1,1,51,0,0.0251326,"\int x^5 \cot ^{-1}(a x) \, dx","Int[x^5*ArcCot[a*x],x]","-\frac{x^3}{18 a^3}+\frac{x}{6 a^5}-\frac{\tan ^{-1}(a x)}{6 a^6}+\frac{x^5}{30 a}+\frac{1}{6} x^6 \cot ^{-1}(a x)","-\frac{x^3}{18 a^3}+\frac{x}{6 a^5}-\frac{\tan ^{-1}(a x)}{6 a^6}+\frac{x^5}{30 a}+\frac{1}{6} x^6 \cot ^{-1}(a x)",1,"x/(6*a^5) - x^3/(18*a^3) + x^5/(30*a) + (x^6*ArcCot[a*x])/6 - ArcTan[a*x]/(6*a^6)","A",4,3,8,0.3750,1,"{4853, 302, 203}"
2,1,49,0,0.0363023,"\int x^4 \cot ^{-1}(a x) \, dx","Int[x^4*ArcCot[a*x],x]","-\frac{x^2}{10 a^3}+\frac{\log \left(a^2 x^2+1\right)}{10 a^5}+\frac{x^4}{20 a}+\frac{1}{5} x^5 \cot ^{-1}(a x)","-\frac{x^2}{10 a^3}+\frac{\log \left(a^2 x^2+1\right)}{10 a^5}+\frac{x^4}{20 a}+\frac{1}{5} x^5 \cot ^{-1}(a x)",1,"-x^2/(10*a^3) + x^4/(20*a) + (x^5*ArcCot[a*x])/5 + Log[1 + a^2*x^2]/(10*a^5)","A",4,3,8,0.3750,1,"{4853, 266, 43}"
3,1,41,0,0.0208234,"\int x^3 \cot ^{-1}(a x) \, dx","Int[x^3*ArcCot[a*x],x]","-\frac{x}{4 a^3}+\frac{\tan ^{-1}(a x)}{4 a^4}+\frac{x^3}{12 a}+\frac{1}{4} x^4 \cot ^{-1}(a x)","-\frac{x}{4 a^3}+\frac{\tan ^{-1}(a x)}{4 a^4}+\frac{x^3}{12 a}+\frac{1}{4} x^4 \cot ^{-1}(a x)",1,"-x/(4*a^3) + x^3/(12*a) + (x^4*ArcCot[a*x])/4 + ArcTan[a*x]/(4*a^4)","A",4,3,8,0.3750,1,"{4853, 302, 203}"
4,1,39,0,0.0260955,"\int x^2 \cot ^{-1}(a x) \, dx","Int[x^2*ArcCot[a*x],x]","-\frac{\log \left(a^2 x^2+1\right)}{6 a^3}+\frac{x^2}{6 a}+\frac{1}{3} x^3 \cot ^{-1}(a x)","-\frac{\log \left(a^2 x^2+1\right)}{6 a^3}+\frac{x^2}{6 a}+\frac{1}{3} x^3 \cot ^{-1}(a x)",1,"x^2/(6*a) + (x^3*ArcCot[a*x])/3 - Log[1 + a^2*x^2]/(6*a^3)","A",4,3,8,0.3750,1,"{4853, 266, 43}"
5,1,31,0,0.0115336,"\int x \cot ^{-1}(a x) \, dx","Int[x*ArcCot[a*x],x]","-\frac{\tan ^{-1}(a x)}{2 a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)+\frac{x}{2 a}","-\frac{\tan ^{-1}(a x)}{2 a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)+\frac{x}{2 a}",1,"x/(2*a) + (x^2*ArcCot[a*x])/2 - ArcTan[a*x]/(2*a^2)","A",3,3,6,0.5000,1,"{4853, 321, 203}"
6,1,24,0,0.0059009,"\int \cot ^{-1}(a x) \, dx","Int[ArcCot[a*x],x]","\frac{\log \left(a^2 x^2+1\right)}{2 a}+x \cot ^{-1}(a x)","\frac{\log \left(a^2 x^2+1\right)}{2 a}+x \cot ^{-1}(a x)",1,"x*ArcCot[a*x] + Log[1 + a^2*x^2]/(2*a)","A",2,2,4,0.5000,1,"{4847, 260}"
7,1,37,0,0.0252205,"\int \frac{\cot ^{-1}(a x)}{x} \, dx","Int[ArcCot[a*x]/x,x]","\frac{1}{2} i \text{PolyLog}\left(2,\frac{i}{a x}\right)-\frac{1}{2} i \text{PolyLog}\left(2,-\frac{i}{a x}\right)","\frac{1}{2} i \text{PolyLog}\left(2,\frac{i}{a x}\right)-\frac{1}{2} i \text{PolyLog}\left(2,-\frac{i}{a x}\right)",1,"(-I/2)*PolyLog[2, (-I)/(a*x)] + (I/2)*PolyLog[2, I/(a*x)]","A",3,2,8,0.2500,1,"{4849, 2391}"
8,1,30,0,0.0178592,"\int \frac{\cot ^{-1}(a x)}{x^2} \, dx","Int[ArcCot[a*x]/x^2,x]","\frac{1}{2} a \log \left(a^2 x^2+1\right)-a \log (x)-\frac{\cot ^{-1}(a x)}{x}","\frac{1}{2} a \log \left(a^2 x^2+1\right)-a \log (x)-\frac{\cot ^{-1}(a x)}{x}",1,"-(ArcCot[a*x]/x) - a*Log[x] + (a*Log[1 + a^2*x^2])/2","A",5,5,8,0.6250,1,"{4853, 266, 36, 29, 31}"
9,1,31,0,0.0148004,"\int \frac{\cot ^{-1}(a x)}{x^3} \, dx","Int[ArcCot[a*x]/x^3,x]","\frac{1}{2} a^2 \tan ^{-1}(a x)-\frac{\cot ^{-1}(a x)}{2 x^2}+\frac{a}{2 x}","\frac{1}{2} a^2 \tan ^{-1}(a x)-\frac{\cot ^{-1}(a x)}{2 x^2}+\frac{a}{2 x}",1,"a/(2*x) - ArcCot[a*x]/(2*x^2) + (a^2*ArcTan[a*x])/2","A",3,3,8,0.3750,1,"{4853, 325, 203}"
10,1,46,0,0.0271508,"\int \frac{\cot ^{-1}(a x)}{x^4} \, dx","Int[ArcCot[a*x]/x^4,x]","-\frac{1}{6} a^3 \log \left(a^2 x^2+1\right)+\frac{1}{3} a^3 \log (x)+\frac{a}{6 x^2}-\frac{\cot ^{-1}(a x)}{3 x^3}","-\frac{1}{6} a^3 \log \left(a^2 x^2+1\right)+\frac{1}{3} a^3 \log (x)+\frac{a}{6 x^2}-\frac{\cot ^{-1}(a x)}{3 x^3}",1,"a/(6*x^2) - ArcCot[a*x]/(3*x^3) + (a^3*Log[x])/3 - (a^3*Log[1 + a^2*x^2])/6","A",4,3,8,0.3750,1,"{4853, 266, 44}"
11,1,41,0,0.018415,"\int \frac{\cot ^{-1}(a x)}{x^5} \, dx","Int[ArcCot[a*x]/x^5,x]","-\frac{a^3}{4 x}-\frac{1}{4} a^4 \tan ^{-1}(a x)+\frac{a}{12 x^3}-\frac{\cot ^{-1}(a x)}{4 x^4}","-\frac{a^3}{4 x}-\frac{1}{4} a^4 \tan ^{-1}(a x)+\frac{a}{12 x^3}-\frac{\cot ^{-1}(a x)}{4 x^4}",1,"a/(12*x^3) - a^3/(4*x) - ArcCot[a*x]/(4*x^4) - (a^4*ArcTan[a*x])/4","A",4,3,8,0.3750,1,"{4853, 325, 203}"
12,1,104,0,0.2214754,"\int x^5 \cot ^{-1}(a x)^2 \, dx","Int[x^5*ArcCot[a*x]^2,x]","\frac{x^4}{60 a^2}-\frac{4 x^2}{45 a^4}+\frac{23 \log \left(a^2 x^2+1\right)}{90 a^6}-\frac{x^3 \cot ^{-1}(a x)}{9 a^3}+\frac{x \cot ^{-1}(a x)}{3 a^5}+\frac{\cot ^{-1}(a x)^2}{6 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^2+\frac{x^5 \cot ^{-1}(a x)}{15 a}","\frac{x^4}{60 a^2}-\frac{4 x^2}{45 a^4}+\frac{23 \log \left(a^2 x^2+1\right)}{90 a^6}-\frac{x^3 \cot ^{-1}(a x)}{9 a^3}+\frac{x \cot ^{-1}(a x)}{3 a^5}+\frac{\cot ^{-1}(a x)^2}{6 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^2+\frac{x^5 \cot ^{-1}(a x)}{15 a}",1,"(-4*x^2)/(45*a^4) + x^4/(60*a^2) + (x*ArcCot[a*x])/(3*a^5) - (x^3*ArcCot[a*x])/(9*a^3) + (x^5*ArcCot[a*x])/(15*a) + ArcCot[a*x]^2/(6*a^6) + (x^6*ArcCot[a*x]^2)/6 + (23*Log[1 + a^2*x^2])/(90*a^6)","A",15,7,10,0.7000,1,"{4853, 4917, 266, 43, 4847, 260, 4885}"
13,1,135,0,0.2108671,"\int x^4 \cot ^{-1}(a x)^2 \, dx","Int[x^4*ArcCot[a*x]^2,x]","\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{5 a^5}+\frac{x^3}{30 a^2}-\frac{x^2 \cot ^{-1}(a x)}{5 a^3}-\frac{3 x}{10 a^4}+\frac{3 \tan ^{-1}(a x)}{10 a^5}+\frac{i \cot ^{-1}(a x)^2}{5 a^5}-\frac{2 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^2+\frac{x^4 \cot ^{-1}(a x)}{10 a}","\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{5 a^5}+\frac{x^3}{30 a^2}-\frac{x^2 \cot ^{-1}(a x)}{5 a^3}-\frac{3 x}{10 a^4}+\frac{3 \tan ^{-1}(a x)}{10 a^5}+\frac{i \cot ^{-1}(a x)^2}{5 a^5}-\frac{2 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^2+\frac{x^4 \cot ^{-1}(a x)}{10 a}",1,"(-3*x)/(10*a^4) + x^3/(30*a^2) - (x^2*ArcCot[a*x])/(5*a^3) + (x^4*ArcCot[a*x])/(10*a) + ((I/5)*ArcCot[a*x]^2)/a^5 + (x^5*ArcCot[a*x]^2)/5 + (3*ArcTan[a*x])/(10*a^5) - (2*ArcCot[a*x]*Log[2/(1 + I*a*x)])/(5*a^5) + ((I/5)*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^5","A",14,9,10,0.9000,1,"{4853, 4917, 302, 203, 321, 4921, 4855, 2402, 2315}"
14,1,80,0,0.1456589,"\int x^3 \cot ^{-1}(a x)^2 \, dx","Int[x^3*ArcCot[a*x]^2,x]","\frac{x^2}{12 a^2}-\frac{\log \left(a^2 x^2+1\right)}{3 a^4}-\frac{x \cot ^{-1}(a x)}{2 a^3}-\frac{\cot ^{-1}(a x)^2}{4 a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^2+\frac{x^3 \cot ^{-1}(a x)}{6 a}","\frac{x^2}{12 a^2}-\frac{\log \left(a^2 x^2+1\right)}{3 a^4}-\frac{x \cot ^{-1}(a x)}{2 a^3}-\frac{\cot ^{-1}(a x)^2}{4 a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^2+\frac{x^3 \cot ^{-1}(a x)}{6 a}",1,"x^2/(12*a^2) - (x*ArcCot[a*x])/(2*a^3) + (x^3*ArcCot[a*x])/(6*a) - ArcCot[a*x]^2/(4*a^4) + (x^4*ArcCot[a*x]^2)/4 - Log[1 + a^2*x^2]/(3*a^4)","A",10,7,10,0.7000,1,"{4853, 4917, 266, 43, 4847, 260, 4885}"
15,1,111,0,0.1401064,"\int x^2 \cot ^{-1}(a x)^2 \, dx","Int[x^2*ArcCot[a*x]^2,x]","-\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{3 a^3}+\frac{x}{3 a^2}-\frac{\tan ^{-1}(a x)}{3 a^3}-\frac{i \cot ^{-1}(a x)^2}{3 a^3}+\frac{2 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{3 a^3}+\frac{1}{3} x^3 \cot ^{-1}(a x)^2+\frac{x^2 \cot ^{-1}(a x)}{3 a}","-\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{3 a^3}+\frac{x}{3 a^2}-\frac{\tan ^{-1}(a x)}{3 a^3}-\frac{i \cot ^{-1}(a x)^2}{3 a^3}+\frac{2 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{3 a^3}+\frac{1}{3} x^3 \cot ^{-1}(a x)^2+\frac{x^2 \cot ^{-1}(a x)}{3 a}",1,"x/(3*a^2) + (x^2*ArcCot[a*x])/(3*a) - ((I/3)*ArcCot[a*x]^2)/a^3 + (x^3*ArcCot[a*x]^2)/3 - ArcTan[a*x]/(3*a^3) + (2*ArcCot[a*x]*Log[2/(1 + I*a*x)])/(3*a^3) - ((I/3)*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3","A",9,8,10,0.8000,1,"{4853, 4917, 321, 203, 4921, 4855, 2402, 2315}"
16,1,53,0,0.0719063,"\int x \cot ^{-1}(a x)^2 \, dx","Int[x*ArcCot[a*x]^2,x]","\frac{\log \left(a^2 x^2+1\right)}{2 a^2}+\frac{\cot ^{-1}(a x)^2}{2 a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)^2+\frac{x \cot ^{-1}(a x)}{a}","\frac{\log \left(a^2 x^2+1\right)}{2 a^2}+\frac{\cot ^{-1}(a x)^2}{2 a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)^2+\frac{x \cot ^{-1}(a x)}{a}",1,"(x*ArcCot[a*x])/a + ArcCot[a*x]^2/(2*a^2) + (x^2*ArcCot[a*x]^2)/2 + Log[1 + a^2*x^2]/(2*a^2)","A",5,5,8,0.6250,1,"{4853, 4917, 4847, 260, 4885}"
17,1,67,0,0.074329,"\int \cot ^{-1}(a x)^2 \, dx","Int[ArcCot[a*x]^2,x]","\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a}+x \cot ^{-1}(a x)^2+\frac{i \cot ^{-1}(a x)^2}{a}-\frac{2 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{a}","\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a}+x \cot ^{-1}(a x)^2+\frac{i \cot ^{-1}(a x)^2}{a}-\frac{2 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{a}",1,"(I*ArcCot[a*x]^2)/a + x*ArcCot[a*x]^2 - (2*ArcCot[a*x]*Log[2/(1 + I*a*x)])/a + (I*PolyLog[2, 1 - 2/(1 + I*a*x)])/a","A",5,5,6,0.8333,1,"{4847, 4921, 4855, 2402, 2315}"
18,1,116,0,0.2129245,"\int \frac{\cot ^{-1}(a x)^2}{x} \, dx","Int[ArcCot[a*x]^2/x,x]","-\frac{1}{2} \text{PolyLog}\left(3,1-\frac{2 i}{a x+i}\right)+\frac{1}{2} \text{PolyLog}\left(3,1-\frac{2 a x}{a x+i}\right)-i \cot ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2 i}{a x+i}\right)+i \cot ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2 a x}{a x+i}\right)+2 \cot ^{-1}(a x)^2 \coth ^{-1}\left(1-\frac{2}{1+i a x}\right)","-\frac{1}{2} \text{PolyLog}\left(3,1-\frac{2 i}{a x+i}\right)+\frac{1}{2} \text{PolyLog}\left(3,1-\frac{2 a x}{a x+i}\right)-i \cot ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2 i}{a x+i}\right)+i \cot ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2 a x}{a x+i}\right)+2 \cot ^{-1}(a x)^2 \coth ^{-1}\left(1-\frac{2}{1+i a x}\right)",1,"2*ArcCot[a*x]^2*ArcCoth[1 - 2/(1 + I*a*x)] - I*ArcCot[a*x]*PolyLog[2, 1 - (2*I)/(I + a*x)] + I*ArcCot[a*x]*PolyLog[2, 1 - (2*a*x)/(I + a*x)] - PolyLog[3, 1 - (2*I)/(I + a*x)]/2 + PolyLog[3, 1 - (2*a*x)/(I + a*x)]/2","A",6,5,10,0.5000,1,"{4851, 4989, 4885, 4993, 6610}"
19,1,66,0,0.1052411,"\int \frac{\cot ^{-1}(a x)^2}{x^2} \, dx","Int[ArcCot[a*x]^2/x^2,x]","-i a \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-i a \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{x}-2 a \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)","-i a \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-i a \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{x}-2 a \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)",1,"(-I)*a*ArcCot[a*x]^2 - ArcCot[a*x]^2/x - 2*a*ArcCot[a*x]*Log[2 - 2/(1 - I*a*x)] - I*a*PolyLog[2, -1 + 2/(1 - I*a*x)]","A",4,4,10,0.4000,1,"{4853, 4925, 4869, 2447}"
20,1,59,0,0.087978,"\int \frac{\cot ^{-1}(a x)^2}{x^3} \, dx","Int[ArcCot[a*x]^2/x^3,x]","-\frac{1}{2} a^2 \log \left(a^2 x^2+1\right)+a^2 \log (x)-\frac{1}{2} a^2 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{2 x^2}+\frac{a \cot ^{-1}(a x)}{x}","-\frac{1}{2} a^2 \log \left(a^2 x^2+1\right)+a^2 \log (x)-\frac{1}{2} a^2 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{2 x^2}+\frac{a \cot ^{-1}(a x)}{x}",1,"(a*ArcCot[a*x])/x - (a^2*ArcCot[a*x]^2)/2 - ArcCot[a*x]^2/(2*x^2) + a^2*Log[x] - (a^2*Log[1 + a^2*x^2])/2","A",8,7,10,0.7000,1,"{4853, 4919, 266, 36, 29, 31, 4885}"
21,1,113,0,0.1613921,"\int \frac{\cot ^{-1}(a x)^2}{x^4} \, dx","Int[ArcCot[a*x]^2/x^4,x]","\frac{1}{3} i a^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-\frac{a^2}{3 x}-\frac{1}{3} a^3 \tan ^{-1}(a x)+\frac{1}{3} i a^3 \cot ^{-1}(a x)^2+\frac{2}{3} a^3 \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)+\frac{a \cot ^{-1}(a x)}{3 x^2}-\frac{\cot ^{-1}(a x)^2}{3 x^3}","\frac{1}{3} i a^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-\frac{a^2}{3 x}-\frac{1}{3} a^3 \tan ^{-1}(a x)+\frac{1}{3} i a^3 \cot ^{-1}(a x)^2+\frac{2}{3} a^3 \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)+\frac{a \cot ^{-1}(a x)}{3 x^2}-\frac{\cot ^{-1}(a x)^2}{3 x^3}",1,"-a^2/(3*x) + (a*ArcCot[a*x])/(3*x^2) + (I/3)*a^3*ArcCot[a*x]^2 - ArcCot[a*x]^2/(3*x^3) - (a^3*ArcTan[a*x])/3 + (2*a^3*ArcCot[a*x]*Log[2 - 2/(1 - I*a*x)])/3 + (I/3)*a^3*PolyLog[2, -1 + 2/(1 - I*a*x)]","A",8,7,10,0.7000,1,"{4853, 4919, 325, 203, 4925, 4869, 2447}"
22,1,89,0,0.1551203,"\int \frac{\cot ^{-1}(a x)^2}{x^5} \, dx","Int[ArcCot[a*x]^2/x^5,x]","-\frac{a^2}{12 x^2}+\frac{1}{3} a^4 \log \left(a^2 x^2+1\right)-\frac{2}{3} a^4 \log (x)+\frac{1}{4} a^4 \cot ^{-1}(a x)^2-\frac{a^3 \cot ^{-1}(a x)}{2 x}+\frac{a \cot ^{-1}(a x)}{6 x^3}-\frac{\cot ^{-1}(a x)^2}{4 x^4}","-\frac{a^2}{12 x^2}+\frac{1}{3} a^4 \log \left(a^2 x^2+1\right)-\frac{2}{3} a^4 \log (x)+\frac{1}{4} a^4 \cot ^{-1}(a x)^2-\frac{a^3 \cot ^{-1}(a x)}{2 x}+\frac{a \cot ^{-1}(a x)}{6 x^3}-\frac{\cot ^{-1}(a x)^2}{4 x^4}",1,"-a^2/(12*x^2) + (a*ArcCot[a*x])/(6*x^3) - (a^3*ArcCot[a*x])/(2*x) + (a^4*ArcCot[a*x]^2)/4 - ArcCot[a*x]^2/(4*x^4) - (2*a^4*Log[x])/3 + (a^4*Log[1 + a^2*x^2])/3","A",13,8,10,0.8000,1,"{4853, 4919, 266, 44, 36, 29, 31, 4885}"
23,1,194,0,0.6666362,"\int x^5 \cot ^{-1}(a x)^3 \, dx","Int[x^5*ArcCot[a*x]^3,x]","\frac{23 i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{30 a^6}+\frac{x^3}{60 a^3}+\frac{x^4 \cot ^{-1}(a x)}{20 a^2}-\frac{x^3 \cot ^{-1}(a x)^2}{6 a^3}-\frac{4 x^2 \cot ^{-1}(a x)}{15 a^4}-\frac{19 x}{60 a^5}+\frac{19 \tan ^{-1}(a x)}{60 a^6}+\frac{x \cot ^{-1}(a x)^2}{2 a^5}+\frac{\cot ^{-1}(a x)^3}{6 a^6}+\frac{23 i \cot ^{-1}(a x)^2}{30 a^6}-\frac{23 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{15 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^3+\frac{x^5 \cot ^{-1}(a x)^2}{10 a}","\frac{23 i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{30 a^6}+\frac{x^3}{60 a^3}+\frac{x^4 \cot ^{-1}(a x)}{20 a^2}-\frac{x^3 \cot ^{-1}(a x)^2}{6 a^3}-\frac{4 x^2 \cot ^{-1}(a x)}{15 a^4}-\frac{19 x}{60 a^5}+\frac{19 \tan ^{-1}(a x)}{60 a^6}+\frac{x \cot ^{-1}(a x)^2}{2 a^5}+\frac{\cot ^{-1}(a x)^3}{6 a^6}+\frac{23 i \cot ^{-1}(a x)^2}{30 a^6}-\frac{23 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{15 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^3+\frac{x^5 \cot ^{-1}(a x)^2}{10 a}",1,"(-19*x)/(60*a^5) + x^3/(60*a^3) - (4*x^2*ArcCot[a*x])/(15*a^4) + (x^4*ArcCot[a*x])/(20*a^2) + (((23*I)/30)*ArcCot[a*x]^2)/a^6 + (x*ArcCot[a*x]^2)/(2*a^5) - (x^3*ArcCot[a*x]^2)/(6*a^3) + (x^5*ArcCot[a*x]^2)/(10*a) + ArcCot[a*x]^3/(6*a^6) + (x^6*ArcCot[a*x]^3)/6 + (19*ArcTan[a*x])/(60*a^6) - (23*ArcCot[a*x]*Log[2/(1 + I*a*x)])/(15*a^6) + (((23*I)/30)*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^6","A",33,11,10,1.100,1,"{4853, 4917, 302, 203, 321, 4921, 4855, 2402, 2315, 4847, 4885}"
24,1,205,0,0.5174345,"\int x^4 \cot ^{-1}(a x)^3 \, dx","Int[x^4*ArcCot[a*x]^3,x]","-\frac{3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{10 a^5}+\frac{3 i \cot ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{5 a^5}+\frac{x^2}{20 a^3}-\frac{\log \left(a^2 x^2+1\right)}{2 a^5}+\frac{x^3 \cot ^{-1}(a x)}{10 a^2}-\frac{3 x^2 \cot ^{-1}(a x)^2}{10 a^3}-\frac{9 x \cot ^{-1}(a x)}{10 a^4}+\frac{i \cot ^{-1}(a x)^3}{5 a^5}-\frac{9 \cot ^{-1}(a x)^2}{20 a^5}-\frac{3 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)^2}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^3+\frac{3 x^4 \cot ^{-1}(a x)^2}{20 a}","-\frac{3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{10 a^5}+\frac{3 i \cot ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{5 a^5}+\frac{x^2}{20 a^3}-\frac{\log \left(a^2 x^2+1\right)}{2 a^5}+\frac{x^3 \cot ^{-1}(a x)}{10 a^2}-\frac{3 x^2 \cot ^{-1}(a x)^2}{10 a^3}-\frac{9 x \cot ^{-1}(a x)}{10 a^4}+\frac{i \cot ^{-1}(a x)^3}{5 a^5}-\frac{9 \cot ^{-1}(a x)^2}{20 a^5}-\frac{3 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)^2}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^3+\frac{3 x^4 \cot ^{-1}(a x)^2}{20 a}",1,"x^2/(20*a^3) - (9*x*ArcCot[a*x])/(10*a^4) + (x^3*ArcCot[a*x])/(10*a^2) - (9*ArcCot[a*x]^2)/(20*a^5) - (3*x^2*ArcCot[a*x]^2)/(10*a^3) + (3*x^4*ArcCot[a*x]^2)/(20*a) + ((I/5)*ArcCot[a*x]^3)/a^5 + (x^5*ArcCot[a*x]^3)/5 - (3*ArcCot[a*x]^2*Log[2/(1 + I*a*x)])/(5*a^5) - Log[1 + a^2*x^2]/(2*a^5) + (((3*I)/5)*ArcCot[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^5 - (3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(10*a^5)","A",22,11,10,1.100,1,"{4853, 4917, 266, 43, 4847, 260, 4885, 4921, 4855, 4995, 6610}"
25,1,148,0,0.3855438,"\int x^3 \cot ^{-1}(a x)^3 \, dx","Int[x^3*ArcCot[a*x]^3,x]","-\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^4}+\frac{x^2 \cot ^{-1}(a x)}{4 a^2}+\frac{x}{4 a^3}-\frac{\tan ^{-1}(a x)}{4 a^4}-\frac{3 x \cot ^{-1}(a x)^2}{4 a^3}-\frac{\cot ^{-1}(a x)^3}{4 a^4}-\frac{i \cot ^{-1}(a x)^2}{a^4}+\frac{2 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^3+\frac{x^3 \cot ^{-1}(a x)^2}{4 a}","-\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^4}+\frac{x^2 \cot ^{-1}(a x)}{4 a^2}+\frac{x}{4 a^3}-\frac{\tan ^{-1}(a x)}{4 a^4}-\frac{3 x \cot ^{-1}(a x)^2}{4 a^3}-\frac{\cot ^{-1}(a x)^3}{4 a^4}-\frac{i \cot ^{-1}(a x)^2}{a^4}+\frac{2 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^3+\frac{x^3 \cot ^{-1}(a x)^2}{4 a}",1,"x/(4*a^3) + (x^2*ArcCot[a*x])/(4*a^2) - (I*ArcCot[a*x]^2)/a^4 - (3*x*ArcCot[a*x]^2)/(4*a^3) + (x^3*ArcCot[a*x]^2)/(4*a) - ArcCot[a*x]^3/(4*a^4) + (x^4*ArcCot[a*x]^3)/4 - ArcTan[a*x]/(4*a^4) + (2*ArcCot[a*x]*Log[2/(1 + I*a*x)])/a^4 - (I*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4","A",18,10,10,1.000,1,"{4853, 4917, 321, 203, 4921, 4855, 2402, 2315, 4847, 4885}"
