 1975, 480, 583, 522, 203, 205}"
349,1,120,0,0.343942,"\int \frac{\cot ^6(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Int[Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2),x]","-\frac{\left(a^2+3 a b+3 b^2\right) \cot (e+f x)}{f (a+b)^3}+\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a f (a+b)^{7/2}}-\frac{\cot ^5(e+f x)}{5 f (a+b)}+\frac{(a+2 b) \cot ^3(e+f x)}{3 f (a+b)^2}-\frac{x}{a}","-\frac{\left(a^2+3 a b+3 b^2\right) \cot (e+f x)}{f (a+b)^3}+\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{a f (a+b)^{7/2}}-\frac{\cot ^5(e+f x)}{5 f (a+b)}+\frac{(a+2 b) \cot ^3(e+f x)}{3 f (a+b)^2}-\frac{x}{a}",1,"-(x/a) + (b^(7/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a*(a + b)^(7/2)*f) - ((a^2 + 3*a*b + 3*b^2)*Cot[e + f*x])/((a + b)^3*f) + ((a + 2*b)*Cot[e + f*x]^3)/(3*(a + b)^2*f) - Cot[e + f*x]^5/(5*(a + b)*f)","A",8,7,23,0.3043,1,"{4141, 1975, 480, 583, 522, 203, 205}"
350,1,77,0,0.1058139,"\int \frac{\tan ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{\left(\frac{1}{a^2}-\frac{1}{b^2}\right) \log \left(a \cos ^2(e+f x)+b\right)}{2 f}-\frac{(a+b)^2}{2 a^2 b f \left(a \cos ^2(e+f x)+b\right)}-\frac{\log (\cos (e+f x))}{b^2 f}","-\frac{\left(\frac{1}{a^2}-\frac{1}{b^2}\right) \log \left(a \cos ^2(e+f x)+b\right)}{2 f}-\frac{(a+b)^2}{2 a^2 b f \left(a \cos ^2(e+f x)+b\right)}-\frac{\log (\cos (e+f x))}{b^2 f}",1,"-(a + b)^2/(2*a^2*b*f*(b + a*Cos[e + f*x]^2)) - Log[Cos[e + f*x]]/(b^2*f) - ((a^(-2) - b^(-2))*Log[b + a*Cos[e + f*x]^2])/(2*f)","A",4,3,23,0.1304,1,"{4138, 446, 88}"
351,1,51,0,0.0823833,"\int \frac{\tan ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Tan[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2,x]","\frac{a+b}{2 a^2 f \left(a \cos ^2(e+f x)+b\right)}+\frac{\log \left(a \cos ^2(e+f x)+b\right)}{2 a^2 f}","\frac{a+b}{2 a^2 f \left(a \cos ^2(e+f x)+b\right)}+\frac{\log \left(a \cos ^2(e+f x)+b\right)}{2 a^2 f}",1,"(a + b)/(2*a^2*f*(b + a*Cos[e + f*x]^2)) + Log[b + a*Cos[e + f*x]^2]/(2*a^2*f)","A",4,3,23,0.1304,1,"{4138, 444, 43}"
352,1,49,0,0.0556289,"\int \frac{\tan (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Tan[e + f*x]/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{b}{2 a^2 f \left(a \cos ^2(e+f x)+b\right)}-\frac{\log \left(a \cos ^2(e+f x)+b\right)}{2 a^2 f}","-\frac{b}{2 a^2 f \left(a \cos ^2(e+f x)+b\right)}-\frac{\log \left(a \cos ^2(e+f x)+b\right)}{2 a^2 f}",1,"-b/(2*a^2*f*(b + a*Cos[e + f*x]^2)) - Log[b + a*Cos[e + f*x]^2]/(2*a^2*f)","A",4,3,21,0.1429,1,"{4138, 266, 43}"
353,1,83,0,0.1136568,"\int \frac{\cot (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Cot[e + f*x]/(a + b*Sec[e + f*x]^2)^2,x]","\frac{b^2}{2 a^2 f (a+b) \left(a \cos ^2(e+f x)+b\right)}+\frac{b (2 a+b) \log \left(a \cos ^2(e+f x)+b\right)}{2 a^2 f (a+b)^2}+\frac{\log (\sin (e+f x))}{f (a+b)^2}","\frac{b^2}{2 a^2 f (a+b) \left(a \cos ^2(e+f x)+b\right)}+\frac{b (2 a+b) \log \left(a \cos ^2(e+f x)+b\right)}{2 a^2 f (a+b)^2}+\frac{\log (\sin (e+f x))}{f (a+b)^2}",1,"b^2/(2*a^2*(a + b)*f*(b + a*Cos[e + f*x]^2)) + (b*(2*a + b)*Log[b + a*Cos[e + f*x]^2])/(2*a^2*(a + b)^2*f) + Log[Sin[e + f*x]]/((a + b)^2*f)","A",4,3,21,0.1429,1,"{4138, 446, 88}"
354,1,111,0,0.1571093,"\int \frac{\cot ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{b^3}{2 a^2 f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)}-\frac{b^2 (3 a+b) \log \left(a \cos ^2(e+f x)+b\right)}{2 a^2 f (a+b)^3}-\frac{\csc ^2(e+f x)}{2 f (a+b)^2}-\frac{(a+3 b) \log (\sin (e+f x))}{f (a+b)^3}","-\frac{b^3}{2 a^2 f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)}-\frac{b^2 (3 a+b) \log \left(a \cos ^2(e+f x)+b\right)}{2 a^2 f (a+b)^3}-\frac{\csc ^2(e+f x)}{2 f (a+b)^2}-\frac{(a+3 b) \log (\sin (e+f x))}{f (a+b)^3}",1,"-b^3/(2*a^2*(a + b)^2*f*(b + a*Cos[e + f*x]^2)) - Csc[e + f*x]^2/(2*(a + b)^2*f) - (b^2*(3*a + b)*Log[b + a*Cos[e + f*x]^2])/(2*a^2*(a + b)^3*f) - ((a + 3*b)*Log[Sin[e + f*x]])/((a + b)^3*f)","A",4,3,23,0.1304,1,"{4138, 446, 88}"
355,1,140,0,0.1972531,"\int \frac{\cot ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2,x]","\frac{b^4}{2 a^2 f (a+b)^3 \left(a \cos ^2(e+f x)+b\right)}+\frac{\left(a^2+4 a b+6 b^2\right) \log (\sin (e+f x))}{f (a+b)^4}+\frac{b^3 (4 a+b) \log \left(a \cos ^2(e+f x)+b\right)}{2 a^2 f (a+b)^4}-\frac{\csc ^4(e+f x)}{4 f (a+b)^2}+\frac{(a+2 b) \csc ^2(e+f x)}{f (a+b)^3}","\frac{b^4}{2 a^2 f (a+b)^3 \left(a \cos ^2(e+f x)+b\right)}+\frac{\left(a^2+4 a b+6 b^2\right) \log (\sin (e+f x))}{f (a+b)^4}+\frac{b^3 (4 a+b) \log \left(a \cos ^2(e+f x)+b\right)}{2 a^2 f (a+b)^4}-\frac{\csc ^4(e+f x)}{4 f (a+b)^2}+\frac{(a+2 b) \csc ^2(e+f x)}{f (a+b)^3}",1,"b^4/(2*a^2*(a + b)^3*f*(b + a*Cos[e + f*x]^2)) + ((a + 2*b)*Csc[e + f*x]^2)/((a + b)^3*f) - Csc[e + f*x]^4/(4*(a + b)^2*f) + (b^3*(4*a + b)*Log[b + a*Cos[e + f*x]^2])/(2*a^2*(a + b)^4*f) + ((a^2 + 4*a*b + 6*b^2)*Log[Sin[e + f*x]])/((a + b)^4*f)","A",4,3,23,0.1304,1,"{4138, 446, 88}"
356,1,119,0,0.2665039,"\int \frac{\tan ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Tan[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{(3 a-2 b) (a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 b^{5/2} f}-\frac{x}{a^2}+\frac{(3 a+b) \tan (e+f x)}{2 a b^2 f}-\frac{(a+b) \tan ^3(e+f x)}{2 a b f \left(a+b \tan ^2(e+f x)+b\right)}","-\frac{(3 a-2 b) (a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 b^{5/2} f}-\frac{x}{a^2}+\frac{(3 a+b) \tan (e+f x)}{2 a b^2 f}-\frac{(a+b) \tan ^3(e+f x)}{2 a b f \left(a+b \tan ^2(e+f x)+b\right)}",1,"-(x/a^2) - ((3*a - 2*b)*(a + b)^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*b^(5/2)*f) + ((3*a + b)*Tan[e + f*x])/(2*a*b^2*f) - ((a + b)*Tan[e + f*x]^3)/(2*a*b*f*(a + b + b*Tan[e + f*x]^2))","A",7,7,23,0.3043,1,"{4141, 1975, 470, 582, 522, 203, 205}"
357,1,90,0,0.1764485,"\int \frac{\tan ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Tan[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]","\frac{(a-2 b) \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 b^{3/2} f}+\frac{x}{a^2}-\frac{(a+b) \tan (e+f x)}{2 a b f \left(a+b \tan ^2(e+f x)+b\right)}","\frac{(a-2 b) \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 b^{3/2} f}+\frac{x}{a^2}-\frac{(a+b) \tan (e+f x)}{2 a b f \left(a+b \tan ^2(e+f x)+b\right)}",1,"x/a^2 + ((a - 2*b)*Sqrt[a + b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*b^(3/2)*f) - ((a + b)*Tan[e + f*x])/(2*a*b*f*(a + b + b*Tan[e + f*x]^2))","A",6,6,23,0.2609,1,"{4141, 1975, 470, 522, 203, 205}"
358,1,85,0,0.1556603,"\int \frac{\tan ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Tan[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2,x]","\frac{(a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 \sqrt{b} f \sqrt{a+b}}-\frac{x}{a^2}+\frac{\tan (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)}","\frac{(a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 \sqrt{b} f \sqrt{a+b}}-\frac{x}{a^2}+\frac{\tan (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)}",1,"-(x/a^2) + ((a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*Sqrt[b]*Sqrt[a + b]*f) + Tan[e + f*x]/(2*a*f*(a + b + b*Tan[e + f*x]^2))","A",6,6,23,0.2609,1,"{4141, 1975, 471, 522, 203, 205}"
359,1,92,0,0.0854052,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[(a + b*Sec[e + f*x]^2)^(-2),x]","-\frac{\sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 f (a+b)^{3/2}}+\frac{x}{a^2}-\frac{b \tan (e+f x)}{2 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}","-\frac{\sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 f (a+b)^{3/2}}+\frac{x}{a^2}-\frac{b \tan (e+f x)}{2 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}",1,"x/a^2 - (Sqrt[b]*(3*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(3/2)*f) - (b*Tan[e + f*x])/(2*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2))","A",5,5,14,0.3571,1,"{4128, 414, 522, 203, 205}"
360,1,121,0,0.2537723,"\int \frac{\cot ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Cot[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2,x]","\frac{b^{3/2} (5 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 f (a+b)^{5/2}}-\frac{x}{a^2}-\frac{(2 a-b) \cot (e+f x)}{2 a f (a+b)^2}-\frac{b \cot (e+f x)}{2 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}","\frac{b^{3/2} (5 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 f (a+b)^{5/2}}-\frac{x}{a^2}-\frac{(2 a-b) \cot (e+f x)}{2 a f (a+b)^2}-\frac{b \cot (e+f x)}{2 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}",1,"-(x/a^2) + (b^(3/2)*(5*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(5/2)*f) - ((2*a - b)*Cot[e + f*x])/(2*a*(a + b)^2*f) - (b*Cot[e + f*x])/(2*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2))","A",7,7,23,0.3043,1,"{4141, 1975, 472, 583, 522, 203, 205}"
361,1,160,0,0.3537136,"\int \frac{\cot ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{b^{5/2} (7 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 f (a+b)^{7/2}}+\frac{\left(2 a^2+6 a b-b^2\right) \cot (e+f x)}{2 a f (a+b)^3}+\frac{x}{a^2}-\frac{(2 a-3 b) \cot ^3(e+f x)}{6 a f (a+b)^2}-\frac{b \cot ^3(e+f x)}{2 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}","-\frac{b^{5/2} (7 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 f (a+b)^{7/2}}+\frac{\left(2 a^2+6 a b-b^2\right) \cot (e+f x)}{2 a f (a+b)^3}+\frac{x}{a^2}-\frac{(2 a-3 b) \cot ^3(e+f x)}{6 a f (a+b)^2}-\frac{b \cot ^3(e+f x)}{2 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}",1,"x/a^2 - (b^(5/2)*(7*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(7/2)*f) + ((2*a^2 + 6*a*b - b^2)*Cot[e + f*x])/(2*a*(a + b)^3*f) - ((2*a - 3*b)*Cot[e + f*x]^3)/(6*a*(a + b)^2*f) - (b*Cot[e + f*x]^3)/(2*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2))","A",8,7,23,0.3043,1,"{4141, 1975, 472, 583, 522, 203, 205}"
362,1,207,0,0.4370148,"\int \frac{\cot ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Int[Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2,x]","\frac{b^{7/2} (9 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 f (a+b)^{9/2}}+\frac{\left(2 a^2+6 a b-3 b^2\right) \cot ^3(e+f x)}{6 a f (a+b)^3}-\frac{\left(8 a^2 b+2 a^3+12 a b^2-b^3\right) \cot (e+f x)}{2 a f (a+b)^4}-\frac{x}{a^2}-\frac{(2 a-5 b) \cot ^5(e+f x)}{10 a f (a+b)^2}-\frac{b \cot ^5(e+f x)}{2 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}","\frac{b^{7/2} (9 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{2 a^2 f (a+b)^{9/2}}+\frac{\left(2 a^2+6 a b-3 b^2\right) \cot ^3(e+f x)}{6 a f (a+b)^3}-\frac{\left(8 a^2 b+2 a^3+12 a b^2-b^3\right) \cot (e+f x)}{2 a f (a+b)^4}-\frac{x}{a^2}-\frac{(2 a-5 b) \cot ^5(e+f x)}{10 a f (a+b)^2}-\frac{b \cot ^5(e+f x)}{2 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}",1,"-(x/a^2) + (b^(7/2)*(9*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(9/2)*f) - ((2*a^3 + 8*a^2*b + 12*a*b^2 - b^3)*Cot[e + f*x])/(2*a*(a + b)^4*f) + ((2*a^2 + 6*a*b - 3*b^2)*Cot[e + f*x]^3)/(6*a*(a + b)^3*f) - ((2*a - 5*b)*Cot[e + f*x]^5)/(10*a*(a + b)^2*f) - (b*Cot[e + f*x]^5)/(2*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2))","A",9,7,23,0.3043,1,"{4141, 1975, 472, 583, 522, 203, 205}"
363,1,78,0,0.1077025,"\int \frac{\tan ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]","\frac{(a+b)^2}{4 a^3 f \left(a \cos ^2(e+f x)+b\right)^2}-\frac{a+b}{a^3 f \left(a \cos ^2(e+f x)+b\right)}-\frac{\log \left(a \cos ^2(e+f x)+b\right)}{2 a^3 f}","\frac{(a+b)^2}{4 a^3 f \left(a \cos ^2(e+f x)+b\right)^2}-\frac{a+b}{a^3 f \left(a \cos ^2(e+f x)+b\right)}-\frac{\log \left(a \cos ^2(e+f x)+b\right)}{2 a^3 f}",1,"(a + b)^2/(4*a^3*f*(b + a*Cos[e + f*x]^2)^2) - (a + b)/(a^3*f*(b + a*Cos[e + f*x]^2)) - Log[b + a*Cos[e + f*x]^2]/(2*a^3*f)","A",4,3,23,0.1304,1,"{4138, 444, 43}"
364,1,81,0,0.1116365,"\int \frac{\tan ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Tan[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{b (a+b)}{4 a^3 f \left(a \cos ^2(e+f x)+b\right)^2}+\frac{a+2 b}{2 a^3 f \left(a \cos ^2(e+f x)+b\right)}+\frac{\log \left(a \cos ^2(e+f x)+b\right)}{2 a^3 f}","-\frac{b (a+b)}{4 a^3 f \left(a \cos ^2(e+f x)+b\right)^2}+\frac{a+2 b}{2 a^3 f \left(a \cos ^2(e+f x)+b\right)}+\frac{\log \left(a \cos ^2(e+f x)+b\right)}{2 a^3 f}",1,"-(b*(a + b))/(4*a^3*f*(b + a*Cos[e + f*x]^2)^2) + (a + 2*b)/(2*a^3*f*(b + a*Cos[e + f*x]^2)) + Log[b + a*Cos[e + f*x]^2]/(2*a^3*f)","A",4,3,23,0.1304,1,"{4138, 446, 77}"
