1,1,165,196,2.1298896,"\int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx","Integrate[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5,x]","\frac{a^2 c^5 \sec ^6(e+f x) \left(-210 \sin (e+f x)-120 \sin (2 (e+f x))+865 \sin (3 (e+f x))-768 \sin (4 (e+f x))+435 \sin (5 (e+f x))-88 \sin (6 (e+f x))+1800 (e+f x) \cos (2 (e+f x))+720 e \cos (4 (e+f x))+720 f x \cos (4 (e+f x))+120 e \cos (6 (e+f x))+120 f x \cos (6 (e+f x))-4560 \cos ^6(e+f x) \tanh ^{-1}(\sin (e+f x))+1200 e+1200 f x\right)}{3840 f}","\frac{3 a^2 c^5 \tan ^5(e+f x)}{5 f}+\frac{a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac{a^2 c^5 \tan (e+f x)}{f}-\frac{19 a^2 c^5 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac{a^2 c^5 \tan ^3(e+f x) \sec ^3(e+f x)}{6 f}+\frac{a^2 c^5 \tan (e+f x) \sec ^3(e+f x)}{8 f}-\frac{3 a^2 c^5 \tan ^3(e+f x) \sec (e+f x)}{4 f}+\frac{17 a^2 c^5 \tan (e+f x) \sec (e+f x)}{16 f}+a^2 c^5 x",1,"(a^2*c^5*Sec[e + f*x]^6*(1200*e + 1200*f*x - 4560*ArcTanh[Sin[e + f*x]]*Cos[e + f*x]^6 + 1800*(e + f*x)*Cos[2*(e + f*x)] + 720*e*Cos[4*(e + f*x)] + 720*f*x*Cos[4*(e + f*x)] + 120*e*Cos[6*(e + f*x)] + 120*f*x*Cos[6*(e + f*x)] - 210*Sin[e + f*x] - 120*Sin[2*(e + f*x)] + 865*Sin[3*(e + f*x)] - 768*Sin[4*(e + f*x)] + 435*Sin[5*(e + f*x)] - 88*Sin[6*(e + f*x)]))/(3840*f)","A",1
2,1,146,140,1.1659075,"\int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4 \, dx","Integrate[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^4,x]","\frac{a^2 c^4 \sec ^5(e+f x) \left(40 \sin (e+f x)+60 \sin (2 (e+f x))-220 \sin (3 (e+f x))+150 \sin (4 (e+f x))-68 \sin (5 (e+f x))+600 (e+f x) \cos (e+f x)+300 e \cos (3 (e+f x))+300 f x \cos (3 (e+f x))+60 e \cos (5 (e+f x))+60 f x \cos (5 (e+f x))-720 \cos ^5(e+f x) \tanh ^{-1}(\sin (e+f x))\right)}{960 f}","\frac{a^2 c^4 \tan ^5(e+f x)}{5 f}+\frac{a^2 c^4 \tan ^3(e+f x)}{3 f}-\frac{a^2 c^4 \tan (e+f x)}{f}-\frac{3 a^2 c^4 \tanh ^{-1}(\sin (e+f x))}{4 f}-\frac{a^2 c^4 \tan ^3(e+f x) \sec (e+f x)}{2 f}+\frac{3 a^2 c^4 \tan (e+f x) \sec (e+f x)}{4 f}+a^2 c^4 x",1,"(a^2*c^4*Sec[e + f*x]^5*(600*(e + f*x)*Cos[e + f*x] - 720*ArcTanh[Sin[e + f*x]]*Cos[e + f*x]^5 + 300*e*Cos[3*(e + f*x)] + 300*f*x*Cos[3*(e + f*x)] + 60*e*Cos[5*(e + f*x)] + 60*f*x*Cos[5*(e + f*x)] + 40*Sin[e + f*x] + 60*Sin[2*(e + f*x)] - 220*Sin[3*(e + f*x)] + 150*Sin[4*(e + f*x)] - 68*Sin[5*(e + f*x)]))/(960*f)","A",1
3,1,122,97,0.7349107,"\int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3 \, dx","Integrate[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^3,x]","\frac{a^2 c^3 \sec ^4(e+f x) \left(-18 \sin (e+f x)-32 \sin (2 (e+f x))+30 \sin (3 (e+f x))-32 \sin (4 (e+f x))+96 (e+f x) \cos (2 (e+f x))+24 e \cos (4 (e+f x))+24 f x \cos (4 (e+f x))-72 \cos ^4(e+f x) \tanh ^{-1}(\sin (e+f x))+72 e+72 f x\right)}{192 f}","-\frac{3 a^2 c^3 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a^2 \tan ^3(e+f x) \left(4 c^3-3 c^3 \sec (e+f x)\right)}{12 f}-\frac{a^2 \tan (e+f x) \left(8 c^3-3 c^3 \sec (e+f x)\right)}{8 f}+a^2 c^3 x",1,"(a^2*c^3*Sec[e + f*x]^4*(72*e + 72*f*x - 72*ArcTanh[Sin[e + f*x]]*Cos[e + f*x]^4 + 96*(e + f*x)*Cos[2*(e + f*x)] + 24*e*Cos[4*(e + f*x)] + 24*f*x*Cos[4*(e + f*x)] - 18*Sin[e + f*x] - 32*Sin[2*(e + f*x)] + 30*Sin[3*(e + f*x)] - 32*Sin[4*(e + f*x)]))/(192*f)","A",1
4,1,45,47,0.0311083,"\int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^2 \, dx","Integrate[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^2,x]","a^2 c^2 \left(\frac{\tan ^{-1}(\tan (e+f x))}{f}+\frac{\tan ^3(e+f x)}{3 f}-\frac{\tan (e+f x)}{f}\right)","\frac{a^2 c^2 \tan ^3(e+f x)}{3 f}-\frac{a^2 c^2 \tan (e+f x)}{f}+a^2 c^2 x",1,"a^2*c^2*(ArcTan[Tan[e + f*x]]/f - Tan[e + f*x]/f + Tan[e + f*x]^3/(3*f))","A",1
5,1,72,55,0.289765,"\int (a+a \sec (e+f x))^2 (c-c \sec (e+f x)) \, dx","Integrate[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x]),x]","\frac{a^2 c \sec ^2(e+f x) \left(-\sin (e+f x)-\sin (2 (e+f x))+(e+f x) \cos (2 (e+f x))+\cos ^2(e+f x) \tanh ^{-1}(\sin (e+f x))+e+f x\right)}{2 f}","\frac{a^2 c \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac{c \tan (e+f x) \left(a^2 \sec (e+f x)+2 a^2\right)}{2 f}+a^2 c x",1,"(a^2*c*Sec[e + f*x]^2*(e + f*x + ArcTanh[Sin[e + f*x]]*Cos[e + f*x]^2 + (e + f*x)*Cos[2*(e + f*x)] - Sin[e + f*x] - Sin[2*(e + f*x)]))/(2*f)","A",1
6,1,169,56,0.2928132,"\int \frac{(a+a \sec (e+f x))^2}{c-c \sec (e+f x)} \, dx","Integrate[(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x]),x]","\frac{a^2 \csc \left(\frac{e}{2}\right) \sin \left(\frac{1}{2} (e+f x)\right) \left(-\cos \left(\frac{f x}{2}\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+f x\right)+\cos \left(e+\frac{f x}{2}\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+f x\right)+8 \sin \left(\frac{f x}{2}\right)\right)}{c f (\cos (e+f x)-1)}","-\frac{a^2 \tanh ^{-1}(\sin (e+f x))}{c f}-\frac{4 a^2 \tan (e+f x)}{c f (1-\sec (e+f x))}+\frac{a^2 x}{c}",1,"(a^2*Csc[e/2]*(-(Cos[(f*x)/2]*(f*x + Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])) + Cos[e + (f*x)/2]*(f*x + Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + 8*Sin[(f*x)/2])*Sin[(e + f*x)/2])/(c*f*(-1 + Cos[e + f*x]))","B",1
7,1,53,71,0.0584746,"\int \frac{(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^2} \, dx","Integrate[(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^2,x]","-\frac{2 a^2 \cot ^3\left(\frac{e}{2}+\frac{f x}{2}\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2\left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{3 c^2 f}","-\frac{4 a^2 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))}-\frac{4 a^2 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))^2}+\frac{a^2 x}{c^2}",1,"(-2*a^2*Cot[e/2 + (f*x)/2]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[e/2 + (f*x)/2]^2])/(3*c^2*f)","C",1
8,1,171,102,0.6290268,"\int \frac{(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^3} \, dx","Integrate[(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^3,x]","\frac{a^2 \csc \left(\frac{e}{2}\right) \csc ^5\left(\frac{1}{2} (e+f x)\right) \left(-360 \sin \left(e+\frac{f x}{2}\right)+280 \sin \left(e+\frac{3 f x}{2}\right)+150 \sin \left(2 e+\frac{3 f x}{2}\right)-86 \sin \left(2 e+\frac{5 f x}{2}\right)-150 f x \cos \left(e+\frac{f x}{2}\right)-75 f x \cos \left(e+\frac{3 f x}{2}\right)+75 f x \cos \left(2 e+\frac{3 f x}{2}\right)+15 f x \cos \left(2 e+\frac{5 f x}{2}\right)-15 f x \cos \left(3 e+\frac{5 f x}{2}\right)-500 \sin \left(\frac{f x}{2}\right)+150 f x \cos \left(\frac{f x}{2}\right)\right)}{480 c^3 f}","-\frac{23 a^2 \tan (e+f x)}{15 c^3 f (1-\sec (e+f x))}-\frac{8 a^2 \tan (e+f x)}{15 c^3 f (1-\sec (e+f x))^2}-\frac{4 a^2 \tan (e+f x)}{5 c^3 f (1-\sec (e+f x))^3}+\frac{a^2 x}{c^3}",1,"(a^2*Csc[e/2]*Csc[(e + f*x)/2]^5*(150*f*x*Cos[(f*x)/2] - 150*f*x*Cos[e + (f*x)/2] - 75*f*x*Cos[e + (3*f*x)/2] + 75*f*x*Cos[2*e + (3*f*x)/2] + 15*f*x*Cos[2*e + (5*f*x)/2] - 15*f*x*Cos[3*e + (5*f*x)/2] - 500*Sin[(f*x)/2] - 360*Sin[e + (f*x)/2] + 280*Sin[e + (3*f*x)/2] + 150*Sin[2*e + (3*f*x)/2] - 86*Sin[2*e + (5*f*x)/2]))/(480*c^3*f)","A",1
9,1,227,133,0.6350841,"\int \frac{(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^4} \, dx","Integrate[(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^4,x]","\frac{a^2 \csc \left(\frac{e}{2}\right) \csc ^7\left(\frac{1}{2} (e+f x)\right) \left(-10430 \sin \left(e+\frac{f x}{2}\right)+8568 \sin \left(e+\frac{3 f x}{2}\right)+4830 \sin \left(2 e+\frac{3 f x}{2}\right)-3206 \sin \left(2 e+\frac{5 f x}{2}\right)-1260 \sin \left(3 e+\frac{5 f x}{2}\right)+638 \sin \left(3 e+\frac{7 f x}{2}\right)-3675 f x \cos \left(e+\frac{f x}{2}\right)-2205 f x \cos \left(e+\frac{3 f x}{2}\right)+2205 f x \cos \left(2 e+\frac{3 f x}{2}\right)+735 f x \cos \left(2 e+\frac{5 f x}{2}\right)-735 f x \cos \left(3 e+\frac{5 f x}{2}\right)-105 f x \cos \left(3 e+\frac{7 f x}{2}\right)+105 f x \cos \left(4 e+\frac{7 f x}{2}\right)-11900 \sin \left(\frac{f x}{2}\right)+3675 f x \cos \left(\frac{f x}{2}\right)\right)}{13440 c^4 f}","-\frac{164 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))}-\frac{59 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}-\frac{12 a^2 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac{4 a^2 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}+\frac{a^2 x}{c^4}",1,"(a^2*Csc[e/2]*Csc[(e + f*x)/2]^7*(3675*f*x*Cos[(f*x)/2] - 3675*f*x*Cos[e + (f*x)/2] - 2205*f*x*Cos[e + (3*f*x)/2] + 2205*f*x*Cos[2*e + (3*f*x)/2] + 735*f*x*Cos[2*e + (5*f*x)/2] - 735*f*x*Cos[3*e + (5*f*x)/2] - 105*f*x*Cos[3*e + (7*f*x)/2] + 105*f*x*Cos[4*e + (7*f*x)/2] - 11900*Sin[(f*x)/2] - 10430*Sin[e + (f*x)/2] + 8568*Sin[e + (3*f*x)/2] + 4830*Sin[2*e + (3*f*x)/2] - 3206*Sin[2*e + (5*f*x)/2] - 1260*Sin[3*e + (5*f*x)/2] + 638*Sin[3*e + (7*f*x)/2]))/(13440*c^4*f)","A",1
10,1,283,164,0.99494,"\int \frac{(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^5} \, dx","Integrate[(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^5,x]","\frac{a^2 \csc \left(\frac{e}{2}\right) \csc ^9\left(\frac{1}{2} (e+f x)\right) \left(-117810 \sin \left(e+\frac{f x}{2}\right)+100002 \sin \left(e+\frac{3 f x}{2}\right)+68670 \sin \left(2 e+\frac{3 f x}{2}\right)-48978 \sin \left(2 e+\frac{5 f x}{2}\right)-23310 \sin \left(3 e+\frac{5 f x}{2}\right)+13662 \sin \left(3 e+\frac{7 f x}{2}\right)+4410 \sin \left(4 e+\frac{7 f x}{2}\right)-2008 \sin \left(4 e+\frac{9 f x}{2}\right)-39690 f x \cos \left(e+\frac{f x}{2}\right)-26460 f x \cos \left(e+\frac{3 f x}{2}\right)+26460 f x \cos \left(2 e+\frac{3 f x}{2}\right)+11340 f x \cos \left(2 e+\frac{5 f x}{2}\right)-11340 f x \cos \left(3 e+\frac{5 f x}{2}\right)-2835 f x \cos \left(3 e+\frac{7 f x}{2}\right)+2835 f x \cos \left(4 e+\frac{7 f x}{2}\right)+315 f x \cos \left(4 e+\frac{9 f x}{2}\right)-315 f x \cos \left(5 e+\frac{9 f x}{2}\right)-135198 \sin \left(\frac{f x}{2}\right)+39690 f x \cos \left(\frac{f x}{2}\right)\right)}{161280 c^5 f}","-\frac{494 a^2 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))}-\frac{179 a^2 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}-\frac{37 a^2 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}-\frac{16 a^2 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac{4 a^2 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}+\frac{a^2 x}{c^5}",1,"(a^2*Csc[e/2]*Csc[(e + f*x)/2]^9*(39690*f*x*Cos[(f*x)/2] - 39690*f*x*Cos[e + (f*x)/2] - 26460*f*x*Cos[e + (3*f*x)/2] + 26460*f*x*Cos[2*e + (3*f*x)/2] + 11340*f*x*Cos[2*e + (5*f*x)/2] - 11340*f*x*Cos[3*e + (5*f*x)/2] - 2835*f*x*Cos[3*e + (7*f*x)/2] + 2835*f*x*Cos[4*e + (7*f*x)/2] + 315*f*x*Cos[4*e + (9*f*x)/2] - 315*f*x*Cos[5*e + (9*f*x)/2] - 135198*Sin[(f*x)/2] - 117810*Sin[e + (f*x)/2] + 100002*Sin[e + (3*f*x)/2] + 68670*Sin[2*e + (3*f*x)/2] - 48978*Sin[2*e + (5*f*x)/2] - 23310*Sin[3*e + (5*f*x)/2] + 13662*Sin[3*e + (7*f*x)/2] + 4410*Sin[4*e + (7*f*x)/2] - 2008*Sin[4*e + (9*f*x)/2]))/(161280*c^5*f)","A",1
11,1,189,188,2.2355016,"\int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx","Integrate[(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5,x]","\frac{a^3 c^5 \sec ^7(e+f x) \left(-4200 \sin (e+f x)+2975 \sin (2 (e+f x))-2184 \sin (3 (e+f x))+980 \sin (4 (e+f x))-2408 \sin (5 (e+f x))+1155 \sin (6 (e+f x))-584 \sin (7 (e+f x))+14700 (e+f x) \cos (e+f x)+8820 e \cos (3 (e+f x))+8820 f x \cos (3 (e+f x))+2940 e \cos (5 (e+f x))+2940 f x \cos (5 (e+f x))+420 e \cos (7 (e+f x))+420 f x \cos (7 (e+f x))-16800 \cos ^7(e+f x) \tanh ^{-1}(\sin (e+f x))\right)}{26880 f}","-\frac{a^3 c^5 \tan ^7(e+f x)}{7 f}-\frac{a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac{a^3 c^5 \tan ^3(e+f x)}{3 f}-\frac{a^3 c^5 \tan (e+f x)}{f}-\frac{5 a^3 c^5 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{a^3 c^5 \tan ^5(e+f x) \sec (e+f x)}{3 f}-\frac{5 a^3 c^5 \tan ^3(e+f x) \sec (e+f x)}{12 f}+\frac{5 a^3 c^5 \tan (e+f x) \sec (e+f x)}{8 f}+a^3 c^5 x",1,"(a^3*c^5*Sec[e + f*x]^7*(14700*(e + f*x)*Cos[e + f*x] - 16800*ArcTanh[Sin[e + f*x]]*Cos[e + f*x]^7 + 8820*e*Cos[3*(e + f*x)] + 8820*f*x*Cos[3*(e + f*x)] + 2940*e*Cos[5*(e + f*x)] + 2940*f*x*Cos[5*(e + f*x)] + 420*e*Cos[7*(e + f*x)] + 420*f*x*Cos[7*(e + f*x)] - 4200*Sin[e + f*x] + 2975*Sin[2*(e + f*x)] - 2184*Sin[3*(e + f*x)] + 980*Sin[4*(e + f*x)] - 2408*Sin[5*(e + f*x)] + 1155*Sin[6*(e + f*x)] - 584*Sin[7*(e + f*x)]))/(26880*f)","A",1
12,1,165,132,1.8706078,"\int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4 \, dx","Integrate[(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^4,x]","\frac{a^3 c^4 \sec ^6(e+f x) \left(450 \sin (e+f x)-600 \sin (2 (e+f x))-25 \sin (3 (e+f x))-384 \sin (4 (e+f x))+165 \sin (5 (e+f x))-184 \sin (6 (e+f x))+1800 (e+f x) \cos (2 (e+f x))+720 e \cos (4 (e+f x))+720 f x \cos (4 (e+f x))+120 e \cos (6 (e+f x))+120 f x \cos (6 (e+f x))-1200 \cos ^6(e+f x) \tanh ^{-1}(\sin (e+f x))+1200 e+1200 f x\right)}{3840 f}","-\frac{5 a^3 c^4 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac{a^3 \tan ^5(e+f x) \left(6 c^4-5 c^4 \sec (e+f x)\right)}{30 f}+\frac{a^3 \tan ^3(e+f x) \left(8 c^4-5 c^4 \sec (e+f x)\right)}{24 f}-\frac{a^3 \tan (e+f x) \left(16 c^4-5 c^4 \sec (e+f x)\right)}{16 f}+a^3 c^4 x",1,"(a^3*c^4*Sec[e + f*x]^6*(1200*e + 1200*f*x - 1200*ArcTanh[Sin[e + f*x]]*Cos[e + f*x]^6 + 1800*(e + f*x)*Cos[2*(e + f*x)] + 720*e*Cos[4*(e + f*x)] + 720*f*x*Cos[4*(e + f*x)] + 120*e*Cos[6*(e + f*x)] + 120*f*x*Cos[6*(e + f*x)] + 450*Sin[e + f*x] - 600*Sin[2*(e + f*x)] - 25*Sin[3*(e + f*x)] - 384*Sin[4*(e + f*x)] + 165*Sin[5*(e + f*x)] - 184*Sin[6*(e + f*x)]))/(3840*f)","A",1
13,1,61,68,0.0392427,"\int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx","Integrate[(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^3,x]","-a^3 c^3 \left(-\frac{\tan ^{-1}(\tan (e+f x))}{f}+\frac{\tan ^5(e+f x)}{5 f}-\frac{\tan ^3(e+f x)}{3 f}+\frac{\tan (e+f x)}{f}\right)","-\frac{a^3 c^3 \tan ^5(e+f x)}{5 f}+\frac{a^3 c^3 \tan ^3(e+f x)}{3 f}-\frac{a^3 c^3 \tan (e+f x)}{f}+a^3 c^3 x",1,"-(a^3*c^3*(-(ArcTan[Tan[e + f*x]]/f) + Tan[e + f*x]/f - Tan[e + f*x]^3/(3*f) + Tan[e + f*x]^5/(5*f)))","A",1
14,1,122,97,0.7978699,"\int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2 \, dx","Integrate[(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^2,x]","\frac{a^3 c^2 \sec ^4(e+f x) \left(18 \sin (e+f x)-32 \sin (2 (e+f x))-30 \sin (3 (e+f x))-32 \sin (4 (e+f x))+96 (e+f x) \cos (2 (e+f x))+24 e \cos (4 (e+f x))+24 f x \cos (4 (e+f x))+72 \cos ^4(e+f x) \tanh ^{-1}(\sin (e+f x))+72 e+72 f x\right)}{192 f}","\frac{3 a^3 c^2 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac{c^2 \tan ^3(e+f x) \left(3 a^3 \sec (e+f x)+4 a^3\right)}{12 f}-\frac{c^2 \tan (e+f x) \left(3 a^3 \sec (e+f x)+8 a^3\right)}{8 f}+a^3 c^2 x",1,"(a^3*c^2*Sec[e + f*x]^4*(72*e + 72*f*x + 72*ArcTanh[Sin[e + f*x]]*Cos[e + f*x]^4 + 96*(e + f*x)*Cos[2*(e + f*x)] + 24*e*Cos[4*(e + f*x)] + 24*f*x*Cos[4*(e + f*x)] + 18*Sin[e + f*x] - 32*Sin[2*(e + f*x)] - 30*Sin[3*(e + f*x)] - 32*Sin[4*(e + f*x)]))/(192*f)","A",1
15,1,101,77,0.4525946,"\int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx","Integrate[(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x]),x]","\frac{a^3 c \sec ^3(e+f x) \left(-6 \sin (e+f x)-6 \sin (2 (e+f x))-2 \sin (3 (e+f x))+9 (e+f x) \cos (e+f x)+3 e \cos (3 (e+f x))+3 f x \cos (3 (e+f x))+12 \cos ^3(e+f x) \tanh ^{-1}(\sin (e+f x))\right)}{12 f}","-\frac{a^3 c \tan ^3(e+f x)}{3 f}-\frac{a^3 c \tan (e+f x)}{f}+\frac{a^3 c \tanh ^{-1}(\sin (e+f x))}{f}-\frac{a^3 c \tan (e+f x) \sec (e+f x)}{f}+a^3 c x",1,"(a^3*c*Sec[e + f*x]^3*(9*(e + f*x)*Cos[e + f*x] + 12*ArcTanh[Sin[e + f*x]]*Cos[e + f*x]^3 + 3*e*Cos[3*(e + f*x)] + 3*f*x*Cos[3*(e + f*x)] - 6*Sin[e + f*x] - 6*Sin[2*(e + f*x)] - 2*Sin[3*(e + f*x)]))/(12*f)","A",1
16,1,240,78,2.5796293,"\int \frac{(a+a \sec (e+f x))^3}{c-c \sec (e+f x)} \, dx","Integrate[(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x]),x]","\frac{a^3 \cos ^2(e+f x) \tan \left(\frac{1}{2} (e+f x)\right) \sec ^4\left(\frac{1}{2} (e+f x)\right) (\sec (e+f x)+1)^3 \left(8 \csc \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sec \left(\frac{1}{2} (e+f x)\right)+\tan \left(\frac{1}{2} (e+f x)\right) \left(\frac{\sin (f x)}{\left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}-4 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+4 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-f x\right)\right)}{4 f (c-c \sec (e+f x))}","-\frac{a^3 \tan (e+f x)}{c f}+\frac{8 a^3 \cot (e+f x)}{c f}+\frac{8 a^3 \csc (e+f x)}{c f}-\frac{4 a^3 \tanh ^{-1}(\sin (e+f x))}{c f}+\frac{a^3 x}{c}",1,"(a^3*Cos[e + f*x]^2*Sec[(e + f*x)/2]^4*(1 + Sec[e + f*x])^3*Tan[(e + f*x)/2]*(8*Csc[e/2]*Sec[(e + f*x)/2]*Sin[(f*x)/2] + (-(f*x) - 4*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 4*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + Sin[f*x]/((Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))*Tan[(e + f*x)/2]))/(4*f*(c - c*Sec[e + f*x]))","B",1
17,1,177,88,1.1820546,"\int \frac{(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^2} \, dx","Integrate[(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^2,x]","\frac{a^3 (\cos (e+f x)+1)^3 \tan \left(\frac{1}{2} (e+f x)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right) \left(-4 \cot \left(\frac{e}{2}\right) \tan \left(\frac{1}{2} (e+f x)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right)+4 \csc \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \sec \left(\frac{1}{2} (e+f x)\right)+3 \tan ^3\left(\frac{1}{2} (e+f x)\right) \left(-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+f x\right)\right)}{6 c^2 f (\cos (e+f x)-1)^2}","\frac{a^3 \tanh ^{-1}(\sin (e+f x))}{c^2 f}+\frac{4 a^3 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))}-\frac{8 a^3 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))^2}+\frac{a^3 x}{c^2}",1,"(a^3*(1 + Cos[e + f*x])^3*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]*(4*Csc[e/2]*Sec[(e + f*x)/2]*Sin[(f*x)/2] - 4*Cot[e/2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(f*x - Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Tan[(e + f*x)/2]^3))/(6*c^2*f*(-1 + Cos[e + f*x])^2)","B",1
18,1,53,102,0.099049,"\int \frac{(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^3} \, dx","Integrate[(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^3,x]","\frac{2 a^3 \cot ^5\left(\frac{e}{2}+\frac{f x}{2}\right) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-\tan ^2\left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{5 c^3 f}","-\frac{26 a^3 \tan (e+f x)}{15 c^3 f (1-\sec (e+f x))}+\frac{4 a^3 \tan (e+f x)}{15 c^3 f (1-\sec (e+f x))^2}-\frac{8 a^3 \tan (e+f x)}{5 c^3 f (1-\sec (e+f x))^3}+\frac{a^3 x}{c^3}",1,"(2*a^3*Cot[e/2 + (f*x)/2]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[e/2 + (f*x)/2]^2])/(5*c^3*f)","C",1
19,1,227,133,0.5992814,"\int \frac{(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx","Integrate[(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^4,x]","\frac{a^3 \csc \left(\frac{e}{2}\right) \csc ^7\left(\frac{1}{2} (e+f x)\right) \left(-11270 \sin \left(e+\frac{f x}{2}\right)+9114 \sin \left(e+\frac{3 f x}{2}\right)+5040 \sin \left(2 e+\frac{3 f x}{2}\right)-3248 \sin \left(2 e+\frac{5 f x}{2}\right)-1470 \sin \left(3 e+\frac{5 f x}{2}\right)+674 \sin \left(3 e+\frac{7 f x}{2}\right)-3675 f x \cos \left(e+\frac{f x}{2}\right)-2205 f x \cos \left(e+\frac{3 f x}{2}\right)+2205 f x \cos \left(2 e+\frac{3 f x}{2}\right)+735 f x \cos \left(2 e+\frac{5 f x}{2}\right)-735 f x \cos \left(3 e+\frac{5 f x}{2}\right)-105 f x \cos \left(3 e+\frac{7 f x}{2}\right)+105 f x \cos \left(4 e+\frac{7 f x}{2}\right)-12320 \sin \left(\frac{f x}{2}\right)+3675 f x \cos \left(\frac{f x}{2}\right)\right)}{13440 c^4 f}","-\frac{167 a^3 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))}-\frac{62 a^3 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}+\frac{4 a^3 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac{8 a^3 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}+\frac{a^3 x}{c^4}",1,"(a^3*Csc[e/2]*Csc[(e + f*x)/2]^7*(3675*f*x*Cos[(f*x)/2] - 3675*f*x*Cos[e + (f*x)/2] - 2205*f*x*Cos[e + (3*f*x)/2] + 2205*f*x*Cos[2*e + (3*f*x)/2] + 735*f*x*Cos[2*e + (5*f*x)/2] - 735*f*x*Cos[3*e + (5*f*x)/2] - 105*f*x*Cos[3*e + (7*f*x)/2] + 105*f*x*Cos[4*e + (7*f*x)/2] - 12320*Sin[(f*x)/2] - 11270*Sin[e + (f*x)/2] + 9114*Sin[e + (3*f*x)/2] + 5040*Sin[2*e + (3*f*x)/2] - 3248*Sin[2*e + (5*f*x)/2] - 1470*Sin[3*e + (5*f*x)/2] + 674*Sin[3*e + (7*f*x)/2]))/(13440*c^4*f)","A",1
20,1,283,164,0.9019894,"\int \frac{(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^5} \, dx","Integrate[(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^5,x]","\frac{a^3 \csc \left(\frac{e}{2}\right) \csc ^9\left(\frac{1}{2} (e+f x)\right) \left(-122850 \sin \left(e+\frac{f x}{2}\right)+103278 \sin \left(e+\frac{3 f x}{2}\right)+73290 \sin \left(2 e+\frac{3 f x}{2}\right)-51102 \sin \left(2 e+\frac{5 f x}{2}\right)-24570 \sin \left(3 e+\frac{5 f x}{2}\right)+13878 \sin \left(3 e+\frac{7 f x}{2}\right)+5040 \sin \left(4 e+\frac{7 f x}{2}\right)-2102 \sin \left(4 e+\frac{9 f x}{2}\right)-39690 f x \cos \left(e+\frac{f x}{2}\right)-26460 f x \cos \left(e+\frac{3 f x}{2}\right)+26460 f x \cos \left(2 e+\frac{3 f x}{2}\right)+11340 f x \cos \left(2 e+\frac{5 f x}{2}\right)-11340 f x \cos \left(3 e+\frac{5 f x}{2}\right)-2835 f x \cos \left(3 e+\frac{7 f x}{2}\right)+2835 f x \cos \left(4 e+\frac{7 f x}{2}\right)+315 f x \cos \left(4 e+\frac{9 f x}{2}\right)-315 f x \cos \left(5 e+\frac{9 f x}{2}\right)-142002 \sin \left(\frac{f x}{2}\right)+39690 f x \cos \left(\frac{f x}{2}\right)\right)}{161280 c^5 f}","-\frac{496 a^3 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))}-\frac{181 a^3 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}-\frac{38 a^3 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}+\frac{4 a^3 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac{8 a^3 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}+\frac{a^3 x}{c^5}",1,"(a^3*Csc[e/2]*Csc[(e + f*x)/2]^9*(39690*f*x*Cos[(f*x)/2] - 39690*f*x*Cos[e + (f*x)/2] - 26460*f*x*Cos[e + (3*f*x)/2] + 26460*f*x*Cos[2*e + (3*f*x)/2] + 11340*f*x*Cos[2*e + (5*f*x)/2] - 11340*f*x*Cos[3*e + (5*f*x)/2] - 2835*f*x*Cos[3*e + (7*f*x)/2] + 2835*f*x*Cos[4*e + (7*f*x)/2] + 315*f*x*Cos[4*e + (9*f*x)/2] - 315*f*x*Cos[5*e + (9*f*x)/2] - 142002*Sin[(f*x)/2] - 122850*Sin[e + (f*x)/2] + 103278*Sin[e + (3*f*x)/2] + 73290*Sin[2*e + (3*f*x)/2] - 51102*Sin[2*e + (5*f*x)/2] - 24570*Sin[3*e + (5*f*x)/2] + 13878*Sin[3*e + (7*f*x)/2] + 5040*Sin[4*e + (7*f*x)/2] - 2102*Sin[4*e + (9*f*x)/2]))/(161280*c^5*f)","A",1
21,1,384,136,3.1510354,"\int \frac{(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx","Integrate[(c - c*Sec[e + f*x])^5/(a + a*Sec[e + f*x])^2,x]","\frac{\cos ^3(e+f x) \cot \left(\frac{1}{2} (e+f x)\right) \csc ^6\left(\frac{1}{2} (e+f x)\right) (c-c \sec (e+f x))^5 \left(-\frac{64 \tan \left(\frac{e}{2}\right) \cot \left(\frac{1}{2} (e+f x)\right) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{f}-\frac{64 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \csc ^3\left(\frac{1}{2} (e+f x)\right)}{f}+3 \cot ^3\left(\frac{1}{2} (e+f x)\right) \left(-\frac{28 \sin (f x)}{f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{1}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{1}{f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{94 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f}+\frac{94 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f}-4 x\right)-\frac{320 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \cot ^2\left(\frac{1}{2} (e+f x)\right) \csc \left(\frac{1}{2} (e+f x)\right)}{f}\right)}{96 a^2 (\sec (e+f x)+1)^2}","\frac{13 c^5 \tan (e+f x)}{2 a^2 f}-\frac{47 c^5 \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}+\frac{112 c^5 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}+\frac{\tan (e+f x) \left(c^5-c^5 \sec (e+f x)\right)}{2 a^2 f}+\frac{c^5 x}{a^2}-\frac{32 c^5 \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2}",1,"(Cos[e + f*x]^3*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^6*(c - c*Sec[e + f*x])^5*((-320*Cot[(e + f*x)/2]^2*Csc[(e + f*x)/2]*Sec[e/2]*Sin[(f*x)/2])/f - (64*Csc[(e + f*x)/2]^3*Sec[e/2]*Sin[(f*x)/2])/f + 3*Cot[(e + f*x)/2]^3*(-4*x - (94*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]])/f + (94*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])/f + 1/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) - 1/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) - (28*Sin[f*x])/(f*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))) - (64*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2*Tan[e/2])/f))/(96*a^2*(1 + Sec[e + f*x])^2)","B",1
22,1,753,102,6.2619366,"\int \frac{(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx","Integrate[(c - c*Sec[e + f*x])^4/(a + a*Sec[e + f*x])^2,x]","\frac{x \cos ^2(e+f x) \cot ^4\left(\frac{e}{2}+\frac{f x}{2}\right) \csc ^4\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4}{4 (a \sec (e+f x)+a)^2}+\frac{\sin \left(\frac{f x}{2}\right) \cos ^2(e+f x) \cot ^4\left(\frac{e}{2}+\frac{f x}{2}\right) \csc ^4\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4}{4 f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) (a \sec (e+f x)+a)^2 \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}+\frac{\sin \left(\frac{f x}{2}\right) \cos ^2(e+f x) \cot ^4\left(\frac{e}{2}+\frac{f x}{2}\right) \csc ^4\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4}{4 f \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) (a \sec (e+f x)+a)^2 \left(\sin \left(\frac{e}{2}+\frac{f x}{2}\right)+\cos \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}+\frac{4 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \cos ^2(e+f x) \cot ^3\left(\frac{e}{2}+\frac{f x}{2}\right) \csc ^5\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4}{3 f (a \sec (e+f x)+a)^2}+\frac{2 \tan \left(\frac{e}{2}\right) \cos ^2(e+f x) \cot ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \csc ^6\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4}{3 f (a \sec (e+f x)+a)^2}+\frac{2 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \cos ^2(e+f x) \cot \left(\frac{e}{2}+\frac{f x}{2}\right) \csc ^7\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4}{3 f (a \sec (e+f x)+a)^2}+\frac{3 \cos ^2(e+f x) \cot ^4\left(\frac{e}{2}+\frac{f x}{2}\right) \csc ^4\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4 \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)-\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{2 f (a \sec (e+f x)+a)^2}-\frac{3 \cos ^2(e+f x) \cot ^4\left(\frac{e}{2}+\frac{f x}{2}\right) \csc ^4\left(\frac{e}{2}+\frac{f x}{2}\right) (c-c \sec (e+f x))^4 \log \left(\sin \left(\frac{e}{2}+\frac{f x}{2}\right)+\cos \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{2 f (a \sec (e+f x)+a)^2}","\frac{c^4 \tan (e+f x)}{a^2 f}-\frac{32 c^4 \cot ^3(e+f x)}{3 a^2 f}-\frac{16 c^4 \cot (e+f x)}{a^2 f}+\frac{32 c^4 \csc ^3(e+f x)}{3 a^2 f}-\frac{6 c^4 \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac{c^4 x}{a^2}",1,"(x*Cos[e + f*x]^2*Cot[e/2 + (f*x)/2]^4*Csc[e/2 + (f*x)/2]^4*(c - c*Sec[e + f*x])^4)/(4*(a + a*Sec[e + f*x])^2) + (3*Cos[e + f*x]^2*Cot[e/2 + (f*x)/2]^4*Csc[e/2 + (f*x)/2]^4*Log[Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2]]*(c - c*Sec[e + f*x])^4)/(2*f*(a + a*Sec[e + f*x])^2) - (3*Cos[e + f*x]^2*Cot[e/2 + (f*x)/2]^4*Csc[e/2 + (f*x)/2]^4*Log[Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2]]*(c - c*Sec[e + f*x])^4)/(2*f*(a + a*Sec[e + f*x])^2) + (4*Cos[e + f*x]^2*Cot[e/2 + (f*x)/2]^3*Csc[e/2 + (f*x)/2]^5*Sec[e/2]*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(3*f*(a + a*Sec[e + f*x])^2) + (2*Cos[e + f*x]^2*Cot[e/2 + (f*x)/2]*Csc[e/2 + (f*x)/2]^7*Sec[e/2]*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(3*f*(a + a*Sec[e + f*x])^2) + (Cos[e + f*x]^2*Cot[e/2 + (f*x)/2]^4*Csc[e/2 + (f*x)/2]^4*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(4*f*(a + a*Sec[e + f*x])^2*(Cos[e/2] - Sin[e/2])*(Cos[e/2 + (f*x)/2] - Sin[e/2 + (f*x)/2])) + (Cos[e + f*x]^2*Cot[e/2 + (f*x)/2]^4*Csc[e/2 + (f*x)/2]^4*(c - c*Sec[e + f*x])^4*Sin[(f*x)/2])/(4*f*(a + a*Sec[e + f*x])^2*(Cos[e/2] + Sin[e/2])*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])) + (2*Cos[e + f*x]^2*Cot[e/2 + (f*x)/2]^2*Csc[e/2 + (f*x)/2]^6*(c - c*Sec[e + f*x])^4*Tan[e/2])/(3*f*(a + a*Sec[e + f*x])^2)","B",1
23,1,216,85,1.096294,"\int \frac{(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^2} \, dx","Integrate[(c - c*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^2,x]","-\frac{c^3 (\cos (e+f x)-1)^3 \cot \left(\frac{1}{2} (e+f x)\right) \csc ^2\left(\frac{1}{2} (e+f x)\right) \left(4 \tan \left(\frac{e}{2}\right) \cot \left(\frac{1}{2} (e+f x)\right) \csc ^2\left(\frac{1}{2} (e+f x)\right)+4 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \csc ^3\left(\frac{1}{2} (e+f x)\right)+3 \cot ^3\left(\frac{1}{2} (e+f x)\right) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+f x\right)-4 \sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) \cot ^2\left(\frac{1}{2} (e+f x)\right) \csc \left(\frac{1}{2} (e+f x)\right)\right)}{6 a^2 f (\cos (e+f x)+1)^2}","-\frac{c^3 \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac{4 c^3 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{8 c^3 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)^2}+\frac{c^3 x}{a^2}",1,"-1/6*(c^3*(-1 + Cos[e + f*x])^3*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2*(3*Cot[(e + f*x)/2]^3*(f*x + Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - 4*Cot[(e + f*x)/2]^2*Csc[(e + f*x)/2]*Sec[e/2]*Sin[(f*x)/2] + 4*Csc[(e + f*x)/2]^3*Sec[e/2]*Sin[(f*x)/2] + 4*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2*Tan[e/2]))/(a^2*f*(1 + Cos[e + f*x])^2)","B",1
24,1,67,67,0.0505182,"\int \frac{(c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^2} \, dx","Integrate[(c - c*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^2,x]","\frac{c^2 \left(\frac{2 \tan ^{-1}\left(\tan \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{f}+\frac{2 \tan ^3\left(\frac{e}{2}+\frac{f x}{2}\right)}{3 f}-\frac{2 \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{f}\right)}{a^2}","-\frac{4 c^2 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{4 c^2 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)^2}+\frac{c^2 x}{a^2}",1,"(c^2*((2*ArcTan[Tan[e/2 + (f*x)/2]])/f - (2*Tan[e/2 + (f*x)/2])/f + (2*Tan[e/2 + (f*x)/2]^3)/(3*f)))/a^2","A",1
25,1,113,61,0.3115124,"\int \frac{c-c \sec (e+f x)}{(a+a \sec (e+f x))^2} \, dx","Integrate[(c - c*Sec[e + f*x])/(a + a*Sec[e + f*x])^2,x]","\frac{c \sec \left(\frac{e}{2}\right) \sec ^3\left(\frac{1}{2} (e+f x)\right) \left(18 \sin \left(e+\frac{f x}{2}\right)-14 \sin \left(e+\frac{3 f x}{2}\right)+9 f x \cos \left(e+\frac{f x}{2}\right)+3 f x \cos \left(e+\frac{3 f x}{2}\right)+3 f x \cos \left(2 e+\frac{3 f x}{2}\right)-24 \sin \left(\frac{f x}{2}\right)+9 f x \cos \left(\frac{f x}{2}\right)\right)}{24 a^2 f}","-\frac{5 c \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{2 c \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)^2}+\frac{c x}{a^2}",1,"(c*Sec[e/2]*Sec[(e + f*x)/2]^3*(9*f*x*Cos[(f*x)/2] + 9*f*x*Cos[e + (f*x)/2] + 3*f*x*Cos[e + (3*f*x)/2] + 3*f*x*Cos[2*e + (3*f*x)/2] - 24*Sin[(f*x)/2] + 18*Sin[e + (f*x)/2] - 14*Sin[e + (3*f*x)/2]))/(24*a^2*f)","A",1
26,1,135,69,0.5481033,"\int \frac{1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))} \, dx","Integrate[1/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])),x]","\frac{\csc \left(\frac{e}{2}\right) \sec \left(\frac{e}{2}\right) \csc \left(\frac{1}{2} (e+f x)\right) \sec ^3\left(\frac{1}{2} (e+f x)\right) (10 \sin (e+f x)+5 \sin (2 (e+f x))-6 \sin (2 e+f x)-8 \sin (e+2 f x)-6 f x \cos (2 e+f x)+3 f x \cos (e+2 f x)-3 f x \cos (3 e+2 f x)-10 \sin (f x)+6 f x \cos (f x))}{96 a^2 c f}","-\frac{\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}+\frac{\cot (e+f x) (3-2 \sec (e+f x))}{3 a^2 c f}+\frac{x}{a^2 c}",1,"(Csc[e/2]*Csc[(e + f*x)/2]*Sec[e/2]*Sec[(e + f*x)/2]^3*(6*f*x*Cos[f*x] - 6*f*x*Cos[2*e + f*x] + 3*f*x*Cos[e + 2*f*x] - 3*f*x*Cos[3*e + 2*f*x] - 10*Sin[f*x] + 10*Sin[e + f*x] + 5*Sin[2*(e + f*x)] - 6*Sin[2*e + f*x] - 8*Sin[e + 2*f*x]))/(96*a^2*c*f)","A",1
27,1,39,46,0.0487004,"\int \frac{1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^2} \, dx","Integrate[1/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^2),x]","-\frac{\cot ^3(e+f x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(e+f x)\right)}{3 a^2 c^2 f}","-\frac{\cot ^3(e+f x)}{3 a^2 c^2 f}+\frac{\cot (e+f x)}{a^2 c^2 f}+\frac{x}{a^2 c^2}",1,"-1/3*(Cot[e + f*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[e + f*x]^2])/(a^2*c^2*f)","C",1
28,1,257,98,1.4109395,"\int \frac{1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx","Integrate[1/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^3),x]","\frac{\csc \left(\frac{e}{2}\right) \sec \left(\frac{e}{2}\right) \csc ^5\left(\frac{1}{2} (e+f x)\right) \sec ^3\left(\frac{1}{2} (e+f x)\right) (-534 \sin (e+f x)+178 \sin (2 (e+f x))+178 \sin (3 (e+f x))-89 \sin (4 (e+f x))-520 \sin (2 e+f x)+248 \sin (e+2 f x)+120 \sin (3 e+2 f x)+248 \sin (2 e+3 f x)+120 \sin (4 e+3 f x)-184 \sin (3 e+4 f x)-360 f x \cos (2 e+f x)-120 f x \cos (e+2 f x)+120 f x \cos (3 e+2 f x)-120 f x \cos (2 e+3 f x)+120 f x \cos (4 e+3 f x)+60 f x \cos (3 e+4 f x)-60 f x \cos (5 e+4 f x)+200 \sin (e)-584 \sin (f x)+360 f x \cos (f x))}{30720 a^2 c^3 f}","\frac{\cot ^5(e+f x) (\sec (e+f x)+1)}{5 a^2 c^3 f}-\frac{\cot ^3(e+f x) (4 \sec (e+f x)+5)}{15 a^2 c^3 f}+\frac{\cot (e+f x) (8 \sec (e+f x)+15)}{15 a^2 c^3 f}+\frac{x}{a^2 c^3}",1,"(Csc[e/2]*Csc[(e + f*x)/2]^5*Sec[e/2]*Sec[(e + f*x)/2]^3*(360*f*x*Cos[f*x] - 360*f*x*Cos[2*e + f*x] - 120*f*x*Cos[e + 2*f*x] + 120*f*x*Cos[3*e + 2*f*x] - 120*f*x*Cos[2*e + 3*f*x] + 120*f*x*Cos[4*e + 3*f*x] + 60*f*x*Cos[3*e + 4*f*x] - 60*f*x*Cos[5*e + 4*f*x] + 200*Sin[e] - 584*Sin[f*x] - 534*Sin[e + f*x] + 178*Sin[2*(e + f*x)] + 178*Sin[3*(e + f*x)] - 89*Sin[4*(e + f*x)] - 520*Sin[2*e + f*x] + 248*Sin[e + 2*f*x] + 120*Sin[3*e + 2*f*x] + 248*Sin[2*e + 3*f*x] + 120*Sin[4*e + 3*f*x] - 184*Sin[3*e + 4*f*x]))/(30720*a^2*c^3*f)","B",1
29,1,315,166,1.2711738,"\int \frac{1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4} \, dx","Integrate[1/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^4),x]","\frac{\csc \left(\frac{e}{2}\right) \sec \left(\frac{e}{2}\right) \csc ^7\left(\frac{1}{2} (e+f x)\right) \sec ^3\left(\frac{1}{2} (e+f x)\right) (-16002 \sin (e+f x)+9144 \sin (2 (e+f x))+3429 \sin (3 (e+f x))-4572 \sin (4 (e+f x))+1143 \sin (5 (e+f x))-11760 \sin (2 e+f x)+8864 \sin (e+2 f x)+3360 \sin (3 e+2 f x)+2064 \sin (2 e+3 f x)+2520 \sin (4 e+3 f x)-4432 \sin (3 e+4 f x)-1680 \sin (5 e+4 f x)+1528 \sin (4 e+5 f x)-5880 f x \cos (2 e+f x)-3360 f x \cos (e+2 f x)+3360 f x \cos (3 e+2 f x)-1260 f x \cos (2 e+3 f x)+1260 f x \cos (4 e+3 f x)+1680 f x \cos (3 e+4 f x)-1680 f x \cos (5 e+4 f x)-420 f x \cos (4 e+5 f x)+420 f x \cos (6 e+5 f x)+4032 \sin (e)-9632 \sin (f x)+5880 f x \cos (f x))}{860160 a^2 c^4 f}","-\frac{2 \cot ^7(e+f x)}{7 a^2 c^4 f}+\frac{\cot ^5(e+f x)}{5 a^2 c^4 f}-\frac{\cot ^3(e+f x)}{3 a^2 c^4 f}+\frac{\cot (e+f x)}{a^2 c^4 f}-\frac{2 \csc ^7(e+f x)}{7 a^2 c^4 f}+\frac{6 \csc ^5(e+f x)}{5 a^2 c^4 f}-\frac{2 \csc ^3(e+f x)}{a^2 c^4 f}+\frac{2 \csc (e+f x)}{a^2 c^4 f}+\frac{x}{a^2 c^4}",1,"(Csc[e/2]*Csc[(e + f*x)/2]^7*Sec[e/2]*Sec[(e + f*x)/2]^3*(5880*f*x*Cos[f*x] - 5880*f*x*Cos[2*e + f*x] - 3360*f*x*Cos[e + 2*f*x] + 3360*f*x*Cos[3*e + 2*f*x] - 1260*f*x*Cos[2*e + 3*f*x] + 1260*f*x*Cos[4*e + 3*f*x] + 1680*f*x*Cos[3*e + 4*f*x] - 1680*f*x*Cos[5*e + 4*f*x] - 420*f*x*Cos[4*e + 5*f*x] + 420*f*x*Cos[6*e + 5*f*x] + 4032*Sin[e] - 9632*Sin[f*x] - 16002*Sin[e + f*x] + 9144*Sin[2*(e + f*x)] + 3429*Sin[3*(e + f*x)] - 4572*Sin[4*(e + f*x)] + 1143*Sin[5*(e + f*x)] - 11760*Sin[2*e + f*x] + 8864*Sin[e + 2*f*x] + 3360*Sin[3*e + 2*f*x] + 2064*Sin[2*e + 3*f*x] + 2520*Sin[4*e + 3*f*x] - 4432*Sin[3*e + 4*f*x] - 1680*Sin[5*e + 4*f*x] + 1528*Sin[4*e + 5*f*x]))/(860160*a^2*c^4*f)","A",1
30,1,383,210,1.1986793,"\int \frac{1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5} \, dx","Integrate[1/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5),x]","\frac{\csc \left(\frac{e}{2}\right) \sec \left(\frac{e}{2}\right) \tan (e+f x) \sec ^6(e+f x) (-675036 \sin (e+f x)+506277 \sin (2 (e+f x))+37502 \sin (3 (e+f x))-225012 \sin (4 (e+f x))+112506 \sin (5 (e+f x))-18751 \sin (6 (e+f x))-431424 \sin (2 e+f x)+375552 \sin (e+2 f x)+201600 \sin (3 e+2 f x)-41248 \sin (2 e+3 f x)+84000 \sin (4 e+3 f x)-155712 \sin (3 e+4 f x)-100800 \sin (5 e+4 f x)+98016 \sin (4 e+5 f x)+30240 \sin (6 e+5 f x)-21376 \sin (5 e+6 f x)-181440 f x \cos (2 e+f x)-136080 f x \cos (e+2 f x)+136080 f x \cos (3 e+2 f x)-10080 f x \cos (2 e+3 f x)+10080 f x \cos (4 e+3 f x)+60480 f x \cos (3 e+4 f x)-60480 f x \cos (5 e+4 f x)-30240 f x \cos (4 e+5 f x)+30240 f x \cos (6 e+5 f x)+5040 f x \cos (5 e+6 f x)-5040 f x \cos (7 e+6 f x)+169344 \sin (e)-338112 \sin (f x)+181440 f x \cos (f x))}{645120 a^2 c^5 f (\sec (e+f x)-1)^5 (\sec (e+f x)+1)^2}","\frac{4 \cot ^9(e+f x)}{9 a^2 c^5 f}-\frac{\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac{\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac{\cot ^3(e+f x)}{3 a^2 c^5 f}+\frac{\cot (e+f x)}{a^2 c^5 f}+\frac{4 \csc ^9(e+f x)}{9 a^2 c^5 f}-\frac{15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac{21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac{13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac{3 \csc (e+f x)}{a^2 c^5 f}+\frac{x}{a^2 c^5}",1,"(Csc[e/2]*Sec[e/2]*Sec[e + f*x]^6*(181440*f*x*Cos[f*x] - 181440*f*x*Cos[2*e + f*x] - 136080*f*x*Cos[e + 2*f*x] + 136080*f*x*Cos[3*e + 2*f*x] - 10080*f*x*Cos[2*e + 3*f*x] + 10080*f*x*Cos[4*e + 3*f*x] + 60480*f*x*Cos[3*e + 4*f*x] - 60480*f*x*Cos[5*e + 4*f*x] - 30240*f*x*Cos[4*e + 5*f*x] + 30240*f*x*Cos[6*e + 5*f*x] + 5040*f*x*Cos[5*e + 6*f*x] - 5040*f*x*Cos[7*e + 6*f*x] + 169344*Sin[e] - 338112*Sin[f*x] - 675036*Sin[e + f*x] + 506277*Sin[2*(e + f*x)] + 37502*Sin[3*(e + f*x)] - 225012*Sin[4*(e + f*x)] + 112506*Sin[5*(e + f*x)] - 18751*Sin[6*(e + f*x)] - 431424*Sin[2*e + f*x] + 375552*Sin[e + 2*f*x] + 201600*Sin[3*e + 2*f*x] - 41248*Sin[2*e + 3*f*x] + 84000*Sin[4*e + 3*f*x] - 155712*Sin[3*e + 4*f*x] - 100800*Sin[5*e + 4*f*x] + 98016*Sin[4*e + 5*f*x] + 30240*Sin[6*e + 5*f*x] - 21376*Sin[5*e + 6*f*x])*Tan[e + f*x])/(645120*a^2*c^5*f*(-1 + Sec[e + f*x])^5*(1 + Sec[e + f*x])^2)","A",1
31,1,557,162,5.8804505,"\int \frac{(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx","Integrate[(c - c*Sec[e + f*x])^5/(a + a*Sec[e + f*x])^3,x]","-\frac{c^5 \sec \left(\frac{e}{2}\right) (\cos (e+f x)-1)^5 \cot \left(\frac{1}{2} (e+f x)\right) \csc ^4\left(\frac{1}{2} (e+f x)\right) \left(1016 \sin \left(\frac{f x}{2}\right) \cot ^6\left(\frac{1}{2} (e+f x)\right) \csc \left(\frac{1}{2} (e+f x)\right)+\sec ^2\left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) (-210 \cos (e+f x)-84 \cos (2 (e+f x))-14 \cos (3 (e+f x))+131 \cos (2 e+f x)+66 \cos (e+2 f x)+66 \cos (3 e+2 f x)+21 \cos (2 e+3 f x)+21 \cos (4 e+3 f x)+76 \cos (e)+131 \cos (f x)-140) \csc ^7\left(\frac{1}{2} (e+f x)\right)+48 \left(\sin \left(\frac{e}{2}\right)-\sin \left(\frac{3 e}{2}\right)\right) \sec ^2\left(\frac{e}{2}\right) \cot \left(\frac{1}{2} (e+f x)\right) \csc ^4\left(\frac{1}{2} (e+f x)\right)-60 \cos (e) \sec \left(\frac{e}{2}\right) \cot ^7\left(\frac{1}{2} (e+f x)\right) \left(-8 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+8 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+f x\right)+8 \left(\sin \left(\frac{e}{2}\right)-\sin \left(\frac{3 e}{2}\right)\right) \sec ^2\left(\frac{e}{2}\right) (\cos (e+f x)-7) \cot ^3\left(\frac{1}{2} (e+f x)\right) \csc ^4\left(\frac{1}{2} (e+f x)\right)+2 \sec \left(\frac{e}{2}\right) \cot ^5\left(\frac{1}{2} (e+f x)\right) \left(30 \cos (e) \left(-8 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+8 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+f x\right)-\tan \left(\frac{e}{2}\right) (15 (\cos (e+f x)+\cos (f x)-1)+\cos (e)) \csc ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}{240 a^3 f \left(\tan \left(\frac{e}{2}\right)-1\right) \left(\tan \left(\frac{e}{2}\right)+1\right) (\cos (e+f x)+1)^3 \left(\cot \left(\frac{1}{2} (e+f x)\right)-1\right) \left(\cot \left(\frac{1}{2} (e+f x)\right)+1\right)}","-\frac{c^5 \tan (e+f x)}{a^3 f}+\frac{128 c^5 \cot ^5(e+f x)}{5 a^3 f}+\frac{128 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac{32 c^5 \cot (e+f x)}{a^3 f}-\frac{128 c^5 \csc ^5(e+f x)}{5 a^3 f}+\frac{64 c^5 \csc ^3(e+f x)}{3 a^3 f}-\frac{16 c^5 \csc (e+f x)}{a^3 f}+\frac{8 c^5 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac{c^5 x}{a^3}",1,"-1/240*(c^5*(-1 + Cos[e + f*x])^5*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^4*Sec[e/2]*(-60*Cos[e]*Cot[(e + f*x)/2]^7*(f*x - 8*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 8*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sec[e/2] + 48*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^4*Sec[e/2]^2*(Sin[e/2] - Sin[(3*e)/2]) + 8*(-7 + Cos[e + f*x])*Cot[(e + f*x)/2]^3*Csc[(e + f*x)/2]^4*Sec[e/2]^2*(Sin[e/2] - Sin[(3*e)/2]) + 1016*Cot[(e + f*x)/2]^6*Csc[(e + f*x)/2]*Sin[(f*x)/2] + (-140 + 76*Cos[e] + 131*Cos[f*x] - 210*Cos[e + f*x] - 84*Cos[2*(e + f*x)] - 14*Cos[3*(e + f*x)] + 131*Cos[2*e + f*x] + 66*Cos[e + 2*f*x] + 66*Cos[3*e + 2*f*x] + 21*Cos[2*e + 3*f*x] + 21*Cos[4*e + 3*f*x])*Csc[(e + f*x)/2]^7*Sec[e/2]^2*Sin[(f*x)/2] + 2*Cot[(e + f*x)/2]^5*Sec[e/2]*(30*Cos[e]*(f*x - 8*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 8*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - (Cos[e] + 15*(-1 + Cos[f*x] + Cos[e + f*x]))*Csc[(e + f*x)/2]^2*Tan[e/2])))/(a^3*f*(1 + Cos[e + f*x])^3*(-1 + Cot[(e + f*x)/2])*(1 + Cot[(e + f*x)/2])*(-1 + Tan[e/2])*(1 + Tan[e/2]))","B",1
