1,1,37,42,0.0176379,"\int x^3 \tan ^{-1}\left(a+b x^4\right) \, dx","Integrate[x^3*ArcTan[a + b*x^4],x]","-\frac{\log \left(\left(a+b x^4\right)^2+1\right)-2 \left(a+b x^4\right) \tan ^{-1}\left(a+b x^4\right)}{8 b}","\frac{\left(a+b x^4\right) \tan ^{-1}\left(a+b x^4\right)}{4 b}-\frac{\log \left(\left(a+b x^4\right)^2+1\right)}{8 b}",1,"-1/8*(-2*(a + b*x^4)*ArcTan[a + b*x^4] + Log[1 + (a + b*x^4)^2])/b","A",1
2,1,40,45,0.0452799,"\int x^{-1+n} \tan ^{-1}\left(a+b x^n\right) \, dx","Integrate[x^(-1 + n)*ArcTan[a + b*x^n],x]","-\frac{\log \left(\left(a+b x^n\right)^2+1\right)-2 \left(a+b x^n\right) \tan ^{-1}\left(a+b x^n\right)}{2 b n}","\frac{\left(a+b x^n\right) \tan ^{-1}\left(a+b x^n\right)}{b n}-\frac{\log \left(\left(a+b x^n\right)^2+1\right)}{2 b n}",1,"-1/2*(-2*(a + b*x^n)*ArcTan[a + b*x^n] + Log[1 + (a + b*x^n)^2])/(b*n)","A",1
3,1,86,144,0.0869842,"\int x^5 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right) \, dx","Integrate[x^5*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]","\frac{3 \left(5 d^3+16 e^3 x^6\right) \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)+\sqrt{-e} x \sqrt{d+e x^2} \left(-15 d^2+10 d e x^2-8 e^2 x^4\right)}{288 e^3}","\frac{5 d^3 \sqrt{-e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{96 e^{7/2}}+\frac{5 d^2 x \sqrt{d+e x^2}}{96 (-e)^{5/2}}+\frac{1}{6} x^6 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)+\frac{x^5 \sqrt{d+e x^2}}{36 \sqrt{-e}}+\frac{5 d x^3 \sqrt{d+e x^2}}{144 (-e)^{3/2}}",1,"(Sqrt[-e]*x*Sqrt[d + e*x^2]*(-15*d^2 + 10*d*e*x^2 - 8*e^2*x^4) + 3*(5*d^3 + 16*e^3*x^6)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(288*e^3)","A",1
4,1,74,116,0.0615685,"\int x^3 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right) \, dx","Integrate[x^3*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]","\frac{\left(8 e^2 x^4-3 d^2\right) \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)+\sqrt{-e} x \sqrt{d+e x^2} \left(3 d-2 e x^2\right)}{32 e^2}","-\frac{3 d^2 \sqrt{-e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{32 e^{5/2}}+\frac{3 d x \sqrt{d+e x^2}}{32 (-e)^{3/2}}+\frac{1}{4} x^4 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)+\frac{x^3 \sqrt{d+e x^2}}{16 \sqrt{-e}}",1,"(Sqrt[-e]*x*(3*d - 2*e*x^2)*Sqrt[d + e*x^2] + (-3*d^2 + 8*e^2*x^4)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(32*e^2)","A",1
5,1,59,88,0.0445096,"\int x \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right) \, dx","Integrate[x*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]","\frac{\left(d+2 e x^2\right) \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)-\sqrt{-e} x \sqrt{d+e x^2}}{4 e}","\frac{d \sqrt{-e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{4 e^{3/2}}+\frac{x \sqrt{d+e x^2}}{4 \sqrt{-e}}+\frac{1}{2} x^2 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)",1,"(-(Sqrt[-e]*x*Sqrt[d + e*x^2]) + (d + 2*e*x^2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(4*e)","A",1
6,1,171,288,2.8719846,"\int \frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x} \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x,x]","\frac{\sqrt{-e} \sqrt{\frac{e x^2}{d}+1} \left(-\text{Li}_2\left(e^{-2 \sinh ^{-1}\left(\sqrt{\frac{e}{d}} x\right)}\right)-2 \log (x) \log \left(\sqrt{\frac{e x^2}{d}+1}+x \sqrt{\frac{e}{d}}\right)+\sinh ^{-1}\left(x \sqrt{\frac{e}{d}}\right)^2+2 \sinh ^{-1}\left(x \sqrt{\frac{e}{d}}\right) \log \left(1-e^{-2 \sinh ^{-1}\left(x \sqrt{\frac{e}{d}}\right)}\right)\right)}{2 \sqrt{\frac{e}{d}} \sqrt{d+e x^2}}+\log (x) \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)","\frac{\sqrt{d} \sqrt{-e} \sqrt{\frac{e x^2}{d}+1} \text{Li}_2\left(e^{2 \sinh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}\right)}{2 \sqrt{e} \sqrt{d+e x^2}}-\frac{\sqrt{d} \sqrt{-e} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)^2}{2 \sqrt{e} \sqrt{d+e x^2}}+\log (x) \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)+\frac{\sqrt{d} \sqrt{-e} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \log \left(1-e^{2 \sinh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}\right)}{\sqrt{e} \sqrt{d+e x^2}}-\frac{\sqrt{d} \sqrt{-e} \log (x) \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e} \sqrt{d+e x^2}}",1,"ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]*Log[x] + (Sqrt[-e]*Sqrt[1 + (e*x^2)/d]*(ArcSinh[Sqrt[e/d]*x]^2 + 2*ArcSinh[Sqrt[e/d]*x]*Log[1 - E^(-2*ArcSinh[Sqrt[e/d]*x])] - 2*Log[x]*Log[Sqrt[e/d]*x + Sqrt[1 + (e*x^2)/d]] - PolyLog[2, E^(-2*ArcSinh[Sqrt[e/d]*x])]))/(2*Sqrt[e/d]*Sqrt[d + e*x^2])","A",0
7,1,54,57,0.0409474,"\int \frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x^3} \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^3,x]","-\frac{\sqrt{-e} x \sqrt{d+e x^2}+d \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{2 d x^2}","-\frac{\sqrt{-e} \sqrt{d+e x^2}}{2 d x}-\frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{2 x^2}",1,"-1/2*(Sqrt[-e]*x*Sqrt[d + e*x^2] + d*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(d*x^2)","A",1
8,1,67,85,0.0508618,"\int \frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x^5} \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^5,x]","\frac{\sqrt{-e} x \sqrt{d+e x^2} \left(2 e x^2-d\right)-3 d^2 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{12 d^2 x^4}","-\frac{(-e)^{3/2} \sqrt{d+e x^2}}{6 d^2 x}-\frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{4 x^4}-\frac{\sqrt{-e} \sqrt{d+e x^2}}{12 d x^3}",1,"(Sqrt[-e]*x*Sqrt[d + e*x^2]*(-d + 2*e*x^2) - 3*d^2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(12*d^2*x^4)","A",1
9,1,78,113,0.0670937,"\int \frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x^7} \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^7,x]","\frac{\sqrt{-e} x \sqrt{d+e x^2} \left(-3 d^2+4 d e x^2-8 e^2 x^4\right)-15 d^3 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{90 d^3 x^6}","-\frac{4 (-e)^{5/2} \sqrt{d+e x^2}}{45 d^3 x}-\frac{2 (-e)^{3/2} \sqrt{d+e x^2}}{45 d^2 x^3}-\frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{6 x^6}-\frac{\sqrt{-e} \sqrt{d+e x^2}}{30 d x^5}",1,"(Sqrt[-e]*x*Sqrt[d + e*x^2]*(-3*d^2 + 4*d*e*x^2 - 8*e^2*x^4) - 15*d^3*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(90*d^3*x^6)","A",1
10,1,89,141,0.0747083,"\int \frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x^9} \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^9,x]","\frac{\sqrt{-e} x \sqrt{d+e x^2} \left(-5 d^3+6 d^2 e x^2-8 d e^2 x^4+16 e^3 x^6\right)-35 d^4 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{280 d^4 x^8}","-\frac{2 (-e)^{7/2} \sqrt{d+e x^2}}{35 d^4 x}-\frac{(-e)^{5/2} \sqrt{d+e x^2}}{35 d^3 x^3}-\frac{3 (-e)^{3/2} \sqrt{d+e x^2}}{140 d^2 x^5}-\frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{8 x^8}-\frac{\sqrt{-e} \sqrt{d+e x^2}}{56 d x^7}",1,"(Sqrt[-e]*x*Sqrt[d + e*x^2]*(-5*d^3 + 6*d^2*e*x^2 - 8*d*e^2*x^4 + 16*e^3*x^6) - 35*d^4*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(280*d^4*x^8)","A",1
11,1,83,124,0.1189645,"\int x^6 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right) \, dx","Integrate[x^6*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]","\frac{\sqrt{d+e x^2} \left(16 d^3-8 d^2 e x^2+6 d e^2 x^4-5 e^3 x^6\right)}{245 (-e)^{7/2}}+\frac{1}{7} x^7 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)","\frac{d^3 \sqrt{d+e x^2}}{7 (-e)^{7/2}}-\frac{d^2 \left(d+e x^2\right)^{3/2}}{7 (-e)^{7/2}}-\frac{\left(d+e x^2\right)^{7/2}}{49 (-e)^{7/2}}+\frac{3 d \left(d+e x^2\right)^{5/2}}{35 (-e)^{7/2}}+\frac{1}{7} x^7 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)",1,"(Sqrt[d + e*x^2]*(16*d^3 - 8*d^2*e*x^2 + 6*d*e^2*x^4 - 5*e^3*x^6))/(245*(-e)^(7/2)) + (x^7*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/7","A",1
12,1,72,99,0.1014726,"\int x^4 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right) \, dx","Integrate[x^4*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]","\frac{\sqrt{d+e x^2} \left(8 d^2-4 d e x^2+3 e^2 x^4\right)}{75 (-e)^{5/2}}+\frac{1}{5} x^5 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)","\frac{d^2 \sqrt{d+e x^2}}{5 (-e)^{5/2}}+\frac{\left(d+e x^2\right)^{5/2}}{25 (-e)^{5/2}}-\frac{2 d \left(d+e x^2\right)^{3/2}}{15 (-e)^{5/2}}+\frac{1}{5} x^5 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)",1,"(Sqrt[d + e*x^2]*(8*d^2 - 4*d*e*x^2 + 3*e^2*x^4))/(75*(-e)^(5/2)) + (x^5*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/5","A",1
13,1,60,74,0.0914835,"\int x^2 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right) \, dx","Integrate[x^2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]","\frac{1}{9} \left(\frac{\left(2 d-e x^2\right) \sqrt{d+e x^2}}{(-e)^{3/2}}+3 x^3 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)\right)","-\frac{\left(d+e x^2\right)^{3/2}}{9 (-e)^{3/2}}+\frac{d \sqrt{d+e x^2}}{3 (-e)^{3/2}}+\frac{1}{3} x^3 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)",1,"(((2*d - e*x^2)*Sqrt[d + e*x^2])/(-e)^(3/2) + 3*x^3*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/9","A",1
14,1,43,43,0.0236577,"\int \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right) \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]","\frac{\sqrt{d+e x^2}}{\sqrt{-e}}+x \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)","\frac{\sqrt{d+e x^2}}{\sqrt{-e}}+x \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)",1,"Sqrt[d + e*x^2]/Sqrt[-e] + x*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]","A",1
15,1,86,59,0.0655952,"\int \frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x^2} \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^2,x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x}+\frac{i \sqrt{e} \log \left(-\frac{2 \sqrt{-e} \sqrt{d+e x^2}}{e x}+\frac{2 i \sqrt{d}}{\sqrt{e} x}\right)}{\sqrt{d}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x}-\frac{\sqrt{-e} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{\sqrt{d}}",1,"-(ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x) + (I*Sqrt[e]*Log[((2*I)*Sqrt[d])/(Sqrt[e]*x) - (2*Sqrt[-e]*Sqrt[d + e*x^2])/(e*x)])/Sqrt[d]","C",1
16,1,101,91,0.1057097,"\int \frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x^4} \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^4,x]","\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{-e}}{\sqrt{e} \sqrt{d+e x^2}}\right)}{6 d^{3/2}}-\frac{\sqrt{-e} \sqrt{d+e x^2}}{6 d x^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{3 x^3}","-\frac{(-e)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{6 d^{3/2}}-\frac{\sqrt{-e} \sqrt{d+e x^2}}{6 d x^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{3 x^3}",1,"-1/6*(Sqrt[-e]*Sqrt[d + e*x^2])/(d*x^2) + (e^(3/2)*ArcTan[(Sqrt[d]*Sqrt[-e])/(Sqrt[e]*Sqrt[d + e*x^2])])/(6*d^(3/2)) - ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/(3*x^3)","A",1
17,1,114,119,0.1307212,"\int \frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x^6} \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^6,x]","-\frac{3 e^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{-e}}{\sqrt{e} \sqrt{d+e x^2}}\right)}{40 d^{5/2}}+\sqrt{-e} \left(\frac{3 e}{40 d^2 x^2}-\frac{1}{20 d x^4}\right) \sqrt{d+e x^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{5 x^5}","-\frac{3 (-e)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{40 d^{5/2}}-\frac{3 (-e)^{3/2} \sqrt{d+e x^2}}{40 d^2 x^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{5 x^5}-\frac{\sqrt{-e} \sqrt{d+e x^2}}{20 d x^4}",1,"Sqrt[-e]*(-1/20*1/(d*x^4) + (3*e)/(40*d^2*x^2))*Sqrt[d + e*x^2] - (3*e^(5/2)*ArcTan[(Sqrt[d]*Sqrt[-e])/(Sqrt[e]*Sqrt[d + e*x^2])])/(40*d^(5/2)) - ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/(5*x^5)","A",1
18,1,170,211,0.6248908,"\int x^{9/2} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right) \, dx","Integrate[x^(9/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]","-\frac{60 i d^3 x \sqrt{\frac{d}{e x^2}+1} F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{d}}{\sqrt{e}}}}{\sqrt{x}}\right)\right|-1\right)}{847 (-e)^{5/2} \sqrt{\frac{i \sqrt{d}}{\sqrt{e}}} \sqrt{d+e x^2}}+\frac{4 \sqrt{x} \sqrt{d+e x^2} \left(15 d^2-9 d e x^2+7 e^2 x^4\right)}{847 (-e)^{5/2}}+\frac{2}{11} x^{11/2} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)","\frac{30 d^{11/4} \sqrt{-e} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{847 e^{13/4} \sqrt{d+e x^2}}+\frac{60 d^2 \sqrt{x} \sqrt{d+e x^2}}{847 (-e)^{5/2}}+\frac{4 x^{9/2} \sqrt{d+e x^2}}{121 \sqrt{-e}}+\frac{36 d x^{5/2} \sqrt{d+e x^2}}{847 (-e)^{3/2}}+\frac{2}{11} x^{11/2} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)",1,"(4*Sqrt[x]*Sqrt[d + e*x^2]*(15*d^2 - 9*d*e*x^2 + 7*e^2*x^4))/(847*(-e)^(5/2)) + (2*x^(11/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/11 - (((60*I)/847)*d^3*Sqrt[1 + d/(e*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d])/Sqrt[e]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[d])/Sqrt[e]]*(-e)^(5/2)*Sqrt[d + e*x^2])","C",1
19,1,158,181,0.4627236,"\int x^{5/2} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right) \, dx","Integrate[x^(5/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]","\frac{2}{147} \sqrt{x} \left(\frac{2 \left(5 d-3 e x^2\right) \sqrt{d+e x^2}}{(-e)^{3/2}}+21 x^3 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)\right)-\frac{20 i d^2 x \sqrt{\frac{d}{e x^2}+1} F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{d}}{\sqrt{e}}}}{\sqrt{x}}\right)\right|-1\right)}{147 (-e)^{3/2} \sqrt{\frac{i \sqrt{d}}{\sqrt{e}}} \sqrt{d+e x^2}}","-\frac{10 d^{7/4} \sqrt{-e} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{147 e^{9/4} \sqrt{d+e x^2}}+\frac{20 d \sqrt{x} \sqrt{d+e x^2}}{147 (-e)^{3/2}}+\frac{4 x^{5/2} \sqrt{d+e x^2}}{49 \sqrt{-e}}+\frac{2}{7} x^{7/2} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)",1,"(2*Sqrt[x]*((2*(5*d - 3*e*x^2)*Sqrt[d + e*x^2])/(-e)^(3/2) + 21*x^3*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]))/147 - (((20*I)/147)*d^2*Sqrt[1 + d/(e*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d])/Sqrt[e]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[d])/Sqrt[e]]*(-e)^(3/2)*Sqrt[d + e*x^2])","C",1
20,1,147,153,0.3096906,"\int \sqrt{x} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right) \, dx","Integrate[Sqrt[x]*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]","\frac{4 \sqrt{x} \sqrt{d+e x^2}}{9 \sqrt{-e}}-\frac{4 i d x \sqrt{\frac{d}{e x^2}+1} F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{d}}{\sqrt{e}}}}{\sqrt{x}}\right)\right|-1\right)}{9 \sqrt{-e} \sqrt{\frac{i \sqrt{d}}{\sqrt{e}}} \sqrt{d+e x^2}}+\frac{2}{3} x^{3/2} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)","\frac{2 d^{3/4} \sqrt{-e} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{9 e^{5/4} \sqrt{d+e x^2}}+\frac{4 \sqrt{x} \sqrt{d+e x^2}}{9 \sqrt{-e}}+\frac{2}{3} x^{3/2} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)",1,"(4*Sqrt[x]*Sqrt[d + e*x^2])/(9*Sqrt[-e]) + (2*x^(3/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/3 - (((4*I)/9)*d*Sqrt[1 + d/(e*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d])/Sqrt[e]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[d])/Sqrt[e]]*Sqrt[-e]*Sqrt[d + e*x^2])","C",1
21,1,115,122,0.1298686,"\int \frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x^{3/2}} \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(3/2),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{\sqrt{x}}+\frac{4 i \sqrt{-e} x \sqrt{\frac{d}{e x^2}+1} F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{d}}{\sqrt{e}}}}{\sqrt{x}}\right)\right|-1\right)}{\sqrt{\frac{i \sqrt{d}}{\sqrt{e}}} \sqrt{d+e x^2}}","\frac{2 \sqrt{-e} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{\sqrt[4]{d} \sqrt[4]{e} \sqrt{d+e x^2}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{\sqrt{x}}",1,"(-2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/Sqrt[x] + ((4*I)*Sqrt[-e]*Sqrt[1 + d/(e*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d])/Sqrt[e]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[d])/Sqrt[e]]*Sqrt[d + e*x^2])","C",1
22,1,150,156,0.2852983,"\int \frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x^{7/2}} \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(7/2),x]","-\frac{2 \left(2 \sqrt{-e} x \sqrt{d+e x^2}+3 d \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)\right)}{15 d x^{5/2}}+\frac{4 i (-e)^{3/2} x \sqrt{\frac{d}{e x^2}+1} F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{d}}{\sqrt{e}}}}{\sqrt{x}}\right)\right|-1\right)}{15 d \sqrt{\frac{i \sqrt{d}}{\sqrt{e}}} \sqrt{d+e x^2}}","-\frac{2 \sqrt{-e} e^{3/4} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{15 d^{5/4} \sqrt{d+e x^2}}-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{15 d x^{3/2}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{5 x^{5/2}}",1,"(-2*(2*Sqrt[-e]*x*Sqrt[d + e*x^2] + 3*d*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]))/(15*d*x^(5/2)) + (((4*I)/15)*(-e)^(3/2)*Sqrt[1 + d/(e*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d])/Sqrt[e]]/Sqrt[x]], -1])/(d*Sqrt[(I*Sqrt[d])/Sqrt[e]]*Sqrt[d + e*x^2])","C",1
23,1,162,186,0.3424064,"\int \frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x^{11/2}} \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(11/2),x]","\frac{4 \sqrt{-e} x \sqrt{d+e x^2} \left(5 e x^2-3 d\right)-42 d^2 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{189 d^2 x^{9/2}}+\frac{20 i (-e)^{5/2} x \sqrt{\frac{d}{e x^2}+1} F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{d}}{\sqrt{e}}}}{\sqrt{x}}\right)\right|-1\right)}{189 d^2 \sqrt{\frac{i \sqrt{d}}{\sqrt{e}}} \sqrt{d+e x^2}}","\frac{10 \sqrt{-e} e^{7/4} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{189 d^{9/4} \sqrt{d+e x^2}}-\frac{20 (-e)^{3/2} \sqrt{d+e x^2}}{189 d^2 x^{3/2}}-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{63 d x^{7/2}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{9 x^{9/2}}",1,"(4*Sqrt[-e]*x*Sqrt[d + e*x^2]*(-3*d + 5*e*x^2) - 42*d^2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(189*d^2*x^(9/2)) + (((20*I)/189)*(-e)^(5/2)*Sqrt[1 + d/(e*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d])/Sqrt[e]]/Sqrt[x]], -1])/(d^2*Sqrt[(I*Sqrt[d])/Sqrt[e]]*Sqrt[d + e*x^2])","C",1
24,1,171,216,0.6146928,"\int \frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x^{15/2}} \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(15/2),x]","\frac{2 \left(\frac{30 i (-e)^{7/2} x^{15/2} \sqrt{\frac{d}{e x^2}+1} F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{d}}{\sqrt{e}}}}{\sqrt{x}}\right)\right|-1\right)}{d^3 \sqrt{\frac{i \sqrt{d}}{\sqrt{e}}} \sqrt{d+e x^2}}-\frac{2 \sqrt{-e} \sqrt{d+e x^2} \left(7 d^2 x-9 d e x^3+15 e^2 x^5\right)}{d^3}-77 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)\right)}{1001 x^{13/2}}","-\frac{30 \sqrt{-e} e^{11/4} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{1001 d^{13/4} \sqrt{d+e x^2}}-\frac{60 (-e)^{5/2} \sqrt{d+e x^2}}{1001 d^3 x^{3/2}}-\frac{36 (-e)^{3/2} \sqrt{d+e x^2}}{1001 d^2 x^{7/2}}-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{143 d x^{11/2}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{13 x^{13/2}}",1,"(2*((-2*Sqrt[-e]*Sqrt[d + e*x^2]*(7*d^2*x - 9*d*e*x^3 + 15*e^2*x^5))/d^3 - 77*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]] + ((30*I)*(-e)^(7/2)*Sqrt[1 + d/(e*x^2)]*x^(15/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d])/Sqrt[e]]/Sqrt[x]], -1])/(d^3*Sqrt[(I*Sqrt[d])/Sqrt[e]]*Sqrt[d + e*x^2])))/(1001*x^(13/2))","C",1
