1,1,120,83,0.0840092,"\int \left(a+b \sec ^2(e+f x)\right) \sin ^7(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^7,x]","-\frac{35 a \cos (e+f x)}{64 f}+\frac{7 a \cos (3 (e+f x))}{64 f}-\frac{7 a \cos (5 (e+f x))}{320 f}+\frac{a \cos (7 (e+f x))}{448 f}+\frac{19 b \cos (e+f x)}{8 f}-\frac{3 b \cos (3 (e+f x))}{16 f}+\frac{b \cos (5 (e+f x))}{80 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,"(-35*a*Cos[e + f*x])/(64*f) + (19*b*Cos[e + f*x])/(8*f) + (7*a*Cos[3*(e + f*x)])/(64*f) - (3*b*Cos[3*(e + f*x)])/(16*f) - (7*a*Cos[5*(e + f*x)])/(320*f) + (b*Cos[5*(e + f*x)])/(80*f) + (a*Cos[7*(e + f*x)])/(448*f) + (b*Sec[e + f*x])/f","A",1
2,1,88,66,0.0403302,"\int \left(a+b \sec ^2(e+f x)\right) \sin ^5(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^5,x]","-\frac{5 a \cos (e+f x)}{8 f}+\frac{5 a \cos (3 (e+f x))}{48 f}-\frac{a \cos (5 (e+f x))}{80 f}+\frac{7 b \cos (e+f x)}{4 f}-\frac{b \cos (3 (e+f x))}{12 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,"(-5*a*Cos[e + f*x])/(8*f) + (7*b*Cos[e + f*x])/(4*f) + (5*a*Cos[3*(e + f*x)])/(48*f) - (b*Cos[3*(e + f*x)])/(12*f) - (a*Cos[5*(e + f*x)])/(80*f) + (b*Sec[e + f*x])/f","A",1
3,1,53,44,0.030258,"\int \left(a+b \sec ^2(e+f x)\right) \sin ^3(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^3,x]","-\frac{3 a \cos (e+f x)}{4 f}+\frac{a \cos (3 (e+f x))}{12 f}+\frac{b \cos (e+f x)}{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,"(-3*a*Cos[e + f*x])/(4*f) + (b*Cos[e + f*x])/f + (a*Cos[3*(e + f*x)])/(12*f) + (b*Sec[e + f*x])/f","A",1
4,1,35,24,0.0157567,"\int \left(a+b \sec ^2(e+f x)\right) \sin (e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)*Sin[e + f*x],x]","\frac{a \sin (e) \sin (f x)}{f}-\frac{a \cos (e) \cos (f x)}{f}+\frac{b \sec (e+f x)}{f}","\frac{b \sec (e+f x)}{f}-\frac{a \cos (e+f x)}{f}",1,"-((a*Cos[e]*Cos[f*x])/f) + (b*Sec[e + f*x])/f + (a*Sin[e]*Sin[f*x])/f","A",1
5,1,84,27,0.0465001,"\int \csc (e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Csc[e + f*x]*(a + b*Sec[e + f*x]^2),x]","\frac{a \log \left(\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{f}-\frac{a \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{f}+\frac{b \sec (e+f x)}{f}+\frac{b \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f}-\frac{b \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f}","\frac{b \sec (e+f x)}{f}-\frac{(a+b) \tanh ^{-1}(\cos (e+f x))}{f}",1,"-((a*Log[Cos[e/2 + (f*x)/2]])/f) - (b*Log[Cos[(e + f*x)/2]])/f + (a*Log[Sin[e/2 + (f*x)/2]])/f + (b*Log[Sin[(e + f*x)/2]])/f + (b*Sec[e + f*x])/f","B",1
6,1,236,53,0.3795333,"\int \csc ^3(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2),x]","-\frac{a \csc ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{a \sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{a \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}-\frac{a \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}-\frac{b \csc ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{b \sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{3 b \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}-\frac{3 b \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}+\frac{b \sin \left(\frac{1}{2} (e+f x)\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}-\frac{b \sin \left(\frac{1}{2} (e+f x)\right)}{f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{(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,"-1/8*(a*Csc[(e + f*x)/2]^2)/f - (b*Csc[(e + f*x)/2]^2)/(8*f) - (a*Log[Cos[(e + f*x)/2]])/(2*f) - (3*b*Log[Cos[(e + f*x)/2]])/(2*f) + (a*Log[Sin[(e + f*x)/2]])/(2*f) + (3*b*Log[Sin[(e + f*x)/2]])/(2*f) + (a*Sec[(e + f*x)/2]^2)/(8*f) + (b*Sec[(e + f*x)/2]^2)/(8*f) + (b*Sin[(e + f*x)/2])/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])) - (b*Sin[(e + f*x)/2])/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","B",1
7,1,198,81,1.8266979,"\int \csc ^5(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Csc[e + f*x]^5*(a + b*Sec[e + f*x]^2),x]","\frac{-\left((a+b) \csc ^4\left(\frac{1}{2} (e+f x)\right)\right)-2 (3 a+7 b) \csc ^2\left(\frac{1}{2} (e+f x)\right)+\frac{-(a+b) \sec ^4\left(\frac{1}{2} (e+f x)\right)+\tan ^2\left(\frac{1}{2} (e+f x)\right) \sec ^4\left(\frac{1}{2} (e+f x)\right) ((3 a+7 b) \cos (e+f x)+4 (a+2 b))+2 \sec ^2\left(\frac{1}{2} (e+f x)\right) \left(4 \cos (e+f x) \left(-3 (a+5 b) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)+3 (a+5 b) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)+8 b\right)-3 (a+13 b)\right)}{\tan ^2\left(\frac{1}{2} (e+f x)\right)-1}}{64 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,"(-2*(3*a + 7*b)*Csc[(e + f*x)/2]^2 - (a + b)*Csc[(e + f*x)/2]^4 + (2*(-3*(a + 13*b) + 4*Cos[e + f*x]*(8*b + 3*(a + 5*b)*Log[Cos[(e + f*x)/2]] - 3*(a + 5*b)*Log[Sin[(e + f*x)/2]]))*Sec[(e + f*x)/2]^2 - (a + b)*Sec[(e + f*x)/2]^4 + (4*(a + 2*b) + (3*a + 7*b)*Cos[e + f*x])*Sec[(e + f*x)/2]^4*Tan[(e + f*x)/2]^2)/(-1 + Tan[(e + f*x)/2]^2))/(64*f)","B",1
8,1,78,98,0.2964912,"\int \left(a+b \sec ^2(e+f x)\right) \sin ^6(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^6,x]","\frac{(96 b-45 a) \sin (2 (e+f x))+(9 a-6 b) \sin (4 (e+f x))-a \sin (6 (e+f x))+60 a e+60 a f x+192 b \tan (e+f x)-360 b e-360 b f x}{192 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,"(60*a*e - 360*b*e + 60*a*f*x - 360*b*f*x + (-45*a + 96*b)*Sin[2*(e + f*x)] + (9*a - 6*b)*Sin[4*(e + f*x)] - a*Sin[6*(e + f*x)] + 192*b*Tan[e + f*x])/(192*f)","A",1
9,1,54,70,0.296706,"\int \left(a+b \sec ^2(e+f x)\right) \sin ^4(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^4,x]","\frac{12 (a-4 b) (e+f x)-8 (a-b) \sin (2 (e+f x))+a \sin (4 (e+f x))+32 b \tan (e+f x)}{32 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,"(12*(a - 4*b)*(e + f*x) - 8*(a - b)*Sin[2*(e + f*x)] + a*Sin[4*(e + f*x)] + 32*b*Tan[e + f*x])/(32*f)","A",1
10,1,54,42,0.0957937,"\int \left(a+b \sec ^2(e+f x)\right) \sin ^2(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^2,x]","\frac{a (e+f x)}{2 f}-\frac{a \sin (2 (e+f x))}{4 f}-\frac{b \tan ^{-1}(\tan (e+f x))}{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*(e + f*x))/(2*f) - (b*ArcTan[Tan[e + f*x]])/f - (a*Sin[2*(e + f*x)])/(4*f) + (b*Tan[e + f*x])/f","A",1
11,1,15,15,0.0024462,"\int \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[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",1
12,1,36,26,0.0607941,"\int \csc ^2(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Csc[e + f*x]^2*(a + b*Sec[e + f*x]^2),x]","-\frac{a \cot (e+f x)}{f}+\frac{b \tan (e+f x)}{f}-\frac{b \cot (e+f x)}{f}","\frac{b \tan (e+f x)}{f}-\frac{(a+b) \cot (e+f x)}{f}",1,"-((a*Cot[e + f*x])/f) - (b*Cot[e + f*x])/f + (b*Tan[e + f*x])/f","A",1
13,1,84,46,0.0441726,"\int \csc ^4(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Csc[e + f*x]^4*(a + b*Sec[e + f*x]^2),x]","-\frac{2 a \cot (e+f x)}{3 f}-\frac{a \cot (e+f x) \csc ^2(e+f x)}{3 f}+\frac{b \tan (e+f x)}{f}-\frac{5 b \cot (e+f x)}{3 f}-\frac{b \cot (e+f x) \csc ^2(e+f x)}{3 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,"(-2*a*Cot[e + f*x])/(3*f) - (5*b*Cot[e + f*x])/(3*f) - (a*Cot[e + f*x]*Csc[e + f*x]^2)/(3*f) - (b*Cot[e + f*x]*Csc[e + f*x]^2)/(3*f) + (b*Tan[e + f*x])/f","A",1
14,1,128,68,0.0421948,"\int \csc ^6(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Csc[e + f*x]^6*(a + b*Sec[e + f*x]^2),x]","-\frac{8 a \cot (e+f x)}{15 f}-\frac{a \cot (e+f x) \csc ^4(e+f x)}{5 f}-\frac{4 a \cot (e+f x) \csc ^2(e+f x)}{15 f}+\frac{b \tan (e+f x)}{f}-\frac{11 b \cot (e+f x)}{5 f}-\frac{b \cot (e+f x) \csc ^4(e+f x)}{5 f}-\frac{3 b \cot (e+f x) \csc ^2(e+f x)}{5 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,"(-8*a*Cot[e + f*x])/(15*f) - (11*b*Cot[e + f*x])/(5*f) - (4*a*Cot[e + f*x]*Csc[e + f*x]^2)/(15*f) - (3*b*Cot[e + f*x]*Csc[e + f*x]^2)/(5*f) - (a*Cot[e + f*x]*Csc[e + f*x]^4)/(5*f) - (b*Cot[e + f*x]*Csc[e + f*x]^4)/(5*f) + (b*Tan[e + f*x])/f","A",1
15,1,118,97,0.6144778,"\int \left(a+b \sec ^2(e+f x)\right)^2 \sin ^5(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^5,x]","-\frac{\sec ^3(e+f x) \left(24 \left(22 a^2-215 a b+120 b^2\right) \cos (2 (e+f x))+12 \left(7 a^2-60 a b+20 b^2\right) \cos (4 (e+f x))-16 a^2 \cos (6 (e+f x))+3 a^2 \cos (8 (e+f x))+425 a^2+40 a b \cos (6 (e+f x))-4400 a b+2000 b^2\right)}{1920 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,"-1/1920*((425*a^2 - 4400*a*b + 2000*b^2 + 24*(22*a^2 - 215*a*b + 120*b^2)*Cos[2*(e + f*x)] + 12*(7*a^2 - 60*a*b + 20*b^2)*Cos[4*(e + f*x)] - 16*a^2*Cos[6*(e + f*x)] + 40*a*b*Cos[6*(e + f*x)] + 3*a^2*Cos[8*(e + f*x)])*Sec[e + f*x]^3)/f","A",1
16,1,83,72,0.4418515,"\int \left(a+b \sec ^2(e+f x)\right)^2 \sin ^3(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^3,x]","\frac{\sec ^3(e+f x) \left(-3 \left(11 a^2-64 a b+16 b^2\right) \cos (2 (e+f x))+a^2 \cos (6 (e+f x))-26 a^2-6 a (a-4 b) \cos (4 (e+f x))+168 a b-16 b^2\right)}{96 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,"((-26*a^2 + 168*a*b - 16*b^2 - 3*(11*a^2 - 64*a*b + 16*b^2)*Cos[2*(e + f*x)] - 6*a*(a - 4*b)*Cos[4*(e + f*x)] + a^2*Cos[6*(e + f*x)])*Sec[e + f*x]^3)/(96*f)","A",1
17,1,75,46,0.112516,"\int \left(a+b \sec ^2(e+f x)\right)^2 \sin (e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x],x]","\frac{4 \sec ^3(e+f x) \left(a \cos ^2(e+f x)+b\right)^2 \left(-3 a^2 \cos ^4(e+f x)+6 a b \cos ^2(e+f x)+b^2\right)}{3 f (a \cos (2 (e+f x))+a+2 b)^2}","-\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,"(4*(b + a*Cos[e + f*x]^2)^2*(b^2 + 6*a*b*Cos[e + f*x]^2 - 3*a^2*Cos[e + f*x]^4)*Sec[e + f*x]^3)/(3*f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","A",1
18,1,108,52,0.512598,"\int \csc (e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Csc[e + f*x]*(a + b*Sec[e + f*x]^2)^2,x]","-\frac{4 \sec ^3(e+f x) \left(a \cos ^2(e+f x)+b\right)^2 \left(-3 b (2 a+b) \cos ^2(e+f x)+3 (a+b)^2 \cos ^3(e+f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)-b^2\right)}{3 f (a \cos (2 (e+f x))+a+2 b)^2}","\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,"(-4*(b + a*Cos[e + f*x]^2)^2*(-b^2 - 3*b*(2*a + b)*Cos[e + f*x]^2 + 3*(a + b)^2*Cos[e + f*x]^3*(Log[Cos[(e + f*x)/2]] - Log[Sin[(e + f*x)/2]]))*Sec[e + f*x]^3)/(3*f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","B",1
19,1,1021,104,6.5887467,"\int \csc ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2,x]","\frac{\left(-a^2-2 b a-b^2\right) \csc ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \left(b \sec ^2(e+f x)+a\right)^2 \cos ^4(e+f x)}{2 f (\cos (2 e+2 f x) a+a+2 b)^2}+\frac{\left(a^2+2 b a+b^2\right) \sec ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \left(b \sec ^2(e+f x)+a\right)^2 \cos ^4(e+f x)}{2 f (\cos (2 e+2 f x) a+a+2 b)^2}-\frac{2 \left(a^2+6 b a+5 b^2\right) \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)\right) \left(b \sec ^2(e+f x)+a\right)^2 \cos ^4(e+f x)}{f (\cos (2 e+2 f x) a+a+2 b)^2}+\frac{2 \left(a^2+6 b a+5 b^2\right) \log \left(\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right) \left(b \sec ^2(e+f x)+a\right)^2 \cos ^4(e+f x)}{f (\cos (2 e+2 f x) a+a+2 b)^2}+\frac{2 b (12 a+13 b) \sec (e) \left(b \sec ^2(e+f x)+a\right)^2 \cos ^4(e+f x)}{3 f (\cos (2 e+2 f x) a+a+2 b)^2}+\frac{2 \left(b \sec ^2(e+f x)+a\right)^2 \left(13 \sin \left(\frac{f x}{2}\right) b^2+12 a \sin \left(\frac{f x}{2}\right) b\right) \cos ^4(e+f x)}{3 f (\cos (2 e+2 f x) a+a+2 b)^2 \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}-\frac{2 \left(b \sec ^2(e+f x)+a\right)^2 \left(13 \sin \left(\frac{f x}{2}\right) b^2+12 a \sin \left(\frac{f x}{2}\right) b\right) \cos ^4(e+f x)}{3 f (\cos (2 e+2 f x) a+a+2 b)^2 \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}+\frac{\left(b \sec ^2(e+f x)+a\right)^2 \left(\cos \left(\frac{e}{2}\right) b^2+\sin \left(\frac{e}{2}\right) b^2\right) \cos ^4(e+f x)}{3 f (\cos (2 e+2 f x) a+a+2 b)^2 \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{\left(b \sec ^2(e+f x)+a\right)^2 \left(b^2 \cos \left(\frac{e}{2}\right)-b^2 \sin \left(\frac{e}{2}\right)\right) \cos ^4(e+f x)}{3 f (\cos (2 e+2 f x) a+a+2 b)^2 \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{2 b^2 \left(b \sec ^2(e+f x)+a\right)^2 \sin \left(\frac{f x}{2}\right) \cos ^4(e+f x)}{3 f (\cos (2 e+2 f x) a+a+2 b)^2 \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^3}-\frac{2 b^2 \left(b \sec ^2(e+f x)+a\right)^2 \sin \left(\frac{f x}{2}\right) \cos ^4(e+f x)}{3 f (\cos (2 e+2 f x) a+a+2 b)^2 \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^3}","-\frac{\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^2 - 2*a*b - b^2)*Cos[e + f*x]^4*Csc[e/2 + (f*x)/2]^2*(a + b*Sec[e + f*x]^2)^2)/(2*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2) - (2*(a^2 + 6*a*b + 5*b^2)*Cos[e + f*x]^4*Log[Cos[e/2 + (f*x)/2]]*(a + b*Sec[e + f*x]^2)^2)/(f*(a + 2*b + a*Cos[2*e + 2*f*x])^2) + (2*(a^2 + 6*a*b + 5*b^2)*Cos[e + f*x]^4*Log[Sin[e/2 + (f*x)/2]]*(a + b*Sec[e + f*x]^2)^2)/(f*(a + 2*b + a*Cos[2*e + 2*f*x])^2) + (2*b*(12*a + 13*b)*Cos[e + f*x]^4*Sec[e]*(a + b*Sec[e + f*x]^2)^2)/(3*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2) + ((a^2 + 2*a*b + b^2)*Cos[e + f*x]^4*Sec[e/2 + (f*x)/2]^2*(a + b*Sec[e + f*x]^2)^2)/(2*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2) + (2*b^2*Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2*Sin[(f*x)/2])/(3*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2*(Cos[e/2] - Sin[e/2])*(Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2])^3) + (Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2*(b^2*Cos[e/2] + b^2*Sin[e/2]))/(3*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2*(Cos[e/2] - Sin[e/2])*(Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2])^2) + (2*Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2*(12*a*b*Sin[(f*x)/2] + 13*b^2*Sin[(f*x)/2]))/(3*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2*(Cos[e/2] - Sin[e/2])*(Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2])) - (2*b^2*Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2*Sin[(f*x)/2])/(3*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2*(Cos[e/2] + Sin[e/2])*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^3) + (Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2*(b^2*Cos[e/2] - b^2*Sin[e/2]))/(3*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2*(Cos[e/2] + Sin[e/2])*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) - (2*Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2*(12*a*b*Sin[(f*x)/2] + 13*b^2*Sin[(f*x)/2]))/(3*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2*(Cos[e/2] + Sin[e/2])*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2]))","B",1
20,1,218,141,1.8496572,"\int \csc ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Csc[e + f*x]^5*(a + b*Sec[e + f*x]^2)^2,x]","-\frac{\sec ^4(e+f x) \left(a \cos ^2(e+f x)+b\right)^2 \left(\frac{1}{2} \left(105 a^2+282 a b+329 b^2\right) (\cos (e+f x)+\cos (3 (e+f x))) \csc ^4(e+f x)+96 \left(3 a^2+30 a b+35 b^2\right) \cos ^4(e+f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)+\cot (e+f x) \csc ^3(e+f x) \left(\left(6 a^2+60 a b+70 b^2\right) \cos (4 (e+f x))-3 \left(3 a^2+30 a b+35 b^2\right) \cos (6 (e+f x))+90 a^2+132 a b-102 b^2\right)\right)}{192 f (a \cos (2 (e+f x))+a+2 b)^2}","-\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,"-1/192*((b + a*Cos[e + f*x]^2)^2*((90*a^2 + 132*a*b - 102*b^2 + (6*a^2 + 60*a*b + 70*b^2)*Cos[4*(e + f*x)] - 3*(3*a^2 + 30*a*b + 35*b^2)*Cos[6*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x]^3 + ((105*a^2 + 282*a*b + 329*b^2)*(Cos[e + f*x] + Cos[3*(e + f*x)])*Csc[e + f*x]^4)/2 + 96*(3*a^2 + 30*a*b + 35*b^2)*Cos[e + f*x]^4*(Log[Cos[(e + f*x)/2]] - Log[Sin[(e + f*x)/2]]))*Sec[e + f*x]^4)/(f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","A",1
21,1,499,148,1.3512682,"\int \left(a+b \sec ^2(e+f x)\right)^2 \sin ^6(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^6,x]","\frac{\sec (e) \sec ^3(e+f x) \left(a \cos ^2(e+f x)+b\right)^2 \left(360 f x \left(a^2-12 a b+8 b^2\right) \cos (2 e+f x)+360 f x \left(a^2-12 a b+8 b^2\right) \cos (f x)-81 a^2 \sin (2 e+f x)-109 a^2 \sin (2 e+3 f x)-109 a^2 \sin (4 e+3 f x)-21 a^2 \sin (4 e+5 f x)-21 a^2 \sin (6 e+5 f x)+6 a^2 \sin (6 e+7 f x)+6 a^2 \sin (8 e+7 f x)-a^2 \sin (8 e+9 f x)-a^2 \sin (10 e+9 f x)+120 a^2 f x \cos (2 e+3 f x)+120 a^2 f x \cos (4 e+3 f x)-81 a^2 \sin (f x)-1164 a b \sin (2 e+f x)+2076 a b \sin (2 e+3 f x)+540 a b \sin (4 e+3 f x)+156 a b \sin (4 e+5 f x)+156 a b \sin (6 e+5 f x)-12 a b \sin (6 e+7 f x)-12 a b \sin (8 e+7 f x)-1440 a b f x \cos (2 e+3 f x)-1440 a b f x \cos (4 e+3 f x)+3444 a b \sin (f x)+2208 b^2 \sin (2 e+f x)-1936 b^2 \sin (2 e+3 f x)-144 b^2 \sin (4 e+3 f x)-48 b^2 \sin (4 e+5 f x)-48 b^2 \sin (6 e+5 f x)+960 b^2 f x \cos (2 e+3 f x)+960 b^2 f x \cos (4 e+3 f x)-3168 b^2 \sin (f x)\right)}{768 f (a \cos (2 (e+f x))+a+2 b)^2}","-\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,"((b + a*Cos[e + f*x]^2)^2*Sec[e]*Sec[e + f*x]^3*(360*(a^2 - 12*a*b + 8*b^2)*f*x*Cos[f*x] + 360*(a^2 - 12*a*b + 8*b^2)*f*x*Cos[2*e + f*x] + 120*a^2*f*x*Cos[2*e + 3*f*x] - 1440*a*b*f*x*Cos[2*e + 3*f*x] + 960*b^2*f*x*Cos[2*e + 3*f*x] + 120*a^2*f*x*Cos[4*e + 3*f*x] - 1440*a*b*f*x*Cos[4*e + 3*f*x] + 960*b^2*f*x*Cos[4*e + 3*f*x] - 81*a^2*Sin[f*x] + 3444*a*b*Sin[f*x] - 3168*b^2*Sin[f*x] - 81*a^2*Sin[2*e + f*x] - 1164*a*b*Sin[2*e + f*x] + 2208*b^2*Sin[2*e + f*x] - 109*a^2*Sin[2*e + 3*f*x] + 2076*a*b*Sin[2*e + 3*f*x] - 1936*b^2*Sin[2*e + 3*f*x] - 109*a^2*Sin[4*e + 3*f*x] + 540*a*b*Sin[4*e + 3*f*x] - 144*b^2*Sin[4*e + 3*f*x] - 21*a^2*Sin[4*e + 5*f*x] + 156*a*b*Sin[4*e + 5*f*x] - 48*b^2*Sin[4*e + 5*f*x] - 21*a^2*Sin[6*e + 5*f*x] + 156*a*b*Sin[6*e + 5*f*x] - 48*b^2*Sin[6*e + 5*f*x] + 6*a^2*Sin[6*e + 7*f*x] - 12*a*b*Sin[6*e + 7*f*x] + 6*a^2*Sin[8*e + 7*f*x] - 12*a*b*Sin[8*e + 7*f*x] - a^2*Sin[8*e + 9*f*x] - a^2*Sin[10*e + 9*f*x]))/(768*f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","B",1
22,1,153,114,1.6188893,"\int \left(a+b \sec ^2(e+f x)\right)^2 \sin ^4(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^4,x]","\frac{\sec ^3(e+f x) \left(a \cos ^2(e+f x)+b\right)^2 \left(3 \cos ^3(e+f x) \left(4 f x \left(3 a^2-24 a b+8 b^2\right)+a^2 \sin (4 (e+f x))-8 a (a-2 b) \sin (2 (e+f x))\right)+64 b (3 a-2 b) \sec (e) \sin (f x) \cos ^2(e+f x)+32 b^2 \tan (e) \cos (e+f x)+32 b^2 \sec (e) \sin (f x)\right)}{24 f (a \cos (2 (e+f x))+a+2 b)^2}","-\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,"((b + a*Cos[e + f*x]^2)^2*Sec[e + f*x]^3*(32*b^2*Sec[e]*Sin[f*x] + 64*(3*a - 2*b)*b*Cos[e + f*x]^2*Sec[e]*Sin[f*x] + 3*Cos[e + f*x]^3*(4*(3*a^2 - 24*a*b + 8*b^2)*f*x - 8*a*(a - 2*b)*Sin[2*(e + f*x)] + a^2*Sin[4*(e + f*x)]) + 32*b^2*Cos[e + f*x]*Tan[e]))/(24*f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","A",1
23,1,126,73,1.0037904,"\int \left(a+b \sec ^2(e+f x)\right)^2 \sin ^2(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^2,x]","-\frac{\sec ^3(e+f x) \left(a \cos ^2(e+f x)+b\right)^2 \left(3 a \cos ^3(e+f x) (a \sin (2 (e+f x))-2 f x (a-4 b))-4 b (6 a-b) \sec (e) \sin (f x) \cos ^2(e+f x)-4 b^2 \tan (e) \cos (e+f x)-4 b^2 \sec (e) \sin (f x)\right)}{3 f (a \cos (2 (e+f x))+a+2 b)^2}","\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,"-1/3*((b + a*Cos[e + f*x]^2)^2*Sec[e + f*x]^3*(-4*b^2*Sec[e]*Sin[f*x] - 4*(6*a - b)*b*Cos[e + f*x]^2*Sec[e]*Sin[f*x] + 3*a*Cos[e + f*x]^3*(-2*(a - 4*b)*f*x + a*Sin[2*(e + f*x)]) - 4*b^2*Cos[e + f*x]*Tan[e]))/(f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","A",1
24,1,106,40,0.3640197,"\int \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[(a + b*Sec[e + f*x]^2)^2,x]","\frac{4 \sec ^3(e+f x) \left(a \cos ^2(e+f x)+b\right)^2 \left(3 a^2 f x \cos ^3(e+f x)+2 b (3 a+b) \sec (e) \sin (f x) \cos ^2(e+f x)+b^2 \tan (e) \cos (e+f x)+b^2 \sec (e) \sin (f x)\right)}{3 f (a \cos (2 (e+f x))+a+2 b)^2}","a^2 x+\frac{b (2 a+b) \tan (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"(4*(b + a*Cos[e + f*x]^2)^2*Sec[e + f*x]^3*(3*a^2*f*x*Cos[e + f*x]^3 + b^2*Sec[e]*Sin[f*x] + 2*b*(3*a + b)*Cos[e + f*x]^2*Sec[e]*Sin[f*x] + b^2*Cos[e + f*x]*Tan[e]))/(3*f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","B",1
25,1,109,50,1.1397229,"\int \csc ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Csc[e + f*x]^2*(a + b*Sec[e + f*x]^2)^2,x]","\frac{4 \sec ^3(e+f x) \left(a \cos ^2(e+f x)+b\right)^2 \left(\sin (f x) \cos ^2(e+f x) \left(3 (a+b)^2 \csc (e) \cot (e+f x)+b (6 a+5 b) \sec (e)\right)+b^2 \tan (e) \cos (e+f x)+b^2 \sec (e) \sin (f x)\right)}{3 f (a \cos (2 (e+f x))+a+2 b)^2}","\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,"(4*(b + a*Cos[e + f*x]^2)^2*Sec[e + f*x]^3*(b^2*Sec[e]*Sin[f*x] + Cos[e + f*x]^2*(3*(a + b)^2*Cot[e + f*x]*Csc[e] + b*(6*a + 5*b)*Sec[e])*Sin[f*x] + b^2*Cos[e + f*x]*Tan[e]))/(3*f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","B",1
26,1,151,76,1.3550981,"\int \csc ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Csc[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2,x]","-\frac{\csc (2 e) \csc ^3(2 (e+f x)) \left(-3 a^2 \sin (2 (e+f x))+a^2 \sin (6 (e+f x))+3 a^2 \sin (4 e+2 f x)+a^2 \sin (4 e+6 f x)-6 a b \sin (2 (e+f x))+2 a b \sin (6 (e+f x))+8 a b \sin (4 e+6 f x)+8 a (a+2 b) \sin (2 e)-6 (a+2 b)^2 \sin (2 f x)+8 b^2 \sin (4 e+6 f x)\right)}{6 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,"-1/6*(Csc[2*e]*Csc[2*(e + f*x)]^3*(8*a*(a + 2*b)*Sin[2*e] - 6*(a + 2*b)^2*Sin[2*f*x] - 3*a^2*Sin[2*(e + f*x)] - 6*a*b*Sin[2*(e + f*x)] + a^2*Sin[6*(e + f*x)] + 2*a*b*Sin[6*(e + f*x)] + 3*a^2*Sin[4*e + 2*f*x] + a^2*Sin[4*e + 6*f*x] + 8*a*b*Sin[4*e + 6*f*x] + 8*b^2*Sin[4*e + 6*f*x]))/f","A",1
27,1,353,103,1.5681729,"\int \csc ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Csc[e + f*x]^6*(a + b*Sec[e + f*x]^2)^2,x]","-\frac{\csc (e) \sec (e) \csc ^5(e+f x) \sec ^3(e+f x) \left(-32 \left(2 a^2+9 a b+12 b^2\right) \sin (2 f x)-24 a^2 \sin (2 (e+f x))+8 a^2 \sin (4 (e+f x))+8 a^2 \sin (6 (e+f x))-4 a^2 \sin (8 (e+f x))+8 a^2 \sin (2 (e+2 f x))+40 a^2 \sin (4 e+2 f x)+8 a^2 \sin (4 e+6 f x)-4 a^2 \sin (6 e+8 f x)-108 a b \sin (2 (e+f x))+36 a b \sin (4 (e+f x))+36 a b \sin (6 (e+f x))-18 a b \sin (8 (e+f x))+96 a b \sin (2 (e+2 f x))+96 a b \sin (4 e+6 f x)-48 a b \sin (6 e+8 f x)+20 a (5 a+12 b) \sin (2 e)-54 b^2 \sin (2 (e+f x))+18 b^2 \sin (4 (e+f x))+18 b^2 \sin (6 (e+f x))-9 b^2 \sin (8 (e+f x))+128 b^2 \sin (2 (e+2 f x))+128 b^2 \sin (4 e+6 f x)-64 b^2 \sin (6 e+8 f x)\right)}{1920 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,"-1/1920*(Csc[e]*Csc[e + f*x]^5*Sec[e]*Sec[e + f*x]^3*(20*a*(5*a + 12*b)*Sin[2*e] - 32*(2*a^2 + 9*a*b + 12*b^2)*Sin[2*f*x] - 24*a^2*Sin[2*(e + f*x)] - 108*a*b*Sin[2*(e + f*x)] - 54*b^2*Sin[2*(e + f*x)] + 8*a^2*Sin[4*(e + f*x)] + 36*a*b*Sin[4*(e + f*x)] + 18*b^2*Sin[4*(e + f*x)] + 8*a^2*Sin[6*(e + f*x)] + 36*a*b*Sin[6*(e + f*x)] + 18*b^2*Sin[6*(e + f*x)] - 4*a^2*Sin[8*(e + f*x)] - 18*a*b*Sin[8*(e + f*x)] - 9*b^2*Sin[8*(e + f*x)] + 8*a^2*Sin[2*(e + 2*f*x)] + 96*a*b*Sin[2*(e + 2*f*x)] + 128*b^2*Sin[2*(e + 2*f*x)] + 40*a^2*Sin[4*e + 2*f*x] + 8*a^2*Sin[4*e + 6*f*x] + 96*a*b*Sin[4*e + 6*f*x] + 128*b^2*Sin[4*e + 6*f*x] - 4*a^2*Sin[6*e + 8*f*x] - 48*a*b*Sin[6*e + 8*f*x] - 64*b^2*Sin[6*e + 8*f*x]))/f","B",1
28,1,425,98,3.2579185,"\int \frac{\sin ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^5/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(-75 a^3 \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)-75 a^3 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{a}}{\sqrt{b}}\right)-8 \sqrt{a} \sqrt{b} \cos (e+f x) \left(3 a^2 \cos (4 (e+f x))+89 a^2-4 a (7 a+5 b) \cos (2 (e+f x))+220 a b+120 b^2\right)+15 \left(5 a^3+64 a^2 b+128 a b^2+64 b^3\right) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)+15 \left(5 a^3+64 a^2 b+128 a b^2+64 b^3\right) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)\right)}{1920 a^{7/2} \sqrt{b} f \left(a+b \sec ^2(e+f x)\right)}","\frac{\sqrt{b} (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{a^{7/2} f}-\frac{(a+b)^2 \cos (e+f x)}{a^3 f}+\frac{(2 a+b) \cos ^3(e+f x)}{3 a^2 f}-\frac{\cos ^5(e+f x)}{5 a f}",1,"((a + 2*b + a*Cos[2*(e + f*x)])*(15*(5*a^3 + 64*a^2*b + 128*a*b^2 + 64*b^3)*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]] + 15*(5*a^3 + 64*a^2*b + 128*a*b^2 + 64*b^3)*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]] - 75*a^3*ArcTan[(Sqrt[a] - Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]] - 75*a^3*ArcTan[(Sqrt[a] + Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]] - 8*Sqrt[a]*Sqrt[b]*Cos[e + f*x]*(89*a^2 + 220*a*b + 120*b^2 - 4*a*(7*a + 5*b)*Cos[2*(e + f*x)] + 3*a^2*Cos[4*(e + f*x)]))*Sec[e + f*x]^2)/(1920*a^(7/2)*Sqrt[b]*f*(a + b*Sec[e + f*x]^2))","C",1
29,1,376,71,1.3924953,"\int \frac{\sin ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(3 \left(a^2+8 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)+3 \left(a^2+8 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)-3 a^2 \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)-3 a^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{a}}{\sqrt{b}}\right)+4 \sqrt{a} \sqrt{b} \cos (e+f x) (a \cos (2 (e+f x))-5 a-6 b)\right)}{48 a^{5/2} \sqrt{b} f \left(a+b \sec ^2(e+f x)\right)}","\frac{\sqrt{b} (a+b) \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{a^{5/2} f}-\frac{(a+b) \cos (e+f x)}{a^2 f}+\frac{\cos ^3(e+f x)}{3 a f}",1,"((a + 2*b + a*Cos[2*(e + f*x)])*(3*(a^2 + 8*a*b + 8*b^2)*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]] + 3*(a^2 + 8*a*b + 8*b^2)*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]] - 3*a^2*ArcTan[(Sqrt[a] - Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]] - 3*a^2*ArcTan[(Sqrt[a] + Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]] + 4*Sqrt[a]*Sqrt[b]*Cos[e + f*x]*(-5*a - 6*b + a*Cos[2*(e + f*x)]))*Sec[e + f*x]^2)/(48*a^(5/2)*Sqrt[b]*f*(a + b*Sec[e + f*x]^2))","C",1
30,1,329,47,0.545599,"\int \frac{\sin (e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]/(a + b*Sec[e + f*x]^2),x]","\frac{(a \cos (2 (e+f x))+a+2 b) \left(-4 \sqrt{a} \sqrt{b} \cos (e+f x)-a \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)-a \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{a}}{\sqrt{b}}\right)+(a+4 b) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)+(a+4 b) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)\right)}{8 a^{3/2} \sqrt{b} f \left(a \cos ^2(e+f x)+b\right)}","\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,"(((a + 4*b)*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]] + (a + 4*b)*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]] - a*ArcTan[(Sqrt[a] - Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]] - a*ArcTan[(Sqrt[a] + Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]] - 4*Sqrt[a]*Sqrt[b]*Cos[e + f*x])*(a + 2*b + a*Cos[2*(e + f*x)]))/(8*a^(3/2)*Sqrt[b]*f*(b + a*Cos[e + f*x]^2))","C",1
31,1,239,55,0.7640288,"\int \frac{\csc (e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]/(a + b*Sec[e + f*x]^2),x]","\frac{\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{\sqrt{a}}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{\sqrt{a}}+\log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{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] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]])/Sqrt[a] + (Sqrt[b]*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]])/Sqrt[a] - Log[Cos[(e + f*x)/2]] + Log[Sin[(e + f*x)/2]])/((a + b)*f)","C",1
32,1,371,86,1.5503453,"\int \frac{\csc ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^3/(a + b*Sec[e + f*x]^2),x]","-\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(-8 \sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)-8 \sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)+a \csc ^2\left(\frac{1}{2} (e+f x)\right)-a \sec ^2\left(\frac{1}{2} (e+f x)\right)-4 a \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)+4 a \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)+b \csc ^2\left(\frac{1}{2} (e+f x)\right)-b \sec ^2\left(\frac{1}{2} (e+f x)\right)+4 b \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-4 b \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{16 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)}","\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,"-1/16*((a + 2*b + a*Cos[2*(e + f*x)])*(-8*Sqrt[a]*Sqrt[b]*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]] - 8*Sqrt[a]*Sqrt[b]*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]] + a*Csc[(e + f*x)/2]^2 + b*Csc[(e + f*x)/2]^2 + 4*a*Log[Cos[(e + f*x)/2]] - 4*b*Log[Cos[(e + f*x)/2]] - 4*a*Log[Sin[(e + f*x)/2]] + 4*b*Log[Sin[(e + f*x)/2]] - a*Sec[(e + f*x)/2]^2 - b*Sec[(e + f*x)/2]^2)*Sec[e + f*x]^2)/((a + b)^2*f*(a + b*Sec[e + f*x]^2))","C",1
33,1,549,129,4.7023998,"\int \frac{\csc ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2),x]","-\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(-64 a^{3/2} \sqrt{b} \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)-64 a^{3/2} \sqrt{b} \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)+a^2 \csc ^4\left(\frac{1}{2} (e+f x)\right)+6 a^2 \csc ^2\left(\frac{1}{2} (e+f x)\right)-a^2 \sec ^4\left(\frac{1}{2} (e+f x)\right)-6 a^2 \sec ^2\left(\frac{1}{2} (e+f x)\right)-24 a^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)+24 a^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)+2 a b \csc ^4\left(\frac{1}{2} (e+f x)\right)+4 a b \csc ^2\left(\frac{1}{2} (e+f x)\right)-2 a b \sec ^4\left(\frac{1}{2} (e+f x)\right)-4 a b \sec ^2\left(\frac{1}{2} (e+f x)\right)+48 a b \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-48 a b \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)+b^2 \csc ^4\left(\frac{1}{2} (e+f x)\right)-2 b^2 \csc ^2\left(\frac{1}{2} (e+f x)\right)-b^2 \sec ^4\left(\frac{1}{2} (e+f x)\right)+2 b^2 \sec ^2\left(\frac{1}{2} (e+f x)\right)+8 b^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-8 b^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{128 f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)}","\frac{a^{3/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)}{f (a+b)^3}-\frac{\left(3 a^2-6 a b-b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 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,"-1/128*((a + 2*b + a*Cos[2*(e + f*x)])*(-64*a^(3/2)*Sqrt[b]*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]] - 64*a^(3/2)*Sqrt[b]*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]] + 6*a^2*Csc[(e + f*x)/2]^2 + 4*a*b*Csc[(e + f*x)/2]^2 - 2*b^2*Csc[(e + f*x)/2]^2 + a^2*Csc[(e + f*x)/2]^4 + 2*a*b*Csc[(e + f*x)/2]^4 + b^2*Csc[(e + f*x)/2]^4 + 24*a^2*Log[Cos[(e + f*x)/2]] - 48*a*b*Log[Cos[(e + f*x)/2]] - 8*b^2*Log[Cos[(e + f*x)/2]] - 24*a^2*Log[Sin[(e + f*x)/2]] + 48*a*b*Log[Sin[(e + f*x)/2]] + 8*b^2*Log[Sin[(e + f*x)/2]] - 6*a^2*Sec[(e + f*x)/2]^2 - 4*a*b*Sec[(e + f*x)/2]^2 + 2*b^2*Sec[(e + f*x)/2]^2 - a^2*Sec[(e + f*x)/2]^4 - 2*a*b*Sec[(e + f*x)/2]^4 - b^2*Sec[(e + f*x)/2]^4)*Sec[e + f*x]^2)/((a + b)^3*f*(a + b*Sec[e + f*x]^2))","C",1
34,1,357,166,4.0862011,"\int \frac{\sin ^6(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^6/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\sqrt{b (\cos (e)-i \sin (e))^4} \left(3 a^3 (9 a+8 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)+2 \sqrt{b} \sqrt{a+b} \left(-a^3 \sin (6 (e+f x))-12 a^3 e+60 a^3 f x-3 a \left(15 a^2+32 a b+16 b^2\right) \sin (2 (e+f x))+3 a^2 (3 a+2 b) \sin (4 (e+f x))+360 a^2 b f x+480 a b^2 f x+192 b^3 f x\right)\right)+3 \sqrt{b} \left(9 a^4+136 a^3 b+384 a^2 b^2+384 a b^3+128 b^4\right) (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)\right)}{768 a^4 \sqrt{b} f \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4} \left(a+b \sec ^2(e+f x)\right)}","-\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{\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(5 a^3+30 a^2 b+40 a b^2+16 b^3\right)}{16 a^4}+\frac{\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f}",1,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(3*Sqrt[b]*(9*a^4 + 136*a^3*b + 384*a^2*b^2 + 384*a*b^3 + 128*b^4)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]) + Sqrt[b*(Cos[e] - I*Sin[e])^4]*(3*a^3*(9*a + 8*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]] + 2*Sqrt[b]*Sqrt[a + b]*(-12*a^3*e + 60*a^3*f*x + 360*a^2*b*f*x + 480*a*b^2*f*x + 192*b^3*f*x - 3*a*(15*a^2 + 32*a*b + 16*b^2)*Sin[2*(e + f*x)] + 3*a^2*(3*a + 2*b)*Sin[4*(e + f*x)] - a^3*Sin[6*(e + f*x)]))))/(768*a^4*Sqrt[b]*Sqrt[a + b]*f*(a + b*Sec[e + f*x]^2)*Sqrt[b*(Cos[e] - I*Sin[e])^4])","C",0
35,1,303,117,1.9354319,"\int \frac{\sin ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^4/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\sqrt{b (\cos (e)-i \sin (e))^4} \left(\sqrt{b} \sqrt{a+b} \left(a^2 \sin (4 (e+f x))-2 a^2 e+12 a^2 f x-8 a (a+b) \sin (2 (e+f x))+48 a b f x+32 b^2 f x\right)+a^2 (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)\right)+\sqrt{b} \left(3 a^3+34 a^2 b+64 a b^2+32 b^3\right) (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)\right)}{64 a^3 \sqrt{b} f \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4} \left(a+b \sec ^2(e+f x)\right)}","-\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{x \left(3 a^2+12 a b+8 b^2\right)}{8 a^3}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f}",1,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(Sqrt[b]*(3*a^3 + 34*a^2*b + 64*a*b^2 + 32*b^3)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]) + Sqrt[b*(Cos[e] - I*Sin[e])^4]*(a^2*(3*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]] + Sqrt[b]*Sqrt[a + b]*(-2*a^2*e + 12*a^2*f*x + 48*a*b*f*x + 32*b^2*f*x - 8*a*(a + b)*Sin[2*(e + f*x)] + a^2*Sin[4*(e + f*x)]))))/(64*a^3*Sqrt[b]*Sqrt[a + b]*f*(a + b*Sec[e + f*x]^2)*Sqrt[b*(Cos[e] - I*Sin[e])^4])","C",1
36,1,245,76,0.8425595,"\int \frac{\sin ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{b} f \sqrt{a+b}}-\frac{-\frac{\left(a^2+8 a b+8 b^2\right) (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{f \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}-4 x (a+2 b)+\frac{2 a \sin (2 e) \cos (2 f x)}{f}+\frac{2 a \cos (2 e) \sin (2 f x)}{f}}{a^2}\right)}{16 \left(a+b \sec ^2(e+f x)\right)}","-\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 + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b]*f) - (-4*(a + 2*b)*x - ((a^2 + 8*a*b + 8*b^2)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (2*a*Cos[2*f*x]*Sin[2*e])/f + (2*a*Cos[2*e]*Sin[2*f*x])/f)/a^2))/(16*(a + b*Sec[e + f*x]^2))","C",1
37,1,182,45,0.2863083,"\int \frac{1}{a+b \sec ^2(e+f x)} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(-1),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(f x \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}+b (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)\right)}{2 a f \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4} \left(a+b \sec ^2(e+f x)\right)}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(Sqrt[a + b]*f*x*Sqrt[b*(Cos[e] - I*Sin[e])^4] + b*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e])))/(2*a*Sqrt[a + b]*f*(a + b*Sec[e + f*x]^2)*Sqrt[b*(Cos[e] - I*Sin[e])^4])","C",1
38,1,189,54,0.641765,"\int \frac{\csc ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^2/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\sqrt{a+b} \csc (e) \sin (f x) \sqrt{b (\cos (e)-i \sin (e))^4} \csc (e+f x)+b (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)\right)}{2 f (a+b)^{3/2} \sqrt{b (\cos (e)-i \sin (e))^4} \left(a+b \sec ^2(e+f x)\right)}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(b*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]) + Sqrt[a + b]*Csc[e]*Csc[e + f*x]*Sqrt[b*(Cos[e] - I*Sin[e])^4]*Sin[f*x]))/(2*(a + b)^(3/2)*f*(a + b*Sec[e + f*x]^2)*Sqrt[b*(Cos[e] - I*Sin[e])^4])","C",1
39,1,226,76,2.1000203,"\int \frac{\csc ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{1}{4} \sqrt{a+b} \csc (e) \sqrt{b (\cos (e)-i \sin (e))^4} \csc ^3(e+f x) ((b-2 a) \sin (2 e+3 f x)+6 a \sin (f x)-3 b \sin (2 e+f x))+3 a b (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)\right)}{6 f (a+b)^{5/2} \sqrt{b (\cos (e)-i \sin (e))^4} \left(a+b \sec ^2(e+f x)\right)}","-\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 + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(3*a*b*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]) + (Sqrt[a + b]*Csc[e]*Csc[e + f*x]^3*Sqrt[b*(Cos[e] - I*Sin[e])^4]*(6*a*Sin[f*x] - 3*b*Sin[2*e + f*x] + (-2*a + b)*Sin[2*e + 3*f*x]))/4))/(6*(a + b)^(5/2)*f*(a + b*Sec[e + f*x]^2)*Sqrt[b*(Cos[e] - I*Sin[e])^4])","C",1
40,1,318,105,1.7176567,"\int \frac{\csc ^6(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^6/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\sqrt{a+b} \csc (e) \sqrt{b (\cos (e)-i \sin (e))^4} \csc ^5(e+f x) \left(10 \left(8 a^2+b^2\right) \sin (f x)-40 a^2 \sin (2 e+3 f x)+8 a^2 \sin (4 e+5 f x)+30 a b \sin (2 e+3 f x)+15 a b \sin (4 e+3 f x)-9 a b \sin (4 e+5 f x)-30 b (3 a+b) \sin (2 e+f x)+10 b^2 \sin (2 e+3 f x)-2 b^2 \sin (4 e+5 f x)\right)+240 a^2 b (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)\right)}{480 f (a+b)^{7/2} \sqrt{b (\cos (e)-i \sin (e))^4} \left(a+b \sec ^2(e+f x)\right)}","-\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*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(240*a^2*b*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]) + Sqrt[a + b]*Csc[e]*Csc[e + f*x]^5*Sqrt[b*(Cos[e] - I*Sin[e])^4]*(10*(8*a^2 + b^2)*Sin[f*x] - 30*b*(3*a + b)*Sin[2*e + f*x] - 40*a^2*Sin[2*e + 3*f*x] + 30*a*b*Sin[2*e + 3*f*x] + 10*b^2*Sin[2*e + 3*f*x] + 15*a*b*Sin[4*e + 3*f*x] + 8*a^2*Sin[4*e + 5*f*x] - 9*a*b*Sin[4*e + 5*f*x] - 2*b^2*Sin[4*e + 5*f*x])))/(480*(a + b)^(7/2)*f*(a + b*Sec[e + f*x]^2)*Sqrt[b*(Cos[e] - I*Sin[e])^4])","C",1
41,1,454,161,6.3045432,"\int \frac{\sin ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Sin[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2,x]","\frac{-\frac{45 a^4 \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{b^{3/2}}-\frac{45 a^4 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{a}}{\sqrt{b}}\right)}{b^{3/2}}+\frac{15 \left(3 a^4+384 a^2 b^2+1280 a b^3+896 b^4\right) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{b^{3/2}}+\frac{15 \left(3 a^4+384 a^2 b^2+1280 a b^3+896 b^4\right) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{b^{3/2}}-\frac{16 \sqrt{a} \cos (e+f x) \left(3 a^3 \cos (6 (e+f x))+150 a^3+a \left(125 a^2+688 a b+560 b^2\right) \cos (2 (e+f x))-2 a^2 (11 a+14 b) \cos (4 (e+f x))+1436 a^2 b+2960 a b^2+1680 b^3\right)}{a \cos (2 (e+f x))+a+2 b}}{3840 a^{9/2} 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{(a+b) (3 a+7 b) \cos (e+f x)}{2 a^4 f}+\frac{(a+b) (3 a+7 b) \cos ^3(e+f x)}{6 a^3 b 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{\cos ^5(e+f x)}{5 a^2 f}",1,"((15*(3*a^4 + 384*a^2*b^2 + 1280*a*b^3 + 896*b^4)*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]])/b^(3/2) + (15*(3*a^4 + 384*a^2*b^2 + 1280*a*b^3 + 896*b^4)*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]])/b^(3/2) - (45*a^4*ArcTan[(Sqrt[a] - Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]])/b^(3/2) - (45*a^4*ArcTan[(Sqrt[a] + Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]])/b^(3/2) - (16*Sqrt[a]*Cos[e + f*x]*(150*a^3 + 1436*a^2*b + 2960*a*b^2 + 1680*b^3 + a*(125*a^2 + 688*a*b + 560*b^2)*Cos[2*(e + f*x)] - 2*a^2*(11*a + 14*b)*Cos[4*(e + f*x)] + 3*a^3*Cos[6*(e + f*x)]))/(a + 2*b + a*Cos[2*(e + f*x)]))/(3840*a^(9/2)*f)","C",1
42,1,403,114,3.4705351,"\int \frac{\sin ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2,x]","\frac{-\frac{9 a^3 \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{b^{3/2}}-\frac{9 a^3 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{a}}{\sqrt{b}}\right)}{b^{3/2}}+\frac{3 \left(3 a^3+192 a b^2+320 b^3\right) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{b^{3/2}}+\frac{3 \left(3 a^3+192 a b^2+320 b^3\right) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{b^{3/2}}-\frac{32 \sqrt{a} \cos (e+f x) \left(a^2 (-\cos (4 (e+f x)))+9 a^2+4 a (2 a+5 b) \cos (2 (e+f x))+56 a b+60 b^2\right)}{a \cos (2 (e+f x))+a+2 b}}{384 a^{7/2} 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{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{\cos ^3(e+f x)}{3 a^2 f}",1,"((3*(3*a^3 + 192*a*b^2 + 320*b^3)*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]])/b^(3/2) + (3*(3*a^3 + 192*a*b^2 + 320*b^3)*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]])/b^(3/2) - (9*a^3*ArcTan[(Sqrt[a] - Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]])/b^(3/2) - (9*a^3*ArcTan[(Sqrt[a] + Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]])/b^(3/2) - (32*Sqrt[a]*Cos[e + f*x]*(9*a^2 + 56*a*b + 60*b^2 + 4*a*(2*a + 5*b)*Cos[2*(e + f*x)] - a^2*Cos[4*(e + f*x)]))/(a + 2*b + a*Cos[2*(e + f*x)]))/(384*a^(7/2)*f)","C",1
43,1,393,84,3.1387735,"\int \frac{\sin (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Sin[e + f*x]/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b)^2 \left(-\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{b^{3/2}}-\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{a}}{\sqrt{b}}\right)}{b^{3/2}}+\frac{\left(a^2+24 b^2\right) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{b^{3/2}}+\frac{\left(a^2+24 b^2\right) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{b^{3/2}}-\frac{16 \sqrt{a} \cos (e+f x) (a \cos (2 (e+f x))+a+3 b)}{a \cos (2 (e+f x))+a+2 b}\right)}{64 a^{5/2} f \left(a+b \sec ^2(e+f x)\right)^2}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])^2*(((a^2 + 24*b^2)*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]])/b^(3/2) + ((a^2 + 24*b^2)*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]])/b^(3/2) - (a^2*ArcTan[(Sqrt[a] - Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]])/b^(3/2) - (a^2*ArcTan[(Sqrt[a] + Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]])/b^(3/2) - (16*Sqrt[a]*Cos[e + f*x]*(a + 3*b + a*Cos[2*(e + f*x)]))/(a + 2*b + a*Cos[2*(e + f*x)]))*Sec[e + f*x]^4)/(64*a^(5/2)*f*(a + b*Sec[e + f*x]^2)^2)","C",1
44,1,384,99,1.2744056,"\int \frac{\csc (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Csc[e + f*x]/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{\sqrt{b} (3 a+b) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{a^{3/2}}+\frac{\sqrt{b} (3 a+b) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{a^{3/2}}-2 \sec (e+f x) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right) (a \cos (2 (e+f x))+a+2 b)+2 \sec (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right) (a \cos (2 (e+f x))+a+2 b)-\frac{2 b (a+b)}{a}\right)}{8 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^3*((-2*b*(a + b))/a + (Sqrt[b]*(3*a + b)*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x])/a^(3/2) + (Sqrt[b]*(3*a + b)*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x])/a^(3/2) - 2*(a + 2*b + a*Cos[2*(e + f*x)])*Log[Cos[(e + f*x)/2]]*Sec[e + f*x] + 2*(a + 2*b + a*Cos[2*(e + f*x)])*Log[Sin[(e + f*x)/2]]*Sec[e + f*x]))/(8*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^2)","C",1
45,1,468,147,1.940412,"\int \frac{\csc ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Csc[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left((a+b) \sec ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b)-4 (a-3 b) \sec (e+f x) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right) (a \cos (2 (e+f x))+a+2 b)-\left((a+b) \csc ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b)\right)+4 (a-3 b) \sec (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right) (a \cos (2 (e+f x))+a+2 b)-\frac{4 \sqrt{b} (b-3 a) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{\sqrt{a}}-\frac{4 \sqrt{b} (b-3 a) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{\sqrt{a}}-8 b (a+b)\right)}{32 f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)^2}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^3*(-8*b*(a + b) - (4*Sqrt[b]*(-3*a + b)*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x])/Sqrt[a] - (4*Sqrt[b]*(-3*a + b)*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x])/Sqrt[a] - (a + b)*(a + 2*b + a*Cos[2*(e + f*x)])*Csc[(e + f*x)/2]^2*Sec[e + f*x] - 4*(a - 3*b)*(a + 2*b + a*Cos[2*(e + f*x)])*Log[Cos[(e + f*x)/2]]*Sec[e + f*x] + 4*(a - 3*b)*(a + 2*b + a*Cos[2*(e + f*x)])*Log[Sin[(e + f*x)/2]]*Sec[e + f*x] + (a + b)*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^2*Sec[e + f*x]))/(32*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^2)","C",1
46,1,450,197,2.4355932,"\int \frac{\csc ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(-24 \left(a^2-6 a b+b^2\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right) (a \cos (2 (e+f x))+a+2 b)+24 \left(a^2-6 a b+b^2\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right) (a \cos (2 (e+f x))+a+2 b)-2 (a+b) \cot (e+f x) \csc ^3(e+f x) \left(4 \left(2 a^2-5 a b+5 b^2\right) \cos (2 (e+f x))+11 a^2-3 a (a-3 b) \cos (4 (e+f x))+43 a b-4 b^2\right)+96 \sqrt{a} \sqrt{b} (a-b) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)+96 \sqrt{a} \sqrt{b} (a-b) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)\right)}{256 f (a+b)^4 \left(a+b \sec ^2(e+f x)\right)^2}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])*(96*Sqrt[a]*(a - b)*Sqrt[b]*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)]) + 96*Sqrt[a]*(a - b)*Sqrt[b]*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)]) - 2*(a + b)*(11*a^2 + 43*a*b - 4*b^2 + 4*(2*a^2 - 5*a*b + 5*b^2)*Cos[2*(e + f*x)] - 3*a*(a - 3*b)*Cos[4*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x]^3 - 24*(a^2 - 6*a*b + b^2)*(a + 2*b + a*Cos[2*(e + f*x)])*Log[Cos[(e + f*x)/2]] + 24*(a^2 - 6*a*b + b^2)*(a + 2*b + a*Cos[2*(e + f*x)])*Log[Sin[(e + f*x)/2]])*Sec[e + f*x]^4)/(256*(a + b)^4*f*(a + b*Sec[e + f*x]^2)^2)","C",1
47,1,2738,267,23.5118158,"\int \frac{\sin ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Sin[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2,x]","\text{Result too large to show}","-\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{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(5 a^3+60 a^2 b+120 a b^2+64 b^3\right)}{16 a^5}+\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,"-1/512*((a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*(16*x + ((-a^3 + 6*a^2*b + 24*a*b^2 + 16*b^3)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(b*(a + b)^(3/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + ((a^2 + 8*a*b + 8*b^2)*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(b*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(a^2*(a + b*Sec[e + f*x]^2)^2) + (3*(a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*(-64*(a + 2*b)*x + ((a^4 - 16*a^3*b - 144*a^2*b^2 - 256*a*b^3 - 128*b^4)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(b*(a + b)^(3/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (16*a*Cos[2*f*x]*Sin[2*e])/f + (16*a*Cos[2*e]*Sin[2*f*x])/f - ((a^3 + 18*a^2*b + 48*a*b^2 + 32*b^3)*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(b*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(4096*a^3*(a + b*Sec[e + f*x]^2)^2) + (3*(a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*(((a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(3/2) - (a*Sqrt[b]*Sin[2*(e + f*x)])/((a + b)*(a + 2*b + a*Cos[2*(e + f*x)]))))/(2048*b^(3/2)*f*(a + b*Sec[e + f*x]^2)^2) - ((a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*(-((a*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(3/2)) + (Sqrt[b]*(a + 2*b)*Sin[2*(e + f*x)])/((a + b)*(a + 2*b + a*Cos[2*(e + f*x)]))))/(2048*b^(3/2)*f*(a + b*Sec[e + f*x]^2)^2) + ((a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*(-(((a^5 - 30*a^4*b - 480*a^3*b^2 - 1600*a^2*b^3 - 1920*a*b^4 - 768*b^5)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])) + (Sec[2*e]*(32*b*(5*a^4 + 39*a^3*b + 106*a^2*b^2 + 120*a*b^3 + 48*b^4)*f*x*Cos[2*e] + 16*a*b*(5*a^3 + 29*a^2*b + 48*a*b^2 + 24*b^3)*f*x*Cos[2*f*x] + 80*a^4*b*f*x*Cos[4*e + 2*f*x] + 464*a^3*b^2*f*x*Cos[4*e + 2*f*x] + 768*a^2*b^3*f*x*Cos[4*e + 2*f*x] + 384*a*b^4*f*x*Cos[4*e + 2*f*x] + a^5*Sin[2*e] + 34*a^4*b*Sin[2*e] + 224*a^3*b^2*Sin[2*e] + 576*a^2*b^3*Sin[2*e] + 640*a*b^4*Sin[2*e] + 256*b^5*Sin[2*e] - a^5*Sin[2*f*x] - 62*a^4*b*Sin[2*f*x] - 318*a^3*b^2*Sin[2*f*x] - 512*a^2*b^3*Sin[2*f*x] - 256*a*b^4*Sin[2*f*x] - 12*a^4*b*Sin[2*(e + 2*f*x)] - 36*a^3*b^2*Sin[2*(e + 2*f*x)] - 24*a^2*b^3*Sin[2*(e + 2*f*x)] - 30*a^4*b*Sin[4*e + 2*f*x] - 158*a^3*b^2*Sin[4*e + 2*f*x] - 256*a^2*b^3*Sin[4*e + 2*f*x] - 128*a*b^4*Sin[4*e + 2*f*x] - 12*a^4*b*Sin[6*e + 4*f*x] - 36*a^3*b^2*Sin[6*e + 4*f*x] - 24*a^2*b^3*Sin[6*e + 4*f*x] + 2*a^4*b*Sin[4*e + 6*f*x] + 2*a^3*b^2*Sin[4*e + 6*f*x] + 2*a^4*b*Sin[8*e + 6*f*x] + 2*a^3*b^2*Sin[8*e + 6*f*x]))/(a + 2*b + a*Cos[2*(e + f*x)])))/(2048*a^4*b*(a + b)*f*(a + b*Sec[e + f*x]^2)^2) + ((a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*(-(((a^6 - 48*a^5*b - 1200*a^4*b^2 - 6400*a^3*b^3 - 13440*a^2*b^4 - 12288*a*b^5 - 4096*b^6)*((ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Cos[2*e])/(8*a^5*b*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/8)*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Sin[2*e])/(a^5*b*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]])))/(a + b)) - (Sec[2*e]*(-960*a^5*b*f*x*Cos[2*e] - 10944*a^4*b^2*f*x*Cos[2*e] - 44544*a^3*b^3*f*x*Cos[2*e] - 83712*a^2*b^4*f*x*Cos[2*e] - 73728*a*b^5*f*x*Cos[2*e] - 24576*b^6*f*x*Cos[2*e] - 480*a^5*b*f*x*Cos[2*f*x] - 4512*a^4*b^2*f*x*Cos[2*f*x] - 13248*a^3*b^3*f*x*Cos[2*f*x] - 15360*a^2*b^4*f*x*Cos[2*f*x] - 6144*a*b^5*f*x*Cos[2*f*x] - 480*a^5*b*f*x*Cos[4*e + 2*f*x] - 4512*a^4*b^2*f*x*Cos[4*e + 2*f*x] - 13248*a^3*b^3*f*x*Cos[4*e + 2*f*x] - 15360*a^2*b^4*f*x*Cos[4*e + 2*f*x] - 6144*a*b^5*f*x*Cos[4*e + 2*f*x] - 3*a^6*Sin[2*e] - 156*a^5*b*Sin[2*e] - 1500*a^4*b^2*Sin[2*e] - 5760*a^3*b^3*Sin[2*e] - 10560*a^2*b^4*Sin[2*e] - 9216*a*b^5*Sin[2*e] - 3072*b^6*Sin[2*e] + 3*a^6*Sin[2*f*x] + 366*a^5*b*Sin[2*f*x] + 3000*a^4*b^2*Sin[2*f*x] + 8400*a^3*b^3*Sin[2*f*x] + 9600*a^2*b^4*Sin[2*f*x] + 3840*a*b^5*Sin[2*f*x] + 216*a^5*b*Sin[4*e + 2*f*x] + 1800*a^4*b^2*Sin[4*e + 2*f*x] + 5040*a^3*b^3*Sin[4*e + 2*f*x] + 5760*a^2*b^4*Sin[4*e + 2*f*x] + 2304*a*b^5*Sin[4*e + 2*f*x] + 76*a^5*b*Sin[2*e + 4*f*x] + 460*a^4*b^2*Sin[2*e + 4*f*x] + 768*a^3*b^3*Sin[2*e + 4*f*x] + 384*a^2*b^4*Sin[2*e + 4*f*x] + 76*a^5*b*Sin[6*e + 4*f*x] + 460*a^4*b^2*Sin[6*e + 4*f*x] + 768*a^3*b^3*Sin[6*e + 4*f*x] + 384*a^2*b^4*Sin[6*e + 4*f*x] - 16*a^5*b*Sin[4*e + 6*f*x] - 48*a^4*b^2*Sin[4*e + 6*f*x] - 32*a^3*b^3*Sin[4*e + 6*f*x] - 16*a^5*b*Sin[8*e + 6*f*x] - 48*a^4*b^2*Sin[8*e + 6*f*x] - 32*a^3*b^3*Sin[8*e + 6*f*x] + 4*a^5*b*Sin[6*e + 8*f*x] + 4*a^4*b^2*Sin[6*e + 8*f*x] + 4*a^5*b*Sin[10*e + 8*f*x] + 4*a^4*b^2*Sin[10*e + 8*f*x]))/(24*a^5*b*(a + b)*f*(a + 2*b + a*Cos[2*e + 2*f*x]))))/(512*(a + b*Sec[e + f*x]^2)^2)","C",0
48,1,1105,191,13.2416967,"\int \frac{\sin ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Sin[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{(\cos (2 e+2 f x) a+a+2 b)^2 \left(16 x+\frac{\left(-a^3+6 b a^2+24 b^2 a+16 b^3\right) \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 e)-i \sin (2 e))}{b (a+b)^{3/2} f \sqrt{b (\cos (e)-i \sin (e))^4}}+\frac{\left(a^2+8 b a+8 b^2\right) ((a+2 b) \sin (2 e)-a \sin (2 f x))}{b (a+b) f (\cos (2 (e+f x)) a+a+2 b) (\cos (e)-\sin (e)) (\cos (e)+\sin (e))}\right) \sec ^4(e+f x)}{256 a^2 \left(b \sec ^2(e+f x)+a\right)^2}+\frac{3 (\cos (2 e+2 f x) a+a+2 b)^2 \left(\frac{(a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}-\frac{a \sqrt{b} \sin (2 (e+f x))}{(a+b) (\cos (2 (e+f x)) a+a+2 b)}\right) \sec ^4(e+f x)}{1024 b^{3/2} f \left(b \sec ^2(e+f x)+a\right)^2}+\frac{(\cos (2 e+2 f x) a+a+2 b)^2 \left(\frac{\sec (2 e) \left(\sin (2 e) a^5-\sin (2 f x) a^5+80 b f x \cos (4 e+2 f x) a^4+34 b \sin (2 e) a^4-62 b \sin (2 f x) a^4-12 b \sin (2 (e+2 f x)) a^4-30 b \sin (4 e+2 f x) a^4-12 b \sin (6 e+4 f x) a^4+2 b \sin (4 e+6 f x) a^4+2 b \sin (8 e+6 f x) a^4+464 b^2 f x \cos (4 e+2 f x) a^3+224 b^2 \sin (2 e) a^3-318 b^2 \sin (2 f x) a^3-36 b^2 \sin (2 (e+2 f x)) a^3-158 b^2 \sin (4 e+2 f x) a^3-36 b^2 \sin (6 e+4 f x) a^3+2 b^2 \sin (4 e+6 f x) a^3+2 b^2 \sin (8 e+6 f x) a^3+768 b^3 f x \cos (4 e+2 f x) a^2+576 b^3 \sin (2 e) a^2-512 b^3 \sin (2 f x) a^2-24 b^3 \sin (2 (e+2 f x)) a^2-256 b^3 \sin (4 e+2 f x) a^2-24 b^3 \sin (6 e+4 f x) a^2+16 b \left(5 a^3+29 b a^2+48 b^2 a+24 b^3\right) f x \cos (2 f x) a+384 b^4 f x \cos (4 e+2 f x) a+640 b^4 \sin (2 e) a-256 b^4 \sin (2 f x) a-128 b^4 \sin (4 e+2 f x) a+32 b \left(5 a^4+39 b a^3+106 b^2 a^2+120 b^3 a+48 b^4\right) f x \cos (2 e)+256 b^5 \sin (2 e)\right)}{\cos (2 (e+f x)) a+a+2 b}-\frac{\left(a^5-30 b a^4-480 b^2 a^3-1600 b^3 a^2-1920 b^4 a-768 b^5\right) \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 e)-i \sin (2 e))}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) \sec ^4(e+f x)}{1024 a^4 b (a+b) f \left(b \sec ^2(e+f x)+a\right)^2}","-\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{3 x \left(a^2+8 a b+8 b^2\right)}{8 a^4}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)}",1,"-1/256*((a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*(16*x + ((-a^3 + 6*a^2*b + 24*a*b^2 + 16*b^3)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(b*(a + b)^(3/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + ((a^2 + 8*a*b + 8*b^2)*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(b*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(a^2*(a + b*Sec[e + f*x]^2)^2) + (3*(a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*(((a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(3/2) - (a*Sqrt[b]*Sin[2*(e + f*x)])/((a + b)*(a + 2*b + a*Cos[2*(e + f*x)]))))/(1024*b^(3/2)*f*(a + b*Sec[e + f*x]^2)^2) + ((a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*(-(((a^5 - 30*a^4*b - 480*a^3*b^2 - 1600*a^2*b^3 - 1920*a*b^4 - 768*b^5)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])) + (Sec[2*e]*(32*b*(5*a^4 + 39*a^3*b + 106*a^2*b^2 + 120*a*b^3 + 48*b^4)*f*x*Cos[2*e] + 16*a*b*(5*a^3 + 29*a^2*b + 48*a*b^2 + 24*b^3)*f*x*Cos[2*f*x] + 80*a^4*b*f*x*Cos[4*e + 2*f*x] + 464*a^3*b^2*f*x*Cos[4*e + 2*f*x] + 768*a^2*b^3*f*x*Cos[4*e + 2*f*x] + 384*a*b^4*f*x*Cos[4*e + 2*f*x] + a^5*Sin[2*e] + 34*a^4*b*Sin[2*e] + 224*a^3*b^2*Sin[2*e] + 576*a^2*b^3*Sin[2*e] + 640*a*b^4*Sin[2*e] + 256*b^5*Sin[2*e] - a^5*Sin[2*f*x] - 62*a^4*b*Sin[2*f*x] - 318*a^3*b^2*Sin[2*f*x] - 512*a^2*b^3*Sin[2*f*x] - 256*a*b^4*Sin[2*f*x] - 12*a^4*b*Sin[2*(e + 2*f*x)] - 36*a^3*b^2*Sin[2*(e + 2*f*x)] - 24*a^2*b^3*Sin[2*(e + 2*f*x)] - 30*a^4*b*Sin[4*e + 2*f*x] - 158*a^3*b^2*Sin[4*e + 2*f*x] - 256*a^2*b^3*Sin[4*e + 2*f*x] - 128*a*b^4*Sin[4*e + 2*f*x] - 12*a^4*b*Sin[6*e + 4*f*x] - 36*a^3*b^2*Sin[6*e + 4*f*x] - 24*a^2*b^3*Sin[6*e + 4*f*x] + 2*a^4*b*Sin[4*e + 6*f*x] + 2*a^3*b^2*Sin[4*e + 6*f*x] + 2*a^4*b*Sin[8*e + 6*f*x] + 2*a^3*b^2*Sin[8*e + 6*f*x]))/(a + 2*b + a*Cos[2*(e + f*x)])))/(1024*a^4*b*(a + b)*f*(a + b*Sec[e + f*x]^2)^2)","C",0
49,1,825,130,11.3068148,"\int \frac{\sin ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{(\cos (2 e+2 f x) a+a+2 b)^2 \left(16 x+\frac{\left(-a^3+6 b a^2+24 b^2 a+16 b^3\right) \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 e)-i \sin (2 e))}{b (a+b)^{3/2} f \sqrt{b (\cos (e)-i \sin (e))^4}}+\frac{\left(a^2+8 b a+8 b^2\right) ((a+2 b) \sin (2 e)-a \sin (2 f x))}{b (a+b) f (\cos (2 (e+f x)) a+a+2 b) (\cos (e)-\sin (e)) (\cos (e)+\sin (e))}\right) \sec ^4(e+f x)}{128 a^2 \left(b \sec ^2(e+f x)+a\right)^2}-\frac{(\cos (2 e+2 f x) a+a+2 b)^2 \left(-64 (a+2 b) x+\frac{\left(a^4-16 b a^3-144 b^2 a^2-256 b^3 a-128 b^4\right) \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 e)-i \sin (2 e))}{b (a+b)^{3/2} f \sqrt{b (\cos (e)-i \sin (e))^4}}+\frac{16 a \cos (2 f x) \sin (2 e)}{f}+\frac{16 a \cos (2 e) \sin (2 f x)}{f}-\frac{\left(a^3+18 b a^2+48 b^2 a+32 b^3\right) ((a+2 b) \sin (2 e)-a \sin (2 f x))}{b (a+b) f (\cos (2 (e+f x)) a+a+2 b) (\cos (e)-\sin (e)) (\cos (e)+\sin (e))}\right) \sec ^4(e+f x)}{256 a^3 \left(b \sec ^2(e+f x)+a\right)^2}+\frac{(\cos (2 e+2 f x) a+a+2 b)^2 \left(\frac{(a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}-\frac{a \sqrt{b} \sin (2 (e+f x))}{(a+b) (\cos (2 (e+f x)) a+a+2 b)}\right) \sec ^4(e+f x)}{128 b^{3/2} f \left(b \sec ^2(e+f x)+a\right)^2}+\frac{(\cos (2 e+2 f x) a+a+2 b)^2 \left(\frac{\sqrt{b} (a+2 b) \sin (2 (e+f x))}{(a+b) (\cos (2 (e+f x)) a+a+2 b)}-\frac{a \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}\right) \sec ^4(e+f x)}{256 b^{3/2} f \left(b \sec ^2(e+f x)+a\right)^2}","-\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{x (a+4 b)}{2 a^3}-\frac{b \tan (e+f x)}{a^2 f \left(a+b \tan ^2(e+f x)+b\right)}-\frac{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)}",1,"-1/128*((a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*(16*x + ((-a^3 + 6*a^2*b + 24*a*b^2 + 16*b^3)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(b*(a + b)^(3/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + ((a^2 + 8*a*b + 8*b^2)*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(b*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(a^2*(a + b*Sec[e + f*x]^2)^2) - ((a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*(-64*(a + 2*b)*x + ((a^4 - 16*a^3*b - 144*a^2*b^2 - 256*a*b^3 - 128*b^4)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(b*(a + b)^(3/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (16*a*Cos[2*f*x]*Sin[2*e])/f + (16*a*Cos[2*e]*Sin[2*f*x])/f - ((a^3 + 18*a^2*b + 48*a*b^2 + 32*b^3)*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(b*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(256*a^3*(a + b*Sec[e + f*x]^2)^2) + ((a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*(((a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(3/2) - (a*Sqrt[b]*Sin[2*(e + f*x)])/((a + b)*(a + 2*b + a*Cos[2*(e + f*x)]))))/(128*b^(3/2)*f*(a + b*Sec[e + f*x]^2)^2) + ((a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*(-((a*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(3/2)) + (Sqrt[b]*(a + 2*b)*Sin[2*(e + f*x)])/((a + b)*(a + 2*b + a*Cos[2*(e + f*x)]))))/(256*b^(3/2)*f*(a + b*Sec[e + f*x]^2)^2)","C",0
50,1,240,92,1.9840874,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(-2),x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(2 x (a \cos (2 (e+f x))+a+2 b)+\frac{b ((a+2 b) \sin (2 e)-a \sin (2 f x))}{f (a+b) (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}+\frac{b (3 a+2 b) (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{f (a+b)^{3/2} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{8 a^2 \left(a+b \sec ^2(e+f x)\right)^2}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^4*(2*x*(a + 2*b + a*Cos[2*(e + f*x)]) + (b*(3*a + 2*b)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[2*e] - I*Sin[2*e]))/((a + b)^(3/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (b*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/((a + b)*f*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(8*a^2*(a + b*Sec[e + f*x]^2)^2)","C",1
51,1,242,91,2.2162357,"\int \frac{\csc ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Csc[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{b ((a+2 b) \sin (2 e)-a \sin (2 f x))}{a (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}+2 \csc (e) \sin (f x) \csc (e+f x) (a \cos (2 (e+f x))+a+2 b)+\frac{3 b (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{8 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^2}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^4*((3*b*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + 2*(a + 2*b + a*Cos[2*(e + f*x)])*Csc[e]*Csc[e + f*x]*Sin[f*x] + (b*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(a*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(8*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^2)","C",1
52,1,303,123,6.1799258,"\int \frac{\csc ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(-3 a b \sec (2 e) \sin (2 f x)-2 (a+b) \cot (e) \csc ^2(e+f x) (a \cos (2 (e+f x))+a+2 b)+2 (a+b) \csc (e) \sin (f x) \csc ^3(e+f x) (a \cos (2 (e+f x))+a+2 b)+4 (a-2 b) \csc (e) \sin (f x) \csc (e+f x) (a \cos (2 (e+f x))+a+2 b)+\frac{3 b (3 a-2 b) (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}+3 b (a+2 b) \tan (2 e)\right)}{24 f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)^2}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^4*(-2*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])*Cot[e]*Csc[e + f*x]^2 + (3*(3*a - 2*b)*b*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + 4*(a - 2*b)*(a + 2*b + a*Cos[2*(e + f*x)])*Csc[e]*Csc[e + f*x]*Sin[f*x] + 2*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])*Csc[e]*Csc[e + f*x]^3*Sin[f*x] - 3*a*b*Sec[2*e]*Sin[2*f*x] + 3*b*(a + 2*b)*Tan[2*e]))/(24*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^2)","C",1
53,1,777,188,3.2179811,"\int \frac{\csc ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Csc[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(-\csc (e) \sec (2 e) \csc ^5(e+f x) \left(240 a^3 \sin (2 e-f x)-240 a^3 \sin (2 e+f x)+160 a^3 \sin (4 e+f x)-16 a^3 \sin (4 e+3 f x)+48 a^3 \sin (2 e+5 f x)+48 a^3 \sin (6 e+5 f x)-16 a^3 \sin (4 e+7 f x)-16 a^3 \sin (8 e+7 f x)+10 a \left(16 a^2+34 a b+123 b^2\right) \sin (f x)-a \left(16 a^2-223 a b+1336 b^2\right) \sin (3 f x)+640 a^2 b \sin (2 e-f x)-715 a^2 b \sin (2 e+f x)+415 a^2 b \sin (4 e+f x)+165 a^2 b \sin (2 e+3 f x)+208 a^2 b \sin (4 e+3 f x)+180 a^2 b \sin (6 e+3 f x)-268 a^2 b \sin (2 e+5 f x)-223 a^2 b \sin (6 e+5 f x)-45 a^2 b \sin (8 e+5 f x)+83 a^2 b \sin (4 e+7 f x)-15 a^2 b \sin (6 e+7 f x)+68 a^2 b \sin (8 e+7 f x)-1460 a b^2 \sin (2 e-f x)+860 a b^2 \sin (2 e+f x)+1830 a b^2 \sin (4 e+f x)-30 a b^2 \sin (2 e+3 f x)-1036 a b^2 \sin (4 e+3 f x)-330 a b^2 \sin (6 e+3 f x)+290 a b^2 \sin (2 e+5 f x)+230 a b^2 \sin (6 e+5 f x)+60 a b^2 \sin (8 e+5 f x)-6 a b^2 \sin (4 e+7 f x)-6 a b^2 \sin (8 e+7 f x)+240 b^3 \sin (2 e-f x)-240 b^3 \sin (2 e+f x)+120 b^3 \sin (2 e+3 f x)+120 b^3 \sin (6 e+3 f x)-24 b^3 \sin (2 e+5 f x)-24 b^3 \sin (6 e+5 f x)\right)+\frac{960 a b (3 a-4 b) (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{7680 f (a+b)^4 \left(a+b \sec ^2(e+f x)\right)^2}","-\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 + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^4*((960*a*(3*a - 4*b)*b*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) - Csc[e]*Csc[e + f*x]^5*Sec[2*e]*(10*a*(16*a^2 + 34*a*b + 123*b^2)*Sin[f*x] - a*(16*a^2 - 223*a*b + 1336*b^2)*Sin[3*f*x] + 240*a^3*Sin[2*e - f*x] + 640*a^2*b*Sin[2*e - f*x] - 1460*a*b^2*Sin[2*e - f*x] + 240*b^3*Sin[2*e - f*x] - 240*a^3*Sin[2*e + f*x] - 715*a^2*b*Sin[2*e + f*x] + 860*a*b^2*Sin[2*e + f*x] - 240*b^3*Sin[2*e + f*x] + 160*a^3*Sin[4*e + f*x] + 415*a^2*b*Sin[4*e + f*x] + 1830*a*b^2*Sin[4*e + f*x] + 165*a^2*b*Sin[2*e + 3*f*x] - 30*a*b^2*Sin[2*e + 3*f*x] + 120*b^3*Sin[2*e + 3*f*x] - 16*a^3*Sin[4*e + 3*f*x] + 208*a^2*b*Sin[4*e + 3*f*x] - 1036*a*b^2*Sin[4*e + 3*f*x] + 180*a^2*b*Sin[6*e + 3*f*x] - 330*a*b^2*Sin[6*e + 3*f*x] + 120*b^3*Sin[6*e + 3*f*x] + 48*a^3*Sin[2*e + 5*f*x] - 268*a^2*b*Sin[2*e + 5*f*x] + 290*a*b^2*Sin[2*e + 5*f*x] - 24*b^3*Sin[2*e + 5*f*x] + 48*a^3*Sin[6*e + 5*f*x] - 223*a^2*b*Sin[6*e + 5*f*x] + 230*a*b^2*Sin[6*e + 5*f*x] - 24*b^3*Sin[6*e + 5*f*x] - 45*a^2*b*Sin[8*e + 5*f*x] + 60*a*b^2*Sin[8*e + 5*f*x] - 16*a^3*Sin[4*e + 7*f*x] + 83*a^2*b*Sin[4*e + 7*f*x] - 6*a*b^2*Sin[4*e + 7*f*x] - 15*a^2*b*Sin[6*e + 7*f*x] - 16*a^3*Sin[8*e + 7*f*x] + 68*a^2*b*Sin[8*e + 7*f*x] - 6*a*b^2*Sin[8*e + 7*f*x])))/(7680*(a + b)^4*f*(a + b*Sec[e + f*x]^2)^2)","C",0
54,1,1641,214,10.5573796,"\int \frac{\sin ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Sin[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]","\frac{(\cos (2 (e+f x)) a+a+2 b) \sec ^6(e+f x) \left(-1935360 \sqrt{a} \cos (e+f x) b^{13/2}-3763200 a^{3/2} \cos (e+f x) b^{11/2}-403200 a^{3/2} \csc (e+f x) \sin (4 (e+f x)) b^{11/2}+483840 \tan ^{-1}\left(\frac{\left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right) \sin (e) \tan \left(\frac{f x}{2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2 b^5+483840 \tan ^{-1}\left(\frac{\left(i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}-\sqrt{a}\right) \sin (e) \tan \left(\frac{f x}{2}\right)+\cos (e) \left(\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)+\sqrt{a}\right)}{\sqrt{b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2 b^5-2803072 a^{5/2} \cos (e+f x) b^{9/2}-129024 a^{5/2} \cos (e+f x) \cos (4 (e+f x)) b^{9/2}-577024 a^{5/2} \csc (e+f x) \sin (4 (e+f x)) b^{9/2}+537600 a \tan ^{-1}\left(\frac{\left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right) \sin (e) \tan \left(\frac{f x}{2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2 b^4+537600 a \tan ^{-1}\left(\frac{\left(i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}-\sqrt{a}\right) \sin (e) \tan \left(\frac{f x}{2}\right)+\cos (e) \left(\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)+\sqrt{a}\right)}{\sqrt{b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2 b^4-936000 a^{7/2} \cos (e+f x) b^{7/2}+43200 a^{5/2} \cos (e+f x) (\cos (2 (e+f x)) a+a+2 b) b^{7/2}-115712 a^{7/2} \cos (e+f x) \cos (4 (e+f x)) b^{7/2}+4608 a^{7/2} \cos (e+f x) \cos (6 (e+f x)) b^{7/2}-252080 a^{7/2} \csc (e+f x) \sin (4 (e+f x)) b^{7/2}+115200 a^2 \tan ^{-1}\left(\frac{\left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right) \sin (e) \tan \left(\frac{f x}{2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2 b^3+115200 a^2 \tan ^{-1}\left(\frac{\left(i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}-\sqrt{a}\right) \sin (e) \tan \left(\frac{f x}{2}\right)+\cos (e) \left(\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)+\sqrt{a}\right)}{\sqrt{b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2 b^3+19200 a^{5/2} \cos (e) \cos (f x) (\cos (2 (e+f x)) a+a+2 b)^2 b^{5/2}-109000 a^{9/2} \cos (e+f x) b^{5/2}+24000 a^{7/2} \cos (e+f x) (\cos (2 (e+f x)) a+a+2 b) b^{5/2}-20352 a^{9/2} \cos (e+f x) \cos (4 (e+f x)) b^{5/2}+2048 a^{9/2} \cos (e+f x) \cos (6 (e+f x)) b^{5/2}-384 a^{9/2} \cos (e+f x) \cos (8 (e+f x)) b^{5/2}-19200 a^{5/2} (\cos (2 (e+f x)) a+a+2 b)^2 \sin (e) \sin (f x) b^{5/2}-32496 a^{9/2} \csc (e+f x) \sin (4 (e+f x)) b^{5/2}-900 a^{11/2} \cos (e+f x) b^{3/2}-900 a^{11/2} \cos (e+f x) \cos (2 (e+f x)) b^{3/2}+900 a^{9/2} \cos (e+f x) (\cos (2 (e+f x)) a+a+2 b) b^{3/2}+225 a^5 \tan ^{-1}\left(\frac{\left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right) \sin (e) \tan \left(\frac{f x}{2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2+225 a^5 \tan ^{-1}\left(\frac{\left(i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}-\sqrt{a}\right) \sin (e) \tan \left(\frac{f x}{2}\right)+\cos (e) \left(\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)+\sqrt{a}\right)}{\sqrt{b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2-225 a^5 \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2-225 a^5 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{a}}{\sqrt{b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2\right)}{491520 a^{11/2} b^{5/2} f \left(b \sec ^2(e+f x)+a\right)^3}","-\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{(a+3 b) (3 a+5 b) \cos ^3(e+f x)}{12 a^4 b f}-\frac{\cos ^5(e+f x)}{5 a^3 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{\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{\left(3 a^2+14 a b+13 b^2\right) \cos (e+f x)}{2 a^5 f}",1,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^6*(-900*a^(11/2)*b^(3/2)*Cos[e + f*x] - 109000*a^(9/2)*b^(5/2)*Cos[e + f*x] - 936000*a^(7/2)*b^(7/2)*Cos[e + f*x] - 2803072*a^(5/2)*b^(9/2)*Cos[e + f*x] - 3763200*a^(3/2)*b^(11/2)*Cos[e + f*x] - 1935360*Sqrt[a]*b^(13/2)*Cos[e + f*x] - 900*a^(11/2)*b^(3/2)*Cos[e + f*x]*Cos[2*(e + f*x)] + 900*a^(9/2)*b^(3/2)*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)]) + 24000*a^(7/2)*b^(5/2)*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)]) + 43200*a^(5/2)*b^(7/2)*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)]) + 225*a^5*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 + 115200*a^2*b^3*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 + 537600*a*b^4*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 + 483840*b^5*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 + 225*a^5*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 + 115200*a^2*b^3*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 + 537600*a*b^4*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 + 483840*b^5*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 - 225*a^5*ArcTan[(Sqrt[a] - Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 - 225*a^5*ArcTan[(Sqrt[a] + Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 + 19200*a^(5/2)*b^(5/2)*Cos[e]*Cos[f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^2 - 20352*a^(9/2)*b^(5/2)*Cos[e + f*x]*Cos[4*(e + f*x)] - 115712*a^(7/2)*b^(7/2)*Cos[e + f*x]*Cos[4*(e + f*x)] - 129024*a^(5/2)*b^(9/2)*Cos[e + f*x]*Cos[4*(e + f*x)] + 2048*a^(9/2)*b^(5/2)*Cos[e + f*x]*Cos[6*(e + f*x)] + 4608*a^(7/2)*b^(7/2)*Cos[e + f*x]*Cos[6*(e + f*x)] - 384*a^(9/2)*b^(5/2)*Cos[e + f*x]*Cos[8*(e + f*x)] - 19200*a^(5/2)*b^(5/2)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Sin[e]*Sin[f*x] - 32496*a^(9/2)*b^(5/2)*Csc[e + f*x]*Sin[4*(e + f*x)] - 252080*a^(7/2)*b^(7/2)*Csc[e + f*x]*Sin[4*(e + f*x)] - 577024*a^(5/2)*b^(9/2)*Csc[e + f*x]*Sin[4*(e + f*x)] - 403200*a^(3/2)*b^(11/2)*Csc[e + f*x]*Sin[4*(e + f*x)]))/(491520*a^(11/2)*b^(5/2)*f*(a + b*Sec[e + f*x]^2)^3)","C",1
55,1,1153,154,9.7750261,"\int \frac{\sin ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3,x]","\frac{(\cos (2 (e+f x)) a+a+2 b)^3 \sec ^6(e+f x) \left(3 \left(9 a^4+1920 b^3 a+4480 b^4\right) \tan ^{-1}\left(\frac{\left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right) \sin (e) \tan \left(\frac{f x}{2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)-\frac{-27 \sqrt{b} \cos (e+f x) a^{11/2}-27 \sqrt{b} \cos (e+f x) \cos (2 (e+f x)) a^{11/2}+162 b^{3/2} \cos (e+f x) a^{9/2}+27 \sqrt{b} \cos (e+f x) (\cos (2 (e+f x)) a+a+2 b) a^{9/2}+54 b^{3/2} \csc (e+f x) \sin (4 (e+f x)) a^{9/2}-27 \tan ^{-1}\left(\frac{\left(i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}-\sqrt{a}\right) \sin (e) \tan \left(\frac{f x}{2}\right)+\cos (e) \left(\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)+\sqrt{a}\right)}{\sqrt{b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2 a^4+27 \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2 a^4+27 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{a}}{\sqrt{b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2 a^4+10816 b^{5/2} \cos (e+f x) a^{7/2}-216 b^{3/2} \cos (e+f x) (\cos (2 (e+f x)) a+a+2 b) a^{7/2}+1920 b^{5/2} \cos (e+f x) \cos (4 (e+f x)) a^{7/2}-128 b^{5/2} \cos (e+f x) \cos (6 (e+f x)) a^{7/2}+3108 b^{5/2} \csc (e+f x) \sin (4 (e+f x)) a^{7/2}+51552 b^{7/2} \cos (e+f x) a^{5/2}+47936 b^{7/2} \cos (e+f x) \cos (2 (e+f x)) a^{5/2}-3600 b^{5/2} \cos (e+f x) (\cos (2 (e+f x)) a+a+2 b) a^{5/2}+3584 b^{7/2} \cos (e+f x) \cos (4 (e+f x)) a^{5/2}-2304 b^{5/2} \cos (e) \cos (f x) (\cos (2 (e+f x)) a+a+2 b)^2 a^{3/2}+87424 b^{9/2} \cos (e+f x) a^{3/2}+44800 b^{9/2} \cos (e+f x) \cos (2 (e+f x)) a^{3/2}-5184 b^{7/2} \cos (e+f x) (\cos (2 (e+f x)) a+a+2 b) a^{3/2}+2304 b^{5/2} (\cos (2 (e+f x)) a+a+2 b)^2 \sin (e) \sin (f x) a^{3/2}-5760 b^3 \tan ^{-1}\left(\frac{\left(i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}-\sqrt{a}\right) \sin (e) \tan \left(\frac{f x}{2}\right)+\cos (e) \left(\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)+\sqrt{a}\right)}{\sqrt{b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2 a+53760 b^{11/2} \cos (e+f x) \sqrt{a}-13440 b^4 \tan ^{-1}\left(\frac{\left(i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}-\sqrt{a}\right) \sin (e) \tan \left(\frac{f x}{2}\right)+\cos (e) \left(\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)+\sqrt{a}\right)}{\sqrt{b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2}{(\cos (2 (e+f x)) a+a+2 b)^2}\right)}{24576 a^{9/2} b^{5/2} f \left(b \sec ^2(e+f x)+a\right)^3}","\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{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{\cos ^3(e+f x)}{3 a^3 f}",1,"((a + 2*b + a*Cos[2*(e + f*x)])^3*Sec[e + f*x]^6*(3*(9*a^4 + 1920*a*b^3 + 4480*b^4)*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]] - (-27*a^(11/2)*Sqrt[b]*Cos[e + f*x] + 162*a^(9/2)*b^(3/2)*Cos[e + f*x] + 10816*a^(7/2)*b^(5/2)*Cos[e + f*x] + 51552*a^(5/2)*b^(7/2)*Cos[e + f*x] + 87424*a^(3/2)*b^(9/2)*Cos[e + f*x] + 53760*Sqrt[a]*b^(11/2)*Cos[e + f*x] - 27*a^(11/2)*Sqrt[b]*Cos[e + f*x]*Cos[2*(e + f*x)] + 47936*a^(5/2)*b^(7/2)*Cos[e + f*x]*Cos[2*(e + f*x)] + 44800*a^(3/2)*b^(9/2)*Cos[e + f*x]*Cos[2*(e + f*x)] + 27*a^(9/2)*Sqrt[b]*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)]) - 216*a^(7/2)*b^(3/2)*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)]) - 3600*a^(5/2)*b^(5/2)*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)]) - 5184*a^(3/2)*b^(7/2)*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)]) - 27*a^4*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 - 5760*a*b^3*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 - 13440*b^4*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 + 27*a^4*ArcTan[(Sqrt[a] - Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 + 27*a^4*ArcTan[(Sqrt[a] + Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 - 2304*a^(3/2)*b^(5/2)*Cos[e]*Cos[f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^2 + 1920*a^(7/2)*b^(5/2)*Cos[e + f*x]*Cos[4*(e + f*x)] + 3584*a^(5/2)*b^(7/2)*Cos[e + f*x]*Cos[4*(e + f*x)] - 128*a^(7/2)*b^(5/2)*Cos[e + f*x]*Cos[6*(e + f*x)] + 2304*a^(3/2)*b^(5/2)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Sin[e]*Sin[f*x] + 54*a^(9/2)*b^(3/2)*Csc[e + f*x]*Sin[4*(e + f*x)] + 3108*a^(7/2)*b^(5/2)*Csc[e + f*x]*Sin[4*(e + f*x)])/(a + 2*b + a*Cos[2*(e + f*x)])^2))/(24576*a^(9/2)*b^(5/2)*f*(a + b*Sec[e + f*x]^2)^3)","C",1
56,1,656,116,5.7449227,"\int \frac{\sin (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Sin[e + f*x]/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b)^3 \left(15 \left(a^3+64 b^3\right) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)+15 \left(a^3+64 b^3\right) \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)+\frac{\sqrt{a} \left(-15 a^{5/2} (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)-15 a^{5/2} (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{a}}{\sqrt{b}}\right)+24 a^4 \sqrt{b} \cos (e+f x)+6 a^4 \sqrt{b} \sin (4 (e+f x)) \csc (e+f x)-24 a^3 b^{3/2} \cos (e+f x)-72 a^3 b^{3/2} \cos (e+f x) \cos (2 (e+f x))-24 a^3 \sqrt{b} \cos (e+f x) (a \cos (2 (e+f x))+a+2 b)-144 a^2 b^{5/2} \cos (e+f x)+72 a^2 b^{3/2} \cos (e+f x) (a \cos (2 (e+f x))+a+2 b)-1152 b^{7/2} \cos (e+f x) (a \cos (2 (e+f x))+a+2 b)-512 b^{5/2} \cos (e) \cos (f x) (a \cos (2 (e+f x))+a+2 b)^2+512 b^{5/2} \sin (e) \sin (f x) (a \cos (2 (e+f x))+a+2 b)^2+512 b^{9/2} \cos (e+f x)\right)}{(a \cos (2 (e+f x))+a+2 b)^2}\right)}{4096 a^{7/2} b^{5/2} f \left(a+b \sec ^2(e+f x)\right)^3}","\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{5 \cos ^3(e+f x)}{8 a^2 f \left(a \cos ^2(e+f x)+b\right)}+\frac{\cos ^5(e+f x)}{4 a f \left(a \cos ^2(e+f x)+b\right)^2}",1,"((a + 2*b + a*Cos[2*(e + f*x)])^3*Sec[e + f*x]^6*(15*(a^3 + 64*b^3)*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]] + 15*(a^3 + 64*b^3)*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]] + (Sqrt[a]*(24*a^4*Sqrt[b]*Cos[e + f*x] - 24*a^3*b^(3/2)*Cos[e + f*x] - 144*a^2*b^(5/2)*Cos[e + f*x] + 512*b^(9/2)*Cos[e + f*x] - 72*a^3*b^(3/2)*Cos[e + f*x]*Cos[2*(e + f*x)] - 24*a^3*Sqrt[b]*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)]) + 72*a^2*b^(3/2)*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)]) - 1152*b^(7/2)*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)]) - 15*a^(5/2)*ArcTan[(Sqrt[a] - Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 - 15*a^(5/2)*ArcTan[(Sqrt[a] + Sqrt[a + b]*Tan[(e + f*x)/2])/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 - 512*b^(5/2)*Cos[e]*Cos[f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^2 + 512*b^(5/2)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Sin[e]*Sin[f*x] + 6*a^4*Sqrt[b]*Csc[e + f*x]*Sin[4*(e + f*x)]))/(a + 2*b + a*Cos[2*(e + f*x)])^2))/(4096*a^(7/2)*b^(5/2)*f*(a + b*Sec[e + f*x]^2)^3)","C",1
57,1,447,154,2.4032881,"\int \frac{\csc (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Csc[e + f*x]/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\sec ^5(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{8 b^2 (a+b)^2}{a^2}-\frac{2 b (9 a+5 b) (a+b) (a \cos (2 (e+f x))+a+2 b)}{a^2}+\frac{\sqrt{b} \left(15 a^2+10 a b+3 b^2\right) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{a^{5/2}}+\frac{\sqrt{b} \left(15 a^2+10 a b+3 b^2\right) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{a^{5/2}}-8 \sec (e+f x) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right) (a \cos (2 (e+f x))+a+2 b)^2+8 \sec (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right) (a \cos (2 (e+f x))+a+2 b)^2\right)}{64 f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)^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{\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 \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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^5*((8*b^2*(a + b)^2)/a^2 - (2*b*(a + b)*(9*a + 5*b)*(a + 2*b + a*Cos[2*(e + f*x)]))/a^2 + (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2*Sec[e + f*x])/a^(5/2) + (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2*Sec[e + f*x])/a^(5/2) - 8*(a + 2*b + a*Cos[2*(e + f*x)])^2*Log[Cos[(e + f*x)/2]]*Sec[e + f*x] + 8*(a + 2*b + a*Cos[2*(e + f*x)])^2*Log[Sin[(e + f*x)/2]]*Sec[e + f*x]))/(64*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^3)","C",1
58,1,532,213,3.5703283,"\int \frac{\csc ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Csc[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\sec ^5(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(-\frac{\sqrt{b} \left(-15 a^2+10 a b+b^2\right) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{a^{3/2}}-\frac{\sqrt{b} \left(-15 a^2+10 a b+b^2\right) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{a^{3/2}}+\frac{8 b^2 (a+b)^2}{a}-\frac{2 b (9 a+b) (a+b) (a \cos (2 (e+f x))+a+2 b)}{a}+(a+b) \sec ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b)^2-4 (a-5 b) \sec (e+f x) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right) (a \cos (2 (e+f x))+a+2 b)^2-(a+b) \csc ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b)^2+4 (a-5 b) \sec (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right) (a \cos (2 (e+f x))+a+2 b)^2\right)}{64 f (a+b)^4 \left(a+b \sec ^2(e+f x)\right)^3}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^5*((8*b^2*(a + b)^2)/a - (2*b*(a + b)*(9*a + b)*(a + 2*b + a*Cos[2*(e + f*x)]))/a - (Sqrt[b]*(-15*a^2 + 10*a*b + b^2)*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2*Sec[e + f*x])/a^(3/2) - (Sqrt[b]*(-15*a^2 + 10*a*b + b^2)*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2*Sec[e + f*x])/a^(3/2) - (a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Csc[(e + f*x)/2]^2*Sec[e + f*x] - 4*(a - 5*b)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Log[Cos[(e + f*x)/2]]*Sec[e + f*x] + 4*(a - 5*b)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Log[Sin[(e + f*x)/2]]*Sec[e + f*x] + (a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Sec[(e + f*x)/2]^2*Sec[e + f*x]))/(64*(a + b)^4*f*(a + b*Sec[e + f*x]^2)^3)","C",1
