1,1,79,94,0.3999496,"\int \sec ^{10}(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Sec[c + d*x]^10*(a + I*a*Tan[c + d*x]),x]","\frac{a \left(\frac{1}{9} \tan ^9(c+d x)+\frac{4}{7} \tan ^7(c+d x)+\frac{6}{5} \tan ^5(c+d x)+\frac{4}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{i a \sec ^{10}(c+d x)}{10 d}","\frac{a \tan ^9(c+d x)}{9 d}+\frac{4 a \tan ^7(c+d x)}{7 d}+\frac{6 a \tan ^5(c+d x)}{5 d}+\frac{4 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{i a \sec ^{10}(c+d x)}{10 d}",1,"((I/10)*a*Sec[c + d*x]^10)/d + (a*(Tan[c + d*x] + (4*Tan[c + d*x]^3)/3 + (6*Tan[c + d*x]^5)/5 + (4*Tan[c + d*x]^7)/7 + Tan[c + d*x]^9/9))/d","A",1
2,1,63,75,0.1176116,"\int \sec ^8(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Sec[c + d*x]^8*(a + I*a*Tan[c + d*x]),x]","\frac{a \left(\frac{1}{7} \tan ^7(c+d x)+\frac{3}{5} \tan ^5(c+d x)+\tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{i a \sec ^8(c+d x)}{8 d}","\frac{a \tan ^7(c+d x)}{7 d}+\frac{3 a \tan ^5(c+d x)}{5 d}+\frac{a \tan ^3(c+d x)}{d}+\frac{a \tan (c+d x)}{d}+\frac{i a \sec ^8(c+d x)}{8 d}",1,"((I/8)*a*Sec[c + d*x]^8)/d + (a*(Tan[c + d*x] + Tan[c + d*x]^3 + (3*Tan[c + d*x]^5)/5 + Tan[c + d*x]^7/7))/d","A",1
3,1,55,62,0.1295986,"\int \sec ^6(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x]),x]","\frac{a \left(\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{i a \sec ^6(c+d x)}{6 d}","\frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{i a \sec ^6(c+d x)}{6 d}",1,"((I/6)*a*Sec[c + d*x]^6)/d + (a*(Tan[c + d*x] + (2*Tan[c + d*x]^3)/3 + Tan[c + d*x]^5/5))/d","A",1
4,1,43,46,0.0475588,"\int \sec ^4(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Sec[c + d*x]^4*(a + I*a*Tan[c + d*x]),x]","\frac{a \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{i a \sec ^4(c+d x)}{4 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{i a \sec ^4(c+d x)}{4 d}",1,"((I/4)*a*Sec[c + d*x]^4)/d + (a*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
5,1,30,27,0.0142571,"\int \sec ^2(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + I*a*Tan[c + d*x]),x]","\frac{a \tan (c+d x)}{d}+\frac{i a \sec ^2(c+d x)}{2 d}","-\frac{i (a+i a \tan (c+d x))^2}{2 a d}",1,"((I/2)*a*Sec[c + d*x]^2)/d + (a*Tan[c + d*x])/d","A",1
6,1,19,19,0.0062841,"\int (a+i a \tan (c+d x)) \, dx","Integrate[a + I*a*Tan[c + d*x],x]","a x-\frac{i a \log (\cos (c+d x))}{d}","a x-\frac{i a \log (\cos (c+d x))}{d}",1,"a*x - (I*a*Log[Cos[c + d*x]])/d","A",1
7,1,48,45,0.0442714,"\int \cos ^2(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + I*a*Tan[c + d*x]),x]","\frac{a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}-\frac{i a \cos ^2(c+d x)}{2 d}","-\frac{i a \cos ^2(c+d x)}{2 d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}",1,"(a*(c + d*x))/(2*d) - ((I/2)*a*Cos[c + d*x]^2)/d + (a*Sin[2*(c + d*x)])/(4*d)","A",1
8,1,46,67,0.0457301,"\int \cos ^4(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + I*a*Tan[c + d*x]),x]","\frac{a \left(8 \sin (2 (c+d x))+\sin (4 (c+d x))-8 i \cos ^4(c+d x)+12 c+12 d x\right)}{32 d}","-\frac{i a \cos ^4(c+d x)}{4 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}",1,"(a*(12*c + 12*d*x - (8*I)*Cos[c + d*x]^4 + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(32*d)","A",1
9,1,56,89,0.0634174,"\int \cos ^6(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x]),x]","\frac{a \left(45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x))-32 i \cos ^6(c+d x)+60 c+60 d x\right)}{192 d}","-\frac{i a \cos ^6(c+d x)}{6 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}",1,"(a*(60*c + 60*d*x - (32*I)*Cos[c + d*x]^6 + 45*Sin[2*(c + d*x)] + 9*Sin[4*(c + d*x)] + Sin[6*(c + d*x)]))/(192*d)","A",1
10,1,68,111,0.1283787,"\int \cos ^8(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cos[c + d*x]^8*(a + I*a*Tan[c + d*x]),x]","\frac{a \left(672 \sin (2 (c+d x))+168 \sin (4 (c+d x))+32 \sin (6 (c+d x))+3 \sin (8 (c+d x))-384 i \cos ^8(c+d x)+840 c+840 d x\right)}{3072 d}","-\frac{i a \cos ^8(c+d x)}{8 d}+\frac{a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a x}{128}",1,"(a*(840*c + 840*d*x - (384*I)*Cos[c + d*x]^8 + 672*Sin[2*(c + d*x)] + 168*Sin[4*(c + d*x)] + 32*Sin[6*(c + d*x)] + 3*Sin[8*(c + d*x)]))/(3072*d)","A",1
11,1,61,98,0.3122467,"\int \sec ^7(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Sec[c + d*x]^7*(a + I*a*Tan[c + d*x]),x]","\frac{a \left(3360 \tanh ^{-1}(\sin (c+d x))+(1981 \sin (2 (c+d x))+700 \sin (4 (c+d x))+105 \sin (6 (c+d x))+1536 i) \sec ^7(c+d x)\right)}{10752 d}","\frac{i a \sec ^7(c+d x)}{7 d}+\frac{5 a \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{5 a \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{5 a \tan (c+d x) \sec (c+d x)}{16 d}",1,"(a*(3360*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]^7*(1536*I + 1981*Sin[2*(c + d*x)] + 700*Sin[4*(c + d*x)] + 105*Sin[6*(c + d*x)])))/(10752*d)","A",1
12,1,70,76,0.1674389,"\int \sec ^5(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Sec[c + d*x]^5*(a + I*a*Tan[c + d*x]),x]","\frac{i a \sec ^5(c+d x)}{5 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{8 d}","\frac{i a \sec ^5(c+d x)}{5 d}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}",1,"((I/5)*a*Sec[c + d*x]^5)/d + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*a*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(8*d)","A",1
13,1,54,54,0.0164418,"\int \sec ^3(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + I*a*Tan[c + d*x]),x]","\frac{i a \sec ^3(c+d x)}{3 d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}","\frac{i a \sec ^3(c+d x)}{3 d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + ((I/3)*a*Sec[c + d*x]^3)/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
14,1,27,27,0.0093525,"\int \sec (c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + I*a*Tan[c + d*x]),x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{i a \sec (c+d x)}{d}","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{i a \sec (c+d x)}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/d + (I*a*Sec[c + d*x])/d","A",1
15,1,51,26,0.0228448,"\int \cos (c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + I*a*Tan[c + d*x]),x]","\frac{i a \sin (c) \sin (d x)}{d}-\frac{i a \cos (c) \cos (d x)}{d}+\frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}","\frac{a \sin (c+d x)}{d}-\frac{i a \cos (c+d x)}{d}",1,"((-I)*a*Cos[c]*Cos[d*x])/d + (a*Cos[d*x]*Sin[c])/d + (a*Cos[c]*Sin[d*x])/d + (I*a*Sin[c]*Sin[d*x])/d","A",1
16,1,46,46,0.01081,"\int \cos ^3(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + I*a*Tan[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^3(c+d x)}{3 d}","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^3(c+d x)}{3 d}",1,"((-1/3*I)*a*Cos[c + d*x]^3)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d)","A",1
17,1,62,62,0.0184284,"\int \cos ^5(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + I*a*Tan[c + d*x]),x]","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^5(c+d x)}{5 d}","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^5(c+d x)}{5 d}",1,"((-1/5*I)*a*Cos[c + d*x]^5)/d + (a*Sin[c + d*x])/d - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)","A",1
18,1,76,76,0.0522,"\int \cos ^7(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x]),x]","-\frac{a \sin ^7(c+d x)}{7 d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^7(c+d x)}{7 d}","-\frac{a \sin ^7(c+d x)}{7 d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^7(c+d x)}{7 d}",1,"((-1/7*I)*a*Cos[c + d*x]^7)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^7)/(7*d)","A",1
19,1,99,109,1.4283464,"\int \sec ^8(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 \sec (c) \sec ^9(c+d x) (-63 \sin (2 c+d x)+84 \sin (2 c+3 d x)+36 \sin (4 c+5 d x)+9 \sin (6 c+7 d x)+\sin (8 c+9 d x)+63 i \cos (2 c+d x)+63 \sin (d x)+63 i \cos (d x))}{504 d}","\frac{i (a+i a \tan (c+d x))^9}{9 a^7 d}-\frac{3 i (a+i a \tan (c+d x))^8}{4 a^6 d}+\frac{12 i (a+i a \tan (c+d x))^7}{7 a^5 d}-\frac{4 i (a+i a \tan (c+d x))^6}{3 a^4 d}",1,"(a^2*Sec[c]*Sec[c + d*x]^9*((63*I)*Cos[d*x] + (63*I)*Cos[2*c + d*x] + 63*Sin[d*x] - 63*Sin[2*c + d*x] + 84*Sin[2*c + 3*d*x] + 36*Sin[4*c + 5*d*x] + 9*Sin[6*c + 7*d*x] + Sin[8*c + 9*d*x]))/(504*d)","A",1
20,1,90,82,1.2111394,"\int \sec ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 \sec (c) \sec ^7(c+d x) (-35 \sin (2 c+d x)+42 \sin (2 c+3 d x)+14 \sin (4 c+5 d x)+2 \sin (6 c+7 d x)+35 i \cos (2 c+d x)+35 \sin (d x)+35 i \cos (d x))}{210 d}","-\frac{i (a+i a \tan (c+d x))^7}{7 a^5 d}+\frac{2 i (a+i a \tan (c+d x))^6}{3 a^4 d}-\frac{4 i (a+i a \tan (c+d x))^5}{5 a^3 d}",1,"(a^2*Sec[c]*Sec[c + d*x]^7*((35*I)*Cos[d*x] + (35*I)*Cos[2*c + d*x] + 35*Sin[d*x] - 35*Sin[2*c + d*x] + 42*Sin[2*c + 3*d*x] + 14*Sin[4*c + 5*d*x] + 2*Sin[6*c + 7*d*x]))/(210*d)","A",1
21,1,77,55,0.5047678,"\int \sec ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 \sec (c) \sec ^5(c+d x) (-5 \sin (2 c+d x)+5 \sin (2 c+3 d x)+\sin (4 c+5 d x)+5 i \cos (2 c+d x)+5 \sin (d x)+5 i \cos (d x))}{20 d}","\frac{i (a+i a \tan (c+d x))^5}{5 a^3 d}-\frac{i (a+i a \tan (c+d x))^4}{2 a^2 d}",1,"(a^2*Sec[c]*Sec[c + d*x]^5*((5*I)*Cos[d*x] + (5*I)*Cos[2*c + d*x] + 5*Sin[d*x] - 5*Sin[2*c + d*x] + 5*Sin[2*c + 3*d*x] + Sin[4*c + 5*d*x]))/(20*d)","A",1
22,1,68,27,0.4094678,"\int \sec ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 \sec (c) \sec ^3(c+d x) (-3 \sin (2 c+d x)+2 \sin (2 c+3 d x)+3 i \cos (2 c+d x)+3 \sin (d x)+3 i \cos (d x))}{6 d}","-\frac{i (a+i a \tan (c+d x))^3}{3 a d}",1,"(a^2*Sec[c]*Sec[c + d*x]^3*((3*I)*Cos[d*x] + (3*I)*Cos[2*c + d*x] + 3*Sin[d*x] - 3*Sin[2*c + d*x] + 2*Sin[2*c + 3*d*x]))/(6*d)","B",1
23,1,100,38,0.7754495,"\int (a+i a \tan (c+d x))^2 \, dx","Integrate[(a + I*a*Tan[c + d*x])^2,x]","-\frac{a^2 \sec (c) \sec (c+d x) \left(-4 d x \cos (2 c+d x)+\cos (d x) \left(-4 d x+i \log \left(\cos ^2(c+d x)\right)\right)+i \cos (2 c+d x) \log \left(\cos ^2(c+d x)\right)+4 \cos (c) \cos (c+d x) \tan ^{-1}(\tan (3 c+d x))+2 \sin (d x)\right)}{2 d}","-\frac{a^2 \tan (c+d x)}{d}-\frac{2 i a^2 \log (\cos (c+d x))}{d}+2 a^2 x",1,"-1/2*(a^2*Sec[c]*Sec[c + d*x]*(4*ArcTan[Tan[3*c + d*x]]*Cos[c]*Cos[c + d*x] - 4*d*x*Cos[2*c + d*x] + Cos[d*x]*(-4*d*x + I*Log[Cos[c + d*x]^2]) + I*Cos[2*c + d*x]*Log[Cos[c + d*x]^2] + 2*Sin[d*x]))/d","B",1
24,1,31,25,0.0718024,"\int \cos ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^2,x]","-\frac{i a^2 (\cos (c+d x)+i \sin (c+d x))^2}{2 d}","-\frac{i a^3}{d (a-i a \tan (c+d x))}",1,"((-1/2*I)*a^2*(Cos[c + d*x] + I*Sin[c + d*x])^2)/d","A",1
25,1,86,63,0.5386239,"\int \cos ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 ((1-4 i d x) \sin (2 (c+d x))+(4 d x-i) \cos (2 (c+d x))-4 i) (\cos (2 (c+2 d x))+i \sin (2 (c+2 d x)))}{16 d (\cos (d x)+i \sin (d x))^2}","-\frac{i a^4}{4 d (a-i a \tan (c+d x))^2}-\frac{i a^3}{4 d (a-i a \tan (c+d x))}+\frac{a^2 x}{4}",1,"(a^2*(-4*I + (-I + 4*d*x)*Cos[2*(c + d*x)] + (1 - (4*I)*d*x)*Sin[2*(c + d*x)])*(Cos[2*(c + 2*d*x)] + I*Sin[2*(c + 2*d*x)]))/(16*d*(Cos[d*x] + I*Sin[d*x])^2)","A",1
26,1,116,117,0.5499227,"\int \cos ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 (-12 i d x \sin (2 (c+d x))+3 \sin (2 (c+d x))+2 \sin (4 (c+d x))+3 (4 d x-i) \cos (2 (c+d x))+i \cos (4 (c+d x))-9 i) (\cos (2 (c+2 d x))+i \sin (2 (c+2 d x)))}{48 d (\cos (d x)+i \sin (d x))^2}","-\frac{i a^5}{12 d (a-i a \tan (c+d x))^3}-\frac{i a^4}{8 d (a-i a \tan (c+d x))^2}-\frac{3 i a^3}{16 d (a-i a \tan (c+d x))}+\frac{i a^3}{16 d (a+i a \tan (c+d x))}+\frac{a^2 x}{4}",1,"(a^2*(-9*I + 3*(-I + 4*d*x)*Cos[2*(c + d*x)] + I*Cos[4*(c + d*x)] + 3*Sin[2*(c + d*x)] - (12*I)*d*x*Sin[2*(c + d*x)] + 2*Sin[4*(c + d*x)])*(Cos[2*(c + 2*d*x)] + I*Sin[2*(c + 2*d*x)]))/(48*d*(Cos[d*x] + I*Sin[d*x])^2)","A",1
27,1,138,171,0.5883787,"\int \cos ^8(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^8*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 (-120 i d x \sin (2 (c+d x))+30 \sin (2 (c+d x))+32 \sin (4 (c+d x))+3 \sin (6 (c+d x))+30 (4 d x-i) \cos (2 (c+d x))+16 i \cos (4 (c+d x))+i \cos (6 (c+d x))-80 i) (\cos (2 (c+2 d x))+i \sin (2 (c+2 d x)))}{512 d (\cos (d x)+i \sin (d x))^2}","-\frac{i a^6}{32 d (a-i a \tan (c+d x))^4}-\frac{i a^5}{16 d (a-i a \tan (c+d x))^3}-\frac{3 i a^4}{32 d (a-i a \tan (c+d x))^2}+\frac{i a^4}{64 d (a+i a \tan (c+d x))^2}-\frac{5 i a^3}{32 d (a-i a \tan (c+d x))}+\frac{5 i a^3}{64 d (a+i a \tan (c+d x))}+\frac{15 a^2 x}{64}",1,"(a^2*(-80*I + 30*(-I + 4*d*x)*Cos[2*(c + d*x)] + (16*I)*Cos[4*(c + d*x)] + I*Cos[6*(c + d*x)] + 30*Sin[2*(c + d*x)] - (120*I)*d*x*Sin[2*(c + d*x)] + 32*Sin[4*(c + d*x)] + 3*Sin[6*(c + d*x)])*(Cos[2*(c + 2*d*x)] + I*Sin[2*(c + 2*d*x)]))/(512*d*(Cos[d*x] + I*Sin[d*x])^2)","A",1
28,1,159,118,1.262259,"\int \sec ^5(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^5*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 (\cos (2 c)-i \sin (2 c)) (\tan (c+d x)-i)^2 \sec ^4(c+d x) \left(150 \sin (c+d x)-35 (17 \sin (3 (c+d x))+3 \sin (5 (c+d x)))-1536 i \cos (c+d x)+1680 \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)}{3840 d (\cos (d x)+i \sin (d x))^2}","\frac{7 i a^2 \sec ^5(c+d x)}{30 d}+\frac{7 a^2 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{i \sec ^5(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{6 d}+\frac{7 a^2 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{7 a^2 \tan (c+d x) \sec (c+d x)}{16 d}",1,"(a^2*Sec[c + d*x]^4*(Cos[2*c] - I*Sin[2*c])*((-1536*I)*Cos[c + d*x] + 1680*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]]) + 150*Sin[c + d*x] - 35*(17*Sin[3*(c + d*x)] + 3*Sin[5*(c + d*x)]))*(-I + Tan[c + d*x])^2)/(3840*d*(Cos[d*x] + I*Sin[d*x])^2)","A",1
29,1,215,94,1.2917569,"\int \sec ^3(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^3*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 \sec ^4(c+d x) \left(-18 \sin (c+d x)+30 \sin (3 (c+d x))+128 i \cos (c+d x)-45 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-60 \cos (2 (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)-15 \cos (4 (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)+45 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 d}","\frac{5 i a^2 \sec ^3(c+d x)}{12 d}+\frac{5 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{i \sec ^3(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{4 d}+\frac{5 a^2 \tan (c+d x) \sec (c+d x)}{8 d}",1,"(a^2*Sec[c + d*x]^4*((128*I)*Cos[c + d*x] - 45*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 60*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 15*Cos[4*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 45*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 18*Sin[c + d*x] + 30*Sin[3*(c + d*x)]))/(192*d)","B",1
30,1,146,68,0.9542511,"\int \sec (c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]*(a + I*a*Tan[c + d*x])^2,x]","-\frac{a^2 \sec ^2(c+d x) \left(2 \sin (c+d x)-8 i \cos (c+d x)+3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 \cos (2 (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)-3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d}","\frac{3 i a^2 \sec (c+d x)}{2 d}+\frac{3 a^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{i \sec (c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{2 d}",1,"-1/4*(a^2*Sec[c + d*x]^2*((-8*I)*Cos[c + d*x] + 3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 2*Sin[c + d*x]))/d","B",1
31,1,180,46,0.2717905,"\int \cos (c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 \left(\cos \left(\frac{1}{2} (c+5 d x)\right)+i \sin \left(\frac{1}{2} (c+5 d x)\right)\right) \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)-2 i\right)+\sin \left(\frac{1}{2} (c+d x)\right) \left(-i \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+i \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2\right)\right)}{d (\cos (d x)+i \sin (d x))^2}","-\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 i \cos (c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{d}",1,"(a^2*(Cos[(c + d*x)/2]*(-2*I + Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (2 - I*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + I*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sin[(c + d*x)/2])*(Cos[(c + 5*d*x)/2] + I*Sin[(c + 5*d*x)/2]))/(d*(Cos[d*x] + I*Sin[d*x])^2)","B",1
32,1,50,51,0.2501578,"\int \cos ^3(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 (2 \cos (c+d x)-i \sin (c+d x)) (\sin (2 (c+d x))-i \cos (2 (c+d x)))}{3 d}","\frac{a^2 \sin (c+d x)}{3 d}-\frac{2 i \cos ^3(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{3 d}",1,"(a^2*(2*Cos[c + d*x] - I*Sin[c + d*x])*((-I)*Cos[2*(c + d*x)] + Sin[2*(c + d*x)]))/(3*d)","A",1
33,1,72,69,0.5242845,"\int \cos ^5(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 (\sin (2 (c+d x))-i \cos (2 (c+d x))) (-5 i \sin (c+d x)+3 i \sin (3 (c+d x))+10 \cos (c+d x)-2 \cos (3 (c+d x)))}{20 d}","-\frac{a^2 \sin ^3(c+d x)}{5 d}+\frac{3 a^2 \sin (c+d x)}{5 d}-\frac{2 i \cos ^5(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{5 d}",1,"(a^2*((-I)*Cos[2*(c + d*x)] + Sin[2*(c + d*x)])*(10*Cos[c + d*x] - 2*Cos[3*(c + d*x)] - (5*I)*Sin[c + d*x] + (3*I)*Sin[3*(c + d*x)]))/(20*d)","A",1
34,1,111,87,0.499461,"\int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 (-70 \sin (c+d x)+63 \sin (3 (c+d x))+5 \sin (5 (c+d x))-140 i \cos (c+d x)+42 i \cos (3 (c+d x))+2 i \cos (5 (c+d x))) (\cos (2 (c+2 d x))+i \sin (2 (c+2 d x)))}{336 d (\cos (d x)+i \sin (d x))^2}","\frac{a^2 \sin ^5(c+d x)}{7 d}-\frac{10 a^2 \sin ^3(c+d x)}{21 d}+\frac{5 a^2 \sin (c+d x)}{7 d}-\frac{2 i \cos ^7(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{7 d}",1,"(a^2*((-140*I)*Cos[c + d*x] + (42*I)*Cos[3*(c + d*x)] + (2*I)*Cos[5*(c + d*x)] - 70*Sin[c + d*x] + 63*Sin[3*(c + d*x)] + 5*Sin[5*(c + d*x)])*(Cos[2*(c + 2*d*x)] + I*Sin[2*(c + 2*d*x)]))/(336*d*(Cos[d*x] + I*Sin[d*x])^2)","A",1
35,1,133,105,1.355121,"\int \cos ^9(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 (-525 \sin (c+d x)+567 \sin (3 (c+d x))+75 \sin (5 (c+d x))+7 \sin (7 (c+d x))-1050 i \cos (c+d x)+378 i \cos (3 (c+d x))+30 i \cos (5 (c+d x))+2 i \cos (7 (c+d x))) (\cos (2 (c+2 d x))+i \sin (2 (c+2 d x)))}{2880 d (\cos (d x)+i \sin (d x))^2}","-\frac{a^2 \sin ^7(c+d x)}{9 d}+\frac{7 a^2 \sin ^5(c+d x)}{15 d}-\frac{7 a^2 \sin ^3(c+d x)}{9 d}+\frac{7 a^2 \sin (c+d x)}{9 d}-\frac{2 i \cos ^9(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{9 d}",1,"(a^2*((-1050*I)*Cos[c + d*x] + (378*I)*Cos[3*(c + d*x)] + (30*I)*Cos[5*(c + d*x)] + (2*I)*Cos[7*(c + d*x)] - 525*Sin[c + d*x] + 567*Sin[3*(c + d*x)] + 75*Sin[5*(c + d*x)] + 7*Sin[7*(c + d*x)])*(Cos[2*(c + 2*d*x)] + I*Sin[2*(c + 2*d*x)]))/(2880*d*(Cos[d*x] + I*Sin[d*x])^2)","A",1
36,1,117,109,2.069488,"\int \sec ^8(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \sec (c) \sec ^{10}(c+d x) (105 \sin (c+2 d x)-105 \sin (3 c+2 d x)+120 \sin (3 c+4 d x)+45 \sin (5 c+6 d x)+10 \sin (7 c+8 d x)+\sin (9 c+10 d x)+105 i \cos (c+2 d x)+105 i \cos (3 c+2 d x)-126 \sin (c)+126 i \cos (c))}{840 d}","\frac{i (a+i a \tan (c+d x))^{10}}{10 a^7 d}-\frac{2 i (a+i a \tan (c+d x))^9}{3 a^6 d}+\frac{3 i (a+i a \tan (c+d x))^8}{2 a^5 d}-\frac{8 i (a+i a \tan (c+d x))^7}{7 a^4 d}",1,"(a^3*Sec[c]*Sec[c + d*x]^10*((126*I)*Cos[c] + (105*I)*Cos[c + 2*d*x] + (105*I)*Cos[3*c + 2*d*x] - 126*Sin[c] + 105*Sin[c + 2*d*x] - 105*Sin[3*c + 2*d*x] + 120*Sin[3*c + 4*d*x] + 45*Sin[5*c + 6*d*x] + 10*Sin[7*c + 8*d*x] + Sin[9*c + 10*d*x]))/(840*d)","A",1
37,1,106,82,1.569782,"\int \sec ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \sec (c) \sec ^8(c+d x) (28 \sin (c+2 d x)-28 \sin (3 c+2 d x)+28 \sin (3 c+4 d x)+8 \sin (5 c+6 d x)+\sin (7 c+8 d x)+28 i \cos (c+2 d x)+28 i \cos (3 c+2 d x)-35 \sin (c)+35 i \cos (c))}{168 d}","-\frac{i (a+i a \tan (c+d x))^8}{8 a^5 d}+\frac{4 i (a+i a \tan (c+d x))^7}{7 a^4 d}-\frac{2 i (a+i a \tan (c+d x))^6}{3 a^3 d}",1,"(a^3*Sec[c]*Sec[c + d*x]^8*((35*I)*Cos[c] + (28*I)*Cos[c + 2*d*x] + (28*I)*Cos[3*c + 2*d*x] - 35*Sin[c] + 28*Sin[c + 2*d*x] - 28*Sin[3*c + 2*d*x] + 28*Sin[3*c + 4*d*x] + 8*Sin[5*c + 6*d*x] + Sin[7*c + 8*d*x]))/(168*d)","A",1
38,1,97,55,1.3025935,"\int \sec ^4(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \sec (c) \sec ^6(c+d x) (15 \sin (c+2 d x)-15 \sin (3 c+2 d x)+12 \sin (3 c+4 d x)+2 \sin (5 c+6 d x)+15 i \cos (c+2 d x)+15 i \cos (3 c+2 d x)-20 \sin (c)+20 i \cos (c))}{60 d}","\frac{i (a+i a \tan (c+d x))^6}{6 a^3 d}-\frac{2 i (a+i a \tan (c+d x))^5}{5 a^2 d}",1,"(a^3*Sec[c]*Sec[c + d*x]^6*((20*I)*Cos[c] + (15*I)*Cos[c + 2*d*x] + (15*I)*Cos[3*c + 2*d*x] - 20*Sin[c] + 15*Sin[c + 2*d*x] - 15*Sin[3*c + 2*d*x] + 12*Sin[3*c + 4*d*x] + 2*Sin[5*c + 6*d*x]))/(60*d)","A",1
39,1,84,27,0.7030575,"\int \sec ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \sec (c) \sec ^4(c+d x) (2 \sin (c+2 d x)-2 \sin (3 c+2 d x)+\sin (3 c+4 d x)+2 i \cos (c+2 d x)+2 i \cos (3 c+2 d x)-3 \sin (c)+3 i \cos (c))}{4 d}","-\frac{i (a+i a \tan (c+d x))^4}{4 a d}",1,"(a^3*Sec[c]*Sec[c + d*x]^4*((3*I)*Cos[c] + (2*I)*Cos[c + 2*d*x] + (2*I)*Cos[3*c + 2*d*x] - 3*Sin[c] + 2*Sin[c + 2*d*x] - 2*Sin[3*c + 2*d*x] + Sin[3*c + 4*d*x]))/(4*d)","B",1
40,1,119,63,1.0407964,"\int (a+i a \tan (c+d x))^3 \, dx","Integrate[(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \sec (c) \sec ^2(c+d x) \left(-3 \sin (c+2 d x)+2 d x \cos (3 c+2 d x)-i \cos (3 c+2 d x) \log \left(\cos ^2(c+d x)\right)+\cos (c+2 d x) \left(2 d x-i \log \left(\cos ^2(c+d x)\right)\right)+\cos (c) \left(-2 i \log \left(\cos ^2(c+d x)\right)+4 d x-i\right)+3 \sin (c)\right)}{2 d}","-\frac{2 a^3 \tan (c+d x)}{d}-\frac{4 i a^3 \log (\cos (c+d x))}{d}+4 a^3 x+\frac{i a (a+i a \tan (c+d x))^2}{2 d}",1,"(a^3*Sec[c]*Sec[c + d*x]^2*(2*d*x*Cos[3*c + 2*d*x] + Cos[c + 2*d*x]*(2*d*x - I*Log[Cos[c + d*x]^2]) + Cos[c]*(-I + 4*d*x - (2*I)*Log[Cos[c + d*x]^2]) - I*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] + 3*Sin[c] - 3*Sin[c + 2*d*x]))/(2*d)","A",1
41,1,99,49,0.3805773,"\int \cos ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^3,x]","-\frac{a^3 (\cos (c+4 d x)+i \sin (c+4 d x)) \left(\cos (c+d x) \left(-i \log \left(\cos ^2(c+d x)\right)+2 d x+2 i\right)+\sin (c+d x) \left(-\log \left(\cos ^2(c+d x)\right)-2 i d x-2\right)\right)}{2 d (\cos (d x)+i \sin (d x))^3}","-\frac{2 i a^4}{d (a-i a \tan (c+d x))}+\frac{i a^3 \log (\cos (c+d x))}{d}-a^3 x",1,"-1/2*(a^3*(Cos[c + d*x]*(2*I + 2*d*x - I*Log[Cos[c + d*x]^2]) + (-2 - (2*I)*d*x - Log[Cos[c + d*x]^2])*Sin[c + d*x])*(Cos[c + 4*d*x] + I*Sin[c + 4*d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3)","B",1
42,1,50,27,0.3025997,"\int \cos ^4(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 (3 \cos (c+d x)-i \sin (c+d x)) (\sin (3 (c+d x))-i \cos (3 (c+d x)))}{8 d}","-\frac{i a^5}{2 d (a-i a \tan (c+d x))^2}",1,"(a^3*(3*Cos[c + d*x] - I*Sin[c + d*x])*((-I)*Cos[3*(c + d*x)] + Sin[3*(c + d*x)]))/(8*d)","A",1
43,1,109,90,0.6941679,"\int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 (-9 \sin (c+d x)-12 i d x \sin (3 (c+d x))+2 \sin (3 (c+d x))-27 i \cos (c+d x)+2 (6 d x-i) \cos (3 (c+d x))) (\cos (3 (c+2 d x))+i \sin (3 (c+2 d x)))}{96 d (\cos (d x)+i \sin (d x))^3}","-\frac{i a^6}{6 d (a-i a \tan (c+d x))^3}-\frac{i a^5}{8 d (a-i a \tan (c+d x))^2}-\frac{i a^4}{8 d (a-i a \tan (c+d x))}+\frac{a^3 x}{8}",1,"(a^3*((-27*I)*Cos[c + d*x] + 2*(-I + 6*d*x)*Cos[3*(c + d*x)] - 9*Sin[c + d*x] + 2*Sin[3*(c + d*x)] - (12*I)*d*x*Sin[3*(c + d*x)])*(Cos[3*(c + 2*d*x)] + I*Sin[3*(c + 2*d*x)]))/(96*d*(Cos[d*x] + I*Sin[d*x])^3)","A",1
44,1,131,144,0.7394058,"\int \cos ^8(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^8*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 (-60 \sin (c+d x)-120 i d x \sin (3 (c+d x))+20 \sin (3 (c+d x))+15 \sin (5 (c+d x))-180 i \cos (c+d x)+20 (6 d x-i) \cos (3 (c+d x))+9 i \cos (5 (c+d x))) (\cos (3 (c+2 d x))+i \sin (3 (c+2 d x)))}{768 d (\cos (d x)+i \sin (d x))^3}","-\frac{i a^7}{16 d (a-i a \tan (c+d x))^4}-\frac{i a^6}{12 d (a-i a \tan (c+d x))^3}-\frac{3 i a^5}{32 d (a-i a \tan (c+d x))^2}-\frac{i a^4}{8 d (a-i a \tan (c+d x))}+\frac{i a^4}{32 d (a+i a \tan (c+d x))}+\frac{5 a^3 x}{32}",1,"(a^3*((-180*I)*Cos[c + d*x] + 20*(-I + 6*d*x)*Cos[3*(c + d*x)] + (9*I)*Cos[5*(c + d*x)] - 60*Sin[c + d*x] + 20*Sin[3*(c + d*x)] - (120*I)*d*x*Sin[3*(c + d*x)] + 15*Sin[5*(c + d*x)])*(Cos[3*(c + 2*d*x)] + I*Sin[3*(c + 2*d*x)]))/(768*d*(Cos[d*x] + I*Sin[d*x])^3)","A",1
45,1,102,127,0.8544529,"\int \sec ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^3*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 (\cos (3 d x)+i \sin (3 d x)) \left(1680 \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)+\sec ^5(c+d x) (-150 \sin (2 (c+d x))+105 \sin (4 (c+d x))+640 i \cos (2 (c+d x))+448 i)\right)}{960 d (\cos (d x)+i \sin (d x))^3}","\frac{7 i a^3 \sec ^3(c+d x)}{12 d}+\frac{7 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{7 i \sec ^3(c+d x) \left(a^3+i a^3 \tan (c+d x)\right)}{20 d}+\frac{7 a^3 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{i a \sec ^3(c+d x) (a+i a \tan (c+d x))^2}{5 d}",1,"(a^3*(Cos[3*d*x] + I*Sin[3*d*x])*(1680*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]] + Sec[c + d*x]^5*(448*I + (640*I)*Cos[2*(c + d*x)] - 150*Sin[2*(c + d*x)] + 105*Sin[4*(c + d*x)])))/(960*d*(Cos[d*x] + I*Sin[d*x])^3)","A",1
46,1,93,99,0.6167439,"\int \sec (c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 (\cos (3 d x)+i \sin (3 d x)) \left(60 \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)+i \sec ^3(c+d x) (9 i \sin (2 (c+d x))+24 \cos (2 (c+d x))+20)\right)}{12 d (\cos (d x)+i \sin (d x))^3}","\frac{5 i a^3 \sec (c+d x)}{2 d}+\frac{5 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{5 i \sec (c+d x) \left(a^3+i a^3 \tan (c+d x)\right)}{6 d}+\frac{i a \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}",1,"(a^3*(Cos[3*d*x] + I*Sin[3*d*x])*(60*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]] + I*Sec[c + d*x]^3*(20 + 24*Cos[2*(c + d*x)] + (9*I)*Sin[2*(c + d*x)])))/(12*d*(Cos[d*x] + I*Sin[d*x])^3)","A",1
47,1,123,61,0.817554,"\int \cos (c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \cos ^2(c+d x) (\tan (c+d x)-i)^3 \left((-\cos (2 c-d x)+i \sin (2 c-d x)) (5 \cos (c+d x)-i \sin (c+d x))+6 (\sin (3 c)+i \cos (3 c)) \cos (c+d x) \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)\right)}{d (\cos (d x)+i \sin (d x))^3}","-\frac{3 i a^3 \sec (c+d x)}{d}-\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^2}{d}",1,"(a^3*Cos[c + d*x]^2*(6*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]]*Cos[c + d*x]*(I*Cos[3*c] + Sin[3*c]) + (-Cos[2*c - d*x] + I*Sin[2*c - d*x])*(5*Cos[c + d*x] - I*Sin[c + d*x]))*(-I + Tan[c + d*x])^3)/(d*(Cos[d*x] + I*Sin[d*x])^3)","B",1
48,1,31,32,0.0930694,"\int \cos ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^3,x]","-\frac{i a^3 (\cos (c+d x)+i \sin (c+d x))^3}{3 d}","-\frac{i \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}",1,"((-1/3*I)*a^3*(Cos[c + d*x] + I*Sin[c + d*x])^3)/d","A",1
49,1,55,88,0.6071689,"\int \cos ^5(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 (-6 i \sin (2 (c+d x))+9 \cos (2 (c+d x))+5) (\sin (3 (c+d x))-i \cos (3 (c+d x)))}{30 d}","-\frac{a^3 \sin ^3(c+d x)}{15 d}+\frac{a^3 \sin (c+d x)}{5 d}-\frac{i a^3 \cos ^3(c+d x)}{15 d}-\frac{2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d}",1,"(a^3*(5 + 9*Cos[2*(c + d*x)] - (6*I)*Sin[2*(c + d*x)])*((-I)*Cos[3*(c + d*x)] + Sin[3*(c + d*x)]))/(30*d)","A",1
50,1,77,106,0.7298833,"\int \cos ^7(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 (\sin (3 (c+d x))-i \cos (3 (c+d x))) (-56 i \sin (2 (c+d x))+20 i \sin (4 (c+d x))+84 \cos (2 (c+d x))-15 \cos (4 (c+d x))+35)}{280 d}","\frac{3 a^3 \sin ^5(c+d x)}{35 d}-\frac{2 a^3 \sin ^3(c+d x)}{7 d}+\frac{3 a^3 \sin (c+d x)}{7 d}-\frac{3 i a^3 \cos ^5(c+d x)}{35 d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{7 d}",1,"(a^3*((-I)*Cos[3*(c + d*x)] + Sin[3*(c + d*x)])*(35 + 84*Cos[2*(c + d*x)] - 15*Cos[4*(c + d*x)] - (56*I)*Sin[2*(c + d*x)] + (20*I)*Sin[4*(c + d*x)]))/(280*d)","A",1
51,1,116,124,0.8467889,"\int \cos ^9(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 (-378 i \sin (2 (c+d x))+216 i \sin (4 (c+d x))+14 i \sin (6 (c+d x))+567 \cos (2 (c+d x))-162 \cos (4 (c+d x))-7 \cos (6 (c+d x))+210) (\sin (3 (c+2 d x))-i \cos (3 (c+2 d x)))}{2016 d (\cos (d x)+i \sin (d x))^3}","-\frac{5 a^3 \sin ^7(c+d x)}{63 d}+\frac{a^3 \sin ^5(c+d x)}{3 d}-\frac{5 a^3 \sin ^3(c+d x)}{9 d}+\frac{5 a^3 \sin (c+d x)}{9 d}-\frac{5 i a^3 \cos ^7(c+d x)}{63 d}-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{9 d}",1,"(a^3*(210 + 567*Cos[2*(c + d*x)] - 162*Cos[4*(c + d*x)] - 7*Cos[6*(c + d*x)] - (378*I)*Sin[2*(c + d*x)] + (216*I)*Sin[4*(c + d*x)] + (14*I)*Sin[6*(c + d*x)])*((-I)*Cos[3*(c + 2*d*x)] + Sin[3*(c + 2*d*x)]))/(2016*d*(Cos[d*x] + I*Sin[d*x])^3)","A",1
52,1,171,163,2.0900145,"\int \sec ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^3*(a + I*a*Tan[c + d*x])^4,x]","-\frac{a^4 (\cos (4 c)-i \sin (4 c)) (\tan (c+d x)-i)^4 \sec ^2(c+d x) \left(-4608 i \cos (c+d x)+5 (90 \sin (c+d x)+155 \sin (3 (c+d x))-63 \sin (5 (c+d x))-512 i \cos (3 (c+d x)))+5040 \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)}{3840 d (\cos (d x)+i \sin (d x))^4}","\frac{7 i a^4 \sec ^3(c+d x)}{8 d}+\frac{21 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{21 i \sec ^3(c+d x) \left(a^4+i a^4 \tan (c+d x)\right)}{40 d}+\frac{21 a^4 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{3 i \sec ^3(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{10 d}+\frac{i a \sec ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}",1,"-1/3840*(a^4*Sec[c + d*x]^2*(Cos[4*c] - I*Sin[4*c])*((-4608*I)*Cos[c + d*x] + 5040*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]]) + 5*((-512*I)*Cos[3*(c + d*x)] + 90*Sin[c + d*x] + 155*Sin[3*(c + d*x)] - 63*Sin[5*(c + d*x)]))*(-I + Tan[c + d*x])^4)/(d*(Cos[d*x] + I*Sin[d*x])^4)","A",1
53,1,237,133,1.4809498,"\int \sec (c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Sec[c + d*x]*(a + I*a*Tan[c + d*x])^4,x]","-\frac{a^4 \sec ^4(c+d x) \left(3 \left(42 \sin (c+d x)+58 \sin (3 (c+d x))-128 i \cos (3 (c+d x))+35 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+105 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+140 \cos (2 (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)-35 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-105 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-896 i \cos (c+d x)\right)}{192 d}","\frac{35 i a^4 \sec (c+d x)}{8 d}+\frac{35 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{35 i \sec (c+d x) \left(a^4+i a^4 \tan (c+d x)\right)}{24 d}+\frac{7 i \sec (c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{12 d}+\frac{i a \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}",1,"-1/192*(a^4*Sec[c + d*x]^4*((-896*I)*Cos[c + d*x] + 3*((-128*I)*Cos[3*(c + d*x)] + 105*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 35*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 140*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 105*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 35*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 42*Sin[c + d*x] + 58*Sin[3*(c + d*x)])))/d","A",1
54,1,906,97,6.6517278,"\int \cos (c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Cos[c + d*x]*(a + I*a*Tan[c + d*x])^4,x]","\frac{\cos ^4(c+d x) (8 \cos (3 c)-8 i \sin (3 c)) \sin (d x) (i \tan (c+d x) a+a)^4}{d (\cos (d x)+i \sin (d x))^4}-\frac{i \cos ^4(c+d x) (4 \cos (4 c)-4 i \sin (4 c)) \sin \left(\frac{d x}{2}\right) (i \tan (c+d x) a+a)^4}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x))^4 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{i \cos ^4(c+d x) (4 \cos (4 c)-4 i \sin (4 c)) \sin \left(\frac{d x}{2}\right) (i \tan (c+d x) a+a)^4}{d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x))^4 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^4(c+d x) \left(\frac{1}{4} \cos (4 c)-\frac{1}{4} i \sin (4 c)\right) (i \tan (c+d x) a+a)^4}{d (\cos (d x)+i \sin (d x))^4 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos ^4(c+d x) \left(\frac{1}{4} i \sin (4 c)-\frac{1}{4} \cos (4 c)\right) (i \tan (c+d x) a+a)^4}{d (\cos (d x)+i \sin (d x))^4 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{15 \cos (4 c) \cos ^4(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) (i \tan (c+d x) a+a)^4}{2 d (\cos (d x)+i \sin (d x))^4}-\frac{15 \cos (4 c) \cos ^4(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) (i \tan (c+d x) a+a)^4}{2 d (\cos (d x)+i \sin (d x))^4}+\frac{\cos (d x) \cos ^4(c+d x) (-8 i \cos (3 c)-8 \sin (3 c)) (i \tan (c+d x) a+a)^4}{d (\cos (d x)+i \sin (d x))^4}+\frac{\cos ^4(c+d x) \sec (c) (-4 i \cos (4 c)-4 \sin (4 c)) (i \tan (c+d x) a+a)^4}{d (\cos (d x)+i \sin (d x))^4}-\frac{15 i \cos ^4(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) \sin (4 c) (i \tan (c+d x) a+a)^4}{2 d (\cos (d x)+i \sin (d x))^4}+\frac{15 i \cos ^4(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) \sin (4 c) (i \tan (c+d x) a+a)^4}{2 d (\cos (d x)+i \sin (d x))^4}","-\frac{15 i a^4 \sec (c+d x)}{2 d}-\frac{15 a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{5 i \sec (c+d x) \left(a^4+i a^4 \tan (c+d x)\right)}{2 d}-\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^3}{d}",1,"(15*Cos[4*c]*Cos[c + d*x]^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(a + I*a*Tan[c + d*x])^4)/(2*d*(Cos[d*x] + I*Sin[d*x])^4) - (15*Cos[4*c]*Cos[c + d*x]^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(a + I*a*Tan[c + d*x])^4)/(2*d*(Cos[d*x] + I*Sin[d*x])^4) + (Cos[d*x]*Cos[c + d*x]^4*((-8*I)*Cos[3*c] - 8*Sin[3*c])*(a + I*a*Tan[c + d*x])^4)/(d*(Cos[d*x] + I*Sin[d*x])^4) + (Cos[c + d*x]^4*Sec[c]*((-4*I)*Cos[4*c] - 4*Sin[4*c])*(a + I*a*Tan[c + d*x])^4)/(d*(Cos[d*x] + I*Sin[d*x])^4) - (((15*I)/2)*Cos[c + d*x]^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sin[4*c]*(a + I*a*Tan[c + d*x])^4)/(d*(Cos[d*x] + I*Sin[d*x])^4) + (((15*I)/2)*Cos[c + d*x]^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sin[4*c]*(a + I*a*Tan[c + d*x])^4)/(d*(Cos[d*x] + I*Sin[d*x])^4) + (Cos[c + d*x]^4*(8*Cos[3*c] - (8*I)*Sin[3*c])*Sin[d*x]*(a + I*a*Tan[c + d*x])^4)/(d*(Cos[d*x] + I*Sin[d*x])^4) + (Cos[c + d*x]^4*(Cos[4*c]/4 - (I/4)*Sin[4*c])*(a + I*a*Tan[c + d*x])^4)/(d*(Cos[d*x] + I*Sin[d*x])^4*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) - (I*Cos[c + d*x]^4*(4*Cos[4*c] - (4*I)*Sin[4*c])*Sin[(d*x)/2]*(a + I*a*Tan[c + d*x])^4)/(d*(Cos[c/2] - Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^4*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (Cos[c + d*x]^4*(-1/4*Cos[4*c] + (I/4)*Sin[4*c])*(a + I*a*Tan[c + d*x])^4)/(d*(Cos[d*x] + I*Sin[d*x])^4*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (I*Cos[c + d*x]^4*(4*Cos[4*c] - (4*I)*Sin[4*c])*Sin[(d*x)/2]*(a + I*a*Tan[c + d*x])^4)/(d*(Cos[c/2] + Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^4*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
55,1,246,78,0.6674803,"\int \cos ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^4,x]","\frac{a^4 (\cos (c+d x)+i \sin (c+d x))^4 \left(6 i \sin (3 c) \sin (d x)-2 i \sin (c) \sin (3 d x)-2 \sin (c) \cos (3 d x)+6 \sin (3 c) \cos (d x)+\cos (3 c) (-6 \sin (d x)+6 i \cos (d x))+2 \cos (c) (\sin (3 d x)-i \cos (3 d x))-3 \cos (4 c) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 i \sin (4 c) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 \cos (4 c) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-3 i \sin (4 c) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{3 d (\cos (d x)+i \sin (d x))^4}","\frac{a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 i \cos (c+d x) \left(a^4+i a^4 \tan (c+d x)\right)}{d}-\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}",1,"(a^4*(-3*Cos[4*c]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3*Cos[4*c]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2*Cos[3*d*x]*Sin[c] + 6*Cos[d*x]*Sin[3*c] + (3*I)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[4*c] - (3*I)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[4*c] + Cos[3*c]*((6*I)*Cos[d*x] - 6*Sin[d*x]) + (6*I)*Sin[3*c]*Sin[d*x] - (2*I)*Sin[c]*Sin[3*d*x] + 2*Cos[c]*((-I)*Cos[3*d*x] + Sin[3*d*x]))*(Cos[c + d*x] + I*Sin[c + d*x])^4)/(3*d*(Cos[d*x] + I*Sin[d*x])^4)","B",1
56,1,50,66,0.4067727,"\int \cos ^5(c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^4,x]","\frac{a^4 (4 \cos (c+d x)-i \sin (c+d x)) (\sin (4 (c+d x))-i \cos (4 (c+d x)))}{15 d}","-\frac{i \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{5 d}-\frac{i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{15 d}",1,"(a^4*(4*Cos[c + d*x] - I*Sin[c + d*x])*((-I)*Cos[4*(c + d*x)] + Sin[4*(c + d*x)]))/(15*d)","A",1
57,1,73,102,0.852511,"\int \cos ^7(c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^4,x]","\frac{a^4 (-i (7 \sin (c+d x)+15 \sin (3 (c+d x)))+28 \cos (c+d x)+20 \cos (3 (c+d x))) (\sin (4 (c+d x))-i \cos (4 (c+d x)))}{140 d}","-\frac{a^4 \sin ^3(c+d x)}{35 d}+\frac{3 a^4 \sin (c+d x)}{35 d}-\frac{2 i \cos ^5(c+d x) \left(a^4+i a^4 \tan (c+d x)\right)}{35 d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^3}{7 d}",1,"(a^4*(28*Cos[c + d*x] + 20*Cos[3*(c + d*x)] - I*(7*Sin[c + d*x] + 15*Sin[3*(c + d*x)]))*((-I)*Cos[4*(c + d*x)] + Sin[4*(c + d*x)]))/(140*d)","A",1
58,1,111,120,0.8659627,"\int \cos ^9(c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^4,x]","\frac{a^4 (-42 \sin (c+d x)-135 \sin (3 (c+d x))+35 \sin (5 (c+d x))-168 i \cos (c+d x)-180 i \cos (3 (c+d x))+28 i \cos (5 (c+d x))) (\cos (4 (c+2 d x))+i \sin (4 (c+2 d x)))}{1008 d (\cos (d x)+i \sin (d x))^4}","\frac{a^4 \sin ^5(c+d x)}{21 d}-\frac{10 a^4 \sin ^3(c+d x)}{63 d}+\frac{5 a^4 \sin (c+d x)}{21 d}-\frac{2 i \cos ^7(c+d x) \left(a^4+i a^4 \tan (c+d x)\right)}{21 d}-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}",1,"(a^4*((-168*I)*Cos[c + d*x] - (180*I)*Cos[3*(c + d*x)] + (28*I)*Cos[5*(c + d*x)] - 42*Sin[c + d*x] - 135*Sin[3*(c + d*x)] + 35*Sin[5*(c + d*x)])*(Cos[4*(c + 2*d*x)] + I*Sin[4*(c + 2*d*x)]))/(1008*d*(Cos[d*x] + I*Sin[d*x])^4)","A",1
59,1,167,109,3.8805979,"\int \sec ^8(c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^5,x]","\frac{a^5 \sec (c) \sec ^{12}(c+d x) (792 \sin (c+2 d x)-792 \sin (3 c+2 d x)+495 \sin (3 c+4 d x)-495 \sin (5 c+4 d x)+440 \sin (5 c+6 d x)+132 \sin (7 c+8 d x)+24 \sin (9 c+10 d x)+2 \sin (11 c+12 d x)+792 i \cos (c+2 d x)+792 i \cos (3 c+2 d x)+495 i \cos (3 c+4 d x)+495 i \cos (5 c+4 d x)-924 \sin (c)+924 i \cos (c))}{3960 d}","\frac{i (a+i a \tan (c+d x))^{12}}{12 a^7 d}-\frac{6 i (a+i a \tan (c+d x))^{11}}{11 a^6 d}+\frac{6 i (a+i a \tan (c+d x))^{10}}{5 a^5 d}-\frac{8 i (a+i a \tan (c+d x))^9}{9 a^4 d}",1,"(a^5*Sec[c]*Sec[c + d*x]^12*((924*I)*Cos[c] + (792*I)*Cos[c + 2*d*x] + (792*I)*Cos[3*c + 2*d*x] + (495*I)*Cos[3*c + 4*d*x] + (495*I)*Cos[5*c + 4*d*x] - 924*Sin[c] + 792*Sin[c + 2*d*x] - 792*Sin[3*c + 2*d*x] + 495*Sin[3*c + 4*d*x] - 495*Sin[5*c + 4*d*x] + 440*Sin[5*c + 6*d*x] + 132*Sin[7*c + 8*d*x] + 24*Sin[9*c + 10*d*x] + 2*Sin[11*c + 12*d*x]))/(3960*d)","A",1
60,1,154,82,2.8359252,"\int \sec ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^5,x]","\frac{a^5 \sec (c) \sec ^{10}(c+d x) (105 \sin (c+2 d x)-105 \sin (3 c+2 d x)+60 \sin (3 c+4 d x)-60 \sin (5 c+4 d x)+45 \sin (5 c+6 d x)+10 \sin (7 c+8 d x)+\sin (9 c+10 d x)+105 i \cos (c+2 d x)+105 i \cos (3 c+2 d x)+60 i \cos (3 c+4 d x)+60 i \cos (5 c+4 d x)-126 \sin (c)+126 i \cos (c))}{360 d}","-\frac{i (a+i a \tan (c+d x))^{10}}{10 a^5 d}+\frac{4 i (a+i a \tan (c+d x))^9}{9 a^4 d}-\frac{i (a+i a \tan (c+d x))^8}{2 a^3 d}",1,"(a^5*Sec[c]*Sec[c + d*x]^10*((126*I)*Cos[c] + (105*I)*Cos[c + 2*d*x] + (105*I)*Cos[3*c + 2*d*x] + (60*I)*Cos[3*c + 4*d*x] + (60*I)*Cos[5*c + 4*d*x] - 126*Sin[c] + 105*Sin[c + 2*d*x] - 105*Sin[3*c + 2*d*x] + 60*Sin[3*c + 4*d*x] - 60*Sin[5*c + 4*d*x] + 45*Sin[5*c + 6*d*x] + 10*Sin[7*c + 8*d*x] + Sin[9*c + 10*d*x]))/(360*d)","A",1
61,1,143,55,2.0922331,"\int \sec ^4(c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^5,x]","\frac{a^5 \sec (c) \sec ^8(c+d x) (28 \sin (c+2 d x)-28 \sin (3 c+2 d x)+14 \sin (3 c+4 d x)-14 \sin (5 c+4 d x)+8 \sin (5 c+6 d x)+\sin (7 c+8 d x)+28 i \cos (c+2 d x)+28 i \cos (3 c+2 d x)+14 i \cos (3 c+4 d x)+14 i \cos (5 c+4 d x)-35 \sin (c)+35 i \cos (c))}{56 d}","\frac{i (a+i a \tan (c+d x))^8}{8 a^3 d}-\frac{2 i (a+i a \tan (c+d x))^7}{7 a^2 d}",1,"(a^5*Sec[c]*Sec[c + d*x]^8*((35*I)*Cos[c] + (28*I)*Cos[c + 2*d*x] + (28*I)*Cos[3*c + 2*d*x] + (14*I)*Cos[3*c + 4*d*x] + (14*I)*Cos[5*c + 4*d*x] - 35*Sin[c] + 28*Sin[c + 2*d*x] - 28*Sin[3*c + 2*d*x] + 14*Sin[3*c + 4*d*x] - 14*Sin[5*c + 4*d*x] + 8*Sin[5*c + 6*d*x] + Sin[7*c + 8*d*x]))/(56*d)","B",1
62,1,134,27,1.9133652,"\int \sec ^2(c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^5,x]","\frac{a^5 \sec (c) \sec ^6(c+d x) (15 \sin (c+2 d x)-15 \sin (3 c+2 d x)+6 \sin (3 c+4 d x)-6 \sin (5 c+4 d x)+2 \sin (5 c+6 d x)+15 i \cos (c+2 d x)+15 i \cos (3 c+2 d x)+6 i \cos (3 c+4 d x)+6 i \cos (5 c+4 d x)-20 \sin (c)+20 i \cos (c))}{12 d}","-\frac{i (a+i a \tan (c+d x))^6}{6 a d}",1,"(a^5*Sec[c]*Sec[c + d*x]^6*((20*I)*Cos[c] + (15*I)*Cos[c + 2*d*x] + (15*I)*Cos[3*c + 2*d*x] + (6*I)*Cos[3*c + 4*d*x] + (6*I)*Cos[5*c + 4*d*x] - 20*Sin[c] + 15*Sin[c + 2*d*x] - 15*Sin[3*c + 2*d*x] + 6*Sin[3*c + 4*d*x] - 6*Sin[5*c + 4*d*x] + 2*Sin[5*c + 6*d*x]))/(12*d)","B",1
63,1,228,117,2.8734614,"\int (a+i a \tan (c+d x))^5 \, dx","Integrate[(a + I*a*Tan[c + d*x])^5,x]","\frac{a^5 \sec (c) \sec ^4(c+d x) \left(-70 \sin (c+2 d x)+30 \sin (3 c+2 d x)-25 \sin (3 c+4 d x)+48 d x \cos (3 c+2 d x)-18 i \cos (3 c+2 d x)+12 d x \cos (3 c+4 d x)+12 d x \cos (5 c+4 d x)-24 i \cos (3 c+2 d x) \log \left(\cos ^2(c+d x)\right)+6 \cos (c+2 d x) \left(-4 i \log \left(\cos ^2(c+d x)\right)+8 d x-3 i\right)+\cos (c) \left(-36 i \log \left(\cos ^2(c+d x)\right)+72 d x-33 i\right)-6 i \cos (3 c+4 d x) \log \left(\cos ^2(c+d x)\right)-6 i \cos (5 c+4 d x) \log \left(\cos ^2(c+d x)\right)+75 \sin (c)\right)}{12 d}","-\frac{8 a^5 \tan (c+d x)}{d}-\frac{16 i a^5 \log (\cos (c+d x))}{d}+16 a^5 x+\frac{2 i a^2 (a+i a \tan (c+d x))^3}{3 d}+\frac{2 i a \left(a^2+i a^2 \tan (c+d x)\right)^2}{d}+\frac{i a (a+i a \tan (c+d x))^4}{4 d}",1,"(a^5*Sec[c]*Sec[c + d*x]^4*((-18*I)*Cos[3*c + 2*d*x] + 48*d*x*Cos[3*c + 2*d*x] + 12*d*x*Cos[3*c + 4*d*x] + 12*d*x*Cos[5*c + 4*d*x] + 6*Cos[c + 2*d*x]*(-3*I + 8*d*x - (4*I)*Log[Cos[c + d*x]^2]) + Cos[c]*(-33*I + 72*d*x - (36*I)*Log[Cos[c + d*x]^2]) - (24*I)*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] - (6*I)*Cos[3*c + 4*d*x]*Log[Cos[c + d*x]^2] - (6*I)*Cos[5*c + 4*d*x]*Log[Cos[c + d*x]^2] + 75*Sin[c] - 70*Sin[c + 2*d*x] + 30*Sin[3*c + 2*d*x] - 25*Sin[3*c + 4*d*x]))/(12*d)","A",1
64,1,649,83,6.8430496,"\int \cos ^2(c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^5,x]","\frac{(4 \cos (3 c)-4 i \sin (3 c)) \sin (2 d x) \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{d (\cos (d x)+i \sin (d x))^5}-\frac{12 x \cos (5 c) \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{(\cos (d x)+i \sin (d x))^5}+\frac{(-4 \sin (3 c)-4 i \cos (3 c)) \cos (2 d x) \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{d (\cos (d x)+i \sin (d x))^5}+\frac{12 i x \sin (5 c) \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{(\cos (d x)+i \sin (d x))^5}+\frac{(5 \cos (5 c)-5 i \sin (5 c)) \sin (d x) \cos ^4(c+d x) (a+i a \tan (c+d x))^5}{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) (\cos (d x)+i \sin (d x))^5}+\frac{\left(\frac{1}{2} \sin (5 c)+\frac{1}{2} i \cos (5 c)\right) \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{d (\cos (d x)+i \sin (d x))^5}+\frac{x \cos ^5(c+d x) \left(36 i \sin ^5(c)+24 i \sin ^3(c)-6 \cos ^5(c)+6 \cos ^3(c)+6 \sin ^5(c) \tan (c)+6 \sin ^3(c) \tan (c)+36 i \sin (c) \cos ^4(c)+90 \sin ^2(c) \cos ^3(c)-120 i \sin ^3(c) \cos ^2(c)-24 i \sin (c) \cos ^2(c)-90 \sin ^4(c) \cos (c)-36 \sin ^2(c) \cos (c)-i \tan (c) (12 \cos (5 c)-12 i \sin (5 c))\right) (a+i a \tan (c+d x))^5}{(\cos (d x)+i \sin (d x))^5}+\frac{6 i \cos (5 c) \cos ^5(c+d x) (a+i a \tan (c+d x))^5 \log \left(\cos ^2(c+d x)\right)}{d (\cos (d x)+i \sin (d x))^5}+\frac{6 \sin (5 c) \cos ^5(c+d x) (a+i a \tan (c+d x))^5 \log \left(\cos ^2(c+d x)\right)}{d (\cos (d x)+i \sin (d x))^5}","-\frac{8 i a^6}{d (a-i a \tan (c+d x))}+\frac{i a^5 \tan ^2(c+d x)}{2 d}+\frac{5 a^5 \tan (c+d x)}{d}+\frac{12 i a^5 \log (\cos (c+d x))}{d}-12 a^5 x",1,"(-12*x*Cos[5*c]*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^5)/(Cos[d*x] + I*Sin[d*x])^5 + ((6*I)*Cos[5*c]*Cos[c + d*x]^5*Log[Cos[c + d*x]^2]*(a + I*a*Tan[c + d*x])^5)/(d*(Cos[d*x] + I*Sin[d*x])^5) + (Cos[2*d*x]*Cos[c + d*x]^5*((-4*I)*Cos[3*c] - 4*Sin[3*c])*(a + I*a*Tan[c + d*x])^5)/(d*(Cos[d*x] + I*Sin[d*x])^5) + (Cos[c + d*x]^3*((I/2)*Cos[5*c] + Sin[5*c]/2)*(a + I*a*Tan[c + d*x])^5)/(d*(Cos[d*x] + I*Sin[d*x])^5) + ((12*I)*x*Cos[c + d*x]^5*Sin[5*c]*(a + I*a*Tan[c + d*x])^5)/(Cos[d*x] + I*Sin[d*x])^5 + (6*Cos[c + d*x]^5*Log[Cos[c + d*x]^2]*Sin[5*c]*(a + I*a*Tan[c + d*x])^5)/(d*(Cos[d*x] + I*Sin[d*x])^5) + (Cos[c + d*x]^4*(5*Cos[5*c] - (5*I)*Sin[5*c])*Sin[d*x]*(a + I*a*Tan[c + d*x])^5)/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^5) + (Cos[c + d*x]^5*(4*Cos[3*c] - (4*I)*Sin[3*c])*Sin[2*d*x]*(a + I*a*Tan[c + d*x])^5)/(d*(Cos[d*x] + I*Sin[d*x])^5) + (x*Cos[c + d*x]^5*(6*Cos[c]^3 - 6*Cos[c]^5 - (24*I)*Cos[c]^2*Sin[c] + (36*I)*Cos[c]^4*Sin[c] - 36*Cos[c]*Sin[c]^2 + 90*Cos[c]^3*Sin[c]^2 + (24*I)*Sin[c]^3 - (120*I)*Cos[c]^2*Sin[c]^3 - 90*Cos[c]*Sin[c]^4 + (36*I)*Sin[c]^5 + 6*Sin[c]^3*Tan[c] + 6*Sin[c]^5*Tan[c] - I*(12*Cos[5*c] - (12*I)*Sin[5*c])*Tan[c])*(a + I*a*Tan[c + d*x])^5)/(Cos[d*x] + I*Sin[d*x])^5","B",1
65,1,110,73,0.7747634,"\int \cos ^4(c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^5,x]","\frac{a^5 (\cos (2 c+7 d x)+i \sin (2 c+7 d x)) \left(\cos (2 (c+d x)) \left(-i \log \left(\cos ^2(c+d x)\right)+2 d x-i\right)+\sin (2 (c+d x)) \left(-\log \left(\cos ^2(c+d x)\right)-2 i d x+1\right)+2 i\right)}{2 d (\cos (d x)+i \sin (d x))^5}","-\frac{2 i a^7}{d (a-i a \tan (c+d x))^2}+\frac{4 i a^6}{d (a-i a \tan (c+d x))}-\frac{i a^5 \log (\cos (c+d x))}{d}+a^5 x",1,"(a^5*(2*I + Cos[2*(c + d*x)]*(-I + 2*d*x - I*Log[Cos[c + d*x]^2]) + (1 - (2*I)*d*x - Log[Cos[c + d*x]^2])*Sin[2*(c + d*x)])*(Cos[2*c + 7*d*x] + I*Sin[2*c + 7*d*x]))/(2*d*(Cos[d*x] + I*Sin[d*x])^5)","A",1
66,1,50,55,0.5621752,"\int \cos ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^5,x]","\frac{a^5 (5 \cos (c+d x)-i \sin (c+d x)) (\sin (5 (c+d x))-i \cos (5 (c+d x)))}{24 d}","\frac{i a^7}{2 d (a-i a \tan (c+d x))^2}-\frac{2 i a^8}{3 d (a-i a \tan (c+d x))^3}",1,"(a^5*(5*Cos[c + d*x] - I*Sin[c + d*x])*((-I)*Cos[5*(c + d*x)] + Sin[5*(c + d*x)]))/(24*d)","A",1
67,1,73,27,1.1142231,"\int \cos ^8(c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Cos[c + d*x]^8*(a + I*a*Tan[c + d*x])^5,x]","\frac{a^5 (-i (2 \sin (c+d x)+3 \sin (3 (c+d x)))+10 \cos (c+d x)+5 \cos (3 (c+d x))) (\sin (5 (c+d x))-i \cos (5 (c+d x)))}{64 d}","-\frac{i a^9}{4 d (a-i a \tan (c+d x))^4}",1,"(a^5*(10*Cos[c + d*x] + 5*Cos[3*(c + d*x)] - I*(2*Sin[c + d*x] + 3*Sin[3*(c + d*x)]))*((-I)*Cos[5*(c + d*x)] + Sin[5*(c + d*x)]))/(64*d)","B",1
68,1,137,144,1.249361,"\int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Cos[c + d*x]^10*(a + I*a*Tan[c + d*x])^5,x]","\frac{a^5 (-100 \sin (c+d x)-225 \sin (3 (c+d x))-120 i d x \sin (5 (c+d x))+12 \sin (5 (c+d x))-500 i \cos (c+d x)-375 i \cos (3 (c+d x))+120 d x \cos (5 (c+d x))-12 i \cos (5 (c+d x))) (\cos (5 (c+2 d x))+i \sin (5 (c+2 d x)))}{3840 d (\cos (d x)+i \sin (d x))^5}","-\frac{i a^{10}}{10 d (a-i a \tan (c+d x))^5}-\frac{i a^9}{16 d (a-i a \tan (c+d x))^4}-\frac{i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac{i a^7}{32 d (a-i a \tan (c+d x))^2}-\frac{i a^6}{32 d (a-i a \tan (c+d x))}+\frac{a^5 x}{32}",1,"(a^5*((-500*I)*Cos[c + d*x] - (375*I)*Cos[3*(c + d*x)] - (12*I)*Cos[5*(c + d*x)] + 120*d*x*Cos[5*(c + d*x)] - 100*Sin[c + d*x] - 225*Sin[3*(c + d*x)] + 12*Sin[5*(c + d*x)] - (120*I)*d*x*Sin[5*(c + d*x)])*(Cos[5*(c + 2*d*x)] + I*Sin[5*(c + 2*d*x)]))/(3840*d*(Cos[d*x] + I*Sin[d*x])^5)","A",1
69,1,159,198,3.1314561,"\int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Cos[c + d*x]^12*(a + I*a*Tan[c + d*x])^5,x]","\frac{a^5 (-350 \sin (c+d x)-945 \sin (3 (c+d x))-840 i d x \sin (5 (c+d x))+84 \sin (5 (c+d x))+70 \sin (7 (c+d x))-1750 i \cos (c+d x)-1575 i \cos (3 (c+d x))+840 d x \cos (5 (c+d x))-84 i \cos (5 (c+d x))+50 i \cos (7 (c+d x))) (\cos (5 (c+2 d x))+i \sin (5 (c+2 d x)))}{15360 d (\cos (d x)+i \sin (d x))^5}","-\frac{i a^{11}}{24 d (a-i a \tan (c+d x))^6}-\frac{i a^{10}}{20 d (a-i a \tan (c+d x))^5}-\frac{3 i a^9}{64 d (a-i a \tan (c+d x))^4}-\frac{i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac{5 i a^7}{128 d (a-i a \tan (c+d x))^2}-\frac{3 i a^6}{64 d (a-i a \tan (c+d x))}+\frac{i a^6}{128 d (a+i a \tan (c+d x))}+\frac{7 a^5 x}{128}",1,"(a^5*((-1750*I)*Cos[c + d*x] - (1575*I)*Cos[3*(c + d*x)] - (84*I)*Cos[5*(c + d*x)] + 840*d*x*Cos[5*(c + d*x)] + (50*I)*Cos[7*(c + d*x)] - 350*Sin[c + d*x] - 945*Sin[3*(c + d*x)] + 84*Sin[5*(c + d*x)] - (840*I)*d*x*Sin[5*(c + d*x)] + 70*Sin[7*(c + d*x)])*(Cos[5*(c + 2*d*x)] + I*Sin[5*(c + 2*d*x)]))/(15360*d*(Cos[d*x] + I*Sin[d*x])^5)","A",1
70,1,115,167,1.275667,"\int \sec (c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Sec[c + d*x]*(a + I*a*Tan[c + d*x])^5,x]","\frac{a^5 (\cos (5 d x)+i \sin (5 d x)) \left(5040 \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)+i \sec ^5(c+d x) (450 i \sin (2 (c+d x))+325 i \sin (4 (c+d x))+1920 \cos (2 (c+d x))+640 \cos (4 (c+d x))+1344)\right)}{320 d (\cos (d x)+i \sin (d x))^5}","\frac{63 i a^5 \sec (c+d x)}{8 d}+\frac{63 a^5 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{21 i \sec (c+d x) \left(a^5+i a^5 \tan (c+d x)\right)}{8 d}+\frac{9 i a^2 \sec (c+d x) (a+i a \tan (c+d x))^3}{20 d}+\frac{21 i a \sec (c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{20 d}+\frac{i a \sec (c+d x) (a+i a \tan (c+d x))^4}{5 d}",1,"(a^5*(Cos[5*d*x] + I*Sin[5*d*x])*(5040*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]] + I*Sec[c + d*x]^5*(1344 + 1920*Cos[2*(c + d*x)] + 640*Cos[4*(c + d*x)] + (450*I)*Sin[2*(c + d*x)] + (325*I)*Sin[4*(c + d*x)])))/(320*d*(Cos[d*x] + I*Sin[d*x])^5)","A",1
71,1,151,130,1.6948299,"\int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Cos[c + d*x]*(a + I*a*Tan[c + d*x])^5,x]","\frac{a^5 \cos ^2(c+d x) (\tan (c+d x)-i)^5 \left((\cos (4 c-d x)-i \sin (4 c-d x)) (-i (49 \sin (c+d x)+57 \sin (3 (c+d x)))+511 \cos (c+d x)+153 \cos (3 (c+d x)))-840 i (\cos (5 c)-i \sin (5 c)) \cos ^3(c+d x) \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)\right)}{24 d (\cos (d x)+i \sin (d x))^5}","-\frac{35 i a^5 \sec (c+d x)}{2 d}-\frac{35 a^5 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{35 i \sec (c+d x) \left(a^5+i a^5 \tan (c+d x)\right)}{6 d}-\frac{7 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}-\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}",1,"(a^5*Cos[c + d*x]^2*((-840*I)*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]]*Cos[c + d*x]^3*(Cos[5*c] - I*Sin[5*c]) + (Cos[4*c - d*x] - I*Sin[4*c - d*x])*(511*Cos[c + d*x] + 153*Cos[3*(c + d*x)] - I*(49*Sin[c + d*x] + 57*Sin[3*(c + d*x)])))*(-I + Tan[c + d*x])^5)/(24*d*(Cos[d*x] + I*Sin[d*x])^5)","A",1
72,1,130,98,1.8773354,"\int \cos ^3(c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^5,x]","\frac{a^5 \cos ^4(c+d x) (\tan (c+d x)-i)^5 \left(30 (\sin (5 c)+i \cos (5 c)) \cos (c+d x) \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)-(\cos (3 c-2 d x)-i \sin (3 c-2 d x)) (-17 i \sin (2 (c+d x))+13 \cos (2 (c+d x))+10)\right)}{3 d (\cos (d x)+i \sin (d x))^5}","\frac{5 i a^5 \sec (c+d x)}{d}+\frac{5 a^5 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{10 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^2}{3 d}-\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^4}{3 d}",1,"(a^5*Cos[c + d*x]^4*(30*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]]*Cos[c + d*x]*(I*Cos[5*c] + Sin[5*c]) - (Cos[3*c - 2*d*x] - I*Sin[3*c - 2*d*x])*(10 + 13*Cos[2*(c + d*x)] - (17*I)*Sin[2*(c + d*x)]))*(-I + Tan[c + d*x])^5)/(3*d*(Cos[d*x] + I*Sin[d*x])^5)","A",1
73,1,31,32,0.1643071,"\int \cos ^5(c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^5,x]","-\frac{i a^5 (\cos (c+d x)+i \sin (c+d x))^5}{5 d}","-\frac{i \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}",1,"((-1/5*I)*a^5*(Cos[c + d*x] + I*Sin[c + d*x])^5)/d","A",1
74,1,55,101,0.9950612,"\int \cos ^7(c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^5,x]","\frac{a^5 (-10 i \sin (2 (c+d x))+25 \cos (2 (c+d x))+21) (\sin (5 (c+d x))-i \cos (5 (c+d x)))}{210 d}","-\frac{2 i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{105 d}-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d}-\frac{2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{35 d}",1,"(a^5*(21 + 25*Cos[2*(c + d*x)] - (10*I)*Sin[2*(c + d*x)])*((-I)*Cos[5*(c + d*x)] + Sin[5*(c + d*x)]))/(210*d)","A",1
75,1,94,141,0.9782407,"\int \cos ^9(c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^5,x]","\frac{a^5 (-120 i \sin (2 (c+d x))-140 i \sin (4 (c+d x))+300 \cos (2 (c+d x))+175 \cos (4 (c+d x))+189) (\sin (5 (c+2 d x))-i \cos (5 (c+2 d x)))}{2520 d (\cos (d x)+i \sin (d x))^5}","\frac{a^5 \sin ^5(c+d x)}{105 d}-\frac{2 a^5 \sin ^3(c+d x)}{63 d}+\frac{a^5 \sin (c+d x)}{21 d}-\frac{i a^5 \cos ^5(c+d x)}{105 d}-\frac{2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d}",1,"(a^5*(189 + 300*Cos[2*(c + d*x)] + 175*Cos[4*(c + d*x)] - (120*I)*Sin[2*(c + d*x)] - (140*I)*Sin[4*(c + d*x)])*((-I)*Cos[5*(c + 2*d*x)] + Sin[5*(c + 2*d*x)]))/(2520*d*(Cos[d*x] + I*Sin[d*x])^5)","A",1
76,1,118,159,1.4278196,"\int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^5 \, dx","Integrate[Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^5,x]","\frac{i a^5 (330 i \sin (2 (c+d x))+616 i \sin (4 (c+d x))-126 i \sin (6 (c+d x))-825 \cos (2 (c+d x))-770 \cos (4 (c+d x))+105 \cos (6 (c+d x))-462) (\cos (5 (c+2 d x))+i \sin (5 (c+2 d x)))}{7392 d (\cos (d x)+i \sin (d x))^5}","-\frac{5 a^5 \sin ^7(c+d x)}{231 d}+\frac{a^5 \sin ^5(c+d x)}{11 d}-\frac{5 a^5 \sin ^3(c+d x)}{33 d}+\frac{5 a^5 \sin (c+d x)}{33 d}-\frac{5 i a^5 \cos ^7(c+d x)}{231 d}-\frac{2 i a^3 \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{33 d}-\frac{2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4}{11 d}",1,"((I/7392)*a^5*(-462 - 825*Cos[2*(c + d*x)] - 770*Cos[4*(c + d*x)] + 105*Cos[6*(c + d*x)] + (330*I)*Sin[2*(c + d*x)] + (616*I)*Sin[4*(c + d*x)] - (126*I)*Sin[6*(c + d*x)])*(Cos[5*(c + 2*d*x)] + I*Sin[5*(c + 2*d*x)]))/(d*(Cos[d*x] + I*Sin[d*x])^5)","A",1
77,1,245,109,9.5010163,"\int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 \sec (c) \sec ^{15}(c+d x) (-6435 \sin (2 c+d x)+5005 \sin (2 c+3 d x)-5005 \sin (4 c+3 d x)+3003 \sin (4 c+5 d x)-3003 \sin (6 c+5 d x)+1365 \sin (6 c+7 d x)-1365 \sin (8 c+7 d x)+910 \sin (8 c+9 d x)+210 \sin (10 c+11 d x)+30 \sin (12 c+13 d x)+2 \sin (14 c+15 d x)+6435 i \cos (2 c+d x)+5005 i \cos (2 c+3 d x)+5005 i \cos (4 c+3 d x)+3003 i \cos (4 c+5 d x)+3003 i \cos (6 c+5 d x)+1365 i \cos (6 c+7 d x)+1365 i \cos (8 c+7 d x)+6435 \sin (d x)+6435 i \cos (d x))}{10920 d}","\frac{i (a+i a \tan (c+d x))^{15}}{15 a^7 d}-\frac{3 i (a+i a \tan (c+d x))^{14}}{7 a^6 d}+\frac{12 i (a+i a \tan (c+d x))^{13}}{13 a^5 d}-\frac{2 i (a+i a \tan (c+d x))^{12}}{3 a^4 d}",1,"(a^8*Sec[c]*Sec[c + d*x]^15*((6435*I)*Cos[d*x] + (6435*I)*Cos[2*c + d*x] + (5005*I)*Cos[2*c + 3*d*x] + (5005*I)*Cos[4*c + 3*d*x] + (3003*I)*Cos[4*c + 5*d*x] + (3003*I)*Cos[6*c + 5*d*x] + (1365*I)*Cos[6*c + 7*d*x] + (1365*I)*Cos[8*c + 7*d*x] + 6435*Sin[d*x] - 6435*Sin[2*c + d*x] + 5005*Sin[2*c + 3*d*x] - 5005*Sin[4*c + 3*d*x] + 3003*Sin[4*c + 5*d*x] - 3003*Sin[6*c + 5*d*x] + 1365*Sin[6*c + 7*d*x] - 1365*Sin[8*c + 7*d*x] + 910*Sin[8*c + 9*d*x] + 210*Sin[10*c + 11*d*x] + 30*Sin[12*c + 13*d*x] + 2*Sin[14*c + 15*d*x]))/(10920*d)","B",1
78,1,234,82,7.0439449,"\int \sec ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 \sec (c) \sec ^{13}(c+d x) (-1716 \sin (2 c+d x)+1287 \sin (2 c+3 d x)-1287 \sin (4 c+3 d x)+715 \sin (4 c+5 d x)-715 \sin (6 c+5 d x)+286 \sin (6 c+7 d x)-286 \sin (8 c+7 d x)+156 \sin (8 c+9 d x)+26 \sin (10 c+11 d x)+2 \sin (12 c+13 d x)+1716 i \cos (2 c+d x)+1287 i \cos (2 c+3 d x)+1287 i \cos (4 c+3 d x)+715 i \cos (4 c+5 d x)+715 i \cos (6 c+5 d x)+286 i \cos (6 c+7 d x)+286 i \cos (8 c+7 d x)+1716 \sin (d x)+1716 i \cos (d x))}{1716 d}","-\frac{i (a+i a \tan (c+d x))^{13}}{13 a^5 d}+\frac{i (a+i a \tan (c+d x))^{12}}{3 a^4 d}-\frac{4 i (a+i a \tan (c+d x))^{11}}{11 a^3 d}",1,"(a^8*Sec[c]*Sec[c + d*x]^13*((1716*I)*Cos[d*x] + (1716*I)*Cos[2*c + d*x] + (1287*I)*Cos[2*c + 3*d*x] + (1287*I)*Cos[4*c + 3*d*x] + (715*I)*Cos[4*c + 5*d*x] + (715*I)*Cos[6*c + 5*d*x] + (286*I)*Cos[6*c + 7*d*x] + (286*I)*Cos[8*c + 7*d*x] + 1716*Sin[d*x] - 1716*Sin[2*c + d*x] + 1287*Sin[2*c + 3*d*x] - 1287*Sin[4*c + 3*d*x] + 715*Sin[4*c + 5*d*x] - 715*Sin[6*c + 5*d*x] + 286*Sin[6*c + 7*d*x] - 286*Sin[8*c + 7*d*x] + 156*Sin[8*c + 9*d*x] + 26*Sin[10*c + 11*d*x] + 2*Sin[12*c + 13*d*x]))/(1716*d)","B",1
79,1,223,55,4.9423068,"\int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 \sec (c) \sec ^{11}(c+d x) (-462 \sin (2 c+d x)+330 \sin (2 c+3 d x)-330 \sin (4 c+3 d x)+165 \sin (4 c+5 d x)-165 \sin (6 c+5 d x)+55 \sin (6 c+7 d x)-55 \sin (8 c+7 d x)+22 \sin (8 c+9 d x)+2 \sin (10 c+11 d x)+462 i \cos (2 c+d x)+330 i \cos (2 c+3 d x)+330 i \cos (4 c+3 d x)+165 i \cos (4 c+5 d x)+165 i \cos (6 c+5 d x)+55 i \cos (6 c+7 d x)+55 i \cos (8 c+7 d x)+462 \sin (d x)+462 i \cos (d x))}{220 d}","\frac{i (a+i a \tan (c+d x))^{11}}{11 a^3 d}-\frac{i (a+i a \tan (c+d x))^{10}}{5 a^2 d}",1,"(a^8*Sec[c]*Sec[c + d*x]^11*((462*I)*Cos[d*x] + (462*I)*Cos[2*c + d*x] + (330*I)*Cos[2*c + 3*d*x] + (330*I)*Cos[4*c + 3*d*x] + (165*I)*Cos[4*c + 5*d*x] + (165*I)*Cos[6*c + 5*d*x] + (55*I)*Cos[6*c + 7*d*x] + (55*I)*Cos[8*c + 7*d*x] + 462*Sin[d*x] - 462*Sin[2*c + d*x] + 330*Sin[2*c + 3*d*x] - 330*Sin[4*c + 3*d*x] + 165*Sin[4*c + 5*d*x] - 165*Sin[6*c + 5*d*x] + 55*Sin[6*c + 7*d*x] - 55*Sin[8*c + 7*d*x] + 22*Sin[8*c + 9*d*x] + 2*Sin[10*c + 11*d*x]))/(220*d)","B",1
80,1,212,27,3.6501669,"\int \sec ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 \sec (c) \sec ^9(c+d x) (-126 \sin (2 c+d x)+84 \sin (2 c+3 d x)-84 \sin (4 c+3 d x)+36 \sin (4 c+5 d x)-36 \sin (6 c+5 d x)+9 \sin (6 c+7 d x)-9 \sin (8 c+7 d x)+2 \sin (8 c+9 d x)+126 i \cos (2 c+d x)+84 i \cos (2 c+3 d x)+84 i \cos (4 c+3 d x)+36 i \cos (4 c+5 d x)+36 i \cos (6 c+5 d x)+9 i \cos (6 c+7 d x)+9 i \cos (8 c+7 d x)+126 \sin (d x)+126 i \cos (d x))}{18 d}","-\frac{i (a+i a \tan (c+d x))^9}{9 a d}",1,"(a^8*Sec[c]*Sec[c + d*x]^9*((126*I)*Cos[d*x] + (126*I)*Cos[2*c + d*x] + (84*I)*Cos[2*c + 3*d*x] + (84*I)*Cos[4*c + 3*d*x] + (36*I)*Cos[4*c + 5*d*x] + (36*I)*Cos[6*c + 5*d*x] + (9*I)*Cos[6*c + 7*d*x] + (9*I)*Cos[8*c + 7*d*x] + 126*Sin[d*x] - 126*Sin[2*c + d*x] + 84*Sin[2*c + 3*d*x] - 84*Sin[4*c + 3*d*x] + 36*Sin[4*c + 5*d*x] - 36*Sin[6*c + 5*d*x] + 9*Sin[6*c + 7*d*x] - 9*Sin[8*c + 7*d*x] + 2*Sin[8*c + 9*d*x]))/(18*d)","B",1
81,1,383,200,4.990534,"\int (a+i a \tan (c+d x))^8 \, dx","Integrate[(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 \sec (c) \sec ^7(c+d x) \left(70 \cos (d x) \left(-105 i \log \left(\cos ^2(c+d x)\right)+210 d x-139 i\right)+70 \cos (2 c+d x) \left(-105 i \log \left(\cos ^2(c+d x)\right)+210 d x-139 i\right)+3 \left(5740 \sin (2 c+d x)-4963 \sin (2 c+3 d x)+2660 \sin (4 c+3 d x)-1981 \sin (4 c+5 d x)+560 \sin (6 c+5 d x)-363 \sin (6 c+7 d x)+980 d x \cos (4 c+5 d x)-420 i \cos (4 c+5 d x)+980 d x \cos (6 c+5 d x)-420 i \cos (6 c+5 d x)+140 d x \cos (6 c+7 d x)+140 d x \cos (8 c+7 d x)-490 i \cos (4 c+5 d x) \log \left(\cos ^2(c+d x)\right)+70 \cos (2 c+3 d x) \left(-21 i \log \left(\cos ^2(c+d x)\right)+42 d x-25 i\right)+70 \cos (4 c+3 d x) \left(-21 i \log \left(\cos ^2(c+d x)\right)+42 d x-25 i\right)-490 i \cos (6 c+5 d x) \log \left(\cos ^2(c+d x)\right)-70 i \cos (6 c+7 d x) \log \left(\cos ^2(c+d x)\right)-70 i \cos (8 c+7 d x) \log \left(\cos ^2(c+d x)\right)-6965 \sin (d x)\right)\right)}{420 d}","-\frac{64 a^8 \tan (c+d x)}{d}-\frac{128 i a^8 \log (\cos (c+d x))}{d}+128 a^8 x+\frac{16 i \left(a^4+i a^4 \tan (c+d x)\right)^2}{d}+\frac{4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac{i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac{16 i a^2 \left(a^2+i a^2 \tan (c+d x)\right)^3}{3 d}+\frac{2 i \left(a^2+i a^2 \tan (c+d x)\right)^4}{d}+\frac{i a (a+i a \tan (c+d x))^7}{7 d}",1,"(a^8*Sec[c]*Sec[c + d*x]^7*(70*Cos[d*x]*(-139*I + 210*d*x - (105*I)*Log[Cos[c + d*x]^2]) + 70*Cos[2*c + d*x]*(-139*I + 210*d*x - (105*I)*Log[Cos[c + d*x]^2]) + 3*((-420*I)*Cos[4*c + 5*d*x] + 980*d*x*Cos[4*c + 5*d*x] - (420*I)*Cos[6*c + 5*d*x] + 980*d*x*Cos[6*c + 5*d*x] + 140*d*x*Cos[6*c + 7*d*x] + 140*d*x*Cos[8*c + 7*d*x] + 70*Cos[2*c + 3*d*x]*(-25*I + 42*d*x - (21*I)*Log[Cos[c + d*x]^2]) + 70*Cos[4*c + 3*d*x]*(-25*I + 42*d*x - (21*I)*Log[Cos[c + d*x]^2]) - (490*I)*Cos[4*c + 5*d*x]*Log[Cos[c + d*x]^2] - (490*I)*Cos[6*c + 5*d*x]*Log[Cos[c + d*x]^2] - (70*I)*Cos[6*c + 7*d*x]*Log[Cos[c + d*x]^2] - (70*I)*Cos[8*c + 7*d*x]*Log[Cos[c + d*x]^2] - 6965*Sin[d*x] + 5740*Sin[2*c + d*x] - 4963*Sin[2*c + 3*d*x] + 2660*Sin[4*c + 3*d*x] - 1981*Sin[4*c + 5*d*x] + 560*Sin[6*c + 5*d*x] - 363*Sin[6*c + 7*d*x])))/(420*d)","A",1
82,1,321,133,7.0072036,"\int \cos ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^8,x]","\frac{\cos ^3(c+d x) (a+i a \tan (c+d x))^8 \left(-960 d x \cos (8 c) \cos ^5(c+d x)-160 i (\cos (6 c)-i \sin (6 c)) \cos (2 d x) \cos ^5(c+d x)+960 i d x \sin (8 c) \cos ^5(c+d x)+160 (\cos (6 c)-i \sin (6 c)) \sin (2 d x) \cos ^5(c+d x)+480 i \cos (8 c) \cos ^5(c+d x) \log \left(\cos ^2(c+d x)\right)+696 \sec (c) (\cos (8 c)-i \sin (8 c)) \sin (d x) \cos ^4(c+d x)-4 (13 \tan (c)-50 i) (\cos (8 c)-i \sin (8 c)) \cos ^3(c+d x)-52 \sec (c) (\cos (8 c)-i \sin (8 c)) \sin (d x) \cos ^2(c+d x)+480 \sin (8 c) \cos ^5(c+d x) \log \left(\cos ^2(c+d x)\right)+(\tan (c)-10 i) (\cos (8 c)-i \sin (8 c)) \cos (c+d x)+\sec (c) (\cos (8 c)-i \sin (8 c)) \sin (d x)\right)}{5 d (\cos (d x)+i \sin (d x))^8}","-\frac{64 i a^9}{d (a-i a \tan (c+d x))}+\frac{a^8 \tan ^5(c+d x)}{5 d}-\frac{2 i a^8 \tan ^4(c+d x)}{d}-\frac{10 a^8 \tan ^3(c+d x)}{d}+\frac{36 i a^8 \tan ^2(c+d x)}{d}+\frac{129 a^8 \tan (c+d x)}{d}+\frac{192 i a^8 \log (\cos (c+d x))}{d}-192 a^8 x",1,"(Cos[c + d*x]^3*(-960*d*x*Cos[8*c]*Cos[c + d*x]^5 + (480*I)*Cos[8*c]*Cos[c + d*x]^5*Log[Cos[c + d*x]^2] - (160*I)*Cos[2*d*x]*Cos[c + d*x]^5*(Cos[6*c] - I*Sin[6*c]) + (960*I)*d*x*Cos[c + d*x]^5*Sin[8*c] + 480*Cos[c + d*x]^5*Log[Cos[c + d*x]^2]*Sin[8*c] + Sec[c]*(Cos[8*c] - I*Sin[8*c])*Sin[d*x] - 52*Cos[c + d*x]^2*Sec[c]*(Cos[8*c] - I*Sin[8*c])*Sin[d*x] + 696*Cos[c + d*x]^4*Sec[c]*(Cos[8*c] - I*Sin[8*c])*Sin[d*x] + 160*Cos[c + d*x]^5*(Cos[6*c] - I*Sin[6*c])*Sin[2*d*x] + Cos[c + d*x]*(Cos[8*c] - I*Sin[8*c])*(-10*I + Tan[c]) - 4*Cos[c + d*x]^3*(Cos[8*c] - I*Sin[8*c])*(-50*I + 13*Tan[c]))*(a + I*a*Tan[c + d*x])^8)/(5*d*(Cos[d*x] + I*Sin[d*x])^8)","B",1
83,1,566,124,2.7199975,"\int \cos ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 \sec (c) \sec ^3(c+d x) (\cos (2 (c+5 d x))+i \sin (2 (c+5 d x))) \left(-120 i d x \sin (2 c+d x)+87 \sin (2 c+d x)-180 i d x \sin (2 c+3 d x)-96 \sin (2 c+3 d x)-180 i d x \sin (4 c+3 d x)+45 \sin (4 c+3 d x)-60 i d x \sin (4 c+5 d x)-44 \sin (4 c+5 d x)-60 i d x \sin (6 c+5 d x)+3 \sin (6 c+5 d x)+180 d x \cos (2 c+3 d x)-66 i \cos (2 c+3 d x)+180 d x \cos (4 c+3 d x)+75 i \cos (4 c+3 d x)+60 d x \cos (4 c+5 d x)-50 i \cos (4 c+5 d x)+60 d x \cos (6 c+5 d x)-3 i \cos (6 c+5 d x)-90 i \cos (2 c+3 d x) \log \left(\cos ^2(c+d x)\right)+3 \cos (2 c+d x) \left(-40 i \log \left(\cos ^2(c+d x)\right)+80 d x+71 i\right)+\cos (d x) \left(-120 i \log \left(\cos ^2(c+d x)\right)+240 d x+119 i\right)-90 i \cos (4 c+3 d x) \log \left(\cos ^2(c+d x)\right)-30 i \cos (4 c+5 d x) \log \left(\cos ^2(c+d x)\right)-30 i \cos (6 c+5 d x) \log \left(\cos ^2(c+d x)\right)-60 \sin (d x) \log \left(\cos ^2(c+d x)\right)-60 \sin (2 c+d x) \log \left(\cos ^2(c+d x)\right)-90 \sin (2 c+3 d x) \log \left(\cos ^2(c+d x)\right)-90 \sin (4 c+3 d x) \log \left(\cos ^2(c+d x)\right)-30 \sin (4 c+5 d x) \log \left(\cos ^2(c+d x)\right)-30 \sin (6 c+5 d x) \log \left(\cos ^2(c+d x)\right)-120 i d x \sin (d x)-101 \sin (d x)\right)}{12 d (\cos (d x)+i \sin (d x))^8}","-\frac{16 i a^{10}}{d (a-i a \tan (c+d x))^2}+\frac{80 i a^9}{d (a-i a \tan (c+d x))}+\frac{a^8 \tan ^3(c+d x)}{3 d}-\frac{4 i a^8 \tan ^2(c+d x)}{d}-\frac{31 a^8 \tan (c+d x)}{d}-\frac{80 i a^8 \log (\cos (c+d x))}{d}+80 a^8 x",1,"(a^8*Sec[c]*Sec[c + d*x]^3*(Cos[2*(c + 5*d*x)] + I*Sin[2*(c + 5*d*x)])*((-66*I)*Cos[2*c + 3*d*x] + 180*d*x*Cos[2*c + 3*d*x] + (75*I)*Cos[4*c + 3*d*x] + 180*d*x*Cos[4*c + 3*d*x] - (50*I)*Cos[4*c + 5*d*x] + 60*d*x*Cos[4*c + 5*d*x] - (3*I)*Cos[6*c + 5*d*x] + 60*d*x*Cos[6*c + 5*d*x] + 3*Cos[2*c + d*x]*(71*I + 80*d*x - (40*I)*Log[Cos[c + d*x]^2]) + Cos[d*x]*(119*I + 240*d*x - (120*I)*Log[Cos[c + d*x]^2]) - (90*I)*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]^2] - (90*I)*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] - (30*I)*Cos[4*c + 5*d*x]*Log[Cos[c + d*x]^2] - (30*I)*Cos[6*c + 5*d*x]*Log[Cos[c + d*x]^2] - 101*Sin[d*x] - (120*I)*d*x*Sin[d*x] - 60*Log[Cos[c + d*x]^2]*Sin[d*x] + 87*Sin[2*c + d*x] - (120*I)*d*x*Sin[2*c + d*x] - 60*Log[Cos[c + d*x]^2]*Sin[2*c + d*x] - 96*Sin[2*c + 3*d*x] - (180*I)*d*x*Sin[2*c + 3*d*x] - 90*Log[Cos[c + d*x]^2]*Sin[2*c + 3*d*x] + 45*Sin[4*c + 3*d*x] - (180*I)*d*x*Sin[4*c + 3*d*x] - 90*Log[Cos[c + d*x]^2]*Sin[4*c + 3*d*x] - 44*Sin[4*c + 5*d*x] - (60*I)*d*x*Sin[4*c + 5*d*x] - 30*Log[Cos[c + d*x]^2]*Sin[4*c + 5*d*x] + 3*Sin[6*c + 5*d*x] - (60*I)*d*x*Sin[6*c + 5*d*x] - 30*Log[Cos[c + d*x]^2]*Sin[6*c + 5*d*x]))/(12*d*(Cos[d*x] + I*Sin[d*x])^8)","B",1
84,1,414,114,2.70857,"\int \cos ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^8,x]","-\frac{a^8 \sec (c) \sec (c+d x) (\cos (3 c+11 d x)+i \sin (3 c+11 d x)) \left(-12 i d x \sin (c+2 d x)+11 \sin (c+2 d x)-12 i d x \sin (3 c+2 d x)+14 \sin (3 c+2 d x)-12 i d x \sin (3 c+4 d x)-4 \sin (3 c+4 d x)-12 i d x \sin (5 c+4 d x)-\sin (5 c+4 d x)+12 d x \cos (3 c+2 d x)+10 i \cos (3 c+2 d x)+12 d x \cos (3 c+4 d x)-2 i \cos (3 c+4 d x)+12 d x \cos (5 c+4 d x)+i \cos (5 c+4 d x)+\cos (c+2 d x) \left(-6 i \log \left(\cos ^2(c+d x)\right)+12 d x+7 i\right)-6 i \cos (3 c+2 d x) \log \left(\cos ^2(c+d x)\right)-6 i \cos (3 c+4 d x) \log \left(\cos ^2(c+d x)\right)-6 i \cos (5 c+4 d x) \log \left(\cos ^2(c+d x)\right)-6 \sin (c+2 d x) \log \left(\cos ^2(c+d x)\right)-6 \sin (3 c+2 d x) \log \left(\cos ^2(c+d x)\right)-6 \sin (3 c+4 d x) \log \left(\cos ^2(c+d x)\right)-6 \sin (5 c+4 d x) \log \left(\cos ^2(c+d x)\right)+12 i \cos (c)\right)}{6 d (\cos (d x)+i \sin (d x))^8}","-\frac{16 i a^{11}}{3 d (a-i a \tan (c+d x))^3}+\frac{16 i a^{10}}{d (a-i a \tan (c+d x))^2}-\frac{24 i a^9}{d (a-i a \tan (c+d x))}+\frac{a^8 \tan (c+d x)}{d}+\frac{8 i a^8 \log (\cos (c+d x))}{d}-8 a^8 x",1,"-1/6*(a^8*Sec[c]*Sec[c + d*x]*((12*I)*Cos[c] + (10*I)*Cos[3*c + 2*d*x] + 12*d*x*Cos[3*c + 2*d*x] - (2*I)*Cos[3*c + 4*d*x] + 12*d*x*Cos[3*c + 4*d*x] + I*Cos[5*c + 4*d*x] + 12*d*x*Cos[5*c + 4*d*x] + Cos[c + 2*d*x]*(7*I + 12*d*x - (6*I)*Log[Cos[c + d*x]^2]) - (6*I)*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] - (6*I)*Cos[3*c + 4*d*x]*Log[Cos[c + d*x]^2] - (6*I)*Cos[5*c + 4*d*x]*Log[Cos[c + d*x]^2] + 11*Sin[c + 2*d*x] - (12*I)*d*x*Sin[c + 2*d*x] - 6*Log[Cos[c + d*x]^2]*Sin[c + 2*d*x] + 14*Sin[3*c + 2*d*x] - (12*I)*d*x*Sin[3*c + 2*d*x] - 6*Log[Cos[c + d*x]^2]*Sin[3*c + 2*d*x] - 4*Sin[3*c + 4*d*x] - (12*I)*d*x*Sin[3*c + 4*d*x] - 6*Log[Cos[c + d*x]^2]*Sin[3*c + 4*d*x] - Sin[5*c + 4*d*x] - (12*I)*d*x*Sin[5*c + 4*d*x] - 6*Log[Cos[c + d*x]^2]*Sin[5*c + 4*d*x])*(Cos[3*c + 11*d*x] + I*Sin[3*c + 11*d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^8)","B",1
85,1,31,43,0.3289137,"\int \cos ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^8*(a + I*a*Tan[c + d*x])^8,x]","-\frac{i a^8 (\cos (c+d x)+i \sin (c+d x))^8}{8 d}","-\frac{i \left(a^3+i a^3 \tan (c+d x)\right)^4}{8 d (a-i a \tan (c+d x))^4}",1,"((-1/8*I)*a^8*(Cos[c + d*x] + I*Sin[c + d*x])^8)/d","A",1
86,1,55,80,1.4547391,"\int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^10*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 (-4 i \sin (2 (c+d x))+16 \cos (2 (c+d x))+15) (\sin (8 (c+d x))-i \cos (8 (c+d x)))}{240 d}","-\frac{4 i a^{13}}{5 d (a-i a \tan (c+d x))^5}+\frac{i a^{12}}{d (a-i a \tan (c+d x))^4}-\frac{i a^{11}}{3 d (a-i a \tan (c+d x))^3}",1,"(a^8*(15 + 16*Cos[2*(c + d*x)] - (4*I)*Sin[2*(c + d*x)])*((-I)*Cos[8*(c + d*x)] + Sin[8*(c + d*x)]))/(240*d)","A",1
87,1,77,55,1.4678632,"\int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^12*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 (-16 i \sin (2 (c+d x))-10 i \sin (4 (c+d x))+64 \cos (2 (c+d x))+20 \cos (4 (c+d x))+45) (\sin (8 (c+d x))-i \cos (8 (c+d x)))}{960 d}","\frac{i a^{13}}{5 d (a-i a \tan (c+d x))^5}-\frac{i a^{14}}{3 d (a-i a \tan (c+d x))^6}",1,"(a^8*(45 + 64*Cos[2*(c + d*x)] + 20*Cos[4*(c + d*x)] - (16*I)*Sin[2*(c + d*x)] - (10*I)*Sin[4*(c + d*x)])*((-I)*Cos[8*(c + d*x)] + Sin[8*(c + d*x)]))/(960*d)","A",1
88,1,116,27,2.0148869,"\int \cos ^{14}(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^14*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 (-14 i \sin (2 (c+d x))-14 i \sin (4 (c+d x))-6 i \sin (6 (c+d x))+56 \cos (2 (c+d x))+28 \cos (4 (c+d x))+8 \cos (6 (c+d x))+35) (\sin (8 (c+2 d x))-i \cos (8 (c+2 d x)))}{896 d (\cos (d x)+i \sin (d x))^8}","-\frac{i a^{15}}{7 d (a-i a \tan (c+d x))^7}",1,"(a^8*(35 + 56*Cos[2*(c + d*x)] + 28*Cos[4*(c + d*x)] + 8*Cos[6*(c + d*x)] - (14*I)*Sin[2*(c + d*x)] - (14*I)*Sin[4*(c + d*x)] - (6*I)*Sin[6*(c + d*x)])*((-I)*Cos[8*(c + 2*d*x)] + Sin[8*(c + 2*d*x)]))/(896*d*(Cos[d*x] + I*Sin[d*x])^8)","B",1
89,1,166,225,6.7671703,"\int \cos ^{16}(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^16*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 (-6272 \sin (2 (c+d x))-7840 \sin (4 (c+d x))-5760 \sin (6 (c+d x))-1680 i d x \sin (8 (c+d x))+105 \sin (8 (c+d x))-25088 i \cos (2 (c+d x))-15680 i \cos (4 (c+d x))-7680 i \cos (6 (c+d x))+1680 d x \cos (8 (c+d x))-105 i \cos (8 (c+d x))-14700 i) (\cos (8 (c+2 d x))+i \sin (8 (c+2 d x)))}{430080 d (\cos (d x)+i \sin (d x))^8}","-\frac{i a^{16}}{16 d (a-i a \tan (c+d x))^8}-\frac{i a^{15}}{28 d (a-i a \tan (c+d x))^7}-\frac{i a^{14}}{48 d (a-i a \tan (c+d x))^6}-\frac{i a^{13}}{80 d (a-i a \tan (c+d x))^5}-\frac{i a^{12}}{128 d (a-i a \tan (c+d x))^4}-\frac{i a^{11}}{192 d (a-i a \tan (c+d x))^3}-\frac{i a^{10}}{256 d (a-i a \tan (c+d x))^2}-\frac{i a^9}{256 d (a-i a \tan (c+d x))}+\frac{a^8 x}{256}",1,"(a^8*(-14700*I - (25088*I)*Cos[2*(c + d*x)] - (15680*I)*Cos[4*(c + d*x)] - (7680*I)*Cos[6*(c + d*x)] - (105*I)*Cos[8*(c + d*x)] + 1680*d*x*Cos[8*(c + d*x)] - 6272*Sin[2*(c + d*x)] - 7840*Sin[4*(c + d*x)] - 5760*Sin[6*(c + d*x)] + 105*Sin[8*(c + d*x)] - (1680*I)*d*x*Sin[8*(c + d*x)])*(Cos[8*(c + 2*d*x)] + I*Sin[8*(c + 2*d*x)]))/(430080*d*(Cos[d*x] + I*Sin[d*x])^8)","A",1
90,1,188,279,7.9743324,"\int \cos ^{18}(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^18*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 (-7056 \sin (2 (c+d x))-10080 \sin (4 (c+d x))-9720 \sin (6 (c+d x))-5040 i d x \sin (8 (c+d x))+315 \sin (8 (c+d x))+280 \sin (10 (c+d x))-28224 i \cos (2 (c+d x))-20160 i \cos (4 (c+d x))-12960 i \cos (6 (c+d x))+5040 d x \cos (8 (c+d x))-315 i \cos (8 (c+d x))+224 i \cos (10 (c+d x))-15876 i) (\cos (8 (c+2 d x))+i \sin (8 (c+2 d x)))}{516096 d (\cos (d x)+i \sin (d x))^8}","-\frac{i a^{17}}{36 d (a-i a \tan (c+d x))^9}-\frac{i a^{16}}{32 d (a-i a \tan (c+d x))^8}-\frac{3 i a^{15}}{112 d (a-i a \tan (c+d x))^7}-\frac{i a^{14}}{48 d (a-i a \tan (c+d x))^6}-\frac{i a^{13}}{64 d (a-i a \tan (c+d x))^5}-\frac{3 i a^{12}}{256 d (a-i a \tan (c+d x))^4}-\frac{7 i a^{11}}{768 d (a-i a \tan (c+d x))^3}-\frac{i a^{10}}{128 d (a-i a \tan (c+d x))^2}-\frac{9 i a^9}{1024 d (a-i a \tan (c+d x))}+\frac{i a^9}{1024 d (a+i a \tan (c+d x))}+\frac{5 a^8 x}{512}",1,"(a^8*(-15876*I - (28224*I)*Cos[2*(c + d*x)] - (20160*I)*Cos[4*(c + d*x)] - (12960*I)*Cos[6*(c + d*x)] - (315*I)*Cos[8*(c + d*x)] + 5040*d*x*Cos[8*(c + d*x)] + (224*I)*Cos[10*(c + d*x)] - 7056*Sin[2*(c + d*x)] - 10080*Sin[4*(c + d*x)] - 9720*Sin[6*(c + d*x)] + 315*Sin[8*(c + d*x)] - (5040*I)*d*x*Sin[8*(c + d*x)] + 280*Sin[10*(c + d*x)])*(Cos[8*(c + 2*d*x)] + I*Sin[8*(c + 2*d*x)]))/(516096*d*(Cos[d*x] + I*Sin[d*x])^8)","A",1
91,1,205,235,3.3766363,"\int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 (\cos (8 c)-i \sin (8 c)) \cos ^2(c+d x) (\tan (c+d x)-i)^8 \left(-658944 i \cos (c+d x)+5 (12870 \sin (c+d x)+22165 \sin (3 (c+d x))+10959 \sin (5 (c+d x))+1536 \sin (7 (c+d x))-73216 i \cos (3 (c+d x))-19968 i \cos (5 (c+d x))-1536 i \cos (7 (c+d x)))+720720 \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)}{3840 d (\cos (d x)+i \sin (d x))^8}","-\frac{3003 i a^8 \sec (c+d x)}{16 d}-\frac{3003 a^8 \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac{1001 i \sec (c+d x) \left(a^8+i a^8 \tan (c+d x)\right)}{16 d}-\frac{1001 i \sec (c+d x) \left(a^4+i a^4 \tan (c+d x)\right)^2}{40 d}-\frac{13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac{429 i a^2 \sec (c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^3}{40 d}-\frac{143 i \sec (c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^4}{30 d}-\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}",1,"(a^8*Cos[c + d*x]^2*(Cos[8*c] - I*Sin[8*c])*((-658944*I)*Cos[c + d*x] + 720720*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]]) + 5*((-73216*I)*Cos[3*(c + d*x)] - (19968*I)*Cos[5*(c + d*x)] - (1536*I)*Cos[7*(c + d*x)] + 12870*Sin[c + d*x] + 22165*Sin[3*(c + d*x)] + 10959*Sin[5*(c + d*x)] + 1536*Sin[7*(c + d*x)]))*(-I + Tan[c + d*x])^8)/(3840*d*(Cos[d*x] + I*Sin[d*x])^8)","A",1
92,1,1540,205,7.4500464,"\int \cos ^3(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^8,x]","\frac{\cos ^8(c+d x) (160 i \sin (7 c)-160 \cos (7 c)) \sin (d x) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8}+\frac{\cos ^8(c+d x) \left(\frac{32}{3} \cos (5 c)-\frac{32}{3} i \sin (5 c)\right) \sin (3 d x) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8}+\frac{i \cos ^8(c+d x) \left(\frac{236}{3} \cos (8 c)-\frac{236}{3} i \sin (8 c)\right) \sin \left(\frac{d x}{2}\right) (i \tan (c+d x) a+a)^8}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x))^8 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}-\frac{i \cos ^8(c+d x) \left(\frac{236}{3} \cos (8 c)-\frac{236}{3} i \sin (8 c)\right) \sin \left(\frac{d x}{2}\right) (i \tan (c+d x) a+a)^8}{d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x))^8 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^8(c+d x) \left((375-32 i) \sin \left(\frac{c}{2}\right)-(375+32 i) \cos \left(\frac{c}{2}\right)\right) \left(\frac{1}{48} \cos (8 c)-\frac{1}{48} i \sin (8 c)\right) (i \tan (c+d x) a+a)^8}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x))^8 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos ^8(c+d x) \left((375-32 i) \cos \left(\frac{c}{2}\right)+(375+32 i) \sin \left(\frac{c}{2}\right)\right) \left(\frac{1}{48} \cos (8 c)-\frac{1}{48} i \sin (8 c)\right) (i \tan (c+d x) a+a)^8}{d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x))^8 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}-\frac{i \cos ^8(c+d x) \left(\frac{4}{3} \cos (8 c)-\frac{4}{3} i \sin (8 c)\right) \sin \left(\frac{d x}{2}\right) (i \tan (c+d x) a+a)^8}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x))^8 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{i \cos ^8(c+d x) \left(\frac{4}{3} \cos (8 c)-\frac{4}{3} i \sin (8 c)\right) \sin \left(\frac{d x}{2}\right) (i \tan (c+d x) a+a)^8}{d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x))^8 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{\cos ^8(c+d x) \left(\frac{1}{16} \cos (8 c)-\frac{1}{16} i \sin (8 c)\right) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}+\frac{\cos ^8(c+d x) \left(\frac{1}{16} i \sin (8 c)-\frac{1}{16} \cos (8 c)\right) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}-\frac{1155 \cos (8 c) \cos ^8(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) (i \tan (c+d x) a+a)^8}{8 d (\cos (d x)+i \sin (d x))^8}+\frac{1155 \cos (8 c) \cos ^8(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) (i \tan (c+d x) a+a)^8}{8 d (\cos (d x)+i \sin (d x))^8}+\frac{\cos (3 d x) \cos ^8(c+d x) \left(-\frac{32}{3} i \cos (5 c)-\frac{32}{3} \sin (5 c)\right) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8}+\frac{\cos (d x) \cos ^8(c+d x) (160 i \cos (7 c)+160 \sin (7 c)) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8}+\frac{1155 i \cos ^8(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) \sin (8 c) (i \tan (c+d x) a+a)^8}{8 d (\cos (d x)+i \sin (d x))^8}-\frac{1155 i \cos ^8(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) \sin (8 c) (i \tan (c+d x) a+a)^8}{8 d (\cos (d x)+i \sin (d x))^8}+\frac{\cos ^8(c+d x) \sec (c) \left(\frac{236}{3} i \cos (8 c)+\frac{236}{3} \sin (8 c)\right) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8}","\frac{1155 i a^8 \sec (c+d x)}{8 d}+\frac{1155 a^8 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{385 i \sec (c+d x) \left(a^8+i a^8 \tan (c+d x)\right)}{8 d}+\frac{77 i \sec (c+d x) \left(a^4+i a^4 \tan (c+d x)\right)^2}{4 d}+\frac{22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}+\frac{33 i a^2 \sec (c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^3}{4 d}-\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}",1,"(-1155*Cos[8*c]*Cos[c + d*x]^8*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(a + I*a*Tan[c + d*x])^8)/(8*d*(Cos[d*x] + I*Sin[d*x])^8) + (1155*Cos[8*c]*Cos[c + d*x]^8*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(a + I*a*Tan[c + d*x])^8)/(8*d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[3*d*x]*Cos[c + d*x]^8*(((-32*I)/3)*Cos[5*c] - (32*Sin[5*c])/3)*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[d*x]*Cos[c + d*x]^8*((160*I)*Cos[7*c] + 160*Sin[7*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (((1155*I)/8)*Cos[c + d*x]^8*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sin[8*c]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) - (((1155*I)/8)*Cos[c + d*x]^8*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sin[8*c]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*Sec[c]*(((236*I)/3)*Cos[8*c] + (236*Sin[8*c])/3)*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*(-160*Cos[7*c] + (160*I)*Sin[7*c])*Sin[d*x]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*((32*Cos[5*c])/3 - ((32*I)/3)*Sin[5*c])*Sin[3*d*x]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*(Cos[8*c]/16 - (I/16)*Sin[8*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^4) - (I*Cos[c + d*x]^8*((4*Cos[8*c])/3 - ((4*I)/3)*Sin[8*c])*Sin[(d*x)/2]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] - Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^8*((-375 - 32*I)*Cos[c/2] + (375 - 32*I)*Sin[c/2])*(Cos[8*c]/48 - (I/48)*Sin[8*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] - Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (I*Cos[c + d*x]^8*((236*Cos[8*c])/3 - ((236*I)/3)*Sin[8*c])*Sin[(d*x)/2]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] - Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (Cos[c + d*x]^8*(-1/16*Cos[8*c] + (I/16)*Sin[8*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + (I*Cos[c + d*x]^8*((4*Cos[8*c])/3 - ((4*I)/3)*Sin[8*c])*Sin[(d*x)/2]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] + Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^8*((375 - 32*I)*Cos[c/2] + (375 + 32*I)*Sin[c/2])*(Cos[8*c]/48 - (I/48)*Sin[8*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] + Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) - (I*Cos[c + d*x]^8*((236*Cos[8*c])/3 - ((236*I)/3)*Sin[8*c])*Sin[(d*x)/2]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] + Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",0
93,1,1162,173,7.4905559,"\int \cos ^5(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^8,x]","\frac{\cos ^8(c+d x) (48 \cos (7 c)-48 i \sin (7 c)) \sin (d x) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8}+\frac{\cos ^8(c+d x) (8 i \sin (5 c)-8 \cos (5 c)) \sin (3 d x) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8}+\frac{\cos ^8(c+d x) \left(\frac{8}{5} \cos (3 c)-\frac{8}{5} i \sin (3 c)\right) \sin (5 d x) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8}-\frac{i \cos ^8(c+d x) (8 \cos (8 c)-8 i \sin (8 c)) \sin \left(\frac{d x}{2}\right) (i \tan (c+d x) a+a)^8}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x))^8 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{i \cos ^8(c+d x) (8 \cos (8 c)-8 i \sin (8 c)) \sin \left(\frac{d x}{2}\right) (i \tan (c+d x) a+a)^8}{d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x))^8 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^8(c+d x) \left(\frac{1}{4} \cos (8 c)-\frac{1}{4} i \sin (8 c)\right) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos ^8(c+d x) \left(\frac{1}{4} i \sin (8 c)-\frac{1}{4} \cos (8 c)\right) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8 \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{63 \cos (8 c) \cos ^8(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) (i \tan (c+d x) a+a)^8}{2 d (\cos (d x)+i \sin (d x))^8}-\frac{63 \cos (8 c) \cos ^8(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) (i \tan (c+d x) a+a)^8}{2 d (\cos (d x)+i \sin (d x))^8}+\frac{\cos (5 d x) \cos ^8(c+d x) \left(-\frac{8}{5} i \cos (3 c)-\frac{8}{5} \sin (3 c)\right) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8}+\frac{\cos (3 d x) \cos ^8(c+d x) (8 i \cos (5 c)+8 \sin (5 c)) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8}+\frac{\cos (d x) \cos ^8(c+d x) (-48 i \cos (7 c)-48 \sin (7 c)) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8}+\frac{\cos ^8(c+d x) \sec (c) (-8 i \cos (8 c)-8 \sin (8 c)) (i \tan (c+d x) a+a)^8}{d (\cos (d x)+i \sin (d x))^8}-\frac{63 i \cos ^8(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) \sin (8 c) (i \tan (c+d x) a+a)^8}{2 d (\cos (d x)+i \sin (d x))^8}+\frac{63 i \cos ^8(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) \sin (8 c) (i \tan (c+d x) a+a)^8}{2 d (\cos (d x)+i \sin (d x))^8}","-\frac{63 i a^8 \sec (c+d x)}{2 d}-\frac{63 a^8 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{21 i \sec (c+d x) \left(a^8+i a^8 \tan (c+d x)\right)}{2 d}+\frac{6 i a^3 \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac{42 i a^2 \cos (c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^3}{5 d}-\frac{2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d}",1,"(63*Cos[8*c]*Cos[c + d*x]^8*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(a + I*a*Tan[c + d*x])^8)/(2*d*(Cos[d*x] + I*Sin[d*x])^8) - (63*Cos[8*c]*Cos[c + d*x]^8*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(a + I*a*Tan[c + d*x])^8)/(2*d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[5*d*x]*Cos[c + d*x]^8*(((-8*I)/5)*Cos[3*c] - (8*Sin[3*c])/5)*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[3*d*x]*Cos[c + d*x]^8*((8*I)*Cos[5*c] + 8*Sin[5*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[d*x]*Cos[c + d*x]^8*((-48*I)*Cos[7*c] - 48*Sin[7*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*Sec[c]*((-8*I)*Cos[8*c] - 8*Sin[8*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) - (((63*I)/2)*Cos[c + d*x]^8*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sin[8*c]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (((63*I)/2)*Cos[c + d*x]^8*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sin[8*c]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*(48*Cos[7*c] - (48*I)*Sin[7*c])*Sin[d*x]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*(-8*Cos[5*c] + (8*I)*Sin[5*c])*Sin[3*d*x]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*((8*Cos[3*c])/5 - ((8*I)/5)*Sin[3*c])*Sin[5*d*x]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*(Cos[8*c]/4 - (I/4)*Sin[8*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) - (I*Cos[c + d*x]^8*(8*Cos[8*c] - (8*I)*Sin[8*c])*Sin[(d*x)/2]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] - Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (Cos[c + d*x]^8*(-1/4*Cos[8*c] + (I/4)*Sin[8*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (I*Cos[c + d*x]^8*(8*Cos[8*c] - (8*I)*Sin[8*c])*Sin[(d*x)/2]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] + Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
94,1,305,152,2.4666232,"\int \cos ^7(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 \left(\cos \left(\frac{1}{2} (7 c+23 d x)\right)+i \sin \left(\frac{1}{2} (7 c+23 d x)\right)\right) \left(-70 \sin \left(\frac{1}{2} (c+d x)\right)-42 \sin \left(\frac{3}{2} (c+d x)\right)+210 \sin \left(\frac{5}{2} (c+d x)\right)+30 \sin \left(\frac{7}{2} (c+d x)\right)-70 i \cos \left(\frac{1}{2} (c+d x)\right)+42 i \cos \left(\frac{3}{2} (c+d x)\right)+210 i \cos \left(\frac{5}{2} (c+d x)\right)-30 i \cos \left(\frac{7}{2} (c+d x)\right)-105 \cos \left(\frac{7}{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)+105 \cos \left(\frac{7}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+105 i \sin \left(\frac{7}{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)-105 i \sin \left(\frac{7}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{105 d (\cos (d x)+i \sin (d x))^8}","\frac{a^8 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 i \cos (c+d x) \left(a^8+i a^8 \tan (c+d x)\right)}{d}+\frac{2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac{2 i a^2 \cos ^3(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^3}{3 d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}",1,"(a^8*((-70*I)*Cos[(c + d*x)/2] + (42*I)*Cos[(3*(c + d*x))/2] + (210*I)*Cos[(5*(c + d*x))/2] - (30*I)*Cos[(7*(c + d*x))/2] - 105*Cos[(7*(c + d*x))/2]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 105*Cos[(7*(c + d*x))/2]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 70*Sin[(c + d*x)/2] - 42*Sin[(3*(c + d*x))/2] + 210*Sin[(5*(c + d*x))/2] + 30*Sin[(7*(c + d*x))/2] + (105*I)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[(7*(c + d*x))/2] - (105*I)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[(7*(c + d*x))/2])*(Cos[(7*c + 23*d*x)/2] + I*Sin[(7*c + 23*d*x)/2]))/(105*d*(Cos[d*x] + I*Sin[d*x])^8)","B",1
95,1,50,66,0.7556638,"\int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 (8 \cos (c+d x)-i \sin (c+d x)) (\sin (8 (c+d x))-i \cos (8 (c+d x)))}{63 d}","-\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d}-\frac{i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{63 d}",1,"(a^8*(8*Cos[c + d*x] - I*Sin[c + d*x])*((-I)*Cos[8*(c + d*x)] + Sin[8*(c + d*x)]))/(63*d)","A",1
96,1,73,136,1.6820632,"\int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 (-i (55 \sin (c+d x)+63 \sin (3 (c+d x)))+440 \cos (c+d x)+168 \cos (3 (c+d x))) (\sin (8 (c+d x))-i \cos (8 (c+d x)))}{4620 d}","-\frac{2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{1155 d}-\frac{2 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^6}{231 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}-\frac{i a \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{33 d}",1,"(a^8*(440*Cos[c + d*x] + 168*Cos[3*(c + d*x)] - I*(55*Sin[c + d*x] + 63*Sin[3*(c + d*x)]))*((-I)*Cos[8*(c + d*x)] + Sin[8*(c + d*x)]))/(4620*d)","A",1
97,1,111,211,1.6524436,"\int \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^13*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 (-1430 i \sin (c+d x)-2457 i \sin (3 (c+d x))-1155 i \sin (5 (c+d x))+11440 \cos (c+d x)+6552 \cos (3 (c+d x))+1848 \cos (5 (c+d x))) (\sin (8 (c+2 d x))-i \cos (8 (c+2 d x)))}{144144 d (\cos (d x)+i \sin (d x))^8}","-\frac{20 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{3003 d}-\frac{20 i a^2 \cos ^9(c+d x) (a+i a \tan (c+d x))^6}{1287 d}-\frac{8 i \cos ^5(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^4}{3003 d}-\frac{8 i a^2 \cos ^3(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^3}{9009 d}-\frac{i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}-\frac{5 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{143 d}",1,"(a^8*(11440*Cos[c + d*x] + 6552*Cos[3*(c + d*x)] + 1848*Cos[5*(c + d*x)] - (1430*I)*Sin[c + d*x] - (2457*I)*Sin[3*(c + d*x)] - (1155*I)*Sin[5*(c + d*x)])*((-I)*Cos[8*(c + 2*d*x)] + Sin[8*(c + 2*d*x)]))/(144144*d*(Cos[d*x] + I*Sin[d*x])^8)","A",1
98,1,133,212,4.179899,"\int \cos ^{15}(c+d x) (a+i a \tan (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^15*(a + I*a*Tan[c + d*x])^8,x]","\frac{a^8 (-3575 i \sin (c+d x)-7371 i \sin (3 (c+d x))-5775 i \sin (5 (c+d x))-3003 i \sin (7 (c+d x))+28600 \cos (c+d x)+19656 \cos (3 (c+d x))+9240 \cos (5 (c+d x))+3432 \cos (7 (c+d x))) (\sin (8 (c+2 d x))-i \cos (8 (c+2 d x)))}{411840 d (\cos (d x)+i \sin (d x))^8}","-\frac{a^8 \sin ^7(c+d x)}{1287 d}+\frac{7 a^8 \sin ^5(c+d x)}{2145 d}-\frac{7 a^8 \sin ^3(c+d x)}{1287 d}+\frac{7 a^8 \sin (c+d x)}{1287 d}-\frac{2 i \cos ^9(c+d x) \left(a^8+i a^8 \tan (c+d x)\right)}{1287 d}-\frac{2 i a^3 \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{195 d}-\frac{2 i a^2 \cos ^{11}(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^3}{715 d}-\frac{2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}",1,"(a^8*(28600*Cos[c + d*x] + 19656*Cos[3*(c + d*x)] + 9240*Cos[5*(c + d*x)] + 3432*Cos[7*(c + d*x)] - (3575*I)*Sin[c + d*x] - (7371*I)*Sin[3*(c + d*x)] - (5775*I)*Sin[5*(c + d*x)] - (3003*I)*Sin[7*(c + d*x)])*((-I)*Cos[8*(c + 2*d*x)] + Sin[8*(c + 2*d*x)]))/(411840*d*(Cos[d*x] + I*Sin[d*x])^8)","A",1
99,1,71,107,0.4906523,"\int \frac{\sec ^{10}(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^10/(a + I*a*Tan[c + d*x]),x]","\frac{\sec (c) \sec ^8(c+d x) (56 \sin (c+2 d x)+28 \sin (3 c+4 d x)+8 \sin (5 c+6 d x)+\sin (7 c+8 d x)-35 \sin (c)-35 i \cos (c))}{280 a d}","-\frac{i (a-i a \tan (c+d x))^8}{8 a^9 d}+\frac{6 i (a-i a \tan (c+d x))^7}{7 a^8 d}-\frac{2 i (a-i a \tan (c+d x))^6}{a^7 d}+\frac{8 i (a-i a \tan (c+d x))^5}{5 a^6 d}",1,"(Sec[c]*Sec[c + d*x]^8*((-35*I)*Cos[c] - 35*Sin[c] + 56*Sin[c + 2*d*x] + 28*Sin[3*c + 4*d*x] + 8*Sin[5*c + 6*d*x] + Sin[7*c + 8*d*x]))/(280*a*d)","A",1
100,1,60,80,0.3748309,"\int \frac{\sec ^8(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^8/(a + I*a*Tan[c + d*x]),x]","\frac{\sec (c) \sec ^6(c+d x) (15 \sin (c+2 d x)+6 \sin (3 c+4 d x)+\sin (5 c+6 d x)-10 \sin (c)-10 i \cos (c))}{60 a d}","\frac{i (a-i a \tan (c+d x))^6}{6 a^7 d}-\frac{4 i (a-i a \tan (c+d x))^5}{5 a^6 d}+\frac{i (a-i a \tan (c+d x))^4}{a^5 d}",1,"(Sec[c]*Sec[c + d*x]^6*((-10*I)*Cos[c] - 10*Sin[c] + 15*Sin[c + 2*d*x] + 6*Sin[3*c + 4*d*x] + Sin[5*c + 6*d*x]))/(60*a*d)","A",1
101,1,49,55,0.2136871,"\int \frac{\sec ^6(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x]),x]","\frac{\sec (c) \sec ^4(c+d x) (4 \sin (c+2 d x)+\sin (3 c+4 d x)-3 \sin (c)-3 i \cos (c))}{12 a d}","\frac{2 i (a-i a \tan (c+d x))^3}{3 a^4 d}-\frac{i (a-i a \tan (c+d x))^4}{4 a^5 d}",1,"(Sec[c]*Sec[c + d*x]^4*((-3*I)*Cos[c] - 3*Sin[c] + 4*Sin[c + 2*d*x] + Sin[3*c + 4*d*x]))/(12*a*d)","A",1
102,1,35,34,0.2107801,"\int \frac{\sec ^4(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^4/(a + I*a*Tan[c + d*x]),x]","\frac{\sec (c+d x) (2 \sec (c) \sin (d x)-i \sec (c+d x))}{2 a d}","\frac{\tan (c+d x)}{a d}-\frac{i \tan ^2(c+d x)}{2 a d}",1,"(Sec[c + d*x]*((-I)*Sec[c + d*x] + 2*Sec[c]*Sin[d*x]))/(2*a*d)","A",1