25,1,139,326,0.1595764,"\int x^{7/2} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right) \, dx","Integrate[x^(7/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]","\frac{2 x^{3/2} \left(2 \sqrt{-e} \left(7 d^2+2 d e x^2-5 e^2 x^4\right)-14 d^2 \sqrt{-e} \sqrt{\frac{e x^2}{d}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{e x^2}{d}\right)+45 e^2 x^3 \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)\right)}{405 e^2 \sqrt{d+e x^2}}","-\frac{14 d^{9/4} \sqrt{-e} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{135 e^{11/4} \sqrt{d+e x^2}}+\frac{28 d^{9/4} \sqrt{-e} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{135 e^{11/4} \sqrt{d+e x^2}}-\frac{28 d^2 \sqrt{-e} \sqrt{x} \sqrt{d+e x^2}}{135 e^{5/2} \left(\sqrt{d}+\sqrt{e} x\right)}+\frac{4 x^{7/2} \sqrt{d+e x^2}}{81 \sqrt{-e}}+\frac{28 d x^{3/2} \sqrt{d+e x^2}}{405 (-e)^{3/2}}+\frac{2}{9} x^{9/2} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)",1,"(2*x^(3/2)*(2*Sqrt[-e]*(7*d^2 + 2*d*e*x^2 - 5*e^2*x^4) + 45*e^2*x^3*Sqrt[d + e*x^2]*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]] - 14*d^2*Sqrt[-e]*Sqrt[1 + (e*x^2)/d]*Hypergeometric2F1[1/2, 3/4, 7/4, -((e*x^2)/d)]))/(405*e^2*Sqrt[d + e*x^2])","C",1
26,1,119,296,0.1225644,"\int x^{3/2} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right) \, dx","Integrate[x^(3/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]","\frac{2 x^{3/2} \left(2 d \sqrt{-e} \sqrt{\frac{e x^2}{d}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{e x^2}{d}\right)-2 \sqrt{-e} \left(d+e x^2\right)+5 e x \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)\right)}{25 e \sqrt{d+e x^2}}","\frac{6 d^{5/4} \sqrt{-e} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{25 e^{7/4} \sqrt{d+e x^2}}-\frac{12 d^{5/4} \sqrt{-e} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{25 e^{7/4} \sqrt{d+e x^2}}+\frac{12 d \sqrt{-e} \sqrt{x} \sqrt{d+e x^2}}{25 e^{3/2} \left(\sqrt{d}+\sqrt{e} x\right)}+\frac{4 x^{3/2} \sqrt{d+e x^2}}{25 \sqrt{-e}}+\frac{2}{5} x^{5/2} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)",1,"(2*x^(3/2)*(-2*Sqrt[-e]*(d + e*x^2) + 5*e*x*Sqrt[d + e*x^2]*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]] + 2*d*Sqrt[-e]*Sqrt[1 + (e*x^2)/d]*Hypergeometric2F1[1/2, 3/4, 7/4, -((e*x^2)/d)]))/(25*e*Sqrt[d + e*x^2])","C",1
27,1,89,260,0.1186212,"\int \frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{\sqrt{x}} \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/Sqrt[x],x]","2 \sqrt{x} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)-\frac{4 \sqrt{-e} x^{3/2} \sqrt{\frac{e x^2}{d}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{e x^2}{d}\right)}{3 \sqrt{d+e x^2}}","-\frac{2 \sqrt[4]{d} \sqrt{-e} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{e^{3/4} \sqrt{d+e x^2}}+\frac{4 \sqrt[4]{d} \sqrt{-e} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{e^{3/4} \sqrt{d+e x^2}}-\frac{4 \sqrt{-e} \sqrt{x} \sqrt{d+e x^2}}{\sqrt{e} \left(\sqrt{d}+\sqrt{e} x\right)}+2 \sqrt{x} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)",1,"2*Sqrt[x]*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]] - (4*Sqrt[-e]*x^(3/2)*Sqrt[1 + (e*x^2)/d]*Hypergeometric2F1[1/2, 3/4, 7/4, -((e*x^2)/d)])/(3*Sqrt[d + e*x^2])","C",1
28,1,121,298,0.1536809,"\int \frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x^{5/2}} \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(5/2),x]","-\frac{2 \left(2 (-e)^{3/2} x^3 \sqrt{\frac{e x^2}{d}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{e x^2}{d}\right)+6 \sqrt{-e} x \left(d+e x^2\right)+3 d \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)\right)}{9 d x^{3/2} \sqrt{d+e x^2}}","\frac{2 \sqrt{-e} \sqrt[4]{e} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{3 d^{3/4} \sqrt{d+e x^2}}-\frac{4 \sqrt{-e} \sqrt[4]{e} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{3 d^{3/4} \sqrt{d+e x^2}}+\frac{4 \sqrt{-e^2} \sqrt{x} \sqrt{d+e x^2}}{3 d \left(\sqrt{d}+\sqrt{e} x\right)}-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{3 d \sqrt{x}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{3 x^{3/2}}",1,"(-2*(6*Sqrt[-e]*x*(d + e*x^2) + 3*d*Sqrt[d + e*x^2]*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]] + 2*(-e)^(3/2)*x^3*Sqrt[1 + (e*x^2)/d]*Hypergeometric2F1[1/2, 3/4, 7/4, -((e*x^2)/d)]))/(9*d*x^(3/2)*Sqrt[d + e*x^2])","C",1
29,1,137,331,0.1355997,"\int \frac{\tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{x^{9/2}} \, dx","Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(9/2),x]","\frac{4 \sqrt{-e} x \left(-d^2+2 d e x^2+3 e^2 x^4\right)-10 d^2 \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)-4 (-e)^{5/2} x^5 \sqrt{\frac{e x^2}{d}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{e x^2}{d}\right)}{35 d^2 x^{7/2} \sqrt{d+e x^2}}","-\frac{6 e^{5/4} \sqrt{-e} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{35 d^{7/4} \sqrt{d+e x^2}}+\frac{12 e^{5/4} \sqrt{-e} \left(\sqrt{d}+\sqrt{e} x\right) \sqrt{\frac{d+e x^2}{\left(\sqrt{d}+\sqrt{e} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{e} \sqrt{x}}{\sqrt[4]{d}}\right)|\frac{1}{2}\right)}{35 d^{7/4} \sqrt{d+e x^2}}-\frac{12 e^{3/2} \sqrt{-e} \sqrt{x} \sqrt{d+e x^2}}{35 d^2 \left(\sqrt{d}+\sqrt{e} x\right)}-\frac{12 (-e)^{3/2} \sqrt{d+e x^2}}{35 d^2 \sqrt{x}}-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{35 d x^{5/2}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right)}{7 x^{7/2}}",1,"(4*Sqrt[-e]*x*(-d^2 + 2*d*e*x^2 + 3*e^2*x^4) - 10*d^2*Sqrt[d + e*x^2]*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]] - 4*(-e)^(5/2)*x^5*Sqrt[1 + (e*x^2)/d]*Hypergeometric2F1[1/2, 3/4, 7/4, -((e*x^2)/d)])/(35*d^2*x^(7/2)*Sqrt[d + e*x^2])","C",1
30,1,50,50,0.0174761,"\int \frac{\tan ^{-1}\left(1+x+x^2\right)}{x^2} \, dx","Integrate[ArcTan[1 + x + x^2]/x^2,x]","-\frac{1}{2} \log \left(x^2+1\right)+\frac{1}{4} \log \left(x^2+2 x+2\right)-\frac{\tan ^{-1}\left(x^2+x+1\right)}{x}+\frac{\log (x)}{2}+\frac{1}{2} \tan ^{-1}(x+1)","-\frac{1}{2} \log \left(x^2+1\right)+\frac{1}{4} \log \left(x^2+2 x+2\right)-\frac{\tan ^{-1}\left(x^2+x+1\right)}{x}+\frac{\log (x)}{2}+\frac{1}{2} \tan ^{-1}(x+1)",1,"ArcTan[1 + x]/2 - ArcTan[1 + x + x^2]/x + Log[x]/2 - Log[1 + x^2]/2 + Log[2 + 2*x + x^2]/4","A",1
31,0,0,43,0.1012109,"\int \frac{\left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^n}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2),x]","\int \frac{\left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^n}{1-c^2 x^2} \, dx","\text{Int}\left(\frac{\left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^n}{1-c^2 x^2},x\right)",0,"Integrate[(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x]","A",-1
32,1,530,431,0.2049597,"\int \frac{\left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^3}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2),x]","-\frac{6 b^2 \text{Li}_3\left(-\frac{\sqrt{1-c x}+i \sqrt{c x+1}}{\sqrt{1-c x}-i \sqrt{c x+1}}\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)-6 b^2 \text{Li}_3\left(\frac{\sqrt{1-c x}+i \sqrt{c x+1}}{\sqrt{1-c x}-i \sqrt{c x+1}}\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)+6 i b \text{Li}_2\left(-\frac{\sqrt{1-c x}+i \sqrt{c x+1}}{\sqrt{1-c x}-i \sqrt{c x+1}}\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2-6 i b \text{Li}_2\left(\frac{\sqrt{1-c x}+i \sqrt{c x+1}}{\sqrt{1-c x}-i \sqrt{c x+1}}\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2+8 \tanh ^{-1}\left(1-\frac{2 i}{-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^3-3 i b^3 \text{Li}_4\left(-\frac{\sqrt{1-c x}+i \sqrt{c x+1}}{\sqrt{1-c x}-i \sqrt{c x+1}}\right)+3 i b^3 \text{Li}_4\left(\frac{\sqrt{1-c x}+i \sqrt{c x+1}}{\sqrt{1-c x}-i \sqrt{c x+1}}\right)}{4 c}","\frac{3 b^2 \text{Li}_3\left(1-\frac{2}{\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}+1}\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{2 c}-\frac{3 b^2 \text{Li}_3\left(\frac{2}{\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}+1}-1\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{2 c}+\frac{3 i b \text{Li}_2\left(1-\frac{2}{\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}+1}\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{2 c}-\frac{3 i b \text{Li}_2\left(\frac{2}{\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}+1}-1\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{2 c}-\frac{2 \tanh ^{-1}\left(1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^3}{c}-\frac{3 i b^3 \text{Li}_4\left(1-\frac{2}{\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}+1}\right)}{4 c}+\frac{3 i b^3 \text{Li}_4\left(\frac{2}{\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}+1}-1\right)}{4 c}",1,"-1/4*(8*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*ArcTanh[1 - (2*I)/(I - Sqrt[1 - c*x]/Sqrt[1 + c*x])] + (6*I)*b*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, -((Sqrt[1 - c*x] + I*Sqrt[1 + c*x])/(Sqrt[1 - c*x] - I*Sqrt[1 + c*x]))] - (6*I)*b*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, (Sqrt[1 - c*x] + I*Sqrt[1 + c*x])/(Sqrt[1 - c*x] - I*Sqrt[1 + c*x])] + 6*b^2*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, -((Sqrt[1 - c*x] + I*Sqrt[1 + c*x])/(Sqrt[1 - c*x] - I*Sqrt[1 + c*x]))] - 6*b^2*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, (Sqrt[1 - c*x] + I*Sqrt[1 + c*x])/(Sqrt[1 - c*x] - I*Sqrt[1 + c*x])] - (3*I)*b^3*PolyLog[4, -((Sqrt[1 - c*x] + I*Sqrt[1 + c*x])/(Sqrt[1 - c*x] - I*Sqrt[1 + c*x]))] + (3*I)*b^3*PolyLog[4, (Sqrt[1 - c*x] + I*Sqrt[1 + c*x])/(Sqrt[1 - c*x] - I*Sqrt[1 + c*x])])/c","A",1
33,1,354,283,0.1345536,"\int \frac{\left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2),x]","-\frac{2 i b \text{Li}_2\left(-\frac{\sqrt{1-c x}+i \sqrt{c x+1}}{\sqrt{1-c x}-i \sqrt{c x+1}}\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)-2 i b \text{Li}_2\left(\frac{\sqrt{1-c x}+i \sqrt{c x+1}}{\sqrt{1-c x}-i \sqrt{c x+1}}\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)+4 \tanh ^{-1}\left(1-\frac{2 i}{-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2+b^2 \text{Li}_3\left(-\frac{\sqrt{1-c x}+i \sqrt{c x+1}}{\sqrt{1-c x}-i \sqrt{c x+1}}\right)-b^2 \text{Li}_3\left(\frac{\sqrt{1-c x}+i \sqrt{c x+1}}{\sqrt{1-c x}-i \sqrt{c x+1}}\right)}{2 c}","\frac{i b \text{Li}_2\left(1-\frac{2}{\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}+1}\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}-\frac{i b \text{Li}_2\left(\frac{2}{\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}+1}-1\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}-\frac{2 \tanh ^{-1}\left(1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{c}+\frac{b^2 \text{Li}_3\left(1-\frac{2}{\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}+1}\right)}{2 c}-\frac{b^2 \text{Li}_3\left(\frac{2}{\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}+1}-1\right)}{2 c}",1,"-1/2*(4*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*ArcTanh[1 - (2*I)/(I - Sqrt[1 - c*x]/Sqrt[1 + c*x])] + (2*I)*b*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, -((Sqrt[1 - c*x] + I*Sqrt[1 + c*x])/(Sqrt[1 - c*x] - I*Sqrt[1 + c*x]))] - (2*I)*b*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, (Sqrt[1 - c*x] + I*Sqrt[1 + c*x])/(Sqrt[1 - c*x] - I*Sqrt[1 + c*x])] + b^2*PolyLog[3, -((Sqrt[1 - c*x] + I*Sqrt[1 + c*x])/(Sqrt[1 - c*x] - I*Sqrt[1 + c*x]))] - b^2*PolyLog[3, (Sqrt[1 - c*x] + I*Sqrt[1 + c*x])/(Sqrt[1 - c*x] - I*Sqrt[1 + c*x])])/c","A",1
34,1,93,98,0.0336551,"\int \frac{a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(1 - c^2*x^2),x]","-\frac{a \log \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)+\frac{1}{2} i b \text{Li}_2\left(-\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}\right)-\frac{1}{2} i b \text{Li}_2\left(\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}\right)}{c}","-\frac{a \log \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}{c}-\frac{i b \text{Li}_2\left(-\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}\right)}{2 c}+\frac{i b \text{Li}_2\left(\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}\right)}{2 c}",1,"-((a*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]] + (I/2)*b*PolyLog[2, ((-I)*Sqrt[1 - c*x])/Sqrt[1 + c*x]] - (I/2)*b*PolyLog[2, (I*Sqrt[1 - c*x])/Sqrt[1 + c*x]])/c)","A",1
35,0,0,43,0.1013319,"\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)} \, dx","Integrate[1/((1 - c^2*x^2)*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])),x]","\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)} \, dx","\text{Int}\left(\frac{1}{\left(1-c^2 x^2\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)},x\right)",0,"Integrate[1/((1 - c^2*x^2)*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), x]","A",-1
36,0,0,43,0.8900636,"\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2} \, dx","Integrate[1/((1 - c^2*x^2)*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2),x]","\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(1-c^2 x^2\right) \left(a+b \tan ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2},x\right)",0,"Integrate[1/((1 - c^2*x^2)*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]","A",-1
37,1,34,37,0.0642814,"\int x^m \tan ^{-1}(\tan (a+b x)) \, dx","Integrate[x^m*ArcTan[Tan[a + b*x]],x]","x^m \left(\frac{x \left(\tan ^{-1}(\tan (a+b x))-b x\right)}{m+1}+\frac{b x^2}{m+2}\right)","\frac{x^{m+1} \tan ^{-1}(\tan (a+b x))}{m+1}-\frac{b x^{m+2}}{m^2+3 m+2}",1,"x^m*((b*x^2)/(2 + m) + (x*(-(b*x) + ArcTan[Tan[a + b*x]]))/(1 + m))","A",1
38,1,20,23,0.0173969,"\int x^2 \tan ^{-1}(\tan (a+b x)) \, dx","Integrate[x^2*ArcTan[Tan[a + b*x]],x]","-\frac{1}{12} x^3 \left(b x-4 \tan ^{-1}(\tan (a+b x))\right)","\frac{1}{3} x^3 \tan ^{-1}(\tan (a+b x))-\frac{b x^4}{12}",1,"-1/12*(x^3*(b*x - 4*ArcTan[Tan[a + b*x]]))","A",1
39,1,20,23,0.0169694,"\int x \tan ^{-1}(\tan (a+b x)) \, dx","Integrate[x*ArcTan[Tan[a + b*x]],x]","-\frac{1}{6} x^2 \left(b x-3 \tan ^{-1}(\tan (a+b x))\right)","\frac{1}{2} x^2 \tan ^{-1}(\tan (a+b x))-\frac{b x^3}{6}",1,"-1/6*(x^2*(b*x - 3*ArcTan[Tan[a + b*x]]))","A",1
40,1,18,16,0.0090724,"\int \tan ^{-1}(\tan (a+b x)) \, dx","Integrate[ArcTan[Tan[a + b*x]],x]","x \tan ^{-1}(\tan (a+b x))-\frac{b x^2}{2}","\frac{\tan ^{-1}(\tan (a+b x))^2}{2 b}",1,"-1/2*(b*x^2) + x*ArcTan[Tan[a + b*x]]","A",1
41,1,19,21,0.0173958,"\int \frac{\tan ^{-1}(\tan (a+b x))}{x} \, dx","Integrate[ArcTan[Tan[a + b*x]]/x,x]","\log (x) \left(\tan ^{-1}(\tan (a+b x))-b x\right)+b x","b x-\log (x) \left(b x-\tan ^{-1}(\tan (a+b x))\right)",1,"b*x + (-(b*x) + ArcTan[Tan[a + b*x]])*Log[x]","A",1
42,1,31,36,0.0540728,"\int x^m \tan ^{-1}(\cot (a+b x)) \, dx","Integrate[x^m*ArcTan[Cot[a + b*x]],x]","\frac{x^{m+1} \left((m+2) \tan ^{-1}(\cot (a+b x))+b x\right)}{(m+1) (m+2)}","\frac{x^{m+1} \tan ^{-1}(\cot (a+b x))}{m+1}+\frac{b x^{m+2}}{m^2+3 m+2}",1,"(x^(1 + m)*(b*x + (2 + m)*ArcTan[Cot[a + b*x]]))/((1 + m)*(2 + m))","A",1
43,1,20,23,0.0177644,"\int x^2 \tan ^{-1}(\cot (a+b x)) \, dx","Integrate[x^2*ArcTan[Cot[a + b*x]],x]","\frac{1}{12} x^3 \left(4 \tan ^{-1}(\cot (a+b x))+b x\right)","\frac{1}{3} x^3 \tan ^{-1}(\cot (a+b x))+\frac{b x^4}{12}",1,"(x^3*(b*x + 4*ArcTan[Cot[a + b*x]]))/12","A",1
44,1,20,23,0.0162698,"\int x \tan ^{-1}(\cot (a+b x)) \, dx","Integrate[x*ArcTan[Cot[a + b*x]],x]","\frac{1}{6} x^2 \left(3 \tan ^{-1}(\cot (a+b x))+b x\right)","\frac{1}{2} x^2 \tan ^{-1}(\cot (a+b x))+\frac{b x^3}{6}",1,"(x^2*(b*x + 3*ArcTan[Cot[a + b*x]]))/6","A",1
45,1,18,16,0.0088831,"\int \tan ^{-1}(\cot (a+b x)) \, dx","Integrate[ArcTan[Cot[a + b*x]],x]","x \tan ^{-1}(\cot (a+b x))+\frac{b x^2}{2}","-\frac{\tan ^{-1}(\cot (a+b x))^2}{2 b}",1,"(b*x^2)/2 + x*ArcTan[Cot[a + b*x]]","A",1
46,1,19,19,0.0190269,"\int \frac{\tan ^{-1}(\cot (a+b x))}{x} \, dx","Integrate[ArcTan[Cot[a + b*x]]/x,x]","\log (x) \left(\tan ^{-1}(\cot (a+b x))+b x\right)-b x","\log (x) \left(\tan ^{-1}(\cot (a+b x))+b x\right)-b x",1,"-(b*x) + (b*x + ArcTan[Cot[a + b*x]])*Log[x]","A",1
47,1,18,16,0.0006124,"\int \tan ^{-1}(\tan (a+b x)) \, dx","Integrate[ArcTan[Tan[a + b*x]],x]","x \tan ^{-1}(\tan (a+b x))-\frac{b x^2}{2}","\frac{\tan ^{-1}(\tan (a+b x))^2}{2 b}",1,"-1/2*(b*x^2) + x*ArcTan[Tan[a + b*x]]","A",1
48,1,363,403,1.0777762,"\int x^2 \tan ^{-1}(c+d \tan (a+b x)) \, dx","Integrate[x^2*ArcTan[c + d*Tan[a + b*x]],x]","\frac{1}{3} x^3 \tan ^{-1}(d \tan (a+b x)+c)+\frac{4 i b^3 x^3 \log \left(1+\frac{(c-i (d+1)) e^{2 i (a+b x)}}{c+i (d-1)}\right)-4 i b^3 x^3 \log \left(1+\frac{(c-i d+i) e^{2 i (a+b x)}}{c+i (d+1)}\right)+6 b^2 x^2 \text{Li}_2\left(-\frac{(c-i (d+1)) e^{2 i (a+b x)}}{c+i (d-1)}\right)-6 b^2 x^2 \text{Li}_2\left(-\frac{(c-i d+i) e^{2 i (a+b x)}}{c+i (d+1)}\right)+6 i b x \text{Li}_3\left(-\frac{(c-i (d+1)) e^{2 i (a+b x)}}{c+i (d-1)}\right)-6 i b x \text{Li}_3\left(-\frac{(c-i d+i) e^{2 i (a+b x)}}{c+i (d+1)}\right)-3 \text{Li}_4\left(-\frac{(c-i (d+1)) e^{2 i (a+b x)}}{c+i (d-1)}\right)+3 \text{Li}_4\left(-\frac{(c-i d+i) e^{2 i (a+b x)}}{c+i (d+1)}\right)}{24 b^3}","-\frac{\text{Li}_4\left(-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{8 b^3}+\frac{\text{Li}_4\left(-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{8 b^3}+\frac{i x \text{Li}_3\left(-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{4 b^2}-\frac{i x \text{Li}_3\left(-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{4 b^2}+\frac{x^2 \text{Li}_2\left(-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{4 b}-\frac{x^2 \text{Li}_2\left(-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{4 b}+\frac{1}{6} i x^3 \log \left(1+\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)-\frac{1}{6} i x^3 \log \left(1+\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)+\frac{1}{3} x^3 \tan ^{-1}(d \tan (a+b x)+c)",1,"(x^3*ArcTan[c + d*Tan[a + b*x]])/3 + ((4*I)*b^3*x^3*Log[1 + ((c - I*(1 + d))*E^((2*I)*(a + b*x)))/(c + I*(-1 + d))] - (4*I)*b^3*x^3*Log[1 + ((I + c - I*d)*E^((2*I)*(a + b*x)))/(c + I*(1 + d))] + 6*b^2*x^2*PolyLog[2, -(((c - I*(1 + d))*E^((2*I)*(a + b*x)))/(c + I*(-1 + d)))] - 6*b^2*x^2*PolyLog[2, -(((I + c - I*d)*E^((2*I)*(a + b*x)))/(c + I*(1 + d)))] + (6*I)*b*x*PolyLog[3, -(((c - I*(1 + d))*E^((2*I)*(a + b*x)))/(c + I*(-1 + d)))] - (6*I)*b*x*PolyLog[3, -(((I + c - I*d)*E^((2*I)*(a + b*x)))/(c + I*(1 + d)))] - 3*PolyLog[4, -(((c - I*(1 + d))*E^((2*I)*(a + b*x)))/(c + I*(-1 + d)))] + 3*PolyLog[4, -(((I + c - I*d)*E^((2*I)*(a + b*x)))/(c + I*(1 + d)))])/(24*b^3)","A",1