59,1,549,257,5.1047914,"\int \frac{\csc ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(-48 \left(a^2-10 a b+5 b^2\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right) (a \cos (2 (e+f x))+a+2 b)^2+48 \left(a^2-10 a b+5 b^2\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right) (a \cos (2 (e+f x))+a+2 b)^2+\frac{48 \sqrt{b} \left(5 a^2-10 a b+b^2\right) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{\sqrt{a}}+\frac{48 \sqrt{b} \left(5 a^2-10 a b+b^2\right) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left(\frac{\sin (e) \tan \left(\frac{f x}{2}\right) \left(-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right)+\cos (e) \left(\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left(\frac{f x}{2}\right)\right)}{\sqrt{b}}\right)}{\sqrt{a}}-2 (a+b) \cot (e+f x) \csc ^3(e+f x) \left(-3 a^3 \cos (6 (e+f x))+30 a^3+18 a^2 b \cos (6 (e+f x))+112 a^2 b+\left(35 a^3+78 a^2 b-93 a b^2+224 b^3\right) \cos (2 (e+f x))+2 \left(a^3-8 a^2 b+53 a b^2-10 b^3\right) \cos (4 (e+f x))-3 a b^2 \cos (6 (e+f x))+182 a b^2-140 b^3\right)\right)}{1024 f (a+b)^5 \left(a+b \sec ^2(e+f x)\right)^3}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])*((48*Sqrt[b]*(5*a^2 - 10*a*b + b^2)*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2)/Sqrt[a] + (48*Sqrt[b]*(5*a^2 - 10*a*b + b^2)*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2)/Sqrt[a] - 2*(a + b)*(30*a^3 + 112*a^2*b + 182*a*b^2 - 140*b^3 + (35*a^3 + 78*a^2*b - 93*a*b^2 + 224*b^3)*Cos[2*(e + f*x)] + 2*(a^3 - 8*a^2*b + 53*a*b^2 - 10*b^3)*Cos[4*(e + f*x)] - 3*a^3*Cos[6*(e + f*x)] + 18*a^2*b*Cos[6*(e + f*x)] - 3*a*b^2*Cos[6*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x]^3 - 48*(a^2 - 10*a*b + 5*b^2)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Log[Cos[(e + f*x)/2]] + 48*(a^2 - 10*a*b + 5*b^2)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Log[Sin[(e + f*x)/2]])*Sec[e + f*x]^6)/(1024*(a + b)^5*f*(a + b*Sec[e + f*x]^2)^3)","C",1
60,1,1639,314,18.8705602,"\int \frac{\sin ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Sin[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]","\frac{5 (\cos (2 e+2 f x) a+a+2 b)^3 \left(\frac{\left(3 a^2+8 b a+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}-\frac{a \sqrt{b} \left(3 a^2+16 b a+3 (a+2 b) \cos (2 (e+f x)) a+16 b^2\right) \sin (2 (e+f x))}{(a+b)^2 (\cos (2 (e+f x)) a+a+2 b)^2}\right) \sec ^6(e+f x)}{65536 b^{5/2} f \left(b \sec ^2(e+f x)+a\right)^3}-\frac{15 (\cos (2 e+2 f x) a+a+2 b)^3 \left(\frac{a \sec (2 e) \left(\left(-9 a^4-16 b a^3+48 b^2 a^2+128 b^3 a+64 b^4\right) \sin (2 f x)+a \left(-3 a^3+2 b a^2+24 b^2 a+16 b^3\right) \sin (2 (e+2 f x))+\left(3 a^4-64 b^2 a^2-128 b^3 a-64 b^4\right) \sin (4 e+2 f x)\right)+\left(9 a^5+18 b a^4-64 b^2 a^3-256 b^3 a^2-320 b^4 a-128 b^5\right) \tan (2 e)}{a^2 (\cos (2 (e+f x)) a+a+2 b)^2}-\frac{6 a^2 \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 e)-i \sin (2 e))}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) \sec ^6(e+f x)}{262144 b^2 (a+b)^2 f \left(b \sec ^2(e+f x)+a\right)^3}+\frac{3 (\cos (2 e+2 f x) a+a+2 b)^3 \left(-1536 (a+2 b) x-\frac{3 \left(a^6-8 b a^5+120 b^2 a^4+1280 b^3 a^3+3200 b^4 a^2+3072 b^5 a+1024 b^6\right) \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 e)-i \sin (2 e))}{b^2 (a+b)^{5/2} f \sqrt{b (\cos (e)-i \sin (e))^4}}+\frac{4 \left(a^4+32 b a^3+160 b^2 a^2+256 b^3 a+128 b^4\right) \sec (2 e) ((a+2 b) \sin (2 e)-a \sin (2 f x))}{b (a+b) f (\cos (2 (e+f x)) a+a+2 b)^2}+\frac{256 a \sin (2 (e+f x))}{f}+\frac{a \left(-3 a^5+26 b a^4+736 b^2 a^3+2624 b^3 a^2+3200 b^4 a+1280 b^5\right) \sec (2 e) \sin (2 f x)+\left(3 a^6-24 b a^5-920 b^2 a^4-4864 b^3 a^3-10112 b^4 a^2-9216 b^5 a-3072 b^6\right) \tan (2 e)}{b^2 (a+b)^2 f (\cos (2 (e+f x)) a+a+2 b)}\right) \sec ^6(e+f x)}{65536 a^4 \left(b \sec ^2(e+f x)+a\right)^3}-\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \left(\frac{256 \sin (6 (e+f x)) a^3}{f}+\frac{192 (a+2 b) (-6 i \cos (4 (e+f x))-6 \sin (4 (e+f x))) a^2}{f}+\frac{1152 i (a+2 b) (\cos (4 (e+f x))+i \sin (4 (e+f x))) a^2}{f}+\frac{1152 \left(7 a^2+32 b a+32 b^2\right) (\sin (2 (e+f x))-i \cos (2 (e+f x))) a}{f}+\frac{1152 \left(7 a^2+32 b a+32 b^2\right) (i \cos (2 (e+f x))+\sin (2 (e+f x))) a}{f}-6144 \left(7 a^3+54 b a^2+120 b^2 a+80 b^3\right) x-\frac{3 \left(3 a^8-64 b a^7+2240 b^2 a^6+53760 b^3 a^5+313600 b^4 a^4+802816 b^5 a^3+1032192 b^6 a^2+655360 b^7 a+163840 b^8\right) \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 e)-i \sin (2 e))}{b^2 (a+b)^{5/2} f \sqrt{b (\cos (e)-i \sin (e))^4}}+\frac{12 \left(a^6+72 b a^5+840 b^2 a^4+3584 b^3 a^3+6912 b^4 a^2+6144 b^5 a+2048 b^6\right) \sec (2 e) ((a+2 b) \sin (2 e)-a \sin (2 f x))}{b (a+b) f (\cos (2 (e+f x)) a+a+2 b)^2}+\frac{3 \left(3 a \left(-a^7+22 b a^6+1352 b^2 a^5+11312 b^3 a^4+37120 b^4 a^3+57856 b^5 a^2+43008 b^6 a+12288 b^7\right) \sec (2 e) \sin (2 f x)+\left(3 a^8-64 b a^7-4480 b^2 a^6-45696 b^3 a^5-196928 b^4 a^4-438272 b^5 a^3-528384 b^6 a^2-327680 b^7 a-81920 b^8\right) \tan (2 e)\right)}{b^2 (a+b)^2 f (\cos (2 (e+f x)) a+a+2 b)}\right) \sec ^6(e+f x)}{393216 a^6 \left(b \sec ^2(e+f x)+a\right)^3}","-\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{5 x (a+2 b) \left(a^2+16 a b+16 b^2\right)}{16 a^6}-\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{\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*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(((3*a^2 + 8*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(65536*b^(5/2)*f*(a + b*Sec[e + f*x]^2)^3) - (15*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((-6*a^2*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (a*Sec[2*e]*((-9*a^4 - 16*a^3*b + 48*a^2*b^2 + 128*a*b^3 + 64*b^4)*Sin[2*f*x] + a*(-3*a^3 + 2*a^2*b + 24*a*b^2 + 16*b^3)*Sin[2*(e + 2*f*x)] + (3*a^4 - 64*a^2*b^2 - 128*a*b^3 - 64*b^4)*Sin[4*e + 2*f*x]) + (9*a^5 + 18*a^4*b - 64*a^3*b^2 - 256*a^2*b^3 - 320*a*b^4 - 128*b^5)*Tan[2*e])/(a^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(262144*b^2*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^3) + (3*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(-1536*(a + 2*b)*x - (3*(a^6 - 8*a^5*b + 120*a^4*b^2 + 1280*a^3*b^3 + 3200*a^2*b^4 + 3072*a*b^5 + 1024*b^6)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(b^2*(a + b)^(5/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (4*(a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4)*Sec[2*e]*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(b*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)])^2) + (256*a*Sin[2*(e + f*x)])/f + (a*(-3*a^5 + 26*a^4*b + 736*a^3*b^2 + 2624*a^2*b^3 + 3200*a*b^4 + 1280*b^5)*Sec[2*e]*Sin[2*f*x] + (3*a^6 - 24*a^5*b - 920*a^4*b^2 - 4864*a^3*b^3 - 10112*a^2*b^4 - 9216*a*b^5 - 3072*b^6)*Tan[2*e])/(b^2*(a + b)^2*f*(a + 2*b + a*Cos[2*(e + f*x)]))))/(65536*a^4*(a + b*Sec[e + f*x]^2)^3) - ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(-6144*(7*a^3 + 54*a^2*b + 120*a*b^2 + 80*b^3)*x - (3*(3*a^8 - 64*a^7*b + 2240*a^6*b^2 + 53760*a^5*b^3 + 313600*a^4*b^4 + 802816*a^3*b^5 + 1032192*a^2*b^6 + 655360*a*b^7 + 163840*b^8)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(b^2*(a + b)^(5/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (12*(a^6 + 72*a^5*b + 840*a^4*b^2 + 3584*a^3*b^3 + 6912*a^2*b^4 + 6144*a*b^5 + 2048*b^6)*Sec[2*e]*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(b*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)])^2) + (1152*a*(7*a^2 + 32*a*b + 32*b^2)*((-I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)]))/f + (1152*a*(7*a^2 + 32*a*b + 32*b^2)*(I*Cos[2*(e + f*x)] + Sin[2*(e + f*x)]))/f + (192*a^2*(a + 2*b)*((-6*I)*Cos[4*(e + f*x)] - 6*Sin[4*(e + f*x)]))/f + ((1152*I)*a^2*(a + 2*b)*(Cos[4*(e + f*x)] + I*Sin[4*(e + f*x)]))/f + (256*a^3*Sin[6*(e + f*x)])/f + (3*(3*a*(-a^7 + 22*a^6*b + 1352*a^5*b^2 + 11312*a^4*b^3 + 37120*a^3*b^4 + 57856*a^2*b^5 + 43008*a*b^6 + 12288*b^7)*Sec[2*e]*Sin[2*f*x] + (3*a^8 - 64*a^7*b - 4480*a^6*b^2 - 45696*a^5*b^3 - 196928*a^4*b^4 - 438272*a^3*b^5 - 528384*a^2*b^6 - 327680*a*b^7 - 81920*b^8)*Tan[2*e]))/(b^2*(a + b)^2*f*(a + 2*b + a*Cos[2*(e + f*x)]))))/(393216*a^6*(a + b*Sec[e + f*x]^2)^3)","C",0
61,1,2469,238,25.0112661,"\int \frac{\sin ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Sin[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]","\text{Result too large to show}","-\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{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{\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*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(((3*a^2 + 8*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(16384*b^(5/2)*f*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((-3*a*(a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(5/2) + (Sqrt[b]*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a*b + 4*b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(16384*b^(5/2)*f*(a + b*Sec[e + f*x]^2)^3) - (3*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((2*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 + 256*b^5)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (Sec[2*e]*(256*b^2*(a + b)^2*(3*a^2 + 8*a*b + 8*b^2)*f*x*Cos[2*e] + 512*a*b^2*(a + b)^2*(a + 2*b)*f*x*Cos[2*f*x] + 128*a^4*b^2*f*x*Cos[2*(e + 2*f*x)] + 256*a^3*b^3*f*x*Cos[2*(e + 2*f*x)] + 128*a^2*b^4*f*x*Cos[2*(e + 2*f*x)] + 512*a^4*b^2*f*x*Cos[4*e + 2*f*x] + 2048*a^3*b^3*f*x*Cos[4*e + 2*f*x] + 2560*a^2*b^4*f*x*Cos[4*e + 2*f*x] + 1024*a*b^5*f*x*Cos[4*e + 2*f*x] + 128*a^4*b^2*f*x*Cos[6*e + 4*f*x] + 256*a^3*b^3*f*x*Cos[6*e + 4*f*x] + 128*a^2*b^4*f*x*Cos[6*e + 4*f*x] - 9*a^6*Sin[2*e] + 12*a^5*b*Sin[2*e] + 684*a^4*b^2*Sin[2*e] + 2880*a^3*b^3*Sin[2*e] + 5280*a^2*b^4*Sin[2*e] + 4608*a*b^5*Sin[2*e] + 1536*b^6*Sin[2*e] + 9*a^6*Sin[2*f*x] - 14*a^5*b*Sin[2*f*x] - 608*a^4*b^2*Sin[2*f*x] - 2112*a^3*b^3*Sin[2*f*x] - 2560*a^2*b^4*Sin[2*f*x] - 1024*a*b^5*Sin[2*f*x] + 3*a^6*Sin[2*(e + 2*f*x)] - 12*a^5*b*Sin[2*(e + 2*f*x)] - 204*a^4*b^2*Sin[2*(e + 2*f*x)] - 384*a^3*b^3*Sin[2*(e + 2*f*x)] - 192*a^2*b^4*Sin[2*(e + 2*f*x)] - 3*a^6*Sin[4*e + 2*f*x] + 10*a^5*b*Sin[4*e + 2*f*x] + 304*a^4*b^2*Sin[4*e + 2*f*x] + 1056*a^3*b^3*Sin[4*e + 2*f*x] + 1280*a^2*b^4*Sin[4*e + 2*f*x] + 512*a*b^5*Sin[4*e + 2*f*x]))/(a + 2*b + a*Cos[2*(e + f*x)])^2))/(65536*a^3*b^2*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^3) - ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((-6*a^2*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (a*Sec[2*e]*((-9*a^4 - 16*a^3*b + 48*a^2*b^2 + 128*a*b^3 + 64*b^4)*Sin[2*f*x] + a*(-3*a^3 + 2*a^2*b + 24*a*b^2 + 16*b^3)*Sin[2*(e + 2*f*x)] + (3*a^4 - 64*a^2*b^2 - 128*a*b^3 - 64*b^4)*Sin[4*e + 2*f*x]) + (9*a^5 + 18*a^4*b - 64*a^3*b^2 - 256*a^2*b^3 - 320*a*b^4 - 128*b^5)*Tan[2*e])/(a^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(8192*b^2*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(-1536*(a + 2*b)*x - (3*(a^6 - 8*a^5*b + 120*a^4*b^2 + 1280*a^3*b^3 + 3200*a^2*b^4 + 3072*a*b^5 + 1024*b^6)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(b^2*(a + b)^(5/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (4*(a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4)*Sec[2*e]*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(b*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)])^2) + (256*a*Sin[2*(e + f*x)])/f + (a*(-3*a^5 + 26*a^4*b + 736*a^3*b^2 + 2624*a^2*b^3 + 3200*a*b^4 + 1280*b^5)*Sec[2*e]*Sin[2*f*x] + (3*a^6 - 24*a^5*b - 920*a^4*b^2 - 4864*a^3*b^3 - 10112*a^2*b^4 - 9216*a*b^5 - 3072*b^6)*Tan[2*e])/(b^2*(a + b)^2*f*(a + 2*b + a*Cos[2*(e + f*x)]))))/(16384*a^4*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(768*(7*a^2 + 32*a*b + 32*b^2)*x + (3*(a^7 - 14*a^6*b + 336*a^5*b^2 + 5600*a^4*b^3 + 22400*a^3*b^4 + 37632*a^2*b^5 + 28672*a*b^6 + 8192*b^7)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(b^2*(a + b)^(5/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) - (4*(a^5 + 50*a^4*b + 400*a^3*b^2 + 1120*a^2*b^3 + 1280*a*b^4 + 512*b^5)*Sec[2*e]*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(b*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)])^2) - ((768*I)*a*(a + 2*b)*(Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)]))/f + ((768*I)*a*(a + 2*b)*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]))/f + (128*a^2*Sin[4*(e + f*x)])/f + (a*(3*a^6 - 44*a^5*b - 1900*a^4*b^2 - 10880*a^3*b^3 - 23360*a^2*b^4 - 21504*a*b^5 - 7168*b^6)*Sec[2*e]*Sin[2*f*x] + (-3*a^7 + 42*a^6*b + 2192*a^5*b^2 + 16480*a^4*b^3 + 51200*a^3*b^4 + 77824*a^2*b^5 + 57344*a*b^6 + 16384*b^7)*Tan[2*e])/(b^2*(a + b)^2*f*(a + 2*b + a*Cos[2*(e + f*x)]))))/(32768*a^5*(a + b*Sec[e + f*x]^2)^3)","C",0
62,1,1915,184,17.644284,"\int \frac{\sin ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3,x]","\frac{5 (\cos (2 e+2 f x) a+a+2 b)^3 \left(\frac{\left(3 a^2+8 b a+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}-\frac{a \sqrt{b} \left(3 a^2+16 b a+3 (a+2 b) \cos (2 (e+f x)) a+16 b^2\right) \sin (2 (e+f x))}{(a+b)^2 (\cos (2 (e+f x)) a+a+2 b)^2}\right) \sec ^6(e+f x)}{8192 b^{5/2} f \left(b \sec ^2(e+f x)+a\right)^3}+\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \left(\frac{\sqrt{b} \left(3 a^3+14 b a^2+24 b^2 a+\left(3 a^2+4 b a+4 b^2\right) \cos (2 (e+f x)) a+16 b^3\right) \sin (2 (e+f x))}{(a+b)^2 (\cos (2 (e+f x)) a+a+2 b)^2}-\frac{3 a (a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}\right) \sec ^6(e+f x)}{2048 b^{5/2} f \left(b \sec ^2(e+f x)+a\right)^3}-\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \left(\frac{2 \left(3 a^5-10 b a^4+80 b^2 a^3+480 b^3 a^2+640 b^4 a+256 b^5\right) \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 e)-i \sin (2 e))}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}+\frac{\sec (2 e) \left(-9 \sin (2 e) a^6+9 \sin (2 f x) a^6+3 \sin (2 (e+2 f x)) a^6-3 \sin (4 e+2 f x) a^6+12 b \sin (2 e) a^5-14 b \sin (2 f x) a^5-12 b \sin (2 (e+2 f x)) a^5+10 b \sin (4 e+2 f x) a^5+128 b^2 f x \cos (2 (e+2 f x)) a^4+512 b^2 f x \cos (4 e+2 f x) a^4+128 b^2 f x \cos (6 e+4 f x) a^4+684 b^2 \sin (2 e) a^4-608 b^2 \sin (2 f x) a^4-204 b^2 \sin (2 (e+2 f x)) a^4+304 b^2 \sin (4 e+2 f x) a^4+256 b^3 f x \cos (2 (e+2 f x)) a^3+2048 b^3 f x \cos (4 e+2 f x) a^3+256 b^3 f x \cos (6 e+4 f x) a^3+2880 b^3 \sin (2 e) a^3-2112 b^3 \sin (2 f x) a^3-384 b^3 \sin (2 (e+2 f x)) a^3+1056 b^3 \sin (4 e+2 f x) a^3+128 b^4 f x \cos (2 (e+2 f x)) a^2+2560 b^4 f x \cos (4 e+2 f x) a^2+128 b^4 f x \cos (6 e+4 f x) a^2+5280 b^4 \sin (2 e) a^2-2560 b^4 \sin (2 f x) a^2-192 b^4 \sin (2 (e+2 f x)) a^2+1280 b^4 \sin (4 e+2 f x) a^2+512 b^2 (a+b)^2 (a+2 b) f x \cos (2 f x) a+1024 b^5 f x \cos (4 e+2 f x) a+4608 b^5 \sin (2 e) a-1024 b^5 \sin (2 f x) a+512 b^5 \sin (4 e+2 f x) a+256 b^2 (a+b)^2 \left(3 a^2+8 b a+8 b^2\right) f x \cos (2 e)+1536 b^6 \sin (2 e)\right)}{(\cos (2 (e+f x)) a+a+2 b)^2}\right) \sec ^6(e+f x)}{4096 a^3 b^2 (a+b)^2 f \left(b \sec ^2(e+f x)+a\right)^3}-\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \left(\frac{a \sec (2 e) \left(\left(-9 a^4-16 b a^3+48 b^2 a^2+128 b^3 a+64 b^4\right) \sin (2 f x)+a \left(-3 a^3+2 b a^2+24 b^2 a+16 b^3\right) \sin (2 (e+2 f x))+\left(3 a^4-64 b^2 a^2-128 b^3 a-64 b^4\right) \sin (4 e+2 f x)\right)+\left(9 a^5+18 b a^4-64 b^2 a^3-256 b^3 a^2-320 b^4 a-128 b^5\right) \tan (2 e)}{a^2 (\cos (2 (e+f x)) a+a+2 b)^2}-\frac{6 a^2 \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 e)-i \sin (2 e))}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) \sec ^6(e+f x)}{4096 b^2 (a+b)^2 f \left(b \sec ^2(e+f x)+a\right)^3}-\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \left(-1536 (a+2 b) x-\frac{3 \left(a^6-8 b a^5+120 b^2 a^4+1280 b^3 a^3+3200 b^4 a^2+3072 b^5 a+1024 b^6\right) \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 e)-i \sin (2 e))}{b^2 (a+b)^{5/2} f \sqrt{b (\cos (e)-i \sin (e))^4}}+\frac{4 \left(a^4+32 b a^3+160 b^2 a^2+256 b^3 a+128 b^4\right) \sec (2 e) ((a+2 b) \sin (2 e)-a \sin (2 f x))}{b (a+b) f (\cos (2 (e+f x)) a+a+2 b)^2}+\frac{256 a \sin (2 (e+f x))}{f}+\frac{a \left(-3 a^5+26 b a^4+736 b^2 a^3+2624 b^3 a^2+3200 b^4 a+1280 b^5\right) \sec (2 e) \sin (2 f x)+\left(3 a^6-24 b a^5-920 b^2 a^4-4864 b^3 a^3-10112 b^4 a^2-9216 b^5 a-3072 b^6\right) \tan (2 e)}{b^2 (a+b)^2 f (\cos (2 (e+f x)) a+a+2 b)}\right) \sec ^6(e+f x)}{8192 a^4 \left(b \sec ^2(e+f x)+a\right)^3}","\frac{x (a+6 b)}{2 a^4}-\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{\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{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"(5*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(((3*a^2 + 8*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(8192*b^(5/2)*f*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((-3*a*(a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(5/2) + (Sqrt[b]*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a*b + 4*b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(2048*b^(5/2)*f*(a + b*Sec[e + f*x]^2)^3) - ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((2*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 + 256*b^5)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (Sec[2*e]*(256*b^2*(a + b)^2*(3*a^2 + 8*a*b + 8*b^2)*f*x*Cos[2*e] + 512*a*b^2*(a + b)^2*(a + 2*b)*f*x*Cos[2*f*x] + 128*a^4*b^2*f*x*Cos[2*(e + 2*f*x)] + 256*a^3*b^3*f*x*Cos[2*(e + 2*f*x)] + 128*a^2*b^4*f*x*Cos[2*(e + 2*f*x)] + 512*a^4*b^2*f*x*Cos[4*e + 2*f*x] + 2048*a^3*b^3*f*x*Cos[4*e + 2*f*x] + 2560*a^2*b^4*f*x*Cos[4*e + 2*f*x] + 1024*a*b^5*f*x*Cos[4*e + 2*f*x] + 128*a^4*b^2*f*x*Cos[6*e + 4*f*x] + 256*a^3*b^3*f*x*Cos[6*e + 4*f*x] + 128*a^2*b^4*f*x*Cos[6*e + 4*f*x] - 9*a^6*Sin[2*e] + 12*a^5*b*Sin[2*e] + 684*a^4*b^2*Sin[2*e] + 2880*a^3*b^3*Sin[2*e] + 5280*a^2*b^4*Sin[2*e] + 4608*a*b^5*Sin[2*e] + 1536*b^6*Sin[2*e] + 9*a^6*Sin[2*f*x] - 14*a^5*b*Sin[2*f*x] - 608*a^4*b^2*Sin[2*f*x] - 2112*a^3*b^3*Sin[2*f*x] - 2560*a^2*b^4*Sin[2*f*x] - 1024*a*b^5*Sin[2*f*x] + 3*a^6*Sin[2*(e + 2*f*x)] - 12*a^5*b*Sin[2*(e + 2*f*x)] - 204*a^4*b^2*Sin[2*(e + 2*f*x)] - 384*a^3*b^3*Sin[2*(e + 2*f*x)] - 192*a^2*b^4*Sin[2*(e + 2*f*x)] - 3*a^6*Sin[4*e + 2*f*x] + 10*a^5*b*Sin[4*e + 2*f*x] + 304*a^4*b^2*Sin[4*e + 2*f*x] + 1056*a^3*b^3*Sin[4*e + 2*f*x] + 1280*a^2*b^4*Sin[4*e + 2*f*x] + 512*a*b^5*Sin[4*e + 2*f*x]))/(a + 2*b + a*Cos[2*(e + f*x)])^2))/(4096*a^3*b^2*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^3) - ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((-6*a^2*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (a*Sec[2*e]*((-9*a^4 - 16*a^3*b + 48*a^2*b^2 + 128*a*b^3 + 64*b^4)*Sin[2*f*x] + a*(-3*a^3 + 2*a^2*b + 24*a*b^2 + 16*b^3)*Sin[2*(e + 2*f*x)] + (3*a^4 - 64*a^2*b^2 - 128*a*b^3 - 64*b^4)*Sin[4*e + 2*f*x]) + (9*a^5 + 18*a^4*b - 64*a^3*b^2 - 256*a^2*b^3 - 320*a*b^4 - 128*b^5)*Tan[2*e])/(a^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(4096*b^2*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^3) - ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(-1536*(a + 2*b)*x - (3*(a^6 - 8*a^5*b + 120*a^4*b^2 + 1280*a^3*b^3 + 3200*a^2*b^4 + 3072*a*b^5 + 1024*b^6)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(b^2*(a + b)^(5/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (4*(a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4)*Sec[2*e]*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(b*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)])^2) + (256*a*Sin[2*(e + f*x)])/f + (a*(-3*a^5 + 26*a^4*b + 736*a^3*b^2 + 2624*a^2*b^3 + 3200*a*b^4 + 1280*b^5)*Sec[2*e]*Sin[2*f*x] + (3*a^6 - 24*a^5*b - 920*a^4*b^2 - 4864*a^3*b^3 - 10112*a^2*b^4 - 9216*a*b^5 - 3072*b^6)*Tan[2*e])/(b^2*(a + b)^2*f*(a + 2*b + a*Cos[2*(e + f*x)]))))/(8192*a^4*(a + b*Sec[e + f*x]^2)^3)","C",0
63,1,332,144,5.6247619,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(-3),x]","\frac{\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{b \left(\left(9 a^2+28 a b+16 b^2\right) \sin (2 e)-3 a (3 a+2 b) \sin (2 f x)\right) (a \cos (2 (e+f x))+a+2 b)}{f (a+b)^2 (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}+\frac{b \left(15 a^2+20 a b+8 b^2\right) (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{f (a+b)^{5/2} \sqrt{b (\cos (e)-i \sin (e))^4}}-\frac{4 b^2 ((a+2 b) \sin (2 e)-a \sin (2 f x))}{f (a+b) (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}+8 x (a \cos (2 (e+f x))+a+2 b)^2\right)}{64 a^3 \left(a+b \sec ^2(e+f x)\right)^3}","\frac{x}{a^3}-\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{\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 \tan (e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^6*(8*x*(a + 2*b + a*Cos[2*(e + f*x)])^2 + (b*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])^2*(Cos[2*e] - I*Sin[2*e]))/((a + b)^(5/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) - (4*b^2*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/((a + b)*f*(Cos[e] - Sin[e])*(Cos[e] + Sin[e])) + (b*(a + 2*b + a*Cos[2*(e + f*x)])*((9*a^2 + 28*a*b + 16*b^2)*Sin[2*e] - 3*a*(3*a + 2*b)*Sin[2*f*x]))/((a + b)^2*f*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(64*a^3*(a + b*Sec[e + f*x]^2)^3)","C",0
64,1,987,124,6.8200802,"\int \frac{\csc ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Csc[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3,x]","\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \left(\frac{15 b \tan ^{-1}\left(\sec (f x) \left(\frac{\cos (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}-\frac{i \sin (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}\right) (-a \sin (f x)-2 b \sin (f x)+a \sin (2 e+f x))\right) \cos (2 e)}{64 \sqrt{a+b} f \sqrt{b \cos (4 e)-i b \sin (4 e)}}-\frac{15 i b \tan ^{-1}\left(\sec (f x) \left(\frac{\cos (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}-\frac{i \sin (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}\right) (-a \sin (f x)-2 b \sin (f x)+a \sin (2 e+f x))\right) \sin (2 e)}{64 \sqrt{a+b} f \sqrt{b \cos (4 e)-i b \sin (4 e)}}\right) \sec ^6(e+f x)}{(a+b)^3 \left(b \sec ^2(e+f x)+a\right)^3}+\frac{(\cos (2 e+2 f x) a+a+2 b) \csc (e) \csc (e+f x) \sec (2 e) \left(-32 \sin (f x) a^4+32 \sin (3 f x) a^4-48 \sin (2 e-f x) a^4+48 \sin (2 e+f x) a^4-32 \sin (4 e+f x) a^4-8 \sin (2 e+3 f x) a^4+32 \sin (4 e+3 f x) a^4-8 \sin (6 e+3 f x) a^4+8 \sin (2 e+5 f x) a^4+8 \sin (6 e+5 f x) a^4-64 b \sin (f x) a^3+46 b \sin (3 f x) a^3-128 b \sin (2 e-f x) a^3+146 b \sin (2 e+f x) a^3-82 b \sin (4 e+f x) a^3+18 b \sin (2 e+3 f x) a^3+73 b \sin (4 e+3 f x) a^3-9 b \sin (6 e+3 f x) a^3-9 b \sin (2 e+5 f x) a^3+9 b \sin (4 e+5 f x) a^3+22 b^2 \sin (f x) a^2-54 b^2 \sin (3 f x) a^2-106 b^2 \sin (2 e-f x) a^2+182 b^2 \sin (2 e+f x) a^2-54 b^2 \sin (4 e+f x) a^2+54 b^2 \sin (2 e+3 f x) a^2+24 b^2 \sin (4 e+3 f x) a^2-24 b^2 \sin (6 e+3 f x) a^2-2 b^2 \sin (2 e+5 f x) a^2+2 b^2 \sin (4 e+5 f x) a^2+80 b^3 \sin (f x) a-8 b^3 \sin (3 f x) a+80 b^3 \sin (2 e-f x) a+80 b^3 \sin (2 e+f x) a-80 b^3 \sin (4 e+f x) a+8 b^3 \sin (2 e+3 f x) a+8 b^3 \sin (4 e+3 f x) a-8 b^3 \sin (6 e+3 f x) a+16 b^4 \sin (f x)+16 b^4 \sin (2 e-f x)+16 b^4 \sin (2 e+f x)-16 b^4 \sin (4 e+f x)\right) \sec ^6(e+f x)}{512 a^2 (a+b)^3 f \left(b \sec ^2(e+f x)+a\right)^3}","-\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,"((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((15*b*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Cos[2*e])/(64*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - (((15*I)/64)*b*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Sin[2*e])/(Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]])))/((a + b)^3*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])*Csc[e]*Csc[e + f*x]*Sec[2*e]*Sec[e + f*x]^6*(-32*a^4*Sin[f*x] - 64*a^3*b*Sin[f*x] + 22*a^2*b^2*Sin[f*x] + 80*a*b^3*Sin[f*x] + 16*b^4*Sin[f*x] + 32*a^4*Sin[3*f*x] + 46*a^3*b*Sin[3*f*x] - 54*a^2*b^2*Sin[3*f*x] - 8*a*b^3*Sin[3*f*x] - 48*a^4*Sin[2*e - f*x] - 128*a^3*b*Sin[2*e - f*x] - 106*a^2*b^2*Sin[2*e - f*x] + 80*a*b^3*Sin[2*e - f*x] + 16*b^4*Sin[2*e - f*x] + 48*a^4*Sin[2*e + f*x] + 146*a^3*b*Sin[2*e + f*x] + 182*a^2*b^2*Sin[2*e + f*x] + 80*a*b^3*Sin[2*e + f*x] + 16*b^4*Sin[2*e + f*x] - 32*a^4*Sin[4*e + f*x] - 82*a^3*b*Sin[4*e + f*x] - 54*a^2*b^2*Sin[4*e + f*x] - 80*a*b^3*Sin[4*e + f*x] - 16*b^4*Sin[4*e + f*x] - 8*a^4*Sin[2*e + 3*f*x] + 18*a^3*b*Sin[2*e + 3*f*x] + 54*a^2*b^2*Sin[2*e + 3*f*x] + 8*a*b^3*Sin[2*e + 3*f*x] + 32*a^4*Sin[4*e + 3*f*x] + 73*a^3*b*Sin[4*e + 3*f*x] + 24*a^2*b^2*Sin[4*e + 3*f*x] + 8*a*b^3*Sin[4*e + 3*f*x] - 8*a^4*Sin[6*e + 3*f*x] - 9*a^3*b*Sin[6*e + 3*f*x] - 24*a^2*b^2*Sin[6*e + 3*f*x] - 8*a*b^3*Sin[6*e + 3*f*x] + 8*a^4*Sin[2*e + 5*f*x] - 9*a^3*b*Sin[2*e + 5*f*x] - 2*a^2*b^2*Sin[2*e + 5*f*x] + 9*a^3*b*Sin[4*e + 5*f*x] + 2*a^2*b^2*Sin[4*e + 5*f*x] + 8*a^4*Sin[6*e + 5*f*x]))/(512*a^2*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^3)","C",0
65,1,994,164,4.0541086,"\int \frac{\csc ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]","\frac{(\cos (2 (e+f x)) a+a+2 b) \sec ^6(e+f x) \left(\frac{480 (3 a-4 b) b \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 (e+f x)) a+a+2 b)^2 (\cos (2 e)-i \sin (2 e))}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}-\frac{\csc (e) \csc ^3(e+f x) \sec (2 e) \left(224 \sin (2 e-f x) a^4-224 \sin (2 e+f x) a^4+176 \sin (4 e+f x) a^4+48 \sin (2 e+3 f x) a^4-96 \sin (4 e+3 f x) a^4+48 \sin (6 e+3 f x) a^4+16 \sin (2 e+5 f x) a^4+16 \sin (6 e+5 f x) a^4+16 \sin (4 e+7 f x) a^4+16 \sin (8 e+7 f x) a^4+576 b \sin (2 e-f x) a^3-657 b \sin (2 e+f x) a^3+569 b \sin (4 e+f x) a^3+111 b \sin (2 e+3 f x) a^3-152 b \sin (4 e+3 f x) a^3+192 b \sin (6 e+3 f x) a^3+72 b \sin (4 e+5 f x) a^3+27 b \sin (6 e+5 f x) a^3+45 b \sin (8 e+5 f x) a^3-83 b \sin (4 e+7 f x) a^3+27 b \sin (6 e+7 f x) a^3-56 b \sin (8 e+7 f x) a^3+124 b^2 \sin (2 e-f x) a^2-538 b^2 \sin (2 e+f x) a^2+666 b^2 \sin (4 e+f x) a^2+360 b^2 \sin (2 e+3 f x) a^2+146 b^2 \sin (4 e+3 f x) a^2+558 b^2 \sin (6 e+3 f x) a^2-598 b^2 \sin (2 e+5 f x) a^2+150 b^2 \sin (4 e+5 f x) a^2-388 b^2 \sin (6 e+5 f x) a^2-60 b^2 \sin (8 e+5 f x) a^2+6 b^2 \sin (4 e+7 f x) a^2-6 b^2 \sin (6 e+7 f x) a^2-2184 b^3 \sin (2 e-f x) a+984 b^3 \sin (2 e+f x) a+1704 b^3 \sin (4 e+f x) a+312 b^3 \sin (2 e+3 f x) a-728 b^3 \sin (4 e+3 f x) a-168 b^3 \sin (6 e+3 f x) a+48 b^3 \sin (2 e+5 f x) a-48 b^3 \sin (4 e+5 f x) a+4 \left(44 a^4+122 b a^3+63 b^2 a^2+126 b^3 a+36 b^4\right) \sin (f x)+\left(-96 a^4-71 b a^3+344 b^2 a^2-1208 b^3 a+48 b^4\right) \sin (3 f x)+144 b^4 \sin (2 e-f x)+144 b^4 \sin (2 e+f x)-144 b^4 \sin (4 e+f x)-48 b^4 \sin (2 e+3 f x)-48 b^4 \sin (4 e+3 f x)+48 b^4 \sin (6 e+3 f x)\right)}{a}\right)}{6144 (a+b)^4 f \left(b \sec ^2(e+f x)+a\right)^3}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^6*((480*(3*a - 4*b)*b*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])^2*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) - (Csc[e]*Csc[e + f*x]^3*Sec[2*e]*(4*(44*a^4 + 122*a^3*b + 63*a^2*b^2 + 126*a*b^3 + 36*b^4)*Sin[f*x] + (-96*a^4 - 71*a^3*b + 344*a^2*b^2 - 1208*a*b^3 + 48*b^4)*Sin[3*f*x] + 224*a^4*Sin[2*e - f*x] + 576*a^3*b*Sin[2*e - f*x] + 124*a^2*b^2*Sin[2*e - f*x] - 2184*a*b^3*Sin[2*e - f*x] + 144*b^4*Sin[2*e - f*x] - 224*a^4*Sin[2*e + f*x] - 657*a^3*b*Sin[2*e + f*x] - 538*a^2*b^2*Sin[2*e + f*x] + 984*a*b^3*Sin[2*e + f*x] + 144*b^4*Sin[2*e + f*x] + 176*a^4*Sin[4*e + f*x] + 569*a^3*b*Sin[4*e + f*x] + 666*a^2*b^2*Sin[4*e + f*x] + 1704*a*b^3*Sin[4*e + f*x] - 144*b^4*Sin[4*e + f*x] + 48*a^4*Sin[2*e + 3*f*x] + 111*a^3*b*Sin[2*e + 3*f*x] + 360*a^2*b^2*Sin[2*e + 3*f*x] + 312*a*b^3*Sin[2*e + 3*f*x] - 48*b^4*Sin[2*e + 3*f*x] - 96*a^4*Sin[4*e + 3*f*x] - 152*a^3*b*Sin[4*e + 3*f*x] + 146*a^2*b^2*Sin[4*e + 3*f*x] - 728*a*b^3*Sin[4*e + 3*f*x] - 48*b^4*Sin[4*e + 3*f*x] + 48*a^4*Sin[6*e + 3*f*x] + 192*a^3*b*Sin[6*e + 3*f*x] + 558*a^2*b^2*Sin[6*e + 3*f*x] - 168*a*b^3*Sin[6*e + 3*f*x] + 48*b^4*Sin[6*e + 3*f*x] + 16*a^4*Sin[2*e + 5*f*x] - 598*a^2*b^2*Sin[2*e + 5*f*x] + 48*a*b^3*Sin[2*e + 5*f*x] + 72*a^3*b*Sin[4*e + 5*f*x] + 150*a^2*b^2*Sin[4*e + 5*f*x] - 48*a*b^3*Sin[4*e + 5*f*x] + 16*a^4*Sin[6*e + 5*f*x] + 27*a^3*b*Sin[6*e + 5*f*x] - 388*a^2*b^2*Sin[6*e + 5*f*x] + 45*a^3*b*Sin[8*e + 5*f*x] - 60*a^2*b^2*Sin[8*e + 5*f*x] + 16*a^4*Sin[4*e + 7*f*x] - 83*a^3*b*Sin[4*e + 7*f*x] + 6*a^2*b^2*Sin[4*e + 7*f*x] + 27*a^3*b*Sin[6*e + 7*f*x] - 6*a^2*b^2*Sin[6*e + 7*f*x] + 16*a^4*Sin[8*e + 7*f*x] - 56*a^3*b*Sin[8*e + 7*f*x]))/a))/(6144*(a + b)^4*f*(a + b*Sec[e + f*x]^2)^3)","C",0
66,1,479,242,5.768504,"\int \frac{\csc ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Csc[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(8 \left(8 a^2-59 a b+23 b^2\right) \csc (e) \sin (f x) \csc (e+f x) (a \cos (2 (e+f x))+a+2 b)^2+15 b \sec (2 e) \left(\left(9 a^2+16 a b-8 b^2\right) \sin (2 e)+3 a (2 b-3 a) \sin (2 f x)\right) (a \cos (2 (e+f x))+a+2 b)+\frac{15 b \left(15 a^2-40 a b+8 b^2\right) (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}-60 b^2 (a+b) \sec (2 e) ((a+2 b) \sin (2 e)-a \sin (2 f x))-24 (a+b)^2 \cot (e) \csc ^4(e+f x) (a \cos (2 (e+f x))+a+2 b)^2-8 (4 a-11 b) (a+b) \cot (e) \csc ^2(e+f x) (a \cos (2 (e+f x))+a+2 b)^2+24 (a+b)^2 \csc (e) \sin (f x) \csc ^5(e+f x) (a \cos (2 (e+f x))+a+2 b)^2+8 (4 a-11 b) (a+b) \csc (e) \sin (f x) \csc ^3(e+f x) (a \cos (2 (e+f x))+a+2 b)^2\right)}{960 f (a+b)^5 \left(a+b \sec ^2(e+f x)\right)^3}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^6*(-8*(4*a - 11*b)*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Cot[e]*Csc[e + f*x]^2 - 24*(a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2*Cot[e]*Csc[e + f*x]^4 + (15*b*(15*a^2 - 40*a*b + 8*b^2)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])^2*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + 8*(8*a^2 - 59*a*b + 23*b^2)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Csc[e]*Csc[e + f*x]*Sin[f*x] + 8*(4*a - 11*b)*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Csc[e]*Csc[e + f*x]^3*Sin[f*x] + 24*(a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2*Csc[e]*Csc[e + f*x]^5*Sin[f*x] - 60*b^2*(a + b)*Sec[2*e]*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]) + 15*b*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[2*e]*((9*a^2 + 16*a*b - 8*b^2)*Sin[2*e] + 3*a*(-3*a + 2*b)*Sin[2*f*x])))/(960*(a + b)^5*f*(a + b*Sec[e + f*x]^2)^3)","C",0
67,1,152,139,0.8466147,"\int \sqrt{a+b \sec ^2(e+f x)} \sin ^5(e+f x) \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^5,x]","-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(\frac{2 \left(a \cos ^2(e+f x)+b\right)^{5/2}}{5 a^2}-\frac{2 (2 a+b) \left(a \cos ^2(e+f x)+b\right)^{3/2}}{3 a^2}+2 \sqrt{a \cos ^2(e+f x)+b}-2 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)+b}}{\sqrt{b}}\right)\right)}{\sqrt{2} f \sqrt{a \cos (2 e+2 f x)+a+2 b}}","\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,"-((Cos[e + f*x]*(-2*Sqrt[b]*ArcTanh[Sqrt[b + a*Cos[e + f*x]^2]/Sqrt[b]] + 2*Sqrt[b + a*Cos[e + f*x]^2] - (2*(2*a + b)*(b + a*Cos[e + f*x]^2)^(3/2))/(3*a^2) + (2*(b + a*Cos[e + f*x]^2)^(5/2))/(5*a^2))*Sqrt[a + b*Sec[e + f*x]^2])/(Sqrt[2]*f*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]))","A",1
68,1,120,100,0.3766837,"\int \sqrt{a+b \sec ^2(e+f x)} \sin ^3(e+f x) \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^3,x]","\frac{\sqrt{2} \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(\sqrt{a \cos ^2(e+f x)+b} \left(a \cos ^2(e+f x)-3 a+b\right)+3 a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)+b}}{\sqrt{b}}\right)\right)}{3 a f \sqrt{a \cos (2 (e+f x))+a+2 b}}","\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[2]*Cos[e + f*x]*(3*a*Sqrt[b]*ArcTanh[Sqrt[b + a*Cos[e + f*x]^2]/Sqrt[b]] + Sqrt[b + a*Cos[e + f*x]^2]*(-3*a + b + a*Cos[e + f*x]^2))*Sqrt[a + b*Sec[e + f*x]^2])/(3*a*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])","A",1
69,1,98,66,0.1349346,"\int \sqrt{a+b \sec ^2(e+f x)} \sin (e+f x) \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x],x]","\frac{\sqrt{2} \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)+b}}{\sqrt{b}}\right)-\sqrt{a \cos ^2(e+f x)+b}\right)}{f \sqrt{a \cos (2 (e+f x))+a+2 b}}","\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[2]*Cos[e + f*x]*(Sqrt[b]*ArcTanh[Sqrt[b + a*Cos[e + f*x]^2]/Sqrt[b]] - Sqrt[b + a*Cos[e + f*x]^2])*Sqrt[a + b*Sec[e + f*x]^2])/(f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])","A",1
70,1,119,82,0.1289375,"\int \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{2} \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{b}}\right)-\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{a+b}}\right)\right)}{f \sqrt{a \cos (2 e+2 f x)+a+2 b}}","\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[2]*(Sqrt[b]*ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[b]] - Sqrt[a + b]*ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[a + b]])*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(f*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])","A",1
71,1,163,124,0.4292105,"\int \csc ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(2 \sqrt{b} (a+b) \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{b}}\right)-\sqrt{a+b} (a+2 b) \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{a+b}}\right)-\left((a+b) \csc ^2(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b}\right)\right)}{\sqrt{2} f (a+b) \sqrt{a \cos (2 (e+f x))+a+2 b}}","\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,"(Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2]*(2*Sqrt[b]*(a + b)*ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[b]] - Sqrt[a + b]*(a + 2*b)*ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[a + b]] - (a + b)*Csc[e + f*x]^2*Sqrt[a + b - a*Sin[e + f*x]^2]))/(Sqrt[2]*(a + b)*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])","A",1
72,1,198,183,1.3821984,"\int \csc ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(-\left(3 a^2+12 a b+8 b^2\right) \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{a+b}}\right)+8 \sqrt{b} (a+b)^2 \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{b}}\right)-(a+b) \csc ^2(e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \left(2 (a+b) \csc ^2(e+f x)+3 a+4 b\right)\right)}{4 \sqrt{2} f (a+b)^2 \sqrt{a \cos (2 (e+f x))+a+2 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,"(Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2]*(8*Sqrt[b]*(a + b)^2*ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[b]] - Sqrt[a + b]*(3*a^2 + 12*a*b + 8*b^2)*ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[a + b]] - (a + b)*Csc[e + f*x]^2*(3*a + 4*b + 2*(a + b)*Csc[e + f*x]^2)*Sqrt[a + b - a*Sin[e + f*x]^2]))/(4*Sqrt[2]*(a + b)^2*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])","A",1
73,0,0,240,8.9105867,"\int \sqrt{a+b \sec ^2(e+f x)} \sin ^6(e+f x) \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^6,x]","\int \sqrt{a+b \sec ^2(e+f x)} \sin ^6(e+f x) \, dx","-\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{\left(5 a^3-15 a^2 b-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{\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,"Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^6, x]","F",-1
74,0,0,181,5.3488269,"\int \sqrt{a+b \sec ^2(e+f x)} \sin ^4(e+f x) \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^4,x]","\int \sqrt{a+b \sec ^2(e+f x)} \sin ^4(e+f x) \, dx","\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,"Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^4, x]","F",-1
75,1,432,123,5.7957389,"\int \sqrt{a+b \sec ^2(e+f x)} \sin ^2(e+f x) \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^2,x]","\frac{e^{-i (e+f x)} \cos (e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{2 e^{2 i (e+f x)} \left(-i (a-b) \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)+i (a-b) \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)-4 \sqrt{a} \sqrt{b} \log \left(\frac{f \left(i \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}-\sqrt{b} \left(-1+e^{2 i (e+f x)}\right)\right)}{2 b \left(1+e^{2 i (e+f x)}\right)}\right)+2 a f x-2 b f x\right)}{\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}+i \left(-1+e^{2 i (e+f x)}\right)\right) \sqrt{a+b \sec ^2(e+f x)}}{4 \sqrt{2} f \sqrt{a \cos (2 e+2 f x)+a+2 b}}","\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,"(Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]*(I*(-1 + E^((2*I)*(e + f*x))) + (2*E^((2*I)*(e + f*x))*(2*a*f*x - 2*b*f*x - I*(a - b)*Log[a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] + I*(a - b)*Log[a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - 4*Sqrt[a]*Sqrt[b]*Log[((-(Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))) + I*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*f)/(2*b*(1 + E^((2*I)*(e + f*x))))]))/(Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]))*Sqrt[a + b*Sec[e + f*x]^2])/(4*Sqrt[2]*E^(I*(e + f*x))*f*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])","C",1
76,0,0,79,1.6368781,"\int \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]^2],x]","\int \sqrt{a+b \sec ^2(e+f x)} \, dx","\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,"Integrate[Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
77,1,61,68,0.2015045,"\int \csc ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\cot (e+f x) \sqrt{a+b \sec ^2(e+f x)} \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \sin ^2(e+f x)}{-a \sin ^2(e+f x)+a+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{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f}",1,"-((Cot[e + f*x]*Hypergeometric2F1[-1/2, 1, 1/2, (b*Sin[e + f*x]^2)/(a + b - a*Sin[e + f*x]^2)]*Sqrt[a + b*Sec[e + f*x]^2])/f)","C",1
78,1,285,105,7.2169876,"\int \csc ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\sqrt{2} \cot (e+f x) \csc ^2(e+f x) \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right) \sqrt{a+b \sec ^2(e+f x)} \left(\frac{4 b \tan ^2(e+f x) \sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)^2 \sqrt{\frac{a+b \sec ^2(e+f x)}{a+b}} \, _2F_1\left(2,2;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)}{(a+b)^2}+\left(2 a \sin ^2(e+f x)+a+b\right) \left(\sqrt{\frac{a+b \sec ^2(e+f x)}{a+b}}+\sqrt{-\frac{b \tan ^2(e+f x)}{a+b}} \sin ^{-1}\left(\sqrt{-\frac{b \tan ^2(e+f x)}{a+b}}\right)\right)\right)}{3 f \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos (2 e+2 f x)+a+2 b} \sqrt{\frac{a+b \sec ^2(e+f x)}{a+b}}}","\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,"-1/3*(Sqrt[2]*Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2]*(1 - (a*Sin[e + f*x]^2)/(a + b))*((4*b*Hypergeometric2F1[2, 2, 3/2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Sqrt[(a + b*Sec[e + f*x]^2)/(a + b)]*(a + b - a*Sin[e + f*x]^2)^2*Tan[e + f*x]^2)/(a + b)^2 + (a + b + 2*a*Sin[e + f*x]^2)*(Sqrt[(a + b*Sec[e + f*x]^2)/(a + b)] + ArcSin[Sqrt[-((b*Tan[e + f*x]^2)/(a + b))]]*Sqrt[-((b*Tan[e + f*x]^2)/(a + b))])))/(f*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sqrt[(a + b*Sec[e + f*x]^2)/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])","C",0
79,1,422,149,7.9830544,"\int \csc ^6(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{2} e^{i (e+f x)} \cos (e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(-\frac{i \left(8 a^2 \left(-6 e^{2 i (e+f x)}+16 e^{4 i (e+f x)}-6 e^{6 i (e+f x)}+e^{8 i (e+f x)}+1\right)+a b \left(-136 e^{2 i (e+f x)}+318 e^{4 i (e+f x)}-136 e^{6 i (e+f x)}+25 e^{8 i (e+f x)}+25\right)+b^2 \left(-80 e^{2 i (e+f x)}+178 e^{4 i (e+f x)}-80 e^{6 i (e+f x)}+15 e^{8 i (e+f x)}+15\right)\right)}{(a+b)^2 \left(-1+e^{2 i (e+f x)}\right)^5}-\frac{15 \sqrt{b} \log \left(\frac{4 i f \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}-4 \sqrt{b} f \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}\right) \sqrt{a+b \sec ^2(e+f x)}}{15 f \sqrt{a \cos (2 e+2 f x)+a+2 b}}","\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[2]*E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]*(((-I)*(8*a^2*(1 - 6*E^((2*I)*(e + f*x)) + 16*E^((4*I)*(e + f*x)) - 6*E^((6*I)*(e + f*x)) + E^((8*I)*(e + f*x))) + b^2*(15 - 80*E^((2*I)*(e + f*x)) + 178*E^((4*I)*(e + f*x)) - 80*E^((6*I)*(e + f*x)) + 15*E^((8*I)*(e + f*x))) + a*b*(25 - 136*E^((2*I)*(e + f*x)) + 318*E^((4*I)*(e + f*x)) - 136*E^((6*I)*(e + f*x)) + 25*E^((8*I)*(e + f*x)))))/((a + b)^2*(-1 + E^((2*I)*(e + f*x)))^5) - (15*Sqrt[b]*Log[(-4*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*f + (4*I)*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]*f)/(1 + E^((2*I)*(e + f*x)))])/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*Sqrt[a + b*Sec[e + f*x]^2])/(15*f*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])","C",1
80,1,188,196,1.3073899,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \sin ^5(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^5,x]","\frac{\sqrt{2} \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \left(-5 (3 a-4 b) \left(\sqrt{-a \sin ^2(e+f x)+a+b} \left(-a \sin ^2(e+f x)+a+4 b\right)-3 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{b}}\right)\right)-\frac{6 b \left(-a \sin ^2(e+f x)+a+b\right)^{5/2}}{a}+15 \sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)^{5/2}\right)}{15 b f (a \cos (2 (e+f x))+a+2 b)^{3/2}}","\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,"(Sqrt[2]*Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2)*((-6*b*(a + b - a*Sin[e + f*x]^2)^(5/2))/a + 15*Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)^(5/2) - 5*(3*a - 4*b)*(-3*b^(3/2)*ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[b]] + Sqrt[a + b - a*Sin[e + f*x]^2]*(a + 4*b - a*Sin[e + f*x]^2))))/(15*b*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2))","A",1
81,1,164,162,0.7558684,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \sin ^3(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^3,x]","\frac{\sqrt{2} \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \left(3 \sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)^{5/2}-(3 a-2 b) \left(\sqrt{-a \sin ^2(e+f x)+a+b} \left(-a \sin ^2(e+f x)+a+4 b\right)-3 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{b}}\right)\right)\right)}{3 b f (a \cos (2 (e+f x))+a+2 b)^{3/2}}","\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,"(Sqrt[2]*Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2)*(3*Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)^(5/2) - (3*a - 2*b)*(-3*b^(3/2)*ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[b]] + Sqrt[a + b - a*Sin[e + f*x]^2]*(a + 4*b - a*Sin[e + f*x]^2))))/(3*b*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2))","A",1
82,1,73,100,0.5990407,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \sin (e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x],x]","-\frac{a \cos (e+f x) (a \cos (2 (e+f x))+a+2 b)^2 \sqrt{a+b \sec ^2(e+f x)} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{a \cos ^2(e+f x)}{b}+1\right)}{20 b^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,"-1/20*(a*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^2*Hypergeometric2F1[2, 5/2, 7/2, 1 + (a*Cos[e + f*x]^2)/b]*Sqrt[a + b*Sec[e + f*x]^2])/(b^2*f)","C",1
83,1,171,122,0.5484531,"\int \csc (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sec (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(\sqrt{2} b \sqrt{a \cos (2 (e+f x))+a+2 b}+2 \sqrt{b} (3 a+2 b) \cos ^2(e+f x) \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{b}}\right)-4 (a+b)^{3/2} \cos ^2(e+f x) \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{a+b}}\right)\right)}{2 \sqrt{2} f \sqrt{a \cos (2 (e+f x))+a+2 b}}","\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,"((2*Sqrt[b]*(3*a + 2*b)*ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[b]]*Cos[e + f*x]^2 - 4*(a + b)^(3/2)*ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[a + b]]*Cos[e + f*x]^2 + Sqrt[2]*b*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(2*Sqrt[2]*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])","A",1
84,1,202,161,1.4206935,"\int \csc ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\csc ^2(e+f x) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(\sqrt{2} \sqrt{a \cos (2 (e+f x))+a+2 b} ((a+2 b) \cos (2 (e+f x))+a)-\sqrt{b} (3 a+4 b) \sin ^2(2 (e+f x)) \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{b}}\right)+\sqrt{a+b} (a+4 b) \sin ^2(2 (e+f x)) \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{a+b}}\right)\right)}{4 \sqrt{2} f \sqrt{a \cos (2 (e+f x))+a+2 b}}","\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,"-1/4*(Csc[e + f*x]^2*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2]*(Sqrt[2]*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + (a + 2*b)*Cos[2*(e + f*x)]) - Sqrt[b]*(3*a + 4*b)*ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[b]]*Sin[2*(e + f*x)]^2 + Sqrt[a + b]*(a + 4*b)*ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[a + b]]*Sin[2*(e + f*x)]^2))/(Sqrt[2]*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])","A",1
85,1,262,218,3.3530784,"\int \csc ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\sec (e+f x) \left(a \cos ^2(e+f x)+b\right) \sqrt{a+b \sec ^2(e+f x)} \left(3 b \sqrt{a+b} \left(a^2+8 a b+8 b^2\right) \cos ^2(e+f x) \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{a+b}}\right)-12 b^{3/2} \left(a^2+3 a b+2 b^2\right) \cos ^2(e+f x) \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{b}}\right)+\frac{b (a+b) \csc ^4(e+f x) \sqrt{a \cos (2 (e+f x))+a+2 b} (8 (a+3 b) \cos (2 (e+f x))-3 (a+4 b) \cos (4 (e+f x))+11 a+4 b)}{8 \sqrt{2}}\right)}{2 \sqrt{2} b f (a+b) (a \cos (2 (e+f x))+a+2 b)^{3/2}}","-\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,"-1/2*((b + a*Cos[e + f*x]^2)*(-12*b^(3/2)*(a^2 + 3*a*b + 2*b^2)*ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[b]]*Cos[e + f*x]^2 + 3*b*Sqrt[a + b]*(a^2 + 8*a*b + 8*b^2)*ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[a + b]]*Cos[e + f*x]^2 + (b*(a + b)*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(11*a + 4*b + 8*(a + 3*b)*Cos[2*(e + f*x)] - 3*(a + 4*b)*Cos[4*(e + f*x)])*Csc[e + f*x]^4)/(8*Sqrt[2]))*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(Sqrt[2]*b*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2))","A",1
86,0,0,298,9.9910174,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \sin ^6(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^6,x]","\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \sin ^6(e+f x) \, dx","-\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{\left(5 a^3-45 a^2 b+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{(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,"Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^6, x]","F",-1
87,1,211,217,5.306137,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \sin ^4(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^4,x]","\frac{3 \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \left(\frac{\left(a^2-6 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{a}}+4 \sqrt{b} (a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)\right)}{2 \sqrt{2} f (a \cos (2 e+2 f x)+a+2 b)^{3/2}}+\frac{\tan (e+f x) ((10 b-6 a) \cos (2 (e+f x))+a \cos (4 (e+f x))-7 a+26 b) \sqrt{a+b \sec ^2(e+f x)}}{32 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]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[a] + 4*(a - b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])*Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2))/(2*Sqrt[2]*f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)) + ((-7*a + 26*b + (-6*a + 10*b)*Cos[2*(e + f*x)] + a*Cos[4*(e + f*x)])*Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x])/(32*f)","A",1
88,1,493,161,5.7619101,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \sin ^2(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^2,x]","\frac{e^{-i (e+f x)} \cos ^3(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{i \left(-1+e^{2 i (e+f x)}\right) \left(a \left(1+e^{2 i (e+f x)}\right)^2-4 b e^{2 i (e+f x)}\right)}{\left(1+e^{2 i (e+f x)}\right)^2}+\frac{2 e^{2 i (e+f x)} \left(-i \sqrt{a} (a-3 b) \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)+i \sqrt{a} (a-3 b) \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)+2 \sqrt{b} (b-3 a) \log \left(\frac{f \left(\sqrt{b} \left(-1+e^{2 i (e+f x)}\right)-i \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}\right)}{b (b-3 a) \left(1+e^{2 i (e+f x)}\right)}\right)+2 \sqrt{a} f x (a-3 b)\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}\right) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{2 \sqrt{2} f (a \cos (2 e+2 f x)+a+2 b)^{3/2}}","\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[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]^3*((I*(-1 + E^((2*I)*(e + f*x)))*(-4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2))/(1 + E^((2*I)*(e + f*x)))^2 + (2*E^((2*I)*(e + f*x))*(2*Sqrt[a]*(a - 3*b)*f*x - I*Sqrt[a]*(a - 3*b)*Log[a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] + I*Sqrt[a]*(a - 3*b)*Log[a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] + 2*Sqrt[b]*(-3*a + b)*Log[((Sqrt[b]*(-1 + E^((2*I)*(e + f*x))) - I*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*f)/(b*(-3*a + b)*(1 + E^((2*I)*(e + f*x))))]))/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*(a + b*Sec[e + f*x]^2)^(3/2))/(2*Sqrt[2]*E^(I*(e + f*x))*f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2))","C",1
89,1,527,118,4.9861623,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} e^{i (e+f x)} \cos ^3(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{-i a^{3/2} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)+i a^{3/2} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)+2 a^{3/2} f x-b^{3/2} \log \left(\frac{2 i f \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}-2 \sqrt{b} f \left(-1+e^{2 i (e+f x)}\right)}{b (3 a+b) \left(1+e^{2 i (e+f x)}\right)}\right)-3 a \sqrt{b} \log \left(\frac{2 i f \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}-2 \sqrt{b} f \left(-1+e^{2 i (e+f x)}\right)}{b (3 a+b) \left(1+e^{2 i (e+f x)}\right)}\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}-\frac{i b \left(-1+e^{2 i (e+f x)}\right)}{\left(1+e^{2 i (e+f x)}\right)^2}\right) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{f (a \cos (2 e+2 f x)+a+2 b)^{3/2}}","\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,"(Sqrt[2]*E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]^3*(((-I)*b*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))^2 + (2*a^(3/2)*f*x - I*a^(3/2)*Log[a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] + I*a^(3/2)*Log[a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - 3*a*Sqrt[b]*Log[(-2*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*f + (2*I)*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]*f)/(b*(3*a + b)*(1 + E^((2*I)*(e + f*x))))] - b^(3/2)*Log[(-2*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*f + (2*I)*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]*f)/(b*(3*a + b)*(1 + E^((2*I)*(e + f*x))))])/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*(a + b*Sec[e + f*x]^2)^(3/2))/(f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2))","C",1
90,1,64,105,0.1823083,"\int \csc ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]^2*(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{(a+b) \cot (e+f x) \sqrt{a+b \sec ^2(e+f x)} \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{b \sin ^2(e+f x)}{-a \sin ^2(e+f x)+a+b}\right)}{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,"-(((a + b)*Cot[e + f*x]*Hypergeometric2F1[-1/2, 2, 1/2, (b*Sin[e + f*x]^2)/(a + b - a*Sin[e + f*x]^2)]*Sqrt[a + b*Sec[e + f*x]^2])/f)","C",1
91,1,135,172,8.0949043,"\int \csc ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]^4*(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\csc (e+f x) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b) \left(a+b \sec ^2(e+f x)\right)^{3/2} \left((a+b) \left((a+b) \csc ^2(e+f x)+2 a\right) \, _2F_1\left(1,2;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)+8 b \left(a+b \sec ^2(e+f x)\right) \, _2F_1\left(2,3;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)\right)}{6 f (a+b)^3}","\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,"-1/6*((a + 2*b + a*Cos[2*(e + f*x)])*Csc[e + f*x]*Sec[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2)*((a + b)*(2*a + (a + b)*Csc[e + f*x]^2)*Hypergeometric2F1[1, 2, 1/2, -((b*Tan[e + f*x]^2)/(a + b))] + 8*b*Hypergeometric2F1[2, 3, 3/2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + b*Sec[e + f*x]^2)))/((a + b)^3*f)","C",0
92,1,512,209,10.3269678,"\int \csc ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]^6*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} e^{i (e+f x)} \cos ^3(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(-\frac{i \left(16 a^2 \left(1+e^{2 i (e+f x)}\right)^2 \left(-6 e^{2 i (e+f x)}+16 e^{4 i (e+f x)}-6 e^{6 i (e+f x)}+e^{8 i (e+f x)}+1\right)+a b \left(-402 e^{2 i (e+f x)}+317 e^{4 i (e+f x)}+708 e^{6 i (e+f x)}+317 e^{8 i (e+f x)}-402 e^{10 i (e+f x)}+115 e^{12 i (e+f x)}+115\right)+b^2 \left(-350 e^{2 i (e+f x)}+231 e^{4 i (e+f x)}+412 e^{6 i (e+f x)}+231 e^{8 i (e+f x)}-350 e^{10 i (e+f x)}+105 e^{12 i (e+f x)}+105\right)\right)}{(a+b) \left(-1+e^{2 i (e+f x)}\right)^5 \left(1+e^{2 i (e+f x)}\right)^2}-\frac{15 \sqrt{b} (3 a+7 b) \log \left(\frac{4 i f \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}-4 \sqrt{b} f \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}\right) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{15 f (a \cos (2 (e+f x))+a+2 b)^{3/2}}","\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[2]*E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]^3*(((-I)*(16*a^2*(1 + E^((2*I)*(e + f*x)))^2*(1 - 6*E^((2*I)*(e + f*x)) + 16*E^((4*I)*(e + f*x)) - 6*E^((6*I)*(e + f*x)) + E^((8*I)*(e + f*x))) + b^2*(105 - 350*E^((2*I)*(e + f*x)) + 231*E^((4*I)*(e + f*x)) + 412*E^((6*I)*(e + f*x)) + 231*E^((8*I)*(e + f*x)) - 350*E^((10*I)*(e + f*x)) + 105*E^((12*I)*(e + f*x))) + a*b*(115 - 402*E^((2*I)*(e + f*x)) + 317*E^((4*I)*(e + f*x)) + 708*E^((6*I)*(e + f*x)) + 317*E^((8*I)*(e + f*x)) - 402*E^((10*I)*(e + f*x)) + 115*E^((12*I)*(e + f*x)))))/((a + b)*(-1 + E^((2*I)*(e + f*x)))^5*(1 + E^((2*I)*(e + f*x)))^2) - (15*Sqrt[b]*(3*a + 7*b)*Log[(-4*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*f + (4*I)*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]*f)/(1 + E^((2*I)*(e + f*x)))])/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*(a + b*Sec[e + f*x]^2)^(3/2))/(15*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2))","C",1
93,1,93,123,0.933296,"\int \frac{\sin ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Sin[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\sec (e+f x) (a \cos (2 (e+f x))+a+2 b) \left(3 a^2 \cos (4 (e+f x))+89 a^2-4 a (7 a+4 b) \cos (2 (e+f x))+144 a b+64 b^2\right)}{240 a^3 f \sqrt{a+b \sec ^2(e+f x)}}","\frac{2 (5 a+2 b) \cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{15 a^2 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{\cos ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)}}{5 a f}",1,"-1/240*((a + 2*b + a*Cos[2*(e + f*x)])*(89*a^2 + 144*a*b + 64*b^2 - 4*a*(7*a + 4*b)*Cos[2*(e + f*x)] + 3*a^2*Cos[4*(e + f*x)])*Sec[e + f*x])/(a^3*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
94,1,64,74,0.2765438,"\int \frac{\sin ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Sin[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sec (e+f x) (a \cos (2 (e+f x))-5 a-4 b) (a \cos (2 (e+f x))+a+2 b)}{12 a^2 f \sqrt{a+b \sec ^2(e+f 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}",1,"((-5*a - 4*b + a*Cos[2*(e + f*x)])*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x])/(12*a^2*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
95,1,48,30,0.1126798,"\int \frac{\sin (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Sin[e + f*x]/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\sec (e+f x) (a \cos (2 e+2 f x)+a+2 b)}{2 a f \sqrt{a+b \sec ^2(e+f x)}}","-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{a f}",1,"-1/2*((a + 2*b + a*Cos[2*e + 2*f*x])*Sec[e + f*x])/(a*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
96,1,86,43,0.0999915,"\int \frac{\csc (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\sec (e+f x) \sqrt{a \cos (2 e+2 f x)+a+2 b} \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{a+b}}\right)}{\sqrt{2} f \sqrt{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 \sqrt{a+b}}",1,"-((ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[a + b]]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sec[e + f*x])/(Sqrt[2]*Sqrt[a + b]*f*Sqrt[a + b*Sec[e + f*x]^2]))","A",1
97,1,140,87,0.9898072,"\int \frac{\csc ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{a \sec (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a \cos (2 e+2 f x)+a+2 b} \left(\frac{(a+b) \csc ^2(e+f x)}{a}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}\right)}{\sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}\right)}{2 \sqrt{2} f (a+b)^2 \sqrt{a+b \sec ^2(e+f 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)}",1,"-1/2*(a*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sec[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2]*(((a + b)*Csc[e + f*x]^2)/a + ArcTanh[Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]]/Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]))/(Sqrt[2]*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
98,1,78,138,0.189728,"\int \frac{\csc ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{a^2 \sec (e+f x) (a \cos (2 (e+f x))+a+2 b) \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\frac{a \sin ^2(e+f x)}{a+b}\right)}{2 f (a+b)^3 \sqrt{a+b \sec ^2(e+f 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}",1,"-1/2*(a^2*(a + 2*b + a*Cos[2*(e + f*x)])*Hypergeometric2F1[1/2, 3, 3/2, 1 - (a*Sin[e + f*x]^2)/(a + b)]*Sec[e + f*x])/((a + b)^3*f*Sqrt[a + b*Sec[e + f*x]^2])","C",1
99,1,163,193,1.3883765,"\int \frac{\sin ^6(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Sin[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sec (e+f x) \sqrt{a \cos (2 (e+f x))+a+2 b} \left(15 (a+b)^3 \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)-\sqrt{a} \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \left(8 a^2 \sin ^4(e+f x)+10 a (a+b) \sin ^2(e+f x)+15 (a+b)^2\right)\right)}{48 \sqrt{2} a^{7/2} f \sqrt{a+b \sec ^2(e+f x)}}","\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{\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{\sin ^3(e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{6 a f}",1,"(Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*Sec[e + f*x]*(15*(a + b)^3*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]] - Sqrt[a]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2]*(15*(a + b)^2 + 10*a*(a + b)*Sin[e + f*x]^2 + 8*a^2*Sin[e + f*x]^4)))/(48*Sqrt[2]*a^(7/2)*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
100,1,145,135,0.4544682,"\int \frac{\sin ^4(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Sin[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sec (e+f x) \sqrt{a \cos (2 (e+f x))+a+2 b} \left(3 (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)-\sqrt{a} \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \left(3 (a+b)+2 a \sin ^2(e+f x)\right)\right)}{8 \sqrt{2} a^{5/2} f \sqrt{a+b \sec ^2(e+f 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}",1,"(Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*Sec[e + f*x]*(3*(a + b)^2*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]] - Sqrt[a]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2]*(3*(a + b) + 2*a*Sin[e + f*x]^2)))/(8*Sqrt[2]*a^(5/2)*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
101,1,125,85,0.2345748,"\int \frac{\sin ^2(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Sin[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sec (e+f x) \sqrt{a \cos (2 (e+f x))+a+2 b} \left((a+b) \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)-\sqrt{a} \sin (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b}\right)}{2 \sqrt{2} a^{3/2} f \sqrt{a+b \sec ^2(e+f 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}",1,"(Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*Sec[e + f*x]*((a + b)*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]] - Sqrt[a]*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2]))/(2*Sqrt[2]*a^(3/2)*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
102,1,87,39,0.0659503,"\int \frac{1}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[1/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sec (e+f x) \sqrt{a \cos (2 e+2 f x)+a+2 b} \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{2} \sqrt{a} f \sqrt{a+b \sec ^2(e+f 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}",1,"(ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sec[e + f*x])/(Sqrt[2]*Sqrt[a]*f*Sqrt[a + b*Sec[e + f*x]^2])","B",1
103,1,55,33,0.1080736,"\int \frac{\csc ^2(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\csc (e+f x) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b)}{2 f (a+b) \sqrt{a+b \sec ^2(e+f x)}}","-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f (a+b)}",1,"-1/2*((a + 2*b + a*Cos[2*(e + f*x)])*Csc[e + f*x]*Sec[e + f*x])/((a + b)*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
104,1,74,78,0.193614,"\int \frac{\csc ^4(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\csc ^3(e+f x) \sec (e+f x) (a \cos (2 (e+f x))-2 a-b) (a \cos (2 (e+f x))+a+2 b)}{6 f (a+b)^2 \sqrt{a+b \sec ^2(e+f 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}",1,"((-2*a - b + a*Cos[2*(e + f*x)])*(a + 2*b + a*Cos[2*(e + f*x)])*Csc[e + f*x]^3*Sec[e + f*x])/(6*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
105,1,100,132,0.3245903,"\int \frac{\csc ^6(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\csc ^5(e+f x) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b) \left(a^2 \cos (4 (e+f x))+8 a^2-2 a (3 a+b) \cos (2 (e+f x))+8 a b+3 b^2\right)}{30 f (a+b)^3 \sqrt{a+b \sec ^2(e+f 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}",1,"-1/30*((a + 2*b + a*Cos[2*(e + f*x)])*(8*a^2 + 8*a*b + 3*b^2 - 2*a*(3*a + b)*Cos[2*(e + f*x)] + a^2*Cos[4*(e + f*x)])*Csc[e + f*x]^5*Sec[e + f*x])/((a + b)^3*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
106,1,432,171,7.5057103,"\int \frac{\sin ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+2 a+4 b) (a \cos (2 e+2 f x)+a+2 b)^{3/2}}{32 a^2 f \sqrt{a \cos (2 (e+f x))+a+2 b} \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\sec ^3(e+f x) \left(-2 a^2 \cos (4 (e+f x))+27 a^2+16 a (a+2 b) \cos (2 (e+f x))+128 a b+128 b^2\right) (a \cos (2 e+2 f x)+a+2 b)^{3/2}}{192 a^3 f \sqrt{a \cos (2 (e+f x))+a+2 b} \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\sec ^3(e+f x) \left(a^3 \cos (6 (e+f x))+40 a^3+a \left(25 a^2+128 a b+128 b^2\right) \cos (2 (e+f x))-4 a^2 (a+2 b) \cos (4 (e+f x))+336 a^2 b+768 a b^2+512 b^3\right) (a \cos (2 e+2 f x)+a+2 b)^{3/2}}{320 a^4 f \sqrt{a \cos (2 (e+f x))+a+2 b} \left(a+b \sec ^2(e+f x)\right)^{3/2}}+\frac{3 \sec ^3(e+f x) (a \cos (2 e+2 f x)+a+2 b)^{3/2}}{64 a f \sqrt{a \cos (2 (e+f x))+a+2 b} \left(a+b \sec ^2(e+f x)\right)^{3/2}}","\frac{2 (5 a+3 b) \cos ^3(e+f x)}{15 a^2 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{\cos ^5(e+f x)}{5 a f \sqrt{a+b \sec ^2(e+f x)}}",1,"(3*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sec[e + f*x]^3)/(64*a*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b*Sec[e + f*x]^2)^(3/2)) + ((2*a + 4*b + a*Cos[2*(e + f*x)])*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sec[e + f*x]^3)/(32*a^2*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b*Sec[e + f*x]^2)^(3/2)) - ((27*a^2 + 128*a*b + 128*b^2 + 16*a*(a + 2*b)*Cos[2*(e + f*x)] - 2*a^2*Cos[4*(e + f*x)])*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sec[e + f*x]^3)/(192*a^3*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b*Sec[e + f*x]^2)^(3/2)) - ((40*a^3 + 336*a^2*b + 768*a*b^2 + 512*b^3 + a*(25*a^2 + 128*a*b + 128*b^2)*Cos[2*(e + f*x)] - 4*a^2*(a + 2*b)*Cos[4*(e + f*x)] + a^3*Cos[6*(e + f*x)])*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sec[e + f*x]^3)/(320*a^4*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b*Sec[e + f*x]^2)^(3/2))","B",1
107,1,93,114,3.3992833,"\int \frac{\sin ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(a^2 (-\cos (4 (e+f x)))+9 a^2+8 a (a+2 b) \cos (2 (e+f x))+64 a b+64 b^2\right)}{48 a^3 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}","-\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,"-1/48*((a + 2*b + a*Cos[2*(e + f*x)])*(9*a^2 + 64*a*b + 64*b^2 + 8*a*(a + 2*b)*Cos[2*(e + f*x)] - a^2*Cos[4*(e + f*x)])*Sec[e + f*x]^3)/(a^3*f*(a + b*Sec[e + f*x]^2)^(3/2))","A",1
108,1,64,62,1.2713909,"\int \frac{\sin (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) (a \cos (2 (e+f x))+a+4 b)}{4 a^2 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}","-\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,"-1/4*((a + 2*b + a*Cos[2*(e + f*x)])*(a + 4*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^3)/(a^2*f*(a + b*Sec[e + f*x]^2)^(3/2))","A",1
109,1,113,80,0.7364917,"\int \frac{\csc (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csc[e + f*x]/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(a \sqrt{-a \sin ^2(e+f x)+a+b} \tanh ^{-1}\left(\frac{\sqrt{-a \sin ^2(e+f x)+a+b}}{\sqrt{a+b}}\right)+b \sqrt{a+b}\right)}{2 a f (a+b)^{3/2} \left(a+b \sec ^2(e+f x)\right)^{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,"-1/2*((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^3*(b*Sqrt[a + b] + a*ArcTanh[Sqrt[a + b - a*Sin[e + f*x]^2]/Sqrt[a + b]]*Sqrt[a + b - a*Sin[e + f*x]^2]))/(a*(a + b)^(3/2)*f*(a + b*Sec[e + f*x]^2)^(3/2))","A",1
110,1,97,126,0.3101018,"\int \frac{\csc ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csc[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left((a+b) \csc ^2(e+f x)-(a-2 b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1-\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}{4 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^{3/2}}","-\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,"-1/4*((a + 2*b + a*Cos[2*(e + f*x)])*((a + b)*Csc[e + f*x]^2 - (a - 2*b)*Hypergeometric2F1[-1/2, 1, 1/2, 1 - (a*Sin[e + f*x]^2)/(a + b)])*Sec[e + f*x]^3)/((a + b)^2*f*(a + b*Sec[e + f*x]^2)^(3/2))","C",1
111,1,100,177,0.4679505,"\int \frac{\csc ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left((a+b)^2 \csc ^4(e+f x)-a (a-4 b) \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};1-\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}{8 f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)^{3/2}}","-\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,"-1/8*((a + 2*b + a*Cos[2*(e + f*x)])*((a + b)^2*Csc[e + f*x]^4 - a*(a - 4*b)*Hypergeometric2F1[-1/2, 2, 1/2, 1 - (a*Sin[e + f*x]^2)/(a + b)])*Sec[e + f*x]^3)/((a + b)^3*f*(a + b*Sec[e + f*x]^2)^(3/2))","C",1
112,1,256,242,8.1676063,"\int \frac{\sin ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(120 (a+b)^2 (a+7 b) \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) (a \cos (2 (e+f x))+a+2 b)-2 \sqrt{2} \sqrt{a} \sqrt{a+b} \sin (e+f x) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}} \left(a^3 \cos (6 (e+f x))+37 a^3+a \left(29 a^2+108 a b+70 b^2\right) \cos (2 (e+f x))-7 a^2 (a+b) \cos (4 (e+f x))+439 a^2 b+830 a b^2+420 b^3\right)\right)}{1536 a^{9/2} f \sqrt{a+b} \sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}} \left(a+b \sec ^2(e+f x)\right)^{3/2}}","\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{(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{(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{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{\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f \sqrt{a+b \tan ^2(e+f x)+b}}",1,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^3*(120*(a + b)^2*(a + 7*b)*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*(a + 2*b + a*Cos[2*(e + f*x)]) - 2*Sqrt[2]*Sqrt[a]*Sqrt[a + b]*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*(37*a^3 + 439*a^2*b + 830*a*b^2 + 420*b^3 + a*(29*a^2 + 108*a*b + 70*b^2)*Cos[2*(e + f*x)] - 7*a^2*(a + b)*Cos[4*(e + f*x)] + a^3*Cos[6*(e + f*x)])*Sin[e + f*x]))/(1536*a^(9/2)*Sqrt[a + b]*f*(a + b*Sec[e + f*x]^2)^(3/2)*Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)])","A",1
113,1,229,175,3.4334795,"\int \frac{\sin ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(24 \left(a^2+6 a b+5 b^2\right) \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) (a \cos (2 (e+f x))+a+2 b)-2 \sqrt{2} \sqrt{a} \sqrt{a+b} \sin (e+f x) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}} \left(a^2 (-\cos (4 (e+f x)))+7 a^2+2 a (3 a+5 b) \cos (2 (e+f x))+62 a b+60 b^2\right)\right)}{256 a^{7/2} f \sqrt{a+b} \sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}} \left(a+b \sec ^2(e+f x)\right)^{3/2}}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^3*(24*(a^2 + 6*a*b + 5*b^2)*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*(a + 2*b + a*Cos[2*(e + f*x)]) - 2*Sqrt[2]*Sqrt[a]*Sqrt[a + b]*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*(7*a^2 + 62*a*b + 60*b^2 + 2*a*(3*a + 5*b)*Cos[2*(e + f*x)] - a^2*Cos[4*(e + f*x)])*Sin[e + f*x]))/(256*a^(7/2)*Sqrt[a + b]*f*(a + b*Sec[e + f*x]^2)^(3/2)*Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)])","A",1
114,1,190,121,1.1668144,"\int \frac{\sin ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(4 (a+3 b) \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) (a \cos (2 (e+f x))+a+2 b)-2 \sqrt{2} \sqrt{a} \sqrt{a+b} \sin (e+f x) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}} (a \cos (2 (e+f x))+a+6 b)\right)}{32 a^{5/2} f \sqrt{a+b} \sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}} \left(a+b \sec ^2(e+f x)\right)^{3/2}}","\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 + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^3*(4*(a + 3*b)*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*(a + 2*b + a*Cos[2*(e + f*x)]) - 2*Sqrt[2]*Sqrt[a]*Sqrt[a + b]*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*(a + 6*b + a*Cos[2*(e + f*x)])*Sin[e + f*x]))/(32*a^(5/2)*Sqrt[a + b]*f*(a + b*Sec[e + f*x]^2)^(3/2)*Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)])","A",1
115,1,168,77,1.3069008,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(-3/2),x]","\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\sqrt{a+b} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) (a \cos (2 (e+f x))+a+2 b)-\sqrt{2} \sqrt{a} b \sin (e+f x) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}}\right)}{4 a^{3/2} f (a+b) \sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}} \left(a+b \sec ^2(e+f x)\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^{3/2} f}-\frac{b \tan (e+f x)}{a f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}",1,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^3*(Sqrt[a + b]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*(a + 2*b + a*Cos[2*(e + f*x)]) - Sqrt[2]*Sqrt[a]*b*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*Sin[e + f*x]))/(4*a^(3/2)*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)*Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)])","B",1
116,1,76,68,1.6167082,"\int \frac{\csc ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csc[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\csc (e+f x) \sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) ((a-b) \cos (2 (e+f x))+a+3 b)}{4 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^{3/2}}","-\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,"-1/4*((a + 2*b + a*Cos[2*(e + f*x)])*(a + 3*b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]*Sec[e + f*x]^3)/((a + b)^2*f*(a + b*Sec[e + f*x]^2)^(3/2))","A",1
117,1,102,123,0.6182111,"\int \frac{\csc ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\tan (e+f x) \sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\left(a^2-2 a b-3 b^2\right) \csc ^2(e+f x)+(a+b)^2 \csc ^4(e+f x)-2 a (a-3 b)\right)}{6 f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)^{3/2}}","-\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,"-1/6*((a + 2*b + a*Cos[2*(e + f*x)])*(-2*a*(a - 3*b) + (a^2 - 2*a*b - 3*b^2)*Csc[e + f*x]^2 + (a + b)^2*Csc[e + f*x]^4)*Sec[e + f*x]^2*Tan[e + f*x])/((a + b)^3*f*(a + b*Sec[e + f*x]^2)^(3/2))","A",1
118,1,126,183,0.9190204,"\int \frac{\csc ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csc[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\tan (e+f x) \sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(4 a \left(a^2-4 a b-5 b^2\right) \csc ^2(e+f x)-8 a^2 (a-5 b)+3 (a+b)^3 \csc ^6(e+f x)+(a-5 b) (a+b)^2 \csc ^4(e+f x)\right)}{30 f (a+b)^4 \left(a+b \sec ^2(e+f x)\right)^{3/2}}","-\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,"-1/30*((a + 2*b + a*Cos[2*(e + f*x)])*(-8*a^2*(a - 5*b) + 4*a*(a^2 - 4*a*b - 5*b^2)*Csc[e + f*x]^2 + (a - 5*b)*(a + b)^2*Csc[e + f*x]^4 + 3*(a + b)^3*Csc[e + f*x]^6)*Sec[e + f*x]^2*Tan[e + f*x])/((a + b)^4*f*(a + b*Sec[e + f*x]^2)^(3/2))","A",1
119,1,182,219,3.2140981,"\int \frac{\sin ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{\sec ^5(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(-16 a^4 \cos (6 (e+f x))+3 a^4 \cos (8 (e+f x))+425 a^4-32 a^3 b \cos (6 (e+f x))+6400 a^3 b+12 a^2 \left(7 a^2+64 a b+64 b^2\right) \cos (4 (e+f x))+22784 a^2 b^2+48 a \left(11 a^3+150 a^2 b+384 a b^2+256 b^3\right) \cos (2 (e+f x))+32768 a b^3+16384 b^4\right)}{3840 a^5 f \left(a+b \sec ^2(e+f x)\right)^{5/2}}","\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{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{\cos ^5(e+f x)}{5 a f \left(a+b \sec ^2(e+f x)\right)^{3/2}}",1,"-1/3840*((a + 2*b + a*Cos[2*(e + f*x)])*(425*a^4 + 6400*a^3*b + 22784*a^2*b^2 + 32768*a*b^3 + 16384*b^4 + 48*a*(11*a^3 + 150*a^2*b + 384*a*b^2 + 256*b^3)*Cos[2*(e + f*x)] + 12*a^2*(7*a^2 + 64*a*b + 64*b^2)*Cos[4*(e + f*x)] - 16*a^4*Cos[6*(e + f*x)] - 32*a^3*b*Cos[6*(e + f*x)] + 3*a^4*Cos[8*(e + f*x)])*Sec[e + f*x]^5)/(a^5*f*(a + b*Sec[e + f*x]^2)^(5/2))","A",1
120,1,129,146,2.3145943,"\int \frac{\sin ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{\sec ^5(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(a^3 (-\cos (6 (e+f x)))+26 a^3+3 a \left(11 a^2+96 a b+128 b^2\right) \cos (2 (e+f x))+6 a^2 (a+4 b) \cos (4 (e+f x))+264 a^2 b+640 a b^2+512 b^3\right)}{192 a^4 f \left(a+b \sec ^2(e+f x)\right)^{5/2}}","-\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,"-1/192*((a + 2*b + a*Cos[2*(e + f*x)])*(26*a^3 + 264*a^2*b + 640*a*b^2 + 512*b^3 + 3*a*(11*a^2 + 96*a*b + 128*b^2)*Cos[2*(e + f*x)] + 6*a^2*(a + 4*b)*Cos[4*(e + f*x)] - a^3*Cos[6*(e + f*x)])*Sec[e + f*x]^5)/(a^4*f*(a + b*Sec[e + f*x]^2)^(5/2))","A",1
121,1,88,97,1.3218946,"\int \frac{\sin (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{\sec ^5(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(3 a^2 \cos (4 (e+f x))+12 a (a+4 b) \cos (2 (e+f x))+(3 a+8 b)^2\right)}{48 a^3 f \left(a+b \sec ^2(e+f x)\right)^{5/2}}","-\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,"-1/48*((a + 2*b + a*Cos[2*(e + f*x)])*((3*a + 8*b)^2 + 12*a*(a + 4*b)*Cos[2*(e + f*x)] + 3*a^2*Cos[4*(e + f*x)])*Sec[e + f*x]^5)/(a^3*f*(a + b*Sec[e + f*x]^2)^(5/2))","A",1
122,1,108,127,5.1112346,"\int \frac{\csc (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csc[e + f*x]/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\sec ^5(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(a^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};1-\frac{a \sin ^2(e+f x)}{a+b}\right)+(a+b) \left(3 a \sin ^2(e+f x)-2 (2 a+b)\right)\right)}{6 a^2 f (a+b) \left(a+b \sec ^2(e+f x)\right)^{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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^5*(a^2*Hypergeometric2F1[-3/2, 1, -1/2, 1 - (a*Sin[e + f*x]^2)/(a + b)] + (a + b)*(-2*(2*a + b) + 3*a*Sin[e + f*x]^2)))/(6*a^2*(a + b)*f*(a + b*Sec[e + f*x]^2)^(5/2))","C",1
123,1,151,171,1.4000559,"\int \frac{\csc ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csc[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{\sec ^5(e+f x) (a \cos (2 (e+f x))+a+2 b) \left((a+b) \csc ^2(e+f x) \left(\left(3 a^2+2 b^2\right) \cos (2 (e+f x))+3 a^2+6 a b-2 b^2\right)-3 a (a-4 b) (a \cos (2 (e+f x))+a+2 b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1-\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}{24 a f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)^{5/2}}","-\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,"-1/24*((a + 2*b + a*Cos[2*(e + f*x)])*((a + b)*(3*a^2 + 6*a*b - 2*b^2 + (3*a^2 + 2*b^2)*Cos[2*(e + f*x)])*Csc[e + f*x]^2 - 3*a*(a - 4*b)*(a + 2*b + a*Cos[2*(e + f*x)])*Hypergeometric2F1[-1/2, 1, 1/2, 1 - (a*Sin[e + f*x]^2)/(a + b)])*Sec[e + f*x]^5)/(a*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^(5/2))","C",1
124,1,129,234,1.7987742,"\int \frac{\csc ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{\sec ^5(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(3 (a+b) \csc ^4(e+f x) ((a+8 b) \cos (2 (e+f x))+3 a-4 b)-2 \left(3 a^2-24 a b+8 b^2\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};1-\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}{96 f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)^{5/2}}","-\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,"-1/96*((a + 2*b + a*Cos[2*(e + f*x)])*(3*(a + b)*(3*a - 4*b + (a + 8*b)*Cos[2*(e + f*x)])*Csc[e + f*x]^4 - 2*(3*a^2 - 24*a*b + 8*b^2)*Hypergeometric2F1[-3/2, 1, -1/2, 1 - (a*Sin[e + f*x]^2)/(a + b)])*Sec[e + f*x]^5)/((a + b)^3*f*(a + b*Sec[e + f*x]^2)^(5/2))","C",1