365,1,74,0,0.07414,"\int \frac{\tan (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Tan[e + f*x]/(a + b*Sec[e + f*x]^2)^3,x]","\frac{b^2}{4 a^3 f \left(a \cos ^2(e+f x)+b\right)^2}-\frac{b}{a^3 f \left(a \cos ^2(e+f x)+b\right)}-\frac{\log \left(a \cos ^2(e+f x)+b\right)}{2 a^3 f}","\frac{b^2}{4 a^3 f \left(a \cos ^2(e+f x)+b\right)^2}-\frac{b}{a^3 f \left(a \cos ^2(e+f x)+b\right)}-\frac{\log \left(a \cos ^2(e+f x)+b\right)}{2 a^3 f}",1,"b^2/(4*a^3*f*(b + a*Cos[e + f*x]^2)^2) - b/(a^3*f*(b + a*Cos[e + f*x]^2)) - Log[b + a*Cos[e + f*x]^2]/(2*a^3*f)","A",4,3,21,0.1429,1,"{4138, 266, 43}"
366,1,130,0,0.1650653,"\int \frac{\cot (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Cot[e + f*x]/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{b^3}{4 a^3 f (a+b) \left(a \cos ^2(e+f x)+b\right)^2}+\frac{b^2 (3 a+2 b)}{2 a^3 f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)}+\frac{b \left(3 a^2+3 a b+b^2\right) \log \left(a \cos ^2(e+f x)+b\right)}{2 a^3 f (a+b)^3}+\frac{\log (\sin (e+f x))}{f (a+b)^3}","-\frac{b^3}{4 a^3 f (a+b) \left(a \cos ^2(e+f x)+b\right)^2}+\frac{b^2 (3 a+2 b)}{2 a^3 f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)}+\frac{b \left(3 a^2+3 a b+b^2\right) \log \left(a \cos ^2(e+f x)+b\right)}{2 a^3 f (a+b)^3}+\frac{\log (\sin (e+f x))}{f (a+b)^3}",1,"-b^3/(4*a^3*(a + b)*f*(b + a*Cos[e + f*x]^2)^2) + (b^2*(3*a + 2*b))/(2*a^3*(a + b)^2*f*(b + a*Cos[e + f*x]^2)) + (b*(3*a^2 + 3*a*b + b^2)*Log[b + a*Cos[e + f*x]^2])/(2*a^3*(a + b)^3*f) + Log[Sin[e + f*x]]/((a + b)^3*f)","A",4,3,21,0.1429,1,"{4138, 446, 88}"
367,1,154,0,0.2088229,"\int \frac{\cot ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3,x]","\frac{b^4}{4 a^3 f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)^2}-\frac{b^3 (2 a+b)}{a^3 f (a+b)^3 \left(a \cos ^2(e+f x)+b\right)}-\frac{b^2 \left(6 a^2+4 a b+b^2\right) \log \left(a \cos ^2(e+f x)+b\right)}{2 a^3 f (a+b)^4}-\frac{\csc ^2(e+f x)}{2 f (a+b)^3}-\frac{(a+4 b) \log (\sin (e+f x))}{f (a+b)^4}","\frac{b^4}{4 a^3 f (a+b)^2 \left(a \cos ^2(e+f x)+b\right)^2}-\frac{b^3 (2 a+b)}{a^3 f (a+b)^3 \left(a \cos ^2(e+f x)+b\right)}-\frac{b^2 \left(6 a^2+4 a b+b^2\right) \log \left(a \cos ^2(e+f x)+b\right)}{2 a^3 f (a+b)^4}-\frac{\csc ^2(e+f x)}{2 f (a+b)^3}-\frac{(a+4 b) \log (\sin (e+f x))}{f (a+b)^4}",1,"b^4/(4*a^3*(a + b)^2*f*(b + a*Cos[e + f*x]^2)^2) - (b^3*(2*a + b))/(a^3*(a + b)^3*f*(b + a*Cos[e + f*x]^2)) - Csc[e + f*x]^2/(2*(a + b)^3*f) - (b^2*(6*a^2 + 4*a*b + b^2)*Log[b + a*Cos[e + f*x]^2])/(2*a^3*(a + b)^4*f) - ((a + 4*b)*Log[Sin[e + f*x]])/((a + b)^4*f)","A",4,3,23,0.1304,1,"{4138, 446, 88}"
368,1,192,0,0.269266,"\int \frac{\cot ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{b^5}{4 a^3 f (a+b)^3 \left(a \cos ^2(e+f x)+b\right)^2}+\frac{b^4 (5 a+2 b)}{2 a^3 f (a+b)^4 \left(a \cos ^2(e+f x)+b\right)}+\frac{\left(a^2+5 a b+10 b^2\right) \log (\sin (e+f x))}{f (a+b)^5}+\frac{b^3 \left(10 a^2+5 a b+b^2\right) \log \left(a \cos ^2(e+f x)+b\right)}{2 a^3 f (a+b)^5}-\frac{\csc ^4(e+f x)}{4 f (a+b)^3}+\frac{(2 a+5 b) \csc ^2(e+f x)}{2 f (a+b)^4}","-\frac{b^5}{4 a^3 f (a+b)^3 \left(a \cos ^2(e+f x)+b\right)^2}+\frac{b^4 (5 a+2 b)}{2 a^3 f (a+b)^4 \left(a \cos ^2(e+f x)+b\right)}+\frac{\left(a^2+5 a b+10 b^2\right) \log (\sin (e+f x))}{f (a+b)^5}+\frac{b^3 \left(10 a^2+5 a b+b^2\right) \log \left(a \cos ^2(e+f x)+b\right)}{2 a^3 f (a+b)^5}-\frac{\csc ^4(e+f x)}{4 f (a+b)^3}+\frac{(2 a+5 b) \csc ^2(e+f x)}{2 f (a+b)^4}",1,"-b^5/(4*a^3*(a + b)^3*f*(b + a*Cos[e + f*x]^2)^2) + (b^4*(5*a + 2*b))/(2*a^3*(a + b)^4*f*(b + a*Cos[e + f*x]^2)) + ((2*a + 5*b)*Csc[e + f*x]^2)/(2*(a + b)^4*f) - Csc[e + f*x]^4/(4*(a + b)^3*f) + (b^3*(10*a^2 + 5*a*b + b^2)*Log[b + a*Cos[e + f*x]^2])/(2*a^3*(a + b)^5*f) + ((a^2 + 5*a*b + 10*b^2)*Log[Sin[e + f*x]])/((a + b)^5*f)","A",4,3,23,0.1304,1,"{4138, 446, 88}"
369,1,147,0,0.2924075,"\int \frac{\tan ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Tan[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{(3 a-4 b) (a+b) \tan (e+f x)}{8 a^2 b^2 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\sqrt{a+b} \left(3 a^2-4 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 b^{5/2} f}-\frac{x}{a^3}-\frac{(a+b) \tan ^3(e+f x)}{4 a b f \left(a+b \tan ^2(e+f x)+b\right)^2}","-\frac{(3 a-4 b) (a+b) \tan (e+f x)}{8 a^2 b^2 f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{\sqrt{a+b} \left(3 a^2-4 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 b^{5/2} f}-\frac{x}{a^3}-\frac{(a+b) \tan ^3(e+f x)}{4 a b f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"-(x/a^3) + (Sqrt[a + b]*(3*a^2 - 4*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*b^(5/2)*f) - ((a + b)*Tan[e + f*x]^3)/(4*a*b*f*(a + b + b*Tan[e + f*x]^2)^2) - ((3*a - 4*b)*(a + b)*Tan[e + f*x])/(8*a^2*b^2*f*(a + b + b*Tan[e + f*x]^2))","A",7,7,23,0.3043,1,"{4141, 1975, 470, 578, 522, 203, 205}"
370,1,137,0,0.2573513,"\int \frac{\tan ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Tan[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\left(a^2-4 a b-8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 b^{3/2} f \sqrt{a+b}}+\frac{(a-4 b) \tan (e+f x)}{8 a^2 b f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x}{a^3}-\frac{(a+b) \tan (e+f x)}{4 a b f \left(a+b \tan ^2(e+f x)+b\right)^2}","\frac{\left(a^2-4 a b-8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 b^{3/2} f \sqrt{a+b}}+\frac{(a-4 b) \tan (e+f x)}{8 a^2 b f \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x}{a^3}-\frac{(a+b) \tan (e+f x)}{4 a b f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"x/a^3 + ((a^2 - 4*a*b - 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*b^(3/2)*Sqrt[a + b]*f) - ((a + b)*Tan[e + f*x])/(4*a*b*f*(a + b + b*Tan[e + f*x]^2)^2) + ((a - 4*b)*Tan[e + f*x])/(8*a^2*b*f*(a + b + b*Tan[e + f*x]^2))","A",7,7,23,0.3043,1,"{4141, 1975, 470, 527, 522, 203, 205}"
371,1,138,0,0.2271715,"\int \frac{\tan ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Tan[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\left(3 a^2+12 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 \sqrt{b} f (a+b)^{3/2}}+\frac{(3 a+4 b) \tan (e+f x)}{8 a^2 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}-\frac{x}{a^3}+\frac{\tan (e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)^2}","\frac{\left(3 a^2+12 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 \sqrt{b} f (a+b)^{3/2}}+\frac{(3 a+4 b) \tan (e+f x)}{8 a^2 f (a+b) \left(a+b \tan ^2(e+f x)+b\right)}-\frac{x}{a^3}+\frac{\tan (e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"-(x/a^3) + ((3*a^2 + 12*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*Sqrt[b]*(a + b)^(3/2)*f) + Tan[e + f*x]/(4*a*f*(a + b + b*Tan[e + f*x]^2)^2) + ((3*a + 4*b)*Tan[e + f*x])/(8*a^2*(a + b)*f*(a + b + b*Tan[e + f*x]^2))","A",7,7,23,0.3043,1,"{4141, 1975, 471, 527, 522, 203, 205}"
372,1,144,0,0.1779442,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[(a + b*Sec[e + f*x]^2)^(-3),x]","-\frac{\sqrt{b} \left(15 a^2+20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 f (a+b)^{5/2}}-\frac{b (7 a+4 b) \tan (e+f x)}{8 a^2 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x}{a^3}-\frac{b \tan (e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}","-\frac{\sqrt{b} \left(15 a^2+20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 f (a+b)^{5/2}}-\frac{b (7 a+4 b) \tan (e+f x)}{8 a^2 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x}{a^3}-\frac{b \tan (e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"x/a^3 - (Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(5/2)*f) - (b*Tan[e + f*x])/(4*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(7*a + 4*b)*Tan[e + f*x])/(8*a^2*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))","A",6,6,14,0.4286,1,"{4128, 414, 527, 522, 203, 205}"
373,1,181,0,0.3803873,"\int \frac{\cot ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Cot[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3,x]","\frac{b^{3/2} \left(35 a^2+28 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 f (a+b)^{7/2}}-\frac{\left(8 a^2-11 a b-4 b^2\right) \cot (e+f x)}{8 a^2 f (a+b)^3}-\frac{b (9 a+4 b) \cot (e+f x)}{8 a^2 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{x}{a^3}-\frac{b \cot (e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}","\frac{b^{3/2} \left(35 a^2+28 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 f (a+b)^{7/2}}-\frac{\left(8 a^2-11 a b-4 b^2\right) \cot (e+f x)}{8 a^2 f (a+b)^3}-\frac{b (9 a+4 b) \cot (e+f x)}{8 a^2 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{x}{a^3}-\frac{b \cot (e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"-(x/a^3) + (b^(3/2)*(35*a^2 + 28*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(7/2)*f) - ((8*a^2 - 11*a*b - 4*b^2)*Cot[e + f*x])/(8*a^2*(a + b)^3*f) - (b*Cot[e + f*x])/(4*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(9*a + 4*b)*Cot[e + f*x])/(8*a^2*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))","A",8,8,23,0.3478,1,"{4141, 1975, 472, 579, 583, 522, 203, 205}"
374,1,230,0,0.4611201,"\int \frac{\cot ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{b^{5/2} \left(63 a^2+36 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 f (a+b)^{9/2}}-\frac{\left(8 a^2-39 a b-12 b^2\right) \cot ^3(e+f x)}{24 a^2 f (a+b)^3}+\frac{\left(32 a^2 b+8 a^3-15 a b^2-4 b^3\right) \cot (e+f x)}{8 a^2 f (a+b)^4}-\frac{b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x}{a^3}-\frac{b \cot ^3(e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}","-\frac{b^{5/2} \left(63 a^2+36 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 f (a+b)^{9/2}}-\frac{\left(8 a^2-39 a b-12 b^2\right) \cot ^3(e+f x)}{24 a^2 f (a+b)^3}+\frac{\left(32 a^2 b+8 a^3-15 a b^2-4 b^3\right) \cot (e+f x)}{8 a^2 f (a+b)^4}-\frac{b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}+\frac{x}{a^3}-\frac{b \cot ^3(e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"x/a^3 - (b^(5/2)*(63*a^2 + 36*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(9/2)*f) + ((8*a^3 + 32*a^2*b - 15*a*b^2 - 4*b^3)*Cot[e + f*x])/(8*a^2*(a + b)^4*f) - ((8*a^2 - 39*a*b - 12*b^2)*Cot[e + f*x]^3)/(24*a^2*(a + b)^3*f) - (b*Cot[e + f*x]^3)/(4*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(11*a + 4*b)*Cot[e + f*x]^3)/(8*a^2*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))","A",9,8,23,0.3478,1,"{4141, 1975, 472, 579, 583, 522, 203, 205}"
375,1,285,0,0.6067204,"\int \frac{\cot ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Int[Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]","\frac{b^{7/2} \left(99 a^2+44 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 f (a+b)^{11/2}}-\frac{\left(8 a^2-75 a b-20 b^2\right) \cot ^5(e+f x)}{40 a^2 f (a+b)^3}+\frac{\left(32 a^2 b+8 a^3-51 a b^2-12 b^3\right) \cot ^3(e+f x)}{24 a^2 f (a+b)^4}-\frac{\left(80 a^2 b^2+40 a^3 b+8 a^4-19 a b^3-4 b^4\right) \cot (e+f x)}{8 a^2 f (a+b)^5}-\frac{b (13 a+4 b) \cot ^5(e+f x)}{8 a^2 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{x}{a^3}-\frac{b \cot ^5(e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}","\frac{b^{7/2} \left(99 a^2+44 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{8 a^3 f (a+b)^{11/2}}-\frac{\left(8 a^2-75 a b-20 b^2\right) \cot ^5(e+f x)}{40 a^2 f (a+b)^3}+\frac{\left(32 a^2 b+8 a^3-51 a b^2-12 b^3\right) \cot ^3(e+f x)}{24 a^2 f (a+b)^4}-\frac{\left(80 a^2 b^2+40 a^3 b+8 a^4-19 a b^3-4 b^4\right) \cot (e+f x)}{8 a^2 f (a+b)^5}-\frac{b (13 a+4 b) \cot ^5(e+f x)}{8 a^2 f (a+b)^2 \left(a+b \tan ^2(e+f x)+b\right)}-\frac{x}{a^3}-\frac{b \cot ^5(e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"-(x/a^3) + (b^(7/2)*(99*a^2 + 44*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(11/2)*f) - ((8*a^4 + 40*a^3*b + 80*a^2*b^2 - 19*a*b^3 - 4*b^4)*Cot[e + f*x])/(8*a^2*(a + b)^5*f) + ((8*a^3 + 32*a^2*b - 51*a*b^2 - 12*b^3)*Cot[e + f*x]^3)/(24*a^2*(a + b)^4*f) - ((8*a^2 - 75*a*b - 20*b^2)*Cot[e + f*x]^5)/(40*a^2*(a + b)^3*f) - (b*Cot[e + f*x]^5)/(4*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(13*a + 4*b)*Cot[e + f*x]^5)/(8*a^2*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))","A",10,8,23,0.3478,1,"{4141, 1975, 472, 579, 583, 522, 203, 205}"
376,1,111,0,0.1411123,"\int \sqrt{a+b \sec ^2(e+f x)} \tan ^5(e+f x) \, dx","Int[Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^5,x]","\frac{\left(a+b \sec ^2(e+f x)\right)^{5/2}}{5 b^2 f}-\frac{(a+2 b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 b^2 f}+\frac{\sqrt{a+b \sec ^2(e+f x)}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}","\frac{\left(a+b \sec ^2(e+f x)\right)^{5/2}}{5 b^2 f}-\frac{(a+2 b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 b^2 f}+\frac{\sqrt{a+b \sec ^2(e+f x)}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}",1,"-((Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f) + Sqrt[a + b*Sec[e + f*x]^2]/f - ((a + 2*b)*(a + b*Sec[e + f*x]^2)^(3/2))/(3*b^2*f) + (a + b*Sec[e + f*x]^2)^(5/2)/(5*b^2*f)","A",7,6,25,0.2400,1,"{4139, 446, 88, 50, 63, 208}"