103,1,31,23,0.1157557,"\int \frac{\sec ^2(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a + I*a*Tan[c + d*x]),x]","\frac{2 \tan ^{-1}(\tan (d x))+i \log \left(\cos ^2(c+d x)\right)}{2 a d}","\frac{x}{a}+\frac{i \log (\cos (c+d x))}{a d}",1,"(2*ArcTan[Tan[d*x]] + I*Log[Cos[c + d*x]^2])/(2*a*d)","A",1
104,1,45,33,0.113953,"\int \frac{1}{a+i a \tan (c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-1),x]","\frac{(2 d x-i) \tan (c+d x)-2 i d x+1}{4 a d (\tan (c+d x)-i)}","\frac{x}{2 a}+\frac{i}{2 d (a+i a \tan (c+d x))}",1,"(1 - (2*I)*d*x + (-I + 2*d*x)*Tan[c + d*x])/(4*a*d*(-I + Tan[c + d*x]))","A",1
105,1,78,82,0.2778495,"\int \frac{\cos ^2(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a + I*a*Tan[c + d*x]),x]","-\frac{2 \cos (2 (c+d x))-12 d x \tan (c+d x)+6 i \tan (c+d x)+3 i \sin (3 (c+d x)) \sec (c+d x)+12 i d x-7}{32 a d (\tan (c+d x)-i)}","\frac{i a}{8 d (a+i a \tan (c+d x))^2}-\frac{i}{8 d (a-i a \tan (c+d x))}+\frac{i}{4 d (a+i a \tan (c+d x))}+\frac{3 x}{8 a}",1,"-1/32*(-7 + (12*I)*d*x + 2*Cos[2*(c + d*x)] + (3*I)*Sec[c + d*x]*Sin[3*(c + d*x)] + (6*I)*Tan[c + d*x] - 12*d*x*Tan[c + d*x])/(a*d*(-I + Tan[c + d*x]))","A",1
106,1,109,134,0.251408,"\int \frac{\cos ^4(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]^4/(a + I*a*Tan[c + d*x]),x]","-\frac{\sec (c+d x) (-120 d x \sin (c+d x)+60 i \sin (c+d x)+45 i \sin (3 (c+d x))+5 i \sin (5 (c+d x))+60 i (2 d x+i) \cos (c+d x)+15 \cos (3 (c+d x))+\cos (5 (c+d x)))}{384 a d (\tan (c+d x)-i)}","\frac{i a^2}{24 d (a+i a \tan (c+d x))^3}-\frac{i a}{32 d (a-i a \tan (c+d x))^2}+\frac{3 i a}{32 d (a+i a \tan (c+d x))^2}-\frac{i}{8 d (a-i a \tan (c+d x))}+\frac{3 i}{16 d (a+i a \tan (c+d x))}+\frac{5 x}{16 a}",1,"-1/384*(Sec[c + d*x]*((60*I)*(I + 2*d*x)*Cos[c + d*x] + 15*Cos[3*(c + d*x)] + Cos[5*(c + d*x)] + (60*I)*Sin[c + d*x] - 120*d*x*Sin[c + d*x] + (45*I)*Sin[3*(c + d*x)] + (5*I)*Sin[5*(c + d*x)]))/(a*d*(-I + Tan[c + d*x]))","A",1
107,1,60,84,0.4049831,"\int \frac{\sec ^7(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^7/(a + I*a*Tan[c + d*x]),x]","\frac{240 \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)+(70 \sin (2 (c+d x))+15 \sin (4 (c+d x))-64 i) \sec ^5(c+d x)}{320 a d}","-\frac{i \sec ^5(c+d x)}{5 a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{8 a d}",1,"(240*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]] + Sec[c + d*x]^5*(-64*I + 70*Sin[2*(c + d*x)] + 15*Sin[4*(c + d*x)]))/(320*a*d)","A",1
108,1,50,60,0.2489355,"\int \frac{\sec ^5(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^5/(a + I*a*Tan[c + d*x]),x]","\frac{12 \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)+(3 \sin (2 (c+d x))-4 i) \sec ^3(c+d x)}{12 a d}","-\frac{i \sec ^3(c+d x)}{3 a d}+\frac{\tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{\tan (c+d x) \sec (c+d x)}{2 a d}",1,"(12*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]] + Sec[c + d*x]^3*(-4*I + 3*Sin[2*(c + d*x)]))/(12*a*d)","A",1
109,1,34,31,0.169785,"\int \frac{\sec ^3(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^3/(a + I*a*Tan[c + d*x]),x]","\frac{2 \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)-i \sec (c+d x)}{a d}","\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{i \sec (c+d x)}{a d}",1,"(2*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]] - I*Sec[c + d*x])/(a*d)","A",1
110,1,25,28,0.0298748,"\int \frac{\sec (c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]/(a + I*a*Tan[c + d*x]),x]","\frac{\sec (c+d x)}{a d (\tan (c+d x)-i)}","\frac{i \sec (c+d x)}{d (a+i a \tan (c+d x))}",1,"Sec[c + d*x]/(a*d*(-I + Tan[c + d*x]))","A",1
111,1,50,47,0.131053,"\int \frac{\cos (c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]/(a + I*a*Tan[c + d*x]),x]","-\frac{\sec (c+d x) (2 i \sin (2 (c+d x))+\cos (2 (c+d x))-3)}{6 a d (\tan (c+d x)-i)}","\frac{2 \sin (c+d x)}{3 a d}+\frac{i \cos (c+d x)}{3 d (a+i a \tan (c+d x))}",1,"-1/6*(Sec[c + d*x]*(-3 + Cos[2*(c + d*x)] + (2*I)*Sin[2*(c + d*x)]))/(a*d*(-I + Tan[c + d*x]))","A",1
112,1,72,67,0.1848734,"\int \frac{\cos ^3(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x]),x]","-\frac{\sec (c+d x) (40 i \sin (2 (c+d x))+4 i \sin (4 (c+d x))+20 \cos (2 (c+d x))+\cos (4 (c+d x))-45)}{120 a d (\tan (c+d x)-i)}","-\frac{4 \sin ^3(c+d x)}{15 a d}+\frac{4 \sin (c+d x)}{5 a d}+\frac{i \cos ^3(c+d x)}{5 d (a+i a \tan (c+d x))}",1,"-1/120*(Sec[c + d*x]*(-45 + 20*Cos[2*(c + d*x)] + Cos[4*(c + d*x)] + (40*I)*Sin[2*(c + d*x)] + (4*I)*Sin[4*(c + d*x)]))/(a*d*(-I + Tan[c + d*x]))","A",1
113,1,94,85,0.2250291,"\int \frac{\cos ^5(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]^5/(a + I*a*Tan[c + d*x]),x]","-\frac{\sec (c+d x) (350 i \sin (2 (c+d x))+56 i \sin (4 (c+d x))+6 i \sin (6 (c+d x))+175 \cos (2 (c+d x))+14 \cos (4 (c+d x))+\cos (6 (c+d x))-350)}{1120 a d (\tan (c+d x)-i)}","\frac{6 \sin ^5(c+d x)}{35 a d}-\frac{4 \sin ^3(c+d x)}{7 a d}+\frac{6 \sin (c+d x)}{7 a d}+\frac{i \cos ^5(c+d x)}{7 d (a+i a \tan (c+d x))}",1,"-1/1120*(Sec[c + d*x]*(-350 + 175*Cos[2*(c + d*x)] + 14*Cos[4*(c + d*x)] + Cos[6*(c + d*x)] + (350*I)*Sin[2*(c + d*x)] + (56*I)*Sin[4*(c + d*x)] + (6*I)*Sin[6*(c + d*x)]))/(a*d*(-I + Tan[c + d*x]))","A",1
114,1,90,82,0.5498867,"\int \frac{\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^10/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec (c) \sec ^7(c+d x) (-35 \sin (2 c+d x)+42 \sin (2 c+3 d x)+14 \sin (4 c+5 d x)+2 \sin (6 c+7 d x)-35 i \cos (2 c+d x)+35 \sin (d x)-35 i \cos (d x))}{210 a^2 d}","\frac{i (a-i a \tan (c+d x))^7}{7 a^9 d}-\frac{2 i (a-i a \tan (c+d x))^6}{3 a^8 d}+\frac{4 i (a-i a \tan (c+d x))^5}{5 a^7 d}",1,"(Sec[c]*Sec[c + d*x]^7*((-35*I)*Cos[d*x] - (35*I)*Cos[2*c + d*x] + 35*Sin[d*x] - 35*Sin[2*c + d*x] + 42*Sin[2*c + 3*d*x] + 14*Sin[4*c + 5*d*x] + 2*Sin[6*c + 7*d*x]))/(210*a^2*d)","A",1
115,1,77,55,0.4187266,"\int \frac{\sec ^8(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec (c) \sec ^5(c+d x) (-5 \sin (2 c+d x)+5 \sin (2 c+3 d x)+\sin (4 c+5 d x)-5 i \cos (2 c+d x)+5 \sin (d x)-5 i \cos (d x))}{20 a^2 d}","\frac{i (a-i a \tan (c+d x))^4}{2 a^6 d}-\frac{i (a-i a \tan (c+d x))^5}{5 a^7 d}",1,"(Sec[c]*Sec[c + d*x]^5*((-5*I)*Cos[d*x] - (5*I)*Cos[2*c + d*x] + 5*Sin[d*x] - 5*Sin[2*c + d*x] + 5*Sin[2*c + 3*d*x] + Sin[4*c + 5*d*x]))/(20*a^2*d)","A",1
116,1,68,27,0.2447654,"\int \frac{\sec ^6(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec (c) \sec ^3(c+d x) (-3 \sin (2 c+d x)+2 \sin (2 c+3 d x)-3 i \cos (2 c+d x)+3 \sin (d x)-3 i \cos (d x))}{6 a^2 d}","\frac{i (a-i a \tan (c+d x))^3}{3 a^5 d}",1,"(Sec[c]*Sec[c + d*x]^3*((-3*I)*Cos[d*x] - (3*I)*Cos[2*c + d*x] + 3*Sin[d*x] - 3*Sin[2*c + d*x] + 2*Sin[2*c + 3*d*x]))/(6*a^2*d)","B",1
117,1,71,38,0.4391147,"\int \frac{\sec ^4(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^2,x]","\frac{4 \tan ^{-1}(\tan (d x))+i \sec (c) \sec (c+d x) \left(\cos (d x) \log \left(\cos ^2(c+d x)\right)+\cos (2 c+d x) \log \left(\cos ^2(c+d x)\right)+2 i \sin (d x)\right)}{2 a^2 d}","-\frac{\tan (c+d x)}{a^2 d}+\frac{2 i \log (\cos (c+d x))}{a^2 d}+\frac{2 x}{a^2}",1,"(4*ArcTan[Tan[d*x]] + I*Sec[c]*Sec[c + d*x]*(Cos[d*x]*Log[Cos[c + d*x]^2] + Cos[2*c + d*x]*Log[Cos[c + d*x]^2] + (2*I)*Sin[d*x]))/(2*a^2*d)","A",1
118,1,32,26,0.0552979,"\int \frac{\sec ^2(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^2,x]","\frac{i \sec ^2(c+d x)}{2 d (a+i a \tan (c+d x))^2}","\frac{i}{d \left(a^2+i a^2 \tan (c+d x)\right)}",1,"((I/2)*Sec[c + d*x]^2)/(d*(a + I*a*Tan[c + d*x])^2)","A",1
119,1,68,61,0.2176732,"\int \frac{1}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-2),x]","-\frac{\sec ^2(c+d x) ((1+4 i d x) \sin (2 (c+d x))+(4 d x+i) \cos (2 (c+d x))+4 i)}{16 a^2 d (\tan (c+d x)-i)^2}","\frac{i}{4 d \left(a^2+i a^2 \tan (c+d x)\right)}+\frac{x}{4 a^2}+\frac{i}{4 d (a+i a \tan (c+d x))^2}",1,"-1/16*(Sec[c + d*x]^2*(4*I + (I + 4*d*x)*Cos[2*(c + d*x)] + (1 + (4*I)*d*x)*Sin[2*(c + d*x)]))/(a^2*d*(-I + Tan[c + d*x])^2)","A",1
120,1,95,114,0.3001382,"\int \frac{\cos ^2(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^2,x]","\frac{i \sec ^2(c+d x) (-12 d x \sin (2 (c+d x))+3 i \sin (2 (c+d x))+2 i \sin (4 (c+d x))+(-3+12 i d x) \cos (2 (c+d x))+\cos (4 (c+d x))-9)}{48 a^2 d (\tan (c+d x)-i)^2}","-\frac{i}{16 d \left(a^2-i a^2 \tan (c+d x)\right)}+\frac{3 i}{16 d \left(a^2+i a^2 \tan (c+d x)\right)}+\frac{x}{4 a^2}+\frac{i a}{12 d (a+i a \tan (c+d x))^3}+\frac{i}{8 d (a+i a \tan (c+d x))^2}",1,"((I/48)*Sec[c + d*x]^2*(-9 + (-3 + (12*I)*d*x)*Cos[2*(c + d*x)] + Cos[4*(c + d*x)] + (3*I)*Sin[2*(c + d*x)] - 12*d*x*Sin[2*(c + d*x)] + (2*I)*Sin[4*(c + d*x)]))/(a^2*d*(-I + Tan[c + d*x])^2)","A",1
121,1,120,165,0.343264,"\int \frac{\cos ^4(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^2,x]","\frac{i \sec ^2(c+d x) (-120 d x \sin (2 (c+d x))+30 i \sin (2 (c+d x))+32 i \sin (4 (c+d x))+3 i \sin (6 (c+d x))+30 i (4 d x+i) \cos (2 (c+d x))+16 \cos (4 (c+d x))+\cos (6 (c+d x))-80)}{512 a^2 d (\tan (c+d x)-i)^2}","\frac{i a^2}{32 d (a+i a \tan (c+d x))^4}-\frac{5 i}{64 d \left(a^2-i a^2 \tan (c+d x)\right)}+\frac{5 i}{32 d \left(a^2+i a^2 \tan (c+d x)\right)}+\frac{15 x}{64 a^2}+\frac{i a}{16 d (a+i a \tan (c+d x))^3}-\frac{i}{64 d (a-i a \tan (c+d x))^2}+\frac{3 i}{32 d (a+i a \tan (c+d x))^2}",1,"((I/512)*Sec[c + d*x]^2*(-80 + (30*I)*(I + 4*d*x)*Cos[2*(c + d*x)] + 16*Cos[4*(c + d*x)] + Cos[6*(c + d*x)] + (30*I)*Sin[2*(c + d*x)] - 120*d*x*Sin[2*(c + d*x)] + (32*I)*Sin[4*(c + d*x)] + (3*I)*Sin[6*(c + d*x)]))/(a^2*d*(-I + Tan[c + d*x])^2)","A",1
122,1,294,124,2.1451948,"\int \frac{\sec ^9(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^2,x]","-\frac{\sec ^6(c+d x) \left(5 \left(60 \sin (c+d x)-238 \sin (3 (c+d x))-42 \sin (5 (c+d x))+21 \cos (6 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+210 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+315 \cos (2 (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)+126 \cos (4 (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)-21 \cos (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-210 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+3072 i \cos (c+d x)\right)}{7680 a^2 d}","\frac{7 \tanh ^{-1}(\sin (c+d x))}{16 a^2 d}-\frac{2 i \sec ^7(c+d x)}{5 d \left(a^2+i a^2 \tan (c+d x)\right)}+\frac{7 \tan (c+d x) \sec ^5(c+d x)}{30 a^2 d}+\frac{7 \tan (c+d x) \sec ^3(c+d x)}{24 a^2 d}+\frac{7 \tan (c+d x) \sec (c+d x)}{16 a^2 d}",1,"-1/7680*(Sec[c + d*x]^6*((3072*I)*Cos[c + d*x] + 5*(210*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 21*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 315*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 126*Cos[4*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 210*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 21*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 60*Sin[c + d*x] - 238*Sin[3*(c + d*x)] - 42*Sin[5*(c + d*x)])))/(a^2*d)","B",1
123,1,215,100,1.1117293,"\int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^2,x]","-\frac{\sec ^4(c+d x) \left(18 \sin (c+d x)-30 \sin (3 (c+d x))+128 i \cos (c+d x)+45 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+60 \cos (2 (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)+15 \cos (4 (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)-45 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 a^2 d}","\frac{5 \tanh ^{-1}(\sin (c+d x))}{8 a^2 d}-\frac{2 i \sec ^5(c+d x)}{3 d \left(a^2+i a^2 \tan (c+d x)\right)}+\frac{5 \tan (c+d x) \sec ^3(c+d x)}{12 a^2 d}+\frac{5 \tan (c+d x) \sec (c+d x)}{8 a^2 d}",1,"-1/192*(Sec[c + d*x]^4*((128*I)*Cos[c + d*x] + 45*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 60*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 15*Cos[4*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 45*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 18*Sin[c + d*x] - 30*Sin[3*(c + d*x)]))/(a^2*d)","B",1
124,1,146,74,0.472376,"\int \frac{\sec ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^2,x]","-\frac{\sec ^2(c+d x) \left(2 \sin (c+d x)+8 i \cos (c+d x)+3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 \cos (2 (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)-3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 a^2 d}","\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac{2 i \sec ^3(c+d x)}{d \left(a^2+i a^2 \tan (c+d x)\right)}+\frac{3 \tan (c+d x) \sec (c+d x)}{2 a^2 d}",1,"-1/4*(Sec[c + d*x]^2*((8*I)*Cos[c + d*x] + 3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 2*Sin[c + d*x]))/(a^2*d)","A",1
125,1,184,48,0.2089551,"\int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^2,x]","-\frac{\sec ^2(c+d x) \left(\cos \left(\frac{3}{2} (c+d x)\right)+i \sin \left(\frac{3}{2} (c+d x)\right)\right) \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)+2 i\right)+\sin \left(\frac{1}{2} (c+d x)\right) \left(i \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-i \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2\right)\right)}{a^2 d (\tan (c+d x)-i)^2}","-\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{2 i \sec (c+d x)}{d \left(a^2+i a^2 \tan (c+d x)\right)}",1,"-((Sec[c + d*x]^2*(Cos[(c + d*x)/2]*(2*I + Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (2 + I*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - I*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sin[(c + d*x)/2])*(Cos[(3*(c + d*x))/2] + I*Sin[(3*(c + d*x))/2]))/(a^2*d*(-I + Tan[c + d*x])^2))","B",1
126,1,38,65,0.1096672,"\int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]/(a + I*a*Tan[c + d*x])^2,x]","\frac{(\tan (c+d x)-2 i) \sec (c+d x)}{3 a^2 d (\tan (c+d x)-i)^2}","\frac{i \sec (c+d x)}{3 d \left(a^2+i a^2 \tan (c+d x)\right)}+\frac{i \sec (c+d x)}{3 d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]*(-2*I + Tan[c + d*x]))/(3*a^2*d*(-I + Tan[c + d*x])^2)","A",1
127,1,68,71,0.3299137,"\int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Cos[c + d*x]/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec (c+d x) (4 i \cos (2 (c+d x))+5 \tan (c+d x)-3 \sin (3 (c+d x)) \sec (c+d x)-12 i)}{20 a^2 d (\tan (c+d x)-i)^2}","-\frac{\sin ^3(c+d x)}{5 a^2 d}+\frac{3 \sin (c+d x)}{5 a^2 d}+\frac{2 i \cos ^3(c+d x)}{5 d \left(a^2+i a^2 \tan (c+d x)\right)}",1,"(Sec[c + d*x]*(-12*I + (4*I)*Cos[2*(c + d*x)] - 3*Sec[c + d*x]*Sin[3*(c + d*x)] + 5*Tan[c + d*x]))/(20*a^2*d*(-I + Tan[c + d*x])^2)","A",1
128,1,95,89,0.2428575,"\int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^2,x]","\frac{i \sec ^2(c+d x) (-70 i \sin (c+d x)+63 i \sin (3 (c+d x))+5 i \sin (5 (c+d x))-140 \cos (c+d x)+42 \cos (3 (c+d x))+2 \cos (5 (c+d x)))}{336 a^2 d (\tan (c+d x)-i)^2}","\frac{\sin ^5(c+d x)}{7 a^2 d}-\frac{10 \sin ^3(c+d x)}{21 a^2 d}+\frac{5 \sin (c+d x)}{7 a^2 d}+\frac{2 i \cos ^5(c+d x)}{7 d \left(a^2+i a^2 \tan (c+d x)\right)}",1,"((I/336)*Sec[c + d*x]^2*(-140*Cos[c + d*x] + 42*Cos[3*(c + d*x)] + 2*Cos[5*(c + d*x)] - (70*I)*Sin[c + d*x] + (63*I)*Sin[3*(c + d*x)] + (5*I)*Sin[5*(c + d*x)]))/(a^2*d*(-I + Tan[c + d*x])^2)","A",1
129,1,117,107,0.4847381,"\int \frac{\cos ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^5/(a + I*a*Tan[c + d*x])^2,x]","\frac{i \sec ^2(c+d x) (-525 i \sin (c+d x)+567 i \sin (3 (c+d x))+75 i \sin (5 (c+d x))+7 i \sin (7 (c+d x))-1050 \cos (c+d x)+378 \cos (3 (c+d x))+30 \cos (5 (c+d x))+2 \cos (7 (c+d x)))}{2880 a^2 d (\tan (c+d x)-i)^2}","-\frac{\sin ^7(c+d x)}{9 a^2 d}+\frac{7 \sin ^5(c+d x)}{15 a^2 d}-\frac{7 \sin ^3(c+d x)}{9 a^2 d}+\frac{7 \sin (c+d x)}{9 a^2 d}+\frac{2 i \cos ^7(c+d x)}{9 d \left(a^2+i a^2 \tan (c+d x)\right)}",1,"((I/2880)*Sec[c + d*x]^2*(-1050*Cos[c + d*x] + 378*Cos[3*(c + d*x)] + 30*Cos[5*(c + d*x)] + 2*Cos[7*(c + d*x)] - (525*I)*Sin[c + d*x] + (567*I)*Sin[3*(c + d*x)] + (75*I)*Sin[5*(c + d*x)] + (7*I)*Sin[7*(c + d*x)]))/(a^2*d*(-I + Tan[c + d*x])^2)","A",1
130,1,117,109,1.0199005,"\int \frac{\sec ^{14}(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^14/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec (c) \sec ^{10}(c+d x) (105 \sin (c+2 d x)-105 \sin (3 c+2 d x)+120 \sin (3 c+4 d x)+45 \sin (5 c+6 d x)+10 \sin (7 c+8 d x)+\sin (9 c+10 d x)-105 i \cos (c+2 d x)-105 i \cos (3 c+2 d x)-126 \sin (c)-126 i \cos (c))}{840 a^3 d}","-\frac{i (a-i a \tan (c+d x))^{10}}{10 a^{13} d}+\frac{2 i (a-i a \tan (c+d x))^9}{3 a^{12} d}-\frac{3 i (a-i a \tan (c+d x))^8}{2 a^{11} d}+\frac{8 i (a-i a \tan (c+d x))^7}{7 a^{10} d}",1,"(Sec[c]*Sec[c + d*x]^10*((-126*I)*Cos[c] - (105*I)*Cos[c + 2*d*x] - (105*I)*Cos[3*c + 2*d*x] - 126*Sin[c] + 105*Sin[c + 2*d*x] - 105*Sin[3*c + 2*d*x] + 120*Sin[3*c + 4*d*x] + 45*Sin[5*c + 6*d*x] + 10*Sin[7*c + 8*d*x] + Sin[9*c + 10*d*x]))/(840*a^3*d)","A",1
131,1,106,82,0.6648478,"\int \frac{\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^12/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec (c) \sec ^8(c+d x) (28 \sin (c+2 d x)-28 \sin (3 c+2 d x)+28 \sin (3 c+4 d x)+8 \sin (5 c+6 d x)+\sin (7 c+8 d x)-28 i \cos (c+2 d x)-28 i \cos (3 c+2 d x)-35 \sin (c)-35 i \cos (c))}{168 a^3 d}","\frac{i (a-i a \tan (c+d x))^8}{8 a^{11} d}-\frac{4 i (a-i a \tan (c+d x))^7}{7 a^{10} d}+\frac{2 i (a-i a \tan (c+d x))^6}{3 a^9 d}",1,"(Sec[c]*Sec[c + d*x]^8*((-35*I)*Cos[c] - (28*I)*Cos[c + 2*d*x] - (28*I)*Cos[3*c + 2*d*x] - 35*Sin[c] + 28*Sin[c + 2*d*x] - 28*Sin[3*c + 2*d*x] + 28*Sin[3*c + 4*d*x] + 8*Sin[5*c + 6*d*x] + Sin[7*c + 8*d*x]))/(168*a^3*d)","A",1
132,1,97,55,0.5984903,"\int \frac{\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^10/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec (c) \sec ^6(c+d x) (15 \sin (c+2 d x)-15 \sin (3 c+2 d x)+12 \sin (3 c+4 d x)+2 \sin (5 c+6 d x)-15 i \cos (c+2 d x)-15 i \cos (3 c+2 d x)-20 \sin (c)-20 i \cos (c))}{60 a^3 d}","\frac{2 i (a-i a \tan (c+d x))^5}{5 a^8 d}-\frac{i (a-i a \tan (c+d x))^6}{6 a^9 d}",1,"(Sec[c]*Sec[c + d*x]^6*((-20*I)*Cos[c] - (15*I)*Cos[c + 2*d*x] - (15*I)*Cos[3*c + 2*d*x] - 20*Sin[c] + 15*Sin[c + 2*d*x] - 15*Sin[3*c + 2*d*x] + 12*Sin[3*c + 4*d*x] + 2*Sin[5*c + 6*d*x]))/(60*a^3*d)","A",1
133,1,84,27,0.4658889,"\int \frac{\sec ^8(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec (c) \sec ^4(c+d x) (2 \sin (c+2 d x)-2 \sin (3 c+2 d x)+\sin (3 c+4 d x)-2 i \cos (c+2 d x)-2 i \cos (3 c+2 d x)-3 \sin (c)-3 i \cos (c))}{4 a^3 d}","\frac{i (a-i a \tan (c+d x))^4}{4 a^7 d}",1,"(Sec[c]*Sec[c + d*x]^4*((-3*I)*Cos[c] - (2*I)*Cos[c + 2*d*x] - (2*I)*Cos[3*c + 2*d*x] - 3*Sin[c] + 2*Sin[c + 2*d*x] - 2*Sin[3*c + 2*d*x] + Sin[3*c + 4*d*x]))/(4*a^3*d)","B",1
134,1,113,58,0.4724085,"\int \frac{\sec ^6(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec (c) \sec ^2(c+d x) (-3 \sin (c+2 d x)+2 d x \cos (3 c+2 d x)+2 i \cos (3 c+2 d x) \log (\cos (c+d x))+2 \cos (c+2 d x) (d x+i \log (\cos (c+d x)))+\cos (c) (4 i \log (\cos (c+d x))+4 d x+i)+3 \sin (c))}{2 a^3 d}","\frac{i \tan ^2(c+d x)}{2 a^3 d}-\frac{3 \tan (c+d x)}{a^3 d}+\frac{4 i \log (\cos (c+d x))}{a^3 d}+\frac{4 x}{a^3}",1,"(Sec[c]*Sec[c + d*x]^2*(2*d*x*Cos[3*c + 2*d*x] + 2*Cos[c + 2*d*x]*(d*x + I*Log[Cos[c + d*x]]) + Cos[c]*(I + 4*d*x + (4*I)*Log[Cos[c + d*x]]) + (2*I)*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]] + 3*Sin[c] - 3*Sin[c + 2*d*x]))/(2*a^3*d)","A",1
135,1,88,50,0.2502401,"\int \frac{\sec ^4(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x))) (\log (\cos (c+d x))+\tan (c+d x) (i \log (\cos (c+d x))+d x+i)-i d x-1)}{a^3 d (\tan (c+d x)-i)^3}","\frac{2 i}{d \left(a^3+i a^3 \tan (c+d x)\right)}-\frac{i \log (\cos (c+d x))}{a^3 d}-\frac{x}{a^3}",1,"(Sec[c + d*x]^2*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])*(-1 - I*d*x + Log[Cos[c + d*x]] + (I + d*x + I*Log[Cos[c + d*x]])*Tan[c + d*x]))/(a^3*d*(-I + Tan[c + d*x])^3)","A",1
136,1,42,27,0.102482,"\int \frac{\sec ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^3,x]","-\frac{i (\tan (c+d x)-3 i) \sec ^2(c+d x)}{8 a^3 d (\tan (c+d x)-i)^3}","\frac{i}{2 a d (a+i a \tan (c+d x))^2}",1,"((-1/8*I)*Sec[c + d*x]^2*(-3*I + Tan[c + d*x]))/(a^3*d*(-I + Tan[c + d*x])^3)","A",1
137,1,93,88,0.2466858,"\int \frac{1}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-3),x]","\frac{i \sec ^3(c+d x) (-9 \sin (c+d x)+12 i d x \sin (3 (c+d x))+2 \sin (3 (c+d x))+27 i \cos (c+d x)+2 (6 d x+i) \cos (3 (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3}","\frac{i}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{x}{8 a^3}+\frac{i}{8 a d (a+i a \tan (c+d x))^2}+\frac{i}{6 d (a+i a \tan (c+d x))^3}",1,"((I/96)*Sec[c + d*x]^3*((27*I)*Cos[c + d*x] + 2*(I + 6*d*x)*Cos[3*(c + d*x)] - 9*Sin[c + d*x] + 2*Sin[3*(c + d*x)] + (12*I)*d*x*Sin[3*(c + d*x)]))/(a^3*d*(-I + Tan[c + d*x])^3)","A",1
138,1,115,141,0.2965498,"\int \frac{\cos ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (-60 i \sin (c+d x)-120 d x \sin (3 (c+d x))+20 i \sin (3 (c+d x))+15 i \sin (5 (c+d x))-180 \cos (c+d x)+20 i (6 d x+i) \cos (3 (c+d x))+9 \cos (5 (c+d x)))}{768 a^3 d (\tan (c+d x)-i)^3}","-\frac{i}{32 d \left(a^3-i a^3 \tan (c+d x)\right)}+\frac{i}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{5 x}{32 a^3}+\frac{i a}{16 d (a+i a \tan (c+d x))^4}+\frac{i}{12 d (a+i a \tan (c+d x))^3}+\frac{3 i}{32 a d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^3*(-180*Cos[c + d*x] + (20*I)*(I + 6*d*x)*Cos[3*(c + d*x)] + 9*Cos[5*(c + d*x)] - (60*I)*Sin[c + d*x] + (20*I)*Sin[3*(c + d*x)] - 120*d*x*Sin[3*(c + d*x)] + (15*I)*Sin[5*(c + d*x)]))/(768*a^3*d*(-I + Tan[c + d*x])^3)","A",1
139,1,137,195,0.5275905,"\int \frac{\cos ^4(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (-350 i \sin (c+d x)-840 d x \sin (3 (c+d x))+140 i \sin (3 (c+d x))+175 i \sin (5 (c+d x))+14 i \sin (7 (c+d x))-1050 \cos (c+d x)+140 i (6 d x+i) \cos (3 (c+d x))+105 \cos (5 (c+d x))+6 \cos (7 (c+d x)))}{5120 a^3 d (\tan (c+d x)-i)^3}","-\frac{3 i}{64 d \left(a^3-i a^3 \tan (c+d x)\right)}+\frac{15 i}{128 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{21 x}{128 a^3}+\frac{i a^2}{40 d (a+i a \tan (c+d x))^5}+\frac{3 i a}{64 d (a+i a \tan (c+d x))^4}+\frac{i}{16 d (a+i a \tan (c+d x))^3}-\frac{i}{128 a d (a-i a \tan (c+d x))^2}+\frac{5 i}{64 a d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^3*(-1050*Cos[c + d*x] + (140*I)*(I + 6*d*x)*Cos[3*(c + d*x)] + 105*Cos[5*(c + d*x)] + 6*Cos[7*(c + d*x)] - (350*I)*Sin[c + d*x] + (140*I)*Sin[3*(c + d*x)] - 840*d*x*Sin[3*(c + d*x)] + (175*I)*Sin[5*(c + d*x)] + (14*I)*Sin[7*(c + d*x)]))/(5120*a^3*d*(-I + Tan[c + d*x])^3)","A",1
140,1,113,119,0.4899983,"\int \frac{\sec ^9(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^8(c+d x) (\cos (3 (c+d x))+i \sin (3 (c+d x))) \left(-150 i \sin (2 (c+d x))+105 i \sin (4 (c+d x))+640 \cos (2 (c+d x))+1680 i \cos ^5(c+d x) \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)+448\right)}{960 a^3 d (\tan (c+d x)-i)^3}","-\frac{7 i \sec ^5(c+d x)}{15 a^3 d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}+\frac{7 \tan (c+d x) \sec ^3(c+d x)}{12 a^3 d}+\frac{7 \tan (c+d x) \sec (c+d x)}{8 a^3 d}-\frac{2 i \sec ^7(c+d x)}{3 a d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^8*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)])*(448 + (1680*I)*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]]*Cos[c + d*x]^5 + 640*Cos[2*(c + d*x)] - (150*I)*Sin[2*(c + d*x)] + (105*I)*Sin[4*(c + d*x)]))/(960*a^3*d*(-I + Tan[c + d*x])^3)","A",1
141,1,63,93,0.4343324,"\int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^3,x]","\frac{60 \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)-i \sec ^3(c+d x) (-9 i \sin (2 (c+d x))+24 \cos (2 (c+d x))+20)}{12 a^3 d}","-\frac{5 i \sec ^3(c+d x)}{3 a^3 d}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{5 \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{2 i \sec ^5(c+d x)}{a d (a+i a \tan (c+d x))^2}",1,"(60*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]] - I*Sec[c + d*x]^3*(20 + 24*Cos[2*(c + d*x)] - (9*I)*Sin[2*(c + d*x)]))/(12*a^3*d)","A",1
142,1,108,65,0.3386102,"\int \frac{\sec ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (-\sin (d x)+i \cos (d x))^3 \left((\tan (c+d x)-5 i) (\cos (2 c-d x)+i \sin (2 c-d x))+6 (\cos (3 c)+i \sin (3 c)) \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)\right)}{a^3 d (\tan (c+d x)-i)^3}","\frac{3 i \sec (c+d x)}{a^3 d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{2 i \sec ^3(c+d x)}{a d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^3*(I*Cos[d*x] - Sin[d*x])^3*(6*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]]*(Cos[3*c] + I*Sin[3*c]) + (Cos[2*c - d*x] + I*Sin[2*c - d*x])*(-5*I + Tan[c + d*x])))/(a^3*d*(-I + Tan[c + d*x])^3)","A",1
143,1,32,32,0.0659225,"\int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^3,x]","\frac{i \sec ^3(c+d x)}{3 d (a+i a \tan (c+d x))^3}","\frac{i \sec ^3(c+d x)}{3 d (a+i a \tan (c+d x))^3}",1,"((I/3)*Sec[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^3)","A",1
144,1,54,98,0.1824459,"\int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]/(a + I*a*Tan[c + d*x])^3,x]","-\frac{\sec ^3(c+d x) (6 i \sin (2 (c+d x))+9 \cos (2 (c+d x))+5)}{30 a^3 d (\tan (c+d x)-i)^3}","\frac{2 i \sec (c+d x)}{15 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{2 i \sec (c+d x)}{15 a d (a+i a \tan (c+d x))^2}+\frac{i \sec (c+d x)}{5 d (a+i a \tan (c+d x))^3}",1,"-1/30*(Sec[c + d*x]^3*(5 + 9*Cos[2*(c + d*x)] + (6*I)*Sin[2*(c + d*x)]))/(a^3*d*(-I + Tan[c + d*x])^3)","A",1
145,1,76,101,0.2227903,"\int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Cos[c + d*x]/(a + I*a*Tan[c + d*x])^3,x]","-\frac{\sec ^3(c+d x) (56 i \sin (2 (c+d x))-20 i \sin (4 (c+d x))+84 \cos (2 (c+d x))-15 \cos (4 (c+d x))+35)}{280 a^3 d (\tan (c+d x)-i)^3}","-\frac{4 \sin ^3(c+d x)}{35 a^3 d}+\frac{12 \sin (c+d x)}{35 a^3 d}+\frac{8 i \cos ^3(c+d x)}{35 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{i \cos (c+d x)}{7 d (a+i a \tan (c+d x))^3}",1,"-1/280*(Sec[c + d*x]^3*(35 + 84*Cos[2*(c + d*x)] - 15*Cos[4*(c + d*x)] + (56*I)*Sin[2*(c + d*x)] - (20*I)*Sin[4*(c + d*x)]))/(a^3*d*(-I + Tan[c + d*x])^3)","A",1
146,1,98,121,0.3029047,"\int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (-378 i \sin (2 (c+d x))+216 i \sin (4 (c+d x))+14 i \sin (6 (c+d x))-567 \cos (2 (c+d x))+162 \cos (4 (c+d x))+7 \cos (6 (c+d x))-210)}{2016 a^3 d (\tan (c+d x)-i)^3}","\frac{2 \sin ^5(c+d x)}{21 a^3 d}-\frac{20 \sin ^3(c+d x)}{63 a^3 d}+\frac{10 \sin (c+d x)}{21 a^3 d}+\frac{4 i \cos ^5(c+d x)}{21 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{i \cos ^3(c+d x)}{9 d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^3*(-210 - 567*Cos[2*(c + d*x)] + 162*Cos[4*(c + d*x)] + 7*Cos[6*(c + d*x)] - (378*I)*Sin[2*(c + d*x)] + (216*I)*Sin[4*(c + d*x)] + (14*I)*Sin[6*(c + d*x)]))/(2016*a^3*d*(-I + Tan[c + d*x])^3)","A",1
147,1,120,139,0.7124373,"\int \frac{\cos ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^5/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (-11088 i \sin (2 (c+d x))+7920 i \sin (4 (c+d x))+880 i \sin (6 (c+d x))+72 i \sin (8 (c+d x))-16632 \cos (2 (c+d x))+5940 \cos (4 (c+d x))+440 \cos (6 (c+d x))+27 \cos (8 (c+d x))-5775)}{63360 a^3 d (\tan (c+d x)-i)^3}","-\frac{8 \sin ^7(c+d x)}{99 a^3 d}+\frac{56 \sin ^5(c+d x)}{165 a^3 d}-\frac{56 \sin ^3(c+d x)}{99 a^3 d}+\frac{56 \sin (c+d x)}{99 a^3 d}+\frac{16 i \cos ^7(c+d x)}{99 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{i \cos ^5(c+d x)}{11 d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^3*(-5775 - 16632*Cos[2*(c + d*x)] + 5940*Cos[4*(c + d*x)] + 440*Cos[6*(c + d*x)] + 27*Cos[8*(c + d*x)] - (11088*I)*Sin[2*(c + d*x)] + (7920*I)*Sin[4*(c + d*x)] + (880*I)*Sin[6*(c + d*x)] + (72*I)*Sin[8*(c + d*x)]))/(63360*a^3*d*(-I + Tan[c + d*x])^3)","A",1
148,1,136,82,0.7873334,"\int \frac{\sec ^{14}(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^14/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec (c) \sec ^9(c+d x) (-63 \sin (2 c+d x)+42 \sin (2 c+3 d x)-42 \sin (4 c+3 d x)+36 \sin (4 c+5 d x)+9 \sin (6 c+7 d x)+\sin (8 c+9 d x)-63 i \cos (2 c+d x)-42 i \cos (2 c+3 d x)-42 i \cos (4 c+3 d x)+63 \sin (d x)-63 i \cos (d x))}{252 a^4 d}","\frac{i (a-i a \tan (c+d x))^9}{9 a^{13} d}-\frac{i (a-i a \tan (c+d x))^8}{2 a^{12} d}+\frac{4 i (a-i a \tan (c+d x))^7}{7 a^{11} d}",1,"(Sec[c]*Sec[c + d*x]^9*((-63*I)*Cos[d*x] - (63*I)*Cos[2*c + d*x] - (42*I)*Cos[2*c + 3*d*x] - (42*I)*Cos[4*c + 3*d*x] + 63*Sin[d*x] - 63*Sin[2*c + d*x] + 42*Sin[2*c + 3*d*x] - 42*Sin[4*c + 3*d*x] + 36*Sin[4*c + 5*d*x] + 9*Sin[6*c + 7*d*x] + Sin[8*c + 9*d*x]))/(252*a^4*d)","A",1
149,1,127,55,0.5277564,"\int \frac{\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^12/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec (c) \sec ^7(c+d x) (-35 \sin (2 c+d x)+21 \sin (2 c+3 d x)-21 \sin (4 c+3 d x)+14 \sin (4 c+5 d x)+2 \sin (6 c+7 d x)-35 i \cos (2 c+d x)-21 i \cos (2 c+3 d x)-21 i \cos (4 c+3 d x)+35 \sin (d x)-35 i \cos (d x))}{84 a^4 d}","\frac{i (a-i a \tan (c+d x))^6}{3 a^{10} d}-\frac{i (a-i a \tan (c+d x))^7}{7 a^{11} d}",1,"(Sec[c]*Sec[c + d*x]^7*((-35*I)*Cos[d*x] - (35*I)*Cos[2*c + d*x] - (21*I)*Cos[2*c + 3*d*x] - (21*I)*Cos[4*c + 3*d*x] + 35*Sin[d*x] - 35*Sin[2*c + d*x] + 21*Sin[2*c + 3*d*x] - 21*Sin[4*c + 3*d*x] + 14*Sin[4*c + 5*d*x] + 2*Sin[6*c + 7*d*x]))/(84*a^4*d)","B",1
150,1,116,27,0.4494422,"\int \frac{\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^10/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec (c) \sec ^5(c+d x) (-10 \sin (2 c+d x)+5 \sin (2 c+3 d x)-5 \sin (4 c+3 d x)+2 \sin (4 c+5 d x)-10 i \cos (2 c+d x)-5 i \cos (2 c+3 d x)-5 i \cos (4 c+3 d x)+10 \sin (d x)-10 i \cos (d x))}{10 a^4 d}","\frac{i (a-i a \tan (c+d x))^5}{5 a^9 d}",1,"(Sec[c]*Sec[c + d*x]^5*((-10*I)*Cos[d*x] - (10*I)*Cos[2*c + d*x] - (5*I)*Cos[2*c + 3*d*x] - (5*I)*Cos[4*c + 3*d*x] + 10*Sin[d*x] - 10*Sin[2*c + d*x] + 5*Sin[2*c + 3*d*x] - 5*Sin[4*c + 3*d*x] + 2*Sin[4*c + 5*d*x]))/(10*a^4*d)","B",1
151,1,168,90,0.7801391,"\int \frac{\sec ^8(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec (c) \sec ^3(c+d x) (12 \sin (2 c+d x)-11 \sin (2 c+3 d x)+6 d x \cos (2 c+3 d x)+6 d x \cos (4 c+3 d x)+6 i \cos (2 c+3 d x) \log (\cos (c+d x))+6 \cos (d x) (3 i \log (\cos (c+d x))+3 d x+i)+6 \cos (2 c+d x) (3 i \log (\cos (c+d x))+3 d x+i)+6 i \cos (4 c+3 d x) \log (\cos (c+d x))-21 \sin (d x))}{6 a^4 d}","-\frac{i (a-i a \tan (c+d x))^3}{3 a^7 d}-\frac{i (a-i a \tan (c+d x))^2}{a^6 d}-\frac{4 \tan (c+d x)}{a^4 d}+\frac{8 i \log (\cos (c+d x))}{a^4 d}+\frac{8 x}{a^4}",1,"(Sec[c]*Sec[c + d*x]^3*(6*d*x*Cos[2*c + 3*d*x] + 6*d*x*Cos[4*c + 3*d*x] + 6*Cos[d*x]*(I + 3*d*x + (3*I)*Log[Cos[c + d*x]]) + 6*Cos[2*c + d*x]*(I + 3*d*x + (3*I)*Log[Cos[c + d*x]]) + (6*I)*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]] + (6*I)*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]] - 21*Sin[d*x] + 12*Sin[2*c + d*x] - 11*Sin[2*c + 3*d*x]))/(6*a^4*d)","A",1
152,1,214,63,0.7214082,"\int \frac{\sec ^6(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec (c) \sec (c+d x) (-\cos (c+d x)+i \sin (c+d x)) (2 i d x \sin (c+2 d x)-2 \sin (c+2 d x)+2 i d x \sin (3 c+2 d x)-\sin (3 c+2 d x)+2 d x \cos (3 c+2 d x)-i \cos (3 c+2 d x)+2 i \cos (3 c+2 d x) \log (\cos (c+d x))+2 \cos (c+2 d x) (d x+i \log (\cos (c+d x)))+\cos (c) (4 i \log (\cos (c+d x))+4 d x-3 i)-2 \sin (c+2 d x) \log (\cos (c+d x))-2 \sin (3 c+2 d x) \log (\cos (c+d x))+\sin (c))}{2 a^4 d}","\frac{\tan (c+d x)}{a^4 d}+\frac{4 i}{d \left(a^4+i a^4 \tan (c+d x)\right)}-\frac{4 i \log (\cos (c+d x))}{a^4 d}-\frac{4 x}{a^4}",1,"(Sec[c]*Sec[c + d*x]*(-Cos[c + d*x] + I*Sin[c + d*x])*((-I)*Cos[3*c + 2*d*x] + 2*d*x*Cos[3*c + 2*d*x] + 2*Cos[c + 2*d*x]*(d*x + I*Log[Cos[c + d*x]]) + Cos[c]*(-3*I + 4*d*x + (4*I)*Log[Cos[c + d*x]]) + (2*I)*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]] + Sin[c] - 2*Sin[c + 2*d*x] + (2*I)*d*x*Sin[c + 2*d*x] - 2*Log[Cos[c + d*x]]*Sin[c + 2*d*x] - Sin[3*c + 2*d*x] + (2*I)*d*x*Sin[3*c + 2*d*x] - 2*Log[Cos[c + d*x]]*Sin[3*c + 2*d*x]))/(2*a^4*d)","B",1
153,1,32,29,0.0728713,"\int \frac{\sec ^4(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^4,x]","\frac{i \sec ^4(c+d x)}{4 d (a+i a \tan (c+d x))^4}","\frac{\tan (c+d x)}{d \left(a^2+i a^2 \tan (c+d x)\right)^2}",1,"((I/4)*Sec[c + d*x]^4)/(d*(a + I*a*Tan[c + d*x])^4)","A",1
154,1,56,27,0.1902955,"\int \frac{\sec ^2(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^4,x]","\frac{i \sec ^4(c+d x) (2 i \sin (2 (c+d x))+4 \cos (2 (c+d x))+3)}{24 a^4 d (\tan (c+d x)-i)^4}","\frac{i}{3 a d (a+i a \tan (c+d x))^3}",1,"((I/24)*Sec[c + d*x]^4*(3 + 4*Cos[2*(c + d*x)] + (2*I)*Sin[2*(c + d*x)]))/(a^4*d*(-I + Tan[c + d*x])^4)","B",1
155,1,98,116,0.2306108,"\int \frac{1}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-4),x]","\frac{\sec ^4(c+d x) (-32 \sin (2 (c+d x))+24 i d x \sin (4 (c+d x))+3 \sin (4 (c+d x))+64 i \cos (2 (c+d x))+3 (8 d x+i) \cos (4 (c+d x))+36 i)}{384 a^4 d (\tan (c+d x)-i)^4}","\frac{i}{16 d \left(a^4+i a^4 \tan (c+d x)\right)}+\frac{x}{16 a^4}+\frac{i}{16 d \left(a^2+i a^2 \tan (c+d x)\right)^2}+\frac{i}{12 a d (a+i a \tan (c+d x))^3}+\frac{i}{8 d (a+i a \tan (c+d x))^4}",1,"(Sec[c + d*x]^4*(36*I + (64*I)*Cos[2*(c + d*x)] + 3*(I + 8*d*x)*Cos[4*(c + d*x)] - 32*Sin[2*(c + d*x)] + 3*Sin[4*(c + d*x)] + (24*I)*d*x*Sin[4*(c + d*x)]))/(384*a^4*d*(-I + Tan[c + d*x])^4)","A",1
156,1,120,169,0.3438356,"\int \frac{\cos ^2(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (-100 \sin (2 (c+d x))+120 i d x \sin (4 (c+d x))+15 \sin (4 (c+d x))+12 \sin (6 (c+d x))+200 i \cos (2 (c+d x))+15 (8 d x+i) \cos (4 (c+d x))-8 i \cos (6 (c+d x))+100 i)}{1280 a^4 d (\tan (c+d x)-i)^4}","-\frac{i}{64 d \left(a^4-i a^4 \tan (c+d x)\right)}+\frac{5 i}{64 d \left(a^4+i a^4 \tan (c+d x)\right)}+\frac{3 x}{32 a^4}+\frac{i}{16 d \left(a^2+i a^2 \tan (c+d x)\right)^2}+\frac{i a}{20 d (a+i a \tan (c+d x))^5}+\frac{i}{16 d (a+i a \tan (c+d x))^4}+\frac{i}{16 a d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^4*(100*I + (200*I)*Cos[2*(c + d*x)] + 15*(I + 8*d*x)*Cos[4*(c + d*x)] - (8*I)*Cos[6*(c + d*x)] - 100*Sin[2*(c + d*x)] + 15*Sin[4*(c + d*x)] + (120*I)*d*x*Sin[4*(c + d*x)] + 12*Sin[6*(c + d*x)]))/(1280*a^4*d*(-I + Tan[c + d*x])^4)","A",1
157,1,142,224,0.7574894,"\int \frac{\cos ^4(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (-560 \sin (2 (c+d x))+840 i d x \sin (4 (c+d x))+105 \sin (4 (c+d x))+144 \sin (6 (c+d x))+10 \sin (8 (c+d x))+1120 i \cos (2 (c+d x))+105 (8 d x+i) \cos (4 (c+d x))-96 i \cos (6 (c+d x))-5 i \cos (8 (c+d x))+525 i)}{7680 a^4 d (\tan (c+d x)-i)^4}","-\frac{7 i}{256 d \left(a^4-i a^4 \tan (c+d x)\right)}+\frac{21 i}{256 d \left(a^4+i a^4 \tan (c+d x)\right)}+\frac{7 x}{64 a^4}+\frac{i a^2}{48 d (a+i a \tan (c+d x))^6}-\frac{i}{256 d \left(a^2-i a^2 \tan (c+d x)\right)^2}+\frac{15 i}{256 d \left(a^2+i a^2 \tan (c+d x)\right)^2}+\frac{3 i a}{80 d (a+i a \tan (c+d x))^5}+\frac{3 i}{64 d (a+i a \tan (c+d x))^4}+\frac{5 i}{96 a d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^4*(525*I + (1120*I)*Cos[2*(c + d*x)] + 105*(I + 8*d*x)*Cos[4*(c + d*x)] - (96*I)*Cos[6*(c + d*x)] - (5*I)*Cos[8*(c + d*x)] - 560*Sin[2*(c + d*x)] + 105*Sin[4*(c + d*x)] + (840*I)*d*x*Sin[4*(c + d*x)] + 144*Sin[6*(c + d*x)] + 10*Sin[8*(c + d*x)]))/(7680*a^4*d*(-I + Tan[c + d*x])^4)","A",1
158,1,237,133,1.199285,"\int \frac{\sec ^9(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^4,x]","-\frac{\sec ^4(c+d x) \left(896 i \cos (c+d x)+3 \left(42 \sin (c+d x)+58 \sin (3 (c+d x))+128 i \cos (3 (c+d x))+35 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+105 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+140 \cos (2 (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)-35 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-105 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{192 a^4 d}","\frac{35 \tanh ^{-1}(\sin (c+d x))}{8 a^4 d}-\frac{14 i \sec ^5(c+d x)}{3 d \left(a^4+i a^4 \tan (c+d x)\right)}+\frac{35 \tan (c+d x) \sec ^3(c+d x)}{12 a^4 d}+\frac{35 \tan (c+d x) \sec (c+d x)}{8 a^4 d}-\frac{2 i \sec ^7(c+d x)}{a d (a+i a \tan (c+d x))^3}",1,"-1/192*(Sec[c + d*x]^4*((896*I)*Cos[c + d*x] + 3*((128*I)*Cos[3*(c + d*x)] + 105*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 35*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 140*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 105*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 35*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 42*Sin[c + d*x] + 58*Sin[3*(c + d*x)])))/(a^4*d)","A",1
159,1,988,107,6.1875793,"\int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^4,x]","\frac{15 \cos (4 c) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^4(c+d x) (\cos (d x)+i \sin (d x))^4}{2 d (i \tan (c+d x) a+a)^4}-\frac{15 \cos (4 c) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^4(c+d x) (\cos (d x)+i \sin (d x))^4}{2 d (i \tan (c+d x) a+a)^4}+\frac{\cos (d x) \sec ^4(c+d x) (8 i \cos (3 c)-8 \sin (3 c)) (\cos (d x)+i \sin (d x))^4}{d (i \tan (c+d x) a+a)^4}+\frac{\sec (c) \sec ^4(c+d x) (4 i \cos (4 c)-4 \sin (4 c)) (\cos (d x)+i \sin (d x))^4}{d (i \tan (c+d x) a+a)^4}+\frac{15 i \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^4(c+d x) \sin (4 c) (\cos (d x)+i \sin (d x))^4}{2 d (i \tan (c+d x) a+a)^4}-\frac{15 i \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^4(c+d x) \sin (4 c) (\cos (d x)+i \sin (d x))^4}{2 d (i \tan (c+d x) a+a)^4}+\frac{\sec ^4(c+d x) (8 \cos (3 c)+8 i \sin (3 c)) \sin (d x) (\cos (d x)+i \sin (d x))^4}{d (i \tan (c+d x) a+a)^4}+\frac{4 \sec ^4(c+d x) \left(\frac{1}{2} \cos \left(4 c-\frac{d x}{2}\right)-\frac{1}{2} \cos \left(4 c+\frac{d x}{2}\right)+\frac{1}{2} i \sin \left(4 c-\frac{d x}{2}\right)-\frac{1}{2} i \sin \left(4 c+\frac{d x}{2}\right)\right) (\cos (d x)+i \sin (d x))^4}{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) (i \tan (c+d x) a+a)^4}+\frac{4 \sec ^4(c+d x) \left(-\frac{1}{2} \cos \left(4 c-\frac{d x}{2}\right)+\frac{1}{2} \cos \left(4 c+\frac{d x}{2}\right)-\frac{1}{2} i \sin \left(4 c-\frac{d x}{2}\right)+\frac{1}{2} i \sin \left(4 c+\frac{d x}{2}\right)\right) (\cos (d x)+i \sin (d x))^4}{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) (i \tan (c+d x) a+a)^4}+\frac{\sec ^4(c+d x) \left(\frac{1}{4} \cos (4 c)+\frac{1}{4} i \sin (4 c)\right) (\cos (d x)+i \sin (d x))^4}{d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2 (i \tan (c+d x) a+a)^4}+\frac{\sec ^4(c+d x) \left(-\frac{1}{4} \cos (4 c)-\frac{1}{4} i \sin (4 c)\right) (\cos (d x)+i \sin (d x))^4}{d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2 (i \tan (c+d x) a+a)^4}","-\frac{15 \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{10 i \sec ^3(c+d x)}{d \left(a^4+i a^4 \tan (c+d x)\right)}-\frac{15 \tan (c+d x) \sec (c+d x)}{2 a^4 d}+\frac{2 i \sec ^5(c+d x)}{a d (a+i a \tan (c+d x))^3}",1,"(15*Cos[4*c]*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^4*(Cos[d*x] + I*Sin[d*x])^4)/(2*d*(a + I*a*Tan[c + d*x])^4) - (15*Cos[4*c]*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^4*(Cos[d*x] + I*Sin[d*x])^4)/(2*d*(a + I*a*Tan[c + d*x])^4) + (Cos[d*x]*Sec[c + d*x]^4*((8*I)*Cos[3*c] - 8*Sin[3*c])*(Cos[d*x] + I*Sin[d*x])^4)/(d*(a + I*a*Tan[c + d*x])^4) + (Sec[c]*Sec[c + d*x]^4*((4*I)*Cos[4*c] - 4*Sin[4*c])*(Cos[d*x] + I*Sin[d*x])^4)/(d*(a + I*a*Tan[c + d*x])^4) + (((15*I)/2)*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^4*Sin[4*c]*(Cos[d*x] + I*Sin[d*x])^4)/(d*(a + I*a*Tan[c + d*x])^4) - (((15*I)/2)*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^4*Sin[4*c]*(Cos[d*x] + I*Sin[d*x])^4)/(d*(a + I*a*Tan[c + d*x])^4) + (Sec[c + d*x]^4*(8*Cos[3*c] + (8*I)*Sin[3*c])*(Cos[d*x] + I*Sin[d*x])^4*Sin[d*x])/(d*(a + I*a*Tan[c + d*x])^4) + (Sec[c + d*x]^4*(Cos[4*c]/4 + (I/4)*Sin[4*c])*(Cos[d*x] + I*Sin[d*x])^4)/(d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2*(a + I*a*Tan[c + d*x])^4) + (Sec[c + d*x]^4*(-1/4*Cos[4*c] - (I/4)*Sin[4*c])*(Cos[d*x] + I*Sin[d*x])^4)/(d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2*(a + I*a*Tan[c + d*x])^4) + (4*Sec[c + d*x]^4*(Cos[d*x] + I*Sin[d*x])^4*(Cos[4*c - (d*x)/2]/2 - Cos[4*c + (d*x)/2]/2 + (I/2)*Sin[4*c - (d*x)/2] - (I/2)*Sin[4*c + (d*x)/2]))/(d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])*(a + I*a*Tan[c + d*x])^4) + (4*Sec[c + d*x]^4*(Cos[d*x] + I*Sin[d*x])^4*(-1/2*Cos[4*c - (d*x)/2] + Cos[4*c + (d*x)/2]/2 - (I/2)*Sin[4*c - (d*x)/2] + (I/2)*Sin[4*c + (d*x)/2]))/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])*(a + I*a*Tan[c + d*x])^4)","B",1
160,1,247,82,0.3110016,"\int \frac{\sec ^5(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (\cos (d x)+i \sin (d x))^4 \left(-6 i \sin (3 c) \sin (d x)+2 i \sin (c) \sin (3 d x)-2 \sin (c) \cos (3 d x)+6 \sin (3 c) \cos (d x)+\cos (3 c) (-6 \sin (d x)-6 i \cos (d x))+2 \cos (c) (\sin (3 d x)+i \cos (3 d x))-3 \cos (4 c) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 i \sin (4 c) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 \cos (4 c) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+3 i \sin (4 c) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{3 a^4 d (\tan (c+d x)-i)^4}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{2 i \sec (c+d x)}{d \left(a^4+i a^4 \tan (c+d x)\right)}+\frac{2 i \sec ^3(c+d x)}{3 a d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^4*(Cos[d*x] + I*Sin[d*x])^4*(-3*Cos[4*c]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3*Cos[4*c]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2*Cos[3*d*x]*Sin[c] + 6*Cos[d*x]*Sin[3*c] - (3*I)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[4*c] + (3*I)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[4*c] + Cos[3*c]*((-6*I)*Cos[d*x] - 6*Sin[d*x]) - (6*I)*Sin[3*c]*Sin[d*x] + (2*I)*Sin[c]*Sin[3*d*x] + 2*Cos[c]*(I*Cos[3*d*x] + Sin[3*d*x])))/(3*a^4*d*(-I + Tan[c + d*x])^4)","B",1
161,1,40,68,0.0994716,"\int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^4,x]","-\frac{(\tan (c+d x)-4 i) \sec ^3(c+d x)}{15 a^4 d (\tan (c+d x)-i)^4}","\frac{i \sec ^3(c+d x)}{15 a d (a+i a \tan (c+d x))^3}+\frac{i \sec ^3(c+d x)}{5 d (a+i a \tan (c+d x))^4}",1,"-1/15*(Sec[c + d*x]^3*(-4*I + Tan[c + d*x]))/(a^4*d*(-I + Tan[c + d*x])^4)","A",1
162,1,73,132,0.2244843,"\int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Sec[c + d*x]/(a + I*a*Tan[c + d*x])^4,x]","\frac{i \sec ^4(c+d x) (7 i \sin (c+d x)+15 i \sin (3 (c+d x))+28 \cos (c+d x)+20 \cos (3 (c+d x)))}{140 a^4 d (\tan (c+d x)-i)^4}","\frac{2 i \sec (c+d x)}{35 d \left(a^4+i a^4 \tan (c+d x)\right)}+\frac{2 i \sec (c+d x)}{35 d \left(a^2+i a^2 \tan (c+d x)\right)^2}+\frac{3 i \sec (c+d x)}{35 a d (a+i a \tan (c+d x))^3}+\frac{i \sec (c+d x)}{7 d (a+i a \tan (c+d x))^4}",1,"((I/140)*Sec[c + d*x]^4*(28*Cos[c + d*x] + 20*Cos[3*(c + d*x)] + (7*I)*Sin[c + d*x] + (15*I)*Sin[3*(c + d*x)]))/(a^4*d*(-I + Tan[c + d*x])^4)","A",1
163,1,95,134,0.2449653,"\int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Cos[c + d*x]/(a + I*a*Tan[c + d*x])^4,x]","-\frac{i \sec ^4(c+d x) (-42 i \sin (c+d x)-135 i \sin (3 (c+d x))+35 i \sin (5 (c+d x))-168 \cos (c+d x)-180 \cos (3 (c+d x))+28 \cos (5 (c+d x)))}{1008 a^4 d (\tan (c+d x)-i)^4}","-\frac{4 \sin ^3(c+d x)}{63 a^4 d}+\frac{4 \sin (c+d x)}{21 a^4 d}+\frac{8 i \cos ^3(c+d x)}{63 d \left(a^4+i a^4 \tan (c+d x)\right)}+\frac{5 i \cos (c+d x)}{63 a d (a+i a \tan (c+d x))^3}+\frac{i \cos (c+d x)}{9 d (a+i a \tan (c+d x))^4}",1,"((-1/1008*I)*Sec[c + d*x]^4*(-168*Cos[c + d*x] - 180*Cos[3*(c + d*x)] + 28*Cos[5*(c + d*x)] - (42*I)*Sin[c + d*x] - (135*I)*Sin[3*(c + d*x)] + (35*I)*Sin[5*(c + d*x)]))/(a^4*d*(-I + Tan[c + d*x])^4)","A",1
164,1,117,156,0.5080152,"\int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^4,x]","-\frac{i \sec ^4(c+d x) (-231 i \sin (c+d x)-891 i \sin (3 (c+d x))+385 i \sin (5 (c+d x))+21 i \sin (7 (c+d x))-924 \cos (c+d x)-1188 \cos (3 (c+d x))+308 \cos (5 (c+d x))+12 \cos (7 (c+d x)))}{6336 a^4 d (\tan (c+d x)-i)^4}","\frac{2 \sin ^5(c+d x)}{33 a^4 d}-\frac{20 \sin ^3(c+d x)}{99 a^4 d}+\frac{10 \sin (c+d x)}{33 a^4 d}+\frac{4 i \cos ^5(c+d x)}{33 d \left(a^4+i a^4 \tan (c+d x)\right)}+\frac{7 i \cos ^3(c+d x)}{99 a d (a+i a \tan (c+d x))^3}+\frac{i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}",1,"((-1/6336*I)*Sec[c + d*x]^4*(-924*Cos[c + d*x] - 1188*Cos[3*(c + d*x)] + 308*Cos[5*(c + d*x)] + 12*Cos[7*(c + d*x)] - (231*I)*Sin[c + d*x] - (891*I)*Sin[3*(c + d*x)] + (385*I)*Sin[5*(c + d*x)] + (21*I)*Sin[7*(c + d*x)]))/(a^4*d*(-I + Tan[c + d*x])^4)","A",1
165,1,139,174,0.9018529,"\int \frac{\cos ^5(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^5/(a + I*a*Tan[c + d*x])^4,x]","-\frac{i \sec ^4(c+d x) (-6006 i \sin (c+d x)-25740 i \sin (3 (c+d x))+14300 i \sin (5 (c+d x))+1365 i \sin (7 (c+d x))+99 i \sin (9 (c+d x))-24024 \cos (c+d x)-34320 \cos (3 (c+d x))+11440 \cos (5 (c+d x))+780 \cos (7 (c+d x))+44 \cos (9 (c+d x)))}{183040 a^4 d (\tan (c+d x)-i)^4}","-\frac{8 \sin ^7(c+d x)}{143 a^4 d}+\frac{168 \sin ^5(c+d x)}{715 a^4 d}-\frac{56 \sin ^3(c+d x)}{143 a^4 d}+\frac{56 \sin (c+d x)}{143 a^4 d}+\frac{16 i \cos ^7(c+d x)}{143 d \left(a^4+i a^4 \tan (c+d x)\right)}+\frac{9 i \cos ^5(c+d x)}{143 a d (a+i a \tan (c+d x))^3}+\frac{i \cos ^5(c+d x)}{13 d (a+i a \tan (c+d x))^4}",1,"((-1/183040*I)*Sec[c + d*x]^4*(-24024*Cos[c + d*x] - 34320*Cos[3*(c + d*x)] + 11440*Cos[5*(c + d*x)] + 780*Cos[7*(c + d*x)] + 44*Cos[9*(c + d*x)] - (6006*I)*Sin[c + d*x] - (25740*I)*Sin[3*(c + d*x)] + (14300*I)*Sin[5*(c + d*x)] + (1365*I)*Sin[7*(c + d*x)] + (99*I)*Sin[9*(c + d*x)]))/(a^4*d*(-I + Tan[c + d*x])^4)","A",1
166,1,599,134,2.8666379,"\int \frac{\sec ^{14}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^14/(a + I*a*Tan[c + d*x])^8,x]","\frac{\sec (c) \sec ^{13}(c+d x) (-\cos (7 (c+d x))-i \sin (7 (c+d x))) (300 i d x \sin (c+2 d x)-985 \sin (c+2 d x)+300 i d x \sin (3 c+2 d x)+320 \sin (3 c+2 d x)+240 i d x \sin (3 c+4 d x)-512 \sin (3 c+4 d x)+240 i d x \sin (5 c+4 d x)+10 \sin (5 c+4 d x)+60 i d x \sin (5 c+6 d x)-97 \sin (5 c+6 d x)+60 i d x \sin (7 c+6 d x)-10 \sin (7 c+6 d x)+900 d x \cos (3 c+2 d x)-220 i \cos (3 c+2 d x)+360 d x \cos (3 c+4 d x)+238 i \cos (3 c+4 d x)+360 d x \cos (5 c+4 d x)-110 i \cos (5 c+4 d x)+60 d x \cos (5 c+6 d x)+77 i \cos (5 c+6 d x)+60 d x \cos (7 c+6 d x)-10 i \cos (7 c+6 d x)+900 i \cos (3 c+2 d x) \log (\cos (c+d x))+10 \cos (c) (120 i \log (\cos (c+d x))+120 d x-7 i)+5 \cos (c+2 d x) (180 i \log (\cos (c+d x))+180 d x+43 i)+360 i \cos (3 c+4 d x) \log (\cos (c+d x))+360 i \cos (5 c+4 d x) \log (\cos (c+d x))+60 i \cos (5 c+6 d x) \log (\cos (c+d x))+60 i \cos (7 c+6 d x) \log (\cos (c+d x))-300 \sin (c+2 d x) \log (\cos (c+d x))-300 \sin (3 c+2 d x) \log (\cos (c+d x))-240 \sin (3 c+4 d x) \log (\cos (c+d x))-240 \sin (5 c+4 d x) \log (\cos (c+d x))-60 \sin (5 c+6 d x) \log (\cos (c+d x))-60 \sin (7 c+6 d x) \log (\cos (c+d x))+870 \sin (c))}{20 a^8 d (\tan (c+d x)-i)^8}","\frac{\tan ^5(c+d x)}{5 a^8 d}+\frac{2 i \tan ^4(c+d x)}{a^8 d}-\frac{10 \tan ^3(c+d x)}{a^8 d}-\frac{36 i \tan ^2(c+d x)}{a^8 d}+\frac{129 \tan (c+d x)}{a^8 d}+\frac{64 i}{d \left(a^8+i a^8 \tan (c+d x)\right)}-\frac{192 i \log (\cos (c+d x))}{a^8 d}-\frac{192 x}{a^8}",1,"(Sec[c]*Sec[c + d*x]^13*(-Cos[7*(c + d*x)] - I*Sin[7*(c + d*x)])*((-220*I)*Cos[3*c + 2*d*x] + 900*d*x*Cos[3*c + 2*d*x] + (238*I)*Cos[3*c + 4*d*x] + 360*d*x*Cos[3*c + 4*d*x] - (110*I)*Cos[5*c + 4*d*x] + 360*d*x*Cos[5*c + 4*d*x] + (77*I)*Cos[5*c + 6*d*x] + 60*d*x*Cos[5*c + 6*d*x] - (10*I)*Cos[7*c + 6*d*x] + 60*d*x*Cos[7*c + 6*d*x] + 10*Cos[c]*(-7*I + 120*d*x + (120*I)*Log[Cos[c + d*x]]) + 5*Cos[c + 2*d*x]*(43*I + 180*d*x + (180*I)*Log[Cos[c + d*x]]) + (900*I)*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]] + (360*I)*Cos[3*c + 4*d*x]*Log[Cos[c + d*x]] + (360*I)*Cos[5*c + 4*d*x]*Log[Cos[c + d*x]] + (60*I)*Cos[5*c + 6*d*x]*Log[Cos[c + d*x]] + (60*I)*Cos[7*c + 6*d*x]*Log[Cos[c + d*x]] + 870*Sin[c] - 985*Sin[c + 2*d*x] + (300*I)*d*x*Sin[c + 2*d*x] - 300*Log[Cos[c + d*x]]*Sin[c + 2*d*x] + 320*Sin[3*c + 2*d*x] + (300*I)*d*x*Sin[3*c + 2*d*x] - 300*Log[Cos[c + d*x]]*Sin[3*c + 2*d*x] - 512*Sin[3*c + 4*d*x] + (240*I)*d*x*Sin[3*c + 4*d*x] - 240*Log[Cos[c + d*x]]*Sin[3*c + 4*d*x] + 10*Sin[5*c + 4*d*x] + (240*I)*d*x*Sin[5*c + 4*d*x] - 240*Log[Cos[c + d*x]]*Sin[5*c + 4*d*x] - 97*Sin[5*c + 6*d*x] + (60*I)*d*x*Sin[5*c + 6*d*x] - 60*Log[Cos[c + d*x]]*Sin[5*c + 6*d*x] - 10*Sin[7*c + 6*d*x] + (60*I)*d*x*Sin[7*c + 6*d*x] - 60*Log[Cos[c + d*x]]*Sin[7*c + 6*d*x]))/(20*a^8*d*(-I + Tan[c + d*x])^8)","B",1
167,1,537,126,1.5545519,"\int \frac{\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^12/(a + I*a*Tan[c + d*x])^8,x]","\frac{\sec (c) \sec ^{11}(c+d x) (\cos (6 (c+d x))+i \sin (6 (c+d x))) (120 i d x \sin (2 c+d x)+87 \sin (2 c+d x)+180 i d x \sin (2 c+3 d x)-96 \sin (2 c+3 d x)+180 i d x \sin (4 c+3 d x)+45 \sin (4 c+3 d x)+60 i d x \sin (4 c+5 d x)-44 \sin (4 c+5 d x)+60 i d x \sin (6 c+5 d x)+3 \sin (6 c+5 d x)+180 d x \cos (2 c+3 d x)+66 i \cos (2 c+3 d x)+180 d x \cos (4 c+3 d x)-75 i \cos (4 c+3 d x)+60 d x \cos (4 c+5 d x)+50 i \cos (4 c+5 d x)+60 d x \cos (6 c+5 d x)+3 i \cos (6 c+5 d x)+180 i \cos (2 c+3 d x) \log (\cos (c+d x))+3 \cos (2 c+d x) (80 i \log (\cos (c+d x))+80 d x-71 i)+\cos (d x) (240 i \log (\cos (c+d x))+240 d x-119 i)+180 i \cos (4 c+3 d x) \log (\cos (c+d x))+60 i \cos (4 c+5 d x) \log (\cos (c+d x))+60 i \cos (6 c+5 d x) \log (\cos (c+d x))-120 \sin (d x) \log (\cos (c+d x))-120 \sin (2 c+d x) \log (\cos (c+d x))-180 \sin (2 c+3 d x) \log (\cos (c+d x))-180 \sin (4 c+3 d x) \log (\cos (c+d x))-60 \sin (4 c+5 d x) \log (\cos (c+d x))-60 \sin (6 c+5 d x) \log (\cos (c+d x))+120 i d x \sin (d x)-101 \sin (d x))}{12 a^8 d (\tan (c+d x)-i)^8}","\frac{\tan ^3(c+d x)}{3 a^8 d}+\frac{4 i \tan ^2(c+d x)}{a^8 d}-\frac{31 \tan (c+d x)}{a^8 d}-\frac{80 i}{d \left(a^8+i a^8 \tan (c+d x)\right)}+\frac{80 i \log (\cos (c+d x))}{a^8 d}+\frac{80 x}{a^8}+\frac{16 i}{d \left(a^4+i a^4 \tan (c+d x)\right)^2}",1,"(Sec[c]*Sec[c + d*x]^11*(Cos[6*(c + d*x)] + I*Sin[6*(c + d*x)])*((66*I)*Cos[2*c + 3*d*x] + 180*d*x*Cos[2*c + 3*d*x] - (75*I)*Cos[4*c + 3*d*x] + 180*d*x*Cos[4*c + 3*d*x] + (50*I)*Cos[4*c + 5*d*x] + 60*d*x*Cos[4*c + 5*d*x] + (3*I)*Cos[6*c + 5*d*x] + 60*d*x*Cos[6*c + 5*d*x] + 3*Cos[2*c + d*x]*(-71*I + 80*d*x + (80*I)*Log[Cos[c + d*x]]) + Cos[d*x]*(-119*I + 240*d*x + (240*I)*Log[Cos[c + d*x]]) + (180*I)*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]] + (180*I)*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]] + (60*I)*Cos[4*c + 5*d*x]*Log[Cos[c + d*x]] + (60*I)*Cos[6*c + 5*d*x]*Log[Cos[c + d*x]] - 101*Sin[d*x] + (120*I)*d*x*Sin[d*x] - 120*Log[Cos[c + d*x]]*Sin[d*x] + 87*Sin[2*c + d*x] + (120*I)*d*x*Sin[2*c + d*x] - 120*Log[Cos[c + d*x]]*Sin[2*c + d*x] - 96*Sin[2*c + 3*d*x] + (180*I)*d*x*Sin[2*c + 3*d*x] - 180*Log[Cos[c + d*x]]*Sin[2*c + 3*d*x] + 45*Sin[4*c + 3*d*x] + (180*I)*d*x*Sin[4*c + 3*d*x] - 180*Log[Cos[c + d*x]]*Sin[4*c + 3*d*x] - 44*Sin[4*c + 5*d*x] + (60*I)*d*x*Sin[4*c + 5*d*x] - 60*Log[Cos[c + d*x]]*Sin[4*c + 5*d*x] + 3*Sin[6*c + 5*d*x] + (60*I)*d*x*Sin[6*c + 5*d*x] - 60*Log[Cos[c + d*x]]*Sin[6*c + 5*d*x]))/(12*a^8*d*(-I + Tan[c + d*x])^8)","B",1
168,1,397,116,1.0553227,"\int \frac{\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^10/(a + I*a*Tan[c + d*x])^8,x]","\frac{\sec (c) \sec ^9(c+d x) (-\cos (5 (c+d x))-i \sin (5 (c+d x))) (12 i d x \sin (c+2 d x)+11 \sin (c+2 d x)+12 i d x \sin (3 c+2 d x)+14 \sin (3 c+2 d x)+12 i d x \sin (3 c+4 d x)-4 \sin (3 c+4 d x)+12 i d x \sin (5 c+4 d x)-\sin (5 c+4 d x)+12 d x \cos (3 c+2 d x)-10 i \cos (3 c+2 d x)+12 d x \cos (3 c+4 d x)+2 i \cos (3 c+4 d x)+12 d x \cos (5 c+4 d x)-i \cos (5 c+4 d x)+\cos (c+2 d x) (12 i \log (\cos (c+d x))+12 d x-7 i)+12 i \cos (3 c+2 d x) \log (\cos (c+d x))+12 i \cos (3 c+4 d x) \log (\cos (c+d x))+12 i \cos (5 c+4 d x) \log (\cos (c+d x))-12 \sin (c+2 d x) \log (\cos (c+d x))-12 \sin (3 c+2 d x) \log (\cos (c+d x))-12 \sin (3 c+4 d x) \log (\cos (c+d x))-12 \sin (5 c+4 d x) \log (\cos (c+d x))-12 i \cos (c))}{6 a^8 d (\tan (c+d x)-i)^8}","\frac{\tan (c+d x)}{a^8 d}+\frac{24 i}{d \left(a^8+i a^8 \tan (c+d x)\right)}-\frac{8 i \log (\cos (c+d x))}{a^8 d}-\frac{8 x}{a^8}+\frac{16 i}{3 a^5 d (a+i a \tan (c+d x))^3}-\frac{16 i}{d \left(a^4+i a^4 \tan (c+d x)\right)^2}",1,"(Sec[c]*Sec[c + d*x]^9*(-Cos[5*(c + d*x)] - I*Sin[5*(c + d*x)])*((-12*I)*Cos[c] - (10*I)*Cos[3*c + 2*d*x] + 12*d*x*Cos[3*c + 2*d*x] + (2*I)*Cos[3*c + 4*d*x] + 12*d*x*Cos[3*c + 4*d*x] - I*Cos[5*c + 4*d*x] + 12*d*x*Cos[5*c + 4*d*x] + Cos[c + 2*d*x]*(-7*I + 12*d*x + (12*I)*Log[Cos[c + d*x]]) + (12*I)*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]] + (12*I)*Cos[3*c + 4*d*x]*Log[Cos[c + d*x]] + (12*I)*Cos[5*c + 4*d*x]*Log[Cos[c + d*x]] + 11*Sin[c + 2*d*x] + (12*I)*d*x*Sin[c + 2*d*x] - 12*Log[Cos[c + d*x]]*Sin[c + 2*d*x] + 14*Sin[3*c + 2*d*x] + (12*I)*d*x*Sin[3*c + 2*d*x] - 12*Log[Cos[c + d*x]]*Sin[3*c + 2*d*x] - 4*Sin[3*c + 4*d*x] + (12*I)*d*x*Sin[3*c + 4*d*x] - 12*Log[Cos[c + d*x]]*Sin[3*c + 4*d*x] - Sin[5*c + 4*d*x] + (12*I)*d*x*Sin[5*c + 4*d*x] - 12*Log[Cos[c + d*x]]*Sin[5*c + 4*d*x]))/(6*a^8*d*(-I + Tan[c + d*x])^8)","B",1
169,1,32,43,0.0720904,"\int \frac{\sec ^8(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^8,x]","\frac{i \sec ^8(c+d x)}{8 d (a+i a \tan (c+d x))^8}","\frac{i (a-i a \tan (c+d x))^4}{8 d \left(a^3+i a^3 \tan (c+d x)\right)^4}",1,"((I/8)*Sec[c + d*x]^8)/(d*(a + I*a*Tan[c + d*x])^8)","A",1
170,1,56,81,0.2785601,"\int \frac{\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^8,x]","\frac{i \sec ^8(c+d x) (4 i \sin (2 (c+d x))+16 \cos (2 (c+d x))+15)}{240 a^8 d (\tan (c+d x)-i)^8}","\frac{i}{3 a^5 d (a+i a \tan (c+d x))^3}+\frac{4 i}{5 a^3 d (a+i a \tan (c+d x))^5}-\frac{i}{d \left(a^2+i a^2 \tan (c+d x)\right)^4}",1,"((I/240)*Sec[c + d*x]^8*(15 + 16*Cos[2*(c + d*x)] + (4*I)*Sin[2*(c + d*x)]))/(a^8*d*(-I + Tan[c + d*x])^8)","A",1
171,1,78,55,0.2661245,"\int \frac{\sec ^4(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^8,x]","\frac{i \sec ^8(c+d x) (16 i \sin (2 (c+d x))+10 i \sin (4 (c+d x))+64 \cos (2 (c+d x))+20 \cos (4 (c+d x))+45)}{960 a^8 d (\tan (c+d x)-i)^8}","\frac{i}{3 a^2 d (a+i a \tan (c+d x))^6}-\frac{i}{5 a^3 d (a+i a \tan (c+d x))^5}",1,"((I/960)*Sec[c + d*x]^8*(45 + 64*Cos[2*(c + d*x)] + 20*Cos[4*(c + d*x)] + (16*I)*Sin[2*(c + d*x)] + (10*I)*Sin[4*(c + d*x)]))/(a^8*d*(-I + Tan[c + d*x])^8)","A",1
172,1,100,27,0.3085428,"\int \frac{\sec ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^8,x]","\frac{i \sec ^8(c+d x) (14 i \sin (2 (c+d x))+14 i \sin (4 (c+d x))+6 i \sin (6 (c+d x))+56 \cos (2 (c+d x))+28 \cos (4 (c+d x))+8 \cos (6 (c+d x))+35)}{896 a^8 d (\tan (c+d x)-i)^8}","\frac{i}{7 a d (a+i a \tan (c+d x))^7}",1,"((I/896)*Sec[c + d*x]^8*(35 + 56*Cos[2*(c + d*x)] + 28*Cos[4*(c + d*x)] + 8*Cos[6*(c + d*x)] + (14*I)*Sin[2*(c + d*x)] + (14*I)*Sin[4*(c + d*x)] + (6*I)*Sin[6*(c + d*x)]))/(a^8*d*(-I + Tan[c + d*x])^8)","B",1
173,1,148,229,0.717962,"\int \frac{1}{(a+i a \tan (c+d x))^8} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-8),x]","\frac{\sec ^8(c+d x) (-6272 \sin (2 (c+d x))-7840 \sin (4 (c+d x))-5760 \sin (6 (c+d x))+1680 i d x \sin (8 (c+d x))+105 \sin (8 (c+d x))+25088 i \cos (2 (c+d x))+15680 i \cos (4 (c+d x))+7680 i \cos (6 (c+d x))+1680 d x \cos (8 (c+d x))+105 i \cos (8 (c+d x))+14700 i)}{430080 a^8 d (\tan (c+d x)-i)^8}","\frac{i}{256 d \left(a^8+i a^8 \tan (c+d x)\right)}+\frac{x}{256 a^8}+\frac{i}{256 d \left(a^4+i a^4 \tan (c+d x)\right)^2}+\frac{i}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac{i}{192 a^2 d \left(a^2+i a^2 \tan (c+d x)\right)^3}+\frac{i}{128 d \left(a^2+i a^2 \tan (c+d x)\right)^4}+\frac{i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac{i}{28 a d (a+i a \tan (c+d x))^7}+\frac{i}{16 d (a+i a \tan (c+d x))^8}",1,"(Sec[c + d*x]^8*(14700*I + (25088*I)*Cos[2*(c + d*x)] + (15680*I)*Cos[4*(c + d*x)] + (7680*I)*Cos[6*(c + d*x)] + (105*I)*Cos[8*(c + d*x)] + 1680*d*x*Cos[8*(c + d*x)] - 6272*Sin[2*(c + d*x)] - 7840*Sin[4*(c + d*x)] - 5760*Sin[6*(c + d*x)] + 105*Sin[8*(c + d*x)] + (1680*I)*d*x*Sin[8*(c + d*x)]))/(430080*a^8*d*(-I + Tan[c + d*x])^8)","A",1
174,1,170,278,1.2349023,"\int \frac{\cos ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^8,x]","\frac{\sec ^8(c+d x) (-7056 \sin (2 (c+d x))-10080 \sin (4 (c+d x))-9720 \sin (6 (c+d x))+5040 i d x \sin (8 (c+d x))+315 \sin (8 (c+d x))+280 \sin (10 (c+d x))+28224 i \cos (2 (c+d x))+20160 i \cos (4 (c+d x))+12960 i \cos (6 (c+d x))+5040 d x \cos (8 (c+d x))+315 i \cos (8 (c+d x))-224 i \cos (10 (c+d x))+15876 i)}{516096 a^8 d (\tan (c+d x)-i)^8}","-\frac{i}{1024 d \left(a^8-i a^8 \tan (c+d x)\right)}+\frac{9 i}{1024 d \left(a^8+i a^8 \tan (c+d x)\right)}+\frac{5 x}{512 a^8}+\frac{7 i}{768 a^5 d (a+i a \tan (c+d x))^3}+\frac{i}{128 d \left(a^4+i a^4 \tan (c+d x)\right)^2}+\frac{i}{64 a^3 d (a+i a \tan (c+d x))^5}+\frac{3 i}{256 d \left(a^2+i a^2 \tan (c+d x)\right)^4}+\frac{i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac{i a}{36 d (a+i a \tan (c+d x))^9}+\frac{i}{32 d (a+i a \tan (c+d x))^8}+\frac{3 i}{112 a d (a+i a \tan (c+d x))^7}",1,"(Sec[c + d*x]^8*(15876*I + (28224*I)*Cos[2*(c + d*x)] + (20160*I)*Cos[4*(c + d*x)] + (12960*I)*Cos[6*(c + d*x)] + (315*I)*Cos[8*(c + d*x)] + 5040*d*x*Cos[8*(c + d*x)] - (224*I)*Cos[10*(c + d*x)] - 7056*Sin[2*(c + d*x)] - 10080*Sin[4*(c + d*x)] - 9720*Sin[6*(c + d*x)] + 315*Sin[8*(c + d*x)] + (5040*I)*d*x*Sin[8*(c + d*x)] + 280*Sin[10*(c + d*x)]))/(516096*a^8*d*(-I + Tan[c + d*x])^8)","A",1
175,1,192,333,1.7284457,"\int \frac{\cos ^4(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^8,x]","\frac{\sec ^8(c+d x) (-44352 \sin (2 (c+d x))-69300 \sin (4 (c+d x))-79200 \sin (6 (c+d x))+55440 i d x \sin (8 (c+d x))+3465 \sin (8 (c+d x))+5600 \sin (10 (c+d x))+252 \sin (12 (c+d x))+177408 i \cos (2 (c+d x))+138600 i \cos (4 (c+d x))+105600 i \cos (6 (c+d x))+55440 d x \cos (8 (c+d x))+3465 i \cos (8 (c+d x))-4480 i \cos (10 (c+d x))-168 i \cos (12 (c+d x))+97020 i)}{3440640 a^8 d (\tan (c+d x)-i)^8}","-\frac{11 i}{4096 d \left(a^8-i a^8 \tan (c+d x)\right)}+\frac{55 i}{4096 d \left(a^8+i a^8 \tan (c+d x)\right)}+\frac{33 x}{2048 a^8}+\frac{3 i}{256 a^5 d (a+i a \tan (c+d x))^3}-\frac{i}{4096 d \left(a^4-i a^4 \tan (c+d x)\right)^2}+\frac{45 i}{4096 d \left(a^4+i a^4 \tan (c+d x)\right)^2}+\frac{21 i}{1280 a^3 d (a+i a \tan (c+d x))^5}+\frac{i a^2}{80 d (a+i a \tan (c+d x))^{10}}+\frac{7 i}{512 d \left(a^2+i a^2 \tan (c+d x)\right)^4}+\frac{5 i}{256 a^2 d (a+i a \tan (c+d x))^6}+\frac{i a}{48 d (a+i a \tan (c+d x))^9}+\frac{3 i}{128 d (a+i a \tan (c+d x))^8}+\frac{5 i}{224 a d (a+i a \tan (c+d x))^7}",1,"(Sec[c + d*x]^8*(97020*I + (177408*I)*Cos[2*(c + d*x)] + (138600*I)*Cos[4*(c + d*x)] + (105600*I)*Cos[6*(c + d*x)] + (3465*I)*Cos[8*(c + d*x)] + 55440*d*x*Cos[8*(c + d*x)] - (4480*I)*Cos[10*(c + d*x)] - (168*I)*Cos[12*(c + d*x)] - 44352*Sin[2*(c + d*x)] - 69300*Sin[4*(c + d*x)] - 79200*Sin[6*(c + d*x)] + 3465*Sin[8*(c + d*x)] + (55440*I)*d*x*Sin[8*(c + d*x)] + 5600*Sin[10*(c + d*x)] + 252*Sin[12*(c + d*x)]))/(3440640*a^8*d*(-I + Tan[c + d*x])^8)","A",1
176,1,1704,205,6.4033594,"\int \frac{\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^13/(a + I*a*Tan[c + d*x])^8,x]","-\frac{1155 \cos (8 c) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8}{8 d (i \tan (c+d x) a+a)^8}+\frac{1155 \cos (8 c) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8}{8 d (i \tan (c+d x) a+a)^8}+\frac{\cos (3 d x) \sec ^8(c+d x) \left(\frac{32}{3} i \cos (5 c)-\frac{32}{3} \sin (5 c)\right) (\cos (d x)+i \sin (d x))^8}{d (i \tan (c+d x) a+a)^8}+\frac{\cos (d x) \sec ^8(c+d x) (160 \sin (7 c)-160 i \cos (7 c)) (\cos (d x)+i \sin (d x))^8}{d (i \tan (c+d x) a+a)^8}-\frac{1155 i \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^8(c+d x) \sin (8 c) (\cos (d x)+i \sin (d x))^8}{8 d (i \tan (c+d x) a+a)^8}+\frac{1155 i \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^8(c+d x) \sin (8 c) (\cos (d x)+i \sin (d x))^8}{8 d (i \tan (c+d x) a+a)^8}+\frac{\sec (c) \sec ^8(c+d x) \left(\frac{236}{3} \sin (8 c)-\frac{236}{3} i \cos (8 c)\right) (\cos (d x)+i \sin (d x))^8}{d (i \tan (c+d x) a+a)^8}+\frac{\sec ^8(c+d x) (-160 \cos (7 c)-160 i \sin (7 c)) \sin (d x) (\cos (d x)+i \sin (d x))^8}{d (i \tan (c+d x) a+a)^8}+\frac{\sec ^8(c+d x) \left(\frac{32}{3} \cos (5 c)+\frac{32}{3} i \sin (5 c)\right) \sin (3 d x) (\cos (d x)+i \sin (d x))^8}{d (i \tan (c+d x) a+a)^8}+\frac{236 \sec ^8(c+d x) \left(\frac{1}{2} \cos \left(8 c-\frac{d x}{2}\right)-\frac{1}{2} \cos \left(8 c+\frac{d x}{2}\right)+\frac{1}{2} i \sin \left(8 c-\frac{d x}{2}\right)-\frac{1}{2} i \sin \left(8 c+\frac{d x}{2}\right)\right) (\cos (d x)+i \sin (d x))^8}{3 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) (i \tan (c+d x) a+a)^8}+\frac{4 \sec ^8(c+d x) \left(\frac{1}{2} \cos \left(8 c-\frac{d x}{2}\right)-\frac{1}{2} \cos \left(8 c+\frac{d x}{2}\right)+\frac{1}{2} i \sin \left(8 c-\frac{d x}{2}\right)-\frac{1}{2} i \sin \left(8 c+\frac{d x}{2}\right)\right) (\cos (d x)+i \sin (d x))^8}{3 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 (i \tan (c+d x) a+a)^8}+\frac{236 \sec ^8(c+d x) \left(-\frac{1}{2} \cos \left(8 c-\frac{d x}{2}\right)+\frac{1}{2} \cos \left(8 c+\frac{d x}{2}\right)-\frac{1}{2} i \sin \left(8 c-\frac{d x}{2}\right)+\frac{1}{2} i \sin \left(8 c+\frac{d x}{2}\right)\right) (\cos (d x)+i \sin (d x))^8}{3 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) (i \tan (c+d x) a+a)^8}+\frac{4 \sec ^8(c+d x) \left(-\frac{1}{2} \cos \left(8 c-\frac{d x}{2}\right)+\frac{1}{2} \cos \left(8 c+\frac{d x}{2}\right)-\frac{1}{2} i \sin \left(8 c-\frac{d x}{2}\right)+\frac{1}{2} i \sin \left(8 c+\frac{d x}{2}\right)\right) (\cos (d x)+i \sin (d x))^8}{3 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 (i \tan (c+d x) a+a)^8}-\frac{\left(\frac{1}{96}+\frac{i}{96}\right) \sec ^8(c+d x) \left(-407 i \cos \left(\frac{15 c}{2}\right)+343 \cos \left(\frac{17 c}{2}\right)+407 \sin \left(\frac{15 c}{2}\right)+343 i \sin \left(\frac{17 c}{2}\right)\right) (\cos (d x)+i \sin (d x))^8}{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 (i \tan (c+d x) a+a)^8}+\frac{\left(\frac{1}{96}+\frac{i}{96}\right) \sec ^8(c+d x) \left(407 \cos \left(\frac{15 c}{2}\right)-343 i \cos \left(\frac{17 c}{2}\right)+407 i \sin \left(\frac{15 c}{2}\right)+343 \sin \left(\frac{17 c}{2}\right)\right) (\cos (d x)+i \sin (d x))^8}{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 (i \tan (c+d x) a+a)^8}+\frac{\sec ^8(c+d x) \left(\frac{1}{16} \cos (8 c)+\frac{1}{16} i \sin (8 c)\right) (\cos (d x)+i \sin (d x))^8}{d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4 (i \tan (c+d x) a+a)^8}+\frac{\sec ^8(c+d x) \left(-\frac{1}{16} \cos (8 c)-\frac{1}{16} i \sin (8 c)\right) (\cos (d x)+i \sin (d x))^8}{d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4 (i \tan (c+d x) a+a)^8}","\frac{1155 \tanh ^{-1}(\sin (c+d x))}{8 a^8 d}-\frac{154 i \sec ^5(c+d x)}{d \left(a^8+i a^8 \tan (c+d x)\right)}+\frac{385 \tan (c+d x) \sec ^3(c+d x)}{4 a^8 d}+\frac{1155 \tan (c+d x) \sec (c+d x)}{8 a^8 d}-\frac{22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}-\frac{66 i \sec ^7(c+d x)}{a^2 d \left(a^2+i a^2 \tan (c+d x)\right)^3}+\frac{2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}",1,"(-1155*Cos[8*c]*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8)/(8*d*(a + I*a*Tan[c + d*x])^8) + (1155*Cos[8*c]*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8)/(8*d*(a + I*a*Tan[c + d*x])^8) + (Cos[3*d*x]*Sec[c + d*x]^8*(((32*I)/3)*Cos[5*c] - (32*Sin[5*c])/3)*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (Cos[d*x]*Sec[c + d*x]^8*((-160*I)*Cos[7*c] + 160*Sin[7*c])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) - (((1155*I)/8)*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^8*Sin[8*c]*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (((1155*I)/8)*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^8*Sin[8*c]*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (Sec[c]*Sec[c + d*x]^8*(((-236*I)/3)*Cos[8*c] + (236*Sin[8*c])/3)*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*(-160*Cos[7*c] - (160*I)*Sin[7*c])*(Cos[d*x] + I*Sin[d*x])^8*Sin[d*x])/(d*(a + I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*((32*Cos[5*c])/3 + ((32*I)/3)*Sin[5*c])*(Cos[d*x] + I*Sin[d*x])^8*Sin[3*d*x])/(d*(a + I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*(Cos[8*c]/16 + (I/16)*Sin[8*c])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^4*(a + I*a*Tan[c + d*x])^8) - ((1/96 + I/96)*Sec[c + d*x]^8*((-407*I)*Cos[(15*c)/2] + 343*Cos[(17*c)/2] + 407*Sin[(15*c)/2] + (343*I)*Sin[(17*c)/2])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2*(a + I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*(-1/16*Cos[8*c] - (I/16)*Sin[8*c])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4*(a + I*a*Tan[c + d*x])^8) + ((1/96 + I/96)*Sec[c + d*x]^8*(407*Cos[(15*c)/2] - (343*I)*Cos[(17*c)/2] + (407*I)*Sin[(15*c)/2] + 343*Sin[(17*c)/2])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2*(a + I*a*Tan[c + d*x])^8) + (236*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8*(Cos[8*c - (d*x)/2]/2 - Cos[8*c + (d*x)/2]/2 + (I/2)*Sin[8*c - (d*x)/2] - (I/2)*Sin[8*c + (d*x)/2]))/(3*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])*(a + I*a*Tan[c + d*x])^8) + (4*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8*(Cos[8*c - (d*x)/2]/2 - Cos[8*c + (d*x)/2]/2 + (I/2)*Sin[8*c - (d*x)/2] - (I/2)*Sin[8*c + (d*x)/2]))/(3*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3*(a + I*a*Tan[c + d*x])^8) + (4*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8*(-1/2*Cos[8*c - (d*x)/2] + Cos[8*c + (d*x)/2]/2 - (I/2)*Sin[8*c - (d*x)/2] + (I/2)*Sin[8*c + (d*x)/2]))/(3*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3*(a + I*a*Tan[c + d*x])^8) + (236*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8*(-1/2*Cos[8*c - (d*x)/2] + Cos[8*c + (d*x)/2]/2 - (I/2)*Sin[8*c - (d*x)/2] + (I/2)*Sin[8*c + (d*x)/2]))/(3*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])*(a + I*a*Tan[c + d*x])^8)","B",0
177,1,1244,183,6.2824556,"\int \frac{\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^11/(a + I*a*Tan[c + d*x])^8,x]","\frac{63 \cos (8 c) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8}{2 d (i \tan (c+d x) a+a)^8}-\frac{63 \cos (8 c) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8}{2 d (i \tan (c+d x) a+a)^8}+\frac{\cos (5 d x) \sec ^8(c+d x) \left(\frac{8}{5} i \cos (3 c)-\frac{8}{5} \sin (3 c)\right) (\cos (d x)+i \sin (d x))^8}{d (i \tan (c+d x) a+a)^8}+\frac{\cos (3 d x) \sec ^8(c+d x) (8 \sin (5 c)-8 i \cos (5 c)) (\cos (d x)+i \sin (d x))^8}{d (i \tan (c+d x) a+a)^8}+\frac{\cos (d x) \sec ^8(c+d x) (48 i \cos (7 c)-48 \sin (7 c)) (\cos (d x)+i \sin (d x))^8}{d (i \tan (c+d x) a+a)^8}+\frac{\sec (c) \sec ^8(c+d x) (8 i \cos (8 c)-8 \sin (8 c)) (\cos (d x)+i \sin (d x))^8}{d (i \tan (c+d x) a+a)^8}+\frac{63 i \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^8(c+d x) \sin (8 c) (\cos (d x)+i \sin (d x))^8}{2 d (i \tan (c+d x) a+a)^8}-\frac{63 i \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^8(c+d x) \sin (8 c) (\cos (d x)+i \sin (d x))^8}{2 d (i \tan (c+d x) a+a)^8}+\frac{\sec ^8(c+d x) (48 \cos (7 c)+48 i \sin (7 c)) \sin (d x) (\cos (d x)+i \sin (d x))^8}{d (i \tan (c+d x) a+a)^8}+\frac{\sec ^8(c+d x) (-8 \cos (5 c)-8 i \sin (5 c)) \sin (3 d x) (\cos (d x)+i \sin (d x))^8}{d (i \tan (c+d x) a+a)^8}+\frac{\sec ^8(c+d x) \left(\frac{8}{5} \cos (3 c)+\frac{8}{5} i \sin (3 c)\right) \sin (5 d x) (\cos (d x)+i \sin (d x))^8}{d (i \tan (c+d x) a+a)^8}+\frac{8 \sec ^8(c+d x) \left(\frac{1}{2} \cos \left(8 c-\frac{d x}{2}\right)-\frac{1}{2} \cos \left(8 c+\frac{d x}{2}\right)+\frac{1}{2} i \sin \left(8 c-\frac{d x}{2}\right)-\frac{1}{2} i \sin \left(8 c+\frac{d x}{2}\right)\right) (\cos (d x)+i \sin (d x))^8}{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) (i \tan (c+d x) a+a)^8}+\frac{8 \sec ^8(c+d x) \left(-\frac{1}{2} \cos \left(8 c-\frac{d x}{2}\right)+\frac{1}{2} \cos \left(8 c+\frac{d x}{2}\right)-\frac{1}{2} i \sin \left(8 c-\frac{d x}{2}\right)+\frac{1}{2} i \sin \left(8 c+\frac{d x}{2}\right)\right) (\cos (d x)+i \sin (d x))^8}{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) (i \tan (c+d x) a+a)^8}+\frac{\sec ^8(c+d x) \left(\frac{1}{4} \cos (8 c)+\frac{1}{4} i \sin (8 c)\right) (\cos (d x)+i \sin (d x))^8}{d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2 (i \tan (c+d x) a+a)^8}+\frac{\sec ^8(c+d x) \left(-\frac{1}{4} \cos (8 c)-\frac{1}{4} i \sin (8 c)\right) (\cos (d x)+i \sin (d x))^8}{d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2 (i \tan (c+d x) a+a)^8}","-\frac{63 \tanh ^{-1}(\sin (c+d x))}{2 a^8 d}+\frac{42 i \sec ^3(c+d x)}{d \left(a^8+i a^8 \tan (c+d x)\right)}-\frac{63 \tan (c+d x) \sec (c+d x)}{2 a^8 d}-\frac{6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac{42 i \sec ^5(c+d x)}{5 a^2 d \left(a^2+i a^2 \tan (c+d x)\right)^3}+\frac{2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}",1,"(63*Cos[8*c]*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8)/(2*d*(a + I*a*Tan[c + d*x])^8) - (63*Cos[8*c]*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8)/(2*d*(a + I*a*Tan[c + d*x])^8) + (Cos[5*d*x]*Sec[c + d*x]^8*(((8*I)/5)*Cos[3*c] - (8*Sin[3*c])/5)*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (Cos[3*d*x]*Sec[c + d*x]^8*((-8*I)*Cos[5*c] + 8*Sin[5*c])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (Cos[d*x]*Sec[c + d*x]^8*((48*I)*Cos[7*c] - 48*Sin[7*c])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (Sec[c]*Sec[c + d*x]^8*((8*I)*Cos[8*c] - 8*Sin[8*c])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (((63*I)/2)*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^8*Sin[8*c]*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) - (((63*I)/2)*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^8*Sin[8*c]*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*(48*Cos[7*c] + (48*I)*Sin[7*c])*(Cos[d*x] + I*Sin[d*x])^8*Sin[d*x])/(d*(a + I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*(-8*Cos[5*c] - (8*I)*Sin[5*c])*(Cos[d*x] + I*Sin[d*x])^8*Sin[3*d*x])/(d*(a + I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*((8*Cos[3*c])/5 + ((8*I)/5)*Sin[3*c])*(Cos[d*x] + I*Sin[d*x])^8*Sin[5*d*x])/(d*(a + I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*(Cos[8*c]/4 + (I/4)*Sin[8*c])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2*(a + I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*(-1/4*Cos[8*c] - (I/4)*Sin[8*c])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2*(a + I*a*Tan[c + d*x])^8) + (8*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8*(Cos[8*c - (d*x)/2]/2 - Cos[8*c + (d*x)/2]/2 + (I/2)*Sin[8*c - (d*x)/2] - (I/2)*Sin[8*c + (d*x)/2]))/(d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])*(a + I*a*Tan[c + d*x])^8) + (8*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8*(-1/2*Cos[8*c - (d*x)/2] + Cos[8*c + (d*x)/2]/2 - (I/2)*Sin[8*c - (d*x)/2] + (I/2)*Sin[8*c + (d*x)/2]))/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])*(a + I*a*Tan[c + d*x])^8)","B",1
178,1,304,156,1.0797654,"\int \frac{\sec ^9(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^8,x]","\frac{\sec ^8(c+d x) \left(\cos \left(\frac{9}{2} (c+d x)\right)+i \sin \left(\frac{9}{2} (c+d x)\right)\right) \left(-70 \sin \left(\frac{1}{2} (c+d x)\right)-42 \sin \left(\frac{3}{2} (c+d x)\right)+210 \sin \left(\frac{5}{2} (c+d x)\right)+30 \sin \left(\frac{7}{2} (c+d x)\right)+70 i \cos \left(\frac{1}{2} (c+d x)\right)-42 i \cos \left(\frac{3}{2} (c+d x)\right)-210 i \cos \left(\frac{5}{2} (c+d x)\right)+30 i \cos \left(\frac{7}{2} (c+d x)\right)-105 \cos \left(\frac{7}{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)+105 \cos \left(\frac{7}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-105 i \sin \left(\frac{7}{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)+105 i \sin \left(\frac{7}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{105 a^8 d (\tan (c+d x)-i)^8}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^8 d}-\frac{2 i \sec (c+d x)}{d \left(a^8+i a^8 \tan (c+d x)\right)}-\frac{2 i \sec ^5(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac{2 i \sec ^3(c+d x)}{3 a^2 d \left(a^2+i a^2 \tan (c+d x)\right)^3}+\frac{2 i \sec ^7(c+d x)}{7 a d (a+i a \tan (c+d x))^7}",1,"(Sec[c + d*x]^8*((70*I)*Cos[(c + d*x)/2] - (42*I)*Cos[(3*(c + d*x))/2] - (210*I)*Cos[(5*(c + d*x))/2] + (30*I)*Cos[(7*(c + d*x))/2] - 105*Cos[(7*(c + d*x))/2]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 105*Cos[(7*(c + d*x))/2]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 70*Sin[(c + d*x)/2] - 42*Sin[(3*(c + d*x))/2] + 210*Sin[(5*(c + d*x))/2] + 30*Sin[(7*(c + d*x))/2] - (105*I)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[(7*(c + d*x))/2] + (105*I)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[(7*(c + d*x))/2])*(Cos[(9*(c + d*x))/2] + I*Sin[(9*(c + d*x))/2]))/(105*a^8*d*(-I + Tan[c + d*x])^8)","A",1
179,1,40,68,0.1247121,"\int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^8,x]","-\frac{(\tan (c+d x)-8 i) \sec ^7(c+d x)}{63 a^8 d (\tan (c+d x)-i)^8}","\frac{i \sec ^7(c+d x)}{63 a d (a+i a \tan (c+d x))^7}+\frac{i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8}",1,"-1/63*(Sec[c + d*x]^7*(-8*I + Tan[c + d*x]))/(a^8*d*(-I + Tan[c + d*x])^8)","A",1
180,1,73,138,0.3199978,"\int \frac{\sec ^5(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^8,x]","\frac{i \sec ^8(c+d x) (55 i \sin (c+d x)+63 i \sin (3 (c+d x))+440 \cos (c+d x)+168 \cos (3 (c+d x)))}{4620 a^8 d (\tan (c+d x)-i)^8}","\frac{2 i \sec ^5(c+d x)}{1155 a^3 d (a+i a \tan (c+d x))^5}+\frac{2 i \sec ^5(c+d x)}{231 a^2 d (a+i a \tan (c+d x))^6}+\frac{i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac{i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}",1,"((I/4620)*Sec[c + d*x]^8*(440*Cos[c + d*x] + 168*Cos[3*(c + d*x)] + (55*I)*Sin[c + d*x] + (63*I)*Sin[3*(c + d*x)]))/(a^8*d*(-I + Tan[c + d*x])^8)","A",1
181,1,95,213,0.3238577,"\int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^8,x]","\frac{i \sec ^8(c+d x) (1430 i \sin (c+d x)+2457 i \sin (3 (c+d x))+1155 i \sin (5 (c+d x))+11440 \cos (c+d x)+6552 \cos (3 (c+d x))+1848 \cos (5 (c+d x)))}{144144 a^8 d (\tan (c+d x)-i)^8}","\frac{20 i \sec ^3(c+d x)}{3003 a^3 d (a+i a \tan (c+d x))^5}+\frac{8 i \sec ^3(c+d x)}{9009 a^2 d \left(a^2+i a^2 \tan (c+d x)\right)^3}+\frac{8 i \sec ^3(c+d x)}{3003 d \left(a^2+i a^2 \tan (c+d x)\right)^4}+\frac{20 i \sec ^3(c+d x)}{1287 a^2 d (a+i a \tan (c+d x))^6}+\frac{5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac{i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}",1,"((I/144144)*Sec[c + d*x]^8*(11440*Cos[c + d*x] + 6552*Cos[3*(c + d*x)] + 1848*Cos[5*(c + d*x)] + (1430*I)*Sin[c + d*x] + (2457*I)*Sin[3*(c + d*x)] + (1155*I)*Sin[5*(c + d*x)]))/(a^8*d*(-I + Tan[c + d*x])^8)","A",1
182,1,117,269,0.6403423,"\int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Sec[c + d*x]/(a + I*a*Tan[c + d*x])^8,x]","\frac{i \sec ^8(c+d x) (3575 i \sin (c+d x)+7371 i \sin (3 (c+d x))+5775 i \sin (5 (c+d x))+3003 i \sin (7 (c+d x))+28600 \cos (c+d x)+19656 \cos (3 (c+d x))+9240 \cos (5 (c+d x))+3432 \cos (7 (c+d x)))}{411840 a^8 d (\tan (c+d x)-i)^8}","\frac{16 i \sec (c+d x)}{6435 d \left(a^8+i a^8 \tan (c+d x)\right)}+\frac{16 i \sec (c+d x)}{6435 d \left(a^4+i a^4 \tan (c+d x)\right)^2}+\frac{14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac{8 i \sec (c+d x)}{2145 a^2 d \left(a^2+i a^2 \tan (c+d x)\right)^3}+\frac{8 i \sec (c+d x)}{1287 d \left(a^2+i a^2 \tan (c+d x)\right)^4}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}",1,"((I/411840)*Sec[c + d*x]^8*(28600*Cos[c + d*x] + 19656*Cos[3*(c + d*x)] + 9240*Cos[5*(c + d*x)] + 3432*Cos[7*(c + d*x)] + (3575*I)*Sin[c + d*x] + (7371*I)*Sin[3*(c + d*x)] + (5775*I)*Sin[5*(c + d*x)] + (3003*I)*Sin[7*(c + d*x)]))/(a^8*d*(-I + Tan[c + d*x])^8)","A",1
183,1,139,271,1.137571,"\int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Cos[c + d*x]/(a + I*a*Tan[c + d*x])^8,x]","-\frac{i \sec ^8(c+d x) (-24310 i \sin (c+d x)-55692 i \sin (3 (c+d x))-56100 i \sin (5 (c+d x))-51051 i \sin (7 (c+d x))+6435 i \sin (9 (c+d x))-194480 \cos (c+d x)-148512 \cos (3 (c+d x))-89760 \cos (5 (c+d x))-58344 \cos (7 (c+d x))+5720 \cos (9 (c+d x)))}{3111680 a^8 d (\tan (c+d x)-i)^8}","-\frac{64 \sin ^3(c+d x)}{12155 a^8 d}+\frac{192 \sin (c+d x)}{12155 a^8 d}+\frac{128 i \cos ^3(c+d x)}{12155 d \left(a^8+i a^8 \tan (c+d x)\right)}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{16 i \cos (c+d x)}{2431 a^2 d \left(a^2+i a^2 \tan (c+d x)\right)^3}+\frac{112 i \cos (c+d x)}{12155 d \left(a^2+i a^2 \tan (c+d x)\right)^4}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}",1,"((-1/3111680*I)*Sec[c + d*x]^8*(-194480*Cos[c + d*x] - 148512*Cos[3*(c + d*x)] - 89760*Cos[5*(c + d*x)] - 58344*Cos[7*(c + d*x)] + 5720*Cos[9*(c + d*x)] - (24310*I)*Sin[c + d*x] - (55692*I)*Sin[3*(c + d*x)] - (56100*I)*Sin[5*(c + d*x)] - (51051*I)*Sin[7*(c + d*x)] + (6435*I)*Sin[9*(c + d*x)]))/(a^8*d*(-I + Tan[c + d*x])^8)","A",1
184,1,161,301,1.5249649,"\int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^8,x]","-\frac{i \sec ^8(c+d x) (-92378 i \sin (c+d x)-226746 i \sin (3 (c+d x))-266475 i \sin (5 (c+d x))-323323 i \sin (7 (c+d x))+73359 i \sin (9 (c+d x))+2431 i \sin (11 (c+d x))-739024 \cos (c+d x)-604656 \cos (3 (c+d x))-426360 \cos (5 (c+d x))-369512 \cos (7 (c+d x))+65208 \cos (9 (c+d x))+1768 \cos (11 (c+d x)))}{12899328 a^8 d (\tan (c+d x)-i)^8}","\frac{32 \sin ^5(c+d x)}{4199 a^8 d}-\frac{320 \sin ^3(c+d x)}{12597 a^8 d}+\frac{160 \sin (c+d x)}{4199 a^8 d}+\frac{64 i \cos ^5(c+d x)}{4199 d \left(a^8+i a^8 \tan (c+d x)\right)}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{112 i \cos ^3(c+d x)}{12597 a^2 d \left(a^2+i a^2 \tan (c+d x)\right)^3}+\frac{48 i \cos ^3(c+d x)}{4199 d \left(a^2+i a^2 \tan (c+d x)\right)^4}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}",1,"((-1/12899328*I)*Sec[c + d*x]^8*(-739024*Cos[c + d*x] - 604656*Cos[3*(c + d*x)] - 426360*Cos[5*(c + d*x)] - 369512*Cos[7*(c + d*x)] + 65208*Cos[9*(c + d*x)] + 1768*Cos[11*(c + d*x)] - (92378*I)*Sin[c + d*x] - (226746*I)*Sin[3*(c + d*x)] - (266475*I)*Sin[5*(c + d*x)] - (323323*I)*Sin[7*(c + d*x)] + (73359*I)*Sin[9*(c + d*x)] + (2431*I)*Sin[11*(c + d*x)]))/(a^8*d*(-I + Tan[c + d*x])^8)","A",1
185,1,156,123,2.2141112,"\int (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx","Integrate[(e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x]),x]","\frac{a e e^{-i d x} (\cos (d x)-i \sin (d x)) (e \sec (c+d x))^{5/2} (\cos (c+3 d x)+i \sin (c+3 d x)) \left(7 i e^{-2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-28 i \cos (2 (c+d x))+27 \tan (c+d x)+7 \sin (3 (c+d x)) \sec (c+d x)-36 i\right)}{70 d}","-\frac{6 a e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{6 a e^3 \sin (c+d x) \sqrt{e \sec (c+d x)}}{5 d}+\frac{2 i a (e \sec (c+d x))^{7/2}}{7 d}+\frac{2 a e \sin (c+d x) (e \sec (c+d x))^{5/2}}{5 d}",1,"(a*e*(e*Sec[c + d*x])^(5/2)*(Cos[d*x] - I*Sin[d*x])*(Cos[c + 3*d*x] + I*Sin[c + 3*d*x])*(-36*I - (28*I)*Cos[2*(c + d*x)] + ((7*I)*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^((2*I)*(c + d*x)) + 7*Sec[c + d*x]*Sin[3*(c + d*x)] + 27*Tan[c + d*x]))/(70*d*E^(I*d*x))","C",1
186,1,57,94,0.5507429,"\int (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x)) \, dx","Integrate[(e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x]),x]","\frac{a (e \sec (c+d x))^{5/2} \left(5 \sin (2 (c+d x))+10 \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 i\right)}{15 d}","\frac{2 a e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{3 d}+\frac{2 i a (e \sec (c+d x))^{5/2}}{5 d}+\frac{2 a e \sin (c+d x) (e \sec (c+d x))^{3/2}}{3 d}",1,"(a*(e*Sec[c + d*x])^(5/2)*(6*I + 10*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 5*Sin[2*(c + d*x)]))/(15*d)","A",1
187,1,102,90,0.8460761,"\int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx","Integrate[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x]),x]","\frac{2 a e e^{-2 i d x} \sqrt{e \sec (c+d x)} \left(i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\tan (c+d x)-2 i\right) (\cos (c+3 d x)+i \sin (c+3 d x))}{3 d}","-\frac{2 a e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 i a (e \sec (c+d x))^{3/2}}{3 d}+\frac{2 a e \sin (c+d x) \sqrt{e \sec (c+d x)}}{d}",1,"(2*a*e*Sqrt[e*Sec[c + d*x]]*(Cos[c + 3*d*x] + I*Sin[c + 3*d*x])*(-2*I + I*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Tan[c + d*x]))/(3*d*E^((2*I)*d*x))","C",1
188,1,44,60,0.2646097,"\int \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx","Integrate[Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x]),x]","\frac{2 a \sqrt{e \sec (c+d x)} \left(\sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+i\right)}{d}","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{d}+\frac{2 i a \sqrt{e \sec (c+d x)}}{d}",1,"(2*a*(I + Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])*Sqrt[e*Sec[c + d*x]])/d","A",1
189,1,73,60,0.3642681,"\int \frac{a+i a \tan (c+d x)}{\sqrt{e \sec (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])/Sqrt[e*Sec[c + d*x]],x]","-\frac{4 i a e^{2 i (c+d x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)}{3 d \sqrt{1+e^{2 i (c+d x)}} \sqrt{e \sec (c+d x)}}","\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{2 i a}{d \sqrt{e \sec (c+d x)}}",1,"(((-4*I)/3)*a*E^((2*I)*(c + d*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/(d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[e*Sec[c + d*x]])","C",1
190,1,62,96,0.4243679,"\int \frac{a+i a \tan (c+d x)}{(e \sec (c+d x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])/(e*Sec[c + d*x])^(3/2),x]","\frac{2 a \left(\sin (c+d x)-i \cos (c+d x)+\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{\sqrt{\cos (c+d x)}}\right)}{3 d e \sqrt{e \sec (c+d x)}}","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{3 d e^2}-\frac{2 i a}{3 d (e \sec (c+d x))^{3/2}}+\frac{2 a \sin (c+d x)}{3 d e \sqrt{e \sec (c+d x)}}",1,"(2*a*((-I)*Cos[c + d*x] + EllipticF[(c + d*x)/2, 2]/Sqrt[Cos[c + d*x]] + Sin[c + d*x]))/(3*d*e*Sqrt[e*Sec[c + d*x]])","A",1
191,1,99,96,0.8005231,"\int \frac{a+i a \tan (c+d x)}{(e \sec (c+d x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])/(e*Sec[c + d*x])^(5/2),x]","-\frac{a (\tan (c+d x)-i) \left(-2 \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 i \sin (2 (c+d x))+2 \cos (2 (c+d x))+2\right)}{5 d e^2 \sqrt{e \sec (c+d x)}}","\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d e^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{2 i a}{5 d (e \sec (c+d x))^{5/2}}+\frac{2 a \sin (c+d x)}{5 d e (e \sec (c+d x))^{3/2}}",1,"-1/5*(a*(2 + 2*Cos[2*(c + d*x)] - 2*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] - (3*I)*Sin[2*(c + d*x)])*(-I + Tan[c + d*x]))/(d*e^2*Sqrt[e*Sec[c + d*x]])","C",1
192,1,121,125,0.7614005,"\int \frac{a+i a \tan (c+d x)}{(e \sec (c+d x))^{7/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])/(e*Sec[c + d*x])^(7/2),x]","\frac{a \sqrt{e \sec (c+d x)} (\cos (c+d x)+i \sin (c+d x)) \left(5 \sin (c+d x)+5 \sin (3 (c+d x))-14 i \cos (c+d x)+2 i \cos (3 (c+d x))+20 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (c+d x)-i \sin (c+d x))\right)}{42 d e^4}","\frac{10 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{21 d e^4}+\frac{10 a \sin (c+d x)}{21 d e^3 \sqrt{e \sec (c+d x)}}-\frac{2 i a}{7 d (e \sec (c+d x))^{7/2}}+\frac{2 a \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}",1,"(a*Sqrt[e*Sec[c + d*x]]*(Cos[c + d*x] + I*Sin[c + d*x])*((-14*I)*Cos[c + d*x] + (2*I)*Cos[3*(c + d*x)] + 20*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[c + d*x] - I*Sin[c + d*x]) + 5*Sin[c + d*x] + 5*Sin[3*(c + d*x)]))/(42*d*e^4)","A",1
193,1,267,138,2.6716986,"\int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2 \, dx","Integrate[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^2,x]","\frac{(a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2} \left(\frac{1}{2} \csc (c) (\cos (2 c)-i \sin (2 c)) \sec ^{\frac{5}{2}}(c+d x) (20 i \sin (2 c+d x)+27 \cos (2 c+d x)+21 \cos (2 c+3 d x)-20 i \sin (d x)+36 \cos (d x))-\frac{14 i \sqrt{2} \left(3 \sqrt{1+e^{2 i (c+d x)}}-\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}\right)}{15 d \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^2}","-\frac{14 a^2 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{14 i a^2 (e \sec (c+d x))^{3/2}}{15 d}+\frac{14 a^2 e \sin (c+d x) \sqrt{e \sec (c+d x)}}{5 d}+\frac{2 i \left(a^2+i a^2 \tan (c+d x)\right) (e \sec (c+d x))^{3/2}}{5 d}",1,"((e*Sec[c + d*x])^(3/2)*(((-14*I)*Sqrt[2]*(3*Sqrt[1 + E^((2*I)*(c + d*x))] - E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/((-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + (Csc[c]*Sec[c + d*x]^(5/2)*(Cos[2*c] - I*Sin[2*c])*(36*Cos[d*x] + 27*Cos[2*c + d*x] + 21*Cos[2*c + 3*d*x] - (20*I)*Sin[d*x] + (20*I)*Sin[2*c + d*x]))/2)*(a + I*a*Tan[c + d*x])^2)/(15*d*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^2)","C",1
194,1,67,106,0.7422186,"\int \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^2 \, dx","Integrate[Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^2,x]","\frac{2 a^2 (e \sec (c+d x))^{3/2} \left(-\sin (c+d x)+6 i \cos (c+d x)+5 \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d e}","\frac{10 i a^2 \sqrt{e \sec (c+d x)}}{3 d}+\frac{2 i \left(a^2+i a^2 \tan (c+d x)\right) \sqrt{e \sec (c+d x)}}{3 d}+\frac{10 a^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{3 d}",1,"(2*a^2*(e*Sec[c + d*x])^(3/2)*((6*I)*Cos[c + d*x] + 5*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - Sin[c + d*x]))/(3*d*e)","A",1
195,1,132,107,1.1873709,"\int \frac{(a+i a \tan (c+d x))^2}{\sqrt{e \sec (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^2/Sqrt[e*Sec[c + d*x]],x]","-\frac{2 i \sqrt{2} a^2 e^{2 i (c+d x)} \left(\left(1+e^{2 i (c+d x)}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-\sqrt{1+e^{2 i (c+d x)}}\right)}{d \left(1+e^{2 i (c+d x)}\right)^{3/2} \sqrt{\frac{e e^{i (c+d x)}}{1+e^{2 i (c+d x)}}}}","-\frac{6 a^2 \sin (c+d x) \sqrt{e \sec (c+d x)}}{d e}-\frac{4 i \left(a^2+i a^2 \tan (c+d x)\right)}{d \sqrt{e \sec (c+d x)}}+\frac{6 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}",1,"((-2*I)*Sqrt[2]*a^2*E^((2*I)*(c + d*x))*(-Sqrt[1 + E^((2*I)*(c + d*x))] + (1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(d*Sqrt[(e*E^(I*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))^(3/2))","C",1
196,1,114,85,0.5890279,"\int \frac{(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^2/(e*Sec[c + d*x])^(3/2),x]","-\frac{2 a^2 \sec ^2(c+d x) (\cos (c+3 d x)+i \sin (c+3 d x)) \left(2 i \cos (c+d x)+\sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (c+d x)-i \sin (c+d x))\right)}{3 d (\cos (d x)+i \sin (d x))^2 (e \sec (c+d x))^{3/2}}","-\frac{2 a^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{3 d e^2}-\frac{4 i \left(a^2+i a^2 \tan (c+d x)\right)}{3 d (e \sec (c+d x))^{3/2}}",1,"(-2*a^2*Sec[c + d*x]^2*((2*I)*Cos[c + d*x] + Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[c + d*x] - I*Sin[c + d*x]))*(Cos[c + 3*d*x] + I*Sin[c + 3*d*x]))/(3*d*(e*Sec[c + d*x])^(3/2)*(Cos[d*x] + I*Sin[d*x])^2)","A",1
197,1,114,85,1.1540371,"\int \frac{(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^2/(e*Sec[c + d*x])^(5/2),x]","-\frac{i \sqrt{2} a^2 \left(1+e^{2 i (c+d x)}\right)^{3/2} \left(\frac{e e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{3/2} \left(2 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+3 \sqrt{1+e^{2 i (c+d x)}}\right)}{15 d e^4}","\frac{2 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d e^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{4 i \left(a^2+i a^2 \tan (c+d x)\right)}{5 d (e \sec (c+d x))^{5/2}}",1,"((-1/15*I)*Sqrt[2]*a^2*((e*E^(I*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^(3/2)*(3*Sqrt[1 + E^((2*I)*(c + d*x))] + 2*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(d*e^4)","C",1
198,1,133,116,1.0357695,"\int \frac{(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{7/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^2/(e*Sec[c + d*x])^(7/2),x]","\frac{a^2 \sqrt{e \sec (c+d x)} (\cos (2 (c+2 d x))+i \sin (2 (c+2 d x))) \left(-\sin (2 (c+d x))-2 i \cos (2 (c+d x))+2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (2 (c+d x))-i \sin (2 (c+d x)))-2 i\right)}{7 d e^4 (\cos (d x)+i \sin (d x))^2}","\frac{2 a^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{7 d e^4}+\frac{2 a^2 \sin (c+d x)}{7 d e^3 \sqrt{e \sec (c+d x)}}-\frac{4 i \left(a^2+i a^2 \tan (c+d x)\right)}{7 d (e \sec (c+d x))^{7/2}}",1,"(a^2*Sqrt[e*Sec[c + d*x]]*(-2*I - (2*I)*Cos[2*(c + d*x)] + 2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)]) - Sin[2*(c + d*x)])*(Cos[2*(c + 2*d*x)] + I*Sin[2*(c + 2*d*x)]))/(7*d*e^4*(Cos[d*x] + I*Sin[d*x])^2)","A",1
199,1,133,116,1.8795952,"\int \frac{(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{9/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^2/(e*Sec[c + d*x])^(9/2),x]","\frac{i a^2 \left(-\frac{8 e^{2 i (c+d x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-4 e^{2 i (c+d x)}-e^{4 i (c+d x)}+9\right)}{18 \sqrt{2} d e^4 \sqrt{\frac{e e^{i (c+d x)}}{1+e^{2 i (c+d x)}}}}","\frac{2 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^4 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 a^2 \sin (c+d x)}{9 d e^3 (e \sec (c+d x))^{3/2}}-\frac{4 i \left(a^2+i a^2 \tan (c+d x)\right)}{9 d (e \sec (c+d x))^{9/2}}",1,"((I/18)*a^2*(9 - 4*E^((2*I)*(c + d*x)) - E^((4*I)*(c + d*x)) - (8*E^((2*I)*(c + d*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))]))/(Sqrt[2]*d*e^4*Sqrt[(e*E^(I*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])","C",1
200,1,155,147,1.3870226,"\int \frac{(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{11/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^2/(e*Sec[c + d*x])^(11/2),x]","\frac{a^2 \sqrt{e \sec (c+d x)} (\cos (2 (c+2 d x))+i \sin (2 (c+2 d x))) \left(-6 \sin (2 (c+d x))+7 \sin (4 (c+d x))-24 i \cos (2 (c+d x))+4 i \cos (4 (c+d x))+40 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (2 (c+d x))-i \sin (2 (c+d x)))-28 i\right)}{132 d e^6 (\cos (d x)+i \sin (d x))^2}","\frac{10 a^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{33 d e^6}+\frac{10 a^2 \sin (c+d x)}{33 d e^5 \sqrt{e \sec (c+d x)}}+\frac{2 a^2 \sin (c+d x)}{11 d e^3 (e \sec (c+d x))^{5/2}}-\frac{4 i \left(a^2+i a^2 \tan (c+d x)\right)}{11 d (e \sec (c+d x))^{11/2}}",1,"(a^2*Sqrt[e*Sec[c + d*x]]*(-28*I - (24*I)*Cos[2*(c + d*x)] + (4*I)*Cos[4*(c + d*x)] + 40*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)]) - 6*Sin[2*(c + d*x)] + 7*Sin[4*(c + d*x)])*(Cos[2*(c + 2*d*x)] + I*Sin[2*(c + 2*d*x)]))/(132*d*e^6*(Cos[d*x] + I*Sin[d*x])^2)","A",1