125,1,1705,288,19.164016,"\int \frac{\sin ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{\left(\frac{\cos (2 (e+f x)) a+a+2 b}{a+b}\right)^{3/2} (\cos (2 e+2 f x) a+a+2 b)^{5/2} \left(\sqrt{a} \sin (e+f x) \sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}} \left(192 a^3 (a+b)^2 \sin ^6(e+f x)+672 a^2 b (a+b)^2 \sin ^4(e+f x)-2 a \left(459 a^4+3180 b a^3+7200 b^2 a^2+6720 b^3 a+2240 b^4\right) \sin ^2(e+f x)+3 \left(239 a^5+1839 b a^4+5200 b^2 a^3+6960 b^3 a^2+4480 b^4 a+1120 b^5\right)\right)-60 \sqrt{a+b} \left(3 a^3+17 b a^2+28 b^2 a+14 b^3\right) \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2\right) \sec ^5(e+f x)}{3072 \sqrt{2} a^{9/2} f (\cos (2 (e+f x)) a+a+2 b)^{7/2} \left(b \sec ^2(e+f x)+a\right)^{5/2}}+\frac{\left(\frac{\cos (2 (e+f x)) a+a+2 b}{a+b}\right)^{3/2} (\cos (2 e+2 f x) a+a+2 b)^{5/2} \left(420 \sqrt{a+b} \left(a^4+9 b a^3+26 b^2 a^2+30 b^3 a+12 b^4\right) \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2-\sqrt{a} \sin (e+f x) \sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}} \left(512 a^4 (a+b)^2 \sin ^8(e+f x)-576 a^3 (a-2 b) (a+b)^2 \sin ^6(e+f x)+672 a^2 (a+b)^2 \left(a^2+3 b a+6 b^2\right) \sin ^4(e+f x)-2 a \left(1151 a^5+11230 b a^4+39200 b^2 a^3+62720 b^3 a^2+47040 b^4 a+13440 b^5\right) \sin ^2(e+f x)+3 \left(561 a^6+6161 b a^5+25200 b^2 a^4+50960 b^3 a^3+54880 b^4 a^2+30240 b^5 a+6720 b^6\right)\right)\right) \sec ^5(e+f x)}{3072 \sqrt{2} a^{11/2} f (\cos (2 (e+f x)) a+a+2 b)^{7/2} \left(b \sec ^2(e+f x)+a\right)^{5/2}}-\frac{5 (\cos (2 e+2 f x) a+a+2 b)^{5/2} \csc (e+f x) \left(-\frac{12 \sin ^4(e+f x)}{a+b}+\frac{(\cos (2 (e+f x)) a+a+2 b) \sin ^2(e+f x)}{(a+b)^2}+\frac{\sin ^2(e+f x)}{a+b}+\frac{16 \left(-a \sin ^2(e+f x)+a+b\right) \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\frac{a^2 (a+b) \sin ^4(e+f x)}{\left(-a \sin ^2(e+f x)+a+b\right)^2}-\frac{6 a (a+b) \sin ^2(e+f x)}{\cos (2 (e+f x)) a+a+2 b}+\frac{3 \sqrt{a} \sqrt{a+b} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) \sin (e+f x)}{\sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}}}\right)}{a^3}\right) \sec ^5(e+f x)}{12288 \sqrt{2} f \left(b \sec ^2(e+f x)+a\right)^{5/2} \left(-a \sin ^2(e+f x)+a+b\right)^{3/2}}+\frac{5 (\cos (2 e+2 f x) a+a+2 b)^{5/2} \csc (e+f x) \left(\frac{96 \sin ^6(e+f x)}{a}-\frac{24 \sin ^4(e+f x)}{a+b}+\frac{(\cos (2 (e+f x)) a+a+2 b) \sin ^2(e+f x)}{(a+b)^2}+\frac{\sin ^2(e+f x)}{a+b}+\frac{80 \left(-a \sin ^2(e+f x)+a+b\right) \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\frac{a^2 (a+b) \sin ^4(e+f x)}{\left(-a \sin ^2(e+f x)+a+b\right)^2}-\frac{6 a (a+b) \sin ^2(e+f x)}{\cos (2 (e+f x)) a+a+2 b}+\frac{3 \sqrt{a} \sqrt{a+b} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) \sin (e+f x)}{\sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}}}\right)}{a^3}-\frac{160 \left(-a \sin ^2(e+f x)+a+b\right) \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\frac{a^2 \sin ^4(e+f x)}{\left(\frac{a \sin ^2(e+f x)}{a+b}-1\right)^2}-\frac{6 a (a+b)^2 \sin ^2(e+f x)}{\cos (2 (e+f x)) a+a+2 b}+\frac{3 \sqrt{a} (a+b)^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) \sin (e+f x)}{\sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}}}\right)}{a^4}\right) \sec ^5(e+f x)}{12288 \sqrt{2} f \left(b \sec ^2(e+f x)+a\right)^{5/2} \left(-a \sin ^2(e+f x)+a+b\right)^{3/2}}+\frac{5 (\cos (2 (e+f x)) a+2 a+3 b) (\cos (2 e+2 f x) a+a+2 b)^{5/2} \tan (e+f x) \sec ^4(e+f x)}{3072 (a+b)^2 f (\cos (2 (e+f x)) a+a+2 b)^{3/2} \left(b \sec ^2(e+f x)+a\right)^{5/2}}-\frac{5 (b+(3 a+2 b) \cos (2 (e+f x))) (\cos (2 e+2 f x) a+a+2 b)^{5/2} \tan (e+f x) \sec ^4(e+f x)}{3072 (a+b)^2 f (\cos (2 (e+f x)) a+a+2 b)^{3/2} \left(b \sec ^2(e+f x)+a\right)^{5/2}}","-\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{(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{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{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{\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,"-1/3072*(((a + 2*b + a*Cos[2*(e + f*x)])/(a + b))^(3/2)*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^5*(-60*Sqrt[a + b]*(3*a^3 + 17*a^2*b + 28*a*b^2 + 14*b^3)*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 + Sqrt[a]*Sin[e + f*x]*Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)]*(3*(239*a^5 + 1839*a^4*b + 5200*a^3*b^2 + 6960*a^2*b^3 + 4480*a*b^4 + 1120*b^5) - 2*a*(459*a^4 + 3180*a^3*b + 7200*a^2*b^2 + 6720*a*b^3 + 2240*b^4)*Sin[e + f*x]^2 + 672*a^2*b*(a + b)^2*Sin[e + f*x]^4 + 192*a^3*(a + b)^2*Sin[e + f*x]^6)))/(Sqrt[2]*a^(9/2)*f*(a + 2*b + a*Cos[2*(e + f*x)])^(7/2)*(a + b*Sec[e + f*x]^2)^(5/2)) + (((a + 2*b + a*Cos[2*(e + f*x)])/(a + b))^(3/2)*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^5*(420*Sqrt[a + b]*(a^4 + 9*a^3*b + 26*a^2*b^2 + 30*a*b^3 + 12*b^4)*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 - Sqrt[a]*Sin[e + f*x]*Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)]*(3*(561*a^6 + 6161*a^5*b + 25200*a^4*b^2 + 50960*a^3*b^3 + 54880*a^2*b^4 + 30240*a*b^5 + 6720*b^6) - 2*a*(1151*a^5 + 11230*a^4*b + 39200*a^3*b^2 + 62720*a^2*b^3 + 47040*a*b^4 + 13440*b^5)*Sin[e + f*x]^2 + 672*a^2*(a + b)^2*(a^2 + 3*a*b + 6*b^2)*Sin[e + f*x]^4 - 576*a^3*(a - 2*b)*(a + b)^2*Sin[e + f*x]^6 + 512*a^4*(a + b)^2*Sin[e + f*x]^8)))/(3072*Sqrt[2]*a^(11/2)*f*(a + 2*b + a*Cos[2*(e + f*x)])^(7/2)*(a + b*Sec[e + f*x]^2)^(5/2)) - (5*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Csc[e + f*x]*Sec[e + f*x]^5*(Sin[e + f*x]^2/(a + b) + ((a + 2*b + a*Cos[2*(e + f*x)])*Sin[e + f*x]^2)/(a + b)^2 - (12*Sin[e + f*x]^4)/(a + b) + (16*(a + b - a*Sin[e + f*x]^2)*(1 - (a*Sin[e + f*x]^2)/(a + b))*((-6*a*(a + b)*Sin[e + f*x]^2)/(a + 2*b + a*Cos[2*(e + f*x)]) + (a^2*(a + b)*Sin[e + f*x]^4)/(a + b - a*Sin[e + f*x]^2)^2 + (3*Sqrt[a]*Sqrt[a + b]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)]))/a^3))/(12288*Sqrt[2]*f*(a + b*Sec[e + f*x]^2)^(5/2)*(a + b - a*Sin[e + f*x]^2)^(3/2)) + (5*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Csc[e + f*x]*Sec[e + f*x]^5*(Sin[e + f*x]^2/(a + b) + ((a + 2*b + a*Cos[2*(e + f*x)])*Sin[e + f*x]^2)/(a + b)^2 - (24*Sin[e + f*x]^4)/(a + b) + (96*Sin[e + f*x]^6)/a + (80*(a + b - a*Sin[e + f*x]^2)*(1 - (a*Sin[e + f*x]^2)/(a + b))*((-6*a*(a + b)*Sin[e + f*x]^2)/(a + 2*b + a*Cos[2*(e + f*x)]) + (a^2*(a + b)*Sin[e + f*x]^4)/(a + b - a*Sin[e + f*x]^2)^2 + (3*Sqrt[a]*Sqrt[a + b]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)]))/a^3 - (160*(a + b - a*Sin[e + f*x]^2)*(1 - (a*Sin[e + f*x]^2)/(a + b))*((-6*a*(a + b)^2*Sin[e + f*x]^2)/(a + 2*b + a*Cos[2*(e + f*x)]) + (3*Sqrt[a]*(a + b)^(3/2)*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)] + (a^2*Sin[e + f*x]^4)/(-1 + (a*Sin[e + f*x]^2)/(a + b))^2))/a^4))/(12288*Sqrt[2]*f*(a + b*Sec[e + f*x]^2)^(5/2)*(a + b - a*Sin[e + f*x]^2)^(3/2)) + (5*(2*a + 3*b + a*Cos[2*(e + f*x)])*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^4*Tan[e + f*x])/(3072*(a + b)^2*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*(a + b*Sec[e + f*x]^2)^(5/2)) - (5*(b + (3*a + 2*b)*Cos[2*(e + f*x)])*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^4*Tan[e + f*x])/(3072*(a + b)^2*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*(a + b*Sec[e + f*x]^2)^(5/2))","B",0
126,1,1315,227,13.9500942,"\int \frac{\sin ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{\left(\frac{\cos (2 (e+f x)) a+a+2 b}{a+b}\right)^{3/2} (\cos (2 e+2 f x) a+a+2 b)^{5/2} \left(\sqrt{a} \sin (e+f x) \sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}} \left(192 a^3 (a+b)^2 \sin ^6(e+f x)+672 a^2 b (a+b)^2 \sin ^4(e+f x)-2 a \left(459 a^4+3180 b a^3+7200 b^2 a^2+6720 b^3 a+2240 b^4\right) \sin ^2(e+f x)+3 \left(239 a^5+1839 b a^4+5200 b^2 a^3+6960 b^3 a^2+4480 b^4 a+1120 b^5\right)\right)-60 \sqrt{a+b} \left(3 a^3+17 b a^2+28 b^2 a+14 b^3\right) \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) (\cos (2 (e+f x)) a+a+2 b)^2\right) \sec ^5(e+f x)}{768 \sqrt{2} a^{9/2} f (\cos (2 (e+f x)) a+a+2 b)^{7/2} \left(b \sec ^2(e+f x)+a\right)^{5/2}}-\frac{(\cos (2 e+2 f x) a+a+2 b)^{5/2} \csc (e+f x) \left(-\frac{12 \sin ^4(e+f x)}{a+b}+\frac{(\cos (2 (e+f x)) a+a+2 b) \sin ^2(e+f x)}{(a+b)^2}+\frac{\sin ^2(e+f x)}{a+b}+\frac{16 \left(-a \sin ^2(e+f x)+a+b\right) \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\frac{a^2 (a+b) \sin ^4(e+f x)}{\left(-a \sin ^2(e+f x)+a+b\right)^2}-\frac{6 a (a+b) \sin ^2(e+f x)}{\cos (2 (e+f x)) a+a+2 b}+\frac{3 \sqrt{a} \sqrt{a+b} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) \sin (e+f x)}{\sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}}}\right)}{a^3}\right) \sec ^5(e+f x)}{768 \sqrt{2} f \left(b \sec ^2(e+f x)+a\right)^{5/2} \left(-a \sin ^2(e+f x)+a+b\right)^{3/2}}+\frac{(\cos (2 e+2 f x) a+a+2 b)^{5/2} \csc (e+f x) \left(\frac{96 \sin ^6(e+f x)}{a}-\frac{24 \sin ^4(e+f x)}{a+b}+\frac{(\cos (2 (e+f x)) a+a+2 b) \sin ^2(e+f x)}{(a+b)^2}+\frac{\sin ^2(e+f x)}{a+b}+\frac{80 \left(-a \sin ^2(e+f x)+a+b\right) \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\frac{a^2 (a+b) \sin ^4(e+f x)}{\left(-a \sin ^2(e+f x)+a+b\right)^2}-\frac{6 a (a+b) \sin ^2(e+f x)}{\cos (2 (e+f x)) a+a+2 b}+\frac{3 \sqrt{a} \sqrt{a+b} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) \sin (e+f x)}{\sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}}}\right)}{a^3}-\frac{160 \left(-a \sin ^2(e+f x)+a+b\right) \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\frac{a^2 \sin ^4(e+f x)}{\left(\frac{a \sin ^2(e+f x)}{a+b}-1\right)^2}-\frac{6 a (a+b)^2 \sin ^2(e+f x)}{\cos (2 (e+f x)) a+a+2 b}+\frac{3 \sqrt{a} (a+b)^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) \sin (e+f x)}{\sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}}}\right)}{a^4}\right) \sec ^5(e+f x)}{3072 \sqrt{2} f \left(b \sec ^2(e+f x)+a\right)^{5/2} \left(-a \sin ^2(e+f x)+a+b\right)^{3/2}}+\frac{(\cos (2 (e+f x)) a+2 a+3 b) (\cos (2 e+2 f x) a+a+2 b)^{5/2} \tan (e+f x) \sec ^4(e+f x)}{256 (a+b)^2 f (\cos (2 (e+f x)) a+a+2 b)^{3/2} \left(b \sec ^2(e+f x)+a\right)^{5/2}}-\frac{(b+(3 a+2 b) \cos (2 (e+f x))) (\cos (2 e+2 f x) a+a+2 b)^{5/2} \tan (e+f x) \sec ^4(e+f x)}{384 (a+b)^2 f (\cos (2 (e+f x)) a+a+2 b)^{3/2} \left(b \sec ^2(e+f x)+a\right)^{5/2}}","-\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{\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{\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,"-1/768*(((a + 2*b + a*Cos[2*(e + f*x)])/(a + b))^(3/2)*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^5*(-60*Sqrt[a + b]*(3*a^3 + 17*a^2*b + 28*a*b^2 + 14*b^3)*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 + Sqrt[a]*Sin[e + f*x]*Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)]*(3*(239*a^5 + 1839*a^4*b + 5200*a^3*b^2 + 6960*a^2*b^3 + 4480*a*b^4 + 1120*b^5) - 2*a*(459*a^4 + 3180*a^3*b + 7200*a^2*b^2 + 6720*a*b^3 + 2240*b^4)*Sin[e + f*x]^2 + 672*a^2*b*(a + b)^2*Sin[e + f*x]^4 + 192*a^3*(a + b)^2*Sin[e + f*x]^6)))/(Sqrt[2]*a^(9/2)*f*(a + 2*b + a*Cos[2*(e + f*x)])^(7/2)*(a + b*Sec[e + f*x]^2)^(5/2)) - ((a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Csc[e + f*x]*Sec[e + f*x]^5*(Sin[e + f*x]^2/(a + b) + ((a + 2*b + a*Cos[2*(e + f*x)])*Sin[e + f*x]^2)/(a + b)^2 - (12*Sin[e + f*x]^4)/(a + b) + (16*(a + b - a*Sin[e + f*x]^2)*(1 - (a*Sin[e + f*x]^2)/(a + b))*((-6*a*(a + b)*Sin[e + f*x]^2)/(a + 2*b + a*Cos[2*(e + f*x)]) + (a^2*(a + b)*Sin[e + f*x]^4)/(a + b - a*Sin[e + f*x]^2)^2 + (3*Sqrt[a]*Sqrt[a + b]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)]))/a^3))/(768*Sqrt[2]*f*(a + b*Sec[e + f*x]^2)^(5/2)*(a + b - a*Sin[e + f*x]^2)^(3/2)) + ((a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Csc[e + f*x]*Sec[e + f*x]^5*(Sin[e + f*x]^2/(a + b) + ((a + 2*b + a*Cos[2*(e + f*x)])*Sin[e + f*x]^2)/(a + b)^2 - (24*Sin[e + f*x]^4)/(a + b) + (96*Sin[e + f*x]^6)/a + (80*(a + b - a*Sin[e + f*x]^2)*(1 - (a*Sin[e + f*x]^2)/(a + b))*((-6*a*(a + b)*Sin[e + f*x]^2)/(a + 2*b + a*Cos[2*(e + f*x)]) + (a^2*(a + b)*Sin[e + f*x]^4)/(a + b - a*Sin[e + f*x]^2)^2 + (3*Sqrt[a]*Sqrt[a + b]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)]))/a^3 - (160*(a + b - a*Sin[e + f*x]^2)*(1 - (a*Sin[e + f*x]^2)/(a + b))*((-6*a*(a + b)^2*Sin[e + f*x]^2)/(a + 2*b + a*Cos[2*(e + f*x)]) + (3*Sqrt[a]*(a + b)^(3/2)*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)] + (a^2*Sin[e + f*x]^4)/(-1 + (a*Sin[e + f*x]^2)/(a + b))^2))/a^4))/(3072*Sqrt[2]*f*(a + b*Sec[e + f*x]^2)^(5/2)*(a + b - a*Sin[e + f*x]^2)^(3/2)) + ((2*a + 3*b + a*Cos[2*(e + f*x)])*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^4*Tan[e + f*x])/(256*(a + b)^2*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*(a + b*Sec[e + f*x]^2)^(5/2)) - ((b + (3*a + 2*b)*Cos[2*(e + f*x)])*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^4*Tan[e + f*x])/(384*(a + b)^2*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*(a + b*Sec[e + f*x]^2)^(5/2))","B",0
127,1,983,167,10.5279865,"\int \frac{\sin ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{(\cos (2 e+2 f x) a+a+2 b)^{5/2} \csc (e+f x) \left(-\frac{12 \sin ^4(e+f x)}{a+b}+\frac{(\cos (2 (e+f x)) a+a+2 b) \sin ^2(e+f x)}{(a+b)^2}+\frac{\sin ^2(e+f x)}{a+b}+\frac{16 \left(-a \sin ^2(e+f x)+a+b\right) \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\frac{a^2 (a+b) \sin ^4(e+f x)}{\left(-a \sin ^2(e+f x)+a+b\right)^2}-\frac{6 a (a+b) \sin ^2(e+f x)}{\cos (2 (e+f x)) a+a+2 b}+\frac{3 \sqrt{a} \sqrt{a+b} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) \sin (e+f x)}{\sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}}}\right)}{a^3}\right) \sec ^5(e+f x)}{256 \sqrt{2} f \left(b \sec ^2(e+f x)+a\right)^{5/2} \left(-a \sin ^2(e+f x)+a+b\right)^{3/2}}-\frac{(\cos (2 e+2 f x) a+a+2 b)^{5/2} \csc (e+f x) \left(\frac{96 \sin ^6(e+f x)}{a}-\frac{24 \sin ^4(e+f x)}{a+b}+\frac{(\cos (2 (e+f x)) a+a+2 b) \sin ^2(e+f x)}{(a+b)^2}+\frac{\sin ^2(e+f x)}{a+b}+\frac{80 \left(-a \sin ^2(e+f x)+a+b\right) \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\frac{a^2 (a+b) \sin ^4(e+f x)}{\left(-a \sin ^2(e+f x)+a+b\right)^2}-\frac{6 a (a+b) \sin ^2(e+f x)}{\cos (2 (e+f x)) a+a+2 b}+\frac{3 \sqrt{a} \sqrt{a+b} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) \sin (e+f x)}{\sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}}}\right)}{a^3}-\frac{160 \left(-a \sin ^2(e+f x)+a+b\right) \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\frac{a^2 \sin ^4(e+f x)}{\left(\frac{a \sin ^2(e+f x)}{a+b}-1\right)^2}-\frac{6 a (a+b)^2 \sin ^2(e+f x)}{\cos (2 (e+f x)) a+a+2 b}+\frac{3 \sqrt{a} (a+b)^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) \sin (e+f x)}{\sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}}}\right)}{a^4}\right) \sec ^5(e+f x)}{768 \sqrt{2} f \left(b \sec ^2(e+f x)+a\right)^{5/2} \left(-a \sin ^2(e+f x)+a+b\right)^{3/2}}+\frac{5 (\cos (2 (e+f x)) a+2 a+3 b) (\cos (2 e+2 f x) a+a+2 b)^{5/2} \tan (e+f x) \sec ^4(e+f x)}{384 (a+b)^2 f (\cos (2 (e+f x)) a+a+2 b)^{3/2} \left(b \sec ^2(e+f x)+a\right)^{5/2}}-\frac{(b+(3 a+2 b) \cos (2 (e+f x))) (\cos (2 e+2 f x) a+a+2 b)^{5/2} \tan (e+f x) \sec ^4(e+f x)}{384 (a+b)^2 f (\cos (2 (e+f x)) a+a+2 b)^{3/2} \left(b \sec ^2(e+f x)+a\right)^{5/2}}","\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,"-1/256*((a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Csc[e + f*x]*Sec[e + f*x]^5*(Sin[e + f*x]^2/(a + b) + ((a + 2*b + a*Cos[2*(e + f*x)])*Sin[e + f*x]^2)/(a + b)^2 - (12*Sin[e + f*x]^4)/(a + b) + (16*(a + b - a*Sin[e + f*x]^2)*(1 - (a*Sin[e + f*x]^2)/(a + b))*((-6*a*(a + b)*Sin[e + f*x]^2)/(a + 2*b + a*Cos[2*(e + f*x)]) + (a^2*(a + b)*Sin[e + f*x]^4)/(a + b - a*Sin[e + f*x]^2)^2 + (3*Sqrt[a]*Sqrt[a + b]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)]))/a^3))/(Sqrt[2]*f*(a + b*Sec[e + f*x]^2)^(5/2)*(a + b - a*Sin[e + f*x]^2)^(3/2)) - ((a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Csc[e + f*x]*Sec[e + f*x]^5*(Sin[e + f*x]^2/(a + b) + ((a + 2*b + a*Cos[2*(e + f*x)])*Sin[e + f*x]^2)/(a + b)^2 - (24*Sin[e + f*x]^4)/(a + b) + (96*Sin[e + f*x]^6)/a + (80*(a + b - a*Sin[e + f*x]^2)*(1 - (a*Sin[e + f*x]^2)/(a + b))*((-6*a*(a + b)*Sin[e + f*x]^2)/(a + 2*b + a*Cos[2*(e + f*x)]) + (a^2*(a + b)*Sin[e + f*x]^4)/(a + b - a*Sin[e + f*x]^2)^2 + (3*Sqrt[a]*Sqrt[a + b]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)]))/a^3 - (160*(a + b - a*Sin[e + f*x]^2)*(1 - (a*Sin[e + f*x]^2)/(a + b))*((-6*a*(a + b)^2*Sin[e + f*x]^2)/(a + 2*b + a*Cos[2*(e + f*x)]) + (3*Sqrt[a]*(a + b)^(3/2)*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)] + (a^2*Sin[e + f*x]^4)/(-1 + (a*Sin[e + f*x]^2)/(a + b))^2))/a^4))/(768*Sqrt[2]*f*(a + b*Sec[e + f*x]^2)^(5/2)*(a + b - a*Sin[e + f*x]^2)^(3/2)) + (5*(2*a + 3*b + a*Cos[2*(e + f*x)])*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^4*Tan[e + f*x])/(384*(a + b)^2*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*(a + b*Sec[e + f*x]^2)^(5/2)) - ((b + (3*a + 2*b)*Cos[2*(e + f*x)])*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^4*Tan[e + f*x])/(384*(a + b)^2*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*(a + b*Sec[e + f*x]^2)^(5/2))","B",1
128,1,1927,125,16.9461371,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(-5/2),x]","\frac{3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^4(e+f x) \sin (e+f x)}{4 \sqrt{2} f \left(b \sec ^2(e+f x)+a\right)^{5/2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \left(\frac{3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^5(e+f x)}{4 \sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}+\frac{15 a (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin ^2(e+f x) \cos ^5(e+f x)}{4 \sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{7/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin (e+f x) \left(\left(5 a \left(\frac{21 a f F_1\left(\frac{5}{2};-2,\frac{9}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{5 (a+b)}-\frac{12}{5} f F_1\left(\frac{5}{2};-1,\frac{7}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)-4 (a+b) \left(\frac{3 a f F_1\left(\frac{5}{2};-1,\frac{7}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{a+b}-\frac{6 (a+b)^3 f \cot (e+f x) \csc ^4(e+f x) \left(\frac{a \sin ^2(e+f x)}{a+b}-1\right)^2 \left(\frac{a^2 \sin ^4(e+f x)}{3 (a+b)^2 \left(\frac{a \sin ^2(e+f x)}{a+b}-1\right)^2}+\frac{a \sin ^2(e+f x)}{(a+b) \left(\frac{a \sin ^2(e+f x)}{a+b}-1\right)}+\frac{\sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) \sin (e+f x)}{\sqrt{a+b} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}\right)}{a^3 \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)^{3/2}}\right)\right) \sin ^2(e+f x)+2 f \left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \cos (e+f x) \sin (e+f x)+3 (a+b) \left(\frac{5 a f F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac{4}{3} f F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)\right) \cos ^4(e+f x)}{4 \sqrt{2} f \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right){}^2}+\frac{3 (a+b) \sin (e+f x) \left(\frac{5 a f F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac{4}{3} f F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right) \cos ^4(e+f x)}{4 \sqrt{2} f \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin ^2(e+f x) \cos ^3(e+f x)}{\sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}\right)}","\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,"(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^4*Sin[e + f*x])/(4*Sqrt[2]*f*(a + b*Sec[e + f*x]^2)^(5/2)*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)*((15*a*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^5*Sin[e + f*x]^2)/(4*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(7/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^5)/(4*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^3*Sin[e + f*x]^2)/(Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*Cos[e + f*x]^4*Sin[e + f*x]*((5*a*f*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - (4*f*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3))/(4*Sqrt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^4*Sin[e + f*x]*(2*f*(5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Cos[e + f*x]*Sin[e + f*x] + 3*(a + b)*((5*a*f*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - (4*f*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3) + Sin[e + f*x]^2*(5*a*((21*a*f*AppellF1[5/2, -2, 9/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(5*(a + b)) - (12*f*AppellF1[5/2, -1, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5) - 4*(a + b)*((3*a*f*AppellF1[5/2, -1, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - (6*(a + b)^3*f*Cot[e + f*x]*Csc[e + f*x]^4*(-1 + (a*Sin[e + f*x]^2)/(a + b))^2*((Sqrt[a]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/(Sqrt[a + b]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + (a^2*Sin[e + f*x]^4)/(3*(a + b)^2*(-1 + (a*Sin[e + f*x]^2)/(a + b))^2) + (a*Sin[e + f*x]^2)/((a + b)*(-1 + (a*Sin[e + f*x]^2)/(a + b)))))/(a^3*(1 - (a*Sin[e + f*x]^2)/(a + b))^(3/2))))))/(4*Sqrt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)^2)))","C",0
129,1,108,106,1.9743004,"\int \frac{\csc ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csc[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{\tan ^3(e+f x) \sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(-6 \left(a^2-b^2\right) \csc ^2(e+f x)+3 a^2+3 (a+b)^2 \csc ^4(e+f x)-6 a b-b^2\right)}{6 f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)^{5/2}}","-\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,"-1/6*((a + 2*b + a*Cos[2*(e + f*x)])*(3*a^2 - 6*a*b - b^2 - 6*(a^2 - b^2)*Csc[e + f*x]^2 + 3*(a + b)^2*Csc[e + f*x]^4)*Sec[e + f*x]^2*Tan[e + f*x]^3)/((a + b)^3*f*(a + b*Sec[e + f*x]^2)^(5/2))","A",1
130,1,138,158,4.5912737,"\int \frac{\csc ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\tan (e+f x) \sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b)^3 \left(\frac{4 b^2 (a+b)}{(a \cos (2 (e+f x))+a+2 b)^2}+\frac{4 b (b-3 a)}{a \cos (2 (e+f x))+a+2 b}-\left((a+b) \csc ^4(e+f x)\right)-2 (a-3 b) \csc ^2(e+f x)\right)}{24 f (a+b)^4 \left(a+b \sec ^2(e+f x)\right)^{5/2}}","-\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 + 2*b + a*Cos[2*(e + f*x)])^3*((4*b^2*(a + b))/(a + 2*b + a*Cos[2*(e + f*x)])^2 + (4*b*(-3*a + b))/(a + 2*b + a*Cos[2*(e + f*x)]) - 2*(a - 3*b)*Csc[e + f*x]^2 - (a + b)*Csc[e + f*x]^4)*Sec[e + f*x]^4*Tan[e + f*x])/(24*(a + b)^4*f*(a + b*Sec[e + f*x]^2)^(5/2))","A",1
131,1,173,226,7.4124348,"\int \frac{\csc ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csc[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\tan (e+f x) \sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b)^3 \left(\left(-8 a^2+50 a b-15 b^2\right) \csc ^2(e+f x)+\frac{20 a b^2 (a+b)}{(a \cos (2 (e+f x))+a+2 b)^2}+\frac{10 a b (5 b-6 a)}{a \cos (2 (e+f x))+a+2 b}-3 (a+b)^2 \csc ^6(e+f x)+2 (a+b) (5 b-2 a) \csc ^4(e+f x)\right)}{120 f (a+b)^5 \left(a+b \sec ^2(e+f x)\right)^{5/2}}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])^3*((20*a*b^2*(a + b))/(a + 2*b + a*Cos[2*(e + f*x)])^2 + (10*a*b*(-6*a + 5*b))/(a + 2*b + a*Cos[2*(e + f*x)]) + (-8*a^2 + 50*a*b - 15*b^2)*Csc[e + f*x]^2 + 2*(a + b)*(-2*a + 5*b)*Csc[e + f*x]^4 - 3*(a + b)^2*Csc[e + f*x]^6)*Sec[e + f*x]^4*Tan[e + f*x])/(120*(a + b)^5*f*(a + b*Sec[e + f*x]^2)^(5/2))","A",1
132,1,286,123,4.00485,"\int \left(a+b \sec ^2(e+f x)\right)^p (d \sin (e+f x))^m \, dx","Integrate[(a + b*Sec[e + f*x]^2)^p*(d*Sin[e + f*x])^m,x]","\frac{\sin (e+f x) \cos (e+f x) (d \sin (e+f x))^m \left(a+b \sec ^2(e+f x)\right)^p F_1\left(\frac{m+1}{2};\frac{m+2}{2},-p;\frac{m+3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}{f (m+1) \left(F_1\left(\frac{m+1}{2};\frac{m+2}{2},-p;\frac{m+3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-\frac{\tan ^2(e+f x) \left((m+2) (a+b) F_1\left(\frac{m+3}{2};\frac{m+4}{2},-p;\frac{m+5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{m+3}{2};\frac{m+2}{2},1-p;\frac{m+5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right)}{(m+3) (a+b)}\right)}","\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,"(AppellF1[(1 + m)/2, (2 + m)/2, -p, (3 + m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]*(d*Sin[e + f*x])^m)/(f*(1 + m)*(AppellF1[(1 + m)/2, (2 + m)/2, -p, (3 + m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - ((-2*b*p*AppellF1[(3 + m)/2, (2 + m)/2, 1 - p, (5 + m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*(2 + m)*AppellF1[(3 + m)/2, (4 + m)/2, -p, (5 + m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)/((a + b)*(3 + m))))","B",0
133,1,253,182,7.7695978,"\int \left(a+b \sec ^2(e+f x)\right)^p \sin ^5(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^5,x]","\frac{2 \sin ^4(e+f x) \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \left(4 \left(15 a^2+10 a b (1-2 p)+b^2 \left(4 p^2-8 p+3\right)\right) \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a}\right)+(a \cos (2 (e+f x))+a+2 b) (3 a \cos (2 (e+f x))-17 a+4 b p-6 b) \left(\frac{a+b \tan ^2(e+f x)+b}{a}\right)^p\right)}{15 a^2 f \left(4 \cos (2 (e+f x)) \left(\frac{a+b \tan ^2(e+f x)+b}{a}\right)^p-2^{-p} \left(2^p \cos (4 (e+f x)) \left(\frac{a+b \tan ^2(e+f x)+b}{a}\right)^p+3 \left(\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b)}{a}\right)^p\right)\right)}","-\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,"(2*Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^4*(4*(15*a^2 + 10*a*b*(1 - 2*p) + b^2*(3 - 8*p + 4*p^2))*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/a)] + (a + 2*b + a*Cos[2*(e + f*x)])*(-17*a - 6*b + 4*b*p + 3*a*Cos[2*(e + f*x)])*((a + b + b*Tan[e + f*x]^2)/a)^p))/(15*a^2*f*(4*Cos[2*(e + f*x)]*((a + b + b*Tan[e + f*x]^2)/a)^p - (3*(((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2)/a)^p + 2^p*Cos[4*(e + f*x)]*((a + b + b*Tan[e + f*x]^2)/a)^p)/2^p))","A",0
134,1,178,117,3.8853471,"\int \left(a+b \sec ^2(e+f x)\right)^p \sin ^3(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^3,x]","-\frac{\sin ^2(e+f x) \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \left((a \cos (2 (e+f x))+a+2 b) \left(\frac{a+b \tan ^2(e+f x)+b}{a}\right)^p-2 (3 a-2 b p+b) \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a}\right)\right)}{3 a f \left(\left(\frac{a+b \tan ^2(e+f x)+b}{a}\right)^p-2 \left(\frac{b \sec ^2(e+f x)}{a}+1\right)^p+\cos (2 (e+f x)) \left(\frac{a+b \tan ^2(e+f x)+b}{a}\right)^p\right)}","\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,"-1/3*(Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^2*(-2*(3*a + b - 2*b*p)*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/a)] + (a + 2*b + a*Cos[2*(e + f*x)])*((a + b + b*Tan[e + f*x]^2)/a)^p))/(a*f*(-2*(1 + (b*Sec[e + f*x]^2)/a)^p + ((a + b + b*Tan[e + f*x]^2)/a)^p + Cos[2*(e + f*x)]*((a + b + b*Tan[e + f*x]^2)/a)^p))","A",1
135,1,68,68,1.6504057,"\int \left(a+b \sec ^2(e+f x)\right)^p \sin (e+f x) \, dx","Integrate[(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",1
136,1,1532,77,16.8780364,"\int \csc (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Csc[e + f*x]*(a + b*Sec[e + f*x]^2)^p,x]","\frac{(\cos (2 (e+f x)) a+a+2 b)^p \csc (e+f x) \sec ^2(e+f x)^p \left(b \sec ^2(e+f x)+a\right)^p \left(\frac{2 F_1\left(-p-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2}-p;-\cot ^2(e+f x),-\frac{(a+b) \cot ^2(e+f x)}{b}\right) \left(\frac{(a+b) \cot ^2(e+f x)}{b}+1\right)^{-p} \sqrt{\sec ^2(e+f x)}}{(2 p+1) \sqrt{\csc ^2(e+f x)}}-F_1\left(1;\frac{1}{2},-p;2;-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan ^2(e+f x) \left(\frac{b \tan ^2(e+f x)+a+b}{a+b}\right)^{-p}\right)}{2 f \left(-a p \sec ^2(e+f x)^p \sin (2 (e+f x)) \left(\frac{2 F_1\left(-p-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2}-p;-\cot ^2(e+f x),-\frac{(a+b) \cot ^2(e+f x)}{b}\right) \left(\frac{(a+b) \cot ^2(e+f x)}{b}+1\right)^{-p} \sqrt{\sec ^2(e+f x)}}{(2 p+1) \sqrt{\csc ^2(e+f x)}}-F_1\left(1;\frac{1}{2},-p;2;-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan ^2(e+f x) \left(\frac{b \tan ^2(e+f x)+a+b}{a+b}\right)^{-p}\right) (\cos (2 (e+f x)) a+a+2 b)^{p-1}+p \sec ^2(e+f x)^p \tan (e+f x) \left(\frac{2 F_1\left(-p-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2}-p;-\cot ^2(e+f x),-\frac{(a+b) \cot ^2(e+f x)}{b}\right) \left(\frac{(a+b) \cot ^2(e+f x)}{b}+1\right)^{-p} \sqrt{\sec ^2(e+f x)}}{(2 p+1) \sqrt{\csc ^2(e+f x)}}-F_1\left(1;\frac{1}{2},-p;2;-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan ^2(e+f x) \left(\frac{b \tan ^2(e+f x)+a+b}{a+b}\right)^{-p}\right) (\cos (2 (e+f x)) a+a+2 b)^p+\frac{1}{2} \sec ^2(e+f x)^p \left(\frac{4 (a+b) p F_1\left(-p-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2}-p;-\cot ^2(e+f x),-\frac{(a+b) \cot ^2(e+f x)}{b}\right) \cot (e+f x) \sqrt{\csc ^2(e+f x)} \sqrt{\sec ^2(e+f x)} \left(\frac{(a+b) \cot ^2(e+f x)}{b}+1\right)^{-p-1}}{b (2 p+1)}+\frac{2 F_1\left(-p-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2}-p;-\cot ^2(e+f x),-\frac{(a+b) \cot ^2(e+f x)}{b}\right) \sqrt{\sec ^2(e+f x)} \tan (e+f x) \left(\frac{(a+b) \cot ^2(e+f x)}{b}+1\right)^{-p}}{(2 p+1) \sqrt{\csc ^2(e+f x)}}+\frac{2 \left(-\frac{2 (a+b) \left(-p-\frac{1}{2}\right) p F_1\left(\frac{1}{2}-p;-\frac{1}{2},1-p;\frac{3}{2}-p;-\cot ^2(e+f x),-\frac{(a+b) \cot ^2(e+f x)}{b}\right) \cot (e+f x) \csc ^2(e+f x)}{b \left(\frac{1}{2}-p\right)}-\frac{\left(-p-\frac{1}{2}\right) F_1\left(\frac{1}{2}-p;\frac{1}{2},-p;\frac{3}{2}-p;-\cot ^2(e+f x),-\frac{(a+b) \cot ^2(e+f x)}{b}\right) \cot (e+f x) \csc ^2(e+f x)}{\frac{1}{2}-p}\right) \sqrt{\sec ^2(e+f x)} \left(\frac{(a+b) \cot ^2(e+f x)}{b}+1\right)^{-p}}{(2 p+1) \sqrt{\csc ^2(e+f x)}}+\frac{2 F_1\left(-p-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2}-p;-\cot ^2(e+f x),-\frac{(a+b) \cot ^2(e+f x)}{b}\right) \cot (e+f x) \sqrt{\sec ^2(e+f x)} \left(\frac{(a+b) \cot ^2(e+f x)}{b}+1\right)^{-p}}{(2 p+1) \sqrt{\csc ^2(e+f x)}}+\frac{2 b p F_1\left(1;\frac{1}{2},-p;2;-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan ^3(e+f x) \left(\frac{b \tan ^2(e+f x)+a+b}{a+b}\right)^{-p-1}}{a+b}-2 F_1\left(1;\frac{1}{2},-p;2;-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x) \left(\frac{b \tan ^2(e+f x)+a+b}{a+b}\right)^{-p}-\tan ^2(e+f x) \left(\frac{b p F_1\left(2;\frac{1}{2},1-p;3;-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{a+b}-\frac{1}{2} F_1\left(2;\frac{3}{2},-p;3;-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right) \left(\frac{b \tan ^2(e+f x)+a+b}{a+b}\right)^{-p}\right) (\cos (2 (e+f x)) a+a+2 b)^p\right)}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])^p*Csc[e + f*x]*(Sec[e + f*x]^2)^p*(a + b*Sec[e + f*x]^2)^p*((2*AppellF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b)]*Sqrt[Sec[e + f*x]^2])/((1 + 2*p)*(1 + ((a + b)*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e + f*x]^2]) - (AppellF1[1, 1/2, -p, 2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2)/((a + b + b*Tan[e + f*x]^2)/(a + b))^p))/(2*f*(-(a*p*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^p*Sin[2*(e + f*x)]*((2*AppellF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b)]*Sqrt[Sec[e + f*x]^2])/((1 + 2*p)*(1 + ((a + b)*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e + f*x]^2]) - (AppellF1[1, 1/2, -p, 2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2)/((a + b + b*Tan[e + f*x]^2)/(a + b))^p)) + p*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Tan[e + f*x]*((2*AppellF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b)]*Sqrt[Sec[e + f*x]^2])/((1 + 2*p)*(1 + ((a + b)*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e + f*x]^2]) - (AppellF1[1, 1/2, -p, 2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2)/((a + b + b*Tan[e + f*x]^2)/(a + b))^p) + ((a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*((2*AppellF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b)]*Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])/((1 + 2*p)*(1 + ((a + b)*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e + f*x]^2]) + (4*(a + b)*p*AppellF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b)]*Cot[e + f*x]*(1 + ((a + b)*Cot[e + f*x]^2)/b)^(-1 - p)*Sqrt[Csc[e + f*x]^2]*Sqrt[Sec[e + f*x]^2])/(b*(1 + 2*p)) + (2*((-2*(a + b)*(-1/2 - p)*p*AppellF1[1/2 - p, -1/2, 1 - p, 3/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b)]*Cot[e + f*x]*Csc[e + f*x]^2)/(b*(1/2 - p)) - ((-1/2 - p)*AppellF1[1/2 - p, 1/2, -p, 3/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b)]*Cot[e + f*x]*Csc[e + f*x]^2)/(1/2 - p))*Sqrt[Sec[e + f*x]^2])/((1 + 2*p)*(1 + ((a + b)*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e + f*x]^2]) + (2*AppellF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b)]*Sqrt[Sec[e + f*x]^2]*Tan[e + f*x])/((1 + 2*p)*(1 + ((a + b)*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e + f*x]^2]) + (2*b*p*AppellF1[1, 1/2, -p, 2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x]^3*((a + b + b*Tan[e + f*x]^2)/(a + b))^(-1 - p))/(a + b) - (2*AppellF1[1, 1/2, -p, 2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/((a + b + b*Tan[e + f*x]^2)/(a + b))^p - (Tan[e + f*x]^2*((b*p*AppellF1[2, 1/2, 1 - p, 3, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(a + b) - (AppellF1[2, 3/2, -p, 3, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/2))/((a + b + b*Tan[e + f*x]^2)/(a + b))^p))/2))","B",0
137,1,266,81,4.4071651,"\int \csc ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\csc ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^p F_1\left(\frac{1}{2}-p;-\frac{1}{2},-p;\frac{3}{2}-p;-\cot ^2(e+f x),-\frac{(a+b) \cot ^2(e+f x)}{b}\right)}{f (2 p-1) \left(\sec (e+f x) F_1\left(\frac{1}{2}-p;-\frac{1}{2},-p;\frac{3}{2}-p;-\cot ^2(e+f x),-\frac{(a+b) \cot ^2(e+f x)}{b}\right)-\frac{\cot (e+f x) \csc (e+f x) \left(2 p (a+b) F_1\left(\frac{3}{2}-p;-\frac{1}{2},1-p;\frac{5}{2}-p;-\cot ^2(e+f x),-\frac{(a+b) \cot ^2(e+f x)}{b}\right)+b F_1\left(\frac{3}{2}-p;\frac{1}{2},-p;\frac{5}{2}-p;-\cot ^2(e+f x),-\frac{(a+b) \cot ^2(e+f x)}{b}\right)\right)}{b (2 p-3)}\right)}","\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[1/2 - p, -1/2, -p, 3/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b)]*Csc[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p)/(f*(-1 + 2*p)*(-(((2*(a + b)*p*AppellF1[3/2 - p, -1/2, 1 - p, 5/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b)] + b*AppellF1[3/2 - p, 1/2, -p, 5/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b)])*Cot[e + f*x]*Csc[e + f*x])/(b*(-3 + 2*p))) + AppellF1[1/2 - p, -1/2, -p, 3/2 - p, -Cot[e + f*x]^2, -(((a + b)*Cot[e + f*x]^2)/b)]*Sec[e + f*x]))","B",0
138,1,5878,88,25.7227903,"\int \left(a+b \sec ^2(e+f x)\right)^p \sin ^4(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^4,x]","\text{Result too large to show}","\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,"Result too large to show","B",0
139,1,3781,88,21.1326488,"\int \left(a+b \sec ^2(e+f x)\right)^p \sin ^2(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^2,x]","\text{Result too large to show}","\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,"(3*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-2 + p)*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^2*Tan[e + f*x]*(AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]/(-3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(-(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]) + 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)))/(f*(3*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-1 + p)*(AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]/(-3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(-(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]) + 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) - 6*a*(a + b)*p*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^(-2 + p)*Sin[2*(e + f*x)]*Tan[e + f*x]*(AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]/(-3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(-(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]) + 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) + 6*(a + b)*(-2 + p)*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-2 + p)*Tan[e + f*x]^2*(AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]/(-3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(-(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]) + 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) + 3*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-2 + p)*Tan[e + f*x]*(((2*b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (4*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3)/(-3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(-(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]) + 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (2*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (Sec[e + f*x]^2*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(4*(-(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]) + 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Sec[e + f*x]^2*Tan[e + f*x] - 3*(a + b)*((2*b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (4*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(-(b*p*((-6*b*(1 - p)*AppellF1[5/2, 2, 2 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (12*AppellF1[5/2, 3, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5)) + 2*(a + b)*((6*b*p*AppellF1[5/2, 3, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (18*AppellF1[5/2, 4, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(-3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(-(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]) + 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)^2 - (AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*(4*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Sec[e + f*x]^2*Tan[e + f*x] + 3*(a + b)*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(b*p*((-6*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5 - (6*b*(1 - p)*AppellF1[5/2, 2 - p, 1, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b))) - (a + b)*((6*b*p*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (12*AppellF1[5/2, -p, 3, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)^2)))","B",0
140,1,2137,83,15.1676874,"\int \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[(a + b*Sec[e + f*x]^2)^p,x]","\text{Result too large to show}","\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,"(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x])/(f*(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)*((3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-1 + p))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Sin[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (6*(a + b)*p*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Sin[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (6*a*(a + b)*p*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^p*Sin[e + f*x]*Sin[2*(e + f*x)])/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (3*(a + b)*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Sin[e + f*x]*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Sin[e + f*x]*(4*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Sec[e + f*x]^2*Tan[e + f*x] + 3*(a + b)*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(b*p*((-6*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5 - (6*b*(1 - p)*AppellF1[5/2, 2 - p, 1, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b))) - (a + b)*((6*b*p*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (12*AppellF1[5/2, -p, 3, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)^2))","B",0
141,1,72,73,1.0884459,"\int \csc ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Csc[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p,x]","-\frac{\cot (e+f x) \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \left(a+b \sec ^2(e+f x)\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*Sec[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p))","A",1
142,1,132,128,2.2461196,"\int \csc ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Csc[e + f*x]^4*(a + b*Sec[e + f*x]^2)^p,x]","-\frac{\cot (e+f x) \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \left(a+b \sec ^2(e+f x)\right)^p \left((3 a+2 b (p+1)) \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)+\cot ^2(e+f x) \left(a+b \tan ^2(e+f x)+b\right) \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^p\right)}{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,"-1/3*(Cot[e + f*x]*(a + b*Sec[e + f*x]^2)^p*((3*a + 2*b*(1 + p))*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))] + Cot[e + f*x]^2*(a + b + b*Tan[e + f*x]^2)*(1 + (b*Tan[e + f*x]^2)/(a + b))^p))/((a + b)*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",1
143,1,149,192,2.0012348,"\int \csc ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Csc[e + f*x]^6*(a + b*Sec[e + f*x]^2)^p,x]","-\frac{\cot (e+f x) \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \left(a+b \sec ^2(e+f x)\right)^p \left(15 \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)+3 \cot ^4(e+f x) \, _2F_1\left(-\frac{5}{2},-p;-\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)+10 \cot ^2(e+f x) \, _2F_1\left(-\frac{3}{2},-p;-\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)\right)}{15 f}","-\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,"-1/15*(Cot[e + f*x]*(3*Cot[e + f*x]^4*Hypergeometric2F1[-5/2, -p, -3/2, -((b*Tan[e + f*x]^2)/(a + b))] + 10*Cot[e + f*x]^2*Hypergeometric2F1[-3/2, -p, -1/2, -((b*Tan[e + f*x]^2)/(a + b))] + 15*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))])*(a + b*Sec[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",1
144,1,72,74,0.0405715,"\int \left(a-a \sec ^2(c+d x)\right)^4 \, dx","Integrate[(a - a*Sec[c + d*x]^2)^4,x]","a^4 \left(\frac{\tan ^{-1}(\tan (c+d x))}{d}+\frac{\tan ^7(c+d x)}{7 d}-\frac{\tan ^5(c+d x)}{5 d}+\frac{\tan ^3(c+d x)}{3 d}-\frac{\tan (c+d x)}{d}\right)","\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*(ArcTan[Tan[c + d*x]]/d - Tan[c + d*x]/d + Tan[c + d*x]^3/(3*d) - Tan[c + d*x]^5/(5*d) + Tan[c + d*x]^7/(7*d))","A",1
145,1,58,56,0.0303478,"\int \left(a-a \sec ^2(c+d x)\right)^3 \, dx","Integrate[(a - a*Sec[c + d*x]^2)^3,x]","-a^3 \left(-\frac{\tan ^{-1}(\tan (c+d x))}{d}+\frac{\tan ^5(c+d x)}{5 d}-\frac{\tan ^3(c+d x)}{3 d}+\frac{\tan (c+d x)}{d}\right)","-\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*(-(ArcTan[Tan[c + d*x]]/d) + Tan[c + d*x]/d - Tan[c + d*x]^3/(3*d) + Tan[c + d*x]^5/(5*d)))","A",1
146,1,42,38,0.0188609,"\int \left(a-a \sec ^2(c+d x)\right)^2 \, dx","Integrate[(a - a*Sec[c + d*x]^2)^2,x]","a^2 \left(\frac{\tan ^{-1}(\tan (c+d x))}{d}+\frac{\tan ^3(c+d x)}{3 d}-\frac{\tan (c+d x)}{d}\right)","\frac{a^2 \tan ^3(c+d x)}{3 d}-\frac{a^2 \tan (c+d x)}{d}+a^2 x",1,"a^2*(ArcTan[Tan[c + d*x]]/d - Tan[c + d*x]/d + Tan[c + d*x]^3/(3*d))","A",1
147,1,26,16,0.0061998,"\int \left(a-a \sec ^2(c+d x)\right) \, dx","Integrate[a - a*Sec[c + d*x]^2,x]","-a \left(\frac{\tan (c+d x)}{d}-\frac{\tan ^{-1}(\tan (c+d x))}{d}\right)","a x-\frac{a \tan (c+d x)}{d}",1,"-(a*(-(ArcTan[Tan[c + d*x]]/d) + Tan[c + d*x]/d))","A",1
148,1,31,19,0.0303948,"\int \frac{1}{a-a \sec ^2(c+d x)} \, dx","Integrate[(a - a*Sec[c + d*x]^2)^(-1),x]","\frac{\cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{a d}","\frac{\cot (c+d x)}{a d}+\frac{x}{a}",1,"(Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/(a*d)","C",1
149,1,36,37,0.0256276,"\int \frac{1}{\left(a-a \sec ^2(c+d x)\right)^2} \, dx","Integrate[(a - a*Sec[c + d*x]^2)^(-2),x]","-\frac{\cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)}{3 a^2 d}","-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{\cot (c+d x)}{a^2 d}+\frac{x}{a^2}",1,"-1/3*(Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(a^2*d)","C",1
150,1,36,55,0.05001,"\int \frac{1}{\left(a-a \sec ^2(c+d x)\right)^3} \, dx","Integrate[(a - a*Sec[c + d*x]^2)^(-3),x]","\frac{\cot ^5(c+d x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(c+d x)\right)}{5 a^3 d}","\frac{\cot ^5(c+d x)}{5 a^3 d}-\frac{\cot ^3(c+d x)}{3 a^3 d}+\frac{\cot (c+d x)}{a^3 d}+\frac{x}{a^3}",1,"(Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/(5*a^3*d)","C",1
151,1,36,73,0.0174091,"\int \frac{1}{\left(a-a \sec ^2(c+d x)\right)^4} \, dx","Integrate[(a - a*Sec[c + d*x]^2)^(-4),x]","-\frac{\cot ^7(c+d x) \, _2F_1\left(-\frac{7}{2},1;-\frac{5}{2};-\tan ^2(c+d x)\right)}{7 a^4 d}","-\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,"-1/7*(Cot[c + d*x]^7*Hypergeometric2F1[-7/2, 1, -5/2, -Tan[c + d*x]^2])/(a^4*d)","C",1
152,1,75,98,0.3391366,"\int \sec ^5(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]^5*(a + b*Sec[e + f*x]^2),x]","\frac{3 (6 a+5 b) \tanh ^{-1}(\sin (e+f x))+\tan (e+f x) \sec (e+f x) \left(2 (6 a+5 b) \sec ^2(e+f x)+3 (6 a+5 b)+8 b \sec ^4(e+f x)\right)}{48 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,"(3*(6*a + 5*b)*ArcTanh[Sin[e + f*x]] + Sec[e + f*x]*(3*(6*a + 5*b) + 2*(6*a + 5*b)*Sec[e + f*x]^2 + 8*b*Sec[e + f*x]^4)*Tan[e + f*x])/(48*f)","A",1
153,1,54,70,0.1313134,"\int \sec ^3(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2),x]","\frac{(4 a+3 b) \tanh ^{-1}(\sin (e+f x))+\tan (e+f x) \sec (e+f x) \left(4 a+2 b \sec ^2(e+f x)+3 b\right)}{8 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]] + Sec[e + f*x]*(4*a + 3*b + 2*b*Sec[e + f*x]^2)*Tan[e + f*x])/(8*f)","A",1
154,1,48,40,0.0146975,"\int \sec (e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]*(a + b*Sec[e + f*x]^2),x]","\frac{a \tanh ^{-1}(\sin (e+f x))}{f}+\frac{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,"(a*ArcTanh[Sin[e + f*x]])/f + (b*ArcTanh[Sin[e + f*x]])/(2*f) + (b*Sec[e + f*x]*Tan[e + f*x])/(2*f)","A",1
155,1,35,24,0.0162863,"\int \cos (e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Cos[e + f*x]*(a + b*Sec[e + f*x]^2),x]","\frac{a \sin (e) \cos (f x)}{f}+\frac{a \cos (e) \sin (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*Cos[f*x]*Sin[e])/f + (a*Cos[e]*Sin[f*x])/f","A",1
156,1,50,30,0.0214348,"\int \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2),x]","-\frac{a \sin ^3(e+f x)}{3 f}+\frac{a \sin (e+f x)}{f}+\frac{b \sin (e) \cos (f x)}{f}+\frac{b \cos (e) \sin (f x)}{f}","\frac{(a+b) \sin (e+f x)}{f}-\frac{a \sin ^3(e+f x)}{3 f}",1,"(b*Cos[f*x]*Sin[e])/f + (b*Cos[e]*Sin[f*x])/f + (a*Sin[e + f*x])/f - (a*Sin[e + f*x]^3)/(3*f)","A",1
157,1,71,50,0.0207904,"\int \cos ^5(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2),x]","\frac{a \sin ^5(e+f x)}{5 f}-\frac{2 a \sin ^3(e+f x)}{3 f}+\frac{a \sin (e+f x)}{f}-\frac{b \sin ^3(e+f x)}{3 f}+\frac{b \sin (e+f x)}{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*Sin[e + f*x])/f + (b*Sin[e + f*x])/f - (2*a*Sin[e + f*x]^3)/(3*f) - (b*Sin[e + f*x]^3)/(3*f) + (a*Sin[e + f*x]^5)/(5*f)","A",1
158,1,81,87,0.2911946,"\int \sec ^6(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]^6*(a + b*Sec[e + f*x]^2),x]","\frac{a \left(\frac{1}{5} \tan ^5(e+f x)+\frac{2}{3} \tan ^3(e+f x)+\tan (e+f x)\right)}{f}+\frac{b \left(\frac{1}{7} \tan ^7(e+f x)+\frac{3}{5} \tan ^5(e+f x)+\tan ^3(e+f x)+\tan (e+f x)\right)}{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,"(a*(Tan[e + f*x] + (2*Tan[e + f*x]^3)/3 + Tan[e + f*x]^5/5))/f + (b*(Tan[e + f*x] + Tan[e + f*x]^3 + (3*Tan[e + f*x]^5)/5 + Tan[e + f*x]^7/7))/f","A",1
159,1,61,65,0.2033141,"\int \sec ^4(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]^4*(a + b*Sec[e + f*x]^2),x]","\frac{a \left(\frac{1}{3} \tan ^3(e+f x)+\tan (e+f x)\right)}{f}+\frac{b \left(\frac{1}{5} \tan ^5(e+f x)+\frac{2}{3} \tan ^3(e+f x)+\tan (e+f x)\right)}{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,"(a*(Tan[e + f*x] + Tan[e + f*x]^3/3))/f + (b*(Tan[e + f*x] + (2*Tan[e + f*x]^3)/3 + Tan[e + f*x]^5/5))/f","A",1
160,1,36,43,0.0882991,"\int \sec ^2(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]^2*(a + b*Sec[e + f*x]^2),x]","\frac{a \tan (e+f x)}{f}+\frac{b \left(\frac{1}{3} \tan ^3(e+f x)+\tan (e+f x)\right)}{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,"(a*Tan[e + f*x])/f + (b*(Tan[e + f*x] + Tan[e + f*x]^3/3))/f","A",1
161,1,15,15,0.0025408,"\int \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[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",1
162,1,33,31,0.0290255,"\int \cos ^2(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Cos[e + f*x]^2*(a + b*Sec[e + f*x]^2),x]","\frac{a (e+f x)}{2 f}+\frac{a \sin (2 (e+f x))}{4 f}+b x","\frac{1}{2} x (a+2 b)+\frac{a \sin (e+f x) \cos (e+f x)}{2 f}",1,"b*x + (a*(e + f*x))/(2*f) + (a*Sin[2*(e + f*x)])/(4*f)","A",1
163,1,45,61,0.0891336,"\int \cos ^4(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2),x]","\frac{4 (3 a+4 b) (e+f x)+8 (a+b) \sin (2 (e+f x))+a \sin (4 (e+f x))}{32 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,"(4*(3*a + 4*b)*(e + f*x) + 8*(a + b)*Sin[2*(e + f*x)] + a*Sin[4*(e + f*x)])/(32*f)","A",1
164,1,68,89,0.1055879,"\int \cos ^6(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Cos[e + f*x]^6*(a + b*Sec[e + f*x]^2),x]","\frac{(45 a+48 b) \sin (2 (e+f x))+(9 a+6 b) \sin (4 (e+f x))+a \sin (6 (e+f x))+60 a e+60 a f x+72 b e+72 b f x}{192 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,"(60*a*e + 72*b*e + 60*a*f*x + 72*b*f*x + (45*a + 48*b)*Sin[2*(e + f*x)] + (9*a + 6*b)*Sin[4*(e + f*x)] + a*Sin[6*(e + f*x)])/(192*f)","A",1
165,1,119,165,0.5326756,"\int \sec ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Sec[e + f*x]^5*(a + b*Sec[e + f*x]^2)^2,x]","\frac{3 \left(48 a^2+80 a b+35 b^2\right) \tanh ^{-1}(\sin (e+f x))+\tan (e+f x) \sec (e+f x) \left(2 \left(48 a^2+80 a b+35 b^2\right) \sec ^2(e+f x)+3 \left(48 a^2+80 a b+35 b^2\right)+8 b (16 a+7 b) \sec ^4(e+f x)+48 b^2 \sec ^6(e+f x)\right)}{384 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,"(3*(48*a^2 + 80*a*b + 35*b^2)*ArcTanh[Sin[e + f*x]] + Sec[e + f*x]*(3*(48*a^2 + 80*a*b + 35*b^2) + 2*(48*a^2 + 80*a*b + 35*b^2)*Sec[e + f*x]^2 + 8*b*(16*a + 7*b)*Sec[e + f*x]^4 + 48*b^2*Sec[e + f*x]^6)*Tan[e + f*x])/(384*f)","A",1
166,1,94,129,0.3895954,"\int \sec ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2,x]","\frac{3 \left(8 a^2+12 a b+5 b^2\right) \tanh ^{-1}(\sin (e+f x))+\tan (e+f x) \sec (e+f x) \left(3 \left(8 a^2+12 a b+5 b^2\right)+2 b (12 a+5 b) \sec ^2(e+f x)+8 b^2 \sec ^4(e+f x)\right)}{48 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,"(3*(8*a^2 + 12*a*b + 5*b^2)*ArcTanh[Sin[e + f*x]] + Sec[e + f*x]*(3*(8*a^2 + 12*a*b + 5*b^2) + 2*b*(12*a + 5*b)*Sec[e + f*x]^2 + 8*b^2*Sec[e + f*x]^4)*Tan[e + f*x])/(48*f)","A",1
167,1,63,91,0.1363403,"\int \sec (e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[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))+b \tan (e+f x) \sec (e+f x) \left(8 a+2 b \sec ^2(e+f x)+3 b\right)}{8 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]] + b*Sec[e + f*x]*(8*a + 3*b + 2*b*Sec[e + f*x]^2)*Tan[e + f*x])/(8*f)","A",1
168,1,80,56,0.0306682,"\int \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^2,x]","\frac{a^2 \sin (e) \cos (f x)}{f}+\frac{a^2 \cos (e) \sin (f x)}{f}+\frac{2 a b \tanh ^{-1}(\sin (e+f x))}{f}+\frac{b^2 \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,"(2*a*b*ArcTanh[Sin[e + f*x]])/f + (b^2*ArcTanh[Sin[e + f*x]])/(2*f) + (a^2*Cos[f*x]*Sin[e])/f + (a^2*Cos[e]*Sin[f*x])/f + (b^2*Sec[e + f*x]*Tan[e + f*x])/(2*f)","A",1
169,1,72,49,0.0226612,"\int \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[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^2 \sin (e+f x)}{f}+\frac{2 a b \sin (e) \cos (f x)}{f}+\frac{2 a b \cos (e) \sin (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 + (2*a*b*Cos[f*x]*Sin[e])/f + (2*a*b*Cos[e]*Sin[f*x])/f + (a^2*Sin[e + f*x])/f - (a^2*Sin[e + f*x]^3)/(3*f)","A",1
170,1,106,53,0.0245631,"\int \cos ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[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^2 \sin ^3(e+f x)}{3 f}+\frac{a^2 \sin (e+f x)}{f}-\frac{2 a b \sin ^3(e+f x)}{3 f}+\frac{2 a b \sin (e+f x)}{f}+\frac{b^2 \sin (e) \cos (f x)}{f}+\frac{b^2 \cos (e) \sin (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,"(b^2*Cos[f*x]*Sin[e])/f + (b^2*Cos[e]*Sin[f*x])/f + (a^2*Sin[e + f*x])/f + (2*a*b*Sin[e + f*x])/f - (2*a^2*Sin[e + f*x]^3)/(3*f) - (2*a*b*Sin[e + f*x]^3)/(3*f) + (a^2*Sin[e + f*x]^5)/(5*f)","A",1
171,1,96,106,0.4045786,"\int \sec ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Sec[e + f*x]^6*(a + b*Sec[e + f*x]^2)^2,x]","\frac{63 \left(a^2+6 a b+6 b^2\right) \tan ^5(e+f x)+210 \left(a^2+3 a b+2 b^2\right) \tan ^3(e+f x)+90 b (a+2 b) \tan ^7(e+f x)+315 (a+b)^2 \tan (e+f x)+35 b^2 \tan ^9(e+f x)}{315 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,"(315*(a + b)^2*Tan[e + f*x] + 210*(a^2 + 3*a*b + 2*b^2)*Tan[e + f*x]^3 + 63*(a^2 + 6*a*b + 6*b^2)*Tan[e + f*x]^5 + 90*b*(a + 2*b)*Tan[e + f*x]^7 + 35*b^2*Tan[e + f*x]^9)/(315*f)","A",1
172,1,75,80,0.3698312,"\int \sec ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Sec[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2,x]","\frac{35 \left(a^2+4 a b+3 b^2\right) \tan ^3(e+f x)+21 b (2 a+3 b) \tan ^5(e+f x)+105 (a+b)^2 \tan (e+f x)+15 b^2 \tan ^7(e+f x)}{105 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,"(105*(a + b)^2*Tan[e + f*x] + 35*(a^2 + 4*a*b + 3*b^2)*Tan[e + f*x]^3 + 21*b*(2*a + 3*b)*Tan[e + f*x]^5 + 15*b^2*Tan[e + f*x]^7)/(105*f)","A",1
173,1,48,53,0.2619777,"\int \sec ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Sec[e + f*x]^2*(a + b*Sec[e + f*x]^2)^2,x]","\frac{10 b (a+b) \tan ^3(e+f x)+15 (a+b)^2 \tan (e+f x)+3 b^2 \tan ^5(e+f x)}{15 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,"(15*(a + b)^2*Tan[e + f*x] + 10*b*(a + b)*Tan[e + f*x]^3 + 3*b^2*Tan[e + f*x]^5)/(15*f)","A",1
174,1,106,40,0.3838503,"\int \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[(a + b*Sec[e + f*x]^2)^2,x]","\frac{4 \sec ^3(e+f x) \left(a \cos ^2(e+f x)+b\right)^2 \left(3 a^2 f x \cos ^3(e+f x)+2 b (3 a+b) \sec (e) \sin (f x) \cos ^2(e+f x)+b^2 \tan (e) \cos (e+f x)+b^2 \sec (e) \sin (f x)\right)}{3 f (a \cos (2 (e+f x))+a+2 b)^2}","a^2 x+\frac{b (2 a+b) \tan (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"(4*(b + a*Cos[e + f*x]^2)^2*Sec[e + f*x]^3*(3*a^2*f*x*Cos[e + f*x]^3 + b^2*Sec[e]*Sin[f*x] + 2*b*(3*a + b)*Cos[e + f*x]^2*Sec[e]*Sin[f*x] + b^2*Cos[e + f*x]*Tan[e]))/(3*f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","B",1
175,1,52,47,0.1501165,"\int \cos ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Cos[e + f*x]^2*(a + b*Sec[e + f*x]^2)^2,x]","\frac{a^2 (e+f x)}{2 f}+\frac{a^2 \sin (2 (e+f x))}{4 f}+2 a b x+\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,"2*a*b*x + (a^2*(e + f*x))/(2*f) + (a^2*Sin[2*(e + f*x)])/(4*f) + (b^2*Tan[e + f*x])/f","A",1
176,1,58,81,0.1130652,"\int \cos ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2,x]","\frac{4 \left(3 a^2+8 a b+8 b^2\right) (e+f x)+a^2 \sin (4 (e+f x))+8 a (a+2 b) \sin (2 (e+f x))}{32 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,"(4*(3*a^2 + 8*a*b + 8*b^2)*(e + f*x) + 8*a*(a + 2*b)*Sin[2*(e + f*x)] + a^2*Sin[4*(e + f*x)])/(32*f)","A",1
177,1,99,119,0.1913103,"\int \cos ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Cos[e + f*x]^6*(a + b*Sec[e + f*x]^2)^2,x]","\frac{\left(45 a^2+96 a b+48 b^2\right) \sin (2 (e+f x))+a^2 \sin (6 (e+f x))+60 a^2 e+60 a^2 f x+3 a (3 a+4 b) \sin (4 (e+f x))+144 a b e+144 a b f x+96 b^2 e+96 b^2 f x}{192 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,"(60*a^2*e + 144*a*b*e + 96*b^2*e + 60*a^2*f*x + 144*a*b*f*x + 96*b^2*f*x + (45*a^2 + 96*a*b + 48*b^2)*Sin[2*(e + f*x)] + 3*a*(3*a + 4*b)*Sin[4*(e + f*x)] + a^2*Sin[6*(e + f*x)])/(192*f)","A",1
178,1,268,73,1.0185329,"\int \left(a+b \sec ^2(c+d x)\right)^3 \, dx","Integrate[(a + b*Sec[c + d*x]^2)^3,x]","\frac{\sec (c) \sec ^5(c+d x) \left(150 a^3 d x \cos (2 c+d x)+75 a^3 d x \cos (2 c+3 d x)+75 a^3 d x \cos (4 c+3 d x)+15 a^3 d x \cos (4 c+5 d x)+15 a^3 d x \cos (6 c+5 d x)+150 a^3 d x \cos (d x)-360 a^2 b \sin (2 c+d x)+360 a^2 b \sin (2 c+3 d x)-90 a^2 b \sin (4 c+3 d x)+90 a^2 b \sin (4 c+5 d x)+540 a^2 b \sin (d x)-180 a b^2 \sin (2 c+d x)+300 a b^2 \sin (2 c+3 d x)+60 a b^2 \sin (4 c+5 d x)+420 a b^2 \sin (d x)+80 b^3 \sin (2 c+3 d x)+16 b^3 \sin (4 c+5 d x)+160 b^3 \sin (d x)\right)}{480 d}","a^3 x+\frac{b \left(3 a^2+3 a b+b^2\right) \tan (c+d x)}{d}+\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,"(Sec[c]*Sec[c + d*x]^5*(150*a^3*d*x*Cos[d*x] + 150*a^3*d*x*Cos[2*c + d*x] + 75*a^3*d*x*Cos[2*c + 3*d*x] + 75*a^3*d*x*Cos[4*c + 3*d*x] + 15*a^3*d*x*Cos[4*c + 5*d*x] + 15*a^3*d*x*Cos[6*c + 5*d*x] + 540*a^2*b*Sin[d*x] + 420*a*b^2*Sin[d*x] + 160*b^3*Sin[d*x] - 360*a^2*b*Sin[2*c + d*x] - 180*a*b^2*Sin[2*c + d*x] + 360*a^2*b*Sin[2*c + 3*d*x] + 300*a*b^2*Sin[2*c + 3*d*x] + 80*b^3*Sin[2*c + 3*d*x] - 90*a^2*b*Sin[4*c + 3*d*x] + 90*a^2*b*Sin[4*c + 5*d*x] + 60*a*b^2*Sin[4*c + 5*d*x] + 16*b^3*Sin[4*c + 5*d*x]))/(480*d)","B",1
179,1,455,111,1.5739194,"\int \left(a+b \sec ^2(c+d x)\right)^4 \, dx","Integrate[(a + b*Sec[c + d*x]^2)^4,x]","\frac{\sec (c) \sec ^7(c+d x) \left(3675 a^4 d x \cos (2 c+d x)+2205 a^4 d x \cos (2 c+3 d x)+2205 a^4 d x \cos (4 c+3 d x)+735 a^4 d x \cos (4 c+5 d x)+735 a^4 d x \cos (6 c+5 d x)+105 a^4 d x \cos (6 c+7 d x)+105 a^4 d x \cos (8 c+7 d x)+3675 a^4 d x \cos (d x)-12600 a^3 b \sin (2 c+d x)+12600 a^3 b \sin (2 c+3 d x)-5040 a^3 b \sin (4 c+3 d x)+5040 a^3 b \sin (4 c+5 d x)-840 a^3 b \sin (6 c+5 d x)+840 a^3 b \sin (6 c+7 d x)+16800 a^3 b \sin (d x)-10920 a^2 b^2 \sin (2 c+d x)+15120 a^2 b^2 \sin (2 c+3 d x)-2520 a^2 b^2 \sin (4 c+3 d x)+5880 a^2 b^2 \sin (4 c+5 d x)+840 a^2 b^2 \sin (6 c+7 d x)+18480 a^2 b^2 \sin (d x)-4480 a b^3 \sin (2 c+d x)+9408 a b^3 \sin (2 c+3 d x)+3136 a b^3 \sin (4 c+5 d x)+448 a b^3 \sin (6 c+7 d x)+11200 a b^3 \sin (d x)+2016 b^4 \sin (2 c+3 d x)+672 b^4 \sin (4 c+5 d x)+96 b^4 \sin (6 c+7 d x)+3360 b^4 \sin (d x)\right)}{13440 d}","a^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}+\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,"(Sec[c]*Sec[c + d*x]^7*(3675*a^4*d*x*Cos[d*x] + 3675*a^4*d*x*Cos[2*c + d*x] + 2205*a^4*d*x*Cos[2*c + 3*d*x] + 2205*a^4*d*x*Cos[4*c + 3*d*x] + 735*a^4*d*x*Cos[4*c + 5*d*x] + 735*a^4*d*x*Cos[6*c + 5*d*x] + 105*a^4*d*x*Cos[6*c + 7*d*x] + 105*a^4*d*x*Cos[8*c + 7*d*x] + 16800*a^3*b*Sin[d*x] + 18480*a^2*b^2*Sin[d*x] + 11200*a*b^3*Sin[d*x] + 3360*b^4*Sin[d*x] - 12600*a^3*b*Sin[2*c + d*x] - 10920*a^2*b^2*Sin[2*c + d*x] - 4480*a*b^3*Sin[2*c + d*x] + 12600*a^3*b*Sin[2*c + 3*d*x] + 15120*a^2*b^2*Sin[2*c + 3*d*x] + 9408*a*b^3*Sin[2*c + 3*d*x] + 2016*b^4*Sin[2*c + 3*d*x] - 5040*a^3*b*Sin[4*c + 3*d*x] - 2520*a^2*b^2*Sin[4*c + 3*d*x] + 5040*a^3*b*Sin[4*c + 5*d*x] + 5880*a^2*b^2*Sin[4*c + 5*d*x] + 3136*a*b^3*Sin[4*c + 5*d*x] + 672*b^4*Sin[4*c + 5*d*x] - 840*a^3*b*Sin[6*c + 5*d*x] + 840*a^3*b*Sin[6*c + 7*d*x] + 840*a^2*b^2*Sin[6*c + 7*d*x] + 448*a*b^3*Sin[6*c + 7*d*x] + 96*b^4*Sin[6*c + 7*d*x]))/(13440*d)","B",1
180,1,1195,86,6.1190444,"\int \frac{\sec ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2),x]","\frac{(\cos (2 (e+f x)) a+a+2 b) \sec ^2(e+f x) \left(\frac{2 i \tan ^{-1}\left(\frac{2 \sin (e) \left(\sin (2 e) a+i a-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}+\sqrt{a+b} \cos (f x) \sqrt{(\cos (e)-i \sin (e))^2} \sqrt{a}-\sqrt{a+b} \cos (2 e+f x) \sqrt{(\cos (e)-i \sin (e))^2} \sqrt{a}+i b+i (a+b) \cos (2 e)+b \sin (2 e)\right)}{i (a+3 b) \cos (e)+i (a+b) \cos (3 e)+i a \cos (e+2 f x)+i a \cos (3 e+2 f x)+3 a \sin (e)+b \sin (e)+a \sin (3 e)+b \sin (3 e)+a \sin (e+2 f x)-a \sin (3 e+2 f x)}\right) \sqrt{(\cos (e)-i \sin (e))^2} (\cos (e)+i \sin (e)) a^{3/2}}{\sqrt{a+b}}-\frac{i \log \left(-\cos (2 (e+f x)) a-2 i \sin (2 e) a+a+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}+2 (a+b) \cos (2 e)-2 i b \sin (2 e)\right) \sin (e) a^{3/2}}{\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}}+\frac{i \log \left(\cos (2 (e+f x)) a+2 i \sin (2 e) a-a+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}-2 (a+b) \cos (2 e)+2 i b \sin (2 e)\right) \sin (e) a^{3/2}}{\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}}+\frac{2 \tan ^{-1}\left(\frac{(a+b) \sin (e)}{(a+b) \cos (e)-\sqrt{a} \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} (\cos (2 e)+i \sin (2 e)) \sin (e+f x)}\right) \sqrt{(\cos (e)-i \sin (e))^2} (\sin (e)-i \cos (e)) a^{3/2}}{\sqrt{a+b}}+\frac{\cos (e) \log \left(-\cos (2 (e+f x)) a-2 i \sin (2 e) a+a+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}+2 (a+b) \cos (2 e)-2 i b \sin (2 e)\right) a^{3/2}}{\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}}-\frac{\cos (e) \log \left(\cos (2 (e+f x)) a+2 i \sin (2 e) a-a+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}-2 (a+b) \cos (2 e)+2 i b \sin (2 e)\right) a^{3/2}}{\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}}+4 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) a-4 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) a-2 b \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+2 b \log \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)+\frac{b}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{b}{\left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}\right)}{8 b^2 f \left(b \sec ^2(e+f x)+a\right)}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(4*a*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 2*b*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 4*a*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 2*b*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (a^(3/2)*Cos[e]*Log[a + 2*(a + b)*Cos[2*e] - a*Cos[2*(e + f*x)] - (2*I)*a*Sin[2*e] - (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]])/(Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]) - (a^(3/2)*Cos[e]*Log[-a - 2*(a + b)*Cos[2*e] + a*Cos[2*(e + f*x)] + (2*I)*a*Sin[2*e] + (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]])/(Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]) + ((2*I)*a^(3/2)*ArcTan[(2*Sin[e]*(I*a + I*b + I*(a + b)*Cos[2*e] + Sqrt[a]*Sqrt[a + b]*Cos[f*x]*Sqrt[(Cos[e] - I*Sin[e])^2] - Sqrt[a]*Sqrt[a + b]*Cos[2*e + f*x]*Sqrt[(Cos[e] - I*Sin[e])^2] + a*Sin[2*e] + b*Sin[2*e] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]))/(I*(a + 3*b)*Cos[e] + I*(a + b)*Cos[3*e] + I*a*Cos[e + 2*f*x] + I*a*Cos[3*e + 2*f*x] + 3*a*Sin[e] + b*Sin[e] + a*Sin[3*e] + b*Sin[3*e] + a*Sin[e + 2*f*x] - a*Sin[3*e + 2*f*x])]*Sqrt[(Cos[e] - I*Sin[e])^2]*(Cos[e] + I*Sin[e]))/Sqrt[a + b] - (I*a^(3/2)*Log[a + 2*(a + b)*Cos[2*e] - a*Cos[2*(e + f*x)] - (2*I)*a*Sin[2*e] - (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]]*Sin[e])/(Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]) + (I*a^(3/2)*Log[-a - 2*(a + b)*Cos[2*e] + a*Cos[2*(e + f*x)] + (2*I)*a*Sin[2*e] + (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]]*Sin[e])/(Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]) + (2*a^(3/2)*ArcTan[((a + b)*Sin[e])/((a + b)*Cos[e] - Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*(Cos[2*e] + I*Sin[2*e])*Sin[e + f*x])]*Sqrt[(Cos[e] - I*Sin[e])^2]*((-I)*Cos[e] + Sin[e]))/Sqrt[a + b] + b/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 - b/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2))/(8*b^2*f*(a + b*Sec[e + f*x]^2))","C",0
181,1,1022,55,1.3493362,"\int \frac{\sec ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Sec[e + f*x]^3/(a + b*Sec[e + f*x]^2),x]","\frac{(\cos (2 (e+f x)) a+a+2 b) \sec ^2(e+f x) \left(-4 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\sqrt{a} \cos (e) \log \left(-\cos (2 (e+f x)) a-2 i \sin (2 e) a+a+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}+2 (a+b) \cos (2 e)-2 i b \sin (2 e)\right)+\sqrt{a} \cos (e) \log \left(\cos (2 (e+f x)) a+2 i \sin (2 e) a-a+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}-2 (a+b) \cos (2 e)+2 i b \sin (2 e)\right)-2 i \sqrt{a} \tan ^{-1}\left(\frac{2 \sin (e) \left(\sin (2 e) a+i a-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}+\sqrt{a+b} \cos (f x) \sqrt{(\cos (e)-i \sin (e))^2} \sqrt{a}-\sqrt{a+b} \cos (2 e+f x) \sqrt{(\cos (e)-i \sin (e))^2} \sqrt{a}+i b+i (a+b) \cos (2 e)+b \sin (2 e)\right)}{i (a+3 b) \cos (e)+i (a+b) \cos (3 e)+i a \cos (e+2 f x)+i a \cos (3 e+2 f x)+3 a \sin (e)+b \sin (e)+a \sin (3 e)+b \sin (3 e)+a \sin (e+2 f x)-a \sin (3 e+2 f x)}\right) (\cos (e)-i \sin (e))+i \sqrt{a} \log \left(-\cos (2 (e+f x)) a-2 i \sin (2 e) a+a+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}+2 (a+b) \cos (2 e)-2 i b \sin (2 e)\right) \sin (e)-i \sqrt{a} \log \left(\cos (2 (e+f x)) a+2 i \sin (2 e) a-a+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}-2 (a+b) \cos (2 e)+2 i b \sin (2 e)\right) \sin (e)+2 \sqrt{a} \tan ^{-1}\left(\frac{(a+b) \sin (e)}{(a+b) \cos (e)-\sqrt{a} \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} (\cos (2 e)+i \sin (2 e)) \sin (e+f x)}\right) (i \cos (e)+\sin (e))+4 \sqrt{a+b} \log \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{(\cos (e)-i \sin (e))^2}\right)}{8 b \sqrt{a+b} f \left(b \sec ^2(e+f x)+a\right) \sqrt{(\cos (e)-i \sin (e))^2}}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(-(Sqrt[a]*Cos[e]*Log[a + 2*(a + b)*Cos[2*e] - a*Cos[2*(e + f*x)] - (2*I)*a*Sin[2*e] - (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]]) + Sqrt[a]*Cos[e]*Log[-a - 2*(a + b)*Cos[2*e] + a*Cos[2*(e + f*x)] + (2*I)*a*Sin[2*e] + (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]] - (2*I)*Sqrt[a]*ArcTan[(2*Sin[e]*(I*a + I*b + I*(a + b)*Cos[2*e] + Sqrt[a]*Sqrt[a + b]*Cos[f*x]*Sqrt[(Cos[e] - I*Sin[e])^2] - Sqrt[a]*Sqrt[a + b]*Cos[2*e + f*x]*Sqrt[(Cos[e] - I*Sin[e])^2] + a*Sin[2*e] + b*Sin[2*e] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]))/(I*(a + 3*b)*Cos[e] + I*(a + b)*Cos[3*e] + I*a*Cos[e + 2*f*x] + I*a*Cos[3*e + 2*f*x] + 3*a*Sin[e] + b*Sin[e] + a*Sin[3*e] + b*Sin[3*e] + a*Sin[e + 2*f*x] - a*Sin[3*e + 2*f*x])]*(Cos[e] - I*Sin[e]) - 4*Sqrt[a + b]*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sqrt[(Cos[e] - I*Sin[e])^2] + 4*Sqrt[a + b]*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sqrt[(Cos[e] - I*Sin[e])^2] + I*Sqrt[a]*Log[a + 2*(a + b)*Cos[2*e] - a*Cos[2*(e + f*x)] - (2*I)*a*Sin[2*e] - (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]]*Sin[e] - I*Sqrt[a]*Log[-a - 2*(a + b)*Cos[2*e] + a*Cos[2*(e + f*x)] + (2*I)*a*Sin[2*e] + (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]]*Sin[e] + 2*Sqrt[a]*ArcTan[((a + b)*Sin[e])/((a + b)*Cos[e] - Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*(Cos[2*e] + I*Sin[2*e])*Sin[e + f*x])]*(I*Cos[e] + Sin[e])))/(8*b*Sqrt[a + b]*f*(a + b*Sec[e + f*x]^2)*Sqrt[(Cos[e] - I*Sin[e])^2])","C",0
182,1,36,36,0.0758197,"\int \frac{\sec (e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[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",1
183,1,52,52,0.0965658,"\int \frac{\cos (e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]/(a + b*Sec[e + f*x]^2),x]","\frac{\sqrt{a} \sin (e+f x)-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{a+b}}}{a^{3/2} f}","\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]])/Sqrt[a + b]) + Sqrt[a]*Sin[e + f*x])/(a^(3/2)*f)","A",1
184,1,105,76,0.28307,"\int \frac{\cos ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2),x]","\frac{a^{3/2} \sin (3 (e+f x))+\frac{6 b^2 \left(\log \left(\sqrt{a+b}+\sqrt{a} \sin (e+f x)\right)-\log \left(\sqrt{a+b}-\sqrt{a} \sin (e+f x)\right)\right)}{\sqrt{a+b}}+3 \sqrt{a} (3 a-4 b) \sin (e+f x)}{12 a^{5/2} 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,"((6*b^2*(-Log[Sqrt[a + b] - Sqrt[a]*Sin[e + f*x]] + Log[Sqrt[a + b] + Sqrt[a]*Sin[e + f*x]]))/Sqrt[a + b] + 3*Sqrt[a]*(3*a - 4*b)*Sin[e + f*x] + a^(3/2)*Sin[3*(e + f*x)])/(12*a^(5/2)*f)","A",1
185,1,136,108,0.6850061,"\int \frac{\cos ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2),x]","\frac{5 a^{3/2} (5 a-4 b) \sin (3 (e+f x))+3 a^{5/2} \sin (5 (e+f x))+30 \sqrt{a} \left(5 a^2-6 a b+8 b^2\right) \sin (e+f x)+\frac{120 b^3 \left(\log \left(\sqrt{a+b}-\sqrt{a} \sin (e+f x)\right)-\log \left(\sqrt{a+b}+\sqrt{a} \sin (e+f x)\right)\right)}{\sqrt{a+b}}}{240 a^{7/2} 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{\left(a^2-a b+b^2\right) \sin (e+f x)}{a^3 f}+\frac{\sin ^5(e+f x)}{5 a f}",1,"((120*b^3*(Log[Sqrt[a + b] - Sqrt[a]*Sin[e + f*x]] - Log[Sqrt[a + b] + Sqrt[a]*Sin[e + f*x]]))/Sqrt[a + b] + 30*Sqrt[a]*(5*a^2 - 6*a*b + 8*b^2)*Sin[e + f*x] + 5*a^(3/2)*(5*a - 4*b)*Sin[3*(e + f*x)] + 3*a^(5/2)*Sin[5*(e + f*x)])/(240*a^(7/2)*f)","A",1
186,1,224,77,2.2587813,"\int \frac{\sec ^6(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Sec[e + f*x]^6/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\sqrt{a+b} \sqrt{b (\sin (e)+i \cos (e))^4} \sec (e+f x) \left(\sec (e) \sin (f x) \left(-3 a+b \sec ^2(e+f x)+2 b\right)+b \tan (e) \sec (e+f x)\right)-3 a^2 (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)\right)}{6 b^2 f \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4} \left(a+b \sec ^2(e+f x)\right)}","\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*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(-3*a^2*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]) + Sqrt[a + b]*Sec[e + f*x]*Sqrt[b*(I*Cos[e] + Sin[e])^4]*(Sec[e]*(-3*a + 2*b + b*Sec[e + f*x]^2)*Sin[f*x] + b*Sec[e + f*x]*Tan[e])))/(6*b^2*Sqrt[a + b]*f*(a + b*Sec[e + f*x]^2)*Sqrt[b*(Cos[e] - I*Sin[e])^4])","C",1
187,1,192,52,0.6266682,"\int \frac{\sec ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Sec[e + f*x]^4/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\sqrt{a+b} \sec (e) \sin (f x) \sqrt{b (\sin (e)+i \cos (e))^4} \sec (e+f x)+a (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)\right)}{2 b f \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4} \left(a+b \sec ^2(e+f x)\right)}","\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 + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(a*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]) + Sqrt[a + b]*Sec[e]*Sec[e + f*x]*Sqrt[b*(I*Cos[e] + Sin[e])^4]*Sin[f*x]))/(2*b*Sqrt[a + b]*f*(a + b*Sec[e + f*x]^2)*Sqrt[b*(Cos[e] - I*Sin[e])^4])","C",1
188,1,36,36,0.0745774,"\int \frac{\sec ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[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",1
189,1,182,45,0.2857421,"\int \frac{1}{a+b \sec ^2(e+f x)} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(-1),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(f x \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}+b (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)\right)}{2 a f \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4} \left(a+b \sec ^2(e+f x)\right)}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(Sqrt[a + b]*f*x*Sqrt[b*(Cos[e] - I*Sin[e])^4] + b*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e])))/(2*a*Sqrt[a + b]*f*(a + b*Sec[e + f*x]^2)*Sqrt[b*(Cos[e] - I*Sin[e])^4])","C",1
190,1,67,75,0.2321626,"\int \frac{\cos ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2),x]","\frac{\frac{4 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{a+b}}+2 (a-2 b) (e+f x)+a \sin (2 (e+f x))}{4 a^2 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,"(2*(a - 2*b)*(e + f*x) + (4*b^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/Sqrt[a + b] + a*Sin[2*(e + f*x)])/(4*a^2*f)","A",1
191,1,95,117,0.399059,"\int \frac{\cos ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]^4/(a + b*Sec[e + f*x]^2),x]","\frac{4 \left(3 a^2-4 a b+8 b^2\right) (e+f x)+a^2 \sin (4 (e+f x))-\frac{32 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{a+b}}+8 a (a-b) \sin (2 (e+f x))}{32 a^3 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{(3 a-4 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f}+\frac{x \left(3 a^2-4 a b+8 b^2\right)}{8 a^3}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f}",1,"(4*(3*a^2 - 4*a*b + 8*b^2)*(e + f*x) - (32*b^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/Sqrt[a + b] + 8*a*(a - b)*Sin[2*(e + f*x)] + a^2*Sin[4*(e + f*x)])/(32*a^3*f)","A",1
192,1,133,163,0.8531247,"\int \frac{\cos ^6(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2),x]","\frac{a^3 \sin (6 (e+f x))+3 a \left(15 a^2-16 a b+16 b^2\right) \sin (2 (e+f x))+3 a^2 (3 a-2 b) \sin (4 (e+f x))+12 \left(5 a^3-6 a^2 b+8 a b^2-16 b^3\right) (e+f x)+\frac{192 b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{a+b}}}{192 a^4 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{(5 a-6 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f}+\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(5 a^3-6 a^2 b+8 a b^2-16 b^3\right)}{16 a^4}+\frac{\sin (e+f x) \cos ^5(e+f x)}{6 a f}",1,"(12*(5*a^3 - 6*a^2*b + 8*a*b^2 - 16*b^3)*(e + f*x) + (192*b^(7/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/Sqrt[a + b] + 3*a*(15*a^2 - 16*a*b + 16*b^2)*Sin[2*(e + f*x)] + 3*a^2*(3*a - 2*b)*Sin[4*(e + f*x)] + a^3*Sin[6*(e + f*x)])/(192*a^4*f)","A",1
193,1,980,102,3.8903586,"\int \frac{\sec ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2,x]","\frac{(\cos (2 (e+f x)) a+a+2 b) \sec ^3(e+f x) \left(-8 b \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan (e+f x) a^{3/2}-2 i (2 a+3 b) \tan ^{-1}\left(\frac{2 \sin (e) \left(\sin (2 e) a+i a-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}+\sqrt{a+b} \cos (f x) \sqrt{(\cos (e)-i \sin (e))^2} \sqrt{a}-\sqrt{a+b} \cos (2 e+f x) \sqrt{(\cos (e)-i \sin (e))^2} \sqrt{a}+i b+i (a+b) \cos (2 e)+b \sin (2 e)\right)}{i (a+3 b) \cos (e)+i (a+b) \cos (3 e)+i a \cos (e+2 f x)+i a \cos (3 e+2 f x)+3 a \sin (e)+b \sin (e)+a \sin (3 e)+b \sin (3 e)+a \sin (e+2 f x)-a \sin (3 e+2 f x)}\right) (\cos (2 (e+f x)) a+a+2 b) \sec (e+f x) (\cos (e)-i \sin (e)) a-(2 a+3 b) (\cos (2 (e+f x)) a+a+2 b) \log \left(-\cos (2 (e+f x)) a-2 i \sin (2 e) a+a+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}+2 (a+b) \cos (2 e)-2 i b \sin (2 e)\right) \sec (e+f x) (\cos (e)-i \sin (e)) a+(2 a+3 b) (\cos (2 (e+f x)) a+a+2 b) \log \left(\cos (2 (e+f x)) a+2 i \sin (2 e) a-a+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}-2 (a+b) \cos (2 e)+2 i b \sin (2 e)\right) \sec (e+f x) (\cos (e)-i \sin (e)) a+2 (2 a+3 b) \tan ^{-1}\left(\frac{(a+b) \sin (e)}{(a+b) \cos (e)-\sqrt{a} \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} (\cos (2 e)+i \sin (2 e)) \sin (e+f x)}\right) (\cos (2 (e+f x)) a+a+2 b) \sec (e+f x) (i \cos (e)+\sin (e)) a-8 (a+b)^{3/2} (\cos (2 (e+f x)) a+a+2 b) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \sec (e+f x) \sqrt{(\cos (e)-i \sin (e))^2} \sqrt{a}+8 (a+b)^{3/2} (\cos (2 (e+f x)) a+a+2 b) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sec (e+f x) \sqrt{(\cos (e)-i \sin (e))^2} \sqrt{a}\right)}{32 \sqrt{a} b^2 (a+b)^{3/2} f \left(b \sec ^2(e+f x)+a\right)^2 \sqrt{(\cos (e)-i \sin (e))^2}}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^3*((-2*I)*a*(2*a + 3*b)*ArcTan[(2*Sin[e]*(I*a + I*b + I*(a + b)*Cos[2*e] + Sqrt[a]*Sqrt[a + b]*Cos[f*x]*Sqrt[(Cos[e] - I*Sin[e])^2] - Sqrt[a]*Sqrt[a + b]*Cos[2*e + f*x]*Sqrt[(Cos[e] - I*Sin[e])^2] + a*Sin[2*e] + b*Sin[2*e] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]))/(I*(a + 3*b)*Cos[e] + I*(a + b)*Cos[3*e] + I*a*Cos[e + 2*f*x] + I*a*Cos[3*e + 2*f*x] + 3*a*Sin[e] + b*Sin[e] + a*Sin[3*e] + b*Sin[3*e] + a*Sin[e + 2*f*x] - a*Sin[3*e + 2*f*x])]*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]*(Cos[e] - I*Sin[e]) - a*(2*a + 3*b)*(a + 2*b + a*Cos[2*(e + f*x)])*Log[a + 2*(a + b)*Cos[2*e] - a*Cos[2*(e + f*x)] - (2*I)*a*Sin[2*e] - (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]]*Sec[e + f*x]*(Cos[e] - I*Sin[e]) + a*(2*a + 3*b)*(a + 2*b + a*Cos[2*(e + f*x)])*Log[-a - 2*(a + b)*Cos[2*e] + a*Cos[2*(e + f*x)] + (2*I)*a*Sin[2*e] + (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]]*Sec[e + f*x]*(Cos[e] - I*Sin[e]) - 8*Sqrt[a]*(a + b)^(3/2)*(a + 2*b + a*Cos[2*(e + f*x)])*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sec[e + f*x]*Sqrt[(Cos[e] - I*Sin[e])^2] + 8*Sqrt[a]*(a + b)^(3/2)*(a + 2*b + a*Cos[2*(e + f*x)])*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sec[e + f*x]*Sqrt[(Cos[e] - I*Sin[e])^2] + 2*a*(2*a + 3*b)*ArcTan[((a + b)*Sin[e])/((a + b)*Cos[e] - Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*(Cos[2*e] + I*Sin[2*e])*Sin[e + f*x])]*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]*(I*Cos[e] + Sin[e]) - 8*a^(3/2)*b*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[e + f*x]))/(32*Sqrt[a]*b^2*(a + b)^(3/2)*f*(a + b*Sec[e + f*x]^2)^2*Sqrt[(Cos[e] - I*Sin[e])^2])","C",0
194,1,88,74,0.272459,"\int \frac{\sec ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Sec[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sqrt{a} \sqrt{a+b} \sin (e+f x)+\left(-a \sin ^2(e+f x)+a+b\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{a} f (a+b)^{3/2} (a \cos (2 (e+f x))+a+2 b)}","\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,"(Sqrt[a]*Sqrt[a + b]*Sin[e + f*x] + ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*(a + b - a*Sin[e + f*x]^2))/(Sqrt[a]*(a + b)^(3/2)*f*(a + 2*b + a*Cos[2*(e + f*x)]))","A",1
195,1,82,83,0.3780309,"\int \frac{\sec (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Sec[e + f*x]/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}-\frac{2 \sqrt{a} b \sin (e+f x)}{(a+b) (a \cos (2 (e+f x))+a+2 b)}}{2 a^{3/2} f}","\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]])/(a + b)^(3/2) - (2*Sqrt[a]*b*Sin[e + f*x])/((a + b)*(a + 2*b + a*Cos[2*(e + f*x)])))/(2*a^(3/2)*f)","A",1
196,1,945,101,3.0052122,"\int \frac{\cos (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Cos[e + f*x]/(a + b*Sec[e + f*x]^2)^2,x]","\frac{(\cos (2 (e+f x)) a+a+2 b) \sec ^3(e+f x) \left(8 \sqrt{a} \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan (e+f x) b^2-2 i (4 a+3 b) \tan ^{-1}\left(\frac{2 \sin (e) \left(\sin (2 e) a+i a-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}+\sqrt{a+b} \cos (f x) \sqrt{(\cos (e)-i \sin (e))^2} \sqrt{a}-\sqrt{a+b} \cos (2 e+f x) \sqrt{(\cos (e)-i \sin (e))^2} \sqrt{a}+i b+i (a+b) \cos (2 e)+b \sin (2 e)\right)}{i (a+3 b) \cos (e)+i (a+b) \cos (3 e)+i a \cos (e+2 f x)+i a \cos (3 e+2 f x)+3 a \sin (e)+b \sin (e)+a \sin (3 e)+b \sin (3 e)+a \sin (e+2 f x)-a \sin (3 e+2 f x)}\right) (\cos (2 (e+f x)) a+a+2 b) \sec (e+f x) (\cos (e)-i \sin (e)) b-(4 a+3 b) (\cos (2 (e+f x)) a+a+2 b) \log \left(-\cos (2 (e+f x)) a-2 i \sin (2 e) a+a+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}+2 (a+b) \cos (2 e)-2 i b \sin (2 e)\right) \sec (e+f x) (\cos (e)-i \sin (e)) b+(4 a+3 b) (\cos (2 (e+f x)) a+a+2 b) \log \left(\cos (2 (e+f x)) a+2 i \sin (2 e) a-a+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}-2 (a+b) \cos (2 e)+2 i b \sin (2 e)\right) \sec (e+f x) (\cos (e)-i \sin (e)) b+2 (4 a+3 b) \tan ^{-1}\left(\frac{(a+b) \sin (e)}{(a+b) \cos (e)-\sqrt{a} \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} (\cos (2 e)+i \sin (2 e)) \sin (e+f x)}\right) (\cos (2 (e+f x)) a+a+2 b) \sec (e+f x) (i \cos (e)+\sin (e)) b+8 \sqrt{a} (a+b)^{3/2} \cos (f x) (\cos (2 (e+f x)) a+a+2 b) \sec (e+f x) \sqrt{(\cos (e)-i \sin (e))^2} \sin (e)+8 \sqrt{a} (a+b)^{3/2} \cos (e) (\cos (2 (e+f x)) a+a+2 b) \sec (e+f x) \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x)\right)}{32 a^{5/2} (a+b)^{3/2} f \left(b \sec ^2(e+f x)+a\right)^2 \sqrt{(\cos (e)-i \sin (e))^2}}","-\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{b^2 \sin (e+f x)}{2 a^2 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{\sin (e+f x)}{a^2 f}",1,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^3*((-2*I)*b*(4*a + 3*b)*ArcTan[(2*Sin[e]*(I*a + I*b + I*(a + b)*Cos[2*e] + Sqrt[a]*Sqrt[a + b]*Cos[f*x]*Sqrt[(Cos[e] - I*Sin[e])^2] - Sqrt[a]*Sqrt[a + b]*Cos[2*e + f*x]*Sqrt[(Cos[e] - I*Sin[e])^2] + a*Sin[2*e] + b*Sin[2*e] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]))/(I*(a + 3*b)*Cos[e] + I*(a + b)*Cos[3*e] + I*a*Cos[e + 2*f*x] + I*a*Cos[3*e + 2*f*x] + 3*a*Sin[e] + b*Sin[e] + a*Sin[3*e] + b*Sin[3*e] + a*Sin[e + 2*f*x] - a*Sin[3*e + 2*f*x])]*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]*(Cos[e] - I*Sin[e]) - b*(4*a + 3*b)*(a + 2*b + a*Cos[2*(e + f*x)])*Log[a + 2*(a + b)*Cos[2*e] - a*Cos[2*(e + f*x)] - (2*I)*a*Sin[2*e] - (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]]*Sec[e + f*x]*(Cos[e] - I*Sin[e]) + b*(4*a + 3*b)*(a + 2*b + a*Cos[2*(e + f*x)])*Log[-a - 2*(a + b)*Cos[2*e] + a*Cos[2*(e + f*x)] + (2*I)*a*Sin[2*e] + (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]]*Sec[e + f*x]*(Cos[e] - I*Sin[e]) + 8*Sqrt[a]*(a + b)^(3/2)*Cos[f*x]*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[e] + 2*b*(4*a + 3*b)*ArcTan[((a + b)*Sin[e])/((a + b)*Cos[e] - Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*(Cos[2*e] + I*Sin[2*e])*Sin[e + f*x])]*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]*(I*Cos[e] + Sin[e]) + 8*Sqrt[a]*(a + b)^(3/2)*Cos[e]*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] + 8*Sqrt[a]*b^2*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[e + f*x]))/(32*a^(5/2)*(a + b)^(3/2)*f*(a + b*Sec[e + f*x]^2)^2*Sqrt[(Cos[e] - I*Sin[e])^2])","C",0
197,1,139,126,1.1136531,"\int \frac{\cos ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2,x]","\frac{a^{3/2} \sin (3 (e+f x))+3 \sqrt{a} \sin (e+f x) \left(-\frac{4 b^3}{(a+b) (a \cos (2 (e+f x))+a+2 b)}+3 a-8 b\right)-\frac{3 b^2 (6 a+5 b) \left(\log \left(\sqrt{a+b}-\sqrt{a} \sin (e+f x)\right)-\log \left(\sqrt{a+b}+\sqrt{a} \sin (e+f x)\right)\right)}{(a+b)^{3/2}}}{12 a^{7/2} f}","\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{b^3 \sin (e+f x)}{2 a^3 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{(a-2 b) \sin (e+f x)}{a^3 f}-\frac{\sin ^3(e+f x)}{3 a^2 f}",1,"((-3*b^2*(6*a + 5*b)*(Log[Sqrt[a + b] - Sqrt[a]*Sin[e + f*x]] - Log[Sqrt[a + b] + Sqrt[a]*Sin[e + f*x]]))/(a + b)^(3/2) + 3*Sqrt[a]*(3*a - 8*b - (4*b^3)/((a + b)*(a + 2*b + a*Cos[2*(e + f*x)])))*Sin[e + f*x] + a^(3/2)*Sin[3*(e + f*x)])/(12*a^(7/2)*f)","A",1