377,1,80,0,0.1046519,"\int \sqrt{a+b \sec ^2(e+f x)} \tan ^3(e+f x) \, dx","Int[Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^3,x]","\frac{\left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 b f}-\frac{\sqrt{a+b \sec ^2(e+f x)}}{f}+\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}","\frac{\left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 b f}-\frac{\sqrt{a+b \sec ^2(e+f x)}}{f}+\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}",1,"(Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f - Sqrt[a + b*Sec[e + f*x]^2]/f + (a + b*Sec[e + f*x]^2)^(3/2)/(3*b*f)","A",6,6,25,0.2400,1,"{4139, 446, 80, 50, 63, 208}"
378,1,54,0,0.0647038,"\int \sqrt{a+b \sec ^2(e+f x)} \tan (e+f x) \, dx","Int[Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x],x]","\frac{\sqrt{a+b \sec ^2(e+f x)}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}","\frac{\sqrt{a+b \sec ^2(e+f x)}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}",1,"-((Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f) + Sqrt[a + b*Sec[e + f*x]^2]/f","A",5,5,23,0.2174,1,"{4139, 266, 50, 63, 208}"
379,1,70,0,0.1100147,"\int \cot (e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Cot[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}-\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{f}","\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}-\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{f}",1,"(Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f - (Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/f","A",7,5,23,0.2174,1,"{4139, 446, 83, 63, 208}"
380,1,109,0,0.148657,"\int \cot ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Cot[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\cot ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f \sqrt{a+b}}","-\frac{\cot ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f \sqrt{a+b}}",1,"-((Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f) + ((2*a + b)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(2*Sqrt[a + b]*f) - (Cot[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2])/(2*f)","A",8,6,25,0.2400,1,"{4139, 446, 99, 156, 63, 208}"
381,1,161,0,0.2292693,"\int \cot ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Cot[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\left(8 a^2+12 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{3/2}}-\frac{\cot ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{4 f}+\frac{(4 a+3 b) \cot ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{8 f (a+b)}+\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}","-\frac{\left(8 a^2+12 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{3/2}}-\frac{\cot ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{4 f}+\frac{(4 a+3 b) \cot ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{8 f (a+b)}+\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}",1,"(Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f - ((8*a^2 + 12*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(3/2)*f) + ((4*a + 3*b)*Cot[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2])/(8*(a + b)*f) - (Cot[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2])/(4*f)","A",9,7,25,0.2800,1,"{4139, 446, 99, 151, 156, 63, 208}"
382,1,219,0,0.435785,"\int \sqrt{a+b \sec ^2(e+f x)} \tan ^6(e+f x) \, dx","Int[Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^6,x]","\frac{\left(5 a^2 b+a^3+15 a b^2-5 b^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 b^{5/2} f}-\frac{(a-b) (a+5 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{16 b^2 f}+\frac{\tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{6 f}+\frac{(a-5 b) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{24 b f}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}","\frac{\left(5 a^2 b+a^3+15 a b^2-5 b^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 b^{5/2} f}-\frac{(a-b) (a+5 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{16 b^2 f}+\frac{\tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{6 f}+\frac{(a-5 b) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{24 b f}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}",1,"-((Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) + ((a^3 + 5*a^2*b + 15*a*b^2 - 5*b^3)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*b^(5/2)*f) - ((a - b)*(a + 5*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(16*b^2*f) + ((a - 5*b)*Tan[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(24*b*f) + (Tan[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(6*f)","A",10,9,25,0.3600,1,"{4141, 1975, 478, 582, 523, 217, 206, 377, 203}"
383,1,165,0,0.3140652,"\int \sqrt{a+b \sec ^2(e+f x)} \tan ^4(e+f x) \, dx","Int[Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^4,x]","-\frac{\left(a^2+6 a b-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 b^{3/2} f}+\frac{\tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 f}+\frac{(a-3 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 b f}+\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}","-\frac{\left(a^2+6 a b-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 b^{3/2} f}+\frac{\tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 f}+\frac{(a-3 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 b f}+\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}",1,"(Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - ((a^2 + 6*a*b - 3*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*b^(3/2)*f) + ((a - 3*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*b*f) + (Tan[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(4*f)","A",9,9,25,0.3600,1,"{4141, 1975, 478, 582, 523, 217, 206, 377, 203}"
384,1,118,0,0.2234409,"\int \sqrt{a+b \sec ^2(e+f x)} \tan ^2(e+f x) \, dx","Int[Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^2,x]","-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{\tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}+\frac{(a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 \sqrt{b} f}","-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{\tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}+\frac{(a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 \sqrt{b} f}",1,"-((Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) + ((a - b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*Sqrt[b]*f) + (Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)","A",8,8,25,0.3200,1,"{4141, 1975, 478, 523, 217, 206, 377, 203}"
385,1,79,0,0.0508672,"\int \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}",1,"(Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f","A",6,6,16,0.3750,1,"{4128, 402, 217, 206, 377, 203}"
386,1,69,0,0.1786918,"\int \cot ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Cot[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f}","-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f}",1,"-((Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) - (Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/f","A",6,6,25,0.2400,1,"{4141, 1975, 475, 12, 377, 203}"
387,1,114,0,0.2500146,"\int \cot ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Cot[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f}+\frac{(3 a+2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f (a+b)}","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f}+\frac{(3 a+2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f (a+b)}",1,"(Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + ((3*a + 2*b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*(a + b)*f) - (Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*f)","A",7,7,25,0.2800,1,"{4141, 1975, 475, 583, 12, 377, 203}"
388,1,167,0,0.333206,"\int \cot ^6(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Int[Cot[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\left(15 a^2+25 a b+8 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^2}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 f}-\frac{(b-5 (a+b)) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)}","-\frac{\left(15 a^2+25 a b+8 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^2}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 f}-\frac{(b-5 (a+b)) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)}",1,"-((Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) - ((15*a^2 + 25*a*b + 8*b^2)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*(a + b)^2*f) - ((b - 5*(a + b))*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*(a + b)*f) - (Cot[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(5*f)","A",8,7,25,0.2800,1,"{4141, 1975, 475, 583, 12, 377, 203}"
389,1,135,0,0.1640869,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \tan ^5(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^5,x]","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{\left(a+b \sec ^2(e+f x)\right)^{7/2}}{7 b^2 f}-\frac{(a+2 b) \left(a+b \sec ^2(e+f x)\right)^{5/2}}{5 b^2 f}+\frac{\left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 f}+\frac{a \sqrt{a+b \sec ^2(e+f x)}}{f}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{\left(a+b \sec ^2(e+f x)\right)^{7/2}}{7 b^2 f}-\frac{(a+2 b) \left(a+b \sec ^2(e+f x)\right)^{5/2}}{5 b^2 f}+\frac{\left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 f}+\frac{a \sqrt{a+b \sec ^2(e+f x)}}{f}",1,"-((a^(3/2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f) + (a*Sqrt[a + b*Sec[e + f*x]^2])/f + (a + b*Sec[e + f*x]^2)^(3/2)/(3*f) - ((a + 2*b)*(a + b*Sec[e + f*x]^2)^(5/2))/(5*b^2*f) + (a + b*Sec[e + f*x]^2)^(7/2)/(7*b^2*f)","A",8,6,25,0.2400,1,"{4139, 446, 88, 50, 63, 208}"
390,1,104,0,0.1269544,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \tan ^3(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^3,x]","\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{\left(a+b \sec ^2(e+f x)\right)^{5/2}}{5 b f}-\frac{\left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 f}-\frac{a \sqrt{a+b \sec ^2(e+f x)}}{f}","\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{\left(a+b \sec ^2(e+f x)\right)^{5/2}}{5 b f}-\frac{\left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 f}-\frac{a \sqrt{a+b \sec ^2(e+f x)}}{f}",1,"(a^(3/2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f - (a*Sqrt[a + b*Sec[e + f*x]^2])/f - (a + b*Sec[e + f*x]^2)^(3/2)/(3*f) + (a + b*Sec[e + f*x]^2)^(5/2)/(5*b*f)","A",7,6,25,0.2400,1,"{4139, 446, 80, 50, 63, 208}"
391,1,78,0,0.0824348,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \tan (e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x],x]","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{a \sqrt{a+b \sec ^2(e+f x)}}{f}+\frac{\left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 f}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{a \sqrt{a+b \sec ^2(e+f x)}}{f}+\frac{\left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 f}",1,"-((a^(3/2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f) + (a*Sqrt[a + b*Sec[e + f*x]^2])/f + (a + b*Sec[e + f*x]^2)^(3/2)/(3*f)","A",6,5,23,0.2174,1,"{4139, 266, 50, 63, 208}"
392,1,91,0,0.1361376,"\int \cot (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{b \sqrt{a+b \sec ^2(e+f x)}}{f}-\frac{(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{f}","\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{b \sqrt{a+b \sec ^2(e+f x)}}{f}-\frac{(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{f}",1,"(a^(3/2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f - ((a + b)^(3/2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/f + (b*Sqrt[a + b*Sec[e + f*x]^2])/f","A",8,6,23,0.2609,1,"{4139, 446, 84, 156, 63, 208}"
393,1,114,0,0.1702027,"\int \cot ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}-\frac{(a+b) \cot ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 f}+\frac{(2 a-b) \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}-\frac{(a+b) \cot ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 f}+\frac{(2 a-b) \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f}",1,"-((a^(3/2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f) + ((2*a - b)*Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(2*f) - ((a + b)*Cot[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2])/(2*f)","A",8,6,25,0.2400,1,"{4139, 446, 98, 156, 63, 208}"