49,1,272,305,0.6976528,"\int x \tan ^{-1}(c+d \tan (a+b x)) \, dx","Integrate[x*ArcTan[c + d*Tan[a + b*x]],x]","\frac{1}{2} x^2 \tan ^{-1}(d \tan (a+b x)+c)+\frac{i \left(2 b^2 x^2 \log \left(1+\frac{(c-i (d+1)) e^{2 i (a+b x)}}{c+i (d-1)}\right)-2 b^2 x^2 \log \left(1+\frac{(c-i d+i) e^{2 i (a+b x)}}{c+i (d+1)}\right)-2 i b x \text{Li}_2\left(-\frac{(c-i (d+1)) e^{2 i (a+b x)}}{c+i (d-1)}\right)+2 i b x \text{Li}_2\left(-\frac{(c-i d+i) e^{2 i (a+b x)}}{c+i (d+1)}\right)+\text{Li}_3\left(-\frac{(c-i (d+1)) e^{2 i (a+b x)}}{c+i (d-1)}\right)-\text{Li}_3\left(-\frac{(c-i d+i) e^{2 i (a+b x)}}{c+i (d+1)}\right)\right)}{8 b^2}","\frac{i \text{Li}_3\left(-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{8 b^2}-\frac{i \text{Li}_3\left(-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{8 b^2}+\frac{x \text{Li}_2\left(-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{4 b}-\frac{x \text{Li}_2\left(-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{4 b}+\frac{1}{4} i x^2 \log \left(1+\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)-\frac{1}{4} i x^2 \log \left(1+\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)+\frac{1}{2} x^2 \tan ^{-1}(d \tan (a+b x)+c)",1,"(x^2*ArcTan[c + d*Tan[a + b*x]])/2 + ((I/8)*(2*b^2*x^2*Log[1 + ((c - I*(1 + d))*E^((2*I)*(a + b*x)))/(c + I*(-1 + d))] - 2*b^2*x^2*Log[1 + ((I + c - I*d)*E^((2*I)*(a + b*x)))/(c + I*(1 + d))] - (2*I)*b*x*PolyLog[2, -(((c - I*(1 + d))*E^((2*I)*(a + b*x)))/(c + I*(-1 + d)))] + (2*I)*b*x*PolyLog[2, -(((I + c - I*d)*E^((2*I)*(a + b*x)))/(c + I*(1 + d)))] + PolyLog[3, -(((c - I*(1 + d))*E^((2*I)*(a + b*x)))/(c + I*(-1 + d)))] - PolyLog[3, -(((I + c - I*d)*E^((2*I)*(a + b*x)))/(c + I*(1 + d)))]))/b^2","A",1
50,1,555,198,7.9221356,"\int \tan ^{-1}(c+d \tan (a+b x)) \, dx","Integrate[ArcTan[c + d*Tan[a + b*x]],x]","x \tan ^{-1}(d \tan (a+b x)+c)+\frac{x \left(-i \sqrt{-d^2} \left(\text{Li}_2\left(\frac{d^2 (1-i \tan (a+b x))}{d^2+i c d-i \sqrt{-d^2}}\right)+\log (1-i \tan (a+b x)) \log \left(\frac{d^2 (-\tan (a+b x))-c d+\sqrt{-d^2}}{-c d+i d^2+\sqrt{-d^2}}\right)\right)+i \sqrt{-d^2} \left(\text{Li}_2\left(\frac{d^2 (1-i \tan (a+b x))}{d^2+i c d+i \sqrt{-d^2}}\right)+\log (1-i \tan (a+b x)) \log \left(\frac{d^2 \tan (a+b x)+c d+\sqrt{-d^2}}{c d-i d^2+\sqrt{-d^2}}\right)\right)+i \sqrt{-d^2} \left(\text{Li}_2\left(\frac{d^2 (i \tan (a+b x)+1)}{d^2-i c d+i \sqrt{-d^2}}\right)+\log (1+i \tan (a+b x)) \log \left(\frac{d^2 \tan (a+b x)+c d-\sqrt{-d^2}}{c d+i d^2-\sqrt{-d^2}}\right)\right)-i \sqrt{-d^2} \left(\text{Li}_2\left(\frac{d^2 (i \tan (a+b x)+1)}{d^2-i \left(c d+\sqrt{-d^2}\right)}\right)+\log (1+i \tan (a+b x)) \log \left(\frac{d^2 \tan (a+b x)+c d+\sqrt{-d^2}}{c d+i d^2+\sqrt{-d^2}}\right)\right)-4 a d \tan ^{-1}(d \tan (a+b x)+c)\right)}{2 d (-i \log (1-i \tan (a+b x))+i \log (1+i \tan (a+b x))+2 a)}","\frac{\text{Li}_2\left(-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{4 b}-\frac{\text{Li}_2\left(-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{4 b}+\frac{1}{2} i x \log \left(1+\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)-\frac{1}{2} i x \log \left(1+\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)+x \tan ^{-1}(d \tan (a+b x)+c)",1,"x*ArcTan[c + d*Tan[a + b*x]] + (x*(-4*a*d*ArcTan[c + d*Tan[a + b*x]] - I*Sqrt[-d^2]*(Log[1 - I*Tan[a + b*x]]*Log[(-(c*d) + Sqrt[-d^2] - d^2*Tan[a + b*x])/(-(c*d) + I*d^2 + Sqrt[-d^2])] + PolyLog[2, (d^2*(1 - I*Tan[a + b*x]))/(I*c*d + d^2 - I*Sqrt[-d^2])]) + I*Sqrt[-d^2]*(Log[1 - I*Tan[a + b*x]]*Log[(c*d + Sqrt[-d^2] + d^2*Tan[a + b*x])/(c*d - I*d^2 + Sqrt[-d^2])] + PolyLog[2, (d^2*(1 - I*Tan[a + b*x]))/(I*c*d + d^2 + I*Sqrt[-d^2])]) + I*Sqrt[-d^2]*(Log[1 + I*Tan[a + b*x]]*Log[(c*d - Sqrt[-d^2] + d^2*Tan[a + b*x])/(c*d + I*d^2 - Sqrt[-d^2])] + PolyLog[2, (d^2*(1 + I*Tan[a + b*x]))/((-I)*c*d + d^2 + I*Sqrt[-d^2])]) - I*Sqrt[-d^2]*(Log[1 + I*Tan[a + b*x]]*Log[(c*d + Sqrt[-d^2] + d^2*Tan[a + b*x])/(c*d + I*d^2 + Sqrt[-d^2])] + PolyLog[2, (d^2*(1 + I*Tan[a + b*x]))/(d^2 - I*(c*d + Sqrt[-d^2]))])))/(2*d*(2*a - I*Log[1 - I*Tan[a + b*x]] + I*Log[1 + I*Tan[a + b*x]]))","B",0
51,0,0,18,5.24243,"\int \frac{\tan ^{-1}(c+d \tan (a+b x))}{x} \, dx","Integrate[ArcTan[c + d*Tan[a + b*x]]/x,x]","\int \frac{\tan ^{-1}(c+d \tan (a+b x))}{x} \, dx","\text{Int}\left(\frac{\tan ^{-1}(d \tan (a+b x)+c)}{x},x\right)",0,"Integrate[ArcTan[c + d*Tan[a + b*x]]/x, x]","A",-1
52,1,140,154,0.4846678,"\int x^2 \tan ^{-1}(c+(1+i c) \tan (a+b x)) \, dx","Integrate[x^2*ArcTan[c + (1 + I*c)*Tan[a + b*x]],x]","\frac{1}{3} x^3 \tan ^{-1}(c+(1+i c) \tan (a+b x))-\frac{4 i b^3 x^3 \log \left(1+\frac{i e^{-2 i (a+b x)}}{c}\right)-6 b^2 x^2 \text{Li}_2\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)+6 i b x \text{Li}_3\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)+3 \text{Li}_4\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)}{24 b^3}","\frac{\text{Li}_4\left(i c e^{2 i a+2 i b x}\right)}{8 b^3}-\frac{i x \text{Li}_3\left(i c e^{2 i a+2 i b x}\right)}{4 b^2}-\frac{x^2 \text{Li}_2\left(i c e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{6} i x^3 \log \left(1-i c e^{2 i a+2 i b x}\right)+\frac{1}{3} x^3 \tan ^{-1}(c+(1+i c) \tan (a+b x))-\frac{b x^4}{12}",1,"(x^3*ArcTan[c + (1 + I*c)*Tan[a + b*x]])/3 - ((4*I)*b^3*x^3*Log[1 + I/(c*E^((2*I)*(a + b*x)))] - 6*b^2*x^2*PolyLog[2, (-I)/(c*E^((2*I)*(a + b*x)))] + (6*I)*b*x*PolyLog[3, (-I)/(c*E^((2*I)*(a + b*x)))] + 3*PolyLog[4, (-I)/(c*E^((2*I)*(a + b*x)))])/(24*b^3)","A",1
53,1,110,123,0.3578947,"\int x \tan ^{-1}(c+(1+i c) \tan (a+b x)) \, dx","Integrate[x*ArcTan[c + (1 + I*c)*Tan[a + b*x]],x]","\frac{1}{2} x^2 \tan ^{-1}(c+(1+i c) \tan (a+b x))-\frac{i \left(2 b^2 x^2 \log \left(1+\frac{i e^{-2 i (a+b x)}}{c}\right)+2 i b x \text{Li}_2\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)+\text{Li}_3\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)\right)}{8 b^2}","-\frac{i \text{Li}_3\left(i c e^{2 i a+2 i b x}\right)}{8 b^2}-\frac{x \text{Li}_2\left(i c e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{4} i x^2 \log \left(1-i c e^{2 i a+2 i b x}\right)+\frac{1}{2} x^2 \tan ^{-1}(c+(1+i c) \tan (a+b x))-\frac{b x^3}{6}",1,"(x^2*ArcTan[c + (1 + I*c)*Tan[a + b*x]])/2 - ((I/8)*(2*b^2*x^2*Log[1 + I/(c*E^((2*I)*(a + b*x)))] + (2*I)*b*x*PolyLog[2, (-I)/(c*E^((2*I)*(a + b*x)))] + PolyLog[3, (-I)/(c*E^((2*I)*(a + b*x)))]))/b^2","A",1
54,1,967,85,18.7515421,"\int \tan ^{-1}(c+(1+i c) \tan (a+b x)) \, dx","Integrate[ArcTan[c + (1 + I*c)*Tan[a + b*x]],x]","\frac{i x \left(2 i b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))-\log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c+i) \cos (a+b x)+(i c+1) \sin (a+b x))}{2 c}\right) \log (1-i \tan (b x))+\log \left(\frac{\sec (b x) ((1-i c) \cos (a+b x)+(c-i) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\right) \log (i \tan (b x)+1)-\text{Li}_2(i \sin (2 b x)-\cos (2 b x))-\text{Li}_2\left(\frac{\sec (b x) ((c-i) \cos (a)+i (c+i) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)+\text{Li}_2\left(\frac{1}{2} \sec (b x) ((i c+1) \cos (a)-(c+i) \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right)\right) (\cos (b x)+i \sin (b x)) (i \cos (b x)+\sin (b x)) ((1-i c) \cos (a+b x)+(c-i) \sin (a+b x)) \sec ^2(a+b x)}{((c+i) \cos (a+b x)+(i c+1) \sin (a+b x)) \left(\log (i \tan (b x)+1) \tan (b x) \cos ^2(a)+2 b x-i \log \left(1-\frac{\sec (b x) ((c-i) \cos (a)+i (c+i) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)-i \log \left(\frac{1}{2} \sec (b x) ((-i c-1) \cos (a)+(c+i) \sin (a)) (\cos (a+b x)+i \sin (a+b x))+1\right)+\log (i \tan (b x)+1) \sin ^2(a) \tan (b x)-2 i b x \tan (b x)+\log \left(1-\frac{\sec (b x) ((c-i) \cos (a)+i (c+i) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right) \tan (b x)-\log \left(\frac{1}{2} \sec (b x) ((-i c-1) \cos (a)+(c+i) \sin (a)) (\cos (a+b x)+i \sin (a+b x))+1\right) \tan (b x)-\log (1-i \tan (b x)) \tan (b x)-\frac{i (c-i) \cos (a+b x) (\log (1-i \tan (b x))-\log (i \tan (b x)+1))}{(c+i) \cos (a+b x)+(i c+1) \sin (a+b x)}+\frac{(c+i) (\log (1-i \tan (b x))-\log (i \tan (b x)+1)) \sin (a+b x)}{(c+i) \cos (a+b x)+(i c+1) \sin (a+b x)}+\frac{\log \left(\frac{\sec (b x) ((1-i c) \cos (a+b x)+(c-i) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\right) \sec ^2(b x)}{\tan (b x)-i}-\frac{\log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c+i) \cos (a+b x)+(i c+1) \sin (a+b x))}{2 c}\right) \sec ^2(b x)}{\tan (b x)+i}\right) (\tan (a+b x)-i) (-i c+(c-i) \tan (a+b x)+1)}+x \tan ^{-1}(c+(i c+1) \tan (a+b x))","-\frac{\text{Li}_2\left(i c e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{2} i x \log \left(1-i c e^{2 i a+2 i b x}\right)+x \tan ^{-1}(c+(1+i c) \tan (a+b x))-\frac{b x^2}{2}",1,"x*ArcTan[c + (1 + I*c)*Tan[a + b*x]] + (I*x*((2*I)*b*x*Log[2*Cos[b*x]*(Cos[b*x] - I*Sin[b*x])] - Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/(2*c)]*Log[1 - I*Tan[b*x]] + Log[(Sec[b*x]*((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]))/(2*Cos[a] - (2*I)*Sin[a])]*Log[1 + I*Tan[b*x]] - PolyLog[2, -Cos[2*b*x] + I*Sin[2*b*x]] - PolyLog[2, (Sec[b*x]*((-I + c)*Cos[a] + I*(I + c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] + PolyLog[2, (Sec[b*x]*((1 + I*c)*Cos[a] - (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2])*Sec[a + b*x]^2*(Cos[b*x] + I*Sin[b*x])*(I*Cos[b*x] + Sin[b*x])*((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]))/(((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x])*(2*b*x - I*Log[1 - (Sec[b*x]*((-I + c)*Cos[a] + I*(I + c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] - I*Log[1 + (Sec[b*x]*((-1 - I*c)*Cos[a] + (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2] - (I*(-I + c)*Cos[a + b*x]*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]]))/((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]) + ((I + c)*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]])*Sin[a + b*x])/((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]) - (2*I)*b*x*Tan[b*x] + Log[1 - (Sec[b*x]*((-I + c)*Cos[a] + I*(I + c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)]*Tan[b*x] - Log[1 + (Sec[b*x]*((-1 - I*c)*Cos[a] + (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2]*Tan[b*x] - Log[1 - I*Tan[b*x]]*Tan[b*x] + Cos[a]^2*Log[1 + I*Tan[b*x]]*Tan[b*x] + Log[1 + I*Tan[b*x]]*Sin[a]^2*Tan[b*x] + (Log[(Sec[b*x]*((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]))/(2*Cos[a] - (2*I)*Sin[a])]*Sec[b*x]^2)/(-I + Tan[b*x]) - (Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/(2*c)]*Sec[b*x]^2)/(I + Tan[b*x]))*(-I + Tan[a + b*x])*(1 - I*c + (-I + c)*Tan[a + b*x]))","B",0
55,0,0,24,0.7351189,"\int \frac{\tan ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx","Integrate[ArcTan[c + (1 + I*c)*Tan[a + b*x]]/x,x]","\int \frac{\tan ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx","\text{Int}\left(\frac{\tan ^{-1}(c+(1+i c) \tan (a+b x))}{x},x\right)",0,"Integrate[ArcTan[c + (1 + I*c)*Tan[a + b*x]]/x, x]","A",-1
56,1,137,155,0.512663,"\int x^2 \tan ^{-1}(c+(-1+i c) \tan (a+b x)) \, dx","Integrate[x^2*ArcTan[c + (-1 + I*c)*Tan[a + b*x]],x]","\frac{1}{24} \left(\frac{3 \text{Li}_4\left(\frac{i e^{-2 i (a+b x)}}{c}\right)}{b^3}+\frac{6 i x \text{Li}_3\left(\frac{i e^{-2 i (a+b x)}}{c}\right)}{b^2}-\frac{6 x^2 \text{Li}_2\left(\frac{i e^{-2 i (a+b x)}}{c}\right)}{b}+4 i x^3 \log \left(1-\frac{i e^{-2 i (a+b x)}}{c}\right)+8 x^3 \tan ^{-1}(c+i (c+i) \tan (a+b x))\right)","-\frac{\text{Li}_4\left(-i c e^{2 i a+2 i b x}\right)}{8 b^3}+\frac{i x \text{Li}_3\left(-i c e^{2 i a+2 i b x}\right)}{4 b^2}+\frac{x^2 \text{Li}_2\left(-i c e^{2 i a+2 i b x}\right)}{4 b}+\frac{1}{6} i x^3 \log \left(1+i c e^{2 i a+2 i b x}\right)+\frac{1}{3} x^3 \tan ^{-1}(c-(1-i c) \tan (a+b x))+\frac{b x^4}{12}",1,"(8*x^3*ArcTan[c + I*(I + c)*Tan[a + b*x]] + (4*I)*x^3*Log[1 - I/(c*E^((2*I)*(a + b*x)))] - (6*x^2*PolyLog[2, I/(c*E^((2*I)*(a + b*x)))])/b + ((6*I)*x*PolyLog[3, I/(c*E^((2*I)*(a + b*x)))])/b^2 + (3*PolyLog[4, I/(c*E^((2*I)*(a + b*x)))])/b^3)/24","A",1
57,1,111,124,0.3909147,"\int x \tan ^{-1}(c+(-1+i c) \tan (a+b x)) \, dx","Integrate[x*ArcTan[c + (-1 + I*c)*Tan[a + b*x]],x]","\frac{i \left(2 b^2 x^2 \log \left(1-\frac{i e^{-2 i (a+b x)}}{c}\right)+2 i b x \text{Li}_2\left(\frac{i e^{-2 i (a+b x)}}{c}\right)+\text{Li}_3\left(\frac{i e^{-2 i (a+b x)}}{c}\right)\right)}{8 b^2}+\frac{1}{2} x^2 \tan ^{-1}(c+i (c+i) \tan (a+b x))","\frac{i \text{Li}_3\left(-i c e^{2 i a+2 i b x}\right)}{8 b^2}+\frac{x \text{Li}_2\left(-i c e^{2 i a+2 i b x}\right)}{4 b}+\frac{1}{4} i x^2 \log \left(1+i c e^{2 i a+2 i b x}\right)+\frac{1}{2} x^2 \tan ^{-1}(c-(1-i c) \tan (a+b x))+\frac{b x^3}{6}",1,"(x^2*ArcTan[c + I*(I + c)*Tan[a + b*x]])/2 + ((I/8)*(2*b^2*x^2*Log[1 - I/(c*E^((2*I)*(a + b*x)))] + (2*I)*b*x*PolyLog[2, I/(c*E^((2*I)*(a + b*x)))] + PolyLog[3, I/(c*E^((2*I)*(a + b*x)))]))/b^2","A",1
58,1,847,86,17.5826238,"\int \tan ^{-1}(c+(-1+i c) \tan (a+b x)) \, dx","Integrate[ArcTan[c + (-1 + I*c)*Tan[a + b*x]],x]","x \tan ^{-1}(c+i (c+i) \tan (a+b x))+\frac{i x \left(-2 i b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))+\log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c-i) \cos (a+b x)+i (c+i) \sin (a+b x))}{2 c}\right) \log (1-i \tan (b x))-\log \left(\frac{1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((i c+1) \cos (a+b x)-(c+i) \sin (a+b x))\right) \log (i \tan (b x)+1)+\text{Li}_2(i \sin (2 b x)-\cos (2 b x))+\text{Li}_2\left(\frac{\sec (b x) ((c+i) \cos (a)+(i c+1) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)-\text{Li}_2\left(\frac{1}{2} (\cos (a)+i \sin (a)) ((c+i) \cos (a)+(i c+1) \sin (a)) (\tan (b x)-i)\right)\right) \sec (a+b x) (\cos (b x)+i \sin (b x)) (i \cos (b x)+\sin (b x))}{((c-i) \cos (a+b x)+i (c+i) \sin (a+b x)) \left(-\frac{\log \left(\frac{1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((i c+1) \cos (a+b x)-(c+i) \sin (a+b x))\right) \sec ^2(b x)}{\tan (b x)-i}+\frac{\log \left(1-\frac{1}{2} (\cos (a)+i \sin (a)) ((c+i) \cos (a)+(i c+1) \sin (a)) (\tan (b x)-i)\right) \sec ^2(b x)}{\tan (b x)-i}+\frac{\log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c-i) \cos (a+b x)+i (c+i) \sin (a+b x))}{2 c}\right) \sec ^2(b x)}{\tan (b x)+i}-2 b x+i \log \left(1-\frac{\sec (b x) ((c+i) \cos (a)+(i c+1) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)+2 i b x \tan (b x)-\log \left(1-\frac{\sec (b x) ((c+i) \cos (a)+(i c+1) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right) \tan (b x)+\log (1-i \tan (b x)) \tan (b x)-\log (i \tan (b x)+1) \tan (b x)+\frac{i (c+i) \cos (a+b x) (\log (1-i \tan (b x))-\log (i \tan (b x)+1))}{(c-i) \cos (a+b x)+i (c+i) \sin (a+b x)}+\frac{(i c+1) (\log (1-i \tan (b x))-\log (i \tan (b x)+1)) \sin (a+b x)}{(-i c-1) \cos (a+b x)+(c+i) \sin (a+b x)}\right) (\tan (a+b x)-i)}","\frac{\text{Li}_2\left(-i c e^{2 i a+2 i b x}\right)}{4 b}+\frac{1}{2} i x \log \left(1+i c e^{2 i a+2 i b x}\right)+x \tan ^{-1}(c-(1-i c) \tan (a+b x))+\frac{b x^2}{2}",1,"x*ArcTan[c + I*(I + c)*Tan[a + b*x]] + (I*x*((-2*I)*b*x*Log[2*Cos[b*x]*(Cos[b*x] - I*Sin[b*x])] + Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]))/(2*c)]*Log[1 - I*Tan[b*x]] - Log[(Sec[b*x]*(Cos[a] + I*Sin[a])*((1 + I*c)*Cos[a + b*x] - (I + c)*Sin[a + b*x]))/2]*Log[1 + I*Tan[b*x]] + PolyLog[2, -Cos[2*b*x] + I*Sin[2*b*x]] + PolyLog[2, (Sec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] - PolyLog[2, ((Cos[a] + I*Sin[a])*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(-I + Tan[b*x]))/2])*Sec[a + b*x]*(Cos[b*x] + I*Sin[b*x])*(I*Cos[b*x] + Sin[b*x]))/(((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x])*(-2*b*x + I*Log[1 - (Sec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] + (I*(I + c)*Cos[a + b*x]*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]]))/((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]) + ((1 + I*c)*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]])*Sin[a + b*x])/((-1 - I*c)*Cos[a + b*x] + (I + c)*Sin[a + b*x]) + (2*I)*b*x*Tan[b*x] - Log[1 - (Sec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)]*Tan[b*x] + Log[1 - I*Tan[b*x]]*Tan[b*x] - Log[1 + I*Tan[b*x]]*Tan[b*x] - (Log[(Sec[b*x]*(Cos[a] + I*Sin[a])*((1 + I*c)*Cos[a + b*x] - (I + c)*Sin[a + b*x]))/2]*Sec[b*x]^2)/(-I + Tan[b*x]) + (Log[1 - ((Cos[a] + I*Sin[a])*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(-I + Tan[b*x]))/2]*Sec[b*x]^2)/(-I + Tan[b*x]) + (Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]))/(2*c)]*Sec[b*x]^2)/(I + Tan[b*x]))*(-I + Tan[a + b*x]))","B",0
59,0,0,24,1.136031,"\int \frac{\tan ^{-1}(c+(-1+i c) \tan (a+b x))}{x} \, dx","Integrate[ArcTan[c + (-1 + I*c)*Tan[a + b*x]]/x,x]","\int \frac{\tan ^{-1}(c+(-1+i c) \tan (a+b x))}{x} \, dx","\text{Int}\left(\frac{\tan ^{-1}(c+(-1+i c) \tan (a+b x))}{x},x\right)",0,"Integrate[ArcTan[c + (-1 + I*c)*Tan[a + b*x]]/x, x]","A",-1
60,1,18,16,0.0093771,"\int \tan ^{-1}(\cot (a+b x)) \, dx","Integrate[ArcTan[Cot[a + b*x]],x]","x \tan ^{-1}(\cot (a+b x))+\frac{b x^2}{2}","-\frac{\tan ^{-1}(\cot (a+b x))^2}{2 b}",1,"(b*x^2)/2 + x*ArcTan[Cot[a + b*x]]","A",1
61,1,359,399,1.0056904,"\int x^2 \tan ^{-1}(c+d \cot (a+b x)) \, dx","Integrate[x^2*ArcTan[c + d*Cot[a + b*x]],x]","\frac{1}{3} x^3 \tan ^{-1}(d \cot (a+b x)+c)+\frac{4 i b^3 x^3 \log \left(1-\frac{(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right)-4 i b^3 x^3 \log \left(1-\frac{(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right)+6 b^2 x^2 \text{Li}_2\left(\frac{(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right)-6 b^2 x^2 \text{Li}_2\left(\frac{(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right)+6 i b x \text{Li}_3\left(\frac{(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right)-6 i b x \text{Li}_3\left(\frac{(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right)-3 \text{Li}_4\left(\frac{(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right)+3 \text{Li}_4\left(\frac{(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right)}{24 b^3}","-\frac{\text{Li}_4\left(\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{8 b^3}+\frac{\text{Li}_4\left(\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{8 b^3}+\frac{i x \text{Li}_3\left(\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{4 b^2}-\frac{i x \text{Li}_3\left(\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{4 b^2}+\frac{x^2 \text{Li}_2\left(\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{4 b}-\frac{x^2 \text{Li}_2\left(\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{4 b}+\frac{1}{6} i x^3 \log \left(1-\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)-\frac{1}{6} i x^3 \log \left(1-\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)+\frac{1}{3} x^3 \tan ^{-1}(d \cot (a+b x)+c)",1,"(x^3*ArcTan[c + d*Cot[a + b*x]])/3 + ((4*I)*b^3*x^3*Log[1 - ((c + I*(-1 + d))*E^((2*I)*(a + b*x)))/(c - I*(1 + d))] - (4*I)*b^3*x^3*Log[1 - ((c + I*(1 + d))*E^((2*I)*(a + b*x)))/(I + c - I*d)] + 6*b^2*x^2*PolyLog[2, ((c + I*(-1 + d))*E^((2*I)*(a + b*x)))/(c - I*(1 + d))] - 6*b^2*x^2*PolyLog[2, ((c + I*(1 + d))*E^((2*I)*(a + b*x)))/(I + c - I*d)] + (6*I)*b*x*PolyLog[3, ((c + I*(-1 + d))*E^((2*I)*(a + b*x)))/(c - I*(1 + d))] - (6*I)*b*x*PolyLog[3, ((c + I*(1 + d))*E^((2*I)*(a + b*x)))/(I + c - I*d)] - 3*PolyLog[4, ((c + I*(-1 + d))*E^((2*I)*(a + b*x)))/(c - I*(1 + d))] + 3*PolyLog[4, ((c + I*(1 + d))*E^((2*I)*(a + b*x)))/(I + c - I*d)])/(24*b^3)","A",1
