1,1,134,151,0.4787194,"\int (a+a \sec (c+d x)) \tan ^9(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Tan[c + d*x]^9,x]","\frac{a \sec ^9(c+d x)}{9 d}-\frac{4 a \sec ^7(c+d x)}{7 d}+\frac{6 a \sec ^5(c+d x)}{5 d}-\frac{4 a \sec ^3(c+d x)}{3 d}+\frac{a \sec (c+d x)}{d}-\frac{a \left(-3 \tan ^8(c+d x)+4 \tan ^6(c+d x)-6 \tan ^4(c+d x)+12 \tan ^2(c+d x)+24 \log (\cos (c+d x))\right)}{24 d}","\frac{a \sec ^9(c+d x)}{9 d}+\frac{a \sec ^8(c+d x)}{8 d}-\frac{4 a \sec ^7(c+d x)}{7 d}-\frac{2 a \sec ^6(c+d x)}{3 d}+\frac{6 a \sec ^5(c+d x)}{5 d}+\frac{3 a \sec ^4(c+d x)}{2 d}-\frac{4 a \sec ^3(c+d x)}{3 d}-\frac{2 a \sec ^2(c+d x)}{d}+\frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"(a*Sec[c + d*x])/d - (4*a*Sec[c + d*x]^3)/(3*d) + (6*a*Sec[c + d*x]^5)/(5*d) - (4*a*Sec[c + d*x]^7)/(7*d) + (a*Sec[c + d*x]^9)/(9*d) - (a*(24*Log[Cos[c + d*x]] + 12*Tan[c + d*x]^2 - 6*Tan[c + d*x]^4 + 4*Tan[c + d*x]^6 - 3*Tan[c + d*x]^8))/(24*d)","A",1
2,1,106,118,0.466807,"\int (a+a \sec (c+d x)) \tan ^7(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Tan[c + d*x]^7,x]","\frac{a \sec ^7(c+d x)}{7 d}-\frac{3 a \sec ^5(c+d x)}{5 d}+\frac{a \sec ^3(c+d x)}{d}-\frac{a \sec (c+d x)}{d}+\frac{a \left(2 \tan ^6(c+d x)-3 \tan ^4(c+d x)+6 \tan ^2(c+d x)+12 \log (\cos (c+d x))\right)}{12 d}","\frac{a \sec ^7(c+d x)}{7 d}+\frac{a \sec ^6(c+d x)}{6 d}-\frac{3 a \sec ^5(c+d x)}{5 d}-\frac{3 a \sec ^4(c+d x)}{4 d}+\frac{a \sec ^3(c+d x)}{d}+\frac{3 a \sec ^2(c+d x)}{2 d}-\frac{a \sec (c+d x)}{d}+\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Sec[c + d*x])/d) + (a*Sec[c + d*x]^3)/d - (3*a*Sec[c + d*x]^5)/(5*d) + (a*Sec[c + d*x]^7)/(7*d) + (a*(12*Log[Cos[c + d*x]] + 6*Tan[c + d*x]^2 - 3*Tan[c + d*x]^4 + 2*Tan[c + d*x]^6))/(12*d)","A",1
3,1,82,87,0.3660363,"\int (a+a \sec (c+d x)) \tan ^5(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Tan[c + d*x]^5,x]","\frac{a \sec ^5(c+d x)}{5 d}-\frac{2 a \sec ^3(c+d x)}{3 d}+\frac{a \sec (c+d x)}{d}-\frac{a \left(-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right)}{4 d}","\frac{a \sec ^5(c+d x)}{5 d}+\frac{a \sec ^4(c+d x)}{4 d}-\frac{2 a \sec ^3(c+d x)}{3 d}-\frac{a \sec ^2(c+d x)}{d}+\frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"(a*Sec[c + d*x])/d - (2*a*Sec[c + d*x]^3)/(3*d) + (a*Sec[c + d*x]^5)/(5*d) - (a*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/(4*d)","A",1
4,1,55,57,0.1150254,"\int (a+a \sec (c+d x)) \tan ^3(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Tan[c + d*x]^3,x]","\frac{a \sec ^3(c+d x)}{3 d}-\frac{a \sec (c+d x)}{d}+\frac{a \left(\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)}{2 d}","\frac{a \sec ^3(c+d x)}{3 d}+\frac{a \sec ^2(c+d x)}{2 d}-\frac{a \sec (c+d x)}{d}+\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Sec[c + d*x])/d) + (a*Sec[c + d*x]^3)/(3*d) + (a*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/(2*d)","A",1
5,1,25,25,0.0199516,"\int (a+a \sec (c+d x)) \tan (c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Tan[c + d*x],x]","\frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}","\frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Log[Cos[c + d*x]])/d) + (a*Sec[c + d*x])/d","A",1
6,1,29,16,0.0251516,"\int \cot (c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + a*Sec[c + d*x]),x]","\frac{2 a \left(\log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d}","\frac{a \log (1-\cos (c+d x))}{d}",1,"(2*a*(Log[Cos[(c + d*x)/2]] + Log[Tan[(c + d*x)/2]]))/d","A",1
7,1,114,57,0.7829114,"\int \cot ^3(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + a*Sec[c + d*x]),x]","-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{a \left(\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right)}{2 d}","-\frac{a}{2 d (1-\cos (c+d x))}-\frac{3 a \log (1-\cos (c+d x))}{4 d}-\frac{a \log (\cos (c+d x)+1)}{4 d}",1,"-1/8*(a*Csc[(c + d*x)/2]^2)/d + (a*Log[Cos[(c + d*x)/2]])/(2*d) - (a*Log[Sin[(c + d*x)/2]])/(2*d) - (a*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]]))/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d)","A",1
8,1,127,95,0.5437893,"\int \cot ^5(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + a*Sec[c + d*x]),x]","\frac{a \left(-16 \cot ^4(c+d x)+32 \cot ^2(c+d x)-\csc ^4\left(\frac{1}{2} (c+d x)\right)+10 \csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^4\left(\frac{1}{2} (c+d x)\right)-10 \sec ^2\left(\frac{1}{2} (c+d x)\right)+24 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+64 \log (\tan (c+d x))-24 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+64 \log (\cos (c+d x))\right)}{64 d}","\frac{3 a}{4 d (1-\cos (c+d x))}+\frac{a}{8 d (\cos (c+d x)+1)}-\frac{a}{8 d (1-\cos (c+d x))^2}+\frac{11 a \log (1-\cos (c+d x))}{16 d}+\frac{5 a \log (\cos (c+d x)+1)}{16 d}",1,"(a*(32*Cot[c + d*x]^2 - 16*Cot[c + d*x]^4 + 10*Csc[(c + d*x)/2]^2 - Csc[(c + d*x)/2]^4 - 24*Log[Cos[(c + d*x)/2]] + 64*Log[Cos[c + d*x]] + 24*Log[Sin[(c + d*x)/2]] + 64*Log[Tan[c + d*x]] - 10*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^4))/(64*d)","A",1
9,1,165,133,0.4148018,"\int \cot ^7(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]^7*(a + a*Sec[c + d*x]),x]","-\frac{a \left(64 \cot ^6(c+d x)-96 \cot ^4(c+d x)+192 \cot ^2(c+d x)+\csc ^6\left(\frac{1}{2} (c+d x)\right)-12 \csc ^4\left(\frac{1}{2} (c+d x)\right)+66 \csc ^2\left(\frac{1}{2} (c+d x)\right)-\sec ^6\left(\frac{1}{2} (c+d x)\right)+12 \sec ^4\left(\frac{1}{2} (c+d x)\right)-66 \sec ^2\left(\frac{1}{2} (c+d x)\right)+120 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+384 \log (\tan (c+d x))-120 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+384 \log (\cos (c+d x))\right)}{384 d}","-\frac{15 a}{16 d (1-\cos (c+d x))}-\frac{a}{4 d (\cos (c+d x)+1)}+\frac{9 a}{32 d (1-\cos (c+d x))^2}+\frac{a}{32 d (\cos (c+d x)+1)^2}-\frac{a}{24 d (1-\cos (c+d x))^3}-\frac{21 a \log (1-\cos (c+d x))}{32 d}-\frac{11 a \log (\cos (c+d x)+1)}{32 d}",1,"-1/384*(a*(192*Cot[c + d*x]^2 - 96*Cot[c + d*x]^4 + 64*Cot[c + d*x]^6 + 66*Csc[(c + d*x)/2]^2 - 12*Csc[(c + d*x)/2]^4 + Csc[(c + d*x)/2]^6 - 120*Log[Cos[(c + d*x)/2]] + 384*Log[Cos[c + d*x]] + 120*Log[Sin[(c + d*x)/2]] + 384*Log[Tan[c + d*x]] - 66*Sec[(c + d*x)/2]^2 + 12*Sec[(c + d*x)/2]^4 - Sec[(c + d*x)/2]^6))/d","A",1
10,1,115,129,1.8941822,"\int (a+a \sec (c+d x)) \tan ^8(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Tan[c + d*x]^8,x]","\frac{a \left(13440 \tan ^{-1}(\tan (c+d x))+3675 \tanh ^{-1}(\sin (c+d x))-\frac{1}{32} (223232 \cos (c+d x)+75915 \cos (2 (c+d x))+147968 \cos (3 (c+d x))+12950 \cos (4 (c+d x))+47616 \cos (5 (c+d x))+9765 \cos (6 (c+d x))+11264 \cos (7 (c+d x))+18970) \tan (c+d x) \sec ^7(c+d x)\right)}{13440 d}","\frac{35 a \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{\tan ^7(c+d x) (7 a \sec (c+d x)+8 a)}{56 d}-\frac{\tan ^5(c+d x) (35 a \sec (c+d x)+48 a)}{240 d}+\frac{\tan ^3(c+d x) (35 a \sec (c+d x)+64 a)}{192 d}-\frac{\tan (c+d x) (35 a \sec (c+d x)+128 a)}{128 d}+a x",1,"(a*(13440*ArcTan[Tan[c + d*x]] + 3675*ArcTanh[Sin[c + d*x]] - ((18970 + 223232*Cos[c + d*x] + 75915*Cos[2*(c + d*x)] + 147968*Cos[3*(c + d*x)] + 12950*Cos[4*(c + d*x)] + 47616*Cos[5*(c + d*x)] + 9765*Cos[6*(c + d*x)] + 11264*Cos[7*(c + d*x)])*Sec[c + d*x]^7*Tan[c + d*x])/32))/(13440*d)","A",1
11,1,95,102,1.2470155,"\int (a+a \sec (c+d x)) \tan ^6(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Tan[c + d*x]^6,x]","-\frac{a \left(240 \tan ^{-1}(\tan (c+d x))+75 \tanh ^{-1}(\sin (c+d x))-\frac{1}{8} (1168 \cos (c+d x)+140 \cos (2 (c+d x))+568 \cos (3 (c+d x))+165 \cos (4 (c+d x))+184 \cos (5 (c+d x))+295) \tan (c+d x) \sec ^5(c+d x)\right)}{240 d}","-\frac{5 a \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\tan ^5(c+d x) (5 a \sec (c+d x)+6 a)}{30 d}-\frac{\tan ^3(c+d x) (5 a \sec (c+d x)+8 a)}{24 d}+\frac{\tan (c+d x) (5 a \sec (c+d x)+16 a)}{16 d}-a x",1,"-1/240*(a*(240*ArcTan[Tan[c + d*x]] + 75*ArcTanh[Sin[c + d*x]] - ((295 + 1168*Cos[c + d*x] + 140*Cos[2*(c + d*x)] + 568*Cos[3*(c + d*x)] + 165*Cos[4*(c + d*x)] + 184*Cos[5*(c + d*x)])*Sec[c + d*x]^5*Tan[c + d*x])/8))/d","A",1
12,1,75,73,0.4398419,"\int (a+a \sec (c+d x)) \tan ^4(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Tan[c + d*x]^4,x]","\frac{a \left(24 \tan ^{-1}(\tan (c+d x))+9 \tanh ^{-1}(\sin (c+d x))-\frac{1}{2} (32 \cos (c+d x)+15 \cos (2 (c+d x))+16 \cos (3 (c+d x))+3) \tan (c+d x) \sec ^3(c+d x)\right)}{24 d}","\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan ^3(c+d x) (3 a \sec (c+d x)+4 a)}{12 d}-\frac{\tan (c+d x) (3 a \sec (c+d x)+8 a)}{8 d}+a x",1,"(a*(24*ArcTan[Tan[c + d*x]] + 9*ArcTanh[Sin[c + d*x]] - ((3 + 32*Cos[c + d*x] + 15*Cos[2*(c + d*x)] + 16*Cos[3*(c + d*x)])*Sec[c + d*x]^3*Tan[c + d*x])/2))/(24*d)","A",1
13,1,60,45,0.0297322,"\int (a+a \sec (c+d x)) \tan ^2(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Tan[c + d*x]^2,x]","-\frac{a \tan ^{-1}(\tan (c+d x))}{d}+\frac{a \tan (c+d x)}{d}-\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}","-\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\tan (c+d x) (a \sec (c+d x)+2 a)}{2 d}-a x",1,"-((a*ArcTan[Tan[c + d*x]])/d) - (a*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Tan[c + d*x])/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
14,1,43,26,0.0266467,"\int \cot ^2(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + a*Sec[c + d*x]),x]","-\frac{a \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}-\frac{a \csc (c+d x)}{d}","-\frac{\cot (c+d x) (a \sec (c+d x)+a)}{d}-a x",1,"-((a*Csc[c + d*x])/d) - (a*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d","C",1
15,1,62,55,0.0376209,"\int \cot ^4(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + a*Sec[c + d*x]),x]","-\frac{a \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)}{3 d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{a \csc (c+d x)}{d}","-\frac{\cot ^3(c+d x) (a \sec (c+d x)+a)}{3 d}+\frac{\cot (c+d x) (2 a \sec (c+d x)+3 a)}{3 d}+a x",1,"(a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d)","C",1
16,1,79,84,0.055636,"\int \cot ^6(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]^6*(a + a*Sec[c + d*x]),x]","-\frac{a \cot ^5(c+d x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(c+d x)\right)}{5 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{2 a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}","-\frac{\cot ^5(c+d x) (a \sec (c+d x)+a)}{5 d}+\frac{\cot ^3(c+d x) (4 a \sec (c+d x)+5 a)}{15 d}-\frac{\cot (c+d x) (8 a \sec (c+d x)+15 a)}{15 d}-a x",1,"-((a*Csc[c + d*x])/d) + (2*a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d) - (a*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/(5*d)","C",1
17,1,92,111,0.0550523,"\int \cot ^8(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]^8*(a + a*Sec[c + d*x]),x]","-\frac{a \cot ^7(c+d x) \, _2F_1\left(-\frac{7}{2},1;-\frac{5}{2};-\tan ^2(c+d x)\right)}{7 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{3 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{d}+\frac{a \csc (c+d x)}{d}","-\frac{\cot ^7(c+d x) (a \sec (c+d x)+a)}{7 d}+\frac{\cot ^5(c+d x) (6 a \sec (c+d x)+7 a)}{35 d}-\frac{\cot ^3(c+d x) (24 a \sec (c+d x)+35 a)}{105 d}+\frac{\cot (c+d x) (16 a \sec (c+d x)+35 a)}{35 d}+a x",1,"(a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d) - (a*Cot[c + d*x]^7*Hypergeometric2F1[-7/2, 1, -5/2, -Tan[c + d*x]^2])/(7*d)","C",1
18,1,111,140,0.0824627,"\int \cot ^{10}(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]^10*(a + a*Sec[c + d*x]),x]","-\frac{a \cot ^9(c+d x) \, _2F_1\left(-\frac{9}{2},1;-\frac{7}{2};-\tan ^2(c+d x)\right)}{9 d}-\frac{a \csc ^9(c+d x)}{9 d}+\frac{4 a \csc ^7(c+d x)}{7 d}-\frac{6 a \csc ^5(c+d x)}{5 d}+\frac{4 a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}","-\frac{\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}+\frac{\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{63 d}-\frac{\cot ^5(c+d x) (16 a \sec (c+d x)+21 a)}{105 d}+\frac{\cot ^3(c+d x) (64 a \sec (c+d x)+105 a)}{315 d}-\frac{\cot (c+d x) (128 a \sec (c+d x)+315 a)}{315 d}-a x",1,"-((a*Csc[c + d*x])/d) + (4*a*Csc[c + d*x]^3)/(3*d) - (6*a*Csc[c + d*x]^5)/(5*d) + (4*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d) - (a*Cot[c + d*x]^9*Hypergeometric2F1[-9/2, 1, -7/2, -Tan[c + d*x]^2])/(9*d)","C",1
19,1,140,192,0.5092724,"\int (a+a \sec (c+d x))^2 \tan ^9(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Tan[c + d*x]^9,x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(-252 \sec ^{10}(c+d x)-560 \sec ^9(c+d x)+945 \sec ^8(c+d x)+2880 \sec ^7(c+d x)-840 \sec ^6(c+d x)-6048 \sec ^5(c+d x)-1260 \sec ^4(c+d x)+6720 \sec ^3(c+d x)+3780 \sec ^2(c+d x)-5040 \sec (c+d x)+2520 \log (\cos (c+d x))\right)}{10080 d}","\frac{a^2 \sec ^{10}(c+d x)}{10 d}+\frac{2 a^2 \sec ^9(c+d x)}{9 d}-\frac{3 a^2 \sec ^8(c+d x)}{8 d}-\frac{8 a^2 \sec ^7(c+d x)}{7 d}+\frac{a^2 \sec ^6(c+d x)}{3 d}+\frac{12 a^2 \sec ^5(c+d x)}{5 d}+\frac{a^2 \sec ^4(c+d x)}{2 d}-\frac{8 a^2 \sec ^3(c+d x)}{3 d}-\frac{3 a^2 \sec ^2(c+d x)}{2 d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d}",1,"-1/10080*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(2520*Log[Cos[c + d*x]] - 5040*Sec[c + d*x] + 3780*Sec[c + d*x]^2 + 6720*Sec[c + d*x]^3 - 1260*Sec[c + d*x]^4 - 6048*Sec[c + d*x]^5 - 840*Sec[c + d*x]^6 + 2880*Sec[c + d*x]^7 + 945*Sec[c + d*x]^8 - 560*Sec[c + d*x]^9 - 252*Sec[c + d*x]^10))/d","A",1
20,1,110,132,0.3064988,"\int (a+a \sec (c+d x))^2 \tan ^7(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Tan[c + d*x]^7,x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(105 \sec ^8(c+d x)+240 \sec ^7(c+d x)-280 \sec ^6(c+d x)-1008 \sec ^5(c+d x)+1680 \sec ^3(c+d x)+840 \sec ^2(c+d x)-1680 \sec (c+d x)+840 \log (\cos (c+d x))\right)}{3360 d}","\frac{a^2 \sec ^8(c+d x)}{8 d}+\frac{2 a^2 \sec ^7(c+d x)}{7 d}-\frac{a^2 \sec ^6(c+d x)}{3 d}-\frac{6 a^2 \sec ^5(c+d x)}{5 d}+\frac{2 a^2 \sec ^3(c+d x)}{d}+\frac{a^2 \sec ^2(c+d x)}{d}-\frac{2 a^2 \sec (c+d x)}{d}+\frac{a^2 \log (\cos (c+d x))}{d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(840*Log[Cos[c + d*x]] - 1680*Sec[c + d*x] + 840*Sec[c + d*x]^2 + 1680*Sec[c + d*x]^3 - 1008*Sec[c + d*x]^5 - 280*Sec[c + d*x]^6 + 240*Sec[c + d*x]^7 + 105*Sec[c + d*x]^8))/(3360*d)","A",1
21,1,125,120,0.4165191,"\int (a+a \sec (c+d x))^2 \tan ^5(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Tan[c + d*x]^5,x]","\frac{a^2 \sec ^6(c+d x) (312 \cos (c+d x)-5 (-28 \cos (3 (c+d x))+6 \cos (4 (c+d x))-12 \cos (5 (c+d x))+18 \cos (4 (c+d x)) \log (\cos (c+d x))+3 \cos (6 (c+d x)) \log (\cos (c+d x))+30 \log (\cos (c+d x))+9 \cos (2 (c+d x)) (5 \log (\cos (c+d x))+4)+14))}{480 d}","\frac{a^2 \sec ^6(c+d x)}{6 d}+\frac{2 a^2 \sec ^5(c+d x)}{5 d}-\frac{a^2 \sec ^4(c+d x)}{4 d}-\frac{4 a^2 \sec ^3(c+d x)}{3 d}-\frac{a^2 \sec ^2(c+d x)}{2 d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d}",1,"(a^2*(312*Cos[c + d*x] - 5*(14 - 28*Cos[3*(c + d*x)] + 6*Cos[4*(c + d*x)] - 12*Cos[5*(c + d*x)] + 30*Log[Cos[c + d*x]] + 18*Cos[4*(c + d*x)]*Log[Cos[c + d*x]] + 3*Cos[6*(c + d*x)]*Log[Cos[c + d*x]] + 9*Cos[2*(c + d*x)]*(4 + 5*Log[Cos[c + d*x]])))*Sec[c + d*x]^6)/(480*d)","A",1
22,1,83,65,0.210472,"\int (a+a \sec (c+d x))^2 \tan ^3(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Tan[c + d*x]^3,x]","\frac{a^2 \sec ^4(c+d x) (3 (-4 \cos (3 (c+d x))+4 \cos (2 (c+d x)) \log (\cos (c+d x))+\cos (4 (c+d x)) \log (\cos (c+d x))+3 \log (\cos (c+d x))+2)-20 \cos (c+d x))}{24 d}","\frac{a^2 \sec ^4(c+d x)}{4 d}+\frac{2 a^2 \sec ^3(c+d x)}{3 d}-\frac{2 a^2 \sec (c+d x)}{d}+\frac{a^2 \log (\cos (c+d x))}{d}",1,"(a^2*(-20*Cos[c + d*x] + 3*(2 - 4*Cos[3*(c + d*x)] + 3*Log[Cos[c + d*x]] + 4*Cos[2*(c + d*x)]*Log[Cos[c + d*x]] + Cos[4*(c + d*x)]*Log[Cos[c + d*x]]))*Sec[c + d*x]^4)/(24*d)","A",1
23,1,51,48,0.1456099,"\int (a+a \sec (c+d x))^2 \tan (c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Tan[c + d*x],x]","-\frac{a^2 \sec ^2(c+d x) (-4 \cos (c+d x)+\cos (2 (c+d x)) \log (\cos (c+d x))+\log (\cos (c+d x))-1)}{2 d}","\frac{a^2 \sec ^2(c+d x)}{2 d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d}",1,"-1/2*(a^2*(-1 - 4*Cos[c + d*x] + Log[Cos[c + d*x]] + Cos[2*(c + d*x)]*Log[Cos[c + d*x]])*Sec[c + d*x]^2)/d","A",1
24,1,29,35,0.0377274,"\int \cot (c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]*(a + a*Sec[c + d*x])^2,x]","-\frac{a^2 \left(\log (\cos (c+d x))-4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d}","\frac{2 a^2 \log (1-\cos (c+d x))}{d}-\frac{a^2 \log (\cos (c+d x))}{d}",1,"-((a^2*(Log[Cos[c + d*x]] - 4*Log[Sin[(c + d*x)/2]]))/d)","A",1
25,1,56,40,0.0706371,"\int \cot ^3(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^3*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \csc ^2\left(\frac{1}{2} (c+d x)\right) \left(-2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \cos (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-1\right)}{2 d}","-\frac{a^2}{d (1-\cos (c+d x))}-\frac{a^2 \log (1-\cos (c+d x))}{d}",1,"(a^2*Csc[(c + d*x)/2]^2*(-1 - 2*Log[Sin[(c + d*x)/2]] + 2*Cos[c + d*x]*Log[Sin[(c + d*x)/2]]))/(2*d)","A",1
26,1,86,85,0.2682271,"\int \cot ^5(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^2,x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\csc ^4\left(\frac{1}{2} (c+d x)\right)-10 \csc ^2\left(\frac{1}{2} (c+d x)\right)-4 \left(7 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{64 d}","\frac{5 a^2}{4 d (1-\cos (c+d x))}-\frac{a^2}{4 d (1-\cos (c+d x))^2}+\frac{7 a^2 \log (1-\cos (c+d x))}{8 d}+\frac{a^2 \log (\cos (c+d x)+1)}{8 d}",1,"-1/64*(a^2*(1 + Cos[c + d*x])^2*(-10*Csc[(c + d*x)/2]^2 + Csc[(c + d*x)/2]^4 - 4*(Log[Cos[(c + d*x)/2]] + 7*Log[Sin[(c + d*x)/2]]))*Sec[(c + d*x)/2]^4)/d","A",1
27,1,114,127,0.249639,"\int \cot ^7(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^7*(a + a*Sec[c + d*x])^2,x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\csc ^6\left(\frac{1}{2} (c+d x)\right)-12 \csc ^4\left(\frac{1}{2} (c+d x)\right)+69 \csc ^2\left(\frac{1}{2} (c+d x)\right)+3 \left(\sec ^2\left(\frac{1}{2} (c+d x)\right)+52 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{384 d}","-\frac{23 a^2}{16 d (1-\cos (c+d x))}-\frac{a^2}{16 d (\cos (c+d x)+1)}+\frac{a^2}{2 d (1-\cos (c+d x))^2}-\frac{a^2}{12 d (1-\cos (c+d x))^3}-\frac{13 a^2 \log (1-\cos (c+d x))}{16 d}-\frac{3 a^2 \log (\cos (c+d x)+1)}{16 d}",1,"-1/384*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(69*Csc[(c + d*x)/2]^2 - 12*Csc[(c + d*x)/2]^4 + Csc[(c + d*x)/2]^6 + 3*(12*Log[Cos[(c + d*x)/2]] + 52*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2)))/d","A",1
28,1,146,169,0.3448075,"\int \cot ^9(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^9*(a + a*Sec[c + d*x])^2,x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(3 \csc ^8\left(\frac{1}{2} (c+d x)\right)-44 \csc ^6\left(\frac{1}{2} (c+d x)\right)+288 \csc ^4\left(\frac{1}{2} (c+d x)\right)-1224 \csc ^2\left(\frac{1}{2} (c+d x)\right)-6 \left(-\sec ^4\left(\frac{1}{2} (c+d x)\right)+18 \sec ^2\left(\frac{1}{2} (c+d x)\right)+396 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+116 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{6144 d}","\frac{51 a^2}{32 d (1-\cos (c+d x))}+\frac{9 a^2}{64 d (\cos (c+d x)+1)}-\frac{3 a^2}{4 d (1-\cos (c+d x))^2}-\frac{a^2}{64 d (\cos (c+d x)+1)^2}+\frac{11 a^2}{48 d (1-\cos (c+d x))^3}-\frac{a^2}{32 d (1-\cos (c+d x))^4}+\frac{99 a^2 \log (1-\cos (c+d x))}{128 d}+\frac{29 a^2 \log (\cos (c+d x)+1)}{128 d}",1,"-1/6144*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(-1224*Csc[(c + d*x)/2]^2 + 288*Csc[(c + d*x)/2]^4 - 44*Csc[(c + d*x)/2]^6 + 3*Csc[(c + d*x)/2]^8 - 6*(116*Log[Cos[(c + d*x)/2]] + 396*Log[Sin[(c + d*x)/2]] + 18*Sec[(c + d*x)/2]^2 - Sec[(c + d*x)/2]^4)))/d","A",1
29,1,337,161,1.5804998,"\int (a+a \sec (c+d x))^2 \tan ^6(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Tan[c + d*x]^6,x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec ^7(c+d x) \left(33600 \cos ^7(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sec (c) (-16240 \sin (2 c+d x)+2975 \sin (c+2 d x)+2975 \sin (3 c+2 d x)+14448 \sin (2 c+3 d x)-10080 \sin (4 c+3 d x)+980 \sin (3 c+4 d x)+980 \sin (5 c+4 d x)+6496 \sin (4 c+5 d x)-1680 \sin (6 c+5 d x)+1155 \sin (5 c+6 d x)+1155 \sin (7 c+6 d x)+1168 \sin (6 c+7 d x)-14700 d x \cos (2 c+d x)-8820 d x \cos (2 c+3 d x)-8820 d x \cos (4 c+3 d x)-2940 d x \cos (4 c+5 d x)-2940 d x \cos (6 c+5 d x)-420 d x \cos (6 c+7 d x)-420 d x \cos (8 c+7 d x)+24640 \sin (d x)-14700 d x \cos (d x))\right)}{215040 d}","\frac{a^2 \tan ^7(c+d x)}{7 d}+\frac{a^2 \tan ^5(c+d x)}{5 d}-\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}-\frac{5 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \tan ^5(c+d x) \sec (c+d x)}{3 d}-\frac{5 a^2 \tan ^3(c+d x) \sec (c+d x)}{12 d}+\frac{5 a^2 \tan (c+d x) \sec (c+d x)}{8 d}-a^2 x",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*Sec[c + d*x]^7*(33600*Cos[c + d*x]^7*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*(-14700*d*x*Cos[d*x] - 14700*d*x*Cos[2*c + d*x] - 8820*d*x*Cos[2*c + 3*d*x] - 8820*d*x*Cos[4*c + 3*d*x] - 2940*d*x*Cos[4*c + 5*d*x] - 2940*d*x*Cos[6*c + 5*d*x] - 420*d*x*Cos[6*c + 7*d*x] - 420*d*x*Cos[8*c + 7*d*x] + 24640*Sin[d*x] - 16240*Sin[2*c + d*x] + 2975*Sin[c + 2*d*x] + 2975*Sin[3*c + 2*d*x] + 14448*Sin[2*c + 3*d*x] - 10080*Sin[4*c + 3*d*x] + 980*Sin[3*c + 4*d*x] + 980*Sin[5*c + 4*d*x] + 6496*Sin[4*c + 5*d*x] - 1680*Sin[6*c + 5*d*x] + 1155*Sin[5*c + 6*d*x] + 1155*Sin[7*c + 6*d*x] + 1168*Sin[6*c + 7*d*x])))/(215040*d)","B",1
30,1,558,119,5.6318891,"\int (a+a \sec (c+d x))^2 \tan ^4(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Tan[c + d*x]^4,x]","\frac{1}{960} a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{\cos \left(\frac{c}{2}\right) \left(\frac{151}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{36}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{151}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{36}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}\right)}{d}+\frac{149 \sin \left(\frac{c}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{149 \sin \left(\frac{c}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{24 \sin \left(\frac{c}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{24 \sin \left(\frac{c}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{180 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{180 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{\sec (c) \sin \left(\frac{d x}{2}\right) \left(333 \cos \left(2 c+\frac{3 d x}{2}\right)+287 \cos \left(2 c+\frac{5 d x}{2}\right)+67 \cos \left(4 c+\frac{7 d x}{2}\right)+68 \cos \left(4 c+\frac{9 d x}{2}\right)+293 \cos \left(\frac{d x}{2}\right)\right) \sec ^5(c+d x)}{2 d}+240 x\right)","\frac{a^2 \tan ^5(c+d x)}{5 d}+\frac{a^2 \tan ^3(c+d x)}{3 d}-\frac{a^2 \tan (c+d x)}{d}+\frac{3 a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^2 \tan ^3(c+d x) \sec (c+d x)}{2 d}-\frac{3 a^2 \tan (c+d x) \sec (c+d x)}{4 d}+a^2 x",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(240*x - (180*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (180*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d - ((293*Cos[(d*x)/2] + 333*Cos[2*c + (3*d*x)/2] + 287*Cos[2*c + (5*d*x)/2] + 67*Cos[4*c + (7*d*x)/2] + 68*Cos[4*c + (9*d*x)/2])*Sec[c]*Sec[c + d*x]^5*Sin[(d*x)/2])/(2*d) - (24*Sin[c/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + (149*Sin[c/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - (24*Sin[c/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + (149*Sin[c/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (Cos[c/2]*(36/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) - 151/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - 36/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + 151/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)))/d))/960","B",1
31,1,773,72,6.3296297,"\int (a+a \sec (c+d x))^2 \tan ^2(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Tan[c + d*x]^2,x]","-\frac{1}{4} x \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2+\frac{\sin \left(\frac{d x}{2}\right) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2}{6 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\sin \left(\frac{d x}{2}\right) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2}{6 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\left(7 \cos \left(\frac{c}{2}\right)-5 \sin \left(\frac{c}{2}\right)\right) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2}{48 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\left(-5 \sin \left(\frac{c}{2}\right)-7 \cos \left(\frac{c}{2}\right)\right) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2}{48 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\sin \left(\frac{d x}{2}\right) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2}{24 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{\sin \left(\frac{d x}{2}\right) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2}{24 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{\cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{4 d}-\frac{\cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{4 d}","\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{d}-a^2 x",1,"-1/4*(x*Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2) + (Cos[c + d*x]^2*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(4*d) - (Cos[c + d*x]^2*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(4*d) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(24*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(7*Cos[c/2] - 5*Sin[c/2]))/(48*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(6*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(24*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(-7*Cos[c/2] - 5*Sin[c/2]))/(48*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(6*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
32,1,46,35,0.0365653,"\int \cot ^2(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*(a + a*Sec[c + d*x])^2,x]","-\frac{2 a^2 \cot \left(\frac{c}{2}+\frac{d x}{2}\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}","-\frac{2 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc (c+d x)}{d}+a^2 (-x)",1,"(-2*a^2*Cot[c/2 + (d*x)/2]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c/2 + (d*x)/2]^2])/d","C",1
33,1,112,69,0.2714811,"\int \cot ^4(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \csc \left(\frac{c}{2}\right) \csc ^3\left(\frac{1}{2} (c+d x)\right) \left(-12 \sin \left(c+\frac{d x}{2}\right)+10 \sin \left(c+\frac{3 d x}{2}\right)-9 d x \cos \left(c+\frac{d x}{2}\right)-3 d x \cos \left(c+\frac{3 d x}{2}\right)+3 d x \cos \left(2 c+\frac{3 d x}{2}\right)-18 \sin \left(\frac{d x}{2}\right)+9 d x \cos \left(\frac{d x}{2}\right)\right)}{24 d}","-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}+\frac{2 a^2 \csc (c+d x)}{d}+a^2 x",1,"(a^2*Csc[c/2]*Csc[(c + d*x)/2]^3*(9*d*x*Cos[(d*x)/2] - 9*d*x*Cos[c + (d*x)/2] - 3*d*x*Cos[c + (3*d*x)/2] + 3*d*x*Cos[2*c + (3*d*x)/2] - 18*Sin[(d*x)/2] - 12*Sin[c + (d*x)/2] + 10*Sin[c + (3*d*x)/2]))/(24*d)","A",1
34,1,194,107,0.811878,"\int \cot ^6(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc ^5\left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) (445 \sin (c+d x)-356 \sin (2 (c+d x))+89 \sin (3 (c+d x))+240 \sin (2 c+d x)-296 \sin (c+2 d x)-120 \sin (3 c+2 d x)+104 \sin (2 c+3 d x)+150 d x \cos (2 c+d x)+120 d x \cos (c+2 d x)-120 d x \cos (3 c+2 d x)-30 d x \cos (2 c+3 d x)+30 d x \cos (4 c+3 d x)-80 \sin (c)+280 \sin (d x)-150 d x \cos (d x))}{3840 d}","-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}+\frac{4 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}-a^2 x",1,"(a^2*Csc[c/2]*Csc[(c + d*x)/2]^5*Sec[c/2]*Sec[(c + d*x)/2]*(-150*d*x*Cos[d*x] + 150*d*x*Cos[2*c + d*x] + 120*d*x*Cos[c + 2*d*x] - 120*d*x*Cos[3*c + 2*d*x] - 30*d*x*Cos[2*c + 3*d*x] + 30*d*x*Cos[4*c + 3*d*x] - 80*Sin[c] + 280*Sin[d*x] + 445*Sin[c + d*x] - 356*Sin[2*(c + d*x)] + 89*Sin[3*(c + d*x)] + 240*Sin[2*c + d*x] - 296*Sin[c + 2*d*x] - 120*Sin[3*c + 2*d*x] + 104*Sin[2*c + 3*d*x]))/(3840*d)","A",1
35,1,312,139,1.1521735,"\int \cot ^8(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^8*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc ^7\left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) (-16002 \sin (c+d x)+9144 \sin (2 (c+d x))+3429 \sin (3 (c+d x))-4572 \sin (4 (c+d x))+1143 \sin (5 (c+d x))-11760 \sin (2 c+d x)+8864 \sin (c+2 d x)+3360 \sin (3 c+2 d x)+2064 \sin (2 c+3 d x)+2520 \sin (4 c+3 d x)-4432 \sin (3 c+4 d x)-1680 \sin (5 c+4 d x)+1528 \sin (4 c+5 d x)-5880 d x \cos (2 c+d x)-3360 d x \cos (c+2 d x)+3360 d x \cos (3 c+2 d x)-1260 d x \cos (2 c+3 d x)+1260 d x \cos (4 c+3 d x)+1680 d x \cos (3 c+4 d x)-1680 d x \cos (5 c+4 d x)-420 d x \cos (4 c+5 d x)+420 d x \cos (6 c+5 d x)+4032 \sin (c)-9632 \sin (d x)+5880 d x \cos (d x))}{860160 d}","-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}+\frac{6 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{d}+\frac{2 a^2 \csc (c+d x)}{d}+a^2 x",1,"(a^2*Csc[c/2]*Csc[(c + d*x)/2]^7*Sec[c/2]*Sec[(c + d*x)/2]^3*(5880*d*x*Cos[d*x] - 5880*d*x*Cos[2*c + d*x] - 3360*d*x*Cos[c + 2*d*x] + 3360*d*x*Cos[3*c + 2*d*x] - 1260*d*x*Cos[2*c + 3*d*x] + 1260*d*x*Cos[4*c + 3*d*x] + 1680*d*x*Cos[3*c + 4*d*x] - 1680*d*x*Cos[5*c + 4*d*x] - 420*d*x*Cos[4*c + 5*d*x] + 420*d*x*Cos[6*c + 5*d*x] + 4032*Sin[c] - 9632*Sin[d*x] - 16002*Sin[c + d*x] + 9144*Sin[2*(c + d*x)] + 3429*Sin[3*(c + d*x)] - 4572*Sin[4*(c + d*x)] + 1143*Sin[5*(c + d*x)] - 11760*Sin[2*c + d*x] + 8864*Sin[c + 2*d*x] + 3360*Sin[3*c + 2*d*x] + 2064*Sin[2*c + 3*d*x] + 2520*Sin[4*c + 3*d*x] - 4432*Sin[3*c + 4*d*x] - 1680*Sin[5*c + 4*d*x] + 1528*Sin[4*c + 5*d*x]))/(860160*d)","B",1
36,1,428,179,1.986995,"\int \cot ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^10*(a + a*Sec[c + d*x])^2,x]","-\frac{a^2 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc ^9\left(\frac{1}{2} (c+d x)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) (-1152405 \sin (c+d x)+512180 \sin (2 (c+d x))+486571 \sin (3 (c+d x))-409744 \sin (4 (c+d x))-25609 \sin (5 (c+d x))+102436 \sin (6 (c+d x))-25609 \sin (7 (c+d x))-825216 \sin (2 c+d x)+622976 \sin (c+2 d x)+142464 \sin (3 c+2 d x)+297088 \sin (2 c+3 d x)+430080 \sin (4 c+3 d x)-424192 \sin (3 c+4 d x)-188160 \sin (5 c+4 d x)+2048 \sin (4 c+5 d x)-40320 \sin (6 c+5 d x)+112768 \sin (5 c+6 d x)+40320 \sin (7 c+6 d x)-38272 \sin (6 c+7 d x)-453600 d x \cos (2 c+d x)-201600 d x \cos (c+2 d x)+201600 d x \cos (3 c+2 d x)-191520 d x \cos (2 c+3 d x)+191520 d x \cos (4 c+3 d x)+161280 d x \cos (3 c+4 d x)-161280 d x \cos (5 c+4 d x)+10080 d x \cos (4 c+5 d x)-10080 d x \cos (6 c+5 d x)-40320 d x \cos (5 c+6 d x)+40320 d x \cos (7 c+6 d x)+10080 d x \cos (6 c+7 d x)-10080 d x \cos (8 c+7 d x)+259584 \sin (c)-897024 \sin (d x)+453600 d x \cos (d x))}{330301440 d}","-\frac{2 a^2 \cot ^9(c+d x)}{9 d}+\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^9(c+d x)}{9 d}+\frac{8 a^2 \csc ^7(c+d x)}{7 d}-\frac{12 a^2 \csc ^5(c+d x)}{5 d}+\frac{8 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}-a^2 x",1,"-1/330301440*(a^2*Csc[c/2]*Csc[(c + d*x)/2]^9*Sec[c/2]*Sec[(c + d*x)/2]^5*(453600*d*x*Cos[d*x] - 453600*d*x*Cos[2*c + d*x] - 201600*d*x*Cos[c + 2*d*x] + 201600*d*x*Cos[3*c + 2*d*x] - 191520*d*x*Cos[2*c + 3*d*x] + 191520*d*x*Cos[4*c + 3*d*x] + 161280*d*x*Cos[3*c + 4*d*x] - 161280*d*x*Cos[5*c + 4*d*x] + 10080*d*x*Cos[4*c + 5*d*x] - 10080*d*x*Cos[6*c + 5*d*x] - 40320*d*x*Cos[5*c + 6*d*x] + 40320*d*x*Cos[7*c + 6*d*x] + 10080*d*x*Cos[6*c + 7*d*x] - 10080*d*x*Cos[8*c + 7*d*x] + 259584*Sin[c] - 897024*Sin[d*x] - 1152405*Sin[c + d*x] + 512180*Sin[2*(c + d*x)] + 486571*Sin[3*(c + d*x)] - 409744*Sin[4*(c + d*x)] - 25609*Sin[5*(c + d*x)] + 102436*Sin[6*(c + d*x)] - 25609*Sin[7*(c + d*x)] - 825216*Sin[2*c + d*x] + 622976*Sin[c + 2*d*x] + 142464*Sin[3*c + 2*d*x] + 297088*Sin[2*c + 3*d*x] + 430080*Sin[4*c + 3*d*x] - 424192*Sin[3*c + 4*d*x] - 188160*Sin[5*c + 4*d*x] + 2048*Sin[4*c + 5*d*x] - 40320*Sin[6*c + 5*d*x] + 112768*Sin[5*c + 6*d*x] + 40320*Sin[7*c + 6*d*x] - 38272*Sin[6*c + 7*d*x]))/d","B",1
37,1,214,210,0.883263,"\int (a+a \sec (c+d x))^3 \tan ^9(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^9,x]","-\frac{a^3 \sec ^{11}(c+d x) (-1613260 \cos (2 (c+d x))+960960 \cos (3 (c+d x))-1131504 \cos (4 (c+d x))+314160 \cos (5 (c+d x))-432894 \cos (6 (c+d x))+145530 \cos (7 (c+d x))-106260 \cos (8 (c+d x))+6930 \cos (9 (c+d x))-20790 \cos (10 (c+d x))+1143450 \cos (3 (c+d x)) \log (\cos (c+d x))+571725 \cos (5 (c+d x)) \log (\cos (c+d x))+190575 \cos (7 (c+d x)) \log (\cos (c+d x))+38115 \cos (9 (c+d x)) \log (\cos (c+d x))+3465 \cos (11 (c+d x)) \log (\cos (c+d x))+462 \cos (c+d x) (3465 \log (\cos (c+d x))+2606)-1151740)}{3548160 d}","\frac{a^3 \sec ^{11}(c+d x)}{11 d}+\frac{3 a^3 \sec ^{10}(c+d x)}{10 d}-\frac{a^3 \sec ^9(c+d x)}{9 d}-\frac{11 a^3 \sec ^8(c+d x)}{8 d}-\frac{6 a^3 \sec ^7(c+d x)}{7 d}+\frac{7 a^3 \sec ^6(c+d x)}{3 d}+\frac{14 a^3 \sec ^5(c+d x)}{5 d}-\frac{3 a^3 \sec ^4(c+d x)}{2 d}-\frac{11 a^3 \sec ^3(c+d x)}{3 d}-\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d}",1,"-1/3548160*(a^3*(-1151740 - 1613260*Cos[2*(c + d*x)] + 960960*Cos[3*(c + d*x)] - 1131504*Cos[4*(c + d*x)] + 314160*Cos[5*(c + d*x)] - 432894*Cos[6*(c + d*x)] + 145530*Cos[7*(c + d*x)] - 106260*Cos[8*(c + d*x)] + 6930*Cos[9*(c + d*x)] - 20790*Cos[10*(c + d*x)] + 1143450*Cos[3*(c + d*x)]*Log[Cos[c + d*x]] + 571725*Cos[5*(c + d*x)]*Log[Cos[c + d*x]] + 190575*Cos[7*(c + d*x)]*Log[Cos[c + d*x]] + 38115*Cos[9*(c + d*x)]*Log[Cos[c + d*x]] + 3465*Cos[11*(c + d*x)]*Log[Cos[c + d*x]] + 462*Cos[c + d*x]*(2606 + 3465*Log[Cos[c + d*x]]))*Sec[c + d*x]^11)/d","A",1
38,1,110,137,0.4383014,"\int (a+a \sec (c+d x))^3 \tan ^7(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^7,x]","\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(40 \sec ^9(c+d x)+135 \sec ^8(c+d x)-480 \sec ^6(c+d x)-432 \sec ^5(c+d x)+540 \sec ^4(c+d x)+960 \sec ^3(c+d x)-1080 \sec (c+d x)+360 \log (\cos (c+d x))\right)}{2880 d}","\frac{a^3 \sec ^9(c+d x)}{9 d}+\frac{3 a^3 \sec ^8(c+d x)}{8 d}-\frac{4 a^3 \sec ^6(c+d x)}{3 d}-\frac{6 a^3 \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \sec ^4(c+d x)}{2 d}+\frac{8 a^3 \sec ^3(c+d x)}{3 d}-\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d}",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(360*Log[Cos[c + d*x]] - 1080*Sec[c + d*x] + 960*Sec[c + d*x]^3 + 540*Sec[c + d*x]^4 - 432*Sec[c + d*x]^5 - 480*Sec[c + d*x]^6 + 135*Sec[c + d*x]^8 + 40*Sec[c + d*x]^9))/(2880*d)","A",1
39,1,140,138,0.4244554,"\int (a+a \sec (c+d x))^3 \tan ^5(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^5,x]","-\frac{a^3 \sec ^7(c+d x) (-4522 \cos (2 (c+d x))+1050 \cos (3 (c+d x))-2380 \cos (4 (c+d x))-210 \cos (5 (c+d x))-630 \cos (6 (c+d x))+2205 \cos (3 (c+d x)) \log (\cos (c+d x))+735 \cos (5 (c+d x)) \log (\cos (c+d x))+105 \cos (7 (c+d x)) \log (\cos (c+d x))+105 \cos (c+d x) (35 \log (\cos (c+d x))+8)-3732)}{6720 d}","\frac{a^3 \sec ^7(c+d x)}{7 d}+\frac{a^3 \sec ^6(c+d x)}{2 d}+\frac{a^3 \sec ^5(c+d x)}{5 d}-\frac{5 a^3 \sec ^4(c+d x)}{4 d}-\frac{5 a^3 \sec ^3(c+d x)}{3 d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d}",1,"-1/6720*(a^3*(-3732 - 4522*Cos[2*(c + d*x)] + 1050*Cos[3*(c + d*x)] - 2380*Cos[4*(c + d*x)] - 210*Cos[5*(c + d*x)] - 630*Cos[6*(c + d*x)] + 2205*Cos[3*(c + d*x)]*Log[Cos[c + d*x]] + 735*Cos[5*(c + d*x)]*Log[Cos[c + d*x]] + 105*Cos[7*(c + d*x)]*Log[Cos[c + d*x]] + 105*Cos[c + d*x]*(8 + 35*Log[Cos[c + d*x]]))*Sec[c + d*x]^7)/d","A",1
40,1,92,99,0.3011468,"\int (a+a \sec (c+d x))^3 \tan ^3(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^3,x]","-\frac{a^3 \sec ^5(c+d x) (280 \cos (2 (c+d x))+90 \cos (4 (c+d x))+\cos (3 (c+d x)) (60-75 \log (\cos (c+d x)))-150 \cos (c+d x) \log (\cos (c+d x))-15 \cos (5 (c+d x)) \log (\cos (c+d x))+142)}{240 d}","\frac{a^3 \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \sec ^4(c+d x)}{4 d}+\frac{2 a^3 \sec ^3(c+d x)}{3 d}-\frac{a^3 \sec ^2(c+d x)}{d}-\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d}",1,"-1/240*(a^3*(142 + 280*Cos[2*(c + d*x)] + 90*Cos[4*(c + d*x)] + Cos[3*(c + d*x)]*(60 - 75*Log[Cos[c + d*x]]) - 150*Cos[c + d*x]*Log[Cos[c + d*x]] - 15*Cos[5*(c + d*x)]*Log[Cos[c + d*x]])*Sec[c + d*x]^5)/d","A",1
41,1,64,66,0.1618736,"\int (a+a \sec (c+d x))^3 \tan (c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x],x]","-\frac{a^3 \sec ^3(c+d x) (-18 \cos (2 (c+d x))+9 \cos (c+d x) (\log (\cos (c+d x))-2)+3 \cos (3 (c+d x)) \log (\cos (c+d x))-22)}{12 d}","\frac{a^3 \sec ^3(c+d x)}{3 d}+\frac{3 a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d}",1,"-1/12*(a^3*(-22 - 18*Cos[2*(c + d*x)] + 9*Cos[c + d*x]*(-2 + Log[Cos[c + d*x]]) + 3*Cos[3*(c + d*x)]*Log[Cos[c + d*x]])*Sec[c + d*x]^3)/d","A",1
42,1,36,48,0.0905394,"\int \cot (c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cot[c + d*x]*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 \left(\sec (c+d x)+8 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 \log (\cos (c+d x))\right)}{d}","\frac{a^3 \sec (c+d x)}{d}+\frac{4 a^3 \log (1-\cos (c+d x))}{d}-\frac{3 a^3 \log (\cos (c+d x))}{d}",1,"(a^3*(-3*Log[Cos[c + d*x]] + 8*Log[Sin[(c + d*x)/2]] + Sec[c + d*x]))/d","A",1
43,1,46,40,0.1326992,"\int \cot ^3(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^3*(a + a*Sec[c + d*x])^3,x]","-\frac{a^3 \left(\cot ^2\left(\frac{1}{2} (c+d x)\right)+2 \left(\log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{d}","-\frac{2 a^3}{d (1-\cos (c+d x))}-\frac{a^3 \log (1-\cos (c+d x))}{d}",1,"-((a^3*(Cot[(c + d*x)/2]^2 + 2*(Log[Cos[(c + d*x)/2]] + Log[Tan[(c + d*x)/2]])))/d)","A",1
44,1,72,61,0.1691594,"\int \cot ^5(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^3,x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(\csc ^4\left(\frac{1}{2} (c+d x)\right)-8 \csc ^2\left(\frac{1}{2} (c+d x)\right)-16 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{64 d}","\frac{2 a^3}{d (1-\cos (c+d x))}-\frac{a^3}{2 d (1-\cos (c+d x))^2}+\frac{a^3 \log (1-\cos (c+d x))}{d}",1,"-1/64*(a^3*(1 + Cos[c + d*x])^3*(-8*Csc[(c + d*x)/2]^2 + Csc[(c + d*x)/2]^4 - 16*Log[Sin[(c + d*x)/2]])*Sec[(c + d*x)/2]^6)/d","A",1
45,1,102,107,0.6789823,"\int \cot ^7(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^7*(a + a*Sec[c + d*x])^3,x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(2 \csc ^6\left(\frac{1}{2} (c+d x)\right)-21 \csc ^4\left(\frac{1}{2} (c+d x)\right)+102 \csc ^2\left(\frac{1}{2} (c+d x)\right)+12 \left(15 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{768 d}","-\frac{17 a^3}{8 d (1-\cos (c+d x))}+\frac{7 a^3}{8 d (1-\cos (c+d x))^2}-\frac{a^3}{6 d (1-\cos (c+d x))^3}-\frac{15 a^3 \log (1-\cos (c+d x))}{16 d}-\frac{a^3 \log (\cos (c+d x)+1)}{16 d}",1,"-1/768*(a^3*(1 + Cos[c + d*x])^3*(102*Csc[(c + d*x)/2]^2 - 21*Csc[(c + d*x)/2]^4 + 2*Csc[(c + d*x)/2]^6 + 12*(Log[Cos[(c + d*x)/2]] + 15*Log[Sin[(c + d*x)/2]]))*Sec[(c + d*x)/2]^6)/d","A",1
46,1,130,149,0.399498,"\int \cot ^9(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^9*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(-3 \csc ^8\left(\frac{1}{2} (c+d x)\right)+40 \csc ^6\left(\frac{1}{2} (c+d x)\right)-234 \csc ^4\left(\frac{1}{2} (c+d x)\right)+864 \csc ^2\left(\frac{1}{2} (c+d x)\right)+12 \left(\sec ^2\left(\frac{1}{2} (c+d x)\right)+114 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+14 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{6144 d}","\frac{9 a^3}{4 d (1-\cos (c+d x))}+\frac{a^3}{32 d (\cos (c+d x)+1)}-\frac{39 a^3}{32 d (1-\cos (c+d x))^2}+\frac{5 a^3}{12 d (1-\cos (c+d x))^3}-\frac{a^3}{16 d (1-\cos (c+d x))^4}+\frac{57 a^3 \log (1-\cos (c+d x))}{64 d}+\frac{7 a^3 \log (\cos (c+d x)+1)}{64 d}",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(864*Csc[(c + d*x)/2]^2 - 234*Csc[(c + d*x)/2]^4 + 40*Csc[(c + d*x)/2]^6 - 3*Csc[(c + d*x)/2]^8 + 12*(14*Log[Cos[(c + d*x)/2]] + 114*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2)))/(6144*d)","A",1
47,1,363,237,2.1776767,"\int (a+a \sec (c+d x))^3 \tan ^6(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^6,x]","\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^8(c+d x) \left(1680000 \cos ^8(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-133175 \sin (2 c+d x)-544768 \sin (c+2 d x)+286720 \sin (3 c+2 d x)-63595 \sin (2 c+3 d x)-63595 \sin (4 c+3 d x)-254464 \sin (3 c+4 d x)+161280 \sin (5 c+4 d x)-65135 \sin (4 c+5 d x)-65135 \sin (6 c+5 d x)-118784 \sin (5 c+6 d x)-27195 \sin (6 c+7 d x)-27195 \sin (8 c+7 d x)-14848 \sin (7 c+8 d x)+470400 d x \cos (c)+376320 d x \cos (c+2 d x)+376320 d x \cos (3 c+2 d x)+188160 d x \cos (3 c+4 d x)+188160 d x \cos (5 c+4 d x)+53760 d x \cos (5 c+6 d x)+53760 d x \cos (7 c+6 d x)+6720 d x \cos (7 c+8 d x)+6720 d x \cos (9 c+8 d x)+519680 \sin (c)-133175 \sin (d x))\right)}{13762560 d}","\frac{3 a^3 \tan ^7(c+d x)}{7 d}+\frac{a^3 \tan ^5(c+d x)}{5 d}-\frac{a^3 \tan ^3(c+d x)}{3 d}+\frac{a^3 \tan (c+d x)}{d}-\frac{125 a^3 \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{a^3 \tan ^5(c+d x) \sec ^3(c+d x)}{8 d}-\frac{5 a^3 \tan ^3(c+d x) \sec ^3(c+d x)}{48 d}+\frac{5 a^3 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac{a^3 \tan ^5(c+d x) \sec (c+d x)}{2 d}-\frac{5 a^3 \tan ^3(c+d x) \sec (c+d x)}{8 d}+\frac{115 a^3 \tan (c+d x) \sec (c+d x)}{128 d}-a^3 x",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]^8*(1680000*Cos[c + d*x]^8*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(470400*d*x*Cos[c] + 376320*d*x*Cos[c + 2*d*x] + 376320*d*x*Cos[3*c + 2*d*x] + 188160*d*x*Cos[3*c + 4*d*x] + 188160*d*x*Cos[5*c + 4*d*x] + 53760*d*x*Cos[5*c + 6*d*x] + 53760*d*x*Cos[7*c + 6*d*x] + 6720*d*x*Cos[7*c + 8*d*x] + 6720*d*x*Cos[9*c + 8*d*x] + 519680*Sin[c] - 133175*Sin[d*x] - 133175*Sin[2*c + d*x] - 544768*Sin[c + 2*d*x] + 286720*Sin[3*c + 2*d*x] - 63595*Sin[2*c + 3*d*x] - 63595*Sin[4*c + 3*d*x] - 254464*Sin[3*c + 4*d*x] + 161280*Sin[5*c + 4*d*x] - 65135*Sin[4*c + 5*d*x] - 65135*Sin[6*c + 5*d*x] - 118784*Sin[5*c + 6*d*x] - 27195*Sin[6*c + 7*d*x] - 27195*Sin[8*c + 7*d*x] - 14848*Sin[7*c + 8*d*x])))/(13762560*d)","A",1
48,1,303,169,1.2922477,"\int (a+a \sec (c+d x))^3 \tan ^4(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^4,x]","\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \left(\sec (c) (210 \sin (2 c+d x)-1440 \sin (c+2 d x)+1200 \sin (3 c+2 d x)-865 \sin (2 c+3 d x)-865 \sin (4 c+3 d x)-1296 \sin (3 c+4 d x)-240 \sin (5 c+4 d x)-435 \sin (4 c+5 d x)-435 \sin (6 c+5 d x)-176 \sin (5 c+6 d x)+2400 d x \cos (c)+1800 d x \cos (c+2 d x)+1800 d x \cos (3 c+2 d x)+720 d x \cos (3 c+4 d x)+720 d x \cos (5 c+4 d x)+120 d x \cos (5 c+6 d x)+120 d x \cos (7 c+6 d x)+1760 \sin (c)+210 \sin (d x))-9120 \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{61440 d}","\frac{3 a^3 \tan ^5(c+d x)}{5 d}+\frac{a^3 \tan ^3(c+d x)}{3 d}-\frac{a^3 \tan (c+d x)}{d}+\frac{19 a^3 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 \tan ^3(c+d x) \sec ^3(c+d x)}{6 d}-\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac{3 a^3 \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac{17 a^3 \tan (c+d x) \sec (c+d x)}{16 d}+a^3 x",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]^6*(-9120*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*(2400*d*x*Cos[c] + 1800*d*x*Cos[c + 2*d*x] + 1800*d*x*Cos[3*c + 2*d*x] + 720*d*x*Cos[3*c + 4*d*x] + 720*d*x*Cos[5*c + 4*d*x] + 120*d*x*Cos[5*c + 6*d*x] + 120*d*x*Cos[7*c + 6*d*x] + 1760*Sin[c] + 210*Sin[d*x] + 210*Sin[2*c + d*x] - 1440*Sin[c + 2*d*x] + 1200*Sin[3*c + 2*d*x] - 865*Sin[2*c + 3*d*x] - 865*Sin[4*c + 3*d*x] - 1296*Sin[3*c + 4*d*x] - 240*Sin[5*c + 4*d*x] - 435*Sin[4*c + 5*d*x] - 435*Sin[6*c + 5*d*x] - 176*Sin[5*c + 6*d*x])))/(61440*d)","A",1
49,1,230,98,0.8491761,"\int (a+a \sec (c+d x))^3 \tan ^2(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^2,x]","-\frac{a^3 \sec ^4(c+d x) \left(-38 \sin (c+d x)-32 \sin (2 (c+d x))-22 \sin (3 (c+d x))-39 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+39 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 \cos (2 (c+d x)) \left(-13 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+13 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 d x\right)+\cos (4 (c+d x)) \left(-13 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+13 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 d x\right)+24 d x\right)}{64 d}","\frac{a^3 \tan ^3(c+d x)}{d}+\frac{a^3 \tan (c+d x)}{d}-\frac{13 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{11 a^3 \tan (c+d x) \sec (c+d x)}{8 d}-a^3 x",1,"-1/64*(a^3*Sec[c + d*x]^4*(24*d*x - 39*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 39*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4*Cos[2*(c + d*x)]*(8*d*x - 13*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 13*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[4*(c + d*x)]*(8*d*x - 13*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 13*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 38*Sin[c + d*x] - 32*Sin[2*(c + d*x)] - 22*Sin[3*(c + d*x)]))/d","B",1
50,1,109,49,0.2289562,"\int \cot ^2(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*(a + a*Sec[c + d*x])^3,x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(-4 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc \left(\frac{1}{2} (c+d x)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+d x\right)}{8 d}","-\frac{4 a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+a^3 (-x)",1,"-1/8*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(d*x + Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 4*Csc[c/2]*Csc[(c + d*x)/2]*Sin[(d*x)/2]))/d","B",1
51,1,112,69,0.2413319,"\int \cot ^4(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 \csc \left(\frac{c}{2}\right) \csc ^3\left(\frac{1}{2} (c+d x)\right) \left(-18 \sin \left(c+\frac{d x}{2}\right)+14 \sin \left(c+\frac{3 d x}{2}\right)-9 d x \cos \left(c+\frac{d x}{2}\right)-3 d x \cos \left(c+\frac{3 d x}{2}\right)+3 d x \cos \left(2 c+\frac{3 d x}{2}\right)-24 \sin \left(\frac{d x}{2}\right)+9 d x \cos \left(\frac{d x}{2}\right)\right)}{24 d}","-\frac{4 a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x",1,"(a^3*Csc[c/2]*Csc[(c + d*x)/2]^3*(9*d*x*Cos[(d*x)/2] - 9*d*x*Cos[c + (d*x)/2] - 3*d*x*Cos[c + (3*d*x)/2] + 3*d*x*Cos[2*c + (3*d*x)/2] - 24*Sin[(d*x)/2] - 18*Sin[c + (d*x)/2] + 14*Sin[c + (3*d*x)/2]))/(24*d)","A",1
52,1,112,107,0.6735665,"\int \cot ^6(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^6*(a + a*Sec[c + d*x])^3,x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(\cot \left(\frac{c}{2}\right) (13 \cos (c+d x)-10) \csc ^4\left(\frac{1}{2} (c+d x)\right)+\csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) (51 \cos (c+d x)-16 \cos (2 (c+d x))-38) \csc ^5\left(\frac{1}{2} (c+d x)\right)+60 d x\right)}{480 d}","-\frac{4 a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^5(c+d x)}{5 d}+\frac{7 a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc (c+d x)}{d}-a^3 x",1,"-1/480*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(60*d*x + (-10 + 13*Cos[c + d*x])*Cot[c/2]*Csc[(c + d*x)/2]^4 + (-38 + 51*Cos[c + d*x] - 16*Cos[2*(c + d*x)])*Csc[c/2]*Csc[(c + d*x)/2]^5*Sin[(d*x)/2]))/d","A",1
53,1,252,141,0.9830194,"\int \cot ^8(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^8*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc ^7\left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) (-23282 \sin (c+d x)+23282 \sin (2 (c+d x))-9978 \sin (3 (c+d x))+1663 \sin (4 (c+d x))-13720 \sin (2 c+d x)+15512 \sin (c+2 d x)+9240 \sin (3 c+2 d x)-8088 \sin (2 c+3 d x)-2520 \sin (4 c+3 d x)+1768 \sin (3 c+4 d x)-5880 d x \cos (2 c+d x)-5880 d x \cos (c+2 d x)+5880 d x \cos (3 c+2 d x)+2520 d x \cos (2 c+3 d x)-2520 d x \cos (4 c+3 d x)-420 d x \cos (3 c+4 d x)+420 d x \cos (5 c+4 d x)+4200 \sin (c)-11032 \sin (d x)+5880 d x \cos (d x))}{215040 d}","-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^7(c+d x)}{7 d}+\frac{11 a^3 \csc ^5(c+d x)}{5 d}-\frac{10 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x",1,"(a^3*Csc[c/2]*Csc[(c + d*x)/2]^7*Sec[c/2]*Sec[(c + d*x)/2]*(5880*d*x*Cos[d*x] - 5880*d*x*Cos[2*c + d*x] - 5880*d*x*Cos[c + 2*d*x] + 5880*d*x*Cos[3*c + 2*d*x] + 2520*d*x*Cos[2*c + 3*d*x] - 2520*d*x*Cos[4*c + 3*d*x] - 420*d*x*Cos[3*c + 4*d*x] + 420*d*x*Cos[5*c + 4*d*x] + 4200*Sin[c] - 11032*Sin[d*x] - 23282*Sin[c + d*x] + 23282*Sin[2*(c + d*x)] - 9978*Sin[3*(c + d*x)] + 1663*Sin[4*(c + d*x)] - 13720*Sin[2*c + d*x] + 15512*Sin[c + 2*d*x] + 9240*Sin[3*c + 2*d*x] - 8088*Sin[2*c + 3*d*x] - 2520*Sin[4*c + 3*d*x] + 1768*Sin[3*c + 4*d*x]))/(215040*d)","A",1
54,1,370,179,1.4151814,"\int \cot ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^10*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc ^9\left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) (675036 \sin (c+d x)-506277 \sin (2 (c+d x))-37502 \sin (3 (c+d x))+225012 \sin (4 (c+d x))-112506 \sin (5 (c+d x))+18751 \sin (6 (c+d x))+431424 \sin (2 c+d x)-375552 \sin (c+2 d x)-201600 \sin (3 c+2 d x)+41248 \sin (2 c+3 d x)-84000 \sin (4 c+3 d x)+155712 \sin (3 c+4 d x)+100800 \sin (5 c+4 d x)-98016 \sin (4 c+5 d x)-30240 \sin (6 c+5 d x)+21376 \sin (5 c+6 d x)+181440 d x \cos (2 c+d x)+136080 d x \cos (c+2 d x)-136080 d x \cos (3 c+2 d x)+10080 d x \cos (2 c+3 d x)-10080 d x \cos (4 c+3 d x)-60480 d x \cos (3 c+4 d x)+60480 d x \cos (5 c+4 d x)+30240 d x \cos (4 c+5 d x)-30240 d x \cos (6 c+5 d x)-5040 d x \cos (5 c+6 d x)+5040 d x \cos (7 c+6 d x)-169344 \sin (c)+338112 \sin (d x)-181440 d x \cos (d x))}{41287680 d}","-\frac{4 a^3 \cot ^9(c+d x)}{9 d}+\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^9(c+d x)}{9 d}+\frac{15 a^3 \csc ^7(c+d x)}{7 d}-\frac{21 a^3 \csc ^5(c+d x)}{5 d}+\frac{13 a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc (c+d x)}{d}-a^3 x",1,"(a^3*Csc[c/2]*Csc[(c + d*x)/2]^9*Sec[c/2]*Sec[(c + d*x)/2]^3*(-181440*d*x*Cos[d*x] + 181440*d*x*Cos[2*c + d*x] + 136080*d*x*Cos[c + 2*d*x] - 136080*d*x*Cos[3*c + 2*d*x] + 10080*d*x*Cos[2*c + 3*d*x] - 10080*d*x*Cos[4*c + 3*d*x] - 60480*d*x*Cos[3*c + 4*d*x] + 60480*d*x*Cos[5*c + 4*d*x] + 30240*d*x*Cos[4*c + 5*d*x] - 30240*d*x*Cos[6*c + 5*d*x] - 5040*d*x*Cos[5*c + 6*d*x] + 5040*d*x*Cos[7*c + 6*d*x] - 169344*Sin[c] + 338112*Sin[d*x] + 675036*Sin[c + d*x] - 506277*Sin[2*(c + d*x)] - 37502*Sin[3*(c + d*x)] + 225012*Sin[4*(c + d*x)] - 112506*Sin[5*(c + d*x)] + 18751*Sin[6*(c + d*x)] + 431424*Sin[2*c + d*x] - 375552*Sin[c + 2*d*x] - 201600*Sin[3*c + 2*d*x] + 41248*Sin[2*c + 3*d*x] - 84000*Sin[4*c + 3*d*x] + 155712*Sin[3*c + 4*d*x] + 100800*Sin[5*c + 4*d*x] - 98016*Sin[4*c + 5*d*x] - 30240*Sin[6*c + 5*d*x] + 21376*Sin[5*c + 6*d*x]))/(41287680*d)","B",1
55,1,268,213,6.0474819,"\int \cot ^{12}(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^12*(a + a*Sec[c + d*x])^3,x]","-\frac{a^3 \tan \left(\frac{c}{2}\right) (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(20 \cot ^2\left(\frac{c}{2}\right) (-4528480 \cos (c+d x)+2388316 \cos (2 (c+d x))-750112 \cos (3 (c+d x))+112229 \cos (4 (c+d x))+2786111) \csc ^{10}\left(\frac{1}{2} (c+d x)\right)+7392 \csc \left(\frac{c}{2}\right) \left(-3060 \sin \left(c+\frac{d x}{2}\right)+2860 \sin \left(c+\frac{3 d x}{2}\right)-855 \sin \left(2 c+\frac{3 d x}{2}\right)+743 \sin \left(2 c+\frac{5 d x}{2}\right)+4370 \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)-5 \cot \left(\frac{c}{2}\right) \left(\csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) (54812150 \cos (c+d x)-32118776 \cos (2 (c+d x))+12626567 \cos (3 (c+d x))-3023754 \cos (4 (c+d x))+347267 \cos (5 (c+d x))-32611198) \csc ^{11}\left(\frac{1}{2} (c+d x)\right)+90832896 d x\right)\right)}{3633315840 d}","-\frac{4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^{11}(c+d x)}{11 d}+\frac{19 a^3 \csc ^9(c+d x)}{9 d}-\frac{36 a^3 \csc ^7(c+d x)}{7 d}+\frac{34 a^3 \csc ^5(c+d x)}{5 d}-\frac{16 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x",1,"-1/3633315840*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(20*(2786111 - 4528480*Cos[c + d*x] + 2388316*Cos[2*(c + d*x)] - 750112*Cos[3*(c + d*x)] + 112229*Cos[4*(c + d*x)])*Cot[c/2]^2*Csc[(c + d*x)/2]^10 - 5*Cot[c/2]*(90832896*d*x + (-32611198 + 54812150*Cos[c + d*x] - 32118776*Cos[2*(c + d*x)] + 12626567*Cos[3*(c + d*x)] - 3023754*Cos[4*(c + d*x)] + 347267*Cos[5*(c + d*x)])*Csc[c/2]*Csc[(c + d*x)/2]^11*Sin[(d*x)/2]) + 7392*Csc[c/2]*Sec[(c + d*x)/2]^5*(4370*Sin[(d*x)/2] - 3060*Sin[c + (d*x)/2] + 2860*Sin[c + (3*d*x)/2] - 855*Sin[2*c + (3*d*x)/2] + 743*Sin[2*c + (5*d*x)/2]))*Tan[c/2])/d","A",1
56,1,137,135,0.5619135,"\int \frac{\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]^9/(a + a*Sec[c + d*x]),x]","-\frac{\sec ^7(c+d x) (35 \cos (c+d x) (105 \log (\cos (c+d x))+104)+3 (602 \cos (2 (c+d x))+140 \cos (4 (c+d x))+210 \cos (5 (c+d x))+70 \cos (6 (c+d x))+245 \cos (5 (c+d x)) \log (\cos (c+d x))+35 \cos (7 (c+d x)) \log (\cos (c+d x))+105 \cos (3 (c+d x)) (7 \log (\cos (c+d x))+6)+212))}{6720 a d}","\frac{\sec ^7(c+d x)}{7 a d}-\frac{\sec ^6(c+d x)}{6 a d}-\frac{3 \sec ^5(c+d x)}{5 a d}+\frac{3 \sec ^4(c+d x)}{4 a d}+\frac{\sec ^3(c+d x)}{a d}-\frac{3 \sec ^2(c+d x)}{2 a d}-\frac{\sec (c+d x)}{a d}-\frac{\log (\cos (c+d x))}{a d}",1,"-1/6720*((35*Cos[c + d*x]*(104 + 105*Log[Cos[c + d*x]]) + 3*(212 + 602*Cos[2*(c + d*x)] + 140*Cos[4*(c + d*x)] + 210*Cos[5*(c + d*x)] + 70*Cos[6*(c + d*x)] + 245*Cos[5*(c + d*x)]*Log[Cos[c + d*x]] + 35*Cos[7*(c + d*x)]*Log[Cos[c + d*x]] + 105*Cos[3*(c + d*x)]*(6 + 7*Log[Cos[c + d*x]])))*Sec[c + d*x]^7)/(a*d)","A",1
57,1,103,97,0.2728139,"\int \frac{\tan ^7(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]^7/(a + a*Sec[c + d*x]),x]","\frac{\sec ^5(c+d x) (40 \cos (2 (c+d x))+60 \cos (3 (c+d x))+30 \cos (4 (c+d x))+75 \cos (3 (c+d x)) \log (\cos (c+d x))+15 \cos (5 (c+d x)) \log (\cos (c+d x))+30 \cos (c+d x) (5 \log (\cos (c+d x))+4)+58)}{240 a d}","\frac{\sec ^5(c+d x)}{5 a d}-\frac{\sec ^4(c+d x)}{4 a d}-\frac{2 \sec ^3(c+d x)}{3 a d}+\frac{\sec ^2(c+d x)}{a d}+\frac{\sec (c+d x)}{a d}+\frac{\log (\cos (c+d x))}{a d}",1,"((58 + 40*Cos[2*(c + d*x)] + 60*Cos[3*(c + d*x)] + 30*Cos[4*(c + d*x)] + 75*Cos[3*(c + d*x)]*Log[Cos[c + d*x]] + 15*Cos[5*(c + d*x)]*Log[Cos[c + d*x]] + 30*Cos[c + d*x]*(4 + 5*Log[Cos[c + d*x]]))*Sec[c + d*x]^5)/(240*a*d)","A",1
58,1,65,66,0.1937918,"\int \frac{\tan ^5(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]^5/(a + a*Sec[c + d*x]),x]","-\frac{\sec ^3(c+d x) (6 \cos (2 (c+d x))+3 \cos (3 (c+d x)) \log (\cos (c+d x))+\cos (c+d x) (9 \log (\cos (c+d x))+6)+2)}{12 a d}","\frac{\sec ^3(c+d x)}{3 a d}-\frac{\sec ^2(c+d x)}{2 a d}-\frac{\sec (c+d x)}{a d}-\frac{\log (\cos (c+d x))}{a d}",1,"-1/12*((2 + 6*Cos[2*(c + d*x)] + 3*Cos[3*(c + d*x)]*Log[Cos[c + d*x]] + Cos[c + d*x]*(6 + 9*Log[Cos[c + d*x]]))*Sec[c + d*x]^3)/(a*d)","A",1
59,1,21,28,0.0716173,"\int \frac{\tan ^3(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]^3/(a + a*Sec[c + d*x]),x]","\frac{\sec (c+d x)+\log (\cos (c+d x))}{a d}","\frac{\sec (c+d x)}{a d}+\frac{\log (\cos (c+d x))}{a d}",1,"(Log[Cos[c + d*x]] + Sec[c + d*x])/(a*d)","A",1
60,1,19,17,0.0184405,"\int \frac{\tan (c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]/(a + a*Sec[c + d*x]),x]","-\frac{2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a d}","-\frac{\log (\cos (c+d x)+1)}{a d}",1,"(-2*Log[Cos[(c + d*x)/2]])/(a*d)","A",1
61,1,67,61,0.1197341,"\int \frac{\cot (c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]/(a + a*Sec[c + d*x]),x]","\frac{\sec (c+d x) \left(2 \cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+1\right)}{2 a d (\sec (c+d x)+1)}","\frac{1}{2 a d (\cos (c+d x)+1)}+\frac{\log (1-\cos (c+d x))}{4 a d}+\frac{3 \log (\cos (c+d x)+1)}{4 a d}",1,"((1 + 2*Cos[(c + d*x)/2]^2*(3*Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]]))*Sec[c + d*x])/(2*a*d*(1 + Sec[c + d*x]))","A",1