32,1,231,148,1.2016231,"\int \frac{(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx","Integrate[(c - c*Sec[e + f*x])^4/(a + a*Sec[e + f*x])^3,x]","\frac{c^4 (\cos (e+f x)-1)^4 \cot \left(\frac{1}{2} (e+f x)\right) \csc ^2\left(\frac{1}{2} (e+f x)\right) \left(8 \tan \left(\frac{e}{2}\right) \cot ^3\left(\frac{1}{2} (e+f x)\right) \csc ^2\left(\frac{1}{2} (e+f x)\right)-4 \tan \left(\frac{e}{2}\right) \cot \left(\frac{1}{2} (e+f x)\right) \csc ^4\left(\frac{1}{2} (e+f x)\right)+5 \cot ^5\left(\frac{1}{2} (e+f x)\right) \left(-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+f x\right)-\sec \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right) (8 \cos (e+f x)+3 \cos (2 (e+f x))+9) \csc ^5\left(\frac{1}{2} (e+f x)\right)\right)}{10 a^3 f (\cos (e+f x)+1)^3}","\frac{c^4 \tanh ^{-1}(\sin (e+f x))}{a^3 f}-\frac{c^4 \tan (e+f x) \sec ^2(e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}-\frac{23 c^4 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)}+\frac{14 c^4 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^2}-\frac{3 c^4 \tan (e+f x)}{a^3 f (\sec (e+f x)+1)^3}+\frac{c^4 x}{a^3}",1,"(c^4*(-1 + Cos[e + f*x])^4*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2*(5*Cot[(e + f*x)/2]^5*(f*x - Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - (9 + 8*Cos[e + f*x] + 3*Cos[2*(e + f*x)])*Csc[(e + f*x)/2]^5*Sec[e/2]*Sin[(f*x)/2] + 8*Cot[(e + f*x)/2]^3*Csc[(e + f*x)/2]^2*Tan[e/2] - 4*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^4*Tan[e/2]))/(10*a^3*f*(1 + Cos[e + f*x])^3)","A",1
33,1,90,96,0.0778646,"\int \frac{(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^3} \, dx","Integrate[(c - c*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^3,x]","-\frac{c^3 \left(-\frac{2 \tan ^{-1}\left(\tan \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{f}+\frac{2 \tan ^5\left(\frac{e}{2}+\frac{f x}{2}\right)}{5 f}-\frac{2 \tan ^3\left(\frac{e}{2}+\frac{f x}{2}\right)}{3 f}+\frac{2 \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{f}\right)}{a^3}","-\frac{26 c^3 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)}+\frac{4 c^3 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)^2}-\frac{8 c^3 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}+\frac{c^3 x}{a^3}",1,"-((c^3*((-2*ArcTan[Tan[e/2 + (f*x)/2]])/f + (2*Tan[e/2 + (f*x)/2])/f - (2*Tan[e/2 + (f*x)/2]^3)/(3*f) + (2*Tan[e/2 + (f*x)/2]^5)/(5*f)))/a^3)","A",1
34,1,171,96,0.4658445,"\int \frac{(c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^3} \, dx","Integrate[(c - c*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^3,x]","\frac{c^2 \sec \left(\frac{e}{2}\right) \sec ^5\left(\frac{1}{2} (e+f x)\right) \left(360 \sin \left(e+\frac{f x}{2}\right)-280 \sin \left(e+\frac{3 f x}{2}\right)+150 \sin \left(2 e+\frac{3 f x}{2}\right)-86 \sin \left(2 e+\frac{5 f x}{2}\right)+150 f x \cos \left(e+\frac{f x}{2}\right)+75 f x \cos \left(e+\frac{3 f x}{2}\right)+75 f x \cos \left(2 e+\frac{3 f x}{2}\right)+15 f x \cos \left(2 e+\frac{5 f x}{2}\right)+15 f x \cos \left(3 e+\frac{5 f x}{2}\right)-500 \sin \left(\frac{f x}{2}\right)+150 f x \cos \left(\frac{f x}{2}\right)\right)}{480 a^3 f}","-\frac{23 c^2 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)}-\frac{8 c^2 \tan (e+f x)}{15 a^3 f (\sec (e+f x)+1)^2}-\frac{4 c^2 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}+\frac{c^2 x}{a^3}",1,"(c^2*Sec[e/2]*Sec[(e + f*x)/2]^5*(150*f*x*Cos[(f*x)/2] + 150*f*x*Cos[e + (f*x)/2] + 75*f*x*Cos[e + (3*f*x)/2] + 75*f*x*Cos[2*e + (3*f*x)/2] + 15*f*x*Cos[2*e + (5*f*x)/2] + 15*f*x*Cos[3*e + (5*f*x)/2] - 500*Sin[(f*x)/2] + 360*Sin[e + (f*x)/2] - 280*Sin[e + (3*f*x)/2] + 150*Sin[2*e + (3*f*x)/2] - 86*Sin[2*e + (5*f*x)/2]))/(480*a^3*f)","A",1
35,1,169,88,0.4599906,"\int \frac{c-c \sec (e+f x)}{(a+a \sec (e+f x))^3} \, dx","Integrate[(c - c*Sec[e + f*x])/(a + a*Sec[e + f*x])^3,x]","\frac{c \sec \left(\frac{e}{2}\right) \sec ^5\left(\frac{1}{2} (e+f x)\right) \left(110 \sin \left(e+\frac{f x}{2}\right)-90 \sin \left(e+\frac{3 f x}{2}\right)+40 \sin \left(2 e+\frac{3 f x}{2}\right)-26 \sin \left(2 e+\frac{5 f x}{2}\right)+50 f x \cos \left(e+\frac{f x}{2}\right)+25 f x \cos \left(e+\frac{3 f x}{2}\right)+25 f x \cos \left(2 e+\frac{3 f x}{2}\right)+5 f x \cos \left(2 e+\frac{5 f x}{2}\right)+5 f x \cos \left(3 e+\frac{5 f x}{2}\right)-150 \sin \left(\frac{f x}{2}\right)+50 f x \cos \left(\frac{f x}{2}\right)\right)}{160 a^3 f}","-\frac{8 c \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)}-\frac{3 c \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^2}-\frac{2 c \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}+\frac{c x}{a^3}",1,"(c*Sec[e/2]*Sec[(e + f*x)/2]^5*(50*f*x*Cos[(f*x)/2] + 50*f*x*Cos[e + (f*x)/2] + 25*f*x*Cos[e + (3*f*x)/2] + 25*f*x*Cos[2*e + (3*f*x)/2] + 5*f*x*Cos[2*e + (5*f*x)/2] + 5*f*x*Cos[3*e + (5*f*x)/2] - 150*Sin[(f*x)/2] + 110*Sin[e + (f*x)/2] - 90*Sin[e + (3*f*x)/2] + 40*Sin[2*e + (3*f*x)/2] - 26*Sin[2*e + (5*f*x)/2]))/(160*a^3*f)","A",1
36,1,197,126,0.9877843,"\int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))} \, dx","Integrate[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])),x]","-\frac{\csc \left(\frac{e}{2}\right) \sec \left(\frac{e}{2}\right) \csc \left(\frac{1}{2} (e+f x)\right) \sec ^5\left(\frac{1}{2} (e+f x)\right) (-445 \sin (e+f x)-356 \sin (2 (e+f x))-89 \sin (3 (e+f x))+240 \sin (2 e+f x)+296 \sin (e+2 f x)+120 \sin (3 e+2 f x)+104 \sin (2 e+3 f x)+150 f x \cos (2 e+f x)-120 f x \cos (e+2 f x)+120 f x \cos (3 e+2 f x)-30 f x \cos (2 e+3 f x)+30 f x \cos (4 e+3 f x)+80 \sin (e)+280 \sin (f x)-150 f x \cos (f x))}{3840 a^3 c f}","\frac{2 \cot ^5(e+f x)}{5 a^3 c f}-\frac{\cot ^3(e+f x)}{3 a^3 c f}+\frac{\cot (e+f x)}{a^3 c f}-\frac{2 \csc ^5(e+f x)}{5 a^3 c f}+\frac{4 \csc ^3(e+f x)}{3 a^3 c f}-\frac{2 \csc (e+f x)}{a^3 c f}+\frac{x}{a^3 c}",1,"-1/3840*(Csc[e/2]*Csc[(e + f*x)/2]*Sec[e/2]*Sec[(e + f*x)/2]^5*(-150*f*x*Cos[f*x] + 150*f*x*Cos[2*e + f*x] - 120*f*x*Cos[e + 2*f*x] + 120*f*x*Cos[3*e + 2*f*x] - 30*f*x*Cos[2*e + 3*f*x] + 30*f*x*Cos[4*e + 3*f*x] + 80*Sin[e] + 280*Sin[f*x] - 445*Sin[e + f*x] - 356*Sin[2*(e + f*x)] - 89*Sin[3*(e + f*x)] + 240*Sin[2*e + f*x] + 296*Sin[e + 2*f*x] + 120*Sin[3*e + 2*f*x] + 104*Sin[2*e + 3*f*x]))/(a^3*c*f)","A",1
37,1,257,100,1.0084713,"\int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2} \, dx","Integrate[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^2),x]","\frac{\csc \left(\frac{e}{2}\right) \sec \left(\frac{e}{2}\right) \csc ^3\left(\frac{1}{2} (e+f x)\right) \sec ^5\left(\frac{1}{2} (e+f x)\right) (534 \sin (e+f x)+178 \sin (2 (e+f x))-178 \sin (3 (e+f x))-89 \sin (4 (e+f x))-520 \sin (2 e+f x)-248 \sin (e+2 f x)-120 \sin (3 e+2 f x)+248 \sin (2 e+3 f x)+120 \sin (4 e+3 f x)+184 \sin (3 e+4 f x)-360 f x \cos (2 e+f x)+120 f x \cos (e+2 f x)-120 f x \cos (3 e+2 f x)-120 f x \cos (2 e+3 f x)+120 f x \cos (4 e+3 f x)-60 f x \cos (3 e+4 f x)+60 f x \cos (5 e+4 f x)-200 \sin (e)-584 \sin (f x)+360 f x \cos (f x))}{30720 a^3 c^2 f}","\frac{\cot ^5(e+f x) (1-\sec (e+f x))}{5 a^3 c^2 f}-\frac{\cot ^3(e+f x) (5-4 \sec (e+f x))}{15 a^3 c^2 f}+\frac{\cot (e+f x) (15-8 \sec (e+f x))}{15 a^3 c^2 f}+\frac{x}{a^3 c^2}",1,"(Csc[e/2]*Csc[(e + f*x)/2]^3*Sec[e/2]*Sec[(e + f*x)/2]^5*(360*f*x*Cos[f*x] - 360*f*x*Cos[2*e + f*x] + 120*f*x*Cos[e + 2*f*x] - 120*f*x*Cos[3*e + 2*f*x] - 120*f*x*Cos[2*e + 3*f*x] + 120*f*x*Cos[4*e + 3*f*x] - 60*f*x*Cos[3*e + 4*f*x] + 60*f*x*Cos[5*e + 4*f*x] - 200*Sin[e] - 584*Sin[f*x] + 534*Sin[e + f*x] + 178*Sin[2*(e + f*x)] - 178*Sin[3*(e + f*x)] - 89*Sin[4*(e + f*x)] - 520*Sin[2*e + f*x] - 248*Sin[e + 2*f*x] - 120*Sin[3*e + 2*f*x] + 248*Sin[2*e + 3*f*x] + 120*Sin[4*e + 3*f*x] + 184*Sin[3*e + 4*f*x]))/(30720*a^3*c^2*f)","B",1
38,1,39,67,0.0699411,"\int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3} \, dx","Integrate[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^3),x]","\frac{\cot ^5(e+f x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(e+f x)\right)}{5 a^3 c^3 f}","\frac{\cot ^5(e+f x)}{5 a^3 c^3 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^3 f}+\frac{\cot (e+f x)}{a^3 c^3 f}+\frac{x}{a^3 c^3}",1,"(Cot[e + f*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[e + f*x]^2])/(5*a^3*c^3*f)","C",1
39,1,362,129,1.4054362,"\int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx","Integrate[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^4),x]","\frac{\csc \left(\frac{e}{2}\right) \sec \left(\frac{e}{2}\right) \csc ^7\left(\frac{1}{2} (e+f x)\right) \sec ^5\left(\frac{1}{2} (e+f x)\right) (-22860 \sin (e+f x)+5715 \sin (2 (e+f x))+11430 \sin (3 (e+f x))-4572 \sin (4 (e+f x))-2286 \sin (5 (e+f x))+1143 \sin (6 (e+f x))-26208 \sin (2 e+f x)+14080 \sin (e+2 f x)+16400 \sin (2 e+3 f x)+11760 \sin (4 e+3 f x)-7904 \sin (3 e+4 f x)-3360 \sin (5 e+4 f x)-3952 \sin (4 e+5 f x)-1680 \sin (6 e+5 f x)+2816 \sin (5 e+6 f x)-16800 f x \cos (2 e+f x)-4200 f x \cos (e+2 f x)+4200 f x \cos (3 e+2 f x)-8400 f x \cos (2 e+3 f x)+8400 f x \cos (4 e+3 f x)+3360 f x \cos (3 e+4 f x)-3360 f x \cos (5 e+4 f x)+1680 f x \cos (4 e+5 f x)-1680 f x \cos (6 e+5 f x)-840 f x \cos (5 e+6 f x)+840 f x \cos (7 e+6 f x)+3136 \sin (e)-30112 \sin (f x)+16800 f x \cos (f x))}{6881280 a^3 c^4 f}","-\frac{\cot ^7(e+f x) (\sec (e+f x)+1)}{7 a^3 c^4 f}+\frac{\cot ^5(e+f x) (6 \sec (e+f x)+7)}{35 a^3 c^4 f}-\frac{\cot ^3(e+f x) (24 \sec (e+f x)+35)}{105 a^3 c^4 f}+\frac{\cot (e+f x) (16 \sec (e+f x)+35)}{35 a^3 c^4 f}+\frac{x}{a^3 c^4}",1,"(Csc[e/2]*Csc[(e + f*x)/2]^7*Sec[e/2]*Sec[(e + f*x)/2]^5*(16800*f*x*Cos[f*x] - 16800*f*x*Cos[2*e + f*x] - 4200*f*x*Cos[e + 2*f*x] + 4200*f*x*Cos[3*e + 2*f*x] - 8400*f*x*Cos[2*e + 3*f*x] + 8400*f*x*Cos[4*e + 3*f*x] + 3360*f*x*Cos[3*e + 4*f*x] - 3360*f*x*Cos[5*e + 4*f*x] + 1680*f*x*Cos[4*e + 5*f*x] - 1680*f*x*Cos[6*e + 5*f*x] - 840*f*x*Cos[5*e + 6*f*x] + 840*f*x*Cos[7*e + 6*f*x] + 3136*Sin[e] - 30112*Sin[f*x] - 22860*Sin[e + f*x] + 5715*Sin[2*(e + f*x)] + 11430*Sin[3*(e + f*x)] - 4572*Sin[4*(e + f*x)] - 2286*Sin[5*(e + f*x)] + 1143*Sin[6*(e + f*x)] - 26208*Sin[2*e + f*x] + 14080*Sin[e + 2*f*x] + 16400*Sin[2*e + 3*f*x] + 11760*Sin[4*e + 3*f*x] - 7904*Sin[3*e + 4*f*x] - 3360*Sin[5*e + 4*f*x] - 3952*Sin[4*e + 5*f*x] - 1680*Sin[6*e + 5*f*x] + 2816*Sin[5*e + 6*f*x]))/(6881280*a^3*c^4*f)","B",1
40,1,441,210,1.7423348,"\int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx","Integrate[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5),x]","\frac{\csc \left(\frac{e}{2}\right) \sec \left(\frac{e}{2}\right) \tan (e+f x) \sec ^7(e+f x) (-1152405 \sin (e+f x)+512180 \sin (2 (e+f x))+486571 \sin (3 (e+f x))-409744 \sin (4 (e+f x))-25609 \sin (5 (e+f x))+102436 \sin (6 (e+f x))-25609 \sin (7 (e+f x))-825216 \sin (2 e+f x)+622976 \sin (e+2 f x)+142464 \sin (3 e+2 f x)+297088 \sin (2 e+3 f x)+430080 \sin (4 e+3 f x)-424192 \sin (3 e+4 f x)-188160 \sin (5 e+4 f x)+2048 \sin (4 e+5 f x)-40320 \sin (6 e+5 f x)+112768 \sin (5 e+6 f x)+40320 \sin (7 e+6 f x)-38272 \sin (6 e+7 f x)-453600 f x \cos (2 e+f x)-201600 f x \cos (e+2 f x)+201600 f x \cos (3 e+2 f x)-191520 f x \cos (2 e+3 f x)+191520 f x \cos (4 e+3 f x)+161280 f x \cos (3 e+4 f x)-161280 f x \cos (5 e+4 f x)+10080 f x \cos (4 e+5 f x)-10080 f x \cos (6 e+5 f x)-40320 f x \cos (5 e+6 f x)+40320 f x \cos (7 e+6 f x)+10080 f x \cos (6 e+7 f x)-10080 f x \cos (8 e+7 f x)+259584 \sin (e)-897024 \sin (f x)+453600 f x \cos (f x))}{2580480 a^3 c^5 f (\sec (e+f x)-1)^5 (\sec (e+f x)+1)^3}","\frac{2 \cot ^9(e+f x)}{9 a^3 c^5 f}-\frac{\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac{\cot (e+f x)}{a^3 c^5 f}+\frac{2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac{8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac{12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac{8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac{2 \csc (e+f x)}{a^3 c^5 f}+\frac{x}{a^3 c^5}",1,"(Csc[e/2]*Sec[e/2]*Sec[e + f*x]^7*(453600*f*x*Cos[f*x] - 453600*f*x*Cos[2*e + f*x] - 201600*f*x*Cos[e + 2*f*x] + 201600*f*x*Cos[3*e + 2*f*x] - 191520*f*x*Cos[2*e + 3*f*x] + 191520*f*x*Cos[4*e + 3*f*x] + 161280*f*x*Cos[3*e + 4*f*x] - 161280*f*x*Cos[5*e + 4*f*x] + 10080*f*x*Cos[4*e + 5*f*x] - 10080*f*x*Cos[6*e + 5*f*x] - 40320*f*x*Cos[5*e + 6*f*x] + 40320*f*x*Cos[7*e + 6*f*x] + 10080*f*x*Cos[6*e + 7*f*x] - 10080*f*x*Cos[8*e + 7*f*x] + 259584*Sin[e] - 897024*Sin[f*x] - 1152405*Sin[e + f*x] + 512180*Sin[2*(e + f*x)] + 486571*Sin[3*(e + f*x)] - 409744*Sin[4*(e + f*x)] - 25609*Sin[5*(e + f*x)] + 102436*Sin[6*(e + f*x)] - 25609*Sin[7*(e + f*x)] - 825216*Sin[2*e + f*x] + 622976*Sin[e + 2*f*x] + 142464*Sin[3*e + 2*f*x] + 297088*Sin[2*e + 3*f*x] + 430080*Sin[4*e + 3*f*x] - 424192*Sin[3*e + 4*f*x] - 188160*Sin[5*e + 4*f*x] + 2048*Sin[4*e + 5*f*x] - 40320*Sin[6*e + 5*f*x] + 112768*Sin[5*e + 6*f*x] + 40320*Sin[7*e + 6*f*x] - 38272*Sin[6*e + 7*f*x])*Tan[e + f*x])/(2580480*a^3*c^5*f*(-1 + Sec[e + f*x])^5*(1 + Sec[e + f*x])^3)","B",1
41,1,499,252,2.3621549,"\int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx","Integrate[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^6),x]","\frac{\csc \left(\frac{e}{2}\right) \sec \left(\frac{e}{2}\right) \tan (e+f x) \sec ^8(e+f x) (-86058610 \sin (e+f x)+51635166 \sin (2 (e+f x))+26599934 \sin (3 (e+f x))-39117550 \sin (4 (e+f x))+7823510 \sin (5 (e+f x))+7823510 \sin (6 (e+f x))-4694106 \sin (7 (e+f x))+782351 \sin (8 (e+f x))-55651200 \sin (2 e+f x)+47971968 \sin (e+2 f x)+14990976 \sin (3 e+2 f x)+8100992 \sin (2 e+3 f x)+24334464 \sin (4 e+3 f x)-28627840 \sin (3 e+4 f x)-19071360 \sin (5 e+4 f x)+9687680 \sin (4 e+5 f x)-147840 \sin (6 e+5 f x)+5548160 \sin (5 e+6 f x)+3991680 \sin (7 e+6 f x)-4393344 \sin (6 e+7 f x)-1330560 \sin (8 e+7 f x)+953984 \sin (7 e+8 f x)-24393600 f x \cos (2 e+f x)-14636160 f x \cos (e+2 f x)+14636160 f x \cos (3 e+2 f x)-7539840 f x \cos (2 e+3 f x)+7539840 f x \cos (4 e+3 f x)+11088000 f x \cos (3 e+4 f x)-11088000 f x \cos (5 e+4 f x)-2217600 f x \cos (4 e+5 f x)+2217600 f x \cos (6 e+5 f x)-2217600 f x \cos (5 e+6 f x)+2217600 f x \cos (7 e+6 f x)+1330560 f x \cos (6 e+7 f x)-1330560 f x \cos (8 e+7 f x)-221760 f x \cos (7 e+8 f x)+221760 f x \cos (9 e+8 f x)+17677440 \sin (e)-49287040 \sin (f x)+24393600 f x \cos (f x))}{113541120 a^3 c^6 f (\sec (e+f x)-1)^6 (\sec (e+f x)+1)^3}","-\frac{4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac{\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac{\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac{\cot (e+f x)}{a^3 c^6 f}-\frac{4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac{19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac{36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac{34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac{16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac{3 \csc (e+f x)}{a^3 c^6 f}+\frac{x}{a^3 c^6}",1,"(Csc[e/2]*Sec[e/2]*Sec[e + f*x]^8*(24393600*f*x*Cos[f*x] - 24393600*f*x*Cos[2*e + f*x] - 14636160*f*x*Cos[e + 2*f*x] + 14636160*f*x*Cos[3*e + 2*f*x] - 7539840*f*x*Cos[2*e + 3*f*x] + 7539840*f*x*Cos[4*e + 3*f*x] + 11088000*f*x*Cos[3*e + 4*f*x] - 11088000*f*x*Cos[5*e + 4*f*x] - 2217600*f*x*Cos[4*e + 5*f*x] + 2217600*f*x*Cos[6*e + 5*f*x] - 2217600*f*x*Cos[5*e + 6*f*x] + 2217600*f*x*Cos[7*e + 6*f*x] + 1330560*f*x*Cos[6*e + 7*f*x] - 1330560*f*x*Cos[8*e + 7*f*x] - 221760*f*x*Cos[7*e + 8*f*x] + 221760*f*x*Cos[9*e + 8*f*x] + 17677440*Sin[e] - 49287040*Sin[f*x] - 86058610*Sin[e + f*x] + 51635166*Sin[2*(e + f*x)] + 26599934*Sin[3*(e + f*x)] - 39117550*Sin[4*(e + f*x)] + 7823510*Sin[5*(e + f*x)] + 7823510*Sin[6*(e + f*x)] - 4694106*Sin[7*(e + f*x)] + 782351*Sin[8*(e + f*x)] - 55651200*Sin[2*e + f*x] + 47971968*Sin[e + 2*f*x] + 14990976*Sin[3*e + 2*f*x] + 8100992*Sin[2*e + 3*f*x] + 24334464*Sin[4*e + 3*f*x] - 28627840*Sin[3*e + 4*f*x] - 19071360*Sin[5*e + 4*f*x] + 9687680*Sin[4*e + 5*f*x] - 147840*Sin[6*e + 5*f*x] + 5548160*Sin[5*e + 6*f*x] + 3991680*Sin[7*e + 6*f*x] - 4393344*Sin[6*e + 7*f*x] - 1330560*Sin[8*e + 7*f*x] + 953984*Sin[7*e + 8*f*x])*Tan[e + f*x])/(113541120*a^3*c^6*f*(-1 + Sec[e + f*x])^6*(1 + Sec[e + f*x])^3)","A",1
42,1,121,175,1.047609,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^4 \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^4,x]","\frac{2 c^4 \tan \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) \sqrt{a (\sec (e+f x)+1)} \left((-198 \cos (e+f x)+61 \cos (2 (e+f x))-44 \cos (3 (e+f x))+76) \sqrt{\sec (e+f x)-1}+105 \cos ^3(e+f x) \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)\right)}{105 f \sqrt{\sec (e+f x)-1}}","\frac{2 a^4 c^4 \tan ^7(e+f x)}{7 f (a \sec (e+f x)+a)^{7/2}}-\frac{2 a^3 c^4 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^2 c^4 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 \sqrt{a} c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a c^4 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*c^4*(105*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x]^3 + (76 - 198*Cos[e + f*x] + 61*Cos[2*(e + f*x)] - 44*Cos[3*(e + f*x)])*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]^3*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(105*f*Sqrt[-1 + Sec[e + f*x]])","A",1
43,1,111,140,1.1946169,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^3 \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^3,x]","\frac{c^3 \tan \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \left((22 \cos (e+f x)-23 \cos (2 (e+f x))-29) \sqrt{\sec (e+f x)-1}+30 \cos ^2(e+f x) \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)\right)}{15 f \sqrt{\sec (e+f x)-1}}","-\frac{2 a^3 c^3 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^2 c^3 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 \sqrt{a} c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a c^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(c^3*(30*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x]^2 + (-29 + 22*Cos[e + f*x] - 23*Cos[2*(e + f*x)])*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]^2*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(15*f*Sqrt[-1 + Sec[e + f*x]])","A",1
44,1,97,105,0.885251,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^2 \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^2,x]","-\frac{2 c^2 \tan \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{a (\sec (e+f x)+1)} \left((4 \cos (e+f x)-1) \sqrt{\sec (e+f x)-1}-3 \cos (e+f x) \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)\right)}{3 f \sqrt{\sec (e+f x)-1}}","\frac{2 a^2 c^2 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 \sqrt{a} c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a c^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(-2*c^2*(-3*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x] + (-1 + 4*Cos[e + f*x])*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(3*f*Sqrt[-1 + Sec[e + f*x]])","A",1
45,1,70,66,0.3208111,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x)) \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x]),x]","-\frac{2 c \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(\sqrt{\sec (e+f x)-1}-\tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)\right)}{f \sqrt{\sec (e+f x)-1}}","\frac{2 \sqrt{a} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a c \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(-2*c*(-ArcTan[Sqrt[-1 + Sec[e + f*x]]] + Sqrt[-1 + Sec[e + f*x]])*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(f*Sqrt[-1 + Sec[e + f*x]])","A",1
46,1,90,69,0.5878915,"\int \frac{\sqrt{a+a \sec (e+f x)}}{c-c \sec (e+f x)} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x]),x]","\frac{2 a \tan (e+f x) \sec (e+f x) \left(\cos (e+f x) \sqrt{\sec (e+f x)-1}-(\cos (e+f x)-1) \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)\right)}{c f (\sec (e+f x)-1)^{3/2} \sqrt{a (\sec (e+f x)+1)}}","\frac{2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c f}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}",1,"(2*a*(-(ArcTan[Sqrt[-1 + Sec[e + f*x]]]*(-1 + Cos[e + f*x])) + Cos[e + f*x]*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]*Tan[e + f*x])/(c*f*(-1 + Sec[e + f*x])^(3/2)*Sqrt[a*(1 + Sec[e + f*x])])","A",1
47,1,78,104,0.2415949,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^2} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^2,x]","-\frac{2 \sqrt{\cos (e+f x)} \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)}{3 c^2 f (\cos (e+f x)-1)^2}","-\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a c^2 f}+\frac{2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^2 f}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^2 f}",1,"(-2*Sqrt[Cos[e + f*x]]*Hypergeometric2F1[-3/2, -3/2, -1/2, 2*Sin[(e + f*x)/2]^2]*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(3*c^2*f*(-1 + Cos[e + f*x])^2)","C",1
48,1,78,139,0.2647636,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^3} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^3,x]","-\frac{2 \sqrt{\cos (e+f x)} \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \, _2F_1\left(-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)}{5 c^3 f (\cos (e+f x)-1)^3}","\frac{2 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a^2 c^3 f}-\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a c^3 f}+\frac{2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^3 f}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^3 f}",1,"(-2*Sqrt[Cos[e + f*x]]*Hypergeometric2F1[-5/2, -5/2, -3/2, 2*Sin[(e + f*x)/2]^2]*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(5*c^3*f*(-1 + Cos[e + f*x])^3)","C",1
49,1,78,174,0.2443717,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^4} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^4,x]","-\frac{2 \sqrt{\cos (e+f x)} \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \, _2F_1\left(-\frac{7}{2},-\frac{7}{2};-\frac{5}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)}{7 c^4 f (\cos (e+f x)-1)^4}","-\frac{2 \cot ^7(e+f x) (a \sec (e+f x)+a)^{7/2}}{7 a^3 c^4 f}+\frac{2 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a^2 c^4 f}-\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a c^4 f}+\frac{2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^4 f}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^4 f}",1,"(-2*Sqrt[Cos[e + f*x]]*Hypergeometric2F1[-7/2, -7/2, -5/2, 2*Sin[(e + f*x)/2]^2]*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(7*c^4*f*(-1 + Cos[e + f*x])^4)","C",1
50,1,122,177,0.9485972,"\int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^3 \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^3,x]","\frac{a c^3 \tan \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) \sqrt{a (\sec (e+f x)+1)} \left((-171 \cos (e+f x)+32 \cos (2 (e+f x))-73 \cos (3 (e+f x))+2) \sqrt{\sec (e+f x)-1}+210 \cos ^3(e+f x) \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)\right)}{105 f \sqrt{\sec (e+f x)-1}}","\frac{2 a^{3/2} c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^5 c^3 \tan ^7(e+f x)}{7 f (a \sec (e+f x)+a)^{7/2}}-\frac{2 a^4 c^3 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^3 c^3 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}-\frac{2 a^2 c^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(a*c^3*(210*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x]^3 + (2 - 171*Cos[e + f*x] + 32*Cos[2*(e + f*x)] - 73*Cos[3*(e + f*x)])*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]^3*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(105*f*Sqrt[-1 + Sec[e + f*x]])","A",1
51,1,112,142,0.9082274,"\int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^2 \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^2,x]","-\frac{a c^2 \tan \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \left((2 \cos (e+f x)+17 \cos (2 (e+f x))+11) \sqrt{\sec (e+f x)-1}-30 \cos ^2(e+f x) \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)\right)}{15 f \sqrt{\sec (e+f x)-1}}","\frac{2 a^{3/2} c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}+\frac{2 a^4 c^2 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^3 c^2 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}-\frac{2 a^2 c^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"-1/15*(a*c^2*(-30*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x]^2 + (11 + 2*Cos[e + f*x] + 17*Cos[2*(e + f*x)])*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]^2*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(f*Sqrt[-1 + Sec[e + f*x]])","A",1
52,1,96,101,0.6765233,"\int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x)) \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x]),x]","-\frac{2 a c \tan \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{a (\sec (e+f x)+1)} \left((2 \cos (e+f x)+1) \sqrt{\sec (e+f x)-1}-3 \cos (e+f x) \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)\right)}{3 f \sqrt{\sec (e+f x)-1}}","\frac{2 a^{3/2} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^3 c \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}-\frac{2 a^2 c \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(-2*a*c*(-3*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x] + (1 + 2*Cos[e + f*x])*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(3*f*Sqrt[-1 + Sec[e + f*x]])","A",1
53,1,93,70,0.5836909,"\int \frac{(a+a \sec (e+f x))^{3/2}}{c-c \sec (e+f x)} \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x]),x]","\frac{2 a^2 \tan (e+f x) \sec (e+f x) \left(2 \cos (e+f x) \sqrt{\sec (e+f x)-1}-(\cos (e+f x)-1) \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)\right)}{c f (\sec (e+f x)-1)^{3/2} \sqrt{a (\sec (e+f x)+1)}}","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}+\frac{4 a \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c f}",1,"(2*a^2*(-(ArcTan[Sqrt[-1 + Sec[e + f*x]]]*(-1 + Cos[e + f*x])) + 2*Cos[e + f*x]*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]*Tan[e + f*x])/(c*f*(-1 + Sec[e + f*x])^(3/2)*Sqrt[a*(1 + Sec[e + f*x])])","A",1
54,1,113,102,0.642668,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^2} \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^2,x]","-\frac{2 a \sqrt{\cos (e+f x)} \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(\sqrt{\cos (e+f x)} (5 \cos (e+f x)-3)-6 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right) \sin ^3\left(\frac{1}{2} (e+f x)\right)\right)}{3 c^2 f (\cos (e+f x)-1)^2}","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^2 f}-\frac{4 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^2 f}+\frac{2 a \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^2 f}",1,"(-2*a*Sqrt[Cos[e + f*x]]*Sqrt[a*(1 + Sec[e + f*x])]*(Sqrt[Cos[e + f*x]]*(-3 + 5*Cos[e + f*x]) - 6*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]]*Sin[(e + f*x)/2]^3)*Tan[(e + f*x)/2])/(3*c^2*f*(-1 + Cos[e + f*x])^2)","A",1
55,1,102,137,0.8037416,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^3} \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^3,x]","\frac{2 a \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(5 (\cos (e+f x)-1) \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)+6 \cos ^{\frac{5}{2}}(e+f x)\right)}{15 c^3 f \cos ^{\frac{5}{2}}(e+f x) (\sec (e+f x)-1)^3}","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^3 f}+\frac{4 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a c^3 f}-\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^3 f}+\frac{2 a \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^3 f}",1,"(2*a*(6*Cos[e + f*x]^(5/2) + 5*(-1 + Cos[e + f*x])*Hypergeometric2F1[-3/2, -3/2, -1/2, 2*Sin[(e + f*x)/2]^2])*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(15*c^3*f*Cos[e + f*x]^(5/2)*(-1 + Sec[e + f*x])^3)","C",1
56,1,102,172,1.2905464,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^4} \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^4,x]","-\frac{2 a \sqrt{\cos (e+f x)} \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(7 (\cos (e+f x)-1) \, _2F_1\left(-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)+10 \cos ^{\frac{7}{2}}(e+f x)\right)}{35 c^4 f (\cos (e+f x)-1)^4}","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^4 f}-\frac{4 \cot ^7(e+f x) (a \sec (e+f x)+a)^{7/2}}{7 a^2 c^4 f}+\frac{2 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a c^4 f}-\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^4 f}+\frac{2 a \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^4 f}",1,"(-2*a*Sqrt[Cos[e + f*x]]*(10*Cos[e + f*x]^(7/2) + 7*(-1 + Cos[e + f*x])*Hypergeometric2F1[-5/2, -5/2, -3/2, 2*Sin[(e + f*x)/2]^2])*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(35*c^4*f*(-1 + Cos[e + f*x])^4)","C",1
57,1,134,212,1.3053241,"\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^3 \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^3,x]","-\frac{a^2 c^3 \tan \left(\frac{1}{2} (e+f x)\right) \sec ^4(e+f x) \sqrt{a (\sec (e+f x)+1)} \left((164 \cos (e+f x)+1004 \cos (2 (e+f x))+68 \cos (3 (e+f x))+383 \cos (4 (e+f x))+901) \sqrt{\sec (e+f x)-1}-2520 \cos ^4(e+f x) \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)\right)}{1260 f \sqrt{\sec (e+f x)-1}}","\frac{2 a^{5/2} c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^7 c^3 \tan ^9(e+f x)}{9 f (a \sec (e+f x)+a)^{9/2}}-\frac{6 a^6 c^3 \tan ^7(e+f x)}{7 f (a \sec (e+f x)+a)^{7/2}}-\frac{2 a^5 c^3 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^4 c^3 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}-\frac{2 a^3 c^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"-1/1260*(a^2*c^3*(-2520*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x]^4 + (901 + 164*Cos[e + f*x] + 1004*Cos[2*(e + f*x)] + 68*Cos[3*(e + f*x)] + 383*Cos[4*(e + f*x)])*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]^4*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(f*Sqrt[-1 + Sec[e + f*x]])","A",1
58,1,124,177,0.9241484,"\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^2 \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^2,x]","-\frac{2 a^2 c^2 \tan \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) \sqrt{a (\sec (e+f x)+1)} \left((51 \cos (e+f x)+23 \cos (2 (e+f x))+23 \cos (3 (e+f x))+8) \sqrt{\sec (e+f x)-1}-105 \cos ^3(e+f x) \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)\right)}{105 f \sqrt{\sec (e+f x)-1}}","\frac{2 a^{5/2} c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}+\frac{2 a^6 c^2 \tan ^7(e+f x)}{7 f (a \sec (e+f x)+a)^{7/2}}+\frac{6 a^5 c^2 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^4 c^2 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}-\frac{2 a^3 c^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(-2*a^2*c^2*(-105*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x]^3 + (8 + 51*Cos[e + f*x] + 23*Cos[2*(e + f*x)] + 23*Cos[3*(e + f*x)])*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]^3*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(105*f*Sqrt[-1 + Sec[e + f*x]])","A",1
59,1,110,132,0.8132388,"\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x)) \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x]),x]","-\frac{a^2 c \tan \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \left((6 \cos (e+f x)+\cos (2 (e+f x))+3) \sqrt{\sec (e+f x)-1}-10 \cos ^2(e+f x) \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)\right)}{5 f \sqrt{\sec (e+f x)-1}}","\frac{2 a^{5/2} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}-\frac{2 a^5 c \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}-\frac{2 a^4 c \tan ^3(e+f x)}{f (a \sec (e+f x)+a)^{3/2}}-\frac{2 a^3 c \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"-1/5*(a^2*c*(-10*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x]^2 + (3 + 6*Cos[e + f*x] + Cos[2*(e + f*x)])*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]^2*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(f*Sqrt[-1 + Sec[e + f*x]])","A",1
60,1,96,103,0.7188532,"\int \frac{(a+a \sec (e+f x))^{5/2}}{c-c \sec (e+f x)} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x]),x]","\frac{2 a^3 \tan (e+f x) \sec (e+f x) \left((5 \cos (e+f x)-1) \sqrt{\sec (e+f x)-1}-(\cos (e+f x)-1) \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)\right)}{c f (\sec (e+f x)-1)^{3/2} \sqrt{a (\sec (e+f x)+1)}}","\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}-\frac{2 a^3 \tan (e+f x)}{c f \sqrt{a \sec (e+f x)+a}}+\frac{8 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c f}",1,"(2*a^3*(-(ArcTan[Sqrt[-1 + Sec[e + f*x]]]*(-1 + Cos[e + f*x])) + (-1 + 5*Cos[e + f*x])*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]*Tan[e + f*x])/(c*f*(-1 + Sec[e + f*x])^(3/2)*Sqrt[a*(1 + Sec[e + f*x])])","A",1
61,1,102,74,4.223507,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^2} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^2,x]","-\frac{\cos ^{\frac{5}{2}}(e+f x) \csc ^3\left(\frac{1}{2} (e+f x)\right) \sec ^5\left(\frac{1}{2} (e+f x)\right) (a (\sec (e+f x)+1))^{5/2} \left(4 \cos ^{\frac{3}{2}}(e+f x)-3 (1-\cos (e+f x))^{3/2} \sin ^{-1}\left(\sqrt{1-\cos (e+f x)}\right)\right)}{24 c^2 f}","\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^2 f}-\frac{8 a \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^2 f}",1,"-1/24*(Cos[e + f*x]^(5/2)*(-3*ArcSin[Sqrt[1 - Cos[e + f*x]]]*(1 - Cos[e + f*x])^(3/2) + 4*Cos[e + f*x]^(3/2))*Csc[(e + f*x)/2]^3*Sec[(e + f*x)/2]^5*(a*(1 + Sec[e + f*x]))^(5/2))/(c^2*f)","A",1
62,1,196,104,5.3308971,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^3} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^3,x]","\frac{a^2 \sqrt{\cos (e+f x)} \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(4 (20 \cos (e+f x)-15 \cos (2 (e+f x))-29) \, _2F_1\left(-\frac{5}{2},-\frac{1}{2};\frac{1}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)+30 \sqrt{1-\cos (e+f x)} (7 \cos (e+f x)-1) \cos ^2\left(\frac{1}{2} (e+f x)\right) \sin ^{-1}\left(\sqrt{1-\cos (e+f x)}\right)+5 \sin ^2(e+f x) \sqrt{\cos (e+f x)} (11 \cos (e+f x)+3 \cos (2 (e+f x)))\right)}{60 c^3 f (\cos (e+f x)-1)^3}","\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^3 f}+\frac{2 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^3 f}+\frac{8 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 c^3 f}",1,"(a^2*Sqrt[Cos[e + f*x]]*Sqrt[a*(1 + Sec[e + f*x])]*(30*ArcSin[Sqrt[1 - Cos[e + f*x]]]*Cos[(e + f*x)/2]^2*Sqrt[1 - Cos[e + f*x]]*(-1 + 7*Cos[e + f*x]) + 4*(-29 + 20*Cos[e + f*x] - 15*Cos[2*(e + f*x)])*Hypergeometric2F1[-5/2, -1/2, 1/2, 2*Sin[(e + f*x)/2]^2] + 5*Sqrt[Cos[e + f*x]]*(11*Cos[e + f*x] + 3*Cos[2*(e + f*x)])*Sin[e + f*x]^2)*Tan[(e + f*x)/2])/(60*c^3*f*(-1 + Cos[e + f*x])^3)","C",0
63,1,361,140,7.9771895,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^4} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^4,x]","-\frac{\sin ^8\left(\frac{e}{2}+\frac{f x}{2}\right) \sqrt{\frac{1}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}} \sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)} \csc ^7\left(\frac{1}{2} (e+f x)\right) \sec ^5\left(\frac{1}{2} (e+f x)\right) \sec ^{\frac{3}{2}}(e+f x) (a (\sec (e+f x)+1))^{5/2} \left(336 \sin ^2\left(\frac{1}{2} (e+f x)\right) \left(5 \sin ^4\left(\frac{1}{2} (e+f x)\right)-8 \sin ^2\left(\frac{1}{2} (e+f x)\right)+3\right) \, _2F_1\left(-\frac{5}{2},-\frac{1}{2};\frac{1}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)+4 \left(35 \sin ^4\left(\frac{1}{2} (e+f x)\right)-42 \sin ^2\left(\frac{1}{2} (e+f x)\right)+15\right) \, _2F_1\left(-\frac{7}{2},-\frac{3}{2};-\frac{1}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)-105 \left(3 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sqrt{\sin ^2\left(\frac{1}{2} (e+f x)\right)}\right) \sin ^2\left(\frac{1}{2} (e+f x)\right)^{3/2}+2 \left(5-4 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right) \sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)} \sin ^4\left(\frac{1}{2} (e+f x)\right)\right) \cos ^4\left(\frac{1}{2} (e+f x)\right)\right)}{210 f (c-c \sec (e+f x))^4}","\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^4 f}+\frac{2 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^4 f}-\frac{8 \cot ^7(e+f x) (a \sec (e+f x)+a)^{7/2}}{7 a c^4 f}-\frac{2 a \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^4 f}",1,"-1/210*(Csc[(e + f*x)/2]^7*Sec[(e + f*x)/2]^5*Sec[e + f*x]^(3/2)*(a*(1 + Sec[e + f*x]))^(5/2)*Sin[e/2 + (f*x)/2]^8*Sqrt[(1 - 2*Sin[(e + f*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[(e + f*x)/2]^2]*(336*Hypergeometric2F1[-5/2, -1/2, 1/2, 2*Sin[(e + f*x)/2]^2]*Sin[(e + f*x)/2]^2*(3 - 8*Sin[(e + f*x)/2]^2 + 5*Sin[(e + f*x)/2]^4) + 4*Hypergeometric2F1[-7/2, -3/2, -1/2, 2*Sin[(e + f*x)/2]^2]*(15 - 42*Sin[(e + f*x)/2]^2 + 35*Sin[(e + f*x)/2]^4) - 105*Cos[(e + f*x)/2]^4*(3*Sqrt[2]*ArcSin[Sqrt[2]*Sqrt[Sin[(e + f*x)/2]^2]]*(Sin[(e + f*x)/2]^2)^(3/2) + 2*Sin[(e + f*x)/2]^4*(5 - 4*Sin[(e + f*x)/2]^2)*Sqrt[1 - 2*Sin[(e + f*x)/2]^2])))/(f*(c - c*Sec[e + f*x])^4)","C",0
64,1,205,172,3.5607998,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^5} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^5,x]","\frac{a^2 \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(-15 (\cos (e+f x)-2 \cos (2 (e+f x))-\cos (3 (e+f x))+2) \, _3F_2\left(-\frac{7}{2},-\frac{3}{2},2;-\frac{1}{2},1;2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)+240 \sin ^2(e+f x) (2 \cos (e+f x)+1) \, _2F_1\left(-\frac{7}{2},-\frac{3}{2};-\frac{1}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)+(108 \cos (e+f x)+63 \cos (2 (e+f x))+109) \, _2F_1\left(-\frac{9}{2},-\frac{5}{2};-\frac{3}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}{315 c^5 f \cos ^{\frac{9}{2}}(e+f x) (\sec (e+f x)-1)^5}","\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c^5 f}+\frac{8 \cot ^9(e+f x) (a \sec (e+f x)+a)^{9/2}}{9 a^2 c^5 f}+\frac{2 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^5 f}+\frac{2 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 c^5 f}-\frac{2 a \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^5 f}",1,"(a^2*Sqrt[a*(1 + Sec[e + f*x])]*((109 + 108*Cos[e + f*x] + 63*Cos[2*(e + f*x)])*Hypergeometric2F1[-9/2, -5/2, -3/2, 2*Sin[(e + f*x)/2]^2] - 15*(2 + Cos[e + f*x] - 2*Cos[2*(e + f*x)] - Cos[3*(e + f*x)])*HypergeometricPFQ[{-7/2, -3/2, 2}, {-1/2, 1}, 2*Sin[(e + f*x)/2]^2] + 240*(1 + 2*Cos[e + f*x])*Hypergeometric2F1[-7/2, -3/2, -1/2, 2*Sin[(e + f*x)/2]^2]*Sin[e + f*x]^2)*Tan[(e + f*x)/2])/(315*c^5*f*Cos[e + f*x]^(9/2)*(-1 + Sec[e + f*x])^5)","C",0
65,1,153,185,1.4521192,"\int \frac{(c-c \sec (e+f x))^4}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(c - c*Sec[e + f*x])^4/Sqrt[a + a*Sec[e + f*x]],x]","\frac{c^4 \cot \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) \left(-155 \cos (e+f x)+96 \cos (2 (e+f x))-41 \cos (3 (e+f x))+20 \cos ^3(e+f x) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)-160 \sqrt{2} \cos ^3(e+f x) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)+100\right)}{10 f \sqrt{a (\sec (e+f x)+1)}}","\frac{2 a^2 c^4 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{16 \sqrt{2} c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{2 a c^4 \tan ^3(e+f x)}{f (a \sec (e+f x)+a)^{3/2}}+\frac{14 c^4 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(c^4*Cot[(e + f*x)/2]*(100 - 155*Cos[e + f*x] + 96*Cos[2*(e + f*x)] - 41*Cos[3*(e + f*x)] + 20*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x]^3*Sqrt[-1 + Sec[e + f*x]] - 160*Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Cos[e + f*x]^3*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]^3)/(10*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
66,1,166,152,1.331301,"\int \frac{(c-c \sec (e+f x))^3}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(c - c*Sec[e + f*x])^3/Sqrt[a + a*Sec[e + f*x]],x]","\frac{4 c^3 \cos \left(\frac{e}{2}\right) \cos (e) \cot \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \left(11 \cos (e+f x)-5 \cos (2 (e+f x))+3 \cos ^2(e+f x) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)-12 \sqrt{2} \cos ^2(e+f x) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)-6\right)}{3 f \left(\cos \left(\frac{e}{2}\right)+\cos \left(\frac{3 e}{2}\right)\right) \sqrt{a (\sec (e+f x)+1)}}","\frac{2 c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{8 \sqrt{2} c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{2 a c^3 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{6 c^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(4*c^3*Cos[e/2]*Cos[e]*Cot[(e + f*x)/2]*(-6 + 11*Cos[e + f*x] - 5*Cos[2*(e + f*x)] + 3*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x]^2*Sqrt[-1 + Sec[e + f*x]] - 12*Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Cos[e + f*x]^2*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]^2)/(3*f*(Cos[e/2] + Cos[(3*e)/2])*Sqrt[a*(1 + Sec[e + f*x])])","A",1
67,1,124,119,0.5062494,"\int \frac{(c-c \sec (e+f x))^2}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(c - c*Sec[e + f*x])^2/Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 c^2 \cot \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \left(-\cos (e+f x)+\cos (e+f x) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)-2 \sqrt{2} \cos (e+f x) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)+1\right)}{f \sqrt{a (\sec (e+f x)+1)}}","\frac{2 c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{4 \sqrt{2} c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}+\frac{2 c^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*c^2*Cot[(e + f*x)/2]*(1 - Cos[e + f*x] + ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x]*Sqrt[-1 + Sec[e + f*x]] - 2*Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Cos[e + f*x]*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x])/(f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