62,1,270,303,0.6279203,"\int x \tan ^{-1}(c+d \cot (a+b x)) \, dx","Integrate[x*ArcTan[c + d*Cot[a + b*x]],x]","\frac{1}{2} x^2 \tan ^{-1}(d \cot (a+b x)+c)+\frac{i \left(2 b^2 x^2 \log \left(1-\frac{(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right)-2 b^2 x^2 \log \left(1-\frac{(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right)-2 i b x \text{Li}_2\left(\frac{(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right)+2 i b x \text{Li}_2\left(\frac{(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right)+\text{Li}_3\left(\frac{(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right)-\text{Li}_3\left(\frac{(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right)\right)}{8 b^2}","\frac{i \text{Li}_3\left(\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{8 b^2}-\frac{i \text{Li}_3\left(\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{8 b^2}+\frac{x \text{Li}_2\left(\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{4 b}-\frac{x \text{Li}_2\left(\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{4 b}+\frac{1}{4} i x^2 \log \left(1-\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)-\frac{1}{4} i x^2 \log \left(1-\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)+\frac{1}{2} x^2 \tan ^{-1}(d \cot (a+b x)+c)",1,"(x^2*ArcTan[c + d*Cot[a + b*x]])/2 + ((I/8)*(2*b^2*x^2*Log[1 - ((c + I*(-1 + d))*E^((2*I)*(a + b*x)))/(c - I*(1 + d))] - 2*b^2*x^2*Log[1 - ((c + I*(1 + d))*E^((2*I)*(a + b*x)))/(I + c - I*d)] - (2*I)*b*x*PolyLog[2, ((c + I*(-1 + d))*E^((2*I)*(a + b*x)))/(c - I*(1 + d))] + (2*I)*b*x*PolyLog[2, ((c + I*(1 + d))*E^((2*I)*(a + b*x)))/(I + c - I*d)] + PolyLog[3, ((c + I*(-1 + d))*E^((2*I)*(a + b*x)))/(c - I*(1 + d))] - PolyLog[3, ((c + I*(1 + d))*E^((2*I)*(a + b*x)))/(I + c - I*d)]))/b^2","A",1
63,1,1648,198,22.19162,"\int \tan ^{-1}(c+d \cot (a+b x)) \, dx","Integrate[ArcTan[c + d*Cot[a + b*x]],x]","x \tan ^{-1}(c+d \cot (a+b x))+\frac{d \left(4 a \sqrt{-d^2} \tan ^{-1}\left(\frac{\tan (a+b x) c^2+d c+\tan (a+b x)}{d}\right)+i d \log (i \tan (a+b x)+1) \log \left(\frac{\tan (a+b x) c^2+d c+\tan (a+b x)-\sqrt{-d^2}}{i c^2+d c-\sqrt{-d^2}+i}\right)+i d \log (1-i \tan (a+b x)) \log \left(\frac{\tan (a+b x) c^2+d c+\tan (a+b x)+\sqrt{-d^2}}{-i c^2+d c+\sqrt{-d^2}-i}\right)-i d \log (i \tan (a+b x)+1) \log \left(\frac{\tan (a+b x) c^2+d c+\tan (a+b x)+\sqrt{-d^2}}{i c^2+d c+\sqrt{-d^2}+i}\right)-i d \log (1-i \tan (a+b x)) \log \left(\frac{-c d-\left(c^2+1\right) \tan (a+b x)+\sqrt{-d^2}}{i c^2-d c+\sqrt{-d^2}+i}\right)-i d \text{Li}_2\left(\frac{\left(c^2+1\right) (1-i \tan (a+b x))}{c^2+i d c-i \sqrt{-d^2}+1}\right)+i d \text{Li}_2\left(\frac{\left(c^2+1\right) (1-i \tan (a+b x))}{c^2+i d c+i \sqrt{-d^2}+1}\right)-i d \text{Li}_2\left(\frac{\left(c^2+1\right) (i \tan (a+b x)+1)}{c^2-i d c-i \sqrt{-d^2}+1}\right)+i d \text{Li}_2\left(\frac{\left(c^2+1\right) (i \tan (a+b x)+1)}{c^2-i d c+i \sqrt{-d^2}+1}\right)\right) \left(\frac{2 a}{b \left(\cos (2 (a+b x)) c^2-c^2-2 d \sin (2 (a+b x)) c-d^2-d^2 \cos (2 (a+b x))+\cos (2 (a+b x))-1\right)}-\frac{2 (a+b x)}{b \left(\cos (2 (a+b x)) c^2-c^2-2 d \sin (2 (a+b x)) c-d^2-d^2 \cos (2 (a+b x))+\cos (2 (a+b x))-1\right)}\right)}{\frac{d \log \left(1-\frac{\left(c^2+1\right) (1-i \tan (a+b x))}{c^2+i d c-i \sqrt{-d^2}+1}\right) \sec ^2(a+b x)}{1-i \tan (a+b x)}-\frac{d \log \left(1-\frac{\left(c^2+1\right) (1-i \tan (a+b x))}{c^2+i d c+i \sqrt{-d^2}+1}\right) \sec ^2(a+b x)}{1-i \tan (a+b x)}+\frac{d \log \left(\frac{\tan (a+b x) c^2+d c+\tan (a+b x)+\sqrt{-d^2}}{-i c^2+d c+\sqrt{-d^2}-i}\right) \sec ^2(a+b x)}{1-i \tan (a+b x)}-\frac{d \log \left(\frac{-c d-\left(c^2+1\right) \tan (a+b x)+\sqrt{-d^2}}{i c^2-d c+\sqrt{-d^2}+i}\right) \sec ^2(a+b x)}{1-i \tan (a+b x)}-\frac{d \log \left(1-\frac{\left(c^2+1\right) (i \tan (a+b x)+1)}{c^2-i d c-i \sqrt{-d^2}+1}\right) \sec ^2(a+b x)}{i \tan (a+b x)+1}+\frac{d \log \left(1-\frac{\left(c^2+1\right) (i \tan (a+b x)+1)}{c^2-i d c+i \sqrt{-d^2}+1}\right) \sec ^2(a+b x)}{i \tan (a+b x)+1}-\frac{d \log \left(\frac{\tan (a+b x) c^2+d c+\tan (a+b x)-\sqrt{-d^2}}{i c^2+d c-\sqrt{-d^2}+i}\right) \sec ^2(a+b x)}{i \tan (a+b x)+1}+\frac{d \log \left(\frac{\tan (a+b x) c^2+d c+\tan (a+b x)+\sqrt{-d^2}}{i c^2+d c+\sqrt{-d^2}+i}\right) \sec ^2(a+b x)}{i \tan (a+b x)+1}+\frac{i \left(c^2+1\right) d \log (1-i \tan (a+b x)) \sec ^2(a+b x)}{-c d-\left(c^2+1\right) \tan (a+b x)+\sqrt{-d^2}}+\frac{i d \log (i \tan (a+b x)+1) \left(c^2 \sec ^2(a+b x)+\sec ^2(a+b x)\right)}{\tan (a+b x) c^2+d c+\tan (a+b x)-\sqrt{-d^2}}+\frac{i d \log (1-i \tan (a+b x)) \left(c^2 \sec ^2(a+b x)+\sec ^2(a+b x)\right)}{\tan (a+b x) c^2+d c+\tan (a+b x)+\sqrt{-d^2}}-\frac{i d \log (i \tan (a+b x)+1) \left(c^2 \sec ^2(a+b x)+\sec ^2(a+b x)\right)}{\tan (a+b x) c^2+d c+\tan (a+b x)+\sqrt{-d^2}}+\frac{4 a \sqrt{-d^2} \left(c^2 \sec ^2(a+b x)+\sec ^2(a+b x)\right)}{d \left(\frac{\left(\tan (a+b x) c^2+d c+\tan (a+b x)\right)^2}{d^2}+1\right)}}","\frac{\text{Li}_2\left(\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{4 b}-\frac{\text{Li}_2\left(\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{4 b}+\frac{1}{2} i x \log \left(1-\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)-\frac{1}{2} i x \log \left(1-\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)+x \tan ^{-1}(d \cot (a+b x)+c)",1,"x*ArcTan[c + d*Cot[a + b*x]] + (d*(4*a*Sqrt[-d^2]*ArcTan[(c*d + Tan[a + b*x] + c^2*Tan[a + b*x])/d] + I*d*Log[1 + I*Tan[a + b*x]]*Log[(c*d - Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(I + I*c^2 + c*d - Sqrt[-d^2])] + I*d*Log[1 - I*Tan[a + b*x]]*Log[(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(-I - I*c^2 + c*d + Sqrt[-d^2])] - I*d*Log[1 + I*Tan[a + b*x]]*Log[(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(I + I*c^2 + c*d + Sqrt[-d^2])] - I*d*Log[1 - I*Tan[a + b*x]]*Log[(-(c*d) + Sqrt[-d^2] - (1 + c^2)*Tan[a + b*x])/(I + I*c^2 - c*d + Sqrt[-d^2])] - I*d*PolyLog[2, ((1 + c^2)*(1 - I*Tan[a + b*x]))/(1 + c^2 + I*c*d - I*Sqrt[-d^2])] + I*d*PolyLog[2, ((1 + c^2)*(1 - I*Tan[a + b*x]))/(1 + c^2 + I*c*d + I*Sqrt[-d^2])] - I*d*PolyLog[2, ((1 + c^2)*(1 + I*Tan[a + b*x]))/(1 + c^2 - I*c*d - I*Sqrt[-d^2])] + I*d*PolyLog[2, ((1 + c^2)*(1 + I*Tan[a + b*x]))/(1 + c^2 - I*c*d + I*Sqrt[-d^2])])*((2*a)/(b*(-1 - c^2 - d^2 + Cos[2*(a + b*x)] + c^2*Cos[2*(a + b*x)] - d^2*Cos[2*(a + b*x)] - 2*c*d*Sin[2*(a + b*x)])) - (2*(a + b*x))/(b*(-1 - c^2 - d^2 + Cos[2*(a + b*x)] + c^2*Cos[2*(a + b*x)] - d^2*Cos[2*(a + b*x)] - 2*c*d*Sin[2*(a + b*x)]))))/((d*Log[1 - ((1 + c^2)*(1 - I*Tan[a + b*x]))/(1 + c^2 + I*c*d - I*Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 - I*Tan[a + b*x]) - (d*Log[1 - ((1 + c^2)*(1 - I*Tan[a + b*x]))/(1 + c^2 + I*c*d + I*Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 - I*Tan[a + b*x]) + (d*Log[(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(-I - I*c^2 + c*d + Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 - I*Tan[a + b*x]) - (d*Log[(-(c*d) + Sqrt[-d^2] - (1 + c^2)*Tan[a + b*x])/(I + I*c^2 - c*d + Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 - I*Tan[a + b*x]) - (d*Log[1 - ((1 + c^2)*(1 + I*Tan[a + b*x]))/(1 + c^2 - I*c*d - I*Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 + I*Tan[a + b*x]) + (d*Log[1 - ((1 + c^2)*(1 + I*Tan[a + b*x]))/(1 + c^2 - I*c*d + I*Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 + I*Tan[a + b*x]) - (d*Log[(c*d - Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(I + I*c^2 + c*d - Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 + I*Tan[a + b*x]) + (d*Log[(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(I + I*c^2 + c*d + Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 + I*Tan[a + b*x]) + (I*d*Log[1 + I*Tan[a + b*x]]*(Sec[a + b*x]^2 + c^2*Sec[a + b*x]^2))/(c*d - Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x]) + (I*d*Log[1 - I*Tan[a + b*x]]*(Sec[a + b*x]^2 + c^2*Sec[a + b*x]^2))/(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x]) - (I*d*Log[1 + I*Tan[a + b*x]]*(Sec[a + b*x]^2 + c^2*Sec[a + b*x]^2))/(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x]) + (I*(1 + c^2)*d*Log[1 - I*Tan[a + b*x]]*Sec[a + b*x]^2)/(-(c*d) + Sqrt[-d^2] - (1 + c^2)*Tan[a + b*x]) + (4*a*Sqrt[-d^2]*(Sec[a + b*x]^2 + c^2*Sec[a + b*x]^2))/(d*(1 + (c*d + Tan[a + b*x] + c^2*Tan[a + b*x])^2/d^2)))","B",0
64,0,0,18,5.1294706,"\int \frac{\tan ^{-1}(c+d \cot (a+b x))}{x} \, dx","Integrate[ArcTan[c + d*Cot[a + b*x]]/x,x]","\int \frac{\tan ^{-1}(c+d \cot (a+b x))}{x} \, dx","\text{Int}\left(\frac{\tan ^{-1}(d \cot (a+b x)+c)}{x},x\right)",0,"Integrate[ArcTan[c + d*Cot[a + b*x]]/x, x]","A",-1
65,1,136,154,0.3833929,"\int x^2 \tan ^{-1}(c+(1-i c) \cot (a+b x)) \, dx","Integrate[x^2*ArcTan[c + (1 - I*c)*Cot[a + b*x]],x]","\frac{1}{24} \left(\frac{3 \text{Li}_4\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)}{b^3}+\frac{6 i x \text{Li}_3\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)}{b^2}-\frac{6 x^2 \text{Li}_2\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)}{b}+4 i x^3 \log \left(1+\frac{i e^{-2 i (a+b x)}}{c}\right)+8 x^3 \tan ^{-1}(c+(1-i c) \cot (a+b x))\right)","-\frac{\text{Li}_4\left(i c e^{2 i a+2 i b x}\right)}{8 b^3}+\frac{i x \text{Li}_3\left(i c e^{2 i a+2 i b x}\right)}{4 b^2}+\frac{x^2 \text{Li}_2\left(i c e^{2 i a+2 i b x}\right)}{4 b}+\frac{1}{6} i x^3 \log \left(1-i c e^{2 i a+2 i b x}\right)+\frac{1}{3} x^3 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{b x^4}{12}",1,"(8*x^3*ArcTan[c + (1 - I*c)*Cot[a + b*x]] + (4*I)*x^3*Log[1 + I/(c*E^((2*I)*(a + b*x)))] - (6*x^2*PolyLog[2, (-I)/(c*E^((2*I)*(a + b*x)))])/b + ((6*I)*x*PolyLog[3, (-I)/(c*E^((2*I)*(a + b*x)))])/b^2 + (3*PolyLog[4, (-I)/(c*E^((2*I)*(a + b*x)))])/b^3)/24","A",1
66,1,110,123,0.2335026,"\int x \tan ^{-1}(c+(1-i c) \cot (a+b x)) \, dx","Integrate[x*ArcTan[c + (1 - I*c)*Cot[a + b*x]],x]","\frac{i \left(2 b^2 x^2 \log \left(1+\frac{i e^{-2 i (a+b x)}}{c}\right)+2 i b x \text{Li}_2\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)+\text{Li}_3\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)\right)}{8 b^2}+\frac{1}{2} x^2 \tan ^{-1}(c+(1-i c) \cot (a+b x))","\frac{i \text{Li}_3\left(i c e^{2 i a+2 i b x}\right)}{8 b^2}+\frac{x \text{Li}_2\left(i c e^{2 i a+2 i b x}\right)}{4 b}+\frac{1}{4} i x^2 \log \left(1-i c e^{2 i a+2 i b x}\right)+\frac{1}{2} x^2 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{b x^3}{6}",1,"(x^2*ArcTan[c + (1 - I*c)*Cot[a + b*x]])/2 + ((I/8)*(2*b^2*x^2*Log[1 + I/(c*E^((2*I)*(a + b*x)))] + (2*I)*b*x*PolyLog[2, (-I)/(c*E^((2*I)*(a + b*x)))] + PolyLog[3, (-I)/(c*E^((2*I)*(a + b*x)))]))/b^2","A",1
67,1,929,85,20.2684492,"\int \tan ^{-1}(c+(1-i c) \cot (a+b x)) \, dx","Integrate[ArcTan[c + (1 - I*c)*Cot[a + b*x]],x]","x \tan ^{-1}(c+(1-i c) \cot (a+b x))-\frac{i x \csc ^2(a+b x) \left(2 b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))+i \log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c+i) \cos (a+b x)+(i c+1) \sin (a+b x))}{2 c}\right) \log (1-i \tan (b x))-i \log \left(\frac{\sec (b x) ((1-i c) \cos (a+b x)+(c-i) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\right) \log (i \tan (b x)+1)+i \text{Li}_2(i \sin (2 b x)-\cos (2 b x))+i \text{Li}_2\left(\frac{\sec (b x) ((c-i) \cos (a)+i (c+i) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)-i \text{Li}_2\left(\frac{1}{2} \sec (b x) ((i c+1) \cos (a)-(c+i) \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right)\right) (\cos (b x)-i \sin (b x)) (\cos (b x)+i \sin (b x))}{(\cot (a+b x)+i) (i c+(c+i) \cot (a+b x)+1) \left(i \log (i \tan (b x)+1) \tan (b x) \cos ^2(a)+2 i b x+\log \left(1-\frac{\sec (b x) ((c-i) \cos (a)+i (c+i) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)+\log \left(\frac{1}{2} \sec (b x) ((-i c-1) \cos (a)+(c+i) \sin (a)) (\cos (a+b x)+i \sin (a+b x))+1\right)+i \log (i \tan (b x)+1) \sin ^2(a) \tan (b x)+2 b x \tan (b x)+i \log \left(1-\frac{\sec (b x) ((c-i) \cos (a)+i (c+i) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right) \tan (b x)-i \log \left(\frac{1}{2} \sec (b x) ((-i c-1) \cos (a)+(c+i) \sin (a)) (\cos (a+b x)+i \sin (a+b x))+1\right) \tan (b x)-i \log (1-i \tan (b x)) \tan (b x)+\frac{(c-i) \cos (a+b x) (\log (1-i \tan (b x))-\log (i \tan (b x)+1))}{(c+i) \cos (a+b x)+(i c+1) \sin (a+b x)}+\frac{(c+i) (\log (1-i \tan (b x))-\log (i \tan (b x)+1)) \sin (a+b x)}{(1-i c) \cos (a+b x)+(c-i) \sin (a+b x)}+\frac{i \log \left(\frac{\sec (b x) ((1-i c) \cos (a+b x)+(c-i) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\right) \sec ^2(b x)}{\tan (b x)-i}-\frac{i \log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c+i) \cos (a+b x)+(i c+1) \sin (a+b x))}{2 c}\right) \sec ^2(b x)}{\tan (b x)+i}\right)}","\frac{\text{Li}_2\left(i c e^{2 i a+2 i b x}\right)}{4 b}+\frac{1}{2} i x \log \left(1-i c e^{2 i a+2 i b x}\right)+x \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{b x^2}{2}",1,"x*ArcTan[c + (1 - I*c)*Cot[a + b*x]] - (I*x*Csc[a + b*x]^2*(2*b*x*Log[2*Cos[b*x]*(Cos[b*x] - I*Sin[b*x])] + I*Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/(2*c)]*Log[1 - I*Tan[b*x]] - I*Log[(Sec[b*x]*((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]))/(2*Cos[a] - (2*I)*Sin[a])]*Log[1 + I*Tan[b*x]] + I*PolyLog[2, -Cos[2*b*x] + I*Sin[2*b*x]] + I*PolyLog[2, (Sec[b*x]*((-I + c)*Cos[a] + I*(I + c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] - I*PolyLog[2, (Sec[b*x]*((1 + I*c)*Cos[a] - (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2])*(Cos[b*x] - I*Sin[b*x])*(Cos[b*x] + I*Sin[b*x]))/((I + Cot[a + b*x])*(1 + I*c + (I + c)*Cot[a + b*x])*((2*I)*b*x + Log[1 - (Sec[b*x]*((-I + c)*Cos[a] + I*(I + c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] + Log[1 + (Sec[b*x]*((-1 - I*c)*Cos[a] + (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2] + ((-I + c)*Cos[a + b*x]*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]]))/((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]) + ((I + c)*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]])*Sin[a + b*x])/((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]) + 2*b*x*Tan[b*x] + I*Log[1 - (Sec[b*x]*((-I + c)*Cos[a] + I*(I + c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)]*Tan[b*x] - I*Log[1 + (Sec[b*x]*((-1 - I*c)*Cos[a] + (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2]*Tan[b*x] - I*Log[1 - I*Tan[b*x]]*Tan[b*x] + I*Cos[a]^2*Log[1 + I*Tan[b*x]]*Tan[b*x] + I*Log[1 + I*Tan[b*x]]*Sin[a]^2*Tan[b*x] + (I*Log[(Sec[b*x]*((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]))/(2*Cos[a] - (2*I)*Sin[a])]*Sec[b*x]^2)/(-I + Tan[b*x]) - (I*Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/(2*c)]*Sec[b*x]^2)/(I + Tan[b*x])))","B",0
68,0,0,24,0.7640078,"\int \frac{\tan ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx","Integrate[ArcTan[c + (1 - I*c)*Cot[a + b*x]]/x,x]","\int \frac{\tan ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx","\text{Int}\left(\frac{\tan ^{-1}(c+(1-i c) \cot (a+b x))}{x},x\right)",0,"Integrate[ArcTan[c + (1 - I*c)*Cot[a + b*x]]/x, x]","A",-1
69,1,140,155,0.381124,"\int x^2 \tan ^{-1}(c+(-1-i c) \cot (a+b x)) \, dx","Integrate[x^2*ArcTan[c + (-1 - I*c)*Cot[a + b*x]],x]","\frac{1}{3} x^3 \tan ^{-1}(c+(-1-i c) \cot (a+b x))-\frac{4 i b^3 x^3 \log \left(1-\frac{i e^{-2 i (a+b x)}}{c}\right)-6 b^2 x^2 \text{Li}_2\left(\frac{i e^{-2 i (a+b x)}}{c}\right)+6 i b x \text{Li}_3\left(\frac{i e^{-2 i (a+b x)}}{c}\right)+3 \text{Li}_4\left(\frac{i e^{-2 i (a+b x)}}{c}\right)}{24 b^3}","\frac{\text{Li}_4\left(-i c e^{2 i a+2 i b x}\right)}{8 b^3}-\frac{i x \text{Li}_3\left(-i c e^{2 i a+2 i b x}\right)}{4 b^2}-\frac{x^2 \text{Li}_2\left(-i c e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{6} i x^3 \log \left(1+i c e^{2 i a+2 i b x}\right)+\frac{1}{3} x^3 \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac{b x^4}{12}",1,"(x^3*ArcTan[c + (-1 - I*c)*Cot[a + b*x]])/3 - ((4*I)*b^3*x^3*Log[1 - I/(c*E^((2*I)*(a + b*x)))] - 6*b^2*x^2*PolyLog[2, I/(c*E^((2*I)*(a + b*x)))] + (6*I)*b*x*PolyLog[3, I/(c*E^((2*I)*(a + b*x)))] + 3*PolyLog[4, I/(c*E^((2*I)*(a + b*x)))])/(24*b^3)","A",1
70,1,110,124,0.2231159,"\int x \tan ^{-1}(c+(-1-i c) \cot (a+b x)) \, dx","Integrate[x*ArcTan[c + (-1 - I*c)*Cot[a + b*x]],x]","\frac{1}{2} x^2 \tan ^{-1}(c+(-1-i c) \cot (a+b x))-\frac{i \left(2 b^2 x^2 \log \left(1-\frac{i e^{-2 i (a+b x)}}{c}\right)+2 i b x \text{Li}_2\left(\frac{i e^{-2 i (a+b x)}}{c}\right)+\text{Li}_3\left(\frac{i e^{-2 i (a+b x)}}{c}\right)\right)}{8 b^2}","-\frac{i \text{Li}_3\left(-i c e^{2 i a+2 i b x}\right)}{8 b^2}-\frac{x \text{Li}_2\left(-i c e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{4} i x^2 \log \left(1+i c e^{2 i a+2 i b x}\right)+\frac{1}{2} x^2 \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac{b x^3}{6}",1,"(x^2*ArcTan[c + (-1 - I*c)*Cot[a + b*x]])/2 - ((I/8)*(2*b^2*x^2*Log[1 - I/(c*E^((2*I)*(a + b*x)))] + (2*I)*b*x*PolyLog[2, I/(c*E^((2*I)*(a + b*x)))] + PolyLog[3, I/(c*E^((2*I)*(a + b*x)))]))/b^2","A",1