62,1,107,103,0.6209898,"\int \frac{\cot ^3(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^3/(a + a*Sec[c + d*x]),x]","-\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(2 \csc ^2\left(\frac{1}{2} (c+d x)\right)-\sec ^4\left(\frac{1}{2} (c+d x)\right)+12 \sec ^2\left(\frac{1}{2} (c+d x)\right)+20 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+44 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{16 a d (\sec (c+d x)+1)}","-\frac{1}{8 a d (1-\cos (c+d x))}-\frac{3}{4 a d (\cos (c+d x)+1)}+\frac{1}{8 a d (\cos (c+d x)+1)^2}-\frac{5 \log (1-\cos (c+d x))}{16 a d}-\frac{11 \log (\cos (c+d x)+1)}{16 a d}",1,"-1/16*(Cos[(c + d*x)/2]^2*(2*Csc[(c + d*x)/2]^2 + 44*Log[Cos[(c + d*x)/2]] + 20*Log[Sin[(c + d*x)/2]] + 12*Sec[(c + d*x)/2]^2 - Sec[(c + d*x)/2]^4)*Sec[c + d*x])/(a*d*(1 + Sec[c + d*x]))","A",1
63,1,135,145,0.5384542,"\int \frac{\cot ^5(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^5/(a + a*Sec[c + d*x]),x]","-\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(3 \csc ^4\left(\frac{1}{2} (c+d x)\right)-48 \csc ^2\left(\frac{1}{2} (c+d x)\right)-2 \sec ^6\left(\frac{1}{2} (c+d x)\right)+27 \sec ^4\left(\frac{1}{2} (c+d x)\right)-180 \sec ^2\left(\frac{1}{2} (c+d x)\right)-264 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-504 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 a d (\sec (c+d x)+1)}","\frac{1}{4 a d (1-\cos (c+d x))}+\frac{15}{16 a d (\cos (c+d x)+1)}-\frac{1}{32 a d (1-\cos (c+d x))^2}-\frac{9}{32 a d (\cos (c+d x)+1)^2}+\frac{1}{24 a d (\cos (c+d x)+1)^3}+\frac{11 \log (1-\cos (c+d x))}{32 a d}+\frac{21 \log (\cos (c+d x)+1)}{32 a d}",1,"-1/192*(Cos[(c + d*x)/2]^2*(-48*Csc[(c + d*x)/2]^2 + 3*Csc[(c + d*x)/2]^4 - 504*Log[Cos[(c + d*x)/2]] - 264*Log[Sin[(c + d*x)/2]] - 180*Sec[(c + d*x)/2]^2 + 27*Sec[(c + d*x)/2]^4 - 2*Sec[(c + d*x)/2]^6)*Sec[c + d*x])/(a*d*(1 + Sec[c + d*x]))","A",1
64,1,301,105,0.8986547,"\int \frac{\tan ^8(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]^8/(a + a*Sec[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(2400 \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sec (c) \sec ^6(c+d x) (450 \sin (2 c+d x)-3360 \sin (c+2 d x)+2160 \sin (3 c+2 d x)-25 \sin (2 c+3 d x)-25 \sin (4 c+3 d x)-1488 \sin (3 c+4 d x)+720 \sin (5 c+4 d x)+165 \sin (4 c+5 d x)+165 \sin (6 c+5 d x)-368 \sin (5 c+6 d x)+2400 d x \cos (c)+1800 d x \cos (c+2 d x)+1800 d x \cos (3 c+2 d x)+720 d x \cos (3 c+4 d x)+720 d x \cos (5 c+4 d x)+120 d x \cos (5 c+6 d x)+120 d x \cos (7 c+6 d x)+3680 \sin (c)+450 \sin (d x))\right)}{3840 a d (\sec (c+d x)+1)}","-\frac{5 \tanh ^{-1}(\sin (c+d x))}{16 a d}-\frac{\tan ^5(c+d x) (6-5 \sec (c+d x))}{30 a d}+\frac{\tan ^3(c+d x) (8-5 \sec (c+d x))}{24 a d}-\frac{\tan (c+d x) (16-5 \sec (c+d x))}{16 a d}+\frac{x}{a}",1,"(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(2400*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*Sec[c + d*x]^6*(2400*d*x*Cos[c] + 1800*d*x*Cos[c + 2*d*x] + 1800*d*x*Cos[3*c + 2*d*x] + 720*d*x*Cos[3*c + 4*d*x] + 720*d*x*Cos[5*c + 4*d*x] + 120*d*x*Cos[5*c + 6*d*x] + 120*d*x*Cos[7*c + 6*d*x] + 3680*Sin[c] + 450*Sin[d*x] + 450*Sin[2*c + d*x] - 3360*Sin[c + 2*d*x] + 2160*Sin[3*c + 2*d*x] - 25*Sin[2*c + 3*d*x] - 25*Sin[4*c + 3*d*x] - 1488*Sin[3*c + 4*d*x] + 720*Sin[5*c + 4*d*x] + 165*Sin[4*c + 5*d*x] + 165*Sin[6*c + 5*d*x] - 368*Sin[5*c + 6*d*x])))/(3840*a*d*(1 + Sec[c + d*x]))","B",1
65,1,893,78,6.449604,"\int \frac{\tan ^6(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]^6/(a + a*Sec[c + d*x]),x]","-\frac{2 x \sec (c+d x) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sec (c+d x) a+a}-\frac{3 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec (c+d x) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (\sec (c+d x) a+a)}+\frac{3 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec (c+d x) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (\sec (c+d x) a+a)}+\frac{8 \sec (c+d x) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\sec (c+d x) a+a) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{8 \sec (c+d x) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\sec (c+d x) a+a) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\sec (c+d x) \left(11 \sin \left(\frac{c}{2}\right)-19 \cos \left(\frac{c}{2}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (\sec (c+d x) a+a) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\sec (c+d x) \left(19 \cos \left(\frac{c}{2}\right)+11 \sin \left(\frac{c}{2}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (\sec (c+d x) a+a) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}-\frac{\sec (c+d x) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\sec (c+d x) a+a) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}-\frac{\sec (c+d x) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\sec (c+d x) a+a) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{\sec (c+d x) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (\sec (c+d x) a+a) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}-\frac{\sec (c+d x) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (\sec (c+d x) a+a) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}","\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac{\tan ^3(c+d x) (4-3 \sec (c+d x))}{12 a d}+\frac{\tan (c+d x) (8-3 \sec (c+d x))}{8 a d}-\frac{x}{a}",1,"(-2*x*Cos[c/2 + (d*x)/2]^2*Sec[c + d*x])/(a + a*Sec[c + d*x]) - (3*Cos[c/2 + (d*x)/2]^2*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x])/(4*d*(a + a*Sec[c + d*x])) + (3*Cos[c/2 + (d*x)/2]^2*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x])/(4*d*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Sec[c + d*x])/(8*d*(a + a*Sec[c + d*x])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^4) - (Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]*Sin[(d*x)/2])/(3*d*(a + a*Sec[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]*(-19*Cos[c/2] + 11*Sin[c/2]))/(24*d*(a + a*Sec[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (8*Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]*Sin[(d*x)/2])/(3*d*(a + a*Sec[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - (Cos[c/2 + (d*x)/2]^2*Sec[c + d*x])/(8*d*(a + a*Sec[c + d*x])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) - (Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]*Sin[(d*x)/2])/(3*d*(a + a*Sec[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]*(19*Cos[c/2] + 11*Sin[c/2]))/(24*d*(a + a*Sec[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (8*Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]*Sin[(d*x)/2])/(3*d*(a + a*Sec[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",0
66,1,241,49,0.8777428,"\int \frac{\tan ^4(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]^4/(a + a*Sec[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(-\frac{4 \sin (d x)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+4 x\right)}{2 a (\sec (c+d x)+1)}","-\frac{\tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{\tan (c+d x) (2-\sec (c+d x))}{2 a d}+\frac{x}{a}",1,"(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(4*x + (2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d - (2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + 1/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - 1/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - (4*Sin[d*x])/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(2*a*(1 + Sec[c + d*x]))","B",1
67,1,60,21,0.0939644,"\int \frac{\tan ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]^2/(a + a*Sec[c + d*x]),x]","-\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+d x}{a d}","\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{x}{a}",1,"-((d*x + Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(a*d))","B",1
68,1,100,61,0.7988089,"\int \frac{\cot ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^2/(a + a*Sec[c + d*x]),x]","\frac{\sec (c+d x) \left(-12 d x \cos ^2\left(\frac{1}{2} (c+d x)\right)-\tan \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{d x}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(3 \csc \left(\frac{c}{2}\right) \cot \left(\frac{1}{2} (c+d x)\right)+13 \sec \left(\frac{c}{2}\right)\right)\right)}{6 a d (\sec (c+d x)+1)}","\frac{\cot ^3(c+d x) (1-\sec (c+d x))}{3 a d}-\frac{\cot (c+d x) (3-2 \sec (c+d x))}{3 a d}-\frac{x}{a}",1,"(Sec[c + d*x]*(-12*d*x*Cos[(c + d*x)/2]^2 + Cos[(c + d*x)/2]*(3*Cot[(c + d*x)/2]*Csc[c/2] + 13*Sec[c/2])*Sin[(d*x)/2] - Tan[(c + d*x)/2]))/(6*a*d*(1 + Sec[c + d*x]))","A",1
69,1,254,88,0.900773,"\int \frac{\cot ^4(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^4/(a + a*Sec[c + d*x]),x]","\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc ^3(c+d x) \sec (c+d x) (534 \sin (c+d x)+178 \sin (2 (c+d x))-178 \sin (3 (c+d x))-89 \sin (4 (c+d x))-520 \sin (2 c+d x)-248 \sin (c+2 d x)-120 \sin (3 c+2 d x)+248 \sin (2 c+3 d x)+120 \sin (4 c+3 d x)+184 \sin (3 c+4 d x)-360 d x \cos (2 c+d x)+120 d x \cos (c+2 d x)-120 d x \cos (3 c+2 d x)-120 d x \cos (2 c+3 d x)+120 d x \cos (4 c+3 d x)-60 d x \cos (3 c+4 d x)+60 d x \cos (5 c+4 d x)-200 \sin (c)-584 \sin (d x)+360 d x \cos (d x))}{1920 a d (\sec (c+d x)+1)}","\frac{\cot ^5(c+d x) (1-\sec (c+d x))}{5 a d}-\frac{\cot ^3(c+d x) (5-4 \sec (c+d x))}{15 a d}+\frac{\cot (c+d x) (15-8 \sec (c+d x))}{15 a d}+\frac{x}{a}",1,"(Csc[c/2]*Csc[c + d*x]^3*Sec[c/2]*Sec[c + d*x]*(360*d*x*Cos[d*x] - 360*d*x*Cos[2*c + d*x] + 120*d*x*Cos[c + 2*d*x] - 120*d*x*Cos[3*c + 2*d*x] - 120*d*x*Cos[2*c + 3*d*x] + 120*d*x*Cos[4*c + 3*d*x] - 60*d*x*Cos[3*c + 4*d*x] + 60*d*x*Cos[5*c + 4*d*x] - 200*Sin[c] - 584*Sin[d*x] + 534*Sin[c + d*x] + 178*Sin[2*(c + d*x)] - 178*Sin[3*(c + d*x)] - 89*Sin[4*(c + d*x)] - 520*Sin[2*c + d*x] - 248*Sin[c + 2*d*x] - 120*Sin[3*c + 2*d*x] + 248*Sin[2*c + 3*d*x] + 120*Sin[4*c + 3*d*x] + 184*Sin[3*c + 4*d*x]))/(1920*a*d*(1 + Sec[c + d*x]))","B",1
70,1,359,117,1.0970697,"\int \frac{\cot ^6(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^6/(a + a*Sec[c + d*x]),x]","\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc ^5(c+d x) \sec (c+d x) (-22860 \sin (c+d x)-5715 \sin (2 (c+d x))+11430 \sin (3 (c+d x))+4572 \sin (4 (c+d x))-2286 \sin (5 (c+d x))-1143 \sin (6 (c+d x))+26208 \sin (2 c+d x)+14080 \sin (c+2 d x)-16400 \sin (2 c+3 d x)-11760 \sin (4 c+3 d x)-7904 \sin (3 c+4 d x)-3360 \sin (5 c+4 d x)+3952 \sin (4 c+5 d x)+1680 \sin (6 c+5 d x)+2816 \sin (5 c+6 d x)+16800 d x \cos (2 c+d x)-4200 d x \cos (c+2 d x)+4200 d x \cos (3 c+2 d x)+8400 d x \cos (2 c+3 d x)-8400 d x \cos (4 c+3 d x)+3360 d x \cos (3 c+4 d x)-3360 d x \cos (5 c+4 d x)-1680 d x \cos (4 c+5 d x)+1680 d x \cos (6 c+5 d x)-840 d x \cos (5 c+6 d x)+840 d x \cos (7 c+6 d x)+3136 \sin (c)+30112 \sin (d x)-16800 d x \cos (d x))}{107520 a d (\sec (c+d x)+1)}","\frac{\cot ^7(c+d x) (1-\sec (c+d x))}{7 a d}-\frac{\cot ^5(c+d x) (7-6 \sec (c+d x))}{35 a d}+\frac{\cot ^3(c+d x) (35-24 \sec (c+d x))}{105 a d}-\frac{\cot (c+d x) (35-16 \sec (c+d x))}{35 a d}-\frac{x}{a}",1,"(Csc[c/2]*Csc[c + d*x]^5*Sec[c/2]*Sec[c + d*x]*(-16800*d*x*Cos[d*x] + 16800*d*x*Cos[2*c + d*x] - 4200*d*x*Cos[c + 2*d*x] + 4200*d*x*Cos[3*c + 2*d*x] + 8400*d*x*Cos[2*c + 3*d*x] - 8400*d*x*Cos[4*c + 3*d*x] + 3360*d*x*Cos[3*c + 4*d*x] - 3360*d*x*Cos[5*c + 4*d*x] - 1680*d*x*Cos[4*c + 5*d*x] + 1680*d*x*Cos[6*c + 5*d*x] - 840*d*x*Cos[5*c + 6*d*x] + 840*d*x*Cos[7*c + 6*d*x] + 3136*Sin[c] + 30112*Sin[d*x] - 22860*Sin[c + d*x] - 5715*Sin[2*(c + d*x)] + 11430*Sin[3*(c + d*x)] + 4572*Sin[4*(c + d*x)] - 2286*Sin[5*(c + d*x)] - 1143*Sin[6*(c + d*x)] + 26208*Sin[2*c + d*x] + 14080*Sin[c + 2*d*x] - 16400*Sin[2*c + 3*d*x] - 11760*Sin[4*c + 3*d*x] - 7904*Sin[3*c + 4*d*x] - 3360*Sin[5*c + 4*d*x] + 3952*Sin[4*c + 5*d*x] + 1680*Sin[6*c + 5*d*x] + 2816*Sin[5*c + 6*d*x]))/(107520*a*d*(1 + Sec[c + d*x]))","B",1
71,1,125,120,0.5257677,"\int \frac{\tan ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^9/(a + a*Sec[c + d*x])^2,x]","-\frac{\sec ^6(c+d x) (312 \cos (c+d x)+5 (28 \cos (3 (c+d x))+6 \cos (4 (c+d x))+12 \cos (5 (c+d x))+18 \cos (4 (c+d x)) \log (\cos (c+d x))+3 \cos (6 (c+d x)) \log (\cos (c+d x))+30 \log (\cos (c+d x))+9 \cos (2 (c+d x)) (5 \log (\cos (c+d x))+4)+14))}{480 a^2 d}","\frac{\sec ^6(c+d x)}{6 a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{\sec ^4(c+d x)}{4 a^2 d}+\frac{4 \sec ^3(c+d x)}{3 a^2 d}-\frac{\sec ^2(c+d x)}{2 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d}",1,"-1/480*((312*Cos[c + d*x] + 5*(14 + 28*Cos[3*(c + d*x)] + 6*Cos[4*(c + d*x)] + 12*Cos[5*(c + d*x)] + 30*Log[Cos[c + d*x]] + 18*Cos[4*(c + d*x)]*Log[Cos[c + d*x]] + 3*Cos[6*(c + d*x)]*Log[Cos[c + d*x]] + 9*Cos[2*(c + d*x)]*(4 + 5*Log[Cos[c + d*x]])))*Sec[c + d*x]^6)/(a^2*d)","A",1
72,1,83,65,0.2003687,"\int \frac{\tan ^7(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^7/(a + a*Sec[c + d*x])^2,x]","\frac{\sec ^4(c+d x) (20 \cos (c+d x)+3 (4 \cos (3 (c+d x))+4 \cos (2 (c+d x)) \log (\cos (c+d x))+\cos (4 (c+d x)) \log (\cos (c+d x))+3 \log (\cos (c+d x))+2))}{24 a^2 d}","\frac{\sec ^4(c+d x)}{4 a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}+\frac{2 \sec (c+d x)}{a^2 d}+\frac{\log (\cos (c+d x))}{a^2 d}",1,"((20*Cos[c + d*x] + 3*(2 + 4*Cos[3*(c + d*x)] + 3*Log[Cos[c + d*x]] + 4*Cos[2*(c + d*x)]*Log[Cos[c + d*x]] + Cos[4*(c + d*x)]*Log[Cos[c + d*x]]))*Sec[c + d*x]^4)/(24*a^2*d)","A",1
73,1,51,48,0.1226999,"\int \frac{\tan ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^5/(a + a*Sec[c + d*x])^2,x]","-\frac{\sec ^2(c+d x) (4 \cos (c+d x)+\cos (2 (c+d x)) \log (\cos (c+d x))+\log (\cos (c+d x))-1)}{2 a^2 d}","\frac{\sec ^2(c+d x)}{2 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d}",1,"-1/2*((-1 + 4*Cos[c + d*x] + Log[Cos[c + d*x]] + Cos[2*(c + d*x)]*Log[Cos[c + d*x]])*Sec[c + d*x]^2)/(a^2*d)","A",1
74,1,30,33,0.0682348,"\int \frac{\tan ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^3/(a + a*Sec[c + d*x])^2,x]","\frac{4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log (\cos (c+d x))}{a^2 d}","\frac{2 \log (\cos (c+d x)+1)}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d}",1,"(4*Log[Cos[(c + d*x)/2]] - Log[Cos[c + d*x]])/(a^2*d)","A",1
75,1,56,36,0.1333965,"\int \frac{\tan (c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]/(a + a*Sec[c + d*x])^2,x]","-\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) \left(2 \cos (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+1\right)}{2 a^2 d}","-\frac{1}{a^2 d (\cos (c+d x)+1)}-\frac{\log (\cos (c+d x)+1)}{a^2 d}",1,"-1/2*((1 + 2*Log[Cos[(c + d*x)/2]] + 2*Cos[c + d*x]*Log[Cos[(c + d*x)/2]])*Sec[(c + d*x)/2]^2)/(a^2*d)","A",1
76,1,83,81,0.1914985,"\int \frac{\cot (c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Cot[c + d*x]/(a + a*Sec[c + d*x])^2,x]","\frac{\sec ^2(c+d x) \left(10 \cos ^2\left(\frac{1}{2} (c+d x)\right)+4 \cos ^4\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+7 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-1\right)}{4 a^2 d (\sec (c+d x)+1)^2}","\frac{5}{4 a^2 d (\cos (c+d x)+1)}-\frac{1}{4 a^2 d (\cos (c+d x)+1)^2}+\frac{\log (1-\cos (c+d x))}{8 a^2 d}+\frac{7 \log (\cos (c+d x)+1)}{8 a^2 d}",1,"((-1 + 10*Cos[(c + d*x)/2]^2 + 4*Cos[(c + d*x)/2]^4*(7*Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]]))*Sec[c + d*x]^2)/(4*a^2*d*(1 + Sec[c + d*x])^2)","A",1
77,1,121,123,0.3933826,"\int \frac{\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^3/(a + a*Sec[c + d*x])^2,x]","-\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(3 \csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^6\left(\frac{1}{2} (c+d x)\right)-12 \sec ^4\left(\frac{1}{2} (c+d x)\right)+69 \sec ^2\left(\frac{1}{2} (c+d x)\right)+36 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+156 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{24 a^2 d (\sec (c+d x)+1)^2}","-\frac{1}{16 a^2 d (1-\cos (c+d x))}-\frac{23}{16 a^2 d (\cos (c+d x)+1)}+\frac{1}{2 a^2 d (\cos (c+d x)+1)^2}-\frac{1}{12 a^2 d (\cos (c+d x)+1)^3}-\frac{3 \log (1-\cos (c+d x))}{16 a^2 d}-\frac{13 \log (\cos (c+d x)+1)}{16 a^2 d}",1,"-1/24*(Cos[(c + d*x)/2]^4*(3*Csc[(c + d*x)/2]^2 + 156*Log[Cos[(c + d*x)/2]] + 36*Log[Sin[(c + d*x)/2]] + 69*Sec[(c + d*x)/2]^2 - 12*Sec[(c + d*x)/2]^4 + Sec[(c + d*x)/2]^6)*Sec[c + d*x]^2)/(a^2*d*(1 + Sec[c + d*x])^2)","A",1
78,1,154,165,0.819576,"\int \frac{\cot ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^2,x]","-\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(6 \csc ^4\left(\frac{1}{2} (c+d x)\right)-108 \csc ^2\left(\frac{1}{2} (c+d x)\right)+3 \sec ^8\left(\frac{1}{2} (c+d x)\right)-44 \sec ^6\left(\frac{1}{2} (c+d x)\right)+288 \sec ^4\left(\frac{1}{2} (c+d x)\right)-1224 \sec ^2\left(\frac{1}{2} (c+d x)\right)-24 \left(29 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+99 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{384 a^2 d (\sec (c+d x)+1)^2}","\frac{9}{64 a^2 d (1-\cos (c+d x))}+\frac{51}{32 a^2 d (\cos (c+d x)+1)}-\frac{1}{64 a^2 d (1-\cos (c+d x))^2}-\frac{3}{4 a^2 d (\cos (c+d x)+1)^2}+\frac{11}{48 a^2 d (\cos (c+d x)+1)^3}-\frac{1}{32 a^2 d (\cos (c+d x)+1)^4}+\frac{29 \log (1-\cos (c+d x))}{128 a^2 d}+\frac{99 \log (\cos (c+d x)+1)}{128 a^2 d}",1,"-1/384*(Cos[(c + d*x)/2]^4*(-108*Csc[(c + d*x)/2]^2 + 6*Csc[(c + d*x)/2]^4 - 24*(99*Log[Cos[(c + d*x)/2]] + 29*Log[Sin[(c + d*x)/2]]) - 1224*Sec[(c + d*x)/2]^2 + 288*Sec[(c + d*x)/2]^4 - 44*Sec[(c + d*x)/2]^6 + 3*Sec[(c + d*x)/2]^8)*Sec[c + d*x]^2)/(a^2*d*(1 + Sec[c + d*x])^2)","A",1
79,1,495,119,5.9051295,"\int \frac{\tan ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^8/(a + a*Sec[c + d*x])^2,x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(-\frac{151 \sin \left(\frac{c}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{151 \sin \left(\frac{c}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{36 \sin \left(\frac{c}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{36 \sin \left(\frac{c}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{180 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{180 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{\sec (c) \sin \left(\frac{d x}{2}\right) \left(333 \cos \left(2 c+\frac{3 d x}{2}\right)+287 \cos \left(2 c+\frac{5 d x}{2}\right)+67 \cos \left(4 c+\frac{7 d x}{2}\right)+68 \cos \left(4 c+\frac{9 d x}{2}\right)+293 \cos \left(\frac{d x}{2}\right)\right) \sec ^5(c+d x)}{2 d}+\frac{\cos \left(\frac{c}{2}\right) \sec (c) \left(-43 \sin \left(\frac{c}{2}+d x\right)-43 \sin \left(\frac{3 c}{2}+d x\right)-346 \sin \left(\frac{3 c}{2}+2 d x\right)+346 \sin \left(\frac{5 c}{2}+2 d x\right)+149 \sin \left(\frac{5 c}{2}+3 d x\right)+149 \sin \left(\frac{7 c}{2}+3 d x\right)+308 \sin \left(\frac{c}{2}\right)\right) \sec ^4(c+d x)}{4 d}+240 x\right)}{60 a^2 (\sec (c+d x)+1)^2}","\frac{\tan ^5(c+d x)}{5 a^2 d}+\frac{\tan ^3(c+d x)}{3 a^2 d}-\frac{\tan (c+d x)}{a^2 d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac{\tan ^3(c+d x) \sec (c+d x)}{2 a^2 d}+\frac{3 \tan (c+d x) \sec (c+d x)}{4 a^2 d}+\frac{x}{a^2}",1,"(Cos[(c + d*x)/2]^4*Sec[c + d*x]^2*(240*x + (180*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d - (180*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d - ((293*Cos[(d*x)/2] + 333*Cos[2*c + (3*d*x)/2] + 287*Cos[2*c + (5*d*x)/2] + 67*Cos[4*c + (7*d*x)/2] + 68*Cos[4*c + (9*d*x)/2])*Sec[c]*Sec[c + d*x]^5*Sin[(d*x)/2])/(2*d) + (36*Sin[c/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) - (151*Sin[c/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (36*Sin[c/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) - (151*Sin[c/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (Cos[c/2]*Sec[c]*Sec[c + d*x]^4*(308*Sin[c/2] - 43*Sin[c/2 + d*x] - 43*Sin[(3*c)/2 + d*x] - 346*Sin[(3*c)/2 + 2*d*x] + 346*Sin[(5*c)/2 + 2*d*x] + 149*Sin[(5*c)/2 + 3*d*x] + 149*Sin[(7*c)/2 + 3*d*x]))/(4*d)))/(60*a^2*(1 + Sec[c + d*x])^2)","B",1
80,1,767,72,6.3062784,"\int \frac{\tan ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^6/(a + a*Sec[c + d*x])^2,x]","-\frac{4 x \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x)}{(a \sec (c+d x)+a)^2}+\frac{8 \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x)}{3 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) (a \sec (c+d x)+a)^2 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{8 \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x)}{3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) (a \sec (c+d x)+a)^2 \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\left(7 \sin \left(\frac{c}{2}\right)-5 \cos \left(\frac{c}{2}\right)\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x)}{3 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) (a \sec (c+d x)+a)^2 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\left(7 \sin \left(\frac{c}{2}\right)+5 \cos \left(\frac{c}{2}\right)\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x)}{3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) (a \sec (c+d x)+a)^2 \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{2 \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x)}{3 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) (a \sec (c+d x)+a)^2 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{2 \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x)}{3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) (a \sec (c+d x)+a)^2 \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}-\frac{4 \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \sec (c+d x)+a)^2}+\frac{4 \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x) \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \sec (c+d x)+a)^2}","\frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{\tan (c+d x)}{a^2 d}+\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{\tan (c+d x) \sec (c+d x)}{a^2 d}-\frac{x}{a^2}",1,"(-4*x*Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2 - (4*Cos[c/2 + (d*x)/2]^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^2)/(d*(a + a*Sec[c + d*x])^2) + (4*Cos[c/2 + (d*x)/2]^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^2)/(d*(a + a*Sec[c + d*x])^2) + (2*Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*Sin[(d*x)/2])/(3*d*(a + a*Sec[c + d*x])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*(-5*Cos[c/2] + 7*Sin[c/2]))/(3*d*(a + a*Sec[c + d*x])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (8*Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*Sin[(d*x)/2])/(3*d*(a + a*Sec[c + d*x])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (2*Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*Sin[(d*x)/2])/(3*d*(a + a*Sec[c + d*x])^2*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*(5*Cos[c/2] + 7*Sin[c/2]))/(3*d*(a + a*Sec[c + d*x])^2*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (8*Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*Sin[(d*x)/2])/(3*d*(a + a*Sec[c + d*x])^2*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
81,1,177,34,0.5233864,"\int \frac{\tan ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^4/(a + a*Sec[c + d*x])^2,x]","\frac{4 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(\frac{\sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+d x\right)}{a^2 d (\sec (c+d x)+1)^2}","\frac{\tan (c+d x)}{a^2 d}-\frac{2 \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{x}{a^2}",1,"(4*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2*(d*x + 2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sin[d*x]/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(a^2*d*(1 + Sec[c + d*x])^2)","B",1
82,1,42,33,0.0200512,"\int \frac{\tan ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^2/(a + a*Sec[c + d*x])^2,x]","\frac{\frac{2 \tan \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{2 \tan ^{-1}\left(\tan \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}}{a^2}","\frac{2 \tan (c+d x)}{a d (a \sec (c+d x)+a)}-\frac{x}{a^2}",1,"((-2*ArcTan[Tan[c/2 + (d*x)/2]])/d + (2*Tan[c/2 + (d*x)/2])/d)/a^2","A",1
83,1,149,107,1.3483773,"\int \frac{\cot ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^2/(a + a*Sec[c + d*x])^2,x]","\frac{\sec ^2(c+d x) \left(-120 d x \cos ^4\left(\frac{1}{2} (c+d x)\right)+3 \tan \left(\frac{1}{2} (c+d x)\right)-31 \tan \left(\frac{c}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)-31 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{d x}{2}\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(15 \csc \left(\frac{c}{2}\right) \cot \left(\frac{1}{2} (c+d x)\right)+193 \sec \left(\frac{c}{2}\right)\right)\right)}{30 a^2 d (\sec (c+d x)+1)^2}","-\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{\cot (c+d x)}{a^2 d}+\frac{2 \csc ^5(c+d x)}{5 a^2 d}-\frac{4 \csc ^3(c+d x)}{3 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}-\frac{x}{a^2}",1,"(Sec[c + d*x]^2*(-120*d*x*Cos[(c + d*x)/2]^4 - 31*Cos[(c + d*x)/2]*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]^3*(15*Cot[(c + d*x)/2]*Csc[c/2] + 193*Sec[c/2])*Sin[(d*x)/2] - 31*Cos[(c + d*x)/2]^2*Tan[c/2] + 3*Tan[(c + d*x)/2]))/(30*a^2*d*(1 + Sec[c + d*x])^2)","A",1
84,1,314,139,1.0671176,"\int \frac{\cot ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^4/(a + a*Sec[c + d*x])^2,x]","\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc ^3(c+d x) \sec ^2(c+d x) (16002 \sin (c+d x)+9144 \sin (2 (c+d x))-3429 \sin (3 (c+d x))-4572 \sin (4 (c+d x))-1143 \sin (5 (c+d x))-11760 \sin (2 c+d x)-8864 \sin (c+2 d x)-3360 \sin (3 c+2 d x)+2064 \sin (2 c+3 d x)+2520 \sin (4 c+3 d x)+4432 \sin (3 c+4 d x)+1680 \sin (5 c+4 d x)+1528 \sin (4 c+5 d x)-5880 d x \cos (2 c+d x)+3360 d x \cos (c+2 d x)-3360 d x \cos (3 c+2 d x)-1260 d x \cos (2 c+3 d x)+1260 d x \cos (4 c+3 d x)-1680 d x \cos (3 c+4 d x)+1680 d x \cos (5 c+4 d x)-420 d x \cos (4 c+5 d x)+420 d x \cos (6 c+5 d x)-4032 \sin (c)-9632 \sin (d x)+5880 d x \cos (d x))}{26880 a^2 d (\sec (c+d x)+1)^2}","-\frac{2 \cot ^7(c+d x)}{7 a^2 d}+\frac{\cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{\cot (c+d x)}{a^2 d}+\frac{2 \csc ^7(c+d x)}{7 a^2 d}-\frac{6 \csc ^5(c+d x)}{5 a^2 d}+\frac{2 \csc ^3(c+d x)}{a^2 d}-\frac{2 \csc (c+d x)}{a^2 d}+\frac{x}{a^2}",1,"(Csc[c/2]*Csc[c + d*x]^3*Sec[c/2]*Sec[c + d*x]^2*(5880*d*x*Cos[d*x] - 5880*d*x*Cos[2*c + d*x] + 3360*d*x*Cos[c + 2*d*x] - 3360*d*x*Cos[3*c + 2*d*x] - 1260*d*x*Cos[2*c + 3*d*x] + 1260*d*x*Cos[4*c + 3*d*x] - 1680*d*x*Cos[3*c + 4*d*x] + 1680*d*x*Cos[5*c + 4*d*x] - 420*d*x*Cos[4*c + 5*d*x] + 420*d*x*Cos[6*c + 5*d*x] - 4032*Sin[c] - 9632*Sin[d*x] + 16002*Sin[c + d*x] + 9144*Sin[2*(c + d*x)] - 3429*Sin[3*(c + d*x)] - 4572*Sin[4*(c + d*x)] - 1143*Sin[5*(c + d*x)] - 11760*Sin[2*c + d*x] - 8864*Sin[c + 2*d*x] - 3360*Sin[3*c + 2*d*x] + 2064*Sin[2*c + 3*d*x] + 2520*Sin[4*c + 3*d*x] + 4432*Sin[3*c + 4*d*x] + 1680*Sin[5*c + 4*d*x] + 1528*Sin[4*c + 5*d*x]))/(26880*a^2*d*(1 + Sec[c + d*x])^2)","B",1
85,1,802,179,6.5713292,"\int \frac{\cot ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \sin \left(\frac{d x}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{288 d (\sec (c+d x) a+a)^2}+\frac{\sec ^2(c+d x) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{288 d (\sec (c+d x) a+a)^2}-\frac{109 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \sin \left(\frac{d x}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{2016 d (\sec (c+d x) a+a)^2}-\frac{109 \sec ^2(c+d x) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2016 d (\sec (c+d x) a+a)^2}+\frac{313 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{840 d (\sec (c+d x) a+a)^2}-\frac{17 \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \cot ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \csc \left(\frac{c}{2}\right) \sec ^2(c+d x) \sin \left(\frac{d x}{2}\right)}{160 d (\sec (c+d x) a+a)^2}+\frac{201 \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \cot \left(\frac{c}{2}+\frac{d x}{2}\right) \csc \left(\frac{c}{2}\right) \sec ^2(c+d x) \sin \left(\frac{d x}{2}\right)}{160 d (\sec (c+d x) a+a)^2}+\frac{\cot ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \csc \left(\frac{c}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x) \sin \left(\frac{d x}{2}\right)}{160 d (\sec (c+d x) a+a)^2}+\frac{63881 \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \sin \left(\frac{d x}{2}\right)}{10080 d (\sec (c+d x) a+a)^2}-\frac{7891 \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \sin \left(\frac{d x}{2}\right)}{5040 d (\sec (c+d x) a+a)^2}-\frac{7891 \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x) \tan \left(\frac{c}{2}\right)}{5040 d (\sec (c+d x) a+a)^2}+\frac{313 \sec ^2(c+d x) \tan \left(\frac{c}{2}\right)}{840 d (\sec (c+d x) a+a)^2}-\frac{4 x \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x)}{(\sec (c+d x) a+a)^2}-\frac{\cot \left(\frac{c}{2}\right) \cot ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x)}{160 d (\sec (c+d x) a+a)^2}+\frac{17 \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \cot \left(\frac{c}{2}\right) \cot ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x)}{160 d (\sec (c+d x) a+a)^2}","-\frac{2 \cot ^9(c+d x)}{9 a^2 d}+\frac{\cot ^7(c+d x)}{7 a^2 d}-\frac{\cot ^5(c+d x)}{5 a^2 d}+\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{\cot (c+d x)}{a^2 d}+\frac{2 \csc ^9(c+d x)}{9 a^2 d}-\frac{8 \csc ^7(c+d x)}{7 a^2 d}+\frac{12 \csc ^5(c+d x)}{5 a^2 d}-\frac{8 \csc ^3(c+d x)}{3 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}-\frac{x}{a^2}",1,"(-4*x*Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2 + (17*Cos[c/2 + (d*x)/2]^2*Cot[c/2]*Cot[c/2 + (d*x)/2]^2*Sec[c + d*x]^2)/(160*d*(a + a*Sec[c + d*x])^2) - (Cot[c/2]*Cot[c/2 + (d*x)/2]^4*Sec[c + d*x]^2)/(160*d*(a + a*Sec[c + d*x])^2) + (201*Cos[c/2 + (d*x)/2]^3*Cot[c/2 + (d*x)/2]*Csc[c/2]*Sec[c + d*x]^2*Sin[(d*x)/2])/(160*d*(a + a*Sec[c + d*x])^2) - (17*Cos[c/2 + (d*x)/2]*Cot[c/2 + (d*x)/2]^3*Csc[c/2]*Sec[c + d*x]^2*Sin[(d*x)/2])/(160*d*(a + a*Sec[c + d*x])^2) + (Cot[c/2 + (d*x)/2]^4*Csc[c/2]*Csc[c/2 + (d*x)/2]*Sec[c + d*x]^2*Sin[(d*x)/2])/(160*d*(a + a*Sec[c + d*x])^2) - (7891*Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c + d*x]^2*Sin[(d*x)/2])/(5040*d*(a + a*Sec[c + d*x])^2) + (63881*Cos[c/2 + (d*x)/2]^3*Sec[c/2]*Sec[c + d*x]^2*Sin[(d*x)/2])/(10080*d*(a + a*Sec[c + d*x])^2) + (313*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sec[c + d*x]^2*Sin[(d*x)/2])/(840*d*(a + a*Sec[c + d*x])^2) - (109*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*Sec[c + d*x]^2*Sin[(d*x)/2])/(2016*d*(a + a*Sec[c + d*x])^2) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^5*Sec[c + d*x]^2*Sin[(d*x)/2])/(288*d*(a + a*Sec[c + d*x])^2) + (313*Sec[c + d*x]^2*Tan[c/2])/(840*d*(a + a*Sec[c + d*x])^2) - (7891*Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]^2*Tan[c/2])/(5040*d*(a + a*Sec[c + d*x])^2) - (109*Sec[c/2 + (d*x)/2]^2*Sec[c + d*x]^2*Tan[c/2])/(2016*d*(a + a*Sec[c + d*x])^2) + (Sec[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*Tan[c/2])/(288*d*(a + a*Sec[c + d*x])^2)","B",1
86,1,140,137,0.3453666,"\int \frac{\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^11/(a + a*Sec[c + d*x])^3,x]","\frac{\sec ^7(c+d x) (4522 \cos (2 (c+d x))+1050 \cos (3 (c+d x))+2380 \cos (4 (c+d x))-210 \cos (5 (c+d x))+630 \cos (6 (c+d x))+2205 \cos (3 (c+d x)) \log (\cos (c+d x))+735 \cos (5 (c+d x)) \log (\cos (c+d x))+105 \cos (7 (c+d x)) \log (\cos (c+d x))+105 \cos (c+d x) (35 \log (\cos (c+d x))+8)+3732)}{6720 a^3 d}","\frac{\sec ^7(c+d x)}{7 a^3 d}-\frac{\sec ^6(c+d x)}{2 a^3 d}+\frac{\sec ^5(c+d x)}{5 a^3 d}+\frac{5 \sec ^4(c+d x)}{4 a^3 d}-\frac{5 \sec ^3(c+d x)}{3 a^3 d}-\frac{\sec ^2(c+d x)}{2 a^3 d}+\frac{3 \sec (c+d x)}{a^3 d}+\frac{\log (\cos (c+d x))}{a^3 d}",1,"((3732 + 4522*Cos[2*(c + d*x)] + 1050*Cos[3*(c + d*x)] + 2380*Cos[4*(c + d*x)] - 210*Cos[5*(c + d*x)] + 630*Cos[6*(c + d*x)] + 2205*Cos[3*(c + d*x)]*Log[Cos[c + d*x]] + 735*Cos[5*(c + d*x)]*Log[Cos[c + d*x]] + 105*Cos[7*(c + d*x)]*Log[Cos[c + d*x]] + 105*Cos[c + d*x]*(8 + 35*Log[Cos[c + d*x]]))*Sec[c + d*x]^7)/(6720*a^3*d)","A",1
87,1,93,99,0.3664696,"\int \frac{\tan ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^9/(a + a*Sec[c + d*x])^3,x]","-\frac{\sec ^5(c+d x) (280 \cos (2 (c+d x))+90 \cos (4 (c+d x))+150 \cos (c+d x) \log (\cos (c+d x))+15 \cos (5 (c+d x)) \log (\cos (c+d x))+15 \cos (3 (c+d x)) (5 \log (\cos (c+d x))-4)+142)}{240 a^3 d}","\frac{\sec ^5(c+d x)}{5 a^3 d}-\frac{3 \sec ^4(c+d x)}{4 a^3 d}+\frac{2 \sec ^3(c+d x)}{3 a^3 d}+\frac{\sec ^2(c+d x)}{a^3 d}-\frac{3 \sec (c+d x)}{a^3 d}-\frac{\log (\cos (c+d x))}{a^3 d}",1,"-1/240*((142 + 280*Cos[2*(c + d*x)] + 90*Cos[4*(c + d*x)] + 150*Cos[c + d*x]*Log[Cos[c + d*x]] + 15*Cos[5*(c + d*x)]*Log[Cos[c + d*x]] + 15*Cos[3*(c + d*x)]*(-4 + 5*Log[Cos[c + d*x]]))*Sec[c + d*x]^5)/(a^3*d)","A",1
88,1,64,65,0.1808329,"\int \frac{\tan ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^7/(a + a*Sec[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (18 \cos (2 (c+d x))+9 \cos (c+d x) (\log (\cos (c+d x))-2)+3 \cos (3 (c+d x)) \log (\cos (c+d x))+22)}{12 a^3 d}","\frac{\sec ^3(c+d x)}{3 a^3 d}-\frac{3 \sec ^2(c+d x)}{2 a^3 d}+\frac{3 \sec (c+d x)}{a^3 d}+\frac{\log (\cos (c+d x))}{a^3 d}",1,"((22 + 18*Cos[2*(c + d*x)] + 9*Cos[c + d*x]*(-2 + Log[Cos[c + d*x]]) + 3*Cos[3*(c + d*x)]*Log[Cos[c + d*x]])*Sec[c + d*x]^3)/(12*a^3*d)","A",1
89,1,36,46,0.1164798,"\int \frac{\tan ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]","\frac{\sec (c+d x)-8 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+3 \log (\cos (c+d x))}{a^3 d}","\frac{\sec (c+d x)}{a^3 d}+\frac{3 \log (\cos (c+d x))}{a^3 d}-\frac{4 \log (\cos (c+d x)+1)}{a^3 d}",1,"(-8*Log[Cos[(c + d*x)/2]] + 3*Log[Cos[c + d*x]] + Sec[c + d*x])/(a^3*d)","A",1
90,1,33,35,0.0585627,"\int \frac{\tan ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^3/(a + a*Sec[c + d*x])^3,x]","\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^3 d}","\frac{2}{a^3 d (\cos (c+d x)+1)}+\frac{\log (\cos (c+d x)+1)}{a^3 d}",1,"(2*Log[Cos[(c + d*x)/2]] + Tan[(c + d*x)/2]^2)/(a^3*d)","A",1
91,1,79,56,0.1308977,"\int \frac{\tan (c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Tan[c + d*x]/(a + a*Sec[c + d*x])^3,x]","-\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(8 \cos ^2\left(\frac{1}{2} (c+d x)\right)+16 \cos ^4\left(\frac{1}{2} (c+d x)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-1\right)}{a^3 d (\sec (c+d x)+1)^3}","-\frac{2}{a^3 d (\cos (c+d x)+1)}+\frac{1}{2 a^3 d (\cos (c+d x)+1)^2}-\frac{\log (\cos (c+d x)+1)}{a^3 d}",1,"-((Cos[(c + d*x)/2]^2*(-1 + 8*Cos[(c + d*x)/2]^2 + 16*Cos[(c + d*x)/2]^4*Log[Cos[(c + d*x)/2]])*Sec[c + d*x]^3)/(a^3*d*(1 + Sec[c + d*x])^3))","A",1
92,1,97,101,0.3257088,"\int \frac{\cot (c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Cot[c + d*x]/(a + a*Sec[c + d*x])^3,x]","\frac{\sec ^3(c+d x) \left(102 \cos ^4\left(\frac{1}{2} (c+d x)\right)-21 \cos ^2\left(\frac{1}{2} (c+d x)\right)+12 \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+15 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+2\right)}{12 a^3 d (\sec (c+d x)+1)^3}","\frac{17}{8 a^3 d (\cos (c+d x)+1)}-\frac{7}{8 a^3 d (\cos (c+d x)+1)^2}+\frac{1}{6 a^3 d (\cos (c+d x)+1)^3}+\frac{\log (1-\cos (c+d x))}{16 a^3 d}+\frac{15 \log (\cos (c+d x)+1)}{16 a^3 d}",1,"((2 - 21*Cos[(c + d*x)/2]^2 + 102*Cos[(c + d*x)/2]^4 + 12*Cos[(c + d*x)/2]^6*(15*Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]]))*Sec[c + d*x]^3)/(12*a^3*d*(1 + Sec[c + d*x])^3)","A",1
93,1,140,143,0.6390622,"\int \frac{\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^3/(a + a*Sec[c + d*x])^3,x]","-\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(12 \csc ^2\left(\frac{1}{2} (c+d x)\right)-3 \sec ^8\left(\frac{1}{2} (c+d x)\right)+40 \sec ^6\left(\frac{1}{2} (c+d x)\right)-234 \sec ^4\left(\frac{1}{2} (c+d x)\right)+864 \sec ^2\left(\frac{1}{2} (c+d x)\right)+24 \left(7 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+57 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{96 a^3 d (\sec (c+d x)+1)^3}","-\frac{1}{32 a^3 d (1-\cos (c+d x))}-\frac{9}{4 a^3 d (\cos (c+d x)+1)}+\frac{39}{32 a^3 d (\cos (c+d x)+1)^2}-\frac{5}{12 a^3 d (\cos (c+d x)+1)^3}+\frac{1}{16 a^3 d (\cos (c+d x)+1)^4}-\frac{7 \log (1-\cos (c+d x))}{64 a^3 d}-\frac{57 \log (\cos (c+d x)+1)}{64 a^3 d}",1,"-1/96*(Cos[(c + d*x)/2]^6*(12*Csc[(c + d*x)/2]^2 + 24*(57*Log[Cos[(c + d*x)/2]] + 7*Log[Sin[(c + d*x)/2]]) + 864*Sec[(c + d*x)/2]^2 - 234*Sec[(c + d*x)/2]^4 + 40*Sec[(c + d*x)/2]^6 - 3*Sec[(c + d*x)/2]^8)*Sec[c + d*x]^3)/(a^3*d*(1 + Sec[c + d*x])^3)","A",1
94,1,169,185,1.2424181,"\int \frac{\cot ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]","\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(1400 \cos ^4\left(\frac{1}{2} (c+d x)\right)-195 \cos ^2\left(\frac{1}{2} (c+d x)\right)+60 \cos ^8\left(\frac{1}{2} (c+d x)\right) \left(10 \cot ^2\left(\frac{1}{2} (c+d x)\right)+303\right)-30 \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\cot ^4\left(\frac{1}{2} (c+d x)\right)+198\right)+120 \cos ^{10}\left(\frac{1}{2} (c+d x)\right) \left(37 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+219 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+12\right)}{1920 a^3 d (\sec (c+d x)+1)^3}","\frac{5}{64 a^3 d (1-\cos (c+d x))}+\frac{303}{128 a^3 d (\cos (c+d x)+1)}-\frac{1}{128 a^3 d (1-\cos (c+d x))^2}-\frac{99}{64 a^3 d (\cos (c+d x)+1)^2}+\frac{35}{48 a^3 d (\cos (c+d x)+1)^3}-\frac{13}{64 a^3 d (\cos (c+d x)+1)^4}+\frac{1}{40 a^3 d (\cos (c+d x)+1)^5}+\frac{37 \log (1-\cos (c+d x))}{256 a^3 d}+\frac{219 \log (\cos (c+d x)+1)}{256 a^3 d}",1,"((12 - 195*Cos[(c + d*x)/2]^2 + 1400*Cos[(c + d*x)/2]^4 + 60*Cos[(c + d*x)/2]^8*(303 + 10*Cot[(c + d*x)/2]^2) - 30*Cos[(c + d*x)/2]^6*(198 + Cot[(c + d*x)/2]^4) + 120*Cos[(c + d*x)/2]^10*(219*Log[Cos[(c + d*x)/2]] + 37*Log[Sin[(c + d*x)/2]]))*Sec[(c + d*x)/2]^4*Sec[c + d*x]^3)/(1920*a^3*d*(1 + Sec[c + d*x])^3)","A",1
95,1,362,237,1.3391842,"\int \frac{\tan ^{12}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^12/(a + a*Sec[c + d*x])^3,x]","\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(1680000 \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sec (c) \sec ^8(c+d x) (133175 \sin (2 c+d x)-544768 \sin (c+2 d x)+286720 \sin (3 c+2 d x)+63595 \sin (2 c+3 d x)+63595 \sin (4 c+3 d x)-254464 \sin (3 c+4 d x)+161280 \sin (5 c+4 d x)+65135 \sin (4 c+5 d x)+65135 \sin (6 c+5 d x)-118784 \sin (5 c+6 d x)+27195 \sin (6 c+7 d x)+27195 \sin (8 c+7 d x)-14848 \sin (7 c+8 d x)+470400 d x \cos (c)+376320 d x \cos (c+2 d x)+376320 d x \cos (3 c+2 d x)+188160 d x \cos (3 c+4 d x)+188160 d x \cos (5 c+4 d x)+53760 d x \cos (5 c+6 d x)+53760 d x \cos (7 c+6 d x)+6720 d x \cos (7 c+8 d x)+6720 d x \cos (9 c+8 d x)+519680 \sin (c)+133175 \sin (d x))\right)}{215040 a^3 d (\sec (c+d x)+1)^3}","-\frac{3 \tan ^7(c+d x)}{7 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{\tan (c+d x)}{a^3 d}-\frac{125 \tanh ^{-1}(\sin (c+d x))}{128 a^3 d}+\frac{\tan ^5(c+d x) \sec ^3(c+d x)}{8 a^3 d}-\frac{5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a^3 d}+\frac{5 \tan (c+d x) \sec ^3(c+d x)}{64 a^3 d}+\frac{\tan ^5(c+d x) \sec (c+d x)}{2 a^3 d}-\frac{5 \tan ^3(c+d x) \sec (c+d x)}{8 a^3 d}+\frac{115 \tan (c+d x) \sec (c+d x)}{128 a^3 d}+\frac{x}{a^3}",1,"(Cos[(c + d*x)/2]^6*Sec[c + d*x]^3*(1680000*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*Sec[c + d*x]^8*(470400*d*x*Cos[c] + 376320*d*x*Cos[c + 2*d*x] + 376320*d*x*Cos[3*c + 2*d*x] + 188160*d*x*Cos[3*c + 4*d*x] + 188160*d*x*Cos[5*c + 4*d*x] + 53760*d*x*Cos[5*c + 6*d*x] + 53760*d*x*Cos[7*c + 6*d*x] + 6720*d*x*Cos[7*c + 8*d*x] + 6720*d*x*Cos[9*c + 8*d*x] + 519680*Sin[c] + 133175*Sin[d*x] + 133175*Sin[2*c + d*x] - 544768*Sin[c + 2*d*x] + 286720*Sin[3*c + 2*d*x] + 63595*Sin[2*c + 3*d*x] + 63595*Sin[4*c + 3*d*x] - 254464*Sin[3*c + 4*d*x] + 161280*Sin[5*c + 4*d*x] + 65135*Sin[4*c + 5*d*x] + 65135*Sin[6*c + 5*d*x] - 118784*Sin[5*c + 6*d*x] + 27195*Sin[6*c + 7*d*x] + 27195*Sin[8*c + 7*d*x] - 14848*Sin[7*c + 8*d*x])))/(215040*a^3*d*(1 + Sec[c + d*x])^3)","A",1
96,1,303,169,0.9190421,"\int \frac{\tan ^{10}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^10/(a + a*Sec[c + d*x])^3,x]","-\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(9120 \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sec (c) \sec ^6(c+d x) (-210 \sin (2 c+d x)-1440 \sin (c+2 d x)+1200 \sin (3 c+2 d x)+865 \sin (2 c+3 d x)+865 \sin (4 c+3 d x)-1296 \sin (3 c+4 d x)-240 \sin (5 c+4 d x)+435 \sin (4 c+5 d x)+435 \sin (6 c+5 d x)-176 \sin (5 c+6 d x)+2400 d x \cos (c)+1800 d x \cos (c+2 d x)+1800 d x \cos (3 c+2 d x)+720 d x \cos (3 c+4 d x)+720 d x \cos (5 c+4 d x)+120 d x \cos (5 c+6 d x)+120 d x \cos (7 c+6 d x)+1760 \sin (c)-210 \sin (d x))\right)}{960 a^3 d (\sec (c+d x)+1)^3}","-\frac{3 \tan ^5(c+d x)}{5 a^3 d}-\frac{\tan ^3(c+d x)}{3 a^3 d}+\frac{\tan (c+d x)}{a^3 d}+\frac{19 \tanh ^{-1}(\sin (c+d x))}{16 a^3 d}+\frac{\tan ^3(c+d x) \sec ^3(c+d x)}{6 a^3 d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{8 a^3 d}+\frac{3 \tan ^3(c+d x) \sec (c+d x)}{4 a^3 d}-\frac{17 \tan (c+d x) \sec (c+d x)}{16 a^3 d}-\frac{x}{a^3}",1,"-1/960*(Cos[(c + d*x)/2]^6*Sec[c + d*x]^3*(9120*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*Sec[c + d*x]^6*(2400*d*x*Cos[c] + 1800*d*x*Cos[c + 2*d*x] + 1800*d*x*Cos[3*c + 2*d*x] + 720*d*x*Cos[3*c + 4*d*x] + 720*d*x*Cos[5*c + 4*d*x] + 120*d*x*Cos[5*c + 6*d*x] + 120*d*x*Cos[7*c + 6*d*x] + 1760*Sin[c] - 210*Sin[d*x] - 210*Sin[2*c + d*x] - 1440*Sin[c + 2*d*x] + 1200*Sin[3*c + 2*d*x] + 865*Sin[2*c + 3*d*x] + 865*Sin[4*c + 3*d*x] - 1296*Sin[3*c + 4*d*x] - 240*Sin[5*c + 4*d*x] + 435*Sin[4*c + 5*d*x] + 435*Sin[6*c + 5*d*x] - 176*Sin[5*c + 6*d*x])))/(a^3*d*(1 + Sec[c + d*x])^3)","A",1
97,1,230,99,0.7570976,"\int \frac{\tan ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^8/(a + a*Sec[c + d*x])^3,x]","\frac{\sec ^4(c+d x) \left(38 \sin (c+d x)-32 \sin (2 (c+d x))+22 \sin (3 (c+d x))+39 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \cos (2 (c+d x)) \left(13 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-13 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 d x\right)+\cos (4 (c+d x)) \left(13 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-13 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 d x\right)-39 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+24 d x\right)}{64 a^3 d}","-\frac{\tan ^3(c+d x)}{a^3 d}-\frac{\tan (c+d x)}{a^3 d}-\frac{13 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{4 a^3 d}+\frac{11 \tan (c+d x) \sec (c+d x)}{8 a^3 d}+\frac{x}{a^3}",1,"(Sec[c + d*x]^4*(24*d*x + 39*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*Cos[2*(c + d*x)]*(8*d*x + 13*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 13*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[4*(c + d*x)]*(8*d*x + 13*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 13*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 39*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 38*Sin[c + d*x] - 32*Sin[2*(c + d*x)] + 22*Sin[3*(c + d*x)]))/(64*a^3*d)","B",1
98,1,241,66,0.9615086,"\int \frac{\tan ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(-\frac{12 \sin (d x)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{14 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{14 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}-4 x\right)}{a^3 (\sec (c+d x)+1)^3}","-\frac{5 \tan (c+d x)}{2 a^3 d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{\tan (c+d x) (1-\sec (c+d x))}{2 a^3 d}-\frac{x}{a^3}",1,"(2*Cos[(c + d*x)/2]^6*Sec[c + d*x]^3*(-4*x - (14*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (14*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + 1/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - 1/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - (12*Sin[d*x])/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(a^3*(1 + Sec[c + d*x])^3)","B",1
99,1,117,46,0.267794,"\int \frac{\tan ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^4/(a + a*Sec[c + d*x])^3,x]","\frac{8 \cos ^5\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(\cos \left(\frac{1}{2} (c+d x)\right) \left(-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+d x\right)-4 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)\right)}{a^3 d (\sec (c+d x)+1)^3}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{x}{a^3}-\frac{4 \tan (c+d x)}{a^2 d (a \sec (c+d x)+a)}",1,"(8*Cos[(c + d*x)/2]^5*Sec[c + d*x]^3*(Cos[(c + d*x)/2]*(d*x - Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 4*Sec[c/2]*Sin[(d*x)/2]))/(a^3*d*(1 + Sec[c + d*x])^3)","B",1
100,1,125,60,0.4027923,"\int \frac{\tan ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^2/(a + a*Sec[c + d*x])^3,x]","-\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(351 \sin \left(c+\frac{d x}{2}\right)-277 \sin \left(c+\frac{3 d x}{2}\right)-3 \sin \left(2 c+\frac{3 d x}{2}\right)+180 d x \cos \left(c+\frac{d x}{2}\right)+60 d x \cos \left(c+\frac{3 d x}{2}\right)+60 d x \cos \left(2 c+\frac{3 d x}{2}\right)-471 \sin \left(\frac{d x}{2}\right)+180 d x \cos \left(\frac{d x}{2}\right)\right)}{480 a^3 d}","-\frac{x}{a^3}+\frac{2 \tan (c+d x)}{a^2 d (a \sec (c+d x)+a)}-\frac{\tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^3}",1,"-1/480*(Sec[c/2]*Sec[(c + d*x)/2]^3*(180*d*x*Cos[(d*x)/2] + 180*d*x*Cos[c + (d*x)/2] + 60*d*x*Cos[c + (3*d*x)/2] + 60*d*x*Cos[2*c + (3*d*x)/2] - 471*Sin[(d*x)/2] + 351*Sin[c + (d*x)/2] - 277*Sin[c + (3*d*x)/2] - 3*Sin[2*c + (3*d*x)/2]))/(a^3*d)","B",1
101,1,252,143,1.2334147,"\int \frac{\cot ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^2/(a + a*Sec[c + d*x])^3,x]","\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc \left(\frac{1}{2} (c+d x)\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) (-23282 \sin (c+d x)-23282 \sin (2 (c+d x))-9978 \sin (3 (c+d x))-1663 \sin (4 (c+d x))+13720 \sin (2 c+d x)+15512 \sin (c+2 d x)+9240 \sin (3 c+2 d x)+8088 \sin (2 c+3 d x)+2520 \sin (4 c+3 d x)+1768 \sin (3 c+4 d x)+5880 d x \cos (2 c+d x)-5880 d x \cos (c+2 d x)+5880 d x \cos (3 c+2 d x)-2520 d x \cos (2 c+3 d x)+2520 d x \cos (4 c+3 d x)-420 d x \cos (3 c+4 d x)+420 d x \cos (5 c+4 d x)+4200 \sin (c)+11032 \sin (d x)-5880 d x \cos (d x))}{215040 a^3 d}","\frac{4 \cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \csc ^7(c+d x)}{7 a^3 d}+\frac{11 \csc ^5(c+d x)}{5 a^3 d}-\frac{10 \csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{x}{a^3}",1,"(Csc[c/2]*Csc[(c + d*x)/2]*Sec[c/2]*Sec[(c + d*x)/2]^7*(-5880*d*x*Cos[d*x] + 5880*d*x*Cos[2*c + d*x] - 5880*d*x*Cos[c + 2*d*x] + 5880*d*x*Cos[3*c + 2*d*x] - 2520*d*x*Cos[2*c + 3*d*x] + 2520*d*x*Cos[4*c + 3*d*x] - 420*d*x*Cos[3*c + 4*d*x] + 420*d*x*Cos[5*c + 4*d*x] + 4200*Sin[c] + 11032*Sin[d*x] - 23282*Sin[c + d*x] - 23282*Sin[2*(c + d*x)] - 9978*Sin[3*(c + d*x)] - 1663*Sin[4*(c + d*x)] + 13720*Sin[2*c + d*x] + 15512*Sin[c + 2*d*x] + 9240*Sin[3*c + 2*d*x] + 8088*Sin[2*c + 3*d*x] + 2520*Sin[4*c + 3*d*x] + 1768*Sin[3*c + 4*d*x]))/(215040*a^3*d)","A",1
102,1,366,177,1.1844642,"\int \frac{\cot ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^4/(a + a*Sec[c + d*x])^3,x]","\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc ^3(2 (c+d x)) (675036 \sin (c+d x)+506277 \sin (2 (c+d x))-37502 \sin (3 (c+d x))-225012 \sin (4 (c+d x))-112506 \sin (5 (c+d x))-18751 \sin (6 (c+d x))-431424 \sin (2 c+d x)-375552 \sin (c+2 d x)-201600 \sin (3 c+2 d x)-41248 \sin (2 c+3 d x)+84000 \sin (4 c+3 d x)+155712 \sin (3 c+4 d x)+100800 \sin (5 c+4 d x)+98016 \sin (4 c+5 d x)+30240 \sin (6 c+5 d x)+21376 \sin (5 c+6 d x)-181440 d x \cos (2 c+d x)+136080 d x \cos (c+2 d x)-136080 d x \cos (3 c+2 d x)-10080 d x \cos (2 c+3 d x)+10080 d x \cos (4 c+3 d x)-60480 d x \cos (3 c+4 d x)+60480 d x \cos (5 c+4 d x)-30240 d x \cos (4 c+5 d x)+30240 d x \cos (6 c+5 d x)-5040 d x \cos (5 c+6 d x)+5040 d x \cos (7 c+6 d x)-169344 \sin (c)-338112 \sin (d x)+181440 d x \cos (d x))}{80640 a^3 d (\sec (c+d x)+1)^3}","\frac{4 \cot ^9(c+d x)}{9 a^3 d}-\frac{\cot ^7(c+d x)}{7 a^3 d}+\frac{\cot ^5(c+d x)}{5 a^3 d}-\frac{\cot ^3(c+d x)}{3 a^3 d}+\frac{\cot (c+d x)}{a^3 d}-\frac{4 \csc ^9(c+d x)}{9 a^3 d}+\frac{15 \csc ^7(c+d x)}{7 a^3 d}-\frac{21 \csc ^5(c+d x)}{5 a^3 d}+\frac{13 \csc ^3(c+d x)}{3 a^3 d}-\frac{3 \csc (c+d x)}{a^3 d}+\frac{x}{a^3}",1,"(Csc[c/2]*Csc[2*(c + d*x)]^3*Sec[c/2]*(181440*d*x*Cos[d*x] - 181440*d*x*Cos[2*c + d*x] + 136080*d*x*Cos[c + 2*d*x] - 136080*d*x*Cos[3*c + 2*d*x] - 10080*d*x*Cos[2*c + 3*d*x] + 10080*d*x*Cos[4*c + 3*d*x] - 60480*d*x*Cos[3*c + 4*d*x] + 60480*d*x*Cos[5*c + 4*d*x] - 30240*d*x*Cos[4*c + 5*d*x] + 30240*d*x*Cos[6*c + 5*d*x] - 5040*d*x*Cos[5*c + 6*d*x] + 5040*d*x*Cos[7*c + 6*d*x] - 169344*Sin[c] - 338112*Sin[d*x] + 675036*Sin[c + d*x] + 506277*Sin[2*(c + d*x)] - 37502*Sin[3*(c + d*x)] - 225012*Sin[4*(c + d*x)] - 112506*Sin[5*(c + d*x)] - 18751*Sin[6*(c + d*x)] - 431424*Sin[2*c + d*x] - 375552*Sin[c + 2*d*x] - 201600*Sin[3*c + 2*d*x] - 41248*Sin[2*c + 3*d*x] + 84000*Sin[4*c + 3*d*x] + 155712*Sin[3*c + 4*d*x] + 100800*Sin[5*c + 4*d*x] + 98016*Sin[4*c + 5*d*x] + 30240*Sin[6*c + 5*d*x] + 21376*Sin[5*c + 6*d*x]))/(80640*a^3*d*(1 + Sec[c + d*x])^3)","B",1
103,1,394,215,3.7250986,"\int \frac{\cot ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]","-\frac{\tan \left(\frac{c}{2}\right) \cos ^6\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(315 \sec ^{10}\left(\frac{1}{2} (c+d x)\right)-5425 \sec ^8\left(\frac{1}{2} (c+d x)\right)+41320 \sec ^6\left(\frac{1}{2} (c+d x)\right)-184650 \sec ^4\left(\frac{1}{2} (c+d x)\right)+561145 \sec ^2\left(\frac{1}{2} (c+d x)\right)+6468 \sin (c) \csc ^3\left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc ^3\left(\frac{1}{2} (c+d x)\right)+231 \cot ^2\left(\frac{c}{2}\right) (28 \cos (c+d x)-25) \csc ^4\left(\frac{1}{2} (c+d x)\right)+231 \cot \left(\frac{c}{2}\right) \left(3840 d x-\csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc \left(\frac{1}{2} (c+d x)\right) \left(3 \csc ^4\left(\frac{1}{2} (c+d x)\right)+743\right)\right)+315 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^{11}\left(\frac{1}{2} (c+d x)\right)-5425 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^9\left(\frac{1}{2} (c+d x)\right)+41320 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right)-184650 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)+561145 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)-1736335 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)\right)}{110880 a^3 d (\sec (c+d x)+1)^3}","\frac{4 \cot ^{11}(c+d x)}{11 a^3 d}-\frac{\cot ^9(c+d x)}{9 a^3 d}+\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac{19 \csc ^9(c+d x)}{9 a^3 d}-\frac{36 \csc ^7(c+d x)}{7 a^3 d}+\frac{34 \csc ^5(c+d x)}{5 a^3 d}-\frac{16 \csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{x}{a^3}",1,"-1/110880*(Cos[(c + d*x)/2]^6*Sec[c + d*x]^3*(231*(-25 + 28*Cos[c + d*x])*Cot[c/2]^2*Csc[(c + d*x)/2]^4 + 561145*Sec[(c + d*x)/2]^2 - 184650*Sec[(c + d*x)/2]^4 + 41320*Sec[(c + d*x)/2]^6 - 5425*Sec[(c + d*x)/2]^8 + 315*Sec[(c + d*x)/2]^10 - 1736335*Csc[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] + 561145*Csc[c/2]*Sec[(c + d*x)/2]^3*Sin[(d*x)/2] - 184650*Csc[c/2]*Sec[(c + d*x)/2]^5*Sin[(d*x)/2] + 41320*Csc[c/2]*Sec[(c + d*x)/2]^7*Sin[(d*x)/2] - 5425*Csc[c/2]*Sec[(c + d*x)/2]^9*Sin[(d*x)/2] + 315*Csc[c/2]*Sec[(c + d*x)/2]^11*Sin[(d*x)/2] + 6468*Csc[c/2]^3*Csc[(c + d*x)/2]^3*Sin[c]*Sin[(d*x)/2] + 231*Cot[c/2]*(3840*d*x - Csc[c/2]*Csc[(c + d*x)/2]*(743 + 3*Csc[(c + d*x)/2]^4)*Sin[(d*x)/2]))*Tan[c/2])/(a^3*d*(1 + Sec[c + d*x])^3)","A",1
104,1,186,310,2.3871458,"\int (a+a \sec (c+d x)) (e \tan (c+d x))^{5/2} \, dx","Integrate[(a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2),x]","\frac{a (\cos (c+d x)+1) \csc (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (e \tan (c+d x))^{5/2} \left(\frac{24 \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(c+d x)\right)}{\sqrt{\sec ^2(c+d x)}}-36 \cos ^2(c+d x)+20 \cos (c+d x)+15 \sqrt{\sin (2 (c+d x))} \cot ^2(c+d x) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+15 \sqrt{\sin (2 (c+d x))} \cot ^2(c+d x) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+12\right)}{60 d}","\frac{a e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}-\frac{a e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}-\frac{a e^{5/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{a e^{5/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{6 a e^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 d \sqrt{\sin (2 c+2 d x)}}-\frac{6 a e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac{2 e (3 a \sec (c+d x)+5 a) (e \tan (c+d x))^{3/2}}{15 d}",1,"(a*(1 + Cos[c + d*x])*Csc[c + d*x]*Sec[(c + d*x)/2]^2*(12 + 20*Cos[c + d*x] - 36*Cos[c + d*x]^2 + (24*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[c + d*x]^2])/Sqrt[Sec[c + d*x]^2] + 15*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Cot[c + d*x]^2*Sqrt[Sin[2*(c + d*x)]] + 15*Cot[c + d*x]^2*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]])*(e*Tan[c + d*x])^(5/2))/(60*d)","C",1
105,1,214,282,2.1073411,"\int (a+a \sec (c+d x)) (e \tan (c+d x))^{3/2} \, dx","Integrate[(a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2),x]","-\frac{a e \cos (2 (c+d x)) \csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec ^2(c+d x)} \sqrt{e \tan (c+d x)} \left(\sqrt{\sec ^2(c+d x)} \left(12 \sin (c+d x)+4 \tan (c+d x)+3 \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-3 \sqrt{\sin (2 (c+d x))} \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)+4 \sqrt[4]{-1} \sqrt{\tan (c+d x)} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (c+d x)}\right)\right|-1\right)\right)}{12 d \left(\tan ^2(c+d x)-1\right)}","\frac{a e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}-\frac{a e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}+\frac{a e^{3/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{a e^{3/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{a e^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d \sqrt{e \tan (c+d x)}}+\frac{2 e (a \sec (c+d x)+3 a) \sqrt{e \tan (c+d x)}}{3 d}",1,"-1/12*(a*e*Cos[2*(c + d*x)]*Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]^2]*Sqrt[e*Tan[c + d*x]]*(4*(-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[c + d*x]]], -1]*Sqrt[Tan[c + d*x]] + Sqrt[Sec[c + d*x]^2]*(12*Sin[c + d*x] + 3*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]] - 3*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] + 4*Tan[c + d*x])))/(d*(-1 + Tan[c + d*x]^2))","C",1
106,1,182,272,1.5705604,"\int (a+a \sec (c+d x)) \sqrt{e \tan (c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]],x]","-\frac{a (\cos (c+d x)+1) \csc (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{e \tan (c+d x)} \left(8 \tan ^2(c+d x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(c+d x)\right)+3 \sqrt{\sec ^2(c+d x)} \left(-4 \sin ^2(c+d x)+\sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+\sqrt{\sin (2 (c+d x))} \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{12 d \sqrt{\sec ^2(c+d x)}}","-\frac{a \sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}+\frac{a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}+\frac{a \sqrt{e} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{a \sqrt{e} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{2 a \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac{2 a \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{d \sqrt{\sin (2 c+2 d x)}}",1,"-1/12*(a*(1 + Cos[c + d*x])*Csc[c + d*x]*Sec[(c + d*x)/2]^2*Sqrt[e*Tan[c + d*x]]*(3*Sqrt[Sec[c + d*x]^2]*(-4*Sin[c + d*x]^2 + ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]] + Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]]) + 8*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^2))/(d*Sqrt[Sec[c + d*x]^2])","C",1
107,1,220,244,1.7642405,"\int \frac{a+a \sec (c+d x)}{\sqrt{e \tan (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])/Sqrt[e*Tan[c + d*x]],x]","\frac{20 a \sin (c+d x) \cos ^2\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{d \sqrt{e \tan (c+d x)} \left(2 (\cos (c+d x)-1) \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)+5 (\cos (c+d x)+1) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}","-\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \sqrt{e}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \sqrt{e}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{a \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{d \sqrt{e \tan (c+d x)}}",1,"(20*a*AppellF1[1/4, 1/2, 1, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*(1 + Sec[c + d*x])*Sin[c + d*x])/(d*(2*(2*AppellF1[5/4, 1/2, 2, 9/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[5/4, 3/2, 1, 9/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(-1 + Cos[c + d*x]) + 5*AppellF1[1/4, 1/2, 1, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x]))*Sqrt[e*Tan[c + d*x]])","C",0
