1,1,86,91,0.8791898,"\int \tan ^2(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{a \left(-6 (A-i B) \tan ^{-1}(\tan (c+d x))+3 (B+i A) \tan ^2(c+d x)+6 (A-i B) \tan (c+d x)+6 (B+i A) \log (\cos (c+d x))+2 i B \tan ^3(c+d x)\right)}{6 d}","\frac{a (B+i A) \tan ^2(c+d x)}{2 d}+\frac{a (A-i B) \tan (c+d x)}{d}+\frac{a (B+i A) \log (\cos (c+d x))}{d}-a x (A-i B)+\frac{i a B \tan ^3(c+d x)}{3 d}",1,"(a*(-6*(A - I*B)*ArcTan[Tan[c + d*x]] + 6*(I*A + B)*Log[Cos[c + d*x]] + 6*(A - I*B)*Tan[c + d*x] + 3*(I*A + B)*Tan[c + d*x]^2 + (2*I)*B*Tan[c + d*x]^3))/(6*d)","A",1
2,1,70,69,0.3149595,"\int \tan (c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{a \left((-2 B-2 i A) \tan ^{-1}(\tan (c+d x))+2 (B+i A) \tan (c+d x)-2 (A-i B) \log (\cos (c+d x))+i B \tan ^2(c+d x)\right)}{2 d}","\frac{a (B+i A) \tan (c+d x)}{d}-\frac{a (A-i B) \log (\cos (c+d x))}{d}-a x (B+i A)+\frac{i a B \tan ^2(c+d x)}{2 d}",1,"(a*(((-2*I)*A - 2*B)*ArcTan[Tan[c + d*x]] - 2*(A - I*B)*Log[Cos[c + d*x]] + 2*(I*A + B)*Tan[c + d*x] + I*B*Tan[c + d*x]^2))/(2*d)","A",1
3,1,66,46,0.02635,"\int (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","-\frac{i a A \log (\cos (c+d x))}{d}+a A x-\frac{i a B \tan ^{-1}(\tan (c+d x))}{d}+\frac{i a B \tan (c+d x)}{d}-\frac{a B \log (\cos (c+d x))}{d}","-\frac{a (B+i A) \log (\cos (c+d x))}{d}+a x (A-i B)+\frac{i a B \tan (c+d x)}{d}",1,"a*A*x - (I*a*B*ArcTan[Tan[c + d*x]])/d - (I*a*A*Log[Cos[c + d*x]])/d - (a*B*Log[Cos[c + d*x]])/d + (I*a*B*Tan[c + d*x])/d","A",1
4,1,49,40,0.0611423,"\int \cot (c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{a A (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+i a A x-\frac{i a B \log (\cos (c+d x))}{d}+a B x","a x (B+i A)+\frac{a A \log (\sin (c+d x))}{d}-\frac{i a B \log (\cos (c+d x))}{d}",1,"I*a*A*x + a*B*x - (I*a*B*Log[Cos[c + d*x]])/d + (a*A*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","A",1
5,1,84,44,0.190395,"\int \cot ^2(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","-\frac{a A \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}+\frac{i a A (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+\frac{a B (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+i a B x","\frac{a (B+i A) \log (\sin (c+d x))}{d}-a x (A-i B)-\frac{a A \cot (c+d x)}{d}",1,"I*a*B*x - (a*A*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d + (I*a*A*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d + (a*B*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","C",1
6,1,76,68,0.3735406,"\int \cot ^3(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","-\frac{a \left(2 (B+i A) \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)+2 (A-i B) (\log (\tan (c+d x))+\log (\cos (c+d x)))+A \cot ^2(c+d x)\right)}{2 d}","-\frac{a (B+i A) \cot (c+d x)}{d}-\frac{a (A-i B) \log (\sin (c+d x))}{d}-a x (B+i A)-\frac{a A \cot ^2(c+d x)}{2 d}",1,"-1/2*(a*(A*Cot[c + d*x]^2 + 2*(I*A + B)*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2] + 2*(A - I*B)*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]])))/d","C",1
7,1,102,89,0.722492,"\int \cot ^4(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","-\frac{a \left(3 (B+i A) \left(\cot ^2(c+d x)+2 (\log (\tan (c+d x))+\log (\cos (c+d x)))\right)+2 A \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)+6 i B \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)\right)}{6 d}","-\frac{a (B+i A) \cot ^2(c+d x)}{2 d}+\frac{a (A-i B) \cot (c+d x)}{d}-\frac{a (B+i A) \log (\sin (c+d x))}{d}+a x (A-i B)-\frac{a A \cot ^3(c+d x)}{3 d}",1,"-1/6*(a*(2*A*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2] + (6*I)*B*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2] + 3*(I*A + B)*(Cot[c + d*x]^2 + 2*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))))/d","C",1
8,1,96,111,0.8783209,"\int \cot ^5(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","-\frac{a \left(4 (B+i A) \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)-6 (A-i B) \cot ^2(c+d x)-12 (A-i B) (\log (\tan (c+d x))+\log (\cos (c+d x)))+3 A \cot ^4(c+d x)\right)}{12 d}","-\frac{a (B+i A) \cot ^3(c+d x)}{3 d}+\frac{a (A-i B) \cot ^2(c+d x)}{2 d}+\frac{a (B+i A) \cot (c+d x)}{d}+\frac{a (A-i B) \log (\sin (c+d x))}{d}+a x (B+i A)-\frac{a A \cot ^4(c+d x)}{4 d}",1,"-1/12*(a*(-6*(A - I*B)*Cot[c + d*x]^2 + 3*A*Cot[c + d*x]^4 + 4*(I*A + B)*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2] - 12*(A - I*B)*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]])))/d","C",1
9,1,305,141,6.49243,"\int \tan ^2(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \left(-4 d x (A-i B) (\cos (2 c)-i \sin (2 c)) \cos ^3(c+d x)+2 (A-i B) (\cos (2 c)-i \sin (2 c)) \cos ^3(c+d x) \tan ^{-1}(\tan (3 c+d x))+\frac{1}{3} (7 A-8 i B) \sec (c) (\cos (2 c)-i \sin (2 c)) \sin (d x) \cos ^2(c+d x)+(B+i A) (\cos (2 c)-i \sin (2 c)) \cos ^3(c+d x) \log \left(\cos ^2(c+d x)\right)-\frac{1}{6} (\cos (2 c)-i \sin (2 c)) (2 (A-2 i B) \tan (c)-6 i A-9 B) \cos (c+d x)+\frac{1}{3} (A-2 i B) \cos (c) (\tan (c)+i)^2 \sin (d x)-\frac{1}{4} B (\cos (2 c)-i \sin (2 c)) \sec (c+d x)\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{a^2 (4 A-5 i B) \tan ^3(c+d x)}{12 d}+\frac{a^2 (B+i A) \tan ^2(c+d x)}{d}+\frac{2 a^2 (A-i B) \tan (c+d x)}{d}+\frac{2 a^2 (B+i A) \log (\cos (c+d x))}{d}-2 a^2 x (A-i B)+\frac{i B \tan ^3(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{4 d}",1,"((-4*(A - I*B)*d*x*Cos[c + d*x]^3*(Cos[2*c] - I*Sin[2*c]) + 2*(A - I*B)*ArcTan[Tan[3*c + d*x]]*Cos[c + d*x]^3*(Cos[2*c] - I*Sin[2*c]) + (I*A + B)*Cos[c + d*x]^3*Log[Cos[c + d*x]^2]*(Cos[2*c] - I*Sin[2*c]) - (B*Sec[c + d*x]*(Cos[2*c] - I*Sin[2*c]))/4 + ((7*A - (8*I)*B)*Cos[c + d*x]^2*Sec[c]*(Cos[2*c] - I*Sin[2*c])*Sin[d*x])/3 + ((A - (2*I)*B)*Cos[c]*Sin[d*x]*(I + Tan[c])^2)/3 - (Cos[c + d*x]*(Cos[2*c] - I*Sin[2*c])*((-6*I)*A - 9*B + 2*(A - (2*I)*B)*Tan[c]))/6)*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
10,1,273,107,4.3405383,"\int \tan (c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \left((A-i B) \cos ^3(c+d x) (-4 d x \sin (2 c)-4 i d x \cos (2 c))+2 (B+i A) (\cos (2 c)-i \sin (2 c)) \cos ^3(c+d x) \tan ^{-1}(\tan (3 c+d x))+\frac{1}{3} (6 A-7 i B) \sec (c) (\sin (2 c)+i \cos (2 c)) \sin (d x) \cos ^2(c+d x)-(A-i B) (\cos (2 c)-i \sin (2 c)) \cos ^3(c+d x) \log \left(\cos ^2(c+d x)\right)-\frac{1}{6} (\cos (2 c)-i \sin (2 c)) (3 A+2 B \tan (c)-6 i B) \cos (c+d x)+\frac{1}{3} B \cos (c) (\tan (c)+i)^2 \sin (d x)\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{a^2 (B+i A) \tan (c+d x)}{d}-\frac{2 a^2 (A-i B) \log (\cos (c+d x))}{d}-2 a^2 x (B+i A)+\frac{A (a+i a \tan (c+d x))^2}{2 d}-\frac{i B (a+i a \tan (c+d x))^3}{3 a d}",1,"((2*(I*A + B)*ArcTan[Tan[3*c + d*x]]*Cos[c + d*x]^3*(Cos[2*c] - I*Sin[2*c]) - (A - I*B)*Cos[c + d*x]^3*Log[Cos[c + d*x]^2]*(Cos[2*c] - I*Sin[2*c]) + (A - I*B)*Cos[c + d*x]^3*((-4*I)*d*x*Cos[2*c] - 4*d*x*Sin[2*c]) + ((6*A - (7*I)*B)*Cos[c + d*x]^2*Sec[c]*(I*Cos[2*c] + Sin[2*c])*Sin[d*x])/3 + (B*Cos[c]*Sin[d*x]*(I + Tan[c])^2)/3 - (Cos[c + d*x]*(Cos[2*c] - I*Sin[2*c])*(3*A - (6*I)*B + 2*B*Tan[c]))/6)*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
11,1,263,80,2.5844712,"\int (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{a^2 \sec (c) \sec ^2(c+d x) (\cos (2 d x)+i \sin (2 d x)) \left(-8 (A-i B) \cos (c) \cos ^2(c+d x) \tan ^{-1}(\tan (3 c+d x))-i \left((B+i A) \cos (c+2 d x) \left(4 d x-i \log \left(\cos ^2(c+d x)\right)\right)+2 \cos (c) \left((A-i B) \log \left(\cos ^2(c+d x)\right)+4 i A d x+4 B d x-i B\right)-2 i A \sin (c+2 d x)+4 i A d x \cos (3 c+2 d x)+A \cos (3 c+2 d x) \log \left(\cos ^2(c+d x)\right)+2 i A \sin (c)-4 B \sin (c+2 d x)+4 B d x \cos (3 c+2 d x)-i B \cos (3 c+2 d x) \log \left(\cos ^2(c+d x)\right)+4 B \sin (c)\right)\right)}{4 d (\cos (d x)+i \sin (d x))^2}","-\frac{a^2 (A-i B) \tan (c+d x)}{d}-\frac{2 a^2 (B+i A) \log (\cos (c+d x))}{d}+2 a^2 x (A-i B)+\frac{B (a+i a \tan (c+d x))^2}{2 d}",1,"(a^2*Sec[c]*Sec[c + d*x]^2*(Cos[2*d*x] + I*Sin[2*d*x])*(-8*(A - I*B)*ArcTan[Tan[3*c + d*x]]*Cos[c]*Cos[c + d*x]^2 - I*((4*I)*A*d*x*Cos[3*c + 2*d*x] + 4*B*d*x*Cos[3*c + 2*d*x] + (I*A + B)*Cos[c + 2*d*x]*(4*d*x - I*Log[Cos[c + d*x]^2]) + A*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] - I*B*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] + 2*Cos[c]*((-I)*B + (4*I)*A*d*x + 4*B*d*x + (A - I*B)*Log[Cos[c + d*x]^2]) + (2*I)*A*Sin[c] + 4*B*Sin[c] - (2*I)*A*Sin[c + 2*d*x] - 4*B*Sin[c + 2*d*x])))/(4*d*(Cos[d*x] + I*Sin[d*x])^2)","B",1
12,1,201,75,3.1609093,"\int \cot (c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{a^2 (\cos (2 d x)+i \sin (2 d x)) (A+B \tan (c+d x)) \left(\sec (c) \left(\cos (d x) \left((A-2 i B) \log \left(\cos ^2(c+d x)\right)+8 d x (B+i A)+A \log \left(\sin ^2(c+d x)\right)\right)+\cos (2 c+d x) \left((A-2 i B) \log \left(\cos ^2(c+d x)\right)+8 d x (B+i A)+A \log \left(\sin ^2(c+d x)\right)\right)-4 B \sin (d x)\right)-8 i (A-i B) \cos (c+d x) \tan ^{-1}(\tan (3 c+d x))\right)}{4 d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{a^2 (A-2 i B) \log (\cos (c+d x))}{d}+2 a^2 x (B+i A)+\frac{a^2 A \log (\sin (c+d x))}{d}+\frac{i B \left(a^2+i a^2 \tan (c+d x)\right)}{d}",1,"(a^2*((-8*I)*(A - I*B)*ArcTan[Tan[3*c + d*x]]*Cos[c + d*x] + Sec[c]*(Cos[d*x]*(8*(I*A + B)*d*x + (A - (2*I)*B)*Log[Cos[c + d*x]^2] + A*Log[Sin[c + d*x]^2]) + Cos[2*c + d*x]*(8*(I*A + B)*d*x + (A - (2*I)*B)*Log[Cos[c + d*x]^2] + A*Log[Sin[c + d*x]^2]) - 4*B*Sin[d*x]))*(Cos[2*d*x] + I*Sin[2*d*x])*(A + B*Tan[c + d*x]))/(4*d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
13,1,202,79,3.6358867,"\int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{a^2 (\cos (2 d x)+i \sin (2 d x)) (A \cot (c+d x)+B) \left(8 (A-i B) \sin (c+d x) \tan ^{-1}(\tan (3 c+d x))+\csc (c) \left(\cos (2 c+d x) \left((-B-2 i A) \log \left(\sin ^2(c+d x)\right)+8 d x (A-i B)-B \log \left(\cos ^2(c+d x)\right)\right)+\cos (d x) \left((B+2 i A) \log \left(\sin ^2(c+d x)\right)-8 d x (A-i B)+B \log \left(\cos ^2(c+d x)\right)\right)+4 A \sin (d x)\right)\right)}{4 d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{a^2 (B+2 i A) \log (\sin (c+d x))}{d}-2 a^2 x (A-i B)-\frac{A \cot (c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{d}+\frac{a^2 B \log (\cos (c+d x))}{d}",1,"(a^2*(B + A*Cot[c + d*x])*(Cos[2*d*x] + I*Sin[2*d*x])*(Csc[c]*(Cos[2*c + d*x]*(8*(A - I*B)*d*x - B*Log[Cos[c + d*x]^2] + ((-2*I)*A - B)*Log[Sin[c + d*x]^2]) + Cos[d*x]*(-8*(A - I*B)*d*x + B*Log[Cos[c + d*x]^2] + ((2*I)*A + B)*Log[Sin[c + d*x]^2]) + 4*A*Sin[d*x]) + 8*(A - I*B)*ArcTan[Tan[3*c + d*x]]*Sin[c + d*x]))/(4*d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
14,1,302,94,2.8374189,"\int \cot ^3(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{a^2 \csc (c) \csc ^2(c+d x) (\cos (2 d x)+i \sin (2 d x)) \left(8 (B+i A) \sin (c) \sin ^2(c+d x) \tan ^{-1}(\tan (3 c+d x))+2 (B+2 i A) \cos (c)-8 i A d x \sin (c)-4 i A d x \sin (c+2 d x)+4 i A d x \sin (3 c+2 d x)-4 i A \cos (c+2 d x)-2 A \sin (c) \log \left(\sin ^2(c+d x)\right)-A \sin (c+2 d x) \log \left(\sin ^2(c+d x)\right)+A \sin (3 c+2 d x) \log \left(\sin ^2(c+d x)\right)-2 A \sin (c)-8 B d x \sin (c)-4 B d x \sin (c+2 d x)+4 B d x \sin (3 c+2 d x)-2 B \cos (c+2 d x)+2 i B \sin (c) \log \left(\sin ^2(c+d x)\right)+i B \sin (c+2 d x) \log \left(\sin ^2(c+d x)\right)-i B \sin (3 c+2 d x) \log \left(\sin ^2(c+d x)\right)\right)}{4 d (\cos (d x)+i \sin (d x))^2}","-\frac{a^2 (2 B+3 i A) \cot (c+d x)}{2 d}-\frac{2 a^2 (A-i B) \log (\sin (c+d x))}{d}-2 a^2 x (B+i A)-\frac{A \cot ^2(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{2 d}",1,"(a^2*Csc[c]*Csc[c + d*x]^2*(Cos[2*d*x] + I*Sin[2*d*x])*(2*((2*I)*A + B)*Cos[c] - (4*I)*A*Cos[c + 2*d*x] - 2*B*Cos[c + 2*d*x] - 2*A*Sin[c] - (8*I)*A*d*x*Sin[c] - 8*B*d*x*Sin[c] - 2*A*Log[Sin[c + d*x]^2]*Sin[c] + (2*I)*B*Log[Sin[c + d*x]^2]*Sin[c] + 8*(I*A + B)*ArcTan[Tan[3*c + d*x]]*Sin[c]*Sin[c + d*x]^2 - (4*I)*A*d*x*Sin[c + 2*d*x] - 4*B*d*x*Sin[c + 2*d*x] - A*Log[Sin[c + d*x]^2]*Sin[c + 2*d*x] + I*B*Log[Sin[c + d*x]^2]*Sin[c + 2*d*x] + (4*I)*A*d*x*Sin[3*c + 2*d*x] + 4*B*d*x*Sin[3*c + 2*d*x] + A*Log[Sin[c + d*x]^2]*Sin[3*c + 2*d*x] - I*B*Log[Sin[c + d*x]^2]*Sin[3*c + 2*d*x]))/(4*d*(Cos[d*x] + I*Sin[d*x])^2)","B",1
15,1,435,117,3.8725659,"\int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{a^2 \csc (c) \csc ^3(c+d x) (\cos (2 d x)+i \sin (2 d x)) \left(-48 (A-i B) \sin (c) \sin ^3(c+d x) \tan ^{-1}(\tan (3 c+d x))+3 \cos (d x) \left((-3 B-3 i A) \log \left(\sin ^2(c+d x)\right)+4 A (3 d x-i)+2 B (-1-6 i d x)\right)-18 A \sin (2 c+d x)+14 A \sin (2 c+3 d x)+12 i A \cos (2 c+d x)-36 A d x \cos (2 c+d x)-12 A d x \cos (2 c+3 d x)+12 A d x \cos (4 c+3 d x)+9 i A \cos (2 c+d x) \log \left(\sin ^2(c+d x)\right)+3 i A \cos (2 c+3 d x) \log \left(\sin ^2(c+d x)\right)-3 i A \cos (4 c+3 d x) \log \left(\sin ^2(c+d x)\right)-24 A \sin (d x)+12 i B \sin (2 c+d x)-12 i B \sin (2 c+3 d x)+6 B \cos (2 c+d x)+36 i B d x \cos (2 c+d x)+12 i B d x \cos (2 c+3 d x)-12 i B d x \cos (4 c+3 d x)+9 B \cos (2 c+d x) \log \left(\sin ^2(c+d x)\right)+3 B \cos (2 c+3 d x) \log \left(\sin ^2(c+d x)\right)-3 B \cos (4 c+3 d x) \log \left(\sin ^2(c+d x)\right)+24 i B \sin (d x)\right)}{24 d (\cos (d x)+i \sin (d x))^2}","-\frac{a^2 (3 B+4 i A) \cot ^2(c+d x)}{6 d}+\frac{2 a^2 (A-i B) \cot (c+d x)}{d}-\frac{2 a^2 (B+i A) \log (\sin (c+d x))}{d}+2 a^2 x (A-i B)-\frac{A \cot ^3(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{3 d}",1,"(a^2*Csc[c]*Csc[c + d*x]^3*(Cos[2*d*x] + I*Sin[2*d*x])*((12*I)*A*Cos[2*c + d*x] + 6*B*Cos[2*c + d*x] - 36*A*d*x*Cos[2*c + d*x] + (36*I)*B*d*x*Cos[2*c + d*x] - 12*A*d*x*Cos[2*c + 3*d*x] + (12*I)*B*d*x*Cos[2*c + 3*d*x] + 12*A*d*x*Cos[4*c + 3*d*x] - (12*I)*B*d*x*Cos[4*c + 3*d*x] + (9*I)*A*Cos[2*c + d*x]*Log[Sin[c + d*x]^2] + 9*B*Cos[2*c + d*x]*Log[Sin[c + d*x]^2] + (3*I)*A*Cos[2*c + 3*d*x]*Log[Sin[c + d*x]^2] + 3*B*Cos[2*c + 3*d*x]*Log[Sin[c + d*x]^2] - (3*I)*A*Cos[4*c + 3*d*x]*Log[Sin[c + d*x]^2] - 3*B*Cos[4*c + 3*d*x]*Log[Sin[c + d*x]^2] + 3*Cos[d*x]*(2*B*(-1 - (6*I)*d*x) + 4*A*(-I + 3*d*x) + ((-3*I)*A - 3*B)*Log[Sin[c + d*x]^2]) - 24*A*Sin[d*x] + (24*I)*B*Sin[d*x] - 48*(A - I*B)*ArcTan[Tan[3*c + d*x]]*Sin[c]*Sin[c + d*x]^3 - 18*A*Sin[2*c + d*x] + (12*I)*B*Sin[2*c + d*x] + 14*A*Sin[2*c + 3*d*x] - (12*I)*B*Sin[2*c + 3*d*x]))/(24*d*(Cos[d*x] + I*Sin[d*x])^2)","B",1
16,1,902,139,9.1469603,"\int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","a^2 \left(\frac{(\cot (c+d x)+i)^2 (B+A \cot (c+d x)) (A \cos (c)-i B \cos (c)-i A \sin (c)-B \sin (c)) \left(-2 i \tan ^{-1}(\tan (3 c+d x)) \cos (c)-2 \tan ^{-1}(\tan (3 c+d x)) \sin (c)\right) \sin ^3(c+d x)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^2 (B+A \cot (c+d x)) (A \cos (c)-i B \cos (c)-i A \sin (c)-B \sin (c)) \left(\cos (c) \log \left(\sin ^2(c+d x)\right)-i \log \left(\sin ^2(c+d x)\right) \sin (c)\right) \sin ^3(c+d x)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{x (\cot (c+d x)+i)^2 (B+A \cot (c+d x)) \left(6 i A \cos ^2(c)+6 B \cos ^2(c)-2 A \cot (c) \cos ^2(c)+2 i B \cot (c) \cos ^2(c)+6 A \sin (c) \cos (c)-6 i B \sin (c) \cos (c)-2 i A \sin ^2(c)-2 B \sin ^2(c)+(A-i B) \cot (c) (2 \cos (2 c)-2 i \sin (2 c))\right) \sin ^3(c+d x)}{(\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(i A+B) (\cot (c+d x)+i)^2 (B+A \cot (c+d x)) (2 d x \cos (2 c)-2 i d x \sin (2 c)) \sin ^3(c+d x)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^2 (B+A \cot (c+d x)) \csc (c) \left(\frac{1}{3} \cos (2 c)-\frac{1}{3} i \sin (2 c)\right) (-8 i A \sin (d x)-7 B \sin (d x)) \sin ^2(c+d x)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^2 (B+A \cot (c+d x)) \csc (c) (-4 i A \cos (c)-2 B \cos (c)+9 A \sin (c)-6 i B \sin (c)) \left(\frac{1}{6} \cos (2 c)-\frac{1}{6} i \sin (2 c)\right) \sin (c+d x)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^2 (B+A \cot (c+d x)) \csc (c) \left(\frac{1}{3} \cos (2 c)-\frac{1}{3} i \sin (2 c)\right) (2 i A \sin (d x)+B \sin (d x))}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^2 (B+A \cot (c+d x)) \csc (c+d x) \left(\frac{1}{4} i A \sin (2 c)-\frac{1}{4} A \cos (2 c)\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}\right)","-\frac{a^2 (4 B+5 i A) \cot ^3(c+d x)}{12 d}+\frac{a^2 (A-i B) \cot ^2(c+d x)}{d}+\frac{2 a^2 (B+i A) \cot (c+d x)}{d}+\frac{2 a^2 (A-i B) \log (\sin (c+d x))}{d}+2 a^2 x (B+i A)-\frac{A \cot ^4(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{4 d}",1,"a^2*(((I + Cot[c + d*x])^2*(B + A*Cot[c + d*x])*Csc[c + d*x]*(-1/4*(A*Cos[2*c]) + (I/4)*A*Sin[2*c]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^2*(B + A*Cot[c + d*x])*Csc[c]*(Cos[2*c]/3 - (I/3)*Sin[2*c])*((2*I)*A*Sin[d*x] + B*Sin[d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^2*(B + A*Cot[c + d*x])*Csc[c]*((-4*I)*A*Cos[c] - 2*B*Cos[c] + 9*A*Sin[c] - (6*I)*B*Sin[c])*(Cos[2*c]/6 - (I/6)*Sin[2*c])*Sin[c + d*x])/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^2*(B + A*Cot[c + d*x])*Csc[c]*(Cos[2*c]/3 - (I/3)*Sin[2*c])*((-8*I)*A*Sin[d*x] - 7*B*Sin[d*x])*Sin[c + d*x]^2)/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^2*(B + A*Cot[c + d*x])*(A*Cos[c] - I*B*Cos[c] - I*A*Sin[c] - B*Sin[c])*((-2*I)*ArcTan[Tan[3*c + d*x]]*Cos[c] - 2*ArcTan[Tan[3*c + d*x]]*Sin[c])*Sin[c + d*x]^3)/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^2*(B + A*Cot[c + d*x])*(A*Cos[c] - I*B*Cos[c] - I*A*Sin[c] - B*Sin[c])*(Cos[c]*Log[Sin[c + d*x]^2] - I*Log[Sin[c + d*x]^2]*Sin[c])*Sin[c + d*x]^3)/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (x*(I + Cot[c + d*x])^2*(B + A*Cot[c + d*x])*((6*I)*A*Cos[c]^2 + 6*B*Cos[c]^2 - 2*A*Cos[c]^2*Cot[c] + (2*I)*B*Cos[c]^2*Cot[c] + 6*A*Cos[c]*Sin[c] - (6*I)*B*Cos[c]*Sin[c] - (2*I)*A*Sin[c]^2 - 2*B*Sin[c]^2 + (A - I*B)*Cot[c]*(2*Cos[2*c] - (2*I)*Sin[2*c]))*Sin[c + d*x]^3)/((Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I*A + B)*(I + Cot[c + d*x])^2*(B + A*Cot[c + d*x])*(2*d*x*Cos[2*c] - (2*I)*d*x*Sin[2*c])*Sin[c + d*x]^3)/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x])))","B",1
17,1,847,182,8.6015472,"\int \tan ^2(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{x \left(-2 A \cos ^3(c)+2 i B \cos ^3(c)+8 i A \sin (c) \cos ^2(c)+8 B \sin (c) \cos ^2(c)+12 A \sin ^2(c) \cos (c)-12 i B \sin ^2(c) \cos (c)+2 A \cos (c)-2 i B \cos (c)-8 i A \sin ^3(c)-8 B \sin ^3(c)-4 i A \sin (c)-4 B \sin (c)-2 A \sin ^3(c) \tan (c)+2 i B \sin ^3(c) \tan (c)-2 A \sin (c) \tan (c)+2 i B \sin (c) \tan (c)-i (A-i B) (4 \cos (3 c)-4 i \sin (3 c)) \tan (c)\right) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x)) \cos ^4(c+d x)}{(\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\left(i A \cos \left(\frac{3 c}{2}\right)+B \cos \left(\frac{3 c}{2}\right)+A \sin \left(\frac{3 c}{2}\right)-i B \sin \left(\frac{3 c}{2}\right)\right) \left(2 \cos \left(\frac{3 c}{2}\right) \log \left(\cos ^2(c+d x)\right)-2 i \log \left(\cos ^2(c+d x)\right) \sin \left(\frac{3 c}{2}\right)\right) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x)) \cos ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\sec (c) \sec (c+d x) \left(\frac{1}{240} \cos (3 c)-\frac{1}{240} i \sin (3 c)\right) (195 i A \cos (d x)+225 B \cos (d x)-300 A d x \cos (d x)+300 i B d x \cos (d x)+195 i A \cos (2 c+d x)+225 B \cos (2 c+d x)-300 A d x \cos (2 c+d x)+300 i B d x \cos (2 c+d x)+75 i A \cos (2 c+3 d x)+105 B \cos (2 c+3 d x)-150 A d x \cos (2 c+3 d x)+150 i B d x \cos (2 c+3 d x)+75 i A \cos (4 c+3 d x)+105 B \cos (4 c+3 d x)-150 A d x \cos (4 c+3 d x)+150 i B d x \cos (4 c+3 d x)-30 A d x \cos (4 c+5 d x)+30 i B d x \cos (4 c+5 d x)-30 A d x \cos (6 c+5 d x)+30 i B d x \cos (6 c+5 d x)+420 A \sin (d x)-470 i B \sin (d x)-330 A \sin (2 c+d x)+360 i B \sin (2 c+d x)+270 A \sin (2 c+3 d x)-280 i B \sin (2 c+3 d x)-105 A \sin (4 c+3 d x)+135 i B \sin (4 c+3 d x)+75 A \sin (4 c+5 d x)-83 i B \sin (4 c+5 d x)) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{a^3 (45 A-47 i B) \tan ^3(c+d x)}{60 d}-\frac{(5 A-7 i B) \tan ^3(c+d x) \left(a^3+i a^3 \tan (c+d x)\right)}{20 d}+\frac{2 a^3 (B+i A) \tan ^2(c+d x)}{d}+\frac{4 a^3 (A-i B) \tan (c+d x)}{d}+\frac{4 a^3 (B+i A) \log (\cos (c+d x))}{d}-4 a^3 x (A-i B)+\frac{i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^2}{5 d}",1,"(Cos[c + d*x]^4*(I*A*Cos[(3*c)/2] + B*Cos[(3*c)/2] + A*Sin[(3*c)/2] - I*B*Sin[(3*c)/2])*(2*Cos[(3*c)/2]*Log[Cos[c + d*x]^2] - (2*I)*Log[Cos[c + d*x]^2]*Sin[(3*c)/2])*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Sec[c]*Sec[c + d*x]*(Cos[3*c]/240 - (I/240)*Sin[3*c])*((195*I)*A*Cos[d*x] + 225*B*Cos[d*x] - 300*A*d*x*Cos[d*x] + (300*I)*B*d*x*Cos[d*x] + (195*I)*A*Cos[2*c + d*x] + 225*B*Cos[2*c + d*x] - 300*A*d*x*Cos[2*c + d*x] + (300*I)*B*d*x*Cos[2*c + d*x] + (75*I)*A*Cos[2*c + 3*d*x] + 105*B*Cos[2*c + 3*d*x] - 150*A*d*x*Cos[2*c + 3*d*x] + (150*I)*B*d*x*Cos[2*c + 3*d*x] + (75*I)*A*Cos[4*c + 3*d*x] + 105*B*Cos[4*c + 3*d*x] - 150*A*d*x*Cos[4*c + 3*d*x] + (150*I)*B*d*x*Cos[4*c + 3*d*x] - 30*A*d*x*Cos[4*c + 5*d*x] + (30*I)*B*d*x*Cos[4*c + 5*d*x] - 30*A*d*x*Cos[6*c + 5*d*x] + (30*I)*B*d*x*Cos[6*c + 5*d*x] + 420*A*Sin[d*x] - (470*I)*B*Sin[d*x] - 330*A*Sin[2*c + d*x] + (360*I)*B*Sin[2*c + d*x] + 270*A*Sin[2*c + 3*d*x] - (280*I)*B*Sin[2*c + 3*d*x] - 105*A*Sin[4*c + 3*d*x] + (135*I)*B*Sin[4*c + 3*d*x] + 75*A*Sin[4*c + 5*d*x] - (83*I)*B*Sin[4*c + 5*d*x])*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (x*Cos[c + d*x]^4*(2*A*Cos[c] - (2*I)*B*Cos[c] - 2*A*Cos[c]^3 + (2*I)*B*Cos[c]^3 - (4*I)*A*Sin[c] - 4*B*Sin[c] + (8*I)*A*Cos[c]^2*Sin[c] + 8*B*Cos[c]^2*Sin[c] + 12*A*Cos[c]*Sin[c]^2 - (12*I)*B*Cos[c]*Sin[c]^2 - (8*I)*A*Sin[c]^3 - 8*B*Sin[c]^3 - 2*A*Sin[c]*Tan[c] + (2*I)*B*Sin[c]*Tan[c] - 2*A*Sin[c]^3*Tan[c] + (2*I)*B*Sin[c]^3*Tan[c] - I*(A - I*B)*(4*Cos[3*c] - (4*I)*Sin[3*c])*Tan[c])*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/((Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
18,1,980,138,8.0951554,"\int \tan (c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{x \left(-2 i A \cos ^3(c)-2 B \cos ^3(c)-8 A \sin (c) \cos ^2(c)+8 i B \sin (c) \cos ^2(c)+12 i A \sin ^2(c) \cos (c)+12 B \sin ^2(c) \cos (c)+2 i A \cos (c)+2 B \cos (c)+8 A \sin ^3(c)-8 i B \sin ^3(c)+4 A \sin (c)-4 i B \sin (c)-2 i A \sin ^3(c) \tan (c)-2 B \sin ^3(c) \tan (c)-2 i A \sin (c) \tan (c)-2 B \sin (c) \tan (c)+(A-i B) (4 \cos (3 c)-4 i \sin (3 c)) \tan (c)\right) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x)) \cos ^4(c+d x)}{(\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\left(A \cos \left(\frac{3 c}{2}\right)-i B \cos \left(\frac{3 c}{2}\right)-i A \sin \left(\frac{3 c}{2}\right)-B \sin \left(\frac{3 c}{2}\right)\right) \left(2 i \log \left(\cos ^2(c+d x)\right) \sin \left(\frac{3 c}{2}\right)-2 \cos \left(\frac{3 c}{2}\right) \log \left(\cos ^2(c+d x)\right)\right) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x)) \cos ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(A-i B) (-4 i d x \cos (3 c)-4 d x \sin (3 c)) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x)) \cos ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\left(\frac{1}{3} \cos (3 c)-\frac{1}{3} i \sin (3 c)\right) (13 i A \sin (d x)+15 B \sin (d x)) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x)) \cos ^3(c+d x)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(-9 A \cos (c)+15 i B \cos (c)-2 i A \sin (c)-6 B \sin (c)) \left(\frac{1}{6} \cos (3 c)-\frac{1}{6} i \sin (3 c)\right) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x)) \cos ^2(c+d x)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\left(\frac{1}{3} \cos (3 c)-\frac{1}{3} i \sin (3 c)\right) (-i A \sin (d x)-3 B \sin (d x)) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x)) \cos (c+d x)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\left(-\frac{1}{4} i B \cos (3 c)-\frac{1}{4} B \sin (3 c)\right) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","\frac{2 a^3 (B+i A) \tan (c+d x)}{d}-\frac{4 a^3 (A-i B) \log (\cos (c+d x))}{d}-4 a^3 x (B+i A)+\frac{a (A-i B) (a+i a \tan (c+d x))^2}{2 d}+\frac{A (a+i a \tan (c+d x))^3}{3 d}-\frac{i B (a+i a \tan (c+d x))^4}{4 a d}",1,"(Cos[c + d*x]^4*(A*Cos[(3*c)/2] - I*B*Cos[(3*c)/2] - I*A*Sin[(3*c)/2] - B*Sin[(3*c)/2])*(-2*Cos[(3*c)/2]*Log[Cos[c + d*x]^2] + (2*I)*Log[Cos[c + d*x]^2]*Sin[(3*c)/2])*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^2*(-9*A*Cos[c] + (15*I)*B*Cos[c] - (2*I)*A*Sin[c] - 6*B*Sin[c])*(Cos[3*c]/6 - (I/6)*Sin[3*c])*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (((-1/4*I)*B*Cos[3*c] - (B*Sin[3*c])/4)*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((A - I*B)*Cos[c + d*x]^4*((-4*I)*d*x*Cos[3*c] - 4*d*x*Sin[3*c])*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]*(Cos[3*c]/3 - (I/3)*Sin[3*c])*((-I)*A*Sin[d*x] - 3*B*Sin[d*x])*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^3*(Cos[3*c]/3 - (I/3)*Sin[3*c])*((13*I)*A*Sin[d*x] + 15*B*Sin[d*x])*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (x*Cos[c + d*x]^4*((2*I)*A*Cos[c] + 2*B*Cos[c] - (2*I)*A*Cos[c]^3 - 2*B*Cos[c]^3 + 4*A*Sin[c] - (4*I)*B*Sin[c] - 8*A*Cos[c]^2*Sin[c] + (8*I)*B*Cos[c]^2*Sin[c] + (12*I)*A*Cos[c]*Sin[c]^2 + 12*B*Cos[c]*Sin[c]^2 + 8*A*Sin[c]^3 - (8*I)*B*Sin[c]^3 - (2*I)*A*Sin[c]*Tan[c] - 2*B*Sin[c]*Tan[c] - (2*I)*A*Sin[c]^3*Tan[c] - 2*B*Sin[c]^3*Tan[c] + (A - I*B)*(4*Cos[3*c] - (4*I)*Sin[3*c])*Tan[c])*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/((Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
19,1,331,110,4.3512225,"\int (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{a^3 \sec (c) \sec ^3(c+d x) \left(3 \cos (d x) \left((-3 B-3 i A) \log \left(\cos ^2(c+d x)\right)+6 A d x-i A-6 i B d x-3 B\right)+3 \cos (2 c+d x) \left((-3 B-3 i A) \log \left(\cos ^2(c+d x)\right)+6 A d x-i A-6 i B d x-3 B\right)+9 A \sin (2 c+d x)-9 A \sin (2 c+3 d x)+6 A d x \cos (2 c+3 d x)+6 A d x \cos (4 c+3 d x)-3 i A \cos (2 c+3 d x) \log \left(\cos ^2(c+d x)\right)-3 i A \cos (4 c+3 d x) \log \left(\cos ^2(c+d x)\right)-18 A \sin (d x)-15 i B \sin (2 c+d x)+13 i B \sin (2 c+3 d x)-6 i B d x \cos (2 c+3 d x)-6 i B d x \cos (4 c+3 d x)-3 B \cos (2 c+3 d x) \log \left(\cos ^2(c+d x)\right)-3 B \cos (4 c+3 d x) \log \left(\cos ^2(c+d x)\right)+24 i B \sin (d x)\right)}{12 d}","-\frac{2 a^3 (A-i B) \tan (c+d x)}{d}-\frac{4 a^3 (B+i A) \log (\cos (c+d x))}{d}+4 a^3 x (A-i B)+\frac{a (B+i A) (a+i a \tan (c+d x))^2}{2 d}+\frac{B (a+i a \tan (c+d x))^3}{3 d}",1,"(a^3*Sec[c]*Sec[c + d*x]^3*(6*A*d*x*Cos[2*c + 3*d*x] - (6*I)*B*d*x*Cos[2*c + 3*d*x] + 6*A*d*x*Cos[4*c + 3*d*x] - (6*I)*B*d*x*Cos[4*c + 3*d*x] - (3*I)*A*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]^2] - 3*B*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]^2] - (3*I)*A*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] - 3*B*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] + 3*Cos[d*x]*((-I)*A - 3*B + 6*A*d*x - (6*I)*B*d*x + ((-3*I)*A - 3*B)*Log[Cos[c + d*x]^2]) + 3*Cos[2*c + d*x]*((-I)*A - 3*B + 6*A*d*x - (6*I)*B*d*x + ((-3*I)*A - 3*B)*Log[Cos[c + d*x]^2]) - 18*A*Sin[d*x] + (24*I)*B*Sin[d*x] + 9*A*Sin[2*c + d*x] - (15*I)*B*Sin[2*c + d*x] - 9*A*Sin[2*c + 3*d*x] + (13*I)*B*Sin[2*c + 3*d*x]))/(12*d)","B",1
20,1,281,107,8.4802356,"\int \cot (c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{a^3 \sec (c) \sec ^2(c+d x) (\cos (3 d x)+i \sin (3 d x)) \left(2 \cos (c) \left((3 A-4 i B) \log \left(\cos ^2(c+d x)\right)+A \log \left(\sin ^2(c+d x)\right)+8 i A d x+8 B d x-2 i B\right)+\cos (c+2 d x) \left((3 A-4 i B) \log \left(\cos ^2(c+d x)\right)+8 d x (B+i A)+A \log \left(\sin ^2(c+d x)\right)\right)-4 i A \sin (c+2 d x)+8 i A d x \cos (3 c+2 d x)+3 A \cos (3 c+2 d x) \log \left(\cos ^2(c+d x)\right)+A \cos (3 c+2 d x) \log \left(\sin ^2(c+d x)\right)+4 i A \sin (c)-12 B \sin (c+2 d x)+8 B d x \cos (3 c+2 d x)-4 i B \cos (3 c+2 d x) \log \left(\cos ^2(c+d x)\right)+12 B \sin (c)\right)}{8 d (\cos (d x)+i \sin (d x))^3}","-\frac{(A-2 i B) \left(a^3+i a^3 \tan (c+d x)\right)}{d}+\frac{a^3 (3 A-4 i B) \log (\cos (c+d x))}{d}+4 a^3 x (B+i A)+\frac{a^3 A \log (\sin (c+d x))}{d}+\frac{i a B (a+i a \tan (c+d x))^2}{2 d}",1,"(a^3*Sec[c]*Sec[c + d*x]^2*(Cos[3*d*x] + I*Sin[3*d*x])*((8*I)*A*d*x*Cos[3*c + 2*d*x] + 8*B*d*x*Cos[3*c + 2*d*x] + 3*A*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] - (4*I)*B*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] + A*Cos[3*c + 2*d*x]*Log[Sin[c + d*x]^2] + 2*Cos[c]*((-2*I)*B + (8*I)*A*d*x + 8*B*d*x + (3*A - (4*I)*B)*Log[Cos[c + d*x]^2] + A*Log[Sin[c + d*x]^2]) + Cos[c + 2*d*x]*(8*(I*A + B)*d*x + (3*A - (4*I)*B)*Log[Cos[c + d*x]^2] + A*Log[Sin[c + d*x]^2]) + (4*I)*A*Sin[c] + 12*B*Sin[c] - (4*I)*A*Sin[c + 2*d*x] - 12*B*Sin[c + 2*d*x]))/(8*d*(Cos[d*x] + I*Sin[d*x])^3)","B",1
21,1,291,116,5.269966,"\int \cot ^2(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{a^3 \csc (c) \sec (c) \csc (c+d x) \sec (c+d x) \left(4 (3 A-i B) \sin (2 c) \sin (2 (c+d x)) \tan ^{-1}(\tan (4 c+d x))+\cos (2 d x) \left((B+3 i A) \log \left(\sin ^2(c+d x)\right)+(3 B+i A) \log \left(\cos ^2(c+d x)\right)+2 d x (-7 A+5 i B)\right)+4 A \sin (2 (c+d x))+14 A d x \cos (4 c+2 d x)-i A \cos (4 c+2 d x) \log \left(\cos ^2(c+d x)\right)-3 i A \cos (4 c+2 d x) \log \left(\sin ^2(c+d x)\right)-4 A \sin (2 c)+4 A \sin (2 d x)+4 i B \sin (2 (c+d x))-10 i B d x \cos (4 c+2 d x)-3 B \cos (4 c+2 d x) \log \left(\cos ^2(c+d x)\right)-B \cos (4 c+2 d x) \log \left(\sin ^2(c+d x)\right)-4 i B \sin (2 c)-4 i B \sin (2 d x)\right)}{16 d}","\frac{(-B+i A) \left(a^3+i a^3 \tan (c+d x)\right)}{d}+\frac{a^3 (B+3 i A) \log (\sin (c+d x))}{d}+\frac{a^3 (3 B+i A) \log (\cos (c+d x))}{d}-4 a^3 x (A-i B)-\frac{a A \cot (c+d x) (a+i a \tan (c+d x))^2}{d}",1,"(a^3*Csc[c]*Csc[c + d*x]*Sec[c]*Sec[c + d*x]*(14*A*d*x*Cos[4*c + 2*d*x] - (10*I)*B*d*x*Cos[4*c + 2*d*x] - I*A*Cos[4*c + 2*d*x]*Log[Cos[c + d*x]^2] - 3*B*Cos[4*c + 2*d*x]*Log[Cos[c + d*x]^2] - (3*I)*A*Cos[4*c + 2*d*x]*Log[Sin[c + d*x]^2] - B*Cos[4*c + 2*d*x]*Log[Sin[c + d*x]^2] + Cos[2*d*x]*(2*(-7*A + (5*I)*B)*d*x + (I*A + 3*B)*Log[Cos[c + d*x]^2] + ((3*I)*A + B)*Log[Sin[c + d*x]^2]) - 4*A*Sin[2*c] - (4*I)*B*Sin[2*c] + 4*A*Sin[2*d*x] - (4*I)*B*Sin[2*d*x] + 4*A*Sin[2*(c + d*x)] + (4*I)*B*Sin[2*(c + d*x)] + 4*(3*A - I*B)*ArcTan[Tan[4*c + d*x]]*Sin[2*c]*Sin[2*(c + d*x)]))/(16*d)","B",1
22,1,1010,123,9.4503077,"\int \cot ^3(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","a^3 \left(\frac{x (\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left(-16 i A \cos ^3(c)-\frac{25}{2} B \cos ^3(c)+4 A \cot (c) \cos ^3(c)-3 i B \cot (c) \cos ^3(c)-24 A \sin (c) \cos ^2(c)+20 i B \sin (c) \cos ^2(c)+16 i A \sin ^2(c) \cos (c)+15 B \sin ^2(c) \cos (c)+\frac{1}{2} B \cos (c)+4 A \sin ^3(c)-5 i B \sin ^3(c)-i B \sin (c)+(2 \cos (2 c) A+2 A-i B-2 i B \cos (2 c)) \csc (c) \sec (c) (i \sin (3 c)-\cos (3 c))-\frac{1}{2} B \sin ^3(c) \tan (c)-\frac{1}{2} B \sin (c) \tan (c)\right) \sin ^4(c+d x)}{(\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{i B \cos (3 c) (\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \log \left(\cos ^2(c+d x)\right) \sin ^4(c+d x)}{2 d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left(4 A \cos \left(\frac{3 c}{2}\right)-3 i B \cos \left(\frac{3 c}{2}\right)-4 i A \sin \left(\frac{3 c}{2}\right)-3 B \sin \left(\frac{3 c}{2}\right)\right) \left(i \tan ^{-1}(\tan (4 c+d x)) \cos \left(\frac{3 c}{2}\right)+\tan ^{-1}(\tan (4 c+d x)) \sin \left(\frac{3 c}{2}\right)\right) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left(4 A \cos \left(\frac{3 c}{2}\right)-3 i B \cos \left(\frac{3 c}{2}\right)-4 i A \sin \left(\frac{3 c}{2}\right)-3 B \sin \left(\frac{3 c}{2}\right)\right) \left(\frac{1}{2} i \log \left(\sin ^2(c+d x)\right) \sin \left(\frac{3 c}{2}\right)-\frac{1}{2} \cos \left(\frac{3 c}{2}\right) \log \left(\sin ^2(c+d x)\right)\right) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{B (\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \log \left(\cos ^2(c+d x)\right) \sin (3 c) \sin ^4(c+d x)}{2 d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(A-i B) (\cot (c+d x)+i)^3 (B+A \cot (c+d x)) (-4 i d x \cos (3 c)-4 d x \sin (3 c)) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{1}{2} \cos (3 c)-\frac{1}{2} i \sin (3 c)\right) (3 i A \sin (d x)+B \sin (d x)) \sin ^3(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left(\frac{1}{2} i A \sin (3 c)-\frac{1}{2} A \cos (3 c)\right) \sin ^2(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}\right)","-\frac{a^3 (4 A-3 i B) \log (\sin (c+d x))}{d}-\frac{(B+2 i A) \cot (c+d x) \left(a^3+i a^3 \tan (c+d x)\right)}{d}-4 a^3 x (B+i A)+\frac{i a^3 B \log (\cos (c+d x))}{d}-\frac{a A \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}",1,"a^3*(((I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*(-1/2*(A*Cos[3*c]) + (I/2)*A*Sin[3*c])*Sin[c + d*x]^2)/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*Csc[c/2]*Sec[c/2]*(Cos[3*c]/2 - (I/2)*Sin[3*c])*((3*I)*A*Sin[d*x] + B*Sin[d*x])*Sin[c + d*x]^3)/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I/2)*B*Cos[3*c]*(I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*Log[Cos[c + d*x]^2]*Sin[c + d*x]^4)/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*(4*A*Cos[(3*c)/2] - (3*I)*B*Cos[(3*c)/2] - (4*I)*A*Sin[(3*c)/2] - 3*B*Sin[(3*c)/2])*(I*ArcTan[Tan[4*c + d*x]]*Cos[(3*c)/2] + ArcTan[Tan[4*c + d*x]]*Sin[(3*c)/2])*Sin[c + d*x]^4)/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*(4*A*Cos[(3*c)/2] - (3*I)*B*Cos[(3*c)/2] - (4*I)*A*Sin[(3*c)/2] - 3*B*Sin[(3*c)/2])*(-1/2*(Cos[(3*c)/2]*Log[Sin[c + d*x]^2]) + (I/2)*Log[Sin[c + d*x]^2]*Sin[(3*c)/2])*Sin[c + d*x]^4)/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (B*(I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*Log[Cos[c + d*x]^2]*Sin[3*c]*Sin[c + d*x]^4)/(2*d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((A - I*B)*(I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*((-4*I)*d*x*Cos[3*c] - 4*d*x*Sin[3*c])*Sin[c + d*x]^4)/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (x*(I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*Sin[c + d*x]^4*((B*Cos[c])/2 - (16*I)*A*Cos[c]^3 - (25*B*Cos[c]^3)/2 + 4*A*Cos[c]^3*Cot[c] - (3*I)*B*Cos[c]^3*Cot[c] - I*B*Sin[c] - 24*A*Cos[c]^2*Sin[c] + (20*I)*B*Cos[c]^2*Sin[c] + (16*I)*A*Cos[c]*Sin[c]^2 + 15*B*Cos[c]*Sin[c]^2 + 4*A*Sin[c]^3 - (5*I)*B*Sin[c]^3 + (2*A - I*B + 2*A*Cos[2*c] - (2*I)*B*Cos[2*c])*Csc[c]*Sec[c]*(-Cos[3*c] + I*Sin[3*c]) - (B*Sin[c]*Tan[c])/2 - (B*Sin[c]^3*Tan[c])/2))/((Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])))","B",1
23,1,442,134,5.3851237,"\int \cot ^4(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{a^3 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc ^3(c+d x) (\cos (3 d x)+i \sin (3 d x)) \left(-48 (A-i B) \sin (c) \sin ^3(c+d x) \tan ^{-1}(\tan (4 c+d x))+\cos (d x) \left((-9 B-9 i A) \log \left(\sin ^2(c+d x)\right)+36 A d x-9 i A-36 i B d x-3 B\right)-15 A \sin (2 c+d x)+13 A \sin (2 c+3 d x)+9 i A \cos (2 c+d x)-36 A d x \cos (2 c+d x)-12 A d x \cos (2 c+3 d x)+12 A d x \cos (4 c+3 d x)+9 i A \cos (2 c+d x) \log \left(\sin ^2(c+d x)\right)+3 i A \cos (2 c+3 d x) \log \left(\sin ^2(c+d x)\right)-3 i A \cos (4 c+3 d x) \log \left(\sin ^2(c+d x)\right)-24 A \sin (d x)+9 i B \sin (2 c+d x)-9 i B \sin (2 c+3 d x)+3 B \cos (2 c+d x)+36 i B d x \cos (2 c+d x)+12 i B d x \cos (2 c+3 d x)-12 i B d x \cos (4 c+3 d x)+9 B \cos (2 c+d x) \log \left(\sin ^2(c+d x)\right)+3 B \cos (2 c+3 d x) \log \left(\sin ^2(c+d x)\right)-3 B \cos (4 c+3 d x) \log \left(\sin ^2(c+d x)\right)+18 i B \sin (d x)\right)}{24 d (\cos (d x)+i \sin (d x))^3}","\frac{a^3 (17 A-15 i B) \cot (c+d x)}{6 d}-\frac{4 a^3 (B+i A) \log (\sin (c+d x))}{d}-\frac{(3 B+5 i A) \cot ^2(c+d x) \left(a^3+i a^3 \tan (c+d x)\right)}{6 d}+4 a^3 x (A-i B)-\frac{a A \cot ^3(c+d x) (a+i a \tan (c+d x))^2}{3 d}",1,"(a^3*Csc[c/2]*Csc[c + d*x]^3*Sec[c/2]*(Cos[3*d*x] + I*Sin[3*d*x])*((9*I)*A*Cos[2*c + d*x] + 3*B*Cos[2*c + d*x] - 36*A*d*x*Cos[2*c + d*x] + (36*I)*B*d*x*Cos[2*c + d*x] - 12*A*d*x*Cos[2*c + 3*d*x] + (12*I)*B*d*x*Cos[2*c + 3*d*x] + 12*A*d*x*Cos[4*c + 3*d*x] - (12*I)*B*d*x*Cos[4*c + 3*d*x] + (9*I)*A*Cos[2*c + d*x]*Log[Sin[c + d*x]^2] + 9*B*Cos[2*c + d*x]*Log[Sin[c + d*x]^2] + (3*I)*A*Cos[2*c + 3*d*x]*Log[Sin[c + d*x]^2] + 3*B*Cos[2*c + 3*d*x]*Log[Sin[c + d*x]^2] - (3*I)*A*Cos[4*c + 3*d*x]*Log[Sin[c + d*x]^2] - 3*B*Cos[4*c + 3*d*x]*Log[Sin[c + d*x]^2] + Cos[d*x]*((-9*I)*A - 3*B + 36*A*d*x - (36*I)*B*d*x + ((-9*I)*A - 9*B)*Log[Sin[c + d*x]^2]) - 24*A*Sin[d*x] + (18*I)*B*Sin[d*x] - 48*(A - I*B)*ArcTan[Tan[4*c + d*x]]*Sin[c]*Sin[c + d*x]^3 - 15*A*Sin[2*c + d*x] + (9*I)*B*Sin[2*c + d*x] + 13*A*Sin[2*c + 3*d*x] - (9*I)*B*Sin[2*c + 3*d*x]))/(24*d*(Cos[d*x] + I*Sin[d*x])^3)","B",1
24,1,1007,157,9.1183058,"\int \cot ^5(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","a^3 \left(\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left(A \cos \left(\frac{3 c}{2}\right)-i B \cos \left(\frac{3 c}{2}\right)-i A \sin \left(\frac{3 c}{2}\right)-B \sin \left(\frac{3 c}{2}\right)\right) \left(-4 i \tan ^{-1}(\tan (4 c+d x)) \cos \left(\frac{3 c}{2}\right)-4 \tan ^{-1}(\tan (4 c+d x)) \sin \left(\frac{3 c}{2}\right)\right) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left(A \cos \left(\frac{3 c}{2}\right)-i B \cos \left(\frac{3 c}{2}\right)-i A \sin \left(\frac{3 c}{2}\right)-B \sin \left(\frac{3 c}{2}\right)\right) \left(2 \cos \left(\frac{3 c}{2}\right) \log \left(\sin ^2(c+d x)\right)-2 i \log \left(\sin ^2(c+d x)\right) \sin \left(\frac{3 c}{2}\right)\right) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{x (\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left(16 i A \cos ^3(c)+16 B \cos ^3(c)-4 A \cot (c) \cos ^3(c)+4 i B \cot (c) \cos ^3(c)+24 A \sin (c) \cos ^2(c)-24 i B \sin (c) \cos ^2(c)-16 i A \sin ^2(c) \cos (c)-16 B \sin ^2(c) \cos (c)-4 A \sin ^3(c)+4 i B \sin ^3(c)+(A-i B) \cot (c) (4 \cos (3 c)-4 i \sin (3 c))\right) \sin ^4(c+d x)}{(\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(i A+B) (\cot (c+d x)+i)^3 (B+A \cot (c+d x)) (4 d x \cos (3 c)-4 i d x \sin (3 c)) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{1}{6} \cos (3 c)-\frac{1}{6} i \sin (3 c)\right) (-15 i A \sin (d x)-13 B \sin (d x)) \sin ^3(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (-6 i A \cos (c)-2 B \cos (c)+15 A \sin (c)-9 i B \sin (c)) \left(\frac{1}{12} \cos (3 c)-\frac{1}{12} i \sin (3 c)\right) \sin ^2(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{1}{6} \cos (3 c)-\frac{1}{6} i \sin (3 c)\right) (3 i A \sin (d x)+B \sin (d x)) \sin (c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left(\frac{1}{4} i A \sin (3 c)-\frac{1}{4} A \cos (3 c)\right)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}\right)","\frac{a^3 (15 A-14 i B) \cot ^2(c+d x)}{12 d}+\frac{4 a^3 (B+i A) \cot (c+d x)}{d}+\frac{4 a^3 (A-i B) \log (\sin (c+d x))}{d}-\frac{(2 B+3 i A) \cot ^3(c+d x) \left(a^3+i a^3 \tan (c+d x)\right)}{6 d}+4 a^3 x (B+i A)-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^2}{4 d}",1,"a^3*(((I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*(-1/4*(A*Cos[3*c]) + (I/4)*A*Sin[3*c]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*Csc[c/2]*Sec[c/2]*(Cos[3*c]/6 - (I/6)*Sin[3*c])*((3*I)*A*Sin[d*x] + B*Sin[d*x])*Sin[c + d*x])/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*Csc[c/2]*Sec[c/2]*((-6*I)*A*Cos[c] - 2*B*Cos[c] + 15*A*Sin[c] - (9*I)*B*Sin[c])*(Cos[3*c]/12 - (I/12)*Sin[3*c])*Sin[c + d*x]^2)/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*Csc[c/2]*Sec[c/2]*(Cos[3*c]/6 - (I/6)*Sin[3*c])*((-15*I)*A*Sin[d*x] - 13*B*Sin[d*x])*Sin[c + d*x]^3)/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*(A*Cos[(3*c)/2] - I*B*Cos[(3*c)/2] - I*A*Sin[(3*c)/2] - B*Sin[(3*c)/2])*((-4*I)*ArcTan[Tan[4*c + d*x]]*Cos[(3*c)/2] - 4*ArcTan[Tan[4*c + d*x]]*Sin[(3*c)/2])*Sin[c + d*x]^4)/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*(A*Cos[(3*c)/2] - I*B*Cos[(3*c)/2] - I*A*Sin[(3*c)/2] - B*Sin[(3*c)/2])*(2*Cos[(3*c)/2]*Log[Sin[c + d*x]^2] - (2*I)*Log[Sin[c + d*x]^2]*Sin[(3*c)/2])*Sin[c + d*x]^4)/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (x*(I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*((16*I)*A*Cos[c]^3 + 16*B*Cos[c]^3 - 4*A*Cos[c]^3*Cot[c] + (4*I)*B*Cos[c]^3*Cot[c] + 24*A*Cos[c]^2*Sin[c] - (24*I)*B*Cos[c]^2*Sin[c] - (16*I)*A*Cos[c]*Sin[c]^2 - 16*B*Cos[c]*Sin[c]^2 - 4*A*Sin[c]^3 + (4*I)*B*Sin[c]^3 + (A - I*B)*Cot[c]*(4*Cos[3*c] - (4*I)*Sin[3*c]))*Sin[c + d*x]^4)/((Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I*A + B)*(I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*(4*d*x*Cos[3*c] - (4*I)*d*x*Sin[3*c])*Sin[c + d*x]^4)/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])))","B",1
25,1,943,180,9.5104599,"\int \cot ^6(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","a^3 \left(\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left(i A \cos \left(\frac{3 c}{2}\right)+B \cos \left(\frac{3 c}{2}\right)+A \sin \left(\frac{3 c}{2}\right)-i B \sin \left(\frac{3 c}{2}\right)\right) \left(-4 i \tan ^{-1}(\tan (4 c+d x)) \cos \left(\frac{3 c}{2}\right)-4 \tan ^{-1}(\tan (4 c+d x)) \sin \left(\frac{3 c}{2}\right)\right) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left(i A \cos \left(\frac{3 c}{2}\right)+B \cos \left(\frac{3 c}{2}\right)+A \sin \left(\frac{3 c}{2}\right)-i B \sin \left(\frac{3 c}{2}\right)\right) \left(2 \cos \left(\frac{3 c}{2}\right) \log \left(\sin ^2(c+d x)\right)-2 i \log \left(\sin ^2(c+d x)\right) \sin \left(\frac{3 c}{2}\right)\right) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{x (\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left(-16 A \cos ^3(c)+16 i B \cos ^3(c)-4 i A \cot (c) \cos ^3(c)-4 B \cot (c) \cos ^3(c)+24 i A \sin (c) \cos ^2(c)+24 B \sin (c) \cos ^2(c)+16 A \sin ^2(c) \cos (c)-16 i B \sin ^2(c) \cos (c)-4 i A \sin ^3(c)-4 B \sin ^3(c)+(i A+B) \cot (c) (4 \cos (3 c)-4 i \sin (3 c))\right) \sin ^4(c+d x)}{(\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \csc (c) \csc (c+d x) \left(\frac{1}{240} \cos (3 c)-\frac{1}{240} i \sin (3 c)\right) (225 i A \cos (d x)+195 B \cos (d x)-300 A d x \cos (d x)+300 i B d x \cos (d x)-225 i A \cos (2 c+d x)-195 B \cos (2 c+d x)+300 A d x \cos (2 c+d x)-300 i B d x \cos (2 c+d x)-105 i A \cos (2 c+3 d x)-75 B \cos (2 c+3 d x)+150 A d x \cos (2 c+3 d x)-150 i B d x \cos (2 c+3 d x)+105 i A \cos (4 c+3 d x)+75 B \cos (4 c+3 d x)-150 A d x \cos (4 c+3 d x)+150 i B d x \cos (4 c+3 d x)-30 A d x \cos (4 c+5 d x)+30 i B d x \cos (4 c+5 d x)+30 A d x \cos (6 c+5 d x)-30 i B d x \cos (6 c+5 d x)+470 A \sin (d x)-420 i B \sin (d x)+360 A \sin (2 c+d x)-330 i B \sin (2 c+d x)-280 A \sin (2 c+3 d x)+270 i B \sin (2 c+3 d x)-135 A \sin (4 c+3 d x)+105 i B \sin (4 c+3 d x)+83 A \sin (4 c+5 d x)-75 i B \sin (4 c+5 d x))}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}\right)","\frac{a^3 (47 A-45 i B) \cot ^3(c+d x)}{60 d}+\frac{2 a^3 (B+i A) \cot ^2(c+d x)}{d}-\frac{4 a^3 (A-i B) \cot (c+d x)}{d}+\frac{4 a^3 (B+i A) \log (\sin (c+d x))}{d}-\frac{(5 B+7 i A) \cot ^4(c+d x) \left(a^3+i a^3 \tan (c+d x)\right)}{20 d}-4 a^3 x (A-i B)-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d}",1,"a^3*(((I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*(I*A*Cos[(3*c)/2] + B*Cos[(3*c)/2] + A*Sin[(3*c)/2] - I*B*Sin[(3*c)/2])*((-4*I)*ArcTan[Tan[4*c + d*x]]*Cos[(3*c)/2] - 4*ArcTan[Tan[4*c + d*x]]*Sin[(3*c)/2])*Sin[c + d*x]^4)/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*(I*A*Cos[(3*c)/2] + B*Cos[(3*c)/2] + A*Sin[(3*c)/2] - I*B*Sin[(3*c)/2])*(2*Cos[(3*c)/2]*Log[Sin[c + d*x]^2] - (2*I)*Log[Sin[c + d*x]^2]*Sin[(3*c)/2])*Sin[c + d*x]^4)/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (x*(I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*(-16*A*Cos[c]^3 + (16*I)*B*Cos[c]^3 - (4*I)*A*Cos[c]^3*Cot[c] - 4*B*Cos[c]^3*Cot[c] + (24*I)*A*Cos[c]^2*Sin[c] + 24*B*Cos[c]^2*Sin[c] + 16*A*Cos[c]*Sin[c]^2 - (16*I)*B*Cos[c]*Sin[c]^2 - (4*I)*A*Sin[c]^3 - 4*B*Sin[c]^3 + (I*A + B)*Cot[c]*(4*Cos[3*c] - (4*I)*Sin[3*c]))*Sin[c + d*x]^4)/((Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*Csc[c]*Csc[c + d*x]*(Cos[3*c]/240 - (I/240)*Sin[3*c])*((225*I)*A*Cos[d*x] + 195*B*Cos[d*x] - 300*A*d*x*Cos[d*x] + (300*I)*B*d*x*Cos[d*x] - (225*I)*A*Cos[2*c + d*x] - 195*B*Cos[2*c + d*x] + 300*A*d*x*Cos[2*c + d*x] - (300*I)*B*d*x*Cos[2*c + d*x] - (105*I)*A*Cos[2*c + 3*d*x] - 75*B*Cos[2*c + 3*d*x] + 150*A*d*x*Cos[2*c + 3*d*x] - (150*I)*B*d*x*Cos[2*c + 3*d*x] + (105*I)*A*Cos[4*c + 3*d*x] + 75*B*Cos[4*c + 3*d*x] - 150*A*d*x*Cos[4*c + 3*d*x] + (150*I)*B*d*x*Cos[4*c + 3*d*x] - 30*A*d*x*Cos[4*c + 5*d*x] + (30*I)*B*d*x*Cos[4*c + 5*d*x] + 30*A*d*x*Cos[6*c + 5*d*x] - (30*I)*B*d*x*Cos[6*c + 5*d*x] + 470*A*Sin[d*x] - (420*I)*B*Sin[d*x] + 360*A*Sin[2*c + d*x] - (330*I)*B*Sin[2*c + d*x] - 280*A*Sin[2*c + 3*d*x] + (270*I)*B*Sin[2*c + 3*d*x] - 135*A*Sin[4*c + 3*d*x] + (105*I)*B*Sin[4*c + 3*d*x] + 83*A*Sin[4*c + 5*d*x] - (75*I)*B*Sin[4*c + 5*d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])))","B",1
26,1,951,225,9.1559858,"\int \tan ^2(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{x \left(-4 A \cos ^4(c)+4 i B \cos ^4(c)+20 i A \sin (c) \cos ^3(c)+20 B \sin (c) \cos ^3(c)+40 A \sin ^2(c) \cos ^2(c)-40 i B \sin ^2(c) \cos ^2(c)+4 A \cos ^2(c)-4 i B \cos ^2(c)-40 i A \sin ^3(c) \cos (c)-40 B \sin ^3(c) \cos (c)-12 i A \sin (c) \cos (c)-12 B \sin (c) \cos (c)-20 A \sin ^4(c)+20 i B \sin ^4(c)-12 A \sin ^2(c)+12 i B \sin ^2(c)+4 i A \sin ^4(c) \tan (c)+4 B \sin ^4(c) \tan (c)+4 i A \sin ^2(c) \tan (c)+4 B \sin ^2(c) \tan (c)-i (A-i B) (8 \cos (4 c)-8 i \sin (4 c)) \tan (c)\right) (i \tan (c+d x) a+a)^4 (A+B \tan (c+d x)) \cos ^5(c+d x)}{(\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(i A \cos (2 c)+B \cos (2 c)+A \sin (2 c)-i B \sin (2 c)) \left(4 \cos (2 c) \log \left(\cos ^2(c+d x)\right)-4 i \log \left(\cos ^2(c+d x)\right) \sin (2 c)\right) (i \tan (c+d x) a+a)^4 (A+B \tan (c+d x)) \cos ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\sec (c) \sec (c+d x) \left(\frac{1}{240} \cos (4 c)-\frac{1}{240} i \sin (4 c)\right) (420 i A \cos (c)+490 B \cos (c)-600 A d x \cos (c)+600 i B d x \cos (c)+300 i A \cos (c+2 d x)+345 B \cos (c+2 d x)-450 A d x \cos (c+2 d x)+450 i B d x \cos (c+2 d x)+300 i A \cos (3 c+2 d x)+345 B \cos (3 c+2 d x)-450 A d x \cos (3 c+2 d x)+450 i B d x \cos (3 c+2 d x)+90 i A \cos (3 c+4 d x)+120 B \cos (3 c+4 d x)-180 A d x \cos (3 c+4 d x)+180 i B d x \cos (3 c+4 d x)+90 i A \cos (5 c+4 d x)+120 B \cos (5 c+4 d x)-180 A d x \cos (5 c+4 d x)+180 i B d x \cos (5 c+4 d x)-30 A d x \cos (5 c+6 d x)+30 i B d x \cos (5 c+6 d x)-30 A d x \cos (7 c+6 d x)+30 i B d x \cos (7 c+6 d x)-790 A \sin (c)+860 i B \sin (c)+720 A \sin (c+2 d x)-780 i B \sin (c+2 d x)-465 A \sin (3 c+2 d x)+510 i B \sin (3 c+2 d x)+354 A \sin (3 c+4 d x)-366 i B \sin (3 c+4 d x)-120 A \sin (5 c+4 d x)+150 i B \sin (5 c+4 d x)+79 A \sin (5 c+6 d x)-86 i B \sin (5 c+6 d x)) (i \tan (c+d x) a+a)^4 (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{a^4 (92 A-93 i B) \tan ^3(c+d x)}{60 d}-\frac{(12 A-13 i B) \tan ^3(c+d x) \left(a^4+i a^4 \tan (c+d x)\right)}{20 d}+\frac{4 a^4 (B+i A) \tan ^2(c+d x)}{d}+\frac{8 a^4 (A-i B) \tan (c+d x)}{d}+\frac{8 a^4 (B+i A) \log (\cos (c+d x))}{d}-8 a^4 x (A-i B)-\frac{(2 A-3 i B) \tan ^3(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{10 d}+\frac{i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}",1,"(Cos[c + d*x]^5*(I*A*Cos[2*c] + B*Cos[2*c] + A*Sin[2*c] - I*B*Sin[2*c])*(4*Cos[2*c]*Log[Cos[c + d*x]^2] - (4*I)*Log[Cos[c + d*x]^2]*Sin[2*c])*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Sec[c]*Sec[c + d*x]*(Cos[4*c]/240 - (I/240)*Sin[4*c])*((420*I)*A*Cos[c] + 490*B*Cos[c] - 600*A*d*x*Cos[c] + (600*I)*B*d*x*Cos[c] + (300*I)*A*Cos[c + 2*d*x] + 345*B*Cos[c + 2*d*x] - 450*A*d*x*Cos[c + 2*d*x] + (450*I)*B*d*x*Cos[c + 2*d*x] + (300*I)*A*Cos[3*c + 2*d*x] + 345*B*Cos[3*c + 2*d*x] - 450*A*d*x*Cos[3*c + 2*d*x] + (450*I)*B*d*x*Cos[3*c + 2*d*x] + (90*I)*A*Cos[3*c + 4*d*x] + 120*B*Cos[3*c + 4*d*x] - 180*A*d*x*Cos[3*c + 4*d*x] + (180*I)*B*d*x*Cos[3*c + 4*d*x] + (90*I)*A*Cos[5*c + 4*d*x] + 120*B*Cos[5*c + 4*d*x] - 180*A*d*x*Cos[5*c + 4*d*x] + (180*I)*B*d*x*Cos[5*c + 4*d*x] - 30*A*d*x*Cos[5*c + 6*d*x] + (30*I)*B*d*x*Cos[5*c + 6*d*x] - 30*A*d*x*Cos[7*c + 6*d*x] + (30*I)*B*d*x*Cos[7*c + 6*d*x] - 790*A*Sin[c] + (860*I)*B*Sin[c] + 720*A*Sin[c + 2*d*x] - (780*I)*B*Sin[c + 2*d*x] - 465*A*Sin[3*c + 2*d*x] + (510*I)*B*Sin[3*c + 2*d*x] + 354*A*Sin[3*c + 4*d*x] - (366*I)*B*Sin[3*c + 4*d*x] - 120*A*Sin[5*c + 4*d*x] + (150*I)*B*Sin[5*c + 4*d*x] + 79*A*Sin[5*c + 6*d*x] - (86*I)*B*Sin[5*c + 6*d*x])*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (x*Cos[c + d*x]^5*(4*A*Cos[c]^2 - (4*I)*B*Cos[c]^2 - 4*A*Cos[c]^4 + (4*I)*B*Cos[c]^4 - (12*I)*A*Cos[c]*Sin[c] - 12*B*Cos[c]*Sin[c] + (20*I)*A*Cos[c]^3*Sin[c] + 20*B*Cos[c]^3*Sin[c] - 12*A*Sin[c]^2 + (12*I)*B*Sin[c]^2 + 40*A*Cos[c]^2*Sin[c]^2 - (40*I)*B*Cos[c]^2*Sin[c]^2 - (40*I)*A*Cos[c]*Sin[c]^3 - 40*B*Cos[c]*Sin[c]^3 - 20*A*Sin[c]^4 + (20*I)*B*Sin[c]^4 + (4*I)*A*Sin[c]^2*Tan[c] + 4*B*Sin[c]^2*Tan[c] + (4*I)*A*Sin[c]^4*Tan[c] + 4*B*Sin[c]^4*Tan[c] - I*(A - I*B)*(8*Cos[4*c] - (8*I)*Sin[4*c])*Tan[c])*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]))/((Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
27,1,589,168,5.2444994,"\int \tan (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{a^4 \sec (c) \sec ^5(c+d x) \left(-15 i \cos (d x) \left(-10 i (A-i B) \log \left(\cos ^2(c+d x)\right)+20 A d x-11 i A-20 i B d x-14 B\right)-15 i \cos (2 c+d x) \left(-10 i (A-i B) \log \left(\cos ^2(c+d x)\right)+20 A d x-11 i A-20 i B d x-14 B\right)-300 i A \sin (2 c+d x)+260 i A \sin (2 c+3 d x)-90 i A \sin (4 c+3 d x)+70 i A \sin (4 c+5 d x)-60 A \cos (2 c+3 d x)-150 i A d x \cos (2 c+3 d x)-60 A \cos (4 c+3 d x)-150 i A d x \cos (4 c+3 d x)-30 i A d x \cos (4 c+5 d x)-30 i A d x \cos (6 c+5 d x)-75 A \cos (2 c+3 d x) \log \left(\cos ^2(c+d x)\right)-75 A \cos (4 c+3 d x) \log \left(\cos ^2(c+d x)\right)-15 A \cos (4 c+5 d x) \log \left(\cos ^2(c+d x)\right)-15 A \cos (6 c+5 d x) \log \left(\cos ^2(c+d x)\right)+400 i A \sin (d x)-345 B \sin (2 c+d x)+275 B \sin (2 c+3 d x)-120 B \sin (4 c+3 d x)+79 B \sin (4 c+5 d x)+90 i B \cos (2 c+3 d x)-150 B d x \cos (2 c+3 d x)+90 i B \cos (4 c+3 d x)-150 B d x \cos (4 c+3 d x)-30 B d x \cos (4 c+5 d x)-30 B d x \cos (6 c+5 d x)+75 i B \cos (2 c+3 d x) \log \left(\cos ^2(c+d x)\right)+75 i B \cos (4 c+3 d x) \log \left(\cos ^2(c+d x)\right)+15 i B \cos (4 c+5 d x) \log \left(\cos ^2(c+d x)\right)+15 i B \cos (6 c+5 d x) \log \left(\cos ^2(c+d x)\right)+445 B \sin (d x)\right)}{120 d}","\frac{4 a^4 (B+i A) \tan (c+d x)}{d}-\frac{8 a^4 (A-i B) \log (\cos (c+d x))}{d}-8 a^4 x (B+i A)+\frac{(A-i B) \left(a^2+i a^2 \tan (c+d x)\right)^2}{d}+\frac{a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac{A (a+i a \tan (c+d x))^4}{4 d}-\frac{i B (a+i a \tan (c+d x))^5}{5 a d}",1,"(a^4*Sec[c]*Sec[c + d*x]^5*(-60*A*Cos[2*c + 3*d*x] + (90*I)*B*Cos[2*c + 3*d*x] - (150*I)*A*d*x*Cos[2*c + 3*d*x] - 150*B*d*x*Cos[2*c + 3*d*x] - 60*A*Cos[4*c + 3*d*x] + (90*I)*B*Cos[4*c + 3*d*x] - (150*I)*A*d*x*Cos[4*c + 3*d*x] - 150*B*d*x*Cos[4*c + 3*d*x] - (30*I)*A*d*x*Cos[4*c + 5*d*x] - 30*B*d*x*Cos[4*c + 5*d*x] - (30*I)*A*d*x*Cos[6*c + 5*d*x] - 30*B*d*x*Cos[6*c + 5*d*x] - 75*A*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]^2] + (75*I)*B*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]^2] - 75*A*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] + (75*I)*B*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] - 15*A*Cos[4*c + 5*d*x]*Log[Cos[c + d*x]^2] + (15*I)*B*Cos[4*c + 5*d*x]*Log[Cos[c + d*x]^2] - 15*A*Cos[6*c + 5*d*x]*Log[Cos[c + d*x]^2] + (15*I)*B*Cos[6*c + 5*d*x]*Log[Cos[c + d*x]^2] - (15*I)*Cos[d*x]*((-11*I)*A - 14*B + 20*A*d*x - (20*I)*B*d*x - (10*I)*(A - I*B)*Log[Cos[c + d*x]^2]) - (15*I)*Cos[2*c + d*x]*((-11*I)*A - 14*B + 20*A*d*x - (20*I)*B*d*x - (10*I)*(A - I*B)*Log[Cos[c + d*x]^2]) + (400*I)*A*Sin[d*x] + 445*B*Sin[d*x] - (300*I)*A*Sin[2*c + d*x] - 345*B*Sin[2*c + d*x] + (260*I)*A*Sin[2*c + 3*d*x] + 275*B*Sin[2*c + 3*d*x] - (90*I)*A*Sin[4*c + 3*d*x] - 120*B*Sin[4*c + 3*d*x] + (70*I)*A*Sin[4*c + 5*d*x] + 79*B*Sin[4*c + 5*d*x]))/(120*d)","B",1
28,1,448,140,3.6963007,"\int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{a^4 \sec (c) \sec ^4(c+d x) \left(3 \cos (c) \left((-6 B-6 i A) \log \left(\cos ^2(c+d x)\right)+12 A d x-4 i A-12 i B d x-7 B\right)+6 \cos (c+2 d x) \left((-2 B-2 i A) \log \left(\cos ^2(c+d x)\right)+4 A d x-i A-4 i B d x-2 B\right)-32 A \sin (c+2 d x)+12 A \sin (3 c+2 d x)-11 A \sin (3 c+4 d x)-6 i A \cos (3 c+2 d x)+24 A d x \cos (3 c+2 d x)+6 A d x \cos (3 c+4 d x)+6 A d x \cos (5 c+4 d x)-12 i A \cos (3 c+2 d x) \log \left(\cos ^2(c+d x)\right)-3 i A \cos (3 c+4 d x) \log \left(\cos ^2(c+d x)\right)-3 i A \cos (5 c+4 d x) \log \left(\cos ^2(c+d x)\right)+33 A \sin (c)+38 i B \sin (c+2 d x)-18 i B \sin (3 c+2 d x)+14 i B \sin (3 c+4 d x)-12 B \cos (3 c+2 d x)-24 i B d x \cos (3 c+2 d x)-6 i B d x \cos (3 c+4 d x)-6 i B d x \cos (5 c+4 d x)-12 B \cos (3 c+2 d x) \log \left(\cos ^2(c+d x)\right)-3 B \cos (3 c+4 d x) \log \left(\cos ^2(c+d x)\right)-3 B \cos (5 c+4 d x) \log \left(\cos ^2(c+d x)\right)-42 i B \sin (c)\right)}{12 d}","-\frac{4 a^4 (A-i B) \tan (c+d x)}{d}-\frac{8 a^4 (B+i A) \log (\cos (c+d x))}{d}+8 a^4 x (A-i B)+\frac{(B+i A) \left(a^2+i a^2 \tan (c+d x)\right)^2}{d}+\frac{a (B+i A) (a+i a \tan (c+d x))^3}{3 d}+\frac{B (a+i a \tan (c+d x))^4}{4 d}",1,"(a^4*Sec[c]*Sec[c + d*x]^4*((-6*I)*A*Cos[3*c + 2*d*x] - 12*B*Cos[3*c + 2*d*x] + 24*A*d*x*Cos[3*c + 2*d*x] - (24*I)*B*d*x*Cos[3*c + 2*d*x] + 6*A*d*x*Cos[3*c + 4*d*x] - (6*I)*B*d*x*Cos[3*c + 4*d*x] + 6*A*d*x*Cos[5*c + 4*d*x] - (6*I)*B*d*x*Cos[5*c + 4*d*x] - (12*I)*A*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] - 12*B*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] - (3*I)*A*Cos[3*c + 4*d*x]*Log[Cos[c + d*x]^2] - 3*B*Cos[3*c + 4*d*x]*Log[Cos[c + d*x]^2] - (3*I)*A*Cos[5*c + 4*d*x]*Log[Cos[c + d*x]^2] - 3*B*Cos[5*c + 4*d*x]*Log[Cos[c + d*x]^2] + 3*Cos[c]*((-4*I)*A - 7*B + 12*A*d*x - (12*I)*B*d*x + ((-6*I)*A - 6*B)*Log[Cos[c + d*x]^2]) + 6*Cos[c + 2*d*x]*((-I)*A - 2*B + 4*A*d*x - (4*I)*B*d*x + ((-2*I)*A - 2*B)*Log[Cos[c + d*x]^2]) + 33*A*Sin[c] - (42*I)*B*Sin[c] - 32*A*Sin[c + 2*d*x] + (38*I)*B*Sin[c + 2*d*x] + 12*A*Sin[3*c + 2*d*x] - (18*I)*B*Sin[3*c + 2*d*x] - 11*A*Sin[3*c + 4*d*x] + (14*I)*B*Sin[3*c + 4*d*x]))/(12*d)","B",1
29,1,429,142,7.9062481,"\int \cot (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{a^4 \sec (c) \sec ^3(c+d x) (\cos (4 d x)+i \sin (4 d x)) \left(3 \cos (d x) \left(3 (7 A-8 i B) \log \left(\cos ^2(c+d x)\right)+3 A \log \left(\sin ^2(c+d x)\right)+48 i A d x+4 A+48 B d x-16 i B\right)+3 \cos (2 c+d x) \left(3 (7 A-8 i B) \log \left(\cos ^2(c+d x)\right)+3 A \log \left(\sin ^2(c+d x)\right)+48 i A d x+4 A+48 B d x-16 i B\right)+48 i A \sin (2 c+d x)-48 i A \sin (2 c+3 d x)+48 i A d x \cos (2 c+3 d x)+48 i A d x \cos (4 c+3 d x)+21 A \cos (2 c+3 d x) \log \left(\cos ^2(c+d x)\right)+21 A \cos (4 c+3 d x) \log \left(\cos ^2(c+d x)\right)+3 A \cos (2 c+3 d x) \log \left(\sin ^2(c+d x)\right)+3 A \cos (4 c+3 d x) \log \left(\sin ^2(c+d x)\right)-96 i A \sin (d x)+96 B \sin (2 c+d x)-88 B \sin (2 c+3 d x)+48 B d x \cos (2 c+3 d x)+48 B d x \cos (4 c+3 d x)-24 i B \cos (2 c+3 d x) \log \left(\cos ^2(c+d x)\right)-24 i B \cos (4 c+3 d x) \log \left(\cos ^2(c+d x)\right)-168 B \sin (d x)\right)}{48 d (\cos (d x)+i \sin (d x))^4}","-\frac{(3 A-4 i B) \left(a^4+i a^4 \tan (c+d x)\right)}{d}+\frac{a^4 (7 A-8 i B) \log (\cos (c+d x))}{d}+8 a^4 x (B+i A)+\frac{a^4 A \log (\sin (c+d x))}{d}-\frac{(A-2 i B) \left(a^2+i a^2 \tan (c+d x)\right)^2}{2 d}+\frac{i a B (a+i a \tan (c+d x))^3}{3 d}",1,"(a^4*Sec[c]*Sec[c + d*x]^3*(Cos[4*d*x] + I*Sin[4*d*x])*((48*I)*A*d*x*Cos[2*c + 3*d*x] + 48*B*d*x*Cos[2*c + 3*d*x] + (48*I)*A*d*x*Cos[4*c + 3*d*x] + 48*B*d*x*Cos[4*c + 3*d*x] + 21*A*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]^2] - (24*I)*B*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]^2] + 21*A*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] - (24*I)*B*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] + 3*A*Cos[2*c + 3*d*x]*Log[Sin[c + d*x]^2] + 3*A*Cos[4*c + 3*d*x]*Log[Sin[c + d*x]^2] + 3*Cos[d*x]*(4*A - (16*I)*B + (48*I)*A*d*x + 48*B*d*x + 3*(7*A - (8*I)*B)*Log[Cos[c + d*x]^2] + 3*A*Log[Sin[c + d*x]^2]) + 3*Cos[2*c + d*x]*(4*A - (16*I)*B + (48*I)*A*d*x + 48*B*d*x + 3*(7*A - (8*I)*B)*Log[Cos[c + d*x]^2] + 3*A*Log[Sin[c + d*x]^2]) - (96*I)*A*Sin[d*x] - 168*B*Sin[d*x] + (48*I)*A*Sin[2*c + d*x] + 96*B*Sin[2*c + d*x] - (48*I)*A*Sin[2*c + 3*d*x] - 88*B*Sin[2*c + 3*d*x]))/(48*d*(Cos[d*x] + I*Sin[d*x])^4)","B",1
30,1,1122,144,11.5316017,"\int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","a^4 \left(\frac{x (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \left(-22 A \cos ^4(c)+\frac{17}{2} i B \cos ^4(c)-4 i A \cot (c) \cos ^4(c)-B \cot (c) \cos ^4(c)+50 i A \sin (c) \cos ^3(c)+\frac{55}{2} B \sin (c) \cos ^3(c)+60 A \sin ^2(c) \cos ^2(c)-45 i B \sin ^2(c) \cos ^2(c)+2 A \cos ^2(c)-\frac{7}{2} i B \cos ^2(c)-40 i A \sin ^3(c) \cos (c)-40 B \sin ^3(c) \cos (c)-6 i A \sin (c) \cos (c)-\frac{21}{2} B \sin (c) \cos (c)-14 A \sin ^4(c)+\frac{37}{2} i B \sin ^4(c)-6 A \sin ^2(c)+\frac{21}{2} i B \sin ^2(c)+(4 \cos (2 c) B-3 B+4 i A \cos (2 c)) \csc (c) \sec (c) (\cos (4 c)-i \sin (4 c))+2 i A \sin ^4(c) \tan (c)+\frac{7}{2} B \sin ^4(c) \tan (c)+2 i A \sin ^2(c) \tan (c)+\frac{7}{2} B \sin ^2(c) \tan (c)\right) \sin ^5(c+d x)}{(\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (4 i A \cos (2 c)+B \cos (2 c)+4 A \sin (2 c)-i B \sin (2 c)) \left(-i \tan ^{-1}(\tan (5 c+d x)) \cos (2 c)-\tan ^{-1}(\tan (5 c+d x)) \sin (2 c)\right) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (4 i A \cos (2 c)+7 B \cos (2 c)+4 A \sin (2 c)-7 i B \sin (2 c)) \left(\frac{1}{2} \cos (2 c) \log \left(\cos ^2(c+d x)\right)-\frac{1}{2} i \log \left(\cos ^2(c+d x)\right) \sin (2 c)\right) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (4 i A \cos (2 c)+B \cos (2 c)+4 A \sin (2 c)-i B \sin (2 c)) \left(\frac{1}{2} \cos (2 c) \log \left(\sin ^2(c+d x)\right)-\frac{1}{2} i \log \left(\sin ^2(c+d x)\right) \sin (2 c)\right) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(A-i B) (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (8 i d x \sin (4 c)-8 d x \cos (4 c)) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \sec (c) (\cos (4 c)-i \sin (4 c)) (A \sin (d x)-4 i B \sin (d x)) \tan (c+d x) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{A (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \csc (c) (\cos (4 c)-i \sin (4 c)) \sin (d x) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \left(\frac{1}{2} B \cos (4 c)-\frac{1}{2} i B \sin (4 c)\right) \tan ^2(c+d x) \sin ^3(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}\right)","\frac{a^4 (B+4 i A) \log (\sin (c+d x))}{d}+\frac{a^4 (7 B+4 i A) \log (\cos (c+d x))}{d}-8 a^4 x (A-i B)-\frac{3 B \left(a^4+i a^4 \tan (c+d x)\right)}{d}+\frac{(-B+2 i A) \left(a^2+i a^2 \tan (c+d x)\right)^2}{2 d}-\frac{a A \cot (c+d x) (a+i a \tan (c+d x))^3}{d}",1,"a^4*((A*(I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*Csc[c]*(Cos[4*c] - I*Sin[4*c])*Sin[d*x]*Sin[c + d*x]^4)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*((4*I)*A*Cos[2*c] + B*Cos[2*c] + 4*A*Sin[2*c] - I*B*Sin[2*c])*((-I)*ArcTan[Tan[5*c + d*x]]*Cos[2*c] - ArcTan[Tan[5*c + d*x]]*Sin[2*c])*Sin[c + d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*((4*I)*A*Cos[2*c] + 7*B*Cos[2*c] + 4*A*Sin[2*c] - (7*I)*B*Sin[2*c])*((Cos[2*c]*Log[Cos[c + d*x]^2])/2 - (I/2)*Log[Cos[c + d*x]^2]*Sin[2*c])*Sin[c + d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*((4*I)*A*Cos[2*c] + B*Cos[2*c] + 4*A*Sin[2*c] - I*B*Sin[2*c])*((Cos[2*c]*Log[Sin[c + d*x]^2])/2 - (I/2)*Log[Sin[c + d*x]^2]*Sin[2*c])*Sin[c + d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((A - I*B)*(I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*(-8*d*x*Cos[4*c] + (8*I)*d*x*Sin[4*c])*Sin[c + d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (x*(I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*Sin[c + d*x]^5*(2*A*Cos[c]^2 - ((7*I)/2)*B*Cos[c]^2 - 22*A*Cos[c]^4 + ((17*I)/2)*B*Cos[c]^4 - (4*I)*A*Cos[c]^4*Cot[c] - B*Cos[c]^4*Cot[c] - (6*I)*A*Cos[c]*Sin[c] - (21*B*Cos[c]*Sin[c])/2 + (50*I)*A*Cos[c]^3*Sin[c] + (55*B*Cos[c]^3*Sin[c])/2 - 6*A*Sin[c]^2 + ((21*I)/2)*B*Sin[c]^2 + 60*A*Cos[c]^2*Sin[c]^2 - (45*I)*B*Cos[c]^2*Sin[c]^2 - (40*I)*A*Cos[c]*Sin[c]^3 - 40*B*Cos[c]*Sin[c]^3 - 14*A*Sin[c]^4 + ((37*I)/2)*B*Sin[c]^4 + (-3*B + (4*I)*A*Cos[2*c] + 4*B*Cos[2*c])*Csc[c]*Sec[c]*(Cos[4*c] - I*Sin[4*c]) + (2*I)*A*Sin[c]^2*Tan[c] + (7*B*Sin[c]^2*Tan[c])/2 + (2*I)*A*Sin[c]^4*Tan[c] + (7*B*Sin[c]^4*Tan[c])/2))/((Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*Sec[c]*(Cos[4*c] - I*Sin[4*c])*(A*Sin[d*x] - (4*I)*B*Sin[d*x])*Sin[c + d*x]^4*Tan[c + d*x])/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*((B*Cos[4*c])/2 - (I/2)*B*Sin[4*c])*Sin[c + d*x]^3*Tan[c + d*x]^2)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])))","B",1
31,1,1116,156,11.5044548,"\int \cot ^3(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","a^4 \left(\frac{x (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \left(-\frac{71}{2} i A \cos ^4(c)-22 B \cos ^4(c)+7 A \cot (c) \cos ^4(c)-4 i B \cot (c) \cos ^4(c)-\frac{145}{2} A \sin (c) \cos ^3(c)+50 i B \sin (c) \cos ^3(c)+75 i A \sin ^2(c) \cos ^2(c)+60 B \sin ^2(c) \cos ^2(c)+\frac{1}{2} i A \cos ^2(c)+2 B \cos ^2(c)+40 A \sin ^3(c) \cos (c)-40 i B \sin ^3(c) \cos (c)+\frac{3}{2} A \sin (c) \cos (c)-6 i B \sin (c) \cos (c)-\frac{19}{2} i A \sin ^4(c)-14 B \sin ^4(c)-\frac{3}{2} i A \sin ^2(c)-6 B \sin ^2(c)+(4 \cos (2 c) A+3 A-4 i B \cos (2 c)) \csc (c) \sec (c) (i \sin (4 c)-\cos (4 c))-\frac{1}{2} A \sin ^4(c) \tan (c)+2 i B \sin ^4(c) \tan (c)-\frac{1}{2} A \sin ^2(c) \tan (c)+2 i B \sin ^2(c) \tan (c)\right) \sin ^5(c+d x)}{(\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (7 A \cos (2 c)-4 i B \cos (2 c)-7 i A \sin (2 c)-4 B \sin (2 c)) \left(i \tan ^{-1}(\tan (5 c+d x)) \cos (2 c)+\tan ^{-1}(\tan (5 c+d x)) \sin (2 c)\right) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (A \cos (2 c)-4 i B \cos (2 c)-i A \sin (2 c)-4 B \sin (2 c)) \left(\frac{1}{2} i \log \left(\cos ^2(c+d x)\right) \sin (2 c)-\frac{1}{2} \cos (2 c) \log \left(\cos ^2(c+d x)\right)\right) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (7 A \cos (2 c)-4 i B \cos (2 c)-7 i A \sin (2 c)-4 B \sin (2 c)) \left(\frac{1}{2} i \log \left(\sin ^2(c+d x)\right) \sin (2 c)-\frac{1}{2} \cos (2 c) \log \left(\sin ^2(c+d x)\right)\right) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(A-i B) (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (-8 i d x \cos (4 c)-8 d x \sin (4 c)) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{B (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \sec (c) (\cos (4 c)-i \sin (4 c)) \sin (d x) \tan (c+d x) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \csc (c) (\cos (4 c)-i \sin (4 c)) (4 i A \sin (d x)+B \sin (d x)) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \left(\frac{1}{2} i A \sin (4 c)-\frac{1}{2} A \cos (4 c)\right) \sin ^3(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}\right)","-\frac{a^4 (7 A-4 i B) \log (\sin (c+d x))}{d}-\frac{a^4 (A-4 i B) \log (\cos (c+d x))}{d}-8 a^4 x (B+i A)-\frac{3 A \left(a^4+i a^4 \tan (c+d x)\right)}{d}-\frac{(2 B+5 i A) \cot (c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{2 d}-\frac{a A \cot ^2(c+d x) (a+i a \tan (c+d x))^3}{2 d}",1,"a^4*(((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*(-1/2*(A*Cos[4*c]) + (I/2)*A*Sin[4*c])*Sin[c + d*x]^3)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*Csc[c]*(Cos[4*c] - I*Sin[4*c])*((4*I)*A*Sin[d*x] + B*Sin[d*x])*Sin[c + d*x]^4)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*(7*A*Cos[2*c] - (4*I)*B*Cos[2*c] - (7*I)*A*Sin[2*c] - 4*B*Sin[2*c])*(I*ArcTan[Tan[5*c + d*x]]*Cos[2*c] + ArcTan[Tan[5*c + d*x]]*Sin[2*c])*Sin[c + d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*(A*Cos[2*c] - (4*I)*B*Cos[2*c] - I*A*Sin[2*c] - 4*B*Sin[2*c])*(-1/2*(Cos[2*c]*Log[Cos[c + d*x]^2]) + (I/2)*Log[Cos[c + d*x]^2]*Sin[2*c])*Sin[c + d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*(7*A*Cos[2*c] - (4*I)*B*Cos[2*c] - (7*I)*A*Sin[2*c] - 4*B*Sin[2*c])*(-1/2*(Cos[2*c]*Log[Sin[c + d*x]^2]) + (I/2)*Log[Sin[c + d*x]^2]*Sin[2*c])*Sin[c + d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((A - I*B)*(I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*((-8*I)*d*x*Cos[4*c] - 8*d*x*Sin[4*c])*Sin[c + d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (x*(I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*Sin[c + d*x]^5*((I/2)*A*Cos[c]^2 + 2*B*Cos[c]^2 - ((71*I)/2)*A*Cos[c]^4 - 22*B*Cos[c]^4 + 7*A*Cos[c]^4*Cot[c] - (4*I)*B*Cos[c]^4*Cot[c] + (3*A*Cos[c]*Sin[c])/2 - (6*I)*B*Cos[c]*Sin[c] - (145*A*Cos[c]^3*Sin[c])/2 + (50*I)*B*Cos[c]^3*Sin[c] - ((3*I)/2)*A*Sin[c]^2 - 6*B*Sin[c]^2 + (75*I)*A*Cos[c]^2*Sin[c]^2 + 60*B*Cos[c]^2*Sin[c]^2 + 40*A*Cos[c]*Sin[c]^3 - (40*I)*B*Cos[c]*Sin[c]^3 - ((19*I)/2)*A*Sin[c]^4 - 14*B*Sin[c]^4 + (3*A + 4*A*Cos[2*c] - (4*I)*B*Cos[2*c])*Csc[c]*Sec[c]*(-Cos[4*c] + I*Sin[4*c]) - (A*Sin[c]^2*Tan[c])/2 + (2*I)*B*Sin[c]^2*Tan[c] - (A*Sin[c]^4*Tan[c])/2 + (2*I)*B*Sin[c]^4*Tan[c]))/((Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (B*(I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*Sec[c]*(Cos[4*c] - I*Sin[4*c])*Sin[d*x]*Sin[c + d*x]^4*Tan[c + d*x])/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])))","B",1
32,1,1138,163,10.6069231,"\int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","a^4 \left(\frac{x (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \left(40 A \cos ^4(c)-\frac{71}{2} i B \cos ^4(c)+8 i A \cot (c) \cos ^4(c)+7 B \cot (c) \cos ^4(c)-80 i A \sin (c) \cos ^3(c)-\frac{145}{2} B \sin (c) \cos ^3(c)-80 A \sin ^2(c) \cos ^2(c)+75 i B \sin ^2(c) \cos ^2(c)+\frac{1}{2} i B \cos ^2(c)+40 i A \sin ^3(c) \cos (c)+40 B \sin ^3(c) \cos (c)+\frac{3}{2} B \sin (c) \cos (c)+8 A \sin ^4(c)-\frac{19}{2} i B \sin ^4(c)-\frac{3}{2} i B \sin ^2(c)-i (4 \cos (2 c) A+4 A-3 i B-4 i B \cos (2 c)) \csc (c) \sec (c) (\cos (4 c)-i \sin (4 c))-\frac{1}{2} B \sin ^4(c) \tan (c)-\frac{1}{2} B \sin ^2(c) \tan (c)\right) \sin ^5(c+d x)}{(\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}-\frac{B \cos (4 c) (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \log \left(\cos ^2(c+d x)\right) \sin ^5(c+d x)}{2 d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (8 A \cos (2 c)-7 i B \cos (2 c)-8 i A \sin (2 c)-7 B \sin (2 c)) \left(i \tan ^{-1}(\tan (5 c+d x)) \sin (2 c)-\tan ^{-1}(\tan (5 c+d x)) \cos (2 c)\right) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (8 A \cos (2 c)-7 i B \cos (2 c)-8 i A \sin (2 c)-7 B \sin (2 c)) \left(-\frac{1}{2} i \cos (2 c) \log \left(\sin ^2(c+d x)\right)-\frac{1}{2} \sin (2 c) \log \left(\sin ^2(c+d x)\right)\right) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{i B (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \log \left(\cos ^2(c+d x)\right) \sin (4 c) \sin ^5(c+d x)}{2 d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(A-i B) (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (8 d x \cos (4 c)-8 i d x \sin (4 c)) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \csc (c) \left(\frac{2}{3} i \sin (4 c)-\frac{2}{3} \cos (4 c)\right) (11 A \sin (d x)-6 i B \sin (d x)) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \csc (c) (-2 A \cos (c)-12 i A \sin (c)-3 B \sin (c)) \left(\frac{1}{6} \cos (4 c)-\frac{1}{6} i \sin (4 c)\right) \sin ^3(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{A (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \csc (c) \left(\frac{1}{3} \cos (4 c)-\frac{1}{3} i \sin (4 c)\right) \sin (d x) \sin ^2(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}\right)","-\frac{a^4 (7 B+8 i A) \log (\sin (c+d x))}{d}+\frac{(4 A-3 i B) \cot (c+d x) \left(a^4+i a^4 \tan (c+d x)\right)}{d}+8 a^4 x (A-i B)-\frac{a^4 B \log (\cos (c+d x))}{d}-\frac{(B+2 i A) \cot ^2(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{2 d}-\frac{a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}",1,"a^4*((A*(I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*Csc[c]*(Cos[4*c]/3 - (I/3)*Sin[4*c])*Sin[d*x]*Sin[c + d*x]^2)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*Csc[c]*(-2*A*Cos[c] - (12*I)*A*Sin[c] - 3*B*Sin[c])*(Cos[4*c]/6 - (I/6)*Sin[4*c])*Sin[c + d*x]^3)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*Csc[c]*((-2*Cos[4*c])/3 + ((2*I)/3)*Sin[4*c])*(11*A*Sin[d*x] - (6*I)*B*Sin[d*x])*Sin[c + d*x]^4)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) - (B*Cos[4*c]*(I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*Log[Cos[c + d*x]^2]*Sin[c + d*x]^5)/(2*d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*(8*A*Cos[2*c] - (7*I)*B*Cos[2*c] - (8*I)*A*Sin[2*c] - 7*B*Sin[2*c])*(-(ArcTan[Tan[5*c + d*x]]*Cos[2*c]) + I*ArcTan[Tan[5*c + d*x]]*Sin[2*c])*Sin[c + d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*(8*A*Cos[2*c] - (7*I)*B*Cos[2*c] - (8*I)*A*Sin[2*c] - 7*B*Sin[2*c])*((-1/2*I)*Cos[2*c]*Log[Sin[c + d*x]^2] - (Log[Sin[c + d*x]^2]*Sin[2*c])/2)*Sin[c + d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I/2)*B*(I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*Log[Cos[c + d*x]^2]*Sin[4*c]*Sin[c + d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((A - I*B)*(I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*(8*d*x*Cos[4*c] - (8*I)*d*x*Sin[4*c])*Sin[c + d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (x*(I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*Sin[c + d*x]^5*((I/2)*B*Cos[c]^2 + 40*A*Cos[c]^4 - ((71*I)/2)*B*Cos[c]^4 + (8*I)*A*Cos[c]^4*Cot[c] + 7*B*Cos[c]^4*Cot[c] + (3*B*Cos[c]*Sin[c])/2 - (80*I)*A*Cos[c]^3*Sin[c] - (145*B*Cos[c]^3*Sin[c])/2 - ((3*I)/2)*B*Sin[c]^2 - 80*A*Cos[c]^2*Sin[c]^2 + (75*I)*B*Cos[c]^2*Sin[c]^2 + (40*I)*A*Cos[c]*Sin[c]^3 + 40*B*Cos[c]*Sin[c]^3 + 8*A*Sin[c]^4 - ((19*I)/2)*B*Sin[c]^4 - I*(4*A - (3*I)*B + 4*A*Cos[2*c] - (4*I)*B*Cos[2*c])*Csc[c]*Sec[c]*(Cos[4*c] - I*Sin[4*c]) - (B*Sin[c]^2*Tan[c])/2 - (B*Sin[c]^4*Tan[c])/2))/((Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])))","B",1
33,1,319,177,6.3644916,"\int \cot ^5(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{a^4 \sin (c+d x) (\cot (c+d x)+i)^4 (A \cot (c+d x)+B) \left(192 d x (A-i B) (\sin (4 c)+i \cos (4 c)) \sin ^4(c+d x)+48 (A-i B) (\cos (4 c)-i \sin (4 c)) \sin ^4(c+d x) \log \left(\sin ^2(c+d x)\right)-96 i (A-i B) (\cos (4 c)-i \sin (4 c)) \sin ^4(c+d x) \tan ^{-1}(\tan (5 c+d x))+4 (\cos (4 c)-i \sin (4 c)) ((-B-4 i A) \cot (c)+12 A-6 i B) \sin ^2(c+d x)-8 i (14 A-11 i B) \csc (c) (\cos (4 c)-i \sin (4 c)) \sin (d x) \sin ^3(c+d x)+4 (B+4 i A) \csc (c) (\cos (4 c)-i \sin (4 c)) \sin (d x) \sin (c+d x)+3 i A \sin (4 c)-3 A \cos (4 c)\right)}{12 d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}","\frac{a^4 (64 B+67 i A) \cot (c+d x)}{12 d}+\frac{8 a^4 (A-i B) \log (\sin (c+d x))}{d}+\frac{(19 A-16 i B) \cot ^2(c+d x) \left(a^4+i a^4 \tan (c+d x)\right)}{12 d}+8 a^4 x (B+i A)-\frac{(4 B+7 i A) \cot ^3(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{12 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^3}{4 d}",1,"(a^4*(I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*Sin[c + d*x]*(-3*A*Cos[4*c] + (3*I)*A*Sin[4*c] + 4*((4*I)*A + B)*Csc[c]*(Cos[4*c] - I*Sin[4*c])*Sin[d*x]*Sin[c + d*x] + 4*(12*A - (6*I)*B + ((-4*I)*A - B)*Cot[c])*(Cos[4*c] - I*Sin[4*c])*Sin[c + d*x]^2 - (8*I)*(14*A - (11*I)*B)*Csc[c]*(Cos[4*c] - I*Sin[4*c])*Sin[d*x]*Sin[c + d*x]^3 - (96*I)*(A - I*B)*ArcTan[Tan[5*c + d*x]]*(Cos[4*c] - I*Sin[4*c])*Sin[c + d*x]^4 + 48*(A - I*B)*Log[Sin[c + d*x]^2]*(Cos[4*c] - I*Sin[4*c])*Sin[c + d*x]^4 + 192*(A - I*B)*d*x*(I*Cos[4*c] + Sin[4*c])*Sin[c + d*x]^4))/(12*d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
34,1,542,200,8.8973862,"\int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{a^4 (\cot (c+d x)+i)^4 (A \cot (c+d x)+B) \left(-8 d x (A-i B) (\cos (4 c)-i \sin (4 c)) \sin ^5(c+d x)+4 (A-i B) (\sin (4 c)+i \cos (4 c)) \sin ^5(c+d x) \log \left(\sin ^2(c+d x)\right)+8 (A-i B) (\cos (4 c)-i \sin (4 c)) \sin ^5(c+d x) \tan ^{-1}(\tan (5 c+d x))+\frac{1}{120} \csc (c) (\cos (4 c)-i \sin (4 c)) (15 (20 A d x-14 i A-20 i B d x-11 B) \cos (2 c+d x)+15 \cos (d x) (A (-20 d x+14 i)+B (11+20 i d x))+345 A \sin (2 c+d x)-275 A \sin (2 c+3 d x)-120 A \sin (4 c+3 d x)+79 A \sin (4 c+5 d x)-90 i A \cos (2 c+3 d x)+150 A d x \cos (2 c+3 d x)+90 i A \cos (4 c+3 d x)-150 A d x \cos (4 c+3 d x)-30 A d x \cos (4 c+5 d x)+30 A d x \cos (6 c+5 d x)+445 A \sin (d x)-300 i B \sin (2 c+d x)+260 i B \sin (2 c+3 d x)+90 i B \sin (4 c+3 d x)-70 i B \sin (4 c+5 d x)-60 B \cos (2 c+3 d x)-150 i B d x \cos (2 c+3 d x)+60 B \cos (4 c+3 d x)+150 i B d x \cos (4 c+3 d x)+30 i B d x \cos (4 c+5 d x)-30 i B d x \cos (6 c+5 d x)-400 i B \sin (d x))\right)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}","\frac{a^4 (145 B+148 i A) \cot ^2(c+d x)}{60 d}-\frac{8 a^4 (A-i B) \cot (c+d x)}{d}+\frac{8 a^4 (B+i A) \log (\sin (c+d x))}{d}+\frac{(28 A-25 i B) \cot ^3(c+d x) \left(a^4+i a^4 \tan (c+d x)\right)}{30 d}-8 a^4 x (A-i B)-\frac{(5 B+8 i A) \cot ^4(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{20 d}-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}",1,"(a^4*(I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*(-8*(A - I*B)*d*x*(Cos[4*c] - I*Sin[4*c])*Sin[c + d*x]^5 + 8*(A - I*B)*ArcTan[Tan[5*c + d*x]]*(Cos[4*c] - I*Sin[4*c])*Sin[c + d*x]^5 + 4*(A - I*B)*Log[Sin[c + d*x]^2]*(I*Cos[4*c] + Sin[4*c])*Sin[c + d*x]^5 + (Csc[c]*(Cos[4*c] - I*Sin[4*c])*(15*(A*(14*I - 20*d*x) + B*(11 + (20*I)*d*x))*Cos[d*x] + 15*((-14*I)*A - 11*B + 20*A*d*x - (20*I)*B*d*x)*Cos[2*c + d*x] - (90*I)*A*Cos[2*c + 3*d*x] - 60*B*Cos[2*c + 3*d*x] + 150*A*d*x*Cos[2*c + 3*d*x] - (150*I)*B*d*x*Cos[2*c + 3*d*x] + (90*I)*A*Cos[4*c + 3*d*x] + 60*B*Cos[4*c + 3*d*x] - 150*A*d*x*Cos[4*c + 3*d*x] + (150*I)*B*d*x*Cos[4*c + 3*d*x] - 30*A*d*x*Cos[4*c + 5*d*x] + (30*I)*B*d*x*Cos[4*c + 5*d*x] + 30*A*d*x*Cos[6*c + 5*d*x] - (30*I)*B*d*x*Cos[6*c + 5*d*x] + 445*A*Sin[d*x] - (400*I)*B*Sin[d*x] + 345*A*Sin[2*c + d*x] - (300*I)*B*Sin[2*c + d*x] - 275*A*Sin[2*c + 3*d*x] + (260*I)*B*Sin[2*c + 3*d*x] - 120*A*Sin[4*c + 3*d*x] + (90*I)*B*Sin[4*c + 3*d*x] + 79*A*Sin[4*c + 5*d*x] - (70*I)*B*Sin[4*c + 5*d*x]))/120))/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
35,1,1009,223,10.1521155,"\int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^7*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","a^4 \left(\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (A \cos (2 c)-i B \cos (2 c)-i A \sin (2 c)-B \sin (2 c)) \left(8 i \tan ^{-1}(\tan (5 c+d x)) \cos (2 c)+8 \tan ^{-1}(\tan (5 c+d x)) \sin (2 c)\right) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (A \cos (2 c)-i B \cos (2 c)-i A \sin (2 c)-B \sin (2 c)) \left(4 i \log \left(\sin ^2(c+d x)\right) \sin (2 c)-4 \cos (2 c) \log \left(\sin ^2(c+d x)\right)\right) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{x (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \left(-40 i A \cos ^4(c)-40 B \cos ^4(c)+8 A \cot (c) \cos ^4(c)-8 i B \cot (c) \cos ^4(c)-80 A \sin (c) \cos ^3(c)+80 i B \sin (c) \cos ^3(c)+80 i A \sin ^2(c) \cos ^2(c)+80 B \sin ^2(c) \cos ^2(c)+40 A \sin ^3(c) \cos (c)-40 i B \sin ^3(c) \cos (c)-8 i A \sin ^4(c)-8 B \sin ^4(c)+(A-i B) \cot (c) (8 i \sin (4 c)-8 \cos (4 c))\right) \sin ^5(c+d x)}{(\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \csc (c) \csc (c+d x) \left(\frac{1}{240} \cos (4 c)-\frac{1}{240} i \sin (4 c)\right) (860 i A \cos (c)+790 B \cos (c)-780 i A \cos (c+2 d x)-720 B \cos (c+2 d x)-510 i A \cos (3 c+2 d x)-465 B \cos (3 c+2 d x)+366 i A \cos (3 c+4 d x)+354 B \cos (3 c+4 d x)+150 i A \cos (5 c+4 d x)+120 B \cos (5 c+4 d x)-86 i A \cos (5 c+6 d x)-79 B \cos (5 c+6 d x)-490 A \sin (c)+420 i B \sin (c)-600 i A d x \sin (c)-600 B d x \sin (c)-345 A \sin (c+2 d x)+300 i B \sin (c+2 d x)-450 i A d x \sin (c+2 d x)-450 B d x \sin (c+2 d x)+345 A \sin (3 c+2 d x)-300 i B \sin (3 c+2 d x)+450 i A d x \sin (3 c+2 d x)+450 B d x \sin (3 c+2 d x)+120 A \sin (3 c+4 d x)-90 i B \sin (3 c+4 d x)+180 i A d x \sin (3 c+4 d x)+180 B d x \sin (3 c+4 d x)-120 A \sin (5 c+4 d x)+90 i B \sin (5 c+4 d x)-180 i A d x \sin (5 c+4 d x)-180 B d x \sin (5 c+4 d x)-30 i A d x \sin (5 c+6 d x)-30 B d x \sin (5 c+6 d x)+30 i A d x \sin (7 c+6 d x)+30 B d x \sin (7 c+6 d x))}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}\right)","\frac{a^4 (92 B+93 i A) \cot ^3(c+d x)}{60 d}-\frac{4 a^4 (A-i B) \cot ^2(c+d x)}{d}-\frac{8 a^4 (B+i A) \cot (c+d x)}{d}-\frac{8 a^4 (A-i B) \log (\sin (c+d x))}{d}+\frac{(13 A-12 i B) \cot ^4(c+d x) \left(a^4+i a^4 \tan (c+d x)\right)}{20 d}-8 a^4 x (B+i A)-\frac{(2 B+3 i A) \cot ^5(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{10 d}-\frac{a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}",1,"a^4*(((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*(A*Cos[2*c] - I*B*Cos[2*c] - I*A*Sin[2*c] - B*Sin[2*c])*((8*I)*ArcTan[Tan[5*c + d*x]]*Cos[2*c] + 8*ArcTan[Tan[5*c + d*x]]*Sin[2*c])*Sin[c + d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*(A*Cos[2*c] - I*B*Cos[2*c] - I*A*Sin[2*c] - B*Sin[2*c])*(-4*Cos[2*c]*Log[Sin[c + d*x]^2] + (4*I)*Log[Sin[c + d*x]^2]*Sin[2*c])*Sin[c + d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (x*(I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*((-40*I)*A*Cos[c]^4 - 40*B*Cos[c]^4 + 8*A*Cos[c]^4*Cot[c] - (8*I)*B*Cos[c]^4*Cot[c] - 80*A*Cos[c]^3*Sin[c] + (80*I)*B*Cos[c]^3*Sin[c] + (80*I)*A*Cos[c]^2*Sin[c]^2 + 80*B*Cos[c]^2*Sin[c]^2 + 40*A*Cos[c]*Sin[c]^3 - (40*I)*B*Cos[c]*Sin[c]^3 - (8*I)*A*Sin[c]^4 - 8*B*Sin[c]^4 + (A - I*B)*Cot[c]*(-8*Cos[4*c] + (8*I)*Sin[4*c]))*Sin[c + d*x]^5)/((Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*Csc[c]*Csc[c + d*x]*(Cos[4*c]/240 - (I/240)*Sin[4*c])*((860*I)*A*Cos[c] + 790*B*Cos[c] - (780*I)*A*Cos[c + 2*d*x] - 720*B*Cos[c + 2*d*x] - (510*I)*A*Cos[3*c + 2*d*x] - 465*B*Cos[3*c + 2*d*x] + (366*I)*A*Cos[3*c + 4*d*x] + 354*B*Cos[3*c + 4*d*x] + (150*I)*A*Cos[5*c + 4*d*x] + 120*B*Cos[5*c + 4*d*x] - (86*I)*A*Cos[5*c + 6*d*x] - 79*B*Cos[5*c + 6*d*x] - 490*A*Sin[c] + (420*I)*B*Sin[c] - (600*I)*A*d*x*Sin[c] - 600*B*d*x*Sin[c] - 345*A*Sin[c + 2*d*x] + (300*I)*B*Sin[c + 2*d*x] - (450*I)*A*d*x*Sin[c + 2*d*x] - 450*B*d*x*Sin[c + 2*d*x] + 345*A*Sin[3*c + 2*d*x] - (300*I)*B*Sin[3*c + 2*d*x] + (450*I)*A*d*x*Sin[3*c + 2*d*x] + 450*B*d*x*Sin[3*c + 2*d*x] + 120*A*Sin[3*c + 4*d*x] - (90*I)*B*Sin[3*c + 4*d*x] + (180*I)*A*d*x*Sin[3*c + 4*d*x] + 180*B*d*x*Sin[3*c + 4*d*x] - 120*A*Sin[5*c + 4*d*x] + (90*I)*B*Sin[5*c + 4*d*x] - (180*I)*A*d*x*Sin[5*c + 4*d*x] - 180*B*d*x*Sin[5*c + 4*d*x] - (30*I)*A*d*x*Sin[5*c + 6*d*x] - 30*B*d*x*Sin[5*c + 6*d*x] + (30*I)*A*d*x*Sin[7*c + 6*d*x] + 30*B*d*x*Sin[7*c + 6*d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])))","B",1
36,1,898,129,7.3130414,"\int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]),x]","\frac{\left(\frac{1}{2} B \sin (c)-\frac{1}{2} i B \cos (c)\right) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \sec ^2(c+d x)}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{(\cos (d x)+i \sin (d x)) (A \cos (c-d x)+i B \cos (c-d x)-A \cos (c+d x)-i B \cos (c+d x)+i A \sin (c-d x)-B \sin (c-d x)-i A \sin (c+d x)+B \sin (c+d x)) (A+B \tan (c+d x)) \sec (c+d x)}{2 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{x (\cos (d x)+i \sin (d x)) (-i A \sec (c)+2 B \sec (c)+(A+2 i B) (\cos (c)+i \sin (c)) \tan (c)) (A+B \tan (c+d x))}{(A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{\left(A \cos \left(\frac{c}{2}\right)+2 i B \cos \left(\frac{c}{2}\right)+i A \sin \left(\frac{c}{2}\right)-2 B \sin \left(\frac{c}{2}\right)\right) \left(i \tan ^{-1}(\tan (d x)) \cos \left(\frac{c}{2}\right)-\tan ^{-1}(\tan (d x)) \sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x))}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{\left(A \cos \left(\frac{c}{2}\right)+2 i B \cos \left(\frac{c}{2}\right)+i A \sin \left(\frac{c}{2}\right)-2 B \sin \left(\frac{c}{2}\right)\right) \left(-\frac{1}{2} \cos \left(\frac{c}{2}\right) \log \left(\cos ^2(c+d x)\right)-\frac{1}{2} i \sin \left(\frac{c}{2}\right) \log \left(\cos ^2(c+d x)\right)\right) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x))}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{(A+i B) \cos (2 d x) \left(\frac{\cos (c)}{4}-\frac{1}{4} i \sin (c)\right) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x))}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{(A+i B) \left(\frac{3}{2} i d x \cos (c)-\frac{3}{2} d x \sin (c)\right) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x))}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{(B-i A) \left(\frac{\cos (c)}{4}-\frac{1}{4} i \sin (c)\right) (\cos (d x)+i \sin (d x)) \sin (2 d x) (A+B \tan (c+d x))}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}","\frac{(-B+i A) \tan ^3(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{(A+2 i B) \tan ^2(c+d x)}{2 a d}-\frac{3 (-B+i A) \tan (c+d x)}{2 a d}-\frac{(A+2 i B) \log (\cos (c+d x))}{a d}+\frac{3 x (-B+i A)}{2 a}",1,"((A*Cos[c/2] + (2*I)*B*Cos[c/2] + I*A*Sin[c/2] - 2*B*Sin[c/2])*(I*ArcTan[Tan[d*x]]*Cos[c/2] - ArcTan[Tan[d*x]]*Sin[c/2])*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + ((A*Cos[c/2] + (2*I)*B*Cos[c/2] + I*A*Sin[c/2] - 2*B*Sin[c/2])*(-1/2*(Cos[c/2]*Log[Cos[c + d*x]^2]) - (I/2)*Log[Cos[c + d*x]^2]*Sin[c/2])*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + ((A + I*B)*Cos[2*d*x]*(Cos[c]/4 - (I/4)*Sin[c])*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + (Sec[c + d*x]^2*((-1/2*I)*B*Cos[c] + (B*Sin[c])/2)*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + ((A + I*B)*(((3*I)/2)*d*x*Cos[c] - (3*d*x*Sin[c])/2)*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + (((-I)*A + B)*(Cos[c]/4 - (I/4)*Sin[c])*(Cos[d*x] + I*Sin[d*x])*Sin[2*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + (Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])*(A*Cos[c - d*x] + I*B*Cos[c - d*x] - A*Cos[c + d*x] - I*B*Cos[c + d*x] + I*A*Sin[c - d*x] - B*Sin[c - d*x] - I*A*Sin[c + d*x] + B*Sin[c + d*x])*(A + B*Tan[c + d*x]))/(2*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + (x*(Cos[d*x] + I*Sin[d*x])*((-I)*A*Sec[c] + 2*B*Sec[c] + (A + (2*I)*B)*(Cos[c] + I*Sin[c])*Tan[c])*(A + B*Tan[c + d*x]))/((A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x]))","B",1
37,1,240,101,4.9771122,"\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]),x]","\frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left((B-i A) (\cos (c)-i \sin (c)) \cos (2 d x)+2 d x (A+3 i B) (\cos (c)+i \sin (c))+(A+i B) (-\cos (c)+i \sin (c)) \sin (2 d x)+2 i (A+i B) (\cos (c)+i \sin (c)) \log \left(\cos ^2(c+d x)\right)+4 (A+i B) (\cos (c)+i \sin (c)) \tan ^{-1}(\tan (d x))+4 d x (A+i B) \tan (c) (\sin (c)-i \cos (c))-4 A d x \sec (c)-4 i B d x \sec (c)+4 B (\tan (c)-i) \sin (d x) \sec (c+d x)\right)}{4 d (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))}","\frac{(-B+i A) \tan ^2(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{(A+3 i B) \tan (c+d x)}{2 a d}+\frac{(-B+i A) \log (\cos (c+d x))}{a d}+\frac{x (A+3 i B)}{2 a}",1,"((Cos[d*x] + I*Sin[d*x])*(-4*A*d*x*Sec[c] - (4*I)*B*d*x*Sec[c] + ((-I)*A + B)*Cos[2*d*x]*(Cos[c] - I*Sin[c]) + 2*(A + (3*I)*B)*d*x*(Cos[c] + I*Sin[c]) + 4*(A + I*B)*ArcTan[Tan[d*x]]*(Cos[c] + I*Sin[c]) + (2*I)*(A + I*B)*Log[Cos[c + d*x]^2]*(Cos[c] + I*Sin[c]) + (A + I*B)*(-Cos[c] + I*Sin[c])*Sin[2*d*x] + 4*(A + I*B)*d*x*((-I)*Cos[c] + Sin[c])*Tan[c] + 4*B*Sec[c + d*x]*Sin[d*x]*(-I + Tan[c]))*(A + B*Tan[c + d*x]))/(4*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x]))","B",1
38,1,148,67,1.0828371,"\int \frac{\tan (c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]),x]","\frac{\cos (c+d x) (A+B \tan (c+d x)) \left(\tan (c+d x) \left(-2 i A d x+A+2 i B \log \left(\cos ^2(c+d x)\right)-2 B d x+i B\right)-2 A d x+i A+4 B \tan ^{-1}(\tan (d x)) (\tan (c+d x)-i)+2 B \log \left(\cos ^2(c+d x)\right)+2 i B d x-B\right)}{4 a d (\tan (c+d x)-i) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{A+i B}{2 a d (1+i \tan (c+d x))}-\frac{x (-B+i A)}{2 a}+\frac{i B \log (\cos (c+d x))}{a d}",1,"(Cos[c + d*x]*(A + B*Tan[c + d*x])*(I*A - B - 2*A*d*x + (2*I)*B*d*x + 2*B*Log[Cos[c + d*x]^2] + (A + I*B - (2*I)*A*d*x - 2*B*d*x + (2*I)*B*Log[Cos[c + d*x]^2])*Tan[c + d*x] + 4*B*ArcTan[Tan[d*x]]*(-I + Tan[c + d*x])))/(4*a*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(-I + Tan[c + d*x]))","B",1
39,1,102,47,0.5486603,"\int \frac{A+B \tan (c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x]),x]","\frac{\cos (c+d x) (A+B \tan (c+d x)) ((A (2 d x-i)-2 i B d x+B) \tan (c+d x)-2 i A d x+A+B (-2 d x+i))}{4 a d (\tan (c+d x)-i) (A \cos (c+d x)+B \sin (c+d x))}","\frac{-B+i A}{2 d (a+i a \tan (c+d x))}+\frac{x (A-i B)}{2 a}",1,"(Cos[c + d*x]*(A + B*Tan[c + d*x])*(A - (2*I)*A*d*x + B*(I - 2*d*x) + (B - (2*I)*B*d*x + A*(-I + 2*d*x))*Tan[c + d*x]))/(4*a*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(-I + Tan[c + d*x]))","B",1
40,1,150,62,1.139655,"\int \frac{\cot (c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]),x]","\frac{\cos (c+d x) (A+B \tan (c+d x)) \left(\tan (c+d x) \left(2 A \log \left(\sin ^2(c+d x)\right)+2 i A d x-A+2 B d x-i B\right)-4 i A \tan ^{-1}(\tan (d x)) (\tan (c+d x)-i)-2 i A \log \left(\sin ^2(c+d x)\right)+2 A d x-i A-2 i B d x+B\right)}{4 a d (\tan (c+d x)-i) (A \cos (c+d x)+B \sin (c+d x))}","\frac{A+i B}{2 d (a+i a \tan (c+d x))}-\frac{x (-B+i A)}{2 a}+\frac{A \log (\sin (c+d x))}{a d}",1,"(Cos[c + d*x]*(A + B*Tan[c + d*x])*((-I)*A + B + 2*A*d*x - (2*I)*B*d*x - (2*I)*A*Log[Sin[c + d*x]^2] + (-A - I*B + (2*I)*A*d*x + 2*B*d*x + 2*A*Log[Sin[c + d*x]^2])*Tan[c + d*x] - (4*I)*A*ArcTan[Tan[d*x]]*(-I + Tan[c + d*x])))/(4*a*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(-I + Tan[c + d*x]))","B",1
41,1,225,102,3.1674383,"\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]),x]","\frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left(\frac{1}{2} (B-i A) (\cos (c)-i \sin (c)) \cos (2 d x)-\frac{1}{2} (A+i B) (\cos (c)-i \sin (c)) \sin (2 d x)+2 d x (A+i B) (\cos (c)+i \sin (c))-d x (3 A+i B) (\cos (c)+i \sin (c))+(B-i A) (\cos (c)+i \sin (c)) \log \left(\sin ^2(c+d x)\right)-2 (A+i B) (\cos (c)+i \sin (c)) \tan ^{-1}(\tan (d x))+2 A (\cot (c)+i) \sin (d x) \csc (c+d x)\right)}{2 d (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{(3 A+i B) \cot (c+d x)}{2 a d}-\frac{(-B+i A) \log (\sin (c+d x))}{a d}+\frac{(A+i B) \cot (c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{x (3 A+i B)}{2 a}",1,"((Cos[d*x] + I*Sin[d*x])*((((-I)*A + B)*Cos[2*d*x]*(Cos[c] - I*Sin[c]))/2 + 2*(A + I*B)*d*x*(Cos[c] + I*Sin[c]) - (3*A + I*B)*d*x*(Cos[c] + I*Sin[c]) - 2*(A + I*B)*ArcTan[Tan[d*x]]*(Cos[c] + I*Sin[c]) + ((-I)*A + B)*Log[Sin[c + d*x]^2]*(Cos[c] + I*Sin[c]) + 2*A*(I + Cot[c])*Csc[c + d*x]*Sin[d*x] - ((A + I*B)*(Cos[c] - I*Sin[c])*Sin[2*d*x])/2)*(A + B*Tan[c + d*x]))/(2*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x]))","B",1
42,1,902,131,7.4658439,"\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]),x]","\frac{\left(-\frac{1}{2} A \cos (c)-\frac{1}{2} i A \sin (c)\right) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \csc ^2(c+d x)}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (\cos (d x)+i \sin (d x)) \left(\frac{1}{2} A \cos (c-d x)+\frac{1}{2} i B \cos (c-d x)-\frac{1}{2} A \cos (c+d x)-\frac{1}{2} i B \cos (c+d x)+\frac{1}{2} i A \sin (c-d x)-\frac{1}{2} B \sin (c-d x)-\frac{1}{2} i A \sin (c+d x)+\frac{1}{2} B \sin (c+d x)\right) (A+B \tan (c+d x)) \csc (c+d x)}{2 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{\left(2 A \cos \left(\frac{c}{2}\right)+i B \cos \left(\frac{c}{2}\right)+2 i A \sin \left(\frac{c}{2}\right)-B \sin \left(\frac{c}{2}\right)\right) \left(i \tan ^{-1}(\tan (d x)) \cos \left(\frac{c}{2}\right)-\tan ^{-1}(\tan (d x)) \sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x))}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{\left(2 A \cos \left(\frac{c}{2}\right)+i B \cos \left(\frac{c}{2}\right)+2 i A \sin \left(\frac{c}{2}\right)-B \sin \left(\frac{c}{2}\right)\right) \left(-\frac{1}{2} \cos \left(\frac{c}{2}\right) \log \left(\sin ^2(c+d x)\right)-\frac{1}{2} i \sin \left(\frac{c}{2}\right) \log \left(\sin ^2(c+d x)\right)\right) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x))}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{x (2 A \csc (c)+i B \csc (c)+(2 A+i B) \cot (c) (-\cos (c)-i \sin (c))) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x))}{(A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{(A+i B) \cos (2 d x) \left(\frac{1}{4} i \sin (c)-\frac{\cos (c)}{4}\right) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x))}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{(A+i B) \left(\frac{3}{2} i d x \cos (c)-\frac{3}{2} d x \sin (c)\right) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x))}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{(A+i B) \left(\frac{1}{4} i \cos (c)+\frac{\sin (c)}{4}\right) (\cos (d x)+i \sin (d x)) \sin (2 d x) (A+B \tan (c+d x))}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}","-\frac{(2 A+i B) \cot ^2(c+d x)}{2 a d}+\frac{3 (-B+i A) \cot (c+d x)}{2 a d}-\frac{(2 A+i B) \log (\sin (c+d x))}{a d}+\frac{(A+i B) \cot ^2(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac{3 x (-B+i A)}{2 a}",1,"((2*A*Cos[c/2] + I*B*Cos[c/2] + (2*I)*A*Sin[c/2] - B*Sin[c/2])*(I*ArcTan[Tan[d*x]]*Cos[c/2] - ArcTan[Tan[d*x]]*Sin[c/2])*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + ((2*A*Cos[c/2] + I*B*Cos[c/2] + (2*I)*A*Sin[c/2] - B*Sin[c/2])*(-1/2*(Cos[c/2]*Log[Sin[c + d*x]^2]) - (I/2)*Log[Sin[c + d*x]^2]*Sin[c/2])*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + (x*(2*A*Csc[c] + I*B*Csc[c] + (2*A + I*B)*Cot[c]*(-Cos[c] - I*Sin[c]))*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/((A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + ((A + I*B)*Cos[2*d*x]*(-1/4*Cos[c] + (I/4)*Sin[c])*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + (Csc[c + d*x]^2*(-1/2*(A*Cos[c]) - (I/2)*A*Sin[c])*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + ((A + I*B)*(((3*I)/2)*d*x*Cos[c] - (3*d*x*Sin[c])/2)*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + ((A + I*B)*((I/4)*Cos[c] + Sin[c]/4)*(Cos[d*x] + I*Sin[d*x])*Sin[2*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + (Csc[c/2]*Csc[c + d*x]*Sec[c/2]*(Cos[d*x] + I*Sin[d*x])*((A*Cos[c - d*x])/2 + (I/2)*B*Cos[c - d*x] - (A*Cos[c + d*x])/2 - (I/2)*B*Cos[c + d*x] + (I/2)*A*Sin[c - d*x] - (B*Sin[c - d*x])/2 - (I/2)*A*Sin[c + d*x] + (B*Sin[c + d*x])/2)*(A + B*Tan[c + d*x]))/(2*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x]))","B",1
43,1,1062,155,7.6779178,"\int \frac{\cot ^4(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]),x]","\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (\cos (d x)+i \sin (d x)) \left(\frac{1}{2} i A \cos (c-d x)-\frac{1}{2} i A \cos (c+d x)-\frac{1}{2} A \sin (c-d x)+\frac{1}{2} A \sin (c+d x)\right) (A+B \tan (c+d x)) \csc ^3(c+d x)}{6 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(-\frac{\cos (c)}{12}-\frac{1}{12} i \sin (c)\right) (2 A \cos (c)-3 i A \sin (c)+3 B \sin (c)) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \csc ^2(c+d x)}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (\cos (d x)+i \sin (d x)) \left(-\frac{7}{2} i A \cos (c-d x)+\frac{3}{2} B \cos (c-d x)+\frac{7}{2} i A \cos (c+d x)-\frac{3}{2} B \cos (c+d x)+\frac{7}{2} A \sin (c-d x)+\frac{3}{2} i B \sin (c-d x)-\frac{7}{2} A \sin (c+d x)-\frac{3}{2} i B \sin (c+d x)\right) (A+B \tan (c+d x)) \csc (c+d x)}{6 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{\left(A \cos \left(\frac{c}{2}\right)+i B \cos \left(\frac{c}{2}\right)+i A \sin \left(\frac{c}{2}\right)-B \sin \left(\frac{c}{2}\right)\right) \left(2 \tan ^{-1}(\tan (d x)) \cos \left(\frac{c}{2}\right)+2 i \tan ^{-1}(\tan (d x)) \sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x))}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{\left(A \cos \left(\frac{c}{2}\right)+i B \cos \left(\frac{c}{2}\right)+i A \sin \left(\frac{c}{2}\right)-B \sin \left(\frac{c}{2}\right)\right) \left(i \cos \left(\frac{c}{2}\right) \log \left(\sin ^2(c+d x)\right)-\log \left(\sin ^2(c+d x)\right) \sin \left(\frac{c}{2}\right)\right) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x))}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{x (-2 i A \csc (c)+2 B \csc (c)+i (A+i B) \cot (c) (2 \cos (c)+2 i \sin (c))) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x))}{(A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{(A+i B) \cos (2 d x) \left(\frac{1}{4} i \cos (c)+\frac{\sin (c)}{4}\right) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x))}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{(5 A+3 i B) \left(\frac{1}{2} d x \cos (c)+\frac{1}{2} i d x \sin (c)\right) (\cos (d x)+i \sin (d x)) (A+B \tan (c+d x))}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}+\frac{(A+i B) \left(\frac{\cos (c)}{4}-\frac{1}{4} i \sin (c)\right) (\cos (d x)+i \sin (d x)) \sin (2 d x) (A+B \tan (c+d x))}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)}","-\frac{(5 A+3 i B) \cot ^3(c+d x)}{6 a d}+\frac{(-B+i A) \cot ^2(c+d x)}{a d}+\frac{(5 A+3 i B) \cot (c+d x)}{2 a d}+\frac{2 (-B+i A) \log (\sin (c+d x))}{a d}+\frac{(A+i B) \cot ^3(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac{x (5 A+3 i B)}{2 a}",1,"((A*Cos[c/2] + I*B*Cos[c/2] + I*A*Sin[c/2] - B*Sin[c/2])*(2*ArcTan[Tan[d*x]]*Cos[c/2] + (2*I)*ArcTan[Tan[d*x]]*Sin[c/2])*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + ((A*Cos[c/2] + I*B*Cos[c/2] + I*A*Sin[c/2] - B*Sin[c/2])*(I*Cos[c/2]*Log[Sin[c + d*x]^2] - Log[Sin[c + d*x]^2]*Sin[c/2])*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + (x*((-2*I)*A*Csc[c] + 2*B*Csc[c] + I*(A + I*B)*Cot[c]*(2*Cos[c] + (2*I)*Sin[c]))*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/((A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + ((A + I*B)*Cos[2*d*x]*((I/4)*Cos[c] + Sin[c]/4)*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + (Csc[c/2]*Csc[c + d*x]^2*Sec[c/2]*(-1/12*Cos[c] - (I/12)*Sin[c])*(2*A*Cos[c] - (3*I)*A*Sin[c] + 3*B*Sin[c])*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + ((5*A + (3*I)*B)*((d*x*Cos[c])/2 + (I/2)*d*x*Sin[c])*(Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + ((A + I*B)*(Cos[c]/4 - (I/4)*Sin[c])*(Cos[d*x] + I*Sin[d*x])*Sin[2*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + (Csc[c/2]*Csc[c + d*x]^3*Sec[c/2]*(Cos[d*x] + I*Sin[d*x])*((I/2)*A*Cos[c - d*x] - (I/2)*A*Cos[c + d*x] - (A*Sin[c - d*x])/2 + (A*Sin[c + d*x])/2)*(A + B*Tan[c + d*x]))/(6*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])) + (Csc[c/2]*Csc[c + d*x]*Sec[c/2]*(Cos[d*x] + I*Sin[d*x])*(((-7*I)/2)*A*Cos[c - d*x] + (3*B*Cos[c - d*x])/2 + ((7*I)/2)*A*Cos[c + d*x] - (3*B*Cos[c + d*x])/2 + (7*A*Sin[c - d*x])/2 + ((3*I)/2)*B*Sin[c - d*x] - (7*A*Sin[c + d*x])/2 - ((3*I)/2)*B*Sin[c + d*x])*(A + B*Tan[c + d*x]))/(6*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x]))","B",1
44,1,956,142,7.0875832,"\int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2,x]","\frac{i \sec (c) \sec ^2(c+d x) (-B \cos (2 c-d x)+B \cos (2 c+d x)-i B \sin (2 c-d x)+i B \sin (2 c+d x)) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{2 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{x \sec (c+d x) (-\tan (c) A+i A-2 B-2 i B \tan (c)+(A+2 i B) (-\cos (2 c)-i \sin (2 c)) \tan (c)) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{(A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}-\frac{(2 A+3 i B) \cos (2 d x) \sec (c+d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{4 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{\sec (c+d x) (A \cos (c)+2 i B \cos (c)+i A \sin (c)-2 B \sin (c)) \left(\tan ^{-1}(\tan (d x)) \sin (c)-i \tan ^{-1}(\tan (d x)) \cos (c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{\sec (c+d x) (A \cos (c)+2 i B \cos (c)+i A \sin (c)-2 B \sin (c)) \left(\frac{1}{2} \cos (c) \log \left(\cos ^2(c+d x)\right)+\frac{1}{2} i \sin (c) \log \left(\cos ^2(c+d x)\right)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{(A+i B) \cos (4 d x) \sec (c+d x) \left(\frac{1}{16} \cos (2 c)-\frac{1}{16} i \sin (2 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{(3 B-i A) \sec (c+d x) \left(\frac{3}{4} d x \cos (2 c)+\frac{3}{4} i d x \sin (2 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{i (2 A+3 i B) \sec (c+d x) \sin (2 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{4 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{(B-i A) \sec (c+d x) \left(\frac{1}{16} \cos (2 c)-\frac{1}{16} i \sin (2 c)\right) \sin (4 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}","\frac{(A+2 i B) \tan ^2(c+d x)}{2 a^2 d (1+i \tan (c+d x))}+\frac{3 (-3 B+i A) \tan (c+d x)}{4 a^2 d}+\frac{(A+2 i B) \log (\cos (c+d x))}{a^2 d}-\frac{3 x (-3 B+i A)}{4 a^2}+\frac{(-B+i A) \tan ^3(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"-1/4*((2*A + (3*I)*B)*Cos[2*d*x]*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + (Sec[c + d*x]*(A*Cos[c] + (2*I)*B*Cos[c] + I*A*Sin[c] - 2*B*Sin[c])*((-I)*ArcTan[Tan[d*x]]*Cos[c] + ArcTan[Tan[d*x]]*Sin[c])*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + (Sec[c + d*x]*(A*Cos[c] + (2*I)*B*Cos[c] + I*A*Sin[c] - 2*B*Sin[c])*((Cos[c]*Log[Cos[c + d*x]^2])/2 + (I/2)*Log[Cos[c + d*x]^2]*Sin[c])*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + ((A + I*B)*Cos[4*d*x]*Sec[c + d*x]*(Cos[2*c]/16 - (I/16)*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + (((-I)*A + 3*B)*Sec[c + d*x]*((3*d*x*Cos[2*c])/4 + ((3*I)/4)*d*x*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + ((I/4)*(2*A + (3*I)*B)*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*Sin[2*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + (((-I)*A + B)*Sec[c + d*x]*(Cos[2*c]/16 - (I/16)*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^2*Sin[4*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + ((I/2)*Sec[c]*Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^2*(-(B*Cos[2*c - d*x]) + B*Cos[2*c + d*x] - I*B*Sin[2*c - d*x] + I*B*Sin[2*c + d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + (x*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*(I*A - 2*B - A*Tan[c] - (2*I)*B*Tan[c] + (A + (2*I)*B)*(-Cos[2*c] - I*Sin[2*c])*Tan[c])*(A + B*Tan[c + d*x]))/((A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2)","B",1
45,1,185,103,1.0677987,"\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec ^2(c+d x) \left(\cos (2 (c+d x)) \left(4 A d x+i A-8 B \log \left(\cos ^2(c+d x)\right)-4 i B d x-B\right)+4 i A d x \sin (2 (c+d x))+A \sin (2 (c+d x))-4 i A+i B \sin (2 (c+d x))+4 B d x \sin (2 (c+d x))-8 i B \sin (2 (c+d x)) \log \left(\cos ^2(c+d x)\right)+16 i B \tan ^{-1}(\tan (d x)) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+8 B\right)}{16 a^2 d (\tan (c+d x)-i)^2}","\frac{-3 B+i A}{4 a^2 d (1+i \tan (c+d x))}-\frac{x (A+3 i B)}{4 a^2}+\frac{B \log (\cos (c+d x))}{a^2 d}+\frac{(-B+i A) \tan ^2(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^2*((-4*I)*A + 8*B + Cos[2*(c + d*x)]*(I*A - B + 4*A*d*x - (4*I)*B*d*x - 8*B*Log[Cos[c + d*x]^2]) + (16*I)*B*ArcTan[Tan[d*x]]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + A*Sin[2*(c + d*x)] + I*B*Sin[2*(c + d*x)] + (4*I)*A*d*x*Sin[2*(c + d*x)] + 4*B*d*x*Sin[2*(c + d*x)] - (8*I)*B*Log[Cos[c + d*x]^2]*Sin[2*(c + d*x)]))/(16*a^2*d*(-I + Tan[c + d*x])^2)","A",1
46,1,92,76,0.5856013,"\int \frac{\tan (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec ^2(c+d x) ((-4 A d x-i A+4 i B d x+B) \sin (2 (c+d x))+(4 i A d x+A+B (4 d x+i)) \cos (2 (c+d x))-4 i B)}{16 a^2 d (\tan (c+d x)-i)^2}","\frac{A+3 i B}{4 a^2 d (1+i \tan (c+d x))}-\frac{x (B+i A)}{4 a^2}-\frac{A+i B}{4 d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^2*((-4*I)*B + (A + (4*I)*A*d*x + B*(I + 4*d*x))*Cos[2*(c + d*x)] + ((-I)*A + B - 4*A*d*x + (4*I)*B*d*x)*Sin[2*(c + d*x)]))/(16*a^2*d*(-I + Tan[c + d*x])^2)","A",1
47,1,94,80,0.6362986,"\int \frac{A+B \tan (c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^2,x]","-\frac{\sec ^2(c+d x) ((4 i A d x+A+4 B d x+i B) \sin (2 (c+d x))+(A (4 d x+i)+B (-1-4 i d x)) \cos (2 (c+d x))+4 i A)}{16 a^2 d (\tan (c+d x)-i)^2}","\frac{B+i A}{4 d \left(a^2+i a^2 \tan (c+d x)\right)}+\frac{x (A-i B)}{4 a^2}+\frac{-B+i A}{4 d (a+i a \tan (c+d x))^2}",1,"-1/16*(Sec[c + d*x]^2*((4*I)*A + (B*(-1 - (4*I)*d*x) + A*(I + 4*d*x))*Cos[2*(c + d*x)] + (A + I*B + (4*I)*A*d*x + 4*B*d*x)*Sin[2*(c + d*x)]))/(a^2*d*(-I + Tan[c + d*x])^2)","A",1
48,1,184,95,1.1836519,"\int \frac{\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2,x]","-\frac{i \sec ^2(c+d x) \left(\cos (2 (c+d x)) \left(-8 i A \log \left(\sin ^2(c+d x)\right)+4 A d x-i A-4 i B d x+B\right)+4 i A d x \sin (2 (c+d x))-A \sin (2 (c+d x))+8 A \sin (2 (c+d x)) \log \left(\sin ^2(c+d x)\right)-16 A \tan ^{-1}(\tan (d x)) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-8 i A-i B \sin (2 (c+d x))+4 B d x \sin (2 (c+d x))+4 B\right)}{16 a^2 d (\tan (c+d x)-i)^2}","\frac{3 A+i B}{4 a^2 d (1+i \tan (c+d x))}-\frac{x (-B+3 i A)}{4 a^2}+\frac{A \log (\sin (c+d x))}{a^2 d}+\frac{A+i B}{4 d (a+i a \tan (c+d x))^2}",1,"((-1/16*I)*Sec[c + d*x]^2*((-8*I)*A + 4*B + Cos[2*(c + d*x)]*((-I)*A + B + 4*A*d*x - (4*I)*B*d*x - (8*I)*A*Log[Sin[c + d*x]^2]) - 16*A*ArcTan[Tan[d*x]]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) - A*Sin[2*(c + d*x)] - I*B*Sin[2*(c + d*x)] + (4*I)*A*d*x*Sin[2*(c + d*x)] + 4*B*d*x*Sin[2*(c + d*x)] + 8*A*Log[Sin[c + d*x]^2]*Sin[2*(c + d*x)]))/(a^2*d*(-I + Tan[c + d*x])^2)","A",1
49,1,960,141,7.1922782,"\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2,x]","\frac{\csc (c) \csc (c+d x) \sec (c+d x) \left(\frac{1}{2} i A \cos (2 c-d x)-\frac{1}{2} i A \cos (2 c+d x)-\frac{1}{2} A \sin (2 c-d x)+\frac{1}{2} A \sin (2 c+d x)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{(2 B-3 i A) \cos (2 d x) \sec (c+d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{4 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{\sec (c+d x) (-2 i A \cos (c)+B \cos (c)+2 A \sin (c)+i B \sin (c)) \left(\tan ^{-1}(\tan (d x)) \sin (c)-i \tan ^{-1}(\tan (d x)) \cos (c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{\sec (c+d x) (-2 i A \cos (c)+B \cos (c)+2 A \sin (c)+i B \sin (c)) \left(\frac{1}{2} \cos (c) \log \left(\sin ^2(c+d x)\right)+\frac{1}{2} i \sin (c) \log \left(\sin ^2(c+d x)\right)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{x \sec (c+d x) (2 i \cot (c) A-2 A-i B-B \cot (c)+(B-2 i A) \cot (c) (\cos (2 c)+i \sin (2 c))) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{(A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{(B-i A) \cos (4 d x) \sec (c+d x) \left(\frac{1}{16} \cos (2 c)-\frac{1}{16} i \sin (2 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{(3 A+i B) \sec (c+d x) \left(-\frac{3}{4} d x \cos (2 c)-\frac{3}{4} i d x \sin (2 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}-\frac{(3 A+2 i B) \sec (c+d x) \sin (2 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{4 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{(A+i B) \sec (c+d x) \left(\frac{1}{16} i \sin (2 c)-\frac{1}{16} \cos (2 c)\right) \sin (4 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}","-\frac{3 (3 A+i B) \cot (c+d x)}{4 a^2 d}-\frac{(-B+2 i A) \log (\sin (c+d x))}{a^2 d}+\frac{(2 A+i B) \cot (c+d x)}{2 a^2 d (1+i \tan (c+d x))}-\frac{3 x (3 A+i B)}{4 a^2}+\frac{(A+i B) \cot (c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"(((-3*I)*A + 2*B)*Cos[2*d*x]*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(4*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + (Sec[c + d*x]*((-2*I)*A*Cos[c] + B*Cos[c] + 2*A*Sin[c] + I*B*Sin[c])*((-I)*ArcTan[Tan[d*x]]*Cos[c] + ArcTan[Tan[d*x]]*Sin[c])*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + (Sec[c + d*x]*((-2*I)*A*Cos[c] + B*Cos[c] + 2*A*Sin[c] + I*B*Sin[c])*((Cos[c]*Log[Sin[c + d*x]^2])/2 + (I/2)*Log[Sin[c + d*x]^2]*Sin[c])*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + (x*Sec[c + d*x]*(-2*A - I*B + (2*I)*A*Cot[c] - B*Cot[c] + ((-2*I)*A + B)*Cot[c]*(Cos[2*c] + I*Sin[2*c]))*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/((A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + (((-I)*A + B)*Cos[4*d*x]*Sec[c + d*x]*(Cos[2*c]/16 - (I/16)*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + ((3*A + I*B)*Sec[c + d*x]*((-3*d*x*Cos[2*c])/4 - ((3*I)/4)*d*x*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) - ((3*A + (2*I)*B)*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*Sin[2*d*x]*(A + B*Tan[c + d*x]))/(4*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + ((A + I*B)*Sec[c + d*x]*(-1/16*Cos[2*c] + (I/16)*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^2*Sin[4*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + (Csc[c]*Csc[c + d*x]*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*((I/2)*A*Cos[2*c - d*x] - (I/2)*A*Cos[2*c + d*x] - (A*Sin[2*c - d*x])/2 + (A*Sin[2*c + d*x])/2)*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2)","B",1
50,1,1112,170,7.4523272,"\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2,x]","\frac{\csc (c) \csc (c+d x) \sec (c+d x) \left(A \cos (2 c-d x)+\frac{1}{2} i B \cos (2 c-d x)-A \cos (2 c+d x)-\frac{1}{2} i B \cos (2 c+d x)+i A \sin (2 c-d x)-\frac{1}{2} B \sin (2 c-d x)-i A \sin (2 c+d x)+\frac{1}{2} B \sin (2 c+d x)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}-\frac{(4 A+3 i B) \cos (2 d x) \sec (c+d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{4 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{\sec (c+d x) (2 A \cos (c)+i B \cos (c)+2 i A \sin (c)-B \sin (c)) \left(2 i \tan ^{-1}(\tan (d x)) \cos (c)-2 \tan ^{-1}(\tan (d x)) \sin (c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{\sec (c+d x) (2 A \cos (c)+i B \cos (c)+2 i A \sin (c)-B \sin (c)) \left(-\cos (c) \log \left(\sin ^2(c+d x)\right)-i \sin (c) \log \left(\sin ^2(c+d x)\right)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{x \sec (c+d x) (4 \cot (c) A+4 i A-2 B+2 i B \cot (c)+(2 A+i B) \cot (c) (-2 \cos (2 c)-2 i \sin (2 c))) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{(A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{(A+i B) \cos (4 d x) \sec (c+d x) \left(\frac{1}{16} i \sin (2 c)-\frac{1}{16} \cos (2 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{\csc ^2(c+d x) \sec (c+d x) \left(-\frac{1}{2} A \cos (2 c)-\frac{1}{2} i A \sin (2 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{(5 A+3 i B) \sec (c+d x) \left(\frac{3}{4} i d x \cos (2 c)-\frac{3}{4} d x \sin (2 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{i (4 A+3 i B) \sec (c+d x) \sin (2 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{4 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}+\frac{(A+i B) \sec (c+d x) \left(\frac{1}{16} i \cos (2 c)+\frac{1}{16} \sin (2 c)\right) \sin (4 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^2}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^2}","-\frac{(2 A+i B) \cot ^2(c+d x)}{a^2 d}+\frac{3 (-3 B+5 i A) \cot (c+d x)}{4 a^2 d}-\frac{2 (2 A+i B) \log (\sin (c+d x))}{a^2 d}+\frac{(5 A+3 i B) \cot ^2(c+d x)}{4 a^2 d (1+i \tan (c+d x))}+\frac{3 x (-3 B+5 i A)}{4 a^2}+\frac{(A+i B) \cot ^2(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"-1/4*((4*A + (3*I)*B)*Cos[2*d*x]*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + (Sec[c + d*x]*(2*A*Cos[c] + I*B*Cos[c] + (2*I)*A*Sin[c] - B*Sin[c])*((2*I)*ArcTan[Tan[d*x]]*Cos[c] - 2*ArcTan[Tan[d*x]]*Sin[c])*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + (Sec[c + d*x]*(2*A*Cos[c] + I*B*Cos[c] + (2*I)*A*Sin[c] - B*Sin[c])*(-(Cos[c]*Log[Sin[c + d*x]^2]) - I*Log[Sin[c + d*x]^2]*Sin[c])*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + (x*Sec[c + d*x]*((4*I)*A - 2*B + 4*A*Cot[c] + (2*I)*B*Cot[c] + (2*A + I*B)*Cot[c]*(-2*Cos[2*c] - (2*I)*Sin[2*c]))*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/((A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + ((A + I*B)*Cos[4*d*x]*Sec[c + d*x]*(-1/16*Cos[2*c] + (I/16)*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + (Csc[c + d*x]^2*Sec[c + d*x]*(-1/2*(A*Cos[2*c]) - (I/2)*A*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + ((5*A + (3*I)*B)*Sec[c + d*x]*(((3*I)/4)*d*x*Cos[2*c] - (3*d*x*Sin[2*c])/4)*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + ((I/4)*(4*A + (3*I)*B)*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*Sin[2*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + ((A + I*B)*Sec[c + d*x]*((I/16)*Cos[2*c] + Sin[2*c]/16)*(Cos[d*x] + I*Sin[d*x])^2*Sin[4*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2) + (Csc[c]*Csc[c + d*x]*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[2*c - d*x] + (I/2)*B*Cos[2*c - d*x] - A*Cos[2*c + d*x] - (I/2)*B*Cos[2*c + d*x] + I*A*Sin[2*c - d*x] - (B*Sin[2*c - d*x])/2 - I*A*Sin[2*c + d*x] + (B*Sin[2*c + d*x])/2)*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2)","B",1
51,1,1251,191,7.2589104,"\int \frac{\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (-B \cos (3 c-d x)+B \cos (3 c+d x)-i B \sin (3 c-d x)+i B \sin (3 c+d x)) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{2 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{x \sec ^2(c+d x) \left(-\frac{1}{2} A \cos ^3(c)-\frac{3}{2} i B \cos ^3(c)-2 i A \sin (c) \cos ^2(c)+6 B \sin (c) \cos ^2(c)+3 A \sin ^2(c) \cos (c)+9 i B \sin ^2(c) \cos (c)+\frac{1}{2} A \cos (c)+\frac{3}{2} i B \cos (c)+2 i A \sin ^3(c)-6 B \sin ^3(c)+i A \sin (c)-3 B \sin (c)-\frac{1}{2} A \sin ^3(c) \tan (c)-\frac{3}{2} i B \sin ^3(c) \tan (c)-\frac{1}{2} A \sin (c) \tan (c)-\frac{3}{2} i B \sin (c) \tan (c)+i (A+3 i B) (\cos (3 c)+i \sin (3 c)) \tan (c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{(A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(11 A+23 i B) \cos (2 d x) \sec ^2(c+d x) \left(\frac{1}{16} i \cos (c)-\frac{\sin (c)}{16}\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(7 B-5 i A) \cos (4 d x) \sec ^2(c+d x) \left(\frac{\cos (c)}{32}-\frac{1}{32} i \sin (c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{\sec ^2(c+d x) \left(-i A \cos \left(\frac{3 c}{2}\right)+3 B \cos \left(\frac{3 c}{2}\right)+A \sin \left(\frac{3 c}{2}\right)+3 i B \sin \left(\frac{3 c}{2}\right)\right) \left(\cos \left(\frac{3 c}{2}\right) \log (\cos (c+d x))+i \sin \left(\frac{3 c}{2}\right) \log (\cos (c+d x))\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(A+i B) \cos (6 d x) \sec ^2(c+d x) \left(\frac{1}{48} i \cos (3 c)+\frac{1}{48} \sin (3 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(7 A+25 i B) \sec ^2(c+d x) \left(-\frac{1}{8} d x \cos (3 c)-\frac{1}{8} i d x \sin (3 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(11 A+23 i B) \sec ^2(c+d x) \left(\frac{\cos (c)}{16}+\frac{1}{16} i \sin (c)\right) \sin (2 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(5 A+7 i B) \sec ^2(c+d x) \left(\frac{1}{32} i \sin (c)-\frac{\cos (c)}{32}\right) \sin (4 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(A+i B) \sec ^2(c+d x) \left(\frac{1}{48} \cos (3 c)-\frac{1}{48} i \sin (3 c)\right) \sin (6 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}","-\frac{(-3 B+i A) \tan ^2(c+d x)}{2 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{(7 A+25 i B) \tan (c+d x)}{8 a^3 d}-\frac{(-3 B+i A) \log (\cos (c+d x))}{a^3 d}-\frac{x (7 A+25 i B)}{8 a^3}+\frac{(-B+i A) \tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(5 A+11 i B) \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}",1,"((11*A + (23*I)*B)*Cos[2*d*x]*Sec[c + d*x]^2*((I/16)*Cos[c] - Sin[c]/16)*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (((-5*I)*A + 7*B)*Cos[4*d*x]*Sec[c + d*x]^2*(Cos[c]/32 - (I/32)*Sin[c])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (Sec[c + d*x]^2*((-I)*A*Cos[(3*c)/2] + 3*B*Cos[(3*c)/2] + A*Sin[(3*c)/2] + (3*I)*B*Sin[(3*c)/2])*(Cos[(3*c)/2]*Log[Cos[c + d*x]] + I*Log[Cos[c + d*x]]*Sin[(3*c)/2])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((A + I*B)*Cos[6*d*x]*Sec[c + d*x]^2*((I/48)*Cos[3*c] + Sin[3*c]/48)*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((7*A + (25*I)*B)*Sec[c + d*x]^2*(-1/8*(d*x*Cos[3*c]) - (I/8)*d*x*Sin[3*c])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((11*A + (23*I)*B)*Sec[c + d*x]^2*(Cos[c]/16 + (I/16)*Sin[c])*(Cos[d*x] + I*Sin[d*x])^3*Sin[2*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((5*A + (7*I)*B)*Sec[c + d*x]^2*(-1/32*Cos[c] + (I/32)*Sin[c])*(Cos[d*x] + I*Sin[d*x])^3*Sin[4*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((A + I*B)*Sec[c + d*x]^2*(Cos[3*c]/48 - (I/48)*Sin[3*c])*(Cos[d*x] + I*Sin[d*x])^3*Sin[6*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^3*(-(B*Cos[3*c - d*x]) + B*Cos[3*c + d*x] - I*B*Sin[3*c - d*x] + I*B*Sin[3*c + d*x])*(A + B*Tan[c + d*x]))/(2*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (x*Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^3*((A*Cos[c])/2 + ((3*I)/2)*B*Cos[c] - (A*Cos[c]^3)/2 - ((3*I)/2)*B*Cos[c]^3 + I*A*Sin[c] - 3*B*Sin[c] - (2*I)*A*Cos[c]^2*Sin[c] + 6*B*Cos[c]^2*Sin[c] + 3*A*Cos[c]*Sin[c]^2 + (9*I)*B*Cos[c]*Sin[c]^2 + (2*I)*A*Sin[c]^3 - 6*B*Sin[c]^3 - (A*Sin[c]*Tan[c])/2 - ((3*I)/2)*B*Sin[c]*Tan[c] - (A*Sin[c]^3*Tan[c])/2 - ((3*I)/2)*B*Sin[c]^3*Tan[c] + I*(A + (3*I)*B)*(Cos[3*c] + I*Sin[3*c])*Tan[c])*(A + B*Tan[c + d*x]))/((A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3)","B",1
52,1,178,148,1.4870305,"\int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) ((-51 B+9 i A) \cos (c+d x)-2 \cos (3 (c+d x)) (6 A d x-i A-48 B \log (\cos (c+d x))+42 i B d x+B)-27 A \sin (c+d x)+2 A \sin (3 (c+d x))-12 i A d x \sin (3 (c+d x))-81 i B \sin (c+d x)+2 i B \sin (3 (c+d x))+84 B d x \sin (3 (c+d x))+96 i B \sin (3 (c+d x)) \log (\cos (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3}","\frac{A+7 i B}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{x (-7 B+i A)}{8 a^3}-\frac{i B \log (\cos (c+d x))}{a^3 d}+\frac{(-B+i A) \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(A+3 i B) \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^3*(((9*I)*A - 51*B)*Cos[c + d*x] - 2*Cos[3*(c + d*x)]*((-I)*A + B + 6*A*d*x + (42*I)*B*d*x - 48*B*Log[Cos[c + d*x]]) - 27*A*Sin[c + d*x] - (81*I)*B*Sin[c + d*x] + 2*A*Sin[3*(c + d*x)] + (2*I)*B*Sin[3*(c + d*x)] - (12*I)*A*d*x*Sin[3*(c + d*x)] + 84*B*d*x*Sin[3*(c + d*x)] + (96*I)*B*Log[Cos[c + d*x]]*Sin[3*(c + d*x)]))/(96*a^3*d*(-I + Tan[c + d*x])^3)","A",1
53,1,147,124,1.2150186,"\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (-9 (A-i B) \cos (c+d x)+2 (-6 i A d x+A-6 B d x+i B) \cos (3 (c+d x))-3 i A \sin (c+d x)-2 i A \sin (3 (c+d x))+12 A d x \sin (3 (c+d x))-27 B \sin (c+d x)+2 B \sin (3 (c+d x))-12 i B d x \sin (3 (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3}","\frac{17 B+i A}{24 d \left(a^3+i a^3 \tan (c+d x)\right)}-\frac{x (A-i B)}{8 a^3}+\frac{(-B+i A) \tan ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{-7 B+i A}{24 a d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^3*(-9*(A - I*B)*Cos[c + d*x] + 2*(A + I*B - (6*I)*A*d*x - 6*B*d*x)*Cos[3*(c + d*x)] - (3*I)*A*Sin[c + d*x] - 27*B*Sin[c + d*x] - (2*I)*A*Sin[3*(c + d*x)] + 2*B*Sin[3*(c + d*x)] + 12*A*d*x*Sin[3*(c + d*x)] - (12*I)*B*d*x*Sin[3*(c + d*x)]))/(96*a^3*d*(-I + Tan[c + d*x])^3)","A",1
54,1,148,110,1.5475614,"\int \frac{\tan (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]","\frac{(\cos (3 (c+d x))-i \sin (3 (c+d x))) (3 (A+3 i B) \cos (c+d x)-2 (6 i A d x+A+B (6 d x+i)) \cos (3 (c+d x))+9 i A \sin (c+d x)+2 i A \sin (3 (c+d x))+12 A d x \sin (3 (c+d x))-3 B \sin (c+d x)-2 B \sin (3 (c+d x))-12 i B d x \sin (3 (c+d x)))}{96 a^3 d}","\frac{A-i B}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}-\frac{x (B+i A)}{8 a^3}+\frac{A+3 i B}{8 a d (a+i a \tan (c+d x))^2}-\frac{A+i B}{6 d (a+i a \tan (c+d x))^3}",1,"((Cos[3*(c + d*x)] - I*Sin[3*(c + d*x)])*(3*(A + (3*I)*B)*Cos[c + d*x] - 2*(A + (6*I)*A*d*x + B*(I + 6*d*x))*Cos[3*(c + d*x)] + (9*I)*A*Sin[c + d*x] - 3*B*Sin[c + d*x] + (2*I)*A*Sin[3*(c + d*x)] - 2*B*Sin[3*(c + d*x)] + 12*A*d*x*Sin[3*(c + d*x)] - (12*I)*B*d*x*Sin[3*(c + d*x)]))/(96*a^3*d)","A",1
55,1,150,112,0.9384609,"\int \frac{A+B \tan (c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) ((-27 A+3 i B) \cos (c+d x)+2 (6 i A d x-A+6 B d x-i B) \cos (3 (c+d x))-9 i A \sin (c+d x)+2 i A \sin (3 (c+d x))-12 A d x \sin (3 (c+d x))-9 B \sin (c+d x)-2 B \sin (3 (c+d x))+12 i B d x \sin (3 (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3}","\frac{B+i A}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{x (A-i B)}{8 a^3}+\frac{-B+i A}{6 d (a+i a \tan (c+d x))^3}+\frac{B+i A}{8 a d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^3*((-27*A + (3*I)*B)*Cos[c + d*x] + 2*(-A - I*B + (6*I)*A*d*x + 6*B*d*x)*Cos[3*(c + d*x)] - (9*I)*A*Sin[c + d*x] - 9*B*Sin[c + d*x] + (2*I)*A*Sin[3*(c + d*x)] - 2*B*Sin[3*(c + d*x)] - 12*A*d*x*Sin[3*(c + d*x)] + (12*I)*B*d*x*Sin[3*(c + d*x)]))/(96*a^3*d*(-I + Tan[c + d*x])^3)","A",1
56,1,180,131,1.3334093,"\int \frac{\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) ((-27 B+81 i A) \cos (c+d x)+2 \cos (3 (c+d x)) (48 i A \log (\sin (c+d x))+42 A d x+i A+6 i B d x-B)-51 A \sin (c+d x)+2 A \sin (3 (c+d x))+84 i A d x \sin (3 (c+d x))-96 A \sin (3 (c+d x)) \log (\sin (c+d x))-9 i B \sin (c+d x)+2 i B \sin (3 (c+d x))-12 B d x \sin (3 (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3}","\frac{7 A+i B}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}-\frac{x (-B+7 i A)}{8 a^3}+\frac{A \log (\sin (c+d x))}{a^3 d}+\frac{A+i B}{6 d (a+i a \tan (c+d x))^3}+\frac{3 A+i B}{8 a d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^3*(((81*I)*A - 27*B)*Cos[c + d*x] + 2*Cos[3*(c + d*x)]*(I*A - B + 42*A*d*x + (6*I)*B*d*x + (48*I)*A*Log[Sin[c + d*x]]) - 51*A*Sin[c + d*x] - (9*I)*B*Sin[c + d*x] + 2*A*Sin[3*(c + d*x)] + (2*I)*B*Sin[3*(c + d*x)] + (84*I)*A*d*x*Sin[3*(c + d*x)] - 12*B*d*x*Sin[3*(c + d*x)] - 96*A*Log[Sin[c + d*x]]*Sin[3*(c + d*x)]))/(96*a^3*d*(-I + Tan[c + d*x])^3)","A",1
57,1,1282,183,7.3854352,"\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]","\frac{\csc \left(\frac{c}{2}\right) \csc (c+d x) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(\frac{1}{2} i A \cos (3 c-d x)-\frac{1}{2} i A \cos (3 c+d x)-\frac{1}{2} A \sin (3 c-d x)+\frac{1}{2} A \sin (3 c+d x)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{2 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(5 B-7 i A) \cos (4 d x) \sec ^2(c+d x) \left(\frac{\cos (c)}{32}-\frac{1}{32} i \sin (c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(11 B-23 i A) \cos (2 d x) \sec ^2(c+d x) \left(\frac{\cos (c)}{16}+\frac{1}{16} i \sin (c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{\sec ^2(c+d x) \left(-3 i A \cos \left(\frac{3 c}{2}\right)+B \cos \left(\frac{3 c}{2}\right)+3 A \sin \left(\frac{3 c}{2}\right)+i B \sin \left(\frac{3 c}{2}\right)\right) \left(\tan ^{-1}(\tan (d x)) \sin \left(\frac{3 c}{2}\right)-i \tan ^{-1}(\tan (d x)) \cos \left(\frac{3 c}{2}\right)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{\sec ^2(c+d x) \left(-3 i A \cos \left(\frac{3 c}{2}\right)+B \cos \left(\frac{3 c}{2}\right)+3 A \sin \left(\frac{3 c}{2}\right)+i B \sin \left(\frac{3 c}{2}\right)\right) \left(\frac{1}{2} \cos \left(\frac{3 c}{2}\right) \log \left(\sin ^2(c+d x)\right)+\frac{1}{2} i \sin \left(\frac{3 c}{2}\right) \log \left(\sin ^2(c+d x)\right)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{x \sec ^2(c+d x) (-6 A \cos (c)-2 i B \cos (c)+3 i A \cot (c) \cos (c)-B \cot (c) \cos (c)-3 i A \sin (c)+B \sin (c)+(B-3 i A) \cot (c) (\cos (3 c)+i \sin (3 c))) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{(A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(B-i A) \cos (6 d x) \sec ^2(c+d x) \left(\frac{1}{48} \cos (3 c)-\frac{1}{48} i \sin (3 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(25 A+7 i B) \sec ^2(c+d x) \left(-\frac{1}{8} d x \cos (3 c)-\frac{1}{8} i d x \sin (3 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(23 A+11 i B) \sec ^2(c+d x) \left(-\frac{\cos (c)}{16}-\frac{1}{16} i \sin (c)\right) \sin (2 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(7 A+5 i B) \sec ^2(c+d x) \left(\frac{1}{32} i \sin (c)-\frac{\cos (c)}{32}\right) \sin (4 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(A+i B) \sec ^2(c+d x) \left(\frac{1}{48} i \sin (3 c)-\frac{1}{48} \cos (3 c)\right) \sin (6 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}","-\frac{(25 A+7 i B) \cot (c+d x)}{8 a^3 d}-\frac{(-B+3 i A) \log (\sin (c+d x))}{a^3 d}+\frac{(3 A+i B) \cot (c+d x)}{2 d \left(a^3+i a^3 \tan (c+d x)\right)}-\frac{x (25 A+7 i B)}{8 a^3}+\frac{(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}",1,"(((-7*I)*A + 5*B)*Cos[4*d*x]*Sec[c + d*x]^2*(Cos[c]/32 - (I/32)*Sin[c])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (((-23*I)*A + 11*B)*Cos[2*d*x]*Sec[c + d*x]^2*(Cos[c]/16 + (I/16)*Sin[c])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (Sec[c + d*x]^2*((-3*I)*A*Cos[(3*c)/2] + B*Cos[(3*c)/2] + 3*A*Sin[(3*c)/2] + I*B*Sin[(3*c)/2])*((-I)*ArcTan[Tan[d*x]]*Cos[(3*c)/2] + ArcTan[Tan[d*x]]*Sin[(3*c)/2])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (Sec[c + d*x]^2*((-3*I)*A*Cos[(3*c)/2] + B*Cos[(3*c)/2] + 3*A*Sin[(3*c)/2] + I*B*Sin[(3*c)/2])*((Cos[(3*c)/2]*Log[Sin[c + d*x]^2])/2 + (I/2)*Log[Sin[c + d*x]^2]*Sin[(3*c)/2])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (x*Sec[c + d*x]^2*(-6*A*Cos[c] - (2*I)*B*Cos[c] + (3*I)*A*Cos[c]*Cot[c] - B*Cos[c]*Cot[c] - (3*I)*A*Sin[c] + B*Sin[c] + ((-3*I)*A + B)*Cot[c]*(Cos[3*c] + I*Sin[3*c]))*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/((A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (((-I)*A + B)*Cos[6*d*x]*Sec[c + d*x]^2*(Cos[3*c]/48 - (I/48)*Sin[3*c])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((25*A + (7*I)*B)*Sec[c + d*x]^2*(-1/8*(d*x*Cos[3*c]) - (I/8)*d*x*Sin[3*c])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((23*A + (11*I)*B)*Sec[c + d*x]^2*(-1/16*Cos[c] - (I/16)*Sin[c])*(Cos[d*x] + I*Sin[d*x])^3*Sin[2*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((7*A + (5*I)*B)*Sec[c + d*x]^2*(-1/32*Cos[c] + (I/32)*Sin[c])*(Cos[d*x] + I*Sin[d*x])^3*Sin[4*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((A + I*B)*Sec[c + d*x]^2*(-1/48*Cos[3*c] + (I/48)*Sin[3*c])*(Cos[d*x] + I*Sin[d*x])^3*Sin[6*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (Csc[c/2]*Csc[c + d*x]*Sec[c/2]*Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^3*((I/2)*A*Cos[3*c - d*x] - (I/2)*A*Cos[3*c + d*x] - (A*Sin[3*c - d*x])/2 + (A*Sin[3*c + d*x])/2)*(A + B*Tan[c + d*x]))/(2*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3)","B",1
58,1,1448,216,7.6165325,"\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]","\frac{\csc \left(\frac{c}{2}\right) \csc (c+d x) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(\frac{3}{2} A \cos (3 c-d x)+\frac{1}{2} i B \cos (3 c-d x)-\frac{3}{2} A \cos (3 c+d x)-\frac{1}{2} i B \cos (3 c+d x)+\frac{3}{2} i A \sin (3 c-d x)-\frac{1}{2} B \sin (3 c-d x)-\frac{3}{2} i A \sin (3 c+d x)+\frac{1}{2} B \sin (3 c+d x)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{2 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(9 A+7 i B) \cos (4 d x) \sec ^2(c+d x) \left(\frac{1}{32} i \sin (c)-\frac{\cos (c)}{32}\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(39 A+23 i B) \cos (2 d x) \sec ^2(c+d x) \left(-\frac{\cos (c)}{16}-\frac{1}{16} i \sin (c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{\sec ^2(c+d x) \left(7 A \cos \left(\frac{3 c}{2}\right)+3 i B \cos \left(\frac{3 c}{2}\right)+7 i A \sin \left(\frac{3 c}{2}\right)-3 B \sin \left(\frac{3 c}{2}\right)\right) \left(i \tan ^{-1}(\tan (d x)) \cos \left(\frac{3 c}{2}\right)-\tan ^{-1}(\tan (d x)) \sin \left(\frac{3 c}{2}\right)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{\sec ^2(c+d x) \left(7 A \cos \left(\frac{3 c}{2}\right)+3 i B \cos \left(\frac{3 c}{2}\right)+7 i A \sin \left(\frac{3 c}{2}\right)-3 B \sin \left(\frac{3 c}{2}\right)\right) \left(-\frac{1}{2} \cos \left(\frac{3 c}{2}\right) \log \left(\sin ^2(c+d x)\right)-\frac{1}{2} i \sin \left(\frac{3 c}{2}\right) \log \left(\sin ^2(c+d x)\right)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{x \sec ^2(c+d x) (14 i A \cos (c)-6 B \cos (c)+7 A \cot (c) \cos (c)+3 i B \cot (c) \cos (c)-7 A \sin (c)-3 i B \sin (c)+(7 A+3 i B) \cot (c) (-\cos (3 c)-i \sin (3 c))) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{(A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(A+i B) \cos (6 d x) \sec ^2(c+d x) \left(\frac{1}{48} i \sin (3 c)-\frac{1}{48} \cos (3 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{\csc ^2(c+d x) \sec ^2(c+d x) \left(-\frac{1}{2} A \cos (3 c)-\frac{1}{2} i A \sin (3 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(11 A+5 i B) \sec ^2(c+d x) \left(\frac{5}{8} i d x \cos (3 c)-\frac{5}{8} d x \sin (3 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(39 A+23 i B) \sec ^2(c+d x) \left(\frac{1}{16} i \cos (c)-\frac{\sin (c)}{16}\right) \sin (2 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(9 A+7 i B) \sec ^2(c+d x) \left(\frac{1}{32} i \cos (c)+\frac{\sin (c)}{32}\right) \sin (4 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(A+i B) \sec ^2(c+d x) \left(\frac{1}{48} i \cos (3 c)+\frac{1}{48} \sin (3 c)\right) \sin (6 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}","-\frac{(7 A+3 i B) \cot ^2(c+d x)}{2 a^3 d}+\frac{5 (-5 B+11 i A) \cot (c+d x)}{8 a^3 d}-\frac{(7 A+3 i B) \log (\sin (c+d x))}{a^3 d}+\frac{5 (11 A+5 i B) \cot ^2(c+d x)}{24 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{5 x (-5 B+11 i A)}{8 a^3}+\frac{(13 A+7 i B) \cot ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{(A+i B) \cot ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}",1,"((9*A + (7*I)*B)*Cos[4*d*x]*Sec[c + d*x]^2*(-1/32*Cos[c] + (I/32)*Sin[c])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((39*A + (23*I)*B)*Cos[2*d*x]*Sec[c + d*x]^2*(-1/16*Cos[c] - (I/16)*Sin[c])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (Sec[c + d*x]^2*(7*A*Cos[(3*c)/2] + (3*I)*B*Cos[(3*c)/2] + (7*I)*A*Sin[(3*c)/2] - 3*B*Sin[(3*c)/2])*(I*ArcTan[Tan[d*x]]*Cos[(3*c)/2] - ArcTan[Tan[d*x]]*Sin[(3*c)/2])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (Sec[c + d*x]^2*(7*A*Cos[(3*c)/2] + (3*I)*B*Cos[(3*c)/2] + (7*I)*A*Sin[(3*c)/2] - 3*B*Sin[(3*c)/2])*(-1/2*(Cos[(3*c)/2]*Log[Sin[c + d*x]^2]) - (I/2)*Log[Sin[c + d*x]^2]*Sin[(3*c)/2])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (x*Sec[c + d*x]^2*((14*I)*A*Cos[c] - 6*B*Cos[c] + 7*A*Cos[c]*Cot[c] + (3*I)*B*Cos[c]*Cot[c] - 7*A*Sin[c] - (3*I)*B*Sin[c] + (7*A + (3*I)*B)*Cot[c]*(-Cos[3*c] - I*Sin[3*c]))*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/((A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((A + I*B)*Cos[6*d*x]*Sec[c + d*x]^2*(-1/48*Cos[3*c] + (I/48)*Sin[3*c])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (Csc[c + d*x]^2*Sec[c + d*x]^2*(-1/2*(A*Cos[3*c]) - (I/2)*A*Sin[3*c])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((11*A + (5*I)*B)*Sec[c + d*x]^2*(((5*I)/8)*d*x*Cos[3*c] - (5*d*x*Sin[3*c])/8)*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((39*A + (23*I)*B)*Sec[c + d*x]^2*((I/16)*Cos[c] - Sin[c]/16)*(Cos[d*x] + I*Sin[d*x])^3*Sin[2*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((9*A + (7*I)*B)*Sec[c + d*x]^2*((I/32)*Cos[c] + Sin[c]/32)*(Cos[d*x] + I*Sin[d*x])^3*Sin[4*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((A + I*B)*Sec[c + d*x]^2*((I/48)*Cos[3*c] + Sin[3*c]/48)*(Cos[d*x] + I*Sin[d*x])^3*Sin[6*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (Csc[c/2]*Csc[c + d*x]*Sec[c/2]*Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^3*((3*A*Cos[3*c - d*x])/2 + (I/2)*B*Cos[3*c - d*x] - (3*A*Cos[3*c + d*x])/2 - (I/2)*B*Cos[3*c + d*x] + ((3*I)/2)*A*Sin[3*c - d*x] - (B*Sin[3*c - d*x])/2 - ((3*I)/2)*A*Sin[3*c + d*x] + (B*Sin[3*c + d*x])/2)*(A + B*Tan[c + d*x]))/(2*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3)","B",1
59,1,195,185,1.4379312,"\int \frac{\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (16 (21 B-4 i A) \cos (2 (c+d x))+3 \cos (4 (c+d x)) (8 A d x+i A-128 B \log (\cos (c+d x))+120 i B d x-B)+32 A \sin (2 (c+d x))+24 i A d x \sin (4 (c+d x))+3 A \sin (4 (c+d x))+36 i A+288 i B \sin (2 (c+d x))+3 i B \sin (4 (c+d x))-360 B d x \sin (4 (c+d x))-384 i B \sin (4 (c+d x)) \log (\cos (c+d x))-96 B)}{384 a^4 d (\tan (c+d x)-i)^4}","-\frac{(-7 B+i A) \tan ^2(c+d x)}{16 a^4 d (1+i \tan (c+d x))^2}-\frac{-15 B+i A}{16 a^4 d (1+i \tan (c+d x))}+\frac{x (A+15 i B)}{16 a^4}-\frac{B \log (\cos (c+d x))}{a^4 d}+\frac{(-B+i A) \tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{(A+3 i B) \tan ^3(c+d x)}{12 a d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^4*((36*I)*A - 96*B + 16*((-4*I)*A + 21*B)*Cos[2*(c + d*x)] + 3*Cos[4*(c + d*x)]*(I*A - B + 8*A*d*x + (120*I)*B*d*x - 128*B*Log[Cos[c + d*x]]) + 32*A*Sin[2*(c + d*x)] + (288*I)*B*Sin[2*(c + d*x)] + 3*A*Sin[4*(c + d*x)] + (3*I)*B*Sin[4*(c + d*x)] + (24*I)*A*d*x*Sin[4*(c + d*x)] - 360*B*d*x*Sin[4*(c + d*x)] - (384*I)*B*Log[Cos[c + d*x]]*Sin[4*(c + d*x)]))/(384*a^4*d*(-I + Tan[c + d*x])^4)","A",1
60,1,158,159,1.5284257,"\int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (16 (A-4 i B) \cos (2 (c+d x))+3 (8 i A d x+A+8 B d x+i B) \cos (4 (c+d x))+32 i A \sin (2 (c+d x))-3 i A \sin (4 (c+d x))-24 A d x \sin (4 (c+d x))+32 B \sin (2 (c+d x))+24 i B d x \sin (4 (c+d x))+3 B \sin (4 (c+d x))+36 i B)}{384 a^4 d (\tan (c+d x)-i)^4}","\frac{5 A-29 i B}{48 a^4 d (1+i \tan (c+d x))}-\frac{A-13 i B}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{x (B+i A)}{16 a^4}+\frac{(-B+i A) \tan ^3(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{(A+5 i B) \tan ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^4*((36*I)*B + 16*(A - (4*I)*B)*Cos[2*(c + d*x)] + 3*(A + I*B + (8*I)*A*d*x + 8*B*d*x)*Cos[4*(c + d*x)] + (32*I)*A*Sin[2*(c + d*x)] + 32*B*Sin[2*(c + d*x)] - (3*I)*A*Sin[4*(c + d*x)] + 3*B*Sin[4*(c + d*x)] - 24*A*d*x*Sin[4*(c + d*x)] + (24*I)*B*d*x*Sin[4*(c + d*x)]))/(384*a^4*d*(-I + Tan[c + d*x])^4)","A",1
61,1,144,145,1.6822359,"\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4,x]","-\frac{(\cos (4 (c+d x))-i \sin (4 (c+d x))) (3 (8 A d x+i A-8 i B d x-B) \cos (4 (c+d x))+24 i A d x \sin (4 (c+d x))+3 A \sin (4 (c+d x))-12 i A-32 i B \sin (2 (c+d x))+3 i B \sin (4 (c+d x))+24 B d x \sin (4 (c+d x))-16 B \cos (2 (c+d x)))}{384 a^4 d}","-\frac{B+i A}{16 a^4 d (1+i \tan (c+d x))}+\frac{5 B+i A}{16 a^4 d (1+i \tan (c+d x))^2}-\frac{x (A-i B)}{16 a^4}+\frac{(-B+i A) \tan ^2(c+d x)}{8 d (a+i a \tan (c+d x))^4}-\frac{B}{6 a d (a+i a \tan (c+d x))^3}",1,"-1/384*((Cos[4*(c + d*x)] - I*Sin[4*(c + d*x)])*((-12*I)*A - 16*B*Cos[2*(c + d*x)] + 3*(I*A - B + 8*A*d*x - (8*I)*B*d*x)*Cos[4*(c + d*x)] - (32*I)*B*Sin[2*(c + d*x)] + 3*A*Sin[4*(c + d*x)] + (3*I)*B*Sin[4*(c + d*x)] + (24*I)*A*d*x*Sin[4*(c + d*x)] + 24*B*d*x*Sin[4*(c + d*x)]))/(a^4*d)","A",1
62,1,141,143,1.5094972,"\int \frac{\tan (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (-3 (8 i A d x+A+B (8 d x+i)) \cos (4 (c+d x))+32 i A \sin (2 (c+d x))+3 i A \sin (4 (c+d x))+24 A d x \sin (4 (c+d x))+16 A \cos (2 (c+d x))-24 i B d x \sin (4 (c+d x))-3 B \sin (4 (c+d x))+12 i B)}{384 a^4 d (\tan (c+d x)-i)^4}","\frac{A-i B}{16 d \left(a^4+i a^4 \tan (c+d x)\right)}-\frac{x (B+i A)}{16 a^4}+\frac{A-i B}{16 d \left(a^2+i a^2 \tan (c+d x)\right)^2}+\frac{A+3 i B}{12 a d (a+i a \tan (c+d x))^3}-\frac{A+i B}{8 d (a+i a \tan (c+d x))^4}",1,"(Sec[c + d*x]^4*((12*I)*B + 16*A*Cos[2*(c + d*x)] - 3*(A + (8*I)*A*d*x + B*(I + 8*d*x))*Cos[4*(c + d*x)] + (32*I)*A*Sin[2*(c + d*x)] + (3*I)*A*Sin[4*(c + d*x)] - 3*B*Sin[4*(c + d*x)] + 24*A*d*x*Sin[4*(c + d*x)] - (24*I)*B*d*x*Sin[4*(c + d*x)]))/(384*a^4*d*(-I + Tan[c + d*x])^4)","A",1
63,1,160,145,1.0331224,"\int \frac{A+B \tan (c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (16 (B+4 i A) \cos (2 (c+d x))+3 (8 A d x+i A-8 i B d x-B) \cos (4 (c+d x))-32 A \sin (2 (c+d x))+24 i A d x \sin (4 (c+d x))+3 A \sin (4 (c+d x))+36 i A+32 i B \sin (2 (c+d x))+3 i B \sin (4 (c+d x))+24 B d x \sin (4 (c+d x)))}{384 a^4 d (\tan (c+d x)-i)^4}","\frac{B+i A}{16 d \left(a^4+i a^4 \tan (c+d x)\right)}+\frac{x (A-i B)}{16 a^4}+\frac{B+i A}{16 d \left(a^2+i a^2 \tan (c+d x)\right)^2}+\frac{-B+i A}{8 d (a+i a \tan (c+d x))^4}+\frac{B+i A}{12 a d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^4*((36*I)*A + 16*((4*I)*A + B)*Cos[2*(c + d*x)] + 3*(I*A - B + 8*A*d*x - (8*I)*B*d*x)*Cos[4*(c + d*x)] - 32*A*Sin[2*(c + d*x)] + (32*I)*B*Sin[2*(c + d*x)] + 3*A*Sin[4*(c + d*x)] + (3*I)*B*Sin[4*(c + d*x)] + (24*I)*A*d*x*Sin[4*(c + d*x)] + 24*B*d*x*Sin[4*(c + d*x)]))/(384*a^4*d*(-I + Tan[c + d*x])^4)","A",1
64,1,193,162,1.428722,"\int \frac{\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (16 (21 A+4 i B) \cos (2 (c+d x))+3 \cos (4 (c+d x)) (128 A \log (\sin (c+d x))-120 i A d x+A+8 B d x+i B)+288 i A \sin (2 (c+d x))+360 A d x \sin (4 (c+d x))-3 i A \sin (4 (c+d x))+384 i A \sin (4 (c+d x)) \log (\sin (c+d x))+96 A-32 B \sin (2 (c+d x))+3 B \sin (4 (c+d x))+24 i B d x \sin (4 (c+d x))+36 i B)}{384 a^4 d (\tan (c+d x)-i)^4}","\frac{15 A+i B}{16 a^4 d (1+i \tan (c+d x))}+\frac{7 A+i B}{16 a^4 d (1+i \tan (c+d x))^2}-\frac{x (-B+15 i A)}{16 a^4}+\frac{A \log (\sin (c+d x))}{a^4 d}+\frac{A+i B}{8 d (a+i a \tan (c+d x))^4}+\frac{3 A+i B}{12 a d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^4*(96*A + (36*I)*B + 16*(21*A + (4*I)*B)*Cos[2*(c + d*x)] + 3*Cos[4*(c + d*x)]*(A + I*B - (120*I)*A*d*x + 8*B*d*x + 128*A*Log[Sin[c + d*x]]) + (288*I)*A*Sin[2*(c + d*x)] - 32*B*Sin[2*(c + d*x)] - (3*I)*A*Sin[4*(c + d*x)] + 3*B*Sin[4*(c + d*x)] + 360*A*d*x*Sin[4*(c + d*x)] + (24*I)*B*d*x*Sin[4*(c + d*x)] + (384*I)*A*Log[Sin[c + d*x]]*Sin[4*(c + d*x)]))/(384*a^4*d*(-I + Tan[c + d*x])^4)","A",1
65,1,1466,220,7.3111863,"\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4,x]","\frac{\csc (c) \csc (c+d x) \sec ^3(c+d x) \left(\frac{1}{2} i A \cos (4 c-d x)-\frac{1}{2} i A \cos (4 c+d x)-\frac{1}{2} A \sin (4 c-d x)+\frac{1}{2} A \sin (4 c+d x)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{(8 B-15 i A) \cos (4 d x) \sec ^3(c+d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{32 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{(3 B-4 i A) \cos (6 d x) \sec ^3(c+d x) \left(\frac{1}{48} \cos (2 c)-\frac{1}{48} i \sin (2 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{(13 B-36 i A) \cos (2 d x) \sec ^3(c+d x) \left(\frac{1}{16} \cos (2 c)+\frac{1}{16} i \sin (2 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{\sec ^3(c+d x) (-4 i A \cos (2 c)+B \cos (2 c)+4 A \sin (2 c)+i B \sin (2 c)) \left(\tan ^{-1}(\tan (d x)) \sin (2 c)-i \tan ^{-1}(\tan (d x)) \cos (2 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{\sec ^3(c+d x) (-4 i A \cos (2 c)+B \cos (2 c)+4 A \sin (2 c)+i B \sin (2 c)) \left(\frac{1}{2} \cos (2 c) \log \left(\sin ^2(c+d x)\right)+\frac{1}{2} i \sin (2 c) \log \left(\sin ^2(c+d x)\right)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{x \sec ^3(c+d x) \left(-12 A \cos ^2(c)-3 i B \cos ^2(c)+4 i A \cot (c) \cos ^2(c)-B \cot (c) \cos ^2(c)-12 i A \sin (c) \cos (c)+3 B \sin (c) \cos (c)+4 A \sin ^2(c)+i B \sin ^2(c)+(B-4 i A) \cot (c) (\cos (4 c)+i \sin (4 c))\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{(A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{(B-i A) \cos (8 d x) \sec ^3(c+d x) \left(\frac{1}{128} \cos (4 c)-\frac{1}{128} i \sin (4 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{(13 A+3 i B) \sec ^3(c+d x) \left(-\frac{5}{16} d x \cos (4 c)-\frac{5}{16} i d x \sin (4 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{(36 A+13 i B) \sec ^3(c+d x) \left(-\frac{1}{16} \cos (2 c)-\frac{1}{16} i \sin (2 c)\right) \sin (2 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}-\frac{(15 A+8 i B) \sec ^3(c+d x) \sin (4 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{32 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{(4 A+3 i B) \sec ^3(c+d x) \left(\frac{1}{48} i \sin (2 c)-\frac{1}{48} \cos (2 c)\right) \sin (6 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{(A+i B) \sec ^3(c+d x) \left(\frac{1}{128} i \sin (4 c)-\frac{1}{128} \cos (4 c)\right) \sin (8 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}","-\frac{5 (13 A+3 i B) \cot (c+d x)}{16 a^4 d}-\frac{(-B+4 i A) \log (\sin (c+d x))}{a^4 d}+\frac{(4 A+i B) \cot (c+d x)}{2 a^4 d (1+i \tan (c+d x))}+\frac{(31 A+9 i B) \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}-\frac{5 x (13 A+3 i B)}{16 a^4}+\frac{(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}",1,"(((-15*I)*A + 8*B)*Cos[4*d*x]*Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(32*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (((-4*I)*A + 3*B)*Cos[6*d*x]*Sec[c + d*x]^3*(Cos[2*c]/48 - (I/48)*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (((-36*I)*A + 13*B)*Cos[2*d*x]*Sec[c + d*x]^3*(Cos[2*c]/16 + (I/16)*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (Sec[c + d*x]^3*((-4*I)*A*Cos[2*c] + B*Cos[2*c] + 4*A*Sin[2*c] + I*B*Sin[2*c])*((-I)*ArcTan[Tan[d*x]]*Cos[2*c] + ArcTan[Tan[d*x]]*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (Sec[c + d*x]^3*((-4*I)*A*Cos[2*c] + B*Cos[2*c] + 4*A*Sin[2*c] + I*B*Sin[2*c])*((Cos[2*c]*Log[Sin[c + d*x]^2])/2 + (I/2)*Log[Sin[c + d*x]^2]*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (x*Sec[c + d*x]^3*(-12*A*Cos[c]^2 - (3*I)*B*Cos[c]^2 + (4*I)*A*Cos[c]^2*Cot[c] - B*Cos[c]^2*Cot[c] - (12*I)*A*Cos[c]*Sin[c] + 3*B*Cos[c]*Sin[c] + 4*A*Sin[c]^2 + I*B*Sin[c]^2 + ((-4*I)*A + B)*Cot[c]*(Cos[4*c] + I*Sin[4*c]))*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/((A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (((-I)*A + B)*Cos[8*d*x]*Sec[c + d*x]^3*(Cos[4*c]/128 - (I/128)*Sin[4*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + ((13*A + (3*I)*B)*Sec[c + d*x]^3*((-5*d*x*Cos[4*c])/16 - ((5*I)/16)*d*x*Sin[4*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + ((36*A + (13*I)*B)*Sec[c + d*x]^3*(-1/16*Cos[2*c] - (I/16)*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^4*Sin[2*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) - ((15*A + (8*I)*B)*Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^4*Sin[4*d*x]*(A + B*Tan[c + d*x]))/(32*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + ((4*A + (3*I)*B)*Sec[c + d*x]^3*(-1/48*Cos[2*c] + (I/48)*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^4*Sin[6*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + ((A + I*B)*Sec[c + d*x]^3*(-1/128*Cos[4*c] + (I/128)*Sin[4*c])*(Cos[d*x] + I*Sin[d*x])^4*Sin[8*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (Csc[c]*Csc[c + d*x]*Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^4*((I/2)*A*Cos[4*c - d*x] - (I/2)*A*Cos[4*c + d*x] - (A*Sin[4*c - d*x])/2 + (A*Sin[4*c + d*x])/2)*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4)","B",1