198,1,171,157,2.0311989,"\int \frac{\cos ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2,x]","\frac{5 a^{3/2} (5 a-8 b) \sin (3 (e+f x))+3 a^{5/2} \sin (5 (e+f x))+30 \sqrt{a} \sin (e+f x) \left(5 a^2+8 b^2 \left(\frac{b^2}{(a+b) (a \cos (2 (e+f x))+a+2 b)}+3\right)-12 a b\right)+\frac{60 b^3 (8 a+7 b) \left(\log \left(\sqrt{a+b}-\sqrt{a} \sin (e+f x)\right)-\log \left(\sqrt{a+b}+\sqrt{a} \sin (e+f x)\right)\right)}{(a+b)^{3/2}}}{240 a^{9/2} 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{b^4 \sin (e+f x)}{2 a^4 f (a+b) \left(-a \sin ^2(e+f x)+a+b\right)}-\frac{2 (a-b) \sin ^3(e+f x)}{3 a^3 f}+\frac{\sin ^5(e+f x)}{5 a^2 f}+\frac{\left(a^2-2 a b+3 b^2\right) \sin (e+f x)}{a^4 f}",1,"((60*b^3*(8*a + 7*b)*(Log[Sqrt[a + b] - Sqrt[a]*Sin[e + f*x]] - Log[Sqrt[a + b] + Sqrt[a]*Sin[e + f*x]]))/(a + b)^(3/2) + 30*Sqrt[a]*(5*a^2 - 12*a*b + 8*b^2*(3 + b^2/((a + b)*(a + 2*b + a*Cos[2*(e + f*x)]))))*Sin[e + f*x] + 5*a^(3/2)*(5*a - 8*b)*Sin[3*(e + f*x)] + 3*a^(5/2)*Sin[5*(e + f*x)])/(240*a^(9/2)*f)","A",1
199,1,248,100,2.3715435,"\int \frac{\sec ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Sec[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{a (a \sin (2 f x)-(a+2 b) \sin (2 e))}{(a+b) (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}+2 \sec (e) \sin (f x) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b)+\frac{a (3 a+4 b) (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{(a+b)^{3/2} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{8 b^2 f \left(a+b \sec ^2(e+f x)\right)^2}","\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 + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^4*((a*(3*a + 4*b)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[2*e] - I*Sin[2*e]))/((a + b)^(3/2)*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + 2*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[e]*Sec[e + f*x]*Sin[f*x] + (a*(-((a + 2*b)*Sin[2*e]) + a*Sin[2*f*x]))/((a + b)*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(8*b^2*f*(a + b*Sec[e + f*x]^2)^2)","C",0
200,1,84,82,0.2814583,"\int \frac{\sec ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Sec[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\frac{(a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}-\frac{a \sqrt{b} \sin (2 (e+f x))}{(a+b) (a \cos (2 (e+f x))+a+2 b)}}{2 b^{3/2} f}","\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]])/(a + b)^(3/2) - (a*Sqrt[b]*Sin[2*(e + f*x)])/((a + b)*(a + 2*b + a*Cos[2*(e + f*x)])))/(2*b^(3/2)*f)","A",1
201,1,211,73,0.8889741,"\int \frac{\sec ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{a \sin (2 f x)-(a+2 b) \sin (2 e)}{a (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}-\frac{(\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{8 f (a+b) \left(a+b \sec ^2(e+f x)\right)^2}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^4*(-((ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])) + (-((a + 2*b)*Sin[2*e]) + a*Sin[2*f*x])/(a*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(8*(a + b)*f*(a + b*Sec[e + f*x]^2)^2)","C",1
202,1,240,92,1.9143019,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(-2),x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(2 x (a \cos (2 (e+f x))+a+2 b)+\frac{b ((a+2 b) \sin (2 e)-a \sin (2 f x))}{f (a+b) (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}+\frac{b (3 a+2 b) (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{f (a+b)^{3/2} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{8 a^2 \left(a+b \sec ^2(e+f x)\right)^2}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^4*(2*x*(a + 2*b + a*Cos[2*(e + f*x)]) + (b*(3*a + 2*b)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[2*e] - I*Sin[2*e]))/((a + b)^(3/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (b*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/((a + b)*f*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(8*a^2*(a + b*Sec[e + f*x]^2)^2)","C",1
203,1,103,142,1.3290165,"\int \frac{\cos ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\frac{2 b^{3/2} (5 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}+\sin (2 (e+f x)) \left(\frac{2 a b^2}{(a+b) (a \cos (2 (e+f x))+a+2 b)}+a\right)+2 (a-4 b) (e+f x)}{4 a^3 f}","\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{x (a-4 b)}{2 a^3}+\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{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)}",1,"(2*(a - 4*b)*(e + f*x) + (2*b^(3/2)*(5*a + 4*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(3/2) + (a + (2*a*b^2)/((a + b)*(a + 2*b + a*Cos[2*(e + f*x)])))*Sin[2*(e + f*x)])/(4*a^3*f)","A",1
204,1,138,203,1.620231,"\int \frac{\cos ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Cos[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]","\frac{4 \left(3 a^2-8 a b+24 b^2\right) (e+f x)+a^2 \sin (4 (e+f x))-\frac{16 b^{5/2} (7 a+6 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}-\frac{16 a b^3 \sin (2 (e+f x))}{(a+b) (a \cos (2 (e+f x))+a+2 b)}+8 a (a-2 b) \sin (2 (e+f x))}{32 a^4 f}","-\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{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{x \left(3 a^2-8 a b+24 b^2\right)}{8 a^4}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)}",1,"(4*(3*a^2 - 8*a*b + 24*b^2)*(e + f*x) - (16*b^(5/2)*(7*a + 6*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(3/2) + 8*a*(a - 2*b)*Sin[2*(e + f*x)] - (16*a*b^3*Sin[2*(e + f*x)])/((a + b)*(a + 2*b + a*Cos[2*(e + f*x)])) + a^2*Sin[4*(e + f*x)])/(32*a^4*f)","A",1
205,1,499,278,4.4190309,"\int \frac{\cos ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{a^3 \sin (6 e) \cos (6 f x) (a \cos (2 (e+f x))+a+2 b)}{f}+\frac{a^3 \cos (6 e) \sin (6 f x) (a \cos (2 (e+f x))+a+2 b)}{f}+\frac{3 a \left(15 a^2-32 a b+48 b^2\right) \sin (2 e) \cos (2 f x) (a \cos (2 (e+f x))+a+2 b)}{f}+\frac{3 a \left(15 a^2-32 a b+48 b^2\right) \cos (2 e) \sin (2 f x) (a \cos (2 (e+f x))+a+2 b)}{f}+\frac{3 a^2 (3 a-4 b) \sin (4 e) \cos (4 f x) (a \cos (2 (e+f x))+a+2 b)}{f}+\frac{3 a^2 (3 a-4 b) \cos (4 e) \sin (4 f x) (a \cos (2 (e+f x))+a+2 b)}{f}+12 x \left(5 a^3-12 a^2 b+24 a b^2-64 b^3\right) (a \cos (2 (e+f x))+a+2 b)-\frac{96 b^4 ((a+2 b) \sin (2 e)-a \sin (2 f x))}{f (a+b) (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}-\frac{96 b^4 (9 a+8 b) (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{f (a+b)^{3/2} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{768 a^5 \left(a+b \sec ^2(e+f x)\right)^2}","\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{(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{\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(5 a^3-12 a^2 b+24 a b^2-64 b^3\right)}{16 a^5}+\frac{b \left(5 a^3-7 a^2 b+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{\sin (e+f x) \cos ^5(e+f x)}{6 a f \left(a+b \tan ^2(e+f x)+b\right)}",1,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^4*(12*(5*a^3 - 12*a^2*b + 24*a*b^2 - 64*b^3)*x*(a + 2*b + a*Cos[2*(e + f*x)]) - (96*b^4*(9*a + 8*b)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[2*e] - I*Sin[2*e]))/((a + b)^(3/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (3*a*(15*a^2 - 32*a*b + 48*b^2)*Cos[2*f*x]*(a + 2*b + a*Cos[2*(e + f*x)])*Sin[2*e])/f + (3*a^2*(3*a - 4*b)*Cos[4*f*x]*(a + 2*b + a*Cos[2*(e + f*x)])*Sin[4*e])/f + (a^3*Cos[6*f*x]*(a + 2*b + a*Cos[2*(e + f*x)])*Sin[6*e])/f + (3*a*(15*a^2 - 32*a*b + 48*b^2)*Cos[2*e]*(a + 2*b + a*Cos[2*(e + f*x)])*Sin[2*f*x])/f - (96*b^4*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/((a + b)*f*(Cos[e] - Sin[e])*(Cos[e] + Sin[e])) + (3*a^2*(3*a - 4*b)*Cos[4*e]*(a + 2*b + a*Cos[2*(e + f*x)])*Sin[4*f*x])/f + (a^3*Cos[6*e]*(a + 2*b + a*Cos[2*(e + f*x)])*Sin[6*f*x])/f))/(768*a^5*(a + b*Sec[e + f*x]^2)^2)","C",0
206,1,128,108,0.475782,"\int \frac{\sec ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b)^3 \left(\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{a} (a+b)^{5/2}}+\frac{4 \sin (e+f x) \left(5 (a+b)-3 a \sin ^2(e+f x)\right)}{(a+b)^2 (a \cos (2 (e+f x))+a+2 b)^2}\right)}{64 f \left(a+b \sec ^2(e+f x)\right)^3}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])^3*Sec[e + f*x]^6*((3*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(5/2)) + (4*Sin[e + f*x]*(5*(a + b) - 3*a*Sin[e + f*x]^2))/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(64*f*(a + b*Sec[e + f*x]^2)^3)","A",1
207,1,163,125,0.608279,"\int \frac{\sec ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Sec[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b)^3 \left(\frac{8 \sin (e+f x)}{\left(-a \sin ^2(e+f x)+a+b\right)^2}-(4 a+b) \left(\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{a} (a+b)^{5/2}}+\frac{4 \sin (e+f x) \left(5 (a+b)-3 a \sin ^2(e+f x)\right)}{(a+b)^2 (a \cos (2 (e+f x))+a+2 b)^2}\right)\right)}{192 a f \left(a+b \sec ^2(e+f x)\right)^3}","\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,"-1/192*((a + 2*b + a*Cos[2*(e + f*x)])^3*Sec[e + f*x]^6*((8*Sin[e + f*x])/(a + b - a*Sin[e + f*x]^2)^2 - (4*a + b)*((3*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(5/2)) + (4*Sin[e + f*x]*(5*(a + b) - 3*a*Sin[e + f*x]^2))/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2))))/(a*f*(a + b*Sec[e + f*x]^2)^3)","A",1
208,1,927,144,6.8241978,"\int \frac{\sec (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Sec[e + f*x]/(a + b*Sec[e + f*x]^2)^3,x]","\frac{(\cos (2 (e+f x)) a+a+2 b) \sec ^5(e+f x) \left(32 \sqrt{a} (a+b)^{3/2} \sqrt{(\cos (e)-i \sin (e))^2} \tan (e+f x) b^2-8 \sqrt{a} \sqrt{a+b} (8 a+5 b) (\cos (2 (e+f x)) a+a+2 b) \sqrt{(\cos (e)-i \sin (e))^2} \tan (e+f x) b-2 i \left(8 a^2+8 b a+3 b^2\right) \tan ^{-1}\left(\frac{(a+b) \sin (e)}{(a+b) \cos (e)-\sqrt{a} \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} (\cos (2 e)+i \sin (2 e)) \sin (e+f x)}\right) (\cos (2 (e+f x)) a+a+2 b)^2 \sec (e+f x) (\cos (e)-i \sin (e))+\left(8 a^2+8 b a+3 b^2\right) (\cos (2 (e+f x)) a+a+2 b)^2 \log \left(-\cos (2 (e+f x)) a-2 i \sin (2 e) a+a+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}+2 (a+b) \cos (2 e)-2 i b \sin (2 e)\right) \sec (e+f x) (\cos (e)-i \sin (e))-\left(8 a^2+8 b a+3 b^2\right) (\cos (2 (e+f x)) a+a+2 b)^2 \log \left(\cos (2 (e+f x)) a+2 i \sin (2 e) a-a+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}+2 \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}-2 (a+b) \cos (2 e)+2 i b \sin (2 e)\right) \sec (e+f x) (\cos (e)-i \sin (e))+2 \left(8 a^2+8 b a+3 b^2\right) \tan ^{-1}\left(\frac{2 \sin (e) \left(\sin (2 e) a+i a-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt{a}+\sqrt{a+b} \cos (f x) \sqrt{(\cos (e)-i \sin (e))^2} \sqrt{a}-\sqrt{a+b} \cos (2 e+f x) \sqrt{(\cos (e)-i \sin (e))^2} \sqrt{a}+i b+i (a+b) \cos (2 e)+b \sin (2 e)\right)}{i (a+3 b) \cos (e)+i (a+b) \cos (3 e)+i a \cos (e+2 f x)+i a \cos (3 e+2 f x)+3 a \sin (e)+b \sin (e)+a \sin (3 e)+b \sin (3 e)+a \sin (e+2 f x)-a \sin (3 e+2 f x)}\right) (\cos (2 (e+f x)) a+a+2 b)^2 \sec (e+f x) (i \cos (e)+\sin (e))\right)}{256 a^{5/2} (a+b)^{5/2} f \left(b \sec ^2(e+f x)+a\right)^3 \sqrt{(\cos (e)-i \sin (e))^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{\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{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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^5*((-2*I)*(8*a^2 + 8*a*b + 3*b^2)*ArcTan[((a + b)*Sin[e])/((a + b)*Cos[e] - Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*(Cos[2*e] + I*Sin[2*e])*Sin[e + f*x])]*(a + 2*b + a*Cos[2*(e + f*x)])^2*Sec[e + f*x]*(Cos[e] - I*Sin[e]) + (8*a^2 + 8*a*b + 3*b^2)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Log[a + 2*(a + b)*Cos[2*e] - a*Cos[2*(e + f*x)] - (2*I)*a*Sin[2*e] - (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]]*Sec[e + f*x]*(Cos[e] - I*Sin[e]) - (8*a^2 + 8*a*b + 3*b^2)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Log[-a - 2*(a + b)*Cos[2*e] + a*Cos[2*(e + f*x)] + (2*I)*a*Sin[2*e] + (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]]*Sec[e + f*x]*(Cos[e] - I*Sin[e]) + 2*(8*a^2 + 8*a*b + 3*b^2)*ArcTan[(2*Sin[e]*(I*a + I*b + I*(a + b)*Cos[2*e] + Sqrt[a]*Sqrt[a + b]*Cos[f*x]*Sqrt[(Cos[e] - I*Sin[e])^2] - Sqrt[a]*Sqrt[a + b]*Cos[2*e + f*x]*Sqrt[(Cos[e] - I*Sin[e])^2] + a*Sin[2*e] + b*Sin[2*e] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[f*x] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Sin[2*e + f*x]))/(I*(a + 3*b)*Cos[e] + I*(a + b)*Cos[3*e] + I*a*Cos[e + 2*f*x] + I*a*Cos[3*e + 2*f*x] + 3*a*Sin[e] + b*Sin[e] + a*Sin[3*e] + b*Sin[3*e] + a*Sin[e + 2*f*x] - a*Sin[3*e + 2*f*x])]*(a + 2*b + a*Cos[2*(e + f*x)])^2*Sec[e + f*x]*(I*Cos[e] + Sin[e]) + 32*Sqrt[a]*b^2*(a + b)^(3/2)*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[e + f*x] - 8*Sqrt[a]*b*Sqrt[a + b]*(8*a + 5*b)*(a + 2*b + a*Cos[2*(e + f*x)])*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[e + f*x]))/(256*a^(5/2)*(a + b)^(5/2)*f*(a + b*Sec[e + f*x]^2)^3*Sqrt[(Cos[e] - I*Sin[e])^2])","C",0
209,1,2382,156,7.5109489,"\int \frac{\cos (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Cos[e + f*x]/(a + b*Sec[e + f*x]^2)^3,x]","\text{Result too large to show}","-\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{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{\sin (e+f x)}{a^3 f}",1,"(Cos[f*x]*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*Sin[e])/(8*a^3*f*(a + b*Sec[e + f*x]^2)^3) + ((8*a^2*b + 12*a*b^2 + 5*b^3)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((((-3*I)/128)*ArcTan[((-I)*a*Cos[e] - I*b*Cos[e] + I*a*Cos[3*e] + I*b*Cos[3*e] + a*Sin[e] + b*Sin[e] - Sqrt[a]*Sqrt[a + b]*Cos[e - f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + Sqrt[a]*Sqrt[a + b]*Cos[3*e + f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + a*Sin[3*e] + b*Sin[3*e] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e - f*x] - (2*I)*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e + f*x] + I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[3*e + f*x])/(a*Cos[e] + 3*b*Cos[e] + a*Cos[3*e] + b*Cos[3*e] + a*Cos[e + 2*f*x] + a*Cos[3*e + 2*f*x] - (3*I)*a*Sin[e] - I*b*Sin[e] - I*a*Sin[3*e] - I*b*Sin[3*e] - I*a*Sin[e + 2*f*x] + I*a*Sin[3*e + 2*f*x])]*Cos[e])/(a^(7/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (3*ArcTan[((-I)*a*Cos[e] - I*b*Cos[e] + I*a*Cos[3*e] + I*b*Cos[3*e] + a*Sin[e] + b*Sin[e] - Sqrt[a]*Sqrt[a + b]*Cos[e - f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + Sqrt[a]*Sqrt[a + b]*Cos[3*e + f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + a*Sin[3*e] + b*Sin[3*e] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e - f*x] - (2*I)*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e + f*x] + I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[3*e + f*x])/(a*Cos[e] + 3*b*Cos[e] + a*Cos[3*e] + b*Cos[3*e] + a*Cos[e + 2*f*x] + a*Cos[3*e + 2*f*x] - (3*I)*a*Sin[e] - I*b*Sin[e] - I*a*Sin[3*e] - I*b*Sin[3*e] - I*a*Sin[e + 2*f*x] + I*a*Sin[3*e + 2*f*x])]*Sin[e])/(128*a^(7/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((a + b)^2*(a + b*Sec[e + f*x]^2)^3) + ((8*a^2*b + 12*a*b^2 + 5*b^3)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((3*ArcTanh[(2*(a + b)*Sin[e])/((-2*I)*a*Cos[e] - (2*I)*b*Cos[e] - Sqrt[a]*Sqrt[a + b]*Cos[e - f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + Sqrt[a]*Sqrt[a + b]*Cos[3*e + f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e - f*x] + I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[3*e + f*x])]*Cos[e])/(128*a^(7/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (((3*I)/128)*ArcTanh[(2*(a + b)*Sin[e])/((-2*I)*a*Cos[e] - (2*I)*b*Cos[e] - Sqrt[a]*Sqrt[a + b]*Cos[e - f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + Sqrt[a]*Sqrt[a + b]*Cos[3*e + f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e - f*x] + I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[3*e + f*x])]*Sin[e])/(a^(7/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((a + b)^2*(a + b*Sec[e + f*x]^2)^3) + ((8*a^2*b + 12*a*b^2 + 5*b^3)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((-3*Cos[e]*Log[a + 2*a*Cos[2*e] + 2*b*Cos[2*e] - a*Cos[2*e + 2*f*x] - (2*I)*a*Sin[2*e] - (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[2*e + f*x]])/(256*a^(7/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) + (((3*I)/256)*Log[a + 2*a*Cos[2*e] + 2*b*Cos[2*e] - a*Cos[2*e + 2*f*x] - (2*I)*a*Sin[2*e] - (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[2*e + f*x]]*Sin[e])/(a^(7/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((a + b)^2*(a + b*Sec[e + f*x]^2)^3) + ((8*a^2*b + 12*a*b^2 + 5*b^3)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((3*Cos[e]*Log[-a - 2*a*Cos[2*e] - 2*b*Cos[2*e] + a*Cos[2*e + 2*f*x] + (2*I)*a*Sin[2*e] + (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[2*e + f*x]])/(256*a^(7/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (((3*I)/256)*Log[-a - 2*a*Cos[2*e] - 2*b*Cos[2*e] + a*Cos[2*e + 2*f*x] + (2*I)*a*Sin[2*e] + (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[2*e + f*x]]*Sin[e])/(a^(7/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((a + b)^2*(a + b*Sec[e + f*x]^2)^3) + (Cos[e]*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*Sin[f*x])/(8*a^3*f*(a + b*Sec[e + f*x]^2)^3) + (3*(a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^6*(4*a*b^2*Sin[e + f*x] + 3*b^3*Sin[e + f*x]))/(32*a^3*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^3) - (b^3*(a + 2*b + a*Cos[2*e + 2*f*x])*Sec[e + f*x]^5*Tan[e + f*x])/(8*a^3*(a + b)*f*(a + b*Sec[e + f*x]^2)^3)","C",0
210,1,194,181,4.3856246,"\int \frac{\cos ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3,x]","\frac{4 a^{3/2} \sin (3 (e+f x))-\frac{3 b^2 \left(48 a^2+80 a b+35 b^2\right) \left(\log \left(\sqrt{a+b}-\sqrt{a} \sin (e+f x)\right)-\log \left(\sqrt{a+b}+\sqrt{a} \sin (e+f x)\right)\right)}{(a+b)^{5/2}}+12 \sqrt{a} \sin (e+f x) \left(-\frac{b^4 (13 a \cos (2 (e+f x))+9 a+22 b)}{(a+b)^2 (a \cos (2 (e+f x))+a+2 b)^2}+a \left(3-\frac{16 b^3}{(a+b)^2 (a \cos (2 (e+f x))+a+2 b)}\right)-12 b\right)}{48 a^{9/2} 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{(a-3 b) \sin (e+f x)}{a^4 f}-\frac{\sin ^3(e+f x)}{3 a^3 f}+\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}}",1,"((-3*b^2*(48*a^2 + 80*a*b + 35*b^2)*(Log[Sqrt[a + b] - Sqrt[a]*Sin[e + f*x]] - Log[Sqrt[a + b] + Sqrt[a]*Sin[e + f*x]]))/(a + b)^(5/2) + 12*Sqrt[a]*(-12*b - (b^4*(9*a + 22*b + 13*a*Cos[2*(e + f*x)]))/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2) + a*(3 - (16*b^3)/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)]))))*Sin[e + f*x] + 4*a^(3/2)*Sin[3*(e + f*x)])/(48*a^(9/2)*f)","A",1
211,1,2670,214,7.5968301,"\int \frac{\cos ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]","\text{Result too large to show}","-\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{(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^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{\left(a^2-3 a b+6 b^2\right) \sin (e+f x)}{a^5 f}",1,"((5*a^2 - 18*a*b + 48*b^2)*Cos[f*x]*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*Sin[e])/(64*a^5*f*(a + b*Sec[e + f*x]^2)^3) + ((-80*a^2*b^3 - 140*a*b^4 - 63*b^5)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(((I/128)*ArcTan[((-I)*a*Cos[e] - I*b*Cos[e] + I*a*Cos[3*e] + I*b*Cos[3*e] + a*Sin[e] + b*Sin[e] - Sqrt[a]*Sqrt[a + b]*Cos[e - f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + Sqrt[a]*Sqrt[a + b]*Cos[3*e + f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + a*Sin[3*e] + b*Sin[3*e] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e - f*x] - (2*I)*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e + f*x] + I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[3*e + f*x])/(a*Cos[e] + 3*b*Cos[e] + a*Cos[3*e] + b*Cos[3*e] + a*Cos[e + 2*f*x] + a*Cos[3*e + 2*f*x] - (3*I)*a*Sin[e] - I*b*Sin[e] - I*a*Sin[3*e] - I*b*Sin[3*e] - I*a*Sin[e + 2*f*x] + I*a*Sin[3*e + 2*f*x])]*Cos[e])/(a^(11/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) + (ArcTan[((-I)*a*Cos[e] - I*b*Cos[e] + I*a*Cos[3*e] + I*b*Cos[3*e] + a*Sin[e] + b*Sin[e] - Sqrt[a]*Sqrt[a + b]*Cos[e - f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + Sqrt[a]*Sqrt[a + b]*Cos[3*e + f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + a*Sin[3*e] + b*Sin[3*e] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e - f*x] - (2*I)*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e + f*x] + I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[3*e + f*x])/(a*Cos[e] + 3*b*Cos[e] + a*Cos[3*e] + b*Cos[3*e] + a*Cos[e + 2*f*x] + a*Cos[3*e + 2*f*x] - (3*I)*a*Sin[e] - I*b*Sin[e] - I*a*Sin[3*e] - I*b*Sin[3*e] - I*a*Sin[e + 2*f*x] + I*a*Sin[3*e + 2*f*x])]*Sin[e])/(128*a^(11/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((a + b)^2*(a + b*Sec[e + f*x]^2)^3) + ((80*a^2*b^3 + 140*a*b^4 + 63*b^5)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((ArcTanh[(2*(a + b)*Sin[e])/((-2*I)*a*Cos[e] - (2*I)*b*Cos[e] - Sqrt[a]*Sqrt[a + b]*Cos[e - f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + Sqrt[a]*Sqrt[a + b]*Cos[3*e + f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e - f*x] + I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[3*e + f*x])]*Cos[e])/(128*a^(11/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) - ((I/128)*ArcTanh[(2*(a + b)*Sin[e])/((-2*I)*a*Cos[e] - (2*I)*b*Cos[e] - Sqrt[a]*Sqrt[a + b]*Cos[e - f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + Sqrt[a]*Sqrt[a + b]*Cos[3*e + f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e - f*x] + I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[3*e + f*x])]*Sin[e])/(a^(11/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((a + b)^2*(a + b*Sec[e + f*x]^2)^3) + ((-80*a^2*b^3 - 140*a*b^4 - 63*b^5)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((Cos[e]*Log[a + 2*a*Cos[2*e] + 2*b*Cos[2*e] - a*Cos[2*e + 2*f*x] - (2*I)*a*Sin[2*e] - (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[2*e + f*x]])/(256*a^(11/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) - ((I/256)*Log[a + 2*a*Cos[2*e] + 2*b*Cos[2*e] - a*Cos[2*e + 2*f*x] - (2*I)*a*Sin[2*e] - (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[2*e + f*x]]*Sin[e])/(a^(11/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((a + b)^2*(a + b*Sec[e + f*x]^2)^3) + ((80*a^2*b^3 + 140*a*b^4 + 63*b^5)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((Cos[e]*Log[-a - 2*a*Cos[2*e] - 2*b*Cos[2*e] + a*Cos[2*e + 2*f*x] + (2*I)*a*Sin[2*e] + (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[2*e + f*x]])/(256*a^(11/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) - ((I/256)*Log[-a - 2*a*Cos[2*e] - 2*b*Cos[2*e] + a*Cos[2*e + 2*f*x] + (2*I)*a*Sin[2*e] + (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[2*e + f*x]]*Sin[e])/(a^(11/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((a + b)^2*(a + b*Sec[e + f*x]^2)^3) + ((5*a - 12*b)*Cos[3*f*x]*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*Sin[3*e])/(384*a^4*f*(a + b*Sec[e + f*x]^2)^3) + (Cos[5*f*x]*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*Sin[5*e])/(640*a^3*f*(a + b*Sec[e + f*x]^2)^3) + ((5*a^2 - 18*a*b + 48*b^2)*Cos[e]*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*Sin[f*x])/(64*a^5*f*(a + b*Sec[e + f*x]^2)^3) + ((5*a - 12*b)*Cos[3*e]*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*Sin[3*f*x])/(384*a^4*f*(a + b*Sec[e + f*x]^2)^3) + (Cos[5*e]*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*Sin[5*f*x])/(640*a^3*f*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^6*(20*a*b^4*Sin[e + f*x] + 17*b^5*Sin[e + f*x]))/(32*a^5*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^3) - (b^5*(a + 2*b + a*Cos[2*e + 2*f*x])*Sec[e + f*x]^5*Tan[e + f*x])/(8*a^5*(a + b)*f*(a + b*Sec[e + f*x]^2)^3)","C",0
212,1,125,142,0.8986629,"\int \frac{\sec ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Sec[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\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)}{(a+b)^{5/2}}-\frac{a \sqrt{b} \sin (2 (e+f x)) \left(3 a^2+3 a (a+2 b) \cos (2 (e+f x))+16 a b+16 b^2\right)}{(a+b)^2 (a \cos (2 (e+f x))+a+2 b)^2}}{8 b^{5/2} f}","\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]])/(a + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2))/(8*b^(5/2)*f)","A",1
213,1,283,123,3.5019365,"\int \frac{\sec ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Sec[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{((a+4 b) \sin (2 e)-(a-2 b) \sin (2 f x)) (a \cos (2 (e+f x))+a+2 b)}{b (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}-\frac{4 (a+b) ((a+2 b) \sin (2 e)-a \sin (2 f x))}{a (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}-\frac{(a+4 b) (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{b \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{64 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^3}","\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 + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^6*(-(((a + 4*b)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])^2*(Cos[2*e] - I*Sin[2*e]))/(b*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])) - (4*(a + b)*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(a*(Cos[e] - Sin[e])*(Cos[e] + Sin[e])) + ((a + 2*b + a*Cos[2*(e + f*x)])*((a + 4*b)*Sin[2*e] - (a - 2*b)*Sin[2*f*x]))/(b*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(64*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^3)","C",1
214,1,265,106,2.5411041,"\int \frac{\sec ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{\sec (2 e) \left(a (5 a+2 b) \sin (2 f x)-\left(5 a^2+16 a b+8 b^2\right) \sin (2 e)\right) (a \cos (2 (e+f x))+a+2 b)}{a^2}+\frac{4 b (a+b) \sec (2 e) ((a+2 b) \sin (2 e)-a \sin (2 f x))}{a^2}-\frac{3 (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{64 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^3}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^6*((-3*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])^2*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (4*b*(a + b)*Sec[2*e]*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/a^2 + ((a + 2*b + a*Cos[2*(e + f*x)])*Sec[2*e]*(-((5*a^2 + 16*a*b + 8*b^2)*Sin[2*e]) + a*(5*a + 2*b)*Sin[2*f*x]))/a^2))/(64*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^3)","C",1
215,1,332,144,5.5673086,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(-3),x]","\frac{\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{b \left(\left(9 a^2+28 a b+16 b^2\right) \sin (2 e)-3 a (3 a+2 b) \sin (2 f x)\right) (a \cos (2 (e+f x))+a+2 b)}{f (a+b)^2 (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}+\frac{b \left(15 a^2+20 a b+8 b^2\right) (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{f (a+b)^{5/2} \sqrt{b (\cos (e)-i \sin (e))^4}}-\frac{4 b^2 ((a+2 b) \sin (2 e)-a \sin (2 f x))}{f (a+b) (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}+8 x (a \cos (2 (e+f x))+a+2 b)^2\right)}{64 a^3 \left(a+b \sec ^2(e+f x)\right)^3}","\frac{x}{a^3}-\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{\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 \tan (e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^6*(8*x*(a + 2*b + a*Cos[2*(e + f*x)])^2 + (b*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])^2*(Cos[2*e] - I*Sin[2*e]))/((a + b)^(5/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) - (4*b^2*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/((a + b)*f*(Cos[e] - Sin[e])*(Cos[e] + Sin[e])) + (b*(a + 2*b + a*Cos[2*(e + f*x)])*((9*a^2 + 28*a*b + 16*b^2)*Sin[2*e] - 3*a*(3*a + 2*b)*Sin[2*f*x]))/((a + b)^2*f*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(64*a^3*(a + b*Sec[e + f*x]^2)^3)","C",0
216,1,156,201,3.469039,"\int \frac{\cos ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\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)}{(a+b)^{5/2}}+a \sin (2 (e+f x)) \left(\frac{2 b^3 (5 a \cos (2 (e+f x))+3 a+8 b)}{(a+b)^2 (a \cos (2 (e+f x))+a+2 b)^2}+\frac{13 a b^2}{(a+b)^2 (a \cos (2 (e+f x))+a+2 b)}+2\right)+4 (a-6 b) (e+f x)}{8 a^4 f}","\frac{x (a-6 b)}{2 a^4}+\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{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{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"(4*(a - 6*b)*(e + f*x) + (b^(3/2)*(35*a^2 + 56*a*b + 24*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(5/2) + a*(2 + (13*a*b^2)/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])) + (2*b^3*(3*a + 8*b + 5*a*Cos[2*(e + f*x)]))/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2))*Sin[2*(e + f*x)])/(8*a^4*f)","A",1
217,1,1430,269,6.5309056,"\int \frac{\cos ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Cos[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\left(21 a^2+36 b a+16 b^2\right) (\cos (2 e+2 f x) a+a+2 b)^3 \left(\frac{3 b^3 \tan ^{-1}\left(\sec (f x) \left(\frac{\cos (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}-\frac{i \sin (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}\right) (-a \sin (f x)-2 b \sin (f x)+a \sin (2 e+f x))\right) \cos (2 e)}{64 a^5 \sqrt{a+b} f \sqrt{b \cos (4 e)-i b \sin (4 e)}}-\frac{3 i b^3 \tan ^{-1}\left(\sec (f x) \left(\frac{\cos (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}-\frac{i \sin (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}\right) (-a \sin (f x)-2 b \sin (f x)+a \sin (2 e+f x))\right) \sin (2 e)}{64 a^5 \sqrt{a+b} f \sqrt{b \cos (4 e)-i b \sin (4 e)}}\right) \sec ^6(e+f x)}{(a+b)^2 \left(b \sec ^2(e+f x)+a\right)^3}+\frac{(\cos (2 e+2 f x) a+a+2 b) \sec (2 e) \left(144 f x \cos (2 e) a^6+96 f x \cos (2 f x) a^6+96 f x \cos (4 e+2 f x) a^6+24 f x \cos (2 e+4 f x) a^6+24 f x \cos (6 e+4 f x) a^6+44 \sin (2 f x) a^6+44 \sin (4 e+2 f x) a^6+38 \sin (2 e+4 f x) a^6+38 \sin (6 e+4 f x) a^6+12 \sin (4 e+6 f x) a^6+12 \sin (8 e+6 f x) a^6+\sin (6 e+8 f x) a^6+\sin (10 e+8 f x) a^6+96 b f x \cos (2 e) a^5-48 b f x \cos (2 e+4 f x) a^5-48 b f x \cos (6 e+4 f x) a^5+104 b \sin (2 f x) a^5+104 b \sin (4 e+2 f x) a^5+60 b \sin (2 e+4 f x) a^5+60 b \sin (6 e+4 f x) a^5+8 b \sin (4 e+6 f x) a^5+8 b \sin (8 e+6 f x) a^5+2 b \sin (6 e+8 f x) a^5+2 b \sin (10 e+8 f x) a^5+912 b^2 f x \cos (2 e) a^4+480 b^2 f x \cos (2 f x) a^4+480 b^2 f x \cos (4 e+2 f x) a^4+216 b^2 f x \cos (2 e+4 f x) a^4+216 b^2 f x \cos (6 e+4 f x) a^4-180 b^2 \sin (2 f x) a^4-180 b^2 \sin (4 e+2 f x) a^4-170 b^2 \sin (2 e+4 f x) a^4-170 b^2 \sin (6 e+4 f x) a^4-20 b^2 \sin (4 e+6 f x) a^4-20 b^2 \sin (8 e+6 f x) a^4+b^2 \sin (6 e+8 f x) a^4+b^2 \sin (10 e+8 f x) a^4+6720 b^3 f x \cos (2 e) a^3+4416 b^3 f x \cos (2 f x) a^3+4416 b^3 f x \cos (4 e+2 f x) a^3+672 b^3 f x \cos (2 e+4 f x) a^3+672 b^3 f x \cos (6 e+4 f x) a^3+816 b^3 \sin (2 e) a^3-1696 b^3 \sin (2 f x) a^3-608 b^3 \sin (4 e+2 f x) a^3-640 b^3 \sin (2 e+4 f x) a^3-368 b^3 \sin (6 e+4 f x) a^3-16 b^3 \sin (4 e+6 f x) a^3-16 b^3 \sin (8 e+6 f x) a^3+16512 b^4 f x \cos (2 e) a^2+6912 b^4 f x \cos (2 f x) a^2+6912 b^4 f x \cos (4 e+2 f x) a^2+384 b^4 f x \cos (2 e+4 f x) a^2+384 b^4 f x \cos (6 e+4 f x) a^2+2848 b^4 \sin (2 e) a^2-3264 b^4 \sin (2 f x) a^2-192 b^4 \sin (4 e+2 f x) a^2-400 b^4 \sin (2 e+4 f x) a^2-176 b^4 \sin (6 e+4 f x) a^2+16896 b^5 f x \cos (2 e) a+3072 b^5 f x \cos (2 f x) a+3072 b^5 f x \cos (4 e+2 f x) a+3968 b^5 \sin (2 e) a-1664 b^5 \sin (2 f x) a+128 b^5 \sin (4 e+2 f x) a+6144 b^6 f x \cos (2 e)+1792 b^6 \sin (2 e)\right) \sec ^6(e+f x)}{2048 a^5 (a+b)^2 f \left(b \sec ^2(e+f x)+a\right)^3}","\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{3 x \left(a^2-4 a b+16 b^2\right)}{8 a^5}-\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{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"((21*a^2 + 36*a*b + 16*b^2)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((3*b^3*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Cos[2*e])/(64*a^5*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - (((3*I)/64)*b^3*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Sin[2*e])/(a^5*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]])))/((a + b)^2*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])*Sec[2*e]*Sec[e + f*x]^6*(144*a^6*f*x*Cos[2*e] + 96*a^5*b*f*x*Cos[2*e] + 912*a^4*b^2*f*x*Cos[2*e] + 6720*a^3*b^3*f*x*Cos[2*e] + 16512*a^2*b^4*f*x*Cos[2*e] + 16896*a*b^5*f*x*Cos[2*e] + 6144*b^6*f*x*Cos[2*e] + 96*a^6*f*x*Cos[2*f*x] + 480*a^4*b^2*f*x*Cos[2*f*x] + 4416*a^3*b^3*f*x*Cos[2*f*x] + 6912*a^2*b^4*f*x*Cos[2*f*x] + 3072*a*b^5*f*x*Cos[2*f*x] + 96*a^6*f*x*Cos[4*e + 2*f*x] + 480*a^4*b^2*f*x*Cos[4*e + 2*f*x] + 4416*a^3*b^3*f*x*Cos[4*e + 2*f*x] + 6912*a^2*b^4*f*x*Cos[4*e + 2*f*x] + 3072*a*b^5*f*x*Cos[4*e + 2*f*x] + 24*a^6*f*x*Cos[2*e + 4*f*x] - 48*a^5*b*f*x*Cos[2*e + 4*f*x] + 216*a^4*b^2*f*x*Cos[2*e + 4*f*x] + 672*a^3*b^3*f*x*Cos[2*e + 4*f*x] + 384*a^2*b^4*f*x*Cos[2*e + 4*f*x] + 24*a^6*f*x*Cos[6*e + 4*f*x] - 48*a^5*b*f*x*Cos[6*e + 4*f*x] + 216*a^4*b^2*f*x*Cos[6*e + 4*f*x] + 672*a^3*b^3*f*x*Cos[6*e + 4*f*x] + 384*a^2*b^4*f*x*Cos[6*e + 4*f*x] + 816*a^3*b^3*Sin[2*e] + 2848*a^2*b^4*Sin[2*e] + 3968*a*b^5*Sin[2*e] + 1792*b^6*Sin[2*e] + 44*a^6*Sin[2*f*x] + 104*a^5*b*Sin[2*f*x] - 180*a^4*b^2*Sin[2*f*x] - 1696*a^3*b^3*Sin[2*f*x] - 3264*a^2*b^4*Sin[2*f*x] - 1664*a*b^5*Sin[2*f*x] + 44*a^6*Sin[4*e + 2*f*x] + 104*a^5*b*Sin[4*e + 2*f*x] - 180*a^4*b^2*Sin[4*e + 2*f*x] - 608*a^3*b^3*Sin[4*e + 2*f*x] - 192*a^2*b^4*Sin[4*e + 2*f*x] + 128*a*b^5*Sin[4*e + 2*f*x] + 38*a^6*Sin[2*e + 4*f*x] + 60*a^5*b*Sin[2*e + 4*f*x] - 170*a^4*b^2*Sin[2*e + 4*f*x] - 640*a^3*b^3*Sin[2*e + 4*f*x] - 400*a^2*b^4*Sin[2*e + 4*f*x] + 38*a^6*Sin[6*e + 4*f*x] + 60*a^5*b*Sin[6*e + 4*f*x] - 170*a^4*b^2*Sin[6*e + 4*f*x] - 368*a^3*b^3*Sin[6*e + 4*f*x] - 176*a^2*b^4*Sin[6*e + 4*f*x] + 12*a^6*Sin[4*e + 6*f*x] + 8*a^5*b*Sin[4*e + 6*f*x] - 20*a^4*b^2*Sin[4*e + 6*f*x] - 16*a^3*b^3*Sin[4*e + 6*f*x] + 12*a^6*Sin[8*e + 6*f*x] + 8*a^5*b*Sin[8*e + 6*f*x] - 20*a^4*b^2*Sin[8*e + 6*f*x] - 16*a^3*b^3*Sin[8*e + 6*f*x] + a^6*Sin[6*e + 8*f*x] + 2*a^5*b*Sin[6*e + 8*f*x] + a^4*b^2*Sin[6*e + 8*f*x] + a^6*Sin[10*e + 8*f*x] + 2*a^5*b*Sin[10*e + 8*f*x] + a^4*b^2*Sin[10*e + 8*f*x]))/(2048*a^5*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^3)","C",0
218,1,1770,352,6.6125406,"\int \frac{\cos ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\left(99 a^2+176 b a+80 b^2\right) (\cos (2 e+2 f x) a+a+2 b)^3 \left(\frac{i b^4 \tan ^{-1}\left(\sec (f x) \left(\frac{\cos (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}-\frac{i \sin (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}\right) (-a \sin (f x)-2 b \sin (f x)+a \sin (2 e+f x))\right) \sin (2 e)}{64 a^6 \sqrt{a+b} f \sqrt{b \cos (4 e)-i b \sin (4 e)}}-\frac{b^4 \tan ^{-1}\left(\sec (f x) \left(\frac{\cos (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}-\frac{i \sin (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}\right) (-a \sin (f x)-2 b \sin (f x)+a \sin (2 e+f x))\right) \cos (2 e)}{64 a^6 \sqrt{a+b} f \sqrt{b \cos (4 e)-i b \sin (4 e)}}\right) \sec ^6(e+f x)}{(a+b)^2 \left(b \sec ^2(e+f x)+a\right)^3}+\frac{(\cos (2 e+2 f x) a+a+2 b) \sec (2 e) \left(720 f x \cos (2 e) a^7+480 f x \cos (2 f x) a^7+480 f x \cos (4 e+2 f x) a^7+120 f x \cos (2 e+4 f x) a^7+120 f x \cos (6 e+4 f x) a^7+262 \sin (2 f x) a^7+262 \sin (4 e+2 f x) a^7+238 \sin (2 e+4 f x) a^7+238 \sin (6 e+4 f x) a^7+87 \sin (4 e+6 f x) a^7+87 \sin (8 e+6 f x) a^7+13 \sin (6 e+8 f x) a^7+13 \sin (10 e+8 f x) a^7+\sin (8 e+10 f x) a^7+\sin (12 e+10 f x) a^7+768 b f x \cos (2 e) a^6+192 b f x \cos (2 f x) a^6+192 b f x \cos (4 e+2 f x) a^6-192 b f x \cos (2 e+4 f x) a^6-192 b f x \cos (6 e+4 f x) a^6+524 b \sin (2 f x) a^6+524 b \sin (4 e+2 f x) a^6+304 b \sin (2 e+4 f x) a^6+304 b \sin (6 e+4 f x) a^6+46 b \sin (4 e+6 f x) a^6+46 b \sin (8 e+6 f x) a^6+16 b \sin (6 e+8 f x) a^6+16 b \sin (10 e+8 f x) a^6+2 b \sin (8 e+10 f x) a^6+2 b \sin (12 e+10 f x) a^6+1296 b^2 f x \cos (2 e) a^5+96 b^2 f x \cos (2 f x) a^5+96 b^2 f x \cos (4 e+2 f x) a^5+408 b^2 f x \cos (2 e+4 f x) a^5+408 b^2 f x \cos (6 e+4 f x) a^5-26 b^2 \sin (2 f x) a^5-26 b^2 \sin (4 e+2 f x) a^5-250 b^2 \sin (2 e+4 f x) a^5-250 b^2 \sin (6 e+4 f x) a^5-9 b^2 \sin (4 e+6 f x) a^5-9 b^2 \sin (8 e+6 f x) a^5-7 b^2 \sin (6 e+8 f x) a^5-7 b^2 \sin (10 e+8 f x) a^5+b^2 \sin (8 e+10 f x) a^5+b^2 \sin (12 e+10 f x) a^5-8352 b^3 f x \cos (2 e) a^4-4608 b^3 f x \cos (2 f x) a^4-4608 b^3 f x \cos (4 e+2 f x) a^4-1968 b^3 f x \cos (2 e+4 f x) a^4-1968 b^3 f x \cos (6 e+4 f x) a^4+1728 b^3 \sin (2 f x) a^4+1728 b^3 \sin (4 e+2 f x) a^4+1556 b^3 \sin (2 e+4 f x) a^4+1556 b^3 \sin (6 e+4 f x) a^4+192 b^3 \sin (4 e+6 f x) a^4+192 b^3 \sin (8 e+6 f x) a^4-10 b^3 \sin (6 e+8 f x) a^4-10 b^3 \sin (10 e+8 f x) a^4-64128 b^4 f x \cos (2 e) a^3-41856 b^4 f x \cos (2 f x) a^3-41856 b^4 f x \cos (4 e+2 f x) a^3-6528 b^4 f x \cos (2 e+4 f x) a^3-6528 b^4 f x \cos (6 e+4 f x) a^3-6048 b^4 \sin (2 e) a^3+14976 b^4 \sin (2 f x) a^3+6912 b^4 \sin (4 e+2 f x) a^3+5904 b^4 \sin (2 e+4 f x) a^3+3888 b^4 \sin (6 e+4 f x) a^3+160 b^4 \sin (4 e+6 f x) a^3+160 b^4 \sin (8 e+6 f x) a^3-158976 b^5 f x \cos (2 e) a^2-67584 b^5 f x \cos (2 f x) a^2-67584 b^5 f x \cos (4 e+2 f x) a^2-3840 b^5 f x \cos (2 e+4 f x) a^2-3840 b^5 f x \cos (6 e+4 f x) a^2-21312 b^5 \sin (2 e) a^2+28416 b^5 \sin (2 f x) a^2+5376 b^5 \sin (4 e+2 f x) a^2+3744 b^5 \sin (2 e+4 f x) a^2+2016 b^5 \sin (6 e+4 f x) a^2-165888 b^6 f x \cos (2 e) a-30720 b^6 f x \cos (2 f x) a-30720 b^6 f x \cos (4 e+2 f x) a-29952 b^6 \sin (2 e) a+14592 b^6 \sin (2 f x) a+768 b^6 \sin (4 e+2 f x) a-61440 b^7 f x \cos (2 e)-13824 b^7 \sin (2 e)\right) \sec ^6(e+f x)}{12288 a^6 (a+b)^2 f \left(b \sec ^2(e+f x)+a\right)^3}","\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{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{\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(5 a^3-18 a^2 b+48 a b^2-160 b^3\right)}{16 a^6}+\frac{b \left(15 a^3-29 a^2 b+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{b \left(5 a^4-8 a^3 b+17 a^2 b^2+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{\sin (e+f x) \cos ^5(e+f x)}{6 a f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"((99*a^2 + 176*a*b + 80*b^2)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(-1/64*(b^4*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Cos[2*e])/(a^6*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) + ((I/64)*b^4*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Sin[2*e])/(a^6*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]])))/((a + b)^2*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])*Sec[2*e]*Sec[e + f*x]^6*(720*a^7*f*x*Cos[2*e] + 768*a^6*b*f*x*Cos[2*e] + 1296*a^5*b^2*f*x*Cos[2*e] - 8352*a^4*b^3*f*x*Cos[2*e] - 64128*a^3*b^4*f*x*Cos[2*e] - 158976*a^2*b^5*f*x*Cos[2*e] - 165888*a*b^6*f*x*Cos[2*e] - 61440*b^7*f*x*Cos[2*e] + 480*a^7*f*x*Cos[2*f*x] + 192*a^6*b*f*x*Cos[2*f*x] + 96*a^5*b^2*f*x*Cos[2*f*x] - 4608*a^4*b^3*f*x*Cos[2*f*x] - 41856*a^3*b^4*f*x*Cos[2*f*x] - 67584*a^2*b^5*f*x*Cos[2*f*x] - 30720*a*b^6*f*x*Cos[2*f*x] + 480*a^7*f*x*Cos[4*e + 2*f*x] + 192*a^6*b*f*x*Cos[4*e + 2*f*x] + 96*a^5*b^2*f*x*Cos[4*e + 2*f*x] - 4608*a^4*b^3*f*x*Cos[4*e + 2*f*x] - 41856*a^3*b^4*f*x*Cos[4*e + 2*f*x] - 67584*a^2*b^5*f*x*Cos[4*e + 2*f*x] - 30720*a*b^6*f*x*Cos[4*e + 2*f*x] + 120*a^7*f*x*Cos[2*e + 4*f*x] - 192*a^6*b*f*x*Cos[2*e + 4*f*x] + 408*a^5*b^2*f*x*Cos[2*e + 4*f*x] - 1968*a^4*b^3*f*x*Cos[2*e + 4*f*x] - 6528*a^3*b^4*f*x*Cos[2*e + 4*f*x] - 3840*a^2*b^5*f*x*Cos[2*e + 4*f*x] + 120*a^7*f*x*Cos[6*e + 4*f*x] - 192*a^6*b*f*x*Cos[6*e + 4*f*x] + 408*a^5*b^2*f*x*Cos[6*e + 4*f*x] - 1968*a^4*b^3*f*x*Cos[6*e + 4*f*x] - 6528*a^3*b^4*f*x*Cos[6*e + 4*f*x] - 3840*a^2*b^5*f*x*Cos[6*e + 4*f*x] - 6048*a^3*b^4*Sin[2*e] - 21312*a^2*b^5*Sin[2*e] - 29952*a*b^6*Sin[2*e] - 13824*b^7*Sin[2*e] + 262*a^7*Sin[2*f*x] + 524*a^6*b*Sin[2*f*x] - 26*a^5*b^2*Sin[2*f*x] + 1728*a^4*b^3*Sin[2*f*x] + 14976*a^3*b^4*Sin[2*f*x] + 28416*a^2*b^5*Sin[2*f*x] + 14592*a*b^6*Sin[2*f*x] + 262*a^7*Sin[4*e + 2*f*x] + 524*a^6*b*Sin[4*e + 2*f*x] - 26*a^5*b^2*Sin[4*e + 2*f*x] + 1728*a^4*b^3*Sin[4*e + 2*f*x] + 6912*a^3*b^4*Sin[4*e + 2*f*x] + 5376*a^2*b^5*Sin[4*e + 2*f*x] + 768*a*b^6*Sin[4*e + 2*f*x] + 238*a^7*Sin[2*e + 4*f*x] + 304*a^6*b*Sin[2*e + 4*f*x] - 250*a^5*b^2*Sin[2*e + 4*f*x] + 1556*a^4*b^3*Sin[2*e + 4*f*x] + 5904*a^3*b^4*Sin[2*e + 4*f*x] + 3744*a^2*b^5*Sin[2*e + 4*f*x] + 238*a^7*Sin[6*e + 4*f*x] + 304*a^6*b*Sin[6*e + 4*f*x] - 250*a^5*b^2*Sin[6*e + 4*f*x] + 1556*a^4*b^3*Sin[6*e + 4*f*x] + 3888*a^3*b^4*Sin[6*e + 4*f*x] + 2016*a^2*b^5*Sin[6*e + 4*f*x] + 87*a^7*Sin[4*e + 6*f*x] + 46*a^6*b*Sin[4*e + 6*f*x] - 9*a^5*b^2*Sin[4*e + 6*f*x] + 192*a^4*b^3*Sin[4*e + 6*f*x] + 160*a^3*b^4*Sin[4*e + 6*f*x] + 87*a^7*Sin[8*e + 6*f*x] + 46*a^6*b*Sin[8*e + 6*f*x] - 9*a^5*b^2*Sin[8*e + 6*f*x] + 192*a^4*b^3*Sin[8*e + 6*f*x] + 160*a^3*b^4*Sin[8*e + 6*f*x] + 13*a^7*Sin[6*e + 8*f*x] + 16*a^6*b*Sin[6*e + 8*f*x] - 7*a^5*b^2*Sin[6*e + 8*f*x] - 10*a^4*b^3*Sin[6*e + 8*f*x] + 13*a^7*Sin[10*e + 8*f*x] + 16*a^6*b*Sin[10*e + 8*f*x] - 7*a^5*b^2*Sin[10*e + 8*f*x] - 10*a^4*b^3*Sin[10*e + 8*f*x] + a^7*Sin[8*e + 10*f*x] + 2*a^6*b*Sin[8*e + 10*f*x] + a^5*b^2*Sin[8*e + 10*f*x] + a^7*Sin[12*e + 10*f*x] + 2*a^6*b*Sin[12*e + 10*f*x] + a^5*b^2*Sin[12*e + 10*f*x]))/(12288*a^6*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^3)","C",0
219,1,1411,204,6.863832,"\int \frac{1}{\left(a+b \sec ^2(c+d x)\right)^4} \, dx","Integrate[(a + b*Sec[c + d*x]^2)^(-4),x]","\frac{\left(35 a^3+70 b a^2+56 b^2 a+16 b^3\right) (\cos (2 c+2 d x) a+a+2 b)^4 \left(\frac{b \tan ^{-1}\left(\sec (d x) \left(\frac{\cos (2 c)}{2 \sqrt{a+b} \sqrt{b \cos (4 c)-i b \sin (4 c)}}-\frac{i \sin (2 c)}{2 \sqrt{a+b} \sqrt{b \cos (4 c)-i b \sin (4 c)}}\right) (-a \sin (d x)-2 b \sin (d x)+a \sin (2 c+d x))\right) \cos (2 c)}{256 a^4 \sqrt{a+b} d \sqrt{b \cos (4 c)-i b \sin (4 c)}}-\frac{i b \tan ^{-1}\left(\sec (d x) \left(\frac{\cos (2 c)}{2 \sqrt{a+b} \sqrt{b \cos (4 c)-i b \sin (4 c)}}-\frac{i \sin (2 c)}{2 \sqrt{a+b} \sqrt{b \cos (4 c)-i b \sin (4 c)}}\right) (-a \sin (d x)-2 b \sin (d x)+a \sin (2 c+d x))\right) \sin (2 c)}{256 a^4 \sqrt{a+b} d \sqrt{b \cos (4 c)-i b \sin (4 c)}}\right) \sec ^8(c+d x)}{(a+b)^3 \left(b \sec ^2(c+d x)+a\right)^4}+\frac{(\cos (2 c+2 d x) a+a+2 b) \sec (2 c) \left(480 d x \cos (2 c) a^6+360 d x \cos (2 d x) a^6+360 d x \cos (4 c+2 d x) a^6+144 d x \cos (2 c+4 d x) a^6+144 d x \cos (6 c+4 d x) a^6+24 d x \cos (4 c+6 d x) a^6+24 d x \cos (8 c+6 d x) a^6+3168 b d x \cos (2 c) a^5+2232 b d x \cos (2 d x) a^5+2232 b d x \cos (4 c+2 d x) a^5+720 b d x \cos (2 c+4 d x) a^5+720 b d x \cos (6 c+4 d x) a^5+72 b d x \cos (4 c+6 d x) a^5+72 b d x \cos (8 c+6 d x) a^5+870 b \sin (2 c) a^5-870 b \sin (2 d x) a^5+435 b \sin (4 c+2 d x) a^5-435 b \sin (2 c+4 d x) a^5+87 b \sin (6 c+4 d x) a^5-87 b \sin (4 c+6 d x) a^5+8928 b^2 d x \cos (2 c) a^4+5688 b^2 d x \cos (2 d x) a^4+5688 b^2 d x \cos (4 c+2 d x) a^4+1296 b^2 d x \cos (2 c+4 d x) a^4+1296 b^2 d x \cos (6 c+4 d x) a^4+72 b^2 d x \cos (4 c+6 d x) a^4+72 b^2 d x \cos (8 c+6 d x) a^4+4292 b^2 \sin (2 c) a^4-3792 b^2 \sin (2 d x) a^4+2124 b^2 \sin (4 c+2 d x) a^4-1374 b^2 \sin (2 c+4 d x) a^4+366 b^2 \sin (6 c+4 d x) a^4-116 b^2 \sin (4 c+6 d x) a^4+14112 b^3 d x \cos (2 c) a^3+7272 b^3 d x \cos (2 d x) a^3+7272 b^3 d x \cos (4 c+2 d x) a^3+1008 b^3 d x \cos (2 c+4 d x) a^3+1008 b^3 d x \cos (6 c+4 d x) a^3+24 b^3 d x \cos (4 c+6 d x) a^3+24 b^3 d x \cos (8 c+6 d x) a^3+8792 b^3 \sin (2 c) a^3-6432 b^3 \sin (2 d x) a^3+3972 b^3 \sin (4 c+2 d x) a^3-1248 b^3 \sin (2 c+4 d x) a^3+408 b^3 \sin (6 c+4 d x) a^3-44 b^3 \sin (4 c+6 d x) a^3+13248 b^4 d x \cos (2 c) a^2+4608 b^4 d x \cos (2 d x) a^2+4608 b^4 d x \cos (4 c+2 d x) a^2+288 b^4 d x \cos (2 c+4 d x) a^2+288 b^4 d x \cos (6 c+4 d x) a^2+9936 b^4 \sin (2 c) a^2-4608 b^4 \sin (2 d x) a^2+3072 b^4 \sin (4 c+2 d x) a^2-384 b^4 \sin (2 c+4 d x) a^2+144 b^4 \sin (6 c+4 d x) a^2+6912 b^5 d x \cos (2 c) a+1152 b^5 d x \cos (2 d x) a+1152 b^5 d x \cos (4 c+2 d x) a+5824 b^5 \sin (2 c) a-1248 b^5 \sin (2 d x) a+864 b^5 \sin (4 c+2 d x) a+1536 b^6 d x \cos (2 c)+1408 b^6 \sin (2 c)\right) \sec ^8(c+d x)}{3072 a^4 (a+b)^3 d \left(b \sec ^2(c+d x)+a\right)^4}","\frac{x}{a^4}-\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{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{\sqrt{b} \left(35 a^3+70 a^2 b+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 \tan (c+d x)}{6 a d (a+b) \left(a+b \tan ^2(c+d x)+b\right)^3}",1,"((35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*(a + 2*b + a*Cos[2*c + 2*d*x])^4*Sec[c + d*x]^8*((b*ArcTan[Sec[d*x]*(Cos[2*c]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*c] - I*b*Sin[4*c]]) - ((I/2)*Sin[2*c])/(Sqrt[a + b]*Sqrt[b*Cos[4*c] - I*b*Sin[4*c]]))*(-(a*Sin[d*x]) - 2*b*Sin[d*x] + a*Sin[2*c + d*x])]*Cos[2*c])/(256*a^4*Sqrt[a + b]*d*Sqrt[b*Cos[4*c] - I*b*Sin[4*c]]) - ((I/256)*b*ArcTan[Sec[d*x]*(Cos[2*c]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*c] - I*b*Sin[4*c]]) - ((I/2)*Sin[2*c])/(Sqrt[a + b]*Sqrt[b*Cos[4*c] - I*b*Sin[4*c]]))*(-(a*Sin[d*x]) - 2*b*Sin[d*x] + a*Sin[2*c + d*x])]*Sin[2*c])/(a^4*Sqrt[a + b]*d*Sqrt[b*Cos[4*c] - I*b*Sin[4*c]])))/((a + b)^3*(a + b*Sec[c + d*x]^2)^4) + ((a + 2*b + a*Cos[2*c + 2*d*x])*Sec[2*c]*Sec[c + d*x]^8*(480*a^6*d*x*Cos[2*c] + 3168*a^5*b*d*x*Cos[2*c] + 8928*a^4*b^2*d*x*Cos[2*c] + 14112*a^3*b^3*d*x*Cos[2*c] + 13248*a^2*b^4*d*x*Cos[2*c] + 6912*a*b^5*d*x*Cos[2*c] + 1536*b^6*d*x*Cos[2*c] + 360*a^6*d*x*Cos[2*d*x] + 2232*a^5*b*d*x*Cos[2*d*x] + 5688*a^4*b^2*d*x*Cos[2*d*x] + 7272*a^3*b^3*d*x*Cos[2*d*x] + 4608*a^2*b^4*d*x*Cos[2*d*x] + 1152*a*b^5*d*x*Cos[2*d*x] + 360*a^6*d*x*Cos[4*c + 2*d*x] + 2232*a^5*b*d*x*Cos[4*c + 2*d*x] + 5688*a^4*b^2*d*x*Cos[4*c + 2*d*x] + 7272*a^3*b^3*d*x*Cos[4*c + 2*d*x] + 4608*a^2*b^4*d*x*Cos[4*c + 2*d*x] + 1152*a*b^5*d*x*Cos[4*c + 2*d*x] + 144*a^6*d*x*Cos[2*c + 4*d*x] + 720*a^5*b*d*x*Cos[2*c + 4*d*x] + 1296*a^4*b^2*d*x*Cos[2*c + 4*d*x] + 1008*a^3*b^3*d*x*Cos[2*c + 4*d*x] + 288*a^2*b^4*d*x*Cos[2*c + 4*d*x] + 144*a^6*d*x*Cos[6*c + 4*d*x] + 720*a^5*b*d*x*Cos[6*c + 4*d*x] + 1296*a^4*b^2*d*x*Cos[6*c + 4*d*x] + 1008*a^3*b^3*d*x*Cos[6*c + 4*d*x] + 288*a^2*b^4*d*x*Cos[6*c + 4*d*x] + 24*a^6*d*x*Cos[4*c + 6*d*x] + 72*a^5*b*d*x*Cos[4*c + 6*d*x] + 72*a^4*b^2*d*x*Cos[4*c + 6*d*x] + 24*a^3*b^3*d*x*Cos[4*c + 6*d*x] + 24*a^6*d*x*Cos[8*c + 6*d*x] + 72*a^5*b*d*x*Cos[8*c + 6*d*x] + 72*a^4*b^2*d*x*Cos[8*c + 6*d*x] + 24*a^3*b^3*d*x*Cos[8*c + 6*d*x] + 870*a^5*b*Sin[2*c] + 4292*a^4*b^2*Sin[2*c] + 8792*a^3*b^3*Sin[2*c] + 9936*a^2*b^4*Sin[2*c] + 5824*a*b^5*Sin[2*c] + 1408*b^6*Sin[2*c] - 870*a^5*b*Sin[2*d*x] - 3792*a^4*b^2*Sin[2*d*x] - 6432*a^3*b^3*Sin[2*d*x] - 4608*a^2*b^4*Sin[2*d*x] - 1248*a*b^5*Sin[2*d*x] + 435*a^5*b*Sin[4*c + 2*d*x] + 2124*a^4*b^2*Sin[4*c + 2*d*x] + 3972*a^3*b^3*Sin[4*c + 2*d*x] + 3072*a^2*b^4*Sin[4*c + 2*d*x] + 864*a*b^5*Sin[4*c + 2*d*x] - 435*a^5*b*Sin[2*c + 4*d*x] - 1374*a^4*b^2*Sin[2*c + 4*d*x] - 1248*a^3*b^3*Sin[2*c + 4*d*x] - 384*a^2*b^4*Sin[2*c + 4*d*x] + 87*a^5*b*Sin[6*c + 4*d*x] + 366*a^4*b^2*Sin[6*c + 4*d*x] + 408*a^3*b^3*Sin[6*c + 4*d*x] + 144*a^2*b^4*Sin[6*c + 4*d*x] - 87*a^5*b*Sin[4*c + 6*d*x] - 116*a^4*b^2*Sin[4*c + 6*d*x] - 44*a^3*b^3*Sin[4*c + 6*d*x]))/(3072*a^4*(a + b)^3*d*(a + b*Sec[c + d*x]^2)^4)","C",0
220,1,70,134,2.1608328,"\int \left(a-a \sec ^2(c+d x)\right)^{7/2} \, dx","Integrate[(a - a*Sec[c + d*x]^2)^(7/2),x]","\frac{\cot ^7(c+d x) \left(-a \tan ^2(c+d x)\right)^{7/2} \left(2 \tan ^6(c+d x)-3 \tan ^4(c+d x)+6 \tan ^2(c+d x)+12 \log (\cos (c+d x))\right)}{12 d}","-\frac{a^3 \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 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 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d}",1,"(Cot[c + d*x]^7*(-(a*Tan[c + d*x]^2))^(7/2)*(12*Log[Cos[c + d*x]] + 6*Tan[c + d*x]^2 - 3*Tan[c + d*x]^4 + 2*Tan[c + d*x]^6))/(12*d)","A",1
221,1,60,101,0.5431747,"\int \left(a-a \sec ^2(c+d x)\right)^{5/2} \, dx","Integrate[(a - a*Sec[c + d*x]^2)^(5/2),x]","-\frac{\cot ^5(c+d x) \left(-a \tan ^2(c+d x)\right)^{5/2} \left(-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right)}{4 d}","-\frac{a^2 \tan (c+d x) \sqrt{-a \tan ^2(c+d x)}}{2 d}+\frac{a^2 \tan ^3(c+d x) \sqrt{-a \tan ^2(c+d x)}}{4 d}-\frac{a^2 \cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d}",1,"-1/4*(Cot[c + d*x]^5*(-(a*Tan[c + d*x]^2))^(5/2)*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/d","A",1
222,1,48,64,0.1063463,"\int \left(a-a \sec ^2(c+d x)\right)^{3/2} \, dx","Integrate[(a - a*Sec[c + d*x]^2)^(3/2),x]","\frac{\cot ^3(c+d x) \left(-a \tan ^2(c+d x)\right)^{3/2} \left(\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)}{2 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,"(Cot[c + d*x]^3*(-(a*Tan[c + d*x]^2))^(3/2)*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/(2*d)","A",1
223,1,33,33,0.047802,"\int \sqrt{a-a \sec ^2(c+d x)} \, dx","Integrate[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",1
224,1,40,32,0.0587383,"\int \frac{1}{\sqrt{a-a \sec ^2(c+d x)}} \, dx","Integrate[1/Sqrt[a - a*Sec[c + d*x]^2],x]","\frac{\tan (c+d x) (\log (\tan (c+d x))+\log (\cos (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[Cos[c + d*x]] + Log[Tan[c + d*x]])*Tan[c + d*x])/(d*Sqrt[-(a*Tan[c + d*x]^2)])","A",1
225,1,57,67,0.1435565,"\int \frac{1}{\left(a-a \sec ^2(c+d x)\right)^{3/2}} \, dx","Integrate[(a - a*Sec[c + d*x]^2)^(-3/2),x]","-\frac{\tan ^3(c+d x) \left(\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right)}{2 d \left(-a \tan ^2(c+d x)\right)^{3/2}}","\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,"-1/2*((Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]])*Tan[c + d*x]^3)/(d*(-(a*Tan[c + d*x]^2))^(3/2))","A",1
226,1,69,100,0.2819424,"\int \frac{1}{\left(a-a \sec ^2(c+d x)\right)^{5/2}} \, dx","Integrate[(a - a*Sec[c + d*x]^2)^(-5/2),x]","\frac{\tan ^5(c+d x) \left(-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right)}{4 d \left(-a \tan ^2(c+d x)\right)^{5/2}}","-\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,"((2*Cot[c + d*x]^2 - Cot[c + d*x]^4 + 4*Log[Cos[c + d*x]] + 4*Log[Tan[c + d*x]])*Tan[c + d*x]^5)/(4*d*(-(a*Tan[c + d*x]^2))^(5/2))","A",1
227,1,79,133,0.3364525,"\int \frac{1}{\left(a-a \sec ^2(c+d x)\right)^{7/2}} \, dx","Integrate[(a - a*Sec[c + d*x]^2)^(-7/2),x]","-\frac{\tan ^7(c+d x) \left(2 \cot ^6(c+d x)-3 \cot ^4(c+d x)+6 \cot ^2(c+d x)+12 \log (\tan (c+d x))+12 \log (\cos (c+d x))\right)}{12 d \left(-a \tan ^2(c+d x)\right)^{7/2}}","\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,"-1/12*((6*Cot[c + d*x]^2 - 3*Cot[c + d*x]^4 + 2*Cot[c + d*x]^6 + 12*Log[Cos[c + d*x]] + 12*Log[Tan[c + d*x]])*Tan[c + d*x]^7)/(d*(-(a*Tan[c + d*x]^2))^(7/2))","A",1
228,0,0,372,27.0321921,"\int \sec ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Sec[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2],x]","\int \sec ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","-\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{(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{\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-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,"Integrate[Sec[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
229,0,0,288,11.5470353,"\int \sec ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Sec[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2],x]","\int \sec ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","\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,"Integrate[Sec[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
230,0,0,218,11.6591028,"\int \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2],x]","\int \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","\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,"Integrate[Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
231,1,69,80,0.2550714,"\int \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{2} \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} E\left(e+f x\left|\frac{a}{a+b}\right.\right)}{f \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+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[2]*Cos[e + f*x]*EllipticE[e + f*x, a/(a + b)]*Sqrt[a + b*Sec[e + f*x]^2])/(f*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)])","A",1
232,1,539,246,8.4997862,"\int \cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(\frac{(2 a+2 b) \sqrt{\frac{a \cos (2 e+2 f x)+a+2 b}{2 a+2 b}} E\left(\frac{1}{2} (2 e+2 f x)|\frac{2 a}{2 a+2 b}\right)}{f \sqrt{a \cos (2 e+2 f x)+a+2 b}}+\frac{\sin (2 e+2 f x) \cos (2 (e+f x)) \sec \left(2 \left(\frac{1}{2} \left(\cos ^{-1}(\cos (2 e+2 f x))-2 e\right)+e\right)\right) \left(-\sqrt{-\frac{1}{a+b}} (a \cos (2 e+2 f x)-a) (a \cos (2 e+2 f x)+a) \sqrt{a \cos (2 e+2 f x)+a+2 b}-i a b \sqrt{\frac{-2 (a \cos (2 e+2 f x)+a+2 b)+4 a+4 b}{a+b}} \sqrt{2-\frac{a \cos (2 e+2 f x)+a+2 b}{b}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{a+b}} \sqrt{\cos (2 e+2 f x) a+a+2 b}}{\sqrt{2}}\right)|\frac{a+b}{b}\right)-i b (a+2 b) \sqrt{\frac{a-a \cos (2 e+2 f x)}{a+b}} \sqrt{4-\frac{2 (a \cos (2 e+2 f x)+a+2 b)}{b}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{a+b}} \sqrt{\cos (2 e+2 f x) a+a+2 b}}{\sqrt{2}}\right)|\frac{a+b}{b}\right)\right)}{3 a^2 f \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(2 e+2 f x)} \sqrt{\frac{(a-a \cos (2 e+2 f x)) (a \cos (2 e+2 f x)+a)}{a^2}}}\right)}{2 \sqrt{a \cos (2 e+2 f x)+a+2 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]*Sqrt[a + b*Sec[e + f*x]^2]*(((2*a + 2*b)*Sqrt[(a + 2*b + a*Cos[2*e + 2*f*x])/(2*a + 2*b)]*EllipticE[(2*e + 2*f*x)/2, (2*a)/(2*a + 2*b)])/(f*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]) + (Cos[2*(e + f*x)]*(-(Sqrt[-(a + b)^(-1)]*(-a + a*Cos[2*e + 2*f*x])*(a + a*Cos[2*e + 2*f*x])*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]) - I*b*(a + 2*b)*Sqrt[(a - a*Cos[2*e + 2*f*x])/(a + b)]*Sqrt[4 - (2*(a + 2*b + a*Cos[2*e + 2*f*x]))/b]*EllipticE[I*ArcSinh[(Sqrt[-(a + b)^(-1)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])/Sqrt[2]], (a + b)/b] - I*a*b*Sqrt[(4*a + 4*b - 2*(a + 2*b + a*Cos[2*e + 2*f*x]))/(a + b)]*Sqrt[2 - (a + 2*b + a*Cos[2*e + 2*f*x])/b]*EllipticF[I*ArcSinh[(Sqrt[-(a + b)^(-1)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])/Sqrt[2]], (a + b)/b])*Sec[2*(e + (-2*e + ArcCos[Cos[2*e + 2*f*x]])/2)]*Sin[2*e + 2*f*x])/(3*a^2*Sqrt[-(a + b)^(-1)]*f*Sqrt[((a - a*Cos[2*e + 2*f*x])*(a + a*Cos[2*e + 2*f*x]))/a^2]*Sqrt[1 - Cos[2*e + 2*f*x]^2])))/(2*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])","C",1
233,0,0,338,11.5092792,"\int \cos ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2],x]","\int \cos ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","\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,"Integrate[Cos[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
234,1,968,186,11.3987576,"\int \sec ^6(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Sec[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{i e^{i (e+f x)} \sqrt{a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2+4 b} \left(\frac{15 (a+b) \tan ^{-1}\left(\frac{\sqrt{b} \left(-1+e^{2 i (e+f x)}\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}\right) a^2}{8 b^{3/2}}+\frac{15 \left(-1+e^{2 i (e+f x)}\right) \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}} a^2}{8 b \left(1+e^{2 i (e+f x)}\right)^2}-\frac{15 (a+b) \tan ^{-1}\left(\frac{\sqrt{b} \left(-1+e^{2 i (e+f x)}\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}\right) a}{4 \sqrt{b}}-\frac{15 \left(-1+e^{2 i (e+f x)}\right) \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}} a}{4 \left(1+e^{2 i (e+f x)}\right)^2}+\frac{7 (7 a+15 b) \left(a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}\right)^{3/2}}{4 b \left(1+e^{2 i (e+f x)}\right)^3}-\frac{2 (8 a+35 b) \left(a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}\right)^{3/2}}{b \left(1+e^{2 i (e+f x)}\right)^3}+\frac{50 \left(a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}\right)^{3/2}}{\left(1+e^{2 i (e+f x)}\right)^3}+\frac{3 (5 a+21 b) \left(a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}\right)^{3/2}}{2 b \left(1+e^{2 i (e+f x)}\right)^4}-\frac{24 \left(a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}\right)^{3/2}}{\left(1+e^{2 i (e+f x)}\right)^4}-\frac{60 \left(a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}\right)^{3/2}}{\left(1+e^{2 i (e+f x)}\right)^5}+\frac{40 \left(a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}\right)^{3/2}}{\left(1+e^{2 i (e+f x)}\right)^6}+\frac{75}{8} \sqrt{b} (a+b) \tan ^{-1}\left(\frac{\sqrt{b} \left(-1+e^{2 i (e+f x)}\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}\right)+\frac{75 b \left(-1+e^{2 i (e+f x)}\right) \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}{8 \left(1+e^{2 i (e+f x)}\right)^2}\right) \cos (e+f x) \sqrt{b \sec ^2(e+f x)+a}}{15 \sqrt{2} b \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}} f \sqrt{\cos (2 e+2 f x) a+a+2 b}}","\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,"((-1/15*I)*E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*((-15*a*(-1 + E^((2*I)*(e + f*x)))*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])/(4*(1 + E^((2*I)*(e + f*x)))^2) + (15*a^2*(-1 + E^((2*I)*(e + f*x)))*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])/(8*b*(1 + E^((2*I)*(e + f*x)))^2) + (75*b*(-1 + E^((2*I)*(e + f*x)))*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])/(8*(1 + E^((2*I)*(e + f*x)))^2) + (40*(4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2)^(3/2))/(1 + E^((2*I)*(e + f*x)))^6 - (60*(4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2)^(3/2))/(1 + E^((2*I)*(e + f*x)))^5 - (24*(4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2)^(3/2))/(1 + E^((2*I)*(e + f*x)))^4 + (3*(5*a + 21*b)*(4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2)^(3/2))/(2*b*(1 + E^((2*I)*(e + f*x)))^4) + (50*(4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2)^(3/2))/(1 + E^((2*I)*(e + f*x)))^3 + (7*(7*a + 15*b)*(4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2)^(3/2))/(4*b*(1 + E^((2*I)*(e + f*x)))^3) - (2*(8*a + 35*b)*(4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2)^(3/2))/(b*(1 + E^((2*I)*(e + f*x)))^3) + (15*a^2*(a + b)*ArcTan[(Sqrt[b]*(-1 + E^((2*I)*(e + f*x))))/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]])/(8*b^(3/2)) - (15*a*(a + b)*ArcTan[(Sqrt[b]*(-1 + E^((2*I)*(e + f*x))))/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]])/(4*Sqrt[b]) + (75*Sqrt[b]*(a + b)*ArcTan[(Sqrt[b]*(-1 + E^((2*I)*(e + f*x))))/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]])/8)*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(Sqrt[2]*b*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]*f*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])","C",1
235,1,390,122,8.1331433,"\int \sec ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Sec[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{i e^{i (e+f x)} \cos (e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{\left(a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}\right)^{3/2}}{\left(1+e^{2 i (e+f x)}\right)^3}-\frac{2 \left(a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}\right)^{3/2}}{\left(1+e^{2 i (e+f x)}\right)^4}+\frac{1}{2} (3 b-a) \left(\frac{\left(-1+e^{2 i (e+f x)}\right) \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}{\left(1+e^{2 i (e+f x)}\right)^2}+\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \left(-1+e^{2 i (e+f x)}\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}\right)}{\sqrt{b}}\right)\right) \sqrt{a+b \sec ^2(e+f x)}}{2 \sqrt{2} b f \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}} \sqrt{a \cos (2 e+2 f x)+a+2 b}}","-\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,"((-1/2*I)*E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*((-2*(4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2)^(3/2))/(1 + E^((2*I)*(e + f*x)))^4 + (4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2)^(3/2)/(1 + E^((2*I)*(e + f*x)))^3 + ((-a + 3*b)*(((-1 + E^((2*I)*(e + f*x)))*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])/(1 + E^((2*I)*(e + f*x)))^2 + ((a + b)*ArcTan[(Sqrt[b]*(-1 + E^((2*I)*(e + f*x))))/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]])/Sqrt[b]))/2)*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(Sqrt[2]*b*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]*f*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])","C",1
236,1,210,76,1.6074744,"\int \sec ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Sec[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\tan (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \sqrt{a+b \sec ^2(e+f x)} \left(\sqrt{\frac{b \sin ^2(e+f x)}{a+b}} (a \cos (2 (e+f x))+a+2 b)+\sqrt{2} (a+b) \cos ^2(e+f x) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}} \tanh ^{-1}\left(\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a+b}}}{\sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}}}\right)\right)}{\sqrt{2} f \sqrt{\frac{b \sin ^2(e+f x)}{a+b}} (a \cos (2 (e+f x))+a+2 b)^{3/2}}","\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 + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]*(Sqrt[2]*(a + b)*ArcTanh[Sqrt[(b*Sin[e + f*x]^2)/(a + b)]/Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)]]*Cos[e + f*x]^2*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)] + (a + 2*b + a*Cos[2*(e + f*x)])*Sqrt[(b*Sin[e + f*x]^2)/(a + b)])*Tan[e + f*x])/(Sqrt[2]*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*Sqrt[(b*Sin[e + f*x]^2)/(a + b)])","B",1
237,0,0,79,0.0910015,"\int \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]^2],x]","\int \sqrt{a+b \sec ^2(e+f x)} \, dx","\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,"Integrate[Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
238,1,136,82,0.6664814,"\int \cos ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(2 \sqrt{a+b} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)+\sqrt{2} \sqrt{a} \sin (e+f x) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}}\right)}{2 \sqrt{2} \sqrt{a} f \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}}}","\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,"(Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2]*(2*Sqrt[a + b]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]] + Sqrt[2]*Sqrt[a]*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*Sin[e + f*x]))/(2*Sqrt[2]*Sqrt[a]*f*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)])","A",1
239,1,152,140,1.2329069,"\int \cos ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(\sqrt{2} (3 a-b) \sqrt{a+b} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)+\sqrt{a} \sin (e+f x) (a \cos (2 (e+f x))+4 a+b) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}}\right)}{8 a^{3/2} f \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}}}","\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,"(Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2]*(Sqrt[2]*(3*a - b)*Sqrt[a + b]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]] + Sqrt[a]*(4*a + b + a*Cos[2*(e + f*x)])*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*Sin[e + f*x]))/(8*a^(3/2)*f*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)])","A",1
240,1,1902,196,16.3677333,"\int \cos ^6(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{3 (a+b) F_1\left(\frac{1}{2};-2,-\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^{10}(e+f x) \sqrt{\cos (2 (e+f x)) a+a+2 b} \sqrt{b \sec ^2(e+f x)+a} \sin (e+f x)}{f \left(3 (a+b) F_1\left(\frac{1}{2};-2,-\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-\left(a F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)+4 (a+b) F_1\left(\frac{3}{2};-1,-\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)\right) \left(\frac{3 (a+b) F_1\left(\frac{1}{2};-2,-\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sqrt{\cos (2 (e+f x)) a+a+2 b} \cos ^5(e+f x)}{3 (a+b) F_1\left(\frac{1}{2};-2,-\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-\left(a F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)+4 (a+b) F_1\left(\frac{3}{2};-1,-\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)}-\frac{3 (a+b) F_1\left(\frac{1}{2};-2,-\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sqrt{\cos (2 (e+f x)) a+a+2 b} \sin (e+f x) \left(-\left(\left(a \left(\frac{3 a f F_1\left(\frac{5}{2};-2,\frac{3}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{5 (a+b)}-\frac{12}{5} f F_1\left(\frac{5}{2};-1,\frac{1}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)+4 (a+b) \left(-\frac{6}{5} f \cos (e+f x) \sin (e+f x) \left(\frac{5 (a+b)^3 \left(-\frac{4 a^2 \sin ^4(e+f x)}{3 (a+b)^2}-\frac{2 a \sin ^2(e+f x)}{a+b}+\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) \sin (e+f x)}{\sqrt{a+b} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}\right) \csc ^6(e+f x)}{32 a^3 \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)}+\frac{5}{6 \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)}\right) \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)^{3/2}-\frac{3 a f F_1\left(\frac{5}{2};-1,\frac{1}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{5 (a+b)}\right)\right) \sin ^2(e+f x)\right)-2 f \left(a F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)+4 (a+b) F_1\left(\frac{3}{2};-1,-\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \cos (e+f x) \sin (e+f x)+3 (a+b) \left(-\frac{a f F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac{4}{3} f F_1\left(\frac{3}{2};-1,-\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)\right) \cos ^4(e+f x)}{f \left(3 (a+b) F_1\left(\frac{1}{2};-2,-\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-\left(a F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)+4 (a+b) F_1\left(\frac{3}{2};-1,-\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)\right){}^2}-\frac{3 a (a+b) F_1\left(\frac{1}{2};-2,-\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin (e+f x) \sin (2 (e+f x)) \cos ^4(e+f x)}{\sqrt{\cos (2 (e+f x)) a+a+2 b} \left(3 (a+b) F_1\left(\frac{1}{2};-2,-\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-\left(a F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)+4 (a+b) F_1\left(\frac{3}{2};-1,-\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)\right)}+\frac{3 (a+b) \sqrt{\cos (2 (e+f x)) a+a+2 b} \sin (e+f x) \left(-\frac{a f F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac{4}{3} f F_1\left(\frac{3}{2};-1,-\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right) \cos ^4(e+f x)}{f \left(3 (a+b) F_1\left(\frac{1}{2};-2,-\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-\left(a F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)+4 (a+b) F_1\left(\frac{3}{2};-1,-\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)\right)}-\frac{12 (a+b) F_1\left(\frac{1}{2};-2,-\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sqrt{\cos (2 (e+f x)) a+a+2 b} \sin ^2(e+f x) \cos ^3(e+f x)}{3 (a+b) F_1\left(\frac{1}{2};-2,-\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-\left(a F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)+4 (a+b) F_1\left(\frac{3}{2};-1,-\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)}\right)}","\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{(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{\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,"(3*(a + b)*AppellF1[1/2, -2, -1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^10*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x])/(f*(3*(a + b)*AppellF1[1/2, -2, -1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - (a*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 4*(a + b)*AppellF1[3/2, -1, -1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)*((3*(a + b)*AppellF1[1/2, -2, -1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^5*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])/(3*(a + b)*AppellF1[1/2, -2, -1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - (a*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 4*(a + b)*AppellF1[3/2, -1, -1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2) - (12*(a + b)*AppellF1[1/2, -2, -1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^3*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*Sin[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -2, -1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - (a*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 4*(a + b)*AppellF1[3/2, -1, -1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2) + (3*(a + b)*Cos[e + f*x]^4*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*Sin[e + f*x]*(-1/3*(a*f*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - (4*f*AppellF1[3/2, -1, -1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3))/(f*(3*(a + b)*AppellF1[1/2, -2, -1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - (a*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 4*(a + b)*AppellF1[3/2, -1, -1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*AppellF1[1/2, -2, -1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^4*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*Sin[e + f*x]*(-2*f*(a*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 4*(a + b)*AppellF1[3/2, -1, -1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Cos[e + f*x]*Sin[e + f*x] + 3*(a + b)*(-1/3*(a*f*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - (4*f*AppellF1[3/2, -1, -1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3) - Sin[e + f*x]^2*(a*((3*a*f*AppellF1[5/2, -2, 3/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(5*(a + b)) - (12*f*AppellF1[5/2, -1, 1/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5) + 4*(a + b)*((-3*a*f*AppellF1[5/2, -1, 1/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(5*(a + b)) - (6*f*Cos[e + f*x]*Sin[e + f*x]*(1 - (a*Sin[e + f*x]^2)/(a + b))^(3/2)*(5/(6*(1 - (a*Sin[e + f*x]^2)/(a + b))) + (5*(a + b)^3*Csc[e + f*x]^6*((-2*a*Sin[e + f*x]^2)/(a + b) - (4*a^2*Sin[e + f*x]^4)/(3*(a + b)^2) + (2*Sqrt[a]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/(Sqrt[a + b]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])))/(32*a^3*(1 - (a*Sin[e + f*x]^2)/(a + b)))))/5))))/(f*(3*(a + b)*AppellF1[1/2, -2, -1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - (a*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 4*(a + b)*AppellF1[3/2, -1, -1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)^2) - (3*a*(a + b)*AppellF1[1/2, -2, -1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^4*Sin[e + f*x]*Sin[2*(e + f*x)])/(Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(3*(a + b)*AppellF1[1/2, -2, -1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - (a*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 4*(a + b)*AppellF1[3/2, -1, -1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2))))","C",0
241,0,0,450,9.6832601,"\int \sec ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sec[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2),x]","\int \sec ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","-\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,"Integrate[Sec[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2), x]","F",-1
242,0,0,371,16.2399093,"\int \sec ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2),x]","\int \sec ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","\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{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{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{(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,"Integrate[Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2), x]","F",-1
243,0,0,290,11.4822168,"\int \sec (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sec[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2),x]","\int \sec (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","\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,"Integrate[Sec[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2), x]","F",-1
244,0,0,224,12.5808731,"\int \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2),x]","\int \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","\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,"Integrate[Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2), x]","F",-1
245,1,179,241,1.8789445,"\int \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(4 \sqrt{2} \left(a^2+3 a b+2 b^2\right) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}} E\left(e+f x\left|\frac{a}{a+b}\right.\right)+a \sin (2 (e+f x)) (a \cos (2 (e+f x))+a+2 b)-2 \sqrt{2} b (a+b) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}} F\left(e+f x\left|\frac{a}{a+b}\right.\right)\right)}{6 f (a \cos (2 (e+f x))+a+2 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,"(Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2]*(4*Sqrt[2]*(a^2 + 3*a*b + 2*b^2)*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*EllipticE[e + f*x, a/(a + b)] - 2*Sqrt[2]*b*(a + b)*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*EllipticF[e + f*x, a/(a + b)] + a*(a + 2*b + a*Cos[2*(e + f*x)])*Sin[2*(e + f*x)]))/(6*f*(a + 2*b + a*Cos[2*(e + f*x)]))","A",1
246,1,350,319,8.9799694,"\int \cos ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\cos ^3(e+f x) \csc (2 (e+f x)) \left(a+b \sec ^2(e+f x)\right)^{3/2} \left(a \left(a \sqrt{-\frac{1}{a+b}} \sin ^2(2 (e+f x)) \sqrt{a \cos (2 (e+f x))+a+2 b} (3 a \cos (2 (e+f x))+11 a+12 b)+16 i \sqrt{2} b (2 a+3 b) \sqrt{\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{-\frac{a \cos ^2(e+f x)}{b}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{a+b}} \sqrt{\cos (2 (e+f x)) a+a+2 b}}{\sqrt{2}}\right)|\frac{a+b}{b}\right)\right)-8 i \sqrt{2} b \left(8 a^2+13 a b+3 b^2\right) \sqrt{\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{-\frac{a \cos ^2(e+f x)}{b}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{a+b}} \sqrt{\cos (2 (e+f x)) a+a+2 b}}{\sqrt{2}}\right)|\frac{a+b}{b}\right)\right)}{30 a^2 f \sqrt{-\frac{1}{a+b}} (a \cos (2 (e+f x))+a+2 b)^{3/2}}","\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,"(Cos[e + f*x]^3*Csc[2*(e + f*x)]*(a + b*Sec[e + f*x]^2)^(3/2)*((-8*I)*Sqrt[2]*b*(8*a^2 + 13*a*b + 3*b^2)*Sqrt[-((a*Cos[e + f*x]^2)/b)]*EllipticE[I*ArcSinh[(Sqrt[-(a + b)^(-1)]*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])/Sqrt[2]], (a + b)/b]*Sqrt[(a*Sin[e + f*x]^2)/(a + b)] + a*((16*I)*Sqrt[2]*b*(2*a + 3*b)*Sqrt[-((a*Cos[e + f*x]^2)/b)]*EllipticF[I*ArcSinh[(Sqrt[-(a + b)^(-1)]*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])/Sqrt[2]], (a + b)/b]*Sqrt[(a*Sin[e + f*x]^2)/(a + b)] + a*Sqrt[-(a + b)^(-1)]*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(11*a + 12*b + 3*a*Cos[2*(e + f*x)])*Sin[2*(e + f*x)]^2)))/(30*a^2*Sqrt[-(a + b)^(-1)]*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2))","C",1
247,1,512,243,10.8544015,"\int \sec ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sec[e + f*x]^6*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{e^{i (e+f x)} \cos ^3(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(-\frac{3 \left(3 a^2-10 a b+35 b^2\right) (a+b)^2 \log \left(\frac{4 i f \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}-4 \sqrt{b} f \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}-\frac{i \sqrt{b} \left(-1+e^{2 i (e+f x)}\right) \left(-9 a^3 \left(1+e^{2 i (e+f x)}\right)^6+3 a^2 b \left(18 e^{2 i (e+f x)}+5 e^{4 i (e+f x)}+5\right) \left(1+e^{2 i (e+f x)}\right)^4+a b^2 \left(948 e^{2 i (e+f x)}+2758 e^{4 i (e+f x)}+948 e^{6 i (e+f x)}+145 e^{8 i (e+f x)}+145\right) \left(1+e^{2 i (e+f x)}\right)^2+b^3 \left(910 e^{2 i (e+f x)}+3591 e^{4 i (e+f x)}+8644 e^{6 i (e+f x)}+3591 e^{8 i (e+f x)}+910 e^{10 i (e+f x)}+105 e^{12 i (e+f x)}+105\right)\right)}{\left(1+e^{2 i (e+f x)}\right)^8}\right) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{96 \sqrt{2} b^{5/2} f (a \cos (2 e+2 f x)+a+2 b)^{3/2}}","\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,"(E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]^3*(((-I)*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*(-9*a^3*(1 + E^((2*I)*(e + f*x)))^6 + 3*a^2*b*(1 + E^((2*I)*(e + f*x)))^4*(5 + 18*E^((2*I)*(e + f*x)) + 5*E^((4*I)*(e + f*x))) + a*b^2*(1 + E^((2*I)*(e + f*x)))^2*(145 + 948*E^((2*I)*(e + f*x)) + 2758*E^((4*I)*(e + f*x)) + 948*E^((6*I)*(e + f*x)) + 145*E^((8*I)*(e + f*x))) + b^3*(105 + 910*E^((2*I)*(e + f*x)) + 3591*E^((4*I)*(e + f*x)) + 8644*E^((6*I)*(e + f*x)) + 3591*E^((8*I)*(e + f*x)) + 910*E^((10*I)*(e + f*x)) + 105*E^((12*I)*(e + f*x)))))/(1 + E^((2*I)*(e + f*x)))^8 - (3*(a + b)^2*(3*a^2 - 10*a*b + 35*b^2)*Log[(-4*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*f + (4*I)*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]*f)/(1 + E^((2*I)*(e + f*x)))])/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*(a + b*Sec[e + f*x]^2)^(3/2))/(96*Sqrt[2]*b^(5/2)*f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2))","C",1
248,1,400,165,9.1721888,"\int \sec ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sec[e + f*x]^4*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{e^{i (e+f x)} \cos ^3(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{3 (a-5 b) (a+b)^2 \log \left(\frac{4 i f \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}-4 \sqrt{b} f \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}-\frac{i \sqrt{b} \left(-1+e^{2 i (e+f x)}\right) \left(3 a^2 \left(1+e^{2 i (e+f x)}\right)^4+2 a b \left(50 e^{2 i (e+f x)}+11 e^{4 i (e+f x)}+11\right) \left(1+e^{2 i (e+f x)}\right)^2+b^2 \left(100 e^{2 i (e+f x)}+298 e^{4 i (e+f x)}+100 e^{6 i (e+f x)}+15 e^{8 i (e+f x)}+15\right)\right)}{\left(1+e^{2 i (e+f x)}\right)^6}\right) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{12 \sqrt{2} b^{3/2} f (a \cos (2 e+2 f x)+a+2 b)^{3/2}}","-\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,"(E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]^3*(((-I)*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*(3*a^2*(1 + E^((2*I)*(e + f*x)))^4 + 2*a*b*(1 + E^((2*I)*(e + f*x)))^2*(11 + 50*E^((2*I)*(e + f*x)) + 11*E^((4*I)*(e + f*x))) + b^2*(15 + 100*E^((2*I)*(e + f*x)) + 298*E^((4*I)*(e + f*x)) + 100*E^((6*I)*(e + f*x)) + 15*E^((8*I)*(e + f*x)))))/(1 + E^((2*I)*(e + f*x)))^6 + (3*(a - 5*b)*(a + b)^2*Log[(-4*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*f + (4*I)*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]*f)/(1 + E^((2*I)*(e + f*x)))])/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*(a + b*Sec[e + f*x]^2)^(3/2))/(12*Sqrt[2]*b^(3/2)*f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2))","C",1
249,1,84,111,0.2450634,"\int \sec ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sec[e + f*x]^2*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{(a+b)^2 \sin (2 (e+f x)) \sqrt{a+b \sec ^2(e+f x)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{b \sin ^2(e+f x)}{-a \sin ^2(e+f x)+a+b}\right)}{f (a \cos (2 (e+f x))+a+2 b)}","\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,"((a + b)^2*Hypergeometric2F1[1/2, 3, 3/2, (b*Sin[e + f*x]^2)/(a + b - a*Sin[e + f*x]^2)]*Sqrt[a + b*Sec[e + f*x]^2]*Sin[2*(e + f*x)])/(f*(a + 2*b + a*Cos[2*(e + f*x)]))","C",1
250,1,527,118,1.8746346,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} e^{i (e+f x)} \cos ^3(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{-i a^{3/2} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)+i a^{3/2} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)+2 a^{3/2} f x-b^{3/2} \log \left(\frac{2 i f \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}-2 \sqrt{b} f \left(-1+e^{2 i (e+f x)}\right)}{b (3 a+b) \left(1+e^{2 i (e+f x)}\right)}\right)-3 a \sqrt{b} \log \left(\frac{2 i f \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}-2 \sqrt{b} f \left(-1+e^{2 i (e+f x)}\right)}{b (3 a+b) \left(1+e^{2 i (e+f x)}\right)}\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}-\frac{i b \left(-1+e^{2 i (e+f x)}\right)}{\left(1+e^{2 i (e+f x)}\right)^2}\right) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{f (a \cos (2 e+2 f x)+a+2 b)^{3/2}}","\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,"(Sqrt[2]*E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]^3*(((-I)*b*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))^2 + (2*a^(3/2)*f*x - I*a^(3/2)*Log[a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] + I*a^(3/2)*Log[a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - 3*a*Sqrt[b]*Log[(-2*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*f + (2*I)*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]*f)/(b*(3*a + b)*(1 + E^((2*I)*(e + f*x))))] - b^(3/2)*Log[(-2*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*f + (2*I)*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]*f)/(b*(3*a + b)*(1 + E^((2*I)*(e + f*x))))])/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*(a + b*Sec[e + f*x]^2)^(3/2))/(f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2))","C",1