108,1,196,305,2.663195,"\int \frac{a+a \sec (c+d x)}{(e \tan (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sec[c + d*x])/(e*Tan[c + d*x])^(3/2),x]","-\frac{a (\cos (c+d x)+1) \csc (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{e \tan (c+d x)} \left(8 \tan ^2(c+d x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(c+d x)\right)+3 \sqrt{\sec ^2(c+d x)} \left(4 \cos (c+d x)+2 \cos (2 (c+d x))-\sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-\sqrt{\sin (2 (c+d x))} \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+2\right)\right)}{12 d e^2 \sqrt{\sec ^2(c+d x)}}","\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{3/2}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{3/2}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}+\frac{2 a \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e^3}-\frac{2 a \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{d e^2 \sqrt{\sin (2 c+2 d x)}}-\frac{2 (a \sec (c+d x)+a)}{d e \sqrt{e \tan (c+d x)}}",1,"-1/12*(a*(1 + Cos[c + d*x])*Csc[c + d*x]*Sec[(c + d*x)/2]^2*Sqrt[e*Tan[c + d*x]]*(3*Sqrt[Sec[c + d*x]^2]*(2 + 4*Cos[c + d*x] + 2*Cos[2*(c + d*x)] - ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]] - Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]]) + 8*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^2))/(d*e^2*Sqrt[Sec[c + d*x]^2])","C",1
109,1,200,282,1.5262593,"\int \frac{a+a \sec (c+d x)}{(e \tan (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sec[c + d*x])/(e*Tan[c + d*x])^(5/2),x]","-\frac{a \csc (c+d x) \sqrt{e \tan (c+d x)} \left(\sqrt{\sec ^2(c+d x)} \left(2 \cot \left(\frac{1}{2} (c+d x)\right)-3 \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+2 \cos \left(\frac{3}{2} (c+d x)\right) \csc \left(\frac{1}{2} (c+d x)\right)+3 \sqrt{\sin (2 (c+d x))} \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)-4 \sqrt[4]{-1} \sqrt{\tan (c+d x)} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (c+d x)}\right)\right|-1\right)\right)}{6 d e^3 \sqrt{\sec ^2(c+d x)}}","\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{5/2}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{5/2}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}-\frac{a \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d e^2 \sqrt{e \tan (c+d x)}}-\frac{2 (a \sec (c+d x)+a)}{3 d e (e \tan (c+d x))^{3/2}}",1,"-1/6*(a*Csc[c + d*x]*(Sqrt[Sec[c + d*x]^2]*(2*Cot[(c + d*x)/2] + 2*Cos[(3*(c + d*x))/2]*Csc[(c + d*x)/2] - 3*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]] + 3*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]]) - 4*(-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[c + d*x]]], -1]*Sqrt[Tan[c + d*x]])*Sqrt[e*Tan[c + d*x]])/(d*e^3*Sqrt[Sec[c + d*x]^2])","C",1
110,1,254,346,2.5578227,"\int \frac{a+a \sec (c+d x)}{(e \tan (c+d x))^{7/2}} \, dx","Integrate[(a + a*Sec[c + d*x])/(e*Tan[c + d*x])^(7/2),x]","-\frac{a \csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1) \left(-8 \sin ^2(c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec ^2(c+d x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(c+d x)\right)-19 \sin (c+d x)+2 \cot \left(\frac{1}{2} (c+d x)\right)+12 \sin ^2(c+d x) \tan \left(\frac{1}{2} (c+d x)\right)+5 \sin (c+d x) \tan ^2\left(\frac{1}{2} (c+d x)\right)+5 \sqrt{\sin (2 (c+d x))} \tan \left(\frac{1}{2} (c+d x)\right) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+5 \sqrt{\sin (2 (c+d x))} \tan \left(\frac{1}{2} (c+d x)\right) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)}{20 d e^3 \sqrt{e \tan (c+d x)}}","-\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{7/2}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{7/2}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}-\frac{6 a \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d e^5}+\frac{6 a \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 d e^4 \sqrt{\sin (2 c+2 d x)}}+\frac{2 (3 a \sec (c+d x)+5 a)}{5 d e^3 \sqrt{e \tan (c+d x)}}-\frac{2 (a \sec (c+d x)+a)}{5 d e (e \tan (c+d x))^{5/2}}",1,"-1/20*(a*Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(1 + Sec[c + d*x])*(2*Cot[(c + d*x)/2] - 19*Sin[c + d*x] + 12*Sin[c + d*x]^2*Tan[(c + d*x)/2] - 8*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[c + d*x]^2]*Sqrt[Sec[c + d*x]^2]*Sin[c + d*x]^2*Tan[(c + d*x)/2] + 5*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]]*Tan[(c + d*x)/2] + 5*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]]*Tan[(c + d*x)/2] + 5*Sin[c + d*x]*Tan[(c + d*x)/2]^2))/(d*e^3*Sqrt[e*Tan[c + d*x]])","C",1
111,1,117,366,6.2123229,"\int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx","Integrate[(a + a*Sec[c + d*x])^2*(e*Tan[c + d*x])^(5/2),x]","\frac{2 a^2 e \cos ^4\left(\frac{1}{2} (c+d x)\right) (e \tan (c+d x))^{3/2} \sec ^4\left(\frac{1}{2} \tan ^{-1}(\tan (c+d x))\right) \left(-42 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2(c+d x)\right)-35 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)+15 \tan ^2(c+d x)+42 \sqrt{\sec ^2(c+d x)}+35\right)}{105 d}","\frac{a^2 e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}-\frac{a^2 e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}-\frac{a^2 e^{5/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{a^2 e^{5/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{12 a^2 e^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 d \sqrt{\sin (2 c+2 d x)}}+\frac{2 a^2 (e \tan (c+d x))^{7/2}}{7 d e}+\frac{2 a^2 e (e \tan (c+d x))^{3/2}}{3 d}-\frac{12 a^2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac{4 a^2 e \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 d}",1,"(2*a^2*e*Cos[(c + d*x)/2]^4*Sec[ArcTan[Tan[c + d*x]]/2]^4*(e*Tan[c + d*x])^(3/2)*(35 - 42*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] - 35*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2] + 42*Sqrt[Sec[c + d*x]^2] + 15*Tan[c + d*x]^2))/(105*d)","C",0
112,1,257,335,11.8870651,"\int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{3/2} \, dx","Integrate[(a + a*Sec[c + d*x])^2*(e*Tan[c + d*x])^(3/2),x]","\frac{a^2 \cos ^4\left(\frac{1}{2} (c+d x)\right) (e \tan (c+d x))^{3/2} \sec ^4\left(\frac{1}{2} \tan ^{-1}(\tan (c+d x))\right) \left(-80 \sqrt{\tan (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(c+d x)\right)+30 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-30 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+24 \tan ^{\frac{5}{2}}(c+d x)+120 \sqrt{\tan (c+d x)}+15 \sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-15 \sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+80 \sqrt{\tan (c+d x)} \sqrt{\sec ^2(c+d x)}\right)}{60 d \tan ^{\frac{3}{2}}(c+d x)}","\frac{a^2 e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}-\frac{a^2 e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}+\frac{a^2 e^{3/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{a^2 e^{3/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{2 a^2 e^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d \sqrt{e \tan (c+d x)}}+\frac{2 a^2 (e \tan (c+d x))^{5/2}}{5 d e}+\frac{2 a^2 e \sqrt{e \tan (c+d x)}}{d}+\frac{4 a^2 e \sec (c+d x) \sqrt{e \tan (c+d x)}}{3 d}",1,"(a^2*Cos[(c + d*x)/2]^4*Sec[ArcTan[Tan[c + d*x]]/2]^4*(e*Tan[c + d*x])^(3/2)*(30*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 30*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + 15*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 15*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 120*Sqrt[Tan[c + d*x]] - 80*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[c + d*x]^2]*Sqrt[Tan[c + d*x]] + 80*Sqrt[Sec[c + d*x]^2]*Sqrt[Tan[c + d*x]] + 24*Tan[c + d*x]^(5/2)))/(60*d*Tan[c + d*x]^(3/2))","C",0
113,1,106,309,1.2689778,"\int (a+a \sec (c+d x))^2 \sqrt{e \tan (c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]],x]","\frac{4 a^2 \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{e \tan (c+d x)} \left(2 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2(c+d x)\right)+\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)+1\right) \sec ^4\left(\frac{1}{2} \tan ^{-1}(\tan (c+d x))\right)}{3 d}","-\frac{a^2 \sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}+\frac{a^2 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}+\frac{2 a^2 (e \tan (c+d x))^{3/2}}{3 d e}+\frac{a^2 \sqrt{e} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{a^2 \sqrt{e} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{4 a^2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac{4 a^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{d \sqrt{\sin (2 c+2 d x)}}",1,"(4*a^2*Cos[(c + d*x)/2]^5*(1 + 2*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] + Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2])*Sec[c + d*x]*Sec[ArcTan[Tan[c + d*x]]/2]^4*Sin[(c + d*x)/2]*Sqrt[e*Tan[c + d*x]])/(3*d)","C",0
114,1,220,278,2.4667516,"\int \frac{(a+a \sec (c+d x))^2}{\sqrt{e \tan (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^2/Sqrt[e*Tan[c + d*x]],x]","\frac{a^2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\tan (c+d x)} \sec ^4\left(\frac{1}{2} \tan ^{-1}(\tan (c+d x))\right) \left(16 \sqrt{\tan (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(c+d x)\right)-2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+8 \sqrt{\tan (c+d x)}-\sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 d \sqrt{e \tan (c+d x)}}","-\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \sqrt{e}}+\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \sqrt{e}}+\frac{2 a^2 \sqrt{e \tan (c+d x)}}{d e}-\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{d \sqrt{e \tan (c+d x)}}",1,"(a^2*Cos[(c + d*x)/2]^4*Sec[ArcTan[Tan[c + d*x]]/2]^4*(-2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] - Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 8*Sqrt[Tan[c + d*x]] + 16*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[c + d*x]^2]*Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/(4*d*Sqrt[e*Tan[c + d*x]])","C",0
115,1,238,310,7.5913412,"\int \frac{(a+a \sec (c+d x))^2}{(e \tan (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sec[c + d*x])^2/(e*Tan[c + d*x])^(3/2),x]","\frac{a^2 \left(8 e^{3 i (c+d x)} \sqrt{1-e^{4 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{4 i (c+d x)}\right)-24 \left(e^{i (c+d x)}+e^{2 i (c+d x)}+e^{3 i (c+d x)}+1\right)-3 \sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (c+d x)}}\right)+6 \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right)}{6 d e \left(1+e^{2 i (c+d x)}\right) \sqrt{e \tan (c+d x)}}","\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{3/2}}-\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{3/2}}-\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}+\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}-\frac{4 a^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{d e^2 \sqrt{\sin (2 c+2 d x)}}-\frac{4 a^2}{d e \sqrt{e \tan (c+d x)}}-\frac{4 a^2 \cos (c+d x)}{d e \sqrt{e \tan (c+d x)}}",1,"(a^2*(-24*(1 + E^(I*(c + d*x)) + E^((2*I)*(c + d*x)) + E^((3*I)*(c + d*x))) - 3*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 6*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]] + 8*E^((3*I)*(c + d*x))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))]))/(6*d*e*(1 + E^((2*I)*(c + d*x)))*Sqrt[e*Tan[c + d*x]])","C",0
116,1,224,316,5.160212,"\int \frac{(a+a \sec (c+d x))^2}{(e \tan (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sec[c + d*x])^2/(e*Tan[c + d*x])^(5/2),x]","-\frac{a^2 \cos ^2\left(\frac{1}{2} (c+d x)\right) \cos (c+d x) \cot \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} \tan ^{-1}(\tan (c+d x))\right) \left(16 \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\tan ^2(c+d x)\right)+16 \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(c+d x)\right)+3 \sqrt{2} \tan ^{\frac{3}{2}}(c+d x) \left(2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)\right)}{24 d e^2 \sqrt{e \tan (c+d x)}}","\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{5/2}}-\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{5/2}}+\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}-\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}-\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d e^2 \sqrt{e \tan (c+d x)}}-\frac{4 a^2}{3 d e (e \tan (c+d x))^{3/2}}-\frac{4 a^2 \sec (c+d x)}{3 d e (e \tan (c+d x))^{3/2}}",1,"-1/24*(a^2*Cos[(c + d*x)/2]^2*Cos[c + d*x]*Cot[(c + d*x)/2]*Sec[ArcTan[Tan[c + d*x]]/2]^4*(16*Hypergeometric2F1[-3/4, 1/2, 1/4, -Tan[c + d*x]^2] + 16*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2] + 3*Sqrt[2]*(2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])*Tan[c + d*x]^(3/2)))/(d*e^2*Sqrt[e*Tan[c + d*x]])","C",0
117,1,2820,370,14.2764478,"\int \frac{(a+a \sec (c+d x))^2}{(e \tan (c+d x))^{7/2}} \, dx","Integrate[(a + a*Sec[c + d*x])^2/(e*Tan[c + d*x])^(7/2),x]","\text{Result too large to show}","-\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{7/2}}+\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{7/2}}+\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}-\frac{a^2 \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}+\frac{12 a^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 d e^4 \sqrt{\sin (2 c+2 d x)}}+\frac{2 a^2}{d e^3 \sqrt{e \tan (c+d x)}}+\frac{12 a^2 \cos (c+d x)}{5 d e^3 \sqrt{e \tan (c+d x)}}-\frac{4 a^2}{5 d e (e \tan (c+d x))^{5/2}}-\frac{4 a^2 \sec (c+d x)}{5 d e (e \tan (c+d x))^{5/2}}",1,"(Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*((7*Cot[c/2])/(10*d) - (Cot[c/2]*Csc[c/2 + (d*x)/2]^2)/(20*d) - (3*(4*Cos[c/2] - Cos[(3*c)/2] + Cos[(5*c)/2])*Cos[d*x]*Sec[2*c]*Sin[c/2])/(10*d*(-1 + 2*Cos[c])) - (7*Csc[c/2]*Csc[c/2 + (d*x)/2]*Sin[(d*x)/2])/(10*d) + (Csc[c/2]*Csc[c/2 + (d*x)/2]^3*Sin[(d*x)/2])/(20*d) - (3*(2 - 5*Cos[c] + 6*Cos[2*c] + Cos[3*c])*Sec[2*c]*Sin[d*x])/(20*d*(-1 + 2*Cos[c])))*Sin[c + d*x]^2*Tan[c + d*x]^2)/(e*Tan[c + d*x])^(7/2) + ((E^((2*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c + d*x]^2*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(16*d*E^(I*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) + ((-(E^((4*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]]) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c + d*x]^2*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(16*d*E^((2*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) - ((-(E^((6*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]]) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c + d*x]^2*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(16*d*E^((3*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) + ((Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*E^((2*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c + d*x]^2*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(16*d*E^(I*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) - ((Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*E^((4*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c + d*x]^2*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(16*d*E^((2*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) + ((Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*E^((6*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c + d*x]^2*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(16*d*E^((3*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) - (Cos[c + d*x]^2*(3 - 3*E^((4*I)*(c + d*x)) + E^((4*I)*(c + d*x))*(1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(20*d*E^(I*(2*c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) - (Cos[c + d*x]^2*(3 - 3*E^((4*I)*(c + d*x)) + E^((2*I)*(c + 2*d*x))*(1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(20*d*E^(I*d*x)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) + (E^(I*(c - d*x))*Cos[c + d*x]^2*(3 - 3*E^((4*I)*(c + d*x)) + E^((4*I)*d*x)*(1 + E^((4*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(8*d*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) - (Cos[c + d*x]^2*(3 - 3*E^((4*I)*(c + d*x)) + E^((4*I)*(c + d*x))*(1 + E^((4*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(40*d*E^(I*(3*c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) - (Cos[c + d*x]^2*(-3*E^((2*I)*c)*(-1 + E^((4*I)*(c + d*x))) + E^((4*I)*d*x)*(1 + E^((6*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(10*d*E^(I*d*x)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2))","C",0
118,1,332,330,21.9431388,"\int \frac{(e \tan (c+d x))^{11/2}}{a+a \sec (c+d x)} \, dx","Integrate[(e*Tan[c + d*x])^(11/2)/(a + a*Sec[c + d*x]),x]","\frac{e^5 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\sqrt{\sec ^2(c+d x)}+1\right) \sqrt{e \tan (c+d x)} \left(-320 \sqrt{\tan (c+d x)} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\tan ^2(c+d x)\right)+280 \sqrt{\tan (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(c+d x)\right)+70 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-70 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-56 \tan ^{\frac{5}{2}}(c+d x)+280 \sqrt{\tan (c+d x)}+35 \sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-35 \sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+40 \tan ^{\frac{5}{2}}(c+d x) \sqrt{\sec ^2(c+d x)}+40 \sqrt{\tan (c+d x)} \sqrt{\sec ^2(c+d x)}\right)}{70 a d \sqrt{\tan (c+d x)} (\sec (c+d x)+1)^2}","\frac{e^{11/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}-\frac{e^{11/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}+\frac{e^{11/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{e^{11/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{5 e^6 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{21 a d \sqrt{e \tan (c+d x)}}+\frac{2 e^5 (21-5 \sec (c+d x)) \sqrt{e \tan (c+d x)}}{21 a d}-\frac{2 e^3 (7-5 \sec (c+d x)) (e \tan (c+d x))^{5/2}}{35 a d}",1,"(e^5*Cos[(c + d*x)/2]^2*Sec[c + d*x]*(1 + Sqrt[Sec[c + d*x]^2])*Sqrt[e*Tan[c + d*x]]*(70*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 70*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + 35*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 35*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 280*Sqrt[Tan[c + d*x]] - 320*Hypergeometric2F1[-1/2, 1/4, 5/4, -Tan[c + d*x]^2]*Sqrt[Tan[c + d*x]] + 280*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[c + d*x]^2]*Sqrt[Tan[c + d*x]] + 40*Sqrt[Sec[c + d*x]^2]*Sqrt[Tan[c + d*x]] - 56*Tan[c + d*x]^(5/2) + 40*Sqrt[Sec[c + d*x]^2]*Tan[c + d*x]^(5/2)))/(70*a*d*(1 + Sec[c + d*x])^2*Sqrt[Tan[c + d*x]])","C",0
119,1,129,326,13.0121419,"\int \frac{(e \tan (c+d x))^{9/2}}{a+a \sec (c+d x)} \, dx","Integrate[(e*Tan[c + d*x])^(9/2)/(a + a*Sec[c + d*x]),x]","\frac{4 e^3 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\sqrt{\sec ^2(c+d x)}+1\right) (e \tan (c+d x))^{3/2} \left(\, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2(c+d x)\right)-\, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2(c+d x)\right)+\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)-1\right)}{3 a d (\sec (c+d x)+1)^2}","-\frac{e^{9/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}+\frac{e^{9/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}+\frac{e^{9/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{e^{9/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{6 e^4 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 a d \sqrt{\sin (2 c+2 d x)}}-\frac{6 e^3 \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 a d}-\frac{2 e^3 (5-3 \sec (c+d x)) (e \tan (c+d x))^{3/2}}{15 a d}",1,"(4*e^3*Cos[(c + d*x)/2]^2*(-1 + Hypergeometric2F1[-1/2, 3/4, 7/4, -Tan[c + d*x]^2] - Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] + Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2])*Sec[c + d*x]*(1 + Sqrt[Sec[c + d*x]^2])*(e*Tan[c + d*x])^(3/2))/(3*a*d*(1 + Sec[c + d*x])^2)","C",0
120,1,271,295,54.6009032,"\int \frac{(e \tan (c+d x))^{7/2}}{a+a \sec (c+d x)} \, dx","Integrate[(e*Tan[c + d*x])^(7/2)/(a + a*Sec[c + d*x]),x]","\frac{e^3 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\sqrt{\sec ^2(c+d x)}+1\right) \sqrt{e \tan (c+d x)} \left(8 \sqrt{\tan (c+d x)} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\tan ^2(c+d x)\right)-8 \sqrt{\tan (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(c+d x)\right)-2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-8 \sqrt{\tan (c+d x)}-\sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{2 a d \sqrt{\tan (c+d x)} (\sec (c+d x)+1)^2}","-\frac{e^{7/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}+\frac{e^{7/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}-\frac{e^{7/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{e^{7/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{e^4 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a d \sqrt{e \tan (c+d x)}}-\frac{2 e^3 (3-\sec (c+d x)) \sqrt{e \tan (c+d x)}}{3 a d}",1,"(e^3*Cos[(c + d*x)/2]^2*Sec[c + d*x]*(1 + Sqrt[Sec[c + d*x]^2])*(-2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] - Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 8*Sqrt[Tan[c + d*x]] + 8*Hypergeometric2F1[-1/2, 1/4, 5/4, -Tan[c + d*x]^2]*Sqrt[Tan[c + d*x]] - 8*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[c + d*x]^2]*Sqrt[Tan[c + d*x]])*Sqrt[e*Tan[c + d*x]])/(2*a*d*(1 + Sec[c + d*x])^2*Sqrt[Tan[c + d*x]])","C",0
121,1,105,285,5.508356,"\int \frac{(e \tan (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx","Integrate[(e*Tan[c + d*x])^(5/2)/(a + a*Sec[c + d*x]),x]","\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \csc (c+d x) \left(\sqrt{\sec ^2(c+d x)}+1\right) (e \tan (c+d x))^{5/2} \left(\, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2(c+d x)\right)-\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)\right)}{3 a d (\sec (c+d x)+1)^2}","\frac{e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}-\frac{e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}-\frac{e^{5/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{e^{5/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{2 e^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{a d \sqrt{\sin (2 c+2 d x)}}+\frac{2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{a d}",1,"(4*Cos[(c + d*x)/2]^2*Csc[c + d*x]*(Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] - Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2])*(1 + Sqrt[Sec[c + d*x]^2])*(e*Tan[c + d*x])^(5/2))/(3*a*d*(1 + Sec[c + d*x])^2)","C",0
122,1,1211,257,13.206305,"\int \frac{(e \tan (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx","Integrate[(e*Tan[c + d*x])^(3/2)/(a + a*Sec[c + d*x]),x]","-\frac{4 \sqrt[4]{-1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (c+d x)}\right)\right|-1\right) (e \tan (c+d x))^{3/2} \sec ^4(c+d x)}{d (\sec (c+d x) a+a) \tan ^{\frac{3}{2}}(c+d x) \left(\tan ^2(c+d x)+1\right)^{3/2}}-\frac{2 e^{-i (c+d x)} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(1+e^{2 i (c+d x)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec (2 c) (e \tan (c+d x))^{3/2} \sec (c+d x)}{d (\sec (c+d x) a+a) \tan ^{\frac{3}{2}}(c+d x)}-\frac{e^{-2 i c} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(e^{4 i c} \sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (c+d x)}}\right)+2 \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec (2 c) (e \tan (c+d x))^{3/2} \sec (c+d x)}{2 d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a) \tan ^{\frac{3}{2}}(c+d x)}-\frac{e^{-2 i c} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(\sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (c+d x)}}\right)+2 e^{4 i c} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec (2 c) (e \tan (c+d x))^{3/2} \sec (c+d x)}{2 d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a) \tan ^{\frac{3}{2}}(c+d x)}+\frac{e^{-i (2 c+d x)} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(3 \left(-1+e^{4 i (c+d x)}\right)+e^{4 i (c+d x)} \left(-1+e^{2 i c}\right) \sqrt{1-e^{4 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{4 i (c+d x)}\right)\right) \sec (2 c) (e \tan (c+d x))^{3/2} \sec (c+d x)}{3 d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a) \tan ^{\frac{3}{2}}(c+d x)}-\frac{e^{-i d x} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(e^{2 i (c+2 d x)} \left(-1+e^{2 i c}\right) \sqrt{1-e^{4 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{4 i (c+d x)}\right)-3 e^{4 i (c+d x)}+3\right) \sec (2 c) (e \tan (c+d x))^{3/2} \sec (c+d x)}{3 d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a) \tan ^{\frac{3}{2}}(c+d x)}+\frac{\cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \csc (c+d x) \left(\frac{8 \cos (c) \cos (d x) \sec (2 c) \sin ^2\left(\frac{c}{2}\right)}{d}-\frac{16 \cos \left(\frac{c}{2}\right) \sec (2 c) \sin ^3\left(\frac{c}{2}\right) \sin (d x)}{d}\right) (e \tan (c+d x))^{3/2}}{\sec (c+d x) a+a}","\frac{e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}-\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}+\frac{e^{3/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{e^{3/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{e^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a d \sqrt{e \tan (c+d x)}}",1,"(Cos[c/2 + (d*x)/2]^2*Csc[c + d*x]*((8*Cos[c]*Cos[d*x]*Sec[2*c]*Sin[c/2]^2)/d - (16*Cos[c/2]*Sec[2*c]*Sin[c/2]^3*Sin[d*x])/d)*(e*Tan[c + d*x])^(3/2))/(a + a*Sec[c + d*x]) - (2*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*(e*Tan[c + d*x])^(3/2))/(d*E^(I*(c + d*x))*(a + a*Sec[c + d*x])*Tan[c + d*x]^(3/2)) - (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(E^((4*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*(e*Tan[c + d*x])^(3/2))/(2*d*E^((2*I)*c)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])*Tan[c + d*x]^(3/2)) - (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 2*E^((4*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*(e*Tan[c + d*x])^(3/2))/(2*d*E^((2*I)*c)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])*Tan[c + d*x]^(3/2)) + (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*Cos[c/2 + (d*x)/2]^2*(3*(-1 + E^((4*I)*(c + d*x))) + E^((4*I)*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]*(e*Tan[c + d*x])^(3/2))/(3*d*E^(I*(2*c + d*x))*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])*Tan[c + d*x]^(3/2)) - (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*Cos[c/2 + (d*x)/2]^2*(3 - 3*E^((4*I)*(c + d*x)) + E^((2*I)*(c + 2*d*x))*(-1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]*(e*Tan[c + d*x])^(3/2))/(3*d*E^(I*d*x)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])*Tan[c + d*x]^(3/2)) - (4*(-1)^(1/4)*Cos[c/2 + (d*x)/2]^2*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[c + d*x]]], -1]*Sec[c + d*x]^4*(e*Tan[c + d*x])^(3/2))/(d*(a + a*Sec[c + d*x])*Tan[c + d*x]^(3/2)*(1 + Tan[c + d*x]^2)^(3/2))","C",0
123,1,2715,315,8.0890573,"\int \frac{\sqrt{e \tan (c+d x)}}{a+a \sec (c+d x)} \, dx","Integrate[Sqrt[e*Tan[c + d*x]]/(a + a*Sec[c + d*x]),x]","\text{Result too large to show}","-\frac{\sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}+\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}+\frac{2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{a d e}+\frac{2 e (1-\sec (c+d x))}{a d \sqrt{e \tan (c+d x)}}-\frac{2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{a d \sqrt{\sin (2 c+2 d x)}}",1,"(Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]*((-2*Cos[c/2]*Cos[d*x]*Sec[2*c]*(4*Sin[c/2] + Sin[(3*c)/2] + Sin[(5*c)/2]))/(d*(1 + 2*Cos[c])) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/d - ((-2 - 5*Cos[c] - 6*Cos[2*c] + Cos[3*c])*Sec[2*c]*Sin[d*x])/(d*(1 + 2*Cos[c])) - (4*Tan[c/2])/d)*Sqrt[e*Tan[c + d*x]])/(a + a*Sec[c + d*x]) + ((E^((2*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*Sqrt[e*Tan[c + d*x]])/(2*d*E^(I*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])*Sqrt[Tan[c + d*x]]) - ((-(E^((4*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]]) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*Sqrt[e*Tan[c + d*x]])/(2*d*E^((2*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])*Sqrt[Tan[c + d*x]]) - ((-(E^((6*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]]) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*Sqrt[e*Tan[c + d*x]])/(2*d*E^((3*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])*Sqrt[Tan[c + d*x]]) + ((Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*E^((2*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*Sqrt[e*Tan[c + d*x]])/(2*d*E^(I*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])*Sqrt[Tan[c + d*x]]) + ((Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*E^((4*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*Sqrt[e*Tan[c + d*x]])/(2*d*E^((2*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])*Sqrt[Tan[c + d*x]]) + ((Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*E^((6*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*Sqrt[e*Tan[c + d*x]])/(2*d*E^((3*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])*Sqrt[Tan[c + d*x]]) + (Cos[c/2 + (d*x)/2]^2*(3 - 3*E^((4*I)*(c + d*x)) + E^((4*I)*(c + d*x))*(1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]*Sqrt[e*Tan[c + d*x]])/(3*d*E^(I*(2*c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])*Sqrt[Tan[c + d*x]]) + (Cos[c/2 + (d*x)/2]^2*(3 - 3*E^((4*I)*(c + d*x)) + E^((2*I)*(c + 2*d*x))*(1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]*Sqrt[e*Tan[c + d*x]])/(3*d*E^(I*d*x)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])*Sqrt[Tan[c + d*x]]) + (5*E^(I*(c - d*x))*Cos[c/2 + (d*x)/2]^2*(3 - 3*E^((4*I)*(c + d*x)) + E^((4*I)*d*x)*(1 + E^((4*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]*Sqrt[e*Tan[c + d*x]])/(6*d*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])*Sqrt[Tan[c + d*x]]) - (Cos[c/2 + (d*x)/2]^2*(3 - 3*E^((4*I)*(c + d*x)) + E^((4*I)*(c + d*x))*(1 + E^((4*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]*Sqrt[e*Tan[c + d*x]])/(6*d*E^(I*(3*c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])*Sqrt[Tan[c + d*x]]) + (2*Cos[c/2 + (d*x)/2]^2*(-3*E^((2*I)*c)*(-1 + E^((4*I)*(c + d*x))) + E^((4*I)*d*x)*(1 + E^((6*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]*Sqrt[e*Tan[c + d*x]])/(3*d*E^(I*d*x)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])*Sqrt[Tan[c + d*x]])","C",0
124,1,1253,290,9.0675456,"\int \frac{1}{(a+a \sec (c+d x)) \sqrt{e \tan (c+d x)}} \, dx","Integrate[1/((a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]),x]","\frac{4 \sqrt[4]{-1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (c+d x)}\right)\right|-1\right) \sqrt{\tan (c+d x)} \sec ^4(c+d x)}{3 d (\sec (c+d x) a+a) \sqrt{e \tan (c+d x)} \left(\tan ^2(c+d x)+1\right)^{3/2}}+\frac{\cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{2 \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (-2 \cos (c)+3 \cos (2 c)+3) \cos (d x) \sec (2 c)}{3 d}-\frac{2 \sec (2 c) (3 \sin (2 c)-2 \sin (c)) \sin (d x)}{3 d}-\frac{4}{3 d}\right) \tan (c+d x) \sec (c+d x)}{(\sec (c+d x) a+a) \sqrt{e \tan (c+d x)}}+\frac{2 e^{-i (c+d x)} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(1+e^{2 i (c+d x)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec (2 c) \sqrt{\tan (c+d x)} \sec (c+d x)}{3 d (\sec (c+d x) a+a) \sqrt{e \tan (c+d x)}}+\frac{e^{-2 i c} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(e^{4 i c} \sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (c+d x)}}\right)+2 \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec (2 c) \sqrt{\tan (c+d x)} \sec (c+d x)}{2 d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a) \sqrt{e \tan (c+d x)}}+\frac{e^{-2 i c} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(\sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (c+d x)}}\right)+2 e^{4 i c} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec (2 c) \sqrt{\tan (c+d x)} \sec (c+d x)}{2 d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a) \sqrt{e \tan (c+d x)}}-\frac{e^{-i (2 c+d x)} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(3 \left(-1+e^{4 i (c+d x)}\right)+e^{4 i (c+d x)} \left(-1+e^{2 i c}\right) \sqrt{1-e^{4 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{4 i (c+d x)}\right)\right) \sec (2 c) \sqrt{\tan (c+d x)} \sec (c+d x)}{3 d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a) \sqrt{e \tan (c+d x)}}+\frac{e^{-i d x} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(e^{2 i (c+2 d x)} \left(-1+e^{2 i c}\right) \sqrt{1-e^{4 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{4 i (c+d x)}\right)-3 e^{4 i (c+d x)}+3\right) \sec (2 c) \sqrt{\tan (c+d x)} \sec (c+d x)}{3 d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a) \sqrt{e \tan (c+d x)}}","-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d \sqrt{e}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d \sqrt{e}}-\frac{\log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d \sqrt{e}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d \sqrt{e}}+\frac{2 e (1-\sec (c+d x))}{3 a d (e \tan (c+d x))^{3/2}}-\frac{\sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a d \sqrt{e \tan (c+d x)}}",1,"(2*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*Sqrt[Tan[c + d*x]])/(3*d*E^(I*(c + d*x))*(a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]) + (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(E^((4*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*Sqrt[Tan[c + d*x]])/(2*d*E^((2*I)*c)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]) + (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 2*E^((4*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*Sqrt[Tan[c + d*x]])/(2*d*E^((2*I)*c)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]) - (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*Cos[c/2 + (d*x)/2]^2*(3*(-1 + E^((4*I)*(c + d*x))) + E^((4*I)*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]*Sqrt[Tan[c + d*x]])/(3*d*E^(I*(2*c + d*x))*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]) + (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*Cos[c/2 + (d*x)/2]^2*(3 - 3*E^((4*I)*(c + d*x)) + E^((2*I)*(c + 2*d*x))*(-1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]*Sqrt[Tan[c + d*x]])/(3*d*E^(I*d*x)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]) + (Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]*(-4/(3*d) + (2*(3 - 2*Cos[c] + 3*Cos[2*c])*Cos[d*x]*Sec[2*c])/(3*d) + (2*Sec[c/2 + (d*x)/2]^2)/(3*d) - (2*Sec[2*c]*(-2*Sin[c] + 3*Sin[2*c])*Sin[d*x])/(3*d))*Tan[c + d*x])/((a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]) + (4*(-1)^(1/4)*Cos[c/2 + (d*x)/2]^2*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[c + d*x]]], -1]*Sec[c + d*x]^4*Sqrt[Tan[c + d*x]])/(3*d*(a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]*(1 + Tan[c + d*x]^2)^(3/2))","C",0
125,1,180,359,13.9663282,"\int \frac{1}{(a+a \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx","Integrate[1/((a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2)),x]","-\frac{4 \sin ^2\left(\frac{1}{2} (c+d x)\right) \csc (c+d x) \left(\sqrt{\sec ^2(c+d x)}+1\right) \sqrt{e \tan (c+d x)} \left(-5 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2(c+d x)\right)+5 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)+3 \cot ^4(c+d x) \, _2F_1\left(-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\tan ^2(c+d x)\right)-15 \cot ^2(c+d x) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\tan ^2(c+d x)\right)-3 \cot ^4(c+d x)+15 \cot ^2(c+d x)\right)}{15 a d e^2}","\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d e^{3/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d e^{3/2}}-\frac{\log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d e^{3/2}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d e^{3/2}}-\frac{6 \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 a d e^3}+\frac{6 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 a d e^2 \sqrt{\sin (2 c+2 d x)}}-\frac{2 (5-3 \sec (c+d x))}{5 a d e \sqrt{e \tan (c+d x)}}+\frac{2 e (1-\sec (c+d x))}{5 a d (e \tan (c+d x))^{5/2}}",1,"(-4*Csc[c + d*x]*(15*Cot[c + d*x]^2 - 3*Cot[c + d*x]^4 + 3*Cot[c + d*x]^4*Hypergeometric2F1[-5/4, -1/2, -1/4, -Tan[c + d*x]^2] - 15*Cot[c + d*x]^2*Hypergeometric2F1[-1/2, -1/4, 3/4, -Tan[c + d*x]^2] - 5*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] + 5*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2])*(1 + Sqrt[Sec[c + d*x]^2])*Sin[(c + d*x)/2]^2*Sqrt[e*Tan[c + d*x]])/(15*a*d*e^2)","C",0
126,1,1299,328,9.5737089,"\int \frac{1}{(a+a \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx","Integrate[1/((a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2)),x]","-\frac{20 \sqrt[4]{-1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (c+d x)}\right)\right|-1\right) \tan ^{\frac{5}{2}}(c+d x) \sec ^4(c+d x)}{21 d (\sec (c+d x) a+a) (e \tan (c+d x))^{5/2} \left(\tan ^2(c+d x)+1\right)^{3/2}}+\frac{\cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{\sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{14 d}-\frac{13 \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{14 d}-\frac{\csc ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d}-\frac{2 (-10 \cos (c)+21 \cos (2 c)+21) \cos (d x) \sec (2 c)}{21 d}+\frac{2 \sec (2 c) (21 \sin (2 c)-10 \sin (c)) \sin (d x)}{21 d}+\frac{40}{21 d}\right) \tan ^3(c+d x) \sec (c+d x)}{(\sec (c+d x) a+a) (e \tan (c+d x))^{5/2}}-\frac{10 e^{-i (c+d x)} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(1+e^{2 i (c+d x)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec (2 c) \tan ^{\frac{5}{2}}(c+d x) \sec (c+d x)}{21 d (\sec (c+d x) a+a) (e \tan (c+d x))^{5/2}}-\frac{e^{-2 i c} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(e^{4 i c} \sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (c+d x)}}\right)+2 \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec (2 c) \tan ^{\frac{5}{2}}(c+d x) \sec (c+d x)}{2 d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a) (e \tan (c+d x))^{5/2}}-\frac{e^{-2 i c} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(\sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (c+d x)}}\right)+2 e^{4 i c} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec (2 c) \tan ^{\frac{5}{2}}(c+d x) \sec (c+d x)}{2 d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a) (e \tan (c+d x))^{5/2}}+\frac{e^{-i (2 c+d x)} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(3 \left(-1+e^{4 i (c+d x)}\right)+e^{4 i (c+d x)} \left(-1+e^{2 i c}\right) \sqrt{1-e^{4 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{4 i (c+d x)}\right)\right) \sec (2 c) \tan ^{\frac{5}{2}}(c+d x) \sec (c+d x)}{3 d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a) (e \tan (c+d x))^{5/2}}-\frac{e^{-i d x} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(e^{2 i (c+2 d x)} \left(-1+e^{2 i c}\right) \sqrt{1-e^{4 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{4 i (c+d x)}\right)-3 e^{4 i (c+d x)}+3\right) \sec (2 c) \tan ^{\frac{5}{2}}(c+d x) \sec (c+d x)}{3 d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a) (e \tan (c+d x))^{5/2}}","\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d e^{5/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d e^{5/2}}+\frac{\log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d e^{5/2}}-\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d e^{5/2}}+\frac{5 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{21 a d e^2 \sqrt{e \tan (c+d x)}}-\frac{2 (7-5 \sec (c+d x))}{21 a d e (e \tan (c+d x))^{3/2}}+\frac{2 e (1-\sec (c+d x))}{7 a d (e \tan (c+d x))^{7/2}}",1,"(-10*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*Tan[c + d*x]^(5/2))/(21*d*E^(I*(c + d*x))*(a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2)) - (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(E^((4*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*Tan[c + d*x]^(5/2))/(2*d*E^((2*I)*c)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2)) - (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 2*E^((4*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*Tan[c + d*x]^(5/2))/(2*d*E^((2*I)*c)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2)) + (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*Cos[c/2 + (d*x)/2]^2*(3*(-1 + E^((4*I)*(c + d*x))) + E^((4*I)*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]*Tan[c + d*x]^(5/2))/(3*d*E^(I*(2*c + d*x))*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2)) - (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*Cos[c/2 + (d*x)/2]^2*(3 - 3*E^((4*I)*(c + d*x)) + E^((2*I)*(c + 2*d*x))*(-1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]*Tan[c + d*x]^(5/2))/(3*d*E^(I*d*x)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2)) + (Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]*(40/(21*d) - Csc[c/2 + (d*x)/2]^2/(6*d) - (2*(21 - 10*Cos[c] + 21*Cos[2*c])*Cos[d*x]*Sec[2*c])/(21*d) - (13*Sec[c/2 + (d*x)/2]^2)/(14*d) + Sec[c/2 + (d*x)/2]^4/(14*d) + (2*Sec[2*c]*(-10*Sin[c] + 21*Sin[2*c])*Sin[d*x])/(21*d))*Tan[c + d*x]^3)/((a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2)) - (20*(-1)^(1/4)*Cos[c/2 + (d*x)/2]^2*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[c + d*x]]], -1]*Sec[c + d*x]^4*Tan[c + d*x]^(5/2))/(21*d*(a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2)*(1 + Tan[c + d*x]^2)^(3/2))","C",0
127,0,0,372,13.8072543,"\int \frac{(e \tan (c+d x))^{13/2}}{(a+a \sec (c+d x))^2} \, dx","Integrate[(e*Tan[c + d*x])^(13/2)/(a + a*Sec[c + d*x])^2,x]","\int \frac{(e \tan (c+d x))^{13/2}}{(a+a \sec (c+d x))^2} \, dx","\frac{e^{13/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}-\frac{e^{13/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}-\frac{e^{13/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{e^{13/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{12 e^6 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 a^2 d \sqrt{\sin (2 c+2 d x)}}+\frac{2 e^5 (e \tan (c+d x))^{3/2}}{3 a^2 d}+\frac{12 e^5 \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 a^2 d}-\frac{4 e^5 \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 a^2 d}+\frac{2 e^3 (e \tan (c+d x))^{7/2}}{7 a^2 d}",1,"Integrate[(e*Tan[c + d*x])^(13/2)/(a + a*Sec[c + d*x])^2, x]","F",-1
128,0,0,339,74.2126664,"\int \frac{(e \tan (c+d x))^{11/2}}{(a+a \sec (c+d x))^2} \, dx","Integrate[(e*Tan[c + d*x])^(11/2)/(a + a*Sec[c + d*x])^2,x]","\int \frac{(e \tan (c+d x))^{11/2}}{(a+a \sec (c+d x))^2} \, dx","\frac{e^{11/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}-\frac{e^{11/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}+\frac{e^{11/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{e^{11/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{2 e^6 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a^2 d \sqrt{e \tan (c+d x)}}+\frac{2 e^5 \sqrt{e \tan (c+d x)}}{a^2 d}-\frac{4 e^5 \sec (c+d x) \sqrt{e \tan (c+d x)}}{3 a^2 d}+\frac{2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d}",1,"Integrate[(e*Tan[c + d*x])^(11/2)/(a + a*Sec[c + d*x])^2, x]","F",-1
129,0,0,312,4.6010896,"\int \frac{(e \tan (c+d x))^{9/2}}{(a+a \sec (c+d x))^2} \, dx","Integrate[(e*Tan[c + d*x])^(9/2)/(a + a*Sec[c + d*x])^2,x]","\int \frac{(e \tan (c+d x))^{9/2}}{(a+a \sec (c+d x))^2} \, dx","-\frac{e^{9/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}+\frac{e^{9/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}+\frac{e^{9/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{e^{9/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{4 e^4 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{a^2 d \sqrt{\sin (2 c+2 d x)}}+\frac{2 e^3 (e \tan (c+d x))^{3/2}}{3 a^2 d}-\frac{4 e^3 \cos (c+d x) (e \tan (c+d x))^{3/2}}{a^2 d}",1,"Integrate[(e*Tan[c + d*x])^(9/2)/(a + a*Sec[c + d*x])^2, x]","F",-1
130,0,0,281,4.1156516,"\int \frac{(e \tan (c+d x))^{7/2}}{(a+a \sec (c+d x))^2} \, dx","Integrate[(e*Tan[c + d*x])^(7/2)/(a + a*Sec[c + d*x])^2,x]","\int \frac{(e \tan (c+d x))^{7/2}}{(a+a \sec (c+d x))^2} \, dx","-\frac{e^{7/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}+\frac{e^{7/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}-\frac{e^{7/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{e^{7/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{2 e^4 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a^2 d \sqrt{e \tan (c+d x)}}+\frac{2 e^3 \sqrt{e \tan (c+d x)}}{a^2 d}",1,"Integrate[(e*Tan[c + d*x])^(7/2)/(a + a*Sec[c + d*x])^2, x]","F",-1
131,1,812,310,6.7098373,"\int \frac{(e \tan (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx","Integrate[(e*Tan[c + d*x])^(5/2)/(a + a*Sec[c + d*x])^2,x]","\frac{\csc ^2(c+d x) \left(\frac{32 \cos \left(\frac{c}{2}\right) \cos (d x) \sec (2 c) \sin \left(\frac{c}{2}\right)}{d}+\frac{16 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \sin \left(\frac{d x}{2}\right)}{d}-\frac{16 \cos (c) \sec (2 c) \sin (d x)}{d}+\frac{16 \tan \left(\frac{c}{2}\right)}{d}\right) (e \tan (c+d x))^{5/2} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\sec (c+d x) a+a)^2}+\frac{e^{-2 i c} \left(2 \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)-e^{4 i c} \sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (c+d x)}}\right)\right) \sec (2 c) \sec ^2(c+d x) (e \tan (c+d x))^{5/2} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a)^2 \tan ^{\frac{5}{2}}(c+d x)}-\frac{e^{-2 i c} \left(\sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (c+d x)}}\right)-2 e^{4 i c} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right) \sec (2 c) \sec ^2(c+d x) (e \tan (c+d x))^{5/2} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a)^2 \tan ^{\frac{5}{2}}(c+d x)}-\frac{8 e^{i (c-d x)} \left(e^{4 i d x} \left(1+e^{4 i c}\right) \sqrt{1-e^{4 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{4 i (c+d x)}\right)-3 e^{4 i (c+d x)}+3\right) \sec (2 c) \sec ^2(c+d x) (e \tan (c+d x))^{5/2} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a)^2 \tan ^{\frac{5}{2}}(c+d x)}","\frac{e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}-\frac{e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}-\frac{e^{5/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}+\frac{e^{5/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{4 e^3}{a^2 d \sqrt{e \tan (c+d x)}}+\frac{4 e^3 \cos (c+d x)}{a^2 d \sqrt{e \tan (c+d x)}}+\frac{4 e^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{a^2 d \sqrt{\sin (2 c+2 d x)}}",1,"(Cos[c/2 + (d*x)/2]^4*Csc[c + d*x]^2*((32*Cos[c/2]*Cos[d*x]*Sec[2*c]*Sin[c/2])/d + (16*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/d - (16*Cos[c]*Sec[2*c]*Sin[d*x])/d + (16*Tan[c/2])/d)*(e*Tan[c + d*x])^(5/2))/(a + a*Sec[c + d*x])^2 + ((-(E^((4*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]]) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c + d*x]^2*(e*Tan[c + d*x])^(5/2))/(d*E^((2*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(5/2)) - ((Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*E^((4*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c + d*x]^2*(e*Tan[c + d*x])^(5/2))/(d*E^((2*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(5/2)) - (8*E^(I*(c - d*x))*Cos[c/2 + (d*x)/2]^4*(3 - 3*E^((4*I)*(c + d*x)) + E^((4*I)*d*x)*(1 + E^((4*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]^2*(e*Tan[c + d*x])^(5/2))/(3*d*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(5/2))","C",0
132,0,0,316,16.5191267,"\int \frac{(e \tan (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx","Integrate[(e*Tan[c + d*x])^(3/2)/(a + a*Sec[c + d*x])^2,x]","\int \frac{(e \tan (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx","\frac{e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}-\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}+\frac{e^{3/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{e^{3/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{4 e^3}{3 a^2 d (e \tan (c+d x))^{3/2}}+\frac{4 e^3 \sec (c+d x)}{3 a^2 d (e \tan (c+d x))^{3/2}}+\frac{2 e^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a^2 d \sqrt{e \tan (c+d x)}}",1,"Integrate[(e*Tan[c + d*x])^(3/2)/(a + a*Sec[c + d*x])^2, x]","F",-1
133,1,2792,363,8.4503537,"\int \frac{\sqrt{e \tan (c+d x)}}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sqrt[e*Tan[c + d*x]]/(a + a*Sec[c + d*x])^2,x]","\text{Result too large to show}","-\frac{4 e^3}{5 a^2 d (e \tan (c+d x))^{5/2}}+\frac{4 e^3 \sec (c+d x)}{5 a^2 d (e \tan (c+d x))^{5/2}}-\frac{\sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d}+\frac{2 e}{a^2 d \sqrt{e \tan (c+d x)}}+\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d}-\frac{12 e \cos (c+d x)}{5 a^2 d \sqrt{e \tan (c+d x)}}-\frac{12 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{5 a^2 d \sqrt{\sin (2 c+2 d x)}}",1,"(Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*((-24*Cos[c/2]*Cos[d*x]*Sec[2*c]*(4*Sin[c/2] + Sin[(3*c)/2] + Sin[(5*c)/2]))/(5*d*(1 + 2*Cos[c])) - (56*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*Sin[(d*x)/2])/(5*d) - (12*(-2 - 5*Cos[c] - 6*Cos[2*c] + Cos[3*c])*Sec[2*c]*Sin[d*x])/(5*d*(1 + 2*Cos[c])) - (56*Tan[c/2])/(5*d) + (4*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(5*d))*Sqrt[e*Tan[c + d*x]])/(a + a*Sec[c + d*x])^2 + ((E^((2*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c + d*x]^2*Sqrt[e*Tan[c + d*x]])/(d*E^(I*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])^2*Sqrt[Tan[c + d*x]]) - ((-(E^((4*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]]) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c + d*x]^2*Sqrt[e*Tan[c + d*x]])/(d*E^((2*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])^2*Sqrt[Tan[c + d*x]]) - ((-(E^((6*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]]) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c + d*x]^2*Sqrt[e*Tan[c + d*x]])/(d*E^((3*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])^2*Sqrt[Tan[c + d*x]]) + ((Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*E^((2*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c + d*x]^2*Sqrt[e*Tan[c + d*x]])/(d*E^(I*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])^2*Sqrt[Tan[c + d*x]]) + ((Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*E^((4*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c + d*x]^2*Sqrt[e*Tan[c + d*x]])/(d*E^((2*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])^2*Sqrt[Tan[c + d*x]]) + ((Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*E^((6*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c + d*x]^2*Sqrt[e*Tan[c + d*x]])/(d*E^((3*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])^2*Sqrt[Tan[c + d*x]]) + (4*Cos[c/2 + (d*x)/2]^4*(3 - 3*E^((4*I)*(c + d*x)) + E^((4*I)*(c + d*x))*(1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]^2*Sqrt[e*Tan[c + d*x]])/(5*d*E^(I*(2*c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])^2*Sqrt[Tan[c + d*x]]) + (4*Cos[c/2 + (d*x)/2]^4*(3 - 3*E^((4*I)*(c + d*x)) + E^((2*I)*(c + 2*d*x))*(1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]^2*Sqrt[e*Tan[c + d*x]])/(5*d*E^(I*d*x)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])^2*Sqrt[Tan[c + d*x]]) + (2*E^(I*(c - d*x))*Cos[c/2 + (d*x)/2]^4*(3 - 3*E^((4*I)*(c + d*x)) + E^((4*I)*d*x)*(1 + E^((4*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]^2*Sqrt[e*Tan[c + d*x]])/(d*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])^2*Sqrt[Tan[c + d*x]]) - (2*Cos[c/2 + (d*x)/2]^4*(3 - 3*E^((4*I)*(c + d*x)) + E^((4*I)*(c + d*x))*(1 + E^((4*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]^2*Sqrt[e*Tan[c + d*x]])/(5*d*E^(I*(3*c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])^2*Sqrt[Tan[c + d*x]]) + (8*Cos[c/2 + (d*x)/2]^4*(-3*E^((2*I)*c)*(-1 + E^((4*I)*(c + d*x))) + E^((4*I)*d*x)*(1 + E^((6*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]^2*Sqrt[e*Tan[c + d*x]])/(5*d*E^(I*d*x)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])^2*Sqrt[Tan[c + d*x]])","C",0
134,1,1281,365,8.9140732,"\int \frac{1}{(a+a \sec (c+d x))^2 \sqrt{e \tan (c+d x)}} \, dx","Integrate[1/((a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]),x]","\frac{80 \sqrt[4]{-1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (c+d x)}\right)\right|-1\right) \sqrt{\tan (c+d x)} \sec ^5(c+d x)}{21 d (\sec (c+d x) a+a)^2 \sqrt{e \tan (c+d x)} \left(\tan ^2(c+d x)+1\right)^{3/2}}+\frac{\cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \left(-\frac{2 \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d}+\frac{64 \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d}+\frac{4 (-20 \cos (c)+21 \cos (2 c)+21) \cos (d x) \sec (2 c)}{21 d}-\frac{4 \sec (2 c) (21 \sin (2 c)-20 \sin (c)) \sin (d x)}{21 d}-\frac{104}{21 d}\right) \tan (c+d x) \sec ^2(c+d x)}{(\sec (c+d x) a+a)^2 \sqrt{e \tan (c+d x)}}+\frac{40 e^{-i (c+d x)} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(1+e^{2 i (c+d x)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec (2 c) \sqrt{\tan (c+d x)} \sec ^2(c+d x)}{21 d (\sec (c+d x) a+a)^2 \sqrt{e \tan (c+d x)}}+\frac{e^{-2 i c} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(e^{4 i c} \sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (c+d x)}}\right)+2 \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec (2 c) \sqrt{\tan (c+d x)} \sec ^2(c+d x)}{d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a)^2 \sqrt{e \tan (c+d x)}}+\frac{e^{-2 i c} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(\sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (c+d x)}}\right)+2 e^{4 i c} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec (2 c) \sqrt{\tan (c+d x)} \sec ^2(c+d x)}{d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a)^2 \sqrt{e \tan (c+d x)}}-\frac{2 e^{-i (2 c+d x)} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \left(3 \left(-1+e^{4 i (c+d x)}\right)+e^{4 i (c+d x)} \left(-1+e^{2 i c}\right) \sqrt{1-e^{4 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{4 i (c+d x)}\right)\right) \sec (2 c) \sqrt{\tan (c+d x)} \sec ^2(c+d x)}{3 d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a)^2 \sqrt{e \tan (c+d x)}}+\frac{2 e^{-i d x} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \left(e^{2 i (c+2 d x)} \left(-1+e^{2 i c}\right) \sqrt{1-e^{4 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{4 i (c+d x)}\right)-3 e^{4 i (c+d x)}+3\right) \sec (2 c) \sqrt{\tan (c+d x)} \sec ^2(c+d x)}{3 d \left(-1+e^{2 i (c+d x)}\right) (\sec (c+d x) a+a)^2 \sqrt{e \tan (c+d x)}}","-\frac{4 e^3}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac{4 e^3 \sec (c+d x)}{7 a^2 d (e \tan (c+d x))^{7/2}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a^2 d \sqrt{e}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a^2 d \sqrt{e}}+\frac{2 e}{3 a^2 d (e \tan (c+d x))^{3/2}}-\frac{\log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d \sqrt{e}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a^2 d \sqrt{e}}-\frac{20 e \sec (c+d x)}{21 a^2 d (e \tan (c+d x))^{3/2}}-\frac{10 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{21 a^2 d \sqrt{e \tan (c+d x)}}",1,"(40*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*Cos[c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c + d*x]^2*Sqrt[Tan[c + d*x]])/(21*d*E^(I*(c + d*x))*(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]) + (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(E^((4*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c + d*x]^2*Sqrt[Tan[c + d*x]])/(d*E^((2*I)*c)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]) + (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 2*E^((4*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c + d*x]^2*Sqrt[Tan[c + d*x]])/(d*E^((2*I)*c)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]) - (2*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*Cos[c/2 + (d*x)/2]^4*(3*(-1 + E^((4*I)*(c + d*x))) + E^((4*I)*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]^2*Sqrt[Tan[c + d*x]])/(3*d*E^(I*(2*c + d*x))*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]) + (2*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*Cos[c/2 + (d*x)/2]^4*(3 - 3*E^((4*I)*(c + d*x)) + E^((2*I)*(c + 2*d*x))*(-1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]^2*Sqrt[Tan[c + d*x]])/(3*d*E^(I*d*x)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]) + (Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*(-104/(21*d) + (4*(21 - 20*Cos[c] + 21*Cos[2*c])*Cos[d*x]*Sec[2*c])/(21*d) + (64*Sec[c/2 + (d*x)/2]^2)/(21*d) - (2*Sec[c/2 + (d*x)/2]^4)/(7*d) - (4*Sec[2*c]*(-20*Sin[c] + 21*Sin[2*c])*Sin[d*x])/(21*d))*Tan[c + d*x])/((a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]) + (80*(-1)^(1/4)*Cos[c/2 + (d*x)/2]^4*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[c + d*x]]], -1]*Sec[c + d*x]^5*Sqrt[Tan[c + d*x]])/(21*d*(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]*(1 + Tan[c + d*x]^2)^(3/2))","C",0