66,1,1625,255,7.6334601,"\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4,x]","\frac{\csc (c) \csc (c+d x) \sec ^3(c+d x) \left(2 A \cos (4 c-d x)+\frac{1}{2} i B \cos (4 c-d x)-2 A \cos (4 c+d x)-\frac{1}{2} i B \cos (4 c+d x)+2 i A \sin (4 c-d x)-\frac{1}{2} B \sin (4 c-d x)-2 i A \sin (4 c+d x)+\frac{1}{2} B \sin (4 c+d x)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}-\frac{3 (8 A+5 i B) \cos (4 d x) \sec ^3(c+d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{32 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{(5 A+4 i B) \cos (6 d x) \sec ^3(c+d x) \left(\frac{1}{48} i \sin (2 c)-\frac{1}{48} \cos (2 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{(25 A+12 i B) \cos (2 d x) \sec ^3(c+d x) \left(-\frac{3}{16} \cos (2 c)-\frac{3}{16} i \sin (2 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{\sec ^3(c+d x) (11 A \cos (2 c)+4 i B \cos (2 c)+11 i A \sin (2 c)-4 B \sin (2 c)) \left(i \tan ^{-1}(\tan (d x)) \cos (2 c)-\tan ^{-1}(\tan (d x)) \sin (2 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{\sec ^3(c+d x) (11 A \cos (2 c)+4 i B \cos (2 c)+11 i A \sin (2 c)-4 B \sin (2 c)) \left(-\frac{1}{2} \cos (2 c) \log \left(\sin ^2(c+d x)\right)-\frac{1}{2} i \sin (2 c) \log \left(\sin ^2(c+d x)\right)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{x \sec ^3(c+d x) \left(33 i A \cos ^2(c)-12 B \cos ^2(c)+11 A \cot (c) \cos ^2(c)+4 i B \cot (c) \cos ^2(c)-33 A \sin (c) \cos (c)-12 i B \sin (c) \cos (c)-11 i A \sin ^2(c)+4 B \sin ^2(c)+(11 A+4 i B) \cot (c) (-\cos (4 c)-i \sin (4 c))\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{(A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{(A+i B) \cos (8 d x) \sec ^3(c+d x) \left(\frac{1}{128} i \sin (4 c)-\frac{1}{128} \cos (4 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{\csc ^2(c+d x) \sec ^3(c+d x) \left(-\frac{1}{2} A \cos (4 c)-\frac{1}{2} i A \sin (4 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{(35 A+13 i B) \sec ^3(c+d x) \left(\frac{5}{16} i d x \cos (4 c)-\frac{5}{16} d x \sin (4 c)\right) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{(25 A+12 i B) \sec ^3(c+d x) \left(\frac{3}{16} i \cos (2 c)-\frac{3}{16} \sin (2 c)\right) \sin (2 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{3 i (8 A+5 i B) \sec ^3(c+d x) \sin (4 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{32 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{(5 A+4 i B) \sec ^3(c+d x) \left(\frac{1}{48} i \cos (2 c)+\frac{1}{48} \sin (2 c)\right) \sin (6 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}+\frac{(A+i B) \sec ^3(c+d x) \left(\frac{1}{128} i \cos (4 c)+\frac{1}{128} \sin (4 c)\right) \sin (8 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^4}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}","-\frac{(11 A+4 i B) \cot ^2(c+d x)}{2 a^4 d}+\frac{5 (-13 B+35 i A) \cot (c+d x)}{16 a^4 d}-\frac{(11 A+4 i B) \log (\sin (c+d x))}{a^4 d}+\frac{5 (35 A+13 i B) \cot ^2(c+d x)}{48 a^4 d (1+i \tan (c+d x))}+\frac{(43 A+17 i B) \cot ^2(c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{5 x (-13 B+35 i A)}{16 a^4}+\frac{(2 A+i B) \cot ^2(c+d x)}{6 a d (a+i a \tan (c+d x))^3}+\frac{(A+i B) \cot ^2(c+d x)}{8 d (a+i a \tan (c+d x))^4}",1,"(-3*(8*A + (5*I)*B)*Cos[4*d*x]*Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(32*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + ((5*A + (4*I)*B)*Cos[6*d*x]*Sec[c + d*x]^3*(-1/48*Cos[2*c] + (I/48)*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + ((25*A + (12*I)*B)*Cos[2*d*x]*Sec[c + d*x]^3*((-3*Cos[2*c])/16 - ((3*I)/16)*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (Sec[c + d*x]^3*(11*A*Cos[2*c] + (4*I)*B*Cos[2*c] + (11*I)*A*Sin[2*c] - 4*B*Sin[2*c])*(I*ArcTan[Tan[d*x]]*Cos[2*c] - ArcTan[Tan[d*x]]*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (Sec[c + d*x]^3*(11*A*Cos[2*c] + (4*I)*B*Cos[2*c] + (11*I)*A*Sin[2*c] - 4*B*Sin[2*c])*(-1/2*(Cos[2*c]*Log[Sin[c + d*x]^2]) - (I/2)*Log[Sin[c + d*x]^2]*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (x*Sec[c + d*x]^3*((33*I)*A*Cos[c]^2 - 12*B*Cos[c]^2 + 11*A*Cos[c]^2*Cot[c] + (4*I)*B*Cos[c]^2*Cot[c] - 33*A*Cos[c]*Sin[c] - (12*I)*B*Cos[c]*Sin[c] - (11*I)*A*Sin[c]^2 + 4*B*Sin[c]^2 + (11*A + (4*I)*B)*Cot[c]*(-Cos[4*c] - I*Sin[4*c]))*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/((A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + ((A + I*B)*Cos[8*d*x]*Sec[c + d*x]^3*(-1/128*Cos[4*c] + (I/128)*Sin[4*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (Csc[c + d*x]^2*Sec[c + d*x]^3*(-1/2*(A*Cos[4*c]) - (I/2)*A*Sin[4*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + ((35*A + (13*I)*B)*Sec[c + d*x]^3*(((5*I)/16)*d*x*Cos[4*c] - (5*d*x*Sin[4*c])/16)*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + ((25*A + (12*I)*B)*Sec[c + d*x]^3*(((3*I)/16)*Cos[2*c] - (3*Sin[2*c])/16)*(Cos[d*x] + I*Sin[d*x])^4*Sin[2*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (((3*I)/32)*(8*A + (5*I)*B)*Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^4*Sin[4*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + ((5*A + (4*I)*B)*Sec[c + d*x]^3*((I/48)*Cos[2*c] + Sin[2*c]/48)*(Cos[d*x] + I*Sin[d*x])^4*Sin[6*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + ((A + I*B)*Sec[c + d*x]^3*((I/128)*Cos[4*c] + Sin[4*c]/128)*(Cos[d*x] + I*Sin[d*x])^4*Sin[8*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (Csc[c]*Csc[c + d*x]*Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^4*(2*A*Cos[4*c - d*x] + (I/2)*B*Cos[4*c - d*x] - 2*A*Cos[4*c + d*x] - (I/2)*B*Cos[4*c + d*x] + (2*I)*A*Sin[4*c - d*x] - (B*Sin[4*c - d*x])/2 - (2*I)*A*Sin[4*c + d*x] + (B*Sin[4*c + d*x])/2)*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4)","B",1
67,1,201,194,3.5656853,"\int \tan ^3(c+d x) \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{\sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \left(\frac{\sqrt{2} (A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)}{\sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}+\frac{2}{105} \sqrt{\sec (c+d x)} \left((-46 B-7 i A) \tan (c+d x)+3 \sec ^2(c+d x) (7 A+5 B \tan (c+d x)-i B)-112 A+46 i B\right)\right)}{d \sec ^{\frac{3}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","\frac{2 (7 A-i B) \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}-\frac{2 (7 A-31 i B) (a+i a \tan (c+d x))^{3/2}}{105 a d}-\frac{8 (7 A-i B) \sqrt{a+i a \tan (c+d x)}}{35 d}+\frac{\sqrt{2} \sqrt{a} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{2 B \tan ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}",1,"(Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x])*((Sqrt[2]*(A - I*B)*ArcSinh[E^(I*(c + d*x))])/(Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + (2*Sqrt[Sec[c + d*x]]*(-112*A + (46*I)*B + ((-7*I)*A - 46*B)*Tan[c + d*x] + 3*Sec[c + d*x]^2*(7*A - I*B + 5*B*Tan[c + d*x])))/105))/(d*Sec[c + d*x]^(3/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
68,1,184,143,2.9002314,"\int \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{\sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \left(\frac{\sqrt{2} (B+i A) \sinh ^{-1}\left(e^{i (c+d x)}\right)}{\sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}+\frac{2}{15} \sqrt{\sec (c+d x)} \left((5 A-i B) \tan (c+d x)-5 i A+3 B \sec ^2(c+d x)-16 B\right)\right)}{d \sec ^{\frac{3}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{2 (B+5 i A) (a+i a \tan (c+d x))^{3/2}}{15 a d}+\frac{\sqrt{2} \sqrt{a} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{2 B \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}-\frac{8 B \sqrt{a+i a \tan (c+d x)}}{5 d}",1,"(Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x])*((Sqrt[2]*(I*A + B)*ArcSinh[E^(I*(c + d*x))])/(Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + (2*Sqrt[Sec[c + d*x]]*((-5*I)*A - 16*B + 3*B*Sec[c + d*x]^2 + (5*A - I*B)*Tan[c + d*x]))/15))/(d*Sec[c + d*x]^(3/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
69,1,132,105,1.4386124,"\int \tan (c+d x) \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{e^{-i (c+d x)} \sqrt{a+i a \tan (c+d x)} \left(-3 (A-i B) \left(1+e^{2 i (c+d x)}\right)^{3/2} \sinh ^{-1}\left(e^{i (c+d x)}\right)+6 A e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)-4 i B e^{3 i (c+d x)}\right)}{3 d \left(1+e^{2 i (c+d x)}\right)}","-\frac{\sqrt{2} \sqrt{a} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{2 A \sqrt{a+i a \tan (c+d x)}}{d}-\frac{2 i B (a+i a \tan (c+d x))^{3/2}}{3 a d}",1,"(((-4*I)*B*E^((3*I)*(c + d*x)) + 6*A*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x))) - 3*(A - I*B)*(1 + E^((2*I)*(c + d*x)))^(3/2)*ArcSinh[E^(I*(c + d*x))])*Sqrt[a + I*a*Tan[c + d*x]])/(3*d*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x))))","A",1
70,1,87,75,1.2632742,"\int \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{e^{-i (c+d x)} \sqrt{a+i a \tan (c+d x)} \left(2 B e^{i (c+d x)}-i (A-i B) \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{d}","\frac{2 B \sqrt{a+i a \tan (c+d x)}}{d}-\frac{\sqrt{2} \sqrt{a} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}",1,"((2*B*E^(I*(c + d*x)) - I*(A - I*B)*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
71,1,113,86,1.8321737,"\int \cot (c+d x) \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sqrt{a+i a \tan (c+d x)} \left((A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)-\sqrt{2} A \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)\right)}{d}","\frac{\sqrt{2} \sqrt{a} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 \sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{d}",1,"(Sqrt[1 + E^((2*I)*(c + d*x))]*((A - I*B)*ArcSinh[E^(I*(c + d*x))] - Sqrt[2]*A*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
72,1,293,123,4.5698895,"\int \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{\sqrt{a+i a \tan (c+d x)} \left(-8 A \cot (c+d x)+e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(\sqrt{2} (2 B+i A) \left(\log \left(\left(-1+e^{i (c+d x)}\right)^2\right)-\log \left(\left(1+e^{i (c+d x)}\right)^2\right)+\log \left(-2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)-\log \left(2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)\right)+8 (B+i A) \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)\right)}{8 d}","-\frac{\sqrt{a} (2 B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{d}+\frac{\sqrt{2} \sqrt{a} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{A \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{d}",1,"((-8*A*Cot[c + d*x] + (Sqrt[1 + E^((2*I)*(c + d*x))]*(8*(I*A + B)*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*(I*A + 2*B)*(Log[(-1 + E^(I*(c + d*x)))^2] - Log[(1 + E^(I*(c + d*x)))^2] + Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] - 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])))/E^(I*(c + d*x)))*Sqrt[a + I*a*Tan[c + d*x]])/(8*d)","B",1
73,1,230,169,3.6295986,"\int \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{\sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \left(\frac{2 (7 A-4 i B) \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)-8 \sqrt{2} (A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)}{\sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}-\frac{2 \csc (c+d x) (2 A \csc (c+d x)+(4 B+i A) \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)}\right)}{8 d \sec ^{\frac{3}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","\frac{\sqrt{a} (7 A-4 i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{4 d}-\frac{\sqrt{2} \sqrt{a} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{(4 B+i A) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 d}-\frac{A \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 d}",1,"(((-8*Sqrt[2]*(A - I*B)*ArcSinh[E^(I*(c + d*x))] + 2*(7*A - (4*I)*B)*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]])/(Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) - (2*Csc[c + d*x]*(2*A*Csc[c + d*x] + (I*A + 4*B)*Sec[c + d*x]))/Sec[c + d*x]^(3/2))*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(8*d*Sec[c + d*x]^(3/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
74,1,414,210,4.9618842,"\int \cot ^4(c+d x) \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{\sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \left(-\frac{2 i \left((9 A-14 i B) \left(\log \left(\left(-1+e^{i (c+d x)}\right)^2\right)-\log \left(\left(1+e^{i (c+d x)}\right)^2\right)+\log \left(-2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)-\log \left(2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)\right)+32 \sqrt{2} (A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{\sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}-\frac{4 \csc ^3(c+d x) (2 (6 B+i A) \sin (2 (c+d x))+(29 A-6 i B) \cos (2 (c+d x))-13 A+6 i B)}{3 \sqrt{\sec (c+d x)}}\right)}{64 d \sec ^{\frac{3}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","\frac{\sqrt{a} (14 B+9 i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{8 d}-\frac{\sqrt{2} \sqrt{a} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{(6 B+i A) \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d}+\frac{(7 A-2 i B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{8 d}-\frac{A \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{3 d}",1,"((((-2*I)*(32*Sqrt[2]*(A - I*B)*ArcSinh[E^(I*(c + d*x))] + (9*A - (14*I)*B)*(Log[(-1 + E^(I*(c + d*x)))^2] - Log[(1 + E^(I*(c + d*x)))^2] + Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] - 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])))/(Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) - (4*Csc[c + d*x]^3*(-13*A + (6*I)*B + (29*A - (6*I)*B)*Cos[2*(c + d*x)] + 2*(I*A + 6*B)*Sin[2*(c + d*x)]))/(3*Sqrt[Sec[c + d*x]]))*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(64*d*Sec[c + d*x]^(3/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",0
75,1,239,197,4.4457484,"\int \tan ^2(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left(\frac{2 \sqrt{2} (B+i A) \sinh ^{-1}\left(e^{i (c+d x)}\right)}{\left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{3/2} \left(1+e^{2 i (c+d x)}\right)^{3/2}}-\frac{1}{210} (\tan (c+d x)+i) \sec ^{\frac{5}{2}}(c+d x) (21 (17 A-18 i B) \cos (c+d x)+(147 A-158 i B) \cos (3 (c+d x))+42 i A \sin (c+d x)+42 i A \sin (3 (c+d x))-7 B \sin (c+d x)+53 B \sin (3 (c+d x)))\right)}{d \sec ^{\frac{5}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","\frac{2 \sqrt{2} a^{3/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{2 a (8 B+7 i A) \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}-\frac{4 (19 B+21 i A) (a+i a \tan (c+d x))^{3/2}}{105 d}-\frac{8 a (8 B+7 i A) \sqrt{a+i a \tan (c+d x)}}{35 d}+\frac{2 i a B \tan ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}",1,"((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x])*((2*Sqrt[2]*(I*A + B)*ArcSinh[E^(I*(c + d*x))])/((E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^(3/2)) - (Sec[c + d*x]^(5/2)*(21*(17*A - (18*I)*B)*Cos[c + d*x] + (147*A - (158*I)*B)*Cos[3*(c + d*x)] + (42*I)*A*Sin[c + d*x] - 7*B*Sin[c + d*x] + (42*I)*A*Sin[3*(c + d*x)] + 53*B*Sin[3*(c + d*x)])*(I + Tan[c + d*x]))/210))/(d*Sec[c + d*x]^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
76,1,204,137,4.0674716,"\int \tan (c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left(\frac{1}{15} (\tan (c+d x)+i) \sec ^{\frac{3}{2}}(c+d x) ((5 A-6 i B) \sin (2 (c+d x))+(-21 B-20 i A) \cos (2 (c+d x))-20 i A-15 B)-\frac{2 \sqrt{2} (A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)}{\left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{3/2} \left(1+e^{2 i (c+d x)}\right)^{3/2}}\right)}{d \sec ^{\frac{5}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{2 \sqrt{2} a^{3/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{2 a (A-i B) \sqrt{a+i a \tan (c+d x)}}{d}+\frac{2 A (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac{2 i B (a+i a \tan (c+d x))^{5/2}}{5 a d}",1,"((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x])*((-2*Sqrt[2]*(A - I*B)*ArcSinh[E^(I*(c + d*x))])/((E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^(3/2)) + (Sec[c + d*x]^(3/2)*((-20*I)*A - 15*B + ((-20*I)*A - 21*B)*Cos[2*(c + d*x)] + (5*A - (6*I)*B)*Sin[2*(c + d*x)])*(I + Tan[c + d*x]))/15))/(d*Sec[c + d*x]^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
77,1,190,107,2.8533045,"\int (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left(\frac{2}{3} (\cos (c)-i \sin (c)) \sqrt{\sec (c+d x)} (\sin (d x)+i \cos (d x)) (3 A+B \tan (c+d x)-4 i B)-\frac{2 i \sqrt{2} (A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)}{\left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{3/2} \left(1+e^{2 i (c+d x)}\right)^{3/2}}\right)}{d \sec ^{\frac{5}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{2 \sqrt{2} a^{3/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{2 a (B+i A) \sqrt{a+i a \tan (c+d x)}}{d}+\frac{2 B (a+i a \tan (c+d x))^{3/2}}{3 d}",1,"((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x])*(((-2*I)*Sqrt[2]*(A - I*B)*ArcSinh[E^(I*(c + d*x))])/((E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^(3/2)) + (2*Sqrt[Sec[c + d*x]]*(Cos[c] - I*Sin[c])*(I*Cos[d*x] + Sin[d*x])*(3*A - (4*I)*B + B*Tan[c + d*x]))/3))/(d*Sec[c + d*x]^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
78,1,157,113,2.2577596,"\int \cot (c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{\sqrt{2} a e^{-i (c+d x)} \sqrt{a+i a \tan (c+d x)} \left(\sqrt{2} (A-i B) \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)-A \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)+i \sqrt{2} B e^{i (c+d x)}\right)}{d}","\frac{2 \sqrt{2} a^{3/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{d}+\frac{2 i a B \sqrt{a+i a \tan (c+d x)}}{d}",1,"(Sqrt[2]*a*(I*Sqrt[2]*B*E^(I*(c + d*x)) + Sqrt[2]*(A - I*B)*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))] - A*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
79,1,201,125,3.3161244,"\int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","-\frac{a e^{-\frac{1}{2} i (4 c+5 d x)} \left(1+e^{2 i (c+d x)}\right)^{3/2} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sec (c+d x) \left(\cos \left(\frac{d x}{2}\right)+i \sin \left(\frac{d x}{2}\right)\right) \left((-4 B-4 i A) \sinh ^{-1}\left(e^{i (c+d x)}\right)+\sqrt{2} (2 B+3 i A) \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)+A \sqrt{1+e^{2 i (c+d x)}} \csc (c+d x)\right)}{2 \sqrt{2} d}","-\frac{a^{3/2} (2 B+3 i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{d}+\frac{2 \sqrt{2} a^{3/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{a A \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{d}",1,"-1/2*(a*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))^(3/2)*(((-4*I)*A - 4*B)*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*((3*I)*A + 2*B)*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]] + A*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c + d*x])*Sec[c + d*x]*(Cos[(d*x)/2] + I*Sin[(d*x)/2]))/(Sqrt[2]*d*E^((I/2)*(4*c + 5*d*x)))","A",1
80,1,400,171,6.2206097,"\int \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left(\frac{-2 (11 A-12 i B) \left(\log \left(\left(-1+e^{i (c+d x)}\right)^2\right)-\log \left(\left(1+e^{i (c+d x)}\right)^2\right)+\log \left(-2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)-\log \left(2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)\right)-64 \sqrt{2} (A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)}{\left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{3/2} \left(1+e^{2 i (c+d x)}\right)^{3/2}}+\frac{8 i (\tan (c+d x)+i) \csc (c+d x) (2 A \csc (c+d x)+(4 B+5 i A) \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)}\right)}{32 d \sec ^{\frac{5}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","\frac{a^{3/2} (11 A-12 i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{4 d}-\frac{2 \sqrt{2} a^{3/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{a (4 B+5 i A) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 d}-\frac{a A \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 d}",1,"((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x])*((-64*Sqrt[2]*(A - I*B)*ArcSinh[E^(I*(c + d*x))] - 2*(11*A - (12*I)*B)*(Log[(-1 + E^(I*(c + d*x)))^2] - Log[(1 + E^(I*(c + d*x)))^2] + Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] - 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])]))/((E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^(3/2)) + ((8*I)*Csc[c + d*x]*(2*A*Csc[c + d*x] + ((5*I)*A + 4*B)*Sec[c + d*x])*(I + Tan[c + d*x]))/Sec[c + d*x]^(5/2)))/(32*d*Sec[c + d*x]^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
81,1,439,213,6.7990463,"\int \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left(-\frac{2 i \left((23 A-22 i B) \left(\log \left(\left(-1+e^{i (c+d x)}\right)^2\right)-\log \left(\left(1+e^{i (c+d x)}\right)^2\right)+\log \left(-2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)-\log \left(2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)\right)+64 \sqrt{2} (A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{\left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{3/2} \left(1+e^{2 i (c+d x)}\right)^{3/2}}-\frac{4 (\cos (c)-i \sin (c)) \csc ^3(c+d x) (2 (6 B+7 i A) \sin (2 (c+d x))+5 (7 A-6 i B) \cos (2 (c+d x))-19 A+30 i B)}{\sqrt{\sec (c+d x)} (3 \cos (d x)+3 i \sin (d x))}\right)}{64 d \sec ^{\frac{5}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","\frac{a^{3/2} (22 B+23 i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{8 d}-\frac{2 \sqrt{2} a^{3/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{a (6 B+7 i A) \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d}+\frac{a (9 A-10 i B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{8 d}-\frac{a A \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{3 d}",1,"((((-2*I)*(64*Sqrt[2]*(A - I*B)*ArcSinh[E^(I*(c + d*x))] + (23*A - (22*I)*B)*(Log[(-1 + E^(I*(c + d*x)))^2] - Log[(1 + E^(I*(c + d*x)))^2] + Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] - 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])))/((E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^(3/2)) - (4*Csc[c + d*x]^3*(Cos[c] - I*Sin[c])*(-19*A + (30*I)*B + 5*(7*A - (6*I)*B)*Cos[2*(c + d*x)] + 2*((7*I)*A + 6*B)*Sin[2*(c + d*x)]))/(Sqrt[Sec[c + d*x]]*(3*Cos[d*x] + (3*I)*Sin[d*x])))*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/(64*d*Sec[c + d*x]^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
82,1,284,246,6.0140009,"\int \tan ^2(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(4 \sqrt{2} (B+i A) e^{-3 i (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)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)-\frac{i (\cos (2 c)-i \sin (2 c)) \sec ^{\frac{9}{2}}(c+d x) (12 (260 A-251 i B) \cos (2 (c+d x))+(915 A-961 i B) \cos (4 (c+d x))+390 i A \sin (2 (c+d x))+285 i A \sin (4 (c+d x))+2205 A+282 B \sin (2 (c+d x))+331 B \sin (4 (c+d x))-2331 i B)}{1260 (\cos (d x)+i \sin (d x))^2}\right)}{d \sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","\frac{4 \sqrt{2} a^{5/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 a^2 (3 A-4 i B) \tan ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}+\frac{2 a^2 (46 B+45 i A) \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{8 a^2 (46 B+45 i A) \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{8 a (59 B+60 i A) (a+i a \tan (c+d x))^{3/2}}{315 d}+\frac{2 i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}",1,"(((4*Sqrt[2]*(I*A + B)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])/E^((3*I)*(c + d*x)) - ((I/1260)*Sec[c + d*x]^(9/2)*(Cos[2*c] - I*Sin[2*c])*(2205*A - (2331*I)*B + 12*(260*A - (251*I)*B)*Cos[2*(c + d*x)] + (915*A - (961*I)*B)*Cos[4*(c + d*x)] + (390*I)*A*Sin[2*(c + d*x)] + 282*B*Sin[2*(c + d*x)] + (285*I)*A*Sin[4*(c + d*x)] + 331*B*Sin[4*(c + d*x)]))/(Cos[d*x] + I*Sin[d*x])^2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*Sec[c + d*x]^(7/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
83,1,268,171,4.4738677,"\int \tan (c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(\frac{(\cos (2 c)-i \sin (2 c)) \sec ^{\frac{7}{2}}(c+d x) (21 (37 A-35 i B) \cos (c+d x)+(287 A-305 i B) \cos (3 (c+d x))+77 i A \sin (c+d x)+77 i A \sin (3 (c+d x))+35 B \sin (c+d x)+95 B \sin (3 (c+d x)))}{210 (\cos (d x)+i \sin (d x))^2}-4 \sqrt{2} (A-i B) e^{-3 i (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)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{d \sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{4 \sqrt{2} a^{5/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{4 a^2 (A-i B) \sqrt{a+i a \tan (c+d x)}}{d}+\frac{2 a (A-i B) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{2 A (a+i a \tan (c+d x))^{5/2}}{5 d}-\frac{2 i B (a+i a \tan (c+d x))^{7/2}}{7 a d}",1,"(((-4*Sqrt[2]*(A - I*B)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])/E^((3*I)*(c + d*x)) + (Sec[c + d*x]^(7/2)*(Cos[2*c] - I*Sin[2*c])*(21*(37*A - (35*I)*B)*Cos[c + d*x] + (287*A - (305*I)*B)*Cos[3*(c + d*x)] + (77*I)*A*Sin[c + d*x] + 35*B*Sin[c + d*x] + (77*I)*A*Sin[3*(c + d*x)] + 95*B*Sin[3*(c + d*x)]))/(210*(Cos[d*x] + I*Sin[d*x])^2))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*Sec[c + d*x]^(7/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
84,1,236,141,3.4560305,"\int (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(\frac{(\cos (2 c)-i \sin (2 c)) \sec ^{\frac{5}{2}}(c+d x) ((-5 A+11 i B) \sin (2 (c+d x))+(41 B+35 i A) \cos (2 (c+d x))+35 (B+i A))}{15 (\cos (d x)+i \sin (d x))^2}-4 i \sqrt{2} (A-i B) e^{-3 i (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)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{d \sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{4 \sqrt{2} a^{5/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{4 a^2 (B+i A) \sqrt{a+i a \tan (c+d x)}}{d}+\frac{2 a (B+i A) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{2 B (a+i a \tan (c+d x))^{5/2}}{5 d}",1,"((((-4*I)*Sqrt[2]*(A - I*B)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])/E^((3*I)*(c + d*x)) + (Sec[c + d*x]^(5/2)*(Cos[2*c] - I*Sin[2*c])*(35*(I*A + B) + ((35*I)*A + 41*B)*Cos[2*(c + d*x)] + (-5*A + (11*I)*B)*Sin[2*(c + d*x)]))/(15*(Cos[d*x] + I*Sin[d*x])^2))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*Sec[c + d*x]^(7/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
85,1,429,147,8.2726227,"\int \cot (c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{\cos ^3(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left((3 A-8 i B) \left(-\frac{2}{3} \cos (2 c)+\frac{2}{3} i \sin (2 c)\right)+\sec (c+d x) \left(-\frac{2}{3} B \sin (3 c+d x)-\frac{2}{3} i B \cos (3 c+d x)\right)\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{e^{-2 i c} \sqrt{e^{i d x}} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(8 (A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)+\sqrt{2} A \left(\log \left(1-e^{i (c+d x)}\right)-\log \left(1+e^{i (c+d x)}\right)+\log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-\log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+e^{i (c+d x)}+1\right)\right)\right)}{\sqrt{2} d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","\frac{4 \sqrt{2} a^{5/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{d}-\frac{2 a^2 (A-2 i B) \sqrt{a+i a \tan (c+d x)}}{d}+\frac{2 i a B (a+i a \tan (c+d x))^{3/2}}{3 d}",1,"(Sqrt[E^(I*d*x)]*(8*(A - I*B)*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*A*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))] + Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] - Log[1 + E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]]))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(Sqrt[2]*d*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))]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^3*((3*A - (8*I)*B)*((-2*Cos[2*c])/3 + ((2*I)/3)*Sin[2*c]) + Sec[c + d*x]*(((-2*I)/3)*B*Cos[3*c + d*x] - (2*B*Sin[3*c + d*x])/3))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
86,1,413,158,7.5544264,"\int \cot ^2(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(e^{-3 i (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(2 (2 B+5 i A) \left(\log \left(\left(-1+e^{i (c+d x)}\right)^2\right)-\log \left(\left(1+e^{i (c+d x)}\right)^2\right)+\log \left(-2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)-\log \left(2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)\right)+32 \sqrt{2} (B+i A) \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)-\frac{8 (\cos (2 c)-i \sin (2 c)) (A \csc (c+d x)+2 B \sec (c+d x))}{\sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x))^2}\right)}{8 d \sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{a^{5/2} (2 B+5 i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{d}+\frac{4 \sqrt{2} a^{5/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{a^2 (-2 B+i A) \sqrt{a+i a \tan (c+d x)}}{d}-\frac{a A \cot (c+d x) (a+i a \tan (c+d x))^{3/2}}{d}",1,"(((Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(32*Sqrt[2]*(I*A + B)*ArcSinh[E^(I*(c + d*x))] + 2*((5*I)*A + 2*B)*(Log[(-1 + E^(I*(c + d*x)))^2] - Log[(1 + E^(I*(c + d*x)))^2] + Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] - 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])))/E^((3*I)*(c + d*x)) - (8*(A*Csc[c + d*x] + 2*B*Sec[c + d*x])*(Cos[2*c] - I*Sin[2*c]))/(Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])^2))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(8*d*Sec[c + d*x]^(7/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
87,1,577,173,9.3381879,"\int \cot ^3(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{\cos ^3(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(\csc (c) \left(\frac{1}{4} \cos (2 c)-\frac{1}{4} i \sin (2 c)\right) \csc (c+d x) (4 B \sin (d x)+9 i A \sin (d x))-i \csc (c) \left(\frac{1}{4} \cos (2 c)-\frac{1}{4} i \sin (2 c)\right) (2 i A \sin (c)+9 A \cos (c)-4 i B \cos (c))+\left(-\frac{1}{2} A \cos (2 c)+\frac{1}{2} i A \sin (2 c)\right) \csc ^2(c+d x)\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}-\frac{e^{-2 i c} \sqrt{e^{i d x}} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(\sqrt{2} (23 A-20 i B) \left(\log \left(\left(-1+e^{i (c+d x)}\right)^2\right)-\log \left(\left(1+e^{i (c+d x)}\right)^2\right)+\log \left(-2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)-\log \left(2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)\right)+128 (A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{16 \sqrt{2} d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","\frac{a^{5/2} (23 A-20 i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{4 d}-\frac{4 \sqrt{2} a^{5/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{a^2 (4 B+7 i A) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 d}-\frac{a A \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2}}{2 d}",1,"-1/16*(Sqrt[E^(I*d*x)]*(128*(A - I*B)*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*(23*A - (20*I)*B)*(Log[(-1 + E^(I*(c + d*x)))^2] - Log[(1 + E^(I*(c + d*x)))^2] + Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] - 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])]))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(Sqrt[2]*d*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))]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^3*((-I)*Csc[c]*(9*A*Cos[c] - (4*I)*B*Cos[c] + (2*I)*A*Sin[c])*(Cos[2*c]/4 - (I/4)*Sin[2*c]) + Csc[c + d*x]^2*(-1/2*(A*Cos[2*c]) + (I/2)*A*Sin[2*c]) + Csc[c]*Csc[c + d*x]*(Cos[2*c]/4 - (I/4)*Sin[2*c])*((9*I)*A*Sin[d*x] + 4*B*Sin[d*x]))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
88,1,634,217,9.1604918,"\int \cot ^4(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{\cos ^3(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(\csc (c) \left(\frac{1}{12} \cos (2 c)-\frac{1}{12} i \sin (2 c)\right) \csc ^2(c+d x) (-13 i A \sin (c)-4 A \cos (c)-6 B \sin (c))+\csc (c) \left(\frac{1}{24} \cos (2 c)-\frac{1}{24} i \sin (2 c)\right) \csc (c+d x) (-65 A \sin (d x)+54 i B \sin (d x))+\csc (c) \left(\frac{1}{24} \cos (2 c)-\frac{1}{24} i \sin (2 c)\right) (26 i A \sin (c)+65 A \cos (c)+12 B \sin (c)-54 i B \cos (c))+A \csc (c) \left(\frac{1}{3} \cos (2 c)-\frac{1}{3} i \sin (2 c)\right) \sin (d x) \csc ^3(c+d x)\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}-\frac{i e^{-2 i c} \sqrt{e^{i d x}} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(\sqrt{2} (45 A-46 i B) \left(\log \left(\left(-1+e^{i (c+d x)}\right)^2\right)-\log \left(\left(1+e^{i (c+d x)}\right)^2\right)+\log \left(-2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)-\log \left(2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)\right)+256 (A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{32 \sqrt{2} d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","\frac{a^{5/2} (46 B+45 i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{8 d}-\frac{4 \sqrt{2} a^{5/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{a^2 (2 B+3 i A) \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{a^2 (19 A-18 i B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{8 d}-\frac{a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}",1,"((-1/32*I)*Sqrt[E^(I*d*x)]*(256*(A - I*B)*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*(45*A - (46*I)*B)*(Log[(-1 + E^(I*(c + d*x)))^2] - Log[(1 + E^(I*(c + d*x)))^2] + Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] - 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])]))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(Sqrt[2]*d*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))]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^3*(Csc[c]*(65*A*Cos[c] - (54*I)*B*Cos[c] + (26*I)*A*Sin[c] + 12*B*Sin[c])*(Cos[2*c]/24 - (I/24)*Sin[2*c]) + Csc[c]*Csc[c + d*x]^2*(-4*A*Cos[c] - (13*I)*A*Sin[c] - 6*B*Sin[c])*(Cos[2*c]/12 - (I/12)*Sin[2*c]) + A*Csc[c]*Csc[c + d*x]^3*(Cos[2*c]/3 - (I/3)*Sin[2*c])*Sin[d*x] + Csc[c]*Csc[c + d*x]*(Cos[2*c]/24 - (I/24)*Sin[2*c])*(-65*A*Sin[d*x] + (54*I)*B*Sin[d*x]))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
89,1,698,261,9.368616,"\int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{\cos ^3(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(\csc (c) \left(\frac{1}{24} \cos (2 c)-\frac{1}{24} i \sin (2 c)\right) \csc ^3(c+d x) (8 B \sin (d x)+17 i A \sin (d x))+\csc (c) \left(\frac{1}{192} \cos (3 c)-\frac{1}{192} i \sin (3 c)\right) \csc ^2(c+d x) (223 A \sin (2 c)-223 i A \cos (2 c)+87 i A-136 i B \sin (2 c)-136 B \cos (2 c)+72 B)+\csc (c) \left(\frac{1}{192} \cos (2 c)-\frac{1}{192} i \sin (2 c)\right) \csc (c+d x) (-520 B \sin (d x)-583 i A \sin (d x))+\csc (c) \left(\frac{1}{192} \cos (2 c)-\frac{1}{192} i \sin (2 c)\right) (-262 A \sin (c)+583 i A \cos (c)+208 i B \sin (c)+520 B \cos (c))+\left(-\frac{1}{4} A \cos (2 c)+\frac{1}{4} i A \sin (2 c)\right) \csc ^4(c+d x)\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{e^{-2 i c} \sqrt{e^{i d x}} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(3 \sqrt{2} (121 A-120 i B) \left(\log \left(\left(-1+e^{i (c+d x)}\right)^2\right)-\log \left(\left(1+e^{i (c+d x)}\right)^2\right)+\log \left(-2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)-\log \left(2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)\right)+2048 (A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{256 \sqrt{2} d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","-\frac{3 a^{5/2} (121 A-120 i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{64 d}+\frac{4 \sqrt{2} a^{5/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{24 d}+\frac{a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{96 d}+\frac{a^2 (152 B+149 i A) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{64 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}",1,"(Sqrt[E^(I*d*x)]*(2048*(A - I*B)*ArcSinh[E^(I*(c + d*x))] + 3*Sqrt[2]*(121*A - (120*I)*B)*(Log[(-1 + E^(I*(c + d*x)))^2] - Log[(1 + E^(I*(c + d*x)))^2] + Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] - 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])]))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(256*Sqrt[2]*d*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))]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^3*(Csc[c]*((583*I)*A*Cos[c] + 520*B*Cos[c] - 262*A*Sin[c] + (208*I)*B*Sin[c])*(Cos[2*c]/192 - (I/192)*Sin[2*c]) + Csc[c + d*x]^4*(-1/4*(A*Cos[2*c]) + (I/4)*A*Sin[2*c]) + Csc[c]*Csc[c + d*x]^2*((87*I)*A + 72*B - (223*I)*A*Cos[2*c] - 136*B*Cos[2*c] + 223*A*Sin[2*c] - (136*I)*B*Sin[2*c])*(Cos[3*c]/192 - (I/192)*Sin[3*c]) + Csc[c]*Csc[c + d*x]*(Cos[2*c]/192 - (I/192)*Sin[2*c])*((-583*I)*A*Sin[d*x] - 520*B*Sin[d*x]) + Csc[c]*Csc[c + d*x]^3*(Cos[2*c]/24 - (I/24)*Sin[2*c])*((17*I)*A*Sin[d*x] + 8*B*Sin[d*x]))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
90,1,176,205,3.8579129,"\int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{(A+B \tan (c+d x)) \left((A-i B) \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+\frac{1}{30} \sec ^2(c+d x) (5 (23 A+37 i B) \cos (c+d x)+(25 A+59 i B) \cos (3 (c+d x))+4 i \sin (c+d x) ((5 A+22 i B) \cos (2 (c+d x))+5 A+16 i B))\right)}{2 d \sqrt{a+i a \tan (c+d x)} (A \cos (c+d x)+B \sin (c+d x))}","-\frac{(25 A+23 i B) (a+i a \tan (c+d x))^{3/2}}{15 a^2 d}+\frac{(-B+i A) \tan ^3(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}-\frac{(5 A+7 i B) \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 a d}+\frac{4 (5 A+7 i B) \sqrt{a+i a \tan (c+d x)}}{5 a d}+\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}",1,"(((A - I*B)*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))] + (Sec[c + d*x]^2*(5*(23*A + (37*I)*B)*Cos[c + d*x] + (25*A + (59*I)*B)*Cos[3*(c + d*x)] + (4*I)*(5*A + (16*I)*B + (5*A + (22*I)*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/30)*(A + B*Tan[c + d*x]))/(2*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
91,1,147,159,2.7898022,"\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{(A+B \tan (c+d x)) \left((B+i A) \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+\frac{1}{3} \sec (c+d x) ((6 A+2 i B) \sin (2 (c+d x))+(5 B-9 i A) \cos (2 (c+d x))+9 (B-i A))\right)}{2 d \sqrt{a+i a \tan (c+d x)} (A \cos (c+d x)+B \sin (c+d x))}","\frac{(-5 B+3 i A) (a+i a \tan (c+d x))^{3/2}}{3 a^2 d}+\frac{(-B+i A) \tan ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}-\frac{4 (-B+i A) \sqrt{a+i a \tan (c+d x)}}{a d}+\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}",1,"(((I*A + B)*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))] + (Sec[c + d*x]*(9*((-I)*A + B) + ((-9*I)*A + 5*B)*Cos[2*(c + d*x)] + (6*A + (2*I)*B)*Sin[2*(c + d*x)]))/3)*(A + B*Tan[c + d*x]))/(2*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
92,1,140,109,1.7069732,"\int \frac{\tan (c+d x) (A+B \tan (c+d x))}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{e^{-2 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left((A-i B) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+A \left(1+e^{2 i (c+d x)}\right)+i B \left(1+5 e^{2 i (c+d x)}\right)\right)}{\sqrt{2} a d}","-\frac{A+i B}{d \sqrt{a+i a \tan (c+d x)}}-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}-\frac{2 i B \sqrt{a+i a \tan (c+d x)}}{a d}",1,"-((Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(A*(1 + E^((2*I)*(c + d*x))) + I*B*(1 + 5*E^((2*I)*(c + d*x))) + (A - I*B)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))]))/(Sqrt[2]*a*d*E^((2*I)*(c + d*x))))","A",1
93,1,129,82,1.3263393,"\int \frac{A+B \tan (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(A + B*Tan[c + d*x])/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{i e^{-2 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left((A+i B) \left(1+e^{2 i (c+d x)}\right)-(A-i B) 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} a d}","\frac{-B+i A}{d \sqrt{a+i a \tan (c+d x)}}-\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}",1,"(I*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*((A + I*B)*(1 + E^((2*I)*(c + d*x))) - (A - I*B)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))]))/(Sqrt[2]*a*d*E^((2*I)*(c + d*x)))","A",1
94,1,208,114,2.7292498,"\int \frac{\cot (c+d x) (A+B \tan (c+d x))}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left((A+i B) \sqrt{1+e^{2 i (c+d x)}}+(A-i B) e^{i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)-2 \sqrt{2} A e^{i (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)\right)}{2 d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","\frac{A+i B}{d \sqrt{a+i a \tan (c+d x)}}+\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}-\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"(((A + I*B)*Sqrt[1 + E^((2*I)*(c + d*x))] + (A - I*B)*E^(I*(c + d*x))*ArcSinh[E^(I*(c + d*x))] - 2*Sqrt[2]*A*E^(I*(c + d*x))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[Sec[c + d*x]])/(2*d*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))])","A",1
95,1,224,167,4.5012674,"\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{(A \cot (c+d x)+B) \left(2 (B-2 i A) \sin (c+d x)+\frac{\left(-1+e^{2 i (c+d x)}\right) \sqrt{\sec (c+d x)} \left((A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)+\sqrt{2} (A+2 i B) \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)\right)}{\sqrt{2} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}-2 A \cos (c+d x)\right)}{2 d \sqrt{a+i a \tan (c+d x)} (A \cos (c+d x)+B \sin (c+d x))}","\frac{(-2 B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}+\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}-\frac{(2 A+i B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{a d}+\frac{(A+i B) \cot (c+d x)}{d \sqrt{a+i a \tan (c+d x)}}",1,"((B + A*Cot[c + d*x])*(-2*A*Cos[c + d*x] + ((-1 + E^((2*I)*(c + d*x)))*((A - I*B)*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*(A + (2*I)*B)*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[Sec[c + d*x]])/(Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + 2*((-2*I)*A + B)*Sin[c + d*x]))/(2*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",0
96,1,363,219,5.1114016,"\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{(A+B \tan (c+d x)) \left(4 \cot (c+d x) \csc (c+d x) (i (A+4 i B) \sin (2 (c+d x))+(5 A+8 i B) \cos (2 (c+d x))-9 A-8 i B)-\sqrt{1+e^{2 i (c+d x)}} \left(\sqrt{2} (11 A+4 i B) \left(\log \left(\left(-1+e^{i (c+d x)}\right)^2\right)-\log \left(\left(1+e^{i (c+d x)}\right)^2\right)+\log \left(-2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)-\log \left(2 e^{i (c+d x)} \left(1+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)}+2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+3\right)\right)+16 (A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)\right)}{32 d \sqrt{a+i a \tan (c+d x)} (A \cos (c+d x)+B \sin (c+d x))}","\frac{(11 A+4 i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{4 \sqrt{a} d}-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}-\frac{(3 A+2 i B) \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a d}+\frac{(A+i B) \cot ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}+\frac{(-8 B+7 i A) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 a d}",1,"((-(Sqrt[1 + E^((2*I)*(c + d*x))]*(16*(A - I*B)*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*(11*A + (4*I)*B)*(Log[(-1 + E^(I*(c + d*x)))^2] - Log[(1 + E^(I*(c + d*x)))^2] + Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] - 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])]))) + 4*Cot[c + d*x]*Csc[c + d*x]*(-9*A - (8*I)*B + (5*A + (8*I)*B)*Cos[2*(c + d*x)] + I*(A + (4*I)*B)*Sin[2*(c + d*x)]))*(A + B*Tan[c + d*x]))/(32*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
97,1,176,209,4.7650998,"\int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{i \sec ^3(c+d x) (21 (3 A+5 i B) \cos (c+d x)+(37 A+51 i B) \cos (3 (c+d x))+2 i \sin (c+d x) ((39 A+53 i B) \cos (2 (c+d x))+39 A+61 i B))-\frac{24 i (A-i B) e^{3 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)}{\left(1+e^{2 i (c+d x)}\right)^{3/2}}}{24 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(11 A+21 i B) (a+i a \tan (c+d x))^{3/2}}{6 a^3 d}-\frac{2 (3 A+5 i B) \sqrt{a+i a \tan (c+d x)}}{a^2 d}+\frac{(-B+i A) \tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(3 A+5 i B) \tan ^2(c+d x)}{2 a d \sqrt{a+i a \tan (c+d x)}}",1,"(((-24*I)*(A - I*B)*E^((3*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])/(1 + E^((2*I)*(c + d*x)))^(3/2) + I*Sec[c + d*x]^3*(21*(3*A + (5*I)*B)*Cos[c + d*x] + (37*A + (51*I)*B)*Cos[3*(c + d*x)] + (2*I)*(39*A + (61*I)*B + (39*A + (53*I)*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(24*a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
98,1,167,167,3.3309528,"\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{3 (A-i B) e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+A \left(7 e^{2 i (c+d x)}+8 e^{4 i (c+d x)}-1\right)+i B \left(13 e^{2 i (c+d x)}+38 e^{4 i (c+d x)}-1\right)}{3 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{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(-7 B+i A) \sqrt{a+i a \tan (c+d x)}}{3 a^2 d}+\frac{(-B+i A) \tan ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{-11 B+5 i A}{6 a d \sqrt{a+i a \tan (c+d x)}}",1,"(A*(-1 + 7*E^((2*I)*(c + d*x)) + 8*E^((4*I)*(c + d*x))) + I*B*(-1 + 13*E^((2*I)*(c + d*x)) + 38*E^((4*I)*(c + d*x))) + 3*(A - I*B)*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])/(3*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
99,1,145,119,2.6640735,"\int \frac{\tan (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{\sqrt{1+e^{2 i (c+d x)}} \left(B \left(-1+8 e^{2 i (c+d x)}\right)-i A \left(-1+2 e^{2 i (c+d x)}\right)\right)+3 (B+i A) e^{3 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)}{3 a d \left(1+e^{2 i (c+d x)}\right)^{3/2} (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{A+i B}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{A+3 i B}{2 a d \sqrt{a+i a \tan (c+d x)}}",1,"(Sqrt[1 + E^((2*I)*(c + d*x))]*((-I)*A*(-1 + 2*E^((2*I)*(c + d*x))) + B*(-1 + 8*E^((2*I)*(c + d*x)))) + 3*(I*A + B)*E^((3*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])/(3*a*d*(1 + E^((2*I)*(c + d*x)))^(3/2)*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
100,1,143,121,2.586179,"\int \frac{A+B \tan (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{\sqrt{1+e^{2 i (c+d x)}} \left(4 A e^{2 i (c+d x)}+A-i B \left(-1+2 e^{2 i (c+d x)}\right)\right)-3 (A-i B) e^{3 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)}{3 a d \left(1+e^{2 i (c+d x)}\right)^{3/2} (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","-\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{-B+i A}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{B+i A}{2 a d \sqrt{a+i a \tan (c+d x)}}",1,"(Sqrt[1 + E^((2*I)*(c + d*x))]*(A + 4*A*E^((2*I)*(c + d*x)) - I*B*(-1 + 2*E^((2*I)*(c + d*x)))) - 3*(A - I*B)*E^((3*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])/(3*a*d*(1 + E^((2*I)*(c + d*x)))^(3/2)*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
101,1,192,156,4.3705321,"\int \frac{\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{-\frac{12 i (A-i B) e^{3 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)}{\left(1+e^{2 i (c+d x)}\right)^{3/2}}+18 A \tan (c+d x)+\frac{48 i \sqrt{2} A e^{3 i (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{\left(1+e^{2 i (c+d x)}\right)^{3/2}}-22 i A+6 i B \tan (c+d x)+10 B}{12 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{A+i B}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{3 A+i B}{2 a d \sqrt{a+i a \tan (c+d x)}}",1,"((-22*I)*A + 10*B - ((12*I)*(A - I*B)*E^((3*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])/(1 + E^((2*I)*(c + d*x)))^(3/2) + ((48*I)*Sqrt[2]*A*E^((3*I)*(c + d*x))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]])/(1 + E^((2*I)*(c + d*x)))^(3/2) + 18*A*Tan[c + d*x] + (6*I)*B*Tan[c + d*x])/(12*a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
102,1,259,217,5.3853831,"\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{\sqrt{\sec (c+d x)} (A+B \tan (c+d x)) \left(\sqrt{2} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{3/2} \left(1+e^{2 i (c+d x)}\right)^{3/2} \left((B+i A) \sinh ^{-1}\left(e^{i (c+d x)}\right)+2 \sqrt{2} (-2 B+3 i A) \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)\right)-\frac{\csc (c+d x) ((-11 B+29 i A) \sin (2 (c+d x))+9 (3 A+i B) \cos (2 (c+d x))-3 (5 A+3 i B))}{3 \sqrt{\sec (c+d x)}}\right)}{4 d (a+i a \tan (c+d x))^{3/2} (A \cos (c+d x)+B \sin (c+d x))}","\frac{(-2 B+3 i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(7 A+3 i B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a^2 d}+\frac{(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}+\frac{(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}",1,"(Sqrt[Sec[c + d*x]]*(Sqrt[2]*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^(3/2)*((I*A + B)*ArcSinh[E^(I*(c + d*x))] + 2*Sqrt[2]*((3*I)*A - 2*B)*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]]) - (Csc[c + d*x]*(-3*(5*A + (3*I)*B) + 9*(3*A + I*B)*Cos[2*(c + d*x)] + ((29*I)*A - 11*B)*Sin[2*(c + d*x)]))/(3*Sqrt[Sec[c + d*x]]))*(A + B*Tan[c + d*x]))/(4*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^(3/2))","A",1
103,1,283,268,6.1306311,"\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{\sqrt{\sec (c+d x)} (A+B \tan (c+d x)) \left(\sqrt{2} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{3/2} \left(1+e^{2 i (c+d x)}\right)^{3/2} \left(\sqrt{2} (23 A+12 i B) \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)-2 (A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)+\frac{\csc ^2(c+d x) (-((50 A+29 i B) \cos (c+d x))+(38 A+29 i B) \cos (3 (c+d x))+6 \sin (c+d x) ((-9 B+12 i A) \cos (2 (c+d x))-9 i A+5 B))}{3 \sqrt{\sec (c+d x)}}\right)}{8 d (a+i a \tan (c+d x))^{3/2} (A \cos (c+d x)+B \sin (c+d x))}","\frac{(23 A+12 i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{3/2} d}-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(22 A+13 i B) \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d}+\frac{7 (-2 B+3 i A) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 a^2 d}+\frac{(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}+\frac{(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}",1,"(Sqrt[Sec[c + d*x]]*(Sqrt[2]*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^(3/2)*(-2*(A - I*B)*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*(23*A + (12*I)*B)*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]]) + (Csc[c + d*x]^2*(-((50*A + (29*I)*B)*Cos[c + d*x]) + (38*A + (29*I)*B)*Cos[3*(c + d*x)] + 6*((-9*I)*A + 5*B + ((12*I)*A - 9*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(3*Sqrt[Sec[c + d*x]]))*(A + B*Tan[c + d*x]))/(8*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^(3/2))","A",1
104,1,191,255,5.9603608,"\int \frac{\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{\frac{120 (B+i A) e^{5 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)}{\left(1+e^{2 i (c+d x)}\right)^{5/2}}+\sec ^4(c+d x) ((747 B-317 i A) \cos (2 (c+d x))+(493 B-233 i A) \cos (4 (c+d x))+340 A \sin (2 (c+d x))+230 A \sin (4 (c+d x))-84 i A+780 i B \sin (2 (c+d x))+490 i B \sin (4 (c+d x))+174 B)}{120 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","-\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}-\frac{(-361 B+151 i A) (a+i a \tan (c+d x))^{3/2}}{60 a^4 d}+\frac{(-89 B+39 i A) \sqrt{a+i a \tan (c+d x)}}{5 a^3 d}-\frac{(-89 B+39 i A) \tan ^2(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{(-B+i A) \tan ^4(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{(11 A+21 i B) \tan ^3(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}",1,"((120*(I*A + B)*E^((5*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])/(1 + E^((2*I)*(c + d*x)))^(5/2) + Sec[c + d*x]^4*((-84*I)*A + 174*B + ((-317*I)*A + 747*B)*Cos[2*(c + d*x)] + ((-233*I)*A + 493*B)*Cos[4*(c + d*x)] + 340*A*Sin[2*(c + d*x)] + (780*I)*B*Sin[2*(c + d*x)] + 230*A*Sin[4*(c + d*x)] + (490*I)*B*Sin[4*(c + d*x)]))/(120*a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
105,1,193,211,5.0084397,"\int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{15 (A-i B) e^{5 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+A \left(-16 e^{2 i (c+d x)}+64 e^{4 i (c+d x)}+83 e^{6 i (c+d x)}+3\right)+i B \left(-26 e^{2 i (c+d x)}+194 e^{4 i (c+d x)}+463 e^{6 i (c+d x)}+3\right)}{15 a^2 d \left(1+e^{2 i (c+d x)}\right)^3 (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}+\frac{(13 A+83 i B) \sqrt{a+i a \tan (c+d x)}}{30 a^3 d}+\frac{41 A+151 i B}{60 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{(-B+i A) \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{(7 A+17 i B) \tan ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}",1,"-1/15*(A*(3 - 16*E^((2*I)*(c + d*x)) + 64*E^((4*I)*(c + d*x)) + 83*E^((6*I)*(c + d*x))) + I*B*(3 - 26*E^((2*I)*(c + d*x)) + 194*E^((4*I)*(c + d*x)) + 463*E^((6*I)*(c + d*x))) + 15*(A - I*B)*E^((5*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])/(a^2*d*(1 + E^((2*I)*(c + d*x)))^3*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
106,1,176,167,3.731666,"\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{e^{-6 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(B \left(-19 e^{2 i (c+d x)}+83 e^{4 i (c+d x)}+3\right)-3 i A \left(-3 e^{2 i (c+d x)}+e^{4 i (c+d x)}+1\right)\right)+15 (B+i A) e^{5 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{240 a^2 d \sqrt{a+i a \tan (c+d x)}}","\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}-\frac{-31 B+i A}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{(-B+i A) \tan ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{-13 B+3 i A}{30 a d (a+i a \tan (c+d x))^{3/2}}",1,"((1 + E^((2*I)*(c + d*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(c + d*x))]*((-3*I)*A*(1 - 3*E^((2*I)*(c + d*x)) + E^((4*I)*(c + d*x))) + B*(3 - 19*E^((2*I)*(c + d*x)) + 83*E^((4*I)*(c + d*x)))) + 15*(I*A + B)*E^((5*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])*Sec[c + d*x]^2)/(240*a^2*d*E^((6*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
107,1,176,153,3.2301755,"\int \frac{\tan (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{e^{-6 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(A \left(-e^{2 i (c+d x)}+17 e^{4 i (c+d x)}-3\right)-3 i B \left(-3 e^{2 i (c+d x)}+e^{4 i (c+d x)}+1\right)\right)-15 (A-i B) e^{5 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{240 a^2 d \sqrt{a+i a \tan (c+d x)}}","-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}+\frac{A-i B}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{A+3 i B}{6 a d (a+i a \tan (c+d x))^{3/2}}-\frac{A+i B}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"((1 + E^((2*I)*(c + d*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(c + d*x))]*((-3*I)*B*(1 - 3*E^((2*I)*(c + d*x)) + E^((4*I)*(c + d*x))) + A*(-3 - E^((2*I)*(c + d*x)) + 17*E^((4*I)*(c + d*x)))) - 15*(A - I*B)*E^((5*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])*Sec[c + d*x]^2)/(240*a^2*d*E^((6*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
108,1,176,155,2.9329542,"\int \frac{A+B \tan (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{e^{-6 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(B \left(e^{2 i (c+d x)}-17 e^{4 i (c+d x)}+3\right)-i A \left(11 e^{2 i (c+d x)}+23 e^{4 i (c+d x)}+3\right)\right)+15 (B+i A) e^{5 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{240 a^2 d \sqrt{a+i a \tan (c+d x)}}","-\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}+\frac{B+i A}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{-B+i A}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{B+i A}{6 a d (a+i a \tan (c+d x))^{3/2}}",1,"-1/240*((1 + E^((2*I)*(c + d*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(B*(3 + E^((2*I)*(c + d*x)) - 17*E^((4*I)*(c + d*x))) - I*A*(3 + 11*E^((2*I)*(c + d*x)) + 23*E^((4*I)*(c + d*x)))) + 15*(I*A + B)*E^((5*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])*Sec[c + d*x]^2)/(a^2*d*E^((6*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
109,1,233,192,4.7729548,"\int \frac{\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{e^{-6 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(3 A \left(7 e^{2 i (c+d x)}+41 e^{4 i (c+d x)}+1\right)+i B \left(11 e^{2 i (c+d x)}+23 e^{4 i (c+d x)}+3\right)\right)+15 (A-i B) e^{5 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)-120 \sqrt{2} A e^{5 i (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)\right)}{240 a^2 d \sqrt{a+i a \tan (c+d x)}}","\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}-\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{7 A+i B}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{A+i B}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{3 A+i B}{6 a d (a+i a \tan (c+d x))^{3/2}}",1,"((1 + E^((2*I)*(c + d*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(I*B*(3 + 11*E^((2*I)*(c + d*x)) + 23*E^((4*I)*(c + d*x))) + 3*A*(1 + 7*E^((2*I)*(c + d*x)) + 41*E^((4*I)*(c + d*x)))) + 15*(A - I*B)*E^((5*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))] - 120*Sqrt[2]*A*E^((5*I)*(c + d*x))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^2)/(240*a^2*d*E^((6*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
110,1,287,259,8.6607085,"\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x)) \left(\sqrt{2} e^{2 i (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((B+i A) \sinh ^{-1}\left(e^{i (c+d x)}\right)+4 \sqrt{2} (-2 B+5 i A) \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)\right)+\frac{14 \sec (c+d x) (2 (9 B-29 i A) \cos (2 (c+d x))-13 i A+3 B)-40 \csc (c+d x) ((20 A+6 i B) \cos (2 (c+d x))-17 A-6 i B)}{15 \sec ^{\frac{3}{2}}(c+d x)}\right)}{8 d (a+i a \tan (c+d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","\frac{(-2 B+5 i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}-\frac{7 (3 A+i B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 a^3 d}+\frac{(41 A+15 i B) \cot (c+d x)}{12 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{(19 A+9 i B) \cot (c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{(A+i B) \cot (c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"(Sec[c + d*x]^(3/2)*(Sqrt[2]*E^((2*I)*(c + d*x))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*((I*A + B)*ArcSinh[E^(I*(c + d*x))] + 4*Sqrt[2]*((5*I)*A - 2*B)*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]]) + (-40*(-17*A - (6*I)*B + (20*A + (6*I)*B)*Cos[2*(c + d*x)])*Csc[c + d*x] + 14*((-13*I)*A + 3*B + 2*((-29*I)*A + 9*B)*Cos[2*(c + d*x)])*Sec[c + d*x])/(15*Sec[c + d*x]^(3/2)))*(A + B*Tan[c + d*x]))/(8*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^(5/2))","A",1
111,1,317,312,10.3799347,"\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x)) \left(\sqrt{2} e^{2 i (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(\sqrt{2} (43 A+20 i B) \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)-(A-i B) \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)+\frac{\csc ^2(c+d x) (-15 (44 A+21 i B) \cos (2 (c+d x))+(388 A+203 i B) \cos (4 (c+d x))-695 i A \sin (2 (c+d x))+385 i A \sin (4 (c+d x))+212 A+340 B \sin (2 (c+d x))-200 B \sin (4 (c+d x))+112 i B)}{15 \sqrt{\sec (c+d x)}}\right)}{8 d (a+i a \tan (c+d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","\frac{(43 A+20 i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{5/2} d}-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}-\frac{(85 A+41 i B) \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 a^3 d}+\frac{21 (-B+2 i A) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 a^3 d}+\frac{(337 A+167 i B) \cot ^2(c+d x)}{60 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{(23 A+13 i B) \cot ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{(A+i B) \cot ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"(Sec[c + d*x]^(3/2)*(Sqrt[2]*E^((2*I)*(c + d*x))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(-((A - I*B)*ArcSinh[E^(I*(c + d*x))]) + Sqrt[2]*(43*A + (20*I)*B)*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]]) + (Csc[c + d*x]^2*(212*A + (112*I)*B - 15*(44*A + (21*I)*B)*Cos[2*(c + d*x)] + (388*A + (203*I)*B)*Cos[4*(c + d*x)] - (695*I)*A*Sin[2*(c + d*x)] + 340*B*Sin[2*(c + d*x)] + (385*I)*A*Sin[4*(c + d*x)] - 200*B*Sin[4*(c + d*x)]))/(15*Sqrt[Sec[c + d*x]]))*(A + B*Tan[c + d*x]))/(8*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^(5/2))","A",1
112,1,280,130,4.628663,"\int \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{\cos ^2(c+d x) (\cos (d x)-i \sin (d x)) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \left(\frac{2 e^{-i c} (B+i A) \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}-\frac{1}{105} \cos (c) (\tan (c)+i) \sqrt{\tan (c+d x)} \sec ^2(c+d x) (5 (4 B+7 i A) \tan (c+d x)+\cos (2 (c+d x)) (5 (10 B+7 i A) \tan (c+d x)+126 (A-i B))+84 (A-i B))\right)}{d (A \cos (c+d x)+B \sin (c+d x))}","-\frac{2 \sqrt[4]{-1} a (B+i A) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}+\frac{2 a (B+i A) \tan ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a (A-i B) \tan ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{2 a (B+i A) \sqrt{\tan (c+d x)}}{d}+\frac{2 i a B \tan ^{\frac{7}{2}}(c+d x)}{7 d}",1,"(Cos[c + d*x]^2*(Cos[d*x] - I*Sin[d*x])*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x])*((2*(I*A + B)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/(E^(I*c)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]) - (Cos[c]*Sec[c + d*x]^2*(I + Tan[c])*Sqrt[Tan[c + d*x]]*(84*(A - I*B) + 5*((7*I)*A + 4*B)*Tan[c + d*x] + Cos[2*(c + d*x)]*(126*(A - I*B) + 5*((7*I)*A + 10*B)*Tan[c + d*x])))/105))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
113,1,266,105,3.1029476,"\int \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{\cos ^2(c+d x) (\cos (d x)-i \sin (d x)) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \left(\frac{2 e^{-i c} (B+i A) \sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}+\frac{1}{15} (\cos (c)-i \sin (c)) \sqrt{\tan (c+d x)} \sec ^2(c+d x) (5 (B+i A) \sin (2 (c+d x))+3 (5 A-6 i B) \cos (2 (c+d x))+3 (5 A-4 i B))\right)}{d (A \cos (c+d x)+B \sin (c+d x))}","\frac{2 \sqrt[4]{-1} a (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}+\frac{2 a (B+i A) \tan ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a (A-i B) \sqrt{\tan (c+d x)}}{d}+\frac{2 i a B \tan ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(Cos[c + d*x]^2*(Cos[d*x] - I*Sin[d*x])*((2*(I*A + B)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/(E^(I*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) + (Sec[c + d*x]^2*(Cos[c] - I*Sin[c])*(3*(5*A - (4*I)*B) + 3*(5*A - (6*I)*B)*Cos[2*(c + d*x)] + 5*(I*A + B)*Sin[2*(c + d*x)])*Sqrt[Tan[c + d*x]])/15)*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
114,1,112,80,2.0894524,"\int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{2 a \sqrt{\tan (c+d x)} \left(\sqrt{i \tan (c+d x)} (3 i A+i B \tan (c+d x)+3 B)+(-3 B-3 i A) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right)}{3 d \sqrt{i \tan (c+d x)}}","\frac{2 \sqrt[4]{-1} a (B+i A) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}+\frac{2 a (B+i A) \sqrt{\tan (c+d x)}}{d}+\frac{2 i a B \tan ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(2*a*Sqrt[Tan[c + d*x]]*(((-3*I)*A - 3*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]] + Sqrt[I*Tan[c + d*x]]*((3*I)*A + 3*B + I*B*Tan[c + d*x])))/(3*d*Sqrt[I*Tan[c + d*x]])","A",1
115,1,92,55,1.8831829,"\int \frac{(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","Integrate[((a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]],x]","\frac{2 a \sqrt{\tan (c+d x)} \left((A-i B) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+i B \sqrt{i \tan (c+d x)}\right)}{d \sqrt{i \tan (c+d x)}}","\frac{2 i a B \sqrt{\tan (c+d x)}}{d}-\frac{2 \sqrt[4]{-1} a (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}",1,"(2*a*((A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]] + I*B*Sqrt[I*Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/(d*Sqrt[I*Tan[c + d*x]])","A",1
116,1,76,53,2.5656265,"\int \frac{(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2),x]","\frac{2 a \left(-A+(A-i B) \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right)}{d \sqrt{\tan (c+d x)}}","-\frac{2 a A}{d \sqrt{\tan (c+d x)}}-\frac{2 \sqrt[4]{-1} a (B+i A) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}",1,"(2*a*(-A + (A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/(d*Sqrt[Tan[c + d*x]])","A",1
117,1,94,78,2.0583215,"\int \frac{(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2),x]","-\frac{2 a \left(-3 i (A-i B) \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+A \cot (c+d x)+3 i A+3 B\right)}{3 d \sqrt{\tan (c+d x)}}","\frac{2 \sqrt[4]{-1} a (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{2 a (B+i A)}{d \sqrt{\tan (c+d x)}}-\frac{2 a A}{3 d \tan ^{\frac{3}{2}}(c+d x)}",1,"(-2*a*((3*I)*A + 3*B + A*Cot[c + d*x] - (3*I)*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/(3*d*Sqrt[Tan[c + d*x]])","A",1
118,1,265,103,4.2628474,"\int \frac{(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2),x]","\frac{\cos ^2(c+d x) (\cos (d x)-i \sin (d x)) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \left(-\frac{2 i e^{-i c} (A-i B) \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}-\frac{(\cos (c)-i \sin (c)) \csc ^2(c+d x) (5 (B+i A) \sin (2 (c+d x))+3 (6 A-5 i B) \cos (2 (c+d x))-12 A+15 i B)}{15 \sqrt{\tan (c+d x)}}\right)}{d (A \cos (c+d x)+B \sin (c+d x))}","\frac{2 \sqrt[4]{-1} a (B+i A) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{2 a (B+i A)}{3 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 a (A-i B)}{d \sqrt{\tan (c+d x)}}-\frac{2 a A}{5 d \tan ^{\frac{5}{2}}(c+d x)}",1,"(Cos[c + d*x]^2*(Cos[d*x] - I*Sin[d*x])*(((-2*I)*(A - I*B)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/(E^(I*c)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]) - (Csc[c + d*x]^2*(Cos[c] - I*Sin[c])*(-12*A + (15*I)*B + 3*(6*A - (5*I)*B)*Cos[2*(c + d*x)] + 5*(I*A + B)*Sin[2*(c + d*x)]))/(15*Sqrt[Tan[c + d*x]]))*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
119,1,315,183,6.9362809,"\int \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{\cos ^3(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \left(\frac{4 e^{-2 i c} (B+i A) \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}-\frac{i (\cos (2 c)-i \sin (2 c)) \sqrt{\tan (c+d x)} \sec ^4(c+d x) (30 (8 B+11 i A) \sin (2 (c+d x))+15 (20 B+17 i A) \sin (4 (c+d x))+140 (18 A-17 i B) \cos (2 (c+d x))+(756 A-791 i B) \cos (4 (c+d x))+21 (84 A-89 i B))}{1260}\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{4 \sqrt[4]{-1} a^2 (B+i A) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{2 a^2 (9 A-11 i B) \tan ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{4 a^2 (B+i A) \tan ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{4 a^2 (A-i B) \tan ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{4 a^2 (B+i A) \sqrt{\tan (c+d x)}}{d}+\frac{2 i B \tan ^{\frac{7}{2}}(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{9 d}",1,"(Cos[c + d*x]^3*((4*(I*A + B)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/(E^((2*I)*c)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]) - (I/1260)*Sec[c + d*x]^4*(Cos[2*c] - I*Sin[2*c])*(21*(84*A - (89*I)*B) + 140*(18*A - (17*I)*B)*Cos[2*(c + d*x)] + (756*A - (791*I)*B)*Cos[4*(c + d*x)] + 30*((11*I)*A + 8*B)*Sin[2*(c + d*x)] + 15*((17*I)*A + 20*B)*Sin[4*(c + d*x)])*Sqrt[Tan[c + d*x]])*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
120,1,307,156,5.4564871,"\int \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{\cos ^3(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \left(\frac{4 e^{-2 i c} (B+i A) \sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}+\frac{1}{210} (\cos (2 c)-i \sin (2 c)) \sqrt{\tan (c+d x)} \sec ^3(c+d x) (21 (29 A-28 i B) \cos (c+d x)+21 (11 A-12 i B) \cos (3 (c+d x))+70 i A \sin (c+d x)+70 i A \sin (3 (c+d x))+25 B \sin (c+d x)+85 B \sin (3 (c+d x)))\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{4 \sqrt[4]{-1} a^2 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{2 a^2 (7 A-9 i B) \tan ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{4 a^2 (B+i A) \tan ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 a^2 (A-i B) \sqrt{\tan (c+d x)}}{d}+\frac{2 i B \tan ^{\frac{5}{2}}(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{7 d}",1,"(Cos[c + d*x]^3*((4*(I*A + B)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/(E^((2*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) + (Sec[c + d*x]^3*(Cos[2*c] - I*Sin[2*c])*(21*(29*A - (28*I)*B)*Cos[c + d*x] + 21*(11*A - (12*I)*B)*Cos[3*(c + d*x)] + (70*I)*A*Sin[c + d*x] + 25*B*Sin[c + d*x] + (70*I)*A*Sin[3*(c + d*x)] + 85*B*Sin[3*(c + d*x)])*Sqrt[Tan[c + d*x]])/210)*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
121,1,272,129,5.6175573,"\int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{\cos ^3(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \left(\frac{1}{15} (\cos (2 c)-i \sin (2 c)) \sqrt{\tan (c+d x)} \sec ^2(c+d x) (-5 (A-2 i B) \sin (2 (c+d x))+(33 B+30 i A) \cos (2 (c+d x))+30 i A+27 B)-\frac{4 i e^{-2 i c} (A-i B) \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{4 \sqrt[4]{-1} a^2 (B+i A) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{2 a^2 (5 A-7 i B) \tan ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{4 a^2 (B+i A) \sqrt{\tan (c+d x)}}{d}+\frac{2 i B \tan ^{\frac{3}{2}}(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)}{5 d}",1,"(Cos[c + d*x]^3*(((-4*I)*(A - I*B)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/(E^((2*I)*c)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]) + (Sec[c + d*x]^2*(Cos[2*c] - I*Sin[2*c])*((30*I)*A + 27*B + ((30*I)*A + 33*B)*Cos[2*(c + d*x)] - 5*(A - (2*I)*B)*Sin[2*(c + d*x)])*Sqrt[Tan[c + d*x]])/15)*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
122,1,110,104,4.0097838,"\int \frac{(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","Integrate[((a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]],x]","-\frac{2 a^2 \sqrt{\tan (c+d x)} \left(\sqrt{i \tan (c+d x)} (3 A+B \tan (c+d x)-6 i B)-6 (A-i B) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right)}{3 d \sqrt{i \tan (c+d x)}}","-\frac{4 \sqrt[4]{-1} a^2 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{2 a^2 (3 A-5 i B) \sqrt{\tan (c+d x)}}{3 d}+\frac{2 i B \sqrt{\tan (c+d x)} \left(a^2+i a^2 \tan (c+d x)\right)}{3 d}",1,"(-2*a^2*Sqrt[Tan[c + d*x]]*(-6*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]] + Sqrt[I*Tan[c + d*x]]*(3*A - (6*I)*B + B*Tan[c + d*x])))/(3*d*Sqrt[I*Tan[c + d*x]])","A",1
123,1,85,98,3.9471221,"\int \frac{(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2),x]","-\frac{2 a^2 \left(-2 (A-i B) \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+A+B \tan (c+d x)\right)}{d \sqrt{\tan (c+d x)}}","-\frac{4 \sqrt[4]{-1} a^2 (B+i A) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}+\frac{2 a^2 (-B+i A) \sqrt{\tan (c+d x)}}{d}-\frac{2 A \left(a^2+i a^2 \tan (c+d x)\right)}{d \sqrt{\tan (c+d x)}}",1,"(-2*a^2*(A - 2*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]] + B*Tan[c + d*x]))/(d*Sqrt[Tan[c + d*x]])","A",1
124,1,96,102,3.8141991,"\int \frac{(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2),x]","-\frac{2 a^2 \left(-6 i (A-i B) \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+A \cot (c+d x)+6 i A+3 B\right)}{3 d \sqrt{\tan (c+d x)}}","\frac{4 \sqrt[4]{-1} a^2 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{2 a^2 (3 B+5 i A)}{3 d \sqrt{\tan (c+d x)}}-\frac{2 A \left(a^2+i a^2 \tan (c+d x)\right)}{3 d \tan ^{\frac{3}{2}}(c+d x)}",1,"(-2*a^2*((6*I)*A + 3*B + A*Cot[c + d*x] - (6*I)*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/(3*d*Sqrt[Tan[c + d*x]])","A",1
125,1,272,127,5.5621848,"\int \frac{(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2),x]","\frac{\cos ^3(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \left(-\frac{4 i e^{-2 i c} (A-i B) \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}-\frac{(\cos (2 c)-i \sin (2 c)) \csc ^2(c+d x) (5 (B+2 i A) \sin (2 (c+d x))+(33 A-30 i B) \cos (2 (c+d x))-27 A+30 i B)}{15 \sqrt{\tan (c+d x)}}\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{4 \sqrt[4]{-1} a^2 (B+i A) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{2 a^2 (5 B+7 i A)}{15 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (A-i B)}{d \sqrt{\tan (c+d x)}}-\frac{2 A \left(a^2+i a^2 \tan (c+d x)\right)}{5 d \tan ^{\frac{5}{2}}(c+d x)}",1,"(Cos[c + d*x]^3*(((-4*I)*(A - I*B)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/(E^((2*I)*c)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]) - (Csc[c + d*x]^2*(Cos[2*c] - I*Sin[2*c])*(-27*A + (30*I)*B + (33*A - (30*I)*B)*Cos[2*(c + d*x)] + 5*((2*I)*A + B)*Sin[2*(c + d*x)]))/(15*Sqrt[Tan[c + d*x]]))*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
126,1,296,154,8.1128867,"\int \frac{(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2),x]","\frac{\cos ^3(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \left(\frac{4 e^{-2 i c} (A-i B) \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}-\frac{(\cos (2 c)-i \sin (2 c)) \csc ^3(c+d x) ((-25 A+70 i B) \cos (c+d x)+(85 A-70 i B) \cos (3 (c+d x))+42 \sin (c+d x) ((11 B+12 i A) \cos (2 (c+d x))-8 i A-9 B))}{210 \sqrt{\tan (c+d x)}}\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{4 \sqrt[4]{-1} a^2 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}+\frac{4 a^2 (A-i B)}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 a^2 (7 B+9 i A)}{35 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (B+i A)}{d \sqrt{\tan (c+d x)}}-\frac{2 A \left(a^2+i a^2 \tan (c+d x)\right)}{7 d \tan ^{\frac{7}{2}}(c+d x)}",1,"(Cos[c + d*x]^3*((4*(A - I*B)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/(E^((2*I)*c)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]) - (Csc[c + d*x]^3*(Cos[2*c] - I*Sin[2*c])*((-25*A + (70*I)*B)*Cos[c + d*x] + (85*A - (70*I)*B)*Cos[3*(c + d*x)] + 42*((-8*I)*A - 9*B + ((12*I)*A + 11*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(210*Sqrt[Tan[c + d*x]]))*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
127,1,496,198,10.412885,"\int \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{\cos ^4(c+d x) \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \left(\sec (c) \left(\frac{2}{7} \cos (3 c)-\frac{2}{7} i \sin (3 c)\right) \sec ^3(c+d x) (-3 B \sin (d x)-i A \sin (d x))+\sec (c) \left(-\frac{2}{315} \cos (3 c)+\frac{2}{315} i \sin (3 c)\right) \sec ^2(c+d x) (45 i A \sin (c)+189 A \cos (c)+135 B \sin (c)-322 i B \cos (c))+\sec (c) \left(\frac{2}{21} \cos (3 c)-\frac{2}{21} i \sin (3 c)\right) \sec (c+d x) (37 B \sin (d x)+31 i A \sin (d x))+\sec (c) \left(\frac{2}{315} \cos (3 c)-\frac{2}{315} i \sin (3 c)\right) (465 i A \sin (c)+1449 A \cos (c)+555 B \sin (c)-1547 i B \cos (c))+\left(-\frac{2}{9} B \sin (3 c)-\frac{2}{9} i B \cos (3 c)\right) \sec ^4(c+d x)\right)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}-\frac{8 e^{-3 i c} (A-i B) \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{d \sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","\frac{8 \sqrt[4]{-1} a^3 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{16 a^3 (18 A-19 i B) \tan ^{\frac{5}{2}}(c+d x)}{315 d}+\frac{8 a^3 (B+i A) \tan ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{2 (9 A-13 i B) \tan ^{\frac{5}{2}}(c+d x) \left(a^3+i a^3 \tan (c+d x)\right)}{63 d}+\frac{8 a^3 (A-i B) \sqrt{\tan (c+d x)}}{d}+\frac{2 i a B \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2}{9 d}",1,"(-8*(A - I*B)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*E^((3*I)*c)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^4*(Sec[c]*(1449*A*Cos[c] - (1547*I)*B*Cos[c] + (465*I)*A*Sin[c] + 555*B*Sin[c])*((2*Cos[3*c])/315 - ((2*I)/315)*Sin[3*c]) + Sec[c]*Sec[c + d*x]^2*(189*A*Cos[c] - (322*I)*B*Cos[c] + (45*I)*A*Sin[c] + 135*B*Sin[c])*((-2*Cos[3*c])/315 + ((2*I)/315)*Sin[3*c]) + Sec[c + d*x]^4*(((-2*I)/9)*B*Cos[3*c] - (2*B*Sin[3*c])/9) + Sec[c]*Sec[c + d*x]^3*((2*Cos[3*c])/7 - ((2*I)/7)*Sin[3*c])*((-I)*A*Sin[d*x] - 3*B*Sin[d*x]) + Sec[c]*Sec[c + d*x]*((2*Cos[3*c])/21 - ((2*I)/21)*Sin[3*c])*((31*I)*A*Sin[d*x] + 37*B*Sin[d*x]))*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
128,1,452,171,10.3907821,"\int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{\cos ^4(c+d x) \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \left(\sec (c) \left(-\frac{2}{35} \sin (3 c)-\frac{2}{35} i \cos (3 c)\right) \sec ^2(c+d x) (7 A \cos (c)+5 B \sin (c)-21 i B \cos (c))+\sec (c) \left(-\frac{2}{21} \cos (3 c)+\frac{2}{21} i \sin (3 c)\right) \sec (c+d x) (21 A \sin (d x)-31 i B \sin (d x))+\sec (c) \left(\frac{2}{105} \cos (3 c)-\frac{2}{105} i \sin (3 c)\right) (-105 A \sin (c)+441 i A \cos (c)+155 i B \sin (c)+483 B \cos (c))-i B \sec (c) \left(\frac{2}{7} \cos (3 c)-\frac{2}{7} i \sin (3 c)\right) \sin (d x) \sec ^3(c+d x)\right)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}-\frac{8 i e^{-3 i c} (A-i B) \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{d \sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","\frac{8 \sqrt[4]{-1} a^3 (B+i A) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{8 a^3 (21 A-23 i B) \tan ^{\frac{3}{2}}(c+d x)}{105 d}-\frac{2 (7 A-11 i B) \tan ^{\frac{3}{2}}(c+d x) \left(a^3+i a^3 \tan (c+d x)\right)}{35 d}+\frac{8 a^3 (B+i A) \sqrt{\tan (c+d x)}}{d}+\frac{2 i a B \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2}{7 d}",1,"((-8*I)*(A - I*B)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*E^((3*I)*c)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^4*(Sec[c]*Sec[c + d*x]^2*(7*A*Cos[c] - (21*I)*B*Cos[c] + 5*B*Sin[c])*(((-2*I)/35)*Cos[3*c] - (2*Sin[3*c])/35) + Sec[c]*((441*I)*A*Cos[c] + 483*B*Cos[c] - 105*A*Sin[c] + (155*I)*B*Sin[c])*((2*Cos[3*c])/105 - ((2*I)/105)*Sin[3*c]) - I*B*Sec[c]*Sec[c + d*x]^3*((2*Cos[3*c])/7 - ((2*I)/7)*Sin[3*c])*Sin[d*x] + Sec[c]*Sec[c + d*x]*((-2*Cos[3*c])/21 + ((2*I)/21)*Sin[3*c])*(21*A*Sin[d*x] - (31*I)*B*Sin[d*x]))*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
129,1,273,146,7.7723474,"\int \frac{(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","Integrate[((a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]],x]","\frac{\cos ^4(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \left(\frac{8 e^{-3 i c} (A-i B) \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}-\frac{1}{15} (\cos (3 c)-i \sin (3 c)) \sqrt{\tan (c+d x)} \sec ^2(c+d x) (5 (3 B+i A) \sin (2 (c+d x))+9 (5 A-7 i B) \cos (2 (c+d x))+45 A-57 i B)\right)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{8 \sqrt[4]{-1} a^3 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{16 a^3 (5 A-6 i B) \sqrt{\tan (c+d x)}}{15 d}-\frac{2 (5 A-9 i B) \sqrt{\tan (c+d x)} \left(a^3+i a^3 \tan (c+d x)\right)}{15 d}+\frac{2 i a B \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^2}{5 d}",1,"(Cos[c + d*x]^4*((8*(A - I*B)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/(E^((3*I)*c)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]) - (Sec[c + d*x]^2*(Cos[3*c] - I*Sin[3*c])*(45*A - (57*I)*B + 9*(5*A - (7*I)*B)*Cos[2*(c + d*x)] + 5*(I*A + 3*B)*Sin[2*(c + d*x)])*Sqrt[Tan[c + d*x]])/15)*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
130,1,151,134,6.930217,"\int \frac{(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2),x]","-\frac{a^3 \sqrt{i \tan (c+d x)} \sqrt{\tan (c+d x)} \csc ^2(c+d x) \left(\sqrt{i \tan (c+d x)} (3 (A-3 i B) \sin (2 (c+d x))+(-B-3 i A) \cos (2 (c+d x))-3 i A+B)-12 (A-i B) \sin (2 (c+d x)) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right)}{3 d}","-\frac{8 \sqrt[4]{-1} a^3 (B+i A) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}+\frac{2 (-B+3 i A) \sqrt{\tan (c+d x)} \left(a^3+i a^3 \tan (c+d x)\right)}{3 d}-\frac{16 a^3 B \sqrt{\tan (c+d x)}}{3 d}-\frac{2 a A (a+i a \tan (c+d x))^2}{d \sqrt{\tan (c+d x)}}",1,"-1/3*(a^3*Csc[c + d*x]^2*(-12*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sin[2*(c + d*x)] + ((-3*I)*A + B + ((-3*I)*A - B)*Cos[2*(c + d*x)] + 3*(A - (3*I)*B)*Sin[2*(c + d*x)])*Sqrt[I*Tan[c + d*x]])*Sqrt[I*Tan[c + d*x]]*Sqrt[Tan[c + d*x]])/d","A",1
131,1,266,136,7.217243,"\int \frac{(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2),x]","\frac{\cos ^4(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \left(-\frac{8 e^{-3 i c} (A-i B) \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}-\frac{1}{3} (\cos (3 c)-i \sin (3 c)) \sqrt{\tan (c+d x)} \csc ^2(c+d x) (3 (B+3 i A) \sin (2 (c+d x))+(A-3 i B) \cos (2 (c+d x))+A+3 i B)\right)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","\frac{8 \sqrt[4]{-1} a^3 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{2 (3 B+7 i A) \left(a^3+i a^3 \tan (c+d x)\right)}{3 d \sqrt{\tan (c+d x)}}-\frac{16 a^3 A \sqrt{\tan (c+d x)}}{3 d}-\frac{2 a A (a+i a \tan (c+d x))^2}{3 d \tan ^{\frac{3}{2}}(c+d x)}",1,"(Cos[c + d*x]^4*((-8*(A - I*B)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/(E^((3*I)*c)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]) - (Csc[c + d*x]^2*(Cos[3*c] - I*Sin[3*c])*(A + (3*I)*B + (A - (3*I)*B)*Cos[2*(c + d*x)] + 3*((3*I)*A + B)*Sin[2*(c + d*x)])*Sqrt[Tan[c + d*x]])/3)*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
132,1,273,144,8.9422859,"\int \frac{(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2),x]","\frac{\cos ^4(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \left(-\frac{8 i e^{-3 i c} (A-i B) \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}-\frac{(\cos (3 c)-i \sin (3 c)) \csc ^2(c+d x) (5 (B+3 i A) \sin (2 (c+d x))+9 (7 A-5 i B) \cos (2 (c+d x))-57 A+45 i B)}{15 \sqrt{\tan (c+d x)}}\right)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","\frac{8 \sqrt[4]{-1} a^3 (B+i A) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{2 (5 B+9 i A) \left(a^3+i a^3 \tan (c+d x)\right)}{15 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{16 a^3 (6 A-5 i B)}{15 d \sqrt{\tan (c+d x)}}-\frac{2 a A (a+i a \tan (c+d x))^2}{5 d \tan ^{\frac{5}{2}}(c+d x)}",1,"(Cos[c + d*x]^4*(((-8*I)*(A - I*B)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/(E^((3*I)*c)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]) - (Csc[c + d*x]^2*(Cos[3*c] - I*Sin[3*c])*(-57*A + (45*I)*B + 9*(7*A - (5*I)*B)*Cos[2*(c + d*x)] + 5*((3*I)*A + B)*Sin[2*(c + d*x)]))/(15*Sqrt[Tan[c + d*x]]))*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
133,1,495,169,12.5788358,"\int \frac{(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2),x]","\frac{8 e^{-3 i c} (A-i B) \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{d \sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\cos ^4(c+d x) \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \left(\csc (c) \left(\frac{2}{5} \cos (3 c)-\frac{2}{5} i \sin (3 c)\right) \csc ^3(c+d x) (B \sin (d x)+3 i A \sin (d x))+\csc (c) \left(\frac{2}{105} \cos (3 c)-\frac{2}{105} i \sin (3 c)\right) \csc ^2(c+d x) (170 A \sin (c)-63 i A \cos (c)-105 i B \sin (c)-21 B \cos (c))+\csc (c) \left(\frac{2}{5} \cos (3 c)-\frac{2}{5} i \sin (3 c)\right) \csc (c+d x) (-21 B \sin (d x)-23 i A \sin (d x))+\csc (c) \left(\frac{2}{105} \cos (3 c)-\frac{2}{105} i \sin (3 c)\right) (-155 A \sin (c)+483 i A \cos (c)+105 i B \sin (c)+441 B \cos (c))+\left(-\frac{2}{7} A \cos (3 c)+\frac{2}{7} i A \sin (3 c)\right) \csc ^4(c+d x)\right)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{8 \sqrt[4]{-1} a^3 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}+\frac{8 a^3 (23 A-21 i B)}{105 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 (7 B+11 i A) \left(a^3+i a^3 \tan (c+d x)\right)}{35 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{8 a^3 (B+i A)}{d \sqrt{\tan (c+d x)}}-\frac{2 a A (a+i a \tan (c+d x))^2}{7 d \tan ^{\frac{7}{2}}(c+d x)}",1,"(8*(A - I*B)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*E^((3*I)*c)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^4*(Csc[c]*Csc[c + d*x]^2*((-63*I)*A*Cos[c] - 21*B*Cos[c] + 170*A*Sin[c] - (105*I)*B*Sin[c])*((2*Cos[3*c])/105 - ((2*I)/105)*Sin[3*c]) + Csc[c]*((483*I)*A*Cos[c] + 441*B*Cos[c] - 155*A*Sin[c] + (105*I)*B*Sin[c])*((2*Cos[3*c])/105 - ((2*I)/105)*Sin[3*c]) + Csc[c + d*x]^4*((-2*A*Cos[3*c])/7 + ((2*I)/7)*A*Sin[3*c]) + Csc[c]*Csc[c + d*x]*((2*Cos[3*c])/5 - ((2*I)/5)*Sin[3*c])*((-23*I)*A*Sin[d*x] - 21*B*Sin[d*x]) + Csc[c]*Csc[c + d*x]^3*((2*Cos[3*c])/5 - ((2*I)/5)*Sin[3*c])*((3*I)*A*Sin[d*x] + B*Sin[d*x]))*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
134,1,248,306,3.1261836,"\int \frac{\tan ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]),x]","\frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left(\frac{2}{3} \tan (c+d x) \sec (c+d x) (\cos (d x)-i \sin (d x)) (4 (3 A+2 i B) \sin (2 (c+d x))+(11 B-15 i A) \cos (2 (c+d x))-15 i A+19 B)-(1+i) (\cos (c)+i \sin (c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left(((4+i) A+(1+6 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+((6+i) B-(1+4 i) A) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{8 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{\left(\frac{1}{4}+\frac{i}{4}\right) ((4+i) A+(1+6 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d}+\frac{\left(\frac{1}{4}+\frac{i}{4}\right) ((4+i) A+(1+6 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{(-B+i A) \tan ^{\frac{5}{2}}(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{(3 A+7 i B) \tan ^{\frac{3}{2}}(c+d x)}{6 a d}-\frac{5 (-B+i A) \sqrt{\tan (c+d x)}}{2 a d}-\frac{\left(\frac{1}{8}+\frac{i}{8}\right) ((1+4 i) A-(6+i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d}-\frac{((3-5 i) A+(5+7 i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{8 \sqrt{2} a d}",1,"((Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x])*((-1 - I)*(((4 + I)*A + (1 + 6*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + ((-1 - 4*I)*A + (6 + I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(Cos[c] + I*Sin[c])*Sqrt[Sin[2*(c + d*x)]] + (2*Sec[c + d*x]*(Cos[d*x] - I*Sin[d*x])*((-15*I)*A + 19*B + ((-15*I)*A + 11*B)*Cos[2*(c + d*x)] + 4*(3*A + (2*I)*B)*Sin[2*(c + d*x)])*Tan[c + d*x])/3))/(8*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x]))","A",1
135,1,220,275,2.4020447,"\int \frac{\tan ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]),x]","\frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left(\tan (c+d x) (-4 \cos (d x)+4 i \sin (d x)) (-4 B \sin (c+d x)+(A+5 i B) \cos (c+d x))-(\cos (c)+i \sin (c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left(((1-3 i) A+(3+5 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((2+i) A+(1+4 i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{8 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{((1-3 i) A+(3+5 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{4 \sqrt{2} a d}-\frac{\left(\frac{1}{4}+\frac{i}{4}\right) ((1+2 i) A-(4+i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{(-B+i A) \tan ^{\frac{3}{2}}(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{(A+5 i B) \sqrt{\tan (c+d x)}}{2 a d}-\frac{\left(\frac{1}{8}+\frac{i}{8}\right) ((2+i) A+(1+4 i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) ((2+i) A+(1+4 i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d}",1,"((Cos[d*x] + I*Sin[d*x])*(A + B*Tan[c + d*x])*(-((((1 - 3*I)*A + (3 + 5*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] - (1 + I)*((2 + I)*A + (1 + 4*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(Cos[c] + I*Sin[c])*Sqrt[Sin[2*(c + d*x)]]) + (-4*Cos[d*x] + (4*I)*Sin[d*x])*((A + (5*I)*B)*Cos[c + d*x] - 4*B*Sin[c + d*x])*Tan[c + d*x]))/(8*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x]))","A",1
136,1,198,236,1.8302769,"\int \frac{\sqrt{\tan (c+d x)} (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx","Integrate[(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]),x]","\frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left(4 (A+i B) \sin (c+d x) (\sin (d x)+i \cos (d x))+(1+i) (-\sin (c)+i \cos (c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left((A+(2-i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+i (A-(2+i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{8 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) (A+(2-i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) (A+(2-i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{(-B+i A) \sqrt{\tan (c+d x)}}{2 d (a+i a \tan (c+d x))}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) (A-(2+i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d}-\frac{\left(\frac{1}{8}+\frac{i}{8}\right) (A-(2+i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d}",1,"((Cos[d*x] + I*Sin[d*x])*(4*(A + I*B)*(I*Cos[d*x] + Sin[d*x])*Sin[c + d*x] + (1 + I)*((A + (2 - I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + I*(A - (2 + I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(I*Cos[c] - Sin[c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(8*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x]))","A",1
137,1,199,234,2.3020256,"\int \frac{A+B \tan (c+d x)}{\sqrt{\tan (c+d x)} (a+i a \tan (c+d x))} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])),x]","\frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left(4 (A+i B) \sin (c+d x) (\cos (d x)-i \sin (d x))+(1+i) (-\sin (c)+i \cos (c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left((B+(2+i) A) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+(i B-(1+2 i) A) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{8 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) (B+(2+i) A) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a d}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) (B+(2+i) A) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{(A+i B) \sqrt{\tan (c+d x)}}{2 d (a+i a \tan (c+d x))}-\frac{((3+i) A-(1+i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{8 \sqrt{2} a d}+\frac{((3+i) A-(1+i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{8 \sqrt{2} a d}",1,"((Cos[d*x] + I*Sin[d*x])*(4*(A + I*B)*(Cos[d*x] - I*Sin[d*x])*Sin[c + d*x] + (1 + I)*(((2 + I)*A + B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + ((-1 - 2*I)*A + I*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(I*Cos[c] - Sin[c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(8*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x]))","A",1
138,1,217,267,2.4384444,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])),x]","\frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left((-4 \cos (d x)+4 i \sin (d x)) (4 A \cos (c+d x)+(-B+5 i A) \sin (c+d x))+(-\sin (c)+i \cos (c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left(((3-5 i) A+(1+3 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((4+i) A+(1+2 i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{8 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))}","\frac{((5+3 i) A-(3-i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{4 \sqrt{2} a d}+\frac{((3-i) B-(5+3 i) A) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{4 \sqrt{2} a d}+\frac{A+i B}{2 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))}-\frac{5 A+i B}{2 a d \sqrt{\tan (c+d x)}}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) ((4+i) A+(1+2 i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{((5-3 i) A+(3+i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{8 \sqrt{2} a d}",1,"((Cos[d*x] + I*Sin[d*x])*((-4*Cos[d*x] + (4*I)*Sin[d*x])*(4*A*Cos[c + d*x] + ((5*I)*A - B)*Sin[c + d*x]) + (((3 - 5*I)*A + (1 + 3*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] - (1 + I)*((4 + I)*A + (1 + 2*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(I*Cos[c] - Sin[c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(8*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x]))","A",1
139,1,241,296,3.063879,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])),x]","\frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left(\frac{2}{3} \csc (c+d x) (\cos (d x)-i \sin (d x)) ((-12 B+8 i A) \sin (2 (c+d x))+(11 A+15 i B) \cos (2 (c+d x))-19 A-15 i B)+(1-i) (\cos (c)+i \sin (c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left(((6+i) A+(1+4 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+((4+i) B-(1+6 i) A) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{8 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))}","\frac{((7-5 i) A+(5+3 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{4 \sqrt{2} a d}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) ((6+i) A+(1+4 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{A+i B}{2 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))}-\frac{7 A+3 i B}{6 a d \tan ^{\frac{3}{2}}(c+d x)}+\frac{5 (-B+i A)}{2 a d \sqrt{\tan (c+d x)}}+\frac{((7+5 i) A-(5-3 i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{8 \sqrt{2} a d}+\frac{((5-3 i) B-(7+5 i) A) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{8 \sqrt{2} a d}",1,"((Cos[d*x] + I*Sin[d*x])*((1 - I)*(((6 + I)*A + (1 + 4*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + ((-1 - 6*I)*A + (4 + I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(Cos[c] + I*Sin[c])*Sqrt[Sin[2*(c + d*x)]] + (2*Csc[c + d*x]*(Cos[d*x] - I*Sin[d*x])*(-19*A - (15*I)*B + (11*A + (15*I)*B)*Cos[2*(c + d*x)] + ((8*I)*A - 12*B)*Sin[2*(c + d*x)]))/3)*(A + B*Tan[c + d*x]))/(8*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x]))","A",1
140,1,255,316,2.5022958,"\int \frac{\tan ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec (c+d x) (\cos (d x)+i \sin (d x))^2 (A+B \tan (c+d x)) \left(2 \tan (c+d x) (\sin (2 d x)+i \cos (2 d x)) ((-43 B+7 i A) \sin (2 (c+d x))+(5 A+41 i B) \cos (2 (c+d x))+5 A+9 i B)+(-\sin (2 c)+i \cos (2 c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left(((5-9 i) A+(21+25 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((7+2 i) A+(2+23 i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{((9+5 i) A-(25-21 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{16 \sqrt{2} a^2 d}-\frac{((9+5 i) A-(25-21 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{16 \sqrt{2} a^2 d}+\frac{(3 A+7 i B) \tan ^{\frac{3}{2}}(c+d x)}{8 a^2 d (1+i \tan (c+d x))}+\frac{5 (-5 B+i A) \sqrt{\tan (c+d x)}}{8 a^2 d}-\frac{\left(\frac{1}{32}-\frac{i}{32}\right) ((7+2 i) A+(2+23 i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{\left(\frac{1}{32}-\frac{i}{32}\right) ((7+2 i) A+(2+23 i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{(-B+i A) \tan ^{\frac{5}{2}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x])*((((5 - 9*I)*A + (21 + 25*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] - (1 + I)*((7 + 2*I)*A + (2 + 23*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(I*Cos[2*c] - Sin[2*c])*Sqrt[Sin[2*(c + d*x)]] + 2*(I*Cos[2*d*x] + Sin[2*d*x])*(5*A + (9*I)*B + (5*A + (41*I)*B)*Cos[2*(c + d*x)] + ((7*I)*A - 43*B)*Sin[2*(c + d*x)])*Tan[c + d*x]))/(32*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^2)","A",1
141,1,243,277,2.4955192,"\int \frac{\tan ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec (c+d x) (\cos (d x)+i \sin (d x))^2 (A+B \tan (c+d x)) \left(4 \sin (c+d x) (\sin (2 d x)+i \cos (2 d x)) ((3 A+7 i B) \sin (c+d x)+(5 B-i A) \cos (c+d x))-(1+i) (-\sin (2 c)+i \cos (2 c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left(((2+7 i) B-(1-2 i) A) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+((7+2 i) B-(2-i) A) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{((1+3 i) A+(9+5 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{16 \sqrt{2} a^2 d}-\frac{((1+3 i) A+(9+5 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{16 \sqrt{2} a^2 d}+\frac{(A+5 i B) \sqrt{\tan (c+d x)}}{8 a^2 d (1+i \tan (c+d x))}+\frac{((1-3 i) A-(9-5 i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^2 d}-\frac{((1-3 i) A-(9-5 i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^2 d}+\frac{(-B+i A) \tan ^{\frac{3}{2}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*(4*(I*Cos[2*d*x] + Sin[2*d*x])*Sin[c + d*x]*(((-I)*A + 5*B)*Cos[c + d*x] + (3*A + (7*I)*B)*Sin[c + d*x]) - (1 + I)*(((-1 + 2*I)*A + (2 + 7*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + ((-2 + I)*A + (7 + 2*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(I*Cos[2*c] - Sin[2*c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(32*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^2)","A",1
142,1,241,279,2.2062382,"\int \frac{\sqrt{\tan (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec (c+d x) (\cos (d x)+i \sin (d x))^2 (A+B \tan (c+d x)) \left((1-i) (-\sin (2 c)+i \cos (2 c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left(((1+2 i) A+(2+i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+((1+2 i) B-(2+i) A) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)-4 \sin (c+d x) (\cos (2 d x)-i \sin (2 d x)) ((A-3 i B) \sin (c+d x)+(-B-3 i A) \cos (c+d x))\right)}{32 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{((1+3 i) B-(1-3 i) A) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{16 \sqrt{2} a^2 d}-\frac{((1+3 i) B-(1-3 i) A) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{16 \sqrt{2} a^2 d}+\frac{(3 B+i A) \sqrt{\tan (c+d x)}}{8 a^2 d (1+i \tan (c+d x))}+\frac{((1+3 i) A+(1-3 i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^2 d}-\frac{((1+3 i) A+(1-3 i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^2 d}+\frac{(-B+i A) \sqrt{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*(-4*(Cos[2*d*x] - I*Sin[2*d*x])*Sin[c + d*x]*(((-3*I)*A - B)*Cos[c + d*x] + (A - (3*I)*B)*Sin[c + d*x]) + (1 - I)*(((1 + 2*I)*A + (2 + I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + ((-2 - I)*A + (1 + 2*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(I*Cos[2*c] - Sin[2*c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(32*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^2)","A",1
143,1,243,285,2.4578682,"\int \frac{A+B \tan (c+d x)}{\sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^2} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^2),x]","\frac{\sec (c+d x) (\cos (d x)+i \sin (d x))^2 (A+B \tan (c+d x)) \left(4 \sin (c+d x) (\sin (2 d x)+i \cos (2 d x)) ((5 A+i B) \sin (c+d x)+(3 B-7 i A) \cos (c+d x))+(-\sin (2 c)+i \cos (2 c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left(((5+9 i) A+(3+i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((2+7 i) A+(1-2 i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{\left(\frac{1}{16}+\frac{i}{16}\right) ((1+2 i) B-(2-7 i) A) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d}+\frac{((9-5 i) A+(1-3 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{16 \sqrt{2} a^2 d}+\frac{(5 A+i B) \sqrt{\tan (c+d x)}}{8 a^2 d (1+i \tan (c+d x))}+\frac{\left(\frac{1}{32}+\frac{i}{32}\right) ((2+i) B-(7-2 i) A) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{((9+5 i) A-(1+3 i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^2 d}+\frac{(A+i B) \sqrt{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*(4*(I*Cos[2*d*x] + Sin[2*d*x])*Sin[c + d*x]*(((-7*I)*A + 3*B)*Cos[c + d*x] + (5*A + I*B)*Sin[c + d*x]) + (((5 + 9*I)*A + (3 + I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] - (1 + I)*((2 + 7*I)*A + (1 - 2*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(I*Cos[2*c] - Sin[2*c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(32*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^2)","A",1
144,1,250,318,2.6337542,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2),x]","\frac{\sec (c+d x) (\cos (d x)+i \sin (d x))^2 (A+B \tan (c+d x)) \left((-2 \cos (2 d x)+2 i \sin (2 d x)) ((-7 B+43 i A) \sin (2 (c+d x))+(41 A+5 i B) \cos (2 (c+d x))-9 A-5 i B)+(-\sin (2 c)+i \cos (2 c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left(((21-25 i) A+(5+9 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((23+2 i) A+(2+7 i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{((25+21 i) A-(9-5 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{16 \sqrt{2} a^2 d}-\frac{\left(\frac{1}{16}-\frac{i}{16}\right) ((2+23 i) A-(7+2 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d}-\frac{5 (5 A+i B)}{8 a^2 d \sqrt{\tan (c+d x)}}+\frac{7 A+3 i B}{8 a^2 d (1+i \tan (c+d x)) \sqrt{\tan (c+d x)}}-\frac{\left(\frac{1}{32}-\frac{i}{32}\right) ((23+2 i) A+(2+7 i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{\left(\frac{1}{32}-\frac{i}{32}\right) ((23+2 i) A+(2+7 i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{A+i B}{4 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*((((21 - 25*I)*A + (5 + 9*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] - (1 + I)*((23 + 2*I)*A + (2 + 7*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(I*Cos[2*c] - Sin[2*c])*Sqrt[Sin[2*(c + d*x)]] + (-2*Cos[2*d*x] + (2*I)*Sin[2*d*x])*(-9*A - (5*I)*B + (41*A + (5*I)*B)*Cos[2*(c + d*x)] + ((43*I)*A - 7*B)*Sin[2*(c + d*x)]))*(A + B*Tan[c + d*x]))/(32*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^2)","A",1
145,1,282,347,3.5224782,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2),x]","-\frac{i \sec (c+d x) (\cos (d x)+i \sin (d x))^2 (A+B \tan (c+d x)) \left(\frac{1}{3} \csc (c+d x) (\cos (2 d x)-i \sin (2 d x)) ((129 B-269 i A) \cos (c+d x)+(-129 B+205 i A) \cos (3 (c+d x))-2 \sin (c+d x) ((199 A+123 i B) \cos (2 (c+d x))-71 A-27 i B))+(1+i) (\cos (2 c)+i \sin (2 c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left(((47+2 i) A+(2+23 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+((23+2 i) B-(2+47 i) A) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{\left(\frac{1}{16}-\frac{i}{16}\right) ((47+2 i) A+(2+23 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^2 d}-\frac{\left(\frac{1}{16}-\frac{i}{16}\right) ((47+2 i) A+(2+23 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d}-\frac{7 (7 A+3 i B)}{24 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{9 A+5 i B}{8 a^2 d (1+i \tan (c+d x)) \tan ^{\frac{3}{2}}(c+d x)}+\frac{5 (-5 B+9 i A)}{8 a^2 d \sqrt{\tan (c+d x)}}+\frac{((49+45 i) A-(25-21 i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^2 d}-\frac{\left(\frac{1}{32}-\frac{i}{32}\right) ((2+47 i) A-(23+2 i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{A+i B}{4 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2}",1,"((-1/32*I)*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*((Csc[c + d*x]*(Cos[2*d*x] - I*Sin[2*d*x])*(((-269*I)*A + 129*B)*Cos[c + d*x] + ((205*I)*A - 129*B)*Cos[3*(c + d*x)] - 2*(-71*A - (27*I)*B + (199*A + (123*I)*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/3 + (1 + I)*(((47 + 2*I)*A + (2 + 23*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + ((-2 - 47*I)*A + (23 + 2*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(Cos[2*c] + I*Sin[2*c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^2)","A",1
146,1,300,393,5.366004,"\int \frac{\tan ^{\frac{9}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^(9/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (\cos (d x)+i \sin (d x))^3 (A+B \tan (c+d x)) \left(3 (-\sin (3 c)+i \cos (3 c)) \sqrt{\sin (2 (c+d x))} \left(((30-28 i) A+(77+75 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+(1+i) ((1-76 i) B-(29-i) A) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)+\tan (c+d x) (\sin (3 d x)+i \cos (3 d x)) (2 (90 A+241 i B) \cos (2 (c+d x))+(147 A+349 i B) \cos (4 (c+d x))+194 i A \sin (2 (c+d x))+145 i A \sin (4 (c+d x))+33 A-502 B \sin (2 (c+d x))-347 B \sin (4 (c+d x))+69 i B)\right)}{96 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","\frac{\left(\frac{1}{16}+\frac{i}{16}\right) ((29+i) A+(1+76 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^3 d}-\frac{\left(\frac{1}{16}+\frac{i}{16}\right) ((29+i) A+(1+76 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^3 d}-\frac{3 (-5 B+2 i A) \tan ^{\frac{5}{2}}(c+d x)}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{7 (4 A+11 i B) \tan ^{\frac{3}{2}}(c+d x)}{24 a^3 d}+\frac{15 (-5 B+2 i A) \sqrt{\tan (c+d x)}}{8 a^3 d}-\frac{((28-30 i) A+(75+77 i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}-\frac{\left(\frac{1}{32}+\frac{i}{32}\right) ((1+29 i) A-(76+i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^3 d}+\frac{(-B+i A) \tan ^{\frac{9}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(A+2 i B) \tan ^{\frac{7}{2}}(c+d x)}{4 a d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x])*(3*(((30 - 28*I)*A + (77 + 75*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + (1 + I)*((-29 + I)*A + (1 - 76*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(I*Cos[3*c] - Sin[3*c])*Sqrt[Sin[2*(c + d*x)]] + (I*Cos[3*d*x] + Sin[3*d*x])*(33*A + (69*I)*B + 2*(90*A + (241*I)*B)*Cos[2*(c + d*x)] + (147*A + (349*I)*B)*Cos[4*(c + d*x)] + (194*I)*A*Sin[2*(c + d*x)] - 502*B*Sin[2*(c + d*x)] + (145*I)*A*Sin[4*(c + d*x)] - 347*B*Sin[4*(c + d*x)])*Tan[c + d*x]))/(96*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^3)","A",1
147,1,286,364,3.7565182,"\int \frac{\tan ^{\frac{7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^(7/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^3 (A+B \tan (c+d x)) \left(\frac{2}{3} \tan (c+d x) (\cos (3 d x)-i \sin (3 d x)) ((9 A+33 i B) \cos (c+d x)+21 (A+7 i B) \cos (3 (c+d x))+2 i \sin (c+d x) ((19 A+145 i B) \cos (2 (c+d x))+19 A+97 i B))-i (\cos (3 c)+i \sin (3 c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left(((7+5 i) A-(30-28 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+(1-i) ((6+i) A+(1+29 i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{\left(\frac{1}{16}+\frac{i}{16}\right) ((1+6 i) A-(29+i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^3 d}-\frac{((5-7 i) A+(28+30 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{16 \sqrt{2} a^3 d}-\frac{7 (-4 B+i A) \tan ^{\frac{3}{2}}(c+d x)}{24 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{5 (A+6 i B) \sqrt{\tan (c+d x)}}{8 a^3 d}+\frac{\left(\frac{1}{32}+\frac{i}{32}\right) ((6+i) A+(1+29 i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^3 d}-\frac{\left(\frac{1}{32}+\frac{i}{32}\right) ((6+i) A+(1+29 i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^3 d}+\frac{(-B+i A) \tan ^{\frac{7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(2 A+5 i B) \tan ^{\frac{5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x])*((-I)*(((7 + 5*I)*A - (30 - 28*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + (1 - I)*((6 + I)*A + (1 + 29*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(Cos[3*c] + I*Sin[3*c])*Sqrt[Sin[2*(c + d*x)]] + (2*(Cos[3*d*x] - I*Sin[3*d*x])*((9*A + (33*I)*B)*Cos[c + d*x] + 21*(A + (7*I)*B)*Cos[3*(c + d*x)] + (2*I)*(19*A + (97*I)*B + (19*A + (145*I)*B)*Cos[2*(c + d*x)])*Sin[c + d*x])*Tan[c + d*x])/3))/(32*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^3)","A",1
148,1,254,307,2.8022497,"\int \frac{\tan ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^3 (A+B \tan (c+d x)) \left(\frac{4}{3} \sin (c+d x) (\cos (3 d x)-i \sin (3 d x)) ((A+19 i B) \sin (2 (c+d x))+3 (7 B-i A) \cos (2 (c+d x))+3 i A-6 B)+(\cos (3 c)+i \sin (3 c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left((2 A+(5-7 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+(1-i) ((1+i) A+(1-6 i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","\frac{(2 A+(5-7 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{16 \sqrt{2} a^3 d}-\frac{(2 A+(5-7 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{16 \sqrt{2} a^3 d}-\frac{(2 A-(5+7 i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}+\frac{(2 A-(5+7 i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}+\frac{5 B \sqrt{\tan (c+d x)}}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{(-B+i A) \tan ^{\frac{5}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(A+4 i B) \tan ^{\frac{3}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^3*(((2*A + (5 - 7*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + (1 - I)*((1 + I)*A + (1 - 6*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(Cos[3*c] + I*Sin[3*c])*Sqrt[Sin[2*(c + d*x)]] + (4*(Cos[3*d*x] - I*Sin[3*d*x])*Sin[c + d*x]*((3*I)*A - 6*B + 3*((-I)*A + 7*B)*Cos[2*(c + d*x)] + (A + (19*I)*B)*Sin[2*(c + d*x)]))/3)*(A + B*Tan[c + d*x]))/(32*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^3)","A",1
149,1,274,309,4.0618829,"\int \frac{\tan ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]","\frac{e^{-4 i (c+d x)} \sqrt{\tan (c+d x)} \csc (c+d x) (\cos (3 (c+d x))-i \sin (3 (c+d x))) \left(\left(-2 e^{2 i (c+d x)}-e^{4 i (c+d x)}+2 e^{6 i (c+d x)}+1\right) \left(-i A \left(1+2 e^{2 i (c+d x)}\right)+B \left(-e^{2 i (c+d x)}\right)+B\right)+6 (B+i A) e^{6 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)-3 B e^{6 i (c+d x)} \sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (c+d x)}}\right)\right)}{96 a^3 d}","\frac{(2 B+(1+i) A) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{16 \sqrt{2} a^3 d}-\frac{(2 B+(1+i) A) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{16 \sqrt{2} a^3 d}+\frac{(A-2 i B) \sqrt{\tan (c+d x)}}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}-\frac{(2 B-(1-i) A) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}+\frac{(2 B-(1-i) A) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}+\frac{(-B+i A) \tan ^{\frac{3}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{i B \sqrt{\tan (c+d x)}}{4 a d (a+i a \tan (c+d x))^2}",1,"(((1 - 2*E^((2*I)*(c + d*x)) - E^((4*I)*(c + d*x)) + 2*E^((6*I)*(c + d*x)))*(B - B*E^((2*I)*(c + d*x)) - I*A*(1 + 2*E^((2*I)*(c + d*x)))) - 3*B*E^((6*I)*(c + d*x))*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 6*(I*A + B)*E^((6*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Csc[c + d*x]*(Cos[3*(c + d*x)] - I*Sin[3*(c + d*x)])*Sqrt[Tan[c + d*x]])/(96*a^3*d*E^((4*I)*(c + d*x)))","A",0
150,1,272,317,3.6462689,"\int \frac{\sqrt{\tan (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]","\frac{e^{-4 i (c+d x)} \sec (c+d x) (\cos (3 (c+d x))-i \sin (3 (c+d x))) \left(\left(-2 e^{2 i (c+d x)}+e^{4 i (c+d x)}+2 e^{6 i (c+d x)}-1\right) \left(A e^{2 i (c+d x)}+A-2 i B e^{2 i (c+d x)}+i B\right)-6 (A-i B) e^{6 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)-3 A e^{6 i (c+d x)} \sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (c+d x)}}\right)\right)}{96 a^3 d \sqrt{\tan (c+d x)}}","\frac{\left(\frac{1}{16}+\frac{i}{16}\right) (B+(1+i) A) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^3 d}-\frac{\left(\frac{1}{16}+\frac{i}{16}\right) (B+(1+i) A) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^3 d}+\frac{(2 i A+(1-i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}-\frac{(2 i A+(1-i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}+\frac{B \sqrt{\tan (c+d x)}}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{(-B+i A) \sqrt{\tan (c+d x)}}{6 d (a+i a \tan (c+d x))^3}+\frac{(2 B+i A) \sqrt{\tan (c+d x)}}{12 a d (a+i a \tan (c+d x))^2}",1,"(((A + I*B + A*E^((2*I)*(c + d*x)) - (2*I)*B*E^((2*I)*(c + d*x)))*(-1 - 2*E^((2*I)*(c + d*x)) + E^((4*I)*(c + d*x)) + 2*E^((6*I)*(c + d*x))) - 3*A*E^((6*I)*(c + d*x))*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 6*(A - I*B)*E^((6*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Sec[c + d*x]*(Cos[3*(c + d*x)] - I*Sin[3*(c + d*x)]))/(96*a^3*d*E^((4*I)*(c + d*x))*Sqrt[Tan[c + d*x]])","A",0
151,1,258,315,3.4147246,"\int \frac{A+B \tan (c+d x)}{\sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^3),x]","\frac{\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^3 (A+B \tan (c+d x)) \left(\frac{4}{3} \sin (c+d x) (\cos (3 d x)-i \sin (3 d x)) ((-B+19 i A) \sin (2 (c+d x))+3 (7 A+i B) \cos (2 (c+d x))+6 A+3 i B)+(-\sin (3 c)+i \cos (3 c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left((2 B+(5+7 i) A) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((1+6 i) A+(1-i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{((7-5 i) A-2 i B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{16 \sqrt{2} a^3 d}+\frac{((7-5 i) A-2 i B) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{16 \sqrt{2} a^3 d}-\frac{((7+5 i) A-2 i B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}+\frac{((7+5 i) A-2 i B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}+\frac{5 A \sqrt{\tan (c+d x)}}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{(4 A+i B) \sqrt{\tan (c+d x)}}{12 a d (a+i a \tan (c+d x))^2}+\frac{(A+i B) \sqrt{\tan (c+d x)}}{6 d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^3*((((5 + 7*I)*A + 2*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] - (1 + I)*((1 + 6*I)*A + (1 - I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(I*Cos[3*c] - Sin[3*c])*Sqrt[Sin[2*(c + d*x)]] + (4*(Cos[3*d*x] - I*Sin[3*d*x])*Sin[c + d*x]*(6*A + (3*I)*B + 3*(7*A + I*B)*Cos[2*(c + d*x)] + ((19*I)*A - B)*Sin[2*(c + d*x)]))/3)*(A + B*Tan[c + d*x]))/(32*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^3)","A",1
152,1,278,364,3.5145602,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^3),x]","\frac{\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^3 (A+B \tan (c+d x)) \left(\frac{2}{3} (\cos (3 d x)-i \sin (3 d x)) ((49 A+19 i B) \cos (c+d x)-(145 A+19 i B) \cos (3 (c+d x))+6 \sin (c+d x) (7 (B-7 i A) \cos (2 (c+d x))-19 i A+2 B))+(-\sin (3 c)+i \cos (3 c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left(((28-30 i) A+(5+7 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((29+i) A+(1+6 i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","\frac{((30+28 i) A-(7-5 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{16 \sqrt{2} a^3 d}-\frac{\left(\frac{1}{16}-\frac{i}{16}\right) ((1+29 i) A-(6+i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^3 d}-\frac{5 (6 A+i B)}{8 a^3 d \sqrt{\tan (c+d x)}}+\frac{7 (4 A+i B)}{24 d \sqrt{\tan (c+d x)} \left(a^3+i a^3 \tan (c+d x)\right)}-\frac{\left(\frac{1}{32}-\frac{i}{32}\right) ((29+i) A+(1+6 i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^3 d}+\frac{\left(\frac{1}{32}-\frac{i}{32}\right) ((29+i) A+(1+6 i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^3 d}+\frac{A+i B}{6 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3}+\frac{5 A+2 i B}{12 a d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^3*((2*(Cos[3*d*x] - I*Sin[3*d*x])*((49*A + (19*I)*B)*Cos[c + d*x] - (145*A + (19*I)*B)*Cos[3*(c + d*x)] + 6*((-19*I)*A + 2*B + 7*((-7*I)*A + B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/3 + (((28 - 30*I)*A + (5 + 7*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] - (1 + I)*((29 + I)*A + (1 + 6*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(I*Cos[3*c] - Sin[3*c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(32*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^3)","A",1
153,1,306,393,4.3669449,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^3),x]","\frac{\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^3 (A+B \tan (c+d x)) \left(\frac{1}{3} \csc (c+d x) (\cos (3 d x)-i \sin (3 d x)) (-2 (241 A+90 i B) \cos (2 (c+d x))+(349 A+147 i B) \cos (4 (c+d x))-502 i A \sin (2 (c+d x))+347 i A \sin (4 (c+d x))+69 A+194 B \sin (2 (c+d x))-145 B \sin (4 (c+d x))+33 i B)+(1-i) (\cos (3 c)+i \sin (3 c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left(((76+i) A+(1+29 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+((29+i) B-(1+76 i) A) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","\frac{\left(\frac{1}{16}-\frac{i}{16}\right) ((76+i) A+(1+29 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} a^3 d}-\frac{\left(\frac{1}{16}-\frac{i}{16}\right) ((76+i) A+(1+29 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^3 d}+\frac{3 (5 A+2 i B)}{8 d \tan ^{\frac{3}{2}}(c+d x) \left(a^3+i a^3 \tan (c+d x)\right)}-\frac{7 (11 A+4 i B)}{24 a^3 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{15 (-2 B+5 i A)}{8 a^3 d \sqrt{\tan (c+d x)}}+\frac{((77+75 i) A-(30-28 i) B) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}-\frac{\left(\frac{1}{32}-\frac{i}{32}\right) ((1+76 i) A-(29+i) B) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} a^3 d}+\frac{2 A+i B}{4 a d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2}+\frac{A+i B}{6 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^3*((1 - I)*(((76 + I)*A + (1 + 29*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + ((-1 - 76*I)*A + (29 + I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(Cos[3*c] + I*Sin[3*c])*Sqrt[Sin[2*(c + d*x)]] + (Csc[c + d*x]*(Cos[3*d*x] - I*Sin[3*d*x])*(69*A + (33*I)*B - 2*(241*A + (90*I)*B)*Cos[2*(c + d*x)] + (349*A + (147*I)*B)*Cos[4*(c + d*x)] - (502*I)*A*Sin[2*(c + d*x)] + 194*B*Sin[2*(c + d*x)] + (347*I)*A*Sin[4*(c + d*x)] - 145*B*Sin[4*(c + d*x)]))/3)*(A + B*Tan[c + d*x]))/(32*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^3)","A",1
154,0,0,200,8.0258826,"\int \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\int \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","\frac{(-1)^{3/4} \sqrt{a} (7 B+4 i A) \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{4 d}+\frac{(4 A-i B) \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{(1+i) \sqrt{a} (B+i A) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{B \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 d}",1,"Integrate[Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x]","F",-1
155,1,560,152,4.7289346,"\int \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","-\frac{e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \sqrt{a+i a \tan (c+d x)} \left(8 (B+i A) \left(1+e^{2 i (c+d x)}\right) \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)-i \sqrt{2} (2 A-i B) \left(1+e^{2 i (c+d x)}\right) \log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)+2 i \sqrt{2} A e^{2 i (c+d x)} \log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)+2 i \sqrt{2} A \log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-8 B e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}+\sqrt{2} B e^{2 i (c+d x)} \log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)+\sqrt{2} B \log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)}{4 \sqrt{2} d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\sec (c+d x)}}","-\frac{(-1)^{3/4} \sqrt{a} (2 A-i B) \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{(1+i) \sqrt{a} (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{B \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{d}",1,"-1/4*(Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(-8*B*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))] + 8*(I*A + B)*(1 + E^((2*I)*(c + d*x)))*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] - I*Sqrt[2]*(2*A - I*B)*(1 + E^((2*I)*(c + d*x)))*Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] + (2*I)*Sqrt[2]*A*Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*B*Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] + (2*I)*Sqrt[2]*A*E^((2*I)*(c + d*x))*Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*B*E^((2*I)*(c + d*x))*Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[2]*d*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[Sec[c + d*x]])","B",1
156,1,238,112,4.0844857,"\int \frac{\sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","Integrate[(Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]],x]","\frac{\sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \cos (c+d x) \sqrt{a+i a \tan (c+d x)} \left(4 (A-i B) \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)+i \sqrt{2} B \left(\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)\right)}{2 d \sqrt{-1+e^{2 i (c+d x)}}}","-\frac{(1+i) \sqrt{a} (B+i A) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 (-1)^{3/4} \sqrt{a} B \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}",1,"(Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*Cos[c + d*x]*(4*(A - I*B)*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] + I*Sqrt[2]*B*(Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] - Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]]))*Sqrt[a + I*a*Tan[c + d*x]])/(2*d*Sqrt[-1 + E^((2*I)*(c + d*x))])","B",1
157,1,156,90,5.7646078,"\int \frac{\sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2),x]","\frac{(A-i B) e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right) \sqrt{a+i a \tan (c+d x)}}{d \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}-\frac{2 A \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}","\frac{(1+i) \sqrt{a} (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 A \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}",1,"((A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) - (2*A*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])","A",1
158,1,174,135,6.2076042,"\int \frac{\sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2),x]","\frac{(B+i A) e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right) \sqrt{a+i a \tan (c+d x)}}{d \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}-\frac{2 \sqrt{a+i a \tan (c+d x)} (A \cot (c+d x)+i A+3 B)}{3 d \sqrt{\tan (c+d x)}}","-\frac{2 (3 B+i A) \sqrt{a+i a \tan (c+d x)}}{3 d \sqrt{\tan (c+d x)}}+\frac{(1+i) \sqrt{a} (B+i A) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 A \sqrt{a+i a \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}",1,"((I*A + B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) - (2*(I*A + 3*B + A*Cot[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(3*d*Sqrt[Tan[c + d*x]])","A",1
159,1,211,178,6.9677348,"\int \frac{\sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2),x]","-\frac{(A-i B) e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right) \sqrt{a+i a \tan (c+d x)}}{d \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}-\frac{\csc ^2(c+d x) \sqrt{a+i a \tan (c+d x)} ((5 B+i A) \sin (2 (c+d x))+(16 A-5 i B) \cos (2 (c+d x))-10 A+5 i B)}{15 d \sqrt{\tan (c+d x)}}","-\frac{2 (5 B+i A) \sqrt{a+i a \tan (c+d x)}}{15 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 (13 A-5 i B) \sqrt{a+i a \tan (c+d x)}}{15 d \sqrt{\tan (c+d x)}}-\frac{(1+i) \sqrt{a} (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 A \sqrt{a+i a \tan (c+d x)}}{5 d \tan ^{\frac{5}{2}}(c+d x)}",1,"-(((A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))])) - (Csc[c + d*x]^2*(-10*A + (5*I)*B + (16*A - (5*I)*B)*Cos[2*(c + d*x)] + (I*A + 5*B)*Sin[2*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(15*d*Sqrt[Tan[c + d*x]])","A",1
160,1,239,221,10.1815004,"\int \frac{\sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2),x]","-\frac{i (A-i B) e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right) \sqrt{a+i a \tan (c+d x)}}{d \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}-\frac{\csc ^3(c+d x) \sqrt{a+i a \tan (c+d x)} (7 (2 A+i B) \cos (c+d x)+(46 A-7 i B) \cos (3 (c+d x))+4 \sin (c+d x) ((56 B+23 i A) \cos (2 (c+d x))-20 i A-35 B))}{210 d \sqrt{\tan (c+d x)}}","\frac{2 (31 A-7 i B) \sqrt{a+i a \tan (c+d x)}}{105 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 (7 B+i A) \sqrt{a+i a \tan (c+d x)}}{35 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 (91 B+43 i A) \sqrt{a+i a \tan (c+d x)}}{105 d \sqrt{\tan (c+d x)}}+\frac{(1-i) \sqrt{a} (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 A \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}",1,"((-I)*(A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) - (Csc[c + d*x]^3*(7*(2*A + I*B)*Cos[c + d*x] + (46*A - (7*I)*B)*Cos[3*(c + d*x)] + 4*((-20*I)*A - 35*B + ((23*I)*A + 56*B)*Cos[2*(c + d*x)])*Sin[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(210*d*Sqrt[Tan[c + d*x]])","A",1
161,1,420,248,7.9537538,"\int \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left(\frac{\sqrt{2} e^{-i (c+d x)} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(\sqrt{2} (22 A-23 i B) \left(\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)-128 (A-i B) \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)\right)}{\sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}}}+\frac{4 (\cos (c)-i \sin (c)) \sqrt{\tan (c+d x)} \sec ^{\frac{5}{2}}(c+d x) (2 (7 B+6 i A) \sin (2 (c+d x))+5 (6 A-7 i B) \cos (2 (c+d x))+30 A-19 i B)}{3 \cos (d x)+3 i \sin (d x)}\right)}{64 d \sec ^{\frac{5}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","\frac{(-1)^{3/4} a^{3/2} (23 B+22 i A) \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{8 d}+\frac{(2+2 i) a^{3/2} (B+i A) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{a (7 B+6 i A) \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d}+\frac{a (10 A-9 i B) \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{8 d}+\frac{i a B \tan ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{3 d}",1,"(((Sqrt[2]*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(-128*(A - I*B)*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*(22*A - (23*I)*B)*(Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] - Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]])))/(E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]) + (4*Sec[c + d*x]^(5/2)*(Cos[c] - I*Sin[c])*(30*A - (19*I)*B + 5*(6*A - (7*I)*B)*Cos[2*(c + d*x)] + 2*((6*I)*A + 7*B)*Sin[2*(c + d*x)])*Sqrt[Tan[c + d*x]])/(3*Cos[d*x] + (3*I)*Sin[d*x]))*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/(64*d*Sec[c + d*x]^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
162,1,389,204,6.5574705,"\int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left(\frac{\sqrt{2} e^{-i (c+d x)} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(\sqrt{2} (11 B+12 i A) \left(\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)-64 i (A-i B) \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)\right)}{\sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}}}+8 (\sin (c)+i \cos (c)) \sqrt{\tan (c+d x)} \sqrt{\sec (c+d x)} (\cos (d x)-i \sin (d x)) (4 A+2 B \tan (c+d x)-5 i B)\right)}{32 d \sec ^{\frac{5}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{(-1)^{3/4} a^{3/2} (12 A-11 i B) \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{4 d}-\frac{(2+2 i) a^{3/2} (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{a (5 B+4 i A) \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{i a B \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 d}",1,"((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x])*((Sqrt[2]*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*((-64*I)*(A - I*B)*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*((12*I)*A + 11*B)*(Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] - Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]])))/(E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]) + 8*Sqrt[Sec[c + d*x]]*(I*Cos[c] + Sin[c])*(Cos[d*x] - I*Sin[d*x])*Sqrt[Tan[c + d*x]]*(4*A - (5*I)*B + 2*B*Tan[c + d*x])))/(32*d*Sec[c + d*x]^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
163,1,221,156,3.8496873,"\int \frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]],x]","\frac{a e^{-i (c+d x)} \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)} \left(2 \sqrt{2} (A-i B) \left(1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+i \left((3 B+2 i A) \left(1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+\sqrt{2} B e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}\right)\right)}{\sqrt{2} d \sqrt{-1+e^{2 i (c+d x)}}}","-\frac{(-1)^{3/4} a^{3/2} (3 B+2 i A) \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{(2+2 i) a^{3/2} (B+i A) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{i a B \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{d}",1,"(a*(2*Sqrt[2]*(A - I*B)*(1 + E^((2*I)*(c + d*x)))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + I*(Sqrt[2]*B*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))] + ((2*I)*A + 3*B)*(1 + E^((2*I)*(c + d*x)))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[2]*d*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))])","A",1
164,1,234,146,4.5556205,"\int \frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2),x]","\frac{a e^{-\frac{1}{2} i (4 c+5 d x)} \left(1+e^{2 i (c+d x)}\right)^2 \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\tan (c+d x)} \sec (c+d x) \left(\cos \left(\frac{d x}{2}\right)+i \sin \left(\frac{d x}{2}\right)\right) \left(2 (B+i A) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-A \sqrt{-1+e^{2 i (c+d x)}} \csc (c+d x)-\sqrt{2} B \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{\sqrt{2} d \sqrt{-1+e^{2 i (c+d x)}}}","\frac{(2+2 i) a^{3/2} (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{2 \sqrt[4]{-1} a^{3/2} B \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a A \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}",1,"(a*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))^2*(2*(I*A + B)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] - Sqrt[2]*B*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]] - A*Sqrt[-1 + E^((2*I)*(c + d*x))]*Csc[c + d*x])*Sec[c + d*x]*(Cos[(d*x)/2] + I*Sin[(d*x)/2])*Sqrt[Tan[c + d*x]])/(Sqrt[2]*d*E^((I/2)*(4*c + 5*d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))])","A",1
165,1,221,137,6.3815815,"\int \frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2),x]","\frac{a e^{-3 i (c+d x)} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(1+e^{2 i (c+d x)}\right)^2 (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)} \left(e^{i (c+d x)} \sqrt{1-e^{2 i (c+d x)}} \left(i A \left(-3+5 e^{2 i (c+d x)}\right)+3 B \left(-1+e^{2 i (c+d x)}\right)\right)+3 (B+i A) \left(-1+e^{2 i (c+d x)}\right)^2 \sin ^{-1}\left(e^{i (c+d x)}\right)\right)}{3 d \left(1-e^{2 i (c+d x)}\right)^{5/2}}","\frac{(2+2 i) a^{3/2} (B+i A) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a (3 B+4 i A) \sqrt{a+i a \tan (c+d x)}}{3 d \sqrt{\tan (c+d x)}}-\frac{2 a A \sqrt{a+i a \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}",1,"(a*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))^2*(E^(I*(c + d*x))*Sqrt[1 - E^((2*I)*(c + d*x))]*(3*B*(-1 + E^((2*I)*(c + d*x))) + I*A*(-3 + 5*E^((2*I)*(c + d*x)))) + 3*(I*A + B)*(-1 + E^((2*I)*(c + d*x)))^2*ArcSin[E^(I*(c + d*x))])*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(3*d*E^((3*I)*(c + d*x))*(1 - E^((2*I)*(c + d*x)))^(5/2))","A",1
166,1,237,181,9.8496575,"\int \frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2),x]","\frac{a (A-i B) e^{-3 i (c+d x)} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(1+e^{2 i (c+d x)}\right)^2 (\tan (c+d x)-i) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right) \sqrt{a+i a \tan (c+d x)}}{d \sqrt{-1+e^{2 i (c+d x)}}}-\frac{a \csc ^2(c+d x) \sqrt{a+i a \tan (c+d x)} ((5 B+6 i A) \sin (2 (c+d x))+(21 A-20 i B) \cos (2 (c+d x))-15 A+20 i B)}{15 d \sqrt{\tan (c+d x)}}","-\frac{(2+2 i) a^{3/2} (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a (5 B+6 i A) \sqrt{a+i a \tan (c+d x)}}{15 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{4 a (9 A-10 i B) \sqrt{a+i a \tan (c+d x)}}{15 d \sqrt{\tan (c+d x)}}-\frac{2 a A \sqrt{a+i a \tan (c+d x)}}{5 d \tan ^{\frac{5}{2}}(c+d x)}",1,"-1/15*(a*Csc[c + d*x]^2*(-15*A + (20*I)*B + (21*A - (20*I)*B)*Cos[2*(c + d*x)] + ((6*I)*A + 5*B)*Sin[2*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]) + (a*(A - I*B)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))^2*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))])","A",1
167,1,261,225,12.8210763,"\int \frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2),x]","\frac{a (B+i A) e^{-3 i (c+d x)} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(1+e^{2 i (c+d x)}\right)^2 (\tan (c+d x)-i) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right) \sqrt{a+i a \tan (c+d x)}}{d \sqrt{-1+e^{2 i (c+d x)}}}-\frac{a \csc ^3(c+d x) \sqrt{a+i a \tan (c+d x)} (7 (A+6 i B) \cos (c+d x)+(53 A-42 i B) \cos (3 (c+d x))+2 \sin (c+d x) ((147 B+158 i A) \cos (2 (c+d x))-110 i A-105 B))}{210 d \sqrt{\tan (c+d x)}}","\frac{(2-2 i) a^{3/2} (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{4 a (19 A-21 i B) \sqrt{a+i a \tan (c+d x)}}{105 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 a (7 B+8 i A) \sqrt{a+i a \tan (c+d x)}}{35 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{4 a (63 B+67 i A) \sqrt{a+i a \tan (c+d x)}}{105 d \sqrt{\tan (c+d x)}}-\frac{2 a A \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}",1,"-1/210*(a*Csc[c + d*x]^3*(7*(A + (6*I)*B)*Cos[c + d*x] + (53*A - (42*I)*B)*Cos[3*(c + d*x)] + 2*((-110*I)*A - 105*B + ((158*I)*A + 147*B)*Cos[2*(c + d*x)])*Sin[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]) + (a*(I*A + B)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))^2*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))])","A",1
168,1,242,269,15.6969516,"\int \frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(11/2),x]","\frac{a \sqrt{a+i a \tan (c+d x)} \left(\frac{2520 (A-i B) e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)}{\sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}+\frac{\csc ^4(c+d x) (12 (117 A-134 i B) \cos (2 (c+d x))+(-487 A+474 i B) \cos (4 (c+d x))+144 i A \sin (2 (c+d x))-172 i A \sin (4 (c+d x))-1197 A+138 B \sin (2 (c+d x))-159 B \sin (4 (c+d x))+1134 i B)}{\sqrt{\tan (c+d x)}}\right)}{1260 d}","\frac{(2+2 i) a^{3/2} (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{4 a (57 B+61 i A) \sqrt{a+i a \tan (c+d x)}}{315 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{4 a (11 A-12 i B) \sqrt{a+i a \tan (c+d x)}}{105 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 a (9 B+10 i A) \sqrt{a+i a \tan (c+d x)}}{63 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{4 a (193 A-201 i B) \sqrt{a+i a \tan (c+d x)}}{315 d \sqrt{\tan (c+d x)}}-\frac{2 a A \sqrt{a+i a \tan (c+d x)}}{9 d \tan ^{\frac{9}{2}}(c+d x)}",1,"(a*((2520*(A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/(E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) + (Csc[c + d*x]^4*(-1197*A + (1134*I)*B + 12*(117*A - (134*I)*B)*Cos[2*(c + d*x)] + (-487*A + (474*I)*B)*Cos[4*(c + d*x)] + (144*I)*A*Sin[2*(c + d*x)] + 138*B*Sin[2*(c + d*x)] - (172*I)*A*Sin[4*(c + d*x)] - 159*B*Sin[4*(c + d*x)]))/Sqrt[Tan[c + d*x]])*Sqrt[a + I*a*Tan[c + d*x]])/(1260*d)","A",1
169,1,581,298,10.4105409,"\int \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{\cos ^3(c+d x) \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left((8 A-23 i B) \left(-\frac{1}{24} \cos (2 c)+\frac{1}{24} i \sin (2 c)\right) \sec ^2(c+d x)+(104 A-131 i B) \sec (c+d x) \left(-\frac{1}{96} \cos (3 c+d x)+\frac{1}{96} i \sin (3 c+d x)\right)+(56 A-65 i B) \left(\frac{13}{192} \cos (2 c)-\frac{13}{192} i \sin (2 c)\right)+\sec ^3(c+d x) \left(-\frac{1}{4} B \sin (3 c+d x)-\frac{1}{4} i B \cos (3 c+d x)\right)\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{e^{-2 i c} \sqrt{e^{i d x}} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} (a+i a \tan (c+d x))^{5/2} \left(3 \sqrt{2} (120 A-121 i B) \left(\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)-2048 (A-i B) \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)\right) (A+B \tan (c+d x))}{256 \sqrt{2} d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","\frac{3 (-1)^{3/4} a^{5/2} (121 B+120 i A) \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{64 d}+\frac{(4+4 i) a^{5/2} (B+i A) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{a^2 (8 A-11 i B) \tan ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{24 d}+\frac{a^2 (107 B+104 i A) \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{96 d}+\frac{a^2 (152 A-149 i B) \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{64 d}+\frac{i a B \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}",1,"(Sqrt[E^(I*d*x)]*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(-2048*(A - I*B)*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] + 3*Sqrt[2]*(120*A - (121*I)*B)*(Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] - Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]]))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(256*Sqrt[2]*d*E^((2*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^3*((8*A - (23*I)*B)*Sec[c + d*x]^2*(-1/24*Cos[2*c] + (I/24)*Sin[2*c]) + (56*A - (65*I)*B)*((13*Cos[2*c])/192 - ((13*I)/192)*Sin[2*c]) + (104*A - (131*I)*B)*Sec[c + d*x]*(-1/96*Cos[3*c + d*x] + (I/96)*Sin[3*c + d*x]) + Sec[c + d*x]^3*((-1/4*I)*B*Cos[3*c + d*x] - (B*Sin[3*c + d*x])/4))*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",0
170,1,537,252,9.7003073,"\int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{\cos ^3(c+d x) \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left((6 A-13 i B) \sec (c+d x) \left(-\frac{1}{12} \sin (3 c+d x)-\frac{1}{12} i \cos (3 c+d x)\right)+(91 B+66 i A) \left(\frac{1}{24} \cos (2 c)-\frac{1}{24} i \sin (2 c)\right)+\left(-\frac{1}{3} B \cos (2 c)+\frac{1}{3} i B \sin (2 c)\right) \sec ^2(c+d x)\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{e^{-2 i c} \sqrt{e^{i d x}} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} (a+i a \tan (c+d x))^{5/2} \left(\sqrt{2} (45 B+46 i A) \left(\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)-256 i (A-i B) \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)\right) (A+B \tan (c+d x))}{32 \sqrt{2} d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","-\frac{(-1)^{3/4} a^{5/2} (46 A-45 i B) \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{8 d}-\frac{(4+4 i) a^{5/2} (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{a^2 (2 A-3 i B) \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{a^2 (19 B+18 i A) \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{8 d}+\frac{i a B \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}",1,"(Sqrt[E^(I*d*x)]*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*((-256*I)*(A - I*B)*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*((46*I)*A + 45*B)*(Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] - Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]]))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(32*Sqrt[2]*d*E^((2*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^3*(((66*I)*A + 91*B)*(Cos[2*c]/24 - (I/24)*Sin[2*c]) + Sec[c + d*x]^2*(-1/3*(B*Cos[2*c]) + (I/3)*B*Sin[2*c]) + (6*A - (13*I)*B)*Sec[c + d*x]*((-1/12*I)*Cos[3*c + d*x] - Sin[3*c + d*x]/12))*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