394,1,159,0,0.2428696,"\int \cot ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\left(8 a^2+4 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f \sqrt{a+b}}+\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}-\frac{(a+b) \cot ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{4 f}+\frac{(4 a-b) \cot ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{8 f}","-\frac{\left(8 a^2+4 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f \sqrt{a+b}}+\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{f}-\frac{(a+b) \cot ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{4 f}+\frac{(4 a-b) \cot ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{8 f}",1,"(a^(3/2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f - ((8*a^2 + 4*a*b - b^2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(8*Sqrt[a + b]*f) + ((4*a - b)*Cot[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2])/(8*f) - ((a + b)*Cot[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2])/(4*f)","A",9,7,25,0.2800,1,"{4139, 446, 98, 151, 156, 63, 208}"
395,1,290,0,0.5705781,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \tan ^6(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^6,x]","\frac{\left(3 a^2-50 a b-5 b^2\right) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{192 b f}-\frac{\left(17 a^2 b+3 a^3-55 a b^2-5 b^3\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{128 b^2 f}+\frac{\left(90 a^2 b^2+20 a^3 b+3 a^4-60 a b^3-5 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{128 b^{5/2} f}-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{b \tan ^7(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 f}+\frac{(9 a+b) \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{48 f}","\frac{\left(3 a^2-50 a b-5 b^2\right) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{192 b f}-\frac{\left(17 a^2 b+3 a^3-55 a b^2-5 b^3\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{128 b^2 f}+\frac{\left(90 a^2 b^2+20 a^3 b+3 a^4-60 a b^3-5 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{128 b^{5/2} f}-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{b \tan ^7(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 f}+\frac{(9 a+b) \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{48 f}",1,"-((a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) + ((3*a^4 + 20*a^3*b + 90*a^2*b^2 - 60*a*b^3 - 5*b^4)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(128*b^(5/2)*f) - ((3*a^3 + 17*a^2*b - 55*a*b^2 - 5*b^3)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(128*b^2*f) + ((3*a^2 - 50*a*b - 5*b^2)*Tan[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(192*b*f) + ((9*a + b)*Tan[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(48*f) + (b*Tan[e + f*x]^7*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*f)","A",11,9,25,0.3600,1,"{4141, 1975, 477, 582, 523, 217, 206, 377, 203}"
396,1,214,0,0.4782176,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \tan ^4(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^4,x]","\frac{\left(a^2-8 a b-b^2\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{16 b f}-\frac{(a-b) \left(a^2+10 a b+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 b^{3/2} f}+\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{b \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{6 f}+\frac{(7 a+b) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{24 f}","\frac{\left(a^2-8 a b-b^2\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{16 b f}-\frac{(a-b) \left(a^2+10 a b+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{16 b^{3/2} f}+\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{b \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{6 f}+\frac{(7 a+b) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{24 f}",1,"(a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - ((a - b)*(a^2 + 10*a*b + b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*b^(3/2)*f) + ((a^2 - 8*a*b - b^2)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(16*b*f) + ((7*a + b)*Tan[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(24*f) + (b*Tan[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(6*f)","A",10,9,25,0.3600,1,"{4141, 1975, 477, 582, 523, 217, 206, 377, 203}"
397,1,166,0,0.3649263,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \tan ^2(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^2,x]","\frac{\left(3 a^2-6 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 \sqrt{b} f}-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{b \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 f}+\frac{(5 a+b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 f}","\frac{\left(3 a^2-6 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 \sqrt{b} f}-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{b \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 f}+\frac{(5 a+b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 f}",1,"-((a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) + ((3*a^2 - 6*a*b - b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*Sqrt[b]*f) + ((5*a + b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*f) + (b*Tan[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(4*f)","A",9,9,25,0.3600,1,"{4141, 1975, 477, 582, 523, 217, 206, 377, 203}"
398,1,118,0,0.0959582,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}+\frac{\sqrt{b} (3 a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 f}+\frac{\sqrt{b} (3 a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 f}",1,"(a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*(3*a + b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (b*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)","A",7,7,16,0.4375,1,"{4128, 416, 523, 217, 206, 377, 203}"
399,1,111,0,0.2275516,"\int \cot ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]^2*(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{(a+b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f}","-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}+\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{(a+b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f}",1,"-((a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) + (b^(3/2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - ((a + b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/f","A",8,8,25,0.3200,1,"{4141, 1975, 474, 523, 217, 206, 377, 203}"
400,1,112,0,0.2751489,"\int \cot ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]^4*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{(a+b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f}+\frac{(3 a-b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f}","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{(a+b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f}+\frac{(3 a-b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f}",1,"(a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + ((3*a - b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*f) - ((a + b)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*f)","A",7,7,25,0.2800,1,"{4141, 1975, 474, 583, 12, 377, 203}"
401,1,165,0,0.3551792,"\int \cot ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]^6*(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\left(15 a^2+10 a b-2 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)}-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{(a+b) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 f}+\frac{(5 a-b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f}","-\frac{\left(15 a^2+10 a b-2 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)}-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{f}-\frac{(a+b) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 f}+\frac{(5 a-b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f}",1,"-((a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) - ((15*a^2 + 10*a*b - 2*b^2)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*(a + b)*f) + ((5*a - b)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*f) - ((a + b)*Cot[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(5*f)","A",8,7,25,0.2800,1,"{4141, 1975, 474, 583, 12, 377, 203}"
402,1,89,0,0.1330722,"\int \frac{\tan ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 b^2 f}-\frac{(a+2 b) \sqrt{a+b \sec ^2(e+f x)}}{b^2 f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}","\frac{\left(a+b \sec ^2(e+f x)\right)^{3/2}}{3 b^2 f}-\frac{(a+2 b) \sqrt{a+b \sec ^2(e+f x)}}{b^2 f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}",1,"-(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f)) - ((a + 2*b)*Sqrt[a + b*Sec[e + f*x]^2])/(b^2*f) + (a + b*Sec[e + f*x]^2)^(3/2)/(3*b^2*f)","A",6,5,25,0.2000,1,"{4139, 446, 88, 63, 208}"
403,1,56,0,0.0954163,"\int \frac{\tan ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{a+b \sec ^2(e+f x)}}{b f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}","\frac{\sqrt{a+b \sec ^2(e+f x)}}{b f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}",1,"ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f) + Sqrt[a + b*Sec[e + f*x]^2]/(b*f)","A",5,5,25,0.2000,1,"{4139, 446, 80, 63, 208}"
404,1,33,0,0.0546615,"\int \frac{\tan (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Tan[e + f*x]/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}",1,"-(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f))","A",4,4,23,0.1739,1,"{4139, 266, 63, 208}"
405,1,70,0,0.1064415,"\int \frac{\cot (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Cot[e + f*x]/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{f \sqrt{a+b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{f \sqrt{a+b}}",1,"ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f) - ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]]/(Sqrt[a + b]*f)","A",7,5,23,0.2174,1,"{4139, 446, 86, 63, 208}"
406,1,116,0,0.164005,"\int \frac{\cot ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\cot ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 f (a+b)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}+\frac{(2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f (a+b)^{3/2}}","-\frac{\cot ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{2 f (a+b)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}+\frac{(2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f (a+b)^{3/2}}",1,"-(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f)) + ((2*a + 3*b)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(2*(a + b)^(3/2)*f) - (Cot[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2])/(2*(a + b)*f)","A",8,6,25,0.2400,1,"{4139, 446, 103, 156, 63, 208}"
407,1,166,0,0.2441554,"\int \frac{\cot ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\left(8 a^2+20 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{5/2}}-\frac{\cot ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{4 f (a+b)}+\frac{(4 a+7 b) \cot ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{8 f (a+b)^2}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}","-\frac{\left(8 a^2+20 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{5/2}}-\frac{\cot ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{4 f (a+b)}+\frac{(4 a+7 b) \cot ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{8 f (a+b)^2}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}",1,"ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f) - ((8*a^2 + 20*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(5/2)*f) + ((4*a + 7*b)*Cot[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2])/(8*(a + b)^2*f) - (Cot[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2])/(4*(a + b)*f)","A",9,7,25,0.2800,1,"{4139, 446, 103, 151, 156, 63, 208}"
408,1,173,0,0.3187187,"\int \frac{\tan ^6(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\left(3 a^2+10 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 b^{5/2} f}-\frac{(3 a+7 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 b^2 f}+\frac{\tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 b f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}","\frac{\left(3 a^2+10 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{8 b^{5/2} f}-\frac{(3 a+7 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 b^2 f}+\frac{\tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 b f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}",1,"-(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f)) + ((3*a^2 + 10*a*b + 15*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*b^(5/2)*f) - ((3*a + 7*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*b^2*f) + (Tan[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(4*b*f)","A",9,9,25,0.3600,1,"{4141, 1975, 479, 582, 523, 217, 206, 377, 203}"