201,1,442,202,7.8494634,"\int (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))^3 \, dx","Integrate[(e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])^3,x]","\frac{2 i \sqrt{2} e^{-i (2 c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{7/2}}{3 \left(-1+e^{2 i c}\right) d \sec ^{\frac{13}{2}}(c+d x) (\cos (d x)+i \sin (d x))^3}+\frac{\cos ^6(c+d x) (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{7/2} \left(\csc (c) (2 \cos (3 c)-2 i \sin (3 c)) \cos (d x)+\left(-\frac{2}{11} \sin (3 c)-\frac{2}{11} i \cos (3 c)\right) \sec ^5(c+d x)+\sec (c) \left(-\frac{2}{3} \cos (3 c)+\frac{2}{3} i \sin (3 c)\right) \sin (d x) \sec ^4(c+d x)+\sec (c) (12 \cos (c)+7 i \sin (c)) \left(\frac{2}{21} \sin (3 c)+\frac{2}{21} i \cos (3 c)\right) \sec ^3(c+d x)+\sec (c) \left(\frac{2}{3} \cos (3 c)-\frac{2}{3} i \sin (3 c)\right) \sin (d x) \sec ^2(c+d x)+\tan (c) \left(\frac{2}{3} \cos (3 c)-\frac{2}{3} i \sin (3 c)\right) \sec (c+d x)\right)}{d (\cos (d x)+i \sin (d x))^3}","-\frac{2 a^3 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 a^3 e^3 \sin (c+d x) \sqrt{e \sec (c+d x)}}{d}+\frac{10 i a^3 (e \sec (c+d x))^{7/2}}{21 d}+\frac{2 a^3 e \sin (c+d x) (e \sec (c+d x))^{5/2}}{3 d}+\frac{10 i \left(a^3+i a^3 \tan (c+d x)\right) (e \sec (c+d x))^{7/2}}{33 d}+\frac{2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{7/2}}{11 d}",1,"(((2*I)/3)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])^3)/(d*E^(I*(2*c + d*x))*(-1 + E^((2*I)*c))*Sec[c + d*x]^(13/2)*(Cos[d*x] + I*Sin[d*x])^3) + (Cos[c + d*x]^6*(e*Sec[c + d*x])^(7/2)*(Sec[c + d*x]^5*(((-2*I)/11)*Cos[3*c] - (2*Sin[3*c])/11) + Cos[d*x]*Csc[c]*(2*Cos[3*c] - (2*I)*Sin[3*c]) + Sec[c]*Sec[c + d*x]^3*(12*Cos[c] + (7*I)*Sin[c])*(((2*I)/21)*Cos[3*c] + (2*Sin[3*c])/21) + Sec[c]*Sec[c + d*x]^2*((2*Cos[3*c])/3 - ((2*I)/3)*Sin[3*c])*Sin[d*x] + Sec[c]*Sec[c + d*x]^4*((-2*Cos[3*c])/3 + ((2*I)/3)*Sin[3*c])*Sin[d*x] + Sec[c + d*x]*((2*Cos[3*c])/3 - ((2*I)/3)*Sin[3*c])*Tan[c])*(a + I*a*Tan[c + d*x])^3)/(d*(Cos[d*x] + I*Sin[d*x])^3)","C",1
202,1,89,175,1.9286916,"\int (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^3 \, dx","Integrate[(e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \sec ^2(c+d x) (e \sec (c+d x))^{5/2} \left(-150 \sin (2 (c+d x))+195 \sin (4 (c+d x))+1008 i \cos (2 (c+d x))+1560 \cos ^{\frac{9}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+728 i\right)}{1260 d}","\frac{26 a^3 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{21 d}+\frac{26 i a^3 (e \sec (c+d x))^{5/2}}{35 d}+\frac{26 a^3 e \sin (c+d x) (e \sec (c+d x))^{3/2}}{21 d}+\frac{26 i \left(a^3+i a^3 \tan (c+d x)\right) (e \sec (c+d x))^{5/2}}{63 d}+\frac{2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{5/2}}{9 d}",1,"(a^3*Sec[c + d*x]^2*(e*Sec[c + d*x])^(5/2)*(728*I + (1008*I)*Cos[2*(c + d*x)] + 1560*Cos[c + d*x]^(9/2)*EllipticF[(c + d*x)/2, 2] - 150*Sin[2*(c + d*x)] + 195*Sin[4*(c + d*x)]))/(1260*d)","A",1
203,1,129,175,2.7308005,"\int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3 \, dx","Integrate[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 (1+i \tan (c+d x)) (e \sec (c+d x))^{3/2} \left(77 i e^{-2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-308 i \cos (2 (c+d x))+17 \tan (c+d x)+77 \sin (3 (c+d x)) \sec (c+d x)-116 i\right)}{210 d}","-\frac{22 a^3 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac{22 a^3 e \sin (c+d x) \sqrt{e \sec (c+d x)}}{5 d}+\frac{22 i \left(a^3+i a^3 \tan (c+d x)\right) (e \sec (c+d x))^{3/2}}{35 d}+\frac{2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d}",1,"(a^3*(e*Sec[c + d*x])^(3/2)*(1 + I*Tan[c + d*x])*(-116*I - (308*I)*Cos[2*(c + d*x)] + ((77*I)*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^((2*I)*(c + d*x)) + 77*Sec[c + d*x]*Sin[3*(c + d*x)] + 17*Tan[c + d*x]))/(210*d)","C",1
204,1,79,139,1.4503055,"\int \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^3 \, dx","Integrate[Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \sec ^2(c+d x) \sqrt{e \sec (c+d x)} \left(-5 \sin (2 (c+d x))+20 i \cos (2 (c+d x))+30 \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+18 i\right)}{5 d}","\frac{6 i a^3 \sqrt{e \sec (c+d x)}}{d}+\frac{6 i \left(a^3+i a^3 \tan (c+d x)\right) \sqrt{e \sec (c+d x)}}{5 d}+\frac{6 a^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{d}+\frac{2 i a (a+i a \tan (c+d x))^2 \sqrt{e \sec (c+d x)}}{5 d}",1,"(a^3*Sec[c + d*x]^2*Sqrt[e*Sec[c + d*x]]*(18*I + (20*I)*Cos[2*(c + d*x)] + 30*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] - 5*Sin[2*(c + d*x)]))/(5*d)","A",1
205,1,101,124,1.8114976,"\int \frac{(a+i a \tan (c+d x))^3}{\sqrt{e \sec (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^3/Sqrt[e*Sec[c + d*x]],x]","\frac{2 a^3 (\cos (c)+i \sin (c)) (\sin (d x)-i \cos (d x)) \sqrt{e \sec (c+d x)} \left(7 \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-i \tan (c+d x)-8\right)}{3 d e}","-\frac{26 i a^3}{3 d \sqrt{e \sec (c+d x)}}-\frac{2 i a^3 \tan ^2(c+d x)}{3 d \sqrt{e \sec (c+d x)}}-\frac{6 a^3 \tan (c+d x)}{d \sqrt{e \sec (c+d x)}}+\frac{14 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}",1,"(2*a^3*Sqrt[e*Sec[c + d*x]]*(Cos[c] + I*Sin[c])*((-I)*Cos[d*x] + Sin[d*x])*(-8 + 7*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] - I*Tan[c + d*x]))/(3*d*e)","C",1
206,1,123,111,0.8065387,"\int \frac{(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(3/2),x]","-\frac{2 a^3 \sec ^2(c+d x) (\cos (c+4 d x)+i \sin (c+4 d x)) \left(3 \sin (c+d x)+7 i \cos (c+d x)+5 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (c+d x)-i \sin (c+d x))\right)}{3 d (\cos (d x)+i \sin (d x))^3 (e \sec (c+d x))^{3/2}}","-\frac{10 i a^3 \sqrt{e \sec (c+d x)}}{3 d e^2}-\frac{10 a^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{3 d e^2}-\frac{4 i a (a+i a \tan (c+d x))^2}{3 d (e \sec (c+d x))^{3/2}}",1,"(-2*a^3*Sec[c + d*x]^2*((7*I)*Cos[c + d*x] + 5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[c + d*x] - I*Sin[c + d*x]) + 3*Sin[c + d*x])*(Cos[c + 4*d*x] + I*Sin[c + 4*d*x]))/(3*d*(e*Sec[c + d*x])^(3/2)*(Cos[d*x] + I*Sin[d*x])^3)","A",1
207,1,108,111,1.4471523,"\int \frac{(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(5/2),x]","-\frac{4 i a^3 e^{2 i (c+d x)} \left(-\sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+e^{2 i (c+d x)}+1\right)}{5 d e^2 \left(1+e^{2 i (c+d x)}\right) \sqrt{e \sec (c+d x)}}","\frac{6 i a^3}{5 d e^2 \sqrt{e \sec (c+d x)}}-\frac{6 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d e^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{4 i a (a+i a \tan (c+d x))^2}{5 d (e \sec (c+d x))^{5/2}}",1,"(((-4*I)/5)*a^3*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)) - Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(d*e^2*(1 + E^((2*I)*(c + d*x)))*Sqrt[e*Sec[c + d*x]])","C",1
208,1,133,124,1.0300431,"\int \frac{(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{7/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(7/2),x]","-\frac{a^3 \sqrt{e \sec (c+d x)} (\cos (2 c+5 d x)+i \sin (2 c+5 d x)) \left(-\sin (2 (c+d x))+5 i \cos (2 (c+d x))+2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (2 (c+d x))-i \sin (2 (c+d x)))+5 i\right)}{21 d e^4 (\cos (d x)+i \sin (d x))^3}","-\frac{2 a^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{21 d e^4}-\frac{4 i \left(a^3+i a^3 \tan (c+d x)\right)}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac{2 i (a+i a \tan (c+d x))^3}{7 d (e \sec (c+d x))^{7/2}}",1,"-1/21*(a^3*Sqrt[e*Sec[c + d*x]]*(5*I + (5*I)*Cos[2*(c + d*x)] + 2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)]) - Sin[2*(c + d*x)])*(Cos[2*c + 5*d*x] + I*Sin[2*c + 5*d*x]))/(d*e^4*(Cos[d*x] + I*Sin[d*x])^3)","A",1
209,1,118,124,2.670658,"\int \frac{(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(9/2),x]","-\frac{a^3 e^{-2 i (c+d x)} \left(4 \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+16 e^{2 i (c+d x)}+5 e^{4 i (c+d x)}+11\right) (\tan (c+d x)-i)^3}{90 d e^2 (e \sec (c+d x))^{5/2}}","\frac{2 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d e^4 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{4 i \left(a^3+i a^3 \tan (c+d x)\right)}{15 d e^2 (e \sec (c+d x))^{5/2}}-\frac{2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}",1,"-1/90*(a^3*(11 + 16*E^((2*I)*(c + d*x)) + 5*E^((4*I)*(c + d*x)) + 4*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(-I + Tan[c + d*x])^3)/(d*e^2*E^((2*I)*(c + d*x))*(e*Sec[c + d*x])^(5/2))","C",1
210,1,148,155,1.3769687,"\int \frac{(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{11/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(11/2),x]","\frac{a^3 \sqrt{e \sec (c+d x)} (\cos (3 (c+2 d x))+i \sin (3 (c+2 d x))) \left(-15 \sin (c+d x)-15 \sin (3 (c+d x))-46 i \cos (c+d x)-22 i \cos (3 (c+d x))+20 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (3 (c+d x))-i \sin (3 (c+d x)))\right)}{154 d e^6 (\cos (d x)+i \sin (d x))^3}","\frac{10 a^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{77 d e^6}+\frac{10 a^3 \sin (c+d x)}{77 d e^5 \sqrt{e \sec (c+d x)}}-\frac{20 i \left(a^3+i a^3 \tan (c+d x)\right)}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac{2 i (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}}",1,"(a^3*Sqrt[e*Sec[c + d*x]]*((-46*I)*Cos[c + d*x] - (22*I)*Cos[3*(c + d*x)] - 15*Sin[c + d*x] + 20*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[3*(c + d*x)] - I*Sin[3*(c + d*x)]) - 15*Sin[3*(c + d*x)])*(Cos[3*(c + 2*d*x)] + I*Sin[3*(c + 2*d*x)]))/(154*d*e^6*(Cos[d*x] + I*Sin[d*x])^3)","A",1
211,1,155,155,6.6725775,"\int \frac{(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{13/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(13/2),x]","-\frac{a^3 e^{-4 i (c+d x)} \left(112 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-34 e^{2 i (c+d x)}+124 e^{4 i (c+d x)}+50 e^{6 i (c+d x)}+9 e^{8 i (c+d x)}-117\right) (\tan (c+d x)-i)^3}{936 d e^4 (e \sec (c+d x))^{5/2}}","\frac{14 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{39 d e^6 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{14 a^3 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac{28 i \left(a^3+i a^3 \tan (c+d x)\right)}{117 d e^2 (e \sec (c+d x))^{9/2}}-\frac{2 i (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}",1,"-1/936*(a^3*(-117 - 34*E^((2*I)*(c + d*x)) + 124*E^((4*I)*(c + d*x)) + 50*E^((6*I)*(c + d*x)) + 9*E^((8*I)*(c + d*x)) + 112*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(-I + Tan[c + d*x])^3)/(d*e^4*E^((4*I)*(c + d*x))*(e*Sec[c + d*x])^(5/2))","C",1
212,1,170,186,2.0328953,"\int \frac{(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{15/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(15/2),x]","\frac{a^3 \sqrt{e \sec (c+d x)} (\cos (3 (c+2 d x))+i \sin (3 (c+2 d x))) \left(-114 \sin (c+d x)-81 \sin (3 (c+d x))+33 \sin (5 (c+d x))-332 i \cos (c+d x)-154 i \cos (3 (c+d x))+22 i \cos (5 (c+d x))+240 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (3 (c+d x))-i \sin (3 (c+d x)))\right)}{1320 d e^8 (\cos (d x)+i \sin (d x))^3}","\frac{2 a^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{11 d e^8}+\frac{2 a^3 \sin (c+d x)}{11 d e^7 \sqrt{e \sec (c+d x)}}+\frac{6 a^3 \sin (c+d x)}{55 d e^5 (e \sec (c+d x))^{5/2}}-\frac{12 i \left(a^3+i a^3 \tan (c+d x)\right)}{55 d e^2 (e \sec (c+d x))^{11/2}}-\frac{2 i (a+i a \tan (c+d x))^3}{15 d (e \sec (c+d x))^{15/2}}",1,"(a^3*Sqrt[e*Sec[c + d*x]]*((-332*I)*Cos[c + d*x] - (154*I)*Cos[3*(c + d*x)] + (22*I)*Cos[5*(c + d*x)] - 114*Sin[c + d*x] + 240*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[3*(c + d*x)] - I*Sin[3*(c + d*x)]) - 81*Sin[3*(c + d*x)] + 33*Sin[5*(c + d*x)])*(Cos[3*(c + 2*d*x)] + I*Sin[3*(c + 2*d*x)]))/(1320*d*e^8*(Cos[d*x] + I*Sin[d*x])^3)","A",1
213,1,429,215,7.8226672,"\int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx","Integrate[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^4,x]","\frac{22 i \sqrt{2} e^{-i (3 c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (a+i a \tan (c+d x))^4 (e \sec (c+d x))^{3/2}}{9 \left(-1+e^{2 i c}\right) d \sec ^{\frac{11}{2}}(c+d x) (\cos (d x)+i \sin (d x))^4}+\frac{\cos ^5(c+d x) (a+i a \tan (c+d x))^4 (e \sec (c+d x))^{3/2} \left(\csc (c) \left(\frac{22}{3} \cos (4 c)-\frac{22}{3} i \sin (4 c)\right) \cos (d x)+\sec (c) \left(\frac{2}{9} \cos (4 c)-\frac{2}{9} i \sin (4 c)\right) \sin (d x) \sec ^4(c+d x)+\sec (c) (36 \cos (c)+7 i \sin (c)) \left(-\frac{2}{63} \sin (4 c)-\frac{2}{63} i \cos (4 c)\right) \sec ^3(c+d x)+\sec (c) \left(-\frac{26}{9} \cos (4 c)+\frac{26}{9} i \sin (4 c)\right) \sin (d x) \sec ^2(c+d x)+\sec (c) (24 \cos (c)+13 i \sin (c)) \left(\frac{2}{9} \sin (4 c)+\frac{2}{9} i \cos (4 c)\right) \sec (c+d x)\right)}{d (\cos (d x)+i \sin (d x))^4}","-\frac{22 a^4 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{22 i a^4 (e \sec (c+d x))^{3/2}}{9 d}+\frac{22 a^4 e \sin (c+d x) \sqrt{e \sec (c+d x)}}{3 d}+\frac{22 i \left(a^4+i a^4 \tan (c+d x)\right) (e \sec (c+d x))^{3/2}}{21 d}+\frac{10 i \left(a^2+i a^2 \tan (c+d x)\right)^2 (e \sec (c+d x))^{3/2}}{21 d}+\frac{2 i a (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}{9 d}",1,"(((22*I)/9)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^4)/(d*E^(I*(3*c + d*x))*(-1 + E^((2*I)*c))*Sec[c + d*x]^(11/2)*(Cos[d*x] + I*Sin[d*x])^4) + (Cos[c + d*x]^5*(e*Sec[c + d*x])^(3/2)*(Sec[c]*Sec[c + d*x]^3*(36*Cos[c] + (7*I)*Sin[c])*(((-2*I)/63)*Cos[4*c] - (2*Sin[4*c])/63) + Cos[d*x]*Csc[c]*((22*Cos[4*c])/3 - ((22*I)/3)*Sin[4*c]) + Sec[c]*Sec[c + d*x]*(24*Cos[c] + (13*I)*Sin[c])*(((2*I)/9)*Cos[4*c] + (2*Sin[4*c])/9) + Sec[c]*Sec[c + d*x]^4*((2*Cos[4*c])/9 - ((2*I)/9)*Sin[4*c])*Sin[d*x] + Sec[c]*Sec[c + d*x]^2*((-26*Cos[4*c])/9 + ((26*I)/9)*Sin[4*c])*Sin[d*x])*(a + I*a*Tan[c + d*x])^4)/(d*(Cos[d*x] + I*Sin[d*x])^4)","C",1
214,1,101,183,1.7574486,"\int \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^4 \, dx","Integrate[Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^4,x]","\frac{a^4 \sec ^4(c+d x) \sqrt{e \sec (c+d x)} \left(-150 \sin (2 (c+d x))-85 \sin (4 (c+d x))+1008 i \cos (2 (c+d x))+280 i \cos (4 (c+d x))+1560 \cos ^{\frac{9}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+728 i\right)}{140 d}","\frac{78 i a^4 \sqrt{e \sec (c+d x)}}{7 d}+\frac{78 i \left(a^4+i a^4 \tan (c+d x)\right) \sqrt{e \sec (c+d x)}}{35 d}+\frac{78 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{7 d}+\frac{26 i \left(a^2+i a^2 \tan (c+d x)\right)^2 \sqrt{e \sec (c+d x)}}{35 d}+\frac{2 i a (a+i a \tan (c+d x))^3 \sqrt{e \sec (c+d x)}}{7 d}",1,"(a^4*Sec[c + d*x]^4*Sqrt[e*Sec[c + d*x]]*(728*I + (1008*I)*Cos[2*(c + d*x)] + (280*I)*Cos[4*(c + d*x)] + 1560*Cos[c + d*x]^(9/2)*EllipticF[(c + d*x)/2, 2] - 150*Sin[2*(c + d*x)] - 85*Sin[4*(c + d*x)]))/(140*d)","A",1
215,1,123,178,4.3573551,"\int \frac{(a+i a \tan (c+d x))^4}{\sqrt{e \sec (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^4/Sqrt[e*Sec[c + d*x]],x]","-\frac{2 i a^4 e^{i (c+d x)} \left(77 \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-176 e^{2 i (c+d x)}-111 e^{4 i (c+d x)}-77\right) \sqrt{e \sec (c+d x)}}{15 d e \left(1+e^{2 i (c+d x)}\right)^2}","-\frac{154 i a^4 (e \sec (c+d x))^{3/2}}{15 d e^2}-\frac{22 i \left(a^4+i a^4 \tan (c+d x)\right) (e \sec (c+d x))^{3/2}}{5 d e^2}-\frac{154 a^4 \sin (c+d x) \sqrt{e \sec (c+d x)}}{5 d e}+\frac{154 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{4 i a (a+i a \tan (c+d x))^3}{d \sqrt{e \sec (c+d x)}}",1,"(((-2*I)/15)*a^4*E^(I*(c + d*x))*(-77 - 176*E^((2*I)*(c + d*x)) - 111*E^((4*I)*(c + d*x)) + 77*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sqrt[e*Sec[c + d*x]])/(d*e*(1 + E^((2*I)*(c + d*x)))^2)","C",1
216,1,130,146,1.3904057,"\int \frac{(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(3/2),x]","\frac{a^4 \sec ^3(c+d x) (\sin (c+5 d x)-i \cos (c+5 d x)) \left(-11 i \sin (2 (c+d x))+19 \cos (2 (c+d x))-30 i \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (c+d x)-i \sin (c+d x))+21\right)}{3 d (\cos (d x)+i \sin (d x))^4 (e \sec (c+d x))^{3/2}}","-\frac{10 i a^4 \sqrt{e \sec (c+d x)}}{d e^2}-\frac{2 i \left(a^4+i a^4 \tan (c+d x)\right) \sqrt{e \sec (c+d x)}}{d e^2}-\frac{10 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{d e^2}-\frac{4 i a (a+i a \tan (c+d x))^3}{3 d (e \sec (c+d x))^{3/2}}",1,"(a^4*Sec[c + d*x]^3*(21 + 19*Cos[2*(c + d*x)] - (30*I)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2]*(Cos[c + d*x] - I*Sin[c + d*x]) - (11*I)*Sin[2*(c + d*x)])*((-I)*Cos[c + 5*d*x] + Sin[c + 5*d*x]))/(3*d*(e*Sec[c + d*x])^(3/2)*(Cos[d*x] + I*Sin[d*x])^4)","A",1
217,1,110,156,2.7857439,"\int \frac{(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(5/2),x]","-\frac{4 i a^4 e^{2 i (c+d x)} \left(-7 \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 e^{2 i (c+d x)}+7\right)}{5 d e^2 \left(1+e^{2 i (c+d x)}\right) \sqrt{e \sec (c+d x)}}","\frac{42 a^4 \sin (c+d x) \sqrt{e \sec (c+d x)}}{5 d e^3}+\frac{28 i \left(a^4+i a^4 \tan (c+d x)\right)}{5 d e^2 \sqrt{e \sec (c+d x)}}-\frac{42 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d e^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{4 i a (a+i a \tan (c+d x))^3}{5 d (e \sec (c+d x))^{5/2}}",1,"(((-4*I)/5)*a^4*E^((2*I)*(c + d*x))*(7 + 2*E^((2*I)*(c + d*x)) - 7*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(d*e^2*(1 + E^((2*I)*(c + d*x)))*Sqrt[e*Sec[c + d*x]])","C",1
218,1,133,125,1.2272089,"\int \frac{(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{7/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(7/2),x]","\frac{2 a^4 \sqrt{e \sec (c+d x)} (\cos (2 (c+3 d x))+i \sin (2 (c+3 d x))) \left(8 \sin (2 (c+d x))+2 i \cos (2 (c+d x))+5 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (2 (c+d x))-i \sin (2 (c+d x)))+2 i\right)}{21 d e^4 (\cos (d x)+i \sin (d x))^4}","\frac{10 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{21 d e^4}+\frac{20 i \left(a^4+i a^4 \tan (c+d x)\right)}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac{4 i a (a+i a \tan (c+d x))^3}{7 d (e \sec (c+d x))^{7/2}}",1,"(2*a^4*Sqrt[e*Sec[c + d*x]]*(2*I + (2*I)*Cos[2*(c + d*x)] + 5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)]) + 8*Sin[2*(c + d*x)])*(Cos[2*(c + 3*d*x)] + I*Sin[2*(c + 3*d*x)]))/(21*d*e^4*(Cos[d*x] + I*Sin[d*x])^4)","A",1
219,1,108,125,3.9472563,"\int \frac{(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{9/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(9/2),x]","-\frac{i a^4 e^{i (c+d x)} \left(-2 \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+7 e^{2 i (c+d x)}+5 e^{4 i (c+d x)}+2\right) \sqrt{e \sec (c+d x)}}{45 d e^5}","-\frac{2 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d e^4 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{4 i \left(a^4+i a^4 \tan (c+d x)\right)}{15 d e^2 (e \sec (c+d x))^{5/2}}-\frac{4 i a (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}",1,"((-1/45*I)*a^4*E^(I*(c + d*x))*(2 + 7*E^((2*I)*(c + d*x)) + 5*E^((4*I)*(c + d*x)) - 2*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sqrt[e*Sec[c + d*x]])/(d*e^5)","C",1
220,1,148,156,1.6517192,"\int \frac{(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{11/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(11/2),x]","-\frac{a^4 \sqrt{e \sec (c+d x)} (\cos (3 c+7 d x)+i \sin (3 c+7 d x)) \left(-3 \sin (c+d x)-3 \sin (3 (c+d x))+37 i \cos (c+d x)+11 i \cos (3 (c+d x))+4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (3 (c+d x))-i \sin (3 (c+d x)))\right)}{154 d e^6 (\cos (d x)+i \sin (d x))^4}","-\frac{2 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{77 d e^6}-\frac{2 a^4 \sin (c+d x)}{77 d e^5 \sqrt{e \sec (c+d x)}}+\frac{4 i \left(a^4+i a^4 \tan (c+d x)\right)}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac{4 i a (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}}",1,"-1/154*(a^4*Sqrt[e*Sec[c + d*x]]*((37*I)*Cos[c + d*x] + (11*I)*Cos[3*(c + d*x)] - 3*Sin[c + d*x] + 4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[3*(c + d*x)] - I*Sin[3*(c + d*x)]) - 3*Sin[3*(c + d*x)])*(Cos[3*c + 7*d*x] + I*Sin[3*c + 7*d*x]))/(d*e^6*(Cos[d*x] + I*Sin[d*x])^4)","A",1
221,1,450,156,7.2163486,"\int \frac{(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{13/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(13/2),x]","\frac{\sec ^3(c+d x) (a+i a \tan (c+d x))^4 \left(\left(-\frac{59 \sin (c)}{468}-\frac{59}{468} i \cos (c)\right) \cos (3 d x)+\left(\frac{37 \sin (c)}{468}-\frac{37}{468} i \cos (c)\right) \cos (5 d x)+\left(\frac{1}{52} \sin (3 c)-\frac{1}{52} i \cos (3 c)\right) \cos (7 d x)+\left(\frac{55}{468} \cos (3 c)-\frac{55}{468} i \sin (3 c)\right) \sin (d x)+\left(\frac{59 \cos (c)}{468}-\frac{59}{468} i \sin (c)\right) \sin (3 d x)+\left(\frac{37 \cos (c)}{468}+\frac{37}{468} i \sin (c)\right) \sin (5 d x)+\left(\frac{1}{52} \cos (3 c)+\frac{1}{52} i \sin (3 c)\right) \sin (7 d x)+\csc (c) (24 \cos (c)+31 i \sin (c)) \left(-\frac{1}{468} \cos (3 c)+\frac{1}{468} i \sin (3 c)\right) \cos (d x)\right)}{d (\cos (d x)+i \sin (d x))^4 (e \sec (c+d x))^{13/2}}-\frac{2 i \sqrt{2} e^{-i (3 c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^4}{117 \left(-1+e^{2 i c}\right) d (\cos (d x)+i \sin (d x))^4 (e \sec (c+d x))^{13/2}}","\frac{2 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{39 d e^6 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 a^4 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac{4 i \left(a^4+i a^4 \tan (c+d x)\right)}{117 d e^2 (e \sec (c+d x))^{9/2}}-\frac{4 i a (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}",1,"(((-2*I)/117)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^4)/(d*E^(I*(3*c + d*x))*(-1 + E^((2*I)*c))*(e*Sec[c + d*x])^(13/2)*(Cos[d*x] + I*Sin[d*x])^4) + (Sec[c + d*x]^3*(Cos[3*d*x]*(((-59*I)/468)*Cos[c] - (59*Sin[c])/468) + Cos[5*d*x]*(((-37*I)/468)*Cos[c] + (37*Sin[c])/468) + Cos[d*x]*Csc[c]*(24*Cos[c] + (31*I)*Sin[c])*(-1/468*Cos[3*c] + (I/468)*Sin[3*c]) + Cos[7*d*x]*((-1/52*I)*Cos[3*c] + Sin[3*c]/52) + ((55*Cos[3*c])/468 - ((55*I)/468)*Sin[3*c])*Sin[d*x] + ((59*Cos[c])/468 - ((59*I)/468)*Sin[c])*Sin[3*d*x] + ((37*Cos[c])/468 + ((37*I)/468)*Sin[c])*Sin[5*d*x] + (Cos[3*c]/52 + (I/52)*Sin[3*c])*Sin[7*d*x])*(a + I*a*Tan[c + d*x])^4)/(d*(e*Sec[c + d*x])^(13/2)*(Cos[d*x] + I*Sin[d*x])^4)","C",0
222,1,155,187,2.324084,"\int \frac{(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{15/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(15/2),x]","-\frac{i a^4 \sqrt{e \sec (c+d x)} (\cos (4 (c+2 d x))+i \sin (4 (c+2 d x))) \left(-54 i \sin (2 (c+d x))-37 i \sin (4 (c+d x))+112 \cos (2 (c+d x))+48 \cos (4 (c+d x))+40 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sin (4 (c+d x))+i \cos (4 (c+d x)))+64\right)}{660 d e^8 (\cos (d x)+i \sin (d x))^4}","\frac{2 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{33 d e^8}+\frac{2 a^4 \sin (c+d x)}{33 d e^7 \sqrt{e \sec (c+d x)}}+\frac{2 a^4 \sin (c+d x)}{55 d e^5 (e \sec (c+d x))^{5/2}}-\frac{4 i \left(a^4+i a^4 \tan (c+d x)\right)}{55 d e^2 (e \sec (c+d x))^{11/2}}-\frac{4 i a (a+i a \tan (c+d x))^3}{15 d (e \sec (c+d x))^{15/2}}",1,"((-1/660*I)*a^4*Sqrt[e*Sec[c + d*x]]*(64 + 112*Cos[2*(c + d*x)] + 48*Cos[4*(c + d*x)] - (54*I)*Sin[2*(c + d*x)] - (37*I)*Sin[4*(c + d*x)] + 40*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(I*Cos[4*(c + d*x)] + Sin[4*(c + d*x)]))*(Cos[4*(c + 2*d*x)] + I*Sin[4*(c + 2*d*x)]))/(d*e^8*(Cos[d*x] + I*Sin[d*x])^4)","A",1
223,1,128,136,1.5933231,"\int \frac{(e \sec (c+d x))^{11/2}}{a+i a \tan (c+d x)} \, dx","Integrate[(e*Sec[c + d*x])^(11/2)/(a + I*a*Tan[c + d*x]),x]","\frac{e^4 (\tan (c+d x)-i) (e \sec (c+d x))^{3/2} \left(-7 e^{-2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+28 \cos (2 (c+d x))-13 i \tan (c+d x)+7 i \sin (3 (c+d x)) \sec (c+d x)+76\right)}{70 a d}","-\frac{6 e^6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{6 e^5 \sin (c+d x) \sqrt{e \sec (c+d x)}}{5 a d}+\frac{2 e^3 \sin (c+d x) (e \sec (c+d x))^{5/2}}{5 a d}-\frac{2 i e^2 (e \sec (c+d x))^{7/2}}{7 a d}",1,"(e^4*(e*Sec[c + d*x])^(3/2)*(76 + 28*Cos[2*(c + d*x)] - (7*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^((2*I)*(c + d*x)) + (7*I)*Sec[c + d*x]*Sin[3*(c + d*x)] - (13*I)*Tan[c + d*x])*(-I + Tan[c + d*x]))/(70*a*d)","C",1
224,1,62,105,0.6937861,"\int \frac{(e \sec (c+d x))^{9/2}}{a+i a \tan (c+d x)} \, dx","Integrate[(e*Sec[c + d*x])^(9/2)/(a + I*a*Tan[c + d*x]),x]","\frac{e^2 (e \sec (c+d x))^{5/2} \left(5 \sin (2 (c+d x))+10 \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 i\right)}{15 a d}","\frac{2 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{3 a d}+\frac{2 e^3 \sin (c+d x) (e \sec (c+d x))^{3/2}}{3 a d}-\frac{2 i e^2 (e \sec (c+d x))^{5/2}}{5 a d}",1,"(e^2*(e*Sec[c + d*x])^(5/2)*(-6*I + 10*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 5*Sin[2*(c + d*x)]))/(15*a*d)","A",1
225,1,102,101,0.8419414,"\int \frac{(e \sec (c+d x))^{7/2}}{a+i a \tan (c+d x)} \, dx","Integrate[(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x]),x]","\frac{2 i e^3 (\cos (c)+i \sin (c)) (\cos (d x)+i \sin (d x)) \sqrt{e \sec (c+d x)} \left(\sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+i \tan (c+d x)-4\right)}{3 a d}","-\frac{2 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 e^3 \sin (c+d x) \sqrt{e \sec (c+d x)}}{a d}-\frac{2 i e^2 (e \sec (c+d x))^{3/2}}{3 a d}",1,"(((2*I)/3)*e^3*Sqrt[e*Sec[c + d*x]]*(Cos[c] + I*Sin[c])*(Cos[d*x] + I*Sin[d*x])*(-4 + Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + I*Tan[c + d*x]))/(a*d)","C",1
226,1,49,70,0.3767306,"\int \frac{(e \sec (c+d x))^{5/2}}{a+i a \tan (c+d x)} \, dx","Integrate[(e*Sec[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x]),x]","\frac{2 e^2 \sqrt{e \sec (c+d x)} \left(\sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-i\right)}{a d}","\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{a d}-\frac{2 i e^2 \sqrt{e \sec (c+d x)}}{a d}",1,"(2*e^2*(-I + Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])*Sqrt[e*Sec[c + d*x]])/(a*d)","A",1
227,1,74,70,0.4201809,"\int \frac{(e \sec (c+d x))^{3/2}}{a+i a \tan (c+d x)} \, dx","Integrate[(e*Sec[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x]),x]","\frac{2 i e e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right) \sqrt{e \sec (c+d x)}}{a d}","\frac{2 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 i e^2}{a d \sqrt{e \sec (c+d x)}}",1,"((2*I)*e*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))]*Sqrt[e*Sec[c + d*x]])/(a*d*E^(I*(c + d*x)))","C",1
228,1,83,80,0.3402527,"\int \frac{\sqrt{e \sec (c+d x)}}{a+i a \tan (c+d x)} \, dx","Integrate[Sqrt[e*Sec[c + d*x]]/(a + I*a*Tan[c + d*x]),x]","\frac{2 (e \sec (c+d x))^{3/2} \left(\cos (c+d x)+\sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sin (c+d x)-i \cos (c+d x))\right)}{3 a d e (\tan (c+d x)-i)}","\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{3 a d}+\frac{2 i \sqrt{e \sec (c+d x)}}{3 d (a+i a \tan (c+d x))}",1,"(2*(e*Sec[c + d*x])^(3/2)*(Cos[c + d*x] + Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*((-I)*Cos[c + d*x] + Sin[c + d*x])))/(3*a*d*e*(-I + Tan[c + d*x]))","A",1
229,1,109,80,0.8768387,"\int \frac{1}{\sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx","Integrate[1/(Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])),x]","\frac{(\tan (c+d x)+i) \left(-2 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+3 i \sin (2 (c+d x))+4 \cos (2 (c+d x))+4\right)}{5 a d \sqrt{e \sec (c+d x)}}","\frac{6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 i}{5 d (a+i a \tan (c+d x)) \sqrt{e \sec (c+d x)}}",1,"((4 + 4*Cos[2*(c + d*x)] - 2*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + (3*I)*Sin[2*(c + d*x)])*(I + Tan[c + d*x]))/(5*a*d*Sqrt[e*Sec[c + d*x]])","C",1
230,1,125,114,0.599373,"\int \frac{1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))} \, dx","Integrate[1/((e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])),x]","-\frac{\sec ^3(c+d x) \left(5 i \sin (c+d x)+5 i \sin (3 (c+d x))-14 \cos (c+d x)+2 \cos (3 (c+d x))+20 i \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (c+d x)+i \sin (c+d x))\right)}{42 a d (\tan (c+d x)-i) (e \sec (c+d x))^{3/2}}","\frac{10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{21 a d e^2}+\frac{10 \sin (c+d x)}{21 a d e \sqrt{e \sec (c+d x)}}+\frac{2 i}{7 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{3/2}}",1,"-1/42*(Sec[c + d*x]^3*(-14*Cos[c + d*x] + 2*Cos[3*(c + d*x)] + (20*I)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[c + d*x] + I*Sin[c + d*x]) + (5*I)*Sin[c + d*x] + (5*I)*Sin[3*(c + d*x)]))/(a*d*(e*Sec[c + d*x])^(3/2)*(-I + Tan[c + d*x]))","A",1
231,1,134,114,1.1436187,"\int \frac{1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))} \, dx","Integrate[1/((e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])),x]","\frac{(\tan (c+d x)+i) \left(-56 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+70 i \sin (2 (c+d x))-7 i \sin (4 (c+d x))+104 \cos (2 (c+d x))-2 \cos (4 (c+d x))+106\right)}{180 a d e^2 \sqrt{e \sec (c+d x)}}","\frac{14 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 a d e^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{14 \sin (c+d x)}{45 a d e (e \sec (c+d x))^{3/2}}+\frac{2 i}{9 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{5/2}}",1,"((106 + 104*Cos[2*(c + d*x)] - 2*Cos[4*(c + d*x)] - 56*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + (70*I)*Sin[2*(c + d*x)] - (7*I)*Sin[4*(c + d*x)])*(I + Tan[c + d*x]))/(180*a*d*e^2*Sqrt[e*Sec[c + d*x]])","C",1
232,1,142,145,0.8516936,"\int \frac{1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx","Integrate[1/((e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])),x]","-\frac{(e \sec (c+d x))^{3/2} \left(78 i \sin (c+d x)+87 i \sin (3 (c+d x))+9 i \sin (5 (c+d x))-148 \cos (c+d x)+34 \cos (3 (c+d x))+2 \cos (5 (c+d x))+240 i \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (c+d x)+i \sin (c+d x))\right)}{616 a d e^5 (\tan (c+d x)-i)}","\frac{30 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{77 a d e^4}+\frac{30 \sin (c+d x)}{77 a d e^3 \sqrt{e \sec (c+d x)}}+\frac{18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac{2 i}{11 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}}",1,"-1/616*((e*Sec[c + d*x])^(3/2)*(-148*Cos[c + d*x] + 34*Cos[3*(c + d*x)] + 2*Cos[5*(c + d*x)] + (240*I)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[c + d*x] + I*Sin[c + d*x]) + (78*I)*Sin[c + d*x] + (87*I)*Sin[3*(c + d*x)] + (9*I)*Sin[5*(c + d*x)]))/(a*d*e^5*(-I + Tan[c + d*x]))","A",1
233,1,302,183,2.4682445,"\int \frac{(e \sec (c+d x))^{15/2}}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(e*Sec[c + d*x])^(15/2)/(a + I*a*Tan[c + d*x])^2,x]","\frac{(\cos (d x)+i \sin (d x))^2 (e \sec (c+d x))^{15/2} \left(\frac{22 i \sqrt{2} e^{3 i c-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)}{-1+e^{2 i c}}+\frac{1}{56} \csc (c) (\cos (2 c)+i \sin (2 c)) \sec ^{\frac{9}{2}}(c+d x) (-720 i \sin (2 c+d x)+1050 \cos (2 c+d x)+1078 \cos (2 c+3 d x)+77 \cos (4 c+3 d x)+231 \cos (4 c+5 d x)+720 i \sin (d x)+1260 \cos (d x))\right)}{45 d \sec ^{\frac{11}{2}}(c+d x) (a+i a \tan (c+d x))^2}","-\frac{22 e^8 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 a^2 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{22 e^7 \sin (c+d x) \sqrt{e \sec (c+d x)}}{15 a^2 d}+\frac{22 e^5 \sin (c+d x) (e \sec (c+d x))^{5/2}}{45 a^2 d}+\frac{22 e^3 \sin (c+d x) (e \sec (c+d x))^{9/2}}{63 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{11/2}}{7 d \left(a^2+i a^2 \tan (c+d x)\right)}",1,"((e*Sec[c + d*x])^(15/2)*(Cos[d*x] + I*Sin[d*x])^2*(((22*I)*Sqrt[2]*E^((3*I)*c - I*d*x)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(-1 + E^((2*I)*c)) + (Csc[c]*Sec[c + d*x]^(9/2)*(Cos[2*c] + I*Sin[2*c])*(1260*Cos[d*x] + 1050*Cos[2*c + d*x] + 1078*Cos[2*c + 3*d*x] + 77*Cos[4*c + 3*d*x] + 231*Cos[4*c + 5*d*x] + (720*I)*Sin[d*x] - (720*I)*Sin[2*c + d*x]))/56))/(45*d*Sec[c + d*x]^(11/2)*(a + I*a*Tan[c + d*x])^2)","C",1
234,1,85,152,0.5809279,"\int \frac{(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(e*Sec[c + d*x])^(13/2)/(a + I*a*Tan[c + d*x])^2,x]","\frac{e^6 \sec ^3(c+d x) \sqrt{e \sec (c+d x)} \left(-5 \sin (c+d x)+15 \sin (3 (c+d x))-56 i \cos (c+d x)+60 \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{70 a^2 d}","\frac{6 e^6 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{7 a^2 d}+\frac{6 e^5 \sin (c+d x) (e \sec (c+d x))^{3/2}}{7 a^2 d}+\frac{18 e^3 \sin (c+d x) (e \sec (c+d x))^{7/2}}{35 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left(a^2+i a^2 \tan (c+d x)\right)}",1,"(e^6*Sec[c + d*x]^3*Sqrt[e*Sec[c + d*x]]*((-56*I)*Cos[c + d*x] + 60*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] - 5*Sin[c + d*x] + 15*Sin[3*(c + d*x)]))/(70*a^2*d)","A",1
235,1,123,152,0.9798805,"\int \frac{(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(e*Sec[c + d*x])^(11/2)/(a + I*a*Tan[c + d*x])^2,x]","\frac{2 i e^5 e^{i (c+d x)} \left(7 \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-56 e^{2 i (c+d x)}-21 e^{4 i (c+d x)}-47\right) \sqrt{e \sec (c+d x)}}{15 a^2 d \left(1+e^{2 i (c+d x)}\right)^2}","-\frac{14 e^6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{14 e^5 \sin (c+d x) \sqrt{e \sec (c+d x)}}{5 a^2 d}+\frac{14 e^3 \sin (c+d x) (e \sec (c+d x))^{5/2}}{15 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{7/2}}{3 d \left(a^2+i a^2 \tan (c+d x)\right)}",1,"(((2*I)/15)*e^5*E^(I*(c + d*x))*(-47 - 56*E^((2*I)*(c + d*x)) - 21*E^((4*I)*(c + d*x)) + 7*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sqrt[e*Sec[c + d*x]])/(a^2*d*(1 + E^((2*I)*(c + d*x)))^2)","C",1
236,1,67,119,0.3679457,"\int \frac{(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(e*Sec[c + d*x])^(9/2)/(a + I*a*Tan[c + d*x])^2,x]","\frac{2 e^3 (e \sec (c+d x))^{3/2} \left(-\sin (c+d x)-6 i \cos (c+d x)+5 \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 a^2 d}","\frac{10 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{3 a^2 d}+\frac{10 e^3 \sin (c+d x) (e \sec (c+d x))^{3/2}}{3 a^2 d}-\frac{4 i e^2 (e \sec (c+d x))^{5/2}}{d \left(a^2+i a^2 \tan (c+d x)\right)}",1,"(2*e^3*(e*Sec[c + d*x])^(3/2)*((-6*I)*Cos[c + d*x] + 5*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - Sin[c + d*x]))/(3*a^2*d)","A",1
237,1,80,115,0.5336254,"\int \frac{(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^2,x]","\frac{2 i e^3 e^{-i (c+d x)} \left(-1+3 \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)\right) \sqrt{e \sec (c+d x)}}{a^2 d}","\frac{6 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{6 e^3 \sin (c+d x) \sqrt{e \sec (c+d x)}}{a^2 d}+\frac{4 i e^2 (e \sec (c+d x))^{3/2}}{d \left(a^2+i a^2 \tan (c+d x)\right)}",1,"((2*I)*e^3*(-1 + 3*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])*Sqrt[e*Sec[c + d*x]])/(a^2*d*E^(I*(c + d*x)))","C",1
238,1,101,90,0.426947,"\int \frac{(e \sec (c+d x))^{5/2}}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(e*Sec[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x])^2,x]","\frac{2 (e \sec (c+d x))^{5/2} (\cos (c+d x)+i \sin (c+d x)) \left(\sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (c+d x)+i \sin (c+d x))-2 i \cos (c+d x)\right)}{3 a^2 d (\tan (c+d x)-i)^2}","-\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{3 a^2 d}+\frac{4 i e^2 \sqrt{e \sec (c+d x)}}{3 d \left(a^2+i a^2 \tan (c+d x)\right)}",1,"(2*(e*Sec[c + d*x])^(5/2)*((-2*I)*Cos[c + d*x] + Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[c + d*x] + I*Sin[c + d*x]))*(Cos[c + d*x] + I*Sin[c + d*x]))/(3*a^2*d*(-I + Tan[c + d*x])^2)","A",1
239,1,102,90,0.5768224,"\int \frac{(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(e*Sec[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x])^2,x]","\frac{i e e^{-3 i (c+d x)} \left(2 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+e^{2 i (c+d x)}+1\right) \sqrt{e \sec (c+d x)}}{5 a^2 d}","\frac{2 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{4 i e^2}{5 d \left(a^2+i a^2 \tan (c+d x)\right) \sqrt{e \sec (c+d x)}}",1,"((I/5)*e*(1 + E^((2*I)*(c + d*x)) + 2*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])*Sqrt[e*Sec[c + d*x]])/(a^2*d*E^((3*I)*(c + d*x)))","C",1
240,1,112,116,0.4629026,"\int \frac{\sqrt{e \sec (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Sqrt[e*Sec[c + d*x]]/(a + I*a*Tan[c + d*x])^2,x]","-\frac{\sec ^2(c+d x) \sqrt{e \sec (c+d x)} \left(-\sin (2 (c+d x))+2 i \cos (2 (c+d x))+2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+2 i\right)}{7 a^2 d (\tan (c+d x)-i)^2}","\frac{4 i e^2}{7 d \left(a^2+i a^2 \tan (c+d x)\right) (e \sec (c+d x))^{3/2}}+\frac{2 e \sin (c+d x)}{7 a^2 d \sqrt{e \sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{7 a^2 d}",1,"-1/7*(Sec[c + d*x]^2*Sqrt[e*Sec[c + d*x]]*(2*I + (2*I)*Cos[2*(c + d*x)] + 2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) - Sin[2*(c + d*x)]))/(a^2*d*(-I + Tan[c + d*x])^2)","A",1
241,1,123,116,1.4513757,"\int \frac{1}{\sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^2} \, dx","Integrate[1/(Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^2),x]","\frac{(\sin (2 (c+d x))+i \cos (2 (c+d x))) \left(2 (7 i \sin (2 (c+d x))+8 \cos (2 (c+d x))+2)-\frac{8 e^{4 i (c+d x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{18 a^2 d \sqrt{e \sec (c+d x)}}","\frac{4 i e^2}{9 d \left(a^2+i a^2 \tan (c+d x)\right) (e \sec (c+d x))^{5/2}}+\frac{2 e \sin (c+d x)}{9 a^2 d (e \sec (c+d x))^{3/2}}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}",1,"(((-8*E^((4*I)*(c + d*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + 2*(2 + 8*Cos[2*(c + d*x)] + (7*I)*Sin[2*(c + d*x)]))*(I*Cos[2*(c + d*x)] + Sin[2*(c + d*x)]))/(18*a^2*d*Sqrt[e*Sec[c + d*x]])","C",1
242,1,134,150,0.6124036,"\int \frac{1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2} \, dx","Integrate[1/((e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^2),x]","-\frac{\sec ^4(c+d x) \left(-6 \sin (2 (c+d x))+7 \sin (4 (c+d x))+24 i \cos (2 (c+d x))-4 i \cos (4 (c+d x))+40 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+28 i\right)}{132 a^2 d (\tan (c+d x)-i)^2 (e \sec (c+d x))^{3/2}}","\frac{4 i e^2}{11 d \left(a^2+i a^2 \tan (c+d x)\right) (e \sec (c+d x))^{7/2}}+\frac{10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{33 a^2 d e^2}+\frac{2 e \sin (c+d x)}{11 a^2 d (e \sec (c+d x))^{5/2}}+\frac{10 \sin (c+d x)}{33 a^2 d e \sqrt{e \sec (c+d x)}}",1,"-1/132*(Sec[c + d*x]^4*(28*I + (24*I)*Cos[2*(c + d*x)] - (4*I)*Cos[4*(c + d*x)] + 40*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) - 6*Sin[2*(c + d*x)] + 7*Sin[4*(c + d*x)]))/(a^2*d*(e*Sec[c + d*x])^(3/2)*(-I + Tan[c + d*x])^2)","A",1
243,1,149,150,2.2212441,"\int \frac{1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx","Integrate[1/((e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^2),x]","\frac{(\cos (2 (c+d x))-i \sin (2 (c+d x))) \left(-\frac{224 i e^{4 i (c+d x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-356 \sin (2 (c+d x))+18 \sin (4 (c+d x))+416 i \cos (2 (c+d x))-8 i \cos (4 (c+d x))+88 i\right)}{520 a^2 d e^2 \sqrt{e \sec (c+d x)}}","\frac{4 i e^2}{13 d \left(a^2+i a^2 \tan (c+d x)\right) (e \sec (c+d x))^{9/2}}+\frac{42 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{65 a^2 d e^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac{14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}}",1,"((Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)])*(88*I + (416*I)*Cos[2*(c + d*x)] - (8*I)*Cos[4*(c + d*x)] - ((224*I)*E^((4*I)*(c + d*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] - 356*Sin[2*(c + d*x)] + 18*Sin[4*(c + d*x)]))/(520*a^2*d*e^2*Sqrt[e*Sec[c + d*x]])","C",1
244,1,151,181,0.9996125,"\int \frac{1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))^2} \, dx","Integrate[1/((e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])^2),x]","-\frac{(e \sec (c+d x))^{5/2} \left(-17 \sin (2 (c+d x))+128 \sin (4 (c+d x))+11 \sin (6 (c+d x))+228 i \cos (2 (c+d x))-72 i \cos (4 (c+d x))-4 i \cos (6 (c+d x))+480 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+296 i\right)}{1680 a^2 d e^6 (\tan (c+d x)-i)^2}","\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{7 a^2 d e^4}+\frac{2 \sin (c+d x)}{7 a^2 d e^3 \sqrt{e \sec (c+d x)}}+\frac{4 i e^2}{15 d \left(a^2+i a^2 \tan (c+d x)\right) (e \sec (c+d x))^{11/2}}+\frac{2 e \sin (c+d x)}{15 a^2 d (e \sec (c+d x))^{9/2}}+\frac{6 \sin (c+d x)}{35 a^2 d e (e \sec (c+d x))^{5/2}}",1,"-1/1680*((e*Sec[c + d*x])^(5/2)*(296*I + (228*I)*Cos[2*(c + d*x)] - (72*I)*Cos[4*(c + d*x)] - (4*I)*Cos[6*(c + d*x)] + 480*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) - 17*Sin[2*(c + d*x)] + 128*Sin[4*(c + d*x)] + 11*Sin[6*(c + d*x)]))/(a^2*d*e^6*(-I + Tan[c + d*x])^2)","A",1
245,1,128,178,1.6494253,"\int \frac{(e \sec (c+d x))^{15/2}}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(e*Sec[c + d*x])^(15/2)/(a + I*a*Tan[c + d*x])^3,x]","-\frac{e^6 (\tan (c+d x)-i) (e \sec (c+d x))^{3/2} \left(77 e^{-2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-868 \cos (2 (c+d x))+143 i \tan (c+d x)+203 i \sin (3 (c+d x)) \sec (c+d x)-556\right)}{210 a^3 d}","-\frac{22 e^8 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^3 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{22 e^7 \sin (c+d x) \sqrt{e \sec (c+d x)}}{5 a^3 d}+\frac{22 e^5 \sin (c+d x) (e \sec (c+d x))^{5/2}}{15 a^3 d}-\frac{22 i e^4 (e \sec (c+d x))^{7/2}}{21 a^3 d}-\frac{4 i e^2 (e \sec (c+d x))^{11/2}}{3 a d (a+i a \tan (c+d x))^2}",1,"-1/210*(e^6*(e*Sec[c + d*x])^(3/2)*(-556 - 868*Cos[2*(c + d*x)] + (77*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^((2*I)*(c + d*x)) + (203*I)*Sec[c + d*x]*Sin[3*(c + d*x)] + (143*I)*Tan[c + d*x])*(-I + Tan[c + d*x]))/(a^3*d)","C",1
246,1,74,141,0.6770452,"\int \frac{(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(e*Sec[c + d*x])^(13/2)/(a + I*a*Tan[c + d*x])^3,x]","\frac{e^4 (e \sec (c+d x))^{5/2} \left(-5 \sin (2 (c+d x))-20 i \cos (2 (c+d x))+30 \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-18 i\right)}{5 a^3 d}","\frac{6 e^6 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{a^3 d}+\frac{6 e^5 \sin (c+d x) (e \sec (c+d x))^{3/2}}{a^3 d}-\frac{18 i e^4 (e \sec (c+d x))^{5/2}}{5 a^3 d}-\frac{4 i e^2 (e \sec (c+d x))^{9/2}}{a d (a+i a \tan (c+d x))^2}",1,"(e^4*(e*Sec[c + d*x])^(5/2)*(-18*I - (20*I)*Cos[2*(c + d*x)] + 30*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] - 5*Sin[2*(c + d*x)]))/(5*a^3*d)","A",1
247,1,93,141,1.0562784,"\int \frac{(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(e*Sec[c + d*x])^(11/2)/(a + I*a*Tan[c + d*x])^3,x]","\frac{i e^4 (e \sec (c+d x))^{3/2} \left(-7 \left(1+e^{2 i (c+d x)}\right)^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+9 i \sin (2 (c+d x))+33 \cos (2 (c+d x))+35\right)}{3 a^3 d}","\frac{14 e^6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{14 e^5 \sin (c+d x) \sqrt{e \sec (c+d x)}}{a^3 d}+\frac{14 i e^4 (e \sec (c+d x))^{3/2}}{3 a^3 d}+\frac{4 i e^2 (e \sec (c+d x))^{7/2}}{a d (a+i a \tan (c+d x))^2}",1,"((I/3)*e^4*(e*Sec[c + d*x])^(3/2)*(35 + 33*Cos[2*(c + d*x)] - 7*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + (9*I)*Sin[2*(c + d*x)]))/(a^3*d)","C",1
248,1,125,116,0.4539919,"\int \frac{(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(e*Sec[c + d*x])^(9/2)/(a + I*a*Tan[c + d*x])^3,x]","\frac{2 e^4 \sec ^3(c+d x) \sqrt{e \sec (c+d x)} (\sin (2 (c+d x))-i \cos (2 (c+d x))) \left(3 \sin (c+d x)-7 i \cos (c+d x)+5 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (c+d x)+i \sin (c+d x))\right)}{3 a^3 d (\tan (c+d x)-i)^3}","\frac{10 i e^4 \sqrt{e \sec (c+d x)}}{3 a^3 d}-\frac{10 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{3 a^3 d}+\frac{4 i e^2 (e \sec (c+d x))^{5/2}}{3 a d (a+i a \tan (c+d x))^2}",1,"(2*e^4*Sec[c + d*x]^3*Sqrt[e*Sec[c + d*x]]*((-7*I)*Cos[c + d*x] + 5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[c + d*x] + I*Sin[c + d*x]) + 3*Sin[c + d*x])*((-I)*Cos[2*(c + d*x)] + Sin[2*(c + d*x)]))/(3*a^3*d*(-I + Tan[c + d*x])^3)","A",1
249,1,117,116,0.6454912,"\int \frac{(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^3,x]","\frac{2 e e^{-i d x} \left(-2+\frac{6 e^{2 i (c+d x)} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right) (e \sec (c+d x))^{5/2} (\cos (c+2 d x)+i \sin (c+2 d x))}{5 a^3 d (\tan (c+d x)-i)^3}","-\frac{6 i e^4}{5 a^3 d \sqrt{e \sec (c+d x)}}-\frac{6 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^3 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{4 i e^2 (e \sec (c+d x))^{3/2}}{5 a d (a+i a \tan (c+d x))^2}",1,"(2*e*(-2 + (6*E^((2*I)*(c + d*x))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))])*(e*Sec[c + d*x])^(5/2)*(Cos[c + 2*d*x] + I*Sin[c + 2*d*x]))/(5*a^3*d*E^(I*d*x)*(-I + Tan[c + d*x])^3)","C",1
250,1,104,132,0.642873,"\int \frac{(e \sec (c+d x))^{5/2}}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(e*Sec[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x])^3,x]","\frac{(e \sec (c+d x))^{5/2} \left(-\sin (2 (c+d x))-5 i \cos (2 (c+d x))+2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-5 i\right)}{21 a^3 d (\tan (c+d x)-i)^2}","-\frac{2 i e^2 \sqrt{e \sec (c+d x)}}{21 d \left(a^3+i a^3 \tan (c+d x)\right)}-\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{21 a^3 d}+\frac{4 i e^2 \sqrt{e \sec (c+d x)}}{7 a d (a+i a \tan (c+d x))^2}",1,"((e*Sec[c + d*x])^(5/2)*(-5*I - (5*I)*Cos[2*(c + d*x)] + 2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) - Sin[2*(c + d*x)]))/(21*a^3*d*(-I + Tan[c + d*x])^2)","A",1
251,1,140,132,0.8502883,"\int \frac{(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(e*Sec[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x])^3,x]","-\frac{e^{-i d x} \sec ^2(c+d x) (\cos (d x)+i \sin (d x)) (e \sec (c+d x))^{3/2} \left(6 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+3 i \sin (2 (c+d x))+8 \cos (2 (c+d x))+8\right)}{45 a^3 d (\tan (c+d x)-i)^3}","\frac{2 i e^2}{45 d \left(a^3+i a^3 \tan (c+d x)\right) \sqrt{e \sec (c+d x)}}+\frac{2 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 a^3 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{4 i e^2}{9 a d (a+i a \tan (c+d x))^2 \sqrt{e \sec (c+d x)}}",1,"-1/45*(Sec[c + d*x]^2*(e*Sec[c + d*x])^(3/2)*(Cos[d*x] + I*Sin[d*x])*(8 + 8*Cos[2*(c + d*x)] + 6*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + (3*I)*Sin[2*(c + d*x)]))/(a^3*d*E^(I*d*x)*(-I + Tan[c + d*x])^3)","C",1
252,1,129,152,0.5232396,"\int \frac{\sqrt{e \sec (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Sqrt[e*Sec[c + d*x]]/(a + I*a*Tan[c + d*x])^3,x]","\frac{i \sec ^3(c+d x) \sqrt{e \sec (c+d x)} \left(-15 \sin (c+d x)-15 \sin (3 (c+d x))+46 i \cos (c+d x)+22 i \cos (3 (c+d x))+20 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (3 (c+d x))+i \sin (3 (c+d x)))\right)}{154 a^3 d (\tan (c+d x)-i)^3}","\frac{20 i e^2}{77 d \left(a^3+i a^3 \tan (c+d x)\right) (e \sec (c+d x))^{3/2}}+\frac{10 e \sin (c+d x)}{77 a^3 d \sqrt{e \sec (c+d x)}}+\frac{10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{77 a^3 d}+\frac{2 i \sqrt{e \sec (c+d x)}}{11 d (a+i a \tan (c+d x))^3}",1,"((I/154)*Sec[c + d*x]^3*Sqrt[e*Sec[c + d*x]]*((46*I)*Cos[c + d*x] + (22*I)*Cos[3*(c + d*x)] - 15*Sin[c + d*x] + 20*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)]) - 15*Sin[3*(c + d*x)]))/(a^3*d*(-I + Tan[c + d*x])^3)","A",1
253,1,145,152,1.4499825,"\int \frac{1}{\sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^3} \, dx","Integrate[1/(Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^3),x]","\frac{\sqrt{e \sec (c+d x)} (\sin (3 (c+d x))+i \cos (3 (c+d x))) \left(-56 e^{4 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+126 i \sin (2 (c+d x))+105 i \sin (4 (c+d x))+176 \cos (2 (c+d x))+114 \cos (4 (c+d x))+62\right)}{468 a^3 d e}","\frac{28 i e^2}{117 d \left(a^3+i a^3 \tan (c+d x)\right) (e \sec (c+d x))^{5/2}}+\frac{14 e \sin (c+d x)}{117 a^3 d (e \sec (c+d x))^{3/2}}+\frac{14 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{39 a^3 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 i}{13 d (a+i a \tan (c+d x))^3 \sqrt{e \sec (c+d x)}}",1,"(Sqrt[e*Sec[c + d*x]]*(I*Cos[3*(c + d*x)] + Sin[3*(c + d*x)])*(62 + 176*Cos[2*(c + d*x)] + 114*Cos[4*(c + d*x)] - 56*E^((4*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + (126*I)*Sin[2*(c + d*x)] + (105*I)*Sin[4*(c + d*x)]))/(468*a^3*d*e)","C",1
254,1,151,186,0.7338266,"\int \frac{1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3} \, dx","Integrate[1/((e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^3),x]","\frac{\sec ^5(c+d x) \left(-114 i \sin (c+d x)-81 i \sin (3 (c+d x))+33 i \sin (5 (c+d x))-332 \cos (c+d x)-154 \cos (3 (c+d x))+22 \cos (5 (c+d x))+240 i \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (3 (c+d x))+i \sin (3 (c+d x)))\right)}{1320 a^3 d (\tan (c+d x)-i)^3 (e \sec (c+d x))^{3/2}}","\frac{12 i e^2}{55 d \left(a^3+i a^3 \tan (c+d x)\right) (e \sec (c+d x))^{7/2}}+\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{11 a^3 d e^2}+\frac{6 e \sin (c+d x)}{55 a^3 d (e \sec (c+d x))^{5/2}}+\frac{2 \sin (c+d x)}{11 a^3 d e \sqrt{e \sec (c+d x)}}+\frac{2 i}{15 d (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}",1,"(Sec[c + d*x]^5*(-332*Cos[c + d*x] - 154*Cos[3*(c + d*x)] + 22*Cos[5*(c + d*x)] - (114*I)*Sin[c + d*x] + (240*I)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)]) - (81*I)*Sin[3*(c + d*x)] + (33*I)*Sin[5*(c + d*x)]))/(1320*a^3*d*(e*Sec[c + d*x])^(3/2)*(-I + Tan[c + d*x])^3)","A",1
255,1,124,192,1.4338517,"\int \frac{(e \sec (c+d x))^{15/2}}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(e*Sec[c + d*x])^(15/2)/(a + I*a*Tan[c + d*x])^4,x]","-\frac{i e^5 (e \sec (c+d x))^{5/2} \left(77 e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-1133 \cos (c+d x)-3 (33 i \sin (c+d x)+37 i \sin (3 (c+d x))+117 \cos (3 (c+d x)))\right)}{30 a^4 d}","\frac{154 e^8 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^4 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{154 e^7 \sin (c+d x) \sqrt{e \sec (c+d x)}}{5 a^4 d}-\frac{154 e^5 \sin (c+d x) (e \sec (c+d x))^{5/2}}{15 a^4 d}+\frac{44 i e^4 (e \sec (c+d x))^{7/2}}{3 d \left(a^4+i a^4 \tan (c+d x)\right)}+\frac{4 i e^2 (e \sec (c+d x))^{11/2}}{a d (a+i a \tan (c+d x))^3}",1,"((-1/30*I)*e^5*(e*Sec[c + d*x])^(5/2)*(-1133*Cos[c + d*x] + (77*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x)) - 3*(117*Cos[3*(c + d*x)] + (33*I)*Sin[c + d*x] + (37*I)*Sin[3*(c + d*x)])))/(a^4*d)","C",1
256,1,134,157,0.5866594,"\int \frac{(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(e*Sec[c + d*x])^(13/2)/(a + I*a*Tan[c + d*x])^4,x]","\frac{i e^6 \sec ^5(c+d x) \sqrt{e \sec (c+d x)} (\cos (3 (c+d x))+i \sin (3 (c+d x))) \left(11 i \sin (2 (c+d x))+19 \cos (2 (c+d x))+30 i \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (c+d x)+i \sin (c+d x))+21\right)}{3 a^4 d (\tan (c+d x)-i)^4}","-\frac{10 e^6 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{a^4 d}-\frac{10 e^5 \sin (c+d x) (e \sec (c+d x))^{3/2}}{a^4 d}+\frac{12 i e^4 (e \sec (c+d x))^{5/2}}{d \left(a^4+i a^4 \tan (c+d x)\right)}+\frac{4 i e^2 (e \sec (c+d x))^{9/2}}{3 a d (a+i a \tan (c+d x))^3}",1,"((I/3)*e^6*Sec[c + d*x]^5*Sqrt[e*Sec[c + d*x]]*(21 + 19*Cos[2*(c + d*x)] + (30*I)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2]*(Cos[c + d*x] + I*Sin[c + d*x]) + (11*I)*Sin[2*(c + d*x)])*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)]))/(a^4*d*(-I + Tan[c + d*x])^4)","A",1
257,1,106,163,0.6257057,"\int \frac{(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(e*Sec[c + d*x])^(11/2)/(a + I*a*Tan[c + d*x])^4,x]","-\frac{2 i e^5 e^{-3 i (c+d x)} \left(21 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-7 e^{2 i (c+d x)}-2\right) \sqrt{e \sec (c+d x)}}{5 a^4 d}","-\frac{42 e^6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^4 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{42 e^5 \sin (c+d x) \sqrt{e \sec (c+d x)}}{5 a^4 d}-\frac{28 i e^4 (e \sec (c+d x))^{3/2}}{5 d \left(a^4+i a^4 \tan (c+d x)\right)}+\frac{4 i e^2 (e \sec (c+d x))^{7/2}}{5 a d (a+i a \tan (c+d x))^3}",1,"(((-2*I)/5)*e^5*(-2 - 7*E^((2*I)*(c + d*x)) + 21*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])*Sqrt[e*Sec[c + d*x]])/(a^4*d*E^((3*I)*(c + d*x)))","C",1
258,1,137,132,0.5558543,"\int \frac{(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(e*Sec[c + d*x])^(9/2)/(a + I*a*Tan[c + d*x])^4,x]","\frac{2 e^4 \sec ^4(c+d x) \sqrt{e \sec (c+d x)} (\cos (2 (c+d x))+i \sin (2 (c+d x))) \left(5 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-2 i (4 i \sin (2 (c+d x))+\cos (2 (c+d x))+1)\right)}{21 a^4 d (\tan (c+d x)-i)^4}","-\frac{20 i e^4 \sqrt{e \sec (c+d x)}}{21 d \left(a^4+i a^4 \tan (c+d x)\right)}+\frac{10 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{21 a^4 d}+\frac{4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3}",1,"(2*e^4*Sec[c + d*x]^4*Sqrt[e*Sec[c + d*x]]*(5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) - (2*I)*(1 + Cos[2*(c + d*x)] + (4*I)*Sin[2*(c + d*x)]))*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]))/(21*a^4*d*(-I + Tan[c + d*x])^4)","A",1
259,1,149,132,0.8033974,"\int \frac{(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^4,x]","\frac{e^3 e^{-i d x} \sec ^4(c+d x) \sqrt{e \sec (c+d x)} (\sin (c+2 d x)-i \cos (c+2 d x)) \left(6 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+3 i \sin (2 (c+d x))-7 \cos (2 (c+d x))-7\right)}{45 a^4 d (\tan (c+d x)-i)^4}","-\frac{4 i e^4}{15 d \left(a^4+i a^4 \tan (c+d x)\right) \sqrt{e \sec (c+d x)}}-\frac{2 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 a^4 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{4 i e^2 (e \sec (c+d x))^{3/2}}{9 a d (a+i a \tan (c+d x))^3}",1,"(e^3*Sec[c + d*x]^4*Sqrt[e*Sec[c + d*x]]*(-7 - 7*Cos[2*(c + d*x)] + 6*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + (3*I)*Sin[2*(c + d*x)])*((-I)*Cos[c + 2*d*x] + Sin[c + 2*d*x]))/(45*a^4*d*E^(I*d*x)*(-I + Tan[c + d*x])^4)","C",1
260,1,144,163,0.6098958,"\int \frac{(e \sec (c+d x))^{5/2}}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(e*Sec[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec ^2(c+d x) (e \sec (c+d x))^{5/2} (\cos (c+d x)+i \sin (c+d x)) \left(3 \sin (c+d x)+3 \sin (3 (c+d x))+37 i \cos (c+d x)+11 i \cos (3 (c+d x))-4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (3 (c+d x))+i \sin (3 (c+d x)))\right)}{154 a^4 d (\tan (c+d x)-i)^4}","-\frac{4 i e^4}{77 d \left(a^4+i a^4 \tan (c+d x)\right) (e \sec (c+d x))^{3/2}}-\frac{2 e^3 \sin (c+d x)}{77 a^4 d \sqrt{e \sec (c+d x)}}-\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{77 a^4 d}+\frac{4 i e^2 \sqrt{e \sec (c+d x)}}{11 a d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^2*(e*Sec[c + d*x])^(5/2)*(Cos[c + d*x] + I*Sin[c + d*x])*((37*I)*Cos[c + d*x] + (11*I)*Cos[3*(c + d*x)] + 3*Sin[c + d*x] - 4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)]) + 3*Sin[3*(c + d*x)]))/(154*a^4*d*(-I + Tan[c + d*x])^4)","A",1
261,1,142,163,1.5240722,"\int \frac{(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(e*Sec[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x])^4,x]","\frac{i e^{-i d x} \sec ^2(c+d x) (\cos (d x)+i \sin (d x)) (e \sec (c+d x))^{3/2} \left(\frac{24 e^{4 i (c+d x)} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+22 i \sin (2 (c+d x))+40 \cos (2 (c+d x))+28\right)}{234 a^4 d (\tan (c+d x)-i)^4}","\frac{4 i e^4}{117 d \left(a^4+i a^4 \tan (c+d x)\right) (e \sec (c+d x))^{5/2}}+\frac{2 e^3 \sin (c+d x)}{117 a^4 d (e \sec (c+d x))^{3/2}}+\frac{2 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{39 a^4 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{4 i e^2}{13 a d (a+i a \tan (c+d x))^3 \sqrt{e \sec (c+d x)}}",1,"((I/234)*Sec[c + d*x]^2*(e*Sec[c + d*x])^(3/2)*(Cos[d*x] + I*Sin[d*x])*(28 + 40*Cos[2*(c + d*x)] + (24*E^((4*I)*(c + d*x))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + (22*I)*Sin[2*(c + d*x)]))/(a^4*d*E^(I*d*x)*(-I + Tan[c + d*x])^4)","C",1
262,1,137,191,0.6279215,"\int \frac{\sqrt{e \sec (c+d x)}}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Sqrt[e*Sec[c + d*x]]/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) \sqrt{e \sec (c+d x)} \left(i (54 i \sin (2 (c+d x))+37 i \sin (4 (c+d x))+112 \cos (2 (c+d x))+48 \cos (4 (c+d x))+64)+40 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (4 (c+d x))+i \sin (4 (c+d x)))\right)}{660 a^4 d (\tan (c+d x)-i)^4}","\frac{4 i e^2}{33 d \left(a^4+i a^4 \tan (c+d x)\right) (e \sec (c+d x))^{3/2}}+\frac{2 e \sin (c+d x)}{33 a^4 d \sqrt{e \sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \sec (c+d x)}}{33 a^4 d}+\frac{14 i \sqrt{e \sec (c+d x)}}{165 a d (a+i a \tan (c+d x))^3}+\frac{2 i \sqrt{e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4}",1,"(Sec[c + d*x]^4*Sqrt[e*Sec[c + d*x]]*(40*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[4*(c + d*x)] + I*Sin[4*(c + d*x)]) + I*(64 + 112*Cos[2*(c + d*x)] + 48*Cos[4*(c + d*x)] + (54*I)*Sin[2*(c + d*x)] + (37*I)*Sin[4*(c + d*x)])))/(660*a^4*d*(-I + Tan[c + d*x])^4)","A",1
263,1,104,69,1.2667016,"\int (d \sec (e+f x))^{5/3} (a+i a \tan (e+f x)) \, dx","Integrate[(d*Sec[e + f*x])^(5/3)*(a + I*a*Tan[e + f*x]),x]","\frac{3 a d e^{-2 i f x} (d \sec (e+f x))^{2/3} \left(i \left(1+e^{2 i (e+f x)}\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i (e+f x)}\right)+2 \tan (e+f x)-3 i\right) (\cos (e+3 f x)+i \sin (e+3 f x))}{10 f}","\frac{6 i 2^{5/6} a (d \sec (e+f x))^{5/3} \, _2F_1\left(-\frac{5}{6},\frac{5}{6};\frac{11}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{5 f (1+i \tan (e+f x))^{5/6}}",1,"(3*a*d*(d*Sec[e + f*x])^(2/3)*(Cos[e + 3*f*x] + I*Sin[e + 3*f*x])*(-3*I + I*(1 + E^((2*I)*(e + f*x)))^(2/3)*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(e + f*x))] + 2*Tan[e + f*x]))/(10*E^((2*I)*f*x)*f)","A",1
264,1,92,67,0.5958044,"\int \sqrt[3]{d \sec (e+f x)} (a+i a \tan (e+f x)) \, dx","Integrate[(d*Sec[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x]),x]","\frac{3 a d e^{-i e} \left(-1+\sqrt[3]{1+e^{2 i (e+f x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (e+f x)}\right)\right) (\tan (e+f x)-i) (\cos (f x)-i \sin (f x))}{f (d \sec (e+f x))^{2/3}}","\frac{6 i \sqrt[6]{2} a \sqrt[3]{d \sec (e+f x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{6};\frac{7}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{f \sqrt[6]{1+i \tan (e+f x)}}",1,"(3*a*d*(-1 + (1 + E^((2*I)*(e + f*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(e + f*x))])*(Cos[f*x] - I*Sin[f*x])*(-I + Tan[e + f*x]))/(E^(I*e)*f*(d*Sec[e + f*x])^(2/3))","A",1
265,1,98,67,0.4679874,"\int \frac{a+i a \tan (e+f x)}{\sqrt[3]{d \sec (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(d*Sec[e + f*x])^(1/3),x]","-\frac{3 i 2^{2/3} a e^{2 i (e+f x)} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i (e+f x)}\right)}{5 f \sqrt[3]{1+e^{2 i (e+f x)}} \sqrt[3]{\frac{d e^{i (e+f x)}}{1+e^{2 i (e+f x)}}}}","-\frac{3 i 2^{5/6} a \sqrt[6]{1+i \tan (e+f x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{6};\frac{5}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{f \sqrt[3]{d \sec (e+f x)}}",1,"(((-3*I)/5)*2^(2/3)*a*E^((2*I)*(e + f*x))*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(e + f*x))])/(((d*E^(I*(e + f*x)))/(1 + E^((2*I)*(e + f*x))))^(1/3)*(1 + E^((2*I)*(e + f*x)))^(1/3)*f)","A",1
266,1,106,69,0.481863,"\int \frac{a+i a \tan (e+f x)}{(d \sec (e+f x))^{5/3}} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(d*Sec[e + f*x])^(5/3),x]","-\frac{3 i a e^{i (e+f x)} \left(4 \sqrt[3]{1+e^{2 i (e+f x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (e+f x)}\right)+e^{2 i (e+f x)}+1\right)}{5 d f \left(1+e^{2 i (e+f x)}\right) (d \sec (e+f x))^{2/3}}","-\frac{3 i \sqrt[6]{2} a (1+i \tan (e+f x))^{5/6} \, _2F_1\left(-\frac{5}{6},\frac{5}{6};\frac{1}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{5 f (d \sec (e+f x))^{5/3}}",1,"(((-3*I)/5)*a*E^(I*(e + f*x))*(1 + E^((2*I)*(e + f*x)) + 4*(1 + E^((2*I)*(e + f*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(e + f*x))]))/(d*(1 + E^((2*I)*(e + f*x)))*f*(d*Sec[e + f*x])^(2/3))","A",1
267,1,267,71,2.8203208,"\int (d \sec (e+f x))^{5/3} (a+i a \tan (e+f x))^2 \, dx","Integrate[(d*Sec[e + f*x])^(5/3)*(a + I*a*Tan[e + f*x])^2,x]","\frac{(a+i a \tan (e+f x))^2 (d \sec (e+f x))^{5/3} \left(\frac{3}{4} \csc (e) (\cos (2 e)-i \sin (2 e)) \sec ^{\frac{8}{3}}(e+f x) (64 i \sin (2 e+f x)+75 \cos (2 e+f x)+55 \cos (2 e+3 f x)-64 i \sin (f x)+90 \cos (f x))-\frac{33 i 2^{2/3} \left(5 \sqrt[3]{1+e^{2 i (e+f x)}}-\left(-1+e^{2 i e}\right) e^{2 i f x} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i (e+f x)}\right)\right)}{\left(-1+e^{2 i e}\right) \sqrt[3]{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt[3]{1+e^{2 i (e+f x)}}}\right)}{80 f \sec ^{\frac{11}{3}}(e+f x) (\cos (f x)+i \sin (f x))^2}","\frac{12 i 2^{5/6} a^2 (d \sec (e+f x))^{5/3} \, _2F_1\left(-\frac{11}{6},\frac{5}{6};\frac{11}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{5 f (1+i \tan (e+f x))^{5/6}}",1,"((d*Sec[e + f*x])^(5/3)*(((-33*I)*2^(2/3)*(5*(1 + E^((2*I)*(e + f*x)))^(1/3) - E^((2*I)*f*x)*(-1 + E^((2*I)*e))*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(e + f*x))]))/((-1 + E^((2*I)*e))*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(1/3)*(1 + E^((2*I)*(e + f*x)))^(1/3)) + (3*Csc[e]*Sec[e + f*x]^(8/3)*(Cos[2*e] - I*Sin[2*e])*(90*Cos[f*x] + 75*Cos[2*e + f*x] + 55*Cos[2*e + 3*f*x] - (64*I)*Sin[f*x] + (64*I)*Sin[2*e + f*x]))/4)*(a + I*a*Tan[e + f*x])^2)/(80*f*Sec[e + f*x]^(11/3)*(Cos[f*x] + I*Sin[f*x])^2)","B",1
268,1,128,69,1.0755799,"\int \sqrt[3]{d \sec (e+f x)} (a+i a \tan (e+f x))^2 \, dx","Integrate[(d*Sec[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^2,x]","-\frac{3 a^2 e^{-2 i e} \sqrt[3]{d \sec (e+f x)} (\cos (2 (e+f x))+i \sin (2 (e+f x))) \left(7 i \sqrt[3]{1+e^{2 i (e+f x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (e+f x)}\right)+\sec (e) \sin (f x) \sec (e+f x)+\tan (e)-8 i\right)}{4 f (\cos (f x)+i \sin (f x))^2}","\frac{12 i \sqrt[6]{2} a^2 \sqrt[3]{d \sec (e+f x)} \, _2F_1\left(-\frac{7}{6},\frac{1}{6};\frac{7}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{f \sqrt[6]{1+i \tan (e+f x)}}",1,"(-3*a^2*(d*Sec[e + f*x])^(1/3)*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*(-8*I + (7*I)*(1 + E^((2*I)*(e + f*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(e + f*x))] + Sec[e]*Sec[e + f*x]*Sin[f*x] + Tan[e]))/(4*E^((2*I)*e)*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
269,1,132,83,1.2881977,"\int \frac{(a+i a \tan (e+f x))^2}{\sqrt[3]{d \sec (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(d*Sec[e + f*x])^(1/3),x]","-\frac{3 i a^2 e^{2 i (e+f x)} \left(\left(1+e^{2 i (e+f x)}\right) \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i (e+f x)}\right)-\sqrt[3]{1+e^{2 i (e+f x)}}\right)}{\sqrt[3]{2} f \left(1+e^{2 i (e+f x)}\right)^{4/3} \sqrt[3]{\frac{d e^{i (e+f x)}}{1+e^{2 i (e+f x)}}}}","-\frac{6 i 2^{5/6} \left(a^2+i a^2 \tan (e+f x)\right) \, _2F_1\left(-\frac{5}{6},-\frac{1}{6};\frac{5}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{f (1+i \tan (e+f x))^{5/6} \sqrt[3]{d \sec (e+f x)}}",1,"((-3*I)*a^2*E^((2*I)*(e + f*x))*(-(1 + E^((2*I)*(e + f*x)))^(1/3) + (1 + E^((2*I)*(e + f*x)))*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(e + f*x))]))/(2^(1/3)*((d*E^(I*(e + f*x)))/(1 + E^((2*I)*(e + f*x))))^(1/3)*(1 + E^((2*I)*(e + f*x)))^(4/3)*f)","A",1
270,1,105,85,0.7318018,"\int \frac{(a+i a \tan (e+f x))^2}{(d \sec (e+f x))^{5/3}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(d*Sec[e + f*x])^(5/3),x]","-\frac{12 i a^2 e^{2 i (e+f x)} \left(-\sqrt[3]{1+e^{2 i (e+f x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (e+f x)}\right)+e^{2 i (e+f x)}+1\right)}{5 f \left(1+e^{2 i (e+f x)}\right)^2 (d \sec (e+f x))^{5/3}}","-\frac{6 i \sqrt[6]{2} \left(a^2+i a^2 \tan (e+f x)\right) \, _2F_1\left(-\frac{5}{6},-\frac{1}{6};\frac{1}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{5 f \sqrt[6]{1+i \tan (e+f x)} (d \sec (e+f x))^{5/3}}",1,"(((-12*I)/5)*a^2*E^((2*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)) - (1 + E^((2*I)*(e + f*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(e + f*x))]))/((1 + E^((2*I)*(e + f*x)))^2*f*(d*Sec[e + f*x])^(5/3))","A",1
271,1,84,83,0.5085948,"\int \frac{(d \sec (e+f x))^{5/3}}{a+i a \tan (e+f x)} \, dx","Integrate[(d*Sec[e + f*x])^(5/3)/(a + I*a*Tan[e + f*x]),x]","\frac{6 d e^{i (e+f x)} \, _2F_1\left(-\frac{1}{6},\frac{2}{3};\frac{5}{6};-e^{2 i (e+f x)}\right) (d \sec (e+f x))^{2/3}}{a f \sqrt[3]{1+e^{2 i (e+f x)}} (\tan (e+f x)-i)}","\frac{3 i \sqrt[6]{1+i \tan (e+f x)} (d \sec (e+f x))^{5/3} \, _2F_1\left(\frac{5}{6},\frac{7}{6};\frac{11}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{5 \sqrt[6]{2} f (a+i a \tan (e+f x))}",1,"(6*d*E^(I*(e + f*x))*Hypergeometric2F1[-1/6, 2/3, 5/6, -E^((2*I)*(e + f*x))]*(d*Sec[e + f*x])^(2/3))/(a*(1 + E^((2*I)*(e + f*x)))^(1/3)*f*(-I + Tan[e + f*x]))","A",1
272,1,103,81,0.4856705,"\int \frac{\sqrt[3]{d \sec (e+f x)}}{a+i a \tan (e+f x)} \, dx","Integrate[(d*Sec[e + f*x])^(1/3)/(a + I*a*Tan[e + f*x]),x]","-\frac{3 i e^{-2 i (e+f x)} \left(4 e^{2 i (e+f x)} \sqrt[3]{1+e^{2 i (e+f x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (e+f x)}\right)-e^{2 i (e+f x)}-1\right) \sqrt[3]{d \sec (e+f x)}}{10 a f}","\frac{3 i (1+i \tan (e+f x))^{5/6} \sqrt[3]{d \sec (e+f x)} \, _2F_1\left(\frac{1}{6},\frac{11}{6};\frac{7}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{2^{5/6} f (a+i a \tan (e+f x))}",1,"(((-3*I)/10)*(-1 - E^((2*I)*(e + f*x)) + 4*E^((2*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(e + f*x))])*(d*Sec[e + f*x])^(1/3))/(a*E^((2*I)*(e + f*x))*f)","A",1
273,1,112,71,0.9884281,"\int \frac{1}{\sqrt[3]{d \sec (e+f x)} (a+i a \tan (e+f x))} \, dx","Integrate[1/((d*Sec[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])),x]","\frac{3 (\tan (e+f x)+i) \left(5 (4 i \sin (2 (e+f x))+5 \cos (2 (e+f x))+5)-8 e^{2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i (e+f x)}\right)\right)}{70 a f \sqrt[3]{d \sec (e+f x)}}","-\frac{3 i \sqrt[6]{1+i \tan (e+f x)} \, _2F_1\left(-\frac{1}{6},\frac{13}{6};\frac{5}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{2 \sqrt[6]{2} a f \sqrt[3]{d \sec (e+f x)}}",1,"(3*(-8*E^((2*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^(2/3)*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(e + f*x))] + 5*(5 + 5*Cos[2*(e + f*x)] + (4*I)*Sin[2*(e + f*x)]))*(I + Tan[e + f*x]))/(70*a*f*(d*Sec[e + f*x])^(1/3))","A",1
274,1,119,71,0.9496507,"\int \frac{1}{(d \sec (e+f x))^{5/3} (a+i a \tan (e+f x))} \, dx","Integrate[1/((d*Sec[e + f*x])^(5/3)*(a + I*a*Tan[e + f*x])),x]","-\frac{3 \sec ^2(e+f x) \left(\frac{128 e^{2 i (e+f x)} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (e+f x)}\right)}{\left(1+e^{2 i (e+f x)}\right)^{2/3}}+16 i \sin (2 (e+f x))+6 \cos (2 (e+f x))-26\right)}{220 a f (\tan (e+f x)-i) (d \sec (e+f x))^{5/3}}","-\frac{3 i (1+i \tan (e+f x))^{5/6} \, _2F_1\left(-\frac{5}{6},\frac{17}{6};\frac{1}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{10\ 2^{5/6} a f (d \sec (e+f x))^{5/3}}",1,"(-3*Sec[e + f*x]^2*(-26 + 6*Cos[2*(e + f*x)] + (128*E^((2*I)*(e + f*x))*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(e + f*x))])/(1 + E^((2*I)*(e + f*x)))^(2/3) + (16*I)*Sin[2*(e + f*x)]))/(220*a*f*(d*Sec[e + f*x])^(5/3)*(-I + Tan[e + f*x]))","A",1
275,1,128,87,0.7643839,"\int \frac{(d \sec (e+f x))^{5/3}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(d*Sec[e + f*x])^(5/3)/(a + I*a*Tan[e + f*x])^2,x]","-\frac{3 e^{-i (4 e+5 f x)} \left(1+e^{2 i (e+f x)}\right) \left(2 e^{2 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{2}{3};\frac{5}{6};-e^{2 i (e+f x)}\right)+e^{2 i (e+f x)}+1\right) (\sin (f x)-i \cos (f x)) (d \sec (e+f x))^{5/3}}{28 a^2 f}","\frac{3 i \sqrt[6]{1+i \tan (e+f x)} (d \sec (e+f x))^{5/3} \, _2F_1\left(\frac{5}{6},\frac{13}{6};\frac{11}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{10 \sqrt[6]{2} f \left(a^2+i a^2 \tan (e+f x)\right)}",1,"(-3*(1 + E^((2*I)*(e + f*x)))*(1 + E^((2*I)*(e + f*x)) + 2*E^((2*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^(2/3)*Hypergeometric2F1[-1/6, 2/3, 5/6, -E^((2*I)*(e + f*x))])*(d*Sec[e + f*x])^(5/3)*((-I)*Cos[f*x] + Sin[f*x]))/(28*a^2*E^(I*(4*e + 5*f*x))*f)","A",1
276,1,121,87,0.6149789,"\int \frac{\sqrt[3]{d \sec (e+f x)}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(d*Sec[e + f*x])^(1/3)/(a + I*a*Tan[e + f*x])^2,x]","\frac{3 \sec ^2(e+f x) \sqrt[3]{d \sec (e+f x)} \left(4 i e^{2 i (e+f x)} \sqrt[3]{1+e^{2 i (e+f x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (e+f x)}\right)+\sin (2 (e+f x))-2 i \cos (2 (e+f x))-2 i\right)}{22 a^2 f (\tan (e+f x)-i)^2}","\frac{3 i (1+i \tan (e+f x))^{5/6} \sqrt[3]{d \sec (e+f x)} \, _2F_1\left(\frac{1}{6},\frac{17}{6};\frac{7}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{2\ 2^{5/6} f \left(a^2+i a^2 \tan (e+f x)\right)}",1,"(3*Sec[e + f*x]^2*(d*Sec[e + f*x])^(1/3)*(-2*I - (2*I)*Cos[2*(e + f*x)] + (4*I)*E^((2*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(e + f*x))] + Sin[2*(e + f*x)]))/(22*a^2*f*(-I + Tan[e + f*x])^2)","A",1
277,1,141,71,1.4461309,"\int \frac{1}{\sqrt[3]{d \sec (e+f x)} (a+i a \tan (e+f x))^2} \, dx","Integrate[1/((d*Sec[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^2),x]","\frac{(d \sec (e+f x))^{2/3} (-3 \sin (2 (e+f x))-3 i \cos (2 (e+f x))) \left(16 e^{3 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i (e+f x)}\right)-10 \left(7 \cos (e+f x)+5 \cos (3 (e+f x))+18 i \sin (e+f x) \cos ^2(e+f x)\right)\right)}{260 a^2 d f}","-\frac{3 i \sqrt[6]{1+i \tan (e+f x)} \, _2F_1\left(-\frac{1}{6},\frac{19}{6};\frac{5}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{4 \sqrt[6]{2} a^2 f \sqrt[3]{d \sec (e+f x)}}",1,"((d*Sec[e + f*x])^(2/3)*(16*E^((3*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^(2/3)*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(e + f*x))] - 10*(7*Cos[e + f*x] + 5*Cos[3*(e + f*x)] + (18*I)*Cos[e + f*x]^2*Sin[e + f*x]))*((-3*I)*Cos[2*(e + f*x)] - 3*Sin[2*(e + f*x)]))/(260*a^2*d*f)","A",1
278,1,143,71,0.9114535,"\int \frac{1}{(d \sec (e+f x))^{5/3} (a+i a \tan (e+f x))^2} \, dx","Integrate[1/((d*Sec[e + f*x])^(5/3)*(a + I*a*Tan[e + f*x])^2),x]","\frac{3 i \sec ^4(e+f x) \left(128 e^{2 i (e+f x)} \sqrt[3]{1+e^{2 i (e+f x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (e+f x)}\right)-10 i \sin (2 (e+f x))+11 i \sin (4 (e+f x))-40 \cos (2 (e+f x))+6 \cos (4 (e+f x))-46\right)}{680 a^2 f (\tan (e+f x)-i)^2 (d \sec (e+f x))^{5/3}}","-\frac{3 i (1+i \tan (e+f x))^{5/6} \, _2F_1\left(-\frac{5}{6},\frac{23}{6};\frac{1}{6};\frac{1}{2} (1-i \tan (e+f x))\right)}{20\ 2^{5/6} a^2 f (d \sec (e+f x))^{5/3}}",1,"(((3*I)/680)*Sec[e + f*x]^4*(-46 - 40*Cos[2*(e + f*x)] + 6*Cos[4*(e + f*x)] + 128*E^((2*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(e + f*x))] - (10*I)*Sin[2*(e + f*x)] + (11*I)*Sin[4*(e + f*x)]))/(a^2*f*(d*Sec[e + f*x])^(5/3)*(-I + Tan[e + f*x])^2)","B",1
279,1,95,117,0.8835763,"\int \sec ^8(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^8*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 \sec ^7(c+d x) \sqrt{a+i a \tan (c+d x)} (-3 i (90 \sin (c+d x)+233 \sin (3 (c+d x)))+510 \cos (c+d x)+731 \cos (3 (c+d x))) (\sin (4 (c+d x))-i \cos (4 (c+d x)))}{6435 d}","\frac{2 i (a+i a \tan (c+d x))^{15/2}}{15 a^7 d}-\frac{12 i (a+i a \tan (c+d x))^{13/2}}{13 a^6 d}+\frac{24 i (a+i a \tan (c+d x))^{11/2}}{11 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{9/2}}{9 a^4 d}",1,"(2*Sec[c + d*x]^7*(510*Cos[c + d*x] + 731*Cos[3*(c + d*x)] - (3*I)*(90*Sin[c + d*x] + 233*Sin[3*(c + d*x)]))*((-I)*Cos[4*(c + d*x)] + Sin[4*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(6435*d)","A",1
280,1,77,88,0.4373504,"\int \sec ^6(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^6*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 \sec ^5(c+d x) \sqrt{a+i a \tan (c+d x)} (-91 i \sin (2 (c+d x))+107 \cos (2 (c+d x))+44) (\sin (3 (c+d x))-i \cos (3 (c+d x)))}{693 d}","-\frac{2 i (a+i a \tan (c+d x))^{11/2}}{11 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{9/2}}{9 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{7/2}}{7 a^3 d}",1,"(2*Sec[c + d*x]^5*(44 + 107*Cos[2*(c + d*x)] - (91*I)*Sin[2*(c + d*x)])*((-I)*Cos[3*(c + d*x)] + Sin[3*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(693*d)","A",1
281,1,58,59,0.3525425,"\int \sec ^4(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^4*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 \sqrt{a+i a \tan (c+d x)} \left(8 (\tan (c+d x)-i)+(5 \tan (c+d x)-i) \sec ^2(c+d x)\right)}{35 d}","\frac{2 i (a+i a \tan (c+d x))^{7/2}}{7 a^3 d}-\frac{4 i (a+i a \tan (c+d x))^{5/2}}{5 a^2 d}",1,"(2*Sqrt[a + I*a*Tan[c + d*x]]*(8*(-I + Tan[c + d*x]) + Sec[c + d*x]^2*(-I + 5*Tan[c + d*x])))/(35*d)","A",1
282,1,34,29,0.181598,"\int \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}{3 d}","-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a d}",1,"(2*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(3*d)","A",1
283,1,105,120,0.4942221,"\int \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^{-2 i (c+d x)} \left(-e^{2 i (c+d x)}+e^{4 i (c+d x)}+3 e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)-2\right) \sqrt{a+i a \tan (c+d x)}}{8 d}","-\frac{i a^2}{2 d (a-i a \tan (c+d x)) \sqrt{a+i a \tan (c+d x)}}+\frac{3 i a}{4 d \sqrt{a+i a \tan (c+d x)}}-\frac{3 i \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} d}",1,"((-1/8*I)*(-2 - E^((2*I)*(c + d*x)) + E^((4*I)*(c + d*x)) + 3*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^((2*I)*(c + d*x)))","A",1
284,1,133,193,0.4589826,"\int \cos ^4(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]^4*Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^{-4 i (c+d x)} \left(-88 e^{2 i (c+d x)}-41 e^{4 i (c+d x)}+45 e^{6 i (c+d x)}+6 e^{8 i (c+d x)}+105 e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)-8\right) \sqrt{a+i a \tan (c+d x)}}{384 d}","-\frac{i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac{7 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}+\frac{35 i a^2}{96 d (a+i a \tan (c+d x))^{3/2}}+\frac{35 i a}{64 d \sqrt{a+i a \tan (c+d x)}}-\frac{35 i \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{64 \sqrt{2} d}",1,"((-1/384*I)*(-8 - 88*E^((2*I)*(c + d*x)) - 41*E^((4*I)*(c + d*x)) + 45*E^((6*I)*(c + d*x)) + 6*E^((8*I)*(c + d*x)) + 105*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^((4*I)*(c + d*x)))","A",1
285,1,159,266,0.7774681,"\int \cos ^6(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]^6*Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^{-6 i (c+d x)} \left(-464 e^{2 i (c+d x)}-3184 e^{4 i (c+d x)}-1433 e^{6 i (c+d x)}+1645 e^{8 i (c+d x)}+350 e^{10 i (c+d x)}+40 e^{12 i (c+d x)}+3465 e^{5 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)-48\right) \sqrt{a+i a \tan (c+d x)}}{15360 d}","-\frac{i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{5/2}}-\frac{11 i a^5}{48 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{5/2}}-\frac{33 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{5/2}}+\frac{231 i a^3}{640 d (a+i a \tan (c+d x))^{5/2}}+\frac{77 i a^2}{256 d (a+i a \tan (c+d x))^{3/2}}+\frac{231 i a}{512 d \sqrt{a+i a \tan (c+d x)}}-\frac{231 i \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{512 \sqrt{2} d}",1,"((-1/15360*I)*(-48 - 464*E^((2*I)*(c + d*x)) - 3184*E^((4*I)*(c + d*x)) - 1433*E^((6*I)*(c + d*x)) + 1645*E^((8*I)*(c + d*x)) + 350*E^((10*I)*(c + d*x)) + 40*E^((12*I)*(c + d*x)) + 3465*E^((5*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^((6*I)*(c + d*x)))","A",1
286,1,95,147,0.6725218,"\int \sec ^7(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^7*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 \sec ^6(c+d x) \sqrt{a+i a \tan (c+d x)} (7 i (26 \sin (c+d x)+59 \sin (3 (c+d x)))+390 \cos (c+d x)+445 \cos (3 (c+d x))) (\sin (4 (c+d x))+i \cos (4 (c+d x)))}{3003 d}","\frac{256 i a^4 \sec ^7(c+d x)}{3003 d (a+i a \tan (c+d x))^{7/2}}+\frac{64 i a^3 \sec ^7(c+d x)}{429 d (a+i a \tan (c+d x))^{5/2}}+\frac{24 i a^2 \sec ^7(c+d x)}{143 d (a+i a \tan (c+d x))^{3/2}}+\frac{2 i a \sec ^7(c+d x)}{13 d \sqrt{a+i a \tan (c+d x)}}",1,"(2*Sec[c + d*x]^6*(390*Cos[c + d*x] + 445*Cos[3*(c + d*x)] + (7*I)*(26*Sin[c + d*x] + 59*Sin[3*(c + d*x)]))*(I*Cos[4*(c + d*x)] + Sin[4*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(3003*d)","A",1
287,1,77,110,0.4496638,"\int \sec ^5(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^5*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 \sec ^4(c+d x) \sqrt{a+i a \tan (c+d x)} (55 i \sin (2 (c+d x))+71 \cos (2 (c+d x))+36) (\sin (3 (c+d x))+i \cos (3 (c+d x)))}{315 d}","\frac{64 i a^3 \sec ^5(c+d x)}{315 d (a+i a \tan (c+d x))^{5/2}}+\frac{16 i a^2 \sec ^5(c+d x)}{63 d (a+i a \tan (c+d x))^{3/2}}+\frac{2 i a \sec ^5(c+d x)}{9 d \sqrt{a+i a \tan (c+d x)}}",1,"(2*Sec[c + d*x]^4*(36 + 71*Cos[2*(c + d*x)] + (55*I)*Sin[2*(c + d*x)])*(I*Cos[3*(c + d*x)] + Sin[3*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(315*d)","A",1
288,1,63,73,0.2801788,"\int \sec ^3(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{2 (3 \tan (c+d x)-7 i) \sec (c+d x) \sqrt{a+i a \tan (c+d x)} (\cos (2 (c+d x))-i \sin (2 (c+d x)))}{15 d}","\frac{8 i a^2 \sec ^3(c+d x)}{15 d (a+i a \tan (c+d x))^{3/2}}+\frac{2 i a \sec ^3(c+d x)}{5 d \sqrt{a+i a \tan (c+d x)}}",1,"(-2*Sec[c + d*x]*(Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)])*(-7*I + 3*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(15*d)","A",1
289,1,39,31,0.1822972,"\int \sec (c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 \sqrt{a+i a \tan (c+d x)} (\sin (c+d x)+i \cos (c+d x))}{d}","\frac{2 i a \sec (c+d x)}{d \sqrt{a+i a \tan (c+d x)}}",1,"(2*(I*Cos[c + d*x] + Sin[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/d","A",1
290,1,87,83,0.567353,"\int \cos (c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^{-i (c+d x)} \left(e^{2 i (c+d x)}-\sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+1\right) \sqrt{a+i a \tan (c+d x)}}{2 d}","\frac{i \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{2} d}-\frac{i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{d}",1,"((-1/2*I)*(1 + E^((2*I)*(c + d*x)) - Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
291,1,126,154,0.535212,"\int \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^{-3 i (c+d x)} \left(11 e^{2 i (c+d x)}+16 e^{4 i (c+d x)}+2 e^{6 i (c+d x)}-15 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-3\right) \sqrt{a+i a \tan (c+d x)}}{48 d}","-\frac{i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{3 d}-\frac{5 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{8 d}+\frac{5 i a \cos (c+d x)}{12 d \sqrt{a+i a \tan (c+d x)}}+\frac{5 i \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{8 \sqrt{2} d}",1,"((-1/48*I)*(-3 + 11*E^((2*I)*(c + d*x)) + 16*E^((4*I)*(c + d*x)) + 2*E^((6*I)*(c + d*x)) - 15*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^((3*I)*(c + d*x)))","A",1
292,1,152,223,0.6444339,"\int \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]^5*Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^{-5 i (c+d x)} \left(-95 e^{2 i (c+d x)}+203 e^{4 i (c+d x)}+344 e^{6 i (c+d x)}+64 e^{8 i (c+d x)}+8 e^{10 i (c+d x)}-315 e^{4 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-10\right) \sqrt{a+i a \tan (c+d x)}}{1280 d}","-\frac{i \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}-\frac{21 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{80 d}+\frac{9 i a \cos ^3(c+d x)}{40 d \sqrt{a+i a \tan (c+d x)}}-\frac{63 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{128 d}+\frac{21 i a \cos (c+d x)}{64 d \sqrt{a+i a \tan (c+d x)}}+\frac{63 i \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{128 \sqrt{2} d}",1,"((-1/1280*I)*(-10 - 95*E^((2*I)*(c + d*x)) + 203*E^((4*I)*(c + d*x)) + 344*E^((6*I)*(c + d*x)) + 64*E^((8*I)*(c + d*x)) + 8*E^((10*I)*(c + d*x)) - 315*E^((4*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^((5*I)*(c + d*x)))","A",1
293,1,111,117,1.1943774,"\int \sec ^8(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{2 a \sec ^8(c+d x) (\cos (d x)-i \sin (d x)) \sqrt{a+i a \tan (c+d x)} (-11 i (34 \sin (c+d x)+99 \sin (3 (c+d x)))+646 \cos (c+d x)+1121 \cos (3 (c+d x))) (\sin (5 c+6 d x)-i \cos (5 c+6 d x))}{12155 d}","\frac{2 i (a+i a \tan (c+d x))^{17/2}}{17 a^7 d}-\frac{4 i (a+i a \tan (c+d x))^{15/2}}{5 a^6 d}+\frac{24 i (a+i a \tan (c+d x))^{13/2}}{13 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{11/2}}{11 a^4 d}",1,"(2*a*Sec[c + d*x]^8*(Cos[d*x] - I*Sin[d*x])*(646*Cos[c + d*x] + 1121*Cos[3*(c + d*x)] - (11*I)*(34*Sin[c + d*x] + 99*Sin[3*(c + d*x)]))*((-I)*Cos[5*c + 6*d*x] + Sin[5*c + 6*d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(12155*d)","A",1
294,1,93,88,0.6128989,"\int \sec ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{2 a \sec ^6(c+d x) (\cos (d x)-i \sin (d x)) \sqrt{a+i a \tan (c+d x)} (-135 i \sin (2 (c+d x))+151 \cos (2 (c+d x))+52) (\sin (4 c+5 d x)-i \cos (4 c+5 d x))}{1287 d}","-\frac{2 i (a+i a \tan (c+d x))^{13/2}}{13 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{11/2}}{11 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{9/2}}{9 a^3 d}",1,"(2*a*Sec[c + d*x]^6*(Cos[d*x] - I*Sin[d*x])*(52 + 151*Cos[2*(c + d*x)] - (135*I)*Sin[2*(c + d*x)])*((-I)*Cos[4*c + 5*d*x] + Sin[4*c + 5*d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(1287*d)","A",1
295,1,81,59,0.602368,"\int \sec ^4(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{2 a (7 \tan (c+d x)+11 i) \sec ^3(c+d x) (\cos (d x)-i \sin (d x)) \sqrt{a+i a \tan (c+d x)} (\cos (3 c+4 d x)+i \sin (3 c+4 d x))}{63 d}","\frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a^3 d}-\frac{4 i (a+i a \tan (c+d x))^{7/2}}{7 a^2 d}",1,"(-2*a*Sec[c + d*x]^3*(Cos[d*x] - I*Sin[d*x])*(Cos[3*c + 4*d*x] + I*Sin[3*c + 4*d*x])*(11*I + 7*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(63*d)","A",1
296,1,69,29,0.3436508,"\int \sec ^2(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{2 a \sec ^2(c+d x) (\cos (d x)-i \sin (d x)) \sqrt{a+i a \tan (c+d x)} (\sin (2 c+3 d x)-i \cos (2 c+3 d x))}{5 d}","-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{5 a d}",1,"(2*a*Sec[c + d*x]^2*(Cos[d*x] - I*Sin[d*x])*((-I)*Cos[2*c + 3*d*x] + Sin[2*c + 3*d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(5*d)","B",1
297,1,97,93,0.6374698,"\int \cos ^2(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{i a e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}}+\sinh ^{-1}\left(e^{i (c+d x)}\right)\right) \sqrt{a+i a \tan (c+d x)}}{4 d}","-\frac{i a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} d}-\frac{i a^2 \sqrt{a+i a \tan (c+d x)}}{2 d (a-i a \tan (c+d x))}",1,"((-1/4*I)*a*Sqrt[1 + E^((2*I)*(c + d*x))]*(E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))] + ArcSinh[E^(I*(c + d*x))])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
298,1,143,166,0.8411385,"\int \cos ^4(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{a e^{-2 i (c+d x)} \cos ^2(c+d x) (\tan (c+d x)-i) \left(\sqrt{1+e^{2 i (c+d x)}} \left(9 e^{2 i (c+d x)}+2 e^{4 i (c+d x)}-8\right)+15 e^{i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right) \sqrt{a+i a \tan (c+d x)}}{32 d \sqrt{1+e^{2 i (c+d x)}}}","-\frac{15 i a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{32 \sqrt{2} d}-\frac{i a^4}{4 d (a-i a \tan (c+d x))^2 \sqrt{a+i a \tan (c+d x)}}-\frac{5 i a^3}{16 d (a-i a \tan (c+d x)) \sqrt{a+i a \tan (c+d x)}}+\frac{15 i a^2}{32 d \sqrt{a+i a \tan (c+d x)}}",1,"(a*(Sqrt[1 + E^((2*I)*(c + d*x))]*(-8 + 9*E^((2*I)*(c + d*x)) + 2*E^((4*I)*(c + d*x))) + 15*E^(I*(c + d*x))*ArcSinh[E^(I*(c + d*x))])*Cos[c + d*x]^2*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(32*d*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))])","A",1
299,1,169,239,1.1121312,"\int \cos ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{a e^{-4 i (c+d x)} \cos ^2(c+d x) (\tan (c+d x)-i) \left(\sqrt{1+e^{2 i (c+d x)}} \left(-208 e^{2 i (c+d x)}+165 e^{4 i (c+d x)}+50 e^{6 i (c+d x)}+8 e^{8 i (c+d x)}-16\right)+315 e^{3 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right) \sqrt{a+i a \tan (c+d x)}}{768 d \sqrt{1+e^{2 i (c+d x)}}}","-\frac{105 i a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{256 \sqrt{2} d}-\frac{i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{3/2}}-\frac{3 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{3/2}}-\frac{21 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}+\frac{35 i a^3}{128 d (a+i a \tan (c+d x))^{3/2}}+\frac{105 i a^2}{256 d \sqrt{a+i a \tan (c+d x)}}",1,"(a*(Sqrt[1 + E^((2*I)*(c + d*x))]*(-16 - 208*E^((2*I)*(c + d*x)) + 165*E^((4*I)*(c + d*x)) + 50*E^((6*I)*(c + d*x)) + 8*E^((8*I)*(c + d*x))) + 315*E^((3*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])*Cos[c + d*x]^2*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(768*d*E^((4*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))])","A",1
300,1,109,147,1.0579772,"\int \sec ^5(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^5*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{2 a \sec ^4(c+d x) (\cos (d x)-i \sin (d x)) \sqrt{a+i a \tan (c+d x)} (\sin (3 c+2 d x)+i \cos (3 c+2 d x)) (494 \cos (2 (c+d x))+110 i \tan (c+d x)+215 i \sin (3 (c+d x)) \sec (c+d x)+39)}{1155 d}","\frac{256 i a^4 \sec ^5(c+d x)}{1155 d (a+i a \tan (c+d x))^{5/2}}+\frac{64 i a^3 \sec ^5(c+d x)}{231 d (a+i a \tan (c+d x))^{3/2}}+\frac{8 i a^2 \sec ^5(c+d x)}{33 d \sqrt{a+i a \tan (c+d x)}}+\frac{2 i a \sec ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{11 d}",1,"(2*a*Sec[c + d*x]^4*(Cos[d*x] - I*Sin[d*x])*(I*Cos[3*c + 2*d*x] + Sin[3*c + 2*d*x])*(39 + 494*Cos[2*(c + d*x)] + (215*I)*Sec[c + d*x]*Sin[3*(c + d*x)] + (110*I)*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(1155*d)","A",1
301,1,91,110,0.5148951,"\int \sec ^3(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{2 a \sec ^3(c+d x) (\cos (d x)-i \sin (d x)) \sqrt{a+i a \tan (c+d x)} (27 i \sin (2 (c+d x))+43 \cos (2 (c+d x))+28) (\sin (2 c+d x)+i \cos (2 c+d x))}{105 d}","\frac{64 i a^3 \sec ^3(c+d x)}{105 d (a+i a \tan (c+d x))^{3/2}}+\frac{16 i a^2 \sec ^3(c+d x)}{35 d \sqrt{a+i a \tan (c+d x)}}+\frac{2 i a \sec ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}",1,"(2*a*Sec[c + d*x]^3*(Cos[d*x] - I*Sin[d*x])*(28 + 43*Cos[2*(c + d*x)] + (27*I)*Sin[2*(c + d*x)])*(I*Cos[2*c + d*x] + Sin[2*c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(105*d)","A",1
302,1,57,69,0.3182621,"\int \sec (c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]*(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{2 a (\cos (c)-i \sin (c)) (\tan (c+d x)-5 i) (\cos (d x)-i \sin (d x)) \sqrt{a+i a \tan (c+d x)}}{3 d}","\frac{8 i a^2 \sec (c+d x)}{3 d \sqrt{a+i a \tan (c+d x)}}+\frac{2 i a \sec (c+d x) \sqrt{a+i a \tan (c+d x)}}{3 d}",1,"(-2*a*(Cos[c] - I*Sin[c])*(Cos[d*x] - I*Sin[d*x])*(-5*I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(3*d)","A",1
303,1,31,31,0.150577,"\int \cos (c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]*(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{2 i a \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{d}","-\frac{2 i a \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{d}",1,"((-2*I)*a*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d","A",1
304,1,101,122,0.9607428,"\int \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{i a e^{-i (c+d x)} \left(5 e^{2 i (c+d x)}+e^{4 i (c+d x)}-3 \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+4\right) \sqrt{a+i a \tan (c+d x)}}{12 d}","\frac{i a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{2 \sqrt{2} d}-\frac{i \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac{i a \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{2 d}",1,"((-1/12*I)*a*(4 + 5*E^((2*I)*(c + d*x)) + E^((4*I)*(c + d*x)) - 3*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
305,1,160,192,1.5026211,"\int \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{i a e^{-3 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(101 e^{2 i (c+d x)}+148 e^{4 i (c+d x)}+38 e^{6 i (c+d x)}+6 e^{8 i (c+d x)}-105 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-15\right)}{240 \sqrt{2} d}","\frac{7 i a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{16 \sqrt{2} d}+\frac{7 i a^2 \cos (c+d x)}{24 d \sqrt{a+i a \tan (c+d x)}}-\frac{i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}-\frac{7 i a \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{30 d}-\frac{7 i a \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{16 d}",1,"((-1/240*I)*a*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-15 + 101*E^((2*I)*(c + d*x)) + 148*E^((4*I)*(c + d*x)) + 38*E^((6*I)*(c + d*x)) + 6*E^((8*I)*(c + d*x)) - 105*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*d*E^((3*I)*(c + d*x)))","A",1
306,1,113,117,1.4521461,"\int \sec ^8(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{2 a^2 \sec ^8(c+d x) \sqrt{a+i a \tan (c+d x)} (\cos (6 c+8 d x)+i \sin (6 c+8 d x)) (3262 i \cos (2 (c+d x))+494 \tan (c+d x)+1599 \sin (3 (c+d x)) \sec (c+d x)-833 i)}{20995 d (\cos (d x)+i \sin (d x))^2}","\frac{2 i (a+i a \tan (c+d x))^{19/2}}{19 a^7 d}-\frac{12 i (a+i a \tan (c+d x))^{17/2}}{17 a^6 d}+\frac{8 i (a+i a \tan (c+d x))^{15/2}}{5 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{13/2}}{13 a^4 d}",1,"(-2*a^2*Sec[c + d*x]^8*(Cos[6*c + 8*d*x] + I*Sin[6*c + 8*d*x])*(-833*I + (3262*I)*Cos[2*(c + d*x)] + 1599*Sec[c + d*x]*Sin[3*(c + d*x)] + 494*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(20995*d*(Cos[d*x] + I*Sin[d*x])^2)","A",1
307,1,97,88,0.7878021,"\int \sec ^6(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{2 a^2 \sec ^7(c+d x) \sqrt{a+i a \tan (c+d x)} (-187 i \sin (2 (c+d x))+203 \cos (2 (c+d x))+60) (\sin (5 c+7 d x)-i \cos (5 c+7 d x))}{2145 d (\cos (d x)+i \sin (d x))^2}","-\frac{2 i (a+i a \tan (c+d x))^{15/2}}{15 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{13/2}}{13 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{11/2}}{11 a^3 d}",1,"(2*a^2*Sec[c + d*x]^7*(60 + 203*Cos[2*(c + d*x)] - (187*I)*Sin[2*(c + d*x)])*((-I)*Cos[5*c + 7*d*x] + Sin[5*c + 7*d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(2145*d*(Cos[d*x] + I*Sin[d*x])^2)","A",1
308,1,85,59,0.5905525,"\int \sec ^4(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{2 a^2 (9 \tan (c+d x)+13 i) \sec ^4(c+d x) \sqrt{a+i a \tan (c+d x)} (\cos (4 c+6 d x)+i \sin (4 c+6 d x))}{99 d (\cos (d x)+i \sin (d x))^2}","\frac{2 i (a+i a \tan (c+d x))^{11/2}}{11 a^3 d}-\frac{4 i (a+i a \tan (c+d x))^{9/2}}{9 a^2 d}",1,"(-2*a^2*Sec[c + d*x]^4*(Cos[4*c + 6*d*x] + I*Sin[4*c + 6*d*x])*(13*I + 9*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(99*d*(Cos[d*x] + I*Sin[d*x])^2)","A",1
309,1,73,29,0.4361018,"\int \sec ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{2 a^2 \sec ^3(c+d x) \sqrt{a+i a \tan (c+d x)} (\sin (3 c+5 d x)-i \cos (3 c+5 d x))}{7 d (\cos (d x)+i \sin (d x))^2}","-\frac{2 i (a+i a \tan (c+d x))^{7/2}}{7 a d}",1,"(2*a^2*Sec[c + d*x]^3*((-I)*Cos[3*c + 5*d*x] + Sin[3*c + 5*d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(7*d*(Cos[d*x] + I*Sin[d*x])^2)","B",1
310,1,116,89,0.6562447,"\int \cos ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i e^{-5 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \left(\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{5/2} \left(e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}}-\sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{\sqrt{2} d}","\frac{i a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} d}-\frac{i a^3 \sqrt{a+i a \tan (c+d x)}}{d (a-i a \tan (c+d x))}",1,"((-I)*((a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^(5/2)*(1 + E^((2*I)*(c + d*x)))^(5/2)*(E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))] - ArcSinh[E^(I*(c + d*x))]))/(Sqrt[2]*d*E^((5*I)*(c + d*x)))","A",1
311,1,116,137,0.8382234,"\int \cos ^4(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i a^2 e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(5+2 e^{2 i (c+d x)}\right)+3 \sinh ^{-1}\left(e^{i (c+d x)}\right)\right) \sqrt{a+i a \tan (c+d x)}}{32 d}","-\frac{3 i a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} d}-\frac{i a^4 \sqrt{a+i a \tan (c+d x)}}{4 d (a-i a \tan (c+d x))^2}-\frac{3 i a^3 \sqrt{a+i a \tan (c+d x)}}{16 d (a-i a \tan (c+d x))}",1,"((-1/32*I)*a^2*Sqrt[1 + E^((2*I)*(c + d*x))]*(E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*(5 + 2*E^((2*I)*(c + d*x))) + 3*ArcSinh[E^(I*(c + d*x))])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