135,1,102,147,0.6480706,"\int \sqrt{a+a \sec (c+d x)} \tan ^5(c+d x) \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^5,x]","\frac{2 \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{\sec (c+d x)+1} \left(35 \sec ^4(c+d x)+5 \sec ^3(c+d x)-132 \sec ^2(c+d x)-34 \sec (c+d x)+383\right)-315 \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)\right)}{315 d \sqrt{\sec (c+d x)+1}}","\frac{2 (a \sec (c+d x)+a)^{9/2}}{9 a^4 d}-\frac{6 (a \sec (c+d x)+a)^{7/2}}{7 a^3 d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a^2 d}+\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a d}+\frac{2 \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}",1,"(2*Sqrt[a*(1 + Sec[c + d*x])]*(-315*ArcTanh[Sqrt[1 + Sec[c + d*x]]] + Sqrt[1 + Sec[c + d*x]]*(383 - 34*Sec[c + d*x] - 132*Sec[c + d*x]^2 + 5*Sec[c + d*x]^3 + 35*Sec[c + d*x]^4)))/(315*d*Sqrt[1 + Sec[c + d*x]])","A",1
136,1,80,99,0.167993,"\int \sqrt{a+a \sec (c+d x)} \tan ^3(c+d x) \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^3,x]","\frac{2 \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{\sec (c+d x)+1} \left(3 \sec ^2(c+d x)+\sec (c+d x)-17\right)+15 \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)\right)}{15 d \sqrt{\sec (c+d x)+1}}","\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a^2 d}-\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a d}-\frac{2 \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}",1,"(2*Sqrt[a*(1 + Sec[c + d*x])]*(15*ArcTanh[Sqrt[1 + Sec[c + d*x]]] + Sqrt[1 + Sec[c + d*x]]*(-17 + Sec[c + d*x] + 3*Sec[c + d*x]^2)))/(15*d*Sqrt[1 + Sec[c + d*x]])","A",1
137,1,60,51,0.0478502,"\int \sqrt{a+a \sec (c+d x)} \tan (c+d x) \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x],x]","\frac{\sqrt{a (\sec (c+d x)+1)} \left(2 \sqrt{\sec (c+d x)+1}-2 \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)\right)}{d \sqrt{\sec (c+d x)+1}}","\frac{2 \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}",1,"(Sqrt[a*(1 + Sec[c + d*x])]*(-2*ArcTanh[Sqrt[1 + Sec[c + d*x]]] + 2*Sqrt[1 + Sec[c + d*x]]))/(d*Sqrt[1 + Sec[c + d*x]])","A",1
138,1,72,73,0.0561321,"\int \cot (c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sqrt{a (\sec (c+d x)+1)} \left(2 \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{\sec (c+d x)+1}}{\sqrt{2}}\right)\right)}{d \sqrt{\sec (c+d x)+1}}","\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{\sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}",1,"((2*ArcTanh[Sqrt[1 + Sec[c + d*x]]] - Sqrt[2]*ArcTanh[Sqrt[1 + Sec[c + d*x]]/Sqrt[2]])*Sqrt[a*(1 + Sec[c + d*x])])/(d*Sqrt[1 + Sec[c + d*x]])","A",1
139,1,87,131,0.2967352,"\int \cot ^3(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]],x]","\frac{\cot ^2(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(-7 (\sec (c+d x)-1) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{1}{2} (\sec (c+d x)+1)\right)+8 (\sec (c+d x)-1) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\sec (c+d x)+1\right)-2\right)}{4 d}","\frac{a}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a}{2 d (1-\sec (c+d x)) \sqrt{a \sec (c+d x)+a}}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{7 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} d}",1,"(Cot[c + d*x]^2*(-2 - 7*Hypergeometric2F1[-1/2, 1, 1/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x]) + 8*Hypergeometric2F1[-1/2, 1, 1/2, 1 + Sec[c + d*x]]*(-1 + Sec[c + d*x]))*Sqrt[a*(1 + Sec[c + d*x])])/(4*d)","C",1
140,1,102,193,0.3281297,"\int \cot ^5(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^5*Sqrt[a + a*Sec[c + d*x]],x]","\frac{\cot ^4(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(107 (\sec (c+d x)-1)^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{1}{2} (\sec (c+d x)+1)\right)-2 \left(32 (\sec (c+d x)-1)^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\sec (c+d x)+1\right)-45 \sec (c+d x)+57\right)\right)}{96 d}","\frac{43 a^2}{96 d (a \sec (c+d x)+a)^{3/2}}-\frac{15 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}-\frac{21 a}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{107 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{64 \sqrt{2} d}",1,"(Cot[c + d*x]^4*(-2*(57 + 32*Hypergeometric2F1[-3/2, 1, -1/2, 1 + Sec[c + d*x]]*(-1 + Sec[c + d*x])^2 - 45*Sec[c + d*x]) + 107*Hypergeometric2F1[-3/2, 1, -1/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x])^2)*Sqrt[a*(1 + Sec[c + d*x])])/(96*d)","C",1
141,1,134,222,7.2820068,"\int \sqrt{a+a \sec (c+d x)} \tan ^6(c+d x) \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^6,x]","-\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(792 \sin \left(\frac{1}{2} (c+d x)\right)-1386 \sin \left(\frac{3}{2} (c+d x)\right)+495 \sin \left(\frac{5}{2} (c+d x)\right)-616 \sin \left(\frac{7}{2} (c+d x)\right)-247 \sin \left(\frac{11}{2} (c+d x)\right)+3960 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{11}{2}}(c+d x)\right)}{3960 d}","\frac{2 a^6 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{10 a^5 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{2 a^4 \tan ^7(c+d x)}{d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^3 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a^2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"-1/3960*(Sec[(c + d*x)/2]*Sec[c + d*x]^5*Sqrt[a*(1 + Sec[c + d*x])]*(3960*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(11/2) + 792*Sin[(c + d*x)/2] - 1386*Sin[(3*(c + d*x))/2] + 495*Sin[(5*(c + d*x))/2] - 616*Sin[(7*(c + d*x))/2] - 247*Sin[(11*(c + d*x))/2]))/d","A",1
142,1,110,160,5.9134212,"\int \sqrt{a+a \sec (c+d x)} \tan ^4(c+d x) \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^4,x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(35 \sin \left(\frac{1}{2} (c+d x)\right)-28 \sin \left(\frac{3}{2} (c+d x)\right)-23 \sin \left(\frac{7}{2} (c+d x)\right)+105 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{7}{2}}(c+d x)\right)}{105 d}","\frac{2 a^4 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{6 a^3 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a^2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(Sec[(c + d*x)/2]*Sec[c + d*x]^3*Sqrt[a*(1 + Sec[c + d*x])]*(105*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(7/2) + 35*Sin[(c + d*x)/2] - 28*Sin[(3*(c + d*x))/2] - 23*Sin[(7*(c + d*x))/2]))/(105*d)","A",1
143,1,226,96,4.1538501,"\int \sqrt{a+a \sec (c+d x)} \tan ^2(c+d x) \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^2,x]","\frac{8 \sqrt{2} \tan ^3(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{7/2} \sqrt{a (\sec (c+d x)+1)} \left(-\frac{4}{7} \tan ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-2 \sec (c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)-\frac{\cos (c+d x) (3 \cos (c+d x)+7) \csc ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left((4 \cos (c+d x)-1) \sqrt{1-\sec (c+d x)}-3 \cos (c+d x) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{24 \sqrt{1-\sec (c+d x)}}\right)}{3 d \left(1-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)^{5/2}}","\frac{2 a^2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(8*Sqrt[2]*((1 + Sec[c + d*x])^(-1))^(7/2)*Sqrt[a*(1 + Sec[c + d*x])]*(-1/24*(Cos[c + d*x]*(7 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^4*Sec[(c + d*x)/2]^2*(-3*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*Cos[c + d*x] + (-1 + 4*Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]]))/Sqrt[1 - Sec[c + d*x]] - (4*Hypergeometric2F1[2, 7/2, 9/2, -2*Sec[c + d*x]*Sin[(c + d*x)/2]^2]*Sec[c + d*x]*Tan[(c + d*x)/2]^2)/7)*Tan[c + d*x]^3)/(3*d*(1 - Tan[(c + d*x)/2]^2)^(5/2))","C",0
144,1,5502,109,24.1894512,"\int \cot ^2(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]],x]","\text{Result too large to show}","-\frac{\cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{2} d}",1,"Result too large to show","C",0
145,1,5552,196,24.0076696,"\int \cot ^4(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 a d}+\frac{7 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{8 d}+\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{9 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{8 \sqrt{2} d}-\frac{\cos (c+d x) \cot ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{4 a d}",1,"Result too large to show","C",0
146,1,5594,280,24.3704935,"\int \cot ^6(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^6*Sqrt[a + a*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{87 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{160 a^2 d}-\frac{\cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{16 a^2 d}-\frac{17 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{32 a^2 d}-\frac{23 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{192 a d}-\frac{105 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{128 d}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{151 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{128 \sqrt{2} d}",1,"Result too large to show","C",0
147,1,112,169,0.5531295,"\int (a+a \sec (c+d x))^{3/2} \tan ^5(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^5,x]","\frac{2 (a (\sec (c+d x)+1))^{3/2} \left(\sqrt{\sec (c+d x)+1} \left(105 \sec ^5(c+d x)+140 \sec ^4(c+d x)-325 \sec ^3(c+d x)-534 \sec ^2(c+d x)+327 \sec (c+d x)+1656\right)-1155 \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)\right)}{1155 d (\sec (c+d x)+1)^{3/2}}","-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{2 (a \sec (c+d x)+a)^{11/2}}{11 a^4 d}-\frac{2 (a \sec (c+d x)+a)^{9/2}}{3 a^3 d}+\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a^2 d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a d}+\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac{2 a \sqrt{a \sec (c+d x)+a}}{d}",1,"(2*(a*(1 + Sec[c + d*x]))^(3/2)*(-1155*ArcTanh[Sqrt[1 + Sec[c + d*x]]] + Sqrt[1 + Sec[c + d*x]]*(1656 + 327*Sec[c + d*x] - 534*Sec[c + d*x]^2 - 325*Sec[c + d*x]^3 + 140*Sec[c + d*x]^4 + 105*Sec[c + d*x]^5)))/(1155*d*(1 + Sec[c + d*x])^(3/2))","A",1
148,1,92,121,0.2700826,"\int (a+a \sec (c+d x))^{3/2} \tan ^3(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^3,x]","\frac{2 (a (\sec (c+d x)+1))^{3/2} \left(\sqrt{\sec (c+d x)+1} \left(15 \sec ^3(c+d x)+24 \sec ^2(c+d x)-32 \sec (c+d x)-146\right)+105 \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)\right)}{105 d (\sec (c+d x)+1)^{3/2}}","\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a^2 d}-\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a d}-\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 d}-\frac{2 a \sqrt{a \sec (c+d x)+a}}{d}",1,"(2*(a*(1 + Sec[c + d*x]))^(3/2)*(105*ArcTanh[Sqrt[1 + Sec[c + d*x]]] + Sqrt[1 + Sec[c + d*x]]*(-146 - 32*Sec[c + d*x] + 24*Sec[c + d*x]^2 + 15*Sec[c + d*x]^3)))/(105*d*(1 + Sec[c + d*x])^(3/2))","A",1
149,1,70,73,0.1496415,"\int (a+a \sec (c+d x))^{3/2} \tan (c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x],x]","\frac{2 (a (\sec (c+d x)+1))^{3/2} \left(\sqrt{\sec (c+d x)+1} (\sec (c+d x)+4)-3 \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)\right)}{3 d (\sec (c+d x)+1)^{3/2}}","-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{2 a \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(2*(a*(1 + Sec[c + d*x]))^(3/2)*(-3*ArcTanh[Sqrt[1 + Sec[c + d*x]]] + Sqrt[1 + Sec[c + d*x]]*(4 + Sec[c + d*x])))/(3*d*(1 + Sec[c + d*x])^(3/2))","A",1
150,1,72,73,0.0643491,"\int \cot (c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]*(a + a*Sec[c + d*x])^(3/2),x]","\frac{(a (\sec (c+d x)+1))^{3/2} \left(2 \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{\sec (c+d x)+1}}{\sqrt{2}}\right)\right)}{d (\sec (c+d x)+1)^{3/2}}","\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{2 \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}",1,"((2*ArcTanh[Sqrt[1 + Sec[c + d*x]]] - 2*Sqrt[2]*ArcTanh[Sqrt[1 + Sec[c + d*x]]/Sqrt[2]])*(a*(1 + Sec[c + d*x]))^(3/2))/(d*(1 + Sec[c + d*x])^(3/2))","A",1
151,1,99,109,0.3622811,"\int \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2),x]","\frac{(a (\sec (c+d x)+1))^{3/2} \left(-\frac{2 \sqrt{\sec (c+d x)+1}}{\sec (c+d x)-1}-8 \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)+5 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{\sec (c+d x)+1}}{\sqrt{2}}\right)\right)}{4 d (\sec (c+d x)+1)^{3/2}}","-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{5 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} d}+\frac{a \sqrt{a \sec (c+d x)+a}}{2 d (1-\sec (c+d x))}",1,"((a*(1 + Sec[c + d*x]))^(3/2)*(-8*ArcTanh[Sqrt[1 + Sec[c + d*x]]] + 5*Sqrt[2]*ArcTanh[Sqrt[1 + Sec[c + d*x]]/Sqrt[2]] - (2*Sqrt[1 + Sec[c + d*x]])/(-1 + Sec[c + d*x])))/(4*d*(1 + Sec[c + d*x])^(3/2))","A",1
152,1,104,171,0.3188551,"\int \cot ^5(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2),x]","\frac{a^2 \left(71 (\sec (c+d x)-1)^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{1}{2} (\sec (c+d x)+1)\right)-64 (\sec (c+d x)-1)^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\sec (c+d x)+1\right)+26 \sec (c+d x)-34\right)}{32 d (\sec (c+d x)-1)^2 \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{71 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{32 \sqrt{2} d}+\frac{7 a^2}{32 d \sqrt{a \sec (c+d x)+a}}-\frac{13 a^2}{16 d (1-\sec (c+d x)) \sqrt{a \sec (c+d x)+a}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 \sqrt{a \sec (c+d x)+a}}",1,"(a^2*(-34 + 71*Hypergeometric2F1[-1/2, 1, 1/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x])^2 - 64*Hypergeometric2F1[-1/2, 1, 1/2, 1 + Sec[c + d*x]]*(-1 + Sec[c + d*x])^2 + 26*Sec[c + d*x]))/(32*d*(-1 + Sec[c + d*x])^2*Sqrt[a*(1 + Sec[c + d*x])])","C",1
153,1,147,258,8.6220629,"\int (a+a \sec (c+d x))^{3/2} \tan ^6(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^6,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(164736 \sin \left(\frac{1}{2} (c+d x)\right)+81081 \sin \left(\frac{3}{2} (c+d x)\right)+134849 \sin \left(\frac{5}{2} (c+d x)\right)+98176 \sin \left(\frac{9}{2} (c+d x)\right)+45045 \sin \left(\frac{11}{2} (c+d x)\right)+32429 \sin \left(\frac{13}{2} (c+d x)\right)-1441440 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{13}{2}}(c+d x)\right)}{1441440 d}","-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^8 \tan ^{13}(c+d x)}{13 d (a \sec (c+d x)+a)^{13/2}}+\frac{14 a^7 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{34 a^6 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{30 a^5 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^4 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a^3 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 a^2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(a*Sec[(c + d*x)/2]*Sec[c + d*x]^6*Sqrt[a*(1 + Sec[c + d*x])]*(-1441440*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(13/2) + 164736*Sin[(c + d*x)/2] + 81081*Sin[(3*(c + d*x))/2] + 134849*Sin[(5*(c + d*x))/2] + 98176*Sin[(9*(c + d*x))/2] + 45045*Sin[(11*(c + d*x))/2] + 32429*Sin[(13*(c + d*x))/2]))/(1441440*d)","A",1
154,1,123,194,6.5021481,"\int (a+a \sec (c+d x))^{3/2} \tan ^4(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^4,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(126 \sin \left(\frac{1}{2} (c+d x)\right)-288 \sin \left(\frac{5}{2} (c+d x)\right)-315 \sin \left(\frac{7}{2} (c+d x)\right)-169 \sin \left(\frac{9}{2} (c+d x)\right)+2520 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{9}{2}}(c+d x)\right)}{2520 d}","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^6 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{10 a^5 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{14 a^4 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a^3 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 a^2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(a*Sec[(c + d*x)/2]*Sec[c + d*x]^4*Sqrt[a*(1 + Sec[c + d*x])]*(2520*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(9/2) + 126*Sin[(c + d*x)/2] - 288*Sin[(5*(c + d*x))/2] - 315*Sin[(7*(c + d*x))/2] - 169*Sin[(9*(c + d*x))/2]))/(2520*d)","A",1
155,1,97,128,5.5916562,"\int (a+a \sec (c+d x))^{3/2} \tan ^2(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^2,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(5 \sin \left(\frac{3}{2} (c+d x)\right)+\sin \left(\frac{5}{2} (c+d x)\right)-10 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{5}{2}}(c+d x)\right)}{10 d}","-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^4 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a^3 \tan ^3(c+d x)}{d (a \sec (c+d x)+a)^{3/2}}+\frac{2 a^2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(a*Sec[(c + d*x)/2]*Sec[c + d*x]^2*Sqrt[a*(1 + Sec[c + d*x])]*(-10*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(5/2) + 5*Sin[(3*(c + d*x))/2] + Sin[(5*(c + d*x))/2]))/(10*d)","A",1
156,1,102,64,0.3785214,"\int \cot ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2),x]","-\frac{2 \cot (c+d x) \sqrt{\frac{1}{\sec (c+d x)+1}} (a (\sec (c+d x)+1))^{3/2} \left(\sqrt{\cos (c+d x)} \sqrt{\frac{1}{\cos (c+d x)+1}}+\tan \left(\frac{1}{2} (c+d x)\right) \sin ^{-1}\left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\frac{1}{\cos (c+d x)+1}}}\right)\right)}{d}","-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{2 a \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d}",1,"(-2*Cot[c + d*x]*Sqrt[(1 + Sec[c + d*x])^(-1)]*(a*(1 + Sec[c + d*x]))^(3/2)*(Sqrt[Cos[c + d*x]]*Sqrt[(1 + Cos[c + d*x])^(-1)] + ArcSin[Tan[(c + d*x)/2]/Sqrt[(1 + Cos[c + d*x])^(-1)]]*Tan[(c + d*x)/2]))/d","A",0
157,1,5542,144,24.0045446,"\int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} d}-\frac{\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac{3 a \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}",1,"Result too large to show","C",0
158,1,5582,226,24.1187877,"\int \cot ^6(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^6*(a + a*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{11 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} d}+\frac{3 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 a d}+\frac{5 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{24 d}-\frac{21 a \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{16 d}-\frac{\cos (c+d x) \cot ^5(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{4 a d}",1,"Result too large to show","C",0
159,1,156,193,0.7777472,"\int (a+a \sec (c+d x))^{5/2} \tan ^5(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^5,x]","\frac{(a (\sec (c+d x)+1))^{5/2} \left(\frac{2}{13} (\sec (c+d x)+1)^{13/2}-\frac{6}{11} (\sec (c+d x)+1)^{11/2}+\frac{2}{9} (\sec (c+d x)+1)^{9/2}+\frac{2}{7} (\sec (c+d x)+1)^{7/2}+\frac{2}{5} (\sec (c+d x)+1)^{5/2}+\frac{2}{3} (\sec (c+d x)+1)^{3/2}+2 \sqrt{\sec (c+d x)+1}-2 \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)\right)}{d (\sec (c+d x)+1)^{5/2}}","-\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{2 (a \sec (c+d x)+a)^{13/2}}{13 a^4 d}-\frac{6 (a \sec (c+d x)+a)^{11/2}}{11 a^3 d}+\frac{2 (a \sec (c+d x)+a)^{9/2}}{9 a^2 d}+\frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 d}+\frac{2 a (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"((a*(1 + Sec[c + d*x]))^(5/2)*(-2*ArcTanh[Sqrt[1 + Sec[c + d*x]]] + 2*Sqrt[1 + Sec[c + d*x]] + (2*(1 + Sec[c + d*x])^(3/2))/3 + (2*(1 + Sec[c + d*x])^(5/2))/5 + (2*(1 + Sec[c + d*x])^(7/2))/7 + (2*(1 + Sec[c + d*x])^(9/2))/9 - (6*(1 + Sec[c + d*x])^(11/2))/11 + (2*(1 + Sec[c + d*x])^(13/2))/13))/(d*(1 + Sec[c + d*x])^(5/2))","A",1
160,1,102,145,0.6088859,"\int (a+a \sec (c+d x))^{5/2} \tan ^3(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^3,x]","\frac{2 (a (\sec (c+d x)+1))^{5/2} \left(\sqrt{\sec (c+d x)+1} \left(35 \sec ^4(c+d x)+95 \sec ^3(c+d x)+12 \sec ^2(c+d x)-226 \sec (c+d x)-493\right)+315 \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)\right)}{315 d (\sec (c+d x)+1)^{5/2}}","\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{2 (a \sec (c+d x)+a)^{9/2}}{9 a^2 d}-\frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a d}-\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 d}-\frac{2 a (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(2*(a*(1 + Sec[c + d*x]))^(5/2)*(315*ArcTanh[Sqrt[1 + Sec[c + d*x]]] + Sqrt[1 + Sec[c + d*x]]*(-493 - 226*Sec[c + d*x] + 12*Sec[c + d*x]^2 + 95*Sec[c + d*x]^3 + 35*Sec[c + d*x]^4)))/(315*d*(1 + Sec[c + d*x])^(5/2))","A",1
161,1,82,97,0.235868,"\int (a+a \sec (c+d x))^{5/2} \tan (c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x],x]","\frac{2 (a (\sec (c+d x)+1))^{5/2} \left(\sqrt{\sec (c+d x)+1} \left(3 \sec ^2(c+d x)+11 \sec (c+d x)+23\right)-15 \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)\right)}{15 d (\sec (c+d x)+1)^{5/2}}","-\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 a (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 d}",1,"(2*(a*(1 + Sec[c + d*x]))^(5/2)*(-15*ArcTanh[Sqrt[1 + Sec[c + d*x]]] + Sqrt[1 + Sec[c + d*x]]*(23 + 11*Sec[c + d*x] + 3*Sec[c + d*x]^2)))/(15*d*(1 + Sec[c + d*x])^(5/2))","A",1
162,1,83,95,0.1312408,"\int \cot (c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]*(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 (a (\sec (c+d x)+1))^{5/2} \left(\sqrt{\sec (c+d x)+1}+\tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{\sec (c+d x)+1}}{\sqrt{2}}\right)\right)}{d (\sec (c+d x)+1)^{5/2}}","\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{4 \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}",1,"(2*(a*(1 + Sec[c + d*x]))^(5/2)*(ArcTanh[Sqrt[1 + Sec[c + d*x]]] - 2*Sqrt[2]*ArcTanh[Sqrt[1 + Sec[c + d*x]]/Sqrt[2]] + Sqrt[1 + Sec[c + d*x]]))/(d*(1 + Sec[c + d*x])^(5/2))","A",1
163,1,115,106,0.287527,"\int \cot ^3(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2),x]","-\frac{(a (\sec (c+d x)+1))^{5/2} \left(2 \sqrt{\sec (c+d x)+1}+4 (\sec (c+d x)-1) \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)-3 \sqrt{2} (\sec (c+d x)-1) \tanh ^{-1}\left(\frac{\sqrt{\sec (c+d x)+1}}{\sqrt{2}}\right)\right)}{2 d (\sec (c+d x)-1) (\sec (c+d x)+1)^{5/2}}","-\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}+\frac{3 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} d}+\frac{a^2 \sqrt{a \sec (c+d x)+a}}{d (1-\sec (c+d x))}",1,"-1/2*((a*(1 + Sec[c + d*x]))^(5/2)*(4*ArcTanh[Sqrt[1 + Sec[c + d*x]]]*(-1 + Sec[c + d*x]) - 3*Sqrt[2]*ArcTanh[Sqrt[1 + Sec[c + d*x]]/Sqrt[2]]*(-1 + Sec[c + d*x]) + 2*Sqrt[1 + Sec[c + d*x]]))/(d*(-1 + Sec[c + d*x])*(1 + Sec[c + d*x])^(5/2))","A",1
164,1,138,147,1.3884472,"\int \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2),x]","\frac{(a (\sec (c+d x)+1))^{5/2} \left(\sqrt{\sec (c+d x)+1} (11 \sec (c+d x)-15)+32 (\sec (c+d x)-1)^2 \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)-86 \sqrt{2} \sin ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \tanh ^{-1}\left(\frac{\sqrt{\sec (c+d x)+1}}{\sqrt{2}}\right)\right)}{16 d (\sec (c+d x)-1)^2 (\sec (c+d x)+1)^{5/2}}","\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{d}-\frac{43 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} d}-\frac{11 a^2 \sqrt{a \sec (c+d x)+a}}{16 d (1-\sec (c+d x))}-\frac{a^2 \sqrt{a \sec (c+d x)+a}}{4 d (1-\sec (c+d x))^2}",1,"((a*(1 + Sec[c + d*x]))^(5/2)*(32*ArcTanh[Sqrt[1 + Sec[c + d*x]]]*(-1 + Sec[c + d*x])^2 + Sqrt[1 + Sec[c + d*x]]*(-15 + 11*Sec[c + d*x]) - 86*Sqrt[2]*ArcTanh[Sqrt[1 + Sec[c + d*x]]/Sqrt[2]]*Sec[c + d*x]^2*Sin[(c + d*x)/2]^4))/(16*d*(-1 + Sec[c + d*x])^2*(1 + Sec[c + d*x])^(5/2))","A",1
165,1,173,290,10.3347473,"\int (a+a \sec (c+d x))^{5/2} \tan ^6(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^6,x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^7(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(604890 \sin \left(\frac{1}{2} (c+d x)\right)-87230 \sin \left(\frac{3}{2} (c+d x)\right)+450450 \sin \left(\frac{5}{2} (c+d x)\right)-137670 \sin \left(\frac{7}{2} (c+d x)\right)+210210 \sin \left(\frac{9}{2} (c+d x)\right)+75450 \sin \left(\frac{11}{2} (c+d x)\right)+90090 \sin \left(\frac{13}{2} (c+d x)\right)+16066 \sin \left(\frac{15}{2} (c+d x)\right)-2882880 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{15}{2}}(c+d x)\right)}{2882880 d}","-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^{10} \tan ^{15}(c+d x)}{15 d (a \sec (c+d x)+a)^{15/2}}+\frac{18 a^9 \tan ^{13}(c+d x)}{13 d (a \sec (c+d x)+a)^{13/2}}+\frac{62 a^8 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{98 a^7 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{62 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^5 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 a^3 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]^7*Sqrt[a*(1 + Sec[c + d*x])]*(-2882880*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(15/2) + 604890*Sin[(c + d*x)/2] - 87230*Sin[(3*(c + d*x))/2] + 450450*Sin[(5*(c + d*x))/2] - 137670*Sin[(7*(c + d*x))/2] + 210210*Sin[(9*(c + d*x))/2] + 75450*Sin[(11*(c + d*x))/2] + 90090*Sin[(13*(c + d*x))/2] + 16066*Sin[(15*(c + d*x))/2]))/(2882880*d)","A",1
166,1,149,224,7.4987617,"\int (a+a \sec (c+d x))^{5/2} \tan ^4(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^4,x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(-1386 \sin \left(\frac{1}{2} (c+d x)\right)+1584 \sin \left(\frac{3}{2} (c+d x)\right)-1386 \sin \left(\frac{5}{2} (c+d x)\right)-143 \sin \left(\frac{7}{2} (c+d x)\right)-693 \sin \left(\frac{9}{2} (c+d x)\right)-26 \sin \left(\frac{11}{2} (c+d x)\right)+5544 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{11}{2}}(c+d x)\right)}{5544 d}","\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^8 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{14 a^7 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{34 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{6 a^5 \tan ^5(c+d x)}{d (a \sec (c+d x)+a)^{5/2}}+\frac{2 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 a^3 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]^5*Sqrt[a*(1 + Sec[c + d*x])]*(5544*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(11/2) - 1386*Sin[(c + d*x)/2] + 1584*Sin[(3*(c + d*x))/2] - 1386*Sin[(5*(c + d*x))/2] - 143*Sin[(7*(c + d*x))/2] - 693*Sin[(9*(c + d*x))/2] - 26*Sin[(11*(c + d*x))/2]))/(5544*d)","A",1
167,1,125,160,5.9175152,"\int (a+a \sec (c+d x))^{5/2} \tan ^2(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^2,x]","-\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(-35 \sin \left(\frac{1}{2} (c+d x)\right)+7 \sin \left(\frac{3}{2} (c+d x)\right)-21 \sin \left(\frac{5}{2} (c+d x)\right)+5 \sin \left(\frac{7}{2} (c+d x)\right)+42 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{7}{2}}(c+d x)\right)}{42 d}","-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^5 \tan ^5(c+d x)}{d (a \sec (c+d x)+a)^{5/2}}+\frac{14 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 a^3 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"-1/42*(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]^3*Sqrt[a*(1 + Sec[c + d*x])]*(42*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(7/2) - 35*Sin[(c + d*x)/2] + 7*Sin[(3*(c + d*x))/2] - 21*Sin[(5*(c + d*x))/2] + 5*Sin[(7*(c + d*x))/2]))/d","A",1
168,1,124,66,0.8263227,"\int \cot ^2(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2),x]","-\frac{\sqrt{2} \cot (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{\sec (c+d x)+1}\right)^{3/2} (a (\sec (c+d x)+1))^{5/2} \left(2 \cos (c+d x)-\frac{(\cos (c+d x)-1) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)}{\sqrt{1-\sec (c+d x)}}\right)}{d \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}}","-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{4 a^2 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d}",1,"-((Sqrt[2]*Cot[c + d*x]*Sec[(c + d*x)/2]^2*(2*Cos[c + d*x] - (ArcTanh[Sqrt[1 - Sec[c + d*x]]]*(-1 + Cos[c + d*x]))/Sqrt[1 - Sec[c + d*x]])*((1 + Sec[c + d*x])^(-1))^(3/2)*(a*(1 + Sec[c + d*x]))^(5/2))/(d*Sqrt[1 - Tan[(c + d*x)/2]^2]))","A",1
169,1,81,96,0.2926287,"\int \cot ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2),x]","-\frac{2 \left(\frac{1}{\cos (c+d x)+1}\right)^{3/2} \cot ^3(c+d x) (a (\sec (c+d x)+1))^{5/2} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)}{3 d \sqrt{\frac{1}{\sec (c+d x)+1}}}","\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 a \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(-2*((1 + Cos[c + d*x])^(-1))^(3/2)*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, -3/2, -1/2, 2*Sin[(c + d*x)/2]^2]*(a*(1 + Sec[c + d*x]))^(5/2))/(3*d*Sqrt[(1 + Sec[c + d*x])^(-1)])","C",0
170,1,5562,176,24.1791392,"\int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{2} d}-\frac{7 a^2 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}-\frac{\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}+\frac{a \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d}",1,"Result too large to show","C",0
171,1,88,126,0.1837179,"\int \frac{\tan ^5(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Tan[c + d*x]^5/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \left(15 \sec ^4(c+d x)-3 \sec ^3(c+d x)-64 \sec ^2(c+d x)+46 \sec (c+d x)-105 \sqrt{\sec (c+d x)+1} \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)+92\right)}{105 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a^4 d}-\frac{6 (a \sec (c+d x)+a)^{5/2}}{5 a^3 d}+\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a^2 d}+\frac{2 \sqrt{a \sec (c+d x)+a}}{a d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"(2*(92 + 46*Sec[c + d*x] - 64*Sec[c + d*x]^2 - 3*Sec[c + d*x]^3 + 15*Sec[c + d*x]^4 - 105*ArcTanh[Sqrt[1 + Sec[c + d*x]]]*Sqrt[1 + Sec[c + d*x]]))/(105*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
172,1,66,78,0.0909678,"\int \frac{\tan ^3(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Tan[c + d*x]^3/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \left(\sec ^2(c+d x)-\sec (c+d x)+3 \sqrt{\sec (c+d x)+1} \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)-2\right)}{3 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a^2 d}-\frac{2 \sqrt{a \sec (c+d x)+a}}{a d}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"(2*(-2 - Sec[c + d*x] + Sec[c + d*x]^2 + 3*ArcTanh[Sqrt[1 + Sec[c + d*x]]]*Sqrt[1 + Sec[c + d*x]]))/(3*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
173,1,44,31,0.0403887,"\int \frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Tan[c + d*x]/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{2 \sqrt{\sec (c+d x)+1} \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)}{d \sqrt{a (\sec (c+d x)+1)}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"(-2*ArcTanh[Sqrt[1 + Sec[c + d*x]]]*Sqrt[1 + Sec[c + d*x]])/(d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
174,1,57,92,0.0575471,"\int \frac{\cot (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Cot[c + d*x]/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{1}{2} (\sec (c+d x)+1)\right)-2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\sec (c+d x)+1\right)}{d \sqrt{a (\sec (c+d x)+1)}}","-\frac{1}{d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}",1,"(Hypergeometric2F1[-1/2, 1, 1/2, (1 + Sec[c + d*x])/2] - 2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + Sec[c + d*x]])/(d*Sqrt[a*(1 + Sec[c + d*x])])","C",1
175,1,90,152,0.215185,"\int \frac{\cot ^3(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Cot[c + d*x]^3/Sqrt[a + a*Sec[c + d*x]],x]","\frac{a \left(-9 (\sec (c+d x)-1) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{1}{2} (\sec (c+d x)+1)\right)+8 (\sec (c+d x)-1) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\sec (c+d x)+1\right)-6\right)}{12 d (\sec (c+d x)-1) (a (\sec (c+d x)+1))^{3/2}}","-\frac{a}{12 d (a \sec (c+d x)+a)^{3/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}+\frac{7}{8 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}+\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{8 \sqrt{2} \sqrt{a} d}",1,"(a*(-6 - 9*Hypergeometric2F1[-3/2, 1, -1/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x]) + 8*Hypergeometric2F1[-3/2, 1, -1/2, 1 + Sec[c + d*x]]*(-1 + Sec[c + d*x])))/(12*d*(-1 + Sec[c + d*x])*(a*(1 + Sec[c + d*x]))^(3/2))","C",1
176,1,102,214,0.2820125,"\int \frac{\cot ^5(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Cot[c + d*x]^5/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\cot ^4(c+d x) \left(151 (\sec (c+d x)-1)^2 \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\frac{1}{2} (\sec (c+d x)+1)\right)-2 \left(32 (\sec (c+d x)-1)^2 \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\sec (c+d x)+1\right)-85 \sec (c+d x)+105\right)\right)}{160 d \sqrt{a (\sec (c+d x)+1)}}","\frac{87 a^2}{160 d (a \sec (c+d x)+a)^{5/2}}-\frac{17 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}+\frac{23 a}{192 d (a \sec (c+d x)+a)^{3/2}}-\frac{105}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{\sqrt{a} d}-\frac{151 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{128 \sqrt{2} \sqrt{a} d}",1,"(Cot[c + d*x]^4*(-2*(105 + 32*Hypergeometric2F1[-5/2, 1, -3/2, 1 + Sec[c + d*x]]*(-1 + Sec[c + d*x])^2 - 85*Sec[c + d*x]) + 151*Hypergeometric2F1[-5/2, 1, -3/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x])^2))/(160*d*Sqrt[a*(1 + Sec[c + d*x])])","C",1
177,1,467,189,19.2235731,"\int \frac{\tan ^6(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Tan[c + d*x]^6/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\frac{1532}{315} \sin \left(\frac{1}{2} (c+d x)\right)+\frac{4}{9} \sin \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x)-\frac{4}{63} \sin \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x)-\frac{176}{105} \sin \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x)+\frac{136}{315} \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)}{d \sqrt{a (\sec (c+d x)+1)}}+\frac{16 \left(-3-2 \sqrt{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\left(10-7 \sqrt{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)-5 \sqrt{2}+7}{\cos \left(\frac{1}{2} (c+d x)\right)+1}} \sqrt{\frac{-\left(\left(\sqrt{2}-2\right) \cos \left(\frac{1}{2} (c+d x)\right)\right)+\sqrt{2}-1}{\cos \left(\frac{1}{2} (c+d x)\right)+1}} \left(\left(\sqrt{2}-2\right) \cos \left(\frac{1}{2} (c+d x)\right)-\sqrt{2}+1\right) \cos ^4\left(\frac{1}{4} (c+d x)\right) \sqrt{-\tan ^2\left(\frac{1}{4} (c+d x)\right)-2 \sqrt{2}+3} \sec ^2(c+d x) \sqrt{\left(\left(2+\sqrt{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)-\sqrt{2}-1\right) \sec ^2\left(\frac{1}{4} (c+d x)\right)} \left(F\left(\sin ^{-1}\left(\frac{\tan \left(\frac{1}{4} (c+d 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} (c+d x)\right)}{\sqrt{3-2 \sqrt{2}}}\right)|17-12 \sqrt{2}\right)\right)}{d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^4 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{6 a^3 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^2 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 a \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]*((1532*Sin[(c + d*x)/2])/315 + (136*Sec[c + d*x]*Sin[(c + d*x)/2])/315 - (176*Sec[c + d*x]^2*Sin[(c + d*x)/2])/105 - (4*Sec[c + d*x]^3*Sin[(c + d*x)/2])/63 + (4*Sec[c + d*x]^4*Sin[(c + d*x)/2])/9))/(d*Sqrt[a*(1 + Sec[c + d*x])]) + (16*(-3 - 2*Sqrt[2])*Cos[(c + d*x)/4]^4*Cos[(c + d*x)/2]*Sqrt[(7 - 5*Sqrt[2] + (10 - 7*Sqrt[2])*Cos[(c + d*x)/2])/(1 + Cos[(c + d*x)/2])]*Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[(c + d*x)/2])/(1 + Cos[(c + d*x)/2])]*(1 - Sqrt[2] + (-2 + Sqrt[2])*Cos[(c + d*x)/2])*(EllipticF[ArcSin[Tan[(c + d*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] - 2*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(c + d*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]])*Sqrt[(-1 - Sqrt[2] + (2 + Sqrt[2])*Cos[(c + d*x)/2])*Sec[(c + d*x)/4]^2]*Sec[c + d*x]^2*Sqrt[3 - 2*Sqrt[2] - Tan[(c + d*x)/4]^2])/(d*Sqrt[a*(1 + Sec[c + d*x])])","C",0
178,1,238,125,3.1117037,"\int \frac{\tan ^4(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Tan[c + d*x]^4/Sqrt[a + a*Sec[c + d*x]],x]","\frac{16 \sqrt{2} \tan ^5(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{9/2} \left(-\frac{4}{9} \tan ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};-2 \sec (c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)-\frac{\cos (c+d x) (5 \cos (c+d x)+9) \csc ^6\left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left((22 \cos (c+d x)-23 \cos (2 (c+d x))-29) \sqrt{1-\sec (c+d x)}+30 \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{480 \sqrt{1-\sec (c+d x)}}\right)}{5 d \left(1-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)^{7/2} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^2 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 a \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(16*Sqrt[2]*((1 + Sec[c + d*x])^(-1))^(9/2)*(-1/480*(Cos[c + d*x]*(9 + 5*Cos[c + d*x])*Csc[(c + d*x)/2]^6*Sec[(c + d*x)/2]^2*(30*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*Cos[c + d*x]^2 + (-29 + 22*Cos[c + d*x] - 23*Cos[2*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]))/Sqrt[1 - Sec[c + d*x]] - (4*Hypergeometric2F1[2, 9/2, 11/2, -2*Sec[c + d*x]*Sin[(c + d*x)/2]^2]*Sec[c + d*x]*Tan[(c + d*x)/2]^2)/9)*Tan[c + d*x]^5)/(5*d*Sqrt[a*(1 + Sec[c + d*x])]*(1 - Tan[(c + d*x)/2]^2)^(7/2))","C",0
179,1,119,63,0.7711747,"\int \frac{\tan ^2(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Tan[c + d*x]^2/Sqrt[a + a*Sec[c + d*x]],x]","\frac{16 \cos ^6\left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{5/2} \left(\sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{\frac{1}{\cos (c+d x)+1}}-\cos (c+d x) \sin ^{-1}\left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\frac{1}{\cos (c+d x)+1}}}\right)\right)}{d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(16*Cos[(c + d*x)/2]^6*Sec[c + d*x]^4*((1 + Sec[c + d*x])^(-1))^(5/2)*(-(ArcSin[Tan[(c + d*x)/2]/Sqrt[(1 + Cos[c + d*x])^(-1)]]*Cos[c + d*x]) + Sqrt[Cos[c + d*x]]*Sqrt[(1 + Cos[c + d*x])^(-1)]*Sin[c + d*x]))/(d*Sqrt[a*(1 + Sec[c + d*x])])","A",0
180,1,5534,165,24.2969359,"\int \frac{\cot ^2(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Cot[c + d*x]^2/Sqrt[a + a*Sec[c + d*x]],x]","\text{Result too large to show}","-\frac{\cot (c+d x) \sqrt{a \sec (c+d x)+a}}{4 a d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{7 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{2} \sqrt{a} d}-\frac{\cos (c+d x) \cot (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{4 a d}",1,"Result too large to show","C",0
181,1,5574,251,24.1806245,"\int \frac{\cot ^4(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Cot[c + d*x]^4/Sqrt[a + a*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{43 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{96 a^2 d}-\frac{\cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{16 a^2 d}-\frac{15 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{32 a^2 d}+\frac{21 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{64 a d}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{107 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{64 \sqrt{2} \sqrt{a} d}",1,"Result too large to show","C",0
182,1,5618,335,24.1500711,"\int \frac{\cot ^6(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Cot[c + d*x]^6/Sqrt[a + a*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{579 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{640 a^3 d}-\frac{\cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{48 a^3 d}-\frac{23 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{192 a^3 d}-\frac{101 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{128 a^3 d}-\frac{323 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{768 a^2 d}-\frac{189 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{512 a d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{835 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{512 \sqrt{2} \sqrt{a} d}",1,"Result too large to show","C",0
183,1,79,100,0.1776785,"\int \frac{\tan ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^5/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 \left(\sec ^3(c+d x)-2 \sec ^2(c+d x)-2 \sec (c+d x)-5 \sqrt{\sec (c+d x)+1} \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)+1\right)}{5 a d \sqrt{a (\sec (c+d x)+1)}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a^4 d}-\frac{2 (a \sec (c+d x)+a)^{3/2}}{a^3 d}+\frac{2 \sqrt{a \sec (c+d x)+a}}{a^2 d}",1,"(2*(1 - 2*Sec[c + d*x] - 2*Sec[c + d*x]^2 + Sec[c + d*x]^3 - 5*ArcTanh[Sqrt[1 + Sec[c + d*x]]]*Sqrt[1 + Sec[c + d*x]]))/(5*a*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
184,1,56,54,0.0828288,"\int \frac{\tan ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^3/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 \left(\sec (c+d x)+\sqrt{\sec (c+d x)+1} \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)+1\right)}{a d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{2 \sqrt{a \sec (c+d x)+a}}{a^2 d}",1,"(2*(1 + Sec[c + d*x] + ArcTanh[Sqrt[1 + Sec[c + d*x]]]*Sqrt[1 + Sec[c + d*x]]))/(a*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
185,1,38,54,0.0375719,"\int \frac{\tan (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\sec (c+d x)+1\right)}{a d \sqrt{a (\sec (c+d x)+1)}}","\frac{2}{a d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}",1,"(2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + Sec[c + d*x]])/(a*d*Sqrt[a*(1 + Sec[c + d*x])])","C",1
186,1,60,120,0.0601346,"\int \frac{\cot (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{1}{2} (\sec (c+d x)+1)\right)-2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\sec (c+d x)+1\right)}{3 d (a (\sec (c+d x)+1))^{3/2}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{1}{3 d (a \sec (c+d x)+a)^{3/2}}",1,"(Hypergeometric2F1[-3/2, 1, -1/2, (1 + Sec[c + d*x])/2] - 2*Hypergeometric2F1[-3/2, 1, -1/2, 1 + Sec[c + d*x]])/(3*d*(a*(1 + Sec[c + d*x]))^(3/2))","C",1
187,1,90,176,0.1755582,"\int \frac{\cot ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^3/(a + a*Sec[c + d*x])^(3/2),x]","\frac{a \left(-11 (\sec (c+d x)-1) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\frac{1}{2} (\sec (c+d x)+1)\right)+8 (\sec (c+d x)-1) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\sec (c+d x)+1\right)-10\right)}{20 d (\sec (c+d x)-1) (a (\sec (c+d x)+1))^{5/2}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} d}-\frac{3 a}{20 d (a \sec (c+d x)+a)^{5/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}+\frac{5}{24 d (a \sec (c+d x)+a)^{3/2}}+\frac{21}{16 a d \sqrt{a \sec (c+d x)+a}}",1,"(a*(-10 - 11*Hypergeometric2F1[-5/2, 1, -3/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x]) + 8*Hypergeometric2F1[-5/2, 1, -3/2, 1 + Sec[c + d*x]]*(-1 + Sec[c + d*x])))/(20*d*(-1 + Sec[c + d*x])*(a*(1 + Sec[c + d*x]))^(5/2))","C",1
188,1,99,238,0.3189113,"\int \frac{\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\cot ^4(c+d x) \left(203 (\sec (c+d x)-1)^2 \, _2F_1\left(-\frac{7}{2},1;-\frac{5}{2};\frac{1}{2} (\sec (c+d x)+1)\right)-64 (\sec (c+d x)-1)^2 \, _2F_1\left(-\frac{7}{2},1;-\frac{5}{2};\sec (c+d x)+1\right)+266 \sec (c+d x)-322\right)}{224 d (a (\sec (c+d x)+1))^{3/2}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{3/2} d}-\frac{203 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{256 \sqrt{2} a^{3/2} d}+\frac{139 a^2}{224 d (a \sec (c+d x)+a)^{7/2}}-\frac{19 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}+\frac{15 a}{64 d (a \sec (c+d x)+a)^{5/2}}-\frac{53}{384 d (a \sec (c+d x)+a)^{3/2}}-\frac{309}{256 a d \sqrt{a \sec (c+d x)+a}}",1,"(Cot[c + d*x]^4*(-322 + 203*Hypergeometric2F1[-7/2, 1, -5/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x])^2 - 64*Hypergeometric2F1[-7/2, 1, -5/2, 1 + Sec[c + d*x]]*(-1 + Sec[c + d*x])^2 + 266*Sec[c + d*x]))/(224*d*(a*(1 + Sec[c + d*x]))^(3/2))","C",1
189,1,248,157,2.8797578,"\int \frac{\tan ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^6/(a + a*Sec[c + d*x])^(3/2),x]","\frac{32 \sqrt{2} \tan ^7(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{11/2} \left(\frac{\cos (c+d x) (7 \cos (c+d x)+11) \csc ^8\left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left((-198 \cos (c+d x)+61 \cos (2 (c+d x))-44 \cos (3 (c+d x))+76) \sqrt{1-\sec (c+d x)}+105 \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{3360 \sqrt{1-\sec (c+d x)}}-\frac{4}{11} \tan ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \, _2F_1\left(2,\frac{11}{2};\frac{13}{2};-2 \sec (c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{7 d \left(1-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)^{9/2} (a (\sec (c+d x)+1))^{3/2}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{2 a^2 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 \tan (c+d x)}{a d \sqrt{a \sec (c+d x)+a}}",1,"(32*Sqrt[2]*((1 + Sec[c + d*x])^(-1))^(11/2)*((Cos[c + d*x]*(11 + 7*Cos[c + d*x])*Csc[(c + d*x)/2]^8*Sec[(c + d*x)/2]^2*(105*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*Cos[c + d*x]^3 + (76 - 198*Cos[c + d*x] + 61*Cos[2*(c + d*x)] - 44*Cos[3*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]))/(3360*Sqrt[1 - Sec[c + d*x]]) - (4*Hypergeometric2F1[2, 11/2, 13/2, -2*Sec[c + d*x]*Sin[(c + d*x)/2]^2]*Sec[c + d*x]*Tan[(c + d*x)/2]^2)/11)*Tan[c + d*x]^7)/(7*d*(a*(1 + Sec[c + d*x]))^(3/2)*(1 - Tan[(c + d*x)/2]^2)^(9/2))","C",0
190,1,162,95,4.8265076,"\int \frac{\tan ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^4/(a + a*Sec[c + d*x])^(3/2),x]","\frac{64 \cos ^6\left(\frac{1}{2} (c+d x)\right) \cot ^4\left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{7/2} \left((\sin (c+d x)-2 \sin (2 (c+d x))) \sqrt{\frac{1}{\cos (c+d x)+1}} \sqrt{\cos (c+d x)}+3 \cos ^2(c+d x) \sin ^{-1}\left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\frac{1}{\cos (c+d x)+1}}}\right)\right)}{3 d \left(\cot ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2 (a (\sec (c+d x)+1))^{3/2}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \tan (c+d x)}{a d \sqrt{a \sec (c+d x)+a}}",1,"(64*Cos[(c + d*x)/2]^6*Cot[(c + d*x)/2]^4*Sec[c + d*x]^5*((1 + Sec[c + d*x])^(-1))^(7/2)*(3*ArcSin[Tan[(c + d*x)/2]/Sqrt[(1 + Cos[c + d*x])^(-1)]]*Cos[c + d*x]^2 + Sqrt[Cos[c + d*x]]*Sqrt[(1 + Cos[c + d*x])^(-1)]*(Sin[c + d*x] - 2*Sin[2*(c + d*x)])))/(3*d*(-1 + Cot[(c + d*x)/2]^2)^2*(a*(1 + Sec[c + d*x]))^(3/2))","A",0
191,1,4739,85,23.8442112,"\int \frac{\tan ^2(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^2/(a + a*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}",1,"(-32*Cos[(c + d*x)/4]^2*Cos[(c + d*x)/2]^2*(-2/Sqrt[Sec[c + d*x]] + 2*Sqrt[Sec[c + d*x]])*Sec[c + d*x]^2*((((2 + Sqrt[2])*EllipticF[ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2] - (2 + Sqrt[2])*EllipticPi[-(1/Sqrt[2]), ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2] + (-2 + Sqrt[2])*EllipticPi[1/Sqrt[2], ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2])*Sqrt[(-1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*Sqrt[(1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*(-1 + Sqrt[2] + Tan[(c + d*x)/4])^2*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])])/4 + EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(c + d*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]*Sqrt[3 + 2*Sqrt[2] - Tan[(c + d*x)/4]^2]*Sqrt[(-3 + 2*Sqrt[2])*(-3 + 2*Sqrt[2] + Tan[(c + d*x)/4]^2)]))/(d*(a*(1 + Sec[c + d*x]))^(3/2)*(16*Cos[(c + d*x)/4]*Sqrt[Sec[c + d*x]]*Sin[(c + d*x)/4]*((((2 + Sqrt[2])*EllipticF[ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2] - (2 + Sqrt[2])*EllipticPi[-(1/Sqrt[2]), ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2] + (-2 + Sqrt[2])*EllipticPi[1/Sqrt[2], ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2])*Sqrt[(-1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*Sqrt[(1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*(-1 + Sqrt[2] + Tan[(c + d*x)/4])^2*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])])/4 + EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(c + d*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]*Sqrt[3 + 2*Sqrt[2] - Tan[(c + d*x)/4]^2]*Sqrt[(-3 + 2*Sqrt[2])*(-3 + 2*Sqrt[2] + Tan[(c + d*x)/4]^2)]) - 16*Cos[(c + d*x)/4]^2*Sec[c + d*x]^(3/2)*Sin[c + d*x]*((((2 + Sqrt[2])*EllipticF[ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2] - (2 + Sqrt[2])*EllipticPi[-(1/Sqrt[2]), ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2] + (-2 + Sqrt[2])*EllipticPi[1/Sqrt[2], ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2])*Sqrt[(-1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*Sqrt[(1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*(-1 + Sqrt[2] + Tan[(c + d*x)/4])^2*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])])/4 + EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(c + d*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]*Sqrt[3 + 2*Sqrt[2] - Tan[(c + d*x)/4]^2]*Sqrt[(-3 + 2*Sqrt[2])*(-3 + 2*Sqrt[2] + Tan[(c + d*x)/4]^2)]) - 32*Cos[(c + d*x)/4]^2*Sqrt[Sec[c + d*x]]*((((2 + Sqrt[2])*EllipticF[ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2] - (2 + Sqrt[2])*EllipticPi[-(1/Sqrt[2]), ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2] + (-2 + Sqrt[2])*EllipticPi[1/Sqrt[2], ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2])*Sec[(c + d*x)/4]^2*Sqrt[(-1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*Sqrt[(1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*(-1 + Sqrt[2] + Tan[(c + d*x)/4])*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])])/8 + ((-3 + 2*Sqrt[2])*EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(c + d*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]*Sec[(c + d*x)/4]^2*Tan[(c + d*x)/4]*Sqrt[3 + 2*Sqrt[2] - Tan[(c + d*x)/4]^2])/(4*Sqrt[(-3 + 2*Sqrt[2])*(-3 + 2*Sqrt[2] + Tan[(c + d*x)/4]^2)]) - (EllipticPi[-3 + 2*Sqrt[2], ArcSin[Tan[(c + d*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]]*Sec[(c + d*x)/4]^2*Tan[(c + d*x)/4]*Sqrt[(-3 + 2*Sqrt[2])*(-3 + 2*Sqrt[2] + Tan[(c + d*x)/4]^2)])/(4*Sqrt[3 + 2*Sqrt[2] - Tan[(c + d*x)/4]^2]) + (Sec[(c + d*x)/4]^2*Sqrt[3 + 2*Sqrt[2] - Tan[(c + d*x)/4]^2]*Sqrt[(-3 + 2*Sqrt[2])*(-3 + 2*Sqrt[2] + Tan[(c + d*x)/4]^2)])/(4*Sqrt[3 - 2*Sqrt[2]]*Sqrt[1 - Tan[(c + d*x)/4]^2/(3 - 2*Sqrt[2])]*Sqrt[1 - ((17 - 12*Sqrt[2])*Tan[(c + d*x)/4]^2)/(3 - 2*Sqrt[2])]*(1 - ((-3 + 2*Sqrt[2])*Tan[(c + d*x)/4]^2)/(3 - 2*Sqrt[2]))) + (((2 + Sqrt[2])*EllipticF[ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2] - (2 + Sqrt[2])*EllipticPi[-(1/Sqrt[2]), ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2] + (-2 + Sqrt[2])*EllipticPi[1/Sqrt[2], ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2])*Sqrt[(1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*(-1 + Sqrt[2] + Tan[(c + d*x)/4])^2*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*(-1/4*(Sec[(c + d*x)/4]^2*(-1 - Sqrt[2] + Tan[(c + d*x)/4]))/(-1 + Sqrt[2] + Tan[(c + d*x)/4])^2 + Sec[(c + d*x)/4]^2/(4*(-1 + Sqrt[2] + Tan[(c + d*x)/4]))))/(8*Sqrt[(-1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]) + (((2 + Sqrt[2])*EllipticF[ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2] - (2 + Sqrt[2])*EllipticPi[-(1/Sqrt[2]), ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2] + (-2 + Sqrt[2])*EllipticPi[1/Sqrt[2], ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2])*Sqrt[(-1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*(-1 + Sqrt[2] + Tan[(c + d*x)/4])^2*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*(-1/4*(Sec[(c + d*x)/4]^2*(1 - Sqrt[2] + Tan[(c + d*x)/4]))/(-1 + Sqrt[2] + Tan[(c + d*x)/4])^2 + Sec[(c + d*x)/4]^2/(4*(-1 + Sqrt[2] + Tan[(c + d*x)/4]))))/(8*Sqrt[(1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]) + (((2 + Sqrt[2])*EllipticF[ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2] - (2 + Sqrt[2])*EllipticPi[-(1/Sqrt[2]), ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2] + (-2 + Sqrt[2])*EllipticPi[1/Sqrt[2], ArcSin[2^(1/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]], 1/2])*Sqrt[(-1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*Sqrt[(1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*(-1 + Sqrt[2] + Tan[(c + d*x)/4])^2*(Sec[(c + d*x)/4]^2/(4*(-1 + Sqrt[2] + Tan[(c + d*x)/4])) - (Sec[(c + d*x)/4]^2*(1 + Sqrt[2] + Tan[(c + d*x)/4]))/(4*(-1 + Sqrt[2] + Tan[(c + d*x)/4])^2)))/(8*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]) + (Sqrt[(-1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*Sqrt[(1 - Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*(-1 + Sqrt[2] + Tan[(c + d*x)/4])^2*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(-1 + Sqrt[2] + Tan[(c + d*x)/4])]*(((2 + Sqrt[2])*(-1/4*((1 + Sqrt[2])*Sec[(c + d*x)/4]^2*(1 + Sqrt[2] + Tan[(c + d*x)/4]))/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])^2 + Sec[(c + d*x)/4]^2/(4*(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4]))))/(2^(3/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]*Sqrt[1 - (1 + Sqrt[2] + Tan[(c + d*x)/4])/(Sqrt[2]*(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4]))]*Sqrt[1 - (Sqrt[2]*(1 + Sqrt[2] + Tan[(c + d*x)/4]))/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]) + ((-2 + Sqrt[2])*(-1/4*((1 + Sqrt[2])*Sec[(c + d*x)/4]^2*(1 + Sqrt[2] + Tan[(c + d*x)/4]))/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])^2 + Sec[(c + d*x)/4]^2/(4*(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4]))))/(2^(3/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]*(1 - (1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4]))*Sqrt[1 - (1 + Sqrt[2] + Tan[(c + d*x)/4])/(Sqrt[2]*(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4]))]*Sqrt[1 - (Sqrt[2]*(1 + Sqrt[2] + Tan[(c + d*x)/4]))/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]) - ((2 + Sqrt[2])*(-1/4*((1 + Sqrt[2])*Sec[(c + d*x)/4]^2*(1 + Sqrt[2] + Tan[(c + d*x)/4]))/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])^2 + Sec[(c + d*x)/4]^2/(4*(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4]))))/(2^(3/4)*Sqrt[(1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])]*(1 + (1 + Sqrt[2] + Tan[(c + d*x)/4])/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4]))*Sqrt[1 - (1 + Sqrt[2] + Tan[(c + d*x)/4])/(Sqrt[2]*(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4]))]*Sqrt[1 - (Sqrt[2]*(1 + Sqrt[2] + Tan[(c + d*x)/4]))/(1 + (1 + Sqrt[2])*Tan[(c + d*x)/4])])))/4)))","C",0
192,1,5578,215,24.6478454,"\int \frac{\cot ^2(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^2/(a + a*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{71 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{32 \sqrt{2} a^{3/2} d}+\frac{7 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{32 a^2 d}-\frac{\cos ^2(c+d x) \cot (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{16 a^2 d}-\frac{13 \cos (c+d x) \cot (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{32 a^2 d}",1,"Result too large to show","C",0
193,1,5620,303,24.0082787,"\int \frac{\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^4/(a + a*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{533 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{256 \sqrt{2} a^{3/2} d}+\frac{277 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{384 a^3 d}-\frac{\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{48 a^3 d}-\frac{7 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{64 a^3 d}-\frac{81 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{128 a^3 d}-\frac{21 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{256 a^2 d}",1,"Result too large to show","C",0
194,1,5662,387,24.2091633,"\int \frac{\cot ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{16363 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{8192 \sqrt{2} a^{3/2} d}+\frac{12267 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{10240 a^4 d}-\frac{\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{128 a^4 d}-\frac{29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{768 a^4 d}-\frac{511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{3072 a^4 d}-\frac{2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{2048 a^4 d}-\frac{8171 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12288 a^3 d}-\frac{21 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{8192 a^2 d}",1,"Result too large to show","C",0