171,1,499,206,9.9985856,"\int \frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]],x]","\frac{\cos ^3(c+d x) \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left((4 A-11 i B) \left(-\frac{1}{4} \cos (2 c)+\frac{1}{4} i \sin (2 c)\right)+\sec (c+d x) \left(-\frac{1}{2} B \sin (3 c+d x)-\frac{1}{2} i B \cos (3 c+d x)\right)\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{e^{-2 i c} \sqrt{e^{i d x}} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} (a+i a \tan (c+d x))^{5/2} \left(128 (A-i B) \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)-\sqrt{2} (20 A-23 i B) \left(\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)\right) (A+B \tan (c+d x))}{16 \sqrt{2} d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","-\frac{(-1)^{3/4} a^{5/2} (23 B+20 i A) \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{4 d}+\frac{(4-4 i) a^{5/2} (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{a^2 (4 A-7 i B) \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{i a B \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}{2 d}",1,"(Sqrt[E^(I*d*x)]*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(128*(A - I*B)*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] - Sqrt[2]*(20*A - (23*I)*B)*(Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] - Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]]))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(16*Sqrt[2]*d*E^((2*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^3*((4*A - (11*I)*B)*(-1/4*Cos[2*c] + (I/4)*Sin[2*c]) + Sec[c + d*x]*((-1/2*I)*B*Cos[3*c + d*x] - (B*Sin[3*c + d*x])/2))*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
172,1,493,196,10.3287426,"\int \frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2),x]","\frac{\cos ^3(c+d x) \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) (\csc (c) (-\cos (2 c)+i \sin (2 c)) (2 A \cos (c)+B \sin (c))+A \csc (c) (2 \cos (2 c)-2 i \sin (2 c)) \sin (d x) \csc (c+d x))}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{e^{-2 i c} \sqrt{e^{i d x}} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} (a+i a \tan (c+d x))^{5/2} \left(32 (B+i A) \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)-i \sqrt{2} (2 A-5 i B) \left(\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)\right) (A+B \tan (c+d x))}{4 \sqrt{2} d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","\frac{(-1)^{3/4} a^{5/2} (2 A-5 i B) \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{(4+4 i) a^{5/2} (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{a^2 (-B+2 i A) \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{d}-\frac{2 a A (a+i a \tan (c+d x))^{3/2}}{d \sqrt{\tan (c+d x)}}",1,"(Sqrt[E^(I*d*x)]*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(32*(I*A + B)*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] - I*Sqrt[2]*(2*A - (5*I)*B)*(Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] - Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]]))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(4*Sqrt[2]*d*E^((2*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^3*(Csc[c]*(2*A*Cos[c] + B*Sin[c])*(-Cos[2*c] + I*Sin[2*c]) + A*Csc[c]*Csc[c + d*x]*(2*Cos[2*c] - (2*I)*Sin[2*c])*Sin[d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
173,1,618,190,10.5235256,"\int \frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2),x]","\frac{\cos ^3(c+d x) \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(\csc (c) \left(\frac{2}{3} \cos (2 c)-\frac{2}{3} i \sin (2 c)\right) \csc (c+d x) (3 B \sin (d x)+7 i A \sin (d x))-i \csc (c) \left(\frac{2}{3} \cos (2 c)-\frac{2}{3} i \sin (2 c)\right) (i A \sin (c)+7 A \cos (c)-3 i B \cos (c))+\left(-\frac{2}{3} A \cos (2 c)+\frac{2}{3} i A \sin (2 c)\right) \csc ^2(c+d x)\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\sqrt{e^{i d x}} e^{-i (3 c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} (a+i a \tan (c+d x))^{5/2} \left(8 (B+i A) \log \left(e^{-i c} \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)\right)+\sqrt{2} B \log \left(\frac{2 e^{\frac{7 i c}{2}} \left(2 i \sqrt{-1+e^{2 i (c+d x)}}-i \sqrt{2} e^{i (c+d x)}+\sqrt{2}\right)}{B \left(e^{i (c+d x)}-i\right)}\right)-\sqrt{2} B \log \left(-\frac{2 i e^{\frac{7 i c}{2}} \left(2 \sqrt{-1+e^{2 i (c+d x)}}+\sqrt{2} e^{i (c+d x)}-i \sqrt{2}\right)}{B \left(e^{i (c+d x)}+i\right)}\right)\right) (A+B \tan (c+d x))}{\sqrt{2} d \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","\frac{(4+4 i) a^{5/2} (B+i A) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{2 (-1)^{3/4} a^{5/2} B \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a^2 (B+2 i A) \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}-\frac{2 a A (a+i a \tan (c+d x))^{3/2}}{3 d \tan ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[E^(I*d*x)]*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(Sqrt[2]*B*Log[(2*E^(((7*I)/2)*c)*(Sqrt[2] - I*Sqrt[2]*E^(I*(c + d*x)) + (2*I)*Sqrt[-1 + E^((2*I)*(c + d*x))]))/(B*(-I + E^(I*(c + d*x))))] + 8*(I*A + B)*Log[(E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))])/E^(I*c)] - Sqrt[2]*B*Log[((-2*I)*E^(((7*I)/2)*c)*((-I)*Sqrt[2] + Sqrt[2]*E^(I*(c + d*x)) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]))/(B*(I + E^(I*(c + d*x))))])*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(Sqrt[2]*d*E^(I*(3*c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^3*((-I)*Csc[c]*(7*A*Cos[c] - (3*I)*B*Cos[c] + I*A*Sin[c])*((2*Cos[2*c])/3 - ((2*I)/3)*Sin[2*c]) + Csc[c + d*x]^2*((-2*A*Cos[2*c])/3 + ((2*I)/3)*A*Sin[2*c]) + Csc[c]*Csc[c + d*x]*((2*Cos[2*c])/3 - ((2*I)/3)*Sin[2*c])*((7*I)*A*Sin[d*x] + 3*B*Sin[d*x]))*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
174,1,323,185,11.8726874,"\int \frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2),x]","-\frac{4 \sqrt{2} e^{-2 i c} \sqrt{e^{i d x}} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} (a+i a \tan (c+d x))^{5/2} \left(e^{i (c+d x)} \sqrt{1-e^{2 i (c+d x)}} \left(i A \left(-35 e^{2 i (c+d x)}+26 e^{4 i (c+d x)}+15\right)+5 B \left(-7 e^{2 i (c+d x)}+4 e^{4 i (c+d x)}+3\right)\right)+15 (B+i A) \left(-1+e^{2 i (c+d x)}\right)^3 \sin ^{-1}\left(e^{i (c+d x)}\right)\right) (A+B \tan (c+d x))}{15 d \left(1-e^{2 i (c+d x)}\right)^{7/2} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","-\frac{(4+4 i) a^{5/2} (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a^2 (5 B+8 i A) \sqrt{a+i a \tan (c+d x)}}{15 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (38 A-35 i B) \sqrt{a+i a \tan (c+d x)}}{15 d \sqrt{\tan (c+d x)}}-\frac{2 a A (a+i a \tan (c+d x))^{3/2}}{5 d \tan ^{\frac{5}{2}}(c+d x)}",1,"(-4*Sqrt[2]*Sqrt[E^(I*d*x)]*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(E^(I*(c + d*x))*Sqrt[1 - E^((2*I)*(c + d*x))]*(5*B*(3 - 7*E^((2*I)*(c + d*x)) + 4*E^((4*I)*(c + d*x))) + I*A*(15 - 35*E^((2*I)*(c + d*x)) + 26*E^((4*I)*(c + d*x)))) + 15*(I*A + B)*(-1 + E^((2*I)*(c + d*x)))^3*ArcSin[E^(I*(c + d*x))])*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(15*d*E^((2*I)*c)*(1 - E^((2*I)*(c + d*x)))^(7/2)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",0
175,1,363,231,14.6247742,"\int \frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2),x]","\frac{4 \sqrt{2} e^{-2 i c} \sqrt{e^{i d x}} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} (a+i a \tan (c+d x))^{5/2} \left(e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \left(7 i B \left(50 e^{2 i (c+d x)}-61 e^{4 i (c+d x)}+26 e^{6 i (c+d x)}-15\right)-5 A \left(70 e^{2 i (c+d x)}-77 e^{4 i (c+d x)}+40 e^{6 i (c+d x)}-21\right)\right)+105 (A-i B) \left(-1+e^{2 i (c+d x)}\right)^4 \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)\right) (A+B \tan (c+d x))}{105 d \left(-1+e^{2 i (c+d x)}\right)^{9/2} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","\frac{(4-4 i) a^{5/2} (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{2 a^2 (80 A-77 i B) \sqrt{a+i a \tan (c+d x)}}{105 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 a^2 (7 B+10 i A) \sqrt{a+i a \tan (c+d x)}}{35 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (133 B+130 i A) \sqrt{a+i a \tan (c+d x)}}{105 d \sqrt{\tan (c+d x)}}-\frac{2 a A (a+i a \tan (c+d x))^{3/2}}{7 d \tan ^{\frac{7}{2}}(c+d x)}",1,"(4*Sqrt[2]*Sqrt[E^(I*d*x)]*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*((7*I)*B*(-15 + 50*E^((2*I)*(c + d*x)) - 61*E^((4*I)*(c + d*x)) + 26*E^((6*I)*(c + d*x))) - 5*A*(-21 + 70*E^((2*I)*(c + d*x)) - 77*E^((4*I)*(c + d*x)) + 40*E^((6*I)*(c + d*x)))) + 105*(A - I*B)*(-1 + E^((2*I)*(c + d*x)))^4*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]])*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(105*d*E^((2*I)*c)*(-1 + E^((2*I)*(c + d*x)))^(9/2)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",0
176,1,246,277,16.5667835,"\int \frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(11/2),x]","\frac{a^2 \sqrt{a+i a \tan (c+d x)} \left(\frac{1260 (A-i B) e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)}{\sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}+\frac{\csc ^2(2 (c+d x)) (12 (251 A-260 i B) \cos (2 (c+d x))+(-961 A+915 i B) \cos (4 (c+d x))+282 i A \sin (2 (c+d x))-331 i A \sin (4 (c+d x))-2331 A+390 B \sin (2 (c+d x))-285 B \sin (4 (c+d x))+2205 i B)}{\tan ^{\frac{5}{2}}(c+d x)}\right)}{315 d}","\frac{(4+4 i) a^{5/2} (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{8 a^2 (60 B+59 i A) \sqrt{a+i a \tan (c+d x)}}{315 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (46 A-45 i B) \sqrt{a+i a \tan (c+d x)}}{105 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 a^2 (3 B+4 i A) \sqrt{a+i a \tan (c+d x)}}{21 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{8 a^2 (197 A-195 i B) \sqrt{a+i a \tan (c+d x)}}{315 d \sqrt{\tan (c+d x)}}-\frac{2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac{9}{2}}(c+d x)}",1,"(a^2*((1260*(A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/(E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) + (Csc[2*(c + d*x)]^2*(-2331*A + (2205*I)*B + 12*(251*A - (260*I)*B)*Cos[2*(c + d*x)] + (-961*A + (915*I)*B)*Cos[4*(c + d*x)] + (282*I)*A*Sin[2*(c + d*x)] + 390*B*Sin[2*(c + d*x)] - (331*I)*A*Sin[4*(c + d*x)] - 285*B*Sin[4*(c + d*x)]))/Tan[c + d*x]^(5/2))*Sqrt[a + I*a*Tan[c + d*x]])/(315*d)","A",1
177,1,328,323,20.2368071,"\int \frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(13/2),x]","\frac{4 \sqrt{2} a^2 (B+i A) e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)}{d \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}-\frac{a^2 \csc ^3(c+d x) \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (66 (95 A-47 i B) \cos (c+d x)+(-5225 A+6743 i B) \cos (3 (c+d x))+84810 i A \sin (c+d x)-42185 i A \sin (3 (c+d x))+10925 i A \sin (5 (c+d x))+3995 A \cos (5 (c+d x))+84414 B \sin (c+d x)-43703 B \sin (3 (c+d x))+10571 B \sin (5 (c+d x))-3641 i B \cos (5 (c+d x)))}{27720 d \tan ^{\frac{5}{2}}(c+d x)}","\frac{(4+4 i) a^{5/2} (B+i A) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{8 a^2 (655 A-649 i B) \sqrt{a+i a \tan (c+d x)}}{3465 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (253 B+250 i A) \sqrt{a+i a \tan (c+d x)}}{1155 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 (212 A-209 i B) \sqrt{a+i a \tan (c+d x)}}{693 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 a^2 (11 B+14 i A) \sqrt{a+i a \tan (c+d x)}}{99 d \tan ^{\frac{9}{2}}(c+d x)}-\frac{8 a^2 (2167 B+2155 i A) \sqrt{a+i a \tan (c+d x)}}{3465 d \sqrt{\tan (c+d x)}}-\frac{2 a A (a+i a \tan (c+d x))^{3/2}}{11 d \tan ^{\frac{11}{2}}(c+d x)}",1,"(4*Sqrt[2]*a^2*(I*A + B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/(d*E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) - (a^2*Csc[c + d*x]^3*Sec[c + d*x]^2*(66*(95*A - (47*I)*B)*Cos[c + d*x] + (-5225*A + (6743*I)*B)*Cos[3*(c + d*x)] + 3995*A*Cos[5*(c + d*x)] - (3641*I)*B*Cos[5*(c + d*x)] + (84810*I)*A*Sin[c + d*x] + 84414*B*Sin[c + d*x] - (42185*I)*A*Sin[3*(c + d*x)] - 43703*B*Sin[3*(c + d*x)] + (10925*I)*A*Sin[5*(c + d*x)] + 10571*B*Sin[5*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(27720*d*Tan[c + d*x]^(5/2))","A",1
178,1,485,190,10.731003,"\int \frac{(a+i a \tan (c+d x))^{5/2} \left(\frac{3 b B}{2 a}+B \tan (c+d x)\right)}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(5/2)*((3*b*B)/(2*a) + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2),x]","\frac{\cos ^3(c+d x) \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{5/2} \left(\frac{3 b B}{2 a}+B \tan (c+d x)\right) \left(\csc (c) (2 \cos (2 c)-2 i \sin (2 c)) \csc (c+d x) (2 a \sin (d x)+7 i b \sin (d x))-i \csc (c) (2 \cos (2 c)-2 i \sin (2 c)) (-2 i a \cos (c)+i b \sin (c)+7 b \cos (c))+(-2 b \cos (2 c)+2 i b \sin (2 c)) \csc ^2(c+d x)\right)}{d (\cos (d x)+i \sin (d x))^2 (2 a \sin (c+d x)+3 b \cos (c+d x))}-\frac{2 \sqrt{2} e^{-2 i c} \sqrt{e^{i d x}} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} (a+i a \tan (c+d x))^{5/2} \left((6 b-4 i a) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+i \sqrt{2} a \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \left(\frac{3 b B}{2 a}+B \tan (c+d x)\right)}{d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (2 a \sin (c+d x)+3 b \cos (c+d x))}","\frac{(2+2 i) a^{3/2} B (2 a+3 i b) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{2 (-1)^{3/4} a^{5/2} B \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{b B (a+i a \tan (c+d x))^{3/2}}{d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 a B (a+3 i b) \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}",1,"(-2*Sqrt[2]*Sqrt[E^(I*d*x)]*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(((-4*I)*a + 6*b)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + I*Sqrt[2]*a*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*(a + I*a*Tan[c + d*x])^(5/2)*((3*b*B)/(2*a) + B*Tan[c + d*x]))/(d*E^((2*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(3*b*Cos[c + d*x] + 2*a*Sin[c + d*x])) + (Cos[c + d*x]^3*((-I)*Csc[c]*((-2*I)*a*Cos[c] + 7*b*Cos[c] + I*b*Sin[c])*(2*Cos[2*c] - (2*I)*Sin[2*c]) + Csc[c + d*x]^2*(-2*b*Cos[2*c] + (2*I)*b*Sin[2*c]) + Csc[c]*Csc[c + d*x]*(2*Cos[2*c] - (2*I)*Sin[2*c])*(2*a*Sin[d*x] + (7*I)*b*Sin[d*x]))*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)*((3*b*B)/(2*a) + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(3*b*Cos[c + d*x] + 2*a*Sin[c + d*x]))","B",1
179,1,277,205,5.2892293,"\int \frac{\tan ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{(A+B \tan (c+d x)) \left(\frac{\sqrt{2} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left((B+i A) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+\sqrt{2} (B-2 i A) \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{\sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \sqrt{\sec (c+d x)}}-2 \sqrt{\tan (c+d x)} (-B \sin (c+d x)+(A+2 i B) \cos (c+d x))\right)}{2 d \sqrt{a+i a \tan (c+d x)} (A \cos (c+d x)+B \sin (c+d x))}","\frac{(-1)^{3/4} (-B+2 i A) \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}+\frac{(-B+i A) \tan ^{\frac{3}{2}}(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}-\frac{(A+2 i B) \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{a d}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"(((Sqrt[2]*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*((I*A + B)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*((-2*I)*A + B)*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[Sec[c + d*x]]) - 2*((A + (2*I)*B)*Cos[c + d*x] - B*Sin[c + d*x])*Sqrt[Tan[c + d*x]])*(A + B*Tan[c + d*x]))/(2*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
180,1,183,156,4.0025522,"\int \frac{\sqrt{\tan (c+d x)} (A+B \tan (c+d x))}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{\sqrt{\tan (c+d x)} \left(i (A+i B) \sqrt{-1+e^{2 i (c+d x)}}-i (A-i B) e^{i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+2 \sqrt{2} B e^{i (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{a+i a \tan (c+d x)}}","\frac{(-B+i A) \sqrt{\tan (c+d x)}}{d \sqrt{a+i a \tan (c+d x)}}-\frac{\left(\frac{1}{2}+\frac{i}{2}\right) (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}-\frac{2 \sqrt[4]{-1} B \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"((I*(A + I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))] - I*(A - I*B)*E^(I*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + 2*Sqrt[2]*B*E^(I*(c + d*x))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]])/(d*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
181,1,123,99,3.5722954,"\int \frac{A+B \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{\sqrt{\tan (c+d x)} \left((A+i B) \sqrt{-1+e^{2 i (c+d x)}}+(A-i B) e^{i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{a+i a \tan (c+d x)}}","\frac{(A+i B) \sqrt{\tan (c+d x)}}{d \sqrt{a+i a \tan (c+d x)}}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"(((A + I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))] + (A - I*B)*E^(I*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]])/(d*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
182,1,181,143,3.6929477,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{(A+B \tan (c+d x)) \left(\frac{(A-i B) \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)}{\sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}+\frac{-4 A \cos (c+d x)+2 (B-3 i A) \sin (c+d x)}{\sqrt{\tan (c+d x)}}\right)}{2 d \sqrt{a+i a \tan (c+d x)} (A \cos (c+d x)+B \sin (c+d x))}","\frac{A+i B}{d \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{(3 A+i B) \sqrt{a+i a \tan (c+d x)}}{a d \sqrt{\tan (c+d x)}}+\frac{\left(\frac{1}{2}+\frac{i}{2}\right) (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"((((A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))] + (-4*A*Cos[c + d*x] + 2*((-3*I)*A + B)*Sin[c + d*x])/Sqrt[Tan[c + d*x]])*(A + B*Tan[c + d*x]))/(2*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
183,1,221,191,4.3075161,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{e^{-i (c+d x)} (A+B \tan (c+d x)) \left(3 (B+i A) e^{i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+i A \left(-18 e^{2 i (c+d x)}+7 e^{4 i (c+d x)}+3\right)-3 B \left(-6 e^{2 i (c+d x)}+5 e^{4 i (c+d x)}+1\right)\right)}{6 d \left(-1+e^{2 i (c+d x)}\right) \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (A \cos (c+d x)+B \sin (c+d x))}","-\frac{(5 A+3 i B) \sqrt{a+i a \tan (c+d x)}}{3 a d \tan ^{\frac{3}{2}}(c+d x)}+\frac{A+i B}{d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}+\frac{(-9 B+7 i A) \sqrt{a+i a \tan (c+d x)}}{3 a d \sqrt{\tan (c+d x)}}+\frac{\left(\frac{1}{2}+\frac{i}{2}\right) (B+i A) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"((-3*B*(1 - 6*E^((2*I)*(c + d*x)) + 5*E^((4*I)*(c + d*x))) + I*A*(3 - 18*E^((2*I)*(c + d*x)) + 7*E^((4*I)*(c + d*x))) + 3*(I*A + B)*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*(A + B*Tan[c + d*x]))/(6*d*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
184,1,241,237,5.6164965,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{(A+B \tan (c+d x)) \left(-\frac{(A-i B) \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)}{\sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}-\frac{\csc ^2(c+d x) (-5 (2 A+i B) \cos (c+d x)+(22 A+5 i B) \cos (3 (c+d x))+\sin (c+d x) ((-25 B+59 i A) \cos (2 (c+d x))+9 (5 B-7 i A)))}{15 \sqrt{\tan (c+d x)}}\right)}{2 d \sqrt{a+i a \tan (c+d x)} (A \cos (c+d x)+B \sin (c+d x))}","\frac{(-25 B+23 i A) \sqrt{a+i a \tan (c+d x)}}{15 a d \tan ^{\frac{3}{2}}(c+d x)}-\frac{(7 A+5 i B) \sqrt{a+i a \tan (c+d x)}}{5 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{A+i B}{d \tan ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}+\frac{(61 A+35 i B) \sqrt{a+i a \tan (c+d x)}}{15 a d \sqrt{\tan (c+d x)}}-\frac{\left(\frac{1}{2}+\frac{i}{2}\right) (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"((-(((A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) - (Csc[c + d*x]^2*(-5*(2*A + I*B)*Cos[c + d*x] + (22*A + (5*I)*B)*Cos[3*(c + d*x)] + (9*((-7*I)*A + 5*B) + ((59*I)*A - 25*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(15*Sqrt[Tan[c + d*x]]))*(A + B*Tan[c + d*x]))/(2*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
185,1,285,203,6.7297355,"\int \frac{\tan ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{e^{-i (c+d x)} \sqrt{\tan (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \left(\sqrt{-1+e^{2 i (c+d x)}} \left(-4 i A e^{2 i (c+d x)}+i A+10 B e^{2 i (c+d x)}-B\right)+3 (B+i A) e^{3 i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-12 \sqrt{2} B e^{3 i (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{6 \sqrt{2} a d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{2 (-1)^{3/4} B \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{(-B+i A) \tan ^{\frac{3}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(A+3 i B) \sqrt{\tan (c+d x)}}{2 a d \sqrt{a+i a \tan (c+d x)}}",1,"((Sqrt[-1 + E^((2*I)*(c + d*x))]*(I*A - B - (4*I)*A*E^((2*I)*(c + d*x)) + 10*B*E^((2*I)*(c + d*x))) + 3*(I*A + B)*E^((3*I)*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] - 12*Sqrt[2]*B*E^((3*I)*(c + d*x))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^(3/2)*Sqrt[Tan[c + d*x]])/(6*Sqrt[2]*a*d*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
186,1,228,150,5.3227415,"\int \frac{\sqrt{\tan (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{e^{-i (c+d x)} \sqrt{\tan (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \left(\sqrt{-1+e^{2 i (c+d x)}} \left(2 A e^{2 i (c+d x)}+A-i B \left(-1+4 e^{2 i (c+d x)}\right)\right)-3 (A-i B) e^{3 i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{6 \sqrt{2} a d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","-\frac{\left(\frac{1}{4}+\frac{i}{4}\right) (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{(-B+i A) \sqrt{\tan (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(5 B+i A) \sqrt{\tan (c+d x)}}{6 a d \sqrt{a+i a \tan (c+d x)}}",1,"((Sqrt[-1 + E^((2*I)*(c + d*x))]*(A + 2*A*E^((2*I)*(c + d*x)) - I*B*(-1 + 4*E^((2*I)*(c + d*x)))) - 3*(A - I*B)*E^((3*I)*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^(3/2)*Sqrt[Tan[c + d*x]])/(6*Sqrt[2]*a*d*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
187,1,230,148,5.6928688,"\int \frac{A+B \tan (c+d x)}{\sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)),x]","\frac{e^{-i (c+d x)} \sqrt{\tan (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \left(\sqrt{-1+e^{2 i (c+d x)}} \left(-i A \left(1+8 e^{2 i (c+d x)}\right)+2 B e^{2 i (c+d x)}+B\right)-3 i (A-i B) e^{3 i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{6 \sqrt{2} a d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{\left(\frac{1}{4}-\frac{i}{4}\right) (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{(A+i B) \sqrt{\tan (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(7 A+i B) \sqrt{\tan (c+d x)}}{6 a d \sqrt{a+i a \tan (c+d x)}}",1,"((Sqrt[-1 + E^((2*I)*(c + d*x))]*(B + 2*B*E^((2*I)*(c + d*x)) - I*A*(1 + 8*E^((2*I)*(c + d*x)))) - (3*I)*(A - I*B)*E^((3*I)*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^(3/2)*Sqrt[Tan[c + d*x]])/(6*Sqrt[2]*a*d*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
188,1,237,194,5.4151584,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)),x]","\frac{i e^{-2 i (c+d x)} \sqrt{\tan (c+d x)} \csc (c+d x) \sec (c+d x) \left(\sqrt{-1+e^{2 i (c+d x)}} \left(A \left(-13 e^{2 i (c+d x)}+38 e^{4 i (c+d x)}-1\right)+i B \left(-7 e^{2 i (c+d x)}+8 e^{4 i (c+d x)}-1\right)\right)-3 (A-i B) e^{3 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{12 a d \sqrt{-1+e^{2 i (c+d x)}} (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{\left(\frac{1}{4}+\frac{i}{4}\right) (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}-\frac{(25 A+7 i B) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \sqrt{\tan (c+d x)}}+\frac{A+i B}{3 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac{11 A+5 i B}{6 a d \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}",1,"((I/12)*(Sqrt[-1 + E^((2*I)*(c + d*x))]*(I*B*(-1 - 7*E^((2*I)*(c + d*x)) + 8*E^((4*I)*(c + d*x))) + A*(-1 - 13*E^((2*I)*(c + d*x)) + 38*E^((4*I)*(c + d*x)))) - 3*(A - I*B)*E^((3*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Csc[c + d*x]*Sec[c + d*x]*Sqrt[Tan[c + d*x]])/(a*d*E^((2*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
189,1,244,240,7.2408179,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(3/2)),x]","\frac{i e^{-2 i (c+d x)} \sec ^2(c+d x) \left(-3 i (A-i B) e^{3 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-i A \left(18 e^{2 i (c+d x)}-87 e^{4 i (c+d x)}+52 e^{6 i (c+d x)}+1\right)+B \left(12 e^{2 i (c+d x)}-51 e^{4 i (c+d x)}+38 e^{6 i (c+d x)}+1\right)\right)}{12 a d \left(-1+e^{2 i (c+d x)}\right) \sqrt{\tan (c+d x)} (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{\left(\frac{1}{4}+\frac{i}{4}\right) (B+i A) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}-\frac{(21 A+11 i B) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{(-25 B+39 i A) \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \sqrt{\tan (c+d x)}}+\frac{5 A+3 i B}{2 a d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}+\frac{A+i B}{3 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}",1,"((I/12)*(B*(1 + 12*E^((2*I)*(c + d*x)) - 51*E^((4*I)*(c + d*x)) + 38*E^((6*I)*(c + d*x))) - I*A*(1 + 18*E^((2*I)*(c + d*x)) - 87*E^((4*I)*(c + d*x)) + 52*E^((6*I)*(c + d*x))) - (3*I)*(A - I*B)*E^((3*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^2)/(a*d*E^((2*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*Sqrt[Tan[c + d*x]]*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
190,1,275,249,8.1363692,"\int \frac{\tan ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{e^{-2 i (c+d x)} \sqrt{\tan (c+d x)} \sec ^2(c+d x) \left(\sqrt{-1+e^{2 i (c+d x)}} \left(i A \left(-11 e^{2 i (c+d x)}+23 e^{4 i (c+d x)}+3\right)-3 B \left(-7 e^{2 i (c+d x)}+41 e^{4 i (c+d x)}+1\right)\right)-15 i (A-i B) e^{5 i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+120 \sqrt{2} B e^{5 i (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{60 a^2 d \sqrt{-1+e^{2 i (c+d x)}} (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}+\frac{2 \sqrt[4]{-1} B \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{(-7 B+i A) \sqrt{\tan (c+d x)}}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{(-B+i A) \tan ^{\frac{5}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{(A+3 i B) \tan ^{\frac{3}{2}}(c+d x)}{6 a d (a+i a \tan (c+d x))^{3/2}}",1,"((Sqrt[-1 + E^((2*I)*(c + d*x))]*(I*A*(3 - 11*E^((2*I)*(c + d*x)) + 23*E^((4*I)*(c + d*x))) - 3*B*(1 - 7*E^((2*I)*(c + d*x)) + 41*E^((4*I)*(c + d*x)))) - (15*I)*(A - I*B)*E^((5*I)*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + 120*Sqrt[2]*B*E^((5*I)*(c + d*x))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^2*Sqrt[Tan[c + d*x]])/(60*a^2*d*E^((2*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
191,1,214,194,7.8998182,"\int \frac{\tan ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{e^{-3 i (c+d x)} \sec ^3(c+d x) \left(\left(-1+e^{2 i (c+d x)}\right) \left(i A \left(e^{2 i (c+d x)}+17 e^{4 i (c+d x)}-3\right)+B \left(-11 e^{2 i (c+d x)}+23 e^{4 i (c+d x)}+3\right)\right)-15 i (A-i B) e^{5 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{120 a^2 d \sqrt{\tan (c+d x)} (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}+\frac{(13 A-37 i B) \sqrt{\tan (c+d x)}}{60 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{(-B+i A) \tan ^{\frac{3}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{(A+11 i B) \sqrt{\tan (c+d x)}}{30 a d (a+i a \tan (c+d x))^{3/2}}",1,"(((-1 + E^((2*I)*(c + d*x)))*(I*A*(-3 + E^((2*I)*(c + d*x)) + 17*E^((4*I)*(c + d*x))) + B*(3 - 11*E^((2*I)*(c + d*x)) + 23*E^((4*I)*(c + d*x)))) - (15*I)*(A - I*B)*E^((5*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^3)/(120*a^2*d*E^((3*I)*(c + d*x))*Sqrt[Tan[c + d*x]]*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
192,1,215,196,6.2784586,"\int \frac{\sqrt{\tan (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{e^{-2 i (c+d x)} \sqrt{\tan (c+d x)} \sec ^2(c+d x) \left(\sqrt{-1+e^{2 i (c+d x)}} \left(-3 i A \left(3 e^{2 i (c+d x)}+e^{4 i (c+d x)}+1\right)-B \left(e^{2 i (c+d x)}+17 e^{4 i (c+d x)}-3\right)\right)+15 (B+i A) e^{5 i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{60 a^2 d \sqrt{-1+e^{2 i (c+d x)}} (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","-\frac{\left(\frac{1}{8}+\frac{i}{8}\right) (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{(-13 B+3 i A) \sqrt{\tan (c+d x)}}{60 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{(7 B+3 i A) \sqrt{\tan (c+d x)}}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{(-B+i A) \sqrt{\tan (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"((Sqrt[-1 + E^((2*I)*(c + d*x))]*((-3*I)*A*(1 + 3*E^((2*I)*(c + d*x)) + E^((4*I)*(c + d*x))) - B*(-3 + E^((2*I)*(c + d*x)) + 17*E^((4*I)*(c + d*x)))) + 15*(I*A + B)*E^((5*I)*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^2*Sqrt[Tan[c + d*x]])/(60*a^2*d*E^((2*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
193,1,216,194,6.9072352,"\int \frac{A+B \tan (c+d x)}{\sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)),x]","-\frac{e^{-2 i (c+d x)} \sqrt{\tan (c+d x)} \sec ^2(c+d x) \left(\sqrt{-1+e^{2 i (c+d x)}} \left(A \left(19 e^{2 i (c+d x)}+83 e^{4 i (c+d x)}+3\right)+3 i B \left(3 e^{2 i (c+d x)}+e^{4 i (c+d x)}+1\right)\right)+15 (A-i B) e^{5 i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{60 a^2 d \sqrt{-1+e^{2 i (c+d x)}} (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{\left(\frac{1}{8}-\frac{i}{8}\right) (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}+\frac{(67 A-3 i B) \sqrt{\tan (c+d x)}}{60 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{(A+i B) \sqrt{\tan (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{(13 A+3 i B) \sqrt{\tan (c+d x)}}{30 a d (a+i a \tan (c+d x))^{3/2}}",1,"-1/60*((Sqrt[-1 + E^((2*I)*(c + d*x))]*((3*I)*B*(1 + 3*E^((2*I)*(c + d*x)) + E^((4*I)*(c + d*x))) + A*(3 + 19*E^((2*I)*(c + d*x)) + 83*E^((4*I)*(c + d*x)))) + 15*(A - I*B)*E^((5*I)*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^2*Sqrt[Tan[c + d*x]])/(a^2*d*E^((2*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
194,1,288,240,10.7256861,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)),x]","\frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x)) \left(\frac{\sqrt{2} (B+i A) e^{3 i (c+d x)} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)}{\sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}}}+\frac{2 ((19 B-149 i A) \tan (c+d x)+\cos (2 (c+d x)) ((86 B-466 i A) \tan (c+d x)-20 (23 A+4 i B))+340 A+80 i B)}{15 \sqrt{\tan (c+d x)} \sqrt{\sec (c+d x)}}\right)}{8 d (a+i a \tan (c+d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) (A-i B) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{(317 A+67 i B) \sqrt{a+i a \tan (c+d x)}}{60 a^3 d \sqrt{\tan (c+d x)}}+\frac{151 A+41 i B}{60 a^2 d \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{A+i B}{5 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}+\frac{17 A+7 i B}{30 a d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}",1,"(Sec[c + d*x]^(3/2)*(A + B*Tan[c + d*x])*((Sqrt[2]*(I*A + B)*E^((3*I)*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/(Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]) + (2*(340*A + (80*I)*B + ((-149*I)*A + 19*B)*Tan[c + d*x] + Cos[2*(c + d*x)]*(-20*(23*A + (4*I)*B) + ((-466*I)*A + 86*B)*Tan[c + d*x])))/(15*Sqrt[Sec[c + d*x]]*Sqrt[Tan[c + d*x]])))/(8*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^(5/2))","A",1
195,1,268,286,8.8256157,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2)),x]","\frac{e^{-3 i (c+d x)} \sec ^3(c+d x) \left(-15 i (A-i B) e^{5 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-i A \left(33 e^{2 i (c+d x)}+348 e^{4 i (c+d x)}-1527 e^{6 i (c+d x)}+983 e^{8 i (c+d x)}+3\right)+B \left(23 e^{2 i (c+d x)}+168 e^{4 i (c+d x)}-657 e^{6 i (c+d x)}+463 e^{8 i (c+d x)}+3\right)\right)}{120 a^2 d \left(-1+e^{2 i (c+d x)}\right) \sqrt{\tan (c+d x)} (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) (B+i A) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{(361 A+151 i B) \sqrt{a+i a \tan (c+d x)}}{60 a^3 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{(-317 B+707 i A) \sqrt{a+i a \tan (c+d x)}}{60 a^3 d \sqrt{\tan (c+d x)}}+\frac{89 A+39 i B}{20 a^2 d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}+\frac{21 A+11 i B}{30 a d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{A+i B}{5 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}",1,"((B*(3 + 23*E^((2*I)*(c + d*x)) + 168*E^((4*I)*(c + d*x)) - 657*E^((6*I)*(c + d*x)) + 463*E^((8*I)*(c + d*x))) - I*A*(3 + 33*E^((2*I)*(c + d*x)) + 348*E^((4*I)*(c + d*x)) - 1527*E^((6*I)*(c + d*x)) + 983*E^((8*I)*(c + d*x))) - (15*I)*(A - I*B)*E^((5*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^3)/(120*a^2*d*E^((3*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*Sqrt[Tan[c + d*x]]*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
196,-1,0,201,180.0049807,"\int \sqrt[3]{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[(a + I*a*Tan[c + d*x])^(1/3)*(A + B*Tan[c + d*x]),x]","\text{\$Aborted}","-\frac{\sqrt{3} \sqrt[3]{a} (B+i A) \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2^{2/3} d}+\frac{3 \sqrt[3]{a} (B+i A) \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2\ 2^{2/3} d}+\frac{\sqrt[3]{a} (B+i A) \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{\sqrt[3]{a} x (A-i B)}{2\ 2^{2/3}}+\frac{3 B \sqrt[3]{a+i a \tan (c+d x)}}{d}",1,"$Aborted","F",-1
197,1,104,270,2.9550622,"\int \tan ^2(c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(2/3)*(A + B*Tan[c + d*x]),x]","\frac{3 (a+i a \tan (c+d x))^{2/3} \left(10 (B+i A) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)+(8 A-2 i B) \tan (c+d x)-8 i A+5 B \sec ^2(c+d x)-22 B\right)}{40 d}","-\frac{\sqrt{3} a^{2/3} (B+i A) \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt[3]{2} d}-\frac{3 a^{2/3} (B+i A) \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2 \sqrt[3]{2} d}-\frac{a^{2/3} (B+i A) \log (\cos (c+d x))}{2 \sqrt[3]{2} d}+\frac{a^{2/3} x (A-i B)}{2 \sqrt[3]{2}}-\frac{3 (B+4 i A) (a+i a \tan (c+d x))^{5/3}}{20 a d}+\frac{3 B \tan ^2(c+d x) (a+i a \tan (c+d x))^{2/3}}{8 d}-\frac{9 B (a+i a \tan (c+d x))^{2/3}}{8 d}",1,"(3*(a + I*a*Tan[c + d*x])^(2/3)*((-8*I)*A - 22*B + 10*(I*A + B)*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))] + 5*B*Sec[c + d*x]^2 + (8*A - (2*I)*B)*Tan[c + d*x]))/(40*d)","C",1
198,1,115,232,1.763281,"\int \tan (c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^(2/3)*(A + B*Tan[c + d*x]),x]","\frac{3 \left(e^{i d x}\right)^{2/3} (a+i a \tan (c+d x))^{2/3} \left(-5 (A-i B) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)+10 A+4 B \tan (c+d x)-4 i B\right)}{20 d (\cos (d x)+i \sin (d x))^{2/3}}","\frac{\sqrt{3} a^{2/3} (A-i B) \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt[3]{2} d}+\frac{3 a^{2/3} (A-i B) \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2 \sqrt[3]{2} d}+\frac{a^{2/3} (A-i B) \log (\cos (c+d x))}{2 \sqrt[3]{2} d}+\frac{a^{2/3} x (B+i A)}{2 \sqrt[3]{2}}+\frac{3 A (a+i a \tan (c+d x))^{2/3}}{2 d}-\frac{3 i B (a+i a \tan (c+d x))^{5/3}}{5 a d}",1,"(3*(E^(I*d*x))^(2/3)*(a + I*a*Tan[c + d*x])^(2/3)*(10*A - (4*I)*B - 5*(A - I*B)*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))] + 4*B*Tan[c + d*x]))/(20*d*(Cos[d*x] + I*Sin[d*x])^(2/3))","C",1
199,1,91,202,1.2585693,"\int (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx","Integrate[(a + I*a*Tan[c + d*x])^(2/3)*(A + B*Tan[c + d*x]),x]","-\frac{3 \left(\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left(-2 B+(B+i A) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)\right)}{2 \sqrt[3]{2} d}","\frac{\sqrt{3} a^{2/3} (B+i A) \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt[3]{2} d}+\frac{3 a^{2/3} (B+i A) \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2 \sqrt[3]{2} d}+\frac{a^{2/3} (B+i A) \log (\cos (c+d x))}{2 \sqrt[3]{2} d}-\frac{a^{2/3} x (A-i B)}{2 \sqrt[3]{2}}+\frac{3 B (a+i a \tan (c+d x))^{2/3}}{2 d}",1,"(-3*((a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(-2*B + (I*A + B)*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]))/(2*2^(1/3)*d)","C",1
200,1,127,289,1.7631989,"\int \cot (c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(2/3)*(A + B*Tan[c + d*x]),x]","\frac{3 \left(\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left((A-i B) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)-2 A \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{2 e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)\right)}{2 \sqrt[3]{2} d}","-\frac{\sqrt{3} a^{2/3} (A-i B) \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt[3]{2} d}-\frac{3 a^{2/3} (A-i B) \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2 \sqrt[3]{2} d}-\frac{a^{2/3} (A-i B) \log (\cos (c+d x))}{2 \sqrt[3]{2} d}-\frac{a^{2/3} x (B+i A)}{2 \sqrt[3]{2}}+\frac{\sqrt{3} a^{2/3} A \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{d}-\frac{a^{2/3} A \log (\tan (c+d x))}{2 d}+\frac{3 a^{2/3} A \log \left(\sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2 d}",1,"(3*((a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^(2/3)*((A - I*B)*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))] - 2*A*Hypergeometric2F1[2/3, 1, 5/3, (2*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]))/(2*2^(1/3)*d)","C",1
201,0,0,342,6.948963,"\int \cot ^2(c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(2/3)*(A + B*Tan[c + d*x]),x]","\int \cot ^2(c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx","\frac{a^{2/3} (3 B+2 i A) \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} d}-\frac{\sqrt{3} a^{2/3} (B+i A) \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt[3]{2} d}-\frac{a^{2/3} (3 B+2 i A) \log (\tan (c+d x))}{6 d}+\frac{a^{2/3} (3 B+2 i A) \log \left(\sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2 d}-\frac{3 a^{2/3} (B+i A) \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2 \sqrt[3]{2} d}-\frac{a^{2/3} (B+i A) \log (\cos (c+d x))}{2 \sqrt[3]{2} d}+\frac{a^{2/3} x (A-i B)}{2 \sqrt[3]{2}}-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d}",1,"Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(2/3)*(A + B*Tan[c + d*x]), x]","F",-1
202,1,137,213,1.3016741,"\int \frac{A+B \tan (c+d x)}{\sqrt[3]{a+i a \tan (c+d x)}} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^(1/3),x]","-\frac{3 i e^{-2 i (c+d x)} \left(\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left((A-i B) e^{2 i (c+d x)} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)-2 (A+i B) \left(1+e^{2 i (c+d x)}\right)\right)}{4 \sqrt[3]{2} a d}","\frac{\sqrt{3} (B+i A) \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2 \sqrt[3]{2} \sqrt[3]{a} d}+\frac{3 (-B+i A)}{2 d \sqrt[3]{a+i a \tan (c+d x)}}+\frac{3 (B+i A) \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{4 \sqrt[3]{2} \sqrt[3]{a} d}+\frac{(B+i A) \log (\cos (c+d x))}{4 \sqrt[3]{2} \sqrt[3]{a} d}-\frac{x (A-i B)}{4 \sqrt[3]{2} \sqrt[3]{a}}",1,"(((-3*I)/4)*((a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(-2*(A + I*B)*(1 + E^((2*I)*(c + d*x))) + (A - I*B)*E^((2*I)*(c + d*x))*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]))/(2^(1/3)*a*d*E^((2*I)*(c + d*x)))","C",1
203,0,0,213,0.7865602,"\int \frac{A+B \tan (c+d x)}{(a+i a \tan (c+d x))^{2/3}} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^(2/3),x]","\int \frac{A+B \tan (c+d x)}{(a+i a \tan (c+d x))^{2/3}} \, dx","-\frac{\sqrt{3} (B+i A) \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2\ 2^{2/3} a^{2/3} d}+\frac{3 (B+i A) \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{4\ 2^{2/3} a^{2/3} d}+\frac{(B+i A) \log (\cos (c+d x))}{4\ 2^{2/3} a^{2/3} d}-\frac{x (A-i B)}{4\ 2^{2/3} a^{2/3}}+\frac{3 (-B+i A)}{4 d (a+i a \tan (c+d x))^{2/3}}",1,"Integrate[(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^(2/3), x]","F",-1
204,1,1805,290,14.2259664,"\int \tan ^m(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","-\frac{8 i (A-i B) e^{-4 i c} \left(-1+e^{2 i (c+d x)}\right)^m \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \cos ^5(c+d x) \left(-\frac{\left(1+e^{2 i c}\right) \left(-1+e^{2 i (c+d x)}\right) \, _2F_1\left(1,m+1;m+2;\frac{1-e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) \left(1+e^{2 i (c+d x)}\right)^{-m-1}}{m+1}-\frac{\, _2F_1\left(1,m;m+1;\frac{1-e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) \left(1+e^{2 i (c+d x)}\right)^{-m}}{m}+\frac{2^{-m} \, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)}{m}\right) (i \tan (c+d x) a+a)^4 (A+B \tan (c+d x)) \left(\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{-m}}{d \left(1+e^{2 i c}\right) (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\cos ^5(c+d x) \left(\frac{\sec ^2(c) \left(\frac{1}{2} \cos (4 c)-\frac{1}{2} i \sin (4 c)\right) (B \cos (c-d x)-B \cos (c+d x)-A \sin (c-d x)+4 i B \sin (c-d x)+A \sin (c+d x)-4 i B \sin (c+d x)) \sec ^3(c+d x)}{m+3}+\frac{\sec ^2(c) (-\cos (2 c) B-4 i \sin (2 c) B+B+A \sin (2 c)) \left(\frac{1}{2} \cos (4 c)-\frac{1}{2} i \sin (4 c)\right) \sec ^2(c+d x)}{m+3}+\frac{\sec ^2(c) (\cos (4 c)-i \sin (4 c)) (B \cos (c-d x)-B \cos (c+d x)-A \sin (c-d x)+4 i B \sin (c-d x)+A \sin (c+d x)-4 i B \sin (c+d x)) \sec (c+d x)}{(m+1) (m+3)}+\frac{\sec (c) (A \cos (c)-4 i B \cos (c)+B \sin (c)) (2 \cos (4 c)-2 i \sin (4 c)) \tan (c)}{(m+1) (m+3)}\right) \tan ^m(c+d x) (i \tan (c+d x) a+a)^4 (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\cos ^5(c+d x) \left(\frac{\sec (c+d x) (2 \cos (4 c)-2 i \sin (4 c)) (-i A \cos (c-d x)-2 B \cos (c-d x)+i A \cos (c+d x)+2 B \cos (c+d x)+2 A \sin (c-d x)-3 i B \sin (c-d x)-2 A \sin (c+d x)+3 i B \sin (c+d x)) \sec ^2(c)}{m+1}+\frac{(2 A \cos (c)-3 i B \cos (c)+i A \sin (c)+2 B \sin (c)) (4 i \sin (4 c)-4 \cos (4 c)) \tan (c) \sec (c)}{m+1}\right) \tan ^m(c+d x) (i \tan (c+d x) a+a)^4 (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\cos ^5(c+d x) \left(\frac{(A-2 i B) (-2 m+\cos (2 c)-3) (-2 i \cos (4 c)-2 \sin (4 c)) \sec ^2(c)}{(m+1) (m+2)}+\frac{(i A \cos (c-d x)+2 B \cos (c-d x)-i A \cos (c+d x)-2 B \cos (c+d x)) \sec (c+d x) (2 \cos (4 c)-2 i \sin (4 c)) \sec ^2(c)}{m+1}+\frac{(A-2 i B) \sec ^2(c+d x) (-4 i \cos (4 c)-4 \sin (4 c))}{m+2}\right) \tan ^m(c+d x) (i \tan (c+d x) a+a)^4 (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\cos ^5(c+d x) \left(\frac{(B \cos (4 c)-i B \sin (4 c)) \sec ^4(c+d x)}{m+4}+\frac{(B \cos (c+d x)-B \cos (c-d x)) \sec ^2(c) \left(\frac{1}{2} \cos (4 c)-\frac{1}{2} i \sin (4 c)\right) \sec ^3(c+d x)}{m+3}+\frac{\left(-2 B m^2-9 B m+3 B \cos (2 c) m-8 B+8 B \cos (2 c)\right) \sec ^2(c) \left(\frac{1}{2} \cos (4 c)-\frac{1}{2} i \sin (4 c)\right) \sec ^2(c+d x)}{(m+2) \left(m^2+7 m+12\right)}+\frac{(B \cos (c+d x)-B \cos (c-d x)) \sec ^2(c) (\cos (4 c)-i \sin (4 c)) \sec (c+d x)}{(m+1) (m+3)}+\frac{\left(-2 m^2+2 \cos (2 c) m-10 m+5 \cos (2 c)-11\right) \sec ^2(c) (B \cos (4 c)-i B \sin (4 c))}{(m+1) (m+2) (m+3) (m+4)}\right) \tan ^m(c+d x) (i \tan (c+d x) a+a)^4 (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{2^{3-m} (i A+B) e^{-2 i c} \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \cos ^5(c+d x) \left(2^m \, _2F_1\left(1,m;m+1;-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)-\left(1+e^{2 i (c+d x)}\right)^m \, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)\right) (i \tan (c+d x) a+a)^4 (A+B \tan (c+d x))}{d \left(1+e^{2 i c}\right) m (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}","\frac{8 a^4 (A-i B) \tan ^{m+1}(c+d x) \, _2F_1(1,m+1;m+2;i \tan (c+d x))}{d (m+1)}-\frac{2 \left(A (m+4)^2-i B \left(m^2+8 m+19\right)\right) \left(a^4+i a^4 \tan (c+d x)\right) \tan ^{m+1}(c+d x)}{d (m+2) (m+3) (m+4)}-\frac{2 a^4 \left(A \left(2 m^3+19 m^2+60 m+64\right)-i B \left(2 m^3+19 m^2+60 m+67\right)\right) \tan ^{m+1}(c+d x)}{d (m+1) (m+2) (m+3) (m+4)}-\frac{(A (m+4)-i B (m+7)) \left(a^2+i a^2 \tan (c+d x)\right)^2 \tan ^{m+1}(c+d x)}{d (m+3) (m+4)}+\frac{i a B (a+i a \tan (c+d x))^3 \tan ^{m+1}(c+d x)}{d (m+4)}",1,"(2^(3 - m)*(I*A + B)*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*Cos[c + d*x]^5*(2^m*Hypergeometric2F1[1, m, 1 + m, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))] - (1 + E^((2*I)*(c + d*x)))^m*Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2])*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]))/(d*E^((2*I)*c)*(1 + E^((2*I)*c))*m*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) - ((8*I)*(A - I*B)*(-1 + E^((2*I)*(c + d*x)))^m*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*Cos[c + d*x]^5*(-(Hypergeometric2F1[1, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]/((1 + E^((2*I)*(c + d*x)))^m*m)) - ((1 + E^((2*I)*c))*(-1 + E^((2*I)*(c + d*x)))*(1 + E^((2*I)*(c + d*x)))^(-1 - m)*Hypergeometric2F1[1, 1 + m, 2 + m, (1 - E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])/(1 + m) + Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2]/(2^m*m))*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]))/(d*E^((4*I)*c)*(1 + E^((2*I)*c))*((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^m*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^5*(((A - (2*I)*B)*Sec[c + d*x]^2*((-4*I)*Cos[4*c] - 4*Sin[4*c]))/(2 + m) + ((A - (2*I)*B)*(-3 - 2*m + Cos[2*c])*Sec[c]^2*((-2*I)*Cos[4*c] - 2*Sin[4*c]))/((1 + m)*(2 + m)) + ((I*A*Cos[c - d*x] + 2*B*Cos[c - d*x] - I*A*Cos[c + d*x] - 2*B*Cos[c + d*x])*Sec[c]^2*Sec[c + d*x]*(2*Cos[4*c] - (2*I)*Sin[4*c]))/(1 + m))*Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^5*(((-8*B - 9*B*m - 2*B*m^2 + 8*B*Cos[2*c] + 3*B*m*Cos[2*c])*Sec[c]^2*Sec[c + d*x]^2*(Cos[4*c]/2 - (I/2)*Sin[4*c]))/((2 + m)*(12 + 7*m + m^2)) + ((-(B*Cos[c - d*x]) + B*Cos[c + d*x])*Sec[c]^2*Sec[c + d*x]^3*(Cos[4*c]/2 - (I/2)*Sin[4*c]))/(3 + m) + ((-(B*Cos[c - d*x]) + B*Cos[c + d*x])*Sec[c]^2*Sec[c + d*x]*(Cos[4*c] - I*Sin[4*c]))/((1 + m)*(3 + m)) + ((-11 - 10*m - 2*m^2 + 5*Cos[2*c] + 2*m*Cos[2*c])*Sec[c]^2*(B*Cos[4*c] - I*B*Sin[4*c]))/((1 + m)*(2 + m)*(3 + m)*(4 + m)) + (Sec[c + d*x]^4*(B*Cos[4*c] - I*B*Sin[4*c]))/(4 + m))*Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^5*((Sec[c]^2*Sec[c + d*x]^2*(B - B*Cos[2*c] + A*Sin[2*c] - (4*I)*B*Sin[2*c])*(Cos[4*c]/2 - (I/2)*Sin[4*c]))/(3 + m) + (Sec[c]^2*Sec[c + d*x]^3*(Cos[4*c]/2 - (I/2)*Sin[4*c])*(B*Cos[c - d*x] - B*Cos[c + d*x] - A*Sin[c - d*x] + (4*I)*B*Sin[c - d*x] + A*Sin[c + d*x] - (4*I)*B*Sin[c + d*x]))/(3 + m) + (Sec[c]^2*Sec[c + d*x]*(Cos[4*c] - I*Sin[4*c])*(B*Cos[c - d*x] - B*Cos[c + d*x] - A*Sin[c - d*x] + (4*I)*B*Sin[c - d*x] + A*Sin[c + d*x] - (4*I)*B*Sin[c + d*x]))/((1 + m)*(3 + m)) + (Sec[c]*(A*Cos[c] - (4*I)*B*Cos[c] + B*Sin[c])*(2*Cos[4*c] - (2*I)*Sin[4*c])*Tan[c])/((1 + m)*(3 + m)))*Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^5*((Sec[c]^2*Sec[c + d*x]*(2*Cos[4*c] - (2*I)*Sin[4*c])*((-I)*A*Cos[c - d*x] - 2*B*Cos[c - d*x] + I*A*Cos[c + d*x] + 2*B*Cos[c + d*x] + 2*A*Sin[c - d*x] - (3*I)*B*Sin[c - d*x] - 2*A*Sin[c + d*x] + (3*I)*B*Sin[c + d*x]))/(1 + m) + (Sec[c]*(2*A*Cos[c] - (3*I)*B*Cos[c] + I*A*Sin[c] + 2*B*Sin[c])*(-4*Cos[4*c] + (4*I)*Sin[4*c])*Tan[c])/(1 + m))*Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
205,1,1305,205,12.6375817,"\int \tan ^m(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","-\frac{4 i (A-i B) e^{-3 i c} \left(-1+e^{2 i (c+d x)}\right)^m \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \cos ^4(c+d x) \left(-\frac{\left(1+e^{2 i c}\right) \left(-1+e^{2 i (c+d x)}\right) \, _2F_1\left(1,m+1;m+2;\frac{1-e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) \left(1+e^{2 i (c+d x)}\right)^{-m-1}}{m+1}-\frac{\, _2F_1\left(1,m;m+1;\frac{1-e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) \left(1+e^{2 i (c+d x)}\right)^{-m}}{m}+\frac{2^{-m} \, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)}{m}\right) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x)) \left(\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{-m}}{d \left(1+e^{2 i c}\right) (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\cos ^4(c+d x) \left(\frac{\sec (c+d x) \left(\frac{1}{2} \cos (3 c)-\frac{1}{2} i \sin (3 c)\right) (-i A \cos (c-d x)-3 B \cos (c-d x)+i A \cos (c+d x)+3 B \cos (c+d x)+3 A \sin (c-d x)-5 i B \sin (c-d x)-3 A \sin (c+d x)+5 i B \sin (c+d x)) \sec ^2(c)}{m+1}+\frac{(3 A \cos (c)-5 i B \cos (c)+i A \sin (c)+3 B \sin (c)) (i \sin (3 c)-\cos (3 c)) \tan (c) \sec (c)}{m+1}\right) \tan ^m(c+d x) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\cos ^4(c+d x) \left(-\frac{i B \sec (c) (\cos (3 c)-i \sin (3 c)) \sin (d x) \sec ^3(c+d x)}{m+3}-\frac{i B (\cos (3 c)-i \sin (3 c)) \tan (c) \sec ^2(c+d x)}{m+3}-\frac{i B \sec (c) (2 \cos (3 c)-2 i \sin (3 c)) \sin (d x) \sec (c+d x)}{(m+1) (m+3)}-\frac{i (2 B \cos (3 c)-2 i B \sin (3 c)) \tan (c)}{(m+1) (m+3)}\right) \tan ^m(c+d x) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\cos ^4(c+d x) \left(\frac{(A-3 i B) (-2 m+\cos (2 c)-3) \left(-\frac{1}{2} i \cos (3 c)-\frac{1}{2} \sin (3 c)\right) \sec ^2(c)}{(m+1) (m+2)}+\frac{(i A \cos (c-d x)+3 B \cos (c-d x)-i A \cos (c+d x)-3 B \cos (c+d x)) \sec (c+d x) \left(\frac{1}{2} \cos (3 c)-\frac{1}{2} i \sin (3 c)\right) \sec ^2(c)}{m+1}+\frac{(A-3 i B) \sec ^2(c+d x) (-i \cos (3 c)-\sin (3 c))}{m+2}\right) \tan ^m(c+d x) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{2^{2-m} (i A+B) e^{-i c} \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \cos ^4(c+d x) \left(2^m \, _2F_1\left(1,m;m+1;-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)-\left(1+e^{2 i (c+d x)}\right)^m \, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)\right) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x))}{d \left(1+e^{2 i c}\right) m (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","\frac{4 a^3 (A-i B) \tan ^{m+1}(c+d x) \, _2F_1(1,m+1;m+2;i \tan (c+d x))}{d (m+1)}-\frac{a^3 \left(A \left(2 m^2+11 m+15\right)-i B \left(2 m^2+11 m+17\right)\right) \tan ^{m+1}(c+d x)}{d (m+1) (m+2) (m+3)}-\frac{(A (m+3)-i B (m+5)) \left(a^3+i a^3 \tan (c+d x)\right) \tan ^{m+1}(c+d x)}{d (m+2) (m+3)}+\frac{i a B (a+i a \tan (c+d x))^2 \tan ^{m+1}(c+d x)}{d (m+3)}",1,"(2^(2 - m)*(I*A + B)*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*Cos[c + d*x]^4*(2^m*Hypergeometric2F1[1, m, 1 + m, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))] - (1 + E^((2*I)*(c + d*x)))^m*Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2])*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*E^(I*c)*(1 + E^((2*I)*c))*m*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) - ((4*I)*(A - I*B)*(-1 + E^((2*I)*(c + d*x)))^m*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*Cos[c + d*x]^4*(-(Hypergeometric2F1[1, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]/((1 + E^((2*I)*(c + d*x)))^m*m)) - ((1 + E^((2*I)*c))*(-1 + E^((2*I)*(c + d*x)))*(1 + E^((2*I)*(c + d*x)))^(-1 - m)*Hypergeometric2F1[1, 1 + m, 2 + m, (1 - E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])/(1 + m) + Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2]/(2^m*m))*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*E^((3*I)*c)*(1 + E^((2*I)*c))*((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^m*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^4*(((A - (3*I)*B)*Sec[c + d*x]^2*((-I)*Cos[3*c] - Sin[3*c]))/(2 + m) + ((A - (3*I)*B)*(-3 - 2*m + Cos[2*c])*Sec[c]^2*((-1/2*I)*Cos[3*c] - Sin[3*c]/2))/((1 + m)*(2 + m)) + ((I*A*Cos[c - d*x] + 3*B*Cos[c - d*x] - I*A*Cos[c + d*x] - 3*B*Cos[c + d*x])*Sec[c]^2*Sec[c + d*x]*(Cos[3*c]/2 - (I/2)*Sin[3*c]))/(1 + m))*Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^4*((Sec[c]^2*Sec[c + d*x]*(Cos[3*c]/2 - (I/2)*Sin[3*c])*((-I)*A*Cos[c - d*x] - 3*B*Cos[c - d*x] + I*A*Cos[c + d*x] + 3*B*Cos[c + d*x] + 3*A*Sin[c - d*x] - (5*I)*B*Sin[c - d*x] - 3*A*Sin[c + d*x] + (5*I)*B*Sin[c + d*x]))/(1 + m) + (Sec[c]*(3*A*Cos[c] - (5*I)*B*Cos[c] + I*A*Sin[c] + 3*B*Sin[c])*(-Cos[3*c] + I*Sin[3*c])*Tan[c])/(1 + m))*Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^4*(((-I)*B*Sec[c]*Sec[c + d*x]^3*(Cos[3*c] - I*Sin[3*c])*Sin[d*x])/(3 + m) - (I*B*Sec[c]*Sec[c + d*x]*(2*Cos[3*c] - (2*I)*Sin[3*c])*Sin[d*x])/((1 + m)*(3 + m)) - (I*B*Sec[c + d*x]^2*(Cos[3*c] - I*Sin[3*c])*Tan[c])/(3 + m) - (I*(2*B*Cos[3*c] - (2*I)*B*Sin[3*c])*Tan[c])/((1 + m)*(3 + m)))*Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
206,1,1587,132,10.795489,"\int \tan ^m(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","-\frac{2 i (A-i B) e^{-2 i c} \left(-1+e^{2 i (c+d x)}\right)^m \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \cos ^3(c+d x) \left(-\frac{\left(1+e^{2 i c}\right) \left(-1+e^{2 i (c+d x)}\right) \, _2F_1\left(1,m+1;m+2;\frac{1-e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) \left(1+e^{2 i (c+d x)}\right)^{-m-1}}{m+1}-\frac{\, _2F_1\left(1,m;m+1;\frac{1-e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) \left(1+e^{2 i (c+d x)}\right)^{-m}}{m}+\frac{2^{-m} \, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)}{m}\right) (i \tan (c+d x) a+a)^2 (A+B \tan (c+d x)) \left(\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{-m}}{d \left(1+e^{2 i c}\right) (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\cos ^3(c+d x) \left(\frac{\sec (c+d x) \left(\frac{1}{2} \cos (2 c)-\frac{1}{2} i \sin (2 c)\right) (-B \cos (c-d x)+B \cos (c+d x)+A \sin (c-d x)-2 i B \sin (c-d x)-A \sin (c+d x)+2 i B \sin (c+d x)) \sec ^2(c)}{m+1}+\frac{(A \cos (c)-2 i B \cos (c)+B \sin (c)) (i \sin (2 c)-\cos (2 c)) \tan (c) \sec (c)}{m+1}\right) \tan ^m(c+d x) (i \tan (c+d x) a+a)^2 (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\cos ^3(c+d x) \left(\frac{(B \cos (c-d x)-B \cos (c+d x)) \sec (c+d x) \left(\frac{1}{2} \cos (2 c)-\frac{1}{2} i \sin (2 c)\right) \sec ^2(c)}{m+1}+\frac{(-2 m+\cos (2 c)-3) \left(\frac{1}{2} i B \sin (2 c)-\frac{1}{2} B \cos (2 c)\right) \sec ^2(c)}{(m+1) (m+2)}+\frac{\sec ^2(c+d x) (i B \sin (2 c)-B \cos (2 c))}{m+2}\right) \tan ^m(c+d x) (i \tan (c+d x) a+a)^2 (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{2^{-m} A \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \cos ^3(c+d x) \left(2^m \, _2F_1\left(1,m;m+1;-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)-\left(1+e^{2 i (c+d x)}\right)^m \, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)\right) \tan (c) (i \tan (c+d x) a+a)^2 (A+B \tan (c+d x))}{d m (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}-\frac{i 2^{-m} B \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \cos ^3(c+d x) \left(2^m \, _2F_1\left(1,m;m+1;-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)-\left(1+e^{2 i (c+d x)}\right)^m \, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)\right) \tan (c) (i \tan (c+d x) a+a)^2 (A+B \tan (c+d x))}{d m (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{i 2^{-m} A \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \cos ^3(c+d x) \left(2^m \, _2F_1\left(1,m;m+1;-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)-\left(1+e^{2 i (c+d x)}\right)^m \, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)\right) (i \tan (c+d x) a+a)^2 (A+B \tan (c+d x))}{d m (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{2^{-m} B \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \cos ^3(c+d x) \left(2^m \, _2F_1\left(1,m;m+1;-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)-\left(1+e^{2 i (c+d x)}\right)^m \, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)\right) (i \tan (c+d x) a+a)^2 (A+B \tan (c+d x))}{d m (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{2 a^2 (A-i B) \tan ^{m+1}(c+d x) \, _2F_1(1,m+1;m+2;i \tan (c+d x))}{d (m+1)}+\frac{i a^2 (B+(m+2) (B+i A)) \tan ^{m+1}(c+d x)}{d (m+1) (m+2)}+\frac{i B \left(a^2+i a^2 \tan (c+d x)\right) \tan ^{m+1}(c+d x)}{d (m+2)}",1,"(I*A*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*Cos[c + d*x]^3*(2^m*Hypergeometric2F1[1, m, 1 + m, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))] - (1 + E^((2*I)*(c + d*x)))^m*Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2])*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/(2^m*d*m*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (B*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*Cos[c + d*x]^3*(2^m*Hypergeometric2F1[1, m, 1 + m, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))] - (1 + E^((2*I)*(c + d*x)))^m*Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2])*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/(2^m*d*m*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x])) - ((2*I)*(A - I*B)*(-1 + E^((2*I)*(c + d*x)))^m*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*Cos[c + d*x]^3*(-(Hypergeometric2F1[1, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]/((1 + E^((2*I)*(c + d*x)))^m*m)) - ((1 + E^((2*I)*c))*(-1 + E^((2*I)*(c + d*x)))*(1 + E^((2*I)*(c + d*x)))^(-1 - m)*Hypergeometric2F1[1, 1 + m, 2 + m, (1 - E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])/(1 + m) + Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2]/(2^m*m))*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/(d*E^((2*I)*c)*(1 + E^((2*I)*c))*((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^m*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (A*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*Cos[c + d*x]^3*(2^m*Hypergeometric2F1[1, m, 1 + m, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))] - (1 + E^((2*I)*(c + d*x)))^m*Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2])*Tan[c]*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/(2^m*d*m*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x])) - (I*B*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*Cos[c + d*x]^3*(2^m*Hypergeometric2F1[1, m, 1 + m, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))] - (1 + E^((2*I)*(c + d*x)))^m*Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2])*Tan[c]*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/(2^m*d*m*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^3*(((B*Cos[c - d*x] - B*Cos[c + d*x])*Sec[c]^2*Sec[c + d*x]*(Cos[2*c]/2 - (I/2)*Sin[2*c]))/(1 + m) + ((-3 - 2*m + Cos[2*c])*Sec[c]^2*(-1/2*(B*Cos[2*c]) + (I/2)*B*Sin[2*c]))/((1 + m)*(2 + m)) + (Sec[c + d*x]^2*(-(B*Cos[2*c]) + I*B*Sin[2*c]))/(2 + m))*Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^3*((Sec[c]^2*Sec[c + d*x]*(Cos[2*c]/2 - (I/2)*Sin[2*c])*(-(B*Cos[c - d*x]) + B*Cos[c + d*x] + A*Sin[c - d*x] - (2*I)*B*Sin[c - d*x] - A*Sin[c + d*x] + (2*I)*B*Sin[c + d*x]))/(1 + m) + (Sec[c]*(A*Cos[c] - (2*I)*B*Cos[c] + B*Sin[c])*(-Cos[2*c] + I*Sin[2*c])*Tan[c])/(1 + m))*Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
207,1,190,70,2.5733847,"\int \tan ^m(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","-\frac{i a e^{-i c} 2^{-m-1} \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^{m+1} \cos ^2(c+d x) (1+i \tan (c+d x)) (A+B \tan (c+d x)) \left(-B 2^{m+1}+(B+i A) \left(1+e^{2 i (c+d x)}\right)^{m+1} \, _2F_1\left(m+1,m+1;m+2;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)\right)}{d (m+1) (\cos (d x)+i \sin (d x)) (A \cos (c+d x)+B \sin (c+d x))}","\frac{a (A-i B) \tan ^{m+1}(c+d x) \, _2F_1(1,m+1;m+2;i \tan (c+d x))}{d (m+1)}+\frac{i a B \tan ^{m+1}(c+d x)}{d (m+1)}",1,"((-I)*2^(-1 - m)*a*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^(1 + m)*Cos[c + d*x]^2*(-(2^(1 + m)*B) + (I*A + B)*(1 + E^((2*I)*(c + d*x)))^(1 + m)*Hypergeometric2F1[1 + m, 1 + m, 2 + m, (1 - E^((2*I)*(c + d*x)))/2])*(1 + I*Tan[c + d*x])*(A + B*Tan[c + d*x]))/(d*E^(I*c)*(1 + m)*(Cos[d*x] + I*Sin[d*x])*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
208,0,0,168,7.7950496,"\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]),x]","\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx","\frac{(A (1-m)-i B (m+1)) \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{2 a d (m+1)}+\frac{m (-B+i A) \tan ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)}{2 a d (m+2)}+\frac{(A+i B) \tan ^{m+1}(c+d x)}{2 d (a+i a \tan (c+d x))}",1,"Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]), x]","F",-1
209,1,565,226,9.1939901,"\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2,x]","-\frac{i e^{-2 i c} \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \left(\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{-m} \sec (c+d x) (\cos (d x)+i \sin (d x))^2 (A+B \tan (c+d x)) \left(\frac{e^{4 i c} 2^{1-m} \left(A \left(2 m^2-4 m+1\right)+i B \left(2 m^2-1\right)\right) \left(\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^m \left(\left(1+e^{2 i (c+d x)}\right)^m \left((m+1) \, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+m \left(-1+e^{2 i (c+d x)}\right) \, _2F_1\left(m,m+1;m+2;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)\right)-2^m (m+1) \, _2F_1\left(1,m;m+1;\frac{1-e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)\right)}{m (m+1)}+\frac{e^{4 i c} 2^{2-m} (A (2 m-3)+i (2 B m+B)) \left(-1+e^{2 i (c+d x)}\right)^{m+1} \, _2F_1\left(m-1,m+1;m+2;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)}{m+1}+(A+i B) e^{-4 i d x} \left(-1+e^{2 i (c+d x)}\right)^{m+1} \left(1+e^{2 i (c+d x)}\right)^{2-m}+(A (3-2 m)-i (2 B m+B)) e^{2 i (c-d x)} \left(-1+e^{2 i (c+d x)}\right)^{m+1} \left(1+e^{2 i (c+d x)}\right)^{2-m}\right)}{16 d (a+i a \tan (c+d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{(1-m) (A (1-m)-i B (m+1)) \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{4 a^2 d (m+1)}+\frac{m (B m+i A (2-m)) \tan ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)}{4 a^2 d (m+2)}+\frac{(A (2-m)-i B m) \tan ^{m+1}(c+d x)}{4 a^2 d (1+i \tan (c+d x))}+\frac{(A+i B) \tan ^{m+1}(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"((-1/16*I)*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*(((A + I*B)*(-1 + E^((2*I)*(c + d*x)))^(1 + m)*(1 + E^((2*I)*(c + d*x)))^(2 - m))/E^((4*I)*d*x) + E^((2*I)*(c - d*x))*(-1 + E^((2*I)*(c + d*x)))^(1 + m)*(1 + E^((2*I)*(c + d*x)))^(2 - m)*(A*(3 - 2*m) - I*(B + 2*B*m)) + (2^(2 - m)*E^((4*I)*c)*(-1 + E^((2*I)*(c + d*x)))^(1 + m)*(A*(-3 + 2*m) + I*(B + 2*B*m))*Hypergeometric2F1[-1 + m, 1 + m, 2 + m, (1 - E^((2*I)*(c + d*x)))/2])/(1 + m) + (2^(1 - m)*E^((4*I)*c)*((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^m*(I*B*(-1 + 2*m^2) + A*(1 - 4*m + 2*m^2))*(-(2^m*(1 + m)*Hypergeometric2F1[1, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]) + (1 + E^((2*I)*(c + d*x)))^m*((1 + m)*Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2] + (-1 + E^((2*I)*(c + d*x)))*m*Hypergeometric2F1[m, 1 + m, 2 + m, (1 - E^((2*I)*(c + d*x)))/2])))/(m*(1 + m)))*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*(A + B*Tan[c + d*x]))/(d*E^((2*I)*c)*((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^m*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2)","B",0
210,1,712,308,127.1806695,"\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]","-\frac{i e^{-3 i c} \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \left(\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{-m} \sec ^2(c+d x) (\cos (d x)+i \sin (d x))^3 (A+B \tan (c+d x)) \left(\frac{3 e^{6 i c} 2^{3-m} \left(A \left(-2 m^2+7 m-4\right)+i B \left(-2 m^2+m+2\right)\right) \left(-1+e^{2 i (c+d x)}\right)^{m+1} \, _2F_1\left(m-2,m+1;m+2;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)}{m+1}+\frac{e^{6 i c} 2^{1-m} \left(A \left(4 m^3-18 m^2+20 m-3\right)+i B \left(4 m^3-6 m^2-4 m+3\right)\right) \left(\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^m \left(2^m (m+1) \, _2F_1\left(1,m;m+1;\frac{1-e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)-\left(1+e^{2 i (c+d x)}\right)^m \left(2 m \left(-1+e^{2 i (c+d x)}\right) \, _2F_1\left(m-1,m+1;m+2;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+(m+1) \, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+m \left(-1+e^{2 i (c+d x)}\right) \, _2F_1\left(m,m+1;m+2;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)\right)\right)}{m (m+1)}-2 \left(A \left(-2 m^2+7 m-4\right)+i B \left(-2 m^2+m+2\right)\right) e^{-2 i (d x-2 c)} \left(-1+e^{2 i (c+d x)}\right)^{m+1} \left(1+e^{2 i (c+d x)}\right)^{3-m}+2 (A+i B) e^{-6 i d x} \left(-1+e^{2 i (c+d x)}\right)^{m+1} \left(1+e^{2 i (c+d x)}\right)^{3-m}+(A (5-2 m)-i (2 B m+B)) e^{2 i (c-2 d x)} \left(-1+e^{2 i (c+d x)}\right)^{m+1} \left(1+e^{2 i (c+d x)}\right)^{3-m}\right)}{96 d (a+i a \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{(1-m) \left(-A \left(2 m^2-7 m+3\right)+i B \left(-2 m^2+m+3\right)\right) \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{24 a^3 d (m+1)}+\frac{(2-m) m (i A (5-2 m)+2 B m+B) \tan ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)}{24 a^3 d (m+2)}+\frac{(2-m) (A (5-2 m)-i (2 B m+B)) \tan ^{m+1}(c+d x)}{24 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{(A (7-2 m)+i B (1-2 m)) \tan ^{m+1}(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{(A+i B) \tan ^{m+1}(c+d x)}{6 d (a+i a \tan (c+d x))^3}",1,"((-1/96*I)*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*((2*(A + I*B)*(-1 + E^((2*I)*(c + d*x)))^(1 + m)*(1 + E^((2*I)*(c + d*x)))^(3 - m))/E^((6*I)*d*x) + E^((2*I)*(c - 2*d*x))*(-1 + E^((2*I)*(c + d*x)))^(1 + m)*(1 + E^((2*I)*(c + d*x)))^(3 - m)*(A*(5 - 2*m) - I*(B + 2*B*m)) - (2*(-1 + E^((2*I)*(c + d*x)))^(1 + m)*(1 + E^((2*I)*(c + d*x)))^(3 - m)*(I*B*(2 + m - 2*m^2) + A*(-4 + 7*m - 2*m^2)))/E^((2*I)*(-2*c + d*x)) + (3*2^(3 - m)*E^((6*I)*c)*(-1 + E^((2*I)*(c + d*x)))^(1 + m)*(I*B*(2 + m - 2*m^2) + A*(-4 + 7*m - 2*m^2))*Hypergeometric2F1[-2 + m, 1 + m, 2 + m, (1 - E^((2*I)*(c + d*x)))/2])/(1 + m) + (2^(1 - m)*E^((6*I)*c)*((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^m*(A*(-3 + 20*m - 18*m^2 + 4*m^3) + I*B*(3 - 4*m - 6*m^2 + 4*m^3))*(2^m*(1 + m)*Hypergeometric2F1[1, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))] - (1 + E^((2*I)*(c + d*x)))^m*(2*(-1 + E^((2*I)*(c + d*x)))*m*Hypergeometric2F1[-1 + m, 1 + m, 2 + m, (1 - E^((2*I)*(c + d*x)))/2] + (1 + m)*Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2] + (-1 + E^((2*I)*(c + d*x)))*m*Hypergeometric2F1[m, 1 + m, 2 + m, (1 - E^((2*I)*(c + d*x)))/2])))/(m*(1 + m)))*Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*E^((3*I)*c)*((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^m*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3)","B",0
211,1,921,386,128.4577123,"\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4,x]","-\frac{i e^{-4 i c} \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^m \left(\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{-m} \left(3 (A+i B) e^{-8 i d x} \left(-1+e^{2 i (c+d x)}\right)^{m+1} \left(1+e^{2 i (c+d x)}\right)^{4-m}+e^{2 i (c-3 d x)} \left(-1+e^{2 i (c+d x)}\right)^{m+1} (A (7-2 m)-i (2 m B+B)) \left(1+e^{2 i (c+d x)}\right)^{4-m}-e^{4 i (c-d x)} \left(-1+e^{2 i (c+d x)}\right)^{m+1} \left(i B \left(-2 m^2+2 m+3\right)+A \left(-2 m^2+10 m-9\right)\right) \left(1+e^{2 i (c+d x)}\right)^{4-m}+\frac{e^{-2 i (d x-3 c)} \left(-1+e^{2 i (c+d x)}\right)^{m+1} m (m+1) \left(A \left(-4 m^3+26 m^2-44 m+13\right)-i B \left(4 m^3-10 m^2-4 m+7\right)\right) \left(1+e^{2 i (c+d x)}\right)^{4-m}-2^{5-m} e^{8 i c} \left(-1+e^{2 i (c+d x)}\right)^{m+1} m \left(A \left(-4 m^3+26 m^2-44 m+13\right)-i B \left(4 m^3-10 m^2-4 m+7\right)\right) \, _2F_1\left(m-3,m+1;m+2;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+2^{2-m} e^{8 i c} \left(\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^m \left(A \left(2 m^4-16 m^3+40 m^2-32 m+3\right)+i B \left(2 m^4-8 m^3+4 m^2+8 m-3\right)\right) \left(\left(1+e^{2 i (c+d x)}\right)^m \left(4 \left(-1+e^{2 i (c+d x)}\right) m \, _2F_1\left(m-2,m+1;m+2;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+2 \left(-1+e^{2 i (c+d x)}\right) m \, _2F_1\left(m-1,m+1;m+2;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+m \, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+\, _2F_1\left(m,m;m+1;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+e^{2 i (c+d x)} m \, _2F_1\left(m,m+1;m+2;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)-m \, _2F_1\left(m,m+1;m+2;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)\right)-2^m (m+1) \, _2F_1\left(1,m;m+1;\frac{1-e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)\right)}{m (m+1)}\right) \sec ^3(c+d x) (\cos (d x)+i \sin (d x))^4 (A+B \tan (c+d x))}{384 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^4}","-\frac{\left(m^2-4 m+3\right) \left(-A \left(m^2-4 m+1\right)+i B \left(1-m^2\right)\right) \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{48 a^4 d (m+1)}+\frac{(2-m) m \left(B \left(-m^2+2 m+2\right)+i A \left(m^2-6 m+8\right)\right) \tan ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)}{48 a^4 d (m+2)}-\frac{(2-m) \left(-A \left(m^2-6 m+8\right)+i B \left(-m^2+2 m+2\right)\right) \tan ^{m+1}(c+d x)}{48 a^4 d (1+i \tan (c+d x))}-\frac{\left(-A \left(m^2-7 m+13\right)+i B \left(-m^2+3 m+1\right)\right) \tan ^{m+1}(c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{(A (5-m)+i B (1-m)) \tan ^{m+1}(c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{(A+i B) \tan ^{m+1}(c+d x)}{8 d (a+i a \tan (c+d x))^4}",1,"((-1/384*I)*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^m*((3*(A + I*B)*(-1 + E^((2*I)*(c + d*x)))^(1 + m)*(1 + E^((2*I)*(c + d*x)))^(4 - m))/E^((8*I)*d*x) + E^((2*I)*(c - 3*d*x))*(-1 + E^((2*I)*(c + d*x)))^(1 + m)*(1 + E^((2*I)*(c + d*x)))^(4 - m)*(A*(7 - 2*m) - I*(B + 2*B*m)) - E^((4*I)*(c - d*x))*(-1 + E^((2*I)*(c + d*x)))^(1 + m)*(1 + E^((2*I)*(c + d*x)))^(4 - m)*(I*B*(3 + 2*m - 2*m^2) + A*(-9 + 10*m - 2*m^2)) + (((-1 + E^((2*I)*(c + d*x)))^(1 + m)*(1 + E^((2*I)*(c + d*x)))^(4 - m)*m*(1 + m)*(A*(13 - 44*m + 26*m^2 - 4*m^3) - I*B*(7 - 4*m - 10*m^2 + 4*m^3)))/E^((2*I)*(-3*c + d*x)) - 2^(5 - m)*E^((8*I)*c)*(-1 + E^((2*I)*(c + d*x)))^(1 + m)*m*(A*(13 - 44*m + 26*m^2 - 4*m^3) - I*B*(7 - 4*m - 10*m^2 + 4*m^3))*Hypergeometric2F1[-3 + m, 1 + m, 2 + m, (1 - E^((2*I)*(c + d*x)))/2] + 2^(2 - m)*E^((8*I)*c)*((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^m*(A*(3 - 32*m + 40*m^2 - 16*m^3 + 2*m^4) + I*B*(-3 + 8*m + 4*m^2 - 8*m^3 + 2*m^4))*(-(2^m*(1 + m)*Hypergeometric2F1[1, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]) + (1 + E^((2*I)*(c + d*x)))^m*(4*(-1 + E^((2*I)*(c + d*x)))*m*Hypergeometric2F1[-2 + m, 1 + m, 2 + m, (1 - E^((2*I)*(c + d*x)))/2] + 2*(-1 + E^((2*I)*(c + d*x)))*m*Hypergeometric2F1[-1 + m, 1 + m, 2 + m, (1 - E^((2*I)*(c + d*x)))/2] + Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2] + m*Hypergeometric2F1[m, m, 1 + m, (1 - E^((2*I)*(c + d*x)))/2] - m*Hypergeometric2F1[m, 1 + m, 2 + m, (1 - E^((2*I)*(c + d*x)))/2] + E^((2*I)*(c + d*x))*m*Hypergeometric2F1[m, 1 + m, 2 + m, (1 - E^((2*I)*(c + d*x)))/2])))/(m*(1 + m)))*Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*E^((4*I)*c)*((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^m*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4)","B",0
212,0,0,316,7.5974462,"\int \tan ^m(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\int \tan ^m(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","\frac{4 a^3 (A-i B) \sqrt{1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left(m+1;\frac{1}{2},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right)}{d (m+1) \sqrt{a+i a \tan (c+d x)}}+\frac{2 a^2 \left(2 B \left(4 m^2+17 m+19\right)+i A \left(8 m^2+34 m+35\right)\right) \sqrt{a+i a \tan (c+d x)} \tan ^m(c+d x) (-i \tan (c+d x))^{-m} \, _2F_1\left(\frac{1}{2},-m;\frac{3}{2};i \tan (c+d x)+1\right)}{d (2 m+3) (2 m+5)}+\frac{2 a^2 (-A (2 m+5)+2 i B (m+4)) \sqrt{a+i a \tan (c+d x)} \tan ^{m+1}(c+d x)}{d (2 m+3) (2 m+5)}+\frac{2 i a B (a+i a \tan (c+d x))^{3/2} \tan ^{m+1}(c+d x)}{d (2 m+5)}",1,"Integrate[Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x]","F",-1
213,0,0,227,5.3165637,"\int \tan ^m(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\int \tan ^m(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","\frac{2 a^2 (A-i B) \sqrt{1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left(m+1;\frac{1}{2},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right)}{d (m+1) \sqrt{a+i a \tan (c+d x)}}+\frac{2 a (B+(2 m+3) (B+i A)) \sqrt{a+i a \tan (c+d x)} \tan ^m(c+d x) (-i \tan (c+d x))^{-m} \, _2F_1\left(\frac{1}{2},-m;\frac{3}{2};i \tan (c+d x)+1\right)}{d (2 m+3)}+\frac{2 i a B \sqrt{a+i a \tan (c+d x)} \tan ^{m+1}(c+d x)}{d (2 m+3)}",1,"Integrate[Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x]","F",-1
214,0,0,159,4.2212991,"\int \tan ^m(c+d x) \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\int \tan ^m(c+d x) \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","\frac{a (A-i B) \sqrt{1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left(m+1;\frac{1}{2},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right)}{d (m+1) \sqrt{a+i a \tan (c+d x)}}+\frac{2 B \sqrt{a+i a \tan (c+d x)} \tan ^m(c+d x) (-i \tan (c+d x))^{-m} \, _2F_1\left(\frac{1}{2},-m;\frac{3}{2};i \tan (c+d x)+1\right)}{d}",1,"Integrate[Tan[c + d*x]^m*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x]","F",-1
215,-1,0,214,180.0011445,"\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]],x]","\text{\$Aborted}","\frac{(A-i B) \sqrt{1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left(m+1;\frac{1}{2},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right)}{2 d (m+1) \sqrt{a+i a \tan (c+d x)}}+\frac{(2 m+1) (-B+i A) \sqrt{a+i a \tan (c+d x)} \tan ^m(c+d x) (-i \tan (c+d x))^{-m} \, _2F_1\left(\frac{1}{2},-m;\frac{3}{2};i \tan (c+d x)+1\right)}{a d}+\frac{(A+i B) \tan ^{m+1}(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}",1,"$Aborted","F",-1
216,0,0,285,17.1360085,"\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]","\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx","\frac{(2 m+1) (i A (5-4 m)+4 B m+B) \sqrt{a+i a \tan (c+d x)} \tan ^m(c+d x) (-i \tan (c+d x))^{-m} \, _2F_1\left(\frac{1}{2},-m;\frac{3}{2};i \tan (c+d x)+1\right)}{6 a^2 d}+\frac{(A-i B) \sqrt{1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left(m+1;\frac{1}{2},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right)}{4 a d (m+1) \sqrt{a+i a \tan (c+d x)}}+\frac{(A (5-4 m)-i (4 B m+B)) \tan ^{m+1}(c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}+\frac{(A+i B) \tan ^{m+1}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}",1,"Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2), x]","F",-1
217,0,0,363,78.2887266,"\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2),x]","\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx","\frac{(2 m+1) \left(B \left(-16 m^2+12 m+13\right)+i A \left(16 m^2-52 m+37\right)\right) \sqrt{a+i a \tan (c+d x)} \tan ^m(c+d x) (-i \tan (c+d x))^{-m} \, _2F_1\left(\frac{1}{2},-m;\frac{3}{2};i \tan (c+d x)+1\right)}{60 a^3 d}+\frac{(A-i B) \sqrt{1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left(m+1;\frac{1}{2},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right)}{8 a^2 d (m+1) \sqrt{a+i a \tan (c+d x)}}-\frac{\left(-A \left(16 m^2-52 m+37\right)+i B \left(-16 m^2+12 m+13\right)\right) \tan ^{m+1}(c+d x)}{60 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{(A (11-4 m)+i B (1-4 m)) \tan ^{m+1}(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{(A+i B) \tan ^{m+1}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2), x]","F",-1
218,0,0,167,19.2979358,"\int \tan ^m(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \tan ^m(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","\frac{(A-i B) \tan ^{m+1}(c+d x) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1(m+1;1-n,1;m+2;-i \tan (c+d x),i \tan (c+d x))}{d (m+1)}+\frac{i B \tan ^{m+1}(c+d x) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1(m+1,1-n;m+2;-i \tan (c+d x))}{d (m+1)}",1,"Integrate[Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
219,0,0,245,61.7181007,"\int \tan ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \tan ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","\frac{(A-i B) (a+i a \tan (c+d x))^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (i \tan (c+d x)+1)\right)}{2 d n}-\frac{\left(A n (n+3)-i B \left(n^2+3 n+6\right)\right) (a+i a \tan (c+d x))^{n+1}}{a d (n+1) (n+2) (n+3)}-\frac{(-A (n+3)+i B n) \tan ^2(c+d x) (a+i a \tan (c+d x))^n}{d (n+2) (n+3)}+\frac{2 (-A (n+3)+i B n) (a+i a \tan (c+d x))^n}{d n (n+2) (n+3)}+\frac{B \tan ^3(c+d x) (a+i a \tan (c+d x))^n}{d (n+3)}",1,"Integrate[Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
220,0,0,164,22.6910953,"\int \tan ^2(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \tan ^2(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","\frac{(B+i A) (a+i a \tan (c+d x))^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (i \tan (c+d x)+1)\right)}{2 d n}-\frac{(B n+i A (n+2)) (a+i a \tan (c+d x))^{n+1}}{a d (n+1) (n+2)}+\frac{B \tan ^2(c+d x) (a+i a \tan (c+d x))^n}{d (n+2)}-\frac{2 B (a+i a \tan (c+d x))^n}{d n (n+2)}",1,"Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
221,1,270,111,32.5166488,"\int \tan (c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","2^{n-1} e^{-2 i d n x} \left(e^{i d x}\right)^n \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^n \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n \left(\frac{(A+i B) e^{2 i d n x} \left(1+e^{2 i (c+d x)}\right)^n \, _2F_1\left(n,n+2;n+1;-e^{2 i (c+d x)}\right)}{d n}+\frac{e^{2 i c} \left(-\frac{(A-i B) e^{2 i (c+d (n+2) x)} \left(1+e^{2 i (c+d x)}\right)^n \, _2F_1\left(n+2,n+2;n+3;-e^{2 i (c+d x)}\right)}{n+2}-\frac{2 i B e^{2 i d (n+1) x}}{(n+1) \left(1+e^{2 i (c+d x)}\right)}\right)}{d}\right)","-\frac{(A-i B) (a+i a \tan (c+d x))^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (i \tan (c+d x)+1)\right)}{2 d n}+\frac{A (a+i a \tan (c+d x))^n}{d n}-\frac{i B (a+i a \tan (c+d x))^{n+1}}{a d (n+1)}",1,"(2^(-1 + n)*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(((A + I*B)*E^((2*I)*d*n*x)*(1 + E^((2*I)*(c + d*x)))^n*Hypergeometric2F1[n, 2 + n, 1 + n, -E^((2*I)*(c + d*x))])/(d*n) + (E^((2*I)*c)*(((-2*I)*B*E^((2*I)*d*(1 + n)*x))/((1 + E^((2*I)*(c + d*x)))*(1 + n)) - ((A - I*B)*E^((2*I)*(c + d*(2 + n)*x))*(1 + E^((2*I)*(c + d*x)))^n*Hypergeometric2F1[2 + n, 2 + n, 3 + n, -E^((2*I)*(c + d*x))])/(2 + n)))/d)*(a + I*a*Tan[c + d*x])^n)/(E^((2*I)*d*n*x)*Sec[c + d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)","B",1
222,1,152,78,6.3966259,"\int (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\frac{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 \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n \left((n+1) (B-i A)-i n (A-i B) e^{2 i (c+d x)} \, _2F_1\left(1,1;n+2;-e^{2 i (c+d x)}\right)\right)}{d n (n+1)}","\frac{B (a+i a \tan (c+d x))^n}{d n}-\frac{(B+i A) (a+i a \tan (c+d x))^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (i \tan (c+d x)+1)\right)}{2 d n}",1,"(2^(-1 + n)*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(((-I)*A + B)*(1 + n) - I*(A - I*B)*E^((2*I)*(c + d*x))*n*Hypergeometric2F1[1, 1, 2 + n, -E^((2*I)*(c + d*x))])*(a + I*a*Tan[c + d*x])^n)/(d*n*(1 + n)*Sec[c + d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)","A",0
223,0,0,97,25.0581765,"\int \cot (c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \cot (c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","\frac{(A-i B) (a+i a \tan (c+d x))^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (i \tan (c+d x)+1)\right)}{2 d n}-\frac{A (a+i a \tan (c+d x))^n \, _2F_1(1,n;n+1;i \tan (c+d x)+1)}{d n}",1,"Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
224,0,0,131,47.0288904,"\int \cot ^2(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \cot ^2(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","\frac{(B+i A) (a+i a \tan (c+d x))^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (i \tan (c+d x)+1)\right)}{2 d n}-\frac{(B+i A n) (a+i a \tan (c+d x))^n \, _2F_1(1,n;n+1;i \tan (c+d x)+1)}{d n}-\frac{A \cot (c+d x) (a+i a \tan (c+d x))^n}{d}",1,"Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
225,0,0,185,68.4329014,"\int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \cot ^3(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","-\frac{\left(-A \left(n^2-n+2\right)+2 i B n\right) (a+i a \tan (c+d x))^n \, _2F_1(1,n;n+1;i \tan (c+d x)+1)}{2 d n}-\frac{(A-i B) (a+i a \tan (c+d x))^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (i \tan (c+d x)+1)\right)}{2 d n}-\frac{(2 B+i A n) \cot (c+d x) (a+i a \tan (c+d x))^n}{2 d}-\frac{A \cot ^2(c+d x) (a+i a \tan (c+d x))^n}{2 d}",1,"Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
226,0,0,383,16.3600788,"\int \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","\frac{2 (B+i A) \sqrt{\tan (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d}-\frac{2 \left(4 B n \left(2 n^2+8 n+9\right)+i A \left(8 n^3+32 n^2+36 n+15\right)\right) \sqrt{\tan (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};-i \tan (c+d x)\right)}{d (2 n+1) (2 n+3) (2 n+5)}-\frac{2 \left(B \left(4 n^2+10 n+15\right)+2 i A n (2 n+5)\right) \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^n}{d (2 n+1) (2 n+3) (2 n+5)}-\frac{2 (-A (2 n+5)+2 i B n) \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^n}{d (2 n+3) (2 n+5)}+\frac{2 B \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^n}{d (2 n+5)}",1,"Integrate[Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
227,0,0,291,17.4180122,"\int \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","-\frac{2 (A-i B) \sqrt{\tan (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d}+\frac{2 \left(2 A n (2 n+3)-i B \left(4 n^2+6 n+3\right)\right) \sqrt{\tan (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};-i \tan (c+d x)\right)}{d (2 n+1) (2 n+3)}-\frac{2 (-A (2 n+3)+2 i B n) \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^n}{d (2 n+1) (2 n+3)}+\frac{2 B \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^n}{d (2 n+3)}",1,"Integrate[Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
228,0,0,215,19.6597106,"\int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","-\frac{2 (B+i A) \sqrt{\tan (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d}+\frac{2 (2 B n+i A (2 n+1)) \sqrt{\tan (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};-i \tan (c+d x)\right)}{d (2 n+1)}+\frac{2 B \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^n}{d (2 n+1)}",1,"Integrate[Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
229,0,0,158,20.8048311,"\int \frac{(a+i a \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","Integrate[((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]],x]","\int \frac{(a+i a \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","\frac{2 (A-i B) \sqrt{\tan (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d}+\frac{2 i B \sqrt{\tan (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};-i \tan (c+d x)\right)}{d}",1,"Integrate[((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]], x]","F",-1
230,0,0,194,10.2716945,"\int \frac{(a+i a \tan (c+d x))^n (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2),x]","\int \frac{(a+i a \tan (c+d x))^n (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","\frac{2 (B+i A) \sqrt{\tan (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d}-\frac{2 i A (1-2 n) \sqrt{\tan (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};-i \tan (c+d x)\right)}{d}-\frac{2 A (a+i a \tan (c+d x))^n}{d \sqrt{\tan (c+d x)}}",1,"Integrate[((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2), x]","F",-1
231,0,0,247,13.5335343,"\int \frac{(a+i a \tan (c+d x))^n (A+B \tan (c+d x))}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2),x]","\int \frac{(a+i a \tan (c+d x))^n (A+B \tan (c+d x))}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","-\frac{2 (A-i B) \sqrt{\tan (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d}-\frac{2 (1-2 n) (-2 A n+3 i B) \sqrt{\tan (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};-i \tan (c+d x)\right)}{3 d}-\frac{2 (3 B+2 i A n) (a+i a \tan (c+d x))^n}{3 d \sqrt{\tan (c+d x)}}-\frac{2 A (a+i a \tan (c+d x))^n}{3 d \tan ^{\frac{3}{2}}(c+d x)}",1,"Integrate[((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x]","F",-1
232,1,86,87,0.5971019,"\int \tan ^2(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{(6 b B-6 a A) \tan ^{-1}(\tan (c+d x))+3 (a B+A b) \tan ^2(c+d x)+6 (a A-b B) \tan (c+d x)+6 (a B+A b) \log (\cos (c+d x))+2 b B \tan ^3(c+d x)}{6 d}","\frac{(a B+A b) \tan ^2(c+d x)}{2 d}+\frac{(a A-b B) \tan (c+d x)}{d}+\frac{(a B+A b) \log (\cos (c+d x))}{d}-x (a A-b B)+\frac{b B \tan ^3(c+d x)}{3 d}",1,"((-6*a*A + 6*b*B)*ArcTan[Tan[c + d*x]] + 6*(A*b + a*B)*Log[Cos[c + d*x]] + 6*(a*A - b*B)*Tan[c + d*x] + 3*(A*b + a*B)*Tan[c + d*x]^2 + 2*b*B*Tan[c + d*x]^3)/(6*d)","A",1
233,1,67,65,0.3012779,"\int \tan (c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{-2 (a B+A b) \tan ^{-1}(\tan (c+d x))+2 (a B+A b) \tan (c+d x)+2 (b B-a A) \log (\cos (c+d x))+b B \tan ^2(c+d x)}{2 d}","\frac{(a B+A b) \tan (c+d x)}{d}-\frac{(a A-b B) \log (\cos (c+d x))}{d}-x (a B+A b)+\frac{b B \tan ^2(c+d x)}{2 d}",1,"(-2*(A*b + a*B)*ArcTan[Tan[c + d*x]] + 2*(-(a*A) + b*B)*Log[Cos[c + d*x]] + 2*(A*b + a*B)*Tan[c + d*x] + b*B*Tan[c + d*x]^2)/(2*d)","A",1
234,1,59,42,0.0296542,"\int (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","a A x-\frac{a B \log (\cos (c+d x))}{d}-\frac{A b \log (\cos (c+d x))}{d}-\frac{b B \tan ^{-1}(\tan (c+d x))}{d}+\frac{b B \tan (c+d x)}{d}","-\frac{(a B+A b) \log (\cos (c+d x))}{d}+x (a A-b B)+\frac{b B \tan (c+d x)}{d}",1,"a*A*x - (b*B*ArcTan[Tan[c + d*x]])/d - (A*b*Log[Cos[c + d*x]])/d - (a*B*Log[Cos[c + d*x]])/d + (b*B*Tan[c + d*x])/d","A",1
235,1,44,37,0.075897,"\int \cot (c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{a A (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+a B x+A b x-\frac{b B \log (\cos (c+d x))}{d}","x (a B+A b)+\frac{a A \log (\sin (c+d x))}{d}-\frac{b B \log (\cos (c+d x))}{d}",1,"A*b*x + a*B*x - (b*B*Log[Cos[c + d*x]])/d + (a*A*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","A",1
236,1,78,43,0.179486,"\int \cot ^2(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","-\frac{a A \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}+\frac{a B (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+\frac{A b (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+b B x","\frac{(a B+A b) \log (\sin (c+d x))}{d}-(x (a A-b B))-\frac{a A \cot (c+d x)}{d}",1,"b*B*x - (a*A*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d + (A*b*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d + (a*B*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","C",1
237,1,77,66,0.4808349,"\int \cot ^3(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","-\frac{2 (a B+A b) \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)+2 (a A-b B) (\log (\tan (c+d x))+\log (\cos (c+d x)))+a A \cot ^2(c+d x)}{2 d}","-\frac{(a B+A b) \cot (c+d x)}{d}-\frac{(a A-b B) \log (\sin (c+d x))}{d}-x (a B+A b)-\frac{a A \cot ^2(c+d x)}{2 d}",1,"-1/2*(a*A*Cot[c + d*x]^2 + 2*(A*b + a*B)*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2] + 2*(a*A - b*B)*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","C",1
238,1,101,87,1.058264,"\int \cot ^4(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","-\frac{3 (a B+A b) \left(\cot ^2(c+d x)+2 (\log (\tan (c+d x))+\log (\cos (c+d x)))\right)+2 a A \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)+6 b B \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{6 d}","-\frac{(a B+A b) \cot ^2(c+d x)}{2 d}+\frac{(a A-b B) \cot (c+d x)}{d}-\frac{(a B+A b) \log (\sin (c+d x))}{d}+x (a A-b B)-\frac{a A \cot ^3(c+d x)}{3 d}",1,"-1/6*(2*a*A*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2] + 6*b*B*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2] + 3*(A*b + a*B)*(Cot[c + d*x]^2 + 2*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]])))/d","C",1
239,1,100,108,1.1721907,"\int \cot ^5(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","-\frac{4 (a B+A b) \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)+3 \left((2 b B-2 a A) \cot ^2(c+d x)-4 (a A-b B) (\log (\tan (c+d x))+\log (\cos (c+d x)))+a A \cot ^4(c+d x)\right)}{12 d}","-\frac{(a B+A b) \cot ^3(c+d x)}{3 d}+\frac{(a A-b B) \cot ^2(c+d x)}{2 d}+\frac{(a B+A b) \cot (c+d x)}{d}+\frac{(a A-b B) \log (\sin (c+d x))}{d}+x (a B+A b)-\frac{a A \cot ^4(c+d x)}{4 d}",1,"-1/12*(4*(A*b + a*B)*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2] + 3*((-2*a*A + 2*b*B)*Cot[c + d*x]^2 + a*A*Cot[c + d*x]^4 - 4*(a*A - b*B)*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]])))/d","C",1
240,1,221,148,6.213439,"\int \tan ^2(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{B \tan (c+d x) (a+b \tan (c+d x))^3}{4 b d}+\frac{\frac{(4 A b-a B) (a+b \tan (c+d x))^3}{3 b d}+\frac{2 \left((A b-a B) \left(-i (a-i b)^2 \log (\tan (c+d x)+i)+i (a+i b)^2 \log (-\tan (c+d x)+i)-2 b^2 \tan (c+d x)\right)-B \left(6 a b^2 \tan (c+d x)+(-b+i a)^3 \log (-\tan (c+d x)+i)-(b+i a)^3 \log (\tan (c+d x)+i)+b^3 \tan ^2(c+d x)\right)\right)}{d}}{4 b}","\frac{\left(a^2 B+2 a A b-b^2 B\right) \log (\cos (c+d x))}{d}-x \left(a^2 A-2 a b B-A b^2\right)+\frac{(4 A b-a B) (a+b \tan (c+d x))^3}{12 b^2 d}-\frac{b (a B+A b) \tan (c+d x)}{d}+\frac{B \tan (c+d x) (a+b \tan (c+d x))^3}{4 b d}-\frac{B (a+b \tan (c+d x))^2}{2 d}",1,"(B*Tan[c + d*x]*(a + b*Tan[c + d*x])^3)/(4*b*d) + (((4*A*b - a*B)*(a + b*Tan[c + d*x])^3)/(3*b*d) + (2*((A*b - a*B)*(I*(a + I*b)^2*Log[I - Tan[c + d*x]] - I*(a - I*b)^2*Log[I + Tan[c + d*x]] - 2*b^2*Tan[c + d*x]) - B*((I*a - b)^3*Log[I - Tan[c + d*x]] - (I*a + b)^3*Log[I + Tan[c + d*x]] + 6*a*b^2*Tan[c + d*x] + b^3*Tan[c + d*x]^2)))/d)/(4*b)","C",1
241,1,172,112,1.7934951,"\int \tan (c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{3 (a A+b B) \left(-2 b^2 \tan (c+d x)+i \left((a+i b)^2 \log (-\tan (c+d x)+i)-(a-i b)^2 \log (\tan (c+d x)+i)\right)\right)+3 A \left(6 a b^2 \tan (c+d x)+(-b+i a)^3 \log (-\tan (c+d x)+i)-(b+i a)^3 \log (\tan (c+d x)+i)+b^3 \tan ^2(c+d x)\right)+2 B (a+b \tan (c+d x))^3}{6 b d}","-\frac{\left(a^2 A-2 a b B-A b^2\right) \log (\cos (c+d x))}{d}-x \left(a^2 B+2 a A b-b^2 B\right)+\frac{b (a A-b B) \tan (c+d x)}{d}+\frac{A (a+b \tan (c+d x))^2}{2 d}+\frac{B (a+b \tan (c+d x))^3}{3 b d}",1,"(2*B*(a + b*Tan[c + d*x])^3 + 3*(a*A + b*B)*(I*((a + I*b)^2*Log[I - Tan[c + d*x]] - (a - I*b)^2*Log[I + Tan[c + d*x]]) - 2*b^2*Tan[c + d*x]) + 3*A*((I*a - b)^3*Log[I - Tan[c + d*x]] - (I*a + b)^3*Log[I + Tan[c + d*x]] + 6*a*b^2*Tan[c + d*x] + b^3*Tan[c + d*x]^2))/(6*b*d)","C",1
242,1,96,87,0.434214,"\int (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{2 b (2 a B+A b) \tan (c+d x)+(a-i b)^2 (B+i A) \log (\tan (c+d x)+i)+(a+i b)^2 (B-i A) \log (-\tan (c+d x)+i)+b^2 B \tan ^2(c+d x)}{2 d}","-\frac{\left(a^2 B+2 a A b-b^2 B\right) \log (\cos (c+d x))}{d}+x \left(a^2 A-2 a b B-A b^2\right)+\frac{b (a B+A b) \tan (c+d x)}{d}+\frac{B (a+b \tan (c+d x))^2}{2 d}",1,"((a + I*b)^2*((-I)*A + B)*Log[I - Tan[c + d*x]] + (a - I*b)^2*(I*A + B)*Log[I + Tan[c + d*x]] + 2*b*(A*b + 2*a*B)*Tan[c + d*x] + b^2*B*Tan[c + d*x]^2)/(2*d)","C",1
243,1,93,70,0.2839953,"\int \cot (c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","-\frac{-2 a^2 A \log (\tan (c+d x))+(a+i b)^2 (A+i B) \log (-\tan (c+d x)+i)+(a-i b)^2 (A-i B) \log (\tan (c+d x)+i)-2 b B (a+b \tan (c+d x))}{2 d}","x \left(a^2 B+2 a A b-b^2 B\right)+\frac{a^2 A \log (\sin (c+d x))}{d}-\frac{b (2 a B+A b) \log (\cos (c+d x))}{d}+\frac{b^2 B \tan (c+d x)}{d}",1,"-1/2*((a + I*b)^2*(A + I*B)*Log[I - Tan[c + d*x]] - 2*a^2*A*Log[Tan[c + d*x]] + (a - I*b)^2*(A - I*B)*Log[I + Tan[c + d*x]] - 2*b*B*(a + b*Tan[c + d*x]))/d","C",1
244,1,100,72,0.270874,"\int \cot ^2(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{-2 a^2 A \cot (c+d x)+2 a (a B+2 A b) \log (\tan (c+d x))+i (a+i b)^2 (A+i B) \log (-\tan (c+d x)+i)-(a-i b)^2 (B+i A) \log (\tan (c+d x)+i)}{2 d}","-x \left(a^2 A-2 a b B-A b^2\right)-\frac{a^2 A \cot (c+d x)}{d}+\frac{a (a B+2 A b) \log (\sin (c+d x))}{d}-\frac{b^2 B \log (\cos (c+d x))}{d}",1,"(-2*a^2*A*Cot[c + d*x] + I*(a + I*b)^2*(A + I*B)*Log[I - Tan[c + d*x]] + 2*a*(2*A*b + a*B)*Log[Tan[c + d*x]] - (a - I*b)^2*(I*A + B)*Log[I + Tan[c + d*x]])/(2*d)","C",1
245,1,123,88,0.3568351,"\int \cot ^3(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{-2 \left(a^2 A-2 a b B-A b^2\right) \log (\tan (c+d x))-a^2 A \cot ^2(c+d x)-2 a (a B+2 A b) \cot (c+d x)+(a-i b)^2 (A-i B) \log (\tan (c+d x)+i)+(a+i b)^2 (A+i B) \log (-\tan (c+d x)+i)}{2 d}","-\frac{\left(a^2 A-2 a b B-A b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 A \cot ^2(c+d x)}{2 d}+x \left(b^2 B-a (a B+2 A b)\right)-\frac{a (a B+2 A b) \cot (c+d x)}{d}",1,"(-2*a*(2*A*b + a*B)*Cot[c + d*x] - a^2*A*Cot[c + d*x]^2 + (a + I*b)^2*(A + I*B)*Log[I - Tan[c + d*x]] - 2*(a^2*A - A*b^2 - 2*a*b*B)*Log[Tan[c + d*x]] + (a - I*b)^2*(A - I*B)*Log[I + Tan[c + d*x]])/(2*d)","C",1
246,1,152,118,1.4003466,"\int \cot ^4(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{6 \left(a^2 A-2 a b B-A b^2\right) \cot (c+d x)-6 \left(a^2 B+2 a A b-b^2 B\right) \log (\tan (c+d x))-2 a^2 A \cot ^3(c+d x)-3 a (a B+2 A b) \cot ^2(c+d x)+3 (a+i b)^2 (B-i A) \log (-\tan (c+d x)+i)+3 (a-i b)^2 (B+i A) \log (\tan (c+d x)+i)}{6 d}","\frac{\left(a^2 A-2 a b B-A b^2\right) \cot (c+d x)}{d}+x \left(a^2 A-2 a b B-A b^2\right)-\frac{a^2 A \cot ^3(c+d x)}{3 d}+\frac{\left(b^2 B-a (a B+2 A b)\right) \log (\sin (c+d x))}{d}-\frac{a (a B+2 A b) \cot ^2(c+d x)}{2 d}",1,"(6*(a^2*A - A*b^2 - 2*a*b*B)*Cot[c + d*x] - 3*a*(2*A*b + a*B)*Cot[c + d*x]^2 - 2*a^2*A*Cot[c + d*x]^3 + 3*(a + I*b)^2*((-I)*A + B)*Log[I - Tan[c + d*x]] - 6*(2*a*A*b + a^2*B - b^2*B)*Log[Tan[c + d*x]] + 3*(a - I*b)^2*(I*A + B)*Log[I + Tan[c + d*x]])/(6*d)","C",1
247,1,180,151,2.9191089,"\int \cot ^5(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{6 \left(a^2 A-2 a b B-A b^2\right) \cot ^2(c+d x)+12 \left(a^2 B+2 a A b-b^2 B\right) \cot (c+d x)-6 \left(\left(-2 a^2 A+4 a b B+2 A b^2\right) \log (\tan (c+d x))+(a-i b)^2 (A-i B) \log (\tan (c+d x)+i)+(a+i b)^2 (A+i B) \log (-\tan (c+d x)+i)\right)-3 a^2 A \cot ^4(c+d x)-4 a (a B+2 A b) \cot ^3(c+d x)}{12 d}","\frac{\left(a^2 A-2 a b B-A b^2\right) \cot ^2(c+d x)}{2 d}+\frac{\left(a^2 A-2 a b B-A b^2\right) \log (\sin (c+d x))}{d}+x \left(a^2 B+2 a A b-b^2 B\right)-\frac{a^2 A \cot ^4(c+d x)}{4 d}-\frac{\left(b^2 B-a (a B+2 A b)\right) \cot (c+d x)}{d}-\frac{a (a B+2 A b) \cot ^3(c+d x)}{3 d}",1,"(12*(2*a*A*b + a^2*B - b^2*B)*Cot[c + d*x] + 6*(a^2*A - A*b^2 - 2*a*b*B)*Cot[c + d*x]^2 - 4*a*(2*A*b + a*B)*Cot[c + d*x]^3 - 3*a^2*A*Cot[c + d*x]^4 - 6*((a + I*b)^2*(A + I*B)*Log[I - Tan[c + d*x]] + (-2*a^2*A + 2*A*b^2 + 4*a*b*B)*Log[Tan[c + d*x]] + (a - I*b)^2*(A - I*B)*Log[I + Tan[c + d*x]]))/(12*d)","C",1
248,1,241,201,2.213472,"\int \tan ^2(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{10 B \left(6 b^2 \left(b^2-6 a^2\right) \tan (c+d x)-12 a b^3 \tan ^2(c+d x)-3 i (a-i b)^4 \log (\tan (c+d x)+i)+3 i (a+i b)^4 \log (-\tan (c+d x)+i)-2 b^4 \tan ^3(c+d x)\right)-30 (A b-a B) \left(6 a b^2 \tan (c+d x)+(-b+i a)^3 \log (-\tan (c+d x)+i)-(b+i a)^3 \log (\tan (c+d x)+i)+b^3 \tan ^2(c+d x)\right)+\frac{3 (5 A b-a B) (a+b \tan (c+d x))^4}{b}+12 B \tan (c+d x) (a+b \tan (c+d x))^4}{60 b d}","-\frac{b \left(a^2 B+2 a A b-b^2 B\right) \tan (c+d x)}{d}+\frac{\left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \log (\cos (c+d x))}{d}-x \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right)+\frac{(5 A b-a B) (a+b \tan (c+d x))^4}{20 b^2 d}-\frac{(a B+A b) (a+b \tan (c+d x))^2}{2 d}+\frac{B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d}-\frac{B (a+b \tan (c+d x))^3}{3 d}",1,"((3*(5*A*b - a*B)*(a + b*Tan[c + d*x])^4)/b + 12*B*Tan[c + d*x]*(a + b*Tan[c + d*x])^4 - 30*(A*b - a*B)*((I*a - b)^3*Log[I - Tan[c + d*x]] - (I*a + b)^3*Log[I + Tan[c + d*x]] + 6*a*b^2*Tan[c + d*x] + b^3*Tan[c + d*x]^2) + 10*B*((3*I)*(a + I*b)^4*Log[I - Tan[c + d*x]] - (3*I)*(a - I*b)^4*Log[I + Tan[c + d*x]] + 6*b^2*(-6*a^2 + b^2)*Tan[c + d*x] - 12*a*b^3*Tan[c + d*x]^2 - 2*b^4*Tan[c + d*x]^3))/(60*b*d)","C",1
249,1,209,165,1.494781,"\int \tan (c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{-12 A b^2 \left(b^2-6 a^2\right) \tan (c+d x)+24 a A b^3 \tan ^2(c+d x)-6 (a A+b B) \left(6 a b^2 \tan (c+d x)+(-b+i a)^3 \log (-\tan (c+d x)+i)-(b+i a)^3 \log (\tan (c+d x)+i)+b^3 \tan ^2(c+d x)\right)+6 i A (a-i b)^4 \log (\tan (c+d x)+i)-6 i A (a+i b)^4 \log (-\tan (c+d x)+i)+3 B (a+b \tan (c+d x))^4+4 A b^4 \tan ^3(c+d x)}{12 b d}","\frac{b \left(a^2 A-2 a b B-A b^2\right) \tan (c+d x)}{d}-\frac{\left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right) \log (\cos (c+d x))}{d}-x \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right)+\frac{(a A-b B) (a+b \tan (c+d x))^2}{2 d}+\frac{A (a+b \tan (c+d x))^3}{3 d}+\frac{B (a+b \tan (c+d x))^4}{4 b d}",1,"((-6*I)*A*(a + I*b)^4*Log[I - Tan[c + d*x]] + (6*I)*A*(a - I*b)^4*Log[I + Tan[c + d*x]] - 12*A*b^2*(-6*a^2 + b^2)*Tan[c + d*x] + 24*a*A*b^3*Tan[c + d*x]^2 + 4*A*b^4*Tan[c + d*x]^3 + 3*B*(a + b*Tan[c + d*x])^4 - 6*(a*A + b*B)*((I*a - b)^3*Log[I - Tan[c + d*x]] - (I*a + b)^3*Log[I + Tan[c + d*x]] + 6*a*b^2*Tan[c + d*x] + b^3*Tan[c + d*x]^2))/(12*b*d)","C",1
250,1,130,140,1.0021509,"\int (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{6 b \left(3 a^2 B+3 a A b-b^2 B\right) \tan (c+d x)+3 b^2 (3 a B+A b) \tan ^2(c+d x)+3 (a-i b)^3 (B+i A) \log (\tan (c+d x)+i)+3 (a+i b)^3 (B-i A) \log (-\tan (c+d x)+i)+2 b^3 B \tan ^3(c+d x)}{6 d}","\frac{b \left(a^2 B+2 a A b-b^2 B\right) \tan (c+d x)}{d}-\frac{\left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \log (\cos (c+d x))}{d}+x \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right)+\frac{(a B+A b) (a+b \tan (c+d x))^2}{2 d}+\frac{B (a+b \tan (c+d x))^3}{3 d}",1,"(3*(a + I*b)^3*((-I)*A + B)*Log[I - Tan[c + d*x]] + 3*(a - I*b)^3*(I*A + B)*Log[I + Tan[c + d*x]] + 6*b*(3*a*A*b + 3*a^2*B - b^2*B)*Tan[c + d*x] + 3*b^2*(A*b + 3*a*B)*Tan[c + d*x]^2 + 2*b^3*B*Tan[c + d*x]^3)/(6*d)","C",1
251,1,115,117,0.5960214,"\int \cot (c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{2 a^3 A \log (\tan (c+d x))+2 b^2 (2 a B+A b) \tan (c+d x)-(a+i b)^3 (A+i B) \log (-\tan (c+d x)+i)-(a-i b)^3 (A-i B) \log (\tan (c+d x)+i)+b B (a+b \tan (c+d x))^2}{2 d}","\frac{a^3 A \log (\sin (c+d x))}{d}-\frac{b \left(3 a^2 B+3 a A b-b^2 B\right) \log (\cos (c+d x))}{d}+x \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right)+\frac{b^2 (2 a B+A b) \tan (c+d x)}{d}+\frac{b B (a+b \tan (c+d x))^2}{2 d}",1,"(-((a + I*b)^3*(A + I*B)*Log[I - Tan[c + d*x]]) + 2*a^3*A*Log[Tan[c + d*x]] - (a - I*b)^3*(A - I*B)*Log[I + Tan[c + d*x]] + 2*b^2*(A*b + 2*a*B)*Tan[c + d*x] + b*B*(a + b*Tan[c + d*x])^2)/(2*d)","C",1
252,1,113,119,0.5233211,"\int \cot ^2(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{-2 a^3 A \cot (c+d x)+2 a^2 (a B+3 A b) \log (\tan (c+d x))+i (a+i b)^3 (A+i B) \log (-\tan (c+d x)+i)+(b+i a)^3 (A-i B) \log (\tan (c+d x)+i)+2 b^3 B \tan (c+d x)}{2 d}","\frac{a^2 (a B+3 A b) \log (\sin (c+d x))}{d}-x \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right)+\frac{b^2 (a A+b B) \tan (c+d x)}{d}-\frac{b^2 (3 a B+A b) \log (\cos (c+d x))}{d}-\frac{a A \cot (c+d x) (a+b \tan (c+d x))^2}{d}",1,"(-2*a^3*A*Cot[c + d*x] + I*(a + I*b)^3*(A + I*B)*Log[I - Tan[c + d*x]] + 2*a^2*(3*A*b + a*B)*Log[Tan[c + d*x]] + (I*a + b)^3*(A - I*B)*Log[I + Tan[c + d*x]] + 2*b^3*B*Tan[c + d*x])/(2*d)","C",1
253,1,126,127,0.4416809,"\int \cot ^3(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{a^3 (-A) \cot ^2(c+d x)-2 a \left(a^2 A-3 a b B-3 A b^2\right) \log (\tan (c+d x))-2 a^2 (a B+3 A b) \cot (c+d x)+(a+i b)^3 (A+i B) \log (-\tan (c+d x)+i)+(a-i b)^3 (A-i B) \log (\tan (c+d x)+i)}{2 d}","-\frac{a \left(a^2 A-3 a b B-3 A b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 (a B+2 A b) \cot (c+d x)}{d}-x \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right)-\frac{a A \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{b^3 B \log (\cos (c+d x))}{d}",1,"(-2*a^2*(3*A*b + a*B)*Cot[c + d*x] - a^3*A*Cot[c + d*x]^2 + (a + I*b)^3*(A + I*B)*Log[I - Tan[c + d*x]] - 2*a*(a^2*A - 3*A*b^2 - 3*a*b*B)*Log[Tan[c + d*x]] + (a - I*b)^3*(A - I*B)*Log[I + Tan[c + d*x]])/(2*d)","C",1
254,1,164,154,1.2342138,"\int \cot ^4(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{-2 a^3 A \cot ^3(c+d x)+6 a \left(a^2 A-3 a b B-3 A b^2\right) \cot (c+d x)-3 a^2 (a B+3 A b) \cot ^2(c+d x)-6 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \log (\tan (c+d x))+3 (a-i b)^3 (B+i A) \log (\tan (c+d x)+i)+3 (a+i b)^3 (B-i A) \log (-\tan (c+d x)+i)}{6 d}","\frac{a \left(3 a^2 A-9 a b B-8 A b^2\right) \cot (c+d x)}{3 d}-\frac{a^2 (3 a B+5 A b) \cot ^2(c+d x)}{6 d}-\frac{\left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \log (\sin (c+d x))}{d}+x \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right)-\frac{a A \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d}",1,"(6*a*(a^2*A - 3*A*b^2 - 3*a*b*B)*Cot[c + d*x] - 3*a^2*(3*A*b + a*B)*Cot[c + d*x]^2 - 2*a^3*A*Cot[c + d*x]^3 + 3*(a + I*b)^3*((-I)*A + B)*Log[I - Tan[c + d*x]] - 6*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Log[Tan[c + d*x]] + 3*(a - I*b)^3*(I*A + B)*Log[I + Tan[c + d*x]])/(6*d)","C",1
255,1,199,191,0.7702951,"\int \cot ^5(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{-3 a^3 A \cot ^4(c+d x)+6 a \left(a^2 A-3 a b B-3 A b^2\right) \cot ^2(c+d x)-4 a^2 (a B+3 A b) \cot ^3(c+d x)+12 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \cot (c+d x)+12 \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right) \log (\tan (c+d x))-6 (a+i b)^3 (A+i B) \log (-\tan (c+d x)+i)-6 (a-i b)^3 (A-i B) \log (\tan (c+d x)+i)}{12 d}","\frac{a \left(2 a^2 A-6 a b B-5 A b^2\right) \cot ^2(c+d x)}{4 d}-\frac{a^2 (2 a B+3 A b) \cot ^3(c+d x)}{6 d}+\frac{\left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \cot (c+d x)}{d}+\frac{\left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right) \log (\sin (c+d x))}{d}+x \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right)-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}",1,"(12*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Cot[c + d*x] + 6*a*(a^2*A - 3*A*b^2 - 3*a*b*B)*Cot[c + d*x]^2 - 4*a^2*(3*A*b + a*B)*Cot[c + d*x]^3 - 3*a^3*A*Cot[c + d*x]^4 - 6*(a + I*b)^3*(A + I*B)*Log[I - Tan[c + d*x]] + 12*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*Log[Tan[c + d*x]] - 6*(a - I*b)^3*(A - I*B)*Log[I + Tan[c + d*x]])/(12*d)","C",1
256,1,237,233,1.2137098,"\int \cot ^6(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^6*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{-12 a^3 A \cot ^5(c+d x)+20 a \left(a^2 A-3 a b B-3 A b^2\right) \cot ^3(c+d x)-15 a^2 (a B+3 A b) \cot ^4(c+d x)+30 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \cot ^2(c+d x)-60 \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right) \cot (c+d x)+60 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \log (\tan (c+d x))+30 i (a+i b)^3 (A+i B) \log (-\tan (c+d x)+i)+30 (b+i a)^3 (A-i B) \log (\tan (c+d x)+i)}{60 d}","\frac{a \left(5 a^2 A-15 a b B-12 A b^2\right) \cot ^3(c+d x)}{15 d}-\frac{a^2 (5 a B+7 A b) \cot ^4(c+d x)}{20 d}+\frac{\left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \cot ^2(c+d x)}{2 d}-\frac{\left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right) \cot (c+d x)}{d}+\frac{\left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \log (\sin (c+d x))}{d}-x \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right)-\frac{a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}",1,"(-60*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*Cot[c + d*x] + 30*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Cot[c + d*x]^2 + 20*a*(a^2*A - 3*A*b^2 - 3*a*b*B)*Cot[c + d*x]^3 - 15*a^2*(3*A*b + a*B)*Cot[c + d*x]^4 - 12*a^3*A*Cot[c + d*x]^5 + (30*I)*(a + I*b)^3*(A + I*B)*Log[I - Tan[c + d*x]] + 60*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Log[Tan[c + d*x]] + 30*(I*a + b)^3*(A - I*B)*Log[I + Tan[c + d*x]])/(60*d)","C",1
257,1,290,263,5.9499508,"\int \tan ^2(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{10 (A b-a B) \left(6 b^2 \left(b^2-6 a^2\right) \tan (c+d x)-12 a b^3 \tan ^2(c+d x)-3 i (a-i b)^4 \log (\tan (c+d x)+i)+3 i (a+i b)^4 \log (-\tan (c+d x)+i)-2 b^4 \tan ^3(c+d x)\right)+5 B \left(-60 a b^2 \left(2 a^2-b^2\right) \tan (c+d x)+6 b^3 \left(b^2-10 a^2\right) \tan ^2(c+d x)-20 a b^4 \tan ^3(c+d x)+6 i (a+i b)^5 \log (-\tan (c+d x)+i)-6 (b+i a)^5 \log (\tan (c+d x)+i)-3 b^5 \tan ^4(c+d x)\right)+\frac{2 (6 A b-a B) (a+b \tan (c+d x))^5}{b}+10 B \tan (c+d x) (a+b \tan (c+d x))^5}{60 b d}","-\frac{\left(a^2 B+2 a A b-b^2 B\right) (a+b \tan (c+d x))^2}{2 d}-\frac{b \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \tan (c+d x)}{d}+\frac{\left(a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right) \log (\cos (c+d x))}{d}-x \left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right)+\frac{(6 A b-a B) (a+b \tan (c+d x))^5}{30 b^2 d}-\frac{(a B+A b) (a+b \tan (c+d x))^3}{3 d}+\frac{B \tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac{B (a+b \tan (c+d x))^4}{4 d}",1,"((2*(6*A*b - a*B)*(a + b*Tan[c + d*x])^5)/b + 10*B*Tan[c + d*x]*(a + b*Tan[c + d*x])^5 + 10*(A*b - a*B)*((3*I)*(a + I*b)^4*Log[I - Tan[c + d*x]] - (3*I)*(a - I*b)^4*Log[I + Tan[c + d*x]] + 6*b^2*(-6*a^2 + b^2)*Tan[c + d*x] - 12*a*b^3*Tan[c + d*x]^2 - 2*b^4*Tan[c + d*x]^3) + 5*B*((6*I)*(a + I*b)^5*Log[I - Tan[c + d*x]] - 6*(I*a + b)^5*Log[I + Tan[c + d*x]] - 60*a*b^2*(2*a^2 - b^2)*Tan[c + d*x] + 6*b^3*(-10*a^2 + b^2)*Tan[c + d*x]^2 - 20*a*b^4*Tan[c + d*x]^3 - 3*b^5*Tan[c + d*x]^4))/(60*b*d)","C",1
258,1,257,226,3.9931359,"\int \tan (c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{10 (a A+b B) \left(6 b^2 \left(b^2-6 a^2\right) \tan (c+d x)-12 a b^3 \tan ^2(c+d x)-3 i (a-i b)^4 \log (\tan (c+d x)+i)+3 i (a+i b)^4 \log (-\tan (c+d x)+i)-2 b^4 \tan ^3(c+d x)\right)-5 A \left(-60 a b^2 \left(2 a^2-b^2\right) \tan (c+d x)+6 b^3 \left(b^2-10 a^2\right) \tan ^2(c+d x)-20 a b^4 \tan ^3(c+d x)+6 i (a+i b)^5 \log (-\tan (c+d x)+i)-6 (b+i a)^5 \log (\tan (c+d x)+i)-3 b^5 \tan ^4(c+d x)\right)+12 B (a+b \tan (c+d x))^5}{60 b d}","\frac{\left(a^2 A-2 a b B-A b^2\right) (a+b \tan (c+d x))^2}{2 d}+\frac{b \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right) \tan (c+d x)}{d}-\frac{\left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right) \log (\cos (c+d x))}{d}-x \left(a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right)+\frac{(a A-b B) (a+b \tan (c+d x))^3}{3 d}+\frac{A (a+b \tan (c+d x))^4}{4 d}+\frac{B (a+b \tan (c+d x))^5}{5 b d}",1,"(12*B*(a + b*Tan[c + d*x])^5 + 10*(a*A + b*B)*((3*I)*(a + I*b)^4*Log[I - Tan[c + d*x]] - (3*I)*(a - I*b)^4*Log[I + Tan[c + d*x]] + 6*b^2*(-6*a^2 + b^2)*Tan[c + d*x] - 12*a*b^3*Tan[c + d*x]^2 - 2*b^4*Tan[c + d*x]^3) - 5*A*((6*I)*(a + I*b)^5*Log[I - Tan[c + d*x]] - 6*(I*a + b)^5*Log[I + Tan[c + d*x]] - 60*a*b^2*(2*a^2 - b^2)*Tan[c + d*x] + 6*b^3*(-10*a^2 + b^2)*Tan[c + d*x]^2 - 20*a*b^4*Tan[c + d*x]^3 - 3*b^5*Tan[c + d*x]^4))/(60*b*d)","C",1
259,1,240,202,3.4659429,"\int (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{B \left(60 a b^2 \left(2 a^2-b^2\right) \tan (c+d x)-6 b^3 \left(b^2-10 a^2\right) \tan ^2(c+d x)+20 a b^4 \tan ^3(c+d x)+6 (b-i a)^5 \log (-\tan (c+d x)+i)+6 (b+i a)^5 \log (\tan (c+d x)+i)+3 b^5 \tan ^4(c+d x)\right)-2 (A b-a B) \left(6 b^2 \left(b^2-6 a^2\right) \tan (c+d x)-12 a b^3 \tan ^2(c+d x)-3 i (a-i b)^4 \log (\tan (c+d x)+i)+3 i (a+i b)^4 \log (-\tan (c+d x)+i)-2 b^4 \tan ^3(c+d x)\right)}{12 b d}","\frac{\left(a^2 B+2 a A b-b^2 B\right) (a+b \tan (c+d x))^2}{2 d}+\frac{b \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \tan (c+d x)}{d}-\frac{\left(a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right) \log (\cos (c+d x))}{d}+x \left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right)+\frac{(a B+A b) (a+b \tan (c+d x))^3}{3 d}+\frac{B (a+b \tan (c+d x))^4}{4 d}",1,"(-2*(A*b - a*B)*((3*I)*(a + I*b)^4*Log[I - Tan[c + d*x]] - (3*I)*(a - I*b)^4*Log[I + Tan[c + d*x]] + 6*b^2*(-6*a^2 + b^2)*Tan[c + d*x] - 12*a*b^3*Tan[c + d*x]^2 - 2*b^4*Tan[c + d*x]^3) + B*(6*((-I)*a + b)^5*Log[I - Tan[c + d*x]] + 6*(I*a + b)^5*Log[I + Tan[c + d*x]] + 60*a*b^2*(2*a^2 - b^2)*Tan[c + d*x] - 6*b^3*(-10*a^2 + b^2)*Tan[c + d*x]^2 + 20*a*b^4*Tan[c + d*x]^3 + 3*b^5*Tan[c + d*x]^4))/(12*b*d)","C",1
260,1,149,172,1.4324541,"\int \cot (c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{6 a^4 A \log (\tan (c+d x))+6 b^2 \left(3 a^2 B+3 a A b-b^2 B\right) \tan (c+d x)+3 b (2 a B+A b) (a+b \tan (c+d x))^2-3 (a+i b)^4 (A+i B) \log (-\tan (c+d x)+i)-3 (a-i b)^4 (A-i B) \log (\tan (c+d x)+i)+2 b B (a+b \tan (c+d x))^3}{6 d}","\frac{a^4 A \log (\sin (c+d x))}{d}+\frac{b^2 \left(3 a^2 B+3 a A b-b^2 B\right) \tan (c+d x)}{d}-\frac{b \left(4 a^3 B+6 a^2 A b-4 a b^2 B-A b^3\right) \log (\cos (c+d x))}{d}+x \left(a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right)+\frac{b (2 a B+A b) (a+b \tan (c+d x))^2}{2 d}+\frac{b B (a+b \tan (c+d x))^3}{3 d}",1,"(-3*(a + I*b)^4*(A + I*B)*Log[I - Tan[c + d*x]] + 6*a^4*A*Log[Tan[c + d*x]] - 3*(a - I*b)^4*(A - I*B)*Log[I + Tan[c + d*x]] + 6*b^2*(3*a*A*b + 3*a^2*B - b^2*B)*Tan[c + d*x] + 3*b*(A*b + 2*a*B)*(a + b*Tan[c + d*x])^2 + 2*b*B*(a + b*Tan[c + d*x])^3)/(6*d)","C",1
261,1,134,175,1.0246062,"\int \cot ^2(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{-2 a^4 A \cot (c+d x)+2 a^3 (a B+4 A b) \log (\tan (c+d x))+2 b^3 (4 a B+A b) \tan (c+d x)+i (a+i b)^4 (A+i B) \log (-\tan (c+d x)+i)-(a-i b)^4 (B+i A) \log (\tan (c+d x)+i)+b^4 B \tan ^2(c+d x)}{2 d}","\frac{a^3 (a B+4 A b) \log (\sin (c+d x))}{d}+\frac{b^2 \left(a^2 A+3 a b B+A b^2\right) \tan (c+d x)}{d}-\frac{b^2 \left(6 a^2 B+4 a A b-b^2 B\right) \log (\cos (c+d x))}{d}-x \left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right)+\frac{b (2 a A+b B) (a+b \tan (c+d x))^2}{2 d}-\frac{a A \cot (c+d x) (a+b \tan (c+d x))^3}{d}",1,"(-2*a^4*A*Cot[c + d*x] + I*(a + I*b)^4*(A + I*B)*Log[I - Tan[c + d*x]] + 2*a^3*(4*A*b + a*B)*Log[Tan[c + d*x]] - (a - I*b)^4*(I*A + B)*Log[I + Tan[c + d*x]] + 2*b^3*(A*b + 4*a*B)*Tan[c + d*x] + b^4*B*Tan[c + d*x]^2)/(2*d)","C",1
262,1,140,186,0.6854846,"\int \cot ^3(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{a^4 (-A) \cot ^2(c+d x)-2 a^3 (a B+4 A b) \cot (c+d x)-2 a^2 \left(a^2 A-4 a b B-6 A b^2\right) \log (\tan (c+d x))+(a+i b)^4 (A+i B) \log (-\tan (c+d x)+i)+(a-i b)^4 (A-i B) \log (\tan (c+d x)+i)+2 b^4 B \tan (c+d x)}{2 d}","\frac{b^2 \left(a^2 B+3 a A b+b^2 B\right) \tan (c+d x)}{d}-\frac{a^2 \left(a^2 A-4 a b B-6 A b^2\right) \log (\sin (c+d x))}{d}-x \left(a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right)-\frac{b^3 (4 a B+A b) \log (\cos (c+d x))}{d}-\frac{a (2 a B+5 A b) \cot (c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{a A \cot ^2(c+d x) (a+b \tan (c+d x))^3}{2 d}",1,"(-2*a^3*(4*A*b + a*B)*Cot[c + d*x] - a^4*A*Cot[c + d*x]^2 + (a + I*b)^4*(A + I*B)*Log[I - Tan[c + d*x]] - 2*a^2*(a^2*A - 6*A*b^2 - 4*a*b*B)*Log[Tan[c + d*x]] + (a - I*b)^4*(A - I*B)*Log[I + Tan[c + d*x]] + 2*b^4*B*Tan[c + d*x])/(2*d)","C",1
263,1,167,187,1.0988867,"\int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{-2 a^4 A \cot ^3(c+d x)-3 a^3 (a B+4 A b) \cot ^2(c+d x)+6 a^2 \left(a^2 A-4 a b B-6 A b^2\right) \cot (c+d x)-6 a \left(a^3 B+4 a^2 A b-6 a b^2 B-4 A b^3\right) \log (\tan (c+d x))+3 (a+i b)^4 (B-i A) \log (-\tan (c+d x)+i)+3 (a-i b)^4 (B+i A) \log (\tan (c+d x)+i)}{6 d}","\frac{a^2 \left(a^2 A-3 a b B-3 A b^2\right) \cot (c+d x)}{d}-\frac{a \left(a^3 B+4 a^2 A b-6 a b^2 B-4 A b^3\right) \log (\sin (c+d x))}{d}+x \left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right)-\frac{a (a B+2 A b) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}-\frac{b^4 B \log (\cos (c+d x))}{d}",1,"(6*a^2*(a^2*A - 6*A*b^2 - 4*a*b*B)*Cot[c + d*x] - 3*a^3*(4*A*b + a*B)*Cot[c + d*x]^2 - 2*a^4*A*Cot[c + d*x]^3 + 3*(a + I*b)^4*((-I)*A + B)*Log[I - Tan[c + d*x]] - 6*a*(4*a^2*A*b - 4*A*b^3 + a^3*B - 6*a*b^2*B)*Log[Tan[c + d*x]] + 3*(a - I*b)^4*(I*A + B)*Log[I + Tan[c + d*x]])/(6*d)","C",1
264,1,211,225,0.9394242,"\int \cot ^5(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{-3 a^4 A \cot ^4(c+d x)-4 a^3 (a B+4 A b) \cot ^3(c+d x)+6 a^2 \left(a^2 A-4 a b B-6 A b^2\right) \cot ^2(c+d x)+12 a \left(a^3 B+4 a^2 A b-6 a b^2 B-4 A b^3\right) \cot (c+d x)+12 \left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right) \log (\tan (c+d x))-6 (a-i b)^4 (A-i B) \log (\tan (c+d x)+i)-6 (a+i b)^4 (A+i B) \log (-\tan (c+d x)+i)}{12 d}","\frac{a^2 \left(6 a^2 A-16 a b B-13 A b^2\right) \cot ^2(c+d x)}{12 d}+\frac{a \left(6 a^3 B+24 a^2 A b-34 a b^2 B-19 A b^3\right) \cot (c+d x)}{6 d}+\frac{\left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right) \log (\sin (c+d x))}{d}+x \left(a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right)-\frac{a (4 a B+7 A b) \cot ^3(c+d x) (a+b \tan (c+d x))^2}{12 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d}",1,"(12*a*(4*a^2*A*b - 4*A*b^3 + a^3*B - 6*a*b^2*B)*Cot[c + d*x] + 6*a^2*(a^2*A - 6*A*b^2 - 4*a*b*B)*Cot[c + d*x]^2 - 4*a^3*(4*A*b + a*B)*Cot[c + d*x]^3 - 3*a^4*A*Cot[c + d*x]^4 - 6*(a + I*b)^4*(A + I*B)*Log[I - Tan[c + d*x]] + 12*(a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Log[Tan[c + d*x]] - 6*(a - I*b)^4*(A - I*B)*Log[I + Tan[c + d*x]])/(12*d)","C",1
265,1,257,273,1.6087395,"\int \cot ^6(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^6*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{-12 a^4 A \cot ^5(c+d x)-15 a^3 (a B+4 A b) \cot ^4(c+d x)+20 a^2 \left(a^2 A-4 a b B-6 A b^2\right) \cot ^3(c+d x)+30 a \left(a^3 B+4 a^2 A b-6 a b^2 B-4 A b^3\right) \cot ^2(c+d x)-60 \left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right) \cot (c+d x)+60 \left(a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right) \log (\tan (c+d x))+30 i (a+i b)^4 (A+i B) \log (-\tan (c+d x)+i)-30 (a-i b)^4 (B+i A) \log (\tan (c+d x)+i)}{60 d}","\frac{a^2 \left(10 a^2 A-25 a b B-18 A b^2\right) \cot ^3(c+d x)}{30 d}+\frac{a \left(10 a^3 B+40 a^2 A b-55 a b^2 B-28 A b^3\right) \cot ^2(c+d x)}{20 d}-\frac{\left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right) \cot (c+d x)}{d}+\frac{\left(a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right) \log (\sin (c+d x))}{d}-x \left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right)-\frac{a (5 a B+8 A b) \cot ^4(c+d x) (a+b \tan (c+d x))^2}{20 d}-\frac{a A \cot ^5(c+d x) (a+b \tan (c+d x))^3}{5 d}",1,"(-60*(a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Cot[c + d*x] + 30*a*(4*a^2*A*b - 4*A*b^3 + a^3*B - 6*a*b^2*B)*Cot[c + d*x]^2 + 20*a^2*(a^2*A - 6*A*b^2 - 4*a*b*B)*Cot[c + d*x]^3 - 15*a^3*(4*A*b + a*B)*Cot[c + d*x]^4 - 12*a^4*A*Cot[c + d*x]^5 + (30*I)*(a + I*b)^4*(A + I*B)*Log[I - Tan[c + d*x]] + 60*(4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*Log[Tan[c + d*x]] - 30*(a - I*b)^4*(I*A + B)*Log[I + Tan[c + d*x]])/(60*d)","C",1
266,1,299,323,1.3250024,"\int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^7*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{-10 a^4 A \cot ^6(c+d x)-12 a^3 (a B+4 A b) \cot ^5(c+d x)+15 a^2 \left(a^2 A-4 a b B-6 A b^2\right) \cot ^4(c+d x)+20 a \left(a^3 B+4 a^2 A b-6 a b^2 B-4 A b^3\right) \cot ^3(c+d x)-30 \left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right) \cot ^2(c+d x)-60 \left(a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right) \cot (c+d x)-60 \left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right) \log (\tan (c+d x))+30 (a+i b)^4 (A+i B) \log (-\tan (c+d x)+i)+30 (a-i b)^4 (A-i B) \log (\tan (c+d x)+i)}{60 d}","\frac{a^2 \left(5 a^2 A-12 a b B-8 A b^2\right) \cot ^4(c+d x)}{20 d}+\frac{a \left(5 a^3 B+20 a^2 A b-27 a b^2 B-13 A b^3\right) \cot ^3(c+d x)}{15 d}-\frac{\left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right) \cot ^2(c+d x)}{2 d}-\frac{\left(a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right) \cot (c+d x)}{d}-\frac{\left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right) \log (\sin (c+d x))}{d}-x \left(a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right)-\frac{a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{10 d}-\frac{a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}",1,"(-60*(4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*Cot[c + d*x] - 30*(a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Cot[c + d*x]^2 + 20*a*(4*a^2*A*b - 4*A*b^3 + a^3*B - 6*a*b^2*B)*Cot[c + d*x]^3 + 15*a^2*(a^2*A - 6*A*b^2 - 4*a*b*B)*Cot[c + d*x]^4 - 12*a^3*(4*A*b + a*B)*Cot[c + d*x]^5 - 10*a^4*A*Cot[c + d*x]^6 + 30*(a + I*b)^4*(A + I*B)*Log[I - Tan[c + d*x]] - 60*(a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Log[Tan[c + d*x]] + 30*(a - I*b)^4*(A - I*B)*Log[I + Tan[c + d*x]])/(60*d)","C",1
267,1,138,127,1.5845983,"\int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{\frac{2 a^3 (a B-A b) \log (a+b \tan (c+d x))}{b^2 \left(a^2+b^2\right)}+\frac{2 (A b-a B) \tan (c+d x)}{b}-\frac{b (A+i B) \log (-\tan (c+d x)+i)}{a+i b}-\frac{b (A-i B) \log (\tan (c+d x)+i)}{a-i b}+B \tan ^2(c+d x)}{2 b d}","\frac{(a A+b B) \log (\cos (c+d x))}{d \left(a^2+b^2\right)}-\frac{x (A b-a B)}{a^2+b^2}-\frac{a^3 (A b-a B) \log (a+b \tan (c+d x))}{b^3 d \left(a^2+b^2\right)}+\frac{(A b-a B) \tan (c+d x)}{b^2 d}+\frac{B \tan ^2(c+d x)}{2 b d}",1,"(-((b*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b)) - (b*(A - I*B)*Log[I + Tan[c + d*x]])/(a - I*b) + (2*a^3*(-(A*b) + a*B)*Log[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)) + (2*(A*b - a*B)*Tan[c + d*x])/b + B*Tan[c + d*x]^2)/(2*b*d)","C",1
268,1,118,101,0.592224,"\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{\frac{2 a^2 (A b-a B) \log (a+b \tan (c+d x))}{b^2 \left(a^2+b^2\right)}+\frac{i (A+i B) \log (-\tan (c+d x)+i)}{a+i b}-\frac{(B+i A) \log (\tan (c+d x)+i)}{a-i b}+\frac{2 B \tan (c+d x)}{b}}{2 d}","\frac{a^2 (A b-a B) \log (a+b \tan (c+d x))}{b^2 d \left(a^2+b^2\right)}-\frac{(A b-a B) \log (\cos (c+d x))}{d \left(a^2+b^2\right)}-\frac{x (a A+b B)}{a^2+b^2}+\frac{B \tan (c+d x)}{b d}",1,"((I*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b) - ((I*A + B)*Log[I + Tan[c + d*x]])/(a - I*b) + (2*a^2*(A*b - a*B)*Log[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)) + (2*B*Tan[c + d*x])/b)/(2*d)","C",1
269,1,98,80,0.1809185,"\int \frac{\tan (c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{b (a-i b) (A+i B) \log (-\tan (c+d x)+i)+b (a+i b) (A-i B) \log (\tan (c+d x)+i)+2 a (a B-A b) \log (a+b \tan (c+d x))}{2 b d \left(a^2+b^2\right)}","-\frac{a (A b-a B) \log (a \cos (c+d x)+b \sin (c+d x))}{b d \left(a^2+b^2\right)}+\frac{x (A b-a B)}{a^2+b^2}-\frac{B \log (\cos (c+d x))}{b d}",1,"((a - I*b)*b*(A + I*B)*Log[I - Tan[c + d*x]] + (a + I*b)*b*(A - I*B)*Log[I + Tan[c + d*x]] + 2*a*(-(A*b) + a*B)*Log[a + b*Tan[c + d*x]])/(2*b*(a^2 + b^2)*d)","C",1
270,1,66,58,0.1069223,"\int \frac{A+B \tan (c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x]),x]","\frac{2 (a A+b B) \tan ^{-1}(\tan (c+d x))-(A b-a B) \left(\log \left(\sec ^2(c+d x)\right)-2 \log (a+b \tan (c+d x))\right)}{2 d \left(a^2+b^2\right)}","\frac{(A b-a B) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)}+\frac{x (a A+b B)}{a^2+b^2}",1,"(2*(a*A + b*B)*ArcTan[Tan[c + d*x]] - (A*b - a*B)*(Log[Sec[c + d*x]^2] - 2*Log[a + b*Tan[c + d*x]]))/(2*(a^2 + b^2)*d)","A",1
271,1,113,80,0.385909,"\int \frac{\cot (c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","-\frac{\frac{2 b (A b-a B) \log (a+b \tan (c+d x))}{a \left(a^2+b^2\right)}+\frac{(A+i B) \log (-\tan (c+d x)+i)}{a+i b}+\frac{(A-i B) \log (\tan (c+d x)+i)}{a-i b}-\frac{2 A \log (\tan (c+d x))}{a}}{2 d}","-\frac{b (A b-a B) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left(a^2+b^2\right)}-\frac{x (A b-a B)}{a^2+b^2}+\frac{A \log (\sin (c+d x))}{a d}",1,"-1/2*(((A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b) - (2*A*Log[Tan[c + d*x]])/a + ((A - I*B)*Log[I + Tan[c + d*x]])/(a - I*b) + (2*b*(A*b - a*B)*Log[a + b*Tan[c + d*x]])/(a*(a^2 + b^2)))/d","C",1
272,1,138,103,0.8775772,"\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{\frac{2 b^2 (A b-a B) \log (a+b \tan (c+d x))}{a^2 \left(a^2+b^2\right)}+\frac{2 (a B-A b) \log (\tan (c+d x))}{a^2}+\frac{i (A+i B) \log (-\tan (c+d x)+i)}{a+i b}-\frac{(B+i A) \log (\tan (c+d x)+i)}{a-i b}-\frac{2 A \cot (c+d x)}{a}}{2 d}","\frac{b^2 (A b-a B) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 d \left(a^2+b^2\right)}-\frac{x (a A+b B)}{a^2+b^2}-\frac{(A b-a B) \log (\sin (c+d x))}{a^2 d}-\frac{A \cot (c+d x)}{a d}",1,"((-2*A*Cot[c + d*x])/a + (I*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b) + (2*(-(A*b) + a*B)*Log[Tan[c + d*x]])/a^2 - ((I*A + B)*Log[I + Tan[c + d*x]])/(a - I*b) + (2*b^2*(A*b - a*B)*Log[a + b*Tan[c + d*x]])/(a^2*(a^2 + b^2)))/(2*d)","C",1
273,1,163,137,1.397773,"\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{\frac{2 (A b-a B) \cot (c+d x)}{a^2}-\frac{2 \left(a^2 A+a b B-A b^2\right) \log (\tan (c+d x))}{a^3}+\frac{2 b^3 (a B-A b) \log (a+b \tan (c+d x))}{a^3 \left(a^2+b^2\right)}+\frac{(A+i B) \log (-\tan (c+d x)+i)}{a+i b}+\frac{(A-i B) \log (\tan (c+d x)+i)}{a-i b}-\frac{A \cot ^2(c+d x)}{a}}{2 d}","\frac{x (A b-a B)}{a^2+b^2}+\frac{(A b-a B) \cot (c+d x)}{a^2 d}-\frac{\left(a^2 A+a b B-A b^2\right) \log (\sin (c+d x))}{a^3 d}-\frac{b^3 (A b-a B) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left(a^2+b^2\right)}-\frac{A \cot ^2(c+d x)}{2 a d}",1,"((2*(A*b - a*B)*Cot[c + d*x])/a^2 - (A*Cot[c + d*x]^2)/a + ((A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b) - (2*(a^2*A - A*b^2 + a*b*B)*Log[Tan[c + d*x]])/a^3 + ((A - I*B)*Log[I + Tan[c + d*x]])/(a - I*b) + (2*b^3*(-(A*b) + a*B)*Log[a + b*Tan[c + d*x]])/(a^3*(a^2 + b^2)))/(2*d)","C",1
274,1,194,169,2.5169558,"\int \frac{\cot ^4(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{\frac{6 (a-b) (a+b) (A b-a B) \log (\tan (c+d x))}{a^4}+\frac{3 (A b-a B) \cot ^2(c+d x)}{a^2}+\frac{6 b^4 (A b-a B) \log (a+b \tan (c+d x))}{a^4 \left(a^2+b^2\right)}+\frac{6 \left(a^2 A+a b B-A b^2\right) \cot (c+d x)}{a^3}+\frac{3 (B-i A) \log (-\tan (c+d x)+i)}{a+i b}+\frac{3 (B+i A) \log (\tan (c+d x)+i)}{a-i b}-\frac{2 A \cot ^3(c+d x)}{a}}{6 d}","\frac{x (a A+b B)}{a^2+b^2}+\frac{(A b-a B) \cot ^2(c+d x)}{2 a^2 d}+\frac{\left(a^2-b^2\right) (A b-a B) \log (\sin (c+d x))}{a^4 d}+\frac{b^4 (A b-a B) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left(a^2+b^2\right)}+\frac{\left(a^2 A+a b B-A b^2\right) \cot (c+d x)}{a^3 d}-\frac{A \cot ^3(c+d x)}{3 a d}",1,"((6*(a^2*A - A*b^2 + a*b*B)*Cot[c + d*x])/a^3 + (3*(A*b - a*B)*Cot[c + d*x]^2)/a^2 - (2*A*Cot[c + d*x]^3)/a + (3*((-I)*A + B)*Log[I - Tan[c + d*x]])/(a + I*b) + (6*(a - b)*(a + b)*(A*b - a*B)*Log[Tan[c + d*x]])/a^4 + (3*(I*A + B)*Log[I + Tan[c + d*x]])/(a - I*b) + (6*b^4*(A*b - a*B)*Log[a + b*Tan[c + d*x]])/(a^4*(a^2 + b^2)))/(6*d)","C",1
275,1,444,208,4.1617413,"\int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{2 b^2 B \left(a^2+b^2\right)^2 \tan ^2(c+d x)+2 i a^2 \left(2 a^3 B-a^2 A b+4 a b^2 B-3 A b^3\right) \tan ^{-1}(\tan (c+d x)) (a+b \tan (c+d x))+a \left(2 \left(a^2+b^2\right)^2 (2 a B-A b) \log (\cos (c+d x))+2 (a+i b)^2 (c+d x) \left(-2 i a^3 B+i a^2 b (A+4 i B)+2 a b^2 (A+i B)+b^3 B\right)+a^2 \left(-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)\right)+b \tan (c+d x) \left(2 \left(a^2+b^2\right)^2 (2 a B-A b) \log (\cos (c+d x))+a^2 \left(-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)+2 \left(-2 i a^5 B (c+d x+i)+i a^4 A b (c+d x+i)+a^3 b^2 B (-4 i c-4 i d x+3)+a^2 b^3 (B (c+d x)+i A (3 c+3 d x+i))+a b^4 (B-2 A (c+d x))-b^5 B (c+d x)\right)\right)}{2 b^3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}","\frac{a (A b-a B) \tan ^2(c+d x)}{b d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\left(-2 a^2 B+a A b-b^2 B\right) \tan (c+d x)}{b^2 d \left(a^2+b^2\right)}+\frac{\left(a^2 A+2 a b B-A b^2\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^2}-\frac{x \left(a^2 (-B)+2 a A b+b^2 B\right)}{\left(a^2+b^2\right)^2}+\frac{a^2 \left(-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right) \log (a+b \tan (c+d x))}{b^3 d \left(a^2+b^2\right)^2}",1,"(a*(2*(a + I*b)^2*(2*a*b^2*(A + I*B) + I*a^2*b*(A + (4*I)*B) - (2*I)*a^3*B + b^3*B)*(c + d*x) + 2*(a^2 + b^2)^2*(-(A*b) + 2*a*B)*Log[Cos[c + d*x]] + a^2*(a^2*A*b + 3*A*b^3 - 2*a^3*B - 4*a*b^2*B)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]) + b*(2*(a^3*b^2*B*(3 - (4*I)*c - (4*I)*d*x) - b^5*B*(c + d*x) + I*a^4*A*b*(I + c + d*x) - (2*I)*a^5*B*(I + c + d*x) + a*b^4*(B - 2*A*(c + d*x)) + a^2*b^3*(B*(c + d*x) + I*A*(I + 3*c + 3*d*x))) + 2*(a^2 + b^2)^2*(-(A*b) + 2*a*B)*Log[Cos[c + d*x]] + a^2*(a^2*A*b + 3*A*b^3 - 2*a^3*B - 4*a*b^2*B)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2])*Tan[c + d*x] + 2*b^2*(a^2 + b^2)^2*B*Tan[c + d*x]^2 + (2*I)*a^2*(-(a^2*A*b) - 3*A*b^3 + 2*a^3*B + 4*a*b^2*B)*ArcTan[Tan[c + d*x]]*(a + b*Tan[c + d*x]))/(2*b^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))","C",1
276,1,323,157,2.143153,"\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{-2 i a \left(a B \left(a^2+3 b^2\right)-2 A b^3\right) \tan ^{-1}(\tan (c+d x)) (a+b \tan (c+d x))+a \left(a \left(a B \left(a^2+3 b^2\right)-2 A b^3\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)-2 B \left(a^2+b^2\right)^2 \log (\cos (c+d x))+2 (a+i b)^2 (c+d x) \left(-A b^2+a B (2 b+i a)\right)\right)+b \tan (c+d x) \left(a \left(a B \left(a^2+3 b^2\right)-2 A b^3\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)-2 B \left(a^2+b^2\right)^2 \log (\cos (c+d x))+2 (a+i b) \left(i a^3 B (c+d x+i)+a^2 b (A+B (c+d x+i))-a b^2 (A (c+d x+i)-2 i B (c+d x))-i A b^3 (c+d x)\right)\right)}{2 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}","-\frac{a^2 (A b-a B)}{b^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\left(a^2 (-B)+2 a A b+b^2 B\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^2}-\frac{x \left(a^2 A+2 a b B-A b^2\right)}{\left(a^2+b^2\right)^2}-\frac{a \left(2 A b^3-a B \left(a^2+3 b^2\right)\right) \log (a+b \tan (c+d x))}{b^2 d \left(a^2+b^2\right)^2}",1,"(a*(2*(a + I*b)^2*(-(A*b^2) + a*(I*a + 2*b)*B)*(c + d*x) - 2*(a^2 + b^2)^2*B*Log[Cos[c + d*x]] + a*(-2*A*b^3 + a*(a^2 + 3*b^2)*B)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]) + b*(2*(a + I*b)*((-I)*A*b^3*(c + d*x) + I*a^3*B*(I + c + d*x) - a*b^2*((-2*I)*B*(c + d*x) + A*(I + c + d*x)) + a^2*b*(A + B*(I + c + d*x))) - 2*(a^2 + b^2)^2*B*Log[Cos[c + d*x]] + a*(-2*A*b^3 + a*(a^2 + 3*b^2)*B)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2])*Tan[c + d*x] - (2*I)*a*(-2*A*b^3 + a*(a^2 + 3*b^2)*B)*ArcTan[Tan[c + d*x]]*(a + b*Tan[c + d*x]))/(2*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))","C",1
277,1,140,115,2.1280336,"\int \frac{\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{\frac{2 \left(\left(a^2 (-A)-2 a b B+A b^2\right) \log (a+b \tan (c+d x))-\frac{a \left(a^2+b^2\right) (a B-A b)}{b (a+b \tan (c+d x))}\right)}{\left(a^2+b^2\right)^2}+\frac{(A+i B) \log (-\tan (c+d x)+i)}{(a+i b)^2}+\frac{(A-i B) \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 d}","\frac{a (A b-a B)}{b d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\left(a^2 A+2 a b B-A b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^2}+\frac{x \left(a^2 (-B)+2 a A b+b^2 B\right)}{\left(a^2+b^2\right)^2}",1,"(((A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b)^2 + ((A - I*B)*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (2*((-(a^2*A) + A*b^2 - 2*a*b*B)*Log[a + b*Tan[c + d*x]] - (a*(a^2 + b^2)*(-(A*b) + a*B))/(b*(a + b*Tan[c + d*x]))))/(a^2 + b^2)^2)/(2*d)","C",1
278,1,190,111,2.1291138,"\int \frac{A+B \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^2,x]","\frac{\frac{B ((-b-i a) \log (-\tan (c+d x)+i)+i (a+i b) \log (\tan (c+d x)+i)+2 b \log (a+b \tan (c+d x)))}{a^2+b^2}-(A b-a B) \left(\frac{2 b \left(\frac{a^2+b^2}{a+b \tan (c+d x)}-2 a \log (a+b \tan (c+d x))\right)}{\left(a^2+b^2\right)^2}+\frac{i \log (-\tan (c+d x)+i)}{(a+i b)^2}-\frac{i \log (\tan (c+d x)+i)}{(a-i b)^2}\right)}{2 b d}","-\frac{A b-a B}{d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\left(a^2 (-B)+2 a A b+b^2 B\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^2}+\frac{x \left(a^2 A+2 a b B-A b^2\right)}{\left(a^2+b^2\right)^2}",1,"((B*(((-I)*a - b)*Log[I - Tan[c + d*x]] + I*(a + I*b)*Log[I + Tan[c + d*x]] + 2*b*Log[a + b*Tan[c + d*x]]))/(a^2 + b^2) - (A*b - a*B)*((I*Log[I - Tan[c + d*x]])/(a + I*b)^2 - (I*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (2*b*(-2*a*Log[a + b*Tan[c + d*x]] + (a^2 + b^2)/(a + b*Tan[c + d*x])))/(a^2 + b^2)^2))/(2*b*d)","C",1
279,1,183,137,0.8529218,"\int \frac{\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{\frac{A \left(a^2+b^2\right) \log (\tan (c+d x))}{a}+\frac{b \left(2 a^3 B-3 a^2 A b-A b^3\right) \log (a+b \tan (c+d x))}{a \left(a^2+b^2\right)}+\frac{b (A b-a B)}{a+b \tan (c+d x)}-\frac{a (a-i b) (A+i B) \log (-\tan (c+d x)+i)}{2 (a+i b)}-\frac{a (a+i b) (A-i B) \log (\tan (c+d x)+i)}{2 (a-i b)}}{a d \left(a^2+b^2\right)}","\frac{b (A b-a B)}{a d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{x \left(a^2 (-B)+2 a A b+b^2 B\right)}{\left(a^2+b^2\right)^2}+\frac{A \log (\sin (c+d x))}{a^2 d}-\frac{b \left(-2 a^3 B+3 a^2 A b+A b^3\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 d \left(a^2+b^2\right)^2}",1,"(-1/2*(a*(a - I*b)*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b) + (A*(a^2 + b^2)*Log[Tan[c + d*x]])/a - (a*(a + I*b)*(A - I*B)*Log[I + Tan[c + d*x]])/(2*(a - I*b)) + (b*(-3*a^2*A*b - A*b^3 + 2*a^3*B)*Log[a + b*Tan[c + d*x]])/(a*(a^2 + b^2)) + (b*(A*b - a*B))/(a + b*Tan[c + d*x]))/(a*(a^2 + b^2)*d)","C",1
280,1,193,192,3.456292,"\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{\frac{2 (a B-2 A b) \log (\tan (c+d x))}{a^3}+\frac{2 b^2 (a B-A b)}{a^2 \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{2 A \cot (c+d x)}{a^2}-\frac{2 b^2 \left(3 a^3 B-4 a^2 A b+a b^2 B-2 A b^3\right) \log (a+b \tan (c+d x))}{a^3 \left(a^2+b^2\right)^2}+\frac{i (A+i B) \log (-\tan (c+d x)+i)}{(a+i b)^2}-\frac{(B+i A) \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 d}","-\frac{(2 A b-a B) \log (\sin (c+d x))}{a^3 d}-\frac{b \left(a^2 A-a b B+2 A b^2\right)}{a^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{x \left(a^2 A+2 a b B-A b^2\right)}{\left(a^2+b^2\right)^2}+\frac{b^2 \left(-3 a^3 B+4 a^2 A b-a b^2 B+2 A b^3\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left(a^2+b^2\right)^2}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))}",1,"((-2*A*Cot[c + d*x])/a^2 + (I*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b)^2 + (2*(-2*A*b + a*B)*Log[Tan[c + d*x]])/a^3 - ((I*A + B)*Log[I + Tan[c + d*x]])/(a - I*b)^2 - (2*b^2*(-4*a^2*A*b - 2*A*b^3 + 3*a^3*B + a*b^2*B)*Log[a + b*Tan[c + d*x]])/(a^3*(a^2 + b^2)^2) + (2*b^2*(-(A*b) + a*B))/(a^2*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(2*d)","C",1
281,1,220,250,4.6210964,"\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{-\frac{2 (a B-2 A b) \cot (c+d x)}{a^3}-\frac{A \cot ^2(c+d x)}{a^2}-\frac{2 \left(a^2 A+2 a b B-3 A b^2\right) \log (\tan (c+d x))}{a^4}+\frac{2 b^3 (A b-a B)}{a^3 \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{2 b^3 \left(4 a^3 B-5 a^2 A b+2 a b^2 B-3 A b^3\right) \log (a+b \tan (c+d x))}{a^4 \left(a^2+b^2\right)^2}+\frac{(A+i B) \log (-\tan (c+d x)+i)}{(a+i b)^2}+\frac{(A-i B) \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 d}","\frac{x \left(a^2 (-B)+2 a A b+b^2 B\right)}{\left(a^2+b^2\right)^2}+\frac{(3 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac{\left(a^2 A+2 a b B-3 A b^2\right) \log (\sin (c+d x))}{a^4 d}+\frac{b \left(a^3 (-B)+2 a^2 A b-2 a b^2 B+3 A b^3\right)}{a^3 d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{b^3 \left(-4 a^3 B+5 a^2 A b-2 a b^2 B+3 A b^3\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left(a^2+b^2\right)^2}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}",1,"((-2*(-2*A*b + a*B)*Cot[c + d*x])/a^3 - (A*Cot[c + d*x]^2)/a^2 + ((A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b)^2 - (2*(a^2*A - 3*A*b^2 + 2*a*b*B)*Log[Tan[c + d*x]])/a^4 + ((A - I*B)*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (2*b^3*(-5*a^2*A*b - 3*A*b^3 + 4*a^3*B + 2*a*b^2*B)*Log[a + b*Tan[c + d*x]])/(a^4*(a^2 + b^2)^2) + (2*b^3*(A*b - a*B))/(a^3*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(2*d)","C",1
282,1,1146,331,6.8559443,"\int \frac{\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]","\frac{(a B-A b) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) (A+B \tan (c+d x)) a^4}{2 (a-i b)^2 (a+i b)^2 b^2 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \left(2 B \sin (c+d x) a^5-A b \sin (c+d x) a^4+5 b^2 B \sin (c+d x) a^3-4 A b^3 \sin (c+d x) a^2\right) (A+B \tan (c+d x))}{(a-i b)^2 (a+i b)^2 b^3 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{B \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \tan (c+d x) (A+B \tan (c+d x))}{b^3 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{\left(A a^3+3 b B a^2-3 A b^2 a-b^3 B\right) (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (A+B \tan (c+d x))}{(a-i b)^3 (a+i b)^3 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{\left(6 a^2 A b^{13}+6 i a^3 A b^{12}-10 a^3 B b^{12}+15 a^4 A b^{11}-10 i a^4 B b^{11}+15 i a^5 A b^{10}-29 a^5 B b^{10}+13 a^6 A b^9-29 i a^6 B b^9+13 i a^7 A b^8-31 a^7 B b^8+5 a^8 A b^7-31 i a^8 B b^7+5 i a^9 A b^6-15 a^9 B b^6+a^{10} A b^5-15 i a^{10} B b^5+i a^{11} A b^4-3 a^{11} B b^4-3 i a^{12} B b^3\right) (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (A+B \tan (c+d x))}{(a-i b)^6 (a+i b)^5 b^7 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3}-\frac{i \left(-3 B a^7+A b a^6-9 b^2 B a^5+3 A b^3 a^4-10 b^4 B a^3+6 A b^5 a^2\right) \tan ^{-1}(\tan (c+d x)) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (A+B \tan (c+d x))}{b^4 \left(a^2+b^2\right)^3 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{(3 a B-A b) \log (\cos (c+d x)) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (A+B \tan (c+d x))}{b^4 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{\left(-3 B a^7+A b a^6-9 b^2 B a^5+3 A b^3 a^4-10 b^4 B a^3+6 A b^5 a^2\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (A+B \tan (c+d x))}{2 b^4 \left(a^2+b^2\right)^3 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3}","\frac{a (A b-a B) \tan ^3(c+d x)}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{a \left(-3 a^3 B+a^2 A b-7 a b^2 B+5 A b^3\right) \tan ^2(c+d x)}{2 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{\left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{x \left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right)}{\left(a^2+b^2\right)^3}-\frac{\left(-3 a^4 B+a^3 A b-6 a^2 b^2 B+3 a A b^3-b^4 B\right) \tan (c+d x)}{b^3 d \left(a^2+b^2\right)^2}+\frac{a^2 \left(-3 a^5 B+a^4 A b-9 a^3 b^2 B+3 a^2 A b^3-10 a b^4 B+6 A b^5\right) \log (a+b \tan (c+d x))}{b^4 d \left(a^2+b^2\right)^3}",1,"(a^4*(-(A*b) + a*B)*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])*(A + B*Tan[c + d*x]))/(2*(a - I*b)^2*(a + I*b)^2*b^2*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^3) + ((a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*(c + d*x)*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*(A + B*Tan[c + d*x]))/((a - I*b)^3*(a + I*b)^3*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^3) + ((I*a^11*A*b^4 + a^10*A*b^5 + (5*I)*a^9*A*b^6 + 5*a^8*A*b^7 + (13*I)*a^7*A*b^8 + 13*a^6*A*b^9 + (15*I)*a^5*A*b^10 + 15*a^4*A*b^11 + (6*I)*a^3*A*b^12 + 6*a^2*A*b^13 - (3*I)*a^12*b^3*B - 3*a^11*b^4*B - (15*I)*a^10*b^5*B - 15*a^9*b^6*B - (31*I)*a^8*b^7*B - 31*a^7*b^8*B - (29*I)*a^6*b^9*B - 29*a^5*b^10*B - (10*I)*a^4*b^11*B - 10*a^3*b^12*B)*(c + d*x)*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*(A + B*Tan[c + d*x]))/((a - I*b)^6*(a + I*b)^5*b^7*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^3) - (I*(a^6*A*b + 3*a^4*A*b^3 + 6*a^2*A*b^5 - 3*a^7*B - 9*a^5*b^2*B - 10*a^3*b^4*B)*ArcTan[Tan[c + d*x]]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*(A + B*Tan[c + d*x]))/(b^4*(a^2 + b^2)^3*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^3) + ((-(A*b) + 3*a*B)*Log[Cos[c + d*x]]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*(A + B*Tan[c + d*x]))/(b^4*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^3) + ((a^6*A*b + 3*a^4*A*b^3 + 6*a^2*A*b^5 - 3*a^7*B - 9*a^5*b^2*B - 10*a^3*b^4*B)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*(A + B*Tan[c + d*x]))/(2*b^4*(a^2 + b^2)^3*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^3) + (Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2*(-(a^4*A*b*Sin[c + d*x]) - 4*a^2*A*b^3*Sin[c + d*x] + 2*a^5*B*Sin[c + d*x] + 5*a^3*b^2*B*Sin[c + d*x])*(A + B*Tan[c + d*x]))/((a - I*b)^2*(a + I*b)^2*b^3*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^3) + (B*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*Tan[c + d*x]*(A + B*Tan[c + d*x]))/(b^3*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^3)","C",1
283,1,462,250,4.8900535,"\int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]","\frac{\sec ^2(c+d x) (A+B \tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x)) \left(-2 a b \left(a^2+b^2\right) \left(a B \left(a^2+4 b^2\right)-3 A b^3\right) \sin (c+d x) (a \cos (c+d x)+b \sin (c+d x))-2 B \left(a^2+b^2\right)^3 \log (\cos (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2+a^3 b^2 \left(a^2+b^2\right) (A b-a B)+2 b^3 (c+d x) \left(a^3 B-3 a^2 A b-3 a b^2 B+A b^3\right) (a \cos (c+d x)+b \sin (c+d x))^2+2 i a (c+d x) \left(a^5 B+3 a^3 b^2 B+a^2 A b^3+6 a b^4 B-3 A b^5\right) (a \cos (c+d x)+b \sin (c+d x))^2+a \left(a^5 B+3 a^3 b^2 B+a^2 A b^3+6 a b^4 B-3 A b^5\right) (a \cos (c+d x)+b \sin (c+d x))^2 \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)-2 i a \left(a^5 B+3 a^3 b^2 B+a^2 A b^3+6 a b^4 B-3 A b^5\right) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2\right)}{2 b^3 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","\frac{a (A b-a B) \tan ^2(c+d x)}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{a^2 \left(2 A b^3-a B \left(a^2+3 b^2\right)\right)}{b^3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{\left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^3}-\frac{x \left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right)}{\left(a^2+b^2\right)^3}+\frac{a \left(a^5 B+3 a^3 b^2 B+a^2 A b^3+6 a b^4 B-3 A b^5\right) \log (a+b \tan (c+d x))}{b^3 d \left(a^2+b^2\right)^3}",1,"(Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])*(a^3*b^2*(a^2 + b^2)*(A*b - a*B) - 2*a*b*(a^2 + b^2)*(-3*A*b^3 + a*(a^2 + 4*b^2)*B)*Sin[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x]) + 2*b^3*(-3*a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*(c + d*x)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + (2*I)*a*(a^2*A*b^3 - 3*A*b^5 + a^5*B + 3*a^3*b^2*B + 6*a*b^4*B)*(c + d*x)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 - (2*I)*a*(a^2*A*b^3 - 3*A*b^5 + a^5*B + 3*a^3*b^2*B + 6*a*b^4*B)*ArcTan[Tan[c + d*x]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 - 2*(a^2 + b^2)^3*B*Log[Cos[c + d*x]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + a*(a^2*A*b^3 - 3*A*b^5 + a^5*B + 3*a^3*b^2*B + 6*a*b^4*B)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)*(A + B*Tan[c + d*x]))/(2*b^3*(a^2 + b^2)^3*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^3)","C",1