71,1,872,86,15.6729902,"\int \tan ^{-1}(c+(-1-i c) \cot (a+b x)) \, dx","Integrate[ArcTan[c + (-1 - I*c)*Cot[a + b*x]],x]","x \tan ^{-1}(c+(-i c-1) \cot (a+b x))+\frac{i x \csc (a+b x) \left(2 b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))+i \log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c-i) \cos (a+b x)+i (c+i) \sin (a+b x))}{2 c}\right) \log (1-i \tan (b x))-i \log \left(\frac{1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((i c+1) \cos (a+b x)-(c+i) \sin (a+b x))\right) \log (i \tan (b x)+1)+i \text{Li}_2(i \sin (2 b x)-\cos (2 b x))+i \text{Li}_2\left(\frac{\sec (b x) ((c+i) \cos (a)+(i c+1) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)-i \text{Li}_2\left(\frac{1}{2} (\cos (a)+i \sin (a)) ((c+i) \cos (a)+(i c+1) \sin (a)) (\tan (b x)-i)\right)\right) (\cos (b x)-i \sin (b x)) (\cos (b x)+i \sin (b x))}{(\cot (a+b x)+i) ((c-i) \cos (a+b x)+i (c+i) \sin (a+b x)) \left(\frac{\log \left(\frac{1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((i c+1) \cos (a+b x)-(c+i) \sin (a+b x))\right) \sec ^2(b x)}{i \tan (b x)+1}+\frac{i \log \left(1-\frac{1}{2} (\cos (a)+i \sin (a)) ((c+i) \cos (a)+(i c+1) \sin (a)) (\tan (b x)-i)\right) \sec ^2(b x)}{\tan (b x)-i}+\frac{i \log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c-i) \cos (a+b x)+i (c+i) \sin (a+b x))}{2 c}\right) \sec ^2(b x)}{\tan (b x)+i}-2 i b x-\log \left(1-\frac{\sec (b x) ((c+i) \cos (a)+(i c+1) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)-2 b x \tan (b x)-i \log \left(1-\frac{\sec (b x) ((c+i) \cos (a)+(i c+1) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right) \tan (b x)+i \log (1-i \tan (b x)) \tan (b x)-i \log (i \tan (b x)+1) \tan (b x)-\frac{\log (1-i \tan (b x)) ((c+i) \cos (a+b x)+(i c+1) \sin (a+b x))}{(c-i) \cos (a+b x)+i (c+i) \sin (a+b x)}+\frac{\log (i \tan (b x)+1) ((c+i) \cos (a+b x)+(i c+1) \sin (a+b x))}{(c-i) \cos (a+b x)+i (c+i) \sin (a+b x)}\right)}","-\frac{\text{Li}_2\left(-i c e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{2} i x \log \left(1+i c e^{2 i a+2 i b x}\right)+x \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac{b x^2}{2}",1,"x*ArcTan[c + (-1 - I*c)*Cot[a + b*x]] + (I*x*Csc[a + b*x]*(2*b*x*Log[2*Cos[b*x]*(Cos[b*x] - I*Sin[b*x])] + I*Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]))/(2*c)]*Log[1 - I*Tan[b*x]] - I*Log[(Sec[b*x]*(Cos[a] + I*Sin[a])*((1 + I*c)*Cos[a + b*x] - (I + c)*Sin[a + b*x]))/2]*Log[1 + I*Tan[b*x]] + I*PolyLog[2, -Cos[2*b*x] + I*Sin[2*b*x]] + I*PolyLog[2, (Sec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] - I*PolyLog[2, ((Cos[a] + I*Sin[a])*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(-I + Tan[b*x]))/2])*(Cos[b*x] - I*Sin[b*x])*(Cos[b*x] + I*Sin[b*x]))/((I + Cot[a + b*x])*((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x])*((-2*I)*b*x - Log[1 - (Sec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] - (Log[1 - I*Tan[b*x]]*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]) + (Log[1 + I*Tan[b*x]]*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]) + (Log[(Sec[b*x]*(Cos[a] + I*Sin[a])*((1 + I*c)*Cos[a + b*x] - (I + c)*Sin[a + b*x]))/2]*Sec[b*x]^2)/(1 + I*Tan[b*x]) - 2*b*x*Tan[b*x] - I*Log[1 - (Sec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)]*Tan[b*x] + I*Log[1 - I*Tan[b*x]]*Tan[b*x] - I*Log[1 + I*Tan[b*x]]*Tan[b*x] + (I*Log[1 - ((Cos[a] + I*Sin[a])*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(-I + Tan[b*x]))/2]*Sec[b*x]^2)/(-I + Tan[b*x]) + (I*Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]))/(2*c)]*Sec[b*x]^2)/(I + Tan[b*x])))","B",0
72,0,0,24,0.7461973,"\int \frac{\tan ^{-1}(c+(-1-i c) \cot (a+b x))}{x} \, dx","Integrate[ArcTan[c + (-1 - I*c)*Cot[a + b*x]]/x,x]","\int \frac{\tan ^{-1}(c+(-1-i c) \cot (a+b x))}{x} \, dx","\text{Int}\left(\frac{\tan ^{-1}(c+(-1-i c) \cot (a+b x))}{x},x\right)",0,"Integrate[ArcTan[c + (-1 - I*c)*Cot[a + b*x]]/x, x]","A",-1
73,1,64,39,0.0312113,"\int \tan ^{-1}(\sinh (x)) \, dx","Integrate[ArcTan[Sinh[x]],x]","x \tan ^{-1}(\sinh (x))+i \left(\text{Li}_2\left(-i e^{-x}\right)-\text{Li}_2\left(i e^{-x}\right)+x \left(\log \left(1-i e^{-x}\right)-\log \left(1+i e^{-x}\right)\right)\right)","i \text{Li}_2\left(-i e^x\right)-i \text{Li}_2\left(i e^x\right)-2 x \tan ^{-1}\left(e^x\right)+x \tan ^{-1}(\sinh (x))",1,"x*ArcTan[Sinh[x]] + I*(x*(Log[1 - I/E^x] - Log[1 + I/E^x]) + PolyLog[2, (-I)/E^x] - PolyLog[2, I/E^x])","A",1
74,1,105,74,0.0263441,"\int x \tan ^{-1}(\sinh (x)) \, dx","Integrate[x*ArcTan[Sinh[x]],x]","\frac{1}{2} x^2 \tan ^{-1}(\sinh (x))-\frac{1}{2} i \left(-2 x \left(\text{Li}_2\left(-i e^{-x}\right)-\text{Li}_2\left(i e^{-x}\right)\right)-2 \left(\text{Li}_3\left(-i e^{-x}\right)-\text{Li}_3\left(i e^{-x}\right)\right)-\left(x^2 \left(\log \left(1-i e^{-x}\right)-\log \left(1+i e^{-x}\right)\right)\right)\right)","i x \text{Li}_2\left(-i e^x\right)-i x \text{Li}_2\left(i e^x\right)-i \text{Li}_3\left(-i e^x\right)+i \text{Li}_3\left(i e^x\right)+x^2 \left(-\tan ^{-1}\left(e^x\right)\right)+\frac{1}{2} x^2 \tan ^{-1}(\sinh (x))",1,"(x^2*ArcTan[Sinh[x]])/2 - (I/2)*(-(x^2*(Log[1 - I/E^x] - Log[1 + I/E^x])) - 2*x*(PolyLog[2, (-I)/E^x] - PolyLog[2, I/E^x]) - 2*(PolyLog[3, (-I)/E^x] - PolyLog[3, I/E^x]))","A",1
75,1,356,108,0.1179113,"\int x^2 \tan ^{-1}(\sinh (x)) \, dx","Integrate[x^2*ArcTan[Sinh[x]],x]","\frac{1}{192} i \left(192 x^2 \text{Li}_2\left(-i e^x\right)+192 i \pi  x \text{Li}_2\left(i e^x\right)+384 x \text{Li}_3\left(-i e^{-x}\right)-384 x \text{Li}_3\left(-i e^x\right)-48 (\pi -2 i x)^2 \text{Li}_2\left(-i e^{-x}\right)-48 \pi ^2 \text{Li}_2\left(i e^x\right)+192 i \pi  \text{Li}_3\left(-i e^{-x}\right)-192 i \pi  \text{Li}_3\left(i e^x\right)+384 \text{Li}_4\left(-i e^{-x}\right)+384 \text{Li}_4\left(-i e^x\right)-16 x^4-32 i \pi  x^3-64 x^3 \log \left(1+i e^{-x}\right)+64 x^3 \log \left(1+i e^x\right)-64 i x^3 \tan ^{-1}(\sinh (x))+24 \pi ^2 x^2-96 i \pi  x^2 \log \left(1+i e^{-x}\right)+96 i \pi  x^2 \log \left(1-i e^x\right)+8 i \pi ^3 x+48 \pi ^2 x \log \left(1+i e^{-x}\right)-48 \pi ^2 x \log \left(1-i e^x\right)+8 i \pi ^3 \log \left(1+i e^{-x}\right)-8 i \pi ^3 \log \left(1+i e^x\right)+8 i \pi ^3 \log \left(\tan \left(\frac{1}{4} (\pi +2 i x)\right)\right)+7 \pi ^4\right)","i x^2 \text{Li}_2\left(-i e^x\right)-i x^2 \text{Li}_2\left(i e^x\right)-2 i x \text{Li}_3\left(-i e^x\right)+2 i x \text{Li}_3\left(i e^x\right)+2 i \text{Li}_4\left(-i e^x\right)-2 i \text{Li}_4\left(i e^x\right)-\frac{2}{3} x^3 \tan ^{-1}\left(e^x\right)+\frac{1}{3} x^3 \tan ^{-1}(\sinh (x))",1,"(I/192)*(7*Pi^4 + (8*I)*Pi^3*x + 24*Pi^2*x^2 - (32*I)*Pi*x^3 - 16*x^4 - (64*I)*x^3*ArcTan[Sinh[x]] + (8*I)*Pi^3*Log[1 + I/E^x] + 48*Pi^2*x*Log[1 + I/E^x] - (96*I)*Pi*x^2*Log[1 + I/E^x] - 64*x^3*Log[1 + I/E^x] - 48*Pi^2*x*Log[1 - I*E^x] + (96*I)*Pi*x^2*Log[1 - I*E^x] - (8*I)*Pi^3*Log[1 + I*E^x] + 64*x^3*Log[1 + I*E^x] + (8*I)*Pi^3*Log[Tan[(Pi + (2*I)*x)/4]] - 48*(Pi - (2*I)*x)^2*PolyLog[2, (-I)/E^x] + 192*x^2*PolyLog[2, (-I)*E^x] - 48*Pi^2*PolyLog[2, I*E^x] + (192*I)*Pi*x*PolyLog[2, I*E^x] + (192*I)*Pi*PolyLog[3, (-I)/E^x] + 384*x*PolyLog[3, (-I)/E^x] - 384*x*PolyLog[3, (-I)*E^x] - (192*I)*Pi*PolyLog[3, I*E^x] + 384*PolyLog[4, (-I)/E^x] + 384*PolyLog[4, (-I)*E^x])","B",1
76,1,600,299,5.4938411,"\int (e+f x)^3 \tan ^{-1}(\tanh (a+b x)) \, dx","Integrate[(e + f*x)^3*ArcTan[Tanh[a + b*x]],x]","\frac{1}{4} x \left(4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right) \tan ^{-1}(\tanh (a+b x))-\frac{i \left(8 b^4 e^3 x \log \left(1-i e^{2 (a+b x)}\right)-8 b^4 e^3 x \log \left(1+i e^{2 (a+b x)}\right)+12 b^4 e^2 f x^2 \log \left(1-i e^{2 (a+b x)}\right)-12 b^4 e^2 f x^2 \log \left(1+i e^{2 (a+b x)}\right)+8 b^4 e f^2 x^3 \log \left(1-i e^{2 (a+b x)}\right)-8 b^4 e f^2 x^3 \log \left(1+i e^{2 (a+b x)}\right)+2 b^4 f^3 x^4 \log \left(1-i e^{2 (a+b x)}\right)-2 b^4 f^3 x^4 \log \left(1+i e^{2 (a+b x)}\right)-4 b^3 (e+f x)^3 \text{Li}_2\left(-i e^{2 (a+b x)}\right)+4 b^3 (e+f x)^3 \text{Li}_2\left(i e^{2 (a+b x)}\right)+6 b^2 e^2 f \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b^2 e^2 f \text{Li}_3\left(i e^{2 (a+b x)}\right)+12 b^2 e f^2 x \text{Li}_3\left(-i e^{2 (a+b x)}\right)-12 b^2 e f^2 x \text{Li}_3\left(i e^{2 (a+b x)}\right)+6 b^2 f^3 x^2 \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b^2 f^3 x^2 \text{Li}_3\left(i e^{2 (a+b x)}\right)-6 b e f^2 \text{Li}_4\left(-i e^{2 (a+b x)}\right)+6 b e f^2 \text{Li}_4\left(i e^{2 (a+b x)}\right)-6 b f^3 x \text{Li}_4\left(-i e^{2 (a+b x)}\right)+6 b f^3 x \text{Li}_4\left(i e^{2 (a+b x)}\right)+3 f^3 \text{Li}_5\left(-i e^{2 (a+b x)}\right)-3 f^3 \text{Li}_5\left(i e^{2 (a+b x)}\right)\right)}{16 b^4}","-\frac{3 i f^3 \text{Li}_5\left(-i e^{2 a+2 b x}\right)}{16 b^4}+\frac{3 i f^3 \text{Li}_5\left(i e^{2 a+2 b x}\right)}{16 b^4}+\frac{3 i f^2 (e+f x) \text{Li}_4\left(-i e^{2 a+2 b x}\right)}{8 b^3}-\frac{3 i f^2 (e+f x) \text{Li}_4\left(i e^{2 a+2 b x}\right)}{8 b^3}-\frac{3 i f (e+f x)^2 \text{Li}_3\left(-i e^{2 a+2 b x}\right)}{8 b^2}+\frac{3 i f (e+f x)^2 \text{Li}_3\left(i e^{2 a+2 b x}\right)}{8 b^2}+\frac{i (e+f x)^3 \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}-\frac{i (e+f x)^3 \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}-\frac{(e+f x)^4 \tan ^{-1}\left(e^{2 a+2 b x}\right)}{4 f}+\frac{(e+f x)^4 \tan ^{-1}(\tanh (a+b x))}{4 f}",1,"(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)*ArcTan[Tanh[a + b*x]])/4 - ((I/16)*(8*b^4*e^3*x*Log[1 - I*E^(2*(a + b*x))] + 12*b^4*e^2*f*x^2*Log[1 - I*E^(2*(a + b*x))] + 8*b^4*e*f^2*x^3*Log[1 - I*E^(2*(a + b*x))] + 2*b^4*f^3*x^4*Log[1 - I*E^(2*(a + b*x))] - 8*b^4*e^3*x*Log[1 + I*E^(2*(a + b*x))] - 12*b^4*e^2*f*x^2*Log[1 + I*E^(2*(a + b*x))] - 8*b^4*e*f^2*x^3*Log[1 + I*E^(2*(a + b*x))] - 2*b^4*f^3*x^4*Log[1 + I*E^(2*(a + b*x))] - 4*b^3*(e + f*x)^3*PolyLog[2, (-I)*E^(2*(a + b*x))] + 4*b^3*(e + f*x)^3*PolyLog[2, I*E^(2*(a + b*x))] + 6*b^2*e^2*f*PolyLog[3, (-I)*E^(2*(a + b*x))] + 12*b^2*e*f^2*x*PolyLog[3, (-I)*E^(2*(a + b*x))] + 6*b^2*f^3*x^2*PolyLog[3, (-I)*E^(2*(a + b*x))] - 6*b^2*e^2*f*PolyLog[3, I*E^(2*(a + b*x))] - 12*b^2*e*f^2*x*PolyLog[3, I*E^(2*(a + b*x))] - 6*b^2*f^3*x^2*PolyLog[3, I*E^(2*(a + b*x))] - 6*b*e*f^2*PolyLog[4, (-I)*E^(2*(a + b*x))] - 6*b*f^3*x*PolyLog[4, (-I)*E^(2*(a + b*x))] + 6*b*e*f^2*PolyLog[4, I*E^(2*(a + b*x))] + 6*b*f^3*x*PolyLog[4, I*E^(2*(a + b*x))] + 3*f^3*PolyLog[5, (-I)*E^(2*(a + b*x))] - 3*f^3*PolyLog[5, I*E^(2*(a + b*x))]))/b^4","B",1
77,1,375,229,2.9353174,"\int (e+f x)^2 \tan ^{-1}(\tanh (a+b x)) \, dx","Integrate[(e + f*x)^2*ArcTan[Tanh[a + b*x]],x]","\frac{1}{3} x \left(3 e^2+3 e f x+f^2 x^2\right) \tan ^{-1}(\tanh (a+b x))-\frac{i \left(12 b^3 e^2 x \log \left(1-i e^{2 (a+b x)}\right)-12 b^3 e^2 x \log \left(1+i e^{2 (a+b x)}\right)+12 b^3 e f x^2 \log \left(1-i e^{2 (a+b x)}\right)-12 b^3 e f x^2 \log \left(1+i e^{2 (a+b x)}\right)+4 b^3 f^2 x^3 \log \left(1-i e^{2 (a+b x)}\right)-4 b^3 f^2 x^3 \log \left(1+i e^{2 (a+b x)}\right)-6 b^2 (e+f x)^2 \text{Li}_2\left(-i e^{2 (a+b x)}\right)+6 b^2 (e+f x)^2 \text{Li}_2\left(i e^{2 (a+b x)}\right)+6 b e f \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b e f \text{Li}_3\left(i e^{2 (a+b x)}\right)+6 b f^2 x \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b f^2 x \text{Li}_3\left(i e^{2 (a+b x)}\right)-3 f^2 \text{Li}_4\left(-i e^{2 (a+b x)}\right)+3 f^2 \text{Li}_4\left(i e^{2 (a+b x)}\right)\right)}{24 b^3}","\frac{i f^2 \text{Li}_4\left(-i e^{2 a+2 b x}\right)}{8 b^3}-\frac{i f^2 \text{Li}_4\left(i e^{2 a+2 b x}\right)}{8 b^3}-\frac{i f (e+f x) \text{Li}_3\left(-i e^{2 a+2 b x}\right)}{4 b^2}+\frac{i f (e+f x) \text{Li}_3\left(i e^{2 a+2 b x}\right)}{4 b^2}+\frac{i (e+f x)^2 \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}-\frac{i (e+f x)^2 \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}-\frac{(e+f x)^3 \tan ^{-1}\left(e^{2 a+2 b x}\right)}{3 f}+\frac{(e+f x)^3 \tan ^{-1}(\tanh (a+b x))}{3 f}",1,"(x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcTan[Tanh[a + b*x]])/3 - ((I/24)*(12*b^3*e^2*x*Log[1 - I*E^(2*(a + b*x))] + 12*b^3*e*f*x^2*Log[1 - I*E^(2*(a + b*x))] + 4*b^3*f^2*x^3*Log[1 - I*E^(2*(a + b*x))] - 12*b^3*e^2*x*Log[1 + I*E^(2*(a + b*x))] - 12*b^3*e*f*x^2*Log[1 + I*E^(2*(a + b*x))] - 4*b^3*f^2*x^3*Log[1 + I*E^(2*(a + b*x))] - 6*b^2*(e + f*x)^2*PolyLog[2, (-I)*E^(2*(a + b*x))] + 6*b^2*(e + f*x)^2*PolyLog[2, I*E^(2*(a + b*x))] + 6*b*e*f*PolyLog[3, (-I)*E^(2*(a + b*x))] + 6*b*f^2*x*PolyLog[3, (-I)*E^(2*(a + b*x))] - 6*b*e*f*PolyLog[3, I*E^(2*(a + b*x))] - 6*b*f^2*x*PolyLog[3, I*E^(2*(a + b*x))] - 3*f^2*PolyLog[4, (-I)*E^(2*(a + b*x))] + 3*f^2*PolyLog[4, I*E^(2*(a + b*x))]))/b^3","A",1
78,1,278,159,1.9418508,"\int (e+f x) \tan ^{-1}(\tanh (a+b x)) \, dx","Integrate[(e + f*x)*ArcTan[Tanh[a + b*x]],x]","-\frac{i f \left(2 b^2 x^2 \log \left(1-i e^{2 (a+b x)}\right)-2 b^2 x^2 \log \left(1+i e^{2 (a+b x)}\right)-2 b x \text{Li}_2\left(-i e^{2 (a+b x)}\right)+2 b x \text{Li}_2\left(i e^{2 (a+b x)}\right)+\text{Li}_3\left(-i e^{2 (a+b x)}\right)-\text{Li}_3\left(i e^{2 (a+b x)}\right)\right)}{8 b^2}-\frac{e \left(-2 i \left(\text{Li}_2\left(-i e^{2 (a+b x)}\right)-\text{Li}_2\left(i e^{2 (a+b x)}\right)\right)-\left((-4 i a-4 i b x+\pi ) \left(\log \left(1-i e^{2 (a+b x)}\right)-\log \left(1+i e^{2 (a+b x)}\right)\right)\right)+(\pi -4 i a) \log \left(\cot \left(\frac{1}{4} (4 i a+4 i b x+\pi )\right)\right)\right)}{8 b}+e x \tan ^{-1}(\tanh (a+b x))+\frac{1}{2} f x^2 \tan ^{-1}(\tanh (a+b x))","-\frac{i f \text{Li}_3\left(-i e^{2 a+2 b x}\right)}{8 b^2}+\frac{i f \text{Li}_3\left(i e^{2 a+2 b x}\right)}{8 b^2}+\frac{i (e+f x) \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}-\frac{i (e+f x) \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}-\frac{(e+f x)^2 \tan ^{-1}\left(e^{2 a+2 b x}\right)}{2 f}+\frac{(e+f x)^2 \tan ^{-1}(\tanh (a+b x))}{2 f}",1,"e*x*ArcTan[Tanh[a + b*x]] + (f*x^2*ArcTan[Tanh[a + b*x]])/2 - (e*(-(((-4*I)*a + Pi - (4*I)*b*x)*(Log[1 - I*E^(2*(a + b*x))] - Log[1 + I*E^(2*(a + b*x))])) + ((-4*I)*a + Pi)*Log[Cot[((4*I)*a + Pi + (4*I)*b*x)/4]] - (2*I)*(PolyLog[2, (-I)*E^(2*(a + b*x))] - PolyLog[2, I*E^(2*(a + b*x))])))/(8*b) - ((I/8)*f*(2*b^2*x^2*Log[1 - I*E^(2*(a + b*x))] - 2*b^2*x^2*Log[1 + I*E^(2*(a + b*x))] - 2*b*x*PolyLog[2, (-I)*E^(2*(a + b*x))] + 2*b*x*PolyLog[2, I*E^(2*(a + b*x))] + PolyLog[3, (-I)*E^(2*(a + b*x))] - PolyLog[3, I*E^(2*(a + b*x))]))/b^2","A",1
79,1,132,74,0.0701045,"\int \tan ^{-1}(\tanh (a+b x)) \, dx","Integrate[ArcTan[Tanh[a + b*x]],x]","x \tan ^{-1}(\tanh (a+b x))-\frac{-2 i \left(\text{Li}_2\left(-i e^{2 (a+b x)}\right)-\text{Li}_2\left(i e^{2 (a+b x)}\right)\right)-\left((-4 i a-4 i b x+\pi ) \left(\log \left(1-i e^{2 (a+b x)}\right)-\log \left(1+i e^{2 (a+b x)}\right)\right)\right)+(\pi -4 i a) \log \left(\cot \left(\frac{1}{4} (4 i a+4 i b x+\pi )\right)\right)}{8 b}","\frac{i \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}-\frac{i \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}-x \tan ^{-1}\left(e^{2 a+2 b x}\right)+x \tan ^{-1}(\tanh (a+b x))",1,"x*ArcTan[Tanh[a + b*x]] - (-(((-4*I)*a + Pi - (4*I)*b*x)*(Log[1 - I*E^(2*(a + b*x))] - Log[1 + I*E^(2*(a + b*x))])) + ((-4*I)*a + Pi)*Log[Cot[((4*I)*a + Pi + (4*I)*b*x)/4]] - (2*I)*(PolyLog[2, (-I)*E^(2*(a + b*x))] - PolyLog[2, I*E^(2*(a + b*x))]))/(8*b)","A",1
80,0,0,18,6.6520829,"\int \frac{\tan ^{-1}(\tanh (a+b x))}{e+f x} \, dx","Integrate[ArcTan[Tanh[a + b*x]]/(e + f*x),x]","\int \frac{\tan ^{-1}(\tanh (a+b x))}{e+f x} \, dx","\text{Int}\left(\frac{\tan ^{-1}(\tanh (a+b x))}{e+f x},x\right)",0,"Integrate[ArcTan[Tanh[a + b*x]]/(e + f*x), x]","A",-1
81,1,305,355,5.8079389,"\int x^2 \tan ^{-1}(c+d \tanh (a+b x)) \, dx","Integrate[x^2*ArcTan[c + d*Tanh[a + b*x]],x]","\frac{1}{3} x^3 \tan ^{-1}(d \tanh (a+b x)+c)+\frac{i \left(4 b^3 x^3 \log \left(1+\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-4 b^3 x^3 \log \left(1+\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)+6 b^2 x^2 \text{Li}_2\left(-\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-6 b^2 x^2 \text{Li}_2\left(-\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)-6 b x \text{Li}_3\left(-\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)+6 b x \text{Li}_3\left(-\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)+3 \text{Li}_4\left(-\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-3 \text{Li}_4\left(-\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)\right)}{24 b^3}","\frac{i \text{Li}_4\left(-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^3}-\frac{i \text{Li}_4\left(-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^3}-\frac{i x \text{Li}_3\left(-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b^2}+\frac{i x \text{Li}_3\left(-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b^2}+\frac{i x^2 \text{Li}_2\left(-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}-\frac{i x^2 \text{Li}_2\left(-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b}+\frac{1}{6} i x^3 \log \left(1+\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)-\frac{1}{6} i x^3 \log \left(1+\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)+\frac{1}{3} x^3 \tan ^{-1}(d \tanh (a+b x)+c)",1,"(x^3*ArcTan[c + d*Tanh[a + b*x]])/3 + ((I/24)*(4*b^3*x^3*Log[1 + ((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d)] - 4*b^3*x^3*Log[1 + ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)] + 6*b^2*x^2*PolyLog[2, -(((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d))] - 6*b^2*x^2*PolyLog[2, -(((I + c + d)*E^(2*(a + b*x)))/(I + c - d))] - 6*b*x*PolyLog[3, -(((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d))] + 6*b*x*PolyLog[3, -(((I + c + d)*E^(2*(a + b*x)))/(I + c - d))] + 3*PolyLog[4, -(((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d))] - 3*PolyLog[4, -(((I + c + d)*E^(2*(a + b*x)))/(I + c - d))]))/b^3","A",1
82,1,229,267,4.8709714,"\int x \tan ^{-1}(c+d \tanh (a+b x)) \, dx","Integrate[x*ArcTan[c + d*Tanh[a + b*x]],x]","\frac{1}{2} x^2 \tan ^{-1}(d \tanh (a+b x)+c)+\frac{i \left(2 b^2 x^2 \log \left(1+\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-2 b^2 x^2 \log \left(1+\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)+2 b x \text{Li}_2\left(-\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-2 b x \text{Li}_2\left(-\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)-\text{Li}_3\left(-\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)+\text{Li}_3\left(-\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)\right)}{8 b^2}","-\frac{i \text{Li}_3\left(-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^2}+\frac{i \text{Li}_3\left(-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^2}+\frac{i x \text{Li}_2\left(-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}-\frac{i x \text{Li}_2\left(-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b}+\frac{1}{4} i x^2 \log \left(1+\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)-\frac{1}{4} i x^2 \log \left(1+\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)+\frac{1}{2} x^2 \tan ^{-1}(d \tanh (a+b x)+c)",1,"(x^2*ArcTan[c + d*Tanh[a + b*x]])/2 + ((I/8)*(2*b^2*x^2*Log[1 + ((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d)] - 2*b^2*x^2*Log[1 + ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)] + 2*b*x*PolyLog[2, -(((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d))] - 2*b*x*PolyLog[2, -(((I + c + d)*E^(2*(a + b*x)))/(I + c - d))] - PolyLog[3, -(((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d))] + PolyLog[3, -(((I + c + d)*E^(2*(a + b*x)))/(I + c - d))]))/b^2","A",1
83,1,288,174,4.5204309,"\int \tan ^{-1}(c+d \tanh (a+b x)) \, dx","Integrate[ArcTan[c + d*Tanh[a + b*x]],x]","\frac{d \text{Li}_2\left(-\frac{\left(c^2+2 d c+d^2+1\right) e^{2 (a+b x)}}{c^2-d^2+2 \sqrt{-d^2}+1}\right)-d \text{Li}_2\left(\frac{\left(c^2+2 d c+d^2+1\right) e^{2 (a+b x)}}{-c^2+d^2+2 \sqrt{-d^2}-1}\right)-2 d (a+b x) \log \left(\frac{2 \left((c+d)^2+1\right) e^{2 (a+b x)}}{2 c^2-2 d^2-4 \sqrt{-d^2}+2}+1\right)+2 d (a+b x) \log \left(\frac{\left((c+d)^2+1\right) e^{2 (a+b x)}}{c^2-d^2+2 \sqrt{-d^2}+1}+1\right)+4 a \sqrt{-d^2} \tan ^{-1}\left(\frac{\left(c^2+2 c d+d^2+1\right) e^{2 (a+b x)}+c^2-d^2+1}{2 d}\right)}{4 b \sqrt{-d^2}}+x \tan ^{-1}(d \tanh (a+b x)+c)","\frac{i \text{Li}_2\left(-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}-\frac{i \text{Li}_2\left(-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b}+\frac{1}{2} i x \log \left(1+\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)-\frac{1}{2} i x \log \left(1+\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)+x \tan ^{-1}(d \tanh (a+b x)+c)",1,"x*ArcTan[c + d*Tanh[a + b*x]] + (4*a*Sqrt[-d^2]*ArcTan[(1 + c^2 - d^2 + (1 + c^2 + 2*c*d + d^2)*E^(2*(a + b*x)))/(2*d)] - 2*d*(a + b*x)*Log[1 + (2*(1 + (c + d)^2)*E^(2*(a + b*x)))/(2 + 2*c^2 - 2*d^2 - 4*Sqrt[-d^2])] + 2*d*(a + b*x)*Log[1 + ((1 + (c + d)^2)*E^(2*(a + b*x)))/(1 + c^2 - d^2 + 2*Sqrt[-d^2])] + d*PolyLog[2, -(((1 + c^2 + 2*c*d + d^2)*E^(2*(a + b*x)))/(1 + c^2 - d^2 + 2*Sqrt[-d^2]))] - d*PolyLog[2, ((1 + c^2 + 2*c*d + d^2)*E^(2*(a + b*x)))/(-1 - c^2 + d^2 + 2*Sqrt[-d^2])])/(4*b*Sqrt[-d^2])","A",1