195,1,69,78,0.1226747,"\int \frac{\tan ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^5/(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 \left(\sec ^2(c+d x)-7 \sec (c+d x)-3 \sqrt{\sec (c+d x)+1} \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)-8\right)}{3 a^2 d \sqrt{a (\sec (c+d x)+1)}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a^4 d}-\frac{6 \sqrt{a \sec (c+d x)+a}}{a^3 d}",1,"(2*(-8 - 7*Sec[c + d*x] + Sec[c + d*x]^2 - 3*ArcTanh[Sqrt[1 + Sec[c + d*x]]]*Sqrt[1 + Sec[c + d*x]]))/(3*a^2*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
196,1,50,54,0.0571026,"\int \frac{\tan ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^3/(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 \left(\sqrt{\sec (c+d x)+1} \tanh ^{-1}\left(\sqrt{\sec (c+d x)+1}\right)-2\right)}{a^2 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}-\frac{4}{a^2 d \sqrt{a \sec (c+d x)+a}}",1,"(2*(-2 + ArcTanh[Sqrt[1 + Sec[c + d*x]]]*Sqrt[1 + Sec[c + d*x]]))/(a^2*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
197,1,40,78,0.0588037,"\int \frac{\tan (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]/(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\sec (c+d x)+1\right)}{3 a d (a (\sec (c+d x)+1))^{3/2}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{2}{a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{2}{3 a d (a \sec (c+d x)+a)^{3/2}}",1,"(2*Hypergeometric2F1[-3/2, 1, -1/2, 1 + Sec[c + d*x]])/(3*a*d*(a*(1 + Sec[c + d*x]))^(3/2))","C",1
198,1,60,144,0.0737575,"\int \frac{\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\frac{1}{2} (\sec (c+d x)+1)\right)-2 \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\sec (c+d x)+1\right)}{5 d (a (\sec (c+d x)+1))^{5/2}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}-\frac{7}{4 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{1}{2 a d (a \sec (c+d x)+a)^{3/2}}-\frac{1}{5 d (a \sec (c+d x)+a)^{5/2}}",1,"(Hypergeometric2F1[-5/2, 1, -3/2, (1 + Sec[c + d*x])/2] - 2*Hypergeometric2F1[-5/2, 1, -3/2, 1 + Sec[c + d*x]])/(5*d*(a*(1 + Sec[c + d*x]))^(5/2))","C",1
199,1,90,200,0.2118262,"\int \frac{\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]^3/(a + a*Sec[c + d*x])^(5/2),x]","\frac{a \left(-13 (\sec (c+d x)-1) \, _2F_1\left(-\frac{7}{2},1;-\frac{5}{2};\frac{1}{2} (\sec (c+d x)+1)\right)+8 (\sec (c+d x)-1) \, _2F_1\left(-\frac{7}{2},1;-\frac{5}{2};\sec (c+d x)+1\right)-14\right)}{28 d (\sec (c+d x)-1) (a (\sec (c+d x)+1))^{7/2}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{13 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{32 \sqrt{2} a^{5/2} d}+\frac{51}{32 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{5 a}{28 d (a \sec (c+d x)+a)^{7/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}+\frac{3}{40 d (a \sec (c+d x)+a)^{5/2}}+\frac{19}{48 a d (a \sec (c+d x)+a)^{3/2}}",1,"(a*(-14 - 13*Hypergeometric2F1[-7/2, 1, -5/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x]) + 8*Hypergeometric2F1[-7/2, 1, -5/2, 1 + Sec[c + d*x]]*(-1 + Sec[c + d*x])))/(28*d*(-1 + Sec[c + d*x])*(a*(1 + Sec[c + d*x]))^(7/2))","C",1
200,1,99,262,0.317298,"\int \frac{\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\cot ^4(c+d x) \left(263 (\sec (c+d x)-1)^2 \, _2F_1\left(-\frac{9}{2},1;-\frac{7}{2};\frac{1}{2} (\sec (c+d x)+1)\right)-64 (\sec (c+d x)-1)^2 \, _2F_1\left(-\frac{9}{2},1;-\frac{7}{2};\sec (c+d x)+1\right)+378 \sec (c+d x)-450\right)}{288 d (a (\sec (c+d x)+1))^{5/2}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right)}{a^{5/2} d}-\frac{263 \tanh ^{-1}\left(\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{512 \sqrt{2} a^{5/2} d}+\frac{199 a^2}{288 d (a \sec (c+d x)+a)^{9/2}}-\frac{21 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}-\frac{761}{512 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{135 a}{448 d (a \sec (c+d x)+a)^{7/2}}+\frac{7}{640 d (a \sec (c+d x)+a)^{5/2}}-\frac{83}{256 a d (a \sec (c+d x)+a)^{3/2}}",1,"(Cot[c + d*x]^4*(-450 + 263*Hypergeometric2F1[-9/2, 1, -7/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x])^2 - 64*Hypergeometric2F1[-9/2, 1, -7/2, 1 + Sec[c + d*x]]*(-1 + Sec[c + d*x])^2 + 378*Sec[c + d*x]))/(288*d*(a*(1 + Sec[c + d*x]))^(5/2))","C",1
201,1,447,127,6.0792493,"\int \frac{\tan ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^6/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\sqrt{2} \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(\frac{2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}-1\right)^3 \tan ^7(c+d x) \cot ^8\left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{\sec (c+d x)+1}\right)^{9/2} \left(\frac{2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \left(\frac{2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}-1\right)}+\frac{8 \tan ^6\left(\frac{1}{2} (c+d x)\right)}{5 \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^3 \left(\frac{2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}-1\right)^3}+\frac{4 \tan ^4\left(\frac{1}{2} (c+d x)\right)}{3 \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^2 \left(\frac{2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}-1\right)^2}+\frac{\sqrt{2} \tan \left(\frac{1}{2} (c+d x)\right) \sin ^{-1}\left(\frac{\sqrt{2} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}\right)}{\sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \sqrt{1-\frac{2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}\right)}{d \left(1-\frac{2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^{5/2} (a (\sec (c+d x)+1))^{5/2}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{2 \tan (c+d x)}{a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 \tan ^3(c+d x)}{3 a d (a \sec (c+d x)+a)^{3/2}}",1,"(Sqrt[2]*Cot[(c + d*x)/2]^8*((1 + Sec[c + d*x])^(-1))^(9/2)*Sqrt[1 + Tan[(c + d*x)/2]^2]*(-1 + (2*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2))^3*((Sqrt[2]*ArcSin[(Sqrt[2]*Tan[(c + d*x)/2])/Sqrt[1 + Tan[(c + d*x)/2]^2]]*Tan[(c + d*x)/2])/(Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 - (2*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (8*Tan[(c + d*x)/2]^6)/(5*(1 + Tan[(c + d*x)/2]^2)^3*(-1 + (2*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2))^3) + (4*Tan[(c + d*x)/2]^4)/(3*(1 + Tan[(c + d*x)/2]^2)^2*(-1 + (2*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2))^2) + (2*Tan[(c + d*x)/2]^2)/((1 + Tan[(c + d*x)/2]^2)*(-1 + (2*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2))))*Tan[c + d*x]^7)/(d*(a*(1 + Sec[c + d*x]))^(5/2)*(1 - (2*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2))^(5/2))","B",0
202,1,5491,113,23.7370832,"\int \frac{\tan ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^4/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{2 \tan (c+d x)}{a^2 d \sqrt{a \sec (c+d x)+a}}",1,"Result too large to show","C",0
203,1,5521,127,23.8108601,"\int \frac{\tan ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^2/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{2} a^{5/2} d}+\frac{\sin (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{2 a^2 d \sqrt{a \sec (c+d x)+a}}",1,"Result too large to show","C",0
204,1,5604,265,23.9308906,"\int \frac{\cot ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]^2/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{319 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{128 \sqrt{2} a^{5/2} d}+\frac{63 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{128 a^3 d}-\frac{\cos ^3(c+d x) \cot (c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{48 a^3 d}-\frac{19 \cos ^2(c+d x) \cot (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{192 a^3 d}-\frac{191 \cos (c+d x) \cot (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{a \sec (c+d x)+a}}{384 a^3 d}",1,"Result too large to show","C",0
205,1,5646,355,24.1574376,"\int \frac{\cot ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]^4/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{9683 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{4096 \sqrt{2} a^{5/2} d}+\frac{5587 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{6144 a^4 d}-\frac{\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{128 a^4 d}-\frac{9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{256 a^4 d}-\frac{145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{1024 a^4 d}-\frac{1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{3/2}}{2048 a^4 d}-\frac{1491 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{4096 a^3 d}",1,"Result too large to show","C",0
206,1,5688,439,24.2739018,"\int \frac{\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{74461 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{32768 \sqrt{2} a^{5/2} d}+\frac{58077 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{40960 a^5 d}-\frac{\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{320 a^5 d}-\frac{7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{512 a^5 d}-\frac{155 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{3072 a^5 d}-\frac{2473 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{12288 a^5 d}-\frac{9467 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sec (c+d x)+a)^{5/2}}{8192 a^5 d}-\frac{41693 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{49152 a^4 d}+\frac{8925 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{32768 a^3 d}",1,"Result too large to show","C",0
207,1,5584,177,24.1423638,"\int \frac{\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx","Integrate[Tan[e + f*x]^2/(a + a*Sec[e + f*x])^(9/2),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^{9/2} f}+\frac{91 \tan ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right)}{32 \sqrt{2} a^{9/2} f}+\frac{27 \tan (e+f x)}{32 a^3 f (a \sec (e+f x)+a)^{3/2}}+\frac{11 \tan (e+f x)}{24 a^2 f (a \sec (e+f x)+a)^{5/2}}+\frac{\tan (e+f x)}{3 a f (a \sec (e+f x)+a)^{7/2}}",1,"Result too large to show","C",0
208,0,0,125,1.2887993,"\int (a+a \sec (c+d x))^n (e \tan (c+d x))^m \, dx","Integrate[(a + a*Sec[c + d*x])^n*(e*Tan[c + d*x])^m,x]","\int (a+a \sec (c+d x))^n (e \tan (c+d x))^m \, dx","\frac{2^{m+n+1} (a \sec (c+d x)+a)^n (e \tan (c+d x))^{m+1} \left(\frac{1}{\sec (c+d x)+1}\right)^{m+n+1} F_1\left(\frac{m+1}{2};m+n,1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d e (m+1)}",1,"Integrate[(a + a*Sec[c + d*x])^n*(e*Tan[c + d*x])^m, x]","F",-1
209,1,391,243,6.2273352,"\int (a+a \sec (c+d x))^3 (e \tan (c+d x))^m \, dx","Integrate[(a + a*Sec[c + d*x])^3*(e*Tan[c + d*x])^m,x]","\frac{a^3 e \sec ^6\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^3 \left(-\tan ^2(c+d x)\right)^{\frac{1-m}{2}} (e \tan (c+d x))^{m-1} \left(\, _2F_1\left(\frac{3}{2},\frac{1-m}{2};\frac{5}{2};\sec ^2(c+d x)\right)+9 \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3}{2};\sec ^2(c+d x)\right)\right)}{24 d}+\frac{a^3 2^{-m-4} (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tan ^{-m}(c+d x) (e \tan (c+d x))^m \left(i 2^m (m+1) \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \cos (c+d x) \, _2F_1\left(1,m;m+1;-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)-i (m+1) \left(1+e^{2 i (c+d x)}\right)^m \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \cos (c+d x) \, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+3\ 2^{m+1} m \sin (c+d x) \tan ^m(c+d x)\right)}{d m (m+1)}","\frac{a^3 (e \tan (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d e (m+1)}+\frac{a^3 \sec ^3(c+d x) \cos ^2(c+d x)^{\frac{m+4}{2}} (e \tan (c+d x))^{m+1} \, _2F_1\left(\frac{m+1}{2},\frac{m+4}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{3 a^3 \sec (c+d x) \cos ^2(c+d x)^{\frac{m+2}{2}} (e \tan (c+d x))^{m+1} \, _2F_1\left(\frac{m+1}{2},\frac{m+2}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{3 a^3 (e \tan (c+d x))^{m+1}}{d e (m+1)}",1,"(a^3*e*(9*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (1 - m)/2, 3/2, Sec[c + d*x]^2] + Hypergeometric2F1[3/2, (1 - m)/2, 5/2, Sec[c + d*x]^2])*Sec[(c + d*x)/2]^6*(1 + Sec[c + d*x])^3*(e*Tan[c + d*x])^(-1 + m)*(-Tan[c + d*x]^2)^((1 - m)/2))/(24*d) + (2^(-4 - m)*a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]*(e*Tan[c + d*x])^m*(I*2^m*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*(1 + m)*Cos[c + d*x]*Hypergeometric2F1[1, m, 1 + m, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))] - I*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*(1 + E^((2*I)*(c + d*x)))^m*(1 + m)*Cos[c + d*x]*Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2] + 3*2^(1 + m)*m*Sin[c + d*x]*Tan[c + d*x]^m))/(d*m*(1 + m)*Tan[c + d*x]^m)","C",0
210,1,358,161,3.1965563,"\int (a+a \sec (c+d x))^2 (e \tan (c+d x))^m \, dx","Integrate[(a + a*Sec[c + d*x])^2*(e*Tan[c + d*x])^m,x]","\frac{a^2 (\cos (c+d x)+1)^2 \csc (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(-\tan ^2(c+d x)\right)^{\frac{1-m}{2}} (e \tan (c+d x))^m \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3}{2};\sec ^2(c+d x)\right)}{2 d}+\frac{a^2 2^{-m-3} (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tan ^{-m}(c+d x) (e \tan (c+d x))^m \left(i 2^m (m+1) \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \cos (c+d x) \, _2F_1\left(1,m;m+1;-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)-i (m+1) \left(1+e^{2 i (c+d x)}\right)^m \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \cos (c+d x) \, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+2^{m+1} m \sin (c+d x) \tan ^m(c+d x)\right)}{d m (m+1)}","\frac{a^2 (e \tan (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d e (m+1)}+\frac{2 a^2 \sec (c+d x) \cos ^2(c+d x)^{\frac{m+2}{2}} (e \tan (c+d x))^{m+1} \, _2F_1\left(\frac{m+1}{2},\frac{m+2}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a^2 (e \tan (c+d x))^{m+1}}{d e (m+1)}",1,"(a^2*(1 + Cos[c + d*x])^2*Csc[c + d*x]*Hypergeometric2F1[1/2, (1 - m)/2, 3/2, Sec[c + d*x]^2]*Sec[(c + d*x)/2]^4*(e*Tan[c + d*x])^m*(-Tan[c + d*x]^2)^((1 - m)/2))/(2*d) + (2^(-3 - m)*a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*Sec[c + d*x]*(e*Tan[c + d*x])^m*(I*2^m*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*(1 + m)*Cos[c + d*x]*Hypergeometric2F1[1, m, 1 + m, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))] - I*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*(1 + E^((2*I)*(c + d*x)))^m*(1 + m)*Cos[c + d*x]*Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2] + 2^(1 + m)*m*Sin[c + d*x]*Tan[c + d*x]^m))/(d*m*(1 + m)*Tan[c + d*x]^m)","C",1
211,1,105,129,0.7341055,"\int (a+a \sec (c+d x)) (e \tan (c+d x))^m \, dx","Integrate[(a + a*Sec[c + d*x])*(e*Tan[c + d*x])^m,x]","\frac{a (e \tan (c+d x))^m \left(\frac{\tan (c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{m+1}+\csc (c+d x) \left(-\tan ^2(c+d x)\right)^{\frac{1-m}{2}} \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3}{2};\sec ^2(c+d x)\right)\right)}{d}","\frac{a (e \tan (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d e (m+1)}+\frac{a \sec (c+d x) \cos ^2(c+d x)^{\frac{m+2}{2}} (e \tan (c+d x))^{m+1} \, _2F_1\left(\frac{m+1}{2},\frac{m+2}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}",1,"(a*(e*Tan[c + d*x])^m*((Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x])/(1 + m) + Csc[c + d*x]*Hypergeometric2F1[1/2, (1 - m)/2, 3/2, Sec[c + d*x]^2]*(-Tan[c + d*x]^2)^((1 - m)/2)))/d","A",1
212,0,0,130,0.5329948,"\int \frac{(e \tan (c+d x))^m}{a+a \sec (c+d x)} \, dx","Integrate[(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x]),x]","\int \frac{(e \tan (c+d x))^m}{a+a \sec (c+d x)} \, dx","\frac{e (e \tan (c+d x))^{m-1} \, _2F_1\left(1,\frac{m-1}{2};\frac{m+1}{2};-\tan ^2(c+d x)\right)}{a d (1-m)}-\frac{e \sec (c+d x) \cos ^2(c+d x)^{m/2} (e \tan (c+d x))^{m-1} \, _2F_1\left(\frac{m-1}{2},\frac{m}{2};\frac{m+1}{2};\sin ^2(c+d x)\right)}{a d (1-m)}",1,"Integrate[(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x]), x]","F",-1
213,1,329,169,6.9183083,"\int \frac{(e \tan (c+d x))^m}{(a+a \sec (c+d x))^2} \, dx","Integrate[(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x])^2,x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) (e \tan (c+d x))^m \left((m+1) \tan ^2\left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(m,\frac{m+3}{2};\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-3 (m+3) \, _2F_1\left(m,\frac{m+1}{2};\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^m}{2 a^2 d (m+1) (m+3)}+\frac{i 2^{1-m} \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x) \left(2^m \, _2F_1\left(1,m;m+1;-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)-\left(1+e^{2 i (c+d x)}\right)^m \, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)\right) \tan ^{-m}(c+d x) (e \tan (c+d x))^m}{d m (a \sec (c+d x)+a)^2}","-\frac{e^3 (e \tan (c+d x))^{m-3} \, _2F_1\left(1,\frac{m-3}{2};\frac{m-1}{2};-\tan ^2(c+d x)\right)}{a^2 d (3-m)}+\frac{2 e^3 \sec (c+d x) \cos ^2(c+d x)^{\frac{m-2}{2}} (e \tan (c+d x))^{m-3} \, _2F_1\left(\frac{m-3}{2},\frac{m-2}{2};\frac{m-1}{2};\sin ^2(c+d x)\right)}{a^2 d (3-m)}-\frac{e^3 (e \tan (c+d x))^{m-3}}{a^2 d (3-m)}",1,"((Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m*Tan[(c + d*x)/2]*(-3*(3 + m)*Hypergeometric2F1[m, (1 + m)/2, (3 + m)/2, Tan[(c + d*x)/2]^2] + (1 + m)*Hypergeometric2F1[m, (3 + m)/2, (5 + m)/2, Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)*(e*Tan[c + d*x])^m)/(2*a^2*d*(1 + m)*(3 + m)) + (I*2^(1 - m)*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*Cos[c/2 + (d*x)/2]^4*(2^m*Hypergeometric2F1[1, m, 1 + m, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))] - (1 + E^((2*I)*(c + d*x)))^m*Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2])*Sec[c + d*x]^2*(e*Tan[c + d*x])^m)/(d*m*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^m)","C",0
214,0,0,252,11.6022408,"\int \frac{(e \tan (c+d x))^m}{(a+a \sec (c+d x))^3} \, dx","Integrate[(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x])^3,x]","\int \frac{(e \tan (c+d x))^m}{(a+a \sec (c+d x))^3} \, dx","\frac{e^5 (e \tan (c+d x))^{m-5} \, _2F_1\left(1,\frac{m-5}{2};\frac{m-3}{2};-\tan ^2(c+d x)\right)}{a^3 d (5-m)}-\frac{e^5 \sec ^3(c+d x) \cos ^2(c+d x)^{\frac{m-2}{2}} (e \tan (c+d x))^{m-5} \, _2F_1\left(\frac{m-5}{2},\frac{m-2}{2};\frac{m-3}{2};\sin ^2(c+d x)\right)}{a^3 d (5-m)}-\frac{3 e^5 \sec (c+d x) \cos ^2(c+d x)^{\frac{m-4}{2}} (e \tan (c+d x))^{m-5} \, _2F_1\left(\frac{m-5}{2},\frac{m-4}{2};\frac{m-3}{2};\sin ^2(c+d x)\right)}{a^3 d (5-m)}+\frac{3 e^5 (e \tan (c+d x))^{m-5}}{a^3 d (5-m)}",1,"Integrate[(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x])^3, x]","F",-1
215,0,0,131,3.7571531,"\int (a+a \sec (c+d x))^{3/2} (e \tan (c+d x))^m \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)*(e*Tan[c + d*x])^m,x]","\int (a+a \sec (c+d x))^{3/2} (e \tan (c+d x))^m \, dx","\frac{2^{m+\frac{5}{2}} (a \sec (c+d x)+a)^{3/2} \left(\frac{1}{\sec (c+d x)+1}\right)^{m+\frac{5}{2}} (e \tan (c+d x))^{m+1} F_1\left(\frac{m+1}{2};m+\frac{3}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d e (m+1)}",1,"Integrate[(a + a*Sec[c + d*x])^(3/2)*(e*Tan[c + d*x])^m, x]","F",-1
216,0,0,131,7.9990675,"\int \sqrt{a+a \sec (c+d x)} (e \tan (c+d x))^m \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]*(e*Tan[c + d*x])^m,x]","\int \sqrt{a+a \sec (c+d x)} (e \tan (c+d x))^m \, dx","\frac{2^{m+\frac{3}{2}} \sqrt{a \sec (c+d x)+a} \left(\frac{1}{\sec (c+d x)+1}\right)^{m+\frac{3}{2}} (e \tan (c+d x))^{m+1} F_1\left(\frac{m+1}{2};m+\frac{1}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d e (m+1)}",1,"Integrate[Sqrt[a + a*Sec[c + d*x]]*(e*Tan[c + d*x])^m, x]","F",-1
217,0,0,131,2.349703,"\int \frac{(e \tan (c+d x))^m}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(e*Tan[c + d*x])^m/Sqrt[a + a*Sec[c + d*x]],x]","\int \frac{(e \tan (c+d x))^m}{\sqrt{a+a \sec (c+d x)}} \, dx","\frac{2^{m+\frac{1}{2}} \left(\frac{1}{\sec (c+d x)+1}\right)^{m+\frac{1}{2}} (e \tan (c+d x))^{m+1} F_1\left(\frac{m+1}{2};m-\frac{1}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d e (m+1) \sqrt{a \sec (c+d x)+a}}",1,"Integrate[(e*Tan[c + d*x])^m/Sqrt[a + a*Sec[c + d*x]], x]","F",-1
218,0,0,131,10.4648159,"\int \frac{(e \tan (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x])^(3/2),x]","\int \frac{(e \tan (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx","\frac{2^{m-\frac{1}{2}} \left(\frac{1}{\sec (c+d x)+1}\right)^{m-\frac{1}{2}} (e \tan (c+d x))^{m+1} F_1\left(\frac{m+1}{2};m-\frac{3}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d e (m+1) (a \sec (c+d x)+a)^{3/2}}",1,"Integrate[(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x])^(3/2), x]","F",-1
219,1,87,123,0.456422,"\int (a+a \sec (c+d x))^n \tan ^7(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^7,x]","\frac{(\sec (c+d x)+1)^4 (a (\sec (c+d x)+1))^n \left(\frac{\, _2F_1(1,n+4;n+5;\sec (c+d x)+1)}{n+4}+\frac{(\sec (c+d x)+1)^2}{n+6}-\frac{5 (\sec (c+d x)+1)}{n+5}+\frac{7}{n+4}\right)}{d}","\frac{(a \sec (c+d x)+a)^{n+6}}{a^6 d (n+6)}-\frac{5 (a \sec (c+d x)+a)^{n+5}}{a^5 d (n+5)}+\frac{(a \sec (c+d x)+a)^{n+4} \, _2F_1(1,n+4;n+5;\sec (c+d x)+1)}{a^4 d (n+4)}+\frac{7 (a \sec (c+d x)+a)^{n+4}}{a^4 d (n+4)}",1,"((1 + Sec[c + d*x])^4*(a*(1 + Sec[c + d*x]))^n*(7/(4 + n) + Hypergeometric2F1[1, 4 + n, 5 + n, 1 + Sec[c + d*x]]/(4 + n) - (5*(1 + Sec[c + d*x]))/(5 + n) + (1 + Sec[c + d*x])^2/(6 + n)))/d","A",1
220,1,72,97,0.1701473,"\int (a+a \sec (c+d x))^n \tan ^5(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^5,x]","\frac{(\sec (c+d x)+1)^3 (a (\sec (c+d x)+1))^n (-(n+4) \, _2F_1(1,n+3;n+4;\sec (c+d x)+1)+(n+3) \sec (c+d x)-2 n-9)}{d (n+3) (n+4)}","\frac{(a \sec (c+d x)+a)^{n+4}}{a^4 d (n+4)}-\frac{(a \sec (c+d x)+a)^{n+3} \, _2F_1(1,n+3;n+4;\sec (c+d x)+1)}{a^3 d (n+3)}-\frac{3 (a \sec (c+d x)+a)^{n+3}}{a^3 d (n+3)}",1,"((1 + Sec[c + d*x])^3*(a*(1 + Sec[c + d*x]))^n*(-9 - 2*n - (4 + n)*Hypergeometric2F1[1, 3 + n, 4 + n, 1 + Sec[c + d*x]] + (3 + n)*Sec[c + d*x]))/(d*(3 + n)*(4 + n))","A",1
221,1,49,69,0.0528283,"\int (a+a \sec (c+d x))^n \tan ^3(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^3,x]","\frac{(\sec (c+d x)+1)^2 (a (\sec (c+d x)+1))^n (\, _2F_1(1,n+2;n+3;\sec (c+d x)+1)+1)}{d (n+2)}","\frac{(a \sec (c+d x)+a)^{n+2} \, _2F_1(1,n+2;n+3;\sec (c+d x)+1)}{a^2 d (n+2)}+\frac{(a \sec (c+d x)+a)^{n+2}}{a^2 d (n+2)}",1,"((1 + Hypergeometric2F1[1, 2 + n, 3 + n, 1 + Sec[c + d*x]])*(1 + Sec[c + d*x])^2*(a*(1 + Sec[c + d*x]))^n)/(d*(2 + n))","A",1
222,1,43,43,0.045111,"\int (a+a \sec (c+d x))^n \tan (c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^n*Tan[c + d*x],x]","-\frac{(a (\sec (c+d x)+1))^{n+1} \, _2F_1(1,n+1;n+2;\sec (c+d x)+1)}{a d (n+1)}","-\frac{(a \sec (c+d x)+a)^{n+1} \, _2F_1(1,n+1;n+2;\sec (c+d x)+1)}{a d (n+1)}",1,"-((Hypergeometric2F1[1, 1 + n, 2 + n, 1 + Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(1 + n))/(a*d*(1 + n)))","A",1
223,1,57,74,0.0371344,"\int \cot (c+d x) (a+a \sec (c+d x))^n \, dx","Integrate[Cot[c + d*x]*(a + a*Sec[c + d*x])^n,x]","-\frac{(a (\sec (c+d x)+1))^n \left(\, _2F_1\left(1,n;n+1;\frac{1}{2} (\sec (c+d x)+1)\right)-2 \, _2F_1(1,n;n+1;\sec (c+d x)+1)\right)}{2 d n}","\frac{(a \sec (c+d x)+a)^n \, _2F_1(1,n;n+1;\sec (c+d x)+1)}{d n}-\frac{(a \sec (c+d x)+a)^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (\sec (c+d x)+1)\right)}{2 d n}",1,"-1/2*((Hypergeometric2F1[1, n, 1 + n, (1 + Sec[c + d*x])/2] - 2*Hypergeometric2F1[1, n, 1 + n, 1 + Sec[c + d*x]])*(a*(1 + Sec[c + d*x]))^n)/(d*n)","A",1
224,1,96,127,0.2566084,"\int \cot ^3(c+d x) (a+a \sec (c+d x))^n \, dx","Integrate[Cot[c + d*x]^3*(a + a*Sec[c + d*x])^n,x]","-\frac{a (a (\sec (c+d x)+1))^{n-1} \left((n-4) (\sec (c+d x)-1) \, _2F_1\left(1,n-1;n;\frac{1}{2} (\sec (c+d x)+1)\right)+4 (\sec (c+d x)-1) \, _2F_1(1,n-1;n;\sec (c+d x)+1)+2 n-2\right)}{4 d (n-1) (\sec (c+d x)-1)}","-\frac{a (4-n) (a \sec (c+d x)+a)^{n-1} \, _2F_1\left(1,n-1;n;\frac{1}{2} (\sec (c+d x)+1)\right)}{4 d (1-n)}+\frac{a (a \sec (c+d x)+a)^{n-1} \, _2F_1(1,n-1;n;\sec (c+d x)+1)}{d (1-n)}+\frac{a (a \sec (c+d x)+a)^{n-1}}{2 d (1-\sec (c+d x))}",1,"-1/4*(a*(-2 + 2*n + (-4 + n)*Hypergeometric2F1[1, -1 + n, n, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x]) + 4*Hypergeometric2F1[1, -1 + n, n, 1 + Sec[c + d*x]]*(-1 + Sec[c + d*x]))*(a*(1 + Sec[c + d*x]))^(-1 + n))/(d*(-1 + n)*(-1 + Sec[c + d*x]))","A",1
225,0,0,106,1.3465429,"\int (a+a \sec (c+d x))^n \tan ^4(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^4,x]","\int (a+a \sec (c+d x))^n \tan ^4(c+d x) \, dx","\frac{2^{n+5} \tan ^5(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n+5} (a \sec (c+d x)+a)^n F_1\left(\frac{5}{2};n+4,1;\frac{7}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{5 d}",1,"Integrate[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^4, x]","F",-1
226,1,910,106,11.3270458,"\int (a+a \sec (c+d x))^n \tan ^2(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^2,x]","\frac{(a (\sec (c+d x)+1))^n \left(-\frac{\, _2F_1\left(1-n,n+2;2-n;\frac{1}{2} \left(1-\tan \left(\frac{1}{2} (c+d x)\right)\right)\right) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^n \left(\tan \left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan \left(\frac{1}{2} (c+d x)\right)+1\right)^n (\sec (c+d x)+1)^{-n}}{n-1}-\frac{4 \, _2F_1\left(-n-1,n;-n;\frac{1}{2} \left(1-\tan \left(\frac{1}{2} (c+d x)\right)\right)\right) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^n \left(\tan \left(\frac{1}{2} (c+d x)\right)+1\right)^n (\sec (c+d x)+1)^{-n}}{(n+1) \left(\tan \left(\frac{1}{2} (c+d x)\right)-1\right)}-\frac{120 F_1\left(\frac{1}{2};n,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \cos ^2\left(\frac{1}{2} (c+d x)\right) \cos (c+d x) \sin (c+d x) \left(3 F_1\left(\frac{1}{2};n,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 \left(F_1\left(\frac{3}{2};n,2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-n F_1\left(\frac{3}{2};n+1,1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{45 F_1\left(\frac{1}{2};n,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right){}^2 (-2 \cos (c+d x) n+2 n+\cos (2 (c+d x))+1) \cos ^2\left(\frac{1}{2} (c+d x)\right)+40 \left(F_1\left(\frac{3}{2};n,2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-n F_1\left(\frac{3}{2};n+1,1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right){}^2 \cos (c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)+6 F_1\left(\frac{1}{2};n,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \sin ^2\left(\frac{1}{2} (c+d x)\right) \left(-48 \left(2 F_1\left(\frac{5}{2};n,3;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 n F_1\left(\frac{5}{2};n+1,2;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+n (n+1) F_1\left(\frac{5}{2};n+2,1;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right) \cot (c+d x) \csc (c+d x) \sin ^4\left(\frac{1}{2} (c+d x)\right)-5 F_1\left(\frac{3}{2};n,2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) (2 n-2 (n+2) \cos (c+d x)+\cos (2 (c+d x))+1)+5 n F_1\left(\frac{3}{2};n+1,1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) (2 n-2 (n+2) \cos (c+d x)+\cos (2 (c+d x))+1)\right)}\right)}{4 d}","\frac{2^{n+3} \tan ^3(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n+3} (a \sec (c+d x)+a)^n F_1\left(\frac{3}{2};n+2,1;\frac{5}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{3 d}",1,"((a*(1 + Sec[c + d*x]))^n*((-4*Hypergeometric2F1[-1 - n, n, -n, (1 - Tan[(c + d*x)/2])/2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*(1 + Tan[(c + d*x)/2])^n)/((1 + n)*(1 + Sec[c + d*x])^n*(-1 + Tan[(c + d*x)/2])) - (Hypergeometric2F1[1 - n, 2 + n, 2 - n, (1 - Tan[(c + d*x)/2])/2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*(-1 + Tan[(c + d*x)/2])*(1 + Tan[(c + d*x)/2])^n)/((-1 + n)*(1 + Sec[c + d*x])^n) - (120*AppellF1[1/2, n, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*Cos[c + d*x]*Sin[c + d*x]*(3*AppellF1[1/2, n, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*(AppellF1[3/2, n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - n*AppellF1[3/2, 1 + n, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))/(45*AppellF1[1/2, n, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]^2*Cos[(c + d*x)/2]^2*(1 + 2*n - 2*n*Cos[c + d*x] + Cos[2*(c + d*x)]) + 6*AppellF1[1/2, n, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sin[(c + d*x)/2]^2*(-5*AppellF1[3/2, n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + 2*n - 2*(2 + n)*Cos[c + d*x] + Cos[2*(c + d*x)]) + 5*n*AppellF1[3/2, 1 + n, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + 2*n - 2*(2 + n)*Cos[c + d*x] + Cos[2*(c + d*x)]) - 48*(2*AppellF1[5/2, n, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*n*AppellF1[5/2, 1 + n, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*(1 + n)*AppellF1[5/2, 2 + n, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cot[c + d*x]*Csc[c + d*x]*Sin[(c + d*x)/2]^4) + 40*(AppellF1[3/2, n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - n*AppellF1[3/2, 1 + n, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])^2*Cos[c + d*x]*Sin[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2)))/(4*d)","B",0
227,1,893,102,4.18286,"\int \cot ^2(c+d x) (a+a \sec (c+d x))^n \, dx","Integrate[Cot[c + d*x]^2*(a + a*Sec[c + d*x])^n,x]","\frac{(a (\sec (c+d x)+1))^n \left(-2^n \cot \left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(-\frac{1}{2},n;\frac{1}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^n \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^n (\sec (c+d x)+1)^{-n}+2^n \, _2F_1\left(\frac{1}{2},n;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^n \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^n \tan \left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^{-n}-\frac{60 F_1\left(\frac{1}{2};n,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \cos ^2\left(\frac{1}{2} (c+d x)\right) \cos (c+d x) \sin (c+d x) \left(3 F_1\left(\frac{1}{2};n,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 \left(F_1\left(\frac{3}{2};n,2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-n F_1\left(\frac{3}{2};n+1,1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{45 F_1\left(\frac{1}{2};n,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right){}^2 (-2 \cos (c+d x) n+2 n+\cos (2 (c+d x))+1) \cos ^2\left(\frac{1}{2} (c+d x)\right)+40 \left(F_1\left(\frac{3}{2};n,2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-n F_1\left(\frac{3}{2};n+1,1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right){}^2 \cos (c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)+6 F_1\left(\frac{1}{2};n,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \sin ^2\left(\frac{1}{2} (c+d x)\right) \left(-48 \left(2 F_1\left(\frac{5}{2};n,3;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 n F_1\left(\frac{5}{2};n+1,2;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+n (n+1) F_1\left(\frac{5}{2};n+2,1;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right) \cot (c+d x) \csc (c+d x) \sin ^4\left(\frac{1}{2} (c+d x)\right)-5 F_1\left(\frac{3}{2};n,2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) (2 n-2 (n+2) \cos (c+d x)+\cos (2 (c+d x))+1)+5 n F_1\left(\frac{3}{2};n+1,1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) (2 n-2 (n+2) \cos (c+d x)+\cos (2 (c+d x))+1)\right)}\right)}{2 d}","-\frac{2^{n-1} \cot (c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n-1} (a \sec (c+d x)+a)^n F_1\left(-\frac{1}{2};n-2,1;\frac{1}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d}",1,"((a*(1 + Sec[c + d*x]))^n*(-((2^n*Cot[(c + d*x)/2]*Hypergeometric2F1[-1/2, n, 1/2, Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n)/(1 + Sec[c + d*x])^n) + (2^n*Hypergeometric2F1[1/2, n, 3/2, Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*Tan[(c + d*x)/2])/(1 + Sec[c + d*x])^n - (60*AppellF1[1/2, n, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*Cos[c + d*x]*Sin[c + d*x]*(3*AppellF1[1/2, n, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*(AppellF1[3/2, n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - n*AppellF1[3/2, 1 + n, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))/(45*AppellF1[1/2, n, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]^2*Cos[(c + d*x)/2]^2*(1 + 2*n - 2*n*Cos[c + d*x] + Cos[2*(c + d*x)]) + 6*AppellF1[1/2, n, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sin[(c + d*x)/2]^2*(-5*AppellF1[3/2, n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + 2*n - 2*(2 + n)*Cos[c + d*x] + Cos[2*(c + d*x)]) + 5*n*AppellF1[3/2, 1 + n, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + 2*n - 2*(2 + n)*Cos[c + d*x] + Cos[2*(c + d*x)]) - 48*(2*AppellF1[5/2, n, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*n*AppellF1[5/2, 1 + n, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*(1 + n)*AppellF1[5/2, 2 + n, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cot[c + d*x]*Csc[c + d*x]*Sin[(c + d*x)/2]^4) + 40*(AppellF1[3/2, n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - n*AppellF1[3/2, 1 + n, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])^2*Cos[c + d*x]*Sin[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2)))/(2*d)","B",0
228,0,0,106,1.704569,"\int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx","Integrate[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^n,x]","\int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx","-\frac{2^{n-3} \cot ^3(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n-3} (a \sec (c+d x)+a)^n F_1\left(-\frac{3}{2};n-4,1;-\frac{1}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{3 d}",1,"Integrate[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^n, x]","F",-1
229,1,2072,114,19.0199302,"\int (a+a \sec (c+d x))^n \tan ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^(3/2),x]","\text{Result too large to show}","\frac{2^{n+\frac{7}{2}} \tan ^{\frac{5}{2}}(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n+\frac{5}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{5}{4};n+\frac{3}{2},1;\frac{9}{4};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{5 d}",1,"(2^(1 + n)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*(a*(1 + Sec[c + d*x]))^n*(-1 + Tan[(c + d*x)/2])^(-1/2 - n)*(-2*AppellF1[1/4, 1/2 + n, 1, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/2 + n)*(-1 + Tan[(c + d*x)/2])^(1/2 + n) + (AppellF1[1/2, 1/2 + n, 3/2 + n, 3/2, Tan[(c + d*x)/2], -Tan[(c + d*x)/2]] + AppellF1[1/2, 3/2 + n, 1/2 + n, 3/2, Tan[(c + d*x)/2], -Tan[(c + d*x)/2]])*(1 - Tan[(c + d*x)/2])^(1/2 + n)*(-1 + Tan[(c + d*x)/2]^2)^(1/2 + n))*Tan[c + d*x]^2)/(d*((2^n*Sec[c + d*x]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*(-1 + Tan[(c + d*x)/2])^(-1/2 - n)*(-2*AppellF1[1/4, 1/2 + n, 1, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/2 + n)*(-1 + Tan[(c + d*x)/2])^(1/2 + n) + (AppellF1[1/2, 1/2 + n, 3/2 + n, 3/2, Tan[(c + d*x)/2], -Tan[(c + d*x)/2]] + AppellF1[1/2, 3/2 + n, 1/2 + n, 3/2, Tan[(c + d*x)/2], -Tan[(c + d*x)/2]])*(1 - Tan[(c + d*x)/2])^(1/2 + n)*(-1 + Tan[(c + d*x)/2]^2)^(1/2 + n)))/Sqrt[Tan[c + d*x]] + 2^n*(-1/2 - n)*Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*(-1 + Tan[(c + d*x)/2])^(-3/2 - n)*(-2*AppellF1[1/4, 1/2 + n, 1, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/2 + n)*(-1 + Tan[(c + d*x)/2])^(1/2 + n) + (AppellF1[1/2, 1/2 + n, 3/2 + n, 3/2, Tan[(c + d*x)/2], -Tan[(c + d*x)/2]] + AppellF1[1/2, 3/2 + n, 1/2 + n, 3/2, Tan[(c + d*x)/2], -Tan[(c + d*x)/2]])*(1 - Tan[(c + d*x)/2])^(1/2 + n)*(-1 + Tan[(c + d*x)/2]^2)^(1/2 + n))*Sqrt[Tan[c + d*x]] + 2^(1 + n)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*(-1 + Tan[(c + d*x)/2])^(-1/2 - n)*(-((1/2 + n)*AppellF1[1/4, 1/2 + n, 1, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/2 + n)*(-1 + Tan[(c + d*x)/2])^(-1/2 + n)) - 2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/2 + n)*(-1 + Tan[(c + d*x)/2])^(1/2 + n)*(-1/5*(AppellF1[5/4, 1/2 + n, 2, 9/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + ((1/2 + n)*AppellF1[5/4, 3/2 + n, 1, 9/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) - 2*(1/2 + n)*AppellF1[1/4, 1/2 + n, 1, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(-1/2 + n)*(-1 + Tan[(c + d*x)/2])^(1/2 + n)*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (1/2 + n)*(AppellF1[1/2, 1/2 + n, 3/2 + n, 3/2, Tan[(c + d*x)/2], -Tan[(c + d*x)/2]] + AppellF1[1/2, 3/2 + n, 1/2 + n, 3/2, Tan[(c + d*x)/2], -Tan[(c + d*x)/2]])*Sec[(c + d*x)/2]^2*(1 - Tan[(c + d*x)/2])^(1/2 + n)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)^(-1/2 + n) - ((1/2 + n)*(AppellF1[1/2, 1/2 + n, 3/2 + n, 3/2, Tan[(c + d*x)/2], -Tan[(c + d*x)/2]] + AppellF1[1/2, 3/2 + n, 1/2 + n, 3/2, Tan[(c + d*x)/2], -Tan[(c + d*x)/2]])*Sec[(c + d*x)/2]^2*(1 - Tan[(c + d*x)/2])^(-1/2 + n)*(-1 + Tan[(c + d*x)/2]^2)^(1/2 + n))/2 + (-1/6*((3/2 + n)*AppellF1[3/2, 1/2 + n, 5/2 + n, 5/2, Tan[(c + d*x)/2], -Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2) + ((3/2 + n)*AppellF1[3/2, 5/2 + n, 1/2 + n, 5/2, Tan[(c + d*x)/2], -Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/6)*(1 - Tan[(c + d*x)/2])^(1/2 + n)*(-1 + Tan[(c + d*x)/2]^2)^(1/2 + n))*Sqrt[Tan[c + d*x]] + 2^(1 + n)*n*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(-1 + n)*(-1 + Tan[(c + d*x)/2])^(-1/2 - n)*(-2*AppellF1[1/4, 1/2 + n, 1, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/2 + n)*(-1 + Tan[(c + d*x)/2])^(1/2 + n) + (AppellF1[1/2, 1/2 + n, 3/2 + n, 3/2, Tan[(c + d*x)/2], -Tan[(c + d*x)/2]] + AppellF1[1/2, 3/2 + n, 1/2 + n, 3/2, Tan[(c + d*x)/2], -Tan[(c + d*x)/2]])*(1 - Tan[(c + d*x)/2])^(1/2 + n)*(-1 + Tan[(c + d*x)/2]^2)^(1/2 + n))*Sqrt[Tan[c + d*x]]*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x])))","B",0
230,1,238,114,2.2292415,"\int (a+a \sec (c+d x))^n \sqrt{\tan (c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^n*Sqrt[Tan[c + d*x]],x]","\frac{56 \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\tan (c+d x)} (a (\sec (c+d x)+1))^n F_1\left(\frac{3}{4};n+\frac{1}{2},1;\frac{7}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{d \left(6 (\cos (c+d x)-1) \left(2 F_1\left(\frac{7}{4};n+\frac{1}{2},2;\frac{11}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-(2 n+1) F_1\left(\frac{7}{4};n+\frac{3}{2},1;\frac{11}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)+21 (\cos (c+d x)+1) F_1\left(\frac{3}{4};n+\frac{1}{2},1;\frac{7}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}","\frac{2^{n+\frac{5}{2}} \tan ^{\frac{3}{2}}(c+d x) \left(\frac{1}{\sec (c+d x)+1}\right)^{n+\frac{3}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{3}{4};n+\frac{1}{2},1;\frac{7}{4};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{3 d}",1,"(56*AppellF1[3/4, 1/2 + n, 1, 7/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^3*(a*(1 + Sec[c + d*x]))^n*Sin[(c + d*x)/2]*Sqrt[Tan[c + d*x]])/(d*(6*(2*AppellF1[7/4, 1/2 + n, 2, 11/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - (1 + 2*n)*AppellF1[7/4, 3/2 + n, 1, 11/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(-1 + Cos[c + d*x]) + 21*AppellF1[3/4, 1/2 + n, 1, 7/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x])))","B",0
231,1,229,111,1.6124863,"\int \frac{(a+a \sec (c+d x))^n}{\sqrt{\tan (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^n/Sqrt[Tan[c + d*x]],x]","\frac{10 \cos (c+d x) (\cos (c+d x)+1) \sqrt{\tan (c+d x)} (a (\sec (c+d x)+1))^n F_1\left(\frac{1}{4};n-\frac{1}{2},1;\frac{5}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{d \left(2 (\cos (c+d x)-1) \left(2 F_1\left(\frac{5}{4};n-\frac{1}{2},2;\frac{9}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+(1-2 n) F_1\left(\frac{5}{4};n+\frac{1}{2},1;\frac{9}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)+5 (\cos (c+d x)+1) F_1\left(\frac{1}{4};n-\frac{1}{2},1;\frac{5}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}","\frac{2^{n+\frac{3}{2}} \sqrt{\tan (c+d x)} \left(\frac{1}{\sec (c+d x)+1}\right)^{n+\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{1}{4};n-\frac{1}{2},1;\frac{5}{4};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d}",1,"(10*AppellF1[1/4, -1/2 + n, 1, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[c + d*x]*(1 + Cos[c + d*x])*(a*(1 + Sec[c + d*x]))^n*Sqrt[Tan[c + d*x]])/(d*(2*(2*AppellF1[5/4, -1/2 + n, 2, 9/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (1 - 2*n)*AppellF1[5/4, 1/2 + n, 1, 9/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(-1 + Cos[c + d*x]) + 5*AppellF1[1/4, -1/2 + n, 1, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x])))","B",0
232,1,2164,112,15.8647076,"\int \frac{(a+a \sec (c+d x))^n}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^n/Tan[c + d*x]^(3/2),x]","\text{Result too large to show}","-\frac{2^{n+\frac{1}{2}} \left(\frac{1}{\sec (c+d x)+1}\right)^{n-\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left(-\frac{1}{4};n-\frac{3}{2},1;\frac{3}{4};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right)}{d \sqrt{\tan (c+d x)}}",1,"-1/21*(2^(1/2 + n)*Cot[c + d*x]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*(a*(1 + Sec[c + d*x]))^n*(21*Hypergeometric2F1[-1/4, -1/2 + n, 3/4, Tan[(c + d*x)/2]^2] + 7*AppellF1[3/4, -1/2 + n, 1, 7/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2 + 7*Hypergeometric2F1[3/4, -1/2 + n, 7/4, Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2 - 3*AppellF1[7/4, -1/2 + n, 1, 11/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^4))/(d*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*((2^(-1/2 + n)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*Sec[c + d*x]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*(21*Hypergeometric2F1[-1/4, -1/2 + n, 3/4, Tan[(c + d*x)/2]^2] + 7*AppellF1[3/4, -1/2 + n, 1, 7/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2 + 7*Hypergeometric2F1[3/4, -1/2 + n, 7/4, Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2 - 3*AppellF1[7/4, -1/2 + n, 1, 11/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^4))/(21*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[c + d*x]^(3/2)) + (2^(-1/2 + n)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x]))*(21*Hypergeometric2F1[-1/4, -1/2 + n, 3/4, Tan[(c + d*x)/2]^2] + 7*AppellF1[3/4, -1/2 + n, 1, 7/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2 + 7*Hypergeometric2F1[3/4, -1/2 + n, 7/4, Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2 - 3*AppellF1[7/4, -1/2 + n, 1, 11/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^4))/(21*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*Sqrt[Tan[c + d*x]]) - (2^(1/2 + n)*n*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1 + n)*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(21*Hypergeometric2F1[-1/4, -1/2 + n, 3/4, Tan[(c + d*x)/2]^2] + 7*AppellF1[3/4, -1/2 + n, 1, 7/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2 + 7*Hypergeometric2F1[3/4, -1/2 + n, 7/4, Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2 - 3*AppellF1[7/4, -1/2 + n, 1, 11/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^4))/(21*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Tan[c + d*x]]) - (2^(1/2 + n)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*(7*AppellF1[3/4, -1/2 + n, 1, 7/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 7*Hypergeometric2F1[3/4, -1/2 + n, 7/4, Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - 6*AppellF1[7/4, -1/2 + n, 1, 11/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^3 + 7*Tan[(c + d*x)/2]^2*((-3*AppellF1[7/4, -1/2 + n, 2, 11/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/7 + (3*(-1/2 + n)*AppellF1[7/4, 1/2 + n, 1, 11/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/7) - 3*Tan[(c + d*x)/2]^4*((-7*AppellF1[11/4, -1/2 + n, 2, 15/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/11 + (7*(-1/2 + n)*AppellF1[11/4, 1/2 + n, 1, 15/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/11) + (21*Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(Hypergeometric2F1[-1/4, -1/2 + n, 3/4, Tan[(c + d*x)/2]^2] - (1 - Tan[(c + d*x)/2]^2)^(1/2 - n)))/4 + (21*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]*(-Hypergeometric2F1[3/4, -1/2 + n, 7/4, Tan[(c + d*x)/2]^2] + (1 - Tan[(c + d*x)/2]^2)^(1/2 - n)))/4))/(21*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Tan[c + d*x]]) - (2^(1/2 + n)*n*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(-1 + n)*(21*Hypergeometric2F1[-1/4, -1/2 + n, 3/4, Tan[(c + d*x)/2]^2] + 7*AppellF1[3/4, -1/2 + n, 1, 7/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2 + 7*Hypergeometric2F1[3/4, -1/2 + n, 7/4, Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2 - 3*AppellF1[7/4, -1/2 + n, 1, 11/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^4)*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(21*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Tan[c + d*x]])))","B",0
233,1,185,320,3.7971861,"\int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx","Integrate[(e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x]),x]","-\frac{a \sec (c+d x) (e \cot (c+d x))^{5/2} \left(\sqrt{\cot (c+d x)} \left(-3 \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+4 (\cos (c+d x)+1) \cot (c+d x)+3 \sqrt{\sin (2 (c+d x))} \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)+2 \sqrt[4]{-1} \sin (2 (c+d x)) \sqrt{\csc ^2(c+d x)} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\cot (c+d x)}\right)\right|-1\right)\right)}{6 d \cot ^{\frac{5}{2}}(c+d x)}","\frac{a \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}{\sqrt{2} d}-\frac{a \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}{\sqrt{2} d}+\frac{a \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{a \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 \tan (c+d x) (a \sec (c+d x)+a) (e \cot (c+d x))^{5/2}}{3 d}-\frac{a \sqrt{\sin (2 c+2 d x)} \tan ^2(c+d x) \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) (e \cot (c+d x))^{5/2}}{3 d}",1,"-1/6*(a*(e*Cot[c + d*x])^(5/2)*Sec[c + d*x]*(Sqrt[Cot[c + d*x]]*(4*(1 + Cos[c + d*x])*Cot[c + d*x] - 3*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]] + 3*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]]) + 2*(-1)^(1/4)*Sqrt[Csc[c + d*x]^2]*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Cot[c + d*x]]], -1]*Sin[2*(c + d*x)]))/(d*Cot[c + d*x]^(5/2))","C",1
234,1,191,346,1.1964333,"\int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx","Integrate[(e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x]),x]","\frac{a e (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{e \cot (c+d x)} \left(8 \cot ^2(c+d x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\cot ^2(c+d x)\right)+3 \sqrt{\csc ^2(c+d x)} \left(-4 \cos ^2(c+d x)-4 \cos (c+d x)+\sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+\sqrt{\sin (2 (c+d x))} \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{12 d \sqrt{\csc ^2(c+d x)}}","\frac{a \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} d}-\frac{a \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} d}-\frac{a \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{a \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 a \sin (c+d x) \tan ^2(c+d x) (e \cot (c+d x))^{3/2}}{d}-\frac{2 \tan (c+d x) (a \sec (c+d x)+a) (e \cot (c+d x))^{3/2}}{d}-\frac{2 a \sin (c+d x) \tan (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) (e \cot (c+d x))^{3/2}}{d \sqrt{\sin (2 c+2 d x)}}",1,"(a*e*(1 + Cos[c + d*x])*Sqrt[e*Cot[c + d*x]]*Sec[(c + d*x)/2]^2*Sec[c + d*x]*(8*Cot[c + d*x]^2*Hypergeometric2F1[3/4, 3/2, 7/4, -Cot[c + d*x]^2] + 3*Sqrt[Csc[c + d*x]^2]*(-4*Cos[c + d*x] - 4*Cos[c + d*x]^2 + ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]] + Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]])))/(12*d*Sqrt[Csc[c + d*x]^2])","C",1
235,1,169,274,1.8298155,"\int \sqrt{e \cot (c+d x)} (a+a \sec (c+d x)) \, dx","Integrate[Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x]),x]","\frac{a (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{e \cot (c+d x)} \left(\sqrt{\sin (2 (c+d x))} \sqrt{\csc ^2(c+d x)} \left(\log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)-\sin ^{-1}(\cos (c+d x)-\sin (c+d x))\right)+4 \sqrt[4]{-1} \sqrt{\cot (c+d x)} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\cot (c+d x)}\right)\right|-1\right)\right)}{4 d \sqrt{\csc ^2(c+d x)}}","-\frac{a \sqrt{\tan (c+d x)} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \sqrt{e \cot (c+d x)}}{\sqrt{2} d}+\frac{a \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}{\sqrt{2} d}-\frac{a \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{a \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{a \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \cot (c+d x)}}{d}",1,"(a*(1 + Cos[c + d*x])*Sqrt[e*Cot[c + d*x]]*Sec[(c + d*x)/2]^2*Sec[c + d*x]*(4*(-1)^(1/4)*Sqrt[Cot[c + d*x]]*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Cot[c + d*x]]], -1] + Sqrt[Csc[c + d*x]^2]*(-ArcSin[Cos[c + d*x] - Sin[c + d*x]] + Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sqrt[Sin[2*(c + d*x)]]))/(4*d*Sqrt[Csc[c + d*x]^2])","C",1
236,1,189,299,1.6019368,"\int \frac{a+a \sec (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])/Sqrt[e*Cot[c + d*x]],x]","\frac{a (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(8 \cot ^3(c+d x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\cot ^2(c+d x)\right)-3 \cot (c+d x) \sqrt{\csc ^2(c+d x)} \left(2 \cos (2 (c+d x))+\sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+\sqrt{\sin (2 (c+d x))} \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)-2\right)\right)}{12 d \sqrt{\csc ^2(c+d x)} \sqrt{e \cot (c+d x)}}","\frac{2 a \sin (c+d x)}{d \sqrt{e \cot (c+d x)}}-\frac{a \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{a \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{a \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{a \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{2 a \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{d \sqrt{\sin (2 c+2 d x)} \sqrt{e \cot (c+d x)}}",1,"(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sec[c + d*x]*(8*Cot[c + d*x]^3*Hypergeometric2F1[3/4, 3/2, 7/4, -Cot[c + d*x]^2] - 3*Cot[c + d*x]*Sqrt[Csc[c + d*x]^2]*(-2 + 2*Cos[2*(c + d*x)] + ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]] + Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]])))/(12*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Csc[c + d*x]^2])","C",1
237,1,224,320,2.6009531,"\int \frac{a+a \sec (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sec[c + d*x])/(e*Cot[c + d*x])^(3/2),x]","\frac{a (\cos (c+d x)+1) \cos (2 (c+d x)) \csc (c+d x) \sqrt{\csc ^2(c+d x)} \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{\csc ^2(c+d x)} \left(12 \cos (c+d x)+3 \sqrt{\sin (2 (c+d x))} \cot (c+d x) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-3 \sqrt{\sin (2 (c+d x))} \cot (c+d x) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+4\right)-4 \sqrt[4]{-1} \cot ^{\frac{3}{2}}(c+d x) F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\cot (c+d x)}\right)\right|-1\right)\right)}{12 d \left(\cot ^2(c+d x)-1\right) (e \cot (c+d x))^{3/2}}","\frac{a \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{2 \cot (c+d x) (a \sec (c+d x)+3 a)}{3 d (e \cot (c+d x))^{3/2}}+\frac{a \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a \sqrt{\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d (e \cot (c+d x))^{3/2}}",1,"(a*(1 + Cos[c + d*x])*Cos[2*(c + d*x)]*Csc[c + d*x]*Sqrt[Csc[c + d*x]^2]*Sec[(c + d*x)/2]^2*(-4*(-1)^(1/4)*Cot[c + d*x]^(3/2)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Cot[c + d*x]]], -1] + Sqrt[Csc[c + d*x]^2]*(4 + 12*Cos[c + d*x] + 3*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Cot[c + d*x]*Sqrt[Sin[2*(c + d*x)]] - 3*Cot[c + d*x]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]])))/(12*d*(e*Cot[c + d*x])^(3/2)*(-1 + Cot[c + d*x]^2))","C",1
238,1,93,357,2.3328633,"\int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx","Integrate[(e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2,x]","-\frac{2 a^2 e \cos ^4\left(\frac{1}{2} (c+d x)\right) (e \cot (c+d x))^{3/2} \sec ^4\left(\frac{1}{2} \cot ^{-1}(\cot (c+d x))\right) \left(2 \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\tan ^2(c+d x)\right)-\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)+2\right)}{3 d}","\frac{a^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}{\sqrt{2} d}-\frac{a^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}{\sqrt{2} d}-\frac{4 a^2 \tan (c+d x) (e \cot (c+d x))^{5/2}}{3 d}+\frac{a^2 \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{a^2 \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{4 a^2 \tan (c+d x) \sec (c+d x) (e \cot (c+d x))^{5/2}}{3 d}-\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \tan ^2(c+d x) \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) (e \cot (c+d x))^{5/2}}{3 d}",1,"(-2*a^2*e*Cos[(c + d*x)/2]^4*(e*Cot[c + d*x])^(3/2)*(2 + 2*Hypergeometric2F1[-3/4, 1/2, 1/4, -Tan[c + d*x]^2] - Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])*Sec[ArcCot[Cot[c + d*x]]/2]^4)/(3*d)","C",0
239,1,220,343,5.2474259,"\int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx","Integrate[(e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2,x]","-\frac{a^2 \cos ^4\left(\frac{1}{2} (c+d x)\right) (e \cot (c+d x))^{3/2} \sec ^4\left(\frac{1}{2} \cot ^{-1}(\cot (c+d x))\right) \left(16 \sqrt{\cot (c+d x)} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\tan ^2(c+d x)\right)+16 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{4 d \cot ^{\frac{3}{2}}(c+d x)}","-\frac{4 a^2 \sin (c+d x) (e \cot (c+d x))^{3/2}}{d}+\frac{a^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} d}-\frac{a^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} d}-\frac{4 a^2 \tan (c+d x) (e \cot (c+d x))^{3/2}}{d}-\frac{a^2 \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{a^2 \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{4 a^2 \sin (c+d x) \tan (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) (e \cot (c+d x))^{3/2}}{d \sqrt{\sin (2 c+2 d x)}}",1,"-1/4*(a^2*Cos[(c + d*x)/2]^4*(e*Cot[c + d*x])^(3/2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 16*Sqrt[Cot[c + d*x]] + 16*Sqrt[Cot[c + d*x]]*Hypergeometric2F1[-1/4, 1/2, 3/4, -Tan[c + d*x]^2] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])*Sec[ArcCot[Cot[c + d*x]]/2]^4)/(d*Cot[c + d*x]^(3/2))","C",0
240,1,118,311,1.845112,"\int \sqrt{e \cot (c+d x)} (a+a \sec (c+d x))^2 \, dx","Integrate[Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 e (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} \cot ^{-1}(\cot (c+d x))\right) \left(6 \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(c+d x)\right)-2 \cot ^2(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)+3 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)\right)}{6 d \sqrt{e \cot (c+d x)}}","-\frac{a^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}{\sqrt{2} d}+\frac{a^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}{\sqrt{2} d}+\frac{2 a^2 \tan (c+d x) \sqrt{e \cot (c+d x)}}{d}-\frac{a^2 \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{a^2 \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \cot (c+d x)}}{d}",1,"(a^2*e*(1 + Cos[c + d*x])^2*(3*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + 6*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[c + d*x]^2] - 2*Cot[c + d*x]^2*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])*Sec[ArcCot[Cot[c + d*x]]/2]^4)/(6*d*Sqrt[e*Cot[c + d*x]])","C",0
241,1,221,339,11.4632182,"\int \frac{(a+a \sec (c+d x))^2}{\sqrt{e \cot (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^2/Sqrt[e*Cot[c + d*x]],x]","\frac{a^2 \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sec ^4\left(\frac{1}{2} \cot ^{-1}(\cot (c+d x))\right) \left(8 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2(c+d x)\right)+4 \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)+3 \sqrt{2} \cot ^{\frac{3}{2}}(c+d x) \left(\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{3 d \sqrt{e \cot (c+d x)}}","\frac{4 a^2 \sin (c+d x)}{d \sqrt{e \cot (c+d x)}}-\frac{a^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{a^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{2 a^2 \tan (c+d x)}{3 d \sqrt{e \cot (c+d x)}}+\frac{a^2 \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{a^2 \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{4 a^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{d \sqrt{\sin (2 c+2 d x)} \sqrt{e \cot (c+d x)}}",1,"(a^2*Cos[(c + d*x)/2]^5*(4*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2] + 8*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] + 3*Sqrt[2]*Cot[c + d*x]^(3/2)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))*Sec[c + d*x]*Sec[ArcCot[Cot[c + d*x]]/2]^4*Sin[(c + d*x)/2])/(3*d*Sqrt[e*Cot[c + d*x]])","C",0