68,1,82,87,0.2968811,"\int \frac{c-c \sec (e+f x)}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(c - c*Sec[e + f*x])/Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 c \cot \left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)-1} \left(\tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)\right)}{f \sqrt{a (\sec (e+f x)+1)}}","\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{2 \sqrt{2} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}",1,"(2*c*(ArcTan[Sqrt[-1 + Sec[e + f*x]]] - Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]])*Cot[(e + f*x)/2]*Sqrt[-1 + Sec[e + f*x]])/(f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
69,1,101,121,0.5746094,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx","Integrate[1/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])),x]","\frac{\cot \left(\frac{1}{2} (e+f x)\right) \left(4 \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)-\sqrt{2} \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)+2\right)}{2 c f \sqrt{a (\sec (e+f x)+1)}}","\frac{\cot (e+f x) \sqrt{a \sec (e+f x)+a}}{a c f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} c f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{2} \sqrt{a} c f}",1,"(Cot[(e + f*x)/2]*(2 + 4*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Sqrt[-1 + Sec[e + f*x]] - Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Sqrt[-1 + Sec[e + f*x]]))/(2*c*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
70,1,5576,161,24.1170901,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^2} \, dx","Integrate[1/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^2),x]","\text{Result too large to show}","-\frac{\cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a^2 c^2 f}+\frac{3 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{2 a c^2 f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} c^2 f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{2 \sqrt{2} \sqrt{a} c^2 f}",1,"Result too large to show","C",0
71,1,5592,196,23.7421615,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^3} \, dx","Integrate[1/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^3),x]","\text{Result too large to show}","\frac{\cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 a^3 c^3 f}-\frac{\cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{2 a^2 c^3 f}+\frac{7 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{4 a c^3 f}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} c^3 f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{4 \sqrt{2} \sqrt{a} c^3 f}",1,"Result too large to show","C",0
72,1,196,203,1.5787492,"\int \frac{(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(c - c*Sec[e + f*x])^4/(a + a*Sec[e + f*x])^(3/2),x]","\frac{c^4 \csc \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \left(20 \cos (e+f x)-26 \cos (2 (e+f x))+28 \cos (3 (e+f x))+6 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{3}{2} (e+f x)\right)\right)^2 \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)+36 \sqrt{2} \left(\cos \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{3}{2} (e+f x)\right)\right)^2 \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)-22\right)}{12 a f \sqrt{a (\sec (e+f x)+1)}}","\frac{2 c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}+\frac{12 \sqrt{2} c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}+\frac{8 c^4 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}-\frac{14 c^4 \tan (e+f x)}{a f \sqrt{a \sec (e+f x)+a}}-\frac{a c^4 \sin (e+f x) \tan ^4(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{f (a \sec (e+f x)+a)^{5/2}}",1,"(c^4*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*(-22 + 20*Cos[e + f*x] - 26*Cos[2*(e + f*x)] + 28*Cos[3*(e + f*x)] + 6*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*(Cos[(e + f*x)/2] + Cos[(3*(e + f*x))/2])^2*Sqrt[-1 + Sec[e + f*x]] + 36*Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*(Cos[(e + f*x)/2] + Cos[(3*(e + f*x))/2])^2*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]^2)/(12*a*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
73,1,132,169,1.7724113,"\int \frac{(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(c - c*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^(3/2),x]","\frac{2 c^3 \tan \left(\frac{1}{2} (e+f x)\right) \left(-\sec (e+f x)+\cot ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)+\sqrt{2} \cot ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)-3\right)}{a f \sqrt{a (\sec (e+f x)+1)}}","\frac{2 c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}+\frac{2 \sqrt{2} c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}-\frac{4 c^3 \tan (e+f x)}{a f \sqrt{a \sec (e+f x)+a}}+\frac{c^3 \sin (e+f x) \tan ^2(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{f (a \sec (e+f x)+a)^{3/2}}",1,"(2*c^3*(-3 + ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cot[(e + f*x)/2]^2*Sqrt[-1 + Sec[e + f*x]] + Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Cot[(e + f*x)/2]^2*Sqrt[-1 + Sec[e + f*x]] - Sec[e + f*x])*Tan[(e + f*x)/2])/(a*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
74,1,128,119,1.1139987,"\int \frac{(c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(c - c*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^(3/2),x]","\frac{c^2 \cot \left(\frac{1}{2} (e+f x)\right) \left(\sec ^2\left(\frac{1}{2} (e+f x)\right) \left(\cos (e+f x)+(\cos (e+f x)+1) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)-1\right)-\sqrt{2} \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)\right)}{a f \sqrt{a (\sec (e+f x)+1)}}","\frac{2 c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}-\frac{\sqrt{2} c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}-\frac{2 c^2 \tan (e+f x)}{f (a \sec (e+f x)+a)^{3/2}}",1,"(c^2*Cot[(e + f*x)/2]*(Sec[(e + f*x)/2]^2*(-1 + Cos[e + f*x] + ArcTan[Sqrt[-1 + Sec[e + f*x]]]*(1 + Cos[e + f*x])*Sqrt[-1 + Sec[e + f*x]]) - Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Sqrt[-1 + Sec[e + f*x]]))/(a*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
75,1,130,113,1.0025048,"\int \frac{c-c \sec (e+f x)}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(c - c*Sec[e + f*x])/(a + a*Sec[e + f*x])^(3/2),x]","\frac{c \cot \left(\frac{1}{2} (e+f x)\right) \left(\sec ^2\left(\frac{1}{2} (e+f x)\right) \left(\cos (e+f x)+2 (\cos (e+f x)+1) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)-1\right)-3 \sqrt{2} \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)\right)}{2 a f \sqrt{a (\sec (e+f x)+1)}}","\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}-\frac{3 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{2} a^{3/2} f}-\frac{c \tan (e+f x)}{f (a \sec (e+f x)+a)^{3/2}}",1,"(c*Cot[(e + f*x)/2]*(Sec[(e + f*x)/2]^2*(-1 + Cos[e + f*x] + 2*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*(1 + Cos[e + f*x])*Sqrt[-1 + Sec[e + f*x]]) - 3*Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Sqrt[-1 + Sec[e + f*x]]))/(2*a*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
76,1,154,177,1.1849174,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])),x]","\frac{\sin ^2\left(\frac{1}{2} (e+f x)\right) \tan \left(\frac{1}{2} (e+f x)\right) \left(3 \cos (e+f x)-7 \sqrt{2} \cos ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)+8 (\cos (e+f x)+1) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)+1\right)}{2 a c f (\cos (e+f x)-1)^2 \sqrt{a (\sec (e+f x)+1)}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} c f}-\frac{7 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{4 \sqrt{2} a^{3/2} c f}+\frac{\cot (e+f x) \sqrt{a \sec (e+f x)+a}}{4 a^2 c f}+\frac{\cos (e+f x) \cot (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{a \sec (e+f x)+a}}{4 a^2 c f}",1,"((1 + 3*Cos[e + f*x] - 7*Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Cos[(e + f*x)/2]^2*Sqrt[-1 + Sec[e + f*x]] + 8*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*(1 + Cos[e + f*x])*Sqrt[-1 + Sec[e + f*x]])*Sin[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2*a*c*f*(-1 + Cos[e + f*x])^2*Sqrt[a*(1 + Sec[e + f*x])])","A",1
77,1,5612,214,24.0324146,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^2} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^2),x]","\text{Result too large to show}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} c^2 f}-\frac{9 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{8 \sqrt{2} a^{3/2} c^2 f}+\frac{\cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{12 a^3 c^2 f}-\frac{\cos (e+f x) \cot ^3(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) (a \sec (e+f x)+a)^{3/2}}{4 a^3 c^2 f}+\frac{7 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{8 a^2 c^2 f}",1,"Result too large to show","C",0
78,1,5629,249,23.976544,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^3} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^3),x]","\text{Result too large to show}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} c^3 f}-\frac{11 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{16 \sqrt{2} a^{3/2} c^3 f}-\frac{3 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{20 a^4 c^3 f}+\frac{\cos (e+f x) \cot ^5(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) (a \sec (e+f x)+a)^{5/2}}{4 a^4 c^3 f}-\frac{5 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{24 a^3 c^3 f}+\frac{21 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{16 a^2 c^3 f}",1,"Result too large to show","C",0
79,1,180,260,3.6423824,"\int \frac{(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^{5/2}} \, dx","Integrate[(c - c*Sec[e + f*x])^5/(a + a*Sec[e + f*x])^(5/2),x]","\frac{c^5 \cot \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \left((-30 \cos (e+f x)+52 \cos (2 (e+f x))-66 \cos (3 (e+f x))-37 \cos (4 (e+f x))+81) \sec ^4\left(\frac{1}{2} (e+f x)\right)+96 \cos ^2(e+f x) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)-1104 \sqrt{2} \cos ^2(e+f x) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)\right)}{48 a^2 f \sqrt{a (\sec (e+f x)+1)}}","\frac{2 c^5 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{23 \sqrt{2} c^5 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}+\frac{21 c^5 \tan (e+f x)}{a^2 f \sqrt{a \sec (e+f x)+a}}-\frac{19 c^5 \tan ^3(e+f x)}{6 a f (a \sec (e+f x)+a)^{3/2}}+\frac{a c^5 \sin ^2(e+f x) \tan ^5(e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)}{4 f (a \sec (e+f x)+a)^{7/2}}+\frac{3 c^5 \sin (e+f x) \tan ^4(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{4 f (a \sec (e+f x)+a)^{5/2}}",1,"(c^5*Cot[(e + f*x)/2]*((81 - 30*Cos[e + f*x] + 52*Cos[2*(e + f*x)] - 66*Cos[3*(e + f*x)] - 37*Cos[4*(e + f*x)])*Sec[(e + f*x)/2]^4 + 96*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x]^2*Sqrt[-1 + Sec[e + f*x]] - 1104*Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Cos[e + f*x]^2*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]^2)/(48*a^2*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
80,1,164,229,2.680204,"\int \frac{(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^{5/2}} \, dx","Integrate[(c - c*Sec[e + f*x])^4/(a + a*Sec[e + f*x])^(5/2),x]","\frac{c^4 \cot \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \left((19 \cos (e+f x)-12 \cos (2 (e+f x))-3 \cos (3 (e+f x))-4) \sec ^4\left(\frac{1}{2} (e+f x)\right)+32 \cos (e+f x) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)-88 \sqrt{2} \cos (e+f x) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)\right)}{16 a^2 f \sqrt{a (\sec (e+f x)+1)}}","\frac{2 c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{11 c^4 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{2} a^{5/2} f}+\frac{7 c^4 \tan (e+f x)}{2 a^2 f \sqrt{a \sec (e+f x)+a}}-\frac{c^4 \sin ^2(e+f x) \tan ^3(e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)}{4 f (a \sec (e+f x)+a)^{5/2}}-\frac{c^4 \sin (e+f x) \tan ^2(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{4 a f (a \sec (e+f x)+a)^{3/2}}",1,"(c^4*Cot[(e + f*x)/2]*((-4 + 19*Cos[e + f*x] - 12*Cos[2*(e + f*x)] - 3*Cos[3*(e + f*x)])*Sec[(e + f*x)/2]^4 + 32*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x]*Sqrt[-1 + Sec[e + f*x]] - 88*Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Cos[e + f*x]*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x])/(16*a^2*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
81,1,136,191,1.5903591,"\int \frac{(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^{5/2}} \, dx","Integrate[(c - c*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^(5/2),x]","-\frac{c^3 \cot \left(\frac{1}{2} (e+f x)\right) \left((8 \cos (e+f x)-3 \cos (2 (e+f x))-5) \sec ^4\left(\frac{1}{2} (e+f x)\right)-32 \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)+28 \sqrt{2} \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)\right)}{16 a^2 f \sqrt{a (\sec (e+f x)+1)}}","\frac{2 c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{7 c^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{2 \sqrt{2} a^{5/2} f}-\frac{c^3 \sin (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{4 a^2 f \sqrt{a \sec (e+f x)+a}}+\frac{c^3 \sin ^2(e+f x) \tan (e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)}{4 a f (a \sec (e+f x)+a)^{3/2}}",1,"-1/16*(c^3*Cot[(e + f*x)/2]*((-5 + 8*Cos[e + f*x] - 3*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4 - 32*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Sqrt[-1 + Sec[e + f*x]] + 28*Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Sqrt[-1 + Sec[e + f*x]]))/(a^2*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
82,1,136,189,1.5609952,"\int \frac{(c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx","Integrate[(c - c*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^(5/2),x]","-\frac{c^2 \cot \left(\frac{1}{2} (e+f x)\right) \left((8 \cos (e+f x)-7 \cos (2 (e+f x))-1) \sec ^4\left(\frac{1}{2} (e+f x)\right)-64 \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)+44 \sqrt{2} \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)\right)}{32 a^2 f \sqrt{a (\sec (e+f x)+1)}}","\frac{2 c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{11 c^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{4 \sqrt{2} a^{5/2} f}-\frac{3 c^2 \sin (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 a^2 f \sqrt{a \sec (e+f x)+a}}-\frac{c^2 \sin (e+f x) \cos (e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)}{4 a^2 f \sqrt{a \sec (e+f x)+a}}",1,"-1/32*(c^2*Cot[(e + f*x)/2]*((-1 + 8*Cos[e + f*x] - 7*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4 - 64*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Sqrt[-1 + Sec[e + f*x]] + 44*Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Sqrt[-1 + Sec[e + f*x]]))/(a^2*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
83,1,134,148,1.4901524,"\int \frac{c-c \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx","Integrate[(c - c*Sec[e + f*x])/(a + a*Sec[e + f*x])^(5/2),x]","-\frac{c \cot \left(\frac{1}{2} (e+f x)\right) \left((8 \cos (e+f x)-11 \cos (2 (e+f x))+3) \sec ^4\left(\frac{1}{2} (e+f x)\right)-128 \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)+92 \sqrt{2} \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)\right)}{64 a^2 f \sqrt{a (\sec (e+f x)+1)}}","\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{23 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{8 \sqrt{2} a^{5/2} f}-\frac{7 c \tan (e+f x)}{8 a f (a \sec (e+f x)+a)^{3/2}}-\frac{c \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{5/2}}",1,"-1/64*(c*Cot[(e + f*x)/2]*((3 + 8*Cos[e + f*x] - 11*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4 - 128*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Sqrt[-1 + Sec[e + f*x]] + 92*Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Sqrt[-1 + Sec[e + f*x]]))/(a^2*f*Sqrt[a*(1 + Sec[e + f*x])])","A",1
84,1,158,230,1.5189702,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])),x]","\frac{\tan ^3\left(\frac{1}{2} (e+f x)\right) \left(24 \cos (e+f x)+27 \cos (2 (e+f x))+512 \cos ^4\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\sqrt{\sec (e+f x)-1}\right)-284 \sqrt{2} \cos ^4\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right)+13\right)}{64 a^2 c f (\cos (e+f x)-1)^2 \sqrt{a (\sec (e+f x)+1)}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} c f}-\frac{71 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{32 \sqrt{2} a^{5/2} c f}-\frac{7 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{32 a^3 c f}+\frac{\cos ^2(e+f x) \cot (e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right) \sqrt{a \sec (e+f x)+a}}{16 a^3 c f}+\frac{13 \cos (e+f x) \cot (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{a \sec (e+f x)+a}}{32 a^3 c f}",1,"((13 + 24*Cos[e + f*x] + 27*Cos[2*(e + f*x)] + 512*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[(e + f*x)/2]^4*Sqrt[-1 + Sec[e + f*x]] - 284*Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Cos[(e + f*x)/2]^4*Sqrt[-1 + Sec[e + f*x]])*Tan[(e + f*x)/2]^3)/(64*a^2*c*f*(-1 + Cos[e + f*x])^2*Sqrt[a*(1 + Sec[e + f*x])])","A",1
85,1,5650,269,24.1729172,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^2} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^2),x]","\text{Result too large to show}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} c^2 f}-\frac{107 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{64 \sqrt{2} a^{5/2} c^2 f}+\frac{43 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{96 a^4 c^2 f}-\frac{\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right) (a \sec (e+f x)+a)^{3/2}}{16 a^4 c^2 f}-\frac{15 \cos (e+f x) \cot ^3(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) (a \sec (e+f x)+a)^{3/2}}{32 a^4 c^2 f}+\frac{21 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{64 a^3 c^2 f}",1,"Result too large to show","C",0
86,1,149,185,6.1364829,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2),x]","\frac{c^3 \csc \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)} \left(-18 \cos (2 (e+f x))+3 i f x \cos (3 (e+f x))+9 \left(-\log \left(1+e^{2 i (e+f x)}\right)+i f x+2\right) \cos (e+f x)-3 \log \left(1+e^{2 i (e+f x)}\right) \cos (3 (e+f x))-22\right)}{24 f}","\frac{a c^4 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c^3 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}-\frac{a c^2 \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}}-\frac{a c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt{a \sec (e+f x)+a}}",1,"(c^3*Csc[(e + f*x)/2]*(-22 - 18*Cos[2*(e + f*x)] + (3*I)*f*x*Cos[3*(e + f*x)] + 9*Cos[e + f*x]*(2 + I*f*x - Log[1 + E^((2*I)*(e + f*x))]) - 3*Cos[3*(e + f*x)]*Log[1 + E^((2*I)*(e + f*x))])*Sec[(e + f*x)/2]*Sec[e + f*x]^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(24*f)","C",1
87,1,162,139,2.310522,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2),x]","-\frac{c^2 e^{-3 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^3 \left(\cot \left(\frac{1}{2} (e+f x)\right)+i\right) \sec ^4(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)} \left(\log \left(1+e^{2 i (e+f x)}\right)+4 \cos (e+f x)+\left(\log \left(1+e^{2 i (e+f x)}\right)-i f x\right) \cos (2 (e+f x))-i f x-1\right)}{16 f \left(1+e^{i (e+f x)}\right)}","\frac{a c^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}-\frac{a c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}}",1,"-1/16*(c^2*(1 + E^((2*I)*(e + f*x)))^3*(I + Cot[(e + f*x)/2])*(-1 - I*f*x + 4*Cos[e + f*x] + Log[1 + E^((2*I)*(e + f*x))] + Cos[2*(e + f*x)]*((-I)*f*x + Log[1 + E^((2*I)*(e + f*x))]))*Sec[e + f*x]^4*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(E^((3*I)*(e + f*x))*(1 + E^(I*(e + f*x)))*f)","C",1
88,1,99,93,1.2436907,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2),x]","\frac{i c \left(\cot \left(\frac{1}{2} (e+f x)\right)+i\right) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)} \left(\left(f x+i \log \left(1+e^{2 i (e+f x)}\right)\right) \cos (e+f x)+i\right)}{f \left(1+e^{i (e+f x)}\right)}","\frac{a c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}",1,"(I*c*(I + Cot[(e + f*x)/2])*(I + Cos[e + f*x]*(f*x + I*Log[1 + E^((2*I)*(e + f*x))]))*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/((1 + E^(I*(e + f*x)))*f)","C",1
89,1,102,48,0.5817003,"\int \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]],x]","\frac{i e^{\frac{1}{2} i (e+f x)} \left(f x+i \log \left(1+e^{2 i (e+f x)}\right)\right) \cos (e+f x) \csc \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{f \left(1+e^{i (e+f x)}\right)}","\frac{a c \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(I*E^((I/2)*(e + f*x))*Cos[e + f*x]*Csc[(e + f*x)/2]*(f*x + I*Log[1 + E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/((1 + E^(I*(e + f*x)))*f)","C",1
90,1,86,51,0.9775602,"\int \frac{\sqrt{a+a \sec (e+f x)}}{\sqrt{c-c \sec (e+f x)}} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]/Sqrt[c - c*Sec[e + f*x]],x]","-\frac{\left(-1+e^{i (e+f x)}\right) \left(f x+2 i \log \left(1-e^{i (e+f x)}\right)\right) \sqrt{a (\sec (e+f x)+1)}}{f \left(1+e^{i (e+f x)}\right) \sqrt{c-c \sec (e+f x)}}","\frac{a \tan (e+f x) \log (1-\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"-(((-1 + E^(I*(e + f*x)))*(f*x + (2*I)*Log[1 - E^(I*(e + f*x))])*Sqrt[a*(1 + Sec[e + f*x])])/((1 + E^(I*(e + f*x)))*f*Sqrt[c - c*Sec[e + f*x]]))","C",1
91,1,107,96,1.0423311,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^(3/2),x]","\frac{\tan \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{a (\sec (e+f x)+1)} \left(-2 \log \left(1-e^{i (e+f x)}\right)+\left(2 \log \left(1-e^{i (e+f x)}\right)-i f x\right) \cos (e+f x)+i f x-1\right)}{f (c-c \sec (e+f x))^{3/2}}","\frac{a \tan (e+f x) \log (1-\cos (e+f x))}{c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}",1,"((-1 + I*f*x - 2*Log[1 - E^(I*(e + f*x))] + Cos[e + f*x]*((-I)*f*x + 2*Log[1 - E^(I*(e + f*x))]))*Sec[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(f*(c - c*Sec[e + f*x])^(3/2))","C",1
92,1,152,142,1.2535469,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(6 \log \left(1-e^{i (e+f x)}\right)+\left(-8 \log \left(1-e^{i (e+f x)}\right)+4 i f x-4\right) \cos (e+f x)+\left(2 \log \left(1-e^{i (e+f x)}\right)-i f x\right) \cos (2 (e+f x))-3 i f x+3\right)}{2 c^2 f (\cos (e+f x)-1)^2 \sqrt{c-c \sec (e+f x)}}","\frac{a \tan (e+f x) \log (1-\cos (e+f x))}{c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a \tan (e+f x)}{c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{a \tan (e+f x)}{2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}",1,"((3 - (3*I)*f*x + Cos[e + f*x]*(-4 + (4*I)*f*x - 8*Log[1 - E^(I*(e + f*x))]) + 6*Log[1 - E^(I*(e + f*x))] + Cos[2*(e + f*x)]*((-I)*f*x + 2*Log[1 - E^(I*(e + f*x))]))*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(2*c^2*f*(-1 + Cos[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]])","C",1
93,1,198,188,1.9421617,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{7/2}} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^(7/2),x]","\frac{\tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(-60 \log \left(1-e^{i (e+f x)}\right)-3 i f x \cos (3 (e+f x))+18 i \left(2 i \log \left(1-e^{i (e+f x)}\right)+f x+i\right) \cos (2 (e+f x))+6 \log \left(1-e^{i (e+f x)}\right) \cos (3 (e+f x))+9 \left(10 \log \left(1-e^{i (e+f x)}\right)-5 i f x+6\right) \cos (e+f x)+30 i f x-40\right)}{12 c^3 f (\cos (e+f x)-1)^3 \sqrt{c-c \sec (e+f x)}}","\frac{a \tan (e+f x) \log (1-\cos (e+f x))}{c^3 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a \tan (e+f x)}{c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{a \tan (e+f x)}{2 c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac{a \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}",1,"((-40 + (30*I)*f*x - (3*I)*f*x*Cos[3*(e + f*x)] + (18*I)*Cos[2*(e + f*x)]*(I + f*x + (2*I)*Log[1 - E^(I*(e + f*x))]) - 60*Log[1 - E^(I*(e + f*x))] + 6*Cos[3*(e + f*x)]*Log[1 - E^(I*(e + f*x))] + 9*Cos[e + f*x]*(6 - (5*I)*f*x + 10*Log[1 - E^(I*(e + f*x))]))*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(12*c^3*f*(-1 + Cos[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]])","C",1
94,1,157,190,1.2775828,"\int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{5/2} \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(5/2),x]","\frac{i a c^2 \csc \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)} \left(6 i \cos (2 (e+f x))+3 f x \cos (3 (e+f x))+\left(9 i \log \left(1+e^{2 i (e+f x)}\right)+9 f x+6 i\right) \cos (e+f x)+3 i \log \left(1+e^{2 i (e+f x)}\right) \cos (3 (e+f x))+2 i\right)}{24 f}","\frac{a^2 c^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^2 c^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}-\frac{a^2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}}+\frac{a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt{a \sec (e+f x)+a}}",1,"((I/24)*a*c^2*Csc[(e + f*x)/2]*(2*I + (6*I)*Cos[2*(e + f*x)] + 3*f*x*Cos[3*(e + f*x)] + Cos[e + f*x]*(6*I + 9*f*x + (9*I)*Log[1 + E^((2*I)*(e + f*x))]) + (3*I)*Cos[3*(e + f*x)]*Log[1 + E^((2*I)*(e + f*x))])*Sec[(e + f*x)/2]*Sec[e + f*x]^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/f","C",1
95,1,159,103,1.3911573,"\int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(3/2),x]","\frac{i a c e^{-2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2 \left(\cot \left(\frac{1}{2} (e+f x)\right)+i\right) \sec ^3(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)} \left(i \log \left(1+e^{2 i (e+f x)}\right)+\left(f x+i \log \left(1+e^{2 i (e+f x)}\right)\right) \cos (2 (e+f x))+f x+i\right)}{8 f \left(1+e^{i (e+f x)}\right)}","\frac{a^2 c^2 \tan ^3(e+f x)}{2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{a^2 c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"((I/8)*a*c*(1 + E^((2*I)*(e + f*x)))^2*(I + Cot[(e + f*x)/2])*(I + f*x + Cos[2*(e + f*x)]*(f*x + I*Log[1 + E^((2*I)*(e + f*x))]) + I*Log[1 + E^((2*I)*(e + f*x))])*Sec[e + f*x]^3*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(E^((2*I)*(e + f*x))*(1 + E^(I*(e + f*x)))*f)","C",1
96,1,128,93,0.7838623,"\int (a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)} \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]],x]","\frac{a e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right) \left(\cot \left(\frac{1}{2} (e+f x)\right)+i\right) \sec (e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)} \left(1+\left(i f x-\log \left(1+e^{2 i (e+f x)}\right)\right) \cos (e+f x)\right)}{2 f \left(1+e^{i (e+f x)}\right)}","\frac{a^2 c \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f \sqrt{c-c \sec (e+f x)}}",1,"(a*(1 + E^((2*I)*(e + f*x)))*(I + Cot[(e + f*x)/2])*(1 + Cos[e + f*x]*(I*f*x - Log[1 + E^((2*I)*(e + f*x))]))*Sec[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(2*E^(I*(e + f*x))*(1 + E^(I*(e + f*x)))*f)","C",1
97,1,105,104,1.2013338,"\int \frac{(a+a \sec (e+f x))^{3/2}}{\sqrt{c-c \sec (e+f x)}} \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)/Sqrt[c - c*Sec[e + f*x]],x]","-\frac{a \left(-1+e^{i (e+f x)}\right) \left(4 i \log \left(1-e^{i (e+f x)}\right)-i \log \left(1+e^{2 i (e+f x)}\right)+f x\right) \sqrt{a (\sec (e+f x)+1)}}{f \left(1+e^{i (e+f x)}\right) \sqrt{c-c \sec (e+f x)}}","\frac{2 a^2 \tan (e+f x) \log (1-\sec (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{a^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"-((a*(-1 + E^(I*(e + f*x)))*(f*x + (4*I)*Log[1 - E^(I*(e + f*x))] - I*Log[1 + E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sec[e + f*x])])/((1 + E^(I*(e + f*x)))*f*Sqrt[c - c*Sec[e + f*x]]))","C",1
98,1,115,100,0.684518,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^(3/2),x]","\frac{a \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(-2 \log \left(1-e^{i (e+f x)}\right)+\left(2 \log \left(1-e^{i (e+f x)}\right)-i f x\right) \cos (e+f x)+i f x-2\right)}{c f (\cos (e+f x)-1) \sqrt{c-c \sec (e+f x)}}","\frac{a^2 \tan (e+f x) \log (1-\cos (e+f x))}{c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{2 a^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}",1,"(a*(-2 + I*f*x - 2*Log[1 - E^(I*(e + f*x))] + Cos[e + f*x]*((-I)*f*x + 2*Log[1 - E^(I*(e + f*x))]))*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(c*f*(-1 + Cos[e + f*x])*Sqrt[c - c*Sec[e + f*x]])","C",1
99,1,153,146,1.2720676,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^(5/2),x]","\frac{a \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(6 \log \left(1-e^{i (e+f x)}\right)+\left(-8 \log \left(1-e^{i (e+f x)}\right)+4 i f x-6\right) \cos (e+f x)+\left(2 \log \left(1-e^{i (e+f x)}\right)-i f x\right) \cos (2 (e+f x))-3 i f x+4\right)}{2 c^2 f (\cos (e+f x)-1)^2 \sqrt{c-c \sec (e+f x)}}","\frac{a^2 \tan (e+f x) \log (1-\cos (e+f x))}{c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^2 \tan (e+f x)}{c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{a^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}",1,"(a*(4 - (3*I)*f*x + Cos[e + f*x]*(-6 + (4*I)*f*x - 8*Log[1 - E^(I*(e + f*x))]) + 6*Log[1 - E^(I*(e + f*x))] + Cos[2*(e + f*x)]*((-I)*f*x + 2*Log[1 - E^(I*(e + f*x))]))*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(2*c^2*f*(-1 + Cos[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]])","C",1
100,1,199,196,2.2022669,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{7/2}} \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^(7/2),x]","\frac{a \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(-60 \log \left(1-e^{i (e+f x)}\right)-3 i f x \cos (3 (e+f x))+6 i \left(6 i \log \left(1-e^{i (e+f x)}\right)+3 f x+4 i\right) \cos (2 (e+f x))+6 \log \left(1-e^{i (e+f x)}\right) \cos (3 (e+f x))+\left(90 \log \left(1-e^{i (e+f x)}\right)-45 i f x+66\right) \cos (e+f x)+30 i f x-50\right)}{12 c^3 f (\cos (e+f x)-1)^3 \sqrt{c-c \sec (e+f x)}}","\frac{a^2 \tan (e+f x) \log (1-\cos (e+f x))}{c^3 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^2 \tan (e+f x)}{c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{a^2 \tan (e+f x)}{2 c f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac{2 a^2 \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}",1,"(a*(-50 + (30*I)*f*x - (3*I)*f*x*Cos[3*(e + f*x)] + (6*I)*Cos[2*(e + f*x)]*(4*I + 3*f*x + (6*I)*Log[1 - E^(I*(e + f*x))]) - 60*Log[1 - E^(I*(e + f*x))] + 6*Cos[3*(e + f*x)]*Log[1 - E^(I*(e + f*x))] + Cos[e + f*x]*(66 - (45*I)*f*x + 90*Log[1 - E^(I*(e + f*x))]))*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(12*c^3*f*(-1 + Cos[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]])","C",1
101,1,164,153,1.5355245,"\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{5/2} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(5/2),x]","\frac{i a^2 c^2 \csc \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)} \left(3 i \log \left(1+e^{2 i (e+f x)}\right)+\left(f x+i \log \left(1+e^{2 i (e+f x)}\right)\right) \cos (4 (e+f x))+4 \left(i \log \left(1+e^{2 i (e+f x)}\right)+f x+i\right) \cos (2 (e+f x))+3 f x+2 i\right)}{16 f}","-\frac{a^3 c^3 \tan ^5(e+f x)}{4 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{a^3 c^3 \tan ^3(e+f x)}{2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{a^3 c^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"((I/16)*a^2*c^2*Csc[(e + f*x)/2]*(2*I + 3*f*x + Cos[4*(e + f*x)]*(f*x + I*Log[1 + E^((2*I)*(e + f*x))]) + 4*Cos[2*(e + f*x)]*(I + f*x + I*Log[1 + E^((2*I)*(e + f*x))]) + (3*I)*Log[1 + E^((2*I)*(e + f*x))])*Sec[(e + f*x)/2]*Sec[e + f*x]^3*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/f","C",1
102,1,149,190,1.195886,"\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2),x]","\frac{a^2 c \csc \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)} \left(6 \cos (2 (e+f x))+3 i f x \cos (3 (e+f x))+\left(-9 \log \left(1+e^{2 i (e+f x)}\right)+9 i f x-6\right) \cos (e+f x)-3 \log \left(1+e^{2 i (e+f x)}\right) \cos (3 (e+f x))+2\right)}{24 f}","\frac{a^3 c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^2 c^2 \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f \sqrt{c-c \sec (e+f x)}}-\frac{a c^2 \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sec (e+f x)}}+\frac{c^2 \tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sec (e+f x)}}",1,"(a^2*c*Csc[(e + f*x)/2]*(2 + 6*Cos[2*(e + f*x)] + (3*I)*f*x*Cos[3*(e + f*x)] + Cos[e + f*x]*(-6 + (9*I)*f*x - 9*Log[1 + E^((2*I)*(e + f*x))]) - 3*Cos[3*(e + f*x)]*Log[1 + E^((2*I)*(e + f*x))])*Sec[(e + f*x)/2]*Sec[e + f*x]^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(24*f)","C",1
103,1,164,139,1.4036824,"\int (a+a \sec (e+f x))^{5/2} \sqrt{c-c \sec (e+f x)} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]],x]","\frac{a^2 e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right) \left(\cot \left(\frac{1}{2} (e+f x)\right)+i\right) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)} \left(-\log \left(1+e^{2 i (e+f x)}\right)+4 \cos (e+f x)+\left(i f x-\log \left(1+e^{2 i (e+f x)}\right)\right) \cos (2 (e+f x))+i f x+1\right)}{4 f \left(1+e^{i (e+f x)}\right)}","\frac{a^3 c \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^2 c \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f \sqrt{c-c \sec (e+f x)}}-\frac{a c \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sec (e+f x)}}",1,"(a^2*(1 + E^((2*I)*(e + f*x)))*(I + Cot[(e + f*x)/2])*(1 + I*f*x + 4*Cos[e + f*x] + Cos[2*(e + f*x)]*(I*f*x - Log[1 + E^((2*I)*(e + f*x))]) - Log[1 + E^((2*I)*(e + f*x))])*Sec[e + f*x]^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(4*E^(I*(e + f*x))*(1 + E^(I*(e + f*x)))*f)","C",1
104,1,292,152,6.7638831,"\int \frac{(a+a \sec (e+f x))^{5/2}}{\sqrt{c-c \sec (e+f x)}} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)/Sqrt[c - c*Sec[e + f*x]],x]","\frac{\tan \left(\frac{e}{2}+\frac{f x}{2}\right) \sec (e+f x) (a (\sec (e+f x)+1))^{5/2} \sqrt{(\cos (e+f x)+1) \sec (e+f x)}}{f (\sec (e+f x)+1)^{5/2} \sqrt{c-c \sec (e+f x)}}+\frac{\sqrt{2} e^{\frac{1}{2} i (e+f x)} \sqrt{\frac{\left(1+e^{i (e+f x)}\right)^2}{1+e^{2 i (e+f x)}}} \left(8 \log \left(1-e^{i (e+f x)}\right)-3 \log \left(1+e^{2 i (e+f x)}\right)-i f x\right) \sin \left(\frac{e}{2}+\frac{f x}{2}\right) \sqrt{\sec (e+f x)} (a (\sec (e+f x)+1))^{5/2}}{f \left(1+e^{i (e+f x)}\right) \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} (\sec (e+f x)+1)^{5/2} \sqrt{c-c \sec (e+f x)}}","\frac{a^3 \tan (e+f x) \sec (e+f x)}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{4 a^3 \tan (e+f x) \log (1-\sec (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{a^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(Sqrt[2]*E^((I/2)*(e + f*x))*Sqrt[(1 + E^(I*(e + f*x)))^2/(1 + E^((2*I)*(e + f*x)))]*((-I)*f*x + 8*Log[1 - E^(I*(e + f*x))] - 3*Log[1 + E^((2*I)*(e + f*x))])*Sqrt[Sec[e + f*x]]*(a*(1 + Sec[e + f*x]))^(5/2)*Sin[e/2 + (f*x)/2])/((1 + E^(I*(e + f*x)))*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*f*(1 + Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]]) + (Sec[e + f*x]*Sqrt[(1 + Cos[e + f*x])*Sec[e + f*x]]*(a*(1 + Sec[e + f*x]))^(5/2)*Tan[e/2 + (f*x)/2])/(f*(1 + Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]])","C",1
105,1,111,96,1.2940773,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(3/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(-\log \left(1+e^{2 i (e+f x)}\right)+\left(\log \left(1+e^{2 i (e+f x)}\right)-i f x\right) \cos (e+f x)+i f x-4\right)}{c f (\cos (e+f x)-1) \sqrt{c-c \sec (e+f x)}}","\frac{a^3 \tan (e+f x) \log (\cos (e+f x))}{c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 a^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}",1,"(a^2*(-4 + I*f*x - Log[1 + E^((2*I)*(e + f*x))] + Cos[e + f*x]*((-I)*f*x + Log[1 + E^((2*I)*(e + f*x))]))*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(c*f*(-1 + Cos[e + f*x])*Sqrt[c - c*Sec[e + f*x]])","C",1
106,1,155,100,1.3859572,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{5/2}} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(5/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(6 \log \left(1-e^{i (e+f x)}\right)+\left(-8 \log \left(1-e^{i (e+f x)}\right)+4 i f x-8\right) \cos (e+f x)+\left(2 \log \left(1-e^{i (e+f x)}\right)-i f x\right) \cos (2 (e+f x))-3 i f x+4\right)}{2 c^2 f (\cos (e+f x)-1)^2 \sqrt{c-c \sec (e+f x)}}","\frac{a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{2 a^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}",1,"(a^2*(4 - (3*I)*f*x + Cos[e + f*x]*(-8 + (4*I)*f*x - 8*Log[1 - E^(I*(e + f*x))]) + 6*Log[1 - E^(I*(e + f*x))] + Cos[2*(e + f*x)]*((-I)*f*x + 2*Log[1 - E^(I*(e + f*x))]))*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(2*c^2*f*(-1 + Cos[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]])","C",1
107,1,202,148,2.6696623,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{7/2}} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(7/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(-60 \log \left(1-e^{i (e+f x)}\right)-3 i f x \cos (3 (e+f x))+6 i \left(6 i \log \left(1-e^{i (e+f x)}\right)+3 f x+5 i\right) \cos (2 (e+f x))+6 \log \left(1-e^{i (e+f x)}\right) \cos (3 (e+f x))+9 \left(10 \log \left(1-e^{i (e+f x)}\right)-5 i f x+8\right) \cos (e+f x)+30 i f x-58\right)}{12 c^3 f (\cos (e+f x)-1)^3 \sqrt{c-c \sec (e+f x)}}","\frac{a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^3 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^3 \tan (e+f x)}{c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{4 a^3 \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}",1,"(a^2*(-58 + (30*I)*f*x - (3*I)*f*x*Cos[3*(e + f*x)] + (6*I)*Cos[2*(e + f*x)]*(5*I + 3*f*x + (6*I)*Log[1 - E^(I*(e + f*x))]) - 60*Log[1 - E^(I*(e + f*x))] + 6*Cos[3*(e + f*x)]*Log[1 - E^(I*(e + f*x))] + 9*Cos[e + f*x]*(8 - (5*I)*f*x + 10*Log[1 - E^(I*(e + f*x))]))*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(12*c^3*f*(-1 + Cos[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]])","C",1
108,1,285,194,5.4978961,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{9/2}} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(9/2),x]","\frac{\sin ^9\left(\frac{1}{2} (e+f x)\right) \sec ^{\frac{9}{2}}(e+f x) (a (\sec (e+f x)+1))^{5/2} \left(\frac{(89 \cos (e+f x)-60 \cos (2 (e+f x))+23 \cos (3 (e+f x))-6 \cos (4 (e+f x))-54) \csc ^8\left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)} \sqrt{\sec (e+f x)+1}}{8 f}+\frac{16 \sqrt{2} e^{\frac{1}{2} i (e+f x)} \sqrt{\frac{\left(1+e^{i (e+f x)}\right)^2}{1+e^{2 i (e+f x)}}} \left(2 \log \left(1-e^{i (e+f x)}\right)-i f x\right)}{f \left(1+e^{i (e+f x)}\right) \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}}}\right)}{(\sec (e+f x)+1)^{5/2} (c-c \sec (e+f x))^{9/2}}","\frac{a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^4 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^3 \tan (e+f x)}{c^3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{a^3 \tan (e+f x)}{2 c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac{a^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{9/2}}",1,"(Sec[e + f*x]^(9/2)*(a*(1 + Sec[e + f*x]))^(5/2)*((16*Sqrt[2]*E^((I/2)*(e + f*x))*Sqrt[(1 + E^(I*(e + f*x)))^2/(1 + E^((2*I)*(e + f*x)))]*((-I)*f*x + 2*Log[1 - E^(I*(e + f*x))]))/((1 + E^(I*(e + f*x)))*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*f) + ((-54 + 89*Cos[e + f*x] - 60*Cos[2*(e + f*x)] + 23*Cos[3*(e + f*x)] - 6*Cos[4*(e + f*x)])*Csc[(e + f*x)/2]^8*Sec[(e + f*x)/2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]])/(8*f))*Sin[(e + f*x)/2]^9)/((1 + Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(9/2))","C",1
109,1,299,244,5.9860269,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(11/2),x]","\frac{\sin ^{11}\left(\frac{1}{2} (e+f x)\right) \sec ^{\frac{11}{2}}(e+f x) (a (\sec (e+f x)+1))^{5/2} \left(-\frac{(5612 \cos (e+f x)-5 (736 \cos (2 (e+f x))-367 \cos (3 (e+f x))+111 \cos (4 (e+f x))-21 \cos (5 (e+f x))+625)) \csc ^{10}\left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)} \sqrt{\sec (e+f x)+1}}{240 f}+\frac{32 i \sqrt{2} e^{\frac{1}{2} i (e+f x)} \sqrt{\frac{\left(1+e^{i (e+f x)}\right)^2}{1+e^{2 i (e+f x)}}} \left(f x+2 i \log \left(1-e^{i (e+f x)}\right)\right)}{f \left(1+e^{i (e+f x)}\right) \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}}}\right)}{(\sec (e+f x)+1)^{5/2} (c-c \sec (e+f x))^{11/2}}","\frac{a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^5 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^3 \tan (e+f x)}{c^4 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac{a^3 \tan (e+f x)}{2 c^3 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac{a^3 \tan (e+f x)}{3 c^2 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}-\frac{4 a^3 \tan (e+f x)}{5 f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{11/2}}",1,"(Sec[e + f*x]^(11/2)*(a*(1 + Sec[e + f*x]))^(5/2)*(((32*I)*Sqrt[2]*E^((I/2)*(e + f*x))*Sqrt[(1 + E^(I*(e + f*x)))^2/(1 + E^((2*I)*(e + f*x)))]*(f*x + (2*I)*Log[1 - E^(I*(e + f*x))]))/((1 + E^(I*(e + f*x)))*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*f) - ((5612*Cos[e + f*x] - 5*(625 + 736*Cos[2*(e + f*x)] - 367*Cos[3*(e + f*x)] + 111*Cos[4*(e + f*x)] - 21*Cos[5*(e + f*x)]))*Csc[(e + f*x)/2]^10*Sec[(e + f*x)/2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]])/(240*f))*Sin[(e + f*x)/2]^11)/((1 + Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(11/2))","C",1