251,1,466,124,7.5555989,"\int \cos ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cos[e + f*x]^2*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{e^{-i (e+f x)} \cos ^3(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{2 e^{2 i (e+f x)} \left(2 a^{3/2} f x-4 b^{3/2} \log \left(-\frac{e^{3 i e} f \left(\sqrt{b} \left(-1+e^{2 i (e+f x)}\right)-i \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}\right)}{2 b^2 \left(1+e^{2 i (e+f x)}\right)}\right)-i \sqrt{a} (a+3 b) \log \left(e^{-2 i e} \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)\right)+i \sqrt{a} (a+3 b) \log \left(e^{-2 i e} \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)\right)+6 \sqrt{a} b f x\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}-i a \left(-1+e^{2 i (e+f x)}\right)\right) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{2 \sqrt{2} f (a \cos (2 e+2 f x)+a+2 b)^{3/2}}","\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[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]^3*((-I)*a*(-1 + E^((2*I)*(e + f*x))) + (2*E^((2*I)*(e + f*x))*(2*a^(3/2)*f*x + 6*Sqrt[a]*b*f*x - I*Sqrt[a]*(a + 3*b)*Log[(a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])/E^((2*I)*e)] + I*Sqrt[a]*(a + 3*b)*Log[(a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])/E^((2*I)*e)] - 4*b^(3/2)*Log[-1/2*(E^((3*I)*e)*(Sqrt[b]*(-1 + E^((2*I)*(e + f*x))) - I*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*f)/(b^2*(1 + E^((2*I)*(e + f*x))))]))/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*(a + b*Sec[e + f*x]^2)^(3/2))/(2*Sqrt[2]*E^(I*(e + f*x))*f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2))","C",1
252,1,191,125,1.0247532,"\int \cos ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\cos (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \left(a \cos ^2(e+f x)+b\right) \sqrt{a+b \sec ^2(e+f x)} \left(3 (a+b)^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)+\sqrt{a} \sin (e+f x) \sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}} (a \cos (2 (e+f x))+4 a+5 b)\right)}{2 \sqrt{a} f (a \cos (2 (e+f x))+a+2 b)^{3/2} \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}}}","\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,"(Cos[e + f*x]*(b + a*Cos[e + f*x]^2)*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2]*(3*(a + b)^(3/2)*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]] + Sqrt[a]*(4*a + 5*b + a*Cos[2*(e + f*x)])*Sin[e + f*x]*Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)]))/(2*Sqrt[a]*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)])","A",1
253,1,165,193,2.0062062,"\int \cos ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cos[e + f*x]^6*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(\frac{3 \sqrt{2} \sqrt{a+b} \left(5 a^2+4 a b-b^2\right) \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}}}+\sqrt{a} \sin (e+f x) \left(a^2 \cos (4 (e+f x))+23 a^2+a (9 a+7 b) \cos (2 (e+f x))+29 a b+3 b^2\right)\right)}{48 a^{3/2} 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,"(Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2]*((3*Sqrt[2]*Sqrt[a + b]*(5*a^2 + 4*a*b - b^2)*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)] + Sqrt[a]*(23*a^2 + 29*a*b + 3*b^2 + a*(9*a + 7*b)*Cos[2*(e + f*x)] + a^2*Cos[4*(e + f*x)])*Sin[e + f*x]))/(48*a^(3/2)*f)","A",1
254,1,706,166,10.5140128,"\int \left(a+b \sec ^2(c+d x)\right)^{5/2} \, dx","Integrate[(a + b*Sec[c + d*x]^2)^(5/2),x]","\frac{e^{i (c+d x)} \cos ^5(c+d x) \sqrt{4 b+a e^{-2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^2} \left(\frac{-4 i a^{5/2} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (c+d x)}\right)^2+4 b e^{2 i (c+d x)}}+a e^{2 i (c+d x)}+a+2 b\right)+4 i a^{5/2} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (c+d x)}\right)^2+4 b e^{2 i (c+d x)}}+a e^{2 i (c+d x)}+a+2 b e^{2 i (c+d x)}\right)+8 a^{5/2} d x-15 a^2 \sqrt{b} \log \left(\frac{4 i d \sqrt{a \left(1+e^{2 i (c+d x)}\right)^2+4 b e^{2 i (c+d x)}}-4 \sqrt{b} d \left(-1+e^{2 i (c+d x)}\right)}{b \left(15 a^2+10 a b+3 b^2\right) \left(1+e^{2 i (c+d x)}\right)}\right)-10 a b^{3/2} \log \left(\frac{4 i d \sqrt{a \left(1+e^{2 i (c+d x)}\right)^2+4 b e^{2 i (c+d x)}}-4 \sqrt{b} d \left(-1+e^{2 i (c+d x)}\right)}{b \left(15 a^2+10 a b+3 b^2\right) \left(1+e^{2 i (c+d x)}\right)}\right)-3 b^{5/2} \log \left(\frac{4 i d \sqrt{a \left(1+e^{2 i (c+d x)}\right)^2+4 b e^{2 i (c+d x)}}-4 \sqrt{b} d \left(-1+e^{2 i (c+d x)}\right)}{b \left(15 a^2+10 a b+3 b^2\right) \left(1+e^{2 i (c+d x)}\right)}\right)}{\sqrt{a \left(1+e^{2 i (c+d x)}\right)^2+4 b e^{2 i (c+d x)}}}-\frac{i b \left(-1+e^{2 i (c+d x)}\right) \left(9 a \left(1+e^{2 i (c+d x)}\right)^2+b \left(14 e^{2 i (c+d x)}+3 e^{4 i (c+d x)}+3\right)\right)}{\left(1+e^{2 i (c+d x)}\right)^4}\right) \left(a+b \sec ^2(c+d x)\right)^{5/2}}{\sqrt{2} d (a \cos (2 c+2 d x)+a+2 b)^{5/2}}","\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{\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{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,"(E^(I*(c + d*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(c + d*x)))^2)/E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*(((-I)*b*(-1 + E^((2*I)*(c + d*x)))*(9*a*(1 + E^((2*I)*(c + d*x)))^2 + b*(3 + 14*E^((2*I)*(c + d*x)) + 3*E^((4*I)*(c + d*x)))))/(1 + E^((2*I)*(c + d*x)))^4 + (8*a^(5/2)*d*x - (4*I)*a^(5/2)*Log[a + 2*b + a*E^((2*I)*(c + d*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(c + d*x)) + a*(1 + E^((2*I)*(c + d*x)))^2]] + (4*I)*a^(5/2)*Log[a + a*E^((2*I)*(c + d*x)) + 2*b*E^((2*I)*(c + d*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(c + d*x)) + a*(1 + E^((2*I)*(c + d*x)))^2]] - 15*a^2*Sqrt[b]*Log[(-4*Sqrt[b]*d*(-1 + E^((2*I)*(c + d*x))) + (4*I)*d*Sqrt[4*b*E^((2*I)*(c + d*x)) + a*(1 + E^((2*I)*(c + d*x)))^2])/(b*(15*a^2 + 10*a*b + 3*b^2)*(1 + E^((2*I)*(c + d*x))))] - 10*a*b^(3/2)*Log[(-4*Sqrt[b]*d*(-1 + E^((2*I)*(c + d*x))) + (4*I)*d*Sqrt[4*b*E^((2*I)*(c + d*x)) + a*(1 + E^((2*I)*(c + d*x)))^2])/(b*(15*a^2 + 10*a*b + 3*b^2)*(1 + E^((2*I)*(c + d*x))))] - 3*b^(5/2)*Log[(-4*Sqrt[b]*d*(-1 + E^((2*I)*(c + d*x))) + (4*I)*d*Sqrt[4*b*E^((2*I)*(c + d*x)) + a*(1 + E^((2*I)*(c + d*x)))^2])/(b*(15*a^2 + 10*a*b + 3*b^2)*(1 + E^((2*I)*(c + d*x))))])/Sqrt[4*b*E^((2*I)*(c + d*x)) + a*(1 + E^((2*I)*(c + d*x)))^2])*(a + b*Sec[c + d*x]^2)^(5/2))/(Sqrt[2]*d*(a + 2*b + a*Cos[2*c + 2*d*x])^(5/2))","C",0
255,1,109,42,0.1749715,"\int \left(1+\sec ^2(x)\right)^{3/2} \, dx","Integrate[(1 + Sec[x]^2)^(3/2),x]","\frac{\left(\cos ^2(x)+1\right) \sec (x) \sqrt{\sec ^2(x)+1} \left(\sin (x) \sqrt{\cos (2 x)+3}-2 i \sqrt{2} \cos ^2(x) \log \left(\sqrt{\cos (2 x)+3}+i \sqrt{2} \sin (x)\right)+4 \sqrt{2} \cos ^2(x) \tanh ^{-1}\left(\frac{\sqrt{2} \sin (x)}{\sqrt{\cos (2 x)+3}}\right)\right)}{(\cos (2 x)+3)^{3/2}}","\frac{1}{2} \tan (x) \sqrt{\tan ^2(x)+2}+\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{\tan ^2(x)+2}}\right)+2 \sinh ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)",1,"((1 + Cos[x]^2)*Sec[x]*Sqrt[1 + Sec[x]^2]*(4*Sqrt[2]*ArcTanh[(Sqrt[2]*Sin[x])/Sqrt[3 + Cos[2*x]]]*Cos[x]^2 - (2*I)*Sqrt[2]*Cos[x]^2*Log[Sqrt[3 + Cos[2*x]] + I*Sqrt[2]*Sin[x]] + Sqrt[3 + Cos[2*x]]*Sin[x]))/(3 + Cos[2*x])^(3/2)","C",1
256,1,57,24,0.0493953,"\int \sqrt{1+\sec ^2(x)} \, dx","Integrate[Sqrt[1 + Sec[x]^2],x]","\frac{\sqrt{2} \cos (x) \sqrt{\sec ^2(x)+1} \left(\sin ^{-1}\left(\frac{\sin (x)}{\sqrt{2}}\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sin (x)}{\sqrt{\cos (2 x)+3}}\right)\right)}{\sqrt{\cos (2 x)+3}}","\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{\tan ^2(x)+2}}\right)+\sinh ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)",1,"(Sqrt[2]*(ArcSin[Sin[x]/Sqrt[2]] + ArcTanh[(Sqrt[2]*Sin[x])/Sqrt[3 + Cos[2*x]]])*Cos[x]*Sqrt[1 + Sec[x]^2])/Sqrt[3 + Cos[2*x]]","B",1
257,0,0,330,11.4829798,"\int \frac{\sec ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Sec[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2],x]","\int \frac{\sec ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","-\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,"Integrate[Sec[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
258,0,0,170,10.626973,"\int \frac{\sec ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Sec[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2],x]","\int \frac{\sec ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","\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,"Integrate[Sec[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
259,1,69,80,0.1790013,"\int \frac{\sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Sec[e + f*x]/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sec (e+f x) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}} F\left(e+f x\left|\frac{a}{a+b}\right.\right)}{\sqrt{2} f \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[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*EllipticF[e + f*x, a/(a + b)]*Sec[e + f*x])/(Sqrt[2]*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
260,1,279,105,5.1871195,"\int \frac{\cos (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Cos[e + f*x]/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sin (e+f x) \csc (2 (e+f x)) \left(a^2 \sqrt{-\frac{1}{a+b}} \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}} F\left(e+f x\left|\frac{a}{a+b}\right.\right)-2 i \csc (2 (e+f x)) \sqrt{\frac{a \sin ^2(e+f x)}{a+b}} \sqrt{-\frac{a \cos ^2(e+f x)}{b}} \sqrt{a \cos (2 (e+f x))+a+2 b} \left(a F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{a+b}} \sqrt{\cos (2 (e+f x)) a+a+2 b}}{\sqrt{2}}\right)|\frac{a+b}{b}\right)+2 b E\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{a+b}} \sqrt{\cos (2 (e+f x)) a+a+2 b}}{\sqrt{2}}\right)|\frac{a+b}{b}\right)\right)\right)}{\sqrt{2} a^2 f \sqrt{-\frac{1}{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,"(Csc[2*(e + f*x)]*Sin[e + f*x]*(a^2*Sqrt[-(a + b)^(-1)]*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*EllipticF[e + f*x, a/(a + b)] - (2*I)*Sqrt[-((a*Cos[e + f*x]^2)/b)]*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*Csc[2*(e + f*x)]*(2*b*EllipticE[I*ArcSinh[(Sqrt[-(a + b)^(-1)]*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])/Sqrt[2]], (a + b)/b] + a*EllipticF[I*ArcSinh[(Sqrt[-(a + b)^(-1)]*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])/Sqrt[2]], (a + b)/b])*Sqrt[(a*Sin[e + f*x]^2)/(a + b)]))/(Sqrt[2]*a^2*Sqrt[-(a + b)^(-1)]*f*Sqrt[a + b*Sec[e + f*x]^2])","C",1
261,0,0,255,7.3426589,"\int \frac{\cos ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Cos[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2],x]","\int \frac{\cos ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","-\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,"Integrate[Cos[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
262,0,0,345,14.7450639,"\int \frac{\cos ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Cos[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2],x]","\int \frac{\cos ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","\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{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{\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,"Integrate[Cos[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
263,1,326,137,9.5266274,"\int \frac{\sec ^6(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Sec[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{e^{i (e+f x)} \sec (e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(-\frac{\left(3 a^2-2 a b+3 b^2\right) \log \left(\frac{4 i f \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}-4 \sqrt{b} f \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}-\frac{i \sqrt{b} \left(-1+e^{2 i (e+f x)}\right) \left(b \left(14 e^{2 i (e+f x)}+3 e^{4 i (e+f x)}+3\right)-3 a \left(1+e^{2 i (e+f x)}\right)^2\right)}{\left(1+e^{2 i (e+f x)}\right)^4}\right) \sqrt{a \cos (2 e+2 f x)+a+2 b}}{8 \sqrt{2} b^{5/2} f \sqrt{a+b \sec ^2(e+f 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}",1,"(E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*(((-I)*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*(-3*a*(1 + E^((2*I)*(e + f*x)))^2 + b*(3 + 14*E^((2*I)*(e + f*x)) + 3*E^((4*I)*(e + f*x)))))/(1 + E^((2*I)*(e + f*x)))^4 - ((3*a^2 - 2*a*b + 3*b^2)*Log[(-4*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*f + (4*I)*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]*f)/(1 + E^((2*I)*(e + f*x)))])/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*Sec[e + f*x])/(8*Sqrt[2]*b^(5/2)*f*Sqrt[a + b*Sec[e + f*x]^2])","C",1
264,1,326,81,10.277899,"\int \frac{\sec ^4(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Sec[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\tan (e+f x) \sec ^4(e+f x) \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right) \sqrt{a \cos (2 e+2 f x)+a+2 b} \left(\frac{16 b^2 \tan ^4(e+f x) \left(a \cos ^2(e+f x)+b\right) \, _2F_1\left(2,3;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right) \sqrt{-\frac{b \tan ^2(e+f x) \sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{(a+b)^2}}}{(a+b)^3}+\frac{15 \left(a \left(3-2 \sin ^2(e+f x)\right)+3 b\right) \left(\sin ^{-1}\left(\sqrt{-\frac{b \tan ^2(e+f x)}{a+b}}\right)-\sqrt{-\frac{b \tan ^2(e+f x) \sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{(a+b)^2}}\right)}{a+b}\right)}{30 \sqrt{2} f \sqrt{-a \sin ^2(e+f x)+a+b} \left(-\frac{b \tan ^2(e+f x)}{a+b}\right)^{3/2} \sqrt{a+b \sec ^2(e+f x)} \sqrt{\frac{a+b \sec ^2(e+f x)}{a+b}}}","\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,"(Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sec[e + f*x]^4*(1 - (a*Sin[e + f*x]^2)/(a + b))*Tan[e + f*x]*((16*b^2*(b + a*Cos[e + f*x]^2)*Hypergeometric2F1[2, 3, 7/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^4*Sqrt[-((b*Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)*Tan[e + f*x]^2)/(a + b)^2)])/(a + b)^3 + (15*(3*b + a*(3 - 2*Sin[e + f*x]^2))*(ArcSin[Sqrt[-((b*Tan[e + f*x]^2)/(a + b))]] - Sqrt[-((b*Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)*Tan[e + f*x]^2)/(a + b)^2)]))/(a + b)))/(30*Sqrt[2]*f*Sqrt[a + b*Sec[e + f*x]^2]*Sqrt[(a + b*Sec[e + f*x]^2)/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2]*(-((b*Tan[e + f*x]^2)/(a + b)))^(3/2))","C",0
265,1,87,39,0.1242585,"\int \frac{\sec ^2(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Sec[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sec (e+f x) \sqrt{a \cos (2 e+2 f x)+a+2 b} \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{2} \sqrt{b} f \sqrt{a+b \sec ^2(e+f 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}",1,"(ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sec[e + f*x])/(Sqrt[2]*Sqrt[b]*f*Sqrt[a + b*Sec[e + f*x]^2])","B",1
266,1,87,39,0.0792437,"\int \frac{1}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[1/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sec (e+f x) \sqrt{a \cos (2 e+2 f x)+a+2 b} \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{2} \sqrt{a} f \sqrt{a+b \sec ^2(e+f 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}",1,"(ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sec[e + f*x])/(Sqrt[2]*Sqrt[a]*f*Sqrt[a + b*Sec[e + f*x]^2])","B",1
267,1,126,87,0.2388346,"\int \frac{\cos ^2(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Cos[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sqrt{a \cos (2 (e+f x))+a+2 b} \left(\sqrt{a} \tan (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b}+(a-b) \sec (e+f x) \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)\right)}{2 \sqrt{2} a^{3/2} f \sqrt{a+b \sec ^2(e+f 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}",1,"(Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*((a - b)*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]]*Sec[e + f*x] + Sqrt[a]*Sqrt[a + b - a*Sin[e + f*x]^2]*Tan[e + f*x]))/(2*Sqrt[2]*a^(3/2)*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
268,1,1840,143,16.2282306,"\int \frac{\cos ^4(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Cos[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{3 (a+b) F_1\left(\frac{1}{2};-2,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^8(e+f x) \sin (e+f x)}{f \sqrt{\cos (2 (e+f x)) a+a+2 b} \sqrt{b \sec ^2(e+f x)+a} \left(\left(a F_1\left(\frac{3}{2};-2,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \left(\frac{3 (a+b) F_1\left(\frac{1}{2};-2,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^5(e+f x)}{\sqrt{\cos (2 (e+f x)) a+a+2 b} \left(\left(a F_1\left(\frac{3}{2};-2,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};-2,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin (e+f x) \left(\left(a \left(\frac{9 a f F_1\left(\frac{5}{2};-2,\frac{5}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{5 (a+b)}-\frac{12}{5} f F_1\left(\frac{5}{2};-1,\frac{3}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)-4 (a+b) \left(\frac{3 a f F_1\left(\frac{5}{2};-1,\frac{3}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{5 (a+b)}-\frac{9 (a+b)^3 f \cot (e+f x) \csc ^4(e+f x) \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \left(-\frac{4 a^2 \sin ^4(e+f x)}{3 (a+b)^2}-\frac{2 a \sin ^2(e+f x)}{a+b}+\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) \sin (e+f x)}{\sqrt{a+b} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}\right)}{8 a^3}\right)\right) \sin ^2(e+f x)+2 f \left(a F_1\left(\frac{3}{2};-2,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \cos (e+f x) \sin (e+f x)+3 (a+b) \left(\frac{a f F_1\left(\frac{3}{2};-2,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac{4}{3} f F_1\left(\frac{3}{2};-1,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)\right) \cos ^4(e+f x)}{f \sqrt{\cos (2 (e+f x)) a+a+2 b} \left(\left(a F_1\left(\frac{3}{2};-2,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right){}^2}+\frac{3 a (a+b) F_1\left(\frac{1}{2};-2,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin (e+f x) \sin (2 (e+f x)) \cos ^4(e+f x)}{(\cos (2 (e+f x)) a+a+2 b)^{3/2} \left(\left(a F_1\left(\frac{3}{2};-2,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}+\frac{3 (a+b) \sin (e+f x) \left(\frac{a f F_1\left(\frac{3}{2};-2,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac{4}{3} f F_1\left(\frac{3}{2};-1,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right) \cos ^4(e+f x)}{f \sqrt{\cos (2 (e+f x)) a+a+2 b} \left(\left(a F_1\left(\frac{3}{2};-2,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{12 (a+b) F_1\left(\frac{1}{2};-2,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin ^2(e+f x) \cos ^3(e+f x)}{\sqrt{\cos (2 (e+f x)) a+a+2 b} \left(\left(a F_1\left(\frac{3}{2};-2,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}\right)}","\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{\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{\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)*AppellF1[1/2, -2, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^8*Sin[e + f*x])/(f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*Sqrt[a + b*Sec[e + f*x]^2]*(3*(a + b)*AppellF1[1/2, -2, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)*((3*(a + b)*AppellF1[1/2, -2, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^5)/(Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(3*(a + b)*AppellF1[1/2, -2, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (12*(a + b)*AppellF1[1/2, -2, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^3*Sin[e + f*x]^2)/(Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(3*(a + b)*AppellF1[1/2, -2, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*Cos[e + f*x]^4*Sin[e + f*x]*((a*f*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - (4*f*AppellF1[3/2, -1, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3))/(f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(3*(a + b)*AppellF1[1/2, -2, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*AppellF1[1/2, -2, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^4*Sin[e + f*x]*(2*f*(a*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Cos[e + f*x]*Sin[e + f*x] + 3*(a + b)*((a*f*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - (4*f*AppellF1[3/2, -1, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3) + Sin[e + f*x]^2*(a*((9*a*f*AppellF1[5/2, -2, 5/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(5*(a + b)) - (12*f*AppellF1[5/2, -1, 3/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5) - 4*(a + b)*((3*a*f*AppellF1[5/2, -1, 3/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(5*(a + b)) - (9*(a + b)^3*f*Cot[e + f*x]*Csc[e + f*x]^4*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]*((-2*a*Sin[e + f*x]^2)/(a + b) - (4*a^2*Sin[e + f*x]^4)/(3*(a + b)^2) + (2*Sqrt[a]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/(Sqrt[a + b]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])))/(8*a^3)))))/(f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(3*(a + b)*AppellF1[1/2, -2, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)^2) + (3*a*(a + b)*AppellF1[1/2, -2, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^4*Sin[e + f*x]*Sin[2*(e + f*x)])/((a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*(3*(a + b)*AppellF1[1/2, -2, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2))))","C",0
269,1,1739,204,16.8087848,"\int \frac{\cos ^6(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Cos[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{3 (a+b) F_1\left(\frac{1}{2};-3,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^{12}(e+f x) \sin (e+f x)}{f \sqrt{\cos (2 (e+f x)) a+a+2 b} \sqrt{b \sec ^2(e+f x)+a} \left(\left(a F_1\left(\frac{3}{2};-3,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \left(\frac{3 (a+b) F_1\left(\frac{1}{2};-3,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^7(e+f x)}{\sqrt{\cos (2 (e+f x)) a+a+2 b} \left(\left(a F_1\left(\frac{3}{2};-3,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};-3,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin (e+f x) \left(\left(a \left(\frac{9 a f F_1\left(\frac{5}{2};-3,\frac{5}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{5 (a+b)}-\frac{18}{5} f F_1\left(\frac{5}{2};-2,\frac{3}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)-6 (a+b) \left(\frac{3 a f F_1\left(\frac{5}{2};-2,\frac{3}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{5 (a+b)}-\frac{12}{5} f F_1\left(\frac{5}{2};-1,\frac{1}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)\right) \sin ^2(e+f x)+2 f \left(a F_1\left(\frac{3}{2};-3,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \cos (e+f x) \sin (e+f x)+3 (a+b) \left(\frac{a f F_1\left(\frac{3}{2};-3,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-2 f F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)\right) \cos ^6(e+f x)}{f \sqrt{\cos (2 (e+f x)) a+a+2 b} \left(\left(a F_1\left(\frac{3}{2};-3,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right){}^2}+\frac{3 a (a+b) F_1\left(\frac{1}{2};-3,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin (e+f x) \sin (2 (e+f x)) \cos ^6(e+f x)}{(\cos (2 (e+f x)) a+a+2 b)^{3/2} \left(\left(a F_1\left(\frac{3}{2};-3,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}+\frac{3 (a+b) \sin (e+f x) \left(\frac{a f F_1\left(\frac{3}{2};-3,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-2 f F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right) \cos ^6(e+f x)}{f \sqrt{\cos (2 (e+f x)) a+a+2 b} \left(\left(a F_1\left(\frac{3}{2};-3,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{18 (a+b) F_1\left(\frac{1}{2};-3,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin ^2(e+f x) \cos ^5(e+f x)}{\sqrt{\cos (2 (e+f x)) a+a+2 b} \left(\left(a F_1\left(\frac{3}{2};-3,\frac{3}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{1}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{1}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}\right)}","\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{(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{\sin (e+f x) \cos ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{6 a f}",1,"(3*(a + b)*AppellF1[1/2, -3, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^12*Sin[e + f*x])/(f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*Sqrt[a + b*Sec[e + f*x]^2]*(3*(a + b)*AppellF1[1/2, -3, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)*((3*(a + b)*AppellF1[1/2, -3, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^7)/(Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(3*(a + b)*AppellF1[1/2, -3, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (18*(a + b)*AppellF1[1/2, -3, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^5*Sin[e + f*x]^2)/(Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(3*(a + b)*AppellF1[1/2, -3, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*Cos[e + f*x]^6*Sin[e + f*x]*((a*f*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - 2*f*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x]))/(f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(3*(a + b)*AppellF1[1/2, -3, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*AppellF1[1/2, -3, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^6*Sin[e + f*x]*(2*f*(a*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Cos[e + f*x]*Sin[e + f*x] + 3*(a + b)*((a*f*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - 2*f*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x]) + Sin[e + f*x]^2*(a*((9*a*f*AppellF1[5/2, -3, 5/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(5*(a + b)) - (18*f*AppellF1[5/2, -2, 3/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5) - 6*(a + b)*((3*a*f*AppellF1[5/2, -2, 3/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(5*(a + b)) - (12*f*AppellF1[5/2, -1, 1/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5))))/(f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(3*(a + b)*AppellF1[1/2, -3, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)^2) + (3*a*(a + b)*AppellF1[1/2, -3, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^6*Sin[e + f*x]*Sin[2*(e + f*x)])/((a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*(3*(a + b)*AppellF1[1/2, -3, 1/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 1/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2))))","C",0
270,0,0,289,20.8169852,"\int \frac{\sec ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]","\int \frac{\sec ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","\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,"Integrate[Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2), x]","F",-1
271,1,113,150,2.349584,"\int \frac{\sec ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sec[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\sqrt{2} (a+b) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}} E\left(e+f x\left|\frac{a}{a+b}\right.\right)-a \sin (2 (e+f x))\right)}{4 b f (a+b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}","\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 + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^3*(Sqrt[2]*(a + b)*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*EllipticE[e + f*x, a/(a + b)] - a*Sin[2*(e + f*x)]))/(4*b*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2))","A",1
272,1,822,229,9.6013988,"\int \frac{\sec (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sec[e + f*x]/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{(\cos (2 e+2 f x) a+a+2 b)^{3/2} \sec ^3(e+f x) \left(\frac{\sqrt{-\frac{1}{a+b}} \cos (2 (e+f x)) \left(\sqrt{-\frac{1}{a+b}} (\cos (2 e+2 f x) a+a) \left(-2 a^2+(\cos (2 e+2 f x) a+a-4 b) a+2 b (\cos (2 e+2 f x) a+a)\right)-i b (a+2 b) \sqrt{\frac{a-a \cos (2 e+2 f x)}{a+b}} \sqrt{\cos (2 e+2 f x) a+a+2 b} \sqrt{4-\frac{2 (\cos (2 e+2 f x) a+a+2 b)}{b}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{a+b}} \sqrt{\cos (2 e+2 f x) a+a+2 b}}{\sqrt{2}}\right)|\frac{a+b}{b}\right)-i a b \sqrt{\cos (2 e+2 f x) a+a+2 b} \sqrt{\frac{4 a+4 b-2 (\cos (2 e+2 f x) a+a+2 b)}{a+b}} \sqrt{2-\frac{\cos (2 e+2 f x) a+a+2 b}{b}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{a+b}} \sqrt{\cos (2 e+2 f x) a+a+2 b}}{\sqrt{2}}\right)|\frac{a+b}{b}\right)\right) \sec \left(2 \left(e+\frac{1}{2} \left(\cos ^{-1}(\cos (2 e+2 f x))-2 e\right)\right)\right) \sin (2 e+2 f x)}{4 a^2 b f \sqrt{\frac{(a-a \cos (2 e+2 f x)) (\cos (2 e+2 f x) a+a)}{a^2}} \sqrt{\cos (2 e+2 f x) a+a+2 b} \sqrt{1-\cos ^2(2 e+2 f x)}}-\frac{\left(\frac{2 (a-a \cos (2 e+2 f x)) (\cos (2 e+2 f x) a+a)}{b \sqrt{\cos (2 e+2 f x) a+a+2 b}}+\frac{2 i \sqrt{\frac{a-a \cos (2 e+2 f x)}{a+b}} \sqrt{4-\frac{2 (\cos (2 e+2 f x) a+a+2 b)}{b}} \left(E\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{a+b}} \sqrt{\cos (2 e+2 f x) a+a+2 b}}{\sqrt{2}}\right)|\frac{a+b}{b}\right)-F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{a+b}} \sqrt{\cos (2 e+2 f x) a+a+2 b}}{\sqrt{2}}\right)|\frac{a+b}{b}\right)\right)}{\sqrt{-\frac{1}{a+b}}}\right) \sin (2 e+2 f x)}{8 a (a+b) f \sqrt{\frac{(a-a \cos (2 e+2 f x)) (\cos (2 e+2 f x) a+a)}{a^2}} \sqrt{1-\cos ^2(2 e+2 f x)}}\right)}{2 \left(b \sec ^2(e+f x)+a\right)^{3/2}}","\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,"((a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sec[e + f*x]^3*(-1/8*(((2*(a - a*Cos[2*e + 2*f*x])*(a + a*Cos[2*e + 2*f*x]))/(b*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]) + ((2*I)*Sqrt[(a - a*Cos[2*e + 2*f*x])/(a + b)]*Sqrt[4 - (2*(a + 2*b + a*Cos[2*e + 2*f*x]))/b]*(EllipticE[I*ArcSinh[(Sqrt[-(a + b)^(-1)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])/Sqrt[2]], (a + b)/b] - EllipticF[I*ArcSinh[(Sqrt[-(a + b)^(-1)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])/Sqrt[2]], (a + b)/b]))/Sqrt[-(a + b)^(-1)])*Sin[2*e + 2*f*x])/(a*(a + b)*f*Sqrt[((a - a*Cos[2*e + 2*f*x])*(a + a*Cos[2*e + 2*f*x]))/a^2]*Sqrt[1 - Cos[2*e + 2*f*x]^2]) + (Sqrt[-(a + b)^(-1)]*Cos[2*(e + f*x)]*(Sqrt[-(a + b)^(-1)]*(a + a*Cos[2*e + 2*f*x])*(-2*a^2 + 2*b*(a + a*Cos[2*e + 2*f*x]) + a*(a - 4*b + a*Cos[2*e + 2*f*x])) - I*b*(a + 2*b)*Sqrt[(a - a*Cos[2*e + 2*f*x])/(a + b)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sqrt[4 - (2*(a + 2*b + a*Cos[2*e + 2*f*x]))/b]*EllipticE[I*ArcSinh[(Sqrt[-(a + b)^(-1)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])/Sqrt[2]], (a + b)/b] - I*a*b*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sqrt[(4*a + 4*b - 2*(a + 2*b + a*Cos[2*e + 2*f*x]))/(a + b)]*Sqrt[2 - (a + 2*b + a*Cos[2*e + 2*f*x])/b]*EllipticF[I*ArcSinh[(Sqrt[-(a + b)^(-1)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])/Sqrt[2]], (a + b)/b])*Sec[2*(e + (-2*e + ArcCos[Cos[2*e + 2*f*x]])/2)]*Sin[2*e + 2*f*x])/(4*a^2*b*f*Sqrt[((a - a*Cos[2*e + 2*f*x])*(a + a*Cos[2*e + 2*f*x]))/a^2]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sqrt[1 - Cos[2*e + 2*f*x]^2])))/(2*(a + b*Sec[e + f*x]^2)^(3/2))","C",1
273,0,0,240,14.3707431,"\int \frac{\cos (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cos[e + f*x]/(a + b*Sec[e + f*x]^2)^(3/2),x]","\int \frac{\cos (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","-\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,"Integrate[Cos[e + f*x]/(a + b*Sec[e + f*x]^2)^(3/2), x]","F",-1
274,0,0,335,16.2911003,"\int \frac{\cos ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2),x]","\int \frac{\cos ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","-\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{(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{\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{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,"Integrate[Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2), x]","F",-1
275,0,0,436,15.4659591,"\int \frac{\cos ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]","\int \frac{\cos ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","\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{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(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{\left(8 a^3-9 a^2 b+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{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,"Integrate[Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2), x]","F",-1
276,1,260,138,7.6975702,"\int \frac{\sec ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sec[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\tan (e+f x) \sec ^8(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(4 b \sin ^2(e+f x) (a \cos (2 (e+f x))+a+2 b)^2 \, _3F_2\left(2,2,3;1,\frac{9}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)-7 (a+b) \cos ^2(e+f x) \left(a^2 \cos (4 (e+f x))+8 a^2+2 a (3 a+5 b) \cos (2 (e+f x))+20 a b+15 b^2\right) \, _2F_1\left(1,2;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)+16 b \sin ^2(e+f x) \left(3 a^2 \sin ^4(e+f x)-7 a (a+b) \sin ^2(e+f x)+4 (a+b)^2\right) \, _2F_1\left(2,3;\frac{9}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)\right)}{210 f (a+b)^4 \left(a+b \sec ^2(e+f x)\right)^{3/2}}","-\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{(3 a+b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 b^2 f (a+b)}-\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,"-1/210*((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^8*(-7*(a + b)*Cos[e + f*x]^2*(8*a^2 + 20*a*b + 15*b^2 + 2*a*(3*a + 5*b)*Cos[2*(e + f*x)] + a^2*Cos[4*(e + f*x)])*Hypergeometric2F1[1, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b))] + 4*b*(a + 2*b + a*Cos[2*(e + f*x)])^2*HypergeometricPFQ[{2, 2, 3}, {1, 9/2}, -((b*Tan[e + f*x]^2)/(a + b))]*Sin[e + f*x]^2 + 16*b*Hypergeometric2F1[2, 3, 9/2, -((b*Tan[e + f*x]^2)/(a + b))]*Sin[e + f*x]^2*(4*(a + b)^2 - 7*a*(a + b)*Sin[e + f*x]^2 + 3*a^2*Sin[e + f*x]^4))*Tan[e + f*x])/((a + b)^4*f*(a + b*Sec[e + f*x]^2)^(3/2))","C",0
277,1,405,77,7.9224529,"\int \frac{\sec ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sec[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\tan (e+f x) \sec ^4(e+f x) \sqrt{1-\frac{2 a \sin ^2(e+f x)}{2 a+2 b}} (a \cos (2 e+2 f x)+a+2 b)^{3/2} \left(15 \sec ^2(e+f x) \left(a^2 \left(2 \sin ^4(e+f x)-5 \sin ^2(e+f x)+3\right)+a b \left(6-5 \sin ^2(e+f x)\right)+3 b^2\right) \sin ^{-1}\left(\sqrt{-\frac{b \tan ^2(e+f x)}{a+b}}\right)+4 b (a+b) \sin ^2(e+f x) \, _2F_1\left(2,2;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right) \left(-\frac{b \tan ^2(e+f x) \sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{(a+b)^2}\right)^{3/2}+15 (a+b) \left(a \left(2 \sin ^2(e+f x)-3\right)-3 b\right) \sqrt{-\frac{b \tan ^2(e+f x) \sec ^2(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)}{(a+b)^2}}\right)}{15 f (a+b)^2 (2 a+2 b) \sqrt{-2 a \sin ^2(e+f x)+2 a+2 b} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}} \left(-\frac{b \tan ^2(e+f x)}{a+b}\right)^{3/2} \left(a+b \sec ^2(e+f x)\right)^{3/2} \sqrt{\frac{a+b \sec ^2(e+f x)}{a+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,"-1/15*((a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sec[e + f*x]^4*Sqrt[1 - (2*a*Sin[e + f*x]^2)/(2*a + 2*b)]*Tan[e + f*x]*(15*ArcSin[Sqrt[-((b*Tan[e + f*x]^2)/(a + b))]]*Sec[e + f*x]^2*(3*b^2 + a*b*(6 - 5*Sin[e + f*x]^2) + a^2*(3 - 5*Sin[e + f*x]^2 + 2*Sin[e + f*x]^4)) + 15*(a + b)*(-3*b + a*(-3 + 2*Sin[e + f*x]^2))*Sqrt[-((b*Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)*Tan[e + f*x]^2)/(a + b)^2)] + 4*b*(a + b)*Hypergeometric2F1[2, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b))]*Sin[e + f*x]^2*(-((b*Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)*Tan[e + f*x]^2)/(a + b)^2))^(3/2)))/((a + b)^2*(2*a + 2*b)*f*(a + b*Sec[e + f*x]^2)^(3/2)*Sqrt[(a + b*Sec[e + f*x]^2)/(a + b)]*Sqrt[2*a + 2*b - 2*a*Sin[e + f*x]^2]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]*(-((b*Tan[e + f*x]^2)/(a + b)))^(3/2))","C",0
278,1,57,32,0.6267682,"\int \frac{\sec ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\tan (e+f x) \sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b)}{2 f (a+b) \left(a+b \sec ^2(e+f x)\right)^{3/2}}","\frac{\tan (e+f x)}{f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}",1,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*Tan[e + f*x])/(2*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2))","A",1
279,1,168,77,1.3818303,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(-3/2),x]","\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\sqrt{a+b} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) (a \cos (2 (e+f x))+a+2 b)-\sqrt{2} \sqrt{a} b \sin (e+f x) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}}\right)}{4 a^{3/2} f (a+b) \sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}} \left(a+b \sec ^2(e+f x)\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^{3/2} f}-\frac{b \tan (e+f x)}{a f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}",1,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^3*(Sqrt[a + b]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*(a + 2*b + a*Cos[2*(e + f*x)]) - Sqrt[2]*Sqrt[a]*b*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*Sin[e + f*x]))/(4*a^(3/2)*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)*Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)])","B",1
280,1,2059,131,15.3245279,"\int \frac{\cos ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2),x]","\text{Result too large to show}","\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,"(3*(a + b)*AppellF1[1/2, -2, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^6*Sin[e + f*x])/(2*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b*Sec[e + f*x]^2)^(3/2)*(a + b - a*Sin[e + f*x]^2)*(3*(a + b)*AppellF1[1/2, -2, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (3*a*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)*((3*a*(a + b)*AppellF1[1/2, -2, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^5*Sin[e + f*x]^2)/(Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b - a*Sin[e + f*x]^2)^2*(3*(a + b)*AppellF1[1/2, -2, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (3*a*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*AppellF1[1/2, -2, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^5)/(2*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b - a*Sin[e + f*x]^2)*(3*(a + b)*AppellF1[1/2, -2, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (3*a*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (6*(a + b)*AppellF1[1/2, -2, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^3*Sin[e + f*x]^2)/(Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b - a*Sin[e + f*x]^2)*(3*(a + b)*AppellF1[1/2, -2, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (3*a*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*Cos[e + f*x]^4*Sin[e + f*x]*((a*f*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - (4*f*AppellF1[3/2, -1, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3))/(2*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b - a*Sin[e + f*x]^2)*(3*(a + b)*AppellF1[1/2, -2, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (3*a*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*AppellF1[1/2, -2, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^4*Sin[e + f*x]*(2*f*(3*a*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Cos[e + f*x]*Sin[e + f*x] + 3*(a + b)*((a*f*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - (4*f*AppellF1[3/2, -1, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3) + Sin[e + f*x]^2*(3*a*((3*a*f*AppellF1[5/2, -2, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - (12*f*AppellF1[5/2, -1, 5/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5) - 4*(a + b)*((9*a*f*AppellF1[5/2, -1, 5/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(5*(a + b)) - (6*f*Cos[e + f*x]*Hypergeometric2F1[3/2, 5/2, 7/2, (a*Sin[e + f*x]^2)/(a + b)]*Sin[e + f*x])/5))))/(2*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b - a*Sin[e + f*x]^2)*(3*(a + b)*AppellF1[1/2, -2, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (3*a*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)^2) + (3*a*(a + b)*AppellF1[1/2, -2, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^4*Sin[e + f*x]*Sin[2*(e + f*x)])/(2*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*(a + b - a*Sin[e + f*x]^2)*(3*(a + b)*AppellF1[1/2, -2, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (3*a*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2))))","C",0
281,1,2046,194,16.6682068,"\int \frac{\cos ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cos[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2),x]","\text{Result too large to show}","\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{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{\sin (e+f x) \cos ^3(e+f x)}{4 a f \sqrt{a+b \tan ^2(e+f x)+b}}",1,"((a + b)*AppellF1[1/2, -3, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^10*Sin[e + f*x])/(2*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b*Sec[e + f*x]^2)^(3/2)*(a + b - a*Sin[e + f*x]^2)*((a + b)*AppellF1[1/2, -3, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)*((a*(a + b)*AppellF1[1/2, -3, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^7*Sin[e + f*x]^2)/(Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b - a*Sin[e + f*x]^2)^2*((a + b)*AppellF1[1/2, -3, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + ((a + b)*AppellF1[1/2, -3, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^7)/(2*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b - a*Sin[e + f*x]^2)*((a + b)*AppellF1[1/2, -3, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*AppellF1[1/2, -3, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^5*Sin[e + f*x]^2)/(Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b - a*Sin[e + f*x]^2)*((a + b)*AppellF1[1/2, -3, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + ((a + b)*Cos[e + f*x]^6*Sin[e + f*x]*((a*f*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - 2*f*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x]))/(2*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b - a*Sin[e + f*x]^2)*((a + b)*AppellF1[1/2, -3, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - ((a + b)*AppellF1[1/2, -3, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^6*Sin[e + f*x]*(2*f*(a*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Cos[e + f*x]*Sin[e + f*x] + (a + b)*((a*f*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - 2*f*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x]) + Sin[e + f*x]^2*(a*((3*a*f*AppellF1[5/2, -3, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - (18*f*AppellF1[5/2, -2, 5/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5) - 2*(a + b)*((9*a*f*AppellF1[5/2, -2, 5/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(5*(a + b)) - (12*f*AppellF1[5/2, -1, 3/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5))))/(2*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b - a*Sin[e + f*x]^2)*((a + b)*AppellF1[1/2, -3, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)^2) + (a*(a + b)*AppellF1[1/2, -3, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^6*Sin[e + f*x]*Sin[2*(e + f*x)])/(2*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*(a + b - a*Sin[e + f*x]^2)*((a + b)*AppellF1[1/2, -3, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -2, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2))))","C",0
282,1,2068,271,19.7251929,"\int \frac{\cos ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2),x]","\text{Result too large to show}","\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{\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{\left(5 a^3-9 a^2 b+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(15 a^3-17 a^2 b+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{\sin (e+f x) \cos ^5(e+f x)}{6 a f \sqrt{a+b \tan ^2(e+f x)+b}}",1,"(3*(a + b)*AppellF1[1/2, -4, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^14*Sin[e + f*x])/(2*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b*Sec[e + f*x]^2)^(3/2)*(a + b - a*Sin[e + f*x]^2)*(3*(a + b)*AppellF1[1/2, -4, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (3*a*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 8*(a + b)*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)*((3*a*(a + b)*AppellF1[1/2, -4, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^9*Sin[e + f*x]^2)/(Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b - a*Sin[e + f*x]^2)^2*(3*(a + b)*AppellF1[1/2, -4, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (3*a*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 8*(a + b)*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*AppellF1[1/2, -4, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^9)/(2*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b - a*Sin[e + f*x]^2)*(3*(a + b)*AppellF1[1/2, -4, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (3*a*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 8*(a + b)*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (12*(a + b)*AppellF1[1/2, -4, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^7*Sin[e + f*x]^2)/(Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b - a*Sin[e + f*x]^2)*(3*(a + b)*AppellF1[1/2, -4, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (3*a*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 8*(a + b)*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*Cos[e + f*x]^8*Sin[e + f*x]*((a*f*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - (8*f*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3))/(2*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b - a*Sin[e + f*x]^2)*(3*(a + b)*AppellF1[1/2, -4, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (3*a*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 8*(a + b)*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*AppellF1[1/2, -4, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^8*Sin[e + f*x]*(2*f*(3*a*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 8*(a + b)*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Cos[e + f*x]*Sin[e + f*x] + 3*(a + b)*((a*f*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - (8*f*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3) + Sin[e + f*x]^2*(3*a*((3*a*f*AppellF1[5/2, -4, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - (24*f*AppellF1[5/2, -3, 5/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5) - 8*(a + b)*((9*a*f*AppellF1[5/2, -3, 5/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(5*(a + b)) - (18*f*AppellF1[5/2, -2, 3/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5))))/(2*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(a + b - a*Sin[e + f*x]^2)*(3*(a + b)*AppellF1[1/2, -4, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (3*a*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 8*(a + b)*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)^2) + (3*a*(a + b)*AppellF1[1/2, -4, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^8*Sin[e + f*x]*Sin[2*(e + f*x)])/(2*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*(a + b - a*Sin[e + f*x]^2)*(3*(a + b)*AppellF1[1/2, -4, 3/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (3*a*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 8*(a + b)*AppellF1[3/2, -3, 3/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2))))","C",0
283,1,167,321,2.7634056,"\int \frac{\sec ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\sec ^5(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\sqrt{2} (a+b)^2 \left(\frac{a \cos (2 (e+f x))+a+2 b}{a+b}\right)^{3/2} \left(2 (a+2 b) E\left(e+f x\left|\frac{a}{a+b}\right.\right)-b F\left(e+f x\left|\frac{a}{a+b}\right.\right)\right)-2 a \sin (2 (e+f x)) \left(a^2+a (a+2 b) \cos (2 (e+f x))+5 a b+5 b^2\right)\right)}{24 b^2 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^{5/2}}","-\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 + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^5*(Sqrt[2]*(a + b)^2*((a + 2*b + a*Cos[2*(e + f*x)])/(a + b))^(3/2)*(2*(a + 2*b)*EllipticE[e + f*x, a/(a + b)] - b*EllipticF[e + f*x, a/(a + b)]) - 2*a*(a^2 + 5*a*b + 5*b^2 + a*(a + 2*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)]))/(24*b^2*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^(5/2))","A",1
284,1,1156,319,10.1646913,"\int \frac{\sec ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sec[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{(\cos (2 e+2 f x) a+a+2 b)^{5/2} \sec ^5(e+f x) \left(\frac{\cos (2 (e+f x)) \left(2 i b \left(a^2+b a+b^2\right) \sqrt{\frac{a-a \cos (2 e+2 f x)}{a+b}} \sqrt{4-\frac{2 (\cos (2 e+2 f x) a+a+2 b)}{b}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{a+b}} \sqrt{\cos (2 e+2 f x) a+a+2 b}}{\sqrt{2}}\right)|\frac{a+b}{b}\right) (\cos (2 e+2 f x) a+a+2 b)^{3/2}+i a b (b-a) \sqrt{\frac{a-a \cos (2 e+2 f x)}{a+b}} \sqrt{4-\frac{2 (\cos (2 e+2 f x) a+a+2 b)}{b}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{a+b}} \sqrt{\cos (2 e+2 f x) a+a+2 b}}{\sqrt{2}}\right)|\frac{a+b}{b}\right) (\cos (2 e+2 f x) a+a+2 b)^{3/2}-2 \sqrt{-\frac{1}{a+b}} (-\cos (2 e+2 f x) a-a) \left(4 b^4-(\cos (2 e+2 f x) a+a+2 b)^2 b^2+a \left(10 b^2+(\cos (2 e+2 f x) a+a+2 b) b-(\cos (2 e+2 f x) a+a+2 b)^2\right) b+2 a^3 (\cos (2 e+2 f x) a+a+3 b)+a^2 \left(8 b^2+3 (\cos (2 e+2 f x) a+a+2 b) b-(\cos (2 e+2 f x) a+a+2 b)^2\right)\right)\right) \sec \left(2 \left(e+\frac{1}{2} \left(\cos ^{-1}(\cos (2 e+2 f x))-2 e\right)\right)\right) \sin (2 e+2 f x)}{24 a^2 b^2 \sqrt{-\frac{1}{a+b}} (a+b)^2 f \sqrt{\frac{(a-a \cos (2 e+2 f x)) (\cos (2 e+2 f x) a+a)}{a^2}} (\cos (2 e+2 f x) a+a+2 b)^{3/2} \sqrt{1-\cos ^2(2 e+2 f x)}}-\frac{\left(2 i b (a+2 b) \sqrt{\frac{a-a \cos (2 e+2 f x)}{a+b}} \sqrt{4-\frac{2 (\cos (2 e+2 f x) a+a+2 b)}{b}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{a+b}} \sqrt{\cos (2 e+2 f x) a+a+2 b}}{\sqrt{2}}\right)|\frac{a+b}{b}\right) (\cos (2 e+2 f x) a+a+2 b)^{3/2}-i b (a+3 b) \sqrt{\frac{a-a \cos (2 e+2 f x)}{a+b}} \sqrt{4-\frac{2 (\cos (2 e+2 f x) a+a+2 b)}{b}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{a+b}} \sqrt{\cos (2 e+2 f x) a+a+2 b}}{\sqrt{2}}\right)|\frac{a+b}{b}\right) (\cos (2 e+2 f x) a+a+2 b)^{3/2}-2 \sqrt{-\frac{1}{a+b}} (-\cos (2 e+2 f x) a-a) \left(2 (\cos (2 e+2 f x) a+a+3 b) a^2+\left(4 b^2+5 (\cos (2 e+2 f x) a+a+2 b) b-(\cos (2 e+2 f x) a+a+2 b)^2\right) a+b \left(2 b^2+3 (\cos (2 e+2 f x) a+a+2 b) b-2 (\cos (2 e+2 f x) a+a+2 b)^2\right)\right)\right) \sin (2 e+2 f x)}{24 a b^2 \sqrt{-\frac{1}{a+b}} (a+b)^2 f \sqrt{\frac{(a-a \cos (2 e+2 f x)) (\cos (2 e+2 f x) a+a)}{a^2}} (\cos (2 e+2 f x) a+a+2 b)^{3/2} \sqrt{1-\cos ^2(2 e+2 f x)}}\right)}{2 \left(b \sec ^2(e+f x)+a\right)^{5/2}}","-\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,"((a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^5*(-1/24*((-2*Sqrt[-(a + b)^(-1)]*(-a - a*Cos[2*e + 2*f*x])*(2*a^2*(a + 3*b + a*Cos[2*e + 2*f*x]) + b*(2*b^2 + 3*b*(a + 2*b + a*Cos[2*e + 2*f*x]) - 2*(a + 2*b + a*Cos[2*e + 2*f*x])^2) + a*(4*b^2 + 5*b*(a + 2*b + a*Cos[2*e + 2*f*x]) - (a + 2*b + a*Cos[2*e + 2*f*x])^2)) + (2*I)*b*(a + 2*b)*Sqrt[(a - a*Cos[2*e + 2*f*x])/(a + b)]*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sqrt[4 - (2*(a + 2*b + a*Cos[2*e + 2*f*x]))/b]*EllipticE[I*ArcSinh[(Sqrt[-(a + b)^(-1)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])/Sqrt[2]], (a + b)/b] - I*b*(a + 3*b)*Sqrt[(a - a*Cos[2*e + 2*f*x])/(a + b)]*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sqrt[4 - (2*(a + 2*b + a*Cos[2*e + 2*f*x]))/b]*EllipticF[I*ArcSinh[(Sqrt[-(a + b)^(-1)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])/Sqrt[2]], (a + b)/b])*Sin[2*e + 2*f*x])/(a*b^2*Sqrt[-(a + b)^(-1)]*(a + b)^2*f*Sqrt[((a - a*Cos[2*e + 2*f*x])*(a + a*Cos[2*e + 2*f*x]))/a^2]*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sqrt[1 - Cos[2*e + 2*f*x]^2]) + (Cos[2*(e + f*x)]*(-2*Sqrt[-(a + b)^(-1)]*(-a - a*Cos[2*e + 2*f*x])*(4*b^4 - b^2*(a + 2*b + a*Cos[2*e + 2*f*x])^2 + 2*a^3*(a + 3*b + a*Cos[2*e + 2*f*x]) + a*b*(10*b^2 + b*(a + 2*b + a*Cos[2*e + 2*f*x]) - (a + 2*b + a*Cos[2*e + 2*f*x])^2) + a^2*(8*b^2 + 3*b*(a + 2*b + a*Cos[2*e + 2*f*x]) - (a + 2*b + a*Cos[2*e + 2*f*x])^2)) + (2*I)*b*(a^2 + a*b + b^2)*Sqrt[(a - a*Cos[2*e + 2*f*x])/(a + b)]*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sqrt[4 - (2*(a + 2*b + a*Cos[2*e + 2*f*x]))/b]*EllipticE[I*ArcSinh[(Sqrt[-(a + b)^(-1)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])/Sqrt[2]], (a + b)/b] + I*a*b*(-a + b)*Sqrt[(a - a*Cos[2*e + 2*f*x])/(a + b)]*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sqrt[4 - (2*(a + 2*b + a*Cos[2*e + 2*f*x]))/b]*EllipticF[I*ArcSinh[(Sqrt[-(a + b)^(-1)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])/Sqrt[2]], (a + b)/b])*Sec[2*(e + (-2*e + ArcCos[Cos[2*e + 2*f*x]])/2)]*Sin[2*e + 2*f*x])/(24*a^2*b^2*Sqrt[-(a + b)^(-1)]*(a + b)^2*f*Sqrt[((a - a*Cos[2*e + 2*f*x])*(a + a*Cos[2*e + 2*f*x]))/a^2]*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sqrt[1 - Cos[2*e + 2*f*x]^2])))/(2*(a + b*Sec[e + f*x]^2)^(5/2))","C",0
285,0,0,327,12.2762052,"\int \frac{\sec (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sec[e + f*x]/(a + b*Sec[e + f*x]^2)^(5/2),x]","\int \frac{\sec (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","\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,"Integrate[Sec[e + f*x]/(a + b*Sec[e + f*x]^2)^(5/2), x]","F",-1
286,0,0,349,16.0954625,"\int \frac{\cos (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cos[e + f*x]/(a + b*Sec[e + f*x]^2)^(5/2),x]","\int \frac{\cos (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","-\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{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{\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{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,"Integrate[Cos[e + f*x]/(a + b*Sec[e + f*x]^2)^(5/2), x]","F",-1
287,0,0,441,12.1919895,"\int \frac{\cos ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]","\int \frac{\cos ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","-\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 \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{\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 \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,"Integrate[Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2), x]","F",-1
288,0,0,559,22.2832161,"\int \frac{\cos ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2),x]","\int \frac{\cos ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","-\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{\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(4 a^3-9 a^2 b+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{2 \left(2 a^3-3 a^2 b-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(8 a^4-11 a^3 b+27 a^2 b^2+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{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,"Integrate[Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2), x]","F",-1
289,1,357,133,11.7013778,"\int \frac{\sec ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sec[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{e^{i (e+f x)} \sec ^5(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{i a \sqrt{b} \left(-1+e^{2 i (e+f x)}\right) \left(3 a^2 \left(1+e^{2 i (e+f x)}\right)^2+a b \left(26 e^{2 i (e+f x)}+5 e^{4 i (e+f x)}+5\right)+24 b^2 e^{2 i (e+f x)}\right)}{(a+b)^2 \left(a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}\right)^2}-\frac{3 \log \left(\frac{4 i f \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}-4 \sqrt{b} f \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}\right) (a \cos (2 e+2 f x)+a+2 b)^{5/2}}{12 \sqrt{2} b^{5/2} f \left(a+b \sec ^2(e+f x)\right)^{5/2}}","\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 (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{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,"(E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*((I*a*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*(24*b^2*E^((2*I)*(e + f*x)) + 3*a^2*(1 + E^((2*I)*(e + f*x)))^2 + a*b*(5 + 26*E^((2*I)*(e + f*x)) + 5*E^((4*I)*(e + f*x)))))/((a + b)^2*(4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2)^2) - (3*Log[(-4*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*f + (4*I)*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]*f)/(1 + E^((2*I)*(e + f*x)))])/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*Sec[e + f*x]^5)/(12*Sqrt[2]*b^(5/2)*f*(a + b*Sec[e + f*x]^2)^(5/2))","C",1
290,1,74,79,4.2976918,"\int \frac{\sec ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sec[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\tan (e+f x) \sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) (a \cos (2 (e+f x))+2 a+3 b)}{6 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^{5/2}}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])*(2*a + 3*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^4*Tan[e + f*x])/(6*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^(5/2))","A",1
291,1,215,71,6.115899,"\int \frac{\sec ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{(3 a+b) \sec ^5(e+f x) \left(\frac{2 \sqrt{2} \sin (e+f x)}{(a+b)^2 \sqrt{-a \sin ^2(e+f x)+a+b}}+\frac{\sqrt{2} \sin (e+f x)}{(a+b) \left(-a \sin ^2(e+f x)+a+b\right)^{3/2}}\right) (a \cos (2 e+2 f x)+a+2 b)^{5/2}}{48 a f \left(a+b \sec ^2(e+f x)\right)^{5/2}}-\frac{\tan (e+f x) \sec ^4(e+f x) (a \cos (2 e+2 f x)+a+2 b)^{5/2}}{8 \sqrt{2} a f \left(-a \sin ^2(e+f x)+a+b\right)^{3/2} \left(a+b \sec ^2(e+f x)\right)^{5/2}}","\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,"((3*a + b)*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^5*((Sqrt[2]*Sin[e + f*x])/((a + b)*(a + b - a*Sin[e + f*x]^2)^(3/2)) + (2*Sqrt[2]*Sin[e + f*x])/((a + b)^2*Sqrt[a + b - a*Sin[e + f*x]^2])))/(48*a*f*(a + b*Sec[e + f*x]^2)^(5/2)) - ((a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^4*Tan[e + f*x])/(8*Sqrt[2]*a*f*(a + b*Sec[e + f*x]^2)^(5/2)*(a + b - a*Sin[e + f*x]^2)^(3/2))","B",1
292,1,1927,125,6.4950377,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(-5/2),x]","\frac{3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^4(e+f x) \sin (e+f x)}{4 \sqrt{2} f \left(b \sec ^2(e+f x)+a\right)^{5/2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \left(\frac{3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^5(e+f x)}{4 \sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}+\frac{15 a (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin ^2(e+f x) \cos ^5(e+f x)}{4 \sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{7/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin (e+f x) \left(\left(5 a \left(\frac{21 a f F_1\left(\frac{5}{2};-2,\frac{9}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{5 (a+b)}-\frac{12}{5} f F_1\left(\frac{5}{2};-1,\frac{7}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)-4 (a+b) \left(\frac{3 a f F_1\left(\frac{5}{2};-1,\frac{7}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{a+b}-\frac{6 (a+b)^3 f \cot (e+f x) \csc ^4(e+f x) \left(\frac{a \sin ^2(e+f x)}{a+b}-1\right)^2 \left(\frac{a^2 \sin ^4(e+f x)}{3 (a+b)^2 \left(\frac{a \sin ^2(e+f x)}{a+b}-1\right)^2}+\frac{a \sin ^2(e+f x)}{(a+b) \left(\frac{a \sin ^2(e+f x)}{a+b}-1\right)}+\frac{\sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) \sin (e+f x)}{\sqrt{a+b} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}\right)}{a^3 \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)^{3/2}}\right)\right) \sin ^2(e+f x)+2 f \left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \cos (e+f x) \sin (e+f x)+3 (a+b) \left(\frac{5 a f F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac{4}{3} f F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)\right) \cos ^4(e+f x)}{4 \sqrt{2} f \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right){}^2}+\frac{3 (a+b) \sin (e+f x) \left(\frac{5 a f F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac{4}{3} f F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right) \cos ^4(e+f x)}{4 \sqrt{2} f \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin ^2(e+f x) \cos ^3(e+f x)}{\sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}\right)}","\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,"(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^4*Sin[e + f*x])/(4*Sqrt[2]*f*(a + b*Sec[e + f*x]^2)^(5/2)*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)*((15*a*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^5*Sin[e + f*x]^2)/(4*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(7/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^5)/(4*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^3*Sin[e + f*x]^2)/(Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*Cos[e + f*x]^4*Sin[e + f*x]*((5*a*f*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - (4*f*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3))/(4*Sqrt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^4*Sin[e + f*x]*(2*f*(5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Cos[e + f*x]*Sin[e + f*x] + 3*(a + b)*((5*a*f*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - (4*f*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3) + Sin[e + f*x]^2*(5*a*((21*a*f*AppellF1[5/2, -2, 9/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(5*(a + b)) - (12*f*AppellF1[5/2, -1, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5) - 4*(a + b)*((3*a*f*AppellF1[5/2, -1, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - (6*(a + b)^3*f*Cot[e + f*x]*Csc[e + f*x]^4*(-1 + (a*Sin[e + f*x]^2)/(a + b))^2*((Sqrt[a]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/(Sqrt[a + b]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + (a^2*Sin[e + f*x]^4)/(3*(a + b)^2*(-1 + (a*Sin[e + f*x]^2)/(a + b))^2) + (a*Sin[e + f*x]^2)/((a + b)*(-1 + (a*Sin[e + f*x]^2)/(a + b)))))/(a^3*(1 - (a*Sin[e + f*x]^2)/(a + b))^(3/2))))))/(4*Sqrt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)^2)))","C",0
293,1,1775,187,17.3312479,"\int \frac{\cos ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{3 (a+b) F_1\left(\frac{1}{2};-3,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^8(e+f x) \sin (e+f x)}{4 \sqrt{2} f \left(b \sec ^2(e+f x)+a\right)^{5/2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-3,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \left(\frac{3 (a+b) F_1\left(\frac{1}{2};-3,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^7(e+f x)}{4 \sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-3,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}+\frac{15 a (a+b) F_1\left(\frac{1}{2};-3,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin ^2(e+f x) \cos ^7(e+f x)}{4 \sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{7/2} \left(\left(5 a F_1\left(\frac{3}{2};-3,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};-3,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin (e+f x) \left(\left(5 a \left(\frac{21 a f F_1\left(\frac{5}{2};-3,\frac{9}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{5 (a+b)}-\frac{18}{5} f F_1\left(\frac{5}{2};-2,\frac{7}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)-6 (a+b) \left(\frac{3 a f F_1\left(\frac{5}{2};-2,\frac{7}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{a+b}-\frac{12}{5} f F_1\left(\frac{5}{2};-1,\frac{5}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)\right) \sin ^2(e+f x)+2 f \left(5 a F_1\left(\frac{3}{2};-3,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \cos (e+f x) \sin (e+f x)+3 (a+b) \left(\frac{5 a f F_1\left(\frac{3}{2};-3,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-2 f F_1\left(\frac{3}{2};-2,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)\right) \cos ^6(e+f x)}{4 \sqrt{2} f \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-3,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right){}^2}+\frac{3 (a+b) \sin (e+f x) \left(\frac{5 a f F_1\left(\frac{3}{2};-3,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-2 f F_1\left(\frac{3}{2};-2,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right) \cos ^6(e+f x)}{4 \sqrt{2} f \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-3,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{9 (a+b) F_1\left(\frac{1}{2};-3,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin ^2(e+f x) \cos ^5(e+f x)}{2 \sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-3,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}\right)}","\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{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{\sin (e+f x) \cos (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"(3*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^8*Sin[e + f*x])/(4*Sqrt[2]*f*(a + b*Sec[e + f*x]^2)^(5/2)*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)*((15*a*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^7*Sin[e + f*x]^2)/(4*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(7/2)*(3*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^7)/(4*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (9*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^5*Sin[e + f*x]^2)/(2*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*Cos[e + f*x]^6*Sin[e + f*x]*((5*a*f*AppellF1[3/2, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - 2*f*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x]))/(4*Sqrt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^6*Sin[e + f*x]*(2*f*(5*a*AppellF1[3/2, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Cos[e + f*x]*Sin[e + f*x] + 3*(a + b)*((5*a*f*AppellF1[3/2, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - 2*f*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x]) + Sin[e + f*x]^2*(5*a*((21*a*f*AppellF1[5/2, -3, 9/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(5*(a + b)) - (18*f*AppellF1[5/2, -2, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5) - 6*(a + b)*((3*a*f*AppellF1[5/2, -2, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - (12*f*AppellF1[5/2, -1, 5/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5))))/(4*Sqrt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -3, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -3, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)^2)))","C",0
294,1,1777,261,20.4761402,"\int \frac{\cos ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cos[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{3 (a+b) F_1\left(\frac{1}{2};-4,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^{12}(e+f x) \sin (e+f x)}{4 \sqrt{2} f \left(b \sec ^2(e+f x)+a\right)^{5/2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-4,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-8 (a+b) F_1\left(\frac{3}{2};-3,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-4,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \left(\frac{3 (a+b) F_1\left(\frac{1}{2};-4,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^9(e+f x)}{4 \sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-4,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-8 (a+b) F_1\left(\frac{3}{2};-3,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-4,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}+\frac{15 a (a+b) F_1\left(\frac{1}{2};-4,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin ^2(e+f x) \cos ^9(e+f x)}{4 \sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{7/2} \left(\left(5 a F_1\left(\frac{3}{2};-4,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-8 (a+b) F_1\left(\frac{3}{2};-3,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-4,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};-4,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin (e+f x) \left(\left(5 a \left(\frac{21 a f F_1\left(\frac{5}{2};-4,\frac{9}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{5 (a+b)}-\frac{24}{5} f F_1\left(\frac{5}{2};-3,\frac{7}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)-8 (a+b) \left(\frac{3 a f F_1\left(\frac{5}{2};-3,\frac{7}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{a+b}-\frac{18}{5} f F_1\left(\frac{5}{2};-2,\frac{5}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)\right) \sin ^2(e+f x)+2 f \left(5 a F_1\left(\frac{3}{2};-4,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-8 (a+b) F_1\left(\frac{3}{2};-3,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \cos (e+f x) \sin (e+f x)+3 (a+b) \left(\frac{5 a f F_1\left(\frac{3}{2};-4,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac{8}{3} f F_1\left(\frac{3}{2};-3,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)\right) \cos ^8(e+f x)}{4 \sqrt{2} f \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-4,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-8 (a+b) F_1\left(\frac{3}{2};-3,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-4,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right){}^2}+\frac{3 (a+b) \sin (e+f x) \left(\frac{5 a f F_1\left(\frac{3}{2};-4,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac{8}{3} f F_1\left(\frac{3}{2};-3,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right) \cos ^8(e+f x)}{4 \sqrt{2} f \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-4,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-8 (a+b) F_1\left(\frac{3}{2};-3,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-4,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{3 \sqrt{2} (a+b) F_1\left(\frac{1}{2};-4,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin ^2(e+f x) \cos ^7(e+f x)}{\left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-4,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-8 (a+b) F_1\left(\frac{3}{2};-3,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-4,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}\right)}","\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{\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(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{b \left(9 a^3-15 a^2 b-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{\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 + b)*AppellF1[1/2, -4, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^12*Sin[e + f*x])/(4*Sqrt[2]*f*(a + b*Sec[e + f*x]^2)^(5/2)*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -4, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -4, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 8*(a + b)*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)*((15*a*(a + b)*AppellF1[1/2, -4, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^9*Sin[e + f*x]^2)/(4*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(7/2)*(3*(a + b)*AppellF1[1/2, -4, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -4, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 8*(a + b)*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*AppellF1[1/2, -4, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^9)/(4*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -4, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -4, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 8*(a + b)*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*Sqrt[2]*(a + b)*AppellF1[1/2, -4, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^7*Sin[e + f*x]^2)/((a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -4, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -4, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 8*(a + b)*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*Cos[e + f*x]^8*Sin[e + f*x]*((5*a*f*AppellF1[3/2, -4, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - (8*f*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3))/(4*Sqrt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -4, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -4, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 8*(a + b)*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*AppellF1[1/2, -4, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^8*Sin[e + f*x]*(2*f*(5*a*AppellF1[3/2, -4, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 8*(a + b)*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Cos[e + f*x]*Sin[e + f*x] + 3*(a + b)*((5*a*f*AppellF1[3/2, -4, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - (8*f*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3) + Sin[e + f*x]^2*(5*a*((21*a*f*AppellF1[5/2, -4, 9/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(5*(a + b)) - (24*f*AppellF1[5/2, -3, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5) - 8*(a + b)*((3*a*f*AppellF1[5/2, -3, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - (18*f*AppellF1[5/2, -2, 5/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5))))/(4*Sqrt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -4, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -4, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 8*(a + b)*AppellF1[3/2, -3, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)^2)))","C",0
295,1,1776,332,27.5685292,"\int \frac{\cos ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{3 (a+b) F_1\left(\frac{1}{2};-5,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^{16}(e+f x) \sin (e+f x)}{4 \sqrt{2} f \left(b \sec ^2(e+f x)+a\right)^{5/2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(5 \left(a F_1\left(\frac{3}{2};-5,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-2 (a+b) F_1\left(\frac{3}{2};-4,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-5,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \left(\frac{3 (a+b) F_1\left(\frac{1}{2};-5,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^{11}(e+f x)}{4 \sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(5 \left(a F_1\left(\frac{3}{2};-5,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-2 (a+b) F_1\left(\frac{3}{2};-4,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-5,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}+\frac{15 a (a+b) F_1\left(\frac{1}{2};-5,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin ^2(e+f x) \cos ^{11}(e+f x)}{4 \sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{7/2} \left(5 \left(a F_1\left(\frac{3}{2};-5,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-2 (a+b) F_1\left(\frac{3}{2};-4,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-5,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};-5,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin (e+f x) \left(5 \left(a \left(\frac{21 a f F_1\left(\frac{5}{2};-5,\frac{9}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{5 (a+b)}-6 f F_1\left(\frac{5}{2};-4,\frac{7}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)-2 (a+b) \left(\frac{3 a f F_1\left(\frac{5}{2};-4,\frac{7}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{a+b}-\frac{24}{5} f F_1\left(\frac{5}{2};-3,\frac{5}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)\right) \sin ^2(e+f x)+10 f \left(a F_1\left(\frac{3}{2};-5,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-2 (a+b) F_1\left(\frac{3}{2};-4,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \cos (e+f x) \sin (e+f x)+3 (a+b) \left(\frac{5 a f F_1\left(\frac{3}{2};-5,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac{10}{3} f F_1\left(\frac{3}{2};-4,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)\right) \cos ^{10}(e+f x)}{4 \sqrt{2} f \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(5 \left(a F_1\left(\frac{3}{2};-5,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-2 (a+b) F_1\left(\frac{3}{2};-4,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-5,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right){}^2}+\frac{3 (a+b) \sin (e+f x) \left(\frac{5 a f F_1\left(\frac{3}{2};-5,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac{10}{3} f F_1\left(\frac{3}{2};-4,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right) \cos ^{10}(e+f x)}{4 \sqrt{2} f \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(5 \left(a F_1\left(\frac{3}{2};-5,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-2 (a+b) F_1\left(\frac{3}{2};-4,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-5,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{15 (a+b) F_1\left(\frac{1}{2};-5,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin ^2(e+f x) \cos ^9(e+f x)}{2 \sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(5 \left(a F_1\left(\frac{3}{2};-5,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-2 (a+b) F_1\left(\frac{3}{2};-4,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-5,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}\right)}","\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{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{\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{b \left(15 a^3-25 a^2 b+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{b \left(15 a^4-20 a^3 b+38 a^2 b^2+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{\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,"(3*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^16*Sin[e + f*x])/(4*Sqrt[2]*f*(a + b*Sec[e + f*x]^2)^(5/2)*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 5*(a*AppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)*((15*a*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^11*Sin[e + f*x]^2)/(4*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(7/2)*(3*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 5*(a*AppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^11)/(4*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 5*(a*AppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (15*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^9*Sin[e + f*x]^2)/(2*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 5*(a*AppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*Cos[e + f*x]^10*Sin[e + f*x]*((5*a*f*AppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - (10*f*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3))/(4*Sqrt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 5*(a*AppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^10*Sin[e + f*x]*(10*f*(a*AppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Cos[e + f*x]*Sin[e + f*x] + 3*(a + b)*((5*a*f*AppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - (10*f*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3) + 5*Sin[e + f*x]^2*(a*((21*a*f*AppellF1[5/2, -5, 9/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(5*(a + b)) - 6*f*AppellF1[5/2, -4, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x]) - 2*(a + b)*((3*a*f*AppellF1[5/2, -4, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - (24*f*AppellF1[5/2, -3, 5/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5))))/(4*Sqrt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -5, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + 5*(a*AppellF1[3/2, -5, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 2*(a + b)*AppellF1[3/2, -4, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)^2)))","C",0