84,0,0,18,8.6772036,"\int \frac{\tan ^{-1}(c+d \tanh (a+b x))}{x} \, dx","Integrate[ArcTan[c + d*Tanh[a + b*x]]/x,x]","\int \frac{\tan ^{-1}(c+d \tanh (a+b x))}{x} \, dx","\text{Int}\left(\frac{\tan ^{-1}(d \tanh (a+b x)+c)}{x},x\right)",0,"Integrate[ArcTan[c + d*Tanh[a + b*x]]/x, x]","A",-1
85,1,128,142,5.7566604,"\int x^2 \tan ^{-1}(c+(i+c) \tanh (a+b x)) \, dx","Integrate[x^2*ArcTan[c + (I + c)*Tanh[a + b*x]],x]","\frac{i \left(4 b^3 x^3 \log \left(1-\frac{i e^{-2 (a+b x)}}{c}\right)-6 b^2 x^2 \text{Li}_2\left(\frac{i e^{-2 (a+b x)}}{c}\right)-6 b x \text{Li}_3\left(\frac{i e^{-2 (a+b x)}}{c}\right)-3 \text{Li}_4\left(\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{24 b^3}+\frac{1}{3} x^3 \tan ^{-1}(c+(c+i) \tanh (a+b x))","\frac{i \text{Li}_4\left(-i c e^{2 a+2 b x}\right)}{8 b^3}-\frac{i x \text{Li}_3\left(-i c e^{2 a+2 b x}\right)}{4 b^2}+\frac{i x^2 \text{Li}_2\left(-i c e^{2 a+2 b x}\right)}{4 b}+\frac{1}{6} i x^3 \log \left(1+i c e^{2 a+2 b x}\right)+\frac{1}{3} x^3 \tan ^{-1}(c+(c+i) \tanh (a+b x))-\frac{1}{12} i b x^4",1,"(x^3*ArcTan[c + (I + c)*Tanh[a + b*x]])/3 + ((I/24)*(4*b^3*x^3*Log[1 - I/(c*E^(2*(a + b*x)))] - 6*b^2*x^2*PolyLog[2, I/(c*E^(2*(a + b*x)))] - 6*b*x*PolyLog[3, I/(c*E^(2*(a + b*x)))] - 3*PolyLog[4, I/(c*E^(2*(a + b*x)))]))/b^3","A",1
86,1,102,113,5.7223781,"\int x \tan ^{-1}(c+(i+c) \tanh (a+b x)) \, dx","Integrate[x*ArcTan[c + (I + c)*Tanh[a + b*x]],x]","\frac{i \left(2 b^2 x^2 \log \left(1-\frac{i e^{-2 (a+b x)}}{c}\right)-2 b x \text{Li}_2\left(\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_3\left(\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{8 b^2}+\frac{1}{2} x^2 \tan ^{-1}(c+(c+i) \tanh (a+b x))","-\frac{i \text{Li}_3\left(-i c e^{2 a+2 b x}\right)}{8 b^2}+\frac{i x \text{Li}_2\left(-i c e^{2 a+2 b x}\right)}{4 b}+\frac{1}{4} i x^2 \log \left(1+i c e^{2 a+2 b x}\right)+\frac{1}{2} x^2 \tan ^{-1}(c+(c+i) \tanh (a+b x))-\frac{1}{6} i b x^3",1,"(x^2*ArcTan[c + (I + c)*Tanh[a + b*x]])/2 + ((I/8)*(2*b^2*x^2*Log[1 - I/(c*E^(2*(a + b*x)))] - 2*b*x*PolyLog[2, I/(c*E^(2*(a + b*x)))] - PolyLog[3, I/(c*E^(2*(a + b*x)))]))/b^2","A",1
87,1,71,79,1.9450607,"\int \tan ^{-1}(c+(i+c) \tanh (a+b x)) \, dx","Integrate[ArcTan[c + (I + c)*Tanh[a + b*x]],x]","\frac{i \left(2 b x \log \left(1-\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_2\left(\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{4 b}+x \tan ^{-1}(c+(c+i) \tanh (a+b x))","\frac{i \text{Li}_2\left(-i c e^{2 a+2 b x}\right)}{4 b}+\frac{1}{2} i x \log \left(1+i c e^{2 a+2 b x}\right)+x \tan ^{-1}(c+(c+i) \tanh (a+b x))-\frac{1}{2} i b x^2",1,"x*ArcTan[c + (I + c)*Tanh[a + b*x]] + ((I/4)*(2*b*x*Log[1 - I/(c*E^(2*(a + b*x)))] - PolyLog[2, I/(c*E^(2*(a + b*x)))]))/b","A",1
88,0,0,22,4.0938965,"\int \frac{\tan ^{-1}(c+(i+c) \tanh (a+b x))}{x} \, dx","Integrate[ArcTan[c + (I + c)*Tanh[a + b*x]]/x,x]","\int \frac{\tan ^{-1}(c+(i+c) \tanh (a+b x))}{x} \, dx","\text{Int}\left(\frac{\tan ^{-1}(c+(c+i) \tanh (a+b x))}{x},x\right)",0,"Integrate[ArcTan[c + (I + c)*Tanh[a + b*x]]/x, x]","A",-1
89,1,128,145,5.8126198,"\int x^2 \tan ^{-1}(c-(i-c) \tanh (a+b x)) \, dx","Integrate[x^2*ArcTan[c - (I - c)*Tanh[a + b*x]],x]","\frac{1}{3} x^3 \tan ^{-1}(c+(c-i) \tanh (a+b x))-\frac{i \left(4 b^3 x^3 \log \left(1+\frac{i e^{-2 (a+b x)}}{c}\right)-6 b^2 x^2 \text{Li}_2\left(-\frac{i e^{-2 (a+b x)}}{c}\right)-6 b x \text{Li}_3\left(-\frac{i e^{-2 (a+b x)}}{c}\right)-3 \text{Li}_4\left(-\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{24 b^3}","-\frac{i \text{Li}_4\left(i c e^{2 a+2 b x}\right)}{8 b^3}+\frac{i x \text{Li}_3\left(i c e^{2 a+2 b x}\right)}{4 b^2}-\frac{i x^2 \text{Li}_2\left(i c e^{2 a+2 b x}\right)}{4 b}-\frac{1}{6} i x^3 \log \left(1-i c e^{2 a+2 b x}\right)+\frac{1}{3} x^3 \tan ^{-1}(c-(-c+i) \tanh (a+b x))+\frac{1}{12} i b x^4",1,"(x^3*ArcTan[c + (-I + c)*Tanh[a + b*x]])/3 - ((I/24)*(4*b^3*x^3*Log[1 + I/(c*E^(2*(a + b*x)))] - 6*b^2*x^2*PolyLog[2, (-I)/(c*E^(2*(a + b*x)))] - 6*b*x*PolyLog[3, (-I)/(c*E^(2*(a + b*x)))] - 3*PolyLog[4, (-I)/(c*E^(2*(a + b*x)))]))/b^3","A",1
90,1,102,116,5.676851,"\int x \tan ^{-1}(c-(i-c) \tanh (a+b x)) \, dx","Integrate[x*ArcTan[c - (I - c)*Tanh[a + b*x]],x]","\frac{1}{2} x^2 \tan ^{-1}(c+(c-i) \tanh (a+b x))-\frac{i \left(2 b^2 x^2 \log \left(1+\frac{i e^{-2 (a+b x)}}{c}\right)-2 b x \text{Li}_2\left(-\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_3\left(-\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{8 b^2}","\frac{i \text{Li}_3\left(i c e^{2 a+2 b x}\right)}{8 b^2}-\frac{i x \text{Li}_2\left(i c e^{2 a+2 b x}\right)}{4 b}-\frac{1}{4} i x^2 \log \left(1-i c e^{2 a+2 b x}\right)+\frac{1}{2} x^2 \tan ^{-1}(c-(-c+i) \tanh (a+b x))+\frac{1}{6} i b x^3",1,"(x^2*ArcTan[c + (-I + c)*Tanh[a + b*x]])/2 - ((I/8)*(2*b^2*x^2*Log[1 + I/(c*E^(2*(a + b*x)))] - 2*b*x*PolyLog[2, (-I)/(c*E^(2*(a + b*x)))] - PolyLog[3, (-I)/(c*E^(2*(a + b*x)))]))/b^2","A",1
91,1,71,82,1.8521396,"\int \tan ^{-1}(c-(i-c) \tanh (a+b x)) \, dx","Integrate[ArcTan[c - (I - c)*Tanh[a + b*x]],x]","x \tan ^{-1}(c+(c-i) \tanh (a+b x))-\frac{i \left(2 b x \log \left(1+\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_2\left(-\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{4 b}","-\frac{i \text{Li}_2\left(i c e^{2 a+2 b x}\right)}{4 b}-\frac{1}{2} i x \log \left(1-i c e^{2 a+2 b x}\right)+x \tan ^{-1}(c-(-c+i) \tanh (a+b x))+\frac{1}{2} i b x^2",1,"x*ArcTan[c + (-I + c)*Tanh[a + b*x]] - ((I/4)*(2*b*x*Log[1 + I/(c*E^(2*(a + b*x)))] - PolyLog[2, (-I)/(c*E^(2*(a + b*x)))]))/b","A",1
92,0,0,25,4.1041502,"\int \frac{\tan ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx","Integrate[ArcTan[c - (I - c)*Tanh[a + b*x]]/x,x]","\int \frac{\tan ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx","\text{Int}\left(\frac{\tan ^{-1}(c-(-c+i) \tanh (a+b x))}{x},x\right)",0,"Integrate[ArcTan[c - (I - c)*Tanh[a + b*x]]/x, x]","A",-1
93,1,600,299,6.0548875,"\int (e+f x)^3 \tan ^{-1}(\coth (a+b x)) \, dx","Integrate[(e + f*x)^3*ArcTan[Coth[a + b*x]],x]","\frac{1}{4} x \left(4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right) \tan ^{-1}(\coth (a+b x))+\frac{i \left(8 b^4 e^3 x \log \left(1-i e^{2 (a+b x)}\right)-8 b^4 e^3 x \log \left(1+i e^{2 (a+b x)}\right)+12 b^4 e^2 f x^2 \log \left(1-i e^{2 (a+b x)}\right)-12 b^4 e^2 f x^2 \log \left(1+i e^{2 (a+b x)}\right)+8 b^4 e f^2 x^3 \log \left(1-i e^{2 (a+b x)}\right)-8 b^4 e f^2 x^3 \log \left(1+i e^{2 (a+b x)}\right)+2 b^4 f^3 x^4 \log \left(1-i e^{2 (a+b x)}\right)-2 b^4 f^3 x^4 \log \left(1+i e^{2 (a+b x)}\right)-4 b^3 (e+f x)^3 \text{Li}_2\left(-i e^{2 (a+b x)}\right)+4 b^3 (e+f x)^3 \text{Li}_2\left(i e^{2 (a+b x)}\right)+6 b^2 e^2 f \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b^2 e^2 f \text{Li}_3\left(i e^{2 (a+b x)}\right)+12 b^2 e f^2 x \text{Li}_3\left(-i e^{2 (a+b x)}\right)-12 b^2 e f^2 x \text{Li}_3\left(i e^{2 (a+b x)}\right)+6 b^2 f^3 x^2 \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b^2 f^3 x^2 \text{Li}_3\left(i e^{2 (a+b x)}\right)-6 b e f^2 \text{Li}_4\left(-i e^{2 (a+b x)}\right)+6 b e f^2 \text{Li}_4\left(i e^{2 (a+b x)}\right)-6 b f^3 x \text{Li}_4\left(-i e^{2 (a+b x)}\right)+6 b f^3 x \text{Li}_4\left(i e^{2 (a+b x)}\right)+3 f^3 \text{Li}_5\left(-i e^{2 (a+b x)}\right)-3 f^3 \text{Li}_5\left(i e^{2 (a+b x)}\right)\right)}{16 b^4}","\frac{3 i f^3 \text{Li}_5\left(-i e^{2 a+2 b x}\right)}{16 b^4}-\frac{3 i f^3 \text{Li}_5\left(i e^{2 a+2 b x}\right)}{16 b^4}-\frac{3 i f^2 (e+f x) \text{Li}_4\left(-i e^{2 a+2 b x}\right)}{8 b^3}+\frac{3 i f^2 (e+f x) \text{Li}_4\left(i e^{2 a+2 b x}\right)}{8 b^3}+\frac{3 i f (e+f x)^2 \text{Li}_3\left(-i e^{2 a+2 b x}\right)}{8 b^2}-\frac{3 i f (e+f x)^2 \text{Li}_3\left(i e^{2 a+2 b x}\right)}{8 b^2}-\frac{i (e+f x)^3 \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}+\frac{i (e+f x)^3 \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}+\frac{(e+f x)^4 \tan ^{-1}\left(e^{2 a+2 b x}\right)}{4 f}+\frac{(e+f x)^4 \tan ^{-1}(\coth (a+b x))}{4 f}",1,"(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)*ArcTan[Coth[a + b*x]])/4 + ((I/16)*(8*b^4*e^3*x*Log[1 - I*E^(2*(a + b*x))] + 12*b^4*e^2*f*x^2*Log[1 - I*E^(2*(a + b*x))] + 8*b^4*e*f^2*x^3*Log[1 - I*E^(2*(a + b*x))] + 2*b^4*f^3*x^4*Log[1 - I*E^(2*(a + b*x))] - 8*b^4*e^3*x*Log[1 + I*E^(2*(a + b*x))] - 12*b^4*e^2*f*x^2*Log[1 + I*E^(2*(a + b*x))] - 8*b^4*e*f^2*x^3*Log[1 + I*E^(2*(a + b*x))] - 2*b^4*f^3*x^4*Log[1 + I*E^(2*(a + b*x))] - 4*b^3*(e + f*x)^3*PolyLog[2, (-I)*E^(2*(a + b*x))] + 4*b^3*(e + f*x)^3*PolyLog[2, I*E^(2*(a + b*x))] + 6*b^2*e^2*f*PolyLog[3, (-I)*E^(2*(a + b*x))] + 12*b^2*e*f^2*x*PolyLog[3, (-I)*E^(2*(a + b*x))] + 6*b^2*f^3*x^2*PolyLog[3, (-I)*E^(2*(a + b*x))] - 6*b^2*e^2*f*PolyLog[3, I*E^(2*(a + b*x))] - 12*b^2*e*f^2*x*PolyLog[3, I*E^(2*(a + b*x))] - 6*b^2*f^3*x^2*PolyLog[3, I*E^(2*(a + b*x))] - 6*b*e*f^2*PolyLog[4, (-I)*E^(2*(a + b*x))] - 6*b*f^3*x*PolyLog[4, (-I)*E^(2*(a + b*x))] + 6*b*e*f^2*PolyLog[4, I*E^(2*(a + b*x))] + 6*b*f^3*x*PolyLog[4, I*E^(2*(a + b*x))] + 3*f^3*PolyLog[5, (-I)*E^(2*(a + b*x))] - 3*f^3*PolyLog[5, I*E^(2*(a + b*x))]))/b^4","B",1
94,1,375,229,2.9979292,"\int (e+f x)^2 \tan ^{-1}(\coth (a+b x)) \, dx","Integrate[(e + f*x)^2*ArcTan[Coth[a + b*x]],x]","\frac{1}{3} x \left(3 e^2+3 e f x+f^2 x^2\right) \tan ^{-1}(\coth (a+b x))+\frac{i \left(12 b^3 e^2 x \log \left(1-i e^{2 (a+b x)}\right)-12 b^3 e^2 x \log \left(1+i e^{2 (a+b x)}\right)+12 b^3 e f x^2 \log \left(1-i e^{2 (a+b x)}\right)-12 b^3 e f x^2 \log \left(1+i e^{2 (a+b x)}\right)+4 b^3 f^2 x^3 \log \left(1-i e^{2 (a+b x)}\right)-4 b^3 f^2 x^3 \log \left(1+i e^{2 (a+b x)}\right)-6 b^2 (e+f x)^2 \text{Li}_2\left(-i e^{2 (a+b x)}\right)+6 b^2 (e+f x)^2 \text{Li}_2\left(i e^{2 (a+b x)}\right)+6 b e f \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b e f \text{Li}_3\left(i e^{2 (a+b x)}\right)+6 b f^2 x \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b f^2 x \text{Li}_3\left(i e^{2 (a+b x)}\right)-3 f^2 \text{Li}_4\left(-i e^{2 (a+b x)}\right)+3 f^2 \text{Li}_4\left(i e^{2 (a+b x)}\right)\right)}{24 b^3}","-\frac{i f^2 \text{Li}_4\left(-i e^{2 a+2 b x}\right)}{8 b^3}+\frac{i f^2 \text{Li}_4\left(i e^{2 a+2 b x}\right)}{8 b^3}+\frac{i f (e+f x) \text{Li}_3\left(-i e^{2 a+2 b x}\right)}{4 b^2}-\frac{i f (e+f x) \text{Li}_3\left(i e^{2 a+2 b x}\right)}{4 b^2}-\frac{i (e+f x)^2 \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}+\frac{i (e+f x)^2 \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}+\frac{(e+f x)^3 \tan ^{-1}\left(e^{2 a+2 b x}\right)}{3 f}+\frac{(e+f x)^3 \tan ^{-1}(\coth (a+b x))}{3 f}",1,"(x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcTan[Coth[a + b*x]])/3 + ((I/24)*(12*b^3*e^2*x*Log[1 - I*E^(2*(a + b*x))] + 12*b^3*e*f*x^2*Log[1 - I*E^(2*(a + b*x))] + 4*b^3*f^2*x^3*Log[1 - I*E^(2*(a + b*x))] - 12*b^3*e^2*x*Log[1 + I*E^(2*(a + b*x))] - 12*b^3*e*f*x^2*Log[1 + I*E^(2*(a + b*x))] - 4*b^3*f^2*x^3*Log[1 + I*E^(2*(a + b*x))] - 6*b^2*(e + f*x)^2*PolyLog[2, (-I)*E^(2*(a + b*x))] + 6*b^2*(e + f*x)^2*PolyLog[2, I*E^(2*(a + b*x))] + 6*b*e*f*PolyLog[3, (-I)*E^(2*(a + b*x))] + 6*b*f^2*x*PolyLog[3, (-I)*E^(2*(a + b*x))] - 6*b*e*f*PolyLog[3, I*E^(2*(a + b*x))] - 6*b*f^2*x*PolyLog[3, I*E^(2*(a + b*x))] - 3*f^2*PolyLog[4, (-I)*E^(2*(a + b*x))] + 3*f^2*PolyLog[4, I*E^(2*(a + b*x))]))/b^3","A",1
95,1,278,159,2.1835007,"\int (e+f x) \tan ^{-1}(\coth (a+b x)) \, dx","Integrate[(e + f*x)*ArcTan[Coth[a + b*x]],x]","\frac{i f \left(2 b^2 x^2 \log \left(1-i e^{2 (a+b x)}\right)-2 b^2 x^2 \log \left(1+i e^{2 (a+b x)}\right)-2 b x \text{Li}_2\left(-i e^{2 (a+b x)}\right)+2 b x \text{Li}_2\left(i e^{2 (a+b x)}\right)+\text{Li}_3\left(-i e^{2 (a+b x)}\right)-\text{Li}_3\left(i e^{2 (a+b x)}\right)\right)}{8 b^2}+\frac{e \left(-2 i \left(\text{Li}_2\left(-i e^{2 (a+b x)}\right)-\text{Li}_2\left(i e^{2 (a+b x)}\right)\right)-\left((-4 i a-4 i b x+\pi ) \left(\log \left(1-i e^{2 (a+b x)}\right)-\log \left(1+i e^{2 (a+b x)}\right)\right)\right)+(\pi -4 i a) \log \left(\cot \left(\frac{1}{4} (4 i a+4 i b x+\pi )\right)\right)\right)}{8 b}+e x \tan ^{-1}(\coth (a+b x))+\frac{1}{2} f x^2 \tan ^{-1}(\coth (a+b x))","\frac{i f \text{Li}_3\left(-i e^{2 a+2 b x}\right)}{8 b^2}-\frac{i f \text{Li}_3\left(i e^{2 a+2 b x}\right)}{8 b^2}-\frac{i (e+f x) \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}+\frac{i (e+f x) \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}+\frac{(e+f x)^2 \tan ^{-1}\left(e^{2 a+2 b x}\right)}{2 f}+\frac{(e+f x)^2 \tan ^{-1}(\coth (a+b x))}{2 f}",1,"e*x*ArcTan[Coth[a + b*x]] + (f*x^2*ArcTan[Coth[a + b*x]])/2 + (e*(-(((-4*I)*a + Pi - (4*I)*b*x)*(Log[1 - I*E^(2*(a + b*x))] - Log[1 + I*E^(2*(a + b*x))])) + ((-4*I)*a + Pi)*Log[Cot[((4*I)*a + Pi + (4*I)*b*x)/4]] - (2*I)*(PolyLog[2, (-I)*E^(2*(a + b*x))] - PolyLog[2, I*E^(2*(a + b*x))])))/(8*b) + ((I/8)*f*(2*b^2*x^2*Log[1 - I*E^(2*(a + b*x))] - 2*b^2*x^2*Log[1 + I*E^(2*(a + b*x))] - 2*b*x*PolyLog[2, (-I)*E^(2*(a + b*x))] + 2*b*x*PolyLog[2, I*E^(2*(a + b*x))] + PolyLog[3, (-I)*E^(2*(a + b*x))] - PolyLog[3, I*E^(2*(a + b*x))]))/b^2","A",1
96,1,132,73,0.0846987,"\int \tan ^{-1}(\coth (a+b x)) \, dx","Integrate[ArcTan[Coth[a + b*x]],x]","x \tan ^{-1}(\coth (a+b x))+\frac{-2 i \left(\text{Li}_2\left(-i e^{2 (a+b x)}\right)-\text{Li}_2\left(i e^{2 (a+b x)}\right)\right)-\left((-4 i a-4 i b x+\pi ) \left(\log \left(1-i e^{2 (a+b x)}\right)-\log \left(1+i e^{2 (a+b x)}\right)\right)\right)+(\pi -4 i a) \log \left(\cot \left(\frac{1}{4} (4 i a+4 i b x+\pi )\right)\right)}{8 b}","-\frac{i \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}+\frac{i \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}+x \tan ^{-1}\left(e^{2 a+2 b x}\right)+x \tan ^{-1}(\coth (a+b x))",1,"x*ArcTan[Coth[a + b*x]] + (-(((-4*I)*a + Pi - (4*I)*b*x)*(Log[1 - I*E^(2*(a + b*x))] - Log[1 + I*E^(2*(a + b*x))])) + ((-4*I)*a + Pi)*Log[Cot[((4*I)*a + Pi + (4*I)*b*x)/4]] - (2*I)*(PolyLog[2, (-I)*E^(2*(a + b*x))] - PolyLog[2, I*E^(2*(a + b*x))]))/(8*b)","A",1
97,0,0,18,5.9327454,"\int \frac{\tan ^{-1}(\coth (a+b x))}{e+f x} \, dx","Integrate[ArcTan[Coth[a + b*x]]/(e + f*x),x]","\int \frac{\tan ^{-1}(\coth (a+b x))}{e+f x} \, dx","\text{Int}\left(\frac{\tan ^{-1}(\coth (a+b x))}{e+f x},x\right)",0,"Integrate[ArcTan[Coth[a + b*x]]/(e + f*x), x]","A",-1
98,1,299,351,6.1764779,"\int x^2 \tan ^{-1}(c+d \coth (a+b x)) \, dx","Integrate[x^2*ArcTan[c + d*Coth[a + b*x]],x]","\frac{1}{3} x^3 \tan ^{-1}(d \coth (a+b x)+c)+\frac{i \left(4 b^3 x^3 \log \left(1+\frac{(c+d-i) e^{2 (a+b x)}}{-c+d+i}\right)-4 b^3 x^3 \log \left(1+\frac{(c+d+i) e^{2 (a+b x)}}{-c+d-i}\right)+6 b^2 x^2 \text{Li}_2\left(\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-6 b^2 x^2 \text{Li}_2\left(\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)-6 b x \text{Li}_3\left(\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)+6 b x \text{Li}_3\left(\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)+3 \text{Li}_4\left(\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-3 \text{Li}_4\left(\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)\right)}{24 b^3}","\frac{i \text{Li}_4\left(\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^3}-\frac{i \text{Li}_4\left(\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^3}-\frac{i x \text{Li}_3\left(\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b^2}+\frac{i x \text{Li}_3\left(\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b^2}+\frac{i x^2 \text{Li}_2\left(\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}-\frac{i x^2 \text{Li}_2\left(\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b}+\frac{1}{6} i x^3 \log \left(1-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)-\frac{1}{6} i x^3 \log \left(1-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)+\frac{1}{3} x^3 \tan ^{-1}(d \coth (a+b x)+c)",1,"(x^3*ArcTan[c + d*Coth[a + b*x]])/3 + ((I/24)*(4*b^3*x^3*Log[1 + ((-I + c + d)*E^(2*(a + b*x)))/(I - c + d)] - 4*b^3*x^3*Log[1 + ((I + c + d)*E^(2*(a + b*x)))/(-I - c + d)] + 6*b^2*x^2*PolyLog[2, ((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d)] - 6*b^2*x^2*PolyLog[2, ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)] - 6*b*x*PolyLog[3, ((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d)] + 6*b*x*PolyLog[3, ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)] + 3*PolyLog[4, ((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d)] - 3*PolyLog[4, ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)]))/b^3","A",1
99,1,225,265,4.7631007,"\int x \tan ^{-1}(c+d \coth (a+b x)) \, dx","Integrate[x*ArcTan[c + d*Coth[a + b*x]],x]","\frac{1}{2} x^2 \tan ^{-1}(d \coth (a+b x)+c)+\frac{i \left(2 b^2 x^2 \log \left(1+\frac{(c+d-i) e^{2 (a+b x)}}{-c+d+i}\right)-2 b^2 x^2 \log \left(1+\frac{(c+d+i) e^{2 (a+b x)}}{-c+d-i}\right)+2 b x \text{Li}_2\left(\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-2 b x \text{Li}_2\left(\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)-\text{Li}_3\left(\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)+\text{Li}_3\left(\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)\right)}{8 b^2}","-\frac{i \text{Li}_3\left(\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^2}+\frac{i \text{Li}_3\left(\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^2}+\frac{i x \text{Li}_2\left(\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}-\frac{i x \text{Li}_2\left(\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b}+\frac{1}{4} i x^2 \log \left(1-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)-\frac{1}{4} i x^2 \log \left(1-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)+\frac{1}{2} x^2 \tan ^{-1}(d \coth (a+b x)+c)",1,"(x^2*ArcTan[c + d*Coth[a + b*x]])/2 + ((I/8)*(2*b^2*x^2*Log[1 + ((-I + c + d)*E^(2*(a + b*x)))/(I - c + d)] - 2*b^2*x^2*Log[1 + ((I + c + d)*E^(2*(a + b*x)))/(-I - c + d)] + 2*b*x*PolyLog[2, ((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d)] - 2*b*x*PolyLog[2, ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)] - PolyLog[3, ((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d)] + PolyLog[3, ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)]))/b^2","A",1