110,1,153,204,16.3477414,"\int \frac{(c-c \sec (e+f x))^{7/2}}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(c - c*Sec[e + f*x])^(7/2)/Sqrt[a + a*Sec[e + f*x]],x]","\frac{c^3 \cot \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{c-c \sec (e+f x)} \left(-16 \log \left(1+e^{i (e+f x)}\right)+7 \log \left(1+e^{2 i (e+f x)}\right)+8 \cos (e+f x)+\left(-16 \log \left(1+e^{i (e+f x)}\right)+7 \log \left(1+e^{2 i (e+f x)}\right)+i f x\right) \cos (2 (e+f x))+i f x-1\right)}{2 f \sqrt{a (\sec (e+f x)+1)}}","\frac{c^4 \tan (e+f x) \sec ^2(e+f x)}{2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 c^4 \tan (e+f x) \sec (e+f x)}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{8 c^4 \tan (e+f x) \log (\sec (e+f x)+1)}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{c^4 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(c^3*Cot[(e + f*x)/2]*(-1 + I*f*x + 8*Cos[e + f*x] - 16*Log[1 + E^(I*(e + f*x))] + 7*Log[1 + E^((2*I)*(e + f*x))] + Cos[2*(e + f*x)]*(I*f*x - 16*Log[1 + E^(I*(e + f*x))] + 7*Log[1 + E^((2*I)*(e + f*x))]))*Sec[e + f*x]^2*Sqrt[c - c*Sec[e + f*x]])/(2*f*Sqrt[a*(1 + Sec[e + f*x])])","C",1
111,1,181,151,4.0172374,"\int \frac{(c-c \sec (e+f x))^{5/2}}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(c - c*Sec[e + f*x])^(5/2)/Sqrt[a + a*Sec[e + f*x]],x]","\frac{c^2 e^{-3 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^3 \cos \left(\frac{1}{2} (e+f x)\right) \cot \left(\frac{1}{2} (e+f x)\right) \sec ^4(e+f x) \sqrt{c-c \sec (e+f x)} \left(1+\left(-8 \log \left(1+e^{i (e+f x)}\right)+3 \log \left(1+e^{2 i (e+f x)}\right)+i f x\right) \cos (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+i \sin \left(\frac{1}{2} (e+f x)\right)\right)}{4 f \left(1+e^{i (e+f x)}\right) \sqrt{a (\sec (e+f x)+1)}}","-\frac{c^3 \tan (e+f x) \sec (e+f x)}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{4 c^3 \tan (e+f x) \log (\sec (e+f x)+1)}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{c^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(c^2*(1 + E^((2*I)*(e + f*x)))^3*Cos[(e + f*x)/2]*Cot[(e + f*x)/2]*(1 + Cos[e + f*x]*(I*f*x - 8*Log[1 + E^(I*(e + f*x))] + 3*Log[1 + E^((2*I)*(e + f*x))]))*Sec[e + f*x]^4*Sqrt[c - c*Sec[e + f*x]]*(Cos[(e + f*x)/2] + I*Sin[(e + f*x)/2]))/(4*E^((3*I)*(e + f*x))*(1 + E^(I*(e + f*x)))*f*Sqrt[a*(1 + Sec[e + f*x])])","C",1
112,1,103,102,9.7150463,"\int \frac{(c-c \sec (e+f x))^{3/2}}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(c - c*Sec[e + f*x])^(3/2)/Sqrt[a + a*Sec[e + f*x]],x]","-\frac{c \left(1+e^{i (e+f x)}\right) \left(4 i \log \left(1+e^{i (e+f x)}\right)-i \log \left(1+e^{2 i (e+f x)}\right)+f x\right) \sqrt{c-c \sec (e+f x)}}{f \left(-1+e^{i (e+f x)}\right) \sqrt{a (\sec (e+f x)+1)}}","\frac{2 c^2 \tan (e+f x) \log (\sec (e+f x)+1)}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"-((c*(1 + E^(I*(e + f*x)))*(f*x + (4*I)*Log[1 + E^(I*(e + f*x))] - I*Log[1 + E^((2*I)*(e + f*x))])*Sqrt[c - c*Sec[e + f*x]])/((-1 + E^(I*(e + f*x)))*f*Sqrt[a*(1 + Sec[e + f*x])]))","C",1
113,1,127,49,0.9811299,"\int \frac{\sqrt{c-c \sec (e+f x)}}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[Sqrt[c - c*Sec[e + f*x]]/Sqrt[a + a*Sec[e + f*x]],x]","-\frac{\left(1+e^{i (e+f x)}\right) \sqrt{\frac{c \left(-1+e^{i (e+f x)}\right)^2}{1+e^{2 i (e+f x)}}} \left(f x+2 i \log \left(1+e^{i (e+f x)}\right)\right)}{f \left(-1+e^{i (e+f x)}\right) \sqrt{\frac{a \left(1+e^{i (e+f x)}\right)^2}{1+e^{2 i (e+f x)}}}}","\frac{c \tan (e+f x) \log (\cos (e+f x)+1)}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"-(((1 + E^(I*(e + f*x)))*Sqrt[(c*(-1 + E^(I*(e + f*x)))^2)/(1 + E^((2*I)*(e + f*x)))]*(f*x + (2*I)*Log[1 + E^(I*(e + f*x))]))/((-1 + E^(I*(e + f*x)))*Sqrt[(a*(1 + E^(I*(e + f*x)))^2)/(1 + E^((2*I)*(e + f*x)))]*f))","C",1
114,1,104,46,1.0851613,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}} \, dx","Integrate[1/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]),x]","-\frac{2 \left(-1+e^{i (e+f x)}\right) \left(f x+i \log \left(1-e^{2 i (e+f x)}\right)\right) \cos ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x)}{f \left(1+e^{i (e+f x)}\right) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","\frac{\tan (e+f x) \log (\sin (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(-2*(-1 + E^(I*(e + f*x)))*Cos[(e + f*x)/2]^2*(f*x + I*Log[1 - E^((2*I)*(e + f*x))])*Sec[e + f*x])/((1 + E^(I*(e + f*x)))*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
115,1,143,168,8.6733095,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}} \, dx","Integrate[1/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)),x]","\frac{\tan (e+f x) \left(-3 \log \left(1-e^{i (e+f x)}\right)-\log \left(1+e^{i (e+f x)}\right)+\left(3 \log \left(1-e^{i (e+f x)}\right)+\log \left(1+e^{i (e+f x)}\right)-2 i f x\right) \cos (e+f x)+2 i f x-1\right)}{2 c f (\cos (e+f x)-1) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","\frac{\tan (e+f x)}{2 c f (1-\cos (e+f x)) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{3 \tan (e+f x) \log (1-\cos (e+f x))}{4 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x)+1)}{4 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"((-1 + (2*I)*f*x - 3*Log[1 - E^(I*(e + f*x))] - Log[1 + E^(I*(e + f*x))] + Cos[e + f*x]*((-2*I)*f*x + 3*Log[1 - E^(I*(e + f*x))] + Log[1 + E^(I*(e + f*x))]))*Tan[e + f*x])/(2*c*f*(-1 + Cos[e + f*x])*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
116,1,194,274,1.9299752,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}} \, dx","Integrate[1/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)),x]","\frac{\tan (e+f x) \left(21 \log \left(1-e^{i (e+f x)}\right)+3 \log \left(1+e^{i (e+f x)}\right)+\left(-28 \log \left(1-e^{i (e+f x)}\right)-4 \log \left(1+e^{i (e+f x)}\right)+16 i f x-10\right) \cos (e+f x)+\left(7 \log \left(1-e^{i (e+f x)}\right)+\log \left(1+e^{i (e+f x)}\right)-4 i f x\right) \cos (2 (e+f x))-12 i f x+8\right)}{8 c^2 f (\cos (e+f x)-1)^2 \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","-\frac{3 \tan (e+f x)}{4 c^2 f (1-\sec (e+f x)) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{4 c^2 f (1-\sec (e+f x))^2 \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{7 \tan (e+f x) \log (1-\sec (e+f x))}{8 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\sec (e+f x)+1)}{8 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"((8 - (12*I)*f*x + 21*Log[1 - E^(I*(e + f*x))] + Cos[e + f*x]*(-10 + (16*I)*f*x - 28*Log[1 - E^(I*(e + f*x))] - 4*Log[1 + E^(I*(e + f*x))]) + 3*Log[1 + E^(I*(e + f*x))] + Cos[2*(e + f*x)]*((-4*I)*f*x + 7*Log[1 - E^(I*(e + f*x))] + Log[1 + E^(I*(e + f*x))]))*Tan[e + f*x])/(8*c^2*f*(-1 + Cos[e + f*x])^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
117,1,204,215,2.2746138,"\int \frac{(c-c \sec (e+f x))^{7/2}}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(c - c*Sec[e + f*x])^(7/2)/(a + a*Sec[e + f*x])^(3/2),x]","\frac{c^3 \cot \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{c-c \sec (e+f x)} \left(8 \log \left(1+e^{i (e+f x)}\right)-5 \log \left(1+e^{2 i (e+f x)}\right)+2 \left(8 \log \left(1+e^{i (e+f x)}\right)-5 \log \left(1+e^{2 i (e+f x)}\right)+i f x-9\right) \cos (e+f x)+\left(8 \log \left(1+e^{i (e+f x)}\right)-5 \log \left(1+e^{2 i (e+f x)}\right)+i f x\right) \cos (2 (e+f x))+i f x-2\right)}{2 a f (\cos (e+f x)+1) \sqrt{a (\sec (e+f x)+1)}}","\frac{c^4 \tan (e+f x) \sec (e+f x)}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{8 c^4 \tan (e+f x)}{a f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 c^4 \tan (e+f x) \log (\sec (e+f x)+1)}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{c^4 \tan (e+f x) \log (\cos (e+f x))}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(c^3*Cot[(e + f*x)/2]*(-2 + I*f*x + 8*Log[1 + E^(I*(e + f*x))] + 2*Cos[e + f*x]*(-9 + I*f*x + 8*Log[1 + E^(I*(e + f*x))] - 5*Log[1 + E^((2*I)*(e + f*x))]) + Cos[2*(e + f*x)]*(I*f*x + 8*Log[1 + E^(I*(e + f*x))] - 5*Log[1 + E^((2*I)*(e + f*x))]) - 5*Log[1 + E^((2*I)*(e + f*x))])*Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(2*a*f*(1 + Cos[e + f*x])*Sqrt[a*(1 + Sec[e + f*x])])","C",1
118,1,116,96,0.7437457,"\int \frac{(c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(c - c*Sec[e + f*x])^(5/2)/(a + a*Sec[e + f*x])^(3/2),x]","\frac{i c^2 \cot \left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)} \left(i \log \left(1+e^{2 i (e+f x)}\right)+\left(f x+i \log \left(1+e^{2 i (e+f x)}\right)\right) \cos (e+f x)+f x+4 i\right)}{a f (\cos (e+f x)+1) \sqrt{a (\sec (e+f x)+1)}}","\frac{c^3 \tan (e+f x) \log (\cos (e+f x))}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 c^3 \tan (e+f x)}{f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}",1,"(I*c^2*Cot[(e + f*x)/2]*(4*I + f*x + Cos[e + f*x]*(f*x + I*Log[1 + E^((2*I)*(e + f*x))]) + I*Log[1 + E^((2*I)*(e + f*x))])*Sqrt[c - c*Sec[e + f*x]])/(a*f*(1 + Cos[e + f*x])*Sqrt[a*(1 + Sec[e + f*x])])","C",1
119,1,114,98,1.1398271,"\int \frac{(c-c \sec (e+f x))^{3/2}}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(c - c*Sec[e + f*x])^(3/2)/(a + a*Sec[e + f*x])^(3/2),x]","\frac{i c \cot \left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)} \left(2 i \log \left(1+e^{i (e+f x)}\right)+\left(f x+2 i \log \left(1+e^{i (e+f x)}\right)\right) \cos (e+f x)+f x+2 i\right)}{a f (\cos (e+f x)+1) \sqrt{a (\sec (e+f x)+1)}}","\frac{c^2 \tan (e+f x) \log (\cos (e+f x)+1)}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{2 c^2 \tan (e+f x)}{f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}",1,"(I*c*Cot[(e + f*x)/2]*(2*I + f*x + Cos[e + f*x]*(f*x + (2*I)*Log[1 + E^(I*(e + f*x))]) + (2*I)*Log[1 + E^(I*(e + f*x))])*Sqrt[c - c*Sec[e + f*x]])/(a*f*(1 + Cos[e + f*x])*Sqrt[a*(1 + Sec[e + f*x])])","C",1
120,1,106,94,0.6329489,"\int \frac{\sqrt{c-c \sec (e+f x)}}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[Sqrt[c - c*Sec[e + f*x]]/(a + a*Sec[e + f*x])^(3/2),x]","\frac{i \cot \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{c-c \sec (e+f x)} \left(2 i \log \left(1+e^{i (e+f x)}\right)+\left(f x+2 i \log \left(1+e^{i (e+f x)}\right)\right) \cos (e+f x)+f x+i\right)}{f (a (\sec (e+f x)+1))^{3/2}}","\frac{c \tan (e+f x) \log (\cos (e+f x)+1)}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{c \tan (e+f x)}{f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}",1,"(I*Cot[(e + f*x)/2]*(I + f*x + Cos[e + f*x]*(f*x + (2*I)*Log[1 + E^(I*(e + f*x))]) + (2*I)*Log[1 + E^(I*(e + f*x))])*Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(f*(a*(1 + Sec[e + f*x]))^(3/2))","C",1
121,1,141,215,1.4181668,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)}} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]]),x]","\frac{\tan (e+f x) \left(\log \left(1-e^{i (e+f x)}\right)+3 \log \left(1+e^{i (e+f x)}\right)+\left(\log \left(1-e^{i (e+f x)}\right)+3 \log \left(1+e^{i (e+f x)}\right)-2 i f x\right) \cos (e+f x)-2 i f x+1\right)}{2 a f (\cos (e+f x)+1) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","-\frac{\tan (e+f x)}{2 a f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (1-\sec (e+f x))}{4 a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{3 \tan (e+f x) \log (\sec (e+f x)+1)}{4 a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"((1 - (2*I)*f*x + Log[1 - E^(I*(e + f*x))] + 3*Log[1 + E^(I*(e + f*x))] + Cos[e + f*x]*((-2*I)*f*x + Log[1 - E^(I*(e + f*x))] + 3*Log[1 + E^(I*(e + f*x))]))*Tan[e + f*x])/(2*a*f*(1 + Cos[e + f*x])*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
122,1,121,101,1.5281563,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2}} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(3/2)),x]","\frac{\tan (e+f x) \sec ^2(e+f x) \left(\log \left(1-e^{2 i (e+f x)}\right)+\left(i f x-\log \left(1-e^{2 i (e+f x)}\right)\right) \cos (2 (e+f x))-i f x+1\right)}{2 c f (\sec (e+f x)-1) (a (\sec (e+f x)+1))^{3/2} \sqrt{c-c \sec (e+f x)}}","\frac{\cot (e+f x)}{2 a c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\sin (e+f x))}{a c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"((1 - I*f*x + Cos[2*(e + f*x)]*(I*f*x - Log[1 - E^((2*I)*(e + f*x))]) + Log[1 - E^((2*I)*(e + f*x))])*Sec[e + f*x]^2*Tan[e + f*x])/(2*c*f*(-1 + Sec[e + f*x])*(a*(1 + Sec[e + f*x]))^(3/2)*Sqrt[c - c*Sec[e + f*x]])","C",1
123,1,275,347,2.6188544,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{5/2}} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(5/2)),x]","\frac{\tan (e+f x) \left(22 \log \left(1-e^{i (e+f x)}\right)+10 \log \left(1+e^{i (e+f x)}\right)-8 i f x \cos (3 (e+f x))+11 \log \left(1-e^{i (e+f x)}\right) \cos (3 (e+f x))+\left(-11 \log \left(1-e^{i (e+f x)}\right)-5 \log \left(1+e^{i (e+f x)}\right)+8 i f x-12\right) \cos (e+f x)+2 \left(-11 \log \left(1-e^{i (e+f x)}\right)-5 \log \left(1+e^{i (e+f x)}\right)+8 i f x-5\right) \cos (2 (e+f x))+5 \log \left(1+e^{i (e+f x)}\right) \cos (3 (e+f x))-16 i f x+14\right)}{32 a c^2 f (\cos (e+f x)-1)^2 (\cos (e+f x)+1) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","-\frac{\tan (e+f x)}{2 a c^2 f (1-\sec (e+f x)) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{8 a c^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{8 a c^2 f (1-\sec (e+f x))^2 \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{11 \tan (e+f x) \log (1-\sec (e+f x))}{16 a c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{5 \tan (e+f x) \log (\sec (e+f x)+1)}{16 a c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{a c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"((14 - (16*I)*f*x - (8*I)*f*x*Cos[3*(e + f*x)] + 22*Log[1 - E^(I*(e + f*x))] + 11*Cos[3*(e + f*x)]*Log[1 - E^(I*(e + f*x))] + Cos[e + f*x]*(-12 + (8*I)*f*x - 11*Log[1 - E^(I*(e + f*x))] - 5*Log[1 + E^(I*(e + f*x))]) + 2*Cos[2*(e + f*x)]*(-5 + (8*I)*f*x - 11*Log[1 - E^(I*(e + f*x))] - 5*Log[1 + E^(I*(e + f*x))]) + 10*Log[1 + E^(I*(e + f*x))] + 5*Cos[3*(e + f*x)]*Log[1 + E^(I*(e + f*x))])*Tan[e + f*x])/(32*a*c^2*f*(-1 + Cos[e + f*x])^2*(1 + Cos[e + f*x])*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
124,1,157,220,2.5401085,"\int \frac{(c-c \sec (e+f x))^{7/2}}{(a+a \sec (e+f x))^{5/2}} \, dx","Integrate[(c - c*Sec[e + f*x])^(7/2)/(a + a*Sec[e + f*x])^(5/2),x]","\frac{c^3 \cot \left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)} \left(4 \left(-4 \log \left(1+e^{i (e+f x)}\right)+\log \left(1+e^{2 i (e+f x)}\right)+i f x-2\right) \cos (e+f x)+\left(-4 \log \left(1+e^{i (e+f x)}\right)+\log \left(1+e^{2 i (e+f x)}\right)+i f x\right) (\cos (2 (e+f x))+3)\right)}{2 a^2 f (\cos (e+f x)+1)^2 \sqrt{a (\sec (e+f x)+1)}}","\frac{4 c^4 \tan (e+f x)}{a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{4 c^4 \tan (e+f x)}{a^2 f (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{2 c^4 \tan (e+f x) \log (\sec (e+f x)+1)}{a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{c^4 \tan (e+f x) \log (\cos (e+f x))}{a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(c^3*Cot[(e + f*x)/2]*(4*Cos[e + f*x]*(-2 + I*f*x - 4*Log[1 + E^(I*(e + f*x))] + Log[1 + E^((2*I)*(e + f*x))]) + (3 + Cos[2*(e + f*x)])*(I*f*x - 4*Log[1 + E^(I*(e + f*x))] + Log[1 + E^((2*I)*(e + f*x))]))*Sqrt[c - c*Sec[e + f*x]])/(2*a^2*f*(1 + Cos[e + f*x])^2*Sqrt[a*(1 + Sec[e + f*x])])","C",1
125,1,154,98,1.571839,"\int \frac{(c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^{5/2}} \, dx","Integrate[(c - c*Sec[e + f*x])^(5/2)/(a + a*Sec[e + f*x])^(5/2),x]","\frac{i c^2 \cot \left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)} \left(6 i \log \left(1+e^{i (e+f x)}\right)+\left(f x+2 i \log \left(1+e^{i (e+f x)}\right)\right) \cos (2 (e+f x))+4 \left(2 i \log \left(1+e^{i (e+f x)}\right)+f x+2 i\right) \cos (e+f x)+3 f x+4 i\right)}{2 a^2 f (\cos (e+f x)+1)^2 \sqrt{a (\sec (e+f x)+1)}}","\frac{c^3 \tan (e+f x) \log (\cos (e+f x)+1)}{a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{2 c^3 \tan (e+f x)}{f (a \sec (e+f x)+a)^{5/2} \sqrt{c-c \sec (e+f x)}}",1,"((I/2)*c^2*Cot[(e + f*x)/2]*(4*I + 3*f*x + Cos[2*(e + f*x)]*(f*x + (2*I)*Log[1 + E^(I*(e + f*x))]) + 4*Cos[e + f*x]*(2*I + f*x + (2*I)*Log[1 + E^(I*(e + f*x))]) + (6*I)*Log[1 + E^(I*(e + f*x))])*Sqrt[c - c*Sec[e + f*x]])/(a^2*f*(1 + Cos[e + f*x])^2*Sqrt[a*(1 + Sec[e + f*x])])","C",1
126,1,152,144,0.866076,"\int \frac{(c-c \sec (e+f x))^{3/2}}{(a+a \sec (e+f x))^{5/2}} \, dx","Integrate[(c - c*Sec[e + f*x])^(3/2)/(a + a*Sec[e + f*x])^(5/2),x]","\frac{i c \cot \left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)} \left(6 i \log \left(1+e^{i (e+f x)}\right)+\left(f x+2 i \log \left(1+e^{i (e+f x)}\right)\right) \cos (2 (e+f x))+\left(8 i \log \left(1+e^{i (e+f x)}\right)+4 f x+6 i\right) \cos (e+f x)+3 f x+4 i\right)}{2 a^2 f (\cos (e+f x)+1)^2 \sqrt{a (\sec (e+f x)+1)}}","\frac{c^2 \tan (e+f x) \log (\cos (e+f x)+1)}{a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{c^2 \tan (e+f x)}{a f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}-\frac{c^2 \tan (e+f x)}{f (a \sec (e+f x)+a)^{5/2} \sqrt{c-c \sec (e+f x)}}",1,"((I/2)*c*Cot[(e + f*x)/2]*(4*I + 3*f*x + Cos[2*(e + f*x)]*(f*x + (2*I)*Log[1 + E^(I*(e + f*x))]) + Cos[e + f*x]*(6*I + 4*f*x + (8*I)*Log[1 + E^(I*(e + f*x))]) + (6*I)*Log[1 + E^(I*(e + f*x))])*Sqrt[c - c*Sec[e + f*x]])/(a^2*f*(1 + Cos[e + f*x])^2*Sqrt[a*(1 + Sec[e + f*x])])","C",1
127,1,151,140,0.6749872,"\int \frac{\sqrt{c-c \sec (e+f x)}}{(a+a \sec (e+f x))^{5/2}} \, dx","Integrate[Sqrt[c - c*Sec[e + f*x]]/(a + a*Sec[e + f*x])^(5/2),x]","\frac{i \cot \left(\frac{1}{2} (e+f x)\right) \sqrt{c-c \sec (e+f x)} \left(6 i \log \left(1+e^{i (e+f x)}\right)+\left(f x+2 i \log \left(1+e^{i (e+f x)}\right)\right) \cos (2 (e+f x))+4 \left(2 i \log \left(1+e^{i (e+f x)}\right)+f x+i\right) \cos (e+f x)+3 f x+3 i\right)}{2 a^2 f (\cos (e+f x)+1)^2 \sqrt{a (\sec (e+f x)+1)}}","\frac{c \tan (e+f x) \log (\cos (e+f x)+1)}{a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{c \tan (e+f x)}{a f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}-\frac{c \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{5/2} \sqrt{c-c \sec (e+f x)}}",1,"((I/2)*Cot[(e + f*x)/2]*(3*I + 3*f*x + Cos[2*(e + f*x)]*(f*x + (2*I)*Log[1 + E^(I*(e + f*x))]) + 4*Cos[e + f*x]*(I + f*x + (2*I)*Log[1 + E^(I*(e + f*x))]) + (6*I)*Log[1 + E^(I*(e + f*x))])*Sqrt[c - c*Sec[e + f*x]])/(a^2*f*(1 + Cos[e + f*x])^2*Sqrt[a*(1 + Sec[e + f*x])])","C",1
128,1,195,270,1.6594344,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} \sqrt{c-c \sec (e+f x)}} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]]),x]","\frac{\tan (e+f x) \left(3 \log \left(1-e^{i (e+f x)}\right)+21 \log \left(1+e^{i (e+f x)}\right)+\left(\log \left(1-e^{i (e+f x)}\right)+7 \log \left(1+e^{i (e+f x)}\right)-4 i f x\right) \cos (2 (e+f x))+2 \left(2 \log \left(1-e^{i (e+f x)}\right)+14 \log \left(1+e^{i (e+f x)}\right)-8 i f x+5\right) \cos (e+f x)-12 i f x+8\right)}{8 a^2 f (\cos (e+f x)+1)^2 \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","-\frac{3 \tan (e+f x)}{4 a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{4 a^2 f (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (1-\sec (e+f x))}{8 a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{7 \tan (e+f x) \log (\sec (e+f x)+1)}{8 a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"((8 - (12*I)*f*x + 3*Log[1 - E^(I*(e + f*x))] + 21*Log[1 + E^(I*(e + f*x))] + Cos[2*(e + f*x)]*((-4*I)*f*x + Log[1 - E^(I*(e + f*x))] + 7*Log[1 + E^(I*(e + f*x))]) + 2*Cos[e + f*x]*(5 - (8*I)*f*x + 2*Log[1 - E^(I*(e + f*x))] + 14*Log[1 + E^(I*(e + f*x))]))*Tan[e + f*x])/(8*a^2*f*(1 + Cos[e + f*x])^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
129,1,275,345,2.40036,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2}} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2)),x]","\frac{\tan (e+f x) \left(-10 \log \left(1-e^{i (e+f x)}\right)-22 \log \left(1+e^{i (e+f x)}\right)-8 i f x \cos (3 (e+f x))+5 \log \left(1-e^{i (e+f x)}\right) \cos (3 (e+f x))+\left(-5 \log \left(1-e^{i (e+f x)}\right)-11 \log \left(1+e^{i (e+f x)}\right)+8 i f x-12\right) \cos (e+f x)+11 \log \left(1+e^{i (e+f x)}\right) \cos (3 (e+f x))+2 \left(5 \log \left(1-e^{i (e+f x)}\right)+11 \log \left(1+e^{i (e+f x)}\right)-8 i f x+5\right) \cos (2 (e+f x))+16 i f x-14\right)}{32 a^2 c f (\cos (e+f x)-1) (\cos (e+f x)+1)^2 \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","-\frac{\tan (e+f x)}{8 a^2 c f (1-\sec (e+f x)) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{2 a^2 c f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x)}{8 a^2 c f (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{5 \tan (e+f x) \log (1-\sec (e+f x))}{16 a^2 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{11 \tan (e+f x) \log (\sec (e+f x)+1)}{16 a^2 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\cos (e+f x))}{a^2 c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"((-14 + (16*I)*f*x - (8*I)*f*x*Cos[3*(e + f*x)] - 10*Log[1 - E^(I*(e + f*x))] + 5*Cos[3*(e + f*x)]*Log[1 - E^(I*(e + f*x))] + Cos[e + f*x]*(-12 + (8*I)*f*x - 5*Log[1 - E^(I*(e + f*x))] - 11*Log[1 + E^(I*(e + f*x))]) - 22*Log[1 + E^(I*(e + f*x))] + 11*Cos[3*(e + f*x)]*Log[1 + E^(I*(e + f*x))] + 2*Cos[2*(e + f*x)]*(5 - (8*I)*f*x + 5*Log[1 - E^(I*(e + f*x))] + 11*Log[1 + E^(I*(e + f*x))]))*Tan[e + f*x])/(32*a^2*c*f*(-1 + Cos[e + f*x])*(1 + Cos[e + f*x])^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
130,1,149,151,2.19166,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{5/2}} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(5/2)),x]","\frac{\csc ^3(e+f x) \sec (e+f x) \left(3 \log \left(1-e^{2 i (e+f x)}\right)+\left(-4 \log \left(1-e^{2 i (e+f x)}\right)+4 i f x-4\right) \cos (2 (e+f x))+\left(\log \left(1-e^{2 i (e+f x)}\right)-i f x\right) \cos (4 (e+f x))-3 i f x+2\right)}{8 a^2 c^2 f \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}","-\frac{\cot ^3(e+f x)}{4 a^2 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\cot (e+f x)}{2 a^2 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \log (\sin (e+f x))}{a^2 c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}",1,"(Csc[e + f*x]^3*(2 - (3*I)*f*x + Cos[2*(e + f*x)]*(-4 + (4*I)*f*x - 4*Log[1 - E^((2*I)*(e + f*x))]) + 3*Log[1 - E^((2*I)*(e + f*x))] + Cos[4*(e + f*x)]*((-I)*f*x + Log[1 - E^((2*I)*(e + f*x))]))*Sec[e + f*x])/(8*a^2*c^2*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])","C",1
131,0,0,92,1.1033156,"\int (1+\sec (e+f x))^m (c-c \sec (e+f x))^n \, dx","Integrate[(1 + Sec[e + f*x])^m*(c - c*Sec[e + f*x])^n,x]","\int (1+\sec (e+f x))^m (c-c \sec (e+f x))^n \, dx","\frac{2^{m+\frac{1}{2}} \tan (e+f x) (c-c \sec (e+f x))^n F_1\left(n+\frac{1}{2};\frac{1}{2}-m,1;n+\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{\sec (e+f x)+1}}",1,"Integrate[(1 + Sec[e + f*x])^m*(c - c*Sec[e + f*x])^n, x]","F",-1
132,0,0,109,0.4449151,"\int (a+a \sec (e+f x))^m (c-c \sec (e+f x))^n \, dx","Integrate[(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^n,x]","\int (a+a \sec (e+f x))^m (c-c \sec (e+f x))^n \, dx","\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (1-\sec (e+f x))^{\frac{1}{2}-n} (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{n-1} F_1\left(m+\frac{1}{2};\frac{1}{2}-n,1;m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right)}{f (2 m+1)}",1,"Integrate[(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^n, x]","F",-1
133,0,0,101,3.2419375,"\int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^n \, dx","Integrate[(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^n,x]","\int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^n \, dx","\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (a \sec (e+f x)+a)^3 (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left(\frac{7}{2};\frac{1}{2}-n,1;\frac{9}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right) (c-c \sec (e+f x))^{n-1}}{7 f}",1,"Integrate[(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^n, x]","F",-1
134,0,0,101,1.5260999,"\int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^n \, dx","Integrate[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^n,x]","\int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^n \, dx","\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (a \sec (e+f x)+a)^2 (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left(\frac{5}{2};\frac{1}{2}-n,1;\frac{7}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right) (c-c \sec (e+f x))^{n-1}}{5 f}",1,"Integrate[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^n, x]","F",-1
135,0,0,99,1.5980403,"\int (a+a \sec (e+f x)) (c-c \sec (e+f x))^n \, dx","Integrate[(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^n,x]","\int (a+a \sec (e+f x)) (c-c \sec (e+f x))^n \, dx","\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (a \sec (e+f x)+a) (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left(\frac{3}{2};\frac{1}{2}-n,1;\frac{5}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right) (c-c \sec (e+f x))^{n-1}}{3 f}",1,"Integrate[(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^n, x]","F",-1
136,0,0,99,0.9989467,"\int \frac{(c-c \sec (e+f x))^n}{a+a \sec (e+f x)} \, dx","Integrate[(c - c*Sec[e + f*x])^n/(a + a*Sec[e + f*x]),x]","\int \frac{(c-c \sec (e+f x))^n}{a+a \sec (e+f x)} \, dx","-\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left(-\frac{1}{2};\frac{1}{2}-n,1;\frac{1}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right) (c-c \sec (e+f x))^{n-1}}{f (a \sec (e+f x)+a)}",1,"Integrate[(c - c*Sec[e + f*x])^n/(a + a*Sec[e + f*x]), x]","F",-1
137,0,0,101,1.7263981,"\int \frac{(c-c \sec (e+f x))^n}{(a+a \sec (e+f x))^2} \, dx","Integrate[(c - c*Sec[e + f*x])^n/(a + a*Sec[e + f*x])^2,x]","\int \frac{(c-c \sec (e+f x))^n}{(a+a \sec (e+f x))^2} \, dx","-\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left(-\frac{3}{2};\frac{1}{2}-n,1;-\frac{1}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right) (c-c \sec (e+f x))^{n-1}}{3 f (a \sec (e+f x)+a)^2}",1,"Integrate[(c - c*Sec[e + f*x])^n/(a + a*Sec[e + f*x])^2, x]","F",-1
138,0,0,172,8.5404779,"\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^n \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^n,x]","\int (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^n \, dx","\frac{2 a^3 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{6 a^3 \tan (e+f x) (c-c \sec (e+f x))^n}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}-\frac{2 a^3 \tan (e+f x) (c-c \sec (e+f x))^{n+1}}{c f (2 n+3) \sqrt{a \sec (e+f x)+a}}",1,"Integrate[(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^n, x]","F",-1
139,0,0,119,11.825595,"\int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^n \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^n,x]","\int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^n \, dx","\frac{2 a^2 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (c-c \sec (e+f x))^n}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}",1,"Integrate[(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^n, x]","F",-1
140,0,0,68,0.142538,"\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^n \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^n,x]","\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^n \, dx","\frac{2 a \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}",1,"Integrate[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^n, x]","F",-1
141,0,0,139,1.3628179,"\int \frac{(c-c \sec (e+f x))^n}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(c - c*Sec[e + f*x])^n/Sqrt[a + a*Sec[e + f*x]],x]","\int \frac{(c-c \sec (e+f x))^n}{\sqrt{a+a \sec (e+f x)}} \, dx","\frac{2 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}",1,"Integrate[(c - c*Sec[e + f*x])^n/Sqrt[a + a*Sec[e + f*x]], x]","F",-1
142,0,0,205,1.7649811,"\int \frac{(c-c \sec (e+f x))^n}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(c - c*Sec[e + f*x])^n/(a + a*Sec[e + f*x])^(3/2),x]","\int \frac{(c-c \sec (e+f x))^n}{(a+a \sec (e+f x))^{3/2}} \, dx","-\frac{(5-2 n) \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{4 a f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right)}{a f (2 n+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x) (c-c \sec (e+f x))^n}{2 a f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}",1,"Integrate[(c - c*Sec[e + f*x])^n/(a + a*Sec[e + f*x])^(3/2), x]","F",-1
143,1,133,91,0.6519023,"\int \frac{\sqrt{a+a \sec (e+f x)}}{c+c \sec (e+f x)} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]/(c + c*Sec[e + f*x]),x]","-\frac{i \sqrt{1+e^{2 i (e+f x)}} \sqrt{a (\sec (e+f x)+1)} \left(\sinh ^{-1}\left(e^{i (e+f x)}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{-1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right)-\tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)\right)}{c f \left(1+e^{i (e+f x)}\right)}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}-\frac{\sqrt{2} \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{c f}",1,"((-I)*Sqrt[1 + E^((2*I)*(e + f*x))]*(ArcSinh[E^(I*(e + f*x))] - Sqrt[2]*ArcTanh[(-1 + E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])] - ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]])*Sqrt[a*(1 + Sec[e + f*x])])/(c*(1 + E^(I*(e + f*x)))*f)","C",1
144,1,810,231,18.2281471,"\int \frac{(c+d \sec (e+f x))^{3/2}}{a+a \sec (e+f x)} \, dx","Integrate[(c + d*Sec[e + f*x])^(3/2)/(a + a*Sec[e + f*x]),x]","\frac{(c+d \sec (e+f x))^{3/2} \left(2 \sec \left(\frac{1}{2} (e+f x)\right) \left(d \sin \left(\frac{1}{2} (e+f x)\right)-c \sin \left(\frac{1}{2} (e+f x)\right)\right)-2 (d-c) \sin (e+f x)\right) \cos ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{f (d+c \cos (e+f x)) (\sec (e+f x) a+a)}+\frac{2 (c+d \sec (e+f x))^{3/2} \left(c^2 \tan ^5\left(\frac{1}{2} (e+f x)\right)+d^2 \tan ^5\left(\frac{1}{2} (e+f x)\right)-2 c d \tan ^5\left(\frac{1}{2} (e+f x)\right)-2 c^2 \tan ^3\left(\frac{1}{2} (e+f x)\right)+2 c d \tan ^3\left(\frac{1}{2} (e+f x)\right)-4 c^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{c-d}{c+d}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{-c \tan ^2\left(\frac{1}{2} (e+f x)\right)+d \tan ^2\left(\frac{1}{2} (e+f x)\right)+c+d}{c+d}} \tan ^2\left(\frac{1}{2} (e+f x)\right)+c^2 \tan \left(\frac{1}{2} (e+f x)\right)-d^2 \tan \left(\frac{1}{2} (e+f x)\right)+\left(c^2-d^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{c-d}{c+d}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{-c \tan ^2\left(\frac{1}{2} (e+f x)\right)+d \tan ^2\left(\frac{1}{2} (e+f x)\right)+c+d}{c+d}}+2 c (c-d) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{c-d}{c+d}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{-c \tan ^2\left(\frac{1}{2} (e+f x)\right)+d \tan ^2\left(\frac{1}{2} (e+f x)\right)+c+d}{c+d}}-4 c^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{c-d}{c+d}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{-c \tan ^2\left(\frac{1}{2} (e+f x)\right)+d \tan ^2\left(\frac{1}{2} (e+f x)\right)+c+d}{c+d}}\right) \cos ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{f (d+c \cos (e+f x))^{3/2} \sqrt{\sec (e+f x)} (\sec (e+f x) a+a) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)}} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)^{3/2} \sqrt{\frac{-c \tan ^2\left(\frac{1}{2} (e+f x)\right)+d \tan ^2\left(\frac{1}{2} (e+f x)\right)+c+d}{\tan ^2\left(\frac{1}{2} (e+f x)\right)+1}}}","-\frac{2 c \cot (e+f x) \sqrt{-\frac{d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt{\frac{d (\sec (e+f x)+1)}{c+d \sec (e+f x)}} (c+d \sec (e+f x)) \Pi \left(\frac{c}{c+d};\sin ^{-1}\left(\frac{\sqrt{c+d}}{\sqrt{c+d \sec (e+f x)}}\right)|\frac{c-d}{c+d}\right)}{a f \sqrt{c+d}}-\frac{(c-d) \sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{c+d \sec (e+f x)} E\left(\sin ^{-1}\left(\frac{\tan (e+f x)}{\sec (e+f x)+1}\right)|\frac{c-d}{c+d}\right)}{a f \sqrt{\frac{c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}}",1,"(Cos[e/2 + (f*x)/2]^2*(c + d*Sec[e + f*x])^(3/2)*(2*Sec[(e + f*x)/2]*(-(c*Sin[(e + f*x)/2]) + d*Sin[(e + f*x)/2]) - 2*(-c + d)*Sin[e + f*x]))/(f*(d + c*Cos[e + f*x])*(a + a*Sec[e + f*x])) + (2*Cos[e/2 + (f*x)/2]^2*(c + d*Sec[e + f*x])^(3/2)*(c^2*Tan[(e + f*x)/2] - d^2*Tan[(e + f*x)/2] - 2*c^2*Tan[(e + f*x)/2]^3 + 2*c*d*Tan[(e + f*x)/2]^3 + c^2*Tan[(e + f*x)/2]^5 - 2*c*d*Tan[(e + f*x)/2]^5 + d^2*Tan[(e + f*x)/2]^5 - 4*c^2*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(c + d - c*Tan[(e + f*x)/2]^2 + d*Tan[(e + f*x)/2]^2)/(c + d)] - 4*c^2*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)]*Tan[(e + f*x)/2]^2*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(c + d - c*Tan[(e + f*x)/2]^2 + d*Tan[(e + f*x)/2]^2)/(c + d)] + (c^2 - d^2)*EllipticE[ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(c + d - c*Tan[(e + f*x)/2]^2 + d*Tan[(e + f*x)/2]^2)/(c + d)] + 2*c*(c - d)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(c + d - c*Tan[(e + f*x)/2]^2 + d*Tan[(e + f*x)/2]^2)/(c + d)]))/(f*(d + c*Cos[e + f*x])^(3/2)*Sqrt[Sec[e + f*x]]*(a + a*Sec[e + f*x])*Sqrt[(1 - Tan[(e + f*x)/2]^2)^(-1)]*(-1 + Tan[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]^2)^(3/2)*Sqrt[(c + d - c*Tan[(e + f*x)/2]^2 + d*Tan[(e + f*x)/2]^2)/(1 + Tan[(e + f*x)/2]^2)])","B",1
145,1,178,225,8.7461713,"\int \frac{\sqrt{c+d \sec (e+f x)}}{a+a \sec (e+f x)} \, dx","Integrate[Sqrt[c + d*Sec[e + f*x]]/(a + a*Sec[e + f*x]),x]","-\frac{4 \cos ^4\left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{c+d \sec (e+f x)} \left(2 (c-d) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{c-d}{c+d}\right)+(c+d) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{c-d}{c+d}\right)-4 c \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{c-d}{c+d}\right)\right)}{a f (c+d) (\cos (e+f x)+1)^2 \sqrt{\frac{c \cos (e+f x)+d}{(c+d) (\cos (e+f x)+1)}}}","-\frac{2 \cot (e+f x) \sqrt{-\frac{d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt{\frac{d (\sec (e+f x)+1)}{c+d \sec (e+f x)}} (c+d \sec (e+f x)) \Pi \left(\frac{c}{c+d};\sin ^{-1}\left(\frac{\sqrt{c+d}}{\sqrt{c+d \sec (e+f x)}}\right)|\frac{c-d}{c+d}\right)}{a f \sqrt{c+d}}-\frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{c+d \sec (e+f x)} E\left(\sin ^{-1}\left(\frac{\tan (e+f x)}{\sec (e+f x)+1}\right)|\frac{c-d}{c+d}\right)}{a f \sqrt{\frac{c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}}",1,"(-4*Cos[(e + f*x)/2]^4*((c + d)*EllipticE[ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)] + 2*(c - d)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)] - 4*c*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)])*Sqrt[(1 + Sec[e + f*x])^(-1)]*Sqrt[c + d*Sec[e + f*x]])/(a*(c + d)*f*(1 + Cos[e + f*x])^2*Sqrt[(d + c*Cos[e + f*x])/((c + d)*(1 + Cos[e + f*x]))])","A",1
146,1,187,319,12.7082883,"\int \frac{1}{(a+a \sec (e+f x)) \sqrt{c+d \sec (e+f x)}} \, dx","Integrate[1/((a + a*Sec[e + f*x])*Sqrt[c + d*Sec[e + f*x]]),x]","\frac{2 \cos ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sec (e+f x) \sqrt{\frac{c \cos (e+f x)+d}{(c+d) (\cos (e+f x)+1)}} \left(2 (c-2 d) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{c-d}{c+d}\right)+(c+d) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{c-d}{c+d}\right)+4 (d-c) \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{c-d}{c+d}\right)\right)}{a f (d-c) \sqrt{c+d \sec (e+f x)}}","\frac{2 \sqrt{c+d} \cot (e+f x) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (\sec (e+f x)+1)}{c-d}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right)|\frac{c+d}{c-d}\right)}{a f (c-d)}-\frac{2 \sqrt{c+d} \cot (e+f x) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (\sec (e+f x)+1)}{c-d}} \Pi \left(\frac{c+d}{c};\sin ^{-1}\left(\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right)|\frac{c+d}{c-d}\right)}{a c f}-\frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{c+d \sec (e+f x)} E\left(\sin ^{-1}\left(\frac{\tan (e+f x)}{\sec (e+f x)+1}\right)|\frac{c-d}{c+d}\right)}{a f (c-d) \sqrt{\frac{c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}}",1,"(2*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[(d + c*Cos[e + f*x])/((c + d)*(1 + Cos[e + f*x]))]*((c + d)*EllipticE[ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)] + 2*(c - 2*d)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)] + 4*(-c + d)*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)])*Sec[e + f*x])/(a*(-c + d)*f*Sqrt[c + d*Sec[e + f*x]])","A",1
147,1,587,271,14.2964934,"\int \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^4 \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^4,x]","\frac{\cos ^4(e+f x) \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} (c+d \sec (e+f x))^4 \left(\frac{4}{105} \sec (e+f x) \left(105 c^2 d^2 \sin \left(\frac{1}{2} (e+f x)\right)+56 c d^3 \sin \left(\frac{1}{2} (e+f x)\right)+12 d^4 \sin \left(\frac{1}{2} (e+f x)\right)\right)+\frac{8}{105} d \left(105 c^3+105 c^2 d+56 c d^2+12 d^3\right) \sin \left(\frac{1}{2} (e+f x)\right)+\frac{4}{35} \sec ^2(e+f x) \left(14 c d^3 \sin \left(\frac{1}{2} (e+f x)\right)+3 d^4 \sin \left(\frac{1}{2} (e+f x)\right)\right)+\frac{2}{7} d^4 \sin \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x)\right)}{f (c \cos (e+f x)+d)^4}-\frac{8 \left(-3-2 \sqrt{2}\right) c^4 \cos ^4\left(\frac{1}{4} (e+f x)\right) \sqrt{\frac{\left(10-7 \sqrt{2}\right) \cos \left(\frac{1}{2} (e+f x)\right)-5 \sqrt{2}+7}{\cos \left(\frac{1}{2} (e+f x)\right)+1}} \sqrt{\frac{-\left(\left(\sqrt{2}-2\right) \cos \left(\frac{1}{2} (e+f x)\right)\right)+\sqrt{2}-1}{\cos \left(\frac{1}{2} (e+f x)\right)+1}} \left(\left(\sqrt{2}-2\right) \cos \left(\frac{1}{2} (e+f x)\right)-\sqrt{2}+1\right) \cos ^3(e+f x) \sqrt{-\tan ^2\left(\frac{1}{4} (e+f x)\right)-2 \sqrt{2}+3} \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \sqrt{\left(\left(2+\sqrt{2}\right) \cos \left(\frac{1}{2} (e+f x)\right)-\sqrt{2}-1\right) \sec ^2\left(\frac{1}{4} (e+f x)\right)} (c+d \sec (e+f x))^4 \left(F\left(\sin ^{-1}\left(\frac{\tan \left(\frac{1}{4} (e+f x)\right)}{\sqrt{3-2 \sqrt{2}}}\right)|17-12 \sqrt{2}\right)-2 \Pi \left(-3+2 \sqrt{2};\sin ^{-1}\left(\frac{\tan \left(\frac{1}{4} (e+f x)\right)}{\sqrt{3-2 \sqrt{2}}}\right)|17-12 \sqrt{2}\right)\right)}{f (c \cos (e+f x)+d)^4}","\frac{2 a^{3/2} c^4 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{2 d^4 \tan (e+f x) (a-a \sec (e+f x))^3}{7 a^2 f \sqrt{a \sec (e+f x)+a}}-\frac{2 d^2 \left(6 c^2+8 c d+3 d^2\right) \tan (e+f x) (a-a \sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a d (2 c+d) \left(2 c^2+2 c d+d^2\right) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 (4 c+3 d) \tan (e+f x) (a-a \sec (e+f x))^2}{5 a f \sqrt{a \sec (e+f x)+a}}",1,"(Cos[e + f*x]^4*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])]*(c + d*Sec[e + f*x])^4*((8*d*(105*c^3 + 105*c^2*d + 56*c*d^2 + 12*d^3)*Sin[(e + f*x)/2])/105 + (2*d^4*Sec[e + f*x]^3*Sin[(e + f*x)/2])/7 + (4*Sec[e + f*x]^2*(14*c*d^3*Sin[(e + f*x)/2] + 3*d^4*Sin[(e + f*x)/2]))/35 + (4*Sec[e + f*x]*(105*c^2*d^2*Sin[(e + f*x)/2] + 56*c*d^3*Sin[(e + f*x)/2] + 12*d^4*Sin[(e + f*x)/2]))/105))/(f*(d + c*Cos[e + f*x])^4) - (8*(-3 - 2*Sqrt[2])*c^4*Cos[(e + f*x)/4]^4*Sqrt[(7 - 5*Sqrt[2] + (10 - 7*Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*(1 - Sqrt[2] + (-2 + Sqrt[2])*Cos[(e + f*x)/2])*Cos[e + f*x]^3*(EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 2*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]])*Sqrt[(-1 - Sqrt[2] + (2 + Sqrt[2])*Cos[(e + f*x)/2])*Sec[(e + f*x)/4]^2]*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])]*(c + d*Sec[e + f*x])^4*Sqrt[3 - 2*Sqrt[2] - Tan[(e + f*x)/4]^2])/(f*(d + c*Cos[e + f*x])^4)","C",0
148,1,517,205,14.1862061,"\int \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^3 \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^3,x]","\frac{\cos ^3(e+f x) \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} (c+d \sec (e+f x))^3 \left(\frac{2}{15} d \left(45 c^2+30 c d+8 d^2\right) \sin \left(\frac{1}{2} (e+f x)\right)+\frac{2}{15} \sec (e+f x) \left(15 c d^2 \sin \left(\frac{1}{2} (e+f x)\right)+4 d^3 \sin \left(\frac{1}{2} (e+f x)\right)\right)+\frac{2}{5} d^3 \sin \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x)\right)}{f (c \cos (e+f x)+d)^3}-\frac{8 \left(-3-2 \sqrt{2}\right) c^3 \cos ^4\left(\frac{1}{4} (e+f x)\right) \sqrt{\frac{\left(10-7 \sqrt{2}\right) \cos \left(\frac{1}{2} (e+f x)\right)-5 \sqrt{2}+7}{\cos \left(\frac{1}{2} (e+f x)\right)+1}} \sqrt{\frac{-\left(\left(\sqrt{2}-2\right) \cos \left(\frac{1}{2} (e+f x)\right)\right)+\sqrt{2}-1}{\cos \left(\frac{1}{2} (e+f x)\right)+1}} \left(\left(\sqrt{2}-2\right) \cos \left(\frac{1}{2} (e+f x)\right)-\sqrt{2}+1\right) \cos ^2(e+f x) \sqrt{-\tan ^2\left(\frac{1}{4} (e+f x)\right)-2 \sqrt{2}+3} \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \sqrt{\left(\left(2+\sqrt{2}\right) \cos \left(\frac{1}{2} (e+f x)\right)-\sqrt{2}-1\right) \sec ^2\left(\frac{1}{4} (e+f x)\right)} (c+d \sec (e+f x))^3 \left(F\left(\sin ^{-1}\left(\frac{\tan \left(\frac{1}{4} (e+f x)\right)}{\sqrt{3-2 \sqrt{2}}}\right)|17-12 \sqrt{2}\right)-2 \Pi \left(-3+2 \sqrt{2};\sin ^{-1}\left(\frac{\tan \left(\frac{1}{4} (e+f x)\right)}{\sqrt{3-2 \sqrt{2}}}\right)|17-12 \sqrt{2}\right)\right)}{f (c \cos (e+f x)+d)^3}","\frac{2 a^{3/2} c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a d \left(3 c^2+3 c d+d^2\right) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}-\frac{2 d^2 (3 c+2 d) \tan (e+f x) (a-a \sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 \tan (e+f x) (a-a \sec (e+f x))^2}{5 a f \sqrt{a \sec (e+f x)+a}}",1,"(Cos[e + f*x]^3*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])]*(c + d*Sec[e + f*x])^3*((2*d*(45*c^2 + 30*c*d + 8*d^2)*Sin[(e + f*x)/2])/15 + (2*d^3*Sec[e + f*x]^2*Sin[(e + f*x)/2])/5 + (2*Sec[e + f*x]*(15*c*d^2*Sin[(e + f*x)/2] + 4*d^3*Sin[(e + f*x)/2]))/15))/(f*(d + c*Cos[e + f*x])^3) - (8*(-3 - 2*Sqrt[2])*c^3*Cos[(e + f*x)/4]^4*Sqrt[(7 - 5*Sqrt[2] + (10 - 7*Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*(1 - Sqrt[2] + (-2 + Sqrt[2])*Cos[(e + f*x)/2])*Cos[e + f*x]^2*(EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 2*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]])*Sqrt[(-1 - Sqrt[2] + (2 + Sqrt[2])*Cos[(e + f*x)/2])*Sec[(e + f*x)/4]^2]*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])]*(c + d*Sec[e + f*x])^3*Sqrt[3 - 2*Sqrt[2] - Tan[(e + f*x)/4]^2])/(f*(d + c*Cos[e + f*x])^3)","C",0