296,1,1777,179,18.5392218,"\int \frac{1}{\left(a+b \sec ^2(c+d x)\right)^{7/2}} \, dx","Integrate[(a + b*Sec[c + d*x]^2)^(-7/2),x]","\frac{3 (a+b) F_1\left(\frac{1}{2};-3,\frac{7}{2};\frac{3}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right) \cos ^6(c+d x) \sin (c+d x)}{8 \sqrt{2} d \left(b \sec ^2(c+d x)+a\right)^{7/2} \left(-a \sin ^2(c+d x)+a+b\right)^{7/2} \left(\left(7 a F_1\left(\frac{3}{2};-3,\frac{9}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)\right) \sin ^2(c+d x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{7}{2};\frac{3}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)\right) \left(\frac{3 (a+b) F_1\left(\frac{1}{2};-3,\frac{7}{2};\frac{3}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right) \cos ^7(c+d x)}{8 \sqrt{2} \left(-a \sin ^2(c+d x)+a+b\right)^{7/2} \left(\left(7 a F_1\left(\frac{3}{2};-3,\frac{9}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)\right) \sin ^2(c+d x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{7}{2};\frac{3}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)\right)}+\frac{21 a (a+b) F_1\left(\frac{1}{2};-3,\frac{7}{2};\frac{3}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right) \sin ^2(c+d x) \cos ^7(c+d x)}{8 \sqrt{2} \left(-a \sin ^2(c+d x)+a+b\right)^{9/2} \left(\left(7 a F_1\left(\frac{3}{2};-3,\frac{9}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)\right) \sin ^2(c+d x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{7}{2};\frac{3}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};-3,\frac{7}{2};\frac{3}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right) \sin (c+d x) \left(\left(7 a \left(\frac{27 a d F_1\left(\frac{5}{2};-3,\frac{11}{2};\frac{7}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right) \cos (c+d x) \sin (c+d x)}{5 (a+b)}-\frac{18}{5} d F_1\left(\frac{5}{2};-2,\frac{9}{2};\frac{7}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right) \cos (c+d x) \sin (c+d x)\right)-6 (a+b) \left(\frac{21 a d F_1\left(\frac{5}{2};-2,\frac{9}{2};\frac{7}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right) \cos (c+d x) \sin (c+d x)}{5 (a+b)}-\frac{12}{5} d F_1\left(\frac{5}{2};-1,\frac{7}{2};\frac{7}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right) \cos (c+d x) \sin (c+d x)\right)\right) \sin ^2(c+d x)+2 d \left(7 a F_1\left(\frac{3}{2};-3,\frac{9}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)\right) \cos (c+d x) \sin (c+d x)+3 (a+b) \left(\frac{7 a d F_1\left(\frac{3}{2};-3,\frac{9}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right) \cos (c+d x) \sin (c+d x)}{3 (a+b)}-2 d F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right) \cos (c+d x) \sin (c+d x)\right)\right) \cos ^6(c+d x)}{8 \sqrt{2} d \left(-a \sin ^2(c+d x)+a+b\right)^{7/2} \left(\left(7 a F_1\left(\frac{3}{2};-3,\frac{9}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)\right) \sin ^2(c+d x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{7}{2};\frac{3}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)\right){}^2}+\frac{3 (a+b) \sin (c+d x) \left(\frac{7 a d F_1\left(\frac{3}{2};-3,\frac{9}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right) \cos (c+d x) \sin (c+d x)}{3 (a+b)}-2 d F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right) \cos (c+d x) \sin (c+d x)\right) \cos ^6(c+d x)}{8 \sqrt{2} d \left(-a \sin ^2(c+d x)+a+b\right)^{7/2} \left(\left(7 a F_1\left(\frac{3}{2};-3,\frac{9}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)\right) \sin ^2(c+d x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{7}{2};\frac{3}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)\right)}-\frac{9 (a+b) F_1\left(\frac{1}{2};-3,\frac{7}{2};\frac{3}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right) \sin ^2(c+d x) \cos ^5(c+d x)}{4 \sqrt{2} \left(-a \sin ^2(c+d x)+a+b\right)^{7/2} \left(\left(7 a F_1\left(\frac{3}{2};-3,\frac{9}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)-6 (a+b) F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)\right) \sin ^2(c+d x)+3 (a+b) F_1\left(\frac{1}{2};-3,\frac{7}{2};\frac{3}{2};\sin ^2(c+d x),\frac{a \sin ^2(c+d x)}{a+b}\right)\right)}\right)}","\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 \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{b \tan (c+d x)}{5 a d (a+b) \left(a+b \tan ^2(c+d x)+b\right)^{5/2}}",1,"(3*(a + b)*AppellF1[1/2, -3, 7/2, 3/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)]*Cos[c + d*x]^6*Sin[c + d*x])/(8*Sqrt[2]*d*(a + b*Sec[c + d*x]^2)^(7/2)*(a + b - a*Sin[c + d*x]^2)^(7/2)*(3*(a + b)*AppellF1[1/2, -3, 7/2, 3/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)] + (7*a*AppellF1[3/2, -3, 9/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 7/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)])*Sin[c + d*x]^2)*((21*a*(a + b)*AppellF1[1/2, -3, 7/2, 3/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)]*Cos[c + d*x]^7*Sin[c + d*x]^2)/(8*Sqrt[2]*(a + b - a*Sin[c + d*x]^2)^(9/2)*(3*(a + b)*AppellF1[1/2, -3, 7/2, 3/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)] + (7*a*AppellF1[3/2, -3, 9/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 7/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)])*Sin[c + d*x]^2)) + (3*(a + b)*AppellF1[1/2, -3, 7/2, 3/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)]*Cos[c + d*x]^7)/(8*Sqrt[2]*(a + b - a*Sin[c + d*x]^2)^(7/2)*(3*(a + b)*AppellF1[1/2, -3, 7/2, 3/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)] + (7*a*AppellF1[3/2, -3, 9/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 7/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)])*Sin[c + d*x]^2)) - (9*(a + b)*AppellF1[1/2, -3, 7/2, 3/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)]*Cos[c + d*x]^5*Sin[c + d*x]^2)/(4*Sqrt[2]*(a + b - a*Sin[c + d*x]^2)^(7/2)*(3*(a + b)*AppellF1[1/2, -3, 7/2, 3/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)] + (7*a*AppellF1[3/2, -3, 9/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 7/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)])*Sin[c + d*x]^2)) + (3*(a + b)*Cos[c + d*x]^6*Sin[c + d*x]*((7*a*d*AppellF1[3/2, -3, 9/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)]*Cos[c + d*x]*Sin[c + d*x])/(3*(a + b)) - 2*d*AppellF1[3/2, -2, 7/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)]*Cos[c + d*x]*Sin[c + d*x]))/(8*Sqrt[2]*d*(a + b - a*Sin[c + d*x]^2)^(7/2)*(3*(a + b)*AppellF1[1/2, -3, 7/2, 3/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)] + (7*a*AppellF1[3/2, -3, 9/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 7/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)])*Sin[c + d*x]^2)) - (3*(a + b)*AppellF1[1/2, -3, 7/2, 3/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)]*Cos[c + d*x]^6*Sin[c + d*x]*(2*d*(7*a*AppellF1[3/2, -3, 9/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 7/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)])*Cos[c + d*x]*Sin[c + d*x] + 3*(a + b)*((7*a*d*AppellF1[3/2, -3, 9/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)]*Cos[c + d*x]*Sin[c + d*x])/(3*(a + b)) - 2*d*AppellF1[3/2, -2, 7/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)]*Cos[c + d*x]*Sin[c + d*x]) + Sin[c + d*x]^2*(7*a*((27*a*d*AppellF1[5/2, -3, 11/2, 7/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)]*Cos[c + d*x]*Sin[c + d*x])/(5*(a + b)) - (18*d*AppellF1[5/2, -2, 9/2, 7/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)]*Cos[c + d*x]*Sin[c + d*x])/5) - 6*(a + b)*((21*a*d*AppellF1[5/2, -2, 9/2, 7/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)]*Cos[c + d*x]*Sin[c + d*x])/(5*(a + b)) - (12*d*AppellF1[5/2, -1, 7/2, 7/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)]*Cos[c + d*x]*Sin[c + d*x])/5))))/(8*Sqrt[2]*d*(a + b - a*Sin[c + d*x]^2)^(7/2)*(3*(a + b)*AppellF1[1/2, -3, 7/2, 3/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)] + (7*a*AppellF1[3/2, -3, 9/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)] - 6*(a + b)*AppellF1[3/2, -2, 7/2, 5/2, Sin[c + d*x]^2, (a*Sin[c + d*x]^2)/(a + b)])*Sin[c + d*x]^2)^2)))","C",0
297,1,37,14,0.0274734,"\int \frac{1}{\sqrt{1+\sec ^2(x)}} \, dx","Integrate[1/Sqrt[1 + Sec[x]^2],x]","\frac{\sin ^{-1}\left(\frac{\sin (x)}{\sqrt{2}}\right) \sqrt{\cos (2 x)+3} \sec (x)}{\sqrt{2} \sqrt{\sec ^2(x)+1}}","\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{\tan ^2(x)+2}}\right)",1,"(ArcSin[Sin[x]/Sqrt[2]]*Sqrt[3 + Cos[2*x]]*Sec[x])/(Sqrt[2]*Sqrt[1 + Sec[x]^2])","B",1
298,1,2195,111,18.899912,"\int (d \sec (e+f x))^m \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[(d*Sec[e + f*x])^m*(a + b*Sec[e + f*x]^2)^p,x]","\text{Result too large to show}","\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,"(3*(a + b)*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(d*Sec[e + f*x])^m*(Sec[e + f*x]^2)^(-1 + m/2 + p)*(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x])/(f*(3*(a + b)*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*(-2 + m)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)*((3*(a + b)*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(m/2 + p))/(3*(a + b)*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*(-2 + m)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (6*a*(a + b)*p*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^(-1 + m/2 + p)*Sin[2*(e + f*x)]*Tan[e + f*x])/(3*(a + b)*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*(-2 + m)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (6*(a + b)*(-1 + m/2 + p)*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-1 + m/2 + p)*Tan[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*(-2 + m)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (3*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-1 + m/2 + p)*Tan[e + f*x]*((2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*(1 - m/2)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(3*(a + b)*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*(-2 + m)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (3*(a + b)*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-1 + m/2 + p)*Tan[e + f*x]*(2*(2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*(-2 + m)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Sec[e + f*x]^2*Tan[e + f*x] + 3*(a + b)*((2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*(1 - m/2)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3) + Tan[e + f*x]^2*(2*b*p*((-6*b*(1 - p)*AppellF1[5/2, 1 - m/2, 2 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (6*(1 - m/2)*AppellF1[5/2, 2 - m/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5) + (a + b)*(-2 + m)*((6*b*p*AppellF1[5/2, 2 - m/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (6*(2 - m/2)*AppellF1[5/2, 3 - m/2, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*(a + b)*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*(-2 + m)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)^2))","B",0
299,1,1989,103,17.1536368,"\int \sec ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^p,x]","\frac{3 (a+b) F_1\left(\frac{1}{2};-\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^p \sec ^3(e+f x) \sec ^2(e+f x)^{p+\frac{1}{2}} \left(b \sec ^2(e+f x)+a\right)^p \tan (e+f x)}{f \left(\left(2 b p F_1\left(\frac{3}{2};-\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)+(a+b) F_1\left(\frac{3}{2};\frac{1}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \left(-\frac{6 a (a+b) p F_1\left(\frac{1}{2};-\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x)^{p+\frac{1}{2}} \sin (2 (e+f x)) \tan (e+f x) (\cos (2 (e+f x)) a+a+2 b)^{p-1}}{\left(2 b p F_1\left(\frac{3}{2};-\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)+(a+b) F_1\left(\frac{3}{2};\frac{1}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};-\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x)^{p+\frac{1}{2}} \tan (e+f x) \left(2 \left(2 b p F_1\left(\frac{3}{2};-\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)+(a+b) F_1\left(\frac{3}{2};\frac{1}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan (e+f x) \sec ^2(e+f x)+3 (a+b) \left(\frac{2 b p F_1\left(\frac{3}{2};-\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)}{3 (a+b)}+\frac{1}{3} F_1\left(\frac{3}{2};\frac{1}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)\right)+\tan ^2(e+f x) \left(2 b p \left(\frac{3}{5} F_1\left(\frac{5}{2};\frac{1}{2},1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)-\frac{6 b (1-p) F_1\left(\frac{5}{2};-\frac{1}{2},2-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{5 (a+b)}\right)+(a+b) \left(\frac{6 b p F_1\left(\frac{5}{2};\frac{1}{2},1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{5 (a+b)}-\frac{3}{5} F_1\left(\frac{5}{2};\frac{3}{2},-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right)\right)\right) (\cos (2 (e+f x)) a+a+2 b)^p}{\left(\left(2 b p F_1\left(\frac{3}{2};-\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)+(a+b) F_1\left(\frac{3}{2};\frac{1}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right){}^2}+\frac{3 (a+b) F_1\left(\frac{1}{2};-\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x)^{p+\frac{3}{2}} (\cos (2 (e+f x)) a+a+2 b)^p}{\left(2 b p F_1\left(\frac{3}{2};-\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)+(a+b) F_1\left(\frac{3}{2};\frac{1}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}+\frac{6 (a+b) \left(p+\frac{1}{2}\right) F_1\left(\frac{1}{2};-\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x)^{p+\frac{1}{2}} \tan ^2(e+f x) (\cos (2 (e+f x)) a+a+2 b)^p}{\left(2 b p F_1\left(\frac{3}{2};-\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)+(a+b) F_1\left(\frac{3}{2};\frac{1}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}+\frac{3 (a+b) \sec ^2(e+f x)^{p+\frac{1}{2}} \tan (e+f x) \left(\frac{2 b p F_1\left(\frac{3}{2};-\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)}{3 (a+b)}+\frac{1}{3} F_1\left(\frac{3}{2};\frac{1}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)\right) (\cos (2 (e+f x)) a+a+2 b)^p}{\left(2 b p F_1\left(\frac{3}{2};-\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)+(a+b) F_1\left(\frac{3}{2};\frac{1}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}\right)}","\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,"(3*(a + b)*AppellF1[1/2, -1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*Sec[e + f*x]^3*(Sec[e + f*x]^2)^(1/2 + p)*(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x])/(f*(3*(a + b)*AppellF1[1/2, -1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, -1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*AppellF1[3/2, 1/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)*((3*(a + b)*AppellF1[1/2, -1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(3/2 + p))/(3*(a + b)*AppellF1[1/2, -1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, -1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*AppellF1[3/2, 1/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (6*a*(a + b)*p*AppellF1[1/2, -1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^(1/2 + p)*Sin[2*(e + f*x)]*Tan[e + f*x])/(3*(a + b)*AppellF1[1/2, -1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, -1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*AppellF1[3/2, 1/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (6*(a + b)*(1/2 + p)*AppellF1[1/2, -1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(1/2 + p)*Tan[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, -1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*AppellF1[3/2, 1/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (3*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(1/2 + p)*Tan[e + f*x]*((2*b*p*AppellF1[3/2, -1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) + (AppellF1[3/2, 1/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(3*(a + b)*AppellF1[1/2, -1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, -1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*AppellF1[3/2, 1/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (3*(a + b)*AppellF1[1/2, -1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(1/2 + p)*Tan[e + f*x]*(2*(2*b*p*AppellF1[3/2, -1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*AppellF1[3/2, 1/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Sec[e + f*x]^2*Tan[e + f*x] + 3*(a + b)*((2*b*p*AppellF1[3/2, -1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) + (AppellF1[3/2, 1/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3) + Tan[e + f*x]^2*(2*b*p*((-6*b*(1 - p)*AppellF1[5/2, -1/2, 2 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) + (3*AppellF1[5/2, 1/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5) + (a + b)*((6*b*p*AppellF1[5/2, 1/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (3*AppellF1[5/2, 3/2, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*(a + b)*AppellF1[1/2, -1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, -1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*AppellF1[3/2, 1/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)^2))","B",0
300,1,1995,103,16.5830945,"\int \sec (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Sec[e + f*x]*(a + b*Sec[e + f*x]^2)^p,x]","\frac{3 (a+b) F_1\left(\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^p \sec (e+f x) \sec ^2(e+f x)^{p-\frac{1}{2}} \left(b \sec ^2(e+f x)+a\right)^p \tan (e+f x)}{f \left(\left(2 b p F_1\left(\frac{3}{2};\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-(a+b) F_1\left(\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \left(-\frac{6 a (a+b) p F_1\left(\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x)^{p-\frac{1}{2}} \sin (2 (e+f x)) \tan (e+f x) (\cos (2 (e+f x)) a+a+2 b)^{p-1}}{\left(2 b p F_1\left(\frac{3}{2};\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-(a+b) F_1\left(\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x)^{p-\frac{1}{2}} \tan (e+f x) \left(2 \left(2 b p F_1\left(\frac{3}{2};\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-(a+b) F_1\left(\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan (e+f x) \sec ^2(e+f x)+3 (a+b) \left(\frac{2 b p F_1\left(\frac{3}{2};\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{3 (a+b)}-\frac{1}{3} F_1\left(\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right)+\tan ^2(e+f x) \left(2 b p \left(-\frac{6 b (1-p) F_1\left(\frac{5}{2};\frac{1}{2},2-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)}{5 (a+b)}-\frac{3}{5} F_1\left(\frac{5}{2};\frac{3}{2},1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)\right)-(a+b) \left(\frac{6 b p F_1\left(\frac{5}{2};\frac{3}{2},1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{5 (a+b)}-\frac{9}{5} F_1\left(\frac{5}{2};\frac{5}{2},-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right)\right)\right) (\cos (2 (e+f x)) a+a+2 b)^p}{\left(\left(2 b p F_1\left(\frac{3}{2};\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-(a+b) F_1\left(\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right){}^2}+\frac{3 (a+b) F_1\left(\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x)^{p+\frac{1}{2}} (\cos (2 (e+f x)) a+a+2 b)^p}{\left(2 b p F_1\left(\frac{3}{2};\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-(a+b) F_1\left(\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}+\frac{6 (a+b) \left(p-\frac{1}{2}\right) F_1\left(\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x)^{p-\frac{1}{2}} \tan ^2(e+f x) (\cos (2 (e+f x)) a+a+2 b)^p}{\left(2 b p F_1\left(\frac{3}{2};\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-(a+b) F_1\left(\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}+\frac{3 (a+b) \sec ^2(e+f x)^{p-\frac{1}{2}} \tan (e+f x) \left(\frac{2 b p F_1\left(\frac{3}{2};\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{3 (a+b)}-\frac{1}{3} F_1\left(\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right) (\cos (2 (e+f x)) a+a+2 b)^p}{\left(2 b p F_1\left(\frac{3}{2};\frac{1}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-(a+b) F_1\left(\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}\right)}","\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,"(3*(a + b)*AppellF1[1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*Sec[e + f*x]*(Sec[e + f*x]^2)^(-1/2 + p)*(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x])/(f*(3*(a + b)*AppellF1[1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, 1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - (a + b)*AppellF1[3/2, 3/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)*((3*(a + b)*AppellF1[1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(1/2 + p))/(3*(a + b)*AppellF1[1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, 1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - (a + b)*AppellF1[3/2, 3/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (6*a*(a + b)*p*AppellF1[1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^(-1/2 + p)*Sin[2*(e + f*x)]*Tan[e + f*x])/(3*(a + b)*AppellF1[1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, 1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - (a + b)*AppellF1[3/2, 3/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (6*(a + b)*(-1/2 + p)*AppellF1[1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-1/2 + p)*Tan[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, 1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - (a + b)*AppellF1[3/2, 3/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (3*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-1/2 + p)*Tan[e + f*x]*((2*b*p*AppellF1[3/2, 1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (AppellF1[3/2, 3/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(3*(a + b)*AppellF1[1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, 1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - (a + b)*AppellF1[3/2, 3/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (3*(a + b)*AppellF1[1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-1/2 + p)*Tan[e + f*x]*(2*(2*b*p*AppellF1[3/2, 1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - (a + b)*AppellF1[3/2, 3/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Sec[e + f*x]^2*Tan[e + f*x] + 3*(a + b)*((2*b*p*AppellF1[3/2, 1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (AppellF1[3/2, 3/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3) + Tan[e + f*x]^2*(2*b*p*((-6*b*(1 - p)*AppellF1[5/2, 1/2, 2 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (3*AppellF1[5/2, 3/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5) - (a + b)*((6*b*p*AppellF1[5/2, 3/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (9*AppellF1[5/2, 5/2, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*(a + b)*AppellF1[1/2, 1/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, 1/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - (a + b)*AppellF1[3/2, 3/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)^2))","B",0
301,1,1983,101,16.4999619,"\int \cos (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^p,x]","-\frac{3 (a+b) F_1\left(\frac{1}{2};\frac{3}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^p \sec ^2(e+f x)^{p-\frac{3}{2}} \left(b \sec ^2(e+f x)+a\right)^p \sin (e+f x)}{f \left(\left(3 (a+b) F_1\left(\frac{3}{2};\frac{5}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{3}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{3}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \left(\frac{3 (a+b) F_1\left(\frac{1}{2};\frac{3}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^p \tan (e+f x) \left(2 \left(3 (a+b) F_1\left(\frac{3}{2};\frac{5}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{3}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan (e+f x) \sec ^2(e+f x)-3 (a+b) \left(\frac{2 b p F_1\left(\frac{3}{2};\frac{3}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{3 (a+b)}-F_1\left(\frac{3}{2};\frac{5}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right)+\tan ^2(e+f x) \left(3 (a+b) \left(\frac{6 b p F_1\left(\frac{5}{2};\frac{5}{2},1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{5 (a+b)}-3 F_1\left(\frac{5}{2};\frac{7}{2},-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right)-2 b p \left(-\frac{6 b (1-p) F_1\left(\frac{5}{2};\frac{3}{2},2-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)}{5 (a+b)}-\frac{9}{5} F_1\left(\frac{5}{2};\frac{5}{2},1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)\right)\right)\right) \sec ^2(e+f x)^{p-\frac{3}{2}}}{\left(\left(3 (a+b) F_1\left(\frac{3}{2};\frac{5}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{3}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{3}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right){}^2}-\frac{6 (a+b) \left(p-\frac{3}{2}\right) F_1\left(\frac{1}{2};\frac{3}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^p \tan ^2(e+f x) \sec ^2(e+f x)^{p-\frac{3}{2}}}{\left(3 (a+b) F_1\left(\frac{3}{2};\frac{5}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{3}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{3}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}+\frac{6 a (a+b) p F_1\left(\frac{1}{2};\frac{3}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^{p-1} \sin (2 (e+f x)) \tan (e+f x) \sec ^2(e+f x)^{p-\frac{3}{2}}}{\left(3 (a+b) F_1\left(\frac{3}{2};\frac{5}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{3}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{3}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}-\frac{3 (a+b) (\cos (2 (e+f x)) a+a+2 b)^p \tan (e+f x) \left(\frac{2 b p F_1\left(\frac{3}{2};\frac{3}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{3 (a+b)}-F_1\left(\frac{3}{2};\frac{5}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right) \sec ^2(e+f x)^{p-\frac{3}{2}}}{\left(3 (a+b) F_1\left(\frac{3}{2};\frac{5}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{3}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{3}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};\frac{3}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^p \sec ^2(e+f x)^{p-\frac{1}{2}}}{\left(3 (a+b) F_1\left(\frac{3}{2};\frac{5}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{3}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{3}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}\right)}","\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,"(-3*(a + b)*AppellF1[1/2, 3/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-3/2 + p)*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x])/(f*(-3*(a + b)*AppellF1[1/2, 3/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 3/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 3*(a + b)*AppellF1[3/2, 5/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)*((-3*(a + b)*AppellF1[1/2, 3/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-1/2 + p))/(-3*(a + b)*AppellF1[1/2, 3/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 3/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 3*(a + b)*AppellF1[3/2, 5/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (6*a*(a + b)*p*AppellF1[1/2, 3/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^(-3/2 + p)*Sin[2*(e + f*x)]*Tan[e + f*x])/(-3*(a + b)*AppellF1[1/2, 3/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 3/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 3*(a + b)*AppellF1[3/2, 5/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (6*(a + b)*(-3/2 + p)*AppellF1[1/2, 3/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-3/2 + p)*Tan[e + f*x]^2)/(-3*(a + b)*AppellF1[1/2, 3/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 3/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 3*(a + b)*AppellF1[3/2, 5/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (3*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-3/2 + p)*Tan[e + f*x]*((2*b*p*AppellF1[3/2, 3/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - AppellF1[3/2, 5/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x]))/(-3*(a + b)*AppellF1[1/2, 3/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 3/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 3*(a + b)*AppellF1[3/2, 5/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (3*(a + b)*AppellF1[1/2, 3/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-3/2 + p)*Tan[e + f*x]*(2*(-2*b*p*AppellF1[3/2, 3/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 3*(a + b)*AppellF1[3/2, 5/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Sec[e + f*x]^2*Tan[e + f*x] - 3*(a + b)*((2*b*p*AppellF1[3/2, 3/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - AppellF1[3/2, 5/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x]) + Tan[e + f*x]^2*(-2*b*p*((-6*b*(1 - p)*AppellF1[5/2, 3/2, 2 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (9*AppellF1[5/2, 5/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5) + 3*(a + b)*((6*b*p*AppellF1[5/2, 5/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - 3*AppellF1[5/2, 7/2, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x]))))/(-3*(a + b)*AppellF1[1/2, 3/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 3/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 3*(a + b)*AppellF1[3/2, 5/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)^2))","B",0
302,1,1987,103,17.0358709,"\int \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^p,x]","-\frac{3 (a+b) F_1\left(\frac{1}{2};\frac{5}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^p \sec ^2(e+f x)^{p-\frac{7}{2}} \left(b \sec ^2(e+f x)+a\right)^p \sin (e+f x)}{f \left(\left(5 (a+b) F_1\left(\frac{3}{2};\frac{7}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{5}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{5}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \left(\frac{3 (a+b) F_1\left(\frac{1}{2};\frac{5}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^p \tan (e+f x) \left(2 \left(5 (a+b) F_1\left(\frac{3}{2};\frac{7}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{5}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan (e+f x) \sec ^2(e+f x)-3 (a+b) \left(\frac{2 b p F_1\left(\frac{3}{2};\frac{5}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{3 (a+b)}-\frac{5}{3} F_1\left(\frac{3}{2};\frac{7}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right)+\tan ^2(e+f x) \left(5 (a+b) \left(\frac{6 b p F_1\left(\frac{5}{2};\frac{7}{2},1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{5 (a+b)}-\frac{21}{5} F_1\left(\frac{5}{2};\frac{9}{2},-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right)-2 b p \left(-\frac{6 b (1-p) F_1\left(\frac{5}{2};\frac{5}{2},2-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)}{5 (a+b)}-3 F_1\left(\frac{5}{2};\frac{7}{2},1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)\right)\right)\right) \sec ^2(e+f x)^{p-\frac{5}{2}}}{\left(\left(5 (a+b) F_1\left(\frac{3}{2};\frac{7}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{5}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{5}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right){}^2}-\frac{6 (a+b) \left(p-\frac{5}{2}\right) F_1\left(\frac{1}{2};\frac{5}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^p \tan ^2(e+f x) \sec ^2(e+f x)^{p-\frac{5}{2}}}{\left(5 (a+b) F_1\left(\frac{3}{2};\frac{7}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{5}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{5}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}+\frac{6 a (a+b) p F_1\left(\frac{1}{2};\frac{5}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^{p-1} \sin (2 (e+f x)) \tan (e+f x) \sec ^2(e+f x)^{p-\frac{5}{2}}}{\left(5 (a+b) F_1\left(\frac{3}{2};\frac{7}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{5}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{5}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}-\frac{3 (a+b) (\cos (2 (e+f x)) a+a+2 b)^p \tan (e+f x) \left(\frac{2 b p F_1\left(\frac{3}{2};\frac{5}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{3 (a+b)}-\frac{5}{3} F_1\left(\frac{3}{2};\frac{7}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right) \sec ^2(e+f x)^{p-\frac{5}{2}}}{\left(5 (a+b) F_1\left(\frac{3}{2};\frac{7}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{5}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{5}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};\frac{5}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^p \sec ^2(e+f x)^{p-\frac{3}{2}}}{\left(5 (a+b) F_1\left(\frac{3}{2};\frac{7}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{5}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{5}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}\right)}","\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,"(-3*(a + b)*AppellF1[1/2, 5/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-7/2 + p)*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x])/(f*(-3*(a + b)*AppellF1[1/2, 5/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 5/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 5*(a + b)*AppellF1[3/2, 7/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)*((-3*(a + b)*AppellF1[1/2, 5/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-3/2 + p))/(-3*(a + b)*AppellF1[1/2, 5/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 5/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 5*(a + b)*AppellF1[3/2, 7/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (6*a*(a + b)*p*AppellF1[1/2, 5/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^(-5/2 + p)*Sin[2*(e + f*x)]*Tan[e + f*x])/(-3*(a + b)*AppellF1[1/2, 5/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 5/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 5*(a + b)*AppellF1[3/2, 7/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (6*(a + b)*(-5/2 + p)*AppellF1[1/2, 5/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-5/2 + p)*Tan[e + f*x]^2)/(-3*(a + b)*AppellF1[1/2, 5/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 5/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 5*(a + b)*AppellF1[3/2, 7/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (3*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-5/2 + p)*Tan[e + f*x]*((2*b*p*AppellF1[3/2, 5/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (5*AppellF1[3/2, 7/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(-3*(a + b)*AppellF1[1/2, 5/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 5/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 5*(a + b)*AppellF1[3/2, 7/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (3*(a + b)*AppellF1[1/2, 5/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-5/2 + p)*Tan[e + f*x]*(2*(-2*b*p*AppellF1[3/2, 5/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 5*(a + b)*AppellF1[3/2, 7/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Sec[e + f*x]^2*Tan[e + f*x] - 3*(a + b)*((2*b*p*AppellF1[3/2, 5/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (5*AppellF1[3/2, 7/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3) + Tan[e + f*x]^2*(-2*b*p*((-6*b*(1 - p)*AppellF1[5/2, 5/2, 2 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - 3*AppellF1[5/2, 7/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x]) + 5*(a + b)*((6*b*p*AppellF1[5/2, 7/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (21*AppellF1[5/2, 9/2, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(-3*(a + b)*AppellF1[1/2, 5/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 5/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 5*(a + b)*AppellF1[3/2, 7/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)^2))","B",0
303,1,1997,103,17.407084,"\int \cos ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^p,x]","-\frac{3 (a+b) F_1\left(\frac{1}{2};\frac{7}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \cos ^4(e+f x) (\cos (2 (e+f x)) a+a+2 b)^p \sec ^2(e+f x)^{p-\frac{7}{2}} \left(b \sec ^2(e+f x)+a\right)^p \sin (e+f x)}{f \left(\left(7 (a+b) F_1\left(\frac{3}{2};\frac{9}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{7}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{7}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \left(\frac{3 (a+b) F_1\left(\frac{1}{2};\frac{7}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^p \tan (e+f x) \left(2 \left(7 (a+b) F_1\left(\frac{3}{2};\frac{9}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{7}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan (e+f x) \sec ^2(e+f x)-3 (a+b) \left(\frac{2 b p F_1\left(\frac{3}{2};\frac{7}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{3 (a+b)}-\frac{7}{3} F_1\left(\frac{3}{2};\frac{9}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right)+\tan ^2(e+f x) \left(7 (a+b) \left(\frac{6 b p F_1\left(\frac{5}{2};\frac{9}{2},1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{5 (a+b)}-\frac{27}{5} F_1\left(\frac{5}{2};\frac{11}{2},-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right)-2 b p \left(-\frac{6 b (1-p) F_1\left(\frac{5}{2};\frac{7}{2},2-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)}{5 (a+b)}-\frac{21}{5} F_1\left(\frac{5}{2};\frac{9}{2},1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)\right)\right)\right) \sec ^2(e+f x)^{p-\frac{7}{2}}}{\left(\left(7 (a+b) F_1\left(\frac{3}{2};\frac{9}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{7}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{7}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right){}^2}-\frac{6 (a+b) \left(p-\frac{7}{2}\right) F_1\left(\frac{1}{2};\frac{7}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^p \tan ^2(e+f x) \sec ^2(e+f x)^{p-\frac{7}{2}}}{\left(7 (a+b) F_1\left(\frac{3}{2};\frac{9}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{7}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{7}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}+\frac{6 a (a+b) p F_1\left(\frac{1}{2};\frac{7}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^{p-1} \sin (2 (e+f x)) \tan (e+f x) \sec ^2(e+f x)^{p-\frac{7}{2}}}{\left(7 (a+b) F_1\left(\frac{3}{2};\frac{9}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{7}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{7}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}-\frac{3 (a+b) (\cos (2 (e+f x)) a+a+2 b)^p \tan (e+f x) \left(\frac{2 b p F_1\left(\frac{3}{2};\frac{7}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{3 (a+b)}-\frac{7}{3} F_1\left(\frac{3}{2};\frac{9}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right) \sec ^2(e+f x)^{p-\frac{7}{2}}}{\left(7 (a+b) F_1\left(\frac{3}{2};\frac{9}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{7}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{7}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};\frac{7}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) (\cos (2 (e+f x)) a+a+2 b)^p \sec ^2(e+f x)^{p-\frac{5}{2}}}{\left(7 (a+b) F_1\left(\frac{3}{2};\frac{9}{2},-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 b p F_1\left(\frac{3}{2};\frac{7}{2},1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)-3 (a+b) F_1\left(\frac{1}{2};\frac{7}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)}\right)}","\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,"(-3*(a + b)*AppellF1[1/2, 7/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Cos[e + f*x]^4*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-7/2 + p)*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x])/(f*(-3*(a + b)*AppellF1[1/2, 7/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 7/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 7*(a + b)*AppellF1[3/2, 9/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)*((-3*(a + b)*AppellF1[1/2, 7/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-5/2 + p))/(-3*(a + b)*AppellF1[1/2, 7/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 7/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 7*(a + b)*AppellF1[3/2, 9/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (6*a*(a + b)*p*AppellF1[1/2, 7/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^(-7/2 + p)*Sin[2*(e + f*x)]*Tan[e + f*x])/(-3*(a + b)*AppellF1[1/2, 7/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 7/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 7*(a + b)*AppellF1[3/2, 9/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (6*(a + b)*(-7/2 + p)*AppellF1[1/2, 7/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-7/2 + p)*Tan[e + f*x]^2)/(-3*(a + b)*AppellF1[1/2, 7/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 7/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 7*(a + b)*AppellF1[3/2, 9/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (3*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-7/2 + p)*Tan[e + f*x]*((2*b*p*AppellF1[3/2, 7/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (7*AppellF1[3/2, 9/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(-3*(a + b)*AppellF1[1/2, 7/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 7/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 7*(a + b)*AppellF1[3/2, 9/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (3*(a + b)*AppellF1[1/2, 7/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-7/2 + p)*Tan[e + f*x]*(2*(-2*b*p*AppellF1[3/2, 7/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 7*(a + b)*AppellF1[3/2, 9/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Sec[e + f*x]^2*Tan[e + f*x] - 3*(a + b)*((2*b*p*AppellF1[3/2, 7/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (7*AppellF1[3/2, 9/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3) + Tan[e + f*x]^2*(-2*b*p*((-6*b*(1 - p)*AppellF1[5/2, 7/2, 2 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (21*AppellF1[5/2, 9/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5) + 7*(a + b)*((6*b*p*AppellF1[5/2, 9/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (27*AppellF1[5/2, 11/2, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(-3*(a + b)*AppellF1[1/2, 7/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 7/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 7*(a + b)*AppellF1[3/2, 9/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)^2))","B",0
304,1,149,216,2.1062608,"\int \sec ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Sec[e + f*x]^6*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \left(a+b \sec ^2(e+f x)\right)^p \left(10 \tan ^2(e+f x) \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)+15 \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)+3 \tan ^4(e+f x) \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)\right)}{15 f}","\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,"((a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]*(15*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))] + 10*Hypergeometric2F1[3/2, -p, 5/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2 + 3*Hypergeometric2F1[5/2, -p, 7/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^4))/(15*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",1
305,1,126,129,2.2603999,"\int \sec ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Sec[e + f*x]^4*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \left(a+b \sec ^2(e+f x)\right)^p \left((2 b (p+1)-a) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a+b}\right)+\left(a+b \tan ^2(e+f x)+b\right) \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^p\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,"((a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]*((-a + 2*b*(1 + p))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b + b*Tan[e + f*x]^2)*(1 + (b*Tan[e + f*x]^2)/(a + b))^p))/(b*f*(3 + 2*p)*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",1
306,1,71,72,1.0114604,"\int \sec ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Sec[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(\frac{b \tan ^2(e+f x)}{a+b}+1\right)^{-p} \left(a+b \sec ^2(e+f x)\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))]*(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x])/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)","A",1
307,1,2137,83,6.2561674,"\int \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[(a + b*Sec[e + f*x]^2)^p,x]","\text{Result too large to show}","\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,"(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x])/(f*(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)*((3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-1 + p))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Sin[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (6*(a + b)*p*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Sin[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (6*a*(a + b)*p*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^p*Sin[e + f*x]*Sin[2*(e + f*x)])/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (3*(a + b)*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Sin[e + f*x]*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Sin[e + f*x]*(4*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Sec[e + f*x]^2*Tan[e + f*x] + 3*(a + b)*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(b*p*((-6*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5 - (6*b*(1 - p)*AppellF1[5/2, 2 - p, 1, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b))) - (a + b)*((6*b*p*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (12*AppellF1[5/2, -p, 3, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)^2))","B",0
308,1,1914,83,15.7653115,"\int \cos ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Cos[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p,x]","\frac{3 (a+b) 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) \cos (e+f x) (\cos (2 (e+f x)) a+a+2 b)^p \sec ^2(e+f x)^{p-2} \left(b \sec ^2(e+f x)+a\right)^p \sin (e+f x)}{f \left(2 \left(b p F_1\left(\frac{3}{2};2,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 (a+b) F_1\left(\frac{3}{2};3,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)\right) \left(-\frac{3 (a+b) 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) (\cos (2 (e+f x)) a+a+2 b)^p \tan (e+f x) \left(4 \left(b p F_1\left(\frac{3}{2};2,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 (a+b) F_1\left(\frac{3}{2};3,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan (e+f x) \sec ^2(e+f x)+3 (a+b) \left(\frac{2 b p F_1\left(\frac{3}{2};2,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{3 (a+b)}-\frac{4}{3} F_1\left(\frac{3}{2};3,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right)+2 \tan ^2(e+f x) \left(b p \left(-\frac{6 b (1-p) F_1\left(\frac{5}{2};2,2-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)}{5 (a+b)}-\frac{12}{5} F_1\left(\frac{5}{2};3,1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)\right)-2 (a+b) \left(\frac{6 b p F_1\left(\frac{5}{2};3,1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{5 (a+b)}-\frac{18}{5} F_1\left(\frac{5}{2};4,-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right)\right)\right) \sec ^2(e+f x)^{p-2}}{\left(2 \left(b p F_1\left(\frac{3}{2};2,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 (a+b) F_1\left(\frac{3}{2};3,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)\right){}^2}+\frac{6 (a+b) (p-2) 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) (\cos (2 (e+f x)) a+a+2 b)^p \tan ^2(e+f x) \sec ^2(e+f x)^{p-2}}{2 \left(b p F_1\left(\frac{3}{2};2,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 (a+b) F_1\left(\frac{3}{2};3,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)}-\frac{6 a (a+b) 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) (\cos (2 (e+f x)) a+a+2 b)^{p-1} \sin (2 (e+f x)) \tan (e+f x) \sec ^2(e+f x)^{p-2}}{2 \left(b p F_1\left(\frac{3}{2};2,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 (a+b) F_1\left(\frac{3}{2};3,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)}+\frac{3 (a+b) (\cos (2 (e+f x)) a+a+2 b)^p \tan (e+f x) \left(\frac{2 b p F_1\left(\frac{3}{2};2,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{3 (a+b)}-\frac{4}{3} F_1\left(\frac{3}{2};3,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right) \sec ^2(e+f x)^{p-2}}{2 \left(b p F_1\left(\frac{3}{2};2,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 (a+b) F_1\left(\frac{3}{2};3,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)}+\frac{3 (a+b) 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) (\cos (2 (e+f x)) a+a+2 b)^p \sec ^2(e+f x)^{p-1}}{2 \left(b p F_1\left(\frac{3}{2};2,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-2 (a+b) F_1\left(\frac{3}{2};3,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)}\right)}","\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,"(3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-2 + p)*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x])/(f*(3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)*((3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-1 + p))/(3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (6*a*(a + b)*p*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^(-2 + p)*Sin[2*(e + f*x)]*Tan[e + f*x])/(3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (6*(a + b)*(-2 + p)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-2 + p)*Tan[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (3*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-2 + p)*Tan[e + f*x]*((2*b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (4*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-2 + p)*Tan[e + f*x]*(4*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Sec[e + f*x]^2*Tan[e + f*x] + 3*(a + b)*((2*b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (4*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(b*p*((-6*b*(1 - p)*AppellF1[5/2, 2, 2 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (12*AppellF1[5/2, 3, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5) - 2*(a + b)*((6*b*p*AppellF1[5/2, 3, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (18*AppellF1[5/2, 4, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*(a + b)*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 2*(a + b)*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)^2))","B",0
309,1,1912,83,16.2815915,"\int \cos ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^p,x]","\frac{3 (a+b) 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) \cos ^3(e+f x) (\cos (2 (e+f x)) a+a+2 b)^p \sec ^2(e+f x)^{p-3} \left(b \sec ^2(e+f x)+a\right)^p \sin (e+f x)}{f \left(2 \left(b p F_1\left(\frac{3}{2};3,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-3 (a+b) F_1\left(\frac{3}{2};4,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)\right) \left(-\frac{3 (a+b) 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) (\cos (2 (e+f x)) a+a+2 b)^p \tan (e+f x) \left(4 \left(b p F_1\left(\frac{3}{2};3,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-3 (a+b) F_1\left(\frac{3}{2};4,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan (e+f x) \sec ^2(e+f x)+3 (a+b) \left(\frac{2 b p F_1\left(\frac{3}{2};3,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{3 (a+b)}-2 F_1\left(\frac{3}{2};4,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right)+2 \tan ^2(e+f x) \left(b p \left(-\frac{6 b (1-p) F_1\left(\frac{5}{2};3,2-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)}{5 (a+b)}-\frac{18}{5} F_1\left(\frac{5}{2};4,1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)\right)-3 (a+b) \left(\frac{6 b p F_1\left(\frac{5}{2};4,1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{5 (a+b)}-\frac{24}{5} F_1\left(\frac{5}{2};5,-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right)\right)\right) \sec ^2(e+f x)^{p-3}}{\left(2 \left(b p F_1\left(\frac{3}{2};3,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-3 (a+b) F_1\left(\frac{3}{2};4,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)\right){}^2}+\frac{6 (a+b) (p-3) 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) (\cos (2 (e+f x)) a+a+2 b)^p \tan ^2(e+f x) \sec ^2(e+f x)^{p-3}}{2 \left(b p F_1\left(\frac{3}{2};3,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-3 (a+b) F_1\left(\frac{3}{2};4,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)}-\frac{6 a (a+b) 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) (\cos (2 (e+f x)) a+a+2 b)^{p-1} \sin (2 (e+f x)) \tan (e+f x) \sec ^2(e+f x)^{p-3}}{2 \left(b p F_1\left(\frac{3}{2};3,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-3 (a+b) F_1\left(\frac{3}{2};4,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)}+\frac{3 (a+b) (\cos (2 (e+f x)) a+a+2 b)^p \tan (e+f x) \left(\frac{2 b p F_1\left(\frac{3}{2};3,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{3 (a+b)}-2 F_1\left(\frac{3}{2};4,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right) \sec ^2(e+f x)^{p-3}}{2 \left(b p F_1\left(\frac{3}{2};3,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-3 (a+b) F_1\left(\frac{3}{2};4,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)}+\frac{3 (a+b) 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) (\cos (2 (e+f x)) a+a+2 b)^p \sec ^2(e+f x)^{p-2}}{2 \left(b p F_1\left(\frac{3}{2};3,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-3 (a+b) F_1\left(\frac{3}{2};4,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)}\right)}","\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,"(3*(a + b)*AppellF1[1/2, 3, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Cos[e + f*x]^3*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-3 + p)*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x])/(f*(3*(a + b)*AppellF1[1/2, 3, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 3, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 3*(a + b)*AppellF1[3/2, 4, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)*((3*(a + b)*AppellF1[1/2, 3, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-2 + p))/(3*(a + b)*AppellF1[1/2, 3, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 3, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 3*(a + b)*AppellF1[3/2, 4, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (6*a*(a + b)*p*AppellF1[1/2, 3, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^(-3 + p)*Sin[2*(e + f*x)]*Tan[e + f*x])/(3*(a + b)*AppellF1[1/2, 3, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 3, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 3*(a + b)*AppellF1[3/2, 4, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (6*(a + b)*(-3 + p)*AppellF1[1/2, 3, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-3 + p)*Tan[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, 3, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 3, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 3*(a + b)*AppellF1[3/2, 4, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (3*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-3 + p)*Tan[e + f*x]*((2*b*p*AppellF1[3/2, 3, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - 2*AppellF1[3/2, 4, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x]))/(3*(a + b)*AppellF1[1/2, 3, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 3, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 3*(a + b)*AppellF1[3/2, 4, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (3*(a + b)*AppellF1[1/2, 3, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-3 + p)*Tan[e + f*x]*(4*(b*p*AppellF1[3/2, 3, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 3*(a + b)*AppellF1[3/2, 4, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Sec[e + f*x]^2*Tan[e + f*x] + 3*(a + b)*((2*b*p*AppellF1[3/2, 3, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - 2*AppellF1[3/2, 4, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x]) + 2*Tan[e + f*x]^2*(b*p*((-6*b*(1 - p)*AppellF1[5/2, 3, 2 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (18*AppellF1[5/2, 4, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5) - 3*(a + b)*((6*b*p*AppellF1[5/2, 4, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (24*AppellF1[5/2, 5, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*(a + b)*AppellF1[1/2, 3, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 3, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 3*(a + b)*AppellF1[3/2, 4, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)^2))","B",0
310,1,1914,83,17.1198031,"\int \cos ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Cos[e + f*x]^6*(a + b*Sec[e + f*x]^2)^p,x]","\frac{3 (a+b) 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) \cos ^5(e+f x) (\cos (2 (e+f x)) a+a+2 b)^p \sec ^2(e+f x)^{p-4} \left(b \sec ^2(e+f x)+a\right)^p \sin (e+f x)}{f \left(2 \left(b p F_1\left(\frac{3}{2};4,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};5,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)\right) \left(-\frac{3 (a+b) 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) (\cos (2 (e+f x)) a+a+2 b)^p \tan (e+f x) \left(4 \left(b p F_1\left(\frac{3}{2};4,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};5,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan (e+f x) \sec ^2(e+f x)+3 (a+b) \left(\frac{2 b p F_1\left(\frac{3}{2};4,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{3 (a+b)}-\frac{8}{3} F_1\left(\frac{3}{2};5,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right)+2 \tan ^2(e+f x) \left(b p \left(-\frac{6 b (1-p) F_1\left(\frac{5}{2};4,2-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)}{5 (a+b)}-\frac{24}{5} F_1\left(\frac{5}{2};5,1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \tan (e+f x) \sec ^2(e+f x)\right)-4 (a+b) \left(\frac{6 b p F_1\left(\frac{5}{2};5,1-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{5 (a+b)}-6 F_1\left(\frac{5}{2};6,-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right)\right)\right) \sec ^2(e+f x)^{p-4}}{\left(2 \left(b p F_1\left(\frac{3}{2};4,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};5,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)\right){}^2}+\frac{6 (a+b) (p-4) 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) (\cos (2 (e+f x)) a+a+2 b)^p \tan ^2(e+f x) \sec ^2(e+f x)^{p-4}}{2 \left(b p F_1\left(\frac{3}{2};4,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};5,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)}-\frac{6 a (a+b) 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) (\cos (2 (e+f x)) a+a+2 b)^{p-1} \sin (2 (e+f x)) \tan (e+f x) \sec ^2(e+f x)^{p-4}}{2 \left(b p F_1\left(\frac{3}{2};4,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};5,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)}+\frac{3 (a+b) (\cos (2 (e+f x)) a+a+2 b)^p \tan (e+f x) \left(\frac{2 b p F_1\left(\frac{3}{2};4,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)}{3 (a+b)}-\frac{8}{3} F_1\left(\frac{3}{2};5,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right) \sec ^2(e+f x) \tan (e+f x)\right) \sec ^2(e+f x)^{p-4}}{2 \left(b p F_1\left(\frac{3}{2};4,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};5,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)}+\frac{3 (a+b) 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) (\cos (2 (e+f x)) a+a+2 b)^p \sec ^2(e+f x)^{p-3}}{2 \left(b p F_1\left(\frac{3}{2};4,1-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};5,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a+b}\right)\right) \tan ^2(e+f x)+3 (a+b) 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)}\right)}","\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,"(3*(a + b)*AppellF1[1/2, 4, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Cos[e + f*x]^5*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-4 + p)*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x])/(f*(3*(a + b)*AppellF1[1/2, 4, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 4, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 4*(a + b)*AppellF1[3/2, 5, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)*((3*(a + b)*AppellF1[1/2, 4, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-3 + p))/(3*(a + b)*AppellF1[1/2, 4, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 4, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 4*(a + b)*AppellF1[3/2, 5, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (6*a*(a + b)*p*AppellF1[1/2, 4, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^(-4 + p)*Sin[2*(e + f*x)]*Tan[e + f*x])/(3*(a + b)*AppellF1[1/2, 4, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 4, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 4*(a + b)*AppellF1[3/2, 5, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (6*(a + b)*(-4 + p)*AppellF1[1/2, 4, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-4 + p)*Tan[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, 4, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 4, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 4*(a + b)*AppellF1[3/2, 5, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (3*(a + b)*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-4 + p)*Tan[e + f*x]*((2*b*p*AppellF1[3/2, 4, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (8*AppellF1[3/2, 5, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(3*(a + b)*AppellF1[1/2, 4, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 4, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 4*(a + b)*AppellF1[3/2, 5, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (3*(a + b)*AppellF1[1/2, 4, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-4 + p)*Tan[e + f*x]*(4*(b*p*AppellF1[3/2, 4, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 4*(a + b)*AppellF1[3/2, 5, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Sec[e + f*x]^2*Tan[e + f*x] + 3*(a + b)*((2*b*p*AppellF1[3/2, 4, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (8*AppellF1[3/2, 5, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(b*p*((-6*b*(1 - p)*AppellF1[5/2, 4, 2 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (24*AppellF1[5/2, 5, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/5) - 4*(a + b)*((6*b*p*AppellF1[5/2, 5, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - 6*AppellF1[5/2, 6, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x]))))/(3*(a + b)*AppellF1[1/2, 4, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 2*(b*p*AppellF1[3/2, 4, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] - 4*(a + b)*AppellF1[3/2, 5, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)^2))","B",0
311,1,55,72,0.1548026,"\int \left(a+b \sec ^2(e+f x)\right) \tan ^5(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^5,x]","\frac{b \tan ^6(e+f x)}{6 f}-\frac{a \left(-\tan ^4(e+f x)+2 \tan ^2(e+f x)+4 \log (\cos (e+f x))\right)}{4 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,"(b*Tan[e + f*x]^6)/(6*f) - (a*(4*Log[Cos[e + f*x]] + 2*Tan[e + f*x]^2 - Tan[e + f*x]^4))/(4*f)","A",1
312,1,43,49,0.0776093,"\int \left(a+b \sec ^2(e+f x)\right) \tan ^3(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^3,x]","\frac{a \left(\tan ^2(e+f x)+2 \log (\cos (e+f x))\right)}{2 f}+\frac{b \tan ^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,"(b*Tan[e + f*x]^4)/(4*f) + (a*(2*Log[Cos[e + f*x]] + Tan[e + f*x]^2))/(2*f)","A",1
313,1,30,30,0.0154349,"\int \left(a+b \sec ^2(e+f x)\right) \tan (e+f x) \, dx","Integrate[(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",1
314,1,44,28,0.0285812,"\int \cot (e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Cot[e + f*x]*(a + b*Sec[e + f*x]^2),x]","\frac{a (\log (\tan (e+f x))+\log (\cos (e+f x)))}{f}-\frac{b (\log (\cos (e+f x))-\log (\sin (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]] - Log[Sin[e + f*x]]))/f) + (a*(Log[Cos[e + f*x]] + Log[Tan[e + f*x]]))/f","A",1
315,1,52,32,0.1575328,"\int \cot ^3(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Cot[e + f*x]^3*(a + b*Sec[e + f*x]^2),x]","-\frac{a \left(\cot ^2(e+f x)+2 \log (\tan (e+f x))+2 \log (\cos (e+f x))\right)}{2 f}-\frac{b \csc ^2(e+f x)}{2 f}","-\frac{(a+b) \csc ^2(e+f x)}{2 f}-\frac{a \log (\sin (e+f x))}{f}",1,"-1/2*(b*Csc[e + f*x]^2)/f - (a*(Cot[e + f*x]^2 + 2*Log[Cos[e + f*x]] + 2*Log[Tan[e + f*x]]))/(2*f)","A",1
316,1,64,51,0.1743262,"\int \cot ^5(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Cot[e + f*x]^5*(a + b*Sec[e + f*x]^2),x]","\frac{a \left(-\cot ^4(e+f x)+2 \cot ^2(e+f x)+4 \log (\tan (e+f x))+4 \log (\cos (e+f x))\right)}{4 f}-\frac{b \cot ^4(e+f x)}{4 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,"-1/4*(b*Cot[e + f*x]^4)/f + (a*(2*Cot[e + f*x]^2 - Cot[e + f*x]^4 + 4*Log[Cos[e + f*x]] + 4*Log[Tan[e + f*x]]))/(4*f)","A",1
317,1,73,64,0.0262915,"\int \left(a+b \sec ^2(e+f x)\right) \tan ^6(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^6,x]","-\frac{a \tan ^{-1}(\tan (e+f x))}{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}+\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*ArcTan[Tan[e + f*x]])/f) + (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",1
318,1,57,48,0.0199037,"\int \left(a+b \sec ^2(e+f x)\right) \tan ^4(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^4,x]","\frac{a \tan ^{-1}(\tan (e+f x))}{f}+\frac{a \tan ^3(e+f x)}{3 f}-\frac{a \tan (e+f x)}{f}+\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*ArcTan[Tan[e + f*x]])/f - (a*Tan[e + f*x])/f + (a*Tan[e + f*x]^3)/(3*f) + (b*Tan[e + f*x]^5)/(5*f)","A",1
319,1,41,32,0.0150794,"\int \left(a+b \sec ^2(e+f x)\right) \tan ^2(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^2,x]","-\frac{a \tan ^{-1}(\tan (e+f x))}{f}+\frac{a \tan (e+f x)}{f}+\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*ArcTan[Tan[e + f*x]])/f) + (a*Tan[e + f*x])/f + (b*Tan[e + f*x]^3)/(3*f)","A",1
320,1,15,15,0.0024538,"\int \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[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",1
321,1,43,19,0.0285929,"\int \cot ^2(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Cot[e + f*x]^2*(a + b*Sec[e + f*x]^2),x]","-\frac{a \cot (e+f x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(e+f x)\right)}{f}-\frac{b \cot (e+f x)}{f}","-\frac{(a+b) \cot (e+f x)}{f}-a x",1,"-((b*Cot[e + f*x])/f) - (a*Cot[e + f*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[e + f*x]^2])/f","C",1
322,1,51,33,0.0207173,"\int \cot ^4(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Cot[e + f*x]^4*(a + b*Sec[e + f*x]^2),x]","-\frac{a \cot ^3(e+f x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(e+f x)\right)}{3 f}-\frac{b \cot ^3(e+f x)}{3 f}","-\frac{(a+b) \cot ^3(e+f x)}{3 f}+\frac{a \cot (e+f x)}{f}+a x",1,"-1/3*(b*Cot[e + f*x]^3)/f - (a*Cot[e + f*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[e + f*x]^2])/(3*f)","C",1
323,1,51,51,0.0281786,"\int \cot ^6(e+f x) \left(a+b \sec ^2(e+f x)\right) \, dx","Integrate[Cot[e + f*x]^6*(a + b*Sec[e + f*x]^2),x]","-\frac{a \cot ^5(e+f x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(e+f x)\right)}{5 f}-\frac{b \cot ^5(e+f x)}{5 f}","-\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,"-1/5*(b*Cot[e + f*x]^5)/f - (a*Cot[e + f*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[e + f*x]^2])/(5*f)","C",1
324,1,126,100,0.4305902,"\int \left(a+b \sec ^2(e+f x)\right)^2 \tan ^5(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^5,x]","-\frac{\cos ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \left(-6 \left(a^2-4 a b+b^2\right) \sec ^4(e+f x)+24 a^2 \log (\cos (e+f x))-8 b (a-b) \sec ^6(e+f x)+24 a (a-b) \sec ^2(e+f x)-3 b^2 \sec ^8(e+f x)\right)}{6 f (a \cos (2 e+2 f x)+a+2 b)^2}","\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,"-1/6*(Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2*(24*a^2*Log[Cos[e + f*x]] + 24*a*(a - b)*Sec[e + f*x]^2 - 6*(a^2 - 4*a*b + b^2)*Sec[e + f*x]^4 - 8*(a - b)*b*Sec[e + f*x]^6 - 3*b^2*Sec[e + f*x]^8))/(f*(a + 2*b + a*Cos[2*e + 2*f*x])^2)","A",1
325,1,107,77,0.2467579,"\int \left(a+b \sec ^2(e+f x)\right)^2 \tan ^3(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^3,x]","\frac{\cos ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \left(12 a^2 \log (\cos (e+f x))+3 b (2 a-b) \sec ^4(e+f x)+6 a (a-2 b) \sec ^2(e+f x)+2 b^2 \sec ^6(e+f x)\right)}{3 f (a \cos (2 e+2 f x)+a+2 b)^2}","\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,"(Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2*(12*a^2*Log[Cos[e + f*x]] + 6*a*(a - 2*b)*Sec[e + f*x]^2 + 3*(2*a - b)*b*Sec[e + f*x]^4 + 2*b^2*Sec[e + f*x]^6))/(3*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2)","A",1
326,1,82,48,0.1103379,"\int \left(a+b \sec ^2(e+f x)\right)^2 \tan (e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x],x]","-\frac{\sec ^4(e+f x) \left(a \cos ^2(e+f x)+b\right)^2 \left(4 a^2 \cos ^4(e+f x) \log (\cos (e+f x))-4 a b \cos ^2(e+f x)-b^2\right)}{f (a \cos (2 (e+f x))+a+2 b)^2}","-\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,"-(((b + a*Cos[e + f*x]^2)^2*(-b^2 - 4*a*b*Cos[e + f*x]^2 + 4*a^2*Cos[e + f*x]^4*Log[Cos[e + f*x]])*Sec[e + f*x]^4)/(f*(a + 2*b + a*Cos[2*(e + f*x)])^2))","A",1
327,1,84,53,0.2410615,"\int \cot (e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Cot[e + f*x]*(a + b*Sec[e + f*x]^2)^2,x]","\frac{2 (a \cos (e+f x)+b \sec (e+f x))^2 \left(2 \cos ^2(e+f x) \left((a+b)^2 \log (\sin (e+f x))-b (2 a+b) \log (\cos (e+f x))\right)+b^2\right)}{f (a \cos (2 (e+f x))+a+2 b)^2}","\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,"(2*(b^2 + 2*Cos[e + f*x]^2*(-(b*(2*a + b)*Log[Cos[e + f*x]]) + (a + b)^2*Log[Sin[e + f*x]]))*(a*Cos[e + f*x] + b*Sec[e + f*x])^2)/(f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","A",1
328,1,81,57,0.1761183,"\int \cot ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Cot[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2,x]","-\frac{2 \left(a \cos ^2(e+f x)+b\right)^2 \left(2 \left(a^2-b^2\right) \log (\sin (e+f x))+(a+b)^2 \csc ^2(e+f x)+2 b^2 \log (\cos (e+f x))\right)}{f (a \cos (2 (e+f x))+a+2 b)^2}","-\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,"(-2*(b + a*Cos[e + f*x]^2)^2*((a + b)^2*Csc[e + f*x]^2 + 2*b^2*Log[Cos[e + f*x]] + 2*(a^2 - b^2)*Log[Sin[e + f*x]]))/(f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","A",1
329,1,77,51,0.2471347,"\int \cot ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Cot[e + f*x]^5*(a + b*Sec[e + f*x]^2)^2,x]","-\frac{\left(a \cos ^2(e+f x)+b\right)^2 \left(-4 a^2 \log (\sin (e+f x))+(a+b)^2 \csc ^4(e+f x)-4 a (a+b) \csc ^2(e+f x)\right)}{f (a \cos (2 (e+f x))+a+2 b)^2}","\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,"-(((b + a*Cos[e + f*x]^2)^2*(-4*a*(a + b)*Csc[e + f*x]^2 + (a + b)^2*Csc[e + f*x]^4 - 4*a^2*Log[Sin[e + f*x]]))/(f*(a + 2*b + a*Cos[2*(e + f*x)])^2))","A",1
330,1,275,95,2.0735296,"\int \left(a+b \sec ^2(e+f x)\right)^2 \tan ^6(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^6,x]","-\frac{4 \sec ^9(e+f x) \left(a \cos ^2(e+f x)+b\right)^2 \left(\left(231 a^2-270 a b+5 b^2\right) \tan (e) \cos ^7(e+f x)-3 \left(21 a^2-90 a b+25 b^2\right) \tan (e) \cos ^5(e+f x)-\left(483 a^2-90 a b-10 b^2\right) \sec (e) \sin (f x) \cos ^8(e+f x)+\left(231 a^2-270 a b+5 b^2\right) \sec (e) \sin (f x) \cos ^6(e+f x)-3 \left(21 a^2-90 a b+25 b^2\right) \sec (e) \sin (f x) \cos ^4(e+f x)+315 a^2 f x \cos ^9(e+f x)-5 b (18 a-19 b) \tan (e) \cos ^3(e+f x)-5 b (18 a-19 b) \sec (e) \sin (f x) \cos ^2(e+f x)-35 b^2 \tan (e) \cos (e+f x)-35 b^2 \sec (e) \sin (f x)\right)}{315 f (a \cos (2 (e+f x))+a+2 b)^2}","\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,"(-4*(b + a*Cos[e + f*x]^2)^2*Sec[e + f*x]^9*(315*a^2*f*x*Cos[e + f*x]^9 - 35*b^2*Sec[e]*Sin[f*x] - 5*(18*a - 19*b)*b*Cos[e + f*x]^2*Sec[e]*Sin[f*x] - 3*(21*a^2 - 90*a*b + 25*b^2)*Cos[e + f*x]^4*Sec[e]*Sin[f*x] + (231*a^2 - 270*a*b + 5*b^2)*Cos[e + f*x]^6*Sec[e]*Sin[f*x] - (483*a^2 - 90*a*b - 10*b^2)*Cos[e + f*x]^8*Sec[e]*Sin[f*x] - 35*b^2*Cos[e + f*x]*Tan[e] - 5*(18*a - 19*b)*b*Cos[e + f*x]^3*Tan[e] - 3*(21*a^2 - 90*a*b + 25*b^2)*Cos[e + f*x]^5*Tan[e] + (231*a^2 - 270*a*b + 5*b^2)*Cos[e + f*x]^7*Tan[e]))/(315*f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","B",1
331,1,395,77,1.1011539,"\int \left(a+b \sec ^2(e+f x)\right)^2 \tan ^4(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^4,x]","\frac{\sec (e) \sec ^7(e+f x) \left(4480 a^2 \sin (2 e+f x)-3780 a^2 \sin (2 e+3 f x)+2100 a^2 \sin (4 e+3 f x)-1540 a^2 \sin (4 e+5 f x)+420 a^2 \sin (6 e+5 f x)-280 a^2 \sin (6 e+7 f x)+3675 a^2 f x \cos (2 e+f x)+2205 a^2 f x \cos (2 e+3 f x)+2205 a^2 f x \cos (4 e+3 f x)+735 a^2 f x \cos (4 e+5 f x)+735 a^2 f x \cos (6 e+5 f x)+105 a^2 f x \cos (6 e+7 f x)+105 a^2 f x \cos (8 e+7 f x)-5320 a^2 \sin (f x)+3675 a^2 f x \cos (f x)-1260 a b \sin (2 e+f x)+924 a b \sin (2 e+3 f x)-840 a b \sin (4 e+3 f x)+168 a b \sin (4 e+5 f x)-420 a b \sin (6 e+5 f x)+84 a b \sin (6 e+7 f x)+1680 a b \sin (f x)+420 b^2 \sin (2 e+f x)-168 b^2 \sin (2 e+3 f x)-420 b^2 \sin (4 e+3 f x)+84 b^2 \sin (4 e+5 f x)+12 b^2 \sin (6 e+7 f x)+840 b^2 \sin (f x)\right)}{13440 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,"(Sec[e]*Sec[e + f*x]^7*(3675*a^2*f*x*Cos[f*x] + 3675*a^2*f*x*Cos[2*e + f*x] + 2205*a^2*f*x*Cos[2*e + 3*f*x] + 2205*a^2*f*x*Cos[4*e + 3*f*x] + 735*a^2*f*x*Cos[4*e + 5*f*x] + 735*a^2*f*x*Cos[6*e + 5*f*x] + 105*a^2*f*x*Cos[6*e + 7*f*x] + 105*a^2*f*x*Cos[8*e + 7*f*x] - 5320*a^2*Sin[f*x] + 1680*a*b*Sin[f*x] + 840*b^2*Sin[f*x] + 4480*a^2*Sin[2*e + f*x] - 1260*a*b*Sin[2*e + f*x] + 420*b^2*Sin[2*e + f*x] - 3780*a^2*Sin[2*e + 3*f*x] + 924*a*b*Sin[2*e + 3*f*x] - 168*b^2*Sin[2*e + 3*f*x] + 2100*a^2*Sin[4*e + 3*f*x] - 840*a*b*Sin[4*e + 3*f*x] - 420*b^2*Sin[4*e + 3*f*x] - 1540*a^2*Sin[4*e + 5*f*x] + 168*a*b*Sin[4*e + 5*f*x] + 84*b^2*Sin[4*e + 5*f*x] + 420*a^2*Sin[6*e + 5*f*x] - 420*a*b*Sin[6*e + 5*f*x] - 280*a^2*Sin[6*e + 7*f*x] + 84*a*b*Sin[6*e + 7*f*x] + 12*b^2*Sin[6*e + 7*f*x]))/(13440*f)","B",1
332,1,281,59,0.8026925,"\int \left(a+b \sec ^2(e+f x)\right)^2 \tan ^2(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^2,x]","-\frac{\sec (e) \sec ^5(e+f x) \left(120 a^2 \sin (2 e+f x)-120 a^2 \sin (2 e+3 f x)+30 a^2 \sin (4 e+3 f x)-30 a^2 \sin (4 e+5 f x)+150 a^2 f x \cos (2 e+f x)+75 a^2 f x \cos (2 e+3 f x)+75 a^2 f x \cos (4 e+3 f x)+15 a^2 f x \cos (4 e+5 f x)+15 a^2 f x \cos (6 e+5 f x)-180 a^2 \sin (f x)+150 a^2 f x \cos (f x)-120 a b \sin (2 e+f x)+40 a b \sin (2 e+3 f x)-60 a b \sin (4 e+3 f x)+20 a b \sin (4 e+5 f x)+80 a b \sin (f x)-60 b^2 \sin (2 e+f x)+20 b^2 \sin (2 e+3 f x)+4 b^2 \sin (4 e+5 f x)-20 b^2 \sin (f x)\right)}{480 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,"-1/480*(Sec[e]*Sec[e + f*x]^5*(150*a^2*f*x*Cos[f*x] + 150*a^2*f*x*Cos[2*e + f*x] + 75*a^2*f*x*Cos[2*e + 3*f*x] + 75*a^2*f*x*Cos[4*e + 3*f*x] + 15*a^2*f*x*Cos[4*e + 5*f*x] + 15*a^2*f*x*Cos[6*e + 5*f*x] - 180*a^2*Sin[f*x] + 80*a*b*Sin[f*x] - 20*b^2*Sin[f*x] + 120*a^2*Sin[2*e + f*x] - 120*a*b*Sin[2*e + f*x] - 60*b^2*Sin[2*e + f*x] - 120*a^2*Sin[2*e + 3*f*x] + 40*a*b*Sin[2*e + 3*f*x] + 20*b^2*Sin[2*e + 3*f*x] + 30*a^2*Sin[4*e + 3*f*x] - 60*a*b*Sin[4*e + 3*f*x] - 30*a^2*Sin[4*e + 5*f*x] + 20*a*b*Sin[4*e + 5*f*x] + 4*b^2*Sin[4*e + 5*f*x]))/f","B",1
333,1,106,40,0.3699316,"\int \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[(a + b*Sec[e + f*x]^2)^2,x]","\frac{4 \sec ^3(e+f x) \left(a \cos ^2(e+f x)+b\right)^2 \left(3 a^2 f x \cos ^3(e+f x)+2 b (3 a+b) \sec (e) \sin (f x) \cos ^2(e+f x)+b^2 \tan (e) \cos (e+f x)+b^2 \sec (e) \sin (f x)\right)}{3 f (a \cos (2 (e+f x))+a+2 b)^2}","a^2 x+\frac{b (2 a+b) \tan (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"(4*(b + a*Cos[e + f*x]^2)^2*Sec[e + f*x]^3*(3*a^2*f*x*Cos[e + f*x]^3 + b^2*Sec[e]*Sin[f*x] + 2*b*(3*a + b)*Cos[e + f*x]^2*Sec[e]*Sin[f*x] + b^2*Cos[e + f*x]*Tan[e]))/(3*f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","B",1
334,1,82,36,0.7780259,"\int \cot ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Cot[e + f*x]^2*(a + b*Sec[e + f*x]^2)^2,x]","-\frac{4 \sec (e+f x) \left(a \cos ^2(e+f x)+b\right)^2 \left(a^2 f x \cos (e+f x)-\sin (f x) \left((a+b)^2 \csc (e) \cot (e+f x)+b^2 \sec (e)\right)\right)}{f (a \cos (2 (e+f x))+a+2 b)^2}","a^2 (-x)-\frac{(a+b)^2 \cot (e+f x)}{f}+\frac{b^2 \tan (e+f x)}{f}",1,"(-4*(b + a*Cos[e + f*x]^2)^2*Sec[e + f*x]*(a^2*f*x*Cos[e + f*x] - ((a + b)^2*Cot[e + f*x]*Csc[e] + b^2*Sec[e])*Sin[f*x]))/(f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","B",1
335,1,160,45,0.9375612,"\int \cot ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Cot[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2,x]","\frac{\csc (e) \csc ^3(e+f x) \left(-12 a^2 \sin (2 e+f x)+8 a^2 \sin (2 e+3 f x)-9 a^2 f x \cos (2 e+f x)-3 a^2 f x \cos (2 e+3 f x)+3 a^2 f x \cos (4 e+3 f x)-12 a^2 \sin (f x)+9 a^2 f x \cos (f x)-12 a b \sin (2 e+f x)+4 a b \sin (2 e+3 f x)-4 b^2 \sin (2 e+3 f x)+12 b^2 \sin (f x)\right)}{24 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,"(Csc[e]*Csc[e + f*x]^3*(9*a^2*f*x*Cos[f*x] - 9*a^2*f*x*Cos[2*e + f*x] - 3*a^2*f*x*Cos[2*e + 3*f*x] + 3*a^2*f*x*Cos[4*e + 3*f*x] - 12*a^2*Sin[f*x] + 12*b^2*Sin[f*x] - 12*a^2*Sin[2*e + f*x] - 12*a*b*Sin[2*e + f*x] + 8*a^2*Sin[2*e + 3*f*x] + 4*a*b*Sin[2*e + 3*f*x] - 4*b^2*Sin[2*e + 3*f*x]))/(24*f)","B",1
336,1,256,65,1.0305493,"\int \cot ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^2 \, dx","Integrate[Cot[e + f*x]^6*(a + b*Sec[e + f*x]^2)^2,x]","\frac{\csc (e) \csc ^5(e+f x) \left(180 a^2 \sin (2 e+f x)-140 a^2 \sin (2 e+3 f x)-90 a^2 \sin (4 e+3 f x)+46 a^2 \sin (4 e+5 f x)+150 a^2 f x \cos (2 e+f x)+75 a^2 f x \cos (2 e+3 f x)-75 a^2 f x \cos (4 e+3 f x)-15 a^2 f x \cos (4 e+5 f x)+15 a^2 f x \cos (6 e+5 f x)+280 a^2 \sin (f x)-150 a^2 f x \cos (f x)-60 a b \sin (4 e+3 f x)+12 a b \sin (4 e+5 f x)+120 a b \sin (f x)-60 b^2 \sin (2 e+f x)+20 b^2 \sin (2 e+3 f x)-4 b^2 \sin (4 e+5 f x)+20 b^2 \sin (f x)\right)}{480 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,"(Csc[e]*Csc[e + f*x]^5*(-150*a^2*f*x*Cos[f*x] + 150*a^2*f*x*Cos[2*e + f*x] + 75*a^2*f*x*Cos[2*e + 3*f*x] - 75*a^2*f*x*Cos[4*e + 3*f*x] - 15*a^2*f*x*Cos[4*e + 5*f*x] + 15*a^2*f*x*Cos[6*e + 5*f*x] + 280*a^2*Sin[f*x] + 120*a*b*Sin[f*x] + 20*b^2*Sin[f*x] + 180*a^2*Sin[2*e + f*x] - 60*b^2*Sin[2*e + f*x] - 140*a^2*Sin[2*e + 3*f*x] + 20*b^2*Sin[2*e + 3*f*x] - 90*a^2*Sin[4*e + 3*f*x] - 60*a*b*Sin[4*e + 3*f*x] + 46*a^2*Sin[4*e + 5*f*x] + 12*a*b*Sin[4*e + 5*f*x] - 4*b^2*Sin[4*e + 5*f*x]))/(480*f)","B",1
337,1,99,69,0.2698368,"\int \frac{\tan ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(a b \sec ^2(e+f x)+(a+b)^2 \left(-\log \left(-a \sin ^2(e+f x)+a+b\right)\right)+2 a (a+2 b) \log (\cos (e+f x))\right)}{4 a b^2 f \left(a+b \sec ^2(e+f x)\right)}","-\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 + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(2*a*(a + 2*b)*Log[Cos[e + f*x]] - (a + b)^2*Log[a + b - a*Sin[e + f*x]^2] + a*b*Sec[e + f*x]^2))/(4*a*b^2*f*(a + b*Sec[e + f*x]^2))","A",1
338,1,41,45,0.0993323,"\int \frac{\tan ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[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 \log (\cos (e+f x))}{2 a 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,"(-2*a*Log[Cos[e + f*x]] + (a + b)*Log[b + a*Cos[e + f*x]^2])/(2*a*b*f)","A",1
339,1,26,23,0.1893512,"\int \frac{\tan (e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]/(a + b*Sec[e + f*x]^2),x]","-\frac{\log (a \cos (2 (e+f x))+a+2 b)}{2 a f}","-\frac{\log \left(a \cos ^2(e+f x)+b\right)}{2 a f}",1,"-1/2*Log[a + 2*b + a*Cos[2*(e + f*x)]]/(a*f)","A",1
340,1,43,46,0.0963989,"\int \frac{\cot (e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]/(a + b*Sec[e + f*x]^2),x]","\frac{b \log \left(-a \sin ^2(e+f x)+a+b\right)+2 a \log (\sin (e+f x))}{2 a^2 f+2 a b f}","\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,"(2*a*Log[Sin[e + f*x]] + b*Log[a + b - a*Sin[e + f*x]^2])/(2*a^2*f + 2*a*b*f)","A",1
341,1,100,74,0.227797,"\int \frac{\cot ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2),x]","-\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(b^2 \log \left(-a \sin ^2(e+f x)+a+b\right)+a (a+b) \csc ^2(e+f x)+2 a (a+2 b) \log (\sin (e+f x))\right)}{4 a f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)}","-\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,"-1/4*((a + 2*b + a*Cos[2*(e + f*x)])*(a*(a + b)*Csc[e + f*x]^2 + 2*a*(a + 2*b)*Log[Sin[e + f*x]] + b^2*Log[a + b - a*Sin[e + f*x]^2])*Sec[e + f*x]^2)/(a*(a + b)^2*f*(a + b*Sec[e + f*x]^2))","A",1