100,1,287,174,4.2585648,"\int \tan ^{-1}(c+d \coth (a+b x)) \, dx","Integrate[ArcTan[c + d*Coth[a + b*x]],x]","\frac{d \text{Li}_2\left(\frac{\left(c^2+2 d c+d^2+1\right) e^{2 (a+b x)}}{c^2-d^2+2 \sqrt{-d^2}+1}\right)-d \text{Li}_2\left(-\frac{\left(c^2+2 d c+d^2+1\right) e^{2 (a+b x)}}{-c^2+d^2+2 \sqrt{-d^2}-1}\right)+2 d (a+b x) \log \left(1-\frac{\left((c+d)^2+1\right) e^{2 (a+b x)}}{c^2-d^2+2 \sqrt{-d^2}+1}\right)-2 d (a+b x) \log \left(\frac{\left((c+d)^2+1\right) e^{2 (a+b x)}}{-c^2+d^2+2 \sqrt{-d^2}-1}+1\right)+4 a \sqrt{-d^2} \tan ^{-1}\left(\frac{-\left(c^2+2 c d+d^2+1\right) e^{2 (a+b x)}+c^2-d^2+1}{2 d}\right)}{4 b \sqrt{-d^2}}+x \tan ^{-1}(d \coth (a+b x)+c)","\frac{i \text{Li}_2\left(\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}-\frac{i \text{Li}_2\left(\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b}+\frac{1}{2} i x \log \left(1-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)-\frac{1}{2} i x \log \left(1-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)+x \tan ^{-1}(d \coth (a+b x)+c)",1,"x*ArcTan[c + d*Coth[a + b*x]] + (4*a*Sqrt[-d^2]*ArcTan[(1 + c^2 - d^2 - (1 + c^2 + 2*c*d + d^2)*E^(2*(a + b*x)))/(2*d)] + 2*d*(a + b*x)*Log[1 - ((1 + (c + d)^2)*E^(2*(a + b*x)))/(1 + c^2 - d^2 + 2*Sqrt[-d^2])] - 2*d*(a + b*x)*Log[1 + ((1 + (c + d)^2)*E^(2*(a + b*x)))/(-1 - c^2 + d^2 + 2*Sqrt[-d^2])] + d*PolyLog[2, ((1 + c^2 + 2*c*d + d^2)*E^(2*(a + b*x)))/(1 + c^2 - d^2 + 2*Sqrt[-d^2])] - d*PolyLog[2, -(((1 + c^2 + 2*c*d + d^2)*E^(2*(a + b*x)))/(-1 - c^2 + d^2 + 2*Sqrt[-d^2]))])/(4*b*Sqrt[-d^2])","A",1
101,0,0,18,8.8743156,"\int \frac{\tan ^{-1}(c+d \coth (a+b x))}{x} \, dx","Integrate[ArcTan[c + d*Coth[a + b*x]]/x,x]","\int \frac{\tan ^{-1}(c+d \coth (a+b x))}{x} \, dx","\text{Int}\left(\frac{\tan ^{-1}(d \coth (a+b x)+c)}{x},x\right)",0,"Integrate[ArcTan[c + d*Coth[a + b*x]]/x, x]","A",-1
102,1,128,142,1.8298616,"\int x^2 \tan ^{-1}(c+(i+c) \coth (a+b x)) \, dx","Integrate[x^2*ArcTan[c + (I + c)*Coth[a + b*x]],x]","\frac{i \left(4 b^3 x^3 \log \left(1+\frac{i e^{-2 (a+b x)}}{c}\right)-6 b^2 x^2 \text{Li}_2\left(-\frac{i e^{-2 (a+b x)}}{c}\right)-6 b x \text{Li}_3\left(-\frac{i e^{-2 (a+b x)}}{c}\right)-3 \text{Li}_4\left(-\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{24 b^3}+\frac{1}{3} x^3 \tan ^{-1}(c+(c+i) \coth (a+b x))","\frac{i \text{Li}_4\left(i c e^{2 a+2 b x}\right)}{8 b^3}-\frac{i x \text{Li}_3\left(i c e^{2 a+2 b x}\right)}{4 b^2}+\frac{i x^2 \text{Li}_2\left(i c e^{2 a+2 b x}\right)}{4 b}+\frac{1}{6} i x^3 \log \left(1-i c e^{2 a+2 b x}\right)+\frac{1}{3} x^3 \tan ^{-1}(c+(c+i) \coth (a+b x))-\frac{1}{12} i b x^4",1,"(x^3*ArcTan[c + (I + c)*Coth[a + b*x]])/3 + ((I/24)*(4*b^3*x^3*Log[1 + I/(c*E^(2*(a + b*x)))] - 6*b^2*x^2*PolyLog[2, (-I)/(c*E^(2*(a + b*x)))] - 6*b*x*PolyLog[3, (-I)/(c*E^(2*(a + b*x)))] - 3*PolyLog[4, (-I)/(c*E^(2*(a + b*x)))]))/b^3","A",1
103,1,102,113,1.7927148,"\int x \tan ^{-1}(c+(i+c) \coth (a+b x)) \, dx","Integrate[x*ArcTan[c + (I + c)*Coth[a + b*x]],x]","\frac{i \left(2 b^2 x^2 \log \left(1+\frac{i e^{-2 (a+b x)}}{c}\right)-2 b x \text{Li}_2\left(-\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_3\left(-\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{8 b^2}+\frac{1}{2} x^2 \tan ^{-1}(c+(c+i) \coth (a+b x))","-\frac{i \text{Li}_3\left(i c e^{2 a+2 b x}\right)}{8 b^2}+\frac{i x \text{Li}_2\left(i c e^{2 a+2 b x}\right)}{4 b}+\frac{1}{4} i x^2 \log \left(1-i c e^{2 a+2 b x}\right)+\frac{1}{2} x^2 \tan ^{-1}(c+(c+i) \coth (a+b x))-\frac{1}{6} i b x^3",1,"(x^2*ArcTan[c + (I + c)*Coth[a + b*x]])/2 + ((I/8)*(2*b^2*x^2*Log[1 + I/(c*E^(2*(a + b*x)))] - 2*b*x*PolyLog[2, (-I)/(c*E^(2*(a + b*x)))] - PolyLog[3, (-I)/(c*E^(2*(a + b*x)))]))/b^2","A",1
104,1,71,79,0.7911766,"\int \tan ^{-1}(c+(i+c) \coth (a+b x)) \, dx","Integrate[ArcTan[c + (I + c)*Coth[a + b*x]],x]","\frac{i \left(2 b x \log \left(1+\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_2\left(-\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{4 b}+x \tan ^{-1}(c+(c+i) \coth (a+b x))","\frac{i \text{Li}_2\left(i c e^{2 a+2 b x}\right)}{4 b}+\frac{1}{2} i x \log \left(1-i c e^{2 a+2 b x}\right)+x \tan ^{-1}(c+(c+i) \coth (a+b x))-\frac{1}{2} i b x^2",1,"x*ArcTan[c + (I + c)*Coth[a + b*x]] + ((I/4)*(2*b*x*Log[1 + I/(c*E^(2*(a + b*x)))] - PolyLog[2, (-I)/(c*E^(2*(a + b*x)))]))/b","A",1
105,0,0,22,4.132903,"\int \frac{\tan ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx","Integrate[ArcTan[c + (I + c)*Coth[a + b*x]]/x,x]","\int \frac{\tan ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx","\text{Int}\left(\frac{\tan ^{-1}(c+(c+i) \coth (a+b x))}{x},x\right)",0,"Integrate[ArcTan[c + (I + c)*Coth[a + b*x]]/x, x]","A",-1
106,1,128,145,1.8480258,"\int x^2 \tan ^{-1}(c-(i-c) \coth (a+b x)) \, dx","Integrate[x^2*ArcTan[c - (I - c)*Coth[a + b*x]],x]","\frac{1}{3} x^3 \tan ^{-1}(c+(c-i) \coth (a+b x))-\frac{i \left(4 b^3 x^3 \log \left(1-\frac{i e^{-2 (a+b x)}}{c}\right)-6 b^2 x^2 \text{Li}_2\left(\frac{i e^{-2 (a+b x)}}{c}\right)-6 b x \text{Li}_3\left(\frac{i e^{-2 (a+b x)}}{c}\right)-3 \text{Li}_4\left(\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{24 b^3}","-\frac{i \text{Li}_4\left(-i c e^{2 a+2 b x}\right)}{8 b^3}+\frac{i x \text{Li}_3\left(-i c e^{2 a+2 b x}\right)}{4 b^2}-\frac{i x^2 \text{Li}_2\left(-i c e^{2 a+2 b x}\right)}{4 b}-\frac{1}{6} i x^3 \log \left(1+i c e^{2 a+2 b x}\right)+\frac{1}{3} x^3 \tan ^{-1}(c-(-c+i) \coth (a+b x))+\frac{1}{12} i b x^4",1,"(x^3*ArcTan[c + (-I + c)*Coth[a + b*x]])/3 - ((I/24)*(4*b^3*x^3*Log[1 - I/(c*E^(2*(a + b*x)))] - 6*b^2*x^2*PolyLog[2, I/(c*E^(2*(a + b*x)))] - 6*b*x*PolyLog[3, I/(c*E^(2*(a + b*x)))] - 3*PolyLog[4, I/(c*E^(2*(a + b*x)))]))/b^3","A",1
107,1,102,116,1.6998591,"\int x \tan ^{-1}(c-(i-c) \coth (a+b x)) \, dx","Integrate[x*ArcTan[c - (I - c)*Coth[a + b*x]],x]","\frac{1}{2} x^2 \tan ^{-1}(c+(c-i) \coth (a+b x))-\frac{i \left(2 b^2 x^2 \log \left(1-\frac{i e^{-2 (a+b x)}}{c}\right)-2 b x \text{Li}_2\left(\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_3\left(\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{8 b^2}","\frac{i \text{Li}_3\left(-i c e^{2 a+2 b x}\right)}{8 b^2}-\frac{i x \text{Li}_2\left(-i c e^{2 a+2 b x}\right)}{4 b}-\frac{1}{4} i x^2 \log \left(1+i c e^{2 a+2 b x}\right)+\frac{1}{2} x^2 \tan ^{-1}(c-(-c+i) \coth (a+b x))+\frac{1}{6} i b x^3",1,"(x^2*ArcTan[c + (-I + c)*Coth[a + b*x]])/2 - ((I/8)*(2*b^2*x^2*Log[1 - I/(c*E^(2*(a + b*x)))] - 2*b*x*PolyLog[2, I/(c*E^(2*(a + b*x)))] - PolyLog[3, I/(c*E^(2*(a + b*x)))]))/b^2","A",1
108,1,71,82,0.7951287,"\int \tan ^{-1}(c-(i-c) \coth (a+b x)) \, dx","Integrate[ArcTan[c - (I - c)*Coth[a + b*x]],x]","x \tan ^{-1}(c+(c-i) \coth (a+b x))-\frac{i \left(2 b x \log \left(1-\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_2\left(\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{4 b}","-\frac{i \text{Li}_2\left(-i c e^{2 a+2 b x}\right)}{4 b}-\frac{1}{2} i x \log \left(1+i c e^{2 a+2 b x}\right)+x \tan ^{-1}(c-(-c+i) \coth (a+b x))+\frac{1}{2} i b x^2",1,"x*ArcTan[c + (-I + c)*Coth[a + b*x]] - ((I/4)*(2*b*x*Log[1 - I/(c*E^(2*(a + b*x)))] - PolyLog[2, I/(c*E^(2*(a + b*x)))]))/b","A",1
109,0,0,25,4.2683071,"\int \frac{\tan ^{-1}(c-(i-c) \coth (a+b x))}{x} \, dx","Integrate[ArcTan[c - (I - c)*Coth[a + b*x]]/x,x]","\int \frac{\tan ^{-1}(c-(i-c) \coth (a+b x))}{x} \, dx","\text{Int}\left(\frac{\tan ^{-1}(c-(-c+i) \coth (a+b x))}{x},x\right)",0,"Integrate[ArcTan[c - (I - c)*Coth[a + b*x]]/x, x]","A",-1
110,1,59,31,0.0477188,"\int \tan ^{-1}\left(e^x\right) \, dx","Integrate[ArcTan[E^x],x]","x \tan ^{-1}\left(e^x\right)-\frac{1}{2} i \left(-\text{Li}_2\left(-i e^x\right)+\text{Li}_2\left(i e^x\right)+x \left(\log \left(1-i e^x\right)-\log \left(1+i e^x\right)\right)\right)","\frac{1}{2} i \text{Li}_2\left(-i e^x\right)-\frac{1}{2} i \text{Li}_2\left(i e^x\right)",1,"x*ArcTan[E^x] - (I/2)*(x*(Log[1 - I*E^x] - Log[1 + I*E^x]) - PolyLog[2, (-I)*E^x] + PolyLog[2, I*E^x])","A",1
111,1,50,63,0.0115664,"\int x \tan ^{-1}\left(e^x\right) \, dx","Integrate[x*ArcTan[E^x],x]","\frac{1}{2} i \left(x \text{Li}_2\left(-i e^x\right)-x \text{Li}_2\left(i e^x\right)-\text{Li}_3\left(-i e^x\right)+\text{Li}_3\left(i e^x\right)\right)","\frac{1}{2} i x \text{Li}_2\left(-i e^x\right)-\frac{1}{2} i x \text{Li}_2\left(i e^x\right)-\frac{1}{2} i \text{Li}_3\left(-i e^x\right)+\frac{1}{2} i \text{Li}_3\left(i e^x\right)",1,"(I/2)*(x*PolyLog[2, (-I)*E^x] - x*PolyLog[2, I*E^x] - PolyLog[3, (-I)*E^x] + PolyLog[3, I*E^x])","A",1
112,1,91,91,0.0094015,"\int x^2 \tan ^{-1}\left(e^x\right) \, dx","Integrate[x^2*ArcTan[E^x],x]","\frac{1}{2} i x^2 \text{Li}_2\left(-i e^x\right)-\frac{1}{2} i x^2 \text{Li}_2\left(i e^x\right)-i x \text{Li}_3\left(-i e^x\right)+i x \text{Li}_3\left(i e^x\right)+i \text{Li}_4\left(-i e^x\right)-i \text{Li}_4\left(i e^x\right)","\frac{1}{2} i x^2 \text{Li}_2\left(-i e^x\right)-\frac{1}{2} i x^2 \text{Li}_2\left(i e^x\right)-i x \text{Li}_3\left(-i e^x\right)+i x \text{Li}_3\left(i e^x\right)+i \text{Li}_4\left(-i e^x\right)-i \text{Li}_4\left(i e^x\right)",1,"(I/2)*x^2*PolyLog[2, (-I)*E^x] - (I/2)*x^2*PolyLog[2, I*E^x] - I*x*PolyLog[3, (-I)*E^x] + I*x*PolyLog[3, I*E^x] + I*PolyLog[4, (-I)*E^x] - I*PolyLog[4, I*E^x]","A",1
113,1,83,45,0.1128325,"\int \tan ^{-1}\left(e^{a+b x}\right) \, dx","Integrate[ArcTan[E^(a + b*x)],x]","x \tan ^{-1}\left(e^{a+b x}\right)-\frac{i \left(-\text{Li}_2\left(-i e^{a+b x}\right)+\text{Li}_2\left(i e^{a+b x}\right)+b x \left(\log \left(1-i e^{a+b x}\right)-\log \left(1+i e^{a+b x}\right)\right)\right)}{2 b}","\frac{i \text{Li}_2\left(-i e^{a+b x}\right)}{2 b}-\frac{i \text{Li}_2\left(i e^{a+b x}\right)}{2 b}",1,"x*ArcTan[E^(a + b*x)] - ((I/2)*(b*x*(Log[1 - I*E^(a + b*x)] - Log[1 + I*E^(a + b*x)]) - PolyLog[2, (-I)*E^(a + b*x)] + PolyLog[2, I*E^(a + b*x)]))/b","A",1
114,1,71,91,0.0155093,"\int x \tan ^{-1}\left(e^{a+b x}\right) \, dx","Integrate[x*ArcTan[E^(a + b*x)],x]","\frac{i \left(b x \text{Li}_2\left(-i e^{a+b x}\right)-b x \text{Li}_2\left(i e^{a+b x}\right)-\text{Li}_3\left(-i e^{a+b x}\right)+\text{Li}_3\left(i e^{a+b x}\right)\right)}{2 b^2}","-\frac{i \text{Li}_3\left(-i e^{a+b x}\right)}{2 b^2}+\frac{i \text{Li}_3\left(i e^{a+b x}\right)}{2 b^2}+\frac{i x \text{Li}_2\left(-i e^{a+b x}\right)}{2 b}-\frac{i x \text{Li}_2\left(i e^{a+b x}\right)}{2 b}",1,"((I/2)*(b*x*PolyLog[2, (-I)*E^(a + b*x)] - b*x*PolyLog[2, I*E^(a + b*x)] - PolyLog[3, (-I)*E^(a + b*x)] + PolyLog[3, I*E^(a + b*x)]))/b^2","A",1
115,1,133,133,0.0098719,"\int x^2 \tan ^{-1}\left(e^{a+b x}\right) \, dx","Integrate[x^2*ArcTan[E^(a + b*x)],x]","\frac{i \text{Li}_4\left(-i e^{a+b x}\right)}{b^3}-\frac{i \text{Li}_4\left(i e^{a+b x}\right)}{b^3}-\frac{i x \text{Li}_3\left(-i e^{a+b x}\right)}{b^2}+\frac{i x \text{Li}_3\left(i e^{a+b x}\right)}{b^2}+\frac{i x^2 \text{Li}_2\left(-i e^{a+b x}\right)}{2 b}-\frac{i x^2 \text{Li}_2\left(i e^{a+b x}\right)}{2 b}","\frac{i \text{Li}_4\left(-i e^{a+b x}\right)}{b^3}-\frac{i \text{Li}_4\left(i e^{a+b x}\right)}{b^3}-\frac{i x \text{Li}_3\left(-i e^{a+b x}\right)}{b^2}+\frac{i x \text{Li}_3\left(i e^{a+b x}\right)}{b^2}+\frac{i x^2 \text{Li}_2\left(-i e^{a+b x}\right)}{2 b}-\frac{i x^2 \text{Li}_2\left(i e^{a+b x}\right)}{2 b}",1,"((I/2)*x^2*PolyLog[2, (-I)*E^(a + b*x)])/b - ((I/2)*x^2*PolyLog[2, I*E^(a + b*x)])/b - (I*x*PolyLog[3, (-I)*E^(a + b*x)])/b^2 + (I*x*PolyLog[3, I*E^(a + b*x)])/b^2 + (I*PolyLog[4, (-I)*E^(a + b*x)])/b^3 - (I*PolyLog[4, I*E^(a + b*x)])/b^3","A",1
116,1,167,196,0.195542,"\int \tan ^{-1}\left(a+b f^{c+d x}\right) \, dx","Integrate[ArcTan[a + b*f^(c + d*x)],x]","x \tan ^{-1}\left(a+b f^{c+d x}\right)-\frac{b \left(\text{Li}_2\left(-\frac{b^2 f^{c+d x}}{a b-\sqrt{-b^2}}\right)-\text{Li}_2\left(-\frac{b^2 f^{c+d x}}{a b+\sqrt{-b^2}}\right)+d x \log (f) \left(\log \left(\frac{b^2 f^{c+d x}}{a b-\sqrt{-b^2}}+1\right)-\log \left(\frac{b^2 f^{c+d x}}{a b+\sqrt{-b^2}}+1\right)\right)\right)}{2 \sqrt{-b^2} d \log (f)}","\frac{i \text{Li}_2\left(1-\frac{2}{1-i \left(b f^{c+d x}+a\right)}\right)}{2 d \log (f)}-\frac{i \text{Li}_2\left(1-\frac{2 b f^{c+d x}}{(i-a) \left(1-i \left(b f^{c+d x}+a\right)\right)}\right)}{2 d \log (f)}-\frac{\log \left(\frac{2}{1-i \left(a+b f^{c+d x}\right)}\right) \tan ^{-1}\left(a+b f^{c+d x}\right)}{d \log (f)}+\frac{\log \left(\frac{2 b f^{c+d x}}{(-a+i) \left(1-i \left(a+b f^{c+d x}\right)\right)}\right) \tan ^{-1}\left(a+b f^{c+d x}\right)}{d \log (f)}",1,"x*ArcTan[a + b*f^(c + d*x)] - (b*(d*x*Log[f]*(Log[1 + (b^2*f^(c + d*x))/(a*b - Sqrt[-b^2])] - Log[1 + (b^2*f^(c + d*x))/(a*b + Sqrt[-b^2])]) + PolyLog[2, -((b^2*f^(c + d*x))/(a*b - Sqrt[-b^2]))] - PolyLog[2, -((b^2*f^(c + d*x))/(a*b + Sqrt[-b^2]))]))/(2*Sqrt[-b^2]*d*Log[f])","A",1
117,1,236,232,0.0995948,"\int x \tan ^{-1}\left(a+b f^{c+d x}\right) \, dx","Integrate[x*ArcTan[a + b*f^(c + d*x)],x]","\frac{i \left(d^2 x^2 \log ^2(f) \log \left(-i a-i b f^{c+d x}+1\right)-d^2 x^2 \log ^2(f) \log \left(i a+i b f^{c+d x}+1\right)-d^2 x^2 \log ^2(f) \log \left(\frac{a+b f^{c+d x}+i}{a+i}\right)+d^2 x^2 \log ^2(f) \log \left(1+\frac{b f^{c+d x}}{a-i}\right)-2 \text{Li}_3\left(\frac{b f^{c+d x}}{i-a}\right)+2 \text{Li}_3\left(-\frac{b f^{c+d x}}{a+i}\right)+2 d x \log (f) \text{Li}_2\left(\frac{b f^{c+d x}}{i-a}\right)-2 d x \log (f) \text{Li}_2\left(-\frac{b f^{c+d x}}{a+i}\right)\right)}{4 d^2 \log ^2(f)}","\frac{i \text{Li}_3\left(\frac{i b f^{c+d x}}{1-i a}\right)}{2 d^2 \log ^2(f)}-\frac{i \text{Li}_3\left(-\frac{i b f^{c+d x}}{i a+1}\right)}{2 d^2 \log ^2(f)}-\frac{i x \text{Li}_2\left(\frac{i b f^{c+d x}}{1-i a}\right)}{2 d \log (f)}+\frac{i x \text{Li}_2\left(-\frac{i b f^{c+d x}}{i a+1}\right)}{2 d \log (f)}-\frac{1}{4} i x^2 \log \left(1-\frac{i b f^{c+d x}}{1-i a}\right)+\frac{1}{4} i x^2 \log \left(1+\frac{i b f^{c+d x}}{1+i a}\right)+\frac{1}{2} x^2 \tan ^{-1}\left(a+b f^{c+d x}\right)",1,"((I/4)*(d^2*x^2*Log[f]^2*Log[1 - I*a - I*b*f^(c + d*x)] - d^2*x^2*Log[f]^2*Log[1 + I*a + I*b*f^(c + d*x)] - d^2*x^2*Log[f]^2*Log[(I + a + b*f^(c + d*x))/(I + a)] + d^2*x^2*Log[f]^2*Log[1 + (b*f^(c + d*x))/(-I + a)] + 2*d*x*Log[f]*PolyLog[2, (b*f^(c + d*x))/(I - a)] - 2*d*x*Log[f]*PolyLog[2, -((b*f^(c + d*x))/(I + a))] - 2*PolyLog[3, (b*f^(c + d*x))/(I - a)] + 2*PolyLog[3, -((b*f^(c + d*x))/(I + a))]))/(d^2*Log[f]^2)","A",0
118,1,334,302,0.0155328,"\int x^2 \tan ^{-1}\left(a+b f^{c+d x}\right) \, dx","Integrate[x^2*ArcTan[a + b*f^(c + d*x)],x]","\frac{i \text{Li}_4\left(\frac{b f^{c+d x}}{i-a}\right)}{d^3 \log ^3(f)}-\frac{i \text{Li}_4\left(-\frac{b f^{c+d x}}{a+i}\right)}{d^3 \log ^3(f)}+\frac{i x \text{Li}_3\left(\frac{i b f^{c+d x}}{1-i a}\right)}{d^2 \log ^2(f)}-\frac{i x \text{Li}_3\left(-\frac{i b f^{c+d x}}{i a+1}\right)}{d^2 \log ^2(f)}-\frac{i x^2 \text{Li}_2\left(\frac{i b f^{c+d x}}{1-i a}\right)}{2 d \log (f)}+\frac{i x^2 \text{Li}_2\left(-\frac{i b f^{c+d x}}{i a+1}\right)}{2 d \log (f)}+\frac{1}{6} i x^3 \log \left(-i a-i b f^{c+d x}+1\right)-\frac{1}{6} i x^3 \log \left(i a+i b f^{c+d x}+1\right)-\frac{1}{6} i x^3 \log \left(1-\frac{i b f^{c+d x}}{1-i a}\right)+\frac{1}{6} i x^3 \log \left(1+\frac{i b f^{c+d x}}{1+i a}\right)","-\frac{i \text{Li}_4\left(\frac{i b f^{c+d x}}{1-i a}\right)}{d^3 \log ^3(f)}+\frac{i \text{Li}_4\left(-\frac{i b f^{c+d x}}{i a+1}\right)}{d^3 \log ^3(f)}+\frac{i x \text{Li}_3\left(\frac{i b f^{c+d x}}{1-i a}\right)}{d^2 \log ^2(f)}-\frac{i x \text{Li}_3\left(-\frac{i b f^{c+d x}}{i a+1}\right)}{d^2 \log ^2(f)}-\frac{i x^2 \text{Li}_2\left(\frac{i b f^{c+d x}}{1-i a}\right)}{2 d \log (f)}+\frac{i x^2 \text{Li}_2\left(-\frac{i b f^{c+d x}}{i a+1}\right)}{2 d \log (f)}-\frac{1}{6} i x^3 \log \left(1-\frac{i b f^{c+d x}}{1-i a}\right)+\frac{1}{6} i x^3 \log \left(1+\frac{i b f^{c+d x}}{1+i a}\right)+\frac{1}{3} x^3 \tan ^{-1}\left(a+b f^{c+d x}\right)",1,"(I/6)*x^3*Log[1 - I*a - I*b*f^(c + d*x)] - (I/6)*x^3*Log[1 + I*a + I*b*f^(c + d*x)] - (I/6)*x^3*Log[1 - (I*b*f^(c + d*x))/(1 - I*a)] + (I/6)*x^3*Log[1 + (I*b*f^(c + d*x))/(1 + I*a)] - ((I/2)*x^2*PolyLog[2, (I*b*f^(c + d*x))/(1 - I*a)])/(d*Log[f]) + ((I/2)*x^2*PolyLog[2, ((-I)*b*f^(c + d*x))/(1 + I*a)])/(d*Log[f]) + (I*x*PolyLog[3, (I*b*f^(c + d*x))/(1 - I*a)])/(d^2*Log[f]^2) - (I*x*PolyLog[3, ((-I)*b*f^(c + d*x))/(1 + I*a)])/(d^2*Log[f]^2) + (I*PolyLog[4, (b*f^(c + d*x))/(I - a)])/(d^3*Log[f]^3) - (I*PolyLog[4, -((b*f^(c + d*x))/(I + a))])/(d^3*Log[f]^3)","A",0
119,1,25,25,0.0193739,"\int e^{-x} \tan ^{-1}\left(e^x\right) \, dx","Integrate[ArcTan[E^x]/E^x,x]","x-\frac{1}{2} \log \left(e^{2 x}+1\right)-e^{-x} \tan ^{-1}\left(e^x\right)","x-\frac{1}{2} \log \left(e^{2 x}+1\right)-e^{-x} \tan ^{-1}\left(e^x\right)",1,"x - ArcTan[E^x]/E^x - Log[1 + E^(2*x)]/2","A",1
120,1,35,45,0.0381284,"\int \frac{\tan ^{-1}(x)}{(-1+x)^3} \, dx","Integrate[ArcTan[x]/(-1 + x)^3,x]","\frac{1}{8} \left(\log \left(x^2+1\right)-\frac{2}{x-1}-2 \log (1-x)-\frac{4 \tan ^{-1}(x)}{(x-1)^2}\right)","\frac{1}{8} \log \left(x^2+1\right)+\frac{1}{4 (1-x)}-\frac{1}{4} \log (1-x)-\frac{\tan ^{-1}(x)}{2 (1-x)^2}",1,"(-2/(-1 + x) - (4*ArcTan[x])/(-1 + x)^2 - 2*Log[1 - x] + Log[1 + x^2])/8","A",1
121,1,81,64,0.0472097,"\int \frac{\tan ^{-1}(1+2 x)}{(4+3 x)^3} \, dx","Integrate[ArcTan[1 + 2*x]/(4 + 3*x)^3,x]","\frac{-289 \tan ^{-1}(2 x+1)+(3 x+4) ((-15+8 i) (3 x+4) \log ((1+i) x+i)-(15+8 i) (3 x+4) \log (1+(1+i) x)+90 x \log (3 x+4)+120 \log (3 x+4)-51)}{1734 (3 x+4)^2}","-\frac{5}{578} \log \left(2 x^2+2 x+1\right)-\frac{1}{34 (3 x+4)}+\frac{5}{289} \log (3 x+4)-\frac{\tan ^{-1}(2 x+1)}{6 (3 x+4)^2}+\frac{8}{867} \tan ^{-1}(2 x+1)",1,"(-289*ArcTan[1 + 2*x] + (4 + 3*x)*(-51 - (15 - 8*I)*(4 + 3*x)*Log[I + (1 + I)*x] - (15 + 8*I)*(4 + 3*x)*Log[1 + (1 + I)*x] + 120*Log[4 + 3*x] + 90*x*Log[4 + 3*x]))/(1734*(4 + 3*x)^2)","C",1
122,1,22,30,0.0106078,"\int \tan ^{-1}\left(\sqrt{1+x}\right) \, dx","Integrate[ArcTan[Sqrt[1 + x]],x]","(x+2) \tan ^{-1}\left(\sqrt{x+1}\right)-\sqrt{x+1}","-\sqrt{x+1}+x \tan ^{-1}\left(\sqrt{x+1}\right)+2 \tan ^{-1}\left(\sqrt{x+1}\right)",1,"-Sqrt[1 + x] + (2 + x)*ArcTan[Sqrt[1 + x]]","A",1
123,1,5,5,0.0332538,"\int \frac{1}{\left(1+x^2\right) \left(2+\tan ^{-1}(x)\right)} \, dx","Integrate[1/((1 + x^2)*(2 + ArcTan[x])),x]","\log \left(\tan ^{-1}(x)+2\right)","\log \left(\tan ^{-1}(x)+2\right)",1,"Log[2 + ArcTan[x]]","A",1
124,1,17,17,0.0478864,"\int \frac{1}{\left(a+a x^2\right) \left(b-2 b \tan ^{-1}(x)\right)} \, dx","Integrate[1/((a + a*x^2)*(b - 2*b*ArcTan[x])),x]","-\frac{\log \left(2 \tan ^{-1}(x)-1\right)}{2 a b}","-\frac{\log \left(1-2 \tan ^{-1}(x)\right)}{2 a b}",1,"-1/2*Log[-1 + 2*ArcTan[x]]/(a*b)","A",1
125,1,18,18,0.0334781,"\int \frac{x+x^3+(1+x)^2 \tan ^{-1}(x)}{(1+x)^2 \left(1+x^2\right)} \, dx","Integrate[(x + x^3 + (1 + x)^2*ArcTan[x])/((1 + x)^2*(1 + x^2)),x]","\frac{1}{x+1}+\log (x+1)+\frac{1}{2} \tan ^{-1}(x)^2","\frac{1}{x+1}+\log (x+1)+\frac{1}{2} \tan ^{-1}(x)^2",1,"(1 + x)^(-1) + ArcTan[x]^2/2 + Log[1 + x]","A",1