409,1,120,0,0.2300631,"\int \frac{\tan ^4(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{(a+3 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 b^{3/2} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}+\frac{\tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 b f}","-\frac{(a+3 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 b^{3/2} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}+\frac{\tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 b f}",1,"ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f) - ((a + 3*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*b^(3/2)*f) + (Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*b*f)","A",8,8,25,0.3200,1,"{4141, 1975, 479, 523, 217, 206, 377, 203}"
410,1,80,0,0.1968098,"\int \frac{\tan ^2(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{b} f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{b} f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}",1,"-(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f)) + ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[b]*f)","A",7,7,25,0.2800,1,"{4141, 1975, 483, 217, 206, 377, 203}"
411,1,39,0,0.0290335,"\int \frac{1}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[1/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}",1,"ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f)","A",3,3,16,0.1875,1,"{4128, 377, 203}"
412,1,74,0,0.1917356,"\int \frac{\cot ^2(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f (a+b)}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f (a+b)}",1,"-(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f)) - (Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/((a + b)*f)","A",6,6,25,0.2400,1,"{4141, 1975, 480, 12, 377, 203}"
413,1,119,0,0.2493871,"\int \frac{\cot ^4(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f (a+b)}+\frac{(3 a+5 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f (a+b)^2}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f (a+b)}+\frac{(3 a+5 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f (a+b)^2}",1,"ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f) + ((3*a + 5*b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*(a + b)^2*f) - (Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*(a + b)*f)","A",7,7,25,0.2800,1,"{4141, 1975, 480, 583, 12, 377, 203}"
414,1,172,0,0.3448358,"\int \frac{\cot ^6(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\left(15 a^2+40 a b+33 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^3}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 f (a+b)}+\frac{(5 a+9 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^2}","-\frac{\left(15 a^2+40 a b+33 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^3}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 f (a+b)}+\frac{(5 a+9 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^2}",1,"-(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f)) - ((15*a^2 + 40*a*b + 33*b^2)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*(a + b)^3*f) + ((5*a + 9*b)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*(a + b)^2*f) - (Cot[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(5*(a + b)*f)","A",8,7,25,0.2800,1,"{4141, 1975, 480, 583, 12, 377, 203}"
415,1,88,0,0.1507691,"\int \frac{\tan ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}+\frac{(a+b)^2}{a b^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sqrt{a+b \sec ^2(e+f x)}}{b^2 f}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}+\frac{(a+b)^2}{a b^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sqrt{a+b \sec ^2(e+f x)}}{b^2 f}",1,"-(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f)) + (a + b)^2/(a*b^2*f*Sqrt[a + b*Sec[e + f*x]^2]) + Sqrt[a + b*Sec[e + f*x]^2]/(b^2*f)","A",6,5,25,0.2000,1,"{4139, 446, 87, 63, 208}"
416,1,63,0,0.1164525,"\int \frac{\tan ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}-\frac{a+b}{a b f \sqrt{a+b \sec ^2(e+f x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}-\frac{a+b}{a b f \sqrt{a+b \sec ^2(e+f x)}}",1,"ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f) - (a + b)/(a*b*f*Sqrt[a + b*Sec[e + f*x]^2])","A",5,5,25,0.2000,1,"{4139, 446, 78, 63, 208}"
417,1,57,0,0.0721779,"\int \frac{\tan (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{1}{a f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}","\frac{1}{a f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}",1,"-(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f)) + 1/(a*f*Sqrt[a + b*Sec[e + f*x]^2])","A",5,5,23,0.2174,1,"{4139, 266, 51, 63, 208}"
418,1,100,0,0.147423,"\int \frac{\cot (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}-\frac{b}{a f (a+b) \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{f (a+b)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}-\frac{b}{a f (a+b) \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{f (a+b)^{3/2}}",1,"ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f) - ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]]/((a + b)^(3/2)*f) - b/(a*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2])","A",8,6,23,0.2609,1,"{4139, 446, 85, 156, 63, 208}"
419,1,153,0,0.2423159,"\int \frac{\cot ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}-\frac{b (a-2 b)}{2 a f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}-\frac{\cot ^2(e+f x)}{2 f (a+b) \sqrt{a+b \sec ^2(e+f x)}}+\frac{(2 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f (a+b)^{5/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}-\frac{b (a-2 b)}{2 a f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}-\frac{\cot ^2(e+f x)}{2 f (a+b) \sqrt{a+b \sec ^2(e+f x)}}+\frac{(2 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f (a+b)^{5/2}}",1,"-(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f)) + ((2*a + 5*b)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(2*(a + b)^(5/2)*f) - ((a - 2*b)*b)/(2*a*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]) - Cot[e + f*x]^2/(2*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2])","A",9,7,25,0.2800,1,"{4139, 446, 103, 152, 156, 63, 208}"
420,1,213,0,0.3336893,"\int \frac{\cot ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{b \left(4 a^2+11 a b-8 b^2\right)}{8 a f (a+b)^3 \sqrt{a+b \sec ^2(e+f x)}}-\frac{\left(8 a^2+28 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{7/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}-\frac{\cot ^4(e+f x)}{4 f (a+b) \sqrt{a+b \sec ^2(e+f x)}}+\frac{(4 a+9 b) \cot ^2(e+f x)}{8 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}","\frac{b \left(4 a^2+11 a b-8 b^2\right)}{8 a f (a+b)^3 \sqrt{a+b \sec ^2(e+f x)}}-\frac{\left(8 a^2+28 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{7/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}-\frac{\cot ^4(e+f x)}{4 f (a+b) \sqrt{a+b \sec ^2(e+f x)}}+\frac{(4 a+9 b) \cot ^2(e+f x)}{8 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}",1,"ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f) - ((8*a^2 + 28*a*b + 35*b^2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(7/2)*f) + (b*(4*a^2 + 11*a*b - 8*b^2))/(8*a*(a + b)^3*f*Sqrt[a + b*Sec[e + f*x]^2]) + ((4*a + 9*b)*Cot[e + f*x]^2)/(8*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]) - Cot[e + f*x]^4/(4*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2])","A",10,8,25,0.3200,1,"{4139, 446, 103, 151, 152, 156, 63, 208}"
421,1,172,0,0.3485047,"\int \frac{\tan ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}+\frac{(3 a+2 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 a b^2 f}-\frac{(3 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 b^{5/2} f}-\frac{(a+b) \tan ^3(e+f x)}{a b f \sqrt{a+b \tan ^2(e+f x)+b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}+\frac{(3 a+2 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 a b^2 f}-\frac{(3 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{2 b^{5/2} f}-\frac{(a+b) \tan ^3(e+f x)}{a b f \sqrt{a+b \tan ^2(e+f x)+b}}",1,"-(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f)) - ((3*a + 5*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*b^(5/2)*f) - ((a + b)*Tan[e + f*x]^3)/(a*b*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + ((3*a + 2*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*a*b^2*f)","A",9,9,25,0.3600,1,"{4141, 1975, 470, 582, 523, 217, 206, 377, 203}"
422,1,116,0,0.2490248,"\int \frac{\tan ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{b^{3/2} f}-\frac{(a+b) \tan (e+f x)}{a b f \sqrt{a+b \tan ^2(e+f x)+b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{b^{3/2} f}-\frac{(a+b) \tan (e+f x)}{a b f \sqrt{a+b \tan ^2(e+f x)+b}}",1,"ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f) + ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(b^(3/2)*f) - ((a + b)*Tan[e + f*x])/(a*b*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",8,8,25,0.3200,1,"{4141, 1975, 470, 523, 217, 206, 377, 203}"
423,1,71,0,0.2090046,"\int \frac{\tan ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\tan (e+f x)}{a f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}","\frac{\tan (e+f x)}{a f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}",1,"-(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f)) + Tan[e + f*x]/(a*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",5,5,25,0.2000,1,"{4141, 1975, 471, 377, 203}"
424,1,77,0,0.0464997,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[(a + b*Sec[e + f*x]^2)^(-3/2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}-\frac{b \tan (e+f x)}{a f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}-\frac{b \tan (e+f x)}{a f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}",1,"ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f) - (b*Tan[e + f*x])/(a*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",4,4,16,0.2500,1,"{4128, 382, 377, 203}"
425,1,119,0,0.2721581,"\int \frac{\cot ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}-\frac{(a-b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{a f (a+b)^2}-\frac{b \cot (e+f x)}{a f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}-\frac{(a-b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{a f (a+b)^2}-\frac{b \cot (e+f x)}{a f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}",1,"-(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f)) - (b*Cot[e + f*x])/(a*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - ((a - b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(a*(a + b)^2*f)","A",7,7,25,0.2800,1,"{4141, 1975, 472, 583, 12, 377, 203}"
426,1,174,0,0.3620175,"\int \frac{\cot ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}-\frac{(a-3 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 a f (a+b)^2}-\frac{b \cot ^3(e+f x)}{a f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{(3 a-b) (a+3 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 a f (a+b)^3}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}-\frac{(a-3 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 a f (a+b)^2}-\frac{b \cot ^3(e+f x)}{a f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{(3 a-b) (a+3 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 a f (a+b)^3}",1,"ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f) - (b*Cot[e + f*x]^3)/(a*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + ((3*a - b)*(a + 3*b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*a*(a + b)^3*f) - ((a - 3*b)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*a*(a + b)^2*f)","A",8,7,25,0.2800,1,"{4141, 1975, 472, 583, 12, 377, 203}"