26,1,157,0,0.301527,"\int x^2 \cot ^{-1}(a x)^3 \, dx","Int[x^2*ArcCot[a*x]^3,x]","\frac{\text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^3}-\frac{i \cot ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^3}+\frac{\log \left(a^2 x^2+1\right)}{2 a^3}-\frac{i \cot ^{-1}(a x)^3}{3 a^3}+\frac{\cot ^{-1}(a x)^2}{2 a^3}+\frac{x \cot ^{-1}(a x)}{a^2}+\frac{\log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)^2}{a^3}+\frac{1}{3} x^3 \cot ^{-1}(a x)^3+\frac{x^2 \cot ^{-1}(a x)^2}{2 a}","\frac{\text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^3}-\frac{i \cot ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^3}+\frac{\log \left(a^2 x^2+1\right)}{2 a^3}-\frac{i \cot ^{-1}(a x)^3}{3 a^3}+\frac{\cot ^{-1}(a x)^2}{2 a^3}+\frac{x \cot ^{-1}(a x)}{a^2}+\frac{\log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)^2}{a^3}+\frac{1}{3} x^3 \cot ^{-1}(a x)^3+\frac{x^2 \cot ^{-1}(a x)^2}{2 a}",1,"(x*ArcCot[a*x])/a^2 + ArcCot[a*x]^2/(2*a^3) + (x^2*ArcCot[a*x]^2)/(2*a) - ((I/3)*ArcCot[a*x]^3)/a^3 + (x^3*ArcCot[a*x]^3)/3 + (ArcCot[a*x]^2*Log[2/(1 + I*a*x)])/a^3 + Log[1 + a^2*x^2]/(2*a^3) - (I*ArcCot[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3 + PolyLog[3, 1 - 2/(1 + I*a*x)]/(2*a^3)","A",11,9,10,0.9000,1,"{4853, 4917, 4847, 260, 4885, 4921, 4855, 4995, 6610}"
27,1,103,0,0.1708551,"\int x \cot ^{-1}(a x)^3 \, dx","Int[x*ArcCot[a*x]^3,x]","\frac{3 i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^2}+\frac{\cot ^{-1}(a x)^3}{2 a^2}+\frac{3 i \cot ^{-1}(a x)^2}{2 a^2}-\frac{3 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)^3+\frac{3 x \cot ^{-1}(a x)^2}{2 a}","\frac{3 i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^2}+\frac{\cot ^{-1}(a x)^3}{2 a^2}+\frac{3 i \cot ^{-1}(a x)^2}{2 a^2}-\frac{3 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)^3+\frac{3 x \cot ^{-1}(a x)^2}{2 a}",1,"(((3*I)/2)*ArcCot[a*x]^2)/a^2 + (3*x*ArcCot[a*x]^2)/(2*a) + ArcCot[a*x]^3/(2*a^2) + (x^2*ArcCot[a*x]^3)/2 - (3*ArcCot[a*x]*Log[2/(1 + I*a*x)])/a^2 + (((3*I)/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^2","A",8,8,8,1.000,1,"{4853, 4917, 4847, 4921, 4855, 2402, 2315, 4885}"
28,1,96,0,0.1499116,"\int \cot ^{-1}(a x)^3 \, dx","Int[ArcCot[a*x]^3,x]","-\frac{3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a}+\frac{3 i \cot ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a}+x \cot ^{-1}(a x)^3+\frac{i \cot ^{-1}(a x)^3}{a}-\frac{3 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)^2}{a}","-\frac{3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a}+\frac{3 i \cot ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a}+x \cot ^{-1}(a x)^3+\frac{i \cot ^{-1}(a x)^3}{a}-\frac{3 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)^2}{a}",1,"(I*ArcCot[a*x]^3)/a + x*ArcCot[a*x]^3 - (3*ArcCot[a*x]^2*Log[2/(1 + I*a*x)])/a + ((3*I)*ArcCot[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a - (3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(2*a)","A",5,6,6,1.000,1,"{4847, 4921, 4855, 4885, 4995, 6610}"
29,1,178,0,0.3266946,"\int \frac{\cot ^{-1}(a x)^3}{x} \, dx","Int[ArcCot[a*x]^3/x,x]","\frac{3}{4} i \text{PolyLog}\left(4,1-\frac{2 i}{a x+i}\right)-\frac{3}{4} i \text{PolyLog}\left(4,1-\frac{2 a x}{a x+i}\right)-\frac{3}{2} i \cot ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2 i}{a x+i}\right)+\frac{3}{2} i \cot ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2 a x}{a x+i}\right)-\frac{3}{2} \cot ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2 i}{a x+i}\right)+\frac{3}{2} \cot ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2 a x}{a x+i}\right)+2 \cot ^{-1}(a x)^3 \coth ^{-1}\left(1-\frac{2}{1+i a x}\right)","\frac{3}{4} i \text{PolyLog}\left(4,1-\frac{2 i}{a x+i}\right)-\frac{3}{4} i \text{PolyLog}\left(4,1-\frac{2 a x}{a x+i}\right)-\frac{3}{2} i \cot ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2 i}{a x+i}\right)+\frac{3}{2} i \cot ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2 a x}{a x+i}\right)-\frac{3}{2} \cot ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2 i}{a x+i}\right)+\frac{3}{2} \cot ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2 a x}{a x+i}\right)+2 \cot ^{-1}(a x)^3 \coth ^{-1}\left(1-\frac{2}{1+i a x}\right)",1,"2*ArcCot[a*x]^3*ArcCoth[1 - 2/(1 + I*a*x)] - ((3*I)/2)*ArcCot[a*x]^2*PolyLog[2, 1 - (2*I)/(I + a*x)] + ((3*I)/2)*ArcCot[a*x]^2*PolyLog[2, 1 - (2*a*x)/(I + a*x)] - (3*ArcCot[a*x]*PolyLog[3, 1 - (2*I)/(I + a*x)])/2 + (3*ArcCot[a*x]*PolyLog[3, 1 - (2*a*x)/(I + a*x)])/2 + ((3*I)/4)*PolyLog[4, 1 - (2*I)/(I + a*x)] - ((3*I)/4)*PolyLog[4, 1 - (2*a*x)/(I + a*x)]","A",8,6,10,0.6000,1,"{4851, 4989, 4885, 4993, 4997, 6610}"
30,1,93,0,0.1882423,"\int \frac{\cot ^{-1}(a x)^3}{x^2} \, dx","Int[ArcCot[a*x]^3/x^2,x]","-\frac{3}{2} a \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)-3 i a \cot ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-i a \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{x}-3 a \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)^2","-\frac{3}{2} a \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)-3 i a \cot ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-i a \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{x}-3 a \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)^2",1,"(-I)*a*ArcCot[a*x]^3 - ArcCot[a*x]^3/x - 3*a*ArcCot[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (3*I)*a*ArcCot[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] - (3*a*PolyLog[3, -1 + 2/(1 - I*a*x)])/2","A",5,6,10,0.6000,1,"{4853, 4925, 4869, 4885, 4993, 6610}"
31,1,105,0,0.1994374,"\int \frac{\cot ^{-1}(a x)^3}{x^3} \, dx","Int[ArcCot[a*x]^3/x^3,x]","\frac{3}{2} i a^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-\frac{1}{2} a^2 \cot ^{-1}(a x)^3+\frac{3}{2} i a^2 \cot ^{-1}(a x)^2+3 a^2 \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)-\frac{\cot ^{-1}(a x)^3}{2 x^2}+\frac{3 a \cot ^{-1}(a x)^2}{2 x}","\frac{3}{2} i a^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-\frac{1}{2} a^2 \cot ^{-1}(a x)^3+\frac{3}{2} i a^2 \cot ^{-1}(a x)^2+3 a^2 \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)-\frac{\cot ^{-1}(a x)^3}{2 x^2}+\frac{3 a \cot ^{-1}(a x)^2}{2 x}",1,"((3*I)/2)*a^2*ArcCot[a*x]^2 + (3*a*ArcCot[a*x]^2)/(2*x) - (a^2*ArcCot[a*x]^3)/2 - ArcCot[a*x]^3/(2*x^2) + 3*a^2*ArcCot[a*x]*Log[2 - 2/(1 - I*a*x)] + ((3*I)/2)*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)]","A",7,6,10,0.6000,1,"{4853, 4919, 4925, 4869, 2447, 4885}"
32,1,167,0,0.336865,"\int \frac{\cot ^{-1}(a x)^3}{x^4} \, dx","Int[ArcCot[a*x]^3/x^4,x]","\frac{1}{2} a^3 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)+i a^3 \cot ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+\frac{1}{2} a^3 \log \left(a^2 x^2+1\right)-a^3 \log (x)+\frac{1}{3} i a^3 \cot ^{-1}(a x)^3+\frac{1}{2} a^3 \cot ^{-1}(a x)^2-\frac{a^2 \cot ^{-1}(a x)}{x}+a^3 \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)^2+\frac{a \cot ^{-1}(a x)^2}{2 x^2}-\frac{\cot ^{-1}(a x)^3}{3 x^3}","\frac{1}{2} a^3 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)+i a^3 \cot ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+\frac{1}{2} a^3 \log \left(a^2 x^2+1\right)-a^3 \log (x)+\frac{1}{3} i a^3 \cot ^{-1}(a x)^3+\frac{1}{2} a^3 \cot ^{-1}(a x)^2-\frac{a^2 \cot ^{-1}(a x)}{x}+a^3 \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)^2+\frac{a \cot ^{-1}(a x)^2}{2 x^2}-\frac{\cot ^{-1}(a x)^3}{3 x^3}",1,"-((a^2*ArcCot[a*x])/x) + (a^3*ArcCot[a*x]^2)/2 + (a*ArcCot[a*x]^2)/(2*x^2) + (I/3)*a^3*ArcCot[a*x]^3 - ArcCot[a*x]^3/(3*x^3) - a^3*Log[x] + (a^3*Log[1 + a^2*x^2])/2 + a^3*ArcCot[a*x]^2*Log[2 - 2/(1 - I*a*x)] + I*a^3*ArcCot[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (a^3*PolyLog[3, -1 + 2/(1 - I*a*x)])/2","A",14,11,10,1.100,1,"{4853, 4919, 266, 36, 29, 31, 4885, 4925, 4869, 4993, 6610}"
33,1,152,0,0.4199144,"\int \frac{\cot ^{-1}(a x)^3}{x^5} \, dx","Int[ArcCot[a*x]^3/x^5,x]","-i a^4 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-\frac{a^2 \cot ^{-1}(a x)}{4 x^2}+\frac{a^3}{4 x}+\frac{1}{4} a^4 \tan ^{-1}(a x)+\frac{1}{4} a^4 \cot ^{-1}(a x)^3-i a^4 \cot ^{-1}(a x)^2-\frac{3 a^3 \cot ^{-1}(a x)^2}{4 x}-2 a^4 \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)+\frac{a \cot ^{-1}(a x)^2}{4 x^3}-\frac{\cot ^{-1}(a x)^3}{4 x^4}","-i a^4 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-\frac{a^2 \cot ^{-1}(a x)}{4 x^2}+\frac{a^3}{4 x}+\frac{1}{4} a^4 \tan ^{-1}(a x)+\frac{1}{4} a^4 \cot ^{-1}(a x)^3-i a^4 \cot ^{-1}(a x)^2-\frac{3 a^3 \cot ^{-1}(a x)^2}{4 x}-2 a^4 \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)+\frac{a \cot ^{-1}(a x)^2}{4 x^3}-\frac{\cot ^{-1}(a x)^3}{4 x^4}",1,"a^3/(4*x) - (a^2*ArcCot[a*x])/(4*x^2) - I*a^4*ArcCot[a*x]^2 + (a*ArcCot[a*x]^2)/(4*x^3) - (3*a^3*ArcCot[a*x]^2)/(4*x) + (a^4*ArcCot[a*x]^3)/4 - ArcCot[a*x]^3/(4*x^4) + (a^4*ArcTan[a*x])/4 - 2*a^4*ArcCot[a*x]*Log[2 - 2/(1 - I*a*x)] - I*a^4*PolyLog[2, -1 + 2/(1 - I*a*x)]","A",16,8,10,0.8000,1,"{4853, 4919, 325, 203, 4925, 4869, 2447, 4885}"
34,0,0,0,0.0129573,"\int x^m \cot ^{-1}(a x)^3 \, dx","Int[x^m*ArcCot[a*x]^3,x]","\int x^m \cot ^{-1}(a x)^3 \, dx","\text{Int}\left(x^m \cot ^{-1}(a x)^3,x\right)",0,"Defer[Int][x^m*ArcCot[a*x]^3, x]","A",0,0,0,0,-1,"{}"
35,0,0,0,0.0126703,"\int x^m \cot ^{-1}(a x)^2 \, dx","Int[x^m*ArcCot[a*x]^2,x]","\int x^m \cot ^{-1}(a x)^2 \, dx","\text{Int}\left(x^m \cot ^{-1}(a x)^2,x\right)",0,"Defer[Int][x^m*ArcCot[a*x]^2, x]","A",0,0,0,0,-1,"{}"
36,1,57,0,0.020079,"\int x^m \cot ^{-1}(a x) \, dx","Int[x^m*ArcCot[a*x],x]","\frac{a x^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m^2+3 m+2}+\frac{x^{m+1} \cot ^{-1}(a x)}{m+1}","\frac{a x^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m^2+3 m+2}+\frac{x^{m+1} \cot ^{-1}(a x)}{m+1}",1,"(x^(1 + m)*ArcCot[a*x])/(1 + m) + (a*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(a^2*x^2)])/(2 + 3*m + m^2)","A",2,2,8,0.2500,1,"{4853, 364}"
37,1,40,0,0.0965447,"\int \frac{x^4 \cot ^{-1}(x)}{1+x^2} \, dx","Int[(x^4*ArcCot[x])/(1 + x^2),x]","\frac{x^2}{6}-\frac{2}{3} \log \left(x^2+1\right)+\frac{1}{3} x^3 \cot ^{-1}(x)-x \cot ^{-1}(x)-\frac{1}{2} \cot ^{-1}(x)^2","\frac{x^2}{6}-\frac{2}{3} \log \left(x^2+1\right)+\frac{1}{3} x^3 \cot ^{-1}(x)-x \cot ^{-1}(x)-\frac{1}{2} \cot ^{-1}(x)^2",1,"x^2/6 - x*ArcCot[x] + (x^3*ArcCot[x])/3 - ArcCot[x]^2/2 - (2*Log[1 + x^2])/3","A",9,7,13,0.5385,1,"{4917, 4853, 266, 43, 4847, 260, 4885}"
38,1,67,0,0.0926004,"\int \frac{x^3 \cot ^{-1}(x)}{1+x^2} \, dx","Int[(x^3*ArcCot[x])/(1 + x^2),x]","-\frac{1}{2} i \text{PolyLog}\left(2,1-\frac{2}{1+i x}\right)+\frac{1}{2} x^2 \cot ^{-1}(x)+\frac{x}{2}-\frac{1}{2} \tan ^{-1}(x)-\frac{1}{2} i \cot ^{-1}(x)^2+\log \left(\frac{2}{1+i x}\right) \cot ^{-1}(x)","-\frac{1}{2} i \text{PolyLog}\left(2,1-\frac{2}{1+i x}\right)+\frac{1}{2} x^2 \cot ^{-1}(x)+\frac{x}{2}-\frac{1}{2} \tan ^{-1}(x)-\frac{1}{2} i \cot ^{-1}(x)^2+\log \left(\frac{2}{1+i x}\right) \cot ^{-1}(x)",1,"x/2 + (x^2*ArcCot[x])/2 - (I/2)*ArcCot[x]^2 - ArcTan[x]/2 + ArcCot[x]*Log[2/(1 + I*x)] - (I/2)*PolyLog[2, 1 - 2/(1 + I*x)]","A",8,8,13,0.6154,1,"{4917, 4853, 321, 203, 4921, 4855, 2402, 2315}"
39,1,23,0,0.0481948,"\int \frac{x^2 \cot ^{-1}(x)}{1+x^2} \, dx","Int[(x^2*ArcCot[x])/(1 + x^2),x]","\frac{1}{2} \log \left(x^2+1\right)+\frac{1}{2} \cot ^{-1}(x)^2+x \cot ^{-1}(x)","\frac{1}{2} \log \left(x^2+1\right)+\frac{1}{2} \cot ^{-1}(x)^2+x \cot ^{-1}(x)",1,"x*ArcCot[x] + ArcCot[x]^2/2 + Log[1 + x^2]/2","A",4,4,13,0.3077,1,"{4917, 4847, 260, 4885}"
40,1,48,0,0.0540192,"\int \frac{x \cot ^{-1}(x)}{1+x^2} \, dx","Int[(x*ArcCot[x])/(1 + x^2),x]","\frac{1}{2} i \text{PolyLog}\left(2,1-\frac{2}{1+i x}\right)+\frac{1}{2} i \cot ^{-1}(x)^2-\log \left(\frac{2}{1+i x}\right) \cot ^{-1}(x)","\frac{1}{2} i \text{PolyLog}\left(2,1-\frac{2}{1+i x}\right)+\frac{1}{2} i \cot ^{-1}(x)^2-\log \left(\frac{2}{1+i x}\right) \cot ^{-1}(x)",1,"(I/2)*ArcCot[x]^2 - ArcCot[x]*Log[2/(1 + I*x)] + (I/2)*PolyLog[2, 1 - 2/(1 + I*x)]","A",4,4,11,0.3636,1,"{4921, 4855, 2402, 2315}"
41,1,8,0,0.0115701,"\int \frac{\cot ^{-1}(x)}{1+x^2} \, dx","Int[ArcCot[x]/(1 + x^2),x]","-\frac{1}{2} \cot ^{-1}(x)^2","-\frac{1}{2} \cot ^{-1}(x)^2",1,"-ArcCot[x]^2/2","A",1,1,10,0.1000,1,"{4885}"
42,1,49,0,0.0729478,"\int \frac{\cot ^{-1}(x)}{x \left(1+x^2\right)} \, dx","Int[ArcCot[x]/(x*(1 + x^2)),x]","\frac{1}{2} i \text{PolyLog}\left(2,-1+\frac{2}{1-i x}\right)+\frac{1}{2} i \cot ^{-1}(x)^2+\log \left(2-\frac{2}{1-i x}\right) \cot ^{-1}(x)","\frac{1}{2} i \text{PolyLog}\left(2,-1+\frac{2}{1-i x}\right)+\frac{1}{2} i \cot ^{-1}(x)^2+\log \left(2-\frac{2}{1-i x}\right) \cot ^{-1}(x)",1,"(I/2)*ArcCot[x]^2 + ArcCot[x]*Log[2 - 2/(1 - I*x)] + (I/2)*PolyLog[2, -1 + 2/(1 - I*x)]","A",3,3,13,0.2308,1,"{4925, 4869, 2447}"
43,1,30,0,0.0557802,"\int \frac{\cot ^{-1}(x)}{x^2 \left(1+x^2\right)} \, dx","Int[ArcCot[x]/(x^2*(1 + x^2)),x]","\frac{1}{2} \log \left(x^2+1\right)-\log (x)+\frac{1}{2} \cot ^{-1}(x)^2-\frac{\cot ^{-1}(x)}{x}","\frac{1}{2} \log \left(x^2+1\right)-\log (x)+\frac{1}{2} \cot ^{-1}(x)^2-\frac{\cot ^{-1}(x)}{x}",1,"-(ArcCot[x]/x) + ArcCot[x]^2/2 - Log[x] + Log[1 + x^2]/2","A",7,7,13,0.5385,1,"{4919, 4853, 266, 36, 29, 31, 4885}"
44,1,72,0,0.1146665,"\int \frac{\cot ^{-1}(x)}{x^3 \left(1+x^2\right)} \, dx","Int[ArcCot[x]/(x^3*(1 + x^2)),x]","-\frac{1}{2} i \text{PolyLog}\left(2,-1+\frac{2}{1-i x}\right)-\frac{\cot ^{-1}(x)}{2 x^2}+\frac{1}{2 x}+\frac{1}{2} \tan ^{-1}(x)-\frac{1}{2} i \cot ^{-1}(x)^2-\log \left(2-\frac{2}{1-i x}\right) \cot ^{-1}(x)","-\frac{1}{2} i \text{PolyLog}\left(2,-1+\frac{2}{1-i x}\right)-\frac{\cot ^{-1}(x)}{2 x^2}+\frac{1}{2 x}+\frac{1}{2} \tan ^{-1}(x)-\frac{1}{2} i \cot ^{-1}(x)^2-\log \left(2-\frac{2}{1-i x}\right) \cot ^{-1}(x)",1,"1/(2*x) - ArcCot[x]/(2*x^2) - (I/2)*ArcCot[x]^2 + ArcTan[x]/2 - ArcCot[x]*Log[2 - 2/(1 - I*x)] - (I/2)*PolyLog[2, -1 + 2/(1 - I*x)]","A",7,7,13,0.5385,1,"{4919, 4853, 325, 203, 4925, 4869, 2447}"
45,1,47,0,0.1093913,"\int \frac{\cot ^{-1}(x)}{x^4 \left(1+x^2\right)} \, dx","Int[ArcCot[x]/(x^4*(1 + x^2)),x]","\frac{1}{6 x^2}-\frac{2}{3} \log \left(x^2+1\right)-\frac{\cot ^{-1}(x)}{3 x^3}+\frac{4 \log (x)}{3}-\frac{1}{2} \cot ^{-1}(x)^2+\frac{\cot ^{-1}(x)}{x}","\frac{1}{6 x^2}-\frac{2}{3} \log \left(x^2+1\right)-\frac{\cot ^{-1}(x)}{3 x^3}+\frac{4 \log (x)}{3}-\frac{1}{2} \cot ^{-1}(x)^2+\frac{\cot ^{-1}(x)}{x}",1,"1/(6*x^2) - ArcCot[x]/(3*x^3) + ArcCot[x]/x - ArcCot[x]^2/2 + (4*Log[x])/3 - (2*Log[1 + x^2])/3","A",12,8,13,0.6154,1,"{4919, 4853, 266, 44, 36, 29, 31, 4885}"
46,1,206,0,0.579502,"\int \frac{x^2 \cot ^{-1}(c x)}{1+x^2} \, dx","Int[(x^2*ArcCot[c*x])/(1 + x^2),x]","\frac{1}{4} \text{PolyLog}\left(2,1+\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right)-\frac{1}{4} \text{PolyLog}\left(2,1+\frac{2 i (c x+i)}{(c+1) (1-i x)}\right)+\frac{\log \left(c^2 x^2+1\right)}{2 c}+x \cot ^{-1}(c x)-\frac{1}{2} i \tan ^{-1}(x) \log \left(1-\frac{i}{c x}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(1+\frac{i}{c x}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right)-\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (c x+i)}{(c+1) (1-i x)}\right)","\frac{1}{4} \text{PolyLog}\left(2,1+\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right)-\frac{1}{4} \text{PolyLog}\left(2,1+\frac{2 i (c x+i)}{(c+1) (1-i x)}\right)+\frac{\log \left(c^2 x^2+1\right)}{2 c}+x \cot ^{-1}(c x)-\frac{1}{2} i \tan ^{-1}(x) \log \left(1-\frac{i}{c x}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(1+\frac{i}{c x}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right)-\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (c x+i)}{(c+1) (1-i x)}\right)",1,"x*ArcCot[c*x] - (I/2)*ArcTan[x]*Log[1 - I/(c*x)] + (I/2)*ArcTan[x]*Log[1 + I/(c*x)] + (I/2)*ArcTan[x]*Log[((-2*I)*(I - c*x))/((1 - c)*(1 - I*x))] - (I/2)*ArcTan[x]*Log[((-2*I)*(I + c*x))/((1 + c)*(1 - I*x))] + Log[1 + c^2*x^2]/(2*c) + PolyLog[2, 1 + ((2*I)*(I - c*x))/((1 - c)*(1 - I*x))]/4 - PolyLog[2, 1 + ((2*I)*(I + c*x))/((1 + c)*(1 - I*x))]/4","A",28,15,15,1.000,1,"{4917, 4847, 260, 4909, 203, 2470, 6688, 12, 4876, 4848, 2391, 4856, 2402, 2315, 2447}"
47,1,188,0,0.1841697,"\int \frac{x \cot ^{-1}(c x)}{1+x^2} \, dx","Int[(x*ArcCot[c*x])/(1 + x^2),x]","-\frac{1}{2} i \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)+\frac{1}{4} i \text{PolyLog}\left(2,1-\frac{2 i c (-x+i)}{(1-c) (1-i c x)}\right)+\frac{1}{4} i \text{PolyLog}\left(2,1+\frac{2 i c (x+i)}{(c+1) (1-i c x)}\right)+\log \left(\frac{2}{1-i c x}\right) \left(-\cot ^{-1}(c x)\right)+\frac{1}{2} \log \left(\frac{2 i c (-x+i)}{(1-c) (1-i c x)}\right) \cot ^{-1}(c x)+\frac{1}{2} \log \left(-\frac{2 i c (x+i)}{(c+1) (1-i c x)}\right) \cot ^{-1}(c x)","-\frac{1}{2} i \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)+\frac{1}{4} i \text{PolyLog}\left(2,1-\frac{2 i c (-x+i)}{(1-c) (1-i c x)}\right)+\frac{1}{4} i \text{PolyLog}\left(2,1+\frac{2 i c (x+i)}{(c+1) (1-i c x)}\right)+\log \left(\frac{2}{1-i c x}\right) \left(-\cot ^{-1}(c x)\right)+\frac{1}{2} \log \left(\frac{2 i c (-x+i)}{(1-c) (1-i c x)}\right) \cot ^{-1}(c x)+\frac{1}{2} \log \left(-\frac{2 i c (x+i)}{(c+1) (1-i c x)}\right) \cot ^{-1}(c x)",1,"-(ArcCot[c*x]*Log[2/(1 - I*c*x)]) + (ArcCot[c*x]*Log[((2*I)*c*(I - x))/((1 - c)*(1 - I*c*x))])/2 + (ArcCot[c*x]*Log[((-2*I)*c*(I + x))/((1 + c)*(1 - I*c*x))])/2 - (I/2)*PolyLog[2, 1 - 2/(1 - I*c*x)] + (I/4)*PolyLog[2, 1 - ((2*I)*c*(I - x))/((1 - c)*(1 - I*c*x))] + (I/4)*PolyLog[2, 1 + ((2*I)*c*(I + x))/((1 + c)*(1 - I*c*x))]","A",10,5,13,0.3846,1,"{4929, 4857, 2402, 2315, 2447}"
48,1,183,0,0.4646179,"\int \frac{\cot ^{-1}(c x)}{1+x^2} \, dx","Int[ArcCot[c*x]/(1 + x^2),x]","-\frac{1}{4} \text{PolyLog}\left(2,1+\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right)+\frac{1}{4} \text{PolyLog}\left(2,1+\frac{2 i (c x+i)}{(c+1) (1-i x)}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(1-\frac{i}{c x}\right)-\frac{1}{2} i \tan ^{-1}(x) \log \left(1+\frac{i}{c x}\right)-\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (c x+i)}{(c+1) (1-i x)}\right)","-\frac{1}{4} \text{PolyLog}\left(2,1+\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right)+\frac{1}{4} \text{PolyLog}\left(2,1+\frac{2 i (c x+i)}{(c+1) (1-i x)}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(1-\frac{i}{c x}\right)-\frac{1}{2} i \tan ^{-1}(x) \log \left(1+\frac{i}{c x}\right)-\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (c x+i)}{(c+1) (1-i x)}\right)",1,"(I/2)*ArcTan[x]*Log[1 - I/(c*x)] - (I/2)*ArcTan[x]*Log[1 + I/(c*x)] - (I/2)*ArcTan[x]*Log[((-2*I)*(I - c*x))/((1 - c)*(1 - I*x))] + (I/2)*ArcTan[x]*Log[((-2*I)*(I + c*x))/((1 + c)*(1 - I*x))] - PolyLog[2, 1 + ((2*I)*(I - c*x))/((1 - c)*(1 - I*x))]/4 + PolyLog[2, 1 + ((2*I)*(I + c*x))/((1 + c)*(1 - I*x))]/4","A",25,13,12,1.083,1,"{4909, 203, 2470, 260, 6688, 12, 4876, 4848, 2391, 4856, 2402, 2315, 2447}"