312,1,142,210,0.9576051,"\int \cos ^6(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i a^2 e^{-2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(\sqrt{1+e^{2 i (c+d x)}} \left(87 e^{2 i (c+d x)}+38 e^{4 i (c+d x)}+8 e^{6 i (c+d x)}-48\right)+105 e^{i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right) \sqrt{a+i a \tan (c+d x)}}{768 d}","-\frac{35 i a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{128 \sqrt{2} d}-\frac{i a^6}{6 d (a-i a \tan (c+d x))^3 \sqrt{a+i a \tan (c+d x)}}-\frac{7 i a^5}{48 d (a-i a \tan (c+d x))^2 \sqrt{a+i a \tan (c+d x)}}-\frac{35 i a^4}{192 d (a-i a \tan (c+d x)) \sqrt{a+i a \tan (c+d x)}}+\frac{35 i a^3}{128 d \sqrt{a+i a \tan (c+d x)}}",1,"((-1/768*I)*a^2*Sqrt[1 + E^((2*I)*(c + d*x))]*(Sqrt[1 + E^((2*I)*(c + d*x))]*(-48 + 87*E^((2*I)*(c + d*x)) + 38*E^((4*I)*(c + d*x)) + 8*E^((6*I)*(c + d*x))) + 105*E^(I*(c + d*x))*ArcSinh[E^(I*(c + d*x))])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^((2*I)*(c + d*x)))","A",1
313,1,103,147,0.8430628,"\int \sec ^3(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^3*(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{2 a^2 (\sin (2 c)+i \cos (2 c)) \sec ^3(c+d x) \sqrt{a+i a \tan (c+d x)} (242 \cos (2 (c+d x))+54 i \tan (c+d x)+89 i \sin (3 (c+d x)) \sec (c+d x)+77)}{315 d (\cos (d x)+i \sin (d x))^2}","\frac{256 i a^4 \sec ^3(c+d x)}{315 d (a+i a \tan (c+d x))^{3/2}}+\frac{64 i a^3 \sec ^3(c+d x)}{105 d \sqrt{a+i a \tan (c+d x)}}+\frac{8 i a^2 \sec ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}+\frac{2 i a \sec ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}",1,"(2*a^2*Sec[c + d*x]^3*(I*Cos[2*c] + Sin[2*c])*(77 + 242*Cos[2*(c + d*x)] + (89*I)*Sec[c + d*x]*Sin[3*(c + d*x)] + (54*I)*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(315*d*(Cos[d*x] + I*Sin[d*x])^2)","A",1
314,1,93,104,0.38915,"\int \sec (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]*(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{2 a^2 \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (\sin (c-d x)+i \cos (c-d x)) (7 i \sin (2 (c+d x))+23 \cos (2 (c+d x))+20)}{15 d (\cos (d x)+i \sin (d x))^2}","\frac{64 i a^3 \sec (c+d x)}{15 d \sqrt{a+i a \tan (c+d x)}}+\frac{16 i a^2 \sec (c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}+\frac{2 i a \sec (c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}",1,"(2*a^2*Sec[c + d*x]^2*(I*Cos[c - d*x] + Sin[c - d*x])*(20 + 23*Cos[2*(c + d*x)] + (7*I)*Sin[2*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(15*d*(Cos[d*x] + I*Sin[d*x])^2)","A",1
315,1,46,65,0.2915396,"\int \cos (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]*(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{2 i a^2 \sqrt{a+i a \tan (c+d x)} (3 \cos (c+d x)-i \sin (c+d x))}{d}","\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{d}-\frac{8 i a^2 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{d}",1,"((-2*I)*a^2*(3*Cos[c + d*x] - I*Sin[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/d","A",1
316,1,69,35,0.3409923,"\int \cos ^3(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{2 a^2 \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (\sin (c+3 d x)-i \cos (c+3 d x))}{3 d (\cos (d x)+i \sin (d x))^2}","-\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}",1,"(2*a^2*Cos[c + d*x]^2*((-I)*Cos[c + 3*d*x] + Sin[c + 3*d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(3*d*(Cos[d*x] + I*Sin[d*x])^2)","A",1
317,1,118,159,1.123714,"\int \cos ^5(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i a^2 e^{-i (c+d x)} \left(34 e^{2 i (c+d x)}+14 e^{4 i (c+d x)}+3 e^{6 i (c+d x)}-15 \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+23\right) \sqrt{a+i a \tan (c+d x)}}{120 d}","\frac{i a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{4 \sqrt{2} d}-\frac{i a^2 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 d}-\frac{i \cos ^5(c+d x) (a+i a \tan (c+d x))^{5/2}}{5 d}-\frac{i a \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{6 d}",1,"((-1/120*I)*a^2*(23 + 34*E^((2*I)*(c + d*x)) + 14*E^((4*I)*(c + d*x)) + 3*E^((6*I)*(c + d*x)) - 15*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
318,1,155,231,1.248305,"\int \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i a^2 e^{-3 i (c+d x)} \left(353 e^{2 i (c+d x)}+544 e^{4 i (c+d x)}+214 e^{6 i (c+d x)}+68 e^{8 i (c+d x)}+10 e^{10 i (c+d x)}-315 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-35\right) \sqrt{a+i a \tan (c+d x)}}{2240 d}","\frac{9 i a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{32 \sqrt{2} d}+\frac{3 i a^3 \cos (c+d x)}{16 d \sqrt{a+i a \tan (c+d x)}}-\frac{3 i a^2 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{20 d}-\frac{9 i a^2 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{32 d}-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}-\frac{9 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{70 d}",1,"((-1/2240*I)*a^2*(-35 + 353*E^((2*I)*(c + d*x)) + 544*E^((4*I)*(c + d*x)) + 214*E^((6*I)*(c + d*x)) + 68*E^((8*I)*(c + d*x)) + 10*E^((10*I)*(c + d*x)) - 315*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^((3*I)*(c + d*x)))","A",1
319,1,113,117,1.8475264,"\int \sec ^8(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{2 a^3 \sec ^9(c+d x) \sqrt{a+i a \tan (c+d x)} (\cos (7 c+10 d x)+i \sin (7 c+10 d x)) (4554 i \cos (2 (c+d x))+630 \tan (c+d x)+2245 \sin (3 (c+d x)) \sec (c+d x)-1311 i)}{33915 d (\cos (d x)+i \sin (d x))^3}","\frac{2 i (a+i a \tan (c+d x))^{21/2}}{21 a^7 d}-\frac{12 i (a+i a \tan (c+d x))^{19/2}}{19 a^6 d}+\frac{24 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}",1,"(-2*a^3*Sec[c + d*x]^9*(Cos[7*c + 10*d*x] + I*Sin[7*c + 10*d*x])*(-1311*I + (4554*I)*Cos[2*(c + d*x)] + 2245*Sec[c + d*x]*Sin[3*(c + d*x)] + 630*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(33915*d*(Cos[d*x] + I*Sin[d*x])^3)","A",1
320,1,97,88,0.9536022,"\int \sec ^6(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^(7/2),x]","\frac{2 a^3 \sec ^8(c+d x) \sqrt{a+i a \tan (c+d x)} (-247 i \sin (2 (c+d x))+263 \cos (2 (c+d x))+68) (\sin (6 c+9 d x)-i \cos (6 c+9 d x))}{3315 d (\cos (d x)+i \sin (d x))^3}","-\frac{2 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d}",1,"(2*a^3*Sec[c + d*x]^8*(68 + 263*Cos[2*(c + d*x)] - (247*I)*Sin[2*(c + d*x)])*((-I)*Cos[6*c + 9*d*x] + Sin[6*c + 9*d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(3315*d*(Cos[d*x] + I*Sin[d*x])^3)","A",1
321,1,85,59,0.795295,"\int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{2 a^3 (11 \tan (c+d x)+15 i) \sec ^5(c+d x) \sqrt{a+i a \tan (c+d x)} (\cos (5 c+8 d x)+i \sin (5 c+8 d x))}{143 d (\cos (d x)+i \sin (d x))^3}","\frac{2 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d}-\frac{4 i (a+i a \tan (c+d x))^{11/2}}{11 a^2 d}",1,"(-2*a^3*Sec[c + d*x]^5*(Cos[5*c + 8*d*x] + I*Sin[5*c + 8*d*x])*(15*I + 11*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(143*d*(Cos[d*x] + I*Sin[d*x])^3)","A",1
322,1,73,29,0.4697203,"\int \sec ^2(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^(7/2),x]","\frac{2 a^3 \sec ^4(c+d x) \sqrt{a+i a \tan (c+d x)} (\sin (4 c+7 d x)-i \cos (4 c+7 d x))}{9 d (\cos (d x)+i \sin (d x))^3}","-\frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a d}",1,"(2*a^3*Sec[c + d*x]^4*((-I)*Cos[4*c + 7*d*x] + Sin[4*c + 7*d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(9*d*(Cos[d*x] + I*Sin[d*x])^3)","B",1
323,1,137,116,1.4126998,"\int \cos ^2(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{i \sqrt{2} e^{-4 i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(3 e^{i (c+d x)}+e^{3 i (c+d x)}-3 \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right) (a+i a \tan (c+d x))^{7/2}}{d \sec ^{\frac{7}{2}}(c+d x)}","\frac{3 i \sqrt{2} a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{i a^3 (a+i a \tan (c+d x))^{3/2}}{d (a-i a \tan (c+d x))}-\frac{3 i a^3 \sqrt{a+i a \tan (c+d x)}}{d}",1,"((-I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(3*E^(I*(c + d*x)) + E^((3*I)*(c + d*x)) - 3*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])*(a + I*a*Tan[c + d*x])^(7/2))/(d*E^((4*I)*(c + d*x))*Sec[c + d*x]^(7/2))","A",1
324,1,152,137,1.4927648,"\int \cos ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{i e^{-4 i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(e^{i (c+d x)}+3 e^{3 i (c+d x)}+2 e^{5 i (c+d x)}-\sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right) (a+i a \tan (c+d x))^{7/2}}{8 \sqrt{2} d \sec ^{\frac{7}{2}}(c+d x)}","\frac{i a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{8 \sqrt{2} d}-\frac{i a^5 \sqrt{a+i a \tan (c+d x)}}{2 d (a-i a \tan (c+d x))^2}+\frac{i a^4 \sqrt{a+i a \tan (c+d x)}}{8 d (a-i a \tan (c+d x))}",1,"((-1/8*I)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(E^(I*(c + d*x)) + 3*E^((3*I)*(c + d*x)) + 2*E^((5*I)*(c + d*x)) - Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])*(a + I*a*Tan[c + d*x])^(7/2))/(Sqrt[2]*d*E^((4*I)*(c + d*x))*Sec[c + d*x]^(7/2))","A",1
325,1,129,181,1.1072494,"\int \cos ^6(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{i a^3 e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(26 e^{2 i (c+d x)}+8 e^{4 i (c+d x)}+33\right)+15 \sinh ^{-1}\left(e^{i (c+d x)}\right)\right) \sqrt{a+i a \tan (c+d x)}}{384 d}","-\frac{5 i a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{64 \sqrt{2} d}-\frac{i a^6 \sqrt{a+i a \tan (c+d x)}}{6 d (a-i a \tan (c+d x))^3}-\frac{5 i a^5 \sqrt{a+i a \tan (c+d x)}}{48 d (a-i a \tan (c+d x))^2}-\frac{5 i a^4 \sqrt{a+i a \tan (c+d x)}}{64 d (a-i a \tan (c+d x))}",1,"((-1/384*I)*a^3*Sqrt[1 + E^((2*I)*(c + d*x))]*(E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*(33 + 26*E^((2*I)*(c + d*x)) + 8*E^((4*I)*(c + d*x))) + 15*ArcSinh[E^(I*(c + d*x))])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
326,1,109,139,0.7686593,"\int \sec (c+d x) (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[Sec[c + d*x]*(a + I*a*Tan[c + d*x])^(7/2),x]","\frac{2 a^3 \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (\sin (c-2 d x)+i \cos (c-2 d x)) (102 \cos (2 (c+d x))+14 i \tan (c+d x)+19 i \sin (3 (c+d x)) \sec (c+d x)+75)}{35 d (\cos (d x)+i \sin (d x))^3}","\frac{256 i a^4 \sec (c+d x)}{35 d \sqrt{a+i a \tan (c+d x)}}+\frac{64 i a^3 \sec (c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}+\frac{24 i a^2 \sec (c+d x) (a+i a \tan (c+d x))^{3/2}}{35 d}+\frac{2 i a \sec (c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}",1,"(2*a^3*Sec[c + d*x]^2*(I*Cos[c - 2*d*x] + Sin[c - 2*d*x])*(75 + 102*Cos[2*(c + d*x)] + (19*I)*Sec[c + d*x]*Sin[3*(c + d*x)] + (14*I)*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(35*d*(Cos[d*x] + I*Sin[d*x])^3)","A",1
327,1,59,104,0.4430497,"\int \cos (c+d x) (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]*(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{2 i a^3 \sec (c+d x) \sqrt{a+i a \tan (c+d x)} (-5 i \sin (2 (c+d x))+11 \cos (2 (c+d x))+12)}{3 d}","-\frac{64 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{3 d}+\frac{16 i a^2 \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^{5/2}}{3 d}",1,"(((-2*I)/3)*a^3*Sec[c + d*x]*(12 + 11*Cos[2*(c + d*x)] - (5*I)*Sin[2*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/d","A",1
328,1,86,71,0.4060157,"\int \cos ^3(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^(7/2),x]","\frac{2 a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)} (3 \sin (c+d x)+i \cos (c+d x)) (\cos (c+4 d x)+i \sin (c+4 d x))}{3 d (\cos (d x)+i \sin (d x))^3}","\frac{8 i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^{5/2}}{d}",1,"(2*a^3*Cos[c + d*x]*(I*Cos[c + d*x] + 3*Sin[c + d*x])*(Cos[c + 4*d*x] + I*Sin[c + 4*d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(3*d*(Cos[d*x] + I*Sin[d*x])^3)","A",1
329,1,73,35,0.5722508,"\int \cos ^5(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^(7/2),x]","\frac{2 a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)} (\sin (2 c+5 d x)-i \cos (2 c+5 d x))}{5 d (\cos (d x)+i \sin (d x))^3}","-\frac{2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{5/2}}{5 d}",1,"(2*a^3*Cos[c + d*x]^3*((-I)*Cos[2*c + 5*d*x] + Sin[2*c + 5*d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(5*d*(Cos[d*x] + I*Sin[d*x])^3)","B",1
330,1,131,196,1.8799901,"\int \cos ^7(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{i a^3 e^{-i (c+d x)} \left(298 e^{2 i (c+d x)}+188 e^{4 i (c+d x)}+81 e^{6 i (c+d x)}+15 e^{8 i (c+d x)}-105 \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+176\right) \sqrt{a+i a \tan (c+d x)}}{1680 d}","\frac{i a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{8 \sqrt{2} d}-\frac{i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{8 d}-\frac{i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{12 d}-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^{7/2}}{7 d}-\frac{i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{5/2}}{10 d}",1,"((-1/1680*I)*a^3*(176 + 298*E^((2*I)*(c + d*x)) + 188*E^((4*I)*(c + d*x)) + 81*E^((6*I)*(c + d*x)) + 15*E^((8*I)*(c + d*x)) - 105*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
331,1,188,268,3.3276354,"\int \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{i a^3 e^{-3 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(4303 e^{2 i (c+d x)}+7034 e^{4 i (c+d x)}+3754 e^{6 i (c+d x)}+1798 e^{8 i (c+d x)}+530 e^{10 i (c+d x)}+70 e^{12 i (c+d x)}-3465 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-315\right)}{20160 \sqrt{2} d}","\frac{11 i a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{64 \sqrt{2} d}+\frac{11 i a^4 \cos (c+d x)}{96 d \sqrt{a+i a \tan (c+d x)}}-\frac{11 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{120 d}-\frac{11 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{64 d}-\frac{11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}-\frac{11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}",1,"((-1/20160*I)*a^3*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-315 + 4303*E^((2*I)*(c + d*x)) + 7034*E^((4*I)*(c + d*x)) + 3754*E^((6*I)*(c + d*x)) + 1798*E^((8*I)*(c + d*x)) + 530*E^((10*I)*(c + d*x)) + 70*E^((12*I)*(c + d*x)) - 3465*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*d*E^((3*I)*(c + d*x)))","A",1
332,1,194,342,6.5480702,"\int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{i a^3 e^{-5 i (c+d x)} \left(-7161 e^{2 i (c+d x)}+47413 e^{4 i (c+d x)}+78800 e^{6 i (c+d x)}+38512 e^{8 i (c+d x)}+19552 e^{10 i (c+d x)}+7184 e^{12 i (c+d x)}+1624 e^{14 i (c+d x)}+168 e^{16 i (c+d x)}-45045 e^{4 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-462\right) \sqrt{a+i a \tan (c+d x)}}{473088 d}","\frac{195 i a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{1024 \sqrt{2} d}+\frac{39 i a^4 \cos ^3(c+d x)}{448 d \sqrt{a+i a \tan (c+d x)}}+\frac{65 i a^4 \cos (c+d x)}{512 d \sqrt{a+i a \tan (c+d x)}}-\frac{13 i a^3 \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{168 d}-\frac{13 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{128 d}-\frac{195 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{1024 d}-\frac{65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}-\frac{5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}",1,"((-1/473088*I)*a^3*(-462 - 7161*E^((2*I)*(c + d*x)) + 47413*E^((4*I)*(c + d*x)) + 78800*E^((6*I)*(c + d*x)) + 38512*E^((8*I)*(c + d*x)) + 19552*E^((10*I)*(c + d*x)) + 7184*E^((12*I)*(c + d*x)) + 1624*E^((14*I)*(c + d*x)) + 168*E^((16*I)*(c + d*x)) - 45045*E^((4*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^((5*I)*(c + d*x)))","A",1
333,1,95,117,0.6312124,"\int \frac{\sec ^8(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Sec[c + d*x]^8/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 \sec ^7(c+d x) (-7 i (26 \sin (c+d x)+59 \sin (3 (c+d x)))+390 \cos (c+d x)+445 \cos (3 (c+d x))) (\sin (4 (c+d x))-i \cos (4 (c+d x)))}{3003 d \sqrt{a+i a \tan (c+d x)}}","\frac{2 i (a+i a \tan (c+d x))^{13/2}}{13 a^7 d}-\frac{12 i (a+i a \tan (c+d x))^{11/2}}{11 a^6 d}+\frac{8 i (a+i a \tan (c+d x))^{9/2}}{3 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{7/2}}{7 a^4 d}",1,"(2*Sec[c + d*x]^7*(390*Cos[c + d*x] + 445*Cos[3*(c + d*x)] - (7*I)*(26*Sin[c + d*x] + 59*Sin[3*(c + d*x)]))*((-I)*Cos[4*(c + d*x)] + Sin[4*(c + d*x)]))/(3003*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
334,1,77,88,0.3682309,"\int \frac{\sec ^6(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Sec[c + d*x]^6/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 \sec ^5(c+d x) (-55 i \sin (2 (c+d x))+71 \cos (2 (c+d x))+36) (\sin (3 (c+d x))-i \cos (3 (c+d x)))}{315 d \sqrt{a+i a \tan (c+d x)}}","-\frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{7/2}}{7 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{5/2}}{5 a^3 d}",1,"(2*Sec[c + d*x]^5*(36 + 71*Cos[2*(c + d*x)] - (55*I)*Sin[2*(c + d*x)])*((-I)*Cos[3*(c + d*x)] + Sin[3*(c + d*x)]))/(315*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
335,1,65,59,0.2312944,"\int \frac{\sec ^4(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Sec[c + d*x]^4/Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{2 (3 \tan (c+d x)+7 i) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))}{15 d \sqrt{a+i a \tan (c+d x)}}","\frac{2 i (a+i a \tan (c+d x))^{5/2}}{5 a^3 d}-\frac{4 i (a+i a \tan (c+d x))^{3/2}}{3 a^2 d}",1,"(-2*Sec[c + d*x]^2*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])*(7*I + 3*Tan[c + d*x]))/(15*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
336,1,32,27,0.1795308,"\int \frac{\sec ^2(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Sec[c + d*x]^2/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 (\tan (c+d x)-i)}{d \sqrt{a+i a \tan (c+d x)}}","-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{a d}",1,"(2*(-I + Tan[c + d*x]))/(d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
337,1,126,146,0.6552218,"\int \frac{\cos ^2(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Cos[c + d*x]^2/Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^{-2 i (c+d x)} \left(\sqrt{1+e^{2 i (c+d x)}} \left(-14 e^{2 i (c+d x)}+3 e^{4 i (c+d x)}-2\right)+15 e^{3 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{24 d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a+i a \tan (c+d x)}}","-\frac{i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{3/2}}+\frac{5 i a}{12 d (a+i a \tan (c+d x))^{3/2}}+\frac{5 i}{8 d \sqrt{a+i a \tan (c+d x)}}-\frac{5 i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{8 \sqrt{2} \sqrt{a} d}",1,"((-1/24*I)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(-2 - 14*E^((2*I)*(c + d*x)) + 3*E^((4*I)*(c + d*x))) + 15*E^((3*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))]))/(d*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
338,1,152,219,0.7792073,"\int \frac{\cos ^4(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Cos[c + d*x]^4/Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^{-4 i (c+d x)} \left(\sqrt{1+e^{2 i (c+d x)}} \left(-56 e^{2 i (c+d x)}-288 e^{4 i (c+d x)}+85 e^{6 i (c+d x)}+10 e^{8 i (c+d x)}-8\right)+315 e^{5 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{640 d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a+i a \tan (c+d x)}}","-\frac{i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{5/2}}-\frac{9 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{5/2}}+\frac{63 i a^2}{160 d (a+i a \tan (c+d x))^{5/2}}+\frac{21 i a}{64 d (a+i a \tan (c+d x))^{3/2}}+\frac{63 i}{128 d \sqrt{a+i a \tan (c+d x)}}-\frac{63 i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{128 \sqrt{2} \sqrt{a} d}",1,"((-1/640*I)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(-8 - 56*E^((2*I)*(c + d*x)) - 288*E^((4*I)*(c + d*x)) + 85*E^((6*I)*(c + d*x)) + 10*E^((8*I)*(c + d*x))) + 315*E^((5*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))]))/(d*E^((4*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
339,1,178,292,1.1157994,"\int \frac{\cos ^6(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Cos[c + d*x]^6/Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^{-6 i (c+d x)} \left(\sqrt{1+e^{2 i (c+d x)}} \left(-2064 e^{2 i (c+d x)}-9008 e^{4 i (c+d x)}-40784 e^{6 i (c+d x)}+13755 e^{8 i (c+d x)}+2590 e^{10 i (c+d x)}+280 e^{12 i (c+d x)}-240\right)+45045 e^{7 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{107520 d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a+i a \tan (c+d x)}}","-\frac{i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{7/2}}-\frac{13 i a^5}{48 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac{143 i a^4}{192 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac{429 i a^3}{896 d (a+i a \tan (c+d x))^{7/2}}+\frac{429 i a^2}{1280 d (a+i a \tan (c+d x))^{5/2}}+\frac{143 i a}{512 d (a+i a \tan (c+d x))^{3/2}}+\frac{429 i}{1024 d \sqrt{a+i a \tan (c+d x)}}-\frac{429 i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{1024 \sqrt{2} \sqrt{a} d}",1,"((-1/107520*I)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(-240 - 2064*E^((2*I)*(c + d*x)) - 9008*E^((4*I)*(c + d*x)) - 40784*E^((6*I)*(c + d*x)) + 13755*E^((8*I)*(c + d*x)) + 2590*E^((10*I)*(c + d*x)) + 280*E^((12*I)*(c + d*x))) + 45045*E^((7*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))]))/(d*E^((6*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
340,1,95,147,0.6710032,"\int \frac{\sec ^9(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Sec[c + d*x]^9/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 \sec ^8(c+d x) (3 i (90 \sin (c+d x)+233 \sin (3 (c+d x)))+510 \cos (c+d x)+731 \cos (3 (c+d x))) (\sin (4 (c+d x))+i \cos (4 (c+d x)))}{6435 d \sqrt{a+i a \tan (c+d x)}}","\frac{256 i a^4 \sec ^9(c+d x)}{6435 d (a+i a \tan (c+d x))^{9/2}}+\frac{64 i a^3 \sec ^9(c+d x)}{715 d (a+i a \tan (c+d x))^{7/2}}+\frac{8 i a^2 \sec ^9(c+d x)}{65 d (a+i a \tan (c+d x))^{5/2}}+\frac{2 i a \sec ^9(c+d x)}{15 d (a+i a \tan (c+d x))^{3/2}}",1,"(2*Sec[c + d*x]^8*(510*Cos[c + d*x] + 731*Cos[3*(c + d*x)] + (3*I)*(90*Sin[c + d*x] + 233*Sin[3*(c + d*x)]))*(I*Cos[4*(c + d*x)] + Sin[4*(c + d*x)]))/(6435*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
341,1,77,110,0.4399319,"\int \frac{\sec ^7(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Sec[c + d*x]^7/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 \sec ^6(c+d x) (91 i \sin (2 (c+d x))+107 \cos (2 (c+d x))+44) (\sin (3 (c+d x))+i \cos (3 (c+d x)))}{693 d \sqrt{a+i a \tan (c+d x)}}","\frac{64 i a^3 \sec ^7(c+d x)}{693 d (a+i a \tan (c+d x))^{7/2}}+\frac{16 i a^2 \sec ^7(c+d x)}{99 d (a+i a \tan (c+d x))^{5/2}}+\frac{2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}}",1,"(2*Sec[c + d*x]^6*(44 + 107*Cos[2*(c + d*x)] + (91*I)*Sin[2*(c + d*x)])*(I*Cos[3*(c + d*x)] + Sin[3*(c + d*x)]))/(693*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
342,1,65,73,0.2845634,"\int \frac{\sec ^5(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Sec[c + d*x]^5/Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{2 (5 \tan (c+d x)-9 i) \sec ^3(c+d x) (\cos (2 (c+d x))-i \sin (2 (c+d x)))}{35 d \sqrt{a+i a \tan (c+d x)}}","\frac{8 i a^2 \sec ^5(c+d x)}{35 d (a+i a \tan (c+d x))^{5/2}}+\frac{2 i a \sec ^5(c+d x)}{7 d (a+i a \tan (c+d x))^{3/2}}",1,"(-2*Sec[c + d*x]^3*(Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)])*(-9*I + 5*Tan[c + d*x]))/(35*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
343,1,40,35,0.1706885,"\int \frac{\sec ^3(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Sec[c + d*x]^3/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 (\tan (c+d x)+i) \sec (c+d x)}{3 d \sqrt{a+i a \tan (c+d x)}}","\frac{2 i a \sec ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}",1,"(2*Sec[c + d*x]*(I + Tan[c + d*x]))/(3*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
344,1,70,52,0.3751872,"\int \frac{\sec (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Sec[c + d*x]/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 i e^{i (c+d x)} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)}{d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a+i a \tan (c+d x)}}","\frac{i \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"((2*I)*E^(I*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])/(d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
345,1,96,122,0.5537138,"\int \frac{\cos (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Cos[c + d*x]/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{\sec (c+d x) \left(3 i \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-i (3 i \sin (2 (c+d x))+\cos (2 (c+d x))+1)\right)}{8 d \sqrt{a+i a \tan (c+d x)}}","-\frac{3 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 a d}+\frac{i \cos (c+d x)}{2 d \sqrt{a+i a \tan (c+d x)}}+\frac{3 i \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{4 \sqrt{2} \sqrt{a} d}",1,"(Sec[c + d*x]*((3*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]] - I*(1 + Cos[2*(c + d*x)] + (3*I)*Sin[2*(c + d*x)])))/(8*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
346,1,117,193,0.6677499,"\int \frac{\cos ^3(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Cos[c + d*x]^3/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{\sec (c+d x) \left(133 \sin (2 (c+d x))+14 \sin (4 (c+d x))-43 i \cos (2 (c+d x))-2 i \cos (4 (c+d x))+105 i \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-41 i\right)}{384 d \sqrt{a+i a \tan (c+d x)}}","-\frac{7 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{24 a d}+\frac{i \cos ^3(c+d x)}{4 d \sqrt{a+i a \tan (c+d x)}}-\frac{35 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{64 a d}+\frac{35 i \cos (c+d x)}{96 d \sqrt{a+i a \tan (c+d x)}}+\frac{35 i \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{64 \sqrt{2} \sqrt{a} d}",1,"(Sec[c + d*x]*(-41*I + (105*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]] - (43*I)*Cos[2*(c + d*x)] - (2*I)*Cos[4*(c + d*x)] + 133*Sin[2*(c + d*x)] + 14*Sin[4*(c + d*x)]))/(384*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
347,1,110,117,0.8099836,"\int \frac{\sec ^8(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{2 i \sec ^6(c+d x) (\cos (4 (c+d x))+i \sin (4 (c+d x))) (494 i \cos (2 (c+d x))+110 \tan (c+d x)+215 \sin (3 (c+d x)) \sec (c+d x)+39 i)}{1155 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{2 i (a+i a \tan (c+d x))^{11/2}}{11 a^7 d}-\frac{4 i (a+i a \tan (c+d x))^{9/2}}{3 a^6 d}+\frac{24 i (a+i a \tan (c+d x))^{7/2}}{7 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{5/2}}{5 a^4 d}",1,"(((2*I)/1155)*Sec[c + d*x]^6*(Cos[4*(c + d*x)] + I*Sin[4*(c + d*x)])*(39*I + (494*I)*Cos[2*(c + d*x)] + 215*Sec[c + d*x]*Sin[3*(c + d*x)] + 110*Tan[c + d*x]))/(a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
348,1,92,88,0.3939617,"\int \frac{\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{2 \sec ^5(c+d x) (-27 i \sin (2 (c+d x))+43 \cos (2 (c+d x))+28) (\cos (3 (c+d x))+i \sin (3 (c+d x)))}{105 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","-\frac{2 i (a+i a \tan (c+d x))^{7/2}}{7 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{5/2}}{5 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{3/2}}{3 a^3 d}",1,"(-2*Sec[c + d*x]^5*(28 + 43*Cos[2*(c + d*x)] - (27*I)*Sin[2*(c + d*x)])*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)]))/(105*a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
349,1,80,57,0.256367,"\int \frac{\sec ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{2 i (\tan (c+d x)+5 i) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))}{3 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a^3 d}-\frac{4 i \sqrt{a+i a \tan (c+d x)}}{a^2 d}",1,"(((2*I)/3)*Sec[c + d*x]^2*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])*(5*I + Tan[c + d*x]))/(a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
350,1,27,27,0.1639511,"\int \frac{\sec ^2(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{2 i}{a d \sqrt{a+i a \tan (c+d x)}}","\frac{2 i}{a d \sqrt{a+i a \tan (c+d x)}}",1,"(2*I)/(a*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
351,1,142,175,1.0038077,"\int \frac{\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{i e^{-5 i (c+d x)} \sec (c+d x) \left(-38 e^{2 i (c+d x)}-148 e^{4 i (c+d x)}-101 e^{6 i (c+d x)}+15 e^{8 i (c+d x)}+105 e^{5 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)-6\right)}{480 a d \sqrt{a+i a \tan (c+d x)}}","-\frac{7 i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} d}-\frac{i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{5/2}}+\frac{7 i a}{20 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 i}{24 d (a+i a \tan (c+d x))^{3/2}}+\frac{7 i}{16 a d \sqrt{a+i a \tan (c+d x)}}",1,"((-1/480*I)*(-6 - 38*E^((2*I)*(c + d*x)) - 148*E^((4*I)*(c + d*x)) - 101*E^((6*I)*(c + d*x)) + 15*E^((8*I)*(c + d*x)) + 105*E^((5*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])*Sec[c + d*x])/(a*d*E^((5*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
352,1,168,248,1.2571416,"\int \frac{\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{i e^{-7 i (c+d x)} \sec (c+d x) \left(-328 e^{2 i (c+d x)}-1304 e^{4 i (c+d x)}-4584 e^{6 i (c+d x)}-2833 e^{8 i (c+d x)}+805 e^{10 i (c+d x)}+70 e^{12 i (c+d x)}+3465 e^{7 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)-40\right)}{17920 a d \sqrt{a+i a \tan (c+d x)}}","-\frac{99 i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{256 \sqrt{2} a^{3/2} d}-\frac{i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}-\frac{11 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac{99 i a^2}{224 d (a+i a \tan (c+d x))^{7/2}}+\frac{99 i a}{320 d (a+i a \tan (c+d x))^{5/2}}+\frac{33 i}{128 d (a+i a \tan (c+d x))^{3/2}}+\frac{99 i}{256 a d \sqrt{a+i a \tan (c+d x)}}",1,"((-1/17920*I)*(-40 - 328*E^((2*I)*(c + d*x)) - 1304*E^((4*I)*(c + d*x)) - 4584*E^((6*I)*(c + d*x)) - 2833*E^((8*I)*(c + d*x)) + 805*E^((10*I)*(c + d*x)) + 70*E^((12*I)*(c + d*x)) + 3465*E^((7*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])*Sec[c + d*x])/(a*d*E^((7*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
353,1,203,321,2.0948141,"\int \frac{\cos ^6(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^6/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{i e^{-8 i (c+d x)} \left(1136 e^{2 i (c+d x)}+5440 e^{4 i (c+d x)}+17344 e^{6 i (c+d x)}+57632 e^{8 i (c+d x)}+33301 e^{10 i (c+d x)}-13209 e^{12 i (c+d x)}-1974 e^{14 i (c+d x)}-168 e^{16 i (c+d x)}-45045 e^{9 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+112\right)}{129024 a d \left(1+e^{2 i (c+d x)}\right) \sqrt{a+i a \tan (c+d x)}}","-\frac{715 i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2048 \sqrt{2} a^{3/2} d}-\frac{i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{9/2}}-\frac{5 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{9/2}}-\frac{65 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac{715 i a^3}{1152 d (a+i a \tan (c+d x))^{9/2}}+\frac{715 i a^2}{1792 d (a+i a \tan (c+d x))^{7/2}}+\frac{143 i a}{512 d (a+i a \tan (c+d x))^{5/2}}+\frac{715 i}{3072 d (a+i a \tan (c+d x))^{3/2}}+\frac{715 i}{2048 a d \sqrt{a+i a \tan (c+d x)}}",1,"((I/129024)*(112 + 1136*E^((2*I)*(c + d*x)) + 5440*E^((4*I)*(c + d*x)) + 17344*E^((6*I)*(c + d*x)) + 57632*E^((8*I)*(c + d*x)) + 33301*E^((10*I)*(c + d*x)) - 13209*E^((12*I)*(c + d*x)) - 1974*E^((14*I)*(c + d*x)) - 168*E^((16*I)*(c + d*x)) - 45045*E^((9*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))]))/(a*d*E^((8*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
354,1,108,147,0.9939143,"\int \frac{\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^11/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{2 \sec ^9(c+d x) (\sin (4 (c+d x))+i \cos (4 (c+d x))) (-2242 i \cos (2 (c+d x))+374 \tan (c+d x)+1089 \sin (3 (c+d x)) \sec (c+d x)+475 i)}{12155 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{256 i a^4 \sec ^{11}(c+d x)}{12155 d (a+i a \tan (c+d x))^{11/2}}+\frac{64 i a^3 \sec ^{11}(c+d x)}{1105 d (a+i a \tan (c+d x))^{9/2}}+\frac{8 i a^2 \sec ^{11}(c+d x)}{85 d (a+i a \tan (c+d x))^{7/2}}+\frac{2 i a \sec ^{11}(c+d x)}{17 d (a+i a \tan (c+d x))^{5/2}}",1,"(2*Sec[c + d*x]^9*(I*Cos[4*(c + d*x)] + Sin[4*(c + d*x)])*(475*I - (2242*I)*Cos[2*(c + d*x)] + 1089*Sec[c + d*x]*Sin[3*(c + d*x)] + 374*Tan[c + d*x]))/(12155*a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
355,1,92,110,0.5774726,"\int \frac{\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{2 \sec ^8(c+d x) (135 i \sin (2 (c+d x))+151 \cos (2 (c+d x))+52) (\cos (3 (c+d x))-i \sin (3 (c+d x)))}{1287 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{64 i a^3 \sec ^9(c+d x)}{1287 d (a+i a \tan (c+d x))^{9/2}}+\frac{16 i a^2 \sec ^9(c+d x)}{143 d (a+i a \tan (c+d x))^{7/2}}+\frac{2 i a \sec ^9(c+d x)}{13 d (a+i a \tan (c+d x))^{5/2}}",1,"(2*Sec[c + d*x]^8*(52 + 151*Cos[2*(c + d*x)] + (135*I)*Sin[2*(c + d*x)])*(Cos[3*(c + d*x)] - I*Sin[3*(c + d*x)]))/(1287*a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
356,1,80,73,0.4299895,"\int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{2 (7 \tan (c+d x)-11 i) \sec ^5(c+d x) (\sin (2 (c+d x))+i \cos (2 (c+d x)))}{63 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{8 i a^2 \sec ^7(c+d x)}{63 d (a+i a \tan (c+d x))^{7/2}}+\frac{2 i a \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^{5/2}}",1,"(2*Sec[c + d*x]^5*(I*Cos[2*(c + d*x)] + Sin[2*(c + d*x)])*(-11*I + 7*Tan[c + d*x]))/(63*a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
357,1,59,35,0.26266,"\int \frac{\sec ^5(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{2 (1-i \tan (c+d x)) \sec ^3(c+d x)}{5 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{2 i a \sec ^5(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"(2*Sec[c + d*x]^3*(1 - I*Tan[c + d*x]))/(5*a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
358,1,101,86,0.7993656,"\int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{8 e^{3 i (c+d x)} \left(-1+\sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{a d \left(1+e^{2 i (c+d x)}\right)^2 (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{2 i \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}-\frac{2 i \sec (c+d x)}{a d \sqrt{a+i a \tan (c+d x)}}",1,"(8*E^((3*I)*(c + d*x))*(-1 + Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/(a*d*(1 + E^((2*I)*(c + d*x)))^2*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
359,1,95,87,0.703517,"\int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{\sec (c+d x) \left(2+\frac{2 e^{2 i (c+d x)} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{4 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{i \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{i \sec (c+d x)}{2 d (a+i a \tan (c+d x))^{3/2}}",1,"((2 + (2*E^((2*I)*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])/Sqrt[1 + E^((2*I)*(c + d*x))])*Sec[c + d*x])/(4*a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
360,1,120,157,1.0979649,"\int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{\sec (c+d x) \left(\frac{30 e^{2 i (c+d x)} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-2 (10 i \sin (2 (c+d x))+6 \cos (2 (c+d x))-9)\right)}{64 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{15 i \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{32 \sqrt{2} a^{3/2} d}-\frac{15 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{32 a^2 d}+\frac{5 i \cos (c+d x)}{16 a d \sqrt{a+i a \tan (c+d x)}}+\frac{i \cos (c+d x)}{4 d (a+i a \tan (c+d x))^{3/2}}",1,"(Sec[c + d*x]*((30*E^((2*I)*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])/Sqrt[1 + E^((2*I)*(c + d*x))] - 2*(-9 + 6*Cos[2*(c + d*x)] + (10*I)*Sin[2*(c + d*x)])))/(64*a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
361,1,145,233,1.7045918,"\int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{\sec (c+d x) \left(\frac{630 e^{2 i (c+d x)} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-2 (3 i (86 \sin (2 (c+d x))+8 \sin (4 (c+d x))+55 i)+158 \cos (2 (c+d x))+8 \cos (4 (c+d x)))\right)}{1536 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{105 i \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{256 \sqrt{2} a^{3/2} d}-\frac{7 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{32 a^2 d}-\frac{105 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{256 a^2 d}+\frac{3 i \cos ^3(c+d x)}{16 a d \sqrt{a+i a \tan (c+d x)}}+\frac{i \cos ^3(c+d x)}{6 d (a+i a \tan (c+d x))^{3/2}}+\frac{35 i \cos (c+d x)}{128 a d \sqrt{a+i a \tan (c+d x)}}",1,"(Sec[c + d*x]*((630*E^((2*I)*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])/Sqrt[1 + E^((2*I)*(c + d*x))] - 2*(158*Cos[2*(c + d*x)] + 8*Cos[4*(c + d*x)] + (3*I)*(55*I + 86*Sin[2*(c + d*x)] + 8*Sin[4*(c + d*x)]))))/(1536*a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
362,1,116,146,0.8761307,"\int \frac{\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^10/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{2 \sec ^9(c+d x) (2600 \sin (2 (c+d x))+2875 \sin (4 (c+d x))+4264 i \cos (2 (c+d x))+3131 i \cos (4 (c+d x))+2288 i) (\cos (5 (c+d x))+i \sin (5 (c+d x)))}{15015 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","-\frac{2 i (a+i a \tan (c+d x))^{13/2}}{13 a^9 d}+\frac{16 i (a+i a \tan (c+d x))^{11/2}}{11 a^8 d}-\frac{16 i (a+i a \tan (c+d x))^{9/2}}{3 a^7 d}+\frac{64 i (a+i a \tan (c+d x))^{7/2}}{7 a^6 d}-\frac{32 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d}",1,"(2*Sec[c + d*x]^9*(2288*I + (4264*I)*Cos[2*(c + d*x)] + (3131*I)*Cos[4*(c + d*x)] + 2600*Sin[2*(c + d*x)] + 2875*Sin[4*(c + d*x)])*(Cos[5*(c + d*x)] + I*Sin[5*(c + d*x)]))/(15015*a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
363,1,108,117,0.6979789,"\int \frac{\sec ^8(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{2 \sec ^6(c+d x) (\cos (4 (c+d x))+i \sin (4 (c+d x))) (242 i \cos (2 (c+d x))+54 \tan (c+d x)+89 \sin (3 (c+d x)) \sec (c+d x)+77 i)}{315 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a^7 d}-\frac{12 i (a+i a \tan (c+d x))^{7/2}}{7 a^6 d}+\frac{24 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{3/2}}{3 a^4 d}",1,"(2*Sec[c + d*x]^6*(Cos[4*(c + d*x)] + I*Sin[4*(c + d*x)])*(77*I + (242*I)*Cos[2*(c + d*x)] + 89*Sec[c + d*x]*Sin[3*(c + d*x)] + 54*Tan[c + d*x]))/(315*a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
364,1,94,86,0.4250951,"\int \frac{\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{2 \sec ^5(c+d x) (7 \sin (2 (c+d x))+23 i \cos (2 (c+d x))+20 i) (\cos (3 (c+d x))+i \sin (3 (c+d x)))}{15 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{3/2}}{3 a^4 d}-\frac{8 i \sqrt{a+i a \tan (c+d x)}}{a^3 d}",1,"(2*Sec[c + d*x]^5*(20*I + (23*I)*Cos[2*(c + d*x)] + 7*Sin[2*(c + d*x)])*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)]))/(15*a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
365,1,36,55,0.2949876,"\int \frac{\sec ^4(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{-2 \tan (c+d x)+6 i}{a^2 d \sqrt{a+i a \tan (c+d x)}}","\frac{2 i \sqrt{a+i a \tan (c+d x)}}{a^3 d}+\frac{4 i}{a^2 d \sqrt{a+i a \tan (c+d x)}}",1,"(6*I - 2*Tan[c + d*x])/(a^2*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
366,1,39,29,0.2435115,"\int \frac{\sec ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{2}{3 a^2 d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{2 i}{3 a d (a+i a \tan (c+d x))^{3/2}}",1,"2/(3*a^2*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
367,1,163,204,1.2540813,"\int \frac{\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i e^{-8 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{3/2} \sec ^2(c+d x) \left(\sqrt{1+e^{2 i (c+d x)}} \left(-58 e^{2 i (c+d x)}-156 e^{4 i (c+d x)}-388 e^{6 i (c+d x)}+35 e^{8 i (c+d x)}-10\right)+315 e^{7 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{4480 a^2 d \sqrt{a+i a \tan (c+d x)}}","-\frac{9 i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{32 \sqrt{2} a^{5/2} d}-\frac{i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}+\frac{9 i}{32 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{9 i a}{28 d (a+i a \tan (c+d x))^{7/2}}+\frac{9 i}{40 d (a+i a \tan (c+d x))^{5/2}}+\frac{3 i}{16 a d (a+i a \tan (c+d x))^{3/2}}",1,"((-1/4480*I)*(1 + E^((2*I)*(c + d*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(-10 - 58*E^((2*I)*(c + d*x)) - 156*E^((4*I)*(c + d*x)) - 388*E^((6*I)*(c + d*x)) + 35*E^((8*I)*(c + d*x))) + 315*E^((7*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])*Sec[c + d*x]^2)/(a^2*d*E^((8*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
368,1,189,277,1.7069064,"\int \frac{\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i e^{-10 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{3/2} \sec ^2(c+d x) \left(\sqrt{1+e^{2 i (c+d x)}} \left(-2200 e^{2 i (c+d x)}-7944 e^{4 i (c+d x)}-18808 e^{6 i (c+d x)}-50584 e^{8 i (c+d x)}+7875 e^{10 i (c+d x)}+630 e^{12 i (c+d x)}-280\right)+45045 e^{9 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{645120 a^2 d \sqrt{a+i a \tan (c+d x)}}","-\frac{143 i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{512 \sqrt{2} a^{5/2} d}-\frac{i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{9/2}}-\frac{13 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac{143 i a^2}{288 d (a+i a \tan (c+d x))^{9/2}}+\frac{143 i}{512 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{143 i a}{448 d (a+i a \tan (c+d x))^{7/2}}+\frac{143 i}{640 d (a+i a \tan (c+d x))^{5/2}}+\frac{143 i}{768 a d (a+i a \tan (c+d x))^{3/2}}",1,"((-1/645120*I)*(1 + E^((2*I)*(c + d*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(-280 - 2200*E^((2*I)*(c + d*x)) - 7944*E^((4*I)*(c + d*x)) - 18808*E^((6*I)*(c + d*x)) - 50584*E^((8*I)*(c + d*x)) + 7875*E^((10*I)*(c + d*x)) + 630*E^((12*I)*(c + d*x))) + 45045*E^((9*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])*Sec[c + d*x]^2)/(a^2*d*E^((10*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
369,1,112,147,1.114682,"\int \frac{\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^13/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{\sec ^{12}(c+d x) (13 i (38 \sin (c+d x)+123 \sin (3 (c+d x)))+798 \cos (c+d x)+1631 \cos (3 (c+d x))) (-2 \sin (4 (c+d x))-2 i \cos (4 (c+d x)))}{20995 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{256 i a^4 \sec ^{13}(c+d x)}{20995 d (a+i a \tan (c+d x))^{13/2}}+\frac{64 i a^3 \sec ^{13}(c+d x)}{1615 d (a+i a \tan (c+d x))^{11/2}}+\frac{24 i a^2 \sec ^{13}(c+d x)}{323 d (a+i a \tan (c+d x))^{9/2}}+\frac{2 i a \sec ^{13}(c+d x)}{19 d (a+i a \tan (c+d x))^{7/2}}",1,"(Sec[c + d*x]^12*(798*Cos[c + d*x] + 1631*Cos[3*(c + d*x)] + (13*I)*(38*Sin[c + d*x] + 123*Sin[3*(c + d*x)]))*((-2*I)*Cos[4*(c + d*x)] - 2*Sin[4*(c + d*x)]))/(20995*a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
370,1,94,110,0.757834,"\int \frac{\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^11/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{\sec ^{10}(c+d x) (187 i \sin (2 (c+d x))+203 \cos (2 (c+d x))+60) (-2 \sin (3 (c+d x))-2 i \cos (3 (c+d x)))}{2145 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{64 i a^3 \sec ^{11}(c+d x)}{2145 d (a+i a \tan (c+d x))^{11/2}}+\frac{16 i a^2 \sec ^{11}(c+d x)}{195 d (a+i a \tan (c+d x))^{9/2}}+\frac{2 i a \sec ^{11}(c+d x)}{15 d (a+i a \tan (c+d x))^{7/2}}",1,"(Sec[c + d*x]^10*(60 + 203*Cos[2*(c + d*x)] + (187*I)*Sin[2*(c + d*x)])*((-2*I)*Cos[3*(c + d*x)] - 2*Sin[3*(c + d*x)]))/(2145*a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
371,1,80,73,0.5608026,"\int \frac{\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{2 (9 \tan (c+d x)-13 i) \sec ^7(c+d x) (\cos (2 (c+d x))-i \sin (2 (c+d x)))}{99 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{8 i a^2 \sec ^9(c+d x)}{99 d (a+i a \tan (c+d x))^{9/2}}+\frac{2 i a \sec ^9(c+d x)}{11 d (a+i a \tan (c+d x))^{7/2}}",1,"(2*Sec[c + d*x]^7*(Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)])*(-13*I + 9*Tan[c + d*x]))/(99*a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
372,1,57,35,0.4588799,"\int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{2 (\tan (c+d x)+i) \sec ^5(c+d x)}{7 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{2 i a \sec ^7(c+d x)}{7 d (a+i a \tan (c+d x))^{7/2}}",1,"(-2*Sec[c + d*x]^5*(I + Tan[c + d*x]))/(7*a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
373,1,82,123,0.8656039,"\int \frac{\sec ^5(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{2 \sec (c+d x) \left(\tan (c+d x)-6 i \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+7 i\right)}{3 a^2 d \sqrt{a+i a \tan (c+d x)}}","\frac{4 i \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{4 i \sec (c+d x)}{a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{2 i \sec ^3(c+d x)}{3 a d (a+i a \tan (c+d x))^{3/2}}",1,"(-2*Sec[c + d*x]*(7*I - (6*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]] + Tan[c + d*x]))/(3*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
374,1,149,86,1.0754587,"\int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{i e^{-\frac{1}{2} i (2 c+d x)} \sec ^3(c+d x) \left(-e^{2 i (c+d x)}+e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-1\right) \left(\cos \left(c+\frac{d x}{2}\right)+i \sin \left(c+\frac{d x}{2}\right)\right)}{2 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{i \sec (c+d x)}{a d (a+i a \tan (c+d x))^{3/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{2} a^{5/2} d}",1,"((I/2)*(-1 - E^((2*I)*(c + d*x)) + E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^3*(Cos[c + (d*x)/2] + I*Sin[c + (d*x)/2]))/(a^2*d*E^((I/2)*(2*c + d*x))*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
375,1,121,122,0.87827,"\int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]/(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i \sec ^3(c+d x) \left(3 i \sin (2 (c+d x))+7 \cos (2 (c+d x))+3 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+7\right)}{32 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{3 i \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{3 i \sec (c+d x)}{16 a d (a+i a \tan (c+d x))^{3/2}}+\frac{i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}",1,"((-1/32*I)*Sec[c + d*x]^3*(7 + 3*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]] + 7*Cos[2*(c + d*x)] + (3*I)*Sin[2*(c + d*x)]))/(a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
376,1,143,192,1.373015,"\int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{i \sec ^3(c+d x) \left(7 i \sin (2 (c+d x))+56 i \sin (4 (c+d x))-85 \cos (2 (c+d x))+40 \cos (4 (c+d x))-105 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-125\right)}{768 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{35 i \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{128 \sqrt{2} a^{5/2} d}-\frac{35 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{128 a^3 d}+\frac{35 i \cos (c+d x)}{192 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{7 i \cos (c+d x)}{48 a d (a+i a \tan (c+d x))^{3/2}}+\frac{i \cos (c+d x)}{6 d (a+i a \tan (c+d x))^{5/2}}",1,"((I/768)*Sec[c + d*x]^3*(-125 - 105*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]] - 85*Cos[2*(c + d*x)] + 40*Cos[4*(c + d*x)] + (7*I)*Sin[2*(c + d*x)] + (56*I)*Sin[4*(c + d*x)]))/(a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
377,1,165,270,1.3857425,"\int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{i \sec ^3(c+d x) \left(1111 i \sin (2 (c+d x))+2552 i \sin (4 (c+d x))+176 i \sin (6 (c+d x))-1605 \cos (2 (c+d x))+1800 \cos (4 (c+d x))+80 \cos (6 (c+d x))-3465 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-3325\right)}{24576 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{1155 i \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{4096 \sqrt{2} a^{5/2} d}-\frac{77 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{512 a^3 d}-\frac{1155 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{4096 a^3 d}+\frac{33 i \cos ^3(c+d x)}{256 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{385 i \cos (c+d x)}{2048 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{11 i \cos ^3(c+d x)}{96 a d (a+i a \tan (c+d x))^{3/2}}+\frac{i \cos ^3(c+d x)}{8 d (a+i a \tan (c+d x))^{5/2}}",1,"((I/24576)*Sec[c + d*x]^3*(-3325 - 3465*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]] - 1605*Cos[2*(c + d*x)] + 1800*Cos[4*(c + d*x)] + 80*Cos[6*(c + d*x)] + (1111*I)*Sin[2*(c + d*x)] + (2552*I)*Sin[4*(c + d*x)] + (176*I)*Sin[6*(c + d*x)]))/(a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
378,1,114,146,0.9728641,"\int \frac{\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Sec[c + d*x]^10/(a + I*a*Tan[c + d*x])^(7/2),x]","\frac{2 \sec ^9(c+d x) (-1144 i \sin (2 (c+d x))-1027 i \sin (4 (c+d x))+2552 \cos (2 (c+d x))+1283 \cos (4 (c+d x))+1584) (\cos (5 (c+d x))+i \sin (5 (c+d x)))}{3465 a^3 d (\tan (c+d x)-i)^3 \sqrt{a+i a \tan (c+d x)}}","-\frac{2 i (a+i a \tan (c+d x))^{11/2}}{11 a^9 d}+\frac{16 i (a+i a \tan (c+d x))^{9/2}}{9 a^8 d}-\frac{48 i (a+i a \tan (c+d x))^{7/2}}{7 a^7 d}+\frac{64 i (a+i a \tan (c+d x))^{5/2}}{5 a^6 d}-\frac{32 i (a+i a \tan (c+d x))^{3/2}}{3 a^5 d}",1,"(2*Sec[c + d*x]^9*(1584 + 2552*Cos[2*(c + d*x)] + 1283*Cos[4*(c + d*x)] - (1144*I)*Sin[2*(c + d*x)] - (1027*I)*Sin[4*(c + d*x)])*(Cos[5*(c + d*x)] + I*Sin[5*(c + d*x)]))/(3465*a^3*d*(-I + Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + d*x]])","A",1
379,1,110,113,0.6843585,"\int \frac{\sec ^8(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^(7/2),x]","\frac{2 \sec ^7(c+d x) (-i (14 \sin (c+d x)+19 \sin (3 (c+d x)))+126 \cos (c+d x)+51 \cos (3 (c+d x))) (\cos (4 (c+d x))+i \sin (4 (c+d x)))}{35 a^3 d (\tan (c+d x)-i)^3 \sqrt{a+i a \tan (c+d x)}}","\frac{2 i (a+i a \tan (c+d x))^{7/2}}{7 a^7 d}-\frac{12 i (a+i a \tan (c+d x))^{5/2}}{5 a^6 d}+\frac{8 i (a+i a \tan (c+d x))^{3/2}}{a^5 d}-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{a^4 d}",1,"(2*Sec[c + d*x]^7*(126*Cos[c + d*x] + 51*Cos[3*(c + d*x)] - I*(14*Sin[c + d*x] + 19*Sin[3*(c + d*x)]))*(Cos[4*(c + d*x)] + I*Sin[4*(c + d*x)]))/(35*a^3*d*(-I + Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + d*x]])","A",1
380,1,61,84,0.4904946,"\int \frac{\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^(7/2),x]","\frac{2 i \sec ^2(c+d x) (5 i \sin (2 (c+d x))+11 \cos (2 (c+d x))+12)}{3 a^3 d \sqrt{a+i a \tan (c+d x)}}","-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a^5 d}+\frac{8 i \sqrt{a+i a \tan (c+d x)}}{a^4 d}+\frac{8 i}{a^3 d \sqrt{a+i a \tan (c+d x)}}",1,"(((2*I)/3)*Sec[c + d*x]^2*(12 + 11*Cos[2*(c + d*x)] + (5*I)*Sin[2*(c + d*x)]))/(a^3*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
381,1,80,57,0.2302269,"\int \frac{\sec ^4(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^(7/2),x]","\frac{2 (1+3 i \tan (c+d x)) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))}{3 a^3 d (\tan (c+d x)-i)^3 \sqrt{a+i a \tan (c+d x)}}","\frac{4 i}{3 a^2 d (a+i a \tan (c+d x))^{3/2}}-\frac{2 i}{a^3 d \sqrt{a+i a \tan (c+d x)}}",1,"(2*Sec[c + d*x]^2*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])*(1 + (3*I)*Tan[c + d*x]))/(3*a^3*d*(-I + Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + d*x]])","A",1
382,1,39,29,0.3027717,"\int \frac{\sec ^2(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{2 \sqrt{a+i a \tan (c+d x)}}{5 a^4 d (\tan (c+d x)-i)^3}","\frac{2 i}{5 a d (a+i a \tan (c+d x))^{5/2}}",1,"(-2*Sqrt[a + I*a*Tan[c + d*x]])/(5*a^4*d*(-I + Tan[c + d*x])^3)","A",1
383,1,176,233,1.7066661,"\int \frac{\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{i e^{-11 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \sec ^3(c+d x) \left(\sqrt{1+e^{2 i (c+d x)}} \left(-460 e^{2 i (c+d x)}-1338 e^{4 i (c+d x)}-2416 e^{6 i (c+d x)}-4618 e^{8 i (c+d x)}+315 e^{10 i (c+d x)}-70\right)+3465 e^{9 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{161280 a^3 d \sqrt{a+i a \tan (c+d x)}}","-\frac{11 i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{64 \sqrt{2} a^{7/2} d}+\frac{11 i}{64 a^3 d \sqrt{a+i a \tan (c+d x)}}-\frac{i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac{11 i}{96 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{11 i a}{36 d (a+i a \tan (c+d x))^{9/2}}+\frac{11 i}{56 d (a+i a \tan (c+d x))^{7/2}}+\frac{11 i}{80 a d (a+i a \tan (c+d x))^{5/2}}",1,"((-1/161280*I)*(1 + E^((2*I)*(c + d*x)))^(5/2)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(-70 - 460*E^((2*I)*(c + d*x)) - 1338*E^((4*I)*(c + d*x)) - 2416*E^((6*I)*(c + d*x)) - 4618*E^((8*I)*(c + d*x)) + 315*E^((10*I)*(c + d*x))) + 3465*E^((9*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])*Sec[c + d*x]^3)/(a^3*d*E^((11*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
384,1,202,306,2.423947,"\int \frac{\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{i e^{-13 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \sec ^3(c+d x) \left(\sqrt{1+e^{2 i (c+d x)}} \left(-1456 e^{2 i (c+d x)}-5728 e^{4 i (c+d x)}-13824 e^{6 i (c+d x)}-24688 e^{8 i (c+d x)}-54112 e^{10 i (c+d x)}+6699 e^{12 i (c+d x)}+462 e^{14 i (c+d x)}-168\right)+45045 e^{11 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{1892352 a^3 d \sqrt{a+i a \tan (c+d x)}}","-\frac{195 i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{1024 \sqrt{2} a^{7/2} d}-\frac{i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{11/2}}-\frac{15 i a^3}{16 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{11/2}}+\frac{195 i}{1024 a^3 d \sqrt{a+i a \tan (c+d x)}}+\frac{195 i a^2}{352 d (a+i a \tan (c+d x))^{11/2}}+\frac{65 i}{512 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{65 i a}{192 d (a+i a \tan (c+d x))^{9/2}}+\frac{195 i}{896 d (a+i a \tan (c+d x))^{7/2}}+\frac{39 i}{256 a d (a+i a \tan (c+d x))^{5/2}}",1,"((-1/1892352*I)*(1 + E^((2*I)*(c + d*x)))^(5/2)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(-168 - 1456*E^((2*I)*(c + d*x)) - 5728*E^((4*I)*(c + d*x)) - 13824*E^((6*I)*(c + d*x)) - 24688*E^((8*I)*(c + d*x)) - 54112*E^((10*I)*(c + d*x)) + 6699*E^((12*I)*(c + d*x)) + 462*E^((14*I)*(c + d*x))) + 45045*E^((11*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])*Sec[c + d*x]^3)/(a^3*d*E^((13*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
385,1,92,110,1.0995231,"\int \frac{\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Sec[c + d*x]^13/(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{2 \sec ^{12}(c+d x) (247 i \sin (2 (c+d x))+263 \cos (2 (c+d x))+68) (\cos (3 (c+d x))-i \sin (3 (c+d x)))}{3315 a^3 d (\tan (c+d x)-i)^3 \sqrt{a+i a \tan (c+d x)}}","\frac{64 i a^3 \sec ^{13}(c+d x)}{3315 d (a+i a \tan (c+d x))^{13/2}}+\frac{16 i a^2 \sec ^{13}(c+d x)}{255 d (a+i a \tan (c+d x))^{11/2}}+\frac{2 i a \sec ^{13}(c+d x)}{17 d (a+i a \tan (c+d x))^{9/2}}",1,"(-2*Sec[c + d*x]^12*(68 + 263*Cos[2*(c + d*x)] + (247*I)*Sin[2*(c + d*x)])*(Cos[3*(c + d*x)] - I*Sin[3*(c + d*x)]))/(3315*a^3*d*(-I + Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + d*x]])","A",1
386,1,82,73,0.6048716,"\int \frac{\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Sec[c + d*x]^11/(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{2 i (11 \tan (c+d x)-15 i) \sec ^9(c+d x) (\cos (2 (c+d x))-i \sin (2 (c+d x)))}{143 a^3 d (\tan (c+d x)-i)^3 \sqrt{a+i a \tan (c+d x)}}","\frac{8 i a^2 \sec ^{11}(c+d x)}{143 d (a+i a \tan (c+d x))^{11/2}}+\frac{2 i a \sec ^{11}(c+d x)}{13 d (a+i a \tan (c+d x))^{9/2}}",1,"(((-2*I)/143)*Sec[c + d*x]^9*(Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)])*(-15*I + 11*Tan[c + d*x]))/(a^3*d*(-I + Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + d*x]])","A",1
387,1,59,35,0.4570482,"\int \frac{\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^(7/2),x]","\frac{2 i (\tan (c+d x)+i) \sec ^7(c+d x)}{9 a^3 d (\tan (c+d x)-i)^3 \sqrt{a+i a \tan (c+d x)}}","\frac{2 i a \sec ^9(c+d x)}{9 d (a+i a \tan (c+d x))^{9/2}}",1,"(((2*I)/9)*Sec[c + d*x]^7*(I + Tan[c + d*x]))/(a^3*d*(-I + Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + d*x]])","A",1
388,1,130,160,1.2175356,"\int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{128 e^{7 i (c+d x)} \left(-35 e^{2 i (c+d x)}-15 e^{4 i (c+d x)}+15 \left(1+e^{2 i (c+d x)}\right)^{5/2} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-23\right)}{15 a^3 d \left(1+e^{2 i (c+d x)}\right)^6 (\tan (c+d x)-i)^3 \sqrt{a+i a \tan (c+d x)}}","\frac{8 i \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{a^{7/2} d}-\frac{8 i \sec (c+d x)}{a^3 d \sqrt{a+i a \tan (c+d x)}}-\frac{4 i \sec ^3(c+d x)}{3 a^2 d (a+i a \tan (c+d x))^{3/2}}-\frac{2 i \sec ^5(c+d x)}{5 a d (a+i a \tan (c+d x))^{5/2}}",1,"(-128*E^((7*I)*(c + d*x))*(-23 - 35*E^((2*I)*(c + d*x)) - 15*E^((4*I)*(c + d*x)) + 15*(1 + E^((2*I)*(c + d*x)))^(5/2)*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/(15*a^3*d*(1 + E^((2*I)*(c + d*x)))^6*(-I + Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + d*x]])","A",1
389,1,126,121,0.9525811,"\int \frac{\sec ^5(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^(7/2),x]","\frac{16 e^{5 i (c+d x)} \left(-3 e^{2 i (c+d x)}+3 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-1\right)}{a^3 d \left(1+e^{2 i (c+d x)}\right)^4 (\tan (c+d x)-i)^3 \sqrt{a+i a \tan (c+d x)}}","-\frac{3 i \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{a^{7/2} d}+\frac{6 i \sec (c+d x)}{a^2 d (a+i a \tan (c+d x))^{3/2}}-\frac{2 i \sec ^3(c+d x)}{a d (a+i a \tan (c+d x))^{5/2}}",1,"(16*E^((5*I)*(c + d*x))*(-1 - 3*E^((2*I)*(c + d*x)) + 3*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/(a^3*d*(1 + E^((2*I)*(c + d*x)))^4*(-I + Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + d*x]])","A",1
390,1,120,125,1.0005425,"\int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^(7/2),x]","\frac{i \sec ^3(c+d x) \left(i \sin (2 (c+d x))-3 \cos (2 (c+d x))+e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-3\right)}{16 a^3 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{8 \sqrt{2} a^{7/2} d}-\frac{i \sec (c+d x)}{8 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{i \sec (c+d x)}{2 a d (a+i a \tan (c+d x))^{5/2}}",1,"((I/16)*Sec[c + d*x]^3*(-3 + E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]] - 3*Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]))/(a^3*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
391,1,119,157,1.5057983,"\int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Sec[c + d*x]/(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{\sec ^3(c+d x) \left(50 i \sin (2 (c+d x))+82 \cos (2 (c+d x))+\frac{30 e^{4 i (c+d x)} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+52\right)}{384 a^3 d (\tan (c+d x)-i)^3 \sqrt{a+i a \tan (c+d x)}}","\frac{5 i \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{64 \sqrt{2} a^{7/2} d}+\frac{5 i \sec (c+d x)}{64 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{5 i \sec (c+d x)}{48 a d (a+i a \tan (c+d x))^{5/2}}+\frac{i \sec (c+d x)}{6 d (a+i a \tan (c+d x))^{7/2}}",1,"-1/384*(Sec[c + d*x]^3*(52 + (30*E^((4*I)*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])/Sqrt[1 + E^((2*I)*(c + d*x))] + 82*Cos[2*(c + d*x)] + (50*I)*Sin[2*(c + d*x)]))/(a^3*d*(-I + Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + d*x]])","A",1
392,1,141,227,2.1945437,"\int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Cos[c + d*x]/(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{\sec ^3(c+d x) \left(474 i \sin (2 (c+d x))-288 i \sin (4 (c+d x))+826 \cos (2 (c+d x))-224 \cos (4 (c+d x))+\frac{630 e^{4 i (c+d x)} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+420\right)}{4096 a^3 d (\tan (c+d x)-i)^3 \sqrt{a+i a \tan (c+d x)}}","\frac{315 i \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{2048 \sqrt{2} a^{7/2} d}-\frac{315 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{2048 a^4 d}+\frac{105 i \cos (c+d x)}{1024 a^3 d \sqrt{a+i a \tan (c+d x)}}+\frac{21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac{i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}",1,"-1/4096*(Sec[c + d*x]^3*(420 + (630*E^((4*I)*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])/Sqrt[1 + E^((2*I)*(c + d*x))] + 826*Cos[2*(c + d*x)] - 224*Cos[4*(c + d*x)] + (474*I)*Sin[2*(c + d*x)] - (288*I)*Sin[4*(c + d*x)]))/(a^3*d*(-I + Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + d*x]])","A",1
393,1,175,307,2.4873858,"\int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{\sec ^3(c+d x) \left(20048 e^{-2 i (c+d x)}+71190 e^{2 i (c+d x)}+5856 e^{-4 i (c+d x)}-48640 e^{4 i (c+d x)}+768 e^{-6 i (c+d x)}-2560 e^{6 i (c+d x)}+\frac{90090 e^{4 i (c+d x)} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+42140\right)}{491520 a^3 d (\tan (c+d x)-i)^3 \sqrt{a+i a \tan (c+d x)}}","\frac{3003 i \tanh ^{-1}\left(\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right)}{16384 \sqrt{2} a^{7/2} d}-\frac{1001 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{10240 a^4 d}-\frac{3003 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{16384 a^4 d}+\frac{429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt{a+i a \tan (c+d x)}}+\frac{1001 i \cos (c+d x)}{8192 a^3 d \sqrt{a+i a \tan (c+d x)}}+\frac{143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac{i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}",1,"-1/491520*((42140 + 20048/E^((2*I)*(c + d*x)) + 71190*E^((2*I)*(c + d*x)) + 5856/E^((4*I)*(c + d*x)) - 48640*E^((4*I)*(c + d*x)) + 768/E^((6*I)*(c + d*x)) - 2560*E^((6*I)*(c + d*x)) + (90090*E^((4*I)*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])/Sqrt[1 + E^((2*I)*(c + d*x))])*Sec[c + d*x]^3)/(a^3*d*(-I + Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + d*x]])","A",1
394,1,373,524,1.7522273,"\int (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{e (\cos (c)-i \sin (c)) \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)} \left(\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{-\sin (c)+i \cos (c)+1} \sqrt{\tan \left(\frac{d x}{2}\right)+i} \cos (c+d x) \tanh ^{-1}\left(\frac{\sqrt{-\sin (c)+i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{\sin (c)+i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)+\sqrt{\sin (c)+i \cos (c)-1} \left(\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i} (\sin (d x)+i \cos (d x))-\sqrt{\sin (c)-i \cos (c)+1} \sqrt{\tan \left(\frac{d x}{2}\right)+i} \cos (c+d x) \tanh ^{-1}\left(\frac{\sqrt{\sin (c)-i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)\right)\right)}{d \sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\sin (c)+i \cos (c)-1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}","-\frac{i a^{3/2} e^{3/2} \sec (c+d x) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{\sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i a^{3/2} e^{3/2} \sec (c+d x) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{\sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i a^{3/2} e^{3/2} \sec (c+d x) \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{2 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{i a^{3/2} e^{3/2} \sec (c+d x) \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{2 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i a (e \sec (c+d x))^{3/2}}{d \sqrt{a+i a \tan (c+d x)}}",1,"(e*Sqrt[e*Sec[c + d*x]]*(Cos[c] - I*Sin[c])*(ArcTanh[(Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Cos[c + d*x]*Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]] + Sqrt[-1 + I*Cos[c] + Sin[c]]*(Sqrt[-1 - I*Cos[c] - Sin[c]]*(I*Cos[d*x] + Sin[d*x])*Sqrt[I - Tan[(d*x)/2]] - ArcTanh[(Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Cos[c + d*x]*Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]]))*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I - Tan[(d*x)/2]])","A",1
395,1,277,323,1.6584957,"\int \sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{2 e \sqrt{\tan \left(\frac{d x}{2}\right)+i} \sqrt{a+i a \tan (c+d x)} \left(\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{-\sin (c)+i \cos (c)+1} \tanh ^{-1}\left(\frac{\sqrt{\sin (c)-i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)-\sqrt{\sin (c)-i \cos (c)+1} \sqrt{\sin (c)+i \cos (c)-1} \tanh ^{-1}\left(\frac{\sqrt{-\sin (c)+i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{\sin (c)+i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)\right)}{d \sqrt{i \sin (2 c)+\cos (2 c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i} \sqrt{e \sec (c+d x)}}","\frac{i \sqrt{2} \sqrt{a} \sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{d}-\frac{i \sqrt{2} \sqrt{a} \sqrt{e} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{d}-\frac{i \sqrt{a} \sqrt{e} \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{\sqrt{2} d}+\frac{i \sqrt{a} \sqrt{e} \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{\sqrt{2} d}",1,"(-2*e*(ArcTanh[(Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[1 + I*Cos[c] - Sin[c]] - ArcTanh[(Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[-1 + I*Cos[c] + Sin[c]])*Sqrt[I + Tan[(d*x)/2]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[e*Sec[c + d*x]]*Sqrt[1 + Cos[2*c] + I*Sin[2*c]]*Sqrt[I - Tan[(d*x)/2]])","A",1
396,1,36,36,0.064054,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[e*Sec[c + d*x]],x]","-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{d \sqrt{e \sec (c+d x)}}","-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{d \sqrt{e \sec (c+d x)}}",1,"((-2*I)*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[e*Sec[c + d*x]])","A",1
397,1,48,81,0.1872544,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/(e*Sec[c + d*x])^(3/2),x]","\frac{2 (2 \tan (c+d x)+i) \sqrt{a+i a \tan (c+d x)}}{3 d (e \sec (c+d x))^{3/2}}","\frac{4 i a \sqrt{e \sec (c+d x)}}{3 d e^2 \sqrt{a+i a \tan (c+d x)}}-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{3 d (e \sec (c+d x))^{3/2}}",1,"(2*(I + 2*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(3*d*(e*Sec[c + d*x])^(3/2))","A",1
398,1,63,122,0.2176434,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{5/2}} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/(e*Sec[c + d*x])^(5/2),x]","\frac{i \sqrt{a+i a \tan (c+d x)} (-4 i \sin (2 (c+d x))+\cos (2 (c+d x))-15)}{15 d e^2 \sqrt{e \sec (c+d x)}}","\frac{8 i a}{15 d e^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{15 d e^2 \sqrt{e \sec (c+d x)}}-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{5 d (e \sec (c+d x))^{5/2}}",1,"((I/15)*(-15 + Cos[2*(c + d*x)] - (4*I)*Sin[2*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(d*e^2*Sqrt[e*Sec[c + d*x]])","A",1
399,1,80,164,0.2715899,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{7/2}} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/(e*Sec[c + d*x])^(7/2),x]","\frac{\sqrt{a+i a \tan (c+d x)} (70 \sin (c+d x)+6 \sin (3 (c+d x))+35 i \cos (c+d x)+i \cos (3 (c+d x)))}{70 d e^3 \sqrt{e \sec (c+d x)}}","\frac{32 i a \sqrt{e \sec (c+d x)}}{35 d e^4 \sqrt{a+i a \tan (c+d x)}}+\frac{12 i a}{35 d e^2 \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{35 d e^2 (e \sec (c+d x))^{3/2}}-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{7 d (e \sec (c+d x))^{7/2}}",1,"(((35*I)*Cos[c + d*x] + I*Cos[3*(c + d*x)] + 70*Sin[c + d*x] + 6*Sin[3*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(70*d*e^3*Sqrt[e*Sec[c + d*x]])","A",1
400,1,376,453,4.1488416,"\int (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[(e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{a \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2} \left(2 i \sqrt{i \sin (2 c)+\cos (2 c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i} (14 i \sin (2 c+2 d x)+7 \cos (2 c+2 d x)-9)+84 \sqrt{-\sin (c)-i \cos (c)-1} \sqrt{-\sin (c)+i \cos (c)+1} \sqrt{\tan \left(\frac{d x}{2}\right)+i} \cos ^3(c+d x) \tanh ^{-1}\left(\frac{\sqrt{\sin (c)-i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)-84 \sqrt{\sin (c)-i \cos (c)+1} \sqrt{\sin (c)+i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i} \cos ^3(c+d x) \tanh ^{-1}\left(\frac{\sqrt{-\sin (c)+i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{\sin (c)+i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)\right)}{96 d \sqrt{i \sin (2 c)+\cos (2 c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}","\frac{7 i a^{3/2} e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{8 \sqrt{2} d}-\frac{7 i a^{3/2} e^{5/2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{8 \sqrt{2} d}-\frac{7 i a^{3/2} e^{5/2} \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{16 \sqrt{2} d}+\frac{7 i a^{3/2} e^{5/2} \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{16 \sqrt{2} d}+\frac{7 i a^2 (e \sec (c+d x))^{5/2}}{12 d \sqrt{a+i a \tan (c+d x)}}-\frac{7 i a e^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}{8 d}+\frac{i a \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}",1,"-1/96*(a*(e*Sec[c + d*x])^(5/2)*((2*I)*Sqrt[1 + Cos[2*c] + I*Sin[2*c]]*(-9 + 7*Cos[2*c + 2*d*x] + (14*I)*Sin[2*c + 2*d*x])*Sqrt[I - Tan[(d*x)/2]] + 84*ArcTanh[(Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Cos[c + d*x]^3*Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]] - 84*ArcTanh[(Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Cos[c + d*x]^3*Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[1 + Cos[2*c] + I*Sin[2*c]]*Sqrt[I - Tan[(d*x)/2]])","A",1
401,1,11319,571,56.1312818,"\int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{5 i a^{5/2} e^{3/2} \sec (c+d x) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{4 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{5 i a^{5/2} e^{3/2} \sec (c+d x) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{4 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{5 i a^{5/2} e^{3/2} \sec (c+d x) \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{5 i a^{5/2} e^{3/2} \sec (c+d x) \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{5 i a^2 (e \sec (c+d x))^{3/2}}{4 d \sqrt{a+i a \tan (c+d x)}}+\frac{i a \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}",1,"Result too large to show","B",0
402,1,338,364,3.3518578,"\int \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{a e \sqrt{a+i a \tan (c+d x)} \left(-3 \sqrt{-\sin (c)-i \cos (c)-1} \sqrt{-\sin (c)+i \cos (c)+1} \sqrt{\tan \left(\frac{d x}{2}\right)+i} \tanh ^{-1}\left(\frac{\sqrt{\sin (c)-i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)+3 \sqrt{\sin (c)-i \cos (c)+1} \sqrt{\sin (c)+i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i} \tanh ^{-1}\left(\frac{\sqrt{-\sin (c)+i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{\sin (c)+i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)+i \sqrt{i \sin (2 c)+\cos (2 c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i} \sec (c+d x)\right)}{d \sqrt{i \sin (2 c)+\cos (2 c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i} \sqrt{e \sec (c+d x)}}","\frac{3 i a^{3/2} \sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{\sqrt{2} d}-\frac{3 i a^{3/2} \sqrt{e} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{\sqrt{2} d}-\frac{3 i a^{3/2} \sqrt{e} \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{2 \sqrt{2} d}+\frac{3 i a^{3/2} \sqrt{e} \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{2 \sqrt{2} d}+\frac{i a \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}{d}",1,"(a*e*(I*Sec[c + d*x]*Sqrt[1 + Cos[2*c] + I*Sin[2*c]]*Sqrt[I - Tan[(d*x)/2]] - 3*ArcTanh[(Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]] + 3*ArcTanh[(Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[e*Sec[c + d*x]]*Sqrt[1 + Cos[2*c] + I*Sin[2*c]]*Sqrt[I - Tan[(d*x)/2]])","A",1
403,1,11314,520,6.1684998,"\int \frac{(a+i a \tan (c+d x))^{3/2}}{\sqrt{e \sec (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2)/Sqrt[e*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{i \sqrt{2} a^{5/2} \sec (c+d x) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{d \sqrt{e} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{i \sqrt{2} a^{5/2} \sec (c+d x) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{d \sqrt{e} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{i a^{5/2} \sec (c+d x) \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{\sqrt{2} d \sqrt{e} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i a^{5/2} \sec (c+d x) \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{\sqrt{2} d \sqrt{e} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{4 i a \sqrt{a+i a \tan (c+d x)}}{d \sqrt{e \sec (c+d x)}}",1,"Result too large to show","B",0
404,1,38,38,0.0730857,"\int \frac{(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2)/(e*Sec[c + d*x])^(3/2),x]","-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 d (e \sec (c+d x))^{3/2}}","-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 d (e \sec (c+d x))^{3/2}}",1,"(((-2*I)/3)*(a + I*a*Tan[c + d*x])^(3/2))/(d*(e*Sec[c + d*x])^(3/2))","A",1
405,1,84,81,0.3790871,"\int \frac{(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2)/(e*Sec[c + d*x])^(5/2),x]","-\frac{2 a (2 \tan (c+d x)+3 i) (\cos (d x)-i \sin (d x)) \sqrt{a+i a \tan (c+d x)} (\cos (c+2 d x)+i \sin (c+2 d x))}{5 d e (e \sec (c+d x))^{3/2}}","-\frac{4 i a \sqrt{a+i a \tan (c+d x)}}{5 d e^2 \sqrt{e \sec (c+d x)}}-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{5 d (e \sec (c+d x))^{5/2}}",1,"(-2*a*(Cos[d*x] - I*Sin[d*x])*(Cos[c + 2*d*x] + I*Sin[c + 2*d*x])*(3*I + 2*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(5*d*e*(e*Sec[c + d*x])^(3/2))","A",1
406,1,98,125,0.4599373,"\int \frac{(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2)/(e*Sec[c + d*x])^(7/2),x]","\frac{a (\cos (d x)-i \sin (d x)) \sqrt{a+i a \tan (c+d x)} (12 \sin (2 (c+d x))+9 i \cos (2 (c+d x))-7 i) (\cos (c+2 d x)+i \sin (c+2 d x))}{21 d e^3 \sqrt{e \sec (c+d x)}}","\frac{16 i a^2 \sqrt{e \sec (c+d x)}}{21 d e^4 \sqrt{a+i a \tan (c+d x)}}-\frac{8 i a \sqrt{a+i a \tan (c+d x)}}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{7 d (e \sec (c+d x))^{7/2}}",1,"(a*(Cos[d*x] - I*Sin[d*x])*(-7*I + (9*I)*Cos[2*(c + d*x)] + 12*Sin[2*(c + d*x)])*(Cos[c + 2*d*x] + I*Sin[c + 2*d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(21*d*e^3*Sqrt[e*Sec[c + d*x]])","A",1
407,1,113,167,0.587951,"\int \frac{(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{9/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2)/(e*Sec[c + d*x])^(9/2),x]","\frac{a (\cos (d x)-i \sin (d x)) \sqrt{a+i a \tan (c+d x)} (-54 \sin (c+d x)+10 \sin (3 (c+d x))-81 i \cos (c+d x)+5 i \cos (3 (c+d x))) (\cos (c+2 d x)+i \sin (c+2 d x))}{90 d e^4 \sqrt{e \sec (c+d x)}}","\frac{16 i a^2}{45 d e^4 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{32 i a \sqrt{a+i a \tan (c+d x)}}{45 d e^4 \sqrt{e \sec (c+d x)}}-\frac{4 i a \sqrt{a+i a \tan (c+d x)}}{15 d e^2 (e \sec (c+d x))^{5/2}}-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{9 d (e \sec (c+d x))^{9/2}}",1,"(a*(Cos[d*x] - I*Sin[d*x])*((-81*I)*Cos[c + d*x] + (5*I)*Cos[3*(c + d*x)] - 54*Sin[c + d*x] + 10*Sin[3*(c + d*x)])*(Cos[c + 2*d*x] + I*Sin[c + 2*d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(90*d*e^4*Sqrt[e*Sec[c + d*x]])","A",1
408,1,11411,612,56.5636766,"\int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{15 i a^{7/2} e^{3/2} \sec (c+d x) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{15 i a^{7/2} e^{3/2} \sec (c+d x) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{15 i a^{7/2} e^{3/2} \sec (c+d x) \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{15 i a^{7/2} e^{3/2} \sec (c+d x) \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{15 i a^3 (e \sec (c+d x))^{3/2}}{8 d \sqrt{a+i a \tan (c+d x)}}+\frac{3 i a^2 \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{4 d}+\frac{i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}",1,"Result too large to show","B",0
409,1,387,411,4.1453578,"\int \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{a^2 (\cos (2 d x)+i \sin (2 d x)) \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)} \left(\sqrt{i \sin (2 c)+\cos (2 c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i} (2 \tan (c+d x)-9 i)+21 \sqrt{-\sin (c)-i \cos (c)-1} \sqrt{-\sin (c)+i \cos (c)+1} \sqrt{\tan \left(\frac{d x}{2}\right)+i} \cos (c+d x) \tanh ^{-1}\left(\frac{\sqrt{\sin (c)-i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)-21 \sqrt{\sin (c)-i \cos (c)+1} \sqrt{\sin (c)+i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i} \cos (c+d x) \tanh ^{-1}\left(\frac{\sqrt{-\sin (c)+i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{\sin (c)+i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)\right)}{4 d \sqrt{i \sin (2 c)+\cos (2 c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i} (\cos (d x)+i \sin (d x))^2}","\frac{21 i a^{5/2} \sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{4 \sqrt{2} d}-\frac{21 i a^{5/2} \sqrt{e} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{4 \sqrt{2} d}-\frac{21 i a^{5/2} \sqrt{e} \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{8 \sqrt{2} d}+\frac{21 i a^{5/2} \sqrt{e} \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{8 \sqrt{2} d}+\frac{7 i a^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}{4 d}+\frac{i a (a+i a \tan (c+d x))^{3/2} \sqrt{e \sec (c+d x)}}{2 d}",1,"-1/4*(a^2*Sqrt[e*Sec[c + d*x]]*(Cos[2*d*x] + I*Sin[2*d*x])*Sqrt[a + I*a*Tan[c + d*x]]*(21*ArcTanh[(Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Cos[c + d*x]*Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]] - 21*ArcTanh[(Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Cos[c + d*x]*Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]] + Sqrt[1 + Cos[2*c] + I*Sin[2*c]]*Sqrt[I - Tan[(d*x)/2]]*(-9*I + 2*Tan[c + d*x])))/(d*Sqrt[1 + Cos[2*c] + I*Sin[2*c]]*(Cos[d*x] + I*Sin[d*x])^2*Sqrt[I - Tan[(d*x)/2]])","A",1
410,1,11357,563,6.3299288,"\int \frac{(a+i a \tan (c+d x))^{5/2}}{\sqrt{e \sec (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/2)/Sqrt[e*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{5 i a^{7/2} \sec (c+d x) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{\sqrt{2} d \sqrt{e} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{5 i a^{7/2} \sec (c+d x) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{\sqrt{2} d \sqrt{e} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{5 i a^{7/2} \sec (c+d x) \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{2 \sqrt{2} d \sqrt{e} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{5 i a^{7/2} \sec (c+d x) \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{2 \sqrt{2} d \sqrt{e} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{10 i a^2 \sqrt{a+i a \tan (c+d x)}}{d \sqrt{e \sec (c+d x)}}+\frac{i a (a+i a \tan (c+d x))^{3/2}}{d \sqrt{e \sec (c+d x)}}",1,"Result too large to show","B",0
411,1,343,362,3.6524391,"\int \frac{(a+i a \tan (c+d x))^{5/2}}{(e \sec (c+d x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/2)/(e*Sec[c + d*x])^(3/2),x]","\frac{e (a+i a \tan (c+d x))^{5/2} \left(-\frac{4}{3} i (\cos (c)-i \sin (c)) \cos (d x)+\frac{4}{3} (\cos (c)-i \sin (c)) \sin (d x)+\frac{2 (\cos (2 c)-i \sin (2 c)) \sqrt{\tan \left(\frac{d x}{2}\right)+i} \left(\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{-\sin (c)+i \cos (c)+1} \tanh ^{-1}\left(\frac{\sqrt{\sin (c)-i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)-\sqrt{\sin (c)-i \cos (c)+1} \sqrt{\sin (c)+i \cos (c)-1} \tanh ^{-1}\left(\frac{\sqrt{-\sin (c)+i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{\sin (c)+i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)\right)}{\sqrt{i \sin (2 c)+\cos (2 c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}\right)}{d (\cos (d x)+i \sin (d x))^2 (e \sec (c+d x))^{5/2}}","-\frac{i \sqrt{2} a^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{d e^{3/2}}+\frac{i \sqrt{2} a^{5/2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{d e^{3/2}}+\frac{i a^{5/2} \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{\sqrt{2} d e^{3/2}}-\frac{i a^{5/2} \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{\sqrt{2} d e^{3/2}}-\frac{4 i a (a+i a \tan (c+d x))^{3/2}}{3 d (e \sec (c+d x))^{3/2}}",1,"(e*(((-4*I)/3)*Cos[d*x]*(Cos[c] - I*Sin[c]) + (4*(Cos[c] - I*Sin[c])*Sin[d*x])/3 + (2*(ArcTanh[(Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[1 + I*Cos[c] - Sin[c]] - ArcTanh[(Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[-1 + I*Cos[c] + Sin[c]])*(Cos[2*c] - I*Sin[2*c])*Sqrt[I + Tan[(d*x)/2]])/(Sqrt[1 + Cos[2*c] + I*Sin[2*c]]*Sqrt[I - Tan[(d*x)/2]]))*(a + I*a*Tan[c + d*x])^(5/2))/(d*(e*Sec[c + d*x])^(5/2)*(Cos[d*x] + I*Sin[d*x])^2)","A",1
412,1,38,38,0.1200926,"\int \frac{(a+i a \tan (c+d x))^{5/2}}{(e \sec (c+d x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/2)/(e*Sec[c + d*x])^(5/2),x]","-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{5 d (e \sec (c+d x))^{5/2}}","-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{5 d (e \sec (c+d x))^{5/2}}",1,"(((-2*I)/5)*(a + I*a*Tan[c + d*x])^(5/2))/(d*(e*Sec[c + d*x])^(5/2))","A",1
413,1,92,81,0.4482061,"\int \frac{(a+i a \tan (c+d x))^{5/2}}{(e \sec (c+d x))^{7/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/2)/(e*Sec[c + d*x])^(7/2),x]","-\frac{2 a^2 (2 \tan (c+d x)+5 i) \sqrt{a+i a \tan (c+d x)} (\cos (2 (c+2 d x))+i \sin (2 (c+2 d x)))}{21 d e^2 (\cos (d x)+i \sin (d x))^2 (e \sec (c+d x))^{3/2}}","-\frac{4 i a (a+i a \tan (c+d x))^{3/2}}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{7 d (e \sec (c+d x))^{7/2}}",1,"(-2*a^2*(Cos[2*(c + 2*d*x)] + I*Sin[2*(c + 2*d*x)])*(5*I + 2*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(21*d*e^2*(e*Sec[c + d*x])^(3/2)*(Cos[d*x] + I*Sin[d*x])^2)","A",1
414,1,104,125,0.4847199,"\int \frac{(a+i a \tan (c+d x))^{5/2}}{(e \sec (c+d x))^{9/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/2)/(e*Sec[c + d*x])^(9/2),x]","\frac{a^2 \sqrt{a+i a \tan (c+d x)} (-20 i \sin (2 (c+d x))+25 \cos (2 (c+d x))+9) (\sin (2 (c+2 d x))-i \cos (2 (c+2 d x)))}{45 d e^4 (\cos (d x)+i \sin (d x))^2 \sqrt{e \sec (c+d x)}}","-\frac{16 i a^2 \sqrt{a+i a \tan (c+d x)}}{45 d e^4 \sqrt{e \sec (c+d x)}}-\frac{8 i a (a+i a \tan (c+d x))^{3/2}}{45 d e^2 (e \sec (c+d x))^{5/2}}-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{9 d (e \sec (c+d x))^{9/2}}",1,"(a^2*(9 + 25*Cos[2*(c + d*x)] - (20*I)*Sin[2*(c + d*x)])*((-I)*Cos[2*(c + 2*d*x)] + Sin[2*(c + 2*d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(45*d*e^4*Sqrt[e*Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])^2)","A",1
415,1,121,169,0.7378708,"\int \frac{(a+i a \tan (c+d x))^{5/2}}{(e \sec (c+d x))^{11/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/2)/(e*Sec[c + d*x])^(11/2),x]","\frac{a^2 \sqrt{a+i a \tan (c+d x)} (-22 \sin (c+d x)+42 \sin (3 (c+d x))-55 i \cos (c+d x)+35 i \cos (3 (c+d x))) (\cos (2 (c+2 d x))+i \sin (2 (c+2 d x)))}{154 d e^5 (\cos (d x)+i \sin (d x))^2 \sqrt{e \sec (c+d x)}}","\frac{32 i a^3 \sqrt{e \sec (c+d x)}}{77 d e^6 \sqrt{a+i a \tan (c+d x)}}-\frac{16 i a^2 \sqrt{a+i a \tan (c+d x)}}{77 d e^4 (e \sec (c+d x))^{3/2}}-\frac{12 i a (a+i a \tan (c+d x))^{3/2}}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{11 d (e \sec (c+d x))^{11/2}}",1,"(a^2*((-55*I)*Cos[c + d*x] + (35*I)*Cos[3*(c + d*x)] - 22*Sin[c + d*x] + 42*Sin[3*(c + d*x)])*(Cos[2*(c + 2*d*x)] + I*Sin[2*(c + 2*d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(154*d*e^5*Sqrt[e*Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])^2)","A",1
416,1,350,369,3.6974544,"\int \frac{(e \sec (c+d x))^{5/2}}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(e*Sec[c + d*x])^(5/2)/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{e^3 (\tan (c+d x)-i) \left(-i \sqrt{-\sin (c)-i \cos (c)-1} \sqrt{-\sin (c)+i \cos (c)+1} \sqrt{\tan \left(\frac{d x}{2}\right)+i} \tanh ^{-1}\left(\frac{\sqrt{\sin (c)-i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)+i \sqrt{\sin (c)-i \cos (c)+1} \sqrt{\sin (c)+i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i} \tanh ^{-1}\left(\frac{\sqrt{-\sin (c)+i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{\sin (c)+i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)+\sqrt{i \sin (2 c)+\cos (2 c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i} \sec (c+d x)\right)}{d \sqrt{i \sin (2 c)+\cos (2 c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i} \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}","\frac{i e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{\sqrt{2} \sqrt{a} d}-\frac{i e^{5/2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{\sqrt{2} \sqrt{a} d}-\frac{i e^{5/2} \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{2 \sqrt{2} \sqrt{a} d}+\frac{i e^{5/2} \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{2 \sqrt{2} \sqrt{a} d}-\frac{i e^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}{a d}",1,"(e^3*(Sec[c + d*x]*Sqrt[1 + Cos[2*c] + I*Sin[2*c]]*Sqrt[I - Tan[(d*x)/2]] - I*ArcTanh[(Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]] + I*ArcTanh[(Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])*(-I + Tan[c + d*x]))/(d*Sqrt[e*Sec[c + d*x]]*Sqrt[1 + Cos[2*c] + I*Sin[2*c]]*Sqrt[I - Tan[(d*x)/2]]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
417,1,302,483,1.2951199,"\int \frac{(e \sec (c+d x))^{3/2}}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(e*Sec[c + d*x])^(3/2)/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 e \sqrt{\tan \left(\frac{d x}{2}\right)+i} (\cos (d x)+i \sin (d x)) \sqrt{e \sec (c+d x)} \left(\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{-\sin (c)+i \cos (c)+1} \tanh ^{-1}\left(\frac{\sqrt{-\sin (c)+i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{\sin (c)+i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)-\sqrt{\sin (c)-i \cos (c)+1} \sqrt{\sin (c)+i \cos (c)-1} \tanh ^{-1}\left(\frac{\sqrt{\sin (c)-i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)\right)}{d \sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\sin (c)+i \cos (c)-1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i} \sqrt{a+i a \tan (c+d x)}}","-\frac{i \sqrt{2} \sqrt{a} e^{3/2} \sec (c+d x) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i \sqrt{2} \sqrt{a} e^{3/2} \sec (c+d x) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i \sqrt{a} e^{3/2} \sec (c+d x) \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{\sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{i \sqrt{a} e^{3/2} \sec (c+d x) \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{\sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}",1,"(2*e*Sqrt[e*Sec[c + d*x]]*(ArcTanh[(Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[1 + I*Cos[c] - Sin[c]] - ArcTanh[(Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[-1 + I*Cos[c] + Sin[c]])*(Cos[d*x] + I*Sin[d*x])*Sqrt[I + Tan[(d*x)/2]])/(d*Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I - Tan[(d*x)/2]]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
418,1,36,36,0.0660063,"\int \frac{\sqrt{e \sec (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Sqrt[e*Sec[c + d*x]]/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 i \sqrt{e \sec (c+d x)}}{d \sqrt{a+i a \tan (c+d x)}}","\frac{2 i \sqrt{e \sec (c+d x)}}{d \sqrt{a+i a \tan (c+d x)}}",1,"((2*I)*Sqrt[e*Sec[c + d*x]])/(d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
419,1,48,80,0.1211821,"\int \frac{1}{\sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/(Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{4 \tan (c+d x)-2 i}{3 d \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}","\frac{2 i}{3 d \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{4 i \sqrt{a+i a \tan (c+d x)}}{3 a d \sqrt{e \sec (c+d x)}}",1,"(-2*I + 4*Tan[c + d*x])/(3*d*Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
420,1,68,121,0.2663564,"\int \frac{1}{(e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/((e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]","-\frac{i \sec ^2(c+d x) (4 i \sin (2 (c+d x))+\cos (2 (c+d x))-15)}{15 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}","\frac{16 i \sqrt{e \sec (c+d x)}}{15 d e^2 \sqrt{a+i a \tan (c+d x)}}-\frac{8 i \sqrt{a+i a \tan (c+d x)}}{15 a d (e \sec (c+d x))^{3/2}}+\frac{2 i}{5 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}",1,"((-1/15*I)*Sec[c + d*x]^2*(-15 + Cos[2*(c + d*x)] + (4*I)*Sin[2*(c + d*x)]))/(d*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])","A",1
421,1,79,165,0.3842221,"\int \frac{1}{(e \sec (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/((e*Sec[c + d*x])^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]","-\frac{i (\cos (2 (c+d x))+35 i \tan (c+d x)+3 i \sin (3 (c+d x)) \sec (c+d x)+17)}{35 d e^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}","-\frac{32 i \sqrt{a+i a \tan (c+d x)}}{35 a d e^2 \sqrt{e \sec (c+d x)}}+\frac{16 i}{35 d e^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{12 i \sqrt{a+i a \tan (c+d x)}}{35 a d (e \sec (c+d x))^{5/2}}+\frac{2 i}{7 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}",1,"((-1/35*I)*(17 + Cos[2*(c + d*x)] + (3*I)*Sec[c + d*x]*Sin[3*(c + d*x)] + (35*I)*Tan[c + d*x]))/(d*e^2*Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
422,1,87,206,0.4257025,"\int \frac{1}{(e \sec (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/((e*Sec[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{\sqrt{e \sec (c+d x)} (336 \sin (2 (c+d x))+40 \sin (4 (c+d x))-84 i \cos (2 (c+d x))-5 i \cos (4 (c+d x))+945 i)}{1260 d e^4 \sqrt{a+i a \tan (c+d x)}}","\frac{256 i \sqrt{e \sec (c+d x)}}{315 d e^4 \sqrt{a+i a \tan (c+d x)}}-\frac{128 i \sqrt{a+i a \tan (c+d x)}}{315 a d e^2 (e \sec (c+d x))^{3/2}}+\frac{32 i}{105 d e^2 \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{63 a d (e \sec (c+d x))^{7/2}}+\frac{2 i}{9 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{7/2}}",1,"(Sqrt[e*Sec[c + d*x]]*(945*I - (84*I)*Cos[2*(c + d*x)] - (5*I)*Cos[4*(c + d*x)] + 336*Sin[2*(c + d*x)] + 40*Sin[4*(c + d*x)]))/(1260*d*e^4*Sqrt[a + I*a*Tan[c + d*x]])","A",1
423,1,11282,529,52.0873466,"\int \frac{(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{3 i e^{7/2} \sec (c+d x) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{\sqrt{2} \sqrt{a} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{3 i e^{7/2} \sec (c+d x) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{\sqrt{2} \sqrt{a} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{3 i e^{7/2} \sec (c+d x) \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{2 \sqrt{2} \sqrt{a} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{3 i e^{7/2} \sec (c+d x) \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{2 \sqrt{2} \sqrt{a} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{i e^2 (e \sec (c+d x))^{3/2}}{a d \sqrt{a+i a \tan (c+d x)}}",1,"Result too large to show","B",0
424,1,338,365,4.0300952,"\int \frac{(e \sec (c+d x))^{5/2}}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(e*Sec[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{e (\cos (d x)+i \sin (d x))^2 (e \sec (c+d x))^{3/2} \left((-4 \sin (c)+4 i \cos (c)) \cos (d x)+4 (\cos (c)+i \sin (c)) \sin (d x)+\frac{2 (\cos (2 c)+i \sin (2 c)) \sqrt{\tan \left(\frac{d x}{2}\right)+i} \left(\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{-\sin (c)+i \cos (c)+1} \tanh ^{-1}\left(\frac{\sqrt{\sin (c)-i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)-\sqrt{\sin (c)-i \cos (c)+1} \sqrt{\sin (c)+i \cos (c)-1} \tanh ^{-1}\left(\frac{\sqrt{-\sin (c)+i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{\sin (c)+i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)\right)}{\sqrt{i \sin (2 c)+\cos (2 c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}\right)}{d (a+i a \tan (c+d x))^{3/2}}","-\frac{i \sqrt{2} e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{a^{3/2} d}+\frac{i \sqrt{2} e^{5/2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{a^{3/2} d}+\frac{i e^{5/2} \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{\sqrt{2} a^{3/2} d}-\frac{i e^{5/2} \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{\sqrt{2} a^{3/2} d}+\frac{4 i e^2 \sqrt{e \sec (c+d x)}}{a d \sqrt{a+i a \tan (c+d x)}}",1,"(e*(e*Sec[c + d*x])^(3/2)*(Cos[d*x] + I*Sin[d*x])^2*(Cos[d*x]*((4*I)*Cos[c] - 4*Sin[c]) + 4*(Cos[c] + I*Sin[c])*Sin[d*x] + (2*(ArcTanh[(Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[1 + I*Cos[c] - Sin[c]] - ArcTanh[(Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[-1 + I*Cos[c] + Sin[c]])*(Cos[2*c] + I*Sin[2*c])*Sqrt[I + Tan[(d*x)/2]])/(Sqrt[1 + Cos[2*c] + I*Sin[2*c]]*Sqrt[I - Tan[(d*x)/2]])))/(d*(a + I*a*Tan[c + d*x])^(3/2))","A",1