242,1,127,375,6.2741429,"\int \frac{(a+a \sec (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sec[c + d*x])^2/(e*Cot[c + d*x])^(3/2),x]","\frac{a^2 \sin ^2(c+d x) (\sec (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} \cot ^{-1}(\cot (c+d x))\right) \left(\, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(c+d x)\right)+2 \left(\, _2F_1\left(\frac{1}{2},\frac{5}{4};\frac{9}{4};-\tan ^2(c+d x)\right)+5 \cot ^2(c+d x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)\right)\right)}{10 d e \sqrt{e \cot (c+d x)}}","\frac{2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac{a^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}+\frac{4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}+\frac{a^2 \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{a^2 \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{2 a^2 \sqrt{\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 d (e \cot (c+d x))^{3/2}}",1,"(a^2*(Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2] + 2*(5*Cot[c + d*x]^2*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + Hypergeometric2F1[1/2, 5/4, 9/4, -Tan[c + d*x]^2]))*(1 + Sec[c + d*x])^2*Sec[ArcCot[Cot[c + d*x]]/2]^4*Sin[c + d*x]^2)/(10*d*e*Sqrt[e*Cot[c + d*x]])","C",0
243,1,316,405,4.0482212,"\int \frac{(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx","Integrate[(e*Cot[c + d*x])^(3/2)/(a + a*Sec[c + d*x]),x]","-\frac{e \sin ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\sqrt{\sec ^2(c+d x)}+1\right) \sqrt{e \cot (c+d x)} \left(-40 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2(c+d x)\right)+24 \cot ^4(c+d x) \, _2F_1\left(-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\tan ^2(c+d x)\right)-120 \cot ^2(c+d x) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\tan ^2(c+d x)\right)-24 \cot ^4(c+d x)+120 \cot ^2(c+d x)+15 \sqrt{2} \cot ^{\frac{3}{2}}(c+d x) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-15 \sqrt{2} \cot ^{\frac{3}{2}}(c+d x) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+30 \sqrt{2} \cot ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-30 \sqrt{2} \cot ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{30 a d}","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} a d}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}{\sqrt{2} a d}+\frac{2 \cot (c+d x) (1-\sec (c+d x)) (e \cot (c+d x))^{3/2}}{5 a d}-\frac{\tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d}+\frac{\tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d}-\frac{6 \sin (c+d x) \tan ^2(c+d x) (e \cot (c+d x))^{3/2}}{5 a d}-\frac{2 \tan (c+d x) (5-3 \sec (c+d x)) (e \cot (c+d x))^{3/2}}{5 a d}+\frac{6 \sin (c+d x) \tan (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) (e \cot (c+d x))^{3/2}}{5 a d \sqrt{\sin (2 c+2 d x)}}",1,"-1/30*(e*Sqrt[e*Cot[c + d*x]]*(30*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*Cot[c + d*x]^(3/2) - 30*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*Cot[c + d*x]^(3/2) + 120*Cot[c + d*x]^2 - 24*Cot[c + d*x]^4 + 24*Cot[c + d*x]^4*Hypergeometric2F1[-5/4, -1/2, -1/4, -Tan[c + d*x]^2] - 120*Cot[c + d*x]^2*Hypergeometric2F1[-1/2, -1/4, 3/4, -Tan[c + d*x]^2] - 40*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] + 15*Sqrt[2]*Cot[c + d*x]^(3/2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 15*Sqrt[2]*Cot[c + d*x]^(3/2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])*Sec[c + d*x]*(1 + Sqrt[Sec[c + d*x]^2])*Sin[(c + d*x)/2]^2)/(a*d)","C",0
244,1,135,325,1.7119172,"\int \frac{\sqrt{e \cot (c+d x)}}{a+a \sec (c+d x)} \, dx","Integrate[Sqrt[e*Cot[c + d*x]]/(a + a*Sec[c + d*x]),x]","-\frac{4 \sin ^2\left(\frac{1}{2} (c+d x)\right) \csc (c+d x) \left(\sqrt{\sec ^2(c+d x)}+1\right) \sqrt{e \cot (c+d x)} \left(3 \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(c+d x)\right)+\cot ^2(c+d x) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-1\right)+\cot ^2(c+d x) \, _2F_1\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{4};-\tan ^2(c+d x)\right)\right)}{3 a d}","-\frac{\sqrt{\tan (c+d x)} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \sqrt{e \cot (c+d x)}}{\sqrt{2} a d}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}{\sqrt{2} a d}+\frac{2 \cot (c+d x) (1-\sec (c+d x)) \sqrt{e \cot (c+d x)}}{3 a d}-\frac{\sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d}+\frac{\sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d}-\frac{\sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \cot (c+d x)}}{3 a d}",1,"(-4*Sqrt[e*Cot[c + d*x]]*Csc[c + d*x]*(Cot[c + d*x]^2*Hypergeometric2F1[-3/4, -1/2, 1/4, -Tan[c + d*x]^2] + 3*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[c + d*x]^2] + Cot[c + d*x]^2*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))*(1 + Sqrt[Sec[c + d*x]^2])*Sin[(c + d*x)/2]^2)/(3*a*d)","C",0
245,1,249,347,3.7059914,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+a \sec (c+d x))} \, dx","Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])),x]","-\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right) \csc (c+d x) \left(\sqrt{\sec ^2(c+d x)}+1\right) \left(8 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2(c+d x)\right)+24 \cot ^2(c+d x) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\tan ^2(c+d x)\right)-3 \cot ^{\frac{3}{2}}(c+d x) \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{6 a d \sqrt{e \cot (c+d x)}}","\frac{2 \sin (c+d x)}{a d \sqrt{e \cot (c+d x)}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{2 \cot (c+d x) (1-\sec (c+d x))}{a d \sqrt{e \cot (c+d x)}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a d \sqrt{\sin (2 c+2 d x)} \sqrt{e \cot (c+d x)}}",1,"-1/6*(Csc[c + d*x]*(24*Cot[c + d*x]^2*Hypergeometric2F1[-1/2, -1/4, 3/4, -Tan[c + d*x]^2] + 8*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] - 3*Cot[c + d*x]^(3/2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))*(1 + Sqrt[Sec[c + d*x]^2])*Sin[(c + d*x)/2]^2)/(a*d*Sqrt[e*Cot[c + d*x]])","C",0
246,1,112,290,8.8628295,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))} \, dx","Integrate[1/((e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x])),x]","\frac{4 \sin ^2\left(\frac{1}{2} (c+d x)\right) \cot ^2(c+d x) \csc (c+d x) \left(\sqrt{\sec ^2(c+d x)}+1\right) \left(3 \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(c+d x)\right)+\cot ^2(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)\right)}{3 a d (e \cot (c+d x))^{3/2}}","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{\sqrt{\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a d (e \cot (c+d x))^{3/2}}",1,"(4*Cot[c + d*x]^2*Csc[c + d*x]*(3*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[c + d*x]^2] + Cot[c + d*x]^2*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])*(1 + Sqrt[Sec[c + d*x]^2])*Sin[(c + d*x)/2]^2)/(3*a*d*(e*Cot[c + d*x])^(3/2))","C",0
247,1,194,325,62.6742454,"\int \frac{1}{(e \cot (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx","Integrate[1/((e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])),x]","-\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\sqrt{\sec ^2(c+d x)}+1\right) \sqrt{e \cot (c+d x)} \left(3 \sqrt{2} \cot ^{\frac{3}{2}}(c+d x) \left(\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)-8 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2(c+d x)\right)\right)}{6 a d e^3}","\frac{2 \cos (c+d x) \cot (c+d x)}{a d (e \cot (c+d x))^{5/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}+\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac{2 \cos (c+d x) \cot ^2(c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a d \sqrt{\sin (2 c+2 d x)} (e \cot (c+d x))^{5/2}}",1,"-1/6*(Sqrt[e*Cot[c + d*x]]*(-8*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] + 3*Sqrt[2]*Cot[c + d*x]^(3/2)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))*Sec[c + d*x]*(1 + Sqrt[Sec[c + d*x]^2])*Sin[(c + d*x)/2]^2)/(a*d*e^3)","C",0
248,1,130,335,15.9292254,"\int \frac{1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx","Integrate[1/((e*Cot[c + d*x])^(7/2)*(a + a*Sec[c + d*x])),x]","-\frac{4 \sin ^2\left(\frac{1}{2} (c+d x)\right) \csc (c+d x) \left(\sqrt{\sec ^2(c+d x)}+1\right) \sqrt{e \cot (c+d x)} \left(-3 \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\tan ^2(c+d x)\right)+3 \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(c+d x)\right)+\cot ^2(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)+3\right)}{3 a d e^4}","-\frac{2 \cot ^3(c+d x) (3-\sec (c+d x))}{3 a d (e \cot (c+d x))^{7/2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}-\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}+\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}-\frac{\sqrt{\sin (2 c+2 d x)} \cot ^3(c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a d (e \cot (c+d x))^{7/2}}",1,"(-4*Sqrt[e*Cot[c + d*x]]*Csc[c + d*x]*(3 - 3*Hypergeometric2F1[-1/2, 1/4, 5/4, -Tan[c + d*x]^2] + 3*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[c + d*x]^2] + Cot[c + d*x]^2*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])*(1 + Sqrt[Sec[c + d*x]^2])*Sin[(c + d*x)/2]^2)/(3*a*d*e^4)","C",0
249,1,261,371,19.7426777,"\int \frac{1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx","Integrate[1/((e*Cot[c + d*x])^(9/2)*(a + a*Sec[c + d*x])),x]","\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\sqrt{\sec ^2(c+d x)}+1\right) \sqrt{e \cot (c+d x)} \left(8 \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2(c+d x)\right)-8 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2(c+d x)\right)+3 \sqrt{2} \cot ^{\frac{3}{2}}(c+d x) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-3 \sqrt{2} \cot ^{\frac{3}{2}}(c+d x) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+6 \sqrt{2} \cot ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-6 \sqrt{2} \cot ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-8\right)}{6 a d e^5}","-\frac{6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac{2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{6 \cos (c+d x) \cot ^4(c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{5 a d \sqrt{\sin (2 c+2 d x)} (e \cot (c+d x))^{9/2}}",1,"(Sqrt[e*Cot[c + d*x]]*(-8 + 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*Cot[c + d*x]^(3/2) - 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*Cot[c + d*x]^(3/2) + 8*Hypergeometric2F1[-1/2, 3/4, 7/4, -Tan[c + d*x]^2] - 8*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] + 3*Sqrt[2]*Cot[c + d*x]^(3/2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 3*Sqrt[2]*Cot[c + d*x]^(3/2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])*Sec[c + d*x]*(1 + Sqrt[Sec[c + d*x]^2])*Sin[(c + d*x)/2]^2)/(6*a*d*e^5)","C",0
250,0,0,413,13.8459325,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx","Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","\int \frac{1}{\sqrt{e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx","-\frac{4 \cot ^3(c+d x)}{5 a^2 d \sqrt{e \cot (c+d x)}}+\frac{2 \cot (c+d x)}{a^2 d \sqrt{e \cot (c+d x)}}-\frac{12 \cos (c+d x) \cot (c+d x)}{5 a^2 d \sqrt{e \cot (c+d x)}}+\frac{4 \cot ^2(c+d x) \csc (c+d x)}{5 a^2 d \sqrt{e \cot (c+d x)}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}-\frac{12 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{5 a^2 d \sqrt{\sin (2 c+2 d x)} \sqrt{e \cot (c+d x)}}",1,"Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])^2), x]","F",-1
251,0,0,359,10.1214173,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx","Integrate[1/((e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2),x]","\int \frac{1}{(e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx","-\frac{4 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac{4 \cot ^2(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{3/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac{2 \sqrt{\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a^2 d (e \cot (c+d x))^{3/2}}",1,"Integrate[1/((e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2), x]","F",-1
252,0,0,355,5.2222971,"\int \frac{1}{(e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2} \, dx","Integrate[1/((e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2),x]","\int \frac{1}{(e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2} \, dx","-\frac{4 \cot ^3(c+d x)}{a^2 d (e \cot (c+d x))^{5/2}}+\frac{4 \cos (c+d x) \cot ^3(c+d x)}{a^2 d (e \cot (c+d x))^{5/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}+\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}+\frac{4 \cos (c+d x) \cot ^2(c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a^2 d \sqrt{\sin (2 c+2 d x)} (e \cot (c+d x))^{5/2}}",1,"Integrate[1/((e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2), x]","F",-1
253,0,0,321,5.0990798,"\int \frac{1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx","Integrate[1/((e*Cot[c + d*x])^(7/2)*(a + a*Sec[c + d*x])^2),x]","\int \frac{1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx","\frac{2 \cot ^3(c+d x)}{a^2 d (e \cot (c+d x))^{7/2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}-\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}+\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}-\frac{2 \sqrt{\sin (2 c+2 d x)} \cot ^3(c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a^2 d (e \cot (c+d x))^{7/2}}",1,"Integrate[1/((e*Cot[c + d*x])^(7/2)*(a + a*Sec[c + d*x])^2), x]","F",-1
254,0,0,357,55.4350109,"\int \frac{1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))^2} \, dx","Integrate[1/((e*Cot[c + d*x])^(9/2)*(a + a*Sec[c + d*x])^2),x]","\int \frac{1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))^2} \, dx","\frac{2 \cot ^3(c+d x)}{3 a^2 d (e \cot (c+d x))^{9/2}}-\frac{4 \cos (c+d x) \cot ^3(c+d x)}{a^2 d (e \cot (c+d x))^{9/2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac{4 \cos (c+d x) \cot ^4(c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{a^2 d \sqrt{\sin (2 c+2 d x)} (e \cot (c+d x))^{9/2}}",1,"Integrate[1/((e*Cot[c + d*x])^(9/2)*(a + a*Sec[c + d*x])^2), x]","F",-1
255,0,0,389,13.5337916,"\int \frac{1}{(e \cot (c+d x))^{11/2} (a+a \sec (c+d x))^2} \, dx","Integrate[1/((e*Cot[c + d*x])^(11/2)*(a + a*Sec[c + d*x])^2),x]","\int \frac{1}{(e \cot (c+d x))^{11/2} (a+a \sec (c+d x))^2} \, dx","\frac{2 \cot ^5(c+d x)}{a^2 d (e \cot (c+d x))^{11/2}}+\frac{2 \cot ^3(c+d x)}{5 a^2 d (e \cot (c+d x))^{11/2}}-\frac{4 \cot ^4(c+d x) \csc (c+d x)}{3 a^2 d (e \cot (c+d x))^{11/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d \tan ^{\frac{11}{2}}(c+d x) (e \cot (c+d x))^{11/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d \tan ^{\frac{11}{2}}(c+d x) (e \cot (c+d x))^{11/2}}+\frac{\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{11}{2}}(c+d x) (e \cot (c+d x))^{11/2}}-\frac{\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} a^2 d \tan ^{\frac{11}{2}}(c+d x) (e \cot (c+d x))^{11/2}}+\frac{2 \sqrt{\sin (2 c+2 d x)} \cot ^5(c+d x) \csc (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{3 a^2 d (e \cot (c+d x))^{11/2}}",1,"Integrate[1/((e*Cot[c + d*x])^(11/2)*(a + a*Sec[c + d*x])^2), x]","F",-1
256,1,106,111,0.4706901,"\int (a+b \sec (c+d x)) \tan ^7(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])*Tan[c + d*x]^7,x]","\frac{a \left(2 \tan ^6(c+d x)-3 \tan ^4(c+d x)+6 \tan ^2(c+d x)+12 \log (\cos (c+d x))\right)}{12 d}+\frac{b \sec ^7(c+d x)}{7 d}-\frac{3 b \sec ^5(c+d x)}{5 d}+\frac{b \sec ^3(c+d x)}{d}-\frac{b \sec (c+d x)}{d}","\frac{\tan ^6(c+d x) (7 a+6 b \sec (c+d x))}{42 d}-\frac{\tan ^4(c+d x) (35 a+24 b \sec (c+d x))}{140 d}+\frac{\tan ^2(c+d x) (35 a+16 b \sec (c+d x))}{70 d}+\frac{a \log (\cos (c+d x))}{d}-\frac{16 b \sec (c+d x)}{35 d}",1,"-((b*Sec[c + d*x])/d) + (b*Sec[c + d*x]^3)/d - (3*b*Sec[c + d*x]^5)/(5*d) + (b*Sec[c + d*x]^7)/(7*d) + (a*(12*Log[Cos[c + d*x]] + 6*Tan[c + d*x]^2 - 3*Tan[c + d*x]^4 + 2*Tan[c + d*x]^6))/(12*d)","A",1
257,1,82,84,0.2011841,"\int (a+b \sec (c+d x)) \tan ^5(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])*Tan[c + d*x]^5,x]","-\frac{a \left(-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right)}{4 d}+\frac{b \sec ^5(c+d x)}{5 d}-\frac{2 b \sec ^3(c+d x)}{3 d}+\frac{b \sec (c+d x)}{d}","\frac{\tan ^4(c+d x) (5 a+4 b \sec (c+d x))}{20 d}-\frac{\tan ^2(c+d x) (15 a+8 b \sec (c+d x))}{30 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{8 b \sec (c+d x)}{15 d}",1,"(b*Sec[c + d*x])/d - (2*b*Sec[c + d*x]^3)/(3*d) + (b*Sec[c + d*x]^5)/(5*d) - (a*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/(4*d)","A",1
258,1,55,55,0.1364889,"\int (a+b \sec (c+d x)) \tan ^3(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])*Tan[c + d*x]^3,x]","\frac{a \left(\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)}{2 d}+\frac{b \sec ^3(c+d x)}{3 d}-\frac{b \sec (c+d x)}{d}","\frac{\tan ^2(c+d x) (3 a+2 b \sec (c+d x))}{6 d}+\frac{a \log (\cos (c+d x))}{d}-\frac{2 b \sec (c+d x)}{3 d}",1,"-((b*Sec[c + d*x])/d) + (b*Sec[c + d*x]^3)/(3*d) + (a*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/(2*d)","A",1
259,1,25,25,0.0149828,"\int (a+b \sec (c+d x)) \tan (c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])*Tan[c + d*x],x]","\frac{b \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}","\frac{b \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Log[Cos[c + d*x]])/d) + (b*Sec[c + d*x])/d","A",1
260,1,60,43,0.0379479,"\int \cot (c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + b*Sec[c + d*x]),x]","\frac{a (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+\frac{b \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{b \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}","\frac{(a+b) \log (1-\cos (c+d x))}{2 d}+\frac{(a-b) \log (\cos (c+d x)+1)}{2 d}",1,"-((b*Log[Cos[c/2 + (d*x)/2]])/d) + (b*Log[Sin[c/2 + (d*x)/2]])/d + (a*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","A",1
261,1,114,72,1.3495086,"\int \cot ^3(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + b*Sec[c + d*x]),x]","-\frac{a \left(\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right)}{2 d}-\frac{b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}","-\frac{(2 a+b) \log (1-\cos (c+d x))}{4 d}-\frac{(2 a-b) \log (\cos (c+d x)+1)}{4 d}-\frac{\cot ^2(c+d x) (a+b \sec (c+d x))}{2 d}",1,"-1/8*(b*Csc[(c + d*x)/2]^2)/d + (b*Log[Cos[(c + d*x)/2]])/(2*d) - (b*Log[Sin[(c + d*x)/2]])/(2*d) - (a*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]]))/(2*d) + (b*Sec[(c + d*x)/2]^2)/(8*d)","A",1
262,1,166,102,0.3956825,"\int \cot ^5(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + b*Sec[c + d*x]),x]","\frac{a \left(-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right)}{4 d}-\frac{b \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{5 b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{b \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{3 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{3 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}","\frac{(8 a+3 b) \log (1-\cos (c+d x))}{16 d}+\frac{(8 a-3 b) \log (\cos (c+d x)+1)}{16 d}-\frac{\cot ^4(c+d x) (a+b \sec (c+d x))}{4 d}+\frac{\cot ^2(c+d x) (4 a+3 b \sec (c+d x))}{8 d}",1,"(5*b*Csc[(c + d*x)/2]^2)/(32*d) - (b*Csc[(c + d*x)/2]^4)/(64*d) - (3*b*Log[Cos[(c + d*x)/2]])/(8*d) + (3*b*Log[Sin[(c + d*x)/2]])/(8*d) + (a*(2*Cot[c + d*x]^2 - Cot[c + d*x]^4 + 4*Log[Cos[c + d*x]] + 4*Log[Tan[c + d*x]]))/(4*d) - (5*b*Sec[(c + d*x)/2]^2)/(32*d) + (b*Sec[(c + d*x)/2]^4)/(64*d)","A",1
263,1,216,130,0.617608,"\int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]^7*(a + b*Sec[c + d*x]),x]","-\frac{a \left(2 \cot ^6(c+d x)-3 \cot ^4(c+d x)+6 \cot ^2(c+d x)+12 \log (\tan (c+d x))+12 \log (\cos (c+d x))\right)}{12 d}-\frac{b \csc ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}+\frac{b \csc ^4\left(\frac{1}{2} (c+d x)\right)}{32 d}-\frac{11 b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{b \sec ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}-\frac{b \sec ^4\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{11 b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}+\frac{5 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}","-\frac{(16 a+5 b) \log (1-\cos (c+d x))}{32 d}-\frac{(16 a-5 b) \log (\cos (c+d x)+1)}{32 d}-\frac{\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac{\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac{\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d}",1,"(-11*b*Csc[(c + d*x)/2]^2)/(64*d) + (b*Csc[(c + d*x)/2]^4)/(32*d) - (b*Csc[(c + d*x)/2]^6)/(384*d) + (5*b*Log[Cos[(c + d*x)/2]])/(16*d) - (5*b*Log[Sin[(c + d*x)/2]])/(16*d) - (a*(6*Cot[c + d*x]^2 - 3*Cot[c + d*x]^4 + 2*Cot[c + d*x]^6 + 12*Log[Cos[c + d*x]] + 12*Log[Tan[c + d*x]]))/(12*d) + (11*b*Sec[(c + d*x)/2]^2)/(64*d) - (b*Sec[(c + d*x)/2]^4)/(32*d) + (b*Sec[(c + d*x)/2]^6)/(384*d)","A",1
264,1,103,102,1.0690947,"\int (a+b \sec (c+d x)) \tan ^6(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])*Tan[c + d*x]^6,x]","\frac{\frac{1}{8} \tan (c+d x) \sec ^5(c+d x) (1168 a \cos (c+d x)+568 a \cos (3 (c+d x))+184 a \cos (5 (c+d x))+140 b \cos (2 (c+d x))+165 b \cos (4 (c+d x))+295 b)-240 a \tan ^{-1}(\tan (c+d x))-75 b \tanh ^{-1}(\sin (c+d x))}{240 d}","\frac{\tan ^5(c+d x) (6 a+5 b \sec (c+d x))}{30 d}-\frac{\tan ^3(c+d x) (8 a+5 b \sec (c+d x))}{24 d}+\frac{\tan (c+d x) (16 a+5 b \sec (c+d x))}{16 d}-a x-\frac{5 b \tanh ^{-1}(\sin (c+d x))}{16 d}",1,"(-240*a*ArcTan[Tan[c + d*x]] - 75*b*ArcTanh[Sin[c + d*x]] + ((295*b + 1168*a*Cos[c + d*x] + 140*b*Cos[2*(c + d*x)] + 568*a*Cos[3*(c + d*x)] + 165*b*Cos[4*(c + d*x)] + 184*a*Cos[5*(c + d*x)])*Sec[c + d*x]^5*Tan[c + d*x])/8)/(240*d)","A",1
265,1,79,73,0.5962681,"\int (a+b \sec (c+d x)) \tan ^4(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])*Tan[c + d*x]^4,x]","\frac{-\left(\tan (c+d x) \sec ^3(c+d x) (32 a \cos (c+d x)+16 a \cos (3 (c+d x))+15 b \cos (2 (c+d x))+3 b)\right)+48 a \tan ^{-1}(\tan (c+d x))+18 b \tanh ^{-1}(\sin (c+d x))}{48 d}","\frac{\tan ^3(c+d x) (4 a+3 b \sec (c+d x))}{12 d}-\frac{\tan (c+d x) (8 a+3 b \sec (c+d x))}{8 d}+a x+\frac{3 b \tanh ^{-1}(\sin (c+d x))}{8 d}",1,"(48*a*ArcTan[Tan[c + d*x]] + 18*b*ArcTanh[Sin[c + d*x]] - (3*b + 32*a*Cos[c + d*x] + 15*b*Cos[2*(c + d*x)] + 16*a*Cos[3*(c + d*x)])*Sec[c + d*x]^3*Tan[c + d*x])/(48*d)","A",1
266,1,60,45,0.0257515,"\int (a+b \sec (c+d x)) \tan ^2(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])*Tan[c + d*x]^2,x]","-\frac{a \tan ^{-1}(\tan (c+d x))}{d}+\frac{a \tan (c+d x)}{d}-\frac{b \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b \tan (c+d x) \sec (c+d x)}{2 d}","\frac{\tan (c+d x) (2 a+b \sec (c+d x))}{2 d}-a x-\frac{b \tanh ^{-1}(\sin (c+d x))}{2 d}",1,"-((a*ArcTan[Tan[c + d*x]])/d) - (b*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Tan[c + d*x])/d + (b*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
267,1,43,26,0.0219477,"\int \cot ^2(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + b*Sec[c + d*x]),x]","-\frac{a \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}-\frac{b \csc (c+d x)}{d}","-\frac{\cot (c+d x) (a+b \sec (c+d x))}{d}-a x",1,"-((b*Csc[c + d*x])/d) - (a*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d","C",1
268,1,62,55,0.029707,"\int \cot ^4(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + b*Sec[c + d*x]),x]","-\frac{a \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)}{3 d}-\frac{b \csc ^3(c+d x)}{3 d}+\frac{b \csc (c+d x)}{d}","-\frac{\cot ^3(c+d x) (a+b \sec (c+d x))}{3 d}+\frac{\cot (c+d x) (3 a+2 b \sec (c+d x))}{3 d}+a x",1,"(b*Csc[c + d*x])/d - (b*Csc[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d)","C",1
269,1,79,84,0.0431148,"\int \cot ^6(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]^6*(a + b*Sec[c + d*x]),x]","-\frac{a \cot ^5(c+d x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(c+d x)\right)}{5 d}-\frac{b \csc ^5(c+d x)}{5 d}+\frac{2 b \csc ^3(c+d x)}{3 d}-\frac{b \csc (c+d x)}{d}","-\frac{\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac{\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{15 d}-\frac{\cot (c+d x) (15 a+8 b \sec (c+d x))}{15 d}-a x",1,"-((b*Csc[c + d*x])/d) + (2*b*Csc[c + d*x]^3)/(3*d) - (b*Csc[c + d*x]^5)/(5*d) - (a*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/(5*d)","C",1
270,1,92,111,0.052719,"\int \cot ^8(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cot[c + d*x]^8*(a + b*Sec[c + d*x]),x]","-\frac{a \cot ^7(c+d x) \, _2F_1\left(-\frac{7}{2},1;-\frac{5}{2};-\tan ^2(c+d x)\right)}{7 d}-\frac{b \csc ^7(c+d x)}{7 d}+\frac{3 b \csc ^5(c+d x)}{5 d}-\frac{b \csc ^3(c+d x)}{d}+\frac{b \csc (c+d x)}{d}","-\frac{\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac{\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}-\frac{\cot ^3(c+d x) (35 a+24 b \sec (c+d x))}{105 d}+\frac{\cot (c+d x) (35 a+16 b \sec (c+d x))}{35 d}+a x",1,"(b*Csc[c + d*x])/d - (b*Csc[c + d*x]^3)/d + (3*b*Csc[c + d*x]^5)/(5*d) - (b*Csc[c + d*x]^7)/(7*d) - (a*Cot[c + d*x]^7*Hypergeometric2F1[-7/2, 1, -5/2, -Tan[c + d*x]^2])/(7*d)","C",1
271,1,173,185,0.4248305,"\int (a+b \sec (c+d x))^2 \tan ^9(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^9,x]","\frac{315 \left(a^2-4 b^2\right) \sec ^8(c+d x)-840 \left(2 a^2-3 b^2\right) \sec ^6(c+d x)+1260 \left(3 a^2-2 b^2\right) \sec ^4(c+d x)-1260 \left(4 a^2-b^2\right) \sec ^2(c+d x)-2520 a^2 \log (\cos (c+d x))+560 a b \sec ^9(c+d x)-2880 a b \sec ^7(c+d x)+6048 a b \sec ^5(c+d x)-6720 a b \sec ^3(c+d x)+5040 a b \sec (c+d x)+252 b^2 \sec ^{10}(c+d x)}{2520 d}","\frac{a^2 \sec ^8(c+d x)}{8 d}-\frac{2 a^2 \sec ^6(c+d x)}{3 d}+\frac{3 a^2 \sec ^4(c+d x)}{2 d}-\frac{2 a^2 \sec ^2(c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^9(c+d x)}{9 d}-\frac{8 a b \sec ^7(c+d x)}{7 d}+\frac{12 a b \sec ^5(c+d x)}{5 d}-\frac{8 a b \sec ^3(c+d x)}{3 d}+\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \tan ^{10}(c+d x)}{10 d}",1,"(-2520*a^2*Log[Cos[c + d*x]] + 5040*a*b*Sec[c + d*x] - 1260*(4*a^2 - b^2)*Sec[c + d*x]^2 - 6720*a*b*Sec[c + d*x]^3 + 1260*(3*a^2 - 2*b^2)*Sec[c + d*x]^4 + 6048*a*b*Sec[c + d*x]^5 - 840*(2*a^2 - 3*b^2)*Sec[c + d*x]^6 - 2880*a*b*Sec[c + d*x]^7 + 315*(a^2 - 4*b^2)*Sec[c + d*x]^8 + 560*a*b*Sec[c + d*x]^9 + 252*b^2*Sec[c + d*x]^10)/(2520*d)","A",1
272,1,138,149,0.3524132,"\int (a+b \sec (c+d x))^2 \tan ^7(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^7,x]","\frac{140 \left(a^2-3 b^2\right) \sec ^6(c+d x)-630 \left(a^2-b^2\right) \sec ^4(c+d x)+420 \left(3 a^2-b^2\right) \sec ^2(c+d x)+840 a^2 \log (\cos (c+d x))+240 a b \sec ^7(c+d x)-1008 a b \sec ^5(c+d x)+1680 a b \sec ^3(c+d x)-1680 a b \sec (c+d x)+105 b^2 \sec ^8(c+d x)}{840 d}","\frac{a^2 \sec ^6(c+d x)}{6 d}-\frac{3 a^2 \sec ^4(c+d x)}{4 d}+\frac{3 a^2 \sec ^2(c+d x)}{2 d}+\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^7(c+d x)}{7 d}-\frac{6 a b \sec ^5(c+d x)}{5 d}+\frac{2 a b \sec ^3(c+d x)}{d}-\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \tan ^8(c+d x)}{8 d}",1,"(840*a^2*Log[Cos[c + d*x]] - 1680*a*b*Sec[c + d*x] + 420*(3*a^2 - b^2)*Sec[c + d*x]^2 + 1680*a*b*Sec[c + d*x]^3 - 630*(a^2 - b^2)*Sec[c + d*x]^4 - 1008*a*b*Sec[c + d*x]^5 + 140*(a^2 - 3*b^2)*Sec[c + d*x]^6 + 240*a*b*Sec[c + d*x]^7 + 105*b^2*Sec[c + d*x]^8)/(840*d)","A",1
273,1,105,115,0.2658111,"\int (a+b \sec (c+d x))^2 \tan ^5(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^5,x]","\frac{15 \left(a^2-2 b^2\right) \sec ^4(c+d x)+30 \left(b^2-2 a^2\right) \sec ^2(c+d x)-60 a^2 \log (\cos (c+d x))+24 a b \sec ^5(c+d x)-80 a b \sec ^3(c+d x)+120 a b \sec (c+d x)+10 b^2 \sec ^6(c+d x)}{60 d}","\frac{a^2 \sec ^4(c+d x)}{4 d}-\frac{a^2 \sec ^2(c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^5(c+d x)}{5 d}-\frac{4 a b \sec ^3(c+d x)}{3 d}+\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \tan ^6(c+d x)}{6 d}",1,"(-60*a^2*Log[Cos[c + d*x]] + 120*a*b*Sec[c + d*x] + 30*(-2*a^2 + b^2)*Sec[c + d*x]^2 - 80*a*b*Sec[c + d*x]^3 + 15*(a^2 - 2*b^2)*Sec[c + d*x]^4 + 24*a*b*Sec[c + d*x]^5 + 10*b^2*Sec[c + d*x]^6)/(60*d)","A",1
274,1,74,87,0.4339732,"\int (a+b \sec (c+d x))^2 \tan ^3(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^3,x]","\frac{6 \left(a^2-b^2\right) \sec ^2(c+d x)+12 a^2 \log (\cos (c+d x))+8 a b \sec ^3(c+d x)-24 a b \sec (c+d x)+3 b^2 \sec ^4(c+d x)}{12 d}","\frac{\left(a^2-b^2\right) \sec ^2(c+d x)}{2 d}+\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^3(c+d x)}{3 d}-\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \sec ^4(c+d x)}{4 d}",1,"(12*a^2*Log[Cos[c + d*x]] - 24*a*b*Sec[c + d*x] + 6*(a^2 - b^2)*Sec[c + d*x]^2 + 8*a*b*Sec[c + d*x]^3 + 3*b^2*Sec[c + d*x]^4)/(12*d)","A",1
275,1,42,47,0.0637042,"\int (a+b \sec (c+d x))^2 \tan (c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^2*Tan[c + d*x],x]","\frac{-2 a^2 \log (\cos (c+d x))+4 a b \sec (c+d x)+b^2 \sec ^2(c+d x)}{2 d}","-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \sec ^2(c+d x)}{2 d}",1,"(-2*a^2*Log[Cos[c + d*x]] + 4*a*b*Sec[c + d*x] + b^2*Sec[c + d*x]^2)/(2*d)","A",1
276,1,53,61,0.1083029,"\int \cot (c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]*(a + b*Sec[c + d*x])^2,x]","\frac{(a+b)^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+(a-b)^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-b^2 \log (\cos (c+d x))}{d}","\frac{a^2 \log (\cos (c+d x))}{d}+\frac{(a+b)^2 \log (1-\sec (c+d x))}{2 d}+\frac{(a-b)^2 \log (\sec (c+d x)+1)}{2 d}",1,"((a - b)^2*Log[Cos[(c + d*x)/2]] - b^2*Log[Cos[c + d*x]] + (a + b)^2*Log[Sin[(c + d*x)/2]])/d","A",1
277,1,82,92,0.4698668,"\int \cot ^3(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^3*(a + b*Sec[c + d*x])^2,x]","-\frac{(a+b)^2 \csc ^2\left(\frac{1}{2} (c+d x)\right)+(a-b)^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)+8 a \left((a+b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+(a-b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{8 d}","-\frac{\cot ^2(c+d x) \left(a^2+2 a b \sec (c+d x)+b^2\right)}{2 d}-\frac{a^2 \log (\cos (c+d x))}{d}-\frac{a (a+b) \log (1-\sec (c+d x))}{2 d}-\frac{a (a-b) \log (\sec (c+d x)+1)}{2 d}",1,"-1/8*((a + b)^2*Csc[(c + d*x)/2]^2 + 8*a*((a - b)*Log[Cos[(c + d*x)/2]] + (a + b)*Log[Sin[(c + d*x)/2]]) + (a - b)^2*Sec[(c + d*x)/2]^2)/d","A",1
278,1,148,126,3.190963,"\int \cot ^5(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^5*(a + b*Sec[c + d*x])^2,x]","\frac{2 \left(7 a^2+10 a b+3 b^2\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)+2 \left(7 a^2-10 a b+3 b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)-(a+b)^2 \csc ^4\left(\frac{1}{2} (c+d x)\right)-(a-b)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right)+16 a \left((4 a+3 b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+(4 a-3 b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{64 d}","-\frac{\cot ^4(c+d x) \left(a^2+2 a b \sec (c+d x)+b^2\right)}{4 d}+\frac{a^2 \log (\cos (c+d x))}{d}+\frac{a (4 a+3 b) \log (1-\sec (c+d x))}{8 d}+\frac{a (4 a-3 b) \log (\sec (c+d x)+1)}{8 d}+\frac{a \cot ^2(c+d x) (2 a+3 b \sec (c+d x))}{4 d}",1,"(2*(7*a^2 + 10*a*b + 3*b^2)*Csc[(c + d*x)/2]^2 - (a + b)^2*Csc[(c + d*x)/2]^4 + 16*a*((4*a - 3*b)*Log[Cos[(c + d*x)/2]] + (4*a + 3*b)*Log[Sin[(c + d*x)/2]]) + 2*(7*a^2 - 10*a*b + 3*b^2)*Sec[(c + d*x)/2]^2 - (a - b)^2*Sec[(c + d*x)/2]^4)/(64*d)","A",1
279,1,293,157,1.3508396,"\int (a+b \sec (c+d x))^2 \tan ^6(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^6,x]","\frac{-2100 \sec ^6(c+d x) \left(7 a^2 (c+d x)-\left(a^2+b^2\right) \tan (c+d x)\right)-\left(\sec ^7(c+d x) \left(-3444 a^2 \sin (3 (c+d x))-1988 a^2 \sin (5 (c+d x))-644 a^2 \sin (7 (c+d x))+8820 a^2 (c+d x) \cos (3 (c+d x))+2940 a^2 (c+d x) \cos (5 (c+d x))+420 a^2 c \cos (7 (c+d x))+420 a^2 d x \cos (7 (c+d x))-980 a b \sin (4 (c+d x))-1155 a b \sin (6 (c+d x))+1260 b^2 \sin (3 (c+d x))-420 b^2 \sin (5 (c+d x))+60 b^2 \sin (7 (c+d x))\right)\right)+5950 a b \tan (c+d x) \sec ^5(c+d x)+16800 a b \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{26880 d}","\frac{a^2 \tan ^5(c+d x)}{5 d}-\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}-a^2 x-\frac{5 a b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a b \tan ^5(c+d x) \sec (c+d x)}{3 d}-\frac{5 a b \tan ^3(c+d x) \sec (c+d x)}{12 d}+\frac{5 a b \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b^2 \tan ^7(c+d x)}{7 d}",1,"(16800*a*b*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c + d*x]^7*(8820*a^2*(c + d*x)*Cos[3*(c + d*x)] + 2940*a^2*(c + d*x)*Cos[5*(c + d*x)] + 420*a^2*c*Cos[7*(c + d*x)] + 420*a^2*d*x*Cos[7*(c + d*x)] - 3444*a^2*Sin[3*(c + d*x)] + 1260*b^2*Sin[3*(c + d*x)] - 980*a*b*Sin[4*(c + d*x)] - 1988*a^2*Sin[5*(c + d*x)] - 420*b^2*Sin[5*(c + d*x)] - 1155*a*b*Sin[6*(c + d*x)] - 644*a^2*Sin[7*(c + d*x)] + 60*b^2*Sin[7*(c + d*x)]) + 5950*a*b*Sec[c + d*x]^5*Tan[c + d*x] - 2100*Sec[c + d*x]^6*(7*a^2*(c + d*x) - (a^2 + b^2)*Tan[c + d*x]))/(26880*d)","A",1
280,1,355,116,0.9650721,"\int (a+b \sec (c+d x))^2 \tan ^4(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^4,x]","\frac{\sec ^5(c+d x) \left(-80 a^2 \sin (c+d x)-160 a^2 \sin (3 (c+d x))-80 a^2 \sin (5 (c+d x))+60 a^2 c \cos (5 (c+d x))+60 a^2 d x \cos (5 (c+d x))-60 a b \sin (2 (c+d x))-150 a b \sin (4 (c+d x))-45 a b \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+45 a b \cos (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+150 a \cos (c+d x) \left(4 a (c+d x)-3 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+75 a \cos (3 (c+d x)) \left(4 a (c+d x)-3 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+120 b^2 \sin (c+d x)-60 b^2 \sin (3 (c+d x))+12 b^2 \sin (5 (c+d x))\right)}{960 d}","\frac{a^2 \tan ^3(c+d x)}{3 d}-\frac{a^2 \tan (c+d x)}{d}+a^2 x+\frac{3 a b \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b \tan ^3(c+d x) \sec (c+d x)}{2 d}-\frac{3 a b \tan (c+d x) \sec (c+d x)}{4 d}+\frac{b^2 \tan ^5(c+d x)}{5 d}",1,"(Sec[c + d*x]^5*(60*a^2*c*Cos[5*(c + d*x)] + 60*a^2*d*x*Cos[5*(c + d*x)] - 45*a*b*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 45*a*b*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 150*a*Cos[c + d*x]*(4*a*(c + d*x) - 3*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 75*a*Cos[3*(c + d*x)]*(4*a*(c + d*x) - 3*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 80*a^2*Sin[c + d*x] + 120*b^2*Sin[c + d*x] - 60*a*b*Sin[2*(c + d*x)] - 160*a^2*Sin[3*(c + d*x)] - 60*b^2*Sin[3*(c + d*x)] - 150*a*b*Sin[4*(c + d*x)] - 80*a^2*Sin[5*(c + d*x)] + 12*b^2*Sin[5*(c + d*x)]))/(960*d)","B",1
281,1,201,70,1.2038841,"\int (a+b \sec (c+d x))^2 \tan ^2(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^2,x]","\frac{\sec ^3(c+d x) \left(2 \sin (c+d x) \left(\left(3 a^2-b^2\right) \cos (2 (c+d x))+3 a^2+6 a b \cos (c+d x)+b^2\right)-9 a \cos (c+d x) \left(a (c+d x)-b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-3 a \cos (3 (c+d x)) \left(a (c+d x)-b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{12 d}","\frac{a^2 \tan (c+d x)}{d}-a^2 x-\frac{a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b \tan (c+d x) \sec (c+d x)}{d}+\frac{b^2 \tan ^3(c+d x)}{3 d}",1,"(Sec[c + d*x]^3*(-9*a*Cos[c + d*x]*(a*(c + d*x) - b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 3*a*Cos[3*(c + d*x)]*(a*(c + d*x) - b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 2*(3*a^2 + b^2 + 6*a*b*Cos[c + d*x] + (3*a^2 - b^2)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(12*d)","B",1
282,1,39,48,0.3809633,"\int \cot ^2(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*(a + b*Sec[c + d*x])^2,x]","-\frac{\left(a^2+b^2\right) \cot (c+d x)+a (a (c+d x)+2 b \csc (c+d x))}{d}","-\frac{a^2 \cot (c+d x)}{d}+a^2 (-x)-\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot (c+d x)}{d}",1,"-(((a^2 + b^2)*Cot[c + d*x] + a*(a*(c + d*x) + 2*b*Csc[c + d*x]))/d)","A",1
283,1,122,85,0.4769665,"\int \cot ^4(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*(a + b*Sec[c + d*x])^2,x]","-\frac{\csc ^3(c+d x) \left(-9 a^2 c \sin (c+d x)-9 a^2 d x \sin (c+d x)+3 a^2 c \sin (3 (c+d x))+3 a^2 d x \sin (3 (c+d x))+4 a^2 \cos (3 (c+d x))+12 a b \cos (2 (c+d x))-4 a b+3 b^2 \cos (c+d x)+b^2 \cos (3 (c+d x))\right)}{12 d}","-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+a^2 x-\frac{2 a b \csc ^3(c+d x)}{3 d}+\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot ^3(c+d x)}{3 d}",1,"-1/12*(Csc[c + d*x]^3*(-4*a*b + 3*b^2*Cos[c + d*x] + 12*a*b*Cos[2*(c + d*x)] + 4*a^2*Cos[3*(c + d*x)] + b^2*Cos[3*(c + d*x)] - 9*a^2*c*Sin[c + d*x] - 9*a^2*d*x*Sin[c + d*x] + 3*a^2*c*Sin[3*(c + d*x)] + 3*a^2*d*x*Sin[3*(c + d*x)]))/d","A",1
284,1,198,122,0.5898168,"\int \cot ^6(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*(a + b*Sec[c + d*x])^2,x]","-\frac{\csc ^5(c+d x) \left(10 \left(5 a^2+3 b^2\right) \cos (c+d x)+150 a^2 c \sin (c+d x)+150 a^2 d x \sin (c+d x)-75 a^2 c \sin (3 (c+d x))-75 a^2 d x \sin (3 (c+d x))+15 a^2 c \sin (5 (c+d x))+15 a^2 d x \sin (5 (c+d x))-25 a^2 \cos (3 (c+d x))+23 a^2 \cos (5 (c+d x))-80 a b \cos (2 (c+d x))+60 a b \cos (4 (c+d x))+116 a b+15 b^2 \cos (3 (c+d x))+3 b^2 \cos (5 (c+d x))\right)}{240 d}","-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-a^2 x-\frac{2 a b \csc ^5(c+d x)}{5 d}+\frac{4 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot ^5(c+d x)}{5 d}",1,"-1/240*(Csc[c + d*x]^5*(116*a*b + 10*(5*a^2 + 3*b^2)*Cos[c + d*x] - 80*a*b*Cos[2*(c + d*x)] - 25*a^2*Cos[3*(c + d*x)] + 15*b^2*Cos[3*(c + d*x)] + 60*a*b*Cos[4*(c + d*x)] + 23*a^2*Cos[5*(c + d*x)] + 3*b^2*Cos[5*(c + d*x)] + 150*a^2*c*Sin[c + d*x] + 150*a^2*d*x*Sin[c + d*x] - 75*a^2*c*Sin[3*(c + d*x)] - 75*a^2*d*x*Sin[3*(c + d*x)] + 15*a^2*c*Sin[5*(c + d*x)] + 15*a^2*d*x*Sin[5*(c + d*x)]))/d","A",1
285,1,257,153,0.8377671,"\int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^8*(a + b*Sec[c + d*x])^2,x]","-\frac{\csc ^7(c+d x) \left(-3675 a^2 c \sin (c+d x)-3675 a^2 d x \sin (c+d x)+2205 a^2 c \sin (3 (c+d x))+2205 a^2 d x \sin (3 (c+d x))-735 a^2 c \sin (5 (c+d x))-735 a^2 d x \sin (5 (c+d x))+105 a^2 c \sin (7 (c+d x))+105 a^2 d x \sin (7 (c+d x))+1176 a^2 \cos (3 (c+d x))-392 a^2 \cos (5 (c+d x))+176 a^2 \cos (7 (c+d x))+3612 a b \cos (2 (c+d x))-840 a b \cos (4 (c+d x))+420 a b \cos (6 (c+d x))-1272 a b+525 b^2 \cos (c+d x)+315 b^2 \cos (3 (c+d x))+105 b^2 \cos (5 (c+d x))+15 b^2 \cos (7 (c+d x))\right)}{6720 d}","-\frac{a^2 \cot ^7(c+d x)}{7 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+a^2 x-\frac{2 a b \csc ^7(c+d x)}{7 d}+\frac{6 a b \csc ^5(c+d x)}{5 d}-\frac{2 a b \csc ^3(c+d x)}{d}+\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot ^7(c+d x)}{7 d}",1,"-1/6720*(Csc[c + d*x]^7*(-1272*a*b + 525*b^2*Cos[c + d*x] + 3612*a*b*Cos[2*(c + d*x)] + 1176*a^2*Cos[3*(c + d*x)] + 315*b^2*Cos[3*(c + d*x)] - 840*a*b*Cos[4*(c + d*x)] - 392*a^2*Cos[5*(c + d*x)] + 105*b^2*Cos[5*(c + d*x)] + 420*a*b*Cos[6*(c + d*x)] + 176*a^2*Cos[7*(c + d*x)] + 15*b^2*Cos[7*(c + d*x)] - 3675*a^2*c*Sin[c + d*x] - 3675*a^2*d*x*Sin[c + d*x] + 2205*a^2*c*Sin[3*(c + d*x)] + 2205*a^2*d*x*Sin[3*(c + d*x)] - 735*a^2*c*Sin[5*(c + d*x)] - 735*a^2*d*x*Sin[5*(c + d*x)] + 105*a^2*c*Sin[7*(c + d*x)] + 105*a^2*d*x*Sin[7*(c + d*x)]))/d","A",1
286,1,520,250,6.2352915,"\int \frac{\tan ^9(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]^9/(a + b*Sec[c + d*x]),x]","-\frac{\left(2 b^2-a^2\right) \left(a^4-2 a^2 b^2+2 b^4\right) \sec ^2(c+d x) (a \cos (c+d x)+b)}{b^7 d (a+b \sec (c+d x))}-\frac{a \left(a^4-4 a^2 b^2+6 b^4\right) \sec ^3(c+d x) (a \cos (c+d x)+b)}{2 b^6 d (a+b \sec (c+d x))}+\frac{\left(a^4-4 a^2 b^2+6 b^4\right) \sec ^4(c+d x) (a \cos (c+d x)+b)}{3 b^5 d (a+b \sec (c+d x))}+\frac{\left(a^7-4 a^5 b^2+6 a^3 b^4-4 a b^6\right) \sec (c+d x) \log (\cos (c+d x)) (a \cos (c+d x)+b)}{b^8 d (a+b \sec (c+d x))}+\frac{\left(-a^8+4 a^6 b^2-6 a^4 b^4+4 a^2 b^6-b^8\right) \sec (c+d x) (a \cos (c+d x)+b) \log (a \cos (c+d x)+b)}{a b^8 d (a+b \sec (c+d x))}+\frac{a (2 b-a) (a+2 b) \sec ^5(c+d x) (a \cos (c+d x)+b)}{4 b^4 d (a+b \sec (c+d x))}-\frac{(2 b-a) (a+2 b) \sec ^6(c+d x) (a \cos (c+d x)+b)}{5 b^3 d (a+b \sec (c+d x))}-\frac{a \sec ^7(c+d x) (a \cos (c+d x)+b)}{6 b^2 d (a+b \sec (c+d x))}+\frac{\sec ^8(c+d x) (a \cos (c+d x)+b)}{7 b d (a+b \sec (c+d x))}","-\frac{\left(a^2-b^2\right)^4 \log (a+b \sec (c+d x))}{a b^8 d}-\frac{a \left(a^2-4 b^2\right) \sec ^4(c+d x)}{4 b^4 d}+\frac{\left(a^2-4 b^2\right) \sec ^5(c+d x)}{5 b^3 d}-\frac{a \left(a^4-4 a^2 b^2+6 b^4\right) \sec ^2(c+d x)}{2 b^6 d}+\frac{\left(a^4-4 a^2 b^2+6 b^4\right) \sec ^3(c+d x)}{3 b^5 d}+\frac{\left(a^6-4 a^4 b^2+6 a^2 b^4-4 b^6\right) \sec (c+d x)}{b^7 d}-\frac{a \sec ^6(c+d x)}{6 b^2 d}-\frac{\log (\cos (c+d x))}{a d}+\frac{\sec ^7(c+d x)}{7 b d}",1,"((a^7 - 4*a^5*b^2 + 6*a^3*b^4 - 4*a*b^6)*(b + a*Cos[c + d*x])*Log[Cos[c + d*x]]*Sec[c + d*x])/(b^8*d*(a + b*Sec[c + d*x])) + ((-a^8 + 4*a^6*b^2 - 6*a^4*b^4 + 4*a^2*b^6 - b^8)*(b + a*Cos[c + d*x])*Log[b + a*Cos[c + d*x]]*Sec[c + d*x])/(a*b^8*d*(a + b*Sec[c + d*x])) - ((-a^2 + 2*b^2)*(a^4 - 2*a^2*b^2 + 2*b^4)*(b + a*Cos[c + d*x])*Sec[c + d*x]^2)/(b^7*d*(a + b*Sec[c + d*x])) - (a*(a^4 - 4*a^2*b^2 + 6*b^4)*(b + a*Cos[c + d*x])*Sec[c + d*x]^3)/(2*b^6*d*(a + b*Sec[c + d*x])) + ((a^4 - 4*a^2*b^2 + 6*b^4)*(b + a*Cos[c + d*x])*Sec[c + d*x]^4)/(3*b^5*d*(a + b*Sec[c + d*x])) + (a*(-a + 2*b)*(a + 2*b)*(b + a*Cos[c + d*x])*Sec[c + d*x]^5)/(4*b^4*d*(a + b*Sec[c + d*x])) - ((-a + 2*b)*(a + 2*b)*(b + a*Cos[c + d*x])*Sec[c + d*x]^6)/(5*b^3*d*(a + b*Sec[c + d*x])) - (a*(b + a*Cos[c + d*x])*Sec[c + d*x]^7)/(6*b^2*d*(a + b*Sec[c + d*x])) + ((b + a*Cos[c + d*x])*Sec[c + d*x]^8)/(7*b*d*(a + b*Sec[c + d*x]))","B",0
287,1,371,170,6.1913971,"\int \frac{\tan ^7(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]^7/(a + b*Sec[c + d*x]),x]","\frac{a \left(3 b^2-a^2\right) \sec ^3(c+d x) (a \cos (c+d x)+b)}{2 b^4 d (a+b \sec (c+d x))}+\frac{\left(a^2-3 b^2\right) \sec ^4(c+d x) (a \cos (c+d x)+b)}{3 b^3 d (a+b \sec (c+d x))}+\frac{\left(a^5-3 a^3 b^2+3 a b^4\right) \sec (c+d x) \log (\cos (c+d x)) (a \cos (c+d x)+b)}{b^6 d (a+b \sec (c+d x))}+\frac{\left(a^4-3 a^2 b^2+3 b^4\right) \sec ^2(c+d x) (a \cos (c+d x)+b)}{b^5 d (a+b \sec (c+d x))}+\frac{\left(-a^6+3 a^4 b^2-3 a^2 b^4+b^6\right) \sec (c+d x) (a \cos (c+d x)+b) \log (a \cos (c+d x)+b)}{a b^6 d (a+b \sec (c+d x))}-\frac{a \sec ^5(c+d x) (a \cos (c+d x)+b)}{4 b^2 d (a+b \sec (c+d x))}+\frac{\sec ^6(c+d x) (a \cos (c+d x)+b)}{5 b d (a+b \sec (c+d x))}","-\frac{\left(a^2-b^2\right)^3 \log (a+b \sec (c+d x))}{a b^6 d}-\frac{a \left(a^2-3 b^2\right) \sec ^2(c+d x)}{2 b^4 d}+\frac{\left(a^2-3 b^2\right) \sec ^3(c+d x)}{3 b^3 d}+\frac{\left(a^4-3 a^2 b^2+3 b^4\right) \sec (c+d x)}{b^5 d}-\frac{a \sec ^4(c+d x)}{4 b^2 d}+\frac{\log (\cos (c+d x))}{a d}+\frac{\sec ^5(c+d x)}{5 b d}",1,"((a^5 - 3*a^3*b^2 + 3*a*b^4)*(b + a*Cos[c + d*x])*Log[Cos[c + d*x]]*Sec[c + d*x])/(b^6*d*(a + b*Sec[c + d*x])) + ((-a^6 + 3*a^4*b^2 - 3*a^2*b^4 + b^6)*(b + a*Cos[c + d*x])*Log[b + a*Cos[c + d*x]]*Sec[c + d*x])/(a*b^6*d*(a + b*Sec[c + d*x])) + ((a^4 - 3*a^2*b^2 + 3*b^4)*(b + a*Cos[c + d*x])*Sec[c + d*x]^2)/(b^5*d*(a + b*Sec[c + d*x])) + (a*(-a^2 + 3*b^2)*(b + a*Cos[c + d*x])*Sec[c + d*x]^3)/(2*b^4*d*(a + b*Sec[c + d*x])) + ((a^2 - 3*b^2)*(b + a*Cos[c + d*x])*Sec[c + d*x]^4)/(3*b^3*d*(a + b*Sec[c + d*x])) - (a*(b + a*Cos[c + d*x])*Sec[c + d*x]^5)/(4*b^2*d*(a + b*Sec[c + d*x])) + ((b + a*Cos[c + d*x])*Sec[c + d*x]^6)/(5*b*d*(a + b*Sec[c + d*x]))","B",1
288,1,108,108,0.3997996,"\int \frac{\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]^5/(a + b*Sec[c + d*x]),x]","\frac{-3 a^2 b^2 \sec ^2(c+d x)+6 a b \left(a^2-2 b^2\right) \sec (c+d x)+6 a^2 \left(a^2-2 b^2\right) \log (\cos (c+d x))-6 \left(a^2-b^2\right)^2 \log (a \cos (c+d x)+b)+2 a b^3 \sec ^3(c+d x)}{6 a b^4 d}","-\frac{\left(a^2-b^2\right)^2 \log (a+b \sec (c+d x))}{a b^4 d}+\frac{\left(a^2-2 b^2\right) \sec (c+d x)}{b^3 d}-\frac{a \sec ^2(c+d x)}{2 b^2 d}-\frac{\log (\cos (c+d x))}{a d}+\frac{\sec ^3(c+d x)}{3 b d}",1,"(6*a^2*(a^2 - 2*b^2)*Log[Cos[c + d*x]] - 6*(a^2 - b^2)^2*Log[b + a*Cos[c + d*x]] + 6*a*b*(a^2 - 2*b^2)*Sec[c + d*x] - 3*a^2*b^2*Sec[c + d*x]^2 + 2*a*b^3*Sec[c + d*x]^3)/(6*a*b^4*d)","A",1
289,1,52,59,0.1298922,"\int \frac{\tan ^3(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]^3/(a + b*Sec[c + d*x]),x]","\frac{\left(b^2-a^2\right) \log (a \cos (c+d x)+b)+a^2 \log (\cos (c+d x))+a b \sec (c+d x)}{a b^2 d}","-\frac{\left(a^2-b^2\right) \log (a+b \sec (c+d x))}{a b^2 d}+\frac{\log (\cos (c+d x))}{a d}+\frac{\sec (c+d x)}{b d}",1,"(a^2*Log[Cos[c + d*x]] + (-a^2 + b^2)*Log[b + a*Cos[c + d*x]] + a*b*Sec[c + d*x])/(a*b^2*d)","A",1
290,1,19,35,0.0380854,"\int \frac{\tan (c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]/(a + b*Sec[c + d*x]),x]","-\frac{\log (a \cos (c+d x)+b)}{a d}","-\frac{\log (a+b \sec (c+d x))}{a d}-\frac{\log (\cos (c+d x))}{a d}",1,"-(Log[b + a*Cos[c + d*x]]/(a*d))","A",1
291,1,70,94,0.1074736,"\int \frac{\cot (c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]/(a + b*Sec[c + d*x]),x]","\frac{b^2 (-\log (a \cos (c+d x)+b))+a (a-b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+a (a+b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a d (a-b) (a+b)}","-\frac{b^2 \log (a+b \sec (c+d x))}{a d \left(a^2-b^2\right)}+\frac{\log (1-\sec (c+d x))}{2 d (a+b)}+\frac{\log (\sec (c+d x)+1)}{2 d (a-b)}+\frac{\log (\cos (c+d x))}{a d}",1,"(a*(a + b)*Log[Cos[(c + d*x)/2]] - b^2*Log[b + a*Cos[c + d*x]] + a*(a - b)*Log[Sin[(c + d*x)/2]])/(a*(a - b)*(a + b)*d)","A",1
292,1,141,157,1.0532549,"\int \frac{\cot ^3(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^3/(a + b*Sec[c + d*x]),x]","-\frac{8 b^4 \log (a \cos (c+d x)+b)+a (a-b)^2 (a+b) \csc ^2\left(\frac{1}{2} (c+d x)\right)+a (a-b) (a+b)^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)+4 a (a-b)^2 (2 a+3 b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 a (2 a-3 b) (a+b)^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 a d (a-b)^2 (a+b)^2}","-\frac{b^4 \log (a+b \sec (c+d x))}{a d \left(a^2-b^2\right)^2}+\frac{1}{4 d (a+b) (1-\sec (c+d x))}+\frac{1}{4 d (a-b) (\sec (c+d x)+1)}-\frac{(2 a+3 b) \log (1-\sec (c+d x))}{4 d (a+b)^2}-\frac{(2 a-3 b) \log (\sec (c+d x)+1)}{4 d (a-b)^2}-\frac{\log (\cos (c+d x))}{a d}",1,"-1/8*(a*(a - b)^2*(a + b)*Csc[(c + d*x)/2]^2 + 4*a*(2*a - 3*b)*(a + b)^2*Log[Cos[(c + d*x)/2]] + 8*b^4*Log[b + a*Cos[c + d*x]] + 4*a*(a - b)^2*(2*a + 3*b)*Log[Sin[(c + d*x)/2]] + a*(a - b)*(a + b)^2*Sec[(c + d*x)/2]^2)/(a*(a - b)^2*(a + b)^2*d)","A",1
293,1,625,234,6.2508204,"\int \frac{\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^5/(a + b*Sec[c + d*x]),x]","\frac{\left(-8 a^2+21 a b-15 b^2\right) \sec (c+d x) \log \left(\cos ^2\left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)}{16 d (b-a)^3 (a+b \sec (c+d x))}-\frac{i \left(-8 a^2+21 a b-15 b^2\right) \tan ^{-1}(\tan (c+d x)) \sec (c+d x) (a \cos (c+d x)+b)}{8 d (b-a)^3 (a+b \sec (c+d x))}-\frac{i \left(8 a^2+21 a b+15 b^2\right) \tan ^{-1}(\tan (c+d x)) \sec (c+d x) (a \cos (c+d x)+b)}{8 d (a+b)^3 (a+b \sec (c+d x))}+\frac{\left(8 a^2+21 a b+15 b^2\right) \sec (c+d x) \log \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)}{16 d (a+b)^3 (a+b \sec (c+d x))}+\frac{b^6 \sec (c+d x) (a \cos (c+d x)+b) \log (a \cos (c+d x)+b)}{a d \left(b^2-a^2\right)^3 (a+b \sec (c+d x))}+\frac{2 i \left(a^5-3 a^3 b^2+3 a b^4\right) (c+d x) \sec (c+d x) (a \cos (c+d x)+b)}{d (a-b)^3 (a+b)^3 (a+b \sec (c+d x))}+\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (a \cos (c+d x)+b)}{64 d (b-a) (a+b \sec (c+d x))}+\frac{(7 a-9 b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (a \cos (c+d x)+b)}{32 d (b-a)^2 (a+b \sec (c+d x))}-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (a \cos (c+d x)+b)}{64 d (a+b) (a+b \sec (c+d x))}+\frac{(7 a+9 b) \csc ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (a \cos (c+d x)+b)}{32 d (a+b)^2 (a+b \sec (c+d x))}","\frac{\left(8 a^2+21 a b+15 b^2\right) \log (1-\sec (c+d x))}{16 d (a+b)^3}+\frac{\left(8 a^2-21 a b+15 b^2\right) \log (\sec (c+d x)+1)}{16 d (a-b)^3}-\frac{b^6 \log (a+b \sec (c+d x))}{a d \left(a^2-b^2\right)^3}-\frac{5 a+7 b}{16 d (a+b)^2 (1-\sec (c+d x))}-\frac{5 a-7 b}{16 d (a-b)^2 (\sec (c+d x)+1)}-\frac{1}{16 d (a+b) (1-\sec (c+d x))^2}-\frac{1}{16 d (a-b) (\sec (c+d x)+1)^2}+\frac{\log (\cos (c+d x))}{a d}",1,"((2*I)*(a^5 - 3*a^3*b^2 + 3*a*b^4)*(c + d*x)*(b + a*Cos[c + d*x])*Sec[c + d*x])/((a - b)^3*(a + b)^3*d*(a + b*Sec[c + d*x])) - ((I/8)*(-8*a^2 + 21*a*b - 15*b^2)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x])*Sec[c + d*x])/((-a + b)^3*d*(a + b*Sec[c + d*x])) - ((I/8)*(8*a^2 + 21*a*b + 15*b^2)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x])*Sec[c + d*x])/((a + b)^3*d*(a + b*Sec[c + d*x])) + ((7*a + 9*b)*(b + a*Cos[c + d*x])*Csc[(c + d*x)/2]^2*Sec[c + d*x])/(32*(a + b)^2*d*(a + b*Sec[c + d*x])) - ((b + a*Cos[c + d*x])*Csc[(c + d*x)/2]^4*Sec[c + d*x])/(64*(a + b)*d*(a + b*Sec[c + d*x])) + ((-8*a^2 + 21*a*b - 15*b^2)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2]^2]*Sec[c + d*x])/(16*(-a + b)^3*d*(a + b*Sec[c + d*x])) + (b^6*(b + a*Cos[c + d*x])*Log[b + a*Cos[c + d*x]]*Sec[c + d*x])/(a*(-a^2 + b^2)^3*d*(a + b*Sec[c + d*x])) + ((8*a^2 + 21*a*b + 15*b^2)*(b + a*Cos[c + d*x])*Log[Sin[(c + d*x)/2]^2]*Sec[c + d*x])/(16*(a + b)^3*d*(a + b*Sec[c + d*x])) + ((7*a - 9*b)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sec[c + d*x])/(32*(-a + b)^2*d*(a + b*Sec[c + d*x])) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sec[c + d*x])/(64*(-a + b)*d*(a + b*Sec[c + d*x]))","C",1
294,1,907,198,6.1981595,"\int \frac{\tan ^6(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]^6/(a + b*Sec[c + d*x]),x]","-\frac{2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right) (b+a \cos (c+d x)) \sec (c+d x) \left(b^2-a^2\right)^3}{a b^5 \sqrt{a^2-b^2} d (a+b \sec (c+d x))}-\frac{a (b+a \cos (c+d x)) \sec (c+d x) \sin \left(\frac{1}{2} (c+d x)\right)}{6 b^2 d (a+b \sec (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{(b+a \cos (c+d x)) \sec (c+d x) \left(7 a b^2 \sin \left(\frac{1}{2} (c+d x)\right)-3 a^3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{3 b^4 d (a+b \sec (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{(b+a \cos (c+d x)) \sec (c+d x) \left(7 a b^2 \sin \left(\frac{1}{2} (c+d x)\right)-3 a^3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{3 b^4 d (a+b \sec (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{(c+d x) (b+a \cos (c+d x)) \sec (c+d x)}{a d (a+b \sec (c+d x))}+\frac{\left(-8 a^4+20 b^2 a^2-15 b^4\right) (b+a \cos (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec (c+d x)}{8 b^5 d (a+b \sec (c+d x))}+\frac{\left(8 a^4-20 b^2 a^2+15 b^4\right) (b+a \cos (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec (c+d x)}{8 b^5 d (a+b \sec (c+d x))}+\frac{\left(12 a^2-4 b a-27 b^2\right) (b+a \cos (c+d x)) \sec (c+d x)}{48 b^3 d (a+b \sec (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{\left(-12 a^2+4 b a+27 b^2\right) (b+a \cos (c+d x)) \sec (c+d x)}{48 b^3 d (a+b \sec (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a (b+a \cos (c+d x)) \sec (c+d x) \sin \left(\frac{1}{2} (c+d x)\right)}{6 b^2 d (a+b \sec (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{(b+a \cos (c+d x)) \sec (c+d x)}{16 b d (a+b \sec (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{(b+a \cos (c+d x)) \sec (c+d x)}{16 b d (a+b \sec (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}","-\frac{a \left(a^2-2 b^2\right) \tan (c+d x)}{b^4 d}+\frac{\left(4 a^2-7 b^2\right) \tan (c+d x) \sec (c+d x)}{8 b^3 d}+\frac{\left(8 a^4-20 a^2 b^2+15 b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 b^5 d}-\frac{2 (a-b)^{5/2} (a+b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b^5 d}-\frac{a \tan ^3(c+d x)}{3 b^2 d}-\frac{x}{a}+\frac{\tan ^3(c+d x) \sec (c+d x)}{4 b d}",1,"-(((c + d*x)*(b + a*Cos[c + d*x])*Sec[c + d*x])/(a*d*(a + b*Sec[c + d*x]))) - (2*(-a^2 + b^2)^3*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])*Sec[c + d*x])/(a*b^5*Sqrt[a^2 - b^2]*d*(a + b*Sec[c + d*x])) + ((-8*a^4 + 20*a^2*b^2 - 15*b^4)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sec[c + d*x])/(8*b^5*d*(a + b*Sec[c + d*x])) + ((8*a^4 - 20*a^2*b^2 + 15*b^4)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c + d*x])/(8*b^5*d*(a + b*Sec[c + d*x])) + ((b + a*Cos[c + d*x])*Sec[c + d*x])/(16*b*d*(a + b*Sec[c + d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + ((12*a^2 - 4*a*b - 27*b^2)*(b + a*Cos[c + d*x])*Sec[c + d*x])/(48*b^3*d*(a + b*Sec[c + d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - (a*(b + a*Cos[c + d*x])*Sec[c + d*x]*Sin[(c + d*x)/2])/(6*b^2*d*(a + b*Sec[c + d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) - ((b + a*Cos[c + d*x])*Sec[c + d*x])/(16*b*d*(a + b*Sec[c + d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) - (a*(b + a*Cos[c + d*x])*Sec[c + d*x]*Sin[(c + d*x)/2])/(6*b^2*d*(a + b*Sec[c + d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((-12*a^2 + 4*a*b + 27*b^2)*(b + a*Cos[c + d*x])*Sec[c + d*x])/(48*b^3*d*(a + b*Sec[c + d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + ((b + a*Cos[c + d*x])*Sec[c + d*x]*(-3*a^3*Sin[(c + d*x)/2] + 7*a*b^2*Sin[(c + d*x)/2]))/(3*b^4*d*(a + b*Sec[c + d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + ((b + a*Cos[c + d*x])*Sec[c + d*x]*(-3*a^3*Sin[(c + d*x)/2] + 7*a*b^2*Sin[(c + d*x)/2]))/(3*b^4*d*(a + b*Sec[c + d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
295,1,287,126,2.3200257,"\int \frac{\tan ^4(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]^4/(a + b*Sec[c + d*x]),x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(-\frac{4 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{b^3}+\frac{4 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{b^3}+\frac{8 \left(a^2-b^2\right)^{3/2} \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{a b^3}-\frac{4 a \tan (c+d x)}{b^2}+\frac{4 c}{a}+\frac{4 d x}{a}+\frac{1}{b \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{b \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{b}-\frac{6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{b}\right)}{4 d (a+b \sec (c+d x))}","\frac{\left(2 a^2-3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}-\frac{2 (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b^3 d}-\frac{a \tan (c+d x)}{b^2 d}+\frac{x}{a}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*((4*c)/a + (4*d*x)/a + (8*(a^2 - b^2)^(3/2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*b^3) - (4*a^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/b^3 + (6*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/b + (4*a^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/b^3 - (6*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/b + 1/(b*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - 1/(b*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - (4*a*Tan[c + d*x])/b^2))/(4*d*(a + b*Sec[c + d*x]))","B",1
296,1,115,76,0.1493424,"\int \frac{\tan ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Tan[c + d*x]^2/(a + b*Sec[c + d*x]),x]","-\frac{-2 \sqrt{a^2-b^2} \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)+a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+b c+b d x}{a b d}","-\frac{2 \sqrt{a-b} \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b d}-\frac{x}{a}+\frac{\tanh ^{-1}(\sin (c+d x))}{b d}",1,"-((b*c + b*d*x - 2*Sqrt[a^2 - b^2]*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + a*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - a*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(a*b*d))","A",1