284,1,288,189,5.3949952,"\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]","\frac{(A b-a B) \left(\frac{b \left(\frac{\left(a^2+b^2\right) \left(5 a^2+4 a b \tan (c+d x)+b^2\right)}{(a+b \tan (c+d x))^2}+\left(2 b^2-6 a^2\right) \log (a+b \tan (c+d x))\right)}{\left(a^2+b^2\right)^3}+\frac{i \log (-\tan (c+d x)+i)}{(a+i b)^3}-\frac{\log (\tan (c+d x)+i)}{(b+i a)^3}\right)+B \left(\frac{2 b \left(\frac{a^2+b^2}{a+b \tan (c+d x)}-2 a \log (a+b \tan (c+d x))\right)}{\left(a^2+b^2\right)^2}+\frac{i \log (-\tan (c+d x)+i)}{(a+i b)^2}-\frac{i \log (\tan (c+d x)+i)}{(a-i b)^2}\right)-\frac{a B+A b}{b (a+b \tan (c+d x))^2}-\frac{2 B \tan (c+d x)}{(a+b \tan (c+d x))^2}}{2 b d}","-\frac{a^2 (A b-a B)}{2 b^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{a \left(2 A b^3-a B \left(a^2+3 b^2\right)\right)}{b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{\left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}-\frac{x \left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right)}{\left(a^2+b^2\right)^3}",1,"(-((A*b + a*B)/(b*(a + b*Tan[c + d*x])^2)) - (2*B*Tan[c + d*x])/(a + b*Tan[c + d*x])^2 + B*((I*Log[I - Tan[c + d*x]])/(a + I*b)^2 - (I*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (2*b*(-2*a*Log[a + b*Tan[c + d*x]] + (a^2 + b^2)/(a + b*Tan[c + d*x])))/(a^2 + b^2)^2) + (A*b - a*B)*((I*Log[I - Tan[c + d*x]])/(a + I*b)^3 - Log[I + Tan[c + d*x]]/(I*a + b)^3 + (b*((-6*a^2 + 2*b^2)*Log[a + b*Tan[c + d*x]] + ((a^2 + b^2)*(5*a^2 + b^2 + 4*a*b*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2))/(a^2 + b^2)^3))/(2*b*d)","C",1
285,1,188,179,3.8604087,"\int \frac{\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]","\frac{\frac{a (A b-a B)}{b \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{2 \left(a^2 A+2 a b B-A b^2\right)}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{2 \left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^3}+\frac{(A+i B) \log (-\tan (c+d x)+i)}{(a+i b)^3}+\frac{(A-i B) \log (\tan (c+d x)+i)}{(a-i b)^3}}{2 d}","\frac{a (A b-a B)}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{a^2 A+2 a b B-A b^2}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{\left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{x \left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right)}{\left(a^2+b^2\right)^3}",1,"(((A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b)^3 + ((A - I*B)*Log[I + Tan[c + d*x]])/(a - I*b)^3 - (2*(a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^3 + (a*(A*b - a*B))/(b*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) + (2*(a^2*A - A*b^2 + 2*a*b*B))/((a^2 + b^2)^2*(a + b*Tan[c + d*x])))/(2*d)","C",1
286,1,243,175,4.7659488,"\int \frac{A+B \tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^3,x]","-\frac{(A b-a B) \left(\frac{b \left(\frac{\left(a^2+b^2\right) \left(5 a^2+4 a b \tan (c+d x)+b^2\right)}{(a+b \tan (c+d x))^2}+\left(2 b^2-6 a^2\right) \log (a+b \tan (c+d x))\right)}{\left(a^2+b^2\right)^3}+\frac{i \log (-\tan (c+d x)+i)}{(a+i b)^3}-\frac{\log (\tan (c+d x)+i)}{(b+i a)^3}\right)+B \left(\frac{2 b \left(\frac{a^2+b^2}{a+b \tan (c+d x)}-2 a \log (a+b \tan (c+d x))\right)}{\left(a^2+b^2\right)^2}+\frac{i \log (-\tan (c+d x)+i)}{(a+i b)^2}-\frac{i \log (\tan (c+d x)+i)}{(a-i b)^2}\right)}{2 b d}","-\frac{A b-a B}{2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{a^2 (-B)+2 a A b+b^2 B}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{\left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{x \left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right)}{\left(a^2+b^2\right)^3}",1,"-1/2*(B*((I*Log[I - Tan[c + d*x]])/(a + I*b)^2 - (I*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (2*b*(-2*a*Log[a + b*Tan[c + d*x]] + (a^2 + b^2)/(a + b*Tan[c + d*x])))/(a^2 + b^2)^2) + (A*b - a*B)*((I*Log[I - Tan[c + d*x]])/(a + I*b)^3 - Log[I + Tan[c + d*x]]/(I*a + b)^3 + (b*((-6*a^2 + 2*b^2)*Log[a + b*Tan[c + d*x]] + ((a^2 + b^2)*(5*a^2 + b^2 + 4*a*b*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2))/(a^2 + b^2)^3))/(b*d)","C",1
287,1,254,215,3.5751742,"\int \frac{\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]","\frac{\frac{4 a b (A b-a B)}{\left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{2 A b^2}{a^2+a b \tan (c+d x)}+\frac{2 A \left(a^2+b^2\right) \log (\tan (c+d x))}{a^2}-\frac{2 b \left(-3 a^5 B+6 a^4 A b+a^3 b^2 B+3 a^2 A b^3+A b^5\right) \log (a+b \tan (c+d x))}{a^2 \left(a^2+b^2\right)^2}+\frac{b (A b-a B)}{(a+b \tan (c+d x))^2}-\frac{a (a-i b) (A+i B) \log (-\tan (c+d x)+i)}{(a+i b)^2}-\frac{a (a+i b) (A-i B) \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 a d \left(a^2+b^2\right)}","\frac{A \log (\sin (c+d x))}{a^3 d}+\frac{b (A b-a B)}{2 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{b \left(-2 a^3 B+3 a^2 A b+A b^3\right)}{a^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{x \left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right)}{\left(a^2+b^2\right)^3}-\frac{b \left(-3 a^5 B+6 a^4 A b+a^3 b^2 B+3 a^2 A b^3+A b^5\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left(a^2+b^2\right)^3}",1,"(-((a*(a - I*b)*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b)^2) + (2*A*(a^2 + b^2)*Log[Tan[c + d*x]])/a^2 - (a*(a + I*b)*(A - I*B)*Log[I + Tan[c + d*x]])/(a - I*b)^2 - (2*b*(6*a^4*A*b + 3*a^2*A*b^3 + A*b^5 - 3*a^5*B + a^3*b^2*B)*Log[a + b*Tan[c + d*x]])/(a^2*(a^2 + b^2)^2) + (b*(A*b - a*B))/(a + b*Tan[c + d*x])^2 + (4*a*b*(A*b - a*B))/((a^2 + b^2)*(a + b*Tan[c + d*x])) + (2*A*b^2)/(a^2 + a*b*Tan[c + d*x]))/(2*a*(a^2 + b^2)*d)","C",1
288,1,288,287,6.428592,"\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]","-\frac{(3 A b-a B) \log (\tan (c+d x))}{a^4 d}-\frac{A \cot (c+d x)}{a^3 d}-\frac{b^2 (A b-a B)}{2 a^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{b^2 \left(-3 a^3 B+4 a^2 A b-a b^2 B+2 A b^3\right)}{a^3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{b^2 \left(-6 a^5 B+10 a^4 A b-3 a^3 b^2 B+9 a^2 A b^3-a b^4 B+3 A b^5\right) \log (a+b \tan (c+d x))}{a^4 d \left(a^2+b^2\right)^3}+\frac{(A+i B) \log (-\tan (c+d x)+i)}{2 d (-b+i a)^3}-\frac{(B+i A) \log (\tan (c+d x)+i)}{2 d (a-i b)^3}","-\frac{(3 A b-a B) \log (\sin (c+d x))}{a^4 d}-\frac{b \left(2 a^2 A-a b B+3 A b^2\right)}{2 a^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{x \left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right)}{\left(a^2+b^2\right)^3}-\frac{b \left(a^4 A-3 a^3 b B+6 a^2 A b^2-a b^3 B+3 A b^4\right)}{a^3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{b^2 \left(-6 a^5 B+10 a^4 A b-3 a^3 b^2 B+9 a^2 A b^3-a b^4 B+3 A b^5\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left(a^2+b^2\right)^3}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))^2}",1,"-((A*Cot[c + d*x])/(a^3*d)) + ((A + I*B)*Log[I - Tan[c + d*x]])/(2*(I*a - b)^3*d) - ((3*A*b - a*B)*Log[Tan[c + d*x]])/(a^4*d) - ((I*A + B)*Log[I + Tan[c + d*x]])/(2*(a - I*b)^3*d) + (b^2*(10*a^4*A*b + 9*a^2*A*b^3 + 3*A*b^5 - 6*a^5*B - 3*a^3*b^2*B - a*b^4*B)*Log[a + b*Tan[c + d*x]])/(a^4*(a^2 + b^2)^3*d) - (b^2*(A*b - a*B))/(2*a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (b^2*(4*a^2*A*b + 2*A*b^3 - 3*a^3*B - a*b^2*B))/(a^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))","C",1
289,1,320,352,6.4961672,"\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]","\frac{(3 A b-a B) \cot (c+d x)}{a^4 d}-\frac{A \cot ^2(c+d x)}{2 a^3 d}-\frac{\left(a^2 A+3 a b B-6 A b^2\right) \log (\tan (c+d x))}{a^5 d}+\frac{b^3 (A b-a B)}{2 a^3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{b^3 \left(-4 a^3 B+5 a^2 A b-2 a b^2 B+3 A b^3\right)}{a^4 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{b^3 \left(-10 a^5 B+15 a^4 A b-9 a^3 b^2 B+17 a^2 A b^3-3 a b^4 B+6 A b^5\right) \log (a+b \tan (c+d x))}{a^5 d \left(a^2+b^2\right)^3}+\frac{(A+i B) \log (-\tan (c+d x)+i)}{2 d (a+i b)^3}+\frac{(A-i B) \log (\tan (c+d x)+i)}{2 d (a-i b)^3}","\frac{(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac{\left(a^2 A+3 a b B-6 A b^2\right) \log (\sin (c+d x))}{a^5 d}+\frac{b \left(-2 a^3 B+5 a^2 A b-3 a b^2 B+6 A b^3\right)}{2 a^3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{x \left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right)}{\left(a^2+b^2\right)^3}+\frac{b \left(a^5 (-B)+3 a^4 A b-6 a^3 b^2 B+11 a^2 A b^3-3 a b^4 B+6 A b^5\right)}{a^4 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{b^3 \left(-10 a^5 B+15 a^4 A b-9 a^3 b^2 B+17 a^2 A b^3-3 a b^4 B+6 A b^5\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 d \left(a^2+b^2\right)^3}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}",1,"((3*A*b - a*B)*Cot[c + d*x])/(a^4*d) - (A*Cot[c + d*x]^2)/(2*a^3*d) + ((A + I*B)*Log[I - Tan[c + d*x]])/(2*(a + I*b)^3*d) - ((a^2*A - 6*A*b^2 + 3*a*b*B)*Log[Tan[c + d*x]])/(a^5*d) + ((A - I*B)*Log[I + Tan[c + d*x]])/(2*(a - I*b)^3*d) - (b^3*(15*a^4*A*b + 17*a^2*A*b^3 + 6*A*b^5 - 10*a^5*B - 9*a^3*b^2*B - 3*a*b^4*B)*Log[a + b*Tan[c + d*x]])/(a^5*(a^2 + b^2)^3*d) + (b^3*(A*b - a*B))/(2*a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (b^3*(5*a^2*A*b + 3*A*b^3 - 4*a^3*B - 2*a*b^2*B))/(a^4*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))","C",0
290,1,1812,351,6.9097501,"\int \frac{\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx","Integrate[(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]","\frac{\left(-4 a A b^{17}-4 i a^2 A b^{16}+10 a^2 B b^{16}-8 a^3 A b^{15}+10 i a^3 B b^{15}-8 i a^4 A b^{14}+35 a^4 B b^{14}+35 i a^5 B b^{13}+49 a^6 B b^{12}+8 a^7 A b^{11}+49 i a^7 B b^{11}+8 i a^8 A b^{10}+38 a^8 B b^{10}+4 a^9 A b^9+38 i a^9 B b^9+4 i a^{10} A b^8+20 a^{10} B b^8+20 i a^{11} B b^7+7 a^{12} B b^6+7 i a^{13} B b^5+a^{14} B b^4+i a^{15} B b^3\right) (c+d x) \sec ^3(c+d x) (A+B \tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^4}{(a-i b)^8 (a+i b)^7 b^7 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^4}-\frac{i \left(B a^8+4 b^2 B a^6+5 b^4 B a^4+4 A b^5 a^3+10 b^6 B a^2-4 A b^7 a\right) \tan ^{-1}(\tan (c+d x)) \sec ^3(c+d x) (A+B \tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^4}{b^4 \left(a^2+b^2\right)^4 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^4}-\frac{B \log (\cos (c+d x)) \sec ^3(c+d x) (A+B \tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^4}{b^4 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^4}+\frac{\left(B a^8+4 b^2 B a^6+5 b^4 B a^4+4 A b^5 a^3+10 b^6 B a^2-4 A b^7 a\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right) \sec ^3(c+d x) (A+B \tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^4}{2 b^4 \left(a^2+b^2\right)^4 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^4}+\frac{\sec ^3(c+d x) \left(-3 B \sin (c+d x) a^{10}-3 B \sin (3 (c+d x)) a^{10}-12 b B \cos (c+d x) a^9+6 b B \cos (3 (c+d x)) a^9-33 b^2 B \sin (c+d x) a^8-11 b^2 B \sin (3 (c+d x)) a^8-60 b^3 B \cos (c+d x) a^7+9 A b^3 (c+d x) \cos (c+d x) a^7+28 b^3 B \cos (3 (c+d x)) a^7+3 A b^3 (c+d x) \cos (3 (c+d x)) a^7-4 A b^3 \sin (3 (c+d x)) a^7+12 A b^4 \cos (c+d x) a^6+36 b^4 B (c+d x) \cos (c+d x) a^6+8 A b^4 \cos (3 (c+d x)) a^6+12 b^4 B (c+d x) \cos (3 (c+d x)) a^6-123 b^4 B \sin (c+d x) a^6+9 A b^4 (c+d x) \sin (c+d x) a^6-27 b^4 B \sin (3 (c+d x)) a^6+9 A b^4 (c+d x) \sin (3 (c+d x)) a^6-108 b^5 B \cos (c+d x) a^5-45 A b^5 (c+d x) \cos (c+d x) a^5+82 b^5 B \cos (3 (c+d x)) a^5-27 A b^5 (c+d x) \cos (3 (c+d x)) a^5+30 A b^5 \sin (c+d x) a^5+36 b^5 B (c+d x) \sin (c+d x) a^5+18 A b^5 \sin (3 (c+d x)) a^5+36 b^5 B (c+d x) \sin (3 (c+d x)) a^5+48 A b^6 \cos (c+d x) a^4-28 A b^6 \cos (3 (c+d x)) a^4-48 b^6 B (c+d x) \cos (3 (c+d x)) a^4-183 b^6 B \sin (c+d x) a^4-45 A b^6 (c+d x) \sin (c+d x) a^4+11 b^6 B \sin (3 (c+d x)) a^4-57 A b^6 (c+d x) \sin (3 (c+d x)) a^4-60 b^7 B \cos (c+d x) a^3-45 A b^7 (c+d x) \cos (c+d x) a^3+60 b^7 B \cos (3 (c+d x)) a^3+57 A b^7 (c+d x) \cos (3 (c+d x)) a^3+84 A b^7 \sin (c+d x) a^3+4 A b^7 \sin (3 (c+d x)) a^3-48 b^7 B (c+d x) \sin (3 (c+d x)) a^3+36 A b^8 \cos (c+d x) a^2-36 b^8 B (c+d x) \cos (c+d x) a^2-36 A b^8 \cos (3 (c+d x)) a^2+36 b^8 B (c+d x) \cos (3 (c+d x)) a^2-90 b^8 B \sin (c+d x) a^2-45 A b^8 (c+d x) \sin (c+d x) a^2+30 b^8 B \sin (3 (c+d x)) a^2+27 A b^8 (c+d x) \sin (3 (c+d x)) a^2+9 A b^9 (c+d x) \cos (c+d x) a-9 A b^9 (c+d x) \cos (3 (c+d x)) a+54 A b^9 \sin (c+d x) a-36 b^9 B (c+d x) \sin (c+d x) a-18 A b^9 \sin (3 (c+d x)) a+12 b^9 B (c+d x) \sin (3 (c+d x)) a+9 A b^{10} (c+d x) \sin (c+d x)-3 A b^{10} (c+d x) \sin (3 (c+d x))\right) (A+B \tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{12 (a-i b)^4 (a+i b)^4 b^3 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^4}","\frac{a (A b-a B) \tan ^3(c+d x)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{a \left(2 A b^3-a B \left(a^2+3 b^2\right)\right) \tan ^2(c+d x)}{2 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{a^2 \left(a^5 B+3 a^3 b^2 B+a^2 A b^3+6 a b^4 B-3 A b^5\right)}{b^4 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}+\frac{\left(a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{x \left(a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right)}{\left(a^2+b^2\right)^4}+\frac{a \left(a^7 B+4 a^5 b^2 B+5 a^3 b^4 B+4 a^2 A b^5+10 a b^6 B-4 A b^7\right) \log (a+b \tan (c+d x))}{b^4 d \left(a^2+b^2\right)^4}",1,"(((4*I)*a^10*A*b^8 + 4*a^9*A*b^9 + (8*I)*a^8*A*b^10 + 8*a^7*A*b^11 - (8*I)*a^4*A*b^14 - 8*a^3*A*b^15 - (4*I)*a^2*A*b^16 - 4*a*A*b^17 + I*a^15*b^3*B + a^14*b^4*B + (7*I)*a^13*b^5*B + 7*a^12*b^6*B + (20*I)*a^11*b^7*B + 20*a^10*b^8*B + (38*I)*a^9*b^9*B + 38*a^8*b^10*B + (49*I)*a^7*b^11*B + 49*a^6*b^12*B + (35*I)*a^5*b^13*B + 35*a^4*b^14*B + (10*I)*a^3*b^15*B + 10*a^2*b^16*B)*(c + d*x)*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4*(A + B*Tan[c + d*x]))/((a - I*b)^8*(a + I*b)^7*b^7*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^4) - (I*(4*a^3*A*b^5 - 4*a*A*b^7 + a^8*B + 4*a^6*b^2*B + 5*a^4*b^4*B + 10*a^2*b^6*B)*ArcTan[Tan[c + d*x]]*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4*(A + B*Tan[c + d*x]))/(b^4*(a^2 + b^2)^4*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^4) - (B*Log[Cos[c + d*x]]*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4*(A + B*Tan[c + d*x]))/(b^4*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^4) + ((4*a^3*A*b^5 - 4*a*A*b^7 + a^8*B + 4*a^6*b^2*B + 5*a^4*b^4*B + 10*a^2*b^6*B)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4*(A + B*Tan[c + d*x]))/(2*b^4*(a^2 + b^2)^4*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^4) + (Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*(12*a^6*A*b^4*Cos[c + d*x] + 48*a^4*A*b^6*Cos[c + d*x] + 36*a^2*A*b^8*Cos[c + d*x] - 12*a^9*b*B*Cos[c + d*x] - 60*a^7*b^3*B*Cos[c + d*x] - 108*a^5*b^5*B*Cos[c + d*x] - 60*a^3*b^7*B*Cos[c + d*x] + 9*a^7*A*b^3*(c + d*x)*Cos[c + d*x] - 45*a^5*A*b^5*(c + d*x)*Cos[c + d*x] - 45*a^3*A*b^7*(c + d*x)*Cos[c + d*x] + 9*a*A*b^9*(c + d*x)*Cos[c + d*x] + 36*a^6*b^4*B*(c + d*x)*Cos[c + d*x] - 36*a^2*b^8*B*(c + d*x)*Cos[c + d*x] + 8*a^6*A*b^4*Cos[3*(c + d*x)] - 28*a^4*A*b^6*Cos[3*(c + d*x)] - 36*a^2*A*b^8*Cos[3*(c + d*x)] + 6*a^9*b*B*Cos[3*(c + d*x)] + 28*a^7*b^3*B*Cos[3*(c + d*x)] + 82*a^5*b^5*B*Cos[3*(c + d*x)] + 60*a^3*b^7*B*Cos[3*(c + d*x)] + 3*a^7*A*b^3*(c + d*x)*Cos[3*(c + d*x)] - 27*a^5*A*b^5*(c + d*x)*Cos[3*(c + d*x)] + 57*a^3*A*b^7*(c + d*x)*Cos[3*(c + d*x)] - 9*a*A*b^9*(c + d*x)*Cos[3*(c + d*x)] + 12*a^6*b^4*B*(c + d*x)*Cos[3*(c + d*x)] - 48*a^4*b^6*B*(c + d*x)*Cos[3*(c + d*x)] + 36*a^2*b^8*B*(c + d*x)*Cos[3*(c + d*x)] + 30*a^5*A*b^5*Sin[c + d*x] + 84*a^3*A*b^7*Sin[c + d*x] + 54*a*A*b^9*Sin[c + d*x] - 3*a^10*B*Sin[c + d*x] - 33*a^8*b^2*B*Sin[c + d*x] - 123*a^6*b^4*B*Sin[c + d*x] - 183*a^4*b^6*B*Sin[c + d*x] - 90*a^2*b^8*B*Sin[c + d*x] + 9*a^6*A*b^4*(c + d*x)*Sin[c + d*x] - 45*a^4*A*b^6*(c + d*x)*Sin[c + d*x] - 45*a^2*A*b^8*(c + d*x)*Sin[c + d*x] + 9*A*b^10*(c + d*x)*Sin[c + d*x] + 36*a^5*b^5*B*(c + d*x)*Sin[c + d*x] - 36*a*b^9*B*(c + d*x)*Sin[c + d*x] - 4*a^7*A*b^3*Sin[3*(c + d*x)] + 18*a^5*A*b^5*Sin[3*(c + d*x)] + 4*a^3*A*b^7*Sin[3*(c + d*x)] - 18*a*A*b^9*Sin[3*(c + d*x)] - 3*a^10*B*Sin[3*(c + d*x)] - 11*a^8*b^2*B*Sin[3*(c + d*x)] - 27*a^6*b^4*B*Sin[3*(c + d*x)] + 11*a^4*b^6*B*Sin[3*(c + d*x)] + 30*a^2*b^8*B*Sin[3*(c + d*x)] + 9*a^6*A*b^4*(c + d*x)*Sin[3*(c + d*x)] - 57*a^4*A*b^6*(c + d*x)*Sin[3*(c + d*x)] + 27*a^2*A*b^8*(c + d*x)*Sin[3*(c + d*x)] - 3*A*b^10*(c + d*x)*Sin[3*(c + d*x)] + 36*a^5*b^5*B*(c + d*x)*Sin[3*(c + d*x)] - 48*a^3*b^7*B*(c + d*x)*Sin[3*(c + d*x)] + 12*a*b^9*B*(c + d*x)*Sin[3*(c + d*x)])*(A + B*Tan[c + d*x]))/(12*(a - I*b)^4*(a + I*b)^4*b^3*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + b*Tan[c + d*x])^4)","C",1
291,1,465,298,6.3661569,"\int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx","Integrate[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]","-\frac{B \tan ^2(c+d x)}{b d (a+b \tan (c+d x))^3}-\frac{-\frac{(-2 a B-A b) \tan (c+d x)}{2 b d (a+b \tan (c+d x))^3}-\frac{-\frac{2 a^2 B+a A b-2 b^2 B}{3 b d (a+b \tan (c+d x))^3}+\frac{\frac{\left(6 a A b^3+6 b^4 B\right) \left(-\frac{b \left(3 a^2-b^2\right)}{\left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{a b}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{b}{3 \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{4 a b (a-b) (a+b) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^4}-\frac{i \log (-\tan (c+d x)+i)}{2 (a+i b)^4}+\frac{i \log (\tan (c+d x)+i)}{2 (a-i b)^4}\right)}{b}-6 A b^2 \left(-\frac{2 a b}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{b}{2 \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{b \left(3 a^2-b^2\right) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^3}-\frac{\log (-\tan (c+d x)+i)}{2 (-b+i a)^3}+\frac{\log (\tan (c+d x)+i)}{2 (b+i a)^3}\right)}{3 b d}}{2 b}}{b}","\frac{a (A b-a B) \tan ^2(c+d x)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{a^2 \left(2 a^3 B+a^2 A b+8 a b^2 B-5 A b^3\right)}{6 b^3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{\left(a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}-\frac{x \left(a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right)}{\left(a^2+b^2\right)^4}-\frac{a \left(2 a^5 B+a^4 A b+7 a^3 b^2 B+5 a^2 A b^3+17 a b^4 B-8 A b^5\right)}{3 b^3 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}",1,"-((B*Tan[c + d*x]^2)/(b*d*(a + b*Tan[c + d*x])^3)) - (-1/2*((-(A*b) - 2*a*B)*Tan[c + d*x])/(b*d*(a + b*Tan[c + d*x])^3) - (-1/3*(a*A*b + 2*a^2*B - 2*b^2*B)/(b*d*(a + b*Tan[c + d*x])^3) + (((6*a*A*b^3 + 6*b^4*B)*(((-1/2*I)*Log[I - Tan[c + d*x]])/(a + I*b)^4 + ((I/2)*Log[I + Tan[c + d*x]])/(a - I*b)^4 + (4*a*(a - b)*b*(a + b)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^4 - b/(3*(a^2 + b^2)*(a + b*Tan[c + d*x])^3) - (a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x])^2) - (b*(3*a^2 - b^2))/((a^2 + b^2)^3*(a + b*Tan[c + d*x]))))/b - 6*A*b^2*(-1/2*Log[I - Tan[c + d*x]]/(I*a - b)^3 + Log[I + Tan[c + d*x]]/(2*(I*a + b)^3) + (b*(3*a^2 - b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^3 - b/(2*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - (2*a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x]))))/(3*b*d))/(2*b))/b","C",1
292,1,411,261,6.3193379,"\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx","Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]","-\frac{B \tan (c+d x)}{2 b d (a+b \tan (c+d x))^3}-\frac{\frac{a B+2 A b}{3 b d (a+b \tan (c+d x))^3}+\frac{\frac{\left(6 A b^3-6 a b^2 B\right) \left(-\frac{b \left(3 a^2-b^2\right)}{\left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{a b}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{b}{3 \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{4 a b (a-b) (a+b) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^4}-\frac{i \log (-\tan (c+d x)+i)}{2 (a+i b)^4}+\frac{i \log (\tan (c+d x)+i)}{2 (a-i b)^4}\right)}{b}+6 b B \left(-\frac{2 a b}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{b}{2 \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{b \left(3 a^2-b^2\right) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^3}-\frac{\log (-\tan (c+d x)+i)}{2 (-b+i a)^3}+\frac{\log (\tan (c+d x)+i)}{2 (b+i a)^3}\right)}{3 b d}}{2 b}","-\frac{a^2 (A b-a B)}{3 b^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{a \left(2 A b^3-a B \left(a^2+3 b^2\right)\right)}{2 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{\left(a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}-\frac{x \left(a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right)}{\left(a^2+b^2\right)^4}",1,"-1/2*(B*Tan[c + d*x])/(b*d*(a + b*Tan[c + d*x])^3) - ((2*A*b + a*B)/(3*b*d*(a + b*Tan[c + d*x])^3) + (((6*A*b^3 - 6*a*b^2*B)*(((-1/2*I)*Log[I - Tan[c + d*x]])/(a + I*b)^4 + ((I/2)*Log[I + Tan[c + d*x]])/(a - I*b)^4 + (4*a*(a - b)*b*(a + b)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^4 - b/(3*(a^2 + b^2)*(a + b*Tan[c + d*x])^3) - (a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x])^2) - (b*(3*a^2 - b^2))/((a^2 + b^2)^3*(a + b*Tan[c + d*x]))))/b + 6*b*B*(-1/2*Log[I - Tan[c + d*x]]/(I*a - b)^3 + Log[I + Tan[c + d*x]]/(2*(I*a + b)^3) + (b*(3*a^2 - b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^3 - b/(2*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - (2*a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x]))))/(3*b*d))/(2*b)","C",1
293,1,248,250,1.3272949,"\int \frac{\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx","Integrate[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]","\frac{\frac{2 a (A b-a B)}{b \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{3 \left(a^2 A+2 a b B-A b^2\right)}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{6 \left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right)}{\left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{6 \left(a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^4}+\frac{3 (A+i B) \log (-\tan (c+d x)+i)}{(a+i b)^4}+\frac{3 (A-i B) \log (\tan (c+d x)+i)}{(a-i b)^4}}{6 d}","\frac{a (A b-a B)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{a^2 A+2 a b B-A b^2}{2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{a^3 A+3 a^2 b B-3 a A b^2-b^3 B}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{\left(a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{x \left(a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right)}{\left(a^2+b^2\right)^4}",1,"((3*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b)^4 + (3*(A - I*B)*Log[I + Tan[c + d*x]])/(a - I*b)^4 - (6*(a^4*A - 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B - 4*a*b^3*B)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^4 + (2*a*(A*b - a*B))/(b*(a^2 + b^2)*(a + b*Tan[c + d*x])^3) + (3*(a^2*A - A*b^2 + 2*a*b*B))/((a^2 + b^2)^2*(a + b*Tan[c + d*x])^2) + (6*(a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B))/((a^2 + b^2)^3*(a + b*Tan[c + d*x])))/(6*d)","C",1
294,1,327,247,6.2760253,"\int \frac{A+B \tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^4,x]","-\frac{(A b-a B) \left(\frac{6 b \left(3 a^2-b^2\right)}{\left(a^2+b^2\right)^3 (a+b \tan (c+d x))}+\frac{6 a b}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{2 b}{\left(a^2+b^2\right) (a+b \tan (c+d x))^3}-\frac{24 a b (a-b) (a+b) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^4}+\frac{3 i \log (-\tan (c+d x)+i)}{(a+i b)^4}-\frac{3 i \log (\tan (c+d x)+i)}{(a-i b)^4}\right)}{6 b d}-\frac{B \left(\frac{4 a b}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{b}{\left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{2 b \left(3 a^2-b^2\right) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^3}+\frac{\log (-\tan (c+d x)+i)}{(-b+i a)^3}-\frac{\log (\tan (c+d x)+i)}{(b+i a)^3}\right)}{2 b d}","-\frac{A b-a B}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}-\frac{a^2 (-B)+2 a A b+b^2 B}{2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}+\frac{\left(a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{x \left(a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right)}{\left(a^2+b^2\right)^4}",1,"-1/6*((A*b - a*B)*(((3*I)*Log[I - Tan[c + d*x]])/(a + I*b)^4 - ((3*I)*Log[I + Tan[c + d*x]])/(a - I*b)^4 - (24*a*(a - b)*b*(a + b)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^4 + (2*b)/((a^2 + b^2)*(a + b*Tan[c + d*x])^3) + (6*a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x])^2) + (6*b*(3*a^2 - b^2))/((a^2 + b^2)^3*(a + b*Tan[c + d*x]))))/(b*d) - (B*(Log[I - Tan[c + d*x]]/(I*a - b)^3 - Log[I + Tan[c + d*x]]/(I*a + b)^3 - (2*b*(3*a^2 - b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^3 + b/((a^2 + b^2)*(a + b*Tan[c + d*x])^2) + (4*a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x]))))/(2*b*d)","C",1
295,1,308,302,3.5611545,"\int \frac{\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx","Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]","\frac{\frac{2 a b \left(a^2+b^2\right) (A b-a B)}{(a+b \tan (c+d x))^3}+\frac{3 b \left(-2 a^3 B+3 a^2 A b+A b^3\right)}{(a+b \tan (c+d x))^2}+\frac{6 b \left(-3 a^5 B+6 a^4 A b+a^3 b^2 B+3 a^2 A b^3+A b^5\right)}{a \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{3 \left(-a^4 (a-i b)^4 (A+i B) \log (-\tan (c+d x)+i)-a^4 (a+i b)^4 (A-i B) \log (\tan (c+d x)+i)+2 A \left(a^2+b^2\right)^4 \log (\tan (c+d x))-2 b \left(-4 a^7 B+10 a^6 A b+4 a^5 b^2 B+5 a^4 A b^3+4 a^2 A b^5+A b^7\right) \log (a+b \tan (c+d x))\right)}{a^2 \left(a^2+b^2\right)^2}}{6 a^2 d \left(a^2+b^2\right)^2}","\frac{A \log (\sin (c+d x))}{a^4 d}+\frac{b (A b-a B)}{3 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{b \left(-2 a^3 B+3 a^2 A b+A b^3\right)}{2 a^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{x \left(a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right)}{\left(a^2+b^2\right)^4}+\frac{b \left(-3 a^5 B+6 a^4 A b+a^3 b^2 B+3 a^2 A b^3+A b^5\right)}{a^3 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{b \left(-4 a^7 B+10 a^6 A b+4 a^5 b^2 B+5 a^4 A b^3+4 a^2 A b^5+A b^7\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left(a^2+b^2\right)^4}",1,"((3*(-(a^4*(a - I*b)^4*(A + I*B)*Log[I - Tan[c + d*x]]) + 2*A*(a^2 + b^2)^4*Log[Tan[c + d*x]] - a^4*(a + I*b)^4*(A - I*B)*Log[I + Tan[c + d*x]] - 2*b*(10*a^6*A*b + 5*a^4*A*b^3 + 4*a^2*A*b^5 + A*b^7 - 4*a^7*B + 4*a^5*b^2*B)*Log[a + b*Tan[c + d*x]]))/(a^2*(a^2 + b^2)^2) + (2*a*b*(a^2 + b^2)*(A*b - a*B))/(a + b*Tan[c + d*x])^3 + (3*b*(3*a^2*A*b + A*b^3 - 2*a^3*B))/(a + b*Tan[c + d*x])^2 + (6*b*(6*a^4*A*b + 3*a^2*A*b^3 + A*b^5 - 3*a^5*B + a^3*b^2*B))/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(6*a^2*(a^2 + b^2)^2*d)","C",1
296,1,357,399,6.2719965,"\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx","Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]","\frac{\frac{6 (a B-4 A b) \log (\tan (c+d x))}{a^5}-\frac{6 A \cot (c+d x)}{a^4}+\frac{2 b^2 (a B-A b)}{a^2 \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{3 b^2 \left(3 a^3 B-4 a^2 A b+a b^2 B-2 A b^3\right)}{a^3 \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{6 b^2 \left(6 a^5 B-10 a^4 A b+3 a^3 b^2 B-9 a^2 A b^3+a b^4 B-3 A b^5\right)}{a^4 \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{6 b^2 \left(10 a^7 B-20 a^6 A b+5 a^5 b^2 B-24 a^4 A b^3+4 a^3 b^4 B-16 a^2 A b^5+a b^6 B-4 A b^7\right) \log (a+b \tan (c+d x))}{a^5 \left(a^2+b^2\right)^4}+\frac{3 i (A+i B) \log (-\tan (c+d x)+i)}{(a+i b)^4}-\frac{3 (B+i A) \log (\tan (c+d x)+i)}{(a-i b)^4}}{6 d}","-\frac{(4 A b-a B) \log (\sin (c+d x))}{a^5 d}-\frac{b \left(3 a^2 A-a b B+4 A b^2\right)}{3 a^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}-\frac{b \left(2 a^4 A-3 a^3 b B+8 a^2 A b^2-a b^3 B+4 A b^4\right)}{2 a^3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{x \left(a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right)}{\left(a^2+b^2\right)^4}-\frac{b \left(a^6 A-6 a^5 b B+13 a^4 A b^2-3 a^3 b^3 B+12 a^2 A b^4-a b^5 B+4 A b^6\right)}{a^4 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}+\frac{b^2 \left(-10 a^7 B+20 a^6 A b-5 a^5 b^2 B+24 a^4 A b^3-4 a^3 b^4 B+16 a^2 A b^5-a b^6 B+4 A b^7\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 d \left(a^2+b^2\right)^4}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}",1,"((-6*A*Cot[c + d*x])/a^4 + ((3*I)*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b)^4 + (6*(-4*A*b + a*B)*Log[Tan[c + d*x]])/a^5 - (3*(I*A + B)*Log[I + Tan[c + d*x]])/(a - I*b)^4 - (6*b^2*(-20*a^6*A*b - 24*a^4*A*b^3 - 16*a^2*A*b^5 - 4*A*b^7 + 10*a^7*B + 5*a^5*b^2*B + 4*a^3*b^4*B + a*b^6*B)*Log[a + b*Tan[c + d*x]])/(a^5*(a^2 + b^2)^4) + (2*b^2*(-(A*b) + a*B))/(a^2*(a^2 + b^2)*(a + b*Tan[c + d*x])^3) + (3*b^2*(-4*a^2*A*b - 2*A*b^3 + 3*a^3*B + a*b^2*B))/(a^3*(a^2 + b^2)^2*(a + b*Tan[c + d*x])^2) + (6*b^2*(-10*a^4*A*b - 9*a^2*A*b^3 - 3*A*b^5 + 6*a^5*B + 3*a^3*b^2*B + a*b^4*B))/(a^4*(a^2 + b^2)^3*(a + b*Tan[c + d*x])))/(6*d)","C",1
297,1,417,477,6.6926044,"\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx","Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]","\frac{(4 A b-a B) \cot (c+d x)}{a^5 d}-\frac{A \cot ^2(c+d x)}{2 a^4 d}-\frac{\left(a^2 A+4 a b B-10 A b^2\right) \log (\tan (c+d x))}{a^6 d}+\frac{b^3 (A b-a B)}{3 a^3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{b^3 \left(-4 a^3 B+5 a^2 A b-2 a b^2 B+3 A b^3\right)}{2 a^4 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{b^3 \left(-10 a^5 B+15 a^4 A b-9 a^3 b^2 B+17 a^2 A b^3-3 a b^4 B+6 A b^5\right)}{a^5 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{b^3 \left(-20 a^7 B+35 a^6 A b-24 a^5 b^2 B+56 a^4 A b^3-16 a^3 b^4 B+39 a^2 A b^5-4 a b^6 B+10 A b^7\right) \log (a+b \tan (c+d x))}{a^6 d \left(a^2+b^2\right)^4}+\frac{(A+i B) \log (-\tan (c+d x)+i)}{2 d (a+i b)^4}+\frac{(A-i B) \log (\tan (c+d x)+i)}{2 d (a-i b)^4}","\frac{(5 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))^3}-\frac{\left(a^2 A+4 a b B-10 A b^2\right) \log (\sin (c+d x))}{a^6 d}+\frac{b \left(-3 a^3 B+9 a^2 A b-4 a b^2 B+10 A b^3\right)}{3 a^3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{x \left(a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right)}{\left(a^2+b^2\right)^4}+\frac{b \left(-2 a^5 B+7 a^4 A b-8 a^3 b^2 B+19 a^2 A b^3-4 a b^4 B+10 A b^5\right)}{2 a^4 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{b \left(a^7 (-B)+4 a^6 A b-13 a^5 b^2 B+27 a^4 A b^3-12 a^3 b^4 B+29 a^2 A b^5-4 a b^6 B+10 A b^7\right)}{a^5 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{b^3 \left(-20 a^7 B+35 a^6 A b-24 a^5 b^2 B+56 a^4 A b^3-16 a^3 b^4 B+39 a^2 A b^5-4 a b^6 B+10 A b^7\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^6 d \left(a^2+b^2\right)^4}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^3}",1,"((4*A*b - a*B)*Cot[c + d*x])/(a^5*d) - (A*Cot[c + d*x]^2)/(2*a^4*d) + ((A + I*B)*Log[I - Tan[c + d*x]])/(2*(a + I*b)^4*d) - ((a^2*A - 10*A*b^2 + 4*a*b*B)*Log[Tan[c + d*x]])/(a^6*d) + ((A - I*B)*Log[I + Tan[c + d*x]])/(2*(a - I*b)^4*d) - (b^3*(35*a^6*A*b + 56*a^4*A*b^3 + 39*a^2*A*b^5 + 10*A*b^7 - 20*a^7*B - 24*a^5*b^2*B - 16*a^3*b^4*B - 4*a*b^6*B)*Log[a + b*Tan[c + d*x]])/(a^6*(a^2 + b^2)^4*d) + (b^3*(A*b - a*B))/(3*a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (b^3*(5*a^2*A*b + 3*A*b^3 - 4*a^3*B - 2*a*b^2*B))/(2*a^4*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (b^3*(15*a^4*A*b + 17*a^2*A*b^3 + 6*A*b^5 - 10*a^5*B - 9*a^3*b^2*B - 3*a*b^4*B))/(a^5*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))","C",0
298,1,26,29,0.0294511,"\int \frac{\tan ^3(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]^3*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{B \left(\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)}{2 d}","\frac{B \tan ^2(c+d x)}{2 d}+\frac{B \log (\cos (c+d x))}{d}",1,"(B*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/(2*d)","A",1
299,1,25,16,0.0107654,"\int \frac{\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]^2*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","B \left(\frac{\tan (c+d x)}{d}-\frac{\tan ^{-1}(\tan (c+d x))}{d}\right)","\frac{B \tan (c+d x)}{d}-B x",1,"B*(-(ArcTan[Tan[c + d*x]]/d) + Tan[c + d*x]/d)","A",1
300,1,13,13,0.0077498,"\int \frac{\tan (c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","-\frac{B \log (\cos (c+d x))}{d}","-\frac{B \log (\cos (c+d x))}{d}",1,"-((B*Log[Cos[c + d*x]])/d)","A",1
301,1,3,3,0.0003967,"\int \frac{a B+b B \tan (c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(a + b*Tan[c + d*x]),x]","B x","B x",1,"B*x","A",1
302,1,20,12,0.0108296,"\int \frac{\cot (c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{B (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}","\frac{B \log (\sin (c+d x))}{d}",1,"(B*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","A",1
303,1,30,17,0.0180447,"\int \frac{\cot ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^2*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","-\frac{B \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}","-\frac{B \cot (c+d x)}{d}-B x",1,"-((B*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d)","C",1
304,1,35,30,0.0802562,"\int \frac{\cot ^3(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^3*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","-\frac{B \left(\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right)}{2 d}","-\frac{B \cot ^2(c+d x)}{2 d}-\frac{B \log (\sin (c+d x))}{d}",1,"-1/2*(B*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]]))/d","A",1
305,1,34,31,0.0194093,"\int \frac{\cot ^4(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^4*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","-\frac{B \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)}{3 d}","-\frac{B \cot ^3(c+d x)}{3 d}+\frac{B \cot (c+d x)}{d}+B x",1,"-1/3*(B*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/d","C",1
306,1,108,102,0.4615806,"\int \frac{\tan ^4(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^4*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{B \left(\frac{2 a^4 \log (a+b \tan (c+d x))}{b^3 \left(a^2+b^2\right)}-\frac{2 a \tan (c+d x)}{b^2}+\frac{\log (-\tan (c+d x)+i)}{-b+i a}-\frac{\log (\tan (c+d x)+i)}{b+i a}+\frac{\tan ^2(c+d x)}{b}\right)}{2 d}","\frac{b B \log (\cos (c+d x))}{d \left(a^2+b^2\right)}+\frac{a B x}{a^2+b^2}+\frac{a^4 B \log (a+b \tan (c+d x))}{b^3 d \left(a^2+b^2\right)}-\frac{a B \tan (c+d x)}{b^2 d}+\frac{B \tan ^2(c+d x)}{2 b d}",1,"(B*(Log[I - Tan[c + d*x]]/(I*a - b) - Log[I + Tan[c + d*x]]/(I*a + b) + (2*a^4*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)) - (2*a*Tan[c + d*x])/b^2 + Tan[c + d*x]^2/b))/(2*d)","C",1
307,1,92,83,0.4242097,"\int \frac{\tan ^3(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^3*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","-\frac{B \left(\frac{2 a^3 \log (a+b \tan (c+d x))}{b^2 \left(a^2+b^2\right)}+\frac{\log (-\tan (c+d x)+i)}{a+i b}+\frac{\log (\tan (c+d x)+i)}{a-i b}-\frac{2 \tan (c+d x)}{b}\right)}{2 d}","\frac{a B \log (\cos (c+d x))}{d \left(a^2+b^2\right)}-\frac{b B x}{a^2+b^2}-\frac{a^3 B \log (a+b \tan (c+d x))}{b^2 d \left(a^2+b^2\right)}+\frac{B \tan (c+d x)}{b d}",1,"-1/2*(B*(Log[I - Tan[c + d*x]]/(a + I*b) + Log[I + Tan[c + d*x]]/(a - I*b) + (2*a^3*Log[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)) - (2*Tan[c + d*x])/b))/d","C",1
308,1,79,81,0.1024775,"\int \frac{\tan ^2(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^2*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{B \left(2 a^2 \log (a+b \tan (c+d x))+b (b+i a) \log (-\tan (c+d x)+i)+b (b-i a) \log (\tan (c+d x)+i)\right)}{2 b d \left(a^2+b^2\right)}","\frac{a^2 B \log (a \cos (c+d x)+b \sin (c+d x))}{b d \left(a^2+b^2\right)}+\frac{a^3 B x}{b^2 \left(a^2+b^2\right)}-\frac{a B x}{b^2}-\frac{B \log (\cos (c+d x))}{b d}",1,"(B*(b*(I*a + b)*Log[I - Tan[c + d*x]] + b*((-I)*a + b)*Log[I + Tan[c + d*x]] + 2*a^2*Log[a + b*Tan[c + d*x]]))/(2*b*(a^2 + b^2)*d)","C",1
309,1,67,48,0.1332136,"\int \frac{\tan (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{B \left(2 (b-i a) (c+d x)-a \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)+2 i a \tan ^{-1}(\tan (c+d x))\right)}{2 d \left(a^2+b^2\right)}","\frac{b B x}{a^2+b^2}-\frac{a B \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)}",1,"(B*(2*((-I)*a + b)*(c + d*x) + (2*I)*a*ArcTan[Tan[c + d*x]] - a*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]))/(2*(a^2 + b^2)*d)","C",1
310,1,77,47,0.069377,"\int \frac{a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(a + b*Tan[c + d*x])^2,x]","\frac{B ((-b-i a) \log (-\tan (c+d x)+i)+i (a+i b) \log (\tan (c+d x)+i)+2 b \log (a+b \tan (c+d x)))}{2 d \left(a^2+b^2\right)}","\frac{b B \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)}+\frac{a B x}{a^2+b^2}",1,"(B*(((-I)*a - b)*Log[I - Tan[c + d*x]] + I*(a + I*b)*Log[I + Tan[c + d*x]] + 2*b*Log[a + b*Tan[c + d*x]]))/(2*(a^2 + b^2)*d)","C",1
311,1,79,69,0.1195057,"\int \frac{\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","-\frac{B \left(2 b^2 \log (a \cot (c+d x)+b)+a (a+i b) \log (-\cot (c+d x)+i)+a (a-i b) \log (\cot (c+d x)+i)\right)}{2 a d \left(a^2+b^2\right)}","-\frac{b^2 B \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left(a^2+b^2\right)}-\frac{b B x}{a^2+b^2}+\frac{B \log (\sin (c+d x))}{a d}",1,"-1/2*(B*(a*(a + I*b)*Log[I - Cot[c + d*x]] + a*(a - I*b)*Log[I + Cot[c + d*x]] + 2*b^2*Log[b + a*Cot[c + d*x]]))/(a*(a^2 + b^2)*d)","C",1
312,1,97,85,0.4246125,"\int \frac{\cot ^2(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^2*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","-\frac{B \left(-\frac{b^3 \log (a \cot (c+d x)+b)}{a^2 \left(a^2+b^2\right)}-\frac{\log (-\cot (c+d x)+i)}{2 (b+i a)}+\frac{\log (\cot (c+d x)+i)}{2 (-b+i a)}+\frac{\cot (c+d x)}{a}\right)}{d}","-\frac{a B x}{a^2+b^2}+\frac{b^3 B \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 d \left(a^2+b^2\right)}-\frac{b B \log (\sin (c+d x))}{a^2 d}-\frac{B \cot (c+d x)}{a d}",1,"-((B*(Cot[c + d*x]/a - Log[I - Cot[c + d*x]]/(2*(I*a + b)) + Log[I + Cot[c + d*x]]/(2*(I*a - b)) - (b^3*Log[b + a*Cot[c + d*x]])/(a^2*(a^2 + b^2))))/d)","C",1
313,1,107,112,0.6553528,"\int \frac{\cot ^3(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^3*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","-\frac{B \left(-\frac{2 b \cot (c+d x)}{a^2}+\frac{2 b^4 \log (a \cot (c+d x)+b)}{a^3 \left(a^2+b^2\right)}-\frac{\log (-\cot (c+d x)+i)}{a-i b}-\frac{\log (\cot (c+d x)+i)}{a+i b}+\frac{\cot ^2(c+d x)}{a}\right)}{2 d}","\frac{b B x}{a^2+b^2}+\frac{b B \cot (c+d x)}{a^2 d}-\frac{B \left(a^2-b^2\right) \log (\sin (c+d x))}{a^3 d}-\frac{b^4 B \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left(a^2+b^2\right)}-\frac{B \cot ^2(c+d x)}{2 a d}",1,"-1/2*(B*((-2*b*Cot[c + d*x])/a^2 + Cot[c + d*x]^2/a - Log[I - Cot[c + d*x]]/(a - I*b) - Log[I + Cot[c + d*x]]/(a + I*b) + (2*b^4*Log[b + a*Cot[c + d*x]])/(a^3*(a^2 + b^2))))/d","C",1
314,1,62,25,0.0408577,"\int \frac{3+\tan (c+d x)}{2-\tan (c+d x)} \, dx","Integrate[(3 + Tan[c + d*x])/(2 - Tan[c + d*x]),x]","\frac{\tan ^{-1}(\tan (c+d x))}{d}+\frac{\log \left((2-\tan (c+d x))^2-4 (2-\tan (c+d x))+5\right)}{2 d}-\frac{\log (2-\tan (c+d x))}{d}","x-\frac{\log (2 \cos (c+d x)-\sin (c+d x))}{d}",1,"ArcTan[Tan[c + d*x]]/d + Log[5 - 4*(2 - Tan[c + d*x]) + (2 - Tan[c + d*x])^2]/(2*d) - Log[2 - Tan[c + d*x]]/d","B",1
315,1,65,58,0.111922,"\int \frac{\frac{b B}{a}+B \tan (c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[((b*B)/a + B*Tan[c + d*x])/(a + b*Tan[c + d*x]),x]","\frac{B \left(\left(a^2-b^2\right) \left(\log \left(\sec ^2(c+d x)\right)-2 \log (a+b \tan (c+d x))\right)+4 a b \tan ^{-1}(\tan (c+d x))\right)}{2 a d \left(a^2+b^2\right)}","\frac{2 b B x}{a^2+b^2}-\frac{B \left(a-\frac{b^2}{a}\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)}",1,"(B*(4*a*b*ArcTan[Tan[c + d*x]] + (a^2 - b^2)*(Log[Sec[c + d*x]^2] - 2*Log[a + b*Tan[c + d*x]])))/(2*a*(a^2 + b^2)*d)","A",1
316,1,187,101,2.303414,"\int \frac{a+b \tan (c+d x)}{(b+a \tan (c+d x))^2} \, dx","Integrate[(a + b*Tan[c + d*x])/(b + a*Tan[c + d*x])^2,x]","\frac{\frac{b (-((a+i b) \log (-\tan (c+d x)+i))-(a-i b) \log (\tan (c+d x)+i)+2 a \log (a \tan (c+d x)+b))}{a^2+b^2}+(a-b) (a+b) \left(\frac{2 a \left(2 b \log (a \tan (c+d x)+b)-\frac{a^2+b^2}{a \tan (c+d x)+b}\right)}{\left(a^2+b^2\right)^2}+\frac{i \log (-\tan (c+d x)+i)}{(a-i b)^2}-\frac{i \log (\tan (c+d x)+i)}{(a+i b)^2}\right)}{2 a d}","-\frac{a^2-b^2}{d \left(a^2+b^2\right) (a \tan (c+d x)+b)}+\frac{b \left(3 a^2-b^2\right) \log (a \sin (c+d x)+b \cos (c+d x))}{d \left(a^2+b^2\right)^2}-\frac{a x \left(a^2-3 b^2\right)}{\left(a^2+b^2\right)^2}",1,"((b*(-((a + I*b)*Log[I - Tan[c + d*x]]) - (a - I*b)*Log[I + Tan[c + d*x]] + 2*a*Log[b + a*Tan[c + d*x]]))/(a^2 + b^2) + (a - b)*(a + b)*((I*Log[I - Tan[c + d*x]])/(a - I*b)^2 - (I*Log[I + Tan[c + d*x]])/(a + I*b)^2 + (2*a*(2*b*Log[b + a*Tan[c + d*x]] - (a^2 + b^2)/(b + a*Tan[c + d*x])))/(a^2 + b^2)^2))/(2*a*d)","C",1
317,1,212,233,2.7573411,"\int \tan ^3(c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{2 \sqrt{a+b \tan (c+d x)} \left(8 a^3 B-b \left(4 a^2 B-7 a A b+35 b^2 B\right) \tan (c+d x)-14 a^2 A b+3 b^2 (a B+7 A b) \tan ^2(c+d x)-35 a b^2 B-105 A b^3+15 b^3 B \tan ^3(c+d x)\right)}{105 b^3 d}+\frac{\sqrt{a-i b} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{\sqrt{a+i b} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}","-\frac{2 \left(-8 a^2 B+14 a A b+35 b^2 B\right) (a+b \tan (c+d x))^{3/2}}{105 b^3 d}+\frac{2 (7 A b-4 a B) \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{35 b^2 d}+\frac{\sqrt{a-i b} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{\sqrt{a+i b} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{2 A \sqrt{a+b \tan (c+d x)}}{d}+\frac{2 B \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}",1,"(Sqrt[a - I*b]*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + (Sqrt[a + I*b]*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*Sqrt[a + b*Tan[c + d*x]]*(-14*a^2*A*b - 105*A*b^3 + 8*a^3*B - 35*a*b^2*B - b*(-7*a*A*b + 4*a^2*B + 35*b^2*B)*Tan[c + d*x] + 3*b^2*(7*A*b + a*B)*Tan[c + d*x]^2 + 15*b^3*B*Tan[c + d*x]^3))/(105*b^3*d)","A",1
318,1,169,186,1.8875574,"\int \tan ^2(c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left(-2 a^2 B+b (a B+5 A b) \tan (c+d x)+5 a A b+3 b^2 B \tan ^2(c+d x)-15 b^2 B\right)}{b^2}+15 \sqrt{a-i b} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+15 \sqrt{a+i b} (B-i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{15 d}","\frac{2 (5 A b-2 a B) (a+b \tan (c+d x))^{3/2}}{15 b^2 d}+\frac{\sqrt{a-i b} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{\sqrt{a+i b} (-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}+\frac{2 B \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}-\frac{2 B \sqrt{a+b \tan (c+d x)}}{d}",1,"(15*Sqrt[a - I*b]*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + 15*Sqrt[a + I*b]*((-I)*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + (2*Sqrt[a + b*Tan[c + d*x]]*(5*a*A*b - 2*a^2*B - 15*b^2*B + b*(5*A*b + a*B)*Tan[c + d*x] + 3*b^2*B*Tan[c + d*x]^2))/b^2)/(15*d)","A",1
319,1,140,146,0.499729,"\int \tan (c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{2 \sqrt{a+b \tan (c+d x)} (a B+3 A b+b B \tan (c+d x))-3 b \sqrt{a-i b} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)-3 b \sqrt{a+i b} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{3 b d}","-\frac{\sqrt{a-i b} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{\sqrt{a+i b} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}+\frac{2 A \sqrt{a+b \tan (c+d x)}}{d}+\frac{2 B (a+b \tan (c+d x))^{3/2}}{3 b d}",1,"(-3*Sqrt[a - I*b]*b*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - 3*Sqrt[a + I*b]*b*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*Sqrt[a + b*Tan[c + d*x]]*(3*A*b + a*B + b*B*Tan[c + d*x]))/(3*b*d)","A",1
320,1,120,122,0.1280941,"\int \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{-i \sqrt{a-i b} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+i \sqrt{a+i b} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+2 B \sqrt{a+b \tan (c+d x)}}{d}","-\frac{\sqrt{a-i b} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{\sqrt{a+i b} (-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}+\frac{2 B \sqrt{a+b \tan (c+d x)}}{d}",1,"((-I)*Sqrt[a - I*b]*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + I*Sqrt[a + I*b]*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*B*Sqrt[a + b*Tan[c + d*x]])/d","A",1
321,1,219,131,0.5778814,"\int \cot (c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","-\frac{-\frac{\left(A \left(a \sqrt{-b^2}+b^2\right)+b B \left(a-\sqrt{-b^2}\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{-b^2} \sqrt{a-\sqrt{-b^2}}}+\frac{\left(A \left(b^2-a \sqrt{-b^2}\right)+b B \left(a+\sqrt{-b^2}\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{-b^2} \sqrt{a+\sqrt{-b^2}}}+2 \sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{d}","\frac{\sqrt{a-i b} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{\sqrt{a+i b} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{2 \sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{d}",1,"-((2*Sqrt[a]*A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] - ((A*(b^2 + a*Sqrt[-b^2]) + b*(a - Sqrt[-b^2])*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]) + ((A*(b^2 - a*Sqrt[-b^2]) + b*(a + Sqrt[-b^2])*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a + Sqrt[-b^2]]))/d)","A",1
322,1,235,167,2.3803111,"\int \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{\frac{\frac{\left(A \left(a \sqrt{-b^2}+b^2\right)+b B \left(a-\sqrt{-b^2}\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{a-\sqrt{-b^2}}}+\frac{\left(A \left(b^2-a \sqrt{-b^2}\right)+b B \left(a+\sqrt{-b^2}\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{a+\sqrt{-b^2}}}-A b \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{b}-\frac{(2 a B+A b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}}{d}","-\frac{(2 a B+A b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}+\frac{\sqrt{a-i b} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{\sqrt{a+i b} (-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{A \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{d}",1,"(-(((A*b + 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a]) + (((A*(b^2 + a*Sqrt[-b^2]) + b*(a - Sqrt[-b^2])*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/Sqrt[a - Sqrt[-b^2]] + ((A*(b^2 - a*Sqrt[-b^2]) + b*(a + Sqrt[-b^2])*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/Sqrt[a + Sqrt[-b^2]] - A*b*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/b)/d","A",1
323,1,271,219,4.7264367,"\int \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{\frac{\left(8 a^2 A-4 a b B+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{\frac{4 \left(-a A b+a \sqrt{-b^2} B+A \sqrt{-b^2} b+b^2 B\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{a-\sqrt{-b^2}}}-\frac{4 \left(a A b+a \sqrt{-b^2} B+A \sqrt{-b^2} b+b^2 (-B)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{a+\sqrt{-b^2}}}-\frac{b \cot (c+d x) \sqrt{a+b \tan (c+d x)} (2 a A \cot (c+d x)+4 a B+A b)}{a}}{b}}{4 d}","\frac{\left(8 a^2 A-4 a b B+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{3/2} d}-\frac{\sqrt{a-i b} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{\sqrt{a+i b} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{(4 a B+A b) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 a d}-\frac{A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}",1,"(((8*a^2*A + A*b^2 - 4*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/a^(3/2) + ((4*(-(a*A*b) + A*b*Sqrt[-b^2] + b^2*B + a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/Sqrt[a - Sqrt[-b^2]] - (4*(a*A*b + A*b*Sqrt[-b^2] - b^2*B + a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/Sqrt[a + Sqrt[-b^2]] - (b*Cot[c + d*x]*(A*b + 4*a*B + 2*a*A*Cot[c + d*x])*Sqrt[a + b*Tan[c + d*x]])/a)/b)/(4*d)","A",1
324,1,564,279,6.4432441,"\int \cot ^4(c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{2 b^4 \left(-\frac{(a A-b B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{2 a^{3/2} b^3}-\frac{3 (a B+A b) \left(\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{\cot (c+d x) \sqrt{a+b \tan (c+d x)}}{a b}\right)}{8 a b^2}+\frac{5 A \left(\frac{3 \left(\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{\cot (c+d x) \sqrt{a+b \tan (c+d x)}}{a b}\right)}{a}+\frac{2 \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{a b^2}\right)}{48 b}+\frac{(a B+A b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} b^4}-\frac{(a B+A b) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{4 a b^4}+\frac{(a A-b B) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{2 a b^4}-\frac{A \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{6 b^4}-\frac{\left(a A b+a \sqrt{-b^2} B+A \sqrt{-b^2} b+b^2 (-B)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{2 \left(-b^2\right)^{5/2} \sqrt{a+\sqrt{-b^2}}}+\frac{\left(a A b-a \sqrt{-b^2} B-A \sqrt{-b^2} b+b^2 (-B)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{2 b^4 \sqrt{-b^2} \sqrt{a-\sqrt{-b^2}}}\right)}{d}","\frac{\left(8 a^2 A-2 a b B+A b^2\right) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 a^2 d}+\frac{\left(16 a^3 B+8 a^2 A b+2 a b^2 B-A b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{8 a^{5/2} d}-\frac{\sqrt{a-i b} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{\sqrt{a+i b} (-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{(6 a B+A b) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 a d}-\frac{A \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}",1,"(2*b^4*(((A*b + a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*b^4) - ((a*A - b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(2*a^(3/2)*b^3) + ((a*A*b - A*b*Sqrt[-b^2] - b^2*B - a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(2*b^4*Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]) - ((a*A*b + A*b*Sqrt[-b^2] - b^2*B + a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(2*(-b^2)^(5/2)*Sqrt[a + Sqrt[-b^2]]) + ((a*A - b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(2*a*b^4) - ((A*b + a*B)*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(4*a*b^4) - (A*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]])/(6*b^4) - (3*(A*b + a*B)*(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]]/a^(3/2) - (Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(a*b)))/(8*a*b^2) + (5*A*((2*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(a*b^2) + (3*(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]]/a^(3/2) - (Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(a*b)))/a))/(48*b)))/d","B",1
325,1,252,214,2.4905372,"\int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{\frac{2 (7 A b-2 a B) (a+b \tan (c+d x))^{5/2}}{b}+\frac{35}{3} b (A-i B) \left(3 \sqrt{a-i b} (b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)-i \sqrt{a+b \tan (c+d x)} (4 a+b \tan (c+d x)-3 i b)\right)+\frac{35}{3} b (A+i B) \left(i \sqrt{a+b \tan (c+d x)} (4 a+b \tan (c+d x)+3 i b)+3 \sqrt{a+i b} (b-i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)\right)+10 B \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{35 b d}","\frac{2 (7 A b-2 a B) (a+b \tan (c+d x))^{5/2}}{35 b^2 d}-\frac{2 (a B+A b) \sqrt{a+b \tan (c+d x)}}{d}+\frac{(a-i b)^{3/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{(a+i b)^{3/2} (-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}+\frac{2 B \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac{2 B (a+b \tan (c+d x))^{3/2}}{3 d}",1,"((2*(7*A*b - 2*a*B)*(a + b*Tan[c + d*x])^(5/2))/b + 10*B*Tan[c + d*x]*(a + b*Tan[c + d*x])^(5/2) + (35*b*(A - I*B)*(3*Sqrt[a - I*b]*(I*a + b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - I*Sqrt[a + b*Tan[c + d*x]]*(4*a - (3*I)*b + b*Tan[c + d*x])))/3 + (35*b*(A + I*B)*(3*Sqrt[a + I*b]*((-I)*a + b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + I*Sqrt[a + b*Tan[c + d*x]]*(4*a + (3*I)*b + b*Tan[c + d*x])))/3)/(35*b*d)","A",1
326,1,192,175,1.3935467,"\int \tan (c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{5 (A-i B) \left(\sqrt{a+b \tan (c+d x)} (4 a+b \tan (c+d x)-3 i b)-3 (a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)\right)+5 (A+i B) \left(\sqrt{a+b \tan (c+d x)} (4 a+b \tan (c+d x)+3 i b)-3 (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)\right)+\frac{6 B (a+b \tan (c+d x))^{5/2}}{b}}{15 d}","\frac{2 (a A-b B) \sqrt{a+b \tan (c+d x)}}{d}-\frac{(a-i b)^{3/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{(a+i b)^{3/2} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}+\frac{2 A (a+b \tan (c+d x))^{3/2}}{3 d}+\frac{2 B (a+b \tan (c+d x))^{5/2}}{5 b d}",1,"((6*B*(a + b*Tan[c + d*x])^(5/2))/b + 5*(A - I*B)*(-3*(a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + Sqrt[a + b*Tan[c + d*x]]*(4*a - (3*I)*b + b*Tan[c + d*x])) + 5*(A + I*B)*(-3*(a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Sqrt[a + b*Tan[c + d*x]]*(4*a + (3*I)*b + b*Tan[c + d*x])))/(15*d)","A",1
327,1,140,150,0.5136336,"\int (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{2 \sqrt{a+b \tan (c+d x)} (4 a B+3 A b+b B \tan (c+d x))-3 i (a-i b)^{3/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+3 i (a+i b)^{3/2} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{3 d}","\frac{2 (a B+A b) \sqrt{a+b \tan (c+d x)}}{d}-\frac{(a-i b)^{3/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{(a+i b)^{3/2} (-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}+\frac{2 B (a+b \tan (c+d x))^{3/2}}{3 d}",1,"((-3*I)*(a - I*b)^(3/2)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + (3*I)*(a + I*b)^(3/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*Sqrt[a + b*Tan[c + d*x]]*(3*A*b + 4*a*B + b*B*Tan[c + d*x]))/(3*d)","A",1
328,1,144,152,0.3418209,"\int \cot (c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{-2 a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)+(a-i b)^{3/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+(a+i b)^{3/2} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+2 b B \sqrt{a+b \tan (c+d x)}}{d}","-\frac{2 a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{d}+\frac{(a-i b)^{3/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{(a+i b)^{3/2} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}+\frac{2 b B \sqrt{a+b \tan (c+d x)}}{d}",1,"(-2*a^(3/2)*A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + (a - I*b)^(3/2)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + (a + I*b)^(3/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*b*B*Sqrt[a + b*Tan[c + d*x]])/d","A",1
329,1,282,169,0.5959367,"\int \cot ^2(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{(a-i b)^{3/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)-\sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)+A b \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)-i a A \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)-a A \cot (c+d x) \sqrt{a+b \tan (c+d x)}+i b B \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+a B \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}","\frac{(a-i b)^{3/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{\sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{d}-\frac{(a+i b)^{3/2} (-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{a A \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{d}",1,"(-(Sqrt[a]*(3*A*b + 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]]) + (a - I*b)^(3/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - I*a*A*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + A*Sqrt[a + I*b]*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + a*Sqrt[a + I*b]*B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + I*Sqrt[a + I*b]*b*B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - a*A*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/d","A",1
330,1,195,219,2.5059967,"\int \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{\left(8 a^2 A-12 a b B-3 A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)-\sqrt{a} \left(4 (a-i b)^{3/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+4 (a+i b)^{3/2} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+\cot (c+d x) \sqrt{a+b \tan (c+d x)} (2 a A \cot (c+d x)+4 a B+5 A b)\right)}{4 \sqrt{a} d}","\frac{\left(8 a^2 A-12 a b B-3 A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{4 \sqrt{a} d}-\frac{(a-i b)^{3/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{(a+i b)^{3/2} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{(4 a B+5 A b) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}-\frac{a A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}",1,"((8*a^2*A - 3*A*b^2 - 12*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] - Sqrt[a]*(4*(a - I*b)^(3/2)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + 4*(a + I*b)^(3/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Cot[c + d*x]*(5*A*b + 4*a*B + 2*a*A*Cot[c + d*x])*Sqrt[a + b*Tan[c + d*x]]))/(4*Sqrt[a]*d)","A",1
331,1,241,278,5.6224906,"\int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{3 \left(16 a^3 B+24 a^2 A b-6 a b^2 B+A b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)+\sqrt{a} \left(-\cot (c+d x) \sqrt{a+b \tan (c+d x)} \left(8 a^2 A \cot ^2(c+d x)-24 a^2 A+2 a (6 a B+7 A b) \cot (c+d x)+30 a b B+3 A b^2\right)-24 i a (a-i b)^{3/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+24 i a (a+i b)^{3/2} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)\right)}{24 a^{3/2} d}","\frac{\left(8 a^2 A-10 a b B-A b^2\right) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 a d}+\frac{\left(16 a^3 B+24 a^2 A b-6 a b^2 B+A b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{8 a^{3/2} d}-\frac{(a-i b)^{3/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{(a+i b)^{3/2} (-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{(6 a B+7 A b) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a A \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}",1,"(3*(24*a^2*A*b + A*b^3 + 16*a^3*B - 6*a*b^2*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + Sqrt[a]*((-24*I)*a*(a - I*b)^(3/2)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + (24*I)*a*(a + I*b)^(3/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - Cot[c + d*x]*(-24*a^2*A + 3*A*b^2 + 30*a*b*B + 2*a*(7*A*b + 6*a*B)*Cot[c + d*x] + 8*a^2*A*Cot[c + d*x]^2)*Sqrt[a + b*Tan[c + d*x]]))/(24*a^(3/2)*d)","A",1
332,1,296,252,4.5894332,"\int \tan ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{\frac{2 (9 A b-2 a B) (a+b \tan (c+d x))^{7/2}}{b}-\frac{63}{2} i b (A-i B) \left(\frac{2}{5} (a+b \tan (c+d x))^{5/2}+\frac{2}{3} (a-i b) \left(\sqrt{a+b \tan (c+d x)} (4 a+b \tan (c+d x)-3 i b)-3 (a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)\right)\right)+\frac{63}{2} i b (A+i B) \left(\frac{2}{5} (a+b \tan (c+d x))^{5/2}+\frac{2}{3} (a+i b) \left(\sqrt{a+b \tan (c+d x)} (4 a+b \tan (c+d x)+3 i b)-3 (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)\right)\right)+14 B \tan (c+d x) (a+b \tan (c+d x))^{7/2}}{63 b d}","-\frac{2 \left(a^2 B+2 a A b-b^2 B\right) \sqrt{a+b \tan (c+d x)}}{d}+\frac{2 (9 A b-2 a B) (a+b \tan (c+d x))^{7/2}}{63 b^2 d}-\frac{2 (a B+A b) (a+b \tan (c+d x))^{3/2}}{3 d}+\frac{(a-i b)^{5/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{(a+i b)^{5/2} (-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}+\frac{2 B \tan (c+d x) (a+b \tan (c+d x))^{7/2}}{9 b d}-\frac{2 B (a+b \tan (c+d x))^{5/2}}{5 d}",1,"((2*(9*A*b - 2*a*B)*(a + b*Tan[c + d*x])^(7/2))/b + 14*B*Tan[c + d*x]*(a + b*Tan[c + d*x])^(7/2) - ((63*I)/2)*b*(A - I*B)*((2*(a + b*Tan[c + d*x])^(5/2))/5 + (2*(a - I*b)*(-3*(a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + Sqrt[a + b*Tan[c + d*x]]*(4*a - (3*I)*b + b*Tan[c + d*x])))/3) + ((63*I)/2)*b*(A + I*B)*((2*(a + b*Tan[c + d*x])^(5/2))/5 + (2*(a + I*b)*(-3*(a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Sqrt[a + b*Tan[c + d*x]]*(4*a + (3*I)*b + b*Tan[c + d*x])))/3))/(63*b*d)","A",1
333,1,258,213,1.6102957,"\int \tan (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{-7 i (B+i A) \left(\frac{2}{5} (a+b \tan (c+d x))^{5/2}+\frac{2}{3} (a-i b) \left(\sqrt{a+b \tan (c+d x)} (4 a+b \tan (c+d x)-3 i b)-3 (a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)\right)\right)-7 i (-B+i A) \left(\frac{2}{5} (a+b \tan (c+d x))^{5/2}+\frac{2}{3} (a+i b) \left(\sqrt{a+b \tan (c+d x)} (4 a+b \tan (c+d x)+3 i b)-3 (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)\right)\right)+\frac{4 B (a+b \tan (c+d x))^{7/2}}{b}}{14 d}","\frac{2 \left(a^2 A-2 a b B-A b^2\right) \sqrt{a+b \tan (c+d x)}}{d}+\frac{2 (a A-b B) (a+b \tan (c+d x))^{3/2}}{3 d}-\frac{(a-i b)^{5/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{(a+i b)^{5/2} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}+\frac{2 A (a+b \tan (c+d x))^{5/2}}{5 d}+\frac{2 B (a+b \tan (c+d x))^{7/2}}{7 b d}",1,"((4*B*(a + b*Tan[c + d*x])^(7/2))/b - (7*I)*(I*A + B)*((2*(a + b*Tan[c + d*x])^(5/2))/5 + (2*(a - I*b)*(-3*(a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + Sqrt[a + b*Tan[c + d*x]]*(4*a - (3*I)*b + b*Tan[c + d*x])))/3) - (7*I)*(I*A - B)*((2*(a + b*Tan[c + d*x])^(5/2))/5 + (2*(a + I*b)*(-3*(a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Sqrt[a + b*Tan[c + d*x]]*(4*a + (3*I)*b + b*Tan[c + d*x])))/3))/(14*d)","A",1
334,1,233,188,1.075803,"\int (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{i \left((A-i B) \left(\frac{2}{5} (a+b \tan (c+d x))^{5/2}+\frac{2}{3} (a-i b) \left(\sqrt{a+b \tan (c+d x)} (4 a+b \tan (c+d x)-3 i b)-3 (a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)\right)\right)-(A+i B) \left(\frac{2}{5} (a+b \tan (c+d x))^{5/2}+\frac{2}{3} (a+i b) \left(\sqrt{a+b \tan (c+d x)} (4 a+b \tan (c+d x)+3 i b)-3 (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)\right)\right)\right)}{2 d}","\frac{2 \left(a^2 B+2 a A b-b^2 B\right) \sqrt{a+b \tan (c+d x)}}{d}+\frac{2 (a B+A b) (a+b \tan (c+d x))^{3/2}}{3 d}-\frac{(a-i b)^{5/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{(a+i b)^{5/2} (-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}+\frac{2 B (a+b \tan (c+d x))^{5/2}}{5 d}",1,"((I/2)*((A - I*B)*((2*(a + b*Tan[c + d*x])^(5/2))/5 + (2*(a - I*b)*(-3*(a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + Sqrt[a + b*Tan[c + d*x]]*(4*a - (3*I)*b + b*Tan[c + d*x])))/3) - (A + I*B)*((2*(a + b*Tan[c + d*x])^(5/2))/5 + (2*(a + I*b)*(-3*(a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Sqrt[a + b*Tan[c + d*x]]*(4*a + (3*I)*b + b*Tan[c + d*x])))/3)))/d","A",1
335,1,177,182,1.152273,"\int \cot (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{2 \left(-3 a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)+3 b (2 a B+A b) \sqrt{a+b \tan (c+d x)}+\frac{3}{2} (a-i b)^{5/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+\frac{3}{2} (a+i b)^{5/2} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+b B (a+b \tan (c+d x))^{3/2}\right)}{3 d}","-\frac{2 a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{d}+\frac{2 b (2 a B+A b) \sqrt{a+b \tan (c+d x)}}{d}+\frac{(a-i b)^{5/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{(a+i b)^{5/2} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}+\frac{2 b B (a+b \tan (c+d x))^{3/2}}{3 d}",1,"(2*(-3*a^(5/2)*A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + (3*(a - I*b)^(5/2)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/2 + (3*(a + I*b)^(5/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/2 + 3*b*(A*b + 2*a*B)*Sqrt[a + b*Tan[c + d*x]] + b*B*(a + b*Tan[c + d*x])^(3/2)))/(3*d)","A",1
336,1,400,196,1.0522471,"\int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{2 b B \cot (c+d x) (a+b \tan (c+d x))^{3/2}}{d}+2 \left(-\frac{b (4 a B+A b) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{d}-2 \left(\frac{\left(a^2 A-6 a b B-2 A b^2\right) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}-\frac{-\frac{a^{5/2} (2 a B+5 A b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{4 d}+\frac{i \sqrt{a-i b} \left(-\frac{1}{4} a \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right)+\frac{1}{4} i a \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (-a+i b)}-\frac{i \sqrt{a+i b} \left(-\frac{1}{4} a \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right)-\frac{1}{4} i a \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (-a-i b)}}{a}\right)\right)","-\frac{a^{3/2} (2 a B+5 A b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{d}+\frac{b (a A+2 b B) \sqrt{a+b \tan (c+d x)}}{d}+\frac{(a-i b)^{5/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{(a+i b)^{5/2} (-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{a A \cot (c+d x) (a+b \tan (c+d x))^{3/2}}{d}",1,"(2*b*B*Cot[c + d*x]*(a + b*Tan[c + d*x])^(3/2))/d + 2*(-((b*(A*b + 4*a*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/d) - 2*(-((-1/4*(a^(5/2)*(5*A*b + 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d + (I*Sqrt[a - I*b]*((I/4)*a*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B) - (a*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B))/4)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((-a + I*b)*d) - (I*Sqrt[a + I*b]*((-1/4*I)*a*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B) - (a*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B))/4)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((-a - I*b)*d))/a) + ((a^2*A - 2*A*b^2 - 6*a*b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(4*d)))","B",1
337,1,448,220,2.4195971,"\int \cot ^3(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","-\frac{-\sqrt{a} \left(8 a^2 A-20 a b B-15 A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)+4 a^2 A \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+2 a^2 A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}+4 i a^2 B \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+4 a^2 B \cot (c+d x) \sqrt{a+b \tan (c+d x)}-4 A b^2 \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+4 (a-i b)^{5/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+8 i a A b \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+9 a A b \cot (c+d x) \sqrt{a+b \tan (c+d x)}-4 i b^2 B \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)-8 a b B \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{4 d}","\frac{\sqrt{a} \left(8 a^2 A-20 a b B-15 A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{4 d}-\frac{(a-i b)^{5/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{(a+i b)^{5/2} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{a (4 a B+7 A b) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}-\frac{a A \cot ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{2 d}",1,"-1/4*(-(Sqrt[a]*(8*a^2*A - 15*A*b^2 - 20*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]]) + 4*(a - I*b)^(5/2)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + 4*a^2*A*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + (8*I)*a*A*Sqrt[a + I*b]*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - 4*A*Sqrt[a + I*b]*b^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + (4*I)*a^2*Sqrt[a + I*b]*B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - 8*a*Sqrt[a + I*b]*b*B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - (4*I)*Sqrt[a + I*b]*b^2*B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 9*a*A*b*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]] + 4*a^2*B*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]] + 2*a^2*A*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/d","B",1
338,1,548,277,6.3916171,"\int \cot ^4(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","-\frac{2 b B \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d}-\frac{2}{3} \left(\frac{3 A b^2 \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}-\frac{2}{5} \left(\frac{\left(6 A b^2-5 a (a A-2 b B)\right) \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}-\frac{\frac{15 a \left(6 a^2 B+13 a A b-8 b^2 B\right) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{16 d}-\frac{\frac{45 a^2 \left(8 a^2 A-18 a b B-11 A b^2\right) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{16 d}-\frac{\frac{i \sqrt{a-i b} \left(-\frac{45}{2} a^3 \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right)+\frac{45}{2} i a^3 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (-a+i b)}-\frac{i \sqrt{a+i b} \left(-\frac{45}{2} a^3 \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right)-\frac{45}{2} i a^3 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (-a-i b)}-\frac{45 a^{5/2} \left(16 a^3 B+40 a^2 A b-30 a b^2 B-5 A b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{16 d}}{a}}{2 a}}{3 a}\right)\right)","\frac{\left(8 a^2 A-18 a b B-11 A b^2\right) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 d}+\frac{\left(16 a^3 B+40 a^2 A b-30 a b^2 B-5 A b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{8 \sqrt{a} d}-\frac{(a-i b)^{5/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{(a+i b)^{5/2} (-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{a (2 a B+3 A b) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}-\frac{a A \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d}",1,"(-2*b*B*Cot[c + d*x]^3*(a + b*Tan[c + d*x])^(3/2))/(3*d) - (2*((3*A*b^2*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]])/(5*d) - (2*(((6*A*b^2 - 5*a*(a*A - 2*b*B))*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]])/(4*d) - ((15*a*(13*a*A*b + 6*a^2*B - 8*b^2*B)*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(16*d) - (-(((-45*a^(5/2)*(40*a^2*A*b - 5*A*b^3 + 16*a^3*B - 30*a*b^2*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(16*d) + (I*Sqrt[a - I*b]*(((45*I)/2)*a^3*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B) - (45*a^3*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B))/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((-a + I*b)*d) - (I*Sqrt[a + I*b]*(((-45*I)/2)*a^3*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B) - (45*a^3*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B))/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((-a - I*b)*d))/a) + (45*a^2*(8*a^2*A - 11*A*b^2 - 18*a*b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(16*d))/(2*a))/(3*a)))/5))/3","A",1
339,1,622,342,6.5109384,"\int \cot ^5(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","-\frac{2 b B \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{5 d}-\frac{2}{5} \left(\frac{b (2 a B+5 A b) \cot ^4(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d}-\frac{2}{7} \left(-\frac{\left(35 a^2 A-72 a b B-40 A b^2\right) \cot ^4(c+d x) \sqrt{a+b \tan (c+d x)}}{16 d}-\frac{\frac{7 a \left(40 a^2 B+85 a A b-48 b^2 B\right) \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{24 d}-\frac{\frac{35 a^2 \left(48 a^2 A-104 a b B-59 A b^2\right) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{32 d}-\frac{-\frac{105 a^2 \left(64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{32 d}-\frac{\frac{i \sqrt{a-i b} \left(210 a^4 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right)+210 i a^4 \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (-a+i b)}-\frac{i \sqrt{a+i b} \left(210 a^4 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right)-210 i a^4 \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (-a-i b)}-\frac{105 a^{5/2} \left(128 a^4 A-320 a^3 b B-240 a^2 A b^2+40 a b^3 B-5 A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{32 d}}{a}}{2 a}}{3 a}}{4 a}\right)\right)","\frac{\left(48 a^2 A-104 a b B-59 A b^2\right) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{96 d}+\frac{\left(64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{64 a d}-\frac{\left(128 a^4 A-320 a^3 b B-240 a^2 A b^2+40 a b^3 B-5 A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{64 a^{3/2} d}+\frac{(a-i b)^{5/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{(a+i b)^{5/2} (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{a (8 a B+11 A b) \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}",1,"(-2*b*B*Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(3/2))/(5*d) - (2*((b*(5*A*b + 2*a*B)*Cot[c + d*x]^4*Sqrt[a + b*Tan[c + d*x]])/(7*d) - (2*(-1/16*((35*a^2*A - 40*A*b^2 - 72*a*b*B)*Cot[c + d*x]^4*Sqrt[a + b*Tan[c + d*x]])/d - ((7*a*(85*a*A*b + 40*a^2*B - 48*b^2*B)*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]])/(24*d) - ((35*a^2*(48*a^2*A - 59*A*b^2 - 104*a*b*B)*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(32*d) - (-(((-105*a^(5/2)*(128*a^4*A - 240*a^2*A*b^2 - 5*A*b^4 - 320*a^3*b*B + 40*a*b^3*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(32*d) + (I*Sqrt[a - I*b]*(210*a^4*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B) + (210*I)*a^4*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B))*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((-a + I*b)*d) - (I*Sqrt[a + I*b]*(210*a^4*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B) - (210*I)*a^4*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B))*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((-a - I*b)*d))/a) - (105*a^2*(144*a^2*A*b - 5*A*b^3 + 64*a^3*B - 88*a*b^2*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(32*d))/(2*a))/(3*a))/(4*a)))/7))/5","A",1
340,1,193,151,1.2966701,"\int (-a+b \tan (c+d x)) (a+b \tan (c+d x))^{5/2} \, dx","Integrate[(-a + b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(5/2),x]","\frac{\cos (c+d x) (a-b \tan (c+d x)) \left(2 b \sqrt{a+b \tan (c+d x)} \left(-4 a^2+2 a b \tan (c+d x)+b^2 \tan ^2(c+d x)-5 b^2\right)+5 i (a+i b) (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)-5 i (a+i b)^{5/2} (a-i b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)\right)}{5 d (a \cos (c+d x)-b \sin (c+d x))}","-\frac{2 b \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}{d}+\frac{2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\frac{(-b+i a) (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{(a+i b)^{5/2} (b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"(Cos[c + d*x]*(a - b*Tan[c + d*x])*((5*I)*(a - I*b)^(5/2)*(a + I*b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - (5*I)*(a - I*b)*(a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*b*Sqrt[a + b*Tan[c + d*x]]*(-4*a^2 - 5*b^2 + 2*a*b*Tan[c + d*x] + b^2*Tan[c + d*x]^2)))/(5*d*(a*Cos[c + d*x] - b*Sin[c + d*x]))","A",1
341,1,183,408,0.4651813,"\int (-a+b \tan (c+d x)) (a+b \tan (c+d x))^{3/2} \, dx","Integrate[(-a + b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(3/2),x]","\frac{(a-b \tan (c+d x)) \left(3 i \sqrt{a-i b} \left(a^2+b^2\right) \cos (c+d x) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)-3 i \sqrt{a+i b} \left(a^2+b^2\right) \cos (c+d x) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+2 b \sqrt{a+b \tan (c+d x)} (a \cos (c+d x)+b \sin (c+d x))\right)}{3 d (a \cos (c+d x)-b \sin (c+d x))}","-\frac{b \left(a^2+b^2\right) \log \left(-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \left(a^2+b^2\right) \log \left(\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \left(a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}+\frac{b \left(a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}+\frac{2 b (a+b \tan (c+d x))^{3/2}}{3 d}",1,"((a - b*Tan[c + d*x])*((3*I)*Sqrt[a - I*b]*(a^2 + b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]*Cos[c + d*x] - (3*I)*Sqrt[a + I*b]*(a^2 + b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]*Cos[c + d*x] + 2*b*(a*Cos[c + d*x] + b*Sin[c + d*x])*Sqrt[a + b*Tan[c + d*x]]))/(3*d*(a*Cos[c + d*x] - b*Sin[c + d*x]))","C",1
342,1,157,422,0.2457051,"\int (-a+b \tan (c+d x)) \sqrt{a+b \tan (c+d x)} \, dx","Integrate[(-a + b*Tan[c + d*x])*Sqrt[a + b*Tan[c + d*x]],x]","\frac{\cos (c+d x) (a-b \tan (c+d x)) \left(2 b \sqrt{a+b \tan (c+d x)}+i \sqrt{a-i b} (a+i b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)-i (a-i b) \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)\right)}{d (a \cos (c+d x)-b \sin (c+d x))}","\frac{b \sqrt{a^2+b^2} \log \left(-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \sqrt{a^2+b^2} \log \left(\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}+\frac{b \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}+\frac{2 b \sqrt{a+b \tan (c+d x)}}{d}",1,"(Cos[c + d*x]*(a - b*Tan[c + d*x])*(I*Sqrt[a - I*b]*(a + I*b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - I*(a - I*b)*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*b*Sqrt[a + b*Tan[c + d*x]]))/(d*(a*Cos[c + d*x] - b*Sin[c + d*x]))","C",1
343,1,170,213,4.3194376,"\int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left(8 a^2 B+b (5 A b-4 a B) \tan (c+d x)-10 a A b+3 b^2 B \tan ^2(c+d x)-15 b^2 B\right)}{b^3}+\frac{15 (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}+\frac{15 (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}}{15 d}","-\frac{2 \left(-8 a^2 B+10 a A b+15 b^2 B\right) \sqrt{a+b \tan (c+d x)}}{15 b^3 d}+\frac{2 (5 A b-4 a B) \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{15 b^2 d}+\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}+\frac{(A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}+\frac{2 B \tan ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{5 b d}",1,"((15*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] + (15*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a + I*b] + (2*Sqrt[a + b*Tan[c + d*x]]*(-10*a*A*b + 8*a^2*B - 15*b^2*B + b*(5*A*b - 4*a*B)*Tan[c + d*x] + 3*b^2*B*Tan[c + d*x]^2))/b^3)/(15*d)","A",1
344,1,139,166,1.5866916,"\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)} (-2 a B+3 A b+b B \tan (c+d x))}{b^2}+\frac{3 (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}+\frac{3 (B-i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}}{3 d}","\frac{2 (3 A b-2 a B) \sqrt{a+b \tan (c+d x)}}{3 b^2 d}+\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}-\frac{(-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}+\frac{2 B \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{3 b d}",1,"((3*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] + (3*((-I)*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a + I*b] + (2*Sqrt[a + b*Tan[c + d*x]]*(3*A*b - 2*a*B + b*B*Tan[c + d*x]))/b^2)/(3*d)","A",1
345,1,118,124,0.5296759,"\int \frac{\tan (c+d x) (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]","-\frac{\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}+\frac{(A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}-\frac{2 B \sqrt{a+b \tan (c+d x)}}{b}}{d}","-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}-\frac{(A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}+\frac{2 B \sqrt{a+b \tan (c+d x)}}{b d}",1,"-((((A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] + ((A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a + I*b] - (2*B*Sqrt[a + b*Tan[c + d*x]])/b)/d)","A",1
346,1,101,102,0.1062967,"\int \frac{A+B \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(A + B*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]],x]","\frac{i \left(\frac{(A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}\right)}{d}","\frac{(-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}-\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}",1,"(I*(-(((A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b]) + ((A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a + I*b]))/d","A",1
347,1,170,131,0.7614628,"\int \frac{\cot (c+d x) (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]","-\frac{\frac{\frac{\left(\sqrt{-b^2} B-A b\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{a-\sqrt{-b^2}}}-\frac{\left(A b+\sqrt{-b^2} B\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{a+\sqrt{-b^2}}}}{b}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}}{d}","\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}+\frac{(A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}-\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"-(((2*A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + (((-(A*b) + Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/Sqrt[a - Sqrt[-b^2]] - ((A*b + Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/Sqrt[a + Sqrt[-b^2]])/b)/d)","A",1
348,1,201,169,2.8470436,"\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\frac{b (A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{\left(A \sqrt{-b^2}+b B\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{a-\sqrt{-b^2}}}-\frac{\left(A \sqrt{-b^2}-b B\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{a+\sqrt{-b^2}}}-\frac{A b \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{a}}{b d}","\frac{(A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}-\frac{(-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}-\frac{A \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{a d}",1,"((b*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/a^(3/2) + ((A*Sqrt[-b^2] + b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/Sqrt[a - Sqrt[-b^2]] - ((A*Sqrt[-b^2] - b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/Sqrt[a + Sqrt[-b^2]] - (A*b*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/a)/(b*d)","A",1
349,1,362,224,6.2753404,"\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]","\frac{2 b^3 \left(-\frac{3 A \left(\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{\cot (c+d x) \sqrt{a+b \tan (c+d x)}}{a b}\right)}{8 a b}+\frac{B \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{2 a^{3/2} b^2}+\frac{A \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} b^3}-\frac{A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{4 a b^3}-\frac{b \left(A \sqrt{-b^2}-b B\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{2 \left(-b^2\right)^{5/2} \sqrt{a+\sqrt{-b^2}}}-\frac{\left(A \sqrt{-b^2}+b B\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{2 b^3 \sqrt{-b^2} \sqrt{a-\sqrt{-b^2}}}-\frac{B \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{2 a b^3}\right)}{d}","\frac{(3 A b-4 a B) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 a^2 d}+\frac{\left(8 a^2 A+4 a b B-3 A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{5/2} d}-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}-\frac{(A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}-\frac{A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 a d}",1,"(2*b^3*((A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*b^3) + (B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(2*a^(3/2)*b^2) - ((A*Sqrt[-b^2] + b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(2*b^3*Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]) - (b*(A*Sqrt[-b^2] - b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(2*(-b^2)^(5/2)*Sqrt[a + Sqrt[-b^2]]) - (B*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(2*a*b^3) - (A*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(4*a*b^3) - (3*A*(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]]/a^(3/2) - (Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(a*b)))/(8*a*b)))/d","A",1
350,1,300,264,3.5216325,"\int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","\frac{\frac{3 i (a A+b B) \left((a+i b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)-(a-i b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)\right)}{\left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{2 \left(-8 a^2 B+6 a A b+3 b^2 B\right)}{b^2 \sqrt{a+b \tan (c+d x)}}+\frac{2 (3 A b-4 a B) \tan (c+d x)}{b \sqrt{a+b \tan (c+d x)}}+3 i A \left(\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}\right)+\frac{2 B \tan ^2(c+d x)}{\sqrt{a+b \tan (c+d x)}}}{3 b d}","\frac{2 a (A b-a B) \tan ^2(c+d x)}{b d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{2 \left(-4 a^2 B+3 a A b-b^2 B\right) \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{3 b^2 d \left(a^2+b^2\right)}+\frac{2 \left(-8 a^3 B+6 a^2 A b-5 a b^2 B+3 A b^3\right) \sqrt{a+b \tan (c+d x)}}{3 b^3 d \left(a^2+b^2\right)}+\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}+\frac{(A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"((3*I)*A*(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/Sqrt[a - I*b] - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/Sqrt[a + I*b]) + (2*(6*a*A*b - 8*a^2*B + 3*b^2*B))/(b^2*Sqrt[a + b*Tan[c + d*x]]) + ((3*I)*(a*A + b*B)*((a + I*b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)] - (a - I*b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)]))/((a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]]) + (2*(3*A*b - 4*a*B)*Tan[c + d*x])/(b*Sqrt[a + b*Tan[c + d*x]]) + (2*B*Tan[c + d*x]^2)/Sqrt[a + b*Tan[c + d*x]])/(3*b*d)","C",1
351,1,248,167,1.3385947,"\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","\frac{\frac{(A b-a B) \left((b-i a) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)+(b+i a) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)\right)}{\left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{4 a B-2 A b}{b \sqrt{a+b \tan (c+d x)}}+\frac{2 B \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}}+i B \left(\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}\right)}{b d}","-\frac{2 a^2 (A b-a B)}{b^2 d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}-\frac{(-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}+\frac{2 B \sqrt{a+b \tan (c+d x)}}{b^2 d}",1,"(I*B*(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/Sqrt[a - I*b] - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/Sqrt[a + I*b]) + (-2*A*b + 4*a*B)/(b*Sqrt[a + b*Tan[c + d*x]]) + ((A*b - a*B)*(((-I)*a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)] + (I*a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)]))/((a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]]) + (2*B*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]])/(b*d)","C",1
352,1,229,141,1.4292058,"\int \frac{\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","\frac{\frac{b \left(A \left(b^2-a \sqrt{-b^2}\right)-b B \left(a+\sqrt{-b^2}\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{-b^2} \sqrt{a-\sqrt{-b^2}}}-\frac{b \left(A \left(a \sqrt{-b^2}+b^2\right)+b B \left(\sqrt{-b^2}-a\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{-b^2} \sqrt{a+\sqrt{-b^2}}}+\frac{2 a (A b-a B)}{\sqrt{a+b \tan (c+d x)}}}{b d \left(a^2+b^2\right)}","\frac{2 a (A b-a B)}{b d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}-\frac{(A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"((b*(A*(b^2 - a*Sqrt[-b^2]) - b*(a + Sqrt[-b^2])*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]) - (b*(A*(b^2 + a*Sqrt[-b^2]) + b*(-a + Sqrt[-b^2])*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a + Sqrt[-b^2]]) + (2*a*(A*b - a*B))/Sqrt[a + b*Tan[c + d*x]])/(b*(a^2 + b^2)*d)","A",1
353,1,113,138,0.2108415,"\int \frac{A+B \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^(3/2),x]","\frac{i \left(\frac{(A-i B) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)}{a-i b}-\frac{(A+i B) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)}{a+i b}\right)}{d \sqrt{a+b \tan (c+d x)}}","-\frac{2 (A b-a B)}{d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}+\frac{(-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"(I*(((A - I*B)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)])/(a - I*b) - ((A + I*B)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)])/(a + I*b)))/(d*Sqrt[a + b*Tan[c + d*x]])","C",1
354,1,186,171,1.2452342,"\int \frac{\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","\frac{-\frac{2 A \left(a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{2 b (A b-a B)}{\sqrt{a+b \tan (c+d x)}}+\frac{a (a+i b) (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}+\frac{a (a-i b) (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}}{a d \left(a^2+b^2\right)}","-\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{2 b (A b-a B)}{a d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}+\frac{(A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"((-2*A*(a^2 + b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + (a*(a + I*b)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] + (a*(a - I*b)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a + I*b] + (2*b*(A*b - a*B))/Sqrt[a + b*Tan[c + d*x]])/(a*(a^2 + b^2)*d)","A",1
355,1,208,219,3.8514297,"\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","\frac{-\frac{b \left(a^2 A-2 a b B+3 A b^2\right)}{\left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+a^2 \left(\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{(a-i b)^{3/2}}+\frac{(B-i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{(a+i b)^{3/2}}\right)+\frac{(3 A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{a A \cot (c+d x)}{\sqrt{a+b \tan (c+d x)}}}{a^2 d}","\frac{(3 A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} d}-\frac{b \left(a^2 A-2 a b B+3 A b^2\right)}{a^2 d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}-\frac{(-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}-\frac{A \cot (c+d x)}{a d \sqrt{a+b \tan (c+d x)}}",1,"(((3*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + a^2*(((I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(a - I*b)^(3/2) + (((-I)*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(a + I*b)^(3/2)) - (b*(a^2*A + 3*A*b^2 - 2*a*b*B))/((a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]]) - (a*A*Cot[c + d*x])/Sqrt[a + b*Tan[c + d*x]])/(a^2*d)","A",1
356,1,409,285,6.2535331,"\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","-\frac{A \cot ^2(c+d x)}{2 a d \sqrt{a+b \tan (c+d x)}}-\frac{-\frac{(5 A b-4 a B) \cot (c+d x)}{2 a d \sqrt{a+b \tan (c+d x)}}-\frac{\frac{2 \left(\frac{1}{4} b^2 \left(-8 a^2 A-12 a b B+15 A b^2\right)-a \left(-2 a^2 b B-\frac{3}{4} a b (5 A b-4 a B)\right)\right)}{a d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{2 \left(\frac{i \sqrt{a-i b} \left(a^3 (A b-a B)-i a^3 (a A+b B)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (-a+i b)}-\frac{i \sqrt{a+i b} \left(a^3 (A b-a B)+i a^3 (a A+b B)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (-a-i b)}+\frac{\left(a^2+b^2\right) \left(8 a^2 A+12 a b B-15 A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{4 \sqrt{a} d}\right)}{a \left(a^2+b^2\right)}}{a}}{2 a}","\frac{(5 A b-4 a B) \cot (c+d x)}{4 a^2 d \sqrt{a+b \tan (c+d x)}}+\frac{\left(8 a^2 A+12 a b B-15 A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{7/2} d}+\frac{b \left(-4 a^3 B+7 a^2 A b-12 a b^2 B+15 A b^3\right)}{4 a^3 d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}-\frac{(A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}-\frac{A \cot ^2(c+d x)}{2 a d \sqrt{a+b \tan (c+d x)}}",1,"-1/2*(A*Cot[c + d*x]^2)/(a*d*Sqrt[a + b*Tan[c + d*x]]) - (-1/2*((5*A*b - 4*a*B)*Cot[c + d*x])/(a*d*Sqrt[a + b*Tan[c + d*x]]) - ((2*(((a^2 + b^2)*(8*a^2*A - 15*A*b^2 + 12*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*Sqrt[a]*d) + (I*Sqrt[a - I*b]*(a^3*(A*b - a*B) - I*a^3*(a*A + b*B))*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((-a + I*b)*d) - (I*Sqrt[a + I*b]*(a^3*(A*b - a*B) + I*a^3*(a*A + b*B))*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((-a - I*b)*d)))/(a*(a^2 + b^2)) + (2*((b^2*(-8*a^2*A + 15*A*b^2 - 12*a*b*B))/4 - a*(-2*a^2*b*B - (3*a*b*(5*A*b - 4*a*B))/4)))/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]))/a)/(2*a)","A",1
357,1,450,371,6.349186,"\int \frac{\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]","\frac{2 B \tan ^3(c+d x)}{3 b d (a+b \tan (c+d x))^{3/2}}+\frac{2 \left(\frac{3 (A b-2 a B) \tan ^2(c+d x)}{b d (a+b \tan (c+d x))^{3/2}}+\frac{2 \left(\frac{3 \left(-8 a^2 B+4 a A b+b^2 B\right) \tan (c+d x)}{2 b d (a+b \tan (c+d x))^{3/2}}-\frac{3 \left(-\frac{2 \left(-16 a^3 B+8 a^2 A b+2 a b^2 B+A b^3\right)}{3 b (a+b \tan (c+d x))^{3/2}}+\frac{2 \left(\frac{\left(\frac{3}{2} a b^4 B-\frac{3 A b^5}{2}\right) \left(\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)}{3 (-b+i a) (a+b \tan (c+d x))^{3/2}}-\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)}{3 (b+i a) (a+b \tan (c+d x))^{3/2}}\right)}{b}-\frac{3}{2} b^3 B \left(\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)}{(-b+i a) \sqrt{a+b \tan (c+d x)}}-\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)}{(b+i a) \sqrt{a+b \tan (c+d x)}}\right)\right)}{3 b}\right)}{4 b d}\right)}{b}\right)}{3 b}","\frac{2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left(-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right) \tan ^2(c+d x)}{b^2 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}-\frac{2 \left(-8 a^4 B+4 a^3 A b-15 a^2 b^2 B+10 a A b^3-b^4 B\right) \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{3 b^3 d \left(a^2+b^2\right)^2}+\frac{2 \left(-16 a^5 B+8 a^4 A b-30 a^3 b^2 B+17 a^2 A b^3-8 a b^4 B+3 A b^5\right) \sqrt{a+b \tan (c+d x)}}{3 b^4 d \left(a^2+b^2\right)^2}-\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}+\frac{(-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"(2*B*Tan[c + d*x]^3)/(3*b*d*(a + b*Tan[c + d*x])^(3/2)) + (2*((3*(A*b - 2*a*B)*Tan[c + d*x]^2)/(b*d*(a + b*Tan[c + d*x])^(3/2)) + (2*((3*(4*a*A*b - 8*a^2*B + b^2*B)*Tan[c + d*x])/(2*b*d*(a + b*Tan[c + d*x])^(3/2)) - (3*((-2*(8*a^2*A*b + A*b^3 - 16*a^3*B + 2*a*b^2*B))/(3*b*(a + b*Tan[c + d*x])^(3/2)) + (2*((((-3*A*b^5)/2 + (3*a*b^4*B)/2)*(-1/3*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a - I*b)]/((I*a + b)*(a + b*Tan[c + d*x])^(3/2)) + Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a + I*b)]/(3*(I*a - b)*(a + b*Tan[c + d*x])^(3/2))))/b - (3*b^3*B*(-(Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)]/((I*a + b)*Sqrt[a + b*Tan[c + d*x]])) + Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)]/((I*a - b)*Sqrt[a + b*Tan[c + d*x]])))/2))/(3*b)))/(4*b*d)))/b))/(3*b)","C",1
358,1,309,261,3.5868213,"\int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]","-\frac{-2 (a-i b) (a+i b) \left(8 a^2 B-2 a A b+b^2 B\right)-b^2 (a A+b B) \left(i (a+i b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)-(b+i a) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)\right)+3 A b^2 (a+b \tan (c+d x)) \left(i (a+i b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)-(b+i a) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)\right)-6 b (a-i b) (a+i b) (4 a B-A b) \tan (c+d x)-6 b^2 B (a-i b) (a+i b) \tan ^2(c+d x)}{3 b^3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}","\frac{2 a (A b-a B) \tan ^2(c+d x)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}-\frac{2 \left(-4 a^2 B+a A b-3 b^2 B\right) \sqrt{a+b \tan (c+d x)}}{3 b^3 d \left(a^2+b^2\right)}-\frac{2 a^2 \left(-4 a^3 B+a^2 A b-10 a b^2 B+7 A b^3\right)}{3 b^3 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}+\frac{(A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"-1/3*(-2*(a - I*b)*(a + I*b)*(-2*a*A*b + 8*a^2*B + b^2*B) - b^2*(a*A + b*B)*(I*(a + I*b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a - I*b)] - (I*a + b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a + I*b)]) - 6*(a - I*b)*(a + I*b)*b*(-(A*b) + 4*a*B)*Tan[c + d*x] - 6*(a - I*b)*(a + I*b)*b^2*B*Tan[c + d*x]^2 + 3*A*b^2*(I*(a + I*b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)] - (I*a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)])*(a + b*Tan[c + d*x]))/(b^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2))","C",1
359,1,260,198,1.0752869,"\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]","-\frac{b (A b-a B) \left(i (a+i b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)-(b+i a) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)\right)+2 (a-i b) (a+i b) (2 a B+A b)+3 b B (a+b \tan (c+d x)) \left(i (a+i b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)-(b+i a) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)\right)+6 b B (a-i b) (a+i b) \tan (c+d x)}{3 b^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}","-\frac{2 a^2 (A b-a B)}{3 b^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left(2 A b^3-a B \left(a^2+3 b^2\right)\right)}{b^2 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}-\frac{(-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"-1/3*(2*(a - I*b)*(a + I*b)*(A*b + 2*a*B) + b*(A*b - a*B)*(I*(a + I*b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a - I*b)] - (I*a + b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a + I*b)]) + 6*(a - I*b)*(a + I*b)*b*B*Tan[c + d*x] + 3*b*B*(I*(a + I*b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)] - (I*a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)])*(a + b*Tan[c + d*x]))/(b^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2))","C",1
360,1,325,188,3.657898,"\int \frac{\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]","\frac{\frac{2 a \left(a^2+b^2\right) (A b-a B)}{(a+b \tan (c+d x))^{3/2}}+\frac{6 b \left(a^2 A+2 a b B-A b^2\right)}{\sqrt{a+b \tan (c+d x)}}+\frac{3 b \left(-\left(a^2 \left(A \sqrt{-b^2}+b B\right)\right)+2 a b \left(A b-\sqrt{-b^2} B\right)+b^2 \left(A \sqrt{-b^2}+b B\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{-b^2} \sqrt{a-\sqrt{-b^2}}}-\frac{3 b \left(a^2 A \sqrt{-b^2}-a^2 b B+2 a A b^2+2 a \sqrt{-b^2} b B+A \left(-b^2\right)^{3/2}+b^3 B\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{-b^2} \sqrt{a+\sqrt{-b^2}}}}{3 b d \left(a^2+b^2\right)^2}","\frac{2 a (A b-a B)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{2 \left(a^2 A+2 a b B-A b^2\right)}{d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}-\frac{(A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"((3*b*(-(a^2*(A*Sqrt[-b^2] + b*B)) + b^2*(A*Sqrt[-b^2] + b*B) + 2*a*b*(A*b - Sqrt[-b^2]*B))*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]) - (3*b*(2*a*A*b^2 + a^2*A*Sqrt[-b^2] + A*(-b^2)^(3/2) - a^2*b*B + b^3*B + 2*a*b*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a + Sqrt[-b^2]]) + (2*a*(a^2 + b^2)*(A*b - a*B))/(a + b*Tan[c + d*x])^(3/2) + (6*b*(a^2*A - A*b^2 + 2*a*b*B))/Sqrt[a + b*Tan[c + d*x]])/(3*b*(a^2 + b^2)^2*d)","A",1
361,1,115,185,0.2071073,"\int \frac{A+B \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^(5/2),x]","-\frac{i \left(\frac{(A+i B) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)}{a+i b}-\frac{(A-i B) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)}{a-i b}\right)}{3 d (a+b \tan (c+d x))^{3/2}}","-\frac{2 (A b-a B)}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}-\frac{2 \left(a^2 (-B)+2 a A b+b^2 B\right)}{d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}-\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}+\frac{(-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"((-1/3*I)*(-(((A - I*B)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a - I*b)])/(a - I*b)) + ((A + I*B)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a + I*b)])/(a + I*b)))/(d*(a + b*Tan[c + d*x])^(3/2))","C",1
362,1,242,224,5.2088494,"\int \frac{\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]","\frac{2 \left(-\frac{3 A \left(a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{3 b \left(-2 a^3 B+3 a^2 A b+A b^3\right)}{a \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{b (A b-a B)}{(a+b \tan (c+d x))^{3/2}}+\frac{3 a (a+i b) (A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{2 (a-i b)^{3/2}}+\frac{3 a (a-i b) (A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{2 (a+i b)^{3/2}}\right)}{3 a d \left(a^2+b^2\right)}","-\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{2 b (A b-a B)}{3 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left(-2 a^3 B+3 a^2 A b+A b^3\right)}{a^2 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}+\frac{(A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"(2*((-3*A*(a^2 + b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/a^(3/2) + (3*a*(a + I*b)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(2*(a - I*b)^(3/2)) + (3*a*(a - I*b)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(2*(a + I*b)^(3/2)) + (b*(A*b - a*B))/(a + b*Tan[c + d*x])^(3/2) + (3*b*(3*a^2*A*b + A*b^3 - 2*a^3*B))/(a*(a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]])))/(3*a*(a^2 + b^2)*d)","A",1
363,1,306,289,5.228268,"\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]","\frac{\frac{b \left(-3 a^2 A+2 a b B-5 A b^2\right)}{\left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{3 \left(\frac{a^3 (a+i b)^2 (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}+\frac{a^3 (a-i b)^2 (B-i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}+\frac{\left(a^2+b^2\right)^2 (5 A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{b \left(a^4 A-6 a^3 b B+10 a^2 A b^2-2 a b^3 B+5 A b^4\right)}{\sqrt{a+b \tan (c+d x)}}\right)}{a \left(a^2+b^2\right)^2}-\frac{3 a A \cot (c+d x)}{(a+b \tan (c+d x))^{3/2}}}{3 a^2 d}","\frac{(5 A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{7/2} d}-\frac{b \left(3 a^2 A-2 a b B+5 A b^2\right)}{3 a^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}-\frac{b \left(a^4 A-6 a^3 b B+10 a^2 A b^2-2 a b^3 B+5 A b^4\right)}{a^3 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}-\frac{(-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}",1,"((b*(-3*a^2*A - 5*A*b^2 + 2*a*b*B))/((a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) - (3*a*A*Cot[c + d*x])/(a + b*Tan[c + d*x])^(3/2) + (3*(((a^2 + b^2)^2*(5*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + (a^3*(a + I*b)^2*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] + (a^3*(a - I*b)^2*((-I)*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a + I*b] - (b*(a^4*A + 10*a^2*A*b^2 + 5*A*b^4 - 6*a^3*b*B - 2*a*b^3*B))/Sqrt[a + b*Tan[c + d*x]]))/(a*(a^2 + b^2)^2))/(3*a^2*d)","A",1
364,1,593,364,6.3269956,"\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]","-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac{-\frac{(7 A b-4 a B) \cot (c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac{\frac{2 \left(\frac{1}{4} b^2 \left(-8 a^2 A-20 a b B+35 A b^2\right)-a \left(-2 a^2 b B-\frac{5}{4} a b (7 A b-4 a B)\right)\right)}{3 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{2 \left(\frac{2 \left(-\frac{3}{8} b^2 \left(a^2+b^2\right) \left(8 a^2 A+20 a b B-35 A b^2\right)-a \left(3 a^3 b (A b-a B)-\frac{3}{8} a b \left(-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right)\right)\right)}{a d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{2 \left(\frac{3 \left(a^2+b^2\right)^2 \left(8 a^2 A+20 a b B-35 A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{8 \sqrt{a} d}+\frac{i \sqrt{a-i b} \left(\frac{3}{2} a^4 \left(a^2 (-B)+2 a A b+b^2 B\right)-\frac{3}{2} i a^4 \left(a^2 A+2 a b B-A b^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (-a+i b)}-\frac{i \sqrt{a+i b} \left(\frac{3}{2} a^4 \left(a^2 (-B)+2 a A b+b^2 B\right)+\frac{3}{2} i a^4 \left(a^2 A+2 a b B-A b^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (-a-i b)}\right)}{a \left(a^2+b^2\right)}\right)}{3 a \left(a^2+b^2\right)}}{a}}{2 a}","\frac{(7 A b-4 a B) \cot (c+d x)}{4 a^2 d (a+b \tan (c+d x))^{3/2}}+\frac{\left(8 a^2 A+20 a b B-35 A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{9/2} d}+\frac{b \left(-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right)}{12 a^3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{b \left(-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}-\frac{(A+i B) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}",1,"-1/2*(A*Cot[c + d*x]^2)/(a*d*(a + b*Tan[c + d*x])^(3/2)) - (-1/2*((7*A*b - 4*a*B)*Cot[c + d*x])/(a*d*(a + b*Tan[c + d*x])^(3/2)) - ((2*((b^2*(-8*a^2*A + 35*A*b^2 - 20*a*b*B))/4 - a*(-2*a^2*b*B - (5*a*b*(7*A*b - 4*a*B))/4)))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (2*((2*((3*(a^2 + b^2)^2*(8*a^2*A - 35*A*b^2 + 20*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(8*Sqrt[a]*d) + (I*Sqrt[a - I*b]*(((-3*I)/2)*a^4*(a^2*A - A*b^2 + 2*a*b*B) + (3*a^4*(2*a*A*b - a^2*B + b^2*B))/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((-a + I*b)*d) - (I*Sqrt[a + I*b]*(((3*I)/2)*a^4*(a^2*A - A*b^2 + 2*a*b*B) + (3*a^4*(2*a*A*b - a^2*B + b^2*B))/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((-a - I*b)*d)))/(a*(a^2 + b^2)) + (2*((-3*b^2*(a^2 + b^2)*(8*a^2*A - 35*A*b^2 + 20*a*b*B))/8 - a*(3*a^3*b*(A*b - a*B) - (3*a*b*(27*a^2*A*b + 35*A*b^3 - 12*a^3*B - 20*a*b^2*B))/8)))/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])))/(3*a*(a^2 + b^2)))/a)/(2*a)","A",1
365,1,88,362,0.0974888,"\int \frac{a B+b B \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]],x]","-\frac{i B \left(\sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)-\sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)\right)}{d}","\frac{b B \log \left(-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b B \log \left(\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b B \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{b B \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}",1,"((-I)*B*(Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]))/d","C",1
366,1,88,406,0.0664698,"\int \frac{a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(a + b*Tan[c + d*x])^(3/2),x]","-\frac{i B \left(\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}\right)}{d}","-\frac{b B \log \left(-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b B \log \left(\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b B \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}-\frac{b B \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}",1,"((-I)*B*(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/Sqrt[a - I*b] - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/Sqrt[a + I*b]))/d","C",1
367,1,112,119,0.1539894,"\int \frac{\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Cot[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","\frac{B \left(-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}\right)}{d}","-\frac{2 B \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}+\frac{B \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}+\frac{B \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}",1,"(B*((-2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/Sqrt[a - I*b] + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/Sqrt[a + I*b]))/d","A",1
368,1,106,123,0.1482324,"\int \frac{a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(a + b*Tan[c + d*x])^(5/2),x]","\frac{B \left(i (a+i b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)+(-b-i a) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)\right)}{d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}","-\frac{2 b B}{d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{i B \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}+\frac{i B \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"(B*(I*(a + I*b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)] + ((-I)*a - b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)]))/((a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])","C",1
369,1,166,154,1.2248639,"\int \frac{\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(Cot[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]","\frac{B \left(-\frac{2 \left(a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{2 b^2}{\sqrt{a+b \tan (c+d x)}}+\frac{a (a+i b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}+\frac{a (a-i b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}\right)}{a d \left(a^2+b^2\right)}","-\frac{2 B \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{2 b^2 B}{a d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{B \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}+\frac{B \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"(B*((-2*(a^2 + b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + (a*(a + I*b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] + (a*(a - I*b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a + I*b] + (2*b^2)/Sqrt[a + b*Tan[c + d*x]]))/(a*(a^2 + b^2)*d)","A",1
370,1,109,102,0.177968,"\int \frac{-a+b \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(-a + b*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]],x]","\frac{i \left((a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)-(a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)\right)}{d \sqrt{a-i b} \sqrt{a+i b}}","\frac{(-b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}-\frac{(b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}",1,"(I*((a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - (a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]))/(Sqrt[a - I*b]*Sqrt[a + I*b]*d)","A",1
371,1,154,132,0.369217,"\int \frac{-a+b \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(-a + b*Tan[c + d*x])/(a + b*Tan[c + d*x])^(3/2),x]","-\frac{i \cos (c+d x) (a-b \tan (c+d x)) \left((a+i b)^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)-(a-i b)^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)\right)}{d (a-i b) (a+i b) \sqrt{a+b \tan (c+d x)} (a \cos (c+d x)-b \sin (c+d x))}","\frac{4 a b}{d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{(-b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}-\frac{(b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"((-I)*Cos[c + d*x]*((a + I*b)^2*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)] - (a - I*b)^2*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)])*(a - b*Tan[c + d*x]))/((a - I*b)*(a + I*b)*d*(a*Cos[c + d*x] - b*Sin[c + d*x])*Sqrt[a + b*Tan[c + d*x]])","C",1
372,1,156,174,0.3487247,"\int \frac{-a+b \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(-a + b*Tan[c + d*x])/(a + b*Tan[c + d*x])^(5/2),x]","-\frac{i \cos (c+d x) (a-b \tan (c+d x)) \left((a+i b)^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)-(a-i b)^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)\right)}{3 d (a-i b) (a+i b) (a+b \tan (c+d x))^{3/2} (a \cos (c+d x)-b \sin (c+d x))}","\frac{2 b \left(3 a^2-b^2\right)}{d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{4 a b}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{(-b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}-\frac{(b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"((-1/3*I)*Cos[c + d*x]*((a + I*b)^2*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a - I*b)] - (a - I*b)^2*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a + I*b)])*(a - b*Tan[c + d*x]))/((a - I*b)*(a + I*b)*d*(a*Cos[c + d*x] - b*Sin[c + d*x])*(a + b*Tan[c + d*x])^(3/2))","C",1
373,1,70,45,1.6683086,"\int \frac{1+i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(1 + I*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]],x]","-\frac{2 i \tanh ^{-1}\left(\frac{\sqrt{a-\frac{i b \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}","-\frac{2 i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}",1,"((-2*I)*ArcTanh[Sqrt[a - (I*b*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d)","A",1
374,1,45,45,0.0530521,"\int \frac{1-i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(1 - I*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]],x]","\frac{2 i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}","\frac{2 i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}",1,"((2*I)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)","A",1
375,1,69,30,0.1797356,"\int \frac{3+\tan (x)}{\sqrt{4+3 \tan (x)}} \, dx","Integrate[(3 + Tan[x])/Sqrt[4 + 3*Tan[x]],x]","\left(\frac{1}{5}-\frac{3 i}{5}\right) \sqrt{4-3 i} \tanh ^{-1}\left(\frac{\sqrt{3 \tan (x)+4}}{\sqrt{4-3 i}}\right)+\left(\frac{1}{5}+\frac{3 i}{5}\right) \sqrt{4+3 i} \tanh ^{-1}\left(\frac{\sqrt{3 \tan (x)+4}}{\sqrt{4+3 i}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{1-3 \tan (x)}{\sqrt{2} \sqrt{3 \tan (x)+4}}\right)",1,"(1/5 - (3*I)/5)*Sqrt[4 - 3*I]*ArcTanh[Sqrt[4 + 3*Tan[x]]/Sqrt[4 - 3*I]] + (1/5 + (3*I)/5)*Sqrt[4 + 3*I]*ArcTanh[Sqrt[4 + 3*Tan[x]]/Sqrt[4 + 3*I]]","C",1
376,1,65,27,0.1289458,"\int \frac{1-3 \tan (x)}{\sqrt{4+3 \tan (x)}} \, dx","Integrate[(1 - 3*Tan[x])/Sqrt[4 + 3*Tan[x]],x]","\frac{1}{5} \left((3+i) \sqrt{4-3 i} \tanh ^{-1}\left(\frac{\sqrt{3 \tan (x)+4}}{\sqrt{4-3 i}}\right)+(3-i) \sqrt{4+3 i} \tanh ^{-1}\left(\frac{\sqrt{3 \tan (x)+4}}{\sqrt{4+3 i}}\right)\right)","\sqrt{2} \tanh ^{-1}\left(\frac{\tan (x)+3}{\sqrt{2} \sqrt{3 \tan (x)+4}}\right)",1,"((3 + I)*Sqrt[4 - 3*I]*ArcTanh[Sqrt[4 + 3*Tan[x]]/Sqrt[4 - 3*I]] + (3 - I)*Sqrt[4 + 3*I]*ArcTanh[Sqrt[4 + 3*Tan[x]]/Sqrt[4 + 3*I]])/5","C",1
377,1,75,85,0.0952766,"\int \frac{4-3 \tan (a+b x)}{\sqrt{4+3 \tan (a+b x)}} \, dx","Integrate[(4 - 3*Tan[a + b*x])/Sqrt[4 + 3*Tan[a + b*x]],x]","\frac{(3-4 i) \tanh ^{-1}\left(\frac{\sqrt{3 \tan (a+b x)+4}}{\sqrt{4-3 i}}\right)}{\sqrt{4-3 i} b}+\frac{(3+4 i) \tanh ^{-1}\left(\frac{\sqrt{3 \tan (a+b x)+4}}{\sqrt{4+3 i}}\right)}{\sqrt{4+3 i} b}","\frac{13 \tanh ^{-1}\left(\frac{\tan (a+b x)+3}{\sqrt{2} \sqrt{3 \tan (a+b x)+4}}\right)}{5 \sqrt{2} b}-\frac{9 \tan ^{-1}\left(\frac{1-3 \tan (a+b x)}{\sqrt{2} \sqrt{3 \tan (a+b x)+4}}\right)}{5 \sqrt{2} b}",1,"((3 - 4*I)*ArcTanh[Sqrt[4 + 3*Tan[a + b*x]]/Sqrt[4 - 3*I]])/(Sqrt[4 - 3*I]*b) + ((3 + 4*I)*ArcTanh[Sqrt[4 + 3*Tan[a + b*x]]/Sqrt[4 + 3*I]])/(Sqrt[4 + 3*I]*b)","C",1
378,1,151,278,1.7325337,"\int \tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{-105 \sqrt[4]{-1} (b+i a) (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+2 \sqrt{\tan (c+d x)} \left(21 (a B+A b) \tan ^2(c+d x)+35 (a A-b B) \tan (c+d x)-105 (a B+A b)+15 b B \tan ^3(c+d x)\right)+105 (-1)^{3/4} (a+i b) (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{105 d}","\frac{(a (A-B)-b (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{(a (A-B)-b (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 (a B+A b) \tan ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 (a A-b B) \tan ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{2 (a B+A b) \sqrt{\tan (c+d x)}}{d}-\frac{(a (A+B)+b (A-B)) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a (A+B)+b (A-B)) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b B \tan ^{\frac{7}{2}}(c+d x)}{7 d}",1,"(-105*(-1)^(1/4)*(I*a + b)*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 105*(-1)^(3/4)*(a + I*b)*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(-105*(A*b + a*B) + 35*(a*A - b*B)*Tan[c + d*x] + 21*(A*b + a*B)*Tan[c + d*x]^2 + 15*b*B*Tan[c + d*x]^3))/(105*d)","C",1
379,1,134,254,1.0520132,"\int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{15 \sqrt[4]{-1} (a-i b) (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+2 \sqrt{\tan (c+d x)} \left(5 (a B+A b) \tan (c+d x)+15 (a A-b B)+3 b B \tan ^2(c+d x)\right)+15 \sqrt[4]{-1} (a+i b) (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{15 d}","\frac{(a (A+B)+b (A-B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{(a (A+B)+b (A-B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 (a B+A b) \tan ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 (a A-b B) \sqrt{\tan (c+d x)}}{d}+\frac{(a (A-B)-b (A+B)) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a (A-B)-b (A+B)) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b B \tan ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(15*(-1)^(1/4)*(a - I*b)*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 15*(-1)^(1/4)*(a + I*b)*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(15*(a*A - b*B) + 5*(A*b + a*B)*Tan[c + d*x] + 3*b*B*Tan[c + d*x]^2))/(15*d)","C",1
380,1,114,229,0.423777,"\int \sqrt{\tan (c+d x)} (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{3 \sqrt[4]{-1} (b+i a) (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+2 \sqrt{\tan (c+d x)} (3 a B+3 A b+b B \tan (c+d x))-3 (-1)^{3/4} (a+i b) (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{3 d}","-\frac{(a (A-B)-b (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{(a (A-B)-b (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 (a B+A b) \sqrt{\tan (c+d x)}}{d}+\frac{(a (A+B)+b (A-B)) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a (A+B)+b (A-B)) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b B \tan ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(3*(-1)^(1/4)*(I*a + b)*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - 3*(-1)^(3/4)*(a + I*b)*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(3*A*b + 3*a*B + b*B*Tan[c + d*x]))/(3*d)","C",1
381,1,94,205,0.183579,"\int \frac{(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","Integrate[((a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]],x]","-\frac{\sqrt[4]{-1} (a-i b) (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+\sqrt[4]{-1} (a+i b) (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-2 b B \sqrt{\tan (c+d x)}}{d}","-\frac{(a (A+B)+b (A-B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{(a (A+B)+b (A-B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{(a (A-B)-b (A+B)) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a (A-B)-b (A+B)) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b B \sqrt{\tan (c+d x)}}{d}",1,"-(((-1)^(1/4)*(a - I*b)*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (-1)^(1/4)*(a + I*b)*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - 2*b*B*Sqrt[Tan[c + d*x]])/d)","C",1
382,1,158,205,0.6487062,"\int \frac{(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2),x]","-\frac{-2 \sqrt{2} (a (A-B)-b (A+B)) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\sqrt{2} (a (A+B)+b (A-B)) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\frac{8 a A}{\sqrt{\tan (c+d x)}}}{4 d}","\frac{(a (A-B)-b (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{(a (A-B)-b (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{(a (A+B)+b (A-B)) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a (A+B)+b (A-B)) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a A}{d \sqrt{\tan (c+d x)}}",1,"-1/4*(-2*Sqrt[2]*(a*(A - B) - b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + Sqrt[2]*(b*(A - B) + a*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) + (8*a*A)/Sqrt[Tan[c + d*x]])/d","A",1
383,1,178,229,0.7791319,"\int \frac{(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2),x]","\frac{6 \sqrt{2} (a (A+B)+b (A-B)) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-\frac{24 (a B+A b)}{\sqrt{\tan (c+d x)}}+3 \sqrt{2} (a (A-B)-b (A+B)) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-\frac{8 a A}{\tan ^{\frac{3}{2}}(c+d x)}}{12 d}","\frac{(a (A+B)+b (A-B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{(a (A+B)+b (A-B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 (a B+A b)}{d \sqrt{\tan (c+d x)}}+\frac{(a (A-B)-b (A+B)) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a (A-B)-b (A+B)) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a A}{3 d \tan ^{\frac{3}{2}}(c+d x)}",1,"(6*Sqrt[2]*(b*(A - B) + a*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + 3*Sqrt[2]*(a*(A - B) - b*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) - (8*a*A)/Tan[c + d*x]^(3/2) - (24*(A*b + a*B))/Sqrt[Tan[c + d*x]])/(12*d)","A",1
384,1,198,254,1.1991537,"\int \frac{(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2),x]","-\frac{30 \sqrt{2} (a (A-B)-b (A+B)) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\frac{40 (a B+A b)}{\tan ^{\frac{3}{2}}(c+d x)}-\frac{120 (a A-b B)}{\sqrt{\tan (c+d x)}}-15 \sqrt{2} (a (A+B)+b (A-B)) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\frac{24 a A}{\tan ^{\frac{5}{2}}(c+d x)}}{60 d}","-\frac{(a (A-B)-b (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{(a (A-B)-b (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 (a B+A b)}{3 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 (a A-b B)}{d \sqrt{\tan (c+d x)}}+\frac{(a (A+B)+b (A-B)) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a (A+B)+b (A-B)) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a A}{5 d \tan ^{\frac{5}{2}}(c+d x)}",1,"-1/60*(30*Sqrt[2]*(a*(A - B) - b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) - 15*Sqrt[2]*(b*(A - B) + a*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) + (24*a*A)/Tan[c + d*x]^(5/2) + (40*(A*b + a*B))/Tan[c + d*x]^(3/2) - (120*(a*A - b*B))/Sqrt[Tan[c + d*x]])/d","A",1
385,1,320,394,6.1153759,"\int \tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{2 b B \tan ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}+\frac{2}{9} \left(\frac{i \left(\frac{9}{2} \left(a^2 A-2 a b B-A b^2\right)-\frac{9}{2} i \left(a^2 B+2 a A b-b^2 B\right)\right) \left(\frac{2}{5} \tan ^{\frac{5}{2}}(c+d x)-i \left(\frac{2}{3} \tan ^{\frac{3}{2}}(c+d x)-i \left(2 \sqrt[4]{-1} \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+2 \sqrt{\tan (c+d x)}\right)\right)\right)}{2 d}-\frac{i \left(\frac{9}{2} \left(a^2 A-2 a b B-A b^2\right)+\frac{9}{2} i \left(a^2 B+2 a A b-b^2 B\right)\right) \left(\frac{2}{5} \tan ^{\frac{5}{2}}(c+d x)+i \left(\frac{2}{3} \tan ^{\frac{3}{2}}(c+d x)+i \left(2 \sqrt{\tan (c+d x)}+2 \sqrt[4]{-1} \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)\right)\right)}{2 d}+\frac{b (11 a B+9 A b) \tan ^{\frac{7}{2}}(c+d x)}{7 d}\right)","\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 \left(a^2 B+2 a A b-b^2 B\right) \tan ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 \left(a^2 A-2 a b B-A b^2\right) \tan ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{2 \left(a^2 B+2 a A b-b^2 B\right) \sqrt{\tan (c+d x)}}{d}-\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b (11 a B+9 A b) \tan ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 b B \tan ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}",1,"(2*b*B*Tan[c + d*x]^(7/2)*(a + b*Tan[c + d*x]))/(9*d) + (2*((b*(9*A*b + 11*a*B)*Tan[c + d*x]^(7/2))/(7*d) + ((I/2)*((9*(a^2*A - A*b^2 - 2*a*b*B))/2 - ((9*I)/2)*(2*a*A*b + a^2*B - b^2*B))*((2*Tan[c + d*x]^(5/2))/5 - I*((-I)*(2*(-1)^(1/4)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]) + (2*Tan[c + d*x]^(3/2))/3)))/d - ((I/2)*((9*(a^2*A - A*b^2 - 2*a*b*B))/2 + ((9*I)/2)*(2*a*A*b + a^2*B - b^2*B))*((2*Tan[c + d*x]^(5/2))/5 + I*(I*(2*(-1)^(1/4)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]) + (2*Tan[c + d*x]^(3/2))/3)))/d))/9","C",1
386,1,178,360,2.265013,"\int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{2 \sqrt{\tan (c+d x)} \left(35 \left(a^2 B+2 a A b-b^2 B\right) \tan (c+d x)+105 \left(a^2 A-2 a b B-A b^2\right)+21 b (2 a B+A b) \tan ^2(c+d x)+15 b^2 B \tan ^3(c+d x)\right)+105 \sqrt[4]{-1} (a-i b)^2 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+105 \sqrt[4]{-1} (a+i b)^2 (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{105 d}","\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 \left(a^2 B+2 a A b-b^2 B\right) \tan ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 \left(a^2 A-2 a b B-A b^2\right) \sqrt{\tan (c+d x)}}{d}+\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b (9 a B+7 A b) \tan ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 b B \tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}",1,"(105*(-1)^(1/4)*(a - I*b)^2*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 105*(-1)^(1/4)*(a + I*b)^2*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(105*(a^2*A - A*b^2 - 2*a*b*B) + 35*(2*a*A*b + a^2*B - b^2*B)*Tan[c + d*x] + 21*b*(A*b + 2*a*B)*Tan[c + d*x]^2 + 15*b^2*B*Tan[c + d*x]^3))/(105*d)","C",1
387,1,151,326,1.2425048,"\int \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{2 \sqrt{\tan (c+d x)} \left(15 \left(a^2 B+2 a A b-b^2 B\right)+5 b (2 a B+A b) \tan (c+d x)+3 b^2 B \tan ^2(c+d x)\right)+15 \sqrt[4]{-1} (a-i b)^2 (B+i A) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-15 (-1)^{3/4} (a+i b)^2 (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{15 d}","-\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 \left(a^2 B+2 a A b-b^2 B\right) \sqrt{\tan (c+d x)}}{d}+\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b (7 a B+5 A b) \tan ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 b B \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}{5 d}",1,"(15*(-1)^(1/4)*(a - I*b)^2*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - 15*(-1)^(3/4)*(a + I*b)^2*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(15*(2*a*A*b + a^2*B - b^2*B) + 5*b*(A*b + 2*a*B)*Tan[c + d*x] + 3*b^2*B*Tan[c + d*x]^2))/(15*d)","C",1
388,1,119,294,0.5400589,"\int \frac{(a+b \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","Integrate[((a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]],x]","\frac{-3 \sqrt[4]{-1} (a-i b)^2 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+2 b \sqrt{\tan (c+d x)} (6 a B+3 A b+b B \tan (c+d x))-3 \sqrt[4]{-1} (a+i b)^2 (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{3 d}","-\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b (5 a B+3 A b) \sqrt{\tan (c+d x)}}{3 d}+\frac{2 b B \sqrt{\tan (c+d x)} (a+b \tan (c+d x))}{3 d}",1,"(-3*(-1)^(1/4)*(a - I*b)^2*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - 3*(-1)^(1/4)*(a + I*b)^2*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*b*Sqrt[Tan[c + d*x]]*(3*A*b + 6*a*B + b*B*Tan[c + d*x]))/(3*d)","C",1
389,1,211,276,0.9372489,"\int \frac{(a+b \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2),x]","\frac{-8 \left(a^2 A-2 a b B-A b^2\right) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)-\sqrt{2} \left(a^2 B+2 a A b-b^2 B\right) \sqrt{\tan (c+d x)} \left(2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-8 b (3 a B+A b)+8 b B (a+b \tan (c+d x))}{4 d \sqrt{\tan (c+d x)}}","\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a^2 A}{d \sqrt{\tan (c+d x)}}+\frac{2 b^2 B \sqrt{\tan (c+d x)}}{d}",1,"(-8*b*(A*b + 3*a*B) - 8*(a^2*A - A*b^2 - 2*a*b*B)*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2] - Sqrt[2]*(2*a*A*b + a^2*B - b^2*B)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])*Sqrt[Tan[c + d*x]] + 8*b*B*(a + b*Tan[c + d*x]))/(4*d*Sqrt[Tan[c + d*x]])","C",1
390,1,119,283,0.7413684,"\int \frac{(a+b \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2),x]","\frac{2 \left(a^2 (-A)+2 a b B+A b^2\right) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(c+d x)\right)-6 \left(a^2 B+2 a A b-b^2 B\right) \tan (c+d x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)-2 b (2 a B+A b+3 b B \tan (c+d x))}{3 d \tan ^{\frac{3}{2}}(c+d x)}","\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a^2 A}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 a (a B+2 A b)}{d \sqrt{\tan (c+d x)}}",1,"(2*(-(a^2*A) + A*b^2 + 2*a*b*B)*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2] - 6*(2*a*A*b + a^2*B - b^2*B)*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2]*Tan[c + d*x] - 2*b*(A*b + 2*a*B + 3*b*B*Tan[c + d*x]))/(3*d*Tan[c + d*x]^(3/2))","C",1
391,1,120,317,0.6093297,"\int \frac{(a+b \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2),x]","\frac{2 \left(\left(-3 a^2 A+6 a b B+3 A b^2\right) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\tan ^2(c+d x)\right)-5 \left(a^2 B+2 a A b-b^2 B\right) \tan (c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(c+d x)\right)-b (6 a B+3 A b+5 b B \tan (c+d x))\right)}{15 d \tan ^{\frac{5}{2}}(c+d x)}","-\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 \left(a^2 A-2 a b B-A b^2\right)}{d \sqrt{\tan (c+d x)}}+\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a^2 A}{5 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 a (a B+2 A b)}{3 d \tan ^{\frac{3}{2}}(c+d x)}",1,"(2*((-3*a^2*A + 3*A*b^2 + 6*a*b*B)*Hypergeometric2F1[-5/4, 1, -1/4, -Tan[c + d*x]^2] - 5*(2*a*A*b + a^2*B - b^2*B)*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2]*Tan[c + d*x] - b*(3*A*b + 6*a*B + 5*b*B*Tan[c + d*x])))/(15*d*Tan[c + d*x]^(5/2))","C",1
392,1,221,463,3.528791,"\int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{2 \left(7 b \left(22 a^2 B+27 a A b-9 b^2 B\right) \tan ^{\frac{5}{2}}(c+d x)+5 b^2 (13 a B+9 A b) \tan ^{\frac{7}{2}}(c+d x)+\frac{105}{2} (a-i b)^3 (B+i A) \left(-3 (-1)^{3/4} \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+\sqrt{\tan (c+d x)} (\tan (c+d x)-3 i)\right)+\frac{105}{2} (a+i b)^3 (B-i A) \left(3 (-1)^{3/4} \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+\sqrt{\tan (c+d x)} (\tan (c+d x)+3 i)\right)+35 b B \tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^2\right)}{315 d}","\frac{2 b \left(22 a^2 B+27 a A b-9 b^2 B\right) \tan ^{\frac{5}{2}}(c+d x)}{45 d}+\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \tan ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right) \sqrt{\tan (c+d x)}}{d}+\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b^2 (13 a B+9 A b) \tan ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 b B \tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^2}{9 d}",1,"(2*(7*b*(27*a*A*b + 22*a^2*B - 9*b^2*B)*Tan[c + d*x]^(5/2) + 5*b^2*(9*A*b + 13*a*B)*Tan[c + d*x]^(7/2) + 35*b*B*Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2 + (105*(a - I*b)^3*(I*A + B)*(-3*(-1)^(3/4)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + Sqrt[Tan[c + d*x]]*(-3*I + Tan[c + d*x])))/2 + (105*(a + I*b)^3*((-I)*A + B)*(3*(-1)^(3/4)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + Sqrt[Tan[c + d*x]]*(3*I + Tan[c + d*x])))/2))/(315*d)","C",1
393,1,197,421,2.0031765,"\int \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{2 \left(5 b \left(18 a^2 B+21 a A b-7 b^2 B\right) \tan ^{\frac{3}{2}}(c+d x)+3 b^2 (11 a B+7 A b) \tan ^{\frac{5}{2}}(c+d x)+\frac{105}{2} (a-i b)^3 (B+i A) \left(\sqrt[4]{-1} \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+\sqrt{\tan (c+d x)}\right)+\frac{105}{2} (a+i b)^3 (B-i A) \left(\sqrt{\tan (c+d x)}+\sqrt[4]{-1} \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)+15 b B \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^2\right)}{105 d}","\frac{2 b \left(18 a^2 B+21 a A b-7 b^2 B\right) \tan ^{\frac{3}{2}}(c+d x)}{21 d}-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \sqrt{\tan (c+d x)}}{d}+\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b^2 (11 a B+7 A b) \tan ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 b B \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^2}{7 d}",1,"(2*((105*(a - I*b)^3*(I*A + B)*((-1)^(1/4)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + Sqrt[Tan[c + d*x]]))/2 + (105*(a + I*b)^3*((-I)*A + B)*((-1)^(1/4)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + Sqrt[Tan[c + d*x]]))/2 + 5*b*(21*a*A*b + 18*a^2*B - 7*b^2*B)*Tan[c + d*x]^(3/2) + 3*b^2*(7*A*b + 11*a*B)*Tan[c + d*x]^(5/2) + 15*b*B*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2))/(105*d)","C",1
394,1,153,380,1.4972225,"\int \frac{(a+b \tan (c+d x))^3 (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","Integrate[((a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]],x]","\frac{2 b \sqrt{\tan (c+d x)} \left(15 \left(3 a^2 B+3 a A b-b^2 B\right)+5 b (3 a B+A b) \tan (c+d x)+3 b^2 B \tan ^2(c+d x)\right)-15 \sqrt[4]{-1} (a-i b)^3 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-15 \sqrt[4]{-1} (a+i b)^3 (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{15 d}","\frac{2 b \left(14 a^2 B+15 a A b-5 b^2 B\right) \sqrt{\tan (c+d x)}}{5 d}-\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b^2 (9 a B+5 A b) \tan ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 b B \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^2}{5 d}",1,"(-15*(-1)^(1/4)*(a - I*b)^3*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - 15*(-1)^(1/4)*(a + I*b)^3*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*b*Sqrt[Tan[c + d*x]]*(15*(3*a*A*b + 3*a^2*B - b^2*B) + 5*b*(A*b + 3*a*B)*Tan[c + d*x] + 3*b^2*B*Tan[c + d*x]^2))/(15*d)","C",1
395,1,264,374,2.7832522,"\int \frac{(a+b \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2),x]","\frac{8 b \left(-17 a^2 B-12 a A b+3 b^2 B\right)-3 \left(8 \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)+\sqrt{2} \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \sqrt{\tan (c+d x)} \left(2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)\right)+8 b (7 a B+3 A b) (a+b \tan (c+d x))+8 b B (a+b \tan (c+d x))^2}{12 d \sqrt{\tan (c+d x)}}","\frac{2 b \left(2 a^2 A+3 a b B+A b^2\right) \sqrt{\tan (c+d x)}}{d}+\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b^2 (3 a A+b B) \tan ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{2 a A (a+b \tan (c+d x))^2}{d \sqrt{\tan (c+d x)}}",1,"(8*b*(-12*a*A*b - 17*a^2*B + 3*b^2*B) - 3*(8*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2] + Sqrt[2]*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])*Sqrt[Tan[c + d*x]]) + 8*b*(3*A*b + 7*a*B)*(a + b*Tan[c + d*x]) + 8*b*B*(a + b*Tan[c + d*x])^2)/(12*d*Sqrt[Tan[c + d*x]])","C",1
396,1,165,372,1.4052034,"\int \frac{(a+b \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2),x]","-\frac{2 \left(b \left(3 a^2 B+3 b (3 a B+A b) \tan (c+d x)+3 a A b-3 b^2 B \tan ^2(c+d x)-b^2 B\right)+\left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(c+d x)\right)+3 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \tan (c+d x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)\right)}{3 d \tan ^{\frac{3}{2}}(c+d x)}","-\frac{2 a^2 (3 a B+7 A b)}{3 d \sqrt{\tan (c+d x)}}+\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b^2 (a A+3 b B) \sqrt{\tan (c+d x)}}{3 d}-\frac{2 a A (a+b \tan (c+d x))^2}{3 d \tan ^{\frac{3}{2}}(c+d x)}",1,"(-2*((a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2] + 3*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2]*Tan[c + d*x] + b*(3*a*A*b + 3*a^2*B - b^2*B + 3*b*(A*b + 3*a*B)*Tan[c + d*x] - 3*b^2*B*Tan[c + d*x]^2)))/(3*d*Tan[c + d*x]^(3/2))","C",1
397,1,166,380,1.3850243,"\int \frac{(a+b \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2),x]","-\frac{2 \left(b \left(9 a^2 B+5 b (3 a B+A b) \tan (c+d x)+9 a A b+15 b^2 B \tan ^2(c+d x)-3 b^2 B\right)+3 \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\tan ^2(c+d x)\right)+5 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \tan (c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(c+d x)\right)\right)}{15 d \tan ^{\frac{5}{2}}(c+d x)}","\frac{2 a \left(5 a^2 A-15 a b B-14 A b^2\right)}{5 d \sqrt{\tan (c+d x)}}-\frac{2 a^2 (5 a B+9 A b)}{15 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a A (a+b \tan (c+d x))^2}{5 d \tan ^{\frac{5}{2}}(c+d x)}",1,"(-2*(3*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*Hypergeometric2F1[-5/4, 1, -1/4, -Tan[c + d*x]^2] + 5*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2]*Tan[c + d*x] + b*(9*a*A*b + 9*a^2*B - 3*b^2*B + 5*b*(A*b + 3*a*B)*Tan[c + d*x] + 15*b^2*B*Tan[c + d*x]^2)))/(15*d*Tan[c + d*x]^(5/2))","C",1
398,1,187,325,1.1696687,"\int \frac{\tan ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{6 a^{5/2} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+2 \sqrt{b} \left(a^2+b^2\right) \sqrt{\tan (c+d x)} (-3 a B+3 A b+b B \tan (c+d x))-3 \sqrt[4]{-1} b^{5/2} (a+i b) (B+i A) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+3 \sqrt[4]{-1} b^{5/2} (b+i a) (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{3 b^{5/2} d \left(a^2+b^2\right)}","\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(b (A-B)-a (A+B)) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(b (A-B)-a (A+B)) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{2 a^{5/2} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{5/2} d \left(a^2+b^2\right)}+\frac{2 (A b-a B) \sqrt{\tan (c+d x)}}{b^2 d}+\frac{2 B \tan ^{\frac{3}{2}}(c+d x)}{3 b d}",1,"(-3*(-1)^(1/4)*(a + I*b)*b^(5/2)*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 6*a^(5/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + 3*(-1)^(1/4)*b^(5/2)*(I*a + b)*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[b]*(a^2 + b^2)*Sqrt[Tan[c + d*x]]*(3*A*b - 3*a*B + b*B*Tan[c + d*x]))/(3*b^(5/2)*(a^2 + b^2)*d)","C",1
399,1,165,297,0.3642661,"\int \frac{\tan ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{-2 a^{3/2} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+2 \sqrt{b} B \left(a^2+b^2\right) \sqrt{\tan (c+d x)}+\sqrt[4]{-1} b^{3/2} (a+i b) (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+\sqrt[4]{-1} b^{3/2} (a-i b) (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{b^{3/2} d \left(a^2+b^2\right)}","-\frac{(b (A-B)-a (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(b (A-B)-a (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a (A-B)+b (A+B)) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a (A-B)+b (A+B)) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{2 a^{3/2} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{3/2} d \left(a^2+b^2\right)}+\frac{2 B \sqrt{\tan (c+d x)}}{b d}",1,"((-1)^(1/4)*(a + I*b)*b^(3/2)*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - 2*a^(3/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + (-1)^(1/4)*(a - I*b)*b^(3/2)*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[b]*(a^2 + b^2)*B*Sqrt[Tan[c + d*x]])/(b^(3/2)*(a^2 + b^2)*d)","C",1
400,1,195,278,0.4205319,"\int \frac{\sqrt{\tan (c+d x)} (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","-\frac{2 \sqrt{2} (a (A-B)+b (A+B)) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\frac{8 \sqrt{a} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b}}-\sqrt{2} (a (A+B)+b (B-A)) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 d \left(a^2+b^2\right)}","-\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{2 \sqrt{a} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} d \left(a^2+b^2\right)}-\frac{(b (A-B)-a (A+B)) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(b (A-B)-a (A+B)) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}",1,"-1/4*(2*Sqrt[2]*(a*(A - B) + b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + (8*Sqrt[a]*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/Sqrt[b] - Sqrt[2]*(b*(-A + B) + a*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/((a^2 + b^2)*d)","A",1
401,1,194,278,0.3977449,"\int \frac{A+B \tan (c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])),x]","-\frac{2 \sqrt{2} (a (A+B)+b (B-A)) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\frac{8 \sqrt{b} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}+\sqrt{2} (a (A-B)+b (A+B)) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 d \left(a^2+b^2\right)}","\frac{(b (A-B)-a (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(b (A-B)-a (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{2 \sqrt{b} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d \left(a^2+b^2\right)}-\frac{(a (A-B)+b (A+B)) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(a (A-B)+b (A+B)) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}",1,"-1/4*(2*Sqrt[2]*(b*(-A + B) + a*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + (8*Sqrt[b]*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/Sqrt[a] + Sqrt[2]*(a*(A - B) + b*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/((a^2 + b^2)*d)","A",1
402,1,153,297,0.6016582,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])),x]","\frac{\frac{\sqrt[4]{-1} a \left((b-i a) (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+(b+i a) (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{a^2+b^2}+\frac{2 b^{3/2} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \left(a^2+b^2\right)}-\frac{2 A}{\sqrt{\tan (c+d x)}}}{a d}","\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(b (A-B)-a (A+B)) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(b (A-B)-a (A+B)) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{2 b^{3/2} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d \left(a^2+b^2\right)}-\frac{2 A}{a d \sqrt{\tan (c+d x)}}",1,"((2*b^(3/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)) + ((-1)^(1/4)*a*(((-I)*a + b)*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (I*a + b)*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) - (2*A)/Sqrt[Tan[c + d*x]])/(a*d)","C",1