427,1,241,0,0.471311,"\int \frac{\cot ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\left(5 a^2+14 a b-15 b^2\right) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 a f (a+b)^3}-\frac{\left(55 a^2 b+15 a^3+73 a b^2-15 b^3\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 a f (a+b)^4}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}-\frac{(a-5 b) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 a f (a+b)^2}-\frac{b \cot ^5(e+f x)}{a f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}","\frac{\left(5 a^2+14 a b-15 b^2\right) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 a f (a+b)^3}-\frac{\left(55 a^2 b+15 a^3+73 a b^2-15 b^3\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 a f (a+b)^4}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{3/2} f}-\frac{(a-5 b) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 a f (a+b)^2}-\frac{b \cot ^5(e+f x)}{a f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}",1,"-(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f)) - (b*Cot[e + f*x]^5)/(a*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - ((15*a^3 + 55*a^2*b + 73*a*b^2 - 15*b^3)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*a*(a + b)^4*f) + ((5*a^2 + 14*a*b - 15*b^2)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*a*(a + b)^3*f) - ((a - 5*b)*Cot[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(5*a*(a + b)^2*f)","A",9,7,25,0.2800,1,"{4141, 1975, 472, 583, 12, 377, 203}"
428,1,97,0,0.1660952,"\int \frac{\tan ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\frac{1}{a^2}-\frac{1}{b^2}}{f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}+\frac{(a+b)^2}{3 a b^2 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}","\frac{\frac{1}{a^2}-\frac{1}{b^2}}{f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}+\frac{(a+b)^2}{3 a b^2 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}",1,"-(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f)) + (a + b)^2/(3*a*b^2*f*(a + b*Sec[e + f*x]^2)^(3/2)) + (a^(-2) - b^(-2))/(f*Sqrt[a + b*Sec[e + f*x]^2])","A",6,5,25,0.2000,1,"{4139, 446, 87, 63, 208}"
429,1,89,0,0.1284314,"\int \frac{\tan ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{1}{a^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{a+b}{3 a b f \left(a+b \sec ^2(e+f x)\right)^{3/2}}","-\frac{1}{a^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{a+b}{3 a b f \left(a+b \sec ^2(e+f x)\right)^{3/2}}",1,"ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f) - (a + b)/(3*a*b*f*(a + b*Sec[e + f*x]^2)^(3/2)) - 1/(a^2*f*Sqrt[a + b*Sec[e + f*x]^2])","A",6,6,25,0.2400,1,"{4139, 446, 78, 51, 63, 208}"
430,1,83,0,0.0883175,"\int \frac{\tan (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{1}{a^2 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}+\frac{1}{3 a f \left(a+b \sec ^2(e+f x)\right)^{3/2}}","\frac{1}{a^2 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}+\frac{1}{3 a f \left(a+b \sec ^2(e+f x)\right)^{3/2}}",1,"-(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f)) + 1/(3*a*f*(a + b*Sec[e + f*x]^2)^(3/2)) + 1/(a^2*f*Sqrt[a + b*Sec[e + f*x]^2])","A",6,5,23,0.2174,1,"{4139, 266, 51, 63, 208}"
431,1,137,0,0.2065422,"\int \frac{\cot (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{b (2 a+b)}{a^2 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{b}{3 a f (a+b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{f (a+b)^{5/2}}","-\frac{b (2 a+b)}{a^2 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{b}{3 a f (a+b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{f (a+b)^{5/2}}",1,"ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f) - ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]]/((a + b)^(5/2)*f) - b/(3*a*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (b*(2*a + b))/(a^2*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2])","A",9,7,23,0.3043,1,"{4139, 446, 85, 152, 156, 63, 208}"
432,1,200,0,0.3196283,"\int \frac{\cot ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{b \left(a^2-6 a b-2 b^2\right)}{2 a^2 f (a+b)^3 \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{b (3 a-2 b)}{6 a f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\cot ^2(e+f x)}{2 f (a+b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}+\frac{(2 a+7 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f (a+b)^{7/2}}","-\frac{b \left(a^2-6 a b-2 b^2\right)}{2 a^2 f (a+b)^3 \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{b (3 a-2 b)}{6 a f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\cot ^2(e+f x)}{2 f (a+b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}+\frac{(2 a+7 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f (a+b)^{7/2}}",1,"-(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f)) + ((2*a + 7*b)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(2*(a + b)^(7/2)*f) - ((3*a - 2*b)*b)/(6*a*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^(3/2)) - Cot[e + f*x]^2/(2*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (b*(a^2 - 6*a*b - 2*b^2))/(2*a^2*(a + b)^3*f*Sqrt[a + b*Sec[e + f*x]^2])","A",10,7,25,0.2800,1,"{4139, 446, 103, 152, 156, 63, 208}"
433,1,268,0,0.4487141,"\int \frac{\cot ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{b \left(15 a^2 b+4 a^3-32 a b^2-8 b^3\right)}{8 a^2 f (a+b)^4 \sqrt{a+b \sec ^2(e+f x)}}+\frac{b \left(12 a^2+39 a b-8 b^2\right)}{24 a f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\left(8 a^2+36 a b+63 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{9/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{\cot ^4(e+f x)}{4 f (a+b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}+\frac{(4 a+11 b) \cot ^2(e+f x)}{8 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^{3/2}}","\frac{b \left(15 a^2 b+4 a^3-32 a b^2-8 b^3\right)}{8 a^2 f (a+b)^4 \sqrt{a+b \sec ^2(e+f x)}}+\frac{b \left(12 a^2+39 a b-8 b^2\right)}{24 a f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\left(8 a^2+36 a b+63 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{9/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{\cot ^4(e+f x)}{4 f (a+b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}+\frac{(4 a+11 b) \cot ^2(e+f x)}{8 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^{3/2}}",1,"ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f) - ((8*a^2 + 36*a*b + 63*b^2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(9/2)*f) + (b*(12*a^2 + 39*a*b - 8*b^2))/(24*a*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^(3/2)) + ((4*a + 11*b)*Cot[e + f*x]^2)/(8*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^(3/2)) - Cot[e + f*x]^4/(4*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)) + (b*(4*a^3 + 15*a^2*b - 32*a*b^2 - 8*b^3))/(8*a^2*(a + b)^4*f*Sqrt[a + b*Sec[e + f*x]^2])","A",11,8,25,0.3200,1,"{4139, 446, 103, 151, 152, 156, 63, 208}"
434,1,157,0,0.3446523,"\int \frac{\tan ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\left(\frac{1}{a^2}-\frac{1}{b^2}\right) \tan (e+f x)}{f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{b^{5/2} f}-\frac{(a+b) \tan ^3(e+f x)}{3 a b f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","\frac{\left(\frac{1}{a^2}-\frac{1}{b^2}\right) \tan (e+f x)}{f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{b^{5/2} f}-\frac{(a+b) \tan ^3(e+f x)}{3 a b f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"-(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f)) + ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(b^(5/2)*f) - ((a + b)*Tan[e + f*x]^3)/(3*a*b*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + ((a^(-2) - b^(-2))*Tan[e + f*x])/(f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",9,9,25,0.3600,1,"{4141, 1975, 470, 578, 523, 217, 206, 377, 203}"
435,1,120,0,0.2667934,"\int \frac{\tan ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}+\frac{(a-3 b) \tan (e+f x)}{3 a^2 b f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{(a+b) \tan (e+f x)}{3 a b f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}+\frac{(a-3 b) \tan (e+f x)}{3 a^2 b f \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{(a+b) \tan (e+f x)}{3 a b f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f) - ((a + b)*Tan[e + f*x])/(3*a*b*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + ((a - 3*b)*Tan[e + f*x])/(3*a^2*b*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",7,7,25,0.2800,1,"{4141, 1975, 470, 527, 12, 377, 203}"
436,1,119,0,0.2572747,"\int \frac{\tan ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}+\frac{(2 a+3 b) \tan (e+f x)}{3 a^2 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\tan (e+f x)}{3 a f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}+\frac{(2 a+3 b) \tan (e+f x)}{3 a^2 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\tan (e+f x)}{3 a f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"-(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f)) + Tan[e + f*x]/(3*a*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + ((2*a + 3*b)*Tan[e + f*x])/(3*a^2*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",7,7,25,0.2800,1,"{4141, 1975, 471, 527, 12, 377, 203}"
437,1,125,0,0.0989862,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[(a + b*Sec[e + f*x]^2)^(-5/2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}-\frac{b (5 a+3 b) \tan (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{b \tan (e+f x)}{3 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}-\frac{b (5 a+3 b) \tan (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{b \tan (e+f x)}{3 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f) - (b*Tan[e + f*x])/(3*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(5*a + 3*b)*Tan[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])","A",6,6,16,0.3750,1,"{4128, 414, 527, 12, 377, 203}"
438,1,174,0,0.3779971,"\int \frac{\cot ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}-\frac{(a-3 b) (3 a+b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 a^2 f (a+b)^3}-\frac{b (7 a+3 b) \cot (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{b \cot (e+f x)}{3 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}-\frac{(a-3 b) (3 a+b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 a^2 f (a+b)^3}-\frac{b (7 a+3 b) \cot (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{b \cot (e+f x)}{3 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"-(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f)) - (b*Cot[e + f*x])/(3*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(7*a + 3*b)*Cot[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - ((a - 3*b)*(3*a + b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*a^2*(a + b)^3*f)","A",8,8,25,0.3200,1,"{4141, 1975, 472, 579, 583, 12, 377, 203}"
439,1,236,0,0.4817325,"\int \frac{\cot ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{\left(a^2-10 a b-3 b^2\right) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 a^2 f (a+b)^3}+\frac{(a-b) \left(3 a^2+14 a b+3 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 a^2 f (a+b)^4}+\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}-\frac{b (3 a+b) \cot ^3(e+f x)}{a^2 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{b \cot ^3(e+f x)}{3 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","-\frac{\left(a^2-10 a b-3 b^2\right) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 a^2 f (a+b)^3}+\frac{(a-b) \left(3 a^2+14 a b+3 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 a^2 f (a+b)^4}+\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}-\frac{b (3 a+b) \cot ^3(e+f x)}{a^2 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{b \cot ^3(e+f x)}{3 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f) - (b*Cot[e + f*x]^3)/(3*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(3*a + b)*Cot[e + f*x]^3)/(a^2*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + ((a - b)*(3*a^2 + 14*a*b + 3*b^2)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*a^2*(a + b)^4*f) - ((a^2 - 10*a*b - 3*b^2)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*a^2*(a + b)^3*f)","A",9,8,25,0.3200,1,"{4141, 1975, 472, 579, 583, 12, 377, 203}"
440,1,315,0,0.604344,"\int \frac{\cot ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{\left(a^2-20 a b-5 b^2\right) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 a^2 f (a+b)^3}+\frac{\left(19 a^2 b+5 a^3-65 a b^2-15 b^3\right) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 a^2 f (a+b)^4}-\frac{\left(128 a^2 b^2+70 a^3 b+15 a^4-70 a b^3-15 b^4\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 a^2 f (a+b)^5}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}-\frac{b (11 a+3 b) \cot ^5(e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{b \cot ^5(e+f x)}{3 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}","-\frac{\left(a^2-20 a b-5 b^2\right) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 a^2 f (a+b)^3}+\frac{\left(19 a^2 b+5 a^3-65 a b^2-15 b^3\right) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 a^2 f (a+b)^4}-\frac{\left(128 a^2 b^2+70 a^3 b+15 a^4-70 a b^3-15 b^4\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 a^2 f (a+b)^5}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{a^{5/2} f}-\frac{b (11 a+3 b) \cot ^5(e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{b \cot ^5(e+f x)}{3 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"-(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f)) - (b*Cot[e + f*x]^5)/(3*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(11*a + 3*b)*Cot[e + f*x]^5)/(3*a^2*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - ((15*a^4 + 70*a^3*b + 128*a^2*b^2 - 70*a*b^3 - 15*b^4)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*a^2*(a + b)^5*f) + ((5*a^3 + 19*a^2*b - 65*a*b^2 - 15*b^3)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*a^2*(a + b)^4*f) - ((a^2 - 20*a*b - 5*b^2)*Cot[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(5*a^2*(a + b)^3*f)","A",10,8,25,0.3200,1,"{4141, 1975, 472, 579, 583, 12, 377, 203}"