297,1,147,106,0.4238519,"\int \frac{\cot ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^2/(a + b*Sec[c + d*x]),x]","-\frac{\csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(\sqrt{a^2-b^2} \left(\left(a^2-b^2\right) (c+d x) \sin (c+d x)+a^2 \cos (c+d x)-a b\right)-2 b^3 \sin (c+d x) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)\right)}{2 a d (a-b) (a+b) \sqrt{a^2-b^2}}","-\frac{a \cot (c+d x)}{d \left(a^2-b^2\right)}+\frac{b \csc (c+d x)}{d \left(a^2-b^2\right)}-\frac{2 b^3 \tanh ^{-1}\left(\frac{\sqrt{a^2-b^2} \tan \left(\frac{1}{2} (c+d x)\right)}{a+b}\right)}{a d \left(a^2-b^2\right)^{3/2}}-\frac{x}{a}",1,"-1/2*(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(-2*b^3*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*Sin[c + d*x] + Sqrt[a^2 - b^2]*(-(a*b) + a^2*Cos[c + d*x] + (a^2 - b^2)*(c + d*x)*Sin[c + d*x])))/(a*(a - b)*(a + b)*Sqrt[a^2 - b^2]*d)","A",1
298,1,416,177,6.1995556,"\int \frac{\cot ^4(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^4/(a + b*Sec[c + d*x]),x]","\frac{2 b^5 \sec (c+d x) (a \cos (c+d x)+b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{a d \sqrt{a^2-b^2} \left(b^2-a^2\right)^2 (a+b \sec (c+d x))}+\frac{(c+d x) \sec (c+d x) (a \cos (c+d x)+b)}{a d (a+b \sec (c+d x))}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(8 a \cos \left(\frac{1}{2} (c+d x)\right)+11 b \cos \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)}{12 d (a+b)^2 (a+b \sec (c+d x))}-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (a \cos (c+d x)+b)}{24 d (b-a) (a+b \sec (c+d x))}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(11 b \sin \left(\frac{1}{2} (c+d x)\right)-8 a \sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)}{12 d (b-a)^2 (a+b \sec (c+d x))}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (a \cos (c+d x)+b)}{24 d (a+b) (a+b \sec (c+d x))}","-\frac{a \cot ^3(c+d x)}{3 d \left(a^2-b^2\right)}+\frac{a \left(a^2-2 b^2\right) \cot (c+d x)}{d \left(a^2-b^2\right)^2}+\frac{b \csc ^3(c+d x)}{3 d \left(a^2-b^2\right)}-\frac{b \left(a^2-2 b^2\right) \csc (c+d x)}{d \left(a^2-b^2\right)^2}-\frac{2 b^5 \tanh ^{-1}\left(\frac{\sqrt{a^2-b^2} \tan \left(\frac{1}{2} (c+d x)\right)}{a+b}\right)}{a d \left(a^2-b^2\right)^{5/2}}+\frac{x}{a}",1,"((c + d*x)*(b + a*Cos[c + d*x])*Sec[c + d*x])/(a*d*(a + b*Sec[c + d*x])) + (2*b^5*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])*Sec[c + d*x])/(a*Sqrt[a^2 - b^2]*(-a^2 + b^2)^2*d*(a + b*Sec[c + d*x])) + ((8*a*Cos[(c + d*x)/2] + 11*b*Cos[(c + d*x)/2])*(b + a*Cos[c + d*x])*Csc[(c + d*x)/2]*Sec[c + d*x])/(12*(a + b)^2*d*(a + b*Sec[c + d*x])) - ((b + a*Cos[c + d*x])*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*Sec[c + d*x])/(24*(a + b)*d*(a + b*Sec[c + d*x])) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]*Sec[c + d*x]*(-8*a*Sin[(c + d*x)/2] + 11*b*Sin[(c + d*x)/2]))/(12*(-a + b)^2*d*(a + b*Sec[c + d*x])) - ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sec[c + d*x]*Tan[(c + d*x)/2])/(24*(-a + b)*d*(a + b*Sec[c + d*x]))","B",1
299,1,528,255,6.3138147,"\int \frac{\tan ^9(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^9/(a + b*Sec[c + d*x])^2,x]","-\frac{(b-a)^4 (a+b)^4 \sec ^2(c+d x) (a \cos (c+d x)+b)}{a^2 b^7 d (a+b \sec (c+d x))^2}+\frac{4 a \left(2 b^2-a^2\right) \sec ^5(c+d x) (a \cos (c+d x)+b)^2}{3 b^5 d (a+b \sec (c+d x))^2}+\frac{\left(3 a^2-4 b^2\right) \sec ^6(c+d x) (a \cos (c+d x)+b)^2}{4 b^4 d (a+b \sec (c+d x))^2}-\frac{2 a \left(3 a^4-8 a^2 b^2+6 b^4\right) \sec ^3(c+d x) (a \cos (c+d x)+b)^2}{b^7 d (a+b \sec (c+d x))^2}+\frac{\left(5 a^4-12 a^2 b^2+6 b^4\right) \sec ^4(c+d x) (a \cos (c+d x)+b)^2}{2 b^6 d (a+b \sec (c+d x))^2}+\frac{\left(-7 a^6+20 a^4 b^2-18 a^2 b^4+4 b^6\right) \sec ^2(c+d x) \log (\cos (c+d x)) (a \cos (c+d x)+b)^2}{b^8 d (a+b \sec (c+d x))^2}+\frac{\left(7 a^8-20 a^6 b^2+18 a^4 b^4-4 a^2 b^6-b^8\right) \sec ^2(c+d x) (a \cos (c+d x)+b)^2 \log (a \cos (c+d x)+b)}{a^2 b^8 d (a+b \sec (c+d x))^2}-\frac{2 a \sec ^7(c+d x) (a \cos (c+d x)+b)^2}{5 b^3 d (a+b \sec (c+d x))^2}+\frac{\sec ^8(c+d x) (a \cos (c+d x)+b)^2}{6 b^2 d (a+b \sec (c+d x))^2}","\frac{\left(a^2-b^2\right)^4}{a b^8 d (a+b \sec (c+d x))}+\frac{\left(a^2-b^2\right)^3 \left(7 a^2+b^2\right) \log (a+b \sec (c+d x))}{a^2 b^8 d}-\frac{4 a \left(a^2-2 b^2\right) \sec ^3(c+d x)}{3 b^5 d}+\frac{\left(3 a^2-4 b^2\right) \sec ^4(c+d x)}{4 b^4 d}-\frac{\log (\cos (c+d x))}{a^2 d}-\frac{2 a \left(3 a^4-8 a^2 b^2+6 b^4\right) \sec (c+d x)}{b^7 d}+\frac{\left(5 a^4-12 a^2 b^2+6 b^4\right) \sec ^2(c+d x)}{2 b^6 d}-\frac{2 a \sec ^5(c+d x)}{5 b^3 d}+\frac{\sec ^6(c+d x)}{6 b^2 d}",1,"-(((-a + b)^4*(a + b)^4*(b + a*Cos[c + d*x])*Sec[c + d*x]^2)/(a^2*b^7*d*(a + b*Sec[c + d*x])^2)) + ((-7*a^6 + 20*a^4*b^2 - 18*a^2*b^4 + 4*b^6)*(b + a*Cos[c + d*x])^2*Log[Cos[c + d*x]]*Sec[c + d*x]^2)/(b^8*d*(a + b*Sec[c + d*x])^2) + ((7*a^8 - 20*a^6*b^2 + 18*a^4*b^4 - 4*a^2*b^6 - b^8)*(b + a*Cos[c + d*x])^2*Log[b + a*Cos[c + d*x]]*Sec[c + d*x]^2)/(a^2*b^8*d*(a + b*Sec[c + d*x])^2) - (2*a*(3*a^4 - 8*a^2*b^2 + 6*b^4)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^3)/(b^7*d*(a + b*Sec[c + d*x])^2) + ((5*a^4 - 12*a^2*b^2 + 6*b^4)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^4)/(2*b^6*d*(a + b*Sec[c + d*x])^2) + (4*a*(-a^2 + 2*b^2)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^5)/(3*b^5*d*(a + b*Sec[c + d*x])^2) + ((3*a^2 - 4*b^2)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^6)/(4*b^4*d*(a + b*Sec[c + d*x])^2) - (2*a*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^7)/(5*b^3*d*(a + b*Sec[c + d*x])^2) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^8)/(6*b^2*d*(a + b*Sec[c + d*x])^2)","B",0
300,1,383,179,6.2208507,"\int \frac{\tan ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^7/(a + b*Sec[c + d*x])^2,x]","\frac{(b-a)^3 (a+b)^3 \sec ^2(c+d x) (a \cos (c+d x)+b)}{a^2 b^5 d (a+b \sec (c+d x))^2}+\frac{2 a \left(3 b^2-2 a^2\right) \sec ^3(c+d x) (a \cos (c+d x)+b)^2}{b^5 d (a+b \sec (c+d x))^2}+\frac{\left(-5 a^4+9 a^2 b^2-3 b^4\right) \sec ^2(c+d x) \log (\cos (c+d x)) (a \cos (c+d x)+b)^2}{b^6 d (a+b \sec (c+d x))^2}+\frac{\left(5 a^6-9 a^4 b^2+3 a^2 b^4+b^6\right) \sec ^2(c+d x) (a \cos (c+d x)+b)^2 \log (a \cos (c+d x)+b)}{a^2 b^6 d (a+b \sec (c+d x))^2}-\frac{3 (b-a) (a+b) \sec ^4(c+d x) (a \cos (c+d x)+b)^2}{2 b^4 d (a+b \sec (c+d x))^2}-\frac{2 a \sec ^5(c+d x) (a \cos (c+d x)+b)^2}{3 b^3 d (a+b \sec (c+d x))^2}+\frac{\sec ^6(c+d x) (a \cos (c+d x)+b)^2}{4 b^2 d (a+b \sec (c+d x))^2}","\frac{\left(a^2-b^2\right)^3}{a b^6 d (a+b \sec (c+d x))}+\frac{\left(a^2-b^2\right)^2 \left(5 a^2+b^2\right) \log (a+b \sec (c+d x))}{a^2 b^6 d}-\frac{2 a \left(2 a^2-3 b^2\right) \sec (c+d x)}{b^5 d}+\frac{3 \left(a^2-b^2\right) \sec ^2(c+d x)}{2 b^4 d}+\frac{\log (\cos (c+d x))}{a^2 d}-\frac{2 a \sec ^3(c+d x)}{3 b^3 d}+\frac{\sec ^4(c+d x)}{4 b^2 d}",1,"((-a + b)^3*(a + b)^3*(b + a*Cos[c + d*x])*Sec[c + d*x]^2)/(a^2*b^5*d*(a + b*Sec[c + d*x])^2) + ((-5*a^4 + 9*a^2*b^2 - 3*b^4)*(b + a*Cos[c + d*x])^2*Log[Cos[c + d*x]]*Sec[c + d*x]^2)/(b^6*d*(a + b*Sec[c + d*x])^2) + ((5*a^6 - 9*a^4*b^2 + 3*a^2*b^4 + b^6)*(b + a*Cos[c + d*x])^2*Log[b + a*Cos[c + d*x]]*Sec[c + d*x]^2)/(a^2*b^6*d*(a + b*Sec[c + d*x])^2) + (2*a*(-2*a^2 + 3*b^2)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^3)/(b^5*d*(a + b*Sec[c + d*x])^2) - (3*(-a + b)*(a + b)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^4)/(2*b^4*d*(a + b*Sec[c + d*x])^2) - (2*a*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^5)/(3*b^3*d*(a + b*Sec[c + d*x])^2) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^6)/(4*b^2*d*(a + b*Sec[c + d*x])^2)","B",1
301,1,187,121,0.6013869,"\int \frac{\tan ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]","\frac{b \left(-3 a^3 b \sec (c+d x)+a^2 b^2 \sec ^2(c+d x)-2 \left(3 a^4+a^2 \left(3 a^2-2 b^2\right) \log (\cos (c+d x))-2 a^2 b^2+\left(-3 a^4+2 a^2 b^2+b^4\right) \log (a \cos (c+d x)+b)+b^4\right)\right)-2 a \cos (c+d x) \left(a^2 \left(3 a^2-2 b^2\right) \log (\cos (c+d x))+\left(-3 a^4+2 a^2 b^2+b^4\right) \log (a \cos (c+d x)+b)\right)}{2 a^2 b^4 d (a \cos (c+d x)+b)}","\frac{\left(a^2-b^2\right)^2}{a b^4 d (a+b \sec (c+d x))}+\frac{\left(3 a^2+b^2\right) \left(a^2-b^2\right) \log (a+b \sec (c+d x))}{a^2 b^4 d}-\frac{\log (\cos (c+d x))}{a^2 d}-\frac{2 a \sec (c+d x)}{b^3 d}+\frac{\sec ^2(c+d x)}{2 b^2 d}",1,"(-2*a*Cos[c + d*x]*(a^2*(3*a^2 - 2*b^2)*Log[Cos[c + d*x]] + (-3*a^4 + 2*a^2*b^2 + b^4)*Log[b + a*Cos[c + d*x]]) + b*(-2*(3*a^4 - 2*a^2*b^2 + b^4 + a^2*(3*a^2 - 2*b^2)*Log[Cos[c + d*x]] + (-3*a^4 + 2*a^2*b^2 + b^4)*Log[b + a*Cos[c + d*x]]) - 3*a^3*b*Sec[c + d*x] + a^2*b^2*Sec[c + d*x]^2))/(2*a^2*b^4*d*(b + a*Cos[c + d*x]))","A",1
302,1,62,74,0.2762005,"\int \frac{\tan ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]","-\frac{\frac{b-\frac{b^3}{a^2}}{a \cos (c+d x)+b}-\frac{\left(a^2+b^2\right) \log (a \cos (c+d x)+b)}{a^2}+\log (\cos (c+d x))}{b^2 d}","\frac{a^2-b^2}{a b^2 d (a+b \sec (c+d x))}+\frac{\left(a^2+b^2\right) \log (a+b \sec (c+d x))}{a^2 b^2 d}+\frac{\log (\cos (c+d x))}{a^2 d}",1,"-(((b - b^3/a^2)/(b + a*Cos[c + d*x]) + Log[Cos[c + d*x]] - ((a^2 + b^2)*Log[b + a*Cos[c + d*x]])/a^2)/(b^2*d))","A",1
303,1,54,54,0.0400329,"\int \frac{\tan (c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]/(a + b*Sec[c + d*x])^2,x]","-\frac{b \log (a \cos (c+d x)+b)+a \cos (c+d x) \log (a \cos (c+d x)+b)+b}{a^2 d (a \cos (c+d x)+b)}","-\frac{\log (a+b \sec (c+d x))}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d}+\frac{1}{a d (a+b \sec (c+d x))}",1,"-((b + b*Log[b + a*Cos[c + d*x]] + a*Cos[c + d*x]*Log[b + a*Cos[c + d*x]])/(a^2*d*(b + a*Cos[c + d*x])))","A",1
304,1,189,138,0.3584593,"\int \frac{\cot (c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Cot[c + d*x]/(a + b*Sec[c + d*x])^2,x]","\frac{a \cos (c+d x) \left(\left(b^4-3 a^2 b^2\right) \log (a \cos (c+d x)+b)+a^2 (a-b)^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+a^2 (a+b)^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+b \left((a-b) \left(a^2 (a-b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-b^2 (a+b)\right)+\left(b^4-3 a^2 b^2\right) \log (a \cos (c+d x)+b)+a^2 (a+b)^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{a^2 d (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}","\frac{b^2}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{b^2 \left(3 a^2-b^2\right) \log (a+b \sec (c+d x))}{a^2 d \left(a^2-b^2\right)^2}+\frac{\log (\cos (c+d x))}{a^2 d}+\frac{\log (1-\sec (c+d x))}{2 d (a+b)^2}+\frac{\log (\sec (c+d x)+1)}{2 d (a-b)^2}",1,"(a*Cos[c + d*x]*(a^2*(a + b)^2*Log[Cos[(c + d*x)/2]] + (-3*a^2*b^2 + b^4)*Log[b + a*Cos[c + d*x]] + a^2*(a - b)^2*Log[Sin[(c + d*x)/2]]) + b*(a^2*(a + b)^2*Log[Cos[(c + d*x)/2]] + (-3*a^2*b^2 + b^4)*Log[b + a*Cos[c + d*x]] + (a - b)*(-(b^2*(a + b)) + a^2*(a - b)*Log[Sin[(c + d*x)/2]])))/(a^2*(a - b)^2*(a + b)^2*d*(b + a*Cos[c + d*x]))","A",1
305,1,351,197,2.2722349,"\int \frac{\cot ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b) \left(-\frac{8 b^5}{a^2 (a-b)^2 (a+b)^2}+\frac{8 b^4 \left(b^2-5 a^2\right) (a \cos (c+d x)+b) \log (a \cos (c+d x)+b)}{a^2 \left(a^2-b^2\right)^3}-\frac{16 i \left(a^4-3 a^2 b^2-2 b^4\right) (c+d x) (a \cos (c+d x)+b)}{(a-b)^3 (a+b)^3}+\frac{4 (a-2 b) \log \left(\cos ^2\left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)}{(b-a)^3}+\frac{8 i (a+2 b) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b)}{(a+b)^3}+\frac{8 i (a-2 b) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b)}{(a-b)^3}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a+b)^2}-\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a-b)^2}-\frac{4 (a+2 b) \log \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)}{(a+b)^3}\right)}{8 d (a+b \sec (c+d x))^2}","\frac{b^4}{a d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{b^4 \left(5 a^2-b^2\right) \log (a+b \sec (c+d x))}{a^2 d \left(a^2-b^2\right)^3}-\frac{\log (\cos (c+d x))}{a^2 d}+\frac{1}{4 d (a+b)^2 (1-\sec (c+d x))}+\frac{1}{4 d (a-b)^2 (\sec (c+d x)+1)}-\frac{(a+2 b) \log (1-\sec (c+d x))}{2 d (a+b)^3}-\frac{(a-2 b) \log (\sec (c+d x)+1)}{2 d (a-b)^3}",1,"((b + a*Cos[c + d*x])*((-8*b^5)/(a^2*(a - b)^2*(a + b)^2) - ((16*I)*(a^4 - 3*a^2*b^2 - 2*b^4)*(c + d*x)*(b + a*Cos[c + d*x]))/((a - b)^3*(a + b)^3) + ((8*I)*(a - 2*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x]))/(a - b)^3 + ((8*I)*(a + 2*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x]))/(a + b)^3 - ((b + a*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/(a + b)^2 + (4*(a - 2*b)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2]^2])/(-a + b)^3 + (8*b^4*(-5*a^2 + b^2)*(b + a*Cos[c + d*x])*Log[b + a*Cos[c + d*x]])/(a^2*(a^2 - b^2)^3) - (4*(a + 2*b)*(b + a*Cos[c + d*x])*Log[Sin[(c + d*x)/2]^2])/(a + b)^3 - ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a - b)^2)*Sec[c + d*x]^2)/(8*d*(a + b*Sec[c + d*x])^2)","C",1
306,1,473,278,3.1369704,"\int \frac{\cot ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b) \left(\frac{64 b^7}{a^2 (b-a)^3 (a+b)^3}+\frac{8 \left(4 a^2-13 a b+12 b^2\right) \log \left(\cos ^2\left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)}{(a-b)^4}-\frac{16 i \left(4 a^2-13 a b+12 b^2\right) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b)}{(a-b)^4}-\frac{16 i \left(4 a^2+13 a b+12 b^2\right) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b)}{(a+b)^4}+\frac{8 \left(4 a^2+13 a b+12 b^2\right) \log \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)}{(a+b)^4}+\frac{64 \left(b^8-7 a^2 b^6\right) (a \cos (c+d x)+b) \log (a \cos (c+d x)+b)}{a^2 \left(a^2-b^2\right)^4}+\frac{128 i \left(a^6-4 a^4 b^2+6 a^2 b^4+3 b^6\right) (c+d x) (a \cos (c+d x)+b)}{(a-b)^4 (a+b)^4}-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a+b)^2}+\frac{2 (7 a+11 b) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a+b)^3}-\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a-b)^2}+\frac{2 (7 a-11 b) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a-b)^3}\right)}{64 d (a+b \sec (c+d x))^2}","\frac{\left(4 a^2+13 a b+12 b^2\right) \log (1-\sec (c+d x))}{8 d (a+b)^4}+\frac{\left(4 a^2-13 a b+12 b^2\right) \log (\sec (c+d x)+1)}{8 d (a-b)^4}+\frac{b^6}{a d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{b^6 \left(7 a^2-b^2\right) \log (a+b \sec (c+d x))}{a^2 d \left(a^2-b^2\right)^4}+\frac{\log (\cos (c+d x))}{a^2 d}-\frac{5 a+9 b}{16 d (a+b)^3 (1-\sec (c+d x))}-\frac{5 a-9 b}{16 d (a-b)^3 (\sec (c+d x)+1)}-\frac{1}{16 d (a+b)^2 (1-\sec (c+d x))^2}-\frac{1}{16 d (a-b)^2 (\sec (c+d x)+1)^2}",1,"((b + a*Cos[c + d*x])*((64*b^7)/(a^2*(-a + b)^3*(a + b)^3) + ((128*I)*(a^6 - 4*a^4*b^2 + 6*a^2*b^4 + 3*b^6)*(c + d*x)*(b + a*Cos[c + d*x]))/((a - b)^4*(a + b)^4) - ((16*I)*(4*a^2 - 13*a*b + 12*b^2)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x]))/(a - b)^4 - ((16*I)*(4*a^2 + 13*a*b + 12*b^2)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x]))/(a + b)^4 + (2*(7*a + 11*b)*(b + a*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/(a + b)^3 - ((b + a*Cos[c + d*x])*Csc[(c + d*x)/2]^4)/(a + b)^2 + (8*(4*a^2 - 13*a*b + 12*b^2)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2]^2])/(a - b)^4 + (64*(-7*a^2*b^6 + b^8)*(b + a*Cos[c + d*x])*Log[b + a*Cos[c + d*x]])/(a^2*(a^2 - b^2)^4) + (8*(4*a^2 + 13*a*b + 12*b^2)*(b + a*Cos[c + d*x])*Log[Sin[(c + d*x)/2]^2])/(a + b)^4 + (2*(7*a - 11*b)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a - b)^3 - ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/(a - b)^2)*Sec[c + d*x]^2)/(64*d*(a + b*Sec[c + d*x])^2)","C",1
307,1,865,200,6.2470717,"\int \frac{\tan ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^6/(a + b*Sec[c + d*x])^2,x]","\frac{(b+a \cos (c+d x))^2 \sin \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x)}{6 b^2 d (a+b \sec (c+d x))^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{(b+a \cos (c+d x))^2 \left(9 a^2 \sin \left(\frac{1}{2} (c+d x)\right)-7 b^2 \sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^2(c+d x)}{3 b^4 d (a+b \sec (c+d x))^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{(b+a \cos (c+d x))^2 \left(9 a^2 \sin \left(\frac{1}{2} (c+d x)\right)-7 b^2 \sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^2(c+d x)}{3 b^4 d (a+b \sec (c+d x))^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{(b+a \cos (c+d x)) \left(\sin (c+d x) a^4-2 b^2 \sin (c+d x) a^2+b^4 \sin (c+d x)\right) \sec ^2(c+d x)}{a b^4 d (a+b \sec (c+d x))^2}-\frac{(c+d x) (b+a \cos (c+d x))^2 \sec ^2(c+d x)}{a^2 d (a+b \sec (c+d x))^2}-\frac{2 \left(b^2-a^2\right)^2 \left(4 a^2+b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right) (b+a \cos (c+d x))^2 \sec ^2(c+d x)}{a^2 b^5 \sqrt{a^2-b^2} d (a+b \sec (c+d x))^2}+\frac{\left(4 a^3-5 a b^2\right) (b+a \cos (c+d x))^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^2(c+d x)}{b^5 d (a+b \sec (c+d x))^2}+\frac{\left(5 a b^2-4 a^3\right) (b+a \cos (c+d x))^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^2(c+d x)}{b^5 d (a+b \sec (c+d x))^2}+\frac{(b-6 a) (b+a \cos (c+d x))^2 \sec ^2(c+d x)}{12 b^3 d (a+b \sec (c+d x))^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{(6 a-b) (b+a \cos (c+d x))^2 \sec ^2(c+d x)}{12 b^3 d (a+b \sec (c+d x))^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{(b+a \cos (c+d x))^2 \sin \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x)}{6 b^2 d (a+b \sec (c+d x))^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\frac{a \left(4 a^2-5 b^2\right) \tanh ^{-1}(\sin (c+d x))}{b^5 d}+\frac{2 (a-b)^{3/2} (a+b)^{3/2} \left(4 a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 b^5 d}+\frac{\left(3 a^2-2 b^2\right) \tan (c+d x)}{b^4 d}+\frac{\left(a^2-b^2\right)^2 \sin (c+d x)}{a b^4 d (a \cos (c+d x)+b)}-\frac{x}{a^2}-\frac{a \tan (c+d x) \sec (c+d x)}{b^3 d}+\frac{\tan ^3(c+d x)}{3 b^2 d}",1,"-(((c + d*x)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^2)/(a^2*d*(a + b*Sec[c + d*x])^2)) - (2*(-a^2 + b^2)^2*(4*a^2 + b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^2)/(a^2*b^5*Sqrt[a^2 - b^2]*d*(a + b*Sec[c + d*x])^2) + ((4*a^3 - 5*a*b^2)*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sec[c + d*x]^2)/(b^5*d*(a + b*Sec[c + d*x])^2) + ((-4*a^3 + 5*a*b^2)*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c + d*x]^2)/(b^5*d*(a + b*Sec[c + d*x])^2) + ((-6*a + b)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^2)/(12*b^3*d*(a + b*Sec[c + d*x])^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Sin[(c + d*x)/2])/(6*b^2*d*(a + b*Sec[c + d*x])^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Sin[(c + d*x)/2])/(6*b^2*d*(a + b*Sec[c + d*x])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((6*a - b)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^2)/(12*b^3*d*(a + b*Sec[c + d*x])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*(9*a^2*Sin[(c + d*x)/2] - 7*b^2*Sin[(c + d*x)/2]))/(3*b^4*d*(a + b*Sec[c + d*x])^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*(9*a^2*Sin[(c + d*x)/2] - 7*b^2*Sin[(c + d*x)/2]))/(3*b^4*d*(a + b*Sec[c + d*x])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + ((b + a*Cos[c + d*x])*Sec[c + d*x]^2*(a^4*Sin[c + d*x] - 2*a^2*b^2*Sin[c + d*x] + b^4*Sin[c + d*x]))/(a*b^4*d*(a + b*Sec[c + d*x])^2)","B",1
308,1,327,150,1.5845995,"\int \frac{\tan ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^4/(a + b*Sec[c + d*x])^2,x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b) \left(\frac{\left(a^2-b^2\right) \sin (c+d x)}{a b^2}+\frac{(c+d x) (a \cos (c+d x)+b)}{a^2}+\frac{2 \left(-2 a^4+a^2 b^2+b^4\right) (a \cos (c+d x)+b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{a^2 b^3 \sqrt{a^2-b^2}}+\frac{2 a (a \cos (c+d x)+b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{b^3}-\frac{2 a (a \cos (c+d x)+b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{b^3}+\frac{\sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{b^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{b^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{d (a+b \sec (c+d x))^2}","\frac{\left(2 a^2-b^2\right) \sin (c+d x)}{a b^2 d (a \cos (c+d x)+b)}+\frac{2 \sqrt{a-b} \sqrt{a+b} \left(2 a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 b^3 d}+\frac{x}{a^2}-\frac{2 a \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac{\tan (c+d x)}{b d (a \cos (c+d x)+b)}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^2*(((c + d*x)*(b + a*Cos[c + d*x]))/a^2 + (2*(-2*a^4 + a^2*b^2 + b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x]))/(a^2*b^3*Sqrt[a^2 - b^2]) + (2*a*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/b^3 - (2*a*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/b^3 + ((b + a*Cos[c + d*x])*Sin[(c + d*x)/2])/(b^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + ((b + a*Cos[c + d*x])*Sin[(c + d*x)/2])/(b^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + ((a^2 - b^2)*Sin[c + d*x])/(a*b^2)))/(d*(a + b*Sec[c + d*x])^2)","B",1
309,1,80,85,0.2619172,"\int \frac{\tan ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^2/(a + b*Sec[c + d*x])^2,x]","-\frac{\frac{2 b \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{a \sin (c+d x)}{a \cos (c+d x)+b}+c+d x}{a^2 d}","\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{x}{a^2}+\frac{\tan (c+d x)}{a d (a+b \sec (c+d x))}",1,"-((c + d*x + (2*b*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (a*Sin[c + d*x])/(b + a*Cos[c + d*x]))/(a^2*d))","A",1
310,1,209,227,1.7250611,"\int \frac{\cot ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^2/(a + b*Sec[c + d*x])^2,x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b) \left(-\frac{4 b^3 \left(b^2-4 a^2\right) (a \cos (c+d x)+b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right)^{5/2}}-\frac{2 (c+d x) (a \cos (c+d x)+b)}{a^2}+\frac{2 b^4 \sin (c+d x)}{a (a-b)^2 (a+b)^2}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a-b)^2}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a+b)^2}\right)}{2 d (a+b \sec (c+d x))^2}","-\frac{2 b^5 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{b^4 \sin (c+d x)}{a d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{4 b^3 \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{x}{a^2}-\frac{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^2*((-2*(c + d*x)*(b + a*Cos[c + d*x]))/a^2 - (4*b^3*(-4*a^2 + b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x]))/(a^2*(a^2 - b^2)^(5/2)) - ((b + a*Cos[c + d*x])*Cot[(c + d*x)/2])/(a + b)^2 + (2*b^4*Sin[c + d*x])/(a*(a - b)^2*(a + b)^2) + ((b + a*Cos[c + d*x])*Tan[(c + d*x)/2])/(a - b)^2))/(2*d*(a + b*Sec[c + d*x])^2)","A",1
311,1,303,360,2.4424573,"\int \frac{\cot ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^4/(a + b*Sec[c + d*x])^2,x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b) \left(-\frac{48 b^5 \left(b^2-6 a^2\right) (a \cos (c+d x)+b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right)^{7/2}}+\frac{24 (c+d x) (a \cos (c+d x)+b)}{a^2}+\frac{24 b^6 \sin (c+d x)}{a (a-b)^3 (a+b)^3}+\frac{4 (7 b-4 a) \tan \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a-b)^3}+\frac{4 (4 a+7 b) \cot \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a+b)^3}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a+b)^2}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a-b)^2}\right)}{24 d (a+b \sec (c+d x))^2}","-\frac{2 b^7 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b^6 \sin (c+d x)}{a d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{4 b^5 \left(3 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{x}{a^2}+\frac{(3 a+5 b) \sin (c+d x)}{4 d (a+b)^3 (1-\cos (c+d x))}-\frac{\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)}-\frac{(3 a-5 b) \sin (c+d x)}{4 d (a-b)^3 (\cos (c+d x)+1)}-\frac{\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))^2}+\frac{\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)^2}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^2*((24*(c + d*x)*(b + a*Cos[c + d*x]))/a^2 - (48*b^5*(-6*a^2 + b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x]))/(a^2*(a^2 - b^2)^(7/2)) + (4*(4*a + 7*b)*(b + a*Cos[c + d*x])*Cot[(c + d*x)/2])/(a + b)^3 - ((b + a*Cos[c + d*x])*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(a + b)^2 + (24*b^6*Sin[c + d*x])/(a*(a - b)^3*(a + b)^3) + (4*(-4*a + 7*b)*(b + a*Cos[c + d*x])*Tan[(c + d*x)/2])/(a - b)^3 + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a - b)^2))/(24*d*(a + b*Sec[c + d*x])^2)","A",1
312,1,1846,761,26.4388256,"\int \frac{(e \tan (c+d x))^{5/2}}{a+b \sec (c+d x)} \, dx","Integrate[(e*Tan[c + d*x])^(5/2)/(a + b*Sec[c + d*x]),x]","\frac{2 (b+a \cos (c+d x)) \cot (c+d x) (e \tan (c+d x))^{5/2}}{b d (a+b \sec (c+d x))}-\frac{(b+a \cos (c+d x)) \sec (c+d x) (e \tan (c+d x))^{5/2} \left(\frac{\cos (2 (c+d x)) \left(-\frac{24 a b^2 F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(c+d x),\frac{b^2 \tan ^2(c+d x)}{a^2-b^2}\right) \tan ^{\frac{7}{2}}(c+d x)}{a^2-b^2}+\frac{112 a^3 F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(c+d x),\frac{b^2 \tan ^2(c+d x)}{a^2-b^2}\right) \tan ^{\frac{3}{2}}(c+d x)}{a^2-b^2}-\frac{168 a b^2 F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(c+d x),\frac{b^2 \tan ^2(c+d x)}{a^2-b^2}\right) \tan ^{\frac{3}{2}}(c+d x)}{a^2-b^2}-\frac{168 a \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{\tan ^2(c+d x)+1}}-84 \sqrt{2} b \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+84 \sqrt{2} b \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\frac{(42+42 i) \left(2 b^2-a^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)}{\sqrt{b} \sqrt[4]{a^2-b^2}}+\frac{(42+42 i) \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)}{\sqrt{b} \sqrt[4]{a^2-b^2}}+42 \sqrt{2} b \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-42 \sqrt{2} b \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\frac{(21+21 i) \left(a^2-2 b^2\right) \log \left(i b \tan (c+d x)-(1+i) \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\tan (c+d x)}+\sqrt{a^2-b^2}\right)}{\sqrt{b} \sqrt[4]{a^2-b^2}}+\frac{(21+21 i) \left(2 b^2-a^2\right) \log \left(i b \tan (c+d x)+(1+i) \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\tan (c+d x)}+\sqrt{a^2-b^2}\right)}{\sqrt{b} \sqrt[4]{a^2-b^2}}\right) \left(a+b \sqrt{\tan ^2(c+d x)+1}\right) \sec ^2(c+d x)}{84 a (b+a \cos (c+d x)) \left(\tan ^2(c+d x)-1\right) \sqrt{\tan ^2(c+d x)+1}}+\frac{4 a \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(c+d x),\frac{b^2 \tan ^2(c+d x)}{a^2-b^2}\right) \tan ^{\frac{3}{2}}(c+d x)}{3 a^2-3 b^2}+\frac{-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(b \tan (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\tan (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(b \tan (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\tan (c+d x)}+\sqrt{b^2-a^2}\right)}{4 \sqrt{2} \sqrt{b} \sqrt[4]{b^2-a^2}}\right) \left(a+b \sqrt{\tan ^2(c+d x)+1}\right) \sec ^2(c+d x)}{(b+a \cos (c+d x)) \left(\tan ^2(c+d x)+1\right)^{3/2}}-\frac{b \left(-6 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) a^2-3 \sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right) a^2+3 \sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right) a^2+8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(c+d x),\frac{b^2 \tan ^2(c+d x)}{a^2-b^2}\right) \tan ^{\frac{3}{2}}(c+d x) a+6 \sqrt{2} \left(a^2-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+6 \sqrt{2} b^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-(6+6 i) \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+(6+6 i) \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)+3 \sqrt{2} b^2 \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-3 \sqrt{2} b^2 \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+(3+3 i) \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(i b \tan (c+d x)-(1+i) \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\tan (c+d x)}+\sqrt{a^2-b^2}\right)-(3+3 i) \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(i b \tan (c+d x)+(1+i) \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\tan (c+d x)}+\sqrt{a^2-b^2}\right)\right) \left(a+b \sqrt{\tan ^2(c+d x)+1}\right) \sec (c+d x)}{4 \left(a^3-a b^2\right) (b+a \cos (c+d x)) \left(\tan ^2(c+d x)+1\right)}\right)}{b d (a+b \sec (c+d x)) \tan ^{\frac{5}{2}}(c+d x)}","-\frac{e^{5/2} \left(a^2-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a b^2 d}+\frac{e^{5/2} \left(a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a b^2 d}+\frac{e^{5/2} \left(a^2-b^2\right) \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a b^2 d}-\frac{e^{5/2} \left(a^2-b^2\right) \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a b^2 d}+\frac{a e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} b^2 d}-\frac{a e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} b^2 d}-\frac{a e^{5/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} b^2 d}+\frac{a e^{5/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} b^2 d}+\frac{2 \sqrt{2} e^2 \sqrt{a-b} \sqrt{a+b} \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{a b d \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} e^2 \sqrt{a-b} \sqrt{a+b} \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{a b d \sqrt{\sin (c+d x)}}-\frac{2 e^2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)}}{b d \sqrt{\sin (2 c+2 d x)}}+\frac{2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{b d}",1,"(2*(b + a*Cos[c + d*x])*Cot[c + d*x]*(e*Tan[c + d*x])^(5/2))/(b*d*(a + b*Sec[c + d*x])) - ((b + a*Cos[c + d*x])*Sec[c + d*x]*(e*Tan[c + d*x])^(5/2)*((4*a*Sec[c + d*x]^2*((-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Tan[c + d*x]])/(-a^2 + b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Tan[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Tan[c + d*x]] + b*Tan[c + d*x]] - Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Tan[c + d*x]] + b*Tan[c + d*x]])/(4*Sqrt[2]*Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Tan[c + d*x]^(3/2))/(3*a^2 - 3*b^2))*(a + b*Sqrt[1 + Tan[c + d*x]^2]))/((b + a*Cos[c + d*x])*(1 + Tan[c + d*x]^2)^(3/2)) - (b*Sec[c + d*x]*(6*Sqrt[2]*(a^2 - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 6*Sqrt[2]*a^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + 6*Sqrt[2]*b^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] - (6 + 6*I)*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)] + (6 + 6*I)*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)] - 3*Sqrt[2]*a^2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 3*Sqrt[2]*b^2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 3*Sqrt[2]*a^2*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 3*Sqrt[2]*b^2*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + (3 + 3*I)*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + I*b*Tan[c + d*x]] - (3 + 3*I)*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + I*b*Tan[c + d*x]] + 8*a*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Tan[c + d*x]^(3/2))*(a + b*Sqrt[1 + Tan[c + d*x]^2]))/(4*(a^3 - a*b^2)*(b + a*Cos[c + d*x])*(1 + Tan[c + d*x]^2)) + (Cos[2*(c + d*x)]*Sec[c + d*x]^2*(-84*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 84*Sqrt[2]*b*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + ((42 + 42*I)*(-a^2 + 2*b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)])/(Sqrt[b]*(a^2 - b^2)^(1/4)) + ((42 + 42*I)*(a^2 - 2*b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)])/(Sqrt[b]*(a^2 - b^2)^(1/4)) + 42*Sqrt[2]*b*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 42*Sqrt[2]*b*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + ((21 + 21*I)*(a^2 - 2*b^2)*Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + I*b*Tan[c + d*x]])/(Sqrt[b]*(a^2 - b^2)^(1/4)) + ((21 + 21*I)*(-a^2 + 2*b^2)*Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + I*b*Tan[c + d*x]])/(Sqrt[b]*(a^2 - b^2)^(1/4)) + (112*a^3*AppellF1[3/4, 1/2, 1, 7/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Tan[c + d*x]^(3/2))/(a^2 - b^2) - (168*a*b^2*AppellF1[3/4, 1/2, 1, 7/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Tan[c + d*x]^(3/2))/(a^2 - b^2) - (24*a*b^2*AppellF1[7/4, 1/2, 1, 11/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Tan[c + d*x]^(7/2))/(a^2 - b^2) - (168*a*Tan[c + d*x]^(3/2))/Sqrt[1 + Tan[c + d*x]^2])*(a + b*Sqrt[1 + Tan[c + d*x]^2]))/(84*a*(b + a*Cos[c + d*x])*(-1 + Tan[c + d*x]^2)*Sqrt[1 + Tan[c + d*x]^2])))/(b*d*(a + b*Sec[c + d*x])*Tan[c + d*x]^(5/2))","C",0
313,1,202,740,6.9103559,"\int \frac{(e \tan (c+d x))^{3/2}}{a+b \sec (c+d x)} \, dx","Integrate[(e*Tan[c + d*x])^(3/2)/(a + b*Sec[c + d*x]),x]","\frac{2 e \sqrt{1-\tan \left(\frac{1}{2} (c+d x)\right)} \cot \left(\frac{1}{2} (c+d x)\right) \sqrt{e \tan (c+d x)} \sqrt{\frac{-\sin (c+d x)+\cos (c+d x)-1}{\cos (c+d x)+1}} \left(-\Pi \left(-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\sqrt{-\tan \left(\frac{1}{2} (c+d x)\right)}\right)\right|-1\right)-\Pi \left(\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\sqrt{-\tan \left(\frac{1}{2} (c+d x)\right)}\right)\right|-1\right)+\Pi \left(-i;\left.\sin ^{-1}\left(\sqrt{-\tan \left(\frac{1}{2} (c+d x)\right)}\right)\right|-1\right)+\Pi \left(i;\left.\sin ^{-1}\left(\sqrt{-\tan \left(\frac{1}{2} (c+d x)\right)}\right)\right|-1\right)\right)}{a d}","-\frac{e^{3/2} \left(a^2-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a b^2 d}+\frac{e^{3/2} \left(a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a b^2 d}-\frac{e^{3/2} \left(a^2-b^2\right) \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a b^2 d}+\frac{e^{3/2} \left(a^2-b^2\right) \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a b^2 d}-\frac{2 \sqrt{2} e^2 \sqrt{a^2-b^2} \sqrt{\sin (c+d x)} \Pi \left(\frac{b}{a-\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right)}{a b d \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}+\frac{2 \sqrt{2} e^2 \sqrt{a^2-b^2} \sqrt{\sin (c+d x)} \Pi \left(\frac{b}{a+\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right)}{a b d \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}+\frac{a e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} b^2 d}-\frac{a e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} b^2 d}+\frac{a e^{3/2} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} b^2 d}-\frac{a e^{3/2} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} b^2 d}+\frac{e^2 \sqrt{\sin (2 c+2 d x)} \sec (c+d x) F\left(\left.c+d x-\frac{\pi }{4}\right|2\right)}{b d \sqrt{e \tan (c+d x)}}",1,"(2*e*Cot[(c + d*x)/2]*(EllipticPi[-I, ArcSin[Sqrt[-Tan[(c + d*x)/2]]], -1] + EllipticPi[I, ArcSin[Sqrt[-Tan[(c + d*x)/2]]], -1] - EllipticPi[-(Sqrt[a - b]/Sqrt[a + b]), ArcSin[Sqrt[-Tan[(c + d*x)/2]]], -1] - EllipticPi[Sqrt[a - b]/Sqrt[a + b], ArcSin[Sqrt[-Tan[(c + d*x)/2]]], -1])*Sqrt[(-1 + Cos[c + d*x] - Sin[c + d*x])/(1 + Cos[c + d*x])]*Sqrt[1 - Tan[(c + d*x)/2]]*Sqrt[e*Tan[c + d*x]])/(a*d)","C",1
314,1,224,415,5.4799344,"\int \frac{\sqrt{e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx","Integrate[Sqrt[e*Tan[c + d*x]]/(a + b*Sec[c + d*x]),x]","-\frac{4 \sqrt{\tan \left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) \sqrt{e \tan (c+d x)} (a \cos (c+d x)+b) \left(\frac{b \left(\Pi \left(\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\sqrt{\tan \left(\frac{1}{2} (c+d x)\right)}\right)\right|-1\right)-\Pi \left(-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\sqrt{\tan \left(\frac{1}{2} (c+d x)\right)}\right)\right|-1\right)\right)}{\sqrt{a-b} \sqrt{a+b}}-i \Pi \left(-i;\left.\sin ^{-1}\left(\sqrt{\tan \left(\frac{1}{2} (c+d x)\right)}\right)\right|-1\right)+i \Pi \left(i;\left.\sin ^{-1}\left(\sqrt{\tan \left(\frac{1}{2} (c+d x)\right)}\right)\right|-1\right)\right)}{a d \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \sec (c+d x))}","\frac{2 \sqrt{2} b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{a d \sqrt{a-b} \sqrt{a+b} \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{a d \sqrt{a-b} \sqrt{a+b} \sqrt{\sin (c+d x)}}-\frac{\sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d}+\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}-\frac{\sqrt{e} \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d}",1,"(-4*(b + a*Cos[c + d*x])*Csc[c + d*x]*((-I)*EllipticPi[-I, ArcSin[Sqrt[Tan[(c + d*x)/2]]], -1] + I*EllipticPi[I, ArcSin[Sqrt[Tan[(c + d*x)/2]]], -1] + (b*(-EllipticPi[-(Sqrt[a - b]/Sqrt[a + b]), ArcSin[Sqrt[Tan[(c + d*x)/2]]], -1] + EllipticPi[Sqrt[a - b]/Sqrt[a + b], ArcSin[Sqrt[Tan[(c + d*x)/2]]], -1]))/(Sqrt[a - b]*Sqrt[a + b]))*Sqrt[Tan[(c + d*x)/2]]*Sqrt[e*Tan[c + d*x]])/(a*d*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x]))","C",1
315,1,246,422,12.8035317,"\int \frac{1}{(a+b \sec (c+d x)) \sqrt{e \tan (c+d x)}} \, dx","Integrate[1/((a + b*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]),x]","\frac{4 \left(\left(b^2-a^2\right) \Pi \left(-i;\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{2} (c+d x)\right)}}\right)\right|-1\right)+a^2 \left(-\Pi \left(i;\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{2} (c+d x)\right)}}\right)\right|-1\right)\right)-b^2 \Pi \left(-\frac{\sqrt{a+b}}{\sqrt{a-b}};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{2} (c+d x)\right)}}\right)\right|-1\right)-b^2 \Pi \left(\frac{\sqrt{a+b}}{\sqrt{a-b}};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{2} (c+d x)\right)}}\right)\right|-1\right)+a (a-b) F\left(\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{2} (c+d x)\right)}}\right)\right|-1\right)+b^2 \Pi \left(i;\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{2} (c+d x)\right)}}\right)\right|-1\right)\right)}{a d \left(a^2-b^2\right) \sqrt{\tan \left(\frac{1}{2} (c+d x)\right)-1} \sqrt{\cot \left(\frac{1}{2} (c+d x)\right)+1} \sqrt{e \tan (c+d x)}}","-\frac{2 \sqrt{2} b \sqrt{\sin (c+d x)} \Pi \left(\frac{b}{a-\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right)}{a d \sqrt{a^2-b^2} \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}+\frac{2 \sqrt{2} b \sqrt{\sin (c+d x)} \Pi \left(\frac{b}{a+\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right)}{a d \sqrt{a^2-b^2} \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} a d \sqrt{e}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} a d \sqrt{e}}-\frac{\log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d \sqrt{e}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} a d \sqrt{e}}",1,"(4*(a*(a - b)*EllipticF[ArcSin[1/Sqrt[Tan[(c + d*x)/2]]], -1] + (-a^2 + b^2)*EllipticPi[-I, ArcSin[1/Sqrt[Tan[(c + d*x)/2]]], -1] - a^2*EllipticPi[I, ArcSin[1/Sqrt[Tan[(c + d*x)/2]]], -1] + b^2*EllipticPi[I, ArcSin[1/Sqrt[Tan[(c + d*x)/2]]], -1] - b^2*EllipticPi[-(Sqrt[a + b]/Sqrt[a - b]), ArcSin[1/Sqrt[Tan[(c + d*x)/2]]], -1] - b^2*EllipticPi[Sqrt[a + b]/Sqrt[a - b], ArcSin[1/Sqrt[Tan[(c + d*x)/2]]], -1]))/(a*(a^2 - b^2)*d*Sqrt[1 + Cot[(c + d*x)/2]]*Sqrt[-1 + Tan[(c + d*x)/2]]*Sqrt[e*Tan[c + d*x]])","C",1
316,1,1571,863,27.5346827,"\int \frac{1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx","Integrate[1/((a + b*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2)),x]","\frac{(b+a \cos (c+d x)) \sec (c+d x) \left(\frac{2 b \sin (c+d x)}{b^2-a^2}-\frac{2 (b-a \cos (c+d x)) \csc (c+d x)}{b^2-a^2}\right) \tan ^2(c+d x)}{d (a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}}+\frac{(b+a \cos (c+d x)) \sec (c+d x) \left(\frac{b \cos (2 (c+d x)) \sec ^2(c+d x) \left(-\frac{24 a b^2 F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(c+d x),\frac{b^2 \tan ^2(c+d x)}{a^2-b^2}\right) \tan ^{\frac{7}{2}}(c+d x)}{a^2-b^2}+\frac{112 a^3 F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(c+d x),\frac{b^2 \tan ^2(c+d x)}{a^2-b^2}\right) \tan ^{\frac{3}{2}}(c+d x)}{a^2-b^2}-\frac{168 a b^2 F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(c+d x),\frac{b^2 \tan ^2(c+d x)}{a^2-b^2}\right) \tan ^{\frac{3}{2}}(c+d x)}{a^2-b^2}-\frac{168 a \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{\tan ^2(c+d x)+1}}-84 \sqrt{2} b \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+84 \sqrt{2} b \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\frac{(42+42 i) \left(2 b^2-a^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)}{\sqrt{b} \sqrt[4]{a^2-b^2}}+\frac{(42+42 i) \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)}{\sqrt{b} \sqrt[4]{a^2-b^2}}+42 \sqrt{2} b \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-42 \sqrt{2} b \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\frac{(21+21 i) \left(a^2-2 b^2\right) \log \left(i b \tan (c+d x)-(1+i) \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\tan (c+d x)}+\sqrt{a^2-b^2}\right)}{\sqrt{b} \sqrt[4]{a^2-b^2}}+\frac{(21+21 i) \left(2 b^2-a^2\right) \log \left(i b \tan (c+d x)+(1+i) \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\tan (c+d x)}+\sqrt{a^2-b^2}\right)}{\sqrt{b} \sqrt[4]{a^2-b^2}}\right) \left(a+b \sqrt{\tan ^2(c+d x)+1}\right)}{84 a (b+a \cos (c+d x)) \left(\tan ^2(c+d x)-1\right) \sqrt{\tan ^2(c+d x)+1}}-\frac{\left(3 b^2-a^2\right) \sec (c+d x) \left(-6 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) a^2-3 \sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right) a^2+3 \sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right) a^2+8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(c+d x),\frac{b^2 \tan ^2(c+d x)}{a^2-b^2}\right) \tan ^{\frac{3}{2}}(c+d x) a+6 \sqrt{2} \left(a^2-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+6 \sqrt{2} b^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-(6+6 i) \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+(6+6 i) \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)+3 \sqrt{2} b^2 \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-3 \sqrt{2} b^2 \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+(3+3 i) \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(i b \tan (c+d x)-(1+i) \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\tan (c+d x)}+\sqrt{a^2-b^2}\right)-(3+3 i) \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(i b \tan (c+d x)+(1+i) \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\tan (c+d x)}+\sqrt{a^2-b^2}\right)\right) \left(a+b \sqrt{\tan ^2(c+d x)+1}\right)}{12 \left(a^3-a b^2\right) (b+a \cos (c+d x)) \left(\tan ^2(c+d x)+1\right)}\right) \tan ^{\frac{3}{2}}(c+d x)}{(a-b) (a+b) d (a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}}","\frac{2 \sqrt{2} \sqrt{\cos (c+d x)} \Pi \left(-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right) \sqrt{e \tan (c+d x)} b^3}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} \sqrt{\cos (c+d x)} \Pi \left(\frac{\sqrt{a-b}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right) \sqrt{e \tan (c+d x)} b^3}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt{\sin (c+d x)}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right) b^2}{\sqrt{2} a \left(a^2-b^2\right) d e^{3/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right) b^2}{\sqrt{2} a \left(a^2-b^2\right) d e^{3/2}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right) b^2}{2 \sqrt{2} a \left(a^2-b^2\right) d e^{3/2}}-\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right) b^2}{2 \sqrt{2} a \left(a^2-b^2\right) d e^{3/2}}-\frac{2 \cos (c+d x) (e \tan (c+d x))^{3/2} b}{\left(a^2-b^2\right) d e^3}+\frac{2 \cos (c+d x) E\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sqrt{e \tan (c+d x)} b}{\left(a^2-b^2\right) d e^2 \sqrt{\sin (2 c+2 d x)}}+\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} \left(a^2-b^2\right) d e^{3/2}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} \left(a^2-b^2\right) d e^{3/2}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right)}{2 \sqrt{2} \left(a^2-b^2\right) d e^{3/2}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right)}{2 \sqrt{2} \left(a^2-b^2\right) d e^{3/2}}-\frac{2 (a-b \sec (c+d x))}{\left(a^2-b^2\right) d e \sqrt{e \tan (c+d x)}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*((-2*(b - a*Cos[c + d*x])*Csc[c + d*x])/(-a^2 + b^2) + (2*b*Sin[c + d*x])/(-a^2 + b^2))*Tan[c + d*x]^2)/(d*(a + b*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x]^(3/2)*(-1/12*((-a^2 + 3*b^2)*Sec[c + d*x]*(6*Sqrt[2]*(a^2 - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 6*Sqrt[2]*a^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + 6*Sqrt[2]*b^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] - (6 + 6*I)*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)] + (6 + 6*I)*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)] - 3*Sqrt[2]*a^2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 3*Sqrt[2]*b^2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 3*Sqrt[2]*a^2*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 3*Sqrt[2]*b^2*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + (3 + 3*I)*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + I*b*Tan[c + d*x]] - (3 + 3*I)*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + I*b*Tan[c + d*x]] + 8*a*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Tan[c + d*x]^(3/2))*(a + b*Sqrt[1 + Tan[c + d*x]^2]))/((a^3 - a*b^2)*(b + a*Cos[c + d*x])*(1 + Tan[c + d*x]^2)) + (b*Cos[2*(c + d*x)]*Sec[c + d*x]^2*(-84*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 84*Sqrt[2]*b*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + ((42 + 42*I)*(-a^2 + 2*b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)])/(Sqrt[b]*(a^2 - b^2)^(1/4)) + ((42 + 42*I)*(a^2 - 2*b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)])/(Sqrt[b]*(a^2 - b^2)^(1/4)) + 42*Sqrt[2]*b*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 42*Sqrt[2]*b*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + ((21 + 21*I)*(a^2 - 2*b^2)*Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + I*b*Tan[c + d*x]])/(Sqrt[b]*(a^2 - b^2)^(1/4)) + ((21 + 21*I)*(-a^2 + 2*b^2)*Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + I*b*Tan[c + d*x]])/(Sqrt[b]*(a^2 - b^2)^(1/4)) + (112*a^3*AppellF1[3/4, 1/2, 1, 7/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Tan[c + d*x]^(3/2))/(a^2 - b^2) - (168*a*b^2*AppellF1[3/4, 1/2, 1, 7/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Tan[c + d*x]^(3/2))/(a^2 - b^2) - (24*a*b^2*AppellF1[7/4, 1/2, 1, 11/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Tan[c + d*x]^(7/2))/(a^2 - b^2) - (168*a*Tan[c + d*x]^(3/2))/Sqrt[1 + Tan[c + d*x]^2])*(a + b*Sqrt[1 + Tan[c + d*x]^2]))/(84*a*(b + a*Cos[c + d*x])*(-1 + Tan[c + d*x]^2)*Sqrt[1 + Tan[c + d*x]^2])))/((a - b)*(a + b)*d*(a + b*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2))","C",0
317,1,2169,836,23.9021513,"\int \frac{1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{5/2}} \, dx","Integrate[1/((a + b*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2)),x]","\text{Result too large to show}","-\frac{2 \sqrt{2} \Pi \left(\frac{b}{a-\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right) \sqrt{\sin (c+d x)} b^3}{a \left(a^2-b^2\right)^{3/2} d e^2 \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}+\frac{2 \sqrt{2} \Pi \left(\frac{b}{a+\sqrt{a^2-b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{-\cos (c+d x)}}{\sqrt{\sin (c+d x)+1}}\right)\right|-1\right) \sqrt{\sin (c+d x)} b^3}{a \left(a^2-b^2\right)^{3/2} d e^2 \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right) b^2}{\sqrt{2} a \left(a^2-b^2\right) d e^{5/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right) b^2}{\sqrt{2} a \left(a^2-b^2\right) d e^{5/2}}-\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right) b^2}{2 \sqrt{2} a \left(a^2-b^2\right) d e^{5/2}}+\frac{\log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right) b^2}{2 \sqrt{2} a \left(a^2-b^2\right) d e^{5/2}}+\frac{F\left(\left.c+d x-\frac{\pi }{4}\right|2\right) \sec (c+d x) \sqrt{\sin (2 c+2 d x)} b}{3 \left(a^2-b^2\right) d e^2 \sqrt{e \tan (c+d x)}}+\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} \left(a^2-b^2\right) d e^{5/2}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} \left(a^2-b^2\right) d e^{5/2}}+\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right)}{2 \sqrt{2} \left(a^2-b^2\right) d e^{5/2}}-\frac{a \log \left(\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right)}{2 \sqrt{2} \left(a^2-b^2\right) d e^{5/2}}-\frac{2 (a-b \sec (c+d x))}{3 \left(a^2-b^2\right) d e (e \tan (c+d x))^{3/2}}",1,"((b + a*Cos[c + d*x])*((2*a)/(3*(a^2 - b^2)) - (2*(-a + b*Cos[c + d*x])*Csc[c + d*x]^2)/(3*(-a^2 + b^2)))*Sec[c + d*x]*Tan[c + d*x]^3)/(d*(a + b*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2)) - ((b + a*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x]^(5/2)*((2*(3*a^2 - 5*b^2)*Sec[c + d*x]^3*(a + b*Sqrt[1 + Tan[c + d*x]^2])*(((-1/8 + I/8)*a*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)] + Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + I*b*Tan[c + d*x]] - Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + I*b*Tan[c + d*x]]))/(Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(-a^2 + b^2)*AppellF1[1/4, -1/2, 1, 5/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Tan[c + d*x]]*Sqrt[1 + Tan[c + d*x]^2])/((5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2)*(a^2 - b^2*(1 + Tan[c + d*x]^2)))))/((b + a*Cos[c + d*x])*(1 + Tan[c + d*x]^2)^2) + (8*a*b*Sec[c + d*x]^2*(a + b*Sqrt[1 + Tan[c + d*x]^2])*((Sqrt[b]*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Tan[c + d*x]])/(-a^2 + b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Tan[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Tan[c + d*x]] + b*Tan[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Tan[c + d*x]] + b*Tan[c + d*x]]))/(4*Sqrt[2]*(-a^2 + b^2)^(3/4)) + (5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Tan[c + d*x]])/(Sqrt[1 + Tan[c + d*x]^2]*(-5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2)*(-a^2 + b^2*(1 + Tan[c + d*x]^2)))))/((b + a*Cos[c + d*x])*(1 + Tan[c + d*x]^2)^(3/2)) + ((3*a^2 - 3*b^2)*Cos[2*(c + d*x)]*Sec[c + d*x]^3*(a + b*Sqrt[1 + Tan[c + d*x]^2])*((-20*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/a + (20*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/a + ((10 - 10*I)*(a^2 - 2*b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) - ((10 - 10*I)*(a^2 - 2*b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) - (10*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/a + (10*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/a + ((5 - 5*I)*(a^2 - 2*b^2)*Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + I*b*Tan[c + d*x]])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) - ((5 - 5*I)*(a^2 - 2*b^2)*Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + I*b*Tan[c + d*x]])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) - (8*b*AppellF1[5/4, 1/2, 1, 9/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Tan[c + d*x]^(5/2))/(-a^2 + b^2) - (200*b*(-a^2 + b^2)*AppellF1[1/4, 1/2, 1, 5/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Tan[c + d*x]])/(Sqrt[1 + Tan[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2)*(-a^2 + b^2*(1 + Tan[c + d*x]^2)))))/(20*(b + a*Cos[c + d*x])*(1 - Tan[c + d*x]^2)*(1 + Tan[c + d*x]^2))))/(6*(a - b)*(a + b)*d*(a + b*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2))","C",0
318,1,254,169,6.3069393,"\int \sqrt{a+b \sec (c+d x)} \tan ^5(c+d x) \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x]^5,x]","\frac{\sqrt{a+b \sec (c+d x)} \left(-\frac{4 \left(a^2+21 b^2\right) \sec ^2(c+d x)}{105 b^2}-\frac{4 a \left(21 b^2-4 a^2\right) \sec (c+d x)}{315 b^3}+\frac{2 \left(-16 a^4+84 a^2 b^2+315 b^4\right)}{315 b^4}+\frac{2 a \sec ^3(c+d x)}{63 b}+\frac{2}{9} \sec ^4(c+d x)\right)}{d}-\frac{\sin ^2(c+d x) \sqrt{a \cos (c+d x)} \sqrt{a+b \sec (c+d x)} \left(\log \left(\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}+1\right)-\log \left(1-\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}\right)\right)}{d \left(1-\cos ^2(c+d x)\right) \sqrt{a \cos (c+d x)+b}}","\frac{2 \left(3 a^2-2 b^2\right) (a+b \sec (c+d x))^{5/2}}{5 b^4 d}-\frac{2 a \left(a^2-2 b^2\right) (a+b \sec (c+d x))^{3/2}}{3 b^4 d}+\frac{2 (a+b \sec (c+d x))^{9/2}}{9 b^4 d}-\frac{6 a (a+b \sec (c+d x))^{7/2}}{7 b^4 d}+\frac{2 \sqrt{a+b \sec (c+d x)}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{d}",1,"(Sqrt[a + b*Sec[c + d*x]]*((2*(-16*a^4 + 84*a^2*b^2 + 315*b^4))/(315*b^4) - (4*a*(-4*a^2 + 21*b^2)*Sec[c + d*x])/(315*b^3) - (4*(a^2 + 21*b^2)*Sec[c + d*x]^2)/(105*b^2) + (2*a*Sec[c + d*x]^3)/(63*b) + (2*Sec[c + d*x]^4)/9))/d - (Sqrt[a*Cos[c + d*x]]*(-Log[1 - Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]] + Log[1 + Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]])*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x]^2)/(d*Sqrt[b + a*Cos[c + d*x]]*(1 - Cos[c + d*x]^2))","A",1
319,1,194,100,6.2236946,"\int \sqrt{a+b \sec (c+d x)} \tan ^3(c+d x) \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x]^3,x]","\frac{\sqrt{a+b \sec (c+d x)} \left(-\frac{2 \left(2 a^2+15 b^2\right)}{15 b^2}+\frac{2 a \sec (c+d x)}{15 b}+\frac{2}{5} \sec ^2(c+d x)\right)}{d}+\frac{\sin ^2(c+d x) \sqrt{a \cos (c+d x)} \sqrt{a+b \sec (c+d x)} \left(\log \left(\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}+1\right)-\log \left(1-\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}\right)\right)}{d \left(1-\cos ^2(c+d x)\right) \sqrt{a \cos (c+d x)+b}}","\frac{2 (a+b \sec (c+d x))^{5/2}}{5 b^2 d}-\frac{2 a (a+b \sec (c+d x))^{3/2}}{3 b^2 d}-\frac{2 \sqrt{a+b \sec (c+d x)}}{d}+\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{d}",1,"(Sqrt[a + b*Sec[c + d*x]]*((-2*(2*a^2 + 15*b^2))/(15*b^2) + (2*a*Sec[c + d*x])/(15*b) + (2*Sec[c + d*x]^2)/5))/d + (Sqrt[a*Cos[c + d*x]]*(-Log[1 - Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]] + Log[1 + Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]])*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x]^2)/(d*Sqrt[b + a*Cos[c + d*x]]*(1 - Cos[c + d*x]^2))","A",1
320,1,137,51,0.2844674,"\int \sqrt{a+b \sec (c+d x)} \tan (c+d x) \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x],x]","\frac{\sqrt{a+b \sec (c+d x)} \left(2 \sqrt{a \cos (c+d x)+b}+\sqrt{a \cos (c+d x)} \log \left(1-\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}\right)-\sqrt{a \cos (c+d x)} \log \left(\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}+1\right)\right)}{d \sqrt{a \cos (c+d x)+b}}","\frac{2 \sqrt{a+b \sec (c+d x)}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{d}",1,"((2*Sqrt[b + a*Cos[c + d*x]] + Sqrt[a*Cos[c + d*x]]*Log[1 - Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]] - Sqrt[a*Cos[c + d*x]]*Log[1 + Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]])*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[b + a*Cos[c + d*x]])","B",1
321,1,333,106,4.3796253,"\int \cot (c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]*Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{a+b \sec (c+d x)} \left(\sqrt{b} \sqrt{b-a} \sqrt{\frac{a \cos (c+d x)+b}{b \cos (c+d x)+b}} \sin ^{-1}\left(\frac{\sqrt{b-a} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}}{\sqrt{b}}\right)+2 \sqrt{a} \sqrt{\frac{a \cos (c+d x)+b}{\cos (c+d x)+1}} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}}{\sqrt{\frac{a \cos (c+d x)+b}{\cos (c+d x)+1}}}\right)+\sqrt{-a-b} \sqrt{\frac{-a \cos (c+d x)-b}{\cos (c+d x)+1}} \tanh ^{-1}\left(\frac{\sqrt{-a-b} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)}}{\sqrt{-\left(\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)\right)}}\right)\right)}{d (a \cos (c+d x)+b)}","\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{d}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{d}-\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{d}",1,"(2*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*(Sqrt[-a - b]*ArcTanh[(Sqrt[-a - b]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2])/Sqrt[-((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)]]*Sqrt[(-b - a*Cos[c + d*x])/(1 + Cos[c + d*x])] + 2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])])/Sqrt[(b + a*Cos[c + d*x])/(1 + Cos[c + d*x])]]*Sqrt[(b + a*Cos[c + d*x])/(1 + Cos[c + d*x])] + Sqrt[b]*Sqrt[-a + b]*ArcSin[(Sqrt[-a + b]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])])/Sqrt[b]]*Sqrt[(b + a*Cos[c + d*x])/(b + b*Cos[c + d*x])])*Sqrt[a + b*Sec[c + d*x]])/(d*(b + a*Cos[c + d*x]))","B",1