403,1,174,325,3.4951239,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])),x]","\frac{-\frac{2 ((3 a B-3 A b) \tan (c+d x)+a A)}{a^2 \tan ^{\frac{3}{2}}(c+d x)}+\frac{6 b^{5/2} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} \left(a^2+b^2\right)}+\frac{3 \sqrt[4]{-1} (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{a-i b}+\frac{3 \sqrt[4]{-1} (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{a+i b}}{3 d}","-\frac{(b (A-B)-a (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(b (A-B)-a (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a (A-B)+b (A+B)) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a (A-B)+b (A+B)) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{2 (A b-a B)}{a^2 d \sqrt{\tan (c+d x)}}+\frac{2 b^{5/2} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} d \left(a^2+b^2\right)}-\frac{2 A}{3 a d \tan ^{\frac{3}{2}}(c+d x)}",1,"((3*(-1)^(1/4)*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/(a - I*b) + (6*b^(5/2)*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(5/2)*(a^2 + b^2)) + (3*(-1)^(1/4)*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/(a + I*b) - (2*(a*A + (-3*A*b + 3*a*B)*Tan[c + d*x]))/(a^2*Tan[c + d*x]^(3/2)))/(3*d)","C",1
404,1,275,436,2.4023703,"\int \frac{\tan ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{2 \left(\frac{\left(-3 a^3 B+a^2 A b-4 a b^2 B+2 A b^3\right) \sqrt{\tan (c+d x)}}{2 \left(a^2+b^2\right)}-\frac{(a+b \tan (c+d x)) \left(a^{3/2} \left(3 a^3 B-a^2 A b+7 a b^2 B-5 A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+\sqrt[4]{-1} b^{5/2} (a+i b)^2 (B+i A) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+(-1)^{3/4} b^{5/2} (b+i a)^2 (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{2 \sqrt{b} \left(a^2+b^2\right)^2}+(3 a B-A b) \sqrt{\tan (c+d x)}+b B \tan ^{\frac{3}{2}}(c+d x)\right)}{b^2 d (a+b \tan (c+d x))}","\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{a (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{b d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\left(-3 a^2 B+a A b-2 b^2 B\right) \sqrt{\tan (c+d x)}}{b^2 d \left(a^2+b^2\right)}+\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{a^{3/2} \left(-3 a^3 B+a^2 A b-7 a b^2 B+5 A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{5/2} d \left(a^2+b^2\right)^2}",1,"(2*((-(A*b) + 3*a*B)*Sqrt[Tan[c + d*x]] + ((a^2*A*b + 2*A*b^3 - 3*a^3*B - 4*a*b^2*B)*Sqrt[Tan[c + d*x]])/(2*(a^2 + b^2)) + b*B*Tan[c + d*x]^(3/2) - (((-1)^(1/4)*(a + I*b)^2*b^(5/2)*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + a^(3/2)*(-(a^2*A*b) - 5*A*b^3 + 3*a^3*B + 7*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + (-1)^(3/4)*b^(5/2)*(I*a + b)^2*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])*(a + b*Tan[c + d*x]))/(2*Sqrt[b]*(a^2 + b^2)^2)))/(b^2*d*(a + b*Tan[c + d*x]))","C",1
405,1,230,391,1.8686702,"\int \frac{\tan ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{\frac{\left(a^2 B+a A b+2 b^2 B\right) \sqrt{\tan (c+d x)}}{\left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\sqrt{a} \left(a^3 B+a^2 A b+5 a b^2 B-3 A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+\sqrt[4]{-1} b^{3/2} \left((a+i b)^2 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+(a-i b)^2 (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{\sqrt{b} \left(a^2+b^2\right)^2}-\frac{2 B \sqrt{\tan (c+d x)}}{a+b \tan (c+d x)}}{b d}","-\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{a (A b-a B) \sqrt{\tan (c+d x)}}{b d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\sqrt{a} \left(a^3 B+a^2 A b+5 a b^2 B-3 A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{3/2} d \left(a^2+b^2\right)^2}",1,"((Sqrt[a]*(a^2*A*b - 3*A*b^3 + a^3*B + 5*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + (-1)^(1/4)*b^(3/2)*((a + I*b)^2*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a - I*b)^2*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(Sqrt[b]*(a^2 + b^2)^2) - (2*B*Sqrt[Tan[c + d*x]])/(a + b*Tan[c + d*x]) + ((a*A*b + a^2*B + 2*b^2*B)*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*(a + b*Tan[c + d*x])))/(b*d)","C",1
406,1,220,391,1.4265264,"\int \frac{\sqrt{\tan (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{\frac{\sqrt[4]{-1} a \left((a+i b)^2 (B+i A) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+(a-i b)^2 (B-i A) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{a^2+b^2}+\frac{\sqrt{a} \left(a^3 B-3 a^2 A b-3 a b^2 B+A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \left(a^2+b^2\right)}+\frac{b (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{a+b \tan (c+d x)}+(a B-A b) \sqrt{\tan (c+d x)}}{a d \left(a^2+b^2\right)}","-\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{(A b-a B) \sqrt{\tan (c+d x)}}{d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d \left(a^2+b^2\right)^2}",1,"((Sqrt[a]*(-3*a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[b]*(a^2 + b^2)) + ((-1)^(1/4)*a*((a + I*b)^2*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a - I*b)^2*((-I)*A + B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (-(A*b) + a*B)*Sqrt[Tan[c + d*x]] + (b*(A*b - a*B)*Tan[c + d*x]^(3/2))/(a + b*Tan[c + d*x]))/(a*(a^2 + b^2)*d)","C",1
407,1,204,391,1.1467042,"\int \frac{A+B \tan (c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^2} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2),x]","\frac{\frac{\sqrt[4]{-1} \left(-a (a+i b)^2 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-a (a-i b)^2 (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{a^2+b^2}+\frac{\sqrt{b} \left(-3 a^3 B+5 a^2 A b+a b^2 B+A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \left(a^2+b^2\right)}+\frac{b (A b-a B) \sqrt{\tan (c+d x)}}{a+b \tan (c+d x)}}{a d \left(a^2+b^2\right)}","\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{b (A b-a B) \sqrt{\tan (c+d x)}}{a d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\sqrt{b} \left(-3 a^3 B+5 a^2 A b+a b^2 B+A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d \left(a^2+b^2\right)^2}",1,"((Sqrt[b]*(5*a^2*A*b + A*b^3 - 3*a^3*B + a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)) + ((-1)^(1/4)*(-(a*(a + I*b)^2*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]) - a*(a - I*b)^2*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (b*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(a + b*Tan[c + d*x]))/(a*(a^2 + b^2)*d)","C",1
408,1,239,439,2.3824664,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2),x]","\frac{\frac{-2 a^2 A+a b B-3 A b^2}{a \sqrt{\tan (c+d x)}}+\frac{\sqrt[4]{-1} a \left(i (a-i b)^2 (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-i (a+i b)^2 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{a^2+b^2}+\frac{b^{3/2} \left(5 a^3 B-7 a^2 A b+a b^2 B-3 A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} \left(a^2+b^2\right)}+\frac{b (A b-a B)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))}}{a d \left(a^2+b^2\right)}","\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{b (A b-a B)}{a d \left(a^2+b^2\right) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))}-\frac{2 a^2 A-a b B+3 A b^2}{a^2 d \left(a^2+b^2\right) \sqrt{\tan (c+d x)}}+\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{b^{3/2} \left(-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} d \left(a^2+b^2\right)^2}",1,"((b^(3/2)*(-7*a^2*A*b - 3*A*b^3 + 5*a^3*B + a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*(a^2 + b^2)) + ((-1)^(1/4)*a*((-I)*(a + I*b)^2*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + I*(a - I*b)^2*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (-2*a^2*A - 3*A*b^2 + a*b*B)/(a*Sqrt[Tan[c + d*x]]) + (b*(A*b - a*B))/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])))/(a*(a^2 + b^2)*d)","C",1
409,1,287,493,4.0749679,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2),x]","\frac{\frac{-2 a^2 A+3 a b B-5 A b^2}{a \tan ^{\frac{3}{2}}(c+d x)}+\frac{3 \left(-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right)}{a^2 \sqrt{\tan (c+d x)}}+\frac{3 \left(\sqrt[4]{-1} a^{7/2} (a+i b)^2 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+\sqrt[4]{-1} a^{7/2} (a-i b)^2 (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+b^{5/2} \left(-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)\right)}{a^{5/2} \left(a^2+b^2\right)}+\frac{3 b (A b-a B)}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}}{3 a d \left(a^2+b^2\right)}","-\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{b (A b-a B)}{a d \left(a^2+b^2\right) \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}-\frac{2 a^2 A-3 a b B+5 A b^2}{3 a^2 d \left(a^2+b^2\right) \tan ^{\frac{3}{2}}(c+d x)}+\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3}{a^3 d \left(a^2+b^2\right) \sqrt{\tan (c+d x)}}+\frac{b^{5/2} \left(-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{7/2} d \left(a^2+b^2\right)^2}",1,"((3*((-1)^(1/4)*a^(7/2)*(a + I*b)^2*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + b^(5/2)*(9*a^2*A*b + 5*A*b^3 - 7*a^3*B - 3*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + (-1)^(1/4)*a^(7/2)*(a - I*b)^2*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a^(5/2)*(a^2 + b^2)) + (-2*a^2*A - 5*A*b^2 + 3*a*b*B)/(a*Tan[c + d*x]^(3/2)) + (3*(4*a^2*A*b + 5*A*b^3 - 2*a^3*B - 3*a*b^2*B))/(a^2*Sqrt[Tan[c + d*x]]) + (3*b*(A*b - a*B))/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])))/(3*a*(a^2 + b^2)*d)","C",1
410,1,1563,600,6.346333,"\int \frac{\tan ^{\frac{7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^(7/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]","-\frac{15 B \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right) a^{13/2}}{4 b^{7/2} \left(a^2+b^2\right)^3 d}+\frac{3 A \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right) a^{11/2}}{4 b^{5/2} \left(a^2+b^2\right)^3 d}-\frac{15 B \sqrt{\tan (c+d x)} a^5}{4 b^3 \left(a^2+b^2\right)^2 d (a+b \tan (c+d x))}-\frac{23 B \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right) a^{9/2}}{2 b^{3/2} \left(a^2+b^2\right)^3 d}+\frac{3 A \sqrt{\tan (c+d x)} a^4}{4 b^2 \left(a^2+b^2\right)^2 d (a+b \tan (c+d x))}-\frac{5 B \sqrt{\tan (c+d x)} a^4}{2 b^3 \left(a^2+b^2\right) d (a+b \tan (c+d x))^2}+\frac{3 A \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right) a^{7/2}}{2 \sqrt{b} \left(a^2+b^2\right)^3 d}+\frac{(-1)^{3/4} B \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right) a^3}{\left(a^2+b^2\right)^3 d}-\frac{\sqrt[4]{-1} A \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right) a^3}{\left(a^2+b^2\right)^3 d}-\frac{(-1)^{3/4} B \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right) a^3}{\left(a^2+b^2\right)^3 d}-\frac{\sqrt[4]{-1} A \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right) a^3}{\left(a^2+b^2\right)^3 d}-\frac{31 B \sqrt{\tan (c+d x)} a^3}{4 b \left(a^2+b^2\right)^2 d (a+b \tan (c+d x))}+\frac{A \sqrt{\tan (c+d x)} a^3}{2 b^2 \left(a^2+b^2\right) d (a+b \tan (c+d x))^2}-\frac{63 \sqrt{b} B \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right) a^{5/2}}{4 \left(a^2+b^2\right)^3 d}-\frac{3 \sqrt[4]{-1} b B \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right) a^2}{\left(a^2+b^2\right)^3 d}-\frac{3 (-1)^{3/4} A b \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right) a^2}{\left(a^2+b^2\right)^3 d}-\frac{3 \sqrt[4]{-1} b B \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right) a^2}{\left(a^2+b^2\right)^3 d}+\frac{3 (-1)^{3/4} A b \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right) a^2}{\left(a^2+b^2\right)^3 d}+\frac{3 A \sqrt{\tan (c+d x)} a^2}{4 \left(a^2+b^2\right)^2 d (a+b \tan (c+d x))}-\frac{8 B \sqrt{\tan (c+d x)} a^2}{3 b \left(a^2+b^2\right) d (a+b \tan (c+d x))^2}+\frac{10 B \sqrt{\tan (c+d x)} a^2}{b^3 d (a+b \tan (c+d x))^2}+\frac{35 A b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right) a^{3/2}}{4 \left(a^2+b^2\right)^3 d}-\frac{3 (-1)^{3/4} b^2 B \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right) a}{\left(a^2+b^2\right)^3 d}+\frac{3 \sqrt[4]{-1} A b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right) a}{\left(a^2+b^2\right)^3 d}+\frac{3 (-1)^{3/4} b^2 B \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right) a}{\left(a^2+b^2\right)^3 d}+\frac{3 \sqrt[4]{-1} A b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right) a}{\left(a^2+b^2\right)^3 d}-\frac{6 b B \sqrt{\tan (c+d x)} a}{\left(a^2+b^2\right)^2 d (a+b \tan (c+d x))}+\frac{10 B \tan ^{\frac{3}{2}}(c+d x) a}{b^2 d (a+b \tan (c+d x))^2}-\frac{2 A \sqrt{\tan (c+d x)} a}{b^2 d (a+b \tan (c+d x))^2}+\frac{\sqrt[4]{-1} b^3 B \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{\left(a^2+b^2\right)^3 d}+\frac{(-1)^{3/4} A b^3 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{\left(a^2+b^2\right)^3 d}+\frac{\sqrt[4]{-1} b^3 B \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{\left(a^2+b^2\right)^3 d}-\frac{(-1)^{3/4} A b^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{\left(a^2+b^2\right)^3 d}+\frac{2 A b^2 \sqrt{\tan (c+d x)}}{\left(a^2+b^2\right)^2 d (a+b \tan (c+d x))}+\frac{2 B \tan ^{\frac{5}{2}}(c+d x)}{b d (a+b \tan (c+d x))^2}-\frac{2 A \tan ^{\frac{3}{2}}(c+d x)}{b d (a+b \tan (c+d x))^2}+\frac{2 B \sqrt{\tan (c+d x)}}{3 b d (a+b \tan (c+d x))^2}-\frac{2 b B \sqrt{\tan (c+d x)}}{3 \left(a^2+b^2\right) d (a+b \tan (c+d x))^2}","\frac{a (A b-a B) \tan ^{\frac{5}{2}}(c+d x)}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{a \left(-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right) \tan ^{\frac{3}{2}}(c+d x)}{4 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right) \sqrt{\tan (c+d x)}}{4 b^3 d \left(a^2+b^2\right)^2}+\frac{a^{3/2} \left(-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{4 b^{7/2} d \left(a^2+b^2\right)^3}",1,"-(((-1)^(1/4)*a^3*A*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d)) - (3*(-1)^(3/4)*a^2*A*b*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) + (3*(-1)^(1/4)*a*A*b^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) + ((-1)^(3/4)*A*b^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) + ((-1)^(3/4)*a^3*B*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) - (3*(-1)^(1/4)*a^2*b*B*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) - (3*(-1)^(3/4)*a*b^2*B*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) + ((-1)^(1/4)*b^3*B*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) + (3*a^(11/2)*A*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*b^(5/2)*(a^2 + b^2)^3*d) + (3*a^(7/2)*A*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(2*Sqrt[b]*(a^2 + b^2)^3*d) + (35*a^(3/2)*A*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*(a^2 + b^2)^3*d) - (15*a^(13/2)*B*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*b^(7/2)*(a^2 + b^2)^3*d) - (23*a^(9/2)*B*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(2*b^(3/2)*(a^2 + b^2)^3*d) - (63*a^(5/2)*Sqrt[b]*B*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*(a^2 + b^2)^3*d) - ((-1)^(1/4)*a^3*A*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) + (3*(-1)^(3/4)*a^2*A*b*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) + (3*(-1)^(1/4)*a*A*b^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) - ((-1)^(3/4)*A*b^3*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) - ((-1)^(3/4)*a^3*B*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) - (3*(-1)^(1/4)*a^2*b*B*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) + (3*(-1)^(3/4)*a*b^2*B*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) + ((-1)^(1/4)*b^3*B*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) - (2*a*A*Sqrt[Tan[c + d*x]])/(b^2*d*(a + b*Tan[c + d*x])^2) + (a^3*A*Sqrt[Tan[c + d*x]])/(2*b^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (10*a^2*B*Sqrt[Tan[c + d*x]])/(b^3*d*(a + b*Tan[c + d*x])^2) + (2*B*Sqrt[Tan[c + d*x]])/(3*b*d*(a + b*Tan[c + d*x])^2) - (5*a^4*B*Sqrt[Tan[c + d*x]])/(2*b^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (8*a^2*B*Sqrt[Tan[c + d*x]])/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (2*b*B*Sqrt[Tan[c + d*x]])/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (2*A*Tan[c + d*x]^(3/2))/(b*d*(a + b*Tan[c + d*x])^2) + (10*a*B*Tan[c + d*x]^(3/2))/(b^2*d*(a + b*Tan[c + d*x])^2) + (2*B*Tan[c + d*x]^(5/2))/(b*d*(a + b*Tan[c + d*x])^2) + (3*a^2*A*Sqrt[Tan[c + d*x]])/(4*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])) + (3*a^4*A*Sqrt[Tan[c + d*x]])/(4*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])) + (2*A*b^2*Sqrt[Tan[c + d*x]])/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x])) - (15*a^5*B*Sqrt[Tan[c + d*x]])/(4*b^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])) - (31*a^3*B*Sqrt[Tan[c + d*x]])/(4*b*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])) - (6*a*b*B*Sqrt[Tan[c + d*x]])/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))","B",1
411,1,690,534,6.3646221,"\int \frac{\tan ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]","-\frac{2 B \tan ^{\frac{3}{2}}(c+d x)}{b d (a+b \tan (c+d x))^2}-\frac{2 \left(-\frac{(-3 a B-A b) \sqrt{\tan (c+d x)}}{3 b d (a+b \tan (c+d x))^2}-\frac{2 \left(\frac{\left(\frac{1}{4} a b^2 (3 a B+A b)-a \left(-\frac{1}{4} a \left(3 a^2 B+a A b-3 b^2 B\right)-\frac{3 A b^3}{4}\right)\right) \sqrt{\tan (c+d x)}}{2 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{\frac{\left(\frac{3}{8} a^2 b^2 \left(3 a^2 B+a A b+4 b^2 B\right)-a \left(-\frac{3}{8} a^2 \left(3 a^3 B+a^2 A b+4 A b^3\right)-\frac{3}{2} a b^3 (a A+b B)\right)\right) \sqrt{\tan (c+d x)}}{a d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\frac{2 \left(\frac{3}{2} a^3 b^3 \left(a^2 A+2 a b B-A b^2\right)+\frac{3}{16} a^3 b^2 \left(3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right)+\frac{3}{16} a^4 \left(3 a^4 B+a^3 A b+3 a^2 b^2 B+9 a A b^3+8 b^4 B\right)\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d \left(a^2+b^2\right)}+\frac{-\frac{\sqrt[4]{-1} \left(-\frac{3}{2} a^2 b^2 \left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right)+\frac{3}{2} i a^2 b^2 \left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right)\right) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{\sqrt[4]{-1} \left(-\frac{3}{2} a^2 b^2 \left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right)-\frac{3}{2} i a^2 b^2 \left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right)\right) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}}{a^2+b^2}}{a \left(a^2+b^2\right)}}{2 a \left(a^2+b^2\right)}\right)}{3 b}\right)}{b}","\frac{a (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{a \left(3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right) \sqrt{\tan (c+d x)}}{4 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\sqrt{a} \left(3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{4 b^{5/2} d \left(a^2+b^2\right)^3}",1,"(-2*B*Tan[c + d*x]^(3/2))/(b*d*(a + b*Tan[c + d*x])^2) - (2*(-1/3*((-(A*b) - 3*a*B)*Sqrt[Tan[c + d*x]])/(b*d*(a + b*Tan[c + d*x])^2) - (2*((((a*b^2*(A*b + 3*a*B))/4 - a*((-3*A*b^3)/4 - (a*(a*A*b + 3*a^2*B - 3*b^2*B))/4))*Sqrt[Tan[c + d*x]])/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (((2*((3*a^3*b^3*(a^2*A - A*b^2 + 2*a*b*B))/2 + (3*a^3*b^2*(a^2*A*b - 7*A*b^3 + 3*a^3*B + 11*a*b^2*B))/16 + (3*a^4*(a^3*A*b + 9*a*A*b^3 + 3*a^4*B + 3*a^2*b^2*B + 8*b^4*B))/16)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(a^2 + b^2)*d) + (-(((-1)^(1/4)*((-3*a^2*b^2*(3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B))/2 + ((3*I)/2)*a^2*b^2*(a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B))*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - ((-1)^(1/4)*((-3*a^2*b^2*(3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B))/2 - ((3*I)/2)*a^2*b^2*(a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B))*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d)/(a^2 + b^2))/(a*(a^2 + b^2)) + (((3*a^2*b^2*(a*A*b + 3*a^2*B + 4*b^2*B))/8 - a*((-3*a^2*(a^2*A*b + 4*A*b^3 + 3*a^3*B))/8 - (3*a*b^3*(a*A + b*B))/2))*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(2*a*(a^2 + b^2))))/(3*b)))/b","C",1
412,1,333,533,6.0840946,"\int \frac{\tan ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]","\frac{\frac{\left(a^2 B+3 a A b+4 b^2 B\right) \sqrt{\tan (c+d x)}}{a^2+b^2}-\frac{2 (a+b \tan (c+d x)) \left(-\frac{3}{4} a^{5/2} \sqrt{b} \left(a^2+b^2\right) \left(a^3 B+3 a^2 A b+9 a b^2 B-5 A b^3\right) \sqrt{\tan (c+d x)}+(a+b \tan (c+d x)) \left(-\frac{3}{4} a^2 \left(a^5 B+3 a^4 A b+18 a^3 b^2 B-26 a^2 A b^3-15 a b^4 B+3 A b^5\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)-3 \sqrt[4]{-1} a^{5/2} b^{3/2} \left((a+i b)^3 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+(a-i b)^3 (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)\right)\right)}{a^{5/2} \sqrt{b} \left(a^2+b^2\right)^3}-4 B \sqrt{\tan (c+d x)}}{6 b d (a+b \tan (c+d x))^2}","\frac{a (A b-a B) \sqrt{\tan (c+d x)}}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(a^3 B+3 a^2 A b+9 a b^2 B-5 A b^3\right) \sqrt{\tan (c+d x)}}{4 b d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(a^5 B+3 a^4 A b+18 a^3 b^2 B-26 a^2 A b^3-15 a b^4 B+3 A b^5\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{4 \sqrt{a} b^{3/2} d \left(a^2+b^2\right)^3}",1,"(-4*B*Sqrt[Tan[c + d*x]] + ((3*a*A*b + a^2*B + 4*b^2*B)*Sqrt[Tan[c + d*x]])/(a^2 + b^2) - (2*(a + b*Tan[c + d*x])*((-3*a^(5/2)*Sqrt[b]*(a^2 + b^2)*(3*a^2*A*b - 5*A*b^3 + a^3*B + 9*a*b^2*B)*Sqrt[Tan[c + d*x]])/4 + ((-3*a^2*(3*a^4*A*b - 26*a^2*A*b^3 + 3*A*b^5 + a^5*B + 18*a^3*b^2*B - 15*a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/4 - 3*(-1)^(1/4)*a^(5/2)*b^(3/2)*((a + I*b)^3*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a - I*b)^3*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))*(a + b*Tan[c + d*x])))/(a^(5/2)*Sqrt[b]*(a^2 + b^2)^3))/(6*b*d*(a + b*Tan[c + d*x])^2)","C",1
413,1,552,531,6.2763385,"\int \frac{\sqrt{\tan (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]","\frac{b (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{2 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{-\frac{(A b-a B) \sqrt{\tan (c+d x)}}{d (a+b \tan (c+d x))}-\frac{2 \left(\frac{\left(-a \left(-\frac{3}{4} a^2 b (A b-a B)-a b^2 (a A+b B)\right)-\frac{1}{4} a b^3 (A b-a B)\right) \sqrt{\tan (c+d x)}}{a d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\frac{2 \left(a^3 b^2 \left(a^2 A+2 a b B-A b^2\right)+\frac{1}{8} a^3 b \left(-3 a^3 B+7 a^2 A b+5 a b^2 B-A b^3\right)-\frac{1}{8} a b^3 \left(-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right)\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d \left(a^2+b^2\right)}+\frac{-\frac{\sqrt[4]{-1} \left(-a^2 b \left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right)+i a^2 b \left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right)\right) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{\sqrt[4]{-1} \left(a^2 (-b) \left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right)-i a^2 b \left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right)\right) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}}{a^2+b^2}}{a \left(a^2+b^2\right)}\right)}{b}}{2 a \left(a^2+b^2\right)}","-\frac{(A b-a B) \sqrt{\tan (c+d x)}}{2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(-3 a^3 B+7 a^2 A b+5 a b^2 B-A b^3\right) \sqrt{\tan (c+d x)}}{4 a d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{3/2} \sqrt{b} d \left(a^2+b^2\right)^3}",1,"(b*(A*b - a*B)*Tan[c + d*x]^(3/2))/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (-(((A*b - a*B)*Sqrt[Tan[c + d*x]])/(d*(a + b*Tan[c + d*x]))) - (2*(((2*(a^3*b^2*(a^2*A - A*b^2 + 2*a*b*B) - (a*b^3*(9*a^2*A*b + A*b^3 - 5*a^3*B + 3*a*b^2*B))/8 + (a^3*b*(7*a^2*A*b - A*b^3 - 3*a^3*B + 5*a*b^2*B))/8)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(a^2 + b^2)*d) + (-(((-1)^(1/4)*(-(a^2*b*(3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)) + I*a^2*b*(a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B))*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - ((-1)^(1/4)*(-(a^2*b*(3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)) - I*a^2*b*(a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B))*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d)/(a^2 + b^2))/(a*(a^2 + b^2)) + ((-1/4*(a*b^3*(A*b - a*B)) - a*((-3*a^2*b*(A*b - a*B))/4 - a*b^2*(a*A + b*B)))*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))))/b)/(2*a*(a^2 + b^2))","C",1
414,1,288,534,4.6848441,"\int \frac{A+B \tan (c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^3} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^3),x]","\frac{\frac{b \left(-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right) \sqrt{\tan (c+d x)}}{a \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{2 \left(\frac{1}{2} \sqrt{b} \left(-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)-2 \sqrt[4]{-1} a^{5/2} \left((a+i b)^3 (A-i B) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+(a-i b)^3 (A+i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)\right)}{a^{3/2} \left(a^2+b^2\right)^2}+\frac{2 b (A b-a B) \sqrt{\tan (c+d x)}}{(a+b \tan (c+d x))^2}}{4 a d \left(a^2+b^2\right)}","\frac{b (A b-a B) \sqrt{\tan (c+d x)}}{2 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{b \left(-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right) \sqrt{\tan (c+d x)}}{4 a^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\sqrt{b} \left(-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{5/2} d \left(a^2+b^2\right)^3}",1,"((2*((Sqrt[b]*(35*a^4*A*b + 6*a^2*A*b^3 + 3*A*b^5 - 15*a^5*B + 18*a^3*b^2*B + a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/2 - 2*(-1)^(1/4)*a^(5/2)*((a + I*b)^3*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a - I*b)^3*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])))/(a^(3/2)*(a^2 + b^2)^2) + (2*b*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(a + b*Tan[c + d*x])^2 + (b*(11*a^2*A*b + 3*A*b^3 - 7*a^3*B + a*b^2*B)*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(4*a*(a^2 + b^2)*d)","C",1
415,1,585,601,6.2982622,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3),x]","\frac{b (A b-a B)}{2 a d \left(a^2+b^2\right) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac{\frac{\frac{1}{2} b^2 \left(4 a^2 A-a b B+5 A b^2\right)+\frac{9}{2} a^2 b (A b-a B)}{a d \left(a^2+b^2\right) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))}+\frac{-\frac{8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4}{2 a d \sqrt{\tan (c+d x)}}-\frac{2 \left(\frac{2 \left(a^4 (-b) \left(a^2 A+2 a b B-A b^2\right)+\frac{1}{8} a^2 b \left(8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right)+\frac{1}{8} b^2 \left(-8 a^5 B+24 a^4 A b-3 a^3 b^2 B+31 a^2 A b^3-3 a b^4 B+15 A b^5\right)\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d \left(a^2+b^2\right)}+\frac{-\frac{\sqrt[4]{-1} \left(a^3 \left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right)-i a^3 \left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right)\right) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{\sqrt[4]{-1} \left(a^3 \left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right)+i a^3 \left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right)\right) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}}{a^2+b^2}\right)}{a}}{a \left(a^2+b^2\right)}}{2 a \left(a^2+b^2\right)}","\frac{b (A b-a B)}{2 a d \left(a^2+b^2\right) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{b \left(-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right)}{4 a^2 d \left(a^2+b^2\right)^2 \sqrt{\tan (c+d x)} (a+b \tan (c+d x))}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4}{4 a^3 d \left(a^2+b^2\right)^2 \sqrt{\tan (c+d x)}}-\frac{b^{3/2} \left(-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{7/2} d \left(a^2+b^2\right)^3}",1,"(b*(A*b - a*B))/(2*a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2) + (((-2*((2*(-(a^4*b*(a^2*A - A*b^2 + 2*a*b*B)) + (a^2*b*(8*a^4*A + 31*a^2*A*b^2 + 15*A*b^4 - 11*a^3*b*B - 3*a*b^3*B))/8 + (b^2*(24*a^4*A*b + 31*a^2*A*b^3 + 15*A*b^5 - 8*a^5*B - 3*a^3*b^2*B - 3*a*b^4*B))/8)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(a^2 + b^2)*d) + (-(((-1)^(1/4)*(a^3*(3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B) - I*a^3*(a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B))*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - ((-1)^(1/4)*(a^3*(3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B) + I*a^3*(a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B))*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d)/(a^2 + b^2)))/a - (8*a^4*A + 31*a^2*A*b^2 + 15*A*b^4 - 11*a^3*b*B - 3*a*b^3*B)/(2*a*d*Sqrt[Tan[c + d*x]]))/(a*(a^2 + b^2)) + ((9*a^2*b*(A*b - a*B))/2 + (b^2*(4*a^2*A + 5*A*b^2 - a*b*B))/2)/(a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])))/(2*a*(a^2 + b^2))","C",1
416,1,38,156,0.0442791,"\int \frac{\tan ^{\frac{5}{2}}(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]^(5/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","-\frac{2 B \tan ^{\frac{3}{2}}(c+d x) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)-1\right)}{3 d}","\frac{B \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{B \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 B \tan ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{B \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{B \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}",1,"(-2*B*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2])*Tan[c + d*x]^(3/2))/(3*d)","C",1
417,1,138,154,0.1244035,"\int \frac{\tan ^{\frac{3}{2}}(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]^(3/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{B \left(2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+8 \sqrt{\tan (c+d x)}+\sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 d}","\frac{B \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{B \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 B \sqrt{\tan (c+d x)}}{d}+\frac{B \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{B \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}",1,"(B*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 8*Sqrt[Tan[c + d*x]]))/(4*d)","A",1
418,1,36,138,0.0209788,"\int \frac{\sqrt{\tan (c+d x)} (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Sqrt[Tan[c + d*x]]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{2 B \tan ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)}{3 d}","-\frac{B \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{B \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{B \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{B \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}",1,"(2*B*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2))/(3*d)","C",1
419,1,110,138,0.0328421,"\int \frac{a B+b B \tan (c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])),x]","\frac{B \left(-2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{2 \sqrt{2} d}","-\frac{B \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{B \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{B \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{B \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}",1,"(B*(-2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] - Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(2*Sqrt[2]*d)","A",1
420,1,34,154,0.0244928,"\int \frac{a B+b B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])),x]","-\frac{2 B \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)}{d \sqrt{\tan (c+d x)}}","\frac{B \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{B \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 B}{d \sqrt{\tan (c+d x)}}-\frac{B \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{B \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}",1,"(-2*B*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2])/(d*Sqrt[Tan[c + d*x]])","C",1
421,1,36,156,0.0430301,"\int \frac{a B+b B \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])),x]","-\frac{2 B \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(c+d x)\right)}{3 d \tan ^{\frac{3}{2}}(c+d x)}","\frac{B \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{B \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 B}{3 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{B \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{B \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}",1,"(-2*B*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2])/(3*d*Tan[c + d*x]^(3/2))","C",1
422,1,156,256,0.2050753,"\int \frac{\tan ^{\frac{5}{2}}(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^(5/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{B \left(-2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+2 a^2 \sqrt{b} \sqrt{\tan (c+d x)}+\sqrt[4]{-1} b^{3/2} (b-i a) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+\sqrt[4]{-1} b^{3/2} (b+i a) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+2 b^{5/2} \sqrt{\tan (c+d x)}\right)}{b^{3/2} d \left(a^2+b^2\right)}","\frac{B (a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{B (a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{B (a-b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{B (a-b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{2 a^{5/2} B \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{3/2} d \left(a^2+b^2\right)}+\frac{2 B \sqrt{\tan (c+d x)}}{b d}",1,"(B*((-1)^(1/4)*b^(3/2)*((-I)*a + b)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - 2*a^(5/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + (-1)^(1/4)*b^(3/2)*(I*a + b)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*a^2*Sqrt[b]*Sqrt[Tan[c + d*x]] + 2*b^(5/2)*Sqrt[Tan[c + d*x]]))/(b^(3/2)*(a^2 + b^2)*d)","C",1
423,1,228,237,0.2905832,"\int \frac{\tan ^{\frac{3}{2}}(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^(3/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{B \left(3 a \left(8 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+2 \sqrt{2} \sqrt{b} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-2 \sqrt{2} \sqrt{b} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\sqrt{2} \sqrt{b} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\sqrt{2} \sqrt{b} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+8 b^{3/2} \tan ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)\right)}{12 \sqrt{b} d \left(a^2+b^2\right)}","\frac{B (a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{B (a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{B (a+b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{B (a+b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{2 a^{3/2} B \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} d \left(a^2+b^2\right)}",1,"(B*(3*a*(2*Sqrt[2]*Sqrt[b]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*Sqrt[2]*Sqrt[b]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + 8*Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + Sqrt[2]*Sqrt[b]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Sqrt[b]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) + 8*b^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2)))/(12*Sqrt[b]*(a^2 + b^2)*d)","C",1
424,1,205,237,0.1779403,"\int \frac{\sqrt{\tan (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Sqrt[Tan[c + d*x]]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{B \left(-24 \sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+8 a \tan ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)-6 \sqrt{2} b \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+6 \sqrt{2} b \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-3 \sqrt{2} b \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+3 \sqrt{2} b \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{12 d \left(a^2+b^2\right)}","-\frac{B (a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{B (a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{2 \sqrt{a} \sqrt{b} B \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{d \left(a^2+b^2\right)}+\frac{B (a-b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{B (a-b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}",1,"(B*(-6*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 6*Sqrt[2]*b*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] - 24*Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] - 3*Sqrt[2]*b*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 3*Sqrt[2]*b*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 8*a*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2)))/(12*(a^2 + b^2)*d)","C",1
425,1,226,237,0.1871029,"\int \frac{a B+b B \tan (c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^2} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2),x]","\frac{B \left(-6 \sqrt{2} a^{3/2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+6 \sqrt{2} a^{3/2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-3 \sqrt{2} a^{3/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+3 \sqrt{2} a^{3/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+24 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)-8 \sqrt{a} b \tan ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)\right)}{12 \sqrt{a} d \left(a^2+b^2\right)}","-\frac{B (a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{B (a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{B (a+b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{B (a+b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{2 b^{3/2} B \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d \left(a^2+b^2\right)}",1,"(B*(-6*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 6*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + 24*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] - 3*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 3*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 8*Sqrt[a]*b*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2)))/(12*Sqrt[a]*(a^2 + b^2)*d)","C",1
426,1,132,256,0.532189,"\int \frac{a B+b B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2),x]","\frac{B \left(-\frac{2 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{2 \left(a^2+b^2\right)}{a \sqrt{\tan (c+d x)}}-(-1)^{3/4} (a+i b) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+\sqrt[4]{-1} (b+i a) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{d \left(a^2+b^2\right)}","\frac{B (a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{B (a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{B (a-b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{B (a-b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{2 b^{5/2} B \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d \left(a^2+b^2\right)}-\frac{2 B}{a d \sqrt{\tan (c+d x)}}",1,"(B*(-((-1)^(3/4)*(a + I*b)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]) - (2*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/a^(3/2) + (-1)^(1/4)*(I*a + b)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - (2*(a^2 + b^2))/(a*Sqrt[Tan[c + d*x]])))/((a^2 + b^2)*d)","C",1
427,1,304,264,3.461845,"\int \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{-\sqrt{a} \left(a^2 B-4 a A b+8 b^2 B\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+\sqrt{b} \left(-4 \sqrt[4]{-1} b \sqrt{-a+i b} (B+i A) \sqrt{a+b \tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+4 (-1)^{3/4} b \sqrt{a+i b} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right) \sqrt{a+b \tan (c+d x)}+\sqrt{\tan (c+d x)} (a+b \tan (c+d x)) (a B+4 A b+2 b B \tan (c+d x))\right)}{4 b^{3/2} d \sqrt{a+b \tan (c+d x)}}","\frac{\left(a^2 (-B)+4 a A b-8 b^2 B\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{4 b^{3/2} d}+\frac{\sqrt{-b+i a} (-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(4 A b-a B) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b d}+\frac{\sqrt{b+i a} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{B \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 b d}",1,"(-(Sqrt[a]*(-4*a*A*b + a^2*B + 8*b^2*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a]) + Sqrt[b]*(-4*(-1)^(1/4)*Sqrt[-a + I*b]*b*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + 4*(-1)^(3/4)*Sqrt[a + I*b]*b*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])*(4*A*b + a*B + 2*b*B*Tan[c + d*x])))/(4*b^(3/2)*d*Sqrt[a + b*Tan[c + d*x]])","A",1
428,1,239,201,2.8851028,"\int \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{\sqrt[4]{-1} \sqrt{-a+i b} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt[4]{-1} \sqrt{a+i b} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\frac{(a B+2 A b) \sqrt{a+b \tan (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} \sqrt{\frac{b \tan (c+d x)}{a}+1}}+B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}","\frac{\sqrt{-b+i a} (A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(a B+2 A b) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{b} d}-\frac{\sqrt{b+i a} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}",1,"((-1)^(1/4)*Sqrt[-a + I*b]*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (-1)^(1/4)*Sqrt[a + I*b]*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + B*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + ((2*A*b + a*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[a + b*Tan[c + d*x]])/(Sqrt[a]*Sqrt[b]*Sqrt[1 + (b*Tan[c + d*x])/a]))/d","A",1
429,1,204,169,0.888931,"\int \frac{\sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","Integrate[(Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]],x]","\frac{\frac{2 \sqrt{a} \sqrt{b} B \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a+b \tan (c+d x)}}+\sqrt[4]{-1} \left(\sqrt{-a+i b} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt{a+i b} (B-i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{d}","-\frac{\sqrt{-b+i a} (-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{\sqrt{b+i a} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{2 \sqrt{b} B \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"((-1)^(1/4)*(Sqrt[-a + I*b]*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + Sqrt[a + I*b]*((-I)*A + B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]) + (2*Sqrt[a]*Sqrt[b]*B*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]])/d","A",1
430,1,169,154,0.8544783,"\int \frac{\sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2),x]","-\frac{\sqrt[4]{-1} \sqrt{-a+i b} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt[4]{-1} \sqrt{a+i b} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\frac{2 A \sqrt{a+b \tan (c+d x)}}{\sqrt{\tan (c+d x)}}}{d}","-\frac{\sqrt{-b+i a} (A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{\sqrt{b+i a} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 A \sqrt{a+b \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}",1,"-(((-1)^(1/4)*Sqrt[-a + I*b]*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (-1)^(1/4)*Sqrt[a + I*b]*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (2*A*Sqrt[a + b*Tan[c + d*x]])/Sqrt[Tan[c + d*x]])/d)","A",1
431,1,194,199,1.5348321,"\int \frac{\sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2),x]","\frac{-3 \sqrt[4]{-1} \sqrt{-a+i b} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+3 (-1)^{3/4} \sqrt{a+i b} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-\frac{2 \sqrt{a+b \tan (c+d x)} ((3 a B+A b) \tan (c+d x)+a A)}{a \tan ^{\frac{3}{2}}(c+d x)}}{3 d}","\frac{\sqrt{-b+i a} (-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 (3 a B+A b) \sqrt{a+b \tan (c+d x)}}{3 a d \sqrt{\tan (c+d x)}}+\frac{\sqrt{b+i a} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 A \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}",1,"(-3*(-1)^(1/4)*Sqrt[-a + I*b]*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 3*(-1)^(3/4)*Sqrt[a + I*b]*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - (2*Sqrt[a + b*Tan[c + d*x]]*(a*A + (A*b + 3*a*B)*Tan[c + d*x]))/(a*Tan[c + d*x]^(3/2)))/(3*d)","A",1
432,1,226,250,2.5694927,"\int \frac{\sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2),x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left(\left(15 a^2 A-5 a b B+2 A b^2\right) \tan ^2(c+d x)-3 a^2 A-a (5 a B+A b) \tan (c+d x)\right)}{a^2 \tan ^{\frac{5}{2}}(c+d x)}+15 \sqrt[4]{-1} \sqrt{-a+i b} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+15 \sqrt[4]{-1} \sqrt{a+i b} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{15 d}","\frac{2 \left(15 a^2 A-5 a b B+2 A b^2\right) \sqrt{a+b \tan (c+d x)}}{15 a^2 d \sqrt{\tan (c+d x)}}+\frac{\sqrt{-b+i a} (A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 (5 a B+A b) \sqrt{a+b \tan (c+d x)}}{15 a d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\sqrt{b+i a} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 A \sqrt{a+b \tan (c+d x)}}{5 d \tan ^{\frac{5}{2}}(c+d x)}",1,"(15*(-1)^(1/4)*Sqrt[-a + I*b]*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 15*(-1)^(1/4)*Sqrt[a + I*b]*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (2*Sqrt[a + b*Tan[c + d*x]]*(-3*a^2*A - a*(A*b + 5*a*B)*Tan[c + d*x] + (15*a^2*A + 2*A*b^2 - 5*a*b*B)*Tan[c + d*x]^2))/(a^2*Tan[c + d*x]^(5/2)))/(15*d)","A",1
433,1,265,314,3.9820934,"\int \frac{\sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2),x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left(-15 a^3 A+a \left(35 a^2 A-7 a b B+4 A b^2\right) \tan ^2(c+d x)-3 a^2 (7 a B+A b) \tan (c+d x)+\left(105 a^3 B+35 a^2 A b+14 a b^2 B-8 A b^3\right) \tan ^3(c+d x)\right)}{a^3 \tan ^{\frac{7}{2}}(c+d x)}+105 \sqrt[4]{-1} \sqrt{-a+i b} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-105 (-1)^{3/4} \sqrt{a+i b} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{105 d}","\frac{2 \left(35 a^2 A-7 a b B+4 A b^2\right) \sqrt{a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(105 a^3 B+35 a^2 A b+14 a b^2 B-8 A b^3\right) \sqrt{a+b \tan (c+d x)}}{105 a^3 d \sqrt{\tan (c+d x)}}-\frac{\sqrt{-b+i a} (-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 (7 a B+A b) \sqrt{a+b \tan (c+d x)}}{35 a d \tan ^{\frac{5}{2}}(c+d x)}-\frac{\sqrt{b+i a} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 A \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}",1,"(105*(-1)^(1/4)*Sqrt[-a + I*b]*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 105*(-1)^(3/4)*Sqrt[a + I*b]*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (2*Sqrt[a + b*Tan[c + d*x]]*(-15*a^3*A - 3*a^2*(A*b + 7*a*B)*Tan[c + d*x] + a*(35*a^2*A + 4*A*b^2 - 7*a*b*B)*Tan[c + d*x]^2 + (35*a^2*A*b - 8*A*b^3 + 105*a^3*B + 14*a*b^2*B)*Tan[c + d*x]^3))/(a^3*Tan[c + d*x]^(7/2)))/(105*d)","A",1
434,1,347,323,4.4088828,"\int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{-3 \left(a^2 B-6 a A b+8 b^2 B\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}-\frac{3 \sqrt{a} \left(a^3 B-6 a^2 A b+24 a b^2 B+16 A b^3\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+24 \sqrt[4]{-1} b (-a+i b)^{3/2} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+24 (-1)^{3/4} b (a+i b)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+2 (6 A b-a B) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}+8 B \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{24 b d}","\frac{\left(a^2 (-B)+6 a A b-8 b^2 B\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{8 b d}+\frac{\left(a^3 (-B)+6 a^2 A b-24 a b^2 B-16 A b^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{8 b^{3/2} d}+\frac{(-b+i a)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(6 A b-a B) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{12 b d}+\frac{(b+i a)^{3/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{B \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 b d}",1,"(24*(-1)^(1/4)*(-a + I*b)^(3/2)*b*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 24*(-1)^(3/4)*(a + I*b)^(3/2)*b*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 3*(-6*a*A*b + a^2*B + 8*b^2*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + 2*(6*A*b - a*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2) + 8*B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2) - (3*Sqrt[a]*(-6*a^2*A*b + 16*A*b^3 + a^3*B + 24*a*b^2*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]]))/(24*b*d)","A",1
435,1,290,268,2.5287585,"\int \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{\frac{\sqrt{a} \left(3 a^2 B+12 a A b-8 b^2 B\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}-4 \sqrt[4]{-1} (-a+i b)^{3/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+4 \sqrt[4]{-1} (a+i b)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+(5 a B+4 A b) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}+2 b B \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}","\frac{\left(3 a^2 B+12 a A b-8 b^2 B\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{4 \sqrt{b} d}+\frac{(a+i b)^2 (-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{(5 a B+4 A b) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 d}+\frac{(b+i a)^{3/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b B \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}",1,"(-4*(-1)^(1/4)*(-a + I*b)^(3/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 4*(-1)^(1/4)*(a + I*b)^(3/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (4*A*b + 5*a*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + 2*b*B*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]] + (Sqrt[a]*(12*a*A*b + 3*a^2*B - 8*b^2*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]]))/(4*d)","A",1
436,1,243,204,1.0021097,"\int \frac{(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","Integrate[((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]],x]","\frac{-\sqrt[4]{-1} (-a+i b)^{3/2} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-(-1)^{3/4} (a+i b)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\frac{\sqrt{a} \sqrt{b} (3 a B+2 A b) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a+b \tan (c+d x)}}+b B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}","-\frac{(-b+i a)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{\sqrt{b} (3 a B+2 A b) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{(b+i a)^{3/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}",1,"(-((-1)^(1/4)*(-a + I*b)^(3/2)*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]) - (-1)^(3/4)*(a + I*b)^(3/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + b*B*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + (Sqrt[a]*Sqrt[b]*(2*A*b + 3*a*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]])/d","A",1
437,1,121803,209,39.2475614,"\int \frac{(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2),x]","\text{Result too large to show}","-\frac{(a+i b)^2 (-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{(b+i a)^{3/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \sqrt{a+b \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}+\frac{2 b^{3/2} B \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"Result too large to show","C",0
438,1,238,196,1.0075197,"\int \frac{(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2),x]","\frac{3 \sqrt[4]{-1} \tan ^{\frac{3}{2}}(c+d x) \left((-a+i b)^{3/2} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+i (a+i b)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)-2 (3 a B+4 A b) \tan (c+d x) \sqrt{a+b \tan (c+d x)}+(3 b B-2 a A) \sqrt{a+b \tan (c+d x)}-3 b B \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}","\frac{(-b+i a)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 (3 a B+4 A b) \sqrt{a+b \tan (c+d x)}}{3 d \sqrt{\tan (c+d x)}}+\frac{(b+i a)^{3/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}",1,"(3*(-1)^(1/4)*((-a + I*b)^(3/2)*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + I*(a + I*b)^(3/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])*Tan[c + d*x]^(3/2) - 3*b*B*Sqrt[a + b*Tan[c + d*x]] + (-2*a*A + 3*b*B)*Sqrt[a + b*Tan[c + d*x]] - 2*(4*A*b + 3*a*B)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(3*d*Tan[c + d*x]^(3/2))","A",1
439,1,286,259,2.9979884,"\int \frac{(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2),x]","\frac{4 \left(15 a^2 A-20 a b B-3 A b^2\right) \tan ^2(c+d x) \sqrt{a+b \tan (c+d x)}-30 \sqrt[4]{-1} a \tan ^{\frac{5}{2}}(c+d x) \left((-a+i b)^{3/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-(a+i b)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)-4 a (5 a B+6 A b) \tan (c+d x) \sqrt{a+b \tan (c+d x)}-3 a (4 a A-5 b B) \sqrt{a+b \tan (c+d x)}-15 a b B \sqrt{a+b \tan (c+d x)}}{30 a d \tan ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(15 a^2 A-20 a b B-3 A b^2\right) \sqrt{a+b \tan (c+d x)}}{15 a d \sqrt{\tan (c+d x)}}+\frac{(a+i b)^2 (-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{2 (5 a B+6 A b) \sqrt{a+b \tan (c+d x)}}{15 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{(b+i a)^{3/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \sqrt{a+b \tan (c+d x)}}{5 d \tan ^{\frac{5}{2}}(c+d x)}",1,"(-30*(-1)^(1/4)*a*((-a + I*b)^(3/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - (a + I*b)^(3/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])*Tan[c + d*x]^(5/2) - 15*a*b*B*Sqrt[a + b*Tan[c + d*x]] - 3*a*(4*a*A - 5*b*B)*Sqrt[a + b*Tan[c + d*x]] - 4*a*(6*A*b + 5*a*B)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]] + 4*(15*a^2*A - 3*A*b^2 - 20*a*b*B)*Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(30*a*d*Tan[c + d*x]^(5/2))","A",1
440,1,346,311,5.4010096,"\int \frac{(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2),x]","\frac{-5 a^3 (6 a A-7 b B) \sqrt{a+b \tan (c+d x)}-6 a^3 (7 a B+8 A b) \tan (c+d x) \sqrt{a+b \tan (c+d x)}-35 a^3 b B \sqrt{a+b \tan (c+d x)}+a \tan ^2(c+d x) \left(2 a \left(35 a^2 A-42 a b B-3 A b^2\right) \sqrt{a+b \tan (c+d x)}-105 (-1)^{3/4} a^2 \tan ^{\frac{3}{2}}(c+d x) \left((-a+i b)^{3/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+(a+i b)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)+2 \left(105 a^3 B+140 a^2 A b-21 a b^2 B+6 A b^3\right) \tan (c+d x) \sqrt{a+b \tan (c+d x)}\right)}{105 a^3 d \tan ^{\frac{7}{2}}(c+d x)}","\frac{2 \left(35 a^2 A-42 a b B-3 A b^2\right) \sqrt{a+b \tan (c+d x)}}{105 a d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(105 a^3 B+140 a^2 A b-21 a b^2 B+6 A b^3\right) \sqrt{a+b \tan (c+d x)}}{105 a^2 d \sqrt{\tan (c+d x)}}-\frac{(-b+i a)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 (7 a B+8 A b) \sqrt{a+b \tan (c+d x)}}{35 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{(b+i a)^{3/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}",1,"(-35*a^3*b*B*Sqrt[a + b*Tan[c + d*x]] - 5*a^3*(6*a*A - 7*b*B)*Sqrt[a + b*Tan[c + d*x]] - 6*a^3*(8*A*b + 7*a*B)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]] + a*Tan[c + d*x]^2*(-105*(-1)^(3/4)*a^2*((-a + I*b)^(3/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (a + I*b)^(3/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])*Tan[c + d*x]^(3/2) + 2*a*(35*a^2*A - 3*A*b^2 - 42*a*b*B)*Sqrt[a + b*Tan[c + d*x]] + 2*(140*a^2*A*b + 6*A*b^3 + 105*a^3*B - 21*a*b^2*B)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]]))/(105*a^3*d*Tan[c + d*x]^(7/2))","A",1
441,1,474,382,6.6945354,"\int \frac{(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(11/2),x]","-\frac{b B \sqrt{a+b \tan (c+d x)}}{4 d \tan ^{\frac{9}{2}}(c+d x)}+\frac{1}{4} \left(-\frac{(8 a A-9 b B) \sqrt{a+b \tan (c+d x)}}{9 d \tan ^{\frac{9}{2}}(c+d x)}+\frac{2 \left(-\frac{4 a (9 a B+10 A b) \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 \left(-\frac{6 a \left(21 a^2 A-24 a b B-A b^2\right) \sqrt{a+b \tan (c+d x)}}{5 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 \left(\frac{a \left(105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right) \sqrt{a+b \tan (c+d x)}}{d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(\frac{3 a \left(315 a^4 A-420 a^3 b B-63 a^2 A b^2-18 a b^3 B+8 A b^4\right) \sqrt{a+b \tan (c+d x)}}{2 d \sqrt{\tan (c+d x)}}-\frac{945 a^4 \left(\sqrt[4]{-1} (-a+i b)^{3/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-\sqrt[4]{-1} (a+i b)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{4 d}\right)}{3 a}\right)}{5 a}\right)}{7 a}\right)}{9 a}\right)","\frac{2 \left(21 a^2 A-24 a b B-A b^2\right) \sqrt{a+b \tan (c+d x)}}{105 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right) \sqrt{a+b \tan (c+d x)}}{315 a^2 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(315 a^4 A-420 a^3 b B-63 a^2 A b^2-18 a b^3 B+8 A b^4\right) \sqrt{a+b \tan (c+d x)}}{315 a^3 d \sqrt{\tan (c+d x)}}+\frac{(-b+i a)^{3/2} (-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 (9 a B+10 A b) \sqrt{a+b \tan (c+d x)}}{63 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{(b+i a)^{3/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \sqrt{a+b \tan (c+d x)}}{9 d \tan ^{\frac{9}{2}}(c+d x)}",1,"-1/4*(b*B*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(9/2)) + (-1/9*((8*a*A - 9*b*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(9/2)) + (2*((-4*a*(10*A*b + 9*a*B)*Sqrt[a + b*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (2*((-6*a*(21*a^2*A - A*b^2 - 24*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - (2*((a*(126*a^2*A*b + 4*A*b^3 + 105*a^3*B - 9*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(3/2)) - (2*((-945*a^4*((-1)^(1/4)*(-a + I*b)^(3/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - (-1)^(1/4)*(a + I*b)^(3/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]))/(4*d) + (3*a*(315*a^4*A - 63*a^2*A*b^2 + 8*A*b^4 - 420*a^3*b*B - 18*a*b^3*B)*Sqrt[a + b*Tan[c + d*x]])/(2*d*Sqrt[Tan[c + d*x]])))/(3*a)))/(5*a)))/(7*a)))/(9*a))/4","A",1
442,1,411,397,4.7049646,"\int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{-2 \left(5 a^2 B-40 a A b+48 b^2 B\right) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}-3 \left(5 a^3 B-40 a^2 A b+112 a b^2 B+64 A b^3\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}-\frac{3 \sqrt{a} \left(5 a^4 B-40 a^3 A b+240 a^2 b^2 B+320 a A b^3-128 b^4 B\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}-192 \sqrt[4]{-1} b (-a+i b)^{5/2} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+192 (-1)^{3/4} b (a+i b)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+8 (8 A b-a B) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{5/2}+48 B \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{192 b d}","\frac{\left(-5 a^2 B+40 a A b-48 b^2 B\right) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{96 b d}+\frac{\left(-5 a^3 B+40 a^2 A b-112 a b^2 B-64 A b^3\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{64 b d}+\frac{\left(-5 a^4 B+40 a^3 A b-240 a^2 b^2 B-320 a A b^3+128 b^4 B\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{64 b^{3/2} d}-\frac{(-b+i a)^{5/2} (-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(8 A b-a B) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{24 b d}-\frac{(b+i a)^{5/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{B \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}",1,"(-192*(-1)^(1/4)*(-a + I*b)^(5/2)*b*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 192*(-1)^(3/4)*(a + I*b)^(5/2)*b*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 3*(-40*a^2*A*b + 64*A*b^3 + 5*a^3*B + 112*a*b^2*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] - 2*(-40*a*A*b + 5*a^2*B + 48*b^2*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2) + 8*(8*A*b - a*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2) + 48*B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(7/2) - (3*Sqrt[a]*(-40*a^3*A*b + 320*a*A*b^3 + 5*a^4*B + 240*a^2*b^2*B - 128*b^4*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]]))/(192*b*d)","A",1
443,1,345,316,4.5632055,"\int \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{3 \left(5 a^2 B+14 a A b-8 b^2 B\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}+\frac{3 \sqrt{a} \left(5 a^3 B+30 a^2 A b-40 a b^2 B-16 A b^3\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+24 \sqrt[4]{-1} (-a+i b)^{5/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+24 \sqrt[4]{-1} (a+i b)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+6 (3 a B+2 A b) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}+8 b B \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{24 d}","\frac{\left(5 a^2 B+14 a A b-8 b^2 B\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{8 d}+\frac{\left(5 a^3 B+30 a^2 A b-40 a b^2 B-16 A b^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{8 \sqrt{b} d}-\frac{(-b+i a)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(3 a B+2 A b) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{4 d}+\frac{(b+i a)^{5/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b B \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d}",1,"(24*(-1)^(1/4)*(-a + I*b)^(5/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 24*(-1)^(1/4)*(a + I*b)^(5/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 3*(14*a*A*b + 5*a^2*B - 8*b^2*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + 6*(2*A*b + 3*a*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2) + 8*b*B*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2) + (3*Sqrt[a]*(30*a^2*A*b - 16*A*b^3 + 5*a^3*B - 40*a*b^2*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]]))/(24*d)","A",1
444,1,291,260,2.6744601,"\int \frac{(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","Integrate[((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]],x]","\frac{\frac{\sqrt{a} \sqrt{b} \left(15 a^2 B+20 a A b-8 b^2 B\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a+b \tan (c+d x)}}+4 \sqrt[4]{-1} (-a+i b)^{5/2} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-4 (-1)^{3/4} (a+i b)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+b (7 a B+4 A b) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}+2 b B \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{4 d}","\frac{\sqrt{b} \left(15 a^2 B+20 a A b-8 b^2 B\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{4 d}+\frac{(-b+i a)^{5/2} (-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b (7 a B+4 A b) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 d}+\frac{(b+i a)^{5/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b B \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}",1,"(4*(-1)^(1/4)*(-a + I*b)^(5/2)*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 4*(-1)^(3/4)*(a + I*b)^(5/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + b*(4*A*b + 7*a*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + 2*b*B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2) + (Sqrt[a]*Sqrt[b]*(20*a*A*b + 15*a^2*B - 8*b^2*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]])/(4*d)","A",1
445,1,209298,241,41.1766923,"\int \frac{(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2),x]","\text{Result too large to show}","\frac{b^{3/2} (5 a B+2 A b) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(-b+i a)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b (2 a A+b B) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}-\frac{(b+i a)^{5/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A (a+b \tan (c+d x))^{3/2}}{d \sqrt{\tan (c+d x)}}",1,"Result too large to show","C",0
446,1,139636,240,39.7843943,"\int \frac{(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2),x]","\text{Result too large to show}","-\frac{(-b+i a)^{5/2} (-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a (a B+2 A b) \sqrt{a+b \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}-\frac{(b+i a)^{5/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A (a+b \tan (c+d x))^{3/2}}{3 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 b^{5/2} B \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"Result too large to show","C",0
447,1,321,247,2.0216389,"\int \frac{(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2),x]","\frac{8 \left(15 a^2 A-35 a b B-23 A b^2\right) \tan ^2(c+d x) \sqrt{a+b \tan (c+d x)}-4 \left(10 a^2 B+22 a A b-15 b^2 B\right) \tan (c+d x) \sqrt{a+b \tan (c+d x)}-3 \left(8 a^2 A-15 a b B-10 A b^2\right) \sqrt{a+b \tan (c+d x)}+60 \sqrt[4]{-1} \tan ^{\frac{5}{2}}(c+d x) \left((-a+i b)^{5/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+(a+i b)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)+15 b (a B-2 A b) \sqrt{a+b \tan (c+d x)}-60 b B (a+b \tan (c+d x))^{3/2}}{60 d \tan ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(15 a^2 A-35 a b B-23 A b^2\right) \sqrt{a+b \tan (c+d x)}}{15 d \sqrt{\tan (c+d x)}}-\frac{(-b+i a)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a (5 a B+8 A b) \sqrt{a+b \tan (c+d x)}}{15 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{(b+i a)^{5/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A (a+b \tan (c+d x))^{3/2}}{5 d \tan ^{\frac{5}{2}}(c+d x)}",1,"(60*(-1)^(1/4)*((-a + I*b)^(5/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (a + I*b)^(5/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])*Tan[c + d*x]^(5/2) + 15*b*(-2*A*b + a*B)*Sqrt[a + b*Tan[c + d*x]] - 3*(8*a^2*A - 10*A*b^2 - 15*a*b*B)*Sqrt[a + b*Tan[c + d*x]] - 4*(22*a*A*b + 10*a^2*B - 15*b^2*B)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]] + 8*(15*a^2*A - 23*A*b^2 - 35*a*b*B)*Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]] - 60*b*B*(a + b*Tan[c + d*x])^(3/2))/(60*d*Tan[c + d*x]^(5/2))","A",1
448,1,381,309,6.3658036,"\int \frac{(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2),x]","\frac{8 a \left(35 a^2 A-77 a b B-45 A b^2\right) \tan ^2(c+d x) \sqrt{a+b \tan (c+d x)}-6 a \left(28 a^2 B+60 a A b-35 b^2 B\right) \tan (c+d x) \sqrt{a+b \tan (c+d x)}-5 a \left(24 a^2 A-49 a b B-28 A b^2\right) \sqrt{a+b \tan (c+d x)}+8 \left(105 a^3 B+245 a^2 A b-161 a b^2 B-15 A b^3\right) \tan ^3(c+d x) \sqrt{a+b \tan (c+d x)}+420 \sqrt[4]{-1} a \tan ^{\frac{7}{2}}(c+d x) \left((-a+i b)^{5/2} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+(a+i b)^{5/2} (B-i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)-35 a b (a B+4 A b) \sqrt{a+b \tan (c+d x)}-210 a b B (a+b \tan (c+d x))^{3/2}}{420 a d \tan ^{\frac{7}{2}}(c+d x)}","\frac{2 \left(35 a^2 A-77 a b B-45 A b^2\right) \sqrt{a+b \tan (c+d x)}}{105 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(105 a^3 B+245 a^2 A b-161 a b^2 B-15 A b^3\right) \sqrt{a+b \tan (c+d x)}}{105 a d \sqrt{\tan (c+d x)}}+\frac{(-b+i a)^{5/2} (-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a (7 a B+10 A b) \sqrt{a+b \tan (c+d x)}}{35 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{(b+i a)^{5/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A (a+b \tan (c+d x))^{3/2}}{7 d \tan ^{\frac{7}{2}}(c+d x)}",1,"(420*(-1)^(1/4)*a*((-a + I*b)^(5/2)*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (a + I*b)^(5/2)*((-I)*A + B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])*Tan[c + d*x]^(7/2) - 35*a*b*(4*A*b + a*B)*Sqrt[a + b*Tan[c + d*x]] - 5*a*(24*a^2*A - 28*A*b^2 - 49*a*b*B)*Sqrt[a + b*Tan[c + d*x]] - 6*a*(60*a*A*b + 28*a^2*B - 35*b^2*B)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]] + 8*a*(35*a^2*A - 45*A*b^2 - 77*a*b*B)*Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]] + 8*(245*a^2*A*b - 15*A*b^3 + 105*a^3*B - 161*a*b^2*B)*Tan[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]] - 210*a*b*B*(a + b*Tan[c + d*x])^(3/2))/(420*a*d*Tan[c + d*x]^(7/2))","A",1
449,1,543,378,6.9164234,"\int \frac{(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(11/2),x]","-\frac{b B (a+b \tan (c+d x))^{3/2}}{3 d \tan ^{\frac{9}{2}}(c+d x)}+\frac{1}{3} \left(-\frac{3 b (a B+2 A b) \sqrt{a+b \tan (c+d x)}}{8 d \tan ^{\frac{9}{2}}(c+d x)}+\frac{1}{4} \left(-\frac{\left(16 a^2 A-33 a b B-18 A b^2\right) \sqrt{a+b \tan (c+d x)}}{6 d \tan ^{\frac{9}{2}}(c+d x)}-\frac{2 \left(\frac{6 a \left(18 a^2 B+38 a A b-21 b^2 B\right) \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 \left(\frac{18 a^2 \left(21 a^2 A-45 a b B-25 A b^2\right) \sqrt{a+b \tan (c+d x)}}{5 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 \left(-\frac{3 a^2 \left(105 a^3 B+231 a^2 A b-135 a b^2 B-5 A b^3\right) \sqrt{a+b \tan (c+d x)}}{d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(-\frac{9 a^2 \left(315 a^4 A-735 a^3 b B-483 a^2 A b^2+45 a b^3 B-10 A b^4\right) \sqrt{a+b \tan (c+d x)}}{2 d \sqrt{\tan (c+d x)}}-\frac{2835 a^4 \left(\sqrt[4]{-1} (-a+i b)^{5/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt[4]{-1} (a+i b)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{4 d}\right)}{3 a}\right)}{5 a}\right)}{7 a}\right)}{9 a}\right)\right)","\frac{2 \left(21 a^2 A-45 a b B-25 A b^2\right) \sqrt{a+b \tan (c+d x)}}{105 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(105 a^3 B+231 a^2 A b-135 a b^2 B-5 A b^3\right) \sqrt{a+b \tan (c+d x)}}{315 a d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(315 a^4 A-735 a^3 b B-483 a^2 A b^2+45 a b^3 B-10 A b^4\right) \sqrt{a+b \tan (c+d x)}}{315 a^2 d \sqrt{\tan (c+d x)}}+\frac{(-b+i a)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a (3 a B+4 A b) \sqrt{a+b \tan (c+d x)}}{21 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{(b+i a)^{5/2} (A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A (a+b \tan (c+d x))^{3/2}}{9 d \tan ^{\frac{9}{2}}(c+d x)}",1,"-1/3*(b*B*(a + b*Tan[c + d*x])^(3/2))/(d*Tan[c + d*x]^(9/2)) + ((-3*b*(2*A*b + a*B)*Sqrt[a + b*Tan[c + d*x]])/(8*d*Tan[c + d*x]^(9/2)) + (-1/6*((16*a^2*A - 18*A*b^2 - 33*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(9/2)) - (2*((6*a*(38*a*A*b + 18*a^2*B - 21*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (2*((18*a^2*(21*a^2*A - 25*A*b^2 - 45*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - (2*((-3*a^2*(231*a^2*A*b - 5*A*b^3 + 105*a^3*B - 135*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(3/2)) - (2*((-2835*a^4*((-1)^(1/4)*(-a + I*b)^(5/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (-1)^(1/4)*(a + I*b)^(5/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]))/(4*d) - (9*a^2*(315*a^4*A - 483*a^2*A*b^2 - 10*A*b^4 - 735*a^3*b*B + 45*a*b^3*B)*Sqrt[a + b*Tan[c + d*x]])/(2*d*Sqrt[Tan[c + d*x]])))/(3*a)))/(5*a)))/(7*a)))/(9*a))/4)/3","A",1
450,1,632,460,7.1293577,"\int \frac{(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(13/2),x]","-\frac{b B (a+b \tan (c+d x))^{3/2}}{4 d \tan ^{\frac{11}{2}}(c+d x)}+\frac{1}{4} \left(-\frac{b (5 a B+8 A b) \sqrt{a+b \tan (c+d x)}}{10 d \tan ^{\frac{11}{2}}(c+d x)}+\frac{1}{5} \left(-\frac{\left(80 a^2 A-165 a b B-88 A b^2\right) \sqrt{a+b \tan (c+d x)}}{22 d \tan ^{\frac{11}{2}}(c+d x)}-\frac{2 \left(\frac{5 a \left(88 a^2 B+184 a A b-99 b^2 B\right) \sqrt{a+b \tan (c+d x)}}{18 d \tan ^{\frac{9}{2}}(c+d x)}-\frac{2 \left(\frac{10 a^2 \left(99 a^2 A-209 a b B-113 A b^2\right) \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 \left(-\frac{3 a^2 \left(231 a^3 B+495 a^2 A b-275 a b^2 B-5 A b^3\right) \sqrt{a+b \tan (c+d x)}}{d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 \left(-\frac{5 a^2 \left(1155 a^4 A-2541 a^3 b B-1485 a^2 A b^2+55 a b^3 B-20 A b^4\right) \sqrt{a+b \tan (c+d x)}}{2 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(\frac{15 a^2 \left(3465 a^5 B+8085 a^4 A b-5313 a^3 b^2 B-495 a^2 A b^3-110 a b^4 B+40 A b^5\right) \sqrt{a+b \tan (c+d x)}}{4 d \sqrt{\tan (c+d x)}}+\frac{51975 a^5 \left((-1)^{3/4} (-a+i b)^{5/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-(-1)^{3/4} (a+i b)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{8 d}\right)}{3 a}\right)}{5 a}\right)}{7 a}\right)}{9 a}\right)}{11 a}\right)\right)","\frac{2 \left(99 a^2 A-209 a b B-113 A b^2\right) \sqrt{a+b \tan (c+d x)}}{693 d \tan ^{\frac{7}{2}}(c+d x)}+\frac{2 \left(231 a^3 B+495 a^2 A b-275 a b^2 B-5 A b^3\right) \sqrt{a+b \tan (c+d x)}}{1155 a d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 \left(1155 a^4 A-2541 a^3 b B-1485 a^2 A b^2+55 a b^3 B-20 A b^4\right) \sqrt{a+b \tan (c+d x)}}{3465 a^2 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(3465 a^5 B+8085 a^4 A b-5313 a^3 b^2 B-495 a^2 A b^3-110 a b^4 B+40 A b^5\right) \sqrt{a+b \tan (c+d x)}}{3465 a^3 d \sqrt{\tan (c+d x)}}-\frac{(-b+i a)^{5/2} (-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a (11 a B+14 A b) \sqrt{a+b \tan (c+d x)}}{99 d \tan ^{\frac{9}{2}}(c+d x)}-\frac{(b+i a)^{5/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A (a+b \tan (c+d x))^{3/2}}{11 d \tan ^{\frac{11}{2}}(c+d x)}",1,"-1/4*(b*B*(a + b*Tan[c + d*x])^(3/2))/(d*Tan[c + d*x]^(11/2)) + (-1/10*(b*(8*A*b + 5*a*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(11/2)) + (-1/22*((80*a^2*A - 88*A*b^2 - 165*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(11/2)) - (2*((5*a*(184*a*A*b + 88*a^2*B - 99*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(18*d*Tan[c + d*x]^(9/2)) - (2*((10*a^2*(99*a^2*A - 113*A*b^2 - 209*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (2*((-3*a^2*(495*a^2*A*b - 5*A*b^3 + 231*a^3*B - 275*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(5/2)) - (2*((-5*a^2*(1155*a^4*A - 1485*a^2*A*b^2 - 20*A*b^4 - 2541*a^3*b*B + 55*a*b^3*B)*Sqrt[a + b*Tan[c + d*x]])/(2*d*Tan[c + d*x]^(3/2)) - (2*((51975*a^5*((-1)^(3/4)*(-a + I*b)^(5/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - (-1)^(3/4)*(a + I*b)^(5/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]))/(8*d) + (15*a^2*(8085*a^4*A*b - 495*a^2*A*b^3 + 40*A*b^5 + 3465*a^5*B - 5313*a^3*b^2*B - 110*a*b^4*B)*Sqrt[a + b*Tan[c + d*x]])/(4*d*Sqrt[Tan[c + d*x]])))/(3*a)))/(5*a)))/(7*a)))/(9*a)))/(11*a))/5)/4","A",1
451,1,356,253,4.3602863,"\int \frac{(a+b \tan (c+d x))^{5/2} \left(\frac{3 b B}{2 a}+B \tan (c+d x)\right)}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^(5/2)*((3*b*B)/(2*a) + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2),x]","\frac{B \cos (c+d x) (2 a \tan (c+d x)+3 b) \left(4 \sqrt{a} b^{5/2} \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)-\sqrt{\frac{b \tan (c+d x)}{a}+1} \left(2 a \sqrt{a+b \tan (c+d x)} \left(\left(2 a^2+7 b^2\right) \tan (c+d x)+a b\right)+\sqrt[4]{-1} (2 a+3 i b) (-a+i b)^{5/2} \tan ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt[4]{-1} (a+i b)^{5/2} (2 a-3 i b) \tan ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)\right)}{2 a d \tan ^{\frac{3}{2}}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} (2 a \sin (c+d x)+3 b \cos (c+d x))}","-\frac{2 B \left(a^2+3 b^2\right) \sqrt{a+b \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}+\frac{2 b^{5/2} B \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{B (2 a-3 i b) (-b+i a)^{5/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{2 a d}-\frac{b B (a+b \tan (c+d x))^{3/2}}{d \tan ^{\frac{3}{2}}(c+d x)}-\frac{B (2 a+3 i b) (b+i a)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{2 a d}",1,"(B*Cos[c + d*x]*(3*b + 2*a*Tan[c + d*x])*(4*Sqrt[a]*b^(5/2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]] - Sqrt[1 + (b*Tan[c + d*x])/a]*((-1)^(1/4)*(-a + I*b)^(5/2)*(2*a + (3*I)*b)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(3/2) + (-1)^(1/4)*(a + I*b)^(5/2)*(2*a - (3*I)*b)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(3/2) + 2*a*Sqrt[a + b*Tan[c + d*x]]*(a*b + (2*a^2 + 7*b^2)*Tan[c + d*x]))))/(2*a*d*(3*b*Cos[c + d*x] + 2*a*Sin[c + d*x])*Tan[c + d*x]^(3/2)*Sqrt[1 + (b*Tan[c + d*x])/a])","A",1
452,1,245,206,1.8945068,"\int \frac{\tan ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\frac{\sqrt[4]{-1} b (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{(-1)^{3/4} b (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\frac{\sqrt{a} (2 A b-a B) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{b d}","\frac{(2 A b-a B) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{b^{3/2} d}-\frac{(A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}+\frac{B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{b d}",1,"(((-1)^(1/4)*b*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + ((-1)^(3/4)*b*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] + B*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + (Sqrt[a]*(2*A*b - a*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]]))/(b*d)","A",1
453,1,205,168,1.2817705,"\int \frac{\sqrt{\tan (c+d x)} (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\frac{2 \sqrt{a} B \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+\sqrt[4]{-1} \left(\frac{(A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}-\frac{(A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)}{d}","\frac{(-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}+\frac{2 B \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{b} d}",1,"((-1)^(1/4)*(-(((A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b]) + ((A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]) + (2*Sqrt[a]*B*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]]))/d","A",1
454,1,137,123,0.266198,"\int \frac{A+B \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]),x]","\frac{\sqrt[4]{-1} \left(\frac{(B-i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}-\frac{(B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)}{d}","\frac{(A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"((-1)^(1/4)*(-(((I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b]) + (((-I)*A + B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]))/d","A",1
455,1,172,159,0.4935288,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]),x]","\frac{\frac{\sqrt[4]{-1} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{\sqrt[4]{-1} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}-\frac{2 A \sqrt{a+b \tan (c+d x)}}{a \sqrt{\tan (c+d x)}}}{d}","-\frac{(-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}-\frac{2 A \sqrt{a+b \tan (c+d x)}}{a d \sqrt{\tan (c+d x)}}",1,"(((-1)^(1/4)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] - ((-1)^(1/4)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] - (2*A*Sqrt[a + b*Tan[c + d*x]])/(a*Sqrt[Tan[c + d*x]]))/d","A",1
456,1,195,203,1.8471786,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]]),x]","\frac{-\frac{2 \sqrt{a+b \tan (c+d x)} ((3 a B-2 A b) \tan (c+d x)+a A)}{a^2 \tan ^{\frac{3}{2}}(c+d x)}+\frac{3 \sqrt[4]{-1} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{3 (-1)^{3/4} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}}{3 d}","\frac{2 (2 A b-3 a B) \sqrt{a+b \tan (c+d x)}}{3 a^2 d \sqrt{\tan (c+d x)}}-\frac{(A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}-\frac{2 A \sqrt{a+b \tan (c+d x)}}{3 a d \tan ^{\frac{3}{2}}(c+d x)}",1,"((3*(-1)^(1/4)*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + (3*(-1)^(3/4)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] - (2*Sqrt[a + b*Tan[c + d*x]]*(a*A + (-2*A*b + 3*a*B)*Tan[c + d*x]))/(a^2*Tan[c + d*x]^(3/2)))/(3*d)","A",1
457,1,227,256,5.2933131,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]]),x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left(\left(15 a^2 A+10 a b B-8 A b^2\right) \tan ^2(c+d x)-3 a^2 A-a (5 a B-4 A b) \tan (c+d x)\right)}{a^3 \tan ^{\frac{5}{2}}(c+d x)}-\frac{15 \sqrt[4]{-1} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{15 \sqrt[4]{-1} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}}{15 d}","\frac{2 (4 A b-5 a B) \sqrt{a+b \tan (c+d x)}}{15 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(15 a^2 A+10 a b B-8 A b^2\right) \sqrt{a+b \tan (c+d x)}}{15 a^3 d \sqrt{\tan (c+d x)}}+\frac{(-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}-\frac{2 A \sqrt{a+b \tan (c+d x)}}{5 a d \tan ^{\frac{5}{2}}(c+d x)}",1,"((-15*(-1)^(1/4)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + (15*(-1)^(1/4)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] + (2*Sqrt[a + b*Tan[c + d*x]]*(-3*a^2*A - a*(-4*A*b + 5*a*B)*Tan[c + d*x] + (15*a^2*A - 8*A*b^2 + 10*a*b*B)*Tan[c + d*x]^2))/(a^3*Tan[c + d*x]^(5/2)))/(15*d)","A",1
458,1,177751,219,40.2171521,"\int \frac{\tan ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{2 a (A b-a B) \sqrt{\tan (c+d x)}}{b d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{(-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}-\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}+\frac{2 B \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{b^{3/2} d}",1,"Result too large to show","C",0
459,1,239,170,1.6252493,"\int \frac{\sqrt{\tan (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","\frac{-\frac{\sqrt[4]{-1} a (a+i b) (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{\sqrt[4]{-1} a (a-i b) (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\frac{2 b (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \tan (c+d x)}}+2 (a B-A b) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{a d \left(a^2+b^2\right)}","-\frac{2 (A b-a B) \sqrt{\tan (c+d x)}}{d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{(A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}+\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"(-(((-1)^(1/4)*a*(a + I*b)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b]) + ((-1)^(1/4)*a*(a - I*b)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] + (2*b*(A*b - a*B)*Tan[c + d*x]^(3/2))/Sqrt[a + b*Tan[c + d*x]] + 2*(-(A*b) + a*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(a*(a^2 + b^2)*d)","A",1
460,1,202,175,0.7839073,"\int \frac{A+B \tan (c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)),x]","\frac{\frac{\sqrt[4]{-1} (b-i a) (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{\sqrt[4]{-1} (a-i b) (B-i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\frac{2 b (A b-a B) \sqrt{\tan (c+d x)}}{a \sqrt{a+b \tan (c+d x)}}}{d \left(a^2+b^2\right)}","\frac{2 b (A b-a B) \sqrt{\tan (c+d x)}}{a d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{(-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}+\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"(((-1)^(1/4)*((-I)*a + b)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + ((-1)^(1/4)*(a - I*b)*((-I)*A + B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] + (2*b*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(a*Sqrt[a + b*Tan[c + d*x]]))/((a^2 + b^2)*d)","A",1
461,1,248,216,2.0747465,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)),x]","\frac{\frac{\sqrt[4]{-1} \left(\frac{(a+i b) (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{(a-i b) (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{a^2+b^2}-\frac{2 b \left(a^2 A-a b B+2 A b^2\right) \sqrt{\tan (c+d x)}}{a^2 \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{2 A}{a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}}{d}","-\frac{2 b \left(a^2 A-a b B+2 A b^2\right) \sqrt{\tan (c+d x)}}{a^2 d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{(A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}-\frac{2 A}{a d \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}",1,"(((-1)^(1/4)*(((a + I*b)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] - ((a - I*b)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]))/(a^2 + b^2) - (2*A)/(a*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]) - (2*b*(a^2*A + 2*A*b^2 - a*b*B)*Sqrt[Tan[c + d*x]])/(a^2*(a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]]))/d","A",1
462,1,299,276,2.9006075,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(3/2)),x]","\frac{\frac{3 \sqrt[4]{-1} a \left(\frac{(a+i b) (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{(b+i a) (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{a^2+b^2}+\frac{2 b \left(-3 a^3 B+5 a^2 A b-6 a b^2 B+8 A b^3\right) \sqrt{\tan (c+d x)}}{a^2 \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{8 A b-6 a B}{a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{2 A}{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}}{3 a d}","\frac{2 (4 A b-3 a B)}{3 a^2 d \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 b \left(-3 a^3 B+5 a^2 A b-6 a b^2 B+8 A b^3\right) \sqrt{\tan (c+d x)}}{3 a^3 d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{(-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}-\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}-\frac{2 A}{3 a d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}",1,"((3*(-1)^(1/4)*a*(((a + I*b)*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + ((I*a + b)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]))/(a^2 + b^2) - (2*A)/(Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]) + (8*A*b - 6*a*B)/(a*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]) + (2*b*(5*a^2*A*b + 8*A*b^3 - 3*a^3*B - 6*a*b^2*B)*Sqrt[Tan[c + d*x]])/(a^2*(a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]]))/(3*a*d)","A",1
463,1,265550,282,41.7990133,"\int \frac{\tan ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 a (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left(2 A b^3-a B \left(a^2+3 b^2\right)\right) \sqrt{\tan (c+d x)}}{b^2 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{(-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}-\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}+\frac{2 B \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{b^{5/2} d}",1,"Result too large to show","C",0
464,1,308,244,3.0953352,"\int \frac{\tan ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]","\frac{\frac{\left(a^2 B+2 a A b+3 b^2 B\right) \sqrt{\tan (c+d x)}}{\left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{\frac{2 \left(a^3 B+2 a^2 A b+7 a b^2 B-4 A b^3\right) \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}+3 \sqrt[4]{-1} b \left(\frac{i (a-i b)^2 (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\frac{(a+i b)^2 (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)}{\left(a^2+b^2\right)^2}-\frac{3 B \sqrt{\tan (c+d x)}}{(a+b \tan (c+d x))^{3/2}}}{3 b d}","\frac{2 a (A b-a B) \sqrt{\tan (c+d x)}}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{2 \left(a^3 B+2 a^2 A b+7 a b^2 B-4 A b^3\right) \sqrt{\tan (c+d x)}}{3 b d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{(A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}+\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"((-3*B*Sqrt[Tan[c + d*x]])/(a + b*Tan[c + d*x])^(3/2) + ((2*a*A*b + a^2*B + 3*b^2*B)*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) + (3*(-1)^(1/4)*b*(((a + I*b)^2*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + (I*(a - I*b)^2*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]) + (2*(2*a^2*A*b - 4*A*b^3 + a^3*B + 7*a*b^2*B)*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]])/(a^2 + b^2)^2)/(3*b*d)","A",1
465,1,320,244,3.4158974,"\int \frac{\sqrt{\tan (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]","\frac{\frac{3 \left(2 \left(a^2 B-2 a A b-b^2 B\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}+\frac{\sqrt[4]{-1} a (a-i b)^2 (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}-\frac{\sqrt[4]{-1} a (a+i b)^2 (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)}{a^2+b^2}+\frac{6 b \left(a^2 (-B)+2 a A b+b^2 B\right) \tan ^{\frac{3}{2}}(c+d x)}{\left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{2 b (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}}}{3 a d \left(a^2+b^2\right)}","-\frac{2 (A b-a B) \sqrt{\tan (c+d x)}}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}-\frac{2 \left(-2 a^3 B+5 a^2 A b+4 a b^2 B-A b^3\right) \sqrt{\tan (c+d x)}}{3 a d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}-\frac{(-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}+\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"((2*b*(A*b - a*B)*Tan[c + d*x]^(3/2))/(a + b*Tan[c + d*x])^(3/2) + (6*b*(2*a*A*b - a^2*B + b^2*B)*Tan[c + d*x]^(3/2))/((a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]]) + (3*(-(((-1)^(1/4)*a*(a + I*b)^2*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b]) + ((-1)^(1/4)*a*(a - I*b)^2*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] + 2*(-2*a*A*b + a^2*B - b^2*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]))/(a^2 + b^2))/(3*a*(a^2 + b^2)*d)","A",1
466,1,273,247,2.5737177,"\int \frac{A+B \tan (c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2)),x]","\frac{\frac{2 b \left(a^2+b^2\right) (A b-a B) \sqrt{\tan (c+d x)}}{a (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left(-5 a^3 B+8 a^2 A b+a b^2 B+2 A b^3\right) \sqrt{\tan (c+d x)}}{a^2 \sqrt{a+b \tan (c+d x)}}-3 \sqrt[4]{-1} \left(\frac{i (a-i b)^2 (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\frac{(a+i b)^2 (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)}{3 d \left(a^2+b^2\right)^2}","\frac{2 b (A b-a B) \sqrt{\tan (c+d x)}}{3 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left(-5 a^3 B+8 a^2 A b+a b^2 B+2 A b^3\right) \sqrt{\tan (c+d x)}}{3 a^2 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}-\frac{(A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}-\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"(-3*(-1)^(1/4)*(((a + I*b)^2*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + (I*(a - I*b)^2*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]) + (2*b*(a^2 + b^2)*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(a*(a + b*Tan[c + d*x])^(3/2)) + (2*b*(8*a^2*A*b + 2*A*b^3 - 5*a^3*B + a*b^2*B)*Sqrt[Tan[c + d*x]])/(a^2*Sqrt[a + b*Tan[c + d*x]]))/(3*(a^2 + b^2)^2*d)","A",1
467,1,326,301,3.760403,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(5/2)),x]","\frac{-\frac{2 b \left(3 a^2 A-a b B+4 A b^2\right) \sqrt{\tan (c+d x)}}{\left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{-\frac{2 b \left(3 a^4 A-8 a^3 b B+17 a^2 A b^2-2 a b^3 B+8 A b^4\right) \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}+3 \sqrt[4]{-1} a^3 \left(\frac{(a+i b)^2 (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{(a-i b)^2 (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{a \left(a^2+b^2\right)^2}-\frac{6 a A}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}}{3 a^2 d}","-\frac{2 b \left(3 a^2 A-a b B+4 A b^2\right) \sqrt{\tan (c+d x)}}{3 a^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}-\frac{2 b \left(3 a^4 A-8 a^3 b B+17 a^2 A b^2-2 a b^3 B+8 A b^4\right) \sqrt{\tan (c+d x)}}{3 a^3 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{(-B+i A) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}-\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}-\frac{2 A}{a d \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}",1,"((-6*a*A)/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) - (2*b*(3*a^2*A + 4*A*b^2 - a*b*B)*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) + (3*(-1)^(1/4)*a^3*(((a + I*b)^2*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] - ((a - I*b)^2*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]) - (2*b*(3*a^4*A + 17*a^2*A*b^2 + 8*A*b^4 - 8*a^3*b*B - 2*a*b^3*B)*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]])/(a*(a^2 + b^2)^2))/(3*a^2*d)","A",1
468,1,383,359,3.5319035,"\int \frac{A+B \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(5/2)),x]","\frac{\frac{6 b \left(-3 a^3 B+7 a^2 A b-4 a b^2 B+8 A b^3\right) \sqrt{\tan (c+d x)}}{a^2 \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{\frac{6 b \left(-3 a^5 B+8 a^4 A b-17 a^3 b^2 B+30 a^2 A b^3-8 a b^4 B+16 A b^5\right) \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}+9 (-1)^{3/4} a^4 \left(\frac{(a-i b)^2 (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\frac{(a+i b)^2 (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)}{a^3 \left(a^2+b^2\right)^2}+\frac{6 (6 A b-3 a B)}{a \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac{6 A}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}}{9 a d}","\frac{2 (2 A b-a B)}{a^2 d \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left(-3 a^3 B+7 a^2 A b-4 a b^2 B+8 A b^3\right) \sqrt{\tan (c+d x)}}{3 a^3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left(-3 a^5 B+8 a^4 A b-17 a^3 b^2 B+30 a^2 A b^3-8 a b^4 B+16 A b^5\right) \sqrt{\tan (c+d x)}}{3 a^4 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{(A+i B) \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}+\frac{(A-i B) \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}-\frac{2 A}{3 a d \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}",1,"((-6*A)/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)) + (6*(6*A*b - 3*a*B))/(a*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) + (6*b*(7*a^2*A*b + 8*A*b^3 - 3*a^3*B - 4*a*b^2*B)*Sqrt[Tan[c + d*x]])/(a^2*(a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) + (9*(-1)^(3/4)*a^4*(((a + I*b)^2*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + ((a - I*b)^2*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]) + (6*b*(8*a^4*A*b + 30*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B - 17*a^3*b^2*B - 8*a*b^4*B)*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]])/(a^3*(a^2 + b^2)^2))/(9*a*d)","A",1
469,1,190,155,0.8249556,"\int \frac{\tan ^{\frac{3}{2}}(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Tan[c + d*x]^(3/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","\frac{B \left(\frac{2 \sqrt{a} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+(-1)^{3/4} \left(\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)\right)}{d}","-\frac{B \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{2 B \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{b} d}-\frac{B \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"(B*((-1)^(3/4)*(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[-a + I*b] + ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[a + I*b]) + (2*Sqrt[a]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]])))/d","A",1
470,1,124,117,0.0986653,"\int \frac{\sqrt{\tan (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Tan[c + d*x]]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","\frac{\sqrt[4]{-1} B \left(\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)}{d}","\frac{i B \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{i B \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"((-1)^(1/4)*B*(-(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[-a + I*b]) + ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[a + I*b]))/d","A",1
471,1,125,111,0.1402666,"\int \frac{a B+b B \tan (c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)),x]","\frac{(-1)^{3/4} B \left(-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{d}","\frac{B \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{B \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"((-1)^(3/4)*B*(-(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[-a + I*b]) - ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[a + I*b]))/d","A",1
472,1,158,150,0.4479808,"\int \frac{a B+b B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)),x]","\frac{B \left(\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}-\frac{2 \sqrt{a+b \tan (c+d x)}}{a \sqrt{\tan (c+d x)}}\right)}{d}","-\frac{i B \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{2 B \sqrt{a+b \tan (c+d x)}}{a d \sqrt{\tan (c+d x)}}+\frac{i B \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"(B*(((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] - ((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] - (2*Sqrt[a + b*Tan[c + d*x]])/(a*Sqrt[Tan[c + d*x]])))/d","A",1
473,1,263,379,0.9563003,"\int (a+b \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx","Integrate[(a + b*Tan[c + d*x])^(2/3)*(A + B*Tan[c + d*x]),x]","\frac{i \left((A-i B) \left(3 (a+b \tan (c+d x))^{2/3}+(a-i b)^{2/3} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)+3 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)-\log (\tan (c+d x)+i)\right)\right)-(A+i B) \left(3 (a+b \tan (c+d x))^{2/3}+(a+i b)^{2/3} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)+3 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)-\log (-\tan (c+d x)+i)\right)\right)\right)}{4 d}","\frac{\sqrt{3} (a-i b)^{2/3} (B+i A) \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)}{2 d}-\frac{\sqrt{3} (a+i b)^{2/3} (-B+i A) \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)}{2 d}+\frac{3 (a-i b)^{2/3} (B+i A) \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)}{4 d}-\frac{3 (a+i b)^{2/3} (-B+i A) \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)}{4 d}-\frac{(a+i b)^{2/3} (-B+i A) \log (\cos (c+d x))}{4 d}+\frac{(a-i b)^{2/3} (B+i A) \log (\cos (c+d x))}{4 d}-\frac{1}{4} x (a-i b)^{2/3} (A-i B)-\frac{1}{4} x (a+i b)^{2/3} (A+i B)+\frac{3 B (a+b \tan (c+d x))^{2/3}}{2 d}",1,"((I/4)*((A - I*B)*((a - I*b)^(2/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]] - Log[I + Tan[c + d*x]] + 3*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)]) + 3*(a + b*Tan[c + d*x])^(2/3)) - (A + I*B)*((a + I*b)^(2/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]] - Log[I - Tan[c + d*x]] + 3*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)]) + 3*(a + b*Tan[c + d*x])^(2/3))))/d","A",1
474,1,347,377,0.9366884,"\int \sqrt[3]{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[(a + b*Tan[c + d*x])^(1/3)*(A + B*Tan[c + d*x]),x]","\frac{i \left((A-i B) \left(3 \sqrt[3]{a+b \tan (c+d x)}-\frac{1}{2} \sqrt[3]{a-i b} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)-2 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)+\log \left(\sqrt[3]{a-i b} \sqrt[3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}+(a-i b)^{2/3}\right)\right)\right)-(A+i B) \left(3 \sqrt[3]{a+b \tan (c+d x)}-\frac{1}{2} \sqrt[3]{a+i b} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)-2 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)+\log \left(\sqrt[3]{a+i b} \sqrt[3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}+(a+i b)^{2/3}\right)\right)\right)\right)}{2 d}","-\frac{\sqrt{3} \sqrt[3]{a-i b} (B+i A) \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)}{2 d}+\frac{\sqrt{3} \sqrt[3]{a+i b} (-B+i A) \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)}{2 d}+\frac{3 \sqrt[3]{a-i b} (B+i A) \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)}{4 d}-\frac{3 \sqrt[3]{a+i b} (-B+i A) \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)}{4 d}-\frac{\sqrt[3]{a+i b} (-B+i A) \log (\cos (c+d x))}{4 d}+\frac{\sqrt[3]{a-i b} (B+i A) \log (\cos (c+d x))}{4 d}-\frac{1}{4} x \sqrt[3]{a-i b} (A-i B)-\frac{1}{4} x \sqrt[3]{a+i b} (A+i B)+\frac{3 B \sqrt[3]{a+b \tan (c+d x)}}{d}",1,"((I/2)*((A - I*B)*(-1/2*((a - I*b)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]] - 2*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)] + Log[(a - I*b)^(2/3) + (a - I*b)^(1/3)*(a + b*Tan[c + d*x])^(1/3) + (a + b*Tan[c + d*x])^(2/3)])) + 3*(a + b*Tan[c + d*x])^(1/3)) - (A + I*B)*(-1/2*((a + I*b)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]] - 2*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)] + Log[(a + I*b)^(2/3) + (a + I*b)^(1/3)*(a + b*Tan[c + d*x])^(1/3) + (a + b*Tan[c + d*x])^(2/3)])) + 3*(a + b*Tan[c + d*x])^(1/3))))/d","A",1
475,1,227,357,0.4908169,"\int \frac{A+B \tan (c+d x)}{\sqrt[3]{a+b \tan (c+d x)}} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^(1/3),x]","\frac{i \left(\frac{(A-i B) \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)+3 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)-\log (\tan (c+d x)+i)\right)}{\sqrt[3]{a-i b}}-\frac{(A+i B) \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)+3 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)-\log (-\tan (c+d x)+i)\right)}{\sqrt[3]{a+i b}}\right)}{4 d}","\frac{\sqrt{3} (B+i A) \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)}{2 d \sqrt[3]{a-i b}}-\frac{\sqrt{3} (-B+i A) \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)}{2 d \sqrt[3]{a+i b}}+\frac{3 (B+i A) \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)}{4 d \sqrt[3]{a-i b}}-\frac{3 (-B+i A) \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)}{4 d \sqrt[3]{a+i b}}-\frac{(-B+i A) \log (\cos (c+d x))}{4 d \sqrt[3]{a+i b}}+\frac{(B+i A) \log (\cos (c+d x))}{4 d \sqrt[3]{a-i b}}-\frac{x (A-i B)}{4 \sqrt[3]{a-i b}}-\frac{x (A+i B)}{4 \sqrt[3]{a+i b}}",1,"((I/4)*(((A - I*B)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]] - Log[I + Tan[c + d*x]] + 3*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)]))/(a - I*b)^(1/3) - ((A + I*B)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]] - Log[I - Tan[c + d*x]] + 3*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)]))/(a + I*b)^(1/3)))/d","A",1
476,1,305,357,0.2922401,"\int \frac{A+B \tan (c+d x)}{(a+b \tan (c+d x))^{2/3}} \, dx","Integrate[(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^(2/3),x]","\frac{i \left(\frac{(A+i B) \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)-2 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)+\log \left(\sqrt[3]{a+i b} \sqrt[3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}+(a+i b)^{2/3}\right)\right)}{(a+i b)^{2/3}}-\frac{(A-i B) \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)-2 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)+\log \left(\sqrt[3]{a-i b} \sqrt[3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}+(a-i b)^{2/3}\right)\right)}{(a-i b)^{2/3}}\right)}{4 d}","-\frac{\sqrt{3} (B+i A) \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)}{2 d (a-i b)^{2/3}}+\frac{\sqrt{3} (-B+i A) \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)}{2 d (a+i b)^{2/3}}+\frac{3 (B+i A) \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)}{4 d (a-i b)^{2/3}}-\frac{3 (-B+i A) \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)}{4 d (a+i b)^{2/3}}-\frac{(-B+i A) \log (\cos (c+d x))}{4 d (a+i b)^{2/3}}+\frac{(B+i A) \log (\cos (c+d x))}{4 d (a-i b)^{2/3}}-\frac{x (A-i B)}{4 (a-i b)^{2/3}}-\frac{x (A+i B)}{4 (a+i b)^{2/3}}",1,"((I/4)*(-(((A - I*B)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]] - 2*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)] + Log[(a - I*b)^(2/3) + (a - I*b)^(1/3)*(a + b*Tan[c + d*x])^(1/3) + (a + b*Tan[c + d*x])^(2/3)]))/(a - I*b)^(2/3)) + ((A + I*B)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]] - 2*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)] + Log[(a + I*b)^(2/3) + (a + I*b)^(1/3)*(a + b*Tan[c + d*x])^(1/3) + (a + b*Tan[c + d*x])^(2/3)]))/(a + I*b)^(2/3)))/d","A",1
477,1,109,148,1.828025,"\int \frac{i-\tan (e+f x)}{\sqrt[3]{c+d \tan (e+f x)}} \, dx","Integrate[(I - Tan[e + f*x])/(c + d*Tan[e + f*x])^(1/3),x]","\frac{3 \left(c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)^{2/3} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{i c+\frac{d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}{i c+d}\right)}{2 f (c-i d)}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-i d}}}{\sqrt{3}}\right)}{f \sqrt[3]{c-i d}}-\frac{3 \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c-i d}\right)}{2 f \sqrt[3]{c-i d}}-\frac{\log (\cos (e+f x))}{2 f \sqrt[3]{c-i d}}-\frac{i x}{2 \sqrt[3]{c-i d}}",1,"(3*(c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x))))^(2/3)*Hypergeometric2F1[2/3, 1, 5/3, (I*c + (d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x))))/(I*c + d)])/(2*(c - I*d)*f)","C",1
478,1,330,299,0.470215,"\int \frac{d-c \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}} \, dx","Integrate[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x])^(2/3),x]","\frac{2 \sqrt{3} \sqrt[3]{c-i d} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-i d}}}{\sqrt{3}}\right)+2 \sqrt{3} \sqrt[3]{c+i d} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+i d}}}{\sqrt{3}}\right)-2 \sqrt[3]{c-i d} \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c-i d}\right)-2 \sqrt[3]{c+i d} \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c+i d}\right)+\sqrt[3]{c-i d} \log \left(\sqrt[3]{c-i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right)+\sqrt[3]{c+i d} \log \left(\sqrt[3]{c+i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right)}{4 f}","\frac{\sqrt{3} \sqrt[3]{c-i d} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-i d}}}{\sqrt{3}}\right)}{2 f}+\frac{\sqrt{3} \sqrt[3]{c+i d} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+i d}}}{\sqrt{3}}\right)}{2 f}-\frac{3 \sqrt[3]{c-i d} \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c-i d}\right)}{4 f}-\frac{3 \sqrt[3]{c+i d} \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c+i d}\right)}{4 f}-\frac{\sqrt[3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac{\sqrt[3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac{1}{4} i x \sqrt[3]{c-i d}+\frac{1}{4} i x \sqrt[3]{c+i d}",1,"(2*Sqrt[3]*(c - I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]] + 2*Sqrt[3]*(c + I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]] - 2*(c - I*d)^(1/3)*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] - 2*(c + I*d)^(1/3)*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] + (c - I*d)^(1/3)*Log[(c - I*d)^(2/3) + (c - I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)] + (c + I*d)^(1/3)*Log[(c + I*d)^(2/3) + (c + I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)])/(4*f)","A",1
479,1,355,403,5.7119982,"\int \tan ^m(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]","\frac{\tan ^{m+1}(c+d x) \left(b^2 (m+1) \tan (c+d x) \left(a^2 B \left(m^2+9 m+26\right)+2 a A b (m+4)^2-b^2 B \left(m^2+7 m+12\right)\right)-b (m+2) \left(-2 a^3 B \left(m^2+8 m+19\right)-a^2 A b \left(5 m^2+37 m+68\right)+4 a b^2 B \left(m^2+7 m+12\right)+A b^3 \left(m^2+7 m+12\right)\right)+(m+2) (m+3) (m+4) \left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)+(m+1) (m+3) (m+4) \left(a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right) \tan (c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)+b (m+1) (m+2) (a B (m+7)+A b (m+4)) (a+b \tan (c+d x))^2+b B (m+1) (m+2) (m+3) (a+b \tan (c+d x))^3\right)}{d (m+1) (m+2) (m+3) (m+4)}","\frac{b^2 \left(a^2 B \left(m^2+9 m+26\right)+2 a A b (m+4)^2-b^2 B \left(m^2+7 m+12\right)\right) \tan ^{m+2}(c+d x)}{d (m+2) (m+3) (m+4)}-\frac{b \left(-2 a^3 B \left(m^2+8 m+19\right)-a^2 A b \left(5 m^2+37 m+68\right)+4 a b^2 B \left(m^2+7 m+12\right)+A b^3 \left(m^2+7 m+12\right)\right) \tan ^{m+1}(c+d x)}{d (m+1) (m+3) (m+4)}+\frac{\left(a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right) \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d (m+1)}+\frac{\left(a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right) \tan ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)}{d (m+2)}+\frac{b (a B (m+7)+A b (m+4)) \tan ^{m+1}(c+d x) (a+b \tan (c+d x))^2}{d (m+3) (m+4)}+\frac{b B \tan ^{m+1}(c+d x) (a+b \tan (c+d x))^3}{d (m+4)}",1,"(Tan[c + d*x]^(1 + m)*(-(b*(2 + m)*(A*b^3*(12 + 7*m + m^2) + 4*a*b^2*B*(12 + 7*m + m^2) - 2*a^3*B*(19 + 8*m + m^2) - a^2*A*b*(68 + 37*m + 5*m^2))) + (a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*(2 + m)*(3 + m)*(4 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2] + b^2*(1 + m)*(2*a*A*b*(4 + m)^2 - b^2*B*(12 + 7*m + m^2) + a^2*B*(26 + 9*m + m^2))*Tan[c + d*x] + (4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*(1 + m)*(3 + m)*(4 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x] + b*(1 + m)*(2 + m)*(A*b*(4 + m) + a*B*(7 + m))*(a + b*Tan[c + d*x])^2 + b*B*(1 + m)*(2 + m)*(3 + m)*(a + b*Tan[c + d*x])^3))/(d*(1 + m)*(2 + m)*(3 + m)*(4 + m))","A",1
480,1,232,267,2.7230659,"\int \tan ^m(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{\tan ^{m+1}(c+d x) \left(b (m+2) \left(2 a^2 B (m+4)+3 a A b (m+3)-b^2 B (m+3)\right)+(m+2) (m+3) \left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)+(m+1) (m+3) \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \tan (c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)+b^2 (m+1) \tan (c+d x) (a B (m+5)+A b (m+3))+b B (m+1) (m+2) (a+b \tan (c+d x))^2\right)}{d (m+1) (m+2) (m+3)}","\frac{b \left(2 a^2 B (m+4)+3 a A b (m+3)-b^2 B (m+3)\right) \tan ^{m+1}(c+d x)}{d (m+1) (m+3)}+\frac{\left(a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right) \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d (m+1)}+\frac{\left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \tan ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)}{d (m+2)}+\frac{b^2 (a B (m+5)+A b (m+3)) \tan ^{m+2}(c+d x)}{d (m+2) (m+3)}+\frac{b B \tan ^{m+1}(c+d x) (a+b \tan (c+d x))^2}{d (m+3)}",1,"(Tan[c + d*x]^(1 + m)*(b*(2 + m)*(3*a*A*b*(3 + m) - b^2*B*(3 + m) + 2*a^2*B*(4 + m)) + (a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*(2 + m)*(3 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2] + b^2*(1 + m)*(A*b*(3 + m) + a*B*(5 + m))*Tan[c + d*x] + (3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*(1 + m)*(3 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x] + b*B*(1 + m)*(2 + m)*(a + b*Tan[c + d*x])^2))/(d*(1 + m)*(2 + m)*(3 + m))","A",1
481,1,155,194,0.7443365,"\int \tan ^m(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{\tan ^{m+1}(c+d x) \left(\frac{(m+2) \left(a^2 A-2 a b B-A b^2\right) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{m+1}+\left(a^2 B+2 a A b-b^2 B\right) \tan (c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)+\frac{b (a B (m+3)+A b (m+2))}{m+1}+b B (a+b \tan (c+d x))\right)}{d (m+2)}","\frac{\left(a^2 A-2 a b B-A b^2\right) \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d (m+1)}+\frac{\left(a^2 B+2 a A b-b^2 B\right) \tan ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)}{d (m+2)}+\frac{b (a B (m+3)+A b (m+2)) \tan ^{m+1}(c+d x)}{d (m+1) (m+2)}+\frac{b B \tan ^{m+1}(c+d x) (a+b \tan (c+d x))}{d (m+2)}",1,"(Tan[c + d*x]^(1 + m)*((b*(A*b*(2 + m) + a*B*(3 + m)))/(1 + m) + ((a^2*A - A*b^2 - 2*a*b*B)*(2 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2])/(1 + m) + (2*a*A*b + a^2*B - b^2*B)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x] + b*B*(a + b*Tan[c + d*x])))/(d*(2 + m))","A",1
482,1,108,127,0.4572586,"\int \tan ^m(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{\tan ^{m+1}(c+d x) \left(\frac{(a A-b B) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{m+1}+\frac{(a B+A b) \tan (c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)}{m+2}+\frac{b B}{m+1}\right)}{d}","\frac{(a A-b B) \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d (m+1)}+\frac{(a B+A b) \tan ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)}{d (m+2)}+\frac{b B \tan ^{m+1}(c+d x)}{d (m+1)}",1,"(Tan[c + d*x]^(1 + m)*((b*B)/(1 + m) + ((a*A - b*B)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2])/(1 + m) + ((A*b + a*B)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x])/(2 + m)))/d","A",1
483,1,144,185,0.948634,"\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{\tan ^{m+1}(c+d x) \left((a A+b B) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)+\frac{(A b-a B) \left(b (m+2) \, _2F_1\left(1,m+1;m+2;-\frac{b \tan (c+d x)}{a}\right)-a (m+1) \tan (c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)\right)}{a (m+2)}\right)}{d (m+1) \left(a^2+b^2\right)}","\frac{b (A b-a B) \tan ^{m+1}(c+d x) \, _2F_1\left(1,m+1;m+2;-\frac{b \tan (c+d x)}{a}\right)}{a d (m+1) \left(a^2+b^2\right)}+\frac{(a A+b B) \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d (m+1) \left(a^2+b^2\right)}-\frac{(A b-a B) \tan ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)}{d (m+2) \left(a^2+b^2\right)}",1,"(Tan[c + d*x]^(1 + m)*((a*A + b*B)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2] + ((A*b - a*B)*(b*(2 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*Tan[c + d*x])/a)] - a*(1 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]))/(a*(2 + m))))/((a^2 + b^2)*d*(1 + m))","A",1
484,1,239,282,3.0675107,"\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{\tan ^{m+1}(c+d x) \left(\frac{a \left(\frac{\left(a^2 A+2 a b B-A b^2\right) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{m+1}+\frac{\left(a^2 B-2 a A b-b^2 B\right) \tan (c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)}{m+2}\right)}{a^2+b^2}+\frac{b \left(a^3 B (m-1)-a^2 A b (m-2)+a b^2 B (m+1)-A b^3 m\right) \, _2F_1\left(1,m+1;m+2;-\frac{b \tan (c+d x)}{a}\right)}{a (m+1) \left(a^2+b^2\right)}+\frac{b (A b-a B)}{a+b \tan (c+d x)}\right)}{a d \left(a^2+b^2\right)}","\frac{\left(a^2 A+2 a b B-A b^2\right) \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d (m+1) \left(a^2+b^2\right)^2}-\frac{\left(a^2 (-B)+2 a A b+b^2 B\right) \tan ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)}{d (m+2) \left(a^2+b^2\right)^2}+\frac{b (A b-a B) \tan ^{m+1}(c+d x)}{a d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{b \left(-\left(a^3 (B-B m)\right)+a^2 A b (2-m)+a b^2 B (m+1)-A b^3 m\right) \tan ^{m+1}(c+d x) \, _2F_1\left(1,m+1;m+2;-\frac{b \tan (c+d x)}{a}\right)}{a^2 d (m+1) \left(a^2+b^2\right)^2}",1,"(Tan[c + d*x]^(1 + m)*((b*(-(a^2*A*b*(-2 + m)) + a^3*B*(-1 + m) - A*b^3*m + a*b^2*B*(1 + m))*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*Tan[c + d*x])/a)])/(a*(a^2 + b^2)*(1 + m)) + (b*(A*b - a*B))/(a + b*Tan[c + d*x]) + (a*(((a^2*A - A*b^2 + 2*a*b*B)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2])/(1 + m) + ((-2*a*A*b + a^2*B - b^2*B)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x])/(2 + m)))/(a^2 + b^2)))/(a*(a^2 + b^2)*d)","A",1
485,1,534,438,6.2532167,"\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]","\frac{b (A b-a B) \tan ^{m+1}(c+d x)}{2 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{\frac{\left(b^2 \left(2 a^2 A+a b B (m+1)+A b^2 (1-m)\right)-a (-a b (1-m) (A b-a B)-2 a b (A b-a B))\right) \tan ^{m+1}(c+d x)}{a d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\frac{\left(b^2 \left(\left(a^2-b^2 m\right) \left(2 a^2 A+a b B (m+1)+A b^2 (1-m)\right)-a^2 b (3-m) (m+1) (A b-a B)\right)+2 a^3 b \left(a^2 (-B)+2 a A b+b^2 B\right)-a^2 b m \left(a^3 (-B) (3-m)+a^2 A b (5-m)+a b^2 B (m+1)+A b^3 (1-m)\right)\right) \tan ^{m+1}(c+d x) \, _2F_1\left(1,m+1;m+2;-\frac{b \tan (c+d x)}{a}\right)}{a d (m+1) \left(a^2+b^2\right)}+\frac{\frac{2 a^2 \left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right) \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+1}{2}+1;-\tan ^2(c+d x)\right)}{d (m+1)}-\frac{2 a^2 \left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right) \tan ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+2}{2}+1;-\tan ^2(c+d x)\right)}{d (m+2)}}{a^2+b^2}}{a \left(a^2+b^2\right)}}{2 a \left(a^2+b^2\right)}","\frac{b (A b-a B) \tan ^{m+1}(c+d x)}{2 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{\left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right) \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d (m+1) \left(a^2+b^2\right)^3}-\frac{\left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right) \tan ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)}{d (m+2) \left(a^2+b^2\right)^3}+\frac{b \left(a^3 (-B) (3-m)+a^2 A b (5-m)+a b^2 B (m+1)+A b^3 (1-m)\right) \tan ^{m+1}(c+d x)}{2 a^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{b \left(a^5 B \left(m^2-3 m+2\right)-a^4 A b \left(m^2-5 m+6\right)-2 a^3 b^2 B \left(-m^2+m+3\right)+2 a^2 A b^3 \left(-m^2+3 m+1\right)+a b^4 B m (m+1)+A b^5 (1-m) m\right) \tan ^{m+1}(c+d x) \, _2F_1\left(1,m+1;m+2;-\frac{b \tan (c+d x)}{a}\right)}{2 a^3 d (m+1) \left(a^2+b^2\right)^3}",1,"(b*(A*b - a*B)*Tan[c + d*x]^(1 + m))/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (((-(a*(-2*a*b*(A*b - a*B) - a*b*(A*b - a*B)*(1 - m))) + b^2*(2*a^2*A + A*b^2*(1 - m) + a*b*B*(1 + m)))*Tan[c + d*x]^(1 + m))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) + (((2*a^3*b*(2*a*A*b - a^2*B + b^2*B) - a^2*b*m*(A*b^3*(1 - m) - a^3*B*(3 - m) + a^2*A*b*(5 - m) + a*b^2*B*(1 + m)) + b^2*(-(a^2*b*(A*b - a*B)*(3 - m)*(1 + m)) + (a^2 - b^2*m)*(2*a^2*A + A*b^2*(1 - m) + a*b*B*(1 + m))))*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(1 + m))/(a*(a^2 + b^2)*d*(1 + m)) + ((2*a^2*(a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*Hypergeometric2F1[1, (1 + m)/2, 1 + (1 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) - (2*a^2*(3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*Hypergeometric2F1[1, (2 + m)/2, 1 + (2 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(d*(2 + m)))/(a^2 + b^2))/(a*(a^2 + b^2)))/(2*a*(a^2 + b^2))","A",1
486,1,1901,659,6.2928546,"\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]","\frac{b (A b-a B) \tan ^{m+1}(c+d x)}{3 a \left(a^2+b^2\right) d (a+b \tan (c+d x))^3}+\frac{\frac{\left(b^2 \left(3 A a^2+b B (m+1) a+A b^2 (2-m)\right)-a (-3 a b (A b-a B)-a b (2-m) (A b-a B))\right) \tan ^{m+1}(c+d x)}{2 a \left(a^2+b^2\right) d (a+b \tan (c+d x))^2}+\frac{\frac{\left(b^2 \left(\left(2 a^2+b^2 (1-m)\right) \left(3 A a^2+b B (m+1) a+A b^2 (2-m)\right)-a^2 b (A b-a B) (5-m) (m+1)\right)-a \left(-6 b \left(-B a^2+2 A b a+b^2 B\right) a^2-b (1-m) \left(-B (5-m) a^3+A b (8-m) a^2+b^2 B (m+1) a+A b^3 (2-m)\right) a\right)\right) \tan ^{m+1}(c+d x)}{a \left(a^2+b^2\right) d (a+b \tan (c+d x))}+\frac{\frac{\left(b \left(6 \left(-B a^2+2 A b a+b^2 B\right) a^3-b^2 (1-m) \left(-B (5-m) a^3+A b (8-m) a^2+b^2 B (m+1) a+A b^3 (2-m)\right)+b \left(\left(2 a^2+b^2 (1-m)\right) \left(3 A a^2+b B (m+1) a+A b^2 (2-m)\right)-a^2 b (A b-a B) (5-m) (m+1)\right)\right) a^2-m \left(b^2 \left(\left(2 a^2+b^2 (1-m)\right) \left(3 A a^2+b B (m+1) a+A b^2 (2-m)\right)-a^2 b (A b-a B) (5-m) (m+1)\right)-a \left(-6 b \left(-B a^2+2 A b a+b^2 B\right) a^2-b (1-m) \left(-B (5-m) a^3+A b (8-m) a^2+b^2 B (m+1) a+A b^3 (2-m)\right) a\right)\right) a^2+b^2 \left(\left(a^2-b^2 m\right) \left(\left(2 a^2+b^2 (1-m)\right) \left(3 A a^2+b B (m+1) a+A b^2 (2-m)\right)-a^2 b (A b-a B) (5-m) (m+1)\right)+a (m+1) \left(-6 b \left(-B a^2+2 A b a+b^2 B\right) a^2-b (1-m) \left(-B (5-m) a^3+A b (8-m) a^2+b^2 B (m+1) a+A b^3 (2-m)\right) a\right)\right)\right) \, _2F_1\left(1,m+1;m+2;-\frac{b \tan (c+d x)}{a}\right) \tan ^{m+1}(c+d x)}{a \left(a^2+b^2\right) d (m+1)}+\frac{\frac{\left(a \left(\left(a^2-b^2 m\right) \left(\left(2 a^2+b^2 (1-m)\right) \left(3 A a^2+b B (m+1) a+A b^2 (2-m)\right)-a^2 b (A b-a B) (5-m) (m+1)\right)+a (m+1) \left(-6 b \left(-B a^2+2 A b a+b^2 B\right) a^2-b (1-m) \left(-B (5-m) a^3+A b (8-m) a^2+b^2 B (m+1) a+A b^3 (2-m)\right) a\right)+m \left(b^2 \left(\left(2 a^2+b^2 (1-m)\right) \left(3 A a^2+b B (m+1) a+A b^2 (2-m)\right)-a^2 b (A b-a B) (5-m) (m+1)\right)-a \left(-6 b \left(-B a^2+2 A b a+b^2 B\right) a^2-b (1-m) \left(-B (5-m) a^3+A b (8-m) a^2+b^2 B (m+1) a+A b^3 (2-m)\right) a\right)\right)\right)-a b \left(6 \left(-B a^2+2 A b a+b^2 B\right) a^3-b^2 (1-m) \left(-B (5-m) a^3+A b (8-m) a^2+b^2 B (m+1) a+A b^3 (2-m)\right)+b \left(\left(2 a^2+b^2 (1-m)\right) \left(3 A a^2+b B (m+1) a+A b^2 (2-m)\right)-a^2 b (A b-a B) (5-m) (m+1)\right)\right)\right) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+1}{2}+1;-\tan ^2(c+d x)\right) \tan ^{m+1}(c+d x)}{d (m+1)}+\frac{\left(-\left(\left(6 \left(-B a^2+2 A b a+b^2 B\right) a^3-b^2 (1-m) \left(-B (5-m) a^3+A b (8-m) a^2+b^2 B (m+1) a+A b^3 (2-m)\right)+b \left(\left(2 a^2+b^2 (1-m)\right) \left(3 A a^2+b B (m+1) a+A b^2 (2-m)\right)-a^2 b (A b-a B) (5-m) (m+1)\right)\right) a^2\right)-b \left(\left(a^2-b^2 m\right) \left(\left(2 a^2+b^2 (1-m)\right) \left(3 A a^2+b B (m+1) a+A b^2 (2-m)\right)-a^2 b (A b-a B) (5-m) (m+1)\right)+a (m+1) \left(-6 b \left(-B a^2+2 A b a+b^2 B\right) a^2-b (1-m) \left(-B (5-m) a^3+A b (8-m) a^2+b^2 B (m+1) a+A b^3 (2-m)\right) a\right)+m \left(b^2 \left(\left(2 a^2+b^2 (1-m)\right) \left(3 A a^2+b B (m+1) a+A b^2 (2-m)\right)-a^2 b (A b-a B) (5-m) (m+1)\right)-a \left(-6 b \left(-B a^2+2 A b a+b^2 B\right) a^2-b (1-m) \left(-B (5-m) a^3+A b (8-m) a^2+b^2 B (m+1) a+A b^3 (2-m)\right) a\right)\right)\right)\right) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+2}{2}+1;-\tan ^2(c+d x)\right) \tan ^{m+2}(c+d x)}{d (m+2)}}{a^2+b^2}}{a \left(a^2+b^2\right)}}{2 a \left(a^2+b^2\right)}}{3 a \left(a^2+b^2\right)}","\frac{b (A b-a B) \tan ^{m+1}(c+d x)}{3 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{b \left(a^3 (-B) (5-m)+a^2 A b (8-m)+a b^2 B (m+1)+A b^3 (2-m)\right) \tan ^{m+1}(c+d x)}{6 a^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{\left(a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right) \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d (m+1) \left(a^2+b^2\right)^4}-\frac{\left(a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right) \tan ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(c+d x)\right)}{d (m+2) \left(a^2+b^2\right)^4}+\frac{b \left(a^5 (-B) \left(m^2-6 m+11\right)+a^4 A b \left(m^2-9 m+26\right)+2 a^3 b^2 B \left(-m^2+3 m+7\right)+2 a^2 A b^3 \left(m^2-6 m+2\right)+a b^4 B \left(1-m^2\right)+A b^5 \left(m^2-3 m+2\right)\right) \tan ^{m+1}(c+d x)}{6 a^3 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{b \left(a^7 B \left(-m^3+6 m^2-11 m+6\right)-a^6 A b \left(-m^3+9 m^2-26 m+24\right)-3 a^5 b^2 B \left(m^3-4 m^2-m+12\right)+3 a^4 A b^3 \left(m^3-7 m^2+10 m+8\right)+3 a^3 b^4 B \left(-m^3+2 m^2+5 m+2\right)+3 a^2 A b^5 m \left(m^2-5 m+2\right)+a b^6 B m \left(1-m^2\right)+A b^7 m \left(m^2-3 m+2\right)\right) \tan ^{m+1}(c+d x) \, _2F_1\left(1,m+1;m+2;-\frac{b \tan (c+d x)}{a}\right)}{6 a^4 d (m+1) \left(a^2+b^2\right)^4}",1,"(b*(A*b - a*B)*Tan[c + d*x]^(1 + m))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (((-(a*(-3*a*b*(A*b - a*B) - a*b*(A*b - a*B)*(2 - m))) + b^2*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m)))*Tan[c + d*x]^(1 + m))/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (((b^2*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m))*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m))) - a*(-6*a^2*b*(2*a*A*b - a^2*B + b^2*B) - a*b*(1 - m)*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m))))*Tan[c + d*x]^(1 + m))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) + (((a^2*b*(6*a^3*(2*a*A*b - a^2*B + b^2*B) - b^2*(1 - m)*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m)) + b*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m))*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m)))) - a^2*m*(b^2*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m))*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m))) - a*(-6*a^2*b*(2*a*A*b - a^2*B + b^2*B) - a*b*(1 - m)*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m)))) + b^2*((a^2 - b^2*m)*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m))*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m))) + a*(1 + m)*(-6*a^2*b*(2*a*A*b - a^2*B + b^2*B) - a*b*(1 - m)*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m)))))*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(1 + m))/(a*(a^2 + b^2)*d*(1 + m)) + (((-(a*b*(6*a^3*(2*a*A*b - a^2*B + b^2*B) - b^2*(1 - m)*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m)) + b*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m))*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m))))) + a*((a^2 - b^2*m)*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m))*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m))) + a*(1 + m)*(-6*a^2*b*(2*a*A*b - a^2*B + b^2*B) - a*b*(1 - m)*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m))) + m*(b^2*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m))*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m))) - a*(-6*a^2*b*(2*a*A*b - a^2*B + b^2*B) - a*b*(1 - m)*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m))))))*Hypergeometric2F1[1, (1 + m)/2, 1 + (1 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + ((-(a^2*(6*a^3*(2*a*A*b - a^2*B + b^2*B) - b^2*(1 - m)*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m)) + b*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m))*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m))))) - b*((a^2 - b^2*m)*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m))*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m))) + a*(1 + m)*(-6*a^2*b*(2*a*A*b - a^2*B + b^2*B) - a*b*(1 - m)*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m))) + m*(b^2*(-(a^2*b*(A*b - a*B)*(5 - m)*(1 + m)) + (2*a^2 + b^2*(1 - m))*(3*a^2*A + A*b^2*(2 - m) + a*b*B*(1 + m))) - a*(-6*a^2*b*(2*a*A*b - a^2*B + b^2*B) - a*b*(1 - m)*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m))))))*Hypergeometric2F1[1, (2 + m)/2, 1 + (2 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(d*(2 + m)))/(a^2 + b^2))/(a*(a^2 + b^2)))/(2*a*(a^2 + b^2)))/(3*a*(a^2 + b^2))","B",1
487,0,0,193,26.8783802,"\int \tan ^m(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\int \tan ^m(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","\frac{a^2 (A+i B) \tan ^{m+1}(c+d x) \sqrt{a+b \tan (c+d x)} F_1\left(m+1;-\frac{5}{2},1;m+2;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right)}{2 d (m+1) \sqrt{\frac{b \tan (c+d x)}{a}+1}}+\frac{a^2 (A-i B) \tan ^{m+1}(c+d x) \sqrt{a+b \tan (c+d x)} F_1\left(m+1;-\frac{5}{2},1;m+2;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right)}{2 d (m+1) \sqrt{\frac{b \tan (c+d x)}{a}+1}}",1,"Integrate[Tan[c + d*x]^m*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x]","F",-1
488,0,0,189,15.792682,"\int \tan ^m(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\int \tan ^m(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","\frac{a (A+i B) \tan ^{m+1}(c+d x) \sqrt{a+b \tan (c+d x)} F_1\left(m+1;-\frac{3}{2},1;m+2;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right)}{2 d (m+1) \sqrt{\frac{b \tan (c+d x)}{a}+1}}+\frac{a (A-i B) \tan ^{m+1}(c+d x) \sqrt{a+b \tan (c+d x)} F_1\left(m+1;-\frac{3}{2},1;m+2;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right)}{2 d (m+1) \sqrt{\frac{b \tan (c+d x)}{a}+1}}",1,"Integrate[Tan[c + d*x]^m*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x]","F",-1
489,0,0,187,5.28621,"\int \tan ^m(c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\int \tan ^m(c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","\frac{(A+i B) \tan ^{m+1}(c+d x) \sqrt{a+b \tan (c+d x)} F_1\left(m+1;-\frac{1}{2},1;m+2;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right)}{2 d (m+1) \sqrt{\frac{b \tan (c+d x)}{a}+1}}+\frac{(A-i B) \tan ^{m+1}(c+d x) \sqrt{a+b \tan (c+d x)} F_1\left(m+1;-\frac{1}{2},1;m+2;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right)}{2 d (m+1) \sqrt{\frac{b \tan (c+d x)}{a}+1}}",1,"Integrate[Tan[c + d*x]^m*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x]","F",-1
490,0,0,187,6.1879995,"\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]","\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx","\frac{(A+i B) \tan ^{m+1}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(m+1;\frac{1}{2},1;m+2;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right)}{2 d (m+1) \sqrt{a+b \tan (c+d x)}}+\frac{(A-i B) \tan ^{m+1}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(m+1;\frac{1}{2},1;m+2;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right)}{2 d (m+1) \sqrt{a+b \tan (c+d x)}}",1,"Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]], x]","F",-1
491,0,0,193,16.0876987,"\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","\frac{(A+i B) \tan ^{m+1}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(m+1;\frac{3}{2},1;m+2;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right)}{2 a d (m+1) \sqrt{a+b \tan (c+d x)}}+\frac{(A-i B) \tan ^{m+1}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(m+1;\frac{3}{2},1;m+2;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right)}{2 a d (m+1) \sqrt{a+b \tan (c+d x)}}",1,"Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x]","F",-1
492,0,0,193,76.7525275,"\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]","\int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","\frac{(A+i B) \tan ^{m+1}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(m+1;\frac{5}{2},1;m+2;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right)}{2 a^2 d (m+1) \sqrt{a+b \tan (c+d x)}}+\frac{(A-i B) \tan ^{m+1}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(m+1;\frac{5}{2},1;m+2;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right)}{2 a^2 d (m+1) \sqrt{a+b \tan (c+d x)}}",1,"Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x]","F",-1
493,0,0,183,2.7671851,"\int \tan ^m(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^m*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \tan ^m(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","\frac{(A+i B) \tan ^{m+1}(c+d x) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(m+1;-n,1;m+2;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right)}{2 d (m+1)}+\frac{(A-i B) \tan ^{m+1}(c+d x) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(m+1;-n,1;m+2;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right)}{2 d (m+1)}",1,"Integrate[Tan[c + d*x]^m*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
494,1,451,387,6.2622849,"\int \tan ^4(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^4*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\frac{B \tan ^3(c+d x) (a+b \tan (c+d x))^{n+1}}{b d (n+4)}+\frac{\frac{\tan ^2(c+d x) (A b (n+4)-3 a B) (a+b \tan (c+d x))^{n+1}}{b d (n+3)}+\frac{\frac{\tan (c+d x) \left(6 a^2 B-2 a A b (n+4)-b^2 B (n+3) (n+4)\right) (a+b \tan (c+d x))^{n+1}}{b d (n+2)}+\frac{\frac{\left(-6 a^3 B+2 a^2 A b (n+4)+a b^2 B (n+3) (n+4)-A b^3 (n+2) (n+3) (n+4)\right) (a+b \tan (c+d x))^{n+1}}{b d (n+1)}+\frac{i \left(A b^3 (n+2) (n+3) (n+4)+i b^3 B (n+2) (n+3) (n+4)\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{-i a-i b \tan (c+d x)}{b-i a}\right)}{2 d (n+1) (a+i b)}-\frac{i \left(A b^3 (n+2) (n+3) (n+4)-i b^3 B (n+2) (n+3) (n+4)\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;-\frac{i a+i b \tan (c+d x)}{-i a-b}\right)}{2 d (n+1) (a-i b)}}{b (n+2)}}{b (n+3)}}{b (n+4)}","-\frac{\tan ^2(c+d x) (3 a B-A b (n+4)) (a+b \tan (c+d x))^{n+1}}{b^2 d (n+3) (n+4)}-\frac{\tan (c+d x) \left(b^2 B (n+3) (n+4)-2 a (3 a B-A b (n+4))\right) (a+b \tan (c+d x))^{n+1}}{b^3 d (n+2) (n+3) (n+4)}-\frac{\left(A b^3 (n+2) (n+3) (n+4)-a \left(b^2 B (n+3) (n+4)-2 a (3 a B-A b (n+4))\right)\right) (a+b \tan (c+d x))^{n+1}}{b^4 d (n+1) (n+2) (n+3) (n+4)}+\frac{(A-i B) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{2 d (n+1) (b+i a)}-\frac{(A+i B) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{2 d (n+1) (-b+i a)}+\frac{B \tan ^3(c+d x) (a+b \tan (c+d x))^{n+1}}{b d (n+4)}",1,"(B*Tan[c + d*x]^3*(a + b*Tan[c + d*x])^(1 + n))/(b*d*(4 + n)) + (((-3*a*B + A*b*(4 + n))*Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(1 + n))/(b*d*(3 + n)) + (((6*a^2*B - 2*a*A*b*(4 + n) - b^2*B*(3 + n)*(4 + n))*Tan[c + d*x]*(a + b*Tan[c + d*x])^(1 + n))/(b*d*(2 + n)) + (((-6*a^3*B + 2*a^2*A*b*(4 + n) + a*b^2*B*(3 + n)*(4 + n) - A*b^3*(2 + n)*(3 + n)*(4 + n))*(a + b*Tan[c + d*x])^(1 + n))/(b*d*(1 + n)) + ((I/2)*(A*b^3*(2 + n)*(3 + n)*(4 + n) + I*b^3*B*(2 + n)*(3 + n)*(4 + n))*Hypergeometric2F1[1, 1 + n, 2 + n, ((-I)*a - I*b*Tan[c + d*x])/((-I)*a + b)]*(a + b*Tan[c + d*x])^(1 + n))/((a + I*b)*d*(1 + n)) - ((I/2)*(A*b^3*(2 + n)*(3 + n)*(4 + n) - I*b^3*B*(2 + n)*(3 + n)*(4 + n))*Hypergeometric2F1[1, 1 + n, 2 + n, -((I*a + I*b*Tan[c + d*x])/((-I)*a - b))]*(a + b*Tan[c + d*x])^(1 + n))/((a - I*b)*d*(1 + n)))/(b*(2 + n)))/(b*(3 + n)))/(b*(4 + n))","A",1
495,1,281,291,2.4836895,"\int \tan ^3(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^3*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\frac{(a+b \tan (c+d x))^{n+1} \left(2 (a-i b) (a+i b) \left(2 a^2 B-a A b (n+3)-b^2 B (n+2) (n+3)\right)+b^3 \left(n^2+5 n+6\right) (a+i b) (A-i B) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)+b^3 \left(n^2+5 n+6\right) (a-i b) (A+i B) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)-2 b (n+1) (a-i b) (a+i b) \tan (c+d x) (2 a B-A b (n+3))+2 b^2 B (n+1) (n+2) (a-i b) (a+i b) \tan ^2(c+d x)\right)}{2 b^3 d (n+1) (n+2) (n+3) (a-i b) (a+i b)}","\frac{\left(2 a^2 B-a A b (n+3)-b^2 B \left(n^2+5 n+6\right)\right) (a+b \tan (c+d x))^{n+1}}{b^3 d (n+1) (n+2) (n+3)}-\frac{\tan (c+d x) (2 a B-A b (n+3)) (a+b \tan (c+d x))^{n+1}}{b^2 d (n+2) (n+3)}+\frac{(B+i A) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{2 d (n+1) (b+i a)}+\frac{(A+i B) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{2 d (n+1) (a+i b)}+\frac{B \tan ^2(c+d x) (a+b \tan (c+d x))^{n+1}}{b d (n+3)}",1,"((a + b*Tan[c + d*x])^(1 + n)*(2*(a - I*b)*(a + I*b)*(2*a^2*B - a*A*b*(3 + n) - b^2*B*(2 + n)*(3 + n)) + (a + I*b)*b^3*(A - I*B)*(6 + 5*n + n^2)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)] + (a - I*b)*b^3*(A + I*B)*(6 + 5*n + n^2)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)] - 2*(a - I*b)*(a + I*b)*b*(1 + n)*(2*a*B - A*b*(3 + n))*Tan[c + d*x] + 2*(a - I*b)*(a + I*b)*b^2*B*(1 + n)*(2 + n)*Tan[c + d*x]^2))/(2*(a - I*b)*(a + I*b)*b^3*d*(1 + n)*(2 + n)*(3 + n))","A",1
496,1,169,219,1.4798605,"\int \tan ^2(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\frac{(a+b \tan (c+d x))^{n+1} \left(\frac{b (n+2) (B+i A) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{(n+1) (a-i b)}+\frac{b (n+2) (B-i A) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{(n+1) (a+i b)}+\frac{-2 a B+2 A b n+4 A b}{b n+b}+2 B \tan (c+d x)\right)}{2 b d (n+2)}","-\frac{(a B-A b (n+2)) (a+b \tan (c+d x))^{n+1}}{b^2 d (n+1) (n+2)}+\frac{(B+i A) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{2 d (n+1) (a-i b)}+\frac{(A+i B) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{2 d (n+1) (-b+i a)}+\frac{B \tan (c+d x) (a+b \tan (c+d x))^{n+1}}{b d (n+2)}",1,"((a + b*Tan[c + d*x])^(1 + n)*((4*A*b - 2*a*B + 2*A*b*n)/(b + b*n) + (b*(I*A + B)*(2 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)])/((a - I*b)*(1 + n)) + (b*((-I)*A + B)*(2 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)])/((a + I*b)*(1 + n)) + 2*B*Tan[c + d*x]))/(2*b*d*(2 + n))","A",1
497,1,125,168,0.2636194,"\int \tan (c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\frac{(a+b \tan (c+d x))^{n+1} \left(-\frac{(A-i B) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{a-i b}-\frac{(A+i B) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{a+i b}+\frac{2 B}{b}\right)}{2 d (n+1)}","-\frac{(A-i B) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{2 d (n+1) (a-i b)}-\frac{(A+i B) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{2 d (n+1) (a+i b)}+\frac{B (a+b \tan (c+d x))^{n+1}}{b d (n+1)}",1,"(((2*B)/b - ((A - I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)])/(a - I*b) - ((A + I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)])/(a + I*b))*(a + b*Tan[c + d*x])^(1 + n))/(2*d*(1 + n))","A",1
498,1,120,143,0.1873838,"\int (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\frac{i (a+b \tan (c+d x))^{n+1} \left(\frac{(A+i B) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{a+i b}-\frac{(A-i B) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{a-i b}\right)}{2 d (n+1)}","\frac{(A-i B) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{2 d (n+1) (b+i a)}+\frac{(-B+i A) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{2 d (n+1) (a+i b)}",1,"((I/2)*(-(((A - I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)])/(a - I*b)) + ((A + I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)])/(a + I*b))*(a + b*Tan[c + d*x])^(1 + n))/(d*(1 + n))","A",1
499,1,169,190,0.4234751,"\int \cot (c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\frac{(a+b \tan (c+d x))^{n+1} \left(a (a+i b) (A-i B) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)+(a-i b) \left(a (A+i B) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)-2 A (a+i b) \, _2F_1\left(1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)\right)\right)}{2 a d (n+1) (a-i b) (a+i b)}","\frac{(B+i A) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{2 d (n+1) (b+i a)}+\frac{(A+i B) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{2 d (n+1) (a+i b)}-\frac{A (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)}{a d (n+1)}",1,"((a*(a + I*b)*(A - I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)] + (a - I*b)*(a*(A + I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)] - 2*A*(a + I*b)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]))*(a + b*Tan[c + d*x])^(1 + n))/(2*a*(a - I*b)*(a + I*b)*d*(1 + n))","A",1
500,1,202,228,0.4183325,"\int \cot ^2(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\frac{(a+b \tan (c+d x))^{n+1} \left(a^2 (a+i b) (A-i B) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)-(a-i b) \left(a^2 (A+i B) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)+2 (b-i a) \left(a B \, _2F_1\left(1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)-A b \, _2F_1\left(2,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)\right)\right)\right)}{2 a^2 d (n+1) (a-i b) (b-i a)}","-\frac{(a B+A b n) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)}{a^2 d (n+1)}-\frac{(A-i B) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{2 d (n+1) (b+i a)}+\frac{(A+i B) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{2 d (n+1) (-b+i a)}-\frac{A \cot (c+d x) (a+b \tan (c+d x))^{n+1}}{a d}",1,"((a^2*(a + I*b)*(A - I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)] - (a - I*b)*(a^2*(A + I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)] + 2*((-I)*a + b)*(a*B*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a] - A*b*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a])))*(a + b*Tan[c + d*x])^(1 + n))/(2*a^2*(a - I*b)*((-I)*a + b)*d*(1 + n))","A",1
501,1,230,292,0.547027,"\int \cot ^3(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","-\frac{(a+b \tan (c+d x))^{n+1} \left(a^3 (a+i b) (A-i B) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)+(a-i b) \left(a^3 (A+i B) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)-2 (a+i b) \left(a^2 A \, _2F_1\left(1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)+b \left(a B \, _2F_1\left(2,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)-A b \, _2F_1\left(3,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)\right)\right)\right)\right)}{2 a^3 d (n+1) (a-i b) (a+i b)}","-\frac{\cot (c+d x) (2 a B-A b (1-n)) (a+b \tan (c+d x))^{n+1}}{2 a^2 d}+\frac{\left(2 a^2 A-2 a b B n+A b^2 (1-n) n\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)}{2 a^3 d (n+1)}-\frac{(B+i A) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{2 d (n+1) (b+i a)}-\frac{(A+i B) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{2 d (n+1) (a+i b)}-\frac{A \cot ^2(c+d x) (a+b \tan (c+d x))^{n+1}}{2 a d}",1,"-1/2*((a^3*(a + I*b)*(A - I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)] + (a - I*b)*(a^3*(A + I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)] - 2*(a + I*b)*(a^2*A*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a] + b*(a*B*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a] - A*b*Hypergeometric2F1[3, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]))))*(a + b*Tan[c + d*x])^(1 + n))/(a^3*(a - I*b)*(a + I*b)*d*(1 + n))","A",1
502,1,263,103,3.2550053,"\int \cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{a \sin ^2(c+d x) (\cot (c+d x)+i) (\cos (d x)-i \sin (d x)) (A \cot (c+d x)+B) \left(-\frac{2 i e^{-i c} (A-i B) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}}}-\frac{1}{15} (\cos (c)-i \sin (c)) \sqrt{\cot (c+d x)} \csc ^2(c+d x) (5 (B+i A) \sin (2 (c+d x))+3 (6 A-5 i B) \cos (2 (c+d x))-12 A+15 i B)\right)}{d (A \cos (c+d x)+B \sin (c+d x))}","-\frac{2 a (B+i A) \cot ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a (A-i B) \sqrt{\cot (c+d x)}}{d}+\frac{2 \sqrt[4]{-1} a (A-i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}-\frac{2 a A \cot ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(a*(I + Cot[c + d*x])*(B + A*Cot[c + d*x])*(Cos[d*x] - I*Sin[d*x])*Sin[c + d*x]^2*(((-2*I)*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/(E^(I*c)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]) - (Sqrt[Cot[c + d*x]]*Csc[c + d*x]^2*(Cos[c] - I*Sin[c])*(-12*A + (15*I)*B + 3*(6*A - (5*I)*B)*Cos[2*(c + d*x)] + 5*(I*A + B)*Sin[2*(c + d*x)]))/15))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