149,1,444,144,6.5892468,"\int \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^2 \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2,x]","\frac{\sqrt{\frac{1}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}} \sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)} \csc ^3\left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} (c+d \sec (e+f x))^2 \left(256 \sin ^6\left(\frac{1}{2} (e+f x)\right) \left(-2 c \sin ^2\left(\frac{1}{2} (e+f x)\right)+c+d\right)^2 \, _3F_2\left(\frac{3}{2},2,\frac{7}{2};1,\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)+1024 \sin ^6\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(\frac{3}{2},\frac{7}{2};\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right) \left(c^2 \left(2 \sin ^4\left(\frac{1}{2} (e+f x)\right)-3 \sin ^2\left(\frac{1}{2} (e+f x)\right)+1\right)+c d \left(2-3 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)+d^2\right)-\frac{7 \sqrt{2} \left(\sqrt{2} \sqrt{\sin ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)} \left(4 \sin ^2\left(\frac{1}{2} (e+f x)\right)+3\right)-3 \sin ^{-1}\left(\sqrt{2} \sqrt{\sin ^2\left(\frac{1}{2} (e+f x)\right)}\right)\right) \left(c^2 \left(12 \sin ^4\left(\frac{1}{2} (e+f x)\right)-20 \sin ^2\left(\frac{1}{2} (e+f x)\right)+15\right)+10 c d \left(3-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)+15 d^2\right)}{\sqrt{\sin ^2\left(\frac{1}{2} (e+f x)\right)}}\right)}{672 f \sec ^{\frac{5}{2}}(e+f x) (c \cos (e+f x)+d)^2}","\frac{2 a^{3/2} c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a d (2 c+d) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}-\frac{2 d^2 \tan (e+f x) (a-a \sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}",1,"(Csc[(e + f*x)/2]^3*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])]*(c + d*Sec[e + f*x])^2*Sqrt[(1 - 2*Sin[(e + f*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[(e + f*x)/2]^2]*(256*HypergeometricPFQ[{3/2, 2, 7/2}, {1, 9/2}, 2*Sin[(e + f*x)/2]^2]*Sin[(e + f*x)/2]^6*(c + d - 2*c*Sin[(e + f*x)/2]^2)^2 + 1024*Hypergeometric2F1[3/2, 7/2, 9/2, 2*Sin[(e + f*x)/2]^2]*Sin[(e + f*x)/2]^6*(d^2 + c*d*(2 - 3*Sin[(e + f*x)/2]^2) + c^2*(1 - 3*Sin[(e + f*x)/2]^2 + 2*Sin[(e + f*x)/2]^4)) - (7*Sqrt[2]*(-3*ArcSin[Sqrt[2]*Sqrt[Sin[(e + f*x)/2]^2]] + Sqrt[2]*Sqrt[Sin[(e + f*x)/2]^2]*Sqrt[1 - 2*Sin[(e + f*x)/2]^2]*(3 + 4*Sin[(e + f*x)/2]^2))*(15*d^2 + 10*c*d*(3 - 2*Sin[(e + f*x)/2]^2) + c^2*(15 - 20*Sin[(e + f*x)/2]^2 + 12*Sin[(e + f*x)/2]^4)))/Sqrt[Sin[(e + f*x)/2]^2]))/(672*f*(d + c*Cos[e + f*x])^2*Sec[e + f*x]^(5/2))","C",0
150,1,76,66,0.3149592,"\int \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x)) \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]),x]","\frac{\sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(\sqrt{2} c \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{\cos (e+f x)}+2 d \sin \left(\frac{1}{2} (e+f x)\right)\right)}{f}","\frac{2 \sqrt{a} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}+\frac{2 a d \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])]*(Sqrt[2]*c*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]]*Sqrt[Cos[e + f*x]] + 2*d*Sin[(e + f*x)/2]))/f","A",1
151,1,2650,105,25.0676377,"\int \frac{\sqrt{a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x]),x]","\text{Result too large to show}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}-\frac{2 \sqrt{a} \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{c+d} \sqrt{a \sec (e+f x)+a}}\right)}{c f \sqrt{c+d}}",1,"(-4*Sqrt[2]*Cos[(e + f*x)/4]^2*Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*(d + c*Cos[e + f*x])*(c*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 2*(c + d)*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + d*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sec[(e + f*x)/2]*((Cos[(e + f*x)/2]*Sqrt[Sec[e + f*x]])/(2*(d + c*Cos[e + f*x])) + (Cos[(3*(e + f*x))/2]*Sqrt[Sec[e + f*x]])/(2*(d + c*Cos[e + f*x])))*Sec[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[3 - 2*Sqrt[2] - Tan[(e + f*x)/4]^2])/(c*(c + d)*f*(c + d*Sec[e + f*x])*((Sqrt[2]*Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*(c*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 2*(c + d)*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + d*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Tan[(e + f*x)/4])/(c*(c + d)*Sqrt[3 - 2*Sqrt[2] - Tan[(e + f*x)/4]^2]) + (2*Sqrt[2]*Cos[(e + f*x)/4]*Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*(c*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 2*(c + d)*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + d*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Sin[(e + f*x)/4]*Sqrt[3 - 2*Sqrt[2] - Tan[(e + f*x)/4]^2])/(c*(c + d)) - (2*Sqrt[2]*Cos[(e + f*x)/4]^2*(c*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 2*(c + d)*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + d*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*(((-2 + Sqrt[2])*Sin[(e + f*x)/2])/(2*(1 + Cos[(e + f*x)/2])) + ((-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])*Sin[(e + f*x)/2])/(2*(1 + Cos[(e + f*x)/2])^2))*Sqrt[3 - 2*Sqrt[2] - Tan[(e + f*x)/4]^2])/(c*(c + d)*Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]) - (2*Sqrt[2]*Cos[(e + f*x)/4]^2*Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*(c*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 2*(c + d)*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + d*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sec[e + f*x]^(3/2)*Sin[e + f*x]*Sqrt[3 - 2*Sqrt[2] - Tan[(e + f*x)/4]^2])/(c*(c + d)) - (4*Sqrt[2]*Cos[(e + f*x)/4]^2*Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*Sqrt[Sec[e + f*x]]*Sqrt[3 - 2*Sqrt[2] - Tan[(e + f*x)/4]^2]*((c*Sec[(e + f*x)/4]^2)/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]) - ((c + d)*Sec[(e + f*x)/4]^2)/(2*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 - ((-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2]))) + d*(Sec[(e + f*x)/4]^2/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 + ((-3 + 2*Sqrt[2])*(c + d)*Tan[(e + f*x)/4]^2)/((3 - 2*Sqrt[2])*(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)))) + Sec[(e + f*x)/4]^2/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 - ((-3 + 2*Sqrt[2])*(c + d)*Tan[(e + f*x)/4]^2)/((3 - 2*Sqrt[2])*(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d)))))))/(c*(c + d))))","C",0
152,1,2907,219,28.578362,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c+d \sec (e+f x))^2} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x])^2,x]","\text{Result too large to show}","-\frac{a^{3/2} \sqrt{d} (3 c+2 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 f (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{3/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{a d \tan (e+f x)}{c f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}",1,"((d + c*Cos[e + f*x])^2*Sec[(e + f*x)/2]*Sec[e + f*x]^2*Sqrt[a*(1 + Sec[e + f*x])]*(-((d*Sin[(e + f*x)/2])/(c^2*(c + d))) + (d^2*Sin[(e + f*x)/2])/(c^2*(c + d)*(d + c*Cos[e + f*x]))))/(f*(c + d*Sec[e + f*x])^2) - (2*Sqrt[2]*Cos[(e + f*x)/4]^2*Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*(d + c*Cos[e + f*x])^2*(c*(2*c + d)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 4*(c + d)^2*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + d*(3*c + 2*d)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sec[(e + f*x)/2]*((Cos[(e + f*x)/2]*Sqrt[Sec[e + f*x]])/(2*(c + d)*(d + c*Cos[e + f*x])) + (Cos[(3*(e + f*x))/2]*Sqrt[Sec[e + f*x]])/(2*(c + d)*(d + c*Cos[e + f*x])) + (d*Cos[(3*(e + f*x))/2]*Sqrt[Sec[e + f*x]])/(2*c*(c + d)*(d + c*Cos[e + f*x])))*Sec[e + f*x]^2*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[3 - 2*Sqrt[2] - Tan[(e + f*x)/4]^2])/(c^2*(c + d)^2*f*(c + d*Sec[e + f*x])^2*((Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*(c*(2*c + d)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 4*(c + d)^2*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + d*(3*c + 2*d)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Tan[(e + f*x)/4])/(Sqrt[2]*c^2*(c + d)^2*Sqrt[3 - 2*Sqrt[2] - Tan[(e + f*x)/4]^2]) + (Sqrt[2]*Cos[(e + f*x)/4]*Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*(c*(2*c + d)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 4*(c + d)^2*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + d*(3*c + 2*d)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Sin[(e + f*x)/4]*Sqrt[3 - 2*Sqrt[2] - Tan[(e + f*x)/4]^2])/(c^2*(c + d)^2) - (Sqrt[2]*Cos[(e + f*x)/4]^2*(c*(2*c + d)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 4*(c + d)^2*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + d*(3*c + 2*d)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*(((-2 + Sqrt[2])*Sin[(e + f*x)/2])/(2*(1 + Cos[(e + f*x)/2])) + ((-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])*Sin[(e + f*x)/2])/(2*(1 + Cos[(e + f*x)/2])^2))*Sqrt[3 - 2*Sqrt[2] - Tan[(e + f*x)/4]^2])/(c^2*(c + d)^2*Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]) - (Sqrt[2]*Cos[(e + f*x)/4]^2*Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*(c*(2*c + d)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 4*(c + d)^2*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + d*(3*c + 2*d)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sec[e + f*x]^(3/2)*Sin[e + f*x]*Sqrt[3 - 2*Sqrt[2] - Tan[(e + f*x)/4]^2])/(c^2*(c + d)^2) - (2*Sqrt[2]*Cos[(e + f*x)/4]^2*Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(e + f*x)/2])/(1 + Cos[(e + f*x)/2])]*Sqrt[Sec[e + f*x]]*Sqrt[3 - 2*Sqrt[2] - Tan[(e + f*x)/4]^2]*((c*(2*c + d)*Sec[(e + f*x)/4]^2)/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]) - ((c + d)^2*Sec[(e + f*x)/4]^2)/(Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 - ((-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2]))) + d*(3*c + 2*d)*(Sec[(e + f*x)/4]^2/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 + ((-3 + 2*Sqrt[2])*(c + d)*Tan[(e + f*x)/4]^2)/((3 - 2*Sqrt[2])*(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)))) + Sec[(e + f*x)/4]^2/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 - ((-3 + 2*Sqrt[2])*(c + d)*Tan[(e + f*x)/4]^2)/((3 - 2*Sqrt[2])*(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d)))))))/(c^2*(c + d)^2)))","C",0
153,1,3070,287,24.4927178,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c+d \sec (e+f x))^3} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x])^3,x]","\text{Result too large to show}","\frac{2 a^{3/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{a^{3/2} \sqrt{d} \left(15 c^2+20 c d+8 d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{4 c^3 f (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{a d (7 c+4 d) \tan (e+f x)}{4 c^2 f (c+d)^2 \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}-\frac{a d \tan (e+f x)}{2 c f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))^2}",1,"((d + c*Cos[e + f*x])^3*Sec[(e + f*x)/2]*Sec[e + f*x]^3*Sqrt[a*(1 + Sec[e + f*x])]*((-3*d*(3*c + 2*d)*Sin[(e + f*x)/2])/(4*c^3*(c + d)^2) - (d^3*Sin[(e + f*x)/2])/(2*c^3*(c + d)*(d + c*Cos[e + f*x])^2) + (11*c*d^2*Sin[(e + f*x)/2] + 8*d^3*Sin[(e + f*x)/2])/(4*c^3*(c + d)^2*(d + c*Cos[e + f*x]))))/(f*(c + d*Sec[e + f*x])^3) - (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]^2*(d + c*Cos[e + f*x])^3*(c*(8*c^2 + 9*c*d + 4*d^2)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 16*(c + d)^3*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + d*(15*c^2 + 20*c*d + 8*d^2)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sec[(e + f*x)/2]*((Cos[(e + f*x)/2]*Sqrt[Sec[e + f*x]])/(2*(c + d)^2*(d + c*Cos[e + f*x])) + (d*Cos[(e + f*x)/2]*Sqrt[Sec[e + f*x]])/(8*c*(c + d)^2*(d + c*Cos[e + f*x])) + (Cos[(3*(e + f*x))/2]*Sqrt[Sec[e + f*x]])/(2*(c + d)^2*(d + c*Cos[e + f*x])) + (d*Cos[(3*(e + f*x))/2]*Sqrt[Sec[e + f*x]])/(c*(c + d)^2*(d + c*Cos[e + f*x])) + (d^2*Cos[(3*(e + f*x))/2]*Sqrt[Sec[e + f*x]])/(2*c^2*(c + d)^2*(d + c*Cos[e + f*x])))*Sec[e + f*x]^3*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(2*c^3*(c + d)^3*f*(c + d*Sec[e + f*x])^3*((Sqrt[3 - 2*Sqrt[2]]*(3 + 2*Sqrt[2])*(c*(8*c^2 + 9*c*d + 4*d^2)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 16*(c + d)^3*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + d*(15*c^2 + 20*c*d + 8*d^2)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Tan[(e + f*x)/4]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(8*c^3*(c + d)^3*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]) - (Sqrt[3 - 2*Sqrt[2]]*(-3 + 2*Sqrt[2])*(c*(8*c^2 + 9*c*d + 4*d^2)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 16*(c + d)^3*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + d*(15*c^2 + 20*c*d + 8*d^2)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Tan[(e + f*x)/4]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(8*c^3*(c + d)^3*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]) + (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]*(c*(8*c^2 + 9*c*d + 4*d^2)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 16*(c + d)^3*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + d*(15*c^2 + 20*c*d + 8*d^2)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Sin[(e + f*x)/4]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(4*c^3*(c + d)^3) - (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]^2*(c*(8*c^2 + 9*c*d + 4*d^2)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 16*(c + d)^3*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + d*(15*c^2 + 20*c*d + 8*d^2)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sec[e + f*x]^(3/2)*Sin[e + f*x]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(4*c^3*(c + d)^3) - (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]^2*Sqrt[Sec[e + f*x]]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*((c*(8*c^2 + 9*c*d + 4*d^2)*Sec[(e + f*x)/4]^2)/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]) - (4*(c + d)^3*Sec[(e + f*x)/4]^2)/(Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 - ((-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2]))) + d*(15*c^2 + 20*c*d + 8*d^2)*(Sec[(e + f*x)/4]^2/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 + ((-3 + 2*Sqrt[2])*(c + d)*Tan[(e + f*x)/4]^2)/((3 - 2*Sqrt[2])*(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)))) + Sec[(e + f*x)/4]^2/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 - ((-3 + 2*Sqrt[2])*(c + d)*Tan[(e + f*x)/4]^2)/((3 - 2*Sqrt[2])*(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d)))))))/(2*c^3*(c + d)^3)))","C",0
154,1,219,241,3.8918962,"\int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3 \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^3,x]","\frac{a \sec \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) \sqrt{a (\sec (e+f x)+1)} \left(420 \sqrt{2} c^3 \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right) \cos ^{\frac{7}{2}}(e+f x)+2 \sin \left(\frac{1}{2} (e+f x)\right) \left(105 c^3 \cos (3 (e+f x))+2 d \left(105 c^2+189 c d+52 d^2\right) \cos (2 (e+f x))+525 c^2 d \cos (3 (e+f x))+210 c^2 d+9 \left(35 c^3+175 c^2 d+154 c d^2+52 d^3\right) \cos (e+f x)+378 c d^2 \cos (3 (e+f x))+378 c d^2+104 d^3 \cos (3 (e+f x))+164 d^3\right)\right)}{420 f}","\frac{2 a^{5/2} c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) \left(d \left(24 c^2+111 c d+52 d^2\right) \sec (e+f x)+2 \left(36 c^3+243 c^2 d+189 c d^2+52 d^3\right)\right)}{105 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (c+d \sec (e+f x))^3}{7 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 (6 c+13 d) \tan (e+f x) (c+d \sec (e+f x))^2}{35 f \sqrt{a \sec (e+f x)+a}}",1,"(a*Sec[(e + f*x)/2]*Sec[e + f*x]^3*Sqrt[a*(1 + Sec[e + f*x])]*(420*Sqrt[2]*c^3*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]]*Cos[e + f*x]^(7/2) + 2*(210*c^2*d + 378*c*d^2 + 164*d^3 + 9*(35*c^3 + 175*c^2*d + 154*c*d^2 + 52*d^3)*Cos[e + f*x] + 2*d*(105*c^2 + 189*c*d + 52*d^2)*Cos[2*(e + f*x)] + 105*c^3*Cos[3*(e + f*x)] + 525*c^2*d*Cos[3*(e + f*x)] + 378*c*d^2*Cos[3*(e + f*x)] + 104*d^3*Cos[3*(e + f*x)])*Sin[(e + f*x)/2]))/(420*f)","A",1
155,1,145,176,1.3345561,"\int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^2 \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^2,x]","\frac{a \sec \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \left(2 \sin \left(\frac{1}{2} (e+f x)\right) \left(\left(15 c^2+50 c d+18 d^2\right) \cos (2 (e+f x))+15 c^2+2 d (10 c+9 d) \cos (e+f x)+50 c d+24 d^2\right)+30 \sqrt{2} c^2 \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right) \cos ^{\frac{5}{2}}(e+f x)\right)}{30 f}","\frac{2 a^{5/2} c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) \left(2 \left(6 c^2+25 c d+9 d^2\right)+d (4 c+9 d) \sec (e+f x)\right)}{15 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (c+d \sec (e+f x))^2}{5 f \sqrt{a \sec (e+f x)+a}}",1,"(a*Sec[(e + f*x)/2]*Sec[e + f*x]^2*Sqrt[a*(1 + Sec[e + f*x])]*(30*Sqrt[2]*c^2*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]]*Cos[e + f*x]^(5/2) + 2*(15*c^2 + 50*c*d + 24*d^2 + 2*d*(10*c + 9*d)*Cos[e + f*x] + (15*c^2 + 50*c*d + 18*d^2)*Cos[2*(e + f*x)])*Sin[(e + f*x)/2]))/(30*f)","A",1
156,1,102,105,0.5904677,"\int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x)) \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x]),x]","\frac{a \sec \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{a (\sec (e+f x)+1)} \left(2 \sin \left(\frac{1}{2} (e+f x)\right) ((3 c+5 d) \cos (e+f x)+d)+3 \sqrt{2} c \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right) \cos ^{\frac{3}{2}}(e+f x)\right)}{3 f}","\frac{2 a^{3/2} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}+\frac{2 a^2 (3 c+4 d) \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a d \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{3 f}",1,"(a*Sec[(e + f*x)/2]*Sec[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*(3*Sqrt[2]*c*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]]*Cos[e + f*x]^(3/2) + 2*(d + (3*c + 5*d)*Cos[e + f*x])*Sin[(e + f*x)/2]))/(3*f)","A",1
157,1,135,110,0.4970057,"\int \frac{(a+a \sec (e+f x))^{3/2}}{c+d \sec (e+f x)} \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x]),x]","\frac{\sqrt{2} a \sqrt{\cos (e+f x)} \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(\sqrt{d} \sqrt{c+d} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right)+(c-d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d} \sqrt{\cos (e+f x)}}\right)\right)}{c \sqrt{d} f \sqrt{c+d}}","\frac{2 a^{3/2} (c-d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{c+d} \sqrt{a \sec (e+f x)+a}}\right)}{c \sqrt{d} f \sqrt{c+d}}+\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{c f}",1,"(Sqrt[2]*a*(Sqrt[d]*Sqrt[c + d]*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]] + (c - d)*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(e + f*x)/2])/(Sqrt[c + d]*Sqrt[Cos[e + f*x]])])*Sqrt[Cos[e + f*x]]*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])])/(c*Sqrt[d]*Sqrt[c + d]*f)","A",1
158,1,2862,229,24.9219825,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^2} \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^2,x]","\text{Result too large to show}","\frac{a^{5/2} \left(c^2-3 c d-2 d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 \sqrt{d} f (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{5/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{a^2 (c-d) \tan (e+f x)}{c f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}",1,"((d + c*Cos[e + f*x])^2*Sec[(e + f*x)/2]^3*Sec[e + f*x]*(a*(1 + Sec[e + f*x]))^(3/2)*(((c - d)*Sin[(e + f*x)/2])/(2*c^2*(c + d)) + (-(c*d*Sin[(e + f*x)/2]) + d^2*Sin[(e + f*x)/2])/(2*c^2*(c + d)*(d + c*Cos[e + f*x]))))/(f*(c + d*Sec[e + f*x])^2) - (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]^2*(d + c*Cos[e + f*x])^2*(c*(3*c + d)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 4*(c + d)^2*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - (c^2 - 3*c*d - 2*d^2)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sec[(e + f*x)/2]^3*((Cos[(e + f*x)/2]*Sqrt[Sec[e + f*x]])/(2*(c + d)*(d + c*Cos[e + f*x])) + (Cos[(3*(e + f*x))/2]*Sqrt[Sec[e + f*x]])/(4*(c + d)*(d + c*Cos[e + f*x])) + (d*Cos[(3*(e + f*x))/2]*Sqrt[Sec[e + f*x]])/(4*c*(c + d)*(d + c*Cos[e + f*x])))*Sec[e + f*x]*(a*(1 + Sec[e + f*x]))^(3/2)*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(c^2*(c + d)^2*f*(c + d*Sec[e + f*x])^2*((Sqrt[3 - 2*Sqrt[2]]*(3 + 2*Sqrt[2])*(c*(3*c + d)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 4*(c + d)^2*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - (c^2 - 3*c*d - 2*d^2)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Tan[(e + f*x)/4]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(4*c^2*(c + d)^2*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]) - (Sqrt[3 - 2*Sqrt[2]]*(-3 + 2*Sqrt[2])*(c*(3*c + d)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 4*(c + d)^2*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - (c^2 - 3*c*d - 2*d^2)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Tan[(e + f*x)/4]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(4*c^2*(c + d)^2*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]) + (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]*(c*(3*c + d)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 4*(c + d)^2*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - (c^2 - 3*c*d - 2*d^2)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Sin[(e + f*x)/4]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(2*c^2*(c + d)^2) - (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]^2*(c*(3*c + d)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 4*(c + d)^2*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - (c^2 - 3*c*d - 2*d^2)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sec[e + f*x]^(3/2)*Sin[e + f*x]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(2*c^2*(c + d)^2) - (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]^2*Sqrt[Sec[e + f*x]]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*((c*(3*c + d)*Sec[(e + f*x)/4]^2)/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]) - ((c + d)^2*Sec[(e + f*x)/4]^2)/(Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 - ((-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2]))) - (c^2 - 3*c*d - 2*d^2)*(Sec[(e + f*x)/4]^2/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 + ((-3 + 2*Sqrt[2])*(c + d)*Tan[(e + f*x)/4]^2)/((3 - 2*Sqrt[2])*(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)))) + Sec[(e + f*x)/4]^2/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 - ((-3 + 2*Sqrt[2])*(c + d)*Tan[(e + f*x)/4]^2)/((3 - 2*Sqrt[2])*(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d)))))))/(c^2*(c + d)^2)))","C",0
159,1,3166,310,24.7041104,"\int \frac{(a+a \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^3} \, dx","Integrate[(a + a*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^3,x]","\text{Result too large to show}","\frac{2 a^{5/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{a^{5/2} \left(3 c^3-15 c^2 d-20 c d^2-8 d^3\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{4 c^3 \sqrt{d} f (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{a^2 \left(3 c^2-7 c d-4 d^2\right) \tan (e+f x)}{4 c^2 f (c+d)^2 \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{a^2 (c-d) \tan (e+f x)}{2 c f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))^2}",1,"((d + c*Cos[e + f*x])^3*Sec[(e + f*x)/2]^3*Sec[e + f*x]^2*(a*(1 + Sec[e + f*x]))^(3/2)*(-1/8*((-5*c^2 + 7*c*d + 6*d^2)*Sin[(e + f*x)/2])/(c^3*(c + d)^2) + (c*d^2*Sin[(e + f*x)/2] - d^3*Sin[(e + f*x)/2])/(4*c^3*(c + d)*(d + c*Cos[e + f*x])^2) + (-7*c^2*d*Sin[(e + f*x)/2] + 7*c*d^2*Sin[(e + f*x)/2] + 8*d^3*Sin[(e + f*x)/2])/(8*c^3*(c + d)^2*(d + c*Cos[e + f*x]))))/(f*(c + d*Sec[e + f*x])^3) - (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]^2*(d + c*Cos[e + f*x])^3*(c*(11*c^2 + 9*c*d + 4*d^2)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 16*(c + d)^3*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - (3*c^3 - 15*c^2*d - 20*c*d^2 - 8*d^3)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sec[(e + f*x)/2]^3*((7*Cos[(e + f*x)/2]*Sqrt[Sec[e + f*x]])/(16*(c + d)^2*(d + c*Cos[e + f*x])) + (d*Cos[(e + f*x)/2]*Sqrt[Sec[e + f*x]])/(16*c*(c + d)^2*(d + c*Cos[e + f*x])) + (Cos[(3*(e + f*x))/2]*Sqrt[Sec[e + f*x]])/(4*(c + d)^2*(d + c*Cos[e + f*x])) + (d*Cos[(3*(e + f*x))/2]*Sqrt[Sec[e + f*x]])/(2*c*(c + d)^2*(d + c*Cos[e + f*x])) + (d^2*Cos[(3*(e + f*x))/2]*Sqrt[Sec[e + f*x]])/(4*c^2*(c + d)^2*(d + c*Cos[e + f*x])))*Sec[e + f*x]^2*(a*(1 + Sec[e + f*x]))^(3/2)*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(4*c^3*(c + d)^3*f*(c + d*Sec[e + f*x])^3*((Sqrt[3 - 2*Sqrt[2]]*(3 + 2*Sqrt[2])*(c*(11*c^2 + 9*c*d + 4*d^2)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 16*(c + d)^3*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - (3*c^3 - 15*c^2*d - 20*c*d^2 - 8*d^3)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Tan[(e + f*x)/4]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(16*c^3*(c + d)^3*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]) - (Sqrt[3 - 2*Sqrt[2]]*(-3 + 2*Sqrt[2])*(c*(11*c^2 + 9*c*d + 4*d^2)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 16*(c + d)^3*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - (3*c^3 - 15*c^2*d - 20*c*d^2 - 8*d^3)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Tan[(e + f*x)/4]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(16*c^3*(c + d)^3*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]) + (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]*(c*(11*c^2 + 9*c*d + 4*d^2)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 16*(c + d)^3*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - (3*c^3 - 15*c^2*d - 20*c*d^2 - 8*d^3)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Sin[(e + f*x)/4]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(8*c^3*(c + d)^3) - (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]^2*(c*(11*c^2 + 9*c*d + 4*d^2)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 16*(c + d)^3*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - (3*c^3 - 15*c^2*d - 20*c*d^2 - 8*d^3)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sec[e + f*x]^(3/2)*Sin[e + f*x]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(8*c^3*(c + d)^3) - (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]^2*Sqrt[Sec[e + f*x]]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*((c*(11*c^2 + 9*c*d + 4*d^2)*Sec[(e + f*x)/4]^2)/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]) - (4*(c + d)^3*Sec[(e + f*x)/4]^2)/(Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 - ((-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2]))) - (3*c^3 - 15*c^2*d - 20*c*d^2 - 8*d^3)*(Sec[(e + f*x)/4]^2/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 + ((-3 + 2*Sqrt[2])*(c + d)*Tan[(e + f*x)/4]^2)/((3 - 2*Sqrt[2])*(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)))) + Sec[(e + f*x)/4]^2/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 - ((-3 + 2*Sqrt[2])*(c + d)*Tan[(e + f*x)/4]^2)/((3 - 2*Sqrt[2])*(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d)))))))/(4*c^3*(c + d)^3)))","C",0
160,1,286,336,6.4960927,"\int (a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))^3 \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^3,x]","\frac{a^2 \sec \left(\frac{1}{2} (e+f x)\right) \sec ^4(e+f x) \sqrt{a (\sec (e+f x)+1)} \left(2520 \sqrt{2} c^3 \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right) \cos ^{\frac{9}{2}}(e+f x)+2 \sin \left(\frac{1}{2} (e+f x)\right) \left(210 c^3 \cos (3 (e+f x))+840 c^3 \cos (4 (e+f x))+2520 c^3+1764 c^2 d \cos (3 (e+f x))+2709 c^2 d \cos (4 (e+f x))+8883 c^2 d+\left(630 c^3+5292 c^2 d+7290 c d^2+2792 d^3\right) \cos (e+f x)+4 \left(840 c^3+2898 c^2 d+2610 c d^2+803 d^3\right) \cos (2 (e+f x))+2070 c d^2 \cos (3 (e+f x))+2070 c d^2 \cos (4 (e+f x))+8370 c d^2+584 d^3 \cos (3 (e+f x))+584 d^3 \cos (4 (e+f x))+2908 d^3\right)\right)}{2520 f}","\frac{2 a^{7/2} c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a \sec (e+f x)+a} \sqrt{a-a \sec (e+f x)}}-\frac{2 \left(c^3+12 c^2 d+24 c d^2+12 d^3\right) \tan (e+f x) \left(a^3-a^3 \sec (e+f x)\right)}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^3 \left(3 c^3+12 c^2 d+12 c d^2+4 d^3\right) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}+\frac{2 a d \left(3 c^2+15 c d+13 d^2\right) \tan (e+f x) (a-a \sec (e+f x))^2}{5 f \sqrt{a \sec (e+f x)+a}}-\frac{6 d^2 (c+2 d) \tan (e+f x) (a-a \sec (e+f x))^3}{7 f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 \tan (e+f x) (a-a \sec (e+f x))^4}{9 a f \sqrt{a \sec (e+f x)+a}}",1,"(a^2*Sec[(e + f*x)/2]*Sec[e + f*x]^4*Sqrt[a*(1 + Sec[e + f*x])]*(2520*Sqrt[2]*c^3*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]]*Cos[e + f*x]^(9/2) + 2*(2520*c^3 + 8883*c^2*d + 8370*c*d^2 + 2908*d^3 + (630*c^3 + 5292*c^2*d + 7290*c*d^2 + 2792*d^3)*Cos[e + f*x] + 4*(840*c^3 + 2898*c^2*d + 2610*c*d^2 + 803*d^3)*Cos[2*(e + f*x)] + 210*c^3*Cos[3*(e + f*x)] + 1764*c^2*d*Cos[3*(e + f*x)] + 2070*c*d^2*Cos[3*(e + f*x)] + 584*d^3*Cos[3*(e + f*x)] + 840*c^3*Cos[4*(e + f*x)] + 2709*c^2*d*Cos[4*(e + f*x)] + 2070*c*d^2*Cos[4*(e + f*x)] + 584*d^3*Cos[4*(e + f*x)])*Sin[(e + f*x)/2]))/(2520*f)","A",1
161,1,191,258,2.6693648,"\int (a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))^2 \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^2,x]","\frac{a^2 \sec \left(\frac{1}{2} (e+f x)\right) \sec ^3(e+f x) \sqrt{a (\sec (e+f x)+1)} \left(4 \sin \left(\frac{1}{2} (e+f x)\right) \left(\left(420 c^2+987 c d+465 d^2\right) \cos (e+f x)+\left(35 c^2+196 c d+115 d^2\right) \cos (2 (e+f x))+140 c^2 \cos (3 (e+f x))+35 c^2+301 c d \cos (3 (e+f x))+196 c d+115 d^2 \cos (3 (e+f x))+145 d^2\right)+420 \sqrt{2} c^2 \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right) \cos ^{\frac{7}{2}}(e+f x)\right)}{420 f}","\frac{2 a^{7/2} c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{2 \left(c^2+8 c d+8 d^2\right) \tan (e+f x) \left(a^3-a^3 \sec (e+f x)\right)}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^3 (c+2 d) (3 c+2 d) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}+\frac{2 a d (2 c+5 d) \tan (e+f x) (a-a \sec (e+f x))^2}{5 f \sqrt{a \sec (e+f x)+a}}-\frac{2 d^2 \tan (e+f x) (a-a \sec (e+f x))^3}{7 f \sqrt{a \sec (e+f x)+a}}",1,"(a^2*Sec[(e + f*x)/2]*Sec[e + f*x]^3*Sqrt[a*(1 + Sec[e + f*x])]*(420*Sqrt[2]*c^2*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]]*Cos[e + f*x]^(7/2) + 4*(35*c^2 + 196*c*d + 145*d^2 + (420*c^2 + 987*c*d + 465*d^2)*Cos[e + f*x] + (35*c^2 + 196*c*d + 115*d^2)*Cos[2*(e + f*x)] + 140*c^2*Cos[3*(e + f*x)] + 301*c*d*Cos[3*(e + f*x)] + 115*d^2*Cos[3*(e + f*x)])*Sin[(e + f*x)/2]))/(420*f)","A",1
162,1,128,142,1.026037,"\int (a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x)) \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x]),x]","\frac{a^2 \sec \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \left(2 \sin \left(\frac{1}{2} (e+f x)\right) (2 (5 c+14 d) \cos (e+f x)+(40 c+43 d) \cos (2 (e+f x))+40 c+49 d)+30 \sqrt{2} c \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right) \cos ^{\frac{5}{2}}(e+f x)\right)}{30 f}","\frac{2 a^{5/2} c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}+\frac{2 a^3 (35 c+32 d) \tan (e+f x)}{15 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 (5 c+8 d) \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{15 f}+\frac{2 a d \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{5 f}",1,"(a^2*Sec[(e + f*x)/2]*Sec[e + f*x]^2*Sqrt[a*(1 + Sec[e + f*x])]*(30*Sqrt[2]*c*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]]*Cos[e + f*x]^(5/2) + 2*(40*c + 49*d + 2*(5*c + 14*d)*Cos[e + f*x] + (40*c + 43*d)*Cos[2*(e + f*x)])*Sin[(e + f*x)/2]))/(30*f)","A",1
163,1,343,203,6.5474151,"\int \frac{(a+a \sec (e+f x))^{5/2}}{c+d \sec (e+f x)} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x]),x]","\frac{\cos ^{\frac{3}{2}}(e+f x) \sec ^5\left(\frac{1}{2} (e+f x)\right) (a (\sec (e+f x)+1))^{5/2} (c \cos (e+f x)+d) \left(-\frac{16 d (c-d)^2 \sin ^3\left(\frac{1}{2} (e+f x)\right) (c \cos (e+f x)+d) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\frac{2 d \sec (e+f x) \sin ^2\left(\frac{1}{2} (e+f x)\right)}{c+d}\right)}{(c+d)^3 \cos ^{\frac{5}{2}}(e+f x)}+\frac{20 (3 c-d) \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{\cos (e+f x)}}+\frac{10 (c-d)^2 \csc \left(\frac{1}{2} (e+f x)\right) (2 c \cos (e+f x)+c+3 d) \left(\sqrt{-\frac{d (\sec (e+f x)-1)}{c+d}}-\tanh ^{-1}\left(\sqrt{-\frac{d (\sec (e+f x)-1)}{c+d}}\right)\right)}{d (c+d) \sqrt{\cos (e+f x)} \sqrt{-\frac{d (\sec (e+f x)-1)}{c+d}}}+10 c \left(\sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right)-\frac{2 \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{\cos (e+f x)}}\right)\right)}{40 c^2 f (c+d \sec (e+f x))}","-\frac{2 a^{7/2} (c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c d^{3/2} f \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{7/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^3 \tan (e+f x)}{d f \sqrt{a \sec (e+f x)+a}}",1,"(Cos[e + f*x]^(3/2)*(d + c*Cos[e + f*x])*Sec[(e + f*x)/2]^5*(a*(1 + Sec[e + f*x]))^(5/2)*((10*(c - d)^2*(c + 3*d + 2*c*Cos[e + f*x])*Csc[(e + f*x)/2]*(-ArcTanh[Sqrt[-((d*(-1 + Sec[e + f*x]))/(c + d))]] + Sqrt[-((d*(-1 + Sec[e + f*x]))/(c + d))]))/(d*(c + d)*Sqrt[Cos[e + f*x]]*Sqrt[-((d*(-1 + Sec[e + f*x]))/(c + d))]) + (20*(3*c - d)*Sin[(e + f*x)/2])/Sqrt[Cos[e + f*x]] - (16*(c - d)^2*d*(d + c*Cos[e + f*x])*Hypergeometric2F1[2, 5/2, 7/2, (-2*d*Sec[e + f*x]*Sin[(e + f*x)/2]^2)/(c + d)]*Sin[(e + f*x)/2]^3)/((c + d)^3*Cos[e + f*x]^(5/2)) + 10*c*(Sqrt[2]*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]] - (2*Sin[(e + f*x)/2])/Sqrt[Cos[e + f*x]])))/(40*c^2*f*(c + d*Sec[e + f*x]))","C",0
164,1,280,329,3.6445109,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^2} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^2,x]","\frac{\sqrt{\cos (e+f x)} \sec ^5\left(\frac{1}{2} (e+f x)\right) (a (\sec (e+f x)+1))^{5/2} (c \cos (e+f x)+d)^2 \left(\frac{4 \sqrt{2} (c-d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d} \sqrt{\cos (e+f x)}}\right)}{\sqrt{d} \sqrt{c+d}}-\frac{(c-d)^2 \sin \left(\frac{1}{2} (e+f x)\right) \left(2 c \cos (e+f x)-\frac{2 (c+2 d) (c \cos (e+f x)+d) \tanh ^{-1}\left(\sqrt{-\frac{d (\sec (e+f x)-1)}{c+d}}\right)}{(c+d) \sqrt{-\frac{d (\sec (e+f x)-1)}{c+d}}}\right)}{d (c+d) \sqrt{\cos (e+f x)} (c \cos (e+f x)+d)}+2 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right)\right)}{8 c^2 f (c+d \sec (e+f x))^2}","\frac{2 a^{7/2} (c-d) \sqrt{c+d} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 d^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{7/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{a^{7/2} (c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c d^{3/2} f (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{a^3 (c-d)^2 \tan (e+f x)}{c d f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}",1,"(Sqrt[Cos[e + f*x]]*(d + c*Cos[e + f*x])^2*Sec[(e + f*x)/2]^5*(a*(1 + Sec[e + f*x]))^(5/2)*(2*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]] + (4*Sqrt[2]*(c - d)*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(e + f*x)/2])/(Sqrt[c + d]*Sqrt[Cos[e + f*x]])])/(Sqrt[d]*Sqrt[c + d]) - ((c - d)^2*(2*c*Cos[e + f*x] - (2*(c + 2*d)*ArcTanh[Sqrt[-((d*(-1 + Sec[e + f*x]))/(c + d))]]*(d + c*Cos[e + f*x]))/((c + d)*Sqrt[-((d*(-1 + Sec[e + f*x]))/(c + d))]))*Sin[(e + f*x)/2])/(d*(c + d)*Sqrt[Cos[e + f*x]]*(d + c*Cos[e + f*x]))))/(8*c^2*f*(c + d*Sec[e + f*x])^2)","A",1
165,1,3344,536,26.1147569,"\int \frac{(a+a \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^3} \, dx","Integrate[(a + a*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^3,x]","\text{Result too large to show}","-\frac{2 a^{7/2} \sqrt{d} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^3 f \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{7/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{a^{7/2} (c-d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 d^{3/2} f \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 a^{7/2} (c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{4 c d^{3/2} f (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{a^3 (c-d) \tan (e+f x)}{c^2 d f \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}-\frac{3 a^3 (c-d)^2 \tan (e+f x)}{4 c d f (c+d)^2 \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}-\frac{a^3 (c-d)^2 \tan (e+f x)}{2 c d f (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))^2}",1,"((d + c*Cos[e + f*x])^3*Sec[(e + f*x)/2]^5*Sec[e + f*x]*(a*(1 + Sec[e + f*x]))^(5/2)*(-1/16*((c^3 - 12*c^2*d + 5*c*d^2 + 6*d^3)*Sin[(e + f*x)/2])/(c^3*d*(c + d)^2) + (-(c^2*d*Sin[(e + f*x)/2]) + 2*c*d^2*Sin[(e + f*x)/2] - d^3*Sin[(e + f*x)/2])/(8*c^3*(c + d)*(d + c*Cos[e + f*x])^2) + (3*c^3*Sin[(e + f*x)/2] - 14*c^2*d*Sin[(e + f*x)/2] + 3*c*d^2*Sin[(e + f*x)/2] + 8*d^3*Sin[(e + f*x)/2])/(16*c^3*(c + d)^2*(d + c*Cos[e + f*x]))))/(f*(c + d*Sec[e + f*x])^3) - (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]^2*(d + c*Cos[e + f*x])^3*(c*(c^3 + 18*c^2*d + 9*c*d^2 + 4*d^3)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 16*d*(c + d)^3*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - (c^4 + 10*c^3*d - 15*c^2*d^2 - 20*c*d^3 - 8*d^4)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sec[(e + f*x)/2]^5*((7*Cos[(e + f*x)/2]*Sqrt[Sec[e + f*x]])/(16*(c + d)^2*(d + c*Cos[e + f*x])) + (c*Cos[(e + f*x)/2]*Sqrt[Sec[e + f*x]])/(32*d*(c + d)^2*(d + c*Cos[e + f*x])) + (d*Cos[(e + f*x)/2]*Sqrt[Sec[e + f*x]])/(32*c*(c + d)^2*(d + c*Cos[e + f*x])) + (Cos[(3*(e + f*x))/2]*Sqrt[Sec[e + f*x]])/(8*(c + d)^2*(d + c*Cos[e + f*x])) + (d*Cos[(3*(e + f*x))/2]*Sqrt[Sec[e + f*x]])/(4*c*(c + d)^2*(d + c*Cos[e + f*x])) + (d^2*Cos[(3*(e + f*x))/2]*Sqrt[Sec[e + f*x]])/(8*c^2*(c + d)^2*(d + c*Cos[e + f*x])))*Sec[e + f*x]*(a*(1 + Sec[e + f*x]))^(5/2)*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(8*c^3*d*(c + d)^3*f*(c + d*Sec[e + f*x])^3*((Sqrt[3 - 2*Sqrt[2]]*(3 + 2*Sqrt[2])*(c*(c^3 + 18*c^2*d + 9*c*d^2 + 4*d^3)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 16*d*(c + d)^3*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - (c^4 + 10*c^3*d - 15*c^2*d^2 - 20*c*d^3 - 8*d^4)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Tan[(e + f*x)/4]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(32*c^3*d*(c + d)^3*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]) - (Sqrt[3 - 2*Sqrt[2]]*(-3 + 2*Sqrt[2])*(c*(c^3 + 18*c^2*d + 9*c*d^2 + 4*d^3)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 16*d*(c + d)^3*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - (c^4 + 10*c^3*d - 15*c^2*d^2 - 20*c*d^3 - 8*d^4)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Tan[(e + f*x)/4]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(32*c^3*d*(c + d)^3*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]) + (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]*(c*(c^3 + 18*c^2*d + 9*c*d^2 + 4*d^3)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 16*d*(c + d)^3*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - (c^4 + 10*c^3*d - 15*c^2*d^2 - 20*c*d^3 - 8*d^4)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sqrt[Sec[e + f*x]]*Sin[(e + f*x)/4]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(16*c^3*d*(c + d)^3) - (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]^2*(c*(c^3 + 18*c^2*d + 9*c*d^2 + 4*d^3)*EllipticF[ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 16*d*(c + d)^3*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - (c^4 + 10*c^3*d - 15*c^2*d^2 - 20*c*d^3 - 8*d^4)*(EllipticPi[-(((-3 + 2*Sqrt[2])*(c + d))/(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + EllipticPi[((-3 + 2*Sqrt[2])*(c + d))/(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d), ArcSin[Tan[(e + f*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]))*Sec[e + f*x]^(3/2)*Sin[e + f*x]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2])/(16*c^3*d*(c + d)^3) - (Sqrt[3 - 2*Sqrt[2]]*Cos[(e + f*x)/4]^2*Sqrt[Sec[e + f*x]]*Sqrt[1 + (-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*Sqrt[1 - (3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2]*((c*(c^3 + 18*c^2*d + 9*c*d^2 + 4*d^3)*Sec[(e + f*x)/4]^2)/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]) - (4*d*(c + d)^3*Sec[(e + f*x)/4]^2)/(Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 - ((-3 + 2*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2]))) - (c^4 + 10*c^3*d - 15*c^2*d^2 - 20*c*d^3 - 8*d^4)*(Sec[(e + f*x)/4]^2/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 + ((-3 + 2*Sqrt[2])*(c + d)*Tan[(e + f*x)/4]^2)/((3 - 2*Sqrt[2])*(3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] - d)))) + Sec[(e + f*x)/4]^2/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(e + f*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(e + f*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 - ((-3 + 2*Sqrt[2])*(c + d)*Tan[(e + f*x)/4]^2)/((3 - 2*Sqrt[2])*(-3*c + 2*Sqrt[2]*Sqrt[c*(c - d)] + d)))))))/(8*c^3*d*(c + d)^3)))","C",0