49,1,223,0,0.2471398,"\int \frac{\cot ^{-1}(c x)}{x \left(1+x^2\right)} \, dx","Int[ArcCot[c*x]/(x*(1 + x^2)),x]","-\frac{1}{2} i \text{PolyLog}\left(2,-\frac{i}{c x}\right)+\frac{1}{2} i \text{PolyLog}\left(2,\frac{i}{c x}\right)+\frac{1}{2} i \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)-\frac{1}{4} i \text{PolyLog}\left(2,1-\frac{2 i c (-x+i)}{(1-c) (1-i c x)}\right)-\frac{1}{4} i \text{PolyLog}\left(2,1+\frac{2 i c (x+i)}{(c+1) (1-i c x)}\right)+\log \left(\frac{2}{1-i c x}\right) \cot ^{-1}(c x)-\frac{1}{2} \log \left(\frac{2 i c (-x+i)}{(1-c) (1-i c x)}\right) \cot ^{-1}(c x)-\frac{1}{2} \log \left(-\frac{2 i c (x+i)}{(c+1) (1-i c x)}\right) \cot ^{-1}(c x)","-\frac{1}{2} i \text{PolyLog}\left(2,-\frac{i}{c x}\right)+\frac{1}{2} i \text{PolyLog}\left(2,\frac{i}{c x}\right)+\frac{1}{2} i \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)-\frac{1}{4} i \text{PolyLog}\left(2,1-\frac{2 i c (-x+i)}{(1-c) (1-i c x)}\right)-\frac{1}{4} i \text{PolyLog}\left(2,1+\frac{2 i c (x+i)}{(c+1) (1-i c x)}\right)+\log \left(\frac{2}{1-i c x}\right) \cot ^{-1}(c x)-\frac{1}{2} \log \left(\frac{2 i c (-x+i)}{(1-c) (1-i c x)}\right) \cot ^{-1}(c x)-\frac{1}{2} \log \left(-\frac{2 i c (x+i)}{(c+1) (1-i c x)}\right) \cot ^{-1}(c x)",1,"ArcCot[c*x]*Log[2/(1 - I*c*x)] - (ArcCot[c*x]*Log[((2*I)*c*(I - x))/((1 - c)*(1 - I*c*x))])/2 - (ArcCot[c*x]*Log[((-2*I)*c*(I + x))/((1 + c)*(1 - I*c*x))])/2 - (I/2)*PolyLog[2, (-I)/(c*x)] + (I/2)*PolyLog[2, I/(c*x)] + (I/2)*PolyLog[2, 1 - 2/(1 - I*c*x)] - (I/4)*PolyLog[2, 1 - ((2*I)*c*(I - x))/((1 - c)*(1 - I*c*x))] - (I/4)*PolyLog[2, 1 + ((2*I)*c*(I + x))/((1 + c)*(1 - I*c*x))]","A",15,7,15,0.4667,1,"{4929, 4849, 2391, 4857, 2402, 2315, 2447}"
50,1,212,0,0.5031056,"\int \frac{\cot ^{-1}(c x)}{x^2 \left(1+x^2\right)} \, dx","Int[ArcCot[c*x]/(x^2*(1 + x^2)),x]","\frac{1}{4} \text{PolyLog}\left(2,1+\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right)-\frac{1}{4} \text{PolyLog}\left(2,1+\frac{2 i (c x+i)}{(c+1) (1-i x)}\right)+\frac{1}{2} c \log \left(c^2 x^2+1\right)-c \log (x)-\frac{\cot ^{-1}(c x)}{x}-\frac{1}{2} i \tan ^{-1}(x) \log \left(1-\frac{i}{c x}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(1+\frac{i}{c x}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right)-\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (c x+i)}{(c+1) (1-i x)}\right)","\frac{1}{4} \text{PolyLog}\left(2,1+\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right)-\frac{1}{4} \text{PolyLog}\left(2,1+\frac{2 i (c x+i)}{(c+1) (1-i x)}\right)+\frac{1}{2} c \log \left(c^2 x^2+1\right)-c \log (x)-\frac{\cot ^{-1}(c x)}{x}-\frac{1}{2} i \tan ^{-1}(x) \log \left(1-\frac{i}{c x}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(1+\frac{i}{c x}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right)-\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (c x+i)}{(c+1) (1-i x)}\right)",1,"-(ArcCot[c*x]/x) - (I/2)*ArcTan[x]*Log[1 - I/(c*x)] + (I/2)*ArcTan[x]*Log[1 + I/(c*x)] - c*Log[x] + (I/2)*ArcTan[x]*Log[((-2*I)*(I - c*x))/((1 - c)*(1 - I*x))] - (I/2)*ArcTan[x]*Log[((-2*I)*(I + c*x))/((1 + c)*(1 - I*x))] + (c*Log[1 + c^2*x^2])/2 + PolyLog[2, 1 + ((2*I)*(I - c*x))/((1 - c)*(1 - I*x))]/4 - PolyLog[2, 1 + ((2*I)*(I + c*x))/((1 + c)*(1 - I*x))]/4","A",31,19,15,1.267,1,"{4919, 4853, 266, 36, 29, 31, 4909, 203, 2470, 260, 6688, 12, 4876, 4848, 2391, 4856, 2402, 2315, 2447}"
51,1,5,0,0.0199193,"\int \frac{1}{\left(1+x^2\right) \cot ^{-1}(x)} \, dx","Int[1/((1 + x^2)*ArcCot[x]),x]","-\log \left(\cot ^{-1}(x)\right)","-\log \left(\cot ^{-1}(x)\right)",1,"-Log[ArcCot[x]]","A",1,1,12,0.08333,1,"{4883}"
52,1,13,0,0.0235998,"\int \frac{\cot ^{-1}(x)^n}{1+x^2} \, dx","Int[ArcCot[x]^n/(1 + x^2),x]","-\frac{\cot ^{-1}(x)^{n+1}}{n+1}","-\frac{\cot ^{-1}(x)^{n+1}}{n+1}",1,"-(ArcCot[x]^(1 + n)/(1 + n))","A",1,1,12,0.08333,1,"{4885}"
53,1,244,0,0.1763582,"\int \left(c+d x^2\right)^4 \cot ^{-1}(a x) \, dx","Int[(c + d*x^2)^4*ArcCot[a*x],x]","\frac{d^2 x^4 \left(378 a^4 c^2-180 a^2 c d+35 d^2\right)}{1260 a^5}+\frac{d x^2 \left(-378 a^4 c^2 d+420 a^6 c^3+180 a^2 c d^2-35 d^3\right)}{630 a^7}+\frac{\left(378 a^4 c^2 d^2-420 a^6 c^3 d+315 a^8 c^4-180 a^2 c d^3+35 d^4\right) \log \left(a^2 x^2+1\right)}{630 a^9}+\frac{d^3 x^6 \left(36 a^2 c-7 d\right)}{378 a^3}+\frac{6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cot ^{-1}(a x)+c^4 x \cot ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac{d^4 x^8}{72 a}+\frac{1}{9} d^4 x^9 \cot ^{-1}(a x)","\frac{d^2 x^4 \left(378 a^4 c^2-180 a^2 c d+35 d^2\right)}{1260 a^5}+\frac{d x^2 \left(-378 a^4 c^2 d+420 a^6 c^3+180 a^2 c d^2-35 d^3\right)}{630 a^7}+\frac{\left(378 a^4 c^2 d^2-420 a^6 c^3 d+315 a^8 c^4-180 a^2 c d^3+35 d^4\right) \log \left(a^2 x^2+1\right)}{630 a^9}+\frac{d^3 x^6 \left(36 a^2 c-7 d\right)}{378 a^3}+\frac{6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cot ^{-1}(a x)+c^4 x \cot ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac{d^4 x^8}{72 a}+\frac{1}{9} d^4 x^9 \cot ^{-1}(a x)",1,"(d*(420*a^6*c^3 - 378*a^4*c^2*d + 180*a^2*c*d^2 - 35*d^3)*x^2)/(630*a^7) + (d^2*(378*a^4*c^2 - 180*a^2*c*d + 35*d^2)*x^4)/(1260*a^5) + ((36*a^2*c - 7*d)*d^3*x^6)/(378*a^3) + (d^4*x^8)/(72*a) + c^4*x*ArcCot[a*x] + (4*c^3*d*x^3*ArcCot[a*x])/3 + (6*c^2*d^2*x^5*ArcCot[a*x])/5 + (4*c*d^3*x^7*ArcCot[a*x])/7 + (d^4*x^9*ArcCot[a*x])/9 + ((315*a^8*c^4 - 420*a^6*c^3*d + 378*a^4*c^2*d^2 - 180*a^2*c*d^3 + 35*d^4)*Log[1 + a^2*x^2])/(630*a^9)","A",4,4,14,0.2857,1,"{194, 4913, 1810, 260}"
54,1,168,0,0.1200673,"\int \left(c+d x^2\right)^3 \cot ^{-1}(a x) \, dx","Int[(c + d*x^2)^3*ArcCot[a*x],x]","\frac{d x^2 \left(35 a^4 c^2-21 a^2 c d+5 d^2\right)}{70 a^5}+\frac{\left(-35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2-5 d^3\right) \log \left(a^2 x^2+1\right)}{70 a^7}+\frac{d^2 x^4 \left(21 a^2 c-5 d\right)}{140 a^3}+c^2 d x^3 \cot ^{-1}(a x)+c^3 x \cot ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac{d^3 x^6}{42 a}+\frac{1}{7} d^3 x^7 \cot ^{-1}(a x)","\frac{d x^2 \left(35 a^4 c^2-21 a^2 c d+5 d^2\right)}{70 a^5}+\frac{\left(-35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2-5 d^3\right) \log \left(a^2 x^2+1\right)}{70 a^7}+\frac{d^2 x^4 \left(21 a^2 c-5 d\right)}{140 a^3}+c^2 d x^3 \cot ^{-1}(a x)+c^3 x \cot ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac{d^3 x^6}{42 a}+\frac{1}{7} d^3 x^7 \cot ^{-1}(a x)",1,"(d*(35*a^4*c^2 - 21*a^2*c*d + 5*d^2)*x^2)/(70*a^5) + ((21*a^2*c - 5*d)*d^2*x^4)/(140*a^3) + (d^3*x^6)/(42*a) + c^3*x*ArcCot[a*x] + c^2*d*x^3*ArcCot[a*x] + (3*c*d^2*x^5*ArcCot[a*x])/5 + (d^3*x^7*ArcCot[a*x])/7 + ((35*a^6*c^3 - 35*a^4*c^2*d + 21*a^2*c*d^2 - 5*d^3)*Log[1 + a^2*x^2])/(70*a^7)","A",4,4,14,0.2857,1,"{194, 4913, 1810, 260}"
55,1,109,0,0.1265266,"\int \left(c+d x^2\right)^2 \cot ^{-1}(a x) \, dx","Int[(c + d*x^2)^2*ArcCot[a*x],x]","\frac{\left(15 a^4 c^2-10 a^2 c d+3 d^2\right) \log \left(a^2 x^2+1\right)}{30 a^5}+\frac{d x^2 \left(10 a^2 c-3 d\right)}{30 a^3}+c^2 x \cot ^{-1}(a x)+\frac{2}{3} c d x^3 \cot ^{-1}(a x)+\frac{d^2 x^4}{20 a}+\frac{1}{5} d^2 x^5 \cot ^{-1}(a x)","\frac{\left(15 a^4 c^2-10 a^2 c d+3 d^2\right) \log \left(a^2 x^2+1\right)}{30 a^5}+\frac{d x^2 \left(10 a^2 c-3 d\right)}{30 a^3}+c^2 x \cot ^{-1}(a x)+\frac{2}{3} c d x^3 \cot ^{-1}(a x)+\frac{d^2 x^4}{20 a}+\frac{1}{5} d^2 x^5 \cot ^{-1}(a x)",1,"((10*a^2*c - 3*d)*d*x^2)/(30*a^3) + (d^2*x^4)/(20*a) + c^2*x*ArcCot[a*x] + (2*c*d*x^3*ArcCot[a*x])/3 + (d^2*x^5*ArcCot[a*x])/5 + ((15*a^4*c^2 - 10*a^2*c*d + 3*d^2)*Log[1 + a^2*x^2])/(30*a^5)","A",5,5,14,0.3571,1,"{194, 4913, 1594, 1247, 698}"
56,1,58,0,0.061042,"\int \left(c+d x^2\right) \cot ^{-1}(a x) \, dx","Int[(c + d*x^2)*ArcCot[a*x],x]","\frac{\left(3 a^2 c-d\right) \log \left(a^2 x^2+1\right)}{6 a^3}+c x \cot ^{-1}(a x)+\frac{d x^2}{6 a}+\frac{1}{3} d x^3 \cot ^{-1}(a x)","\frac{\left(3 a^2 c-d\right) \log \left(a^2 x^2+1\right)}{6 a^3}+c x \cot ^{-1}(a x)+\frac{d x^2}{6 a}+\frac{1}{3} d x^3 \cot ^{-1}(a x)",1,"(d*x^2)/(6*a) + c*x*ArcCot[a*x] + (d*x^3*ArcCot[a*x])/3 + ((3*a^2*c - d)*Log[1 + a^2*x^2])/(6*a^3)","A",5,4,12,0.3333,1,"{4913, 1593, 444, 43}"
57,1,403,0,0.9192176,"\int \frac{\cot ^{-1}(a x)}{c+d x^2} \, dx","Int[ArcCot[a*x]/(c + d*x^2),x]","-\frac{\text{PolyLog}\left(2,1-\frac{2 i \sqrt{c} \sqrt{d} (-a x+i)}{\left(a \sqrt{c}-\sqrt{d}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}\right)}{4 \sqrt{c} \sqrt{d}}+\frac{\text{PolyLog}\left(2,1+\frac{2 i \sqrt{c} \sqrt{d} (a x+i)}{\left(a \sqrt{c}+\sqrt{d}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}\right)}{4 \sqrt{c} \sqrt{d}}+\frac{i \log \left(1-\frac{i}{a x}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{c} \sqrt{d}}-\frac{i \log \left(1+\frac{i}{a x}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{c} \sqrt{d}}-\frac{i \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right) \log \left(\frac{2 i \sqrt{c} \sqrt{d} (-a x+i)}{\left(a \sqrt{c}-\sqrt{d}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}\right)}{2 \sqrt{c} \sqrt{d}}+\frac{i \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right) \log \left(-\frac{2 i \sqrt{c} \sqrt{d} (a x+i)}{\left(a \sqrt{c}+\sqrt{d}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}\right)}{2 \sqrt{c} \sqrt{d}}","-\frac{\text{PolyLog}\left(2,1-\frac{2 i \sqrt{c} \sqrt{d} (-a x+i)}{\left(a \sqrt{c}-\sqrt{d}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}\right)}{4 \sqrt{c} \sqrt{d}}+\frac{\text{PolyLog}\left(2,1+\frac{2 i \sqrt{c} \sqrt{d} (a x+i)}{\left(a \sqrt{c}+\sqrt{d}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}\right)}{4 \sqrt{c} \sqrt{d}}+\frac{i \log \left(1-\frac{i}{a x}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{c} \sqrt{d}}-\frac{i \log \left(1+\frac{i}{a x}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{c} \sqrt{d}}-\frac{i \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right) \log \left(\frac{2 i \sqrt{c} \sqrt{d} (-a x+i)}{\left(a \sqrt{c}-\sqrt{d}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}\right)}{2 \sqrt{c} \sqrt{d}}+\frac{i \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right) \log \left(-\frac{2 i \sqrt{c} \sqrt{d} (a x+i)}{\left(a \sqrt{c}+\sqrt{d}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}\right)}{2 \sqrt{c} \sqrt{d}}",1,"((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 - I/(a*x)])/(Sqrt[c]*Sqrt[d]) - ((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 + I/(a*x)])/(Sqrt[c]*Sqrt[d]) - ((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[((2*I)*Sqrt[c]*Sqrt[d]*(I - a*x))/((a*Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(Sqrt[c]*Sqrt[d]) + ((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[((-2*I)*Sqrt[c]*Sqrt[d]*(I + a*x))/((a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(Sqrt[c]*Sqrt[d]) - PolyLog[2, 1 - ((2*I)*Sqrt[c]*Sqrt[d]*(I - a*x))/((a*Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))]/(4*Sqrt[c]*Sqrt[d]) + PolyLog[2, 1 + ((2*I)*Sqrt[c]*Sqrt[d]*(I + a*x))/((a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))]/(4*Sqrt[c]*Sqrt[d])","A",27,13,14,0.9286,1,"{4909, 205, 2470, 12, 260, 6688, 4876, 4848, 2391, 4856, 2402, 2315, 2447}"
58,1,801,0,1.1583388,"\int \frac{\cot ^{-1}(a x)}{\left(c+d x^2\right)^2} \, dx","Int[ArcCot[a*x]/(c + d*x^2)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right) \cot ^{-1}(a x)}{2 c^{3/2} \sqrt{d}}+\frac{x \cot ^{-1}(a x)}{2 c \left(d x^2+c\right)}-\frac{i a \log \left(\frac{\sqrt{d} \left(1-\sqrt{-a^2} x\right)}{i \sqrt{-a^2} \sqrt{c}+\sqrt{d}}\right) \log \left(1-\frac{i \sqrt{d} x}{\sqrt{c}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}+\frac{i a \log \left(-\frac{\sqrt{d} \left(\sqrt{-a^2} x+1\right)}{i \sqrt{-a^2} \sqrt{c}-\sqrt{d}}\right) \log \left(1-\frac{i \sqrt{d} x}{\sqrt{c}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}+\frac{i a \log \left(-\frac{\sqrt{d} \left(1-\sqrt{-a^2} x\right)}{i \sqrt{-a^2} \sqrt{c}-\sqrt{d}}\right) \log \left(\frac{i \sqrt{d} x}{\sqrt{c}}+1\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}-\frac{i a \log \left(\frac{\sqrt{d} \left(\sqrt{-a^2} x+1\right)}{i \sqrt{-a^2} \sqrt{c}+\sqrt{d}}\right) \log \left(\frac{i \sqrt{d} x}{\sqrt{c}}+1\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}+\frac{a \log \left(a^2 x^2+1\right)}{4 c \left(a^2 c-d\right)}-\frac{a \log \left(d x^2+c\right)}{4 c \left(a^2 c-d\right)}-\frac{i a \text{PolyLog}\left(2,\frac{\sqrt{-a^2} \left(\sqrt{c}-i \sqrt{d} x\right)}{\sqrt{-a^2} \sqrt{c}-i \sqrt{d}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}+\frac{i a \text{PolyLog}\left(2,\frac{\sqrt{-a^2} \left(\sqrt{c}-i \sqrt{d} x\right)}{\sqrt{-a^2} \sqrt{c}+i \sqrt{d}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}-\frac{i a \text{PolyLog}\left(2,\frac{\sqrt{-a^2} \left(i \sqrt{d} x+\sqrt{c}\right)}{\sqrt{-a^2} \sqrt{c}-i \sqrt{d}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}+\frac{i a \text{PolyLog}\left(2,\frac{\sqrt{-a^2} \left(i \sqrt{d} x+\sqrt{c}\right)}{\sqrt{-a^2} \sqrt{c}+i \sqrt{d}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}","\frac{\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right) \cot ^{-1}(a x)}{2 c^{3/2} \sqrt{d}}+\frac{x \cot ^{-1}(a x)}{2 c \left(d x^2+c\right)}-\frac{i a \log \left(\frac{\sqrt{d} \left(1-\sqrt{-a^2} x\right)}{i \sqrt{-a^2} \sqrt{c}+\sqrt{d}}\right) \log \left(1-\frac{i \sqrt{d} x}{\sqrt{c}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}+\frac{i a \log \left(-\frac{\sqrt{d} \left(\sqrt{-a^2} x+1\right)}{i \sqrt{-a^2} \sqrt{c}-\sqrt{d}}\right) \log \left(1-\frac{i \sqrt{d} x}{\sqrt{c}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}+\frac{i a \log \left(-\frac{\sqrt{d} \left(1-\sqrt{-a^2} x\right)}{i \sqrt{-a^2} \sqrt{c}-\sqrt{d}}\right) \log \left(\frac{i \sqrt{d} x}{\sqrt{c}}+1\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}-\frac{i a \log \left(\frac{\sqrt{d} \left(\sqrt{-a^2} x+1\right)}{i \sqrt{-a^2} \sqrt{c}+\sqrt{d}}\right) \log \left(\frac{i \sqrt{d} x}{\sqrt{c}}+1\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}+\frac{a \log \left(a^2 x^2+1\right)}{4 c \left(a^2 c-d\right)}-\frac{a \log \left(d x^2+c\right)}{4 c \left(a^2 c-d\right)}-\frac{i a \text{PolyLog}\left(2,\frac{\sqrt{-a^2} \left(\sqrt{c}-i \sqrt{d} x\right)}{\sqrt{-a^2} \sqrt{c}-i \sqrt{d}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}+\frac{i a \text{PolyLog}\left(2,\frac{\sqrt{-a^2} \left(\sqrt{c}-i \sqrt{d} x\right)}{\sqrt{-a^2} \sqrt{c}+i \sqrt{d}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}-\frac{i a \text{PolyLog}\left(2,\frac{\sqrt{-a^2} \left(i \sqrt{d} x+\sqrt{c}\right)}{\sqrt{-a^2} \sqrt{c}-i \sqrt{d}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}+\frac{i a \text{PolyLog}\left(2,\frac{\sqrt{-a^2} \left(i \sqrt{d} x+\sqrt{c}\right)}{\sqrt{-a^2} \sqrt{c}+i \sqrt{d}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}",1,"(x*ArcCot[a*x])/(2*c*(c + d*x^2)) + (ArcCot[a*x]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*Sqrt[d]) - ((I/8)*a*Log[(Sqrt[d]*(1 - Sqrt[-a^2]*x))/(I*Sqrt[-a^2]*Sqrt[c] + Sqrt[d])]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d]) + ((I/8)*a*Log[-((Sqrt[d]*(1 + Sqrt[-a^2]*x))/(I*Sqrt[-a^2]*Sqrt[c] - Sqrt[d]))]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d]) + ((I/8)*a*Log[-((Sqrt[d]*(1 - Sqrt[-a^2]*x))/(I*Sqrt[-a^2]*Sqrt[c] - Sqrt[d]))]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d]) - ((I/8)*a*Log[(Sqrt[d]*(1 + Sqrt[-a^2]*x))/(I*Sqrt[-a^2]*Sqrt[c] + Sqrt[d])]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d]) + (a*Log[1 + a^2*x^2])/(4*c*(a^2*c - d)) - (a*Log[c + d*x^2])/(4*c*(a^2*c - d)) - ((I/8)*a*PolyLog[2, (Sqrt[-a^2]*(Sqrt[c] - I*Sqrt[d]*x))/(Sqrt[-a^2]*Sqrt[c] - I*Sqrt[d])])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d]) + ((I/8)*a*PolyLog[2, (Sqrt[-a^2]*(Sqrt[c] - I*Sqrt[d]*x))/(Sqrt[-a^2]*Sqrt[c] + I*Sqrt[d])])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d]) - ((I/8)*a*PolyLog[2, (Sqrt[-a^2]*(Sqrt[c] + I*Sqrt[d]*x))/(Sqrt[-a^2]*Sqrt[c] - I*Sqrt[d])])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d]) + ((I/8)*a*PolyLog[2, (Sqrt[-a^2]*(Sqrt[c] + I*Sqrt[d]*x))/(Sqrt[-a^2]*Sqrt[c] + I*Sqrt[d])])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d])","A",24,12,14,0.8571,1,"{199, 205, 4913, 6725, 444, 36, 31, 4908, 2409, 2394, 2393, 2391}"
59,0,0,0,0.0186725,"\int \sqrt{c+d x^2} \cot ^{-1}(a x) \, dx","Int[Sqrt[c + d*x^2]*ArcCot[a*x],x]","\int \sqrt{c+d x^2} \cot ^{-1}(a x) \, dx","\text{Int}\left(\cot ^{-1}(a x) \sqrt{c+d x^2},x\right)",0,"Defer[Int][Sqrt[c + d*x^2]*ArcCot[a*x], x]","A",0,0,0,0,-1,"{}"
60,0,0,0,0.0221814,"\int \frac{\cot ^{-1}(a x)}{\sqrt{c+d x^2}} \, dx","Int[ArcCot[a*x]/Sqrt[c + d*x^2],x]","\int \frac{\cot ^{-1}(a x)}{\sqrt{c+d x^2}} \, dx","\text{Int}\left(\frac{\cot ^{-1}(a x)}{\sqrt{c+d x^2}},x\right)",0,"Defer[Int][ArcCot[a*x]/Sqrt[c + d*x^2], x]","A",0,0,0,0,-1,"{}"
61,1,66,0,0.0940461,"\int \frac{\cot ^{-1}(a x)}{\left(c+d x^2\right)^{3/2}} \, dx","Int[ArcCot[a*x]/(c + d*x^2)^(3/2),x]","\frac{x \cot ^{-1}(a x)}{c \sqrt{c+d x^2}}-\frac{\tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{c \sqrt{a^2 c-d}}","\frac{x \cot ^{-1}(a x)}{c \sqrt{c+d x^2}}-\frac{\tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{c \sqrt{a^2 c-d}}",1,"(x*ArcCot[a*x])/(c*Sqrt[c + d*x^2]) - ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d]]/(c*Sqrt[a^2*c - d])","A",5,6,16,0.3750,1,"{191, 4913, 12, 444, 63, 208}"