322,1,937,215,18.9481935,"\int \cot ^3(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{1}{2}-\frac{1}{2} \csc ^2(c+d x)\right)}{d}+\frac{\left(\frac{2 \left(4 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{b+a \cos (c+d x)}}{\sqrt{-a \cos (c+d x)}}\right)-\sqrt{a} \left(\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a-b} \sqrt{-a \cos (c+d x)}}\right)+\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a+b} \sqrt{-a \cos (c+d x)}}\right)\right)\right) \sqrt{-a \cos (c+d x)} \cos (2 (c+d x)) \sqrt{\sec (c+d x)} a^2}{\sqrt{a-b} \sqrt{a+b} \left(a^2-2 b^2-2 (b+a \cos (c+d x))^2+4 b (b+a \cos (c+d x))\right)}+\frac{3 b \left(-\sqrt{-a^2} \sqrt{a+b} \log \left(\sqrt{b+a \cos (c+d x)}-\sqrt{b-a}\right)+\sqrt{-a^2} \sqrt{a+b} \log \left(\sqrt{b-a}+\sqrt{b+a \cos (c+d x)}\right)-a \sqrt{b-a} \log \left(\sqrt{b+a \cos (c+d x)}-\sqrt{a+b}\right)+a \sqrt{b-a} \log \left(\sqrt{a+b}+\sqrt{b+a \cos (c+d x)}\right)+\sqrt{-a^2} \sqrt{a+b} \log \left(b+\sqrt{a} \sqrt{-a \cos (c+d x)}-\sqrt{b-a} \sqrt{b+a \cos (c+d x)}\right)-\sqrt{-a^2} \sqrt{a+b} \log \left(b+\sqrt{a} \sqrt{-a \cos (c+d x)}+\sqrt{b-a} \sqrt{b+a \cos (c+d x)}\right)+a \sqrt{b-a} \log \left(b+\sqrt{-a} \sqrt{-a \cos (c+d x)}-\sqrt{a+b} \sqrt{b+a \cos (c+d x)}\right)-a \sqrt{b-a} \log \left(b+\sqrt{-a} \sqrt{-a \cos (c+d x)}+\sqrt{a+b} \sqrt{b+a \cos (c+d x)}\right)\right) a}{2 (-a)^{3/2} \sqrt{b-a} \sqrt{a+b} \sqrt{-a \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{2 \left(\sqrt{a-b} (a+b) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a-b} \sqrt{-a \cos (c+d x)}}\right)+(a-b) \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a+b} \sqrt{-a \cos (c+d x)}}\right)\right) \sqrt{-a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{a}}{(a-b) (a+b)}\right) \sqrt{a+b \sec (c+d x)}}{4 d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}}","-\frac{\cot ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{d}-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{4 d \sqrt{a-b}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{d \sqrt{a-b}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{d \sqrt{a+b}}+\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{4 d \sqrt{a+b}}",1,"((1/2 - Csc[c + d*x]^2/2)*Sqrt[a + b*Sec[c + d*x]])/d + (((3*a*b*(-(Sqrt[-a^2]*Sqrt[a + b]*Log[-Sqrt[-a + b] + Sqrt[b + a*Cos[c + d*x]]]) + Sqrt[-a^2]*Sqrt[a + b]*Log[Sqrt[-a + b] + Sqrt[b + a*Cos[c + d*x]]] - a*Sqrt[-a + b]*Log[-Sqrt[a + b] + Sqrt[b + a*Cos[c + d*x]]] + a*Sqrt[-a + b]*Log[Sqrt[a + b] + Sqrt[b + a*Cos[c + d*x]]] + Sqrt[-a^2]*Sqrt[a + b]*Log[b + Sqrt[a]*Sqrt[-(a*Cos[c + d*x])] - Sqrt[-a + b]*Sqrt[b + a*Cos[c + d*x]]] - Sqrt[-a^2]*Sqrt[a + b]*Log[b + Sqrt[a]*Sqrt[-(a*Cos[c + d*x])] + Sqrt[-a + b]*Sqrt[b + a*Cos[c + d*x]]] + a*Sqrt[-a + b]*Log[b + Sqrt[-a]*Sqrt[-(a*Cos[c + d*x])] - Sqrt[a + b]*Sqrt[b + a*Cos[c + d*x]]] - a*Sqrt[-a + b]*Log[b + Sqrt[-a]*Sqrt[-(a*Cos[c + d*x])] + Sqrt[a + b]*Sqrt[b + a*Cos[c + d*x]]]))/(2*(-a)^(3/2)*Sqrt[-a + b]*Sqrt[a + b]*Sqrt[-(a*Cos[c + d*x])]*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a]*(Sqrt[a - b]*(a + b)*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a - b]*Sqrt[-(a*Cos[c + d*x])])] + (a - b)*Sqrt[a + b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a + b]*Sqrt[-(a*Cos[c + d*x])])])*Sqrt[-(a*Cos[c + d*x])]*Sqrt[Sec[c + d*x]])/((a - b)*(a + b)) + (2*a^2*(4*Sqrt[a - b]*Sqrt[a + b]*ArcTan[Sqrt[b + a*Cos[c + d*x]]/Sqrt[-(a*Cos[c + d*x])]] - Sqrt[a]*(Sqrt[a + b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a - b]*Sqrt[-(a*Cos[c + d*x])])] + Sqrt[a - b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a + b]*Sqrt[-(a*Cos[c + d*x])])]))*Sqrt[-(a*Cos[c + d*x])]*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(Sqrt[a - b]*Sqrt[a + b]*(a^2 - 2*b^2 + 4*b*(b + a*Cos[c + d*x]) - 2*(b + a*Cos[c + d*x])^2)))*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","B",0
323,1,692,344,17.8809243,"\int \sqrt{a+b \sec (c+d x)} \tan ^2(c+d x) \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x]^2,x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{2 a \sin (c+d x)}{3 b}+\frac{2}{3} \tan (c+d x)\right)}{d}-\frac{2 \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \sqrt{a+b \sec (c+d x)} \left(a \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right) \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2-b \tan ^4\left(\frac{1}{2} (c+d x)\right)+b\right)+2 i b (a-b) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)-i a (a-b) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)-6 i a b \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)\right)}{3 b d \sqrt{\frac{b-a}{a+b}} \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+b} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2-b \tan ^4\left(\frac{1}{2} (c+d x)\right)+b\right)}","-\frac{2 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d}+\frac{2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}-\frac{2 \sqrt{a+b} (a+2 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}",1,"(-2*Sqrt[a + b*Sec[c + d*x]]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*((-I)*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(a - b)*b*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*a*b*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + a*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]*(b - b*Tan[(c + d*x)/2]^4 + a*(-1 + Tan[(c + d*x)/2]^2)^2)))/(3*b*Sqrt[(-a + b)/(a + b)]*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(b - b*Tan[(c + d*x)/2]^4 + a*(-1 + Tan[(c + d*x)/2]^2)^2)) + (Sqrt[a + b*Sec[c + d*x]]*((2*a*Sin[c + d*x])/(3*b) + (2*Tan[c + d*x])/3))/d","C",1
324,1,151,125,0.2600103,"\int \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]],x]","\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \sqrt{a+b \sec (c+d x)} \left((b-a) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 a \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{d (a \cos (c+d x)+b)}","-\frac{2 \cot (c+d x) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (\sec (c+d x)+1)}{a+b \sec (c+d x)}} (a+b \sec (c+d x)) \Pi \left(\frac{a}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right)|\frac{a-b}{a+b}\right)}{d \sqrt{a+b}}",1,"(4*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*((-a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sqrt[a + b*Sec[c + d*x]])/(d*(b + a*Cos[c + d*x]))","A",1
325,1,154,246,3.5807565,"\int \cot ^2(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Cot[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{a+b \sec (c+d x)} \left(-\frac{2 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{\frac{a+b \sec (c+d x)}{(a+b) (\sec (c+d x)+1)}} \left((b-2 a) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+4 a \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{a \cos (c+d x)+b}-\cot (c+d x)\right)}{d}","-\frac{\cot (c+d x) \sqrt{a+b \sec (c+d x)}}{d}+\frac{\sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{2 \cot (c+d x) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (\sec (c+d x)+1)}{a+b \sec (c+d x)}} (a+b \sec (c+d x)) \Pi \left(\frac{a}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right)|\frac{a-b}{a+b}\right)}{d \sqrt{a+b}}",1,"(Sqrt[a + b*Sec[c + d*x]]*(-Cot[c + d*x] - (2*Cos[(c + d*x)/2]^2*((-2*a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 4*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))])/(b + a*Cos[c + d*x])))/d","A",1
326,1,248,148,6.3205066,"\int \frac{\tan ^5(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Tan[c + d*x]^5/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(\frac{8 a \left(35 b^2-12 a^2\right)}{105 b^4}-\frac{4 \left(35 b^2-12 a^2\right) \sec (c+d x)}{105 b^3}-\frac{12 a \sec ^2(c+d x)}{35 b^2}+\frac{2 \sec ^3(c+d x)}{7 b}\right)}{d \sqrt{a+b \sec (c+d x)}}-\frac{\sin (c+d x) \tan (c+d x) \sqrt{a \cos (c+d x)} \sqrt{a \cos (c+d x)+b} \left(\log \left(\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}+1\right)-\log \left(1-\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}\right)\right)}{a d \left(1-\cos ^2(c+d x)\right) \sqrt{a+b \sec (c+d x)}}","\frac{2 \left(3 a^2-2 b^2\right) (a+b \sec (c+d x))^{3/2}}{3 b^4 d}-\frac{2 a \left(a^2-2 b^2\right) \sqrt{a+b \sec (c+d x)}}{b^4 d}+\frac{2 (a+b \sec (c+d x))^{7/2}}{7 b^4 d}-\frac{6 a (a+b \sec (c+d x))^{5/2}}{5 b^4 d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*((8*a*(-12*a^2 + 35*b^2))/(105*b^4) - (4*(-12*a^2 + 35*b^2)*Sec[c + d*x])/(105*b^3) - (12*a*Sec[c + d*x]^2)/(35*b^2) + (2*Sec[c + d*x]^3)/(7*b)))/(d*Sqrt[a + b*Sec[c + d*x]]) - (Sqrt[a*Cos[c + d*x]]*Sqrt[b + a*Cos[c + d*x]]*(-Log[1 - Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]] + Log[1 + Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]])*Sin[c + d*x]*Tan[c + d*x])/(a*d*(1 - Cos[c + d*x]^2)*Sqrt[a + b*Sec[c + d*x]])","A",1
327,1,194,79,1.2250655,"\int \frac{\tan ^3(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Tan[c + d*x]^3/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(\frac{2 \sec (c+d x)}{3 b}-\frac{4 a}{3 b^2}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{\sin (c+d x) \tan (c+d x) \sqrt{a \cos (c+d x)} \sqrt{a \cos (c+d x)+b} \left(\log \left(\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}+1\right)-\log \left(1-\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}\right)\right)}{a d \left(1-\cos ^2(c+d x)\right) \sqrt{a+b \sec (c+d x)}}","\frac{2 (a+b \sec (c+d x))^{3/2}}{3 b^2 d}-\frac{2 a \sqrt{a+b \sec (c+d x)}}{b^2 d}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*((-4*a)/(3*b^2) + (2*Sec[c + d*x])/(3*b)))/(d*Sqrt[a + b*Sec[c + d*x]]) + (Sqrt[a*Cos[c + d*x]]*Sqrt[b + a*Cos[c + d*x]]*(-Log[1 - Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]] + Log[1 + Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]])*Sin[c + d*x]*Tan[c + d*x])/(a*d*(1 - Cos[c + d*x]^2)*Sqrt[a + b*Sec[c + d*x]])","B",1
328,1,108,31,0.2027463,"\int \frac{\tan (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Tan[c + d*x]/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{a \cos (c+d x)+b} \left(\log \left(1-\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}\right)-\log \left(\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}+1\right)\right)}{d \sqrt{a \cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"(Sqrt[b + a*Cos[c + d*x]]*(Log[1 - Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]] - Log[1 + Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]]))/(d*Sqrt[a*Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","B",1
329,1,218,106,5.9793593,"\int \frac{\cot (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Cot[c + d*x]/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{a \cos (c+d x)+b} \left(\sqrt{a} \left(\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{a \cos (c+d x)+b}}{\sqrt{a-b} \sqrt{-a \cos (c+d x)}}\right)+\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{a \cos (c+d x)+b}}{\sqrt{a+b} \sqrt{-a \cos (c+d x)}}\right)\right)-2 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{-a \cos (c+d x)}}\right)\right)}{d \sqrt{a-b} \sqrt{a+b} \sqrt{-a \cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{d \sqrt{a-b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{d \sqrt{a+b}}",1,"((-2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[Sqrt[b + a*Cos[c + d*x]]/Sqrt[-(a*Cos[c + d*x])]] + Sqrt[a]*(Sqrt[a + b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a - b]*Sqrt[-(a*Cos[c + d*x])])] + Sqrt[a - b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a + b]*Sqrt[-(a*Cos[c + d*x])])]))*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a - b]*Sqrt[a + b]*d*Sqrt[-(a*Cos[c + d*x])]*Sqrt[a + b*Sec[c + d*x]])","B",1
330,1,1022,260,6.8922104,"\int \frac{\cot ^3(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Cot[c + d*x]^3/Sqrt[a + b*Sec[c + d*x]],x]","\frac{(b+a \cos (c+d x)) \left(\frac{(a-b \cos (c+d x)) \csc ^2(c+d x)}{2 \left(b^2-a^2\right)}+\frac{a}{2 \left(a^2-b^2\right)}\right) \sec (c+d x)}{d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{b+a \cos (c+d x)} \left(-\frac{b \left(-\sqrt{-a^2} \sqrt{a+b} \log \left(\sqrt{b+a \cos (c+d x)}-\sqrt{b-a}\right)+\sqrt{-a^2} \sqrt{a+b} \log \left(\sqrt{b-a}+\sqrt{b+a \cos (c+d x)}\right)-a \sqrt{b-a} \log \left(\sqrt{b+a \cos (c+d x)}-\sqrt{a+b}\right)+a \sqrt{b-a} \log \left(\sqrt{a+b}+\sqrt{b+a \cos (c+d x)}\right)+\sqrt{-a^2} \sqrt{a+b} \log \left(b+\sqrt{a} \sqrt{-a \cos (c+d x)}-\sqrt{b-a} \sqrt{b+a \cos (c+d x)}\right)-\sqrt{-a^2} \sqrt{a+b} \log \left(b+\sqrt{a} \sqrt{-a \cos (c+d x)}+\sqrt{b-a} \sqrt{b+a \cos (c+d x)}\right)+a \sqrt{b-a} \log \left(b+\sqrt{-a} \sqrt{-a \cos (c+d x)}-\sqrt{a+b} \sqrt{b+a \cos (c+d x)}\right)-a \sqrt{b-a} \log \left(b+\sqrt{-a} \sqrt{-a \cos (c+d x)}+\sqrt{a+b} \sqrt{b+a \cos (c+d x)}\right)\right) a^2}{2 (-a)^{3/2} \sqrt{b-a} \sqrt{a+b} \sqrt{-a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{\left(2 a^2-2 b^2\right) \left(4 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{b+a \cos (c+d x)}}{\sqrt{-a \cos (c+d x)}}\right)-\sqrt{a} \left(\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a-b} \sqrt{-a \cos (c+d x)}}\right)+\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a+b} \sqrt{-a \cos (c+d x)}}\right)\right)\right) \sqrt{-a \cos (c+d x)} \cos (2 (c+d x)) \sqrt{\sec (c+d x)} a}{\sqrt{a-b} \sqrt{a+b} \left(a^2-2 b^2-2 (b+a \cos (c+d x))^2+4 b (b+a \cos (c+d x))\right)}-\frac{\left(2 a^2-3 b^2\right) \left(\sqrt{a-b} (a+b) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a-b} \sqrt{-a \cos (c+d x)}}\right)+(a-b) \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a+b} \sqrt{-a \cos (c+d x)}}\right)\right) \sqrt{-a \cos (c+d x)} \sqrt{\sec (c+d x)}}{(a-b) (a+b) \sqrt{a}}\right) \sqrt{\sec (c+d x)}}{4 (a-b) (a+b) d \sqrt{a+b \sec (c+d x)}}","\frac{\sqrt{a+b \sec (c+d x)}}{4 d (a+b) (1-\sec (c+d x))}+\frac{\sqrt{a+b \sec (c+d x)}}{4 d (a-b) (\sec (c+d x)+1)}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{4 d (a-b)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{d \sqrt{a-b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{d \sqrt{a+b}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{4 d (a+b)^{3/2}}",1,"-1/4*(Sqrt[b + a*Cos[c + d*x]]*(-1/2*(a^2*b*(-(Sqrt[-a^2]*Sqrt[a + b]*Log[-Sqrt[-a + b] + Sqrt[b + a*Cos[c + d*x]]]) + Sqrt[-a^2]*Sqrt[a + b]*Log[Sqrt[-a + b] + Sqrt[b + a*Cos[c + d*x]]] - a*Sqrt[-a + b]*Log[-Sqrt[a + b] + Sqrt[b + a*Cos[c + d*x]]] + a*Sqrt[-a + b]*Log[Sqrt[a + b] + Sqrt[b + a*Cos[c + d*x]]] + Sqrt[-a^2]*Sqrt[a + b]*Log[b + Sqrt[a]*Sqrt[-(a*Cos[c + d*x])] - Sqrt[-a + b]*Sqrt[b + a*Cos[c + d*x]]] - Sqrt[-a^2]*Sqrt[a + b]*Log[b + Sqrt[a]*Sqrt[-(a*Cos[c + d*x])] + Sqrt[-a + b]*Sqrt[b + a*Cos[c + d*x]]] + a*Sqrt[-a + b]*Log[b + Sqrt[-a]*Sqrt[-(a*Cos[c + d*x])] - Sqrt[a + b]*Sqrt[b + a*Cos[c + d*x]]] - a*Sqrt[-a + b]*Log[b + Sqrt[-a]*Sqrt[-(a*Cos[c + d*x])] + Sqrt[a + b]*Sqrt[b + a*Cos[c + d*x]]]))/((-a)^(3/2)*Sqrt[-a + b]*Sqrt[a + b]*Sqrt[-(a*Cos[c + d*x])]*Sqrt[Sec[c + d*x]]) - ((2*a^2 - 3*b^2)*(Sqrt[a - b]*(a + b)*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a - b]*Sqrt[-(a*Cos[c + d*x])])] + (a - b)*Sqrt[a + b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a + b]*Sqrt[-(a*Cos[c + d*x])])])*Sqrt[-(a*Cos[c + d*x])]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*(a - b)*(a + b)) - (a*(2*a^2 - 2*b^2)*(4*Sqrt[a - b]*Sqrt[a + b]*ArcTan[Sqrt[b + a*Cos[c + d*x]]/Sqrt[-(a*Cos[c + d*x])]] - Sqrt[a]*(Sqrt[a + b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a - b]*Sqrt[-(a*Cos[c + d*x])])] + Sqrt[a - b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a + b]*Sqrt[-(a*Cos[c + d*x])])]))*Sqrt[-(a*Cos[c + d*x])]*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(Sqrt[a - b]*Sqrt[a + b]*(a^2 - 2*b^2 + 4*b*(b + a*Cos[c + d*x]) - 2*(b + a*Cos[c + d*x])^2)))*Sqrt[Sec[c + d*x]])/((a - b)*(a + b)*d*Sqrt[a + b*Sec[c + d*x]]) + ((b + a*Cos[c + d*x])*(a/(2*(a^2 - b^2)) + ((a - b*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(-a^2 + b^2)))*Sec[c + d*x])/(d*Sqrt[a + b*Sec[c + d*x]])","B",0
331,1,835,404,16.9411734,"\int \frac{\tan ^4(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Tan[c + d*x]^4/Sqrt[a + b*Sec[c + d*x]],x]","\frac{(b+a \cos (c+d x)) \sec (c+d x) \left(-\frac{2 \left(21 b^2-8 a^2\right) \sin (c+d x)}{15 b^3}+\frac{2 \sec (c+d x) \tan (c+d x)}{5 b}-\frac{8 a \tan (c+d x)}{15 b^2}\right)}{d \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(8 a^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+21 b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-21 a b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-8 a^2 b \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)+42 a b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-30 b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 a^3 \tan \left(\frac{1}{2} (c+d x)\right)-21 b^3 \tan \left(\frac{1}{2} (c+d x)\right)-21 a b^2 \tan \left(\frac{1}{2} (c+d x)\right)+8 a^2 b \tan \left(\frac{1}{2} (c+d x)\right)+\left(8 a^3+8 b a^2-21 b^2 a-21 b^3\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 b \left(4 a^2+b a-18 b^2\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-30 b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{15 b^3 d \sqrt{a+b \sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{2 (a-b) \sqrt{a+b} \left(8 a^2-21 b^2\right) \cot (c+d x) \sqrt{-\frac{b (\sec (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^4 d}+\frac{2 \sqrt{a+b} \left(-8 a^2+2 a b+21 b^2\right) \cot (c+d x) \sqrt{-\frac{b (\sec (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sec (c+d x)+1)}{b-a}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^3 d}-\frac{8 a \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b^2 d}+\frac{2 \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b d}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"(-2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(8*a^3*Tan[(c + d*x)/2] + 8*a^2*b*Tan[(c + d*x)/2] - 21*a*b^2*Tan[(c + d*x)/2] - 21*b^3*Tan[(c + d*x)/2] - 16*a^3*Tan[(c + d*x)/2]^3 + 42*a*b^2*Tan[(c + d*x)/2]^3 + 8*a^3*Tan[(c + d*x)/2]^5 - 8*a^2*b*Tan[(c + d*x)/2]^5 - 21*a*b^2*Tan[(c + d*x)/2]^5 + 21*b^3*Tan[(c + d*x)/2]^5 - 30*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*a^3 + 8*a^2*b - 21*a*b^2 - 21*b^3)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*b*(4*a^2 + a*b - 18*b^2)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(15*b^3*d*Sqrt[a + b*Sec[c + d*x]]*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + ((b + a*Cos[c + d*x])*Sec[c + d*x]*((-2*(-8*a^2 + 21*b^2)*Sin[c + d*x])/(15*b^3) - (8*a*Tan[c + d*x])/(15*b^2) + (2*Sec[c + d*x]*Tan[c + d*x])/(5*b)))/(d*Sqrt[a + b*Sec[c + d*x]])","B",0
332,-1,0,310,0,"\int \frac{\tan ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Tan[c + d*x]^2/Sqrt[a + b*Sec[c + d*x]],x]","\text{\$Aborted}","-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"$Aborted","F",-1
333,1,138,106,0.2298816,"\int \frac{1}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[1/Sqrt[a + b*Sec[c + d*x]],x]","-\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sec (c+d x) \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \left(F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{d \sqrt{a+b \sec (c+d x)}}","-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"(-4*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*(EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/(d*Sqrt[a + b*Sec[c + d*x]])","A",1
334,1,1198,361,19.0248966,"\int \frac{\cot ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Cot[c + d*x]^2/Sqrt[a + b*Sec[c + d*x]],x]","\frac{(b+a \cos (c+d x)) \sec (c+d x) \left(\frac{(a \cos (c+d x)-b) \csc (c+d x)}{b^2-a^2}+\frac{b \sin (c+d x)}{b^2-a^2}\right)}{d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \left(-b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)+4 i a^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-4 i b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) b E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-i \left(2 a^2-b a-b^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+4 i a^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-4 i b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{\sqrt{\frac{b-a}{a+b}} \left(a^2-b^2\right) d \sqrt{a+b \sec (c+d x)} \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(\tan ^4\left(\frac{1}{2} (c+d x)\right)-1\right)}","\frac{b^2 \tan (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{\cot (c+d x)}{d \sqrt{a+b \sec (c+d x)}}-\frac{\cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d \sqrt{a+b}}+\frac{\cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d \sqrt{a+b}}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*(((-b + a*Cos[c + d*x])*Csc[c + d*x])/(-a^2 + b^2) + (b*Sin[c + d*x])/(-a^2 + b^2)))/(d*Sqrt[a + b*Sec[c + d*x]]) - (Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 2*a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + (4*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (4*I)*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (4*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (4*I)*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(2*a^2 - a*b - b^2)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-1 + Tan[(c + d*x)/2]^4))","C",0
335,1,263,148,6.3861277,"\int \frac{\tan ^5(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^5/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b)^2 \left(-\frac{2 \left(b^2-a^2\right)^2}{a^2 b^3 (a \cos (c+d x)+b)}+\frac{2 \left(16 a^4-20 a^2 b^2+5 b^4\right)}{5 a^2 b^4}-\frac{6 a \sec (c+d x)}{5 b^3}+\frac{2 \sec ^2(c+d x)}{5 b^2}\right)}{d (a+b \sec (c+d x))^{3/2}}-\frac{\tan ^2(c+d x) \sqrt{a \cos (c+d x)} (a \cos (c+d x)+b)^{3/2} \left(\log \left(\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}+1\right)-\log \left(1-\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}\right)\right)}{a^2 d \left(1-\cos ^2(c+d x)\right) (a+b \sec (c+d x))^{3/2}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{2 \left(3 a^2-2 b^2\right) \sqrt{a+b \sec (c+d x)}}{b^4 d}+\frac{2 \left(a^2-b^2\right)^2}{a b^4 d \sqrt{a+b \sec (c+d x)}}+\frac{2 (a+b \sec (c+d x))^{5/2}}{5 b^4 d}-\frac{2 a (a+b \sec (c+d x))^{3/2}}{b^4 d}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((2*(16*a^4 - 20*a^2*b^2 + 5*b^4))/(5*a^2*b^4) - (2*(-a^2 + b^2)^2)/(a^2*b^3*(b + a*Cos[c + d*x])) - (6*a*Sec[c + d*x])/(5*b^3) + (2*Sec[c + d*x]^2)/(5*b^2)))/(d*(a + b*Sec[c + d*x])^(3/2)) - (Sqrt[a*Cos[c + d*x]]*(b + a*Cos[c + d*x])^(3/2)*(-Log[1 - Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]] + Log[1 + Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]])*Tan[c + d*x]^2)/(a^2*d*(1 - Cos[c + d*x]^2)*(a + b*Sec[c + d*x])^(3/2))","A",1
336,1,167,88,1.0331133,"\int \frac{\tan ^3(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^3/(a + b*Sec[c + d*x])^(3/2),x]","\frac{4 a^2-\frac{b^2 \sqrt{a \cos (c+d x)+b} \log \left(1-\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}\right)}{\sqrt{a \cos (c+d x)}}+\frac{b^2 \sqrt{a \cos (c+d x)+b} \log \left(\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}+1\right)}{\sqrt{a \cos (c+d x)}}+2 a b \sec (c+d x)-2 b^2}{a b^2 d \sqrt{a+b \sec (c+d x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{2 \left(a^2-b^2\right)}{a b^2 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{a+b \sec (c+d x)}}{b^2 d}",1,"(4*a^2 - 2*b^2 - (b^2*Sqrt[b + a*Cos[c + d*x]]*Log[1 - Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]])/Sqrt[a*Cos[c + d*x]] + (b^2*Sqrt[b + a*Cos[c + d*x]]*Log[1 + Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]])/Sqrt[a*Cos[c + d*x]] + 2*a*b*Sec[c + d*x])/(a*b^2*d*Sqrt[a + b*Sec[c + d*x]])","A",1
337,1,128,54,0.3919002,"\int \frac{\tan (c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\sec (c+d x) \left(\sqrt{a \cos (c+d x)} \sqrt{a \cos (c+d x)+b} \left(\log \left(1-\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}\right)-\log \left(\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}+1\right)\right)+2 a \cos (c+d x)\right)}{a^2 d \sqrt{a+b \sec (c+d x)}}","\frac{2}{a d \sqrt{a+b \sec (c+d x)}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}",1,"((2*a*Cos[c + d*x] + Sqrt[a*Cos[c + d*x]]*Sqrt[b + a*Cos[c + d*x]]*(Log[1 - Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]] - Log[1 + Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]]))*Sec[c + d*x])/(a^2*d*Sqrt[a + b*Sec[c + d*x]])","B",1
338,1,1020,142,6.9432635,"\int \frac{\cot (c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(b+a \cos (c+d x))^2 \left(-\frac{2 b^3}{a^2 \left(a^2-b^2\right) (b+a \cos (c+d x))}-\frac{2 b^2}{a^2 \left(b^2-a^2\right)}\right) \sec ^2(c+d x)}{d (a+b \sec (c+d x))^{3/2}}-\frac{(b+a \cos (c+d x))^{3/2} \left(\frac{b \left(-\sqrt{-a^2} \sqrt{a+b} \log \left(\sqrt{b+a \cos (c+d x)}-\sqrt{b-a}\right)+\sqrt{-a^2} \sqrt{a+b} \log \left(\sqrt{b-a}+\sqrt{b+a \cos (c+d x)}\right)-a \sqrt{b-a} \log \left(\sqrt{b+a \cos (c+d x)}-\sqrt{a+b}\right)+a \sqrt{b-a} \log \left(\sqrt{a+b}+\sqrt{b+a \cos (c+d x)}\right)+\sqrt{-a^2} \sqrt{a+b} \log \left(b+\sqrt{a} \sqrt{-a \cos (c+d x)}-\sqrt{b-a} \sqrt{b+a \cos (c+d x)}\right)-\sqrt{-a^2} \sqrt{a+b} \log \left(b+\sqrt{a} \sqrt{-a \cos (c+d x)}+\sqrt{b-a} \sqrt{b+a \cos (c+d x)}\right)+a \sqrt{b-a} \log \left(b+\sqrt{-a} \sqrt{-a \cos (c+d x)}-\sqrt{a+b} \sqrt{b+a \cos (c+d x)}\right)-a \sqrt{b-a} \log \left(b+\sqrt{-a} \sqrt{-a \cos (c+d x)}+\sqrt{a+b} \sqrt{b+a \cos (c+d x)}\right)\right) a^2}{(-a)^{3/2} \sqrt{b-a} \sqrt{a+b} \sqrt{-a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{\left(a^2-b^2\right) \left(4 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{b+a \cos (c+d x)}}{\sqrt{-a \cos (c+d x)}}\right)-\sqrt{a} \left(\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a-b} \sqrt{-a \cos (c+d x)}}\right)+\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a+b} \sqrt{-a \cos (c+d x)}}\right)\right)\right) \sqrt{-a \cos (c+d x)} \cos (2 (c+d x)) \sqrt{\sec (c+d x)} a}{\sqrt{a-b} \sqrt{a+b} \left(a^2-2 b^2-2 (b+a \cos (c+d x))^2+4 b (b+a \cos (c+d x))\right)}-\frac{\left(a^2+b^2\right) \left(\sqrt{a-b} (a+b) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a-b} \sqrt{-a \cos (c+d x)}}\right)+(a-b) \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a+b} \sqrt{-a \cos (c+d x)}}\right)\right) \sqrt{-a \cos (c+d x)} \sqrt{\sec (c+d x)}}{(a-b) (a+b) \sqrt{a}}\right) \sec ^{\frac{3}{2}}(c+d x)}{2 a (b-a) (a+b) d (a+b \sec (c+d x))^{3/2}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{2 b^2}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{d (a-b)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{d (a+b)^{3/2}}",1,"-1/2*((b + a*Cos[c + d*x])^(3/2)*((a^2*b*(-(Sqrt[-a^2]*Sqrt[a + b]*Log[-Sqrt[-a + b] + Sqrt[b + a*Cos[c + d*x]]]) + Sqrt[-a^2]*Sqrt[a + b]*Log[Sqrt[-a + b] + Sqrt[b + a*Cos[c + d*x]]] - a*Sqrt[-a + b]*Log[-Sqrt[a + b] + Sqrt[b + a*Cos[c + d*x]]] + a*Sqrt[-a + b]*Log[Sqrt[a + b] + Sqrt[b + a*Cos[c + d*x]]] + Sqrt[-a^2]*Sqrt[a + b]*Log[b + Sqrt[a]*Sqrt[-(a*Cos[c + d*x])] - Sqrt[-a + b]*Sqrt[b + a*Cos[c + d*x]]] - Sqrt[-a^2]*Sqrt[a + b]*Log[b + Sqrt[a]*Sqrt[-(a*Cos[c + d*x])] + Sqrt[-a + b]*Sqrt[b + a*Cos[c + d*x]]] + a*Sqrt[-a + b]*Log[b + Sqrt[-a]*Sqrt[-(a*Cos[c + d*x])] - Sqrt[a + b]*Sqrt[b + a*Cos[c + d*x]]] - a*Sqrt[-a + b]*Log[b + Sqrt[-a]*Sqrt[-(a*Cos[c + d*x])] + Sqrt[a + b]*Sqrt[b + a*Cos[c + d*x]]]))/((-a)^(3/2)*Sqrt[-a + b]*Sqrt[a + b]*Sqrt[-(a*Cos[c + d*x])]*Sqrt[Sec[c + d*x]]) - ((a^2 + b^2)*(Sqrt[a - b]*(a + b)*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a - b]*Sqrt[-(a*Cos[c + d*x])])] + (a - b)*Sqrt[a + b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a + b]*Sqrt[-(a*Cos[c + d*x])])])*Sqrt[-(a*Cos[c + d*x])]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*(a - b)*(a + b)) - (a*(a^2 - b^2)*(4*Sqrt[a - b]*Sqrt[a + b]*ArcTan[Sqrt[b + a*Cos[c + d*x]]/Sqrt[-(a*Cos[c + d*x])]] - Sqrt[a]*(Sqrt[a + b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a - b]*Sqrt[-(a*Cos[c + d*x])])] + Sqrt[a - b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a + b]*Sqrt[-(a*Cos[c + d*x])])]))*Sqrt[-(a*Cos[c + d*x])]*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(Sqrt[a - b]*Sqrt[a + b]*(a^2 - 2*b^2 + 4*b*(b + a*Cos[c + d*x]) - 2*(b + a*Cos[c + d*x])^2)))*Sec[c + d*x]^(3/2))/(a*(-a + b)*(a + b)*d*(a + b*Sec[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])^2*((-2*b^2)/(a^2*(-a^2 + b^2)) - (2*b^3)/(a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])))*Sec[c + d*x]^2)/(d*(a + b*Sec[c + d*x])^(3/2))","B",0
339,1,1114,236,7.5259551,"\int \frac{\cot ^3(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^3/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(b+a \cos (c+d x))^2 \left(-\frac{2 b^5}{a^2 \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}+\frac{\left(-a^2+2 b \cos (c+d x) a-b^2\right) \csc ^2(c+d x)}{2 \left(b^2-a^2\right)^2}+\frac{a^4+b^2 a^2+4 b^4}{2 a^2 \left(b^2-a^2\right)^2}\right) \sec ^2(c+d x)}{d (a+b \sec (c+d x))^{3/2}}-\frac{(b+a \cos (c+d x))^{3/2} \left(-\frac{\left(2 a^4-6 b^2 a^2-2 b^4\right) \sqrt{-a \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\sqrt{a-b} (a+b) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a-b} \sqrt{-a \cos (c+d x)}}\right)+(a-b) \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a+b} \sqrt{-a \cos (c+d x)}}\right)\right)}{\sqrt{a} (a-b) (a+b)}-\frac{a \left(2 a^4-4 b^2 a^2+2 b^4\right) \left(4 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{b+a \cos (c+d x)}}{\sqrt{-a \cos (c+d x)}}\right)-\sqrt{a} \left(\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a-b} \sqrt{-a \cos (c+d x)}}\right)+\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b+a \cos (c+d x)}}{\sqrt{a+b} \sqrt{-a \cos (c+d x)}}\right)\right)\right) \sqrt{-a \cos (c+d x)} \cos (2 (c+d x)) \sqrt{\sec (c+d x)}}{\sqrt{a-b} \sqrt{a+b} \left(a^2-2 b^2-2 (b+a \cos (c+d x))^2+4 b (b+a \cos (c+d x))\right)}-\frac{a \left(7 a b^3-a^3 b\right) \left(-\sqrt{-a^2} \sqrt{a+b} \log \left(\sqrt{b+a \cos (c+d x)}-\sqrt{b-a}\right)+\sqrt{-a^2} \sqrt{a+b} \log \left(\sqrt{b-a}+\sqrt{b+a \cos (c+d x)}\right)-a \sqrt{b-a} \log \left(\sqrt{b+a \cos (c+d x)}-\sqrt{a+b}\right)+a \sqrt{b-a} \log \left(\sqrt{a+b}+\sqrt{b+a \cos (c+d x)}\right)+\sqrt{-a^2} \sqrt{a+b} \log \left(b+\sqrt{a} \sqrt{-a \cos (c+d x)}-\sqrt{b-a} \sqrt{b+a \cos (c+d x)}\right)-\sqrt{-a^2} \sqrt{a+b} \log \left(b+\sqrt{a} \sqrt{-a \cos (c+d x)}+\sqrt{b-a} \sqrt{b+a \cos (c+d x)}\right)+a \sqrt{b-a} \log \left(b+\sqrt{-a} \sqrt{-a \cos (c+d x)}-\sqrt{a+b} \sqrt{b+a \cos (c+d x)}\right)-a \sqrt{b-a} \log \left(b+\sqrt{-a} \sqrt{-a \cos (c+d x)}+\sqrt{a+b} \sqrt{b+a \cos (c+d x)}\right)\right)}{2 (-a)^{3/2} \sqrt{b-a} \sqrt{a+b} \sqrt{-a \cos (c+d x)} \sqrt{\sec (c+d x)}}\right) \sec ^{\frac{3}{2}}(c+d x)}{4 a (a-b)^2 (a+b)^2 d (a+b \sec (c+d x))^{3/2}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{2 b^4}{a d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{a+b \sec (c+d x)}}{4 d (a+b)^2 (1-\sec (c+d x))}+\frac{\sqrt{a+b \sec (c+d x)}}{4 d (a-b)^2 (\sec (c+d x)+1)}+\frac{(4 a-7 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a-b}}\right)}{4 d (a-b)^{5/2}}+\frac{(4 a+7 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)}{4 d (a+b)^{5/2}}",1,"-1/4*((b + a*Cos[c + d*x])^(3/2)*(-1/2*(a*(-(a^3*b) + 7*a*b^3)*(-(Sqrt[-a^2]*Sqrt[a + b]*Log[-Sqrt[-a + b] + Sqrt[b + a*Cos[c + d*x]]]) + Sqrt[-a^2]*Sqrt[a + b]*Log[Sqrt[-a + b] + Sqrt[b + a*Cos[c + d*x]]] - a*Sqrt[-a + b]*Log[-Sqrt[a + b] + Sqrt[b + a*Cos[c + d*x]]] + a*Sqrt[-a + b]*Log[Sqrt[a + b] + Sqrt[b + a*Cos[c + d*x]]] + Sqrt[-a^2]*Sqrt[a + b]*Log[b + Sqrt[a]*Sqrt[-(a*Cos[c + d*x])] - Sqrt[-a + b]*Sqrt[b + a*Cos[c + d*x]]] - Sqrt[-a^2]*Sqrt[a + b]*Log[b + Sqrt[a]*Sqrt[-(a*Cos[c + d*x])] + Sqrt[-a + b]*Sqrt[b + a*Cos[c + d*x]]] + a*Sqrt[-a + b]*Log[b + Sqrt[-a]*Sqrt[-(a*Cos[c + d*x])] - Sqrt[a + b]*Sqrt[b + a*Cos[c + d*x]]] - a*Sqrt[-a + b]*Log[b + Sqrt[-a]*Sqrt[-(a*Cos[c + d*x])] + Sqrt[a + b]*Sqrt[b + a*Cos[c + d*x]]]))/((-a)^(3/2)*Sqrt[-a + b]*Sqrt[a + b]*Sqrt[-(a*Cos[c + d*x])]*Sqrt[Sec[c + d*x]]) - ((2*a^4 - 6*a^2*b^2 - 2*b^4)*(Sqrt[a - b]*(a + b)*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a - b]*Sqrt[-(a*Cos[c + d*x])])] + (a - b)*Sqrt[a + b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a + b]*Sqrt[-(a*Cos[c + d*x])])])*Sqrt[-(a*Cos[c + d*x])]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*(a - b)*(a + b)) - (a*(2*a^4 - 4*a^2*b^2 + 2*b^4)*(4*Sqrt[a - b]*Sqrt[a + b]*ArcTan[Sqrt[b + a*Cos[c + d*x]]/Sqrt[-(a*Cos[c + d*x])]] - Sqrt[a]*(Sqrt[a + b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a - b]*Sqrt[-(a*Cos[c + d*x])])] + Sqrt[a - b]*ArcTan[(Sqrt[a]*Sqrt[b + a*Cos[c + d*x]])/(Sqrt[a + b]*Sqrt[-(a*Cos[c + d*x])])]))*Sqrt[-(a*Cos[c + d*x])]*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(Sqrt[a - b]*Sqrt[a + b]*(a^2 - 2*b^2 + 4*b*(b + a*Cos[c + d*x]) - 2*(b + a*Cos[c + d*x])^2)))*Sec[c + d*x]^(3/2))/(a*(a - b)^2*(a + b)^2*d*(a + b*Sec[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])^2*((a^4 + a^2*b^2 + 4*b^4)/(2*a^2*(-a^2 + b^2)^2) - (2*b^5)/(a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x])) + ((-a^2 - b^2 + 2*a*b*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(-a^2 + b^2)^2))*Sec[c + d*x]^2)/(d*(a + b*Sec[c + d*x])^(3/2))","B",0
340,1,859,530,17.1864814,"\int \frac{\tan ^4(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^4/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(b+a \cos (c+d x))^2 \left(\frac{2 \left(3 b^2-8 a^2\right) \sin (c+d x)}{3 a b^3}-\frac{2 \left(b^2 \sin (c+d x)-a^2 \sin (c+d x)\right)}{a b^2 (b+a \cos (c+d x))}+\frac{2 \tan (c+d x)}{3 b^2}\right) \sec ^2(c+d x)}{d (a+b \sec (c+d x))^{3/2}}+\frac{2 (b+a \cos (c+d x))^{3/2} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(8 a^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-8 a^2 b \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 a^3 \tan \left(\frac{1}{2} (c+d x)\right)-3 b^3 \tan \left(\frac{1}{2} (c+d x)\right)-3 a b^2 \tan \left(\frac{1}{2} (c+d x)\right)+8 a^2 b \tan \left(\frac{1}{2} (c+d x)\right)+\left(8 a^3+8 b a^2-3 b^2 a-3 b^3\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 a b (4 a+b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) \sec ^{\frac{3}{2}}(c+d x)}{3 a b^3 d (a+b \sec (c+d x))^{3/2} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{2 a^2 \tan (c+d x) \sec (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{4 a \tan (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(4 a^2-b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}+\frac{2 b^2 \tan (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 (2 a+b) \left(4 a^2+a b-3 b^2\right) \cot (c+d x) \sqrt{-\frac{b (\sec (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a b^3 d \sqrt{a+b}}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}+\frac{2 \left(8 a^4-11 a^2 b^2+3 b^4\right) \cot (c+d x) \sqrt{-\frac{b (\sec (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a b^4 d \sqrt{a+b}}",1,"(2*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(8*a^3*Tan[(c + d*x)/2] + 8*a^2*b*Tan[(c + d*x)/2] - 3*a*b^2*Tan[(c + d*x)/2] - 3*b^3*Tan[(c + d*x)/2] - 16*a^3*Tan[(c + d*x)/2]^3 + 6*a*b^2*Tan[(c + d*x)/2]^3 + 8*a^3*Tan[(c + d*x)/2]^5 - 8*a^2*b*Tan[(c + d*x)/2]^5 - 3*a*b^2*Tan[(c + d*x)/2]^5 + 3*b^3*Tan[(c + d*x)/2]^5 + 6*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*a^3 + 8*a^2*b - 3*a*b^2 - 3*b^3)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*b*(4*a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*a*b^3*d*(a + b*Sec[c + d*x])^(3/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((2*(-8*a^2 + 3*b^2)*Sin[c + d*x])/(3*a*b^3) - (2*(-(a^2*Sin[c + d*x]) + b^2*Sin[c + d*x]))/(a*b^2*(b + a*Cos[c + d*x])) + (2*Tan[c + d*x])/(3*b^2)))/(d*(a + b*Sec[c + d*x])^(3/2))","A",0
341,1,5162,344,23.5909045,"\int \frac{\tan ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^2/(a + b*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b^2 d}+\frac{2 \tan (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d}",1,"Result too large to show","C",0
342,1,1249,347,6.1613872,"\int \frac{1}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(a + b*Sec[c + d*x])^(-3/2),x]","\frac{(b+a \cos (c+d x))^2 \left(\frac{2 \sin (c+d x) b^2}{a \left(a^2-b^2\right) (b+a \cos (c+d x))}+\frac{2 \sin (c+d x) b}{a \left(b^2-a^2\right)}\right) \sec ^2(c+d x)}{d (a+b \sec (c+d x))^{3/2}}+\frac{2 (b+a \cos (c+d x))^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(-b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 i a^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+2 i b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) b E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+i \left(a^2+b a-2 b^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i a^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) \sec ^{\frac{3}{2}}(c+d x)}{a \sqrt{\frac{b-a}{a+b}} \left(a^2-b^2\right) d (a+b \sec (c+d x))^{3/2} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{2 b^2 \tan (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}+\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((2*b*Sin[c + d*x])/(a*(-a^2 + b^2)) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(3/2)) + (2*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 2*a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - (2*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(a^2 + a*b - 2*b^2)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",0
343,1,663,449,13.6402318,"\int \frac{\cot ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^2/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b)^2 \left(-\frac{2 b \left(a^2+b^2\right) \sin (c+d x)}{a \left(a^2-b^2\right)^2}+\frac{\csc (c+d x) \left(a^2 (-\cos (c+d x))+2 a b-b^2 \cos (c+d x)\right)}{\left(b^2-a^2\right)^2}+\frac{2 b^4 \sin (c+d x)}{a \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}\right)}{d (a+b \sec (c+d x))^{3/2}}-\frac{2 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) (a \cos (c+d x)+b) \left(-b \sqrt{\frac{b-a}{a+b}} \left(a^2+b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)-4 i \left(a^2-b^2\right)^2 \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)-2 i b \left(-a^3+a^2 b-a b^2+b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)+i \left(2 a^4-a^3 b-2 a^2 b^2-3 a b^3+4 b^4\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)\right)}{a d \sqrt{\frac{b-a}{a+b}} \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^{3/2}}","\frac{2 b^2 \left(a^2+b^2\right) \tan (c+d x)}{a d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \tan (c+d x)}{d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{\left(a^2-a b+2 b^2\right) \cot (c+d x) \sqrt{-\frac{b (\sec (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d (a-b) (a+b)^{3/2}}+\frac{2 \left(a^2+b^2\right) \cot (c+d x) \sqrt{-\frac{b (\sec (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d (a-b) (a+b)^{3/2}}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}-\frac{\cot (c+d x)}{d (a+b \sec (c+d x))^{3/2}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*(((2*a*b - a^2*Cos[c + d*x] - b^2*Cos[c + d*x])*Csc[c + d*x])/(-a^2 + b^2)^2 - (2*b*(a^2 + b^2)*Sin[c + d*x])/(a*(a^2 - b^2)^2) + (2*b^4*Sin[c + d*x])/(a*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(3/2)) - (2*Cos[(c + d*x)/2]^2*(b + a*Cos[c + d*x])*Sec[c + d*x]^2*((-2*I)*b*(-a^3 + a^2*b - a*b^2 + b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] + I*(2*a^4 - a^3*b - 2*a^2*b^2 - 3*a*b^3 + 4*b^4)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - (4*I)*(a^2 - b^2)^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - b*Sqrt[(-a + b)/(a + b)]*(a^2 + b^2)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(a*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(3/2))","C",1
344,1,238,245,3.5153462,"\int (a+b \sec (e+f x))^3 (d \tan (e+f x))^n \, dx","Integrate[(a + b*Sec[e + f*x])^3*(d*Tan[e + f*x])^n,x]","\frac{d \left(-\tan ^2(e+f x)\right)^{-n/2} (d \tan (e+f x))^{n-1} \left(-3 a^3 \left(-\tan ^2(e+f x)\right)^{\frac{n+2}{2}} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)+9 a^2 b (n+1) \sqrt{-\tan ^2(e+f x)} \sec (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};\sec ^2(e+f x)\right)+9 a b^2 \left(\sqrt{-\tan ^2(e+f x)}-\left(-\tan ^2(e+f x)\right)^{\frac{n+2}{2}}\right)+b^3 (n+1) \sqrt{-\tan ^2(e+f x)} \sec ^3(e+f x) \, _2F_1\left(\frac{3}{2},\frac{1-n}{2};\frac{5}{2};\sec ^2(e+f x)\right)\right)}{3 f (n+1)}","\frac{a^3 (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{d f (n+1)}+\frac{3 a^2 b \sec (e+f x) \cos ^2(e+f x)^{\frac{n+2}{2}} (d \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{d f (n+1)}+\frac{3 a b^2 (d \tan (e+f x))^{n+1}}{d f (n+1)}+\frac{b^3 \sec ^3(e+f x) \cos ^2(e+f x)^{\frac{n+4}{2}} (d \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+4}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{d f (n+1)}",1,"(d*(d*Tan[e + f*x])^(-1 + n)*(9*a^2*b*(1 + n)*Hypergeometric2F1[1/2, (1 - n)/2, 3/2, Sec[e + f*x]^2]*Sec[e + f*x]*Sqrt[-Tan[e + f*x]^2] + b^3*(1 + n)*Hypergeometric2F1[3/2, (1 - n)/2, 5/2, Sec[e + f*x]^2]*Sec[e + f*x]^3*Sqrt[-Tan[e + f*x]^2] - 3*a^3*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(-Tan[e + f*x]^2)^((2 + n)/2) + 9*a*b^2*(Sqrt[-Tan[e + f*x]^2] - (-Tan[e + f*x]^2)^((2 + n)/2))))/(3*f*(1 + n)*(-Tan[e + f*x]^2)^(n/2))","A",1
345,1,178,160,1.2964112,"\int (a+b \sec (e+f x))^2 (d \tan (e+f x))^n \, dx","Integrate[(a + b*Sec[e + f*x])^2*(d*Tan[e + f*x])^n,x]","\frac{d \left(-\tan ^2(e+f x)\right)^{-n/2} (d \tan (e+f x))^{n-1} \left(-a^2 \left(-\tan ^2(e+f x)\right)^{\frac{n+2}{2}} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)+2 a b (n+1) \sqrt{-\tan ^2(e+f x)} \sec (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};\sec ^2(e+f x)\right)+b^2 \left(\sqrt{-\tan ^2(e+f x)}-\left(-\tan ^2(e+f x)\right)^{\frac{n+2}{2}}\right)\right)}{f (n+1)}","\frac{a^2 (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{d f (n+1)}+\frac{2 a b \sec (e+f x) \cos ^2(e+f x)^{\frac{n+2}{2}} (d \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{d f (n+1)}+\frac{b^2 (d \tan (e+f x))^{n+1}}{d f (n+1)}",1,"(d*(d*Tan[e + f*x])^(-1 + n)*(2*a*b*(1 + n)*Hypergeometric2F1[1/2, (1 - n)/2, 3/2, Sec[e + f*x]^2]*Sec[e + f*x]*Sqrt[-Tan[e + f*x]^2] - a^2*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(-Tan[e + f*x]^2)^((2 + n)/2) + b^2*(Sqrt[-Tan[e + f*x]^2] - (-Tan[e + f*x]^2)^((2 + n)/2))))/(f*(1 + n)*(-Tan[e + f*x]^2)^(n/2))","A",1
346,1,106,129,0.6754193,"\int (a+b \sec (e+f x)) (d \tan (e+f x))^n \, dx","Integrate[(a + b*Sec[e + f*x])*(d*Tan[e + f*x])^n,x]","\frac{(d \tan (e+f x))^n \left(\frac{a \tan (e+f x) \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{n+1}+b \csc (e+f x) \left(-\tan ^2(e+f x)\right)^{\frac{1-n}{2}} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};\sec ^2(e+f x)\right)\right)}{f}","\frac{a (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{d f (n+1)}+\frac{b \sec (e+f x) \cos ^2(e+f x)^{\frac{n+2}{2}} (d \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{d f (n+1)}",1,"((d*Tan[e + f*x])^n*((a*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x])/(1 + n) + b*Csc[e + f*x]*Hypergeometric2F1[1/2, (1 - n)/2, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^((1 - n)/2)))/f","A",1
347,1,786,266,4.7353777,"\int \frac{(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx","Integrate[(d*Tan[e + f*x])^n/(a + b*Sec[e + f*x]),x]","\frac{2 \tan \left(\frac{1}{2} (e+f x)\right) (d \tan (e+f x))^n \left((a+b) F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-b F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}\right)\right)}{f (a+b \sec (e+f x)) \left(\sec ^2\left(\frac{1}{2} (e+f x)\right) \left((a+b) F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-b F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}\right)\right)-\frac{2 (n+1) \tan ^2\left(\frac{1}{2} (e+f x)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right) \left((a+b)^2 \left(F_1\left(\frac{n+3}{2};n,2;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n F_1\left(\frac{n+3}{2};n+1,1;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)+b n (a+b) F_1\left(\frac{n+3}{2};n+1,1;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}\right)+b (a-b) F_1\left(\frac{n+3}{2};n,2;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}\right)\right)}{(n+3) (a+b)}+2 n \tan \left(\frac{1}{2} (e+f x)\right) \csc (e+f x) \sec (e+f x) \left((a+b) F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-b F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}\right)\right)-16 n \sin ^5\left(\frac{1}{2} (e+f x)\right) \cos \left(\frac{1}{2} (e+f x)\right) \csc ^3(e+f x) \sec (e+f x) \left((a+b) F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-b F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}\right)\right)\right)}","\frac{d \left(-\tan ^2(e+f x)\right)^{\frac{1-n}{2}+\frac{n-1}{2}} (d \tan (e+f x))^{n-1} \left(-\frac{b (1-\sec (e+f x))}{a+b \sec (e+f x)}\right)^{\frac{1-n}{2}} \left(\frac{b (\sec (e+f x)+1)}{a+b \sec (e+f x)}\right)^{\frac{1-n}{2}} F_1\left(1-n;\frac{1-n}{2},\frac{1-n}{2};2-n;\frac{a+b}{a+b \sec (e+f x)},\frac{a-b}{a+b \sec (e+f x)}\right)}{a f (1-n)}-\frac{d \left(-\tan ^2(e+f x)\right)^{\frac{1-n}{2}+\frac{n+1}{2}} (d \tan (e+f x))^{n-1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{a f (n+1)}",1,"(2*((a + b)*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - b*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Tan[(e + f*x)/2]*(d*Tan[e + f*x])^n)/(f*(a + b*Sec[e + f*x])*(((a + b)*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - b*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Sec[(e + f*x)/2]^2 - 16*n*((a + b)*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - b*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Cos[(e + f*x)/2]*Csc[e + f*x]^3*Sec[e + f*x]*Sin[(e + f*x)/2]^5 + 2*n*((a + b)*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - b*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Csc[e + f*x]*Sec[e + f*x]*Tan[(e + f*x)/2] - (2*(1 + n)*((a - b)*b*AppellF1[(3 + n)/2, n, 2, (5 + n)/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)] + (a + b)^2*(AppellF1[(3 + n)/2, n, 2, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[(3 + n)/2, 1 + n, 1, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]) + b*(a + b)*n*AppellF1[(3 + n)/2, 1 + n, 1, (5 + n)/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]^2)/((a + b)*(3 + n))))","B",0
348,0,0,28,8.0608628,"\int (a+b \sec (c+d x))^{3/2} (e \tan (c+d x))^m \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)*(e*Tan[c + d*x])^m,x]","\int (a+b \sec (c+d x))^{3/2} (e \tan (c+d x))^m \, dx","\text{Int}\left((a+b \sec (c+d x))^{3/2} (e \tan (c+d x))^m,x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(3/2)*(e*Tan[c + d*x])^m, x]","A",-1
349,0,0,28,0.7087221,"\int \sqrt{a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]*(e*Tan[c + d*x])^m,x]","\int \sqrt{a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx","\text{Int}\left(\sqrt{a+b \sec (c+d x)} (e \tan (c+d x))^m,x\right)",0,"Integrate[Sqrt[a + b*Sec[c + d*x]]*(e*Tan[c + d*x])^m, x]","A",-1
350,0,0,28,3.0008734,"\int \frac{(e \tan (c+d x))^m}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(e*Tan[c + d*x])^m/Sqrt[a + b*Sec[c + d*x]],x]","\int \frac{(e \tan (c+d x))^m}{\sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{(e \tan (c+d x))^m}{\sqrt{a+b \sec (c+d x)}},x\right)",0,"Integrate[(e*Tan[c + d*x])^m/Sqrt[a + b*Sec[c + d*x]], x]","A",-1
351,0,0,28,3.893667,"\int \frac{(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(e*Tan[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2),x]","\int \frac{(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}},x\right)",0,"Integrate[(e*Tan[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2), x]","A",-1
352,0,0,26,3.4901038,"\int (a+b \sec (c+d x))^n (e \tan (c+d x))^m \, dx","Integrate[(a + b*Sec[c + d*x])^n*(e*Tan[c + d*x])^m,x]","\int (a+b \sec (c+d x))^n (e \tan (c+d x))^m \, dx","\text{Int}\left((e \tan (c+d x))^m (a+b \sec (c+d x))^n,x\right)",0,"Integrate[(a + b*Sec[c + d*x])^n*(e*Tan[c + d*x])^m, x]","A",-1
353,1,298,177,3.3131104,"\int (a+b \sec (c+d x))^n \tan ^5(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^5,x]","-\frac{\sec ^8\left(\frac{1}{2} (c+d x)\right) (a+b \sec (c+d x))^n \left(n (a \cos (c+d x)+b) \left(3 a^3 \cos (3 (c+d x))+3 a \left(3 a^2+b^2 \left(n^2-n-8\right)\right) \cos (c+d x)+2 b (n+1) \left(b^2 \left(n^2+7 n+12\right)-3 a^2\right) \cos (2 (c+d x))-6 a^2 b n-6 a^2 b-a b^2 n^2 \cos (3 (c+d x))-7 a b^2 n \cos (3 (c+d x))-12 a b^2 \cos (3 (c+d x))+4 b^3 n^2+16 b^3 n+12 b^3\right)-2 b^4 \left(n^4+10 n^3+35 n^2+50 n+24\right) \cos ^4(c+d x) \, _2F_1\left(1,-n;1-n;\frac{a \cos (c+d x)}{b+a \cos (c+d x)}\right)\right)}{2 b^4 d n (n+1) (n+2) (n+3) (n+4) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^4}","-\frac{a \left(a^2-2 b^2\right) (a+b \sec (c+d x))^{n+1}}{b^4 d (n+1)}+\frac{\left(3 a^2-2 b^2\right) (a+b \sec (c+d x))^{n+2}}{b^4 d (n+2)}-\frac{3 a (a+b \sec (c+d x))^{n+3}}{b^4 d (n+3)}+\frac{(a+b \sec (c+d x))^{n+4}}{b^4 d (n+4)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a d (n+1)}",1,"-1/2*((n*(b + a*Cos[c + d*x])*(-6*a^2*b + 12*b^3 - 6*a^2*b*n + 16*b^3*n + 4*b^3*n^2 + 3*a*(3*a^2 + b^2*(-8 - n + n^2))*Cos[c + d*x] + 2*b*(1 + n)*(-3*a^2 + b^2*(12 + 7*n + n^2))*Cos[2*(c + d*x)] + 3*a^3*Cos[3*(c + d*x)] - 12*a*b^2*Cos[3*(c + d*x)] - 7*a*b^2*n*Cos[3*(c + d*x)] - a*b^2*n^2*Cos[3*(c + d*x)]) - 2*b^4*(24 + 50*n + 35*n^2 + 10*n^3 + n^4)*Cos[c + d*x]^4*Hypergeometric2F1[1, -n, 1 - n, (a*Cos[c + d*x])/(b + a*Cos[c + d*x])])*Sec[(c + d*x)/2]^8*(a + b*Sec[c + d*x])^n)/(b^4*d*n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(-1 + Tan[(c + d*x)/2]^2)^4)","A",1
354,1,118,102,1.3460736,"\int (a+b \sec (c+d x))^n \tan ^3(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^3,x]","\frac{\sec ^2(c+d x) (a+b \sec (c+d x))^n \left(n (a \cos (c+d x)+b) (-a \cos (c+d x)+b n+b)-b^2 \left(n^2+3 n+2\right) \cos ^2(c+d x) \, _2F_1\left(1,-n;1-n;\frac{a \cos (c+d x)}{b+a \cos (c+d x)}\right)\right)}{b^2 d n (n+1) (n+2)}","-\frac{a (a+b \sec (c+d x))^{n+1}}{b^2 d (n+1)}+\frac{(a+b \sec (c+d x))^{n+2}}{b^2 d (n+2)}+\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a d (n+1)}",1,"((n*(b + b*n - a*Cos[c + d*x])*(b + a*Cos[c + d*x]) - b^2*(2 + 3*n + n^2)*Cos[c + d*x]^2*Hypergeometric2F1[1, -n, 1 - n, (a*Cos[c + d*x])/(b + a*Cos[c + d*x])])*Sec[c + d*x]^2*(a + b*Sec[c + d*x])^n)/(b^2*d*n*(1 + n)*(2 + n))","A",1
355,1,49,48,0.4481733,"\int (a+b \sec (c+d x))^n \tan (c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^n*Tan[c + d*x],x]","\frac{(a+b \sec (c+d x))^n \, _2F_1\left(1,-n;1-n;\frac{a \cos (c+d x)}{b+a \cos (c+d x)}\right)}{d n}","-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a d (n+1)}",1,"(Hypergeometric2F1[1, -n, 1 - n, (a*Cos[c + d*x])/(b + a*Cos[c + d*x])]*(a + b*Sec[c + d*x])^n)/(d*n)","A",1
356,1,163,162,1.6940199,"\int \cot (c+d x) (a+b \sec (c+d x))^n \, dx","Integrate[Cot[c + d*x]*(a + b*Sec[c + d*x])^n,x]","\frac{(a+b \sec (c+d x))^n \left(-2 \, _2F_1\left(1,-n;1-n;\frac{a \cos (c+d x)}{b+a \cos (c+d x)}\right)+\, _2F_1\left(1,-n;1-n;\frac{(a+b) \cos (c+d x)}{b+a \cos (c+d x)}\right)+2^n \left(\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{b}\right)^{-n} \, _2F_1\left(-n,-n;1-n;\frac{(b-a) \cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{2 b}\right)\right)}{2 d n}","-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{2 d (n+1) (a-b)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{2 d (n+1) (a+b)}+\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a d (n+1)}",1,"((-2*Hypergeometric2F1[1, -n, 1 - n, (a*Cos[c + d*x])/(b + a*Cos[c + d*x])] + Hypergeometric2F1[1, -n, 1 - n, ((a + b)*Cos[c + d*x])/(b + a*Cos[c + d*x])] + (2^n*Hypergeometric2F1[-n, -n, 1 - n, ((-a + b)*Cos[c + d*x]*Sec[(c + d*x)/2]^2)/(2*b)])/(((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/b)^n)*(a + b*Sec[c + d*x])^n)/(2*d*n)","A",1
357,1,256,279,7.0104545,"\int \cot ^3(c+d x) (a+b \sec (c+d x))^n \, dx","Integrate[Cot[c + d*x]^3*(a + b*Sec[c + d*x])^n,x]","\frac{\cot ^2(c+d x) \left(a+b \sqrt{\sec ^2(c+d x)}\right) (a+b \sec (c+d x))^n \left((a-b) \left(a (a-b) (2 a-b (n-2)) \tan ^2(c+d x) \, _2F_1\left(1,n+1;n+2;\frac{a+b \sqrt{\sec ^2(c+d x)}}{a+b}\right)-2 (a+b) \left(2 \left(a^2-b^2\right) \tan ^2(c+d x) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{\sec ^2(c+d x)} b}{a}+1\right)+a (n+1) \left(a-b \sqrt{\sec ^2(c+d x)}\right)\right)\right)+a (a+b)^2 (2 a+b (n-2)) \tan ^2(c+d x) \, _2F_1\left(1,n+1;n+2;\frac{a+b \sqrt{\sec ^2(c+d x)}}{a-b}\right)\right)}{4 a d (n+1) (a-b)^2 (a+b)^2}","\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{2 d (n+1) (a-b)}+\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{2 d (n+1) (a+b)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a d (n+1)}-\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{4 d (n+1) (a-b)^2}+\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{4 d (n+1) (a+b)^2}",1,"(Cot[c + d*x]^2*(a + b*Sec[c + d*x])^n*(a + b*Sqrt[Sec[c + d*x]^2])*(a*(a + b)^2*(2*a + b*(-2 + n))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sqrt[Sec[c + d*x]^2])/(a - b)]*Tan[c + d*x]^2 + (a - b)*(a*(a - b)*(2*a - b*(-2 + n))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sqrt[Sec[c + d*x]^2])/(a + b)]*Tan[c + d*x]^2 - 2*(a + b)*(a*(1 + n)*(a - b*Sqrt[Sec[c + d*x]^2]) + 2*(a^2 - b^2)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Sqrt[Sec[c + d*x]^2])/a]*Tan[c + d*x]^2))))/(4*a*(a - b)^2*(a + b)^2*d*(1 + n))","A",0
358,0,0,24,6.654052,"\int (a+b \sec (c+d x))^n \tan ^4(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^4,x]","\int (a+b \sec (c+d x))^n \tan ^4(c+d x) \, dx","\text{Int}\left(\tan ^4(c+d x) (a+b \sec (c+d x))^n,x\right)",0,"Integrate[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^4, x]","A",-1
359,0,0,237,3.5499348,"\int (a+b \sec (c+d x))^n \tan ^2(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^2,x]","\int (a+b \sec (c+d x))^n \tan ^2(c+d x) \, dx","-\text{Int}\left((a+b \sec (c+d x))^n,x\right)+\frac{\sqrt{2} (a+b) \tan (c+d x) (a+b \sec (c+d x))^n \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n-1;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1}}-\frac{\sqrt{2} a \tan (c+d x) (a+b \sec (c+d x))^n \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1}}",0,"Integrate[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^2, x]","A",-1
360,0,0,24,3.7228332,"\int \cot ^2(c+d x) (a+b \sec (c+d x))^n \, dx","Integrate[Cot[c + d*x]^2*(a + b*Sec[c + d*x])^n,x]","\int \cot ^2(c+d x) (a+b \sec (c+d x))^n \, dx","\text{Int}\left(\cot ^2(c+d x) (a+b \sec (c+d x))^n,x\right)",0,"Integrate[Cot[c + d*x]^2*(a + b*Sec[c + d*x])^n, x]","A",-1
361,0,0,24,6.23866,"\int \cot ^4(c+d x) (a+b \sec (c+d x))^n \, dx","Integrate[Cot[c + d*x]^4*(a + b*Sec[c + d*x])^n,x]","\int \cot ^4(c+d x) (a+b \sec (c+d x))^n \, dx","\text{Int}\left(\cot ^4(c+d x) (a+b \sec (c+d x))^n,x\right)",0,"Integrate[Cot[c + d*x]^4*(a + b*Sec[c + d*x])^n, x]","A",-1
362,0,0,26,4.4083292,"\int (a+b \sec (c+d x))^n \tan ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^(3/2),x]","\int (a+b \sec (c+d x))^n \tan ^{\frac{3}{2}}(c+d x) \, dx","\text{Int}\left(\tan ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^n,x\right)",0,"Integrate[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^(3/2), x]","A",-1
363,0,0,26,5.5962153,"\int (a+b \sec (c+d x))^n \sqrt{\tan (c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^n*Sqrt[Tan[c + d*x]],x]","\int (a+b \sec (c+d x))^n \sqrt{\tan (c+d x)} \, dx","\text{Int}\left(\sqrt{\tan (c+d x)} (a+b \sec (c+d x))^n,x\right)",0,"Integrate[(a + b*Sec[c + d*x])^n*Sqrt[Tan[c + d*x]], x]","A",-1
364,0,0,26,5.0865176,"\int \frac{(a+b \sec (c+d x))^n}{\sqrt{\tan (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])^n/Sqrt[Tan[c + d*x]],x]","\int \frac{(a+b \sec (c+d x))^n}{\sqrt{\tan (c+d x)}} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^n}{\sqrt{\tan (c+d x)}},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^n/Sqrt[Tan[c + d*x]], x]","A",-1
365,0,0,26,6.1836024,"\int \frac{(a+b \sec (c+d x))^n}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^n/Tan[c + d*x]^(3/2),x]","\int \frac{(a+b \sec (c+d x))^n}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^n}{\tan ^{\frac{3}{2}}(c+d x)},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^n/Tan[c + d*x]^(3/2), x]","A",-1