441,1,105,0,0.2023282,"\int \left(a+b \sec ^2(e+f x)\right)^p (d \tan (e+f x))^m \, dx","Int[(a + b*Sec[e + f*x]^2)^p*(d*Tan[e + f*x])^m,x]","\frac{(d \tan (e+f x))^{m+1} \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{m+1}{2};1,-p;\frac{m+3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{d f (m+1)}","\frac{(d \tan (e+f x))^{m+1} \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{m+1}{2};1,-p;\frac{m+3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{d f (m+1)}",1,"(AppellF1[(1 + m)/2, 1, -p, (3 + m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(d*Tan[e + f*x])^(1 + m)*(a + b + b*Tan[e + f*x]^2)^p)/(d*f*(1 + m)*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",4,4,25,0.1600,1,"{4141, 1975, 511, 510}"
442,1,122,0,0.1475603,"\int \left(a+b \sec ^2(e+f x)\right)^p \tan ^5(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^5,x]","-\frac{(a+2 b) \left(a+b \sec ^2(e+f x)\right)^{p+1}}{2 b^2 f (p+1)}+\frac{\left(a+b \sec ^2(e+f x)\right)^{p+2}}{2 b^2 f (p+2)}-\frac{\left(a+b \sec ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)}{a}+1\right)}{2 a f (p+1)}","-\frac{(a+2 b) \left(a+b \sec ^2(e+f x)\right)^{p+1}}{2 b^2 f (p+1)}+\frac{\left(a+b \sec ^2(e+f x)\right)^{p+2}}{2 b^2 f (p+2)}-\frac{\left(a+b \sec ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)}{a}+1\right)}{2 a f (p+1)}",1,"-((a + 2*b)*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*b^2*f*(1 + p)) - (Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sec[e + f*x]^2)/a]*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*a*f*(1 + p)) + (a + b*Sec[e + f*x]^2)^(2 + p)/(2*b^2*f*(2 + p))","A",5,4,23,0.1739,1,"{4139, 446, 88, 65}"
443,1,86,0,0.0906664,"\int \left(a+b \sec ^2(e+f x)\right)^p \tan ^3(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^3,x]","\frac{\left(a+b \sec ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)}{a}+1\right)}{2 a f (p+1)}+\frac{\left(a+b \sec ^2(e+f x)\right)^{p+1}}{2 b f (p+1)}","\frac{\left(a+b \sec ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)}{a}+1\right)}{2 a f (p+1)}+\frac{\left(a+b \sec ^2(e+f x)\right)^{p+1}}{2 b f (p+1)}",1,"(a + b*Sec[e + f*x]^2)^(1 + p)/(2*b*f*(1 + p)) + (Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sec[e + f*x]^2)/a]*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*a*f*(1 + p))","A",4,4,23,0.1739,1,"{4139, 446, 80, 65}"
444,1,54,0,0.0556833,"\int \left(a+b \sec ^2(e+f x)\right)^p \tan (e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x],x]","-\frac{\left(a+b \sec ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)}{a}+1\right)}{2 a f (p+1)}","-\frac{\left(a+b \sec ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)}{a}+1\right)}{2 a f (p+1)}",1,"-(Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sec[e + f*x]^2)/a]*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*a*f*(1 + p))","A",3,3,21,0.1429,1,"{4139, 266, 65}"
445,1,114,0,0.1238277,"\int \cot (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Cot[e + f*x]*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\left(a+b \sec ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)}{a}+1\right)}{2 a f (p+1)}-\frac{\left(a+b \sec ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)+a}{a+b}\right)}{2 f (p+1) (a+b)}","\frac{\left(a+b \sec ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)}{a}+1\right)}{2 a f (p+1)}-\frac{\left(a+b \sec ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)+a}{a+b}\right)}{2 f (p+1) (a+b)}",1,"-(Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sec[e + f*x]^2)/(a + b)]*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*(a + b)*f*(1 + p)) + (Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sec[e + f*x]^2)/a]*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*a*f*(1 + p))","A",5,5,21,0.2381,1,"{4139, 446, 86, 68, 65}"
446,1,157,0,0.174266,"\int \cot ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Cot[e + f*x]^3*(a + b*Sec[e + f*x]^2)^p,x]","\frac{(a-b p+b) \left(a+b \sec ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)+a}{a+b}\right)}{2 f (p+1) (a+b)^2}-\frac{\left(a+b \sec ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)}{a}+1\right)}{2 a f (p+1)}-\frac{\cot ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^{p+1}}{2 f (a+b)}","\frac{(a-b p+b) \left(a+b \sec ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)+a}{a+b}\right)}{2 f (p+1) (a+b)^2}-\frac{\left(a+b \sec ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)}{a}+1\right)}{2 a f (p+1)}-\frac{\cot ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^{p+1}}{2 f (a+b)}",1,"-(Cot[e + f*x]^2*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*(a + b)*f) + ((a + b - b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sec[e + f*x]^2)/(a + b)]*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*(a + b)^2*f*(1 + p)) - (Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sec[e + f*x]^2)/a]*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*a*f*(1 + p))","A",6,6,23,0.2609,1,"{4139, 446, 103, 156, 68, 65}"
447,1,88,0,0.1485733,"\int \left(a+b \sec ^2(e+f x)\right)^p \tan ^4(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^4,x]","\frac{\tan ^5(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{5}{2};1,-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{5 f}","\frac{\tan ^5(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{5}{2};1,-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{5 f}",1,"(AppellF1[5/2, 1, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^5*(a + b + b*Tan[e + f*x]^2)^p)/(5*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",4,4,23,0.1739,1,"{4141, 1975, 511, 510}"
448,1,88,0,0.1466444,"\int \left(a+b \sec ^2(e+f x)\right)^p \tan ^2(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^2,x]","\frac{\tan ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{3}{2};1,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{3 f}","\frac{\tan ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{3}{2};1,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{3 f}",1,"(AppellF1[3/2, 1, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^p)/(3*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",4,4,23,0.1739,1,"{4141, 1975, 511, 510}"
449,1,83,0,0.0533354,"\int \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[(a + b*Sec[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}",1,"(AppellF1[1/2, 1, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",3,3,14,0.2143,1,"{4128, 430, 429}"
450,1,84,0,0.1431795,"\int \cot ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Cot[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p,x]","-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(-\frac{1}{2};1,-p;\frac{1}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}","-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(-\frac{1}{2};1,-p;\frac{1}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f}",1,"-((AppellF1[-1/2, 1, -p, 1/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Cot[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p))","A",4,4,23,0.1739,1,"{4141, 1975, 511, 510}"
451,1,88,0,0.1461085,"\int \cot ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Int[Cot[e + f*x]^4*(a + b*Sec[e + f*x]^2)^p,x]","-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(-\frac{3}{2};1,-p;-\frac{1}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{3 f}","-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)+b\right)^p \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} F_1\left(-\frac{3}{2};1,-p;-\frac{1}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{3 f}",1,"-(AppellF1[-3/2, 1, -p, -1/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Cot[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^p)/(3*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",4,4,23,0.1739,1,"{4141, 1975, 511, 510}"
452,1,92,0,0.0689336,"\int \left(a+b \sec ^3(e+f x)\right) \tan ^5(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^3)*Tan[e + f*x]^5,x]","\frac{a \sec ^4(e+f x)}{4 f}-\frac{a \sec ^2(e+f x)}{f}-\frac{a \log (\cos (e+f x))}{f}+\frac{b \sec ^7(e+f x)}{7 f}-\frac{2 b \sec ^5(e+f x)}{5 f}+\frac{b \sec ^3(e+f x)}{3 f}","\frac{a \sec ^4(e+f x)}{4 f}-\frac{a \sec ^2(e+f x)}{f}-\frac{a \log (\cos (e+f x))}{f}+\frac{b \sec ^7(e+f x)}{7 f}-\frac{2 b \sec ^5(e+f x)}{5 f}+\frac{b \sec ^3(e+f x)}{3 f}",1,"-((a*Log[Cos[e + f*x]])/f) - (a*Sec[e + f*x]^2)/f + (b*Sec[e + f*x]^3)/(3*f) + (a*Sec[e + f*x]^4)/(4*f) - (2*b*Sec[e + f*x]^5)/(5*f) + (b*Sec[e + f*x]^7)/(7*f)","A",3,2,21,0.09524,1,"{4138, 1802}"
453,1,61,0,0.0544285,"\int \left(a+b \sec ^3(e+f x)\right) \tan ^3(e+f x) \, dx","Int[(a + b*Sec[e + f*x]^3)*Tan[e + f*x]^3,x]","\frac{a \sec ^2(e+f x)}{2 f}+\frac{a \log (\cos (e+f x))}{f}+\frac{b \sec ^5(e+f x)}{5 f}-\frac{b \sec ^3(e+f x)}{3 f}","\frac{a \sec ^2(e+f x)}{2 f}+\frac{a \log (\cos (e+f x))}{f}+\frac{b \sec ^5(e+f x)}{5 f}-\frac{b \sec ^3(e+f x)}{3 f}",1,"(a*Log[Cos[e + f*x]])/f + (a*Sec[e + f*x]^2)/(2*f) - (b*Sec[e + f*x]^3)/(3*f) + (b*Sec[e + f*x]^5)/(5*f)","A",3,2,21,0.09524,1,"{4138, 1802}"
454,1,30,0,0.023017,"\int \left(a+b \sec ^3(e+f x)\right) \tan (e+f x) \, dx","Int[(a + b*Sec[e + f*x]^3)*Tan[e + f*x],x]","\frac{b \sec ^3(e+f x)}{3 f}-\frac{a \log (\cos (e+f x))}{f}","\frac{b \sec ^3(e+f x)}{3 f}-\frac{a \log (\cos (e+f x))}{f}",1,"-((a*Log[Cos[e + f*x]])/f) + (b*Sec[e + f*x]^3)/(3*f)","A",3,2,19,0.1053,1,"{4138, 14}"
455,1,54,0,0.069843,"\int \cot (e+f x) \left(a+b \sec ^3(e+f x)\right) \, dx","Int[Cot[e + f*x]*(a + b*Sec[e + f*x]^3),x]","\frac{(a+b) \log (1-\cos (e+f x))}{2 f}+\frac{(a-b) \log (\cos (e+f x)+1)}{2 f}+\frac{b \sec (e+f x)}{f}","\frac{(a+b) \log (1-\cos (e+f x))}{2 f}+\frac{(a-b) \log (\cos (e+f x)+1)}{2 f}+\frac{b \sec (e+f x)}{f}",1,"((a + b)*Log[1 - Cos[e + f*x]])/(2*f) + ((a - b)*Log[1 + Cos[e + f*x]])/(2*f) + (b*Sec[e + f*x])/f","A",3,2,19,0.1053,1,"{4138, 1802}"
456,1,72,0,0.0626343,"\int \cot ^3(e+f x) \left(a+b \sec ^3(e+f x)\right) \, dx","Int[Cot[e + f*x]^3*(a + b*Sec[e + f*x]^3),x]","-\frac{(2 a-b) \log (1-\cos (e+f x))}{4 f}-\frac{(2 a+b) \log (\cos (e+f x)+1)}{4 f}-\frac{\csc ^2(e+f x) (a+b \cos (e+f x))}{2 f}","-\frac{(2 a-b) \log (1-\cos (e+f x))}{4 f}-\frac{(2 a+b) \log (\cos (e+f x)+1)}{4 f}-\frac{\csc ^2(e+f x) (a+b \cos (e+f x))}{2 f}",1,"-((a + b*Cos[e + f*x])*Csc[e + f*x]^2)/(2*f) - ((2*a - b)*Log[1 - Cos[e + f*x]])/(4*f) - ((2*a + b)*Log[1 + Cos[e + f*x]])/(4*f)","A",5,4,21,0.1905,1,"{4138, 1814, 633, 31}"
457,1,219,0,0.320877,"\int \frac{\tan ^5(e+f x)}{a+b \sec ^3(e+f x)} \, dx","Int[Tan[e + f*x]^5/(a + b*Sec[e + f*x]^3),x]","\frac{\left(a^{2/3}-2 b^{2/3}\right) \log \left(a^{2/3} \cos ^2(e+f x)-\sqrt[3]{a} \sqrt[3]{b} \cos (e+f x)+b^{2/3}\right)}{6 \sqrt[3]{a} b^{4/3} f}-\frac{\left(a^{2/3}-2 b^{2/3}\right) \log \left(\sqrt[3]{a} \cos (e+f x)+\sqrt[3]{b}\right)}{3 \sqrt[3]{a} b^{4/3} f}-\frac{\left(a^{2/3}+2 b^{2/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{b}-2 \sqrt[3]{a} \cos (e+f x)}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{a} b^{4/3} f}-\frac{\log \left(a \cos ^3(e+f x)+b\right)}{3 a f}+\frac{\sec (e+f x)}{b f}","\frac{\left(a^{2/3}-2 b^{2/3}\right) \log \left(a^{2/3} \cos ^2(e+f x)-\sqrt[3]{a} \sqrt[3]{b} \cos (e+f x)+b^{2/3}\right)}{6 \sqrt[3]{a} b^{4/3} f}-\frac{\left(a^{2/3}-2 b^{2/3}\right) \log \left(\sqrt[3]{a} \cos (e+f x)+\sqrt[3]{b}\right)}{3 \sqrt[3]{a} b^{4/3} f}-\frac{\left(a^{2/3}+2 b^{2/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{b}-2 \sqrt[3]{a} \cos (e+f x)}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{a} b^{4/3} f}-\frac{\log \left(a \cos ^3(e+f x)+b\right)}{3 a f}+\frac{\sec (e+f x)}{b f}",1,"-(((a^(2/3) + 2*b^(2/3))*ArcTan[(b^(1/3) - 2*a^(1/3)*Cos[e + f*x])/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(1/3)*b^(4/3)*f)) - ((a^(2/3) - 2*b^(2/3))*Log[b^(1/3) + a^(1/3)*Cos[e + f*x]])/(3*a^(1/3)*b^(4/3)*f) + ((a^(2/3) - 2*b^(2/3))*Log[b^(2/3) - a^(1/3)*b^(1/3)*Cos[e + f*x] + a^(2/3)*Cos[e + f*x]^2])/(6*a^(1/3)*b^(4/3)*f) - Log[b + a*Cos[e + f*x]^3]/(3*a*f) + Sec[e + f*x]/(b*f)","A",11,10,23,0.4348,1,"{4138, 1834, 1871, 1860, 31, 634, 617, 204, 628, 260}"