166,1,787,258,7.2396286,"\int \frac{(c+d \sec (e+f x))^3}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(c + d*Sec[e + f*x])^3/Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 \sqrt{\frac{1}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}} \sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)} \cos \left(\frac{1}{2} (e+f x)\right) (c+d \sec (e+f x))^3 \left(-\frac{(c-d)^3 \csc ^5\left(\frac{1}{2} (e+f x)\right) \left(-12 \sin ^8\left(\frac{1}{2} (e+f x)\right) \cos ^4\left(\frac{1}{2} (e+f x)\right) \, _3F_2\left(2,2,\frac{7}{2};1,\frac{9}{2};-\frac{\sin ^2\left(\frac{1}{2} (e+f x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}\right)-12 \left(3 \sin ^4\left(\frac{1}{2} (e+f x)\right)-7 \sin ^2\left(\frac{1}{2} (e+f x)\right)+4\right) \sin ^8\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{\sin ^2\left(\frac{1}{2} (e+f x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}\right)+7 \sqrt{-\frac{\sin ^2\left(\frac{1}{2} (e+f x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}} \left(1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)^3 \left(8 \sin ^4\left(\frac{1}{2} (e+f x)\right)-20 \sin ^2\left(\frac{1}{2} (e+f x)\right)+15\right) \left(\left(3-7 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right) \sqrt{-\frac{\sin ^2\left(\frac{1}{2} (e+f x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}}-3 \left(1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right) \tanh ^{-1}\left(\sqrt{-\frac{\sin ^2\left(\frac{1}{2} (e+f x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}}\right)\right)\right)}{63 \left(1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)^{7/2}}+\frac{1}{3} c^3 \sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)} \left(-\frac{6 \sin ^2\left(\frac{1}{2} (e+f x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}+\frac{3 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right) \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}}+\frac{4 \sin ^4\left(\frac{1}{2} (e+f x)\right)}{\left(1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)^2}\right) \csc \left(\frac{1}{2} (e+f x)\right)+\frac{4 c \left(c^2+3 d^2\right) \sin \left(\frac{1}{2} (e+f x)\right)}{3 \sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}}+\frac{2 c \left(c^2+3 d^2\right) \sin \left(\frac{1}{2} (e+f x)\right)}{3 \left(1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)^{3/2}}-\frac{4 c^2 (c+3 d) \sin ^3\left(\frac{1}{2} (e+f x)\right)}{3 \left(1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)^{3/2}}\right)}{f \sec ^{\frac{5}{2}}(e+f x) \sqrt{a (\sec (e+f x)+1)} (c \cos (e+f x)+d)^3}","\frac{2 \sqrt{a} c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^2 (3 c-d) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \sqrt{a} (c-d)^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{2 d^3 \tan (e+f x) (1-\sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*Cos[(e + f*x)/2]*(c + d*Sec[e + f*x])^3*Sqrt[(1 - 2*Sin[(e + f*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[(e + f*x)/2]^2]*((2*c*(c^2 + 3*d^2)*Sin[(e + f*x)/2])/(3*(1 - 2*Sin[(e + f*x)/2]^2)^(3/2)) - (4*c^2*(c + 3*d)*Sin[(e + f*x)/2]^3)/(3*(1 - 2*Sin[(e + f*x)/2]^2)^(3/2)) + (4*c*(c^2 + 3*d^2)*Sin[(e + f*x)/2])/(3*Sqrt[1 - 2*Sin[(e + f*x)/2]^2]) + (c^3*Csc[(e + f*x)/2]*Sqrt[1 - 2*Sin[(e + f*x)/2]^2]*((4*Sin[(e + f*x)/2]^4)/(1 - 2*Sin[(e + f*x)/2]^2)^2 - (6*Sin[(e + f*x)/2]^2)/(1 - 2*Sin[(e + f*x)/2]^2) + (3*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]]*Sin[(e + f*x)/2])/Sqrt[1 - 2*Sin[(e + f*x)/2]^2]))/3 - ((c - d)^3*Csc[(e + f*x)/2]^5*(-12*Cos[(e + f*x)/2]^4*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, -(Sin[(e + f*x)/2]^2/(1 - 2*Sin[(e + f*x)/2]^2))]*Sin[(e + f*x)/2]^8 - 12*Hypergeometric2F1[2, 7/2, 9/2, -(Sin[(e + f*x)/2]^2/(1 - 2*Sin[(e + f*x)/2]^2))]*Sin[(e + f*x)/2]^8*(4 - 7*Sin[(e + f*x)/2]^2 + 3*Sin[(e + f*x)/2]^4) + 7*Sqrt[-(Sin[(e + f*x)/2]^2/(1 - 2*Sin[(e + f*x)/2]^2))]*(1 - 2*Sin[(e + f*x)/2]^2)^3*(15 - 20*Sin[(e + f*x)/2]^2 + 8*Sin[(e + f*x)/2]^4)*((3 - 7*Sin[(e + f*x)/2]^2)*Sqrt[-(Sin[(e + f*x)/2]^2/(1 - 2*Sin[(e + f*x)/2]^2))] - 3*ArcTanh[Sqrt[-(Sin[(e + f*x)/2]^2/(1 - 2*Sin[(e + f*x)/2]^2))]]*(1 - 2*Sin[(e + f*x)/2]^2))))/(63*(1 - 2*Sin[(e + f*x)/2]^2)^(7/2))))/(f*(d + c*Cos[e + f*x])^3*Sec[e + f*x]^(5/2)*Sqrt[a*(1 + Sec[e + f*x])])","C",0
167,1,295,183,2.4958357,"\int \frac{(c+d \sec (e+f x))^2}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(c + d*Sec[e + f*x])^2/Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \cos ^{\frac{3}{2}}(e+f x) (c+d \sec (e+f x))^2 \left(c^2 \left(\sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right)-\frac{2 \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{\cos (e+f x)}}\right)-\frac{(c-d)^2 \sin \left(\frac{1}{2} (e+f x)\right) \sin ^2(e+f x) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\sec (e+f x) \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)}{10 \cos ^{\frac{5}{2}}(e+f x)}+\frac{4 c d \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{\cos (e+f x)}}-\frac{(c-d)^2 \sqrt{\cos (e+f x)-1} (\cos (e+f x)+2) \csc ^3\left(\frac{1}{2} (e+f x)\right) \left(\sqrt{2-2 \sec (e+f x)}-2 \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (e+f x)\right) (-\sec (e+f x))}\right)\right)}{2 \sqrt{2}}\right)}{f \sqrt{a (\sec (e+f x)+1)} (c \cos (e+f x)+d)^2}","\frac{2 \sqrt{a} c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \sqrt{a} (c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*Cos[(e + f*x)/2]*Cos[e + f*x]^(3/2)*(c + d*Sec[e + f*x])^2*(-1/2*((c - d)^2*Sqrt[-1 + Cos[e + f*x]]*(2 + Cos[e + f*x])*Csc[(e + f*x)/2]^3*(-2*ArcTanh[Sqrt[-(Sec[e + f*x]*Sin[(e + f*x)/2]^2)]] + Sqrt[2 - 2*Sec[e + f*x]]))/Sqrt[2] + (4*c*d*Sin[(e + f*x)/2])/Sqrt[Cos[e + f*x]] + c^2*(Sqrt[2]*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]] - (2*Sin[(e + f*x)/2])/Sqrt[Cos[e + f*x]]) - ((c - d)^2*Hypergeometric2F1[2, 5/2, 7/2, -(Sec[e + f*x]*Sin[(e + f*x)/2]^2)]*Sin[(e + f*x)/2]*Sin[e + f*x]^2)/(10*Cos[e + f*x]^(5/2))))/(f*(d + c*Cos[e + f*x])^2*Sqrt[a*(1 + Sec[e + f*x])])","C",0
168,1,92,91,0.294706,"\int \frac{c+d \sec (e+f x)}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(c + d*Sec[e + f*x])/Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \left((d-c) \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{\cos (e+f x)}}\right)+\sqrt{2} c \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right)\right)}{f \sqrt{\cos (e+f x)} \sqrt{a (\sec (e+f x)+1)}}","\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{\sqrt{2} (c-d) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f}",1,"(2*(Sqrt[2]*c*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]] + (-c + d)*ArcTan[Sin[(e + f*x)/2]/Sqrt[Cos[e + f*x]]])*Cos[(e + f*x)/2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[a*(1 + Sec[e + f*x])])","A",1
169,1,431980,166,39.1284531,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx","Integrate[1/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]","\text{Result too large to show}","\frac{2 d^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{c+d} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} c f (c-d) \sqrt{c+d}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} f (c-d)}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{\sqrt{a} c f}",1,"Result too large to show","C",0
170,1,473385,416,36.1284322,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^2} \, dx","Integrate[1/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2),x]","\text{Result too large to show}","\frac{2 \sqrt{a} d^{3/2} (2 c-d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 f (c-d)^2 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{d^2 \tan (e+f x)}{c f \left(c^2-d^2\right) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{2 \sqrt{a} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{\sqrt{a} d^{3/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c f (c-d) (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \sqrt{a} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"Result too large to show","C",0
171,1,654358,653,38.9922838,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^3} \, dx","Integrate[1/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^3),x]","\text{Result too large to show}","\frac{2 \sqrt{a} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{\sqrt{a} d^{3/2} (2 c-d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^2 f (c-d)^2 (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{d^2 (2 c-d) \tan (e+f x)}{c^2 f (c-d)^2 (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{d^2 \tan (e+f x)}{2 c f \left(c^2-d^2\right) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))^2}+\frac{2 \sqrt{a} d^{3/2} \left(3 c^2-3 c d+d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{c^3 f (c-d)^3 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{3 \sqrt{a} d^{3/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{4 c f (c-d) (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{3 d^2 \tan (e+f x)}{4 c f (c-d) (c+d)^2 \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}-\frac{\sqrt{2} \sqrt{a} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"Result too large to show","C",0
172,1,856,324,6.5747206,"\int \frac{(c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(c + d*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^(3/2),x]","\frac{2 \cos ^3\left(\frac{1}{2} (e+f x)\right) (c+d \sec (e+f x))^3 \sqrt{\frac{1}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}} \sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)} \left(-\frac{2 \left(2 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right) \sin ^2\left(\frac{1}{2} (e+f x)\right)+2 \sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)} \sin \left(\frac{1}{2} (e+f x)\right)-\sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right)\right) c^3}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}-\frac{4 (c-3 d) \sin \left(\frac{1}{2} (e+f x)\right) c^2}{\sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}}-\frac{3}{2} (c-d)^3 \tan ^{-1}\left(\frac{1-2 \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}}\right)+\frac{3}{2} (c-d)^3 \tan ^{-1}\left(\frac{2 \sin \left(\frac{1}{2} (e+f x)\right)+1}{\sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}}\right)-\frac{(c-d)^2 (11 c+d) \sin \left(\frac{1}{2} (e+f x)\right) \left(5 \sqrt{-\frac{\sin ^2\left(\frac{1}{2} (e+f x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}} \left(1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)^2 \left(3-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right) \left(\sqrt{-\frac{\sin ^2\left(\frac{1}{2} (e+f x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}}-\tanh ^{-1}\left(\sqrt{-\frac{\sin ^2\left(\frac{1}{2} (e+f x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}}\right)\right) \csc ^4\left(\frac{1}{2} (e+f x)\right)+\frac{2 \cos ^2\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\frac{\sin ^2\left(\frac{1}{2} (e+f x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}\right) \sin ^2\left(\frac{1}{2} (e+f x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}\right)}{10 \left(1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)^{3/2}}-\frac{(c-d)^3 \sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}}{1-\sin \left(\frac{1}{2} (e+f x)\right)}+\frac{(c-d)^3 \sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}}{\sin \left(\frac{1}{2} (e+f x)\right)+1}-\frac{(c-d)^3 \left(2 \sin \left(\frac{1}{2} (e+f x)\right)+1\right)}{4 \left(1-\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}}+\frac{(c-d)^3 \left(1-2 \sin \left(\frac{1}{2} (e+f x)\right)\right)}{4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{1-2 \sin ^2\left(\frac{1}{2} (e+f x)\right)}}\right)}{f (d+c \cos (e+f x))^3 \sec ^{\frac{3}{2}}(e+f x) (a (\sec (e+f x)+1))^{3/2}}","\frac{2 c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^3 \tan (e+f x)}{2 a f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} \sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} (c-d)^2 (c+2 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 \tan (e+f x)}{a f \sqrt{a \sec (e+f x)+a}}",1,"(2*Cos[(e + f*x)/2]^3*(c + d*Sec[e + f*x])^3*Sqrt[(1 - 2*Sin[(e + f*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[(e + f*x)/2]^2]*((-3*(c - d)^3*ArcTan[(1 - 2*Sin[(e + f*x)/2])/Sqrt[1 - 2*Sin[(e + f*x)/2]^2]])/2 + (3*(c - d)^3*ArcTan[(1 + 2*Sin[(e + f*x)/2])/Sqrt[1 - 2*Sin[(e + f*x)/2]^2]])/2 - (4*c^2*(c - 3*d)*Sin[(e + f*x)/2])/Sqrt[1 - 2*Sin[(e + f*x)/2]^2] + ((c - d)^3*(1 - 2*Sin[(e + f*x)/2]))/(4*(1 + Sin[(e + f*x)/2])*Sqrt[1 - 2*Sin[(e + f*x)/2]^2]) - ((c - d)^3*(1 + 2*Sin[(e + f*x)/2]))/(4*(1 - Sin[(e + f*x)/2])*Sqrt[1 - 2*Sin[(e + f*x)/2]^2]) - ((c - d)^3*Sqrt[1 - 2*Sin[(e + f*x)/2]^2])/(1 - Sin[(e + f*x)/2]) + ((c - d)^3*Sqrt[1 - 2*Sin[(e + f*x)/2]^2])/(1 + Sin[(e + f*x)/2]) - (2*c^3*(-(Sqrt[2]*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]]) + 2*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]]*Sin[(e + f*x)/2]^2 + 2*Sin[(e + f*x)/2]*Sqrt[1 - 2*Sin[(e + f*x)/2]^2]))/(1 - 2*Sin[(e + f*x)/2]^2) - ((c - d)^2*(11*c + d)*Sin[(e + f*x)/2]*((2*Cos[(e + f*x)/2]^2*Hypergeometric2F1[2, 5/2, 7/2, -(Sin[(e + f*x)/2]^2/(1 - 2*Sin[(e + f*x)/2]^2))]*Sin[(e + f*x)/2]^2)/(1 - 2*Sin[(e + f*x)/2]^2) + 5*Csc[(e + f*x)/2]^4*Sqrt[-(Sin[(e + f*x)/2]^2/(1 - 2*Sin[(e + f*x)/2]^2))]*(1 - 2*Sin[(e + f*x)/2]^2)^2*(3 - 2*Sin[(e + f*x)/2]^2)*(-ArcTanh[Sqrt[-(Sin[(e + f*x)/2]^2/(1 - 2*Sin[(e + f*x)/2]^2))]] + Sqrt[-(Sin[(e + f*x)/2]^2/(1 - 2*Sin[(e + f*x)/2]^2))])))/(10*(1 - 2*Sin[(e + f*x)/2]^2)^(3/2))))/(f*(d + c*Cos[e + f*x])^3*Sec[e + f*x]^(3/2)*(a*(1 + Sec[e + f*x]))^(3/2))","C",0
173,1,16153,290,27.7494141,"\int \frac{(c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(c + d*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^(3/2),x]","\text{Result too large to show}","-\frac{\sqrt{2} \left(c^2-d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^2 \tan (e+f x)}{2 a f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} \sqrt{a} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"Result too large to show","C",0
174,1,10105,127,26.6960422,"\int \frac{c+d \sec (e+f x)}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(c + d*Sec[e + f*x])/(a + a*Sec[e + f*x])^(3/2),x]","\text{Result too large to show}","-\frac{(5 c-d) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f}+\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{3/2} f}-\frac{(c-d) \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{3/2}}",1,"Result too large to show","C",0
175,1,378865,394,35.2068439,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])),x]","\text{Result too large to show}","-\frac{2 d^{5/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{\sqrt{a} c f (c-d)^2 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{2 a f (c-d) (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} \sqrt{a} f (c-d) \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} (c-2 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{a} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} c f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"Result too large to show","C",0
176,1,582620,560,38.1031728,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^2} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^2),x]","\text{Result too large to show}","-\frac{2 d^{5/2} (3 c-d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{\sqrt{a} c^2 f (c-d)^3 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{d^{5/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{\sqrt{a} c f (c-d)^2 (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{d^3 \tan (e+f x)}{a c f (c-d)^2 (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}-\frac{\tan (e+f x)}{2 a f (c-d)^2 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} \sqrt{a} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} (c-3 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{a} f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"Result too large to show","C",0
177,1,776222,802,41.2298501,"\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^3),x]","\text{Result too large to show}","-\frac{(3 c-d) \tan (e+f x) d^3}{a c^2 (c-d)^3 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}-\frac{3 \tan (e+f x) d^3}{4 a c \left(c^2-d^2\right)^2 f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}-\frac{\tan (e+f x) d^3}{2 a c (c-d)^2 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))^2}-\frac{2 \left(6 c^2-4 d c+d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{5/2}}{\sqrt{a} c^3 (c-d)^4 \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{(3 c-d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{5/2}}{\sqrt{a} c^2 (c-d)^3 (c+d)^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{5/2}}{4 \sqrt{a} c (c-d)^2 (c+d)^{5/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right) \tan (e+f x)}{\sqrt{a} c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right) \tan (e+f x)}{2 \sqrt{2} \sqrt{a} (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\sqrt{2} (c-4 d) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right) \tan (e+f x)}{\sqrt{a} (c-d)^4 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\tan (e+f x)}{2 a (c-d)^3 f (\sec (e+f x)+1) \sqrt{\sec (e+f x) a+a}}",1,"Result too large to show","C",0
178,1,21194,480,29.3896667,"\int \frac{(c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^{5/2}} \, dx","Integrate[(c + d*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^(5/2),x]","\text{Result too large to show}","-\frac{\sqrt{2} \left(c^3-d^3\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 c^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 (c-d)^3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^2 (c+2 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 (c-d)^3 \tan (e+f x)}{16 a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^2 (c+2 d) \tan (e+f x)}{2 a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^3 \tan (e+f x)}{4 a^2 f (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}",1,"Result too large to show","C",0
179,1,16249,468,27.9533502,"\int \frac{(c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx","Integrate[(c + d*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^(5/2),x]","\text{Result too large to show}","-\frac{\left(c^2-d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} c^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 (c-d)^2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\left(c^2-d^2\right) \tan (e+f x)}{2 a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{3 (c-d)^2 \tan (e+f x)}{16 a^2 f (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-d)^2 \tan (e+f x)}{4 a^2 f (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}",1,"Result too large to show","C",0
180,1,11243,164,26.9472022,"\int \frac{c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx","Integrate[(c + d*Sec[e + f*x])/(a + a*Sec[e + f*x])^(5/2),x]","\text{Result too large to show}","-\frac{(43 c-3 d) \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f}+\frac{2 c \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{a^{5/2} f}-\frac{(11 c-3 d) \tan (e+f x)}{16 a f (a \sec (e+f x)+a)^{3/2}}-\frac{(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}}",1,"Result too large to show","C",0
181,1,486155,592,36.9961249,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])),x]","\text{Result too large to show}","-\frac{\sqrt{2} \left(c^2-3 c d+3 d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{a^{3/2} f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^{7/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{a^{3/2} c f (c-d)^3 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} f (c-d) \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-2 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{a^{3/2} c f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x)}{16 a^2 f (c-d) (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-2 d) \tan (e+f x)}{2 a^2 f (c-d)^2 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{4 a^2 f (c-d) (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}",1,"Result too large to show","C",0
182,1,688080,756,40.0373879,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))^2} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^2),x]","\text{Result too large to show}","\frac{2 d^{7/2} (4 c-d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{a^{3/2} c^2 f (c-d)^4 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \left(c^2-4 c d+6 d^2\right) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{a^{3/2} f (c-d)^4 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right)}{a^{3/2} c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{d^{7/2} \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right)}{a^{3/2} c f (c-d)^3 (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-3 d) \tan (e+f x) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{d^4 \tan (e+f x)}{a^2 c f (c-d)^3 (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}-\frac{3 \tan (e+f x)}{16 a^2 f (c-d)^2 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-3 d) \tan (e+f x)}{2 a^2 f (c-d)^3 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{4 a^2 f (c-d)^2 (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}",1,"Result too large to show","C",0
183,1,893714,999,43.4423468,"\int \frac{1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))^3} \, dx","Integrate[1/((a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^3),x]","\text{Result too large to show}","\frac{(4 c-d) \tan (e+f x) d^4}{a^2 c^2 (c-d)^4 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}+\frac{3 \tan (e+f x) d^4}{4 a^2 c (c-d)^3 (c+d)^2 f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}+\frac{\tan (e+f x) d^4}{2 a^2 c (c-d)^3 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))^2}+\frac{2 \left(10 c^2-5 d c+d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{7/2}}{a^{3/2} c^3 (c-d)^5 \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}+\frac{(4 c-d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{7/2}}{a^{3/2} c^2 (c-d)^4 (c+d)^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right) \tan (e+f x) d^{7/2}}{4 a^{3/2} c (c-d)^3 (c+d)^{5/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right) \tan (e+f x)}{a^{3/2} c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\sqrt{2} \left(c^2-5 d c+10 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right) \tan (e+f x)}{a^{3/2} (c-d)^5 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right) \tan (e+f x)}{16 \sqrt{2} a^{3/2} (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{(c-4 d) \tanh ^{-1}\left(\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right) \tan (e+f x)}{2 \sqrt{2} a^{3/2} (c-d)^4 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{3 \tan (e+f x)}{16 a^2 (c-d)^3 f (\sec (e+f x)+1) \sqrt{\sec (e+f x) a+a}}-\frac{(c-4 d) \tan (e+f x)}{2 a^2 (c-d)^4 f (\sec (e+f x)+1) \sqrt{\sec (e+f x) a+a}}-\frac{\tan (e+f x)}{4 a^2 (c-d)^3 f (\sec (e+f x)+1)^2 \sqrt{\sec (e+f x) a+a}}",1,"Result too large to show","C",0
184,1,240,123,18.421876,"\int \sqrt{a+a \sec (e+f x)} \sqrt{c+d \sec (e+f x)} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]],x]","-\frac{2 \cot (e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c+d \sec (e+f x)} \left(\sqrt{c^2 \sin ^2(e+f x)} \sqrt{c-c \cos (e+f x)} \tan ^{-1}\left(\frac{\sqrt{c (\cos (e+f x)+1)} \sqrt{c \cos (e+f x)+d}}{\sqrt{c^2 \sin ^2(e+f x)}}\right)-2 \sqrt{c} \sqrt{d} \sin ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{c (\cos (e+f x)+1)} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{c \cos (e+f x)+d}}{\sqrt{d} \sqrt{c-c \cos (e+f x)}}\right)\right)}{f \sqrt{c (\cos (e+f x)+1)} \sqrt{c-c \cos (e+f x)} \sqrt{c \cos (e+f x)+d}}","\frac{2 \sqrt{a} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{f}+\frac{2 \sqrt{a} \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{f}",1,"(-2*Cot[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c + d*Sec[e + f*x]]*(-2*Sqrt[c]*Sqrt[d]*ArcTanh[(Sqrt[c]*Sqrt[d + c*Cos[e + f*x]])/(Sqrt[d]*Sqrt[c - c*Cos[e + f*x]])]*Sqrt[c*(1 + Cos[e + f*x])]*Sin[(e + f*x)/2]^2 + ArcTan[(Sqrt[c*(1 + Cos[e + f*x])]*Sqrt[d + c*Cos[e + f*x]])/Sqrt[c^2*Sin[e + f*x]^2]]*Sqrt[c - c*Cos[e + f*x]]*Sqrt[c^2*Sin[e + f*x]^2]))/(f*Sqrt[c*(1 + Cos[e + f*x])]*Sqrt[c - c*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])","A",1
185,1,102,61,0.2165353,"\int \frac{\sqrt{a+a \sec (e+f x)}}{\sqrt{c+d \sec (e+f x)}} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]/Sqrt[c + d*Sec[e + f*x]],x]","\frac{\sqrt{2} \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \sqrt{c \cos (e+f x)+d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c \cos (e+f x)+d}}\right)}{\sqrt{c} f \sqrt{c+d \sec (e+f x)}}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{c} f}",1,"(Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[c]*Sin[(e + f*x)/2])/Sqrt[d + c*Cos[e + f*x]]]*Sqrt[d + c*Cos[e + f*x]]*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])])/(Sqrt[c]*f*Sqrt[c + d*Sec[e + f*x]])","A",1
186,1,135,111,0.9601694,"\int \frac{\sqrt{a+a \sec (e+f x)}}{(c+d \sec (e+f x))^{3/2}} \, dx","Integrate[Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x])^(3/2),x]","-\frac{\sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \left(2 \sqrt{c} d \sin \left(\frac{1}{2} (e+f x)\right)-\sqrt{2} (c+d)^{3/2} \sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right) \sqrt{\frac{c \cos (e+f x)+d}{c+d}}\right)}{c^{3/2} f (c+d) \sqrt{c+d \sec (e+f x)}}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{c^{3/2} f}-\frac{2 a d \tan (e+f x)}{c f (c+d) \sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}",1,"-((Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])]*(-(Sqrt[2]*(c + d)^(3/2)*ArcSin[(Sqrt[2]*Sqrt[c]*Sin[(e + f*x)/2])/Sqrt[c + d]]*Sqrt[(d + c*Cos[e + f*x])/(c + d)]) + 2*Sqrt[c]*d*Sin[(e + f*x)/2]))/(c^(3/2)*(c + d)*f*Sqrt[c + d*Sec[e + f*x]]))","A",1
187,1,184,141,14.6152302,"\int \frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[Sqrt[c + d*Sec[e + f*x]]/Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \sqrt{c+d \sec (e+f x)} \left(\frac{\sqrt{2} \sqrt{c} \sqrt{c+d} \sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right) \sqrt{\frac{c \cos (e+f x)+d}{c+d}}}{\sqrt{c \cos (e+f x)+d}}+\sqrt{d-c} \tanh ^{-1}\left(\frac{\sqrt{d-c} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c \cos (e+f x)+d}}\right)\right)}{f \sqrt{a (\sec (e+f x)+1)} \sqrt{c \cos (e+f x)+d}}","\frac{2 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{a} f}-\frac{\sqrt{2} \sqrt{c-d} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{a} f}",1,"(2*Cos[(e + f*x)/2]*(Sqrt[-c + d]*ArcTanh[(Sqrt[-c + d]*Sin[(e + f*x)/2])/Sqrt[d + c*Cos[e + f*x]]] + (Sqrt[2]*Sqrt[c]*Sqrt[c + d]*ArcSin[(Sqrt[2]*Sqrt[c]*Sin[(e + f*x)/2])/Sqrt[c + d]]*Sqrt[(d + c*Cos[e + f*x])/(c + d)])/Sqrt[d + c*Cos[e + f*x]])*Sqrt[c + d*Sec[e + f*x]])/(f*Sqrt[d + c*Cos[e + f*x]]*Sqrt[a*(1 + Sec[e + f*x])])","A",1
188,1,171,141,0.3497161,"\int \frac{1}{\sqrt{a+a \sec (e+f x)} \sqrt{c+d \sec (e+f x)}} \, dx","Integrate[1/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]),x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{c \cos (e+f x)+d} \left(\sqrt{2} \sqrt{c-d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c \cos (e+f x)+d}}\right)-\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c-d} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c \cos (e+f x)+d}}\right)\right)}{\sqrt{c} f \sqrt{c-d} \sqrt{a (\sec (e+f x)+1)} \sqrt{c+d \sec (e+f x)}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{a} \sqrt{c} f}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right)}{\sqrt{a} f \sqrt{c-d}}",1,"(2*(Sqrt[2]*Sqrt[c - d]*ArcTan[(Sqrt[2]*Sqrt[c]*Sin[(e + f*x)/2])/Sqrt[d + c*Cos[e + f*x]]] - Sqrt[c]*ArcTan[(Sqrt[c - d]*Sin[(e + f*x)/2])/Sqrt[d + c*Cos[e + f*x]]])*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*Sec[e + f*x])/(Sqrt[c]*Sqrt[c - d]*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c + d*Sec[e + f*x]])","A",1
189,1,68,67,0.1651725,"\int \frac{a+b \sec (e+f x)}{c+d \sec (e+f x)} \, dx","Integrate[(a + b*Sec[e + f*x])/(c + d*Sec[e + f*x]),x]","\frac{\frac{2 (a d-b c) \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}+a (e+f x)}{c f}","\frac{2 (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c f \sqrt{c-d} \sqrt{c+d}}+\frac{a x}{c}",1,"(a*(e + f*x) + (2*(-(b*c) + a*d)*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2])/(c*f)","A",1
190,1,155,123,0.6606327,"\int \frac{a+b \sec (e+f x)}{(c+d \sec (e+f x))^2} \, dx","Integrate[(a + b*Sec[e + f*x])/(c + d*Sec[e + f*x])^2,x]","\frac{\frac{-c d (b c-a d) \sin (e+f x)+a d \left(c^2-d^2\right) (e+f x)+a c \left(c^2-d^2\right) (e+f x) \cos (e+f x)}{c \cos (e+f x)+d}-\frac{2 \left(a d \left(d^2-2 c^2\right)+b c^3\right) \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}}{c^2 f (c-d) (c+d)}","-\frac{d (b c-a d) \tan (e+f x)}{c f \left(c^2-d^2\right) (c+d \sec (e+f x))}+\frac{2 \left(-2 a c^2 d+a d^3+b c^3\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^2 f (c-d)^{3/2} (c+d)^{3/2}}+\frac{a x}{c^2}",1,"((-2*(b*c^3 + a*d*(-2*c^2 + d^2))*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] + (a*d*(c^2 - d^2)*(e + f*x) + a*c*(c^2 - d^2)*(e + f*x)*Cos[e + f*x] - c*d*(b*c - a*d)*Sin[e + f*x])/(d + c*Cos[e + f*x]))/(c^2*(c - d)*(c + d)*f)","A",1
191,1,267,204,1.4179194,"\int \frac{a+b \sec (e+f x)}{(c+d \sec (e+f x))^3} \, dx","Integrate[(a + b*Sec[e + f*x])/(c + d*Sec[e + f*x])^3,x]","\frac{\sec ^2(e+f x) (a+b \sec (e+f x)) (c \cos (e+f x)+d) \left(-\frac{c d \left(-6 a c^2 d+3 a d^3+4 b c^3-b c d^2\right) \sin (e+f x) (c \cos (e+f x)+d)}{(c-d)^2 (c+d)^2}-\frac{2 \left(a d \left(-6 c^4+5 c^2 d^2-2 d^4\right)+b c^3 \left(2 c^2+d^2\right)\right) (c \cos (e+f x)+d)^2 \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\left(c^2-d^2\right)^{5/2}}+\frac{c d^2 (b c-a d) \sin (e+f x)}{(c-d) (c+d)}+2 a (e+f x) (c \cos (e+f x)+d)^2\right)}{2 c^3 f (a \cos (e+f x)+b) (c+d \sec (e+f x))^3}","-\frac{d (b c-a d) \tan (e+f x)}{2 c f \left(c^2-d^2\right) (c+d \sec (e+f x))^2}-\frac{d \left(-5 a c^2 d+2 a d^3+3 b c^3\right) \tan (e+f x)}{2 c^2 f \left(c^2-d^2\right)^2 (c+d \sec (e+f x))}+\frac{\left(b c^3 \left(2 c^2+d^2\right)-a d \left(6 c^4-5 c^2 d^2+2 d^4\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^3 f (c-d)^{5/2} (c+d)^{5/2}}+\frac{a x}{c^3}",1,"((d + c*Cos[e + f*x])*Sec[e + f*x]^2*(a + b*Sec[e + f*x])*(2*a*(e + f*x)*(d + c*Cos[e + f*x])^2 - (2*(b*c^3*(2*c^2 + d^2) + a*d*(-6*c^4 + 5*c^2*d^2 - 2*d^4))*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x])^2)/(c^2 - d^2)^(5/2) + (c*d^2*(b*c - a*d)*Sin[e + f*x])/((c - d)*(c + d)) - (c*d*(4*b*c^3 - 6*a*c^2*d - b*c*d^2 + 3*a*d^3)*(d + c*Cos[e + f*x])*Sin[e + f*x])/((c - d)^2*(c + d)^2)))/(2*c^3*f*(b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^3)","A",1
192,1,136,133,0.7937992,"\int \frac{(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^2} \, dx","Integrate[(a + b*Sec[e + f*x])^2/(c + d*Sec[e + f*x])^2,x]","\frac{\frac{2 \left(a^2 \left(2 c^2 d-d^3\right)-2 a b c^3+b^2 c^2 d\right) \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\left(c^2-d^2\right)^{3/2}}+a^2 (e+f x)+\frac{c (b c-a d)^2 \sin (e+f x)}{(c-d) (c+d) (c \cos (e+f x)+d)}}{c^2 f}","\frac{a^2 x}{c^2}+\frac{(b c-a d)^2 \sin (e+f x)}{c f \left(c^2-d^2\right) (c \cos (e+f x)+d)}+\frac{2 (b c-a d) \left(2 a c^2-a d^2-b c d\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^2 f (c-d)^{3/2} (c+d)^{3/2}}",1,"(a^2*(e + f*x) + (2*(-2*a*b*c^3 + b^2*c^2*d + a^2*(2*c^2*d - d^3))*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(3/2) + (c*(b*c - a*d)^2*Sin[e + f*x])/((c - d)*(c + d)*(d + c*Cos[e + f*x])))/(c^2*f)","A",1
193,1,493,237,2.0436952,"\int \frac{(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^3} \, dx","Integrate[(a + b*Sec[e + f*x])^2/(c + d*Sec[e + f*x])^3,x]","\frac{\sec (e+f x) (a+b \sec (e+f x))^2 (c \cos (e+f x)+d) \left(\frac{4 \left(a^2 \left(6 c^4 d-5 c^2 d^3+2 d^5\right)-2 a b c^3 \left(2 c^2+d^2\right)+3 b^2 c^4 d\right) (c \cos (e+f x)+d)^2 \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\left(c^2-d^2\right)^{5/2}}+\frac{2 a^2 c^6 e+2 a^2 c^6 f x+6 a^2 c^4 d^2 \sin (2 (e+f x))+10 a^2 c^3 d^3 \sin (e+f x)-3 a^2 c^2 d^4 \sin (2 (e+f x))-6 a^2 c^2 d^4 e-6 a^2 c^2 d^4 f x+2 a^2 c^2 \left(c^2-d^2\right)^2 (e+f x) \cos (2 (e+f x))+8 a^2 c d \left(c^2-d^2\right)^2 (e+f x) \cos (e+f x)-4 a^2 c d^5 \sin (e+f x)+4 a^2 d^6 e+4 a^2 d^6 f x-8 a b c^5 d \sin (2 (e+f x))-12 a b c^4 d^2 \sin (e+f x)+2 a b c^3 d^3 \sin (2 (e+f x))+2 b^2 c^6 \sin (2 (e+f x))+2 b^2 c^5 d \sin (e+f x)+b^2 c^4 d^2 \sin (2 (e+f x))+4 b^2 c^3 d^3 \sin (e+f x)}{\left(c^2-d^2\right)^2}\right)}{4 c^3 f (a \cos (e+f x)+b)^2 (c+d \sec (e+f x))^3}","-\frac{\left(a^2 \left(6 c^4 d-5 c^2 d^3+2 d^5\right)-2 a b c^3 \left(2 c^2+d^2\right)+3 b^2 c^4 d\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^3 f (c-d)^{5/2} (c+d)^{5/2}}+\frac{a^2 x}{c^3}-\frac{(b c-a d) \left(3 a d \left(2 c^2-d^2\right)-b c \left(2 c^2+d^2\right)\right) \sin (e+f x)}{2 c^2 f \left(c^2-d^2\right)^2 (c \cos (e+f x)+d)}-\frac{d (b c-a d)^2 \sin (e+f x)}{2 c^2 f \left(c^2-d^2\right) (c \cos (e+f x)+d)^2}",1,"((d + c*Cos[e + f*x])*Sec[e + f*x]*(a + b*Sec[e + f*x])^2*((4*(3*b^2*c^4*d - 2*a*b*c^3*(2*c^2 + d^2) + a^2*(6*c^4*d - 5*c^2*d^3 + 2*d^5))*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x])^2)/(c^2 - d^2)^(5/2) + (2*a^2*c^6*e - 6*a^2*c^2*d^4*e + 4*a^2*d^6*e + 2*a^2*c^6*f*x - 6*a^2*c^2*d^4*f*x + 4*a^2*d^6*f*x + 8*a^2*c*d*(c^2 - d^2)^2*(e + f*x)*Cos[e + f*x] + 2*a^2*c^2*(c^2 - d^2)^2*(e + f*x)*Cos[2*(e + f*x)] + 2*b^2*c^5*d*Sin[e + f*x] - 12*a*b*c^4*d^2*Sin[e + f*x] + 10*a^2*c^3*d^3*Sin[e + f*x] + 4*b^2*c^3*d^3*Sin[e + f*x] - 4*a^2*c*d^5*Sin[e + f*x] + 2*b^2*c^6*Sin[2*(e + f*x)] - 8*a*b*c^5*d*Sin[2*(e + f*x)] + 6*a^2*c^4*d^2*Sin[2*(e + f*x)] + b^2*c^4*d^2*Sin[2*(e + f*x)] + 2*a*b*c^3*d^3*Sin[2*(e + f*x)] - 3*a^2*c^2*d^4*Sin[2*(e + f*x)])/(c^2 - d^2)^2))/(4*c^3*f*(b + a*Cos[e + f*x])^2*(c + d*Sec[e + f*x])^3)","B",1
194,1,438,377,3.4508801,"\int \frac{(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^4} \, dx","Integrate[(a + b*Sec[e + f*x])^2/(c + d*Sec[e + f*x])^4,x]","\frac{\sec ^2(e+f x) (a+b \sec (e+f x))^2 (c \cos (e+f x)+d) \left(\frac{c \left(a^2 d^2 \left(36 c^4-32 c^2 d^2+11 d^4\right)-2 a b c d \left(18 c^4-5 c^2 d^2+2 d^4\right)+b^2 \left(6 c^6+10 c^4 d^2-c^2 d^4\right)\right) \sin (e+f x) (c \cos (e+f x)+d)^2}{\left(c^2-d^2\right)^3}-\frac{c d \left(a^2 d^2 \left(12 c^2-7 d^2\right)+a b \left(8 c d^3-18 c^3 d\right)+b^2 \left(6 c^4-c^2 d^2\right)\right) \sin (e+f x) (c \cos (e+f x)+d)}{\left(c^2-d^2\right)^2}+\frac{6 \left(a^2 \left(8 c^6 d-8 c^4 d^3+7 c^2 d^5-2 d^7\right)-2 a b c^5 \left(2 c^2+3 d^2\right)+b^2 c^4 d \left(4 c^2+d^2\right)\right) (c \cos (e+f x)+d)^3 \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\left(c^2-d^2\right)^{7/2}}+6 a^2 (e+f x) (c \cos (e+f x)+d)^3+\frac{2 c d^2 (b c-a d)^2 \sin (e+f x)}{c^2-d^2}\right)}{6 c^4 f (a \cos (e+f x)+b)^2 (c+d \sec (e+f x))^4}","-\frac{\left(-a^2 d^2 \left(34 c^4-28 c^2 d^2+9 d^4\right)+2 a b c d \left(18 c^4-5 c^2 d^2+2 d^4\right)-\left(b^2 \left(6 c^6+10 c^4 d^2-c^2 d^4\right)\right)\right) \sin (e+f x)}{6 c^3 f \left(c^2-d^2\right)^3 (c \cos (e+f x)+d)}-\frac{\left(a^2 \left(8 c^6 d-8 c^4 d^3+7 c^2 d^5-2 d^7\right)-a b \left(4 c^7+6 c^5 d^2\right)+b^2 c^4 d \left(4 c^2+d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^4 f (c-d)^{7/2} (c+d)^{7/2}}+\frac{a^2 x}{c^4}+\frac{d^2 \sin (e+f x) (a \cos (e+f x)+b)^2}{3 c f \left(c^2-d^2\right) (c \cos (e+f x)+d)^3}-\frac{d (b c-a d) \left(-8 a c^2 d+3 a d^3+6 b c^3-b c d^2\right) \sin (e+f x)}{6 c^3 f \left(c^2-d^2\right)^2 (c \cos (e+f x)+d)^2}",1,"((d + c*Cos[e + f*x])*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^2*(6*a^2*(e + f*x)*(d + c*Cos[e + f*x])^3 + (6*(b^2*c^4*d*(4*c^2 + d^2) - 2*a*b*c^5*(2*c^2 + 3*d^2) + a^2*(8*c^6*d - 8*c^4*d^3 + 7*c^2*d^5 - 2*d^7))*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x])^3)/(c^2 - d^2)^(7/2) + (2*c*d^2*(b*c - a*d)^2*Sin[e + f*x])/(c^2 - d^2) - (c*d*(a^2*d^2*(12*c^2 - 7*d^2) + b^2*(6*c^4 - c^2*d^2) + a*b*(-18*c^3*d + 8*c*d^3))*(d + c*Cos[e + f*x])*Sin[e + f*x])/(c^2 - d^2)^2 + (c*(-2*a*b*c*d*(18*c^4 - 5*c^2*d^2 + 2*d^4) + a^2*d^2*(36*c^4 - 32*c^2*d^2 + 11*d^4) + b^2*(6*c^6 + 10*c^4*d^2 - c^2*d^4))*(d + c*Cos[e + f*x])^2*Sin[e + f*x])/(c^2 - d^2)^3))/(6*c^4*f*(b + a*Cos[e + f*x])^2*(c + d*Sec[e + f*x])^4)","A",1
195,1,517,254,2.3033662,"\int \frac{(a+b \sec (e+f x))^3}{(c+d \sec (e+f x))^3} \, dx","Integrate[(a + b*Sec[e + f*x])^3/(c + d*Sec[e + f*x])^3,x]","\frac{\frac{2 a^3 c^6 e+2 a^3 c^6 f x+6 a^3 c^4 d^2 \sin (2 (e+f x))+10 a^3 c^3 d^3 \sin (e+f x)+2 a^3 \left(c^3-c d^2\right)^2 (e+f x) \cos (2 (e+f x))-3 a^3 c^2 d^4 \sin (2 (e+f x))-6 a^3 c^2 d^4 e-6 a^3 c^2 d^4 f x+8 a^3 c d \left(c^2-d^2\right)^2 (e+f x) \cos (e+f x)-4 a^3 c d^5 \sin (e+f x)+4 a^3 d^6 e+4 a^3 d^6 f x-12 a^2 b c^5 d \sin (2 (e+f x))-18 a^2 b c^4 d^2 \sin (e+f x)+3 a^2 b c^3 d^3 \sin (2 (e+f x))+6 a b^2 c^6 \sin (2 (e+f x))+6 a b^2 c^5 d \sin (e+f x)+3 a b^2 c^4 d^2 \sin (2 (e+f x))+12 a b^2 c^3 d^3 \sin (e+f x)+2 b^3 c^6 \sin (e+f x)-3 b^3 c^5 d \sin (2 (e+f x))-8 b^3 c^4 d^2 \sin (e+f x)}{\left(c^2-d^2\right)^2 (c \cos (e+f x)+d)^2}-\frac{4 \left(a^3 \left(-6 c^4 d+5 c^2 d^3-2 d^5\right)+3 a^2 b c^3 \left(2 c^2+d^2\right)-9 a b^2 c^4 d+b^3 c^3 \left(c^2+2 d^2\right)\right) \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\left(c^2-d^2\right)^{5/2}}}{4 c^3 f}","\frac{a^3 x}{c^3}-\frac{(b c-a d) \left(-\left(a^2 \left(6 c^4-5 c^2 d^2+2 d^4\right)\right)+2 a b c d \left(4 c^2-d^2\right)-b^2 c^2 \left(c^2+2 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^3 f (c-d)^{5/2} (c+d)^{5/2}}+\frac{(b c-a d)^2 \left(5 a c^2-2 a d^2-3 b c d\right) \sin (e+f x)}{2 c^2 f \left(c^2-d^2\right)^2 (c \cos (e+f x)+d)}+\frac{(b c-a d)^2 \sin (e+f x) (a \cos (e+f x)+b)}{2 c f \left(c^2-d^2\right) (c \cos (e+f x)+d)^2}",1,"((-4*(-9*a*b^2*c^4*d + 3*a^2*b*c^3*(2*c^2 + d^2) + b^3*c^3*(c^2 + 2*d^2) + a^3*(-6*c^4*d + 5*c^2*d^3 - 2*d^5))*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(5/2) + (2*a^3*c^6*e - 6*a^3*c^2*d^4*e + 4*a^3*d^6*e + 2*a^3*c^6*f*x - 6*a^3*c^2*d^4*f*x + 4*a^3*d^6*f*x + 8*a^3*c*d*(c^2 - d^2)^2*(e + f*x)*Cos[e + f*x] + 2*a^3*(c^3 - c*d^2)^2*(e + f*x)*Cos[2*(e + f*x)] + 2*b^3*c^6*Sin[e + f*x] + 6*a*b^2*c^5*d*Sin[e + f*x] - 18*a^2*b*c^4*d^2*Sin[e + f*x] - 8*b^3*c^4*d^2*Sin[e + f*x] + 10*a^3*c^3*d^3*Sin[e + f*x] + 12*a*b^2*c^3*d^3*Sin[e + f*x] - 4*a^3*c*d^5*Sin[e + f*x] + 6*a*b^2*c^6*Sin[2*(e + f*x)] - 12*a^2*b*c^5*d*Sin[2*(e + f*x)] - 3*b^3*c^5*d*Sin[2*(e + f*x)] + 6*a^3*c^4*d^2*Sin[2*(e + f*x)] + 3*a*b^2*c^4*d^2*Sin[2*(e + f*x)] + 3*a^2*b*c^3*d^3*Sin[2*(e + f*x)] - 3*a^3*c^2*d^4*Sin[2*(e + f*x)])/((c^2 - d^2)^2*(d + c*Cos[e + f*x])^2))/(4*c^3*f)","B",1
196,1,459,412,3.7768824,"\int \frac{(a+b \sec (e+f x))^3}{(c+d \sec (e+f x))^4} \, dx","Integrate[(a + b*Sec[e + f*x])^3/(c + d*Sec[e + f*x])^4,x]","\frac{\sec (e+f x) (a+b \sec (e+f x))^3 (c \cos (e+f x)+d) \left(6 a^3 (e+f x) (c \cos (e+f x)+d)^3+\frac{c \left(a^3 \left(36 c^4 d^2-32 c^2 d^4+11 d^6\right)-3 a^2 b c d \left(18 c^4-5 c^2 d^2+2 d^4\right)+3 a b^2 c^2 \left(6 c^4+10 c^2 d^2-d^4\right)-b^3 c^3 d \left(13 c^2+2 d^2\right)\right) \sin (e+f x) (c \cos (e+f x)+d)^2}{\left(c^2-d^2\right)^3}-\frac{6 \left(a^3 \left(-8 c^6 d+8 c^4 d^3-7 c^2 d^5+2 d^7\right)+a^2 b \left(6 c^7+9 c^5 d^2\right)-3 a b^2 c^4 d \left(4 c^2+d^2\right)+b^3 c^5 \left(c^2+4 d^2\right)\right) (c \cos (e+f x)+d)^3 \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\left(c^2-d^2\right)^{7/2}}-\frac{2 c d (b c-a d)^3 \sin (e+f x)}{c^2-d^2}+\frac{c \left(-12 a c^2 d+7 a d^3+3 b c^3+2 b c d^2\right) (b c-a d)^2 \sin (e+f x) (c \cos (e+f x)+d)}{\left(c^2-d^2\right)^2}\right)}{6 c^4 f (a \cos (e+f x)+b)^3 (c+d \sec (e+f x))^4}","\frac{a^3 x}{c^4}-\frac{(b c-a d) \left(a^2 \left(34 c^4 d-28 c^2 d^3+9 d^5\right)-a b c \left(18 c^4+17 c^2 d^2-5 d^4\right)+b^2 c^2 d \left(13 c^2+2 d^2\right)\right) \sin (e+f x)}{6 c^3 f \left(c^2-d^2\right)^3 (c \cos (e+f x)+d)}-\frac{\left(a^3 \left(8 c^6 d-8 c^4 d^3+7 c^2 d^5-2 d^7\right)-a^2 b \left(6 c^7+9 c^5 d^2\right)+3 a b^2 c^4 d \left(4 c^2+d^2\right)-b^3 c^5 \left(c^2+4 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^4 f \sqrt{c-d} \sqrt{c+d} \left(c^2-d^2\right)^3}-\frac{d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^2}{3 c f \left(c^2-d^2\right) (c \cos (e+f x)+d)^3}+\frac{(b c-a d)^2 \left(-8 a c^2 d+3 a d^3+3 b c^3+2 b c d^2\right) \sin (e+f x)}{6 c^3 f \left(c^2-d^2\right)^2 (c \cos (e+f x)+d)^2}",1,"((d + c*Cos[e + f*x])*Sec[e + f*x]*(a + b*Sec[e + f*x])^3*(6*a^3*(e + f*x)*(d + c*Cos[e + f*x])^3 - (6*(-3*a*b^2*c^4*d*(4*c^2 + d^2) + b^3*c^5*(c^2 + 4*d^2) + a^2*b*(6*c^7 + 9*c^5*d^2) + a^3*(-8*c^6*d + 8*c^4*d^3 - 7*c^2*d^5 + 2*d^7))*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x])^3)/(c^2 - d^2)^(7/2) - (2*c*d*(b*c - a*d)^3*Sin[e + f*x])/(c^2 - d^2) + (c*(b*c - a*d)^2*(3*b*c^3 - 12*a*c^2*d + 2*b*c*d^2 + 7*a*d^3)*(d + c*Cos[e + f*x])*Sin[e + f*x])/(c^2 - d^2)^2 + (c*(-(b^3*c^3*d*(13*c^2 + 2*d^2)) + 3*a*b^2*c^2*(6*c^4 + 10*c^2*d^2 - d^4) - 3*a^2*b*c*d*(18*c^4 - 5*c^2*d^2 + 2*d^4) + a^3*(36*c^4*d^2 - 32*c^2*d^4 + 11*d^6))*(d + c*Cos[e + f*x])^2*Sin[e + f*x])/(c^2 - d^2)^3))/(6*c^4*f*(b + a*Cos[e + f*x])^3*(c + d*Sec[e + f*x])^4)","A",1