503,1,161,78,3.6164108,"\int \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","-\frac{2 a e^{-i c} \sin ^2(c+d x) \sqrt{\cot (c+d x)} (\cot (c+d x)+i) (\cos (d x)-i \sin (d x)) (A \cot (c+d x)+B) \left(-3 i (A-i B) \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+A \cot (c+d x)+3 i A+3 B\right)}{3 d (A \cos (c+d x)+B \sin (c+d x))}","-\frac{2 a (B+i A) \sqrt{\cot (c+d x)}}{d}-\frac{2 \sqrt[4]{-1} a (B+i A) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(-2*a*Sqrt[Cot[c + d*x]]*(I + Cot[c + d*x])*(B + A*Cot[c + d*x])*(Cos[d*x] - I*Sin[d*x])*Sin[c + d*x]^2*((3*I)*A + 3*B + A*Cot[c + d*x] - (3*I)*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/(3*d*E^(I*c)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
504,1,92,53,2.2155794,"\int \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{2 a e^{-i c} (\cos (c)+i \sin (c)) \sqrt{\cot (c+d x)} \left(-A+(A-i B) \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right)}{d}","-\frac{2 a A \sqrt{\cot (c+d x)}}{d}-\frac{2 \sqrt[4]{-1} a (A-i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}",1,"(2*a*Sqrt[Cot[c + d*x]]*(Cos[c] + I*Sin[c])*(-A + (A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/(d*E^(I*c))","A",1
505,1,108,55,3.6227818,"\int \sqrt{\cot (c+d x)} (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{2 a e^{-i c} (\cos (c)+i \sin (c)) \left((A-i B) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+i B \sqrt{i \tan (c+d x)}\right)}{d \sqrt{i \tan (c+d x)} \sqrt{\cot (c+d x)}}","\frac{2 \sqrt[4]{-1} a (B+i A) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}+\frac{2 i a B}{d \sqrt{\cot (c+d x)}}",1,"(2*a*(Cos[c] + I*Sin[c])*((A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]] + I*B*Sqrt[I*Tan[c + d*x]]))/(d*E^(I*c)*Sqrt[Cot[c + d*x]]*Sqrt[I*Tan[c + d*x]])","A",1
506,1,96,80,2.8181887,"\int \frac{(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","Integrate[((a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]","\frac{2 a \left(3 (B+i A) \cot (c+d x)+\frac{3 (A-i B) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{(i \tan (c+d x))^{3/2}}+i B\right)}{3 d \cot ^{\frac{3}{2}}(c+d x)}","\frac{2 a (B+i A)}{d \sqrt{\cot (c+d x)}}+\frac{2 \sqrt[4]{-1} a (A-i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}+\frac{2 i a B}{3 d \cot ^{\frac{3}{2}}(c+d x)}",1,"(2*a*(I*B + 3*(I*A + B)*Cot[c + d*x] + (3*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/(I*Tan[c + d*x])^(3/2)))/(3*d*Cot[c + d*x]^(3/2))","A",1
507,1,133,105,3.8534677,"\int \frac{(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(3/2),x]","\frac{a \left(\sec ^2(c+d x) (5 (B+i A) \sin (2 (c+d x))+3 (5 A-6 i B) \cos (2 (c+d x))+3 (5 A-4 i B))-\frac{30 (A-i B) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{i \tan (c+d x)}}\right)}{15 d \sqrt{\cot (c+d x)}}","\frac{2 a (B+i A)}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 a (A-i B)}{d \sqrt{\cot (c+d x)}}-\frac{2 \sqrt[4]{-1} a (B+i A) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}+\frac{2 i a B}{5 d \cot ^{\frac{5}{2}}(c+d x)}",1,"(a*(Sec[c + d*x]^2*(3*(5*A - (4*I)*B) + 3*(5*A - (6*I)*B)*Cos[2*(c + d*x)] + 5*(I*A + B)*Sin[2*(c + d*x)]) - (30*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/Sqrt[I*Tan[c + d*x]]))/(15*d*Sqrt[Cot[c + d*x]])","A",1
508,1,272,128,6.2082991,"\int \cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{a^2 \sin ^3(c+d x) (\cot (c+d x)+i)^2 (A \cot (c+d x)+B) \left(-\frac{4 i e^{-2 i c} (A-i B) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}}}-\frac{1}{15} (\cos (2 c)-i \sin (2 c)) \sqrt{\cot (c+d x)} \csc ^2(c+d x) (5 (B+2 i A) \sin (2 (c+d x))+(33 A-30 i B) \cos (2 (c+d x))-27 A+30 i B)\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{2 a^2 (5 B+7 i A) \cot ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{4 a^2 (A-i B) \sqrt{\cot (c+d x)}}{d}+\frac{4 \sqrt[4]{-1} a^2 (A-i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x) \left(a^2 \cot (c+d x)+i a^2\right)}{5 d}",1,"(a^2*(I + Cot[c + d*x])^2*(B + A*Cot[c + d*x])*Sin[c + d*x]^3*(((-4*I)*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/(E^((2*I)*c)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]) - (Sqrt[Cot[c + d*x]]*Csc[c + d*x]^2*(Cos[2*c] - I*Sin[2*c])*(-27*A + (30*I)*B + (33*A - (30*I)*B)*Cos[2*(c + d*x)] + 5*((2*I)*A + B)*Sin[2*(c + d*x)]))/15))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
509,1,174,103,5.0939259,"\int \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","-\frac{2 a^2 e^{-2 i c} \sin (c+d x) \sqrt{\cot (c+d x)} (\cos (2 (c+d x))+i \sin (2 (c+d x))) (A \cot (c+d x)+B) \left(-6 i (A-i B) \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+A \cot (c+d x)+6 i A+3 B\right)}{3 d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{2 a^2 (3 B+5 i A) \sqrt{\cot (c+d x)}}{3 d}-\frac{4 \sqrt[4]{-1} a^2 (B+i A) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}-\frac{2 A \sqrt{\cot (c+d x)} \left(a^2 \cot (c+d x)+i a^2\right)}{3 d}",1,"(-2*a^2*Sqrt[Cot[c + d*x]]*(B + A*Cot[c + d*x])*Sin[c + d*x]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])*((6*I)*A + 3*B + A*Cot[c + d*x] - (6*I)*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/(3*d*E^((2*I)*c)*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
510,1,163,99,4.9875256,"\int \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","-\frac{2 a^2 e^{-2 i c} \cos (c+d x) \sqrt{\cot (c+d x)} (\cos (2 (c+d x))+i \sin (2 (c+d x))) (A+B \tan (c+d x)) \left(-2 (A-i B) \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+A+B \tan (c+d x)\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{2 a^2 (A+i B) \sqrt{\cot (c+d x)}}{d}-\frac{4 \sqrt[4]{-1} a^2 (A-i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}+\frac{2 i B \left(a^2 \cot (c+d x)+i a^2\right)}{d \sqrt{\cot (c+d x)}}",1,"(-2*a^2*Cos[c + d*x]*Sqrt[Cot[c + d*x]]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])*(A + B*Tan[c + d*x])*(A - 2*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]] + B*Tan[c + d*x]))/(d*E^((2*I)*c)*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
511,1,254,105,4.3332888,"\int \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{a^2 e^{-i (c-d x)} \sqrt{\cot (c+d x)} \left(A \left(1+e^{2 i (c+d x)}\right)-i B \left(-1+e^{2 i (c+d x)}\right)\right) \left(\left(-1+e^{2 i (c+d x)}\right) \left(3 i A \left(1+e^{2 i (c+d x)}\right)+B \left(5+7 e^{2 i (c+d x)}\right)\right)-6 i (A-i B) \sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(1+e^{2 i (c+d x)}\right)^2 \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right)}{3 d \left(e^{2 i c+3 i d x}+e^{i d x}\right)^2 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{2 a^2 (3 A-5 i B)}{3 d \sqrt{\cot (c+d x)}}+\frac{4 \sqrt[4]{-1} a^2 (B+i A) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}+\frac{2 i B \left(a^2 \cot (c+d x)+i a^2\right)}{3 d \cot ^{\frac{3}{2}}(c+d x)}",1,"(a^2*((-I)*B*(-1 + E^((2*I)*(c + d*x))) + A*(1 + E^((2*I)*(c + d*x))))*((-1 + E^((2*I)*(c + d*x)))*((3*I)*A*(1 + E^((2*I)*(c + d*x))) + B*(5 + 7*E^((2*I)*(c + d*x)))) - (6*I)*(A - I*B)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))^2*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Sqrt[Cot[c + d*x]])/(3*d*E^(I*(c - d*x))*(E^(I*d*x) + E^((2*I)*c + (3*I)*d*x))^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",1
512,1,133,130,7.7838973,"\int \frac{(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","Integrate[((a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]","\frac{a^2 \left(\sec ^2(c+d x) (-5 (A-2 i B) \sin (2 (c+d x))+(33 B+30 i A) \cos (2 (c+d x))+30 i A+27 B)-\frac{60 i (A-i B) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{i \tan (c+d x)}}\right)}{15 d \sqrt{\cot (c+d x)}}","-\frac{2 a^2 (5 A-7 i B)}{15 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (B+i A)}{d \sqrt{\cot (c+d x)}}+\frac{4 \sqrt[4]{-1} a^2 (A-i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}+\frac{2 i B \left(a^2 \cot (c+d x)+i a^2\right)}{5 d \cot ^{\frac{5}{2}}(c+d x)}",1,"(a^2*(Sec[c + d*x]^2*((30*I)*A + 27*B + ((30*I)*A + 33*B)*Cos[2*(c + d*x)] - 5*(A - (2*I)*B)*Sin[2*(c + d*x)]) - ((60*I)*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/Sqrt[I*Tan[c + d*x]]))/(15*d*Sqrt[Cot[c + d*x]])","A",1
513,1,161,171,9.793351,"\int \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{a^3 \sqrt{\cot (c+d x)} \left(-1680 i (A-i B) \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)-\left(\csc ^3(c+d x) ((-95 A+105 i B) \cos (c+d x)+5 (31 A-21 i B) \cos (3 (c+d x))+42 \sin (c+d x) ((21 B+23 i A) \cos (2 (c+d x))-17 i A-19 B))\right)\right)}{210 d}","\frac{8 a^3 (23 A-21 i B) \cot ^{\frac{3}{2}}(c+d x)}{105 d}-\frac{2 (7 B+11 i A) \cot ^{\frac{3}{2}}(c+d x) \left(a^3 \cot (c+d x)+i a^3\right)}{35 d}+\frac{8 a^3 (B+i A) \sqrt{\cot (c+d x)}}{d}+\frac{8 \sqrt[4]{-1} a^3 (B+i A) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (a \cot (c+d x)+i a)^2}{7 d}",1,"(a^3*Sqrt[Cot[c + d*x]]*(-(Csc[c + d*x]^3*((-95*A + (105*I)*B)*Cos[c + d*x] + 5*(31*A - (21*I)*B)*Cos[3*(c + d*x)] + 42*((-17*I)*A - 19*B + ((23*I)*A + 21*B)*Cos[2*(c + d*x)])*Sin[c + d*x])) - (1680*I)*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/(210*d)","A",1
514,1,132,146,7.8670281,"\int \cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","-\frac{a^3 \sqrt{\cot (c+d x)} \left(120 (A-i B) \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+\csc ^2(c+d x) (5 (B+3 i A) \sin (2 (c+d x))+9 (7 A-5 i B) \cos (2 (c+d x))-57 A+45 i B)\right)}{15 d}","\frac{16 a^3 (6 A-5 i B) \sqrt{\cot (c+d x)}}{15 d}-\frac{2 (5 B+9 i A) \sqrt{\cot (c+d x)} \left(a^3 \cot (c+d x)+i a^3\right)}{15 d}+\frac{8 \sqrt[4]{-1} a^3 (A-i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}-\frac{2 a A \sqrt{\cot (c+d x)} (a \cot (c+d x)+i a)^2}{5 d}",1,"-1/15*(a^3*Sqrt[Cot[c + d*x]]*(Csc[c + d*x]^2*(-57*A + (45*I)*B + 9*(7*A - (5*I)*B)*Cos[2*(c + d*x)] + 5*((3*I)*A + B)*Sin[2*(c + d*x)]) + 120*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/d","A",1
515,1,146,138,6.5154043,"\int \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","-\frac{a^3 \sqrt{\cot (c+d x)} \csc (c+d x) \sec (c+d x) \left((A-3 i B) \cos (2 (c+d x))-12 i (A-i B) \sin (2 (c+d x)) \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+9 i A \sin (2 (c+d x))+A+3 B \sin (2 (c+d x))+3 i B\right)}{3 d}","-\frac{2 (A+3 i B) \sqrt{\cot (c+d x)} \left(a^3 \cot (c+d x)+i a^3\right)}{3 d}-\frac{8 \sqrt[4]{-1} a^3 (B+i A) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}-\frac{16 i a^3 A \sqrt{\cot (c+d x)}}{3 d}+\frac{2 i a B (a \cot (c+d x)+i a)^2}{d \sqrt{\cot (c+d x)}}",1,"-1/3*(a^3*Sqrt[Cot[c + d*x]]*Csc[c + d*x]*Sec[c + d*x]*(A + (3*I)*B + (A - (3*I)*B)*Cos[2*(c + d*x)] + (9*I)*A*Sin[2*(c + d*x)] + 3*B*Sin[2*(c + d*x)] - (12*I)*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sin[2*(c + d*x)]*Sqrt[I*Tan[c + d*x]]))/d","A",1
516,1,132,142,6.92475,"\int \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","-\frac{a^3 \sqrt{\cot (c+d x)} \left(\sec ^2(c+d x) ((9 B+3 i A) \sin (2 (c+d x))+(3 A-i B) \cos (2 (c+d x))+3 A+i B)-24 (A-i B) \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right)}{3 d}","-\frac{2 (3 A-7 i B) \left(a^3 \cot (c+d x)+i a^3\right)}{3 d \sqrt{\cot (c+d x)}}-\frac{8 \sqrt[4]{-1} a^3 (A-i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}-\frac{16 i a^3 B \sqrt{\cot (c+d x)}}{3 d}+\frac{2 i a B (a \cot (c+d x)+i a)^2}{3 d \cot ^{\frac{3}{2}}(c+d x)}",1,"-1/3*(a^3*Sqrt[Cot[c + d*x]]*(Sec[c + d*x]^2*(3*A + I*B + (3*A - I*B)*Cos[2*(c + d*x)] + ((3*I)*A + 9*B)*Sin[2*(c + d*x)]) - 24*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/d","A",1
517,1,140,148,7.2273192,"\int \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{a^3 \sec ^2(c+d x) \left(-5 (3 B+i A) \sin (2 (c+d x))-9 (5 A-7 i B) \cos (2 (c+d x))+\frac{120 (A-i B) \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{i \tan (c+d x)}}-45 A+57 i B\right)}{15 d \sqrt{\cot (c+d x)}}","-\frac{2 (5 A-9 i B) \left(a^3 \cot (c+d x)+i a^3\right)}{15 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{16 a^3 (5 A-6 i B)}{15 d \sqrt{\cot (c+d x)}}+\frac{8 \sqrt[4]{-1} a^3 (B+i A) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}+\frac{2 i a B (a \cot (c+d x)+i a)^2}{5 d \cot ^{\frac{5}{2}}(c+d x)}",1,"(a^3*Sec[c + d*x]^2*(-45*A + (57*I)*B - 9*(5*A - (7*I)*B)*Cos[2*(c + d*x)] - 5*(I*A + 3*B)*Sin[2*(c + d*x)] + (120*(A - I*B)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Cos[c + d*x]^2)/Sqrt[I*Tan[c + d*x]]))/(15*d*Sqrt[Cot[c + d*x]])","A",1
518,1,298,173,15.1803475,"\int \frac{(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","Integrate[((a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]","\frac{a^3 \sin ^4(c+d x) (\cot (c+d x)+i)^3 (A \cot (c+d x)+B) \left(\frac{(\sin (3 c)+i \cos (3 c)) \sec ^2(c+d x) (10 ((31 B+21 i A) \cos (2 (c+d x))+21 i A+25 B)+21 (59 A-57 i B) \cot (c+d x)+21 (21 A-23 i B) \cos (3 (c+d x)) \csc (c+d x))}{210 \cot ^{\frac{3}{2}}(c+d x)}-8 e^{-3 i c} (A-i B) \sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{8 a^3 (21 A-23 i B)}{105 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{2 (7 A-11 i B) \left(a^3 \cot (c+d x)+i a^3\right)}{35 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{8 a^3 (B+i A)}{d \sqrt{\cot (c+d x)}}+\frac{8 \sqrt[4]{-1} a^3 (A-i B) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}+\frac{2 i a B (a \cot (c+d x)+i a)^2}{7 d \cot ^{\frac{7}{2}}(c+d x)}",1,"(a^3*(I + Cot[c + d*x])^3*(B + A*Cot[c + d*x])*((-8*(A - I*B)*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/E^((3*I)*c) + ((10*((21*I)*A + 25*B + ((21*I)*A + 31*B)*Cos[2*(c + d*x)]) + 21*(59*A - (57*I)*B)*Cot[c + d*x] + 21*(21*A - (23*I)*B)*Cos[3*(c + d*x)]*Csc[c + d*x])*Sec[c + d*x]^2*(I*Cos[3*c] + Sin[3*c]))/(210*Cot[c + d*x]^(3/2)))*Sin[c + d*x]^4)/(d*(Cos[d*x] + I*Sin[d*x])^3*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
519,1,247,297,3.095767,"\int \frac{\cot ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]),x]","\frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left(\frac{2}{3} \cot (c+d x) \csc (c+d x) (\cos (d x)-i \sin (d x)) ((-12 B+8 i A) \sin (2 (c+d x))+(11 A+15 i B) \cos (2 (c+d x))-19 A-15 i B)+(1-i) (\cos (c)+i \sin (c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left(((6+i) A+(1+4 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+((4+i) B-(1+6 i) A) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{8 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))}","\frac{(A+i B) \cot ^{\frac{5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac{(7 A+3 i B) \cot ^{\frac{3}{2}}(c+d x)}{6 a d}+\frac{5 (-B+i A) \sqrt{\cot (c+d x)}}{2 a d}+\frac{((7+5 i) A-(5-3 i) B) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{8 \sqrt{2} a d}+\frac{((5-3 i) B-(7+5 i) A) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{8 \sqrt{2} a d}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) ((6+i) A+(1+4 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a d}+\frac{((7-5 i) A+(5+3 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{4 \sqrt{2} a d}",1,"((Cos[d*x] + I*Sin[d*x])*((1 - I)*Csc[c + d*x]*(((6 + I)*A + (1 + 4*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + ((-1 - 6*I)*A + (4 + I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(Cos[c] + I*Sin[c])*Sqrt[Sin[2*(c + d*x)]] + (2*Cot[c + d*x]*Csc[c + d*x]*(Cos[d*x] - I*Sin[d*x])*(-19*A - (15*I)*B + (11*A + (15*I)*B)*Cos[2*(c + d*x)] + ((8*I)*A - 12*B)*Sin[2*(c + d*x)]))/3)*(A + B*Tan[c + d*x]))/(8*d*Sqrt[Cot[c + d*x]]*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x]))","A",1
520,1,223,268,2.5641134,"\int \frac{\cot ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]),x]","\frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left(\cot (c+d x) (-4 \cos (d x)+4 i \sin (d x)) (4 A \cos (c+d x)+(-B+5 i A) \sin (c+d x))+(-\sin (c)+i \cos (c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left(((3-5 i) A+(1+3 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((4+i) A+(1+2 i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{8 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))}","\frac{(A+i B) \cot ^{\frac{3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac{(5 A+i B) \sqrt{\cot (c+d x)}}{2 a d}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) ((4+i) A+(1+2 i) B) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{((5-3 i) A+(3+i) B) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{8 \sqrt{2} a d}+\frac{((3-i) B-(5+3 i) A) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{4 \sqrt{2} a d}+\frac{((5+3 i) A-(3-i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{4 \sqrt{2} a d}",1,"((Cos[d*x] + I*Sin[d*x])*(Cot[c + d*x]*(-4*Cos[d*x] + (4*I)*Sin[d*x])*(4*A*Cos[c + d*x] + ((5*I)*A - B)*Sin[c + d*x]) + Csc[c + d*x]*(((3 - 5*I)*A + (1 + 3*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] - (1 + I)*((4 + I)*A + (1 + 2*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(I*Cos[c] - Sin[c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(8*d*Sqrt[Cot[c + d*x]]*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x]))","A",1
521,1,199,235,1.9487849,"\int \frac{\sqrt{\cot (c+d x)} (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx","Integrate[(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]),x]","\frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left(4 (A+i B) \cos (c+d x) (\cos (d x)-i \sin (d x))+(1+i) (-\sin (c)+i \cos (c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left((B+(2+i) A) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+(i B-(1+2 i) A) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{8 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))}","\frac{(A+i B) \sqrt{\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}-\frac{((3+i) A-(1+i) B) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{8 \sqrt{2} a d}+\frac{((3+i) A-(1+i) B) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{8 \sqrt{2} a d}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) (B+(2+i) A) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a d}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) (B+(2+i) A) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}",1,"((Cos[d*x] + I*Sin[d*x])*(4*(A + I*B)*Cos[c + d*x]*(Cos[d*x] - I*Sin[d*x]) + (1 + I)*Csc[c + d*x]*(((2 + I)*A + B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + ((-1 - 2*I)*A + I*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(I*Cos[c] - Sin[c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(8*d*Sqrt[Cot[c + d*x]]*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x]))","A",1
522,1,198,237,2.3254026,"\int \frac{A+B \tan (c+d x)}{\sqrt{\cot (c+d x)} (a+i a \tan (c+d x))} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])),x]","\frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left(4 (A+i B) \cos (c+d x) (\sin (d x)+i \cos (d x))+(1+i) (-\sin (c)+i \cos (c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left((A+(2-i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+i (A-(2+i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{8 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))}","\frac{(-B+i A) \sqrt{\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) (A-(2+i) B) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}-\frac{\left(\frac{1}{8}+\frac{i}{8}\right) (A-(2+i) B) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) (A+(2-i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a d}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) (A+(2-i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}",1,"((Cos[d*x] + I*Sin[d*x])*(4*(A + I*B)*Cos[c + d*x]*(I*Cos[d*x] + Sin[d*x]) + (1 + I)*Csc[c + d*x]*((A + (2 - I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + I*(A - (2 + I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(I*Cos[c] - Sin[c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(8*d*Sqrt[Cot[c + d*x]]*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x]))","A",1
523,1,214,276,2.5450148,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])),x]","\frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left((-4 \cos (d x)+4 i \sin (d x)) (-4 B \sin (c+d x)+(A+5 i B) \cos (c+d x))-(\cos (c)+i \sin (c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left(((1-3 i) A+(3+5 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((2+i) A+(1+4 i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{8 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))}","\frac{-B+i A}{2 d \sqrt{\cot (c+d x)} (a \cot (c+d x)+i a)}-\frac{A+5 i B}{2 a d \sqrt{\cot (c+d x)}}-\frac{\left(\frac{1}{8}+\frac{i}{8}\right) ((2+i) A+(1+4 i) B) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) ((2+i) A+(1+4 i) B) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{((1-3 i) A+(3+5 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{4 \sqrt{2} a d}+\frac{\left(\frac{1}{4}+\frac{i}{4}\right) ((1+2 i) A-(4+i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}",1,"((Cos[d*x] + I*Sin[d*x])*((-4*Cos[d*x] + (4*I)*Sin[d*x])*((A + (5*I)*B)*Cos[c + d*x] - 4*B*Sin[c + d*x]) - Csc[c + d*x]*(((1 - 3*I)*A + (3 + 5*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] - (1 + I)*((2 + I)*A + (1 + 4*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(Cos[c] + I*Sin[c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(8*d*Sqrt[Cot[c + d*x]]*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x]))","A",1
524,1,242,307,3.0169898,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])),x]","\frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left(\frac{2}{3} \sec (c+d x) (\cos (d x)-i \sin (d x)) (4 (3 A+2 i B) \sin (2 (c+d x))+(11 B-15 i A) \cos (2 (c+d x))-15 i A+19 B)-(1+i) (\cos (c)+i \sin (c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left(((4+i) A+(1+6 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+((6+i) B-(1+4 i) A) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{8 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))}","\frac{-B+i A}{2 d \cot ^{\frac{3}{2}}(c+d x) (a \cot (c+d x)+i a)}-\frac{3 A+7 i B}{6 a d \cot ^{\frac{3}{2}}(c+d x)}-\frac{5 (-B+i A)}{2 a d \sqrt{\cot (c+d x)}}+\frac{((3-5 i) A+(5+7 i) B) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{8 \sqrt{2} a d}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) ((1+4 i) A-(6+i) B) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{\left(\frac{1}{4}+\frac{i}{4}\right) ((4+i) A+(1+6 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a d}-\frac{\left(\frac{1}{4}+\frac{i}{4}\right) ((4+i) A+(1+6 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}",1,"((Cos[d*x] + I*Sin[d*x])*((-1 - I)*Csc[c + d*x]*(((4 + I)*A + (1 + 6*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + ((-1 - 4*I)*A + (6 + I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(Cos[c] + I*Sin[c])*Sqrt[Sin[2*(c + d*x)]] + (2*Sec[c + d*x]*(Cos[d*x] - I*Sin[d*x])*((-15*I)*A + 19*B + ((-15*I)*A + 11*B)*Cos[2*(c + d*x)] + 4*(3*A + (2*I)*B)*Sin[2*(c + d*x)]))/3)*(A + B*Tan[c + d*x]))/(8*d*Sqrt[Cot[c + d*x]]*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x]))","A",1
525,1,256,317,2.90106,"\int \frac{\cot ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec (c+d x) (\cos (d x)+i \sin (d x))^2 (A+B \tan (c+d x)) \left(\cot (c+d x) (-2 \cos (2 d x)+2 i \sin (2 d x)) ((-7 B+43 i A) \sin (2 (c+d x))+(41 A+5 i B) \cos (2 (c+d x))-9 A-5 i B)+(-\sin (2 c)+i \cos (2 c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left(((21-25 i) A+(5+9 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((23+2 i) A+(2+7 i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{(7 A+3 i B) \cot ^{\frac{3}{2}}(c+d x)}{8 a^2 d (\cot (c+d x)+i)}-\frac{5 (5 A+i B) \sqrt{\cot (c+d x)}}{8 a^2 d}-\frac{\left(\frac{1}{32}-\frac{i}{32}\right) ((23+2 i) A+(2+7 i) B) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{\left(\frac{1}{32}-\frac{i}{32}\right) ((23+2 i) A+(2+7 i) B) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}-\frac{\left(\frac{1}{16}-\frac{i}{16}\right) ((2+23 i) A-(7+2 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a^2 d}+\frac{((25+21 i) A-(9-5 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{16 \sqrt{2} a^2 d}+\frac{(A+i B) \cot ^{\frac{5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}",1,"(Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*(Csc[c + d*x]*(((21 - 25*I)*A + (5 + 9*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] - (1 + I)*((23 + 2*I)*A + (2 + 7*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(I*Cos[2*c] - Sin[2*c])*Sqrt[Sin[2*(c + d*x)]] + Cot[c + d*x]*(-2*Cos[2*d*x] + (2*I)*Sin[2*d*x])*(-9*A - (5*I)*B + (41*A + (5*I)*B)*Cos[2*(c + d*x)] + ((43*I)*A - 7*B)*Sin[2*(c + d*x)]))*(A + B*Tan[c + d*x]))/(32*d*Sqrt[Cot[c + d*x]]*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2)","A",1
526,1,243,284,2.363979,"\int \frac{\sqrt{\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec (c+d x) (\cos (d x)+i \sin (d x))^2 (A+B \tan (c+d x)) \left(4 \cos (c+d x) (\cos (2 d x)-i \sin (2 d x)) ((-B+5 i A) \sin (c+d x)+(7 A+3 i B) \cos (c+d x))+(-\sin (2 c)+i \cos (2 c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left(((5+9 i) A+(3+i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((2+7 i) A+(1-2 i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{(5 A+i B) \sqrt{\cot (c+d x)}}{8 a^2 d (\cot (c+d x)+i)}+\frac{\left(\frac{1}{32}+\frac{i}{32}\right) ((2+i) B-(7-2 i) A) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{((9+5 i) A-(1+3 i) B) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{32 \sqrt{2} a^2 d}+\frac{((9-5 i) A+(1-3 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{16 \sqrt{2} a^2 d}+\frac{\left(\frac{1}{16}+\frac{i}{16}\right) ((1+2 i) B-(2-7 i) A) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{(A+i B) \cot ^{\frac{3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}",1,"(Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*(4*Cos[c + d*x]*(Cos[2*d*x] - I*Sin[2*d*x])*((7*A + (3*I)*B)*Cos[c + d*x] + ((5*I)*A - B)*Sin[c + d*x]) + Csc[c + d*x]*(((5 + 9*I)*A + (3 + I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] - (1 + I)*((2 + 7*I)*A + (1 - 2*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(I*Cos[2*c] - Sin[2*c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(32*d*Sqrt[Cot[c + d*x]]*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2)","A",1
527,1,243,274,2.2156291,"\int \frac{A+B \tan (c+d x)}{\sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^2} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^2),x]","\frac{\csc (c+d x) (\cos (d x)+i \sin (d x))^2 (A \cot (c+d x)+B) \left(4 \cos (c+d x) (\sin (2 d x)+i \cos (2 d x)) ((3 B+i A) \sin (c+d x)+(3 A-i B) \cos (c+d x))+(1-i) (-\sin (2 c)+i \cos (2 c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left(((1+2 i) A+(2+i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+((1+2 i) B-(2+i) A) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 a^2 d \sqrt{\cot (c+d x)} (\cot (c+d x)+i)^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{(B+3 i A) \sqrt{\cot (c+d x)}}{8 a^2 d (\cot (c+d x)+i)}+\frac{((1+3 i) A+(1-3 i) B) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{32 \sqrt{2} a^2 d}-\frac{((1+3 i) A+(1-3 i) B) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{32 \sqrt{2} a^2 d}-\frac{((1+3 i) B-(1-3 i) A) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{16 \sqrt{2} a^2 d}+\frac{((1+3 i) B-(1-3 i) A) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{16 \sqrt{2} a^2 d}+\frac{(A+i B) \sqrt{\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}",1,"((B + A*Cot[c + d*x])*Csc[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*(4*Cos[c + d*x]*(I*Cos[2*d*x] + Sin[2*d*x])*((3*A - I*B)*Cos[c + d*x] + (I*A + 3*B)*Sin[c + d*x]) + (1 - I)*Csc[c + d*x]*(((1 + 2*I)*A + (2 + I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + ((-2 - I)*A + (1 + 2*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(I*Cos[2*c] - Sin[2*c])*Sqrt[Sin[2*(c + d*x)]]))/(32*a^2*d*Sqrt[Cot[c + d*x]]*(I + Cot[c + d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
528,1,241,284,2.8167575,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2),x]","\frac{\sec (c+d x) (\cos (d x)+i \sin (d x))^2 (A+B \tan (c+d x)) \left(4 \cos (c+d x) (\cos (2 d x)-i \sin (2 d x)) ((-7 B+3 i A) \sin (c+d x)+(A+5 i B) \cos (c+d x))-(1+i) (-\sin (2 c)+i \cos (2 c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left(((2+7 i) B-(1-2 i) A) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+((7+2 i) B-(2-i) A) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{(A+5 i B) \sqrt{\cot (c+d x)}}{8 a^2 d (\cot (c+d x)+i)}+\frac{((1-3 i) A-(9-5 i) B) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{32 \sqrt{2} a^2 d}+\frac{\left(\frac{1}{32}+\frac{i}{32}\right) ((1+2 i) A+(2-7 i) B) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}-\frac{\left(\frac{1}{16}+\frac{i}{16}\right) ((2+i) A+(7-2 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a^2 d}+\frac{((1+3 i) A+(9+5 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{16 \sqrt{2} a^2 d}+\frac{(-B+i A) \sqrt{\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}",1,"(Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*(4*Cos[c + d*x]*(Cos[2*d*x] - I*Sin[2*d*x])*((A + (5*I)*B)*Cos[c + d*x] + ((3*I)*A - 7*B)*Sin[c + d*x]) - (1 + I)*Csc[c + d*x]*(((-1 + 2*I)*A + (2 + 7*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + ((-2 + I)*A + (7 + 2*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(I*Cos[2*c] - Sin[2*c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(32*d*Sqrt[Cot[c + d*x]]*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2)","A",1
529,1,249,319,3.1338324,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2),x]","\frac{\sec (c+d x) (\cos (d x)+i \sin (d x))^2 (A+B \tan (c+d x)) \left(2 (\sin (2 d x)+i \cos (2 d x)) ((-43 B+7 i A) \sin (2 (c+d x))+(5 A+41 i B) \cos (2 (c+d x))+5 A+9 i B)+(-\sin (2 c)+i \cos (2 c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left(((5-9 i) A+(21+25 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((7+2 i) A+(2+23 i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","\frac{5 (-5 B+i A)}{8 a^2 d \sqrt{\cot (c+d x)}}+\frac{3 A+7 i B}{8 a^2 d \sqrt{\cot (c+d x)} (\cot (c+d x)+i)}-\frac{\left(\frac{1}{32}-\frac{i}{32}\right) ((7+2 i) A+(2+23 i) B) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{\left(\frac{1}{32}-\frac{i}{32}\right) ((7+2 i) A+(2+23 i) B) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}-\frac{\left(\frac{1}{16}-\frac{i}{16}\right) ((2+7 i) A-(23+2 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a^2 d}+\frac{((9+5 i) A-(25-21 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{16 \sqrt{2} a^2 d}+\frac{-B+i A}{4 d \sqrt{\cot (c+d x)} (a \cot (c+d x)+i a)^2}",1,"(Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*(Csc[c + d*x]*(((5 - 9*I)*A + (21 + 25*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] - (1 + I)*((7 + 2*I)*A + (2 + 23*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(I*Cos[2*c] - Sin[2*c])*Sqrt[Sin[2*(c + d*x)]] + 2*(I*Cos[2*d*x] + Sin[2*d*x])*(5*A + (9*I)*B + (5*A + (41*I)*B)*Cos[2*(c + d*x)] + ((7*I)*A - 43*B)*Sin[2*(c + d*x)]))*(A + B*Tan[c + d*x]))/(32*d*Sqrt[Cot[c + d*x]]*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^2)","A",1
530,1,284,367,4.0667711,"\int \frac{\cot ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^3 (A+B \tan (c+d x)) \left(\frac{2}{3} \cot (c+d x) (\cos (3 d x)-i \sin (3 d x)) ((49 A+19 i B) \cos (c+d x)-(145 A+19 i B) \cos (3 (c+d x))+6 \sin (c+d x) (7 (B-7 i A) \cos (2 (c+d x))-19 i A+2 B))+(-\sin (3 c)+i \cos (3 c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left(((28-30 i) A+(5+7 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((29+i) A+(1+6 i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","\frac{7 (4 A+i B) \cot ^{\frac{3}{2}}(c+d x)}{24 d \left(a^3 \cot (c+d x)+i a^3\right)}-\frac{5 (6 A+i B) \sqrt{\cot (c+d x)}}{8 a^3 d}-\frac{\left(\frac{1}{32}-\frac{i}{32}\right) ((29+i) A+(1+6 i) B) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^3 d}+\frac{\left(\frac{1}{32}-\frac{i}{32}\right) ((29+i) A+(1+6 i) B) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^3 d}-\frac{\left(\frac{1}{16}-\frac{i}{16}\right) ((1+29 i) A-(6+i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a^3 d}+\frac{((30+28 i) A-(7-5 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{16 \sqrt{2} a^3 d}+\frac{(A+i B) \cot ^{\frac{7}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac{(5 A+2 i B) \cot ^{\frac{5}{2}}(c+d x)}{12 a d (a \cot (c+d x)+i a)^2}",1,"(Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^3*((2*Cot[c + d*x]*(Cos[3*d*x] - I*Sin[3*d*x])*((49*A + (19*I)*B)*Cos[c + d*x] - (145*A + (19*I)*B)*Cos[3*(c + d*x)] + 6*((-19*I)*A + 2*B + 7*((-7*I)*A + B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/3 + Csc[c + d*x]*(((28 - 30*I)*A + (5 + 7*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] - (1 + I)*((29 + I)*A + (1 + 6*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(I*Cos[3*c] - Sin[3*c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(32*d*Sqrt[Cot[c + d*x]]*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3)","A",1
531,1,258,318,2.8335579,"\int \frac{\sqrt{\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^3 (A+B \tan (c+d x)) \left(\frac{4}{3} \cos (c+d x) (\cos (3 d x)-i \sin (3 d x)) ((-B+19 i A) \sin (2 (c+d x))+3 (7 A+i B) \cos (2 (c+d x))+6 A+3 i B)+(-\sin (3 c)+i \cos (3 c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left((2 B+(5+7 i) A) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((1+6 i) A+(1-i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{((7+5 i) A-2 i B) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}+\frac{((7+5 i) A-2 i B) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}-\frac{(2 i B-(7-5 i) A) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{16 \sqrt{2} a^3 d}+\frac{(2 i B-(7-5 i) A) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{16 \sqrt{2} a^3 d}+\frac{5 A \sqrt{\cot (c+d x)}}{8 d \left(a^3 \cot (c+d x)+i a^3\right)}+\frac{(A+i B) \cot ^{\frac{5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac{(4 A+i B) \cot ^{\frac{3}{2}}(c+d x)}{12 a d (a \cot (c+d x)+i a)^2}",1,"(Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^3*(Csc[c + d*x]*(((5 + 7*I)*A + 2*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] - (1 + I)*((1 + 6*I)*A + (1 - I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(I*Cos[3*c] - Sin[3*c])*Sqrt[Sin[2*(c + d*x)]] + (4*Cos[c + d*x]*(Cos[3*d*x] - I*Sin[3*d*x])*(6*A + (3*I)*B + 3*(7*A + I*B)*Cos[2*(c + d*x)] + ((19*I)*A - B)*Sin[2*(c + d*x)]))/3)*(A + B*Tan[c + d*x]))/(32*d*Sqrt[Cot[c + d*x]]*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3)","A",1
532,1,272,316,4.2624695,"\int \frac{A+B \tan (c+d x)}{\sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^3} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^3),x]","\frac{e^{-4 i (c+d x)} \sqrt{\cot (c+d x)} \sec (c+d x) (\cos (3 (c+d x))-i \sin (3 (c+d x))) \left(\left(-2 e^{2 i (c+d x)}+e^{4 i (c+d x)}+2 e^{6 i (c+d x)}-1\right) \left(A e^{2 i (c+d x)}+A-2 i B e^{2 i (c+d x)}+i B\right)-6 (A-i B) e^{6 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)-3 A e^{6 i (c+d x)} \sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (c+d x)}}\right)\right)}{96 a^3 d}","\frac{(B+2 i A) \sqrt{\cot (c+d x)}}{8 d \left(a^3 \cot (c+d x)+i a^3\right)}+\frac{(2 i A+(1-i) B) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}-\frac{(2 i A+(1-i) B) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}-\frac{\left(\frac{1}{16}+\frac{i}{16}\right) (B+(1+i) A) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a^3 d}+\frac{\left(\frac{1}{16}+\frac{i}{16}\right) (B+(1+i) A) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^3 d}+\frac{(A+i B) \cot ^{\frac{3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac{A \sqrt{\cot (c+d x)}}{4 a d (a \cot (c+d x)+i a)^2}",1,"(((A + I*B + A*E^((2*I)*(c + d*x)) - (2*I)*B*E^((2*I)*(c + d*x)))*(-1 - 2*E^((2*I)*(c + d*x)) + E^((4*I)*(c + d*x)) + 2*E^((6*I)*(c + d*x))) - 3*A*E^((6*I)*(c + d*x))*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 6*(A - I*B)*E^((6*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Sqrt[Cot[c + d*x]]*Sec[c + d*x]*(Cos[3*(c + d*x)] - I*Sin[3*(c + d*x)]))/(96*a^3*d*E^((4*I)*(c + d*x)))","A",0
533,1,274,308,3.8085038,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^3),x]","\frac{e^{-4 i (c+d x)} \sqrt{\cot (c+d x)} \sec (c+d x) (\cos (3 (c+d x))-i \sin (3 (c+d x))) \left(\left(-2 e^{2 i (c+d x)}-e^{4 i (c+d x)}+2 e^{6 i (c+d x)}+1\right) \left(-i A \left(1+2 e^{2 i (c+d x)}\right)+B \left(-e^{2 i (c+d x)}\right)+B\right)+6 (B+i A) e^{6 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)-3 B e^{6 i (c+d x)} \sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (c+d x)}}\right)\right)}{96 a^3 d}","-\frac{(2 B-(1-i) A) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}+\frac{(2 B-(1-i) A) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}-\frac{(2 B+(1+i) A) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{16 \sqrt{2} a^3 d}+\frac{(2 B+(1+i) A) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{16 \sqrt{2} a^3 d}+\frac{A \sqrt{\cot (c+d x)}}{8 d \left(a^3 \cot (c+d x)+i a^3\right)}+\frac{(B+2 i A) \sqrt{\cot (c+d x)}}{12 a d (a \cot (c+d x)+i a)^2}+\frac{(A+i B) \sqrt{\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}",1,"(((1 - 2*E^((2*I)*(c + d*x)) - E^((4*I)*(c + d*x)) + 2*E^((6*I)*(c + d*x)))*(B - B*E^((2*I)*(c + d*x)) - I*A*(1 + 2*E^((2*I)*(c + d*x)))) - 3*B*E^((6*I)*(c + d*x))*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 6*(I*A + B)*E^((6*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Sqrt[Cot[c + d*x]]*Sec[c + d*x]*(Cos[3*(c + d*x)] - I*Sin[3*(c + d*x)]))/(96*a^3*d*E^((4*I)*(c + d*x)))","A",0
534,1,415,310,4.7750622,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^3),x]","\frac{\cot ^{\frac{3}{2}}(c+d x) \csc ^2(c+d x) \sec ^3(c+d x) (A \cos (c+d x)+B \sin (c+d x)) \left(-(A+19 i B) \cos (4 (c+d x))+(3+3 i) ((1+i) A+(6-i) B) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\sin (3 (c+d x))-i \cos (3 (c+d x)))+6 i A \sin (2 (c+d x))-3 i A \sin (4 (c+d x))+6 i A \sqrt{\sin (2 (c+d x))} \sin (3 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+6 A \sqrt{\sin (2 (c+d x))} \cos (3 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+A-12 B \sin (2 (c+d x))+21 B \sin (4 (c+d x))+(21-15 i) B \sqrt{\sin (2 (c+d x))} \sin (3 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)-(15+21 i) B \sqrt{\sin (2 (c+d x))} \cos (3 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+19 i B\right)}{96 a^3 d (\cot (c+d x)+i)^3 (A+B \tan (c+d x))}","-\frac{(2 A-(5+7 i) B) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}+\frac{(2 A-(5+7 i) B) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{32 \sqrt{2} a^3 d}-\frac{(2 A+(5-7 i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{16 \sqrt{2} a^3 d}+\frac{(2 A+(5-7 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{16 \sqrt{2} a^3 d}+\frac{5 B \sqrt{\cot (c+d x)}}{8 d \left(a^3 \cot (c+d x)+i a^3\right)}+\frac{(-B+i A) \sqrt{\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}+\frac{(A+4 i B) \sqrt{\cot (c+d x)}}{12 a d (a \cot (c+d x)+i a)^2}",1,"(Cot[c + d*x]^(3/2)*Csc[c + d*x]^2*Sec[c + d*x]^3*(A*Cos[c + d*x] + B*Sin[c + d*x])*(A + (19*I)*B - (A + (19*I)*B)*Cos[4*(c + d*x)] + 6*A*Cos[3*(c + d*x)]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] - (15 + 21*I)*B*Cos[3*(c + d*x)]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] + (6*I)*A*Sin[2*(c + d*x)] - 12*B*Sin[2*(c + d*x)] + (6*I)*A*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]]*Sin[3*(c + d*x)] + (21 - 15*I)*B*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]]*Sin[3*(c + d*x)] + (3 + 3*I)*((1 + I)*A + (6 - I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]]*((-I)*Cos[3*(c + d*x)] + Sin[3*(c + d*x)]) - (3*I)*A*Sin[4*(c + d*x)] + 21*B*Sin[4*(c + d*x)]))/(96*a^3*d*(I + Cot[c + d*x])^3*(A + B*Tan[c + d*x]))","A",1
535,1,280,367,4.8956529,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^3),x]","\frac{\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^3 (A+B \tan (c+d x)) \left(\frac{2}{3} (\cos (3 d x)-i \sin (3 d x)) ((9 A+33 i B) \cos (c+d x)+21 (A+7 i B) \cos (3 (c+d x))+2 i \sin (c+d x) ((19 A+145 i B) \cos (2 (c+d x))+19 A+97 i B))-i (\cos (3 c)+i \sin (3 c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left(((7+5 i) A-(30-28 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+(1-i) ((6+i) A+(1+29 i) B) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)\right)}{32 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{7 (-4 B+i A)}{24 d \sqrt{\cot (c+d x)} \left(a^3 \cot (c+d x)+i a^3\right)}+\frac{5 (A+6 i B)}{8 a^3 d \sqrt{\cot (c+d x)}}+\frac{\left(\frac{1}{32}+\frac{i}{32}\right) ((6+i) A+(1+29 i) B) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^3 d}-\frac{\left(\frac{1}{32}+\frac{i}{32}\right) ((6+i) A+(1+29 i) B) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^3 d}+\frac{\left(\frac{1}{16}+\frac{i}{16}\right) ((1+6 i) A-(29+i) B) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a^3 d}+\frac{((5-7 i) A+(28+30 i) B) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{16 \sqrt{2} a^3 d}+\frac{2 A+5 i B}{12 a d \sqrt{\cot (c+d x)} (a \cot (c+d x)+i a)^2}+\frac{-B+i A}{6 d \sqrt{\cot (c+d x)} (a \cot (c+d x)+i a)^3}",1,"(Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^3*((2*(Cos[3*d*x] - I*Sin[3*d*x])*((9*A + (33*I)*B)*Cos[c + d*x] + 21*(A + (7*I)*B)*Cos[3*(c + d*x)] + (2*I)*(19*A + (97*I)*B + (19*A + (145*I)*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/3 - I*Csc[c + d*x]*(((7 + 5*I)*A - (30 - 28*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + (1 - I)*((6 + I)*A + (1 + 29*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(Cos[3*c] + I*Sin[3*c])*Sqrt[Sin[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(32*d*Sqrt[Cot[c + d*x]]*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3)","A",1
536,1,188,198,3.4281944,"\int \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{e^{-i (c+d x)} \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)} \left(-15 (A-i B) \left(-1+e^{2 i (c+d x)}\right)^{5/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+2 A e^{i (c+d x)} \left(-20 e^{2 i (c+d x)}+17 e^{4 i (c+d x)}+15\right)-20 i B e^{3 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)\right)}{15 d \left(-1+e^{2 i (c+d x)}\right)^2}","-\frac{2 (5 B+i A) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}+\frac{2 (13 A-5 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{15 d}-\frac{(1+i) \sqrt{a} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 A \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}",1,"(((-20*I)*B*E^((3*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x))) + 2*A*E^(I*(c + d*x))*(15 - 20*E^((2*I)*(c + d*x)) + 17*E^((4*I)*(c + d*x))) - 15*(A - I*B)*(-1 + E^((2*I)*(c + d*x)))^(5/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(15*d*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^2)","A",1
537,1,162,155,2.5930991,"\int \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","-\frac{e^{-i (c+d x)} \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)} \left(-3 i (A-i B) \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+4 i A e^{3 i (c+d x)}+6 B e^{i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)\right)}{3 d \left(-1+e^{2 i (c+d x)}\right)}","-\frac{2 (3 B+i A) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{3 d}+\frac{(1+i) \sqrt{a} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{3 d}",1,"-1/3*(((4*I)*A*E^((3*I)*(c + d*x)) + 6*B*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x))) - (3*I)*(A - I*B)*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x))))","A",1
538,1,112,110,2.6409496,"\int \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{e^{-i (c+d x)} \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)} \left((A-i B) \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-2 A e^{i (c+d x)}\right)}{d}","\frac{(1+i) \sqrt{a} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 A \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{d}",1,"((-2*A*E^(I*(c + d*x)) + (A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
539,1,241,152,3.5185181,"\int \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}} \sqrt{a+i a \tan (c+d x)} \left((-4 B-4 i A) \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)+\sqrt{2} B \left(\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)\right)}{4 d}","\frac{(1-i) \sqrt{a} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 (-1)^{3/4} \sqrt{a} B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}",1,"(Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*(((-4*I)*A - 4*B)*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*B*(Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] - Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]]))*Sqrt[a + I*a*Tan[c + d*x]])/(4*d*E^(I*(c + d*x)))","A",1
540,0,0,192,8.2979656,"\int \frac{\sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","Integrate[(Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]","\int \frac{\sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","-\frac{(-1)^{3/4} \sqrt{a} (2 A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{(1+i) \sqrt{a} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{B \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}",1,"Integrate[(Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]], x]","F",-1
541,1,320,245,8.1476826,"\int \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left(-2 i \sqrt{2} (A-i B) e^{-2 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-\frac{1}{210} \sqrt{\cot (c+d x)} \csc ^3(c+d x) \sqrt{\sec (c+d x)} (\cos (c+d x)-i \sin (c+d x)) (7 (A+6 i B) \cos (c+d x)+(53 A-42 i B) \cos (3 (c+d x))+2 \sin (c+d x) ((147 B+158 i A) \cos (2 (c+d x))-110 i A-105 B))\right)}{d \sec ^{\frac{5}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","\frac{(2-2 i) a^{3/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a (7 B+8 i A) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}+\frac{4 a (19 A-21 i B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}+\frac{4 a (63 B+67 i A) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{2 a A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}",1,"((((-2*I)*Sqrt[2]*(A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/E^((2*I)*(c + d*x)) - (Sqrt[Cot[c + d*x]]*Csc[c + d*x]^3*Sqrt[Sec[c + d*x]]*(Cos[c + d*x] - I*Sin[c + d*x])*(7*(A + (6*I)*B)*Cos[c + d*x] + (53*A - (42*I)*B)*Cos[3*(c + d*x)] + 2*((-110*I)*A - 105*B + ((158*I)*A + 147*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/210)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/(d*Sec[c + d*x]^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
542,1,289,201,5.8872283,"\int \cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left(\frac{(-1+i \tan (c+d x)) \sqrt{\cot (c+d x)} \csc ^2(c+d x) ((5 B+6 i A) \sin (2 (c+d x))+(21 A-20 i B) \cos (2 (c+d x))-15 A+20 i B)}{15 \sqrt{\sec (c+d x)}}-2 \sqrt{2} (A-i B) e^{-2 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{d \sec ^{\frac{5}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{(2+2 i) a^{3/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a (5 B+6 i A) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}+\frac{4 a (9 A-10 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{15 d}-\frac{2 a A \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}",1,"(((-2*Sqrt[2]*(A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/E^((2*I)*(c + d*x)) + (Sqrt[Cot[c + d*x]]*Csc[c + d*x]^2*(-15*A + (20*I)*B + (21*A - (20*I)*B)*Cos[2*(c + d*x)] + ((6*I)*A + 5*B)*Sin[2*(c + d*x)])*(-1 + I*Tan[c + d*x]))/(15*Sqrt[Sec[c + d*x]]))*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/(d*Sec[c + d*x]^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
543,1,259,157,5.1817156,"\int \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left(2 \sqrt{2} (B+i A) e^{-2 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-\frac{2 (\cot (c+d x)-i) (A \csc (c+d x)+(3 B+4 i A) \sec (c+d x))}{3 \sqrt{\cot (c+d x)} \sec ^{\frac{3}{2}}(c+d x)}\right)}{d \sec ^{\frac{5}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","\frac{(2+2 i) a^{3/2} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a (3 B+4 i A) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{3 d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{3 d}",1,"(((2*Sqrt[2]*(I*A + B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/E^((2*I)*(c + d*x)) - (2*(-I + Cot[c + d*x])*(A*Csc[c + d*x] + ((4*I)*A + 3*B)*Sec[c + d*x]))/(3*Sqrt[Cot[c + d*x]]*Sec[c + d*x]^(3/2)))*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/(d*Sec[c + d*x]^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
544,1,286,186,5.0622428,"\int \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","-\frac{a \cos (c+d x) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)} \left(-4 \sqrt{2} (A-i B) \sqrt{-1+e^{2 i (c+d x)}} \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)+4 \sqrt{2} A e^{i (c+d x)}-i B \sqrt{-1+e^{2 i (c+d x)}} \log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)+i B \sqrt{-1+e^{2 i (c+d x)}} \log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)}{\sqrt{2} d \left(1+e^{2 i (c+d x)}\right)}","\frac{(2+2 i) a^{3/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{2 \sqrt[4]{-1} a^{3/2} B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a A \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{d}",1,"-((a*Cos[c + d*x]*Sqrt[Cot[c + d*x]]*(4*Sqrt[2]*A*E^(I*(c + d*x)) - 4*Sqrt[2]*(A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] - I*B*Sqrt[-1 + E^((2*I)*(c + d*x))]*Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] + I*B*Sqrt[-1 + E^((2*I)*(c + d*x))]*Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[2]*d*(1 + E^((2*I)*(c + d*x)))))","A",1
545,1,360,196,7.9668863,"\int \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left(\sqrt{2} e^{-2 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}} \left(\sqrt{2} (3 B+2 i A) \left(\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)-16 i (A-i B) \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)\right)+\frac{8 B (\tan (c+d x)+i)}{\sqrt{\cot (c+d x)} \sqrt{\sec (c+d x)}}\right)}{8 d \sec ^{\frac{5}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{(-1)^{3/4} a^{3/2} (3 B+2 i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{(2-2 i) a^{3/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{i a B \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}",1,"((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x])*((Sqrt[2]*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*((-16*I)*(A - I*B)*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*((2*I)*A + 3*B)*(Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] - Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]])))/E^((2*I)*(c + d*x)) + (8*B*(I + Tan[c + d*x]))/(Sqrt[Cot[c + d*x]]*Sqrt[Sec[c + d*x]])))/(8*d*Sec[c + d*x]^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
546,1,441,244,6.9412982,"\int \frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]","\frac{\cos ^2(c+d x) \sqrt{\cot (c+d x)} (\cos (d x)-i \sin (d x)) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left(4 (\sin (c)+i \cos (c)) \tan (c+d x) (4 A+2 B \tan (c+d x)-5 i B)-\sqrt{2} (\cos (2 c+d x)-i \sin (2 c+d x)) \sqrt{i \sin ^2(c+d x) (\cot (c+d x)+i)} \left(\sqrt{2} (12 A-11 i B) \log \left(\frac{2 e^{\frac{5 i c}{2}} \left(2 i \sqrt{-1+e^{2 i (c+d x)}}-i \sqrt{2} e^{i (c+d x)}+\sqrt{2}\right)}{(12 A-11 i B) \left(e^{i (c+d x)}-i\right)}\right)+\sqrt{2} (-12 A+11 i B) \log \left(\frac{2 e^{\frac{5 i c}{2}} \left(2 \sqrt{-1+e^{2 i (c+d x)}}+\sqrt{2} e^{i (c+d x)}-i \sqrt{2}\right)}{(11 B+12 i A) \left(e^{i (c+d x)}+i\right)}\right)+32 (A-i B) \log \left((\cos (c)-i \sin (c)) \left(i \sin (c+d x)+\cos (c+d x)+\sqrt{i \sin (2 (c+d x))+\cos (2 (c+d x))-1}\right)\right)\right)\right)}{16 d (A \cos (c+d x)+B \sin (c+d x))}","-\frac{(-1)^{3/4} a^{3/2} (12 A-11 i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{4 d}-\frac{(2+2 i) a^{3/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{a (5 B+4 i A) \sqrt{a+i a \tan (c+d x)}}{4 d \sqrt{\cot (c+d x)}}+\frac{i a B \sqrt{a+i a \tan (c+d x)}}{2 d \cot ^{\frac{3}{2}}(c+d x)}",1,"(Cos[c + d*x]^2*Sqrt[Cot[c + d*x]]*(Cos[d*x] - I*Sin[d*x])*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x])*(-(Sqrt[2]*(Sqrt[2]*(12*A - (11*I)*B)*Log[(2*E^(((5*I)/2)*c)*(Sqrt[2] - I*Sqrt[2]*E^(I*(c + d*x)) + (2*I)*Sqrt[-1 + E^((2*I)*(c + d*x))]))/((12*A - (11*I)*B)*(-I + E^(I*(c + d*x))))] + Sqrt[2]*(-12*A + (11*I)*B)*Log[(2*E^(((5*I)/2)*c)*((-I)*Sqrt[2] + Sqrt[2]*E^(I*(c + d*x)) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]))/(((12*I)*A + 11*B)*(I + E^(I*(c + d*x))))] + 32*(A - I*B)*Log[(Cos[c] - I*Sin[c])*(Cos[c + d*x] + I*Sin[c + d*x] + Sqrt[-1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]])])*Sqrt[I*(I + Cot[c + d*x])*Sin[c + d*x]^2]*(Cos[2*c + d*x] - I*Sin[2*c + d*x])) + 4*(I*Cos[c] + Sin[c])*Tan[c + d*x]*(4*A - (5*I)*B + 2*B*Tan[c + d*x])))/(16*d*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",0
547,1,354,297,12.0932593,"\int \cot ^{\frac{11}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(11/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(4 \sqrt{2} (A-i B) e^{-3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+\frac{(\cos (2 c)-i \sin (2 c)) \sqrt{\cot (c+d x)} \csc ^4(c+d x) \sqrt{\sec (c+d x)} (12 (251 A-260 i B) \cos (2 (c+d x))+(-961 A+915 i B) \cos (4 (c+d x))+282 i A \sin (2 (c+d x))-331 i A \sin (4 (c+d x))-2331 A+390 B \sin (2 (c+d x))-285 B \sin (4 (c+d x))+2205 i B)}{1260 (\cos (d x)+i \sin (d x))^2}\right)}{d \sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","\frac{(4+4 i) a^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a^2 (3 B+4 i A) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}+\frac{2 a^2 (46 A-45 i B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}+\frac{8 a^2 (60 B+59 i A) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{315 d}-\frac{8 a^2 (197 A-195 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{315 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}",1,"(((4*Sqrt[2]*(A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/E^((3*I)*(c + d*x)) + (Sqrt[Cot[c + d*x]]*Csc[c + d*x]^4*Sqrt[Sec[c + d*x]]*(Cos[2*c] - I*Sin[2*c])*(-2331*A + (2205*I)*B + 12*(251*A - (260*I)*B)*Cos[2*(c + d*x)] + (-961*A + (915*I)*B)*Cos[4*(c + d*x)] + (282*I)*A*Sin[2*(c + d*x)] + 390*B*Sin[2*(c + d*x)] - (331*I)*A*Sin[4*(c + d*x)] - 285*B*Sin[4*(c + d*x)]))/(1260*(Cos[d*x] + I*Sin[d*x])^2))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*Sec[c + d*x]^(7/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
548,1,332,251,10.1997983,"\int \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(-4 i \sqrt{2} (A-i B) e^{-3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-\frac{(\cos (2 c)-i \sin (2 c)) \sqrt{\cot (c+d x)} \csc ^3(c+d x) \sqrt{\sec (c+d x)} ((-35 A+77 i B) \cos (c+d x)+(95 A-77 i B) \cos (3 (c+d x))+2 \sin (c+d x) ((287 B+305 i A) \cos (2 (c+d x))-215 i A-245 B))}{210 (\cos (d x)+i \sin (d x))^2}\right)}{d \sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","\frac{(4-4 i) a^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a^2 (7 B+10 i A) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}+\frac{2 a^2 (80 A-77 i B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}+\frac{4 a^2 (133 B+130 i A) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{2 a A \cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}",1,"((((-4*I)*Sqrt[2]*(A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/E^((3*I)*(c + d*x)) - (Sqrt[Cot[c + d*x]]*Csc[c + d*x]^3*Sqrt[Sec[c + d*x]]*(Cos[2*c] - I*Sin[2*c])*((-35*A + (77*I)*B)*Cos[c + d*x] + (95*A - (77*I)*B)*Cos[3*(c + d*x)] + 2*((-215*I)*A - 245*B + ((305*I)*A + 287*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(210*(Cos[d*x] + I*Sin[d*x])^2))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*Sec[c + d*x]^(7/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
549,1,306,205,7.8977006,"\int \cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(-4 \sqrt{2} (A-i B) e^{-3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-\frac{(\cos (2 c)-i \sin (2 c)) \sqrt{\cot (c+d x)} \csc ^2(c+d x) \sqrt{\sec (c+d x)} ((5 B+11 i A) \sin (2 (c+d x))+(41 A-35 i B) \cos (2 (c+d x))-35 (A-i B))}{15 (\cos (d x)+i \sin (d x))^2}\right)}{d \sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","-\frac{(4+4 i) a^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a^2 (5 B+8 i A) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}+\frac{2 a^2 (38 A-35 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{15 d}-\frac{2 a A \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}",1,"(((-4*Sqrt[2]*(A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/E^((3*I)*(c + d*x)) - (Sqrt[Cot[c + d*x]]*Csc[c + d*x]^2*Sqrt[Sec[c + d*x]]*(Cos[2*c] - I*Sin[2*c])*(-35*(A - I*B) + (41*A - (35*I)*B)*Cos[2*(c + d*x)] + ((11*I)*A + 5*B)*Sin[2*(c + d*x)]))/(15*(Cos[d*x] + I*Sin[d*x])^2))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*Sec[c + d*x]^(7/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",1
550,1,496,230,10.7806026,"\int \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{\cos ^3(c+d x) \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left((8 A-3 i B) \left(-\frac{2}{3} \sin (2 c)-\frac{2}{3} i \cos (2 c)\right)+\csc (c+d x) \left(-\frac{2}{3} A \cos (3 c+d x)+\frac{2}{3} i A \sin (3 c+d x)\right)\right)}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\sqrt{e^{i d x}} e^{-i (3 c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}} (a+i a \tan (c+d x))^{5/2} \left(16 (B+i A) \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)+\sqrt{2} B \left(\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)\right) (A+B \tan (c+d x))}{2 \sqrt{2} d \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","\frac{(4+4 i) a^{5/2} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{2 (-1)^{3/4} a^{5/2} B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a^2 (B+2 i A) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}",1,"(Sqrt[E^(I*d*x)]*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*(16*(I*A + B)*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*B*(-Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] + Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]]))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(2*Sqrt[2]*d*E^(I*(3*c + d*x))*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (Cos[c + d*x]^3*Sqrt[Cot[c + d*x]]*((8*A - (3*I)*B)*(((-2*I)/3)*Cos[2*c] - (2*Sin[2*c])/3) + Csc[c + d*x]*((-2*A*Cos[3*c + d*x])/3 + ((2*I)/3)*A*Sin[3*c + d*x]))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","B",0
551,1,387,236,9.973183,"\int \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(\sqrt{2} e^{-3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}} \left(32 (A-i B) \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)-\sqrt{2} (2 A-5 i B) \left(\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)\right)-\frac{8 (\cos (2 c)-i \sin (2 c)) \sqrt{\sec (c+d x)} (2 A \cot (c+d x)+B)}{\sqrt{\cot (c+d x)} (\cos (d x)+i \sin (d x))^2}\right)}{8 d \sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))}","\frac{(-1)^{3/4} a^{5/2} (2 A-5 i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{(4+4 i) a^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{a^2 (-B+2 i A) \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}-\frac{2 a A \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{3/2}}{d}",1,"(((Sqrt[2]*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*(32*(A - I*B)*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] - Sqrt[2]*(2*A - (5*I)*B)*(Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] - Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]])))/E^((3*I)*(c + d*x)) - (8*(B + 2*A*Cot[c + d*x])*Sqrt[Sec[c + d*x]]*(Cos[2*c] - I*Sin[2*c]))/(Sqrt[Cot[c + d*x]]*(Cos[d*x] + I*Sin[d*x])^2))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(8*d*Sec[c + d*x]^(7/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",0
552,1,447,246,9.1581033,"\int \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{\cos ^3(c+d x) \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(\sqrt{2} (\cos (3 c+d x)-i \sin (3 c+d x)) \sqrt{i \sin ^2(c+d x) (\cot (c+d x)+i)} \left(\sqrt{2} (-23 B-20 i A) \log \left(-\frac{2 e^{\frac{7 i c}{2}} \left(-2 \sqrt{-1+e^{2 i (c+d x)}}+\sqrt{2} e^{i (c+d x)}+i \sqrt{2}\right)}{(20 A-23 i B) \left(e^{i (c+d x)}-i\right)}\right)+\sqrt{2} (23 B+20 i A) \log \left(-\frac{2 e^{\frac{7 i c}{2}} \left(2 \sqrt{-1+e^{2 i (c+d x)}}+\sqrt{2} e^{i (c+d x)}-i \sqrt{2}\right)}{(20 A-23 i B) \left(e^{i (c+d x)}+i\right)}\right)-64 i (A-i B) \log \left((\cos (c)-i \sin (c)) \left(i \sin (c+d x)+\cos (c+d x)+\sqrt{i \sin (2 (c+d x))+\cos (2 (c+d x))-1}\right)\right)\right)-4 (\cos (2 c)-i \sin (2 c)) \tan (c+d x) (4 A+2 B \tan (c+d x)-9 i B)\right)}{16 d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{(-1)^{3/4} a^{5/2} (23 B+20 i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{4 d}+\frac{(4-4 i) a^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{a^2 (4 A-7 i B) \sqrt{a+i a \tan (c+d x)}}{4 d \sqrt{\cot (c+d x)}}+\frac{i a B (a+i a \tan (c+d x))^{3/2}}{2 d \sqrt{\cot (c+d x)}}",1,"(Cos[c + d*x]^3*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x])*(Sqrt[2]*(Sqrt[2]*((-20*I)*A - 23*B)*Log[(-2*E^(((7*I)/2)*c)*(I*Sqrt[2] + Sqrt[2]*E^(I*(c + d*x)) - 2*Sqrt[-1 + E^((2*I)*(c + d*x))]))/((20*A - (23*I)*B)*(-I + E^(I*(c + d*x))))] + Sqrt[2]*((20*I)*A + 23*B)*Log[(-2*E^(((7*I)/2)*c)*((-I)*Sqrt[2] + Sqrt[2]*E^(I*(c + d*x)) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]))/((20*A - (23*I)*B)*(I + E^(I*(c + d*x))))] - (64*I)*(A - I*B)*Log[(Cos[c] - I*Sin[c])*(Cos[c + d*x] + I*Sin[c + d*x] + Sqrt[-1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]])])*Sqrt[I*(I + Cot[c + d*x])*Sin[c + d*x]^2]*(Cos[3*c + d*x] - I*Sin[3*c + d*x]) - 4*(Cos[2*c] - I*Sin[2*c])*Tan[c + d*x]*(4*A - (9*I)*B + 2*B*Tan[c + d*x])))/(16*d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",0
553,1,484,292,10.2114649,"\int \frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","Integrate[((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]","\frac{\cos ^3(c+d x) \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left(\frac{2}{3} (\cos (2 c)-i \sin (2 c)) \tan (c+d x) \sec ^2(c+d x) ((-12 A+26 i B) \sin (2 (c+d x))+(65 B+54 i A) \cos (2 (c+d x))+54 i A+49 B)-\sqrt{2} (\cos (3 c+d x)-i \sin (3 c+d x)) \sqrt{i \sin ^2(c+d x) (\cot (c+d x)+i)} \left(\sqrt{2} (46 A-45 i B) \log \left(\frac{2 e^{\frac{7 i c}{2}} \left(2 i \sqrt{-1+e^{2 i (c+d x)}}-i \sqrt{2} e^{i (c+d x)}+\sqrt{2}\right)}{(46 A-45 i B) \left(e^{i (c+d x)}-i\right)}\right)+\sqrt{2} (-46 A+45 i B) \log \left(\frac{2 e^{\frac{7 i c}{2}} \left(2 \sqrt{-1+e^{2 i (c+d x)}}+\sqrt{2} e^{i (c+d x)}-i \sqrt{2}\right)}{(45 B+46 i A) \left(e^{i (c+d x)}+i\right)}\right)+128 (A-i B) \log \left((\cos (c)-i \sin (c)) \left(i \sin (c+d x)+\cos (c+d x)+\sqrt{i \sin (2 (c+d x))+\cos (2 (c+d x))-1}\right)\right)\right)\right)}{32 d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}","-\frac{(-1)^{3/4} a^{5/2} (46 A-45 i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{8 d}-\frac{(4+4 i) a^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{a^2 (2 A-3 i B) \sqrt{a+i a \tan (c+d x)}}{4 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{a^2 (19 B+18 i A) \sqrt{a+i a \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{i a B (a+i a \tan (c+d x))^{3/2}}{3 d \cot ^{\frac{3}{2}}(c+d x)}",1,"(Cos[c + d*x]^3*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x])*(-(Sqrt[2]*(Sqrt[2]*(46*A - (45*I)*B)*Log[(2*E^(((7*I)/2)*c)*(Sqrt[2] - I*Sqrt[2]*E^(I*(c + d*x)) + (2*I)*Sqrt[-1 + E^((2*I)*(c + d*x))]))/((46*A - (45*I)*B)*(-I + E^(I*(c + d*x))))] + Sqrt[2]*(-46*A + (45*I)*B)*Log[(2*E^(((7*I)/2)*c)*((-I)*Sqrt[2] + Sqrt[2]*E^(I*(c + d*x)) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]))/(((46*I)*A + 45*B)*(I + E^(I*(c + d*x))))] + 128*(A - I*B)*Log[(Cos[c] - I*Sin[c])*(Cos[c + d*x] + I*Sin[c + d*x] + Sqrt[-1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]])])*Sqrt[I*(I + Cot[c + d*x])*Sin[c + d*x]^2]*(Cos[3*c + d*x] - I*Sin[3*c + d*x])) + (2*Sec[c + d*x]^2*(Cos[2*c] - I*Sin[2*c])*((54*I)*A + 49*B + ((54*I)*A + 65*B)*Cos[2*(c + d*x)] + (-12*A + (26*I)*B)*Sin[2*(c + d*x)])*Tan[c + d*x])/3))/(32*d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))","A",0
554,1,166,211,4.4319776,"\int \frac{\cot ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(Cot[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{\sqrt{\cot (c+d x)} \csc (c+d x) \sec (c+d x) \left((5 A+9 i B) \cos (2 (c+d x))+\frac{3}{2} (A-i B) e^{-i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+2 i A \sin (2 (c+d x))-9 A-6 B \sin (2 (c+d x))-9 i B\right)}{6 d \sqrt{a+i a \tan (c+d x)}}","-\frac{(5 A+3 i B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{3 a d}+\frac{(A+i B) \cot ^{\frac{3}{2}}(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}+\frac{(-9 B+7 i A) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{3 a d}+\frac{\left(\frac{1}{2}+\frac{i}{2}\right) (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"(Sqrt[Cot[c + d*x]]*Csc[c + d*x]*Sec[c + d*x]*(-9*A - (9*I)*B + (3*(A - I*B)*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/(2*E^(I*(c + d*x))) + (5*A + (9*I)*B)*Cos[2*(c + d*x)] + (2*I)*A*Sin[2*(c + d*x)] - 6*B*Sin[2*(c + d*x)]))/(6*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
555,1,165,163,3.5156253,"\int \frac{\cot ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{e^{-2 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left((A-i B) e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-5 A e^{2 i (c+d x)}+A-i B \left(-1+e^{2 i (c+d x)}\right)\right)}{\sqrt{2} a d}","\frac{(A+i B) \sqrt{\cot (c+d x)}}{d \sqrt{a+i a \tan (c+d x)}}-\frac{(3 A+i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{a d}+\frac{\left(\frac{1}{2}+\frac{i}{2}\right) (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"(Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(A - 5*A*E^((2*I)*(c + d*x)) - I*B*(-1 + E^((2*I)*(c + d*x))) + (A - I*B)*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*a*d*E^((2*I)*(c + d*x)))","A",1
556,1,156,119,2.6352313,"\int \frac{\sqrt{\cot (c+d x)} (A+B \tan (c+d x))}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{e^{-2 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left((B-i A) \left(-1+e^{2 i (c+d x)}\right)-i (A-i B) e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{\sqrt{2} a d}","\frac{A+i B}{d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"(Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(((-I)*A + B)*(-1 + E^((2*I)*(c + d*x))) - I*(A - I*B)*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*a*d*E^((2*I)*(c + d*x)))","A",1
557,1,227,196,4.5189444,"\int \frac{A+B \tan (c+d x)}{\sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{e^{-2 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left((A+i B) \left(-1+e^{2 i (c+d x)}\right)-(A-i B) e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-2 i \sqrt{2} B e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{\sqrt{2} a d}","\frac{-B+i A}{d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{\left(\frac{1}{2}+\frac{i}{2}\right) (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}-\frac{2 \sqrt[4]{-1} B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"(Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*((A + I*B)*(-1 + E^((2*I)*(c + d*x))) - (A - I*B)*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] - (2*I)*Sqrt[2]*B*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*a*d*E^((2*I)*(c + d*x)))","A",1
558,1,195,214,5.1044023,"\int \frac{\cot ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{\cot ^{\frac{3}{2}}(c+d x) \left(-3 (A-i B) e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+A \left(-13 e^{2 i (c+d x)}+38 e^{4 i (c+d x)}-1\right)+i B \left(-7 e^{2 i (c+d x)}+8 e^{4 i (c+d x)}-1\right)\right)}{3 a d \left(1+e^{2 i (c+d x)}\right)^2 (\cot (c+d x)+i) \sqrt{a+i a \tan (c+d x)}}","\frac{\left(\frac{1}{4}+\frac{i}{4}\right) (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}-\frac{(25 A+7 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{6 a^2 d}+\frac{(A+i B) \sqrt{\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(11 A+5 i B) \sqrt{\cot (c+d x)}}{6 a d \sqrt{a+i a \tan (c+d x)}}",1,"-1/3*((I*B*(-1 - 7*E^((2*I)*(c + d*x)) + 8*E^((4*I)*(c + d*x))) + A*(-1 - 13*E^((2*I)*(c + d*x)) + 38*E^((4*I)*(c + d*x))) - 3*(A - I*B)*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Cot[c + d*x]^(3/2))/(a*d*(1 + E^((2*I)*(c + d*x)))^2*(I + Cot[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
559,1,192,168,4.1787502,"\int \frac{\sqrt{\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{e^{-2 i (c+d x)} \sqrt{\cot (c+d x)} \csc (c+d x) \sec (c+d x) \left(\left(-1+e^{2 i (c+d x)}\right) \left(-i A \left(1+8 e^{2 i (c+d x)}\right)+2 B e^{2 i (c+d x)}+B\right)-3 i (A-i B) e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{12 a d (\cot (c+d x)+i) \sqrt{a+i a \tan (c+d x)}}","\frac{\left(\frac{1}{4}-\frac{i}{4}\right) (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{A+i B}{3 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac{7 A+i B}{6 a d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}",1,"(((-1 + E^((2*I)*(c + d*x)))*(B + 2*B*E^((2*I)*(c + d*x)) - I*A*(1 + 8*E^((2*I)*(c + d*x)))) - (3*I)*(A - I*B)*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]]*Csc[c + d*x]*Sec[c + d*x])/(12*a*d*E^((2*I)*(c + d*x))*(I + Cot[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
560,1,190,170,4.5984639,"\int \frac{A+B \tan (c+d x)}{\sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)),x]","\frac{e^{-2 i (c+d x)} \sqrt{\cot (c+d x)} \csc (c+d x) \sec (c+d x) \left(\left(-1+e^{2 i (c+d x)}\right) \left(2 A e^{2 i (c+d x)}+A-i B \left(-1+4 e^{2 i (c+d x)}\right)\right)-3 (A-i B) e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{12 a d (\cot (c+d x)+i) \sqrt{a+i a \tan (c+d x)}}","-\frac{\left(\frac{1}{4}+\frac{i}{4}\right) (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{-B+i A}{3 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac{5 B+i A}{6 a d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}",1,"(((-1 + E^((2*I)*(c + d*x)))*(A + 2*A*E^((2*I)*(c + d*x)) - I*B*(-1 + 4*E^((2*I)*(c + d*x)))) - 3*(A - I*B)*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]]*Csc[c + d*x]*Sec[c + d*x])/(12*a*d*E^((2*I)*(c + d*x))*(I + Cot[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
561,1,388,243,8.0100188,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)),x]","\frac{e^{-2 i (c+d x)} \sqrt{\cot (c+d x)} \sec (c+d x) \left(3 (B+i A) e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)+5 i A e^{2 i (c+d x)}-4 i A e^{4 i (c+d x)}-i A-11 B e^{2 i (c+d x)}+10 B e^{4 i (c+d x)}-3 \sqrt{2} B e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)+3 \sqrt{2} B e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)+B\right) (A+B \tan (c+d x))}{12 d (a+i a \tan (c+d x))^{3/2} (A \cos (c+d x)+B \sin (c+d x))}","\frac{\left(\frac{1}{4}+\frac{i}{4}\right) (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{2 (-1)^{3/4} B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{-B+i A}{3 d \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{A+3 i B}{2 a d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}",1,"(Sqrt[Cot[c + d*x]]*((-I)*A + B + (5*I)*A*E^((2*I)*(c + d*x)) - 11*B*E^((2*I)*(c + d*x)) - (4*I)*A*E^((4*I)*(c + d*x)) + 10*B*E^((4*I)*(c + d*x)) + 3*(I*A + B)*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] - 3*Sqrt[2]*B*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] + 3*Sqrt[2]*B*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]*(A + B*Tan[c + d*x]))/(12*d*E^((2*I)*(c + d*x))*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^(3/2))","A",1
562,1,200,260,9.2012926,"\int \frac{\cot ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{\cot ^{\frac{3}{2}}(c+d x) \sec (c+d x) \left(-20 \csc (c+d x) ((23 A+4 i B) \cos (2 (c+d x))-17 A-4 i B)+\sec (c+d x) ((86 B-466 i A) \cos (2 (c+d x))-149 i A+19 B)+15 (A-i B) e^{2 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \csc (2 (c+d x)) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{60 a^2 d (\cot (c+d x)+i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{(317 A+67 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}+\frac{(151 A+41 i B) \sqrt{\cot (c+d x)}}{60 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{(A+i B) \sqrt{\cot (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{(17 A+7 i B) \sqrt{\cot (c+d x)}}{30 a d (a+i a \tan (c+d x))^{3/2}}",1,"(Cot[c + d*x]^(3/2)*Sec[c + d*x]*(-20*(-17*A - (4*I)*B + (23*A + (4*I)*B)*Cos[2*(c + d*x)])*Csc[c + d*x] + 15*(A - I*B)*E^((2*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]*Csc[2*(c + d*x)] + ((-149*I)*A + 19*B + ((-466*I)*A + 86*B)*Cos[2*(c + d*x)])*Sec[c + d*x]))/(60*a^2*d*(I + Cot[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
563,1,167,214,6.9642327,"\int \frac{\sqrt{\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{\cot ^{\frac{3}{2}}(c+d x) \sec ^2(c+d x) \left(\frac{30 (A-i B) e^{3 i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)}{\sqrt{-1+e^{2 i (c+d x)}}}+2 ((86 A+6 i B) \cos (2 (c+d x))+80 i A \sin (2 (c+d x))+19 A+9 i B)\right)}{120 a^2 d (\cot (c+d x)+i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{\left(\frac{1}{8}-\frac{i}{8}\right) (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}+\frac{67 A-3 i B}{60 a^2 d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{A+i B}{5 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}}+\frac{13 A+3 i B}{30 a d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{3/2}}",1,"(Cot[c + d*x]^(3/2)*Sec[c + d*x]^2*((30*(A - I*B)*E^((3*I)*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/Sqrt[-1 + E^((2*I)*(c + d*x))] + 2*(19*A + (9*I)*B + (86*A + (6*I)*B)*Cos[2*(c + d*x)] + (80*I)*A*Sin[2*(c + d*x)])))/(120*a^2*d*(I + Cot[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
564,1,168,216,7.5645852,"\int \frac{A+B \tan (c+d x)}{\sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)),x]","\frac{\cot ^{\frac{3}{2}}(c+d x) \sec ^2(c+d x) \left(2 (2 (7 B+3 i A) \cos (2 (c+d x))+9 i A+20 i B \sin (2 (c+d x))+B)-\frac{30 i (A-i B) e^{3 i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)}{\sqrt{-1+e^{2 i (c+d x)}}}\right)}{120 a^2 d (\cot (c+d x)+i)^2 \sqrt{a+i a \tan (c+d x)}}","-\frac{\left(\frac{1}{8}+\frac{i}{8}\right) (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{-13 B+3 i A}{60 a^2 d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{7 B+3 i A}{30 a d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac{-B+i A}{5 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}}",1,"(Cot[c + d*x]^(3/2)*Sec[c + d*x]^2*(((-30*I)*(A - I*B)*E^((3*I)*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/Sqrt[-1 + E^((2*I)*(c + d*x))] + 2*((9*I)*A + B + 2*((3*I)*A + 7*B)*Cos[2*(c + d*x)] + (20*I)*B*Sin[2*(c + d*x)])))/(120*a^2*d*(I + Cot[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
565,1,169,214,7.9559817,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)),x]","\frac{\cot ^{\frac{3}{2}}(c+d x) \sec ^2(c+d x) \left(40 (B+i A) \sin (2 (c+d x))+4 (7 A-13 i B) \cos (2 (c+d x))-\frac{30 (A-i B) e^{3 i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)}{\sqrt{-1+e^{2 i (c+d x)}}}+2 A+22 i B\right)}{120 a^2 d (\cot (c+d x)+i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}+\frac{13 A-37 i B}{60 a^2 d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{-B+i A}{5 d \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}+\frac{A+11 i B}{30 a d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{3/2}}",1,"(Cot[c + d*x]^(3/2)*Sec[c + d*x]^2*(2*A + (22*I)*B - (30*(A - I*B)*E^((3*I)*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/Sqrt[-1 + E^((2*I)*(c + d*x))] + 4*(7*A - (13*I)*B)*Cos[2*(c + d*x)] + 40*(I*A + B)*Sin[2*(c + d*x)]))/(120*a^2*d*(I + Cot[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
566,1,426,289,10.4122239,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2)),x]","\frac{e^{-3 i (c+d x)} \sqrt{\cot (c+d x)} \sec ^2(c+d x) \left(15 (A-i B) e^{5 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)-14 A e^{2 i (c+d x)}+34 A e^{4 i (c+d x)}-23 A e^{6 i (c+d x)}+3 A-24 i B e^{2 i (c+d x)}+144 i B e^{4 i (c+d x)}-123 i B e^{6 i (c+d x)}+30 i \sqrt{2} B e^{5 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-30 i \sqrt{2} B e^{5 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)+3 i B\right) (A+B \tan (c+d x))}{120 d (a+i a \tan (c+d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}+\frac{2 \sqrt[4]{-1} B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{-7 B+i A}{4 a^2 d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{A+3 i B}{6 a d \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{-B+i A}{5 d \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}",1,"(Sqrt[Cot[c + d*x]]*(3*A + (3*I)*B - 14*A*E^((2*I)*(c + d*x)) - (24*I)*B*E^((2*I)*(c + d*x)) + 34*A*E^((4*I)*(c + d*x)) + (144*I)*B*E^((4*I)*(c + d*x)) - 23*A*E^((6*I)*(c + d*x)) - (123*I)*B*E^((6*I)*(c + d*x)) + 15*(A - I*B)*E^((5*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] + (30*I)*Sqrt[2]*B*E^((5*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] - (30*I)*Sqrt[2]*B*E^((5*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^2*(A + B*Tan[c + d*x]))/(120*d*E^((3*I)*(c + d*x))*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^(5/2))","A",1
567,0,0,179,19.6866775,"\int \cot ^m(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^m*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \cot ^m(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","\frac{(A-i B) \cot ^{m-1}(c+d x) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1(1-m;1-n,1;2-m;-i \tan (c+d x),i \tan (c+d x))}{d (1-m)}+\frac{i B \cot ^{m-1}(c+d x) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1(1-m,1-n;2-m;-i \tan (c+d x))}{d (1-m)}",1,"Integrate[Cot[c + d*x]^m*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
568,0,0,247,11.3126914,"\int \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","-\frac{2 (A-i B) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d \sqrt{\cot (c+d x)}}-\frac{2 (1-2 n) (-2 A n+3 i B) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};-i \tan (c+d x)\right)}{3 d \sqrt{\cot (c+d x)}}-\frac{2 (3 B+2 i A n) \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^n}{3 d}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^n}{3 d}",1,"Integrate[Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
569,0,0,194,21.1637323,"\int \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","\frac{2 (B+i A) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d \sqrt{\cot (c+d x)}}-\frac{2 i A (1-2 n) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};-i \tan (c+d x)\right)}{d \sqrt{\cot (c+d x)}}-\frac{2 A \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^n}{d}",1,"Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
570,0,0,158,19.3961652,"\int \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","\frac{2 (A-i B) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d \sqrt{\cot (c+d x)}}+\frac{2 i B (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};-i \tan (c+d x)\right)}{d \sqrt{\cot (c+d x)}}",1,"Integrate[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
571,0,0,215,13.659078,"\int \frac{(a+i a \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","Integrate[((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]","\int \frac{(a+i a \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","-\frac{2 (B+i A) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d \sqrt{\cot (c+d x)}}+\frac{2 (2 B n+i A (2 n+1)) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};-i \tan (c+d x)\right)}{d (2 n+1) \sqrt{\cot (c+d x)}}+\frac{2 B (a+i a \tan (c+d x))^n}{d (2 n+1) \sqrt{\cot (c+d x)}}",1,"Integrate[((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]], x]","F",-1
572,0,0,291,18.2092507,"\int \frac{(a+i a \tan (c+d x))^n (A+B \tan (c+d x))}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(3/2),x]","\int \frac{(a+i a \tan (c+d x))^n (A+B \tan (c+d x))}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","-\frac{2 (A-i B) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d \sqrt{\cot (c+d x)}}+\frac{2 \left(2 A n (2 n+3)-i B \left(4 n^2+6 n+3\right)\right) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};-i \tan (c+d x)\right)}{d (2 n+1) (2 n+3) \sqrt{\cot (c+d x)}}-\frac{2 (-A (2 n+3)+2 i B n) (a+i a \tan (c+d x))^n}{d (2 n+1) (2 n+3) \sqrt{\cot (c+d x)}}+\frac{2 B (a+i a \tan (c+d x))^n}{d (2 n+3) \cot ^{\frac{3}{2}}(c+d x)}",1,"Integrate[((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(3/2), x]","F",-1
573,0,0,383,22.9503324,"\int \frac{(a+i a \tan (c+d x))^n (A+B \tan (c+d x))}{\cot ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(5/2),x]","\int \frac{(a+i a \tan (c+d x))^n (A+B \tan (c+d x))}{\cot ^{\frac{5}{2}}(c+d x)} \, dx","\frac{2 (B+i A) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d \sqrt{\cot (c+d x)}}-\frac{2 \left(4 B n \left(2 n^2+8 n+9\right)+i A \left(8 n^3+32 n^2+36 n+15\right)\right) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};-i \tan (c+d x)\right)}{d (2 n+1) (2 n+3) (2 n+5) \sqrt{\cot (c+d x)}}-\frac{2 \left(B \left(4 n^2+10 n+15\right)+2 i A n (2 n+5)\right) (a+i a \tan (c+d x))^n}{d (2 n+1) (2 n+3) (2 n+5) \sqrt{\cot (c+d x)}}-\frac{2 (-A (2 n+5)+2 i B n) (a+i a \tan (c+d x))^n}{d (2 n+3) (2 n+5) \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 B (a+i a \tan (c+d x))^n}{d (2 n+5) \cot ^{\frac{5}{2}}(c+d x)}",1,"Integrate[((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(5/2), x]","F",-1
574,1,198,229,0.9529712,"\int \cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(6 \sqrt{2} (a (A+B)+b (A-B)) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-\frac{24 (a B+A b)}{\sqrt{\tan (c+d x)}}+3 \sqrt{2} (a (A-B)-b (A+B)) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-\frac{8 a A}{\tan ^{\frac{3}{2}}(c+d x)}\right)}{12 d}","-\frac{2 (a B+A b) \sqrt{\cot (c+d x)}}{d}+\frac{(a (A-B)-b (A+B)) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a (A-B)-b (A+B)) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a (A+B)+b (A-B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{(a (A+B)+b (A-B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(Sqrt[Cot[c + d*x]]*(6*Sqrt[2]*(b*(A - B) + a*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + 3*Sqrt[2]*(a*(A - B) - b*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) - (8*a*A)/Tan[c + d*x]^(3/2) - (24*(A*b + a*B))/Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/(12*d)","A",1
575,1,179,205,0.4873721,"\int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(2 \sqrt{2} (a (A-B)-b (A+B)) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-\sqrt{2} (a (A+B)+b (A-B)) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-\frac{8 a A}{\sqrt{\tan (c+d x)}}\right)}{4 d}","-\frac{(a (A+B)+b (A-B)) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a (A+B)+b (A-B)) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a (A-B)-b (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{(a (A-B)-b (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a A \sqrt{\cot (c+d x)}}{d}",1,"(Sqrt[Cot[c + d*x]]*(2*Sqrt[2]*(a*(A - B) - b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) - Sqrt[2]*(b*(A - B) + a*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) - (8*a*A)/Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/(4*d)","A",1
576,1,178,205,0.2168886,"\int \sqrt{\cot (c+d x)} (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]","-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(2 \sqrt{2} (a (A+B)+b (A-B)) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\sqrt{2} (a (A-B)-b (A+B)) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-8 b B \sqrt{\tan (c+d x)}\right)}{4 d}","-\frac{(a (A-B)-b (A+B)) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a (A-B)-b (A+B)) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a (A+B)+b (A-B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{(a (A+B)+b (A-B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b B}{d \sqrt{\cot (c+d x)}}",1,"-1/4*(Sqrt[Cot[c + d*x]]*(2*Sqrt[2]*(b*(A - B) + a*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + Sqrt[2]*(a*(A - B) - b*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) - 8*b*B*Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/d","A",1
577,1,198,229,0.5176881,"\int \frac{(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","Integrate[((a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]","-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(6 \sqrt{2} (a (A-B)-b (A+B)) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-24 (a B+A b) \sqrt{\tan (c+d x)}-3 \sqrt{2} (a (A+B)+b (A-B)) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-8 b B \tan ^{\frac{3}{2}}(c+d x)\right)}{12 d}","\frac{2 (a B+A b)}{d \sqrt{\cot (c+d x)}}+\frac{(a (A+B)+b (A-B)) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a (A+B)+b (A-B)) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a (A-B)-b (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{(a (A-B)-b (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b B}{3 d \cot ^{\frac{3}{2}}(c+d x)}",1,"-1/12*(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(6*Sqrt[2]*(a*(A - B) - b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) - 3*Sqrt[2]*(b*(A - B) + a*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) - 24*(A*b + a*B)*Sqrt[Tan[c + d*x]] - 8*b*B*Tan[c + d*x]^(3/2)))/d","A",1
578,1,255,326,1.8592573,"\int \cot ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(30 \sqrt{2} \left(a^2 (A-B)-2 a b (A+B)+b^2 (B-A)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-\frac{120 \left(a^2 A-2 a b B-A b^2\right)}{\sqrt{\tan (c+d x)}}-15 \sqrt{2} \left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\frac{24 a^2 A}{\tan ^{\frac{5}{2}}(c+d x)}+\frac{40 a (a B+2 A b)}{\tan ^{\frac{3}{2}}(c+d x)}\right)}{60 d}","\frac{2 \left(a^2 A-2 a b B-A b^2\right) \sqrt{\cot (c+d x)}}{d}+\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a (5 a B+7 A b) \cot ^{\frac{3}{2}}(c+d x)}{15 d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}",1,"-1/60*(Sqrt[Cot[c + d*x]]*(30*Sqrt[2]*(a^2*(A - B) + b^2*(-A + B) - 2*a*b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) - 15*Sqrt[2]*(2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) + (24*a^2*A)/Tan[c + d*x]^(5/2) + (40*a*(2*A*b + a*B))/Tan[c + d*x]^(3/2) - (120*(a^2*A - A*b^2 - 2*a*b*B))/Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/d","A",1
579,1,226,294,1.2124141,"\int \cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(6 \sqrt{2} \left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+3 \sqrt{2} \left(a^2 (A-B)-2 a b (A+B)+b^2 (B-A)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-\frac{8 a^2 A}{\tan ^{\frac{3}{2}}(c+d x)}-\frac{24 a (a B+2 A b)}{\sqrt{\tan (c+d x)}}\right)}{12 d}","\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a (3 a B+5 A b) \sqrt{\cot (c+d x)}}{3 d}-\frac{2 a A \sqrt{\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}",1,"(Sqrt[Cot[c + d*x]]*(6*Sqrt[2]*(2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + 3*Sqrt[2]*(a^2*(A - B) + b^2*(-A + B) - 2*a*b*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) - (8*a^2*A)/Tan[c + d*x]^(3/2) - (24*a*(2*A*b + a*B))/Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/(12*d)","A",1
580,1,221,276,0.8435153,"\int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(2 \sqrt{2} \left(a^2 (A-B)-2 a b (A+B)+b^2 (B-A)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-\sqrt{2} \left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-\frac{8 a^2 A}{\sqrt{\tan (c+d x)}}+8 b^2 B \sqrt{\tan (c+d x)}\right)}{4 d}","-\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a^2 A \sqrt{\cot (c+d x)}}{d}+\frac{2 b^2 B}{d \sqrt{\cot (c+d x)}}",1,"(Sqrt[Cot[c + d*x]]*(2*Sqrt[2]*(a^2*(A - B) + b^2*(-A + B) - 2*a*b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) - Sqrt[2]*(2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) - (8*a^2*A)/Sqrt[Tan[c + d*x]] + 8*b^2*B*Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/(4*d)","A",1
581,1,226,283,0.5204665,"\int \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]","-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(6 \sqrt{2} \left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+3 \sqrt{2} \left(a^2 (A-B)-2 a b (A+B)+b^2 (B-A)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-24 b (2 a B+A b) \sqrt{\tan (c+d x)}-8 b^2 B \tan ^{\frac{3}{2}}(c+d x)\right)}{12 d}","-\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b (2 a B+A b)}{d \sqrt{\cot (c+d x)}}+\frac{2 b^2 B}{3 d \cot ^{\frac{3}{2}}(c+d x)}",1,"-1/12*(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(6*Sqrt[2]*(2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + 3*Sqrt[2]*(a^2*(A - B) + b^2*(-A + B) - 2*a*b*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) - 24*b*(A*b + 2*a*B)*Sqrt[Tan[c + d*x]] - 8*b^2*B*Tan[c + d*x]^(3/2)))/d","A",1
582,1,255,317,1.0858575,"\int \frac{(a+b \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","Integrate[((a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]","-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(30 \sqrt{2} \left(a^2 (A-B)-2 a b (A+B)+b^2 (B-A)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-120 \left(a^2 B+2 a A b-b^2 B\right) \sqrt{\tan (c+d x)}-15 \sqrt{2} \left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-40 b (2 a B+A b) \tan ^{\frac{3}{2}}(c+d x)-24 b^2 B \tan ^{\frac{5}{2}}(c+d x)\right)}{60 d}","\frac{2 \left(a^2 B+2 a A b-b^2 B\right)}{d \sqrt{\cot (c+d x)}}+\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b (2 a B+A b)}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2 B}{5 d \cot ^{\frac{5}{2}}(c+d x)}",1,"-1/60*(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(30*Sqrt[2]*(a^2*(A - B) + b^2*(-A + B) - 2*a*b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) - 15*Sqrt[2]*(2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) - 120*(2*a*A*b + a^2*B - b^2*B)*Sqrt[Tan[c + d*x]] - 40*b*(A*b + 2*a*B)*Tan[c + d*x]^(3/2) - 24*b^2*B*Tan[c + d*x]^(5/2)))/d","A",1
583,1,326,421,3.5727793,"\int \cot ^{\frac{9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{a^3 A}{7 \tan ^{\frac{7}{2}}(c+d x)}+\frac{a \left(a^2 A-3 a b B-3 A b^2\right)}{3 \tan ^{\frac{3}{2}}(c+d x)}-\frac{a^2 (a B+3 A b)}{5 \tan ^{\frac{5}{2}}(c+d x)}-\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)+b^3 (B-A)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{2 \sqrt{2}}+\frac{a^3 B+3 a^2 A b-3 a b^2 B-A b^3}{\sqrt{\tan (c+d x)}}-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)+3 a b^2 (B-A)+b^3 (A+B)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 \sqrt{2}}\right)}{d}","\frac{2 a \left(7 a^2 A-21 a b B-18 A b^2\right) \cot ^{\frac{3}{2}}(c+d x)}{21 d}-\frac{2 a^2 (7 a B+11 A b) \cot ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \sqrt{\cot (c+d x)}}{d}-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}",1,"(2*Sqrt[Cot[c + d*x]]*(-1/2*((3*a^2*b*(A - B) + b^3*(-A + B) + a^3*(A + B) - 3*a*b^2*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/Sqrt[2] - ((a^3*(A - B) + 3*a*b^2*(-A + B) - 3*a^2*b*(A + B) + b^3*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(4*Sqrt[2]) - (a^3*A)/(7*Tan[c + d*x]^(7/2)) - (a^2*(3*A*b + a*B))/(5*Tan[c + d*x]^(5/2)) + (a*(a^2*A - 3*A*b^2 - 3*a*b*B))/(3*Tan[c + d*x]^(3/2)) + (3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)/Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/d","A",1
584,1,286,380,2.3417426,"\int \cot ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{a^3 A}{5 \tan ^{\frac{5}{2}}(c+d x)}+\frac{a \left(a^2 A-3 a b B-3 A b^2\right)}{\sqrt{\tan (c+d x)}}-\frac{a^2 (a B+3 A b)}{3 \tan ^{\frac{3}{2}}(c+d x)}-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)+3 a b^2 (B-A)+b^3 (A+B)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{2 \sqrt{2}}+\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)+b^3 (B-A)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 \sqrt{2}}\right)}{d}","\frac{2 a \left(5 a^2 A-15 a b B-14 A b^2\right) \sqrt{\cot (c+d x)}}{5 d}-\frac{2 a^2 (5 a B+9 A b) \cot ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a A \sqrt{\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}",1,"(2*Sqrt[Cot[c + d*x]]*(-1/2*((a^3*(A - B) + 3*a*b^2*(-A + B) - 3*a^2*b*(A + B) + b^3*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/Sqrt[2] + ((3*a^2*b*(A - B) + b^3*(-A + B) + a^3*(A + B) - 3*a*b^2*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(4*Sqrt[2]) - (a^3*A)/(5*Tan[c + d*x]^(5/2)) - (a^2*(3*A*b + a*B))/(3*Tan[c + d*x]^(3/2)) + (a*(a^2*A - 3*A*b^2 - 3*a*b*B))/Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/d","A",1
585,1,270,374,2.1189544,"\int \cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{a^3 A}{3 \tan ^{\frac{3}{2}}(c+d x)}-\frac{a^2 (a B+3 A b)}{\sqrt{\tan (c+d x)}}+\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)+b^3 (B-A)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{2 \sqrt{2}}+\frac{\left(a^3 (A-B)-3 a^2 b (A+B)+3 a b^2 (B-A)+b^3 (A+B)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 \sqrt{2}}+b^3 B \sqrt{\tan (c+d x)}\right)}{d}","-\frac{2 a \left(a^2 B+3 a A b+2 b^2 B\right) \sqrt{\cot (c+d x)}}{d}-\frac{2 a^2 (a A+3 b B) \cot ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b B (a \cot (c+d x)+b)^2}{d \sqrt{\cot (c+d x)}}",1,"(2*Sqrt[Cot[c + d*x]]*(((3*a^2*b*(A - B) + b^3*(-A + B) + a^3*(A + B) - 3*a*b^2*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(2*Sqrt[2]) + ((a^3*(A - B) + 3*a*b^2*(-A + B) - 3*a^2*b*(A + B) + b^3*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(4*Sqrt[2]) - (a^3*A)/(3*Tan[c + d*x]^(3/2)) - (a^2*(3*A*b + a*B))/Sqrt[Tan[c + d*x]] + b^3*B*Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/d","A",1
586,1,270,372,2.0882318,"\int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{a^3 A}{\sqrt{\tan (c+d x)}}+\frac{\left(a^3 (A-B)-3 a^2 b (A+B)+3 a b^2 (B-A)+b^3 (A+B)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{2 \sqrt{2}}-\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)+b^3 (B-A)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 \sqrt{2}}+b^2 (3 a B+A b) \sqrt{\tan (c+d x)}+\frac{1}{3} b^3 B \tan ^{\frac{3}{2}}(c+d x)\right)}{d}","-\frac{2 a^2 (3 a A+b B) \sqrt{\cot (c+d x)}}{3 d}-\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b^2 (7 a B+3 A b)}{3 d \sqrt{\cot (c+d x)}}+\frac{2 b B (a \cot (c+d x)+b)^2}{3 d \cot ^{\frac{3}{2}}(c+d x)}",1,"(2*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((a^3*(A - B) + 3*a*b^2*(-A + B) - 3*a^2*b*(A + B) + b^3*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(2*Sqrt[2]) - ((3*a^2*b*(A - B) + b^3*(-A + B) + a^3*(A + B) - 3*a*b^2*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(4*Sqrt[2]) - (a^3*A)/Sqrt[Tan[c + d*x]] + b^2*(A*b + 3*a*B)*Sqrt[Tan[c + d*x]] + (b^3*B*Tan[c + d*x]^(3/2))/3))/d","A",1
587,1,287,380,1.2587557,"\int \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]","\frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(b \left(3 a^2 B+3 a A b-b^2 B\right) \sqrt{\tan (c+d x)}-\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)+b^3 (B-A)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{2 \sqrt{2}}-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)+3 a b^2 (B-A)+b^3 (A+B)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 \sqrt{2}}+\frac{1}{3} b^2 (3 a B+A b) \tan ^{\frac{3}{2}}(c+d x)+\frac{1}{5} b^3 B \tan ^{\frac{5}{2}}(c+d x)\right)}{d}","\frac{2 b \left(14 a^2 B+15 a A b-5 b^2 B\right)}{5 d \sqrt{\cot (c+d x)}}-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b^2 (9 a B+5 A b)}{15 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 b B (a \cot (c+d x)+b)^2}{5 d \cot ^{\frac{5}{2}}(c+d x)}",1,"(2*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-1/2*((3*a^2*b*(A - B) + b^3*(-A + B) + a^3*(A + B) - 3*a*b^2*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/Sqrt[2] - ((a^3*(A - B) + 3*a*b^2*(-A + B) - 3*a^2*b*(A + B) + b^3*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(4*Sqrt[2]) + b*(3*a*A*b + 3*a^2*B - b^2*B)*Sqrt[Tan[c + d*x]] + (b^2*(A*b + 3*a*B)*Tan[c + d*x]^(3/2))/3 + (b^3*B*Tan[c + d*x]^(5/2))/5))/d","A",1
588,1,327,421,2.4895128,"\int \frac{(a+b \tan (c+d x))^3 (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","Integrate[((a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]","\frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{1}{3} b \left(3 a^2 B+3 a A b-b^2 B\right) \tan ^{\frac{3}{2}}(c+d x)-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)+3 a b^2 (B-A)+b^3 (A+B)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{2 \sqrt{2}}+\left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \sqrt{\tan (c+d x)}+\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)+b^3 (B-A)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 \sqrt{2}}+\frac{1}{5} b^2 (3 a B+A b) \tan ^{\frac{5}{2}}(c+d x)+\frac{1}{7} b^3 B \tan ^{\frac{7}{2}}(c+d x)\right)}{d}","\frac{2 b \left(18 a^2 B+21 a A b-7 b^2 B\right)}{21 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right)}{d \sqrt{\cot (c+d x)}}+\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b^2 (11 a B+7 A b)}{35 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{2 b B (a \cot (c+d x)+b)^2}{7 d \cot ^{\frac{7}{2}}(c+d x)}",1,"(2*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-1/2*((a^3*(A - B) + 3*a*b^2*(-A + B) - 3*a^2*b*(A + B) + b^3*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/Sqrt[2] + ((3*a^2*b*(A - B) + b^3*(-A + B) + a^3*(A + B) - 3*a*b^2*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(4*Sqrt[2]) + (3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Sqrt[Tan[c + d*x]] + (b*(3*a*A*b + 3*a^2*B - b^2*B)*Tan[c + d*x]^(3/2))/3 + (b^2*(A*b + 3*a*B)*Tan[c + d*x]^(5/2))/5 + (b^3*B*Tan[c + d*x]^(7/2))/7))/d","A",1
589,1,272,325,1.6610598,"\int \frac{\cot ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{6 \sqrt{2} (a (A+B)+b (B-A)) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{a^2+b^2}-\frac{3 \sqrt{2} (a (A-B)+b (A+B)) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{a^2+b^2}+\frac{24 (a B-A b)}{a^2 \sqrt{\tan (c+d x)}}+\frac{24 b^{5/2} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} \left(a^2+b^2\right)}+\frac{8 A}{a \tan ^{\frac{3}{2}}(c+d x)}\right)}{12 d}","\frac{(a (A-B)+b (A+B)) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a (A-B)+b (A+B)) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(b (A-B)-a (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(b (A-B)-a (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{2 (A b-a B) \sqrt{\cot (c+d x)}}{a^2 d}-\frac{2 b^{5/2} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{a^{5/2} d \left(a^2+b^2\right)}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d}",1,"-1/12*(Sqrt[Cot[c + d*x]]*((-6*Sqrt[2]*(b*(-A + B) + a*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (24*b^(5/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(5/2)*(a^2 + b^2)) - (3*Sqrt[2]*(a*(A - B) + b*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(a^2 + b^2) + (8*A)/(a*Tan[c + d*x]^(3/2)) + (24*(-(A*b) + a*B))/(a^2*Sqrt[Tan[c + d*x]]))*Sqrt[Tan[c + d*x]])/d","A",1
590,1,249,297,0.9862616,"\int \frac{\cot ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{2 \sqrt{2} (a (A-B)+b (A+B)) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{a^2+b^2}-\frac{\sqrt{2} (a (A+B)+b (B-A)) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{a^2+b^2}+\frac{8 b^{3/2} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} \left(a^2+b^2\right)}-\frac{8 A}{a \sqrt{\tan (c+d x)}}\right)}{4 d}","\frac{(b (A-B)-a (A+B)) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(b (A-B)-a (A+B)) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{2 b^{3/2} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{a^{3/2} d \left(a^2+b^2\right)}-\frac{2 A \sqrt{\cot (c+d x)}}{a d}",1,"(Sqrt[Cot[c + d*x]]*((2*Sqrt[2]*(a*(A - B) + b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (8*b^(3/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*(a^2 + b^2)) - (Sqrt[2]*(b*(-A + B) + a*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(a^2 + b^2) - (8*A)/(a*Sqrt[Tan[c + d*x]]))*Sqrt[Tan[c + d*x]])/(4*d)","A",1
591,1,215,278,0.4230957,"\int \frac{\sqrt{\cot (c+d x)} (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-2 \sqrt{2} (a (A+B)+b (B-A)) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\frac{8 \sqrt{b} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}-\sqrt{2} (a (A-B)+b (A+B)) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)\right)}{4 d \left(a^2+b^2\right)}","-\frac{(a (A-B)+b (A+B)) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(a (A-B)+b (A+B)) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(b (A-B)-a (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(b (A-B)-a (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{2 \sqrt{b} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{\sqrt{a} d \left(a^2+b^2\right)}",1,"(Sqrt[Cot[c + d*x]]*(-2*Sqrt[2]*(b*(-A + B) + a*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + (8*Sqrt[b]*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/Sqrt[a] - Sqrt[2]*(a*(A - B) + b*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))*Sqrt[Tan[c + d*x]])/(4*(a^2 + b^2)*d)","A",1
592,1,215,278,0.4398023,"\int \frac{A+B \tan (c+d x)}{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])),x]","-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(2 \sqrt{2} (a (A-B)+b (A+B)) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\frac{8 \sqrt{a} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b}}-\sqrt{2} (a (A+B)+b (B-A)) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)\right)}{4 d \left(a^2+b^2\right)}","-\frac{(b (A-B)-a (A+B)) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(b (A-B)-a (A+B)) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{2 \sqrt{a} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} d \left(a^2+b^2\right)}",1,"-1/4*(Sqrt[Cot[c + d*x]]*(2*Sqrt[2]*(a*(A - B) + b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + (8*Sqrt[a]*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/Sqrt[b] - Sqrt[2]*(b*(-A + B) + a*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*d)","A",1
593,1,251,297,0.572384,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])),x]","-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(8 a^{3/2} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)-8 \sqrt{b} B \left(a^2+b^2\right) \sqrt{\tan (c+d x)}+2 \sqrt{2} b^{3/2} (b (A-B)-a (A+B)) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-\sqrt{2} b^{3/2} (a (A-B)+b (A+B)) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)\right)}{4 b^{3/2} d \left(a^2+b^2\right)}","\frac{(a (A-B)+b (A+B)) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a (A-B)+b (A+B)) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(b (A-B)-a (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(b (A-B)-a (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{2 a^{3/2} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d \left(a^2+b^2\right)}+\frac{2 B}{b d \sqrt{\cot (c+d x)}}",1,"-1/4*(Sqrt[Cot[c + d*x]]*(2*Sqrt[2]*b^(3/2)*(b*(A - B) - a*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + 8*a^(3/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] - Sqrt[2]*b^(3/2)*(a*(A - B) + b*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) - 8*Sqrt[b]*(a^2 + b^2)*B*Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/(b^(3/2)*(a^2 + b^2)*d)","A",1
594,1,272,325,1.1205484,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{6 \sqrt{2} (a (A-B)+b (A+B)) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{a^2+b^2}-\frac{3 \sqrt{2} (a (A+B)+b (B-A)) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{a^2+b^2}+\frac{24 a^{5/2} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{5/2} \left(a^2+b^2\right)}+\frac{24 (A b-a B) \sqrt{\tan (c+d x)}}{b^2}+\frac{8 B \tan ^{\frac{3}{2}}(c+d x)}{b}\right)}{12 d}","\frac{(b (A-B)-a (A+B)) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(b (A-B)-a (A+B)) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{2 a^{5/2} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{b^{5/2} d \left(a^2+b^2\right)}+\frac{2 (A b-a B)}{b^2 d \sqrt{\cot (c+d x)}}+\frac{2 B}{3 b d \cot ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((6*Sqrt[2]*(a*(A - B) + b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (24*a^(5/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(5/2)*(a^2 + b^2)) - (3*Sqrt[2]*(b*(-A + B) + a*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(a^2 + b^2) + (24*(A*b - a*B)*Sqrt[Tan[c + d*x]])/b^2 + (8*B*Tan[c + d*x]^(3/2))/b))/(12*d)","A",1
595,1,383,438,5.4222211,"\int \frac{\cot ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{2 \sqrt{2} \left(a^2 (A-B)+2 a b (A+B)+b^2 (B-A)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{\left(a^2+b^2\right)^2}+\frac{4 b^2 (a B-A b) \sqrt{\tan (c+d x)}}{a^2 \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\sqrt{2} \left(a^2 (A+B)+2 a b (B-A)-b^2 (A+B)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{\left(a^2+b^2\right)^2}-\frac{8 A}{a^2 \sqrt{\tan (c+d x)}}+\frac{4 b^{3/2} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} \left(a^2+b^2\right)}-\frac{8 b^{3/2} \left(-2 a^3 B+3 a^2 A b+A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} \left(a^2+b^2\right)^2}\right)}{4 d}","\frac{b (A b-a B) \cot ^{\frac{3}{2}}(c+d x)}{a d \left(a^2+b^2\right) (a \cot (c+d x)+b)}-\frac{\left(2 a^2 A-a b B+3 A b^2\right) \sqrt{\cot (c+d x)}}{a^2 d \left(a^2+b^2\right)}+\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{b^{3/2} \left(-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{a^{5/2} d \left(a^2+b^2\right)^2}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((2*Sqrt[2]*(a^2*(A - B) + b^2*(-A + B) + 2*a*b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2)^2 + (4*b^(3/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(5/2)*(a^2 + b^2)) - (8*b^(3/2)*(3*a^2*A*b + A*b^3 - 2*a^3*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(5/2)*(a^2 + b^2)^2) - (Sqrt[2]*(2*a*b*(-A + B) + a^2*(A + B) - b^2*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(a^2 + b^2)^2 - (8*A)/(a^2*Sqrt[Tan[c + d*x]]) + (4*b^2*(-(A*b) + a*B)*Sqrt[Tan[c + d*x]])/(a^2*(a^2 + b^2)*(a + b*Tan[c + d*x]))))/(4*d)","A",1
596,1,341,392,2.7047748,"\int \frac{\sqrt{\cot (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(2 \sqrt{2} \left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\frac{8 \sqrt{b} \left(a^2 (-B)+2 a A b+b^2 B\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{4 b \left(a^2+b^2\right) (A b-a B) \sqrt{\tan (c+d x)}}{a (a+b \tan (c+d x))}-\sqrt{2} \left(a^2 (A-B)+2 a b (A+B)+b^2 (B-A)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\frac{4 \sqrt{b} \left(a^2+b^2\right) (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2}}\right)}{4 d \left(a^2+b^2\right)^2}","\frac{b (A b-a B) \sqrt{\cot (c+d x)}}{a d \left(a^2+b^2\right) (a \cot (c+d x)+b)}-\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\sqrt{b} \left(-3 a^3 B+5 a^2 A b+a b^2 B+A b^3\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{a^{3/2} d \left(a^2+b^2\right)^2}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(2*Sqrt[2]*(2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + (4*Sqrt[b]*(a^2 + b^2)*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/a^(3/2) + (8*Sqrt[b]*(2*a*A*b - a^2*B + b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/Sqrt[a] - Sqrt[2]*(a^2*(A - B) + b^2*(-A + B) + 2*a*b*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) + (4*b*(a^2 + b^2)*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(a*(a + b*Tan[c + d*x]))))/(4*(a^2 + b^2)^2*d)","A",1
597,1,336,390,3.0010884,"\int \frac{A+B \tan (c+d x)}{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^2} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^2),x]","-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{4 \left(a^2+b^2\right) (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}+2 \sqrt{2} \left(a^2 (A-B)+2 a b (A+B)+b^2 (B-A)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\frac{8 \sqrt{b} \left(a^2 A+2 a b B-A b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{4 \left(a^2+b^2\right) (a B-A b) \sqrt{\tan (c+d x)}}{a+b \tan (c+d x)}+\sqrt{2} \left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)\right)}{4 d \left(a^2+b^2\right)^2}","-\frac{(A b-a B) \sqrt{\cot (c+d x)}}{d \left(a^2+b^2\right) (a \cot (c+d x)+b)}-\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{\sqrt{a} \sqrt{b} d \left(a^2+b^2\right)^2}",1,"-1/4*(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(2*Sqrt[2]*(a^2*(A - B) + b^2*(-A + B) + 2*a*b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) - (4*(a^2 + b^2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) + (8*Sqrt[b]*(a^2*A - A*b^2 + 2*a*b*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/Sqrt[a] + Sqrt[2]*(2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) - (4*(a^2 + b^2)*(-(A*b) + a*B)*Sqrt[Tan[c + d*x]])/(a + b*Tan[c + d*x])))/((a^2 + b^2)^2*d)","A",1
598,1,342,392,2.7964307,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-2 \sqrt{2} \left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\frac{4 a \left(a^2+b^2\right) (A b-a B) \sqrt{\tan (c+d x)}}{b (a+b \tan (c+d x))}+\sqrt{2} \left(a^2 (A-B)+2 a b (A+B)+b^2 (B-A)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\frac{4 \sqrt{a} \left(a^2+b^2\right) (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{3/2}}+\frac{8 \sqrt{a} \left(a B \left(a^2+3 b^2\right)-2 A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{3/2}}\right)}{4 d \left(a^2+b^2\right)^2}","\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{b d \left(a^2+b^2\right) (a \cot (c+d x)+b)}+\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\sqrt{a} \left(a^3 B+a^2 A b+5 a b^2 B-3 A b^3\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d \left(a^2+b^2\right)^2}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-2*Sqrt[2]*(2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + (4*Sqrt[a]*(a^2 + b^2)*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/b^(3/2) + (8*Sqrt[a]*(-2*A*b^3 + a*(a^2 + 3*b^2)*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/b^(3/2) + Sqrt[2]*(a^2*(A - B) + b^2*(-A + B) + 2*a*b*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) + (4*a*(a^2 + b^2)*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(b*(a + b*Tan[c + d*x]))))/(4*(a^2 + b^2)^2*d)","A",1
599,1,390,437,3.3749961,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{2 \sqrt{2} \left(a^2 (A-B)+2 a b (A+B)+b^2 (B-A)\right) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{\left(a^2+b^2\right)^2}+\frac{4 a^2 (a B-A b) \sqrt{\tan (c+d x)}}{b^2 \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\sqrt{2} \left(a^2 (A+B)+2 a b (B-A)-b^2 (A+B)\right) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{\left(a^2+b^2\right)^2}+\frac{4 a^{3/2} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{5/2} \left(a^2+b^2\right)}+\frac{8 a^{3/2} \left(-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{5/2} \left(a^2+b^2\right)^2}+\frac{8 B \sqrt{\tan (c+d x)}}{b^2}\right)}{4 d}","\frac{a (A b-a B)}{b d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} (a \cot (c+d x)+b)}-\frac{-3 a^2 B+a A b-2 b^2 B}{b^2 d \left(a^2+b^2\right) \sqrt{\cot (c+d x)}}+\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(-\left(a^2 (A+B)\right)+2 a b (A-B)+b^2 (A+B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{a^{3/2} \left(-3 a^3 B+a^2 A b-7 a b^2 B+5 A b^3\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{b^{5/2} d \left(a^2+b^2\right)^2}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((2*Sqrt[2]*(a^2*(A - B) + b^2*(-A + B) + 2*a*b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2)^2 + (4*a^(3/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(5/2)*(a^2 + b^2)) + (8*a^(3/2)*(a^2*A*b + 3*A*b^3 - 2*a^3*B - 4*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(5/2)*(a^2 + b^2)^2) - (Sqrt[2]*(2*a*b*(-A + B) + a^2*(A + B) - b^2*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(a^2 + b^2)^2 + (8*B*Sqrt[Tan[c + d*x]])/b^2 + (4*a^2*(-(A*b) + a*B)*Sqrt[Tan[c + d*x]])/(b^2*(a^2 + b^2)*(a + b*Tan[c + d*x]))))/(4*d)","A",1
600,1,602,601,6.4979898,"\int \frac{\cot ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]","\frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{A}{a^3 \sqrt{\tan (c+d x)}}-\frac{b^2 (A b-a B) \sqrt{\tan (c+d x)}}{4 a^2 \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{3 b^2 (A b-a B) \left(\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} \sqrt{b}}+\frac{\sqrt{\tan (c+d x)}}{a (a+b \tan (c+d x))}\right)}{8 a^2 \left(a^2+b^2\right)}+\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 \left(a^2+b^2\right)^3}-\frac{b^2 \left(-2 a^3 B+3 a^2 A b+A b^3\right) \sqrt{\tan (c+d x)}}{2 a^3 \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \left(\sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{8 \left(a^2+b^2\right)^3}-\frac{b^{3/2} \left(-2 a^3 B+3 a^2 A b+A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{2 a^{7/2} \left(a^2+b^2\right)^2}-\frac{b^{3/2} \left(-3 a^5 B+6 a^4 A b+a^3 b^2 B+3 a^2 A b^3+A b^5\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{7/2} \left(a^2+b^2\right)^3}\right)}{d}","\frac{b (A b-a B) \cot ^{\frac{5}{2}}(c+d x)}{2 a d \left(a^2+b^2\right) (a \cot (c+d x)+b)^2}+\frac{b \left(-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right) \cot ^{\frac{3}{2}}(c+d x)}{4 a^2 d \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right) \sqrt{\cot (c+d x)}}{4 a^3 d \left(a^2+b^2\right)^2}+\frac{b^{3/2} \left(-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{4 a^{7/2} d \left(a^2+b^2\right)^3}",1,"(2*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(4*(a^2 + b^2)^3) - (b^(3/2)*(3*a^2*A*b + A*b^3 - 2*a^3*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(2*a^(7/2)*(a^2 + b^2)^2) - (b^(3/2)*(6*a^4*A*b + 3*a^2*A*b^3 + A*b^5 - 3*a^5*B + a^3*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(7/2)*(a^2 + b^2)^3) + ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*(Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(8*(a^2 + b^2)^3) - A/(a^3*Sqrt[Tan[c + d*x]]) - (b^2*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(4*a^2*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - (b^2*(3*a^2*A*b + A*b^3 - 2*a^3*B)*Sqrt[Tan[c + d*x]])/(2*a^3*(a^2 + b^2)^2*(a + b*Tan[c + d*x])) - (3*b^2*(A*b - a*B)*(ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]/(a^(3/2)*Sqrt[b]) + Sqrt[Tan[c + d*x]]/(a*(a + b*Tan[c + d*x]))))/(8*a^2*(a^2 + b^2))))/d","A",0
601,1,566,534,6.3892124,"\int \frac{\sqrt{\cot (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]","\frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{b (A b-a B) \sqrt{\tan (c+d x)}}{4 a \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{b \left(a^2 (-B)+2 a A b+b^2 B\right) \sqrt{\tan (c+d x)}}{2 a \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{3 b (A b-a B) \left(\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} \sqrt{b}}+\frac{\sqrt{\tan (c+d x)}}{a (a+b \tan (c+d x))}\right)}{8 a \left(a^2+b^2\right)}+\frac{\sqrt{b} \left(a^2 (-B)+2 a A b+b^2 B\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{2 a^{3/2} \left(a^2+b^2\right)^2}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 \left(a^2+b^2\right)^3}+\frac{\sqrt{b} \left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \left(a^2+b^2\right)^3}-\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \left(\sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{8 \left(a^2+b^2\right)^3}\right)}{d}","\frac{b (A b-a B) \cot ^{\frac{3}{2}}(c+d x)}{2 a d \left(a^2+b^2\right) (a \cot (c+d x)+b)^2}+\frac{b \left(-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right) \sqrt{\cot (c+d x)}}{4 a^2 d \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}-\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\sqrt{b} \left(-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{4 a^{5/2} d \left(a^2+b^2\right)^3}",1,"(2*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(4*(a^2 + b^2)^3) + (Sqrt[b]*(2*a*A*b - a^2*B + b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(2*a^(3/2)*(a^2 + b^2)^2) + (Sqrt[b]*(3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)^3) - ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*(Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(8*(a^2 + b^2)^3) + (b*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(4*a*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) + (b*(2*a*A*b - a^2*B + b^2*B)*Sqrt[Tan[c + d*x]])/(2*a*(a^2 + b^2)^2*(a + b*Tan[c + d*x])) + (3*b*(A*b - a*B)*(ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]/(a^(3/2)*Sqrt[b]) + Sqrt[Tan[c + d*x]]/(a*(a + b*Tan[c + d*x]))))/(8*a*(a^2 + b^2))))/d","A",1
602,1,558,534,6.3684086,"\int \frac{A+B \tan (c+d x)}{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^3),x]","\frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{(A b-a B) \sqrt{\tan (c+d x)}}{4 \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{b \left(a^2 A+2 a b B-A b^2\right) \sqrt{\tan (c+d x)}}{2 a \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{3 (A b-a B) \left(\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} \sqrt{b}}+\frac{\sqrt{\tan (c+d x)}}{a (a+b \tan (c+d x))}\right)}{8 \left(a^2+b^2\right)}-\frac{\sqrt{b} \left(a^2 A+2 a b B-A b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{2 a^{3/2} \left(a^2+b^2\right)^2}-\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 \left(a^2+b^2\right)^3}-\frac{\sqrt{b} \left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \left(a^2+b^2\right)^3}-\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \left(\sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{8 \left(a^2+b^2\right)^3}\right)}{d}","\frac{b (A b-a B) \sqrt{\cot (c+d x)}}{2 a d \left(a^2+b^2\right) (a \cot (c+d x)+b)^2}-\frac{\left(-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right) \sqrt{\cot (c+d x)}}{4 a d \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}-\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{4 a^{3/2} \sqrt{b} d \left(a^2+b^2\right)^3}",1,"(2*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-1/4*((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2)^3 - (Sqrt[b]*(a^2*A - A*b^2 + 2*a*b*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(2*a^(3/2)*(a^2 + b^2)^2) - (Sqrt[b]*(a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)^3) - ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*(Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(8*(a^2 + b^2)^3) - ((A*b - a*B)*Sqrt[Tan[c + d*x]])/(4*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - (b*(a^2*A - A*b^2 + 2*a*b*B)*Sqrt[Tan[c + d*x]])/(2*a*(a^2 + b^2)^2*(a + b*Tan[c + d*x])) - (3*(A*b - a*B)*(ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]/(a^(3/2)*Sqrt[b]) + Sqrt[Tan[c + d*x]]/(a*(a + b*Tan[c + d*x]))))/(8*(a^2 + b^2))))/d","A",1
603,1,568,530,6.4184162,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3),x]","\frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{3 (A b-a B) \left(\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}+\frac{\sqrt{\tan (c+d x)}}{a+b \tan (c+d x)}\right)}{8 b \left(a^2+b^2\right)}+\frac{a (A b-a B) \sqrt{\tan (c+d x)}}{4 b \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{\left(2 A b^3-a B \left(a^2+3 b^2\right)\right) \sqrt{\tan (c+d x)}}{2 b \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{\left(2 A b^3-a B \left(a^2+3 b^2\right)\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{2 \sqrt{a} b^{3/2} \left(a^2+b^2\right)^2}-\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 \left(a^2+b^2\right)^3}-\frac{\sqrt{b} \left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \left(a^2+b^2\right)^3}+\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \left(\sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{8 \left(a^2+b^2\right)^3}\right)}{d}","-\frac{(A b-a B) \sqrt{\cot (c+d x)}}{2 d \left(a^2+b^2\right) (a \cot (c+d x)+b)^2}+\frac{\left(a^3 (-B)+5 a^2 A b+7 a b^2 B-3 A b^3\right) \sqrt{\cot (c+d x)}}{4 b d \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}+\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(a^5 B+3 a^4 A b+18 a^3 b^2 B-26 a^2 A b^3-15 a b^4 B+3 A b^5\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{4 \sqrt{a} b^{3/2} d \left(a^2+b^2\right)^3}",1,"(2*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-1/4*((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2)^3 - (Sqrt[b]*(3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)^3) - ((2*A*b^3 - a*(a^2 + 3*b^2)*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(2*Sqrt[a]*b^(3/2)*(a^2 + b^2)^2) + ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*(Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(8*(a^2 + b^2)^3) + (a*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(4*b*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - ((2*A*b^3 - a*(a^2 + 3*b^2)*B)*Sqrt[Tan[c + d*x]])/(2*b*(a^2 + b^2)^2*(a + b*Tan[c + d*x])) + (3*(A*b - a*B)*(ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]) + Sqrt[Tan[c + d*x]]/(a + b*Tan[c + d*x])))/(8*b*(a^2 + b^2))))/d","A",1
604,1,592,534,6.4829241,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3),x]","\frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{3 (A b-a B) \left(\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b}}+\frac{a \sqrt{\tan (c+d x)}}{a+b \tan (c+d x)}\right)}{8 b^2 \left(a^2+b^2\right)}-\frac{a^2 (A b-a B) \sqrt{\tan (c+d x)}}{4 b^2 \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 \left(a^2+b^2\right)^3}+\frac{a \left(-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right) \sqrt{\tan (c+d x)}}{2 b^2 \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \left(\sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{8 \left(a^2+b^2\right)^3}+\frac{\sqrt{a} \left(-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{2 b^{5/2} \left(a^2+b^2\right)^2}+\frac{\sqrt{a} \left(a^5 B+3 a^3 b^2 B+a^2 A b^3+6 a b^4 B-3 A b^5\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{5/2} \left(a^2+b^2\right)^3}\right)}{d}","\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{2 b d \left(a^2+b^2\right) (a \cot (c+d x)+b)^2}-\frac{a \left(3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right) \sqrt{\cot (c+d x)}}{4 b^2 d \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\sqrt{a} \left(3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{4 b^{5/2} d \left(a^2+b^2\right)^3}",1,"(2*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(4*(a^2 + b^2)^3) + (Sqrt[a]*(a^2*A*b + 3*A*b^3 - 2*a^3*B - 4*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(2*b^(5/2)*(a^2 + b^2)^2) + (Sqrt[a]*(a^2*A*b^3 - 3*A*b^5 + a^5*B + 3*a^3*b^2*B + 6*a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(5/2)*(a^2 + b^2)^3) + ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*(Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(8*(a^2 + b^2)^3) - (a^2*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(4*b^2*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) + (a*(a^2*A*b + 3*A*b^3 - 2*a^3*B - 4*a*b^2*B)*Sqrt[Tan[c + d*x]])/(2*b^2*(a^2 + b^2)^2*(a + b*Tan[c + d*x])) - (3*(A*b - a*B)*((Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/Sqrt[b] + (a*Sqrt[Tan[c + d*x]])/(a + b*Tan[c + d*x])))/(8*b^2*(a^2 + b^2))))/d","A",1
605,1,621,600,6.4851331,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^3),x]","\frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{3 (A b-a B) \left(\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b}}+\frac{a^2 \sqrt{\tan (c+d x)}}{a+b \tan (c+d x)}\right)}{8 b^3 \left(a^2+b^2\right)}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 \left(a^2+b^2\right)^3}+\frac{a^3 (A b-a B) \sqrt{\tan (c+d x)}}{4 b^3 \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{a^2 \left(-3 a^3 B+2 a^2 A b-5 a b^2 B+4 A b^3\right) \sqrt{\tan (c+d x)}}{2 b^3 \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \left(\sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{8 \left(a^2+b^2\right)^3}-\frac{a^{3/2} \left(-3 a^3 B+2 a^2 A b-5 a b^2 B+4 A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{2 b^{7/2} \left(a^2+b^2\right)^2}+\frac{a^{3/2} \left(-3 a^5 B+a^4 A b-9 a^3 b^2 B+3 a^2 A b^3-10 a b^4 B+6 A b^5\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{7/2} \left(a^2+b^2\right)^3}+\frac{B \sqrt{\tan (c+d x)}}{b^3}\right)}{d}","\frac{a (A b-a B)}{2 b d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} (a \cot (c+d x)+b)^2}+\frac{a \left(-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right)}{4 b^2 d \left(a^2+b^2\right)^2 \sqrt{\cot (c+d x)} (a \cot (c+d x)+b)}-\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(-\left(a^3 (A+B)\right)+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B}{4 b^3 d \left(a^2+b^2\right)^2 \sqrt{\cot (c+d x)}}-\frac{a^{3/2} \left(-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{4 b^{7/2} d \left(a^2+b^2\right)^3}",1,"(2*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(4*(a^2 + b^2)^3) - (a^(3/2)*(2*a^2*A*b + 4*A*b^3 - 3*a^3*B - 5*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(2*b^(7/2)*(a^2 + b^2)^2) + (a^(3/2)*(a^4*A*b + 3*a^2*A*b^3 + 6*A*b^5 - 3*a^5*B - 9*a^3*b^2*B - 10*a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(7/2)*(a^2 + b^2)^3) - ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*(Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(8*(a^2 + b^2)^3) + (B*Sqrt[Tan[c + d*x]])/b^3 + (a^3*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(4*b^3*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - (a^2*(2*a^2*A*b + 4*A*b^3 - 3*a^3*B - 5*a*b^2*B)*Sqrt[Tan[c + d*x]])/(2*b^3*(a^2 + b^2)^2*(a + b*Tan[c + d*x])) + (3*(A*b - a*B)*((a^(3/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/Sqrt[b] + (a^2*Sqrt[Tan[c + d*x]])/(a + b*Tan[c + d*x])))/(8*b^3*(a^2 + b^2))))/d","A",1
606,1,38,156,0.0714634,"\int \frac{\cot ^{\frac{5}{2}}(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^(5/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","\frac{2 B \cot ^{\frac{3}{2}}(c+d x) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-1\right)}{3 d}","-\frac{2 B \cot ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{B \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{B \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{B \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{B \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}",1,"(2*B*Cot[c + d*x]^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(3*d)","C",1
607,1,138,154,0.1504736,"\int \frac{\cot ^{\frac{3}{2}}(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^(3/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","-\frac{B \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{4 d}","-\frac{2 B \sqrt{\cot (c+d x)}}{d}-\frac{B \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{B \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{B \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{B \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}",1,"-1/4*(B*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/d","A",1
608,1,36,138,0.0210869,"\int \frac{\sqrt{\cot (c+d x)} (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx","Integrate[(Sqrt[Cot[c + d*x]]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]","-\frac{2 B \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)}{3 d}","-\frac{B \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{B \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{B \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{B \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}",1,"(-2*B*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/(3*d)","C",1
609,1,110,138,0.0312991,"\int \frac{a B+b B \tan (c+d x)}{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])),x]","\frac{B \left(\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{2 \sqrt{2} d}","\frac{B \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{B \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{B \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{B \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}",1,"(B*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(2*Sqrt[2]*d)","A",1
610,1,34,154,0.0342658,"\int \frac{a B+b B \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])),x]","\frac{2 B \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)}{d \sqrt{\cot (c+d x)}}","\frac{2 B}{d \sqrt{\cot (c+d x)}}+\frac{B \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{B \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{B \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{B \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}",1,"(2*B*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2])/(d*Sqrt[Cot[c + d*x]])","C",1
611,1,36,156,0.0455328,"\int \frac{a B+b B \tan (c+d x)}{\cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])),x]","\frac{2 B \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)}{3 d \cot ^{\frac{3}{2}}(c+d x)}","\frac{2 B}{3 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{B \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{B \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{B \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{B \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}",1,"(2*B*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2])/(3*d*Cot[c + d*x]^(3/2))","C",1
612,1,291,354,4.0241101,"\int \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(9/2)*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{\cot ^{\frac{7}{2}}(c+d x) \left(105 \sqrt[4]{-1} a^3 \sqrt{-a+i b} (B+i A) \tan ^{\frac{7}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-105 (-1)^{3/4} a^3 \sqrt{a+i b} (A+i B) \tan ^{\frac{7}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+2 \sqrt{a+b \tan (c+d x)} \left(-15 a^3 A+a \left(35 a^2 A-7 a b B+4 A b^2\right) \tan ^2(c+d x)-3 a^2 (7 a B+A b) \tan (c+d x)+\left(105 a^3 B+35 a^2 A b+14 a b^2 B-8 A b^3\right) \tan ^3(c+d x)\right)\right)}{105 a^3 d}","\frac{2 \left(35 a^2 A-7 a b B+4 A b^2\right) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a^2 d}+\frac{2 \left(105 a^3 B+35 a^2 A b+14 a b^2 B-8 A b^3\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{105 a^3 d}-\frac{2 (7 a B+A b) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{35 a d}-\frac{\sqrt{-b+i a} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{\sqrt{b+i a} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d}",1,"(Cot[c + d*x]^(7/2)*(105*(-1)^(1/4)*a^3*Sqrt[-a + I*b]*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(7/2) - 105*(-1)^(3/4)*a^3*Sqrt[a + I*b]*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(7/2) + 2*Sqrt[a + b*Tan[c + d*x]]*(-15*a^3*A - 3*a^2*(A*b + 7*a*B)*Tan[c + d*x] + a*(35*a^2*A + 4*A*b^2 - 7*a*b*B)*Tan[c + d*x]^2 + (35*a^2*A*b - 8*A*b^3 + 105*a^3*B + 14*a*b^2*B)*Tan[c + d*x]^3)))/(105*a^3*d)","A",1
613,1,252,290,2.1997451,"\int \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{\cot ^{\frac{5}{2}}(c+d x) \left(2 \sqrt{a+b \tan (c+d x)} \left(\left(15 a^2 A-5 a b B+2 A b^2\right) \tan ^2(c+d x)-3 a^2 A-a (5 a B+A b) \tan (c+d x)\right)+15 \sqrt[4]{-1} a^2 \sqrt{-a+i b} (A-i B) \tan ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+15 \sqrt[4]{-1} a^2 \sqrt{a+i b} (A+i B) \tan ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{15 a^2 d}","\frac{2 \left(15 a^2 A-5 a b B+2 A b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{15 a^2 d}-\frac{2 (5 a B+A b) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{15 a d}+\frac{\sqrt{-b+i a} (A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{\sqrt{b+i a} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 A \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}",1,"(Cot[c + d*x]^(5/2)*(15*(-1)^(1/4)*a^2*Sqrt[-a + I*b]*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(5/2) + 15*(-1)^(1/4)*a^2*Sqrt[a + I*b]*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(5/2) + 2*Sqrt[a + b*Tan[c + d*x]]*(-3*a^2*A - a*(A*b + 5*a*B)*Tan[c + d*x] + (15*a^2*A + 2*A*b^2 - 5*a*b*B)*Tan[c + d*x]^2)))/(15*a^2*d)","A",1
614,1,216,239,1.8644417,"\int \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{\cot ^{\frac{3}{2}}(c+d x) \left(-3 \sqrt[4]{-1} a \sqrt{-a+i b} (B+i A) \tan ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+3 (-1)^{3/4} a \sqrt{a+i b} (A+i B) \tan ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-2 \sqrt{a+b \tan (c+d x)} ((3 a B+A b) \tan (c+d x)+a A)\right)}{3 a d}","\frac{\sqrt{-b+i a} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 (3 a B+A b) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 a d}+\frac{\sqrt{b+i a} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}",1,"(Cot[c + d*x]^(3/2)*(-3*(-1)^(1/4)*a*Sqrt[-a + I*b]*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(3/2) + 3*(-1)^(3/4)*a*Sqrt[a + I*b]*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(3/2) - 2*Sqrt[a + b*Tan[c + d*x]]*(a*A + (A*b + 3*a*B)*Tan[c + d*x])))/(3*a*d)","A",1
615,1,189,194,0.5125728,"\int \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","-\frac{\sqrt{\cot (c+d x)} \left(\sqrt[4]{-1} \sqrt{-a+i b} (A-i B) \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt[4]{-1} \sqrt{a+i b} (A+i B) \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+2 A \sqrt{a+b \tan (c+d x)}\right)}{d}","-\frac{\sqrt{-b+i a} (A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{\sqrt{b+i a} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 A \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}",1,"-((Sqrt[Cot[c + d*x]]*((-1)^(1/4)*Sqrt[-a + I*b]*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] + (-1)^(1/4)*Sqrt[a + I*b]*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] + 2*A*Sqrt[a + b*Tan[c + d*x]]))/d)","A",1
616,1,238,229,0.7779468,"\int \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(2 \sqrt{a} \sqrt{b} B \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+\sqrt[4]{-1} \sqrt{a+b \tan (c+d x)} \left(\sqrt{-a+i b} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt{a+i b} (B-i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)\right)}{d \sqrt{a+b \tan (c+d x)}}","-\frac{\sqrt{-b+i a} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{\sqrt{b+i a} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{2 \sqrt{b} B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-1)^(1/4)*(Sqrt[-a + I*b]*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + Sqrt[a + I*b]*((-I)*A + B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])*Sqrt[a + b*Tan[c + d*x]] + 2*Sqrt[a]*Sqrt[b]*B*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a]))/(d*Sqrt[a + b*Tan[c + d*x]])","A",1
617,1,293,261,3.463114,"\int \frac{\sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","Integrate[(Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\sqrt[4]{-1} \sqrt{-a+i b} (A-i B) \sqrt{a+b \tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt[4]{-1} \sqrt{a+i b} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right) \sqrt{a+b \tan (c+d x)}+\frac{(a B+2 A b) (a+b \tan (c+d x)) \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} \sqrt{\frac{b \tan (c+d x)}{a}+1}}+B \sqrt{\tan (c+d x)} (a+b \tan (c+d x))\right)}{d \sqrt{a+b \tan (c+d x)}}","\frac{\sqrt{-b+i a} (A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(a B+2 A b) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{b} d}-\frac{\sqrt{b+i a} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{B \sqrt{a+b \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-1)^(1/4)*Sqrt[-a + I*b]*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + (-1)^(1/4)*Sqrt[a + I*b]*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x]) + ((2*A*b + a*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*(a + b*Tan[c + d*x]))/(Sqrt[a]*Sqrt[b]*Sqrt[1 + (b*Tan[c + d*x])/a])))/(d*Sqrt[a + b*Tan[c + d*x]])","A",1
618,1,356,324,4.7591608,"\int \frac{\sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x))}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(3/2),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\left(a^2 B-4 a A b+8 b^2 B\right) (a+b \tan (c+d x)) \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+\sqrt{a} \sqrt{b} \sqrt{\frac{b \tan (c+d x)}{a}+1} \left(-4 \sqrt[4]{-1} b \sqrt{-a+i b} (B+i A) \sqrt{a+b \tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+4 (-1)^{3/4} b \sqrt{a+i b} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right) \sqrt{a+b \tan (c+d x)}+\sqrt{\tan (c+d x)} (a+b \tan (c+d x)) (a B+4 A b+2 b B \tan (c+d x))\right)\right)}{4 \sqrt{a} b^{3/2} d \sqrt{a+b \tan (c+d x)} \sqrt{\frac{b \tan (c+d x)}{a}+1}}","\frac{\left(a^2 (-B)+4 a A b-8 b^2 B\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{4 b^{3/2} d}+\frac{\sqrt{-b+i a} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(4 A b-a B) \sqrt{a+b \tan (c+d x)}}{4 b d \sqrt{\cot (c+d x)}}+\frac{\sqrt{b+i a} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{B (a+b \tan (c+d x))^{3/2}}{2 b d \sqrt{\cot (c+d x)}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-((-4*a*A*b + a^2*B + 8*b^2*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*(a + b*Tan[c + d*x])) + Sqrt[a]*Sqrt[b]*Sqrt[1 + (b*Tan[c + d*x])/a]*(-4*(-1)^(1/4)*Sqrt[-a + I*b]*b*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + 4*(-1)^(3/4)*Sqrt[a + I*b]*b*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])*(4*A*b + a*B + 2*b*B*Tan[c + d*x]))))/(4*Sqrt[a]*b^(3/2)*d*Sqrt[a + b*Tan[c + d*x]]*Sqrt[1 + (b*Tan[c + d*x])/a])","A",1