342,1,138,108,0.6168069,"\int \frac{\cot ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 e+2 f x)+a+2 b) \left(\frac{4 \left(a^2+3 a b+3 b^2\right) \log (\sin (e+f x))}{(a+b)^3}+\frac{2 b^3 \log \left(-a \sin ^2(e+f x)+a+b\right)}{a (a+b)^3}-\frac{\csc ^4(e+f x)}{a+b}+\frac{2 (2 a+3 b) \csc ^2(e+f x)}{(a+b)^2}\right)}{8 f \left(a+b \sec ^2(e+f x)\right)}","\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,"((a + 2*b + a*Cos[2*e + 2*f*x])*((2*(2*a + 3*b)*Csc[e + f*x]^2)/(a + b)^2 - Csc[e + f*x]^4/(a + b) + (4*(a^2 + 3*a*b + 3*b^2)*Log[Sin[e + f*x]])/(a + b)^3 + (2*b^3*Log[a + b - a*Sin[e + f*x]^2])/(a*(a + b)^3))*Sec[e + f*x]^2)/(8*f*(a + b*Sec[e + f*x]^2))","A",1
343,1,229,83,2.790158,"\int \frac{\tan ^6(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]^6/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(-\frac{(3 a+7 b) \sec (e) \sin (f x) \sec (e+f x)}{b^2 f}-\frac{3 (a+b)^{5/2} (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{a b^2 f \sqrt{b (\cos (e)-i \sin (e))^4}}-\frac{3 x}{a}+\frac{\sec (e) \sin (f x) \sec ^3(e+f x)}{b f}+\frac{\tan (e) \sec ^2(e+f x)}{b f}\right)}{6 \left(a+b \sec ^2(e+f x)\right)}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*((-3*x)/a - (3*(a + b)^(5/2)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(a*b^2*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) - ((3*a + 7*b)*Sec[e]*Sec[e + f*x]*Sin[f*x])/(b^2*f) + (Sec[e]*Sec[e + f*x]^3*Sin[f*x])/(b*f) + (Sec[e + f*x]^2*Tan[e])/(b*f)))/(6*(a + b*Sec[e + f*x]^2))","C",1
344,1,206,59,1.0974405,"\int \frac{\tan ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]^4/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\sqrt{a+b} \sqrt{b (\sin (e)+i \cos (e))^4} (a \sec (e) \sin (f x) \sec (e+f x)+b f x)+(a+b)^2 (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)\right)}{2 a b f \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4} \left(a+b \sec ^2(e+f x)\right)}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*((a + b)^2*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]) + Sqrt[a + b]*Sqrt[b*(I*Cos[e] + Sin[e])^4]*(b*f*x + a*Sec[e]*Sec[e + f*x]*Sin[f*x])))/(2*a*b*Sqrt[a + b]*f*(a + b*Sec[e + f*x]^2)*Sqrt[b*(Cos[e] - I*Sin[e])^4])","C",1
345,1,184,46,0.2927434,"\int \frac{\tan ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]^2/(a + b*Sec[e + f*x]^2),x]","-\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(f x \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}+(a+b) (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)\right)}{2 a f \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4} \left(a+b \sec ^2(e+f x)\right)}","\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,"-1/2*((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(Sqrt[a + b]*f*x*Sqrt[b*(Cos[e] - I*Sin[e])^4] + (a + b)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e])))/(a*Sqrt[a + b]*f*(a + b*Sec[e + f*x]^2)*Sqrt[b*(Cos[e] - I*Sin[e])^4])","C",1
346,1,182,45,0.2328067,"\int \frac{1}{a+b \sec ^2(e+f x)} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(-1),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(f x \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}+b (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)\right)}{2 a f \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4} \left(a+b \sec ^2(e+f x)\right)}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(Sqrt[a + b]*f*x*Sqrt[b*(Cos[e] - I*Sin[e])^4] + b*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e])))/(2*a*Sqrt[a + b]*f*(a + b*Sec[e + f*x]^2)*Sqrt[b*(Cos[e] - I*Sin[e])^4])","C",1
347,1,204,62,1.3536114,"\int \frac{\cot ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^2/(a + b*Sec[e + f*x]^2),x]","-\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(b^2 (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)+\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4} (f x (a+b)-a \csc (e) \sin (f x) \csc (e+f x))\right)}{2 a f (a+b)^{3/2} \sqrt{b (\cos (e)-i \sin (e))^4} \left(a+b \sec ^2(e+f x)\right)}","\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,"-1/2*((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(b^2*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]) + Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]*((a + b)*f*x - a*Csc[e]*Csc[e + f*x]*Sin[f*x])))/(a*(a + b)^(3/2)*f*(a + b*Sec[e + f*x]^2)*Sqrt[b*(Cos[e] - I*Sin[e])^4])","C",1
348,1,390,86,3.3786919,"\int \frac{\cot ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{1}{8} \sqrt{a+b} \csc (e) \sqrt{b (\cos (e)-i \sin (e))^4} \csc ^3(e+f x) \left(-12 a^2 \sin (2 e+f x)+8 a^2 \sin (2 e+3 f x)-3 a^2 f x \cos (2 e+3 f x)+3 a^2 f x \cos (4 e+3 f x)-12 a^2 \sin (f x)-18 a b \sin (2 e+f x)+14 a b \sin (2 e+3 f x)-6 a b f x \cos (2 e+3 f x)+6 a b f x \cos (4 e+3 f x)-9 f x (a+b)^2 \cos (2 e+f x)-24 a b \sin (f x)+9 f x (a+b)^2 \cos (f x)-3 b^2 f x \cos (2 e+3 f x)+3 b^2 f x \cos (4 e+3 f x)\right)+3 b^3 (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)\right)}{6 a f (a+b)^{5/2} \sqrt{b (\cos (e)-i \sin (e))^4} \left(a+b \sec ^2(e+f x)\right)}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(3*b^3*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]) + (Sqrt[a + b]*Csc[e]*Csc[e + f*x]^3*Sqrt[b*(Cos[e] - I*Sin[e])^4]*(9*(a + b)^2*f*x*Cos[f*x] - 9*(a + b)^2*f*x*Cos[2*e + f*x] - 3*a^2*f*x*Cos[2*e + 3*f*x] - 6*a*b*f*x*Cos[2*e + 3*f*x] - 3*b^2*f*x*Cos[2*e + 3*f*x] + 3*a^2*f*x*Cos[4*e + 3*f*x] + 6*a*b*f*x*Cos[4*e + 3*f*x] + 3*b^2*f*x*Cos[4*e + 3*f*x] - 12*a^2*Sin[f*x] - 24*a*b*Sin[f*x] - 12*a^2*Sin[2*e + f*x] - 18*a*b*Sin[2*e + f*x] + 8*a^2*Sin[2*e + 3*f*x] + 14*a*b*Sin[2*e + 3*f*x]))/8))/(6*a*(a + b)^(5/2)*f*(a + b*Sec[e + f*x]^2)*Sqrt[b*(Cos[e] - I*Sin[e])^4])","C",1
349,1,671,120,2.8426526,"\int \frac{\cot ^6(e+f x)}{a+b \sec ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\csc (e) \csc ^5(e+f x) \left(180 a^3 \sin (2 e+f x)-140 a^3 \sin (2 e+3 f x)-90 a^3 \sin (4 e+3 f x)+46 a^3 \sin (4 e+5 f x)+75 a^3 f x \cos (2 e+3 f x)-75 a^3 f x \cos (4 e+3 f x)-15 a^3 f x \cos (4 e+5 f x)+15 a^3 f x \cos (6 e+5 f x)+280 a^3 \sin (f x)+540 a^2 b \sin (2 e+f x)-420 a^2 b \sin (2 e+3 f x)-240 a^2 b \sin (4 e+3 f x)+132 a^2 b \sin (4 e+5 f x)+225 a^2 b f x \cos (2 e+3 f x)-225 a^2 b f x \cos (4 e+3 f x)-45 a^2 b f x \cos (4 e+5 f x)+45 a^2 b f x \cos (6 e+5 f x)+780 a^2 b \sin (f x)+480 a b^2 \sin (2 e+f x)-400 a b^2 \sin (2 e+3 f x)-180 a b^2 \sin (4 e+3 f x)+116 a b^2 \sin (4 e+5 f x)+225 a b^2 f x \cos (2 e+3 f x)-225 a b^2 f x \cos (4 e+3 f x)-45 a b^2 f x \cos (4 e+5 f x)+45 a b^2 f x \cos (6 e+5 f x)+680 a b^2 \sin (f x)+150 f x (a+b)^3 \cos (2 e+f x)-150 f x (a+b)^3 \cos (f x)+75 b^3 f x \cos (2 e+3 f x)-75 b^3 f x \cos (4 e+3 f x)-15 b^3 f x \cos (4 e+5 f x)+15 b^3 f x \cos (6 e+5 f x)\right)-\frac{480 b^4 (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{960 a f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*((-480*b^4*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + Csc[e]*Csc[e + f*x]^5*(-150*(a + b)^3*f*x*Cos[f*x] + 150*(a + b)^3*f*x*Cos[2*e + f*x] + 75*a^3*f*x*Cos[2*e + 3*f*x] + 225*a^2*b*f*x*Cos[2*e + 3*f*x] + 225*a*b^2*f*x*Cos[2*e + 3*f*x] + 75*b^3*f*x*Cos[2*e + 3*f*x] - 75*a^3*f*x*Cos[4*e + 3*f*x] - 225*a^2*b*f*x*Cos[4*e + 3*f*x] - 225*a*b^2*f*x*Cos[4*e + 3*f*x] - 75*b^3*f*x*Cos[4*e + 3*f*x] - 15*a^3*f*x*Cos[4*e + 5*f*x] - 45*a^2*b*f*x*Cos[4*e + 5*f*x] - 45*a*b^2*f*x*Cos[4*e + 5*f*x] - 15*b^3*f*x*Cos[4*e + 5*f*x] + 15*a^3*f*x*Cos[6*e + 5*f*x] + 45*a^2*b*f*x*Cos[6*e + 5*f*x] + 45*a*b^2*f*x*Cos[6*e + 5*f*x] + 15*b^3*f*x*Cos[6*e + 5*f*x] + 280*a^3*Sin[f*x] + 780*a^2*b*Sin[f*x] + 680*a*b^2*Sin[f*x] + 180*a^3*Sin[2*e + f*x] + 540*a^2*b*Sin[2*e + f*x] + 480*a*b^2*Sin[2*e + f*x] - 140*a^3*Sin[2*e + 3*f*x] - 420*a^2*b*Sin[2*e + 3*f*x] - 400*a*b^2*Sin[2*e + 3*f*x] - 90*a^3*Sin[4*e + 3*f*x] - 240*a^2*b*Sin[4*e + 3*f*x] - 180*a*b^2*Sin[4*e + 3*f*x] + 46*a^3*Sin[4*e + 5*f*x] + 132*a^2*b*Sin[4*e + 5*f*x] + 116*a*b^2*Sin[4*e + 5*f*x])))/(960*a*(a + b)^3*f*(a + b*Sec[e + f*x]^2))","C",1
350,1,109,77,0.4259544,"\int \frac{\tan ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{\sec ^4(e+f x) (a \cos (2 e+2 f x)+a+2 b)^2 \left(\left(\frac{1}{a^2}-\frac{1}{b^2}\right) \log \left(a \cos ^2(e+f x)+b\right)+\frac{(a+b)^2}{a^2 b \left(a \cos ^2(e+f x)+b\right)}+\frac{2 \log (\cos (e+f x))}{b^2}\right)}{8 f \left(a+b \sec ^2(e+f x)\right)^2}","-\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,"-1/8*((a + 2*b + a*Cos[2*e + 2*f*x])^2*((a + b)^2/(a^2*b*(b + a*Cos[e + f*x]^2)) + (2*Log[Cos[e + f*x]])/b^2 + (a^(-2) - b^(-2))*Log[b + a*Cos[e + f*x]^2])*Sec[e + f*x]^4)/(f*(a + b*Sec[e + f*x]^2)^2)","A",1
351,1,81,51,0.7178529,"\int \frac{\tan ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Tan[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2,x]","\frac{(a+2 b) \log (a \cos (2 (e+f x))+a+2 b)+a \cos (2 (e+f x)) \log (a \cos (2 (e+f x))+a+2 b)+2 (a+b)}{2 a^2 f (a \cos (2 (e+f x))+a+2 b)}","\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,"(2*(a + b) + (a + 2*b)*Log[a + 2*b + a*Cos[2*(e + f*x)]] + a*Cos[2*(e + f*x)]*Log[a + 2*b + a*Cos[2*(e + f*x)]])/(2*a^2*f*(a + 2*b + a*Cos[2*(e + f*x)]))","A",1
352,1,79,49,0.4823473,"\int \frac{\tan (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Tan[e + f*x]/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{(a+2 b) \log (a \cos (2 (e+f x))+a+2 b)+a \cos (2 (e+f x)) \log (a \cos (2 (e+f x))+a+2 b)+2 b}{2 a^2 f (a \cos (2 (e+f x))+a+2 b)}","-\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,"-1/2*(2*b + (a + 2*b)*Log[a + 2*b + a*Cos[2*(e + f*x)]] + a*Cos[2*(e + f*x)]*Log[a + 2*b + a*Cos[2*(e + f*x)]])/(a^2*f*(a + 2*b + a*Cos[2*(e + f*x)]))","A",1
353,1,112,83,0.3305868,"\int \frac{\cot (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Cot[e + f*x]/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b)^2 \left(\frac{b^2 (a+b)}{a^2 \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{b (2 a+b) \log \left(-a \sin ^2(e+f x)+a+b\right)}{a^2}+2 \log (\sin (e+f x))\right)}{8 f (a+b)^2 \left(a+b \sec ^2(e+f x)\right)^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,"((a + 2*b + a*Cos[2*(e + f*x)])^2*Sec[e + f*x]^4*(2*Log[Sin[e + f*x]] + (b*(2*a + b)*Log[a + b - a*Sin[e + f*x]^2])/a^2 + (b^2*(a + b))/(a^2*(a + b - a*Sin[e + f*x]^2))))/(8*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^2)","A",1
354,1,130,111,1.3112493,"\int \frac{\cot ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2,x]","-\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b)^2 \left(\frac{b^2 \left(\frac{2 b (a+b)}{a \cos (2 (e+f x))+a+2 b}+(3 a+b) \log \left(-a \sin ^2(e+f x)+a+b\right)\right)}{a^2}+(a+b) \csc ^2(e+f x)+2 (a+3 b) \log (\sin (e+f x))\right)}{8 f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)^2}","-\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,"-1/8*((a + 2*b + a*Cos[2*(e + f*x)])^2*((a + b)*Csc[e + f*x]^2 + 2*(a + 3*b)*Log[Sin[e + f*x]] + (b^2*((2*b*(a + b))/(a + 2*b + a*Cos[2*(e + f*x)]) + (3*a + b)*Log[a + b - a*Sin[e + f*x]^2]))/a^2)*Sec[e + f*x]^4)/((a + b)^3*f*(a + b*Sec[e + f*x]^2)^2)","A",1
355,1,162,140,1.837302,"\int \frac{\cot ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b)^2 \left(\frac{2 b^4 (a+b)}{a^2 \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{2 b^3 (4 a+b) \log \left(-a \sin ^2(e+f x)+a+b\right)}{a^2}+4 \left(a^2+4 a b+6 b^2\right) \log (\sin (e+f x))-(a+b)^2 \csc ^4(e+f x)+4 (a+b) (a+2 b) \csc ^2(e+f x)\right)}{16 f (a+b)^4 \left(a+b \sec ^2(e+f x)\right)^2}","\frac{b^4}{2 a^2 f (a+b)^3 \left(a \cos ^2(e+f x)+b\right)}+\frac{b^3 (4 a+b) \log \left(a \cos ^2(e+f x)+b\right)}{2 a^2 f (a+b)^4}+\frac{\left(a^2+4 a b+6 b^2\right) \log (\sin (e+f x))}{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,"((a + 2*b + a*Cos[2*(e + f*x)])^2*Sec[e + f*x]^4*(4*(a + b)*(a + 2*b)*Csc[e + f*x]^2 - (a + b)^2*Csc[e + f*x]^4 + 4*(a^2 + 4*a*b + 6*b^2)*Log[Sin[e + f*x]] + (2*b^3*(4*a + b)*Log[a + b - a*Sin[e + f*x]^2])/a^2 + (2*b^4*(a + b))/(a^2*(a + b - a*Sin[e + f*x]^2))))/(16*(a + b)^4*f*(a + b*Sec[e + f*x]^2)^2)","A",1
356,1,286,119,4.6256215,"\int \frac{\tan ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Tan[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(-\frac{(a+b)^2 ((a+2 b) \sin (2 e)-a \sin (2 f x))}{a^2 b^2 f (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}+\frac{(3 a-2 b) (a+b)^{3/2} (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{a^2 b^2 f \sqrt{b (\cos (e)-i \sin (e))^4}}-\frac{2 x (a \cos (2 (e+f x))+a+2 b)}{a^2}+\frac{2 \sec (e) \sin (f x) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b)}{b^2 f}\right)}{8 \left(a+b \sec ^2(e+f x)\right)^2}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^4*((-2*x*(a + 2*b + a*Cos[2*(e + f*x)]))/a^2 + ((3*a - 2*b)*(a + b)^(3/2)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[2*e] - I*Sin[2*e]))/(a^2*b^2*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (2*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[e]*Sec[e + f*x]*Sin[f*x])/(b^2*f) - ((a + b)^2*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(a^2*b^2*f*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(8*(a + b*Sec[e + f*x]^2)^2)","C",1
357,1,249,90,2.5601508,"\int \frac{\tan ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Tan[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{\left(-a^2+a b+2 b^2\right) (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{b f \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}+2 x (a \cos (2 (e+f x))+a+2 b)+\frac{(a+b) ((a+2 b) \sin (2 e)-a \sin (2 f x))}{b f (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}\right)}{8 a^2 \left(a+b \sec ^2(e+f x)\right)^2}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^4*(2*x*(a + 2*b + a*Cos[2*(e + f*x)]) + ((-a^2 + a*b + 2*b^2)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[2*e] - I*Sin[2*e]))/(b*Sqrt[a + b]*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + ((a + b)*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(b*f*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(8*a^2*(a + b*Sec[e + f*x]^2)^2)","C",1
358,1,346,85,7.6810939,"\int \frac{\tan ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Tan[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b)^2 \left(\frac{\frac{(a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}-\frac{a \sqrt{b} \sin (2 (e+f x))}{(a+b) (a \cos (2 (e+f x))+a+2 b)}}{b^{3/2} f}-\frac{\frac{\left(a^2+8 a b+8 b^2\right) ((a+2 b) \sin (2 e)-a \sin (2 f x))}{b f (a+b) (\cos (e)-\sin (e)) (\sin (e)+\cos (e)) (a \cos (2 (e+f x))+a+2 b)}+\frac{\left(-a^3+6 a^2 b+24 a b^2+16 b^3\right) (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{b f (a+b)^{3/2} \sqrt{b (\cos (e)-i \sin (e))^4}}+16 x}{a^2}\right)}{64 \left(a+b \sec ^2(e+f x)\right)^2}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])^2*Sec[e + f*x]^4*(-((16*x + ((-a^3 + 6*a^2*b + 24*a*b^2 + 16*b^3)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(b*(a + b)^(3/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + ((a^2 + 8*a*b + 8*b^2)*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(b*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[e] - Sin[e])*(Cos[e] + Sin[e])))/a^2) + (((a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(3/2) - (a*Sqrt[b]*Sin[2*(e + f*x)])/((a + b)*(a + 2*b + a*Cos[2*(e + f*x)])))/(b^(3/2)*f)))/(64*(a + b*Sec[e + f*x]^2)^2)","C",0
359,1,240,92,1.9214628,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(-2),x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(2 x (a \cos (2 (e+f x))+a+2 b)+\frac{b ((a+2 b) \sin (2 e)-a \sin (2 f x))}{f (a+b) (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}+\frac{b (3 a+2 b) (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{f (a+b)^{3/2} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{8 a^2 \left(a+b \sec ^2(e+f x)\right)^2}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^4*(2*x*(a + 2*b + a*Cos[2*(e + f*x)]) + (b*(3*a + 2*b)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[2*e] - I*Sin[2*e]))/((a + b)^(3/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (b*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/((a + b)*f*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(8*a^2*(a + b*Sec[e + f*x]^2)^2)","C",1
360,1,288,121,3.9381301,"\int \frac{\cot ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Cot[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2,x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{b^2 (a \sin (2 f x)-(a+2 b) \sin (2 e))}{a^2 f (a+b)^2 (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}-\frac{b^2 (5 a+2 b) (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{a^2 f (a+b)^{5/2} \sqrt{b (\cos (e)-i \sin (e))^4}}-\frac{2 x (a \cos (2 (e+f x))+a+2 b)}{a^2}+\frac{2 \csc (e) \sin (f x) \csc (e+f x) (a \cos (2 (e+f x))+a+2 b)}{f (a+b)^2}\right)}{8 \left(a+b \sec ^2(e+f x)\right)^2}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^4*((-2*x*(a + 2*b + a*Cos[2*(e + f*x)]))/a^2 - (b^2*(5*a + 2*b)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[2*e] - I*Sin[2*e]))/(a^2*(a + b)^(5/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (2*(a + 2*b + a*Cos[2*(e + f*x)])*Csc[e]*Csc[e + f*x]*Sin[f*x])/((a + b)^2*f) + (b^2*(-((a + 2*b)*Sin[2*e]) + a*Sin[2*f*x]))/(a^2*(a + b)^2*f*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(8*(a + b*Sec[e + f*x]^2)^2)","C",0
361,1,1588,160,6.7995313,"\int \frac{\cot ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2,x]","\frac{(\cos (2 (e+f x)) a+a+2 b) \sec ^4(e+f x) \left(\frac{48 (7 a+2 b) \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 (e+f x)) a+a+2 b) (\cos (2 e)-i \sin (2 e)) b^3}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}+\csc (e) \csc ^3(e+f x) \sec (2 e) \left(6 f x \cos (2 e-f x) a^4+6 f x \cos (2 e+f x) a^4-6 f x \cos (4 e+f x) a^4-3 f x \cos (2 e+3 f x) a^4+3 f x \cos (4 e+3 f x) a^4-3 f x \cos (6 e+3 f x) a^4-3 f x \cos (2 e+5 f x) a^4+3 f x \cos (4 e+5 f x) a^4-3 f x \cos (6 e+5 f x) a^4+3 f x \cos (8 e+5 f x) a^4-12 \sin (f x) a^4+4 \sin (3 f x) a^4+4 \sin (2 e-f x) a^4-4 \sin (2 e+f x) a^4-12 \sin (4 e+f x) a^4-12 \sin (2 e+3 f x) a^4+4 \sin (4 e+3 f x) a^4-12 \sin (6 e+3 f x) a^4+8 \sin (2 e+5 f x) a^4+8 \sin (6 e+5 f x) a^4+54 b f x \cos (2 e-f x) a^3+54 b f x \cos (2 e+f x) a^3-54 b f x \cos (4 e+f x) a^3+3 b f x \cos (2 e+3 f x) a^3-3 b f x \cos (4 e+3 f x) a^3+3 b f x \cos (6 e+3 f x) a^3-9 b f x \cos (2 e+5 f x) a^3+9 b f x \cos (4 e+5 f x) a^3-9 b f x \cos (6 e+5 f x) a^3+9 b f x \cos (8 e+5 f x) a^3-60 b \sin (f x) a^3+36 b \sin (3 f x) a^3+76 b \sin (2 e-f x) a^3-76 b \sin (2 e+f x) a^3-60 b \sin (4 e+f x) a^3-24 b \sin (2 e+3 f x) a^3+36 b \sin (4 e+3 f x) a^3-24 b \sin (6 e+3 f x) a^3+20 b \sin (2 e+5 f x) a^3+20 b \sin (6 e+5 f x) a^3+126 b^2 f x \cos (2 e-f x) a^2+126 b^2 f x \cos (2 e+f x) a^2-126 b^2 f x \cos (4 e+f x) a^2+27 b^2 f x \cos (2 e+3 f x) a^2-27 b^2 f x \cos (4 e+3 f x) a^2+27 b^2 f x \cos (6 e+3 f x) a^2-9 b^2 f x \cos (2 e+5 f x) a^2+9 b^2 f x \cos (4 e+5 f x) a^2-9 b^2 f x \cos (6 e+5 f x) a^2+9 b^2 f x \cos (8 e+5 f x) a^2-96 b^2 \sin (f x) a^2+80 b^2 \sin (3 f x) a^2+144 b^2 \sin (2 e-f x) a^2-144 b^2 \sin (2 e+f x) a^2-96 b^2 \sin (4 e+f x) a^2+80 b^2 \sin (4 e+3 f x) a^2+114 b^3 f x \cos (2 e-f x) a+114 b^3 f x \cos (2 e+f x) a-114 b^3 f x \cos (4 e+f x) a+33 b^3 f x \cos (2 e+3 f x) a-33 b^3 f x \cos (4 e+3 f x) a+33 b^3 f x \cos (6 e+3 f x) a-3 b^3 f x \cos (2 e+5 f x) a+3 b^3 f x \cos (4 e+5 f x) a-3 b^3 f x \cos (6 e+5 f x) a+3 b^3 f x \cos (8 e+5 f x) a-6 b^3 \sin (3 f x) a+6 b^3 \sin (2 e+f x) a-6 b^3 \sin (4 e+f x) a+6 b^3 \sin (2 e+3 f x) a-3 b^3 \sin (4 e+3 f x) a+3 b^3 \sin (6 e+3 f x) a+3 b^3 \sin (2 e+5 f x) a-3 b^3 \sin (4 e+5 f x) a-6 (a+b)^3 (a+6 b) f x \cos (f x)+3 (a-4 b) (a+b)^3 f x \cos (3 f x)+36 b^4 f x \cos (2 e-f x)+36 b^4 f x \cos (2 e+f x)-36 b^4 f x \cos (4 e+f x)+12 b^4 f x \cos (2 e+3 f x)-12 b^4 f x \cos (4 e+3 f x)+12 b^4 f x \cos (6 e+3 f x)+18 b^4 \sin (f x)+6 b^4 \sin (3 f x)+18 b^4 \sin (2 e-f x)+18 b^4 \sin (2 e+f x)-18 b^4 \sin (4 e+f x)-6 b^4 \sin (2 e+3 f x)-6 b^4 \sin (4 e+3 f x)+6 b^4 \sin (6 e+3 f x)\right)\right)}{384 a^2 (a+b)^3 f \left(b \sec ^2(e+f x)+a\right)^2}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^4*((48*b^3*(7*a + 2*b)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + Csc[e]*Csc[e + f*x]^3*Sec[2*e]*(-6*(a + b)^3*(a + 6*b)*f*x*Cos[f*x] + 3*(a - 4*b)*(a + b)^3*f*x*Cos[3*f*x] + 6*a^4*f*x*Cos[2*e - f*x] + 54*a^3*b*f*x*Cos[2*e - f*x] + 126*a^2*b^2*f*x*Cos[2*e - f*x] + 114*a*b^3*f*x*Cos[2*e - f*x] + 36*b^4*f*x*Cos[2*e - f*x] + 6*a^4*f*x*Cos[2*e + f*x] + 54*a^3*b*f*x*Cos[2*e + f*x] + 126*a^2*b^2*f*x*Cos[2*e + f*x] + 114*a*b^3*f*x*Cos[2*e + f*x] + 36*b^4*f*x*Cos[2*e + f*x] - 6*a^4*f*x*Cos[4*e + f*x] - 54*a^3*b*f*x*Cos[4*e + f*x] - 126*a^2*b^2*f*x*Cos[4*e + f*x] - 114*a*b^3*f*x*Cos[4*e + f*x] - 36*b^4*f*x*Cos[4*e + f*x] - 3*a^4*f*x*Cos[2*e + 3*f*x] + 3*a^3*b*f*x*Cos[2*e + 3*f*x] + 27*a^2*b^2*f*x*Cos[2*e + 3*f*x] + 33*a*b^3*f*x*Cos[2*e + 3*f*x] + 12*b^4*f*x*Cos[2*e + 3*f*x] + 3*a^4*f*x*Cos[4*e + 3*f*x] - 3*a^3*b*f*x*Cos[4*e + 3*f*x] - 27*a^2*b^2*f*x*Cos[4*e + 3*f*x] - 33*a*b^3*f*x*Cos[4*e + 3*f*x] - 12*b^4*f*x*Cos[4*e + 3*f*x] - 3*a^4*f*x*Cos[6*e + 3*f*x] + 3*a^3*b*f*x*Cos[6*e + 3*f*x] + 27*a^2*b^2*f*x*Cos[6*e + 3*f*x] + 33*a*b^3*f*x*Cos[6*e + 3*f*x] + 12*b^4*f*x*Cos[6*e + 3*f*x] - 3*a^4*f*x*Cos[2*e + 5*f*x] - 9*a^3*b*f*x*Cos[2*e + 5*f*x] - 9*a^2*b^2*f*x*Cos[2*e + 5*f*x] - 3*a*b^3*f*x*Cos[2*e + 5*f*x] + 3*a^4*f*x*Cos[4*e + 5*f*x] + 9*a^3*b*f*x*Cos[4*e + 5*f*x] + 9*a^2*b^2*f*x*Cos[4*e + 5*f*x] + 3*a*b^3*f*x*Cos[4*e + 5*f*x] - 3*a^4*f*x*Cos[6*e + 5*f*x] - 9*a^3*b*f*x*Cos[6*e + 5*f*x] - 9*a^2*b^2*f*x*Cos[6*e + 5*f*x] - 3*a*b^3*f*x*Cos[6*e + 5*f*x] + 3*a^4*f*x*Cos[8*e + 5*f*x] + 9*a^3*b*f*x*Cos[8*e + 5*f*x] + 9*a^2*b^2*f*x*Cos[8*e + 5*f*x] + 3*a*b^3*f*x*Cos[8*e + 5*f*x] - 12*a^4*Sin[f*x] - 60*a^3*b*Sin[f*x] - 96*a^2*b^2*Sin[f*x] + 18*b^4*Sin[f*x] + 4*a^4*Sin[3*f*x] + 36*a^3*b*Sin[3*f*x] + 80*a^2*b^2*Sin[3*f*x] - 6*a*b^3*Sin[3*f*x] + 6*b^4*Sin[3*f*x] + 4*a^4*Sin[2*e - f*x] + 76*a^3*b*Sin[2*e - f*x] + 144*a^2*b^2*Sin[2*e - f*x] + 18*b^4*Sin[2*e - f*x] - 4*a^4*Sin[2*e + f*x] - 76*a^3*b*Sin[2*e + f*x] - 144*a^2*b^2*Sin[2*e + f*x] + 6*a*b^3*Sin[2*e + f*x] + 18*b^4*Sin[2*e + f*x] - 12*a^4*Sin[4*e + f*x] - 60*a^3*b*Sin[4*e + f*x] - 96*a^2*b^2*Sin[4*e + f*x] - 6*a*b^3*Sin[4*e + f*x] - 18*b^4*Sin[4*e + f*x] - 12*a^4*Sin[2*e + 3*f*x] - 24*a^3*b*Sin[2*e + 3*f*x] + 6*a*b^3*Sin[2*e + 3*f*x] - 6*b^4*Sin[2*e + 3*f*x] + 4*a^4*Sin[4*e + 3*f*x] + 36*a^3*b*Sin[4*e + 3*f*x] + 80*a^2*b^2*Sin[4*e + 3*f*x] - 3*a*b^3*Sin[4*e + 3*f*x] - 6*b^4*Sin[4*e + 3*f*x] - 12*a^4*Sin[6*e + 3*f*x] - 24*a^3*b*Sin[6*e + 3*f*x] + 3*a*b^3*Sin[6*e + 3*f*x] + 6*b^4*Sin[6*e + 3*f*x] + 8*a^4*Sin[2*e + 5*f*x] + 20*a^3*b*Sin[2*e + 5*f*x] + 3*a*b^3*Sin[2*e + 5*f*x] - 3*a*b^3*Sin[4*e + 5*f*x] + 8*a^4*Sin[6*e + 5*f*x] + 20*a^3*b*Sin[6*e + 5*f*x])))/(384*a^2*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^2)","C",0
362,1,3028,207,7.3598979,"\int \frac{\cot ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^2} \, dx","Integrate[Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2,x]","\text{Result too large to show}","\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{x}{a^2}-\frac{\left(2 a^3+8 a^2 b+12 a b^2-b^3\right) \cot (e+f x)}{2 a f (a+b)^4}-\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,"((9*a + 2*b)*(a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*(-1/8*(b^4*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Cos[2*e])/(a^2*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) + ((I/8)*b^4*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Sin[2*e])/(a^2*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]])))/((a + b)^4*(a + b*Sec[e + f*x]^2)^2) + ((a + 2*b + a*Cos[2*e + 2*f*x])*Csc[e]*Csc[e + f*x]^5*Sec[2*e]*Sec[e + f*x]^4*(75*a^5*f*x*Cos[f*x] + 900*a^4*b*f*x*Cos[f*x] + 2850*a^3*b^2*f*x*Cos[f*x] + 3900*a^2*b^3*f*x*Cos[f*x] + 2475*a*b^4*f*x*Cos[f*x] + 600*b^5*f*x*Cos[f*x] - 15*a^5*f*x*Cos[3*f*x] + 240*a^4*b*f*x*Cos[3*f*x] + 1110*a^3*b^2*f*x*Cos[3*f*x] + 1740*a^2*b^3*f*x*Cos[3*f*x] + 1185*a*b^4*f*x*Cos[3*f*x] + 300*b^5*f*x*Cos[3*f*x] - 75*a^5*f*x*Cos[2*e - f*x] - 900*a^4*b*f*x*Cos[2*e - f*x] - 2850*a^3*b^2*f*x*Cos[2*e - f*x] - 3900*a^2*b^3*f*x*Cos[2*e - f*x] - 2475*a*b^4*f*x*Cos[2*e - f*x] - 600*b^5*f*x*Cos[2*e - f*x] - 75*a^5*f*x*Cos[2*e + f*x] - 900*a^4*b*f*x*Cos[2*e + f*x] - 2850*a^3*b^2*f*x*Cos[2*e + f*x] - 3900*a^2*b^3*f*x*Cos[2*e + f*x] - 2475*a*b^4*f*x*Cos[2*e + f*x] - 600*b^5*f*x*Cos[2*e + f*x] + 75*a^5*f*x*Cos[4*e + f*x] + 900*a^4*b*f*x*Cos[4*e + f*x] + 2850*a^3*b^2*f*x*Cos[4*e + f*x] + 3900*a^2*b^3*f*x*Cos[4*e + f*x] + 2475*a*b^4*f*x*Cos[4*e + f*x] + 600*b^5*f*x*Cos[4*e + f*x] + 15*a^5*f*x*Cos[2*e + 3*f*x] - 240*a^4*b*f*x*Cos[2*e + 3*f*x] - 1110*a^3*b^2*f*x*Cos[2*e + 3*f*x] - 1740*a^2*b^3*f*x*Cos[2*e + 3*f*x] - 1185*a*b^4*f*x*Cos[2*e + 3*f*x] - 300*b^5*f*x*Cos[2*e + 3*f*x] - 15*a^5*f*x*Cos[4*e + 3*f*x] + 240*a^4*b*f*x*Cos[4*e + 3*f*x] + 1110*a^3*b^2*f*x*Cos[4*e + 3*f*x] + 1740*a^2*b^3*f*x*Cos[4*e + 3*f*x] + 1185*a*b^4*f*x*Cos[4*e + 3*f*x] + 300*b^5*f*x*Cos[4*e + 3*f*x] + 15*a^5*f*x*Cos[6*e + 3*f*x] - 240*a^4*b*f*x*Cos[6*e + 3*f*x] - 1110*a^3*b^2*f*x*Cos[6*e + 3*f*x] - 1740*a^2*b^3*f*x*Cos[6*e + 3*f*x] - 1185*a*b^4*f*x*Cos[6*e + 3*f*x] - 300*b^5*f*x*Cos[6*e + 3*f*x] + 45*a^5*f*x*Cos[2*e + 5*f*x] + 120*a^4*b*f*x*Cos[2*e + 5*f*x] + 30*a^3*b^2*f*x*Cos[2*e + 5*f*x] - 180*a^2*b^3*f*x*Cos[2*e + 5*f*x] - 195*a*b^4*f*x*Cos[2*e + 5*f*x] - 60*b^5*f*x*Cos[2*e + 5*f*x] - 45*a^5*f*x*Cos[4*e + 5*f*x] - 120*a^4*b*f*x*Cos[4*e + 5*f*x] - 30*a^3*b^2*f*x*Cos[4*e + 5*f*x] + 180*a^2*b^3*f*x*Cos[4*e + 5*f*x] + 195*a*b^4*f*x*Cos[4*e + 5*f*x] + 60*b^5*f*x*Cos[4*e + 5*f*x] + 45*a^5*f*x*Cos[6*e + 5*f*x] + 120*a^4*b*f*x*Cos[6*e + 5*f*x] + 30*a^3*b^2*f*x*Cos[6*e + 5*f*x] - 180*a^2*b^3*f*x*Cos[6*e + 5*f*x] - 195*a*b^4*f*x*Cos[6*e + 5*f*x] - 60*b^5*f*x*Cos[6*e + 5*f*x] - 45*a^5*f*x*Cos[8*e + 5*f*x] - 120*a^4*b*f*x*Cos[8*e + 5*f*x] - 30*a^3*b^2*f*x*Cos[8*e + 5*f*x] + 180*a^2*b^3*f*x*Cos[8*e + 5*f*x] + 195*a*b^4*f*x*Cos[8*e + 5*f*x] + 60*b^5*f*x*Cos[8*e + 5*f*x] - 15*a^5*f*x*Cos[4*e + 7*f*x] - 60*a^4*b*f*x*Cos[4*e + 7*f*x] - 90*a^3*b^2*f*x*Cos[4*e + 7*f*x] - 60*a^2*b^3*f*x*Cos[4*e + 7*f*x] - 15*a*b^4*f*x*Cos[4*e + 7*f*x] + 15*a^5*f*x*Cos[6*e + 7*f*x] + 60*a^4*b*f*x*Cos[6*e + 7*f*x] + 90*a^3*b^2*f*x*Cos[6*e + 7*f*x] + 60*a^2*b^3*f*x*Cos[6*e + 7*f*x] + 15*a*b^4*f*x*Cos[6*e + 7*f*x] - 15*a^5*f*x*Cos[8*e + 7*f*x] - 60*a^4*b*f*x*Cos[8*e + 7*f*x] - 90*a^3*b^2*f*x*Cos[8*e + 7*f*x] - 60*a^2*b^3*f*x*Cos[8*e + 7*f*x] - 15*a*b^4*f*x*Cos[8*e + 7*f*x] + 15*a^5*f*x*Cos[10*e + 7*f*x] + 60*a^4*b*f*x*Cos[10*e + 7*f*x] + 90*a^3*b^2*f*x*Cos[10*e + 7*f*x] + 60*a^2*b^3*f*x*Cos[10*e + 7*f*x] + 15*a*b^4*f*x*Cos[10*e + 7*f*x] - 10*a^5*Sin[f*x] + 860*a^4*b*Sin[f*x] + 3120*a^3*b^2*Sin[f*x] + 3600*a^2*b^3*Sin[f*x] - 300*b^5*Sin[f*x] + 46*a^5*Sin[3*f*x] - 508*a^4*b*Sin[3*f*x] - 2324*a^3*b^2*Sin[3*f*x] - 3120*a^2*b^3*Sin[3*f*x] + 75*a*b^4*Sin[3*f*x] - 150*b^5*Sin[3*f*x] - 240*a^5*Sin[2*e - f*x] - 1840*a^4*b*Sin[2*e - f*x] - 4840*a^3*b^2*Sin[2*e - f*x] - 5040*a^2*b^3*Sin[2*e - f*x] - 300*b^5*Sin[2*e - f*x] + 240*a^5*Sin[2*e + f*x] + 1840*a^4*b*Sin[2*e + f*x] + 4840*a^3*b^2*Sin[2*e + f*x] + 5040*a^2*b^3*Sin[2*e + f*x] - 75*a*b^4*Sin[2*e + f*x] - 300*b^5*Sin[2*e + f*x] - 10*a^5*Sin[4*e + f*x] + 860*a^4*b*Sin[4*e + f*x] + 3120*a^3*b^2*Sin[4*e + f*x] + 3600*a^2*b^3*Sin[4*e + f*x] + 75*a*b^4*Sin[4*e + f*x] + 300*b^5*Sin[4*e + f*x] - 240*a^4*b*Sin[2*e + 3*f*x] - 900*a^3*b^2*Sin[2*e + 3*f*x] - 1200*a^2*b^3*Sin[2*e + 3*f*x] - 75*a*b^4*Sin[2*e + 3*f*x] + 150*b^5*Sin[2*e + 3*f*x] + 46*a^5*Sin[4*e + 3*f*x] - 508*a^4*b*Sin[4*e + 3*f*x] - 2324*a^3*b^2*Sin[4*e + 3*f*x] - 3120*a^2*b^3*Sin[4*e + 3*f*x] + 60*a*b^4*Sin[4*e + 3*f*x] + 150*b^5*Sin[4*e + 3*f*x] - 240*a^4*b*Sin[6*e + 3*f*x] - 900*a^3*b^2*Sin[6*e + 3*f*x] - 1200*a^2*b^3*Sin[6*e + 3*f*x] - 60*a*b^4*Sin[6*e + 3*f*x] - 150*b^5*Sin[6*e + 3*f*x] - 48*a^5*Sin[2*e + 5*f*x] - 32*a^4*b*Sin[2*e + 5*f*x] + 340*a^3*b^2*Sin[2*e + 5*f*x] + 864*a^2*b^3*Sin[2*e + 5*f*x] - 60*a*b^4*Sin[2*e + 5*f*x] + 30*b^5*Sin[2*e + 5*f*x] - 90*a^5*Sin[4*e + 5*f*x] - 300*a^4*b*Sin[4*e + 5*f*x] - 300*a^3*b^2*Sin[4*e + 5*f*x] + 60*a*b^4*Sin[4*e + 5*f*x] - 30*b^5*Sin[4*e + 5*f*x] - 48*a^5*Sin[6*e + 5*f*x] - 32*a^4*b*Sin[6*e + 5*f*x] + 340*a^3*b^2*Sin[6*e + 5*f*x] + 864*a^2*b^3*Sin[6*e + 5*f*x] - 15*a*b^4*Sin[6*e + 5*f*x] - 30*b^5*Sin[6*e + 5*f*x] - 90*a^5*Sin[8*e + 5*f*x] - 300*a^4*b*Sin[8*e + 5*f*x] - 300*a^3*b^2*Sin[8*e + 5*f*x] + 15*a*b^4*Sin[8*e + 5*f*x] + 30*b^5*Sin[8*e + 5*f*x] + 46*a^5*Sin[4*e + 7*f*x] + 172*a^4*b*Sin[4*e + 7*f*x] + 216*a^3*b^2*Sin[4*e + 7*f*x] + 15*a*b^4*Sin[4*e + 7*f*x] - 15*a*b^4*Sin[6*e + 7*f*x] + 46*a^5*Sin[8*e + 7*f*x] + 172*a^4*b*Sin[8*e + 7*f*x] + 216*a^3*b^2*Sin[8*e + 7*f*x]))/(7680*a^2*(a + b)^4*f*(a + b*Sec[e + f*x]^2)^2)","C",0
363,1,136,78,2.1548186,"\int \frac{\tan ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{2 \left(a^2+4 a b+3 b^2\right)+a^2 \cos ^2(2 (e+f x)) \log (a \cos (2 (e+f x))+a+2 b)+(a+2 b)^2 \log (a \cos (2 (e+f x))+a+2 b)+2 a \cos (2 (e+f x)) ((a+2 b) \log (a \cos (2 (e+f x))+a+2 b)+2 (a+b))}{2 a^3 f (a \cos (2 (e+f x))+a+2 b)^2}","\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,"-1/2*(2*(a^2 + 4*a*b + 3*b^2) + (a + 2*b)^2*Log[a + 2*b + a*Cos[2*(e + f*x)]] + a^2*Cos[2*(e + f*x)]^2*Log[a + 2*b + a*Cos[2*(e + f*x)]] + 2*a*Cos[2*(e + f*x)]*(2*(a + b) + (a + 2*b)*Log[a + 2*b + a*Cos[2*(e + f*x)]]))/(a^3*f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","A",1
364,1,131,81,0.9295389,"\int \frac{\tan ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Tan[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3,x]","\frac{2 \left(a^2+3 a b+3 b^2\right)+a^2 \cos ^2(2 (e+f x)) \log (a \cos (2 (e+f x))+a+2 b)+(a+2 b)^2 \log (a \cos (2 (e+f x))+a+2 b)+2 a (a+2 b) \cos (2 (e+f x)) (\log (a \cos (2 (e+f x))+a+2 b)+1)}{2 a^3 f (a \cos (2 (e+f x))+a+2 b)^2}","-\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,"(2*(a^2 + 3*a*b + 3*b^2) + (a + 2*b)^2*Log[a + 2*b + a*Cos[2*(e + f*x)]] + a^2*Cos[2*(e + f*x)]^2*Log[a + 2*b + a*Cos[2*(e + f*x)]] + 2*a*(a + 2*b)*Cos[2*(e + f*x)]*(1 + Log[a + 2*b + a*Cos[2*(e + f*x)]]))/(2*a^3*f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","A",1
365,1,129,74,1.330704,"\int \frac{\tan (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Tan[e + f*x]/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{a^2 \cos ^2(2 (e+f x)) \log (a \cos (2 (e+f x))+a+2 b)+(a+2 b)^2 \log (a \cos (2 (e+f x))+a+2 b)+2 a \cos (2 (e+f x)) ((a+2 b) \log (a \cos (2 (e+f x))+a+2 b)+2 b)+2 b (2 a+3 b)}{2 a^3 f (a \cos (2 (e+f x))+a+2 b)^2}","\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,"-1/2*(2*b*(2*a + 3*b) + (a + 2*b)^2*Log[a + 2*b + a*Cos[2*(e + f*x)]] + a^2*Cos[2*(e + f*x)]^2*Log[a + 2*b + a*Cos[2*(e + f*x)]] + 2*a*Cos[2*(e + f*x)]*(2*b + (a + 2*b)*Log[a + 2*b + a*Cos[2*(e + f*x)]]))/(a^3*f*(a + 2*b + a*Cos[2*(e + f*x)])^2)","A",1
366,1,158,130,1.0011567,"\int \frac{\cot (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Cot[e + f*x]/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b)^3 \left(-\frac{b^3 (a+b)^2}{a^3 \left(-a \sin ^2(e+f x)+a+b\right)^2}+\frac{2 b^2 (a+b) (3 a+2 b)}{a^3 \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{2 b \left(3 a^2+3 a b+b^2\right) \log \left(-a \sin ^2(e+f x)+a+b\right)}{a^3}+4 \log (\sin (e+f x))\right)}{32 f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)^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,"((a + 2*b + a*Cos[2*(e + f*x)])^3*Sec[e + f*x]^6*(4*Log[Sin[e + f*x]] + (2*b*(3*a^2 + 3*a*b + b^2)*Log[a + b - a*Sin[e + f*x]^2])/a^3 - (b^3*(a + b)^2)/(a^3*(a + b - a*Sin[e + f*x]^2)^2) + (2*b^2*(a + b)*(3*a + 2*b))/(a^3*(a + b - a*Sin[e + f*x]^2))))/(32*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^3)","A",1
367,1,176,154,1.7346418,"\int \frac{\cot ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b)^3 \left(-\frac{b^4 (a+b)^2}{a^3 \left(-a \sin ^2(e+f x)+a+b\right)^2}+\frac{4 b^3 (a+b) (2 a+b)}{a^3 \left(-a \sin ^2(e+f x)+a+b\right)}+\frac{2 b^2 \left(6 a^2+4 a b+b^2\right) \log \left(-a \sin ^2(e+f x)+a+b\right)}{a^3}+2 (a+b) \csc ^2(e+f x)+4 (a+4 b) \log (\sin (e+f x))\right)}{32 f (a+b)^4 \left(a+b \sec ^2(e+f x)\right)^3}","\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,"-1/32*((a + 2*b + a*Cos[2*(e + f*x)])^3*Sec[e + f*x]^6*(2*(a + b)*Csc[e + f*x]^2 + 4*(a + 4*b)*Log[Sin[e + f*x]] + (2*b^2*(6*a^2 + 4*a*b + b^2)*Log[a + b - a*Sin[e + f*x]^2])/a^3 - (b^4*(a + b)^2)/(a^3*(a + b - a*Sin[e + f*x]^2)^2) + (4*b^3*(a + b)*(2*a + b))/(a^3*(a + b - a*Sin[e + f*x]^2))))/((a + b)^4*f*(a + b*Sec[e + f*x]^2)^3)","A",1
368,1,208,192,5.2645679,"\int \frac{\cot ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b)^3 \left(-\frac{b^5 (a+b)^2}{a^3 \left(-a \sin ^2(e+f x)+a+b\right)^2}+\frac{2 b^4 (a+b) (5 a+2 b)}{a^3 \left(-a \sin ^2(e+f x)+a+b\right)}+4 \left(a^2+5 a b+10 b^2\right) \log (\sin (e+f x))+\frac{2 b^3 \left(10 a^2+5 a b+b^2\right) \log \left(-a \sin ^2(e+f x)+a+b\right)}{a^3}-(a+b)^2 \csc ^4(e+f x)+2 (a+b) (2 a+5 b) \csc ^2(e+f x)\right)}{32 f (a+b)^5 \left(a+b \sec ^2(e+f x)\right)^3}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])^3*Sec[e + f*x]^6*(2*(a + b)*(2*a + 5*b)*Csc[e + f*x]^2 - (a + b)^2*Csc[e + f*x]^4 + 4*(a^2 + 5*a*b + 10*b^2)*Log[Sin[e + f*x]] + (2*b^3*(10*a^2 + 5*a*b + b^2)*Log[a + b - a*Sin[e + f*x]^2])/a^3 - (b^5*(a + b)^2)/(a^3*(a + b - a*Sin[e + f*x]^2)^2) + (2*b^4*(a + b)*(5*a + 2*b))/(a^3*(a + b - a*Sin[e + f*x]^2))))/(32*(a + b)^5*f*(a + b*Sec[e + f*x]^2)^3)","A",1
369,1,523,147,6.4098125,"\int \frac{\tan ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Tan[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]","-\frac{\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\sec (2 e) \left(3 a^4 \sin (2 (e+2 f x))-3 a^4 \sin (4 e+2 f x)-9 a^4 \sin (2 e)+9 a^4 \sin (2 f x)-3 a^3 b \sin (2 (e+2 f x))+a^3 b \sin (4 e+2 f x)-15 a^3 b \sin (2 e)+13 a^3 b \sin (2 f x)-6 a^2 b^2 \sin (2 (e+2 f x))+20 a^2 b^2 \sin (4 e+2 f x)+4 a^2 b^2 f x \cos (2 (e+2 f x))+16 a^2 b^2 f x \cos (4 e+2 f x)+4 a^2 b^2 f x \cos (6 e+4 f x)+8 b^2 f x \left(3 a^2+8 a b+8 b^2\right) \cos (2 e)+18 a^2 b^2 \sin (2 e)-28 a^2 b^2 \sin (2 f x)+16 a b^3 \sin (4 e+2 f x)+32 a b^3 f x \cos (4 e+2 f x)+72 a b^3 \sin (2 e)-32 a b^3 \sin (2 f x)+16 a b^2 f x (a+2 b) \cos (2 f x)+48 b^4 \sin (2 e)\right)+\frac{2 \left(3 a^3-a^2 b+4 a b^2+8 b^3\right) (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{128 a^3 b^2 f \left(a+b \sec ^2(e+f x)\right)^3}","-\frac{x}{a^3}-\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{(a+b) \tan ^3(e+f x)}{4 a b f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"-1/128*((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^6*((2*(3*a^3 - a^2*b + 4*a*b^2 + 8*b^3)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])^2*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + Sec[2*e]*(8*b^2*(3*a^2 + 8*a*b + 8*b^2)*f*x*Cos[2*e] + 16*a*b^2*(a + 2*b)*f*x*Cos[2*f*x] + 4*a^2*b^2*f*x*Cos[2*(e + 2*f*x)] + 16*a^2*b^2*f*x*Cos[4*e + 2*f*x] + 32*a*b^3*f*x*Cos[4*e + 2*f*x] + 4*a^2*b^2*f*x*Cos[6*e + 4*f*x] - 9*a^4*Sin[2*e] - 15*a^3*b*Sin[2*e] + 18*a^2*b^2*Sin[2*e] + 72*a*b^3*Sin[2*e] + 48*b^4*Sin[2*e] + 9*a^4*Sin[2*f*x] + 13*a^3*b*Sin[2*f*x] - 28*a^2*b^2*Sin[2*f*x] - 32*a*b^3*Sin[2*f*x] + 3*a^4*Sin[2*(e + 2*f*x)] - 3*a^3*b*Sin[2*(e + 2*f*x)] - 6*a^2*b^2*Sin[2*(e + 2*f*x)] - 3*a^4*Sin[4*e + 2*f*x] + a^3*b*Sin[4*e + 2*f*x] + 20*a^2*b^2*Sin[4*e + 2*f*x] + 16*a*b^3*Sin[4*e + 2*f*x])))/(a^3*b^2*f*(a + b*Sec[e + f*x]^2)^3)","C",0
370,1,1473,137,14.1555468,"\int \frac{\tan ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Tan[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]","\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \left(\frac{\left(3 a^2+8 b a+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}-\frac{a \sqrt{b} \left(3 a^2+16 b a+3 (a+2 b) \cos (2 (e+f x)) a+16 b^2\right) \sin (2 (e+f x))}{(a+b)^2 (\cos (2 (e+f x)) a+a+2 b)^2}\right) \sec ^6(e+f x)}{1024 b^{5/2} f \left(b \sec ^2(e+f x)+a\right)^3}-\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \left(\frac{\sqrt{b} \left(3 a^3+14 b a^2+24 b^2 a+\left(3 a^2+4 b a+4 b^2\right) \cos (2 (e+f x)) a+16 b^3\right) \sin (2 (e+f x))}{(a+b)^2 (\cos (2 (e+f x)) a+a+2 b)^2}-\frac{3 a (a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}\right) \sec ^6(e+f x)}{2048 b^{5/2} f \left(b \sec ^2(e+f x)+a\right)^3}+\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \left(\frac{2 \left(3 a^5-10 b a^4+80 b^2 a^3+480 b^3 a^2+640 b^4 a+256 b^5\right) \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 e)-i \sin (2 e))}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}+\frac{\sec (2 e) \left(-9 \sin (2 e) a^6+9 \sin (2 f x) a^6+3 \sin (2 (e+2 f x)) a^6-3 \sin (4 e+2 f x) a^6+12 b \sin (2 e) a^5-14 b \sin (2 f x) a^5-12 b \sin (2 (e+2 f x)) a^5+10 b \sin (4 e+2 f x) a^5+128 b^2 f x \cos (2 (e+2 f x)) a^4+512 b^2 f x \cos (4 e+2 f x) a^4+128 b^2 f x \cos (6 e+4 f x) a^4+684 b^2 \sin (2 e) a^4-608 b^2 \sin (2 f x) a^4-204 b^2 \sin (2 (e+2 f x)) a^4+304 b^2 \sin (4 e+2 f x) a^4+256 b^3 f x \cos (2 (e+2 f x)) a^3+2048 b^3 f x \cos (4 e+2 f x) a^3+256 b^3 f x \cos (6 e+4 f x) a^3+2880 b^3 \sin (2 e) a^3-2112 b^3 \sin (2 f x) a^3-384 b^3 \sin (2 (e+2 f x)) a^3+1056 b^3 \sin (4 e+2 f x) a^3+128 b^4 f x \cos (2 (e+2 f x)) a^2+2560 b^4 f x \cos (4 e+2 f x) a^2+128 b^4 f x \cos (6 e+4 f x) a^2+5280 b^4 \sin (2 e) a^2-2560 b^4 \sin (2 f x) a^2-192 b^4 \sin (2 (e+2 f x)) a^2+1280 b^4 \sin (4 e+2 f x) a^2+512 b^2 (a+b)^2 (a+2 b) f x \cos (2 f x) a+1024 b^5 f x \cos (4 e+2 f x) a+4608 b^5 \sin (2 e) a-1024 b^5 \sin (2 f x) a+512 b^5 \sin (4 e+2 f x) a+256 b^2 (a+b)^2 \left(3 a^2+8 b a+8 b^2\right) f x \cos (2 e)+1536 b^6 \sin (2 e)\right)}{(\cos (2 (e+f x)) a+a+2 b)^2}\right) \sec ^6(e+f x)}{4096 a^3 b^2 (a+b)^2 f \left(b \sec ^2(e+f x)+a\right)^3}-\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \left(\frac{a \sec (2 e) \left(\left(-9 a^4-16 b a^3+48 b^2 a^2+128 b^3 a+64 b^4\right) \sin (2 f x)+a \left(-3 a^3+2 b a^2+24 b^2 a+16 b^3\right) \sin (2 (e+2 f x))+\left(3 a^4-64 b^2 a^2-128 b^3 a-64 b^4\right) \sin (4 e+2 f x)\right)+\left(9 a^5+18 b a^4-64 b^2 a^3-256 b^3 a^2-320 b^4 a-128 b^5\right) \tan (2 e)}{a^2 (\cos (2 (e+f x)) a+a+2 b)^2}-\frac{6 a^2 \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 e)-i \sin (2 e))}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) \sec ^6(e+f x)}{2048 b^2 (a+b)^2 f \left(b \sec ^2(e+f x)+a\right)^3}","\frac{x}{a^3}+\frac{(a-4 b) \tan (e+f x)}{8 a^2 b f \left(a+b \tan ^2(e+f x)+b\right)}+\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+b) \tan (e+f x)}{4 a b f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(((3*a^2 + 8*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(1024*b^(5/2)*f*(a + b*Sec[e + f*x]^2)^3) - ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((-3*a*(a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(5/2) + (Sqrt[b]*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a*b + 4*b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(2048*b^(5/2)*f*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((2*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 + 256*b^5)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (Sec[2*e]*(256*b^2*(a + b)^2*(3*a^2 + 8*a*b + 8*b^2)*f*x*Cos[2*e] + 512*a*b^2*(a + b)^2*(a + 2*b)*f*x*Cos[2*f*x] + 128*a^4*b^2*f*x*Cos[2*(e + 2*f*x)] + 256*a^3*b^3*f*x*Cos[2*(e + 2*f*x)] + 128*a^2*b^4*f*x*Cos[2*(e + 2*f*x)] + 512*a^4*b^2*f*x*Cos[4*e + 2*f*x] + 2048*a^3*b^3*f*x*Cos[4*e + 2*f*x] + 2560*a^2*b^4*f*x*Cos[4*e + 2*f*x] + 1024*a*b^5*f*x*Cos[4*e + 2*f*x] + 128*a^4*b^2*f*x*Cos[6*e + 4*f*x] + 256*a^3*b^3*f*x*Cos[6*e + 4*f*x] + 128*a^2*b^4*f*x*Cos[6*e + 4*f*x] - 9*a^6*Sin[2*e] + 12*a^5*b*Sin[2*e] + 684*a^4*b^2*Sin[2*e] + 2880*a^3*b^3*Sin[2*e] + 5280*a^2*b^4*Sin[2*e] + 4608*a*b^5*Sin[2*e] + 1536*b^6*Sin[2*e] + 9*a^6*Sin[2*f*x] - 14*a^5*b*Sin[2*f*x] - 608*a^4*b^2*Sin[2*f*x] - 2112*a^3*b^3*Sin[2*f*x] - 2560*a^2*b^4*Sin[2*f*x] - 1024*a*b^5*Sin[2*f*x] + 3*a^6*Sin[2*(e + 2*f*x)] - 12*a^5*b*Sin[2*(e + 2*f*x)] - 204*a^4*b^2*Sin[2*(e + 2*f*x)] - 384*a^3*b^3*Sin[2*(e + 2*f*x)] - 192*a^2*b^4*Sin[2*(e + 2*f*x)] - 3*a^6*Sin[4*e + 2*f*x] + 10*a^5*b*Sin[4*e + 2*f*x] + 304*a^4*b^2*Sin[4*e + 2*f*x] + 1056*a^3*b^3*Sin[4*e + 2*f*x] + 1280*a^2*b^4*Sin[4*e + 2*f*x] + 512*a*b^5*Sin[4*e + 2*f*x]))/(a + 2*b + a*Cos[2*(e + f*x)])^2))/(4096*a^3*b^2*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^3) - ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((-6*a^2*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (a*Sec[2*e]*((-9*a^4 - 16*a^3*b + 48*a^2*b^2 + 128*a*b^3 + 64*b^4)*Sin[2*f*x] + a*(-3*a^3 + 2*a^2*b + 24*a*b^2 + 16*b^3)*Sin[2*(e + 2*f*x)] + (3*a^4 - 64*a^2*b^2 - 128*a*b^3 - 64*b^4)*Sin[4*e + 2*f*x]) + (9*a^5 + 18*a^4*b - 64*a^3*b^2 - 256*a^2*b^3 - 320*a*b^4 - 128*b^5)*Tan[2*e])/(a^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(2048*b^2*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^3)","C",0
371,1,1334,138,10.6012585,"\int \frac{\tan ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Tan[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3,x]","\frac{(\cos (2 (e+f x)) a+a+2 b)^3 \sec ^6(e+f x) \left(-\frac{6 a (a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}+\frac{4 \left(3 a^2+8 b a+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}-\frac{4 a \sqrt{b} \left(3 a^2+16 b a+3 (a+2 b) \cos (2 (e+f x)) a+16 b^2\right) \sin (2 (e+f x))}{(a+b)^2 (\cos (2 (e+f x)) a+a+2 b)^2}+\frac{2 \sqrt{b} \left(3 a^3+14 b a^2+24 b^2 a+\left(3 a^2+4 b a+4 b^2\right) \cos (2 (e+f x)) a+16 b^3\right) \sin (2 (e+f x))}{(a+b)^2 (\cos (2 (e+f x)) a+a+2 b)^2}-\frac{\sqrt{b} \left(\frac{2 \left(3 a^5-10 b a^4+80 b^2 a^3+480 b^3 a^2+640 b^4 a+256 b^5\right) \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 e)-i \sin (2 e))}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}+\frac{\sec (2 e) \left(-9 \sin (2 e) a^6+9 \sin (2 f x) a^6+3 \sin (2 (e+2 f x)) a^6-3 \sin (4 e+2 f x) a^6+12 b \sin (2 e) a^5-14 b \sin (2 f x) a^5-12 b \sin (2 (e+2 f x)) a^5+10 b \sin (4 e+2 f x) a^5+128 b^2 f x \cos (2 (e+2 f x)) a^4+512 b^2 f x \cos (4 e+2 f x) a^4+128 b^2 f x \cos (6 e+4 f x) a^4+684 b^2 \sin (2 e) a^4-608 b^2 \sin (2 f x) a^4-204 b^2 \sin (2 (e+2 f x)) a^4+304 b^2 \sin (4 e+2 f x) a^4+256 b^3 f x \cos (2 (e+2 f x)) a^3+2048 b^3 f x \cos (4 e+2 f x) a^3+256 b^3 f x \cos (6 e+4 f x) a^3+2880 b^3 \sin (2 e) a^3-2112 b^3 \sin (2 f x) a^3-384 b^3 \sin (2 (e+2 f x)) a^3+1056 b^3 \sin (4 e+2 f x) a^3+128 b^4 f x \cos (2 (e+2 f x)) a^2+2560 b^4 f x \cos (4 e+2 f x) a^2+128 b^4 f x \cos (6 e+4 f x) a^2+5280 b^4 \sin (2 e) a^2-2560 b^4 \sin (2 f x) a^2-192 b^4 \sin (2 (e+2 f x)) a^2+1280 b^4 \sin (4 e+2 f x) a^2+512 b^2 (a+b)^2 (a+2 b) f x \cos (2 f x) a+1024 b^5 f x \cos (4 e+2 f x) a+4608 b^5 \sin (2 e) a-1024 b^5 \sin (2 f x) a+512 b^5 \sin (4 e+2 f x) a+256 b^2 (a+b)^2 \left(3 a^2+8 b a+8 b^2\right) f x \cos (2 e)+1536 b^6 \sin (2 e)\right)}{(\cos (2 (e+f x)) a+a+2 b)^2}\right)}{a^3 (a+b)^2}-\frac{2 \sqrt{b} \left(\frac{a \sec (2 e) \left(\left(-9 a^4-16 b a^3+48 b^2 a^2+128 b^3 a+64 b^4\right) \sin (2 f x)+a \left(-3 a^3+2 b a^2+24 b^2 a+16 b^3\right) \sin (2 (e+2 f x))+\left(3 a^4-64 b^2 a^2-128 b^3 a-64 b^4\right) \sin (4 e+2 f x)\right)+\left(9 a^5+18 b a^4-64 b^2 a^3-256 b^3 a^2-320 b^4 a-128 b^5\right) \tan (2 e)}{a^2 (\cos (2 (e+f x)) a+a+2 b)^2}-\frac{6 a^2 \tan ^{-1}\left(\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right) (\cos (2 e)-i \sin (2 e))}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{(a+b)^2}\right)}{4096 b^{5/2} f \left(b \sec ^2(e+f x)+a\right)^3}","-\frac{x}{a^3}+\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{\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{\tan (e+f x)}{4 a f \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"((a + 2*b + a*Cos[2*(e + f*x)])^3*Sec[e + f*x]^6*((-6*a*(a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(5/2) + (4*(3*a^2 + 8*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(5/2) - (4*a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2) + (2*Sqrt[b]*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a*b + 4*b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2) - (Sqrt[b]*((2*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 + 256*b^5)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (Sec[2*e]*(256*b^2*(a + b)^2*(3*a^2 + 8*a*b + 8*b^2)*f*x*Cos[2*e] + 512*a*b^2*(a + b)^2*(a + 2*b)*f*x*Cos[2*f*x] + 128*a^4*b^2*f*x*Cos[2*(e + 2*f*x)] + 256*a^3*b^3*f*x*Cos[2*(e + 2*f*x)] + 128*a^2*b^4*f*x*Cos[2*(e + 2*f*x)] + 512*a^4*b^2*f*x*Cos[4*e + 2*f*x] + 2048*a^3*b^3*f*x*Cos[4*e + 2*f*x] + 2560*a^2*b^4*f*x*Cos[4*e + 2*f*x] + 1024*a*b^5*f*x*Cos[4*e + 2*f*x] + 128*a^4*b^2*f*x*Cos[6*e + 4*f*x] + 256*a^3*b^3*f*x*Cos[6*e + 4*f*x] + 128*a^2*b^4*f*x*Cos[6*e + 4*f*x] - 9*a^6*Sin[2*e] + 12*a^5*b*Sin[2*e] + 684*a^4*b^2*Sin[2*e] + 2880*a^3*b^3*Sin[2*e] + 5280*a^2*b^4*Sin[2*e] + 4608*a*b^5*Sin[2*e] + 1536*b^6*Sin[2*e] + 9*a^6*Sin[2*f*x] - 14*a^5*b*Sin[2*f*x] - 608*a^4*b^2*Sin[2*f*x] - 2112*a^3*b^3*Sin[2*f*x] - 2560*a^2*b^4*Sin[2*f*x] - 1024*a*b^5*Sin[2*f*x] + 3*a^6*Sin[2*(e + 2*f*x)] - 12*a^5*b*Sin[2*(e + 2*f*x)] - 204*a^4*b^2*Sin[2*(e + 2*f*x)] - 384*a^3*b^3*Sin[2*(e + 2*f*x)] - 192*a^2*b^4*Sin[2*(e + 2*f*x)] - 3*a^6*Sin[4*e + 2*f*x] + 10*a^5*b*Sin[4*e + 2*f*x] + 304*a^4*b^2*Sin[4*e + 2*f*x] + 1056*a^3*b^3*Sin[4*e + 2*f*x] + 1280*a^2*b^4*Sin[4*e + 2*f*x] + 512*a*b^5*Sin[4*e + 2*f*x]))/(a + 2*b + a*Cos[2*(e + f*x)])^2))/(a^3*(a + b)^2) - (2*Sqrt[b]*((-6*a^2*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (a*Sec[2*e]*((-9*a^4 - 16*a^3*b + 48*a^2*b^2 + 128*a*b^3 + 64*b^4)*Sin[2*f*x] + a*(-3*a^3 + 2*a^2*b + 24*a*b^2 + 16*b^3)*Sin[2*(e + 2*f*x)] + (3*a^4 - 64*a^2*b^2 - 128*a*b^3 - 64*b^4)*Sin[4*e + 2*f*x]) + (9*a^5 + 18*a^4*b - 64*a^3*b^2 - 256*a^2*b^3 - 320*a*b^4 - 128*b^5)*Tan[2*e])/(a^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(a + b)^2))/(4096*b^(5/2)*f*(a + b*Sec[e + f*x]^2)^3)","C",0
372,1,332,144,5.5624506,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(-3),x]","\frac{\sec ^6(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\frac{b \left(\left(9 a^2+28 a b+16 b^2\right) \sin (2 e)-3 a (3 a+2 b) \sin (2 f x)\right) (a \cos (2 (e+f x))+a+2 b)}{f (a+b)^2 (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}+\frac{b \left(15 a^2+20 a b+8 b^2\right) (\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b)^2 \tan ^{-1}\left(\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right)}{f (a+b)^{5/2} \sqrt{b (\cos (e)-i \sin (e))^4}}-\frac{4 b^2 ((a+2 b) \sin (2 e)-a \sin (2 f x))}{f (a+b) (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}+8 x (a \cos (2 (e+f x))+a+2 b)^2\right)}{64 a^3 \left(a+b \sec ^2(e+f x)\right)^3}","\frac{x}{a^3}-\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{\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 \tan (e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^6*(8*x*(a + 2*b + a*Cos[2*(e + f*x)])^2 + (b*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e + f*x)])^2*(Cos[2*e] - I*Sin[2*e]))/((a + b)^(5/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) - (4*b^2*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/((a + b)*f*(Cos[e] - Sin[e])*(Cos[e] + Sin[e])) + (b*(a + 2*b + a*Cos[2*(e + f*x)])*((9*a^2 + 28*a*b + 16*b^2)*Sin[2*e] - 3*a*(3*a + 2*b)*Sin[2*f*x]))/((a + b)^2*f*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]))))/(64*a^3*(a + b*Sec[e + f*x]^2)^3)","C",0
373,1,2089,181,7.0919491,"\int \frac{\cot ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Cot[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3,x]","\text{Result too large to show}","-\frac{x}{a^3}-\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{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{b \cot (e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"((35*a^2 + 28*a*b + 8*b^2)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(-1/64*(b^2*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Cos[2*e])/(a^3*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) + ((I/64)*b^2*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Sin[2*e])/(a^3*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]])))/((a + b)^3*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])*Csc[e]*Csc[e + f*x]*Sec[2*e]*Sec[e + f*x]^6*(8*a^5*f*x*Cos[f*x] + 56*a^4*b*f*x*Cos[f*x] + 184*a^3*b^2*f*x*Cos[f*x] + 296*a^2*b^3*f*x*Cos[f*x] + 224*a*b^4*f*x*Cos[f*x] + 64*b^5*f*x*Cos[f*x] - 12*a^5*f*x*Cos[3*f*x] - 68*a^4*b*f*x*Cos[3*f*x] - 132*a^3*b^2*f*x*Cos[3*f*x] - 108*a^2*b^3*f*x*Cos[3*f*x] - 32*a*b^4*f*x*Cos[3*f*x] - 8*a^5*f*x*Cos[2*e - f*x] - 56*a^4*b*f*x*Cos[2*e - f*x] - 184*a^3*b^2*f*x*Cos[2*e - f*x] - 296*a^2*b^3*f*x*Cos[2*e - f*x] - 224*a*b^4*f*x*Cos[2*e - f*x] - 64*b^5*f*x*Cos[2*e - f*x] - 8*a^5*f*x*Cos[2*e + f*x] - 56*a^4*b*f*x*Cos[2*e + f*x] - 184*a^3*b^2*f*x*Cos[2*e + f*x] - 296*a^2*b^3*f*x*Cos[2*e + f*x] - 224*a*b^4*f*x*Cos[2*e + f*x] - 64*b^5*f*x*Cos[2*e + f*x] + 8*a^5*f*x*Cos[4*e + f*x] + 56*a^4*b*f*x*Cos[4*e + f*x] + 184*a^3*b^2*f*x*Cos[4*e + f*x] + 296*a^2*b^3*f*x*Cos[4*e + f*x] + 224*a*b^4*f*x*Cos[4*e + f*x] + 64*b^5*f*x*Cos[4*e + f*x] + 12*a^5*f*x*Cos[2*e + 3*f*x] + 68*a^4*b*f*x*Cos[2*e + 3*f*x] + 132*a^3*b^2*f*x*Cos[2*e + 3*f*x] + 108*a^2*b^3*f*x*Cos[2*e + 3*f*x] + 32*a*b^4*f*x*Cos[2*e + 3*f*x] - 12*a^5*f*x*Cos[4*e + 3*f*x] - 68*a^4*b*f*x*Cos[4*e + 3*f*x] - 132*a^3*b^2*f*x*Cos[4*e + 3*f*x] - 108*a^2*b^3*f*x*Cos[4*e + 3*f*x] - 32*a*b^4*f*x*Cos[4*e + 3*f*x] + 12*a^5*f*x*Cos[6*e + 3*f*x] + 68*a^4*b*f*x*Cos[6*e + 3*f*x] + 132*a^3*b^2*f*x*Cos[6*e + 3*f*x] + 108*a^2*b^3*f*x*Cos[6*e + 3*f*x] + 32*a*b^4*f*x*Cos[6*e + 3*f*x] - 4*a^5*f*x*Cos[2*e + 5*f*x] - 12*a^4*b*f*x*Cos[2*e + 5*f*x] - 12*a^3*b^2*f*x*Cos[2*e + 5*f*x] - 4*a^2*b^3*f*x*Cos[2*e + 5*f*x] + 4*a^5*f*x*Cos[4*e + 5*f*x] + 12*a^4*b*f*x*Cos[4*e + 5*f*x] + 12*a^3*b^2*f*x*Cos[4*e + 5*f*x] + 4*a^2*b^3*f*x*Cos[4*e + 5*f*x] - 4*a^5*f*x*Cos[6*e + 5*f*x] - 12*a^4*b*f*x*Cos[6*e + 5*f*x] - 12*a^3*b^2*f*x*Cos[6*e + 5*f*x] - 4*a^2*b^3*f*x*Cos[6*e + 5*f*x] + 4*a^5*f*x*Cos[8*e + 5*f*x] + 12*a^4*b*f*x*Cos[8*e + 5*f*x] + 12*a^3*b^2*f*x*Cos[8*e + 5*f*x] + 4*a^2*b^3*f*x*Cos[8*e + 5*f*x] - 32*a^5*Sin[f*x] - 64*a^4*b*Sin[f*x] - 30*a^2*b^3*Sin[f*x] - 120*a*b^4*Sin[f*x] - 48*b^5*Sin[f*x] + 32*a^5*Sin[3*f*x] + 64*a^4*b*Sin[3*f*x] + 26*a^3*b^2*Sin[3*f*x] + 86*a^2*b^3*Sin[3*f*x] + 32*a*b^4*Sin[3*f*x] - 48*a^5*Sin[2*e - f*x] - 128*a^4*b*Sin[2*e - f*x] - 128*a^3*b^2*Sin[2*e - f*x] - 30*a^2*b^3*Sin[2*e - f*x] - 120*a*b^4*Sin[2*e - f*x] - 48*b^5*Sin[2*e - f*x] + 48*a^5*Sin[2*e + f*x] + 128*a^4*b*Sin[2*e + f*x] + 102*a^3*b^2*Sin[2*e + f*x] - 86*a^2*b^3*Sin[2*e + f*x] - 136*a*b^4*Sin[2*e + f*x] - 48*b^5*Sin[2*e + f*x] - 32*a^5*Sin[4*e + f*x] - 64*a^4*b*Sin[4*e + f*x] + 26*a^3*b^2*Sin[4*e + f*x] + 86*a^2*b^3*Sin[4*e + f*x] + 136*a*b^4*Sin[4*e + f*x] + 48*b^5*Sin[4*e + f*x] - 8*a^5*Sin[2*e + 3*f*x] - 26*a^3*b^2*Sin[2*e + 3*f*x] - 86*a^2*b^3*Sin[2*e + 3*f*x] - 32*a*b^4*Sin[2*e + 3*f*x] + 32*a^5*Sin[4*e + 3*f*x] + 64*a^4*b*Sin[4*e + 3*f*x] - 13*a^3*b^2*Sin[4*e + 3*f*x] - 36*a^2*b^3*Sin[4*e + 3*f*x] - 16*a*b^4*Sin[4*e + 3*f*x] - 8*a^5*Sin[6*e + 3*f*x] + 13*a^3*b^2*Sin[6*e + 3*f*x] + 36*a^2*b^3*Sin[6*e + 3*f*x] + 16*a*b^4*Sin[6*e + 3*f*x] + 8*a^5*Sin[2*e + 5*f*x] + 13*a^3*b^2*Sin[2*e + 5*f*x] + 6*a^2*b^3*Sin[2*e + 5*f*x] - 13*a^3*b^2*Sin[4*e + 5*f*x] - 6*a^2*b^3*Sin[4*e + 5*f*x] + 8*a^5*Sin[6*e + 5*f*x]))/(512*a^3*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^3)","C",0
374,1,3340,230,7.7563577,"\int \frac{\cot ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]","\text{Result too large to show}","\frac{x}{a^3}-\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{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{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^3+32 a^2 b-15 a b^2-4 b^3\right) \cot (e+f x)}{8 a^2 f (a+b)^4}-\frac{b \cot ^3(e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"((63*a^2 + 36*a*b + 8*b^2)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((b^3*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Cos[2*e])/(64*a^3*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/64)*b^3*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Sin[2*e])/(a^3*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]])))/((a + b)^4*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])*Csc[e]*Csc[e + f*x]^3*Sec[2*e]*Sec[e + f*x]^6*(-36*a^6*f*x*Cos[f*x] - 336*a^5*b*f*x*Cos[f*x] - 1560*a^4*b^2*f*x*Cos[f*x] - 3600*a^3*b^3*f*x*Cos[f*x] - 4260*a^2*b^4*f*x*Cos[f*x] - 2496*a*b^5*f*x*Cos[f*x] - 576*b^6*f*x*Cos[f*x] + 36*a^6*f*x*Cos[3*f*x] + 240*a^5*b*f*x*Cos[3*f*x] + 408*a^4*b^2*f*x*Cos[3*f*x] - 48*a^3*b^3*f*x*Cos[3*f*x] - 732*a^2*b^4*f*x*Cos[3*f*x] - 672*a*b^5*f*x*Cos[3*f*x] - 192*b^6*f*x*Cos[3*f*x] + 36*a^6*f*x*Cos[2*e - f*x] + 336*a^5*b*f*x*Cos[2*e - f*x] + 1560*a^4*b^2*f*x*Cos[2*e - f*x] + 3600*a^3*b^3*f*x*Cos[2*e - f*x] + 4260*a^2*b^4*f*x*Cos[2*e - f*x] + 2496*a*b^5*f*x*Cos[2*e - f*x] + 576*b^6*f*x*Cos[2*e - f*x] + 36*a^6*f*x*Cos[2*e + f*x] + 336*a^5*b*f*x*Cos[2*e + f*x] + 1560*a^4*b^2*f*x*Cos[2*e + f*x] + 3600*a^3*b^3*f*x*Cos[2*e + f*x] + 4260*a^2*b^4*f*x*Cos[2*e + f*x] + 2496*a*b^5*f*x*Cos[2*e + f*x] + 576*b^6*f*x*Cos[2*e + f*x] - 36*a^6*f*x*Cos[4*e + f*x] - 336*a^5*b*f*x*Cos[4*e + f*x] - 1560*a^4*b^2*f*x*Cos[4*e + f*x] - 3600*a^3*b^3*f*x*Cos[4*e + f*x] - 4260*a^2*b^4*f*x*Cos[4*e + f*x] - 2496*a*b^5*f*x*Cos[4*e + f*x] - 576*b^6*f*x*Cos[4*e + f*x] - 36*a^6*f*x*Cos[2*e + 3*f*x] - 240*a^5*b*f*x*Cos[2*e + 3*f*x] - 408*a^4*b^2*f*x*Cos[2*e + 3*f*x] + 48*a^3*b^3*f*x*Cos[2*e + 3*f*x] + 732*a^2*b^4*f*x*Cos[2*e + 3*f*x] + 672*a*b^5*f*x*Cos[2*e + 3*f*x] + 192*b^6*f*x*Cos[2*e + 3*f*x] + 36*a^6*f*x*Cos[4*e + 3*f*x] + 240*a^5*b*f*x*Cos[4*e + 3*f*x] + 408*a^4*b^2*f*x*Cos[4*e + 3*f*x] - 48*a^3*b^3*f*x*Cos[4*e + 3*f*x] - 732*a^2*b^4*f*x*Cos[4*e + 3*f*x] - 672*a*b^5*f*x*Cos[4*e + 3*f*x] - 192*b^6*f*x*Cos[4*e + 3*f*x] - 36*a^6*f*x*Cos[6*e + 3*f*x] - 240*a^5*b*f*x*Cos[6*e + 3*f*x] - 408*a^4*b^2*f*x*Cos[6*e + 3*f*x] + 48*a^3*b^3*f*x*Cos[6*e + 3*f*x] + 732*a^2*b^4*f*x*Cos[6*e + 3*f*x] + 672*a*b^5*f*x*Cos[6*e + 3*f*x] + 192*b^6*f*x*Cos[6*e + 3*f*x] - 12*a^6*f*x*Cos[2*e + 5*f*x] - 144*a^5*b*f*x*Cos[2*e + 5*f*x] - 456*a^4*b^2*f*x*Cos[2*e + 5*f*x] - 624*a^3*b^3*f*x*Cos[2*e + 5*f*x] - 396*a^2*b^4*f*x*Cos[2*e + 5*f*x] - 96*a*b^5*f*x*Cos[2*e + 5*f*x] + 12*a^6*f*x*Cos[4*e + 5*f*x] + 144*a^5*b*f*x*Cos[4*e + 5*f*x] + 456*a^4*b^2*f*x*Cos[4*e + 5*f*x] + 624*a^3*b^3*f*x*Cos[4*e + 5*f*x] + 396*a^2*b^4*f*x*Cos[4*e + 5*f*x] + 96*a*b^5*f*x*Cos[4*e + 5*f*x] - 12*a^6*f*x*Cos[6*e + 5*f*x] - 144*a^5*b*f*x*Cos[6*e + 5*f*x] - 456*a^4*b^2*f*x*Cos[6*e + 5*f*x] - 624*a^3*b^3*f*x*Cos[6*e + 5*f*x] - 396*a^2*b^4*f*x*Cos[6*e + 5*f*x] - 96*a*b^5*f*x*Cos[6*e + 5*f*x] + 12*a^6*f*x*Cos[8*e + 5*f*x] + 144*a^5*b*f*x*Cos[8*e + 5*f*x] + 456*a^4*b^2*f*x*Cos[8*e + 5*f*x] + 624*a^3*b^3*f*x*Cos[8*e + 5*f*x] + 396*a^2*b^4*f*x*Cos[8*e + 5*f*x] + 96*a*b^5*f*x*Cos[8*e + 5*f*x] - 12*a^6*f*x*Cos[4*e + 7*f*x] - 48*a^5*b*f*x*Cos[4*e + 7*f*x] - 72*a^4*b^2*f*x*Cos[4*e + 7*f*x] - 48*a^3*b^3*f*x*Cos[4*e + 7*f*x] - 12*a^2*b^4*f*x*Cos[4*e + 7*f*x] + 12*a^6*f*x*Cos[6*e + 7*f*x] + 48*a^5*b*f*x*Cos[6*e + 7*f*x] + 72*a^4*b^2*f*x*Cos[6*e + 7*f*x] + 48*a^3*b^3*f*x*Cos[6*e + 7*f*x] + 12*a^2*b^4*f*x*Cos[6*e + 7*f*x] - 12*a^6*f*x*Cos[8*e + 7*f*x] - 48*a^5*b*f*x*Cos[8*e + 7*f*x] - 72*a^4*b^2*f*x*Cos[8*e + 7*f*x] - 48*a^3*b^3*f*x*Cos[8*e + 7*f*x] - 12*a^2*b^4*f*x*Cos[8*e + 7*f*x] + 12*a^6*f*x*Cos[10*e + 7*f*x] + 48*a^5*b*f*x*Cos[10*e + 7*f*x] + 72*a^4*b^2*f*x*Cos[10*e + 7*f*x] + 48*a^3*b^3*f*x*Cos[10*e + 7*f*x] + 12*a^2*b^4*f*x*Cos[10*e + 7*f*x] - 128*a^6*Sin[f*x] - 440*a^5*b*Sin[f*x] - 1152*a^4*b^2*Sin[f*x] - 1920*a^3*b^3*Sin[f*x] + 228*a^2*b^4*Sin[f*x] + 1320*a*b^5*Sin[f*x] + 432*b^6*Sin[f*x] + 48*a^6*Sin[3*f*x] + 104*a^5*b*Sin[3*f*x] + 640*a^4*b^2*Sin[3*f*x] + 1511*a^3*b^3*Sin[3*f*x] - 528*a^2*b^4*Sin[3*f*x] + 264*a*b^5*Sin[3*f*x] + 144*b^6*Sin[3*f*x] - 32*a^6*Sin[2*e - f*x] + 384*a^5*b*Sin[2*e - f*x] + 2048*a^4*b^2*Sin[2*e - f*x] + 3072*a^3*b^3*Sin[2*e - f*x] + 228*a^2*b^4*Sin[2*e - f*x] + 1320*a*b^5*Sin[2*e - f*x] + 432*b^6*Sin[2*e - f*x] + 32*a^6*Sin[2*e + f*x] - 384*a^5*b*Sin[2*e + f*x] - 2048*a^4*b^2*Sin[2*e + f*x] - 2919*a^3*b^3*Sin[2*e + f*x] + 642*a^2*b^4*Sin[2*e + f*x] + 1416*a*b^5*Sin[2*e + f*x] + 432*b^6*Sin[2*e + f*x] - 128*a^6*Sin[4*e + f*x] - 440*a^5*b*Sin[4*e + f*x] - 1152*a^4*b^2*Sin[4*e + f*x] - 2073*a^3*b^3*Sin[4*e + f*x] - 642*a^2*b^4*Sin[4*e + f*x] - 1416*a*b^5*Sin[4*e + f*x] - 432*b^6*Sin[4*e + f*x] - 144*a^6*Sin[2*e + 3*f*x] - 672*a^5*b*Sin[2*e + 3*f*x] - 960*a^4*b^2*Sin[2*e + 3*f*x] + 153*a^3*b^3*Sin[2*e + 3*f*x] + 528*a^2*b^4*Sin[2*e + 3*f*x] - 264*a*b^5*Sin[2*e + 3*f*x] - 144*b^6*Sin[2*e + 3*f*x] + 48*a^6*Sin[4*e + 3*f*x] + 104*a^5*b*Sin[4*e + 3*f*x] + 640*a^4*b^2*Sin[4*e + 3*f*x] + 1664*a^3*b^3*Sin[4*e + 3*f*x] - 66*a^2*b^4*Sin[4*e + 3*f*x] - 408*a*b^5*Sin[4*e + 3*f*x] - 144*b^6*Sin[4*e + 3*f*x] - 144*a^6*Sin[6*e + 3*f*x] - 672*a^5*b*Sin[6*e + 3*f*x] - 960*a^4*b^2*Sin[6*e + 3*f*x] + 66*a^2*b^4*Sin[6*e + 3*f*x] + 408*a*b^5*Sin[6*e + 3*f*x] + 144*b^6*Sin[6*e + 3*f*x] + 80*a^6*Sin[2*e + 5*f*x] + 480*a^5*b*Sin[2*e + 5*f*x] + 832*a^4*b^2*Sin[2*e + 5*f*x] + 294*a^2*b^4*Sin[2*e + 5*f*x] + 96*a*b^5*Sin[2*e + 5*f*x] - 48*a^6*Sin[4*e + 5*f*x] - 120*a^5*b*Sin[4*e + 5*f*x] - 294*a^2*b^4*Sin[4*e + 5*f*x] - 96*a*b^5*Sin[4*e + 5*f*x] + 80*a^6*Sin[6*e + 5*f*x] + 480*a^5*b*Sin[6*e + 5*f*x] + 832*a^4*b^2*Sin[6*e + 5*f*x] - 51*a^3*b^3*Sin[6*e + 5*f*x] - 132*a^2*b^4*Sin[6*e + 5*f*x] - 48*a*b^5*Sin[6*e + 5*f*x] - 48*a^6*Sin[8*e + 5*f*x] - 120*a^5*b*Sin[8*e + 5*f*x] + 51*a^3*b^3*Sin[8*e + 5*f*x] + 132*a^2*b^4*Sin[8*e + 5*f*x] + 48*a*b^5*Sin[8*e + 5*f*x] + 32*a^6*Sin[4*e + 7*f*x] + 104*a^5*b*Sin[4*e + 7*f*x] + 51*a^3*b^3*Sin[4*e + 7*f*x] + 18*a^2*b^4*Sin[4*e + 7*f*x] - 51*a^3*b^3*Sin[6*e + 7*f*x] - 18*a^2*b^4*Sin[6*e + 7*f*x] + 32*a^6*Sin[8*e + 7*f*x] + 104*a^5*b*Sin[8*e + 7*f*x]))/(6144*a^3*(a + b)^4*f*(a + b*Sec[e + f*x]^2)^3)","C",0
375,1,976,285,8.3419723,"\int \frac{\cot ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^3} \, dx","Integrate[Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]","\frac{\left(99 a^2+44 b a+8 b^2\right) (\cos (2 e+2 f x) a+a+2 b)^3 \left(\frac{i b^4 \tan ^{-1}\left(\sec (f x) \left(\frac{\cos (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}-\frac{i \sin (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}\right) (-a \sin (f x)-2 b \sin (f x)+a \sin (2 e+f x))\right) \sin (2 e)}{64 a^3 \sqrt{a+b} f \sqrt{b \cos (4 e)-i b \sin (4 e)}}-\frac{b^4 \tan ^{-1}\left(\sec (f x) \left(\frac{\cos (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}-\frac{i \sin (2 e)}{2 \sqrt{a+b} \sqrt{b \cos (4 e)-i b \sin (4 e)}}\right) (-a \sin (f x)-2 b \sin (f x)+a \sin (2 e+f x))\right) \cos (2 e)}{64 a^3 \sqrt{a+b} f \sqrt{b \cos (4 e)-i b \sin (4 e)}}\right) \sec ^6(e+f x)}{(a+b)^5 \left(b \sec ^2(e+f x)+a\right)^3}+\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \csc (e) \csc ^5(e+f x) \sin (f x) \sec ^6(e+f x)}{40 (a+b)^3 f \left(b \sec ^2(e+f x)+a\right)^3}+\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \csc (e) \csc ^3(e+f x) (-11 a \sin (f x)-26 b \sin (f x)) \sec ^6(e+f x)}{120 (a+b)^4 f \left(b \sec ^2(e+f x)+a\right)^3}+\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \csc (e) \csc (e+f x) \left(23 \sin (f x) a^2+106 b \sin (f x) a+173 b^2 \sin (f x)\right) \sec ^6(e+f x)}{120 (a+b)^5 f \left(b \sec ^2(e+f x)+a\right)^3}+\frac{(\cos (2 e+2 f x) a+a+2 b) \sec (2 e) \left(2 \sin (2 e) b^6+a \sin (2 e) b^5-a \sin (2 f x) b^5\right) \sec ^6(e+f x)}{16 a^3 (a+b)^4 f \left(b \sec ^2(e+f x)+a\right)^3}+\frac{(\cos (2 e+2 f x) a+a+2 b)^2 \sec (2 e) \left(-16 \sin (2 e) b^6-52 a \sin (2 e) b^5+6 a \sin (2 f x) b^5-21 a^2 \sin (2 e) b^4+21 a^2 \sin (2 f x) b^4\right) \sec ^6(e+f x)}{64 a^3 (a+b)^5 f \left(b \sec ^2(e+f x)+a\right)^3}-\frac{(\cos (2 e+2 f x) a+a+2 b)^3 \cot (e) \csc ^4(e+f x) \sec ^6(e+f x)}{40 (a+b)^3 f \left(b \sec ^2(e+f x)+a\right)^3}-\frac{x (\cos (2 e+2 f x) a+a+2 b)^3 \sec ^6(e+f x)}{8 a^3 \left(b \sec ^2(e+f x)+a\right)^3}+\frac{(11 a \cos (e)+26 b \cos (e)) (\cos (2 e+2 f x) a+a+2 b)^3 \csc (e) \csc ^2(e+f x) \sec ^6(e+f x)}{120 (a+b)^4 f \left(b \sec ^2(e+f x)+a\right)^3}","-\frac{x}{a^3}-\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{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{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^3+32 a^2 b-51 a b^2-12 b^3\right) \cot ^3(e+f x)}{24 a^2 f (a+b)^4}-\frac{\left(8 a^4+40 a^3 b+80 a^2 b^2-19 a b^3-4 b^4\right) \cot (e+f x)}{8 a^2 f (a+b)^5}-\frac{b \cot ^5(e+f x)}{4 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^2}",1,"-1/8*(x*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6)/(a^3*(a + b*Sec[e + f*x]^2)^3) + ((11*a*Cos[e] + 26*b*Cos[e])*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Csc[e]*Csc[e + f*x]^2*Sec[e + f*x]^6)/(120*(a + b)^4*f*(a + b*Sec[e + f*x]^2)^3) - ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Cot[e]*Csc[e + f*x]^4*Sec[e + f*x]^6)/(40*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^3) + ((99*a^2 + 44*a*b + 8*b^2)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(-1/64*(b^4*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Cos[2*e])/(a^3*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) + ((I/64)*b^4*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Sin[2*e])/(a^3*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]])))/((a + b)^5*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Csc[e]*Csc[e + f*x]^5*Sec[e + f*x]^6*Sin[f*x])/(40*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Csc[e]*Csc[e + f*x]^3*Sec[e + f*x]^6*(-11*a*Sin[f*x] - 26*b*Sin[f*x]))/(120*(a + b)^4*f*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Csc[e]*Csc[e + f*x]*Sec[e + f*x]^6*(23*a^2*Sin[f*x] + 106*a*b*Sin[f*x] + 173*b^2*Sin[f*x]))/(120*(a + b)^5*f*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])*Sec[2*e]*Sec[e + f*x]^6*(a*b^5*Sin[2*e] + 2*b^6*Sin[2*e] - a*b^5*Sin[2*f*x]))/(16*a^3*(a + b)^4*f*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[2*e]*Sec[e + f*x]^6*(-21*a^2*b^4*Sin[2*e] - 52*a*b^5*Sin[2*e] - 16*b^6*Sin[2*e] + 21*a^2*b^4*Sin[2*f*x] + 6*a*b^5*Sin[2*f*x]))/(64*a^3*(a + b)^5*f*(a + b*Sec[e + f*x]^2)^3)","C",0
376,0,0,111,2.2166015,"\int \sqrt{a+b \sec ^2(e+f x)} \tan ^5(e+f x) \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^5,x]","\int \sqrt{a+b \sec ^2(e+f x)} \tan ^5(e+f x) \, dx","\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,"Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^5, x]","F",-1
377,0,0,80,1.0102335,"\int \sqrt{a+b \sec ^2(e+f x)} \tan ^3(e+f x) \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^3,x]","\int \sqrt{a+b \sec ^2(e+f x)} \tan ^3(e+f x) \, dx","\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,"Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^3, x]","F",-1
378,1,119,54,0.4383778,"\int \sqrt{a+b \sec ^2(e+f x)} \tan (e+f x) \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x],x]","\frac{\sqrt{a+b \sec ^2(e+f x)} \left(\sqrt{2} \sqrt{b} \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{b}}-2 \sqrt{a} \cos (e+f x) \sinh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right)\right)}{\sqrt{2} \sqrt{b} f \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{b}}}","\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,"((-2*Sqrt[a]*ArcSinh[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]]*Cos[e + f*x] + Sqrt[2]*Sqrt[b]*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/b])*Sqrt[a + b*Sec[e + f*x]^2])/(Sqrt[2]*Sqrt[b]*f*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/b])","B",1
379,0,0,70,1.6116418,"\int \cot (e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2],x]","\int \cot (e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","\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,"Integrate[Cot[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
380,1,527,109,5.6805737,"\int \cot ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{e^{i (e+f x)} \cos (e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{1+e^{2 i (e+f x)}}{\left(-1+e^{2 i (e+f x)}\right)^2}-\frac{(2 a+b) \log \left(1-e^{2 i (e+f x)}\right)+\sqrt{a} \sqrt{a+b} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)+\sqrt{a} \sqrt{a+b} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)-2 a \log \left(\sqrt{a+b} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+b e^{2 i (e+f x)}+b\right)-b \log \left(\sqrt{a+b} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+b e^{2 i (e+f x)}+b\right)-2 i \sqrt{a} f x \sqrt{a+b}}{\sqrt{a+b} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}\right) \sqrt{a+b \sec ^2(e+f x)}}{\sqrt{2} f \sqrt{a \cos (2 e+2 f x)+a+2 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,"(E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]*((1 + E^((2*I)*(e + f*x)))/(-1 + E^((2*I)*(e + f*x)))^2 - ((-2*I)*Sqrt[a]*Sqrt[a + b]*f*x + (2*a + b)*Log[1 - E^((2*I)*(e + f*x))] + Sqrt[a]*Sqrt[a + b]*Log[a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] + Sqrt[a]*Sqrt[a + b]*Log[a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - 2*a*Log[a + b + a*E^((2*I)*(e + f*x)) + b*E^((2*I)*(e + f*x)) + Sqrt[a + b]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - b*Log[a + b + a*E^((2*I)*(e + f*x)) + b*E^((2*I)*(e + f*x)) + Sqrt[a + b]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]])/(Sqrt[a + b]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]))*Sqrt[a + b*Sec[e + f*x]^2])/(Sqrt[2]*f*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])","C",0
381,0,0,161,5.0812179,"\int \cot ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2],x]","\int \cot ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","-\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,"Integrate[Cot[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
382,1,263,219,3.5711615,"\int \sqrt{a+b \sec ^2(e+f x)} \tan ^6(e+f x) \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^6,x]","-\frac{\tan (e+f x) \sec ^4(e+f x) \left(4 \left(3 a^2+12 a b-7 b^2\right) \cos (2 (e+f x))+\left(3 a^2+14 a b-33 b^2\right) \cos (4 (e+f x))+9 a^2+34 a b-59 b^2\right) \sqrt{a+b \sec ^2(e+f x)}}{384 b^2 f}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(16 \sqrt{a} b^2 \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)-\frac{\left(a^3+5 a^2 b+15 a b^2-5 b^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{b}}\right)}{8 \sqrt{2} b^2 f \sqrt{a \cos (2 e+2 f x)+a+2 b}}","\frac{\left(a^3+5 a^2 b+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{\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 ^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}",1,"-1/8*((16*Sqrt[a]*b^2*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]] - ((a^3 + 5*a^2*b + 15*a*b^2 - 5*b^3)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[b])*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(Sqrt[2]*b^2*f*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]) - ((9*a^2 + 34*a*b - 59*b^2 + 4*(3*a^2 + 12*a*b - 7*b^2)*Cos[2*(e + f*x)] + (3*a^2 + 14*a*b - 33*b^2)*Cos[4*(e + f*x)])*Sec[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x])/(384*b^2*f)","A",1
383,1,208,165,2.6329949,"\int \sqrt{a+b \sec ^2(e+f x)} \tan ^4(e+f x) \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^4,x]","\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(8 \sqrt{a} b \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)-\frac{\left(a^2+6 a b-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{b}}\right)}{4 \sqrt{2} b f \sqrt{a \cos (2 e+2 f x)+a+2 b}}+\frac{\tan (e+f x) \sec ^2(e+f x) ((a-5 b) \cos (2 (e+f x))+a-b) \sqrt{a+b \sec ^2(e+f x)}}{16 b 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{(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{\tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 f}",1,"((8*Sqrt[a]*b*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]] - ((a^2 + 6*a*b - 3*b^2)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[b])*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(4*Sqrt[2]*b*f*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]) + ((a - b + (a - 5*b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x])/(16*b*f)","A",1
384,1,526,118,4.1930287,"\int \sqrt{a+b \sec ^2(e+f x)} \tan ^2(e+f x) \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^2,x]","\frac{e^{i (e+f x)} \cos (e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{i \sqrt{a} \sqrt{b} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)-i \sqrt{a} \sqrt{b} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)-a \log \left(\frac{2 f \left(\sqrt{b} \left(-1+e^{2 i (e+f x)}\right)-i \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}\right)}{(a-b) \left(1+e^{2 i (e+f x)}\right)}\right)+b \log \left(\frac{2 f \left(\sqrt{b} \left(-1+e^{2 i (e+f x)}\right)-i \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}\right)}{(a-b) \left(1+e^{2 i (e+f x)}\right)}\right)-2 \sqrt{a} \sqrt{b} f x}{\sqrt{b} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}-\frac{i \left(-1+e^{2 i (e+f x)}\right)}{\left(1+e^{2 i (e+f x)}\right)^2}\right) \sqrt{a+b \sec ^2(e+f x)}}{\sqrt{2} f \sqrt{a \cos (2 e+2 f x)+a+2 b}}","\frac{\tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 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{(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,"(E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]*(((-I)*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))^2 + (-2*Sqrt[a]*Sqrt[b]*f*x + I*Sqrt[a]*Sqrt[b]*Log[a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - I*Sqrt[a]*Sqrt[b]*Log[a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - a*Log[(2*(Sqrt[b]*(-1 + E^((2*I)*(e + f*x))) - I*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*f)/((a - b)*(1 + E^((2*I)*(e + f*x))))] + b*Log[(2*(Sqrt[b]*(-1 + E^((2*I)*(e + f*x))) - I*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*f)/((a - b)*(1 + E^((2*I)*(e + f*x))))])/(Sqrt[b]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]))*Sqrt[a + b*Sec[e + f*x]^2])/(Sqrt[2]*f*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])","C",1
385,0,0,79,0.0835965,"\int \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]^2],x]","\int \sqrt{a+b \sec ^2(e+f x)} \, dx","\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,"Integrate[Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
386,1,130,69,0.5715921,"\int \cot ^2(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\cot (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(\sqrt{2} \sqrt{a} \sin (e+f x) \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)+\sqrt{a+b} \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}}\right)}{f \sqrt{a+b} \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{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 (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f}",1,"-((Cot[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2]*(Sqrt[a + b]*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)] + Sqrt[2]*Sqrt[a]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x]))/(Sqrt[a + b]*f*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]))","A",1
387,1,176,114,0.7352165,"\int \cot ^4(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\sqrt{2} \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \left(\frac{\csc ^3(e+f x) \left(-a \sin ^2(e+f x)+a+b\right)^{3/2}}{a+b}-3 \csc (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b} \left(\frac{\sqrt{a} \sin (e+f x) \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{a+b} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}+1\right)\right)}{3 f \sqrt{a \cos (2 e+2 f x)+a+2 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,"-1/3*(Sqrt[2]*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2]*((Csc[e + f*x]^3*(a + b - a*Sin[e + f*x]^2)^(3/2))/(a + b) - 3*Csc[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2]*(1 + (Sqrt[a]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/(Sqrt[a + b]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]))))/(f*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])","A",1
388,1,178,167,1.6508436,"\int \cot ^6(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\cot (e+f x) \left(-\left(11 a^2+21 a b+10 b^2\right) \csc ^2(e+f x)+23 a^2+3 (a+b)^2 \csc ^4(e+f x)+40 a b+15 b^2\right) \sqrt{a+b \sec ^2(e+f x)}}{15 f (a+b)^2}-\frac{\sqrt{2} \sqrt{a} \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{f \sqrt{a \cos (2 e+2 f x)+a+2 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[2]*Sqrt[a]*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]]*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(f*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])) - (Cot[e + f*x]*(23*a^2 + 40*a*b + 15*b^2 - (11*a^2 + 21*a*b + 10*b^2)*Csc[e + f*x]^2 + 3*(a + b)^2*Csc[e + f*x]^4)*Sqrt[a + b*Sec[e + f*x]^2])/(15*(a + b)^2*f)","A",1
389,0,0,135,2.9591062,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \tan ^5(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^5,x]","\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \tan ^5(e+f x) \, dx","-\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,"Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^5, x]","F",-1
390,0,0,104,1.9425543,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \tan ^3(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^3,x]","\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \tan ^3(e+f x) \, dx","\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,"Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^3, x]","F",-1
391,1,84,78,0.2318461,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \tan (e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x],x]","\frac{2 b \left(a+b \sec ^2(e+f x)\right)^{3/2} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{a \cos ^2(e+f x)}{b}\right)}{3 f \sqrt{\frac{a \cos ^2(e+f x)}{b}+1} (a \cos (2 (e+f x))+a+2 b)}","-\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,"(2*b*Hypergeometric2F1[-3/2, -3/2, -1/2, -((a*Cos[e + f*x]^2)/b)]*(a + b*Sec[e + f*x]^2)^(3/2))/(3*f*Sqrt[1 + (a*Cos[e + f*x]^2)/b]*(a + 2*b + a*Cos[2*(e + f*x)]))","C",1
392,1,506,91,5.2450435,"\int \cot (e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} e^{i (e+f x)} \cos ^3(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{a^{3/2} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)+a^{3/2} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)-2 i a^{3/2} f x-2 a \sqrt{a+b} \log \left(\sqrt{a+b} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+b e^{2 i (e+f x)}+b\right)+2 (a+b)^{3/2} \log \left(1-e^{2 i (e+f x)}\right)-2 b \sqrt{a+b} \log \left(\sqrt{a+b} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+b e^{2 i (e+f x)}+b\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}+\frac{2 b}{1+e^{2 i (e+f x)}}\right) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{f (a \cos (2 e+2 f x)+a+2 b)^{3/2}}","\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,"(Sqrt[2]*E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]^3*((2*b)/(1 + E^((2*I)*(e + f*x))) + ((-2*I)*a^(3/2)*f*x + 2*(a + b)^(3/2)*Log[1 - E^((2*I)*(e + f*x))] + a^(3/2)*Log[a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] + a^(3/2)*Log[a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - 2*a*Sqrt[a + b]*Log[a + b + a*E^((2*I)*(e + f*x)) + b*E^((2*I)*(e + f*x)) + Sqrt[a + b]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - 2*b*Sqrt[a + b]*Log[a + b + a*E^((2*I)*(e + f*x)) + b*E^((2*I)*(e + f*x)) + Sqrt[a + b]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]])/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*(a + b*Sec[e + f*x]^2)^(3/2))/(f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2))","C",0
393,1,622,114,5.4396809,"\int \cot ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} e^{i (e+f x)} \cos ^3(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{(a+b) \left(1+e^{2 i (e+f x)}\right)}{\left(-1+e^{2 i (e+f x)}\right)^2}-\frac{a^{3/2} \sqrt{a+b} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)+a^{3/2} \sqrt{a+b} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)-2 i a^{3/2} f x \sqrt{a+b}+\left(2 a^2+a b-b^2\right) \log \left(1-e^{2 i (e+f x)}\right)-2 a^2 \log \left(\sqrt{a+b} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+b e^{2 i (e+f x)}+b\right)+b^2 \log \left(\sqrt{a+b} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+b e^{2 i (e+f x)}+b\right)-a b \log \left(\sqrt{a+b} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+b e^{2 i (e+f x)}+b\right)}{\sqrt{a+b} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}\right) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{f (a \cos (2 e+2 f x)+a+2 b)^{3/2}}","-\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,"(Sqrt[2]*E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]^3*(((a + b)*(1 + E^((2*I)*(e + f*x))))/(-1 + E^((2*I)*(e + f*x)))^2 - ((-2*I)*a^(3/2)*Sqrt[a + b]*f*x + (2*a^2 + a*b - b^2)*Log[1 - E^((2*I)*(e + f*x))] + a^(3/2)*Sqrt[a + b]*Log[a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] + a^(3/2)*Sqrt[a + b]*Log[a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - 2*a^2*Log[a + b + a*E^((2*I)*(e + f*x)) + b*E^((2*I)*(e + f*x)) + Sqrt[a + b]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - a*b*Log[a + b + a*E^((2*I)*(e + f*x)) + b*E^((2*I)*(e + f*x)) + Sqrt[a + b]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] + b^2*Log[a + b + a*E^((2*I)*(e + f*x)) + b*E^((2*I)*(e + f*x)) + Sqrt[a + b]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]])/(Sqrt[a + b]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]))*(a + b*Sec[e + f*x]^2)^(3/2))/(f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2))","C",0
394,1,684,159,5.5858337,"\int \cot ^5(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{e^{i (e+f x)} \cos ^3(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{4 a^{3/2} \sqrt{a+b} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)+4 a^{3/2} \sqrt{a+b} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)-8 i a^{3/2} f x \sqrt{a+b}+\left(8 a^2+4 a b-b^2\right) \log \left(1-e^{2 i (e+f x)}\right)-8 a^2 \log \left(\sqrt{a+b} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+b e^{2 i (e+f x)}+b\right)+b^2 \log \left(\sqrt{a+b} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+b e^{2 i (e+f x)}+b\right)-4 a b \log \left(\sqrt{a+b} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+b e^{2 i (e+f x)}+b\right)}{\sqrt{a+b} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}-\frac{\left(1+e^{2 i (e+f x)}\right) \left(a \left(-4 e^{2 i (e+f x)}+6 e^{4 i (e+f x)}+6\right)+b \left(6 e^{2 i (e+f x)}+e^{4 i (e+f x)}+1\right)\right)}{\left(-1+e^{2 i (e+f x)}\right)^4}\right) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{2 \sqrt{2} f (a \cos (2 e+2 f x)+a+2 b)^{3/2}}","\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{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+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,"(E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]^3*(-(((1 + E^((2*I)*(e + f*x)))*(b*(1 + 6*E^((2*I)*(e + f*x)) + E^((4*I)*(e + f*x))) + a*(6 - 4*E^((2*I)*(e + f*x)) + 6*E^((4*I)*(e + f*x)))))/(-1 + E^((2*I)*(e + f*x)))^4) + ((-8*I)*a^(3/2)*Sqrt[a + b]*f*x + (8*a^2 + 4*a*b - b^2)*Log[1 - E^((2*I)*(e + f*x))] + 4*a^(3/2)*Sqrt[a + b]*Log[a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] + 4*a^(3/2)*Sqrt[a + b]*Log[a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - 8*a^2*Log[a + b + a*E^((2*I)*(e + f*x)) + b*E^((2*I)*(e + f*x)) + Sqrt[a + b]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - 4*a*b*Log[a + b + a*E^((2*I)*(e + f*x)) + b*E^((2*I)*(e + f*x)) + Sqrt[a + b]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] + b^2*Log[a + b + a*E^((2*I)*(e + f*x)) + b*E^((2*I)*(e + f*x)) + Sqrt[a + b]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]])/(Sqrt[a + b]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]))*(a + b*Sec[e + f*x]^2)^(3/2))/(2*Sqrt[2]*f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2))","C",0