62,1,134,0,0.3257417,"\int \frac{\cot ^{-1}(a x)}{\left(c+d x^2\right)^{5/2}} \, dx","Int[ArcCot[a*x]/(c + d*x^2)^(5/2),x]","-\frac{\left(3 a^2 c-2 d\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{3 c^2 \left(a^2 c-d\right)^{3/2}}+\frac{a}{3 c \left(a^2 c-d\right) \sqrt{c+d x^2}}+\frac{2 x \cot ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{x \cot ^{-1}(a x)}{3 c \left(c+d x^2\right)^{3/2}}","-\frac{\left(3 a^2 c-2 d\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{3 c^2 \left(a^2 c-d\right)^{3/2}}+\frac{a}{3 c \left(a^2 c-d\right) \sqrt{c+d x^2}}+\frac{2 x \cot ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{x \cot ^{-1}(a x)}{3 c \left(c+d x^2\right)^{3/2}}",1,"a/(3*c*(a^2*c - d)*Sqrt[c + d*x^2]) + (x*ArcCot[a*x])/(3*c*(c + d*x^2)^(3/2)) + (2*x*ArcCot[a*x])/(3*c^2*Sqrt[c + d*x^2]) - ((3*a^2*c - 2*d)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d]])/(3*c^2*(a^2*c - d)^(3/2))","A",7,9,16,0.5625,1,"{192, 191, 4913, 6688, 12, 571, 78, 63, 208}"
63,1,208,0,0.9347165,"\int \frac{\cot ^{-1}(a x)}{\left(c+d x^2\right)^{7/2}} \, dx","Int[ArcCot[a*x]/(c + d*x^2)^(7/2),x]","-\frac{\left(15 a^4 c^2-20 a^2 c d+8 d^2\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{15 c^3 \left(a^2 c-d\right)^{5/2}}+\frac{a \left(7 a^2 c-4 d\right)}{15 c^2 \left(a^2 c-d\right)^2 \sqrt{c+d x^2}}+\frac{a}{15 c \left(a^2 c-d\right) \left(c+d x^2\right)^{3/2}}+\frac{8 x \cot ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}+\frac{4 x \cot ^{-1}(a x)}{15 c^2 \left(c+d x^2\right)^{3/2}}+\frac{x \cot ^{-1}(a x)}{5 c \left(c+d x^2\right)^{5/2}}","-\frac{\left(15 a^4 c^2-20 a^2 c d+8 d^2\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{15 c^3 \left(a^2 c-d\right)^{5/2}}+\frac{a \left(7 a^2 c-4 d\right)}{15 c^2 \left(a^2 c-d\right)^2 \sqrt{c+d x^2}}+\frac{a}{15 c \left(a^2 c-d\right) \left(c+d x^2\right)^{3/2}}+\frac{8 x \cot ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}+\frac{4 x \cot ^{-1}(a x)}{15 c^2 \left(c+d x^2\right)^{3/2}}+\frac{x \cot ^{-1}(a x)}{5 c \left(c+d x^2\right)^{5/2}}",1,"a/(15*c*(a^2*c - d)*(c + d*x^2)^(3/2)) + (a*(7*a^2*c - 4*d))/(15*c^2*(a^2*c - d)^2*Sqrt[c + d*x^2]) + (x*ArcCot[a*x])/(5*c*(c + d*x^2)^(5/2)) + (4*x*ArcCot[a*x])/(15*c^2*(c + d*x^2)^(3/2)) + (8*x*ArcCot[a*x])/(15*c^3*Sqrt[c + d*x^2]) - ((15*a^4*c^2 - 20*a^2*c*d + 8*d^2)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d]])/(15*c^3*(a^2*c - d)^(5/2))","A",8,9,16,0.5625,1,"{192, 191, 4913, 6688, 12, 6715, 897, 1261, 208}"
64,1,293,0,1.1551009,"\int \frac{\cot ^{-1}(a x)}{\left(c+d x^2\right)^{9/2}} \, dx","Int[ArcCot[a*x]/(c + d*x^2)^(9/2),x]","\frac{a \left(19 a^4 c^2-22 a^2 c d+8 d^2\right)}{35 c^3 \left(a^2 c-d\right)^3 \sqrt{c+d x^2}}-\frac{\left(-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{35 c^4 \left(a^2 c-d\right)^{7/2}}+\frac{a \left(11 a^2 c-6 d\right)}{105 c^2 \left(a^2 c-d\right)^2 \left(c+d x^2\right)^{3/2}}+\frac{a}{35 c \left(a^2 c-d\right) \left(c+d x^2\right)^{5/2}}+\frac{16 x \cot ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{8 x \cot ^{-1}(a x)}{35 c^3 \left(c+d x^2\right)^{3/2}}+\frac{6 x \cot ^{-1}(a x)}{35 c^2 \left(c+d x^2\right)^{5/2}}+\frac{x \cot ^{-1}(a x)}{7 c \left(c+d x^2\right)^{7/2}}","\frac{a \left(19 a^4 c^2-22 a^2 c d+8 d^2\right)}{35 c^3 \left(a^2 c-d\right)^3 \sqrt{c+d x^2}}-\frac{\left(-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{35 c^4 \left(a^2 c-d\right)^{7/2}}+\frac{a \left(11 a^2 c-6 d\right)}{105 c^2 \left(a^2 c-d\right)^2 \left(c+d x^2\right)^{3/2}}+\frac{a}{35 c \left(a^2 c-d\right) \left(c+d x^2\right)^{5/2}}+\frac{16 x \cot ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{8 x \cot ^{-1}(a x)}{35 c^3 \left(c+d x^2\right)^{3/2}}+\frac{6 x \cot ^{-1}(a x)}{35 c^2 \left(c+d x^2\right)^{5/2}}+\frac{x \cot ^{-1}(a x)}{7 c \left(c+d x^2\right)^{7/2}}",1,"a/(35*c*(a^2*c - d)*(c + d*x^2)^(5/2)) + (a*(11*a^2*c - 6*d))/(105*c^2*(a^2*c - d)^2*(c + d*x^2)^(3/2)) + (a*(19*a^4*c^2 - 22*a^2*c*d + 8*d^2))/(35*c^3*(a^2*c - d)^3*Sqrt[c + d*x^2]) + (x*ArcCot[a*x])/(7*c*(c + d*x^2)^(7/2)) + (6*x*ArcCot[a*x])/(35*c^2*(c + d*x^2)^(5/2)) + (8*x*ArcCot[a*x])/(35*c^3*(c + d*x^2)^(3/2)) + (16*x*ArcCot[a*x])/(35*c^4*Sqrt[c + d*x^2]) - ((35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 16*d^3)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d]])/(35*c^4*(a^2*c - d)^(7/2))","A",8,9,16,0.5625,1,"{192, 191, 4913, 6688, 12, 6715, 1619, 63, 208}"
65,1,195,0,0.0718277,"\int \sqrt{a+a x^2} \cot ^{-1}(x) \, dx","Int[Sqrt[a + a*x^2]*ArcCot[x],x]","-\frac{i a \sqrt{x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right)}{2 \sqrt{a x^2+a}}+\frac{i a \sqrt{x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right)}{2 \sqrt{a x^2+a}}+\frac{1}{2} \sqrt{a x^2+a}+\frac{1}{2} x \sqrt{a x^2+a} \cot ^{-1}(x)-\frac{i a \sqrt{x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i x}}{\sqrt{1-i x}}\right) \cot ^{-1}(x)}{\sqrt{a x^2+a}}","-\frac{i a \sqrt{x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right)}{2 \sqrt{a x^2+a}}+\frac{i a \sqrt{x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right)}{2 \sqrt{a x^2+a}}+\frac{1}{2} \sqrt{a x^2+a}+\frac{1}{2} x \sqrt{a x^2+a} \cot ^{-1}(x)-\frac{i a \sqrt{x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i x}}{\sqrt{1-i x}}\right) \cot ^{-1}(x)}{\sqrt{a x^2+a}}",1,"Sqrt[a + a*x^2]/2 + (x*Sqrt[a + a*x^2]*ArcCot[x])/2 - (I*a*Sqrt[1 + x^2]*ArcCot[x]*ArcTan[Sqrt[1 + I*x]/Sqrt[1 - I*x]])/Sqrt[a + a*x^2] - ((I/2)*a*Sqrt[1 + x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*x])/Sqrt[1 - I*x]])/Sqrt[a + a*x^2] + ((I/2)*a*Sqrt[1 + x^2]*PolyLog[2, (I*Sqrt[1 + I*x])/Sqrt[1 - I*x]])/Sqrt[a + a*x^2]","A",3,3,14,0.2143,1,"{4879, 4891, 4887}"
66,1,155,0,0.0447531,"\int \frac{\cot ^{-1}(x)}{\sqrt{a+a x^2}} \, dx","Int[ArcCot[x]/Sqrt[a + a*x^2],x]","-\frac{i \sqrt{x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right)}{\sqrt{a x^2+a}}+\frac{i \sqrt{x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right)}{\sqrt{a x^2+a}}-\frac{2 i \sqrt{x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i x}}{\sqrt{1-i x}}\right) \cot ^{-1}(x)}{\sqrt{a x^2+a}}","-\frac{i \sqrt{x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right)}{\sqrt{a x^2+a}}+\frac{i \sqrt{x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right)}{\sqrt{a x^2+a}}-\frac{2 i \sqrt{x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i x}}{\sqrt{1-i x}}\right) \cot ^{-1}(x)}{\sqrt{a x^2+a}}",1,"((-2*I)*Sqrt[1 + x^2]*ArcCot[x]*ArcTan[Sqrt[1 + I*x]/Sqrt[1 - I*x]])/Sqrt[a + a*x^2] - (I*Sqrt[1 + x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*x])/Sqrt[1 - I*x]])/Sqrt[a + a*x^2] + (I*Sqrt[1 + x^2]*PolyLog[2, (I*Sqrt[1 + I*x])/Sqrt[1 - I*x]])/Sqrt[a + a*x^2]","A",2,2,14,0.1429,1,"{4891, 4887}"
67,1,35,0,0.0203738,"\int \frac{\cot ^{-1}(x)}{\left(a+a x^2\right)^{3/2}} \, dx","Int[ArcCot[x]/(a + a*x^2)^(3/2),x]","\frac{x \cot ^{-1}(x)}{a \sqrt{a x^2+a}}-\frac{1}{a \sqrt{a x^2+a}}","\frac{x \cot ^{-1}(x)}{a \sqrt{a x^2+a}}-\frac{1}{a \sqrt{a x^2+a}}",1,"-(1/(a*Sqrt[a + a*x^2])) + (x*ArcCot[x])/(a*Sqrt[a + a*x^2])","A",1,1,14,0.07143,1,"{4895}"
68,1,79,0,0.0440098,"\int \frac{\cot ^{-1}(x)}{\left(a+a x^2\right)^{5/2}} \, dx","Int[ArcCot[x]/(a + a*x^2)^(5/2),x]","-\frac{2}{3 a^2 \sqrt{a x^2+a}}+\frac{2 x \cot ^{-1}(x)}{3 a^2 \sqrt{a x^2+a}}-\frac{1}{9 a \left(a x^2+a\right)^{3/2}}+\frac{x \cot ^{-1}(x)}{3 a \left(a x^2+a\right)^{3/2}}","-\frac{2}{3 a^2 \sqrt{a x^2+a}}+\frac{2 x \cot ^{-1}(x)}{3 a^2 \sqrt{a x^2+a}}-\frac{1}{9 a \left(a x^2+a\right)^{3/2}}+\frac{x \cot ^{-1}(x)}{3 a \left(a x^2+a\right)^{3/2}}",1,"-1/(9*a*(a + a*x^2)^(3/2)) - 2/(3*a^2*Sqrt[a + a*x^2]) + (x*ArcCot[x])/(3*a*(a + a*x^2)^(3/2)) + (2*x*ArcCot[x])/(3*a^2*Sqrt[a + a*x^2])","A",2,2,14,0.1429,1,"{4897, 4895}"
69,1,118,0,0.0670812,"\int \frac{\cot ^{-1}(x)}{\left(a+a x^2\right)^{7/2}} \, dx","Int[ArcCot[x]/(a + a*x^2)^(7/2),x]","-\frac{8}{15 a^3 \sqrt{a x^2+a}}-\frac{4}{45 a^2 \left(a x^2+a\right)^{3/2}}+\frac{8 x \cot ^{-1}(x)}{15 a^3 \sqrt{a x^2+a}}+\frac{4 x \cot ^{-1}(x)}{15 a^2 \left(a x^2+a\right)^{3/2}}-\frac{1}{25 a \left(a x^2+a\right)^{5/2}}+\frac{x \cot ^{-1}(x)}{5 a \left(a x^2+a\right)^{5/2}}","-\frac{8}{15 a^3 \sqrt{a x^2+a}}-\frac{4}{45 a^2 \left(a x^2+a\right)^{3/2}}+\frac{8 x \cot ^{-1}(x)}{15 a^3 \sqrt{a x^2+a}}+\frac{4 x \cot ^{-1}(x)}{15 a^2 \left(a x^2+a\right)^{3/2}}-\frac{1}{25 a \left(a x^2+a\right)^{5/2}}+\frac{x \cot ^{-1}(x)}{5 a \left(a x^2+a\right)^{5/2}}",1,"-1/(25*a*(a + a*x^2)^(5/2)) - 4/(45*a^2*(a + a*x^2)^(3/2)) - 8/(15*a^3*Sqrt[a + a*x^2]) + (x*ArcCot[x])/(5*a*(a + a*x^2)^(5/2)) + (4*x*ArcCot[x])/(15*a^2*(a + a*x^2)^(3/2)) + (8*x*ArcCot[x])/(15*a^3*Sqrt[a + a*x^2])","A",3,2,14,0.1429,1,"{4897, 4895}"
70,1,32,0,0.0261825,"\int \frac{x \cot ^{-1}(x)}{\left(1+x^2\right)^2} \, dx","Int[(x*ArcCot[x])/(1 + x^2)^2,x]","-\frac{x}{4 \left(x^2+1\right)}-\frac{\cot ^{-1}(x)}{2 \left(x^2+1\right)}-\frac{1}{4} \tan ^{-1}(x)","-\frac{x}{4 \left(x^2+1\right)}-\frac{\cot ^{-1}(x)}{2 \left(x^2+1\right)}-\frac{1}{4} \tan ^{-1}(x)",1,"-x/(4*(1 + x^2)) - ArcCot[x]/(2*(1 + x^2)) - ArcTan[x]/4","A",3,3,11,0.2727,1,"{4931, 199, 203}"
71,1,44,0,0.0288825,"\int \frac{x \cot ^{-1}(x)}{\left(1+x^2\right)^3} \, dx","Int[(x*ArcCot[x])/(1 + x^2)^3,x]","-\frac{3 x}{32 \left(x^2+1\right)}-\frac{x}{16 \left(x^2+1\right)^2}-\frac{\cot ^{-1}(x)}{4 \left(x^2+1\right)^2}-\frac{3}{32} \tan ^{-1}(x)","-\frac{3 x}{32 \left(x^2+1\right)}-\frac{x}{16 \left(x^2+1\right)^2}-\frac{\cot ^{-1}(x)}{4 \left(x^2+1\right)^2}-\frac{3}{32} \tan ^{-1}(x)",1,"-x/(16*(1 + x^2)^2) - (3*x)/(32*(1 + x^2)) - ArcCot[x]/(4*(1 + x^2)^2) - (3*ArcTan[x])/32","A",4,3,11,0.2727,1,"{4931, 199, 203}"
72,1,34,0,0.0145949,"\int \frac{\cot ^{-1}(x)}{\left(1+x^2\right)^2} \, dx","Int[ArcCot[x]/(1 + x^2)^2,x]","-\frac{1}{4 \left(x^2+1\right)}+\frac{x \cot ^{-1}(x)}{2 \left(x^2+1\right)}-\frac{1}{4} \cot ^{-1}(x)^2","-\frac{1}{4 \left(x^2+1\right)}+\frac{x \cot ^{-1}(x)}{2 \left(x^2+1\right)}-\frac{1}{4} \cot ^{-1}(x)^2",1,"-1/(4*(1 + x^2)) + (x*ArcCot[x])/(2*(1 + x^2)) - ArcCot[x]^2/4","A",2,2,10,0.2000,1,"{4893, 261}"
73,1,56,0,0.0438201,"\int \frac{\cot ^{-1}(x)^2}{\left(1+x^2\right)^2} \, dx","Int[ArcCot[x]^2/(1 + x^2)^2,x]","-\frac{x}{4 \left(x^2+1\right)}+\frac{x \cot ^{-1}(x)^2}{2 \left(x^2+1\right)}-\frac{\cot ^{-1}(x)}{2 \left(x^2+1\right)}-\frac{1}{4} \tan ^{-1}(x)-\frac{1}{6} \cot ^{-1}(x)^3","-\frac{x}{4 \left(x^2+1\right)}+\frac{x \cot ^{-1}(x)^2}{2 \left(x^2+1\right)}-\frac{\cot ^{-1}(x)}{2 \left(x^2+1\right)}-\frac{1}{4} \tan ^{-1}(x)-\frac{1}{6} \cot ^{-1}(x)^3",1,"-x/(4*(1 + x^2)) - ArcCot[x]/(2*(1 + x^2)) + (x*ArcCot[x]^2)/(2*(1 + x^2)) - ArcCot[x]^3/6 - ArcTan[x]/4","A",4,4,12,0.3333,1,"{4893, 4931, 199, 203}"
74,1,41,0,0.0253619,"\int x^5 \cot ^{-1}\left(a x^2\right) \, dx","Int[x^5*ArcCot[a*x^2],x]","-\frac{\log \left(a^2 x^4+1\right)}{12 a^3}+\frac{x^4}{12 a}+\frac{1}{6} x^6 \cot ^{-1}\left(a x^2\right)","-\frac{\log \left(a^2 x^4+1\right)}{12 a^3}+\frac{x^4}{12 a}+\frac{1}{6} x^6 \cot ^{-1}\left(a x^2\right)",1,"x^4/(12*a) + (x^6*ArcCot[a*x^2])/6 - Log[1 + a^2*x^4]/(12*a^3)","A",4,3,10,0.3000,1,"{5034, 266, 43}"
75,1,37,0,0.0193072,"\int x^3 \cot ^{-1}\left(a x^2\right) \, dx","Int[x^3*ArcCot[a*x^2],x]","-\frac{\tan ^{-1}\left(a x^2\right)}{4 a^2}+\frac{x^2}{4 a}+\frac{1}{4} x^4 \cot ^{-1}\left(a x^2\right)","-\frac{\tan ^{-1}\left(a x^2\right)}{4 a^2}+\frac{x^2}{4 a}+\frac{1}{4} x^4 \cot ^{-1}\left(a x^2\right)",1,"x^2/(4*a) + (x^4*ArcCot[a*x^2])/4 - ArcTan[a*x^2]/(4*a^2)","A",4,4,10,0.4000,1,"{5034, 275, 321, 203}"
76,1,31,0,0.0091751,"\int x \cot ^{-1}\left(a x^2\right) \, dx","Int[x*ArcCot[a*x^2],x]","\frac{\log \left(a^2 x^4+1\right)}{4 a}+\frac{1}{2} x^2 \cot ^{-1}\left(a x^2\right)","\frac{\log \left(a^2 x^4+1\right)}{4 a}+\frac{1}{2} x^2 \cot ^{-1}\left(a x^2\right)",1,"(x^2*ArcCot[a*x^2])/2 + Log[1 + a^2*x^4]/(4*a)","A",2,2,8,0.2500,1,"{5034, 260}"
77,1,37,0,0.0338291,"\int \frac{\cot ^{-1}\left(a x^2\right)}{x} \, dx","Int[ArcCot[a*x^2]/x,x]","\frac{1}{4} i \text{PolyLog}\left(2,\frac{i}{a x^2}\right)-\frac{1}{4} i \text{PolyLog}\left(2,-\frac{i}{a x^2}\right)","\frac{1}{4} i \text{PolyLog}\left(2,\frac{i}{a x^2}\right)-\frac{1}{4} i \text{PolyLog}\left(2,-\frac{i}{a x^2}\right)",1,"(-I/4)*PolyLog[2, (-I)/(a*x^2)] + (I/4)*PolyLog[2, I/(a*x^2)]","A",4,3,10,0.3000,1,"{5032, 4849, 2391}"
78,1,34,0,0.0173108,"\int \frac{\cot ^{-1}\left(a x^2\right)}{x^3} \, dx","Int[ArcCot[a*x^2]/x^3,x]","\frac{1}{4} a \log \left(a^2 x^4+1\right)-\frac{\cot ^{-1}\left(a x^2\right)}{2 x^2}-a \log (x)","\frac{1}{4} a \log \left(a^2 x^4+1\right)-\frac{\cot ^{-1}\left(a x^2\right)}{2 x^2}-a \log (x)",1,"-ArcCot[a*x^2]/(2*x^2) - a*Log[x] + (a*Log[1 + a^2*x^4])/4","A",5,5,10,0.5000,1,"{5034, 266, 36, 29, 31}"
79,1,35,0,0.0179996,"\int \frac{\cot ^{-1}\left(a x^2\right)}{x^5} \, dx","Int[ArcCot[a*x^2]/x^5,x]","\frac{1}{4} a^2 \tan ^{-1}\left(a x^2\right)+\frac{a}{4 x^2}-\frac{\cot ^{-1}\left(a x^2\right)}{4 x^4}","\frac{1}{4} a^2 \tan ^{-1}\left(a x^2\right)+\frac{a}{4 x^2}-\frac{\cot ^{-1}\left(a x^2\right)}{4 x^4}",1,"a/(4*x^2) - ArcCot[a*x^2]/(4*x^4) + (a^2*ArcTan[a*x^2])/4","A",4,4,10,0.4000,1,"{5034, 275, 325, 203}"
80,1,152,0,0.1042535,"\int x^4 \cot ^{-1}\left(a x^2\right) \, dx","Int[x^4*ArcCot[a*x^2],x]","-\frac{\log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)}{10 \sqrt{2} a^{5/2}}+\frac{\log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)}{10 \sqrt{2} a^{5/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)}{5 \sqrt{2} a^{5/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{5 \sqrt{2} a^{5/2}}+\frac{2 x^3}{15 a}+\frac{1}{5} x^5 \cot ^{-1}\left(a x^2\right)","-\frac{\log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)}{10 \sqrt{2} a^{5/2}}+\frac{\log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)}{10 \sqrt{2} a^{5/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)}{5 \sqrt{2} a^{5/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{5 \sqrt{2} a^{5/2}}+\frac{2 x^3}{15 a}+\frac{1}{5} x^5 \cot ^{-1}\left(a x^2\right)",1,"(2*x^3)/(15*a) + (x^5*ArcCot[a*x^2])/5 + ArcTan[1 - Sqrt[2]*Sqrt[a]*x]/(5*Sqrt[2]*a^(5/2)) - ArcTan[1 + Sqrt[2]*Sqrt[a]*x]/(5*Sqrt[2]*a^(5/2)) - Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2]/(10*Sqrt[2]*a^(5/2)) + Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2]/(10*Sqrt[2]*a^(5/2))","A",11,8,10,0.8000,1,"{5034, 321, 297, 1162, 617, 204, 1165, 628}"
81,1,150,0,0.0942241,"\int x^2 \cot ^{-1}\left(a x^2\right) \, dx","Int[x^2*ArcCot[a*x^2],x]","\frac{\log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)}{6 \sqrt{2} a^{3/2}}-\frac{\log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)}{6 \sqrt{2} a^{3/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)}{3 \sqrt{2} a^{3/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{3 \sqrt{2} a^{3/2}}+\frac{1}{3} x^3 \cot ^{-1}\left(a x^2\right)+\frac{2 x}{3 a}","\frac{\log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)}{6 \sqrt{2} a^{3/2}}-\frac{\log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)}{6 \sqrt{2} a^{3/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)}{3 \sqrt{2} a^{3/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{3 \sqrt{2} a^{3/2}}+\frac{1}{3} x^3 \cot ^{-1}\left(a x^2\right)+\frac{2 x}{3 a}",1,"(2*x)/(3*a) + (x^3*ArcCot[a*x^2])/3 + ArcTan[1 - Sqrt[2]*Sqrt[a]*x]/(3*Sqrt[2]*a^(3/2)) - ArcTan[1 + Sqrt[2]*Sqrt[a]*x]/(3*Sqrt[2]*a^(3/2)) + Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2]/(6*Sqrt[2]*a^(3/2)) - Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2]/(6*Sqrt[2]*a^(3/2))","A",11,8,10,0.8000,1,"{5034, 321, 211, 1165, 628, 1162, 617, 204}"
82,1,132,0,0.0760943,"\int \cot ^{-1}\left(a x^2\right) \, dx","Int[ArcCot[a*x^2],x]","\frac{\log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)}{2 \sqrt{2} \sqrt{a}}-\frac{\log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)}{2 \sqrt{2} \sqrt{a}}+x \cot ^{-1}\left(a x^2\right)-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)}{\sqrt{2} \sqrt{a}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{\sqrt{2} \sqrt{a}}","\frac{\log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)}{2 \sqrt{2} \sqrt{a}}-\frac{\log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)}{2 \sqrt{2} \sqrt{a}}+x \cot ^{-1}\left(a x^2\right)-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)}{\sqrt{2} \sqrt{a}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{\sqrt{2} \sqrt{a}}",1,"x*ArcCot[a*x^2] - ArcTan[1 - Sqrt[2]*Sqrt[a]*x]/(Sqrt[2]*Sqrt[a]) + ArcTan[1 + Sqrt[2]*Sqrt[a]*x]/(Sqrt[2]*Sqrt[a]) + Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2]/(2*Sqrt[2]*Sqrt[a]) - Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2]/(2*Sqrt[2]*Sqrt[a])","A",10,7,6,1.167,1,"{5028, 297, 1162, 617, 204, 1165, 628}"
83,1,135,0,0.0786399,"\int \frac{\cot ^{-1}\left(a x^2\right)}{x^2} \, dx","Int[ArcCot[a*x^2]/x^2,x]","\frac{\sqrt{a} \log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)}{2 \sqrt{2}}-\frac{\sqrt{a} \log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)}{2 \sqrt{2}}-\frac{\cot ^{-1}\left(a x^2\right)}{x}+\frac{\sqrt{a} \tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)}{\sqrt{2}}-\frac{\sqrt{a} \tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{\sqrt{2}}","\frac{\sqrt{a} \log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)}{2 \sqrt{2}}-\frac{\sqrt{a} \log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)}{2 \sqrt{2}}-\frac{\cot ^{-1}\left(a x^2\right)}{x}+\frac{\sqrt{a} \tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)}{\sqrt{2}}-\frac{\sqrt{a} \tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{\sqrt{2}}",1,"-(ArcCot[a*x^2]/x) + (Sqrt[a]*ArcTan[1 - Sqrt[2]*Sqrt[a]*x])/Sqrt[2] - (Sqrt[a]*ArcTan[1 + Sqrt[2]*Sqrt[a]*x])/Sqrt[2] + (Sqrt[a]*Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2])/(2*Sqrt[2]) - (Sqrt[a]*Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2])/(2*Sqrt[2])","A",10,7,10,0.7000,1,"{5034, 211, 1165, 628, 1162, 617, 204}"