619,1,495,422,6.5797529,"\int \cot ^{\frac{11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(11/2)*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{b B \sqrt{a+b \tan (c+d x)}}{4 d \tan ^{\frac{9}{2}}(c+d x)}+\frac{1}{4} \left(-\frac{(8 a A-9 b B) \sqrt{a+b \tan (c+d x)}}{9 d \tan ^{\frac{9}{2}}(c+d x)}+\frac{2 \left(-\frac{4 a (9 a B+10 A b) \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 \left(-\frac{6 a \left(21 a^2 A-24 a b B-A b^2\right) \sqrt{a+b \tan (c+d x)}}{5 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 \left(\frac{a \left(105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right) \sqrt{a+b \tan (c+d x)}}{d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(\frac{3 a \left(315 a^4 A-420 a^3 b B-63 a^2 A b^2-18 a b^3 B+8 A b^4\right) \sqrt{a+b \tan (c+d x)}}{2 d \sqrt{\tan (c+d x)}}-\frac{945 a^4 \left(\sqrt[4]{-1} (-a+i b)^{3/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-\sqrt[4]{-1} (a+i b)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{4 d}\right)}{3 a}\right)}{5 a}\right)}{7 a}\right)}{9 a}\right)\right)","\frac{2 \left(21 a^2 A-24 a b B-A b^2\right) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a d}+\frac{2 \left(105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a^2 d}-\frac{2 \left(315 a^4 A-420 a^3 b B-63 a^2 A b^2-18 a b^3 B+8 A b^4\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^3 d}-\frac{2 (9 a B+10 A b) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}+\frac{(-b+i a)^{3/2} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{(b+i a)^{3/2} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}",1,"Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-1/4*(b*B*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(9/2)) + (-1/9*((8*a*A - 9*b*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(9/2)) + (2*((-4*a*(10*A*b + 9*a*B)*Sqrt[a + b*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (2*((-6*a*(21*a^2*A - A*b^2 - 24*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - (2*((a*(126*a^2*A*b + 4*A*b^3 + 105*a^3*B - 9*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(3/2)) - (2*((-945*a^4*((-1)^(1/4)*(-a + I*b)^(3/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - (-1)^(1/4)*(a + I*b)^(3/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]))/(4*d) + (3*a*(315*a^4*A - 63*a^2*A*b^2 + 8*A*b^4 - 420*a^3*b*B - 18*a*b^3*B)*Sqrt[a + b*Tan[c + d*x]])/(2*d*Sqrt[Tan[c + d*x]])))/(3*a)))/(5*a)))/(7*a)))/(9*a))/4)","A",1
620,1,346,351,4.9532148,"\int \cot ^{\frac{9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","-\frac{\cot ^{\frac{7}{2}}(c+d x) \left(5 a^3 (6 a A-7 b B) \sqrt{a+b \tan (c+d x)}+35 a^3 b B \sqrt{a+b \tan (c+d x)}+a \tan (c+d x) \left(-2 a \left(35 a^2 A-42 a b B-3 A b^2\right) \tan (c+d x) \sqrt{a+b \tan (c+d x)}+105 (-1)^{3/4} a^2 \tan ^{\frac{5}{2}}(c+d x) \left((-a+i b)^{3/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+(a+i b)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)+6 a^2 (7 a B+8 A b) \sqrt{a+b \tan (c+d x)}-2 \left(105 a^3 B+140 a^2 A b-21 a b^2 B+6 A b^3\right) \tan ^2(c+d x) \sqrt{a+b \tan (c+d x)}\right)\right)}{105 a^3 d}","\frac{2 \left(35 a^2 A-42 a b B-3 A b^2\right) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a d}+\frac{2 \left(105 a^3 B+140 a^2 A b-21 a b^2 B+6 A b^3\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{105 a^2 d}-\frac{2 (7 a B+8 A b) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{35 d}-\frac{(-b+i a)^{3/2} (A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{(b+i a)^{3/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d}",1,"-1/105*(Cot[c + d*x]^(7/2)*(35*a^3*b*B*Sqrt[a + b*Tan[c + d*x]] + 5*a^3*(6*a*A - 7*b*B)*Sqrt[a + b*Tan[c + d*x]] + a*Tan[c + d*x]*(105*(-1)^(3/4)*a^2*((-a + I*b)^(3/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (a + I*b)^(3/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])*Tan[c + d*x]^(5/2) + 6*a^2*(8*A*b + 7*a*B)*Sqrt[a + b*Tan[c + d*x]] - 2*a*(35*a^2*A - 3*A*b^2 - 42*a*b*B)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]] - 2*(140*a^2*A*b + 6*A*b^3 + 105*a^3*B - 21*a*b^2*B)*Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])))/(a^3*d)","A",1
621,1,286,299,2.9365678,"\int \cot ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","-\frac{\cot ^{\frac{5}{2}}(c+d x) \left(-4 \left(15 a^2 A-20 a b B-3 A b^2\right) \tan ^2(c+d x) \sqrt{a+b \tan (c+d x)}+30 \sqrt[4]{-1} a \tan ^{\frac{5}{2}}(c+d x) \left((-a+i b)^{3/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-(a+i b)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)+4 a (5 a B+6 A b) \tan (c+d x) \sqrt{a+b \tan (c+d x)}+3 a (4 a A-5 b B) \sqrt{a+b \tan (c+d x)}+15 a b B \sqrt{a+b \tan (c+d x)}\right)}{30 a d}","\frac{2 \left(15 a^2 A-20 a b B-3 A b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{15 a d}-\frac{2 (5 a B+6 A b) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{15 d}+\frac{(a+i b)^2 (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{(b+i a)^{3/2} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}",1,"-1/30*(Cot[c + d*x]^(5/2)*(30*(-1)^(1/4)*a*((-a + I*b)^(3/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - (a + I*b)^(3/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])*Tan[c + d*x]^(5/2) + 15*a*b*B*Sqrt[a + b*Tan[c + d*x]] + 3*a*(4*a*A - 5*b*B)*Sqrt[a + b*Tan[c + d*x]] + 4*a*(6*A*b + 5*a*B)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]] - 4*(15*a^2*A - 3*A*b^2 - 20*a*b*B)*Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]]))/(a*d)","A",1
622,1,244,236,1.0260636,"\int \cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{\sqrt{\cot (c+d x)} \left(3 \sqrt[4]{-1} \sqrt{\tan (c+d x)} \left((-a+i b)^{3/2} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+i (a+i b)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)-2 (3 a B+4 A b) \sqrt{a+b \tan (c+d x)}+(3 b B-2 a A) \cot (c+d x) \sqrt{a+b \tan (c+d x)}-3 b B \cot (c+d x) \sqrt{a+b \tan (c+d x)}\right)}{3 d}","\frac{(-b+i a)^{3/2} (A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 (3 a B+4 A b) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{(b+i a)^{3/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}",1,"(Sqrt[Cot[c + d*x]]*(3*(-1)^(1/4)*((-a + I*b)^(3/2)*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + I*(a + I*b)^(3/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])*Sqrt[Tan[c + d*x]] - 2*(4*A*b + 3*a*B)*Sqrt[a + b*Tan[c + d*x]] - 3*b*B*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]] + (-2*a*A + 3*b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]]))/(3*d)","A",1
623,1,114092,269,31.9980581,"\int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\text{Result too large to show}","-\frac{(a+i b)^2 (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{(b+i a)^{3/2} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}+\frac{2 b^{3/2} B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"Result too large to show","C",0
624,1,263,264,1.3672589,"\int \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\sqrt[4]{-1} (-a+i b)^{3/2} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-(-1)^{3/4} (a+i b)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\frac{\sqrt{a} \sqrt{b} (3 a B+2 A b) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a+b \tan (c+d x)}}+b B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}\right)}{d}","-\frac{(-b+i a)^{3/2} (A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{\sqrt{b} (3 a B+2 A b) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{(b+i a)^{3/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b B \sqrt{a+b \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-((-1)^(1/4)*(-a + I*b)^(3/2)*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]) - (-1)^(3/4)*(a + I*b)^(3/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + b*B*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + (Sqrt[a]*Sqrt[b]*(2*A*b + 3*a*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]]))/d","A",1
625,1,310,328,3.1991636,"\int \frac{(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","Integrate[((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{\sqrt{a} \left(3 a^2 B+12 a A b-8 b^2 B\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}-4 \sqrt[4]{-1} (-a+i b)^{3/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+4 \sqrt[4]{-1} (a+i b)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+(5 a B+4 A b) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}+2 b B \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}\right)}{4 d}","\frac{\left(3 a^2 B+12 a A b-8 b^2 B\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{4 \sqrt{b} d}+\frac{(a+i b)^2 (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{(5 a B+4 A b) \sqrt{a+b \tan (c+d x)}}{4 d \sqrt{\cot (c+d x)}}+\frac{(b+i a)^{3/2} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b B \sqrt{a+b \tan (c+d x)}}{2 d \cot ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-4*(-1)^(1/4)*(-a + I*b)^(3/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 4*(-1)^(1/4)*(a + I*b)^(3/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (4*A*b + 5*a*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + 2*b*B*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]] + (Sqrt[a]*(12*a*A*b + 3*a^2*B - 8*b^2*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]])))/(4*d)","A",1
626,1,367,383,5.6748244,"\int \frac{(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(3/2),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-3 \left(a^2 B-6 a A b+8 b^2 B\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}-\frac{3 \sqrt{a} \left(a^3 B-6 a^2 A b+24 a b^2 B+16 A b^3\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+24 \sqrt[4]{-1} b (-a+i b)^{3/2} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+24 (-1)^{3/4} b (a+i b)^{3/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+2 (6 A b-a B) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}+8 B \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{5/2}\right)}{24 b d}","\frac{\left(a^2 (-B)+6 a A b-8 b^2 B\right) \sqrt{a+b \tan (c+d x)}}{8 b d \sqrt{\cot (c+d x)}}+\frac{\left(a^3 (-B)+6 a^2 A b-24 a b^2 B-16 A b^3\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{8 b^{3/2} d}+\frac{(-b+i a)^{3/2} (A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(6 A b-a B) (a+b \tan (c+d x))^{3/2}}{12 b d \sqrt{\cot (c+d x)}}+\frac{(b+i a)^{3/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{B (a+b \tan (c+d x))^{5/2}}{3 b d \sqrt{\cot (c+d x)}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(24*(-1)^(1/4)*(-a + I*b)^(3/2)*b*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 24*(-1)^(3/4)*(a + I*b)^(3/2)*b*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 3*(-6*a*A*b + a^2*B + 8*b^2*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + 2*(6*A*b - a*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2) + 8*B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2) - (3*Sqrt[a]*(-6*a^2*A*b + 16*A*b^3 + a^3*B + 24*a*b^2*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]])))/(24*b*d)","A",1
627,1,653,500,6.9705575,"\int \cot ^{\frac{13}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(13/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{b B (a+b \tan (c+d x))^{3/2}}{4 d \tan ^{\frac{11}{2}}(c+d x)}+\frac{1}{4} \left(-\frac{b (5 a B+8 A b) \sqrt{a+b \tan (c+d x)}}{10 d \tan ^{\frac{11}{2}}(c+d x)}+\frac{1}{5} \left(-\frac{\left(80 a^2 A-165 a b B-88 A b^2\right) \sqrt{a+b \tan (c+d x)}}{22 d \tan ^{\frac{11}{2}}(c+d x)}-\frac{2 \left(\frac{5 a \left(88 a^2 B+184 a A b-99 b^2 B\right) \sqrt{a+b \tan (c+d x)}}{18 d \tan ^{\frac{9}{2}}(c+d x)}-\frac{2 \left(\frac{10 a^2 \left(99 a^2 A-209 a b B-113 A b^2\right) \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 \left(-\frac{3 a^2 \left(231 a^3 B+495 a^2 A b-275 a b^2 B-5 A b^3\right) \sqrt{a+b \tan (c+d x)}}{d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 \left(-\frac{5 a^2 \left(1155 a^4 A-2541 a^3 b B-1485 a^2 A b^2+55 a b^3 B-20 A b^4\right) \sqrt{a+b \tan (c+d x)}}{2 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(\frac{15 a^2 \left(3465 a^5 B+8085 a^4 A b-5313 a^3 b^2 B-495 a^2 A b^3-110 a b^4 B+40 A b^5\right) \sqrt{a+b \tan (c+d x)}}{4 d \sqrt{\tan (c+d x)}}+\frac{51975 a^5 \left((-1)^{3/4} (-a+i b)^{5/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-(-1)^{3/4} (a+i b)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{8 d}\right)}{3 a}\right)}{5 a}\right)}{7 a}\right)}{9 a}\right)}{11 a}\right)\right)\right)","\frac{2 \left(99 a^2 A-209 a b B-113 A b^2\right) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{693 d}+\frac{2 \left(231 a^3 B+495 a^2 A b-275 a b^2 B-5 A b^3\right) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{1155 a d}-\frac{2 \left(1155 a^4 A-2541 a^3 b B-1485 a^2 A b^2+55 a b^3 B-20 A b^4\right) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3465 a^2 d}-\frac{2 \left(3465 a^5 B+8085 a^4 A b-5313 a^3 b^2 B-495 a^2 A b^3-110 a b^4 B+40 A b^5\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3465 a^3 d}-\frac{2 a (11 a B+14 A b) \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{99 d}-\frac{(-b+i a)^{5/2} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{(b+i a)^{5/2} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \cot ^{\frac{11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{11 d}",1,"Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-1/4*(b*B*(a + b*Tan[c + d*x])^(3/2))/(d*Tan[c + d*x]^(11/2)) + (-1/10*(b*(8*A*b + 5*a*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(11/2)) + (-1/22*((80*a^2*A - 88*A*b^2 - 165*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(11/2)) - (2*((5*a*(184*a*A*b + 88*a^2*B - 99*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(18*d*Tan[c + d*x]^(9/2)) - (2*((10*a^2*(99*a^2*A - 113*A*b^2 - 209*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (2*((-3*a^2*(495*a^2*A*b - 5*A*b^3 + 231*a^3*B - 275*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(5/2)) - (2*((-5*a^2*(1155*a^4*A - 1485*a^2*A*b^2 - 20*A*b^4 - 2541*a^3*b*B + 55*a*b^3*B)*Sqrt[a + b*Tan[c + d*x]])/(2*d*Tan[c + d*x]^(3/2)) - (2*((51975*a^5*((-1)^(3/4)*(-a + I*b)^(5/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - (-1)^(3/4)*(a + I*b)^(5/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]))/(8*d) + (15*a^2*(8085*a^4*A*b - 495*a^2*A*b^3 + 40*A*b^5 + 3465*a^5*B - 5313*a^3*b^2*B - 110*a*b^4*B)*Sqrt[a + b*Tan[c + d*x]])/(4*d*Sqrt[Tan[c + d*x]])))/(3*a)))/(5*a)))/(7*a)))/(9*a)))/(11*a))/5)/4)","A",1
628,1,564,418,6.8236042,"\int \cot ^{\frac{11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(11/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{b B (a+b \tan (c+d x))^{3/2}}{3 d \tan ^{\frac{9}{2}}(c+d x)}+\frac{1}{3} \left(-\frac{3 b (a B+2 A b) \sqrt{a+b \tan (c+d x)}}{8 d \tan ^{\frac{9}{2}}(c+d x)}+\frac{1}{4} \left(-\frac{\left(16 a^2 A-33 a b B-18 A b^2\right) \sqrt{a+b \tan (c+d x)}}{6 d \tan ^{\frac{9}{2}}(c+d x)}-\frac{2 \left(\frac{6 a \left(18 a^2 B+38 a A b-21 b^2 B\right) \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 \left(\frac{18 a^2 \left(21 a^2 A-45 a b B-25 A b^2\right) \sqrt{a+b \tan (c+d x)}}{5 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 \left(-\frac{3 a^2 \left(105 a^3 B+231 a^2 A b-135 a b^2 B-5 A b^3\right) \sqrt{a+b \tan (c+d x)}}{d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(-\frac{9 a^2 \left(315 a^4 A-735 a^3 b B-483 a^2 A b^2+45 a b^3 B-10 A b^4\right) \sqrt{a+b \tan (c+d x)}}{2 d \sqrt{\tan (c+d x)}}-\frac{2835 a^4 \left(\sqrt[4]{-1} (-a+i b)^{5/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt[4]{-1} (a+i b)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{4 d}\right)}{3 a}\right)}{5 a}\right)}{7 a}\right)}{9 a}\right)\right)\right)","\frac{2 \left(21 a^2 A-45 a b B-25 A b^2\right) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 d}+\frac{2 \left(105 a^3 B+231 a^2 A b-135 a b^2 B-5 A b^3\right) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a d}-\frac{2 \left(315 a^4 A-735 a^3 b B-483 a^2 A b^2+45 a b^3 B-10 A b^4\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^2 d}-\frac{2 a (3 a B+4 A b) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{21 d}+\frac{(-b+i a)^{5/2} (A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{(b+i a)^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{9 d}",1,"Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-1/3*(b*B*(a + b*Tan[c + d*x])^(3/2))/(d*Tan[c + d*x]^(9/2)) + ((-3*b*(2*A*b + a*B)*Sqrt[a + b*Tan[c + d*x]])/(8*d*Tan[c + d*x]^(9/2)) + (-1/6*((16*a^2*A - 18*A*b^2 - 33*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(9/2)) - (2*((6*a*(38*a*A*b + 18*a^2*B - 21*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (2*((18*a^2*(21*a^2*A - 25*A*b^2 - 45*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - (2*((-3*a^2*(231*a^2*A*b - 5*A*b^3 + 105*a^3*B - 135*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^(3/2)) - (2*((-2835*a^4*((-1)^(1/4)*(-a + I*b)^(5/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (-1)^(1/4)*(a + I*b)^(5/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]))/(4*d) - (9*a^2*(315*a^4*A - 483*a^2*A*b^2 - 10*A*b^4 - 735*a^3*b*B + 45*a*b^3*B)*Sqrt[a + b*Tan[c + d*x]])/(2*d*Sqrt[Tan[c + d*x]])))/(3*a)))/(5*a)))/(7*a)))/(9*a))/4)/3)","A",1
629,1,382,349,5.1557987,"\int \cot ^{\frac{9}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","-\frac{\cot ^{\frac{7}{2}}(c+d x) \left(6 a \left(28 a^2 B+60 a A b-35 b^2 B\right) \tan (c+d x) \sqrt{a+b \tan (c+d x)}+5 a \left(24 a^2 A-49 a b B-28 A b^2\right) \sqrt{a+b \tan (c+d x)}-4 \tan ^2(c+d x) \left(2 a \left(35 a^2 A-77 a b B-45 A b^2\right) \sqrt{a+b \tan (c+d x)}+2 \left(105 a^3 B+245 a^2 A b-161 a b^2 B-15 A b^3\right) \tan (c+d x) \sqrt{a+b \tan (c+d x)}+105 \sqrt[4]{-1} a \tan ^{\frac{3}{2}}(c+d x) \left((-a+i b)^{5/2} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+(a+i b)^{5/2} (B-i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)\right)+35 a b (a B+4 A b) \sqrt{a+b \tan (c+d x)}+210 a b B (a+b \tan (c+d x))^{3/2}\right)}{420 a d}","\frac{2 \left(35 a^2 A-77 a b B-45 A b^2\right) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 d}+\frac{2 \left(105 a^3 B+245 a^2 A b-161 a b^2 B-15 A b^3\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{105 a d}-\frac{2 a (7 a B+10 A b) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{35 d}+\frac{(-b+i a)^{5/2} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(b+i a)^{5/2} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \cot ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{7 d}",1,"-1/420*(Cot[c + d*x]^(7/2)*(35*a*b*(4*A*b + a*B)*Sqrt[a + b*Tan[c + d*x]] + 5*a*(24*a^2*A - 28*A*b^2 - 49*a*b*B)*Sqrt[a + b*Tan[c + d*x]] + 6*a*(60*a*A*b + 28*a^2*B - 35*b^2*B)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]] + 210*a*b*B*(a + b*Tan[c + d*x])^(3/2) - 4*Tan[c + d*x]^2*(105*(-1)^(1/4)*a*((-a + I*b)^(5/2)*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (a + I*b)^(5/2)*((-I)*A + B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])*Tan[c + d*x]^(3/2) + 2*a*(35*a^2*A - 45*A*b^2 - 77*a*b*B)*Sqrt[a + b*Tan[c + d*x]] + 2*(245*a^2*A*b - 15*A*b^3 + 105*a^3*B - 161*a*b^2*B)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]])))/(a*d)","A",1
630,1,321,287,2.7623007,"\int \cot ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{\cot ^{\frac{5}{2}}(c+d x) \left(8 \left(15 a^2 A-35 a b B-23 A b^2\right) \tan ^2(c+d x) \sqrt{a+b \tan (c+d x)}-4 \left(10 a^2 B+22 a A b-15 b^2 B\right) \tan (c+d x) \sqrt{a+b \tan (c+d x)}-3 \left(8 a^2 A-15 a b B-10 A b^2\right) \sqrt{a+b \tan (c+d x)}+60 \sqrt[4]{-1} \tan ^{\frac{5}{2}}(c+d x) \left((-a+i b)^{5/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+(a+i b)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)+15 b (a B-2 A b) \sqrt{a+b \tan (c+d x)}-60 b B (a+b \tan (c+d x))^{3/2}\right)}{60 d}","\frac{2 \left(15 a^2 A-35 a b B-23 A b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{15 d}-\frac{2 a (5 a B+8 A b) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{15 d}-\frac{(-b+i a)^{5/2} (A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(b+i a)^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{5 d}",1,"(Cot[c + d*x]^(5/2)*(60*(-1)^(1/4)*((-a + I*b)^(5/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (a + I*b)^(5/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])*Tan[c + d*x]^(5/2) + 15*b*(-2*A*b + a*B)*Sqrt[a + b*Tan[c + d*x]] - 3*(8*a^2*A - 10*A*b^2 - 15*a*b*B)*Sqrt[a + b*Tan[c + d*x]] - 4*(22*a*A*b + 10*a^2*B - 15*b^2*B)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]] + 8*(15*a^2*A - 23*A*b^2 - 35*a*b*B)*Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]] - 60*b*B*(a + b*Tan[c + d*x])^(3/2)))/(60*d)","A",1
631,1,130606,300,40.1317973,"\int \cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\text{Result too large to show}","-\frac{(-b+i a)^{5/2} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a (a B+2 A b) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}-\frac{(b+i a)^{5/2} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{3 d}+\frac{2 b^{5/2} B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"Result too large to show","C",0
632,1,196709,301,40.9094995,"\int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\text{Result too large to show}","\frac{b^{3/2} (5 a B+2 A b) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(-b+i a)^{5/2} (A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b (2 a A+b B) \sqrt{a+b \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}-\frac{(b+i a)^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a A \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}{d}",1,"Result too large to show","C",0
633,1,311,320,3.4545732,"\int \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{\sqrt{a} \sqrt{b} \left(15 a^2 B+20 a A b-8 b^2 B\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a+b \tan (c+d x)}}+4 \sqrt[4]{-1} (-a+i b)^{5/2} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-4 (-1)^{3/4} (a+i b)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+b (7 a B+4 A b) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}+2 b B \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}\right)}{4 d}","\frac{\sqrt{b} \left(15 a^2 B+20 a A b-8 b^2 B\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{4 d}+\frac{(-b+i a)^{5/2} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b (7 a B+4 A b) \sqrt{a+b \tan (c+d x)}}{4 d \sqrt{\cot (c+d x)}}+\frac{(b+i a)^{5/2} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b B (a+b \tan (c+d x))^{3/2}}{2 d \sqrt{\cot (c+d x)}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(4*(-1)^(1/4)*(-a + I*b)^(5/2)*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 4*(-1)^(3/4)*(a + I*b)^(5/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + b*(4*A*b + 7*a*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + 2*b*B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2) + (Sqrt[a]*Sqrt[b]*(20*a*A*b + 15*a^2*B - 8*b^2*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]]))/(4*d)","A",1
634,1,365,376,5.9739293,"\int \frac{(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","Integrate[((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(3 \left(5 a^2 B+14 a A b-8 b^2 B\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}+\frac{3 \sqrt{a} \left(5 a^3 B+30 a^2 A b-40 a b^2 B-16 A b^3\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+24 \sqrt[4]{-1} (-a+i b)^{5/2} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+24 \sqrt[4]{-1} (a+i b)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+6 (3 a B+2 A b) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}+8 b B \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}\right)}{24 d}","\frac{\left(5 a^2 B+14 a A b-8 b^2 B\right) \sqrt{a+b \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{\left(5 a^3 B+30 a^2 A b-40 a b^2 B-16 A b^3\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{8 \sqrt{b} d}-\frac{(-b+i a)^{5/2} (A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(3 a B+2 A b) (a+b \tan (c+d x))^{3/2}}{4 d \sqrt{\cot (c+d x)}}+\frac{(b+i a)^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b B (a+b \tan (c+d x))^{3/2}}{3 d \cot ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(24*(-1)^(1/4)*(-a + I*b)^(5/2)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 24*(-1)^(1/4)*(a + I*b)^(5/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 3*(14*a*A*b + 5*a^2*B - 8*b^2*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + 6*(2*A*b + 3*a*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2) + 8*b*B*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2) + (3*Sqrt[a]*(30*a^2*A*b - 16*A*b^3 + 5*a^3*B - 40*a*b^2*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]])))/(24*d)","A",1
635,1,431,457,5.583482,"\int \frac{(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(3/2),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-2 \left(5 a^2 B-40 a A b+48 b^2 B\right) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}-3 \left(5 a^3 B-40 a^2 A b+112 a b^2 B+64 A b^3\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}-\frac{3 \sqrt{a} \left(5 a^4 B-40 a^3 A b+240 a^2 b^2 B+320 a A b^3-128 b^4 B\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}-192 \sqrt[4]{-1} b (-a+i b)^{5/2} (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+192 (-1)^{3/4} b (a+i b)^{5/2} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+8 (8 A b-a B) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{5/2}+48 B \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{7/2}\right)}{192 b d}","\frac{\left(-5 a^2 B+40 a A b-48 b^2 B\right) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt{\cot (c+d x)}}+\frac{\left(-5 a^3 B+40 a^2 A b-112 a b^2 B-64 A b^3\right) \sqrt{a+b \tan (c+d x)}}{64 b d \sqrt{\cot (c+d x)}}+\frac{\left(-5 a^4 B+40 a^3 A b-240 a^2 b^2 B-320 a A b^3+128 b^4 B\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{64 b^{3/2} d}-\frac{(-b+i a)^{5/2} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt{\cot (c+d x)}}-\frac{(b+i a)^{5/2} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt{\cot (c+d x)}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-192*(-1)^(1/4)*(-a + I*b)^(5/2)*b*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 192*(-1)^(3/4)*(a + I*b)^(5/2)*b*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 3*(-40*a^2*A*b + 64*A*b^3 + 5*a^3*B + 112*a*b^2*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] - 2*(-40*a*A*b + 5*a^2*B + 48*b^2*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2) + 8*(8*A*b - a*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2) + 48*B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(7/2) - (3*Sqrt[a]*(-40*a^3*A*b + 320*a*A*b^3 + 5*a^4*B + 240*a^2*b^2*B - 128*b^4*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]])))/(192*b*d)","A",1
636,1,244,296,5.9889007,"\int \frac{\cot ^{\frac{7}{2}}(c+d x) (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(Cot[c + d*x]^(7/2)*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\sqrt{\cot (c+d x)} \left(-\frac{2 \sqrt{a+b \tan (c+d x)} \left(3 a^2 A \cot ^2(c+d x)-15 a^2 A+a (5 a B-4 A b) \cot (c+d x)-10 a b B+8 A b^2\right)}{a^3}-\frac{15 \sqrt[4]{-1} (A-i B) \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{15 \sqrt[4]{-1} (A+i B) \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{15 d}","\frac{2 (4 A b-5 a B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{15 a^2 d}+\frac{2 \left(15 a^2 A+10 a b B-8 A b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{15 a^3 d}+\frac{(-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{(B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}-\frac{2 A \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 a d}",1,"(Sqrt[Cot[c + d*x]]*((-15*(-1)^(1/4)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/Sqrt[-a + I*b] + (15*(-1)^(1/4)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*b] - (2*(-15*a^2*A + 8*A*b^2 - 10*a*b*B + a*(-4*A*b + 5*a*B)*Cot[c + d*x] + 3*a^2*A*Cot[c + d*x]^2)*Sqrt[a + b*Tan[c + d*x]])/a^3))/(15*d)","A",1
637,1,213,243,2.6265799,"\int \frac{\cot ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(Cot[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\sqrt{\cot (c+d x)} \left(-\frac{2 \sqrt{a+b \tan (c+d x)} (a A \cot (c+d x)+3 a B-2 A b)}{a^2}+\frac{3 \sqrt[4]{-1} (B+i A) \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{3 (-1)^{3/4} (A+i B) \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{3 d}","\frac{2 (2 A b-3 a B) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 a^2 d}-\frac{(A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{(A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 a d}",1,"(Sqrt[Cot[c + d*x]]*((3*(-1)^(1/4)*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/Sqrt[-a + I*b] + (3*(-1)^(3/4)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*b] - (2*(-2*A*b + 3*a*B + a*A*Cot[c + d*x])*Sqrt[a + b*Tan[c + d*x]])/a^2))/(3*d)","A",1
638,1,193,199,1.5425727,"\int \frac{\cot ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]","-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{\sqrt[4]{-1} (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{\sqrt[4]{-1} (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\frac{2 A \sqrt{a+b \tan (c+d x)}}{a \sqrt{\tan (c+d x)}}\right)}{d}","-\frac{(-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{(B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}-\frac{2 A \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{a d}",1,"-((Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-(((-1)^(1/4)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b]) + ((-1)^(1/4)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] + (2*A*Sqrt[a + b*Tan[c + d*x]])/(a*Sqrt[Tan[c + d*x]])))/d)","A",1
639,1,157,163,0.4114655,"\int \frac{\sqrt{\cot (c+d x)} (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\sqrt[4]{-1} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{(B-i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}-\frac{(B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)}{d}","\frac{(A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{(A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"((-1)^(1/4)*(-(((I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b]) + (((-I)*A + B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b])*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d","A",1
640,1,225,228,1.4892207,"\int \frac{A+B \tan (c+d x)}{\sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{2 \sqrt{a} B \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+\sqrt[4]{-1} \left(\frac{(A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}-\frac{(A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)\right)}{d}","\frac{(-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{(B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}+\frac{2 B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{b} d}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-1)^(1/4)*(-(((A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b]) + ((A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]) + (2*Sqrt[a]*B*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]])))/d","A",1
641,1,354,266,2.5309508,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\sqrt{b} \left(\sqrt[4]{-1} b \sqrt{a+i b} (B+i A) \sqrt{a+b \tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt{-a+i b} \left((-1)^{3/4} b (A+i B) \sqrt{a+b \tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+B \sqrt{a+i b} \sqrt{\tan (c+d x)} (a+b \tan (c+d x))\right)\right)-\sqrt{a} \sqrt{-a+i b} \sqrt{a+i b} (a B-2 A b) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)\right)}{b^{3/2} d \sqrt{-a+i b} \sqrt{a+i b} \sqrt{a+b \tan (c+d x)}}","\frac{(2 A b-a B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{b^{3/2} d}-\frac{(A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{(A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}+\frac{B \sqrt{a+b \tan (c+d x)}}{b d \sqrt{\cot (c+d x)}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-(Sqrt[a]*Sqrt[-a + I*b]*Sqrt[a + I*b]*(-2*A*b + a*B)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a]) + Sqrt[b]*((-1)^(1/4)*Sqrt[a + I*b]*b*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + Sqrt[-a + I*b]*((-1)^(3/4)*b*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + Sqrt[a + I*b]*B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])))))/(Sqrt[-a + I*b]*Sqrt[a + I*b]*b^(3/2)*d*Sqrt[a + b*Tan[c + d*x]])","A",1
642,1,301,316,4.1427508,"\int \frac{\cot ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Cot[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","\frac{\sqrt{\cot (c+d x)} \left(\frac{3 \sqrt[4]{-1} a \sqrt{\tan (c+d x)} \left(\frac{(a+i b) (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{(b+i a) (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{a^2+b^2}+\frac{2 b \left(-3 a^3 B+5 a^2 A b-6 a b^2 B+8 A b^3\right) \tan (c+d x)}{a^2 \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{8 A b-6 a B}{a \sqrt{a+b \tan (c+d x)}}-\frac{2 A \cot (c+d x)}{\sqrt{a+b \tan (c+d x)}}\right)}{3 a d}","\frac{2 (4 A b-3 a B) \sqrt{\cot (c+d x)}}{3 a^2 d \sqrt{a+b \tan (c+d x)}}+\frac{2 b \left(-3 a^3 B+5 a^2 A b-6 a b^2 B+8 A b^3\right)}{3 a^3 d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{(-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}-\frac{(B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d \sqrt{a+b \tan (c+d x)}}",1,"(Sqrt[Cot[c + d*x]]*((3*(-1)^(1/4)*a*(((a + I*b)*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + ((I*a + b)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b])*Sqrt[Tan[c + d*x]])/(a^2 + b^2) + (8*A*b - 6*a*B)/(a*Sqrt[a + b*Tan[c + d*x]]) - (2*A*Cot[c + d*x])/Sqrt[a + b*Tan[c + d*x]] + (2*b*(5*a^2*A*b + 8*A*b^3 - 3*a^3*B - 6*a*b^2*B)*Tan[c + d*x])/(a^2*(a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]])))/(3*a*d)","A",1
643,1,255,256,2.2255484,"\int \frac{\cot ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","\frac{\sqrt{\cot (c+d x)} \left(\frac{\sqrt[4]{-1} a \sqrt{\tan (c+d x)} \left(\frac{(a+i b) (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{(a-i b) (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{a^2+b^2}-\frac{2 b \left(a^2 A-a b B+2 A b^2\right) \tan (c+d x)}{a \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{2 A}{\sqrt{a+b \tan (c+d x)}}\right)}{a d}","-\frac{2 b \left(a^2 A-a b B+2 A b^2\right)}{a^2 d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{(A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}-\frac{(A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}-\frac{2 A \sqrt{\cot (c+d x)}}{a d \sqrt{a+b \tan (c+d x)}}",1,"(Sqrt[Cot[c + d*x]]*(((-1)^(1/4)*a*(((a + I*b)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] - ((a - I*b)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b])*Sqrt[Tan[c + d*x]])/(a^2 + b^2) - (2*A)/Sqrt[a + b*Tan[c + d*x]] - (2*b*(a^2*A + 2*A*b^2 - a*b*B)*Tan[c + d*x])/(a*(a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]])))/(a*d)","A",1
644,1,222,215,1.210786,"\int \frac{\sqrt{\cot (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{\sqrt[4]{-1} (b-i a) (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{\sqrt[4]{-1} (a-i b) (B-i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\frac{2 b (A b-a B) \sqrt{\tan (c+d x)}}{a \sqrt{a+b \tan (c+d x)}}\right)}{d \left(a^2+b^2\right)}","\frac{2 b (A b-a B)}{a d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{(-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}+\frac{(B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((-1)^(1/4)*((-I)*a + b)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + ((-1)^(1/4)*(a - I*b)*((-I)*A + B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] + (2*b*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(a*Sqrt[a + b*Tan[c + d*x]])))/((a^2 + b^2)*d)","A",1
645,1,259,210,2.1105671,"\int \frac{A+B \tan (c+d x)}{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{\sqrt[4]{-1} a (a+i b) (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{\sqrt[4]{-1} a (a-i b) (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\frac{2 b (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \tan (c+d x)}}+2 (a B-A b) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}\right)}{a d \left(a^2+b^2\right)}","-\frac{2 (A b-a B)}{d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{(A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}+\frac{(A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-(((-1)^(1/4)*a*(a + I*b)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b]) + ((-1)^(1/4)*a*(a - I*b)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] + (2*b*(A*b - a*B)*Tan[c + d*x]^(3/2))/Sqrt[a + b*Tan[c + d*x]] + 2*(-(A*b) + a*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]))/(a*(a^2 + b^2)*d)","A",1
646,1,167374,279,40.1972447,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)),x]","\text{Result too large to show}","\frac{2 a (A b-a B)}{b d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{(-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}-\frac{(B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}+\frac{2 B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{b^{3/2} d}",1,"Result too large to show","C",0
647,1,385,399,3.9729339,"\int \frac{\cot ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(Cot[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]","\frac{\sqrt{\cot (c+d x)} \left(\frac{6 b \left(-3 a^3 B+7 a^2 A b-4 a b^2 B+8 A b^3\right) \tan (c+d x)}{a^2 \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{\sqrt{\tan (c+d x)} \left(\frac{6 b \left(-3 a^5 B+8 a^4 A b-17 a^3 b^2 B+30 a^2 A b^3-8 a b^4 B+16 A b^5\right) \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}+9 (-1)^{3/4} a^4 \left(\frac{(a-i b)^2 (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\frac{(a+i b)^2 (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)\right)}{a^3 \left(a^2+b^2\right)^2}+\frac{6 (6 A b-3 a B)}{a (a+b \tan (c+d x))^{3/2}}-\frac{6 A \cot (c+d x)}{(a+b \tan (c+d x))^{3/2}}\right)}{9 a d}","\frac{2 (2 A b-a B) \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left(-3 a^3 B+7 a^2 A b-4 a b^2 B+8 A b^3\right)}{3 a^3 d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left(-3 a^5 B+8 a^4 A b-17 a^3 b^2 B+30 a^2 A b^3-8 a b^4 B+16 A b^5\right)}{3 a^4 d \left(a^2+b^2\right)^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{(A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}+\frac{(A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}",1,"(Sqrt[Cot[c + d*x]]*((6*(6*A*b - 3*a*B))/(a*(a + b*Tan[c + d*x])^(3/2)) - (6*A*Cot[c + d*x])/(a + b*Tan[c + d*x])^(3/2) + (6*b*(7*a^2*A*b + 8*A*b^3 - 3*a^3*B - 4*a*b^2*B)*Tan[c + d*x])/(a^2*(a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) + (Sqrt[Tan[c + d*x]]*(9*(-1)^(3/4)*a^4*(((a + I*b)^2*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + ((a - I*b)^2*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]) + (6*b*(8*a^4*A*b + 30*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B - 17*a^3*b^2*B - 8*a*b^4*B)*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]))/(a^3*(a^2 + b^2)^2)))/(9*a*d)","A",1
648,1,334,341,4.040364,"\int \frac{\cot ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]","\frac{\sqrt{\cot (c+d x)} \left(-\frac{2 b \left(3 a^2 A-a b B+4 A b^2\right) \tan (c+d x)}{a \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{\sqrt{\tan (c+d x)} \left(-\frac{2 b \left(3 a^4 A-8 a^3 b B+17 a^2 A b^2-2 a b^3 B+8 A b^4\right) \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}+3 \sqrt[4]{-1} a^3 \left(\frac{(a+i b)^2 (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{(a-i b)^2 (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)\right)}{a^2 \left(a^2+b^2\right)^2}-\frac{6 A}{(a+b \tan (c+d x))^{3/2}}\right)}{3 a d}","-\frac{2 b \left(3 a^2 A-a b B+4 A b^2\right)}{3 a^2 d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac{2 b \left(3 a^4 A-8 a^3 b B+17 a^2 A b^2-2 a b^3 B+8 A b^4\right)}{3 a^3 d \left(a^2+b^2\right)^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{(-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}-\frac{(B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}-\frac{2 A \sqrt{\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}",1,"(Sqrt[Cot[c + d*x]]*((-6*A)/(a + b*Tan[c + d*x])^(3/2) - (2*b*(3*a^2*A + 4*A*b^2 - a*b*B)*Tan[c + d*x])/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) + (Sqrt[Tan[c + d*x]]*(3*(-1)^(1/4)*a^3*(((a + I*b)^2*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] - ((a - I*b)^2*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]) - (2*b*(3*a^4*A + 17*a^2*A*b^2 + 8*A*b^4 - 8*a^3*b*B - 2*a*b^3*B)*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]))/(a^2*(a^2 + b^2)^2)))/(3*a*d)","A",1
649,1,293,287,1.836116,"\int \frac{\sqrt{\cot (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{2 b \left(a^2+b^2\right) (A b-a B) \sqrt{\tan (c+d x)}}{a (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left(-5 a^3 B+8 a^2 A b+a b^2 B+2 A b^3\right) \sqrt{\tan (c+d x)}}{a^2 \sqrt{a+b \tan (c+d x)}}-3 \sqrt[4]{-1} \left(\frac{i (a-i b)^2 (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\frac{(a+i b)^2 (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)\right)}{3 d \left(a^2+b^2\right)^2}","\frac{2 b (A b-a B)}{3 a d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left(-5 a^3 B+8 a^2 A b+a b^2 B+2 A b^3\right)}{3 a^2 d \left(a^2+b^2\right)^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{(A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}-\frac{(A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-3*(-1)^(1/4)*(((a + I*b)^2*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + (I*(a - I*b)^2*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]) + (2*b*(a^2 + b^2)*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(a*(a + b*Tan[c + d*x])^(3/2)) + (2*b*(8*a^2*A*b + 2*A*b^3 - 5*a^3*B + a*b^2*B)*Sqrt[Tan[c + d*x]])/(a^2*Sqrt[a + b*Tan[c + d*x]])))/(3*(a^2 + b^2)^2*d)","A",1
650,1,340,284,4.4009885,"\int \frac{A+B \tan (c+d x)}{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(5/2)),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{3 \left(2 \left(a^2 B-2 a A b-b^2 B\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}+\frac{\sqrt[4]{-1} a (a-i b)^2 (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}-\frac{\sqrt[4]{-1} a (a+i b)^2 (A-i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)}{a^2+b^2}+\frac{6 b \left(a^2 (-B)+2 a A b+b^2 B\right) \tan ^{\frac{3}{2}}(c+d x)}{\left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{2 b (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}}\right)}{3 a d \left(a^2+b^2\right)}","-\frac{2 (A b-a B)}{3 d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac{2 \left(-2 a^3 B+5 a^2 A b+4 a b^2 B-A b^3\right)}{3 a d \left(a^2+b^2\right)^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{(-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}+\frac{(B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((2*b*(A*b - a*B)*Tan[c + d*x]^(3/2))/(a + b*Tan[c + d*x])^(3/2) + (6*b*(2*a*A*b - a^2*B + b^2*B)*Tan[c + d*x]^(3/2))/((a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]]) + (3*(-(((-1)^(1/4)*a*(a + I*b)^2*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b]) + ((-1)^(1/4)*a*(a - I*b)^2*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] + 2*(-2*a*A*b + a^2*B - b^2*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]))/(a^2 + b^2)))/(3*a*(a^2 + b^2)*d)","A",1
651,1,328,284,3.7088385,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(5/2)),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{\left(a^2 B+2 a A b+3 b^2 B\right) \sqrt{\tan (c+d x)}}{\left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{\frac{2 \left(a^3 B+2 a^2 A b+7 a b^2 B-4 A b^3\right) \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}+3 \sqrt[4]{-1} b \left(\frac{i (a-i b)^2 (A+i B) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\frac{(a+i b)^2 (B+i A) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)}{\left(a^2+b^2\right)^2}-\frac{3 B \sqrt{\tan (c+d x)}}{(a+b \tan (c+d x))^{3/2}}\right)}{3 b d}","\frac{2 a (A b-a B)}{3 b d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{2 \left(a^3 B+2 a^2 A b+7 a b^2 B-4 A b^3\right)}{3 b d \left(a^2+b^2\right)^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{(A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}+\frac{(A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-3*B*Sqrt[Tan[c + d*x]])/(a + b*Tan[c + d*x])^(3/2) + ((2*a*A*b + a^2*B + 3*b^2*B)*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) + (3*(-1)^(1/4)*b*(((a + I*b)^2*(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + (I*(a - I*b)^2*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]) + (2*(2*a^2*A*b - 4*A*b^3 + a^3*B + 7*a*b^2*B)*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]])/(a^2 + b^2)^2))/(3*b*d)","A",1
652,1,250233,342,32.8332443,"\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(5/2)),x]","\text{Result too large to show}","\frac{2 a (A b-a B)}{3 b d \left(a^2+b^2\right) \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left(2 A b^3-a B \left(a^2+3 b^2\right)\right)}{b^2 d \left(a^2+b^2\right)^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{(-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}-\frac{(B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}+\frac{2 B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{b^{5/2} d}",1,"Result too large to show","C",0
653,1,145,151,0.250091,"\int \frac{\sqrt{\cot (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cot[c + d*x]]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]","\frac{(-1)^{3/4} B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{d}","\frac{B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"((-1)^(3/4)*B*(-(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[-a + I*b]) - ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[a + I*b])*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d","A",1
654,1,144,157,0.2154228,"\int \frac{a B+b B \tan (c+d x)}{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)),x]","\frac{\sqrt[4]{-1} B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)}{d}","\frac{i B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{i B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"((-1)^(1/4)*B*(-(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[-a + I*b]) + ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[a + I*b])*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d","A",1
655,1,210,215,1.073226,"\int \frac{a B+b B \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(a*B + b*B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)),x]","\frac{B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{2 \sqrt{a} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+(-1)^{3/4} \left(\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)\right)}{d}","-\frac{B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{2 B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{b} d}-\frac{B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"(B*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-1)^(3/4)*(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[-a + I*b] + ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[a + I*b]) + (2*Sqrt[a]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]])))/d","A",1
656,0,0,195,7.3764198,"\int \cot ^m(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^m*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \cot ^m(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","\frac{(A+i B) \cot ^{m-1}(c+d x) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(1-m;-n,1;2-m;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right)}{2 d (1-m)}+\frac{(A-i B) \cot ^{m-1}(c+d x) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(1-m;-n,1;2-m;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right)}{2 d (1-m)}",1,"Integrate[Cot[c + d*x]^m*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
657,0,0,169,8.1506621,"\int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","-\frac{(A+i B) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(-\frac{1}{2};1,-n;\frac{1}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d}-\frac{(A-i B) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(-\frac{1}{2};1,-n;\frac{1}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d}",1,"Integrate[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
658,0,0,167,6.9522075,"\int \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","\frac{(A+i B) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{1}{2};1,-n;\frac{3}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d \sqrt{\cot (c+d x)}}+\frac{(A-i B) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{1}{2};1,-n;\frac{3}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d \sqrt{\cot (c+d x)}}",1,"Integrate[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
659,0,0,173,13.3088917,"\int \frac{(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","Integrate[((a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]","\int \frac{(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx","\frac{(A+i B) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{3}{2};1,-n;\frac{5}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{(A-i B) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{3}{2};1,-n;\frac{5}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{3 d \cot ^{\frac{3}{2}}(c+d x)}",1,"Integrate[((a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]], x]","F",-1
660,0,0,173,14.6203035,"\int \frac{(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(3/2),x]","\int \frac{(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","\frac{(A+i B) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{5}{2};1,-n;\frac{7}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{5 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{(A-i B) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{5}{2};1,-n;\frac{7}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{5 d \cot ^{\frac{5}{2}}(c+d x)}",1,"Integrate[((a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(3/2), x]","F",-1
661,0,0,173,2.3889332,"\int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","\frac{(A+i B) \tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{5}{2};1,-n;\frac{7}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{5 d}+\frac{(A-i B) \tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{5}{2};1,-n;\frac{7}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{5 d}",1,"Integrate[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
662,0,0,173,1.3498651,"\int \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]","\int \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx","\frac{(A+i B) \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{3}{2};1,-n;\frac{5}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{3 d}+\frac{(A-i B) \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{3}{2};1,-n;\frac{5}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{3 d}",1,"Integrate[Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]","F",-1
663,0,0,167,1.2076336,"\int \frac{(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","Integrate[((a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]],x]","\int \frac{(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx","\frac{(A+i B) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{1}{2};1,-n;\frac{3}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d}+\frac{(A-i B) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{1}{2};1,-n;\frac{3}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d}",1,"Integrate[((a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]], x]","F",-1
664,0,0,169,2.1128544,"\int \frac{(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2),x]","\int \frac{(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","-\frac{(A+i B) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(-\frac{1}{2};1,-n;\frac{1}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d \sqrt{\tan (c+d x)}}-\frac{(A-i B) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(-\frac{1}{2};1,-n;\frac{1}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d \sqrt{\tan (c+d x)}}",1,"Integrate[((a + b*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2), x]","F",-1
665,1,75,63,4.0407692,"\int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx","Integrate[(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n,x]","\frac{i a (c \sec (e+f x))^n (A n+A+B n \tan (e+f x)-i B) \exp (n (-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x))))}{f n (n+1)}","\frac{a (B+i A) (c-i c \tan (e+f x))^n}{f n}-\frac{a B (c-i c \tan (e+f x))^{n+1}}{c f (n+1)}",1,"(I*a*E^(n*(-Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))*(c*Sec[e + f*x])^n*(A - I*B + A*n + B*n*Tan[e + f*x]))/(f*n*(1 + n))","A",1
666,1,226,59,3.5525545,"\int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx","Integrate[(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4,x]","\frac{a c^4 \sec (e) \sec ^5(e+f x) (5 (3 B-5 i A) \cos (2 e+f x)+5 (3 B-5 i A) \cos (f x)-25 A \sin (2 e+f x)+15 A \sin (2 e+3 f x)-10 A \sin (4 e+3 f x)+5 A \sin (4 e+5 f x)-10 i A \cos (2 e+3 f x)-10 i A \cos (4 e+3 f x)+25 A \sin (f x)-15 i B \sin (2 e+f x)+5 i B \sin (2 e+3 f x)-10 i B \sin (4 e+3 f x)+3 i B \sin (4 e+5 f x)+10 B \cos (2 e+3 f x)+10 B \cos (4 e+3 f x)+15 i B \sin (f x))}{40 f}","\frac{a c^4 (B+i A) (1-i \tan (e+f x))^4}{4 f}-\frac{a B c^4 (1-i \tan (e+f x))^5}{5 f}",1,"(a*c^4*Sec[e]*Sec[e + f*x]^5*(5*((-5*I)*A + 3*B)*Cos[f*x] + 5*((-5*I)*A + 3*B)*Cos[2*e + f*x] - (10*I)*A*Cos[2*e + 3*f*x] + 10*B*Cos[2*e + 3*f*x] - (10*I)*A*Cos[4*e + 3*f*x] + 10*B*Cos[4*e + 3*f*x] + 25*A*Sin[f*x] + (15*I)*B*Sin[f*x] - 25*A*Sin[2*e + f*x] - (15*I)*B*Sin[2*e + f*x] + 15*A*Sin[2*e + 3*f*x] + (5*I)*B*Sin[2*e + 3*f*x] - 10*A*Sin[4*e + 3*f*x] - (10*I)*B*Sin[4*e + 3*f*x] + 5*A*Sin[4*e + 5*f*x] + (3*I)*B*Sin[4*e + 5*f*x]))/(40*f)","B",1
667,1,161,59,3.6670233,"\int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) (c-i c \tan (e+f x))^3 \, dx","Integrate[(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3,x]","\frac{a c^3 \sec (e) \sec ^4(e+f x) (3 (B-i A) \cos (e+2 f x)+3 (B-2 i A) \cos (e)+5 A \sin (e+2 f x)-3 A \sin (3 e+2 f x)+2 A \sin (3 e+4 f x)-3 i A \cos (3 e+2 f x)-6 A \sin (e)+i B \sin (e+2 f x)-3 i B \sin (3 e+2 f x)+i B \sin (3 e+4 f x)+3 B \cos (3 e+2 f x)-3 i B \sin (e))}{12 f}","\frac{a c^3 (B+i A) (1-i \tan (e+f x))^3}{3 f}-\frac{a B c^3 (1-i \tan (e+f x))^4}{4 f}",1,"(a*c^3*Sec[e]*Sec[e + f*x]^4*(3*((-2*I)*A + B)*Cos[e] + 3*((-I)*A + B)*Cos[e + 2*f*x] - (3*I)*A*Cos[3*e + 2*f*x] + 3*B*Cos[3*e + 2*f*x] - 6*A*Sin[e] - (3*I)*B*Sin[e] + 5*A*Sin[e + 2*f*x] + I*B*Sin[e + 2*f*x] - 3*A*Sin[3*e + 2*f*x] - (3*I)*B*Sin[3*e + 2*f*x] + 2*A*Sin[3*e + 4*f*x] + I*B*Sin[3*e + 4*f*x]))/(12*f)","B",1
668,1,109,66,2.5648454,"\int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) (c-i c \tan (e+f x))^2 \, dx","Integrate[(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2,x]","\frac{a c^2 \sec (e) \sec ^3(e+f x) (3 (B-i A) \cos (2 e+f x)+3 (B-i A) \cos (f x)-3 A \sin (2 e+f x)+3 A \sin (2 e+3 f x)+6 A \sin (f x)-3 i B \sin (2 e+f x)+i B \sin (2 e+3 f x))}{12 f}","-\frac{a c^2 (-B+i A) \tan ^2(e+f x)}{2 f}+\frac{a A c^2 \tan (e+f x)}{f}-\frac{i a B c^2 \tan ^3(e+f x)}{3 f}",1,"(a*c^2*Sec[e]*Sec[e + f*x]^3*(3*((-I)*A + B)*Cos[f*x] + 3*((-I)*A + B)*Cos[2*e + f*x] + 6*A*Sin[f*x] - 3*A*Sin[2*e + f*x] - (3*I)*B*Sin[2*e + f*x] + 3*A*Sin[2*e + 3*f*x] + I*B*Sin[2*e + 3*f*x]))/(12*f)","A",1
669,1,32,32,0.0430433,"\int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) (c-i c \tan (e+f x)) \, dx","Integrate[(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x]),x]","\frac{a A c \tan (e+f x)}{f}+\frac{a B c \sec ^2(e+f x)}{2 f}","\frac{a A c \tan (e+f x)}{f}+\frac{a B c \tan ^2(e+f x)}{2 f}",1,"(a*B*c*Sec[e + f*x]^2)/(2*f) + (a*A*c*Tan[e + f*x])/f","A",1
670,1,66,46,0.028473,"\int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) \, dx","Integrate[(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]),x]","-\frac{i a A \log (\cos (e+f x))}{f}+a A x-\frac{i a B \tan ^{-1}(\tan (e+f x))}{f}+\frac{i a B \tan (e+f x)}{f}-\frac{a B \log (\cos (e+f x))}{f}","-\frac{a (B+i A) \log (\cos (e+f x))}{f}+a x (A-i B)+\frac{i a B \tan (e+f x)}{f}",1,"a*A*x - (I*a*B*ArcTan[Tan[e + f*x]])/f - (I*a*A*Log[Cos[e + f*x]])/f - (a*B*Log[Cos[e + f*x]])/f + (I*a*B*Tan[e + f*x])/f","A",1
671,1,123,54,1.9391005,"\int \frac{(a+i a \tan (e+f x)) (A+B \tan (e+f x))}{c-i c \tan (e+f x)} \, dx","Integrate[((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x]),x]","\frac{a (\sin (e+f x)-i \cos (e+f x)) \left(\cos (e+f x) \left(A+i B \log \left(\cos ^2(e+f x)\right)-4 B f x-i B\right)+\sin (e+f x) \left(i A+B \log \left(\cos ^2(e+f x)\right)+4 i B f x+B\right)+2 B \tan ^{-1}(\tan (2 e+f x)) (\cos (e+f x)-i \sin (e+f x))\right)}{2 c f}","\frac{a (A-i B)}{c f (\tan (e+f x)+i)}+\frac{a B \log (\cos (e+f x))}{c f}+\frac{i a B x}{c}",1,"(a*((-I)*Cos[e + f*x] + Sin[e + f*x])*(Cos[e + f*x]*(A - I*B - 4*B*f*x + I*B*Log[Cos[e + f*x]^2]) + 2*B*ArcTan[Tan[2*e + f*x]]*(Cos[e + f*x] - I*Sin[e + f*x]) + (I*A + B + (4*I)*B*f*x + B*Log[Cos[e + f*x]^2])*Sin[e + f*x]))/(2*c*f)","B",1
672,1,62,46,2.0063676,"\int \frac{(a+i a \tan (e+f x)) (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^2} \, dx","Integrate[((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^2,x]","\frac{a (\cos (3 (e+f x))+i \sin (3 (e+f x))) ((B-3 i A) \cos (e+f x)-(A+3 i B) \sin (e+f x))}{8 c^2 f}","\frac{a (A+B \tan (e+f x))^2}{2 c^2 f (B+i A) (1-i \tan (e+f x))^2}",1,"(a*(((-3*I)*A + B)*Cos[e + f*x] - (A + (3*I)*B)*Sin[e + f*x])*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)]))/(8*c^2*f)","A",1
673,1,72,55,1.4067281,"\int \frac{(a+i a \tan (e+f x)) (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^3} \, dx","Integrate[((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^3,x]","\frac{a (\cos (4 (e+f x))+i \sin (4 (e+f x))) (-2 (A+2 i B) \sin (2 (e+f x))+2 (B-2 i A) \cos (2 (e+f x))-3 i A)}{24 c^3 f}","-\frac{a (A-i B)}{3 c^3 f (\tan (e+f x)+i)^3}-\frac{a B}{2 c^3 f (\tan (e+f x)+i)^2}",1,"(a*((-3*I)*A + 2*((-2*I)*A + B)*Cos[2*(e + f*x)] - 2*(A + (2*I)*B)*Sin[2*(e + f*x)])*(Cos[4*(e + f*x)] + I*Sin[4*(e + f*x)]))/(24*c^3*f)","A",1
674,1,97,57,1.5829094,"\int \frac{(a+i a \tan (e+f x)) (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^4} \, dx","Integrate[((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^4,x]","\frac{a (\cos (5 (e+f x))+i \sin (5 (e+f x))) (-(3 A+5 i B) (2 \sin (e+f x)+3 \sin (3 (e+f x)))+2 (B-15 i A) \cos (e+f x)+3 (3 B-5 i A) \cos (3 (e+f x)))}{192 c^4 f}","-\frac{a (B+i A)}{4 c^4 f (\tan (e+f x)+i)^4}-\frac{i a B}{3 c^4 f (\tan (e+f x)+i)^3}",1,"(a*(2*((-15*I)*A + B)*Cos[e + f*x] + 3*((-5*I)*A + 3*B)*Cos[3*(e + f*x)] - (3*A + (5*I)*B)*(2*Sin[e + f*x] + 3*Sin[3*(e + f*x)]))*(Cos[5*(e + f*x)] + I*Sin[5*(e + f*x)]))/(192*c^4*f)","A",1
675,1,124,55,3.0562325,"\int \frac{(a+i a \tan (e+f x)) (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^5} \, dx","Integrate[((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^5,x]","-\frac{i a (\cos (6 (e+f x))+i \sin (6 (e+f x))) (5 (6 A+i B) \cos (2 (e+f x))+4 (3 A+2 i B) \cos (4 (e+f x))-10 i A \sin (2 (e+f x))-8 i A \sin (4 (e+f x))+20 A+15 B \sin (2 (e+f x))+12 B \sin (4 (e+f x)))}{320 c^5 f}","\frac{a (A-i B)}{5 c^5 f (\tan (e+f x)+i)^5}+\frac{a B}{4 c^5 f (\tan (e+f x)+i)^4}",1,"((-1/320*I)*a*(20*A + 5*(6*A + I*B)*Cos[2*(e + f*x)] + 4*(3*A + (2*I)*B)*Cos[4*(e + f*x)] - (10*I)*A*Sin[2*(e + f*x)] + 15*B*Sin[2*(e + f*x)] - (8*I)*A*Sin[4*(e + f*x)] + 12*B*Sin[4*(e + f*x)])*(Cos[6*(e + f*x)] + I*Sin[6*(e + f*x)]))/(c^5*f)","B",1
676,1,146,109,7.0245222,"\int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n,x]","\frac{a^2 \sec ^2(e+f x) (c \sec (e+f x))^n \left(\left(B \left(n^2+2 n+4\right)+i A (n+2)^2\right) \cos (2 (e+f x))-n (A (n+2)-i B (n+4)) \sin (2 (e+f x))+(n+2) (-B (n-2)+i A (n+2))\right) \exp (n (-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x))))}{2 f n (n+1) (n+2)}","\frac{2 a^2 (B+i A) (c-i c \tan (e+f x))^n}{f n}-\frac{a^2 (3 B+i A) (c-i c \tan (e+f x))^{n+1}}{c f (n+1)}+\frac{a^2 B (c-i c \tan (e+f x))^{n+2}}{c^2 f (n+2)}",1,"(a^2*E^(n*(-Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))*Sec[e + f*x]^2*(c*Sec[e + f*x])^n*((2 + n)*(-(B*(-2 + n)) + I*A*(2 + n)) + (I*A*(2 + n)^2 + B*(4 + 2*n + n^2))*Cos[2*(e + f*x)] - n*(A*(2 + n) - I*B*(4 + n))*Sin[2*(e + f*x)]))/(2*f*n*(1 + n)*(2 + n))","A",1
677,1,254,99,9.5732067,"\int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^5 \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^5,x]","\frac{a^2 c^5 \sec (e) \sec ^7(e+f x) (35 (3 B-7 i A) \cos (2 e+f x)+35 (3 B-7 i A) \cos (f x)-245 A \sin (2 e+f x)+189 A \sin (2 e+3 f x)-105 A \sin (4 e+3 f x)+98 A \sin (4 e+5 f x)+14 A \sin (6 e+7 f x)-105 i A \cos (2 e+3 f x)-105 i A \cos (4 e+3 f x)+245 A \sin (f x)-105 i B \sin (2 e+f x)+21 i B \sin (2 e+3 f x)-105 i B \sin (4 e+3 f x)+42 i B \sin (4 e+5 f x)+6 i B \sin (6 e+7 f x)+105 B \cos (2 e+3 f x)+105 B \cos (4 e+3 f x)+105 i B \sin (f x))}{840 f}","-\frac{a^2 c^5 (3 B+i A) (1-i \tan (e+f x))^6}{6 f}+\frac{2 a^2 c^5 (B+i A) (1-i \tan (e+f x))^5}{5 f}+\frac{a^2 B c^5 (1-i \tan (e+f x))^7}{7 f}",1,"(a^2*c^5*Sec[e]*Sec[e + f*x]^7*(35*((-7*I)*A + 3*B)*Cos[f*x] + 35*((-7*I)*A + 3*B)*Cos[2*e + f*x] - (105*I)*A*Cos[2*e + 3*f*x] + 105*B*Cos[2*e + 3*f*x] - (105*I)*A*Cos[4*e + 3*f*x] + 105*B*Cos[4*e + 3*f*x] + 245*A*Sin[f*x] + (105*I)*B*Sin[f*x] - 245*A*Sin[2*e + f*x] - (105*I)*B*Sin[2*e + f*x] + 189*A*Sin[2*e + 3*f*x] + (21*I)*B*Sin[2*e + 3*f*x] - 105*A*Sin[4*e + 3*f*x] - (105*I)*B*Sin[4*e + 3*f*x] + 98*A*Sin[4*e + 5*f*x] + (42*I)*B*Sin[4*e + 5*f*x] + 14*A*Sin[6*e + 7*f*x] + (6*I)*B*Sin[6*e + 7*f*x]))/(840*f)","B",1
678,1,177,99,6.2969812,"\int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4,x]","\frac{a^2 c^4 \sec (e) \sec ^6(e+f x) (15 (B-i A) \cos (e+2 f x)+10 (B-3 i A) \cos (e)+30 A \sin (e+2 f x)-15 A \sin (3 e+2 f x)+18 A \sin (3 e+4 f x)+3 A \sin (5 e+6 f x)-15 i A \cos (3 e+2 f x)-30 A \sin (e)-15 i B \sin (3 e+2 f x)+6 i B \sin (3 e+4 f x)+i B \sin (5 e+6 f x)+15 B \cos (3 e+2 f x)-10 i B \sin (e))}{120 f}","-\frac{a^2 c^4 (3 B+i A) (1-i \tan (e+f x))^5}{5 f}+\frac{a^2 c^4 (B+i A) (1-i \tan (e+f x))^4}{2 f}+\frac{a^2 B c^4 (1-i \tan (e+f x))^6}{6 f}",1,"(a^2*c^4*Sec[e]*Sec[e + f*x]^6*(10*((-3*I)*A + B)*Cos[e] + 15*((-I)*A + B)*Cos[e + 2*f*x] - (15*I)*A*Cos[3*e + 2*f*x] + 15*B*Cos[3*e + 2*f*x] - 30*A*Sin[e] - (10*I)*B*Sin[e] + 30*A*Sin[e + 2*f*x] - 15*A*Sin[3*e + 2*f*x] - (15*I)*B*Sin[3*e + 2*f*x] + 18*A*Sin[3*e + 4*f*x] + (6*I)*B*Sin[3*e + 4*f*x] + 3*A*Sin[5*e + 6*f*x] + I*B*Sin[5*e + 6*f*x]))/(120*f)","A",1
679,1,146,99,6.3420835,"\int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^3 \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3,x]","\frac{a^2 c^3 \sec (e) \sec ^5(e+f x) (15 (B-i A) \cos (2 e+f x)+15 (B-i A) \cos (f x)-15 A \sin (2 e+f x)+25 A \sin (2 e+3 f x)+5 A \sin (4 e+5 f x)+35 A \sin (f x)-15 i B \sin (2 e+f x)+5 i B \sin (2 e+3 f x)+i B \sin (4 e+5 f x)-5 i B \sin (f x))}{120 f}","-\frac{a^2 c^3 (3 B+i A) (1-i \tan (e+f x))^4}{4 f}+\frac{2 a^2 c^3 (B+i A) (1-i \tan (e+f x))^3}{3 f}+\frac{a^2 B c^3 (1-i \tan (e+f x))^5}{5 f}",1,"(a^2*c^3*Sec[e]*Sec[e + f*x]^5*(15*((-I)*A + B)*Cos[f*x] + 15*((-I)*A + B)*Cos[2*e + f*x] + 35*A*Sin[f*x] - (5*I)*B*Sin[f*x] - 15*A*Sin[2*e + f*x] - (15*I)*B*Sin[2*e + f*x] + 25*A*Sin[2*e + 3*f*x] + (5*I)*B*Sin[2*e + 3*f*x] + 5*A*Sin[4*e + 5*f*x] + I*B*Sin[4*e + 5*f*x]))/(120*f)","A",1
680,1,53,62,0.1574922,"\int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^2 \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2,x]","\frac{a^2 A c^2 \left(\frac{1}{3} \tan ^3(e+f x)+\tan (e+f x)\right)}{f}+\frac{a^2 B c^2 \sec ^4(e+f x)}{4 f}","\frac{a^2 A c^2 \tan ^3(e+f x)}{3 f}+\frac{a^2 A c^2 \tan (e+f x)}{f}+\frac{a^2 B c^2 \sec ^4(e+f x)}{4 f}",1,"(a^2*B*c^2*Sec[e + f*x]^4)/(4*f) + (a^2*A*c^2*(Tan[e + f*x] + Tan[e + f*x]^3/3))/f","A",1
681,1,109,64,2.9377821,"\int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x)) \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x]),x]","\frac{a^2 c \sec (e) \sec ^3(e+f x) (3 (B+i A) \cos (2 e+f x)+3 (B+i A) \cos (f x)-3 A \sin (2 e+f x)+3 A \sin (2 e+3 f x)+6 A \sin (f x)+3 i B \sin (2 e+f x)-i B \sin (2 e+3 f x))}{12 f}","\frac{a^2 c (B+i A) \tan ^2(e+f x)}{2 f}+\frac{a^2 A c \tan (e+f x)}{f}+\frac{i a^2 B c \tan ^3(e+f x)}{3 f}",1,"(a^2*c*Sec[e]*Sec[e + f*x]^3*(3*(I*A + B)*Cos[f*x] + 3*(I*A + B)*Cos[2*e + f*x] + 6*A*Sin[f*x] - 3*A*Sin[2*e + f*x] + (3*I)*B*Sin[2*e + f*x] + 3*A*Sin[2*e + 3*f*x] - I*B*Sin[2*e + 3*f*x]))/(12*f)","A",1
682,1,263,80,2.404113,"\int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]),x]","\frac{a^2 \sec (e) \sec ^2(e+f x) (\cos (2 f x)+i \sin (2 f x)) \left(-8 (A-i B) \cos (e) \cos ^2(e+f x) \tan ^{-1}(\tan (3 e+f x))-i \left((B+i A) \cos (e+2 f x) \left(4 f x-i \log \left(\cos ^2(e+f x)\right)\right)+2 \cos (e) \left((A-i B) \log \left(\cos ^2(e+f x)\right)+4 i A f x+4 B f x-i B\right)-2 i A \sin (e+2 f x)+4 i A f x \cos (3 e+2 f x)+A \cos (3 e+2 f x) \log \left(\cos ^2(e+f x)\right)+2 i A \sin (e)-4 B \sin (e+2 f x)+4 B f x \cos (3 e+2 f x)-i B \cos (3 e+2 f x) \log \left(\cos ^2(e+f x)\right)+4 B \sin (e)\right)\right)}{4 f (\cos (f x)+i \sin (f x))^2}","-\frac{a^2 (A-i B) \tan (e+f x)}{f}-\frac{2 a^2 (B+i A) \log (\cos (e+f x))}{f}+2 a^2 x (A-i B)+\frac{B (a+i a \tan (e+f x))^2}{2 f}",1,"(a^2*Sec[e]*Sec[e + f*x]^2*(Cos[2*f*x] + I*Sin[2*f*x])*(-8*(A - I*B)*ArcTan[Tan[3*e + f*x]]*Cos[e]*Cos[e + f*x]^2 - I*((4*I)*A*f*x*Cos[3*e + 2*f*x] + 4*B*f*x*Cos[3*e + 2*f*x] + (I*A + B)*Cos[e + 2*f*x]*(4*f*x - I*Log[Cos[e + f*x]^2]) + A*Cos[3*e + 2*f*x]*Log[Cos[e + f*x]^2] - I*B*Cos[3*e + 2*f*x]*Log[Cos[e + f*x]^2] + 2*Cos[e]*((-I)*B + (4*I)*A*f*x + 4*B*f*x + (A - I*B)*Log[Cos[e + f*x]^2]) + (2*I)*A*Sin[e] + 4*B*Sin[e] - (2*I)*A*Sin[e + 2*f*x] - 4*B*Sin[e + 2*f*x])))/(4*f*(Cos[f*x] + I*Sin[f*x])^2)","B",1
683,1,418,93,5.5708119,"\int \frac{(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{c-i c \tan (e+f x)} \, dx","Integrate[((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x]),x]","\frac{a^2 \sec (e) (\sin (e+f x)-i \cos (e+f x))^2 (A+B \tan (e+f x)) \left(2 f x (A-3 i B) \cos ^3(e) \cos (e+f x)+\cos (e) \cos (e+f x) \left(-i \cos (2 e) \left(6 B f x+(A-3 i B) \log \left(\cos ^2(e+f x)\right)\right)+2 (B+i A) \cos (2 f x)-2 A f x \sin ^2(e)+A f x-2 A \sin (2 f x)+18 i B f x \sin ^2(e)-6 B f x \sin (2 e)+2 i B \sin (2 f x)\right)-2 i \sin (e) \cos ^2(e) \cos (e+f x) \left((-3 B-i A) \log \left(\cos ^2(e+f x)\right)+f x (5 A-9 i B)\right)-2 (A-3 i B) \cos (e) (\cos (2 e)-i \sin (2 e)) \cos (e+f x) \tan ^{-1}(\tan (3 e+f x))+A f x \cos (3 e) \cos (e+f x)+2 i A f x \sin ^3(e) \cos (e+f x)+2 i A f x \sin (e) \cos (2 e) \cos (e+f x)+2 B \sin (2 e) \sin (f x)+6 B f x \sin ^3(e) \cos (e+f x)+6 B f x \sin (e) \cos (2 e) \cos (e+f x)-6 i B f x \sin (e) \sin (2 e) \cos (e+f x)+2 i B \cos (2 e) \sin (f x)\right)}{2 c f (\cos (f x)+i \sin (f x))^2 (A \cos (e+f x)+B \sin (e+f x))}","\frac{2 a^2 (A-i B)}{c f (\tan (e+f x)+i)}+\frac{a^2 (3 B+i A) \log (\cos (e+f x))}{c f}-\frac{a^2 x (A-3 i B)}{c}-\frac{i a^2 B \tan (e+f x)}{c f}",1,"(a^2*Sec[e]*(2*(A - (3*I)*B)*f*x*Cos[e]^3*Cos[e + f*x] + A*f*x*Cos[3*e]*Cos[e + f*x] + (2*I)*A*f*x*Cos[2*e]*Cos[e + f*x]*Sin[e] + 6*B*f*x*Cos[2*e]*Cos[e + f*x]*Sin[e] - (2*I)*Cos[e]^2*Cos[e + f*x]*((5*A - (9*I)*B)*f*x + ((-I)*A - 3*B)*Log[Cos[e + f*x]^2])*Sin[e] + (2*I)*A*f*x*Cos[e + f*x]*Sin[e]^3 + 6*B*f*x*Cos[e + f*x]*Sin[e]^3 - 2*(A - (3*I)*B)*ArcTan[Tan[3*e + f*x]]*Cos[e]*Cos[e + f*x]*(Cos[2*e] - I*Sin[2*e]) - (6*I)*B*f*x*Cos[e + f*x]*Sin[e]*Sin[2*e] + (2*I)*B*Cos[2*e]*Sin[f*x] + 2*B*Sin[2*e]*Sin[f*x] + Cos[e]*Cos[e + f*x]*(A*f*x + 2*(I*A + B)*Cos[2*f*x] - I*Cos[2*e]*(6*B*f*x + (A - (3*I)*B)*Log[Cos[e + f*x]^2]) - 2*A*f*x*Sin[e]^2 + (18*I)*B*f*x*Sin[e]^2 - 6*B*f*x*Sin[2*e] - 2*A*Sin[2*f*x] + (2*I)*B*Sin[2*f*x]))*((-I)*Cos[e + f*x] + Sin[e + f*x])^2*(A + B*Tan[e + f*x]))/(2*c*f*(Cos[f*x] + I*Sin[f*x])^2*(A*Cos[e + f*x] + B*Sin[e + f*x]))","B",1
684,1,184,91,3.6069721,"\int \frac{(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^2} \, dx","Integrate[((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^2,x]","\frac{a^2 (\cos (2 (e+2 f x))+i \sin (2 (e+2 f x))) \left(-i \cos (2 (e+f x)) \left(A-2 i B \log \left(\cos ^2(e+f x)\right)+8 B f x-i B\right)+A \sin (2 (e+f x))-8 B f x \sin (2 (e+f x))-i B \sin (2 (e+f x))+2 i B \sin (2 (e+f x)) \log \left(\cos ^2(e+f x)\right)+4 B \tan ^{-1}(\tan (3 e+f x)) (\sin (2 (e+f x))+i \cos (2 (e+f x)))+4 B\right)}{4 c^2 f (\cos (f x)+i \sin (f x))^2}","-\frac{a^2 (A-3 i B)}{c^2 f (\tan (e+f x)+i)}+\frac{a^2 (B+i A)}{c^2 f (\tan (e+f x)+i)^2}-\frac{a^2 B \log (\cos (e+f x))}{c^2 f}-\frac{i a^2 B x}{c^2}",1,"(a^2*(4*B - I*Cos[2*(e + f*x)]*(A - I*B + 8*B*f*x - (2*I)*B*Log[Cos[e + f*x]^2]) + A*Sin[2*(e + f*x)] - I*B*Sin[2*(e + f*x)] - 8*B*f*x*Sin[2*(e + f*x)] + (2*I)*B*Log[Cos[e + f*x]^2]*Sin[2*(e + f*x)] + 4*B*ArcTan[Tan[3*e + f*x]]*(I*Cos[2*(e + f*x)] + Sin[2*(e + f*x)]))*(Cos[2*(e + 2*f*x)] + I*Sin[2*(e + 2*f*x)]))/(4*c^2*f*(Cos[f*x] + I*Sin[f*x])^2)","B",1
685,1,81,93,2.9863686,"\int \frac{(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^3} \, dx","Integrate[((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^3,x]","\frac{a^2 (\cos (5 e+7 f x)+i \sin (5 e+7 f x)) ((B-5 i A) \cos (e+f x)-(A+5 i B) \sin (e+f x))}{24 c^3 f (\cos (f x)+i \sin (f x))^2}","-\frac{a^2 (3 B+i A)}{2 c^3 f (\tan (e+f x)+i)^2}-\frac{2 a^2 (A-i B)}{3 c^3 f (\tan (e+f x)+i)^3}-\frac{i a^2 B}{c^3 f (\tan (e+f x)+i)}",1,"(a^2*(((-5*I)*A + B)*Cos[e + f*x] - (A + (5*I)*B)*Sin[e + f*x])*(Cos[5*e + 7*f*x] + I*Sin[5*e + 7*f*x]))/(24*c^3*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
686,1,91,91,3.1567709,"\int \frac{(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^4} \, dx","Integrate[((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^4,x]","\frac{a^2 (\cos (6 e+8 f x)+i \sin (6 e+8 f x)) (-3 (A+3 i B) \sin (2 (e+f x))+3 (B-3 i A) \cos (2 (e+f x))-8 i A)}{96 c^4 f (\cos (f x)+i \sin (f x))^2}","\frac{a^2 (A-3 i B)}{3 c^4 f (\tan (e+f x)+i)^3}-\frac{a^2 (B+i A)}{2 c^4 f (\tan (e+f x)+i)^4}+\frac{a^2 B}{2 c^4 f (\tan (e+f x)+i)^2}",1,"(a^2*((-8*I)*A + 3*((-3*I)*A + B)*Cos[2*(e + f*x)] - 3*(A + (3*I)*B)*Sin[2*(e + f*x)])*(Cos[6*e + 8*f*x] + I*Sin[6*e + 8*f*x]))/(96*c^4*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
687,1,116,95,3.9463181,"\int \frac{(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^5} \, dx","Integrate[((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^5,x]","\frac{a^2 (\cos (7 e+9 f x)+i \sin (7 e+9 f x)) (-(3 A+7 i B) (5 \sin (e+f x)+6 \sin (3 (e+f x)))+5 (B-21 i A) \cos (e+f x)+6 (3 B-7 i A) \cos (3 (e+f x)))}{960 c^5 f (\cos (f x)+i \sin (f x))^2}","\frac{a^2 (3 B+i A)}{4 c^5 f (\tan (e+f x)+i)^4}+\frac{2 a^2 (A-i B)}{5 c^5 f (\tan (e+f x)+i)^5}+\frac{i a^2 B}{3 c^5 f (\tan (e+f x)+i)^3}",1,"(a^2*(5*((-21*I)*A + B)*Cos[e + f*x] + 6*((-7*I)*A + 3*B)*Cos[3*(e + f*x)] - (3*A + (7*I)*B)*(5*Sin[e + f*x] + 6*Sin[3*(e + f*x)]))*(Cos[7*e + 9*f*x] + I*Sin[7*e + 9*f*x]))/(960*c^5*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
688,1,143,91,5.4733456,"\int \frac{(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^6} \, dx","Integrate[((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^6,x]","-\frac{i a^2 (\cos (8 e+10 f x)+i \sin (8 e+10 f x)) (8 (8 A+i B) \cos (2 (e+f x))+10 (2 A+i B) \cos (4 (e+f x))-16 i A \sin (2 (e+f x))-10 i A \sin (4 (e+f x))+45 A+32 B \sin (2 (e+f x))+20 B \sin (4 (e+f x)))}{960 c^6 f (\cos (f x)+i \sin (f x))^2}","-\frac{a^2 (A-3 i B)}{5 c^6 f (\tan (e+f x)+i)^5}+\frac{a^2 (B+i A)}{3 c^6 f (\tan (e+f x)+i)^6}-\frac{a^2 B}{4 c^6 f (\tan (e+f x)+i)^4}",1,"((-1/960*I)*a^2*(45*A + 8*(8*A + I*B)*Cos[2*(e + f*x)] + 10*(2*A + I*B)*Cos[4*(e + f*x)] - (16*I)*A*Sin[2*(e + f*x)] + 32*B*Sin[2*(e + f*x)] - (10*I)*A*Sin[4*(e + f*x)] + 20*B*Sin[4*(e + f*x)])*(Cos[8*e + 10*f*x] + I*Sin[8*e + 10*f*x]))/(c^6*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
689,1,822,151,13.7659745,"\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n,x]","\frac{\cos ^4(e+f x) \left(-\frac{i \sec (e) \left(B e^{n (i f x-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x)))-i f n x} \cos (3 e)-i B e^{n (i f x-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x)))-i f n x} \sin (3 e)\right) \sin (f x) \sec ^3(e+f x)}{n+3}+\frac{\sec (e) (3 A \cos (e)-9 i B \cos (e)+A n \cos (e)-2 i B n \cos (e)+2 B \sin (e)+B n \sin (e)) \left(-i e^{n (i f x-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x)))-i f n x} \cos (3 e)-e^{n (i f x-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x)))-i f n x} \sin (3 e)\right) \sec ^2(e+f x)}{(n+2) (n+3)}+\frac{\left(A n^2-i B n^2+6 A n-6 i B n+9 A-13 i B\right) \sec (e) \left(2 i e^{n (i f x-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x)))-i f n x} \sin (3 e)-2 e^{n (i f x-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x)))-i f n x} \cos (3 e)\right) \sin (f x) \sec (e+f x)}{(n+1) (n+2) (n+3)}+\frac{\sec (e) \left(i A \cos (e) n^3+B \cos (e) n^3-A \sin (e) n^3+i B \sin (e) n^3+6 i A \cos (e) n^2+6 B \cos (e) n^2-6 A \sin (e) n^2+6 i B \sin (e) n^2+13 i A \cos (e) n+9 B \cos (e) n-9 A \sin (e) n+13 i B \sin (e) n+12 i A \cos (e)+12 B \cos (e)\right) \left(\frac{2 e^{n (i f x-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x)))-i f n x} \cos (3 e)}{n}-\frac{2 i e^{n (i f x-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x)))-i f n x} \sin (3 e)}{n}\right)}{(n+1) (n+2) (n+3)}\right) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{n-\frac{n (\log (c-i c \tan (e+f x))-\log (c \sec (e+f x)))}{\log (c-i c \tan (e+f x))}}}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}","\frac{a^3 (5 B+i A) (c-i c \tan (e+f x))^{n+2}}{c^2 f (n+2)}+\frac{4 a^3 (B+i A) (c-i c \tan (e+f x))^n}{f n}-\frac{4 a^3 (2 B+i A) (c-i c \tan (e+f x))^{n+1}}{c f (n+1)}-\frac{a^3 B (c-i c \tan (e+f x))^{n+3}}{c^3 f (n+3)}",1,"(Cos[e + f*x]^4*((Sec[e]*Sec[e + f*x]^2*(3*A*Cos[e] - (9*I)*B*Cos[e] + A*n*Cos[e] - (2*I)*B*n*Cos[e] + 2*B*Sin[e] + B*n*Sin[e])*((-I)*E^((-I)*f*n*x + n*(I*f*x - Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))*Cos[3*e] - E^((-I)*f*n*x + n*(I*f*x - Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))*Sin[3*e]))/((2 + n)*(3 + n)) + (Sec[e]*((12*I)*A*Cos[e] + 12*B*Cos[e] + (13*I)*A*n*Cos[e] + 9*B*n*Cos[e] + (6*I)*A*n^2*Cos[e] + 6*B*n^2*Cos[e] + I*A*n^3*Cos[e] + B*n^3*Cos[e] - 9*A*n*Sin[e] + (13*I)*B*n*Sin[e] - 6*A*n^2*Sin[e] + (6*I)*B*n^2*Sin[e] - A*n^3*Sin[e] + I*B*n^3*Sin[e])*((2*E^((-I)*f*n*x + n*(I*f*x - Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))*Cos[3*e])/n - ((2*I)*E^((-I)*f*n*x + n*(I*f*x - Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))*Sin[3*e])/n))/((1 + n)*(2 + n)*(3 + n)) + ((9*A - (13*I)*B + 6*A*n - (6*I)*B*n + A*n^2 - I*B*n^2)*Sec[e]*Sec[e + f*x]*(-2*E^((-I)*f*n*x + n*(I*f*x - Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))*Cos[3*e] + (2*I)*E^((-I)*f*n*x + n*(I*f*x - Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))*Sin[3*e])*Sin[f*x])/((1 + n)*(2 + n)*(3 + n)) - (I*Sec[e]*Sec[e + f*x]^3*(B*E^((-I)*f*n*x + n*(I*f*x - Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))*Cos[3*e] - I*B*E^((-I)*f*n*x + n*(I*f*x - Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))*Sin[3*e])*Sin[f*x])/(3 + n))*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(n - (n*(-Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))/Log[c - I*c*Tan[e + f*x]]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","B",1
690,1,262,135,11.3915842,"\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^6 \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^6,x]","\frac{a^3 c^6 \sec (e) \sec ^9(e+f x) (63 (B-3 i A) \cos (2 e+f x)+63 (B-3 i A) \cos (f x)-189 A \sin (2 e+f x)+168 A \sin (2 e+3 f x)-84 A \sin (4 e+3 f x)+108 A \sin (4 e+5 f x)+27 A \sin (6 e+7 f x)+3 A \sin (8 e+9 f x)-84 i A \cos (2 e+3 f x)-84 i A \cos (4 e+3 f x)+189 A \sin (f x)-63 i B \sin (2 e+f x)-84 i B \sin (4 e+3 f x)+36 i B \sin (4 e+5 f x)+9 i B \sin (6 e+7 f x)+i B \sin (8 e+9 f x)+84 B \cos (2 e+3 f x)+84 B \cos (4 e+3 f x)+63 i B \sin (f x))}{1008 f}","\frac{a^3 c^6 (5 B+i A) (1-i \tan (e+f x))^8}{8 f}-\frac{4 a^3 c^6 (2 B+i A) (1-i \tan (e+f x))^7}{7 f}+\frac{2 a^3 c^6 (B+i A) (1-i \tan (e+f x))^6}{3 f}-\frac{a^3 B c^6 (1-i \tan (e+f x))^9}{9 f}",1,"(a^3*c^6*Sec[e]*Sec[e + f*x]^9*(63*((-3*I)*A + B)*Cos[f*x] + 63*((-3*I)*A + B)*Cos[2*e + f*x] - (84*I)*A*Cos[2*e + 3*f*x] + 84*B*Cos[2*e + 3*f*x] - (84*I)*A*Cos[4*e + 3*f*x] + 84*B*Cos[4*e + 3*f*x] + 189*A*Sin[f*x] + (63*I)*B*Sin[f*x] - 189*A*Sin[2*e + f*x] - (63*I)*B*Sin[2*e + f*x] + 168*A*Sin[2*e + 3*f*x] - 84*A*Sin[4*e + 3*f*x] - (84*I)*B*Sin[4*e + 3*f*x] + 108*A*Sin[4*e + 5*f*x] + (36*I)*B*Sin[4*e + 5*f*x] + 27*A*Sin[6*e + 7*f*x] + (9*I)*B*Sin[6*e + 7*f*x] + 3*A*Sin[8*e + 9*f*x] + I*B*Sin[8*e + 9*f*x]))/(1008*f)","A",1
691,1,215,135,10.8907145,"\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^5 \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^5,x]","\frac{a^3 c^5 \sec (e) \sec ^8(e+f x) (70 (B-i A) \cos (e+2 f x)+35 (B-4 i A) \cos (e)+154 A \sin (e+2 f x)-70 A \sin (3 e+2 f x)+112 A \sin (3 e+4 f x)+32 A \sin (5 e+6 f x)+4 A \sin (7 e+8 f x)-70 i A \cos (3 e+2 f x)-140 A \sin (e)-14 i B \sin (e+2 f x)-70 i B \sin (3 e+2 f x)+28 i B \sin (3 e+4 f x)+8 i B \sin (5 e+6 f x)+i B \sin (7 e+8 f x)+70 B \cos (3 e+2 f x)-35 i B \sin (e))}{840 f}","\frac{a^3 c^5 (5 B+i A) (1-i \tan (e+f x))^7}{7 f}-\frac{2 a^3 c^5 (2 B+i A) (1-i \tan (e+f x))^6}{3 f}+\frac{4 a^3 c^5 (B+i A) (1-i \tan (e+f x))^5}{5 f}-\frac{a^3 B c^5 (1-i \tan (e+f x))^8}{8 f}",1,"(a^3*c^5*Sec[e]*Sec[e + f*x]^8*(35*((-4*I)*A + B)*Cos[e] + 70*((-I)*A + B)*Cos[e + 2*f*x] - (70*I)*A*Cos[3*e + 2*f*x] + 70*B*Cos[3*e + 2*f*x] - 140*A*Sin[e] - (35*I)*B*Sin[e] + 154*A*Sin[e + 2*f*x] - (14*I)*B*Sin[e + 2*f*x] - 70*A*Sin[3*e + 2*f*x] - (70*I)*B*Sin[3*e + 2*f*x] + 112*A*Sin[3*e + 4*f*x] + (28*I)*B*Sin[3*e + 4*f*x] + 32*A*Sin[5*e + 6*f*x] + (8*I)*B*Sin[5*e + 6*f*x] + 4*A*Sin[7*e + 8*f*x] + I*B*Sin[7*e + 8*f*x]))/(840*f)","A",1
692,1,172,132,7.8024544,"\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4,x]","\frac{a^3 c^4 \sec (e) \sec ^7(e+f x) (70 (B-i A) \cos (2 e+f x)+70 (B-i A) \cos (f x)-70 A \sin (2 e+f x)+147 A \sin (2 e+3 f x)+49 A \sin (4 e+5 f x)+7 A \sin (6 e+7 f x)+175 A \sin (f x)-70 i B \sin (2 e+f x)+21 i B \sin (2 e+3 f x)+7 i B \sin (4 e+5 f x)+i B \sin (6 e+7 f x)-35 i B \sin (f x))}{840 f}","\frac{a^3 c^4 (5 B+i A) (1-i \tan (e+f x))^6}{6 f}-\frac{4 a^3 c^4 (2 B+i A) (1-i \tan (e+f x))^5}{5 f}+\frac{a^3 c^4 (B+i A) (1-i \tan (e+f x))^4}{f}-\frac{a^3 B c^4 (1-i \tan (e+f x))^7}{7 f}",1,"(a^3*c^4*Sec[e]*Sec[e + f*x]^7*(70*((-I)*A + B)*Cos[f*x] + 70*((-I)*A + B)*Cos[2*e + f*x] + 175*A*Sin[f*x] - (35*I)*B*Sin[f*x] - 70*A*Sin[2*e + f*x] - (70*I)*B*Sin[2*e + f*x] + 147*A*Sin[2*e + 3*f*x] + (21*I)*B*Sin[2*e + 3*f*x] + 49*A*Sin[4*e + 5*f*x] + (7*I)*B*Sin[4*e + 5*f*x] + 7*A*Sin[6*e + 7*f*x] + I*B*Sin[6*e + 7*f*x]))/(840*f)","A",1
693,1,65,84,0.2691318,"\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^3 \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3,x]","\frac{a^3 A c^3 \left(\frac{1}{5} \tan ^5(e+f x)+\frac{2}{3} \tan ^3(e+f x)+\tan (e+f x)\right)}{f}+\frac{a^3 B c^3 \sec ^6(e+f x)}{6 f}","\frac{a^3 A c^3 \tan ^5(e+f x)}{5 f}+\frac{2 a^3 A c^3 \tan ^3(e+f x)}{3 f}+\frac{a^3 A c^3 \tan (e+f x)}{f}+\frac{a^3 B c^3 \sec ^6(e+f x)}{6 f}",1,"(a^3*B*c^3*Sec[e + f*x]^6)/(6*f) + (a^3*A*c^3*(Tan[e + f*x] + (2*Tan[e + f*x]^3)/3 + Tan[e + f*x]^5/5))/f","A",1
694,1,146,101,5.8984083,"\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^2 \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2,x]","\frac{a^3 c^2 \sec (e) \sec ^5(e+f x) (15 (B+i A) \cos (2 e+f x)+15 (B+i A) \cos (f x)-15 A \sin (2 e+f x)+25 A \sin (2 e+3 f x)+5 A \sin (4 e+5 f x)+35 A \sin (f x)+15 i B \sin (2 e+f x)-5 i B \sin (2 e+3 f x)-i B \sin (4 e+5 f x)+5 i B \sin (f x))}{120 f}","\frac{a^3 c^2 (-3 B+i A) (1+i \tan (e+f x))^4}{4 f}-\frac{2 a^3 c^2 (-B+i A) (1+i \tan (e+f x))^3}{3 f}+\frac{a^3 B c^2 (1+i \tan (e+f x))^5}{5 f}",1,"(a^3*c^2*Sec[e]*Sec[e + f*x]^5*(15*(I*A + B)*Cos[f*x] + 15*(I*A + B)*Cos[2*e + f*x] + 35*A*Sin[f*x] + (5*I)*B*Sin[f*x] - 15*A*Sin[2*e + f*x] + (15*I)*B*Sin[2*e + f*x] + 25*A*Sin[2*e + 3*f*x] - (5*I)*B*Sin[2*e + 3*f*x] + 5*A*Sin[4*e + 5*f*x] - I*B*Sin[4*e + 5*f*x]))/(120*f)","A",1
695,1,161,61,3.9817944,"\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x)) \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x]),x]","\frac{a^3 c \sec (e) \sec ^4(e+f x) (3 (B+i A) \cos (e+2 f x)+3 (B+2 i A) \cos (e)+5 A \sin (e+2 f x)-3 A \sin (3 e+2 f x)+2 A \sin (3 e+4 f x)+3 i A \cos (3 e+2 f x)-6 A \sin (e)-i B \sin (e+2 f x)+3 i B \sin (3 e+2 f x)-i B \sin (3 e+4 f x)+3 B \cos (3 e+2 f x)+3 i B \sin (e))}{12 f}","-\frac{a^3 c (-B+i A) (1+i \tan (e+f x))^3}{3 f}-\frac{a^3 B c (1+i \tan (e+f x))^4}{4 f}",1,"(a^3*c*Sec[e]*Sec[e + f*x]^4*(3*((2*I)*A + B)*Cos[e] + 3*(I*A + B)*Cos[e + 2*f*x] + (3*I)*A*Cos[3*e + 2*f*x] + 3*B*Cos[3*e + 2*f*x] - 6*A*Sin[e] + (3*I)*B*Sin[e] + 5*A*Sin[e + 2*f*x] - I*B*Sin[e + 2*f*x] - 3*A*Sin[3*e + 2*f*x] + (3*I)*B*Sin[3*e + 2*f*x] + 2*A*Sin[3*e + 4*f*x] - I*B*Sin[3*e + 4*f*x]))/(12*f)","B",1
696,1,331,110,4.0931253,"\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]),x]","\frac{a^3 \sec (e) \sec ^3(e+f x) \left(3 \cos (f x) \left((-3 B-3 i A) \log \left(\cos ^2(e+f x)\right)+6 A f x-i A-6 i B f x-3 B\right)+3 \cos (2 e+f x) \left((-3 B-3 i A) \log \left(\cos ^2(e+f x)\right)+6 A f x-i A-6 i B f x-3 B\right)+9 A \sin (2 e+f x)-9 A \sin (2 e+3 f x)+6 A f x \cos (2 e+3 f x)+6 A f x \cos (4 e+3 f x)-3 i A \cos (2 e+3 f x) \log \left(\cos ^2(e+f x)\right)-3 i A \cos (4 e+3 f x) \log \left(\cos ^2(e+f x)\right)-18 A \sin (f x)-15 i B \sin (2 e+f x)+13 i B \sin (2 e+3 f x)-6 i B f x \cos (2 e+3 f x)-6 i B f x \cos (4 e+3 f x)-3 B \cos (2 e+3 f x) \log \left(\cos ^2(e+f x)\right)-3 B \cos (4 e+3 f x) \log \left(\cos ^2(e+f x)\right)+24 i B \sin (f x)\right)}{12 f}","-\frac{2 a^3 (A-i B) \tan (e+f x)}{f}-\frac{4 a^3 (B+i A) \log (\cos (e+f x))}{f}+4 a^3 x (A-i B)+\frac{a (B+i A) (a+i a \tan (e+f x))^2}{2 f}+\frac{B (a+i a \tan (e+f x))^3}{3 f}",1,"(a^3*Sec[e]*Sec[e + f*x]^3*(6*A*f*x*Cos[2*e + 3*f*x] - (6*I)*B*f*x*Cos[2*e + 3*f*x] + 6*A*f*x*Cos[4*e + 3*f*x] - (6*I)*B*f*x*Cos[4*e + 3*f*x] - (3*I)*A*Cos[2*e + 3*f*x]*Log[Cos[e + f*x]^2] - 3*B*Cos[2*e + 3*f*x]*Log[Cos[e + f*x]^2] - (3*I)*A*Cos[4*e + 3*f*x]*Log[Cos[e + f*x]^2] - 3*B*Cos[4*e + 3*f*x]*Log[Cos[e + f*x]^2] + 3*Cos[f*x]*((-I)*A - 3*B + 6*A*f*x - (6*I)*B*f*x + ((-3*I)*A - 3*B)*Log[Cos[e + f*x]^2]) + 3*Cos[2*e + f*x]*((-I)*A - 3*B + 6*A*f*x - (6*I)*B*f*x + ((-3*I)*A - 3*B)*Log[Cos[e + f*x]^2]) - 18*A*Sin[f*x] + (24*I)*B*Sin[f*x] + 9*A*Sin[2*e + f*x] - (15*I)*B*Sin[2*e + f*x] - 9*A*Sin[2*e + 3*f*x] + (13*I)*B*Sin[2*e + 3*f*x]))/(12*f)","B",1
697,1,944,119,10.0112813,"\int \frac{(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{c-i c \tan (e+f x)} \, dx","Integrate[((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x]),x]","\frac{x \left(-\frac{2 A \cos ^3(e)}{c}+\frac{4 i B \cos ^3(e)}{c}+\frac{8 i A \sin (e) \cos ^2(e)}{c}+\frac{16 B \sin (e) \cos ^2(e)}{c}+\frac{12 A \sin ^2(e) \cos (e)}{c}-\frac{24 i B \sin ^2(e) \cos (e)}{c}+\frac{2 A \cos (e)}{c}-\frac{4 i B \cos (e)}{c}-\frac{8 i A \sin ^3(e)}{c}-\frac{16 B \sin ^3(e)}{c}-\frac{4 i A \sin (e)}{c}-\frac{8 B \sin (e)}{c}-\frac{2 A \sin ^3(e) \tan (e)}{c}+\frac{4 i B \sin ^3(e) \tan (e)}{c}-\frac{2 A \sin (e) \tan (e)}{c}+\frac{4 i B \sin (e) \tan (e)}{c}-i (A-2 i B) \left(\frac{4 \cos (3 e)}{c}-\frac{4 i \sin (3 e)}{c}\right) \tan (e)\right) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) \cos ^4(e+f x)}{(\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{(A-i B) \left(\frac{2 \cos (e)}{c}-\frac{2 i \sin (e)}{c}\right) \sin (2 f x) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) \cos ^4(e+f x)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{(A-i B) \cos (2 f x) \left(-\frac{2 i \cos (e)}{c}-\frac{2 \sin (e)}{c}\right) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) \cos ^4(e+f x)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{(A-2 i B) \left(\frac{4 i f x \sin (3 e)}{c}-\frac{4 f x \cos (3 e)}{c}\right) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) \cos ^4(e+f x)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{(i A+2 B) \left(\frac{2 \cos (3 e) \log \left(\cos ^2(e+f x)\right)}{c}-\frac{2 i \log \left(\cos ^2(e+f x)\right) \sin (3 e)}{c}\right) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) \cos ^4(e+f x)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{\left(\frac{\cos (3 e)}{c}-\frac{i \sin (3 e)}{c}\right) (A \sin (f x)-4 i B \sin (f x)) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) \cos ^3(e+f x)}{f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{\left(\frac{B \cos (3 e)}{2 c}-\frac{i B \sin (3 e)}{2 c}\right) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) \cos ^2(e+f x)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}","\frac{a^3 (A-4 i B) \tan (e+f x)}{c f}+\frac{4 a^3 (A-i B)}{c f (\tan (e+f x)+i)}+\frac{4 a^3 (2 B+i A) \log (\cos (e+f x))}{c f}-\frac{4 a^3 x (A-2 i B)}{c}+\frac{a^3 B \tan ^2(e+f x)}{2 c f}",1,"((A - I*B)*Cos[2*f*x]*Cos[e + f*x]^4*(((-2*I)*Cos[e])/c - (2*Sin[e])/c)*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (Cos[e + f*x]^2*((B*Cos[3*e])/(2*c) - ((I/2)*B*Sin[3*e])/c)*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x])) + ((A - (2*I)*B)*Cos[e + f*x]^4*((-4*f*x*Cos[3*e])/c + ((4*I)*f*x*Sin[3*e])/c)*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x])) + ((I*A + 2*B)*Cos[e + f*x]^4*((2*Cos[3*e]*Log[Cos[e + f*x]^2])/c - ((2*I)*Log[Cos[e + f*x]^2]*Sin[3*e])/c)*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (Cos[e + f*x]^3*(Cos[3*e]/c - (I*Sin[3*e])/c)*(A*Sin[f*x] - (4*I)*B*Sin[f*x])*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(f*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x])) + ((A - I*B)*Cos[e + f*x]^4*((2*Cos[e])/c - ((2*I)*Sin[e])/c)*Sin[2*f*x]*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (x*Cos[e + f*x]^4*((2*A*Cos[e])/c - ((4*I)*B*Cos[e])/c - (2*A*Cos[e]^3)/c + ((4*I)*B*Cos[e]^3)/c - ((4*I)*A*Sin[e])/c - (8*B*Sin[e])/c + ((8*I)*A*Cos[e]^2*Sin[e])/c + (16*B*Cos[e]^2*Sin[e])/c + (12*A*Cos[e]*Sin[e]^2)/c - ((24*I)*B*Cos[e]*Sin[e]^2)/c - ((8*I)*A*Sin[e]^3)/c - (16*B*Sin[e]^3)/c - (2*A*Sin[e]*Tan[e])/c + ((4*I)*B*Sin[e]*Tan[e])/c - (2*A*Sin[e]^3*Tan[e])/c + ((4*I)*B*Sin[e]^3*Tan[e])/c - I*(A - (2*I)*B)*((4*Cos[3*e])/c - ((4*I)*Sin[3*e])/c)*Tan[e])*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/((Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","B",1
698,1,1063,123,10.088202,"\int \frac{(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^2} \, dx","Integrate[((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^2,x]","\frac{x \left(\frac{A \cos ^3(e)}{2 c^2}-\frac{5 i B \cos ^3(e)}{2 c^2}-\frac{2 i A \sin (e) \cos ^2(e)}{c^2}-\frac{10 B \sin (e) \cos ^2(e)}{c^2}-\frac{3 A \sin ^2(e) \cos (e)}{c^2}+\frac{15 i B \sin ^2(e) \cos (e)}{c^2}-\frac{A \cos (e)}{2 c^2}+\frac{5 i B \cos (e)}{2 c^2}+\frac{2 i A \sin ^3(e)}{c^2}+\frac{10 B \sin ^3(e)}{c^2}+\frac{i A \sin (e)}{c^2}+\frac{5 B \sin (e)}{c^2}+\frac{A \sin ^3(e) \tan (e)}{2 c^2}-\frac{5 i B \sin ^3(e) \tan (e)}{2 c^2}+\frac{A \sin (e) \tan (e)}{2 c^2}-\frac{5 i B \sin (e) \tan (e)}{2 c^2}+i (A-5 i B) \left(\frac{\cos (3 e)}{c^2}-\frac{i \sin (3 e)}{c^2}\right) \tan (e)\right) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) \cos ^4(e+f x)}{(\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{(A-3 i B) \left(\frac{i \sin (e)}{c^2}-\frac{\cos (e)}{c^2}\right) \sin (2 f x) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) \cos ^4(e+f x)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{(A-i B) \left(\frac{\cos (e)}{2 c^2}+\frac{i \sin (e)}{2 c^2}\right) \sin (4 f x) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) \cos ^4(e+f x)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{(i A+3 B) \cos (2 f x) \left(\frac{\cos (e)}{c^2}-\frac{i \sin (e)}{c^2}\right) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) \cos ^4(e+f x)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{(A-i B) \cos (4 f x) \left(\frac{\sin (e)}{2 c^2}-\frac{i \cos (e)}{2 c^2}\right) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) \cos ^4(e+f x)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{(A-5 i B) \left(\frac{f x \cos (3 e)}{c^2}-\frac{i f x \sin (3 e)}{c^2}\right) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) \cos ^4(e+f x)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{(A-5 i B) \left(-\frac{i \cos (3 e) \log \left(\cos ^2(e+f x)\right)}{2 c^2}-\frac{\sin (3 e) \log \left(\cos ^2(e+f x)\right)}{2 c^2}\right) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) \cos ^4(e+f x)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{i B \left(\frac{\cos (3 e)}{c^2}-\frac{i \sin (3 e)}{c^2}\right) \sin (f x) (i \tan (e+f x) a+a)^3 (A+B \tan (e+f x)) \cos ^3(e+f x)}{f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}","-\frac{4 a^3 (A-2 i B)}{c^2 f (\tan (e+f x)+i)}+\frac{2 a^3 (B+i A)}{c^2 f (\tan (e+f x)+i)^2}-\frac{a^3 (5 B+i A) \log (\cos (e+f x))}{c^2 f}+\frac{a^3 x (A-5 i B)}{c^2}+\frac{i a^3 B \tan (e+f x)}{c^2 f}",1,"((I*A + 3*B)*Cos[2*f*x]*Cos[e + f*x]^4*(Cos[e]/c^2 - (I*Sin[e])/c^2)*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x])) + ((A - I*B)*Cos[4*f*x]*Cos[e + f*x]^4*(((-1/2*I)*Cos[e])/c^2 + Sin[e]/(2*c^2))*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x])) + ((A - (5*I)*B)*Cos[e + f*x]^4*((f*x*Cos[3*e])/c^2 - (I*f*x*Sin[3*e])/c^2)*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x])) + ((A - (5*I)*B)*Cos[e + f*x]^4*(((-1/2*I)*Cos[3*e]*Log[Cos[e + f*x]^2])/c^2 - (Log[Cos[e + f*x]^2]*Sin[3*e])/(2*c^2))*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (I*B*Cos[e + f*x]^3*(Cos[3*e]/c^2 - (I*Sin[3*e])/c^2)*Sin[f*x]*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(f*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x])) + ((A - (3*I)*B)*Cos[e + f*x]^4*(-(Cos[e]/c^2) + (I*Sin[e])/c^2)*Sin[2*f*x]*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x])) + ((A - I*B)*Cos[e + f*x]^4*(Cos[e]/(2*c^2) + ((I/2)*Sin[e])/c^2)*Sin[4*f*x]*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (x*Cos[e + f*x]^4*(-1/2*(A*Cos[e])/c^2 + (((5*I)/2)*B*Cos[e])/c^2 + (A*Cos[e]^3)/(2*c^2) - (((5*I)/2)*B*Cos[e]^3)/c^2 + (I*A*Sin[e])/c^2 + (5*B*Sin[e])/c^2 - ((2*I)*A*Cos[e]^2*Sin[e])/c^2 - (10*B*Cos[e]^2*Sin[e])/c^2 - (3*A*Cos[e]*Sin[e]^2)/c^2 + ((15*I)*B*Cos[e]*Sin[e]^2)/c^2 + ((2*I)*A*Sin[e]^3)/c^2 + (10*B*Sin[e]^3)/c^2 + (A*Sin[e]*Tan[e])/(2*c^2) - (((5*I)/2)*B*Sin[e]*Tan[e])/c^2 + (A*Sin[e]^3*Tan[e])/(2*c^2) - (((5*I)/2)*B*Sin[e]^3*Tan[e])/c^2 + I*(A - (5*I)*B)*(Cos[3*e]/c^2 - (I*Sin[3*e])/c^2)*Tan[e])*(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/((Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","B",1
699,1,167,129,4.7141102,"\int \frac{(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^3} \, dx","Integrate[((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^3,x]","\frac{a^3 (\cos (3 (e+2 f x))+i \sin (3 (e+2 f x))) \left(\cos (3 (e+f x)) \left(-i A+3 B \log \left(\cos ^2(e+f x)\right)+6 i B f x-B\right)+A \sin (3 (e+f x))+9 i B \sin (e+f x)-i B \sin (3 (e+f x))+6 B f x \sin (3 (e+f x))-3 B \cos (e+f x)-3 i B \sin (3 (e+f x)) \log \left(\cos ^2(e+f x)\right)\right)}{6 c^3 f (\cos (f x)+i \sin (f x))^3}","-\frac{a^3 (B+i A) (1+i \tan (e+f x))^3}{6 c^3 f (1-i \tan (e+f x))^3}-\frac{4 i a^3 B}{c^3 f (\tan (e+f x)+i)}-\frac{2 a^3 B}{c^3 f (\tan (e+f x)+i)^2}+\frac{a^3 B \log (\cos (e+f x))}{c^3 f}+\frac{i a^3 B x}{c^3}",1,"(a^3*(-3*B*Cos[e + f*x] + Cos[3*(e + f*x)]*((-I)*A - B + (6*I)*B*f*x + 3*B*Log[Cos[e + f*x]^2]) + (9*I)*B*Sin[e + f*x] + A*Sin[3*(e + f*x)] - I*B*Sin[3*(e + f*x)] + 6*B*f*x*Sin[3*(e + f*x)] - (3*I)*B*Log[Cos[e + f*x]^2]*Sin[3*(e + f*x)])*(Cos[3*(e + 2*f*x)] + I*Sin[3*(e + 2*f*x)]))/(6*c^3*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
700,1,81,99,3.5762225,"\int \frac{(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^4} \, dx","Integrate[((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^4,x]","\frac{a^3 (\cos (7 e+10 f x)+i \sin (7 e+10 f x)) ((B-7 i A) \cos (e+f x)-(A+7 i B) \sin (e+f x))}{48 c^4 f (\cos (f x)+i \sin (f x))^3}","-\frac{a^3 (-7 B+i A) (1+i \tan (e+f x))^3}{48 c^4 f (1-i \tan (e+f x))^3}-\frac{a^3 (B+i A) (1+i \tan (e+f x))^3}{8 c^4 f (1-i \tan (e+f x))^4}",1,"(a^3*(((-7*I)*A + B)*Cos[e + f*x] - (A + (7*I)*B)*Sin[e + f*x])*(Cos[7*e + 10*f*x] + I*Sin[7*e + 10*f*x]))/(48*c^4*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
701,1,91,122,4.4951889,"\int \frac{(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^5} \, dx","Integrate[((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^5,x]","\frac{a^3 (\cos (8 e+11 f x)+i \sin (8 e+11 f x)) (-4 (A+4 i B) \sin (2 (e+f x))+4 (B-4 i A) \cos (2 (e+f x))-15 i A)}{240 c^5 f (\cos (f x)+i \sin (f x))^3}","-\frac{a^3 (A-5 i B)}{3 c^5 f (\tan (e+f x)+i)^3}+\frac{a^3 (2 B+i A)}{c^5 f (\tan (e+f x)+i)^4}+\frac{4 a^3 (A-i B)}{5 c^5 f (\tan (e+f x)+i)^5}-\frac{a^3 B}{2 c^5 f (\tan (e+f x)+i)^2}",1,"(a^3*((-15*I)*A + 4*((-4*I)*A + B)*Cos[2*(e + f*x)] - 4*(A + (4*I)*B)*Sin[2*(e + f*x)])*(Cos[8*e + 11*f*x] + I*Sin[8*e + 11*f*x]))/(240*c^5*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
702,1,112,127,6.7905191,"\int \frac{(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^6} \, dx","Integrate[((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^6,x]","\frac{a^3 (\cos (9 e+12 f x)+i \sin (9 e+12 f x)) (-(A+3 i B) (9 \sin (e+f x)+10 \sin (3 (e+f x)))+3 (B-27 i A) \cos (e+f x)+10 (B-3 i A) \cos (3 (e+f x)))}{960 c^6 f (\cos (f x)+i \sin (f x))^3}","-\frac{a^3 (5 B+i A)}{4 c^6 f (\tan (e+f x)+i)^4}-\frac{4 a^3 (A-2 i B)}{5 c^6 f (\tan (e+f x)+i)^5}+\frac{2 a^3 (B+i A)}{3 c^6 f (\tan (e+f x)+i)^6}-\frac{i a^3 B}{3 c^6 f (\tan (e+f x)+i)^3}",1,"(a^3*(3*((-27*I)*A + B)*Cos[e + f*x] + 10*((-3*I)*A + B)*Cos[3*(e + f*x)] - (A + (3*I)*B)*(9*Sin[e + f*x] + 10*Sin[3*(e + f*x)]))*(Cos[9*e + 12*f*x] + I*Sin[9*e + 12*f*x]))/(960*c^6*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
703,1,143,125,8.3974364,"\int \frac{(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^7} \, dx","Integrate[((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^7,x]","-\frac{i a^3 (\cos (10 e+13 f x)+i \sin (10 e+13 f x)) (35 (10 A+i B) \cos (2 (e+f x))+20 (5 A+2 i B) \cos (4 (e+f x))-70 i A \sin (2 (e+f x))-40 i A \sin (4 (e+f x))+252 A+175 B \sin (2 (e+f x))+100 B \sin (4 (e+f x)))}{6720 c^7 f (\cos (f x)+i \sin (f x))^3}","\frac{a^3 (A-5 i B)}{5 c^7 f (\tan (e+f x)+i)^5}-\frac{2 a^3 (2 B+i A)}{3 c^7 f (\tan (e+f x)+i)^6}-\frac{4 a^3 (A-i B)}{7 c^7 f (\tan (e+f x)+i)^7}+\frac{a^3 B}{4 c^7 f (\tan (e+f x)+i)^4}",1,"((-1/6720*I)*a^3*(252*A + 35*(10*A + I*B)*Cos[2*(e + f*x)] + 20*(5*A + (2*I)*B)*Cos[4*(e + f*x)] - (70*I)*A*Sin[2*(e + f*x)] + 175*B*Sin[2*(e + f*x)] - (40*I)*A*Sin[4*(e + f*x)] + 100*B*Sin[4*(e + f*x)])*(Cos[10*e + 13*f*x] + I*Sin[10*e + 13*f*x]))/(c^7*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
704,1,182,127,10.6833022,"\int \frac{(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx","Integrate[((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^8,x]","-\frac{i a^3 (\cos (11 e+14 f x)+i \sin (11 e+14 f x)) (56 (55 A+i B) \cos (e+f x)+30 (55 A+9 i B) \cos (3 (e+f x))-280 i A \sin (e+f x)-450 i A \sin (3 (e+f x))-175 i A \sin (5 (e+f x))+385 A \cos (5 (e+f x))+616 B \sin (e+f x)+990 B \sin (3 (e+f x))+385 B \sin (5 (e+f x))+175 i B \cos (5 (e+f x)))}{53760 c^8 f (\cos (f x)+i \sin (f x))^3}","\frac{a^3 (5 B+i A)}{6 c^8 f (\tan (e+f x)+i)^6}+\frac{4 a^3 (A-2 i B)}{7 c^8 f (\tan (e+f x)+i)^7}-\frac{a^3 (B+i A)}{2 c^8 f (\tan (e+f x)+i)^8}+\frac{i a^3 B}{5 c^8 f (\tan (e+f x)+i)^5}",1,"((-1/53760*I)*a^3*(56*(55*A + I*B)*Cos[e + f*x] + 30*(55*A + (9*I)*B)*Cos[3*(e + f*x)] + 385*A*Cos[5*(e + f*x)] + (175*I)*B*Cos[5*(e + f*x)] - (280*I)*A*Sin[e + f*x] + 616*B*Sin[e + f*x] - (450*I)*A*Sin[3*(e + f*x)] + 990*B*Sin[3*(e + f*x)] - (175*I)*A*Sin[5*(e + f*x)] + 385*B*Sin[5*(e + f*x)])*(Cos[11*e + 14*f*x] + I*Sin[11*e + 14*f*x]))/(c^8*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
705,1,111,115,16.416066,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^n}{a+i a \tan (e+f x)} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n)/(a + I*a*Tan[e + f*x]),x]","\frac{2^{n-1} \left(\frac{c}{1+e^{2 i (e+f x)}}\right)^n \left((A (n-1)+i B (n+1)) e^{2 i (e+f x)} \, _2F_1\left(1,1-n;2-n;1+e^{2 i (e+f x)}\right)+(n-1) (A+i B)\right)}{a f (n-1) (\tan (e+f x)-i)}","\frac{(B (n+1)+i A (1-n)) (c-i c \tan (e+f x))^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (1-i \tan (e+f x))\right)}{4 a f n}+\frac{(-B+i A) (c-i c \tan (e+f x))^n}{2 a f (1+i \tan (e+f x))}",1,"(2^(-1 + n)*(c/(1 + E^((2*I)*(e + f*x))))^n*((A + I*B)*(-1 + n) + E^((2*I)*(e + f*x))*(A*(-1 + n) + I*B*(1 + n))*Hypergeometric2F1[1, 1 - n, 2 - n, 1 + E^((2*I)*(e + f*x))]))/(a*f*(-1 + n)*(-I + Tan[e + f*x]))","A",1
706,1,260,157,3.8919386,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^4}{a+i a \tan (e+f x)} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4)/(a + I*a*Tan[e + f*x]),x]","\frac{c^4 (\cos (f x)+i \sin (f x)) (A+B \tan (e+f x)) \left(24 (A+i B) (\sin (e)+i \cos (e)) \cos (2 f x)+24 (A+i B) (\cos (e)-i \sin (e)) \sin (2 f x)+12 (5 B-3 i A) \left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right)^2 \log \left(\cos ^2(e+f x)\right)-24 (3 A+5 i B) (\cos (e)+i \sin (e)) \tan ^{-1}(\tan (f x))+\cos (e) (\tan (e)-i) (2 B \tan (e)+3 (A+5 i B)) \sec ^2(e+f x)+2 (15 A+37 i B) (1+i \tan (e)) \sin (f x) \sec (e+f x)+2 B (\tan (e)-i) \sin (f x) \sec ^3(e+f x)\right)}{6 f (a+i a \tan (e+f x)) (A \cos (e+f x)+B \sin (e+f x))}","-\frac{c^4 (-5 B+i A) \tan ^2(e+f x)}{2 a f}+\frac{c^4 (5 A+12 i B) \tan (e+f x)}{a f}-\frac{8 c^4 (A+i B)}{a f (-\tan (e+f x)+i)}-\frac{4 c^4 (-5 B+3 i A) \log (\cos (e+f x))}{a f}-\frac{4 c^4 x (3 A+5 i B)}{a}-\frac{i B c^4 \tan ^3(e+f x)}{3 a f}",1,"(c^4*(Cos[f*x] + I*Sin[f*x])*(12*((-3*I)*A + 5*B)*Log[Cos[e + f*x]^2]*(Cos[e/2] + I*Sin[e/2])^2 - 24*(3*A + (5*I)*B)*ArcTan[Tan[f*x]]*(Cos[e] + I*Sin[e]) + 24*(A + I*B)*Cos[2*f*x]*(I*Cos[e] + Sin[e]) + 24*(A + I*B)*(Cos[e] - I*Sin[e])*Sin[2*f*x] + 2*(15*A + (37*I)*B)*Sec[e + f*x]*Sin[f*x]*(1 + I*Tan[e]) + 2*B*Sec[e + f*x]^3*Sin[f*x]*(-I + Tan[e]) + Cos[e]*Sec[e + f*x]^2*(-I + Tan[e])*(3*(A + (5*I)*B) + 2*B*Tan[e]))*(A + B*Tan[e + f*x]))/(6*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x]))","A",1
707,1,212,121,6.2014302,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^3}{a+i a \tan (e+f x)} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3)/(a + I*a*Tan[e + f*x]),x]","\frac{c^3 (\cos (f x)+i \sin (f x)) (A+B \tan (e+f x)) \left(4 (A+i B) (\sin (e)+i \cos (e)) \cos (2 f x)+4 (A+i B) (\cos (e)-i \sin (e)) \sin (2 f x)+4 (2 B-i A) (\cos (e)+i \sin (e)) \log \left(\cos ^2(e+f x)\right)-8 (A+2 i B) (\cos (e)+i \sin (e)) \tan ^{-1}(\tan (f x))+2 (A+4 i B) (1+i \tan (e)) \sin (f x) \sec (e+f x)+B (\cos (e)+i \sin (e)) \sec ^2(e+f x)\right)}{2 f (a+i a \tan (e+f x)) (A \cos (e+f x)+B \sin (e+f x))}","\frac{c^3 (A+4 i B) \tan (e+f x)}{a f}-\frac{4 c^3 (A+i B)}{a f (-\tan (e+f x)+i)}-\frac{4 c^3 (-2 B+i A) \log (\cos (e+f x))}{a f}-\frac{4 c^3 x (A+2 i B)}{a}+\frac{B c^3 \tan ^2(e+f x)}{2 a f}",1,"(c^3*(Cos[f*x] + I*Sin[f*x])*(-8*(A + (2*I)*B)*ArcTan[Tan[f*x]]*(Cos[e] + I*Sin[e]) + 4*((-I)*A + 2*B)*Log[Cos[e + f*x]^2]*(Cos[e] + I*Sin[e]) + B*Sec[e + f*x]^2*(Cos[e] + I*Sin[e]) + 4*(A + I*B)*Cos[2*f*x]*(I*Cos[e] + Sin[e]) + 4*(A + I*B)*(Cos[e] - I*Sin[e])*Sin[2*f*x] + 2*(A + (4*I)*B)*Sec[e + f*x]*Sin[f*x]*(1 + I*Tan[e]))*(A + B*Tan[e + f*x]))/(2*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x]))","A",1
708,1,184,96,3.7615201,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^2}{a+i a \tan (e+f x)} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2)/(a + I*a*Tan[e + f*x]),x]","\frac{c^2 (\cos (f x)+i \sin (f x)) (A+B \tan (e+f x)) \left(2 (A+i B) (\sin (e)+i \cos (e)) \cos (2 f x)+2 (A+i B) (\cos (e)-i \sin (e)) \sin (2 f x)+(3 B-i A) (\cos (e)+i \sin (e)) \log \left(\cos ^2(e+f x)\right)-2 (A+3 i B) (\cos (e)+i \sin (e)) \tan ^{-1}(\tan (f x))-2 B (\tan (e)-i) \sin (f x) \sec (e+f x)\right)}{2 f (a+i a \tan (e+f x)) (A \cos (e+f x)+B \sin (e+f x))}","-\frac{2 c^2 (A+i B)}{a f (-\tan (e+f x)+i)}-\frac{c^2 (-3 B+i A) \log (\cos (e+f x))}{a f}-\frac{c^2 x (A+3 i B)}{a}+\frac{i B c^2 \tan (e+f x)}{a f}",1,"(c^2*(Cos[f*x] + I*Sin[f*x])*(-2*(A + (3*I)*B)*ArcTan[Tan[f*x]]*(Cos[e] + I*Sin[e]) + ((-I)*A + 3*B)*Log[Cos[e + f*x]^2]*(Cos[e] + I*Sin[e]) + 2*(A + I*B)*Cos[2*f*x]*(I*Cos[e] + Sin[e]) + 2*(A + I*B)*(Cos[e] - I*Sin[e])*Sin[2*f*x] - 2*B*Sec[e + f*x]*Sin[f*x]*(-I + Tan[e]))*(A + B*Tan[e + f*x]))/(2*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x]))","A",1
709,1,124,57,1.5836366,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))}{a+i a \tan (e+f x)} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x]))/(a + I*a*Tan[e + f*x]),x]","\frac{c \cos (e+f x) (A+B \tan (e+f x)) \left(\tan (e+f x) \left(-i A+B \log \left(\cos ^2(e+f x)\right)+B\right)+A-2 i B \tan ^{-1}(\tan (f x)) (\tan (e+f x)-i)-i B \log \left(\cos ^2(e+f x)\right)+i B\right)}{2 a f (\tan (e+f x)-i) (A \cos (e+f x)+B \sin (e+f x))}","-\frac{c (A+i B)}{a f (-\tan (e+f x)+i)}+\frac{B c \log (\cos (e+f x))}{a f}-\frac{i B c x}{a}",1,"(c*Cos[e + f*x]*(A + B*Tan[e + f*x])*(A + I*B - I*B*Log[Cos[e + f*x]^2] + ((-I)*A + B + B*Log[Cos[e + f*x]^2])*Tan[e + f*x] - (2*I)*B*ArcTan[Tan[f*x]]*(-I + Tan[e + f*x])))/(2*a*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(-I + Tan[e + f*x]))","B",1
710,1,102,47,0.5190001,"\int \frac{A+B \tan (e+f x)}{a+i a \tan (e+f x)} \, dx","Integrate[(A + B*Tan[e + f*x])/(a + I*a*Tan[e + f*x]),x]","\frac{\cos (e+f x) (A+B \tan (e+f x)) ((A (2 f x-i)-2 i B f x+B) \tan (e+f x)-2 i A f x+A+B (-2 f x+i))}{4 a f (\tan (e+f x)-i) (A \cos (e+f x)+B \sin (e+f x))}","\frac{-B+i A}{2 f (a+i a \tan (e+f x))}+\frac{x (A-i B)}{2 a}",1,"(Cos[e + f*x]*(A + B*Tan[e + f*x])*(A - (2*I)*A*f*x + B*(I - 2*f*x) + (B - (2*I)*B*f*x + A*(-I + 2*f*x))*Tan[e + f*x]))/(4*a*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(-I + Tan[e + f*x]))","B",1
711,1,43,45,0.0836512,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])),x]","\frac{A (2 (e+f x)+\sin (2 (e+f x)))-2 B \cos ^2(e+f x)}{4 a c f}","\frac{A x}{2 a c}-\frac{\cos ^2(e+f x) (B-A \tan (e+f x))}{2 a c f}",1,"(-2*B*Cos[e + f*x]^2 + A*(2*(e + f*x) + Sin[2*(e + f*x)]))/(4*a*c*f)","A",1
712,1,166,113,2.4870364,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^2} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2),x]","\frac{(\cos (2 (e+f x))+i \sin (2 (e+f x))) (A+B \tan (e+f x)) (2 (A (-3-6 i f x)+B (2 f x+i)) \cos (e+f x)+(A+3 i B) \cos (3 (e+f x))-\sin (e+f x) ((-2 B+6 i A) \cos (2 (e+f x))+12 A f x+9 i A+4 i B f x+B))}{32 a c^2 f (\tan (e+f x)-i) (A \cos (e+f x)+B \sin (e+f x))}","-\frac{A+i B}{8 a c^2 f (-\tan (e+f x)+i)}+\frac{B+i A}{8 a c^2 f (\tan (e+f x)+i)^2}+\frac{x (3 A+i B)}{8 a c^2}+\frac{A}{4 a c^2 f (\tan (e+f x)+i)}",1,"((2*(A*(-3 - (6*I)*f*x) + B*(I + 2*f*x))*Cos[e + f*x] + (A + (3*I)*B)*Cos[3*(e + f*x)] - ((9*I)*A + B + 12*A*f*x + (4*I)*B*f*x + ((6*I)*A - 2*B)*Cos[2*(e + f*x)])*Sin[e + f*x])*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*(A + B*Tan[e + f*x]))/(32*a*c^2*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(-I + Tan[e + f*x]))","A",1
713,1,203,149,2.5247436,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^3} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3),x]","\frac{(\cos (3 (e+f x))+i \sin (3 (e+f x))) (A+B \tan (e+f x)) (3 (A (-2-8 i f x)+B (4 f x+i)) \cos (2 (e+f x))+2 (A+2 i B) \cos (4 (e+f x))-24 A f x \sin (2 (e+f x))-6 i A \sin (2 (e+f x))-4 i A \sin (4 (e+f x))-18 A-3 B \sin (2 (e+f x))-12 i B f x \sin (2 (e+f x))+2 B \sin (4 (e+f x)))}{96 a c^3 f (\tan (e+f x)-i) (A \cos (e+f x)+B \sin (e+f x))}","-\frac{A+i B}{16 a c^3 f (-\tan (e+f x)+i)}+\frac{3 A+i B}{16 a c^3 f (\tan (e+f x)+i)}-\frac{A-i B}{12 a c^3 f (\tan (e+f x)+i)^3}+\frac{x (2 A+i B)}{8 a c^3}+\frac{i A}{8 a c^3 f (\tan (e+f x)+i)^2}",1,"((Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*(-18*A + 3*(A*(-2 - (8*I)*f*x) + B*(I + 4*f*x))*Cos[2*(e + f*x)] + 2*(A + (2*I)*B)*Cos[4*(e + f*x)] - (6*I)*A*Sin[2*(e + f*x)] - 3*B*Sin[2*(e + f*x)] - 24*A*f*x*Sin[2*(e + f*x)] - (12*I)*B*f*x*Sin[2*(e + f*x)] - (4*I)*A*Sin[4*(e + f*x)] + 2*B*Sin[4*(e + f*x)])*(A + B*Tan[e + f*x]))/(96*a*c^3*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(-I + Tan[e + f*x]))","A",1
714,1,221,181,2.7731894,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4),x]","\frac{\sec (e+f x) (\cos (4 (e+f x))+i \sin (4 (e+f x))) (-12 (15 A+i B) \cos (e+f x)+4 (-30 i A f x-5 A+18 B f x+3 i B) \cos (3 (e+f x))+60 i A \sin (e+f x)-20 i A \sin (3 (e+f x))-120 A f x \sin (3 (e+f x))-15 i A \sin (5 (e+f x))+9 A \cos (5 (e+f x))-36 B \sin (e+f x)-12 B \sin (3 (e+f x))-72 i B f x \sin (3 (e+f x))+9 B \sin (5 (e+f x))+15 i B \cos (5 (e+f x)))}{768 a c^4 f (\tan (e+f x)-i)}","-\frac{A+i B}{32 a c^4 f (-\tan (e+f x)+i)}+\frac{2 A+i B}{16 a c^4 f (\tan (e+f x)+i)}+\frac{-B+3 i A}{32 a c^4 f (\tan (e+f x)+i)^2}-\frac{B+i A}{16 a c^4 f (\tan (e+f x)+i)^4}+\frac{x (5 A+3 i B)}{32 a c^4}-\frac{A}{12 a c^4 f (\tan (e+f x)+i)^3}",1,"(Sec[e + f*x]*(Cos[4*(e + f*x)] + I*Sin[4*(e + f*x)])*(-12*(15*A + I*B)*Cos[e + f*x] + 4*(-5*A + (3*I)*B - (30*I)*A*f*x + 18*B*f*x)*Cos[3*(e + f*x)] + 9*A*Cos[5*(e + f*x)] + (15*I)*B*Cos[5*(e + f*x)] + (60*I)*A*Sin[e + f*x] - 36*B*Sin[e + f*x] - (20*I)*A*Sin[3*(e + f*x)] - 12*B*Sin[3*(e + f*x)] - 120*A*f*x*Sin[3*(e + f*x)] - (72*I)*B*f*x*Sin[3*(e + f*x)] - (15*I)*A*Sin[5*(e + f*x)] + 9*B*Sin[5*(e + f*x)]))/(768*a*c^4*f*(-I + Tan[e + f*x]))","A",1
715,0,0,115,102.4964289,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n)/(a + I*a*Tan[e + f*x])^2,x]","\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx","\frac{(B (n+2)+i A (2-n)) (c-i c \tan (e+f x))^n \, _2F_1\left(2,n;n+1;\frac{1}{2} (1-i \tan (e+f x))\right)}{16 a^2 f n}+\frac{(-B+i A) (c-i c \tan (e+f x))^n}{4 a^2 f (1+i \tan (e+f x))^2}",1,"Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n)/(a + I*a*Tan[e + f*x])^2, x]","F",-1
716,1,1357,194,11.3803835,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^5}{(a+i a \tan (e+f x))^2} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^5)/(a + I*a*Tan[e + f*x])^2,x]","\frac{4 (5 B-3 i A) \cos (2 f x) \sec (e+f x) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x)) c^5}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}-\frac{4 (3 A+5 i B) \sec (e+f x) (\cos (f x)+i \sin (f x))^2 \sin (2 f x) (A+B \tan (e+f x)) c^5}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{\sec (e) \sec ^4(e+f x) (\cos (f x)+i \sin (f x))^2 \left(-\frac{1}{2} B \cos (2 e-f x) c^5+\frac{1}{2} B \cos (2 e+f x) c^5-\frac{1}{2} i B \sin (2 e-f x) c^5+\frac{1}{2} i B \sin (2 e+f x) c^5\right) (A+B \tan (e+f x))}{3 f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{\sec (e) \sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left(-\frac{21}{2} i A \cos (2 e-f x) c^5+\frac{73}{2} B \cos (2 e-f x) c^5+\frac{21}{2} i A \cos (2 e+f x) c^5-\frac{73}{2} B \cos (2 e+f x) c^5+\frac{21}{2} A \sin (2 e-f x) c^5+\frac{73}{2} i B \sin (2 e-f x) c^5-\frac{21}{2} A \sin (2 e+f x) c^5-\frac{73}{2} i B \sin (2 e+f x) c^5\right) (A+B \tan (e+f x))}{3 f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{x \sec (e+f x) (\cos (f x)+i \sin (f x))^2 \left(-24 A c^5-56 i B c^5-24 i A \tan (e) c^5+56 B \tan (e) c^5+(7 B-3 i A) \left(8 \cos (2 e) c^5+8 i \sin (2 e) c^5\right) \tan (e)\right) (A+B \tan (e+f x))}{(A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{\sec (e+f x) \left(3 A \cos (e) c^5+7 i B \cos (e) c^5+3 i A \sin (e) c^5-7 B \sin (e) c^5\right) \left(8 \tan ^{-1}(\tan (f x)) \cos (e)+8 i \tan ^{-1}(\tan (f x)) \sin (e)\right) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x))}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{\sec (e+f x) \left(3 A \cos (e) c^5+7 i B \cos (e) c^5+3 i A \sin (e) c^5-7 B \sin (e) c^5\right) \left(4 i \cos (e) \log \left(\cos ^2(e+f x)\right)-4 \log \left(\cos ^2(e+f x)\right) \sin (e)\right) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x))}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{\sec (e) \sec ^3(e+f x) (3 A \cos (e)+21 i B \cos (e)+2 B \sin (e)) \left(\frac{1}{6} i c^5 \cos (2 e)-\frac{1}{6} c^5 \sin (2 e)\right) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x))}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{(A+i B) \cos (4 f x) \sec (e+f x) \left(2 i \cos (2 e) c^5+2 \sin (2 e) c^5\right) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x))}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{(3 A+7 i B) \sec (e+f x) \left(8 f x \cos (2 e) c^5+8 i f x \sin (2 e) c^5\right) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x))}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{(A+i B) \sec (e+f x) \left(2 c^5 \cos (2 e)-2 i c^5 \sin (2 e)\right) (\cos (f x)+i \sin (f x))^2 \sin (4 f x) (A+B \tan (e+f x))}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}","\frac{c^5 (-7 B+i A) \tan ^2(e+f x)}{2 a^2 f}-\frac{c^5 (7 A+24 i B) \tan (e+f x)}{a^2 f}+\frac{16 c^5 (2 A+3 i B)}{a^2 f (-\tan (e+f x)+i)}-\frac{8 c^5 (-B+i A)}{a^2 f (-\tan (e+f x)+i)^2}+\frac{8 c^5 (-7 B+3 i A) \log (\cos (e+f x))}{a^2 f}+\frac{8 c^5 x (3 A+7 i B)}{a^2}+\frac{i B c^5 \tan ^3(e+f x)}{3 a^2 f}",1,"(4*((-3*I)*A + 5*B)*c^5*Cos[2*f*x]*Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (Sec[e + f*x]*(3*A*c^5*Cos[e] + (7*I)*B*c^5*Cos[e] + (3*I)*A*c^5*Sin[e] - 7*B*c^5*Sin[e])*(8*ArcTan[Tan[f*x]]*Cos[e] + (8*I)*ArcTan[Tan[f*x]]*Sin[e])*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (Sec[e + f*x]*(3*A*c^5*Cos[e] + (7*I)*B*c^5*Cos[e] + (3*I)*A*c^5*Sin[e] - 7*B*c^5*Sin[e])*((4*I)*Cos[e]*Log[Cos[e + f*x]^2] - 4*Log[Cos[e + f*x]^2]*Sin[e])*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (Sec[e]*Sec[e + f*x]^3*(3*A*Cos[e] + (21*I)*B*Cos[e] + 2*B*Sin[e])*((I/6)*c^5*Cos[2*e] - (c^5*Sin[2*e])/6)*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + ((A + I*B)*Cos[4*f*x]*Sec[e + f*x]*((2*I)*c^5*Cos[2*e] + 2*c^5*Sin[2*e])*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + ((3*A + (7*I)*B)*Sec[e + f*x]*(8*c^5*f*x*Cos[2*e] + (8*I)*c^5*f*x*Sin[2*e])*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) - (4*(3*A + (5*I)*B)*c^5*Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])^2*Sin[2*f*x]*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + ((A + I*B)*Sec[e + f*x]*(2*c^5*Cos[2*e] - (2*I)*c^5*Sin[2*e])*(Cos[f*x] + I*Sin[f*x])^2*Sin[4*f*x]*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (Sec[e]*Sec[e + f*x]^4*(Cos[f*x] + I*Sin[f*x])^2*(-1/2*(B*c^5*Cos[2*e - f*x]) + (B*c^5*Cos[2*e + f*x])/2 - (I/2)*B*c^5*Sin[2*e - f*x] + (I/2)*B*c^5*Sin[2*e + f*x])*(A + B*Tan[e + f*x]))/(3*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (Sec[e]*Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*(((-21*I)/2)*A*c^5*Cos[2*e - f*x] + (73*B*c^5*Cos[2*e - f*x])/2 + ((21*I)/2)*A*c^5*Cos[2*e + f*x] - (73*B*c^5*Cos[2*e + f*x])/2 + (21*A*c^5*Sin[2*e - f*x])/2 + ((73*I)/2)*B*c^5*Sin[2*e - f*x] - (21*A*c^5*Sin[2*e + f*x])/2 - ((73*I)/2)*B*c^5*Sin[2*e + f*x])*(A + B*Tan[e + f*x]))/(3*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (x*Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])^2*(-24*A*c^5 - (56*I)*B*c^5 - (24*I)*A*c^5*Tan[e] + 56*B*c^5*Tan[e] + ((-3*I)*A + 7*B)*(8*c^5*Cos[2*e] + (8*I)*c^5*Sin[2*e])*Tan[e])*(A + B*Tan[e + f*x]))/((A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2)","B",0
717,1,1079,158,9.3583727,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^2} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4)/(a + I*a*Tan[e + f*x])^2,x]","c^4 \left(\frac{\left(-\frac{1}{2} B \cos (2 e)-\frac{1}{2} i B \sin (2 e)\right) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x)) \sec ^3(e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{\sec (e) (\cos (f x)+i \sin (f x))^2 \left(-\frac{1}{2} i A \cos (2 e-f x)+3 B \cos (2 e-f x)+\frac{1}{2} i A \cos (2 e+f x)-3 B \cos (2 e+f x)+\frac{1}{2} A \sin (2 e-f x)+3 i B \sin (2 e-f x)-\frac{1}{2} A \sin (2 e+f x)-3 i B \sin (2 e+f x)\right) (A+B \tan (e+f x)) \sec ^2(e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{x (\cos (f x)+i \sin (f x))^2 (-6 i \tan (e) A-6 A-18 i B+18 B \tan (e)+(3 B-i A) (6 \cos (2 e)+6 i \sin (2 e)) \tan (e)) (A+B \tan (e+f x)) \sec (e+f x)}{(A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{4 (2 B-i A) \cos (2 f x) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x)) \sec (e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{(A \cos (e)+3 i B \cos (e)+i A \sin (e)-3 B \sin (e)) \left(6 \tan ^{-1}(\tan (f x)) \cos (e)+6 i \tan ^{-1}(\tan (f x)) \sin (e)\right) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x)) \sec (e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{(A \cos (e)+3 i B \cos (e)+i A \sin (e)-3 B \sin (e)) \left(3 i \cos (e) \log \left(\cos ^2(e+f x)\right)-3 \log \left(\cos ^2(e+f x)\right) \sin (e)\right) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x)) \sec (e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{(A+i B) \cos (4 f x) (i \cos (2 e)+\sin (2 e)) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x)) \sec (e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{(A+3 i B) (6 f x \cos (2 e)+6 i f x \sin (2 e)) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x)) \sec (e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}-\frac{4 (A+2 i B) (\cos (f x)+i \sin (f x))^2 \sin (2 f x) (A+B \tan (e+f x)) \sec (e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}+\frac{(A+i B) (\cos (2 e)-i \sin (2 e)) (\cos (f x)+i \sin (f x))^2 \sin (4 f x) (A+B \tan (e+f x)) \sec (e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^2}\right)","-\frac{c^4 (A+6 i B) \tan (e+f x)}{a^2 f}+\frac{4 c^4 (3 A+5 i B)}{a^2 f (-\tan (e+f x)+i)}-\frac{4 c^4 (-B+i A)}{a^2 f (-\tan (e+f x)+i)^2}+\frac{6 c^4 (-3 B+i A) \log (\cos (e+f x))}{a^2 f}+\frac{6 c^4 x (A+3 i B)}{a^2}-\frac{B c^4 \tan ^2(e+f x)}{2 a^2 f}",1,"c^4*((4*((-I)*A + 2*B)*Cos[2*f*x]*Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (Sec[e + f*x]*(A*Cos[e] + (3*I)*B*Cos[e] + I*A*Sin[e] - 3*B*Sin[e])*(6*ArcTan[Tan[f*x]]*Cos[e] + (6*I)*ArcTan[Tan[f*x]]*Sin[e])*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (Sec[e + f*x]*(A*Cos[e] + (3*I)*B*Cos[e] + I*A*Sin[e] - 3*B*Sin[e])*((3*I)*Cos[e]*Log[Cos[e + f*x]^2] - 3*Log[Cos[e + f*x]^2]*Sin[e])*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + ((A + I*B)*Cos[4*f*x]*Sec[e + f*x]*(I*Cos[2*e] + Sin[2*e])*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (Sec[e + f*x]^3*(-1/2*(B*Cos[2*e]) - (I/2)*B*Sin[2*e])*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + ((A + (3*I)*B)*Sec[e + f*x]*(6*f*x*Cos[2*e] + (6*I)*f*x*Sin[2*e])*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) - (4*(A + (2*I)*B)*Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])^2*Sin[2*f*x]*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + ((A + I*B)*Sec[e + f*x]*(Cos[2*e] - I*Sin[2*e])*(Cos[f*x] + I*Sin[f*x])^2*Sin[4*f*x]*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (Sec[e]*Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*((-1/2*I)*A*Cos[2*e - f*x] + 3*B*Cos[2*e - f*x] + (I/2)*A*Cos[2*e + f*x] - 3*B*Cos[2*e + f*x] + (A*Sin[2*e - f*x])/2 + (3*I)*B*Sin[2*e - f*x] - (A*Sin[2*e + f*x])/2 - (3*I)*B*Sin[2*e + f*x])*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (x*Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])^2*(-6*A - (18*I)*B - (6*I)*A*Tan[e] + 18*B*Tan[e] + ((-I)*A + 3*B)*(6*Cos[2*e] + (6*I)*Sin[2*e])*Tan[e])*(A + B*Tan[e + f*x]))/((A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2))","B",1
718,1,413,128,7.3800821,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^2} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3)/(a + I*a*Tan[e + f*x])^2,x]","-\frac{c^3 \sec (e) \sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left(i (A+5 i B) \cos ^3(e) \log \left(\cos ^2(e+f x)\right)+\cos (e) \left(\cos (2 e) (2 f x (A+5 i B)+(A+i B) \sin (4 f x)+i (A+i B) \cos (4 f x))+2 i A f x \sin (2 e)-i A \sin (2 e) \sin (4 f x)+A \sin (2 e) \cos (4 f x)-i A \sin ^2(e) \log \left(\cos ^2(e+f x)\right)-2 A f x-2 A \sin (2 f x)-2 i A \cos (2 f x)-10 B f x \sin (2 e)+B \sin (2 e) \sin (4 f x)+i B \sin (2 e) \cos (4 f x)+5 B \sin ^2(e) \log \left(\cos ^2(e+f x)\right)-10 i B f x-6 i B \sin (2 f x)+6 B \cos (2 f x)\right)-2 (A+5 i B) \sin (e) \cos ^2(e) \log \left(\cos ^2(e+f x)\right)+2 (A+5 i B) \cos (e) (\cos (2 e)+i \sin (2 e)) \tan ^{-1}(\tan (f x))+(\cos (e)+i \sin (e)) \sec (e+f x) (2 \cos (e) (f x (5 B-i A) \sin (2 e+f x)+i \sin (f x) (A f x+B (-1+5 i f x)))+B \cos (e-f x)-B \cos (e+f x))\right)}{2 a^2 f (\tan (e+f x)-i)^2}","\frac{4 c^3 (A+2 i B)}{a^2 f (-\tan (e+f x)+i)}-\frac{2 c^3 (-B+i A)}{a^2 f (-\tan (e+f x)+i)^2}+\frac{c^3 (-5 B+i A) \log (\cos (e+f x))}{a^2 f}+\frac{c^3 x (A+5 i B)}{a^2}-\frac{i B c^3 \tan (e+f x)}{a^2 f}",1,"-1/2*(c^3*Sec[e]*Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*(I*(A + (5*I)*B)*Cos[e]^3*Log[Cos[e + f*x]^2] - 2*(A + (5*I)*B)*Cos[e]^2*Log[Cos[e + f*x]^2]*Sin[e] + 2*(A + (5*I)*B)*ArcTan[Tan[f*x]]*Cos[e]*(Cos[2*e] + I*Sin[2*e]) + Cos[e]*(-2*A*f*x - (10*I)*B*f*x - (2*I)*A*Cos[2*f*x] + 6*B*Cos[2*f*x] - I*A*Log[Cos[e + f*x]^2]*Sin[e]^2 + 5*B*Log[Cos[e + f*x]^2]*Sin[e]^2 + (2*I)*A*f*x*Sin[2*e] - 10*B*f*x*Sin[2*e] + A*Cos[4*f*x]*Sin[2*e] + I*B*Cos[4*f*x]*Sin[2*e] - 2*A*Sin[2*f*x] - (6*I)*B*Sin[2*f*x] - I*A*Sin[2*e]*Sin[4*f*x] + B*Sin[2*e]*Sin[4*f*x] + Cos[2*e]*(2*(A + (5*I)*B)*f*x + I*(A + I*B)*Cos[4*f*x] + (A + I*B)*Sin[4*f*x])) + Sec[e + f*x]*(Cos[e] + I*Sin[e])*(B*Cos[e - f*x] - B*Cos[e + f*x] + 2*Cos[e]*(I*(A*f*x + B*(-1 + (5*I)*f*x))*Sin[f*x] + ((-I)*A + 5*B)*f*x*Sin[2*e + f*x]))))/(a^2*f*(-I + Tan[e + f*x])^2)","B",1
719,1,140,97,2.807878,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^2}{(a+i a \tan (e+f x))^2} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2)/(a + I*a*Tan[e + f*x])^2,x]","\frac{c^2 \sec ^2(e+f x) \left(\cos (2 (e+f x)) \left(-i A+2 B \log \left(\cos ^2(e+f x)\right)+B\right)-A \sin (2 (e+f x))-i B \sin (2 (e+f x))+2 i B \sin (2 (e+f x)) \log \left(\cos ^2(e+f x)\right)+4 B \tan ^{-1}(\tan (f x)) (\sin (2 (e+f x))-i \cos (2 (e+f x)))-4 B\right)}{4 a^2 f (\tan (e+f x)-i)^2}","\frac{c^2 (A+3 i B)}{a^2 f (-\tan (e+f x)+i)}-\frac{c^2 (-B+i A)}{a^2 f (-\tan (e+f x)+i)^2}-\frac{B c^2 \log (\cos (e+f x))}{a^2 f}+\frac{i B c^2 x}{a^2}",1,"(c^2*Sec[e + f*x]^2*(-4*B + Cos[2*(e + f*x)]*((-I)*A + B + 2*B*Log[Cos[e + f*x]^2]) - A*Sin[2*(e + f*x)] - I*B*Sin[2*(e + f*x)] + (2*I)*B*Log[Cos[e + f*x]^2]*Sin[2*(e + f*x)] + 4*B*ArcTan[Tan[f*x]]*((-I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)])))/(4*a^2*f*(-I + Tan[e + f*x])^2)","A",1
720,1,58,48,1.6519991,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))}{(a+i a \tan (e+f x))^2} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x]))/(a + I*a*Tan[e + f*x])^2,x]","\frac{(c-i c \tan (e+f x)) ((A-3 i B) \tan (e+f x)-3 i A-B)}{8 a^2 f (\tan (e+f x)-i)^2}","-\frac{c (A+B \tan (e+f x))^2}{2 a^2 f (-B+i A) (1+i \tan (e+f x))^2}",1,"(((-3*I)*A - B + (A - (3*I)*B)*Tan[e + f*x])*(c - I*c*Tan[e + f*x]))/(8*a^2*f*(-I + Tan[e + f*x])^2)","A",1
721,1,94,80,0.5713013,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(A + B*Tan[e + f*x])/(a + I*a*Tan[e + f*x])^2,x]","-\frac{\sec ^2(e+f x) ((4 i A f x+A+4 B f x+i B) \sin (2 (e+f x))+(A (4 f x+i)+B (-1-4 i f x)) \cos (2 (e+f x))+4 i A)}{16 a^2 f (\tan (e+f x)-i)^2}","\frac{B+i A}{4 f \left(a^2+i a^2 \tan (e+f x)\right)}+\frac{x (A-i B)}{4 a^2}+\frac{-B+i A}{4 f (a+i a \tan (e+f x))^2}",1,"-1/16*(Sec[e + f*x]^2*((4*I)*A + (B*(-1 - (4*I)*f*x) + A*(I + 4*f*x))*Cos[2*(e + f*x)] + (A + I*B + (4*I)*A*f*x + 4*B*f*x)*Sin[2*(e + f*x)]))/(a^2*f*(-I + Tan[e + f*x])^2)","A",1
722,1,129,117,2.3109937,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])),x]","-\frac{2 (A-3 i B) \cos (2 (e+f x))+(B+3 i A) \sin (3 (e+f x)) \sec (e+f x)-12 A f x \tan (e+f x)+6 i A \tan (e+f x)+12 i A f x-7 A-2 B \tan (e+f x)+4 i B f x \tan (e+f x)+4 B f x+i B}{32 a^2 c f (\tan (e+f x)-i)}","\frac{A-i B}{8 a^2 c f (\tan (e+f x)+i)}-\frac{-B+i A}{8 a^2 c f (-\tan (e+f x)+i)^2}+\frac{x (3 A-i B)}{8 a^2 c}-\frac{A}{4 a^2 c f (-\tan (e+f x)+i)}",1,"-1/32*(-7*A + I*B + (12*I)*A*f*x + 4*B*f*x + 2*(A - (3*I)*B)*Cos[2*(e + f*x)] + ((3*I)*A + B)*Sec[e + f*x]*Sin[3*(e + f*x)] + (6*I)*A*Tan[e + f*x] - 2*B*Tan[e + f*x] - 12*A*f*x*Tan[e + f*x] + (4*I)*B*f*x*Tan[e + f*x])/(a^2*c*f*(-I + Tan[e + f*x]))","A",1
723,1,53,71,0.1293411,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^2} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^2),x]","\frac{A (12 (e+f x)+8 \sin (2 (e+f x))+\sin (4 (e+f x)))-8 B \cos ^4(e+f x)}{32 a^2 c^2 f}","-\frac{\cos ^4(e+f x) (B-A \tan (e+f x))}{4 a^2 c^2 f}+\frac{3 A \sin (e+f x) \cos (e+f x)}{8 a^2 c^2 f}+\frac{3 A x}{8 a^2 c^2}",1,"(-8*B*Cos[e + f*x]^4 + A*(12*(e + f*x) + 8*Sin[2*(e + f*x)] + Sin[4*(e + f*x)]))/(32*a^2*c^2*f)","A",1
724,1,217,183,2.4914212,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^3} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^3),x]","\frac{\sec ^2(e+f x) (\cos (3 (e+f x))+i \sin (3 (e+f x))) (12 (A (-10 f x+5 i)-2 i B f x+B) \cos (e+f x)+3 (9 B-5 i A) \cos (3 (e+f x))-60 A \sin (e+f x)+120 i A f x \sin (e+f x)-45 A \sin (3 (e+f x))-5 A \sin (5 (e+f x))-i A \cos (5 (e+f x))+12 i B \sin (e+f x)-24 B f x \sin (e+f x)-9 i B \sin (3 (e+f x))-i B \sin (5 (e+f x))+5 B \cos (5 (e+f x)))}{384 a^2 c^3 f (\tan (e+f x)-i)^2}","-\frac{2 A+i B}{16 a^2 c^3 f (-\tan (e+f x)+i)}-\frac{-B+i A}{32 a^2 c^3 f (-\tan (e+f x)+i)^2}+\frac{B+3 i A}{32 a^2 c^3 f (\tan (e+f x)+i)^2}-\frac{A-i B}{24 a^2 c^3 f (\tan (e+f x)+i)^3}+\frac{x (5 A+i B)}{16 a^2 c^3}+\frac{3 A}{16 a^2 c^3 f (\tan (e+f x)+i)}",1,"(Sec[e + f*x]^2*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*(12*(B - (2*I)*B*f*x + A*(5*I - 10*f*x))*Cos[e + f*x] + 3*((-5*I)*A + 9*B)*Cos[3*(e + f*x)] - I*A*Cos[5*(e + f*x)] + 5*B*Cos[5*(e + f*x)] - 60*A*Sin[e + f*x] + (12*I)*B*Sin[e + f*x] + (120*I)*A*f*x*Sin[e + f*x] - 24*B*f*x*Sin[e + f*x] - 45*A*Sin[3*(e + f*x)] - (9*I)*B*Sin[3*(e + f*x)] - 5*A*Sin[5*(e + f*x)] - I*B*Sin[5*(e + f*x)]))/(384*a^2*c^3*f*(-I + Tan[e + f*x])^2)","A",1
725,1,232,221,2.8473112,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^4} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^4),x]","\frac{\sec ^2(e+f x) (\sin (4 (e+f x))-i \cos (4 (e+f x))) (30 (A (-3-12 i f x)+B (4 f x+i)) \cos (2 (e+f x))+16 (3 A+4 i B) \cos (4 (e+f x))-360 A f x \sin (2 (e+f x))-90 i A \sin (2 (e+f x))-96 i A \sin (4 (e+f x))-9 i A \sin (6 (e+f x))+3 A \cos (6 (e+f x))-240 A-30 B \sin (2 (e+f x))-120 i B f x \sin (2 (e+f x))+32 B \sin (4 (e+f x))+3 B \sin (6 (e+f x))+9 i B \cos (6 (e+f x)))}{1536 a^2 c^4 f (\tan (e+f x)-i)^2}","-\frac{5 A+3 i B}{64 a^2 c^4 f (-\tan (e+f x)+i)}+\frac{5 A+i B}{32 a^2 c^4 f (\tan (e+f x)+i)}-\frac{-B+i A}{64 a^2 c^4 f (-\tan (e+f x)+i)^2}-\frac{3 A-i B}{48 a^2 c^4 f (\tan (e+f x)+i)^3}-\frac{B+i A}{32 a^2 c^4 f (\tan (e+f x)+i)^4}+\frac{5 x (3 A+i B)}{64 a^2 c^4}+\frac{3 i A}{32 a^2 c^4 f (\tan (e+f x)+i)^2}",1,"(Sec[e + f*x]^2*((-I)*Cos[4*(e + f*x)] + Sin[4*(e + f*x)])*(-240*A + 30*(A*(-3 - (12*I)*f*x) + B*(I + 4*f*x))*Cos[2*(e + f*x)] + 16*(3*A + (4*I)*B)*Cos[4*(e + f*x)] + 3*A*Cos[6*(e + f*x)] + (9*I)*B*Cos[6*(e + f*x)] - (90*I)*A*Sin[2*(e + f*x)] - 30*B*Sin[2*(e + f*x)] - 360*A*f*x*Sin[2*(e + f*x)] - (120*I)*B*f*x*Sin[2*(e + f*x)] - (96*I)*A*Sin[4*(e + f*x)] + 32*B*Sin[4*(e + f*x)] - (9*I)*A*Sin[6*(e + f*x)] + 3*B*Sin[6*(e + f*x)]))/(1536*a^2*c^4*f*(-I + Tan[e + f*x])^2)","A",1
726,1,274,251,3.5359059,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^5} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^5),x]","\frac{\sec ^2(e+f x) (\cos (5 (e+f x))+i \sin (5 (e+f x))) (50 i (21 A+i B) \cos (e+f x)+20 (A (-42 f x+7 i)+3 B (1-6 i f x)) \cos (3 (e+f x))+350 A \sin (e+f x)-140 A \sin (3 (e+f x))+840 i A f x \sin (3 (e+f x))-175 A \sin (5 (e+f x))-14 A \sin (7 (e+f x))-105 i A \cos (5 (e+f x))-6 i A \cos (7 (e+f x))+150 i B \sin (e+f x)+60 i B \sin (3 (e+f x))-360 B f x \sin (3 (e+f x))-75 i B \sin (5 (e+f x))-6 i B \sin (7 (e+f x))+125 B \cos (5 (e+f x))+14 B \cos (7 (e+f x)))}{5120 a^2 c^5 f (\tan (e+f x)-i)^2}","-\frac{3 A+2 i B}{64 a^2 c^5 f (-\tan (e+f x)+i)}+\frac{5 (3 A+i B)}{128 a^2 c^5 f (\tan (e+f x)+i)}-\frac{-B+i A}{128 a^2 c^5 f (-\tan (e+f x)+i)^2}+\frac{-B+5 i A}{64 a^2 c^5 f (\tan (e+f x)+i)^2}-\frac{B+3 i A}{64 a^2 c^5 f (\tan (e+f x)+i)^4}+\frac{A-i B}{40 a^2 c^5 f (\tan (e+f x)+i)^5}+\frac{3 x (7 A+3 i B)}{128 a^2 c^5}-\frac{A}{16 a^2 c^5 f (\tan (e+f x)+i)^3}",1,"(Sec[e + f*x]^2*(Cos[5*(e + f*x)] + I*Sin[5*(e + f*x)])*((50*I)*(21*A + I*B)*Cos[e + f*x] + 20*(A*(7*I - 42*f*x) + 3*B*(1 - (6*I)*f*x))*Cos[3*(e + f*x)] - (105*I)*A*Cos[5*(e + f*x)] + 125*B*Cos[5*(e + f*x)] - (6*I)*A*Cos[7*(e + f*x)] + 14*B*Cos[7*(e + f*x)] + 350*A*Sin[e + f*x] + (150*I)*B*Sin[e + f*x] - 140*A*Sin[3*(e + f*x)] + (60*I)*B*Sin[3*(e + f*x)] + (840*I)*A*f*x*Sin[3*(e + f*x)] - 360*B*f*x*Sin[3*(e + f*x)] - 175*A*Sin[5*(e + f*x)] - (75*I)*B*Sin[5*(e + f*x)] - 14*A*Sin[7*(e + f*x)] - (6*I)*B*Sin[7*(e + f*x)]))/(5120*a^2*c^5*f*(-I + Tan[e + f*x])^2)","A",1
727,-1,0,115,180.0060064,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^n}{(a+i a \tan (e+f x))^3} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n)/(a + I*a*Tan[e + f*x])^3,x]","\text{\$Aborted}","\frac{(B (n+3)+i A (3-n)) (c-i c \tan (e+f x))^n \, _2F_1\left(3,n;n+1;\frac{1}{2} (1-i \tan (e+f x))\right)}{48 a^3 f n}+\frac{(-B+i A) (c-i c \tan (e+f x))^n}{6 a^3 f (1+i \tan (e+f x))^3}",1,"$Aborted","F",-1
728,1,1496,191,11.6655926,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^5}{(a+i a \tan (e+f x))^3} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^5)/(a + I*a*Tan[e + f*x])^3,x]","\frac{\left(\frac{1}{2} B \cos (3 e) c^5+\frac{1}{2} i B \sin (3 e) c^5\right) (\cos (f x)+i \sin (f x))^3 (A+B \tan (e+f x)) \sec ^4(e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{(\cos (f x)+i \sin (f x))^3 \left(\frac{1}{2} i A \cos (3 e-f x) c^5-4 B \cos (3 e-f x) c^5-\frac{1}{2} i A \cos (3 e+f x) c^5+4 B \cos (3 e+f x) c^5-\frac{1}{2} A \sin (3 e-f x) c^5-4 i B \sin (3 e-f x) c^5+\frac{1}{2} A \sin (3 e+f x) c^5+4 i B \sin (3 e+f x) c^5\right) (A+B \tan (e+f x)) \sec ^3(e+f x)}{f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{x (\cos (f x)+i \sin (f x))^3 \left(-4 A \cos ^3(e) c^5-16 i B \cos ^3(e) c^5+16 i A \sin ^3(e) c^5-64 B \sin ^3(e) c^5+24 A \cos (e) \sin ^2(e) c^5+96 i B \cos (e) \sin ^2(e) c^5+4 A \cos (e) c^5+16 i B \cos (e) c^5-16 i A \cos ^2(e) \sin (e) c^5+64 B \cos ^2(e) \sin (e) c^5+8 i A \sin (e) c^5-32 B \sin (e) c^5-4 A \sin ^3(e) \tan (e) c^5-16 i B \sin ^3(e) \tan (e) c^5-4 A \sin (e) \tan (e) c^5-16 i B \sin (e) \tan (e) c^5+i (A+4 i B) \left(8 \cos (3 e) c^5+8 i \sin (3 e) c^5\right) \tan (e)\right) (A+B \tan (e+f x)) \sec ^2(e+f x)}{(A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{(A+3 i B) \cos (2 f x) \left(6 i c^5 \cos (e)-6 c^5 \sin (e)\right) (\cos (f x)+i \sin (f x))^3 (A+B \tan (e+f x)) \sec ^2(e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{(2 B-i A) \cos (4 f x) \left(2 c^5 \cos (e)-2 i c^5 \sin (e)\right) (\cos (f x)+i \sin (f x))^3 (A+B \tan (e+f x)) \sec ^2(e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{\left(-i A \cos \left(\frac{3 e}{2}\right) c^5+4 B \cos \left(\frac{3 e}{2}\right) c^5+A \sin \left(\frac{3 e}{2}\right) c^5+4 i B \sin \left(\frac{3 e}{2}\right) c^5\right) \left(8 \cos \left(\frac{3 e}{2}\right) \log (\cos (e+f x))+8 i \sin \left(\frac{3 e}{2}\right) \log (\cos (e+f x))\right) (\cos (f x)+i \sin (f x))^3 (A+B \tan (e+f x)) \sec ^2(e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{(A+i B) \cos (6 f x) \left(\frac{2}{3} i \cos (3 e) c^5+\frac{2}{3} \sin (3 e) c^5\right) (\cos (f x)+i \sin (f x))^3 (A+B \tan (e+f x)) \sec ^2(e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{(A+4 i B) \left(-8 f x \cos (3 e) c^5-8 i f x \sin (3 e) c^5\right) (\cos (f x)+i \sin (f x))^3 (A+B \tan (e+f x)) \sec ^2(e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{(A+3 i B) \left(6 \cos (e) c^5+6 i \sin (e) c^5\right) (\cos (f x)+i \sin (f x))^3 \sin (2 f x) (A+B \tan (e+f x)) \sec ^2(e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{(A+2 i B) \left(2 i c^5 \sin (e)-2 c^5 \cos (e)\right) (\cos (f x)+i \sin (f x))^3 \sin (4 f x) (A+B \tan (e+f x)) \sec ^2(e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{(A+i B) \left(\frac{2}{3} c^5 \cos (3 e)-\frac{2}{3} i c^5 \sin (3 e)\right) (\cos (f x)+i \sin (f x))^3 \sin (6 f x) (A+B \tan (e+f x)) \sec ^2(e+f x)}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}","\frac{c^5 (A+8 i B) \tan (e+f x)}{a^3 f}-\frac{8 c^5 (3 A+7 i B)}{a^3 f (-\tan (e+f x)+i)}+\frac{8 c^5 (-3 B+2 i A)}{a^3 f (-\tan (e+f x)+i)^2}+\frac{16 c^5 (A+i B)}{3 a^3 f (-\tan (e+f x)+i)^3}-\frac{8 c^5 (-4 B+i A) \log (\cos (e+f x))}{a^3 f}-\frac{8 c^5 x (A+4 i B)}{a^3}+\frac{B c^5 \tan ^2(e+f x)}{2 a^3 f}",1,"((A + (3*I)*B)*Cos[2*f*x]*Sec[e + f*x]^2*((6*I)*c^5*Cos[e] - 6*c^5*Sin[e])*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + (((-I)*A + 2*B)*Cos[4*f*x]*Sec[e + f*x]^2*(2*c^5*Cos[e] - (2*I)*c^5*Sin[e])*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + (Sec[e + f*x]^2*((-I)*A*c^5*Cos[(3*e)/2] + 4*B*c^5*Cos[(3*e)/2] + A*c^5*Sin[(3*e)/2] + (4*I)*B*c^5*Sin[(3*e)/2])*(8*Cos[(3*e)/2]*Log[Cos[e + f*x]] + (8*I)*Log[Cos[e + f*x]]*Sin[(3*e)/2])*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + ((A + I*B)*Cos[6*f*x]*Sec[e + f*x]^2*(((2*I)/3)*c^5*Cos[3*e] + (2*c^5*Sin[3*e])/3)*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + (Sec[e + f*x]^4*((B*c^5*Cos[3*e])/2 + (I/2)*B*c^5*Sin[3*e])*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + ((A + (4*I)*B)*Sec[e + f*x]^2*(-8*c^5*f*x*Cos[3*e] - (8*I)*c^5*f*x*Sin[3*e])*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + ((A + (3*I)*B)*Sec[e + f*x]^2*(6*c^5*Cos[e] + (6*I)*c^5*Sin[e])*(Cos[f*x] + I*Sin[f*x])^3*Sin[2*f*x]*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + ((A + (2*I)*B)*Sec[e + f*x]^2*(-2*c^5*Cos[e] + (2*I)*c^5*Sin[e])*(Cos[f*x] + I*Sin[f*x])^3*Sin[4*f*x]*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + ((A + I*B)*Sec[e + f*x]^2*((2*c^5*Cos[3*e])/3 - ((2*I)/3)*c^5*Sin[3*e])*(Cos[f*x] + I*Sin[f*x])^3*Sin[6*f*x]*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + (Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*((I/2)*A*c^5*Cos[3*e - f*x] - 4*B*c^5*Cos[3*e - f*x] - (I/2)*A*c^5*Cos[3*e + f*x] + 4*B*c^5*Cos[3*e + f*x] - (A*c^5*Sin[3*e - f*x])/2 - (4*I)*B*c^5*Sin[3*e - f*x] + (A*c^5*Sin[3*e + f*x])/2 + (4*I)*B*c^5*Sin[3*e + f*x])*(A + B*Tan[e + f*x]))/(f*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + (x*Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^3*(4*A*c^5*Cos[e] + (16*I)*B*c^5*Cos[e] - 4*A*c^5*Cos[e]^3 - (16*I)*B*c^5*Cos[e]^3 + (8*I)*A*c^5*Sin[e] - 32*B*c^5*Sin[e] - (16*I)*A*c^5*Cos[e]^2*Sin[e] + 64*B*c^5*Cos[e]^2*Sin[e] + 24*A*c^5*Cos[e]*Sin[e]^2 + (96*I)*B*c^5*Cos[e]*Sin[e]^2 + (16*I)*A*c^5*Sin[e]^3 - 64*B*c^5*Sin[e]^3 - 4*A*c^5*Sin[e]*Tan[e] - (16*I)*B*c^5*Sin[e]*Tan[e] - 4*A*c^5*Sin[e]^3*Tan[e] - (16*I)*B*c^5*Sin[e]^3*Tan[e] + I*(A + (4*I)*B)*(8*c^5*Cos[3*e] + (8*I)*c^5*Sin[3*e])*Tan[e])*(A + B*Tan[e + f*x]))/((A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3)","B",1
729,1,1239,164,9.4436153,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^3} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4)/(a + I*a*Tan[e + f*x])^3,x]","c^4 \left(\frac{\sec ^3(e+f x) \left(-\frac{1}{2} B \cos (3 e-f x)+\frac{1}{2} B \cos (3 e+f x)-\frac{1}{2} i B \sin (3 e-f x)+\frac{1}{2} i B \sin (3 e+f x)\right) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{x \sec ^2(e+f x) \left(-\frac{1}{2} A \cos ^3(e)-\frac{7}{2} i B \cos ^3(e)-2 i A \sin (e) \cos ^2(e)+14 B \sin (e) \cos ^2(e)+3 A \sin ^2(e) \cos (e)+21 i B \sin ^2(e) \cos (e)+\frac{1}{2} A \cos (e)+\frac{7}{2} i B \cos (e)+2 i A \sin ^3(e)-14 B \sin ^3(e)+i A \sin (e)-7 B \sin (e)-\frac{1}{2} A \sin ^3(e) \tan (e)-\frac{7}{2} i B \sin ^3(e) \tan (e)-\frac{1}{2} A \sin (e) \tan (e)-\frac{7}{2} i B \sin (e) \tan (e)+i (A+7 i B) (\cos (3 e)+i \sin (3 e)) \tan (e)\right) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{(A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{(A+5 i B) \cos (2 f x) \sec ^2(e+f x) (i \cos (e)-\sin (e)) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{(3 B-i A) \cos (4 f x) \sec ^2(e+f x) \left(\frac{\cos (e)}{2}-\frac{1}{2} i \sin (e)\right) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{\sec ^2(e+f x) \left(-i A \cos \left(\frac{3 e}{2}\right)+7 B \cos \left(\frac{3 e}{2}\right)+A \sin \left(\frac{3 e}{2}\right)+7 i B \sin \left(\frac{3 e}{2}\right)\right) \left(\cos \left(\frac{3 e}{2}\right) \log (\cos (e+f x))+i \sin \left(\frac{3 e}{2}\right) \log (\cos (e+f x))\right) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{(A+i B) \cos (6 f x) \sec ^2(e+f x) \left(\frac{1}{3} i \cos (3 e)+\frac{1}{3} \sin (3 e)\right) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{(A+7 i B) \sec ^2(e+f x) (-f x \cos (3 e)-i f x \sin (3 e)) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{(A+5 i B) \sec ^2(e+f x) (\cos (e)+i \sin (e)) \sin (2 f x) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{(A+3 i B) \sec ^2(e+f x) \left(\frac{1}{2} i \sin (e)-\frac{\cos (e)}{2}\right) \sin (4 f x) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}+\frac{(A+i B) \sec ^2(e+f x) \left(\frac{1}{3} \cos (3 e)-\frac{1}{3} i \sin (3 e)\right) \sin (6 f x) (A+B \tan (e+f x)) (\cos (f x)+i \sin (f x))^3}{f (A \cos (e+f x)+B \sin (e+f x)) (i \tan (e+f x) a+a)^3}\right)","-\frac{6 c^4 (A+3 i B)}{a^3 f (-\tan (e+f x)+i)}+\frac{2 c^4 (-5 B+3 i A)}{a^3 f (-\tan (e+f x)+i)^2}+\frac{8 c^4 (A+i B)}{3 a^3 f (-\tan (e+f x)+i)^3}-\frac{c^4 (-7 B+i A) \log (\cos (e+f x))}{a^3 f}-\frac{c^4 x (A+7 i B)}{a^3}+\frac{i B c^4 \tan (e+f x)}{a^3 f}",1,"c^4*(((A + (5*I)*B)*Cos[2*f*x]*Sec[e + f*x]^2*(I*Cos[e] - Sin[e])*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + (((-I)*A + 3*B)*Cos[4*f*x]*Sec[e + f*x]^2*(Cos[e]/2 - (I/2)*Sin[e])*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + (Sec[e + f*x]^2*((-I)*A*Cos[(3*e)/2] + 7*B*Cos[(3*e)/2] + A*Sin[(3*e)/2] + (7*I)*B*Sin[(3*e)/2])*(Cos[(3*e)/2]*Log[Cos[e + f*x]] + I*Log[Cos[e + f*x]]*Sin[(3*e)/2])*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + ((A + I*B)*Cos[6*f*x]*Sec[e + f*x]^2*((I/3)*Cos[3*e] + Sin[3*e]/3)*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + ((A + (7*I)*B)*Sec[e + f*x]^2*(-(f*x*Cos[3*e]) - I*f*x*Sin[3*e])*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + ((A + (5*I)*B)*Sec[e + f*x]^2*(Cos[e] + I*Sin[e])*(Cos[f*x] + I*Sin[f*x])^3*Sin[2*f*x]*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + ((A + (3*I)*B)*Sec[e + f*x]^2*(-1/2*Cos[e] + (I/2)*Sin[e])*(Cos[f*x] + I*Sin[f*x])^3*Sin[4*f*x]*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + ((A + I*B)*Sec[e + f*x]^2*(Cos[3*e]/3 - (I/3)*Sin[3*e])*(Cos[f*x] + I*Sin[f*x])^3*Sin[6*f*x]*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + (Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*(-1/2*(B*Cos[3*e - f*x]) + (B*Cos[3*e + f*x])/2 - (I/2)*B*Sin[3*e - f*x] + (I/2)*B*Sin[3*e + f*x])*(A + B*Tan[e + f*x]))/(f*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + (x*Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^3*((A*Cos[e])/2 + ((7*I)/2)*B*Cos[e] - (A*Cos[e]^3)/2 - ((7*I)/2)*B*Cos[e]^3 + I*A*Sin[e] - 7*B*Sin[e] - (2*I)*A*Cos[e]^2*Sin[e] + 14*B*Cos[e]^2*Sin[e] + 3*A*Cos[e]*Sin[e]^2 + (21*I)*B*Cos[e]*Sin[e]^2 + (2*I)*A*Sin[e]^3 - 14*B*Sin[e]^3 - (A*Sin[e]*Tan[e])/2 - ((7*I)/2)*B*Sin[e]*Tan[e] - (A*Sin[e]^3*Tan[e])/2 - ((7*I)/2)*B*Sin[e]^3*Tan[e] + I*(A + (7*I)*B)*(Cos[3*e] + I*Sin[3*e])*Tan[e])*(A + B*Tan[e + f*x]))/((A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3))","B",1
730,1,145,135,3.9214537,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3)/(a + I*a*Tan[e + f*x])^3,x]","\frac{c^3 \sec ^3(e+f x) (-\cos (3 (e+f x)) (A-6 i B \log (\cos (e+f x))-6 B f x+i B)+i A \sin (3 (e+f x))+9 B \sin (e+f x)-B \sin (3 (e+f x))+6 i B f x \sin (3 (e+f x))-3 i B \cos (e+f x)-6 B \sin (3 (e+f x)) \log (\cos (e+f x)))}{6 a^3 f (\tan (e+f x)-i)^3}","\frac{c^3 (-B+i A) (1-i \tan (e+f x))^3}{6 a^3 f (1+i \tan (e+f x))^3}-\frac{4 i B c^3}{a^3 f (-\tan (e+f x)+i)}-\frac{2 B c^3}{a^3 f (-\tan (e+f x)+i)^2}+\frac{B c^3 \log (\cos (e+f x))}{a^3 f}-\frac{i B c^3 x}{a^3}",1,"(c^3*Sec[e + f*x]^3*((-3*I)*B*Cos[e + f*x] - Cos[3*(e + f*x)]*(A + I*B - 6*B*f*x - (6*I)*B*Log[Cos[e + f*x]]) + 9*B*Sin[e + f*x] + I*A*Sin[3*(e + f*x)] - B*Sin[3*(e + f*x)] + (6*I)*B*f*x*Sin[3*(e + f*x)] - 6*B*Log[Cos[e + f*x]]*Sin[3*(e + f*x)]))/(6*a^3*f*(-I + Tan[e + f*x])^3)","A",1
731,1,79,99,3.1244041,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^2}{(a+i a \tan (e+f x))^3} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2)/(a + I*a*Tan[e + f*x])^3,x]","-\frac{i c^2 \sec ^2(e+f x) (\cos (2 (e+f x))-i \sin (2 (e+f x))) ((A-5 i B) \tan (e+f x)-5 i A-B)}{24 a^3 f (\tan (e+f x)-i)^3}","\frac{c^2 (-3 B+i A)}{2 a^3 f (-\tan (e+f x)+i)^2}+\frac{2 c^2 (A+i B)}{3 a^3 f (-\tan (e+f x)+i)^3}-\frac{i B c^2}{a^3 f (-\tan (e+f x)+i)}",1,"((-1/24*I)*c^2*Sec[e + f*x]^2*(Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)])*((-5*I)*A - B + (A - (5*I)*B)*Tan[e + f*x]))/(a^3*f*(-I + Tan[e + f*x])^3)","A",1
732,1,81,59,1.3935312,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))}{(a+i a \tan (e+f x))^3} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x]))/(a + I*a*Tan[e + f*x])^3,x]","\frac{c (\tan (e+f x)+i) \sec ^2(e+f x) (-2 (A-2 i B) \sin (2 (e+f x))+2 (B+2 i A) \cos (2 (e+f x))+3 i A)}{24 a^3 f (\tan (e+f x)-i)^3}","\frac{c (A+i B)}{3 a^3 f (-\tan (e+f x)+i)^3}-\frac{B c}{2 a^3 f (-\tan (e+f x)+i)^2}",1,"(c*Sec[e + f*x]^2*((3*I)*A + 2*((2*I)*A + B)*Cos[2*(e + f*x)] - 2*(A - (2*I)*B)*Sin[2*(e + f*x)])*(I + Tan[e + f*x]))/(24*a^3*f*(-I + Tan[e + f*x])^3)","A",1