425,1,38,38,0.1008009,"\int \frac{(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(e*Sec[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{2 i (e \sec (c+d x))^{3/2}}{3 d (a+i a \tan (c+d x))^{3/2}}","\frac{2 i (e \sec (c+d x))^{3/2}}{3 d (a+i a \tan (c+d x))^{3/2}}",1,"(((2*I)/3)*(e*Sec[c + d*x])^(3/2))/(d*(a + I*a*Tan[c + d*x])^(3/2))","A",1
426,1,63,80,0.2254393,"\int \frac{\sqrt{e \sec (c+d x)}}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Sqrt[e*Sec[c + d*x]]/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{2 (3+2 i \tan (c+d x)) \sqrt{e \sec (c+d x)}}{5 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{4 i \sqrt{e \sec (c+d x)}}{5 a d \sqrt{a+i a \tan (c+d x)}}+\frac{2 i \sqrt{e \sec (c+d x)}}{5 d (a+i a \tan (c+d x))^{3/2}}",1,"(2*Sqrt[e*Sec[c + d*x]]*(3 + (2*I)*Tan[c + d*x]))/(5*a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
427,1,83,121,0.3094124,"\int \frac{1}{\sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)),x]","-\frac{\sec ^2(c+d x) (12 i \sin (2 (c+d x))+9 \cos (2 (c+d x))-7)}{21 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}","-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{21 a^2 d \sqrt{e \sec (c+d x)}}+\frac{8 i}{21 a d \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 i}{7 d (a+i a \tan (c+d x))^{3/2} \sqrt{e \sec (c+d x)}}",1,"-1/21*(Sec[c + d*x]^2*(-7 + 9*Cos[2*(c + d*x)] + (12*I)*Sin[2*(c + d*x)]))/(a*d*Sqrt[e*Sec[c + d*x]]*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
428,1,100,165,0.4791709,"\int \frac{1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[1/((e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)),x]","-\frac{\sec ^3(c+d x) (-54 i \sin (c+d x)+10 i \sin (3 (c+d x))-81 \cos (c+d x)+5 \cos (3 (c+d x)))}{90 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}","-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{45 a^2 d (e \sec (c+d x))^{3/2}}+\frac{32 i \sqrt{e \sec (c+d x)}}{45 a d e^2 \sqrt{a+i a \tan (c+d x)}}+\frac{4 i}{15 a d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}+\frac{2 i}{9 d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}",1,"-1/90*(Sec[c + d*x]^3*(-81*Cos[c + d*x] + 5*Cos[3*(c + d*x)] - (54*I)*Sin[c + d*x] + (10*I)*Sin[3*(c + d*x)]))/(a*d*(e*Sec[c + d*x])^(3/2)*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
429,1,100,209,0.5521506,"\int \frac{1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[1/((e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^(3/2)),x]","-\frac{(e \sec (c+d x))^{3/2} (880 i \sin (2 (c+d x))+56 i \sin (4 (c+d x))+660 \cos (2 (c+d x))+21 \cos (4 (c+d x))-385)}{1540 a d e^4 (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","-\frac{256 i \sqrt{a+i a \tan (c+d x)}}{385 a^2 d e^2 \sqrt{e \sec (c+d x)}}-\frac{96 i \sqrt{a+i a \tan (c+d x)}}{385 a^2 d (e \sec (c+d x))^{5/2}}+\frac{128 i}{385 a d e^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{16 i}{77 a d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}+\frac{2 i}{11 d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{5/2}}",1,"-1/1540*((e*Sec[c + d*x])^(3/2)*(-385 + 660*Cos[2*(c + d*x)] + 21*Cos[4*(c + d*x)] + (880*I)*Sin[2*(c + d*x)] + (56*I)*Sin[4*(c + d*x)]))/(a*d*e^4*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
430,1,370,411,5.8078269,"\int \frac{(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(e*Sec[c + d*x])^(9/2)/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{e^2 (\cos (d x)+i \sin (d x))^3 (e \sec (c+d x))^{5/2} \left((-8 \sin (2 c)+8 i \cos (2 c)) \cos (d x)+8 (\cos (2 c)+i \sin (2 c)) \sin (d x)+(-\sin (3 c)+i \cos (3 c)) \sec (c+d x)+\frac{5 (\cos (3 c)+i \sin (3 c)) \sqrt{\tan \left(\frac{d x}{2}\right)+i} \left(\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{-\sin (c)+i \cos (c)+1} \tanh ^{-1}\left(\frac{\sqrt{\sin (c)-i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)-\sqrt{\sin (c)-i \cos (c)+1} \sqrt{\sin (c)+i \cos (c)-1} \tanh ^{-1}\left(\frac{\sqrt{-\sin (c)+i \cos (c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}{\sqrt{\sin (c)+i \cos (c)-1} \sqrt{\tan \left(\frac{d x}{2}\right)+i}}\right)\right)}{\sqrt{i \sin (2 c)+\cos (2 c)+1} \sqrt{-\tan \left(\frac{d x}{2}\right)+i}}\right)}{d (a+i a \tan (c+d x))^{5/2}}","-\frac{5 i e^{9/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{\sqrt{2} a^{5/2} d}+\frac{5 i e^{9/2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{\sqrt{2} a^{5/2} d}+\frac{5 i e^{9/2} \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{2 \sqrt{2} a^{5/2} d}-\frac{5 i e^{9/2} \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{2 \sqrt{2} a^{5/2} d}+\frac{5 i e^4 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}{a^3 d}+\frac{4 i e^2 (e \sec (c+d x))^{5/2}}{a d (a+i a \tan (c+d x))^{3/2}}",1,"(e^2*(e*Sec[c + d*x])^(5/2)*(Cos[d*x] + I*Sin[d*x])^3*(Cos[d*x]*((8*I)*Cos[2*c] - 8*Sin[2*c]) + Sec[c + d*x]*(I*Cos[3*c] - Sin[3*c]) + 8*(Cos[2*c] + I*Sin[2*c])*Sin[d*x] + (5*(ArcTanh[(Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[1 + I*Cos[c] - Sin[c]] - ArcTanh[(Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[-1 + I*Cos[c] + Sin[c]])*(Cos[3*c] + I*Sin[3*c])*Sqrt[I + Tan[(d*x)/2]])/(Sqrt[1 + Cos[2*c] + I*Sin[2*c]]*Sqrt[I - Tan[(d*x)/2]])))/(d*(a + I*a*Tan[c + d*x])^(5/2))","A",1
431,1,11295,527,27.2580719,"\int \frac{(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{i \sqrt{2} e^{7/2} \sec (c+d x) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{a^{3/2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{i \sqrt{2} e^{7/2} \sec (c+d x) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{a^{3/2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{i e^{7/2} \sec (c+d x) \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{\sqrt{2} a^{3/2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i e^{7/2} \sec (c+d x) \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{\sqrt{2} a^{3/2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{4 i e^2 (e \sec (c+d x))^{3/2}}{3 a d (a+i a \tan (c+d x))^{3/2}}",1,"Result too large to show","B",0
432,1,38,38,0.2486386,"\int \frac{(e \sec (c+d x))^{5/2}}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(e*Sec[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{2 i (e \sec (c+d x))^{5/2}}{5 d (a+i a \tan (c+d x))^{5/2}}","\frac{2 i (e \sec (c+d x))^{5/2}}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"(((2*I)/5)*(e*Sec[c + d*x])^(5/2))/(d*(a + I*a*Tan[c + d*x])^(5/2))","A",1
433,1,63,80,0.2261929,"\int \frac{(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(e*Sec[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{2 (2 \tan (c+d x)-5 i) (e \sec (c+d x))^{3/2}}{21 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{4 i (e \sec (c+d x))^{3/2}}{21 a d (a+i a \tan (c+d x))^{3/2}}+\frac{2 i (e \sec (c+d x))^{3/2}}{7 d (a+i a \tan (c+d x))^{5/2}}",1,"(2*(e*Sec[c + d*x])^(3/2)*(-5*I + 2*Tan[c + d*x]))/(21*a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
434,1,85,121,0.331926,"\int \frac{\sqrt{e \sec (c+d x)}}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Sqrt[e*Sec[c + d*x]]/(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i \sec ^2(c+d x) \sqrt{e \sec (c+d x)} (20 i \sin (2 (c+d x))+25 \cos (2 (c+d x))+9)}{45 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{16 i \sqrt{e \sec (c+d x)}}{45 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{8 i \sqrt{e \sec (c+d x)}}{45 a d (a+i a \tan (c+d x))^{3/2}}+\frac{2 i \sqrt{e \sec (c+d x)}}{9 d (a+i a \tan (c+d x))^{5/2}}",1,"((-1/45*I)*Sec[c + d*x]^2*Sqrt[e*Sec[c + d*x]]*(9 + 25*Cos[2*(c + d*x)] + (20*I)*Sin[2*(c + d*x)]))/(a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
435,1,102,162,0.4239499,"\int \frac{1}{\sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[1/(Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)),x]","\frac{i \sec ^3(c+d x) (-22 i \sin (c+d x)+42 i \sin (3 (c+d x))-55 \cos (c+d x)+35 \cos (3 (c+d x)))}{154 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}","-\frac{32 i \sqrt{a+i a \tan (c+d x)}}{77 a^3 d \sqrt{e \sec (c+d x)}}+\frac{16 i}{77 a^2 d \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{12 i}{77 a d (a+i a \tan (c+d x))^{3/2} \sqrt{e \sec (c+d x)}}+\frac{2 i}{11 d (a+i a \tan (c+d x))^{5/2} \sqrt{e \sec (c+d x)}}",1,"((I/154)*Sec[c + d*x]^3*(-55*Cos[c + d*x] + 35*Cos[3*(c + d*x)] - (22*I)*Sin[c + d*x] + (42*I)*Sin[3*(c + d*x)]))/(a^2*d*Sqrt[e*Sec[c + d*x]]*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
436,1,107,206,0.5928328,"\int \frac{1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[1/((e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)),x]","\frac{\sec ^4(c+d x) (1040 \sin (2 (c+d x))-120 \sin (4 (c+d x))-1300 i \cos (2 (c+d x))+75 i \cos (4 (c+d x))-351 i)}{2340 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}","-\frac{128 i \sqrt{a+i a \tan (c+d x)}}{585 a^3 d (e \sec (c+d x))^{3/2}}+\frac{256 i \sqrt{e \sec (c+d x)}}{585 a^2 d e^2 \sqrt{a+i a \tan (c+d x)}}+\frac{32 i}{195 a^2 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}+\frac{16 i}{117 a d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}+\frac{2 i}{13 d (a+i a \tan (c+d x))^{5/2} (e \sec (c+d x))^{3/2}}",1,"(Sec[c + d*x]^4*(-351*I - (1300*I)*Cos[2*(c + d*x)] + (75*I)*Cos[4*(c + d*x)] + 1040*Sin[2*(c + d*x)] - 120*Sin[4*(c + d*x)]))/(2340*a^2*d*(e*Sec[c + d*x])^(3/2)*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
437,1,118,86,0.7129326,"\int \frac{(e \sec (c+d x))^{7/3}}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(e*Sec[c + d*x])^(7/3)/Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{3 i \sqrt[3]{2} e e^{i (c+d x)} \left(\frac{e e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{4/3} \left(4+\left(1+e^{2 i (c+d x)}\right)^{5/6} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{5}{3};-e^{2 i (c+d x)}\right)\right)}{5 d \sqrt{a+i a \tan (c+d x)}}","\frac{3 i 2^{2/3} a \sqrt[3]{1+i \tan (c+d x)} (e \sec (c+d x))^{7/3} \, _2F_1\left(\frac{1}{3},\frac{7}{6};\frac{13}{6};\frac{1}{2} (1-i \tan (c+d x))\right)}{7 d (a+i a \tan (c+d x))^{3/2}}",1,"(((-3*I)/5)*2^(1/3)*e*E^(I*(c + d*x))*((e*E^(I*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^(4/3)*(4 + (1 + E^((2*I)*(c + d*x)))^(5/6)*Hypergeometric2F1[2/3, 5/6, 5/3, -E^((2*I)*(c + d*x))]))/(d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
438,1,116,86,0.596602,"\int \frac{(e \sec (c+d x))^{5/3}}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(e*Sec[c + d*x])^(5/3)/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{3 i 2^{2/3} e e^{i (c+d x)} \left(\frac{e e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left(-2+\sqrt[6]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{4}{3};-e^{2 i (c+d x)}\right)\right)}{d \sqrt{a+i a \tan (c+d x)}}","\frac{3 i \sqrt[3]{2} a (1+i \tan (c+d x))^{2/3} (e \sec (c+d x))^{5/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};\frac{1}{2} (1-i \tan (c+d x))\right)}{5 d (a+i a \tan (c+d x))^{3/2}}",1,"((3*I)*2^(2/3)*e*E^(I*(c + d*x))*((e*E^(I*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(-2 + (1 + E^((2*I)*(c + d*x)))^(1/6)*Hypergeometric2F1[1/6, 1/3, 4/3, -E^((2*I)*(c + d*x))]))/(d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
439,1,116,85,0.4629801,"\int \frac{(e \sec (c+d x))^{2/3}}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(e*Sec[c + d*x])^(2/3)/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{3 i \sqrt[6]{2} \sqrt[6]{1+e^{2 i (c+d x)}} \left(\frac{e e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{6};\frac{5}{6};-e^{2 i (c+d x)}\right)}{d \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","\frac{3 i \sqrt[6]{1+i \tan (c+d x)} (e \sec (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{7}{6};\frac{4}{3};\frac{1}{2} (1-i \tan (c+d x))\right)}{2 \sqrt[6]{2} d \sqrt{a+i a \tan (c+d x)}}",1,"((3*I)*2^(1/6)*((e*E^(I*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(1 + E^((2*I)*(c + d*x)))^(1/6)*Hypergeometric2F1[-1/6, 1/6, 5/6, -E^((2*I)*(c + d*x))])/(d*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])","A",1
440,1,95,83,0.5676332,"\int \frac{\sqrt[3]{e \sec (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(e*Sec[c + d*x])^(1/3)/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{3 \left(8 i-\frac{2 i e^{2 i (c+d x)} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{5}{3};-e^{2 i (c+d x)}\right)}{\sqrt[6]{1+e^{2 i (c+d x)}}}\right) \sqrt[3]{e \sec (c+d x)}}{16 d \sqrt{a+i a \tan (c+d x)}}","\frac{3 i \sqrt[3]{1+i \tan (c+d x)} \sqrt[3]{e \sec (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{4}{3};\frac{7}{6};\frac{1}{2} (1-i \tan (c+d x))\right)}{\sqrt[3]{2} d \sqrt{a+i a \tan (c+d x)}}",1,"(3*(8*I - ((2*I)*E^((2*I)*(c + d*x))*Hypergeometric2F1[2/3, 5/6, 5/3, -E^((2*I)*(c + d*x))])/(1 + E^((2*I)*(c + d*x)))^(1/6))*(e*Sec[c + d*x])^(1/3))/(16*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
441,1,95,83,0.8114802,"\int \frac{1}{\sqrt[3]{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/((e*Sec[c + d*x])^(1/3)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{12 i-\frac{30 i e^{2 i (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{4}{3};-e^{2 i (c+d x)}\right)}{\left(1+e^{2 i (c+d x)}\right)^{5/6}}}{16 d \sqrt{a+i a \tan (c+d x)} \sqrt[3]{e \sec (c+d x)}}","-\frac{3 i (1+i \tan (c+d x))^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{5}{3};\frac{5}{6};\frac{1}{2} (1-i \tan (c+d x))\right)}{2^{2/3} d \sqrt{a+i a \tan (c+d x)} \sqrt[3]{e \sec (c+d x)}}",1,"(12*I - ((30*I)*E^((2*I)*(c + d*x))*Hypergeometric2F1[1/6, 1/3, 4/3, -E^((2*I)*(c + d*x))])/(1 + E^((2*I)*(c + d*x)))^(5/6))/(16*d*(e*Sec[c + d*x])^(1/3)*Sqrt[a + I*a*Tan[c + d*x]])","A",1
442,1,112,88,0.7092762,"\int \frac{1}{(e \sec (c+d x))^{4/3} \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/((e*Sec[c + d*x])^(4/3)*Sqrt[a + I*a*Tan[c + d*x]]),x]","-\frac{3 i \sec ^2(c+d x) \left(-55 \sqrt[6]{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{6},\frac{1}{6};\frac{5}{6};-e^{2 i (c+d x)}\right)+11 i \sin (2 (c+d x))+3 \cos (2 (c+d x))+3\right)}{112 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{4/3}}","-\frac{3 i \sqrt[6]{1+i \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} \, _2F_1\left(-\frac{2}{3},\frac{13}{6};\frac{1}{3};\frac{1}{2} (1-i \tan (c+d x))\right)}{8 \sqrt[6]{2} a d (e \sec (c+d x))^{4/3}}",1,"(((-3*I)/112)*Sec[c + d*x]^2*(3 + 3*Cos[2*(c + d*x)] - 55*(1 + E^((2*I)*(c + d*x)))^(1/6)*Hypergeometric2F1[-1/6, 1/6, 5/6, -E^((2*I)*(c + d*x))] + (11*I)*Sin[2*(c + d*x)]))/(d*(e*Sec[c + d*x])^(4/3)*Sqrt[a + I*a*Tan[c + d*x]])","A",1
443,1,240,437,1.8324048,"\int \frac{(d \sec (e+f x))^{2/3}}{(a+i a \tan (e+f x))^{7/3}} \, dx","Integrate[(d*Sec[e + f*x])^(2/3)/(a + I*a*Tan[e + f*x])^(7/3),x]","\frac{e^{-2 i (e+f x)} \sec ^2(e+f x) (d \sec (e+f x))^{2/3} \left(-10 f x e^{4 i (e+f x)} \sqrt[3]{1+e^{2 i (e+f x)}}+33 i e^{2 i (e+f x)}+24 i e^{4 i (e+f x)}-15 i e^{4 i (e+f x)} \sqrt[3]{1+e^{2 i (e+f x)}} \log \left(1-\sqrt[3]{1+e^{2 i (e+f x)}}\right)-10 i \sqrt{3} e^{4 i (e+f x)} \sqrt[3]{1+e^{2 i (e+f x)}} \tan ^{-1}\left(\frac{1+2 \sqrt[3]{1+e^{2 i (e+f x)}}}{\sqrt{3}}\right)+9 i\right)}{144 f (a+i a \tan (e+f x))^{7/3}}","\frac{5 i \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a-i a \tan (e+f x)}}{\sqrt{3} \sqrt[3]{a}}\right) (d \sec (e+f x))^{2/3}}{12\ 2^{2/3} \sqrt{3} a^{5/3} f \sqrt[3]{a-i a \tan (e+f x)} \sqrt[3]{a+i a \tan (e+f x)}}-\frac{5 x (d \sec (e+f x))^{2/3}}{72\ 2^{2/3} a^{5/3} \sqrt[3]{a-i a \tan (e+f x)} \sqrt[3]{a+i a \tan (e+f x)}}-\frac{5 i (d \sec (e+f x))^{2/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a-i a \tan (e+f x)}\right)}{24\ 2^{2/3} a^{5/3} f \sqrt[3]{a-i a \tan (e+f x)} \sqrt[3]{a+i a \tan (e+f x)}}-\frac{5 i (d \sec (e+f x))^{2/3} \log (\cos (e+f x))}{72\ 2^{2/3} a^{5/3} f \sqrt[3]{a-i a \tan (e+f x)} \sqrt[3]{a+i a \tan (e+f x)}}+\frac{5 i (d \sec (e+f x))^{2/3}}{24 f \sqrt[3]{a+i a \tan (e+f x)} \left(a^2+i a^2 \tan (e+f x)\right)}+\frac{i (d \sec (e+f x))^{2/3}}{4 f (a+i a \tan (e+f x))^{7/3}}",1,"((9*I + (33*I)*E^((2*I)*(e + f*x)) + (24*I)*E^((4*I)*(e + f*x)) - 10*E^((4*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^(1/3)*f*x - (10*I)*Sqrt[3]*E^((4*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^(1/3)*ArcTan[(1 + 2*(1 + E^((2*I)*(e + f*x)))^(1/3))/Sqrt[3]] - (15*I)*E^((4*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^(1/3)*Log[1 - (1 + E^((2*I)*(e + f*x)))^(1/3)])*Sec[e + f*x]^2*(d*Sec[e + f*x])^(2/3))/(144*E^((2*I)*(e + f*x))*f*(a + I*a*Tan[e + f*x])^(7/3))","A",1
444,1,220,378,1.3182847,"\int \frac{(d \sec (e+f x))^{2/3}}{(a+i a \tan (e+f x))^{4/3}} \, dx","Integrate[(d*Sec[e + f*x])^(2/3)/(a + I*a*Tan[e + f*x])^(4/3),x]","\frac{e^{-i (e+f x)} (d \sec (e+f x))^{5/3} \left(-2 f x e^{2 i (e+f x)} \sqrt[3]{1+e^{2 i (e+f x)}}+3 i e^{2 i (e+f x)}-3 i e^{2 i (e+f x)} \sqrt[3]{1+e^{2 i (e+f x)}} \log \left(1-\sqrt[3]{1+e^{2 i (e+f x)}}\right)-2 i \sqrt{3} e^{2 i (e+f x)} \sqrt[3]{1+e^{2 i (e+f x)}} \tan ^{-1}\left(\frac{1+2 \sqrt[3]{1+e^{2 i (e+f x)}}}{\sqrt{3}}\right)+3 i\right)}{12 d f (a+i a \tan (e+f x))^{4/3}}","\frac{i \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a-i a \tan (e+f x)}}{\sqrt{3} \sqrt[3]{a}}\right) (d \sec (e+f x))^{2/3}}{2^{2/3} \sqrt{3} a^{2/3} f \sqrt[3]{a-i a \tan (e+f x)} \sqrt[3]{a+i a \tan (e+f x)}}-\frac{x (d \sec (e+f x))^{2/3}}{6\ 2^{2/3} a^{2/3} \sqrt[3]{a-i a \tan (e+f x)} \sqrt[3]{a+i a \tan (e+f x)}}-\frac{i (d \sec (e+f x))^{2/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a-i a \tan (e+f x)}\right)}{2\ 2^{2/3} a^{2/3} f \sqrt[3]{a-i a \tan (e+f x)} \sqrt[3]{a+i a \tan (e+f x)}}-\frac{i (d \sec (e+f x))^{2/3} \log (\cos (e+f x))}{6\ 2^{2/3} a^{2/3} f \sqrt[3]{a-i a \tan (e+f x)} \sqrt[3]{a+i a \tan (e+f x)}}+\frac{i (d \sec (e+f x))^{2/3}}{2 f (a+i a \tan (e+f x))^{4/3}}",1,"((3*I + (3*I)*E^((2*I)*(e + f*x)) - 2*E^((2*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^(1/3)*f*x - (2*I)*Sqrt[3]*E^((2*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^(1/3)*ArcTan[(1 + 2*(1 + E^((2*I)*(e + f*x)))^(1/3))/Sqrt[3]] - (3*I)*E^((2*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^(1/3)*Log[1 - (1 + E^((2*I)*(e + f*x)))^(1/3)])*(d*Sec[e + f*x])^(5/3))/(12*d*E^(I*(e + f*x))*f*(a + I*a*Tan[e + f*x])^(4/3))","A",1
445,1,161,340,0.670081,"\int \frac{(d \sec (e+f x))^{2/3}}{\sqrt[3]{a+i a \tan (e+f x)}} \, dx","Integrate[(d*Sec[e + f*x])^(2/3)/(a + I*a*Tan[e + f*x])^(1/3),x]","-\frac{\sqrt[3]{1+e^{2 i (e+f x)}} \left(\frac{d e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{2/3} \left(3 i \log \left(1-\sqrt[3]{1+e^{2 i (e+f x)}}\right)+2 i \sqrt{3} \tan ^{-1}\left(\frac{1+2 \sqrt[3]{1+e^{2 i (e+f x)}}}{\sqrt{3}}\right)+2 f x\right)}{2\ 2^{2/3} f \sqrt[3]{\frac{a e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}}","\frac{i \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a-i a \tan (e+f x)}}{\sqrt{3} \sqrt[3]{a}}\right) (d \sec (e+f x))^{2/3}}{2^{2/3} f \sqrt[3]{a-i a \tan (e+f x)} \sqrt[3]{a+i a \tan (e+f x)}}-\frac{\sqrt[3]{a} x (d \sec (e+f x))^{2/3}}{2\ 2^{2/3} \sqrt[3]{a-i a \tan (e+f x)} \sqrt[3]{a+i a \tan (e+f x)}}-\frac{3 i \sqrt[3]{a} (d \sec (e+f x))^{2/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a-i a \tan (e+f x)}\right)}{2\ 2^{2/3} f \sqrt[3]{a-i a \tan (e+f x)} \sqrt[3]{a+i a \tan (e+f x)}}-\frac{i \sqrt[3]{a} (d \sec (e+f x))^{2/3} \log (\cos (e+f x))}{2\ 2^{2/3} f \sqrt[3]{a-i a \tan (e+f x)} \sqrt[3]{a+i a \tan (e+f x)}}",1,"-1/2*(((d*E^(I*(e + f*x)))/(1 + E^((2*I)*(e + f*x))))^(2/3)*(1 + E^((2*I)*(e + f*x)))^(1/3)*(2*f*x + (2*I)*Sqrt[3]*ArcTan[(1 + 2*(1 + E^((2*I)*(e + f*x)))^(1/3))/Sqrt[3]] + (3*I)*Log[1 - (1 + E^((2*I)*(e + f*x)))^(1/3)]))/(2^(2/3)*((a*E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x))))^(1/3)*f)","A",1
446,1,47,37,0.3621017,"\int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3} \, dx","Integrate[(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(2/3),x]","\frac{3 d^2 (\tan (e+f x)+i) (a+i a \tan (e+f x))^{2/3}}{f (d \sec (e+f x))^{4/3}}","\frac{3 i a (d \sec (e+f x))^{2/3}}{f \sqrt[3]{a+i a \tan (e+f x)}}",1,"(3*d^2*(I + Tan[e + f*x])*(a + I*a*Tan[e + f*x])^(2/3))/(f*(d*Sec[e + f*x])^(4/3))","A",1
447,1,70,81,0.549301,"\int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{5/3} \, dx","Integrate[(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(5/3),x]","-\frac{3 a d (\cos (e)-i \sin (e)) (\tan (e+f x)-7 i) (\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x))^{2/3}}{4 f \sqrt[3]{d \sec (e+f x)}}","\frac{9 i a^2 (d \sec (e+f x))^{2/3}}{2 f \sqrt[3]{a+i a \tan (e+f x)}}+\frac{3 i a (a+i a \tan (e+f x))^{2/3} (d \sec (e+f x))^{2/3}}{4 f}",1,"(-3*a*d*(Cos[e] - I*Sin[e])*(Cos[f*x] - I*Sin[f*x])*(-7*I + Tan[e + f*x])*(a + I*a*Tan[e + f*x])^(2/3))/(4*f*(d*Sec[e + f*x])^(1/3))","A",1
448,1,100,122,0.6210081,"\int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{8/3} \, dx","Integrate[(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(8/3),x]","\frac{3 a^2 (a+i a \tan (e+f x))^{2/3} (d \sec (e+f x))^{5/3} (\sin (e-f x)+i \cos (e-f x)) (5 i \sin (2 (e+f x))+23 \cos (2 (e+f x))+21)}{14 d f (\cos (f x)+i \sin (f x))^2}","\frac{54 i a^3 (d \sec (e+f x))^{2/3}}{7 f \sqrt[3]{a+i a \tan (e+f x)}}+\frac{9 i a^2 (a+i a \tan (e+f x))^{2/3} (d \sec (e+f x))^{2/3}}{7 f}+\frac{3 i a (a+i a \tan (e+f x))^{5/3} (d \sec (e+f x))^{2/3}}{7 f}",1,"(3*a^2*(d*Sec[e + f*x])^(5/3)*(I*Cos[e - f*x] + Sin[e - f*x])*(21 + 23*Cos[2*(e + f*x)] + (5*I)*Sin[2*(e + f*x)])*(a + I*a*Tan[e + f*x])^(2/3))/(14*d*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
449,1,116,163,1.1472334,"\int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{11/3} \, dx","Integrate[(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(11/3),x]","\frac{3 a^3 (a+i a \tan (e+f x))^{2/3} (d \sec (e+f x))^{5/3} (\sin (e-2 f x)+i \cos (e-2 f x)) (442 \cos (2 (e+f x))+45 i \tan (e+f x)+59 i \sin (3 (e+f x)) \sec (e+f x)+364)}{140 d f (\cos (f x)+i \sin (f x))^3}","\frac{486 i a^4 (d \sec (e+f x))^{2/3}}{35 f \sqrt[3]{a+i a \tan (e+f x)}}+\frac{81 i a^3 (a+i a \tan (e+f x))^{2/3} (d \sec (e+f x))^{2/3}}{35 f}+\frac{27 i a^2 (a+i a \tan (e+f x))^{5/3} (d \sec (e+f x))^{2/3}}{35 f}+\frac{3 i a (a+i a \tan (e+f x))^{8/3} (d \sec (e+f x))^{2/3}}{10 f}",1,"(3*a^3*(d*Sec[e + f*x])^(5/3)*(I*Cos[e - 2*f*x] + Sin[e - 2*f*x])*(364 + 442*Cos[2*(e + f*x)] + (59*I)*Sec[e + f*x]*Sin[3*(e + f*x)] + (45*I)*Tan[e + f*x])*(a + I*a*Tan[e + f*x])^(2/3))/(140*d*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
450,1,1165,86,12.921002,"\int (e \sec (c+d x))^m (a+i a \tan (c+d x))^5 \, dx","Integrate[(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^5,x]","-\frac{i 2^{m+5} e^{-i (c-4 d x)} \left(2+3 e^{2 i c}\right) \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^m \, _2F_1\left(1,-\frac{m}{2}-1;\frac{m+6}{2};-e^{2 i (c+d x)}\right) (e \sec (c+d x))^m (i \tan (c+d x) a+a)^5 \sec ^{-m-5}(c+d x)}{d \left(1+e^{2 i c}\right) \left(1+e^{2 i (c+d x)}\right)^3 (m+4) (\cos (d x)+i \sin (d x))^5}+\frac{i 2^{m+5} e^{i (c-d m x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^m \left(e^{i d (m+4) x} (m+6) \, _2F_1\left(1,-\frac{m}{2}-2;\frac{m+6}{2};-e^{2 i (c+d x)}\right)-e^{i d (m+6) x} (m+4) \, _2F_1\left(1,-\frac{m}{2}-1;\frac{m+8}{2};-e^{2 i (c+d x)}\right)\right) (e \sec (c+d x))^m (i \tan (c+d x) a+a)^5 \sec ^{-m-5}(c+d x)}{d \left(1+e^{2 i c}\right) \left(1+e^{2 i (c+d x)}\right)^4 (m+4) (m+6) (\cos (d x)+i \sin (d x))^5}-\frac{i 2^{m+5} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^m \left(1+e^{2 i (c+d x)}\right) \, _2F_1\left(1,1-\frac{m}{2};\frac{m+2}{2};-e^{2 i (c+d x)}\right) (e \sec (c+d x))^m (i \tan (c+d x) a+a)^5 \sec ^{-m-5}(c+d x)}{d \left(e^{3 i c}+e^{5 i c}\right) m (\cos (d x)+i \sin (d x))^5}+\frac{i 2^{m+5} e^{-i (3 c+d m x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^m \left(e^{i d m x} (m+2) \, _2F_1\left(1,-\frac{m}{2};\frac{m+2}{2};-e^{2 i (c+d x)}\right)-e^{i d (m+2) x} m \, _2F_1\left(1,1-\frac{m}{2};\frac{m+4}{2};-e^{2 i (c+d x)}\right)\right) (e \sec (c+d x))^m (i \tan (c+d x) a+a)^5 \sec ^{-m-5}(c+d x)}{d \left(1+e^{2 i c}\right) m (m+2) (\cos (d x)+i \sin (d x))^5}+\frac{i 2^{m+5} e^{i (d x-4 c)} \left(1+4 e^{2 i c}\right) \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+1} \, _2F_1\left(1,-\frac{m}{2};\frac{m+4}{2};-e^{2 i (c+d x)}\right) (e \sec (c+d x))^m (i \tan (c+d x) a+a)^5 \sec ^{-m-5}(c+d x)}{d \left(1+e^{2 i c}\right) (m+2) (\cos (d x)+i \sin (d x))^5}-\frac{3 i 2^{m+5} e^{-i (c+d m x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^m \left(e^{i d (m+2) x} (m+4) \, _2F_1\left(1,-\frac{m}{2}-1;\frac{m+4}{2};-e^{2 i (c+d x)}\right)-e^{i d (m+4) x} (m+2) \, _2F_1\left(1,-\frac{m}{2};\frac{m+6}{2};-e^{2 i (c+d x)}\right)\right) (e \sec (c+d x))^m (i \tan (c+d x) a+a)^5 \sec ^{-m-5}(c+d x)}{d \left(1+e^{2 i c}\right) \left(1+e^{2 i (c+d x)}\right)^2 (m+2) (m+4) (\cos (d x)+i \sin (d x))^5}","\frac{i a^5 2^{\frac{m}{2}+5} (1+i \tan (c+d x))^{-m/2} (e \sec (c+d x))^m \, _2F_1\left(-\frac{m}{2}-4,\frac{m}{2};\frac{m+2}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d m}",1,"((-I)*2^(5 + m)*(2 + 3*E^((2*I)*c))*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^m*Hypergeometric2F1[1, -1 - m/2, (6 + m)/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-5 - m)*(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^5)/(d*E^(I*(c - 4*d*x))*(1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^3*(4 + m)*(Cos[d*x] + I*Sin[d*x])^5) + (I*2^(5 + m)*E^(I*(c - d*m*x))*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^m*(E^(I*d*(4 + m)*x)*(6 + m)*Hypergeometric2F1[1, -2 - m/2, (6 + m)/2, -E^((2*I)*(c + d*x))] - E^(I*d*(6 + m)*x)*(4 + m)*Hypergeometric2F1[1, -1 - m/2, (8 + m)/2, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^(-5 - m)*(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^5)/(d*(1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^4*(4 + m)*(6 + m)*(Cos[d*x] + I*Sin[d*x])^5) - (I*2^(5 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^m*(1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[1, 1 - m/2, (2 + m)/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-5 - m)*(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^5)/(d*(E^((3*I)*c) + E^((5*I)*c))*m*(Cos[d*x] + I*Sin[d*x])^5) + (I*2^(5 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^m*(-(E^(I*d*(2 + m)*x)*m*Hypergeometric2F1[1, 1 - m/2, (4 + m)/2, -E^((2*I)*(c + d*x))]) + E^(I*d*m*x)*(2 + m)*Hypergeometric2F1[1, -1/2*m, (2 + m)/2, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^(-5 - m)*(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^5)/(d*E^(I*(3*c + d*m*x))*(1 + E^((2*I)*c))*m*(2 + m)*(Cos[d*x] + I*Sin[d*x])^5) + (I*2^(5 + m)*E^(I*(-4*c + d*x))*(1 + 4*E^((2*I)*c))*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1 + m)*Hypergeometric2F1[1, -1/2*m, (4 + m)/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-5 - m)*(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^5)/(d*(1 + E^((2*I)*c))*(2 + m)*(Cos[d*x] + I*Sin[d*x])^5) - ((3*I)*2^(5 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^m*(E^(I*d*(2 + m)*x)*(4 + m)*Hypergeometric2F1[1, -1 - m/2, (4 + m)/2, -E^((2*I)*(c + d*x))] - E^(I*d*(4 + m)*x)*(2 + m)*Hypergeometric2F1[1, -1/2*m, (6 + m)/2, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^(-5 - m)*(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^5)/(d*E^(I*(c + d*m*x))*(1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^2*(2 + m)*(4 + m)*(Cos[d*x] + I*Sin[d*x])^5)","B",0
451,1,430,86,6.1941533,"\int (e \sec (c+d x))^m (a+i a \tan (c+d x))^3 \, dx","Integrate[(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^3,x]","-\frac{i e^{-i c} 2^{m+3} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^m (a+i a \tan (c+d x))^3 \left(\frac{\left(1+e^{2 i (c+d x)}\right) \, _2F_1\left(1,1-\frac{m}{2};\frac{m+2}{2};-e^{2 i (c+d x)}\right)}{m}+\frac{e^{2 i d x} \, _2F_1\left(1,1-\frac{m}{2};\frac{m+4}{2};-e^{2 i (c+d x)}\right)}{m+2}-\frac{\, _2F_1\left(1,-\frac{m}{2};\frac{m+2}{2};-e^{2 i (c+d x)}\right)}{m}-\frac{\left(1+2 e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(1,-\frac{m}{2};\frac{m+4}{2};-e^{2 i (c+d x)}\right)}{(m+2) \left(1+e^{2 i (c+d x)}\right)}+\frac{e^{2 i c-i d m x} \left((m+4) e^{i d (m+2) x} \, _2F_1\left(1,-\frac{m}{2}-1;\frac{m+4}{2};-e^{2 i (c+d x)}\right)-(m+2) e^{i d (m+4) x} \, _2F_1\left(1,-\frac{m}{2};\frac{m+6}{2};-e^{2 i (c+d x)}\right)\right)}{(m+2) (m+4) \left(1+e^{2 i (c+d x)}\right)^2}\right) \sec ^{-m-3}(c+d x) (e \sec (c+d x))^m}{\left(1+e^{2 i c}\right) d (\cos (d x)+i \sin (d x))^3}","\frac{i a^3 2^{\frac{m}{2}+3} (1+i \tan (c+d x))^{-m/2} (e \sec (c+d x))^m \, _2F_1\left(-\frac{m}{2}-2,\frac{m}{2};\frac{m+2}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d m}",1,"((-I)*2^(3 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^m*(((1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[1, 1 - m/2, (2 + m)/2, -E^((2*I)*(c + d*x))])/m + (E^((2*I)*d*x)*Hypergeometric2F1[1, 1 - m/2, (4 + m)/2, -E^((2*I)*(c + d*x))])/(2 + m) - Hypergeometric2F1[1, -1/2*m, (2 + m)/2, -E^((2*I)*(c + d*x))]/m - (E^((2*I)*d*x)*(1 + 2*E^((2*I)*c))*Hypergeometric2F1[1, -1/2*m, (4 + m)/2, -E^((2*I)*(c + d*x))])/((1 + E^((2*I)*(c + d*x)))*(2 + m)) + (E^((2*I)*c - I*d*m*x)*(E^(I*d*(2 + m)*x)*(4 + m)*Hypergeometric2F1[1, -1 - m/2, (4 + m)/2, -E^((2*I)*(c + d*x))] - E^(I*d*(4 + m)*x)*(2 + m)*Hypergeometric2F1[1, -1/2*m, (6 + m)/2, -E^((2*I)*(c + d*x))]))/((1 + E^((2*I)*(c + d*x)))^2*(2 + m)*(4 + m)))*Sec[c + d*x]^(-3 - m)*(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^3)/(d*E^(I*c)*(1 + E^((2*I)*c))*(Cos[d*x] + I*Sin[d*x])^3)","B",0
452,1,141,86,1.1671135,"\int (e \sec (c+d x))^m (a+i a \tan (c+d x))^2 \, dx","Integrate[(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^2,x]","-\frac{i 2^{m+2} e^{i (c+3 d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+1} (a+i a \tan (c+d x))^2 \, _2F_1\left(1,1-\frac{m}{2};\frac{m+6}{2};-e^{2 i (c+d x)}\right) \sec ^{-m-2}(c+d x) (e \sec (c+d x))^m}{d (m+4) (\cos (d x)+i \sin (d x))^2}","\frac{i a^2 2^{\frac{m}{2}+2} (1+i \tan (c+d x))^{-m/2} (e \sec (c+d x))^m \, _2F_1\left(-\frac{m}{2}-1,\frac{m}{2};\frac{m+2}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d m}",1,"((-I)*2^(2 + m)*E^(I*(c + 3*d*x))*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1 + m)*Hypergeometric2F1[1, 1 - m/2, (6 + m)/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-2 - m)*(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^2)/(d*(4 + m)*(Cos[d*x] + I*Sin[d*x])^2)","A",0
453,1,130,82,0.6762883,"\int (e \sec (c+d x))^m (a+i a \tan (c+d x)) \, dx","Integrate[(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x]),x]","\frac{a 2^{m+1} e^{i (c+2 d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^m (\tan (c+d x)-i) (\cos (d x)-i \sin (d x)) \, _2F_1\left(1,1-\frac{m}{2};\frac{m+4}{2};-e^{2 i (c+d x)}\right) \sec ^{-m-1}(c+d x) (e \sec (c+d x))^m}{d (m+2)}","\frac{i a 2^{\frac{m}{2}+1} (1+i \tan (c+d x))^{-m/2} (e \sec (c+d x))^m \, _2F_1\left(-\frac{m}{2},\frac{m}{2};\frac{m+2}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d m}",1,"(2^(1 + m)*a*E^(I*(c + 2*d*x))*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^m*Hypergeometric2F1[1, 1 - m/2, (4 + m)/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-1 - m)*(e*Sec[c + d*x])^m*(Cos[d*x] - I*Sin[d*x])*(-I + Tan[c + d*x]))/(d*(2 + m))","A",0
454,1,150,86,0.7035044,"\int \frac{(e \sec (c+d x))^m}{a+i a \tan (c+d x)} \, dx","Integrate[(e*Sec[c + d*x])^m/(a + I*a*Tan[c + d*x]),x]","-\frac{i 2^{m-1} e^{-i (c+2 d x)} \left(1+e^{2 i (c+d x)}\right)^2 \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^m (\cos (d x)+i \sin (d x)) \, _2F_1\left(1,1-\frac{m}{2};\frac{m}{2};-e^{2 i (c+d x)}\right) \sec ^{1-m}(c+d x) (e \sec (c+d x))^m}{d (m-2) (a+i a \tan (c+d x))}","\frac{i 2^{\frac{m}{2}-1} (1+i \tan (c+d x))^{-m/2} (e \sec (c+d x))^m \, _2F_1\left(2-\frac{m}{2},\frac{m}{2};\frac{m+2}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{a d m}",1,"((-I)*2^(-1 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^m*(1 + E^((2*I)*(c + d*x)))^2*Hypergeometric2F1[1, 1 - m/2, m/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(1 - m)*(e*Sec[c + d*x])^m*(Cos[d*x] + I*Sin[d*x]))/(d*E^(I*(c + 2*d*x))*(-2 + m)*(a + I*a*Tan[c + d*x]))","A",0
455,1,154,86,21.1170697,"\int \frac{(e \sec (c+d x))^m}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(e*Sec[c + d*x])^m/(a + I*a*Tan[c + d*x])^2,x]","-\frac{i 2^{m-2} e^{-2 i (c+2 d x)} \left(1+e^{2 i (c+d x)}\right)^3 \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^m (\cos (d x)+i \sin (d x))^2 \, _2F_1\left(1,1-\frac{m}{2};\frac{m-2}{2};-e^{2 i (c+d x)}\right) \sec ^{2-m}(c+d x) (e \sec (c+d x))^m}{d (m-4) (a+i a \tan (c+d x))^2}","\frac{i 2^{\frac{m}{2}-2} (1+i \tan (c+d x))^{-m/2} (e \sec (c+d x))^m \, _2F_1\left(3-\frac{m}{2},\frac{m}{2};\frac{m+2}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{a^2 d m}",1,"((-I)*2^(-2 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^m*(1 + E^((2*I)*(c + d*x)))^3*Hypergeometric2F1[1, 1 - m/2, (-2 + m)/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(2 - m)*(e*Sec[c + d*x])^m*(Cos[d*x] + I*Sin[d*x])^2)/(d*E^((2*I)*(c + 2*d*x))*(-4 + m)*(a + I*a*Tan[c + d*x])^2)","A",0
456,1,151,86,13.4263904,"\int \frac{(e \sec (c+d x))^m}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(e*Sec[c + d*x])^m/(a + I*a*Tan[c + d*x])^3,x]","\frac{2^{m-3} e^{-3 i (c+2 d x)} \left(1+e^{2 i (c+d x)}\right)^4 \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^m (\cos (d x)+i \sin (d x))^3 \, _2F_1\left(1,1-\frac{m}{2};\frac{m-4}{2};-e^{2 i (c+d x)}\right) \sec ^{3-m}(c+d x) (e \sec (c+d x))^m}{a^3 d (m-6) (\tan (c+d x)-i)^3}","\frac{i 2^{\frac{m}{2}-3} (1+i \tan (c+d x))^{-m/2} (e \sec (c+d x))^m \, _2F_1\left(4-\frac{m}{2},\frac{m}{2};\frac{m+2}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{a^3 d m}",1,"(2^(-3 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^m*(1 + E^((2*I)*(c + d*x)))^4*Hypergeometric2F1[1, 1 - m/2, (-4 + m)/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(3 - m)*(e*Sec[c + d*x])^m*(Cos[d*x] + I*Sin[d*x])^3)/(a^3*d*E^((3*I)*(c + 2*d*x))*(-6 + m)*(-I + Tan[c + d*x])^3)","A",0
457,1,178,109,2.5576325,"\int (e \sec (c+d x))^m (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^(7/2),x]","-\frac{i 2^{m+\frac{7}{2}} \sqrt{e^{i d x}} e^{3 i (c+2 d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+\frac{1}{2}} (a+i a \tan (c+d x))^{7/2} \, _2F_1\left(1,1-\frac{m}{2};\frac{m+9}{2};-e^{2 i (c+d x)}\right) \sec ^{-m-\frac{7}{2}}(c+d x) (e \sec (c+d x))^m}{d (m+7) \left(1+e^{2 i (c+d x)}\right)^2 (\cos (d x)+i \sin (d x))^{7/2}}","\frac{i a^3 2^{\frac{m+7}{2}} \sqrt{a+i a \tan (c+d x)} (1+i \tan (c+d x))^{\frac{1}{2} (-m-1)} (e \sec (c+d x))^m \, _2F_1\left(\frac{1}{2} (-m-5),\frac{m}{2};\frac{m+2}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d m}",1,"((-I)*2^(7/2 + m)*E^((3*I)*(c + 2*d*x))*Sqrt[E^(I*d*x)]*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/2 + m)*Hypergeometric2F1[1, 1 - m/2, (9 + m)/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-7/2 - m)*(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^(7/2))/(d*(1 + E^((2*I)*(c + d*x)))^2*(7 + m)*(Cos[d*x] + I*Sin[d*x])^(7/2))","A",0
458,1,163,109,1.8230186,"\int (e \sec (c+d x))^m (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i 2^{m+\frac{5}{2}} \sqrt{e^{i d x}} e^{i (c+3 d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+\frac{3}{2}} (a+i a \tan (c+d x))^{5/2} \, _2F_1\left(1,1-\frac{m}{2};\frac{m+7}{2};-e^{2 i (c+d x)}\right) \sec ^{-m-\frac{5}{2}}(c+d x) (e \sec (c+d x))^m}{d (m+5) (\cos (d x)+i \sin (d x))^{5/2}}","\frac{i a^2 2^{\frac{m+5}{2}} \sqrt{a+i a \tan (c+d x)} (1+i \tan (c+d x))^{\frac{1}{2} (-m-1)} (e \sec (c+d x))^m \, _2F_1\left(\frac{1}{2} (-m-3),\frac{m}{2};\frac{m+2}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d m}",1,"((-I)*2^(5/2 + m)*E^(I*(c + 3*d*x))*Sqrt[E^(I*d*x)]*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2 + m)*Hypergeometric2F1[1, 1 - m/2, (7 + m)/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-5/2 - m)*(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^(5/2))/(d*(5 + m)*(Cos[d*x] + I*Sin[d*x])^(5/2))","A",0
459,1,163,107,1.1448508,"\int (e \sec (c+d x))^m (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{i 2^{m+\frac{3}{2}} \sqrt{e^{i d x}} e^{i (c+2 d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+\frac{1}{2}} (a+i a \tan (c+d x))^{3/2} \, _2F_1\left(1,1-\frac{m}{2};\frac{m+5}{2};-e^{2 i (c+d x)}\right) \sec ^{-m-\frac{3}{2}}(c+d x) (e \sec (c+d x))^m}{d (m+3) (\cos (d x)+i \sin (d x))^{3/2}}","\frac{i a 2^{\frac{m+3}{2}} \sqrt{a+i a \tan (c+d x)} (1+i \tan (c+d x))^{\frac{1}{2} (-m-1)} (e \sec (c+d x))^m \, _2F_1\left(\frac{1}{2} (-m-1),\frac{m}{2};\frac{m+2}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d m}",1,"((-I)*2^(3/2 + m)*E^(I*(c + 2*d*x))*Sqrt[E^(I*d*x)]*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/2 + m)*Hypergeometric2F1[1, 1 - m/2, (5 + m)/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-3/2 - m)*(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^(3/2))/(d*(3 + m)*(Cos[d*x] + I*Sin[d*x])^(3/2))","A",0
460,1,162,107,0.7345289,"\int (e \sec (c+d x))^m \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[(e*Sec[c + d*x])^m*Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i 2^{m+\frac{1}{2}} \sqrt{e^{i d x}} e^{i (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m-\frac{1}{2}} \sqrt{a+i a \tan (c+d x)} \, _2F_1\left(1,1-\frac{m}{2};\frac{m+3}{2};-e^{2 i (c+d x)}\right) \sec ^{-m-\frac{1}{2}}(c+d x) (e \sec (c+d x))^m}{d (m+1) \sqrt{\cos (d x)+i \sin (d x)}}","\frac{i a 2^{\frac{m+1}{2}} (1+i \tan (c+d x))^{\frac{1-m}{2}} (e \sec (c+d x))^m \, _2F_1\left(\frac{1-m}{2},\frac{m}{2};\frac{m+2}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d m \sqrt{a+i a \tan (c+d x)}}",1,"((-I)*2^(1/2 + m)*E^(I*(c + d*x))*Sqrt[E^(I*d*x)]*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(-1/2 + m)*Hypergeometric2F1[1, 1 - m/2, (3 + m)/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-1/2 - m)*(e*Sec[c + d*x])^m*Sqrt[a + I*a*Tan[c + d*x]])/(d*(1 + m)*Sqrt[Cos[d*x] + I*Sin[d*x]])","A",0
461,1,162,106,1.1954882,"\int \frac{(e \sec (c+d x))^m}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(e*Sec[c + d*x])^m/Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i 2^{m-\frac{1}{2}} e^{i (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m-\frac{3}{2}} \sqrt{\cos (d x)+i \sin (d x)} \, _2F_1\left(1,1-\frac{m}{2};\frac{m+1}{2};-e^{2 i (c+d x)}\right) \sec ^{\frac{1}{2}-m}(c+d x) (e \sec (c+d x))^m}{d (m-1) \sqrt{e^{i d x}} \sqrt{a+i a \tan (c+d x)}}","\frac{i 2^{\frac{m-1}{2}} (1+i \tan (c+d x))^{\frac{1-m}{2}} (e \sec (c+d x))^m \, _2F_1\left(\frac{3-m}{2},\frac{m}{2};\frac{m+2}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d m \sqrt{a+i a \tan (c+d x)}}",1,"((-I)*2^(-1/2 + m)*E^(I*(c + d*x))*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(-3/2 + m)*Hypergeometric2F1[1, 1 - m/2, (1 + m)/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(1/2 - m)*(e*Sec[c + d*x])^m*Sqrt[Cos[d*x] + I*Sin[d*x]])/(d*Sqrt[E^(I*d*x)]*(-1 + m)*Sqrt[a + I*a*Tan[c + d*x]])","A",0
462,1,178,109,1.6622702,"\int \frac{(e \sec (c+d x))^m}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(e*Sec[c + d*x])^m/(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{i 2^{m-\frac{3}{2}} \sqrt{e^{i d x}} e^{-2 i (c+2 d x)} \left(1+e^{2 i (c+d x)}\right)^3 \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+\frac{1}{2}} (\cos (d x)+i \sin (d x))^{3/2} \, _2F_1\left(1,1-\frac{m}{2};\frac{m-1}{2};-e^{2 i (c+d x)}\right) \sec ^{\frac{3}{2}-m}(c+d x) (e \sec (c+d x))^m}{d (m-3) (a+i a \tan (c+d x))^{3/2}}","\frac{i 2^{\frac{m-3}{2}} (1+i \tan (c+d x))^{\frac{1-m}{2}} (e \sec (c+d x))^m \, _2F_1\left(\frac{5-m}{2},\frac{m}{2};\frac{m+2}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{a d m \sqrt{a+i a \tan (c+d x)}}",1,"((-I)*2^(-3/2 + m)*Sqrt[E^(I*d*x)]*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/2 + m)*(1 + E^((2*I)*(c + d*x)))^3*Hypergeometric2F1[1, 1 - m/2, (-1 + m)/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(3/2 - m)*(e*Sec[c + d*x])^m*(Cos[d*x] + I*Sin[d*x])^(3/2))/(d*E^((2*I)*(c + 2*d*x))*(-3 + m)*(a + I*a*Tan[c + d*x])^(3/2))","A",0
463,1,178,109,3.2878637,"\int \frac{(e \sec (c+d x))^m}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(e*Sec[c + d*x])^m/(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i 2^{m-\frac{5}{2}} \sqrt{e^{i d x}} e^{-3 i (c+2 d x)} \left(1+e^{2 i (c+d x)}\right)^4 \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+\frac{1}{2}} (\cos (d x)+i \sin (d x))^{5/2} \, _2F_1\left(1,1-\frac{m}{2};\frac{m-3}{2};-e^{2 i (c+d x)}\right) \sec ^{\frac{5}{2}-m}(c+d x) (e \sec (c+d x))^m}{d (m-5) (a+i a \tan (c+d x))^{5/2}}","\frac{i 2^{\frac{m-5}{2}} (1+i \tan (c+d x))^{\frac{1-m}{2}} (e \sec (c+d x))^m \, _2F_1\left(\frac{7-m}{2},\frac{m}{2};\frac{m+2}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{a^2 d m \sqrt{a+i a \tan (c+d x)}}",1,"((-I)*2^(-5/2 + m)*Sqrt[E^(I*d*x)]*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/2 + m)*(1 + E^((2*I)*(c + d*x)))^4*Hypergeometric2F1[1, 1 - m/2, (-3 + m)/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(5/2 - m)*(e*Sec[c + d*x])^m*(Cos[d*x] + I*Sin[d*x])^(5/2))/(d*E^((3*I)*(c + 2*d*x))*(-5 + m)*(a + I*a*Tan[c + d*x])^(5/2))","A",0
464,1,159,105,8.9969476,"\int (e \sec (c+d x))^m (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{m+n} \left(1+e^{2 i (c+d x)}\right) \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+n} (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n (e \sec (c+d x))^m \, _2F_1\left(1,1-\frac{m}{2};\frac{m}{2}+n+1;-e^{2 i (c+d x)}\right) \sec ^{-m-n}(c+d x)}{d (m+2 n)}","\frac{i 2^{\frac{m}{2}+n} (a+i a \tan (c+d x))^n (e \sec (c+d x))^m (1+i \tan (c+d x))^{-\frac{m}{2}-n} \, _2F_1\left(\frac{m}{2},-\frac{m}{2}-n+1;\frac{m+2}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d m}",1,"((-I)*2^(m + n)*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(m + n)*(1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[1, 1 - m/2, 1 + m/2 + n, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-m - n)*(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^n)/(d*(m + 2*n)*(Cos[d*x] + I*Sin[d*x])^n)","A",0
465,1,171,97,14.2198228,"\int \sec ^6(c+d x) (a+i a \tan (c+d x))^n \, dx","Integrate[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{n+5} e^{6 i (c+d x)} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^n \left(2 (n+5) e^{2 i (c+d x)}+2 e^{4 i (c+d x)}+n^2+9 n+20\right) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (n+3) (n+4) (n+5) \left(1+e^{2 i (c+d x)}\right)^5}","-\frac{i (a+i a \tan (c+d x))^{n+5}}{a^5 d (n+5)}+\frac{4 i (a+i a \tan (c+d x))^{n+4}}{a^4 d (n+4)}-\frac{4 i (a+i a \tan (c+d x))^{n+3}}{a^3 d (n+3)}",1,"((-I)*2^(5 + n)*E^((6*I)*(c + d*x))*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(20 + 2*E^((4*I)*(c + d*x)) + 9*n + n^2 + 2*E^((2*I)*(c + d*x))*(5 + n))*(a + I*a*Tan[c + d*x])^n)/(d*(1 + E^((2*I)*(c + d*x)))^5*(3 + n)*(4 + n)*(5 + n)*Sec[c + d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)","A",1
466,1,143,65,13.5267597,"\int \sec ^4(c+d x) (a+i a \tan (c+d x))^n \, dx","Integrate[Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{n+3} e^{4 i (c+d x)} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^n \left(e^{2 i (c+d x)}+n+3\right) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (n+2) (n+3) \left(1+e^{2 i (c+d x)}\right)^3}","\frac{i (a+i a \tan (c+d x))^{n+3}}{a^3 d (n+3)}-\frac{2 i (a+i a \tan (c+d x))^{n+2}}{a^2 d (n+2)}",1,"((-I)*2^(3 + n)*E^((4*I)*(c + d*x))*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(3 + E^((2*I)*(c + d*x)) + n)*(a + I*a*Tan[c + d*x])^n)/(d*(1 + E^((2*I)*(c + d*x)))^3*(2 + n)*(3 + n)*Sec[c + d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)","B",1
467,1,111,32,13.0474713,"\int \sec ^2(c+d x) (a+i a \tan (c+d x))^n \, dx","Integrate[Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{n+1} e^{i (c+d x)} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n+1} \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (n+1)}","-\frac{i (a+i a \tan (c+d x))^{n+1}}{a d (n+1)}",1,"((-I)*2^(1 + n)*E^(I*(c + d*x))*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1 + n)*(a + I*a*Tan[c + d*x])^n)/(d*(1 + n)*Sec[c + d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)","B",1
468,1,141,56,13.2931787,"\int \cos ^2(c+d x) (a+i a \tan (c+d x))^n \, dx","Integrate[Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{n-3} e^{-2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^3 \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^n \, _2F_1\left(1,2;n;-e^{2 i (c+d x)}\right) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (n-1)}","\frac{i a (a+i a \tan (c+d x))^{n-1} \, _2F_1\left(2,n-1;n;\frac{1}{2} (i \tan (c+d x)+1)\right)}{4 d (1-n)}",1,"((-I)*2^(-3 + n)*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(1 + E^((2*I)*(c + d*x)))^3*Hypergeometric2F1[1, 2, n, -E^((2*I)*(c + d*x))]*(a + I*a*Tan[c + d*x])^n)/(d*E^((2*I)*(c + d*x))*(-1 + n)*Sec[c + d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)","B",0
469,1,143,60,4.4372506,"\int \cos ^4(c+d x) (a+i a \tan (c+d x))^n \, dx","Integrate[Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{n-5} e^{-4 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^5 \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^n \, _2F_1\left(1,3;n-1;-e^{2 i (c+d x)}\right) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (n-2)}","\frac{i a^2 (a+i a \tan (c+d x))^{n-2} \, _2F_1\left(3,n-2;n-1;\frac{1}{2} (i \tan (c+d x)+1)\right)}{8 d (2-n)}",1,"((-I)*2^(-5 + n)*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(1 + E^((2*I)*(c + d*x)))^5*Hypergeometric2F1[1, 3, -1 + n, -E^((2*I)*(c + d*x))]*(a + I*a*Tan[c + d*x])^n)/(d*E^((4*I)*(c + d*x))*(-2 + n)*Sec[c + d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)","B",0
470,1,143,60,5.8321612,"\int \cos ^6(c+d x) (a+i a \tan (c+d x))^n \, dx","Integrate[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{n-7} e^{-6 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^7 \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^n \, _2F_1\left(1,4;n-2;-e^{2 i (c+d x)}\right) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (n-3)}","\frac{i a^3 (a+i a \tan (c+d x))^{n-3} \, _2F_1\left(4,n-3;n-2;\frac{1}{2} (i \tan (c+d x)+1)\right)}{16 d (3-n)}",1,"((-I)*2^(-7 + n)*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(1 + E^((2*I)*(c + d*x)))^7*Hypergeometric2F1[1, 4, -2 + n, -E^((2*I)*(c + d*x))]*(a + I*a*Tan[c + d*x])^n)/(d*E^((6*I)*(c + d*x))*(-3 + n)*Sec[c + d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)","B",0
471,1,149,94,14.0072385,"\int \sec ^5(c+d x) (a+i a \tan (c+d x))^n \, dx","Integrate[Sec[c + d*x]^5*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{n+5} e^{5 i (c+d x)} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^n \, _2F_1\left(-\frac{3}{2},1;n+\frac{7}{2};-e^{2 i (c+d x)}\right) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (2 n+5) \left(1+e^{2 i (c+d x)}\right)^4}","\frac{i a^2 2^{n+\frac{5}{2}} \sec ^5(c+d x) (1+i \tan (c+d x))^{-n-\frac{1}{2}} (a+i a \tan (c+d x))^{n-2} \, _2F_1\left(\frac{5}{2},-n-\frac{3}{2};\frac{7}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{5 d}",1,"((-I)*2^(5 + n)*E^((5*I)*(c + d*x))*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*Hypergeometric2F1[-3/2, 1, 7/2 + n, -E^((2*I)*(c + d*x))]*(a + I*a*Tan[c + d*x])^n)/(d*(1 + E^((2*I)*(c + d*x)))^4*(5 + 2*n)*Sec[c + d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)","A",0
472,1,149,92,13.3009826,"\int \sec ^3(c+d x) (a+i a \tan (c+d x))^n \, dx","Integrate[Sec[c + d*x]^3*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{n+3} e^{3 i (c+d x)} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^n \, _2F_1\left(-\frac{1}{2},1;n+\frac{5}{2};-e^{2 i (c+d x)}\right) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (2 n+3) \left(1+e^{2 i (c+d x)}\right)^2}","\frac{i a 2^{n+\frac{3}{2}} \sec ^3(c+d x) (1+i \tan (c+d x))^{-n-\frac{1}{2}} (a+i a \tan (c+d x))^{n-1} \, _2F_1\left(\frac{3}{2},-n-\frac{1}{2};\frac{5}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{3 d}",1,"((-I)*2^(3 + n)*E^((3*I)*(c + d*x))*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*Hypergeometric2F1[-1/2, 1, 5/2 + n, -E^((2*I)*(c + d*x))]*(a + I*a*Tan[c + d*x])^n)/(d*(1 + E^((2*I)*(c + d*x)))^2*(3 + 2*n)*Sec[c + d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)","A",0
473,1,134,88,8.3530083,"\int \sec (c+d x) (a+i a \tan (c+d x))^n \, dx","Integrate[Sec[c + d*x]*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{n+1} e^{i (c+d x)} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^n \, _2F_1\left(\frac{1}{2},1;n+\frac{3}{2};-e^{2 i (c+d x)}\right) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (2 n+1)}","\frac{i a 2^{n+\frac{1}{2}} \sec (c+d x) (1+i \tan (c+d x))^{\frac{1}{2}-n} (a+i a \tan (c+d x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d}",1,"((-I)*2^(1 + n)*E^(I*(c + d*x))*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*Hypergeometric2F1[1/2, 1, 3/2 + n, -E^((2*I)*(c + d*x))]*(a + I*a*Tan[c + d*x])^n)/(d*(1 + 2*n)*Sec[c + d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)","A",0
474,1,136,85,12.8988892,"\int \cos (c+d x) (a+i a \tan (c+d x))^n \, dx","Integrate[Cos[c + d*x]*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{n-1} e^{i (c+d x)} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n-2} \, _2F_1\left(1,\frac{3}{2};n+\frac{1}{2};-e^{2 i (c+d x)}\right) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (2 n-1)}","-\frac{i 2^{n-\frac{1}{2}} \cos (c+d x) (1+i \tan (c+d x))^{\frac{1}{2}-n} (a+i a \tan (c+d x))^n \, _2F_1\left(-\frac{1}{2},\frac{3}{2}-n;\frac{1}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d}",1,"((-I)*2^(-1 + n)*E^(I*(c + d*x))*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(-2 + n)*Hypergeometric2F1[1, 3/2, 1/2 + n, -E^((2*I)*(c + d*x))]*(a + I*a*Tan[c + d*x])^n)/(d*(-1 + 2*n)*Sec[c + d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)","A",0
475,1,149,94,13.5694993,"\int \cos ^3(c+d x) (a+i a \tan (c+d x))^n \, dx","Integrate[Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{n-3} e^{-3 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^4 \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^n \, _2F_1\left(1,\frac{5}{2};n-\frac{1}{2};-e^{2 i (c+d x)}\right) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (2 n-3)}","-\frac{i 2^{n-\frac{3}{2}} \cos ^3(c+d x) (1+i \tan (c+d x))^{\frac{1}{2}-n} (a+i a \tan (c+d x))^{n+1} \, _2F_1\left(-\frac{3}{2},\frac{5}{2}-n;-\frac{1}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{3 a d}",1,"((-I)*2^(-3 + n)*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(1 + E^((2*I)*(c + d*x)))^4*Hypergeometric2F1[1, 5/2, -1/2 + n, -E^((2*I)*(c + d*x))]*(a + I*a*Tan[c + d*x])^n)/(d*E^((3*I)*(c + d*x))*(-3 + 2*n)*Sec[c + d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)","A",0
476,1,149,94,6.3253285,"\int \cos ^5(c+d x) (a+i a \tan (c+d x))^n \, dx","Integrate[Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{n-5} e^{-5 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^6 \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^n \, _2F_1\left(1,\frac{7}{2};n-\frac{3}{2};-e^{2 i (c+d x)}\right) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (2 n-5)}","-\frac{i 2^{n-\frac{5}{2}} \cos ^5(c+d x) (1+i \tan (c+d x))^{\frac{1}{2}-n} (a+i a \tan (c+d x))^{n+2} \, _2F_1\left(-\frac{5}{2},\frac{7}{2}-n;-\frac{3}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{5 a^2 d}",1,"((-I)*2^(-5 + n)*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(1 + E^((2*I)*(c + d*x)))^6*Hypergeometric2F1[1, 7/2, -3/2 + n, -E^((2*I)*(c + d*x))]*(a + I*a*Tan[c + d*x])^n)/(d*E^((5*I)*(c + d*x))*(-5 + 2*n)*Sec[c + d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)","A",0
477,1,156,96,9.0562274,"\int (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{n+\frac{7}{2}} e^{i (c+d x)} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n+\frac{3}{2}} (e \sec (c+d x))^{5/2} \, _2F_1\left(-\frac{1}{4},1;n+\frac{9}{4};-e^{2 i (c+d x)}\right) \sec ^{-n-\frac{5}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (4 n+5)}","\frac{i a 2^{n+\frac{9}{4}} (e \sec (c+d x))^{5/2} (1+i \tan (c+d x))^{-n-\frac{1}{4}} (a+i a \tan (c+d x))^{n-1} \, _2F_1\left(\frac{5}{4},-n-\frac{1}{4};\frac{9}{4};\frac{1}{2} (1-i \tan (c+d x))\right)}{5 d}",1,"((-I)*2^(7/2 + n)*E^(I*(c + d*x))*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2 + n)*Hypergeometric2F1[-1/4, 1, 9/4 + n, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-5/2 - n)*(e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^n)/(d*(5 + 4*n)*(Cos[d*x] + I*Sin[d*x])^n)","A",0
478,1,156,96,8.9346822,"\int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{n+\frac{5}{2}} e^{i (c+d x)} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n+\frac{1}{2}} (e \sec (c+d x))^{3/2} \, _2F_1\left(\frac{1}{4},1;n+\frac{7}{4};-e^{2 i (c+d x)}\right) \sec ^{-n-\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (4 n+3)}","\frac{i a 2^{n+\frac{7}{4}} (e \sec (c+d x))^{3/2} (1+i \tan (c+d x))^{\frac{1}{4}-n} (a+i a \tan (c+d x))^{n-1} \, _2F_1\left(\frac{3}{4},\frac{1}{4}-n;\frac{7}{4};\frac{1}{2} (1-i \tan (c+d x))\right)}{3 d}",1,"((-I)*2^(5/2 + n)*E^(I*(c + d*x))*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/2 + n)*Hypergeometric2F1[1/4, 1, 7/4 + n, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-3/2 - n)*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^n)/(d*(3 + 4*n)*(Cos[d*x] + I*Sin[d*x])^n)","A",0
479,1,156,94,8.5428857,"\int \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^n \, dx","Integrate[Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{n+\frac{3}{2}} e^{i (c+d x)} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n-\frac{1}{2}} \sqrt{e \sec (c+d x)} \, _2F_1\left(\frac{3}{4},1;n+\frac{5}{4};-e^{2 i (c+d x)}\right) \sec ^{-n-\frac{1}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (4 n+1)}","\frac{i a 2^{n+\frac{5}{4}} \sqrt{e \sec (c+d x)} (1+i \tan (c+d x))^{\frac{3}{4}-n} (a+i a \tan (c+d x))^{n-1} \, _2F_1\left(\frac{1}{4},\frac{3}{4}-n;\frac{5}{4};\frac{1}{2} (1-i \tan (c+d x))\right)}{d}",1,"((-I)*2^(3/2 + n)*E^(I*(c + d*x))*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(-1/2 + n)*Hypergeometric2F1[3/4, 1, 5/4 + n, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-1/2 - n)*Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/(d*(1 + 4*n)*(Cos[d*x] + I*Sin[d*x])^n)","A",0
480,1,129,91,9.7501904,"\int \frac{(a+i a \tan (c+d x))^n}{\sqrt{e \sec (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^n/Sqrt[e*Sec[c + d*x]],x]","-\frac{i 2^{n+\frac{1}{2}} e^{i (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n-\frac{3}{2}} \, _2F_1\left(1,\frac{5}{4};n+\frac{3}{4};-e^{2 i (c+d x)}\right) \sec ^{\frac{1}{2}-n}(c+d x) (a+i a \tan (c+d x))^n}{d (4 n-1) \sqrt{e \sec (c+d x)}}","-\frac{i 2^{n+\frac{3}{4}} (1+i \tan (c+d x))^{\frac{1}{4}-n} (a+i a \tan (c+d x))^n \, _2F_1\left(-\frac{1}{4},\frac{5}{4}-n;\frac{3}{4};\frac{1}{2} (1-i \tan (c+d x))\right)}{d \sqrt{e \sec (c+d x)}}",1,"((-I)*2^(1/2 + n)*E^(I*(c + d*x))*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(-3/2 + n)*Hypergeometric2F1[1, 5/4, 3/4 + n, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(1/2 - n)*(a + I*a*Tan[c + d*x])^n)/(d*(-1 + 4*n)*Sqrt[e*Sec[c + d*x]])","A",0
481,1,129,93,10.3367238,"\int \frac{(a+i a \tan (c+d x))^n}{(e \sec (c+d x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^n/(e*Sec[c + d*x])^(3/2),x]","-\frac{i 2^{n-\frac{1}{2}} e^{i (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n-\frac{5}{2}} \, _2F_1\left(1,\frac{7}{4};n+\frac{1}{4};-e^{2 i (c+d x)}\right) \sec ^{\frac{3}{2}-n}(c+d x) (a+i a \tan (c+d x))^n}{d (4 n-3) (e \sec (c+d x))^{3/2}}","-\frac{i 2^{n+\frac{1}{4}} (1+i \tan (c+d x))^{\frac{3}{4}-n} (a+i a \tan (c+d x))^n \, _2F_1\left(-\frac{3}{4},\frac{7}{4}-n;\frac{1}{4};\frac{1}{2} (1-i \tan (c+d x))\right)}{3 d (e \sec (c+d x))^{3/2}}",1,"((-I)*2^(-1/2 + n)*E^(I*(c + d*x))*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(-5/2 + n)*Hypergeometric2F1[1, 7/4, 1/4 + n, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(3/2 - n)*(a + I*a*Tan[c + d*x])^n)/(d*(-3 + 4*n)*(e*Sec[c + d*x])^(3/2))","A",0
482,1,147,98,10.5921422,"\int \frac{(a+i a \tan (c+d x))^n}{(e \sec (c+d x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^n/(e*Sec[c + d*x])^(5/2),x]","-\frac{i 2^{n-\frac{3}{2}} e^{-3 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^4 \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n+\frac{1}{2}} \, _2F_1\left(1,\frac{9}{4};n-\frac{1}{4};-e^{2 i (c+d x)}\right) \sec ^{\frac{1}{2}-n}(c+d x) (a+i a \tan (c+d x))^n}{d e^2 (4 n-5) \sqrt{e \sec (c+d x)}}","-\frac{i 2^{n-\frac{1}{4}} (1+i \tan (c+d x))^{\frac{1}{4}-n} (a+i a \tan (c+d x))^{n+1} \, _2F_1\left(-\frac{5}{4},\frac{9}{4}-n;-\frac{1}{4};\frac{1}{2} (1-i \tan (c+d x))\right)}{5 a d (e \sec (c+d x))^{5/2}}",1,"((-I)*2^(-3/2 + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/2 + n)*(1 + E^((2*I)*(c + d*x)))^4*Hypergeometric2F1[1, 9/4, -1/4 + n, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(1/2 - n)*(a + I*a*Tan[c + d*x])^n)/(d*e^2*E^((3*I)*(c + d*x))*(-5 + 4*n)*Sqrt[e*Sec[c + d*x]])","A",0
483,1,165,269,0.6837623,"\int (e \sec (c+d x))^{-4-n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(-4 - n)*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n} \left(-8 i n^3 \sin (2 (c+d x))-4 i n^3 \sin (4 (c+d x))+4 \left(n^2-16\right) n^2 \cos (2 (c+d x))+\left(n^2-4\right) n^2 \cos (4 (c+d x))+128 i n \sin (2 (c+d x))+16 i n \sin (4 (c+d x))+3 n^4-60 n^2+192\right)}{8 d e^4 (n-4) (n-2) n (n+2) (n+4)}","-\frac{24 i (a+i a \tan (c+d x))^{n+4} (e \sec (c+d x))^{-n-4}}{a^4 d n \left(n^4-20 n^2+64\right)}+\frac{24 i (a+i a \tan (c+d x))^{n+3} (e \sec (c+d x))^{-n-4}}{a^3 d (4-n) n \left(4-n^2\right)}-\frac{12 i (a+i a \tan (c+d x))^{n+2} (e \sec (c+d x))^{-n-4}}{a^2 d (2-n) (4-n) n}+\frac{4 i (a+i a \tan (c+d x))^{n+1} (e \sec (c+d x))^{-n-4}}{a d \left(n^2-6 n+8\right)}+\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n-4}}{d (4-n)}",1,"((-1/8*I)*(192 - 60*n^2 + 3*n^4 + 4*n^2*(-16 + n^2)*Cos[2*(c + d*x)] + n^2*(-4 + n^2)*Cos[4*(c + d*x)] + (128*I)*n*Sin[2*(c + d*x)] - (8*I)*n^3*Sin[2*(c + d*x)] + (16*I)*n*Sin[4*(c + d*x)] - (4*I)*n^3*Sin[4*(c + d*x)])*(a + I*a*Tan[c + d*x])^n)/(d*e^4*(-4 + n)*(-2 + n)*n*(2 + n)*(4 + n)*(e*Sec[c + d*x])^n)","A",1
484,1,119,205,0.6575596,"\int (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(-3 - n)*(a + I*a*Tan[c + d*x])^n,x]","\frac{(a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n} \left(-3 i n \left(n^2-9\right) \cos (c+d x)-i n \left(n^2-1\right) \cos (3 (c+d x))-6 \sin (c+d x) \left(\left(n^2-1\right) \cos (2 (c+d x))+n^2-5\right)\right)}{4 d e^3 (n-3) (n-1) (n+1) (n+3)}","\frac{6 i (a+i a \tan (c+d x))^{n+3} (e \sec (c+d x))^{-n-3}}{a^3 d \left(n^4-10 n^2+9\right)}-\frac{6 i (a+i a \tan (c+d x))^{n+2} (e \sec (c+d x))^{-n-3}}{a^2 d (3-n) \left(1-n^2\right)}+\frac{3 i (a+i a \tan (c+d x))^{n+1} (e \sec (c+d x))^{-n-3}}{a d \left(n^2-4 n+3\right)}+\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n-3}}{d (3-n)}",1,"(((-3*I)*n*(-9 + n^2)*Cos[c + d*x] - I*n*(-1 + n^2)*Cos[3*(c + d*x)] - 6*(-5 + n^2 + (-1 + n^2)*Cos[2*(c + d*x)])*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^n)/(4*d*e^3*(-3 + n)*(-1 + n)*(1 + n)*(3 + n)*(e*Sec[c + d*x])^n)","A",1
485,1,82,148,0.2588707,"\int (e \sec (c+d x))^{-2-n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(-2 - n)*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n} \left(n^2 \cos (2 (c+d x))-2 i n \sin (2 (c+d x))+n^2-4\right)}{2 d e^2 (n-2) n (n+2)}","\frac{2 i (a+i a \tan (c+d x))^{n+2} (e \sec (c+d x))^{-n-2}}{a^2 d n \left(4-n^2\right)}+\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n-2}}{d (2-n)}-\frac{2 i (a+i a \tan (c+d x))^{n+1} (e \sec (c+d x))^{-n-2}}{a d (2-n) n}",1,"((-1/2*I)*(-4 + n^2 + n^2*Cos[2*(c + d*x)] - (2*I)*n*Sin[2*(c + d*x)])*(a + I*a*Tan[c + d*x])^n)/(d*e^2*(-2 + n)*n*(2 + n)*(e*Sec[c + d*x])^n)","A",1
486,1,58,94,0.2042973,"\int (e \sec (c+d x))^{-1-n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(-1 - n)*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i (n-i \tan (c+d x)) (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n-1}}{d (n-1) (n+1)}","\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n-1}}{d (1-n)}-\frac{i (a+i a \tan (c+d x))^{n+1} (e \sec (c+d x))^{-n-1}}{a d \left(1-n^2\right)}",1,"((-I)*(e*Sec[c + d*x])^(-1 - n)*(n - I*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^n)/(d*(-1 + n)*(1 + n))","A",1
487,1,37,37,0.0414591,"\int (e \sec (c+d x))^{-n} (a+i a \tan (c+d x))^n \, dx","Integrate[(a + I*a*Tan[c + d*x])^n/(e*Sec[c + d*x])^n,x]","-\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n}}{d n}","-\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n}}{d n}",1,"((-I)*(a + I*a*Tan[c + d*x])^n)/(d*n*(e*Sec[c + d*x])^n)","A",1
488,1,87,118,4.6038243,"\int (e \sec (c+d x))^{1-n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(1 - n)*(a + I*a*Tan[c + d*x])^n,x]","-\frac{e (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n} (\, _2F_1(1,n;n+1;i \cos (c+d x)-\sin (c+d x))-\, _2F_1(1,n;n+1;\sin (c+d x)-i \cos (c+d x)))}{d n}","\frac{i 2^{\frac{n+1}{2}} (1+i \tan (c+d x))^{\frac{1}{2} (-n-1)} (a+i a \tan (c+d x))^n (e \sec (c+d x))^{1-n} \, _2F_1\left(\frac{1-n}{2},\frac{1-n}{2};\frac{3-n}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d (1-n)}",1,"-((e*(Hypergeometric2F1[1, n, 1 + n, I*Cos[c + d*x] - Sin[c + d*x]] - Hypergeometric2F1[1, n, 1 + n, (-I)*Cos[c + d*x] + Sin[c + d*x]])*(a + I*a*Tan[c + d*x])^n)/(d*n*(e*Sec[c + d*x])^n))","A",1
489,1,112,113,13.6232156,"\int (e \sec (c+d x))^{2-n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(2 - n)*(a + I*a*Tan[c + d*x])^n,x]","\frac{4 e^2 (\cos (2 c)-i \sin (2 c)) (\tan (d x)+i) (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n} \, _2F_1\left(2,1-\frac{n}{2};2-\frac{n}{2};i \sin (2 (c+d x))-\cos (2 (c+d x))\right)}{d (n-2) (-1-i \tan (d x))}","\frac{i a 2^{\frac{n}{2}+1} (1+i \tan (c+d x))^{-n/2} (a+i a \tan (c+d x))^{n-1} (e \sec (c+d x))^{2-n} \, _2F_1\left(\frac{2-n}{2},-\frac{n}{2};\frac{4-n}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d (2-n)}",1,"(4*e^2*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, -Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]]*(Cos[2*c] - I*Sin[2*c])*(I + Tan[d*x])*(a + I*a*Tan[c + d*x])^n)/(d*(-2 + n)*(e*Sec[c + d*x])^n*(-1 - I*Tan[d*x]))","A",1
490,1,116,121,12.2625663,"\int (e \sec (c+d x))^{3-n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(3 - n)*(a + I*a*Tan[c + d*x])^n,x]","\frac{8 e^3 (\tan (d x)+i) \sec (d x) (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n} \, _2F_1\left(3,\frac{3-n}{2};\frac{5-n}{2};i \sin (2 (c+d x))-\cos (2 (c+d x))\right)}{d (n-3) (\cos (c)+i \sin (c))^3 (\tan (d x)-i)^2}","\frac{i a 2^{\frac{n+3}{2}} (1+i \tan (c+d x))^{\frac{1}{2} (-n-1)} (a+i a \tan (c+d x))^{n-1} (e \sec (c+d x))^{3-n} \, _2F_1\left(\frac{1}{2} (-n-1),\frac{3-n}{2};\frac{5-n}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d (3-n)}",1,"(8*e^3*Hypergeometric2F1[3, (3 - n)/2, (5 - n)/2, -Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]]*Sec[d*x]*(I + Tan[d*x])*(a + I*a*Tan[c + d*x])^n)/(d*(-3 + n)*(e*Sec[c + d*x])^n*(Cos[c] + I*Sin[c])^3*(-I + Tan[d*x])^2)","A",1
491,1,122,156,2.188039,"\int (e \sec (c+d x))^{6-2 n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(6 - 2*n)*(a + I*a*Tan[c + d*x])^n,x]","-\frac{e^6 \sec ^5(c+d x) (\sin (3 (c+d x))+i \cos (3 (c+d x))) (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-2 n} \left(i \left(n^2-9 n+18\right) \sin (2 (c+d x))+\left(n^2-9 n+22\right) \cos (2 (c+d x))-2 (n-5)\right)}{d (n-5) (n-4) (n-3)}","\frac{8 i a^3 (a+i a \tan (c+d x))^{n-3} (e \sec (c+d x))^{6-2 n}}{d (5-n) \left(n^2-7 n+12\right)}+\frac{4 i a^2 (a+i a \tan (c+d x))^{n-2} (e \sec (c+d x))^{6-2 n}}{d \left(n^2-9 n+20\right)}+\frac{i a (a+i a \tan (c+d x))^{n-1} (e \sec (c+d x))^{6-2 n}}{d (5-n)}",1,"-((e^6*Sec[c + d*x]^5*(-2*(-5 + n) + (22 - 9*n + n^2)*Cos[2*(c + d*x)] + I*(18 - 9*n + n^2)*Sin[2*(c + d*x)])*(I*Cos[3*(c + d*x)] + Sin[3*(c + d*x)])*(a + I*a*Tan[c + d*x])^n)/(d*(-5 + n)*(-4 + n)*(-3 + n)*(e*Sec[c + d*x])^(2*n)))","A",1
492,1,166,97,13.8003557,"\int (e \sec (c+d x))^{5-2 n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(5 - 2*n)*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{5-n} e^{5 i (c+d x)} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{-n} \left(1+e^{2 i (c+d x)}\right)^{-n} \, _2F_1\left(\frac{5}{2},5-n;\frac{7}{2};-e^{2 i (c+d x)}\right) \sec ^{n-5}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n (e \sec (c+d x))^{5-2 n}}{5 d}","-\frac{i 2^{\frac{5}{2}-n} (1-i \tan (c+d x))^{n-\frac{5}{2}} (a+i a \tan (c+d x))^n (e \sec (c+d x))^{5-2 n} \, _2F_1\left(\frac{5}{2},\frac{1}{2} (2 n-3);\frac{7}{2};\frac{1}{2} (i \tan (c+d x)+1)\right)}{5 d}",1,"((-1/5*I)*2^(5 - n)*E^((5*I)*(c + d*x))*(E^(I*d*x))^n*Hypergeometric2F1[5/2, 5 - n, 7/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-5 + n)*(e*Sec[c + d*x])^(5 - 2*n)*(a + I*a*Tan[c + d*x])^n)/(d*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(1 + E^((2*I)*(c + d*x)))^n*(Cos[d*x] + I*Sin[d*x])^n)","A",1
493,1,91,98,1.2407761,"\int (e \sec (c+d x))^{4-2 n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(4 - 2*n)*(a + I*a*Tan[c + d*x])^n,x]","\frac{e^4 \sec ^2(c+d x) ((n-2) \tan (c+d x)-i (n-4)) (\cos (2 (c+d x))-i \sin (2 (c+d x))) (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-2 n}}{d (n-3) (n-2)}","\frac{2 i a^2 (a+i a \tan (c+d x))^{n-2} (e \sec (c+d x))^{4-2 n}}{d \left(n^2-5 n+6\right)}+\frac{i a (a+i a \tan (c+d x))^{n-1} (e \sec (c+d x))^{4-2 n}}{d (3-n)}",1,"(e^4*Sec[c + d*x]^2*(Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)])*(a + I*a*Tan[c + d*x])^n*((-I)*(-4 + n) + (-2 + n)*Tan[c + d*x]))/(d*(-3 + n)*(-2 + n)*(e*Sec[c + d*x])^(2*n))","A",1
494,1,166,97,11.9731615,"\int (e \sec (c+d x))^{3-2 n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(3 - 2*n)*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{3-n} e^{3 i (c+d x)} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{-n} \left(1+e^{2 i (c+d x)}\right)^{-n} \, _2F_1\left(\frac{3}{2},3-n;\frac{5}{2};-e^{2 i (c+d x)}\right) \sec ^{n-3}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n (e \sec (c+d x))^{3-2 n}}{3 d}","-\frac{i 2^{\frac{3}{2}-n} (1-i \tan (c+d x))^{n-\frac{3}{2}} (a+i a \tan (c+d x))^n (e \sec (c+d x))^{3-2 n} \, _2F_1\left(\frac{3}{2},\frac{1}{2} (2 n-1);\frac{5}{2};\frac{1}{2} (i \tan (c+d x)+1)\right)}{3 d}",1,"((-1/3*I)*2^(3 - n)*E^((3*I)*(c + d*x))*(E^(I*d*x))^n*Hypergeometric2F1[3/2, 3 - n, 5/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-3 + n)*(e*Sec[c + d*x])^(3 - 2*n)*(a + I*a*Tan[c + d*x])^n)/(d*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(1 + E^((2*I)*(c + d*x)))^n*(Cos[d*x] + I*Sin[d*x])^n)","A",1
495,1,59,46,0.6474688,"\int (e \sec (c+d x))^{2-2 n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(2 - 2*n)*(a + I*a*Tan[c + d*x])^n,x]","-\frac{e^2 (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-2 n} (\sec (c) \sin (d x) \sec (c+d x)+\tan (c)+i)}{d (n-1)}","\frac{i a (a+i a \tan (c+d x))^{n-1} (e \sec (c+d x))^{2-2 n}}{d (1-n)}",1,"-((e^2*(I + Sec[c]*Sec[c + d*x]*Sin[d*x] + Tan[c])*(a + I*a*Tan[c + d*x])^n)/(d*(-1 + n)*(e*Sec[c + d*x])^(2*n)))","A",1
496,1,154,95,8.2163853,"\int (e \sec (c+d x))^{1-2 n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(1 - 2*n)*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i e 2^{1-n} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{1-n} \left(1+e^{2 i (c+d x)}\right)^{1-n} \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};-e^{2 i (c+d x)}\right) \sec ^n(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-2 n}}{d}","-\frac{i 2^{\frac{1}{2}-n} (1-i \tan (c+d x))^{n-\frac{1}{2}} (a+i a \tan (c+d x))^n (e \sec (c+d x))^{1-2 n} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (2 n+1);\frac{3}{2};\frac{1}{2} (i \tan (c+d x)+1)\right)}{d}",1,"((-I)*2^(1 - n)*e*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1 - n)*(1 + E^((2*I)*(c + d*x)))^(1 - n)*Hypergeometric2F1[1/2, 1 - n, 3/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^n*(a + I*a*Tan[c + d*x])^n)/(d*(e*Sec[c + d*x])^(2*n)*(Cos[d*x] + I*Sin[d*x])^n)","A",1
497,1,146,65,1.782674,"\int (e \sec (c+d x))^{-2 n} (a+i a \tan (c+d x))^n \, dx","Integrate[(a + I*a*Tan[c + d*x])^n/(e*Sec[c + d*x])^(2*n),x]","\frac{i 2^{-n-1} \left(1+e^{2 i (c+d x)}\right) \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{-n} \, _2F_1\left(1,n+1;n+2;1+e^{2 i (c+d x)}\right) \sec ^n(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-2 n}}{d (n+1)}","-\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-2 n} \, _2F_1\left(1,-n;1-n;\frac{1}{2} (1-i \tan (c+d x))\right)}{2 d n}",1,"(I*2^(-1 - n)*(E^(I*d*x))^n*(1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + E^((2*I)*(c + d*x))]*Sec[c + d*x]^n*(a + I*a*Tan[c + d*x])^n)/(d*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(1 + n)*(e*Sec[c + d*x])^(2*n)*(Cos[d*x] + I*Sin[d*x])^n)","B",1
498,1,157,95,12.8826039,"\int (e \sec (c+d x))^{-1-2 n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(-1 - 2*n)*(a + I*a*Tan[c + d*x])^n,x]","\frac{i 2^{-n-1} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{-n-1} \left(1+e^{2 i (c+d x)}\right)^{-n-1} \, _2F_1\left(-\frac{1}{2},-n-1;\frac{1}{2};-e^{2 i (c+d x)}\right) \sec ^{n+1}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-2 n-1}}{d}","\frac{i 2^{-n-\frac{1}{2}} (1-i \tan (c+d x))^{n+\frac{1}{2}} (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-2 n-1} \, _2F_1\left(-\frac{1}{2},\frac{1}{2} (2 n+3);\frac{1}{2};\frac{1}{2} (i \tan (c+d x)+1)\right)}{d}",1,"(I*2^(-1 - n)*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(-1 - n)*(1 + E^((2*I)*(c + d*x)))^(-1 - n)*Hypergeometric2F1[-1/2, -1 - n, 1/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(1 + n)*(e*Sec[c + d*x])^(-1 - 2*n)*(a + I*a*Tan[c + d*x])^n)/(d*(Cos[d*x] + I*Sin[d*x])^n)","A",1
499,1,151,74,13.1160844,"\int (e \sec (c+d x))^{-2-2 n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(-2 - 2*n)*(a + I*a*Tan[c + d*x])^n,x]","-\frac{i 2^{-n-3} \left(1+e^{2 i (c+d x)}\right)^3 \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{-n} \, _2F_1\left(2,n+3;n+4;1+e^{2 i (c+d x)}\right) \sec ^n(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-2 n}}{d e^2 (n+3)}","-\frac{i (a+i a \tan (c+d x))^{n+1} (e \sec (c+d x))^{-2 (n+1)} \, _2F_1\left(2,-n-1;-n;\frac{1}{2} (1-i \tan (c+d x))\right)}{4 a d (n+1)}",1,"((-I)*2^(-3 - n)*(E^(I*d*x))^n*(1 + E^((2*I)*(c + d*x)))^3*Hypergeometric2F1[2, 3 + n, 4 + n, 1 + E^((2*I)*(c + d*x))]*Sec[c + d*x]^n*(a + I*a*Tan[c + d*x])^n)/(d*e^2*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(3 + n)*(e*Sec[c + d*x])^(2*n)*(Cos[d*x] + I*Sin[d*x])^n)","B",1
500,1,166,97,13.4744762,"\int (e \sec (c+d x))^{-3-2 n} (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Sec[c + d*x])^(-3 - 2*n)*(a + I*a*Tan[c + d*x])^n,x]","\frac{i 2^{-n-3} e^{-3 i (c+d x)} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{-n} \left(1+e^{2 i (c+d x)}\right)^{-n} \, _2F_1\left(-\frac{3}{2},-n-3;-\frac{1}{2};-e^{2 i (c+d x)}\right) \sec ^{n+3}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-2 n-3}}{3 d}","\frac{i 2^{-n-\frac{3}{2}} (1-i \tan (c+d x))^{n+\frac{3}{2}} (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-2 n-3} \, _2F_1\left(-\frac{3}{2},\frac{1}{2} (2 n+5);-\frac{1}{2};\frac{1}{2} (i \tan (c+d x)+1)\right)}{3 d}",1,"((I/3)*2^(-3 - n)*(E^(I*d*x))^n*Hypergeometric2F1[-3/2, -3 - n, -1/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(3 + n)*(e*Sec[c + d*x])^(-3 - 2*n)*(a + I*a*Tan[c + d*x])^n)/(d*E^((3*I)*(c + d*x))*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(1 + E^((2*I)*(c + d*x)))^n*(Cos[d*x] + I*Sin[d*x])^n)","A",1
501,1,165,66,123.4430979,"\int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-2-n} \, dx","Integrate[(d*Sec[e + f*x])^(2*n)*(a + I*a*Tan[e + f*x])^(-2 - n),x]","-\frac{i e^{2 i e} 2^{n-3} \left(1+e^{2 i (e+f x)}\right)^3 \left(e^{i f x}\right)^{-n} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^n \, _2F_1\left(3,3-n;4-n;1+e^{2 i (e+f x)}\right) \sec ^{2-n}(e+f x) (\cos (f x)+i \sin (f x))^{n+2} (a+i a \tan (e+f x))^{-n-2} (d \sec (e+f x))^{2 n}}{f (n-3)}","\frac{i (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n} \, _2F_1\left(3,n;n+1;\frac{1}{2} (1-i \tan (e+f x))\right)}{8 a^2 f n}",1,"((-I)*2^(-3 + n)*E^((2*I)*e)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^n*(1 + E^((2*I)*(e + f*x)))^3*Hypergeometric2F1[3, 3 - n, 4 - n, 1 + E^((2*I)*(e + f*x))]*Sec[e + f*x]^(2 - n)*(d*Sec[e + f*x])^(2*n)*(Cos[f*x] + I*Sin[f*x])^(2 + n)*(a + I*a*Tan[e + f*x])^(-2 - n))/((E^(I*f*x))^n*f*(-3 + n))","B",1
502,1,165,66,14.2988382,"\int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-1-n} \, dx","Integrate[(d*Sec[e + f*x])^(2*n)*(a + I*a*Tan[e + f*x])^(-1 - n),x]","\frac{i e^{i e} 2^{n-2} \left(1+e^{2 i (e+f x)}\right)^2 \left(e^{i f x}\right)^{-n} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^n \, _2F_1\left(2,2-n;3-n;1+e^{2 i (e+f x)}\right) \sec ^{1-n}(e+f x) (\cos (f x)+i \sin (f x))^{n+1} (a+i a \tan (e+f x))^{-n-1} (d \sec (e+f x))^{2 n}}{f (n-2)}","\frac{i (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n} \, _2F_1\left(2,n;n+1;\frac{1}{2} (1-i \tan (e+f x))\right)}{4 a f n}",1,"(I*2^(-2 + n)*E^(I*e)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^n*(1 + E^((2*I)*(e + f*x)))^2*Hypergeometric2F1[2, 2 - n, 3 - n, 1 + E^((2*I)*(e + f*x))]*Sec[e + f*x]^(1 - n)*(d*Sec[e + f*x])^(2*n)*(Cos[f*x] + I*Sin[f*x])^(1 + n)*(a + I*a*Tan[e + f*x])^(-1 - n))/((E^(I*f*x))^n*f*(-2 + n))","B",1
503,1,150,63,1.1606238,"\int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n} \, dx","Integrate[(d*Sec[e + f*x])^(2*n)/(a + I*a*Tan[e + f*x])^n,x]","-\frac{i 2^{n-1} \left(1+e^{2 i (e+f x)}\right) \left(e^{i f x}\right)^{-n} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^n \, _2F_1\left(1,1-n;2-n;1+e^{2 i (e+f x)}\right) \sec ^{-n}(e+f x) (\cos (f x)+i \sin (f x))^n (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n}}{f (n-1)}","\frac{i (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n} \, _2F_1\left(1,n;n+1;\frac{1}{2} (1-i \tan (e+f x))\right)}{2 f n}",1,"((-I)*2^(-1 + n)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^n*(1 + E^((2*I)*(e + f*x)))*Hypergeometric2F1[1, 1 - n, 2 - n, 1 + E^((2*I)*(e + f*x))]*(d*Sec[e + f*x])^(2*n)*(Cos[f*x] + I*Sin[f*x])^n)/((E^(I*f*x))^n*f*(-1 + n)*Sec[e + f*x]^n*(a + I*a*Tan[e + f*x])^n)","B",1
504,1,40,40,0.5002344,"\int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{1-n} \, dx","Integrate[(d*Sec[e + f*x])^(2*n)*(a + I*a*Tan[e + f*x])^(1 - n),x]","\frac{i a (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n}}{f n}","\frac{i a (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n}}{f n}",1,"(I*a*(d*Sec[e + f*x])^(2*n))/(f*n*(a + I*a*Tan[e + f*x])^n)","A",1
505,1,61,92,1.1657247,"\int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{2-n} \, dx","Integrate[(d*Sec[e + f*x])^(2*n)*(a + I*a*Tan[e + f*x])^(2 - n),x]","-\frac{a^2 (n \tan (e+f x)-i (n+2)) (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n}}{f n (n+1)}","\frac{2 i a^2 (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n}}{f n (n+1)}+\frac{i a (a+i a \tan (e+f x))^{1-n} (d \sec (e+f x))^{2 n}}{f (n+1)}",1,"-((a^2*(d*Sec[e + f*x])^(2*n)*((-I)*(2 + n) + n*Tan[e + f*x]))/(f*n*(1 + n)*(a + I*a*Tan[e + f*x])^n))","A",1
506,1,129,148,1.9637229,"\int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{3-n} \, dx","Integrate[(d*Sec[e + f*x])^(2*n)*(a + I*a*Tan[e + f*x])^(3 - n),x]","\frac{i a^3 \sec ^2(e+f x) (\cos (3 f x)+i \sin (3 f x)) (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n} \left(\left(n^2+3 n+4\right) \cos (2 (e+f x))+i n (n+3) \sin (2 (e+f x))+2 (n+2)\right)}{f n (n+1) (n+2) (\cos (f x)+i \sin (f x))^3}","\frac{8 i a^3 (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n}}{f n \left(n^2+3 n+2\right)}+\frac{4 i a^2 (a+i a \tan (e+f x))^{1-n} (d \sec (e+f x))^{2 n}}{f \left(n^2+3 n+2\right)}+\frac{i a (a+i a \tan (e+f x))^{2-n} (d \sec (e+f x))^{2 n}}{f (n+2)}",1,"(I*a^3*Sec[e + f*x]^2*(d*Sec[e + f*x])^(2*n)*(Cos[3*f*x] + I*Sin[3*f*x])*(2*(2 + n) + (4 + 3*n + n^2)*Cos[2*(e + f*x)] + I*n*(3 + n)*Sin[2*(e + f*x)]))/(f*n*(1 + n)*(2 + n)*(Cos[f*x] + I*Sin[f*x])^3*(a + I*a*Tan[e + f*x])^n)","A",1
507,1,53,60,0.1770047,"\int \sec ^6(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Sec[c + d*x]^6*(a + b*Tan[c + d*x]),x]","\frac{a \left(\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{b \sec ^6(c+d x)}{6 d}","\frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^6(c+d x)}{6 d}",1,"(b*Sec[c + d*x]^6)/(6*d) + (a*(Tan[c + d*x] + (2*Tan[c + d*x]^3)/3 + Tan[c + d*x]^5/5))/d","A",1
508,1,68,74,0.2109517,"\int \sec ^5(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Sec[c + d*x]^5*(a + b*Tan[c + d*x]),x]","\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{8 d}+\frac{b \sec ^5(c+d x)}{5 d}","\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \sec ^5(c+d x)}{5 d}",1,"(b*Sec[c + d*x]^5)/(5*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*a*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(8*d)","A",1
509,1,41,44,0.0863262,"\int \sec ^4(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Sec[c + d*x]^4*(a + b*Tan[c + d*x]),x]","\frac{a \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{b \sec ^4(c+d x)}{4 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^4(c+d x)}{4 d}",1,"(b*Sec[c + d*x]^4)/(4*d) + (a*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
510,1,52,52,0.0163995,"\int \sec ^3(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Tan[c + d*x]),x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b \sec ^3(c+d x)}{3 d}","\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b \sec ^3(c+d x)}{3 d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + (b*Sec[c + d*x]^3)/(3*d) + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
511,1,28,28,0.0142452,"\int \sec ^2(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Tan[c + d*x]),x]","\frac{a \tan (c+d x)}{d}+\frac{b \sec ^2(c+d x)}{2 d}","\frac{a \tan (c+d x)}{d}+\frac{b \sec ^2(c+d x)}{2 d}",1,"(b*Sec[c + d*x]^2)/(2*d) + (a*Tan[c + d*x])/d","A",1
512,1,24,24,0.009833,"\int \sec (c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + b*Tan[c + d*x]),x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b \sec (c+d x)}{d}","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b \sec (c+d x)}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/d + (b*Sec[c + d*x])/d","A",1
513,1,46,24,0.0206601,"\int \cos (c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Tan[c + d*x]),x]","\frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}+\frac{b \sin (c) \sin (d x)}{d}-\frac{b \cos (c) \cos (d x)}{d}","\frac{a \sin (c+d x)}{d}-\frac{b \cos (c+d x)}{d}",1,"-((b*Cos[c]*Cos[d*x])/d) + (a*Cos[d*x]*Sin[c])/d + (a*Cos[c]*Sin[d*x])/d + (b*Sin[c]*Sin[d*x])/d","A",1
514,1,46,43,0.0548106,"\int \cos ^2(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Tan[c + d*x]),x]","\frac{a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}-\frac{b \cos ^2(c+d x)}{2 d}","\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}-\frac{b \cos ^2(c+d x)}{2 d}",1,"(a*(c + d*x))/(2*d) - (b*Cos[c + d*x]^2)/(2*d) + (a*Sin[2*(c + d*x)])/(4*d)","A",1
515,1,44,44,0.0126721,"\int \cos ^3(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Tan[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^3(c+d x)}{3 d}","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^3(c+d x)}{3 d}",1,"-1/3*(b*Cos[c + d*x]^3)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d)","A",1
516,1,62,65,0.0954544,"\int \cos ^4(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Tan[c + d*x]),x]","\frac{3 a (c+d x)}{8 d}+\frac{a \sin (2 (c+d x))}{4 d}+\frac{a \sin (4 (c+d x))}{32 d}-\frac{b \cos ^4(c+d x)}{4 d}","\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}-\frac{b \cos ^4(c+d x)}{4 d}",1,"(3*a*(c + d*x))/(8*d) - (b*Cos[c + d*x]^4)/(4*d) + (a*Sin[2*(c + d*x)])/(4*d) + (a*Sin[4*(c + d*x)])/(32*d)","A",1
517,1,133,119,0.5758625,"\int \sec ^8(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^8*(a + b*Tan[c + d*x])^2,x]","\frac{\tan (c+d x) \left(180 \left(a^2+3 b^2\right) \tan ^6(c+d x)+756 \left(a^2+b^2\right) \tan ^4(c+d x)+420 \left(3 a^2+b^2\right) \tan ^2(c+d x)+1260 a^2+315 a b \tan ^7(c+d x)+1260 a b \tan ^5(c+d x)+1890 a b \tan ^3(c+d x)+1260 a b \tan (c+d x)+140 b^2 \tan ^8(c+d x)\right)}{1260 d}","\frac{\left(a^2+3 b^2\right) \tan ^7(c+d x)}{7 d}+\frac{3 \left(a^2+b^2\right) \tan ^5(c+d x)}{5 d}+\frac{\left(3 a^2+b^2\right) \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{a b \sec ^8(c+d x)}{4 d}+\frac{b^2 \tan ^9(c+d x)}{9 d}",1,"(Tan[c + d*x]*(1260*a^2 + 1260*a*b*Tan[c + d*x] + 420*(3*a^2 + b^2)*Tan[c + d*x]^2 + 1890*a*b*Tan[c + d*x]^3 + 756*(a^2 + b^2)*Tan[c + d*x]^4 + 1260*a*b*Tan[c + d*x]^5 + 180*(a^2 + 3*b^2)*Tan[c + d*x]^6 + 315*a*b*Tan[c + d*x]^7 + 140*b^2*Tan[c + d*x]^8))/(1260*d)","A",1
518,1,104,97,0.6533399,"\int \sec ^6(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^6*(a + b*Tan[c + d*x])^2,x]","\frac{\tan (c+d x) \left(21 \left(a^2+2 b^2\right) \tan ^4(c+d x)+35 \left(2 a^2+b^2\right) \tan ^2(c+d x)+105 a^2+35 a b \tan ^5(c+d x)+105 a b \tan ^3(c+d x)+105 a b \tan (c+d x)+15 b^2 \tan ^6(c+d x)\right)}{105 d}","\frac{\left(a^2+2 b^2\right) \tan ^5(c+d x)}{5 d}+\frac{\left(2 a^2+b^2\right) \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{a b \sec ^6(c+d x)}{3 d}+\frac{b^2 \tan ^7(c+d x)}{7 d}",1,"(Tan[c + d*x]*(105*a^2 + 105*a*b*Tan[c + d*x] + 35*(2*a^2 + b^2)*Tan[c + d*x]^2 + 105*a*b*Tan[c + d*x]^3 + 21*(a^2 + 2*b^2)*Tan[c + d*x]^4 + 35*a*b*Tan[c + d*x]^5 + 15*b^2*Tan[c + d*x]^6))/(105*d)","A",1
519,1,54,75,0.1872453,"\int \sec ^4(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^4*(a + b*Tan[c + d*x])^2,x]","\frac{(a+b \tan (c+d x))^3 \left(a^2-3 a b \tan (c+d x)+6 b^2 \tan ^2(c+d x)+10 b^2\right)}{30 b^3 d}","\frac{\left(a^2+b^2\right) (a+b \tan (c+d x))^3}{3 b^3 d}+\frac{(a+b \tan (c+d x))^5}{5 b^3 d}-\frac{a (a+b \tan (c+d x))^4}{2 b^3 d}",1,"((a + b*Tan[c + d*x])^3*(a^2 + 10*b^2 - 3*a*b*Tan[c + d*x] + 6*b^2*Tan[c + d*x]^2))/(30*b^3*d)","A",1
520,1,46,22,0.0424808,"\int \sec ^2(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^2*(a + b*Tan[c + d*x])^2,x]","\frac{a^2 \tan (c+d x)}{d}+\frac{a b \tan ^2(c+d x)}{d}+\frac{b^2 \tan ^3(c+d x)}{3 d}","\frac{(a+b \tan (c+d x))^3}{3 b d}",1,"(a^2*Tan[c + d*x])/d + (a*b*Tan[c + d*x]^2)/d + (b^2*Tan[c + d*x]^3)/(3*d)","B",1
521,1,52,49,0.1255224,"\int \cos ^2(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*(a + b*Tan[c + d*x])^2,x]","\frac{2 \left(a^2+b^2\right) (c+d x)+\left(a^2-b^2\right) \sin (2 (c+d x))-2 a b \cos (2 (c+d x))}{4 d}","\frac{1}{2} x \left(a^2+b^2\right)-\frac{\cos ^2(c+d x) (b-a \tan (c+d x)) (a+b \tan (c+d x))}{2 d}",1,"(2*(a^2 + b^2)*(c + d*x) - 2*a*b*Cos[2*(c + d*x)] + (a^2 - b^2)*Sin[2*(c + d*x)])/(4*d)","A",1
522,1,216,88,2.9769123,"\int \cos ^4(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*(a + b*Tan[c + d*x])^2,x]","\frac{4 \left(a^2+b^2\right) \cos ^4(c+d x) (a \tan (c+d x)+b) (a+b \tan (c+d x))^3+\frac{\left(3 a^2+b^2\right) \left(-\sqrt{-b^2} \left(b^4-a^4\right) \sin (2 (c+d x))-2 a b \sqrt{-b^2} \left(a^2+b^2\right) \cos (2 (c+d x))+b \left(a^2+b^2\right)^2 \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)-b \left(a^2+b^2\right)^2 \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+2 a b \sqrt{-b^2} \left(2 a^2+b^2\right)\right)}{\sqrt{-b^2}}}{16 d \left(a^2+b^2\right)^2}","-\frac{\cos ^2(c+d x) \left(2 a b-\left(3 a^2+b^2\right) \tan (c+d x)\right)}{8 d}+\frac{1}{8} x \left(3 a^2+b^2\right)-\frac{\cos ^4(c+d x) (b-a \tan (c+d x)) (a+b \tan (c+d x))}{4 d}",1,"(((3*a^2 + b^2)*(2*a*b*Sqrt[-b^2]*(2*a^2 + b^2) - 2*a*b*Sqrt[-b^2]*(a^2 + b^2)*Cos[2*(c + d*x)] + b*(a^2 + b^2)^2*Log[Sqrt[-b^2] - b*Tan[c + d*x]] - b*(a^2 + b^2)^2*Log[Sqrt[-b^2] + b*Tan[c + d*x]] - Sqrt[-b^2]*(-a^4 + b^4)*Sin[2*(c + d*x)]))/Sqrt[-b^2] + 4*(a^2 + b^2)*Cos[c + d*x]^4*(b + a*Tan[c + d*x])*(a + b*Tan[c + d*x])^3)/(16*(a^2 + b^2)^2*d)","B",1
523,1,131,163,0.7906381,"\int \sec ^7(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^7*(a + b*Tan[c + d*x])^2,x]","\frac{105 \left(8 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))+56 \left(8 a^2-b^2\right) \tan (c+d x) \sec ^5(c+d x)+70 \left(8 a^2-b^2\right) \tan (c+d x) \sec ^3(c+d x)+105 \left(8 a^2-b^2\right) \tan (c+d x) \sec (c+d x)+48 b \sec ^7(c+d x) (16 a+7 b \tan (c+d x))}{2688 d}","\frac{5 \left(8 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{\left(8 a^2-b^2\right) \tan (c+d x) \sec ^5(c+d x)}{48 d}+\frac{5 \left(8 a^2-b^2\right) \tan (c+d x) \sec ^3(c+d x)}{192 d}+\frac{5 \left(8 a^2-b^2\right) \tan (c+d x) \sec (c+d x)}{128 d}+\frac{9 a b \sec ^7(c+d x)}{56 d}+\frac{b \sec ^7(c+d x) (a+b \tan (c+d x))}{8 d}",1,"(105*(8*a^2 - b^2)*ArcTanh[Sin[c + d*x]] + 105*(8*a^2 - b^2)*Sec[c + d*x]*Tan[c + d*x] + 70*(8*a^2 - b^2)*Sec[c + d*x]^3*Tan[c + d*x] + 56*(8*a^2 - b^2)*Sec[c + d*x]^5*Tan[c + d*x] + 48*b*Sec[c + d*x]^7*(16*a + 7*b*Tan[c + d*x]))/(2688*d)","A",1
524,1,104,131,0.5309734,"\int \sec ^5(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^5*(a + b*Tan[c + d*x])^2,x]","\frac{15 \left(6 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))+10 \left(6 a^2-b^2\right) \tan (c+d x) \sec ^3(c+d x)+15 \left(6 a^2-b^2\right) \tan (c+d x) \sec (c+d x)+8 b \sec ^5(c+d x) (12 a+5 b \tan (c+d x))}{240 d}","\frac{\left(6 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left(6 a^2-b^2\right) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{\left(6 a^2-b^2\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{7 a b \sec ^5(c+d x)}{30 d}+\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))}{6 d}",1,"(15*(6*a^2 - b^2)*ArcTanh[Sin[c + d*x]] + 15*(6*a^2 - b^2)*Sec[c + d*x]*Tan[c + d*x] + 10*(6*a^2 - b^2)*Sec[c + d*x]^3*Tan[c + d*x] + 8*b*Sec[c + d*x]^5*(12*a + 5*b*Tan[c + d*x]))/(240*d)","A",1
525,1,120,99,0.0596758,"\int \sec ^3(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^3*(a + b*Tan[c + d*x])^2,x]","\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 d}+\frac{2 a b \sec ^3(c+d x)}{3 d}-\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{b^2 \tan (c+d x) \sec (c+d x)}{8 d}","\frac{\left(4 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(4 a^2-b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{5 a b \sec ^3(c+d x)}{12 d}+\frac{b \sec ^3(c+d x) (a+b \tan (c+d x))}{4 d}",1,"(a^2*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a*b*Sec[c + d*x]^3)/(3*d) + (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) - (b^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",1
526,1,67,65,0.0383288,"\int \sec (c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]*(a + b*Tan[c + d*x])^2,x]","\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b \sec (c+d x)}{d}-\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^2 \tan (c+d x) \sec (c+d x)}{2 d}","\frac{\left(2 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{3 a b \sec (c+d x)}{2 d}+\frac{b \sec (c+d x) (a+b \tan (c+d x))}{2 d}",1,"(a^2*ArcTanh[Sin[c + d*x]])/d - (b^2*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a*b*Sec[c + d*x])/d + (b^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
527,1,84,47,0.1371317,"\int \cos (c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*(a + b*Tan[c + d*x])^2,x]","\frac{\left(a^2-b^2\right) \sin (c+d x)-2 a b \cos (c+d x)+b^2 \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d}","\frac{\left(a^2-b^2\right) \sin (c+d x)}{d}-\frac{2 a b \cos (c+d x)}{d}+\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(-2*a*b*Cos[c + d*x] + b^2*(-Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (a^2 - b^2)*Sin[c + d*x])/d","A",1
528,1,64,90,0.4629935,"\int \cos ^3(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^3*(a + b*Tan[c + d*x])^2,x]","\frac{\sin (c+d x) \left(\left(a^2-b^2\right) \cos (2 (c+d x))+5 a^2+b^2\right)-3 a b \cos (c+d x)-a b \cos (3 (c+d x))}{6 d}","-\frac{\left(2 a^2+b^2\right) \sin ^3(c+d x)}{6 d}+\frac{\left(2 a^2+b^2\right) \sin (c+d x)}{2 d}-\frac{a b \cos ^3(c+d x)}{6 d}-\frac{b \cos ^3(c+d x) (a+b \tan (c+d x))}{2 d}",1,"(-3*a*b*Cos[c + d*x] - a*b*Cos[3*(c + d*x)] + (5*a^2 + b^2 + (a^2 - b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])/(6*d)","A",1
529,1,116,114,0.219265,"\int \cos ^5(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^5*(a + b*Tan[c + d*x])^2,x]","\frac{150 a^2 \sin (c+d x)+25 a^2 \sin (3 (c+d x))+3 a^2 \sin (5 (c+d x))-60 a b \cos (c+d x)-30 a b \cos (3 (c+d x))-6 a b \cos (5 (c+d x))+30 b^2 \sin (c+d x)-5 b^2 \sin (3 (c+d x))-3 b^2 \sin (5 (c+d x))}{240 d}","\frac{\left(4 a^2+b^2\right) \sin ^5(c+d x)}{20 d}-\frac{\left(4 a^2+b^2\right) \sin ^3(c+d x)}{6 d}+\frac{\left(4 a^2+b^2\right) \sin (c+d x)}{4 d}-\frac{3 a b \cos ^5(c+d x)}{20 d}-\frac{b \cos ^5(c+d x) (a+b \tan (c+d x))}{4 d}",1,"(-60*a*b*Cos[c + d*x] - 30*a*b*Cos[3*(c + d*x)] - 6*a*b*Cos[5*(c + d*x)] + 150*a^2*Sin[c + d*x] + 30*b^2*Sin[c + d*x] + 25*a^2*Sin[3*(c + d*x)] - 5*b^2*Sin[3*(c + d*x)] + 3*a^2*Sin[5*(c + d*x)] - 3*b^2*Sin[5*(c + d*x)])/(240*d)","A",1
530,1,154,138,0.4368387,"\int \cos ^7(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^7*(a + b*Tan[c + d*x])^2,x]","-\frac{-3675 a^2 \sin (c+d x)-735 a^2 \sin (3 (c+d x))-147 a^2 \sin (5 (c+d x))-15 a^2 \sin (7 (c+d x))+1050 a b \cos (c+d x)+630 a b \cos (3 (c+d x))+210 a b \cos (5 (c+d x))+30 a b \cos (7 (c+d x))-525 b^2 \sin (c+d x)+35 b^2 \sin (3 (c+d x))+63 b^2 \sin (5 (c+d x))+15 b^2 \sin (7 (c+d x))}{6720 d}","-\frac{\left(6 a^2+b^2\right) \sin ^7(c+d x)}{42 d}+\frac{\left(6 a^2+b^2\right) \sin ^5(c+d x)}{10 d}-\frac{\left(6 a^2+b^2\right) \sin ^3(c+d x)}{6 d}+\frac{\left(6 a^2+b^2\right) \sin (c+d x)}{6 d}-\frac{5 a b \cos ^7(c+d x)}{42 d}-\frac{b \cos ^7(c+d x) (a+b \tan (c+d x))}{6 d}",1,"-1/6720*(1050*a*b*Cos[c + d*x] + 630*a*b*Cos[3*(c + d*x)] + 210*a*b*Cos[5*(c + d*x)] + 30*a*b*Cos[7*(c + d*x)] - 3675*a^2*Sin[c + d*x] - 525*b^2*Sin[c + d*x] - 735*a^2*Sin[3*(c + d*x)] + 35*b^2*Sin[3*(c + d*x)] - 147*a^2*Sin[5*(c + d*x)] + 63*b^2*Sin[5*(c + d*x)] - 15*a^2*Sin[7*(c + d*x)] + 15*b^2*Sin[7*(c + d*x)])/d","A",1