84,1,150,0,0.0916708,"\int \frac{\cot ^{-1}\left(a x^2\right)}{x^4} \, dx","Int[ArcCot[a*x^2]/x^4,x]","\frac{a^{3/2} \log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)}{6 \sqrt{2}}-\frac{a^{3/2} \log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)}{6 \sqrt{2}}-\frac{a^{3/2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)}{3 \sqrt{2}}+\frac{a^{3/2} \tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{3 \sqrt{2}}-\frac{\cot ^{-1}\left(a x^2\right)}{3 x^3}+\frac{2 a}{3 x}","\frac{a^{3/2} \log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)}{6 \sqrt{2}}-\frac{a^{3/2} \log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)}{6 \sqrt{2}}-\frac{a^{3/2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)}{3 \sqrt{2}}+\frac{a^{3/2} \tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{3 \sqrt{2}}-\frac{\cot ^{-1}\left(a x^2\right)}{3 x^3}+\frac{2 a}{3 x}",1,"(2*a)/(3*x) - ArcCot[a*x^2]/(3*x^3) - (a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[a]*x])/(3*Sqrt[2]) + (a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[a]*x])/(3*Sqrt[2]) + (a^(3/2)*Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2])/(6*Sqrt[2]) - (a^(3/2)*Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2])/(6*Sqrt[2])","A",11,8,10,0.8000,1,"{5034, 325, 297, 1162, 617, 204, 1165, 628}"
85,1,51,0,0.0124204,"\int x^2 \cot ^{-1}\left(\sqrt{x}\right) \, dx","Int[x^2*ArcCot[Sqrt[x]],x]","\frac{x^{5/2}}{15}-\frac{x^{3/2}}{9}+\frac{1}{3} x^3 \cot ^{-1}\left(\sqrt{x}\right)+\frac{\sqrt{x}}{3}-\frac{1}{3} \tan ^{-1}\left(\sqrt{x}\right)","\frac{x^{5/2}}{15}-\frac{x^{3/2}}{9}+\frac{1}{3} x^3 \cot ^{-1}\left(\sqrt{x}\right)+\frac{\sqrt{x}}{3}-\frac{1}{3} \tan ^{-1}\left(\sqrt{x}\right)",1,"Sqrt[x]/3 - x^(3/2)/9 + x^(5/2)/15 + (x^3*ArcCot[Sqrt[x]])/3 - ArcTan[Sqrt[x]]/3","A",6,4,10,0.4000,1,"{5034, 50, 63, 203}"
86,1,42,0,0.0085725,"\int x \cot ^{-1}\left(\sqrt{x}\right) \, dx","Int[x*ArcCot[Sqrt[x]],x]","\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \cot ^{-1}\left(\sqrt{x}\right)-\frac{\sqrt{x}}{2}+\frac{1}{2} \tan ^{-1}\left(\sqrt{x}\right)","\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \cot ^{-1}\left(\sqrt{x}\right)-\frac{\sqrt{x}}{2}+\frac{1}{2} \tan ^{-1}\left(\sqrt{x}\right)",1,"-Sqrt[x]/2 + x^(3/2)/6 + (x^2*ArcCot[Sqrt[x]])/2 + ArcTan[Sqrt[x]]/2","A",5,4,8,0.5000,1,"{5034, 50, 63, 203}"
87,1,22,0,0.0055551,"\int \cot ^{-1}\left(\sqrt{x}\right) \, dx","Int[ArcCot[Sqrt[x]],x]","\sqrt{x}-\tan ^{-1}\left(\sqrt{x}\right)+x \cot ^{-1}\left(\sqrt{x}\right)","\sqrt{x}-\tan ^{-1}\left(\sqrt{x}\right)+x \cot ^{-1}\left(\sqrt{x}\right)",1,"Sqrt[x] + x*ArcCot[Sqrt[x]] - ArcTan[Sqrt[x]]","A",4,4,6,0.6667,1,"{5028, 50, 63, 203}"
88,1,31,0,0.0322822,"\int \frac{\cot ^{-1}\left(\sqrt{x}\right)}{x} \, dx","Int[ArcCot[Sqrt[x]]/x,x]","i \text{PolyLog}\left(2,\frac{i}{\sqrt{x}}\right)-i \text{PolyLog}\left(2,-\frac{i}{\sqrt{x}}\right)","i \text{PolyLog}\left(2,\frac{i}{\sqrt{x}}\right)-i \text{PolyLog}\left(2,-\frac{i}{\sqrt{x}}\right)",1,"(-I)*PolyLog[2, (-I)/Sqrt[x]] + I*PolyLog[2, I/Sqrt[x]]","A",4,3,10,0.3000,1,"{5032, 4849, 2391}"
89,1,23,0,0.0109688,"\int \frac{\cot ^{-1}\left(\sqrt{x}\right)}{x^2} \, dx","Int[ArcCot[Sqrt[x]]/x^2,x]","\frac{1}{\sqrt{x}}+\tan ^{-1}\left(\sqrt{x}\right)-\frac{\cot ^{-1}\left(\sqrt{x}\right)}{x}","\frac{1}{\sqrt{x}}+\tan ^{-1}\left(\sqrt{x}\right)-\frac{\cot ^{-1}\left(\sqrt{x}\right)}{x}",1,"1/Sqrt[x] - ArcCot[Sqrt[x]]/x + ArcTan[Sqrt[x]]","A",4,4,10,0.4000,1,"{5034, 51, 63, 203}"
90,1,42,0,0.0122351,"\int \frac{\cot ^{-1}\left(\sqrt{x}\right)}{x^3} \, dx","Int[ArcCot[Sqrt[x]]/x^3,x]","\frac{1}{6 x^{3/2}}-\frac{\cot ^{-1}\left(\sqrt{x}\right)}{2 x^2}-\frac{1}{2 \sqrt{x}}-\frac{1}{2} \tan ^{-1}\left(\sqrt{x}\right)","\frac{1}{6 x^{3/2}}-\frac{\cot ^{-1}\left(\sqrt{x}\right)}{2 x^2}-\frac{1}{2 \sqrt{x}}-\frac{1}{2} \tan ^{-1}\left(\sqrt{x}\right)",1,"1/(6*x^(3/2)) - 1/(2*Sqrt[x]) - ArcCot[Sqrt[x]]/(2*x^2) - ArcTan[Sqrt[x]]/2","A",5,4,10,0.4000,1,"{5034, 51, 63, 203}"
91,1,36,0,0.0137792,"\int x^{3/2} \cot ^{-1}\left(\sqrt{x}\right) \, dx","Int[x^(3/2)*ArcCot[Sqrt[x]],x]","\frac{x^2}{10}+\frac{2}{5} x^{5/2} \cot ^{-1}\left(\sqrt{x}\right)-\frac{x}{5}+\frac{1}{5} \log (x+1)","\frac{x^2}{10}+\frac{2}{5} x^{5/2} \cot ^{-1}\left(\sqrt{x}\right)-\frac{x}{5}+\frac{1}{5} \log (x+1)",1,"-x/5 + x^2/10 + (2*x^(5/2)*ArcCot[Sqrt[x]])/5 + Log[1 + x]/5","A",3,2,12,0.1667,1,"{5034, 43}"
92,1,29,0,0.0105779,"\int \sqrt{x} \cot ^{-1}\left(\sqrt{x}\right) \, dx","Int[Sqrt[x]*ArcCot[Sqrt[x]],x]","\frac{2}{3} x^{3/2} \cot ^{-1}\left(\sqrt{x}\right)+\frac{x}{3}-\frac{1}{3} \log (x+1)","\frac{2}{3} x^{3/2} \cot ^{-1}\left(\sqrt{x}\right)+\frac{x}{3}-\frac{1}{3} \log (x+1)",1,"x/3 + (2*x^(3/2)*ArcCot[Sqrt[x]])/3 - Log[1 + x]/3","A",3,2,12,0.1667,1,"{5034, 43}"
93,1,18,0,0.0068054,"\int \frac{\cot ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Int[ArcCot[Sqrt[x]]/Sqrt[x],x]","\log (x+1)+2 \sqrt{x} \cot ^{-1}\left(\sqrt{x}\right)","\log (x+1)+2 \sqrt{x} \cot ^{-1}\left(\sqrt{x}\right)",1,"2*Sqrt[x]*ArcCot[Sqrt[x]] + Log[1 + x]","A",2,2,12,0.1667,1,"{5034, 31}"
94,1,22,0,0.0076322,"\int \frac{\cot ^{-1}\left(\sqrt{x}\right)}{x^{3/2}} \, dx","Int[ArcCot[Sqrt[x]]/x^(3/2),x]","-\log (x)+\log (x+1)-\frac{2 \cot ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}}","-\log (x)+\log (x+1)-\frac{2 \cot ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}}",1,"(-2*ArcCot[Sqrt[x]])/Sqrt[x] - Log[x] + Log[1 + x]","A",4,4,12,0.3333,1,"{5034, 36, 29, 31}"
95,1,37,0,0.0120186,"\int \frac{\cot ^{-1}\left(\sqrt{x}\right)}{x^{5/2}} \, dx","Int[ArcCot[Sqrt[x]]/x^(5/2),x]","-\frac{2 \cot ^{-1}\left(\sqrt{x}\right)}{3 x^{3/2}}+\frac{1}{3 x}+\frac{\log (x)}{3}-\frac{1}{3} \log (x+1)","-\frac{2 \cot ^{-1}\left(\sqrt{x}\right)}{3 x^{3/2}}+\frac{1}{3 x}+\frac{\log (x)}{3}-\frac{1}{3} \log (x+1)",1,"1/(3*x) - (2*ArcCot[Sqrt[x]])/(3*x^(3/2)) + Log[x]/3 - Log[1 + x]/3","A",3,2,12,0.1667,1,"{5034, 44}"
96,1,17,0,0.005577,"\int \cot ^{-1}\left(\frac{1}{x}\right) \, dx","Int[ArcCot[x^(-1)],x]","x \cot ^{-1}\left(\frac{1}{x}\right)-\frac{1}{2} \log \left(x^2+1\right)","x \cot ^{-1}\left(\frac{1}{x}\right)-\frac{1}{2} \log \left(x^2+1\right)",1,"x*ArcCot[x^(-1)] - Log[1 + x^2]/2","A",3,3,4,0.7500,1,"{5028, 263, 260}"
97,1,47,0,0.0351754,"\int \frac{\cot ^{-1}\left(a x^n\right)}{x} \, dx","Int[ArcCot[a*x^n]/x,x]","\frac{i \text{PolyLog}\left(2,\frac{i x^{-n}}{a}\right)}{2 n}-\frac{i \text{PolyLog}\left(2,-\frac{i x^{-n}}{a}\right)}{2 n}","\frac{i \text{PolyLog}\left(2,\frac{i x^{-n}}{a}\right)}{2 n}-\frac{i \text{PolyLog}\left(2,-\frac{i x^{-n}}{a}\right)}{2 n}",1,"((-I/2)*PolyLog[2, (-I)/(a*x^n)])/n + ((I/2)*PolyLog[2, I/(a*x^n)])/n","A",4,3,10,0.3000,1,"{5032, 4849, 2391}"
98,1,37,0,0.0344804,"\int \frac{\cot ^{-1}\left(a x^5\right)}{x} \, dx","Int[ArcCot[a*x^5]/x,x]","\frac{1}{10} i \text{PolyLog}\left(2,\frac{i}{a x^5}\right)-\frac{1}{10} i \text{PolyLog}\left(2,-\frac{i}{a x^5}\right)","\frac{1}{10} i \text{PolyLog}\left(2,\frac{i}{a x^5}\right)-\frac{1}{10} i \text{PolyLog}\left(2,-\frac{i}{a x^5}\right)",1,"(-I/10)*PolyLog[2, (-I)/(a*x^5)] + (I/10)*PolyLog[2, I/(a*x^5)]","A",4,3,10,0.3000,1,"{5032, 4849, 2391}"
99,1,106,0,0.1074261,"\int x^3 \cot ^{-1}(a+b x) \, dx","Int[x^3*ArcCot[a + b*x],x]","-\frac{\left(1-6 a^2\right) x}{4 b^3}+\frac{a \left(1-a^2\right) \log \left((a+b x)^2+1\right)}{2 b^4}+\frac{\left(a^4-6 a^2+1\right) \tan ^{-1}(a+b x)}{4 b^4}+\frac{(a+b x)^3}{12 b^4}-\frac{a (a+b x)^2}{2 b^4}+\frac{1}{4} x^4 \cot ^{-1}(a+b x)","-\frac{\left(1-6 a^2\right) x}{4 b^3}+\frac{a \left(1-a^2\right) \log \left((a+b x)^2+1\right)}{2 b^4}+\frac{\left(a^4-6 a^2+1\right) \tan ^{-1}(a+b x)}{4 b^4}+\frac{(a+b x)^3}{12 b^4}-\frac{a (a+b x)^2}{2 b^4}+\frac{1}{4} x^4 \cot ^{-1}(a+b x)",1,"-((1 - 6*a^2)*x)/(4*b^3) - (a*(a + b*x)^2)/(2*b^4) + (a + b*x)^3/(12*b^4) + (x^4*ArcCot[a + b*x])/4 + ((1 - 6*a^2 + a^4)*ArcTan[a + b*x])/(4*b^4) + (a*(1 - a^2)*Log[1 + (a + b*x)^2])/(2*b^4)","A",7,6,10,0.6000,1,"{5048, 4863, 702, 635, 203, 260}"
100,1,80,0,0.0782526,"\int x^2 \cot ^{-1}(a+b x) \, dx","Int[x^2*ArcCot[a + b*x],x]","-\frac{\left(1-3 a^2\right) \log \left((a+b x)^2+1\right)}{6 b^3}+\frac{a \left(3-a^2\right) \tan ^{-1}(a+b x)}{3 b^3}-\frac{a x}{b^2}+\frac{(a+b x)^2}{6 b^3}+\frac{1}{3} x^3 \cot ^{-1}(a+b x)","-\frac{\left(1-3 a^2\right) \log \left((a+b x)^2+1\right)}{6 b^3}+\frac{a \left(3-a^2\right) \tan ^{-1}(a+b x)}{3 b^3}-\frac{a x}{b^2}+\frac{(a+b x)^2}{6 b^3}+\frac{1}{3} x^3 \cot ^{-1}(a+b x)",1,"-((a*x)/b^2) + (a + b*x)^2/(6*b^3) + (x^3*ArcCot[a + b*x])/3 + (a*(3 - a^2)*ArcTan[a + b*x])/(3*b^3) - ((1 - 3*a^2)*Log[1 + (a + b*x)^2])/(6*b^3)","A",7,6,10,0.6000,1,"{5048, 4863, 702, 635, 203, 260}"
101,1,60,0,0.0553805,"\int x \cot ^{-1}(a+b x) \, dx","Int[x*ArcCot[a + b*x],x]","-\frac{\left(1-a^2\right) \tan ^{-1}(a+b x)}{2 b^2}-\frac{a \log \left((a+b x)^2+1\right)}{2 b^2}+\frac{1}{2} x^2 \cot ^{-1}(a+b x)+\frac{x}{2 b}","-\frac{\left(1-a^2\right) \tan ^{-1}(a+b x)}{2 b^2}-\frac{a \log \left((a+b x)^2+1\right)}{2 b^2}+\frac{1}{2} x^2 \cot ^{-1}(a+b x)+\frac{x}{2 b}",1,"x/(2*b) + (x^2*ArcCot[a + b*x])/2 - ((1 - a^2)*ArcTan[a + b*x])/(2*b^2) - (a*Log[1 + (a + b*x)^2])/(2*b^2)","A",7,6,8,0.7500,1,"{5048, 4863, 702, 635, 203, 260}"
102,1,33,0,0.0115419,"\int \cot ^{-1}(a+b x) \, dx","Int[ArcCot[a + b*x],x]","\frac{\log \left((a+b x)^2+1\right)}{2 b}+\frac{(a+b x) \cot ^{-1}(a+b x)}{b}","\frac{\log \left((a+b x)^2+1\right)}{2 b}+\frac{(a+b x) \cot ^{-1}(a+b x)}{b}",1,"((a + b*x)*ArcCot[a + b*x])/b + Log[1 + (a + b*x)^2]/(2*b)","A",3,3,6,0.5000,1,"{5040, 4847, 260}"
103,1,120,0,0.1083402,"\int \frac{\cot ^{-1}(a+b x)}{x} \, dx","Int[ArcCot[a + b*x]/x,x]","-\frac{1}{2} i \text{PolyLog}\left(2,1-\frac{2}{1-i (a+b x)}\right)+\frac{1}{2} i \text{PolyLog}\left(2,1-\frac{2 b x}{(-a+i) (1-i (a+b x))}\right)+\log \left(\frac{2}{1-i (a+b x)}\right) \left(-\cot ^{-1}(a+b x)\right)+\log \left(\frac{2 b x}{(-a+i) (1-i (a+b x))}\right) \cot ^{-1}(a+b x)","-\frac{1}{2} i \text{PolyLog}\left(2,1-\frac{2}{1-i (a+b x)}\right)+\frac{1}{2} i \text{PolyLog}\left(2,1-\frac{2 b x}{(-a+i) (1-i (a+b x))}\right)+\log \left(\frac{2}{1-i (a+b x)}\right) \left(-\cot ^{-1}(a+b x)\right)+\log \left(\frac{2 b x}{(-a+i) (1-i (a+b x))}\right) \cot ^{-1}(a+b x)",1,"-(ArcCot[a + b*x]*Log[2/(1 - I*(a + b*x))]) + ArcCot[a + b*x]*Log[(2*b*x)/((I - a)*(1 - I*(a + b*x)))] - (I/2)*PolyLog[2, 1 - 2/(1 - I*(a + b*x))] + (I/2)*PolyLog[2, 1 - (2*b*x)/((I - a)*(1 - I*(a + b*x)))]","A",5,5,10,0.5000,1,"{5048, 4857, 2402, 2315, 2447}"
104,1,62,0,0.0393536,"\int \frac{\cot ^{-1}(a+b x)}{x^2} \, dx","Int[ArcCot[a + b*x]/x^2,x]","-\frac{b \log (x)}{a^2+1}+\frac{b \log \left((a+b x)^2+1\right)}{2 \left(a^2+1\right)}+\frac{a b \tan ^{-1}(a+b x)}{a^2+1}-\frac{\cot ^{-1}(a+b x)}{x}","-\frac{b \log (x)}{a^2+1}+\frac{b \log \left((a+b x)^2+1\right)}{2 \left(a^2+1\right)}+\frac{a b \tan ^{-1}(a+b x)}{a^2+1}-\frac{\cot ^{-1}(a+b x)}{x}",1,"-(ArcCot[a + b*x]/x) + (a*b*ArcTan[a + b*x])/(1 + a^2) - (b*Log[x])/(1 + a^2) + (b*Log[1 + (a + b*x)^2])/(2*(1 + a^2))","A",7,7,10,0.7000,1,"{5046, 371, 706, 31, 635, 203, 260}"
105,1,95,0,0.0821167,"\int \frac{\cot ^{-1}(a+b x)}{x^3} \, dx","Int[ArcCot[a + b*x]/x^3,x]","\frac{a b^2 \log (x)}{\left(a^2+1\right)^2}-\frac{a b^2 \log \left((a+b x)^2+1\right)}{2 \left(a^2+1\right)^2}+\frac{\left(1-a^2\right) b^2 \tan ^{-1}(a+b x)}{2 \left(a^2+1\right)^2}+\frac{b}{2 \left(a^2+1\right) x}-\frac{\cot ^{-1}(a+b x)}{2 x^2}","\frac{a b^2 \log (x)}{\left(a^2+1\right)^2}-\frac{a b^2 \log \left((a+b x)^2+1\right)}{2 \left(a^2+1\right)^2}+\frac{\left(1-a^2\right) b^2 \tan ^{-1}(a+b x)}{2 \left(a^2+1\right)^2}+\frac{b}{2 \left(a^2+1\right) x}-\frac{\cot ^{-1}(a+b x)}{2 x^2}",1,"b/(2*(1 + a^2)*x) - ArcCot[a + b*x]/(2*x^2) + ((1 - a^2)*b^2*ArcTan[a + b*x])/(2*(1 + a^2)^2) + (a*b^2*Log[x])/(1 + a^2)^2 - (a*b^2*Log[1 + (a + b*x)^2])/(2*(1 + a^2)^2)","A",8,7,10,0.7000,1,"{5046, 371, 710, 801, 635, 203, 260}"
106,1,129,0,0.1117866,"\int \frac{\cot ^{-1}(a+b x)}{x^4} \, dx","Int[ArcCot[a + b*x]/x^4,x]","-\frac{2 a b^2}{3 \left(a^2+1\right)^2 x}+\frac{\left(1-3 a^2\right) b^3 \log (x)}{3 \left(a^2+1\right)^3}-\frac{\left(1-3 a^2\right) b^3 \log \left((a+b x)^2+1\right)}{6 \left(a^2+1\right)^3}-\frac{a \left(3-a^2\right) b^3 \tan ^{-1}(a+b x)}{3 \left(a^2+1\right)^3}+\frac{b}{6 \left(a^2+1\right) x^2}-\frac{\cot ^{-1}(a+b x)}{3 x^3}","-\frac{2 a b^2}{3 \left(a^2+1\right)^2 x}+\frac{\left(1-3 a^2\right) b^3 \log (x)}{3 \left(a^2+1\right)^3}-\frac{\left(1-3 a^2\right) b^3 \log \left((a+b x)^2+1\right)}{6 \left(a^2+1\right)^3}-\frac{a \left(3-a^2\right) b^3 \tan ^{-1}(a+b x)}{3 \left(a^2+1\right)^3}+\frac{b}{6 \left(a^2+1\right) x^2}-\frac{\cot ^{-1}(a+b x)}{3 x^3}",1,"b/(6*(1 + a^2)*x^2) - (2*a*b^2)/(3*(1 + a^2)^2*x) - ArcCot[a + b*x]/(3*x^3) - (a*(3 - a^2)*b^3*ArcTan[a + b*x])/(3*(1 + a^2)^3) + ((1 - 3*a^2)*b^3*Log[x])/(3*(1 + a^2)^3) - ((1 - 3*a^2)*b^3*Log[1 + (a + b*x)^2])/(6*(1 + a^2)^3)","A",8,7,10,0.7000,1,"{5046, 371, 710, 801, 635, 203, 260}"
107,1,655,0,0.9979568,"\int \frac{\cot ^{-1}(a+b x)}{c+d x^2} \, dx","Int[ArcCot[a + b*x]/(c + d*x^2),x]","\frac{i \text{PolyLog}\left(2,-\frac{\sqrt{d} (-a-b x+i)}{b \sqrt{-c}-(-a+i) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{i \text{PolyLog}\left(2,\frac{\sqrt{d} (-a-b x+i)}{b \sqrt{-c}+(-a+i) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{i \text{PolyLog}\left(2,-\frac{\sqrt{d} (a+b x+i)}{b \sqrt{-c}-(a+i) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{i \text{PolyLog}\left(2,\frac{\sqrt{d} (a+b x+i)}{b \sqrt{-c}+(a+i) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{i \log (a+b x-i) \log \left(\frac{b \left(\sqrt{-c}-\sqrt{d} x\right)}{b \sqrt{-c}-(-a+i) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{i \log (a+b x+i) \log \left(\frac{b \left(\sqrt{-c}-\sqrt{d} x\right)}{b \sqrt{-c}+(a+i) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{i \log (a+b x-i) \log \left(\frac{b \left(\sqrt{-c}+\sqrt{d} x\right)}{b \sqrt{-c}+(-a+i) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{i \log (a+b x+i) \log \left(\frac{b \left(\sqrt{-c}+\sqrt{d} x\right)}{b \sqrt{-c}-(a+i) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{i \left(\log \left(-\frac{-a-b x+i}{a+b x}\right)+\log (a+b x)-\log (a+b x-i)\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{c} \sqrt{d}}-\frac{i \left(\log (a+b x)-\log (a+b x+i)+\log \left(\frac{a+b x+i}{a+b x}\right)\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{c} \sqrt{d}}","\frac{\text{PolyLog}\left(2,-\frac{(-a-b x+i) \left(b \sqrt{c}-i a \sqrt{d}\right)}{(a+b x) \left(b \sqrt{c}-(1+i a) \sqrt{d}\right)}\right)}{4 \sqrt{c} \sqrt{d}}-\frac{\text{PolyLog}\left(2,-\frac{(-a-b x+i) \left(b \sqrt{c}+i a \sqrt{d}\right)}{(a+b x) \left(b \sqrt{c}+(1+i a) \sqrt{d}\right)}\right)}{4 \sqrt{c} \sqrt{d}}-\frac{\text{PolyLog}\left(2,\frac{(a+b x+i) \left(b \sqrt{c}-i a \sqrt{d}\right)}{(a+b x) \left(b \sqrt{c}+(1-i a) \sqrt{d}\right)}\right)}{4 \sqrt{c} \sqrt{d}}+\frac{\text{PolyLog}\left(2,\frac{(a+b x+i) \left(b \sqrt{c}+i a \sqrt{d}\right)}{(a+b x) \left(b \sqrt{c}+i (a+i) \sqrt{d}\right)}\right)}{4 \sqrt{c} \sqrt{d}}-\frac{\log \left(\frac{a+b x+i}{a+b x}\right) \log \left(-\frac{b \left(-\sqrt{d} x+i \sqrt{c}\right)}{(a+b x) \left(b \sqrt{c}+(1-i a) \sqrt{d}\right)}\right)}{4 \sqrt{c} \sqrt{d}}+\frac{\log \left(-\frac{-a-b x+i}{a+b x}\right) \log \left(\frac{i b \left(\sqrt{c}+i \sqrt{d} x\right)}{(a+b x) \left(b \sqrt{c}-(1+i a) \sqrt{d}\right)}\right)}{4 \sqrt{c} \sqrt{d}}-\frac{\log \left(-\frac{-a-b x+i}{a+b x}\right) \log \left(\frac{b \left(\sqrt{d} x+i \sqrt{c}\right)}{(a+b x) \left(b \sqrt{c}+(1+i a) \sqrt{d}\right)}\right)}{4 \sqrt{c} \sqrt{d}}+\frac{\log \left(\frac{a+b x+i}{a+b x}\right) \log \left(-\frac{b \left(\sqrt{d} x+i \sqrt{c}\right)}{(a+b x) \left(b \sqrt{c}+i (a+i) \sqrt{d}\right)}\right)}{4 \sqrt{c} \sqrt{d}}",1,"((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*(Log[-((I - a - b*x)/(a + b*x))] + Log[a + b*x] - Log[-I + a + b*x]))/(Sqrt[c]*Sqrt[d]) - ((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*(Log[a + b*x] - Log[I + a + b*x] + Log[(I + a + b*x)/(a + b*x)]))/(Sqrt[c]*Sqrt[d]) + ((I/4)*Log[-I + a + b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] - (I - a)*Sqrt[d])])/(Sqrt[-c]*Sqrt[d]) - ((I/4)*Log[I + a + b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (I + a)*Sqrt[d])])/(Sqrt[-c]*Sqrt[d]) - ((I/4)*Log[-I + a + b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] + (I - a)*Sqrt[d])])/(Sqrt[-c]*Sqrt[d]) + ((I/4)*Log[I + a + b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (I + a)*Sqrt[d])])/(Sqrt[-c]*Sqrt[d]) + ((I/4)*PolyLog[2, -((Sqrt[d]*(I - a - b*x))/(b*Sqrt[-c] - (I - a)*Sqrt[d]))])/(Sqrt[-c]*Sqrt[d]) - ((I/4)*PolyLog[2, (Sqrt[d]*(I - a - b*x))/(b*Sqrt[-c] + (I - a)*Sqrt[d])])/(Sqrt[-c]*Sqrt[d]) + ((I/4)*PolyLog[2, -((Sqrt[d]*(I + a + b*x))/(b*Sqrt[-c] - (I + a)*Sqrt[d]))])/(Sqrt[-c]*Sqrt[d]) - ((I/4)*PolyLog[2, (Sqrt[d]*(I + a + b*x))/(b*Sqrt[-c] + (I + a)*Sqrt[d])])/(Sqrt[-c]*Sqrt[d])","A",37,7,16,0.4375,0,"{5052, 2513, 2409, 2394, 2393, 2391, 205}"