395,1,353,290,6.1880929,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \tan ^6(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^6,x]","-\frac{\tan (e+f x) \sec ^6(e+f x) \left(9 a^3 \cos (6 (e+f x))+90 a^3+57 a^2 b \cos (6 (e+f x))+498 a^2 b+\left(135 a^3+759 a^2 b-2303 a b^2+513 b^3\right) \cos (2 (e+f x))+2 \left(27 a^3+159 a^2 b-523 a b^2-191 b^3\right) \cos (4 (e+f x))-337 a b^2 \cos (6 (e+f x))-1594 a b^2+15 b^3 \cos (6 (e+f x))-626 b^3\right) \sqrt{a+b \sec ^2(e+f x)}}{12288 b^2 f}-\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \left(128 a^{3/2} b^2 \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)-\frac{\left(3 a^4+20 a^3 b+90 a^2 b^2-60 a b^3-5 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{b}}\right)}{32 \sqrt{2} b^2 f (a \cos (2 e+2 f x)+a+2 b)^{3/2}}","-\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{\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(3 a^3+17 a^2 b-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(3 a^4+20 a^3 b+90 a^2 b^2-60 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{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,"-1/32*((128*a^(3/2)*b^2*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]] - ((3*a^4 + 20*a^3*b + 90*a^2*b^2 - 60*a*b^3 - 5*b^4)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[b])*Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2))/(Sqrt[2]*b^2*f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)) - ((90*a^3 + 498*a^2*b - 1594*a*b^2 - 626*b^3 + (135*a^3 + 759*a^2*b - 2303*a*b^2 + 513*b^3)*Cos[2*(e + f*x)] + 2*(27*a^3 + 159*a^2*b - 523*a*b^2 - 191*b^3)*Cos[4*(e + f*x)] + 9*a^3*Cos[6*(e + f*x)] + 57*a^2*b*Cos[6*(e + f*x)] - 337*a*b^2*Cos[6*(e + f*x)] + 15*b^3*Cos[6*(e + f*x)])*Sec[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x])/(12288*b^2*f)","A",1
396,1,258,214,4.1029048,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \tan ^4(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^4,x]","\frac{\tan (e+f x) \sec ^4(e+f x) \left(4 \left(3 a^2-24 a b-11 b^2\right) \cos (2 (e+f x))+\left(3 a^2-38 a b+3 b^2\right) \cos (4 (e+f x))+9 a^2-58 a b+17 b^2\right) \sqrt{a+b \sec ^2(e+f x)}}{384 b f}+\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \left(16 a^{3/2} b \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)-\frac{(a-b) \left(a^2+10 a b+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{b}}\right)}{4 \sqrt{2} b f (a \cos (2 e+2 f x)+a+2 b)^{3/2}}","\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{\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{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,"((16*a^(3/2)*b*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]] - ((a - b)*(a^2 + 10*a*b + b^2)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[b])*Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2))/(4*Sqrt[2]*b*f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)) + ((9*a^2 - 58*a*b + 17*b^2 + 4*(3*a^2 - 24*a*b - 11*b^2)*Cos[2*(e + f*x)] + (3*a^2 - 38*a*b + 3*b^2)*Cos[4*(e + f*x)])*Sec[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x])/(384*b*f)","A",1
397,1,703,166,6.1807034,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \tan ^2(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^2,x]","\frac{e^{i (e+f x)} \cos ^3(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{4 i a^{3/2} \sqrt{b} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)-4 i a^{3/2} \sqrt{b} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)-8 a^{3/2} \sqrt{b} f x-3 a^2 \log \left(\frac{4 f \left(\sqrt{b} \left(-1+e^{2 i (e+f x)}\right)-i \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}\right)}{\left(3 a^2-6 a b-b^2\right) \left(1+e^{2 i (e+f x)}\right)}\right)+6 a b \log \left(\frac{4 f \left(\sqrt{b} \left(-1+e^{2 i (e+f x)}\right)-i \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}\right)}{\left(3 a^2-6 a b-b^2\right) \left(1+e^{2 i (e+f x)}\right)}\right)+b^2 \log \left(\frac{4 f \left(\sqrt{b} \left(-1+e^{2 i (e+f x)}\right)-i \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}\right)}{\left(3 a^2-6 a b-b^2\right) \left(1+e^{2 i (e+f x)}\right)}\right)}{\sqrt{b} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}-\frac{i \left(-1+e^{2 i (e+f x)}\right) \left(5 a \left(1+e^{2 i (e+f x)}\right)^2-b \left(-6 e^{2 i (e+f x)}+e^{4 i (e+f x)}+1\right)\right)}{\left(1+e^{2 i (e+f x)}\right)^4}\right) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{2 \sqrt{2} f (a \cos (2 e+2 f x)+a+2 b)^{3/2}}","-\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{\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{(5 a+b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{8 f}+\frac{b \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 f}",1,"(E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]^3*(((-I)*(-1 + E^((2*I)*(e + f*x)))*(5*a*(1 + E^((2*I)*(e + f*x)))^2 - b*(1 - 6*E^((2*I)*(e + f*x)) + E^((4*I)*(e + f*x)))))/(1 + E^((2*I)*(e + f*x)))^4 + (-8*a^(3/2)*Sqrt[b]*f*x + (4*I)*a^(3/2)*Sqrt[b]*Log[a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - (4*I)*a^(3/2)*Sqrt[b]*Log[a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - 3*a^2*Log[(4*(Sqrt[b]*(-1 + E^((2*I)*(e + f*x))) - I*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*f)/((3*a^2 - 6*a*b - b^2)*(1 + E^((2*I)*(e + f*x))))] + 6*a*b*Log[(4*(Sqrt[b]*(-1 + E^((2*I)*(e + f*x))) - I*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*f)/((3*a^2 - 6*a*b - b^2)*(1 + E^((2*I)*(e + f*x))))] + b^2*Log[(4*(Sqrt[b]*(-1 + E^((2*I)*(e + f*x))) - I*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*f)/((3*a^2 - 6*a*b - b^2)*(1 + E^((2*I)*(e + f*x))))])/(Sqrt[b]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]))*(a + b*Sec[e + f*x]^2)^(3/2))/(2*Sqrt[2]*f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2))","C",1
398,1,527,118,1.7934104,"\int \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} e^{i (e+f x)} \cos ^3(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{-i a^{3/2} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)+i a^{3/2} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)+2 a^{3/2} f x-b^{3/2} \log \left(\frac{2 i f \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}-2 \sqrt{b} f \left(-1+e^{2 i (e+f x)}\right)}{b (3 a+b) \left(1+e^{2 i (e+f x)}\right)}\right)-3 a \sqrt{b} \log \left(\frac{2 i f \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}-2 \sqrt{b} f \left(-1+e^{2 i (e+f x)}\right)}{b (3 a+b) \left(1+e^{2 i (e+f x)}\right)}\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}-\frac{i b \left(-1+e^{2 i (e+f x)}\right)}{\left(1+e^{2 i (e+f x)}\right)^2}\right) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{f (a \cos (2 e+2 f x)+a+2 b)^{3/2}}","\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,"(Sqrt[2]*E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]^3*(((-I)*b*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))^2 + (2*a^(3/2)*f*x - I*a^(3/2)*Log[a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] + I*a^(3/2)*Log[a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - 3*a*Sqrt[b]*Log[(-2*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*f + (2*I)*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]*f)/(b*(3*a + b)*(1 + E^((2*I)*(e + f*x))))] - b^(3/2)*Log[(-2*Sqrt[b]*(-1 + E^((2*I)*(e + f*x)))*f + (2*I)*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]*f)/(b*(3*a + b)*(1 + E^((2*I)*(e + f*x))))])/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*(a + b*Sec[e + f*x]^2)^(3/2))/(f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2))","C",1
399,1,410,111,6.1242958,"\int \cot ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]^2*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} e^{i (e+f x)} \cos ^3(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{-2 \left(a^{3/2} f x+b^{3/2} \log \left(\frac{f \left(\sqrt{b} \left(-1+e^{2 i (e+f x)}\right)-i \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}\right)}{b^2 \left(1+e^{2 i (e+f x)}\right)}\right)\right)+i a^{3/2} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)-i a^{3/2} \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}-\frac{2 i (a+b)}{-1+e^{2 i (e+f x)}}\right) \left(a+b \sec ^2(e+f x)\right)^{3/2}}{f (a \cos (2 e+2 f x)+a+2 b)^{3/2}}","-\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,"(Sqrt[2]*E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*Cos[e + f*x]^3*(((-2*I)*(a + b))/(-1 + E^((2*I)*(e + f*x))) + (I*a^(3/2)*Log[a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - I*a^(3/2)*Log[a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - 2*(a^(3/2)*f*x + b^(3/2)*Log[((Sqrt[b]*(-1 + E^((2*I)*(e + f*x))) - I*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*f)/(b^2*(1 + E^((2*I)*(e + f*x))))]))/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*(a + b*Sec[e + f*x]^2)^(3/2))/(f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2))","C",1
400,1,100,112,0.3108089,"\int \cot ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]^4*(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{2 (a+b) \cot ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{a \sin ^2(e+f x)}{a+b}\right)}{3 f \sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}} (a \cos (2 (e+f x))+a+2 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 ^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,"(-2*(a + b)*Cot[e + f*x]^3*Hypergeometric2F1[-3/2, -3/2, -1/2, (a*Sin[e + f*x]^2)/(a + b)]*(a + b*Sec[e + f*x]^2)^(3/2))/(3*f*(a + 2*b + a*Cos[2*(e + f*x)])*Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)])","C",1
401,1,139,165,1.3529512,"\int \cot ^6(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]^6*(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{2 \cot ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^{3/2} \left(\frac{5 (a+b)^2 \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{a \sin ^2(e+f x)}{a+b}\right)}{\sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}}}-\frac{3}{4} \csc ^2(e+f x) (a \cos (2 (e+f x))+a+2 b)^2\right)}{15 f (a+b) (a \cos (2 (e+f x))+a+2 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{\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+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,"(2*Cot[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2)*((-3*(a + 2*b + a*Cos[2*(e + f*x)])^2*Csc[e + f*x]^2)/4 + (5*(a + b)^2*Hypergeometric2F1[-3/2, -3/2, -1/2, (a*Sin[e + f*x]^2)/(a + b)])/Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)]))/(15*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)]))","C",1
402,0,0,89,1.9190476,"\int \frac{\tan ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2],x]","\int \frac{\tan ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","\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,"Integrate[Tan[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
403,0,0,56,1.5032845,"\int \frac{\tan ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2],x]","\int \frac{\tan ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","\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,"Integrate[Tan[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
404,0,0,33,0.1205349,"\int \frac{\tan (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]/Sqrt[a + b*Sec[e + f*x]^2],x]","\int \frac{\tan (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}",1,"Integrate[Tan[e + f*x]/Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
405,0,0,70,2.2033546,"\int \frac{\cot (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]/Sqrt[a + b*Sec[e + f*x]^2],x]","\int \frac{\cot (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","\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,"Integrate[Cot[e + f*x]/Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
406,0,0,116,4.7600273,"\int \frac{\cot ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2],x]","\int \frac{\cot ^3(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","-\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,"Integrate[Cot[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
407,0,0,166,7.3014875,"\int \frac{\cot ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2],x]","\int \frac{\cot ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","-\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,"Integrate[Cot[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
408,1,230,173,4.3480283,"\int \frac{\tan ^6(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\sec (e+f x) \sqrt{a \cos (2 e+2 f x)+a+2 b} \left(\frac{8 b^2 \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{a}}-\frac{\left(3 a^2+10 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{b}}\right)}{8 \sqrt{2} b^2 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tan (e+f x) \sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) (3 (a+3 b) \cos (2 (e+f x))+3 a+5 b)}{32 b^2 f \sqrt{a+b \sec ^2(e+f 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 ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right)}{\sqrt{a} f}+\frac{\tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{4 b f}",1,"-1/8*(((8*b^2*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[a] - ((3*a^2 + 10*a*b + 15*b^2)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[b])*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sec[e + f*x])/(Sqrt[2]*b^2*f*Sqrt[a + b*Sec[e + f*x]^2]) - ((a + 2*b + a*Cos[2*(e + f*x)])*(3*a + 5*b + 3*(a + 3*b)*Cos[2*(e + f*x)])*Sec[e + f*x]^4*Tan[e + f*x])/(32*b^2*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
409,1,196,120,2.8083012,"\int \frac{\tan ^4(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\tan (e+f x) \sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b)}{4 b f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sec (e+f x) \sqrt{a \cos (2 e+2 f x)+a+2 b} \left(\frac{2 b \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{a}}-\frac{(a+3 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{b}}\right)}{2 \sqrt{2} b f \sqrt{a+b \sec ^2(e+f 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 (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 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,"(((2*b*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[a] - ((a + 3*b)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[b])*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sec[e + f*x])/(2*Sqrt[2]*b*f*Sqrt[a + b*Sec[e + f*x]^2]) + ((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*Tan[e + f*x])/(4*b*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
410,0,0,80,2.7079543,"\int \frac{\tan ^2(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2],x]","\int \frac{\tan ^2(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","\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,"Integrate[Tan[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2], x]","F",-1
411,1,87,39,0.1064718,"\int \frac{1}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[1/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sec (e+f x) \sqrt{a \cos (2 e+2 f x)+a+2 b} \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{2} \sqrt{a} f \sqrt{a+b \sec ^2(e+f 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}",1,"(ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sec[e + f*x])/(Sqrt[2]*Sqrt[a]*f*Sqrt[a + b*Sec[e + f*x]^2])","B",1
412,1,127,74,0.2037563,"\int \frac{\cot ^2(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\sec (e+f x) \sqrt{a \cos (2 (e+f x))+a+2 b} \left((a+b) \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)+\sqrt{a} \csc (e+f x) \sqrt{-a \sin ^2(e+f x)+a+b}\right)}{\sqrt{2} \sqrt{a} f (a+b) \sqrt{a+b \sec ^2(e+f 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)}",1,"-((Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*Sec[e + f*x]*((a + b)*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]] + Sqrt[a]*Csc[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2]))/(Sqrt[2]*Sqrt[a]*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]))","A",1
413,1,168,119,1.8299856,"\int \frac{\cot ^4(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2],x]","\frac{\sec (e+f x) \sqrt{a \cos (2 e+2 f x)+a+2 b} \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{2} \sqrt{a} f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\csc ^3(e+f x) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b) ((2 a+3 b) \cos (2 (e+f x))-a-2 b)}{6 f (a+b)^2 \sqrt{a+b \sec ^2(e+f 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}",1,"(ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sec[e + f*x])/(Sqrt[2]*Sqrt[a]*f*Sqrt[a + b*Sec[e + f*x]^2]) - ((a + 2*b + a*Cos[2*(e + f*x)])*(-a - 2*b + (2*a + 3*b)*Cos[2*(e + f*x)])*Csc[e + f*x]^3*Sec[e + f*x])/(6*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
414,1,199,172,4.1710623,"\int \frac{\cot ^6(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2],x]","-\frac{\csc (e+f x) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b) \left(-\left(11 a^2+26 a b+15 b^2\right) \csc ^2(e+f x)+23 a^2+3 (a+b)^2 \csc ^4(e+f x)+60 a b+45 b^2\right)}{30 f (a+b)^3 \sqrt{a+b \sec ^2(e+f x)}}-\frac{\sec (e+f x) \sqrt{a \cos (2 e+2 f x)+a+2 b} \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{2} \sqrt{a} f \sqrt{a+b \sec ^2(e+f 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}",1,"-((ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sec[e + f*x])/(Sqrt[2]*Sqrt[a]*f*Sqrt[a + b*Sec[e + f*x]^2])) - ((a + 2*b + a*Cos[2*(e + f*x)])*Csc[e + f*x]*(23*a^2 + 60*a*b + 45*b^2 - (11*a^2 + 26*a*b + 15*b^2)*Csc[e + f*x]^2 + 3*(a + b)^2*Csc[e + f*x]^4)*Sec[e + f*x])/(30*(a + b)^3*f*Sqrt[a + b*Sec[e + f*x]^2])","A",1
415,0,0,88,4.9214369,"\int \frac{\tan ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]","\int \frac{\tan ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","-\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,"Integrate[Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2), x]","F",-1
416,1,187,63,3.9093898,"\int \frac{\tan ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{3 (a+b) \tan ^4(e+f x) F_1\left(2;\frac{1}{2},\frac{3}{2};3;\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)}{2 f \left(a+b \sec ^2(e+f x)\right)^{3/2} \left(\sin ^2(e+f x) \left(3 a F_1\left(3;\frac{1}{2},\frac{5}{2};4;\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)+(a+b) F_1\left(3;\frac{3}{2},\frac{3}{2};4;\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)+6 (a+b) F_1\left(2;\frac{1}{2},\frac{3}{2};3;\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}","\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,"(3*(a + b)*AppellF1[2, 1/2, 3/2, 3, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Tan[e + f*x]^4)/(2*f*(a + b*Sec[e + f*x]^2)^(3/2)*(6*(a + b)*AppellF1[2, 1/2, 3/2, 3, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (3*a*AppellF1[3, 1/2, 5/2, 4, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a + b)*AppellF1[3, 3/2, 3/2, 4, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2))","C",0
417,1,382,57,6.3612162,"\int \frac{\tan (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b)^{3/2} \left(-\frac{2}{b \sqrt{a \cos (2 (e+f x))+a+2 b}}+\frac{\sqrt{2} e^{i (e+f x)} \sec (e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{\sqrt{a} (a+4 b) \left(1+e^{2 i (e+f x)}\right)}{b \left(a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}\right)}+\frac{-2 \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)-2 \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)+4 i f x}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}\right)}{a^{3/2}}\right)}{16 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*Sec[e + f*x]^2*(-2/(b*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]) + (Sqrt[2]*E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*((Sqrt[a]*(a + 4*b)*(1 + E^((2*I)*(e + f*x))))/(b*(4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2)) + ((4*I)*f*x - 2*Log[a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - 2*Log[a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]])/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*Sec[e + f*x])/a^(3/2)))/(16*f*(a + b*Sec[e + f*x]^2)^(3/2))","C",1
418,0,0,100,5.9391399,"\int \frac{\cot (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]/(a + b*Sec[e + f*x]^2)^(3/2),x]","\int \frac{\cot (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","\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,"Integrate[Cot[e + f*x]/(a + b*Sec[e + f*x]^2)^(3/2), x]","F",-1
419,0,0,153,9.9706639,"\int \frac{\cot ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2),x]","\int \frac{\cot ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","-\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,"Integrate[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2), x]","F",-1
420,0,0,213,13.4301112,"\int \frac{\cot ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]","\int \frac{\cot ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}+\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{\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,"Integrate[Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2), x]","F",-1
421,1,247,172,9.6932207,"\int \frac{\tan ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\tan (e+f x) \sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\left(3 a^2+4 a b+2 b^2\right) \cos (2 (e+f x))+3 a^2+6 a b+2 b^2\right)}{8 a b^2 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\sec ^3(e+f x) (a \cos (2 e+2 f x)+a+2 b)^{3/2} \left(\frac{2 b^2 \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{a}}+\frac{a (3 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{b}}\right)}{4 \sqrt{2} a b^2 f \left(a+b \sec ^2(e+f x)\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^{3/2} 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{(3 a+2 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{2 a b^2 f}-\frac{(a+b) \tan ^3(e+f x)}{a b f \sqrt{a+b \tan ^2(e+f x)+b}}",1,"-1/4*(((2*b^2*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[a] + (a*(3*a + 5*b)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[b])*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sec[e + f*x]^3)/(Sqrt[2]*a*b^2*f*(a + b*Sec[e + f*x]^2)^(3/2)) + ((a + 2*b + a*Cos[2*(e + f*x)])*(3*a^2 + 6*a*b + 2*b^2 + (3*a^2 + 4*a*b + 2*b^2)*Cos[2*(e + f*x)])*Sec[e + f*x]^4*Tan[e + f*x])/(8*a*b^2*f*(a + b*Sec[e + f*x]^2)^(3/2))","A",1
422,1,201,116,4.2569791,"\int \frac{\tan ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sec ^3(e+f x) (a \cos (2 e+2 f x)+a+2 b)^{3/2} \left(\frac{b \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{a}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{b}}\right)}{2 \sqrt{2} a b f \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{(a+b) \tan (e+f x) \sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b)}{2 a b f \left(a+b \sec ^2(e+f x)\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^{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,"(((b*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[a] + (a*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[b])*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sec[e + f*x]^3)/(2*Sqrt[2]*a*b*f*(a + b*Sec[e + f*x]^2)^(3/2)) - ((a + b)*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*Tan[e + f*x])/(2*a*b*f*(a + b*Sec[e + f*x]^2)^(3/2))","A",1
423,1,169,71,2.4360602,"\int \frac{\tan ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) (a \cos (2 (e+f x))+a+2 b)-\sqrt{2} \sqrt{a} \sqrt{a+b} \sin (e+f x) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}}\right)}{4 a^{3/2} f \sqrt{a+b} \sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}} \left(a+b \sec ^2(e+f x)\right)^{3/2}}","\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,"-1/4*((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^3*(ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*(a + 2*b + a*Cos[2*(e + f*x)]) - Sqrt[2]*Sqrt[a]*Sqrt[a + b]*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*Sin[e + f*x]))/(a^(3/2)*Sqrt[a + b]*f*(a + b*Sec[e + f*x]^2)^(3/2)*Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)])","B",1
424,1,168,77,1.3623952,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(-3/2),x]","\frac{\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\sqrt{a+b} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) (a \cos (2 (e+f x))+a+2 b)-\sqrt{2} \sqrt{a} b \sin (e+f x) \sqrt{\frac{a \cos (2 (e+f x))+a+2 b}{a+b}}\right)}{4 a^{3/2} f (a+b) \sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}} \left(a+b \sec ^2(e+f x)\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^{3/2} f}-\frac{b \tan (e+f x)}{a f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}",1,"((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^3*(Sqrt[a + b]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*(a + 2*b + a*Cos[2*(e + f*x)]) - Sqrt[2]*Sqrt[a]*b*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*Sin[e + f*x]))/(4*a^(3/2)*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)*Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)])","B",1
425,1,182,119,4.2594826,"\int \frac{\cot ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2),x]","-\frac{\sec ^3(e+f x) (a \cos (2 e+2 f x)+a+2 b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{2 \sqrt{2} a^{3/2} f \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\csc (e+f x) \sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(\left(a^2+b^2\right) \cos (2 (e+f x))+a^2+2 a b-b^2\right)}{4 a f (a+b)^2 \left(a+b \sec ^2(e+f x)\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^{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,"-1/2*(ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]]*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sec[e + f*x]^3)/(Sqrt[2]*a^(3/2)*f*(a + b*Sec[e + f*x]^2)^(3/2)) - ((a + 2*b + a*Cos[2*(e + f*x)])*(a^2 + 2*a*b - b^2 + (a^2 + b^2)*Cos[2*(e + f*x)])*Csc[e + f*x]*Sec[e + f*x]^3)/(4*a*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^(3/2))","A",1
426,1,224,174,5.3968494,"\int \frac{\cot ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\sec ^3(e+f x) (a \cos (2 e+2 f x)+a+2 b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{2 \sqrt{2} a^{3/2} f \left(a+b \sec ^2(e+f x)\right)^{3/2}}+\frac{\sec ^3(e+f x) (a \cos (2 e+2 f x)+a+2 b)^2 \left(-\frac{b^3 \sin (e+f x)}{2 a f (a+b)^3 (a \cos (2 e+2 f x)+a+2 b)}-\frac{\csc ^3(e+f x)}{12 f (a+b)^2}+\frac{(4 a+9 b) \csc (e+f x)}{12 f (a+b)^3}\right)}{\left(a+b \sec ^2(e+f x)\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^{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]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]]*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sec[e + f*x]^3)/(2*Sqrt[2]*a^(3/2)*f*(a + b*Sec[e + f*x]^2)^(3/2)) + ((a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^3*(((4*a + 9*b)*Csc[e + f*x])/(12*(a + b)^3*f) - Csc[e + f*x]^3/(12*(a + b)^2*f) - (b^3*Sin[e + f*x])/(2*a*(a + b)^3*f*(a + 2*b + a*Cos[2*e + 2*f*x]))))/(a + b*Sec[e + f*x]^2)^(3/2)","A",1
427,1,237,241,10.7559751,"\int \frac{\cot ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2),x]","\frac{\tan (e+f x) \sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b)^2 \left(-\left(23 a^2+80 a b+90 b^2\right) \csc ^2(e+f x)+\frac{30 b^4}{a (a \cos (2 (e+f x))+a+2 b)}-3 (a+b)^2 \csc ^6(e+f x)+(a+b) (11 a+20 b) \csc ^4(e+f x)\right)}{60 f (a+b)^4 \left(a+b \sec ^2(e+f x)\right)^{3/2}}-\frac{\sec ^3(e+f x) (a \cos (2 e+2 f x)+a+2 b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{2 \sqrt{2} a^{3/2} f \left(a+b \sec ^2(e+f x)\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^{3/2} f}+\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(15 a^3+55 a^2 b+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{(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,"-1/2*(ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]]*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sec[e + f*x]^3)/(Sqrt[2]*a^(3/2)*f*(a + b*Sec[e + f*x]^2)^(3/2)) + ((a + 2*b + a*Cos[2*(e + f*x)])^2*((30*b^4)/(a*(a + 2*b + a*Cos[2*(e + f*x)])) - (23*a^2 + 80*a*b + 90*b^2)*Csc[e + f*x]^2 + (a + b)*(11*a + 20*b)*Csc[e + f*x]^4 - 3*(a + b)^2*Csc[e + f*x]^6)*Sec[e + f*x]^2*Tan[e + f*x])/(60*(a + b)^4*f*(a + b*Sec[e + f*x]^2)^(3/2))","A",1
428,1,187,97,7.8154209,"\int \frac{\tan ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{4 (a+b) \tan ^6(e+f x) F_1\left(3;\frac{1}{2},\frac{5}{2};4;\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)}{3 f \left(a+b \sec ^2(e+f x)\right)^{5/2} \left(\sin ^2(e+f x) \left(5 a F_1\left(4;\frac{1}{2},\frac{7}{2};5;\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)+(a+b) F_1\left(4;\frac{3}{2},\frac{5}{2};5;\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)+8 (a+b) F_1\left(3;\frac{1}{2},\frac{5}{2};4;\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}+\frac{\frac{1}{a^2}-\frac{1}{b^2}}{f \sqrt{a+b \sec ^2(e+f x)}}+\frac{(a+b)^2}{3 a b^2 f \left(a+b \sec ^2(e+f x)\right)^{3/2}}",1,"(4*(a + b)*AppellF1[3, 1/2, 5/2, 4, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Tan[e + f*x]^6)/(3*f*(a + b*Sec[e + f*x]^2)^(5/2)*(8*(a + b)*AppellF1[3, 1/2, 5/2, 4, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[4, 1/2, 7/2, 5, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (a + b)*AppellF1[4, 3/2, 5/2, 5, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2))","C",0
429,1,613,89,10.1652593,"\int \frac{\tan ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{e^{i (e+f x)} \sec ^5(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{-12 \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)-12 \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)+24 i f x}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}-\frac{\sqrt{a} \left(1+e^{2 i (e+f x)}\right) \left(a^3 \left(1+e^{2 i (e+f x)}\right)^2-6 a^2 b \left(e^{2 i (e+f x)}+e^{4 i (e+f x)}+1\right)-32 a b^2 \left(1+e^{2 i (e+f x)}\right)^2-96 b^3 e^{2 i (e+f x)}\right)}{b^2 \left(a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}\right)^2}\right) (a \cos (2 e+2 f x)+a+2 b)^{5/2}}{96 \sqrt{2} a^{5/2} f \left(a+b \sec ^2(e+f x)\right)^{5/2}}-\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+3 b) (a \cos (2 e+2 f x)+a+2 b)^{5/2}}{48 b^2 f (a \cos (2 (e+f x))+a+2 b)^{3/2} \left(a+b \sec ^2(e+f x)\right)^{5/2}}+\frac{\sec ^4(e+f x) ((a-2 b) \cos (2 (e+f x))+a+b) (a \cos (2 e+2 f x)+a+2 b)^{5/2}}{96 b^2 f (a \cos (2 (e+f x))+a+2 b)^{3/2} \left(a+b \sec ^2(e+f x)\right)^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{1}{a^2 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{a+b}{3 a b f \left(a+b \sec ^2(e+f x)\right)^{3/2}}",1,"-1/48*((a + 3*b + a*Cos[2*(e + f*x)])*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^4)/(b^2*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*(a + b*Sec[e + f*x]^2)^(5/2)) + ((a + b + (a - 2*b)*Cos[2*(e + f*x)])*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^4)/(96*b^2*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*(a + b*Sec[e + f*x]^2)^(5/2)) - (E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*(-((Sqrt[a]*(1 + E^((2*I)*(e + f*x)))*(-96*b^3*E^((2*I)*(e + f*x)) + a^3*(1 + E^((2*I)*(e + f*x)))^2 - 32*a*b^2*(1 + E^((2*I)*(e + f*x)))^2 - 6*a^2*b*(1 + E^((2*I)*(e + f*x)) + E^((4*I)*(e + f*x)))))/(b^2*(4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2)^2)) + ((24*I)*f*x - 12*Log[a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - 12*Log[a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]])/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*Sec[e + f*x]^5)/(96*Sqrt[2]*a^(5/2)*f*(a + b*Sec[e + f*x]^2)^(5/2))","C",1
430,1,613,83,7.4337878,"\int \frac{\tan (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{e^{i (e+f x)} \sec ^5(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2} \left(\frac{-12 \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right)-12 \log \left(\sqrt{a} \sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right)+24 i f x}{\sqrt{a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}}}-\frac{\sqrt{a} \left(1+e^{2 i (e+f x)}\right) \left(a^3 \left(1+e^{2 i (e+f x)}\right)^2-6 a^2 b \left(e^{2 i (e+f x)}+e^{4 i (e+f x)}+1\right)-32 a b^2 \left(1+e^{2 i (e+f x)}\right)^2-96 b^3 e^{2 i (e+f x)}\right)}{b^2 \left(a \left(1+e^{2 i (e+f x)}\right)^2+4 b e^{2 i (e+f x)}\right)^2}\right) (a \cos (2 e+2 f x)+a+2 b)^{5/2}}{96 \sqrt{2} a^{5/2} f \left(a+b \sec ^2(e+f x)\right)^{5/2}}-\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+3 b) (a \cos (2 e+2 f x)+a+2 b)^{5/2}}{48 b^2 f (a \cos (2 (e+f x))+a+2 b)^{3/2} \left(a+b \sec ^2(e+f x)\right)^{5/2}}+\frac{\sec ^4(e+f x) ((a-2 b) \cos (2 (e+f x))+a+b) (a \cos (2 e+2 f x)+a+2 b)^{5/2}}{32 b^2 f (a \cos (2 (e+f x))+a+2 b)^{3/2} \left(a+b \sec ^2(e+f x)\right)^{5/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}+\frac{1}{a^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{1}{3 a f \left(a+b \sec ^2(e+f x)\right)^{3/2}}",1,"-1/48*((a + 3*b + a*Cos[2*(e + f*x)])*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^4)/(b^2*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*(a + b*Sec[e + f*x]^2)^(5/2)) + ((a + b + (a - 2*b)*Cos[2*(e + f*x)])*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^4)/(32*b^2*f*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)*(a + b*Sec[e + f*x]^2)^(5/2)) + (E^(I*(e + f*x))*Sqrt[4*b + (a*(1 + E^((2*I)*(e + f*x)))^2)/E^((2*I)*(e + f*x))]*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*(-((Sqrt[a]*(1 + E^((2*I)*(e + f*x)))*(-96*b^3*E^((2*I)*(e + f*x)) + a^3*(1 + E^((2*I)*(e + f*x)))^2 - 32*a*b^2*(1 + E^((2*I)*(e + f*x)))^2 - 6*a^2*b*(1 + E^((2*I)*(e + f*x)) + E^((4*I)*(e + f*x)))))/(b^2*(4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2)^2)) + ((24*I)*f*x - 12*Log[a + 2*b + a*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] - 12*Log[a + a*E^((2*I)*(e + f*x)) + 2*b*E^((2*I)*(e + f*x)) + Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]])/Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])*Sec[e + f*x]^5)/(96*Sqrt[2]*a^(5/2)*f*(a + b*Sec[e + f*x]^2)^(5/2))","C",1
431,0,0,137,8.7613612,"\int \frac{\cot (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]/(a + b*Sec[e + f*x]^2)^(5/2),x]","\int \frac{\cot (e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{b (2 a+b)}{a^2 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}}-\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,"Integrate[Cot[e + f*x]/(a + b*Sec[e + f*x]^2)^(5/2), x]","F",-1
432,0,0,200,19.5078457,"\int \frac{\cot ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]","\int \frac{\cot ^3(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}-\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{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,"Integrate[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2), x]","F",-1
433,0,0,268,30.3032972,"\int \frac{\cot ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2),x]","\int \frac{\cot ^5(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}+\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{b \left(4 a^3+15 a^2 b-32 a b^2-8 b^3\right)}{8 a^2 f (a+b)^4 \sqrt{a+b \sec ^2(e+f x)}}-\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,"Integrate[Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2), x]","F",-1
434,1,316,157,11.2637891,"\int \frac{\tan ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\sec ^5(e+f x) (a \cos (2 e+2 f x)+a+2 b)^3 \left(\frac{a^2 (-\sin (e+f x))-2 a b \sin (e+f x)-b^2 \sin (e+f x)}{6 a^2 b f (a \cos (2 e+2 f x)+a+2 b)^2}+\frac{-3 a^2 \sin (e+f x)+a b \sin (e+f x)+4 b^2 \sin (e+f x)}{12 a^2 b^2 f (a \cos (2 e+2 f x)+a+2 b)}\right)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}}-\frac{\sec ^5(e+f x) (a \cos (2 e+2 f x)+a+2 b)^{5/2} \left(\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{a}}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{\sqrt{b}}\right)}{4 \sqrt{2} a^2 b^2 f \left(a+b \sec ^2(e+f x)\right)^{5/2}}","-\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{\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{\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,"-1/4*(((b^2*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[a] - (a^2*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[b])*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^5)/(Sqrt[2]*a^2*b^2*f*(a + b*Sec[e + f*x]^2)^(5/2)) + ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^5*((-(a^2*Sin[e + f*x]) - 2*a*b*Sin[e + f*x] - b^2*Sin[e + f*x])/(6*a^2*b*f*(a + 2*b + a*Cos[2*e + 2*f*x])^2) + (-3*a^2*Sin[e + f*x] + a*b*Sin[e + f*x] + 4*b^2*Sin[e + f*x])/(12*a^2*b^2*f*(a + 2*b + a*Cos[2*e + 2*f*x]))))/(a + b*Sec[e + f*x]^2)^(5/2)","B",1
435,1,409,120,6.0202194,"\int \frac{\tan ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b)^{5/2} \left(\frac{\sqrt{2} \csc (e+f x) \sec (e+f x) \left(\frac{16 \left(-a \sin ^2(e+f x)+a+b\right) \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\frac{a^2 (a+b) \sin ^4(e+f x)}{\left(-a \sin ^2(e+f x)+a+b\right)^2}+\frac{3 \sqrt{a} \sqrt{a+b} \sin (e+f x) \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}}}-\frac{6 a (a+b) \sin ^2(e+f x)}{a \cos (2 (e+f x))+a+2 b}\right)}{a^3}-\frac{12 \sin ^4(e+f x)}{a+b}+\frac{\sin ^2(e+f x)}{a+b}+\frac{\sin ^2(e+f x) (a \cos (2 (e+f x))+a+2 b)}{(a+b)^2}\right)}{\left(-a \sin ^2(e+f x)+a+b\right)^{3/2}}+\frac{8 \tan (e+f x) (a \cos (2 (e+f x))+2 a+3 b)}{(a+b)^2 (a \cos (2 (e+f x))+a+2 b)^{3/2}}-\frac{12 \tan (e+f x) ((3 a+2 b) \cos (2 (e+f x))+b)}{(a+b)^2 (a \cos (2 (e+f x))+a+2 b)^{3/2}}\right)}{384 f \left(a+b \sec ^2(e+f x)\right)^{5/2}}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])^(5/2)*Sec[e + f*x]^4*((Sqrt[2]*Csc[e + f*x]*Sec[e + f*x]*(Sin[e + f*x]^2/(a + b) + ((a + 2*b + a*Cos[2*(e + f*x)])*Sin[e + f*x]^2)/(a + b)^2 - (12*Sin[e + f*x]^4)/(a + b) + (16*(a + b - a*Sin[e + f*x]^2)*(1 - (a*Sin[e + f*x]^2)/(a + b))*((-6*a*(a + b)*Sin[e + f*x]^2)/(a + 2*b + a*Cos[2*(e + f*x)]) + (a^2*(a + b)*Sin[e + f*x]^4)/(a + b - a*Sin[e + f*x]^2)^2 + (3*Sqrt[a]*Sqrt[a + b]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)]))/a^3))/(a + b - a*Sin[e + f*x]^2)^(3/2) + (8*(2*a + 3*b + a*Cos[2*(e + f*x)])*Tan[e + f*x])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)) - (12*(b + (3*a + 2*b)*Cos[2*(e + f*x)])*Tan[e + f*x])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2))))/(384*f*(a + b*Sec[e + f*x]^2)^(5/2))","B",1
436,1,410,119,4.5548112,"\int \frac{\tan ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b)^{5/2} \left(-\frac{\sqrt{2} \csc (e+f x) \sec (e+f x) \left(\frac{16 \left(-a \sin ^2(e+f x)+a+b\right) \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right) \left(\frac{a^2 (a+b) \sin ^4(e+f x)}{\left(-a \sin ^2(e+f x)+a+b\right)^2}+\frac{3 \sqrt{a} \sqrt{a+b} \sin (e+f x) \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right)}{\sqrt{\frac{-a \sin ^2(e+f x)+a+b}{a+b}}}-\frac{6 a (a+b) \sin ^2(e+f x)}{a \cos (2 (e+f x))+a+2 b}\right)}{a^3}-\frac{12 \sin ^4(e+f x)}{a+b}+\frac{\sin ^2(e+f x)}{a+b}+\frac{\sin ^2(e+f x) (a \cos (2 (e+f x))+a+2 b)}{(a+b)^2}\right)}{\left(-a \sin ^2(e+f x)+a+b\right)^{3/2}}+\frac{8 \tan (e+f x) (a \cos (2 (e+f x))+2 a+3 b)}{(a+b)^2 (a \cos (2 (e+f x))+a+2 b)^{3/2}}-\frac{4 \tan (e+f x) ((3 a+2 b) \cos (2 (e+f x))+b)}{(a+b)^2 (a \cos (2 (e+f x))+a+2 b)^{3/2}}\right)}{384 f \left(a+b \sec ^2(e+f x)\right)^{5/2}}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])^(5/2)*Sec[e + f*x]^4*(-((Sqrt[2]*Csc[e + f*x]*Sec[e + f*x]*(Sin[e + f*x]^2/(a + b) + ((a + 2*b + a*Cos[2*(e + f*x)])*Sin[e + f*x]^2)/(a + b)^2 - (12*Sin[e + f*x]^4)/(a + b) + (16*(a + b - a*Sin[e + f*x]^2)*(1 - (a*Sin[e + f*x]^2)/(a + b))*((-6*a*(a + b)*Sin[e + f*x]^2)/(a + 2*b + a*Cos[2*(e + f*x)]) + (a^2*(a + b)*Sin[e + f*x]^4)/(a + b - a*Sin[e + f*x]^2)^2 + (3*Sqrt[a]*Sqrt[a + b]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/Sqrt[(a + b - a*Sin[e + f*x]^2)/(a + b)]))/a^3))/(a + b - a*Sin[e + f*x]^2)^(3/2)) + (8*(2*a + 3*b + a*Cos[2*(e + f*x)])*Tan[e + f*x])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2)) - (4*(b + (3*a + 2*b)*Cos[2*(e + f*x)])*Tan[e + f*x])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^(3/2))))/(384*f*(a + b*Sec[e + f*x]^2)^(5/2))","B",1
437,1,1927,125,6.4976096,"\int \frac{1}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[(a + b*Sec[e + f*x]^2)^(-5/2),x]","\frac{3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^4(e+f x) \sin (e+f x)}{4 \sqrt{2} f \left(b \sec ^2(e+f x)+a\right)^{5/2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \left(\frac{3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos ^5(e+f x)}{4 \sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}+\frac{15 a (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin ^2(e+f x) \cos ^5(e+f x)}{4 \sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{7/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin (e+f x) \left(\left(5 a \left(\frac{21 a f F_1\left(\frac{5}{2};-2,\frac{9}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{5 (a+b)}-\frac{12}{5} f F_1\left(\frac{5}{2};-1,\frac{7}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)-4 (a+b) \left(\frac{3 a f F_1\left(\frac{5}{2};-1,\frac{7}{2};\frac{7}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{a+b}-\frac{6 (a+b)^3 f \cot (e+f x) \csc ^4(e+f x) \left(\frac{a \sin ^2(e+f x)}{a+b}-1\right)^2 \left(\frac{a^2 \sin ^4(e+f x)}{3 (a+b)^2 \left(\frac{a \sin ^2(e+f x)}{a+b}-1\right)^2}+\frac{a \sin ^2(e+f x)}{(a+b) \left(\frac{a \sin ^2(e+f x)}{a+b}-1\right)}+\frac{\sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right) \sin (e+f x)}{\sqrt{a+b} \sqrt{1-\frac{a \sin ^2(e+f x)}{a+b}}}\right)}{a^3 \left(1-\frac{a \sin ^2(e+f x)}{a+b}\right)^{3/2}}\right)\right) \sin ^2(e+f x)+2 f \left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \cos (e+f x) \sin (e+f x)+3 (a+b) \left(\frac{5 a f F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac{4}{3} f F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right)\right) \cos ^4(e+f x)}{4 \sqrt{2} f \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right){}^2}+\frac{3 (a+b) \sin (e+f x) \left(\frac{5 a f F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac{4}{3} f F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \cos (e+f x) \sin (e+f x)\right) \cos ^4(e+f x)}{4 \sqrt{2} f \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}-\frac{3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right) \sin ^2(e+f x) \cos ^3(e+f x)}{\sqrt{2} \left(-a \sin ^2(e+f x)+a+b\right)^{5/2} \left(\left(5 a F_1\left(\frac{3}{2};-2,\frac{7}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)-4 (a+b) F_1\left(\frac{3}{2};-1,\frac{5}{2};\frac{5}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right) \sin ^2(e+f x)+3 (a+b) F_1\left(\frac{1}{2};-2,\frac{5}{2};\frac{3}{2};\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right)\right)}\right)}","\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,"(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^4*Sin[e + f*x])/(4*Sqrt[2]*f*(a + b*Sec[e + f*x]^2)^(5/2)*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)*((15*a*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^5*Sin[e + f*x]^2)/(4*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(7/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^5)/(4*Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^3*Sin[e + f*x]^2)/(Sqrt[2]*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) + (3*(a + b)*Cos[e + f*x]^4*Sin[e + f*x]*((5*a*f*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - (4*f*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3))/(4*Sqrt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)) - (3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]^4*Sin[e + f*x]*(2*f*(5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Cos[e + f*x]*Sin[e + f*x] + 3*(a + b)*((5*a*f*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)) - (4*f*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/3) + Sin[e + f*x]^2*(5*a*((21*a*f*AppellF1[5/2, -2, 9/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(5*(a + b)) - (12*f*AppellF1[5/2, -1, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/5) - 4*(a + b)*((3*a*f*AppellF1[5/2, -1, 7/2, 7/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*Cos[e + f*x]*Sin[e + f*x])/(a + b) - (6*(a + b)^3*f*Cot[e + f*x]*Csc[e + f*x]^4*(-1 + (a*Sin[e + f*x]^2)/(a + b))^2*((Sqrt[a]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]*Sin[e + f*x])/(Sqrt[a + b]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + (a^2*Sin[e + f*x]^4)/(3*(a + b)^2*(-1 + (a*Sin[e + f*x]^2)/(a + b))^2) + (a*Sin[e + f*x]^2)/((a + b)*(-1 + (a*Sin[e + f*x]^2)/(a + b)))))/(a^3*(1 - (a*Sin[e + f*x]^2)/(a + b))^(3/2))))))/(4*Sqrt[2]*f*(a + b - a*Sin[e + f*x]^2)^(5/2)*(3*(a + b)*AppellF1[1/2, -2, 5/2, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] + (5*a*AppellF1[3/2, -2, 7/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)] - 4*(a + b)*AppellF1[3/2, -1, 5/2, 5/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)])*Sin[e + f*x]^2)^2)))","C",0
438,1,247,174,7.2115684,"\int \frac{\cot ^2(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2),x]","-\frac{\sec ^5(e+f x) (a \cos (2 e+2 f x)+a+2 b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{4 \sqrt{2} a^{5/2} f \left(a+b \sec ^2(e+f x)\right)^{5/2}}-\frac{\csc (e+f x) \sec ^5(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(a \left(3 a^3+9 a b^2+4 b^3\right) \cos (4 (e+f x))+4 \left(3 a^4+6 a^3 b+8 a b^3+3 b^4\right) \cos (2 (e+f x))+3 \left(3 a^4+8 a^3 b+5 a^2 b^2-12 a b^3-4 b^4\right)\right)}{48 a^2 f (a+b)^3 \left(a+b \sec ^2(e+f x)\right)^{5/2}}","-\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,"-1/4*(ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]]*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^5)/(Sqrt[2]*a^(5/2)*f*(a + b*Sec[e + f*x]^2)^(5/2)) - ((a + 2*b + a*Cos[2*(e + f*x)])*(3*(3*a^4 + 8*a^3*b + 5*a^2*b^2 - 12*a*b^3 - 4*b^4) + 4*(3*a^4 + 6*a^3*b + 8*a*b^3 + 3*b^4)*Cos[2*(e + f*x)] + a*(3*a^3 + 9*a*b^2 + 4*b^3)*Cos[4*(e + f*x)])*Csc[e + f*x]*Sec[e + f*x]^5)/(48*a^2*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^(5/2))","A",1
439,1,234,236,13.9540102,"\int \frac{\cot ^4(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\sec ^5(e+f x) (a \cos (2 e+2 f x)+a+2 b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{4 \sqrt{2} a^{5/2} f \left(a+b \sec ^2(e+f x)\right)^{5/2}}-\frac{\sec ^5(e+f x) (a \cos (2 (e+f x))+a+2 b)^3 \left(\frac{4 b^3 \sin (e+f x) \left(6 a^2+2 a (3 a+b) \cos (2 (e+f x))+13 a b+3 b^2\right)}{a^2 (a \cos (2 (e+f x))+a+2 b)^2}+(a+b) \csc ^3(e+f x)-4 (a+3 b) \csc (e+f x)\right)}{24 f (a+b)^4 \left(a+b \sec ^2(e+f x)\right)^{5/2}}","\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{\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{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]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]]*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^5)/(4*Sqrt[2]*a^(5/2)*f*(a + b*Sec[e + f*x]^2)^(5/2)) - ((a + 2*b + a*Cos[2*(e + f*x)])^3*Sec[e + f*x]^5*(-4*(a + 3*b)*Csc[e + f*x] + (a + b)*Csc[e + f*x]^3 + (4*b^3*(6*a^2 + 13*a*b + 3*b^2 + 2*a*(3*a + b)*Cos[2*(e + f*x)])*Sin[e + f*x])/(a^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(24*(a + b)^4*f*(a + b*Sec[e + f*x]^2)^(5/2))","A",1
440,1,272,315,24.4327554,"\int \frac{\cot ^6(e+f x)}{\left(a+b \sec ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2),x]","\frac{\tan (e+f x) \sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b)^3 \left(-\frac{20 b^5 (a+b)}{a^2 (a \cos (2 (e+f x))+a+2 b)^2}+\frac{10 b^4 (15 a+4 b)}{a^2 (a \cos (2 (e+f x))+a+2 b)}-\left(23 a^2+100 a b+150 b^2\right) \csc ^2(e+f x)-3 (a+b)^2 \csc ^6(e+f x)+(a+b) (11 a+25 b) \csc ^4(e+f x)\right)}{120 f (a+b)^5 \left(a+b \sec ^2(e+f x)\right)^{5/2}}-\frac{\sec ^5(e+f x) (a \cos (2 e+2 f x)+a+2 b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right)}{4 \sqrt{2} a^{5/2} f \left(a+b \sec ^2(e+f x)\right)^{5/2}}","-\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{\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{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{\left(5 a^3+19 a^2 b-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(15 a^4+70 a^3 b+128 a^2 b^2-70 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{b \cot ^5(e+f x)}{3 a f (a+b) \left(a+b \tan ^2(e+f x)+b\right)^{3/2}}",1,"-1/4*(ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]]*(a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^5)/(Sqrt[2]*a^(5/2)*f*(a + b*Sec[e + f*x]^2)^(5/2)) + ((a + 2*b + a*Cos[2*(e + f*x)])^3*((-20*b^5*(a + b))/(a^2*(a + 2*b + a*Cos[2*(e + f*x)])^2) + (10*b^4*(15*a + 4*b))/(a^2*(a + 2*b + a*Cos[2*(e + f*x)])) - (23*a^2 + 100*a*b + 150*b^2)*Csc[e + f*x]^2 + (a + b)*(11*a + 25*b)*Csc[e + f*x]^4 - 3*(a + b)^2*Csc[e + f*x]^6)*Sec[e + f*x]^4*Tan[e + f*x])/(120*(a + b)^5*f*(a + b*Sec[e + f*x]^2)^(5/2))","A",1
441,1,259,105,3.4878094,"\int \left(a+b \sec ^2(e+f x)\right)^p (d \tan (e+f x))^m \, dx","Integrate[(a + b*Sec[e + f*x]^2)^p*(d*Tan[e + f*x])^m,x]","\frac{\sin (e+f x) \cos (e+f x) (d \tan (e+f x))^m \left(a+b \sec ^2(e+f x)\right)^p F_1\left(\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b \tan ^2(e+f x)}{a+b},-\tan ^2(e+f x)\right)}{f (m+1) \left(\frac{2 \tan ^2(e+f x) \left(b p F_1\left(\frac{m+3}{2};1-p,1;\frac{m+5}{2};-\frac{b \tan ^2(e+f x)}{a+b},-\tan ^2(e+f x)\right)-(a+b) F_1\left(\frac{m+3}{2};-p,2;\frac{m+5}{2};-\frac{b \tan ^2(e+f x)}{a+b},-\tan ^2(e+f x)\right)\right)}{(m+3) (a+b)}+F_1\left(\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b \tan ^2(e+f x)}{a+b},-\tan ^2(e+f x)\right)\right)}","\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, -p, 1, (3 + m)/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]*(d*Tan[e + f*x])^m)/(f*(1 + m)*(AppellF1[(1 + m)/2, -p, 1, (3 + m)/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + (2*(b*p*AppellF1[(3 + m)/2, 1 - p, 1, (5 + m)/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[(3 + m)/2, -p, 2, (5 + m)/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)/((a + b)*(3 + m))))","B",0
442,1,94,122,0.5272806,"\int \left(a+b \sec ^2(e+f x)\right)^p \tan ^5(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^5,x]","-\frac{\left(a+b \sec ^2(e+f x)\right)^{p+1} \left(b^2 (p+2) \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)}{a}+1\right)+a \left(a-b (p+1) \sec ^2(e+f x)+2 b (p+2)\right)\right)}{2 a b^2 f (p+1) (p+2)}","-\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,"-1/2*((a + b*Sec[e + f*x]^2)^(1 + p)*(b^2*(2 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sec[e + f*x]^2)/a] + a*(a + 2*b*(2 + p) - b*(1 + p)*Sec[e + f*x]^2)))/(a*b^2*f*(1 + p)*(2 + p))","A",1
443,1,61,86,0.1285702,"\int \left(a+b \sec ^2(e+f x)\right)^p \tan ^3(e+f x) \, dx","Integrate[(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} \left(b \, _2F_1\left(1,p+1;p+2;\frac{b \sec ^2(e+f x)}{a}+1\right)+a\right)}{2 a 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*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*b*f*(1 + p))","A",1
444,1,54,54,0.0459766,"\int \left(a+b \sec ^2(e+f x)\right)^p \tan (e+f x) \, dx","Integrate[(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,"-1/2*(Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sec[e + f*x]^2)/a]*(a + b*Sec[e + f*x]^2)^(1 + p))/(a*f*(1 + p))","A",1
445,1,115,114,2.1254318,"\int \cot (e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Cot[e + f*x]*(a + b*Sec[e + f*x]^2)^p,x]","\frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left(a+b \sec ^2(e+f x)\right)^p \left((a+b) \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a+b}{a}\right)-a \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)}{a+b}+1\right)\right)}{4 a 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,"((a + 2*b + a*Cos[2*(e + f*x)])*((a + b)*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b + b*Tan[e + f*x]^2)/a] - a*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Tan[e + f*x]^2)/(a + b)])*Sec[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p)/(4*a*(a + b)*f*(1 + p))","A",1
446,1,139,157,3.5104134,"\int \cot ^3(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Cot[e + f*x]^3*(a + b*Sec[e + f*x]^2)^p,x]","-\frac{\tan ^2(e+f x) \left((a+b) \cot ^2(e+f x)+b\right) \left(a+b \sec ^2(e+f x)\right)^p \left((a+b)^2 \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a+b}{a}\right)-a (a-b p+b) \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)}{a+b}+1\right)+a (p+1) (a+b) \cot ^2(e+f x)\right)}{2 a f (p+1) (a+b)^2}","\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,"-1/2*((b + (a + b)*Cot[e + f*x]^2)*(a*(a + b)*(1 + p)*Cot[e + f*x]^2 + (a + b)^2*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b + b*Tan[e + f*x]^2)/a] - a*(a + b - b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Tan[e + f*x]^2)/(a + b)])*(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^2)/(a*(a + b)^2*f*(1 + p))","A",1
447,1,2777,88,18.1519461,"\int \left(a+b \sec ^2(e+f x)\right)^p \tan ^4(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^4,x]","\text{Result too large to show}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^5*((9*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (-3*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))] + Hypergeometric2F1[3/2, -p, 5/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2)/(1 + (b*Tan[e + f*x]^2)/(a + b))^p))/(3*f*(((a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(1 + p)*((9*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (-3*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))] + Hypergeometric2F1[3/2, -p, 5/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2)/(1 + (b*Tan[e + f*x]^2)/(a + b))^p))/3 - (2*a*p*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^p*Sin[2*(e + f*x)]*Tan[e + f*x]*((9*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (-3*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))] + Hypergeometric2F1[3/2, -p, 5/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2)/(1 + (b*Tan[e + f*x]^2)/(a + b))^p))/3 + (2*p*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Tan[e + f*x]^2*((9*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (-3*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))] + Hypergeometric2F1[3/2, -p, 5/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2)/(1 + (b*Tan[e + f*x]^2)/(a + b))^p))/3 + ((a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Tan[e + f*x]*((-18*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (9*(a + b)*Cos[e + f*x]^2*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (2*b*p*Sec[e + f*x]^2*Tan[e + f*x]*(1 + (b*Tan[e + f*x]^2)/(a + b))^(-1 - p)*(-3*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))] + Hypergeometric2F1[3/2, -p, 5/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2))/(a + b) - (9*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]^2*(4*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Sec[e + f*x]^2*Tan[e + f*x] + 3*(a + b)*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(b*p*((-6*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5 - (6*b*(1 - p)*AppellF1[5/2, 2 - p, 1, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b))) - (a + b)*((6*b*p*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (12*AppellF1[5/2, -p, 3, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)^2 + (2*Hypergeometric2F1[3/2, -p, 5/2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x] - 3*Csc[e + f*x]*Sec[e + f*x]*(-Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))] + (1 + (b*Tan[e + f*x]^2)/(a + b))^p) + 3*Sec[e + f*x]^2*Tan[e + f*x]*(-Hypergeometric2F1[3/2, -p, 5/2, -((b*Tan[e + f*x]^2)/(a + b))] + (1 + (b*Tan[e + f*x]^2)/(a + b))^p))/(1 + (b*Tan[e + f*x]^2)/(a + b))^p))/3))","B",0
448,1,2465,88,16.539206,"\int \left(a+b \sec ^2(e+f x)\right)^p \tan ^2(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^2,x]","\text{Result too large to show}","\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,"((a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^3*(Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))]/(1 + (b*Tan[e + f*x]^2)/(a + b))^p - (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)))/(f*((a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(1 + p)*(Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))]/(1 + (b*Tan[e + f*x]^2)/(a + b))^p - (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) - 2*a*p*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^p*Sin[2*(e + f*x)]*Tan[e + f*x]*(Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))]/(1 + (b*Tan[e + f*x]^2)/(a + b))^p - (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) + 2*p*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Tan[e + f*x]^2*(Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))]/(1 + (b*Tan[e + f*x]^2)/(a + b))^p - (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) + (a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Tan[e + f*x]*((-2*b*p*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x]*(1 + (b*Tan[e + f*x]^2)/(a + b))^(-1 - p))/(a + b) + (6*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (3*(a + b)*Cos[e + f*x]^2*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (Csc[e + f*x]*Sec[e + f*x]*(-Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))] + (1 + (b*Tan[e + f*x]^2)/(a + b))^p))/(1 + (b*Tan[e + f*x]^2)/(a + b))^p + (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]^2*(4*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Sec[e + f*x]^2*Tan[e + f*x] + 3*(a + b)*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(b*p*((-6*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5 - (6*b*(1 - p)*AppellF1[5/2, 2 - p, 1, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b))) - (a + b)*((6*b*p*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (12*AppellF1[5/2, -p, 3, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)^2)))","B",0
449,1,2137,83,6.2643958,"\int \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[(a + b*Sec[e + f*x]^2)^p,x]","\text{Result too large to show}","\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,"(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x])/(f*(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)*((3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-1 + p))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Sin[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (6*(a + b)*p*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Sin[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (6*a*(a + b)*p*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[e + f*x]^2)^p*Sin[e + f*x]*Sin[2*(e + f*x)])/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (3*(a + b)*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Sin[e + f*x]*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*Sin[e + f*x]*(4*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Sec[e + f*x]^2*Tan[e + f*x] + 3*(a + b)*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(b*p*((-6*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5 - (6*b*(1 - p)*AppellF1[5/2, 2 - p, 1, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b))) - (a + b)*((6*b*p*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (12*AppellF1[5/2, -p, 3, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)^2))","B",0
450,1,2469,84,16.9019665,"\int \cot ^2(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Cot[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p,x]","\text{Result too large to show}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])^p*Cot[e + f*x]^3*(Sec[e + f*x]^2)^p*(a + b*Sec[e + f*x]^2)^p*(-(Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]/(1 + (b*Tan[e + f*x]^2)/(a + b))^p) - (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sin[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)))/(f*(2*p*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^p*(-(Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]/(1 + (b*Tan[e + f*x]^2)/(a + b))^p) - (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sin[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) - (a + 2*b + a*Cos[2*(e + f*x)])^p*Csc[e + f*x]^2*(Sec[e + f*x]^2)^p*(-(Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]/(1 + (b*Tan[e + f*x]^2)/(a + b))^p) - (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sin[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) - 2*a*p*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*Cot[e + f*x]*(Sec[e + f*x]^2)^p*Sin[2*(e + f*x)]*(-(Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]/(1 + (b*Tan[e + f*x]^2)/(a + b))^p) - (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sin[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) + (a + 2*b + a*Cos[2*(e + f*x)])^p*Cot[e + f*x]*(Sec[e + f*x]^2)^p*((2*b*p*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x]*(1 + (b*Tan[e + f*x]^2)/(a + b))^(-1 - p))/(a + b) - (6*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (3*(a + b)*Sin[e + f*x]^2*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (Csc[e + f*x]*Sec[e + f*x]*(Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))] - (1 + (b*Tan[e + f*x]^2)/(a + b))^p))/(1 + (b*Tan[e + f*x]^2)/(a + b))^p + (3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sin[e + f*x]^2*(4*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Sec[e + f*x]^2*Tan[e + f*x] + 3*(a + b)*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(b*p*((-6*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5 - (6*b*(1 - p)*AppellF1[5/2, 2 - p, 1, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b))) - (a + b)*((6*b*p*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (12*AppellF1[5/2, -p, 3, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)^2)))","B",0
451,1,3033,88,18.347107,"\int \cot ^4(e+f x) \left(a+b \sec ^2(e+f x)\right)^p \, dx","Integrate[Cot[e + f*x]^4*(a + b*Sec[e + f*x]^2)^p,x]","\text{Result too large to show}","-\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,"((a + 2*b + a*Cos[2*(e + f*x)])^p*Cot[e + f*x]^7*(Sec[e + f*x]^2)^p*(a + b*Sec[e + f*x]^2)^p*((9*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sin[e + f*x]^2*Tan[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (Hypergeometric2F1[-3/2, -p, -1/2, -((b*Tan[e + f*x]^2)/(a + b))] - 3*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2)/(1 + (b*Tan[e + f*x]^2)/(a + b))^p))/(3*f*((2*p*(a + 2*b + a*Cos[2*(e + f*x)])^p*Cot[e + f*x]^2*(Sec[e + f*x]^2)^p*((9*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sin[e + f*x]^2*Tan[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (Hypergeometric2F1[-3/2, -p, -1/2, -((b*Tan[e + f*x]^2)/(a + b))] - 3*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2)/(1 + (b*Tan[e + f*x]^2)/(a + b))^p))/3 - (a + 2*b + a*Cos[2*(e + f*x)])^p*Cot[e + f*x]^2*Csc[e + f*x]^2*(Sec[e + f*x]^2)^p*((9*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sin[e + f*x]^2*Tan[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (Hypergeometric2F1[-3/2, -p, -1/2, -((b*Tan[e + f*x]^2)/(a + b))] - 3*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2)/(1 + (b*Tan[e + f*x]^2)/(a + b))^p) - (2*a*p*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*Cot[e + f*x]^3*(Sec[e + f*x]^2)^p*Sin[2*(e + f*x)]*((9*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sin[e + f*x]^2*Tan[e + f*x]^2)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (Hypergeometric2F1[-3/2, -p, -1/2, -((b*Tan[e + f*x]^2)/(a + b))] - 3*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2)/(1 + (b*Tan[e + f*x]^2)/(a + b))^p))/3 + ((a + 2*b + a*Cos[2*(e + f*x)])^p*Cot[e + f*x]^3*(Sec[e + f*x]^2)^p*((18*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sin[e + f*x]^2*Tan[e + f*x])/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (18*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Tan[e + f*x]^3)/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (9*(a + b)*Sin[e + f*x]^2*Tan[e + f*x]^2*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (2*b*p*Sec[e + f*x]^2*Tan[e + f*x]*(1 + (b*Tan[e + f*x]^2)/(a + b))^(-1 - p)*(Hypergeometric2F1[-3/2, -p, -1/2, -((b*Tan[e + f*x]^2)/(a + b))] - 3*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^2))/(a + b) - (9*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sin[e + f*x]^2*Tan[e + f*x]^2*(4*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Sec[e + f*x]^2*Tan[e + f*x] + 3*(a + b)*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(b*p*((-6*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5 - (6*b*(1 - p)*AppellF1[5/2, 2 - p, 1, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b))) - (a + b)*((6*b*p*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*(a + b)) - (12*AppellF1[5/2, -p, 3, 7/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*(a + b)*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2] - (a + b)*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/(a + b)), -Tan[e + f*x]^2])*Tan[e + f*x]^2)^2 - (-6*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]*Sec[e + f*x]^2*Tan[e + f*x] - 3*Sec[e + f*x]^2*Tan[e + f*x]*(Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))] - (1 + (b*Tan[e + f*x]^2)/(a + b))^p) - 3*Csc[e + f*x]*Sec[e + f*x]*(-Hypergeometric2F1[-3/2, -p, -1/2, -((b*Tan[e + f*x]^2)/(a + b))] + (1 + (b*Tan[e + f*x]^2)/(a + b))^p))/(1 + (b*Tan[e + f*x]^2)/(a + b))^p))/3))","B",0
452,1,87,92,0.2635046,"\int \left(a+b \sec ^3(e+f x)\right) \tan ^5(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^3)*Tan[e + f*x]^5,x]","-\frac{a \left(-\tan ^4(e+f x)+2 \tan ^2(e+f x)+4 \log (\cos (e+f x))\right)}{4 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,"(b*Sec[e + f*x]^3)/(3*f) - (2*b*Sec[e + f*x]^5)/(5*f) + (b*Sec[e + f*x]^7)/(7*f) - (a*(4*Log[Cos[e + f*x]] + 2*Tan[e + f*x]^2 - Tan[e + f*x]^4))/(4*f)","A",1
453,1,59,61,0.1212943,"\int \left(a+b \sec ^3(e+f x)\right) \tan ^3(e+f x) \, dx","Integrate[(a + b*Sec[e + f*x]^3)*Tan[e + f*x]^3,x]","\frac{a \left(\tan ^2(e+f x)+2 \log (\cos (e+f x))\right)}{2 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,"-1/3*(b*Sec[e + f*x]^3)/f + (b*Sec[e + f*x]^5)/(5*f) + (a*(2*Log[Cos[e + f*x]] + Tan[e + f*x]^2))/(2*f)","A",1
454,1,30,30,0.0138062,"\int \left(a+b \sec ^3(e+f x)\right) \tan (e+f x) \, dx","Integrate[(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",1
455,1,65,54,0.0574649,"\int \cot (e+f x) \left(a+b \sec ^3(e+f x)\right) \, dx","Integrate[Cot[e + f*x]*(a + b*Sec[e + f*x]^3),x]","\frac{a (\log (\tan (e+f x))+\log (\cos (e+f x)))}{f}+\frac{b \sec (e+f x)}{f}+\frac{b \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f}-\frac{b \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{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,"-((b*Log[Cos[(e + f*x)/2]])/f) + (b*Log[Sin[(e + f*x)/2]])/f + (a*(Log[Cos[e + f*x]] + Log[Tan[e + f*x]]))/f + (b*Sec[e + f*x])/f","A",1
456,1,114,72,1.0124158,"\int \cot ^3(e+f x) \left(a+b \sec ^3(e+f x)\right) \, dx","Integrate[Cot[e + f*x]^3*(a + b*Sec[e + f*x]^3),x]","-\frac{a \left(\cot ^2(e+f x)+2 \log (\tan (e+f x))+2 \log (\cos (e+f x))\right)}{2 f}-\frac{b \csc ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{b \sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{b \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}-\frac{b \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{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,"-1/8*(b*Csc[(e + f*x)/2]^2)/f - (b*Log[Cos[(e + f*x)/2]])/(2*f) + (b*Log[Sin[(e + f*x)/2]])/(2*f) - (a*(Cot[e + f*x]^2 + 2*Log[Cos[e + f*x]] + 2*Log[Tan[e + f*x]]))/(2*f) + (b*Sec[(e + f*x)/2]^2)/(8*f)","A",1
457,1,251,219,0.3597437,"\int \frac{\tan ^5(e+f x)}{a+b \sec ^3(e+f x)} \, dx","Integrate[Tan[e + f*x]^5/(a + b*Sec[e + f*x]^3),x]","\frac{-\text{RootSum}\left[\text{$\#$1}^3 a-\text{$\#$1}^3 b-6 \text{$\#$1}^2 a+12 \text{$\#$1} a-8 a\&,\frac{\text{$\#$1}^2 a b \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)-\text{$\#$1}^2 b^2 \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)-4 a^2 \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)+2 \text{$\#$1} a^2 \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)+4 a b \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)-8 \text{$\#$1} a b \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)}{\text{$\#$1}^2 a-\text{$\#$1}^2 b-4 \text{$\#$1} a+4 a}\&\right]+3 a \sec (e+f x)+3 b \log \left(\sec ^2\left(\frac{1}{2} (e+f x)\right)\right)}{3 a b f}","\frac{\left(a^{2/3}-2 b^{2/3}\right) \log \left(a^{2/3} \cos ^2(e+f x)-\sqrt[3]{a} \sqrt[3]{b} \cos (e+f x)+b^{2/3}\right)}{6 \sqrt[3]{a} b^{4/3} f}-\frac{\left(a^{2/3}-2 b^{2/3}\right) \log \left(\sqrt[3]{a} \cos (e+f x)+\sqrt[3]{b}\right)}{3 \sqrt[3]{a} b^{4/3} f}-\frac{\left(a^{2/3}+2 b^{2/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{b}-2 \sqrt[3]{a} \cos (e+f x)}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{a} b^{4/3} f}-\frac{\log \left(a \cos ^3(e+f x)+b\right)}{3 a f}+\frac{\sec (e+f x)}{b f}",1,"(3*b*Log[Sec[(e + f*x)/2]^2] - RootSum[-8*a + 12*a*#1 - 6*a*#1^2 + a*#1^3 - b*#1^3 & , (-4*a^2*Log[1 - #1 + Tan[(e + f*x)/2]^2] + 4*a*b*Log[1 - #1 + Tan[(e + f*x)/2]^2] + 2*a^2*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1 - 8*a*b*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1 + a*b*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1^2 - b^2*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1^2)/(4*a - 4*a*#1 + a*#1^2 - b*#1^2) & ] + 3*a*Sec[e + f*x])/(3*a*b*f)","C",1
458,1,242,166,0.2541625,"\int \frac{\tan ^3(e+f x)}{a+b \sec ^3(e+f x)} \, dx","Integrate[Tan[e + f*x]^3/(a + b*Sec[e + f*x]^3),x]","\frac{\text{RootSum}\left[\text{$\#$1}^3 a-\text{$\#$1}^3 b-3 \text{$\#$1}^2 a-3 \text{$\#$1}^2 b+3 \text{$\#$1} a-3 \text{$\#$1} b-a-b\&,\frac{\text{$\#$1}^2 a \log \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-\text{$\#$1}\right)-\text{$\#$1}^2 b \log \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-\text{$\#$1}\right)-4 \text{$\#$1} a \log \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-\text{$\#$1}\right)-a \log \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-\text{$\#$1}\right)-2 \text{$\#$1} b \log \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-\text{$\#$1}\right)-b \log \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-\text{$\#$1}\right)}{\text{$\#$1}^2 a-\text{$\#$1}^2 b-2 \text{$\#$1} a-2 \text{$\#$1} b+a-b}\&\right]-3 \log \left(\sec ^2\left(\frac{1}{2} (e+f x)\right)\right)}{3 a f}","\frac{\log \left(a^{2/3} \cos ^2(e+f x)-\sqrt[3]{a} \sqrt[3]{b} \cos (e+f x)+b^{2/3}\right)}{6 \sqrt[3]{a} b^{2/3} f}-\frac{\log \left(\sqrt[3]{a} \cos (e+f x)+\sqrt[3]{b}\right)}{3 \sqrt[3]{a} b^{2/3} f}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{b}-2 \sqrt[3]{a} \cos (e+f x)}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{a} b^{2/3} f}+\frac{\log \left(a \cos ^3(e+f x)+b\right)}{3 a f}",1,"(-3*Log[Sec[(e + f*x)/2]^2] + RootSum[-a - b + 3*a*#1 - 3*b*#1 - 3*a*#1^2 - 3*b*#1^2 + a*#1^3 - b*#1^3 & , (-(a*Log[-#1 + Tan[(e + f*x)/2]^2]) - b*Log[-#1 + Tan[(e + f*x)/2]^2] - 4*a*Log[-#1 + Tan[(e + f*x)/2]^2]*#1 - 2*b*Log[-#1 + Tan[(e + f*x)/2]^2]*#1 + a*Log[-#1 + Tan[(e + f*x)/2]^2]*#1^2 - b*Log[-#1 + Tan[(e + f*x)/2]^2]*#1^2)/(a - b - 2*a*#1 - 2*b*#1 + a*#1^2 - b*#1^2) & ])/(3*a*f)","C",1
459,1,23,23,0.018188,"\int \frac{\tan (e+f x)}{a+b \sec ^3(e+f x)} \, dx","Integrate[Tan[e + f*x]/(a + b*Sec[e + f*x]^3),x]","-\frac{\log \left(a \cos ^3(e+f x)+b\right)}{3 a f}","-\frac{\log \left(a \cos ^3(e+f x)+b\right)}{3 a f}",1,"-1/3*Log[b + a*Cos[e + f*x]^3]/(a*f)","A",1
460,1,290,295,0.3985848,"\int \frac{\cot (e+f x)}{a+b \sec ^3(e+f x)} \, dx","Integrate[Cot[e + f*x]/(a + b*Sec[e + f*x]^3),x]","\frac{3 \left(a (a-b) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)+a (a+b) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)+b^2 \log \left(\sec ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)-b \text{RootSum}\left[\text{$\#$1}^3 a-\text{$\#$1}^3 b-6 \text{$\#$1}^2 a+12 \text{$\#$1} a-8 a\&,\frac{\text{$\#$1}^2 a b \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)-\text{$\#$1}^2 b^2 \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)-4 a^2 \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)+2 \text{$\#$1} a^2 \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)+4 a b \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)-2 \text{$\#$1} a b \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)}{\text{$\#$1}^2 a-\text{$\#$1}^2 b-4 \text{$\#$1} a+4 a}\&\right]}{3 a f (a-b) (a+b)}","-\frac{b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{b}-2 \sqrt[3]{a} \cos (e+f x)}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{a} f \left(a^{2/3} b^{2/3}+a^{4/3}+b^{4/3}\right)}-\frac{b^2 \log \left(a \cos ^3(e+f x)+b\right)}{3 a f \left(a^2-b^2\right)}+\frac{b^{2/3} \left(a^{2/3}+b^{2/3}\right) \log \left(a^{2/3} \cos ^2(e+f x)-\sqrt[3]{a} \sqrt[3]{b} \cos (e+f x)+b^{2/3}\right)}{6 \sqrt[3]{a} f \left(a^2-b^2\right)}-\frac{b^{2/3} \left(a^{2/3}+b^{2/3}\right) \log \left(\sqrt[3]{a} \cos (e+f x)+\sqrt[3]{b}\right)}{3 \sqrt[3]{a} f \left(a^2-b^2\right)}+\frac{\log (1-\cos (e+f x))}{2 f (a+b)}+\frac{\log (\cos (e+f x)+1)}{2 f (a-b)}",1,"(3*(a*(a + b)*Log[Cos[(e + f*x)/2]] + b^2*Log[Sec[(e + f*x)/2]^2] + a*(a - b)*Log[Sin[(e + f*x)/2]]) - b*RootSum[-8*a + 12*a*#1 - 6*a*#1^2 + a*#1^3 - b*#1^3 & , (-4*a^2*Log[1 - #1 + Tan[(e + f*x)/2]^2] + 4*a*b*Log[1 - #1 + Tan[(e + f*x)/2]^2] + 2*a^2*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1 - 2*a*b*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1 + a*b*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1^2 - b^2*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1^2)/(4*a - 4*a*#1 + a*#1^2 - b*#1^2) & ])/(3*a*(a - b)*(a + b)*f)","C",1
461,1,336,393,1.9710809,"\int \frac{\cot ^3(e+f x)}{a+b \sec ^3(e+f x)} \, dx","Integrate[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^3),x]","\frac{\frac{8 b^2 \left((b-a) \text{RootSum}\left[\text{$\#$1}^3 a-\text{$\#$1}^3 b-6 \text{$\#$1}^2 a+12 \text{$\#$1} a-8 a\&,\frac{2 \text{$\#$1}^2 a^2 \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)+\text{$\#$1}^2 b^2 \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)+8 a^2 \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)-6 \text{$\#$1} a^2 \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)-4 a b \log \left(-\text{$\#$1}+\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)}{\text{$\#$1}^2 a-\text{$\#$1}^2 b-4 \text{$\#$1} a+4 a}\&\right]+3 \left(2 a^2+b^2\right) \log \left(\sec ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}{a \left(a^2-b^2\right)^2}-\frac{3 \csc ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}-\frac{3 \sec ^2\left(\frac{1}{2} (e+f x)\right)}{a-b}-\frac{12 (2 a+5 b) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{(a+b)^2}+\frac{12 (5 b-2 a) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{(a-b)^2}}{24 f}","-\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,"((-3*Csc[(e + f*x)/2]^2)/(a + b) + (12*(-2*a + 5*b)*Log[Cos[(e + f*x)/2]])/(a - b)^2 - (12*(2*a + 5*b)*Log[Sin[(e + f*x)/2]])/(a + b)^2 + (8*b^2*(3*(2*a^2 + b^2)*Log[Sec[(e + f*x)/2]^2] + (-a + b)*RootSum[-8*a + 12*a*#1 - 6*a*#1^2 + a*#1^3 - b*#1^3 & , (8*a^2*Log[1 - #1 + Tan[(e + f*x)/2]^2] - 4*a*b*Log[1 - #1 + Tan[(e + f*x)/2]^2] - 6*a^2*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1 + 2*a^2*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1^2 + b^2*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1^2)/(4*a - 4*a*#1 + a*#1^2 - b*#1^2) & ]))/(a*(a^2 - b^2)^2) - (3*Sec[(e + f*x)/2]^2)/(a - b))/(24*f)","C",1
462,0,0,30,3.1931624,"\int \left(a+b (c \sec (e+f x))^n\right)^p (d \tan (e+f x))^m \, dx","Integrate[(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,"Integrate[(a + b*(c*Sec[e + f*x])^n)^p*(d*Tan[e + f*x])^m, x]","A",-1
463,1,245,226,10.1230511,"\int \left(a+b (c \sec (e+f x))^n\right)^p \tan ^5(e+f x) \, dx","Integrate[(a + b*(c*Sec[e + f*x])^n)^p*Tan[e + f*x]^5,x]","\frac{\left(a+b (c \sec (e+f x))^n\right)^p \left(\frac{b \left(c \sqrt{\sec ^2(e+f x)}\right)^n}{a}+1\right)^{-p} \left(-4 \left(a+b \left(c \sqrt{\sec ^2(e+f x)}\right)^n\right) \left(\frac{b \left(c \sqrt{\sec ^2(e+f x)}\right)^n}{a}+1\right)^p \, _2F_1\left(1,p+1;p+2;\frac{b \left(c \sqrt{\sec ^2(e+f x)}\right)^n}{a}+1\right)-4 a n (p+1) \sec ^2(e+f x) \, _2F_1\left(\frac{2}{n},-p;\frac{n+2}{n};-\frac{b \left(c \sqrt{\sec ^2(e+f x)}\right)^n}{a}\right)+a n (p+1) \sec ^4(e+f x) \, _2F_1\left(\frac{4}{n},-p;\frac{n+4}{n};-\frac{b \left(c \sqrt{\sec ^2(e+f x)}\right)^n}{a}\right)\right)}{4 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,"((a + b*(c*Sec[e + f*x])^n)^p*(-4*a*n*(1 + p)*Hypergeometric2F1[2/n, -p, (2 + n)/n, -((b*(c*Sqrt[Sec[e + f*x]^2])^n)/a)]*Sec[e + f*x]^2 + a*n*(1 + p)*Hypergeometric2F1[4/n, -p, (4 + n)/n, -((b*(c*Sqrt[Sec[e + f*x]^2])^n)/a)]*Sec[e + f*x]^4 - 4*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*(c*Sqrt[Sec[e + f*x]^2])^n)/a]*(a + b*(c*Sqrt[Sec[e + f*x]^2])^n)*(1 + (b*(c*Sqrt[Sec[e + f*x]^2])^n)/a)^p))/(4*a*f*n*(1 + p)*(1 + (b*(c*Sqrt[Sec[e + f*x]^2])^n)/a)^p)","A",0
464,1,162,143,4.5892679,"\int \left(a+b (c \sec (e+f x))^n\right)^p \tan ^3(e+f x) \, dx","Integrate[(a + b*(c*Sec[e + f*x])^n)^p*Tan[e + f*x]^3,x]","\frac{\left(a+b (c \sec (e+f x))^n\right)^p \left(\sec ^2(e+f x) \left(\frac{b \left(c \sqrt{\sec ^2(e+f x)}\right)^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{2}{n},-p;\frac{n+2}{n};-\frac{b \left(c \sqrt{\sec ^2(e+f x)}\right)^n}{a}\right)+\frac{2 \left(a+b \left(c \sqrt{\sec ^2(e+f x)}\right)^n\right) \, _2F_1\left(1,p+1;p+2;\frac{b \left(c \sqrt{\sec ^2(e+f x)}\right)^n}{a}+1\right)}{a n (p+1)}\right)}{2 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)}{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,"((a + b*(c*Sec[e + f*x])^n)^p*((2*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*(c*Sqrt[Sec[e + f*x]^2])^n)/a]*(a + b*(c*Sqrt[Sec[e + f*x]^2])^n))/(a*n*(1 + p)) + (Hypergeometric2F1[2/n, -p, (2 + n)/n, -((b*(c*Sqrt[Sec[e + f*x]^2])^n)/a)]*Sec[e + f*x]^2)/(1 + (b*(c*Sqrt[Sec[e + f*x]^2])^n)/a)^p))/(2*f)","A",0
465,1,59,59,0.0813472,"\int \left(a+b (c \sec (e+f x))^n\right)^p \tan (e+f x) \, dx","Integrate[(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",1
466,0,0,26,3.7679251,"\int \cot (e+f x) \left(a+b (c \sec (e+f x))^n\right)^p \, dx","Integrate[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,"Integrate[Cot[e + f*x]*(a + b*(c*Sec[e + f*x])^n)^p, x]","A",-1
467,0,0,28,36.1969624,"\int \cot ^3(e+f x) \left(a+b (c \sec (e+f x))^n\right)^p \, dx","Integrate[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,"Integrate[Cot[e + f*x]^3*(a + b*(c*Sec[e + f*x])^n)^p, x]","A",-1
468,0,0,28,2.4385448,"\int \left(a+b (c \sec (e+f x))^n\right)^p \tan ^2(e+f x) \, dx","Integrate[(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,"Integrate[(a + b*(c*Sec[e + f*x])^n)^p*Tan[e + f*x]^2, x]","A",-1
469,0,0,19,1.232223,"\int \left(a+b (c \sec (e+f x))^n\right)^p \, dx","Integrate[(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,"Integrate[(a + b*(c*Sec[e + f*x])^n)^p, x]","A",-1
470,0,0,28,1.8530997,"\int \cot ^2(e+f x) \left(a+b (c \sec (e+f x))^n\right)^p \, dx","Integrate[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,"Integrate[Cot[e + f*x]^2*(a + b*(c*Sec[e + f*x])^n)^p, x]","A",-1