126,1,58,68,0.0641591,"\int -x^3 \tan ^{-1}\left(\sqrt{x}-\sqrt{1+x}\right) \, dx","Integrate[-(x^3*ArcTan[Sqrt[x] - Sqrt[1 + x]]),x]","\frac{1}{8} \tan ^{-1}\left(\sqrt{x}\right)-\frac{1}{840} \sqrt{x} \left(210 x^{7/2} \tan ^{-1}\left(\sqrt{x}-\sqrt{x+1}\right)-15 x^3+21 x^2-35 x+105\right)","\frac{x^{7/2}}{56}-\frac{x^{5/2}}{40}+\frac{x^{3/2}}{24}+\frac{\pi  x^4}{16}-\frac{1}{8} x^4 \tan ^{-1}\left(\sqrt{x}\right)-\frac{\sqrt{x}}{8}+\frac{1}{8} \tan ^{-1}\left(\sqrt{x}\right)",1,"ArcTan[Sqrt[x]]/8 - (Sqrt[x]*(105 - 35*x + 21*x^2 - 15*x^3 + 210*x^(7/2)*ArcTan[Sqrt[x] - Sqrt[1 + x]]))/840","A",1
127,1,53,59,0.0331768,"\int -x^2 \tan ^{-1}\left(\sqrt{x}-\sqrt{1+x}\right) \, dx","Integrate[-(x^2*ArcTan[Sqrt[x] - Sqrt[1 + x]]),x]","\frac{1}{90} \left(-\sqrt{x} \left(30 x^{5/2} \tan ^{-1}\left(\sqrt{x}-\sqrt{x+1}\right)-3 x^2+5 x-15\right)-15 \tan ^{-1}\left(\sqrt{x}\right)\right)","\frac{x^{5/2}}{30}-\frac{x^{3/2}}{18}+\frac{\pi  x^3}{12}-\frac{1}{6} x^3 \tan ^{-1}\left(\sqrt{x}\right)+\frac{\sqrt{x}}{6}-\frac{1}{6} \tan ^{-1}\left(\sqrt{x}\right)",1,"(-15*ArcTan[Sqrt[x]] - Sqrt[x]*(-15 + 5*x - 3*x^2 + 30*x^(5/2)*ArcTan[Sqrt[x] - Sqrt[1 + x]]))/90","A",1
128,1,48,50,0.0303684,"\int -x \tan ^{-1}\left(\sqrt{x}-\sqrt{1+x}\right) \, dx","Integrate[-(x*ArcTan[Sqrt[x] - Sqrt[1 + x]]),x]","\frac{1}{12} \left(3 \tan ^{-1}\left(\sqrt{x}\right)-\sqrt{x} \left(6 x^{3/2} \tan ^{-1}\left(\sqrt{x}-\sqrt{x+1}\right)-x+3\right)\right)","\frac{x^{3/2}}{12}+\frac{\pi  x^2}{8}-\frac{1}{4} x^2 \tan ^{-1}\left(\sqrt{x}\right)-\frac{\sqrt{x}}{4}+\frac{1}{4} \tan ^{-1}\left(\sqrt{x}\right)",1,"(3*ArcTan[Sqrt[x]] - Sqrt[x]*(3 - x + 6*x^(3/2)*ArcTan[Sqrt[x] - Sqrt[1 + x]]))/12","A",1
129,1,39,37,0.4284326,"\int -\tan ^{-1}\left(\sqrt{x}-\sqrt{1+x}\right) \, dx","Integrate[-ArcTan[Sqrt[x] - Sqrt[1 + x]],x]","\frac{\sqrt{x}}{2}-\frac{1}{2} \tan ^{-1}\left(\sqrt{x}\right)-x \tan ^{-1}\left(\sqrt{x}-\sqrt{x+1}\right)","\frac{\pi  x}{4}+\frac{\sqrt{x}}{2}-\frac{1}{2} x \tan ^{-1}\left(\sqrt{x}\right)-\frac{1}{2} \tan ^{-1}\left(\sqrt{x}\right)",1,"Sqrt[x]/2 - ArcTan[Sqrt[x]]/2 - x*ArcTan[Sqrt[x] - Sqrt[1 + x]]","A",1
130,1,84,42,0.2020638,"\int -\frac{\tan ^{-1}\left(\sqrt{x}-\sqrt{1+x}\right)}{x} \, dx","Integrate[-(ArcTan[Sqrt[x] - Sqrt[1 + x]]/x),x]","-\log (x) \tan ^{-1}\left(\sqrt{x}-\sqrt{x+1}\right)+\frac{1}{4} i \left(-2 \text{Li}_2\left(-i \sqrt{x}\right)+2 \text{Li}_2\left(i \sqrt{x}\right)+\left(\log \left(1-i \sqrt{x}\right)-\log \left(1+i \sqrt{x}\right)\right) \log (x)\right)","-\frac{1}{2} i \text{Li}_2\left(-i \sqrt{x}\right)+\frac{1}{2} i \text{Li}_2\left(i \sqrt{x}\right)+\frac{1}{4} \pi  \log (x)",1,"-(ArcTan[Sqrt[x] - Sqrt[1 + x]]*Log[x]) + (I/4)*((Log[1 - I*Sqrt[x]] - Log[1 + I*Sqrt[x]])*Log[x] - 2*PolyLog[2, (-I)*Sqrt[x]] + 2*PolyLog[2, I*Sqrt[x]])","A",1
131,1,40,41,0.0339414,"\int -\frac{\tan ^{-1}\left(\sqrt{x}-\sqrt{1+x}\right)}{x^2} \, dx","Integrate[-(ArcTan[Sqrt[x] - Sqrt[1 + x]]/x^2),x]","\frac{1}{2 \sqrt{x}}+\frac{1}{2} \tan ^{-1}\left(\sqrt{x}\right)+\frac{\tan ^{-1}\left(\sqrt{x}-\sqrt{x+1}\right)}{x}","\frac{1}{2 \sqrt{x}}-\frac{\pi }{4 x}+\frac{\tan ^{-1}\left(\sqrt{x}\right)}{2 x}+\frac{1}{2} \tan ^{-1}\left(\sqrt{x}\right)",1,"1/(2*Sqrt[x]) + ArcTan[Sqrt[x]]/2 + ArcTan[Sqrt[x] - Sqrt[1 + x]]/x","A",1
132,1,48,50,0.0308455,"\int -\frac{\tan ^{-1}\left(\sqrt{x}-\sqrt{1+x}\right)}{x^3} \, dx","Integrate[-(ArcTan[Sqrt[x] - Sqrt[1 + x]]/x^3),x]","-\frac{3 x^2 \tan ^{-1}\left(\sqrt{x}\right)+(3 x-1) \sqrt{x}-6 \tan ^{-1}\left(\sqrt{x}-\sqrt{x+1}\right)}{12 x^2}","\frac{1}{12 x^{3/2}}-\frac{\pi }{8 x^2}+\frac{\tan ^{-1}\left(\sqrt{x}\right)}{4 x^2}-\frac{1}{4 \sqrt{x}}-\frac{1}{4} \tan ^{-1}\left(\sqrt{x}\right)",1,"-1/12*(Sqrt[x]*(-1 + 3*x) + 3*x^2*ArcTan[Sqrt[x]] - 6*ArcTan[Sqrt[x] - Sqrt[1 + x]])/x^2","A",1
133,1,51,59,0.0437363,"\int -\frac{\tan ^{-1}\left(\sqrt{x}-\sqrt{1+x}\right)}{x^4} \, dx","Integrate[-(ArcTan[Sqrt[x] - Sqrt[1 + x]]/x^4),x]","\frac{1}{90} \left(\frac{30 \tan ^{-1}\left(\sqrt{x}-\sqrt{x+1}\right)}{x^3}-\frac{-15 x^2+5 x-3}{x^{5/2}}+15 \tan ^{-1}\left(\sqrt{x}\right)\right)","-\frac{1}{18 x^{3/2}}+\frac{1}{30 x^{5/2}}-\frac{\pi }{12 x^3}+\frac{\tan ^{-1}\left(\sqrt{x}\right)}{6 x^3}+\frac{1}{6 \sqrt{x}}+\frac{1}{6} \tan ^{-1}\left(\sqrt{x}\right)",1,"(-((-3 + 5*x - 15*x^2)/x^(5/2)) + 15*ArcTan[Sqrt[x]] + (30*ArcTan[Sqrt[x] - Sqrt[1 + x]])/x^3)/90","A",1
134,1,63,63,0.0630447,"\int \frac{\tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)^m}{\sqrt{d-\frac{c^2 d x^2}{a}}} \, dx","Integrate[ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^m/Sqrt[d - (c^2*d*x^2)/a],x]","\frac{\sqrt{a-c^2 x^2} \tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)^{m+1}}{c (m+1) \sqrt{d-\frac{c^2 d x^2}{a}}}","\frac{\sqrt{a-c^2 x^2} \tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)^{m+1}}{c (m+1) \sqrt{d-\frac{c^2 d x^2}{a}}}",1,"(Sqrt[a - c^2*x^2]*ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^(1 + m))/(c*(1 + m)*Sqrt[d - (c^2*d*x^2)/a])","A",1
135,1,59,59,0.0307882,"\int \frac{\tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)^2}{\sqrt{d-\frac{c^2 d x^2}{a}}} \, dx","Integrate[ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^2/Sqrt[d - (c^2*d*x^2)/a],x]","\frac{\sqrt{a-c^2 x^2} \tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)^3}{3 c \sqrt{d-\frac{c^2 d x^2}{a}}}","\frac{\sqrt{a-c^2 x^2} \tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)^3}{3 c \sqrt{d-\frac{c^2 d x^2}{a}}}",1,"(Sqrt[a - c^2*x^2]*ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^3)/(3*c*Sqrt[d - (c^2*d*x^2)/a])","A",1
136,1,59,59,0.0223338,"\int \frac{\tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)}{\sqrt{d-\frac{c^2 d x^2}{a}}} \, dx","Integrate[ArcTan[(c*x)/Sqrt[a - c^2*x^2]]/Sqrt[d - (c^2*d*x^2)/a],x]","\frac{\sqrt{a-c^2 x^2} \tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)^2}{2 c \sqrt{d-\frac{c^2 d x^2}{a}}}","\frac{\sqrt{a-c^2 x^2} \tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)^2}{2 c \sqrt{d-\frac{c^2 d x^2}{a}}}",1,"(Sqrt[a - c^2*x^2]*ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^2)/(2*c*Sqrt[d - (c^2*d*x^2)/a])","A",1
137,1,55,55,0.0559309,"\int \frac{1}{\sqrt{d-\frac{c^2 d x^2}{a}} \tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)} \, dx","Integrate[1/(Sqrt[d - (c^2*d*x^2)/a]*ArcTan[(c*x)/Sqrt[a - c^2*x^2]]),x]","\frac{\sqrt{a-c^2 x^2} \log \left(\tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)\right)}{c \sqrt{d-\frac{c^2 d x^2}{a}}}","\frac{\sqrt{a-c^2 x^2} \log \left(\tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)\right)}{c \sqrt{d-\frac{c^2 d x^2}{a}}}",1,"(Sqrt[a - c^2*x^2]*Log[ArcTan[(c*x)/Sqrt[a - c^2*x^2]]])/(c*Sqrt[d - (c^2*d*x^2)/a])","A",1
138,1,57,57,0.0269227,"\int \frac{1}{\sqrt{d-\frac{c^2 d x^2}{a}} \tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)^2} \, dx","Integrate[1/(Sqrt[d - (c^2*d*x^2)/a]*ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^2),x]","-\frac{\sqrt{a-c^2 x^2}}{c \sqrt{d-\frac{c^2 d x^2}{a}} \tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)}","-\frac{\sqrt{a-c^2 x^2}}{c \sqrt{d-\frac{c^2 d x^2}{a}} \tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)}",1,"-(Sqrt[a - c^2*x^2]/(c*Sqrt[d - (c^2*d*x^2)/a]*ArcTan[(c*x)/Sqrt[a - c^2*x^2]]))","A",1
139,1,59,59,0.026845,"\int \frac{1}{\sqrt{d-\frac{c^2 d x^2}{a}} \tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)^3} \, dx","Integrate[1/(Sqrt[d - (c^2*d*x^2)/a]*ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^3),x]","-\frac{\sqrt{a-c^2 x^2}}{2 c \sqrt{d-\frac{c^2 d x^2}{a}} \tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)^2}","-\frac{\sqrt{a-c^2 x^2}}{2 c \sqrt{d-\frac{c^2 d x^2}{a}} \tan ^{-1}\left(\frac{c x}{\sqrt{a-c^2 x^2}}\right)^2}",1,"-1/2*Sqrt[a - c^2*x^2]/(c*Sqrt[d - (c^2*d*x^2)/a]*ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^2)","A",1
140,1,66,72,0.2380399,"\int \frac{\tan ^{-1}\left(\frac{e x}{\sqrt{-\frac{a e^2}{b}-e^2 x^2}}\right)^m}{\sqrt{a+b x^2}} \, dx","Integrate[ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^m/Sqrt[a + b*x^2],x]","\frac{\sqrt{-\frac{e^2 \left(a+b x^2\right)}{b}} \tan ^{-1}\left(\frac{e x}{\sqrt{-\frac{e^2 \left(a+b x^2\right)}{b}}}\right)^{m+1}}{e (m+1) \sqrt{a+b x^2}}","\frac{\sqrt{e^2 \left(-x^2\right)-\frac{a e^2}{b}} \tan ^{-1}\left(\frac{e x}{\sqrt{e^2 \left(-x^2\right)-\frac{a e^2}{b}}}\right)^{m+1}}{e (m+1) \sqrt{a+b x^2}}",1,"(Sqrt[-((e^2*(a + b*x^2))/b)]*ArcTan[(e*x)/Sqrt[-((e^2*(a + b*x^2))/b)]]^(1 + m))/(e*(1 + m)*Sqrt[a + b*x^2])","A",1
141,1,62,68,0.1333993,"\int \frac{\tan ^{-1}\left(\frac{e x}{\sqrt{-\frac{a e^2}{b}-e^2 x^2}}\right)^2}{\sqrt{a+b x^2}} \, dx","Integrate[ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^2/Sqrt[a + b*x^2],x]","\frac{\sqrt{-\frac{e^2 \left(a+b x^2\right)}{b}} \tan ^{-1}\left(\frac{e x}{\sqrt{-\frac{e^2 \left(a+b x^2\right)}{b}}}\right)^3}{3 e \sqrt{a+b x^2}}","\frac{\sqrt{e^2 \left(-x^2\right)-\frac{a e^2}{b}} \tan ^{-1}\left(\frac{e x}{\sqrt{e^2 \left(-x^2\right)-\frac{a e^2}{b}}}\right)^3}{3 e \sqrt{a+b x^2}}",1,"(Sqrt[-((e^2*(a + b*x^2))/b)]*ArcTan[(e*x)/Sqrt[-((e^2*(a + b*x^2))/b)]]^3)/(3*e*Sqrt[a + b*x^2])","A",1
142,1,62,68,0.0622006,"\int \frac{\tan ^{-1}\left(\frac{e x}{\sqrt{-\frac{a e^2}{b}-e^2 x^2}}\right)}{\sqrt{a+b x^2}} \, dx","Integrate[ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]/Sqrt[a + b*x^2],x]","\frac{\sqrt{-\frac{e^2 \left(a+b x^2\right)}{b}} \tan ^{-1}\left(\frac{e x}{\sqrt{-\frac{e^2 \left(a+b x^2\right)}{b}}}\right)^2}{2 e \sqrt{a+b x^2}}","\frac{\sqrt{e^2 \left(-x^2\right)-\frac{a e^2}{b}} \tan ^{-1}\left(\frac{e x}{\sqrt{e^2 \left(-x^2\right)-\frac{a e^2}{b}}}\right)^2}{2 e \sqrt{a+b x^2}}",1,"(Sqrt[-((e^2*(a + b*x^2))/b)]*ArcTan[(e*x)/Sqrt[-((e^2*(a + b*x^2))/b)]]^2)/(2*e*Sqrt[a + b*x^2])","A",1
143,1,58,64,0.1088424,"\int \frac{1}{\sqrt{a+b x^2} \tan ^{-1}\left(\frac{e x}{\sqrt{-\frac{a e^2}{b}-e^2 x^2}}\right)} \, dx","Integrate[1/(Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]),x]","\frac{\sqrt{-\frac{e^2 \left(a+b x^2\right)}{b}} \log \left(\tan ^{-1}\left(\frac{e x}{\sqrt{-\frac{e^2 \left(a+b x^2\right)}{b}}}\right)\right)}{e \sqrt{a+b x^2}}","\frac{\sqrt{e^2 \left(-x^2\right)-\frac{a e^2}{b}} \log \left(\tan ^{-1}\left(\frac{e x}{\sqrt{e^2 \left(-x^2\right)-\frac{a e^2}{b}}}\right)\right)}{e \sqrt{a+b x^2}}",1,"(Sqrt[-((e^2*(a + b*x^2))/b)]*Log[ArcTan[(e*x)/Sqrt[-((e^2*(a + b*x^2))/b)]]])/(e*Sqrt[a + b*x^2])","A",1
144,1,60,66,0.0995529,"\int \frac{1}{\sqrt{a+b x^2} \tan ^{-1}\left(\frac{e x}{\sqrt{-\frac{a e^2}{b}-e^2 x^2}}\right)^2} \, dx","Integrate[1/(Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^2),x]","\frac{e \sqrt{a+b x^2}}{b \sqrt{-\frac{e^2 \left(a+b x^2\right)}{b}} \tan ^{-1}\left(\frac{e x}{\sqrt{-\frac{e^2 \left(a+b x^2\right)}{b}}}\right)}","-\frac{\sqrt{e^2 \left(-x^2\right)-\frac{a e^2}{b}}}{e \sqrt{a+b x^2} \tan ^{-1}\left(\frac{e x}{\sqrt{e^2 \left(-x^2\right)-\frac{a e^2}{b}}}\right)}",1,"(e*Sqrt[a + b*x^2])/(b*Sqrt[-((e^2*(a + b*x^2))/b)]*ArcTan[(e*x)/Sqrt[-((e^2*(a + b*x^2))/b)]])","A",1
145,1,62,68,0.0957321,"\int \frac{1}{\sqrt{a+b x^2} \tan ^{-1}\left(\frac{e x}{\sqrt{-\frac{a e^2}{b}-e^2 x^2}}\right)^3} \, dx","Integrate[1/(Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^3),x]","-\frac{\sqrt{-\frac{e^2 \left(a+b x^2\right)}{b}}}{2 e \sqrt{a+b x^2} \tan ^{-1}\left(\frac{e x}{\sqrt{-\frac{e^2 \left(a+b x^2\right)}{b}}}\right)^2}","-\frac{\sqrt{e^2 \left(-x^2\right)-\frac{a e^2}{b}}}{2 e \sqrt{a+b x^2} \tan ^{-1}\left(\frac{e x}{\sqrt{e^2 \left(-x^2\right)-\frac{a e^2}{b}}}\right)^2}",1,"-1/2*Sqrt[-((e^2*(a + b*x^2))/b)]/(e*Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((e^2*(a + b*x^2))/b)]]^2)","A",1
146,1,166,101,3.0342859,"\int \frac{\tan ^{-1}(c (a+b x)) \log (d (a+b x))}{a+b x} \, dx","Integrate[(ArcTan[c*(a + b*x)]*Log[d*(a + b*x)])/(a + b*x),x]","\frac{2 \log (a+b x) \tan ^{-1}(c (a+b x)) \log (d (a+b x))+i (\text{Li}_2(-i c (a+b x)) \log (d (a+b x))-\text{Li}_2(i c (a+b x)) \log (d (a+b x))+\log (a+b x) \log (i a c+i b c x+1) \log (d (a+b x))-\log (a+b x) \log (1-i c (a+b x)) \log (d (a+b x))-\text{Li}_3(-i c (a+b x))+\text{Li}_3(i c (a+b x)))}{2 b}","\frac{i \text{Li}_2(-i c (a+b x)) \log (d (a+b x))}{2 b}-\frac{i \text{Li}_2(i c (a+b x)) \log (d (a+b x))}{2 b}-\frac{i \text{Li}_3(-i c (a+b x))}{2 b}+\frac{i \text{Li}_3(i c (a+b x))}{2 b}",1,"(2*ArcTan[c*(a + b*x)]*Log[a + b*x]*Log[d*(a + b*x)] + I*(Log[a + b*x]*Log[d*(a + b*x)]*Log[1 + I*a*c + I*b*c*x] - Log[a + b*x]*Log[d*(a + b*x)]*Log[1 - I*c*(a + b*x)] + Log[d*(a + b*x)]*PolyLog[2, (-I)*c*(a + b*x)] - Log[d*(a + b*x)]*PolyLog[2, I*c*(a + b*x)] - PolyLog[3, (-I)*c*(a + b*x)] + PolyLog[3, I*c*(a + b*x)]))/(2*b)","A",0
147,1,61,48,0.1151319,"\int e^{c (a+b x)} \tan ^{-1}(\sinh (a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcTan[Sinh[a*c + b*c*x]],x]","-\frac{\log \left(e^{2 c (a+b x)}+1\right)+e^{c (a+b x)} \tan ^{-1}\left(\frac{1}{2} e^{-c (a+b x)}-\frac{1}{2} e^{c (a+b x)}\right)}{b c}","\frac{e^{a c+b c x} \tan ^{-1}(\sinh (c (a+b x)))}{b c}-\frac{\log \left(e^{2 c (a+b x)}+1\right)}{b c}",1,"-((E^(c*(a + b*x))*ArcTan[1/(2*E^(c*(a + b*x))) - E^(c*(a + b*x))/2] + Log[1 + E^(2*c*(a + b*x))])/(b*c))","A",0
148,1,146,103,0.1568901,"\int e^{c (a+b x)} \tan ^{-1}(\cosh (a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcTan[Cosh[a*c + b*c*x]],x]","\frac{\text{RootSum}\left[\text{$\#$1}^4+6 \text{$\#$1}^2+1\&,\frac{-7 \text{$\#$1}^2 \log \left(e^{c (a+b x)}-\text{$\#$1}\right)+7 \text{$\#$1}^2 a c+7 \text{$\#$1}^2 b c x-\log \left(e^{c (a+b x)}-\text{$\#$1}\right)+a c+b c x}{3 \text{$\#$1}^2+1}\&\right]-4 c (a+b x)+2 e^{c (a+b x)} \tan ^{-1}\left(\frac{1}{2} e^{-c (a+b x)} \left(e^{2 c (a+b x)}+1\right)\right)}{2 b c}","-\frac{\left(1-\sqrt{2}\right) \log \left(e^{2 c (a+b x)}+3-2 \sqrt{2}\right)}{2 b c}-\frac{\left(1+\sqrt{2}\right) \log \left(e^{2 c (a+b x)}+3+2 \sqrt{2}\right)}{2 b c}+\frac{e^{a c+b c x} \tan ^{-1}(\cosh (c (a+b x)))}{b c}",1,"(-4*c*(a + b*x) + 2*E^(c*(a + b*x))*ArcTan[(1 + E^(2*c*(a + b*x)))/(2*E^(c*(a + b*x)))] + RootSum[1 + 6*#1^2 + #1^4 & , (a*c + b*c*x - Log[E^(c*(a + b*x)) - #1] + 7*a*c*#1^2 + 7*b*c*x*#1^2 - 7*Log[E^(c*(a + b*x)) - #1]*#1^2)/(1 + 3*#1^2) & ])/(2*b*c)","C",0
149,1,89,180,0.1258102,"\int e^{c (a+b x)} \tan ^{-1}(\tanh (a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcTan[Tanh[a*c + b*c*x]],x]","\frac{\text{RootSum}\left[\text{$\#$1}^4+1\&,\frac{-\log \left(e^{c (a+b x)}-\text{$\#$1}\right)+a c+b c x}{\text{$\#$1}}\&\right]+2 e^{c (a+b x)} \tan ^{-1}\left(\frac{e^{2 c (a+b x)}-1}{e^{2 c (a+b x)}+1}\right)}{2 b c}","-\frac{\log \left(e^{2 c (a+b x)}-\sqrt{2} e^{a c+b c x}+1\right)}{2 \sqrt{2} b c}+\frac{\log \left(e^{2 c (a+b x)}+\sqrt{2} e^{a c+b c x}+1\right)}{2 \sqrt{2} b c}+\frac{\tan ^{-1}\left(1-\sqrt{2} e^{a c+b c x}\right)}{\sqrt{2} b c}-\frac{\tan ^{-1}\left(\sqrt{2} e^{a c+b c x}+1\right)}{\sqrt{2} b c}+\frac{e^{a c+b c x} \tan ^{-1}(\tanh (c (a+b x)))}{b c}",1,"(2*E^(c*(a + b*x))*ArcTan[(-1 + E^(2*c*(a + b*x)))/(1 + E^(2*c*(a + b*x)))] + RootSum[1 + #1^4 & , (a*c + b*c*x - Log[E^(c*(a + b*x)) - #1])/#1 & ])/(2*b*c)","C",0
150,1,89,180,0.1237507,"\int e^{c (a+b x)} \tan ^{-1}(\coth (a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcTan[Coth[a*c + b*c*x]],x]","\frac{\text{RootSum}\left[\text{$\#$1}^4+1\&,\frac{\log \left(e^{c (a+b x)}-\text{$\#$1}\right)-a c-b c x}{\text{$\#$1}}\&\right]+2 e^{c (a+b x)} \tan ^{-1}\left(\frac{e^{2 c (a+b x)}+1}{e^{2 c (a+b x)}-1}\right)}{2 b c}","\frac{\log \left(e^{2 c (a+b x)}-\sqrt{2} e^{a c+b c x}+1\right)}{2 \sqrt{2} b c}-\frac{\log \left(e^{2 c (a+b x)}+\sqrt{2} e^{a c+b c x}+1\right)}{2 \sqrt{2} b c}-\frac{\tan ^{-1}\left(1-\sqrt{2} e^{a c+b c x}\right)}{\sqrt{2} b c}+\frac{\tan ^{-1}\left(\sqrt{2} e^{a c+b c x}+1\right)}{\sqrt{2} b c}+\frac{e^{a c+b c x} \tan ^{-1}(\coth (c (a+b x)))}{b c}",1,"(2*E^(c*(a + b*x))*ArcTan[(1 + E^(2*c*(a + b*x)))/(-1 + E^(2*c*(a + b*x)))] + RootSum[1 + #1^4 & , (-(a*c) - b*c*x + Log[E^(c*(a + b*x)) - #1])/#1 & ])/(2*b*c)","C",0
151,1,145,103,0.1600343,"\int e^{c (a+b x)} \tan ^{-1}(\text{sech}(a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcTan[Sech[a*c + b*c*x]],x]","\frac{\text{RootSum}\left[\text{$\#$1}^4+6 \text{$\#$1}^2+1\&,\frac{7 \text{$\#$1}^2 \log \left(e^{c (a+b x)}-\text{$\#$1}\right)-7 \text{$\#$1}^2 a c-7 \text{$\#$1}^2 b c x+\log \left(e^{c (a+b x)}-\text{$\#$1}\right)-a c-b c x}{3 \text{$\#$1}^2+1}\&\right]+4 c (a+b x)+2 e^{c (a+b x)} \tan ^{-1}\left(\frac{2 e^{c (a+b x)}}{e^{2 c (a+b x)}+1}\right)}{2 b c}","\frac{\left(1-\sqrt{2}\right) \log \left(e^{2 c (a+b x)}+3-2 \sqrt{2}\right)}{2 b c}+\frac{\left(1+\sqrt{2}\right) \log \left(e^{2 c (a+b x)}+3+2 \sqrt{2}\right)}{2 b c}+\frac{e^{a c+b c x} \tan ^{-1}(\text{sech}(c (a+b x)))}{b c}",1,"(4*c*(a + b*x) + 2*E^(c*(a + b*x))*ArcTan[(2*E^(c*(a + b*x)))/(1 + E^(2*c*(a + b*x)))] + RootSum[1 + 6*#1^2 + #1^4 & , (-(a*c) - b*c*x + Log[E^(c*(a + b*x)) - #1] - 7*a*c*#1^2 - 7*b*c*x*#1^2 + 7*Log[E^(c*(a + b*x)) - #1]*#1^2)/(1 + 3*#1^2) & ])/(2*b*c)","C",0
152,1,57,47,0.1156837,"\int e^{c (a+b x)} \tan ^{-1}(\text{csch}(a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcTan[Csch[a*c + b*c*x]],x]","\frac{\log \left(e^{2 c (a+b x)}+1\right)+e^{c (a+b x)} \tan ^{-1}\left(\frac{2 e^{c (a+b x)}}{e^{2 c (a+b x)}-1}\right)}{b c}","\frac{\log \left(e^{2 c (a+b x)}+1\right)}{b c}+\frac{e^{a c+b c x} \tan ^{-1}(\text{csch}(c (a+b x)))}{b c}",1,"(E^(c*(a + b*x))*ArcTan[(2*E^(c*(a + b*x)))/(-1 + E^(2*c*(a + b*x)))] + Log[1 + E^(2*c*(a + b*x))])/(b*c)","A",0
153,1,116,163,0.2987129,"\int \frac{\left(a+b \tan ^{-1}\left(c x^n\right)\right) \left(d+e \log \left(f x^m\right)\right)}{x} \, dx","Integrate[((a + b*ArcTan[c*x^n])*(d + e*Log[f*x^m]))/x,x]","\frac{b c x^n \left(d+e \log \left(f x^m\right)\right) \, _3F_2\left(\frac{1}{2},\frac{1}{2},1;\frac{3}{2},\frac{3}{2};-c^2 x^{2 n}\right)}{n}-\frac{b c e m x^n \, _4F_3\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},1;\frac{3}{2},\frac{3}{2},\frac{3}{2};-c^2 x^{2 n}\right)}{n^2}+\frac{1}{2} a \log (x) \left(2 d+2 e \log \left(f x^m\right)-e m \log (x)\right)","a d \log (x)+\frac{a e \log ^2\left(f x^m\right)}{2 m}+\frac{i b d \text{Li}_2\left(-i c x^n\right)}{2 n}-\frac{i b d \text{Li}_2\left(i c x^n\right)}{2 n}+\frac{i b e \text{Li}_2\left(-i c x^n\right) \log \left(f x^m\right)}{2 n}-\frac{i b e \text{Li}_2\left(i c x^n\right) \log \left(f x^m\right)}{2 n}-\frac{i b e m \text{Li}_3\left(-i c x^n\right)}{2 n^2}+\frac{i b e m \text{Li}_3\left(i c x^n\right)}{2 n^2}",1,"-((b*c*e*m*x^n*HypergeometricPFQ[{1/2, 1/2, 1/2, 1}, {3/2, 3/2, 3/2}, -(c^2*x^(2*n))])/n^2) + (b*c*x^n*HypergeometricPFQ[{1/2, 1/2, 1}, {3/2, 3/2}, -(c^2*x^(2*n))]*(d + e*Log[f*x^m]))/n + (a*Log[x]*(2*d - e*m*Log[x] + 2*e*Log[f*x^m]))/2","C",1