108,1,152,0,0.1416097,"\int \frac{\cot ^{-1}(a+b x)}{c+d x} \, dx","Int[ArcCot[a + b*x]/(c + d*x),x]","\frac{i \text{PolyLog}\left(2,1-\frac{2 b (c+d x)}{(1-i (a+b x)) (-a d+b c+i d)}\right)}{2 d}-\frac{i \text{PolyLog}\left(2,1-\frac{2}{1-i (a+b x)}\right)}{2 d}+\frac{\cot ^{-1}(a+b x) \log \left(\frac{2 b (c+d x)}{(1-i (a+b x)) (-a d+b c+i d)}\right)}{d}-\frac{\log \left(\frac{2}{1-i (a+b x)}\right) \cot ^{-1}(a+b x)}{d}","\frac{i \text{PolyLog}\left(2,1-\frac{2 b (c+d x)}{(1-i (a+b x)) (-a d+b c+i d)}\right)}{2 d}-\frac{i \text{PolyLog}\left(2,1-\frac{2}{1-i (a+b x)}\right)}{2 d}+\frac{\cot ^{-1}(a+b x) \log \left(\frac{2 b (c+d x)}{(1-i (a+b x)) (-a d+b c+i d)}\right)}{d}-\frac{\log \left(\frac{2}{1-i (a+b x)}\right) \cot ^{-1}(a+b x)}{d}",1,"-((ArcCot[a + b*x]*Log[2/(1 - I*(a + b*x))])/d) + (ArcCot[a + b*x]*Log[(2*b*(c + d*x))/((b*c + I*d - a*d)*(1 - I*(a + b*x)))])/d - ((I/2)*PolyLog[2, 1 - 2/(1 - I*(a + b*x))])/d + ((I/2)*PolyLog[2, 1 - (2*b*(c + d*x))/((b*c + I*d - a*d)*(1 - I*(a + b*x)))])/d","A",5,5,14,0.3571,1,"{5048, 4857, 2402, 2315, 2447}"
109,1,422,0,0.4982247,"\int \frac{\cot ^{-1}(a+b x)}{c+\frac{d}{x}} \, dx","Int[ArcCot[a + b*x]/(c + d/x),x]","-\frac{i d \text{PolyLog}\left(2,\frac{c (-a-b x+i)}{b d+(-a+i) c}\right)}{2 c^2}+\frac{i d \text{PolyLog}\left(2,\frac{c (a+b x+i)}{-b d+(a+i) c}\right)}{2 c^2}-\frac{i d \left(\log \left(-\frac{-a-b x+i}{a+b x}\right)+\log (a+b x)-\log (a+b x-i)\right) \log (c x+d)}{2 c^2}+\frac{i d \left(\log (a+b x)-\log (a+b x+i)+\log \left(\frac{a+b x+i}{a+b x}\right)\right) \log (c x+d)}{2 c^2}+\frac{i d \log (a+b x+i) \log \left(-\frac{b (c x+d)}{-b d+(a+i) c}\right)}{2 c^2}-\frac{i d \log (a+b x-i) \log \left(\frac{b (c x+d)}{b d+(-a+i) c}\right)}{2 c^2}+\frac{i x \left(\log \left(-\frac{-a-b x+i}{a+b x}\right)+\log (a+b x)-\log (a+b x-i)\right)}{2 c}-\frac{i (-a-b x+i) \log (a+b x-i)}{2 b c}-\frac{i (a+b x+i) \log (a+b x+i)}{2 b c}-\frac{i x \left(\log (a+b x)-\log (a+b x+i)+\log \left(\frac{a+b x+i}{a+b x}\right)\right)}{2 c}","-\frac{i d \text{PolyLog}\left(2,-\frac{b (c x+d)}{-b d+(a+i) c}\right)}{2 c^2}+\frac{i d \text{PolyLog}\left(2,\frac{b (c x+d)}{-a c+b d+i c}\right)}{2 c^2}+\frac{i d \log (c x+d) \log \left(\frac{c (-a-b x+i)}{-a c+b d+i c}\right)}{2 c^2}-\frac{i d \log \left(-\frac{-a-b x+i}{a+b x}\right) \log (c x+d)}{2 c^2}-\frac{i d \log (c x+d) \log \left(\frac{c (a+b x+i)}{-b d+(a+i) c}\right)}{2 c^2}+\frac{i d \log \left(\frac{a+b x+i}{a+b x}\right) \log (c x+d)}{2 c^2}+\frac{\log (-a-b x+i)}{2 b c}+\frac{i (a+b x) \log \left(-\frac{-a-b x+i}{a+b x}\right)}{2 b c}+\frac{\log (a+b x+i)}{2 b c}-\frac{i (a+b x) \log \left(\frac{a+b x+i}{a+b x}\right)}{2 b c}",1,"((I/2)*x*(Log[-((I - a - b*x)/(a + b*x))] + Log[a + b*x] - Log[-I + a + b*x]))/c - ((I/2)*(I - a - b*x)*Log[-I + a + b*x])/(b*c) - ((I/2)*(I + a + b*x)*Log[I + a + b*x])/(b*c) - ((I/2)*x*(Log[a + b*x] - Log[I + a + b*x] + Log[(I + a + b*x)/(a + b*x)]))/c - ((I/2)*d*(Log[-((I - a - b*x)/(a + b*x))] + Log[a + b*x] - Log[-I + a + b*x])*Log[d + c*x])/c^2 + ((I/2)*d*(Log[a + b*x] - Log[I + a + b*x] + Log[(I + a + b*x)/(a + b*x)])*Log[d + c*x])/c^2 + ((I/2)*d*Log[I + a + b*x]*Log[-((b*(d + c*x))/((I + a)*c - b*d))])/c^2 - ((I/2)*d*Log[-I + a + b*x]*Log[(b*(d + c*x))/((I - a)*c + b*d)])/c^2 - ((I/2)*d*PolyLog[2, (c*(I - a - b*x))/((I - a)*c + b*d)])/c^2 + ((I/2)*d*PolyLog[2, (c*(I + a + b*x))/((I + a)*c - b*d)])/c^2","A",37,10,16,0.6250,0,"{5052, 2513, 2409, 2389, 2295, 2394, 2393, 2391, 193, 43}"
110,1,818,0,1.5178583,"\int \frac{\cot ^{-1}(a+b x)}{c+\frac{d}{x^2}} \, dx","Int[ArcCot[a + b*x]/(c + d/x^2),x]","\frac{i x \left(\log \left(-\frac{-a-b x+i}{a+b x}\right)+\log (a+b x)-\log (a+b x-i)\right)}{2 c}-\frac{i \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) \left(\log \left(-\frac{-a-b x+i}{a+b x}\right)+\log (a+b x)-\log (a+b x-i)\right)}{2 c^{3/2}}-\frac{i (-a-b x+i) \log (a+b x-i)}{2 b c}-\frac{i (a+b x+i) \log (a+b x+i)}{2 b c}-\frac{i x \left(\log (a+b x)-\log (a+b x+i)+\log \left(\frac{a+b x+i}{a+b x}\right)\right)}{2 c}+\frac{i \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) \left(\log (a+b x)-\log (a+b x+i)+\log \left(\frac{a+b x+i}{a+b x}\right)\right)}{2 c^{3/2}}-\frac{i \sqrt{d} \log (a+b x-i) \log \left(-\frac{b \left(\sqrt{d}-\sqrt{-c} x\right)}{(i-a) \sqrt{-c}-b \sqrt{d}}\right)}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \log (a+b x+i) \log \left(\frac{b \left(\sqrt{d}-\sqrt{-c} x\right)}{\sqrt{-c} (a+i)+b \sqrt{d}}\right)}{4 (-c)^{3/2}}-\frac{i \sqrt{d} \log (a+b x+i) \log \left(-\frac{b \left(\sqrt{-c} x+\sqrt{d}\right)}{(a+i) \sqrt{-c}-b \sqrt{d}}\right)}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \log (a+b x-i) \log \left(\frac{b \left(\sqrt{-c} x+\sqrt{d}\right)}{\sqrt{-c} (i-a)+b \sqrt{d}}\right)}{4 (-c)^{3/2}}-\frac{i \sqrt{d} \text{PolyLog}\left(2,\frac{\sqrt{-c} (-a-b x+i)}{(i-a) \sqrt{-c}-b \sqrt{d}}\right)}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \text{PolyLog}\left(2,\frac{\sqrt{-c} (-a-b x+i)}{\sqrt{-c} (i-a)+b \sqrt{d}}\right)}{4 (-c)^{3/2}}-\frac{i \sqrt{d} \text{PolyLog}\left(2,\frac{\sqrt{-c} (a+b x+i)}{(a+i) \sqrt{-c}-b \sqrt{d}}\right)}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \text{PolyLog}\left(2,\frac{\sqrt{-c} (a+b x+i)}{\sqrt{-c} (a+i)+b \sqrt{d}}\right)}{4 (-c)^{3/2}}","-\frac{\sqrt{d} \text{PolyLog}\left(2,\frac{b \left(\sqrt{d}-i \sqrt{c} x\right)}{b \sqrt{d}+(1+i a) \sqrt{c}}\right)}{4 c^{3/2}}+\frac{\sqrt{d} \text{PolyLog}\left(2,\frac{b \left(\sqrt{d}-i \sqrt{c} x\right)}{b \sqrt{d}+i (a+i) \sqrt{c}}\right)}{4 c^{3/2}}+\frac{\sqrt{d} \text{PolyLog}\left(2,-\frac{b \left(\sqrt{d}+i \sqrt{c} x\right)}{-b \sqrt{d}+(1+i a) \sqrt{c}}\right)}{4 c^{3/2}}-\frac{\sqrt{d} \text{PolyLog}\left(2,\frac{b \left(\sqrt{d}+i \sqrt{c} x\right)}{b \sqrt{d}+(1-i a) \sqrt{c}}\right)}{4 c^{3/2}}-\frac{\sqrt{d} \log \left(1-\frac{i \sqrt{c} x}{\sqrt{d}}\right) \log \left(\frac{\sqrt{c} (-a-b x+i)}{(-a+i) \sqrt{c}+i b \sqrt{d}}\right)}{4 c^{3/2}}+\frac{\sqrt{d} \log \left(1-\frac{i \sqrt{c} x}{\sqrt{d}}\right) \log \left(\frac{\sqrt{c} (a+b x+i)}{(a+i) \sqrt{c}-i b \sqrt{d}}\right)}{4 c^{3/2}}+\frac{\sqrt{d} \log \left(1+\frac{i \sqrt{c} x}{\sqrt{d}}\right) \log \left(\frac{\sqrt{c} (-a-b x+i)}{(-a+i) \sqrt{c}-i b \sqrt{d}}\right)}{4 c^{3/2}}-\frac{\sqrt{d} \log \left(1+\frac{i \sqrt{c} x}{\sqrt{d}}\right) \log \left(\frac{\sqrt{c} (a+b x+i)}{(a+i) \sqrt{c}+i b \sqrt{d}}\right)}{4 c^{3/2}}-\frac{i \sqrt{d} \log \left(-\frac{-a-b x+i}{a+b x}\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)}{2 c^{3/2}}+\frac{i \sqrt{d} \log \left(\frac{a+b x+i}{a+b x}\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)}{2 c^{3/2}}+\frac{\log (-a-b x+i)}{2 b c}+\frac{i (a+b x) \log \left(-\frac{-a-b x+i}{a+b x}\right)}{2 b c}+\frac{\log (a+b x+i)}{2 b c}-\frac{i (a+b x) \log \left(\frac{a+b x+i}{a+b x}\right)}{2 b c}",1,"((I/2)*x*(Log[-((I - a - b*x)/(a + b*x))] + Log[a + b*x] - Log[-I + a + b*x]))/c - ((I/2)*Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*(Log[-((I - a - b*x)/(a + b*x))] + Log[a + b*x] - Log[-I + a + b*x]))/c^(3/2) - ((I/2)*(I - a - b*x)*Log[-I + a + b*x])/(b*c) - ((I/2)*(I + a + b*x)*Log[I + a + b*x])/(b*c) - ((I/2)*x*(Log[a + b*x] - Log[I + a + b*x] + Log[(I + a + b*x)/(a + b*x)]))/c + ((I/2)*Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*(Log[a + b*x] - Log[I + a + b*x] + Log[(I + a + b*x)/(a + b*x)]))/c^(3/2) - ((I/4)*Sqrt[d]*Log[-I + a + b*x]*Log[-((b*(Sqrt[d] - Sqrt[-c]*x))/((I - a)*Sqrt[-c] - b*Sqrt[d]))])/(-c)^(3/2) + ((I/4)*Sqrt[d]*Log[I + a + b*x]*Log[(b*(Sqrt[d] - Sqrt[-c]*x))/((I + a)*Sqrt[-c] + b*Sqrt[d])])/(-c)^(3/2) - ((I/4)*Sqrt[d]*Log[I + a + b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/((I + a)*Sqrt[-c] - b*Sqrt[d]))])/(-c)^(3/2) + ((I/4)*Sqrt[d]*Log[-I + a + b*x]*Log[(b*(Sqrt[d] + Sqrt[-c]*x))/((I - a)*Sqrt[-c] + b*Sqrt[d])])/(-c)^(3/2) - ((I/4)*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I - a - b*x))/((I - a)*Sqrt[-c] - b*Sqrt[d])])/(-c)^(3/2) + ((I/4)*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I - a - b*x))/((I - a)*Sqrt[-c] + b*Sqrt[d])])/(-c)^(3/2) - ((I/4)*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I + a + b*x))/((I + a)*Sqrt[-c] - b*Sqrt[d])])/(-c)^(3/2) + ((I/4)*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I + a + b*x))/((I + a)*Sqrt[-c] + b*Sqrt[d])])/(-c)^(3/2)","A",57,11,16,0.6875,0,"{5052, 2513, 2409, 2389, 2295, 2394, 2393, 2391, 193, 321, 205}"
111,1,693,0,2.0647079,"\int \frac{\cot ^{-1}(a+b x)}{c+d \sqrt{x}} \, dx","Int[ArcCot[a + b*x]/(c + d*Sqrt[x]),x]","-\frac{i c \text{PolyLog}\left(2,\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{-a-i} d}\right)}{d^2}-\frac{i c \text{PolyLog}\left(2,\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{-a-i} d}\right)}{d^2}+\frac{i c \text{PolyLog}\left(2,\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{-a+i} d}\right)}{d^2}+\frac{i c \text{PolyLog}\left(2,\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{-a+i} d}\right)}{d^2}-\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{\sqrt{b} c+\sqrt{-a-i} d}\right)}{d^2}+\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{\sqrt{b} c+\sqrt{-a+i} d}\right)}{d^2}-\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(-\frac{d \left(\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{\sqrt{b} c-\sqrt{-a-i} d}\right)}{d^2}+\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(-\frac{d \left(\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{\sqrt{b} c-\sqrt{-a+i} d}\right)}{d^2}-\frac{i c \log \left(-\frac{-a-b x+i}{a+b x}\right) \log \left(c+d \sqrt{x}\right)}{d^2}+\frac{i c \log \left(\frac{a+b x+i}{a+b x}\right) \log \left(c+d \sqrt{x}\right)}{d^2}+\frac{i \sqrt{x} \log \left(-\frac{-a-b x+i}{a+b x}\right)}{d}-\frac{i \sqrt{x} \log \left(\frac{a+b x+i}{a+b x}\right)}{d}-\frac{2 i \sqrt{a+i} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right)}{\sqrt{b} d}+\frac{2 i \sqrt{-a+i} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right)}{\sqrt{b} d}","-\frac{i c \text{PolyLog}\left(2,\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{-a-i} d}\right)}{d^2}-\frac{i c \text{PolyLog}\left(2,\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{-a-i} d}\right)}{d^2}+\frac{i c \text{PolyLog}\left(2,\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{-a+i} d}\right)}{d^2}+\frac{i c \text{PolyLog}\left(2,\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{-a+i} d}\right)}{d^2}-\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{\sqrt{b} c+\sqrt{-a-i} d}\right)}{d^2}+\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{\sqrt{b} c+\sqrt{-a+i} d}\right)}{d^2}-\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(-\frac{d \left(\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{\sqrt{b} c-\sqrt{-a-i} d}\right)}{d^2}+\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(-\frac{d \left(\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{\sqrt{b} c-\sqrt{-a+i} d}\right)}{d^2}-\frac{i c \log \left(-\frac{-a-b x+i}{a+b x}\right) \log \left(c+d \sqrt{x}\right)}{d^2}+\frac{i c \log \left(\frac{a+b x+i}{a+b x}\right) \log \left(c+d \sqrt{x}\right)}{d^2}+\frac{i \sqrt{x} \log \left(-\frac{-a-b x+i}{a+b x}\right)}{d}-\frac{i \sqrt{x} \log \left(\frac{a+b x+i}{a+b x}\right)}{d}-\frac{2 i \sqrt{a+i} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right)}{\sqrt{b} d}+\frac{2 i \sqrt{-a+i} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right)}{\sqrt{b} d}",1,"((-2*I)*Sqrt[I + a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]])/(Sqrt[b]*d) + ((2*I)*Sqrt[I - a]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[I - a]])/(Sqrt[b]*d) - (I*c*Log[(d*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-I - a]*d)]*Log[c + d*Sqrt[x]])/d^2 + (I*c*Log[(d*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[I - a]*d)]*Log[c + d*Sqrt[x]])/d^2 - (I*c*Log[-((d*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-I - a]*d))]*Log[c + d*Sqrt[x]])/d^2 + (I*c*Log[-((d*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c - Sqrt[I - a]*d))]*Log[c + d*Sqrt[x]])/d^2 + (I*Sqrt[x]*Log[-((I - a - b*x)/(a + b*x))])/d - (I*c*Log[c + d*Sqrt[x]]*Log[-((I - a - b*x)/(a + b*x))])/d^2 - (I*Sqrt[x]*Log[(I + a + b*x)/(a + b*x)])/d + (I*c*Log[c + d*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)])/d^2 - (I*c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-I - a]*d)])/d^2 - (I*c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-I - a]*d)])/d^2 + (I*c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[I - a]*d)])/d^2 + (I*c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[I - a]*d)])/d^2","A",55,16,18,0.8889,1,"{5052, 190, 43, 2528, 2523, 12, 481, 205, 208, 2524, 2418, 260, 2416, 2394, 2393, 2391}"
112,1,830,0,2.3218693,"\int \frac{\cot ^{-1}(a+b x)}{c+\frac{d}{\sqrt{x}}} \, dx","Int[ArcCot[a + b*x]/(c + d/Sqrt[x]),x]","\frac{i \log \left(\frac{c \left(\sqrt{-a-i}-\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-i} c+\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right) d^2}{c^3}-\frac{i \log \left(\frac{c \left(\sqrt{i-a}-\sqrt{b} \sqrt{x}\right)}{\sqrt{i-a} c+\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right) d^2}{c^3}+\frac{i \log \left(\frac{c \left(\sqrt{-a-i}+\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-i} c-\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right) d^2}{c^3}-\frac{i \log \left(\frac{c \left(\sqrt{i-a}+\sqrt{b} \sqrt{x}\right)}{\sqrt{i-a} c-\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right) d^2}{c^3}+\frac{i \log \left(\sqrt{x} c+d\right) \log \left(-\frac{-a-b x+i}{a+b x}\right) d^2}{c^3}-\frac{i \log \left(\sqrt{x} c+d\right) \log \left(\frac{a+b x+i}{a+b x}\right) d^2}{c^3}+\frac{i \text{PolyLog}\left(2,-\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{-a-i} c-\sqrt{b} d}\right) d^2}{c^3}-\frac{i \text{PolyLog}\left(2,-\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{i-a} c-\sqrt{b} d}\right) d^2}{c^3}+\frac{i \text{PolyLog}\left(2,\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{-a-i} c+\sqrt{b} d}\right) d^2}{c^3}-\frac{i \text{PolyLog}\left(2,\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{i-a} c+\sqrt{b} d}\right) d^2}{c^3}+\frac{2 i \sqrt{a+i} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right) d}{\sqrt{b} c^2}-\frac{2 i \sqrt{i-a} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{i-a}}\right) d}{\sqrt{b} c^2}-\frac{i \sqrt{x} \log \left(-\frac{-a-b x+i}{a+b x}\right) d}{c^2}+\frac{i \sqrt{x} \log \left(\frac{a+b x+i}{a+b x}\right) d}{c^2}+\frac{(i a+1) \log (-a-b x+i)}{2 b c}+\frac{i x \log \left(-\frac{-a-b x+i}{a+b x}\right)}{2 c}+\frac{(1-i a) \log (a+b x+i)}{2 b c}-\frac{i x \log \left(\frac{a+b x+i}{a+b x}\right)}{2 c}","\frac{i \log \left(\frac{c \left(\sqrt{-a-i}-\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-i} c+\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right) d^2}{c^3}-\frac{i \log \left(\frac{c \left(\sqrt{i-a}-\sqrt{b} \sqrt{x}\right)}{\sqrt{i-a} c+\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right) d^2}{c^3}+\frac{i \log \left(\frac{c \left(\sqrt{-a-i}+\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-i} c-\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right) d^2}{c^3}-\frac{i \log \left(\frac{c \left(\sqrt{i-a}+\sqrt{b} \sqrt{x}\right)}{\sqrt{i-a} c-\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right) d^2}{c^3}+\frac{i \log \left(\sqrt{x} c+d\right) \log \left(-\frac{-a-b x+i}{a+b x}\right) d^2}{c^3}-\frac{i \log \left(\sqrt{x} c+d\right) \log \left(\frac{a+b x+i}{a+b x}\right) d^2}{c^3}+\frac{i \text{PolyLog}\left(2,-\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{-a-i} c-\sqrt{b} d}\right) d^2}{c^3}-\frac{i \text{PolyLog}\left(2,-\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{i-a} c-\sqrt{b} d}\right) d^2}{c^3}+\frac{i \text{PolyLog}\left(2,\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{-a-i} c+\sqrt{b} d}\right) d^2}{c^3}-\frac{i \text{PolyLog}\left(2,\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{i-a} c+\sqrt{b} d}\right) d^2}{c^3}+\frac{2 i \sqrt{a+i} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right) d}{\sqrt{b} c^2}-\frac{2 i \sqrt{i-a} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{i-a}}\right) d}{\sqrt{b} c^2}-\frac{i \sqrt{x} \log \left(-\frac{-a-b x+i}{a+b x}\right) d}{c^2}+\frac{i \sqrt{x} \log \left(\frac{a+b x+i}{a+b x}\right) d}{c^2}+\frac{(i a+1) \log (-a-b x+i)}{2 b c}+\frac{i x \log \left(-\frac{-a-b x+i}{a+b x}\right)}{2 c}+\frac{(1-i a) \log (a+b x+i)}{2 b c}-\frac{i x \log \left(\frac{a+b x+i}{a+b x}\right)}{2 c}",1,"((2*I)*Sqrt[I + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]])/(Sqrt[b]*c^2) - ((2*I)*Sqrt[I - a]*d*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[I - a]])/(Sqrt[b]*c^2) + (I*d^2*Log[(c*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 - (I*d^2*Log[(c*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 + (I*d^2*Log[(c*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 - (I*d^2*Log[(c*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 + ((1 + I*a)*Log[I - a - b*x])/(2*b*c) - (I*d*Sqrt[x]*Log[-((I - a - b*x)/(a + b*x))])/c^2 + ((I/2)*x*Log[-((I - a - b*x)/(a + b*x))])/c + (I*d^2*Log[d + c*Sqrt[x]]*Log[-((I - a - b*x)/(a + b*x))])/c^3 + ((1 - I*a)*Log[I + a + b*x])/(2*b*c) + (I*d*Sqrt[x]*Log[(I + a + b*x)/(a + b*x)])/c^2 - ((I/2)*x*Log[(I + a + b*x)/(a + b*x)])/c - (I*d^2*Log[d + c*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)])/c^3 + (I*d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d))])/c^3 - (I*d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[I - a]*c - Sqrt[b]*d))])/c^3 + (I*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d)])/c^3 - (I*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)])/c^3","A",65,19,18,1.056,1,"{5052, 190, 44, 2528, 2523, 12, 481, 205, 208, 2525, 446, 72, 2524, 2418, 260, 2416, 2394, 2393, 2391}"