197,1,1285,622,6.926181,"\int \frac{(a+b \sec (e+f x))^3}{(c+d \sec (e+f x))^5} \, dx","Integrate[(a + b*Sec[e + f*x])^3/(c + d*Sec[e + f*x])^5,x]","\frac{a^3 (e+f x) \sec ^2(e+f x) (a+b \sec (e+f x))^3 (d+c \cos (e+f x))^5}{c^5 f (b+a \cos (e+f x))^3 (c+d \sec (e+f x))^5}+\frac{\left(-4 b^3 c^9-24 a^2 b c^9+40 a^3 d c^8+60 a b^2 d c^8-27 b^3 d^2 c^7-72 a^2 b d^2 c^7-40 a^3 d^3 c^6+45 a b^2 d^3 c^6-4 b^3 d^4 c^5-9 a^2 b d^4 c^5+63 a^3 d^5 c^4-36 a^3 d^7 c^2+8 a^3 d^9\right) \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right) \sec ^2(e+f x) (a+b \sec (e+f x))^3 (d+c \cos (e+f x))^5}{4 c^5 \sqrt{c^2-d^2} \left(d^2-c^2\right)^4 f (b+a \cos (e+f x))^3 (c+d \sec (e+f x))^5}+\frac{\sec ^2(e+f x) (a+b \sec (e+f x))^3 \left(72 a b^2 \sin (e+f x) c^8-68 b^3 d \sin (e+f x) c^7-288 a^2 b d \sin (e+f x) c^7+240 a^3 d^2 \sin (e+f x) c^6+252 a b^2 d^2 \sin (e+f x) c^6-39 b^3 d^3 \sin (e+f x) c^5+24 a^2 b d^3 \sin (e+f x) c^5-280 a^3 d^4 \sin (e+f x) c^4-15 a b^2 d^4 \sin (e+f x) c^4+2 b^3 d^5 \sin (e+f x) c^3-69 a^2 b d^5 \sin (e+f x) c^3+195 a^3 d^6 \sin (e+f x) c^2+6 a b^2 d^6 \sin (e+f x) c^2+18 a^2 b d^7 \sin (e+f x) c-50 a^3 d^8 \sin (e+f x)\right) (d+c \cos (e+f x))^4}{24 c^4 \left(c^2-d^2\right)^4 f (b+a \cos (e+f x))^3 (c+d \sec (e+f x))^5}+\frac{\sec ^2(e+f x) (a+b \sec (e+f x))^3 \left(12 b^3 \sin (e+f x) c^7-108 a b^2 d \sin (e+f x) c^6+25 b^3 d^2 \sin (e+f x) c^5+216 a^2 b d^2 \sin (e+f x) c^5-120 a^3 d^3 \sin (e+f x) c^4+9 a b^2 d^3 \sin (e+f x) c^4-2 b^3 d^4 \sin (e+f x) c^3-165 a^2 b d^4 \sin (e+f x) c^3+131 a^3 d^5 \sin (e+f x) c^2-6 a b^2 d^5 \sin (e+f x) c^2+54 a^2 b d^6 \sin (e+f x) c-46 a^3 d^7 \sin (e+f x)\right) (d+c \cos (e+f x))^3}{24 c^4 \left(c^2-d^2\right)^3 f (b+a \cos (e+f x))^3 (c+d \sec (e+f x))^5}+\frac{\sec ^2(e+f x) (a+b \sec (e+f x))^3 \left(-13 a^3 \sin (e+f x) d^6+27 a^2 b c \sin (e+f x) d^5+20 a^3 c^2 \sin (e+f x) d^4-15 a b^2 c^2 \sin (e+f x) d^4+b^3 c^3 \sin (e+f x) d^3-48 a^2 b c^3 \sin (e+f x) d^3+36 a b^2 c^4 \sin (e+f x) d^2-8 b^3 c^5 \sin (e+f x) d\right) (d+c \cos (e+f x))^2}{12 c^4 \left(c^2-d^2\right)^2 f (b+a \cos (e+f x))^3 (c+d \sec (e+f x))^5}+\frac{\sec ^2(e+f x) (a+b \sec (e+f x))^3 \left(-a^3 \sin (e+f x) d^5+3 a^2 b c \sin (e+f x) d^4-3 a b^2 c^2 \sin (e+f x) d^3+b^3 c^3 \sin (e+f x) d^2\right) (d+c \cos (e+f x))}{4 c^4 \left(c^2-d^2\right) f (b+a \cos (e+f x))^3 (c+d \sec (e+f x))^5}","\frac{a^3 x}{c^5}-\frac{(b c-a d) \left(-a^2 d^2 \left(58 c^4-35 c^2 d^2+12 d^4\right)+2 a b c d \left(32 c^4+c^2 d^2+2 d^4\right)-\left(b^2 \left(12 c^6+25 c^4 d^2-2 c^2 d^4\right)\right)\right) \sin (e+f x)}{24 c^4 f \left(c^2-d^2\right)^3 (c \cos (e+f x)+d)^2}-\frac{\left(-\left(a^3 \left(212 c^6 d^2-210 c^4 d^4+139 c^2 d^6-36 d^8\right)\right)+a^2 b c d \left(272 c^6+10 c^4 d^2+49 c^2 d^4-16 d^6\right)-3 a b^2 c^2 \left(24 c^6+84 c^4 d^2-5 c^2 d^4+2 d^6\right)+b^3 c^3 d \left(68 c^4+39 c^2 d^2-2 d^4\right)\right) \sin (e+f x)}{24 c^4 f \left(c^2-d^2\right)^4 (c \cos (e+f x)+d)}-\frac{\left(a^3 \left(40 c^8 d-40 c^6 d^3+63 c^4 d^5-36 c^2 d^7+8 d^9\right)-3 a^2 b c^5 \left(8 c^4+24 c^2 d^2+3 d^4\right)+15 a b^2 c^6 d \left(4 c^2+3 d^2\right)-b^3 c^5 \left(4 c^4+27 c^2 d^2+4 d^4\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{4 c^5 f \sqrt{c-d} \sqrt{c+d} \left(c^2-d^2\right)^4}+\frac{d^2 \sin (e+f x) (a \cos (e+f x)+b)^3}{4 c f \left(c^2-d^2\right) (c \cos (e+f x)+d)^4}-\frac{d \left(-11 a c^2 d+4 a d^3+8 b c^3-b c d^2\right) \sin (e+f x) (a \cos (e+f x)+b)^2}{12 c^2 f \left(c^2-d^2\right)^2 (c \cos (e+f x)+d)^3}",1,"(a^3*(e + f*x)*(d + c*Cos[e + f*x])^5*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^3)/(c^5*f*(b + a*Cos[e + f*x])^3*(c + d*Sec[e + f*x])^5) + ((-24*a^2*b*c^9 - 4*b^3*c^9 + 40*a^3*c^8*d + 60*a*b^2*c^8*d - 72*a^2*b*c^7*d^2 - 27*b^3*c^7*d^2 - 40*a^3*c^6*d^3 + 45*a*b^2*c^6*d^3 - 9*a^2*b*c^5*d^4 - 4*b^3*c^5*d^4 + 63*a^3*c^4*d^5 - 36*a^3*c^2*d^7 + 8*a^3*d^9)*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x])^5*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^3)/(4*c^5*Sqrt[c^2 - d^2]*(-c^2 + d^2)^4*f*(b + a*Cos[e + f*x])^3*(c + d*Sec[e + f*x])^5) + ((d + c*Cos[e + f*x])*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^3*(b^3*c^3*d^2*Sin[e + f*x] - 3*a*b^2*c^2*d^3*Sin[e + f*x] + 3*a^2*b*c*d^4*Sin[e + f*x] - a^3*d^5*Sin[e + f*x]))/(4*c^4*(c^2 - d^2)*f*(b + a*Cos[e + f*x])^3*(c + d*Sec[e + f*x])^5) + ((d + c*Cos[e + f*x])^2*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^3*(-8*b^3*c^5*d*Sin[e + f*x] + 36*a*b^2*c^4*d^2*Sin[e + f*x] - 48*a^2*b*c^3*d^3*Sin[e + f*x] + b^3*c^3*d^3*Sin[e + f*x] + 20*a^3*c^2*d^4*Sin[e + f*x] - 15*a*b^2*c^2*d^4*Sin[e + f*x] + 27*a^2*b*c*d^5*Sin[e + f*x] - 13*a^3*d^6*Sin[e + f*x]))/(12*c^4*(c^2 - d^2)^2*f*(b + a*Cos[e + f*x])^3*(c + d*Sec[e + f*x])^5) + ((d + c*Cos[e + f*x])^3*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^3*(12*b^3*c^7*Sin[e + f*x] - 108*a*b^2*c^6*d*Sin[e + f*x] + 216*a^2*b*c^5*d^2*Sin[e + f*x] + 25*b^3*c^5*d^2*Sin[e + f*x] - 120*a^3*c^4*d^3*Sin[e + f*x] + 9*a*b^2*c^4*d^3*Sin[e + f*x] - 165*a^2*b*c^3*d^4*Sin[e + f*x] - 2*b^3*c^3*d^4*Sin[e + f*x] + 131*a^3*c^2*d^5*Sin[e + f*x] - 6*a*b^2*c^2*d^5*Sin[e + f*x] + 54*a^2*b*c*d^6*Sin[e + f*x] - 46*a^3*d^7*Sin[e + f*x]))/(24*c^4*(c^2 - d^2)^3*f*(b + a*Cos[e + f*x])^3*(c + d*Sec[e + f*x])^5) + ((d + c*Cos[e + f*x])^4*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^3*(72*a*b^2*c^8*Sin[e + f*x] - 288*a^2*b*c^7*d*Sin[e + f*x] - 68*b^3*c^7*d*Sin[e + f*x] + 240*a^3*c^6*d^2*Sin[e + f*x] + 252*a*b^2*c^6*d^2*Sin[e + f*x] + 24*a^2*b*c^5*d^3*Sin[e + f*x] - 39*b^3*c^5*d^3*Sin[e + f*x] - 280*a^3*c^4*d^4*Sin[e + f*x] - 15*a*b^2*c^4*d^4*Sin[e + f*x] - 69*a^2*b*c^3*d^5*Sin[e + f*x] + 2*b^3*c^3*d^5*Sin[e + f*x] + 195*a^3*c^2*d^6*Sin[e + f*x] + 6*a*b^2*c^2*d^6*Sin[e + f*x] + 18*a^2*b*c*d^7*Sin[e + f*x] - 50*a^3*d^8*Sin[e + f*x]))/(24*c^4*(c^2 - d^2)^4*f*(b + a*Cos[e + f*x])^3*(c + d*Sec[e + f*x])^5)","B",1
198,1,913,320,17.9352481,"\int \sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x)) \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x]),x]","\frac{2 d \cos (e+f x) \sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x)) \sin (e+f x)}{f (d+c \cos (e+f x))}+\frac{2 \sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x)) \left(a \sqrt{\frac{b-a}{a+b}} d \tan ^5\left(\frac{1}{2} (e+f x)\right)-b \sqrt{\frac{b-a}{a+b}} d \tan ^5\left(\frac{1}{2} (e+f x)\right)-2 a \sqrt{\frac{b-a}{a+b}} d \tan ^3\left(\frac{1}{2} (e+f x)\right)+2 i a c \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \sqrt{\frac{b-a}{a+b}} d \tan \left(\frac{1}{2} (e+f x)\right)+b \sqrt{\frac{b-a}{a+b}} d \tan \left(\frac{1}{2} (e+f x)\right)-i (a-b) d E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{a+b}}-i (a-b) (c-d) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{a+b}}+2 i a c \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{a+b}}\right)}{\sqrt{\frac{b-a}{a+b}} f \sqrt{b+a \cos (e+f x)} (d+c \cos (e+f x)) \sec ^{\frac{3}{2}}(e+f x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)}} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (e+f x)\right)+1}}}","\frac{2 \sqrt{a+b} (a d+b (c-d)) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b f}-\frac{2 c \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}-\frac{2 d (a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b f}",1,"(2*d*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])*Sin[e + f*x])/(f*(d + c*Cos[e + f*x])) + (2*Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])*(a*Sqrt[(-a + b)/(a + b)]*d*Tan[(e + f*x)/2] + b*Sqrt[(-a + b)/(a + b)]*d*Tan[(e + f*x)/2] - 2*a*Sqrt[(-a + b)/(a + b)]*d*Tan[(e + f*x)/2]^3 + a*Sqrt[(-a + b)/(a + b)]*d*Tan[(e + f*x)/2]^5 - b*Sqrt[(-a + b)/(a + b)]*d*Tan[(e + f*x)/2]^5 + (2*I)*a*c*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + (2*I)*a*c*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Tan[(e + f*x)/2]^2*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - I*(a - b)*d*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - I*(a - b)*(c - d)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)]))/(Sqrt[(-a + b)/(a + b)]*f*Sqrt[b + a*Cos[e + f*x]]*(d + c*Cos[e + f*x])*Sec[e + f*x]^(3/2)*Sqrt[(1 - Tan[(e + f*x)/2]^2)^(-1)]*(-1 + Tan[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(1 + Tan[(e + f*x)/2]^2)])","C",1
199,1,225,220,4.5911898,"\int \frac{\sqrt{a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]]/(c + d*Sec[e + f*x]),x]","\frac{4 \cos ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \sqrt{a+b \sec (e+f x)} \left(2 a \left(c^2-d^2\right) \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)-\left(c (a-b) (c+d) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)\right)+2 d (a d-b c) \Pi \left(\frac{c-d}{c+d};\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{c f (c-d) (c+d) (a \cos (e+f x)+b)}","\frac{2 (b c-a d) \tan (e+f x) \sqrt{\frac{a+b \sec (e+f x)}{a+b}} \Pi \left(\frac{2 d}{c+d};\sin ^{-1}\left(\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right)|\frac{2 b}{a+b}\right)}{c f (c+d) \sqrt{-\tan ^2(e+f x)} \sqrt{a+b \sec (e+f x)}}-\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{c f}",1,"(4*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*(-((a - b)*c*(c + d)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]) + 2*a*(c^2 - d^2)*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] + 2*d*(-(b*c) + a*d)*EllipticPi[(c - d)/(c + d), ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)])*Sqrt[a + b*Sec[e + f*x]])/(c*(c - d)*(c + d)*f*(b + a*Cos[e + f*x]))","A",1
200,1,6063,380,24.7647842,"\int (a+b \sec (e+f x))^{3/2} (c+d \sec (e+f x)) \, dx","Integrate[(a + b*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x]),x]","\text{Result too large to show}","\frac{2 \sqrt{a+b} \left(3 a^2 d+a b (6 c-4 d)-b^2 (3 c-d)\right) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b f}-\frac{2 (a-b) \sqrt{a+b} (4 a d+3 b c) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b f}-\frac{2 a c \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}+\frac{2 b d \tan (e+f x) \sqrt{a+b \sec (e+f x)}}{3 f}",1,"Result too large to show","B",0
201,1,230,326,6.3082796,"\int \frac{(a+b \sec (e+f x))^{3/2}}{c+d \sec (e+f x)} \, dx","Integrate[(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x]),x]","-\frac{4 \cos ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \sqrt{a+b \sec (e+f x)} \left(c (a-b)^2 (c+d) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)-2 \left(a^2 \left(c^2-d^2\right) \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)+(b c-a d)^2 \Pi \left(\frac{c-d}{c+d};\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)\right)\right)}{c f (c-d) (c+d) (a \cos (e+f x)+b)}","-\frac{2 (b c-a d)^2 \tan (e+f x) \sqrt{\frac{a+b \sec (e+f x)}{a+b}} \Pi \left(\frac{2 d}{c+d};\sin ^{-1}\left(\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right)|\frac{2 b}{a+b}\right)}{c d f (c+d) \sqrt{-\tan ^2(e+f x)} \sqrt{a+b \sec (e+f x)}}-\frac{2 a \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{c f}+\frac{2 b \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d f}",1,"(-4*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*((a - b)^2*c*(c + d)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] - 2*(a^2*(c^2 - d^2)*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] + (b*c - a*d)^2*EllipticPi[(c - d)/(c + d), ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]))*Sqrt[a + b*Sec[e + f*x]])/(c*(c - d)*(c + d)*f*(b + a*Cos[e + f*x]))","A",1
202,1,7138,442,25.9950363,"\int (a+b \sec (e+f x))^{5/2} (c+d \sec (e+f x)) \, dx","Integrate[(a + b*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x]),x]","\text{Result too large to show}","-\frac{2 (a-b) \sqrt{a+b} \left(23 a^2 d+35 a b c+9 b^2 d\right) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b f}-\frac{2 a^2 c \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}+\frac{2 \sqrt{a+b} \left(15 a^3 d+a^2 b (45 c-23 d)-a b^2 (35 c-17 d)+b^3 (5 c-9 d)\right) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b f}+\frac{2 b (8 a d+5 b c) \tan (e+f x) \sqrt{a+b \sec (e+f x)}}{15 f}+\frac{2 b d \tan (e+f x) (a+b \sec (e+f x))^{3/2}}{5 f}",1,"Result too large to show","B",0
203,1,145,208,2.6871563,"\int \frac{c+d \sec (e+f x)}{\sqrt{a+b \sec (e+f x)}} \, dx","Integrate[(c + d*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]],x]","\frac{4 \cos ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sec (e+f x) \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \left((d-c) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)+2 c \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{f \sqrt{a+b \sec (e+f x)}}","\frac{2 d \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b f}-\frac{2 c \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a f}",1,"(4*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*((-c + d)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] + 2*c*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)])*Sec[e + f*x])/(f*Sqrt[a + b*Sec[e + f*x]])","A",1
204,1,251,216,8.623689,"\int \frac{1}{\sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx","Integrate[1/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]","-\frac{2 \sec ^{\frac{3}{2}}(e+f x) \sqrt{\sec (e+f x)+1} \sqrt{\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} (c \cos (e+f x)+d) \left(c (c+d) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)-2 \left(\left(c^2-d^2\right) \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)+d^2 \Pi \left(\frac{c-d}{c+d};\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)\right)\right)}{c f (c-d) (c+d) \sqrt{\sec ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x))}","-\frac{2 d \tan (e+f x) \sqrt{\frac{a+b \sec (e+f x)}{a+b}} \Pi \left(\frac{2 d}{c+d};\sin ^{-1}\left(\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right)|\frac{2 b}{a+b}\right)}{c f (c+d) \sqrt{-\tan ^2(e+f x)} \sqrt{a+b \sec (e+f x)}}-\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a c f}",1,"(-2*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*(d + c*Cos[e + f*x])*(c*(c + d)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] - 2*((c^2 - d^2)*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] + d^2*EllipticPi[(c - d)/(c + d), ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]))*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^2]*Sec[e + f*x]^(3/2)*Sqrt[1 + Sec[e + f*x]])/(c*(c - d)*(c + d)*f*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x]))","A",1
205,1,1491,376,14.8191053,"\int \frac{c+d \sec (e+f x)}{(a+b \sec (e+f x))^{3/2}} \, dx","Integrate[(c + d*Sec[e + f*x])/(a + b*Sec[e + f*x])^(3/2),x]","\frac{\sec (e+f x) (c+d \sec (e+f x)) \left(\frac{2 (a d-b c) \sin (e+f x)}{a \left(a^2-b^2\right)}-\frac{2 \left(a b d \sin (e+f x)-b^2 c \sin (e+f x)\right)}{a \left(a^2-b^2\right) (b+a \cos (e+f x))}\right) (b+a \cos (e+f x))^2}{f (d+c \cos (e+f x)) (a+b \sec (e+f x))^{3/2}}+\frac{2 \sqrt{\sec (e+f x)} (c+d \sec (e+f x)) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (e+f x)\right)+1}} \left(-b^2 \sqrt{\frac{b-a}{a+b}} c \tan ^5\left(\frac{1}{2} (e+f x)\right)+a b \sqrt{\frac{b-a}{a+b}} c \tan ^5\left(\frac{1}{2} (e+f x)\right)-a^2 \sqrt{\frac{b-a}{a+b}} d \tan ^5\left(\frac{1}{2} (e+f x)\right)+a b \sqrt{\frac{b-a}{a+b}} d \tan ^5\left(\frac{1}{2} (e+f x)\right)-2 a b \sqrt{\frac{b-a}{a+b}} c \tan ^3\left(\frac{1}{2} (e+f x)\right)+2 a^2 \sqrt{\frac{b-a}{a+b}} d \tan ^3\left(\frac{1}{2} (e+f x)\right)-2 i a^2 c \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (e+f x)\right)+2 i b^2 c \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (e+f x)\right)+b^2 \sqrt{\frac{b-a}{a+b}} c \tan \left(\frac{1}{2} (e+f x)\right)+a b \sqrt{\frac{b-a}{a+b}} c \tan \left(\frac{1}{2} (e+f x)\right)-a^2 \sqrt{\frac{b-a}{a+b}} d \tan \left(\frac{1}{2} (e+f x)\right)-a b \sqrt{\frac{b-a}{a+b}} d \tan \left(\frac{1}{2} (e+f x)\right)+i (a-b) (a d-b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{a+b}}+i (a-b) (2 b c+a (c-d)) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{a+b}}-2 i a^2 c \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{a+b}}+2 i b^2 c \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{a+b}}\right) (b+a \cos (e+f x))^{3/2}}{a \sqrt{\frac{b-a}{a+b}} \left(a^2-b^2\right) f (d+c \cos (e+f x)) (a+b \sec (e+f x))^{3/2} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (e+f x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)}} \left(a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)\right)}","\frac{2 b (b c-a d) \tan (e+f x)}{a f \left(a^2-b^2\right) \sqrt{a+b \sec (e+f x)}}-\frac{2 c \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 f}-\frac{2 (b c-a d) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b f \sqrt{a+b}}+\frac{2 (b c-a d) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b f \sqrt{a+b}}",1,"((b + a*Cos[e + f*x])^2*Sec[e + f*x]*(c + d*Sec[e + f*x])*((2*(-(b*c) + a*d)*Sin[e + f*x])/(a*(a^2 - b^2)) - (2*(-(b^2*c*Sin[e + f*x]) + a*b*d*Sin[e + f*x]))/(a*(a^2 - b^2)*(b + a*Cos[e + f*x]))))/(f*(d + c*Cos[e + f*x])*(a + b*Sec[e + f*x])^(3/2)) + (2*(b + a*Cos[e + f*x])^(3/2)*Sqrt[Sec[e + f*x]]*(c + d*Sec[e + f*x])*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(1 + Tan[(e + f*x)/2]^2)]*(a*b*Sqrt[(-a + b)/(a + b)]*c*Tan[(e + f*x)/2] + b^2*Sqrt[(-a + b)/(a + b)]*c*Tan[(e + f*x)/2] - a^2*Sqrt[(-a + b)/(a + b)]*d*Tan[(e + f*x)/2] - a*b*Sqrt[(-a + b)/(a + b)]*d*Tan[(e + f*x)/2] - 2*a*b*Sqrt[(-a + b)/(a + b)]*c*Tan[(e + f*x)/2]^3 + 2*a^2*Sqrt[(-a + b)/(a + b)]*d*Tan[(e + f*x)/2]^3 + a*b*Sqrt[(-a + b)/(a + b)]*c*Tan[(e + f*x)/2]^5 - b^2*Sqrt[(-a + b)/(a + b)]*c*Tan[(e + f*x)/2]^5 - a^2*Sqrt[(-a + b)/(a + b)]*d*Tan[(e + f*x)/2]^5 + a*b*Sqrt[(-a + b)/(a + b)]*d*Tan[(e + f*x)/2]^5 - (2*I)*a^2*c*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + (2*I)*b^2*c*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - (2*I)*a^2*c*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Tan[(e + f*x)/2]^2*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + (2*I)*b^2*c*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Tan[(e + f*x)/2]^2*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + I*(a - b)*(-(b*c) + a*d)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + I*(a - b)*(2*b*c + a*(c - d))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)]))/(a*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)*f*(d + c*Cos[e + f*x])*(a + b*Sec[e + f*x])^(3/2)*(-1 + Tan[(e + f*x)/2]^2)*Sqrt[(1 + Tan[(e + f*x)/2]^2)/(1 - Tan[(e + f*x)/2]^2)]*(a*(-1 + Tan[(e + f*x)/2]^2) - b*(1 + Tan[(e + f*x)/2]^2)))","C",1
206,1,2083,495,17.4311804,"\int \frac{c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx","Integrate[(c + d*Sec[e + f*x])/(a + b*Sec[e + f*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 c \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 f}+\frac{2 b (b c-a d) \tan (e+f x)}{3 a f \left(a^2-b^2\right) (a+b \sec (e+f x))^{3/2}}+\frac{2 \left(-4 a^3 d+7 a^2 b c-3 b^3 c\right) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 b f (a-b) (a+b)^{3/2}}+\frac{2 b \left(-4 a^3 d+7 a^2 b c-3 b^3 c\right) \tan (e+f x)}{3 a^2 f \left(a^2-b^2\right)^2 \sqrt{a+b \sec (e+f x)}}-\frac{2 \left(-3 a^3 d+6 a^2 b c+a^2 b d-a b^2 c-3 b^3 c\right) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 b f (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[e + f*x])^3*Sec[e + f*x]^2*(c + d*Sec[e + f*x])*((2*(-7*a^2*b*c + 3*b^3*c + 4*a^3*d)*Sin[e + f*x])/(3*a^2*(a^2 - b^2)^2) - (2*(b^3*c*Sin[e + f*x] - a*b^2*d*Sin[e + f*x]))/(3*a^2*(a^2 - b^2)*(b + a*Cos[e + f*x])^2) - (2*(-8*a^2*b^2*c*Sin[e + f*x] + 4*b^4*c*Sin[e + f*x] + 5*a^3*b*d*Sin[e + f*x] - a*b^3*d*Sin[e + f*x]))/(3*a^2*(a^2 - b^2)^2*(b + a*Cos[e + f*x]))))/(f*(d + c*Cos[e + f*x])*(a + b*Sec[e + f*x])^(5/2)) + (2*(b + a*Cos[e + f*x])^(5/2)*Sec[e + f*x]^(3/2)*(c + d*Sec[e + f*x])*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(1 + Tan[(e + f*x)/2]^2)]*(7*a^3*b*Sqrt[(-a + b)/(a + b)]*c*Tan[(e + f*x)/2] + 7*a^2*b^2*Sqrt[(-a + b)/(a + b)]*c*Tan[(e + f*x)/2] - 3*a*b^3*Sqrt[(-a + b)/(a + b)]*c*Tan[(e + f*x)/2] - 3*b^4*Sqrt[(-a + b)/(a + b)]*c*Tan[(e + f*x)/2] - 4*a^4*Sqrt[(-a + b)/(a + b)]*d*Tan[(e + f*x)/2] - 4*a^3*b*Sqrt[(-a + b)/(a + b)]*d*Tan[(e + f*x)/2] - 14*a^3*b*Sqrt[(-a + b)/(a + b)]*c*Tan[(e + f*x)/2]^3 + 6*a*b^3*Sqrt[(-a + b)/(a + b)]*c*Tan[(e + f*x)/2]^3 + 8*a^4*Sqrt[(-a + b)/(a + b)]*d*Tan[(e + f*x)/2]^3 + 7*a^3*b*Sqrt[(-a + b)/(a + b)]*c*Tan[(e + f*x)/2]^5 - 7*a^2*b^2*Sqrt[(-a + b)/(a + b)]*c*Tan[(e + f*x)/2]^5 - 3*a*b^3*Sqrt[(-a + b)/(a + b)]*c*Tan[(e + f*x)/2]^5 + 3*b^4*Sqrt[(-a + b)/(a + b)]*c*Tan[(e + f*x)/2]^5 - 4*a^4*Sqrt[(-a + b)/(a + b)]*d*Tan[(e + f*x)/2]^5 + 4*a^3*b*Sqrt[(-a + b)/(a + b)]*d*Tan[(e + f*x)/2]^5 - (6*I)*a^4*c*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + (12*I)*a^2*b^2*c*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - (6*I)*b^4*c*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - (6*I)*a^4*c*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Tan[(e + f*x)/2]^2*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + (12*I)*a^2*b^2*c*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Tan[(e + f*x)/2]^2*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - (6*I)*b^4*c*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Tan[(e + f*x)/2]^2*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + I*(a - b)*(-7*a^2*b*c + 3*b^3*c + 4*a^3*d)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + I*(a - b)*(-4*a*b^2*c - 6*b^3*c + 3*a^3*(c - d) + a^2*b*(9*c + d))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)]))/(3*a^2*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)^2*f*(d + c*Cos[e + f*x])*(a + b*Sec[e + f*x])^(5/2)*(-1 + Tan[(e + f*x)/2]^2)*Sqrt[(1 + Tan[(e + f*x)/2]^2)/(1 - Tan[(e + f*x)/2]^2)]*(a*(-1 + Tan[(e + f*x)/2]^2) - b*(1 + Tan[(e + f*x)/2]^2)))","C",0
207,1,39925,389,32.9472629,"\int \sqrt{a+b \sec (e+f x)} \sqrt{c+d \sec (e+f x)} \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]],x]","\text{Result too large to show}","\frac{2 \cot (e+f x) (a+b \sec (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{\frac{a+b}{c+d}} \sqrt{c+d \sec (e+f x)}}{\sqrt{a+b \sec (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{f \sqrt{\frac{a+b}{c+d}}}-\frac{2 \sqrt{c+d} \cot (e+f x) (a+b \sec (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \Pi \left(\frac{a (c+d)}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{f \sqrt{a+b}}",1,"Result too large to show","C",0
208,1,336,198,5.8729492,"\int \frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{c+d \sec (e+f x)}} \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]]/Sqrt[c + d*Sec[e + f*x]],x]","\frac{4 \sin ^2\left(\frac{1}{2} (e+f x)\right) \csc (e+f x) \sqrt{a+b \sec (e+f x)} \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(a+b) \csc ^2\left(\frac{1}{2} (e+f x)\right) (c \cos (e+f x)+d)}{a d-b c}} \left(c (a+b) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{a d-b c}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right)-a (c+d) \Pi \left(\frac{b c-a d}{a c+b c};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{a d-b c}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right)\right)}{c f (a+b) \sqrt{c+d \sec (e+f x)} \sqrt{\frac{(c+d) \csc ^2\left(\frac{1}{2} (e+f x)\right) (a \cos (e+f x)+b)}{b c-a d}}}","-\frac{2 \sqrt{c+d} \cot (e+f x) (a+b \sec (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \Pi \left(\frac{a (c+d)}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{c f \sqrt{a+b}}",1,"(4*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(-(b*c) + a*d)]*Csc[e + f*x]*((a + b)*c*EllipticF[ArcSin[Sqrt[((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(-(b*c) + a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))] - a*(c + d)*EllipticPi[(b*c - a*d)/(a*c + b*c), ArcSin[Sqrt[((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(-(b*c) + a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))])*Sqrt[a + b*Sec[e + f*x]]*Sin[(e + f*x)/2]^2)/((a + b)*c*f*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[c + d*Sec[e + f*x]])","A",1
209,1,1708,598,9.2829629,"\int \frac{\sqrt{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{3/2}} \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]]/(c + d*Sec[e + f*x])^(3/2),x]","\frac{\sec (e+f x) \sqrt{a+b \sec (e+f x)} \left(\frac{4 b c (b c-a d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) (c+d) \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}+4 (b c-a d) (a c+b d) \left(\frac{\sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) (c+d) \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}-\frac{\sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) \Pi \left(\frac{b c-a d}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) c \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}\right)+2 a d \left(\frac{\sqrt{\frac{b-a}{a+b}} (a+b) \cos \left(\frac{1}{2} (e+f x)\right) \sqrt{d+c \cos (e+f x)} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b-a}{a+b}} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{\frac{b+a \cos (e+f x)}{a+b}}}\right)|\frac{2 (b c-a d)}{(b-a) (c+d)}\right)}{a c \sqrt{\frac{(a+b) \cos ^2\left(\frac{1}{2} (e+f x)\right)}{b+a \cos (e+f x)}} \sqrt{b+a \cos (e+f x)} \sqrt{\frac{b+a \cos (e+f x)}{a+b}} \sqrt{\frac{(a+b) (d+c \cos (e+f x))}{(c+d) (b+a \cos (e+f x))}}}-\frac{2 (b c-a d) \left(\frac{(b c+(a+b) d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) (c+d) \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}-\frac{(b c+a d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) \Pi \left(\frac{b c-a d}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) c \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}\right)}{a c}+\frac{\sqrt{d+c \cos (e+f x)} \sin (e+f x)}{c \sqrt{b+a \cos (e+f x)}}\right)\right) (d+c \cos (e+f x))^{3/2}}{(c-d) (c+d) f \sqrt{b+a \cos (e+f x)} (c+d \sec (e+f x))^{3/2}}+\frac{2 d \sqrt{a+b \sec (e+f x)} \tan (e+f x) (d+c \cos (e+f x))}{\left(d^2-c^2\right) f (c+d \sec (e+f x))^{3/2}}","-\frac{2 \sqrt{c+d} \cot (e+f x) (a+b \sec (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \Pi \left(\frac{a (c+d)}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{c^2 f \sqrt{a+b}}-\frac{2 d (a-b) \sqrt{a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{c f (c-d) \sqrt{c+d} (b c-a d)}-\frac{2 d \sqrt{a+b} \cot (e+f x) (\sec (e+f x)+1) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{c f (c-d) \sqrt{c+d} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}}}",1,"((d + c*Cos[e + f*x])^(3/2)*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]]*((4*b*c*(b*c - a*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) + 4*(b*c - a*d)*(a*c + b*d)*((Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - (Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])) + 2*a*d*((Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*EllipticE[ArcSin[(Sqrt[(-a + b)/(a + b)]*Sin[(e + f*x)/2])/Sqrt[(b + a*Cos[e + f*x])/(a + b)]], (2*(b*c - a*d))/((-a + b)*(c + d))])/(a*c*Sqrt[((a + b)*Cos[(e + f*x)/2]^2)/(b + a*Cos[e + f*x])]*Sqrt[b + a*Cos[e + f*x]]*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Sqrt[((a + b)*(d + c*Cos[e + f*x]))/((c + d)*(b + a*Cos[e + f*x]))]) - (2*(b*c - a*d)*(((b*c + (a + b)*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])))/(a*c) + (Sqrt[d + c*Cos[e + f*x]]*Sin[e + f*x])/(c*Sqrt[b + a*Cos[e + f*x]]))))/((c - d)*(c + d)*f*Sqrt[b + a*Cos[e + f*x]]*(c + d*Sec[e + f*x])^(3/2)) + (2*d*(d + c*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Tan[e + f*x])/((-c^2 + d^2)*f*(c + d*Sec[e + f*x])^(3/2))","B",0
210,1,1990,899,6.9043791,"\int \frac{\sqrt{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]]/(c + d*Sec[e + f*x])^(5/2),x]","\frac{\sec ^2(e+f x) \sqrt{a+b \sec (e+f x)} \left(\frac{2 d^2 \sin (e+f x)}{3 c \left(c^2-d^2\right) (d+c \cos (e+f x))^2}-\frac{2 \left(3 a \sin (e+f x) d^4-2 b c \sin (e+f x) d^3-7 a c^2 \sin (e+f x) d^2+6 b c^3 \sin (e+f x) d\right)}{3 c (b c-a d) \left(c^2-d^2\right)^2 (d+c \cos (e+f x))}\right) (d+c \cos (e+f x))^3}{f (c+d \sec (e+f x))^{5/2}}+\frac{\sec ^2(e+f x) \sqrt{a+b \sec (e+f x)} \left(\frac{4 (b c-a d) \left(3 b^2 c^4-3 a b d c^3-a^2 d^2 c^2+b^2 d^2 c^2-a b d^3 c+a^2 d^4\right) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) (c+d) \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}+4 (b c-a d) \left(3 a b c^4-3 a^2 d c^3+6 b^2 d c^3-7 a b d^2 c^2-a^2 d^3 c-2 b^2 d^3 c+4 a b d^4\right) \left(\frac{\sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) (c+d) \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}-\frac{\sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) \Pi \left(\frac{b c-a d}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) c \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}\right)+2 \left(3 a^2 d^4-2 a b c d^3-7 a^2 c^2 d^2+6 a b c^3 d\right) \left(\frac{\sqrt{\frac{b-a}{a+b}} (a+b) \cos \left(\frac{1}{2} (e+f x)\right) \sqrt{d+c \cos (e+f x)} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b-a}{a+b}} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{\frac{b+a \cos (e+f x)}{a+b}}}\right)|\frac{2 (b c-a d)}{(b-a) (c+d)}\right)}{a c \sqrt{\frac{(a+b) \cos ^2\left(\frac{1}{2} (e+f x)\right)}{b+a \cos (e+f x)}} \sqrt{b+a \cos (e+f x)} \sqrt{\frac{b+a \cos (e+f x)}{a+b}} \sqrt{\frac{(a+b) (d+c \cos (e+f x))}{(c+d) (b+a \cos (e+f x))}}}-\frac{2 (b c-a d) \left(\frac{(b c+(a+b) d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) (c+d) \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}-\frac{(b c+a d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) \Pi \left(\frac{b c-a d}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) c \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}\right)}{a c}+\frac{\sqrt{d+c \cos (e+f x)} \sin (e+f x)}{c \sqrt{b+a \cos (e+f x)}}\right)\right) (d+c \cos (e+f x))^{5/2}}{3 c (c-d)^2 (c+d)^2 (b c-a d) f \sqrt{b+a \cos (e+f x)} (c+d \sec (e+f x))^{5/2}}","\frac{2 \sqrt{a+b \sec (e+f x)} \sin (e+f x) d^2}{3 c \left(c^2-d^2\right) f (d+c \cos (e+f x)) \sqrt{c+d \sec (e+f x)}}+\frac{2 (a-b) \sqrt{a+b} \left(6 b c^3-7 a d c^2-2 b d^2 c+3 a d^3\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} d}{3 c^2 (c-d)^2 (c+d)^{3/2} (b c-a d)^2 f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{a+b} \left(b c^2 \left(3 c^2+3 d c-2 d^2\right)-a d \left(9 c^3-2 d c^2-6 d^2 c+3 d^3\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{3 c^3 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{c^3 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}",1,"((d + c*Cos[e + f*x])^3*Sec[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]]*((2*d^2*Sin[e + f*x])/(3*c*(c^2 - d^2)*(d + c*Cos[e + f*x])^2) - (2*(6*b*c^3*d*Sin[e + f*x] - 7*a*c^2*d^2*Sin[e + f*x] - 2*b*c*d^3*Sin[e + f*x] + 3*a*d^4*Sin[e + f*x]))/(3*c*(b*c - a*d)*(c^2 - d^2)^2*(d + c*Cos[e + f*x]))))/(f*(c + d*Sec[e + f*x])^(5/2)) + ((d + c*Cos[e + f*x])^(5/2)*Sec[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]]*((4*(b*c - a*d)*(3*b^2*c^4 - 3*a*b*c^3*d - a^2*c^2*d^2 + b^2*c^2*d^2 - a*b*c*d^3 + a^2*d^4)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) + 4*(b*c - a*d)*(3*a*b*c^4 - 3*a^2*c^3*d + 6*b^2*c^3*d - 7*a*b*c^2*d^2 - a^2*c*d^3 - 2*b^2*c*d^3 + 4*a*b*d^4)*((Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - (Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])) + 2*(6*a*b*c^3*d - 7*a^2*c^2*d^2 - 2*a*b*c*d^3 + 3*a^2*d^4)*((Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*EllipticE[ArcSin[(Sqrt[(-a + b)/(a + b)]*Sin[(e + f*x)/2])/Sqrt[(b + a*Cos[e + f*x])/(a + b)]], (2*(b*c - a*d))/((-a + b)*(c + d))])/(a*c*Sqrt[((a + b)*Cos[(e + f*x)/2]^2)/(b + a*Cos[e + f*x])]*Sqrt[b + a*Cos[e + f*x]]*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Sqrt[((a + b)*(d + c*Cos[e + f*x]))/((c + d)*(b + a*Cos[e + f*x]))]) - (2*(b*c - a*d)*(((b*c + (a + b)*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])))/(a*c) + (Sqrt[d + c*Cos[e + f*x]]*Sin[e + f*x])/(c*Sqrt[b + a*Cos[e + f*x]]))))/(3*c*(c - d)^2*(c + d)^2*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*(c + d*Sec[e + f*x])^(5/2))","B",0
211,1,1750,744,9.7218449,"\int \frac{(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx","Integrate[(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^(3/2),x]","\frac{2 (d+c \cos (e+f x)) (a d \sin (e+f x)-b c \sin (e+f x)) (a+b \sec (e+f x))^{3/2}}{\left(d^2-c^2\right) f (b+a \cos (e+f x)) (c+d \sec (e+f x))^{3/2}}+\frac{(d+c \cos (e+f x))^{3/2} \left(\frac{4 (b c-a d) \left(a b c-b^2 d\right) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) (c+d) \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}+4 \left(a^2 c-b^2 c\right) (b c-a d) \left(\frac{\sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) (c+d) \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}-\frac{\sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) \Pi \left(\frac{b c-a d}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) c \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}\right)+2 \left(a^2 d-a b c\right) \left(\frac{\sqrt{\frac{b-a}{a+b}} (a+b) \cos \left(\frac{1}{2} (e+f x)\right) \sqrt{d+c \cos (e+f x)} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b-a}{a+b}} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{\frac{b+a \cos (e+f x)}{a+b}}}\right)|\frac{2 (b c-a d)}{(b-a) (c+d)}\right)}{a c \sqrt{\frac{(a+b) \cos ^2\left(\frac{1}{2} (e+f x)\right)}{b+a \cos (e+f x)}} \sqrt{b+a \cos (e+f x)} \sqrt{\frac{b+a \cos (e+f x)}{a+b}} \sqrt{\frac{(a+b) (d+c \cos (e+f x))}{(c+d) (b+a \cos (e+f x))}}}-\frac{2 (b c-a d) \left(\frac{(b c+(a+b) d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) (c+d) \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}-\frac{(b c+a d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) \Pi \left(\frac{b c-a d}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) c \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}\right)}{a c}+\frac{\sqrt{d+c \cos (e+f x)} \sin (e+f x)}{c \sqrt{b+a \cos (e+f x)}}\right)\right) (a+b \sec (e+f x))^{3/2}}{(c-d) (c+d) f (b+a \cos (e+f x))^{3/2} (c+d \sec (e+f x))^{3/2}}","-\frac{2 \sqrt{a+b} (b c-a (2 c-d)) \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{c^2 f (c-d) \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}-\frac{2 a \sqrt{a+b} \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{c^2 f \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}-\frac{2 (a-b) \sqrt{a+b} \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{c f (c-d) \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}",1,"(2*(d + c*Cos[e + f*x])*(a + b*Sec[e + f*x])^(3/2)*(-(b*c*Sin[e + f*x]) + a*d*Sin[e + f*x]))/((-c^2 + d^2)*f*(b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^(3/2)) + ((d + c*Cos[e + f*x])^(3/2)*(a + b*Sec[e + f*x])^(3/2)*((4*(b*c - a*d)*(a*b*c - b^2*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) + 4*(a^2*c - b^2*c)*(b*c - a*d)*((Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - (Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])) + 2*(-(a*b*c) + a^2*d)*((Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*EllipticE[ArcSin[(Sqrt[(-a + b)/(a + b)]*Sin[(e + f*x)/2])/Sqrt[(b + a*Cos[e + f*x])/(a + b)]], (2*(b*c - a*d))/((-a + b)*(c + d))])/(a*c*Sqrt[((a + b)*Cos[(e + f*x)/2]^2)/(b + a*Cos[e + f*x])]*Sqrt[b + a*Cos[e + f*x]]*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Sqrt[((a + b)*(d + c*Cos[e + f*x]))/((c + d)*(b + a*Cos[e + f*x]))]) - (2*(b*c - a*d)*(((b*c + (a + b)*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])))/(a*c) + (Sqrt[d + c*Cos[e + f*x]]*Sin[e + f*x])/(c*Sqrt[b + a*Cos[e + f*x]]))))/((c - d)*(c + d)*f*(b + a*Cos[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^(3/2))","B",0
212,1,1960,919,6.7361185,"\int \frac{(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{5/2}} \, dx","Integrate[(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^(5/2),x]","\frac{\sec (e+f x) (a+b \sec (e+f x))^{3/2} \left(\frac{2 \left(a d^2 \sin (e+f x)-b c d \sin (e+f x)\right)}{3 c \left(c^2-d^2\right) (d+c \cos (e+f x))^2}+\frac{2 \left(3 b \sin (e+f x) c^3-7 a d \sin (e+f x) c^2+b d^2 \sin (e+f x) c+3 a d^3 \sin (e+f x)\right)}{3 c \left(c^2-d^2\right)^2 (d+c \cos (e+f x))}\right) (d+c \cos (e+f x))^3}{f (b+a \cos (e+f x)) (c+d \sec (e+f x))^{5/2}}+\frac{\sec (e+f x) (a+b \sec (e+f x))^{3/2} \left(\frac{4 (b c-a d) \left(3 a b c^3+a^2 d c^2-4 b^2 d c^2+a b d^2 c-a^2 d^3\right) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) (c+d) \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}+4 (b c-a d) \left(3 a^2 c^3-3 b^2 c^3+4 a b d c^2+a^2 d^2 c-b^2 d^2 c-4 a b d^3\right) \left(\frac{\sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) (c+d) \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}-\frac{\sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) \Pi \left(\frac{b c-a d}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) c \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}\right)+2 \left(-3 a b c^3+7 a^2 d c^2-a b d^2 c-3 a^2 d^3\right) \left(\frac{\sqrt{\frac{b-a}{a+b}} (a+b) \cos \left(\frac{1}{2} (e+f x)\right) \sqrt{d+c \cos (e+f x)} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b-a}{a+b}} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{\frac{b+a \cos (e+f x)}{a+b}}}\right)|\frac{2 (b c-a d)}{(b-a) (c+d)}\right)}{a c \sqrt{\frac{(a+b) \cos ^2\left(\frac{1}{2} (e+f x)\right)}{b+a \cos (e+f x)}} \sqrt{b+a \cos (e+f x)} \sqrt{\frac{b+a \cos (e+f x)}{a+b}} \sqrt{\frac{(a+b) (d+c \cos (e+f x))}{(c+d) (b+a \cos (e+f x))}}}-\frac{2 (b c-a d) \left(\frac{(b c+(a+b) d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) (c+d) \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}-\frac{(b c+a d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) \Pi \left(\frac{b c-a d}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) c \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}\right)}{a c}+\frac{\sqrt{d+c \cos (e+f x)} \sin (e+f x)}{c \sqrt{b+a \cos (e+f x)}}\right)\right) (d+c \cos (e+f x))^{5/2}}{3 c (c-d)^2 (c+d)^2 f (b+a \cos (e+f x))^{3/2} (c+d \sec (e+f x))^{5/2}}","-\frac{2 (a-b) \sqrt{a+b} \left(3 b c^3-7 a d c^2+b d^2 c+3 a d^3\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} (d+c \cos (e+f x))^{3/2}}{3 c^2 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 \sqrt{a+b} \left(b^2 (3 c+d) c^3-2 a b \left(3 c^2+2 d c-d^2\right) c^2+a^2 d \left(9 c^3-2 d c^2-6 d^2 c+3 d^3\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} (d+c \cos (e+f x))^{3/2}}{3 c^3 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 a \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} (d+c \cos (e+f x))^{3/2}}{c^3 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 d (b c-a d) \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{3 c \left(c^2-d^2\right) f \sqrt{c+d \sec (e+f x)} (d+c \cos (e+f x))}",1,"((d + c*Cos[e + f*x])^3*Sec[e + f*x]*(a + b*Sec[e + f*x])^(3/2)*((2*(-(b*c*d*Sin[e + f*x]) + a*d^2*Sin[e + f*x]))/(3*c*(c^2 - d^2)*(d + c*Cos[e + f*x])^2) + (2*(3*b*c^3*Sin[e + f*x] - 7*a*c^2*d*Sin[e + f*x] + b*c*d^2*Sin[e + f*x] + 3*a*d^3*Sin[e + f*x]))/(3*c*(c^2 - d^2)^2*(d + c*Cos[e + f*x]))))/(f*(b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^(5/2)) + ((d + c*Cos[e + f*x])^(5/2)*Sec[e + f*x]*(a + b*Sec[e + f*x])^(3/2)*((4*(b*c - a*d)*(3*a*b*c^3 + a^2*c^2*d - 4*b^2*c^2*d + a*b*c*d^2 - a^2*d^3)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) + 4*(b*c - a*d)*(3*a^2*c^3 - 3*b^2*c^3 + 4*a*b*c^2*d + a^2*c*d^2 - b^2*c*d^2 - 4*a*b*d^3)*((Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - (Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])) + 2*(-3*a*b*c^3 + 7*a^2*c^2*d - a*b*c*d^2 - 3*a^2*d^3)*((Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*EllipticE[ArcSin[(Sqrt[(-a + b)/(a + b)]*Sin[(e + f*x)/2])/Sqrt[(b + a*Cos[e + f*x])/(a + b)]], (2*(b*c - a*d))/((-a + b)*(c + d))])/(a*c*Sqrt[((a + b)*Cos[(e + f*x)/2]^2)/(b + a*Cos[e + f*x])]*Sqrt[b + a*Cos[e + f*x]]*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Sqrt[((a + b)*(d + c*Cos[e + f*x]))/((c + d)*(b + a*Cos[e + f*x]))]) - (2*(b*c - a*d)*(((b*c + (a + b)*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])))/(a*c) + (Sqrt[d + c*Cos[e + f*x]]*Sin[e + f*x])/(c*Sqrt[b + a*Cos[e + f*x]]))))/(3*c*(c - d)^2*(c + d)^2*f*(b + a*Cos[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^(5/2))","B",0