531,1,177,194,2.0378499,"\int \sec ^8(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^8*(a + b*Tan[c + d*x])^3,x]","\frac{\frac{3}{8} \left(5 a^2+b^2\right) (a+b \tan (c+d x))^8-\frac{4}{7} a \left(5 a^2+3 b^2\right) (a+b \tan (c+d x))^7+\frac{1}{2} \left(a^2+b^2\right) \left(5 a^2+b^2\right) (a+b \tan (c+d x))^6-\frac{6}{5} a \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^5+\frac{1}{4} \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^4+\frac{1}{10} (a+b \tan (c+d x))^{10}-\frac{2}{3} a (a+b \tan (c+d x))^9}{b^7 d}","\frac{a^3 \tan (c+d x)}{d}+\frac{a \left(a^2+9 b^2\right) \tan ^7(c+d x)}{7 d}+\frac{3 a \left(a^2+3 b^2\right) \tan ^5(c+d x)}{5 d}+\frac{a \left(a^2+b^2\right) \tan ^3(c+d x)}{d}+\frac{3 a^2 b \sec ^8(c+d x)}{8 d}+\frac{a b^2 \tan ^9(c+d x)}{3 d}+\frac{b^3 \tan ^{10}(c+d x)}{10 d}+\frac{3 b^3 \tan ^8(c+d x)}{8 d}+\frac{b^3 \tan ^6(c+d x)}{2 d}+\frac{b^3 \tan ^4(c+d x)}{4 d}",1,"(((a^2 + b^2)^3*(a + b*Tan[c + d*x])^4)/4 - (6*a*(a^2 + b^2)^2*(a + b*Tan[c + d*x])^5)/5 + ((a^2 + b^2)*(5*a^2 + b^2)*(a + b*Tan[c + d*x])^6)/2 - (4*a*(5*a^2 + 3*b^2)*(a + b*Tan[c + d*x])^7)/7 + (3*(5*a^2 + b^2)*(a + b*Tan[c + d*x])^8)/8 - (2*a*(a + b*Tan[c + d*x])^9)/3 + (a + b*Tan[c + d*x])^10/10)/(b^7*d)","A",1
532,1,115,138,0.5703696,"\int \sec ^6(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^6*(a + b*Tan[c + d*x])^3,x]","\frac{\frac{1}{3} \left(3 a^2+b^2\right) (a+b \tan (c+d x))^6-\frac{4}{5} a \left(a^2+b^2\right) (a+b \tan (c+d x))^5+\frac{1}{4} \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^4+\frac{1}{8} (a+b \tan (c+d x))^8-\frac{4}{7} a (a+b \tan (c+d x))^7}{b^5 d}","\frac{\left(3 a^2+b^2\right) (a+b \tan (c+d x))^6}{3 b^5 d}-\frac{4 a \left(a^2+b^2\right) (a+b \tan (c+d x))^5}{5 b^5 d}+\frac{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))^4}{4 b^5 d}+\frac{(a+b \tan (c+d x))^8}{8 b^5 d}-\frac{4 a (a+b \tan (c+d x))^7}{7 b^5 d}",1,"(((a^2 + b^2)^2*(a + b*Tan[c + d*x])^4)/4 - (4*a*(a^2 + b^2)*(a + b*Tan[c + d*x])^5)/5 + ((3*a^2 + b^2)*(a + b*Tan[c + d*x])^6)/3 - (4*a*(a + b*Tan[c + d*x])^7)/7 + (a + b*Tan[c + d*x])^8/8)/(b^5*d)","A",1
533,1,54,75,0.3355689,"\int \sec ^4(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^4*(a + b*Tan[c + d*x])^3,x]","\frac{(a+b \tan (c+d x))^4 \left(a^2-4 a b \tan (c+d x)+10 b^2 \tan ^2(c+d x)+15 b^2\right)}{60 b^3 d}","\frac{\left(a^2+b^2\right) (a+b \tan (c+d x))^4}{4 b^3 d}+\frac{(a+b \tan (c+d x))^6}{6 b^3 d}-\frac{2 a (a+b \tan (c+d x))^5}{5 b^3 d}",1,"((a + b*Tan[c + d*x])^4*(a^2 + 15*b^2 - 4*a*b*Tan[c + d*x] + 10*b^2*Tan[c + d*x]^2))/(60*b^3*d)","A",1
534,1,57,22,0.1578969,"\int \sec ^2(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^2*(a + b*Tan[c + d*x])^3,x]","\frac{\tan (c+d x) \left(4 a^3+6 a^2 b \tan (c+d x)+4 a b^2 \tan ^2(c+d x)+b^3 \tan ^3(c+d x)\right)}{4 d}","\frac{(a+b \tan (c+d x))^4}{4 b d}",1,"(Tan[c + d*x]*(4*a^3 + 6*a^2*b*Tan[c + d*x] + 4*a*b^2*Tan[c + d*x]^2 + b^3*Tan[c + d*x]^3))/(4*d)","B",1
535,1,401,86,0.777356,"\int \cos ^2(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*(a + b*Tan[c + d*x])^3,x]","\frac{-a^5 \sqrt{-b^2} \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+a^5 \sqrt{-b^2} \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+5 a^4 b^2+4 a^3 \left(-b^2\right)^{3/2} \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)-4 a^3 \left(-b^2\right)^{3/2} \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+2 a^2 b^4+2 a^2 b^4 \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+2 a^2 b^4 \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+a b \left(a^4-2 a^2 b^2-3 b^4\right) \sin (2 (c+d x))+\left(-3 a^4 b^2-2 a^2 b^4+b^6\right) \cos (2 (c+d x))-3 a \left(-b^2\right)^{5/2} \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+3 a \sqrt{-b^2} b^4 \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)-b^6+2 b^6 \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+2 b^6 \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)}{4 b d \left(a^2+b^2\right)}","\frac{1}{2} a x \left(a^2+3 b^2\right)-\frac{a b^2 \tan (c+d x)}{2 d}-\frac{\cos ^2(c+d x) (b-a \tan (c+d x)) (a+b \tan (c+d x))^2}{2 d}-\frac{b^3 \log (\cos (c+d x))}{d}",1,"(5*a^4*b^2 + 2*a^2*b^4 - b^6 + (-3*a^4*b^2 - 2*a^2*b^4 + b^6)*Cos[2*(c + d*x)] + 2*a^2*b^4*Log[Sqrt[-b^2] - b*Tan[c + d*x]] + 2*b^6*Log[Sqrt[-b^2] - b*Tan[c + d*x]] - a^5*Sqrt[-b^2]*Log[Sqrt[-b^2] - b*Tan[c + d*x]] + 4*a^3*(-b^2)^(3/2)*Log[Sqrt[-b^2] - b*Tan[c + d*x]] - 3*a*(-b^2)^(5/2)*Log[Sqrt[-b^2] - b*Tan[c + d*x]] + 2*a^2*b^4*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + 2*b^6*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + a^5*Sqrt[-b^2]*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + 3*a*b^4*Sqrt[-b^2]*Log[Sqrt[-b^2] + b*Tan[c + d*x]] - 4*a^3*(-b^2)^(3/2)*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + a*b*(a^4 - 2*a^2*b^2 - 3*b^4)*Sin[2*(c + d*x)])/(4*b*(a^2 + b^2)*d)","B",1
536,1,257,84,3.5805055,"\int \cos ^4(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*(a + b*Tan[c + d*x])^3,x]","\frac{-24 a^4 b^4 \tan ^2(c+d x)-3 a \sqrt{-b^2} \left(a^2+b^2\right)^3 \left(\log \left(\sqrt{-b^2}-b \tan (c+d x)\right)-\log \left(\sqrt{-b^2}+b \tan (c+d x)\right)\right)+8 a^2 b^2 \cos ^2(c+d x) (a+b \tan (c+d x))^4+2 a b \left(3 a^2-b^2\right) \sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))^4+4 b \left(a^2+b^2\right) (a \tan (c+d x)+b) (a \cos (c+d x)+b \sin (c+d x))^4+2 a b^5 \left(b^2-3 a^2\right) \tan ^3(c+d x)-6 a b^3 \left(6 a^4+3 a^2 b^2+b^4\right) \tan (c+d x)}{16 b d \left(a^2+b^2\right)^2}","\frac{3}{8} a x \left(a^2+b^2\right)-\frac{3 a \cos ^2(c+d x) (b-a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}+\frac{\sin (c+d x) \cos ^3(c+d x) (a+b \tan (c+d x))^3}{4 d}",1,"(-3*a*Sqrt[-b^2]*(a^2 + b^2)^3*(Log[Sqrt[-b^2] - b*Tan[c + d*x]] - Log[Sqrt[-b^2] + b*Tan[c + d*x]]) - 6*a*b^3*(6*a^4 + 3*a^2*b^2 + b^4)*Tan[c + d*x] - 24*a^4*b^4*Tan[c + d*x]^2 + 2*a*b^5*(-3*a^2 + b^2)*Tan[c + d*x]^3 + 4*b*(a^2 + b^2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^4*(b + a*Tan[c + d*x]) + 8*a^2*b^2*Cos[c + d*x]^2*(a + b*Tan[c + d*x])^4 + 2*a*b*(3*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]*(a + b*Tan[c + d*x])^4)/(16*b*(a^2 + b^2)^2*d)","B",1
537,1,637,159,2.1920141,"\int \sec ^5(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^5*(a + b*Tan[c + d*x])^3,x]","\frac{\sec ^7(c+d x) \left(4340 a^3 \sin (2 (c+d x))+2800 a^3 \sin (4 (c+d x))+420 a^3 \sin (6 (c+d x))-4410 a^3 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-1470 a^3 \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)-210 a^3 \cos (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4410 a^3 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+1470 a^3 \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)+210 a^3 \cos (7 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+3584 \left(3 a^2 b-b^3\right) \cos (2 (c+d x))-3675 a \left(2 a^2-b^2\right) \cos (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)+10752 a^2 b+6790 a b^2 \sin (2 (c+d x))-1400 a b^2 \sin (4 (c+d x))-210 a b^2 \sin (6 (c+d x))+2205 a b^2 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+735 a b^2 \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)+105 a b^2 \cos (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2205 a b^2 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-735 a b^2 \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)-105 a b^2 \cos (7 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+1536 b^3\right)}{35840 d}","\frac{3 a \left(2 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{b \sec ^5(c+d x) \left(4 \left(8 a^2-b^2\right)+15 a b \tan (c+d x)\right)}{70 d}+\frac{a \left(2 a^2-b^2\right) \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac{3 a \left(2 a^2-b^2\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}",1,"(Sec[c + d*x]^7*(10752*a^2*b + 1536*b^3 + 3584*(3*a^2*b - b^3)*Cos[2*(c + d*x)] - 4410*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2205*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 1470*a^3*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 735*a*b^2*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 210*a^3*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 105*a*b^2*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 3675*a*(2*a^2 - b^2)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 4410*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2205*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 1470*a^3*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 735*a*b^2*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 210*a^3*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 105*a*b^2*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4340*a^3*Sin[2*(c + d*x)] + 6790*a*b^2*Sin[2*(c + d*x)] + 2800*a^3*Sin[4*(c + d*x)] - 1400*a*b^2*Sin[4*(c + d*x)] + 420*a^3*Sin[6*(c + d*x)] - 210*a*b^2*Sin[6*(c + d*x)]))/(35840*d)","B",1
538,1,464,126,1.3048568,"\int \sec ^3(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^3*(a + b*Tan[c + d*x])^3,x]","\frac{\sec ^5(c+d x) \left(240 a^3 \sin (2 (c+d x))+120 a^3 \sin (4 (c+d x))-300 a^3 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-60 a^3 \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)+300 a^3 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+60 a^3 \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)+320 \left(3 a^2 b-b^3\right) \cos (2 (c+d x))-150 a \left(4 a^2-3 b^2\right) \cos (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)+960 a^2 b+540 a b^2 \sin (2 (c+d x))-90 a b^2 \sin (4 (c+d x))+225 a b^2 \cos (3 (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^2 \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)-225 a b^2 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-45 a b^2 \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)+64 b^3\right)}{1920 d}","\frac{a \left(4 a^2-3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \sec ^3(c+d x) \left(8 \left(6 a^2-b^2\right)+21 a b \tan (c+d x)\right)}{60 d}+\frac{a \left(4 a^2-3 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \sec ^3(c+d x) (a+b \tan (c+d x))^2}{5 d}",1,"(Sec[c + d*x]^5*(960*a^2*b + 64*b^3 + 320*(3*a^2*b - b^3)*Cos[2*(c + d*x)] - 300*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 225*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 60*a^3*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 45*a*b^2*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 150*a*(4*a^2 - 3*b^2)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 300*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 225*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 60*a^3*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 45*a*b^2*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 240*a^3*Sin[2*(c + d*x)] + 540*a*b^2*Sin[2*(c + d*x)] + 120*a^3*Sin[4*(c + d*x)] - 90*a*b^2*Sin[4*(c + d*x)]))/(1920*d)","B",1
539,1,293,91,1.5905616,"\int \sec (c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]*(a + b*Tan[c + d*x])^3,x]","\frac{12 a^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-6 a \left(2 a^2-3 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 b \sin ^2\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(\left(18 a^2-5 b^2\right) \cos (2 (c+d x))+18 a^2+2 b^2 \cos (c+d x)-b^2\right)+36 a^2 b+\frac{9 a b^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{9 a b^2}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-18 a b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{b^3}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{b^3}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-10 b^3}{12 d}","\frac{a \left(2 a^2-3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b \sec (c+d x) \left(4 \left(4 a^2-b^2\right)+5 a b \tan (c+d x)\right)}{6 d}+\frac{b \sec (c+d x) (a+b \tan (c+d x))^2}{3 d}",1,"(36*a^2*b - 10*b^3 - 6*a*(2*a^2 - 3*b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*a^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 18*a*b^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (9*a*b^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + b^3/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + 2*b*(18*a^2 - b^2 + 2*b^2*Cos[c + d*x] + (18*a^2 - 5*b^2)*Cos[2*(c + d*x)])*Sec[c + d*x]^3*Sin[(c + d*x)/2]^2 - (9*a*b^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + b^3/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(12*d)","B",1
540,1,131,84,1.0531554,"\int \cos (c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*(a + b*Tan[c + d*x])^3,x]","\frac{\sec (c+d x) \left(a^3 \sin (2 (c+d x))+\left(b^3-3 a^2 b\right) \cos (2 (c+d x))-3 a^2 b-3 a b^2 \sin (2 (c+d x))-6 a b^2 \cos (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)+3 b^3\right)}{2 d}","-\frac{b \sec (c+d x) \left(2 \left(a^2-b^2\right)+a b \tan (c+d x)\right)}{d}+\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{\cos (c+d x) (b-a \tan (c+d x)) (a+b \tan (c+d x))^2}{d}",1,"(Sec[c + d*x]*(-3*a^2*b + 3*b^3 + (-3*a^2*b + b^3)*Cos[2*(c + d*x)] - 6*a*b^2*Cos[c + 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^3*Sin[2*(c + d*x)] - 3*a*b^2*Sin[2*(c + d*x)]))/(2*d)","A",1
541,1,81,70,0.3751592,"\int \cos ^3(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^3*(a + b*Tan[c + d*x])^3,x]","\frac{\left(b^3-3 a^2 b\right) \cos (3 (c+d x))-9 b \left(a^2+b^2\right) \cos (c+d x)+2 a \sin (c+d x) \left(\left(a^2-3 b^2\right) \cos (2 (c+d x))+5 a^2+3 b^2\right)}{12 d}","-\frac{2 \left(a^2+b^2\right) \cos (c+d x) (b-a \tan (c+d x))}{3 d}-\frac{\cos ^3(c+d x) (b-a \tan (c+d x)) (a+b \tan (c+d x))^2}{3 d}",1,"(-9*b*(a^2 + b^2)*Cos[c + d*x] + (-3*a^2*b + b^3)*Cos[3*(c + d*x)] + 2*a*(5*a^2 + 3*b^2 + (a^2 - 3*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])/(12*d)","A",1
542,1,150,105,0.706685,"\int \cos ^5(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^5*(a + b*Tan[c + d*x])^3,x]","\frac{150 a^3 \sin (c+d x)+25 a^3 \sin (3 (c+d x))+3 a^3 \sin (5 (c+d x))-5 \left(9 a^2 b+b^3\right) \cos (3 (c+d x))-30 b \left(3 a^2+b^2\right) \cos (c+d x)-9 a^2 b \cos (5 (c+d x))+90 a b^2 \sin (c+d x)-15 a b^2 \sin (3 (c+d x))-9 a b^2 \sin (5 (c+d x))+3 b^3 \cos (5 (c+d x))}{240 d}","-\frac{2 \left(4 a^2+b^2\right) \cos (c+d x) (b-a \tan (c+d x))}{15 d}-\frac{\cos ^3(c+d x) (b-4 a \tan (c+d x)) (a+b \tan (c+d x))^2}{15 d}+\frac{\sin (c+d x) \cos ^4(c+d x) (a+b \tan (c+d x))^3}{5 d}",1,"(-30*b*(3*a^2 + b^2)*Cos[c + d*x] - 5*(9*a^2*b + b^3)*Cos[3*(c + d*x)] - 9*a^2*b*Cos[5*(c + d*x)] + 3*b^3*Cos[5*(c + d*x)] + 150*a^3*Sin[c + d*x] + 90*a*b^2*Sin[c + d*x] + 25*a^3*Sin[3*(c + d*x)] - 15*a*b^2*Sin[3*(c + d*x)] + 3*a^3*Sin[5*(c + d*x)] - 9*a*b^2*Sin[5*(c + d*x)])/(240*d)","A",1
543,1,204,142,1.0684843,"\int \cos ^7(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^7*(a + b*Tan[c + d*x])^3,x]","\frac{1225 a^3 \sin (c+d x)+245 a^3 \sin (3 (c+d x))+49 a^3 \sin (5 (c+d x))+5 a^3 \sin (7 (c+d x))-35 \left(9 a^2 b+b^3\right) \cos (3 (c+d x))-105 b \left(5 a^2+b^2\right) \cos (c+d x)-105 a^2 b \cos (5 (c+d x))-15 a^2 b \cos (7 (c+d x))+525 a b^2 \sin (c+d x)-35 a b^2 \sin (3 (c+d x))-63 a b^2 \sin (5 (c+d x))-15 a b^2 \sin (7 (c+d x))+7 b^3 \cos (5 (c+d x))+5 b^3 \cos (7 (c+d x))}{2240 d}","\frac{8 a \left(2 a^2+b^2\right) \sin (c+d x)}{35 d}-\frac{2 \cos ^3(c+d x) \left(b \left(6 a^2+b^2\right)-a \left(4 a^2-b^2\right) \tan (c+d x)\right)}{35 d}-\frac{3 \cos ^5(c+d x) (b-2 a \tan (c+d x)) (a+b \tan (c+d x))^2}{35 d}+\frac{\sin (c+d x) \cos ^6(c+d x) (a+b \tan (c+d x))^3}{7 d}",1,"(-105*b*(5*a^2 + b^2)*Cos[c + d*x] - 35*(9*a^2*b + b^3)*Cos[3*(c + d*x)] - 105*a^2*b*Cos[5*(c + d*x)] + 7*b^3*Cos[5*(c + d*x)] - 15*a^2*b*Cos[7*(c + d*x)] + 5*b^3*Cos[7*(c + d*x)] + 1225*a^3*Sin[c + d*x] + 525*a*b^2*Sin[c + d*x] + 245*a^3*Sin[3*(c + d*x)] - 35*a*b^2*Sin[3*(c + d*x)] + 49*a^3*Sin[5*(c + d*x)] - 63*a*b^2*Sin[5*(c + d*x)] + 5*a^3*Sin[7*(c + d*x)] - 15*a*b^2*Sin[7*(c + d*x)])/(2240*d)","A",1
544,1,99,116,1.1792899,"\int \frac{\sec ^6(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^6/(a + b*Tan[c + d*x]),x]","\frac{6 b^2 \left(a^2+b^2\right) \tan ^2(c+d x)-12 a b \left(a^2+2 b^2\right) \tan (c+d x)+12 \left(a^2+b^2\right)^2 \log (a+b \tan (c+d x))-4 a b^3 \tan ^3(c+d x)+3 b^4 \sec ^4(c+d x)}{12 b^5 d}","\frac{\left(a^2+b^2\right)^2 \log (a+b \tan (c+d x))}{b^5 d}-\frac{a \left(a^2+2 b^2\right) \tan (c+d x)}{b^4 d}+\frac{\left(a^2+2 b^2\right) \tan ^2(c+d x)}{2 b^3 d}-\frac{a \tan ^3(c+d x)}{3 b^2 d}+\frac{\tan ^4(c+d x)}{4 b d}",1,"(12*(a^2 + b^2)^2*Log[a + b*Tan[c + d*x]] + 3*b^4*Sec[c + d*x]^4 - 12*a*b*(a^2 + 2*b^2)*Tan[c + d*x] + 6*b^2*(a^2 + b^2)*Tan[c + d*x]^2 - 4*a*b^3*Tan[c + d*x]^3)/(12*b^5*d)","A",1
545,1,52,59,0.1388163,"\int \frac{\sec ^4(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Tan[c + d*x]),x]","\frac{\left(a^2+b^2\right) \log (a+b \tan (c+d x))-a b \tan (c+d x)+\frac{1}{2} b^2 \tan ^2(c+d x)}{b^3 d}","\frac{\left(a^2+b^2\right) \log (a+b \tan (c+d x))}{b^3 d}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\tan ^2(c+d x)}{2 b d}",1,"((a^2 + b^2)*Log[a + b*Tan[c + d*x]] - a*b*Tan[c + d*x] + (b^2*Tan[c + d*x]^2)/2)/(b^3*d)","A",1
546,1,18,18,0.0174589,"\int \frac{\sec ^2(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Tan[c + d*x]),x]","\frac{\log (a+b \tan (c+d x))}{b d}","\frac{\log (a+b \tan (c+d x))}{b d}",1,"Log[a + b*Tan[c + d*x]]/(b*d)","A",1
547,1,143,93,0.2351944,"\int \frac{\cos ^2(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Tan[c + d*x]),x]","\frac{a^3 \sin (2 (c+d x))+2 a^3 c+2 a^3 d x+b \left(a^2+b^2\right) \cos (2 (c+d x))+2 b^3 \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)+a b^2 \sin (2 (c+d x))+6 a b^2 c+6 a b^2 d x-4 i b^3 \tan ^{-1}(\tan (c+d x))+4 i b^3 c+4 i b^3 d x}{4 d \left(a^2+b^2\right)^2}","\frac{\cos ^2(c+d x) (a \tan (c+d x)+b)}{2 d \left(a^2+b^2\right)}+\frac{a x \left(a^2+3 b^2\right)}{2 \left(a^2+b^2\right)^2}+\frac{b^3 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^2}",1,"(2*a^3*c + 6*a*b^2*c + (4*I)*b^3*c + 2*a^3*d*x + 6*a*b^2*d*x + (4*I)*b^3*d*x - (4*I)*b^3*ArcTan[Tan[c + d*x]] + b*(a^2 + b^2)*Cos[2*(c + d*x)] + 2*b^3*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2] + a^3*Sin[2*(c + d*x)] + a*b^2*Sin[2*(c + d*x)])/(4*(a^2 + b^2)^2*d)","C",1
548,1,218,152,0.4058051,"\int \frac{\cos ^4(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Tan[c + d*x]),x]","\frac{8 a^5 \sin (2 (c+d x))+a^5 \sin (4 (c+d x))+12 a^5 c+12 a^5 d x+24 a^3 b^2 \sin (2 (c+d x))+2 a^3 b^2 \sin (4 (c+d x))+40 a^3 b^2 c+40 a^3 b^2 d x+b \left(a^2+b^2\right)^2 \cos (4 (c+d x))+4 b \left(a^4+4 a^2 b^2+3 b^4\right) \cos (2 (c+d x))+32 b^5 \log (a \cos (c+d x)+b \sin (c+d x))+16 a b^4 \sin (2 (c+d x))+a b^4 \sin (4 (c+d x))+60 a b^4 c+60 a b^4 d x}{32 d \left(a^2+b^2\right)^3}","\frac{\cos ^4(c+d x) (a \tan (c+d x)+b)}{4 d \left(a^2+b^2\right)}+\frac{b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{\cos ^2(c+d x) \left(a \left(3 a^2+7 b^2\right) \tan (c+d x)+4 b^3\right)}{8 d \left(a^2+b^2\right)^2}+\frac{a x \left(3 a^4+10 a^2 b^2+15 b^4\right)}{8 \left(a^2+b^2\right)^3}",1,"(12*a^5*c + 40*a^3*b^2*c + 60*a*b^4*c + 12*a^5*d*x + 40*a^3*b^2*d*x + 60*a*b^4*d*x + 4*b*(a^4 + 4*a^2*b^2 + 3*b^4)*Cos[2*(c + d*x)] + b*(a^2 + b^2)^2*Cos[4*(c + d*x)] + 32*b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]] + 8*a^5*Sin[2*(c + d*x)] + 24*a^3*b^2*Sin[2*(c + d*x)] + 16*a*b^4*Sin[2*(c + d*x)] + a^5*Sin[4*(c + d*x)] + 2*a^3*b^2*Sin[4*(c + d*x)] + a*b^4*Sin[4*(c + d*x)])/(32*(a^2 + b^2)^3*d)","A",1
549,1,321,140,2.0048403,"\int \frac{\sec ^5(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Tan[c + d*x]),x]","\frac{48 \left(a^2+b^2\right)^{3/2} \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)+\sec ^3(c+d x) \left(6 a^3 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 a^3 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 b \left(a^2+b^2\right) \cos (2 (c+d x))+9 a \left(2 a^2+3 b^2\right) \cos (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)+12 a^2 b-6 a b^2 \sin (2 (c+d x))+9 a b^2 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-9 a b^2 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+20 b^3\right)}{24 b^4 d}","-\frac{a \left(2 a^2+3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{\left(a^2+b^2\right)^{3/2} \tanh ^{-1}\left(\frac{\cos (c+d x) (b-a \tan (c+d x))}{\sqrt{a^2+b^2}}\right)}{b^4 d}+\frac{\left(a^2+b^2\right) \sec (c+d x)}{b^3 d}-\frac{a \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac{\sec ^3(c+d x)}{3 b d}",1,"(48*(a^2 + b^2)^(3/2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]] + Sec[c + d*x]^3*(12*a^2*b + 20*b^3 + 12*b*(a^2 + b^2)*Cos[2*(c + d*x)] + 6*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 9*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 9*a*(2*a^2 + 3*b^2)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 6*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 9*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 6*a*b^2*Sin[2*(c + d*x)]))/(24*b^4*d)","B",1
550,1,109,79,0.1603757,"\int \frac{\sec ^3(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Tan[c + d*x]),x]","\frac{2 \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)+a \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)+b \sec (c+d x)}{b^2 d}","-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{\cos (c+d x) (b-a \tan (c+d x))}{\sqrt{a^2+b^2}}\right)}{b^2 d}-\frac{a \tanh ^{-1}(\sin (c+d x))}{b^2 d}+\frac{\sec (c+d x)}{b d}",1,"(2*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]] + a*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + b*Sec[c + d*x])/(b^2*d)","A",1
551,1,45,46,0.0449064,"\int \frac{\sec (c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]/(a + b*Tan[c + d*x]),x]","\frac{2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{d \sqrt{a^2+b^2}}","-\frac{\tanh ^{-1}\left(\frac{\cos (c+d x) (b-a \tan (c+d x))}{\sqrt{a^2+b^2}}\right)}{d \sqrt{a^2+b^2}}",1,"(2*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(Sqrt[a^2 + b^2]*d)","A",1
552,1,79,90,0.3108393,"\int \frac{\cos (c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]/(a + b*Tan[c + d*x]),x]","\frac{\sqrt{a^2+b^2} (a \sin (c+d x)+b \cos (c+d x))+2 b^2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{3/2}}","\frac{a \sin (c+d x)}{d \left(a^2+b^2\right)}+\frac{b \cos (c+d x)}{d \left(a^2+b^2\right)}-\frac{b^2 \tanh ^{-1}\left(\frac{\cos (c+d x) (b-a \tan (c+d x))}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{3/2}}",1,"(2*b^2*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]] + Sqrt[a^2 + b^2]*(b*Cos[c + d*x] + a*Sin[c + d*x]))/((a^2 + b^2)^(3/2)*d)","A",1
553,1,137,165,1.2349028,"\int \frac{\cos ^3(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Tan[c + d*x]),x]","\frac{\sqrt{a^2+b^2} \left(3 b \left(a^2+5 b^2\right) \cos (c+d x)+b \left(a^2+b^2\right) \cos (3 (c+d x))+2 a \sin (c+d x) \left(\left(a^2+b^2\right) \cos (2 (c+d x))+5 a^2+11 b^2\right)\right)+24 b^4 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{12 d \left(a^2+b^2\right)^{5/2}}","-\frac{a \sin ^3(c+d x)}{3 d \left(a^2+b^2\right)}+\frac{a b^2 \sin (c+d x)}{d \left(a^2+b^2\right)^2}+\frac{a \sin (c+d x)}{d \left(a^2+b^2\right)}+\frac{b \cos ^3(c+d x)}{3 d \left(a^2+b^2\right)}-\frac{b^4 \tanh ^{-1}\left(\frac{\cos (c+d x) (b-a \tan (c+d x))}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{5/2}}+\frac{b^3 \cos (c+d x)}{d \left(a^2+b^2\right)^2}",1,"(24*b^4*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]] + Sqrt[a^2 + b^2]*(3*b*(a^2 + 5*b^2)*Cos[c + d*x] + b*(a^2 + b^2)*Cos[3*(c + d*x)] + 2*a*(5*a^2 + 11*b^2 + (a^2 + b^2)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(12*(a^2 + b^2)^(5/2)*d)","A",1
554,1,229,178,2.3721381,"\int \frac{\sec ^8(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^8/(a + b*Tan[c + d*x])^2,x]","\frac{b^4 \sec ^4(c+d x) \left(a^2-3 a b \tan (c+d x)+4 b^2\right)-2 \left(-2 a^2 b^4 \tan ^4(c+d x)+30 a^2 \left(a^2+b^2\right)^2 \log (a+b \tan (c+d x))+8 \left(a^2+b^2\right)^3+a b^3 \left(5 a^2+7 b^2\right) \tan ^3(c+d x)-b^2 \left(15 a^4+29 a^2 b^2+8 b^4\right) \tan ^2(c+d x)+2 a b \tan (c+d x) \left(-11 a^4+15 \left(a^2+b^2\right)^2 \log (a+b \tan (c+d x))-18 a^2 b^2-4 b^4\right)\right)+2 b^6 \sec ^6(c+d x)}{10 b^7 d (a+b \tan (c+d x))}","-\frac{\left(a^2+b^2\right)^3}{b^7 d (a+b \tan (c+d x))}-\frac{6 a \left(a^2+b^2\right)^2 \log (a+b \tan (c+d x))}{b^7 d}-\frac{a \left(2 a^2+3 b^2\right) \tan ^2(c+d x)}{b^5 d}+\frac{\left(a^2+b^2\right) \tan ^3(c+d x)}{b^4 d}+\frac{\left(5 a^4+9 a^2 b^2+3 b^4\right) \tan (c+d x)}{b^6 d}-\frac{a \tan ^4(c+d x)}{2 b^3 d}+\frac{\tan ^5(c+d x)}{5 b^2 d}",1,"(2*b^6*Sec[c + d*x]^6 + b^4*Sec[c + d*x]^4*(a^2 + 4*b^2 - 3*a*b*Tan[c + d*x]) - 2*(8*(a^2 + b^2)^3 + 30*a^2*(a^2 + b^2)^2*Log[a + b*Tan[c + d*x]] + 2*a*b*(-11*a^4 - 18*a^2*b^2 - 4*b^4 + 15*(a^2 + b^2)^2*Log[a + b*Tan[c + d*x]])*Tan[c + d*x] - b^2*(15*a^4 + 29*a^2*b^2 + 8*b^4)*Tan[c + d*x]^2 + a*b^3*(5*a^2 + 7*b^2)*Tan[c + d*x]^3 - 2*a^2*b^4*Tan[c + d*x]^4))/(10*b^7*d*(a + b*Tan[c + d*x]))","A",1
555,1,122,116,2.7870794,"\int \frac{\sec ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^6/(a + b*Tan[c + d*x])^2,x]","\frac{4 b \left(2 a^2+b^2\right) \tan (c+d x)+\frac{b^4 \sec ^4(c+d x)-4 \left(a^2+b^2\right) \left(3 a^2 \log (a+b \tan (c+d x))+a^2+3 a b \tan (c+d x) \log (a+b \tan (c+d x))+b^2\right)}{a+b \tan (c+d x)}-2 a b^2 \tan ^2(c+d x)}{3 b^5 d}","-\frac{\left(a^2+b^2\right)^2}{b^5 d (a+b \tan (c+d x))}-\frac{4 a \left(a^2+b^2\right) \log (a+b \tan (c+d x))}{b^5 d}+\frac{\left(3 a^2+2 b^2\right) \tan (c+d x)}{b^4 d}-\frac{a \tan ^2(c+d x)}{b^3 d}+\frac{\tan ^3(c+d x)}{3 b^2 d}",1,"(4*b*(2*a^2 + b^2)*Tan[c + d*x] - 2*a*b^2*Tan[c + d*x]^2 + (b^4*Sec[c + d*x]^4 - 4*(a^2 + b^2)*(a^2 + b^2 + 3*a^2*Log[a + b*Tan[c + d*x]] + 3*a*b*Log[a + b*Tan[c + d*x]]*Tan[c + d*x]))/(a + b*Tan[c + d*x]))/(3*b^5*d)","A",1
556,1,51,61,0.0700443,"\int \frac{\sec ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Tan[c + d*x])^2,x]","\frac{-\frac{a^2+b^2}{a+b \tan (c+d x)}-2 a \log (a+b \tan (c+d x))+b \tan (c+d x)}{b^3 d}","-\frac{a^2+b^2}{b^3 d (a+b \tan (c+d x))}-\frac{2 a \log (a+b \tan (c+d x))}{b^3 d}+\frac{\tan (c+d x)}{b^2 d}",1,"(-2*a*Log[a + b*Tan[c + d*x]] + b*Tan[c + d*x] - (a^2 + b^2)/(a + b*Tan[c + d*x]))/(b^3*d)","A",1
557,1,32,20,0.0482678,"\int \frac{\sec ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Tan[c + d*x])^2,x]","\frac{\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))}","-\frac{1}{b d (a+b \tan (c+d x))}",1,"Sin[c + d*x]/(a*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))","A",1
558,1,304,152,3.9417235,"\int \frac{\cos ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Tan[c + d*x])^2,x]","\frac{-\frac{a b \left(\left(\sqrt{-b^2}-a\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)-2 \sqrt{-b^2} \log (a+b \tan (c+d x))+\left(a+\sqrt{-b^2}\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)\right)}{\sqrt{-b^2} \left(a^2+b^2\right)}+\frac{b \left(a^2-3 b^2\right) \left(\frac{2 \left(a^2+b^2\right)}{a+b \tan (c+d x)}+\left(\frac{b^2-a^2}{\sqrt{-b^2}}+2 a\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+\left(\frac{a^2-b^2}{\sqrt{-b^2}}+2 a\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)-4 a \log (a+b \tan (c+d x))\right)}{2 \left(a^2+b^2\right)^2}+\frac{\cos ^2(c+d x) (a \tan (c+d x)+b)}{a+b \tan (c+d x)}}{2 d \left(a^2+b^2\right)}","\frac{b \left(a^2-3 b^2\right)}{2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{\cos ^2(c+d x) (a \tan (c+d x)+b)}{2 d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{4 a b^3 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{x \left(a^4+6 a^2 b^2-3 b^4\right)}{2 \left(a^2+b^2\right)^3}",1,"(-((a*b*((-a + Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] - 2*Sqrt[-b^2]*Log[a + b*Tan[c + d*x]] + (a + Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]]))/(Sqrt[-b^2]*(a^2 + b^2))) + (Cos[c + d*x]^2*(b + a*Tan[c + d*x]))/(a + b*Tan[c + d*x]) + (b*(a^2 - 3*b^2)*((2*a + (-a^2 + b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] - 4*a*Log[a + b*Tan[c + d*x]] + (2*a + (a^2 - b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + (2*(a^2 + b^2))/(a + b*Tan[c + d*x])))/(2*(a^2 + b^2)^2))/(2*(a^2 + b^2)*d)","A",1
559,1,416,235,3.3283535,"\int \frac{\cos ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Tan[c + d*x])^2,x]","\frac{\frac{2 b \cos ^2(c+d x) \left(3 a \left(a^2+3 b^2\right) \tan (c+d x)-a^2 b+5 b^3\right)}{a^2+b^2}-\frac{\sqrt{-b^2} \left(6 a \left(a^2+b^2\right) \left(a^2+3 b^2\right) (a+b \tan (c+d x)) \left(\left(a-\sqrt{-b^2}\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+2 \sqrt{-b^2} \log (a+b \tan (c+d x))-\left(a+\sqrt{-b^2}\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)\right)+3 \left(a^4+4 a^2 b^2-5 b^4\right) \left(\left(-a^2+2 a \sqrt{-b^2}+b^2\right) (a+b \tan (c+d x)) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+\left(a^2+2 a \sqrt{-b^2}-b^2\right) (a+b \tan (c+d x)) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+2 \sqrt{-b^2} \left(a^2+b^2\right)-4 a \sqrt{-b^2} (a+b \tan (c+d x)) \log (a+b \tan (c+d x))\right)\right)}{\left(a^2+b^2\right)^3}+4 b \cos ^4(c+d x) (a \tan (c+d x)+b)}{16 b d \left(a^2+b^2\right) (a+b \tan (c+d x))}","\frac{3 b \left(a^2-b^2\right) \left(a^2+5 b^2\right)}{8 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) (a \tan (c+d x)+b)}{4 d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left(b \left(a^2-5 b^2\right)-3 a \left(a^2+3 b^2\right) \tan (c+d x)\right)}{8 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{6 a b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{3 x \left(a^6+5 a^4 b^2+15 a^2 b^4-5 b^6\right)}{8 \left(a^2+b^2\right)^4}",1,"(4*b*Cos[c + d*x]^4*(b + a*Tan[c + d*x]) + (2*b*Cos[c + d*x]^2*(-(a^2*b) + 5*b^3 + 3*a*(a^2 + 3*b^2)*Tan[c + d*x]))/(a^2 + b^2) - (Sqrt[-b^2]*(6*a*(a^2 + b^2)*(a^2 + 3*b^2)*((a - Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] + 2*Sqrt[-b^2]*Log[a + b*Tan[c + d*x]] - (a + Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])*(a + b*Tan[c + d*x]) + 3*(a^4 + 4*a^2*b^2 - 5*b^4)*(2*Sqrt[-b^2]*(a^2 + b^2) + (-a^2 + b^2 + 2*a*Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]]*(a + b*Tan[c + d*x]) - 4*a*Sqrt[-b^2]*Log[a + b*Tan[c + d*x]]*(a + b*Tan[c + d*x]) + (a^2 - b^2 + 2*a*Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]]*(a + b*Tan[c + d*x]))))/(a^2 + b^2)^3)/(16*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))","A",1
560,1,1152,235,6.225104,"\int \frac{\sec ^7(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^7/(a + b*Tan[c + d*x])^2,x]","\frac{10 i a (a+i b) (i a+b) \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{\sqrt{a^2+b^2} \left(a \sin \left(\frac{1}{2} (c+d x)\right)-b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\cos \left(\frac{1}{2} (c+d x)\right) a^2+b^2 \cos \left(\frac{1}{2} (c+d x)\right)}\right) (a \cos (c+d x)+b \sin (c+d x))^2 \sec ^2(c+d x)}{b^6 d (a+b \tan (c+d x))^2}-\frac{5 \left(8 a^4+12 b^2 a^2+3 b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^2 \sec ^2(c+d x)}{8 b^6 d (a+b \tan (c+d x))^2}+\frac{5 \left(8 a^4+12 b^2 a^2+3 b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^2 \sec ^2(c+d x)}{8 b^6 d (a+b \tan (c+d x))^2}-\frac{a \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2 \sec ^2(c+d x)}{3 b^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (a+b \tan (c+d x))^2}+\frac{\left(-12 \sin \left(\frac{1}{2} (c+d x)\right) a^3-13 b^2 \sin \left(\frac{1}{2} (c+d x)\right) a\right) (a \cos (c+d x)+b \sin (c+d x))^2 \sec ^2(c+d x)}{3 b^5 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^2}+\frac{\left(12 \sin \left(\frac{1}{2} (c+d x)\right) a^3+13 b^2 \sin \left(\frac{1}{2} (c+d x)\right) a\right) (a \cos (c+d x)+b \sin (c+d x))^2 \sec ^2(c+d x)}{3 b^5 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^2}-\frac{a \left(12 a^2+13 b^2\right) (a \cos (c+d x)+b \sin (c+d x))^2 \sec ^2(c+d x)}{3 b^5 d (a+b \tan (c+d x))^2}+\frac{\left(36 a^2-8 b a+21 b^2\right) (a \cos (c+d x)+b \sin (c+d x))^2 \sec ^2(c+d x)}{48 b^4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a+b \tan (c+d x))^2}+\frac{\left(-36 a^2-8 b a-21 b^2\right) (a \cos (c+d x)+b \sin (c+d x))^2 \sec ^2(c+d x)}{48 b^4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a+b \tan (c+d x))^2}+\frac{a \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2 \sec ^2(c+d x)}{3 b^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (a+b \tan (c+d x))^2}+\frac{(a \cos (c+d x)+b \sin (c+d x))^2 \sec ^2(c+d x)}{16 b^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 (a+b \tan (c+d x))^2}-\frac{(a \cos (c+d x)+b \sin (c+d x))^2 \sec ^2(c+d x)}{16 b^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 (a+b \tan (c+d x))^2}-\frac{(a-i b)^2 (a+i b)^2 (a \cos (c+d x)+b \sin (c+d x)) \sec ^2(c+d x)}{b^5 d (a+b \tan (c+d x))^2}","\frac{5 a \left(a^2+b^2\right)^{3/2} \sec (c+d x) \tanh ^{-1}\left(\frac{b-a \tan (c+d x)}{\sqrt{a^2+b^2} \sqrt{\sec ^2(c+d x)}}\right)}{b^6 d \sqrt{\sec ^2(c+d x)}}-\frac{5 \sec (c+d x) \left(8 a \left(a^2+b^2\right)-b \left(4 a^2+3 b^2\right) \tan (c+d x)\right)}{8 b^5 d}+\frac{5 \left(8 a^4+12 a^2 b^2+3 b^4\right) \sec (c+d x) \sinh ^{-1}(\tan (c+d x))}{8 b^6 d \sqrt{\sec ^2(c+d x)}}-\frac{5 \sec ^3(c+d x) (4 a-3 b \tan (c+d x))}{12 b^3 d}-\frac{\sec ^5(c+d x)}{b d (a+b \tan (c+d x))}",1,"-(((a - I*b)^2*(a + I*b)^2*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(b^5*d*(a + b*Tan[c + d*x])^2)) - (a*(12*a^2 + 13*b^2)*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(3*b^5*d*(a + b*Tan[c + d*x])^2) + ((10*I)*a*(a + I*b)*(I*a + b)*Sqrt[a^2 + b^2]*ArcTanh[(Sqrt[a^2 + b^2]*(-(b*Cos[(c + d*x)/2]) + a*Sin[(c + d*x)/2]))/(a^2*Cos[(c + d*x)/2] + b^2*Cos[(c + d*x)/2])]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(b^6*d*(a + b*Tan[c + d*x])^2) - (5*(8*a^4 + 12*a^2*b^2 + 3*b^4)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(8*b^6*d*(a + b*Tan[c + d*x])^2) + (5*(8*a^4 + 12*a^2*b^2 + 3*b^4)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(8*b^6*d*(a + b*Tan[c + d*x])^2) + (Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(16*b^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4*(a + b*Tan[c + d*x])^2) + ((36*a^2 - 8*a*b + 21*b^2)*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(48*b^4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a + b*Tan[c + d*x])^2) - (a*Sec[c + d*x]^2*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(3*b^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(a + b*Tan[c + d*x])^2) - (Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(16*b^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(a + b*Tan[c + d*x])^2) + (a*Sec[c + d*x]^2*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(3*b^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a + b*Tan[c + d*x])^2) + ((-36*a^2 - 8*a*b - 21*b^2)*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(48*b^4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a + b*Tan[c + d*x])^2) + (Sec[c + d*x]^2*(-12*a^3*Sin[(c + d*x)/2] - 13*a*b^2*Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(3*b^5*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^2) + (Sec[c + d*x]^2*(12*a^3*Sin[(c + d*x)/2] + 13*a*b^2*Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(3*b^5*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^2)","C",1
561,1,709,176,6.1337767,"\int \frac{\sec ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Tan[c + d*x])^2,x]","-\frac{3 \left(2 a^2+b^2\right) \sec ^2(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{2 b^4 d (a+b \tan (c+d x))^2}+\frac{3 \left(2 a^2+b^2\right) \sec ^2(c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{2 b^4 d (a+b \tan (c+d x))^2}-\frac{6 a \sqrt{a^2+b^2} \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \tanh ^{-1}\left(\frac{\sqrt{a^2+b^2} \left(a \sin \left(\frac{1}{2} (c+d x)\right)-b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^2 \cos \left(\frac{1}{2} (c+d x)\right)+b^2 \cos \left(\frac{1}{2} (c+d x)\right)}\right)}{b^4 d (a+b \tan (c+d x))^2}-\frac{2 a \sin \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{b^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^2}-\frac{2 a \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{b^3 d (a+b \tan (c+d x))^2}+\frac{2 a \sin \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{b^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^2}-\frac{(a-i b) (a+i b) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))}{b^3 d (a+b \tan (c+d x))^2}+\frac{\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{4 b^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a+b \tan (c+d x))^2}-\frac{\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{4 b^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 (a+b \tan (c+d x))^2}","\frac{3 a \sqrt{a^2+b^2} \sec (c+d x) \tanh ^{-1}\left(\frac{b-a \tan (c+d x)}{\sqrt{a^2+b^2} \sqrt{\sec ^2(c+d x)}}\right)}{b^4 d \sqrt{\sec ^2(c+d x)}}+\frac{3 \left(2 a^2+b^2\right) \sec (c+d x) \sinh ^{-1}(\tan (c+d x))}{2 b^4 d \sqrt{\sec ^2(c+d x)}}-\frac{3 \sec (c+d x) (2 a-b \tan (c+d x))}{2 b^3 d}-\frac{\sec ^3(c+d x)}{b d (a+b \tan (c+d x))}",1,"-(((a - I*b)*(a + I*b)*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(b^3*d*(a + b*Tan[c + d*x])^2)) - (2*a*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(b^3*d*(a + b*Tan[c + d*x])^2) - (6*a*Sqrt[a^2 + b^2]*ArcTanh[(Sqrt[a^2 + b^2]*(-(b*Cos[(c + d*x)/2]) + a*Sin[(c + d*x)/2]))/(a^2*Cos[(c + d*x)/2] + b^2*Cos[(c + d*x)/2])]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(b^4*d*(a + b*Tan[c + d*x])^2) - (3*(2*a^2 + b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(2*b^4*d*(a + b*Tan[c + d*x])^2) + (3*(2*a^2 + b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(2*b^4*d*(a + b*Tan[c + d*x])^2) + (Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(4*b^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a + b*Tan[c + d*x])^2) - (2*a*Sec[c + d*x]^2*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(b^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^2) - (Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(4*b^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a + b*Tan[c + d*x])^2) + (2*a*Sec[c + d*x]^2*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(b^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^2)","C",1
562,1,120,91,0.8110261,"\int \frac{\sec ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Tan[c + d*x])^2,x]","-\frac{\frac{2 a \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{\sqrt{a^2+b^2}}+\frac{b \sec (c+d x)}{a+b \tan (c+d x)}+\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)}{b^2 d}","\frac{a \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^2 d \sqrt{a^2+b^2}}-\frac{\sec (c+d x)}{b d (a+b \tan (c+d x))}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^2 d}",1,"-(((2*a*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] + Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b*Sec[c + d*x])/(a + b*Tan[c + d*x]))/(b^2*d))","A",1
563,1,78,82,0.3780242,"\int \frac{\sec (c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]/(a + b*Tan[c + d*x])^2,x]","\frac{\frac{2 a \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{3/2}}-\frac{b \sec (c+d x)}{\left(a^2+b^2\right) (a+b \tan (c+d x))}}{d}","-\frac{b \sec (c+d x)}{d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{a \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{3/2}}",1,"((2*a*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (b*Sec[c + d*x])/((a^2 + b^2)*(a + b*Tan[c + d*x])))/d","A",1
564,1,153,157,0.557764,"\int \frac{\cos (c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Cos[c + d*x]/(a + b*Tan[c + d*x])^2,x]","\frac{\sec (c+d x) \left(\left(a^2+b^2\right) \left(a \left(a^2+b^2\right) \sin (2 (c+d x))+b \left(a^2+b^2\right) \cos (2 (c+d x))+3 b \left(a^2-b^2\right)\right)+12 a b^2 \sqrt{a^2+b^2} (a \cos (c+d x)+b \sin (c+d x)) \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)\right)}{2 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}","\frac{\cos (c+d x) (a \tan (c+d x)+b)}{d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{b \left(a^2-2 b^2\right) \sec (c+d x)}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{3 a b^2 \cos (c+d x) \sqrt{\sec ^2(c+d x)} \tanh ^{-1}\left(\frac{b-a \tan (c+d x)}{\sqrt{a^2+b^2} \sqrt{\sec ^2(c+d x)}}\right)}{d \left(a^2+b^2\right)^{5/2}}",1,"(Sec[c + d*x]*(12*a*b^2*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x]) + (a^2 + b^2)*(3*b*(a^2 - b^2) + b*(a^2 + b^2)*Cos[2*(c + d*x)] + a*(a^2 + b^2)*Sin[2*(c + d*x)])))/(2*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))","A",1
565,1,249,241,1.1778932,"\int \frac{\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Tan[c + d*x])^2,x]","\frac{\sec (c+d x) \left(240 a b^4 \sqrt{a^2+b^2} (a \cos (c+d x)+b \sin (c+d x)) \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)+\left(a^2+b^2\right) \left(10 a^5 \sin (2 (c+d x))+a^5 \sin (4 (c+d x))+15 a^4 b+40 a^3 b^2 \sin (2 (c+d x))+2 a^3 b^2 \sin (4 (c+d x))+90 a^2 b^3+b \left(a^2+b^2\right)^2 \cos (4 (c+d x))+20 b^3 \left(a^2+b^2\right) \cos (2 (c+d x))+30 a b^4 \sin (2 (c+d x))+a b^4 \sin (4 (c+d x))-45 b^5\right)\right)}{24 d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))}","\frac{\cos ^3(c+d x) (a \tan (c+d x)+b)}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\cos (c+d x) \left(b \left(a^2-4 b^2\right)-a \left(2 a^2+7 b^2\right) \tan (c+d x)\right)}{3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{5 a b^4 \cos (c+d x) \sqrt{\sec ^2(c+d x)} \tanh ^{-1}\left(\frac{b-a \tan (c+d x)}{\sqrt{a^2+b^2} \sqrt{\sec ^2(c+d x)}}\right)}{d \left(a^2+b^2\right)^{7/2}}+\frac{b \left(2 a^4+9 a^2 b^2-8 b^4\right) \sec (c+d x)}{3 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}",1,"(Sec[c + d*x]*(240*a*b^4*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x]) + (a^2 + b^2)*(15*a^4*b + 90*a^2*b^3 - 45*b^5 + 20*b^3*(a^2 + b^2)*Cos[2*(c + d*x)] + b*(a^2 + b^2)^2*Cos[4*(c + d*x)] + 10*a^5*Sin[2*(c + d*x)] + 40*a^3*b^2*Sin[2*(c + d*x)] + 30*a*b^4*Sin[2*(c + d*x)] + a^5*Sin[4*(c + d*x)] + 2*a^3*b^2*Sin[4*(c + d*x)] + a*b^4*Sin[4*(c + d*x)])))/(24*(a^2 + b^2)^4*d*(a + b*Tan[c + d*x]))","A",1
566,1,272,185,1.3800494,"\int \frac{\sec ^8(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^8/(a + b*Tan[c + d*x])^3,x]","\frac{4 a^2 b^4 \tan ^4(c+d x)+b^4 \sec ^4(c+d x) \left(a^2-2 a b \tan (c+d x)+3 b^2\right)-20 a b^3 \left(a^2+b^2\right) \tan ^3(c+d x)+4 b^2 \tan ^2(c+d x) \left(-13 a^4-10 a^2 b^2+3 \left(5 a^4+6 a^2 b^2+b^4\right) \log (a+b \tan (c+d x))\right)+2 \left(a^2+b^2\right) \left(19 a^4+6 a^2 \left(5 a^2+b^2\right) \log (a+b \tan (c+d x))+16 a^2 b^2-3 b^4\right)+4 a b \tan (c+d x) \left(4 a^4+17 a^2 b^2+6 \left(5 a^4+6 a^2 b^2+b^4\right) \log (a+b \tan (c+d x))+11 b^4\right)+b^6 \sec ^6(c+d x)}{4 b^7 d (a+b \tan (c+d x))^2}","\frac{6 a \left(a^2+b^2\right)^2}{b^7 d (a+b \tan (c+d x))}-\frac{\left(a^2+b^2\right)^3}{2 b^7 d (a+b \tan (c+d x))^2}+\frac{3 \left(a^2+b^2\right) \left(5 a^2+b^2\right) \log (a+b \tan (c+d x))}{b^7 d}-\frac{a \left(10 a^2+9 b^2\right) \tan (c+d x)}{b^6 d}+\frac{3 \left(2 a^2+b^2\right) \tan ^2(c+d x)}{2 b^5 d}-\frac{a \tan ^3(c+d x)}{b^4 d}+\frac{\tan ^4(c+d x)}{4 b^3 d}",1,"(2*(a^2 + b^2)*(19*a^4 + 16*a^2*b^2 - 3*b^4 + 6*a^2*(5*a^2 + b^2)*Log[a + b*Tan[c + d*x]]) + b^6*Sec[c + d*x]^6 + 4*a*b*(4*a^4 + 17*a^2*b^2 + 11*b^4 + 6*(5*a^4 + 6*a^2*b^2 + b^4)*Log[a + b*Tan[c + d*x]])*Tan[c + d*x] + 4*b^2*(-13*a^4 - 10*a^2*b^2 + 3*(5*a^4 + 6*a^2*b^2 + b^4)*Log[a + b*Tan[c + d*x]])*Tan[c + d*x]^2 - 20*a*b^3*(a^2 + b^2)*Tan[c + d*x]^3 + 4*a^2*b^4*Tan[c + d*x]^4 + b^4*Sec[c + d*x]^4*(a^2 + 3*b^2 - 2*a*b*Tan[c + d*x]))/(4*b^7*d*(a + b*Tan[c + d*x])^2)","A",1
567,1,140,121,3.4828388,"\int \frac{\sec ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^6/(a + b*Tan[c + d*x])^3,x]","\frac{-2 a \left(-\frac{a^2+b^2}{a+b \tan (c+d x)}-2 a \log (a+b \tan (c+d x))+b \tan (c+d x)\right)+2 \left(a^2+b^2\right) \left(\frac{3 a^2+4 a b \tan (c+d x)-b^2}{2 (a+b \tan (c+d x))^2}+\log (a+b \tan (c+d x))\right)+\frac{b^4 \sec ^4(c+d x)}{2 (a+b \tan (c+d x))^2}}{b^5 d}","-\frac{\left(a^2+b^2\right)^2}{2 b^5 d (a+b \tan (c+d x))^2}+\frac{4 a \left(a^2+b^2\right)}{b^5 d (a+b \tan (c+d x))}+\frac{2 \left(3 a^2+b^2\right) \log (a+b \tan (c+d x))}{b^5 d}-\frac{3 a \tan (c+d x)}{b^4 d}+\frac{\tan ^2(c+d x)}{2 b^3 d}",1,"((b^4*Sec[c + d*x]^4)/(2*(a + b*Tan[c + d*x])^2) - 2*a*(-2*a*Log[a + b*Tan[c + d*x]] + b*Tan[c + d*x] - (a^2 + b^2)/(a + b*Tan[c + d*x])) + 2*(a^2 + b^2)*(Log[a + b*Tan[c + d*x]] + (3*a^2 - b^2 + 4*a*b*Tan[c + d*x])/(2*(a + b*Tan[c + d*x])^2)))/(b^5*d)","A",1
568,1,57,69,0.1052676,"\int \frac{\sec ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Tan[c + d*x])^3,x]","\frac{-\frac{a^2+b^2}{2 (a+b \tan (c+d x))^2}+\frac{2 a}{a+b \tan (c+d x)}+\log (a+b \tan (c+d x))}{b^3 d}","-\frac{a^2+b^2}{2 b^3 d (a+b \tan (c+d x))^2}+\frac{2 a}{b^3 d (a+b \tan (c+d x))}+\frac{\log (a+b \tan (c+d x))}{b^3 d}",1,"(Log[a + b*Tan[c + d*x]] - (a^2 + b^2)/(2*(a + b*Tan[c + d*x])^2) + (2*a)/(a + b*Tan[c + d*x]))/(b^3*d)","A",1
569,1,58,22,0.1872539,"\int \frac{\sec ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Tan[c + d*x])^3,x]","\frac{2 \tan (c+d x) (a+b \tan (c+d x))-b \sec ^2(c+d x)}{2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}","-\frac{1}{2 b d (a+b \tan (c+d x))^2}",1,"(-(b*Sec[c + d*x]^2) + 2*Tan[c + d*x]*(a + b*Tan[c + d*x]))/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2)","B",1
570,1,458,202,6.2829272,"\int \frac{\cos ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Tan[c + d*x])^3,x]","\frac{b^3 \left(\frac{\cos ^2(c+d x) \left(a b \tan (c+d x)+b^2\right)}{2 b^4 \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{\left(2 a^2-4 b^2\right) \left(-\frac{2 a}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{1}{2 \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{\left(3 a^2-b^2\right) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^3}-\frac{\left(-\frac{a^3-3 a b^2}{\sqrt{-b^2}}+3 a^2-b^2\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)}{2 \left(a^2+b^2\right)^3}-\frac{\left(\frac{a^3-3 a b^2}{\sqrt{-b^2}}+3 a^2-b^2\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)}{2 \left(a^2+b^2\right)^3}\right)-3 a \left(-\frac{1}{\left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\left(2 a-\frac{a^2-b^2}{\sqrt{-b^2}}\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)}{2 \left(a^2+b^2\right)^2}+\frac{2 a \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^2}-\frac{\left(\frac{a^2-b^2}{\sqrt{-b^2}}+2 a\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)}{2 \left(a^2+b^2\right)^2}\right)}{2 b^2 \left(a^2+b^2\right)}\right)}{d}","\frac{a b \left(a^2-11 b^2\right)}{2 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}+\frac{b \left(a^2-2 b^2\right)}{2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{\cos ^2(c+d x) (a \tan (c+d x)+b)}{2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{2 b^3 \left(5 a^2-b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{a x \left(a^4+10 a^2 b^2-15 b^4\right)}{2 \left(a^2+b^2\right)^4}",1,"(b^3*((Cos[c + d*x]^2*(b^2 + a*b*Tan[c + d*x]))/(2*b^4*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - ((2*a^2 - 4*b^2)*(-1/2*((3*a^2 - b^2 - (a^3 - 3*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/(a^2 + b^2)^3 + ((3*a^2 - b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^3 - ((3*a^2 - b^2 + (a^3 - 3*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/(2*(a^2 + b^2)^3) - 1/(2*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - (2*a)/((a^2 + b^2)^2*(a + b*Tan[c + d*x]))) - 3*a*(-1/2*((2*a - (a^2 - b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/(a^2 + b^2)^2 + (2*a*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^2 - ((2*a + (a^2 - b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/(2*(a^2 + b^2)^2) - 1/((a^2 + b^2)*(a + b*Tan[c + d*x]))))/(2*b^2*(a^2 + b^2))))/d","B",1
571,1,596,295,6.2899508,"\int \frac{\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Tan[c + d*x])^3,x]","\frac{b^5 \left(\frac{\cos ^4(c+d x) \left(a b \tan (c+d x)+b^2\right)}{4 b^6 \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{\frac{\cos ^2(c+d x) \left(b \left(-3 a \left(a^2+2 b^2\right)-5 a b^2\right) \tan (c+d x)+5 a^2 b^2-3 b^2 \left(a^2+2 b^2\right)\right)}{2 b^4 \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{3 a \left(3 a^2+11 b^2\right) \left(-\frac{1}{\left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\left(2 a-\frac{a^2-b^2}{\sqrt{-b^2}}\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)}{2 \left(a^2+b^2\right)^2}+\frac{2 a \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^2}-\frac{\left(\frac{a^2-b^2}{\sqrt{-b^2}}+2 a\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)}{2 \left(a^2+b^2\right)^2}\right)+\left(3 \left(a^4+a^2 b^2+8 b^4\right)-3 a^2 \left(3 a^2+11 b^2\right)\right) \left(-\frac{2 a}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{1}{2 \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{\left(3 a^2-b^2\right) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^3}-\frac{\left(-\frac{a^3-3 a b^2}{\sqrt{-b^2}}+3 a^2-b^2\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)}{2 \left(a^2+b^2\right)^3}-\frac{\left(\frac{a^3-3 a b^2}{\sqrt{-b^2}}+3 a^2-b^2\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)}{2 \left(a^2+b^2\right)^3}\right)}{2 b^2 \left(a^2+b^2\right)}}{4 b^2 \left(a^2+b^2\right)}\right)}{d}","\frac{\cos ^4(c+d x) (a \tan (c+d x)+b)}{4 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{\cos ^2(c+d x) \left(2 b \left(a^2-3 b^2\right)-a \left(3 a^2+11 b^2\right) \tan (c+d x)\right)}{8 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{3 b^5 \left(7 a^2-b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^5}+\frac{3 a b \left(a^4+6 a^2 b^2-27 b^4\right)}{8 d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))}+\frac{3 b \left(a^4+5 a^2 b^2-4 b^4\right)}{8 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^2}+\frac{3 a x \left(a^6+7 a^4 b^2+35 a^2 b^4-35 b^6\right)}{8 \left(a^2+b^2\right)^5}",1,"(b^5*((Cos[c + d*x]^4*(b^2 + a*b*Tan[c + d*x]))/(4*b^6*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - ((Cos[c + d*x]^2*(5*a^2*b^2 - 3*b^2*(a^2 + 2*b^2) + b*(-5*a*b^2 - 3*a*(a^2 + 2*b^2))*Tan[c + d*x]))/(2*b^4*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - ((-3*a^2*(3*a^2 + 11*b^2) + 3*(a^4 + a^2*b^2 + 8*b^4))*(-1/2*((3*a^2 - b^2 - (a^3 - 3*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/(a^2 + b^2)^3 + ((3*a^2 - b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^3 - ((3*a^2 - b^2 + (a^3 - 3*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/(2*(a^2 + b^2)^3) - 1/(2*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - (2*a)/((a^2 + b^2)^2*(a + b*Tan[c + d*x]))) + 3*a*(3*a^2 + 11*b^2)*(-1/2*((2*a - (a^2 - b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/(a^2 + b^2)^2 + (2*a*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^2 - ((2*a + (a^2 - b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/(2*(a^2 + b^2)^2) - 1/((a^2 + b^2)*(a + b*Tan[c + d*x]))))/(2*b^2*(a^2 + b^2)))/(4*b^2*(a^2 + b^2))))/d","B",1
572,1,688,239,2.4763325,"\int \frac{\sec ^7(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^7/(a + b*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left(\frac{6 b^2 \left(a^2+b^2\right)^2 \sin (c+d x)}{a}+2 b \left(36 a^2+13 b^2\right) (a \cos (c+d x)+b \sin (c+d x))^2+\frac{2 b \left(36 a^2+13 b^2\right) \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}-\frac{2 b \left(36 a^2+13 b^2\right) \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{6 b (a-i b) (a+i b) \left(8 a^2-b^2\right) (a \cos (c+d x)+b \sin (c+d x))}{a}+30 a \left(4 a^2+3 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^2-30 a \left(4 a^2+3 b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^2+60 \sqrt{a^2+b^2} \left(4 a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)+\frac{2 b^3 \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (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{2 b^3 \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{b^2 (b-9 a) (a \cos (c+d x)+b \sin (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^2 (9 a+b) (a \cos (c+d x)+b \sin (c+d x))^2}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{12 b^6 d (a+b \tan (c+d x))^3}","-\frac{5 \sqrt{a^2+b^2} \left(4 a^2+b^2\right) \sec (c+d x) \tanh ^{-1}\left(\frac{b-a \tan (c+d x)}{\sqrt{a^2+b^2} \sqrt{\sec ^2(c+d x)}}\right)}{2 b^6 d \sqrt{\sec ^2(c+d x)}}-\frac{5 a \left(4 a^2+3 b^2\right) \sec (c+d x) \sinh ^{-1}(\tan (c+d x))}{2 b^6 d \sqrt{\sec ^2(c+d x)}}+\frac{5 \sec (c+d x) \left(4 a^2-2 a b \tan (c+d x)+b^2\right)}{2 b^5 d}+\frac{5 \sec ^3(c+d x) (4 a+b \tan (c+d x))}{6 b^3 d (a+b \tan (c+d x))}-\frac{\sec ^5(c+d x)}{2 b d (a+b \tan (c+d x))^2}",1,"(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*((6*b^2*(a^2 + b^2)^2*Sin[c + d*x])/a + (6*(a - I*b)*(a + I*b)*b*(8*a^2 - b^2)*(a*Cos[c + d*x] + b*Sin[c + d*x]))/a + 2*b*(36*a^2 + 13*b^2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + 60*Sqrt[a^2 + b^2]*(4*a^2 + b^2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + 30*a*(4*a^2 + 3*b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 - 30*a*(4*a^2 + 3*b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + (b^2*(-9*a + b)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*b^3*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (2*b*(36*a^2 + 13*b^2)*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (2*b^3*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (b^2*(9*a + b)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (2*b*(36*a^2 + 13*b^2)*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(12*b^6*d*(a + b*Tan[c + d*x])^3)","C",1
573,1,396,148,2.4521816,"\int \frac{\sec ^5(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left(\frac{b^2 \left(a^2+b^2\right) \sin (c+d x)}{a}+\frac{6 \left(2 a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{\sqrt{a^2+b^2}}+\frac{2 b \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}-\frac{2 b \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+2 b (a \cos (c+d x)+b \sin (c+d x))^2+\frac{b (2 a-b) (2 a+b) (a \cos (c+d x)+b \sin (c+d x))}{a}+6 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^2-6 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^2\right)}{2 b^4 d (a+b \tan (c+d x))^3}","-\frac{3 \left(2 a^2+b^2\right) \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{2 b^4 d \sqrt{a^2+b^2}}-\frac{3 a \tanh ^{-1}(\sin (c+d x))}{b^4 d}+\frac{3 \sec (c+d x) (2 a+b \tan (c+d x))}{2 b^3 d (a+b \tan (c+d x))}-\frac{\sec ^3(c+d x)}{2 b d (a+b \tan (c+d x))^2}",1,"(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*((b^2*(a^2 + b^2)*Sin[c + d*x])/a + ((2*a - b)*b*(2*a + b)*(a*Cos[c + d*x] + b*Sin[c + d*x]))/a + 2*b*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + (6*(2*a^2 + b^2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/Sqrt[a^2 + b^2] + 6*a*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 - 6*a*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + (2*b*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (2*b*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(2*b^4*d*(a + b*Tan[c + d*x])^3)","B",1
574,1,132,95,0.303773,"\int \frac{\sec ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Tan[c + d*x])^3,x]","\frac{\left(a^2+b^2\right) (a \sin (c+d x)-b \cos (c+d x))+2 \sqrt{a^2+b^2} (a \cos (c+d x)+b \sin (c+d x))^2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{2 d (a-i b)^2 (a+i b)^2 (a \cos (c+d x)+b \sin (c+d x))^2}","-\frac{\sec (c+d x) (b-a \tan (c+d x))}{2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{\tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{2 d \left(a^2+b^2\right)^{3/2}}",1,"((a^2 + b^2)*(-(b*Cos[c + d*x]) + a*Sin[c + d*x]) + 2*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(2*(a - I*b)^2*(a + I*b)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)","C",1
575,1,110,155,0.9684887,"\int \frac{\sec (c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]/(a + b*Tan[c + d*x])^3,x]","\frac{\frac{2 \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}-\frac{b \sec (c+d x) \left(4 a^2+3 a b \tan (c+d x)+b^2\right)}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}}{2 d}","-\frac{3 a b \sec (c+d x)}{2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{b \sec (c+d x)}{2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{\left(2 a^2-b^2\right) \sec (c+d x) \tanh ^{-1}\left(\frac{b-a \tan (c+d x)}{\sqrt{a^2+b^2} \sqrt{\sec ^2(c+d x)}}\right)}{2 d \left(a^2+b^2\right)^{5/2} \sqrt{\sec ^2(c+d x)}}",1,"((2*(2*a^2 - b^2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (b*Sec[c + d*x]*(4*a^2 + b^2 + 3*a*b*Tan[c + d*x]))/((a^2 + b^2)^2*(a + b*Tan[c + d*x])^2))/(2*d)","A",1
576,1,183,221,1.9334381,"\int \frac{\cos (c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Cos[c + d*x]/(a + b*Tan[c + d*x])^3,x]","\frac{\frac{\sec ^2(c+d x) \left(b \left(a^2+b^2\right)^2 \cos (3 (c+d x))+b \left(11 a^4-22 a^2 b^2-3 b^4\right) \cos (c+d x)+2 a \sin (c+d x) \left(a^4+\left(a^2+b^2\right)^2 \cos (2 (c+d x))+4 a^2 b^2-12 b^4\right)\right)}{\left(a^2+b^2\right)^3 (a+b \tan (c+d x))^2}-\frac{12 b^2 \left(b^2-4 a^2\right) \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{7/2}}}{4 d}","\frac{\cos (c+d x) (a \tan (c+d x)+b)}{d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{a b \left(2 a^2-13 b^2\right) \sec (c+d x)}{2 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}+\frac{b \left(2 a^2-3 b^2\right) \sec (c+d x)}{2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{3 b^2 \left(4 a^2-b^2\right) \cos (c+d x) \sqrt{\sec ^2(c+d x)} \tanh ^{-1}\left(\frac{b-a \tan (c+d x)}{\sqrt{a^2+b^2} \sqrt{\sec ^2(c+d x)}}\right)}{2 d \left(a^2+b^2\right)^{7/2}}",1,"((-12*b^2*(-4*a^2 + b^2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (Sec[c + d*x]^2*(b*(11*a^4 - 22*a^2*b^2 - 3*b^4)*Cos[c + d*x] + b*(a^2 + b^2)^2*Cos[3*(c + d*x)] + 2*a*(a^4 + 4*a^2*b^2 - 12*b^4 + (a^2 + b^2)^2*Cos[2*(c + d*x)])*Sin[c + d*x]))/((a^2 + b^2)^3*(a + b*Tan[c + d*x])^2))/(4*d)","A",1
577,1,371,310,1.9953097,"\int \frac{\cos ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Tan[c + d*x])^3,x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left(-\frac{b \left(b^2-3 a^2\right) \cos (c+d x) \cos (3 (c+d x)) (a+b \tan (c+d x))^2}{\left(a^2+b^2\right)^3}+\frac{a \left(a^2-3 b^2\right) \sin (3 (c+d x)) \cos (c+d x) (a+b \tan (c+d x))^2}{\left(a^2+b^2\right)^3}+\frac{6 b^6 \tan (c+d x)}{a \left(a^2+b^2\right)^3}-\frac{6 b^5 \left(12 a^2+b^2\right) (a+b \tan (c+d x))}{a \left(a^2+b^2\right)^4}-\frac{60 b^4 \left(b^2-6 a^2\right) \cos (c+d x) (a+b \tan (c+d x))^2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{9/2}}+\frac{9 b \left(a^4+14 a^2 b^2-3 b^4\right) (a \cos (c+d x)+b \sin (c+d x))^2}{\left(a^2+b^2\right)^4}+\frac{9 a \left(a^4+6 a^2 b^2-11 b^4\right) \tan (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{\left(a^2+b^2\right)^4}\right)}{12 d (a+b \tan (c+d x))^3}","\frac{\cos ^3(c+d x) (a \tan (c+d x)+b)}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{\cos (c+d x) \left(b \left(2 a^2-5 b^2\right)-a \left(2 a^2+9 b^2\right) \tan (c+d x)\right)}{3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{5 b^4 \left(6 a^2-b^2\right) \cos (c+d x) \sqrt{\sec ^2(c+d x)} \tanh ^{-1}\left(\frac{b-a \tan (c+d x)}{\sqrt{a^2+b^2} \sqrt{\sec ^2(c+d x)}}\right)}{2 d \left(a^2+b^2\right)^{9/2}}+\frac{a b \left(4 a^4+28 a^2 b^2-81 b^4\right) \sec (c+d x)}{6 d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))}+\frac{b \left(4 a^4+24 a^2 b^2-15 b^4\right) \sec (c+d x)}{6 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^2}",1,"(Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])*((9*b*(a^4 + 14*a^2*b^2 - 3*b^4)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(a^2 + b^2)^4 + (6*b^6*Tan[c + d*x])/(a*(a^2 + b^2)^3) + (9*a*(a^4 + 6*a^2*b^2 - 11*b^4)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2*Tan[c + d*x])/(a^2 + b^2)^4 - (6*b^5*(12*a^2 + b^2)*(a + b*Tan[c + d*x]))/(a*(a^2 + b^2)^4) - (60*b^4*(-6*a^2 + b^2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*Cos[c + d*x]*(a + b*Tan[c + d*x])^2)/(a^2 + b^2)^(9/2) - (b*(-3*a^2 + b^2)*Cos[c + d*x]*Cos[3*(c + d*x)]*(a + b*Tan[c + d*x])^2)/(a^2 + b^2)^3 + (a*(a^2 - 3*b^2)*Cos[c + d*x]*Sin[3*(c + d*x)]*(a + b*Tan[c + d*x])^2)/(a^2 + b^2)^3))/(12*d*(a + b*Tan[c + d*x])^3)","A",1
578,1,69,121,0.6202462,"\int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x)) \, dx","Integrate[(d*Sec[e + f*x])^(7/2)*(a + b*Tan[e + f*x]),x]","\frac{(d \sec (e+f x))^{7/2} \left(70 a \sin (2 (e+f x))+21 a \sin (4 (e+f x))-168 a \cos ^{\frac{7}{2}}(e+f x) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)+40 b\right)}{140 f}","-\frac{6 a d^4 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{5 f \sqrt{\cos (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{6 a d^3 \sin (e+f x) \sqrt{d \sec (e+f x)}}{5 f}+\frac{2 a d \sin (e+f x) (d \sec (e+f x))^{5/2}}{5 f}+\frac{2 b (d \sec (e+f x))^{7/2}}{7 f}",1,"((d*Sec[e + f*x])^(7/2)*(40*b - 168*a*Cos[e + f*x]^(7/2)*EllipticE[(e + f*x)/2, 2] + 70*a*Sin[2*(e + f*x)] + 21*a*Sin[4*(e + f*x)]))/(140*f)","A",1
579,1,58,92,0.4108646,"\int (d \sec (e+f x))^{5/2} (a+b \tan (e+f x)) \, dx","Integrate[(d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x]),x]","\frac{(d \sec (e+f x))^{5/2} \left(5 a \sin (2 (e+f x))+10 a \cos ^{\frac{5}{2}}(e+f x) F\left(\left.\frac{1}{2} (e+f x)\right|2\right)+6 b\right)}{15 f}","\frac{2 a d^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{d \sec (e+f x)}}{3 f}+\frac{2 a d \sin (e+f x) (d \sec (e+f x))^{3/2}}{3 f}+\frac{2 b (d \sec (e+f x))^{5/2}}{5 f}",1,"((d*Sec[e + f*x])^(5/2)*(6*b + 10*a*Cos[e + f*x]^(5/2)*EllipticF[(e + f*x)/2, 2] + 5*a*Sin[2*(e + f*x)]))/(15*f)","A",1
580,1,58,88,0.2948163,"\int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x)) \, dx","Integrate[(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x]),x]","\frac{(d \sec (e+f x))^{3/2} \left(3 a \sin (2 (e+f x))-6 a \cos ^{\frac{3}{2}}(e+f x) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)+2 b\right)}{3 f}","-\frac{2 a d^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{f \sqrt{\cos (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{2 a d \sin (e+f x) \sqrt{d \sec (e+f x)}}{f}+\frac{2 b (d \sec (e+f x))^{3/2}}{3 f}",1,"((d*Sec[e + f*x])^(3/2)*(2*b - 6*a*Cos[e + f*x]^(3/2)*EllipticE[(e + f*x)/2, 2] + 3*a*Sin[2*(e + f*x)]))/(3*f)","A",1
581,1,42,58,0.2002991,"\int \sqrt{d \sec (e+f x)} (a+b \tan (e+f x)) \, dx","Integrate[Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x]),x]","\frac{2 \sqrt{d \sec (e+f x)} \left(a \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)+b\right)}{f}","\frac{2 a \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{d \sec (e+f x)}}{f}+\frac{2 b \sqrt{d \sec (e+f x)}}{f}",1,"(2*(b + a*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])*Sqrt[d*Sec[e + f*x]])/f","A",1
582,1,54,58,0.2878774,"\int \frac{a+b \tan (e+f x)}{\sqrt{d \sec (e+f x)}} \, dx","Integrate[(a + b*Tan[e + f*x])/Sqrt[d*Sec[e + f*x]],x]","\frac{2 a E\left(\left.\frac{1}{2} (e+f x)\right|2\right)-2 b \sqrt{\cos (e+f x)}}{f \sqrt{\cos (e+f x)} \sqrt{d \sec (e+f x)}}","\frac{2 a E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{f \sqrt{\cos (e+f x)} \sqrt{d \sec (e+f x)}}-\frac{2 b}{f \sqrt{d \sec (e+f x)}}",1,"(-2*b*Sqrt[Cos[e + f*x]] + 2*a*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]])","A",1
583,1,69,94,0.2021788,"\int \frac{a+b \tan (e+f x)}{(d \sec (e+f x))^{3/2}} \, dx","Integrate[(a + b*Tan[e + f*x])/(d*Sec[e + f*x])^(3/2),x]","-\frac{\sqrt{d \sec (e+f x)} \left(-a \sin (2 (e+f x))-2 a \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)+b \cos (2 (e+f x))+b\right)}{3 d^2 f}","\frac{2 a \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{d \sec (e+f x)}}{3 d^2 f}+\frac{2 a \sin (e+f x)}{3 d f \sqrt{d \sec (e+f x)}}-\frac{2 b}{3 f (d \sec (e+f x))^{3/2}}",1,"-1/3*(Sqrt[d*Sec[e + f*x]]*(b + b*Cos[2*(e + f*x)] - 2*a*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] - a*Sin[2*(e + f*x)]))/(d^2*f)","A",1
584,1,74,94,0.6046438,"\int \frac{a+b \tan (e+f x)}{(d \sec (e+f x))^{5/2}} \, dx","Integrate[(a + b*Tan[e + f*x])/(d*Sec[e + f*x])^(5/2),x]","\frac{2 \sqrt{d \sec (e+f x)} \left(\cos ^2(e+f x) (a \sin (e+f x)-b \cos (e+f x))+3 a \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{5 d^3 f}","\frac{6 a E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{5 d^2 f \sqrt{\cos (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{2 a \sin (e+f x)}{5 d f (d \sec (e+f x))^{3/2}}-\frac{2 b}{5 f (d \sec (e+f x))^{5/2}}",1,"(2*Sqrt[d*Sec[e + f*x]]*(3*a*Sqrt[Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2] + Cos[e + f*x]^2*(-(b*Cos[e + f*x]) + a*Sin[e + f*x])))/(5*d^3*f)","A",1
585,1,94,123,0.3507106,"\int \frac{a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx","Integrate[(a + b*Tan[e + f*x])/(d*Sec[e + f*x])^(7/2),x]","\frac{\sqrt{d \sec (e+f x)} \left(26 a \sin (2 (e+f x))+3 a \sin (4 (e+f x))+40 a \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)-12 b \cos (2 (e+f x))-3 b \cos (4 (e+f x))-9 b\right)}{84 d^4 f}","\frac{10 a \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{d \sec (e+f x)}}{21 d^4 f}+\frac{10 a \sin (e+f x)}{21 d^3 f \sqrt{d \sec (e+f x)}}+\frac{2 a \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}-\frac{2 b}{7 f (d \sec (e+f x))^{7/2}}",1,"(Sqrt[d*Sec[e + f*x]]*(-9*b - 12*b*Cos[2*(e + f*x)] - 3*b*Cos[4*(e + f*x)] + 40*a*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] + 26*a*Sin[2*(e + f*x)] + 3*a*Sin[4*(e + f*x)]))/(84*d^4*f)","A",1
586,1,127,143,0.777558,"\int (d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^2 \, dx","Integrate[(d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x])^2,x]","\frac{2 d^2 \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2 \left(\frac{5}{2} \left(7 a^2-2 b^2\right) \sin (2 (e+f x))+5 \left(7 a^2-2 b^2\right) \cos ^{\frac{5}{2}}(e+f x) F\left(\left.\frac{1}{2} (e+f x)\right|2\right)+3 b (14 a+5 b \tan (e+f x))\right)}{105 f (a \cos (e+f x)+b \sin (e+f x))^2}","\frac{2 d^2 \left(7 a^2-2 b^2\right) \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{d \sec (e+f x)}}{21 f}+\frac{2 d \left(7 a^2-2 b^2\right) \sin (e+f x) (d \sec (e+f x))^{3/2}}{21 f}+\frac{18 a b (d \sec (e+f x))^{5/2}}{35 f}+\frac{2 b (d \sec (e+f x))^{5/2} (a+b \tan (e+f x))}{7 f}",1,"(2*d^2*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])^2*(5*(7*a^2 - 2*b^2)*Cos[e + f*x]^(5/2)*EllipticF[(e + f*x)/2, 2] + (5*(7*a^2 - 2*b^2)*Sin[2*(e + f*x)])/2 + 3*b*(14*a + 5*b*Tan[e + f*x])))/(105*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^2)","A",1
587,1,126,143,0.6511974,"\int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2 \, dx","Integrate[(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^2,x]","-\frac{2 d^2 (a+b \tan (e+f x))^2 \left(\left(3 b^2-\frac{15 a^2}{2}\right) \sin (2 (e+f x))+3 \left(5 a^2-2 b^2\right) \cos ^{\frac{3}{2}}(e+f x) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)-b (10 a+3 b \tan (e+f x))\right)}{15 f \sqrt{d \sec (e+f x)} (a \cos (e+f x)+b \sin (e+f x))^2}","-\frac{2 d^2 \left(5 a^2-2 b^2\right) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{5 f \sqrt{\cos (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{2 d \left(5 a^2-2 b^2\right) \sin (e+f x) \sqrt{d \sec (e+f x)}}{5 f}+\frac{14 a b (d \sec (e+f x))^{3/2}}{15 f}+\frac{2 b (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}{5 f}",1,"(-2*d^2*(a + b*Tan[e + f*x])^2*(3*(5*a^2 - 2*b^2)*Cos[e + f*x]^(3/2)*EllipticE[(e + f*x)/2, 2] + ((-15*a^2)/2 + 3*b^2)*Sin[2*(e + f*x)] - b*(10*a + 3*b*Tan[e + f*x])))/(15*f*Sqrt[d*Sec[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2)","A",1
588,1,87,103,0.6333606,"\int \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2 \, dx","Integrate[Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])^2,x]","\frac{2 \sec ^2(e+f x) \sqrt{d \sec (e+f x)} \left(\left(3 a^2-2 b^2\right) \cos ^{\frac{5}{2}}(e+f x) F\left(\left.\frac{1}{2} (e+f x)\right|2\right)+b \cos (e+f x) (6 a \cos (e+f x)+b \sin (e+f x))\right)}{3 f}","\frac{2 \left(3 a^2-2 b^2\right) \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{d \sec (e+f x)}}{3 f}+\frac{10 a b \sqrt{d \sec (e+f x)}}{3 f}+\frac{2 b \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}{3 f}",1,"(2*Sec[e + f*x]^2*Sqrt[d*Sec[e + f*x]]*((3*a^2 - 2*b^2)*Cos[e + f*x]^(5/2)*EllipticF[(e + f*x)/2, 2] + b*Cos[e + f*x]*(6*a*Cos[e + f*x] + b*Sin[e + f*x])))/(3*f)","A",1
589,1,64,95,0.9318607,"\int \frac{(a+b \tan (e+f x))^2}{\sqrt{d \sec (e+f x)}} \, dx","Integrate[(a + b*Tan[e + f*x])^2/Sqrt[d*Sec[e + f*x]],x]","\frac{\frac{2 \left(a^2-2 b^2\right) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{\sqrt{\cos (e+f x)}}+2 b (b \tan (e+f x)-2 a)}{f \sqrt{d \sec (e+f x)}}","\frac{2 \left(a^2-2 b^2\right) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{f \sqrt{\cos (e+f x)} \sqrt{d \sec (e+f x)}}-\frac{6 a b}{f \sqrt{d \sec (e+f x)}}+\frac{2 b (a+b \tan (e+f x))}{f \sqrt{d \sec (e+f x)}}",1,"((2*(a^2 - 2*b^2)*EllipticE[(e + f*x)/2, 2])/Sqrt[Cos[e + f*x]] + 2*b*(-2*a + b*Tan[e + f*x]))/(f*Sqrt[d*Sec[e + f*x]])","A",1
590,1,101,139,0.5353796,"\int \frac{(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{3/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^2/(d*Sec[e + f*x])^(3/2),x]","\frac{\sec ^2(e+f x) \left(2 \left(a^2+2 b^2\right) \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)+a^2 \sin (2 (e+f x))-2 a b \cos (2 (e+f x))-2 a b-b^2 \sin (2 (e+f x))\right)}{3 f (d \sec (e+f x))^{3/2}}","\frac{2 \left(a^2+2 b^2\right) \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{d \sec (e+f x)}}{3 d^2 f}+\frac{2 \left(a^2+2 b^2\right) \sin (e+f x)}{3 d f \sqrt{d \sec (e+f x)}}+\frac{2 a b}{3 f (d \sec (e+f x))^{3/2}}-\frac{2 b (a+b \tan (e+f x))}{f (d \sec (e+f x))^{3/2}}",1,"(Sec[e + f*x]^2*(-2*a*b - 2*a*b*Cos[2*(e + f*x)] + 2*(a^2 + 2*b^2)*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] + a^2*Sin[2*(e + f*x)] - b^2*Sin[2*(e + f*x)]))/(3*f*(d*Sec[e + f*x])^(3/2))","A",1
591,1,92,145,0.9207879,"\int \frac{(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{5/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^2/(d*Sec[e + f*x])^(5/2),x]","\frac{\left(6 a^2+4 b^2\right) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)+2 \cos ^{\frac{3}{2}}(e+f x) \left(\left(a^2-b^2\right) \sin (e+f x)-2 a b \cos (e+f x)\right)}{5 f \cos ^{\frac{5}{2}}(e+f x) (d \sec (e+f x))^{5/2}}","\frac{2 \left(3 a^2+2 b^2\right) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{5 d^2 f \sqrt{\cos (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{2 \left(3 a^2+2 b^2\right) \sin (e+f x)}{15 d f (d \sec (e+f x))^{3/2}}-\frac{2 a b}{15 f (d \sec (e+f x))^{5/2}}-\frac{2 b (a+b \tan (e+f x))}{3 f (d \sec (e+f x))^{5/2}}",1,"((6*a^2 + 4*b^2)*EllipticE[(e + f*x)/2, 2] + 2*Cos[e + f*x]^(3/2)*(-2*a*b*Cos[e + f*x] + (a^2 - b^2)*Sin[e + f*x]))/(5*f*Cos[e + f*x]^(5/2)*(d*Sec[e + f*x])^(5/2))","A",1
592,1,127,184,2.2545604,"\int \frac{(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{7/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^2/(d*Sec[e + f*x])^(7/2),x]","\frac{\frac{4 \left(5 a^2+2 b^2\right) F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{\sqrt{\cos (e+f x)}}+23 a^2 \sin (e+f x)+3 a^2 \sin (3 (e+f x))-18 a b \cos (e+f x)-6 a b \cos (3 (e+f x))+5 b^2 \sin (e+f x)-3 b^2 \sin (3 (e+f x))}{42 d^3 f \sqrt{d \sec (e+f x)}}","\frac{2 \left(5 a^2+2 b^2\right) \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{d \sec (e+f x)}}{21 d^4 f}+\frac{2 \left(5 a^2+2 b^2\right) \sin (e+f x)}{21 d^3 f \sqrt{d \sec (e+f x)}}+\frac{2 \left(5 a^2+2 b^2\right) \sin (e+f x)}{35 d f (d \sec (e+f x))^{5/2}}-\frac{6 a b}{35 f (d \sec (e+f x))^{7/2}}-\frac{2 b (a+b \tan (e+f x))}{5 f (d \sec (e+f x))^{7/2}}",1,"(-18*a*b*Cos[e + f*x] - 6*a*b*Cos[3*(e + f*x)] + (4*(5*a^2 + 2*b^2)*EllipticF[(e + f*x)/2, 2])/Sqrt[Cos[e + f*x]] + 23*a^2*Sin[e + f*x] + 5*b^2*Sin[e + f*x] + 3*a^2*Sin[3*(e + f*x)] - 3*b^2*Sin[3*(e + f*x)])/(42*d^3*f*Sqrt[d*Sec[e + f*x]])","A",1
593,1,126,184,2.8181763,"\int \frac{(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{9/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^2/(d*Sec[e + f*x])^(9/2),x]","\frac{4 \cos (e+f x) \left(2 \sin (e+f x) \left(5 \left(a^2-b^2\right) \cos (2 (e+f x))+19 a^2-b^2\right)-30 a b \cos (e+f x)-10 a b \cos (3 (e+f x))\right)+\frac{48 \left(7 a^2+2 b^2\right) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{\sqrt{\cos (e+f x)}}}{360 d^4 f \sqrt{d \sec (e+f x)}}","\frac{2 \left(7 a^2+2 b^2\right) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{15 d^4 f \sqrt{\cos (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{2 \left(7 a^2+2 b^2\right) \sin (e+f x)}{45 d^3 f (d \sec (e+f x))^{3/2}}+\frac{2 \left(7 a^2+2 b^2\right) \sin (e+f x)}{63 d f (d \sec (e+f x))^{7/2}}-\frac{10 a b}{63 f (d \sec (e+f x))^{9/2}}-\frac{2 b (a+b \tan (e+f x))}{7 f (d \sec (e+f x))^{9/2}}",1,"((48*(7*a^2 + 2*b^2)*EllipticE[(e + f*x)/2, 2])/Sqrt[Cos[e + f*x]] + 4*Cos[e + f*x]*(-30*a*b*Cos[e + f*x] - 10*a*b*Cos[3*(e + f*x)] + 2*(19*a^2 - b^2 + 5*(a^2 - b^2)*Cos[2*(e + f*x)])*Sin[e + f*x]))/(360*d^4*f*Sqrt[d*Sec[e + f*x]])","A",1
594,1,157,198,1.6962141,"\int (d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^3 \, dx","Integrate[(d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x])^3,x]","-\frac{2 d (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^3 \left(63 b \left(b^2-3 a^2\right) \cos ^2(e+f x)-15 a \left(7 a^2-6 b^2\right) \cos ^{\frac{9}{2}}(e+f x) F\left(\left.\frac{1}{2} (e+f x)\right|2\right)-15 a \left(7 a^2-6 b^2\right) \sin (e+f x) \cos ^3(e+f x)-\frac{5}{2} b^2 (27 a \sin (2 (e+f x))+14 b)\right)}{315 f (a \cos (e+f x)+b \sin (e+f x))^3}","\frac{2 b d^2 \sec ^2(e+f x) \sqrt{d \sec (e+f x)} \left(14 \left(11 a^2-2 b^2\right)+65 a b \tan (e+f x)\right)}{315 f}+\frac{2 a d^2 \left(7 a^2-6 b^2\right) \tan (e+f x) \sqrt{d \sec (e+f x)}}{21 f}+\frac{2 a d^2 \left(7 a^2-6 b^2\right) \sqrt{d \sec (e+f x)} F\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{21 f \sqrt[4]{\sec ^2(e+f x)}}+\frac{2 b d^2 \sec ^2(e+f x) \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}{9 f}",1,"(-2*d*(d*Sec[e + f*x])^(3/2)*(63*b*(-3*a^2 + b^2)*Cos[e + f*x]^2 - 15*a*(7*a^2 - 6*b^2)*Cos[e + f*x]^(9/2)*EllipticF[(e + f*x)/2, 2] - 15*a*(7*a^2 - 6*b^2)*Cos[e + f*x]^3*Sin[e + f*x] - (5*b^2*(14*b + 27*a*Sin[2*(e + f*x)]))/2)*(a + b*Tan[e + f*x])^3)/(315*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3)","A",1
595,1,155,176,1.7881776,"\int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^3 \, dx","Integrate[(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^3,x]","-\frac{d \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^3 \left(70 b \left(b^2-3 a^2\right) \cos ^2(e+f x)+42 a \left(5 a^2-6 b^2\right) \cos ^{\frac{7}{2}}(e+f x) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)-42 a \left(5 a^2-6 b^2\right) \sin (e+f x) \cos ^3(e+f x)-3 b^2 (21 a \sin (2 (e+f x))+10 b)\right)}{105 f (a \cos (e+f x)+b \sin (e+f x))^3}","\frac{2 b (d \sec (e+f x))^{3/2} \left(10 \left(9 a^2-2 b^2\right)+33 a b \tan (e+f x)\right)}{105 f}+\frac{2 a \left(5 a^2-6 b^2\right) \sin (e+f x) \cos (e+f x) (d \sec (e+f x))^{3/2}}{5 f}-\frac{2 a \left(5 a^2-6 b^2\right) (d \sec (e+f x))^{3/2} E\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{5 f \sec ^2(e+f x)^{3/4}}+\frac{2 b (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}{7 f}",1,"-1/105*(d*Sqrt[d*Sec[e + f*x]]*(70*b*(-3*a^2 + b^2)*Cos[e + f*x]^2 + 42*a*(5*a^2 - 6*b^2)*Cos[e + f*x]^(7/2)*EllipticE[(e + f*x)/2, 2] - 42*a*(5*a^2 - 6*b^2)*Cos[e + f*x]^3*Sin[e + f*x] - 3*b^2*(10*b + 21*a*Sin[2*(e + f*x)]))*(a + b*Tan[e + f*x])^3)/(f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3)","A",1
596,1,132,129,2.053226,"\int \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^3 \, dx","Integrate[Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])^3,x]","-\frac{2 \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^3 \left(5 b \left(b^2-3 a^2\right) \cos ^3(e+f x)-5 a \left(a^2-2 b^2\right) \cos ^{\frac{7}{2}}(e+f x) F\left(\left.\frac{1}{2} (e+f x)\right|2\right)-\frac{1}{2} b^2 \cos (e+f x) (5 a \sin (2 (e+f x))+2 b)\right)}{5 f (a \cos (e+f x)+b \sin (e+f x))^3}","\frac{2 b \sqrt{d \sec (e+f x)} \left(2 \left(7 a^2-2 b^2\right)+3 a b \tan (e+f x)\right)}{5 f}+\frac{2 a \left(a^2-2 b^2\right) \sqrt{d \sec (e+f x)} F\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{f \sqrt[4]{\sec ^2(e+f x)}}+\frac{2 b \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}{5 f}",1,"(-2*Sqrt[d*Sec[e + f*x]]*(5*b*(-3*a^2 + b^2)*Cos[e + f*x]^3 - 5*a*(a^2 - 2*b^2)*Cos[e + f*x]^(7/2)*EllipticF[(e + f*x)/2, 2] - (b^2*Cos[e + f*x]*(2*b + 5*a*Sin[2*(e + f*x)]))/2)*(a + b*Tan[e + f*x])^3)/(5*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3)","A",1
597,1,130,178,1.924248,"\int \frac{(a+b \tan (e+f x))^3}{\sqrt{d \sec (e+f x)}} \, dx","Integrate[(a + b*Tan[e + f*x])^3/Sqrt[d*Sec[e + f*x]],x]","\frac{d (a+b \tan (e+f x))^3 \left(6 a \left(a^2-6 b^2\right) \cos ^{\frac{3}{2}}(e+f x) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)+b \left(\left(3 b^2-9 a^2\right) \cos (2 (e+f x))-9 a^2+9 a b \sin (2 (e+f x))+5 b^2\right)\right)}{3 f (d \sec (e+f x))^{3/2} (a \cos (e+f x)+b \sin (e+f x))^3}","-\frac{2 b \sec ^2(e+f x) \left(2 \left(3 a^2-2 b^2\right)+3 a b \tan (e+f x)\right)}{3 f \sqrt{d \sec (e+f x)}}-\frac{2 a \left(a^2-6 b^2\right) \tan (e+f x)}{f \sqrt{d \sec (e+f x)}}+\frac{2 a \left(a^2-6 b^2\right) \sqrt[4]{\sec ^2(e+f x)} E\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{f \sqrt{d \sec (e+f x)}}-\frac{2 (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{f \sqrt{d \sec (e+f x)}}",1,"(d*(6*a*(a^2 - 6*b^2)*Cos[e + f*x]^(3/2)*EllipticE[(e + f*x)/2, 2] + b*(-9*a^2 + 5*b^2 + (-9*a^2 + 3*b^2)*Cos[2*(e + f*x)] + 9*a*b*Sin[2*(e + f*x)]))*(a + b*Tan[e + f*x])^3)/(3*f*(d*Sec[e + f*x])^(3/2)*(a*Cos[e + f*x] + b*Sin[e + f*x])^3)","A",1
598,1,117,146,1.3209297,"\int \frac{(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^3/(d*Sec[e + f*x])^(3/2),x]","\frac{\sec ^2(e+f x) \left(a^3 \sin (2 (e+f x))+\left(b^3-3 a^2 b\right) \cos (2 (e+f x))+2 a \left(a^2+6 b^2\right) \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)-3 a^2 b-3 a b^2 \sin (2 (e+f x))+7 b^3\right)}{3 f (d \sec (e+f x))^{3/2}}","-\frac{2 b \sec ^2(e+f x) \left(2 \left(a^2-2 b^2\right)+a b \tan (e+f x)\right)}{3 f (d \sec (e+f x))^{3/2}}+\frac{2 a \left(a^2+6 b^2\right) \sec ^2(e+f x)^{3/4} F\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{3 f (d \sec (e+f x))^{3/2}}-\frac{2 (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{3 f (d \sec (e+f x))^{3/2}}",1,"(Sec[e + f*x]^2*(-3*a^2*b + 7*b^3 + (-3*a^2*b + b^3)*Cos[2*(e + f*x)] + 2*a*(a^2 + 6*b^2)*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] + a^3*Sin[2*(e + f*x)] - 3*a*b^2*Sin[2*(e + f*x)]))/(3*f*(d*Sec[e + f*x])^(3/2))","A",1
599,1,150,204,1.4470886,"\int \frac{(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{5/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^3/(d*Sec[e + f*x])^(5/2),x]","\frac{\sqrt{d \sec (e+f x)} \left(a^3 \sin (e+f x)+a^3 \sin (3 (e+f x))-b \left(9 a^2+17 b^2\right) \cos (e+f x)+12 a \left(a^2+2 b^2\right) \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right)-3 a^2 b \cos (3 (e+f x))-3 a b^2 \sin (e+f x)-3 a b^2 \sin (3 (e+f x))+b^3 \cos (3 (e+f x))\right)}{10 d^3 f}","-\frac{6 a \left(a^2+2 b^2\right) \tan (e+f x)}{5 d^2 f \sqrt{d \sec (e+f x)}}-\frac{2 \left(2 b \left(a^2+2 b^2\right)-a \left(3 a^2+5 b^2\right) \tan (e+f x)\right)}{5 d^2 f \sqrt{d \sec (e+f x)}}+\frac{6 a \left(a^2+2 b^2\right) \sqrt[4]{\sec ^2(e+f x)} E\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{5 d^2 f \sqrt{d \sec (e+f x)}}-\frac{2 \cos ^2(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{5 d^2 f \sqrt{d \sec (e+f x)}}",1,"(Sqrt[d*Sec[e + f*x]]*(-(b*(9*a^2 + 17*b^2)*Cos[e + f*x]) - 3*a^2*b*Cos[3*(e + f*x)] + b^3*Cos[3*(e + f*x)] + 12*a*(a^2 + 2*b^2)*Sqrt[Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2] + a^3*Sin[e + f*x] - 3*a*b^2*Sin[e + f*x] + a^3*Sin[3*(e + f*x)] - 3*a*b^2*Sin[3*(e + f*x)]))/(10*d^3*f)","A",1
600,1,150,170,2.6025518,"\int \frac{(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{7/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^3/(d*Sec[e + f*x])^(7/2),x]","\frac{\sqrt{\cos (e+f x)} \sqrt{d \sec (e+f x)} \left(4 \left(5 a^3+6 a b^2\right) F\left(\left.\frac{1}{2} (e+f x)\right|2\right)+\sqrt{\cos (e+f x)} \left(\left(3 b^3-9 a^2 b\right) \cos (3 (e+f x))-b \left(27 a^2+19 b^2\right) \cos (e+f x)+2 a \sin (e+f x) \left(3 \left(a^2-3 b^2\right) \cos (2 (e+f x))+13 a^2+3 b^2\right)\right)\right)}{42 d^4 f}","-\frac{2 \left(2 b \left(3 a^2+2 b^2\right)-a \left(5 a^2+3 b^2\right) \tan (e+f x)\right)}{21 d^2 f (d \sec (e+f x))^{3/2}}+\frac{2 a \left(5 a^2+6 b^2\right) \sec ^2(e+f x)^{3/4} F\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{21 d^2 f (d \sec (e+f x))^{3/2}}-\frac{2 \cos ^2(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{7 d^2 f (d \sec (e+f x))^{3/2}}",1,"(Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]]*(4*(5*a^3 + 6*a*b^2)*EllipticF[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(-(b*(27*a^2 + 19*b^2)*Cos[e + f*x]) + (-9*a^2*b + 3*b^3)*Cos[3*(e + f*x)] + 2*a*(13*a^2 + 3*b^2 + 3*(a^2 - 3*b^2)*Cos[2*(e + f*x)])*Sin[e + f*x])))/(42*d^4*f)","A",1
601,1,372,176,6.3959628,"\int \frac{(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{9/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^3/(d*Sec[e + f*x])^(9/2),x]","\frac{\sec ^2(e+f x) (a+b \tan (e+f x))^3 \left(\frac{1}{180} a \left(19 a^2-3 b^2\right) \sin (e+f x)+\frac{1}{360} a \left(43 a^2-21 b^2\right) \sin (3 (e+f x))+\frac{1}{72} a \left(a^2-3 b^2\right) \sin (5 (e+f x))-\frac{1}{90} b \left(15 a^2+4 b^2\right) \cos (e+f x)-\frac{1}{360} b \left(75 a^2+11 b^2\right) \cos (3 (e+f x))-\frac{1}{72} b \left(3 a^2-b^2\right) \cos (5 (e+f x))\right)}{f (d \sec (e+f x))^{9/2} (a \cos (e+f x)+b \sin (e+f x))^3}+\frac{\sec ^{\frac{3}{2}}(e+f x) (a+b \tan (e+f x))^3 \left(\frac{2 \left(56 a^3+48 a b^2\right) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{\sqrt{\cos (e+f x)} \sqrt{\sec (e+f x)}}-\frac{2 \left(15 a^2 b+7 b^3\right) \sin ^2(e+f x)}{\sqrt{1-\cos ^2(e+f x)} \sqrt{\sec (e+f x)} \sqrt{\cos ^2(e+f x) \left(\sec ^2(e+f x)-1\right)}}\right)}{120 f (d \sec (e+f x))^{9/2} (a \cos (e+f x)+b \sin (e+f x))^3}","-\frac{2 \cos ^2(e+f x) \left(2 b \left(5 a^2+2 b^2\right)-a \left(7 a^2+b^2\right) \tan (e+f x)\right)}{45 d^4 f \sqrt{d \sec (e+f x)}}+\frac{2 a \left(7 a^2+6 b^2\right) \sqrt[4]{\sec ^2(e+f x)} E\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{15 d^4 f \sqrt{d \sec (e+f x)}}-\frac{2 \cos ^4(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{9 d^4 f \sqrt{d \sec (e+f x)}}",1,"(Sec[e + f*x]^(3/2)*((2*(56*a^3 + 48*a*b^2)*EllipticE[(e + f*x)/2, 2])/(Sqrt[Cos[e + f*x]]*Sqrt[Sec[e + f*x]]) - (2*(15*a^2*b + 7*b^3)*Sin[e + f*x]^2)/(Sqrt[1 - Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]]*Sqrt[Cos[e + f*x]^2*(-1 + Sec[e + f*x]^2)]))*(a + b*Tan[e + f*x])^3)/(120*f*(d*Sec[e + f*x])^(9/2)*(a*Cos[e + f*x] + b*Sin[e + f*x])^3) + (Sec[e + f*x]^2*(-1/90*(b*(15*a^2 + 4*b^2)*Cos[e + f*x]) - (b*(75*a^2 + 11*b^2)*Cos[3*(e + f*x)])/360 - (b*(3*a^2 - b^2)*Cos[5*(e + f*x)])/72 + (a*(19*a^2 - 3*b^2)*Sin[e + f*x])/180 + (a*(43*a^2 - 21*b^2)*Sin[3*(e + f*x)])/360 + (a*(a^2 - 3*b^2)*Sin[5*(e + f*x)])/72)*(a + b*Tan[e + f*x])^3)/(f*(d*Sec[e + f*x])^(9/2)*(a*Cos[e + f*x] + b*Sin[e + f*x])^3)","B",1
602,1,296,218,6.4614607,"\int \frac{(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{11/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^3/(d*Sec[e + f*x])^(11/2),x]","\frac{\sec ^3(e+f x) (a+b \tan (e+f x))^3 \left(\frac{a \left(347 a^2+103 b^2\right) \sin (2 (e+f x))}{1232}+\frac{1}{308} a \left(16 a^2-15 b^2\right) \sin (4 (e+f x))+\frac{1}{176} a \left(a^2-3 b^2\right) \sin (6 (e+f x))-\frac{b \left(315 a^2+71 b^2\right) \cos (2 (e+f x))}{1232}-\frac{1}{616} b \left(63 a^2+b^2\right) \cos (4 (e+f x))-\frac{1}{176} b \left(3 a^2-b^2\right) \cos (6 (e+f x))-\frac{1}{616} b \left(105 a^2+31 b^2\right)\right)}{f (d \sec (e+f x))^{11/2} (a \cos (e+f x)+b \sin (e+f x))^3}+\frac{10 a \left(3 a^2+2 b^2\right) F\left(\left.\frac{1}{2} (e+f x)\right|2\right) (a+b \tan (e+f x))^3}{77 f \cos ^{\frac{5}{2}}(e+f x) (d \sec (e+f x))^{11/2} (a \cos (e+f x)+b \sin (e+f x))^3}","\frac{10 a \left(3 a^2+2 b^2\right) \tan (e+f x)}{77 d^4 f (d \sec (e+f x))^{3/2}}-\frac{2 \cos ^2(e+f x) \left(2 b \left(7 a^2+2 b^2\right)-a \left(9 a^2-b^2\right) \tan (e+f x)\right)}{77 d^4 f (d \sec (e+f x))^{3/2}}+\frac{10 a \left(3 a^2+2 b^2\right) \sec ^2(e+f x)^{3/4} F\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{77 d^4 f (d \sec (e+f x))^{3/2}}-\frac{2 \cos ^4(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{11 d^4 f (d \sec (e+f x))^{3/2}}",1,"(10*a*(3*a^2 + 2*b^2)*EllipticF[(e + f*x)/2, 2]*(a + b*Tan[e + f*x])^3)/(77*f*Cos[e + f*x]^(5/2)*(d*Sec[e + f*x])^(11/2)*(a*Cos[e + f*x] + b*Sin[e + f*x])^3) + (Sec[e + f*x]^3*(-1/616*(b*(105*a^2 + 31*b^2)) - (b*(315*a^2 + 71*b^2)*Cos[2*(e + f*x)])/1232 - (b*(63*a^2 + b^2)*Cos[4*(e + f*x)])/616 - (b*(3*a^2 - b^2)*Cos[6*(e + f*x)])/176 + (a*(347*a^2 + 103*b^2)*Sin[2*(e + f*x)])/1232 + (a*(16*a^2 - 15*b^2)*Sin[4*(e + f*x)])/308 + (a*(a^2 - 3*b^2)*Sin[6*(e + f*x)])/176)*(a + b*Tan[e + f*x])^3)/(f*(d*Sec[e + f*x])^(11/2)*(a*Cos[e + f*x] + b*Sin[e + f*x])^3)","A",1
603,1,7284,456,29.5313547,"\int \frac{(d \sec (e+f x))^{7/2}}{a+b \tan (e+f x)} \, dx","Integrate[(d*Sec[e + f*x])^(7/2)/(a + b*Tan[e + f*x]),x]","\text{Result too large to show}","\frac{d^2 \left(a^2+b^2\right)^{3/4} (d \sec (e+f x))^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{b^{5/2} f \sec ^2(e+f x)^{3/4}}-\frac{d^2 \left(a^2+b^2\right)^{3/4} (d \sec (e+f x))^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{b^{5/2} f \sec ^2(e+f x)^{3/4}}-\frac{a d^2 \sqrt{a^2+b^2} \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{b^3 f \sec ^2(e+f x)^{3/4}}+\frac{a d^2 \sqrt{a^2+b^2} \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{b^3 f \sec ^2(e+f x)^{3/4}}-\frac{2 a d^2 \sin (e+f x) \cos (e+f x) (d \sec (e+f x))^{3/2}}{b^2 f}+\frac{2 a d^2 (d \sec (e+f x))^{3/2} E\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{b^2 f \sec ^2(e+f x)^{3/4}}+\frac{2 d^2 (d \sec (e+f x))^{3/2}}{3 b f}",1,"Result too large to show","B",0
604,1,6607,396,25.0334932,"\int \frac{(d \sec (e+f x))^{5/2}}{a+b \tan (e+f x)} \, dx","Integrate[(d*Sec[e + f*x])^(5/2)/(a + b*Tan[e + f*x]),x]","\text{Result too large to show}","\frac{a d^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{b^2 f \sqrt[4]{\sec ^2(e+f x)}}+\frac{a d^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{b^2 f \sqrt[4]{\sec ^2(e+f x)}}-\frac{d^2 \sqrt[4]{a^2+b^2} \sqrt{d \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{b^{3/2} f \sqrt[4]{\sec ^2(e+f x)}}-\frac{d^2 \sqrt[4]{a^2+b^2} \sqrt{d \sec (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{b^{3/2} f \sqrt[4]{\sec ^2(e+f x)}}-\frac{2 a d^2 \sqrt{d \sec (e+f x)} F\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{b^2 f \sqrt[4]{\sec ^2(e+f x)}}+\frac{2 d^2 \sqrt{d \sec (e+f x)}}{b f}",1,"Result too large to show","B",0
605,1,6301,334,21.9383341,"\int \frac{(d \sec (e+f x))^{3/2}}{a+b \tan (e+f x)} \, dx","Integrate[(d*Sec[e + f*x])^(3/2)/(a + b*Tan[e + f*x]),x]","\text{Result too large to show}","\frac{(d \sec (e+f x))^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{\sqrt{b} f \sqrt[4]{a^2+b^2} \sec ^2(e+f x)^{3/4}}-\frac{(d \sec (e+f x))^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{\sqrt{b} f \sqrt[4]{a^2+b^2} \sec ^2(e+f x)^{3/4}}-\frac{a \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{b f \sqrt{a^2+b^2} \sec ^2(e+f x)^{3/4}}+\frac{a \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{b f \sqrt{a^2+b^2} \sec ^2(e+f x)^{3/4}}",1,"Result too large to show","C",0
606,1,4648,324,24.4075622,"\int \frac{\sqrt{d \sec (e+f x)}}{a+b \tan (e+f x)} \, dx","Integrate[Sqrt[d*Sec[e + f*x]]/(a + b*Tan[e + f*x]),x]","\text{Result too large to show}","-\frac{\sqrt{b} \sqrt{d \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{f \left(a^2+b^2\right)^{3/4} \sqrt[4]{\sec ^2(e+f x)}}-\frac{\sqrt{b} \sqrt{d \sec (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{f \left(a^2+b^2\right)^{3/4} \sqrt[4]{\sec ^2(e+f x)}}+\frac{a \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{f \left(a^2+b^2\right) \sqrt[4]{\sec ^2(e+f x)}}+\frac{a \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{f \left(a^2+b^2\right) \sqrt[4]{\sec ^2(e+f x)}}",1,"(-2*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sqrt[d*Sec[e + f*x]]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2)*(-1 + Tan[(e + f*x)/2]^2)*(EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] + (((-2*I)*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(-a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2])*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(I + Tan[(e + f*x)/2])^2)/(Sqrt[2]*(a - I*b)*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])))/(a*f*Sqrt[Sec[(e + f*x)/2]^2]*(a + b*Tan[e + f*x])*((-2*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2)*Tan[(e + f*x)/2]*(EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] + (((-2*I)*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(-a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2])*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(I + Tan[(e + f*x)/2])^2)/(Sqrt[2]*(a - I*b)*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])))/a + (Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)*(EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] + (((-2*I)*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(-a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2])*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(I + Tan[(e + f*x)/2])^2)/(Sqrt[2]*(a - I*b)*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])))/(a*Sqrt[Sec[(e + f*x)/2]^2]) - ((Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2)*(-(Sec[(e + f*x)/2]^4*Sin[e + f*x]) + 2*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Tan[(e + f*x)/2])*(-1 + Tan[(e + f*x)/2]^2)*(EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] + (((-2*I)*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(-a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2])*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(I + Tan[(e + f*x)/2])^2)/(Sqrt[2]*(a - I*b)*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])))/(a*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]) - (2*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2)*(-1 + Tan[(e + f*x)/2]^2)*((((-2*I)*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(-a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2])*Sec[(e + f*x)/2]^2*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(I + Tan[(e + f*x)/2]))/(Sqrt[2]*(a - I*b)*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]) + (((-2*I)*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(-a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2])*(-Cos[e + f*x] - I*Sin[e + f*x])*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(I + Tan[(e + f*x)/2])^2)/(2*Sqrt[2]*(a - I*b)*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]) + (((-2*I)*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(-a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2])*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*(Cos[e + f*x]*(I*Cos[e + f*x] - Sin[e + f*x]) - (Cos[e + f*x] + I*Sin[e + f*x])*Sin[e + f*x])*(I + Tan[(e + f*x)/2])^2)/(2*Sqrt[2]*(a - I*b)*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]) + (Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(((-I)*b*Sqrt[a^2 + b^2]*(Cos[e + f*x] + I*Sin[e + f*x]))/(Sqrt[2]*Sqrt[1 + (-1 + I*Cos[e + f*x] - Sin[e + f*x])/2]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]) + (a*(a - I*b + Sqrt[a^2 + b^2])*(Cos[e + f*x] + I*Sin[e + f*x]))/(2*Sqrt[2]*Sqrt[1 + (-1 + I*Cos[e + f*x] - Sin[e + f*x])/2]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]*(1 - ((1/2 + I/2)*(a + I*(-b + Sqrt[a^2 + b^2]))*(1 - I*Cos[e + f*x] + Sin[e + f*x]))/(a + b - Sqrt[a^2 + b^2]))) + (a*(-a + I*b + Sqrt[a^2 + b^2])*(Cos[e + f*x] + I*Sin[e + f*x]))/(2*Sqrt[2]*Sqrt[1 + (-1 + I*Cos[e + f*x] - Sin[e + f*x])/2]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]*(1 - ((1/2 + I/2)*(a - I*(b + Sqrt[a^2 + b^2]))*(1 - I*Cos[e + f*x] + Sin[e + f*x]))/(a + b + Sqrt[a^2 + b^2]))))*(I + Tan[(e + f*x)/2])^2)/(Sqrt[2]*(a - I*b)*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]) - (((-2*I)*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(-a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2])*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(I + Tan[(e + f*x)/2])^2*(-(Sec[(e + f*x)/2]^4*Sin[e + f*x]) + 2*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Tan[(e + f*x)/2]))/(2*Sqrt[2]*(a - I*b)*Sqrt[a^2 + b^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^4)^(3/2)) + Sec[(e + f*x)/2]^2/(2*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[1 + Tan[(e + f*x)/2]^2])))/(a*Sqrt[Sec[(e + f*x)/2]^2]) - (3*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*(-1 + Tan[(e + f*x)/2]^2)*(EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] + (((-2*I)*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(-a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2])*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(I + Tan[(e + f*x)/2])^2)/(Sqrt[2]*(a - I*b)*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]))*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(a*Sqrt[Sec[(e + f*x)/2]^2])))","C",0