113,1,367,0,0.6860653,"\int \frac{\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx","Int[ArcCot[d + e*x]/(a + b*x + c*x^2),x]","\frac{i \text{PolyLog}\left(2,1+\frac{2 \left(-e \left(b-\sqrt{b^2-4 a c}\right)-2 c (d+e x)+2 c d\right)}{(1-i (d+e x)) \left(-e \sqrt{b^2-4 a c}+b e-2 c d+2 i c\right)}\right)}{2 \sqrt{b^2-4 a c}}-\frac{i \text{PolyLog}\left(2,1+\frac{2 \left(-e \left(\sqrt{b^2-4 a c}+b\right)-2 c (d+e x)+2 c d\right)}{(1-i (d+e x)) \left(e \left(\sqrt{b^2-4 a c}+b\right)+2 c (-d+i)\right)}\right)}{2 \sqrt{b^2-4 a c}}+\frac{\cot ^{-1}(d+e x) \log \left(\frac{2 e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{(1-i (d+e x)) \left(e \left(b-\sqrt{b^2-4 a c}\right)+2 c (-d+i)\right)}\right)}{\sqrt{b^2-4 a c}}-\frac{\cot ^{-1}(d+e x) \log \left(\frac{2 e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{(1-i (d+e x)) \left(e \left(\sqrt{b^2-4 a c}+b\right)+2 c (-d+i)\right)}\right)}{\sqrt{b^2-4 a c}}","\frac{i \text{PolyLog}\left(2,1+\frac{2 \left(-e \left(b-\sqrt{b^2-4 a c}\right)-2 c (d+e x)+2 c d\right)}{(1-i (d+e x)) \left(-e \sqrt{b^2-4 a c}+b e-2 c d+2 i c\right)}\right)}{2 \sqrt{b^2-4 a c}}-\frac{i \text{PolyLog}\left(2,1+\frac{2 \left(-e \left(\sqrt{b^2-4 a c}+b\right)-2 c (d+e x)+2 c d\right)}{(1-i (d+e x)) \left(e \left(\sqrt{b^2-4 a c}+b\right)+2 c (-d+i)\right)}\right)}{2 \sqrt{b^2-4 a c}}+\frac{\cot ^{-1}(d+e x) \log \left(\frac{2 e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{(1-i (d+e x)) \left(e \left(b-\sqrt{b^2-4 a c}\right)+2 c (-d+i)\right)}\right)}{\sqrt{b^2-4 a c}}-\frac{\cot ^{-1}(d+e x) \log \left(\frac{2 e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{(1-i (d+e x)) \left(e \left(\sqrt{b^2-4 a c}+b\right)+2 c (-d+i)\right)}\right)}{\sqrt{b^2-4 a c}}",1,"(ArcCot[d + e*x]*Log[(2*e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(I - d) + (b - Sqrt[b^2 - 4*a*c])*e)*(1 - I*(d + e*x)))])/Sqrt[b^2 - 4*a*c] - (ArcCot[d + e*x]*Log[(2*e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(I - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1 - I*(d + e*x)))])/Sqrt[b^2 - 4*a*c] + ((I/2)*PolyLog[2, 1 + (2*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/(((2*I)*c - 2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e)*(1 - I*(d + e*x)))])/Sqrt[b^2 - 4*a*c] - ((I/2)*PolyLog[2, 1 + (2*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*c*(I - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1 - I*(d + e*x)))])/Sqrt[b^2 - 4*a*c]","A",12,8,19,0.4211,1,"{618, 206, 6728, 5048, 4857, 2402, 2315, 2447}"
114,1,132,0,0.0964322,"\int \frac{\cot ^{-1}(a+b x)}{\sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx","Int[ArcCot[a + b*x]/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2],x]","-\frac{i \text{PolyLog}\left(2,-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{b}+\frac{i \text{PolyLog}\left(2,\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{b}-\frac{2 i \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right) \cot ^{-1}(a+b x)}{b}","-\frac{i \text{PolyLog}\left(2,-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{b}+\frac{i \text{PolyLog}\left(2,\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{b}-\frac{2 i \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right) \cot ^{-1}(a+b x)}{b}",1,"((-2*I)*ArcCot[a + b*x]*ArcTan[Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]])/b - (I*PolyLog[2, ((-I)*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/b + (I*PolyLog[2, (I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/b","A",2,2,28,0.07143,1,"{5056, 4887}"
115,1,216,0,0.1675857,"\int \frac{\cot ^{-1}(a+b x)}{\sqrt{\left(1+a^2\right) c+2 a b c x+b^2 c x^2}} \, dx","Int[ArcCot[a + b*x]/Sqrt[(1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2],x]","-\frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{b \sqrt{c (a+b x)^2+c}}+\frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{b \sqrt{c (a+b x)^2+c}}-\frac{2 i \sqrt{(a+b x)^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right) \cot ^{-1}(a+b x)}{b \sqrt{c (a+b x)^2+c}}","-\frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{b \sqrt{c (a+b x)^2+c}}+\frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{b \sqrt{c (a+b x)^2+c}}-\frac{2 i \sqrt{(a+b x)^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right) \cot ^{-1}(a+b x)}{b \sqrt{c (a+b x)^2+c}}",1,"((-2*I)*Sqrt[1 + (a + b*x)^2]*ArcCot[a + b*x]*ArcTan[Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a + b*x)^2]) - (I*Sqrt[1 + (a + b*x)^2]*PolyLog[2, ((-I)*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a + b*x)^2]) + (I*Sqrt[1 + (a + b*x)^2]*PolyLog[2, (I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a + b*x)^2])","A",3,3,33,0.09091,1,"{5056, 4891, 4887}"
116,0,0,0,0.0400934,"\int \frac{\cot ^{-1}(a+b x)}{\sqrt[3]{1+a^2+2 a b x+b^2 x^2}} \, dx","Int[ArcCot[a + b*x]/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3),x]","\int \frac{\cot ^{-1}(a+b x)}{\sqrt[3]{1+a^2+2 a b x+b^2 x^2}} \, dx","\text{Int}\left(\frac{\cot ^{-1}(a+b x)}{\sqrt[3]{(a+b x)^2+1}},x\right)",0,"Defer[Subst][Defer[Int][ArcCot[x]/(1 + x^2)^(1/3), x], x, a + b*x]/b","A",0,0,0,0,-1,"{}"
117,0,0,0,0.0533345,"\int \frac{\cot ^{-1}(a+b x)}{\sqrt[3]{\left(1+a^2\right) c+2 a b c x+b^2 c x^2}} \, dx","Int[ArcCot[a + b*x]/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]","\int \frac{\cot ^{-1}(a+b x)}{\sqrt[3]{\left(1+a^2\right) c+2 a b c x+b^2 c x^2}} \, dx","\text{Int}\left(\frac{\cot ^{-1}(a+b x)}{\sqrt[3]{c (a+b x)^2+c}},x\right)",0,"Defer[Subst][Defer[Int][ArcCot[x]/(c + c*x^2)^(1/3), x], x, a + b*x]/b","A",0,0,0,0,-1,"{}"
118,1,187,0,0.2229004,"\int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx","Int[((a + b*x)^2*ArcCot[a + b*x])/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2],x]","\frac{i \text{PolyLog}\left(2,-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{2 b}-\frac{i \text{PolyLog}\left(2,\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{2 b}+\frac{\sqrt{(a+b x)^2+1}}{2 b}+\frac{(a+b x) \sqrt{(a+b x)^2+1} \cot ^{-1}(a+b x)}{2 b}+\frac{i \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right) \cot ^{-1}(a+b x)}{b}","\frac{i \text{PolyLog}\left(2,-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{2 b}-\frac{i \text{PolyLog}\left(2,\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{2 b}+\frac{\sqrt{(a+b x)^2+1}}{2 b}+\frac{(a+b x) \sqrt{(a+b x)^2+1} \cot ^{-1}(a+b x)}{2 b}+\frac{i \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right) \cot ^{-1}(a+b x)}{b}",1,"Sqrt[1 + (a + b*x)^2]/(2*b) + ((a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcCot[a + b*x])/(2*b) + (I*ArcCot[a + b*x]*ArcTan[Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]])/b + ((I/2)*PolyLog[2, ((-I)*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/b - ((I/2)*PolyLog[2, (I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/b","A",4,4,35,0.1143,1,"{5058, 4953, 261, 4887}"
119,1,281,0,0.3485996,"\int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt{\left(1+a^2\right) c+2 a b c x+b^2 c x^2}} \, dx","Int[((a + b*x)^2*ArcCot[a + b*x])/Sqrt[(1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2],x]","\frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{2 b \sqrt{c (a+b x)^2+c}}-\frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{2 b \sqrt{c (a+b x)^2+c}}+\frac{\sqrt{c (a+b x)^2+c}}{2 b c}+\frac{(a+b x) \sqrt{c (a+b x)^2+c} \cot ^{-1}(a+b x)}{2 b c}+\frac{i \sqrt{(a+b x)^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right) \cot ^{-1}(a+b x)}{b \sqrt{c (a+b x)^2+c}}","\frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{2 b \sqrt{c (a+b x)^2+c}}-\frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{2 b \sqrt{c (a+b x)^2+c}}+\frac{\sqrt{c (a+b x)^2+c}}{2 b c}+\frac{(a+b x) \sqrt{c (a+b x)^2+c} \cot ^{-1}(a+b x)}{2 b c}+\frac{i \sqrt{(a+b x)^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right) \cot ^{-1}(a+b x)}{b \sqrt{c (a+b x)^2+c}}",1,"Sqrt[c + c*(a + b*x)^2]/(2*b*c) + ((a + b*x)*Sqrt[c + c*(a + b*x)^2]*ArcCot[a + b*x])/(2*b*c) + (I*Sqrt[1 + (a + b*x)^2]*ArcCot[a + b*x]*ArcTan[Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a + b*x)^2]) + ((I/2)*Sqrt[1 + (a + b*x)^2]*PolyLog[2, ((-I)*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a + b*x)^2]) - ((I/2)*Sqrt[1 + (a + b*x)^2]*PolyLog[2, (I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a + b*x)^2])","A",5,5,40,0.1250,1,"{5058, 4953, 261, 4891, 4887}"
120,0,0,0,0.1370706,"\int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt[3]{1+a^2+2 a b x+b^2 x^2}} \, dx","Int[((a + b*x)^2*ArcCot[a + b*x])/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3),x]","\int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt[3]{1+a^2+2 a b x+b^2 x^2}} \, dx","\text{Int}\left(\frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt[3]{(a+b x)^2+1}},x\right)",0,"Defer[Subst][Defer[Int][(x^2*ArcCot[x])/(1 + x^2)^(1/3), x], x, a + b*x]/b","A",0,0,0,0,-1,"{}"
121,0,0,0,0.1897071,"\int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt[3]{\left(1+a^2\right) c+2 a b c x+b^2 c x^2}} \, dx","Int[((a + b*x)^2*ArcCot[a + b*x])/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]","\int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt[3]{\left(1+a^2\right) c+2 a b c x+b^2 c x^2}} \, dx","\text{Int}\left(\frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt[3]{c (a+b x)^2+c}},x\right)",0,"Defer[Subst][Defer[Int][(x^2*ArcCot[x])/(c + c*x^2)^(1/3), x], x, a + b*x]/b","A",0,0,0,0,-1,"{}"
122,1,52,0,0.0386925,"\int (a+b x)^2 \cot ^{-1}(a+b x) \, dx","Int[(a + b*x)^2*ArcCot[a + b*x],x]","\frac{(a+b x)^2}{6 b}-\frac{\log \left((a+b x)^2+1\right)}{6 b}+\frac{(a+b x)^3 \cot ^{-1}(a+b x)}{3 b}","\frac{(a+b x)^2}{6 b}-\frac{\log \left((a+b x)^2+1\right)}{6 b}+\frac{(a+b x)^3 \cot ^{-1}(a+b x)}{3 b}",1,"(a + b*x)^2/(6*b) + ((a + b*x)^3*ArcCot[a + b*x])/(3*b) - Log[1 + (a + b*x)^2]/(6*b)","A",5,4,14,0.2857,1,"{5044, 4853, 266, 43}"
123,1,39,0,0.0209446,"\int (a+b x) \cot ^{-1}(a+b x) \, dx","Int[(a + b*x)*ArcCot[a + b*x],x]","-\frac{\tan ^{-1}(a+b x)}{2 b}+\frac{(a+b x)^2 \cot ^{-1}(a+b x)}{2 b}+\frac{x}{2}","-\frac{\tan ^{-1}(a+b x)}{2 b}+\frac{(a+b x)^2 \cot ^{-1}(a+b x)}{2 b}+\frac{x}{2}",1,"x/2 + ((a + b*x)^2*ArcCot[a + b*x])/(2*b) - ArcTan[a + b*x]/(2*b)","A",4,4,12,0.3333,1,"{5044, 4853, 321, 203}"
124,1,45,0,0.0414755,"\int \frac{\cot ^{-1}(a+b x)}{a+b x} \, dx","Int[ArcCot[a + b*x]/(a + b*x),x]","\frac{i \text{PolyLog}\left(2,\frac{i}{a+b x}\right)}{2 b}-\frac{i \text{PolyLog}\left(2,-\frac{i}{a+b x}\right)}{2 b}","\frac{i \text{PolyLog}\left(2,\frac{i}{a+b x}\right)}{2 b}-\frac{i \text{PolyLog}\left(2,-\frac{i}{a+b x}\right)}{2 b}",1,"((-I/2)*PolyLog[2, (-I)/(a + b*x)])/b + ((I/2)*PolyLog[2, I/(a + b*x)])/b","A",4,3,14,0.2143,1,"{5044, 4849, 2391}"
125,1,47,0,0.0316369,"\int \frac{\cot ^{-1}(a+b x)}{(a+b x)^2} \, dx","Int[ArcCot[a + b*x]/(a + b*x)^2,x]","-\frac{\log (a+b x)}{b}+\frac{\log \left((a+b x)^2+1\right)}{2 b}-\frac{\cot ^{-1}(a+b x)}{b (a+b x)}","-\frac{\log (a+b x)}{b}+\frac{\log \left((a+b x)^2+1\right)}{2 b}-\frac{\cot ^{-1}(a+b x)}{b (a+b x)}",1,"-(ArcCot[a + b*x]/(b*(a + b*x))) - Log[a + b*x]/b + Log[1 + (a + b*x)^2]/(2*b)","A",6,6,14,0.4286,1,"{5044, 4853, 266, 36, 29, 31}"
126,1,35,0,0.036544,"\int \frac{\cot ^{-1}(1+x)}{2+2 x} \, dx","Int[ArcCot[1 + x]/(2 + 2*x),x]","\frac{1}{4} i \text{PolyLog}\left(2,\frac{i}{x+1}\right)-\frac{1}{4} i \text{PolyLog}\left(2,-\frac{i}{x+1}\right)","\frac{1}{4} i \text{PolyLog}\left(2,\frac{i}{x+1}\right)-\frac{1}{4} i \text{PolyLog}\left(2,-\frac{i}{x+1}\right)",1,"(-I/4)*PolyLog[2, (-I)/(1 + x)] + (I/4)*PolyLog[2, I/(1 + x)]","A",5,4,12,0.3333,1,"{5044, 12, 4849, 2391}"
127,1,45,0,0.0456755,"\int \frac{\cot ^{-1}(a+b x)}{\frac{a d}{b}+d x} \, dx","Int[ArcCot[a + b*x]/((a*d)/b + d*x),x]","\frac{i \text{PolyLog}\left(2,\frac{i}{a+b x}\right)}{2 d}-\frac{i \text{PolyLog}\left(2,-\frac{i}{a+b x}\right)}{2 d}","\frac{i \text{PolyLog}\left(2,\frac{i}{a+b x}\right)}{2 d}-\frac{i \text{PolyLog}\left(2,-\frac{i}{a+b x}\right)}{2 d}",1,"((-I/2)*PolyLog[2, (-I)/(a + b*x)])/d + ((I/2)*PolyLog[2, I/(a + b*x)])/d","A",5,4,19,0.2105,1,"{5044, 12, 4849, 2391}"
128,0,0,0,0.0176178,"\int (a+b x)^2 \sqrt{\cot ^{-1}(a+b x)} \, dx","Int[(a + b*x)^2*Sqrt[ArcCot[a + b*x]],x]","\int (a+b x)^2 \sqrt{\cot ^{-1}(a+b x)} \, dx","\text{Int}\left((a+b x)^2 \sqrt{\cot ^{-1}(a+b x)},x\right)",0,"Defer[Int][(a + b*x)^2*Sqrt[ArcCot[a + b*x]], x]","A",0,0,0,0,-1,"{}"
129,1,233,0,0.356664,"\int (e+f x)^3 \left(a+b \cot ^{-1}(c+d x)\right) \, dx","Int[(e + f*x)^3*(a + b*ArcCot[c + d*x]),x]","\frac{(e+f x)^4 \left(a+b \cot ^{-1}(c+d x)\right)}{4 f}+\frac{b f x \left(-\left(1-6 c^2\right) f^2-12 c d e f+6 d^2 e^2\right)}{4 d^3}+\frac{b \left(-6 \left(1-c^2\right) d^2 e^2 f^2+4 c \left(3-c^2\right) d e f^3+\left(c^4-6 c^2+1\right) f^4-4 c d^3 e^3 f+d^4 e^4\right) \tan ^{-1}(c+d x)}{4 d^4 f}+\frac{b f^2 (c+d x)^2 (d e-c f)}{2 d^4}+\frac{b (d e-c f) (-c f+d e+f) (d e-(c+1) f) \log \left((c+d x)^2+1\right)}{2 d^4}+\frac{b f^3 (c+d x)^3}{12 d^4}","\frac{(e+f x)^4 \left(a+b \cot ^{-1}(c+d x)\right)}{4 f}+\frac{b f x \left(-\left(1-6 c^2\right) f^2-12 c d e f+6 d^2 e^2\right)}{4 d^3}+\frac{b \left(-6 \left(1-c^2\right) d^2 e^2 f^2+4 c \left(3-c^2\right) d e f^3+\left(c^4-6 c^2+1\right) f^4-4 c d^3 e^3 f+d^4 e^4\right) \tan ^{-1}(c+d x)}{4 d^4 f}+\frac{b f^2 (c+d x)^2 (d e-c f)}{2 d^4}+\frac{b (d e-c f) (-c f+d e+f) (d e-(c+1) f) \log \left((c+d x)^2+1\right)}{2 d^4}+\frac{b f^3 (c+d x)^3}{12 d^4}",1,"(b*f*(6*d^2*e^2 - 12*c*d*e*f - (1 - 6*c^2)*f^2)*x)/(4*d^3) + (b*f^2*(d*e - c*f)*(c + d*x)^2)/(2*d^4) + (b*f^3*(c + d*x)^3)/(12*d^4) + ((e + f*x)^4*(a + b*ArcCot[c + d*x]))/(4*f) + (b*(d^4*e^4 - 4*c*d^3*e^3*f - 6*(1 - c^2)*d^2*e^2*f^2 + 4*c*(3 - c^2)*d*e*f^3 + (1 - 6*c^2 + c^4)*f^4)*ArcTan[c + d*x])/(4*d^4*f) + (b*(d*e - c*f)*(d*e + f - c*f)*(d*e - (1 + c)*f)*Log[1 + (c + d*x)^2])/(2*d^4)","A",7,6,18,0.3333,1,"{5048, 4863, 702, 635, 203, 260}"
130,1,154,0,0.185669,"\int (e+f x)^2 \left(a+b \cot ^{-1}(c+d x)\right) \, dx","Int[(e + f*x)^2*(a + b*ArcCot[c + d*x]),x]","\frac{(e+f x)^3 \left(a+b \cot ^{-1}(c+d x)\right)}{3 f}+\frac{b \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \log \left((c+d x)^2+1\right)}{6 d^3}+\frac{b (d e-c f) \left(-\left(3-c^2\right) f^2-2 c d e f+d^2 e^2\right) \tan ^{-1}(c+d x)}{3 d^3 f}+\frac{b f x (d e-c f)}{d^2}+\frac{b f^2 (c+d x)^2}{6 d^3}","\frac{(e+f x)^3 \left(a+b \cot ^{-1}(c+d x)\right)}{3 f}+\frac{b \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \log \left((c+d x)^2+1\right)}{6 d^3}+\frac{b (d e-c f) \left(-\left(3-c^2\right) f^2-2 c d e f+d^2 e^2\right) \tan ^{-1}(c+d x)}{3 d^3 f}+\frac{b f x (d e-c f)}{d^2}+\frac{b f^2 (c+d x)^2}{6 d^3}",1,"(b*f*(d*e - c*f)*x)/d^2 + (b*f^2*(c + d*x)^2)/(6*d^3) + ((e + f*x)^3*(a + b*ArcCot[c + d*x]))/(3*f) + (b*(d*e - c*f)*(d^2*e^2 - 2*c*d*e*f - (3 - c^2)*f^2)*ArcTan[c + d*x])/(3*d^3*f) + (b*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*Log[1 + (c + d*x)^2])/(6*d^3)","A",7,6,18,0.3333,1,"{5048, 4863, 702, 635, 203, 260}"
131,1,97,0,0.1144134,"\int (e+f x) \left(a+b \cot ^{-1}(c+d x)\right) \, dx","Int[(e + f*x)*(a + b*ArcCot[c + d*x]),x]","\frac{(e+f x)^2 \left(a+b \cot ^{-1}(c+d x)\right)}{2 f}+\frac{b (d e-c f) \log \left((c+d x)^2+1\right)}{2 d^2}+\frac{b (-c f+d e+f) (d e-(c+1) f) \tan ^{-1}(c+d x)}{2 d^2 f}+\frac{b f x}{2 d}","\frac{(e+f x)^2 \left(a+b \cot ^{-1}(c+d x)\right)}{2 f}+\frac{b (d e-c f) \log \left((c+d x)^2+1\right)}{2 d^2}+\frac{b (-c f+d e+f) (d e-(c+1) f) \tan ^{-1}(c+d x)}{2 d^2 f}+\frac{b f x}{2 d}",1,"(b*f*x)/(2*d) + ((e + f*x)^2*(a + b*ArcCot[c + d*x]))/(2*f) + (b*(d*e + f - c*f)*(d*e - (1 + c)*f)*ArcTan[c + d*x])/(2*d^2*f) + (b*(d*e - c*f)*Log[1 + (c + d*x)^2])/(2*d^2)","A",7,6,16,0.3750,1,"{5048, 4863, 702, 635, 203, 260}"
132,1,38,0,0.0201107,"\int \left(a+b \cot ^{-1}(c+d x)\right) \, dx","Int[a + b*ArcCot[c + d*x],x]","a x+\frac{b \log \left((c+d x)^2+1\right)}{2 d}+\frac{b (c+d x) \cot ^{-1}(c+d x)}{d}","a x+\frac{b \log \left((c+d x)^2+1\right)}{2 d}+\frac{b (c+d x) \cot ^{-1}(c+d x)}{d}",1,"a*x + (b*(c + d*x)*ArcCot[c + d*x])/d + (b*Log[1 + (c + d*x)^2])/(2*d)","A",4,3,10,0.3000,1,"{5040, 4847, 260}"
133,1,162,0,0.1485961,"\int \frac{a+b \cot ^{-1}(c+d x)}{e+f x} \, dx","Int[(a + b*ArcCot[c + d*x])/(e + f*x),x]","\frac{i b \text{PolyLog}\left(2,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{2 f}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i (c+d x)}\right)}{2 f}+\frac{\left(a+b \cot ^{-1}(c+d x)\right) \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{f}-\frac{\log \left(\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{f}","\frac{i b \text{PolyLog}\left(2,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{2 f}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i (c+d x)}\right)}{2 f}+\frac{\left(a+b \cot ^{-1}(c+d x)\right) \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{f}-\frac{\log \left(\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{f}",1,"-(((a + b*ArcCot[c + d*x])*Log[2/(1 - I*(c + d*x))])/f) + ((a + b*ArcCot[c + d*x])*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f - ((I/2)*b*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/f + ((I/2)*b*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f","A",5,5,18,0.2778,1,"{5048, 4857, 2402, 2315, 2447}"
134,1,153,0,0.1109375,"\int \frac{a+b \cot ^{-1}(c+d x)}{(e+f x)^2} \, dx","Int[(a + b*ArcCot[c + d*x])/(e + f*x)^2,x]","-\frac{a+b \cot ^{-1}(c+d x)}{f (e+f x)}+\frac{b d \log \left(c^2+2 c d x+d^2 x^2+1\right)}{2 \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}-\frac{b d \log (e+f x)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}-\frac{b d (d e-c f) \tan ^{-1}(c+d x)}{f \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}","-\frac{a+b \cot ^{-1}(c+d x)}{f (e+f x)}+\frac{b d \log \left(c^2+2 c d x+d^2 x^2+1\right)}{2 \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}-\frac{b d \log (e+f x)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}-\frac{b d (d e-c f) \tan ^{-1}(c+d x)}{f \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}",1,"-((a + b*ArcCot[c + d*x])/(f*(e + f*x))) - (b*d*(d*e - c*f)*ArcTan[c + d*x])/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (b*d*Log[e + f*x])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (b*d*Log[1 + c^2 + 2*c*d*x + d^2*x^2])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2))","A",8,8,18,0.4444,1,"{5046, 1982, 705, 31, 634, 618, 204, 628}"