213,1,2385,1122,7.4136039,"\int \frac{(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{7/2}} \, dx","Integrate[(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^(7/2),x]","\text{Result too large to show}","\frac{2 (b+a \cos (e+f x)) \sqrt{a+b \sec (e+f x)} \sin (e+f x) d^2}{5 c \left(c^2-d^2\right) f (d+c \cos (e+f x))^2 \sqrt{c+d \sec (e+f x)}}-\frac{2 \left(10 b c^3-13 a d c^2-2 b d^2 c+5 a d^3\right) \sqrt{a+b \sec (e+f x)} \sin (e+f x) d}{15 c^2 \left(c^2-d^2\right)^2 f (d+c \cos (e+f x)) \sqrt{c+d \sec (e+f x)}}+\frac{2 (a-b) \sqrt{a+b} \left(-\left(\left(15 c^6+19 d^2 c^4-2 d^4 c^2\right) b^2\right)+2 a c d \left(35 c^4-8 d^2 c^2+5 d^4\right) b-a^2 d^2 \left(58 c^4-41 d^2 c^2+15 d^4\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{15 c^3 (c-d)^3 (c+d)^{5/2} (b c-a d)^2 f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 \sqrt{a+b} \left(b^2 \left(15 c^3+10 d c^2+9 d^2 c-2 d^3\right) c^3-2 a b \left(15 c^4+20 d c^3-4 d^2 c^2-4 d^3 c+5 d^4\right) c^2+a^2 d \left(60 c^5-2 d c^4-66 d^2 c^3+25 d^3 c^2+30 d^4 c-15 d^5\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{15 c^4 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 a \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{c^4 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}",1,"((d + c*Cos[e + f*x])^4*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^(3/2)*((-2*(-(b*c*d^2*Sin[e + f*x]) + a*d^3*Sin[e + f*x]))/(5*c^2*(c^2 - d^2)*(d + c*Cos[e + f*x])^3) - (4*(5*b*c^3*d*Sin[e + f*x] - 8*a*c^2*d^2*Sin[e + f*x] - b*c*d^3*Sin[e + f*x] + 4*a*d^4*Sin[e + f*x]))/(15*c^2*(c^2 - d^2)^2*(d + c*Cos[e + f*x])^2) + (2*(15*b^2*c^6*Sin[e + f*x] - 70*a*b*c^5*d*Sin[e + f*x] + 58*a^2*c^4*d^2*Sin[e + f*x] + 19*b^2*c^4*d^2*Sin[e + f*x] + 16*a*b*c^3*d^3*Sin[e + f*x] - 41*a^2*c^2*d^4*Sin[e + f*x] - 2*b^2*c^2*d^4*Sin[e + f*x] - 10*a*b*c*d^5*Sin[e + f*x] + 15*a^2*d^6*Sin[e + f*x]))/(15*c^2*(b*c - a*d)*(c^2 - d^2)^3*(d + c*Cos[e + f*x]))))/(f*(b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^(7/2)) + ((d + c*Cos[e + f*x])^(7/2)*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^(3/2)*((4*(b*c - a*d)*(-15*a*b^2*c^6 + 5*a^2*b*c^5*d + 25*b^3*c^5*d + 13*a^3*c^4*d^2 - 38*a*b^2*c^4*d^2 + 25*a^2*b*c^3*d^3 + 7*b^3*c^3*d^3 - 18*a^3*c^2*d^4 - 11*a*b^2*c^2*d^4 + 2*a^2*b*c*d^5 + 5*a^3*d^6)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) + 4*(b*c - a*d)*(-15*a^2*b*c^6 + 15*b^3*c^6 + 15*a^3*c^5*d - 55*a*b^2*c^5*d + 33*a^2*b*c^4*d^2 + 19*b^3*c^4*d^2 + 13*a^3*c^3*d^3 + 35*a*b^2*c^3*d^3 - 70*a^2*b*c^2*d^4 - 2*b^3*c^2*d^4 + 4*a^3*c*d^5 - 12*a*b^2*c*d^5 + 20*a^2*b*d^6)*((Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - (Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])) + 2*(15*a*b^2*c^6 - 70*a^2*b*c^5*d + 58*a^3*c^4*d^2 + 19*a*b^2*c^4*d^2 + 16*a^2*b*c^3*d^3 - 41*a^3*c^2*d^4 - 2*a*b^2*c^2*d^4 - 10*a^2*b*c*d^5 + 15*a^3*d^6)*((Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*EllipticE[ArcSin[(Sqrt[(-a + b)/(a + b)]*Sin[(e + f*x)/2])/Sqrt[(b + a*Cos[e + f*x])/(a + b)]], (2*(b*c - a*d))/((-a + b)*(c + d))])/(a*c*Sqrt[((a + b)*Cos[(e + f*x)/2]^2)/(b + a*Cos[e + f*x])]*Sqrt[b + a*Cos[e + f*x]]*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Sqrt[((a + b)*(d + c*Cos[e + f*x]))/((c + d)*(b + a*Cos[e + f*x]))]) - (2*(b*c - a*d)*(((b*c + (a + b)*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])))/(a*c) + (Sqrt[d + c*Cos[e + f*x]]*Sin[e + f*x])/(c*Sqrt[b + a*Cos[e + f*x]]))))/(15*c^2*(c - d)^3*(c + d)^3*(-(b*c) + a*d)*f*(b + a*Cos[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^(7/2))","B",0
214,1,2026,891,6.7263911,"\int \frac{(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{5/2}} \, dx","Integrate[(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} a^2}{c^3 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 (b c-a d)^2 \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{3 c \left(c^2-d^2\right) f (d+c \cos (e+f x)) \sqrt{c+d \sec (e+f x)}}-\frac{2 (a-b) \sqrt{a+b} \left(7 a c^2-4 b d c-3 a d^2\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{3 c^2 (c-d)^2 (c+d)^{3/2} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{a+b} \left(\left(9 c^3-2 d c^2-6 d^2 c+3 d^3\right) a^2-b c \left(7 c^2+4 d c-3 d^2\right) a+b^2 c^2 (c+3 d)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{3 c^3 (c-d)^2 (c+d)^{3/2} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}",1,"((d + c*Cos[e + f*x])^3*(a + b*Sec[e + f*x])^(5/2)*((2*(b^2*c^2*Sin[e + f*x] - 2*a*b*c*d*Sin[e + f*x] + a^2*d^2*Sin[e + f*x]))/(3*c*(c^2 - d^2)*(d + c*Cos[e + f*x])^2) + (2*(7*a*b*c^3*Sin[e + f*x] - 7*a^2*c^2*d*Sin[e + f*x] - 4*b^2*c^2*d*Sin[e + f*x] + a*b*c*d^2*Sin[e + f*x] + 3*a^2*d^3*Sin[e + f*x]))/(3*c*(c^2 - d^2)^2*(d + c*Cos[e + f*x]))))/(f*(b + a*Cos[e + f*x])^2*(c + d*Sec[e + f*x])^(5/2)) + ((d + c*Cos[e + f*x])^(5/2)*(a + b*Sec[e + f*x])^(5/2)*((4*(b*c - a*d)*(2*a^2*b*c^3 + b^3*c^3 + a^3*c^2*d - 8*a*b^2*c^2*d + 2*a^2*b*c*d^2 + 3*b^3*c*d^2 - a^3*d^3)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) + 4*(b*c - a*d)*(3*a^3*c^3 - 7*a*b^2*c^3 + 4*b^3*c^2*d + a^3*c*d^2 + 3*a*b^2*c*d^2 - 4*a^2*b*d^3)*((Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - (Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])) + 2*(-7*a^2*b*c^3 + 7*a^3*c^2*d + 4*a*b^2*c^2*d - a^2*b*c*d^2 - 3*a^3*d^3)*((Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*EllipticE[ArcSin[(Sqrt[(-a + b)/(a + b)]*Sin[(e + f*x)/2])/Sqrt[(b + a*Cos[e + f*x])/(a + b)]], (2*(b*c - a*d))/((-a + b)*(c + d))])/(a*c*Sqrt[((a + b)*Cos[(e + f*x)/2]^2)/(b + a*Cos[e + f*x])]*Sqrt[b + a*Cos[e + f*x]]*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Sqrt[((a + b)*(d + c*Cos[e + f*x]))/((c + d)*(b + a*Cos[e + f*x]))]) - (2*(b*c - a*d)*(((b*c + (a + b)*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])))/(a*c) + (Sqrt[d + c*Cos[e + f*x]]*Sin[e + f*x])/(c*Sqrt[b + a*Cos[e + f*x]]))))/(3*c*(c - d)^2*(c + d)^2*f*(b + a*Cos[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^(5/2))","B",0
215,1,2344,1150,7.3161312,"\int \frac{(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{7/2}} \, dx","Integrate[(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^(7/2),x]","\text{Result too large to show}","-\frac{2 \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} a^2}{c^4 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 (b c-a d) \left(5 b c^3-13 a d c^2+3 b d^2 c+5 a d^3\right) \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{15 c^2 \left(c^2-d^2\right)^2 f (d+c \cos (e+f x)) \sqrt{c+d \sec (e+f x)}}-\frac{2 d (b c-a d) (b+a \cos (e+f x)) \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{5 c \left(c^2-d^2\right) f (d+c \cos (e+f x))^2 \sqrt{c+d \sec (e+f x)}}+\frac{2 (a-b) \sqrt{a+b} \left(\left(15 d^5-41 c^2 d^3+58 c^4 d\right) a^2-b c \left(35 c^4+34 d^2 c^2-5 d^4\right) a+b^2 c^2 d \left(29 c^2+3 d^2\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{15 c^3 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{a+b} \left(b^3 \left(5 c^2+24 d c+3 d^2\right) c^4-a b^2 \left(35 c^3+42 d c^2+21 d^2 c-2 d^3\right) c^3+a^2 b \left(45 c^4+48 d c^3+d^2 c^2-8 d^3 c+10 d^4\right) c^2-a^3 d \left(60 c^5-2 d c^4-66 d^2 c^3+25 d^3 c^2+30 d^4 c-15 d^5\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{15 c^4 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}",1,"((d + c*Cos[e + f*x])^4*Sec[e + f*x]*(a + b*Sec[e + f*x])^(5/2)*((-2*(b^2*c^2*d*Sin[e + f*x] - 2*a*b*c*d^2*Sin[e + f*x] + a^2*d^3*Sin[e + f*x]))/(5*c^2*(c^2 - d^2)*(d + c*Cos[e + f*x])^3) + (2*(5*b^2*c^4*Sin[e + f*x] - 21*a*b*c^3*d*Sin[e + f*x] + 16*a^2*c^2*d^2*Sin[e + f*x] + 3*b^2*c^2*d^2*Sin[e + f*x] + 5*a*b*c*d^3*Sin[e + f*x] - 8*a^2*d^4*Sin[e + f*x]))/(15*c^2*(c^2 - d^2)^2*(d + c*Cos[e + f*x])^2) + (2*(35*a*b*c^5*Sin[e + f*x] - 58*a^2*c^4*d*Sin[e + f*x] - 29*b^2*c^4*d*Sin[e + f*x] + 34*a*b*c^3*d^2*Sin[e + f*x] + 41*a^2*c^2*d^3*Sin[e + f*x] - 3*b^2*c^2*d^3*Sin[e + f*x] - 5*a*b*c*d^4*Sin[e + f*x] - 15*a^2*d^5*Sin[e + f*x]))/(15*c^2*(c^2 - d^2)^3*(d + c*Cos[e + f*x]))))/(f*(b + a*Cos[e + f*x])^2*(c + d*Sec[e + f*x])^(7/2)) + ((d + c*Cos[e + f*x])^(7/2)*Sec[e + f*x]*(a + b*Sec[e + f*x])^(5/2)*((4*(b*c - a*d)*(10*a^2*b*c^5 + 5*b^3*c^5 + 13*a^3*c^4*d - 48*a*b^2*c^4*d + 15*a^2*b*c^3*d^2 + 27*b^3*c^3*d^2 - 18*a^3*c^2*d^3 - 16*a*b^2*c^2*d^3 + 7*a^2*b*c*d^4 + 5*a^3*d^5)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) + 4*(b*c - a*d)*(15*a^3*c^5 - 35*a*b^2*c^5 + 23*a^2*b*c^4*d + 29*b^3*c^4*d + 13*a^3*c^3*d^2 - 5*a*b^2*c^3*d^2 - 75*a^2*b*c^2*d^3 + 3*b^3*c^2*d^3 + 4*a^3*c*d^4 + 8*a*b^2*c*d^4 + 20*a^2*b*d^5)*((Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - (Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])) + 2*(-35*a^2*b*c^5 + 58*a^3*c^4*d + 29*a*b^2*c^4*d - 34*a^2*b*c^3*d^2 - 41*a^3*c^2*d^3 + 3*a*b^2*c^2*d^3 + 5*a^2*b*c*d^4 + 15*a^3*d^5)*((Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*EllipticE[ArcSin[(Sqrt[(-a + b)/(a + b)]*Sin[(e + f*x)/2])/Sqrt[(b + a*Cos[e + f*x])/(a + b)]], (2*(b*c - a*d))/((-a + b)*(c + d))])/(a*c*Sqrt[((a + b)*Cos[(e + f*x)/2]^2)/(b + a*Cos[e + f*x])]*Sqrt[b + a*Cos[e + f*x]]*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Sqrt[((a + b)*(d + c*Cos[e + f*x]))/((c + d)*(b + a*Cos[e + f*x]))]) - (2*(b*c - a*d)*(((b*c + (a + b)*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])))/(a*c) + (Sqrt[d + c*Cos[e + f*x]]*Sin[e + f*x])/(c*Sqrt[b + a*Cos[e + f*x]]))))/(15*c^2*(c - d)^3*(c + d)^3*f*(b + a*Cos[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^(7/2))","B",0
216,1,2979,1428,8.2852974,"\int \frac{(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{9/2}} \, dx","Integrate[(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^(9/2),x]","\text{Result too large to show}","-\frac{2 \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)} a^2}{c^5 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 \left(-\left(\left(35 c^6+67 d^2 c^4-6 d^4 c^2\right) b^2\right)+2 a c d \left(91 c^4-2 d^2 c^2+7 d^4\right) b-a^2 d^2 \left(162 c^4-101 d^2 c^2+35 d^4\right)\right) \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{105 c^3 \left(c^2-d^2\right)^3 f (d+c \cos (e+f x)) \sqrt{c+d \sec (e+f x)}}-\frac{2 d \left(14 b c^3-19 a d c^2-2 b d^2 c+7 a d^3\right) (b+a \cos (e+f x)) \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{35 c^2 \left(c^2-d^2\right)^2 f (d+c \cos (e+f x))^2 \sqrt{c+d \sec (e+f x)}}+\frac{2 d^2 (b+a \cos (e+f x))^2 \sqrt{a+b \sec (e+f x)} \sin (e+f x)}{7 c \left(c^2-d^2\right) f (d+c \cos (e+f x))^3 \sqrt{c+d \sec (e+f x)}}+\frac{2 (a-b) \sqrt{a+b} \left(-\left(\left(-105 d^8+392 c^2 d^6-485 c^4 d^4+582 c^6 d^2\right) a^3\right)+2 b c d \left(406 c^6+73 d^2 c^4+132 d^4 c^2-35 d^6\right) a^2-b^2 c^2 \left(245 c^6+852 d^2 c^4+41 d^4 c^2+14 d^6\right) a+2 b^3 c^3 d \left(133 c^4+62 d^2 c^2-3 d^4\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{105 c^4 (c-d)^4 (c+d)^{7/2} (b c-a d)^2 f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{a+b} \left(b^3 \left(35 c^4+231 d c^3+67 d^2 c^2+57 d^3 c-6 d^4\right) c^4-a b^2 \left(245 c^5+413 d c^4+439 d^2 c^3+53 d^3 c^2-12 d^4 c+14 d^5\right) c^3+a^2 b \left(315 c^6+497 d c^5+219 d^2 c^4-73 d^3 c^3+208 d^4 c^2+56 d^5 c-70 d^6\right) c^2-a^3 d \left(525 c^7+57 d c^6-699 d^2 c^5+214 d^3 c^4+672 d^4 c^3-280 d^5 c^2-210 d^6 c+105 d^7\right)\right) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sqrt{a+b \sec (e+f x)}}{105 c^5 (c-d)^4 (c+d)^{7/2} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}",1,"((d + c*Cos[e + f*x])^5*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^(5/2)*((2*(b^2*c^2*d^2*Sin[e + f*x] - 2*a*b*c*d^3*Sin[e + f*x] + a^2*d^4*Sin[e + f*x]))/(7*c^3*(c^2 - d^2)*(d + c*Cos[e + f*x])^4) + (2*(-14*b^2*c^4*d*Sin[e + f*x] + 43*a*b*c^3*d^2*Sin[e + f*x] - 29*a^2*c^2*d^3*Sin[e + f*x] + 2*b^2*c^2*d^3*Sin[e + f*x] - 19*a*b*c*d^4*Sin[e + f*x] + 17*a^2*d^5*Sin[e + f*x]))/(35*c^3*(c^2 - d^2)^2*(d + c*Cos[e + f*x])^3) + (2*(35*b^2*c^6*Sin[e + f*x] - 224*a*b*c^5*d*Sin[e + f*x] + 234*a^2*c^4*d^2*Sin[e + f*x] + 67*b^2*c^4*d^2*Sin[e + f*x] + 52*a*b*c^3*d^3*Sin[e + f*x] - 209*a^2*c^2*d^4*Sin[e + f*x] - 6*b^2*c^2*d^4*Sin[e + f*x] - 20*a*b*c*d^5*Sin[e + f*x] + 71*a^2*d^6*Sin[e + f*x]))/(105*c^3*(c^2 - d^2)^3*(d + c*Cos[e + f*x])^2) + (2*(245*a*b^2*c^8*Sin[e + f*x] - 812*a^2*b*c^7*d*Sin[e + f*x] - 266*b^3*c^7*d*Sin[e + f*x] + 582*a^3*c^6*d^2*Sin[e + f*x] + 852*a*b^2*c^6*d^2*Sin[e + f*x] - 146*a^2*b*c^5*d^3*Sin[e + f*x] - 124*b^3*c^5*d^3*Sin[e + f*x] - 485*a^3*c^4*d^4*Sin[e + f*x] + 41*a*b^2*c^4*d^4*Sin[e + f*x] - 264*a^2*b*c^3*d^5*Sin[e + f*x] + 6*b^3*c^3*d^5*Sin[e + f*x] + 392*a^3*c^2*d^6*Sin[e + f*x] + 14*a*b^2*c^2*d^6*Sin[e + f*x] + 70*a^2*b*c*d^7*Sin[e + f*x] - 105*a^3*d^8*Sin[e + f*x]))/(105*c^3*(b*c - a*d)*(c^2 - d^2)^4*(d + c*Cos[e + f*x]))))/(f*(b + a*Cos[e + f*x])^2*(c + d*Sec[e + f*x])^(9/2)) + ((d + c*Cos[e + f*x])^(9/2)*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^(5/2)*((4*(b*c - a*d)*(-70*a^2*b^2*c^8 - 35*b^4*c^8 - 77*a^3*b*c^7*d + 427*a*b^3*c^7*d + 162*a^4*c^6*d^2 - 522*a^2*b^2*c^6*d^2 - 298*b^4*c^6*d^2 + 348*a^3*b*c^5*d^3 + 666*a*b^3*c^5*d^3 - 263*a^4*c^4*d^4 - 586*a^2*b^2*c^4*d^4 - 51*b^4*c^4*d^4 + 127*a^3*b*c^3*d^5 + 59*a*b^3*c^3*d^5 + 136*a^4*c^2*d^6 + 26*a^2*b^2*c^2*d^6 - 14*a^3*b*c*d^7 - 35*a^4*d^8)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) + 4*(b*c - a*d)*(-105*a^3*b*c^8 + 245*a*b^3*c^8 + 105*a^4*c^7*d - 567*a^2*b^2*c^7*d - 266*b^4*c^7*d + 190*a^3*b*c^6*d^2 + 586*a*b^3*c^6*d^2 + 162*a^4*c^5*d^3 + 706*a^2*b^2*c^5*d^3 - 124*b^4*c^5*d^3 - 1261*a^3*b*c^4*d^4 - 83*a*b^3*c^4*d^4 + 145*a^4*c^3*d^5 - 223*a^2*b^2*c^3*d^5 + 6*b^4*c^3*d^5 + 548*a^3*b*c^2*d^6 + 20*a*b^3*c^2*d^6 - 28*a^4*c*d^7 + 84*a^2*b^2*c*d^7 - 140*a^3*b*d^8)*((Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - (Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])) + 2*(245*a^2*b^2*c^8 - 812*a^3*b*c^7*d - 266*a*b^3*c^7*d + 582*a^4*c^6*d^2 + 852*a^2*b^2*c^6*d^2 - 146*a^3*b*c^5*d^3 - 124*a*b^3*c^5*d^3 - 485*a^4*c^4*d^4 + 41*a^2*b^2*c^4*d^4 - 264*a^3*b*c^3*d^5 + 6*a*b^3*c^3*d^5 + 392*a^4*c^2*d^6 + 14*a^2*b^2*c^2*d^6 + 70*a^3*b*c*d^7 - 105*a^4*d^8)*((Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*EllipticE[ArcSin[(Sqrt[(-a + b)/(a + b)]*Sin[(e + f*x)/2])/Sqrt[(b + a*Cos[e + f*x])/(a + b)]], (2*(b*c - a*d))/((-a + b)*(c + d))])/(a*c*Sqrt[((a + b)*Cos[(e + f*x)/2]^2)/(b + a*Cos[e + f*x])]*Sqrt[b + a*Cos[e + f*x]]*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Sqrt[((a + b)*(d + c*Cos[e + f*x]))/((c + d)*(b + a*Cos[e + f*x]))]) - (2*(b*c - a*d)*(((b*c + (a + b)*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])))/(a*c) + (Sqrt[d + c*Cos[e + f*x]]*Sin[e + f*x])/(c*Sqrt[b + a*Cos[e + f*x]]))))/(105*c^3*(c - d)^4*(c + d)^4*(-(b*c) + a*d)*f*(b + a*Cos[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^(9/2))","B",0
217,1,49385,652,32.7849132,"\int \frac{(c+d \sec (e+f x))^{3/2}}{\sqrt{a+b \sec (e+f x)}} \, dx","Integrate[(c + d*Sec[e + f*x])^(3/2)/Sqrt[a + b*Sec[e + f*x]],x]","\text{Result too large to show}","\frac{2 (b c-a d) \cot (e+f x) \sqrt{a+b \sec (e+f x)} \sqrt{c+d \sec (e+f x)} \sqrt{\frac{(b c-a d) (\sec (e+f x)-1)}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} F\left(\sin ^{-1}\left(\sqrt{\frac{(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{a b f \sqrt{\frac{(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}}-\frac{2 c (c+d) \cot (e+f x) (a+b \sec (e+f x))^{3/2} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \sqrt{\frac{(a+b) (b c-a d) (\sec (e+f x)-1) (c+d \sec (e+f x))}{(c+d)^2 (a+b \sec (e+f x))^2}} \Pi \left(\frac{a (c+d)}{(a+b) c};\sin ^{-1}\left(\sqrt{\frac{(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{a f (a+b) \sqrt{c+d \sec (e+f x)}}+\frac{2 d (c+d) \cot (e+f x) (a+b \sec (e+f x))^{3/2} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \sqrt{-\frac{(a+b) (a d-b c) (\sec (e+f x)-1) (c+d \sec (e+f x))}{(c+d)^2 (a+b \sec (e+f x))^2}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\sqrt{\frac{(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b f (a+b) \sqrt{c+d \sec (e+f x)}}",1,"Result too large to show","C",0
218,1,325,198,5.5356788,"\int \frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{a+b \sec (e+f x)}} \, dx","Integrate[Sqrt[c + d*Sec[e + f*x]]/Sqrt[a + b*Sec[e + f*x]],x]","\frac{4 \sin ^2\left(\frac{1}{2} (e+f x)\right) \csc (e+f x) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{a-b}} \sqrt{c+d \sec (e+f x)} \sqrt{\frac{(c+d) \csc ^2\left(\frac{1}{2} (e+f x)\right) (a \cos (e+f x)+b)}{b c-a d}} \left(a (c+d) F\left(\sin ^{-1}\left(\sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{2 b c-2 a d}}\right)|\frac{2 (a d-b c)}{(a-b) (c+d)}\right)-c (a+b) \Pi \left(\frac{a d-b c}{a (c+d)};\sin ^{-1}\left(\sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{2 b c-2 a d}}\right)|\frac{2 (a d-b c)}{(a-b) (c+d)}\right)\right)}{a f (c+d) \sqrt{a+b \sec (e+f x)} \sqrt{\frac{(a+b) \csc ^2\left(\frac{1}{2} (e+f x)\right) (c \cos (e+f x)+d)}{a d-b c}}}","-\frac{2 \sqrt{a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{a f \sqrt{c+d}}",1,"(4*Sqrt[((a + b)*Cot[(e + f*x)/2]^2)/(a - b)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*(a*(c + d)*EllipticF[ArcSin[Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(2*b*c - 2*a*d)]], (2*(-(b*c) + a*d))/((a - b)*(c + d))] - (a + b)*c*EllipticPi[(-(b*c) + a*d)/(a*(c + d)), ArcSin[Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(2*b*c - 2*a*d)]], (2*(-(b*c) + a*d))/((a - b)*(c + d))])*Sqrt[c + d*Sec[e + f*x]]*Sin[(e + f*x)/2]^2)/(a*(c + d)*f*Sqrt[((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(-(b*c) + a*d)]*Sqrt[a + b*Sec[e + f*x]])","A",1
219,1,249,398,2.3721044,"\int \frac{1}{\sqrt{a+b \sec (e+f x)} \sqrt{c+d \sec (e+f x)}} \, dx","Integrate[1/(Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]),x]","\frac{4 i \cos ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \sqrt{\frac{c \cos (e+f x)+d}{(c+d) (\cos (e+f x)+1)}} \left(F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)-2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)\right)}{f \sqrt{\frac{b-a}{a+b}} \sqrt{a+b \sec (e+f x)} \sqrt{c+d \sec (e+f x)}}","-\frac{2 b \sqrt{a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{a f \sqrt{c+d} (b c-a d)}-\frac{2 \sqrt{c+d} \cot (e+f x) (a+b \sec (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \Pi \left(\frac{a (c+d)}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{a c f \sqrt{a+b}}",1,"((4*I)*Cos[(e + f*x)/2]^2*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*Sqrt[(d + c*Cos[e + f*x])/((c + d)*(1 + Cos[e + f*x]))]*(EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], ((a + b)*(c - d))/((a - b)*(c + d))] - 2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], ((a + b)*(c - d))/((a - b)*(c + d))])*Sec[e + f*x])/(Sqrt[(-a + b)/(a + b)]*f*Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])","C",1
220,1,1761,622,9.6419939,"\int \frac{1}{\sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx","Integrate[1/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^(3/2)),x]","\frac{2 (b+a \cos (e+f x)) (d+c \cos (e+f x)) \sec (e+f x) \tan (e+f x) d^2}{(a d-b c) \left(d^2-c^2\right) f \sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}}+\frac{\sqrt{b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2} \sec ^2(e+f x) \left(-\frac{4 b c d (b c-a d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) (c+d) \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}+4 (b c-a d) \left(b c^2-a d c-2 b d^2\right) \left(\frac{\sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) (c+d) \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}-\frac{\sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) \Pi \left(\frac{b c-a d}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) c \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}\right)-2 a d^2 \left(\frac{\sqrt{\frac{b-a}{a+b}} (a+b) \cos \left(\frac{1}{2} (e+f x)\right) \sqrt{d+c \cos (e+f x)} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b-a}{a+b}} \sin \left(\frac{1}{2} (e+f x)\right)}{\sqrt{\frac{b+a \cos (e+f x)}{a+b}}}\right)|\frac{2 (b c-a d)}{(b-a) (c+d)}\right)}{a c \sqrt{\frac{(a+b) \cos ^2\left(\frac{1}{2} (e+f x)\right)}{b+a \cos (e+f x)}} \sqrt{b+a \cos (e+f x)} \sqrt{\frac{b+a \cos (e+f x)}{a+b}} \sqrt{\frac{(a+b) (d+c \cos (e+f x))}{(c+d) (b+a \cos (e+f x))}}}-\frac{2 (b c-a d) \left(\frac{(b c+(a+b) d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) (c+d) \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}-\frac{(b c+a d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} (e+f x)\right)}{c-d}} \sqrt{\frac{(c+d) (b+a \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}} \csc (e+f x) \Pi \left(\frac{b c-a d}{(a+b) c};\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) (d+c \cos (e+f x)) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{b c-a d}}}{\sqrt{2}}\right)|\frac{2 (b c-a d)}{(a+b) (c-d)}\right) \sin ^4\left(\frac{1}{2} (e+f x)\right)}{(a+b) c \sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}}\right)}{a c}+\frac{\sqrt{d+c \cos (e+f x)} \sin (e+f x)}{c \sqrt{b+a \cos (e+f x)}}\right)\right)}{(c-d) (c+d) (b c-a d) f \sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}}","-\frac{2 d \sqrt{a+b} (2 c-d) \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{c^2 f (c-d) \sqrt{c+d} (b c-a d)}-\frac{2 \sqrt{a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} \Pi \left(\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{a c^2 f \sqrt{c+d}}-\frac{2 d^2 (a-b) \sqrt{a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{c f (c-d) \sqrt{c+d} (b c-a d)^2}",1,"(Sqrt[b + a*Cos[e + f*x]]*(d + c*Cos[e + f*x])^(3/2)*Sec[e + f*x]^2*((-4*b*c*d*(b*c - a*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) + 4*(b*c - a*d)*(b*c^2 - a*c*d - 2*b*d^2)*((Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - (Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])) - 2*a*d^2*((Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*EllipticE[ArcSin[(Sqrt[(-a + b)/(a + b)]*Sin[(e + f*x)/2])/Sqrt[(b + a*Cos[e + f*x])/(a + b)]], (2*(b*c - a*d))/((-a + b)*(c + d))])/(a*c*Sqrt[((a + b)*Cos[(e + f*x)/2]^2)/(b + a*Cos[e + f*x])]*Sqrt[b + a*Cos[e + f*x]]*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Sqrt[((a + b)*(d + c*Cos[e + f*x]))/((c + d)*(b + a*Cos[e + f*x]))]) - (2*(b*c - a*d)*(((b*c + (a + b)*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])))/(a*c) + (Sqrt[d + c*Cos[e + f*x]]*Sin[e + f*x])/(c*Sqrt[b + a*Cos[e + f*x]]))))/((c - d)*(c + d)*(b*c - a*d)*f*Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^(3/2)) + (2*d^2*(b + a*Cos[e + f*x])*(d + c*Cos[e + f*x])*Sec[e + f*x]*Tan[e + f*x])/((-(b*c) + a*d)*(-c^2 + d^2)*f*Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^(3/2))","B",0
221,0,0,89,2.367105,"\int \frac{\sqrt[3]{a+b \sec (e+f x)}}{\sqrt[3]{c+d \sec (e+f x)}} \, dx","Integrate[(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(1/3),x]","\int \frac{\sqrt[3]{a+b \sec (e+f x)}}{\sqrt[3]{c+d \sec (e+f x)}} \, dx","\frac{\sqrt[3]{a+b \sec (e+f x)} \sqrt[3]{c \cos (e+f x)+d} \text{Int}\left(\frac{\sqrt[3]{a \cos (e+f x)+b}}{\sqrt[3]{c \cos (e+f x)+d}},x\right)}{\sqrt[3]{a \cos (e+f x)+b} \sqrt[3]{c+d \sec (e+f x)}}",0,"Integrate[(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(1/3), x]","A",-1
222,0,0,32,52.1131038,"\int \frac{\sqrt[3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{4/3}} \, dx","Integrate[(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(4/3),x]","\int \frac{\sqrt[3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{4/3}} \, dx","\text{Int}\left(\frac{\sqrt[3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{4/3}},x\right)",0,"Integrate[(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(4/3), x]","A",-1
223,0,0,32,89.284996,"\int \frac{\sqrt[3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{7/3}} \, dx","Integrate[(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(7/3),x]","\int \frac{\sqrt[3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{7/3}} \, dx","\text{Int}\left(\frac{\sqrt[3]{a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{7/3}},x\right)",0,"Integrate[(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(7/3), x]","A",-1
224,0,0,89,2.6125805,"\int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{2/3}} \, dx","Integrate[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(2/3),x]","\int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{2/3}} \, dx","\frac{(a+b \sec (e+f x))^{2/3} (c \cos (e+f x)+d)^{2/3} \text{Int}\left(\frac{(a \cos (e+f x)+b)^{2/3}}{(c \cos (e+f x)+d)^{2/3}},x\right)}{(a \cos (e+f x)+b)^{2/3} (c+d \sec (e+f x))^{2/3}}",0,"Integrate[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(2/3), x]","A",-1
225,0,0,32,53.3080427,"\int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{5/3}} \, dx","Integrate[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(5/3),x]","\int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{5/3}} \, dx","\text{Int}\left(\frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{5/3}},x\right)",0,"Integrate[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(5/3), x]","A",-1
226,0,0,32,92.8054695,"\int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx","Integrate[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(8/3),x]","\int \frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx","\text{Int}\left(\frac{(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}},x\right)",0,"Integrate[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(8/3), x]","A",-1
227,0,0,89,62.592586,"\int \frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{4/3}} \, dx","Integrate[(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(4/3),x]","\int \frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{4/3}} \, dx","\frac{(a+b \sec (e+f x))^{4/3} (c \cos (e+f x)+d)^{4/3} \text{Int}\left(\frac{(a \cos (e+f x)+b)^{4/3}}{(c \cos (e+f x)+d)^{4/3}},x\right)}{(a \cos (e+f x)+b)^{4/3} (c+d \sec (e+f x))^{4/3}}",0,"Integrate[(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(4/3), x]","A",-1
228,0,0,32,102.0478647,"\int \frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{7/3}} \, dx","Integrate[(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(7/3),x]","\int \frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{7/3}} \, dx","\text{Int}\left(\frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{7/3}},x\right)",0,"Integrate[(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(7/3), x]","A",-1
229,0,0,32,148.1741706,"\int \frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx","Integrate[(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(10/3),x]","\int \frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx","\text{Int}\left(\frac{(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}},x\right)",0,"Integrate[(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(10/3), x]","A",-1
230,1,2425,106,14.6117871,"\int \left(c (d \sec (e+f x))^p\right)^n (a+a \sec (e+f x))^m \, dx","Integrate[(c*(d*Sec[e + f*x])^p)^n*(a + a*Sec[e + f*x])^m,x]","\text{Result too large to show}","-\frac{\tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(n p;\frac{1}{2},\frac{1}{2}-m;n p+1;\sec (e+f x),-\sec (e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p \sqrt{1-\sec (e+f x)}}",1,"(3*2^(1 + m)*AppellF1[1/2, m + n*p, 1 - n*p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n*p)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n*p)*(c*(d*Sec[e + f*x])^p)^n*(a*(1 + Sec[e + f*x]))^m*Tan[(e + f*x)/2])/(f*(3*AppellF1[1/2, m + n*p, 1 - n*p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n*p)*AppellF1[3/2, m + n*p, 2 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n*p)*AppellF1[3/2, 1 + m + n*p, 1 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((3*2^m*AppellF1[1/2, m + n*p, 1 - n*p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(n*p)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n*p))/(3*AppellF1[1/2, m + n*p, 1 - n*p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n*p)*AppellF1[3/2, m + n*p, 2 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n*p)*AppellF1[3/2, 1 + m + n*p, 1 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*2^(1 + m)*(-1 + n*p)*AppellF1[1/2, m + n*p, 1 - n*p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n*p)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n*p)*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, m + n*p, 1 - n*p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n*p)*AppellF1[3/2, m + n*p, 2 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n*p)*AppellF1[3/2, 1 + m + n*p, 1 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*2^(1 + m)*(Sec[(e + f*x)/2]^2)^(-1 + n*p)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n*p)*Tan[(e + f*x)/2]*(-1/3*((1 - n*p)*AppellF1[3/2, m + n*p, 2 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n*p)*AppellF1[3/2, 1 + m + n*p, 1 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, m + n*p, 1 - n*p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n*p)*AppellF1[3/2, m + n*p, 2 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n*p)*AppellF1[3/2, 1 + m + n*p, 1 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*2^(1 + m)*AppellF1[1/2, m + n*p, 1 - n*p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n*p)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n*p)*Tan[(e + f*x)/2]*(2*((-1 + n*p)*AppellF1[3/2, m + n*p, 2 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n*p)*AppellF1[3/2, 1 + m + n*p, 1 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n*p)*AppellF1[3/2, m + n*p, 2 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n*p)*AppellF1[3/2, 1 + m + n*p, 1 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*Tan[(e + f*x)/2]^2*((-1 + n*p)*((-3*(2 - n*p)*AppellF1[5/2, m + n*p, 3 - n*p, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(m + n*p)*AppellF1[5/2, 1 + m + n*p, 2 - n*p, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (m + n*p)*((-3*(1 - n*p)*AppellF1[5/2, 1 + m + n*p, 2 - n*p, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + m + n*p)*AppellF1[5/2, 2 + m + n*p, 1 - n*p, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, m + n*p, 1 - n*p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n*p)*AppellF1[3/2, m + n*p, 2 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n*p)*AppellF1[3/2, 1 + m + n*p, 1 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (3*2^(1 + m)*(m + n*p)*AppellF1[1/2, m + n*p, 1 - n*p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n*p)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + m + n*p)*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, m + n*p, 1 - n*p, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n*p)*AppellF1[3/2, m + n*p, 2 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n*p)*AppellF1[3/2, 1 + m + n*p, 1 - n*p, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
231,1,343,275,2.3050404,"\int \left(c (d \sec (e+f x))^p\right)^n (a+a \sec (e+f x))^3 \, dx","Integrate[(c*(d*Sec[e + f*x])^p)^n*(a + a*Sec[e + f*x])^3,x]","-i a^3 2^{n p-3} \sec ^6\left(\frac{1}{2} (e+f x)\right) (\sec (e+f x)+1)^3 \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{n p} \left(\frac{12 e^{2 i (e+f x)} \, _2F_1\left(1,-\frac{n p}{2};\frac{n p}{2}+2;-e^{2 i (e+f x)}\right)}{f (n p+2) \left(1+e^{2 i (e+f x)}\right)}+\frac{8 e^{3 i (e+f x)} \, _2F_1\left(1,\frac{1}{2} (-n p-1);\frac{1}{2} (n p+5);-e^{2 i (e+f x)}\right)}{f (n p+3) \left(1+e^{2 i (e+f x)}\right)^2}+\frac{6 e^{i (e+f x)} \, _2F_1\left(1,\frac{1}{2} (1-n p);\frac{1}{2} (n p+3);-e^{2 i (e+f x)}\right)}{f n p+f}+\frac{\left(1+e^{2 i (e+f x)}\right) \, _2F_1\left(1,1-\frac{n p}{2};\frac{n p}{2}+1;-e^{2 i (e+f x)}\right)}{f n p}\right) \sec ^{-n p-3}(e+f x) \left(c (d \sec (e+f x))^p\right)^n","-\frac{a^3 (4 n p+1) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f \left(1-n^2 p^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (4 n p+7) \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p (n p+2) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (2 n p+5) \tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+1) (n p+2)}+\frac{\tan (e+f x) \left(a^3 \sec (e+f x)+a^3\right) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+2)}",1,"(-I)*2^(-3 + n*p)*a^3*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(n*p)*((12*E^((2*I)*(e + f*x))*Hypergeometric2F1[1, -1/2*(n*p), 2 + (n*p)/2, -E^((2*I)*(e + f*x))])/((1 + E^((2*I)*(e + f*x)))*f*(2 + n*p)) + (8*E^((3*I)*(e + f*x))*Hypergeometric2F1[1, (-1 - n*p)/2, (5 + n*p)/2, -E^((2*I)*(e + f*x))])/((1 + E^((2*I)*(e + f*x)))^2*f*(3 + n*p)) + (6*E^(I*(e + f*x))*Hypergeometric2F1[1, (1 - n*p)/2, (3 + n*p)/2, -E^((2*I)*(e + f*x))])/(f + f*n*p) + ((1 + E^((2*I)*(e + f*x)))*Hypergeometric2F1[1, 1 - (n*p)/2, 1 + (n*p)/2, -E^((2*I)*(e + f*x))])/(f*n*p))*Sec[(e + f*x)/2]^6*Sec[e + f*x]^(-3 - n*p)*(c*(d*Sec[e + f*x])^p)^n*(1 + Sec[e + f*x])^3","C",0
232,1,299,205,2.8187804,"\int \left(c (d \sec (e+f x))^p\right)^n (a+a \sec (e+f x))^2 \, dx","Integrate[(c*(d*Sec[e + f*x])^p)^n*(a + a*Sec[e + f*x])^2,x]","-\frac{i a^2 2^{n p-2} e^{-i (e+f x)} \sec ^4\left(\frac{1}{2} (e+f x)\right) (\sec (e+f x)+1)^2 \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{n p+1} \left(4 n p (n p+1) e^{2 i (e+f x)} \, _2F_1\left(1,-\frac{n p}{2};\frac{n p}{2}+2;-e^{2 i (e+f x)}\right)+(n p+2) \left(1+e^{2 i (e+f x)}\right) \left(4 n p e^{i (e+f x)} \, _2F_1\left(1,\frac{1}{2} (1-n p);\frac{1}{2} (n p+3);-e^{2 i (e+f x)}\right)+(n p+1) \left(1+e^{2 i (e+f x)}\right) \, _2F_1\left(1,1-\frac{n p}{2};\frac{n p}{2}+1;-e^{2 i (e+f x)}\right)\right)\right) \sec ^{-n p-2}(e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f n p (n p+1) (n p+2)}","-\frac{a^2 (2 n p+1) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f \left(1-n^2 p^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{2 a^2 \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p \sqrt{\sin ^2(e+f x)}}+\frac{a^2 \tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+1)}",1,"((-I)*2^(-2 + n*p)*a^2*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(1 + n*p)*(4*E^((2*I)*(e + f*x))*n*p*(1 + n*p)*Hypergeometric2F1[1, -1/2*(n*p), 2 + (n*p)/2, -E^((2*I)*(e + f*x))] + (1 + E^((2*I)*(e + f*x)))*(2 + n*p)*(4*E^(I*(e + f*x))*n*p*Hypergeometric2F1[1, (1 - n*p)/2, (3 + n*p)/2, -E^((2*I)*(e + f*x))] + (1 + E^((2*I)*(e + f*x)))*(1 + n*p)*Hypergeometric2F1[1, 1 - (n*p)/2, 1 + (n*p)/2, -E^((2*I)*(e + f*x))]))*Sec[(e + f*x)/2]^4*Sec[e + f*x]^(-2 - n*p)*(c*(d*Sec[e + f*x])^p)^n*(1 + Sec[e + f*x])^2)/(E^(I*(e + f*x))*f*n*p*(1 + n*p)*(2 + n*p))","C",0
233,1,124,156,0.2091227,"\int \left(c (d \sec (e+f x))^p\right)^n (a+a \sec (e+f x)) \, dx","Integrate[(c*(d*Sec[e + f*x])^p)^n*(a + a*Sec[e + f*x]),x]","\frac{a \sqrt{-\tan ^2(e+f x)} \csc (e+f x) \left(c (d \sec (e+f x))^p\right)^n \left(n p \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sec ^2(e+f x)\right)+(n p+1) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n p}{2};\frac{n p}{2}+1;\sec ^2(e+f x)\right)\right)}{f n p (n p+1)}","\frac{a \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p \sqrt{\sin ^2(e+f x)}}-\frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f (1-n p) \sqrt{\sin ^2(e+f x)}}",1,"(a*Csc[e + f*x]*((1 + n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (n*p)/2, 1 + (n*p)/2, Sec[e + f*x]^2] + n*p*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sec[e + f*x]^2])*(c*(d*Sec[e + f*x])^p)^n*Sqrt[-Tan[e + f*x]^2])/(f*n*p*(1 + n*p))","A",1
234,0,0,208,1.1777247,"\int \frac{\left(c (d \sec (e+f x))^p\right)^n}{a+a \sec (e+f x)} \, dx","Integrate[(c*(d*Sec[e + f*x])^p)^n/(a + a*Sec[e + f*x]),x]","\int \frac{\left(c (d \sec (e+f x))^p\right)^n}{a+a \sec (e+f x)} \, dx","-\frac{\sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{a f \sqrt{\sin ^2(e+f x)}}+\frac{(1-n p) \sin (e+f x) \cos ^2(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (2-n p);\frac{1}{2} (4-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{a f (2-n p) \sqrt{\sin ^2(e+f x)}}+\frac{\sin (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f (a \sec (e+f x)+a)}",1,"Integrate[(c*(d*Sec[e + f*x])^p)^n/(a + a*Sec[e + f*x]), x]","F",-1
235,0,0,248,10.8271976,"\int \frac{\left(c (d \sec (e+f x))^p\right)^n}{(a+a \sec (e+f x))^2} \, dx","Integrate[(c*(d*Sec[e + f*x])^p)^n/(a + a*Sec[e + f*x])^2,x]","\int \frac{\left(c (d \sec (e+f x))^p\right)^n}{(a+a \sec (e+f x))^2} \, dx","\frac{2 (2-n p) \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{(3-2 n p) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{2 (2-n p) \tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{3 a^2 f (\sec (e+f x)+1)}-\frac{\tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{3 f (a \sec (e+f x)+a)^2}",1,"Integrate[(c*(d*Sec[e + f*x])^p)^n/(a + a*Sec[e + f*x])^2, x]","F",-1
236,0,0,56,2.6913019,"\int \left(c (d \sec (e+f x))^p\right)^n (a+b \sec (e+f x))^m \, dx","Integrate[(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x])^m,x]","\int \left(c (d \sec (e+f x))^p\right)^n (a+b \sec (e+f x))^m \, dx","(d \sec (e+f x))^{-n p} \left(c (d \sec (e+f x))^p\right)^n \text{Int}\left((a+b \sec (e+f x))^m (d \sec (e+f x))^{n p},x\right)",0,"Integrate[(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x])^m, x]","A",-1
237,1,278,296,1.3056262,"\int \left(c (d \sec (e+f x))^p\right)^n (a+b \sec (e+f x))^3 \, dx","Integrate[(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x])^3,x]","-\frac{\left(-\tan ^2(e+f x)\right)^{3/2} \csc ^3(e+f x) \left(c (d \sec (e+f x))^p\right)^n \left(a^3 \left(n^3 p^3+6 n^2 p^2+11 n p+6\right) \cos ^3(e+f x) \, _2F_1\left(\frac{1}{2},\frac{n p}{2};\frac{n p}{2}+1;\sec ^2(e+f x)\right)+b n p \left((n p+2) \left(3 a^2 (n p+3) \cos ^2(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sec ^2(e+f x)\right)+b^2 (n p+1) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+3);\frac{1}{2} (n p+5);\sec ^2(e+f x)\right)\right)+3 a b \left(n^2 p^2+4 n p+3\right) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n p}{2}+1;\frac{n p}{2}+2;\sec ^2(e+f x)\right)\right)\right)}{f n p (n p+1) (n p+2) (n p+3)}","-\frac{a \left(a^2 (n p+1)+3 b^2 n p\right) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f \left(1-n^2 p^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{b \left(3 a^2 (n p+2)+b^2 (n p+1)\right) \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p (n p+2) \sqrt{\sin ^2(e+f x)}}+\frac{a b^2 (2 n p+5) \tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+1) (n p+2)}+\frac{b^2 \tan (e+f x) (a+b \sec (e+f x)) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+2)}",1,"-((Csc[e + f*x]^3*(a^3*(6 + 11*n*p + 6*n^2*p^2 + n^3*p^3)*Cos[e + f*x]^3*Hypergeometric2F1[1/2, (n*p)/2, 1 + (n*p)/2, Sec[e + f*x]^2] + b*n*p*(3*a*b*(3 + 4*n*p + n^2*p^2)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + (n*p)/2, 2 + (n*p)/2, Sec[e + f*x]^2] + (2 + n*p)*(3*a^2*(3 + n*p)*Cos[e + f*x]^2*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sec[e + f*x]^2] + b^2*(1 + n*p)*Hypergeometric2F1[1/2, (3 + n*p)/2, (5 + n*p)/2, Sec[e + f*x]^2])))*(c*(d*Sec[e + f*x])^p)^n*(-Tan[e + f*x]^2)^(3/2))/(f*n*p*(1 + n*p)*(2 + n*p)*(3 + n*p)))","A",1
238,1,200,211,0.5422272,"\int \left(c (d \sec (e+f x))^p\right)^n (a+b \sec (e+f x))^2 \, dx","Integrate[(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x])^2,x]","\frac{\sqrt{-\tan ^2(e+f x)} \csc (e+f x) \sec (e+f x) \left(c (d \sec (e+f x))^p\right)^n \left(a^2 \left(n^2 p^2+3 n p+2\right) \cos ^2(e+f x) \, _2F_1\left(\frac{1}{2},\frac{n p}{2};\frac{n p}{2}+1;\sec ^2(e+f x)\right)+b n p \left(2 a (n p+2) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sec ^2(e+f x)\right)+b (n p+1) \, _2F_1\left(\frac{1}{2},\frac{n p}{2}+1;\frac{n p}{2}+2;\sec ^2(e+f x)\right)\right)\right)}{f n p (n p+1) (n p+2)}","-\frac{\left(a^2 (n p+1)+b^2 n p\right) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f \left(1-n^2 p^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{2 a b \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p \sqrt{\sin ^2(e+f x)}}+\frac{b^2 \tan (e+f x) \left(c (d \sec (e+f x))^p\right)^n}{f (n p+1)}",1,"(Csc[e + f*x]*(a^2*(2 + 3*n*p + n^2*p^2)*Cos[e + f*x]^2*Hypergeometric2F1[1/2, (n*p)/2, 1 + (n*p)/2, Sec[e + f*x]^2] + b*n*p*(b*(1 + n*p)*Hypergeometric2F1[1/2, 1 + (n*p)/2, 2 + (n*p)/2, Sec[e + f*x]^2] + 2*a*(2 + n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sec[e + f*x]^2]))*Sec[e + f*x]*(c*(d*Sec[e + f*x])^p)^n*Sqrt[-Tan[e + f*x]^2])/(f*n*p*(1 + n*p)*(2 + n*p))","A",1
239,1,125,156,0.239665,"\int \left(c (d \sec (e+f x))^p\right)^n (a+b \sec (e+f x)) \, dx","Integrate[(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x]),x]","\frac{\sqrt{-\tan ^2(e+f x)} \csc (e+f x) \left(c (d \sec (e+f x))^p\right)^n \left(a (n p+1) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n p}{2};\frac{n p}{2}+1;\sec ^2(e+f x)\right)+b n p \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sec ^2(e+f x)\right)\right)}{f n p (n p+1)}","\frac{b \sin (e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f n p \sqrt{\sin ^2(e+f x)}}-\frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right) \left(c (d \sec (e+f x))^p\right)^n}{f (1-n p) \sqrt{\sin ^2(e+f x)}}",1,"(Csc[e + f*x]*(a*(1 + n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (n*p)/2, 1 + (n*p)/2, Sec[e + f*x]^2] + b*n*p*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sec[e + f*x]^2])*(c*(d*Sec[e + f*x])^p)^n*Sqrt[-Tan[e + f*x]^2])/(f*n*p*(1 + n*p))","A",1
240,1,5411,206,25.6251315,"\int \frac{\left(c (d \sec (e+f x))^p\right)^n}{a+b \sec (e+f x)} \, dx","Integrate[(c*(d*Sec[e + f*x])^p)^n/(a + b*Sec[e + f*x]),x]","\text{Result too large to show}","\frac{a \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p-1)} \left(c (d \sec (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (n p-1),1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{b \sin (e+f x) \cos ^2(e+f x)^{\frac{n p}{2}} \left(c (d \sec (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{n p}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}",1,"Result too large to show","B",0
241,1,14108,322,45.3749533,"\int \frac{\left(c (d \sec (e+f x))^p\right)^n}{(a+b \sec (e+f x))^2} \, dx","Integrate[(c*(d*Sec[e + f*x])^p)^n/(a + b*Sec[e + f*x])^2,x]","\text{Result too large to show}","-\frac{2 a b \sin (e+f x) \cos ^2(e+f x)^{\frac{n p}{2}} \left(c (d \sec (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (n p-2),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}+\frac{a^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p-1)} \left(c (d \sec (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (n p-3),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}+\frac{b^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p-1)} \left(c (d \sec (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (n p-1),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}",1,"Result too large to show","B",0