607,1,4693,451,31.1059542,"\int \frac{1}{\sqrt{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx","Integrate[1/(Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])),x]","\text{Result too large to show}","-\frac{2 a \tan (e+f x)}{f \left(a^2+b^2\right) \sqrt{d \sec (e+f x)}}+\frac{2 (a \tan (e+f x)+b)}{f \left(a^2+b^2\right) \sqrt{d \sec (e+f x)}}+\frac{2 a \sqrt[4]{\sec ^2(e+f x)} E\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{f \left(a^2+b^2\right) \sqrt{d \sec (e+f x)}}-\frac{a b \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt[4]{\sec ^2(e+f x)} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{f \left(a^2+b^2\right)^{3/2} \sqrt{d \sec (e+f x)}}+\frac{a b \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt[4]{\sec ^2(e+f x)} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{f \left(a^2+b^2\right)^{3/2} \sqrt{d \sec (e+f x)}}+\frac{b^{3/2} \sqrt[4]{\sec ^2(e+f x)} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{f \left(a^2+b^2\right)^{5/4} \sqrt{d \sec (e+f x)}}-\frac{b^{3/2} \sqrt[4]{\sec ^2(e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{f \left(a^2+b^2\right)^{5/4} \sqrt{d \sec (e+f x)}}",1,"(-2*Sec[e + f*x]^(3/2)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2)*(a*Cos[e + f*x] + b*Sin[e + f*x])*(Sqrt[Sec[e + f*x]]/(2*(a*Cos[e + f*x] + b*Sin[e + f*x])) + (Cos[2*(e + f*x)]*Sqrt[Sec[e + f*x]])/(2*(a*Cos[e + f*x] + b*Sin[e + f*x])))*(-1 + Tan[(e + f*x)/2]^2)^2*(-((a*Sqrt[(1 + Cos[e + f*x])^(-1)]*EllipticE[ArcSin[Tan[(e + f*x)/2]], -1])/Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]) + ((a^2 + b^2)*Sqrt[(1 + Cos[e + f*x])^(-1)]*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1])/(a*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]) + (-2*Sqrt[2]*b^2*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]] + a*(b*(a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*Sqrt[(2*I)*Cos[e + f*x] - 2*Sin[e + f*x]] + b*(-a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*Sqrt[(2*I)*Cos[e + f*x] - 2*Sin[e + f*x]] - 2*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(b + a*Tan[(e + f*x)/2])))/(2*a*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])])))/((a^2 + b^2)*f*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])*((-4*Sqrt[Sec[(e + f*x)/2]^2]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)*(-((a*Sqrt[(1 + Cos[e + f*x])^(-1)]*EllipticE[ArcSin[Tan[(e + f*x)/2]], -1])/Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]) + ((a^2 + b^2)*Sqrt[(1 + Cos[e + f*x])^(-1)]*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1])/(a*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]) + (-2*Sqrt[2]*b^2*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]] + a*(b*(a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*Sqrt[(2*I)*Cos[e + f*x] - 2*Sin[e + f*x]] + b*(-a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*Sqrt[(2*I)*Cos[e + f*x] - 2*Sin[e + f*x]] - 2*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(b + a*Tan[(e + f*x)/2])))/(2*a*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])])))/(a^2 + b^2) + ((Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2*(-((a*Sqrt[(1 + Cos[e + f*x])^(-1)]*EllipticE[ArcSin[Tan[(e + f*x)/2]], -1])/Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]) + ((a^2 + b^2)*Sqrt[(1 + Cos[e + f*x])^(-1)]*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1])/(a*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]) + (-2*Sqrt[2]*b^2*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]] + a*(b*(a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*Sqrt[(2*I)*Cos[e + f*x] - 2*Sin[e + f*x]] + b*(-a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*Sqrt[(2*I)*Cos[e + f*x] - 2*Sin[e + f*x]] - 2*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(b + a*Tan[(e + f*x)/2])))/(2*a*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])])))/((a^2 + b^2)*Sqrt[Sec[(e + f*x)/2]^2]) - (2*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2)*(-1 + Tan[(e + f*x)/2]^2)^2*(-1/2*(a*((1 + Cos[e + f*x])^(-1))^(3/2)*EllipticE[ArcSin[Tan[(e + f*x)/2]], -1]*Sin[e + f*x])/Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])] + ((a^2 + b^2)*((1 + Cos[e + f*x])^(-1))^(3/2)*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1]*Sin[e + f*x])/(2*a*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]) + (a*Sqrt[(1 + Cos[e + f*x])^(-1)]*EllipticE[ArcSin[Tan[(e + f*x)/2]], -1]*((Cos[e + f*x]*Sin[e + f*x])/(1 + Cos[e + f*x])^2 - Sin[e + f*x]/(1 + Cos[e + f*x])))/(2*(Cos[e + f*x]/(1 + Cos[e + f*x]))^(3/2)) - ((a^2 + b^2)*Sqrt[(1 + Cos[e + f*x])^(-1)]*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1]*((Cos[e + f*x]*Sin[e + f*x])/(1 + Cos[e + f*x])^2 - Sin[e + f*x]/(1 + Cos[e + f*x])))/(2*a*(Cos[e + f*x]/(1 + Cos[e + f*x]))^(3/2)) + ((a^2 + b^2)*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sec[(e + f*x)/2]^2)/(2*a*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[1 + Tan[(e + f*x)/2]^2]) - (a*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sec[(e + f*x)/2]^2*Sqrt[1 + Tan[(e + f*x)/2]^2])/(2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[1 - Tan[(e + f*x)/2]^2]) - ((Cos[e + f*x]*(I*Cos[e + f*x] - Sin[e + f*x]) - (Cos[e + f*x] + I*Sin[e + f*x])*Sin[e + f*x])*(-2*Sqrt[2]*b^2*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]] + a*(b*(a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*Sqrt[(2*I)*Cos[e + f*x] - 2*Sin[e + f*x]] + b*(-a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*Sqrt[(2*I)*Cos[e + f*x] - 2*Sin[e + f*x]] - 2*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(b + a*Tan[(e + f*x)/2]))))/(4*a*Sqrt[a^2 + b^2]*(Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x]))^(3/2)) + (-((Sqrt[2]*b^2*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*(-Cos[e + f*x] - I*Sin[e + f*x]))/Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]) - (b^2*Sqrt[a^2 + b^2]*(Cos[e + f*x] + I*Sin[e + f*x]))/(Sqrt[1 + (-1 + I*Cos[e + f*x] - Sin[e + f*x])/2]*Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]) + a*(-(a*Sqrt[a^2 + b^2]*Sec[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]) + (b*(a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*(-2*Cos[e + f*x] - (2*I)*Sin[e + f*x]))/(2*Sqrt[(2*I)*Cos[e + f*x] - 2*Sin[e + f*x]]) + (b*(-a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*(-2*Cos[e + f*x] - (2*I)*Sin[e + f*x]))/(2*Sqrt[(2*I)*Cos[e + f*x] - 2*Sin[e + f*x]]) + (b*(a + I*b + Sqrt[a^2 + b^2])*Sqrt[(2*I)*Cos[e + f*x] - 2*Sin[e + f*x]]*(Cos[e + f*x] + I*Sin[e + f*x]))/(2*Sqrt[2]*Sqrt[1 + (-1 + I*Cos[e + f*x] - Sin[e + f*x])/2]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]*(1 - ((1/2 + I/2)*(a + I*(-b + Sqrt[a^2 + b^2]))*(1 - I*Cos[e + f*x] + Sin[e + f*x]))/(a + b - Sqrt[a^2 + b^2]))) + (b*(-a - I*b + Sqrt[a^2 + b^2])*Sqrt[(2*I)*Cos[e + f*x] - 2*Sin[e + f*x]]*(Cos[e + f*x] + I*Sin[e + f*x]))/(2*Sqrt[2]*Sqrt[1 + (-1 + I*Cos[e + f*x] - Sin[e + f*x])/2]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]*(1 - ((1/2 + I/2)*(a - I*(b + Sqrt[a^2 + b^2]))*(1 - I*Cos[e + f*x] + Sin[e + f*x]))/(a + b + Sqrt[a^2 + b^2]))) - (Sqrt[a^2 + b^2]*(Cos[e + f*x]*(I*Cos[e + f*x] - Sin[e + f*x]) - (Cos[e + f*x] + I*Sin[e + f*x])*Sin[e + f*x])*(b + a*Tan[(e + f*x)/2]))/Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]))/(2*a*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])])))/((a^2 + b^2)*Sqrt[Sec[(e + f*x)/2]^2]) - (3*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*(-1 + Tan[(e + f*x)/2]^2)^2*(-((a*Sqrt[(1 + Cos[e + f*x])^(-1)]*EllipticE[ArcSin[Tan[(e + f*x)/2]], -1])/Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]) + ((a^2 + b^2)*Sqrt[(1 + Cos[e + f*x])^(-1)]*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1])/(a*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]) + (-2*Sqrt[2]*b^2*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]] + a*(b*(a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*Sqrt[(2*I)*Cos[e + f*x] - 2*Sin[e + f*x]] + b*(-a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2]*Sqrt[(2*I)*Cos[e + f*x] - 2*Sin[e + f*x]] - 2*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(b + a*Tan[(e + f*x)/2])))/(2*a*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]))*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/((a^2 + b^2)*Sqrt[Sec[(e + f*x)/2]^2])))","C",0
608,1,9313,422,27.4326415,"\int \frac{1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx","Integrate[1/((d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])),x]","\text{Result too large to show}","\frac{2 (a \tan (e+f x)+b)}{3 f \left(a^2+b^2\right) (d \sec (e+f x))^{3/2}}+\frac{2 a \sec ^2(e+f x)^{3/4} F\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{3 f \left(a^2+b^2\right) (d \sec (e+f x))^{3/2}}+\frac{a b^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{f \left(a^2+b^2\right)^2 (d \sec (e+f x))^{3/2}}+\frac{a b^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{f \left(a^2+b^2\right)^2 (d \sec (e+f x))^{3/2}}-\frac{b^{5/2} \sec ^2(e+f x)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{f \left(a^2+b^2\right)^{7/4} (d \sec (e+f x))^{3/2}}-\frac{b^{5/2} \sec ^2(e+f x)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{f \left(a^2+b^2\right)^{7/4} (d \sec (e+f x))^{3/2}}",1,"Result too large to show","C",0
609,1,17838,568,33.6414395,"\int \frac{1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))} \, dx","Integrate[1/((d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x])),x]","\text{Result too large to show}","-\frac{2 a \left(3 a^2+8 b^2\right) \tan (e+f x)}{5 d^2 f \left(a^2+b^2\right)^2 \sqrt{d \sec (e+f x)}}+\frac{2 \cos ^2(e+f x) (a \tan (e+f x)+b)}{5 d^2 f \left(a^2+b^2\right) \sqrt{d \sec (e+f x)}}+\frac{2 a \left(3 a^2+8 b^2\right) \sqrt[4]{\sec ^2(e+f x)} E\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{5 d^2 f \left(a^2+b^2\right)^2 \sqrt{d \sec (e+f x)}}+\frac{b^{7/2} \sqrt[4]{\sec ^2(e+f x)} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{d^2 f \left(a^2+b^2\right)^{9/4} \sqrt{d \sec (e+f x)}}-\frac{b^{7/2} \sqrt[4]{\sec ^2(e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{d^2 f \left(a^2+b^2\right)^{9/4} \sqrt{d \sec (e+f x)}}+\frac{2 \left(a \left(3 a^2+8 b^2\right) \tan (e+f x)+5 b^3\right)}{5 d^2 f \left(a^2+b^2\right)^2 \sqrt{d \sec (e+f x)}}-\frac{a b^3 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt[4]{\sec ^2(e+f x)} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{d^2 f \left(a^2+b^2\right)^{5/2} \sqrt{d \sec (e+f x)}}+\frac{a b^3 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt[4]{\sec ^2(e+f x)} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{d^2 f \left(a^2+b^2\right)^{5/2} \sqrt{d \sec (e+f x)}}",1,"Result too large to show","C",0
610,1,1129,480,23.2541805,"\int \frac{(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx","Integrate[(d*Sec[e + f*x])^(7/2)/(a + b*Tan[e + f*x])^2,x]","\frac{\cos (e+f x) (a \cos (e+f x)+b \sin (e+f x))^2 \left(\frac{3 \cos (e+f x)}{a b}+\frac{3 \sin (e+f x)}{b^2}-\frac{1}{b (a \cos (e+f x)+b \sin (e+f x))}\right) (d \sec (e+f x))^{7/2}}{f (a+b \tan (e+f x))^2}+\frac{3 (a \cos (e+f x)+b \sin (e+f x))^2 \left(-\frac{a \sqrt{\tan ^2\left(\frac{1}{2} (e+f x)\right)+1} E\left(\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)\right|-1\right)}{\sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)}}+\frac{-\Pi \left(\frac{(1+i) \left(a-i \left(b+\sqrt{a^2+b^2}\right)\right)}{a+b+\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(1+i) \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)}{\tan \left(\frac{1}{2} (e+f x)\right)+i}}}{\sqrt{2}}\right)\right|2\right) \sqrt{-\frac{2 i \tan \left(\frac{1}{2} (e+f x)\right)+2}{\tan \left(\frac{1}{2} (e+f x)\right)+i}} a^3+\sqrt{2} \sqrt{a^2+b^2} \Pi \left(\frac{(1+i) \left(a-i \left(b+\sqrt{a^2+b^2}\right)\right)}{a+b+\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(1+i) \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)}{\tan \left(\frac{1}{2} (e+f x)\right)+i}}}{\sqrt{2}}\right)\right|2\right) \sqrt{-\frac{i \tan \left(\frac{1}{2} (e+f x)\right)+1}{\tan \left(\frac{1}{2} (e+f x)\right)+i}} a^2+\left(a+i b+\sqrt{a^2+b^2}\right) \Pi \left(\frac{(1+i) \left(a+i \left(\sqrt{a^2+b^2}-b\right)\right)}{a+b-\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(1+i) \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)}{\tan \left(\frac{1}{2} (e+f x)\right)+i}}}{\sqrt{2}}\right)\right|2\right) \sqrt{-\frac{2 i \tan \left(\frac{1}{2} (e+f x)\right)+2}{\tan \left(\frac{1}{2} (e+f x)\right)+i}} a^2-i b \Pi \left(\frac{(1+i) \left(a-i \left(b+\sqrt{a^2+b^2}\right)\right)}{a+b+\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(1+i) \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)}{\tan \left(\frac{1}{2} (e+f x)\right)+i}}}{\sqrt{2}}\right)\right|2\right) \sqrt{-\frac{2 i \tan \left(\frac{1}{2} (e+f x)\right)+2}{\tan \left(\frac{1}{2} (e+f x)\right)+i}} a^2-2 \sqrt{2} b \sqrt{a^2+b^2} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(1+i) \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)}{\tan \left(\frac{1}{2} (e+f x)\right)+i}}}{\sqrt{2}}\right)\right|2\right) \sqrt{-\frac{i \tan \left(\frac{1}{2} (e+f x)\right)+1}{\tan \left(\frac{1}{2} (e+f x)\right)+i}} a-2 b \sqrt{a^2+b^2} \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (e+f x)\right)-1}{\left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)^2}} a-2 b^2 \sqrt{a^2+b^2} \sqrt{\frac{\tan ^2\left(\frac{1}{2} (e+f x)\right)-1}{\left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)^2}}}{2 b \sqrt{a^2+b^2} \sqrt{\frac{\tan ^2\left(\frac{1}{2} (e+f x)\right)-1}{\left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)^2}}}+\frac{2 a F\left(\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)\right|-1\right) \sqrt{\tan ^2\left(\frac{1}{2} (e+f x)\right)+1}}{\sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right) (d \sec (e+f x))^{7/2}}{a b^2 f \sec ^{\frac{3}{2}}(e+f x) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (e+f x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)}} (a+b \tan (e+f x))^2}","-\frac{3 a d^2 (d \sec (e+f x))^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{2 b^{5/2} f \sqrt[4]{a^2+b^2} \sec ^2(e+f x)^{3/4}}+\frac{3 a d^2 (d \sec (e+f x))^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{2 b^{5/2} f \sqrt[4]{a^2+b^2} \sec ^2(e+f x)^{3/4}}+\frac{3 a^2 d^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{2 b^3 f \sqrt{a^2+b^2} \sec ^2(e+f x)^{3/4}}-\frac{3 a^2 d^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{2 b^3 f \sqrt{a^2+b^2} \sec ^2(e+f x)^{3/4}}-\frac{d^2 (d \sec (e+f x))^{3/2}}{b f (a+b \tan (e+f x))}+\frac{3 d^2 \sin (e+f x) \cos (e+f x) (d \sec (e+f x))^{3/2}}{b^2 f}-\frac{3 d^2 (d \sec (e+f x))^{3/2} E\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{b^2 f \sec ^2(e+f x)^{3/4}}",1,"(Cos[e + f*x]*(d*Sec[e + f*x])^(7/2)*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*((3*Cos[e + f*x])/(a*b) + (3*Sin[e + f*x])/b^2 - 1/(b*(a*Cos[e + f*x] + b*Sin[e + f*x]))))/(f*(a + b*Tan[e + f*x])^2) + (3*(d*Sec[e + f*x])^(7/2)*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(-((a*EllipticE[ArcSin[Tan[(e + f*x)/2]], -1]*Sqrt[1 + Tan[(e + f*x)/2]^2])/Sqrt[1 - Tan[(e + f*x)/2]^2]) + (2*a*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1]*Sqrt[1 + Tan[(e + f*x)/2]^2])/Sqrt[1 - Tan[(e + f*x)/2]^2] + (-2*Sqrt[2]*a*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[((1 + I)*(1 + Tan[(e + f*x)/2]))/(I + Tan[(e + f*x)/2])]/Sqrt[2]], 2]*Sqrt[-((1 + I*Tan[(e + f*x)/2])/(I + Tan[(e + f*x)/2]))] + Sqrt[2]*a^2*Sqrt[a^2 + b^2]*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[((1 + I)*(1 + Tan[(e + f*x)/2]))/(I + Tan[(e + f*x)/2])]/Sqrt[2]], 2]*Sqrt[-((1 + I*Tan[(e + f*x)/2])/(I + Tan[(e + f*x)/2]))] + a^2*(a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[((1 + I)*(1 + Tan[(e + f*x)/2]))/(I + Tan[(e + f*x)/2])]/Sqrt[2]], 2]*Sqrt[-((2 + (2*I)*Tan[(e + f*x)/2])/(I + Tan[(e + f*x)/2]))] - a^3*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[((1 + I)*(1 + Tan[(e + f*x)/2]))/(I + Tan[(e + f*x)/2])]/Sqrt[2]], 2]*Sqrt[-((2 + (2*I)*Tan[(e + f*x)/2])/(I + Tan[(e + f*x)/2]))] - I*a^2*b*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[((1 + I)*(1 + Tan[(e + f*x)/2]))/(I + Tan[(e + f*x)/2])]/Sqrt[2]], 2]*Sqrt[-((2 + (2*I)*Tan[(e + f*x)/2])/(I + Tan[(e + f*x)/2]))] - 2*b^2*Sqrt[a^2 + b^2]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)/(I + Tan[(e + f*x)/2])^2] - 2*a*b*Sqrt[a^2 + b^2]*Tan[(e + f*x)/2]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)/(I + Tan[(e + f*x)/2])^2])/(2*b*Sqrt[a^2 + b^2]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)/(I + Tan[(e + f*x)/2])^2])))/(a*b^2*f*Sec[e + f*x]^(3/2)*Sqrt[(1 + Tan[(e + f*x)/2]^2)/(1 - Tan[(e + f*x)/2]^2)]*(a + b*Tan[e + f*x])^2)","C",0
611,1,3091,440,22.0504041,"\int \frac{(d \sec (e+f x))^{5/2}}{(a+b \tan (e+f x))^2} \, dx","Integrate[(d*Sec[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^2,x]","\text{Result too large to show}","-\frac{a^2 d^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{2 b^2 f \left(a^2+b^2\right) \sqrt[4]{\sec ^2(e+f x)}}-\frac{a^2 d^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{2 b^2 f \left(a^2+b^2\right) \sqrt[4]{\sec ^2(e+f x)}}+\frac{a d^2 \sqrt{d \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{2 b^{3/2} f \left(a^2+b^2\right)^{3/4} \sqrt[4]{\sec ^2(e+f x)}}+\frac{a d^2 \sqrt{d \sec (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{2 b^{3/2} f \left(a^2+b^2\right)^{3/4} \sqrt[4]{\sec ^2(e+f x)}}-\frac{d^2 \sqrt{d \sec (e+f x)}}{b f (a+b \tan (e+f x))}+\frac{d^2 \sqrt{d \sec (e+f x)} F\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{b^2 f \sqrt[4]{\sec ^2(e+f x)}}",1,"((d*Sec[e + f*x])^(5/2)*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(-(1/(a*b)) + Sin[e + f*x]/(a*(a*Cos[e + f*x] + b*Sin[e + f*x]))))/(f*(a + b*Tan[e + f*x])^2) - (((-2*I)*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(-a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2])*(d*Sec[e + f*x])^(5/2)*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*Sin[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])*(I + Tan[(e + f*x)/2])^2)/(4*(a - I*b)*b^3*Sqrt[a^2 + b^2]*f*Sqrt[(1 + Cos[e + f*x])^(-1)]*(a + b*Tan[e + f*x])^2*(-1/2*(((-2*I)*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(-a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2])*Sec[(e + f*x)/2]^2*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(I + Tan[(e + f*x)/2]))/((a - I*b)*b^2*Sqrt[a^2 + b^2]*Sqrt[(1 + Cos[e + f*x])^(-1)]) - (((-2*I)*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(-a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2])*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*(-Cos[e + f*x] - I*Sin[e + f*x])*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(I + Tan[(e + f*x)/2])^2)/(4*(a - I*b)*b^2*Sqrt[a^2 + b^2]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]) + (Sqrt[(1 + Cos[e + f*x])^(-1)]*((-2*I)*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(-a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2])*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*Sin[e + f*x]*(I + Tan[(e + f*x)/2])^2)/(4*(a - I*b)*b^2*Sqrt[a^2 + b^2]) - (((-2*I)*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(-a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2])*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*(Cos[e + f*x]*(I*Cos[e + f*x] - Sin[e + f*x]) - (Cos[e + f*x] + I*Sin[e + f*x])*Sin[e + f*x])*(I + Tan[(e + f*x)/2])^2)/(4*(a - I*b)*b^2*Sqrt[a^2 + b^2]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]) - (Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(((-I)*b*Sqrt[a^2 + b^2]*(Cos[e + f*x] + I*Sin[e + f*x]))/(Sqrt[2]*Sqrt[1 + (-1 + I*Cos[e + f*x] - Sin[e + f*x])/2]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]) + (a*(a - I*b + Sqrt[a^2 + b^2])*(Cos[e + f*x] + I*Sin[e + f*x]))/(2*Sqrt[2]*Sqrt[1 + (-1 + I*Cos[e + f*x] - Sin[e + f*x])/2]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]*(1 - ((1/2 + I/2)*(a + I*(-b + Sqrt[a^2 + b^2]))*(1 - I*Cos[e + f*x] + Sin[e + f*x]))/(a + b - Sqrt[a^2 + b^2]))) + (a*(-a + I*b + Sqrt[a^2 + b^2])*(Cos[e + f*x] + I*Sin[e + f*x]))/(2*Sqrt[2]*Sqrt[1 + (-1 + I*Cos[e + f*x] - Sin[e + f*x])/2]*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]*(1 - ((1/2 + I/2)*(a - I*(b + Sqrt[a^2 + b^2]))*(1 - I*Cos[e + f*x] + Sin[e + f*x]))/(a + b + Sqrt[a^2 + b^2]))))*(I + Tan[(e + f*x)/2])^2)/(2*(a - I*b)*b^2*Sqrt[a^2 + b^2]*Sqrt[(1 + Cos[e + f*x])^(-1)]) - (((-2*I)*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(a - I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2] + a*(-a + I*b + Sqrt[a^2 + b^2])*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[1 - I*Cos[e + f*x] + Sin[e + f*x]]/Sqrt[2]], 2])*Sqrt[I*Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])]*(I + Tan[(e + f*x)/2])^2*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(4*(a - I*b)*b^2*Sqrt[a^2 + b^2]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]])))","C",0
612,1,6560,477,30.9962533,"\int \frac{(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^2} \, dx","Integrate[(d*Sec[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^2,x]","\text{Result too large to show}","\frac{a (d \sec (e+f x))^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{2 \sqrt{b} f \left(a^2+b^2\right)^{5/4} \sec ^2(e+f x)^{3/4}}-\frac{a (d \sec (e+f x))^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{2 \sqrt{b} f \left(a^2+b^2\right)^{5/4} \sec ^2(e+f x)^{3/4}}-\frac{b (d \sec (e+f x))^{3/2}}{f \left(a^2+b^2\right) (a+b \tan (e+f x))}+\frac{\sin (e+f x) \cos (e+f x) (d \sec (e+f x))^{3/2}}{f \left(a^2+b^2\right)}-\frac{(d \sec (e+f x))^{3/2} E\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{f \left(a^2+b^2\right) \sec ^2(e+f x)^{3/4}}-\frac{a^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{2 b f \left(a^2+b^2\right)^{3/2} \sec ^2(e+f x)^{3/4}}+\frac{a^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{2 b f \left(a^2+b^2\right)^{3/2} \sec ^2(e+f x)^{3/4}}",1,"Result too large to show","C",0
613,1,8876,430,26.9177644,"\int \frac{\sqrt{d \sec (e+f x)}}{(a+b \tan (e+f x))^2} \, dx","Integrate[Sqrt[d*Sec[e + f*x]]/(a + b*Tan[e + f*x])^2,x]","\text{Result too large to show}","-\frac{3 a \sqrt{b} \sqrt{d \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{2 f \left(a^2+b^2\right)^{7/4} \sqrt[4]{\sec ^2(e+f x)}}-\frac{3 a \sqrt{b} \sqrt{d \sec (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{2 f \left(a^2+b^2\right)^{7/4} \sqrt[4]{\sec ^2(e+f x)}}-\frac{b \sqrt{d \sec (e+f x)}}{f \left(a^2+b^2\right) (a+b \tan (e+f x))}-\frac{\sqrt{d \sec (e+f x)} F\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{f \left(a^2+b^2\right) \sqrt[4]{\sec ^2(e+f x)}}+\frac{3 a^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{2 f \left(a^2+b^2\right)^2 \sqrt[4]{\sec ^2(e+f x)}}+\frac{3 a^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{2 f \left(a^2+b^2\right)^2 \sqrt[4]{\sec ^2(e+f x)}}",1,"Result too large to show","C",0
614,1,17812,555,33.4419417,"\int \frac{1}{\sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2} \, dx","Integrate[1/(Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])^2),x]","\text{Result too large to show}","\frac{b \left(2 a^2-3 b^2\right) \sec ^2(e+f x)}{f \left(a^2+b^2\right)^2 \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{\left(2 a^2-3 b^2\right) \tan (e+f x)}{f \left(a^2+b^2\right)^2 \sqrt{d \sec (e+f x)}}+\frac{2 (a \tan (e+f x)+b)}{f \left(a^2+b^2\right) \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{\left(2 a^2-3 b^2\right) \sqrt[4]{\sec ^2(e+f x)} E\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{f \left(a^2+b^2\right)^2 \sqrt{d \sec (e+f x)}}-\frac{5 a^2 b \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt[4]{\sec ^2(e+f x)} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{2 f \left(a^2+b^2\right)^{5/2} \sqrt{d \sec (e+f x)}}+\frac{5 a^2 b \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt[4]{\sec ^2(e+f x)} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{2 f \left(a^2+b^2\right)^{5/2} \sqrt{d \sec (e+f x)}}+\frac{5 a b^{3/2} \sqrt[4]{\sec ^2(e+f x)} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{2 f \left(a^2+b^2\right)^{9/4} \sqrt{d \sec (e+f x)}}-\frac{5 a b^{3/2} \sqrt[4]{\sec ^2(e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{2 f \left(a^2+b^2\right)^{9/4} \sqrt{d \sec (e+f x)}}",1,"Result too large to show","C",0
615,1,11962,520,28.0150403,"\int \frac{1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx","Integrate[1/((d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^2),x]","\text{Result too large to show}","\frac{b \left(2 a^2-5 b^2\right) \sec ^2(e+f x)}{3 f \left(a^2+b^2\right)^2 (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{2 (a \tan (e+f x)+b)}{3 f \left(a^2+b^2\right) (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{\left(2 a^2-5 b^2\right) \sec ^2(e+f x)^{3/4} F\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{3 f \left(a^2+b^2\right)^2 (d \sec (e+f x))^{3/2}}+\frac{7 a^2 b^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{2 f \left(a^2+b^2\right)^3 (d \sec (e+f x))^{3/2}}+\frac{7 a^2 b^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{2 f \left(a^2+b^2\right)^3 (d \sec (e+f x))^{3/2}}-\frac{7 a b^{5/2} \sec ^2(e+f x)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{2 f \left(a^2+b^2\right)^{11/4} (d \sec (e+f x))^{3/2}}-\frac{7 a b^{5/2} \sec ^2(e+f x)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{2 f \left(a^2+b^2\right)^{11/4} (d \sec (e+f x))^{3/2}}",1,"Result too large to show","C",0
616,1,18542,700,32.9713565,"\int \frac{1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^2} \, dx","Integrate[1/((d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x])^2),x]","\text{Result too large to show}","-\frac{2 \left(b \left(2 a^2-7 b^2\right)-3 a \left(a^2+4 b^2\right) \tan (e+f x)\right)}{5 d^2 f \left(a^2+b^2\right)^2 \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{2 \cos ^2(e+f x) (a \tan (e+f x)+b)}{5 d^2 f \left(a^2+b^2\right) \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{9 a b^{7/2} \sqrt[4]{\sec ^2(e+f x)} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{2 d^2 f \left(a^2+b^2\right)^{13/4} \sqrt{d \sec (e+f x)}}-\frac{9 a b^{7/2} \sqrt[4]{\sec ^2(e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{2 d^2 f \left(a^2+b^2\right)^{13/4} \sqrt{d \sec (e+f x)}}-\frac{9 a^2 b^3 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt[4]{\sec ^2(e+f x)} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{2 d^2 f \left(a^2+b^2\right)^{7/2} \sqrt{d \sec (e+f x)}}+\frac{9 a^2 b^3 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt[4]{\sec ^2(e+f x)} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{2 d^2 f \left(a^2+b^2\right)^{7/2} \sqrt{d \sec (e+f x)}}+\frac{3 b \left(2 a^4+10 a^2 b^2-7 b^4\right) \sec ^2(e+f x)}{5 d^2 f \left(a^2+b^2\right)^3 \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{3 \left(2 a^4+10 a^2 b^2-7 b^4\right) \tan (e+f x)}{5 d^2 f \left(a^2+b^2\right)^3 \sqrt{d \sec (e+f x)}}+\frac{3 \left(2 a^4+10 a^2 b^2-7 b^4\right) \sqrt[4]{\sec ^2(e+f x)} E\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{5 d^2 f \left(a^2+b^2\right)^3 \sqrt{d \sec (e+f x)}}",1,"Result too large to show","C",0
617,1,14225,583,29.4979541,"\int \frac{(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^3} \, dx","Integrate[(d*Sec[e + f*x])^(7/2)/(a + b*Tan[e + f*x])^3,x]","\text{Result too large to show}","\frac{3 a d^2 (d \sec (e+f x))^{3/2}}{4 b f \left(a^2+b^2\right) (a+b \tan (e+f x))}-\frac{3 a d^2 \sin (e+f x) \cos (e+f x) (d \sec (e+f x))^{3/2}}{4 b^2 f \left(a^2+b^2\right)}+\frac{3 a d^2 (d \sec (e+f x))^{3/2} E\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{4 b^2 f \left(a^2+b^2\right) \sec ^2(e+f x)^{3/4}}+\frac{3 d^2 \left(a^2+2 b^2\right) (d \sec (e+f x))^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{8 b^{5/2} f \left(a^2+b^2\right)^{5/4} \sec ^2(e+f x)^{3/4}}-\frac{3 d^2 \left(a^2+2 b^2\right) (d \sec (e+f x))^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{8 b^{5/2} f \left(a^2+b^2\right)^{5/4} \sec ^2(e+f x)^{3/4}}-\frac{3 a d^2 \left(a^2+2 b^2\right) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{8 b^3 f \left(a^2+b^2\right)^{3/2} \sec ^2(e+f x)^{3/4}}+\frac{3 a d^2 \left(a^2+2 b^2\right) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{8 b^3 f \left(a^2+b^2\right)^{3/2} \sec ^2(e+f x)^{3/4}}-\frac{d^2 (d \sec (e+f x))^{3/2}}{2 b f (a+b \tan (e+f x))^2}",1,"Result too large to show","C",0
618,1,4498,532,26.3932934,"\int \frac{(d \sec (e+f x))^{5/2}}{(a+b \tan (e+f x))^3} \, dx","Integrate[(d*Sec[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^3,x]","\text{Result too large to show}","\frac{a d^2 \sqrt{d \sec (e+f x)}}{4 b f \left(a^2+b^2\right) (a+b \tan (e+f x))}+\frac{a d^2 \sqrt{d \sec (e+f x)} F\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{4 b^2 f \left(a^2+b^2\right) \sqrt[4]{\sec ^2(e+f x)}}-\frac{a d^2 \left(a^2-2 b^2\right) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{8 b^2 f \left(a^2+b^2\right)^2 \sqrt[4]{\sec ^2(e+f x)}}-\frac{a d^2 \left(a^2-2 b^2\right) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{8 b^2 f \left(a^2+b^2\right)^2 \sqrt[4]{\sec ^2(e+f x)}}+\frac{d^2 \left(a^2-2 b^2\right) \sqrt{d \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{8 b^{3/2} f \left(a^2+b^2\right)^{7/4} \sqrt[4]{\sec ^2(e+f x)}}+\frac{d^2 \left(a^2-2 b^2\right) \sqrt{d \sec (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{8 b^{3/2} f \left(a^2+b^2\right)^{7/4} \sqrt[4]{\sec ^2(e+f x)}}-\frac{d^2 \sqrt{d \sec (e+f x)}}{2 b f (a+b \tan (e+f x))^2}",1,"(Sec[e + f*x]*(d*Sec[e + f*x])^(5/2)*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(-1/4*1/((a - I*b)*(a + I*b)*b) - b/(2*(a - I*b)*(a + I*b)*(a*Cos[e + f*x] + b*Sin[e + f*x])^2) + (3*Sin[e + f*x])/(4*(a - I*b)*(a + I*b)*(a*Cos[e + f*x] + b*Sin[e + f*x]))))/(f*(a + b*Tan[e + f*x])^3) - (b^(9/2)*Sqrt[a^2 + b^2]*(-8*a*b^(3/2)*(a^2 + b^2)^(3/2)*Sqrt[b^2*(a^2 + b^2)]*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] + (-a^2 + 2*b^2)*(Sqrt[b^2*(a^2 + b^2)]*(Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*(a^2 + b*(b + Sqrt[a^2 + b^2]))*ArcTan[(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])] + (-a^2 + b*(-b + Sqrt[a^2 + b^2]))*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])]) - 4*a*b^(3/2)*(a^2 + b^2)^(3/2)*EllipticPi[a^2/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1] + 4*a*b^(3/2)*(a^2 + b^2)^(3/2)*EllipticPi[a^2/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])), ArcSin[Tan[(e + f*x)/2]], -1]))*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sqrt[Sec[e + f*x]]*(d*Sec[e + f*x])^(5/2)*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(1/(4*(a - I*b)*(a + I*b)*Sqrt[Sec[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])) + (a*Sqrt[Sec[e + f*x]]*Sin[e + f*x])/(8*(a - I*b)*(a + I*b)*b*(a*Cos[e + f*x] + b*Sin[e + f*x]))))/(16*a^2*(b^2*(a^2 + b^2))^(7/2)*f*Sqrt[Sec[(e + f*x)/2]^2]*(a + b*Tan[e + f*x])^3*((b^(9/2)*Sqrt[a^2 + b^2]*(-8*a*b^(3/2)*(a^2 + b^2)^(3/2)*Sqrt[b^2*(a^2 + b^2)]*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] + (-a^2 + 2*b^2)*(Sqrt[b^2*(a^2 + b^2)]*(Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*(a^2 + b*(b + Sqrt[a^2 + b^2]))*ArcTan[(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])] + (-a^2 + b*(-b + Sqrt[a^2 + b^2]))*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])]) - 4*a*b^(3/2)*(a^2 + b^2)^(3/2)*EllipticPi[a^2/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1] + 4*a*b^(3/2)*(a^2 + b^2)^(3/2)*EllipticPi[a^2/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])), ArcSin[Tan[(e + f*x)/2]], -1]))*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*Tan[(e + f*x)/2])/(32*a^2*(b^2*(a^2 + b^2))^(7/2)*Sqrt[Sec[(e + f*x)/2]^2]) - (b^(9/2)*Sqrt[a^2 + b^2]*(-8*a*b^(3/2)*(a^2 + b^2)^(3/2)*Sqrt[b^2*(a^2 + b^2)]*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] + (-a^2 + 2*b^2)*(Sqrt[b^2*(a^2 + b^2)]*(Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*(a^2 + b*(b + Sqrt[a^2 + b^2]))*ArcTan[(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])] + (-a^2 + b*(-b + Sqrt[a^2 + b^2]))*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])]) - 4*a*b^(3/2)*(a^2 + b^2)^(3/2)*EllipticPi[a^2/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1] + 4*a*b^(3/2)*(a^2 + b^2)^(3/2)*EllipticPi[a^2/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])), ArcSin[Tan[(e + f*x)/2]], -1]))*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*(-(Sec[(e + f*x)/2]^4*Sin[e + f*x]) + 2*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Tan[(e + f*x)/2]))/(32*a^2*(b^2*(a^2 + b^2))^(7/2)*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]) - (b^(9/2)*Sqrt[a^2 + b^2]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*((-4*a*b^(3/2)*(a^2 + b^2)^(3/2)*Sqrt[b^2*(a^2 + b^2)]*Sec[(e + f*x)/2]^2)/(Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[1 + Tan[(e + f*x)/2]^2]) + (-a^2 + 2*b^2)*((-2*a*b^(3/2)*(a^2 + b^2)^(3/2)*Sec[(e + f*x)/2]^2)/(Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[1 + Tan[(e + f*x)/2]^2]*(1 - (a^2*Tan[(e + f*x)/2]^2)/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]))) + (2*a*b^(3/2)*(a^2 + b^2)^(3/2)*Sec[(e + f*x)/2]^2)/(Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[1 + Tan[(e + f*x)/2]^2]*(1 - (a^2*Tan[(e + f*x)/2]^2)/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])))) + Sqrt[b^2*(a^2 + b^2)]*((Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*(a^2 + b*(b + Sqrt[a^2 + b^2]))*(-1/2*((a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]) - ((-(Sec[(e + f*x)/2]^4*Sin[e + f*x]) + 2*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Tan[(e + f*x)/2])*(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2))/(4*Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*(Cos[e + f*x]*Sec[(e + f*x)/2]^4)^(3/2))))/(1 + (Cos[(e + f*x)/2]^4*Sec[e + f*x]*(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)^2)/(4*b*(2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])))) + ((-a^2 + b*(-b + Sqrt[a^2 + b^2]))*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*(-1/2*((a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]) - ((-(Sec[(e + f*x)/2]^4*Sin[e + f*x]) + 2*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Tan[(e + f*x)/2])*(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2))/(4*Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*(Cos[e + f*x]*Sec[(e + f*x)/2]^4)^(3/2))))/(1 + (Cos[(e + f*x)/2]^4*Sec[e + f*x]*(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)^2)/(4*b*(2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2]))))))))/(16*a^2*(b^2*(a^2 + b^2))^(7/2)*Sqrt[Sec[(e + f*x)/2]^2]) - (b^(9/2)*Sqrt[a^2 + b^2]*(-8*a*b^(3/2)*(a^2 + b^2)^(3/2)*Sqrt[b^2*(a^2 + b^2)]*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] + (-a^2 + 2*b^2)*(Sqrt[b^2*(a^2 + b^2)]*(Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*(a^2 + b*(b + Sqrt[a^2 + b^2]))*ArcTan[(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])] + (-a^2 + b*(-b + Sqrt[a^2 + b^2]))*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])]) - 4*a*b^(3/2)*(a^2 + b^2)^(3/2)*EllipticPi[a^2/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1] + 4*a*b^(3/2)*(a^2 + b^2)^(3/2)*EllipticPi[a^2/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])), ArcSin[Tan[(e + f*x)/2]], -1]))*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(32*a^2*(b^2*(a^2 + b^2))^(7/2)*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]])))","C",0
619,1,14364,566,29.6409955,"\int \frac{(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx","Integrate[(d*Sec[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^3,x]","\text{Result too large to show}","\frac{\left(3 a^2-2 b^2\right) (d \sec (e+f x))^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{8 \sqrt{b} f \left(a^2+b^2\right)^{9/4} \sec ^2(e+f x)^{3/4}}-\frac{\left(3 a^2-2 b^2\right) (d \sec (e+f x))^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{8 \sqrt{b} f \left(a^2+b^2\right)^{9/4} \sec ^2(e+f x)^{3/4}}-\frac{5 a b (d \sec (e+f x))^{3/2}}{4 f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))}-\frac{b (d \sec (e+f x))^{3/2}}{2 f \left(a^2+b^2\right) (a+b \tan (e+f x))^2}+\frac{5 a \sin (e+f x) \cos (e+f x) (d \sec (e+f x))^{3/2}}{4 f \left(a^2+b^2\right)^2}-\frac{5 a (d \sec (e+f x))^{3/2} E\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{4 f \left(a^2+b^2\right)^2 \sec ^2(e+f x)^{3/4}}-\frac{a \left(3 a^2-2 b^2\right) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{8 b f \left(a^2+b^2\right)^{5/2} \sec ^2(e+f x)^{3/4}}+\frac{a \left(3 a^2-2 b^2\right) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{8 b f \left(a^2+b^2\right)^{5/2} \sec ^2(e+f x)^{3/4}}",1,"Result too large to show","C",0
620,1,4455,515,26.2468916,"\int \frac{\sqrt{d \sec (e+f x)}}{(a+b \tan (e+f x))^3} \, dx","Integrate[Sqrt[d*Sec[e + f*x]]/(a + b*Tan[e + f*x])^3,x]","\text{Result too large to show}","-\frac{3 \sqrt{b} \left(5 a^2-2 b^2\right) \sqrt{d \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{8 f \left(a^2+b^2\right)^{11/4} \sqrt[4]{\sec ^2(e+f x)}}-\frac{3 \sqrt{b} \left(5 a^2-2 b^2\right) \sqrt{d \sec (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{8 f \left(a^2+b^2\right)^{11/4} \sqrt[4]{\sec ^2(e+f x)}}-\frac{7 a b \sqrt{d \sec (e+f x)}}{4 f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))}-\frac{b \sqrt{d \sec (e+f x)}}{2 f \left(a^2+b^2\right) (a+b \tan (e+f x))^2}-\frac{7 a \sqrt{d \sec (e+f x)} F\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{4 f \left(a^2+b^2\right)^2 \sqrt[4]{\sec ^2(e+f x)}}+\frac{3 a \left(5 a^2-2 b^2\right) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{8 f \left(a^2+b^2\right)^3 \sqrt[4]{\sec ^2(e+f x)}}+\frac{3 a \left(5 a^2-2 b^2\right) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{8 f \left(a^2+b^2\right)^3 \sqrt[4]{\sec ^2(e+f x)}}",1,"(Sec[e + f*x]^3*Sqrt[d*Sec[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*((-9*b)/(4*(a - I*b)^2*(a + I*b)^2) - b^3/(2*(a - I*b)^2*(a + I*b)^2*(a*Cos[e + f*x] + b*Sin[e + f*x])^2) + (11*b^2*Sin[e + f*x])/(4*(a - I*b)^2*(a + I*b)^2*(a*Cos[e + f*x] + b*Sin[e + f*x]))))/(f*(a + b*Tan[e + f*x])^3) - ((8*a*Sqrt[b^2*(a^2 + b^2)]*(-4*a^4 - a^2*b^2 + 3*b^4)*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] - 3*Sqrt[b]*(-5*a^2 + 2*b^2)*(Sqrt[b^2*(a^2 + b^2)]*((b + Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])] + (b - Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])]) - 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1] + 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])), ArcSin[Tan[(e + f*x)/2]], -1]))*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sec[e + f*x]^(5/2)*Sqrt[d*Sec[e + f*x]]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(a^2/((a - I*b)^2*(a + I*b)^2*Sqrt[Sec[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])) - (3*b^2)/(4*(a - I*b)^2*(a + I*b)^2*Sqrt[Sec[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])) - (7*a*b*Sqrt[Sec[e + f*x]]*Sin[e + f*x])/(8*(a - I*b)^2*(a + I*b)^2*(a*Cos[e + f*x] + b*Sin[e + f*x]))))/(16*a^2*(a^2 + b^2)^3*Sqrt[b^2*(a^2 + b^2)]*f*Sqrt[Sec[(e + f*x)/2]^2]*(a + b*Tan[e + f*x])^3*(((8*a*Sqrt[b^2*(a^2 + b^2)]*(-4*a^4 - a^2*b^2 + 3*b^4)*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] - 3*Sqrt[b]*(-5*a^2 + 2*b^2)*(Sqrt[b^2*(a^2 + b^2)]*((b + Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])] + (b - Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])]) - 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1] + 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])), ArcSin[Tan[(e + f*x)/2]], -1]))*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*Tan[(e + f*x)/2])/(32*a^2*(a^2 + b^2)^3*Sqrt[b^2*(a^2 + b^2)]*Sqrt[Sec[(e + f*x)/2]^2]) - ((8*a*Sqrt[b^2*(a^2 + b^2)]*(-4*a^4 - a^2*b^2 + 3*b^4)*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] - 3*Sqrt[b]*(-5*a^2 + 2*b^2)*(Sqrt[b^2*(a^2 + b^2)]*((b + Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])] + (b - Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])]) - 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1] + 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])), ArcSin[Tan[(e + f*x)/2]], -1]))*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*(-(Sec[(e + f*x)/2]^4*Sin[e + f*x]) + 2*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Tan[(e + f*x)/2]))/(32*a^2*(a^2 + b^2)^3*Sqrt[b^2*(a^2 + b^2)]*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]) - (Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*((4*a*Sqrt[b^2*(a^2 + b^2)]*(-4*a^4 - a^2*b^2 + 3*b^4)*Sec[(e + f*x)/2]^2)/(Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[1 + Tan[(e + f*x)/2]^2]) - 3*Sqrt[b]*(-5*a^2 + 2*b^2)*((-2*a*b^(3/2)*(a^2 + b^2)*Sec[(e + f*x)/2]^2)/(Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[1 + Tan[(e + f*x)/2]^2]*(1 - (a^2*Tan[(e + f*x)/2]^2)/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]))) + (2*a*b^(3/2)*(a^2 + b^2)*Sec[(e + f*x)/2]^2)/(Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[1 + Tan[(e + f*x)/2]^2]*(1 - (a^2*Tan[(e + f*x)/2]^2)/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])))) + Sqrt[b^2*(a^2 + b^2)]*(((b + Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*(-1/2*((a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]) - ((-(Sec[(e + f*x)/2]^4*Sin[e + f*x]) + 2*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Tan[(e + f*x)/2])*(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2))/(4*Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*(Cos[e + f*x]*Sec[(e + f*x)/2]^4)^(3/2))))/(1 + (Cos[(e + f*x)/2]^4*Sec[e + f*x]*(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)^2)/(4*b*(2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])))) + ((b - Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*(-1/2*((a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]) - ((-(Sec[(e + f*x)/2]^4*Sin[e + f*x]) + 2*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Tan[(e + f*x)/2])*(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2))/(4*Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*(Cos[e + f*x]*Sec[(e + f*x)/2]^4)^(3/2))))/(1 + (Cos[(e + f*x)/2]^4*Sec[e + f*x]*(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)^2)/(4*b*(2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2]))))))))/(16*a^2*(a^2 + b^2)^3*Sqrt[b^2*(a^2 + b^2)]*Sqrt[Sec[(e + f*x)/2]^2]) - ((8*a*Sqrt[b^2*(a^2 + b^2)]*(-4*a^4 - a^2*b^2 + 3*b^4)*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] - 3*Sqrt[b]*(-5*a^2 + 2*b^2)*(Sqrt[b^2*(a^2 + b^2)]*((b + Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])] + (b - Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])]) - 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1] + 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])), ArcSin[Tan[(e + f*x)/2]], -1]))*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(32*a^2*(a^2 + b^2)^3*Sqrt[b^2*(a^2 + b^2)]*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]])))","C",0
621,1,14652,664,30.068037,"\int \frac{1}{\sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^3} \, dx","Integrate[1/(Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])^3),x]","\text{Result too large to show}","\frac{a b \left(8 a^2-37 b^2\right) \sec ^2(e+f x)}{4 f \left(a^2+b^2\right)^3 \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{b \left(4 a^2-5 b^2\right) \sec ^2(e+f x)}{2 f \left(a^2+b^2\right)^2 \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}-\frac{a \left(8 a^2-37 b^2\right) \tan (e+f x)}{4 f \left(a^2+b^2\right)^3 \sqrt{d \sec (e+f x)}}+\frac{2 (a \tan (e+f x)+b)}{f \left(a^2+b^2\right) \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{a \left(8 a^2-37 b^2\right) \sqrt[4]{\sec ^2(e+f x)} E\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{4 f \left(a^2+b^2\right)^3 \sqrt{d \sec (e+f x)}}-\frac{5 a b \left(7 a^2-2 b^2\right) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt[4]{\sec ^2(e+f x)} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{8 f \left(a^2+b^2\right)^{7/2} \sqrt{d \sec (e+f x)}}+\frac{5 a b \left(7 a^2-2 b^2\right) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt[4]{\sec ^2(e+f x)} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{8 f \left(a^2+b^2\right)^{7/2} \sqrt{d \sec (e+f x)}}+\frac{5 b^{3/2} \left(7 a^2-2 b^2\right) \sqrt[4]{\sec ^2(e+f x)} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{8 f \left(a^2+b^2\right)^{13/4} \sqrt{d \sec (e+f x)}}-\frac{5 b^{3/2} \left(7 a^2-2 b^2\right) \sqrt[4]{\sec ^2(e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{8 f \left(a^2+b^2\right)^{13/4} \sqrt{d \sec (e+f x)}}",1,"Result too large to show","C",0
622,1,4709,620,25.8256217,"\int \frac{1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^3} \, dx","Integrate[1/((d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^3),x]","\text{Result too large to show}","\frac{a b \left(8 a^2-69 b^2\right) \sec ^2(e+f x)}{12 f \left(a^2+b^2\right)^3 (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac{b \left(4 a^2-7 b^2\right) \sec ^2(e+f x)}{6 f \left(a^2+b^2\right)^2 (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac{2 (a \tan (e+f x)+b)}{3 f \left(a^2+b^2\right) (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac{a \left(8 a^2-69 b^2\right) \sec ^2(e+f x)^{3/4} F\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right)}{12 f \left(a^2+b^2\right)^3 (d \sec (e+f x))^{3/2}}+\frac{7 a b^2 \left(9 a^2-2 b^2\right) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{8 f \left(a^2+b^2\right)^4 (d \sec (e+f x))^{3/2}}+\frac{7 a b^2 \left(9 a^2-2 b^2\right) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right)}{8 f \left(a^2+b^2\right)^4 (d \sec (e+f x))^{3/2}}-\frac{7 b^{5/2} \left(9 a^2-2 b^2\right) \sec ^2(e+f x)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{8 f \left(a^2+b^2\right)^{15/4} (d \sec (e+f x))^{3/2}}-\frac{7 b^{5/2} \left(9 a^2-2 b^2\right) \sec ^2(e+f x)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right)}{8 f \left(a^2+b^2\right)^{15/4} (d \sec (e+f x))^{3/2}}",1,"(Sec[e + f*x]^5*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*((b*(12*a^2 - 55*b^2))/(12*(a - I*b)^3*(a + I*b)^3) + (b*(3*a^2 - b^2)*Cos[2*(e + f*x)])/(3*(a - I*b)^3*(a + I*b)^3) - b^5/(2*(a - I*b)^3*(a + I*b)^3*(a*Cos[e + f*x] + b*Sin[e + f*x])^2) + (19*b^4*Sin[e + f*x])/(4*(a - I*b)^3*(a + I*b)^3*(a*Cos[e + f*x] + b*Sin[e + f*x])) + (a*(a^2 - 3*b^2)*Sin[2*(e + f*x)])/(3*(a - I*b)^3*(a + I*b)^3)))/(f*(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^3) - ((8*a*Sqrt[b^2*(a^2 + b^2)]*(-4*a^6 - 64*a^4*b^2 - 39*a^2*b^4 + 21*b^6)*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] - 21*b^(5/2)*(-9*a^2 + 2*b^2)*(Sqrt[b^2*(a^2 + b^2)]*((b + Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])] + (b - Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])]) - 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1] + 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])), ArcSin[Tan[(e + f*x)/2]], -1]))*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sec[e + f*x]^(9/2)*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(a^4/(3*(a - I*b)^3*(a + I*b)^3*Sqrt[Sec[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])) + (5*a^2*b^2)/((a - I*b)^3*(a + I*b)^3*Sqrt[Sec[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])) - (7*b^4)/(4*(a - I*b)^3*(a + I*b)^3*Sqrt[Sec[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])) + (a^3*b*Sqrt[Sec[e + f*x]]*Sin[e + f*x])/(3*(a - I*b)^3*(a + I*b)^3*(a*Cos[e + f*x] + b*Sin[e + f*x])) - (23*a*b^3*Sqrt[Sec[e + f*x]]*Sin[e + f*x])/(8*(a - I*b)^3*(a + I*b)^3*(a*Cos[e + f*x] + b*Sin[e + f*x]))))/(48*a^2*(a^2 + b^2)^4*Sqrt[b^2*(a^2 + b^2)]*f*Sqrt[Sec[(e + f*x)/2]^2]*(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^3*(((8*a*Sqrt[b^2*(a^2 + b^2)]*(-4*a^6 - 64*a^4*b^2 - 39*a^2*b^4 + 21*b^6)*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] - 21*b^(5/2)*(-9*a^2 + 2*b^2)*(Sqrt[b^2*(a^2 + b^2)]*((b + Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])] + (b - Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])]) - 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1] + 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])), ArcSin[Tan[(e + f*x)/2]], -1]))*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*Tan[(e + f*x)/2])/(96*a^2*(a^2 + b^2)^4*Sqrt[b^2*(a^2 + b^2)]*Sqrt[Sec[(e + f*x)/2]^2]) - ((8*a*Sqrt[b^2*(a^2 + b^2)]*(-4*a^6 - 64*a^4*b^2 - 39*a^2*b^4 + 21*b^6)*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] - 21*b^(5/2)*(-9*a^2 + 2*b^2)*(Sqrt[b^2*(a^2 + b^2)]*((b + Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])] + (b - Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])]) - 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1] + 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])), ArcSin[Tan[(e + f*x)/2]], -1]))*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*(-(Sec[(e + f*x)/2]^4*Sin[e + f*x]) + 2*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Tan[(e + f*x)/2]))/(96*a^2*(a^2 + b^2)^4*Sqrt[b^2*(a^2 + b^2)]*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]) - (Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*((4*a*Sqrt[b^2*(a^2 + b^2)]*(-4*a^6 - 64*a^4*b^2 - 39*a^2*b^4 + 21*b^6)*Sec[(e + f*x)/2]^2)/(Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[1 + Tan[(e + f*x)/2]^2]) - 21*b^(5/2)*(-9*a^2 + 2*b^2)*((-2*a*b^(3/2)*(a^2 + b^2)*Sec[(e + f*x)/2]^2)/(Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[1 + Tan[(e + f*x)/2]^2]*(1 - (a^2*Tan[(e + f*x)/2]^2)/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]))) + (2*a*b^(3/2)*(a^2 + b^2)*Sec[(e + f*x)/2]^2)/(Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[1 + Tan[(e + f*x)/2]^2]*(1 - (a^2*Tan[(e + f*x)/2]^2)/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])))) + Sqrt[b^2*(a^2 + b^2)]*(((b + Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*(-1/2*((a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]) - ((-(Sec[(e + f*x)/2]^4*Sin[e + f*x]) + 2*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Tan[(e + f*x)/2])*(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2))/(4*Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*(Cos[e + f*x]*Sec[(e + f*x)/2]^4)^(3/2))))/(1 + (Cos[(e + f*x)/2]^4*Sec[e + f*x]*(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)^2)/(4*b*(2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])))) + ((b - Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*(-1/2*((a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]) - ((-(Sec[(e + f*x)/2]^4*Sin[e + f*x]) + 2*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Tan[(e + f*x)/2])*(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2))/(4*Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*(Cos[e + f*x]*Sec[(e + f*x)/2]^4)^(3/2))))/(1 + (Cos[(e + f*x)/2]^4*Sec[e + f*x]*(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)^2)/(4*b*(2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2]))))))))/(48*a^2*(a^2 + b^2)^4*Sqrt[b^2*(a^2 + b^2)]*Sqrt[Sec[(e + f*x)/2]^2]) - ((8*a*Sqrt[b^2*(a^2 + b^2)]*(-4*a^6 - 64*a^4*b^2 - 39*a^2*b^4 + 21*b^6)*EllipticF[ArcSin[Tan[(e + f*x)/2]], -1] - 21*b^(5/2)*(-9*a^2 + 2*b^2)*(Sqrt[b^2*(a^2 + b^2)]*((b + Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b - Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b - Sqrt[a^2 + b^2]) - a^2*(-2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])] + (b - Sqrt[a^2 + b^2])*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*ArcTan[(a^2 - (a^2 + 2*b*(b + Sqrt[a^2 + b^2]))*Tan[(e + f*x)/2]^2)/(2*Sqrt[b]*Sqrt[2*b^2*(b + Sqrt[a^2 + b^2]) + a^2*(2*b + Sqrt[a^2 + b^2])]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4])]) - 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1] + 4*a*b^(3/2)*(a^2 + b^2)*EllipticPi[a^2/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])), ArcSin[Tan[(e + f*x)/2]], -1]))*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(96*a^2*(a^2 + b^2)^4*Sqrt[b^2*(a^2 + b^2)]*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]])))","C",0
623,1,15481,814,29.672934,"\int \frac{1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^3} \, dx","Integrate[1/((d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x])^3),x]","\text{Result too large to show}","\frac{9 \left(11 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right) \sqrt[4]{\sec ^2(e+f x)} b^{7/2}}{8 \left(a^2+b^2\right)^{17/4} d^2 f \sqrt{d \sec (e+f x)}}-\frac{9 \left(11 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{\sec ^2(e+f x)}}{\sqrt[4]{a^2+b^2}}\right) \sqrt[4]{\sec ^2(e+f x)} b^{7/2}}{8 \left(a^2+b^2\right)^{17/4} d^2 f \sqrt{d \sec (e+f x)}}-\frac{9 a \left(11 a^2-2 b^2\right) \cot (e+f x) \Pi \left(-\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right) \sqrt[4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)} b^3}{8 \left(a^2+b^2\right)^{9/2} d^2 f \sqrt{d \sec (e+f x)}}+\frac{9 a \left(11 a^2-2 b^2\right) \cot (e+f x) \Pi \left(\frac{b}{\sqrt{a^2+b^2}};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(e+f x)}\right)\right|-1\right) \sqrt[4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)} b^3}{8 \left(a^2+b^2\right)^{9/2} d^2 f \sqrt{d \sec (e+f x)}}+\frac{3 a \left(8 a^4+64 b^2 a^2-139 b^4\right) \sec ^2(e+f x) b}{20 \left(a^2+b^2\right)^4 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{3 \left(4 a^4+28 b^2 a^2-15 b^4\right) \sec ^2(e+f x) b}{10 \left(a^2+b^2\right)^3 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}-\frac{3 a \left(8 a^4+64 b^2 a^2-139 b^4\right) \tan (e+f x)}{20 \left(a^2+b^2\right)^4 d^2 f \sqrt{d \sec (e+f x)}}-\frac{2 \left(b \left(4 a^2-9 b^2\right)-a \left(3 a^2+16 b^2\right) \tan (e+f x)\right)}{5 \left(a^2+b^2\right)^2 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{3 a \left(8 a^4+64 b^2 a^2-139 b^4\right) E\left(\left.\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right|2\right) \sqrt[4]{\sec ^2(e+f x)}}{20 \left(a^2+b^2\right)^4 d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left(a^2+b^2\right) d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}",1,"Result too large to show","C",0
624,1,126,78,0.5109864,"\int (d \sec (e+f x))^{5/3} (a+b \tan (e+f x)) \, dx","Integrate[(d*Sec[e + f*x])^(5/3)*(a + b*Tan[e + f*x]),x]","\frac{d (d \sec (e+f x))^{2/3} (a+b \tan (e+f x)) \left(3 \cos ^2(e+f x)^{2/3} (5 a \sin (2 (e+f x))+4 b)-10 a \sin (e+f x) \cos ^3(e+f x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)\right)}{20 f \cos ^2(e+f x)^{2/3} (a \cos (e+f x)+b \sin (e+f x))}","\frac{3 a d \sin (e+f x) (d \sec (e+f x))^{2/3} \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(e+f x)\right)}{2 f \sqrt{\sin ^2(e+f x)}}+\frac{3 b (d \sec (e+f x))^{5/3}}{5 f}",1,"(d*(d*Sec[e + f*x])^(2/3)*(-10*a*Cos[e + f*x]^3*Hypergeometric2F1[1/3, 1/2, 3/2, Sin[e + f*x]^2]*Sin[e + f*x] + 3*(Cos[e + f*x]^2)^(2/3)*(4*b + 5*a*Sin[2*(e + f*x)]))*(a + b*Tan[e + f*x]))/(20*f*(Cos[e + f*x]^2)^(2/3)*(a*Cos[e + f*x] + b*Sin[e + f*x]))","A",1
625,1,58,76,0.1810863,"\int \sqrt[3]{d \sec (e+f x)} (a+b \tan (e+f x)) \, dx","Integrate[(d*Sec[e + f*x])^(1/3)*(a + b*Tan[e + f*x]),x]","\frac{\sqrt[3]{d \sec (e+f x)} \left(a \cos ^2(e+f x)^{2/3} \tan (e+f x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\sin ^2(e+f x)\right)+3 b\right)}{f}","\frac{3 b \sqrt[3]{d \sec (e+f x)}}{f}-\frac{3 a d \sin (e+f x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(e+f x)\right)}{2 f \sqrt{\sin ^2(e+f x)} (d \sec (e+f x))^{2/3}}",1,"((d*Sec[e + f*x])^(1/3)*(3*b + a*(Cos[e + f*x]^2)^(2/3)*Hypergeometric2F1[1/2, 2/3, 3/2, Sin[e + f*x]^2]*Tan[e + f*x]))/f","A",1
626,1,119,76,0.9344935,"\int \frac{a+b \tan (e+f x)}{\sqrt[3]{d \sec (e+f x)}} \, dx","Integrate[(a + b*Tan[e + f*x])/(d*Sec[e + f*x])^(1/3),x]","-\frac{3 (a \cot (e+f x)+b) \left(a \sqrt{\sin ^2(e+f x)} \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sec ^2(e+f x)\right)+b \sin (e+f x)\right)}{f \sqrt[3]{d \sec (e+f x)} \left(a \sqrt{\sin ^2(e+f x)} \cot (e+f x)+b \sin (e+f x)\right)}","-\frac{3 a d \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(e+f x)\right)}{4 f \sqrt{\sin ^2(e+f x)} (d \sec (e+f x))^{4/3}}-\frac{3 b}{f \sqrt[3]{d \sec (e+f x)}}",1,"(-3*(b + a*Cot[e + f*x])*(b*Sin[e + f*x] + a*Cot[e + f*x]*Hypergeometric2F1[-1/6, 1/2, 5/6, Sec[e + f*x]^2]*Sqrt[Sin[e + f*x]^2]*Sqrt[-Tan[e + f*x]^2]))/(f*(d*Sec[e + f*x])^(1/3)*(b*Sin[e + f*x] + a*Cot[e + f*x]*Sqrt[Sin[e + f*x]^2]))","A",0
627,1,94,78,0.4306377,"\int \frac{a+b \tan (e+f x)}{(d \sec (e+f x))^{5/3}} \, dx","Integrate[(a + b*Tan[e + f*x])/(d*Sec[e + f*x])^(5/3),x]","\frac{3 \sqrt[3]{\cos ^2(e+f x)} (a \sin (e+f x)-b \cos (e+f x))+2 a \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\sin ^2(e+f x)\right)}{5 d f \sqrt[3]{\cos ^2(e+f x)} (d \sec (e+f x))^{2/3}}","-\frac{3 a d \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(e+f x)\right)}{8 f \sqrt{\sin ^2(e+f x)} (d \sec (e+f x))^{8/3}}-\frac{3 b}{5 f (d \sec (e+f x))^{5/3}}",1,"(2*a*Hypergeometric2F1[1/2, 2/3, 3/2, Sin[e + f*x]^2]*Sin[e + f*x] + 3*(Cos[e + f*x]^2)^(1/3)*(-(b*Cos[e + f*x]) + a*Sin[e + f*x]))/(5*d*f*(Cos[e + f*x]^2)^(1/3)*(d*Sec[e + f*x])^(2/3))","A",1
628,1,108,119,1.7096813,"\int (d \sec (e+f x))^{5/3} (a+b \tan (e+f x))^2 \, dx","Integrate[(d*Sec[e + f*x])^(5/3)*(a + b*Tan[e + f*x])^2,x]","\frac{(d \sec (e+f x))^{5/3} \left(-5 \left(8 a^2-3 b^2\right) \sin (2 (e+f x)) \sqrt[3]{\cos ^2(e+f x)} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)+15 \left(8 a^2-3 b^2\right) \sin (2 (e+f x))+12 b (16 a+5 b \tan (e+f x))\right)}{160 f}","\frac{3 d \left(8 a^2-3 b^2\right) \sin (e+f x) (d \sec (e+f x))^{2/3} \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(e+f x)\right)}{16 f \sqrt{\sin ^2(e+f x)}}+\frac{33 a b (d \sec (e+f x))^{5/3}}{40 f}+\frac{3 b (d \sec (e+f x))^{5/3} (a+b \tan (e+f x))}{8 f}",1,"((d*Sec[e + f*x])^(5/3)*(15*(8*a^2 - 3*b^2)*Sin[2*(e + f*x)] - 5*(8*a^2 - 3*b^2)*(Cos[e + f*x]^2)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, Sin[e + f*x]^2]*Sin[2*(e + f*x)] + 12*b*(16*a + 5*b*Tan[e + f*x])))/(160*f)","A",1
629,1,83,119,0.4773272,"\int \sqrt[3]{d \sec (e+f x)} (a+b \tan (e+f x))^2 \, dx","Integrate[(d*Sec[e + f*x])^(1/3)*(a + b*Tan[e + f*x])^2,x]","\frac{\sqrt[3]{d \sec (e+f x)} \left(\left(4 a^2-3 b^2\right) \cos ^2(e+f x)^{2/3} \tan (e+f x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\sin ^2(e+f x)\right)+3 b (8 a+b \tan (e+f x))\right)}{4 f}","-\frac{3 d \left(4 a^2-3 b^2\right) \sin (e+f x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(e+f x)\right)}{8 f \sqrt{\sin ^2(e+f x)} (d \sec (e+f x))^{2/3}}+\frac{21 a b \sqrt[3]{d \sec (e+f x)}}{4 f}+\frac{3 b \sqrt[3]{d \sec (e+f x)} (a+b \tan (e+f x))}{4 f}",1,"((d*Sec[e + f*x])^(1/3)*((4*a^2 - 3*b^2)*(Cos[e + f*x]^2)^(2/3)*Hypergeometric2F1[1/2, 2/3, 3/2, Sin[e + f*x]^2]*Tan[e + f*x] + 3*b*(8*a + b*Tan[e + f*x])))/(4*f)","A",1
630,1,209,119,3.9411506,"\int \frac{(a+b \tan (e+f x))^2}{\sqrt[3]{d \sec (e+f x)}} \, dx","Integrate[(a + b*Tan[e + f*x])^2/(d*Sec[e + f*x])^(1/3),x]","\frac{3 d \sin (e+f x) (a+b \tan (e+f x))^2 \left(\frac{\left(\left(2 a^2-3 b^2\right) \cot (e+f x)+4 a b\right) \left(\left(2 a^2-3 b^2\right) \sqrt{\sin ^2(e+f x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sec ^2(e+f x)\right)-4 a b \cos (e+f x) \sqrt{-\tan ^2(e+f x)}\right)}{\sqrt{-\tan ^2(e+f x)} \left(\left(2 a^2-3 b^2\right) \sqrt{\sin ^2(e+f x)} \cot (e+f x)+4 a b \sin (e+f x)\right)}+b^2\right)}{2 f (d \sec (e+f x))^{4/3} (a \cos (e+f x)+b \sin (e+f x))^2}","-\frac{3 d \left(2 a^2-3 b^2\right) \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(e+f x)\right)}{8 f \sqrt{\sin ^2(e+f x)} (d \sec (e+f x))^{4/3}}-\frac{15 a b}{2 f \sqrt[3]{d \sec (e+f x)}}+\frac{3 b (a+b \tan (e+f x))}{2 f \sqrt[3]{d \sec (e+f x)}}",1,"(3*d*Sin[e + f*x]*(a + b*Tan[e + f*x])^2*(b^2 + ((4*a*b + (2*a^2 - 3*b^2)*Cot[e + f*x])*((2*a^2 - 3*b^2)*Hypergeometric2F1[-1/6, 1/2, 5/6, Sec[e + f*x]^2]*Sqrt[Sin[e + f*x]^2] - 4*a*b*Cos[e + f*x]*Sqrt[-Tan[e + f*x]^2]))/((4*a*b*Sin[e + f*x] + (2*a^2 - 3*b^2)*Cot[e + f*x]*Sqrt[Sin[e + f*x]^2])*Sqrt[-Tan[e + f*x]^2])))/(2*f*(d*Sec[e + f*x])^(4/3)*(a*Cos[e + f*x] + b*Sin[e + f*x])^2)","A",0
631,1,119,119,0.5223433,"\int \frac{(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{5/3}} \, dx","Integrate[(a + b*Tan[e + f*x])^2/(d*Sec[e + f*x])^(5/3),x]","\frac{\sec ^2(e+f x) \left(2 \left(2 a^2+3 b^2\right) \cos ^2(e+f x)^{2/3} \tan (e+f x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\sin ^2(e+f x)\right)+3 a^2 \sin (2 (e+f x))-6 a b \cos (2 (e+f x))-6 a b-3 b^2 \sin (2 (e+f x))\right)}{10 f (d \sec (e+f x))^{5/3}}","-\frac{3 d \left(2 a^2+3 b^2\right) \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(e+f x)\right)}{16 f \sqrt{\sin ^2(e+f x)} (d \sec (e+f x))^{8/3}}+\frac{3 a b}{10 f (d \sec (e+f x))^{5/3}}-\frac{3 b (a+b \tan (e+f x))}{2 f (d \sec (e+f x))^{5/3}}",1,"(Sec[e + f*x]^2*(-6*a*b - 6*a*b*Cos[2*(e + f*x)] + 3*a^2*Sin[2*(e + f*x)] - 3*b^2*Sin[2*(e + f*x)] + 2*(2*a^2 + 3*b^2)*(Cos[e + f*x]^2)^(2/3)*Hypergeometric2F1[1/2, 2/3, 3/2, Sin[e + f*x]^2]*Tan[e + f*x]))/(10*f*(d*Sec[e + f*x])^(5/3))","A",1
632,1,276,552,6.035876,"\int \frac{(d \sec (e+f x))^{5/3}}{a+b \tan (e+f x)} \, dx","Integrate[(d*Sec[e + f*x])^(5/3)/(a + b*Tan[e + f*x]),x]","-\frac{24 d^2 (a+b \tan (e+f x)) F_1\left(\frac{1}{3};\frac{1}{6},\frac{1}{6};\frac{4}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right)}{b f \sqrt[3]{d \sec (e+f x)} \left((a+i b) F_1\left(\frac{4}{3};\frac{1}{6},\frac{7}{6};\frac{7}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right)+(a-i b) F_1\left(\frac{4}{3};\frac{7}{6},\frac{1}{6};\frac{7}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right)+8 (a+b \tan (e+f x)) F_1\left(\frac{1}{3};\frac{1}{6},\frac{1}{6};\frac{4}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right)\right)}","\frac{\tan (e+f x) (d \sec (e+f x))^{5/3} F_1\left(\frac{1}{2};1,\frac{1}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{a f \sec ^2(e+f x)^{5/6}}+\frac{(d \sec (e+f x))^{5/3} \log \left(-\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{4 b^{2/3} f \sqrt[6]{a^2+b^2} \sec ^2(e+f x)^{5/6}}-\frac{(d \sec (e+f x))^{5/3} \log \left(\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{4 b^{2/3} f \sqrt[6]{a^2+b^2} \sec ^2(e+f x)^{5/6}}-\frac{\sqrt{3} (d \sec (e+f x))^{5/3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}\right)}{2 b^{2/3} f \sqrt[6]{a^2+b^2} \sec ^2(e+f x)^{5/6}}+\frac{\sqrt{3} (d \sec (e+f x))^{5/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right)}{2 b^{2/3} f \sqrt[6]{a^2+b^2} \sec ^2(e+f x)^{5/6}}-\frac{(d \sec (e+f x))^{5/3} \tanh ^{-1}\left(\frac{\sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt[6]{a^2+b^2}}\right)}{b^{2/3} f \sqrt[6]{a^2+b^2} \sec ^2(e+f x)^{5/6}}",1,"(-24*d^2*AppellF1[1/3, 1/6, 1/6, 4/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(a + b*Tan[e + f*x]))/(b*f*(d*Sec[e + f*x])^(1/3)*((a + I*b)*AppellF1[4/3, 1/6, 7/6, 7/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])] + (a - I*b)*AppellF1[4/3, 7/6, 1/6, 7/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])] + 8*AppellF1[1/3, 1/6, 1/6, 4/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(a + b*Tan[e + f*x])))","C",0
633,1,280,552,4.5764902,"\int \frac{\sqrt[3]{d \sec (e+f x)}}{a+b \tan (e+f x)} \, dx","Integrate[(d*Sec[e + f*x])^(1/3)/(a + b*Tan[e + f*x]),x]","-\frac{48 d^2 (a+b \tan (e+f x)) F_1\left(\frac{5}{3};\frac{5}{6},\frac{5}{6};\frac{8}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right)}{5 b f (d \sec (e+f x))^{5/3} \left(5 (a+i b) F_1\left(\frac{8}{3};\frac{5}{6},\frac{11}{6};\frac{11}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right)+5 (a-i b) F_1\left(\frac{8}{3};\frac{11}{6},\frac{5}{6};\frac{11}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right)+16 (a+b \tan (e+f x)) F_1\left(\frac{5}{3};\frac{5}{6},\frac{5}{6};\frac{8}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right)\right)}","\frac{\tan (e+f x) \sqrt[3]{d \sec (e+f x)} F_1\left(\frac{1}{2};1,\frac{5}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{a f \sqrt[6]{\sec ^2(e+f x)}}+\frac{b^{2/3} \sqrt[3]{d \sec (e+f x)} \log \left(-\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{4 f \left(a^2+b^2\right)^{5/6} \sqrt[6]{\sec ^2(e+f x)}}-\frac{b^{2/3} \sqrt[3]{d \sec (e+f x)} \log \left(\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{4 f \left(a^2+b^2\right)^{5/6} \sqrt[6]{\sec ^2(e+f x)}}+\frac{\sqrt{3} b^{2/3} \sqrt[3]{d \sec (e+f x)} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}\right)}{2 f \left(a^2+b^2\right)^{5/6} \sqrt[6]{\sec ^2(e+f x)}}-\frac{\sqrt{3} b^{2/3} \sqrt[3]{d \sec (e+f x)} \tan ^{-1}\left(\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right)}{2 f \left(a^2+b^2\right)^{5/6} \sqrt[6]{\sec ^2(e+f x)}}-\frac{b^{2/3} \sqrt[3]{d \sec (e+f x)} \tanh ^{-1}\left(\frac{\sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt[6]{a^2+b^2}}\right)}{f \left(a^2+b^2\right)^{5/6} \sqrt[6]{\sec ^2(e+f x)}}",1,"(-48*d^2*AppellF1[5/3, 5/6, 5/6, 8/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(a + b*Tan[e + f*x]))/(5*b*f*(d*Sec[e + f*x])^(5/3)*(5*(a + I*b)*AppellF1[8/3, 5/6, 11/6, 11/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])] + 5*(a - I*b)*AppellF1[8/3, 11/6, 5/6, 11/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])] + 16*AppellF1[5/3, 5/6, 5/6, 8/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(a + b*Tan[e + f*x])))","C",0
634,1,285,579,21.6370313,"\int \frac{1}{\sqrt[3]{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx","Integrate[1/((d*Sec[e + f*x])^(1/3)*(a + b*Tan[e + f*x])),x]","-\frac{60 d F_1\left(\frac{7}{3};\frac{7}{6},\frac{7}{6};\frac{10}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right) (a \cos (e+f x)+b \sin (e+f x))}{7 b f (d \sec (e+f x))^{4/3} \left(7 (a+i b) F_1\left(\frac{10}{3};\frac{7}{6},\frac{13}{6};\frac{13}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right)+7 (a-i b) F_1\left(\frac{10}{3};\frac{13}{6},\frac{7}{6};\frac{13}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right)+20 (a+b \tan (e+f x)) F_1\left(\frac{7}{3};\frac{7}{6},\frac{7}{6};\frac{10}{3};\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right)\right)}","\frac{\tan (e+f x) \sqrt[6]{\sec ^2(e+f x)} F_1\left(\frac{1}{2};1,\frac{7}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{a f \sqrt[3]{d \sec (e+f x)}}+\frac{3 b}{f \left(a^2+b^2\right) \sqrt[3]{d \sec (e+f x)}}+\frac{b^{4/3} \sqrt[6]{\sec ^2(e+f x)} \log \left(-\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{4 f \left(a^2+b^2\right)^{7/6} \sqrt[3]{d \sec (e+f x)}}-\frac{b^{4/3} \sqrt[6]{\sec ^2(e+f x)} \log \left(\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{4 f \left(a^2+b^2\right)^{7/6} \sqrt[3]{d \sec (e+f x)}}-\frac{\sqrt{3} b^{4/3} \sqrt[6]{\sec ^2(e+f x)} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}\right)}{2 f \left(a^2+b^2\right)^{7/6} \sqrt[3]{d \sec (e+f x)}}+\frac{\sqrt{3} b^{4/3} \sqrt[6]{\sec ^2(e+f x)} \tan ^{-1}\left(\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right)}{2 f \left(a^2+b^2\right)^{7/6} \sqrt[3]{d \sec (e+f x)}}-\frac{b^{4/3} \sqrt[6]{\sec ^2(e+f x)} \tanh ^{-1}\left(\frac{\sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt[6]{a^2+b^2}}\right)}{f \left(a^2+b^2\right)^{7/6} \sqrt[3]{d \sec (e+f x)}}",1,"(-60*d*AppellF1[7/3, 7/6, 7/6, 10/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(a*Cos[e + f*x] + b*Sin[e + f*x]))/(7*b*f*(d*Sec[e + f*x])^(4/3)*(7*(a + I*b)*AppellF1[10/3, 7/6, 13/6, 13/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])] + 7*(a - I*b)*AppellF1[10/3, 13/6, 7/6, 13/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])] + 20*AppellF1[7/3, 7/6, 7/6, 10/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(a + b*Tan[e + f*x])))","C",0
635,1,6862,581,32.1183461,"\int \frac{1}{(d \sec (e+f x))^{5/3} (a+b \tan (e+f x))} \, dx","Integrate[1/((d*Sec[e + f*x])^(5/3)*(a + b*Tan[e + f*x])),x]","\text{Result too large to show}","\frac{\tan (e+f x) \sec ^2(e+f x)^{5/6} F_1\left(\frac{1}{2};1,\frac{11}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{a f (d \sec (e+f x))^{5/3}}+\frac{3 b}{5 f \left(a^2+b^2\right) (d \sec (e+f x))^{5/3}}+\frac{b^{8/3} \sec ^2(e+f x)^{5/6} \log \left(-\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{4 f \left(a^2+b^2\right)^{11/6} (d \sec (e+f x))^{5/3}}-\frac{b^{8/3} \sec ^2(e+f x)^{5/6} \log \left(\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{4 f \left(a^2+b^2\right)^{11/6} (d \sec (e+f x))^{5/3}}+\frac{\sqrt{3} b^{8/3} \sec ^2(e+f x)^{5/6} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}\right)}{2 f \left(a^2+b^2\right)^{11/6} (d \sec (e+f x))^{5/3}}-\frac{\sqrt{3} b^{8/3} \sec ^2(e+f x)^{5/6} \tan ^{-1}\left(\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right)}{2 f \left(a^2+b^2\right)^{11/6} (d \sec (e+f x))^{5/3}}-\frac{b^{8/3} \sec ^2(e+f x)^{5/6} \tanh ^{-1}\left(\frac{\sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt[6]{a^2+b^2}}\right)}{f \left(a^2+b^2\right)^{11/6} (d \sec (e+f x))^{5/3}}",1,"Result too large to show","B",0
636,1,3398,687,39.5210549,"\int \frac{(d \sec (e+f x))^{5/3}}{(a+b \tan (e+f x))^2} \, dx","Integrate[(d*Sec[e + f*x])^(5/3)/(a + b*Tan[e + f*x])^2,x]","\text{Result too large to show}","\frac{\tan (e+f x) (d \sec (e+f x))^{5/3} F_1\left(\frac{1}{2};2,\frac{1}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{a^2 f \sec ^2(e+f x)^{5/6}}-\frac{a b (d \sec (e+f x))^{5/3}}{f \left(a^2+b^2\right) \left(a^2-b^2 \tan ^2(e+f x)\right)}+\frac{a (d \sec (e+f x))^{5/3} \log \left(-\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{12 b^{2/3} f \left(a^2+b^2\right)^{7/6} \sec ^2(e+f x)^{5/6}}-\frac{a (d \sec (e+f x))^{5/3} \log \left(\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{12 b^{2/3} f \left(a^2+b^2\right)^{7/6} \sec ^2(e+f x)^{5/6}}-\frac{a (d \sec (e+f x))^{5/3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}\right)}{2 \sqrt{3} b^{2/3} f \left(a^2+b^2\right)^{7/6} \sec ^2(e+f x)^{5/6}}+\frac{a (d \sec (e+f x))^{5/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right)}{2 \sqrt{3} b^{2/3} f \left(a^2+b^2\right)^{7/6} \sec ^2(e+f x)^{5/6}}-\frac{a (d \sec (e+f x))^{5/3} \tanh ^{-1}\left(\frac{\sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt[6]{a^2+b^2}}\right)}{3 b^{2/3} f \left(a^2+b^2\right)^{7/6} \sec ^2(e+f x)^{5/6}}+\frac{b^2 \tan ^3(e+f x) (d \sec (e+f x))^{5/3} F_1\left(\frac{3}{2};2,\frac{1}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{3 a^4 f \sec ^2(e+f x)^{5/6}}",1,"(Sec[e + f*x]*(d*Sec[e + f*x])^(5/3)*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*((b*Cos[e + f*x])/(a*(a - I*b)*(a + I*b)) + Sin[e + f*x]/((a - I*b)*(a + I*b)) - b/((a - I*b)*(a + I*b)*(a*Cos[e + f*x] + b*Sin[e + f*x]))))/(f*(a + b*Tan[e + f*x])^2) - (4*AppellF1[1/3, 1/6, 1/6, 4/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(d*Sec[e + f*x])^(5/3)*(a*Cos[e + f*x] + b*Sin[e + f*x])^2)/(a*b*f*(a + b*Tan[e + f*x])*((a + I*b)*AppellF1[4/3, 1/6, 7/6, 7/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])] + (a - I*b)*AppellF1[4/3, 7/6, 1/6, 7/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])] + 8*AppellF1[1/3, 1/6, 1/6, 4/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(a + b*Tan[e + f*x]))) - (Sec[e + f*x]*(d*Sec[e + f*x])^(5/3)*((6*(b + a*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]))/((a^2 + b^2)*Sec[e + f*x]^(1/3)) + (132*a*b^(2/3)*(5*a^2 + 3*b^2)*AppellF1[5/6, 1/2, 1, 11/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^(8/3) - 240*a*b^(8/3)*AppellF1[11/6, 1/2, 1, 17/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^(14/3) - 55*(-1)^(1/6)*(a^2 - b^2)*(a^2 + b^2)^(5/6)*(-2*ArcTan[Sqrt[3] - (2*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6)] + 2*ArcTan[Sqrt[3] + (2*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6)] + 4*ArcTan[((-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6)] + Sqrt[3]*Log[(a^2 + b^2)^(1/3) - (-1)^(1/6)*Sqrt[3]*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3)] - Sqrt[3]*Log[(a^2 + b^2)^(1/3) + (-1)^(1/6)*Sqrt[3]*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3)])*Sqrt[1 - Sec[e + f*x]^2])/(220*b^(2/3)*(a^2 + b^2)^2*Sqrt[1 - Sec[e + f*x]^2]))*(Cos[e + f*x] - Sin[e + f*x])*(Cos[e + f*x] + Sin[e + f*x])*(a*Cos[e + f*x] + b*Sin[e + f*x]))/(6*a*f*(a + b*Tan[e + f*x])^2*((-2*Sec[e + f*x]^(2/3)*(b + a*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x])*Sin[e + f*x])/(a^2 + b^2) + (Sec[e + f*x]^2*(132*a*b^(2/3)*(5*a^2 + 3*b^2)*AppellF1[5/6, 1/2, 1, 11/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^(8/3) - 240*a*b^(8/3)*AppellF1[11/6, 1/2, 1, 17/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^(14/3) - 55*(-1)^(1/6)*(a^2 - b^2)*(a^2 + b^2)^(5/6)*(-2*ArcTan[Sqrt[3] - (2*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6)] + 2*ArcTan[Sqrt[3] + (2*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6)] + 4*ArcTan[((-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6)] + Sqrt[3]*Log[(a^2 + b^2)^(1/3) - (-1)^(1/6)*Sqrt[3]*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3)] - Sqrt[3]*Log[(a^2 + b^2)^(1/3) + (-1)^(1/6)*Sqrt[3]*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3)])*Sqrt[1 - Sec[e + f*x]^2])*Tan[e + f*x])/(220*b^(2/3)*(a^2 + b^2)^2*(1 - Sec[e + f*x]^2)^(3/2)) + (6*((a*Sin[e + f*x])/Sqrt[1 - Cos[e + f*x]^2] + a*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]*Tan[e + f*x]))/((a^2 + b^2)*Sec[e + f*x]^(1/3)) + ((132*a*b^(2/3)*(5*a^2 + 3*b^2)*AppellF1[5/6, 1/2, 1, 11/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^(5/3)*Sin[e + f*x])/Sqrt[1 - Cos[e + f*x]^2] - (240*a*b^(8/3)*AppellF1[11/6, 1/2, 1, 17/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^(11/3)*Sin[e + f*x])/Sqrt[1 - Cos[e + f*x]^2] + 352*a*b^(2/3)*(5*a^2 + 3*b^2)*AppellF1[5/6, 1/2, 1, 11/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^(11/3)*Sin[e + f*x] - 1120*a*b^(8/3)*AppellF1[11/6, 1/2, 1, 17/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^(17/3)*Sin[e + f*x] - 55*(-1)^(1/6)*(a^2 - b^2)*(a^2 + b^2)^(5/6)*Sqrt[1 - Sec[e + f*x]^2]*((4*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(4/3)*Sin[e + f*x])/(3*(a^2 + b^2)^(1/6)*(1 + (Sqrt[3] - (2*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6))^2)) + (4*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(4/3)*Sin[e + f*x])/(3*(a^2 + b^2)^(1/6)*(1 + (Sqrt[3] + (2*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6))^2)) + (4*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(4/3)*Sin[e + f*x])/(3*(a^2 + b^2)^(1/6)*(1 + ((-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3))/(a^2 + b^2)^(1/3))) + (Sqrt[3]*(-(((-1)^(1/6)*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(4/3)*Sin[e + f*x])/Sqrt[3]) + (2*(-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(5/3)*Sin[e + f*x])/3))/((a^2 + b^2)^(1/3) - (-1)^(1/6)*Sqrt[3]*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3)) - (Sqrt[3]*(((-1)^(1/6)*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(4/3)*Sin[e + f*x])/Sqrt[3] + (2*(-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(5/3)*Sin[e + f*x])/3))/((a^2 + b^2)^(1/3) + (-1)^(1/6)*Sqrt[3]*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3))) + (55*(-1)^(1/6)*(a^2 - b^2)*(a^2 + b^2)^(5/6)*(-2*ArcTan[Sqrt[3] - (2*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6)] + 2*ArcTan[Sqrt[3] + (2*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6)] + 4*ArcTan[((-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6)] + Sqrt[3]*Log[(a^2 + b^2)^(1/3) - (-1)^(1/6)*Sqrt[3]*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3)] - Sqrt[3]*Log[(a^2 + b^2)^(1/3) + (-1)^(1/6)*Sqrt[3]*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3)])*Sec[e + f*x]^2*Tan[e + f*x])/Sqrt[1 - Sec[e + f*x]^2] + 132*a*b^(2/3)*(5*a^2 + 3*b^2)*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^(8/3)*((10*b^2*AppellF1[11/6, 1/2, 2, 17/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/(11*(a^2 + b^2)) + (5*AppellF1[11/6, 3/2, 1, 17/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/11) - 240*a*b^(8/3)*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^(14/3)*((22*b^2*AppellF1[17/6, 1/2, 2, 23/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/(17*(a^2 + b^2)) + (11*AppellF1[17/6, 3/2, 1, 23/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/17))/(220*b^(2/3)*(a^2 + b^2)^2*Sqrt[1 - Sec[e + f*x]^2])))","C",0
637,1,4485,687,25.724187,"\int \frac{\sqrt[3]{d \sec (e+f x)}}{(a+b \tan (e+f x))^2} \, dx","Integrate[(d*Sec[e + f*x])^(1/3)/(a + b*Tan[e + f*x])^2,x]","\text{Result too large to show}","\frac{\tan (e+f x) \sqrt[3]{d \sec (e+f x)} F_1\left(\frac{1}{2};2,\frac{5}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{a^2 f \sqrt[6]{\sec ^2(e+f x)}}-\frac{a b \sqrt[3]{d \sec (e+f x)}}{f \left(a^2+b^2\right) \left(a^2-b^2 \tan ^2(e+f x)\right)}+\frac{5 a b^{2/3} \sqrt[3]{d \sec (e+f x)} \log \left(-\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{12 f \left(a^2+b^2\right)^{11/6} \sqrt[6]{\sec ^2(e+f x)}}-\frac{5 a b^{2/3} \sqrt[3]{d \sec (e+f x)} \log \left(\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{12 f \left(a^2+b^2\right)^{11/6} \sqrt[6]{\sec ^2(e+f x)}}+\frac{5 a b^{2/3} \sqrt[3]{d \sec (e+f x)} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}\right)}{2 \sqrt{3} f \left(a^2+b^2\right)^{11/6} \sqrt[6]{\sec ^2(e+f x)}}-\frac{5 a b^{2/3} \sqrt[3]{d \sec (e+f x)} \tan ^{-1}\left(\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right)}{2 \sqrt{3} f \left(a^2+b^2\right)^{11/6} \sqrt[6]{\sec ^2(e+f x)}}-\frac{5 a b^{2/3} \sqrt[3]{d \sec (e+f x)} \tanh ^{-1}\left(\frac{\sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt[6]{a^2+b^2}}\right)}{3 f \left(a^2+b^2\right)^{11/6} \sqrt[6]{\sec ^2(e+f x)}}+\frac{b^2 \tan ^3(e+f x) \sqrt[3]{d \sec (e+f x)} F_1\left(\frac{3}{2};2,\frac{5}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{3 a^4 f \sqrt[6]{\sec ^2(e+f x)}}",1,"((d*Sec[e + f*x])^(1/3)*((5*(-1)^(5/6)*a*b^(2/3)*(-2*ArcTan[Sqrt[3] - (2*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6)] + 2*ArcTan[Sqrt[3] + (2*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6)] + 4*ArcTan[((-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6)] - Sqrt[3]*Log[(a^2 + b^2)^(1/3) - (-1)^(1/6)*Sqrt[3]*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3)] + Sqrt[3]*Log[(a^2 + b^2)^(1/3) + (-1)^(1/6)*Sqrt[3]*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3)]))/(12*(a - I*b)*(a + I*b)*(a^2 + b^2)^(5/6)) + 3*((-2*b^2*AppellF1[7/6, 1/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^(10/3))/(21*(a^2 + b^2)^2*Sqrt[1 - Sec[e + f*x]^2]) + (Sec[e + f*x]^(1/3)*((-(a*b) + b^2*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x])/(a^2 + b^2) + (7*(3*a^2 - 2*b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x])/((-1 + Sec[e + f*x]^2)*(7*(a^2 + b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + 3*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])*Sec[e + f*x]^2))))/(3*(a^2 - b^2*(-1 + Sec[e + f*x]^2))))))/(f*(a + b*Tan[e + f*x])^2*((5*(-1)^(5/6)*a*b^(2/3)*((4*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(4/3)*Sin[e + f*x])/(3*(a^2 + b^2)^(1/6)*(1 + (Sqrt[3] - (2*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6))^2)) + (4*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(4/3)*Sin[e + f*x])/(3*(a^2 + b^2)^(1/6)*(1 + (Sqrt[3] + (2*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6))^2)) + (4*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(4/3)*Sin[e + f*x])/(3*(a^2 + b^2)^(1/6)*(1 + ((-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3))/(a^2 + b^2)^(1/3))) - (Sqrt[3]*(-(((-1)^(1/6)*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(4/3)*Sin[e + f*x])/Sqrt[3]) + (2*(-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(5/3)*Sin[e + f*x])/3))/((a^2 + b^2)^(1/3) - (-1)^(1/6)*Sqrt[3]*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3)) + (Sqrt[3]*(((-1)^(1/6)*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(4/3)*Sin[e + f*x])/Sqrt[3] + (2*(-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(5/3)*Sin[e + f*x])/3))/((a^2 + b^2)^(1/3) + (-1)^(1/6)*Sqrt[3]*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3))))/(12*(a - I*b)*(a + I*b)*(a^2 + b^2)^(5/6)) + 3*((-2*b^2*AppellF1[7/6, 1/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^(19/3)*Sin[e + f*x])/(21*(a^2 + b^2)^2*(1 - Sec[e + f*x]^2)^(3/2)) - (2*b^2*AppellF1[7/6, 1/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^(7/3)*Sin[e + f*x])/(21*(a^2 + b^2)^2*Sqrt[1 - Cos[e + f*x]^2]*Sqrt[1 - Sec[e + f*x]^2]) - (20*b^2*AppellF1[7/6, 1/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^(13/3)*Sin[e + f*x])/(63*(a^2 + b^2)^2*Sqrt[1 - Sec[e + f*x]^2]) + (2*b^2*Sec[e + f*x]^(10/3)*((-(a*b) + b^2*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x])/(a^2 + b^2) + (7*(3*a^2 - 2*b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x])/((-1 + Sec[e + f*x]^2)*(7*(a^2 + b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + 3*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])*Sec[e + f*x]^2)))*Sin[e + f*x])/(3*(a^2 - b^2*(-1 + Sec[e + f*x]^2))^2) + (Sec[e + f*x]^(4/3)*((-(a*b) + b^2*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x])/(a^2 + b^2) + (7*(3*a^2 - 2*b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x])/((-1 + Sec[e + f*x]^2)*(7*(a^2 + b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + 3*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])*Sec[e + f*x]^2)))*Sin[e + f*x])/(9*(a^2 - b^2*(-1 + Sec[e + f*x]^2))) - (2*b^2*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^(10/3)*((14*b^2*AppellF1[13/6, 1/2, 2, 19/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/(13*(a^2 + b^2)) + (7*AppellF1[13/6, 3/2, 1, 19/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/13))/(21*(a^2 + b^2)^2*Sqrt[1 - Sec[e + f*x]^2]) + (Sec[e + f*x]^(1/3)*((7*(3*a^2 - 2*b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(-1 + Sec[e + f*x]^2)*(7*(a^2 + b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + 3*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])*Sec[e + f*x]^2)) - (14*(3*a^2 - 2*b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^3*Tan[e + f*x])/((-1 + Sec[e + f*x]^2)^2*(7*(a^2 + b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + 3*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])*Sec[e + f*x]^2)) + (7*(3*a^2 - 2*b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]*Tan[e + f*x])/((-1 + Sec[e + f*x]^2)*(7*(a^2 + b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + 3*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])*Sec[e + f*x]^2)) + ((b^2*Sin[e + f*x])/Sqrt[1 - Cos[e + f*x]^2] + b^2*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]*Tan[e + f*x])/(a^2 + b^2) + (7*(3*a^2 - 2*b^2)*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]*((2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/(7*(a^2 + b^2)) + (AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/7))/((-1 + Sec[e + f*x]^2)*(7*(a^2 + b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + 3*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])*Sec[e + f*x]^2)) - (7*(3*a^2 - 2*b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]*(6*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])*Sec[e + f*x]^2*Tan[e + f*x] + 7*(a^2 + b^2)*((2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/(7*(a^2 + b^2)) + (AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/7) + 3*Sec[e + f*x]^2*(2*b^2*((28*b^2*AppellF1[13/6, 1/2, 3, 19/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/(13*(a^2 + b^2)) + (7*AppellF1[13/6, 3/2, 2, 19/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/13) + (a^2 + b^2)*((14*b^2*AppellF1[13/6, 3/2, 2, 19/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/(13*(a^2 + b^2)) + (21*AppellF1[13/6, 5/2, 1, 19/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/13))))/((-1 + Sec[e + f*x]^2)*(7*(a^2 + b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + 3*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])*Sec[e + f*x]^2)^2)))/(3*(a^2 - b^2*(-1 + Sec[e + f*x]^2))))))","C",0
638,1,9626,715,51.9833307,"\int \frac{1}{\sqrt[3]{d \sec (e+f x)} (a+b \tan (e+f x))^2} \, dx","Integrate[1/((d*Sec[e + f*x])^(1/3)*(a + b*Tan[e + f*x])^2),x]","\text{Result too large to show}","\frac{\tan (e+f x) \sqrt[6]{\sec ^2(e+f x)} F_1\left(\frac{1}{2};2,\frac{7}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{a^2 f \sqrt[3]{d \sec (e+f x)}}+\frac{7 a b}{f \left(a^2+b^2\right)^2 \sqrt[3]{d \sec (e+f x)}}-\frac{a b}{f \left(a^2+b^2\right) \sqrt[3]{d \sec (e+f x)} \left(a^2-b^2 \tan ^2(e+f x)\right)}+\frac{7 a b^{4/3} \sqrt[6]{\sec ^2(e+f x)} \log \left(-\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{12 f \left(a^2+b^2\right)^{13/6} \sqrt[3]{d \sec (e+f x)}}-\frac{7 a b^{4/3} \sqrt[6]{\sec ^2(e+f x)} \log \left(\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{12 f \left(a^2+b^2\right)^{13/6} \sqrt[3]{d \sec (e+f x)}}-\frac{7 a b^{4/3} \sqrt[6]{\sec ^2(e+f x)} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}\right)}{2 \sqrt{3} f \left(a^2+b^2\right)^{13/6} \sqrt[3]{d \sec (e+f x)}}+\frac{7 a b^{4/3} \sqrt[6]{\sec ^2(e+f x)} \tan ^{-1}\left(\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right)}{2 \sqrt{3} f \left(a^2+b^2\right)^{13/6} \sqrt[3]{d \sec (e+f x)}}-\frac{7 a b^{4/3} \sqrt[6]{\sec ^2(e+f x)} \tanh ^{-1}\left(\frac{\sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt[6]{a^2+b^2}}\right)}{3 f \left(a^2+b^2\right)^{13/6} \sqrt[3]{d \sec (e+f x)}}+\frac{b^2 \tan ^3(e+f x) \sqrt[6]{\sec ^2(e+f x)} F_1\left(\frac{3}{2};2,\frac{7}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{3 a^4 f \sqrt[3]{d \sec (e+f x)}}",1,"Result too large to show","C",0
639,1,5235,717,27.8418344,"\int \frac{1}{(d \sec (e+f x))^{5/3} (a+b \tan (e+f x))^2} \, dx","Integrate[1/((d*Sec[e + f*x])^(5/3)*(a + b*Tan[e + f*x])^2),x]","\text{Result too large to show}","\frac{\tan (e+f x) \sec ^2(e+f x)^{5/6} F_1\left(\frac{1}{2};2,\frac{11}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{a^2 f (d \sec (e+f x))^{5/3}}+\frac{11 a b}{5 f \left(a^2+b^2\right)^2 (d \sec (e+f x))^{5/3}}-\frac{a b}{f \left(a^2+b^2\right) (d \sec (e+f x))^{5/3} \left(a^2-b^2 \tan ^2(e+f x)\right)}+\frac{11 a b^{8/3} \sec ^2(e+f x)^{5/6} \log \left(-\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{12 f \left(a^2+b^2\right)^{17/6} (d \sec (e+f x))^{5/3}}-\frac{11 a b^{8/3} \sec ^2(e+f x)^{5/6} \log \left(\sqrt[3]{b} \sqrt[6]{a^2+b^2} \sqrt[6]{\sec ^2(e+f x)}+\sqrt[3]{a^2+b^2}+b^{2/3} \sqrt[3]{\sec ^2(e+f x)}\right)}{12 f \left(a^2+b^2\right)^{17/6} (d \sec (e+f x))^{5/3}}+\frac{11 a b^{8/3} \sec ^2(e+f x)^{5/6} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}\right)}{2 \sqrt{3} f \left(a^2+b^2\right)^{17/6} (d \sec (e+f x))^{5/3}}-\frac{11 a b^{8/3} \sec ^2(e+f x)^{5/6} \tan ^{-1}\left(\frac{2 \sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt[6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right)}{2 \sqrt{3} f \left(a^2+b^2\right)^{17/6} (d \sec (e+f x))^{5/3}}-\frac{11 a b^{8/3} \sec ^2(e+f x)^{5/6} \tanh ^{-1}\left(\frac{\sqrt[3]{b} \sqrt[6]{\sec ^2(e+f x)}}{\sqrt[6]{a^2+b^2}}\right)}{3 f \left(a^2+b^2\right)^{17/6} (d \sec (e+f x))^{5/3}}+\frac{b^2 \tan ^3(e+f x) \sec ^2(e+f x)^{5/6} F_1\left(\frac{3}{2};2,\frac{11}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{3 a^4 f (d \sec (e+f x))^{5/3}}",1,"Result too large to show","C",0
640,1,334,173,6.4455503,"\int (d \sec (e+f x))^m (a+b \tan (e+f x))^3 \, dx","Integrate[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^3,x]","\frac{a \left(a^2-3 b^2\right) \sin (e+f x) \cos ^4(e+f x) \cos ^2(e+f x)^{\frac{m}{2}-\frac{1}{2}} (a+b \tan (e+f x))^3 (d \sec (e+f x))^m \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{3}{2};\sin ^2(e+f x)\right)}{f (a \cos (e+f x)+b \sin (e+f x))^3}-\frac{b \left(b^2-3 a^2\right) \cos ^3(e+f x) (a+b \tan (e+f x))^3 (d \sec (e+f x))^m}{f m (a \cos (e+f x)+b \sin (e+f x))^3}+\frac{b^3 \cos (e+f x) (a+b \tan (e+f x))^3 (d \sec (e+f x))^m}{f (m+2) (a \cos (e+f x)+b \sin (e+f x))^3}+\frac{3 a b^2 \sin (e+f x) \cos ^2(e+f x)^{\frac{m+2}{2}+\frac{1}{2}} (a+b \tan (e+f x))^3 (d \sec (e+f x))^m \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{3}{2};\sin ^2(e+f x)\right)}{f (a \cos (e+f x)+b \sin (e+f x))^3}","-\frac{a \left(3 b^2-a^2 (m+1)\right) \tan (e+f x) \sec ^2(e+f x)^{-m/2} (d \sec (e+f x))^m \, _2F_1\left(\frac{1}{2},1-\frac{m}{2};\frac{3}{2};-\tan ^2(e+f x)\right)}{f (m+1)}-\frac{b (d \sec (e+f x))^m \left(2 (m+1) \left(b^2-a^2 (m+3)\right)-a b m (m+4) \tan (e+f x)\right)}{f m \left(m^2+3 m+2\right)}+\frac{b (a+b \tan (e+f x))^2 (d \sec (e+f x))^m}{f (m+2)}",1,"(b^3*Cos[e + f*x]*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^3)/(f*(2 + m)*(a*Cos[e + f*x] + b*Sin[e + f*x])^3) - (b*(-3*a^2 + b^2)*Cos[e + f*x]^3*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^3)/(f*m*(a*Cos[e + f*x] + b*Sin[e + f*x])^3) + (a*(a^2 - 3*b^2)*Cos[e + f*x]^4*(Cos[e + f*x]^2)^(-1/2 + m/2)*Hypergeometric2F1[1/2, (1 + m)/2, 3/2, Sin[e + f*x]^2]*(d*Sec[e + f*x])^m*Sin[e + f*x]*(a + b*Tan[e + f*x])^3)/(f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3) + (3*a*b^2*(Cos[e + f*x]^2)^(1/2 + (2 + m)/2)*Hypergeometric2F1[1/2, (3 + m)/2, 3/2, Sin[e + f*x]^2]*(d*Sec[e + f*x])^m*Sin[e + f*x]*(a + b*Tan[e + f*x])^3)/(f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3)","A",1