458,1,166,0,0.1467331,"\int \frac{\tan ^3(e+f x)}{a+b \sec ^3(e+f x)} \, dx","Int[Tan[e + f*x]^3/(a + b*Sec[e + f*x]^3),x]","\frac{\log \left(a^{2/3} \cos ^2(e+f x)-\sqrt[3]{a} \sqrt[3]{b} \cos (e+f x)+b^{2/3}\right)}{6 \sqrt[3]{a} b^{2/3} f}-\frac{\log \left(\sqrt[3]{a} \cos (e+f x)+\sqrt[3]{b}\right)}{3 \sqrt[3]{a} b^{2/3} f}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{b}-2 \sqrt[3]{a} \cos (e+f x)}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{a} b^{2/3} f}+\frac{\log \left(a \cos ^3(e+f x)+b\right)}{3 a f}","\frac{\log \left(a^{2/3} \cos ^2(e+f x)-\sqrt[3]{a} \sqrt[3]{b} \cos (e+f x)+b^{2/3}\right)}{6 \sqrt[3]{a} b^{2/3} f}-\frac{\log \left(\sqrt[3]{a} \cos (e+f x)+\sqrt[3]{b}\right)}{3 \sqrt[3]{a} b^{2/3} f}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{b}-2 \sqrt[3]{a} \cos (e+f x)}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{a} b^{2/3} f}+\frac{\log \left(a \cos ^3(e+f x)+b\right)}{3 a f}",1,"ArcTan[(b^(1/3) - 2*a^(1/3)*Cos[e + f*x])/(Sqrt[3]*b^(1/3))]/(Sqrt[3]*a^(1/3)*b^(2/3)*f) - Log[b^(1/3) + a^(1/3)*Cos[e + f*x]]/(3*a^(1/3)*b^(2/3)*f) + Log[b^(2/3) - a^(1/3)*b^(1/3)*Cos[e + f*x] + a^(2/3)*Cos[e + f*x]^2]/(6*a^(1/3)*b^(2/3)*f) + Log[b + a*Cos[e + f*x]^3]/(3*a*f)","A",9,9,23,0.3913,1,"{4138, 1871, 200, 31, 634, 617, 204, 628, 260}"
459,1,23,0,0.0307096,"\int \frac{\tan (e+f x)}{a+b \sec ^3(e+f x)} \, dx","Int[Tan[e + f*x]/(a + b*Sec[e + f*x]^3),x]","-\frac{\log \left(a \cos ^3(e+f x)+b\right)}{3 a f}","-\frac{\log \left(a \cos ^3(e+f x)+b\right)}{3 a f}",1,"-Log[b + a*Cos[e + f*x]^3]/(3*a*f)","A",2,2,21,0.09524,1,"{4138, 260}"
460,1,295,0,0.5173645,"\int \frac{\cot (e+f x)}{a+b \sec ^3(e+f x)} \, dx","Int[Cot[e + f*x]/(a + b*Sec[e + f*x]^3),x]","-\frac{b^2 \log \left(a \cos ^3(e+f x)+b\right)}{3 a f \left(a^2-b^2\right)}+\frac{b^{2/3} \left(a^{2/3}+b^{2/3}\right) \log \left(a^{2/3} \cos ^2(e+f x)-\sqrt[3]{a} \sqrt[3]{b} \cos (e+f x)+b^{2/3}\right)}{6 \sqrt[3]{a} f \left(a^2-b^2\right)}-\frac{b^{2/3} \left(a^{2/3}+b^{2/3}\right) \log \left(\sqrt[3]{a} \cos (e+f x)+\sqrt[3]{b}\right)}{3 \sqrt[3]{a} f \left(a^2-b^2\right)}-\frac{b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{b}-2 \sqrt[3]{a} \cos (e+f x)}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{a} f \left(a^{2/3} b^{2/3}+a^{4/3}+b^{4/3}\right)}+\frac{\log (1-\cos (e+f x))}{2 f (a+b)}+\frac{\log (\cos (e+f x)+1)}{2 f (a-b)}","-\frac{b^2 \log \left(a \cos ^3(e+f x)+b\right)}{3 a f \left(a^2-b^2\right)}+\frac{b^{2/3} \left(a^{2/3}+b^{2/3}\right) \log \left(a^{2/3} \cos ^2(e+f x)-\sqrt[3]{a} \sqrt[3]{b} \cos (e+f x)+b^{2/3}\right)}{6 \sqrt[3]{a} f \left(a^2-b^2\right)}-\frac{b^{2/3} \left(a^{2/3}+b^{2/3}\right) \log \left(\sqrt[3]{a} \cos (e+f x)+\sqrt[3]{b}\right)}{3 \sqrt[3]{a} f \left(a^2-b^2\right)}-\frac{b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{b}-2 \sqrt[3]{a} \cos (e+f x)}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{a} f \left(a^{2/3} b^{2/3}+a^{4/3}+b^{4/3}\right)}+\frac{\log (1-\cos (e+f x))}{2 f (a+b)}+\frac{\log (\cos (e+f x)+1)}{2 f (a-b)}",1,"-((b^(2/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*Cos[e + f*x])/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(1/3)*(a^(4/3) + a^(2/3)*b^(2/3) + b^(4/3))*f)) + Log[1 - Cos[e + f*x]]/(2*(a + b)*f) + Log[1 + Cos[e + f*x]]/(2*(a - b)*f) - ((a^(2/3) + b^(2/3))*b^(2/3)*Log[b^(1/3) + a^(1/3)*Cos[e + f*x]])/(3*a^(1/3)*(a^2 - b^2)*f) + ((a^(2/3) + b^(2/3))*b^(2/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*Cos[e + f*x] + a^(2/3)*Cos[e + f*x]^2])/(6*a^(1/3)*(a^2 - b^2)*f) - (b^2*Log[b + a*Cos[e + f*x]^3])/(3*a*(a^2 - b^2)*f)","A",11,10,21,0.4762,1,"{4138, 6725, 1871, 1860, 31, 634, 617, 204, 628, 260}"
461,1,393,0,0.6316674,"\int \frac{\cot ^3(e+f x)}{a+b \sec ^3(e+f x)} \, dx","Int[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^3),x]","-\frac{b^2 \left(2 a^2+b^2\right) \log \left(a \cos ^3(e+f x)+b\right)}{3 a f \left(a^2-b^2\right)^2}+\frac{b^{4/3} \left(3 a^{2/3} b^{4/3}+a^2+2 b^2\right) \log \left(a^{2/3} \cos ^2(e+f x)-\sqrt[3]{a} \sqrt[3]{b} \cos (e+f x)+b^{2/3}\right)}{6 \sqrt[3]{a} f \left(a^2-b^2\right)^2}-\frac{b^{4/3} \left(3 a^{2/3} b^{4/3}+a^2+2 b^2\right) \log \left(\sqrt[3]{a} \cos (e+f x)+\sqrt[3]{b}\right)}{3 \sqrt[3]{a} f \left(a^2-b^2\right)^2}+\frac{b^{4/3} \left(-3 a^{2/3} b^{4/3}+a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{b}-2 \sqrt[3]{a} \cos (e+f x)}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{a} f \left(a^2-b^2\right)^2}-\frac{1}{4 f (a+b) (1-\cos (e+f x))}-\frac{1}{4 f (a-b) (\cos (e+f x)+1)}-\frac{(2 a+5 b) \log (1-\cos (e+f x))}{4 f (a+b)^2}-\frac{(2 a-5 b) \log (\cos (e+f x)+1)}{4 f (a-b)^2}","-\frac{b^2 \left(2 a^2+b^2\right) \log \left(a \cos ^3(e+f x)+b\right)}{3 a f \left(a^2-b^2\right)^2}+\frac{b^{4/3} \left(3 a^{2/3} b^{4/3}+a^2+2 b^2\right) \log \left(a^{2/3} \cos ^2(e+f x)-\sqrt[3]{a} \sqrt[3]{b} \cos (e+f x)+b^{2/3}\right)}{6 \sqrt[3]{a} f \left(a^2-b^2\right)^2}-\frac{b^{4/3} \left(3 a^{2/3} b^{4/3}+a^2+2 b^2\right) \log \left(\sqrt[3]{a} \cos (e+f x)+\sqrt[3]{b}\right)}{3 \sqrt[3]{a} f \left(a^2-b^2\right)^2}+\frac{b^{4/3} \left(-3 a^{2/3} b^{4/3}+a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{b}-2 \sqrt[3]{a} \cos (e+f x)}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{a} f \left(a^2-b^2\right)^2}-\frac{1}{4 f (a+b) (1-\cos (e+f x))}-\frac{1}{4 f (a-b) (\cos (e+f x)+1)}-\frac{(2 a+5 b) \log (1-\cos (e+f x))}{4 f (a+b)^2}-\frac{(2 a-5 b) \log (\cos (e+f x)+1)}{4 f (a-b)^2}",1,"(b^(4/3)*(a^2 - 3*a^(2/3)*b^(4/3) + 2*b^2)*ArcTan[(b^(1/3) - 2*a^(1/3)*Cos[e + f*x])/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(1/3)*(a^2 - b^2)^2*f) - 1/(4*(a + b)*f*(1 - Cos[e + f*x])) - 1/(4*(a - b)*f*(1 + Cos[e + f*x])) - ((2*a + 5*b)*Log[1 - Cos[e + f*x]])/(4*(a + b)^2*f) - ((2*a - 5*b)*Log[1 + Cos[e + f*x]])/(4*(a - b)^2*f) - (b^(4/3)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*Log[b^(1/3) + a^(1/3)*Cos[e + f*x]])/(3*a^(1/3)*(a^2 - b^2)^2*f) + (b^(4/3)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*Log[b^(2/3) - a^(1/3)*b^(1/3)*Cos[e + f*x] + a^(2/3)*Cos[e + f*x]^2])/(6*a^(1/3)*(a^2 - b^2)^2*f) - (b^2*(2*a^2 + b^2)*Log[b + a*Cos[e + f*x]^3])/(3*a*(a^2 - b^2)^2*f)","A",11,10,23,0.4348,1,"{4138, 6725, 1871, 1860, 31, 634, 617, 204, 628, 260}"
462,0,0,0,0.060413,"\int \left(a+b (c \sec (e+f x))^n\right)^p (d \tan (e+f x))^m \, dx","Int[(a + b*(c*Sec[e + f*x])^n)^p*(d*Tan[e + f*x])^m,x]","\int \left(a+b (c \sec (e+f x))^n\right)^p (d \tan (e+f x))^m \, dx","\text{Int}\left((d \tan (e+f x))^m \left(a+b (c \sec (e+f x))^n\right)^p,x\right)",0,"Defer[Int][(a + b*(c*Sec[e + f*x])^n)^p*(d*Tan[e + f*x])^m, x]","A",0,0,0,0,-1,"{}"
463,1,226,0,0.5244995,"\int \left(a+b (c \sec (e+f x))^n\right)^p \tan ^5(e+f x) \, dx","Int[(a + b*(c*Sec[e + f*x])^n)^p*Tan[e + f*x]^5,x]","\frac{\sec ^4(e+f x) \left(a+b (c \sec (e+f x))^n\right)^p \left(\frac{b (c \sec (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{4}{n},-p;\frac{n+4}{n};-\frac{b (c \sec (e+f x))^n}{a}\right)}{4 f}-\frac{\sec ^2(e+f x) \left(a+b (c \sec (e+f x))^n\right)^p \left(\frac{b (c \sec (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{2}{n},-p;\frac{n+2}{n};-\frac{b (c \sec (e+f x))^n}{a}\right)}{f}-\frac{\left(a+b (c \sec (e+f x))^n\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b (c \sec (e+f x))^n}{a}+1\right)}{a f n (p+1)}","\frac{\sec ^4(e+f x) \left(a+b (c \sec (e+f x))^n\right)^p \left(\frac{b (c \sec (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{4}{n},-p;\frac{n+4}{n};-\frac{b (c \sec (e+f x))^n}{a}\right)}{4 f}-\frac{\sec ^2(e+f x) \left(a+b (c \sec (e+f x))^n\right)^p \left(\frac{b (c \sec (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{2}{n},-p;\frac{n+2}{n};-\frac{b (c \sec (e+f x))^n}{a}\right)}{f}-\frac{\left(a+b (c \sec (e+f x))^n\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b (c \sec (e+f x))^n}{a}+1\right)}{a f n (p+1)}",1,"-((Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*(c*Sec[e + f*x])^n)/a]*(a + b*(c*Sec[e + f*x])^n)^(1 + p))/(a*f*n*(1 + p))) - (Hypergeometric2F1[2/n, -p, (2 + n)/n, -((b*(c*Sec[e + f*x])^n)/a)]*Sec[e + f*x]^2*(a + b*(c*Sec[e + f*x])^n)^p)/(f*(1 + (b*(c*Sec[e + f*x])^n)/a)^p) + (Hypergeometric2F1[4/n, -p, (4 + n)/n, -((b*(c*Sec[e + f*x])^n)/a)]*Sec[e + f*x]^4*(a + b*(c*Sec[e + f*x])^n)^p)/(4*f*(1 + (b*(c*Sec[e + f*x])^n)/a)^p)","A",15,8,25,0.3200,1,"{4139, 6742, 367, 12, 266, 65, 365, 364}"
464,1,143,0,0.2939151,"\int \left(a+b (c \sec (e+f x))^n\right)^p \tan ^3(e+f x) \, dx","Int[(a + b*(c*Sec[e + f*x])^n)^p*Tan[e + f*x]^3,x]","\frac{\sec ^2(e+f x) \left(a+b (c \sec (e+f x))^n\right)^p \left(\frac{b (c \sec (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{2}{n},-p;\frac{n+2}{n};-\frac{b (c \sec (e+f x))^n}{a}\right)}{2 f}+\frac{\left(a+b (c \sec (e+f x))^n\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b (c \sec (e+f x))^n}{a}+1\right)}{a f n (p+1)}","\frac{\sec ^2(e+f x) \left(a+b (c \sec (e+f x))^n\right)^p \left(\frac{b (c \sec (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{2}{n},-p;\frac{n+2}{n};-\frac{b (c \sec (e+f x))^n}{a}\right)}{2 f}+\frac{\left(a+b (c \sec (e+f x))^n\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b (c \sec (e+f x))^n}{a}+1\right)}{a f n (p+1)}",1,"(Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*(c*Sec[e + f*x])^n)/a]*(a + b*(c*Sec[e + f*x])^n)^(1 + p))/(a*f*n*(1 + p)) + (Hypergeometric2F1[2/n, -p, (2 + n)/n, -((b*(c*Sec[e + f*x])^n)/a)]*Sec[e + f*x]^2*(a + b*(c*Sec[e + f*x])^n)^p)/(2*f*(1 + (b*(c*Sec[e + f*x])^n)/a)^p)","A",11,8,25,0.3200,1,"{4139, 6742, 367, 12, 266, 65, 365, 364}"
465,1,59,0,0.0769761,"\int \left(a+b (c \sec (e+f x))^n\right)^p \tan (e+f x) \, dx","Int[(a + b*(c*Sec[e + f*x])^n)^p*Tan[e + f*x],x]","-\frac{\left(a+b (c \sec (e+f x))^n\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b (c \sec (e+f x))^n}{a}+1\right)}{a f n (p+1)}","-\frac{\left(a+b (c \sec (e+f x))^n\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b (c \sec (e+f x))^n}{a}+1\right)}{a f n (p+1)}",1,"-((Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*(c*Sec[e + f*x])^n)/a]*(a + b*(c*Sec[e + f*x])^n)^(1 + p))/(a*f*n*(1 + p)))","A",5,5,23,0.2174,1,"{4139, 367, 12, 266, 65}"
466,0,0,0,0.0443204,"\int \cot (e+f x) \left(a+b (c \sec (e+f x))^n\right)^p \, dx","Int[Cot[e + f*x]*(a + b*(c*Sec[e + f*x])^n)^p,x]","\int \cot (e+f x) \left(a+b (c \sec (e+f x))^n\right)^p \, dx","\text{Int}\left(\cot (e+f x) \left(a+b (c \sec (e+f x))^n\right)^p,x\right)",0,"Defer[Int][Cot[e + f*x]*(a + b*(c*Sec[e + f*x])^n)^p, x]","A",0,0,0,0,-1,"{}"
467,0,0,0,0.0560734,"\int \cot ^3(e+f x) \left(a+b (c \sec (e+f x))^n\right)^p \, dx","Int[Cot[e + f*x]^3*(a + b*(c*Sec[e + f*x])^n)^p,x]","\int \cot ^3(e+f x) \left(a+b (c \sec (e+f x))^n\right)^p \, dx","\text{Int}\left(\cot ^3(e+f x) \left(a+b (c \sec (e+f x))^n\right)^p,x\right)",0,"Defer[Int][Cot[e + f*x]^3*(a + b*(c*Sec[e + f*x])^n)^p, x]","A",0,0,0,0,-1,"{}"
468,0,0,0,0.055236,"\int \left(a+b (c \sec (e+f x))^n\right)^p \tan ^2(e+f x) \, dx","Int[(a + b*(c*Sec[e + f*x])^n)^p*Tan[e + f*x]^2,x]","\int \left(a+b (c \sec (e+f x))^n\right)^p \tan ^2(e+f x) \, dx","\text{Int}\left(\tan ^2(e+f x) \left(a+b (c \sec (e+f x))^n\right)^p,x\right)",0,"Defer[Int][(a + b*(c*Sec[e + f*x])^n)^p*Tan[e + f*x]^2, x]","A",0,0,0,0,-1,"{}"
469,0,0,0,0.0152094,"\int \left(a+b (c \sec (e+f x))^n\right)^p \, dx","Int[(a + b*(c*Sec[e + f*x])^n)^p,x]","\int \left(a+b (c \sec (e+f x))^n\right)^p \, dx","\text{Int}\left(\left(a+b (c \sec (e+f x))^n\right)^p,x\right)",0,"Defer[Int][(a + b*(c*Sec[e + f*x])^n)^p, x]","A",0,0,0,0,-1,"{}"
470,0,0,0,0.0553758,"\int \cot ^2(e+f x) \left(a+b (c \sec (e+f x))^n\right)^p \, dx","Int[Cot[e + f*x]^2*(a + b*(c*Sec[e + f*x])^n)^p,x]","\int \cot ^2(e+f x) \left(a+b (c \sec (e+f x))^n\right)^p \, dx","\text{Int}\left(\cot ^2(e+f x) \left(a+b (c \sec (e+f x))^n\right)^p,x\right)",0,"Defer[Int][Cot[e + f*x]^2*(a + b*(c*Sec[e + f*x])^n)^p, x]","A",0,0,0,0,-1,"{}"