733,1,150,112,0.7900561,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(A + B*Tan[e + f*x])/(a + I*a*Tan[e + f*x])^3,x]","\frac{\sec ^3(e+f x) ((-27 A+3 i B) \cos (e+f x)+2 (6 i A f x-A+6 B f x-i B) \cos (3 (e+f x))-9 i A \sin (e+f x)+2 i A \sin (3 (e+f x))-12 A f x \sin (3 (e+f x))-9 B \sin (e+f x)-2 B \sin (3 (e+f x))+12 i B f x \sin (3 (e+f x)))}{96 a^3 f (\tan (e+f x)-i)^3}","\frac{B+i A}{8 f \left(a^3+i a^3 \tan (e+f x)\right)}+\frac{x (A-i B)}{8 a^3}+\frac{-B+i A}{6 f (a+i a \tan (e+f x))^3}+\frac{B+i A}{8 a f (a+i a \tan (e+f x))^2}",1,"(Sec[e + f*x]^3*((-27*A + (3*I)*B)*Cos[e + f*x] + 2*(-A - I*B + (6*I)*A*f*x + 6*B*f*x)*Cos[3*(e + f*x)] - (9*I)*A*Sin[e + f*x] - 9*B*Sin[e + f*x] + (2*I)*A*Sin[3*(e + f*x)] - 2*B*Sin[3*(e + f*x)] - 12*A*f*x*Sin[3*(e + f*x)] + (12*I)*B*f*x*Sin[3*(e + f*x)]))/(96*a^3*f*(-I + Tan[e + f*x])^3)","A",1
734,1,164,153,2.394584,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])),x]","-\frac{\sec ^2(e+f x) (3 (A (8 f x+2 i)+B (-1-4 i f x)) \cos (2 (e+f x))+(-4 B-2 i A) \cos (4 (e+f x))+24 i A f x \sin (2 (e+f x))+6 A \sin (2 (e+f x))+4 A \sin (4 (e+f x))+18 i A+3 i B \sin (2 (e+f x))+12 B f x \sin (2 (e+f x))-2 i B \sin (4 (e+f x)))}{96 a^3 c f (\tan (e+f x)-i)^2}","-\frac{3 A-i B}{16 a^3 c f (-\tan (e+f x)+i)}+\frac{A-i B}{16 a^3 c f (\tan (e+f x)+i)}+\frac{A+i B}{12 a^3 c f (-\tan (e+f x)+i)^3}+\frac{x (2 A-i B)}{8 a^3 c}-\frac{i A}{8 a^3 c f (-\tan (e+f x)+i)^2}",1,"-1/96*(Sec[e + f*x]^2*((18*I)*A + 3*(B*(-1 - (4*I)*f*x) + A*(2*I + 8*f*x))*Cos[2*(e + f*x)] + ((-2*I)*A - 4*B)*Cos[4*(e + f*x)] + 6*A*Sin[2*(e + f*x)] + (3*I)*B*Sin[2*(e + f*x)] + (24*I)*A*f*x*Sin[2*(e + f*x)] + 12*B*f*x*Sin[2*(e + f*x)] + 4*A*Sin[4*(e + f*x)] - (2*I)*B*Sin[4*(e + f*x)]))/(a^3*c*f*(-I + Tan[e + f*x])^2)","A",1
735,1,217,185,2.6894119,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^2} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^2),x]","\frac{\sec ^3(e+f x) (\cos (2 (e+f x))+i \sin (2 (e+f x))) (12 (A (-5+10 i f x)+B (2 f x-i)) \cos (e+f x)+3 (5 A-9 i B) \cos (3 (e+f x))+60 i A \sin (e+f x)-120 A f x \sin (e+f x)+45 i A \sin (3 (e+f x))+5 i A \sin (5 (e+f x))+A \cos (5 (e+f x))-12 B \sin (e+f x)+24 i B f x \sin (e+f x)+9 B \sin (3 (e+f x))+B \sin (5 (e+f x))-5 i B \cos (5 (e+f x)))}{384 a^3 c^2 f (\tan (e+f x)-i)^3}","\frac{2 A-i B}{16 a^3 c^2 f (\tan (e+f x)+i)}-\frac{-B+3 i A}{32 a^3 c^2 f (-\tan (e+f x)+i)^2}+\frac{B+i A}{32 a^3 c^2 f (\tan (e+f x)+i)^2}+\frac{A+i B}{24 a^3 c^2 f (-\tan (e+f x)+i)^3}+\frac{x (5 A-i B)}{16 a^3 c^2}-\frac{3 A}{16 a^3 c^2 f (-\tan (e+f x)+i)}",1,"(Sec[e + f*x]^3*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*(12*(A*(-5 + (10*I)*f*x) + B*(-I + 2*f*x))*Cos[e + f*x] + 3*(5*A - (9*I)*B)*Cos[3*(e + f*x)] + A*Cos[5*(e + f*x)] - (5*I)*B*Cos[5*(e + f*x)] + (60*I)*A*Sin[e + f*x] - 12*B*Sin[e + f*x] - 120*A*f*x*Sin[e + f*x] + (24*I)*B*f*x*Sin[e + f*x] + (45*I)*A*Sin[3*(e + f*x)] + 9*B*Sin[3*(e + f*x)] + (5*I)*A*Sin[5*(e + f*x)] + B*Sin[5*(e + f*x)]))/(384*a^3*c^2*f*(-I + Tan[e + f*x])^3)","A",1
736,1,63,99,0.1632531,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^3} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^3),x]","\frac{A (45 \sin (2 (e+f x))+9 \sin (4 (e+f x))+\sin (6 (e+f x))+60 e+60 f x)-32 B \cos ^6(e+f x)}{192 a^3 c^3 f}","-\frac{\cos ^6(e+f x) (B-A \tan (e+f x))}{6 a^3 c^3 f}+\frac{5 A \sin (e+f x) \cos ^3(e+f x)}{24 a^3 c^3 f}+\frac{5 A \sin (e+f x) \cos (e+f x)}{16 a^3 c^3 f}+\frac{5 A x}{16 a^3 c^3}",1,"(-32*B*Cos[e + f*x]^6 + A*(60*e + 60*f*x + 45*Sin[2*(e + f*x)] + 9*Sin[4*(e + f*x)] + Sin[6*(e + f*x)]))/(192*a^3*c^3*f)","A",1
737,1,267,251,3.5131241,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^4),x]","\frac{\sec ^3(e+f x) (-\cos (4 (e+f x))-i \sin (4 (e+f x))) (60 (A (-7-14 i f x)+B (2 f x+i)) \cos (e+f x)+18 (7 A+9 i B) \cos (3 (e+f x))-420 i A \sin (e+f x)-840 A f x \sin (e+f x)-378 i A \sin (3 (e+f x))-70 i A \sin (5 (e+f x))-7 i A \sin (7 (e+f x))+14 A \cos (5 (e+f x))+A \cos (7 (e+f x))-60 B \sin (e+f x)-120 i B f x \sin (e+f x)+54 B \sin (3 (e+f x))+10 B \sin (5 (e+f x))+B \sin (7 (e+f x))+50 i B \cos (5 (e+f x))+7 i B \cos (7 (e+f x)))}{3072 a^3 c^4 f (\tan (e+f x)-i)^3}","-\frac{5 (3 A+i B)}{128 a^3 c^4 f (-\tan (e+f x)+i)}-\frac{-3 B+5 i A}{128 a^3 c^4 f (-\tan (e+f x)+i)^2}+\frac{B+5 i A}{64 a^3 c^4 f (\tan (e+f x)+i)^2}+\frac{A+i B}{96 a^3 c^4 f (-\tan (e+f x)+i)^3}-\frac{2 A-i B}{48 a^3 c^4 f (\tan (e+f x)+i)^3}-\frac{B+i A}{64 a^3 c^4 f (\tan (e+f x)+i)^4}+\frac{5 x (7 A+i B)}{128 a^3 c^4}+\frac{5 A}{32 a^3 c^4 f (\tan (e+f x)+i)}",1,"(Sec[e + f*x]^3*(-Cos[4*(e + f*x)] - I*Sin[4*(e + f*x)])*(60*(A*(-7 - (14*I)*f*x) + B*(I + 2*f*x))*Cos[e + f*x] + 18*(7*A + (9*I)*B)*Cos[3*(e + f*x)] + 14*A*Cos[5*(e + f*x)] + (50*I)*B*Cos[5*(e + f*x)] + A*Cos[7*(e + f*x)] + (7*I)*B*Cos[7*(e + f*x)] - (420*I)*A*Sin[e + f*x] - 60*B*Sin[e + f*x] - 840*A*f*x*Sin[e + f*x] - (120*I)*B*f*x*Sin[e + f*x] - (378*I)*A*Sin[3*(e + f*x)] + 54*B*Sin[3*(e + f*x)] - (70*I)*A*Sin[5*(e + f*x)] + 10*B*Sin[5*(e + f*x)] - (7*I)*A*Sin[7*(e + f*x)] + B*Sin[7*(e + f*x)]))/(3072*a^3*c^4*f*(-I + Tan[e + f*x])^3)","A",1
738,1,280,287,4.707647,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^5),x]","\frac{\sec ^3(e+f x) (\cos (5 (e+f x))+i \sin (5 (e+f x))) (210 (4 A (1+4 i f x)-B (4 f x+i)) \cos (2 (e+f x))-560 (A+i B) \cos (4 (e+f x))+3360 A f x \sin (2 (e+f x))+840 i A \sin (2 (e+f x))+1120 i A \sin (4 (e+f x))+180 i A \sin (6 (e+f x))+16 i A \sin (8 (e+f x))-60 A \cos (6 (e+f x))-4 A \cos (8 (e+f x))+2100 A+210 B \sin (2 (e+f x))+840 i B f x \sin (2 (e+f x))-280 B \sin (4 (e+f x))-45 B \sin (6 (e+f x))-4 B \sin (8 (e+f x))-135 i B \cos (6 (e+f x))-16 i B \cos (8 (e+f x)))}{15360 a^3 c^5 f (\tan (e+f x)-i)^3}","-\frac{3 (7 A+3 i B)}{256 a^3 c^5 f (-\tan (e+f x)+i)}+\frac{5 (7 A+i B)}{256 a^3 c^5 f (\tan (e+f x)+i)}-\frac{-2 B+3 i A}{128 a^3 c^5 f (-\tan (e+f x)+i)^2}+\frac{A+i B}{192 a^3 c^5 f (-\tan (e+f x)+i)^3}-\frac{5 A-i B}{96 a^3 c^5 f (\tan (e+f x)+i)^3}-\frac{B+2 i A}{64 a^3 c^5 f (\tan (e+f x)+i)^4}+\frac{A-i B}{80 a^3 c^5 f (\tan (e+f x)+i)^5}+\frac{7 x (4 A+i B)}{128 a^3 c^5}+\frac{5 i A}{64 a^3 c^5 f (\tan (e+f x)+i)^2}",1,"(Sec[e + f*x]^3*(Cos[5*(e + f*x)] + I*Sin[5*(e + f*x)])*(2100*A + 210*(4*A*(1 + (4*I)*f*x) - B*(I + 4*f*x))*Cos[2*(e + f*x)] - 560*(A + I*B)*Cos[4*(e + f*x)] - 60*A*Cos[6*(e + f*x)] - (135*I)*B*Cos[6*(e + f*x)] - 4*A*Cos[8*(e + f*x)] - (16*I)*B*Cos[8*(e + f*x)] + (840*I)*A*Sin[2*(e + f*x)] + 210*B*Sin[2*(e + f*x)] + 3360*A*f*x*Sin[2*(e + f*x)] + (840*I)*B*f*x*Sin[2*(e + f*x)] + (1120*I)*A*Sin[4*(e + f*x)] - 280*B*Sin[4*(e + f*x)] + (180*I)*A*Sin[6*(e + f*x)] - 45*B*Sin[6*(e + f*x)] + (16*I)*A*Sin[8*(e + f*x)] - 4*B*Sin[8*(e + f*x)]))/(15360*a^3*c^5*f*(-I + Tan[e + f*x])^3)","A",1
739,1,321,319,5.2783312,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^6),x]","\frac{\sec ^3(e+f x) (-\cos (6 (e+f x))-i \sin (6 (e+f x))) (-210 (27 A+i B) \cos (e+f x)+280 (-18 i A f x-3 A+6 B f x+i B) \cos (3 (e+f x))+1890 i A \sin (e+f x)-840 i A \sin (3 (e+f x))-5040 A f x \sin (3 (e+f x))-1350 i A \sin (5 (e+f x))-189 i A \sin (7 (e+f x))-15 i A \sin (9 (e+f x))+810 A \cos (5 (e+f x))+81 A \cos (7 (e+f x))+5 A \cos (9 (e+f x))-630 B \sin (e+f x)-280 B \sin (3 (e+f x))-1680 i B f x \sin (3 (e+f x))+450 B \sin (5 (e+f x))+63 B \sin (7 (e+f x))+5 B \sin (9 (e+f x))+750 i B \cos (5 (e+f x))+147 i B \cos (7 (e+f x))+15 i B \cos (9 (e+f x)))}{30720 a^3 c^6 f (\tan (e+f x)-i)^3}","-\frac{7 (2 A+i B)}{256 a^3 c^6 f (-\tan (e+f x)+i)}+\frac{7 (4 A+i B)}{256 a^3 c^6 f (\tan (e+f x)+i)}-\frac{-5 B+7 i A}{512 a^3 c^6 f (-\tan (e+f x)+i)^2}+\frac{5 (-B+7 i A)}{512 a^3 c^6 f (\tan (e+f x)+i)^2}+\frac{A+i B}{384 a^3 c^6 f (-\tan (e+f x)+i)^3}-\frac{B+5 i A}{128 a^3 c^6 f (\tan (e+f x)+i)^4}+\frac{2 A-i B}{80 a^3 c^6 f (\tan (e+f x)+i)^5}+\frac{B+i A}{96 a^3 c^6 f (\tan (e+f x)+i)^6}+\frac{7 x (3 A+i B)}{128 a^3 c^6}-\frac{5 A}{96 a^3 c^6 f (\tan (e+f x)+i)^3}",1,"(Sec[e + f*x]^3*(-Cos[6*(e + f*x)] - I*Sin[6*(e + f*x)])*(-210*(27*A + I*B)*Cos[e + f*x] + 280*(-3*A + I*B - (18*I)*A*f*x + 6*B*f*x)*Cos[3*(e + f*x)] + 810*A*Cos[5*(e + f*x)] + (750*I)*B*Cos[5*(e + f*x)] + 81*A*Cos[7*(e + f*x)] + (147*I)*B*Cos[7*(e + f*x)] + 5*A*Cos[9*(e + f*x)] + (15*I)*B*Cos[9*(e + f*x)] + (1890*I)*A*Sin[e + f*x] - 630*B*Sin[e + f*x] - (840*I)*A*Sin[3*(e + f*x)] - 280*B*Sin[3*(e + f*x)] - 5040*A*f*x*Sin[3*(e + f*x)] - (1680*I)*B*f*x*Sin[3*(e + f*x)] - (1350*I)*A*Sin[5*(e + f*x)] + 450*B*Sin[5*(e + f*x)] - (189*I)*A*Sin[7*(e + f*x)] + 63*B*Sin[7*(e + f*x)] - (15*I)*A*Sin[9*(e + f*x)] + 5*B*Sin[9*(e + f*x)]))/(30720*a^3*c^6*f*(-I + Tan[e + f*x])^3)","A",1
740,1,90,62,6.5863835,"\int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2),x]","\frac{2 a c^3 \sec ^3(e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\sin (3 e+2 f x)+i \cos (3 e+2 f x)) (9 A+7 B \tan (e+f x)-2 i B)}{63 f}","\frac{2 a (B+i A) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac{2 a B (c-i c \tan (e+f x))^{9/2}}{9 c f}",1,"(2*a*c^3*Sec[e + f*x]^3*(Cos[f*x] - I*Sin[f*x])*(I*Cos[3*e + 2*f*x] + Sin[3*e + 2*f*x])*(9*A - (2*I)*B + 7*B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(63*f)","A",1
741,1,88,62,4.3820379,"\int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{2 a c^2 \sec ^2(e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\sin (2 e+f x)+i \cos (2 e+f x)) (7 A+5 B \tan (e+f x)-2 i B)}{35 f}","\frac{2 a (B+i A) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac{2 a B (c-i c \tan (e+f x))^{7/2}}{7 c f}",1,"(2*a*c^2*Sec[e + f*x]^2*(Cos[f*x] - I*Sin[f*x])*(I*Cos[2*e + f*x] + Sin[2*e + f*x])*(7*A - (2*I)*B + 5*B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(35*f)","A",1
742,1,97,62,3.3485099,"\int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{2 a c (\cos (e)-i \sin (e)) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (5 i A+3 i B \tan (e+f x)+2 B) (A+B \tan (e+f x))}{15 f (A \cos (e+f x)+B \sin (e+f x))}","\frac{2 a (B+i A) (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac{2 a B (c-i c \tan (e+f x))^{5/2}}{5 c f}",1,"(2*a*c*(Cos[e] - I*Sin[e])*(Cos[f*x] - I*Sin[f*x])*((5*I)*A + 2*B + (3*I)*B*Tan[e + f*x])*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(15*f*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
743,1,45,60,2.5354245,"\int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{2 a \sqrt{c-i c \tan (e+f x)} (3 i A+i B \tan (e+f x)+2 B)}{3 f}","\frac{2 a (B+i A) \sqrt{c-i c \tan (e+f x)}}{f}-\frac{2 a B (c-i c \tan (e+f x))^{3/2}}{3 c f}",1,"(2*a*((3*I)*A + 2*B + I*B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(3*f)","A",1
744,1,82,58,2.6308555,"\int \frac{(a+i a \tan (e+f x)) (A+B \tan (e+f x))}{\sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{2 a (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\sin (e+2 f x)-i \cos (e+2 f x)) (-B \sin (e+f x)+(A-2 i B) \cos (e+f x))}{c f}","-\frac{2 a (B+i A)}{f \sqrt{c-i c \tan (e+f x)}}-\frac{2 a B \sqrt{c-i c \tan (e+f x)}}{c f}",1,"(2*a*(Cos[f*x] - I*Sin[f*x])*((A - (2*I)*B)*Cos[e + f*x] - B*Sin[e + f*x])*((-I)*Cos[e + 2*f*x] + Sin[e + 2*f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(c*f)","A",1
745,1,98,60,4.5090197,"\int \frac{(a+i a \tan (e+f x)) (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{2 a \cos (e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\cos (2 e+3 f x)+i \sin (2 e+3 f x)) ((2 B-i A) \cos (e+f x)-3 i B \sin (e+f x))}{3 c^2 f}","\frac{2 a B}{c f \sqrt{c-i c \tan (e+f x)}}-\frac{2 a (B+i A)}{3 f (c-i c \tan (e+f x))^{3/2}}",1,"(2*a*Cos[e + f*x]*(Cos[f*x] - I*Sin[f*x])*(((-I)*A + 2*B)*Cos[e + f*x] - (3*I)*B*Sin[e + f*x])*(Cos[2*e + 3*f*x] + I*Sin[2*e + 3*f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(3*c^2*f)","A",1
746,1,100,62,7.7311921,"\int \frac{(a+i a \tan (e+f x)) (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{2 a \cos ^2(e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\cos (3 e+4 f x)+i \sin (3 e+4 f x)) ((2 B-3 i A) \cos (e+f x)-5 i B \sin (e+f x))}{15 c^3 f}","\frac{2 a B}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac{2 a (B+i A)}{5 f (c-i c \tan (e+f x))^{5/2}}",1,"(2*a*Cos[e + f*x]^2*(Cos[f*x] - I*Sin[f*x])*(((-3*I)*A + 2*B)*Cos[e + f*x] - (5*I)*B*Sin[e + f*x])*(Cos[3*e + 4*f*x] + I*Sin[3*e + 4*f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(15*c^3*f)","A",1
747,1,100,62,11.1870115,"\int \frac{(a+i a \tan (e+f x)) (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(7/2),x]","\frac{2 a \cos ^3(e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\cos (4 e+5 f x)+i \sin (4 e+5 f x)) ((2 B-5 i A) \cos (e+f x)-7 i B \sin (e+f x))}{35 c^4 f}","\frac{2 a B}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac{2 a (B+i A)}{7 f (c-i c \tan (e+f x))^{7/2}}",1,"(2*a*Cos[e + f*x]^3*(Cos[f*x] - I*Sin[f*x])*(((-5*I)*A + 2*B)*Cos[e + f*x] - (7*I)*B*Sin[e + f*x])*(Cos[4*e + 5*f*x] + I*Sin[4*e + 5*f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(35*c^4*f)","A",1
748,1,119,105,11.3097964,"\int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2),x]","\frac{a^2 c^3 \sec ^5(e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (3 e+f x)-i \sin (3 e+f x)) ((-77 A+105 i B) \sin (2 (e+f x))+(93 B+121 i A) \cos (2 (e+f x))+121 i A-33 B)}{693 f (\cos (f x)+i \sin (f x))^2}","-\frac{2 a^2 (3 B+i A) (c-i c \tan (e+f x))^{9/2}}{9 c f}+\frac{4 a^2 (B+i A) (c-i c \tan (e+f x))^{7/2}}{7 f}+\frac{2 a^2 B (c-i c \tan (e+f x))^{11/2}}{11 c^2 f}",1,"(a^2*c^3*Sec[e + f*x]^5*((121*I)*A - 33*B + ((121*I)*A + 93*B)*Cos[2*(e + f*x)] + (-77*A + (105*I)*B)*Sin[2*(e + f*x)])*(Cos[3*e + f*x] - I*Sin[3*e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(693*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
749,1,112,105,7.4755256,"\int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{a^2 c^2 (\sin (2 e)+i \cos (2 e)) \sec ^4(e+f x) \sqrt{c-i c \tan (e+f x)} (5 (13 B+9 i A) \sin (2 (e+f x))+(81 A-61 i B) \cos (2 (e+f x))+81 A+9 i B)}{315 f (\cos (f x)+i \sin (f x))^2}","-\frac{2 a^2 (3 B+i A) (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac{4 a^2 (B+i A) (c-i c \tan (e+f x))^{5/2}}{5 f}+\frac{2 a^2 B (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}",1,"(a^2*c^2*Sec[e + f*x]^4*(I*Cos[2*e] + Sin[2*e])*(81*A + (9*I)*B + (81*A - (61*I)*B)*Cos[2*(e + f*x)] + 5*((9*I)*A + 13*B)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(315*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
750,1,116,105,6.2485316,"\int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{a^2 c \sec ^3(e+f x) \sqrt{c-i c \tan (e+f x)} (\sin (e-f x)+i \cos (e-f x)) (3 (11 B+7 i A) \sin (2 (e+f x))+(49 A-37 i B) \cos (2 (e+f x))+49 A-7 i B)}{105 f (\cos (f x)+i \sin (f x))^2}","-\frac{2 a^2 (3 B+i A) (c-i c \tan (e+f x))^{5/2}}{5 c f}+\frac{4 a^2 (B+i A) (c-i c \tan (e+f x))^{3/2}}{3 f}+\frac{2 a^2 B (c-i c \tan (e+f x))^{7/2}}{7 c^2 f}",1,"(a^2*c*Sec[e + f*x]^3*(I*Cos[e - f*x] + Sin[e - f*x])*(49*A - (7*I)*B + (49*A - (37*I)*B)*Cos[2*(e + f*x)] + 3*((7*I)*A + 11*B)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(105*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
751,1,83,103,4.9633086,"\int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{a^2 \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} ((-5 A+9 i B) \sin (2 (e+f x))+(21 B+25 i A) \cos (2 (e+f x))+5 (3 B+5 i A))}{15 f}","-\frac{2 a^2 (3 B+i A) (c-i c \tan (e+f x))^{3/2}}{3 c f}+\frac{4 a^2 (B+i A) \sqrt{c-i c \tan (e+f x)}}{f}+\frac{2 a^2 B (c-i c \tan (e+f x))^{5/2}}{5 c^2 f}",1,"(a^2*Sec[e + f*x]^2*(5*((5*I)*A + 3*B) + ((25*I)*A + 21*B)*Cos[2*(e + f*x)] + (-5*A + (9*I)*B)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(15*f)","A",1
752,1,138,101,5.1370612,"\int \frac{(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{\sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{a^2 \sqrt{c-i c \tan (e+f x)} (\sin (e+3 f x)-i \cos (e+3 f x)) (A+B \tan (e+f x)) ((-7 B-3 i A) \sin (2 (e+f x))+(9 A-13 i B) \cos (2 (e+f x))+9 A-15 i B)}{3 c f (\cos (f x)+i \sin (f x))^2 (A \cos (e+f x)+B \sin (e+f x))}","-\frac{2 a^2 (3 B+i A) \sqrt{c-i c \tan (e+f x)}}{c f}-\frac{4 a^2 (B+i A)}{f \sqrt{c-i c \tan (e+f x)}}+\frac{2 a^2 B (c-i c \tan (e+f x))^{3/2}}{3 c^2 f}",1,"(a^2*(9*A - (15*I)*B + (9*A - (13*I)*B)*Cos[2*(e + f*x)] + ((-3*I)*A - 7*B)*Sin[2*(e + f*x)])*((-I)*Cos[e + 3*f*x] + Sin[e + 3*f*x])*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(3*c*f*(Cos[f*x] + I*Sin[f*x])^2*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
753,1,112,101,8.7697364,"\int \frac{(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{a^2 \sqrt{c-i c \tan (e+f x)} (\cos (2 (e+2 f x))+i \sin (2 (e+2 f x))) (3 (A-5 i B) \sin (2 (e+f x))+(13 B+i A) \cos (2 (e+f x))+i A+7 B)}{3 c^2 f (\cos (f x)+i \sin (f x))^2}","\frac{2 a^2 (3 B+i A)}{c f \sqrt{c-i c \tan (e+f x)}}-\frac{4 a^2 (B+i A)}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac{2 a^2 B \sqrt{c-i c \tan (e+f x)}}{c^2 f}",1,"(a^2*(I*A + 7*B + (I*A + 13*B)*Cos[2*(e + f*x)] + 3*(A - (5*I)*B)*Sin[2*(e + f*x)])*(Cos[2*(e + 2*f*x)] + I*Sin[2*(e + 2*f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(3*c^2*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
754,1,118,103,11.9215812,"\int \frac{(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{a^2 \cos (e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (3 e+5 f x)+i \sin (3 e+5 f x)) (5 (A+3 i B) \sin (2 (e+f x))+(-21 B-i A) \cos (2 (e+f x))-i A+9 B)}{15 c^3 f (\cos (f x)+i \sin (f x))^2}","\frac{2 a^2 (3 B+i A)}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac{4 a^2 (B+i A)}{5 f (c-i c \tan (e+f x))^{5/2}}-\frac{2 a^2 B}{c^2 f \sqrt{c-i c \tan (e+f x)}}",1,"(a^2*Cos[e + f*x]*((-I)*A + 9*B + ((-I)*A - 21*B)*Cos[2*(e + f*x)] + 5*(A + (3*I)*B)*Sin[2*(e + f*x)])*(Cos[3*e + 5*f*x] + I*Sin[3*e + 5*f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(15*c^3*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
755,1,122,105,13.0791762,"\int \frac{(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(7/2),x]","\frac{a^2 \cos ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (4 e+6 f x)+i \sin (4 e+6 f x)) (7 (3 A+i B) \sin (2 (e+f x))+(-37 B-9 i A) \cos (2 (e+f x))-9 i A+33 B)}{105 c^4 f (\cos (f x)+i \sin (f x))^2}","\frac{2 a^2 (3 B+i A)}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac{4 a^2 (B+i A)}{7 f (c-i c \tan (e+f x))^{7/2}}-\frac{2 a^2 B}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}",1,"(a^2*Cos[e + f*x]^2*((-9*I)*A + 33*B + ((-9*I)*A - 37*B)*Cos[2*(e + f*x)] + 7*(3*A + I*B)*Sin[2*(e + f*x)])*(Cos[4*e + 6*f*x] + I*Sin[4*e + 6*f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(105*c^4*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
756,1,127,144,13.4514931,"\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2),x]","-\frac{2 a^3 c^3 (\cos (3 e)-i \sin (3 e)) \sec ^5(e+f x) \sqrt{c-i c \tan (e+f x)} (7 (169 A-86 i B) \tan (e+f x)+\cos (2 (e+f x)) (7 (169 A-185 i B) \tan (e+f x)-1391 i A-1279 B)-572 i A+737 B)}{9009 f (\cos (f x)+i \sin (f x))^3}","\frac{2 a^3 (5 B+i A) (c-i c \tan (e+f x))^{11/2}}{11 c^2 f}-\frac{8 a^3 (2 B+i A) (c-i c \tan (e+f x))^{9/2}}{9 c f}+\frac{8 a^3 (B+i A) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac{2 a^3 B (c-i c \tan (e+f x))^{13/2}}{13 c^3 f}",1,"(-2*a^3*c^3*Sec[e + f*x]^5*(Cos[3*e] - I*Sin[3*e])*Sqrt[c - I*c*Tan[e + f*x]]*((-572*I)*A + 737*B + 7*(169*A - (86*I)*B)*Tan[e + f*x] + Cos[2*(e + f*x)]*((-1391*I)*A - 1279*B + 7*(169*A - (185*I)*B)*Tan[e + f*x])))/(9009*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
757,1,139,144,12.0792446,"\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]","-\frac{2 a^3 c^2 \sec ^4(e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (2 e-f x)-i \sin (2 e-f x)) (5 (121 A-74 i B) \tan (e+f x)+\cos (2 (e+f x)) ((605 A-685 i B) \tan (e+f x)-781 i A-701 B)+9 (31 B-44 i A))}{3465 f (\cos (f x)+i \sin (f x))^3}","\frac{2 a^3 (5 B+i A) (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac{8 a^3 (2 B+i A) (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac{8 a^3 (B+i A) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac{2 a^3 B (c-i c \tan (e+f x))^{11/2}}{11 c^3 f}",1,"(-2*a^3*c^2*Sec[e + f*x]^4*(Cos[2*e - f*x] - I*Sin[2*e - f*x])*Sqrt[c - I*c*Tan[e + f*x]]*(9*((-44*I)*A + 31*B) + 5*(121*A - (74*I)*B)*Tan[e + f*x] + Cos[2*(e + f*x)]*((-781*I)*A - 701*B + (605*A - (685*I)*B)*Tan[e + f*x])))/(3465*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
758,1,130,144,8.953067,"\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2),x]","-\frac{2 a^3 c \sec ^3(e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (e-2 f x)-i \sin (e-2 f x)) ((81 A-62 i B) \tan (e+f x)+\cos (2 (e+f x)) ((81 A-97 i B) \tan (e+f x)-129 i A-113 B)+7 (B-12 i A))}{315 f (\cos (f x)+i \sin (f x))^3}","\frac{2 a^3 (5 B+i A) (c-i c \tan (e+f x))^{7/2}}{7 c^2 f}-\frac{8 a^3 (2 B+i A) (c-i c \tan (e+f x))^{5/2}}{5 c f}+\frac{8 a^3 (B+i A) (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac{2 a^3 B (c-i c \tan (e+f x))^{9/2}}{9 c^3 f}",1,"(-2*a^3*c*Sec[e + f*x]^3*(Cos[e - 2*f*x] - I*Sin[e - 2*f*x])*Sqrt[c - I*c*Tan[e + f*x]]*(7*((-12*I)*A + B) + (81*A - (62*I)*B)*Tan[e + f*x] + Cos[2*(e + f*x)]*((-129*I)*A - 113*B + (81*A - (97*I)*B)*Tan[e + f*x])))/(315*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
759,1,124,142,7.4697135,"\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{a^3 \sec ^2(e+f x) (\cos (3 f x)+i \sin (3 f x)) \sqrt{c-i c \tan (e+f x)} ((-98 A+100 i B) \tan (e+f x)+\cos (2 (e+f x)) ((-98 A+130 i B) \tan (e+f x)+322 i A+290 B)+280 i A+170 B)}{105 f (\cos (f x)+i \sin (f x))^3}","\frac{2 a^3 (5 B+i A) (c-i c \tan (e+f x))^{5/2}}{5 c^2 f}-\frac{8 a^3 (2 B+i A) (c-i c \tan (e+f x))^{3/2}}{3 c f}+\frac{8 a^3 (B+i A) \sqrt{c-i c \tan (e+f x)}}{f}-\frac{2 a^3 B (c-i c \tan (e+f x))^{7/2}}{7 c^3 f}",1,"(a^3*Sec[e + f*x]^2*(Cos[3*f*x] + I*Sin[3*f*x])*Sqrt[c - I*c*Tan[e + f*x]]*((280*I)*A + 170*B + (-98*A + (100*I)*B)*Tan[e + f*x] + Cos[2*(e + f*x)]*((322*I)*A + 290*B + (-98*A + (130*I)*B)*Tan[e + f*x])))/(105*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
760,1,152,140,7.5870278,"\int \frac{(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{\sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/Sqrt[c - I*c*Tan[e + f*x]],x]","-\frac{2 a^3 \sqrt{c-i c \tan (e+f x)} (\cos (e+4 f x)+i \sin (e+4 f x)) (A+B \tan (e+f x)) ((25 A-38 i B) \tan (e+f x)+\cos (2 (e+f x)) ((25 A-41 i B) \tan (e+f x)+55 i A+71 B)+60 i A+87 B)}{15 c f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}","\frac{2 a^3 (5 B+i A) (c-i c \tan (e+f x))^{3/2}}{3 c^2 f}-\frac{8 a^3 (2 B+i A) \sqrt{c-i c \tan (e+f x)}}{c f}-\frac{8 a^3 (B+i A)}{f \sqrt{c-i c \tan (e+f x)}}-\frac{2 a^3 B (c-i c \tan (e+f x))^{5/2}}{5 c^3 f}",1,"(-2*a^3*(Cos[e + 4*f*x] + I*Sin[e + 4*f*x])*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]]*((60*I)*A + 87*B + (25*A - (38*I)*B)*Tan[e + f*x] + Cos[2*(e + f*x)]*((55*I)*A + 71*B + (25*A - (41*I)*B)*Tan[e + f*x])))/(15*c*f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
761,1,168,140,12.4283528,"\int \frac{(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{a^3 \sqrt{c-i c \tan (e+f x)} (\cos (2 e+5 f x)+i \sin (2 e+5 f x)) (A+B \tan (e+f x)) (15 (3 B+i A) \cos (e+f x)+(23 B+7 i A) \cos (3 (e+f x))+2 \sin (e+f x) ((9 A-25 i B) \cos (2 (e+f x))+9 A-26 i B))}{3 c^2 f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}","\frac{2 a^3 (5 B+i A) \sqrt{c-i c \tan (e+f x)}}{c^2 f}+\frac{8 a^3 (2 B+i A)}{c f \sqrt{c-i c \tan (e+f x)}}-\frac{8 a^3 (B+i A)}{3 f (c-i c \tan (e+f x))^{3/2}}-\frac{2 a^3 B (c-i c \tan (e+f x))^{3/2}}{3 c^3 f}",1,"(a^3*(15*(I*A + 3*B)*Cos[e + f*x] + ((7*I)*A + 23*B)*Cos[3*(e + f*x)] + 2*(9*A - (26*I)*B + (9*A - (25*I)*B)*Cos[2*(e + f*x)])*Sin[e + f*x])*(Cos[2*e + 5*f*x] + I*Sin[2*e + 5*f*x])*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(3*c^2*f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
762,1,135,140,13.0512727,"\int \frac{(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{a^3 \sqrt{c-i c \tan (e+f x)} (\sin (3 (e+2 f x))-i \cos (3 (e+2 f x))) (3 (A-11 i B) \cos (e+f x)+(11 A-91 i B) \cos (3 (e+f x))-10 i \sin (e+f x) ((A-17 i B) \cos (2 (e+f x))+A-14 i B))}{15 c^3 f (\cos (f x)+i \sin (f x))^3}","-\frac{2 a^3 (5 B+i A)}{c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{8 a^3 (2 B+i A)}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac{8 a^3 (B+i A)}{5 f (c-i c \tan (e+f x))^{5/2}}-\frac{2 a^3 B \sqrt{c-i c \tan (e+f x)}}{c^3 f}",1,"(a^3*(3*(A - (11*I)*B)*Cos[e + f*x] + (11*A - (91*I)*B)*Cos[3*(e + f*x)] - (10*I)*(A - (14*I)*B + (A - (17*I)*B)*Cos[2*(e + f*x)])*Sin[e + f*x])*((-I)*Cos[3*(e + 2*f*x)] + Sin[3*(e + 2*f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(15*c^3*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
763,1,141,142,13.1978597,"\int \frac{(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(7/2),x]","\frac{a^3 \cos (e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (4 e+7 f x)+i \sin (4 e+7 f x)) (i (A+13 i B) \cos (e+f x)+(89 B-23 i A) \cos (3 (e+f x))+14 \sin (e+f x) ((A-17 i B) \cos (2 (e+f x))+A-2 i B))}{105 c^4 f (\cos (f x)+i \sin (f x))^3}","-\frac{2 a^3 (5 B+i A)}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}+\frac{8 a^3 (2 B+i A)}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac{8 a^3 (B+i A)}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac{2 a^3 B}{c^3 f \sqrt{c-i c \tan (e+f x)}}",1,"(a^3*Cos[e + f*x]*(I*(A + (13*I)*B)*Cos[e + f*x] + ((-23*I)*A + 89*B)*Cos[3*(e + f*x)] + 14*(A - (2*I)*B + (A - (17*I)*B)*Cos[2*(e + f*x)])*Sin[e + f*x])*(Cos[4*e + 7*f*x] + I*Sin[4*e + 7*f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(105*c^4*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
764,-1,0,220,180.003452,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{a+i a \tan (e+f x)} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2))/(a + I*a*Tan[e + f*x]),x]","\text{\$Aborted}","-\frac{2 \sqrt{2} c^{7/2} (-9 B+5 i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{a f}+\frac{2 c^3 (-9 B+5 i A) \sqrt{c-i c \tan (e+f x)}}{a f}+\frac{c^2 (-9 B+5 i A) (c-i c \tan (e+f x))^{3/2}}{3 a f}+\frac{c (-9 B+5 i A) (c-i c \tan (e+f x))^{5/2}}{10 a f}+\frac{(-B+i A) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))}",1,"$Aborted","F",-1
765,-1,0,180,180.0046515,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2))/(a + I*a*Tan[e + f*x]),x]","\text{\$Aborted}","-\frac{\sqrt{2} c^{5/2} (-7 B+3 i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{a f}+\frac{c^2 (-7 B+3 i A) \sqrt{c-i c \tan (e+f x)}}{a f}+\frac{c (-7 B+3 i A) (c-i c \tan (e+f x))^{3/2}}{6 a f}+\frac{(-B+i A) (c-i c \tan (e+f x))^{5/2}}{2 a f (1+i \tan (e+f x))}",1,"$Aborted","F",-1
766,-1,0,144,180.0021109,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2))/(a + I*a*Tan[e + f*x]),x]","\text{\$Aborted}","-\frac{c^{3/2} (-5 B+i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{\sqrt{2} a f}+\frac{c (-5 B+i A) \sqrt{c-i c \tan (e+f x)}}{2 a f}+\frac{(-B+i A) (c-i c \tan (e+f x))^{3/2}}{2 a f (1+i \tan (e+f x))}",1,"$Aborted","F",-1
767,1,168,109,2.6852467,"\int \frac{(A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}{a+i a \tan (e+f x)} \, dx","Integrate[((A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(a + I*a*Tan[e + f*x]),x]","\frac{(\cos (f x)+i \sin (f x)) (A+B \tan (e+f x)) \left(2 (A+i B) \cos (e+f x) (\sin (f x)+i \cos (f x)) \sqrt{c-i c \tan (e+f x)}+\sqrt{2} \sqrt{c} (3 B+i A) (\cos (e)+i \sin (e)) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)\right)}{4 f (a+i a \tan (e+f x)) (A \cos (e+f x)+B \sin (e+f x))}","\frac{(-B+i A) \sqrt{c-i c \tan (e+f x)}}{2 a f (1+i \tan (e+f x))}+\frac{\sqrt{c} (3 B+i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{2 \sqrt{2} a f}",1,"((Cos[f*x] + I*Sin[f*x])*(A + B*Tan[e + f*x])*(Sqrt[2]*(I*A + 3*B)*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])]*(Cos[e] + I*Sin[e]) + 2*(A + I*B)*Cos[e + f*x]*(I*Cos[f*x] + Sin[f*x])*Sqrt[c - I*c*Tan[e + f*x]]))/(4*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x]))","A",1
768,1,160,141,3.9163686,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]]),x]","\frac{e^{-2 i (e+f x)} \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \left((B+3 i A) e^{2 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)-i \left(1+e^{2 i (e+f x)}\right) \left(A \left(-1+2 e^{2 i (e+f x)}\right)-i B \left(1+2 e^{2 i (e+f x)}\right)\right)\right)}{4 \sqrt{2} a c f}","\frac{-B+i A}{2 a f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}-\frac{B+3 i A}{4 a f \sqrt{c-i c \tan (e+f x)}}+\frac{(B+3 i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{4 \sqrt{2} a \sqrt{c} f}",1,"(Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*((-I)*(1 + E^((2*I)*(e + f*x)))*(A*(-1 + 2*E^((2*I)*(e + f*x))) - I*B*(1 + 2*E^((2*I)*(e + f*x)))) + ((3*I)*A + B)*E^((2*I)*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]]))/(4*Sqrt[2]*a*c*E^((2*I)*(e + f*x))*f)","A",1
769,1,239,184,6.2223532,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2)),x]","-\frac{e^{-i (e+2 f x)} \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} (\cos (f x)+i \sin (f x)) (A+B \tan (e+f x)) \left(\left(1+e^{2 i (e+f x)}\right) \left(i A \left(14 e^{2 i (e+f x)}+2 e^{4 i (e+f x)}-3\right)+B \left(2 e^{2 i (e+f x)}+2 e^{4 i (e+f x)}+3\right)\right)+3 (B-5 i A) e^{2 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)\right)}{24 \sqrt{2} c^2 f (a+i a \tan (e+f x)) (A \cos (e+f x)+B \sin (e+f x))}","\frac{(-B+5 i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{8 \sqrt{2} a c^{3/2} f}+\frac{-B+i A}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}-\frac{-B+5 i A}{8 a c f \sqrt{c-i c \tan (e+f x)}}-\frac{-B+5 i A}{12 a f (c-i c \tan (e+f x))^{3/2}}",1,"-1/24*(Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*((1 + E^((2*I)*(e + f*x)))*(B*(3 + 2*E^((2*I)*(e + f*x)) + 2*E^((4*I)*(e + f*x))) + I*A*(-3 + 14*E^((2*I)*(e + f*x)) + 2*E^((4*I)*(e + f*x)))) + 3*((-5*I)*A + B)*E^((2*I)*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]])*(Cos[f*x] + I*Sin[f*x])*(A + B*Tan[e + f*x]))/(Sqrt[2]*c^2*E^(I*(e + 2*f*x))*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x]))","A",1
770,1,213,223,8.1908648,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2)),x]","-\frac{e^{-2 i (e+f x)} \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \left(\left(1+e^{2 i (e+f x)}\right) \left(i A \left(116 e^{2 i (e+f x)}+32 e^{4 i (e+f x)}+6 e^{6 i (e+f x)}-15\right)+3 B \left(-8 e^{2 i (e+f x)}+4 e^{4 i (e+f x)}+2 e^{6 i (e+f x)}+5\right)\right)+15 (3 B-7 i A) e^{2 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)\right)}{240 \sqrt{2} a c^3 f}","\frac{(-3 B+7 i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{16 \sqrt{2} a c^{5/2} f}-\frac{-3 B+7 i A}{16 a c^2 f \sqrt{c-i c \tan (e+f x)}}-\frac{-3 B+7 i A}{24 a c f (c-i c \tan (e+f x))^{3/2}}-\frac{-3 B+7 i A}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac{-B+i A}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}",1,"-1/240*(Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*((1 + E^((2*I)*(e + f*x)))*(3*B*(5 - 8*E^((2*I)*(e + f*x)) + 4*E^((4*I)*(e + f*x)) + 2*E^((6*I)*(e + f*x))) + I*A*(-15 + 116*E^((2*I)*(e + f*x)) + 32*E^((4*I)*(e + f*x)) + 6*E^((6*I)*(e + f*x)))) + 15*((-7*I)*A + 3*B)*E^((2*I)*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]]))/(Sqrt[2]*a*c^3*E^((2*I)*(e + f*x))*f)","A",1
771,-1,0,275,180.0037327,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(9/2))/(a + I*a*Tan[e + f*x])^2,x]","\text{\$Aborted}","\frac{7 c^{9/2} (-13 B+5 i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{\sqrt{2} a^2 f}-\frac{7 c^4 (-13 B+5 i A) \sqrt{c-i c \tan (e+f x)}}{2 a^2 f}-\frac{7 c^3 (-13 B+5 i A) (c-i c \tan (e+f x))^{3/2}}{12 a^2 f}-\frac{7 c^2 (-13 B+5 i A) (c-i c \tan (e+f x))^{5/2}}{40 a^2 f}-\frac{c (-13 B+5 i A) (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac{(-B+i A) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2}",1,"$Aborted","F",-1
772,-1,0,238,180.0045611,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2))/(a + I*a*Tan[e + f*x])^2,x]","\text{\$Aborted}","\frac{5 c^{7/2} (-11 B+3 i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{2 \sqrt{2} a^2 f}-\frac{5 c^3 (-11 B+3 i A) \sqrt{c-i c \tan (e+f x)}}{4 a^2 f}-\frac{5 c^2 (-11 B+3 i A) (c-i c \tan (e+f x))^{3/2}}{24 a^2 f}-\frac{c (-11 B+3 i A) (c-i c \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac{(-B+i A) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2}",1,"$Aborted","F",-1
773,-1,0,199,180.0023965,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2))/(a + I*a*Tan[e + f*x])^2,x]","\text{\$Aborted}","\frac{3 c^{5/2} (-9 B+i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{4 \sqrt{2} a^2 f}-\frac{3 c^2 (-9 B+i A) \sqrt{c-i c \tan (e+f x)}}{8 a^2 f}-\frac{c (-9 B+i A) (c-i c \tan (e+f x))^{3/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac{(-B+i A) (c-i c \tan (e+f x))^{5/2}}{4 a^2 f (1+i \tan (e+f x))^2}",1,"$Aborted","F",-1
774,1,205,160,4.3183481,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2))/(a + I*a*Tan[e + f*x])^2,x]","\frac{\sec (e+f x) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x)) \left(\sqrt{2} c^{3/2} (A-7 i B) (\sin (2 e)-i \cos (2 e)) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)+2 c \cos (e+f x) (\cos (2 f x)-i \sin (2 f x)) \sqrt{c-i c \tan (e+f x)} ((A+9 i B) \sin (e+f x)+(5 B+3 i A) \cos (e+f x))\right)}{16 f (a+i a \tan (e+f x))^2 (A \cos (e+f x)+B \sin (e+f x))}","-\frac{c^{3/2} (7 B+i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{8 \sqrt{2} a^2 f}+\frac{c (7 B+i A) \sqrt{c-i c \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}+\frac{(-B+i A) (c-i c \tan (e+f x))^{3/2}}{4 a^2 f (1+i \tan (e+f x))^2}",1,"(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x])*(Sqrt[2]*(A - (7*I)*B)*c^(3/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])]*((-I)*Cos[2*e] + Sin[2*e]) + 2*c*Cos[e + f*x]*(Cos[2*f*x] - I*Sin[2*f*x])*(((3*I)*A + 5*B)*Cos[e + f*x] + (A + (9*I)*B)*Sin[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]]))/(16*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2)","A",1
775,1,206,159,3.3604118,"\int \frac{(A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[((A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(a + I*a*Tan[e + f*x])^2,x]","\frac{\sec (e+f x) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x)) \left(2 \cos (e+f x) (\cos (2 f x)-i \sin (2 f x)) \sqrt{c-i c \tan (e+f x)} ((-3 A+5 i B) \sin (e+f x)+(B+7 i A) \cos (e+f x))+\sqrt{2} \sqrt{c} (5 B+3 i A) (\cos (2 e)+i \sin (2 e)) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)\right)}{32 f (a+i a \tan (e+f x))^2 (A \cos (e+f x)+B \sin (e+f x))}","\frac{(-B+i A) \sqrt{c-i c \tan (e+f x)}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac{(5 B+3 i A) \sqrt{c-i c \tan (e+f x)}}{16 a^2 f (1+i \tan (e+f x))}+\frac{\sqrt{c} (5 B+3 i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{16 \sqrt{2} a^2 f}",1,"(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x])*(Sqrt[2]*((3*I)*A + 5*B)*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])]*(Cos[2*e] + I*Sin[2*e]) + 2*Cos[e + f*x]*(Cos[2*f*x] - I*Sin[2*f*x])*(((7*I)*A + B)*Cos[e + f*x] + (-3*A + (5*I)*B)*Sin[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]]))/(32*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2)","A",1
776,1,160,195,5.1215647,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^2 \sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*Sqrt[c - I*c*Tan[e + f*x]]),x]","\frac{\sqrt{c-i c \tan (e+f x)} (\sin (e+f x)+i \cos (e+f x)) \left(3 (5 A-3 i B) e^{i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)-2 \cos (e+f x) (2 (3 B+5 i A) \sin (2 (e+f x))+2 (3 A-5 i B) \cos (2 (e+f x))-9 A-i B)\right)}{64 a^2 c f}","\frac{-B+i A}{4 a^2 f (1+i \tan (e+f x))^2 \sqrt{c-i c \tan (e+f x)}}-\frac{3 (3 B+5 i A)}{32 a^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{3 B+5 i A}{16 a^2 f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}+\frac{3 (3 B+5 i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{32 \sqrt{2} a^2 \sqrt{c} f}",1,"((I*Cos[e + f*x] + Sin[e + f*x])*(3*(5*A - (3*I)*B)*E^(I*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]] - 2*Cos[e + f*x]*(-9*A - I*B + 2*(3*A - (5*I)*B)*Cos[2*(e + f*x)] + 2*((5*I)*A + 3*B)*Sin[2*(e + f*x)]))*Sqrt[c - I*c*Tan[e + f*x]])/(64*a^2*c*f)","A",1
777,1,204,226,6.4461794,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(3/2)),x]","\frac{e^{-4 i (e+f x)} \sqrt{c-i c \tan (e+f x)} \left(15 (B+7 i A) e^{4 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)-i \left(1+e^{2 i (e+f x)}\right) \left(A \left(-39 e^{2 i (e+f x)}+80 e^{4 i (e+f x)}+8 e^{6 i (e+f x)}-6\right)-i B \left(15 e^{2 i (e+f x)}+32 e^{4 i (e+f x)}+8 e^{6 i (e+f x)}+6\right)\right)\right)}{384 a^2 c^2 f}","\frac{5 (B+7 i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{64 \sqrt{2} a^2 c^{3/2} f}+\frac{-B+i A}{4 a^2 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2}}-\frac{5 (B+7 i A)}{64 a^2 c f \sqrt{c-i c \tan (e+f x)}}-\frac{5 (B+7 i A)}{96 a^2 f (c-i c \tan (e+f x))^{3/2}}+\frac{B+7 i A}{16 a^2 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}",1,"(((-I)*(1 + E^((2*I)*(e + f*x)))*((-I)*B*(6 + 15*E^((2*I)*(e + f*x)) + 32*E^((4*I)*(e + f*x)) + 8*E^((6*I)*(e + f*x))) + A*(-6 - 39*E^((2*I)*(e + f*x)) + 80*E^((4*I)*(e + f*x)) + 8*E^((6*I)*(e + f*x)))) + 15*((7*I)*A + B)*E^((4*I)*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]])*Sqrt[c - I*c*Tan[e + f*x]])/(384*a^2*c^2*E^((4*I)*(e + f*x))*f)","A",1
778,1,209,273,9.8523647,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(5/2)),x]","\frac{\sqrt{c-i c \tan (e+f x)} (\cos (e+f x)+i \sin (e+f x)) \left(105 i (9 A+i B) e^{-i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)+2 \cos (e+f x) ((-223 B+87 i A) \cos (2 (e+f x))+6 i (A+9 i B) \cos (4 (e+f x))+423 A \sin (2 (e+f x))+54 A \sin (4 (e+f x))-864 i A+47 i B \sin (2 (e+f x))+6 i B \sin (4 (e+f x))-64 B)\right)}{3840 a^2 c^3 f}","\frac{7 (-B+9 i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{128 \sqrt{2} a^2 c^{5/2} f}-\frac{7 (-B+9 i A)}{128 a^2 c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{-B+i A}{4 a^2 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2}}-\frac{7 (-B+9 i A)}{192 a^2 c f (c-i c \tan (e+f x))^{3/2}}-\frac{7 (-B+9 i A)}{160 a^2 f (c-i c \tan (e+f x))^{5/2}}+\frac{-B+9 i A}{16 a^2 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}",1,"((Cos[e + f*x] + I*Sin[e + f*x])*(((105*I)*(9*A + I*B)*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]])/E^(I*(e + f*x)) + 2*Cos[e + f*x]*((-864*I)*A - 64*B + ((87*I)*A - 223*B)*Cos[2*(e + f*x)] + (6*I)*(A + (9*I)*B)*Cos[4*(e + f*x)] + 423*A*Sin[2*(e + f*x)] + (47*I)*B*Sin[2*(e + f*x)] + 54*A*Sin[4*(e + f*x)] + (6*I)*B*Sin[4*(e + f*x)]))*Sqrt[c - I*c*Tan[e + f*x]])/(3840*a^2*c^3*f)","A",1
779,-1,0,291,180.0051315,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(9/2))/(a + I*a*Tan[e + f*x])^3,x]","\text{\$Aborted}","-\frac{35 c^{9/2} (-5 B+i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{4 \sqrt{2} a^3 f}+\frac{35 c^4 (-5 B+i A) \sqrt{c-i c \tan (e+f x)}}{8 a^3 f}+\frac{35 c^3 (-5 B+i A) (c-i c \tan (e+f x))^{3/2}}{48 a^3 f}+\frac{7 c^2 (-5 B+i A) (c-i c \tan (e+f x))^{5/2}}{16 a^3 f (1+i \tan (e+f x))}-\frac{c (-5 B+i A) (c-i c \tan (e+f x))^{7/2}}{8 a^3 f (1+i \tan (e+f x))^2}+\frac{(-B+i A) (c-i c \tan (e+f x))^{9/2}}{6 a^3 f (1+i \tan (e+f x))^3}",1,"$Aborted","F",-1
780,-1,0,252,180.0022291,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2))/(a + I*a*Tan[e + f*x])^3,x]","\text{\$Aborted}","-\frac{5 c^{7/2} (-13 B+i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{8 \sqrt{2} a^3 f}+\frac{5 c^3 (-13 B+i A) \sqrt{c-i c \tan (e+f x)}}{16 a^3 f}+\frac{5 c^2 (-13 B+i A) (c-i c \tan (e+f x))^{3/2}}{48 a^3 f (1+i \tan (e+f x))}-\frac{c (-13 B+i A) (c-i c \tan (e+f x))^{5/2}}{24 a^3 f (1+i \tan (e+f x))^2}+\frac{(-B+i A) (c-i c \tan (e+f x))^{7/2}}{6 a^3 f (1+i \tan (e+f x))^3}",1,"$Aborted","F",-1
781,1,227,213,7.9764834,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2))/(a + I*a*Tan[e + f*x])^3,x]","\frac{\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^3 (A+B \tan (e+f x)) \left(\sqrt{2} c^{5/2} (11 B+i A) (\cos (3 e)+i \sin (3 e)) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)+\frac{2}{3} c^2 \cos (e+f x) (\cos (3 f x)-i \sin (3 f x)) \sqrt{c-i c \tan (e+f x)} ((11 A-25 i B) \sin (2 (e+f x))+(-41 B+5 i A) \cos (2 (e+f x))+2 i A+22 B)\right)}{32 f (a+i a \tan (e+f x))^3 (A \cos (e+f x)+B \sin (e+f x))}","\frac{c^{5/2} (11 B+i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{16 \sqrt{2} a^3 f}-\frac{c^2 (11 B+i A) \sqrt{c-i c \tan (e+f x)}}{16 a^3 f (1+i \tan (e+f x))}+\frac{c (11 B+i A) (c-i c \tan (e+f x))^{3/2}}{24 a^3 f (1+i \tan (e+f x))^2}+\frac{(-B+i A) (c-i c \tan (e+f x))^{5/2}}{6 a^3 f (1+i \tan (e+f x))^3}",1,"(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x])*(Sqrt[2]*(I*A + 11*B)*c^(5/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])]*(Cos[3*e] + I*Sin[3*e]) + (2*c^2*Cos[e + f*x]*(Cos[3*f*x] - I*Sin[3*f*x])*((2*I)*A + 22*B + ((5*I)*A - 41*B)*Cos[2*(e + f*x)] + (11*A - (25*I)*B)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/3))/(32*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3)","A",1
782,1,224,211,6.0660422,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2))/(a + I*a*Tan[e + f*x])^3,x]","\frac{\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^3 (A+B \tan (e+f x)) \left(\sqrt{2} c^{3/2} (A-3 i B) (\sin (3 e)-i \cos (3 e)) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)+\frac{2}{3} c \cos (e+f x) (\cos (3 f x)-i \sin (3 f x)) \sqrt{c-i c \tan (e+f x)} ((5 A+17 i B) \sin (2 (e+f x))+(B+11 i A) \cos (2 (e+f x))+2 (5 B+7 i A))\right)}{64 f (a+i a \tan (e+f x))^3 (A \cos (e+f x)+B \sin (e+f x))}","-\frac{c^{3/2} (3 B+i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{32 \sqrt{2} a^3 f}-\frac{c (3 B+i A) \sqrt{c-i c \tan (e+f x)}}{32 a^3 f (1+i \tan (e+f x))}+\frac{c (3 B+i A) \sqrt{c-i c \tan (e+f x)}}{8 a^3 f (1+i \tan (e+f x))^2}+\frac{(-B+i A) (c-i c \tan (e+f x))^{3/2}}{6 a^3 f (1+i \tan (e+f x))^3}",1,"(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x])*(Sqrt[2]*(A - (3*I)*B)*c^(3/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])]*((-I)*Cos[3*e] + Sin[3*e]) + (2*c*Cos[e + f*x]*(Cos[3*f*x] - I*Sin[3*f*x])*(2*((7*I)*A + 5*B) + ((11*I)*A + B)*Cos[2*(e + f*x)] + (5*A + (17*I)*B)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/3))/(64*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3)","A",1
783,1,225,209,4.031313,"\int \frac{(A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[((A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(a + I*a*Tan[e + f*x])^3,x]","\frac{\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^3 (A+B \tan (e+f x)) \left(\frac{2}{3} \cos (e+f x) (\sin (3 f x)+i \cos (3 f x)) \sqrt{c-i c \tan (e+f x)} (5 (7 B+5 i A) \sin (2 (e+f x))+(41 A-19 i B) \cos (2 (e+f x))+26 A+2 i B)+\sqrt{2} \sqrt{c} (7 B+5 i A) (\cos (3 e)+i \sin (3 e)) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)\right)}{128 f (a+i a \tan (e+f x))^3 (A \cos (e+f x)+B \sin (e+f x))}","\frac{(-B+i A) \sqrt{c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac{(7 B+5 i A) \sqrt{c-i c \tan (e+f x)}}{64 a^3 f (1+i \tan (e+f x))}+\frac{(7 B+5 i A) \sqrt{c-i c \tan (e+f x)}}{48 a^3 f (1+i \tan (e+f x))^2}+\frac{\sqrt{c} (7 B+5 i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{64 \sqrt{2} a^3 f}",1,"(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x])*(Sqrt[2]*((5*I)*A + 7*B)*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])]*(Cos[3*e] + I*Sin[3*e]) + (2*Cos[e + f*x]*(I*Cos[3*f*x] + Sin[3*f*x])*(26*A + (2*I)*B + (41*A - (19*I)*B)*Cos[2*(e + f*x)] + 5*((5*I)*A + 7*B)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/3))/(128*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3)","A",1
784,1,181,245,5.83926,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 \sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*Sqrt[c - I*c*Tan[e + f*x]]),x]","\frac{\sqrt{c-i c \tan (e+f x)} (\cos (2 (e+f x))-i \sin (2 (e+f x))) \left(15 (5 B+7 i A) e^{2 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)+2 \cos (e+f x) ((7 A-5 i B) (8 \sin (3 (e+f x))-7 \sin (e+f x))+(7 B+125 i A) \cos (e+f x)+(-56 B-40 i A) \cos (3 (e+f x)))\right)}{768 a^3 c f}","\frac{-B+i A}{6 a^3 f (1+i \tan (e+f x))^3 \sqrt{c-i c \tan (e+f x)}}-\frac{5 (5 B+7 i A)}{128 a^3 f \sqrt{c-i c \tan (e+f x)}}+\frac{5 (5 B+7 i A)}{192 a^3 f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}+\frac{5 B+7 i A}{48 a^3 f (1+i \tan (e+f x))^2 \sqrt{c-i c \tan (e+f x)}}+\frac{5 (5 B+7 i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{128 \sqrt{2} a^3 \sqrt{c} f}",1,"((Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)])*(15*((7*I)*A + 5*B)*E^((2*I)*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]] + 2*Cos[e + f*x]*(((125*I)*A + 7*B)*Cos[e + f*x] + ((-40*I)*A - 56*B)*Cos[3*(e + f*x)] + (7*A - (5*I)*B)*(-7*Sin[e + f*x] + 8*Sin[3*(e + f*x)])))*Sqrt[c - I*c*Tan[e + f*x]])/(768*a^3*c*f)","A",1
785,1,206,274,8.323747,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(3/2)),x]","\frac{\sqrt{c-i c \tan (e+f x)} (\sin (e+f x)+i \cos (e+f x)) \left(105 (3 A-i B) e^{i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)-2 \cos (e+f x) (2 (79 A-69 i B) \cos (2 (e+f x))+8 (A-3 i B) \cos (4 (e+f x))+258 i A \sin (2 (e+f x))+24 i A \sin (4 (e+f x))-165 A+86 B \sin (2 (e+f x))+8 B \sin (4 (e+f x))-9 i B)\right)}{1536 a^3 c^2 f}","\frac{35 (B+3 i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{256 \sqrt{2} a^3 c^{3/2} f}+\frac{-B+i A}{6 a^3 f (1+i \tan (e+f x))^3 (c-i c \tan (e+f x))^{3/2}}-\frac{35 (B+3 i A)}{256 a^3 c f \sqrt{c-i c \tan (e+f x)}}-\frac{35 (B+3 i A)}{384 a^3 f (c-i c \tan (e+f x))^{3/2}}+\frac{7 (B+3 i A)}{64 a^3 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}+\frac{B+3 i A}{16 a^3 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2}}",1,"((I*Cos[e + f*x] + Sin[e + f*x])*(105*(3*A - I*B)*E^(I*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]] - 2*Cos[e + f*x]*(-165*A - (9*I)*B + 2*(79*A - (69*I)*B)*Cos[2*(e + f*x)] + 8*(A - (3*I)*B)*Cos[4*(e + f*x)] + (258*I)*A*Sin[2*(e + f*x)] + 86*B*Sin[2*(e + f*x)] + (24*I)*A*Sin[4*(e + f*x)] + 8*B*Sin[4*(e + f*x)]))*Sqrt[c - I*c*Tan[e + f*x]])/(1536*a^3*c^2*f)","A",1
786,1,256,311,12.378041,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(5/2)),x]","\frac{e^{-6 i (e+f x)} \sqrt{c-i c \tan (e+f x)} \left(315 (B+11 i A) e^{6 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)-i \left(1+e^{2 i (e+f x)}\right) \left(A \left(-310 e^{2 i (e+f x)}-1335 e^{4 i (e+f x)}+2768 e^{6 i (e+f x)}+416 e^{8 i (e+f x)}+48 e^{10 i (e+f x)}-40\right)-i B \left(190 e^{2 i (e+f x)}+315 e^{4 i (e+f x)}+688 e^{6 i (e+f x)}+256 e^{8 i (e+f x)}+48 e^{10 i (e+f x)}+40\right)\right)\right)}{15360 a^3 c^3 f}","\frac{21 (B+11 i A) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{512 \sqrt{2} a^3 c^{5/2} f}-\frac{21 (B+11 i A)}{512 a^3 c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{-B+i A}{6 a^3 f (1+i \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2}}-\frac{7 (B+11 i A)}{256 a^3 c f (c-i c \tan (e+f x))^{3/2}}-\frac{21 (B+11 i A)}{640 a^3 f (c-i c \tan (e+f x))^{5/2}}+\frac{3 (B+11 i A)}{64 a^3 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}+\frac{B+11 i A}{48 a^3 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2}}",1,"(((-I)*(1 + E^((2*I)*(e + f*x)))*((-I)*B*(40 + 190*E^((2*I)*(e + f*x)) + 315*E^((4*I)*(e + f*x)) + 688*E^((6*I)*(e + f*x)) + 256*E^((8*I)*(e + f*x)) + 48*E^((10*I)*(e + f*x))) + A*(-40 - 310*E^((2*I)*(e + f*x)) - 1335*E^((4*I)*(e + f*x)) + 2768*E^((6*I)*(e + f*x)) + 416*E^((8*I)*(e + f*x)) + 48*E^((10*I)*(e + f*x)))) + 315*((11*I)*A + B)*E^((6*I)*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]])*Sqrt[c - I*c*Tan[e + f*x]])/(15360*a^3*c^3*E^((6*I)*(e + f*x))*f)","A",1
787,1,257,272,10.4565799,"\int \sqrt{a+i a \tan (e+f x)} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2),x]","\frac{\sqrt{a+i a \tan (e+f x)} (A+B \tan (e+f x)) \left(\frac{5 c^4 (3 B-4 i A) e^{-i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right)}{\sqrt{\frac{c}{1+e^{2 i (e+f x)}}}}+\frac{1}{24} c^3 \sec ^{\frac{7}{2}}(e+f x) \sqrt{c-i c \tan (e+f x)} (64 (3 B-4 i A) \cos (e+f x)+96 (B-i A) \cos (3 (e+f x))-6 \sin (e+f x) ((12 A+17 i B) \cos (2 (e+f x))+12 A+13 i B))\right)}{4 f \sec ^{\frac{3}{2}}(e+f x) (A \cos (e+f x)+B \sin (e+f x))}","-\frac{5 \sqrt{a} c^{7/2} (-3 B+4 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{4 f}-\frac{5 c^3 (-3 B+4 i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{8 f}-\frac{5 c^2 (-3 B+4 i A) \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac{c (-3 B+4 i A) \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{12 f}+\frac{B \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{7/2}}{4 f}",1,"(Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x])*((5*((-4*I)*A + 3*B)*c^4*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))])/(E^(I*(e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]) + (c^3*Sec[e + f*x]^(7/2)*(64*((-4*I)*A + 3*B)*Cos[e + f*x] + 96*((-I)*A + B)*Cos[3*(e + f*x)] - 6*(12*A + (13*I)*B + (12*A + (17*I)*B)*Cos[2*(e + f*x)])*Sin[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/24))/(4*f*Sec[e + f*x]^(3/2)*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
788,1,226,217,7.0557881,"\int \sqrt{a+i a \tan (e+f x)} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{\sqrt{a+i a \tan (e+f x)} (A+B \tan (e+f x)) \left(\frac{c^3 (2 B-3 i A) e^{-i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right)}{\sqrt{\frac{c}{1+e^{2 i (e+f x)}}}}+\frac{1}{12} c^2 \sec ^{\frac{5}{2}}(e+f x) \sqrt{c-i c \tan (e+f x)} (-3 (A+2 i B) \sin (2 (e+f x))+12 (B-i A) \cos (2 (e+f x))-12 i A+8 B)\right)}{f \sec ^{\frac{3}{2}}(e+f x) (A \cos (e+f x)+B \sin (e+f x))}","-\frac{\sqrt{a} c^{5/2} (-2 B+3 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{f}-\frac{c^2 (-2 B+3 i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 f}-\frac{c (-2 B+3 i A) \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 f}+\frac{B \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{3 f}",1,"(Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x])*((((-3*I)*A + 2*B)*c^3*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))])/(E^(I*(e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]) + (c^2*Sec[e + f*x]^(5/2)*((-12*I)*A + 8*B + 12*((-I)*A + B)*Cos[2*(e + f*x)] - 3*(A + (2*I)*B)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/12))/(f*Sec[e + f*x]^(3/2)*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
789,1,159,164,6.1593439,"\int \sqrt{a+i a \tan (e+f x)} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{c^2 e^{-i e} \left(\sin \left(\frac{e}{2}\right)-i \cos \left(\frac{e}{2}\right)\right) \sqrt{a+i a \tan (e+f x)} \left(\cos \left(\frac{e}{2}+f x\right)-i \sin \left(\frac{e}{2}+f x\right)\right) \left((4 A+2 i B) \tan ^{-1}\left(e^{i (e+f x)}\right)+\sec (e+f x) (2 A+B \sec (e) \sin (f x) \sec (e+f x)+B \tan (e)+2 i B)\right)}{2 \sqrt{2} f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}}}","-\frac{\sqrt{a} c^{3/2} (-B+2 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{f}-\frac{c (-B+2 i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 f}+\frac{B \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 f}",1,"(c^2*((-I)*Cos[e/2] + Sin[e/2])*(Cos[e/2 + f*x] - I*Sin[e/2 + f*x])*((4*A + (2*I)*B)*ArcTan[E^(I*(e + f*x))] + Sec[e + f*x]*(2*A + (2*I)*B + B*Sec[e]*Sec[e + f*x]*Sin[f*x] + B*Tan[e]))*Sqrt[a + I*a*Tan[e + f*x]])/(2*Sqrt[2]*E^(I*e)*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f)","A",1
790,1,102,104,3.7864461,"\int \sqrt{a+i a \tan (e+f x)} (A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{\sqrt{2} e^{-i (e+f x)} \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sqrt{a+i a \tan (e+f x)} \left(B e^{i (e+f x)}-i A \left(1+e^{2 i (e+f x)}\right) \tan ^{-1}\left(e^{i (e+f x)}\right)\right)}{f}","\frac{B \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{f}-\frac{2 i \sqrt{a} A \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{f}",1,"(Sqrt[2]*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*(B*E^(I*(e + f*x)) - I*A*(1 + E^((2*I)*(e + f*x)))*ArcTan[E^(I*(e + f*x))])*Sqrt[a + I*a*Tan[e + f*x]])/(E^(I*(e + f*x))*f)","A",1
791,1,127,109,4.4516943,"\int \frac{\sqrt{a+i a \tan (e+f x)} (A+B \tan (e+f x))}{\sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[(Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x]))/Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{\sqrt{a+i a \tan (e+f x)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-i \sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)-i \cos \left(\frac{1}{2} (e+f x)\right)\right) \left(A+2 B \tan ^{-1}\left(e^{i (e+f x)}\right) (\sin (e+f x)+i \cos (e+f x))-i B\right)}{f \sqrt{c-i c \tan (e+f x)}}","\frac{2 \sqrt{a} B \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{\sqrt{c} f}-\frac{(B+i A) \sqrt{a+i a \tan (e+f x)}}{f \sqrt{c-i c \tan (e+f x)}}",1,"((Cos[(e + f*x)/2] - I*Sin[(e + f*x)/2])*((-I)*Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(A - I*B + 2*B*ArcTan[E^(I*(e + f*x))]*(I*Cos[e + f*x] + Sin[e + f*x]))*Sqrt[a + I*a*Tan[e + f*x]])/(f*Sqrt[c - I*c*Tan[e + f*x]])","A",1
792,1,101,102,5.9077339,"\int \frac{\sqrt{a+i a \tan (e+f x)} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[(Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{\cos (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\cos (2 (e+f x))+i \sin (2 (e+f x))) ((B-2 i A) \cos (e+f x)-(A+2 i B) \sin (e+f x))}{3 c^2 f}","-\frac{(-2 B+i A) \sqrt{a+i a \tan (e+f x)}}{3 c f \sqrt{c-i c \tan (e+f x)}}-\frac{(B+i A) \sqrt{a+i a \tan (e+f x)}}{3 f (c-i c \tan (e+f x))^{3/2}}",1,"(Cos[e + f*x]*(((-2*I)*A + B)*Cos[e + f*x] - (A + (2*I)*B)*Sin[e + f*x])*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(3*c^2*f)","A",1
793,1,114,155,9.8367891,"\int \frac{\sqrt{a+i a \tan (e+f x)} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{\cos (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\cos (3 (e+f x))+i \sin (3 (e+f x))) (-3 (2 A+3 i B) \sin (2 (e+f x))+(6 B-9 i A) \cos (2 (e+f x))-5 i A)}{30 c^3 f}","-\frac{(-3 B+2 i A) \sqrt{a+i a \tan (e+f x)}}{15 c^2 f \sqrt{c-i c \tan (e+f x)}}-\frac{(-3 B+2 i A) \sqrt{a+i a \tan (e+f x)}}{15 c f (c-i c \tan (e+f x))^{3/2}}-\frac{(B+i A) \sqrt{a+i a \tan (e+f x)}}{5 f (c-i c \tan (e+f x))^{5/2}}",1,"(Cos[e + f*x]*((-5*I)*A + ((-9*I)*A + 6*B)*Cos[2*(e + f*x)] - 3*(2*A + (3*I)*B)*Sin[2*(e + f*x)])*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(30*c^3*f)","A",1
794,1,136,208,12.3095848,"\int \frac{\sqrt{a+i a \tan (e+f x)} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx","Integrate[(Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(7/2),x]","\frac{\cos (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\cos (4 (e+f x))+i \sin (4 (e+f x))) (-(3 A+4 i B) (7 \sin (e+f x)+15 \sin (3 (e+f x)))+7 (B-12 i A) \cos (e+f x)+15 (3 B-4 i A) \cos (3 (e+f x)))}{420 c^4 f}","-\frac{2 (-4 B+3 i A) \sqrt{a+i a \tan (e+f x)}}{105 c^3 f \sqrt{c-i c \tan (e+f x)}}-\frac{2 (-4 B+3 i A) \sqrt{a+i a \tan (e+f x)}}{105 c^2 f (c-i c \tan (e+f x))^{3/2}}-\frac{(-4 B+3 i A) \sqrt{a+i a \tan (e+f x)}}{35 c f (c-i c \tan (e+f x))^{5/2}}-\frac{(B+i A) \sqrt{a+i a \tan (e+f x)}}{7 f (c-i c \tan (e+f x))^{7/2}}",1,"(Cos[e + f*x]*(7*((-12*I)*A + B)*Cos[e + f*x] + 15*((-4*I)*A + 3*B)*Cos[3*(e + f*x)] - (3*A + (4*I)*B)*(7*Sin[e + f*x] + 15*Sin[3*(e + f*x)]))*(Cos[4*(e + f*x)] + I*Sin[4*(e + f*x)])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(420*c^4*f)","A",1
795,1,257,279,13.7844834,"\int (a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2),x]","\frac{(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) \left(\frac{c^4 (2 B-5 i A) e^{-2 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right)}{\sqrt{\frac{c}{1+e^{2 i (e+f x)}}}}-\frac{1}{240} c^3 (\tan (e+f x)+i) \sec ^{\frac{7}{2}}(e+f x) \sqrt{c-i c \tan (e+f x)} (30 (6 B+i A) \sin (2 (e+f x))+(5 A+2 i B) (64+15 i \sin (4 (e+f x)))+320 (A+i B) \cos (2 (e+f x)))\right)}{4 f \sec ^{\frac{5}{2}}(e+f x) (A \cos (e+f x)+B \sin (e+f x))}","-\frac{a^{3/2} c^{7/2} (-2 B+5 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{4 f}+\frac{a c^3 (5 A+2 i B) \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{8 f}-\frac{c^2 (-2 B+5 i A) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}-\frac{c (-2 B+5 i A) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac{B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f}",1,"((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x])*((((-5*I)*A + 2*B)*c^4*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))])/(E^((2*I)*(e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]) - (c^3*Sec[e + f*x]^(7/2)*(320*(A + I*B)*Cos[2*(e + f*x)] + 30*(I*A + 6*B)*Sin[2*(e + f*x)] + (5*A + (2*I)*B)*(64 + (15*I)*Sin[4*(e + f*x)]))*(I + Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/240))/(4*f*Sec[e + f*x]^(5/2)*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
796,1,241,226,10.8284442,"\int (a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) \left(\frac{c^3 (B-4 i A) e^{-2 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right)}{\sqrt{\frac{c}{1+e^{2 i (e+f x)}}}}+\frac{1}{12} c^2 (1-i \tan (e+f x)) \sec ^{\frac{3}{2}}(e+f x) \sqrt{c-i c \tan (e+f x)} (3 \tan (e+f x) ((4 A+i B) \cos (2 (e+f x))+4 A-3 i B)+16 (B-i A))\right)}{4 f \sec ^{\frac{5}{2}}(e+f x) (A \cos (e+f x)+B \sin (e+f x))}","-\frac{a^{3/2} c^{5/2} (-B+4 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{4 f}+\frac{a c^2 (4 A+i B) \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{8 f}-\frac{c (-B+4 i A) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}+\frac{B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{4 f}",1,"((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x])*((((-4*I)*A + B)*c^3*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))])/(E^((2*I)*(e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]) + (c^2*Sec[e + f*x]^(3/2)*(1 - I*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]]*(16*((-I)*A + B) + 3*(4*A - (3*I)*B + (4*A + I*B)*Cos[2*(e + f*x)])*Tan[e + f*x]))/12))/(4*f*Sec[e + f*x]^(5/2)*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
797,1,109,157,7.5335021,"\int (a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2),x]","-\frac{i a c^2 (\tan (e+f x)-i) (\tan (e+f x)+i)^2 \sqrt{a+i a \tan (e+f x)} \left(3 A \sin (2 (e+f x))-12 i A \cos ^3(e+f x) \tan ^{-1}\left(e^{i (e+f x)}\right)+4 B\right)}{12 f \sqrt{c-i c \tan (e+f x)}}","-\frac{i a^{3/2} A c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{f}+\frac{a A c \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 f}+\frac{B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{3 f}",1,"((-1/12*I)*a*c^2*(4*B - (12*I)*A*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x]^3 + 3*A*Sin[2*(e + f*x)])*(-I + Tan[e + f*x])*(I + Tan[e + f*x])^2*Sqrt[a + I*a*Tan[e + f*x]])/(f*Sqrt[c - I*c*Tan[e + f*x]])","A",1
798,1,220,160,6.7973511,"\int (a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) \left(\frac{\cos (e) (\tan (e)+i) \sqrt{\sec (e+f x)} \sqrt{c-i c \tan (e+f x)} (2 A+B \tan (e+f x)-2 i B)}{2 \cos (f x)+2 i \sin (f x)}-\frac{i c (2 A-i B) e^{-2 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right)}{\sqrt{\frac{c}{1+e^{2 i (e+f x)}}}}\right)}{f \sec ^{\frac{5}{2}}(e+f x) (A \cos (e+f x)+B \sin (e+f x))}","-\frac{a^{3/2} \sqrt{c} (B+2 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{f}+\frac{a (B+2 i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 f}+\frac{B (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}{2 f}",1,"((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x])*(((-I)*(2*A - I*B)*c*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))])/(E^((2*I)*(e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]) + (Cos[e]*Sqrt[Sec[e + f*x]]*(I + Tan[e])*(2*A - (2*I)*B + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(2*Cos[f*x] + (2*I)*Sin[f*x])))/(f*Sec[e + f*x]^(5/2)*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
799,1,190,169,7.0041079,"\int \frac{(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{\sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x]))/Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{2 a e^{-2 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} (\tan (e+f x)-i) \sqrt{a+i a \tan (e+f x)} \left(e^{i (e+f x)} \left(A \left(1+e^{2 i (e+f x)}\right)-i B \left(2+e^{2 i (e+f x)}\right)\right)-(A-2 i B) \left(1+e^{2 i (e+f x)}\right) \tan ^{-1}\left(e^{i (e+f x)}\right)\right)}{c f \sec ^{\frac{3}{2}}(e+f x)}","\frac{2 a^{3/2} (2 B+i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{\sqrt{c} f}-\frac{a (2 B+i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{c f}-\frac{(B+i A) (a+i a \tan (e+f x))^{3/2}}{f \sqrt{c-i c \tan (e+f x)}}",1,"(2*a*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*(E^(I*(e + f*x))*(A*(1 + E^((2*I)*(e + f*x))) - I*B*(2 + E^((2*I)*(e + f*x)))) - (A - (2*I)*B)*(1 + E^((2*I)*(e + f*x)))*ArcTan[E^(I*(e + f*x))])*(-I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])/(c*E^((2*I)*(e + f*x))*f*Sec[e + f*x]^(3/2))","A",1
800,1,123,155,9.2020033,"\int \frac{(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(3/2),x]","-\frac{a e^{-i (e+f x)} \sqrt{a+i a \tan (e+f x)} \left(i A e^{3 i (e+f x)}+B e^{i (e+f x)} \left(-6+e^{2 i (e+f x)}\right)+6 B \tan ^{-1}\left(e^{i (e+f x)}\right)\right)}{3 \sqrt{2} c f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}}}","-\frac{2 a^{3/2} B \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{c^{3/2} f}-\frac{(B+i A) (a+i a \tan (e+f x))^{3/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac{2 a B \sqrt{a+i a \tan (e+f x)}}{c f \sqrt{c-i c \tan (e+f x)}}",1,"-1/3*(a*(I*A*E^((3*I)*(e + f*x)) + B*E^(I*(e + f*x))*(-6 + E^((2*I)*(e + f*x))) + 6*B*ArcTan[E^(I*(e + f*x))])*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[2]*c*E^(I*(e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f)","A",1
801,1,117,102,11.6787668,"\int \frac{(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{a \cos (e+f x) (\cos (f x)-i \sin (f x)) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\cos (4 e+5 f x)+i \sin (4 e+5 f x)) ((B-4 i A) \cos (e+f x)-(A+4 i B) \sin (e+f x))}{15 c^3 f}","-\frac{(-4 B+i A) (a+i a \tan (e+f x))^{3/2}}{15 c f (c-i c \tan (e+f x))^{3/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{3/2}}{5 f (c-i c \tan (e+f x))^{5/2}}",1,"(a*Cos[e + f*x]*(Cos[f*x] - I*Sin[f*x])*(((-4*I)*A + B)*Cos[e + f*x] - (A + (4*I)*B)*Sin[e + f*x])*(Cos[4*e + 5*f*x] + I*Sin[4*e + 5*f*x])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(15*c^3*f)","A",1
802,1,131,155,12.8299468,"\int \frac{(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(7/2),x]","\frac{a \cos (e+f x) (\cos (f x)-i \sin (f x)) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\cos (5 e+6 f x)+i \sin (5 e+6 f x)) (-5 (2 A+5 i B) \sin (2 (e+f x))+5 (2 B-5 i A) \cos (2 (e+f x))-21 i A)}{210 c^4 f}","-\frac{(-5 B+2 i A) (a+i a \tan (e+f x))^{3/2}}{105 c^2 f (c-i c \tan (e+f x))^{3/2}}-\frac{(-5 B+2 i A) (a+i a \tan (e+f x))^{3/2}}{35 c f (c-i c \tan (e+f x))^{5/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{3/2}}{7 f (c-i c \tan (e+f x))^{7/2}}",1,"(a*Cos[e + f*x]*(Cos[f*x] - I*Sin[f*x])*((-21*I)*A + 5*((-5*I)*A + 2*B)*Cos[2*(e + f*x)] - 5*(2*A + (5*I)*B)*Sin[2*(e + f*x)])*(Cos[5*e + 6*f*x] + I*Sin[5*e + 6*f*x])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(210*c^4*f)","A",1
803,1,148,208,9.940428,"\int \frac{(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{9/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(9/2),x]","\frac{a \cos (e+f x) (\cos (f x)-i \sin (f x)) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\cos (6 e+7 f x)+i \sin (6 e+7 f x)) (-(A+2 i B) (27 \sin (e+f x)+35 \sin (3 (e+f x)))+9 (B-18 i A) \cos (e+f x)+35 (B-2 i A) \cos (3 (e+f x)))}{1260 c^5 f}","-\frac{2 (-2 B+i A) (a+i a \tan (e+f x))^{3/2}}{315 c^3 f (c-i c \tan (e+f x))^{3/2}}-\frac{2 (-2 B+i A) (a+i a \tan (e+f x))^{3/2}}{105 c^2 f (c-i c \tan (e+f x))^{5/2}}-\frac{(-2 B+i A) (a+i a \tan (e+f x))^{3/2}}{21 c f (c-i c \tan (e+f x))^{7/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{3/2}}{9 f (c-i c \tan (e+f x))^{9/2}}",1,"(a*Cos[e + f*x]*(Cos[f*x] - I*Sin[f*x])*(9*((-18*I)*A + B)*Cos[e + f*x] + 35*((-2*I)*A + B)*Cos[3*(e + f*x)] - (A + (2*I)*B)*(27*Sin[e + f*x] + 35*Sin[3*(e + f*x)]))*(Cos[6*e + 7*f*x] + I*Sin[6*e + 7*f*x])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(1260*c^5*f)","A",1
804,1,179,261,13.8322902,"\int \frac{(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(11/2),x]","-\frac{i a \cos (e+f x) (\cos (f x)-i \sin (f x)) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\cos (7 e+8 f x)+i \sin (7 e+8 f x)) (308 (7 A+i B) \cos (2 (e+f x))+105 (7 A+4 i B) \cos (4 (e+f x))-616 i A \sin (2 (e+f x))-420 i A \sin (4 (e+f x))+1485 A+1078 B \sin (2 (e+f x))+735 B \sin (4 (e+f x)))}{27720 c^6 f}","-\frac{2 (-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{3465 c^4 f (c-i c \tan (e+f x))^{3/2}}-\frac{2 (-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{1155 c^3 f (c-i c \tan (e+f x))^{5/2}}-\frac{(-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{231 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac{(-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{3/2}}{11 f (c-i c \tan (e+f x))^{11/2}}",1,"((-1/27720*I)*a*Cos[e + f*x]*(Cos[f*x] - I*Sin[f*x])*(1485*A + 308*(7*A + I*B)*Cos[2*(e + f*x)] + 105*(7*A + (4*I)*B)*Cos[4*(e + f*x)] - (616*I)*A*Sin[2*(e + f*x)] + 1078*B*Sin[2*(e + f*x)] - (420*I)*A*Sin[4*(e + f*x)] + 735*B*Sin[4*(e + f*x)])*(Cos[7*e + 8*f*x] + I*Sin[7*e + 8*f*x])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(c^6*f)","A",1
805,1,568,288,17.3401738,"\int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2),x]","\frac{c^4 (B-6 i A) \sqrt{e^{i f x}} e^{-i (3 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{8 f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{7}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{5/2} (A \cos (e+f x)+B \sin (e+f x))}+\frac{\cos ^3(e+f x) (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left(\sec (e) \left(\frac{1}{30} c^3 \cos (2 e)-\frac{1}{30} i c^3 \sin (2 e)\right) \sec ^4(e+f x) (-6 i A \cos (e)-5 i B \sin (e)+6 B \cos (e))+\sec (e) \left(\frac{1}{24} \cos (2 e)-\frac{1}{24} i \sin (2 e)\right) \sec ^3(e+f x) \left(6 A c^3 \sin (f x)+i B c^3 \sin (f x)\right)+\sec (e) \left(\frac{1}{16} \cos (2 e)-\frac{1}{16} i \sin (2 e)\right) \sec (e+f x) \left(6 A c^3 \sin (f x)+i B c^3 \sin (f x)\right)+(6 A+i B) \tan (e) \left(\frac{1}{24} c^3 \cos (2 e)-\frac{1}{24} i c^3 \sin (2 e)\right) \sec ^2(e+f x)+(6 A+i B) \tan (e) \left(\frac{1}{16} c^3 \cos (2 e)-\frac{1}{16} i c^3 \sin (2 e)\right)-i B c^3 \sec (e) \left(\frac{1}{6} \cos (2 e)-\frac{1}{6} i \sin (2 e)\right) \sin (f x) \sec ^5(e+f x)\right)}{f (\cos (f x)+i \sin (f x))^2 (A \cos (e+f x)+B \sin (e+f x))}","-\frac{a^{5/2} c^{7/2} (-B+6 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{8 f}+\frac{a^2 c^3 (6 A+i B) \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{16 f}+\frac{a c^2 (6 A+i B) \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac{c (-B+6 i A) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac{B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{7/2}}{6 f}",1,"(((-6*I)*A + B)*c^4*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(8*E^(I*(3*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(7/2)*(Cos[f*x] + I*Sin[f*x])^(5/2)*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (Cos[e + f*x]^3*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(Sec[e]*Sec[e + f*x]^4*((-6*I)*A*Cos[e] + 6*B*Cos[e] - (5*I)*B*Sin[e])*((c^3*Cos[2*e])/30 - (I/30)*c^3*Sin[2*e]) - I*B*c^3*Sec[e]*Sec[e + f*x]^5*(Cos[2*e]/6 - (I/6)*Sin[2*e])*Sin[f*x] + Sec[e]*Sec[e + f*x]^3*(Cos[2*e]/24 - (I/24)*Sin[2*e])*(6*A*c^3*Sin[f*x] + I*B*c^3*Sin[f*x]) + Sec[e]*Sec[e + f*x]*(Cos[2*e]/16 - (I/16)*Sin[2*e])*(6*A*c^3*Sin[f*x] + I*B*c^3*Sin[f*x]) + (6*A + I*B)*Sec[e + f*x]^2*((c^3*Cos[2*e])/24 - (I/24)*c^3*Sin[2*e])*Tan[e] + (6*A + I*B)*((c^3*Cos[2*e])/16 - (I/16)*c^3*Sin[2*e])*Tan[e])*(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^2*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
806,1,119,213,12.7595026,"\int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]","-\frac{a^2 c^3 (\tan (e+f x)+i) \sec ^4(e+f x) \sqrt{a+i a \tan (e+f x)} \left(i (70 A \sin (2 (e+f x))+15 A \sin (4 (e+f x))+64 B)+240 A \cos ^5(e+f x) \tan ^{-1}\left(e^{i (e+f x)}\right)\right)}{320 f \sqrt{c-i c \tan (e+f x)}}","-\frac{3 i a^{5/2} A c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{4 f}+\frac{3 a^2 A c^2 \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{8 f}+\frac{a A c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{4 f}+\frac{B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{5 f}",1,"-1/320*(a^2*c^3*Sec[e + f*x]^4*(240*A*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x]^5 + I*(64*B + 70*A*Sin[2*(e + f*x)] + 15*A*Sin[4*(e + f*x)]))*(I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])/(f*Sqrt[c - I*c*Tan[e + f*x]])","A",1
807,1,460,222,13.3349074,"\int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{\cos ^3(e+f x) (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left(\sec (e) \left(\frac{1}{12} c \cos (2 e)-\frac{1}{12} i c \sin (2 e)\right) \sec ^2(e+f x) (4 i A \cos (e)+3 i B \sin (e)+4 B \cos (e))+\sec (e) \left(\frac{1}{8} \cos (2 e)-\frac{1}{8} i \sin (2 e)\right) \sec (e+f x) (4 A c \sin (f x)-i B c \sin (f x))+(4 A-i B) \tan (e) \left(\frac{1}{8} c \cos (2 e)-\frac{1}{8} i c \sin (2 e)\right)+i B c \sec (e) \left(\frac{1}{4} \cos (2 e)-\frac{1}{4} i \sin (2 e)\right) \sin (f x) \sec ^3(e+f x)\right)}{f (\cos (f x)+i \sin (f x))^2 (A \cos (e+f x)+B \sin (e+f x))}-\frac{i c^2 (4 A-i B) \sqrt{e^{i f x}} e^{-i (3 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{4 f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{7}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{5/2} (A \cos (e+f x)+B \sin (e+f x))}","-\frac{a^{5/2} c^{3/2} (B+4 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{4 f}+\frac{a^2 c (4 A-i B) \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{8 f}+\frac{a (B+4 i A) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}+\frac{B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}}{4 f}",1,"((-1/4*I)*(4*A - I*B)*c^2*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(E^(I*(3*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(7/2)*(Cos[f*x] + I*Sin[f*x])^(5/2)*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (Cos[e + f*x]^3*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(Sec[e]*Sec[e + f*x]^2*((4*I)*A*Cos[e] + 4*B*Cos[e] + (3*I)*B*Sin[e])*((c*Cos[2*e])/12 - (I/12)*c*Sin[2*e]) + I*B*c*Sec[e]*Sec[e + f*x]^3*(Cos[2*e]/4 - (I/4)*Sin[2*e])*Sin[f*x] + Sec[e]*Sec[e + f*x]*(Cos[2*e]/8 - (I/8)*Sin[2*e])*(4*A*c*Sin[f*x] - I*B*c*Sin[f*x]) + (4*A - I*B)*((c*Cos[2*e])/8 - (I/8)*c*Sin[2*e])*Tan[e])*(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^2*(A*Cos[e + f*x] + B*Sin[e + f*x]))","B",1
808,1,253,217,9.3482256,"\int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \left(\frac{(\sin (2 e)+i \cos (2 e)) \sec ^{\frac{5}{2}}(e+f x) \sqrt{c-i c \tan (e+f x)} ((6 B+3 i A) \sin (2 (e+f x))+12 (A-i B) \cos (2 (e+f x))+12 A-8 i B)}{12 (\cos (f x)+i \sin (f x))^2}-\frac{i c (3 A-2 i B) e^{-3 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right)}{\sqrt{\frac{c}{1+e^{2 i (e+f x)}}}}\right)}{f \sec ^{\frac{7}{2}}(e+f x) (A \cos (e+f x)+B \sin (e+f x))}","-\frac{a^{5/2} \sqrt{c} (2 B+3 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{f}+\frac{a^2 (2 B+3 i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 f}+\frac{a (2 B+3 i A) (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}{6 f}+\frac{B (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}{3 f}",1,"((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*(((-I)*(3*A - (2*I)*B)*c*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))])/(E^((3*I)*(e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]) + (Sec[e + f*x]^(5/2)*(I*Cos[2*e] + Sin[2*e])*(12*A - (8*I)*B + 12*(A - I*B)*Cos[2*(e + f*x)] + ((3*I)*A + 6*B)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(12*(Cos[f*x] + I*Sin[f*x])^2)))/(f*Sec[e + f*x]^(7/2)*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
809,1,239,227,11.2490368,"\int \frac{(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{\sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \left(\frac{3 (3 B+2 i A) e^{-3 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right)}{\sqrt{\frac{c}{1+e^{2 i (e+f x)}}}}-\frac{(\tan (e+f x)+i) \sqrt{\sec (e+f x)} \sqrt{c-i c \tan (e+f x)} ((-5 B-2 i A) \sin (2 (e+f x))+(10 A-13 i B) \cos (2 (e+f x))+5 (2 A-3 i B))}{4 c}\right)}{f \sec ^{\frac{7}{2}}(e+f x) (A \cos (e+f x)+B \sin (e+f x))}","\frac{3 a^{5/2} (3 B+2 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{\sqrt{c} f}-\frac{3 a^2 (3 B+2 i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 c f}-\frac{a (3 B+2 i A) (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}{2 c f}-\frac{(B+i A) (a+i a \tan (e+f x))^{5/2}}{f \sqrt{c-i c \tan (e+f x)}}",1,"((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*((3*((2*I)*A + 3*B)*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))])/(E^((3*I)*(e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]) - (Sqrt[Sec[e + f*x]]*(5*(2*A - (3*I)*B) + (10*A - (13*I)*B)*Cos[2*(e + f*x)] + ((-2*I)*A - 5*B)*Sin[2*(e + f*x)])*(I + Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(4*c)))/(f*Sec[e + f*x]^(7/2)*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
810,1,227,226,14.3950218,"\int \frac{(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \left(\sqrt{\sec (e+f x)} \sqrt{c-i c \tan (e+f x)} ((4 A-13 i B) \sin (2 (e+f x))+(11 B+2 i A) \cos (2 (e+f x))+2 i A+8 B)-\frac{6 i c (A-4 i B) e^{-3 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right)}{\sqrt{\frac{c}{1+e^{2 i (e+f x)}}}}\right)}{3 c^2 f \sec ^{\frac{7}{2}}(e+f x) (A \cos (e+f x)+B \sin (e+f x))}","-\frac{2 a^{5/2} (4 B+i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{c^{3/2} f}+\frac{a^2 (4 B+i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{c^2 f}+\frac{2 a (4 B+i A) (a+i a \tan (e+f x))^{3/2}}{3 c f \sqrt{c-i c \tan (e+f x)}}-\frac{(B+i A) (a+i a \tan (e+f x))^{5/2}}{3 f (c-i c \tan (e+f x))^{3/2}}",1,"((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*(((-6*I)*(A - (4*I)*B)*c*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))])/(E^((3*I)*(e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]) + Sqrt[Sec[e + f*x]]*((2*I)*A + 8*B + ((2*I)*A + 11*B)*Cos[2*(e + f*x)] + (4*A - (13*I)*B)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]]))/(3*c^2*f*Sec[e + f*x]^(7/2)*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
811,1,203,203,15.3541176,"\int \frac{(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{a^2 \cos ^2(e+f x) (\tan (e+f x)-i)^2 \sqrt{a+i a \tan (e+f x)} \left(\cos \left(\frac{1}{2} (e-2 f x)\right)-i \sin \left(\frac{1}{2} (e-2 f x)\right)\right) \left(\cos \left(\frac{1}{2} (e-2 f x)\right)+i \sin \left(\frac{1}{2} (e-2 f x)\right)\right) \left((33 B+3 i A) \cos (2 (e+f x))-3 A \sin (2 (e+f x))-27 i B \sin (2 (e+f x))-30 B \tan ^{-1}\left(e^{i (e+f x)}\right) (\cos (3 (e+f x))-i \sin (3 (e+f x)))-10 B\right)}{15 c^2 f \sqrt{c-i c \tan (e+f x)}}","\frac{2 a^{5/2} B \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{c^{5/2} f}-\frac{2 a^2 B \sqrt{a+i a \tan (e+f x)}}{c^2 f \sqrt{c-i c \tan (e+f x)}}-\frac{(B+i A) (a+i a \tan (e+f x))^{5/2}}{5 f (c-i c \tan (e+f x))^{5/2}}+\frac{2 a B (a+i a \tan (e+f x))^{3/2}}{3 c f (c-i c \tan (e+f x))^{3/2}}",1,"(a^2*Cos[e + f*x]^2*(Cos[(e - 2*f*x)/2] - I*Sin[(e - 2*f*x)/2])*(Cos[(e - 2*f*x)/2] + I*Sin[(e - 2*f*x)/2])*(-10*B + ((3*I)*A + 33*B)*Cos[2*(e + f*x)] - 3*A*Sin[2*(e + f*x)] - (27*I)*B*Sin[2*(e + f*x)] - 30*B*ArcTan[E^(I*(e + f*x))]*(Cos[3*(e + f*x)] - I*Sin[3*(e + f*x)]))*(-I + Tan[e + f*x])^2*Sqrt[a + I*a*Tan[e + f*x]])/(15*c^2*f*Sqrt[c - I*c*Tan[e + f*x]])","A",1
812,1,121,102,12.0201144,"\int \frac{(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(7/2),x]","\frac{a^2 \cos (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\cos (6 e+8 f x)+i \sin (6 e+8 f x)) ((B-6 i A) \cos (e+f x)-(A+6 i B) \sin (e+f x))}{35 c^4 f (\cos (f x)+i \sin (f x))^2}","-\frac{(-6 B+i A) (a+i a \tan (e+f x))^{5/2}}{35 c f (c-i c \tan (e+f x))^{5/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{5/2}}{7 f (c-i c \tan (e+f x))^{7/2}}",1,"(a^2*Cos[e + f*x]*(((-6*I)*A + B)*Cos[e + f*x] - (A + (6*I)*B)*Sin[e + f*x])*(Cos[6*e + 8*f*x] + I*Sin[6*e + 8*f*x])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(35*c^4*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
813,1,135,155,11.5246617,"\int \frac{(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{9/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(9/2),x]","\frac{a^2 \cos (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\cos (7 e+9 f x)+i \sin (7 e+9 f x)) (-7 (2 A+7 i B) \sin (2 (e+f x))+7 (2 B-7 i A) \cos (2 (e+f x))-45 i A)}{630 c^5 f (\cos (f x)+i \sin (f x))^2}","-\frac{(-7 B+2 i A) (a+i a \tan (e+f x))^{5/2}}{315 c^2 f (c-i c \tan (e+f x))^{5/2}}-\frac{(-7 B+2 i A) (a+i a \tan (e+f x))^{5/2}}{63 c f (c-i c \tan (e+f x))^{7/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{5/2}}{9 f (c-i c \tan (e+f x))^{9/2}}",1,"(a^2*Cos[e + f*x]*((-45*I)*A + 7*((-7*I)*A + 2*B)*Cos[2*(e + f*x)] - 7*(2*A + (7*I)*B)*Sin[2*(e + f*x)])*(Cos[7*e + 9*f*x] + I*Sin[7*e + 9*f*x])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(630*c^5*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
814,1,156,208,14.1082427,"\int \frac{(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(11/2),x]","\frac{a^2 \cos (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\cos (8 e+10 f x)+i \sin (8 e+10 f x)) (-(3 A+8 i B) (55 \sin (e+f x)+63 \sin (3 (e+f x)))+55 (B-24 i A) \cos (e+f x)+63 (3 B-8 i A) \cos (3 (e+f x)))}{13860 c^6 f (\cos (f x)+i \sin (f x))^2}","-\frac{2 (-8 B+3 i A) (a+i a \tan (e+f x))^{5/2}}{3465 c^3 f (c-i c \tan (e+f x))^{5/2}}-\frac{2 (-8 B+3 i A) (a+i a \tan (e+f x))^{5/2}}{693 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac{(-8 B+3 i A) (a+i a \tan (e+f x))^{5/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{5/2}}{11 f (c-i c \tan (e+f x))^{11/2}}",1,"(a^2*Cos[e + f*x]*(55*((-24*I)*A + B)*Cos[e + f*x] + 63*((-8*I)*A + 3*B)*Cos[3*(e + f*x)] - (3*A + (8*I)*B)*(55*Sin[e + f*x] + 63*Sin[3*(e + f*x)]))*(Cos[8*e + 10*f*x] + I*Sin[8*e + 10*f*x])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(13860*c^6*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
815,1,577,261,17.0607298,"\int \frac{(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(13/2),x]","\frac{\cos ^3(e+f x) (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left((B-i A) \cos (4 f x) \left(\frac{\cos (2 e)}{160 c^7}+\frac{i \sin (2 e)}{160 c^7}\right)+(A+i B) \sin (4 f x) \left(\frac{\cos (2 e)}{160 c^7}+\frac{i \sin (2 e)}{160 c^7}\right)+(17 B-27 i A) \cos (6 f x) \left(\frac{\cos (4 e)}{1120 c^7}+\frac{i \sin (4 e)}{1120 c^7}\right)+(3 B-13 i A) \cos (8 f x) \left(\frac{\cos (6 e)}{336 c^7}+\frac{i \sin (6 e)}{336 c^7}\right)+(17 A-3 i B) \cos (10 f x) \left(\frac{\sin (8 e)}{528 c^7}-\frac{i \cos (8 e)}{528 c^7}\right)+(63 A-37 i B) \cos (12 f x) \left(\frac{\sin (10 e)}{4576 c^7}-\frac{i \cos (10 e)}{4576 c^7}\right)+(A-i B) \cos (14 f x) \left(\frac{\sin (12 e)}{416 c^7}-\frac{i \cos (12 e)}{416 c^7}\right)+(27 A+17 i B) \sin (6 f x) \left(\frac{\cos (4 e)}{1120 c^7}+\frac{i \sin (4 e)}{1120 c^7}\right)+(13 A+3 i B) \sin (8 f x) \left(\frac{\cos (6 e)}{336 c^7}+\frac{i \sin (6 e)}{336 c^7}\right)+(17 A-3 i B) \sin (10 f x) \left(\frac{\cos (8 e)}{528 c^7}+\frac{i \sin (8 e)}{528 c^7}\right)+(63 A-37 i B) \sin (12 f x) \left(\frac{\cos (10 e)}{4576 c^7}+\frac{i \sin (10 e)}{4576 c^7}\right)+(A-i B) \sin (14 f x) \left(\frac{\cos (12 e)}{416 c^7}+\frac{i \sin (12 e)}{416 c^7}\right)\right)}{f (\cos (f x)+i \sin (f x))^2 (A \cos (e+f x)+B \sin (e+f x))}","-\frac{2 (-9 B+4 i A) (a+i a \tan (e+f x))^{5/2}}{15015 c^4 f (c-i c \tan (e+f x))^{5/2}}-\frac{2 (-9 B+4 i A) (a+i a \tan (e+f x))^{5/2}}{3003 c^3 f (c-i c \tan (e+f x))^{7/2}}-\frac{(-9 B+4 i A) (a+i a \tan (e+f x))^{5/2}}{429 c^2 f (c-i c \tan (e+f x))^{9/2}}-\frac{(-9 B+4 i A) (a+i a \tan (e+f x))^{5/2}}{143 c f (c-i c \tan (e+f x))^{11/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{5/2}}{13 f (c-i c \tan (e+f x))^{13/2}}",1,"(Cos[e + f*x]^3*(((-I)*A + B)*Cos[4*f*x]*(Cos[2*e]/(160*c^7) + ((I/160)*Sin[2*e])/c^7) + ((-27*I)*A + 17*B)*Cos[6*f*x]*(Cos[4*e]/(1120*c^7) + ((I/1120)*Sin[4*e])/c^7) + ((-13*I)*A + 3*B)*Cos[8*f*x]*(Cos[6*e]/(336*c^7) + ((I/336)*Sin[6*e])/c^7) + (17*A - (3*I)*B)*Cos[10*f*x]*(((-1/528*I)*Cos[8*e])/c^7 + Sin[8*e]/(528*c^7)) + (63*A - (37*I)*B)*Cos[12*f*x]*(((-1/4576*I)*Cos[10*e])/c^7 + Sin[10*e]/(4576*c^7)) + (A - I*B)*Cos[14*f*x]*(((-1/416*I)*Cos[12*e])/c^7 + Sin[12*e]/(416*c^7)) + (A + I*B)*(Cos[2*e]/(160*c^7) + ((I/160)*Sin[2*e])/c^7)*Sin[4*f*x] + (27*A + (17*I)*B)*(Cos[4*e]/(1120*c^7) + ((I/1120)*Sin[4*e])/c^7)*Sin[6*f*x] + (13*A + (3*I)*B)*(Cos[6*e]/(336*c^7) + ((I/336)*Sin[6*e])/c^7)*Sin[8*f*x] + (17*A - (3*I)*B)*(Cos[8*e]/(528*c^7) + ((I/528)*Sin[8*e])/c^7)*Sin[10*f*x] + (63*A - (37*I)*B)*(Cos[10*e]/(4576*c^7) + ((I/4576)*Sin[10*e])/c^7)*Sin[12*f*x] + (A - I*B)*(Cos[12*e]/(416*c^7) + ((I/416)*Sin[12*e])/c^7)*Sin[14*f*x])*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^2*(A*Cos[e + f*x] + B*Sin[e + f*x]))","B",1
816,1,666,350,17.5483436,"\int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(9/2),x]","\frac{5 c^5 (B-8 i A) \sqrt{e^{i f x}} e^{-i (4 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{64 f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))}+\frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left(\sec (e) \left(\frac{1}{56} c^4 \cos (3 e)-\frac{1}{56} i c^4 \sin (3 e)\right) \sec ^6(e+f x) (-8 i A \cos (e)-7 i B \sin (e)+8 B \cos (e))+\sec (e) \left(\frac{1}{48} \cos (3 e)-\frac{1}{48} i \sin (3 e)\right) \sec ^5(e+f x) \left(8 A c^4 \sin (f x)+i B c^4 \sin (f x)\right)+\sec (e) \left(\frac{5}{192} \cos (3 e)-\frac{5}{192} i \sin (3 e)\right) \sec ^3(e+f x) \left(8 A c^4 \sin (f x)+i B c^4 \sin (f x)\right)+\sec (e) \left(\frac{5}{128} \cos (3 e)-\frac{5}{128} i \sin (3 e)\right) \sec (e+f x) \left(8 A c^4 \sin (f x)+i B c^4 \sin (f x)\right)+(8 A+i B) \tan (e) \left(\frac{1}{48} c^4 \cos (3 e)-\frac{1}{48} i c^4 \sin (3 e)\right) \sec ^4(e+f x)+(8 A+i B) \tan (e) \left(\frac{5}{192} c^4 \cos (3 e)-\frac{5}{192} i c^4 \sin (3 e)\right) \sec ^2(e+f x)+(8 A+i B) \tan (e) \left(\frac{5}{128} c^4 \cos (3 e)-\frac{5}{128} i c^4 \sin (3 e)\right)-i B c^4 \sec (e) \left(\frac{1}{8} \cos (3 e)-\frac{1}{8} i \sin (3 e)\right) \sin (f x) \sec ^7(e+f x)\right)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}","-\frac{5 a^{7/2} c^{9/2} (-B+8 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{64 f}+\frac{5 a^3 c^4 (8 A+i B) \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{128 f}+\frac{5 a^2 c^3 (8 A+i B) \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{192 f}+\frac{a c^2 (8 A+i B) \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{48 f}-\frac{c (-B+8 i A) (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{56 f}+\frac{B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{9/2}}{8 f}",1,"(5*((-8*I)*A + B)*c^5*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(64*E^(I*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(9/2)*(Cos[f*x] + I*Sin[f*x])^(7/2)*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (Cos[e + f*x]^4*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(Sec[e]*Sec[e + f*x]^6*((-8*I)*A*Cos[e] + 8*B*Cos[e] - (7*I)*B*Sin[e])*((c^4*Cos[3*e])/56 - (I/56)*c^4*Sin[3*e]) - I*B*c^4*Sec[e]*Sec[e + f*x]^7*(Cos[3*e]/8 - (I/8)*Sin[3*e])*Sin[f*x] + Sec[e]*Sec[e + f*x]^5*(Cos[3*e]/48 - (I/48)*Sin[3*e])*(8*A*c^4*Sin[f*x] + I*B*c^4*Sin[f*x]) + Sec[e]*Sec[e + f*x]^3*((5*Cos[3*e])/192 - ((5*I)/192)*Sin[3*e])*(8*A*c^4*Sin[f*x] + I*B*c^4*Sin[f*x]) + Sec[e]*Sec[e + f*x]*((5*Cos[3*e])/128 - ((5*I)/128)*Sin[3*e])*(8*A*c^4*Sin[f*x] + I*B*c^4*Sin[f*x]) + (8*A + I*B)*Sec[e + f*x]^4*((c^4*Cos[3*e])/48 - (I/48)*c^4*Sin[3*e])*Tan[e] + (8*A + I*B)*Sec[e + f*x]^2*((5*c^4*Cos[3*e])/192 - ((5*I)/192)*c^4*Sin[3*e])*Tan[e] + (8*A + I*B)*((5*c^4*Cos[3*e])/128 - ((5*I)/128)*c^4*Sin[3*e])*Tan[e])*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",0
817,1,535,267,17.0208978,"\int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2),x]","\frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left(A c^3 \sec (e) \left(\frac{1}{6} \cos (3 e)-\frac{1}{6} i \sin (3 e)\right) \sin (f x) \sec ^5(e+f x)+A c^3 \sec (e) \left(\frac{5}{24} \cos (3 e)-\frac{5}{24} i \sin (3 e)\right) \sin (f x) \sec ^3(e+f x)+A c^3 \sec (e) \left(\frac{5}{16} \cos (3 e)-\frac{5}{16} i \sin (3 e)\right) \sin (f x) \sec (e+f x)+\tan (e) \sec ^4(e+f x) \left(\frac{1}{6} A c^3 \cos (3 e)-\frac{1}{6} i A c^3 \sin (3 e)\right)+\tan (e) \sec ^2(e+f x) \left(\frac{5}{24} A c^3 \cos (3 e)-\frac{5}{24} i A c^3 \sin (3 e)\right)+\tan (e) \left(\frac{5}{16} A c^3 \cos (3 e)-\frac{5}{16} i A c^3 \sin (3 e)\right)+\sec ^6(e+f x) \left(\frac{1}{7} B c^3 \cos (3 e)-\frac{1}{7} i B c^3 \sin (3 e)\right)\right)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}-\frac{5 i A c^4 \sqrt{e^{i f x}} e^{-i (4 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{8 f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))}","-\frac{5 i a^{7/2} A c^{7/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{8 f}+\frac{5 a^3 A c^3 \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{16 f}+\frac{5 a^2 A c^2 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac{a A c \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac{B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{7 f}",1,"(((-5*I)/8)*A*c^4*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(E^(I*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(9/2)*(Cos[f*x] + I*Sin[f*x])^(7/2)*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (Cos[e + f*x]^4*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(Sec[e + f*x]^6*((B*c^3*Cos[3*e])/7 - (I/7)*B*c^3*Sin[3*e]) + A*c^3*Sec[e]*Sec[e + f*x]^5*(Cos[3*e]/6 - (I/6)*Sin[3*e])*Sin[f*x] + A*c^3*Sec[e]*Sec[e + f*x]^3*((5*Cos[3*e])/24 - ((5*I)/24)*Sin[3*e])*Sin[f*x] + A*c^3*Sec[e]*Sec[e + f*x]*((5*Cos[3*e])/16 - ((5*I)/16)*Sin[3*e])*Sin[f*x] + Sec[e + f*x]^4*((A*c^3*Cos[3*e])/6 - (I/6)*A*c^3*Sin[3*e])*Tan[e] + Sec[e + f*x]^2*((5*A*c^3*Cos[3*e])/24 - ((5*I)/24)*A*c^3*Sin[3*e])*Tan[e] + ((5*A*c^3*Cos[3*e])/16 - ((5*I)/16)*A*c^3*Sin[3*e])*Tan[e])*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","B",1
818,1,572,284,16.9995443,"\int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left(\sec (e) \left(\frac{1}{30} c^2 \cos (3 e)-\frac{1}{30} i c^2 \sin (3 e)\right) \sec ^4(e+f x) (6 i A \cos (e)+5 i B \sin (e)+6 B \cos (e))+\sec (e) \left(\frac{1}{24} \cos (3 e)-\frac{1}{24} i \sin (3 e)\right) \sec ^3(e+f x) \left(6 A c^2 \sin (f x)-i B c^2 \sin (f x)\right)+\sec (e) \left(\frac{1}{16} \cos (3 e)-\frac{1}{16} i \sin (3 e)\right) \sec (e+f x) \left(6 A c^2 \sin (f x)-i B c^2 \sin (f x)\right)+(6 A-i B) \tan (e) \left(\frac{1}{24} c^2 \cos (3 e)-\frac{1}{24} i c^2 \sin (3 e)\right) \sec ^2(e+f x)+(6 A-i B) \tan (e) \left(\frac{1}{16} c^2 \cos (3 e)-\frac{1}{16} i c^2 \sin (3 e)\right)+i B c^2 \sec (e) \left(\frac{1}{6} \cos (3 e)-\frac{1}{6} i \sin (3 e)\right) \sin (f x) \sec ^5(e+f x)\right)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}-\frac{i c^3 (6 A-i B) \sqrt{e^{i f x}} e^{-i (4 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{8 f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))}","-\frac{a^{7/2} c^{5/2} (B+6 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{8 f}+\frac{a^3 c^2 (6 A-i B) \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{16 f}+\frac{a^2 c (6 A-i B) \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac{a (B+6 i A) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac{B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f}",1,"((-1/8*I)*(6*A - I*B)*c^3*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(E^(I*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(9/2)*(Cos[f*x] + I*Sin[f*x])^(7/2)*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (Cos[e + f*x]^4*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(Sec[e]*Sec[e + f*x]^4*((6*I)*A*Cos[e] + 6*B*Cos[e] + (5*I)*B*Sin[e])*((c^2*Cos[3*e])/30 - (I/30)*c^2*Sin[3*e]) + I*B*c^2*Sec[e]*Sec[e + f*x]^5*(Cos[3*e]/6 - (I/6)*Sin[3*e])*Sin[f*x] + Sec[e]*Sec[e + f*x]^3*(Cos[3*e]/24 - (I/24)*Sin[3*e])*(6*A*c^2*Sin[f*x] - I*B*c^2*Sin[f*x]) + Sec[e]*Sec[e + f*x]*(Cos[3*e]/16 - (I/16)*Sin[3*e])*(6*A*c^2*Sin[f*x] - I*B*c^2*Sin[f*x]) + (6*A - I*B)*Sec[e + f*x]^2*((c^2*Cos[3*e])/24 - (I/24)*c^2*Sin[3*e])*Tan[e] + (6*A - I*B)*((c^2*Cos[3*e])/16 - (I/16)*c^2*Sin[3*e])*Tan[e])*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","B",1
819,1,507,279,15.4176389,"\int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left(\sec (e) \left(\frac{1}{4} \cos (3 e)-\frac{1}{4} i \sin (3 e)\right) \sec ^3(e+f x) (-A c \sin (f x)+2 i B c \sin (f x))+\sec (e) \left(\frac{1}{12} c \cos (3 e)-\frac{1}{12} i c \sin (3 e)\right) \sec ^2(e+f x) (-3 A \sin (e)+8 i A \cos (e)+6 i B \sin (e)+8 B \cos (e))+\sec (e) \left(\frac{1}{8} \cos (3 e)-\frac{1}{8} i \sin (3 e)\right) \sec (e+f x) (5 A c \sin (f x)-2 i B c \sin (f x))+(5 A-2 i B) \tan (e) \left(\frac{1}{8} c \cos (3 e)-\frac{1}{8} i c \sin (3 e)\right)+\sec ^4(e+f x) \left(-\frac{1}{5} B c \cos (3 e)+\frac{1}{5} i B c \sin (3 e)\right)\right)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}-\frac{i c^2 (5 A-2 i B) \sqrt{e^{i f x}} e^{-i (4 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{4 f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))}","-\frac{a^{7/2} c^{3/2} (2 B+5 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{4 f}+\frac{a^3 c (5 A-2 i B) \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{8 f}+\frac{a^2 (2 B+5 i A) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}+\frac{a (2 B+5 i A) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}}{20 f}+\frac{B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{3/2}}{5 f}",1,"((-1/4*I)*(5*A - (2*I)*B)*c^2*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(E^(I*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(9/2)*(Cos[f*x] + I*Sin[f*x])^(7/2)*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (Cos[e + f*x]^4*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(Sec[e]*Sec[e + f*x]^2*((8*I)*A*Cos[e] + 8*B*Cos[e] - 3*A*Sin[e] + (6*I)*B*Sin[e])*((c*Cos[3*e])/12 - (I/12)*c*Sin[3*e]) + Sec[e + f*x]^4*(-1/5*(B*c*Cos[3*e]) + (I/5)*B*c*Sin[3*e]) + Sec[e]*Sec[e + f*x]*(Cos[3*e]/8 - (I/8)*Sin[3*e])*(5*A*c*Sin[f*x] - (2*I)*B*c*Sin[f*x]) + Sec[e]*Sec[e + f*x]^3*(Cos[3*e]/4 - (I/4)*Sin[3*e])*(-(A*c*Sin[f*x]) + (2*I)*B*c*Sin[f*x]) + (5*A - (2*I)*B)*((c*Cos[3*e])/8 - (I/8)*c*Sin[3*e])*Tan[e])*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
820,1,465,272,13.2148227,"\int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left(\sec (e) \left(-\frac{1}{12} \sin (3 e)-\frac{1}{12} i \cos (3 e)\right) \sec ^2(e+f x) (4 A \cos (e)+3 B \sin (e)-12 i B \cos (e))+\sec (e) \left(\frac{1}{8} \cos (3 e)-\frac{1}{8} i \sin (3 e)\right) \sec (e+f x) (-12 A \sin (f x)+17 i B \sin (f x))+\sec (e) \left(\frac{1}{8} \cos (3 e)-\frac{1}{8} i \sin (3 e)\right) (-12 A \sin (e)+32 i A \cos (e)+17 i B \sin (e)+32 B \cos (e))-i B \sec (e) \left(\frac{1}{4} \cos (3 e)-\frac{1}{4} i \sin (3 e)\right) \sin (f x) \sec ^3(e+f x)\right)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}-\frac{5 i c (4 A-3 i B) \sqrt{e^{i f x}} e^{-i (4 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{4 f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))}","-\frac{5 a^{7/2} \sqrt{c} (3 B+4 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{4 f}+\frac{5 a^3 (3 B+4 i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{8 f}+\frac{5 a^2 (3 B+4 i A) (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}{24 f}+\frac{a (3 B+4 i A) (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}{12 f}+\frac{B (a+i a \tan (e+f x))^{7/2} \sqrt{c-i c \tan (e+f x)}}{4 f}",1,"(((-5*I)/4)*(4*A - (3*I)*B)*c*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(E^(I*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(9/2)*(Cos[f*x] + I*Sin[f*x])^(7/2)*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (Cos[e + f*x]^4*(Sec[e]*Sec[e + f*x]^2*(4*A*Cos[e] - (12*I)*B*Cos[e] + 3*B*Sin[e])*((-1/12*I)*Cos[3*e] - Sin[3*e]/12) + Sec[e]*((32*I)*A*Cos[e] + 32*B*Cos[e] - 12*A*Sin[e] + (17*I)*B*Sin[e])*(Cos[3*e]/8 - (I/8)*Sin[3*e]) - I*B*Sec[e]*Sec[e + f*x]^3*(Cos[3*e]/4 - (I/4)*Sin[3*e])*Sin[f*x] + Sec[e]*Sec[e + f*x]*(Cos[3*e]/8 - (I/8)*Sin[3*e])*(-12*A*Sin[f*x] + (17*I)*B*Sin[f*x]))*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
821,1,481,283,15.22519,"\int \frac{(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left((A-i B) \cos (2 f x) \left(-\frac{4 \sin (e)}{c}-\frac{4 i \cos (e)}{c}\right)+(A-i B) \sin (2 f x) \left(\frac{4 \cos (e)}{c}-\frac{4 i \sin (e)}{c}\right)+\sec (e) \left(\frac{\cos (3 e)}{2 c}-\frac{i \sin (3 e)}{2 c}\right) \sec (e+f x) (A \sin (f x)-4 i B \sin (f x))+\sec (e) \left(-\frac{\sin (3 e)}{2 c}-\frac{i \cos (3 e)}{2 c}\right) (i A \sin (e)+16 A \cos (e)+4 B \sin (e)-24 i B \cos (e))+\sec ^2(e+f x) \left(\frac{B \cos (3 e)}{3 c}-\frac{i B \sin (3 e)}{3 c}\right)\right)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{5 (4 B+3 i A) \sqrt{e^{i f x}} e^{-i (4 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))}","\frac{5 a^{7/2} (4 B+3 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{\sqrt{c} f}-\frac{5 a^3 (4 B+3 i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 c f}-\frac{5 a^2 (4 B+3 i A) (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}{6 c f}-\frac{a (4 B+3 i A) (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}{3 c f}-\frac{(B+i A) (a+i a \tan (e+f x))^{7/2}}{f \sqrt{c-i c \tan (e+f x)}}",1,"(5*((3*I)*A + 4*B)*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(E^(I*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(9/2)*(Cos[f*x] + I*Sin[f*x])^(7/2)*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (Cos[e + f*x]^4*((A - I*B)*Cos[2*f*x]*(((-4*I)*Cos[e])/c - (4*Sin[e])/c) + Sec[e]*(16*A*Cos[e] - (24*I)*B*Cos[e] + I*A*Sin[e] + 4*B*Sin[e])*(((-1/2*I)*Cos[3*e])/c - Sin[3*e]/(2*c)) + Sec[e + f*x]^2*((B*Cos[3*e])/(3*c) - ((I/3)*B*Sin[3*e])/c) + Sec[e]*Sec[e + f*x]*(Cos[3*e]/(2*c) - ((I/2)*Sin[3*e])/c)*(A*Sin[f*x] - (4*I)*B*Sin[f*x]) + (A - I*B)*((4*Cos[e])/c - ((4*I)*Sin[e])/c)*Sin[2*f*x])*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",1
822,1,517,285,17.6252705,"\int \frac{(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left((11 B+5 i A) \cos (2 f x) \left(\frac{2 \cos (e)}{3 c^2}-\frac{2 i \sin (e)}{3 c^2}\right)+(A-i B) \cos (4 f x) \left(\frac{2 \sin (e)}{3 c^2}-\frac{2 i \cos (e)}{3 c^2}\right)+(5 A-11 i B) \sin (2 f x) \left(-\frac{2 \cos (e)}{3 c^2}+\frac{2 i \sin (e)}{3 c^2}\right)+(A-i B) \sin (4 f x) \left(\frac{2 \cos (e)}{3 c^2}+\frac{2 i \sin (e)}{3 c^2}\right)+\sec (e) \left(\frac{\cos (3 e)}{2 c^2}-\frac{i \sin (3 e)}{2 c^2}\right) (10 i A \cos (e)+i B \sin (e)+26 B \cos (e))+i B \sec (e) \sin (f x) \left(\frac{\cos (3 e)}{2 c^2}-\frac{i \sin (3 e)}{2 c^2}\right) \sec (e+f x)\right)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}-\frac{5 i (2 A-5 i B) \sqrt{e^{i f x}} e^{-i (4 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{c f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))}","-\frac{5 a^{7/2} (5 B+2 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{c^{3/2} f}+\frac{5 a^3 (5 B+2 i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 c^2 f}+\frac{5 a^2 (5 B+2 i A) (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}{6 c^2 f}+\frac{2 a (5 B+2 i A) (a+i a \tan (e+f x))^{5/2}}{3 c f \sqrt{c-i c \tan (e+f x)}}-\frac{(B+i A) (a+i a \tan (e+f x))^{7/2}}{3 f (c-i c \tan (e+f x))^{3/2}}",1,"((-5*I)*(2*A - (5*I)*B)*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c*E^(I*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(9/2)*(Cos[f*x] + I*Sin[f*x])^(7/2)*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (Cos[e + f*x]^4*(((5*I)*A + 11*B)*Cos[2*f*x]*((2*Cos[e])/(3*c^2) - (((2*I)/3)*Sin[e])/c^2) + (A - I*B)*Cos[4*f*x]*((((-2*I)/3)*Cos[e])/c^2 + (2*Sin[e])/(3*c^2)) + Sec[e]*((10*I)*A*Cos[e] + 26*B*Cos[e] + I*B*Sin[e])*(Cos[3*e]/(2*c^2) - ((I/2)*Sin[3*e])/c^2) + I*B*Sec[e]*Sec[e + f*x]*(Cos[3*e]/(2*c^2) - ((I/2)*Sin[3*e])/c^2)*Sin[f*x] + (5*A - (11*I)*B)*((-2*Cos[e])/(3*c^2) + (((2*I)/3)*Sin[e])/c^2)*Sin[2*f*x] + (A - I*B)*((2*Cos[e])/(3*c^2) + (((2*I)/3)*Sin[e])/c^2)*Sin[4*f*x])*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",0
823,1,528,283,17.7656431,"\int \frac{(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left((A-6 i B) \cos (2 f x) \left(-\frac{2 \sin (e)}{3 c^3}-\frac{2 i \cos (e)}{3 c^3}\right)+(6 B+i A) \cos (4 f x) \left(\frac{2 \cos (e)}{15 c^3}+\frac{2 i \sin (e)}{15 c^3}\right)+(A-i B) \cos (6 f x) \left(\frac{\sin (3 e)}{5 c^3}-\frac{i \cos (3 e)}{5 c^3}\right)+(A-6 i B) \sin (2 f x) \left(\frac{2 \cos (e)}{3 c^3}-\frac{2 i \sin (e)}{3 c^3}\right)+(A-6 i B) \sin (4 f x) \left(-\frac{2 \cos (e)}{15 c^3}-\frac{2 i \sin (e)}{15 c^3}\right)+(A-i B) \sin (6 f x) \left(\frac{\cos (3 e)}{5 c^3}+\frac{i \sin (3 e)}{5 c^3}\right)+(A-6 i B) \left(-\frac{\sin (3 e)}{c^3}-\frac{i \cos (3 e)}{c^3}\right)\right)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{2 (6 B+i A) \sqrt{e^{i f x}} e^{-i (4 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{c^2 f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))}","\frac{2 a^{7/2} (6 B+i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{c^{5/2} f}-\frac{a^3 (6 B+i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{c^3 f}-\frac{2 a^2 (6 B+i A) (a+i a \tan (e+f x))^{3/2}}{3 c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{2 a (6 B+i A) (a+i a \tan (e+f x))^{5/2}}{15 c f (c-i c \tan (e+f x))^{3/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{7/2}}{5 f (c-i c \tan (e+f x))^{5/2}}",1,"(2*(I*A + 6*B)*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c^2*E^(I*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(9/2)*(Cos[f*x] + I*Sin[f*x])^(7/2)*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (Cos[e + f*x]^4*((A - (6*I)*B)*Cos[2*f*x]*((((-2*I)/3)*Cos[e])/c^3 - (2*Sin[e])/(3*c^3)) + (I*A + 6*B)*Cos[4*f*x]*((2*Cos[e])/(15*c^3) + (((2*I)/15)*Sin[e])/c^3) + (A - (6*I)*B)*(((-I)*Cos[3*e])/c^3 - Sin[3*e]/c^3) + (A - I*B)*Cos[6*f*x]*(((-1/5*I)*Cos[3*e])/c^3 + Sin[3*e]/(5*c^3)) + (A - (6*I)*B)*((2*Cos[e])/(3*c^3) - (((2*I)/3)*Sin[e])/c^3)*Sin[2*f*x] + (A - (6*I)*B)*((-2*Cos[e])/(15*c^3) - (((2*I)/15)*Sin[e])/c^3)*Sin[4*f*x] + (A - I*B)*(Cos[3*e]/(5*c^3) + ((I/5)*Sin[3*e])/c^3)*Sin[6*f*x])*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","A",0
824,1,570,251,17.751212,"\int \frac{(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(7/2),x]","\frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left((9 B-5 i A) \cos (6 f x) \left(\frac{\cos (3 e)}{70 c^4}+\frac{i \sin (3 e)}{70 c^4}\right)+(A-i B) \cos (8 f x) \left(\frac{\sin (5 e)}{14 c^4}-\frac{i \cos (5 e)}{14 c^4}\right)+(5 A+9 i B) \sin (6 f x) \left(\frac{\cos (3 e)}{70 c^4}+\frac{i \sin (3 e)}{70 c^4}\right)+(A-i B) \sin (8 f x) \left(\frac{\cos (5 e)}{14 c^4}+\frac{i \sin (5 e)}{14 c^4}\right)+\cos (4 f x) \left(-\frac{2 B \cos (e)}{15 c^4}-\frac{2 i B \sin (e)}{15 c^4}\right)+\cos (2 f x) \left(\frac{2 B \cos (e)}{3 c^4}-\frac{2 i B \sin (e)}{3 c^4}\right)+\sin (2 f x) \left(\frac{2 B \sin (e)}{3 c^4}+\frac{2 i B \cos (e)}{3 c^4}\right)+\sin (4 f x) \left(\frac{2 B \sin (e)}{15 c^4}-\frac{2 i B \cos (e)}{15 c^4}\right)-\frac{i B \sin (3 e)}{c^4}+\frac{B \cos (3 e)}{c^4}\right)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}-\frac{2 B \sqrt{e^{i f x}} e^{-i (4 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{c^3 f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))}","-\frac{2 a^{7/2} B \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{c^{7/2} f}+\frac{2 a^3 B \sqrt{a+i a \tan (e+f x)}}{c^3 f \sqrt{c-i c \tan (e+f x)}}-\frac{2 a^2 B (a+i a \tan (e+f x))^{3/2}}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{7/2}}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac{2 a B (a+i a \tan (e+f x))^{5/2}}{5 c f (c-i c \tan (e+f x))^{5/2}}",1,"(-2*B*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c^3*E^(I*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(9/2)*(Cos[f*x] + I*Sin[f*x])^(7/2)*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (Cos[e + f*x]^4*((B*Cos[3*e])/c^4 + Cos[4*f*x]*((-2*B*Cos[e])/(15*c^4) - (((2*I)/15)*B*Sin[e])/c^4) + Cos[2*f*x]*((2*B*Cos[e])/(3*c^4) - (((2*I)/3)*B*Sin[e])/c^4) - (I*B*Sin[3*e])/c^4 + ((-5*I)*A + 9*B)*Cos[6*f*x]*(Cos[3*e]/(70*c^4) + ((I/70)*Sin[3*e])/c^4) + (A - I*B)*Cos[8*f*x]*(((-1/14*I)*Cos[5*e])/c^4 + Sin[5*e]/(14*c^4)) + ((((2*I)/3)*B*Cos[e])/c^4 + (2*B*Sin[e])/(3*c^4))*Sin[2*f*x] + ((((-2*I)/15)*B*Cos[e])/c^4 + (2*B*Sin[e])/(15*c^4))*Sin[4*f*x] + (5*A + (9*I)*B)*(Cos[3*e]/(70*c^4) + ((I/70)*Sin[3*e])/c^4)*Sin[6*f*x] + (A - I*B)*(Cos[5*e]/(14*c^4) + ((I/14)*Sin[5*e])/c^4)*Sin[8*f*x])*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","B",0
825,1,121,102,12.669741,"\int \frac{(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{9/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(9/2),x]","\frac{a^3 \cos (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\cos (8 e+11 f x)+i \sin (8 e+11 f x)) ((B-8 i A) \cos (e+f x)-(A+8 i B) \sin (e+f x))}{63 c^5 f (\cos (f x)+i \sin (f x))^3}","-\frac{(-8 B+i A) (a+i a \tan (e+f x))^{7/2}}{63 c f (c-i c \tan (e+f x))^{7/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{7/2}}{9 f (c-i c \tan (e+f x))^{9/2}}",1,"(a^3*Cos[e + f*x]*(((-8*I)*A + B)*Cos[e + f*x] - (A + (8*I)*B)*Sin[e + f*x])*(Cos[8*e + 11*f*x] + I*Sin[8*e + 11*f*x])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(63*c^5*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
826,1,417,155,16.7857811,"\int \frac{(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(11/2),x]","\frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left((B-i A) \cos (6 f x) \left(\frac{\cos (3 e)}{56 c^6}+\frac{i \sin (3 e)}{56 c^6}\right)+(A+i B) \sin (6 f x) \left(\frac{\cos (3 e)}{56 c^6}+\frac{i \sin (3 e)}{56 c^6}\right)+(9 B-23 i A) \cos (8 f x) \left(\frac{\cos (5 e)}{504 c^6}+\frac{i \sin (5 e)}{504 c^6}\right)+(31 A-9 i B) \cos (10 f x) \left(\frac{\sin (7 e)}{792 c^6}-\frac{i \cos (7 e)}{792 c^6}\right)+(A-i B) \cos (12 f x) \left(\frac{\sin (9 e)}{88 c^6}-\frac{i \cos (9 e)}{88 c^6}\right)+(23 A+9 i B) \sin (8 f x) \left(\frac{\cos (5 e)}{504 c^6}+\frac{i \sin (5 e)}{504 c^6}\right)+(31 A-9 i B) \sin (10 f x) \left(\frac{\cos (7 e)}{792 c^6}+\frac{i \sin (7 e)}{792 c^6}\right)+(A-i B) \sin (12 f x) \left(\frac{\cos (9 e)}{88 c^6}+\frac{i \sin (9 e)}{88 c^6}\right)\right)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}","-\frac{(-9 B+2 i A) (a+i a \tan (e+f x))^{7/2}}{693 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac{(-9 B+2 i A) (a+i a \tan (e+f x))^{7/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{7/2}}{11 f (c-i c \tan (e+f x))^{11/2}}",1,"(Cos[e + f*x]^4*(((-I)*A + B)*Cos[6*f*x]*(Cos[3*e]/(56*c^6) + ((I/56)*Sin[3*e])/c^6) + ((-23*I)*A + 9*B)*Cos[8*f*x]*(Cos[5*e]/(504*c^6) + ((I/504)*Sin[5*e])/c^6) + (31*A - (9*I)*B)*Cos[10*f*x]*(((-1/792*I)*Cos[7*e])/c^6 + Sin[7*e]/(792*c^6)) + (A - I*B)*Cos[12*f*x]*(((-1/88*I)*Cos[9*e])/c^6 + Sin[9*e]/(88*c^6)) + (A + I*B)*(Cos[3*e]/(56*c^6) + ((I/56)*Sin[3*e])/c^6)*Sin[6*f*x] + (23*A + (9*I)*B)*(Cos[5*e]/(504*c^6) + ((I/504)*Sin[5*e])/c^6)*Sin[8*f*x] + (31*A - (9*I)*B)*(Cos[7*e]/(792*c^6) + ((I/792)*Sin[7*e])/c^6)*Sin[10*f*x] + (A - I*B)*(Cos[9*e]/(88*c^6) + ((I/88)*Sin[9*e])/c^6)*Sin[12*f*x])*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","B",1
827,1,495,208,17.0725319,"\int \frac{(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(13/2),x]","\frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left((B-i A) \cos (6 f x) \left(\frac{\cos (3 e)}{112 c^7}+\frac{i \sin (3 e)}{112 c^7}\right)+(A+i B) \sin (6 f x) \left(\frac{\cos (3 e)}{112 c^7}+\frac{i \sin (3 e)}{112 c^7}\right)+(8 B-15 i A) \cos (8 f x) \left(\frac{\cos (5 e)}{504 c^7}+\frac{i \sin (5 e)}{504 c^7}\right)+(B-30 i A) \cos (10 f x) \left(\frac{\cos (7 e)}{792 c^7}+\frac{i \sin (7 e)}{792 c^7}\right)+(25 A-12 i B) \cos (12 f x) \left(\frac{\sin (9 e)}{1144 c^7}-\frac{i \cos (9 e)}{1144 c^7}\right)+(A-i B) \cos (14 f x) \left(\frac{\sin (11 e)}{208 c^7}-\frac{i \cos (11 e)}{208 c^7}\right)+(15 A+8 i B) \sin (8 f x) \left(\frac{\cos (5 e)}{504 c^7}+\frac{i \sin (5 e)}{504 c^7}\right)+(30 A+i B) \sin (10 f x) \left(\frac{\cos (7 e)}{792 c^7}+\frac{i \sin (7 e)}{792 c^7}\right)+(25 A-12 i B) \sin (12 f x) \left(\frac{\cos (9 e)}{1144 c^7}+\frac{i \sin (9 e)}{1144 c^7}\right)+(A-i B) \sin (14 f x) \left(\frac{\cos (11 e)}{208 c^7}+\frac{i \sin (11 e)}{208 c^7}\right)\right)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}","-\frac{2 (-10 B+3 i A) (a+i a \tan (e+f x))^{7/2}}{9009 c^3 f (c-i c \tan (e+f x))^{7/2}}-\frac{2 (-10 B+3 i A) (a+i a \tan (e+f x))^{7/2}}{1287 c^2 f (c-i c \tan (e+f x))^{9/2}}-\frac{(-10 B+3 i A) (a+i a \tan (e+f x))^{7/2}}{143 c f (c-i c \tan (e+f x))^{11/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{7/2}}{13 f (c-i c \tan (e+f x))^{13/2}}",1,"(Cos[e + f*x]^4*(((-I)*A + B)*Cos[6*f*x]*(Cos[3*e]/(112*c^7) + ((I/112)*Sin[3*e])/c^7) + ((-15*I)*A + 8*B)*Cos[8*f*x]*(Cos[5*e]/(504*c^7) + ((I/504)*Sin[5*e])/c^7) + ((-30*I)*A + B)*Cos[10*f*x]*(Cos[7*e]/(792*c^7) + ((I/792)*Sin[7*e])/c^7) + (25*A - (12*I)*B)*Cos[12*f*x]*(((-1/1144*I)*Cos[9*e])/c^7 + Sin[9*e]/(1144*c^7)) + (A - I*B)*Cos[14*f*x]*(((-1/208*I)*Cos[11*e])/c^7 + Sin[11*e]/(208*c^7)) + (A + I*B)*(Cos[3*e]/(112*c^7) + ((I/112)*Sin[3*e])/c^7)*Sin[6*f*x] + (15*A + (8*I)*B)*(Cos[5*e]/(504*c^7) + ((I/504)*Sin[5*e])/c^7)*Sin[8*f*x] + (30*A + I*B)*(Cos[7*e]/(792*c^7) + ((I/792)*Sin[7*e])/c^7)*Sin[10*f*x] + (25*A - (12*I)*B)*(Cos[9*e]/(1144*c^7) + ((I/1144)*Sin[9*e])/c^7)*Sin[12*f*x] + (A - I*B)*(Cos[11*e]/(208*c^7) + ((I/208)*Sin[11*e])/c^7)*Sin[14*f*x])*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","B",1
828,1,577,261,17.3051111,"\int \frac{(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{15/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(15/2),x]","\frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left((B-i A) \cos (6 f x) \left(\frac{\cos (3 e)}{224 c^8}+\frac{i \sin (3 e)}{224 c^8}\right)+(A+i B) \sin (6 f x) \left(\frac{\cos (3 e)}{224 c^8}+\frac{i \sin (3 e)}{224 c^8}\right)+(23 B-37 i A) \cos (8 f x) \left(\frac{\cos (5 e)}{2016 c^8}+\frac{i \sin (5 e)}{2016 c^8}\right)+(11 B-49 i A) \cos (10 f x) \left(\frac{\cos (7 e)}{1584 c^8}+\frac{i \sin (7 e)}{1584 c^8}\right)+(61 A-11 i B) \cos (12 f x) \left(\frac{\sin (9 e)}{2288 c^8}-\frac{i \cos (9 e)}{2288 c^8}\right)+(73 A-43 i B) \cos (14 f x) \left(\frac{\sin (11 e)}{6240 c^8}-\frac{i \cos (11 e)}{6240 c^8}\right)+(A-i B) \cos (16 f x) \left(\frac{\sin (13 e)}{480 c^8}-\frac{i \cos (13 e)}{480 c^8}\right)+(37 A+23 i B) \sin (8 f x) \left(\frac{\cos (5 e)}{2016 c^8}+\frac{i \sin (5 e)}{2016 c^8}\right)+(49 A+11 i B) \sin (10 f x) \left(\frac{\cos (7 e)}{1584 c^8}+\frac{i \sin (7 e)}{1584 c^8}\right)+(61 A-11 i B) \sin (12 f x) \left(\frac{\cos (9 e)}{2288 c^8}+\frac{i \sin (9 e)}{2288 c^8}\right)+(73 A-43 i B) \sin (14 f x) \left(\frac{\cos (11 e)}{6240 c^8}+\frac{i \sin (11 e)}{6240 c^8}\right)+(A-i B) \sin (16 f x) \left(\frac{\cos (13 e)}{480 c^8}+\frac{i \sin (13 e)}{480 c^8}\right)\right)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}","-\frac{2 (-11 B+4 i A) (a+i a \tan (e+f x))^{7/2}}{45045 c^4 f (c-i c \tan (e+f x))^{7/2}}-\frac{2 (-11 B+4 i A) (a+i a \tan (e+f x))^{7/2}}{6435 c^3 f (c-i c \tan (e+f x))^{9/2}}-\frac{(-11 B+4 i A) (a+i a \tan (e+f x))^{7/2}}{715 c^2 f (c-i c \tan (e+f x))^{11/2}}-\frac{(-11 B+4 i A) (a+i a \tan (e+f x))^{7/2}}{195 c f (c-i c \tan (e+f x))^{13/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{7/2}}{15 f (c-i c \tan (e+f x))^{15/2}}",1,"(Cos[e + f*x]^4*(((-I)*A + B)*Cos[6*f*x]*(Cos[3*e]/(224*c^8) + ((I/224)*Sin[3*e])/c^8) + ((-37*I)*A + 23*B)*Cos[8*f*x]*(Cos[5*e]/(2016*c^8) + ((I/2016)*Sin[5*e])/c^8) + ((-49*I)*A + 11*B)*Cos[10*f*x]*(Cos[7*e]/(1584*c^8) + ((I/1584)*Sin[7*e])/c^8) + (61*A - (11*I)*B)*Cos[12*f*x]*(((-1/2288*I)*Cos[9*e])/c^8 + Sin[9*e]/(2288*c^8)) + (73*A - (43*I)*B)*Cos[14*f*x]*(((-1/6240*I)*Cos[11*e])/c^8 + Sin[11*e]/(6240*c^8)) + (A - I*B)*Cos[16*f*x]*(((-1/480*I)*Cos[13*e])/c^8 + Sin[13*e]/(480*c^8)) + (A + I*B)*(Cos[3*e]/(224*c^8) + ((I/224)*Sin[3*e])/c^8)*Sin[6*f*x] + (37*A + (23*I)*B)*(Cos[5*e]/(2016*c^8) + ((I/2016)*Sin[5*e])/c^8)*Sin[8*f*x] + (49*A + (11*I)*B)*(Cos[7*e]/(1584*c^8) + ((I/1584)*Sin[7*e])/c^8)*Sin[10*f*x] + (61*A - (11*I)*B)*(Cos[9*e]/(2288*c^8) + ((I/2288)*Sin[9*e])/c^8)*Sin[12*f*x] + (73*A - (43*I)*B)*(Cos[11*e]/(6240*c^8) + ((I/6240)*Sin[11*e])/c^8)*Sin[14*f*x] + (A - I*B)*(Cos[13*e]/(480*c^8) + ((I/480)*Sin[13*e])/c^8)*Sin[16*f*x])*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","B",1
829,1,655,314,17.705089,"\int \frac{(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{17/2}} \, dx","Integrate[((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(17/2),x]","\frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left((B-i A) \cos (6 f x) \left(\frac{\cos (3 e)}{448 c^9}+\frac{i \sin (3 e)}{448 c^9}\right)+(A+i B) \sin (6 f x) \left(\frac{\cos (3 e)}{448 c^9}+\frac{i \sin (3 e)}{448 c^9}\right)+(15 B-22 i A) \cos (8 f x) \left(\frac{\cos (5 e)}{2016 c^9}+\frac{i \sin (5 e)}{2016 c^9}\right)+(51 B-145 i A) \cos (10 f x) \left(\frac{\cos (7 e)}{6336 c^9}+\frac{i \sin (7 e)}{6336 c^9}\right)+(B-60 i A) \cos (12 f x) \left(\frac{\cos (9 e)}{2288 c^9}+\frac{i \sin (9 e)}{2288 c^9}\right)+(215 A-69 i B) \cos (14 f x) \left(\frac{\sin (11 e)}{12480 c^9}-\frac{i \cos (11 e)}{12480 c^9}\right)+(50 A-33 i B) \cos (16 f x) \left(\frac{\sin (13 e)}{8160 c^9}-\frac{i \cos (13 e)}{8160 c^9}\right)+(A-i B) \cos (18 f x) \left(\frac{\sin (15 e)}{1088 c^9}-\frac{i \cos (15 e)}{1088 c^9}\right)+(22 A+15 i B) \sin (8 f x) \left(\frac{\cos (5 e)}{2016 c^9}+\frac{i \sin (5 e)}{2016 c^9}\right)+(145 A+51 i B) \sin (10 f x) \left(\frac{\cos (7 e)}{6336 c^9}+\frac{i \sin (7 e)}{6336 c^9}\right)+(60 A+i B) \sin (12 f x) \left(\frac{\cos (9 e)}{2288 c^9}+\frac{i \sin (9 e)}{2288 c^9}\right)+(215 A-69 i B) \sin (14 f x) \left(\frac{\cos (11 e)}{12480 c^9}+\frac{i \sin (11 e)}{12480 c^9}\right)+(50 A-33 i B) \sin (16 f x) \left(\frac{\cos (13 e)}{8160 c^9}+\frac{i \sin (13 e)}{8160 c^9}\right)+(A-i B) \sin (18 f x) \left(\frac{\cos (15 e)}{1088 c^9}+\frac{i \sin (15 e)}{1088 c^9}\right)\right)}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}","-\frac{8 (-12 B+5 i A) (a+i a \tan (e+f x))^{7/2}}{765765 c^5 f (c-i c \tan (e+f x))^{7/2}}-\frac{8 (-12 B+5 i A) (a+i a \tan (e+f x))^{7/2}}{109395 c^4 f (c-i c \tan (e+f x))^{9/2}}-\frac{4 (-12 B+5 i A) (a+i a \tan (e+f x))^{7/2}}{12155 c^3 f (c-i c \tan (e+f x))^{11/2}}-\frac{4 (-12 B+5 i A) (a+i a \tan (e+f x))^{7/2}}{3315 c^2 f (c-i c \tan (e+f x))^{13/2}}-\frac{(-12 B+5 i A) (a+i a \tan (e+f x))^{7/2}}{255 c f (c-i c \tan (e+f x))^{15/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{7/2}}{17 f (c-i c \tan (e+f x))^{17/2}}",1,"(Cos[e + f*x]^4*(((-I)*A + B)*Cos[6*f*x]*(Cos[3*e]/(448*c^9) + ((I/448)*Sin[3*e])/c^9) + ((-22*I)*A + 15*B)*Cos[8*f*x]*(Cos[5*e]/(2016*c^9) + ((I/2016)*Sin[5*e])/c^9) + ((-145*I)*A + 51*B)*Cos[10*f*x]*(Cos[7*e]/(6336*c^9) + ((I/6336)*Sin[7*e])/c^9) + ((-60*I)*A + B)*Cos[12*f*x]*(Cos[9*e]/(2288*c^9) + ((I/2288)*Sin[9*e])/c^9) + (215*A - (69*I)*B)*Cos[14*f*x]*(((-1/12480*I)*Cos[11*e])/c^9 + Sin[11*e]/(12480*c^9)) + (50*A - (33*I)*B)*Cos[16*f*x]*(((-1/8160*I)*Cos[13*e])/c^9 + Sin[13*e]/(8160*c^9)) + (A - I*B)*Cos[18*f*x]*(((-1/1088*I)*Cos[15*e])/c^9 + Sin[15*e]/(1088*c^9)) + (A + I*B)*(Cos[3*e]/(448*c^9) + ((I/448)*Sin[3*e])/c^9)*Sin[6*f*x] + (22*A + (15*I)*B)*(Cos[5*e]/(2016*c^9) + ((I/2016)*Sin[5*e])/c^9)*Sin[8*f*x] + (145*A + (51*I)*B)*(Cos[7*e]/(6336*c^9) + ((I/6336)*Sin[7*e])/c^9)*Sin[10*f*x] + (60*A + I*B)*(Cos[9*e]/(2288*c^9) + ((I/2288)*Sin[9*e])/c^9)*Sin[12*f*x] + (215*A - (69*I)*B)*(Cos[11*e]/(12480*c^9) + ((I/12480)*Sin[11*e])/c^9)*Sin[14*f*x] + (50*A - (33*I)*B)*(Cos[13*e]/(8160*c^9) + ((I/8160)*Sin[13*e])/c^9)*Sin[16*f*x] + (A - I*B)*(Cos[15*e]/(1088*c^9) + ((I/1088)*Sin[15*e])/c^9)*Sin[18*f*x])*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))","B",1
830,1,185,228,8.5105987,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt{a+i a \tan (e+f x)}} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2))/Sqrt[a + I*a*Tan[e + f*x]],x]","-\frac{c^3 \sec (e+f x) (\cos (f x)+i \sin (f x)) \left(6 (3 B-2 i A) (\cos (f x)-i \sin (f x)) \tan ^{-1}(\cos (e+f x)+i \sin (e+f x))+\frac{1}{2} \sec ^2(e+f x) (\cos (e+2 f x)-i \sin (e+2 f x)) ((2 A+5 i B) \sin (2 (e+f x))+(13 B-10 i A) \cos (2 (e+f x))+5 (3 B-2 i A))\right)}{2 f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}","\frac{3 c^{5/2} (-3 B+2 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{\sqrt{a} f}+\frac{3 c^2 (-3 B+2 i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 a f}+\frac{c (-3 B+2 i A) \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a f}+\frac{(-B+i A) (c-i c \tan (e+f x))^{5/2}}{f \sqrt{a+i a \tan (e+f x)}}",1,"-1/2*(c^3*Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*(6*((-2*I)*A + 3*B)*ArcTan[Cos[e + f*x] + I*Sin[e + f*x]]*(Cos[f*x] - I*Sin[f*x]) + (Sec[e + f*x]^2*(5*((-2*I)*A + 3*B) + ((-10*I)*A + 13*B)*Cos[2*(e + f*x)] + (2*A + (5*I)*B)*Sin[2*(e + f*x)])*(Cos[e + 2*f*x] - I*Sin[e + 2*f*x]))/2))/(f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])","A",1
831,1,161,169,6.7707001,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{\sqrt{a+i a \tan (e+f x)}} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2))/Sqrt[a + I*a*Tan[e + f*x]],x]","\frac{c^2 (\cos (f x)+i \sin (f x)) (\sin (f x)+i \cos (f x)) (A+B \tan (e+f x)) \left(\cos (e+f x) (\tan (e+f x)+i) (-2 i A+i B \tan (e+f x)+3 B)+2 (A+2 i B) \tan ^{-1}(\cos (e+f x)+i \sin (e+f x))\right)}{f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (A \cos (e+f x)+B \sin (e+f x))}","\frac{2 c^{3/2} (-2 B+i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{\sqrt{a} f}+\frac{c (-2 B+i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{a f}+\frac{(-B+i A) (c-i c \tan (e+f x))^{3/2}}{f \sqrt{a+i a \tan (e+f x)}}",1,"(c^2*(Cos[f*x] + I*Sin[f*x])*(I*Cos[f*x] + Sin[f*x])*(A + B*Tan[e + f*x])*(2*(A + (2*I)*B)*ArcTan[Cos[e + f*x] + I*Sin[e + f*x]] + Cos[e + f*x]*(I + Tan[e + f*x])*((-2*I)*A + 3*B + I*B*Tan[e + f*x])))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])","A",1
832,1,152,110,4.4876675,"\int \frac{(A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}{\sqrt{a+i a \tan (e+f x)}} \, dx","Integrate[((A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/Sqrt[a + I*a*Tan[e + f*x]],x]","\frac{c \sec (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+i \cos \left(\frac{1}{2} (e+f x)\right)\right) \left((A+i B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-i \sin \left(\frac{1}{2} (e+f x)\right)\right)+2 i B \left(\cos \left(\frac{1}{2} (e+f x)\right)+i \sin \left(\frac{1}{2} (e+f x)\right)\right) \tan ^{-1}(\cos (e+f x)+i \sin (e+f x))\right)}{f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}","\frac{(-B+i A) \sqrt{c-i c \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}}-\frac{2 B \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{\sqrt{a} f}",1,"(c*Sec[e + f*x]*((A + I*B)*(Cos[(e + f*x)/2] - I*Sin[(e + f*x)/2]) + (2*I)*B*ArcTan[Cos[e + f*x] + I*Sin[e + f*x]]*(Cos[(e + f*x)/2] + I*Sin[(e + f*x)/2]))*(I*Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])","A",1
833,1,77,92,4.0474166,"\int \frac{A+B \tan (e+f x)}{\sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[(A + B*Tan[e + f*x])/(Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]]),x]","-\frac{\sqrt{c-i c \tan (e+f x)} (\cos (e+f x)+i \sin (e+f x)) (B \cos (e+f x)-A \sin (e+f x))}{c f \sqrt{a+i a \tan (e+f x)}}","\frac{i A \sqrt{c-i c \tan (e+f x)}}{c f \sqrt{a+i a \tan (e+f x)}}-\frac{B+i A}{f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}",1,"-(((Cos[e + f*x] + I*Sin[e + f*x])*(B*Cos[e + f*x] - A*Sin[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(c*f*Sqrt[a + I*a*Tan[e + f*x]]))","A",1
834,1,103,157,7.001458,"\int \frac{A+B \tan (e+f x)}{\sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[(A + B*Tan[e + f*x])/(Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2)),x]","\frac{i \sqrt{c-i c \tan (e+f x)} (\cos (2 (e+f x))+i \sin (2 (e+f x))) ((B-2 i A) \sin (2 (e+f x))+(A+2 i B) \cos (2 (e+f x))-3 A)}{6 c^2 f \sqrt{a+i a \tan (e+f x)}}","\frac{-B+i A}{f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac{(-B+2 i A) \sqrt{a+i a \tan (e+f x)}}{3 a c f \sqrt{c-i c \tan (e+f x)}}-\frac{(-B+2 i A) \sqrt{a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}",1,"((I/6)*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*(-3*A + (A + (2*I)*B)*Cos[2*(e + f*x)] + ((-2*I)*A + B)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(c^2*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
835,1,128,213,11.2216434,"\int \frac{A+B \tan (e+f x)}{\sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(A + B*Tan[e + f*x])/(Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(5/2)),x]","\frac{\sqrt{c-i c \tan (e+f x)} (\cos (3 (e+f x))+i \sin (3 (e+f x))) ((3 A+2 i B) (3 \sin (3 (e+f x))-5 \sin (e+f x))+5 (B-6 i A) \cos (e+f x)+(-9 B+6 i A) \cos (3 (e+f x)))}{60 c^3 f \sqrt{a+i a \tan (e+f x)}}","-\frac{2 (-2 B+3 i A) \sqrt{a+i a \tan (e+f x)}}{15 a c^2 f \sqrt{c-i c \tan (e+f x)}}-\frac{2 (-2 B+3 i A) \sqrt{a+i a \tan (e+f x)}}{15 a c f (c-i c \tan (e+f x))^{3/2}}-\frac{(-2 B+3 i A) \sqrt{a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}+\frac{-B+i A}{f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}",1,"((Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*(5*((-6*I)*A + B)*Cos[e + f*x] + ((6*I)*A - 9*B)*Cos[3*(e + f*x)] + (3*A + (2*I)*B)*(-5*Sin[e + f*x] + 3*Sin[3*(e + f*x)]))*Sqrt[c - I*c*Tan[e + f*x]])/(60*c^3*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
836,1,255,287,13.4522887,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2))/(a + I*a*Tan[e + f*x])^(3/2),x]","\frac{\sqrt{\sec (e+f x)} (A+B \tan (e+f x)) \left(\frac{5 c^4 (5 B-2 i A) e^{i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right)}{\sqrt{\frac{c}{1+e^{2 i (e+f x)}}}}+\frac{1}{12} c^3 \sec ^{\frac{3}{2}}(e+f x) \sqrt{c-i c \tan (e+f x)} (33 (5 B-2 i A) \cos (e+f x)+(71 B-26 i A) \cos (3 (e+f x))+2 \sin (e+f x) ((34 A+79 i B) \cos (2 (e+f x))+34 A+82 i B))\right)}{f (a+i a \tan (e+f x))^{3/2} (A \cos (e+f x)+B \sin (e+f x))}","-\frac{5 c^{7/2} (-5 B+2 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{a^{3/2} f}-\frac{5 c^3 (-5 B+2 i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 a^2 f}-\frac{5 c^2 (-5 B+2 i A) \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^2 f}-\frac{2 c (-5 B+2 i A) (c-i c \tan (e+f x))^{5/2}}{3 a f \sqrt{a+i a \tan (e+f x)}}+\frac{(-B+i A) (c-i c \tan (e+f x))^{7/2}}{3 f (a+i a \tan (e+f x))^{3/2}}",1,"(Sqrt[Sec[e + f*x]]*(A + B*Tan[e + f*x])*((5*((-2*I)*A + 5*B)*c^4*E^(I*(e + f*x))*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))])/Sqrt[c/(1 + E^((2*I)*(e + f*x)))] + (c^3*Sec[e + f*x]^(3/2)*(33*((-2*I)*A + 5*B)*Cos[e + f*x] + ((-26*I)*A + 71*B)*Cos[3*(e + f*x)] + 2*(34*A + (82*I)*B + (34*A + (79*I)*B)*Cos[2*(e + f*x)])*Sin[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/12))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^(3/2))","A",1
837,1,174,229,11.3818478,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2))/(a + I*a*Tan[e + f*x])^(3/2),x]","-\frac{4 \sqrt{2} \left(\frac{c}{1+e^{2 i (e+f x)}}\right)^{5/2} \left(3 (A+4 i B) e^{3 i (e+f x)} \left(1+e^{2 i (e+f x)}\right) \tan ^{-1}\left(e^{i (e+f x)}\right)+A \left(2 e^{2 i (e+f x)}+3 e^{4 i (e+f x)}-1\right)+i B \left(8 e^{2 i (e+f x)}+12 e^{4 i (e+f x)}-1\right)\right)}{3 a f (\tan (e+f x)-i) \sqrt{a+i a \tan (e+f x)}}","-\frac{2 c^{5/2} (-4 B+i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{a^{3/2} f}-\frac{c^2 (-4 B+i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{a^2 f}-\frac{2 c (-4 B+i A) (c-i c \tan (e+f x))^{3/2}}{3 a f \sqrt{a+i a \tan (e+f x)}}+\frac{(-B+i A) (c-i c \tan (e+f x))^{5/2}}{3 f (a+i a \tan (e+f x))^{3/2}}",1,"(-4*Sqrt[2]*(c/(1 + E^((2*I)*(e + f*x))))^(5/2)*(A*(-1 + 2*E^((2*I)*(e + f*x)) + 3*E^((4*I)*(e + f*x))) + I*B*(-1 + 8*E^((2*I)*(e + f*x)) + 12*E^((4*I)*(e + f*x))) + 3*(A + (4*I)*B)*E^((3*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))*ArcTan[E^(I*(e + f*x))]))/(3*a*f*(-I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])","A",1
838,1,114,157,7.6434715,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2))/(a + I*a*Tan[e + f*x])^(3/2),x]","\frac{\sqrt{2} c e^{-2 i (e+f x)} \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \left(i A+B \left(-1+6 e^{2 i (e+f x)}\right)+6 B e^{3 i (e+f x)} \tan ^{-1}\left(e^{i (e+f x)}\right)\right)}{3 a f \sqrt{a+i a \tan (e+f x)}}","\frac{2 B c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{a^{3/2} f}+\frac{(-B+i A) (c-i c \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac{2 B c \sqrt{c-i c \tan (e+f x)}}{a f \sqrt{a+i a \tan (e+f x)}}",1,"(Sqrt[2]*c*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*(I*A + B*(-1 + 6*E^((2*I)*(e + f*x))) + 6*B*E^((3*I)*(e + f*x))*ArcTan[E^(I*(e + f*x))]))/(3*a*E^((2*I)*(e + f*x))*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
839,1,81,104,4.3715058,"\int \frac{(A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^{3/2}} \, dx","Integrate[((A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(a + I*a*Tan[e + f*x])^(3/2),x]","\frac{\sqrt{c-i c \tan (e+f x)} ((2 B+i A) \tan (e+f x)+2 A-i B)}{3 a f (\tan (e+f x)-i) \sqrt{a+i a \tan (e+f x)}}","\frac{(-B+i A) \sqrt{c-i c \tan (e+f x)}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac{(2 B+i A) \sqrt{c-i c \tan (e+f x)}}{3 a f \sqrt{a+i a \tan (e+f x)}}",1,"((2*A - I*B + (I*A + 2*B)*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(3*a*f*(-I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])","A",1
840,1,85,152,4.9826367,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]]),x]","-\frac{i \sqrt{c-i c \tan (e+f x)} ((B+2 i A) \sin (2 (e+f x))+(A-2 i B) \cos (2 (e+f x))-3 A)}{6 a c f \sqrt{a+i a \tan (e+f x)}}","-\frac{B+i A}{f (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}+\frac{(B+2 i A) \sqrt{c-i c \tan (e+f x)}}{3 a c f \sqrt{a+i a \tan (e+f x)}}+\frac{(B+2 i A) \sqrt{c-i c \tan (e+f x)}}{3 c f (a+i a \tan (e+f x))^{3/2}}",1,"((-1/6*I)*(-3*A + (A - (2*I)*B)*Cos[2*(e + f*x)] + ((2*I)*A + B)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(a*c*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
841,1,120,152,8.505214,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2)),x]","\frac{\sqrt{c-i c \tan (e+f x)} (\sin (2 (e+f x))-i \cos (2 (e+f x))) (9 A \tan (e+f x)+A \sin (3 (e+f x)) \sec (e+f x)-2 B \cos (2 (e+f x))-2 B)}{12 a c^2 f (\tan (e+f x)-i) \sqrt{a+i a \tan (e+f x)}}","\frac{-B-i A}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{2 A \tan (e+f x)}{3 a c f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{i A}{3 c f (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}",1,"(((-I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)])*(-2*B - 2*B*Cos[2*(e + f*x)] + A*Sec[e + f*x]*Sin[3*(e + f*x)] + 9*A*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(12*a*c^2*f*(-I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])","A",1
842,1,170,269,11.9837921,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(5/2)),x]","\frac{\sec (e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (3 (e+f x))+i \sin (3 (e+f x))) (20 (A+i B) \cos (2 (e+f x))+(A+4 i B) \cos (4 (e+f x))-40 i A \sin (2 (e+f x))-4 i A \sin (4 (e+f x))-45 A+10 B \sin (2 (e+f x))+B \sin (4 (e+f x)))}{120 a c^3 f (\tan (e+f x)-i) \sqrt{a+i a \tan (e+f x)}}","-\frac{2 (-B+4 i A) \sqrt{a+i a \tan (e+f x)}}{15 a^2 c^2 f \sqrt{c-i c \tan (e+f x)}}-\frac{2 (-B+4 i A) \sqrt{a+i a \tan (e+f x)}}{15 a^2 c f (c-i c \tan (e+f x))^{3/2}}-\frac{(-B+4 i A) \sqrt{a+i a \tan (e+f x)}}{5 a^2 f (c-i c \tan (e+f x))^{5/2}}+\frac{-B+i A}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}+\frac{-B+4 i A}{3 a f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}",1,"(Sec[e + f*x]*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*(-45*A + 20*(A + I*B)*Cos[2*(e + f*x)] + (A + (4*I)*B)*Cos[4*(e + f*x)] - (40*I)*A*Sin[2*(e + f*x)] + 10*B*Sin[2*(e + f*x)] - (4*I)*A*Sin[4*(e + f*x)] + B*Sin[4*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(120*a*c^3*f*(-I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])","A",1
843,1,247,343,15.1240456,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(9/2))/(a + I*a*Tan[e + f*x])^(5/2),x]","-\frac{\sqrt{2} c^4 e^{-4 i (e+f x)} \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \left(105 (7 B-2 i A) e^{5 i (e+f x)} \left(1+e^{2 i (e+f x)}\right)^2 \tan ^{-1}\left(e^{i (e+f x)}\right)-2 i A \left(-8 e^{2 i (e+f x)}+56 e^{4 i (e+f x)}+175 e^{6 i (e+f x)}+105 e^{8 i (e+f x)}+6\right)+B \left(-56 e^{2 i (e+f x)}+392 e^{4 i (e+f x)}+1225 e^{6 i (e+f x)}+735 e^{8 i (e+f x)}+12\right)\right)}{15 a^2 f \left(1+e^{2 i (e+f x)}\right)^2 \sqrt{a+i a \tan (e+f x)}}","\frac{7 c^{9/2} (-7 B+2 i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{a^{5/2} f}+\frac{7 c^4 (-7 B+2 i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 a^3 f}+\frac{7 c^3 (-7 B+2 i A) \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^3 f}+\frac{14 c^2 (-7 B+2 i A) (c-i c \tan (e+f x))^{5/2}}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}-\frac{2 c (-7 B+2 i A) (c-i c \tan (e+f x))^{7/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{(-B+i A) (c-i c \tan (e+f x))^{9/2}}{5 f (a+i a \tan (e+f x))^{5/2}}",1,"-1/15*(Sqrt[2]*c^4*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*((-2*I)*A*(6 - 8*E^((2*I)*(e + f*x)) + 56*E^((4*I)*(e + f*x)) + 175*E^((6*I)*(e + f*x)) + 105*E^((8*I)*(e + f*x))) + B*(12 - 56*E^((2*I)*(e + f*x)) + 392*E^((4*I)*(e + f*x)) + 1225*E^((6*I)*(e + f*x)) + 735*E^((8*I)*(e + f*x))) + 105*((-2*I)*A + 7*B)*E^((5*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^2*ArcTan[E^(I*(e + f*x))]))/(a^2*E^((4*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^2*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
844,1,205,284,14.2480845,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2))/(a + I*a*Tan[e + f*x])^(5/2),x]","\frac{2 \sqrt{2} c^2 e^{-4 i (e+f x)} \left(\frac{c}{1+e^{2 i (e+f x)}}\right)^{3/2} \left(15 i (A+6 i B) e^{5 i (e+f x)} \left(1+e^{2 i (e+f x)}\right) \tan ^{-1}\left(e^{i (e+f x)}\right)+i A \left(-2 e^{2 i (e+f x)}+10 e^{4 i (e+f x)}+15 e^{6 i (e+f x)}+3\right)-3 B \left(-4 e^{2 i (e+f x)}+20 e^{4 i (e+f x)}+30 e^{6 i (e+f x)}+1\right)\right)}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}","\frac{2 c^{7/2} (-6 B+i A) \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{a^{5/2} f}+\frac{c^3 (-6 B+i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{a^3 f}+\frac{2 c^2 (-6 B+i A) (c-i c \tan (e+f x))^{3/2}}{3 a^2 f \sqrt{a+i a \tan (e+f x)}}-\frac{2 c (-6 B+i A) (c-i c \tan (e+f x))^{5/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{(-B+i A) (c-i c \tan (e+f x))^{7/2}}{5 f (a+i a \tan (e+f x))^{5/2}}",1,"(2*Sqrt[2]*c^2*(c/(1 + E^((2*I)*(e + f*x))))^(3/2)*(I*A*(3 - 2*E^((2*I)*(e + f*x)) + 10*E^((4*I)*(e + f*x)) + 15*E^((6*I)*(e + f*x))) - 3*B*(1 - 4*E^((2*I)*(e + f*x)) + 20*E^((4*I)*(e + f*x)) + 30*E^((6*I)*(e + f*x))) + (15*I)*(A + (6*I)*B)*E^((5*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))*ArcTan[E^(I*(e + f*x))]))/(15*a^2*E^((4*I)*(e + f*x))*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
845,1,129,205,12.7901309,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2))/(a + I*a*Tan[e + f*x])^(5/2),x]","-\frac{\sqrt{2} c^2 e^{-4 i (e+f x)} \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \left(-3 i A+B \left(-10 e^{2 i (e+f x)}+30 e^{4 i (e+f x)}+3\right)+30 B e^{5 i (e+f x)} \tan ^{-1}\left(e^{i (e+f x)}\right)\right)}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}","-\frac{2 B c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{a^{5/2} f}-\frac{2 B c^2 \sqrt{c-i c \tan (e+f x)}}{a^2 f \sqrt{a+i a \tan (e+f x)}}+\frac{(-B+i A) (c-i c \tan (e+f x))^{5/2}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{2 B c (c-i c \tan (e+f x))^{3/2}}{3 a f (a+i a \tan (e+f x))^{3/2}}",1,"-1/15*(Sqrt[2]*c^2*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*((-3*I)*A + B*(3 - 10*E^((2*I)*(e + f*x)) + 30*E^((4*I)*(e + f*x))) + 30*B*E^((5*I)*(e + f*x))*ArcTan[E^(I*(e + f*x))]))/(a^2*E^((4*I)*(e + f*x))*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
846,1,92,104,8.0106319,"\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx","Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2))/(a + I*a*Tan[e + f*x])^(5/2),x]","\frac{c (1-i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)} ((A-4 i B) \tan (e+f x)-4 i A-B)}{15 a^2 f (\tan (e+f x)-i)^2 \sqrt{a+i a \tan (e+f x)}}","\frac{(4 B+i A) (c-i c \tan (e+f x))^{3/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{(-B+i A) (c-i c \tan (e+f x))^{3/2}}{5 f (a+i a \tan (e+f x))^{5/2}}",1,"(c*(1 - I*Tan[e + f*x])*((-4*I)*A - B + (A - (4*I)*B)*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(15*a^2*f*(-I + Tan[e + f*x])^2*Sqrt[a + I*a*Tan[e + f*x]])","A",1
847,1,106,157,5.1893717,"\int \frac{(A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^{5/2}} \, dx","Integrate[((A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(a + I*a*Tan[e + f*x])^(5/2),x]","\frac{\sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} ((6 A-9 i B) \sin (2 (e+f x))+(-6 B-9 i A) \cos (2 (e+f x))-5 i A)}{30 a^2 f (\tan (e+f x)-i)^2 \sqrt{a+i a \tan (e+f x)}}","\frac{(3 B+2 i A) \sqrt{c-i c \tan (e+f x)}}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}+\frac{(-B+i A) \sqrt{c-i c \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{(3 B+2 i A) \sqrt{c-i c \tan (e+f x)}}{15 a f (a+i a \tan (e+f x))^{3/2}}",1,"(Sec[e + f*x]^2*((-5*I)*A + ((-9*I)*A - 6*B)*Cos[2*(e + f*x)] + (6*A - (9*I)*B)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(30*a^2*f*(-I + Tan[e + f*x])^2*Sqrt[a + I*a*Tan[e + f*x]])","A",1
848,1,132,212,6.7803355,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]]),x]","-\frac{\sec (e+f x) \sqrt{c-i c \tan (e+f x)} (-i (3 A-2 i B) (5 \sin (e+f x)-3 \sin (3 (e+f x)))+(-30 A+5 i B) \cos (e+f x)+(6 A-9 i B) \cos (3 (e+f x)))}{60 a^2 c f (\tan (e+f x)-i) \sqrt{a+i a \tan (e+f x)}}","\frac{2 (2 B+3 i A) \sqrt{c-i c \tan (e+f x)}}{15 a^2 c f \sqrt{a+i a \tan (e+f x)}}-\frac{B+i A}{f (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}+\frac{2 (2 B+3 i A) \sqrt{c-i c \tan (e+f x)}}{15 a c f (a+i a \tan (e+f x))^{3/2}}+\frac{(2 B+3 i A) \sqrt{c-i c \tan (e+f x)}}{5 c f (a+i a \tan (e+f x))^{5/2}}",1,"-1/60*(Sec[e + f*x]*((-30*A + (5*I)*B)*Cos[e + f*x] + (6*A - (9*I)*B)*Cos[3*(e + f*x)] - I*(3*A - (2*I)*B)*(5*Sin[e + f*x] - 3*Sin[3*(e + f*x)]))*Sqrt[c - I*c*Tan[e + f*x]])/(a^2*c*f*(-I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])","A",1
849,1,133,218,11.8339032,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(3/2)),x]","-\frac{i \sqrt{c-i c \tan (e+f x)} (20 (A-i B) \cos (2 (e+f x))+(A-4 i B) \cos (4 (e+f x))+40 i A \sin (2 (e+f x))+4 i A \sin (4 (e+f x))-45 A+10 B \sin (2 (e+f x))+B \sin (4 (e+f x)))}{120 a^2 c^2 f \sqrt{a+i a \tan (e+f x)}}","\frac{2 (4 A-i B) \tan (e+f x)}{15 a^2 c f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{-B-i A}{3 f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}}+\frac{B+4 i A}{15 a c f (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}+\frac{B+4 i A}{15 c f (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}",1,"((-1/120*I)*(-45*A + 20*(A - I*B)*Cos[2*(e + f*x)] + (A - (4*I)*B)*Cos[4*(e + f*x)] + (40*I)*A*Sin[2*(e + f*x)] + 10*B*Sin[2*(e + f*x)] + (4*I)*A*Sin[4*(e + f*x)] + B*Sin[4*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(a^2*c^2*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
850,1,151,206,11.7107862,"\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2)),x]","\frac{\sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (3 (e+f x))+i \sin (3 (e+f x))) (-150 A \sin (e+f x)-25 A \sin (3 (e+f x))-3 A \sin (5 (e+f x))+30 B \cos (e+f x)+15 B \cos (3 (e+f x))+3 B \cos (5 (e+f x)))}{240 a^2 c^3 f (\tan (e+f x)-i)^2 \sqrt{a+i a \tan (e+f x)}}","\frac{8 A \tan (e+f x)}{15 a^2 c^2 f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{-B-i A}{5 f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}+\frac{4 A \tan (e+f x)}{15 a c f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{i A}{5 c f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}}",1,"(Sec[e + f*x]^2*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*(30*B*Cos[e + f*x] + 15*B*Cos[3*(e + f*x)] + 3*B*Cos[5*(e + f*x)] - 150*A*Sin[e + f*x] - 25*A*Sin[3*(e + f*x)] - 3*A*Sin[5*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(240*a^2*c^3*f*(-I + Tan[e + f*x])^2*Sqrt[a + I*a*Tan[e + f*x]])","A",1
851,1,197,150,20.9268038,"\int (a+i a \tan (e+f x))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx","Integrate[(a + I*a*Tan[e + f*x])^m*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n,x]","\frac{2^{m+n-1} \left(e^{i f x}\right)^m \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^m \left(\frac{c}{1+e^{2 i (e+f x)}}\right)^n \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m \left((m+1) (B-i A) \, _2F_1\left(1,-n;m+1;-e^{2 i (e+f x)}\right)-i m (A-i B) e^{2 i (e+f x)} \, _2F_1\left(1,1-n;m+2;-e^{2 i (e+f x)}\right)\right)}{f m (m+1)}","\frac{(B+i A) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 f n}-\frac{2^{n-1} (B (m-n)+i A (m+n)) (1-i \tan (e+f x))^{-n} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n \, _2F_1\left(m,-n;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right)}{f m n}",1,"(2^(-1 + m + n)*(E^(I*f*x))^m*(c/(1 + E^((2*I)*(e + f*x))))^n*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*((-I)*(A - I*B)*E^((2*I)*(e + f*x))*m*Hypergeometric2F1[1, 1 - n, 2 + m, -E^((2*I)*(e + f*x))] + ((-I)*A + B)*(1 + m)*Hypergeometric2F1[1, -n, 1 + m, -E^((2*I)*(e + f*x))])*(a + I*a*Tan[e + f*x])^m)/(f*m*(1 + m)*Sec[e + f*x]^m*(Cos[f*x] + I*Sin[f*x])^m)","A",0
852,1,177,147,85.9836249,"\int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(1 + m)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(-1 - m),x]","\frac{a e^{i (e+2 f x)} \left(e^{i f x}\right)^m \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^m (\tan (e+f x)-i) \left(\frac{c}{1+e^{2 i (e+f x)}}\right)^{-m} \sec ^{-m-1}(e+f x) (\cos (f x)+i \sin (f x))^{-m-1} (a+i a \tan (e+f x))^m \left(A+2 i B \, _2F_1\left(1,m+1;m+2;-e^{2 i (e+f x)}\right)-i B\right)}{2 c f (m+1)}","\frac{a B 2^m (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{-m} \, _2F_1\left(-m,-m;1-m;\frac{1}{2} (1-i \tan (e+f x))\right)}{c f m}-\frac{(B+i A) (a+i a \tan (e+f x))^{m+1} (c-i c \tan (e+f x))^{-m-1}}{2 f (m+1)}",1,"(a*E^(I*(e + 2*f*x))*(E^(I*f*x))^m*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*(A - I*B + (2*I)*B*Hypergeometric2F1[1, 1 + m, 2 + m, -E^((2*I)*(e + f*x))])*Sec[e + f*x]^(-1 - m)*(Cos[f*x] + I*Sin[f*x])^(-1 - m)*(-I + Tan[e + f*x])*(a + I*a*Tan[e + f*x])^m)/(2*c*(c/(1 + E^((2*I)*(e + f*x))))^m*f*(1 + m))","A",1
853,1,56,33,4.1834013,"\int \frac{(c-i c \tan (e+f x))^n (-i (2+n)+(-2+n) \tan (e+f x))}{(-i+\tan (e+f x))^2} \, dx","Integrate[((c - I*c*Tan[e + f*x])^n*((-I)*(2 + n) + (-2 + n)*Tan[e + f*x]))/(-I + Tan[e + f*x])^2,x]","\frac{(c \sec (e+f x))^n \exp (n (-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x))))}{f (\tan (e+f x)-i)^2}","\frac{(c-i c \tan (e+f x))^n}{f (-\tan (e+f x)+i)^2}",1,"(E^(n*(-Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))*(c*Sec[e + f*x])^n)/(f*(-I + Tan[e + f*x])^2)","A",1
854,1,201,104,1.8937886,"\int \frac{(A+B \tan (e+f x)) (c+d \tan (e+f x))}{(a+i a \tan (e+f x))^2} \, dx","Integrate[((A + B*Tan[e + f*x])*(c + d*Tan[e + f*x]))/(a + I*a*Tan[e + f*x])^2,x]","-\frac{(A+B \tan (e+f x)) (c+d \tan (e+f x)) (\sin (2 (e+f x)) (A (4 i c f x+c+4 d f x+i d)+B (4 c f x+i c-4 i d f x-d))+\cos (2 (e+f x)) (A (c (4 f x+i)+d (-1-4 i f x))-B (4 i c f x+c+d (4 f x+i)))+4 i (A c+B d))}{16 a^2 f (\tan (e+f x)-i)^2 (A \cos (e+f x)+B \sin (e+f x)) (c \cos (e+f x)+d \sin (e+f x))}","\frac{A (d+i c)+B (c+3 i d)}{4 a^2 f (1+i \tan (e+f x))}+\frac{x (A-i B) (c-i d)}{4 a^2}+\frac{(-B+i A) (c+i d)}{4 f (a+i a \tan (e+f x))^2}",1,"-1/16*(((4*I)*(A*c + B*d) + (A*(d*(-1 - (4*I)*f*x) + c*(I + 4*f*x)) - B*(c + (4*I)*c*f*x + d*(I + 4*f*x)))*Cos[2*(e + f*x)] + (B*(I*c - d + 4*c*f*x - (4*I)*d*f*x) + A*(c + I*d + (4*I)*c*f*x + 4*d*f*x))*Sin[2*(e + f*x)])*(A + B*Tan[e + f*x])*(c + d*Tan[e + f*x]))/(a^2*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(c*Cos[e + f*x] + d*Sin[e + f*x])*(-I + Tan[e + f*x])^2)","A",1
855,1,206,147,5.439511,"\int \frac{(A+B \tan (e+f x)) (c+d \tan (e+f x))}{(a+i a \tan (e+f x))^{3/2}} \, dx","Integrate[((A + B*Tan[e + f*x])*(c + d*Tan[e + f*x]))/(a + I*a*Tan[e + f*x])^(3/2),x]","\frac{(A+B \tan (e+f x)) (c+d \tan (e+f x)) \left(\frac{2}{3} \cos (e+f x) ((A (d+5 i c)+B (c+7 i d)) \cos (e+f x)-3 (A c-i A d-i B c+3 B d) \sin (e+f x))-i (A-i B) (c-i d) e^{i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \sinh ^{-1}\left(e^{i (e+f x)}\right)\right)}{4 f (a+i a \tan (e+f x))^{3/2} (A \cos (e+f x)+B \sin (e+f x)) (c \cos (e+f x)+d \sin (e+f x))}","-\frac{(B+i A) (c-i d) \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (e+f x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} f}+\frac{(-B+i A) (c+i d)}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac{A (d+i c)+B (c+3 i d)}{2 a f \sqrt{a+i a \tan (e+f x)}}",1,"(((-I)*(A - I*B)*(c - I*d)*E^(I*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcSinh[E^(I*(e + f*x))] + (2*Cos[e + f*x]*((B*(c + (7*I)*d) + A*((5*I)*c + d))*Cos[e + f*x] - 3*(A*c - I*B*c - I*A*d + 3*B*d)*Sin[e + f*x]))/3)*(A + B*Tan[e + f*x])*(c + d*Tan[e + f*x]))/(4*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(c*Cos[e + f*x] + d*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^(3/2))","A",1