641,1,11095,147,26.4969219,"\int (d \sec (e+f x))^m (a+b \tan (e+f x))^2 \, dx","Integrate[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^2,x]","\text{Result too large to show}","\frac{d \left(b^2-a^2 (m+1)\right) \sin (e+f x) (d \sec (e+f x))^{m-1} \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3-m}{2};\cos ^2(e+f x)\right)}{f (1-m) (m+1) \sqrt{\sin ^2(e+f x)}}+\frac{a b (m+2) (d \sec (e+f x))^m}{f m (m+1)}+\frac{b (a+b \tan (e+f x)) (d \sec (e+f x))^m}{f (m+1)}",1,"Result too large to show","C",0
642,1,3302,93,16.8028163,"\int (d \sec (e+f x))^m (a+b \tan (e+f x)) \, dx","Integrate[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x]),x]","\text{Result too large to show}","\frac{b (d \sec (e+f x))^m}{f m}-\frac{a d \sin (e+f x) (d \sec (e+f x))^{m-1} \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3-m}{2};\cos ^2(e+f x)\right)}{f (1-m) \sqrt{\sin ^2(e+f x)}}",1,"-((Sec[e + f*x]^(-1 - m)*(d*Sec[e + f*x])^m*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*(a*Sec[e + f*x]^m + b*Sec[e + f*x]^(1 + m)*Sin[e + f*x])*Tan[(e + f*x)/2]*(-(b*AppellF1[1, m, 1 - m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m*Tan[(e + f*x)/2]) - b*AppellF1[1, 1 + m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m*Tan[(e + f*x)/2] - (6*a*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + m))/(3*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))*(a + b*Tan[e + f*x]))/(f*(a*Cos[e + f*x] + b*Sin[e + f*x])*(-1/2*(Sec[(e + f*x)/2]^2*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*(-(b*AppellF1[1, m, 1 - m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m*Tan[(e + f*x)/2]) - b*AppellF1[1, 1 + m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m*Tan[(e + f*x)/2] - (6*a*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + m))/(3*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))) - (Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*Tan[(e + f*x)/2]*(-1/2*(b*AppellF1[1, m, 1 - m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m) - (b*AppellF1[1, 1 + m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m)/2 - b*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m*Tan[(e + f*x)/2]*(-1/2*((1 - m)*AppellF1[2, m, 2 - m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (m*AppellF1[2, 1 + m, 1 - m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/2) - b*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m*Tan[(e + f*x)/2]*((m*AppellF1[2, 1 + m, 1 - m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/2 + ((1 + m)*AppellF1[2, 2 + m, -m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/2) - b*m*AppellF1[1, m, 1 - m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + m)*Tan[(e + f*x)/2]*(-(Sec[(e + f*x)/2]^2*Sin[e + f*x]) + Cos[e + f*x]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) - b*m*AppellF1[1, 1 + m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + m)*Tan[(e + f*x)/2]*(-(Sec[(e + f*x)/2]^2*Sin[e + f*x]) + Cos[e + f*x]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) - (6*a*(-1 + m)*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + m)*Tan[(e + f*x)/2])/(3*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (6*a*(Sec[(e + f*x)/2]^2)^(-1 + m)*(-1/3*((1 - m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (6*a*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + m)*(2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*Tan[(e + f*x)/2]^2*((-1 + m)*((-3*(2 - m)*AppellF1[5/2, m, 3 - m, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*m*AppellF1[5/2, 1 + m, 2 - m, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + m*((-3*(1 - m)*AppellF1[5/2, 1 + m, 2 - m, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + m)*AppellF1[5/2, 2 + m, 1 - m, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2) - m*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + m)*Tan[(e + f*x)/2]*(-(b*AppellF1[1, m, 1 - m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m*Tan[(e + f*x)/2]) - b*AppellF1[1, 1 + m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m*Tan[(e + f*x)/2] - (6*a*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + m))/(3*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))))","C",0
643,1,1158,141,15.0546087,"\int \frac{(d \sec (e+f x))^m}{a+b \tan (e+f x)} \, dx","Integrate[(d*Sec[e + f*x])^m/(a + b*Tan[e + f*x]),x]","\frac{(d \sec (e+f x))^m \left(b F_1\left(-m;-\frac{m}{2},-\frac{m}{2};1-m;\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right) \sec ^2(e+f x)^{m/2} \left(\frac{b (\tan (e+f x)-i)}{a+b \tan (e+f x)}\right)^{-m/2} \left(\frac{b (\tan (e+f x)+i)}{a+b \tan (e+f x)}\right)^{-m/2}-b \sec ^2(e+f x)^{m/2}+b+a m \, _2F_1\left(\frac{1}{2},1-\frac{m}{2};\frac{3}{2};-\tan ^2(e+f x)\right) \tan (e+f x)\right)}{f (a+b \tan (e+f x)) \left(-\frac{1}{2} b m F_1\left(-m;-\frac{m}{2},-\frac{m}{2};1-m;\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right) \sec ^2(e+f x)^{m/2} \left(\frac{b (\tan (e+f x)+i)}{a+b \tan (e+f x)}\right)^{-m/2} \left(\frac{b \sec ^2(e+f x)}{a+b \tan (e+f x)}-\frac{b^2 \sec ^2(e+f x) (\tan (e+f x)-i)}{(a+b \tan (e+f x))^2}\right) \left(\frac{b (\tan (e+f x)-i)}{a+b \tan (e+f x)}\right)^{-\frac{m}{2}-1}+b m F_1\left(-m;-\frac{m}{2},-\frac{m}{2};1-m;\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right) \sec ^2(e+f x)^{m/2} \tan (e+f x) \left(\frac{b (\tan (e+f x)+i)}{a+b \tan (e+f x)}\right)^{-m/2} \left(\frac{b (\tan (e+f x)-i)}{a+b \tan (e+f x)}\right)^{-m/2}+b \sec ^2(e+f x)^{m/2} \left(\frac{b (\tan (e+f x)+i)}{a+b \tan (e+f x)}\right)^{-m/2} \left(-\frac{(a-i b) b m^2 F_1\left(1-m;1-\frac{m}{2},-\frac{m}{2};2-m;\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right) \sec ^2(e+f x)}{2 (1-m) (a+b \tan (e+f x))^2}-\frac{(a+i b) b m^2 F_1\left(1-m;-\frac{m}{2},1-\frac{m}{2};2-m;\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right) \sec ^2(e+f x)}{2 (1-m) (a+b \tan (e+f x))^2}\right) \left(\frac{b (\tan (e+f x)-i)}{a+b \tan (e+f x)}\right)^{-m/2}-\frac{1}{2} b m F_1\left(-m;-\frac{m}{2},-\frac{m}{2};1-m;\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right) \sec ^2(e+f x)^{m/2} \left(\frac{b (\tan (e+f x)+i)}{a+b \tan (e+f x)}\right)^{-\frac{m}{2}-1} \left(\frac{b \sec ^2(e+f x)}{a+b \tan (e+f x)}-\frac{b^2 \sec ^2(e+f x) (\tan (e+f x)+i)}{(a+b \tan (e+f x))^2}\right) \left(\frac{b (\tan (e+f x)-i)}{a+b \tan (e+f x)}\right)^{-m/2}+a m \, _2F_1\left(\frac{1}{2},1-\frac{m}{2};\frac{3}{2};-\tan ^2(e+f x)\right) \sec ^2(e+f x)-b m \sec ^2(e+f x)^{m/2} \tan (e+f x)+a m \sec ^2(e+f x) \left(\left(\tan ^2(e+f x)+1\right)^{\frac{m}{2}-1}-\, _2F_1\left(\frac{1}{2},1-\frac{m}{2};\frac{3}{2};-\tan ^2(e+f x)\right)\right)\right)}","\frac{\tan (e+f x) \sec ^2(e+f x)^{-m/2} (d \sec (e+f x))^m F_1\left(\frac{1}{2};1,1-\frac{m}{2};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{a f}-\frac{b (d \sec (e+f x))^m \, _2F_1\left(1,\frac{m}{2};\frac{m+2}{2};\frac{b^2 \sec ^2(e+f x)}{a^2+b^2}\right)}{f m \left(a^2+b^2\right)}",1,"((d*Sec[e + f*x])^m*(b - b*(Sec[e + f*x]^2)^(m/2) + a*m*Hypergeometric2F1[1/2, 1 - m/2, 3/2, -Tan[e + f*x]^2]*Tan[e + f*x] + (b*AppellF1[-m, -1/2*m, -1/2*m, 1 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(Sec[e + f*x]^2)^(m/2))/(((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2))))/(f*(a + b*Tan[e + f*x])*(a*m*Hypergeometric2F1[1/2, 1 - m/2, 3/2, -Tan[e + f*x]^2]*Sec[e + f*x]^2 - b*m*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x] + (b*m*AppellF1[-m, -1/2*m, -1/2*m, 1 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x])/(((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)) + (b*(Sec[e + f*x]^2)^(m/2)*(-1/2*((a - I*b)*b*m^2*AppellF1[1 - m, 1 - m/2, -1/2*m, 2 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*Sec[e + f*x]^2)/((1 - m)*(a + b*Tan[e + f*x])^2) - ((a + I*b)*b*m^2*AppellF1[1 - m, -1/2*m, 1 - m/2, 2 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*Sec[e + f*x]^2)/(2*(1 - m)*(a + b*Tan[e + f*x])^2)))/(((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)) - (b*m*AppellF1[-m, -1/2*m, -1/2*m, 1 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(Sec[e + f*x]^2)^(m/2)*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(-1 - m/2)*(-((b^2*Sec[e + f*x]^2*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x])^2) + (b*Sec[e + f*x]^2)/(a + b*Tan[e + f*x])))/(2*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)) - (b*m*AppellF1[-m, -1/2*m, -1/2*m, 1 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(Sec[e + f*x]^2)^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(-1 - m/2)*(-((b^2*Sec[e + f*x]^2*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x])^2) + (b*Sec[e + f*x]^2)/(a + b*Tan[e + f*x])))/(2*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)) + a*m*Sec[e + f*x]^2*(-Hypergeometric2F1[1/2, 1 - m/2, 3/2, -Tan[e + f*x]^2] + (1 + Tan[e + f*x]^2)^(-1 + m/2))))","C",0
644,1,2453,227,18.519864,"\int \frac{(d \sec (e+f x))^m}{(a+b \tan (e+f x))^2} \, dx","Integrate[(d*Sec[e + f*x])^m/(a + b*Tan[e + f*x])^2,x]","\text{Result too large to show}","\frac{\tan (e+f x) \sec ^2(e+f x)^{-m/2} (d \sec (e+f x))^m F_1\left(\frac{1}{2};2,1-\frac{m}{2};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{a^2 f}-\frac{2 a b (d \sec (e+f x))^m \, _2F_1\left(2,\frac{m}{2};\frac{m+2}{2};\frac{b^2 \sec ^2(e+f x)}{a^2+b^2}\right)}{f m \left(a^2+b^2\right)^2}+\frac{b^2 \tan ^3(e+f x) \sec ^2(e+f x)^{-m/2} (d \sec (e+f x))^m F_1\left(\frac{3}{2};2,1-\frac{m}{2};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{3 a^4 f}",1,"((d*Sec[e + f*x])^m*((-2*a*b*(-1 + (Sec[e + f*x]^2)^(m/2)))/m + (a^2 - b^2)*Hypergeometric2F1[1/2, 1 - m/2, 3/2, -Tan[e + f*x]^2]*Tan[e + f*x] + (2*a*b*AppellF1[-m, -1/2*m, -1/2*m, 1 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(Sec[e + f*x]^2)^(m/2))/(m*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)) + (b*(a^2 + b^2)*AppellF1[1 - m, -1/2*m, -1/2*m, 2 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(Sec[e + f*x]^2)^(m/2))/((-1 + m)*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*(a + b*Tan[e + f*x]))))/(f*(a + b*Tan[e + f*x])^2*((a^2 - b^2)*Hypergeometric2F1[1/2, 1 - m/2, 3/2, -Tan[e + f*x]^2]*Sec[e + f*x]^2 - 2*a*b*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x] + (2*a*b*AppellF1[-m, -1/2*m, -1/2*m, 1 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x])/(((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)) - (b^2*(a^2 + b^2)*AppellF1[1 - m, -1/2*m, -1/2*m, 2 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(Sec[e + f*x]^2)^(1 + m/2))/((-1 + m)*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*(a + b*Tan[e + f*x])^2) + (b*(a^2 + b^2)*m*AppellF1[1 - m, -1/2*m, -1/2*m, 2 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x])/((-1 + m)*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*(a + b*Tan[e + f*x])) + (2*a*b*(Sec[e + f*x]^2)^(m/2)*(-1/2*((a - I*b)*b*m^2*AppellF1[1 - m, 1 - m/2, -1/2*m, 2 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*Sec[e + f*x]^2)/((1 - m)*(a + b*Tan[e + f*x])^2) - ((a + I*b)*b*m^2*AppellF1[1 - m, -1/2*m, 1 - m/2, 2 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*Sec[e + f*x]^2)/(2*(1 - m)*(a + b*Tan[e + f*x])^2)))/(m*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)) + (b*(a^2 + b^2)*(Sec[e + f*x]^2)^(m/2)*(((a - I*b)*b*(1 - m)*m*AppellF1[2 - m, 1 - m/2, -1/2*m, 3 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*Sec[e + f*x]^2)/(2*(2 - m)*(a + b*Tan[e + f*x])^2) + ((a + I*b)*b*(1 - m)*m*AppellF1[2 - m, -1/2*m, 1 - m/2, 3 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*Sec[e + f*x]^2)/(2*(2 - m)*(a + b*Tan[e + f*x])^2)))/((-1 + m)*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*(a + b*Tan[e + f*x])) - (a*b*AppellF1[-m, -1/2*m, -1/2*m, 1 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(Sec[e + f*x]^2)^(m/2)*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(-1 - m/2)*(-((b^2*Sec[e + f*x]^2*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x])^2) + (b*Sec[e + f*x]^2)/(a + b*Tan[e + f*x])))/((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2) - (b*(a^2 + b^2)*m*AppellF1[1 - m, -1/2*m, -1/2*m, 2 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(Sec[e + f*x]^2)^(m/2)*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(-1 - m/2)*(-((b^2*Sec[e + f*x]^2*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x])^2) + (b*Sec[e + f*x]^2)/(a + b*Tan[e + f*x])))/(2*(-1 + m)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*(a + b*Tan[e + f*x])) - (a*b*AppellF1[-m, -1/2*m, -1/2*m, 1 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(Sec[e + f*x]^2)^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(-1 - m/2)*(-((b^2*Sec[e + f*x]^2*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x])^2) + (b*Sec[e + f*x]^2)/(a + b*Tan[e + f*x])))/((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2) - (b*(a^2 + b^2)*m*AppellF1[1 - m, -1/2*m, -1/2*m, 2 - m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(Sec[e + f*x]^2)^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(-1 - m/2)*(-((b^2*Sec[e + f*x]^2*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x])^2) + (b*Sec[e + f*x]^2)/(a + b*Tan[e + f*x])))/(2*(-1 + m)*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*(a + b*Tan[e + f*x])) + (a^2 - b^2)*Sec[e + f*x]^2*(-Hypergeometric2F1[1/2, 1 - m/2, 3/2, -Tan[e + f*x]^2] + (1 + Tan[e + f*x]^2)^(-1 + m/2))))","C",0
645,1,699,181,6.5508641,"\int (d \sec (e+f x))^m (a+b \tan (e+f x))^n \, dx","Integrate[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n,x]","\frac{2 (d \sec (e+f x))^m (a+b \tan (e+f x))^{n+1} F_1\left(n+1;1-\frac{m}{2},1-\frac{m}{2};n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)}{f \left(2 n (b-a \tan (e+f x)) F_1\left(n+1;1-\frac{m}{2},1-\frac{m}{2};n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)+2 (m+n) \tan (e+f x) (a+b \tan (e+f x)) F_1\left(n+1;1-\frac{m}{2},1-\frac{m}{2};n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)+2 b \sec ^2(e+f x) F_1\left(n+1;1-\frac{m}{2},1-\frac{m}{2};n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)-\frac{b (m-2) (n+1) \sec ^2(e+f x) (a+b \tan (e+f x)) \left((a-i b) F_1\left(n+2;1-\frac{m}{2},2-\frac{m}{2};n+3;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)+(a+i b) F_1\left(n+2;2-\frac{m}{2},1-\frac{m}{2};n+3;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)\right)}{(n+2) (a-i b) (a+i b)}-\frac{m \sec ^2(e+f x) (a+b \tan (e+f x)) F_1\left(n+1;1-\frac{m}{2},1-\frac{m}{2};n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)}{\tan (e+f x)-i}-\frac{m \sec ^2(e+f x) (a+b \tan (e+f x)) F_1\left(n+1;1-\frac{m}{2},1-\frac{m}{2};n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)}{\tan (e+f x)+i}\right)}","\frac{b (d \sec (e+f x))^m \left(\frac{a+b \tan (e+f x)}{\sqrt{-b^2}-a}+1\right)^{-m/2} \left(1-\frac{a+b \tan (e+f x)}{a+\sqrt{-b^2}}\right)^{-m/2} (a+b \tan (e+f x))^{n+1} F_1\left(n+1;1-\frac{m}{2},1-\frac{m}{2};n+2;\frac{a+b \tan (e+f x)}{a+\sqrt{-b^2}},\frac{a+b \tan (e+f x)}{a-\sqrt{-b^2}}\right)}{f (n+1) \left(a^2+b^2\right)}",1,"(2*AppellF1[1 + n, 1 - m/2, 1 - m/2, 2 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(1 + n))/(f*(2*b*AppellF1[1 + n, 1 - m/2, 1 - m/2, 2 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*Sec[e + f*x]^2 + 2*n*AppellF1[1 + n, 1 - m/2, 1 - m/2, 2 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*(b - a*Tan[e + f*x]) - (b*(-2 + m)*(1 + n)*((a - I*b)*AppellF1[2 + n, 1 - m/2, 2 - m/2, 3 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)] + (a + I*b)*AppellF1[2 + n, 2 - m/2, 1 - m/2, 3 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)])*Sec[e + f*x]^2*(a + b*Tan[e + f*x]))/((a - I*b)*(a + I*b)*(2 + n)) + 2*(m + n)*AppellF1[1 + n, 1 - m/2, 1 - m/2, 2 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*Tan[e + f*x]*(a + b*Tan[e + f*x]) - (m*AppellF1[1 + n, 1 - m/2, 1 - m/2, 2 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*Sec[e + f*x]^2*(a + b*Tan[e + f*x]))/(-I + Tan[e + f*x]) - (m*AppellF1[1 + n, 1 - m/2, 1 - m/2, 2 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*Sec[e + f*x]^2*(a + b*Tan[e + f*x]))/(I + Tan[e + f*x])))","C",0
646,1,161,161,3.029878,"\int \sec ^6(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Sec[c + d*x]^6*(a + b*Tan[c + d*x])^n,x]","\frac{(a+b \tan (c+d x))^{n+1} \left(4 \left(a^2+b^2\right) \left(\frac{a^2+b^2}{n+1}+\frac{(a+b \tan (c+d x))^2}{n+3}-\frac{2 a (a+b \tan (c+d x))}{n+2}\right)-4 a (a+b \tan (c+d x)) \left(\frac{a^2+b^2}{n+2}+\frac{(a+b \tan (c+d x))^2}{n+4}-\frac{2 a (a+b \tan (c+d x))}{n+3}\right)+b^4 \sec ^4(c+d x)\right)}{b^5 d (n+5)}","\frac{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))^{n+1}}{b^5 d (n+1)}-\frac{4 a \left(a^2+b^2\right) (a+b \tan (c+d x))^{n+2}}{b^5 d (n+2)}+\frac{2 \left(3 a^2+b^2\right) (a+b \tan (c+d x))^{n+3}}{b^5 d (n+3)}-\frac{4 a (a+b \tan (c+d x))^{n+4}}{b^5 d (n+4)}+\frac{(a+b \tan (c+d x))^{n+5}}{b^5 d (n+5)}",1,"((a + b*Tan[c + d*x])^(1 + n)*(b^4*Sec[c + d*x]^4 + 4*(a^2 + b^2)*((a^2 + b^2)/(1 + n) - (2*a*(a + b*Tan[c + d*x]))/(2 + n) + (a + b*Tan[c + d*x])^2/(3 + n)) - 4*a*(a + b*Tan[c + d*x])*((a^2 + b^2)/(2 + n) - (2*a*(a + b*Tan[c + d*x]))/(3 + n) + (a + b*Tan[c + d*x])^2/(4 + n))))/(b^5*d*(5 + n))","A",1
647,1,71,88,0.2403178,"\int \sec ^4(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Sec[c + d*x]^4*(a + b*Tan[c + d*x])^n,x]","\frac{(a+b \tan (c+d x))^{n+1} \left(\frac{a^2+b^2}{n+1}+\frac{(a+b \tan (c+d x))^2}{n+3}-\frac{2 a (a+b \tan (c+d x))}{n+2}\right)}{b^3 d}","\frac{\left(a^2+b^2\right) (a+b \tan (c+d x))^{n+1}}{b^3 d (n+1)}-\frac{2 a (a+b \tan (c+d x))^{n+2}}{b^3 d (n+2)}+\frac{(a+b \tan (c+d x))^{n+3}}{b^3 d (n+3)}",1,"((a + b*Tan[c + d*x])^(1 + n)*((a^2 + b^2)/(1 + n) - (2*a*(a + b*Tan[c + d*x]))/(2 + n) + (a + b*Tan[c + d*x])^2/(3 + n)))/(b^3*d)","A",1
648,1,26,26,0.1846934,"\int \sec ^2(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Sec[c + d*x]^2*(a + b*Tan[c + d*x])^n,x]","\frac{(a+b \tan (c+d x))^{n+1}}{b d (n+1)}","\frac{(a+b \tan (c+d x))^{n+1}}{b d (n+1)}",1,"(a + b*Tan[c + d*x])^(1 + n)/(b*d*(1 + n))","A",1
649,1,225,272,1.2674041,"\int \cos ^2(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Cos[c + d*x]^2*(a + b*Tan[c + d*x])^n,x]","\frac{(a+b \tan (c+d x))^{n+1} \left(-\frac{\left(\sqrt{-b^2} \left(a^2-b^2 (n-1)\right)-a b^2 n\right) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)}{(n+1) \left(a-\sqrt{-b^2}\right)}+\frac{\left(a^2 \sqrt{-b^2}+a b^2 n+\left(-b^2\right)^{3/2} (n-1)\right) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{(n+1) \left(a+\sqrt{-b^2}\right)}+2 b \cos ^2(c+d x) (a \tan (c+d x)+b)\right)}{4 b d \left(a^2+b^2\right)}","-\frac{\left(\sqrt{-b^2} \left(\frac{a^2}{b^2}-n+1\right)-a n\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)}{4 b d (n+1) \left(\frac{a^2}{b^2}+1\right) \left(a-\sqrt{-b^2}\right)}+\frac{b \left(\sqrt{-b^2} \left(\frac{a^2}{b^2}-n+1\right)+a n\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{4 d (n+1) \left(a^2+b^2\right) \left(a+\sqrt{-b^2}\right)}+\frac{\cos ^2(c+d x) (a \tan (c+d x)+b) (a+b \tan (c+d x))^{n+1}}{2 d \left(a^2+b^2\right)}",1,"((a + b*Tan[c + d*x])^(1 + n)*(-(((Sqrt[-b^2]*(a^2 - b^2*(-1 + n)) - a*b^2*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])])/((a - Sqrt[-b^2])*(1 + n))) + ((a^2*Sqrt[-b^2] + (-b^2)^(3/2)*(-1 + n) + a*b^2*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])])/((a + Sqrt[-b^2])*(1 + n)) + 2*b*Cos[c + d*x]^2*(b + a*Tan[c + d*x])))/(4*b*(a^2 + b^2)*d)","A",1
650,1,360,434,4.5753407,"\int \cos ^4(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Cos[c + d*x]^4*(a + b*Tan[c + d*x])^n,x]","\frac{(a+b \tan (c+d x))^{n+1} \left(-\frac{2 b \cos ^2(c+d x) \left(-a \left(3 a^2+b^2 (5-2 n)\right) \tan (c+d x)-a^2 b (n+1)+b^3 (n-3)\right)}{a^2+b^2}+\frac{\frac{\left(a b^2 n \left(3 a^2+b^2 (5-2 n)\right)+\sqrt{-b^2} \left(-3 a^4+a^2 b^2 \left(n^2+2 n-6\right)-b^4 \left(n^2-4 n+3\right)\right)\right) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)}{a-\sqrt{-b^2}}+\frac{\left(a b^2 n \left(3 a^2+b^2 (5-2 n)\right)+\sqrt{-b^2} \left(3 a^4-a^2 b^2 \left(n^2+2 n-6\right)+b^4 \left(n^2-4 n+3\right)\right)\right) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{a+\sqrt{-b^2}}}{(n+1) \left(a^2+b^2\right)}+4 b \cos ^4(c+d x) (a \tan (c+d x)+b)\right)}{16 b d \left(a^2+b^2\right)}","\frac{\cos ^4(c+d x) (a \tan (c+d x)+b) (a+b \tan (c+d x))^{n+1}}{4 d \left(a^2+b^2\right)}+\frac{b \cos ^2(c+d x) \left(a b \left(\frac{3 a^2}{b^2}-2 n+5\right) \tan (c+d x)+a^2 (n+1)+b^2 (3-n)\right) (a+b \tan (c+d x))^{n+1}}{8 d \left(a^2+b^2\right)^2}+\frac{b \left(\frac{a n \left(\frac{3 a^2}{b^2}-2 n+5\right)}{b^2}-\frac{\sqrt{-b^2} \left(3 a^4+a^2 b^2 \left(-n^2-2 n+6\right)+b^4 \left(n^2-4 n+3\right)\right)}{b^6}\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)}{16 d (n+1) \left(\frac{a^2}{b^2}+1\right)^2 \left(a-\sqrt{-b^2}\right)}+\frac{b \left(\frac{a n \left(\frac{3 a^2}{b^2}-2 n+5\right)}{b^2}+\frac{\sqrt{-b^2} \left(3 a^4+a^2 b^2 \left(-n^2-2 n+6\right)+b^4 \left(n^2-4 n+3\right)\right)}{b^6}\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{16 d (n+1) \left(\frac{a^2}{b^2}+1\right)^2 \left(a+\sqrt{-b^2}\right)}",1,"((a + b*Tan[c + d*x])^(1 + n)*((((a*b^2*(3*a^2 + b^2*(5 - 2*n))*n + Sqrt[-b^2]*(-3*a^4 - b^4*(3 - 4*n + n^2) + a^2*b^2*(-6 + 2*n + n^2)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])])/(a - Sqrt[-b^2]) + ((a*b^2*(3*a^2 + b^2*(5 - 2*n))*n + Sqrt[-b^2]*(3*a^4 + b^4*(3 - 4*n + n^2) - a^2*b^2*(-6 + 2*n + n^2)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])])/(a + Sqrt[-b^2]))/((a^2 + b^2)*(1 + n)) + 4*b*Cos[c + d*x]^4*(b + a*Tan[c + d*x]) - (2*b*Cos[c + d*x]^2*(b^3*(-3 + n) - a^2*b*(1 + n) - a*(3*a^2 + b^2*(5 - 2*n))*Tan[c + d*x]))/(a^2 + b^2)))/(16*b*(a^2 + b^2)*d)","A",1
651,1,306,159,4.0376803,"\int \sec ^3(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Sec[c + d*x]^3*(a + b*Tan[c + d*x])^n,x]","\frac{2 (n+2) (a-i b) (a+i b) \sec (c+d x) (a+b \tan (c+d x))^{n+1} F_1\left(n+1;-\frac{1}{2},-\frac{1}{2};n+2;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)}{b d (n+1) \left(2 (n+2) \left(a^2+b^2\right) F_1\left(n+1;-\frac{1}{2},-\frac{1}{2};n+2;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)-(a+b \tan (c+d x)) \left((a-i b) F_1\left(n+2;-\frac{1}{2},\frac{1}{2};n+3;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)+(a+i b) F_1\left(n+2;\frac{1}{2},-\frac{1}{2};n+3;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)\right)\right)}","\frac{\sec (c+d x) (a+b \tan (c+d x))^{n+1} F_1\left(n+1;-\frac{1}{2},-\frac{1}{2};n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}},\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{b d (n+1) \sqrt{1-\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}} \sqrt{1-\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}}}",1,"(2*(a - I*b)*(a + I*b)*(2 + n)*AppellF1[1 + n, -1/2, -1/2, 2 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)]*Sec[c + d*x]*(a + b*Tan[c + d*x])^(1 + n))/(b*d*(1 + n)*(2*(a^2 + b^2)*(2 + n)*AppellF1[1 + n, -1/2, -1/2, 2 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)] - ((a - I*b)*AppellF1[2 + n, -1/2, 1/2, 3 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)] + (a + I*b)*AppellF1[2 + n, 1/2, -1/2, 3 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)])*(a + b*Tan[c + d*x])))","C",0
652,1,340,159,3.8746892,"\int \sec (c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Sec[c + d*x]*(a + b*Tan[c + d*x])^n,x]","\frac{2 (n+2) \left(a^2+b^2\right)^2 \cos ^3(c+d x) (\tan (c+d x)-i) (\tan (c+d x)+i) (a+b \tan (c+d x))^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)}{b d (n+1) (a-i b) (a+i b) \left(2 (n+2) \left(a^2+b^2\right) F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)+(a+b \tan (c+d x)) \left((a-i b) F_1\left(n+2;\frac{1}{2},\frac{3}{2};n+3;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)+(a+i b) F_1\left(n+2;\frac{3}{2},\frac{1}{2};n+3;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)\right)\right)}","\frac{\cos (c+d x) \sqrt{1-\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}} \sqrt{1-\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}} (a+b \tan (c+d x))^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}},\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{b d (n+1)}",1,"(2*(a^2 + b^2)^2*(2 + n)*AppellF1[1 + n, 1/2, 1/2, 2 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)]*Cos[c + d*x]^3*(-I + Tan[c + d*x])*(I + Tan[c + d*x])*(a + b*Tan[c + d*x])^(1 + n))/((a - I*b)*(a + I*b)*b*d*(1 + n)*(2*(a^2 + b^2)*(2 + n)*AppellF1[1 + n, 1/2, 1/2, 2 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)] + ((a - I*b)*AppellF1[2 + n, 1/2, 3/2, 3 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)] + (a + I*b)*AppellF1[2 + n, 3/2, 1/2, 3 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)])*(a + b*Tan[c + d*x])))","C",0
653,1,341,161,4.9520251,"\int \cos (c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Cos[c + d*x]*(a + b*Tan[c + d*x])^n,x]","\frac{2 (n+2) \left(a^2+b^2\right)^2 \cos ^5(c+d x) (\tan (c+d x)-i) (\tan (c+d x)+i) (a+b \tan (c+d x))^{n+1} F_1\left(n+1;\frac{3}{2},\frac{3}{2};n+2;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)}{b d (n+1) (a-i b) (a+i b) \left(2 (n+2) \left(a^2+b^2\right) F_1\left(n+1;\frac{3}{2},\frac{3}{2};n+2;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)+3 (a+b \tan (c+d x)) \left((a-i b) F_1\left(n+2;\frac{3}{2},\frac{5}{2};n+3;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)+(a+i b) F_1\left(n+2;\frac{5}{2},\frac{3}{2};n+3;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)\right)\right)}","\frac{\cos ^3(c+d x) \left(1-\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)^{3/2} \left(1-\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)^{3/2} (a+b \tan (c+d x))^{n+1} F_1\left(n+1;\frac{3}{2},\frac{3}{2};n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}},\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{b d (n+1)}",1,"(2*(a^2 + b^2)^2*(2 + n)*AppellF1[1 + n, 3/2, 3/2, 2 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)]*Cos[c + d*x]^5*(-I + Tan[c + d*x])*(I + Tan[c + d*x])*(a + b*Tan[c + d*x])^(1 + n))/((a - I*b)*(a + I*b)*b*d*(1 + n)*(2*(a^2 + b^2)*(2 + n)*AppellF1[1 + n, 3/2, 3/2, 2 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)] + 3*((a - I*b)*AppellF1[2 + n, 3/2, 5/2, 3 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)] + (a + I*b)*AppellF1[2 + n, 5/2, 3/2, 3 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)])*(a + b*Tan[c + d*x])))","C",0
654,1,341,161,6.3768179,"\int \cos ^3(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Cos[c + d*x]^3*(a + b*Tan[c + d*x])^n,x]","\frac{2 (n+2) \left(a^2+b^2\right)^2 \cos ^7(c+d x) (\tan (c+d x)-i) (\tan (c+d x)+i) (a+b \tan (c+d x))^{n+1} F_1\left(n+1;\frac{5}{2},\frac{5}{2};n+2;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)}{b d (n+1) (a-i b) (a+i b) \left(2 (n+2) \left(a^2+b^2\right) F_1\left(n+1;\frac{5}{2},\frac{5}{2};n+2;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)+5 (a+b \tan (c+d x)) \left((a-i b) F_1\left(n+2;\frac{5}{2},\frac{7}{2};n+3;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)+(a+i b) F_1\left(n+2;\frac{7}{2},\frac{5}{2};n+3;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right)\right)\right)}","\frac{\cos ^5(c+d x) \left(1-\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)^{5/2} \left(1-\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)^{5/2} (a+b \tan (c+d x))^{n+1} F_1\left(n+1;\frac{5}{2},\frac{5}{2};n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}},\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{b d (n+1)}",1,"(2*(a^2 + b^2)^2*(2 + n)*AppellF1[1 + n, 5/2, 5/2, 2 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)]*Cos[c + d*x]^7*(-I + Tan[c + d*x])*(I + Tan[c + d*x])*(a + b*Tan[c + d*x])^(1 + n))/((a - I*b)*(a + I*b)*b*d*(1 + n)*(2*(a^2 + b^2)*(2 + n)*AppellF1[1 + n, 5/2, 5/2, 2 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)] + 5*((a - I*b)*AppellF1[2 + n, 5/2, 7/2, 3 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)] + (a + I*b)*AppellF1[2 + n, 7/2, 5/2, 3 + n, (a + b*Tan[c + d*x])/(a - I*b), (a + b*Tan[c + d*x])/(a + I*b)])*(a + b*Tan[c + d*x])))","C",0
655,1,133,124,0.683057,"\int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx","Integrate[(e*Cos[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x]),x]","\frac{a e^3 (\cos (d x)-i \sin (d x)) \sqrt{e \cos (c+d x)} (\cos (c+2 d x)+i \sin (c+2 d x)) \left(\sqrt{\cos (c+d x)} (5 \sin (2 (c+d x))+2 i \cos (2 (c+d x))-8 i)+10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (c+d x)-i \sin (c+d x))\right)}{21 d \sqrt{\cos (c+d x)}}","-\frac{2 i a (e \cos (c+d x))^{7/2}}{7 d}+\frac{10 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (e \cos (c+d x))^{7/2}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a \tan (c+d x) (e \cos (c+d x))^{7/2}}{7 d}+\frac{10 a \tan (c+d x) \sec ^2(c+d x) (e \cos (c+d x))^{7/2}}{21 d}",1,"(a*e^3*Sqrt[e*Cos[c + d*x]]*(Cos[d*x] - I*Sin[d*x])*(10*EllipticF[(c + d*x)/2, 2]*(Cos[c + d*x] - I*Sin[c + d*x]) + Sqrt[Cos[c + d*x]]*(-8*I + (2*I)*Cos[2*(c + d*x)] + 5*Sin[2*(c + d*x)]))*(Cos[c + 2*d*x] + I*Sin[c + 2*d*x]))/(21*d*Sqrt[Cos[c + d*x]])","A",1
656,1,387,90,12.3658049,"\int (e \cos (c+d x))^{5/2} (a+i a \tan (c+d x)) \, dx","Integrate[(e*Cos[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x]),x]","\frac{(\cos (d x)-i \sin (d x)) (a+i a \tan (c+d x)) (e \cos (c+d x))^{5/2} \left(\frac{2 \sqrt{2} (\cot (c)-i) e^{-i d x} \left(e^{2 i d x} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+3 e^{2 i (c+d x)}-3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left(e^{i (c+d x)}-i\right)} F\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)+3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left(e^{i (c+d x)}-i\right)} E\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)+3\right)}{5 \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}+\frac{2}{5} \sin (c) \sqrt{\cos (c+d x)} \left((1-i \cot (c)) \cos (2 d x)+\cot (c) (\sin (2 d x)+5 i)-6 \cot ^2(c)+i \sin (2 d x)-1\right)\right)}{2 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{2 i a (e \cos (c+d x))^{5/2}}{5 d}+\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (e \cos (c+d x))^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \tan (c+d x) (e \cos (c+d x))^{5/2}}{5 d}",1,"((e*Cos[c + d*x])^(5/2)*(Cos[d*x] - I*Sin[d*x])*((2*Sqrt[2]*(-I + Cot[c])*(3 + 3*E^((2*I)*(c + d*x)) + 3*Sqrt[1 - I*E^(I*(c + d*x))]*Sqrt[E^(I*(c + d*x))*(-I + E^(I*(c + d*x)))]*EllipticE[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1] - 3*Sqrt[1 - I*E^(I*(c + d*x))]*Sqrt[E^(I*(c + d*x))*(-I + E^(I*(c + d*x)))]*EllipticF[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1] + E^((2*I)*d*x)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(5*E^(I*d*x)*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + (2*Sqrt[Cos[c + d*x]]*Sin[c]*(-1 + Cos[2*d*x]*(1 - I*Cot[c]) - 6*Cot[c]^2 + I*Sin[2*d*x] + Cot[c]*(5*I + Sin[2*d*x])))/5)*(a + I*a*Tan[c + d*x]))/(2*d*Cos[c + d*x]^(3/2))","C",1
657,1,100,90,0.3621352,"\int (e \cos (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx","Integrate[(e*Cos[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x]),x]","\frac{2 a e \sqrt{\cos (c+d x)} (\tan (c+d x)-i) (\cos (d x)-i \sin (d x)) \sqrt{e \cos (c+d x)} \left(\sqrt{\cos (c+d x)} (\cos (d x)+i \sin (d x))+(\sin (c)+i \cos (c)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","-\frac{2 i a (e \cos (c+d x))^{3/2}}{3 d}+\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (e \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \tan (c+d x) (e \cos (c+d x))^{3/2}}{3 d}",1,"(2*a*e*Sqrt[Cos[c + d*x]]*Sqrt[e*Cos[c + d*x]]*(EllipticF[(c + d*x)/2, 2]*(I*Cos[c] + Sin[c]) + Sqrt[Cos[c + d*x]]*(Cos[d*x] + I*Sin[d*x]))*(Cos[d*x] - I*Sin[d*x])*(-I + Tan[c + d*x]))/(3*d)","A",1
658,1,244,60,0.7098926,"\int \sqrt{e \cos (c+d x)} (a+i a \tan (c+d x)) \, dx","Integrate[Sqrt[e*Cos[c + d*x]]*(a + I*a*Tan[c + d*x]),x]","\frac{a e (\cot (c)+i) e^{-i (c+d x)} \left(e^{2 i d x} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left(e^{i (c+d x)}-i\right)} F\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)+3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left(e^{i (c+d x)}-i\right)} E\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)\right)}{3 d \sqrt{e \cos (c+d x)}}","\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{2 i a \sqrt{e \cos (c+d x)}}{d}",1,"(a*e*(I + Cot[c])*(3*Sqrt[1 - I*E^(I*(c + d*x))]*Sqrt[E^(I*(c + d*x))*(-I + E^(I*(c + d*x)))]*EllipticE[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1] - 3*Sqrt[1 - I*E^(I*(c + d*x))]*Sqrt[E^(I*(c + d*x))*(-I + E^(I*(c + d*x)))]*EllipticF[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1] + E^((2*I)*d*x)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(3*d*E^(I*(c + d*x))*Sqrt[e*Cos[c + d*x]])","C",1
659,1,143,60,1.0293808,"\int \frac{a+i a \tan (c+d x)}{\sqrt{e \cos (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])/Sqrt[e*Cos[c + d*x]],x]","-\frac{\sqrt{2} a \sin (c) (\cot (c)-i) (\tan (c+d x)-i) (\cos (d x)-i \sin (d x)) \sqrt{e \cos (c+d x)} \left(\sqrt{2} \sqrt{\csc ^2(c)}+i \csc (c) \cos (c+d x) \sqrt{\cos \left(2 d x-2 \tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)\right)}{d e \sqrt{\csc ^2(c)}}","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{e \cos (c+d x)}}+\frac{2 i a}{d \sqrt{e \cos (c+d x)}}",1,"-((Sqrt[2]*a*Sqrt[e*Cos[c + d*x]]*(-I + Cot[c])*(Sqrt[2]*Sqrt[Csc[c]^2] + I*Cos[c + d*x]*Sqrt[1 + Cos[2*d*x - 2*ArcTan[Cot[c]]]]*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]])*Sin[c]*(Cos[d*x] - I*Sin[d*x])*(-I + Tan[c + d*x]))/(d*e*Sqrt[Csc[c]^2]))","C",0
660,1,369,89,3.787301,"\int \frac{a+i a \tan (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])/(e*Cos[c + d*x])^(3/2),x]","\frac{\cos ^{\frac{5}{2}}(c+d x) (\cos (d x)-i \sin (d x)) (a+i a \tan (c+d x)) \left(\frac{2 \sin (c) (\cot (c)-i) (3 \csc (c) \cos (c+2 d x)+3 \cot (c)+2 i)}{3 \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \sqrt{2} (\cot (c)-i) e^{-i d x} \left(e^{2 i d x} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+3 e^{2 i (c+d x)}-3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left(e^{i (c+d x)}-i\right)} F\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)+3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left(e^{i (c+d x)}-i\right)} E\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)+3\right)}{3 \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{2 d (e \cos (c+d x))^{3/2}}","\frac{2 i a}{3 d (e \cos (c+d x))^{3/2}}-\frac{2 a \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d (e \cos (c+d x))^{3/2}}+\frac{2 a \sin (c+d x)}{d e \sqrt{e \cos (c+d x)}}",1,"(Cos[c + d*x]^(5/2)*((-2*Sqrt[2]*(-I + Cot[c])*(3 + 3*E^((2*I)*(c + d*x)) + 3*Sqrt[1 - I*E^(I*(c + d*x))]*Sqrt[E^(I*(c + d*x))*(-I + E^(I*(c + d*x)))]*EllipticE[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1] - 3*Sqrt[1 - I*E^(I*(c + d*x))]*Sqrt[E^(I*(c + d*x))*(-I + E^(I*(c + d*x)))]*EllipticF[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1] + E^((2*I)*d*x)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(3*E^(I*d*x)*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + (2*(-I + Cot[c])*(2*I + 3*Cot[c] + 3*Cos[c + 2*d*x]*Csc[c])*Sin[c])/(3*Cos[c + d*x]^(3/2)))*(Cos[d*x] - I*Sin[d*x])*(a + I*a*Tan[c + d*x]))/(2*d*(e*Cos[c + d*x])^(3/2))","C",1
661,1,57,96,0.5226778,"\int \frac{a+i a \tan (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])/(e*Cos[c + d*x])^(5/2),x]","\frac{a \left(5 \sin (2 (c+d x))+10 \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 i\right)}{15 d (e \cos (c+d x))^{5/2}}","\frac{2 i a}{5 d (e \cos (c+d x))^{5/2}}+\frac{2 a \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d (e \cos (c+d x))^{5/2}}+\frac{2 a \sin (c+d x) \cos (c+d x)}{3 d (e \cos (c+d x))^{5/2}}",1,"(a*(6*I + 10*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 5*Sin[2*(c + d*x)]))/(15*d*(e*Cos[c + d*x])^(5/2))","A",1
662,1,390,130,6.3315369,"\int \frac{a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])/(e*Cos[c + d*x])^(7/2),x]","\frac{\cos ^{\frac{9}{2}}(c+d x) (\cos (d x)-i \sin (d x)) (a+i a \tan (c+d x)) \left(\frac{(\cot (c)-i) (77 \cos (c+2 d x)+7 \cos (3 c+2 d x)+21 \cos (3 c+4 d x)+40 i \sin (c)+63 \cos (c))}{70 \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 \sqrt{2} (\cot (c)-i) e^{-i d x} \left(e^{2 i d x} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+3 e^{2 i (c+d x)}-3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left(e^{i (c+d x)}-i\right)} F\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)+3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left(e^{i (c+d x)}-i\right)} E\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)+3\right)}{5 \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{2 d (e \cos (c+d x))^{7/2}}","\frac{2 i a}{7 d (e \cos (c+d x))^{7/2}}-\frac{6 a \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d (e \cos (c+d x))^{7/2}}+\frac{6 a \sin (c+d x) \cos ^3(c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac{2 a \sin (c+d x) \cos (c+d x)}{5 d (e \cos (c+d x))^{7/2}}",1,"(Cos[c + d*x]^(9/2)*((-2*Sqrt[2]*(-I + Cot[c])*(3 + 3*E^((2*I)*(c + d*x)) + 3*Sqrt[1 - I*E^(I*(c + d*x))]*Sqrt[E^(I*(c + d*x))*(-I + E^(I*(c + d*x)))]*EllipticE[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1] - 3*Sqrt[1 - I*E^(I*(c + d*x))]*Sqrt[E^(I*(c + d*x))*(-I + E^(I*(c + d*x)))]*EllipticF[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1] + E^((2*I)*d*x)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(5*E^(I*d*x)*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + ((-I + Cot[c])*(63*Cos[c] + 77*Cos[c + 2*d*x] + 7*Cos[3*c + 2*d*x] + 21*Cos[3*c + 4*d*x] + (40*I)*Sin[c]))/(70*Cos[c + d*x]^(7/2)))*(Cos[d*x] - I*Sin[d*x])*(a + I*a*Tan[c + d*x]))/(2*d*(e*Cos[c + d*x])^(7/2))","C",1
663,1,156,190,1.2452979,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^2,x]","\frac{e^3 \sqrt{e \cos (c+d x)} \left(\sqrt{\cos (c+d x)} (134 \sin (c+d x)-117 \sin (3 (c+d x))-11 \sin (5 (c+d x))-296 i \cos (c+d x)+68 i \cos (3 (c+d x))+4 i \cos (5 (c+d x)))-240 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))\right)}{840 a^2 d \cos ^{\frac{5}{2}}(c+d x) (\tan (c+d x)-i)^2}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (e \cos (c+d x))^{7/2}}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 i \cos ^2(c+d x) (e \cos (c+d x))^{7/2}}{15 d \left(a^2+i a^2 \tan (c+d x)\right)}+\frac{2 \sin (c+d x) \cos (c+d x) (e \cos (c+d x))^{7/2}}{15 a^2 d}+\frac{6 \tan (c+d x) (e \cos (c+d x))^{7/2}}{35 a^2 d}+\frac{2 \tan (c+d x) \sec ^2(c+d x) (e \cos (c+d x))^{7/2}}{7 a^2 d}",1,"(e^3*Sqrt[e*Cos[c + d*x]]*(-240*EllipticF[(c + d*x)/2, 2]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + Sqrt[Cos[c + d*x]]*((-296*I)*Cos[c + d*x] + (68*I)*Cos[3*(c + d*x)] + (4*I)*Cos[5*(c + d*x)] + 134*Sin[c + d*x] - 117*Sin[3*(c + d*x)] - 11*Sin[5*(c + d*x)])))/(840*a^2*d*Cos[c + d*x]^(5/2)*(-I + Tan[c + d*x])^2)","A",1
664,1,471,154,3.0512583,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x])^2,x]","\frac{(\cos (d x)+i \sin (d x))^2 (e \cos (c+d x))^{5/2} \left(\frac{14 \sqrt{2} \csc (c) e^{-i d x} (\cos (2 c)+i \sin (2 c)) \left(e^{2 i d x} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+3 e^{2 i (c+d x)}-3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left(e^{i (c+d x)}-i\right)} F\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)+3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left(e^{i (c+d x)}-i\right)} E\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)+3\right)}{65 \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}-\frac{1}{260} \csc (c) \sqrt{\cos (c+d x)} (\cos (2 d x)-i \sin (2 d x)) (208 i \sin (c+2 d x)+128 i \sin (3 c+2 d x)-4 i \sin (3 c+4 d x)+4 i \sin (5 c+4 d x)+178 \cos (c+2 d x)+158 \cos (3 c+2 d x)-9 \cos (3 c+4 d x)+9 \cos (5 c+4 d x)-88 i \sin (c))\right)}{2 d \cos ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^2}","\frac{42 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (e \cos (c+d x))^{5/2}}{65 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 i \cos ^2(c+d x) (e \cos (c+d x))^{5/2}}{13 d \left(a^2+i a^2 \tan (c+d x)\right)}+\frac{2 \sin (c+d x) \cos (c+d x) (e \cos (c+d x))^{5/2}}{13 a^2 d}+\frac{14 \tan (c+d x) (e \cos (c+d x))^{5/2}}{65 a^2 d}",1,"((e*Cos[c + d*x])^(5/2)*(Cos[d*x] + I*Sin[d*x])^2*((14*Sqrt[2]*Csc[c]*(3 + 3*E^((2*I)*(c + d*x)) + 3*Sqrt[1 - I*E^(I*(c + d*x))]*Sqrt[E^(I*(c + d*x))*(-I + E^(I*(c + d*x)))]*EllipticE[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1] - 3*Sqrt[1 - I*E^(I*(c + d*x))]*Sqrt[E^(I*(c + d*x))*(-I + E^(I*(c + d*x)))]*EllipticF[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1] + E^((2*I)*d*x)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(Cos[2*c] + I*Sin[2*c]))/(65*E^(I*d*x)*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - (Sqrt[Cos[c + d*x]]*Csc[c]*(Cos[2*d*x] - I*Sin[2*d*x])*(178*Cos[c + 2*d*x] + 158*Cos[3*c + 2*d*x] - 9*Cos[3*c + 4*d*x] + 9*Cos[5*c + 4*d*x] - (88*I)*Sin[c] + (208*I)*Sin[c + 2*d*x] + (128*I)*Sin[3*c + 2*d*x] - (4*I)*Sin[3*c + 4*d*x] + (4*I)*Sin[5*c + 4*d*x]))/260))/(2*d*Cos[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^2)","C",1
665,1,131,154,0.6787853,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x])^2,x]","\frac{(e \cos (c+d x))^{3/2} \left(\sqrt{\cos (c+d x)} (13 \sin (c+d x)-7 \sin (3 (c+d x))-28 i \cos (c+d x)+4 i \cos (3 (c+d x)))-20 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))\right)}{66 a^2 d \cos ^{\frac{7}{2}}(c+d x) (\tan (c+d x)-i)^2}","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (e \cos (c+d x))^{3/2}}{33 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 i \cos ^2(c+d x) (e \cos (c+d x))^{3/2}}{11 d \left(a^2+i a^2 \tan (c+d x)\right)}+\frac{2 \sin (c+d x) \cos (c+d x) (e \cos (c+d x))^{3/2}}{11 a^2 d}+\frac{10 \tan (c+d x) (e \cos (c+d x))^{3/2}}{33 a^2 d}",1,"((e*Cos[c + d*x])^(3/2)*(-20*EllipticF[(c + d*x)/2, 2]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + Sqrt[Cos[c + d*x]]*((-28*I)*Cos[c + d*x] + (4*I)*Cos[3*(c + d*x)] + 13*Sin[c + d*x] - 7*Sin[3*(c + d*x)])))/(66*a^2*d*Cos[c + d*x]^(7/2)*(-I + Tan[c + d*x])^2)","A",1
666,1,420,120,1.7159409,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]/(a + I*a*Tan[c + d*x])^2,x]","\frac{(\cos (d x)+i \sin (d x))^2 \sqrt{e \cos (c+d x)} \left(\frac{2 \sqrt{2} \csc (c) e^{-i d x} (\cos (2 c)+i \sin (2 c)) \left(e^{2 i d x} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+3 e^{2 i (c+d x)}-3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left(e^{i (c+d x)}-i\right)} F\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)+3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left(e^{i (c+d x)}-i\right)} E\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)+3\right)}{9 \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}-\frac{1}{9} \csc (c) \sqrt{\cos (c+d x)} (\cos (2 d x)-i \sin (2 d x)) (-4 i (-2 \sin (c+2 d x)-\sin (3 c+2 d x)+\sin (c))+7 \cos (c+2 d x)+5 \cos (3 c+2 d x))\right)}{2 d \cos ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2}","\frac{2 i \sqrt{e \cos (c+d x)}}{9 d \left(a^2+i a^2 \tan (c+d x)\right)}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{3 a^2 d \sqrt{\cos (c+d x)}}+\frac{2 i \sqrt{e \cos (c+d x)}}{9 d (a+i a \tan (c+d x))^2}",1,"(Sqrt[e*Cos[c + d*x]]*(Cos[d*x] + I*Sin[d*x])^2*((2*Sqrt[2]*Csc[c]*(3 + 3*E^((2*I)*(c + d*x)) + 3*Sqrt[1 - I*E^(I*(c + d*x))]*Sqrt[E^(I*(c + d*x))*(-I + E^(I*(c + d*x)))]*EllipticE[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1] - 3*Sqrt[1 - I*E^(I*(c + d*x))]*Sqrt[E^(I*(c + d*x))*(-I + E^(I*(c + d*x)))]*EllipticF[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1] + E^((2*I)*d*x)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(Cos[2*c] + I*Sin[2*c]))/(9*E^(I*d*x)*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - (Sqrt[Cos[c + d*x]]*Csc[c]*(Cos[2*d*x] - I*Sin[2*d*x])*(7*Cos[c + 2*d*x] + 5*Cos[3*c + 2*d*x] - (4*I)*(Sin[c] - 2*Sin[c + 2*d*x] - Sin[3*c + 2*d*x])))/9))/(2*d*Cos[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2)","C",1
667,1,158,120,0.6079779,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+i a \tan (c+d x))^2} \, dx","Integrate[1/(Sqrt[e*Cos[c + d*x]]*(a + I*a*Tan[c + d*x])^2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)-i \cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\sqrt{\cos (c+d x)} \left(4 i \sin ^3\left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{3}{2} (c+d x)\right)\right)+2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(\sin \left(\frac{3}{2} (c+d x)\right)-i \cos \left(\frac{3}{2} (c+d x)\right)\right)\right)}{7 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\tan (c+d x)-i)^2 \sqrt{e \cos (c+d x)}}","\frac{2 i}{7 d \left(a^2+i a^2 \tan (c+d x)\right) \sqrt{e \cos (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 a^2 d \sqrt{e \cos (c+d x)}}+\frac{2 i}{7 d (a+i a \tan (c+d x))^2 \sqrt{e \cos (c+d x)}}",1,"(((-I)*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Sqrt[Cos[c + d*x]]*(3*Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2] + (4*I)*Sin[(c + d*x)/2]^3) + 2*EllipticF[(c + d*x)/2, 2]*((-I)*Cos[(3*(c + d*x))/2] + Sin[(3*(c + d*x))/2])))/(7*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[e*Cos[c + d*x]]*(-I + Tan[c + d*x])^2)","A",1
668,1,244,92,0.6860956,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+i a \tan (c+d x))^2} \, dx","Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^2),x]","\frac{(\sin (c+d x)+i \cos (c+d x)) \left(i \sin (2 (c+d x))+3 \cos (2 (c+d x))-2 \sqrt{\sin (c+d x)-i \cos (c+d x)+1} \sqrt{\sin (c+d x)+i \sin (2 (c+d x))-i \cos (c+d x)+\cos (2 (c+d x))} F\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)+2 \sqrt{\sin (c+d x)-i \cos (c+d x)+1} \sqrt{\sin (c+d x)+i \sin (2 (c+d x))-i \cos (c+d x)+\cos (2 (c+d x))} E\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)+3\right)}{5 a^2 d e \sqrt{e \cos (c+d x)}}","\frac{2 \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d (e \cos (c+d x))^{3/2}}+\frac{4 i \cos ^2(c+d x)}{5 d \left(a^2+i a^2 \tan (c+d x)\right) (e \cos (c+d x))^{3/2}}",1,"((I*Cos[c + d*x] + Sin[c + d*x])*(3 + 3*Cos[2*(c + d*x)] + 2*EllipticE[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1]*Sqrt[1 - I*Cos[c + d*x] + Sin[c + d*x]]*Sqrt[(-I)*Cos[c + d*x] + Cos[2*(c + d*x)] + Sin[c + d*x] + I*Sin[2*(c + d*x)]] - 2*EllipticF[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1]*Sqrt[1 - I*Cos[c + d*x] + Sin[c + d*x]]*Sqrt[(-I)*Cos[c + d*x] + Cos[2*(c + d*x)] + Sin[c + d*x] + I*Sin[2*(c + d*x)]] + I*Sin[2*(c + d*x)]))/(5*a^2*d*e*Sqrt[e*Cos[c + d*x]])","B",1
669,1,116,92,0.3892784,"\int \frac{1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx","Integrate[1/((e*Cos[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^2),x]","\frac{2 \sqrt{\cos (c+d x)} (\cos (d x)+i \sin (d x))^2 \left(2 \sqrt{\cos (c+d x)} (\sin (c-d x)-i \cos (c-d x))+(\cos (2 c)+i \sin (2 c)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 a^2 d (\tan (c+d x)-i)^2 (e \cos (c+d x))^{5/2}}","-\frac{2 \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d (e \cos (c+d x))^{5/2}}+\frac{4 i \cos ^2(c+d x)}{3 d \left(a^2+i a^2 \tan (c+d x)\right) (e \cos (c+d x))^{5/2}}",1,"(2*Sqrt[Cos[c + d*x]]*(Cos[d*x] + I*Sin[d*x])^2*(EllipticF[(c + d*x)/2, 2]*(Cos[2*c] + I*Sin[2*c]) + 2*Sqrt[Cos[c + d*x]]*((-I)*Cos[c - d*x] + Sin[c - d*x])))/(3*a^2*d*(e*Cos[c + d*x])^(5/2)*(-I + Tan[c + d*x])^2)","A",1
670,1,255,122,0.9515357,"\int \frac{1}{(e \cos (c+d x))^{7/2} (a+i a \tan (c+d x))^2} \, dx","Integrate[1/((e*Cos[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])^2),x]","\frac{-2 \sin (c+d x)+10 i \cos (c+d x)-6 i (\cos (c+d x)-i \sin (c+d x)) \sqrt{\sin (c+d x)-i \cos (c+d x)+1} \sqrt{\sin (c+d x)+i \sin (2 (c+d x))-i \cos (c+d x)+\cos (2 (c+d x))} F\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)+6 \sqrt{\sin (c+d x)-i \cos (c+d x)+1} (\sin (c+d x)+i \cos (c+d x)) \sqrt{\sin (c+d x)+i \sin (2 (c+d x))-i \cos (c+d x)+\cos (2 (c+d x))} E\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)}{a^2 d e^3 \sqrt{e \cos (c+d x)}}","\frac{6 \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (e \cos (c+d x))^{7/2}}-\frac{6 \sin (c+d x) \cos ^3(c+d x)}{a^2 d (e \cos (c+d x))^{7/2}}+\frac{4 i \cos ^2(c+d x)}{d \left(a^2+i a^2 \tan (c+d x)\right) (e \cos (c+d x))^{7/2}}",1,"((10*I)*Cos[c + d*x] - 2*Sin[c + d*x] - (6*I)*EllipticF[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1]*(Cos[c + d*x] - I*Sin[c + d*x])*Sqrt[1 - I*Cos[c + d*x] + Sin[c + d*x]]*Sqrt[(-I)*Cos[c + d*x] + Cos[2*(c + d*x)] + Sin[c + d*x] + I*Sin[2*(c + d*x)]] + 6*EllipticE[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1]*Sqrt[1 - I*Cos[c + d*x] + Sin[c + d*x]]*(I*Cos[c + d*x] + Sin[c + d*x])*Sqrt[(-I)*Cos[c + d*x] + Cos[2*(c + d*x)] + Sin[c + d*x] + I*Sin[2*(c + d*x)]])/(a^2*d*e^3*Sqrt[e*Cos[c + d*x]])","B",1
671,1,67,126,0.3728089,"\int \frac{1}{(e \cos (c+d x))^{9/2} (a+i a \tan (c+d x))^2} \, dx","Integrate[1/((e*Cos[c + d*x])^(9/2)*(a + I*a*Tan[c + d*x])^2),x]","\frac{2 \left(-\sin (c+d x)-6 i \cos (c+d x)+5 \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 a^2 d e^3 (e \cos (c+d x))^{3/2}}","\frac{10 \cos ^{\frac{9}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d (e \cos (c+d x))^{9/2}}+\frac{10 \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (e \cos (c+d x))^{9/2}}-\frac{4 i \cos ^2(c+d x)}{d \left(a^2+i a^2 \tan (c+d x)\right) (e \cos (c+d x))^{9/2}}",1,"(2*((-6*I)*Cos[c + d*x] + 5*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - Sin[c + d*x]))/(3*a^2*d*e^3*(e*Cos[c + d*x])^(3/2))","A",1
672,1,406,164,5.6871406,"\int \frac{1}{(e \cos (c+d x))^{11/2} (a+i a \tan (c+d x))^2} \, dx","Integrate[1/((e*Cos[c + d*x])^(11/2)*(a + I*a*Tan[c + d*x])^2),x]","\frac{2 \sqrt{2} \csc (c) e^{3 i c+2 i d x} (\cos (2 c)+i \sin (2 c)) \cos ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^2 \left(-\frac{1}{2} e^{-2 i c} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(7 \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\left(-1+e^{2 i c}\right) \left(56 e^{2 i (c+d x)}+21 e^{4 i (c+d x)}+47\right)\right)+42 \sqrt{2-2 i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left(e^{i (c+d x)}-i\right)} \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)-42 \sqrt{2-2 i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left(e^{i (c+d x)}-i\right)} \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\sin ^{-1}\left(\sqrt{\sin (c+d x)-i \cos (c+d x)}\right)\right|-1\right)\right)}{15 d \left(1+e^{2 i (c+d x)}\right)^3 (a+i a \tan (c+d x))^2 (e \cos (c+d x))^{11/2}}","-\frac{14 \cos ^{\frac{11}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d (e \cos (c+d x))^{11/2}}+\frac{14 \sin (c+d x) \cos ^5(c+d x)}{5 a^2 d (e \cos (c+d x))^{11/2}}+\frac{14 \sin (c+d x) \cos ^3(c+d x)}{15 a^2 d (e \cos (c+d x))^{11/2}}-\frac{4 i \cos ^2(c+d x)}{3 d \left(a^2+i a^2 \tan (c+d x)\right) (e \cos (c+d x))^{11/2}}",1,"(2*Sqrt[2]*E^((3*I)*c + (2*I)*d*x)*Cos[c + d*x]^(7/2)*Csc[c]*(-42*Sqrt[2 - (2*I)*E^(I*(c + d*x))]*Sqrt[E^(I*(c + d*x))*(-I + E^(I*(c + d*x)))]*Cos[c + d*x]^(5/2)*EllipticE[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1] + 42*Sqrt[2 - (2*I)*E^(I*(c + d*x))]*Sqrt[E^(I*(c + d*x))*(-I + E^(I*(c + d*x)))]*Cos[c + d*x]^(5/2)*EllipticF[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1] - (Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*((-1 + E^((2*I)*c))*(47 + 56*E^((2*I)*(c + d*x)) + 21*E^((4*I)*(c + d*x))) + 7*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(2*E^((2*I)*c)))*(Cos[2*c] + I*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^2)/(15*d*(1 + E^((2*I)*(c + d*x)))^3*(e*Cos[c + d*x])^(11/2)*(a + I*a*Tan[c + d*x])^2)","C",1
673,1,80,179,0.6010324,"\int (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{e^3 \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)} (70 \sin (c+d x)+6 \sin (3 (c+d x))+35 i \cos (c+d x)+i \cos (3 (c+d x)))}{70 d}","-\frac{2 i \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}{7 d}+\frac{32 i a \sec ^4(c+d x) (e \cos (c+d x))^{7/2}}{35 d \sqrt{a+i a \tan (c+d x)}}-\frac{16 i \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}{35 d}+\frac{12 i a \sec ^2(c+d x) (e \cos (c+d x))^{7/2}}{35 d \sqrt{a+i a \tan (c+d x)}}",1,"(e^3*Sqrt[e*Cos[c + d*x]]*((35*I)*Cos[c + d*x] + I*Cos[3*(c + d*x)] + 70*Sin[c + d*x] + 6*Sin[3*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(70*d)","A",1
674,1,63,132,0.3588847,"\int (e \cos (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^(5/2)*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{i e^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)} (-4 i \sin (2 (c+d x))+\cos (2 (c+d x))-15)}{15 d}","-\frac{2 i \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{5/2}}{5 d}-\frac{16 i \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{5/2}}{15 d}+\frac{8 i a \sec ^2(c+d x) (e \cos (c+d x))^{5/2}}{15 d \sqrt{a+i a \tan (c+d x)}}",1,"((I/15)*e^2*Sqrt[e*Cos[c + d*x]]*(-15 + Cos[2*(c + d*x)] - (4*I)*Sin[2*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/d","A",1
675,1,56,85,0.2228066,"\int (e \cos (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 e \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)} (2 \sin (c+d x)+i \cos (c+d x))}{3 d}","\frac{4 i a e \sec (c+d x) \sqrt{e \cos (c+d x)}}{3 d \sqrt{a+i a \tan (c+d x)}}-\frac{2 i \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2}}{3 d}",1,"(2*e*Sqrt[e*Cos[c + d*x]]*(I*Cos[c + d*x] + 2*Sin[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(3*d)","A",1
676,1,36,36,0.1688313,"\int \sqrt{e \cos (c+d x)} \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{2 i \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}{d}","-\frac{2 i \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}{d}",1,"((-2*I)*Sqrt[e*Cos[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d","A",1
677,1,125,335,0.8915708,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{e \cos (c+d x)}} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[e*Cos[c + d*x]],x]","\frac{i \left(-e^{-2 i c}\right)^{3/4} e^{-\frac{3}{2} i d x} \left(1+e^{2 i (c+d x)}\right) \sqrt{a+i a \tan (c+d x)} \left(\tan ^{-1}\left(\frac{e^{\frac{i d x}{2}}}{\sqrt[4]{-e^{-2 i c}}}\right)-\tanh ^{-1}\left(\frac{e^{\frac{i d x}{2}}}{\sqrt[4]{-e^{-2 i c}}}\right)\right)}{d \sqrt{e \cos (c+d x)}}","\frac{i \sqrt{2} \sqrt{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{d \sqrt{e}}-\frac{i \sqrt{2} \sqrt{a} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{d \sqrt{e}}-\frac{i \sqrt{a} \log \left(-\sqrt{2} \sqrt{a} \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}+\sqrt{e} \cos (c+d x) (a+i a \tan (c+d x))+a \sqrt{e}\right)}{\sqrt{2} d \sqrt{e}}+\frac{i \sqrt{a} \log \left(\sqrt{2} \sqrt{a} \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}+\sqrt{e} \cos (c+d x) (a+i a \tan (c+d x))+a \sqrt{e}\right)}{\sqrt{2} d \sqrt{e}}",1,"(I*(-E^((-2*I)*c))^(3/4)*(1 + E^((2*I)*(c + d*x)))*(ArcTan[E^((I/2)*d*x)/(-E^((-2*I)*c))^(1/4)] - ArcTanh[E^((I/2)*d*x)/(-E^((-2*I)*c))^(1/4)])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(((3*I)/2)*d*x)*Sqrt[e*Cos[c + d*x]])","A",1
678,1,274,524,3.849209,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/(e*Cos[c + d*x])^(3/2),x]","\frac{i e^{-\frac{1}{2} i (c+d x)} \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (\cos (c+d x)+i \sin (c+d x)) \left(-2 i \sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)+2 \sqrt{2} \cos \left(\frac{1}{2} (c+d x)\right)+\log \left(-\sqrt{2} e^{\frac{1}{2} i (c+d x)}+e^{i (c+d x)}+1\right) \cos (c+d x)-\log \left(\sqrt{2} e^{\frac{1}{2} i (c+d x)}+e^{i (c+d x)}+1\right) \cos (c+d x)+2 \cos (c+d x) \tan ^{-1}\left(1-\sqrt{2} e^{\frac{1}{2} i (c+d x)}\right)-2 \cos (c+d x) \tan ^{-1}\left(1+\sqrt{2} e^{\frac{1}{2} i (c+d x)}\right)\right)}{\sqrt{2} d \left(1+e^{2 i (c+d x)}\right) (e \cos (c+d x))^{3/2}}","-\frac{i a^{3/2} \sec (c+d x) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a-i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{\sqrt{2} d e^{3/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i a^{3/2} \sec (c+d x) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{a-i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{\sqrt{2} d e^{3/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i a^{3/2} \sec (c+d x) \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{a-i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}{\sqrt{e}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{2 \sqrt{2} d e^{3/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{i a^{3/2} \sec (c+d x) \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{a-i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}{\sqrt{e}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{2 \sqrt{2} d e^{3/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i a}{d \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2}}",1,"(I*Cos[c + d*x]^2*(2*Sqrt[2]*Cos[(c + d*x)/2] + 2*ArcTan[1 - Sqrt[2]*E^((I/2)*(c + d*x))]*Cos[c + d*x] - 2*ArcTan[1 + Sqrt[2]*E^((I/2)*(c + d*x))]*Cos[c + d*x] + Cos[c + d*x]*Log[1 - Sqrt[2]*E^((I/2)*(c + d*x)) + E^(I*(c + d*x))] - Cos[c + d*x]*Log[1 + Sqrt[2]*E^((I/2)*(c + d*x)) + E^(I*(c + d*x))] - (2*I)*Sqrt[2]*Sin[(c + d*x)/2])*(Cos[c + d*x] + I*Sin[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[2]*d*E^((I/2)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))*(e*Cos[c + d*x])^(3/2))","A",1
679,1,227,512,2.4207514,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/(e*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+i a \tan (c+d x)} \left(-3 i \cos ^{\frac{3}{2}}(c+d x)+2 \sqrt{\cos (c+d x)} (\sin (c+d x)+i \cos (c+d x))+\frac{3 i \left(-e^{-2 i c}\right)^{3/4} e^{-\frac{1}{2} i (2 c+5 d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(1+e^{2 i (c+d x)}\right)^2 \left(\tan ^{-1}\left(\frac{e^{\frac{i d x}{2}}}{\sqrt[4]{-e^{-2 i c}}}\right)-\tanh ^{-1}\left(\frac{e^{\frac{i d x}{2}}}{\sqrt[4]{-e^{-2 i c}}}\right)\right)}{4 \sqrt{2}}\right)}{4 d (e \cos (c+d x))^{5/2}}","\frac{3 i \sqrt{a} e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{4 \sqrt{2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac{3 i \sqrt{a} e^{5/2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{4 \sqrt{2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac{3 i \sqrt{a} e^{5/2} \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{8 \sqrt{2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}+\frac{3 i \sqrt{a} e^{5/2} \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{8 \sqrt{2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac{3 i \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{4 d (e \cos (c+d x))^{5/2}}+\frac{i a}{2 d \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{5/2}}",1,"(Sqrt[Cos[c + d*x]]*((((3*I)/4)*(-E^((-2*I)*c))^(3/4)*(1 + E^((2*I)*(c + d*x)))^2*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(ArcTan[E^((I/2)*d*x)/(-E^((-2*I)*c))^(1/4)] - ArcTanh[E^((I/2)*d*x)/(-E^((-2*I)*c))^(1/4)]))/(Sqrt[2]*E^((I/2)*(2*c + 5*d*x))) - (3*I)*Cos[c + d*x]^(3/2) + 2*Sqrt[Cos[c + d*x]]*(I*Cos[c + d*x] + Sin[c + d*x]))*Sqrt[a + I*a*Tan[c + d*x]])/(4*d*(e*Cos[c + d*x])^(5/2))","A",1
680,1,305,719,2.956769,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/(e*Cos[c + d*x])^(7/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+i a \tan (c+d x)} \left(-\frac{40}{3} i \cos ^{\frac{3}{2}}(c+d x)+\frac{5}{8} i e^{-\frac{7}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(1+e^{2 i (c+d x)}\right)^3 \left(\log \left(-\sqrt{2} e^{\frac{1}{2} i (c+d x)}+e^{i (c+d x)}+1\right)-\log \left(\sqrt{2} e^{\frac{1}{2} i (c+d x)}+e^{i (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} e^{\frac{1}{2} i (c+d x)}\right)-2 \tan ^{-1}\left(1+\sqrt{2} e^{\frac{1}{2} i (c+d x)}\right)\right)+20 \cos ^{\frac{5}{2}}(c+d x) (\sin (c+d x)+i \cos (c+d x))+\frac{32}{3} \sqrt{\cos (c+d x)} (\sin (c+d x)+i \cos (c+d x))\right)}{32 d (e \cos (c+d x))^{7/2}}","-\frac{5 i a^{3/2} e^{7/2} \sec (c+d x) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}+\frac{5 i a^{3/2} e^{7/2} \sec (c+d x) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}+\frac{5 i a^{3/2} e^{7/2} \sec (c+d x) \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}-\frac{5 i a^{3/2} e^{7/2} \sec (c+d x) \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}-\frac{5 i \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac{5 i a \cos ^2(c+d x)}{8 d \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}+\frac{i a}{3 d \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}",1,"(Sqrt[Cos[c + d*x]]*(((-40*I)/3)*Cos[c + d*x]^(3/2) + (((5*I)/8)*(1 + E^((2*I)*(c + d*x)))^3*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(2*ArcTan[1 - Sqrt[2]*E^((I/2)*(c + d*x))] - 2*ArcTan[1 + Sqrt[2]*E^((I/2)*(c + d*x))] + Log[1 - Sqrt[2]*E^((I/2)*(c + d*x)) + E^(I*(c + d*x))] - Log[1 + Sqrt[2]*E^((I/2)*(c + d*x)) + E^(I*(c + d*x))]))/E^(((7*I)/2)*(c + d*x)) + (32*Sqrt[Cos[c + d*x]]*(I*Cos[c + d*x] + Sin[c + d*x]))/3 + 20*Cos[c + d*x]^(5/2)*(I*Cos[c + d*x] + Sin[c + d*x]))*Sqrt[a + I*a*Tan[c + d*x]])/(32*d*(e*Cos[c + d*x])^(7/2))","A",1
681,1,80,175,0.6513191,"\int \frac{(e \cos (c+d x))^{5/2}}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(e*Cos[c + d*x])^(5/2)/Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^3 (70 i \sin (c+d x)+6 i \sin (3 (c+d x))+35 \cos (c+d x)+\cos (3 (c+d x)))}{70 d \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}","-\frac{12 i \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{5/2}}{35 a d}+\frac{2 i (e \cos (c+d x))^{5/2}}{7 d \sqrt{a+i a \tan (c+d x)}}-\frac{32 i \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{5/2}}{35 a d}+\frac{16 i \sec ^2(c+d x) (e \cos (c+d x))^{5/2}}{35 d \sqrt{a+i a \tan (c+d x)}}",1,"((-1/70*I)*e^3*(35*Cos[c + d*x] + Cos[3*(c + d*x)] + (70*I)*Sin[c + d*x] + (6*I)*Sin[3*(c + d*x)]))/(d*Sqrt[e*Cos[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
682,1,63,126,0.3789904,"\int \frac{(e \cos (c+d x))^{3/2}}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)/Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^2 (4 i \sin (2 (c+d x))+\cos (2 (c+d x))-15)}{15 d \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}","-\frac{8 i \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2}}{15 a d}+\frac{2 i (e \cos (c+d x))^{3/2}}{5 d \sqrt{a+i a \tan (c+d x)}}+\frac{16 i \sec ^2(c+d x) (e \cos (c+d x))^{3/2}}{15 d \sqrt{a+i a \tan (c+d x)}}",1,"((-1/15*I)*e^2*(-15 + Cos[2*(c + d*x)] + (4*I)*Sin[2*(c + d*x)]))/(d*Sqrt[e*Cos[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
683,1,48,80,0.2042636,"\int \frac{\sqrt{e \cos (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{2 (2 \tan (c+d x)-i) \sqrt{e \cos (c+d x)}}{3 d \sqrt{a+i a \tan (c+d x)}}","\frac{2 i \sqrt{e \cos (c+d x)}}{3 d \sqrt{a+i a \tan (c+d x)}}-\frac{4 i \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}{3 a d}",1,"(2*Sqrt[e*Cos[c + d*x]]*(-I + 2*Tan[c + d*x]))/(3*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
684,1,36,36,0.1875464,"\int \frac{1}{\sqrt{e \cos (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/(Sqrt[e*Cos[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{2 i}{d \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}","\frac{2 i}{d \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}",1,"(2*I)/(d*Sqrt[e*Cos[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
685,1,209,495,9.9084038,"\int \frac{1}{(e \cos (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/((e*Cos[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{i e^{\frac{1}{2} i (c+d x)} \left(\log \left(-\sqrt{2} e^{\frac{1}{2} i (c+d x)}+e^{i (c+d x)}+1\right)-\log \left(\sqrt{2} e^{\frac{1}{2} i (c+d x)}+e^{i (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} e^{\frac{1}{2} i (c+d x)}\right)-2 \tan ^{-1}\left(1+\sqrt{2} e^{\frac{1}{2} i (c+d x)}\right)\right)}{\sqrt{2} d e \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{e e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}","-\frac{i \sqrt{2} \sqrt{a} \sec (c+d x) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a-i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{d e^{3/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i \sqrt{2} \sqrt{a} \sec (c+d x) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{a-i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{d e^{3/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i \sqrt{a} \sec (c+d x) \log \left(-\sqrt{2} \sqrt{a} \sqrt{a-i a \tan (c+d x)} \sqrt{e \cos (c+d x)}+\sqrt{e} \cos (c+d x) (a-i a \tan (c+d x))+a \sqrt{e}\right)}{\sqrt{2} d e^{3/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{i \sqrt{a} \sec (c+d x) \log \left(\sqrt{2} \sqrt{a} \sqrt{a-i a \tan (c+d x)} \sqrt{e \cos (c+d x)}+\sqrt{e} \cos (c+d x) (a-i a \tan (c+d x))+a \sqrt{e}\right)}{\sqrt{2} d e^{3/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}",1,"(I*E^((I/2)*(c + d*x))*(2*ArcTan[1 - Sqrt[2]*E^((I/2)*(c + d*x))] - 2*ArcTan[1 + Sqrt[2]*E^((I/2)*(c + d*x))] + Log[1 - Sqrt[2]*E^((I/2)*(c + d*x)) + E^(I*(c + d*x))] - Log[1 + Sqrt[2]*E^((I/2)*(c + d*x)) + E^(I*(c + d*x))]))/(Sqrt[2]*d*e*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(e*(1 + E^((2*I)*(c + d*x))))/E^(I*(c + d*x))])","A",1
686,1,250,470,1.7660882,"\int \frac{1}{(e \cos (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/((e*Cos[c + d*x])^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{i e^{i c-\frac{i d x}{2}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(\left(-e^{-2 i c}\right)^{3/4} \left(1+e^{2 i (c+d x)}\right) \tan ^{-1}\left(\frac{e^{\frac{i d x}{2}}}{\sqrt[4]{-e^{-2 i c}}}\right)-\left(-e^{-2 i c}\right)^{3/4} \left(1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{e^{\frac{i d x}{2}}}{\sqrt[4]{-e^{-2 i c}}}\right)-2 e^{\frac{3 i d x}{2}}\right)}{d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \sqrt{\cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{5/2}}","\frac{i e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{\sqrt{2} \sqrt{a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac{i e^{5/2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{\sqrt{2} \sqrt{a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac{i e^{5/2} \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{2 \sqrt{2} \sqrt{a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}+\frac{i e^{5/2} \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right)}{2 \sqrt{2} \sqrt{a} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac{i \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{a d (e \cos (c+d x))^{5/2}}",1,"(I*E^(I*c - (I/2)*d*x)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(-2*E^(((3*I)/2)*d*x) + (-E^((-2*I)*c))^(3/4)*(1 + E^((2*I)*(c + d*x)))*ArcTan[E^((I/2)*d*x)/(-E^((-2*I)*c))^(1/4)] - (-E^((-2*I)*c))^(3/4)*(1 + E^((2*I)*(c + d*x)))*ArcTanh[E^((I/2)*d*x)/(-E^((-2*I)*c))^(1/4)]))/(d*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*Sqrt[Cos[c + d*x]]*(e*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])","A",1
687,1,245,682,1.5364558,"\int \frac{1}{(e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/((e*Cos[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{\sqrt{\cos (c+d x)} \left(\frac{3}{4} i e^{\frac{1}{2} i (c+d x)} \left(e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)\right)^{5/2} \left(\log \left(-\sqrt{2} e^{\frac{1}{2} i (c+d x)}+e^{i (c+d x)}+1\right)-\log \left(\sqrt{2} e^{\frac{1}{2} i (c+d x)}+e^{i (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} e^{\frac{1}{2} i (c+d x)}\right)-2 \tan ^{-1}\left(1+\sqrt{2} e^{\frac{1}{2} i (c+d x)}\right)\right)+4 \sqrt{\cos (c+d x)} (2 \sin (c+d x)+i \cos (c+d x))\right)}{16 d \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}","-\frac{3 i \sqrt{a} e^{7/2} \sec (c+d x) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{4 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}+\frac{3 i \sqrt{a} e^{7/2} \sec (c+d x) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right)}{4 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}+\frac{3 i \sqrt{a} e^{7/2} \sec (c+d x) \log \left(-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}-\frac{3 i \sqrt{a} e^{7/2} \sec (c+d x) \log \left(\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right)}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}-\frac{i \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a d (e \cos (c+d x))^{7/2}}+\frac{3 i \cos ^2(c+d x)}{4 d \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}",1,"(Sqrt[Cos[c + d*x]]*(((3*I)/4)*E^((I/2)*(c + d*x))*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^(5/2)*(2*ArcTan[1 - Sqrt[2]*E^((I/2)*(c + d*x))] - 2*ArcTan[1 + Sqrt[2]*E^((I/2)*(c + d*x))] + Log[1 - Sqrt[2]*E^((I/2)*(c + d*x)) + E^(I*(c + d*x))] - Log[1 + Sqrt[2]*E^((I/2)*(c + d*x)) + E^(I*(c + d*x))]) + 4*Sqrt[Cos[c + d*x]]*(I*Cos[c + d*x] + 2*Sin[c + d*x])))/(16*d*(e*Cos[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]])","A",1
688,1,192,105,13.3213656,"\int (e \cos (c+d x))^m (a+i a \tan (c+d x))^n \, dx","Integrate[(e*Cos[c + d*x])^m*(a + I*a*Tan[c + d*x])^n,x]","\frac{i 2^{n-m} \left(1+e^{2 i (c+d x)}\right) \left(e^{i d x}\right)^n \left(e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)\right)^m \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^n \cos ^{-m}(c+d x) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n (e \cos (c+d x))^m \, _2F_1\left(1,\frac{m+2}{2};-\frac{m}{2}+n+1;-e^{2 i (c+d x)}\right)}{d (m-2 n)}","-\frac{i 2^{n-\frac{m}{2}} (a+i a \tan (c+d x))^n (e \cos (c+d x))^m (1+i \tan (c+d x))^{\frac{1}{2} (m-2 n)} \, _2F_1\left(-\frac{m}{2},\frac{1}{2} (m-2 n+2);1-\frac{m}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d m}",1,"(I*2^(-m + n)*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(1 + E^((2*I)*(c + d*x)))*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^m*(e*Cos[c + d*x])^m*Hypergeometric2F1[1, (2 + m)/2, 1 - m/2 + n, -E^((2*I)*(c + d*x))]*(a + I*a*Tan[c + d*x])^n)/(d*(m - 2*n)*Cos[c + d*x]^m*Sec[c + d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)","A",0
689,1,125,86,1.6940065,"\int (e \cos (c+d x))^m (a+i a \tan (c+d x))^2 \, dx","Integrate[(e*Cos[c + d*x])^m*(a + I*a*Tan[c + d*x])^2,x]","-\frac{i a^2 2^{2-m} e^{i (c+d x)} \left(e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)\right)^{m-1} (\tan (c+d x)-i)^2 \, _2F_1\left(1,\frac{m+2}{2};3-\frac{m}{2};-e^{2 i (c+d x)}\right) \cos ^{2-m}(c+d x) (e \cos (c+d x))^m}{d (m-4)}","-\frac{i a^2 2^{2-\frac{m}{2}} (1+i \tan (c+d x))^{m/2} (e \cos (c+d x))^m \, _2F_1\left(\frac{m-2}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d m}",1,"((-I)*2^(2 - m)*a^2*E^(I*(c + d*x))*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^(-1 + m)*Cos[c + d*x]^(2 - m)*(e*Cos[c + d*x])^m*Hypergeometric2F1[1, (2 + m)/2, 3 - m/2, -E^((2*I)*(c + d*x))]*(-I + Tan[c + d*x])^2)/(d*(-4 + m))","A",0
690,1,131,82,8.1010307,"\int (e \cos (c+d x))^m (a+i a \tan (c+d x)) \, dx","Integrate[(e*Cos[c + d*x])^m*(a + I*a*Tan[c + d*x]),x]","-\frac{a 2^{1-m} e^{i (c+2 d x)} \left(e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)\right)^m (\tan (c+d x)-i) (\cos (d x)-i \sin (d x)) \, _2F_1\left(1,\frac{m+2}{2};2-\frac{m}{2};-e^{2 i (c+d x)}\right) \cos ^{1-m}(c+d x) (e \cos (c+d x))^m}{d (m-2)}","-\frac{i a 2^{1-\frac{m}{2}} (1+i \tan (c+d x))^{m/2} (e \cos (c+d x))^m \, _2F_1\left(-\frac{m}{2},\frac{m}{2};1-\frac{m}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d m}",1,"-((2^(1 - m)*a*E^(I*(c + 2*d*x))*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^m*Cos[c + d*x]^(1 - m)*(e*Cos[c + d*x])^m*Hypergeometric2F1[1, (2 + m)/2, 2 - m/2, -E^((2*I)*(c + d*x))]*(Cos[d*x] - I*Sin[d*x])*(-I + Tan[c + d*x]))/(d*(-2 + m)))","A",0
691,1,147,86,60.0483165,"\int \frac{(e \cos (c+d x))^m}{a+i a \tan (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^m/(a + I*a*Tan[c + d*x]),x]","\frac{2^{-m-1} e^{-i (c+2 d x)} \left(1+e^{2 i (c+d x)}\right)^2 \left(e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)\right)^m (\cos (d x)+i \sin (d x)) \, _2F_1\left(1,\frac{m+2}{2};-\frac{m}{2};-e^{2 i (c+d x)}\right) \cos ^{-m-1}(c+d x) (e \cos (c+d x))^m}{a d (m+2) (\tan (c+d x)-i)}","-\frac{i 2^{-\frac{m}{2}-1} (1+i \tan (c+d x))^{m/2} (e \cos (c+d x))^m \, _2F_1\left(-\frac{m}{2},\frac{m+4}{2};1-\frac{m}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{a d m}",1,"(2^(-1 - m)*(1 + E^((2*I)*(c + d*x)))^2*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^m*Cos[c + d*x]^(-1 - m)*(e*Cos[c + d*x])^m*Hypergeometric2F1[1, (2 + m)/2, -1/2*m, -E^((2*I)*(c + d*x))]*(Cos[d*x] + I*Sin[d*x]))/(a*d*E^(I*(c + 2*d*x))*(2 + m)*(-I + Tan[c + d*x]))","A",0
692,1,154,86,71.0867993,"\int \frac{(e \cos (c+d x))^m}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^m/(a + I*a*Tan[c + d*x])^2,x]","-\frac{i 2^{-m-2} e^{-2 i (c+2 d x)} \left(1+e^{2 i (c+d x)}\right)^3 \left(e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)\right)^m (\cos (d x)+i \sin (d x))^2 \, _2F_1\left(1,\frac{m+2}{2};-\frac{m}{2}-1;-e^{2 i (c+d x)}\right) \cos ^{-m-2}(c+d x) (e \cos (c+d x))^m}{a^2 d (m+4) (\tan (c+d x)-i)^2}","-\frac{i 2^{-\frac{m}{2}-2} (1+i \tan (c+d x))^{m/2} (e \cos (c+d x))^m \, _2F_1\left(-\frac{m}{2},\frac{m+6}{2};1-\frac{m}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{a^2 d m}",1,"((-I)*2^(-2 - m)*(1 + E^((2*I)*(c + d*x)))^3*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^m*Cos[c + d*x]^(-2 - m)*(e*Cos[c + d*x])^m*Hypergeometric2F1[1, (2 + m)/2, -1 - m/2, -E^((2*I)*(c + d*x))]*(Cos[d*x] + I*Sin[d*x])^2)/(a^2*d*E^((2*I)*(c + 2*d*x))*(4 + m)*(-I + Tan[c + d*x])^2)","A",0
693,1,106,105,0.8754767,"\int (e \cos (c+d x))^m \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^m*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{i 2^{-m} \left(1+e^{2 i (c+d x)}\right) \sqrt{a+i a \tan (c+d x)} \left(e e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)\right)^m \, _2F_1\left(1,\frac{m+2}{2};\frac{3-m}{2};-e^{2 i (c+d x)}\right)}{d (m-1)}","-\frac{i a 2^{\frac{1}{2}-\frac{m}{2}} (1+i \tan (c+d x))^{\frac{m+1}{2}} (e \cos (c+d x))^m \, _2F_1\left(-\frac{m}{2},\frac{m+1}{2};1-\frac{m}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d m \sqrt{a+i a \tan (c+d x)}}",1,"(I*(1 + E^((2*I)*(c + d*x)))*((e*(1 + E^((2*I)*(c + d*x))))/E^(I*(c + d*x)))^m*Hypergeometric2F1[1, (2 + m)/2, (3 - m)/2, -E^((2*I)*(c + d*x))]*Sqrt[a + I*a*Tan[c + d*x]])/(2^m*d*(-1 + m))","A",0
694,1,143,104,15.6313079,"\int \frac{(e \cos (c+d x))^m}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(e*Cos[c + d*x])^m/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{i 4^{-m} \left(1+e^{2 i (c+d x)}\right) \left(e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)\right)^m \left(e e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)\right)^m \, _2F_1\left(1,\frac{m+2}{2};\frac{1-m}{2};-e^{2 i (c+d x)}\right) \cos ^{-m}(c+d x)}{d (m+1) \sqrt{a+i a \tan (c+d x)}}","-\frac{i 2^{-\frac{m}{2}-\frac{1}{2}} (1+i \tan (c+d x))^{\frac{m+1}{2}} (e \cos (c+d x))^m \, _2F_1\left(-\frac{m}{2},\frac{m+3}{2};1-\frac{m}{2};\frac{1}{2} (1-i \tan (c+d x))\right)}{d m \sqrt{a+i a \tan (c+d x)}}",1,"(I*(1 + E^((2*I)*(c + d*x)))*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^m*((e*(1 + E^((2*I)*(c + d*x))))/E^(I*(c + d*x)))^m*Hypergeometric2F1[1, (2 + m)/2, (1 - m)/2, -E^((2*I)*(c + d*x))])/(4^m*d*(1 + m)*Cos[c + d*x]^m*Sqrt[a + I*a*Tan[c + d*x]])","A",0
695,1,212,183,3.1558973,"\int (d \cos (e+f x))^m (a+b \tan (e+f x))^3 \, dx","Integrate[(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x])^3,x]","\frac{\cos (e+f x) (a+b \tan (e+f x))^3 (d \cos (e+f x))^m \left(-\frac{a \left(a^2-3 b^2\right) \sin (e+f x) \cos ^3(e+f x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{(m+1) \sqrt{\sin ^2(e+f x)}}+\frac{b \left(b^2-3 a^2\right) \cos ^2(e+f x)}{m}-\frac{3 a b^2 \sin (2 (e+f x)) \, _2F_1\left(\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\cos ^2(e+f x)\right)}{2 (m-1) \sqrt{\sin ^2(e+f x)}}-\frac{b^3}{m-2}\right)}{f (a \cos (e+f x)+b \sin (e+f x))^3}","-\frac{a \left(3 b^2-a^2 (1-m)\right) \tan (e+f x) \sec ^2(e+f x)^{m/2} (d \cos (e+f x))^m \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{3}{2};-\tan ^2(e+f x)\right)}{f (1-m)}+\frac{b (d \cos (e+f x))^m \left(2 (1-m) \left(b^2-a^2 (3-m)\right)+a b (4-m) m \tan (e+f x)\right)}{f m \left(m^2-3 m+2\right)}+\frac{b (a+b \tan (e+f x))^2 (d \cos (e+f x))^m}{f (2-m)}",1,"(Cos[e + f*x]*(d*Cos[e + f*x])^m*(-(b^3/(-2 + m)) + (b*(-3*a^2 + b^2)*Cos[e + f*x]^2)/m - (a*(a^2 - 3*b^2)*Cos[e + f*x]^3*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/((1 + m)*Sqrt[Sin[e + f*x]^2]) - (3*a*b^2*Hypergeometric2F1[1/2, (-1 + m)/2, (1 + m)/2, Cos[e + f*x]^2]*Sin[2*(e + f*x)])/(2*(-1 + m)*Sqrt[Sin[e + f*x]^2]))*(a + b*Tan[e + f*x])^3)/(f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3)","A",1
696,1,330,155,3.7503614,"\int (d \cos (e+f x))^m (a+b \tan (e+f x))^2 \, dx","Integrate[(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x])^2,x]","\frac{\cos (e+f x) (a+b \tan (e+f x))^2 (d \cos (e+f x))^m \left(\sqrt{\sin ^2(e+f x)} \left(-\frac{a^2 \cos (e+f x) \cot (e+f x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{m+1}-\frac{b^2 \csc (e+f x) \, _2F_1\left(-\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\cos ^2(e+f x)\right)}{m-1}\right)-\frac{a b 2^{1-m} \left(e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right)\right)^m \, _2F_1\left(1,\frac{m}{2};1-\frac{m}{2};-e^{2 i (e+f x)}\right) \cos ^{1-m}(e+f x)}{m}+\frac{a b 2^{1-m} e^{2 i (e+f x)} \left(e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right)\right)^m \, _2F_1\left(1,\frac{m+2}{2};2-\frac{m}{2};-e^{2 i (e+f x)}\right) \cos ^{1-m}(e+f x)}{m-2}\right)}{f (a \cos (e+f x)+b \sin (e+f x))^2}","\frac{\left(b^2-a^2 (1-m)\right) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^m \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{f (1-m) (m+1) \sqrt{\sin ^2(e+f x)}}-\frac{a b (2-m) (d \cos (e+f x))^m}{f (1-m) m}+\frac{b (a+b \tan (e+f x)) (d \cos (e+f x))^m}{f (1-m)}",1,"(Cos[e + f*x]*(d*Cos[e + f*x])^m*(-((2^(1 - m)*a*b*((1 + E^((2*I)*(e + f*x)))/E^(I*(e + f*x)))^m*Cos[e + f*x]^(1 - m)*Hypergeometric2F1[1, m/2, 1 - m/2, -E^((2*I)*(e + f*x))])/m) + (2^(1 - m)*a*b*E^((2*I)*(e + f*x))*((1 + E^((2*I)*(e + f*x)))/E^(I*(e + f*x)))^m*Cos[e + f*x]^(1 - m)*Hypergeometric2F1[1, (2 + m)/2, 2 - m/2, -E^((2*I)*(e + f*x))])/(-2 + m) + (-((b^2*Csc[e + f*x]*Hypergeometric2F1[-1/2, (-1 + m)/2, (1 + m)/2, Cos[e + f*x]^2])/(-1 + m)) - (a^2*Cos[e + f*x]*Cot[e + f*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2])/(1 + m))*Sqrt[Sin[e + f*x]^2])*(a + b*Tan[e + f*x])^2)/(f*(a*Cos[e + f*x] + b*Sin[e + f*x])^2)","C",0
697,1,203,90,1.0505913,"\int (d \cos (e+f x))^m (a+b \tan (e+f x)) \, dx","Integrate[(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x]),x]","\frac{(d \cos (e+f x))^m \left(-a (m-2) m \sin (2 (e+f x)) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)-2 b \left(m^2-m-2\right) \sqrt{\sin ^2(e+f x)} \, _2F_1\left(1,\frac{m}{2};1-\frac{m}{2};-e^{2 i (e+f x)}\right)+2 b m (m+1) \sqrt{\sin ^2(e+f x)} \, _2F_1\left(1,\frac{m+2}{2};2-\frac{m}{2};-e^{2 i (e+f x)}\right) (\cos (2 (e+f x))+i \sin (2 (e+f x)))\right)}{2 f (m-2) m (m+1) \sqrt{\sin ^2(e+f x)}}","-\frac{a \sin (e+f x) (d \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{d f (m+1) \sqrt{\sin ^2(e+f x)}}-\frac{b (d \cos (e+f x))^m}{f m}",1,"((d*Cos[e + f*x])^m*(-2*b*(-2 - m + m^2)*Hypergeometric2F1[1, m/2, 1 - m/2, -E^((2*I)*(e + f*x))]*Sqrt[Sin[e + f*x]^2] + 2*b*m*(1 + m)*Hypergeometric2F1[1, (2 + m)/2, 2 - m/2, -E^((2*I)*(e + f*x))]*Sqrt[Sin[e + f*x]^2]*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]) - a*(-2 + m)*m*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[2*(e + f*x)]))/(2*f*(-2 + m)*m*(1 + m)*Sqrt[Sin[e + f*x]^2])","C",0
698,1,1132,140,14.2467926,"\int \frac{(d \cos (e+f x))^m}{a+b \tan (e+f x)} \, dx","Integrate[(d*Cos[e + f*x])^m/(a + b*Tan[e + f*x]),x]","\frac{(d \cos (e+f x))^m \left(-b F_1\left(m;\frac{m}{2},\frac{m}{2};m+1;\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right) \left(\frac{b (\tan (e+f x)-i)}{a+b \tan (e+f x)}\right)^{m/2} \left(\frac{b (\tan (e+f x)+i)}{a+b \tan (e+f x)}\right)^{m/2} \sec ^2(e+f x)^{-m/2}+b \left(\sec ^2(e+f x)^{-m/2}-1\right)+a m \, _2F_1\left(\frac{1}{2},\frac{m}{2}+1;\frac{3}{2};-\tan ^2(e+f x)\right) \tan (e+f x)\right)}{f (a+b \tan (e+f x)) \left(-\frac{1}{2} b m F_1\left(m;\frac{m}{2},\frac{m}{2};m+1;\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right) \sec ^2(e+f x)^{-m/2} \left(\frac{b (\tan (e+f x)+i)}{a+b \tan (e+f x)}\right)^{m/2} \left(\frac{b \sec ^2(e+f x)}{a+b \tan (e+f x)}-\frac{b^2 \sec ^2(e+f x) (\tan (e+f x)-i)}{(a+b \tan (e+f x))^2}\right) \left(\frac{b (\tan (e+f x)-i)}{a+b \tan (e+f x)}\right)^{\frac{m}{2}-1}+b m F_1\left(m;\frac{m}{2},\frac{m}{2};m+1;\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right) \sec ^2(e+f x)^{-m/2} \tan (e+f x) \left(\frac{b (\tan (e+f x)+i)}{a+b \tan (e+f x)}\right)^{m/2} \left(\frac{b (\tan (e+f x)-i)}{a+b \tan (e+f x)}\right)^{m/2}-b \sec ^2(e+f x)^{-m/2} \left(\frac{b (\tan (e+f x)+i)}{a+b \tan (e+f x)}\right)^{m/2} \left(-\frac{(a-i b) b m^2 F_1\left(m+1;\frac{m}{2}+1,\frac{m}{2};m+2;\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right) \sec ^2(e+f x)}{2 (m+1) (a+b \tan (e+f x))^2}-\frac{(a+i b) b m^2 F_1\left(m+1;\frac{m}{2},\frac{m}{2}+1;m+2;\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right) \sec ^2(e+f x)}{2 (m+1) (a+b \tan (e+f x))^2}\right) \left(\frac{b (\tan (e+f x)-i)}{a+b \tan (e+f x)}\right)^{m/2}-\frac{1}{2} b m F_1\left(m;\frac{m}{2},\frac{m}{2};m+1;\frac{a-i b}{a+b \tan (e+f x)},\frac{a+i b}{a+b \tan (e+f x)}\right) \sec ^2(e+f x)^{-m/2} \left(\frac{b (\tan (e+f x)+i)}{a+b \tan (e+f x)}\right)^{\frac{m}{2}-1} \left(\frac{b \sec ^2(e+f x)}{a+b \tan (e+f x)}-\frac{b^2 \sec ^2(e+f x) (\tan (e+f x)+i)}{(a+b \tan (e+f x))^2}\right) \left(\frac{b (\tan (e+f x)-i)}{a+b \tan (e+f x)}\right)^{m/2}+a m \, _2F_1\left(\frac{1}{2},\frac{m}{2}+1;\frac{3}{2};-\tan ^2(e+f x)\right) \sec ^2(e+f x)-b m \sec ^2(e+f x)^{-m/2} \tan (e+f x)+a m \sec ^2(e+f x) \left(\left(\tan ^2(e+f x)+1\right)^{-\frac{m}{2}-1}-\, _2F_1\left(\frac{1}{2},\frac{m}{2}+1;\frac{3}{2};-\tan ^2(e+f x)\right)\right)\right)}","\frac{\tan (e+f x) \sec ^2(e+f x)^{m/2} (d \cos (e+f x))^m F_1\left(\frac{1}{2};1,\frac{m+2}{2};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{a f}+\frac{b (d \cos (e+f x))^m \, _2F_1\left(1,-\frac{m}{2};1-\frac{m}{2};\frac{b^2 \sec ^2(e+f x)}{a^2+b^2}\right)}{f m \left(a^2+b^2\right)}",1,"((d*Cos[e + f*x])^m*(b*(-1 + (Sec[e + f*x]^2)^(-1/2*m)) + a*m*Hypergeometric2F1[1/2, 1 + m/2, 3/2, -Tan[e + f*x]^2]*Tan[e + f*x] - (b*AppellF1[m, m/2, m/2, 1 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2))/(Sec[e + f*x]^2)^(m/2)))/(f*(a + b*Tan[e + f*x])*(a*m*Hypergeometric2F1[1/2, 1 + m/2, 3/2, -Tan[e + f*x]^2]*Sec[e + f*x]^2 - (b*m*Tan[e + f*x])/(Sec[e + f*x]^2)^(m/2) + (b*m*AppellF1[m, m/2, m/2, 1 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*Tan[e + f*x]*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2))/(Sec[e + f*x]^2)^(m/2) - (b*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*(-1/2*((a - I*b)*b*m^2*AppellF1[1 + m, 1 + m/2, m/2, 2 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*Sec[e + f*x]^2)/((1 + m)*(a + b*Tan[e + f*x])^2) - ((a + I*b)*b*m^2*AppellF1[1 + m, m/2, 1 + m/2, 2 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*Sec[e + f*x]^2)/(2*(1 + m)*(a + b*Tan[e + f*x])^2)))/(Sec[e + f*x]^2)^(m/2) - (b*m*AppellF1[m, m/2, m/2, 1 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(-1 + m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*(-((b^2*Sec[e + f*x]^2*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x])^2) + (b*Sec[e + f*x]^2)/(a + b*Tan[e + f*x])))/(2*(Sec[e + f*x]^2)^(m/2)) - (b*m*AppellF1[m, m/2, m/2, 1 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(-1 + m/2)*(-((b^2*Sec[e + f*x]^2*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x])^2) + (b*Sec[e + f*x]^2)/(a + b*Tan[e + f*x])))/(2*(Sec[e + f*x]^2)^(m/2)) + a*m*Sec[e + f*x]^2*(-Hypergeometric2F1[1/2, 1 + m/2, 3/2, -Tan[e + f*x]^2] + (1 + Tan[e + f*x]^2)^(-1 - m/2))))","C",0
699,1,2502,227,17.737006,"\int \frac{(d \cos (e+f x))^m}{(a+b \tan (e+f x))^2} \, dx","Integrate[(d*Cos[e + f*x])^m/(a + b*Tan[e + f*x])^2,x]","\text{Result too large to show}","\frac{\tan (e+f x) \sec ^2(e+f x)^{m/2} (d \cos (e+f x))^m F_1\left(\frac{1}{2};2,\frac{m+2}{2};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{a^2 f}+\frac{2 a b (d \cos (e+f x))^m \, _2F_1\left(2,-\frac{m}{2};1-\frac{m}{2};\frac{b^2 \sec ^2(e+f x)}{a^2+b^2}\right)}{f m \left(a^2+b^2\right)^2}+\frac{b^2 \tan ^3(e+f x) \sec ^2(e+f x)^{m/2} (d \cos (e+f x))^m F_1\left(\frac{3}{2};2,\frac{m+2}{2};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right)}{3 a^4 f}",1,"((d*Cos[e + f*x])^m*((2*a*b*(-1 + (Sec[e + f*x]^2)^(-1/2*m)))/m + a^2*Hypergeometric2F1[1/2, 1 + m/2, 3/2, -Tan[e + f*x]^2]*Tan[e + f*x] - b^2*Hypergeometric2F1[1/2, 1 + m/2, 3/2, -Tan[e + f*x]^2]*Tan[e + f*x] - (2*a*b*AppellF1[m, m/2, m/2, 1 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2))/(m*(Sec[e + f*x]^2)^(m/2)) - (b*(a^2 + b^2)*AppellF1[1 + m, m/2, m/2, 2 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2))/((1 + m)*(Sec[e + f*x]^2)^(m/2)*(a + b*Tan[e + f*x]))))/(f*(a + b*Tan[e + f*x])^2*(a^2*Hypergeometric2F1[1/2, 1 + m/2, 3/2, -Tan[e + f*x]^2]*Sec[e + f*x]^2 - b^2*Hypergeometric2F1[1/2, 1 + m/2, 3/2, -Tan[e + f*x]^2]*Sec[e + f*x]^2 - (2*a*b*Tan[e + f*x])/(Sec[e + f*x]^2)^(m/2) + (2*a*b*AppellF1[m, m/2, m/2, 1 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*Tan[e + f*x]*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2))/(Sec[e + f*x]^2)^(m/2) + (b^2*(a^2 + b^2)*AppellF1[1 + m, m/2, m/2, 2 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(Sec[e + f*x]^2)^(1 - m/2)*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2))/((1 + m)*(a + b*Tan[e + f*x])^2) + (b*(a^2 + b^2)*m*AppellF1[1 + m, m/2, m/2, 2 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*Tan[e + f*x]*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2))/((1 + m)*(Sec[e + f*x]^2)^(m/2)*(a + b*Tan[e + f*x])) - (2*a*b*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*(-1/2*((a - I*b)*b*m^2*AppellF1[1 + m, 1 + m/2, m/2, 2 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*Sec[e + f*x]^2)/((1 + m)*(a + b*Tan[e + f*x])^2) - ((a + I*b)*b*m^2*AppellF1[1 + m, m/2, 1 + m/2, 2 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*Sec[e + f*x]^2)/(2*(1 + m)*(a + b*Tan[e + f*x])^2)))/(m*(Sec[e + f*x]^2)^(m/2)) - (b*(a^2 + b^2)*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*(-1/2*((a - I*b)*b*m*(1 + m)*AppellF1[2 + m, 1 + m/2, m/2, 3 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*Sec[e + f*x]^2)/((2 + m)*(a + b*Tan[e + f*x])^2) - ((a + I*b)*b*m*(1 + m)*AppellF1[2 + m, m/2, 1 + m/2, 3 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*Sec[e + f*x]^2)/(2*(2 + m)*(a + b*Tan[e + f*x])^2)))/((1 + m)*(Sec[e + f*x]^2)^(m/2)*(a + b*Tan[e + f*x])) - (a*b*AppellF1[m, m/2, m/2, 1 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(-1 + m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*(-((b^2*Sec[e + f*x]^2*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x])^2) + (b*Sec[e + f*x]^2)/(a + b*Tan[e + f*x])))/(Sec[e + f*x]^2)^(m/2) - (b*(a^2 + b^2)*m*AppellF1[1 + m, m/2, m/2, 2 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(-1 + m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*(-((b^2*Sec[e + f*x]^2*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x])^2) + (b*Sec[e + f*x]^2)/(a + b*Tan[e + f*x])))/(2*(1 + m)*(Sec[e + f*x]^2)^(m/2)*(a + b*Tan[e + f*x])) - (a*b*AppellF1[m, m/2, m/2, 1 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(-1 + m/2)*(-((b^2*Sec[e + f*x]^2*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x])^2) + (b*Sec[e + f*x]^2)/(a + b*Tan[e + f*x])))/(Sec[e + f*x]^2)^(m/2) - (b*(a^2 + b^2)*m*AppellF1[1 + m, m/2, m/2, 2 + m, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*((b*(-I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(m/2)*((b*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x]))^(-1 + m/2)*(-((b^2*Sec[e + f*x]^2*(I + Tan[e + f*x]))/(a + b*Tan[e + f*x])^2) + (b*Sec[e + f*x]^2)/(a + b*Tan[e + f*x])))/(2*(1 + m)*(Sec[e + f*x]^2)^(m/2)*(a + b*Tan[e + f*x])) + a^2*Sec[e + f*x]^2*(-Hypergeometric2F1[1/2, 1 + m/2, 3/2, -Tan[e + f*x]^2] + (1 + Tan[e + f*x]^2)^(-1 - m/2)) - b^2*Sec[e + f*x]^2*(-Hypergeometric2F1[1/2, 1 + m/2, 3/2, -Tan[e + f*x]^2] + (1 + Tan[e + f*x]^2)^(-1 - m/2))))","C",0
700,1,698,187,23.1583513,"\int (d \cos (e+f x))^m (a+b \tan (e+f x))^n \, dx","Integrate[(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x])^n,x]","\frac{2 (d \cos (e+f x))^m (a+b \tan (e+f x))^{n+1} F_1\left(n+1;\frac{m}{2}+1,\frac{m}{2}+1;n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)}{f \left(2 n (b-a \tan (e+f x)) F_1\left(n+1;\frac{m}{2}+1,\frac{m}{2}+1;n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)+2 (n-m) \tan (e+f x) (a+b \tan (e+f x)) F_1\left(n+1;\frac{m}{2}+1,\frac{m}{2}+1;n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)+2 b \sec ^2(e+f x) F_1\left(n+1;\frac{m}{2}+1,\frac{m}{2}+1;n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)+\frac{b (m+2) (n+1) \sec ^2(e+f x) (a+b \tan (e+f x)) \left((a-i b) F_1\left(n+2;\frac{m}{2}+1,\frac{m}{2}+2;n+3;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)+(a+i b) F_1\left(n+2;\frac{m}{2}+2,\frac{m}{2}+1;n+3;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)\right)}{(n+2) (a-i b) (a+i b)}+\frac{m \sec ^2(e+f x) (a+b \tan (e+f x)) F_1\left(n+1;\frac{m}{2}+1,\frac{m}{2}+1;n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)}{\tan (e+f x)-i}+\frac{m \sec ^2(e+f x) (a+b \tan (e+f x)) F_1\left(n+1;\frac{m}{2}+1,\frac{m}{2}+1;n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right)}{\tan (e+f x)+i}\right)}","\frac{\cos ^2(e+f x) (d \cos (e+f x))^m \left(1-\frac{a+b \tan (e+f x)}{a-\sqrt{-b^2}}\right)^{\frac{m+2}{2}} \left(1-\frac{a+b \tan (e+f x)}{a+\sqrt{-b^2}}\right)^{\frac{m+2}{2}} (a+b \tan (e+f x))^{n+1} F_1\left(n+1;\frac{m+2}{2},\frac{m+2}{2};n+2;\frac{a+b \tan (e+f x)}{a-\sqrt{-b^2}},\frac{a+b \tan (e+f x)}{a+\sqrt{-b^2}}\right)}{b f (n+1)}",1,"(2*AppellF1[1 + n, 1 + m/2, 1 + m/2, 2 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x])^(1 + n))/(f*(2*b*AppellF1[1 + n, 1 + m/2, 1 + m/2, 2 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*Sec[e + f*x]^2 + 2*n*AppellF1[1 + n, 1 + m/2, 1 + m/2, 2 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*(b - a*Tan[e + f*x]) + (b*(2 + m)*(1 + n)*((a - I*b)*AppellF1[2 + n, 1 + m/2, 2 + m/2, 3 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)] + (a + I*b)*AppellF1[2 + n, 2 + m/2, 1 + m/2, 3 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)])*Sec[e + f*x]^2*(a + b*Tan[e + f*x]))/((a - I*b)*(a + I*b)*(2 + n)) + 2*(-m + n)*AppellF1[1 + n, 1 + m/2, 1 + m/2, 2 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*Tan[e + f*x]*(a + b*Tan[e + f*x]) + (m*AppellF1[1 + n, 1 + m/2, 1 + m/2, 2 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*Sec[e + f*x]^2*(a + b*Tan[e + f*x]))/(-I + Tan[e + f*x]) + (m*AppellF1[1 + n, 1 + m/2, 1 + m/2, 2 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*Sec[e + f*x]^2*(a + b*Tan[e + f*x]))/(I + Tan[e + f*x])))","C",0