135,1,228,0,0.2784508,"\int \frac{a+b \cot ^{-1}(c+d x)}{(e+f x)^3} \, dx","Int[(a + b*ArcCot[c + d*x])/(e + f*x)^3,x]","-\frac{a+b \cot ^{-1}(c+d x)}{2 f (e+f x)^2}+\frac{b d^2 (d e-c f) \log \left(c^2+2 c d x+d^2 x^2+1\right)}{2 \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)^2}+\frac{b d}{2 (e+f x) \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}-\frac{b d^2 (d e-c f) \log (e+f x)}{\left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)^2}-\frac{b d^2 (-c f+d e+f) (d e-(c+1) f) \tan ^{-1}(c+d x)}{2 f \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)^2}","-\frac{a+b \cot ^{-1}(c+d x)}{2 f (e+f x)^2}+\frac{b d^2 (d e-c f) \log \left(c^2+2 c d x+d^2 x^2+1\right)}{2 \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)^2}+\frac{b d}{2 (e+f x) \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}-\frac{b d^2 (d e-c f) \log (e+f x)}{\left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)^2}-\frac{b d^2 (-c f+d e+f) (d e-(c+1) f) \tan ^{-1}(c+d x)}{2 f \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)^2}",1,"(b*d)/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*(e + f*x)) - (a + b*ArcCot[c + d*x])/(2*f*(e + f*x)^2) - (b*d^2*(d*e + f - c*f)*(d*e - (1 + c)*f)*ArcTan[c + d*x])/(2*f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^2) - (b*d^2*(d*e - c*f)*Log[e + f*x])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^2 + (b*d^2*(d*e - c*f)*Log[1 + c^2 + 2*c*d*x + d^2*x^2])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^2)","A",9,8,18,0.4444,1,"{5046, 1982, 709, 800, 634, 618, 204, 628}"
136,1,382,0,0.5823919,"\int (e+f x)^2 \left(a+b \cot ^{-1}(c+d x)\right)^2 \, dx","Int[(e + f*x)^2*(a + b*ArcCot[c + d*x])^2,x]","\frac{i b^2 \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right)}{3 d^3}+\frac{i \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{3 d^3}-\frac{(d e-c f) \left(-\left(3-c^2\right) f^2-2 c d e f+d^2 e^2\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{3 d^3 f}-\frac{2 b \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{3 d^3}+\frac{2 a b f x (d e-c f)}{d^2}+\frac{b f^2 (c+d x)^2 \left(a+b \cot ^{-1}(c+d x)\right)}{3 d^3}+\frac{(e+f x)^3 \left(a+b \cot ^{-1}(c+d x)\right)^2}{3 f}+\frac{b^2 f (d e-c f) \log \left((c+d x)^2+1\right)}{d^3}+\frac{2 b^2 f (c+d x) (d e-c f) \cot ^{-1}(c+d x)}{d^3}-\frac{b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}+\frac{b^2 f^2 x}{3 d^2}","\frac{i b^2 \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right)}{3 d^3}+\frac{i \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{3 d^3}-\frac{(d e-c f) \left(-\left(3-c^2\right) f^2-2 c d e f+d^2 e^2\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{3 d^3 f}-\frac{2 b \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{3 d^3}+\frac{2 a b f x (d e-c f)}{d^2}+\frac{b f^2 (c+d x)^2 \left(a+b \cot ^{-1}(c+d x)\right)}{3 d^3}+\frac{(e+f x)^3 \left(a+b \cot ^{-1}(c+d x)\right)^2}{3 f}+\frac{b^2 f (d e-c f) \log \left((c+d x)^2+1\right)}{d^3}+\frac{2 b^2 f (c+d x) (d e-c f) \cot ^{-1}(c+d x)}{d^3}-\frac{b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}+\frac{b^2 f^2 x}{3 d^2}",1,"(b^2*f^2*x)/(3*d^2) + (2*a*b*f*(d*e - c*f)*x)/d^2 + (2*b^2*f*(d*e - c*f)*(c + d*x)*ArcCot[c + d*x])/d^3 + (b*f^2*(c + d*x)^2*(a + b*ArcCot[c + d*x]))/(3*d^3) + ((I/3)*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcCot[c + d*x])^2)/d^3 - ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f - (3 - c^2)*f^2)*(a + b*ArcCot[c + d*x])^2)/(3*d^3*f) + ((e + f*x)^3*(a + b*ArcCot[c + d*x])^2)/(3*f) - (b^2*f^2*ArcTan[c + d*x])/(3*d^3) - (2*b*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcCot[c + d*x])*Log[2/(1 + I*(c + d*x))])/(3*d^3) + (b^2*f*(d*e - c*f)*Log[1 + (c + d*x)^2])/d^3 + ((I/3)*b^2*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d^3","A",16,13,20,0.6500,1,"{5048, 4865, 4847, 260, 4853, 321, 203, 4985, 4885, 4921, 4855, 2402, 2315}"
137,1,220,0,0.3841592,"\int (e+f x) \left(a+b \cot ^{-1}(c+d x)\right)^2 \, dx","Int[(e + f*x)*(a + b*ArcCot[c + d*x])^2,x]","\frac{i b^2 (d e-c f) \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right)}{d^2}+\frac{i (d e-c f) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d^2}-\frac{(-c f+d e+f) (d e-(c+1) f) \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 d^2 f}-\frac{2 b (d e-c f) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d^2}+\frac{(e+f x)^2 \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 f}+\frac{a b f x}{d}+\frac{b^2 f \log \left((c+d x)^2+1\right)}{2 d^2}+\frac{b^2 f (c+d x) \cot ^{-1}(c+d x)}{d^2}","\frac{i b^2 (d e-c f) \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right)}{d^2}+\frac{i (d e-c f) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d^2}-\frac{(-c f+d e+f) (d e-(c+1) f) \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 d^2 f}-\frac{2 b (d e-c f) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d^2}+\frac{(e+f x)^2 \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 f}+\frac{a b f x}{d}+\frac{b^2 f \log \left((c+d x)^2+1\right)}{2 d^2}+\frac{b^2 f (c+d x) \cot ^{-1}(c+d x)}{d^2}",1,"(a*b*f*x)/d + (b^2*f*(c + d*x)*ArcCot[c + d*x])/d^2 + (I*(d*e - c*f)*(a + b*ArcCot[c + d*x])^2)/d^2 - ((d*e + f - c*f)*(d*e - (1 + c)*f)*(a + b*ArcCot[c + d*x])^2)/(2*d^2*f) + ((e + f*x)^2*(a + b*ArcCot[c + d*x])^2)/(2*f) - (2*b*(d*e - c*f)*(a + b*ArcCot[c + d*x])*Log[2/(1 + I*(c + d*x))])/d^2 + (b^2*f*Log[1 + (c + d*x)^2])/(2*d^2) + (I*b^2*(d*e - c*f)*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d^2","A",13,10,18,0.5556,1,"{5048, 4865, 4847, 260, 4985, 4885, 4921, 4855, 2402, 2315}"
138,1,102,0,0.1163433,"\int \left(a+b \cot ^{-1}(c+d x)\right)^2 \, dx","Int[(a + b*ArcCot[c + d*x])^2,x]","\frac{i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right)}{d}+\frac{(c+d x) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d}+\frac{i \left(a+b \cot ^{-1}(c+d x)\right)^2}{d}-\frac{2 b \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d}","\frac{i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right)}{d}+\frac{(c+d x) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d}+\frac{i \left(a+b \cot ^{-1}(c+d x)\right)^2}{d}-\frac{2 b \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d}",1,"(I*(a + b*ArcCot[c + d*x])^2)/d + ((c + d*x)*(a + b*ArcCot[c + d*x])^2)/d - (2*b*(a + b*ArcCot[c + d*x])*Log[2/(1 + I*(c + d*x))])/d + (I*b^2*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d","A",6,6,12,0.5000,1,"{5040, 4847, 4921, 4855, 2402, 2315}"
139,1,261,0,0.1767234,"\int \frac{\left(a+b \cot ^{-1}(c+d x)\right)^2}{e+f x} \, dx","Int[(a + b*ArcCot[c + d*x])^2/(e + f*x),x]","\frac{i b \left(a+b \cot ^{-1}(c+d x)\right) \text{PolyLog}\left(2,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{f}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{f}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{2 f}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1-i (c+d x)}\right)}{2 f}+\frac{\left(a+b \cot ^{-1}(c+d x)\right)^2 \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{f}-\frac{\log \left(\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{f}","\frac{i b \left(a+b \cot ^{-1}(c+d x)\right) \text{PolyLog}\left(2,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{f}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{f}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{2 f}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1-i (c+d x)}\right)}{2 f}+\frac{\left(a+b \cot ^{-1}(c+d x)\right)^2 \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{f}-\frac{\log \left(\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{f}",1,"-(((a + b*ArcCot[c + d*x])^2*Log[2/(1 - I*(c + d*x))])/f) + ((a + b*ArcCot[c + d*x])^2*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f - (I*b*(a + b*ArcCot[c + d*x])*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/f + (I*b*(a + b*ArcCot[c + d*x])*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f - (b^2*PolyLog[3, 1 - 2/(1 - I*(c + d*x))])/(2*f) + (b^2*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(2*f)","A",2,2,20,0.1000,1,"{5048, 4859}"
140,1,567,0,1.3863983,"\int \frac{\left(a+b \cot ^{-1}(c+d x)\right)^2}{(e+f x)^2} \, dx","Int[(a + b*ArcCot[c + d*x])^2/(e + f*x)^2,x]","\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1-i (c+d x)}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}-\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (d e+(-c+i) f)}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}-\frac{2 a b d \log (e+f x)}{(d e-c f)^2+f^2}+\frac{a b d \log \left((c+d x)^2+1\right)}{(d e-c f)^2+f^2}-\frac{2 a b d (d e-c f) \tan ^{-1}(c+d x)}{f \left((d e-c f)^2+f^2\right)}-\frac{\left(a+b \cot ^{-1}(c+d x)\right)^2}{f (e+f x)}+\frac{i b^2 d \cot ^{-1}(c+d x)^2}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}+\frac{2 b^2 d \log \left(\frac{2}{1-i (c+d x)}\right) \cot ^{-1}(c+d x)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}-\frac{2 b^2 d \cot ^{-1}(c+d x) \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (d e+(-c+i) f)}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}-\frac{2 b^2 d \log \left(\frac{2}{1+i (c+d x)}\right) \cot ^{-1}(c+d x)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}","\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1-i (c+d x)}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}-\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}-\frac{2 a b d \log (e+f x)}{(d e-c f)^2+f^2}+\frac{a b d \log \left((c+d x)^2+1\right)}{(d e-c f)^2+f^2}-\frac{2 a b d (d e-c f) \tan ^{-1}(c+d x)}{f \left((d e-c f)^2+f^2\right)}-\frac{\left(a+b \cot ^{-1}(c+d x)\right)^2}{f (e+f x)}+\frac{i b^2 d \cot ^{-1}(c+d x)^2}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}+\frac{2 b^2 d \log \left(\frac{2}{1-i (c+d x)}\right) \cot ^{-1}(c+d x)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}-\frac{2 b^2 d \cot ^{-1}(c+d x) \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}-\frac{2 b^2 d \log \left(\frac{2}{1+i (c+d x)}\right) \cot ^{-1}(c+d x)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}",1,"(I*b^2*d*ArcCot[c + d*x]^2)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (b^2*d*(d*e - c*f)*ArcCot[c + d*x]^2)/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (a + b*ArcCot[c + d*x])^2/(f*(e + f*x)) - (2*a*b*d*(d*e - c*f)*ArcTan[c + d*x])/(f*(f^2 + (d*e - c*f)^2)) - (2*a*b*d*Log[e + f*x])/(f^2 + (d*e - c*f)^2) + (2*b^2*d*ArcCot[c + d*x]*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (2*b^2*d*ArcCot[c + d*x]*Log[(2*d*(e + f*x))/((d*e + (I - c)*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (2*b^2*d*ArcCot[c + d*x]*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (a*b*d*Log[1 + (c + d*x)^2])/(f^2 + (d*e - c*f)^2) + (I*b^2*d*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (I*b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + (I - c)*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (I*b^2*d*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)","A",25,25,20,1.250,1,"{5046, 1982, 705, 31, 634, 618, 204, 628, 6741, 5058, 706, 635, 203, 260, 6688, 12, 6725, 4857, 2402, 2315, 2447, 4985, 4885, 4921, 4855}"
141,1,565,0,0.9634987,"\int (e+f x)^2 \left(a+b \cot ^{-1}(c+d x)\right)^3 \, dx","Int[(e + f*x)^2*(a + b*ArcCot[c + d*x])^3,x]","\frac{i b^2 \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d^3}-\frac{b^3 \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \text{PolyLog}\left(3,1-\frac{2}{1+i (c+d x)}\right)}{2 d^3}+\frac{3 i b^3 f (d e-c f) \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right)}{d^3}-\frac{6 b^2 f (d e-c f) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d^3}+\frac{a b^2 f^2 x}{d^2}+\frac{i \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \left(a+b \cot ^{-1}(c+d x)\right)^3}{3 d^3}-\frac{(d e-c f) \left(-\left(3-c^2\right) f^2-2 c d e f+d^2 e^2\right) \left(a+b \cot ^{-1}(c+d x)\right)^3}{3 d^3 f}-\frac{b \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d^3}+\frac{3 i b f (d e-c f) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d^3}+\frac{3 b f (c+d x) (d e-c f) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d^3}+\frac{b f^2 \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 d^3}+\frac{b f^2 (c+d x)^2 \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 d^3}+\frac{(e+f x)^3 \left(a+b \cot ^{-1}(c+d x)\right)^3}{3 f}+\frac{b^3 f^2 \log \left((c+d x)^2+1\right)}{2 d^3}+\frac{b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3}","\frac{i b^2 \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d^3}-\frac{b^3 \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \text{PolyLog}\left(3,1-\frac{2}{1+i (c+d x)}\right)}{2 d^3}+\frac{3 i b^3 f (d e-c f) \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right)}{d^3}-\frac{6 b^2 f (d e-c f) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d^3}+\frac{a b^2 f^2 x}{d^2}+\frac{i \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \left(a+b \cot ^{-1}(c+d x)\right)^3}{3 d^3}-\frac{(d e-c f) \left(-\left(3-c^2\right) f^2-2 c d e f+d^2 e^2\right) \left(a+b \cot ^{-1}(c+d x)\right)^3}{3 d^3 f}-\frac{b \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d^3}+\frac{3 i b f (d e-c f) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d^3}+\frac{3 b f (c+d x) (d e-c f) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d^3}+\frac{b f^2 \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 d^3}+\frac{b f^2 (c+d x)^2 \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 d^3}+\frac{(e+f x)^3 \left(a+b \cot ^{-1}(c+d x)\right)^3}{3 f}+\frac{b^3 f^2 \log \left((c+d x)^2+1\right)}{2 d^3}+\frac{b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3}",1,"(a*b^2*f^2*x)/d^2 + (b^3*f^2*(c + d*x)*ArcCot[c + d*x])/d^3 + (b*f^2*(a + b*ArcCot[c + d*x])^2)/(2*d^3) + ((3*I)*b*f*(d*e - c*f)*(a + b*ArcCot[c + d*x])^2)/d^3 + (3*b*f*(d*e - c*f)*(c + d*x)*(a + b*ArcCot[c + d*x])^2)/d^3 + (b*f^2*(c + d*x)^2*(a + b*ArcCot[c + d*x])^2)/(2*d^3) + ((I/3)*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcCot[c + d*x])^3)/d^3 - ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f - (3 - c^2)*f^2)*(a + b*ArcCot[c + d*x])^3)/(3*d^3*f) + ((e + f*x)^3*(a + b*ArcCot[c + d*x])^3)/(3*f) - (6*b^2*f*(d*e - c*f)*(a + b*ArcCot[c + d*x])*Log[2/(1 + I*(c + d*x))])/d^3 - (b*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcCot[c + d*x])^2*Log[2/(1 + I*(c + d*x))])/d^3 + (b^3*f^2*Log[1 + (c + d*x)^2])/(2*d^3) + ((3*I)*b^3*f*(d*e - c*f)*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d^3 + (I*b^2*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcCot[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d^3 - (b^3*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*d^3)","A",21,14,20,0.7000,1,"{5048, 4865, 4847, 4921, 4855, 2402, 2315, 4853, 4917, 260, 4885, 4985, 4995, 6610}"
142,1,337,0,0.6638497,"\int (e+f x) \left(a+b \cot ^{-1}(c+d x)\right)^3 \, dx","Int[(e + f*x)*(a + b*ArcCot[c + d*x])^3,x]","\frac{3 i b^2 (d e-c f) \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d^2}-\frac{3 b^3 (d e-c f) \text{PolyLog}\left(3,1-\frac{2}{1+i (c+d x)}\right)}{2 d^2}+\frac{3 i b^3 f \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right)}{2 d^2}-\frac{3 b^2 f \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d^2}+\frac{i (d e-c f) \left(a+b \cot ^{-1}(c+d x)\right)^3}{d^2}-\frac{(-c f+d e+f) (d e-(c+1) f) \left(a+b \cot ^{-1}(c+d x)\right)^3}{2 d^2 f}-\frac{3 b (d e-c f) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d^2}+\frac{3 i b f \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 d^2}+\frac{3 b f (c+d x) \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 d^2}+\frac{(e+f x)^2 \left(a+b \cot ^{-1}(c+d x)\right)^3}{2 f}","\frac{3 i b^2 (d e-c f) \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d^2}-\frac{3 b^3 (d e-c f) \text{PolyLog}\left(3,1-\frac{2}{1+i (c+d x)}\right)}{2 d^2}+\frac{3 i b^3 f \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right)}{2 d^2}-\frac{3 b^2 f \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d^2}+\frac{i (d e-c f) \left(a+b \cot ^{-1}(c+d x)\right)^3}{d^2}-\frac{(-c f+d e+f) (d e-(c+1) f) \left(a+b \cot ^{-1}(c+d x)\right)^3}{2 d^2 f}-\frac{3 b (d e-c f) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d^2}+\frac{3 i b f \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 d^2}+\frac{3 b f (c+d x) \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 d^2}+\frac{(e+f x)^2 \left(a+b \cot ^{-1}(c+d x)\right)^3}{2 f}",1,"(((3*I)/2)*b*f*(a + b*ArcCot[c + d*x])^2)/d^2 + (3*b*f*(c + d*x)*(a + b*ArcCot[c + d*x])^2)/(2*d^2) + (I*(d*e - c*f)*(a + b*ArcCot[c + d*x])^3)/d^2 - ((d*e + f - c*f)*(d*e - (1 + c)*f)*(a + b*ArcCot[c + d*x])^3)/(2*d^2*f) + ((e + f*x)^2*(a + b*ArcCot[c + d*x])^3)/(2*f) - (3*b^2*f*(a + b*ArcCot[c + d*x])*Log[2/(1 + I*(c + d*x))])/d^2 - (3*b*(d*e - c*f)*(a + b*ArcCot[c + d*x])^2*Log[2/(1 + I*(c + d*x))])/d^2 + (((3*I)/2)*b^3*f*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d^2 + ((3*I)*b^2*(d*e - c*f)*(a + b*ArcCot[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d^2 - (3*b^3*(d*e - c*f)*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*d^2)","A",15,11,18,0.6111,1,"{5048, 4865, 4847, 4921, 4855, 2402, 2315, 4985, 4885, 4995, 6610}"
143,1,143,0,0.2169054,"\int \left(a+b \cot ^{-1}(c+d x)\right)^3 \, dx","Int[(a + b*ArcCot[c + d*x])^3,x]","\frac{3 i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d}-\frac{3 b^3 \text{PolyLog}\left(3,1-\frac{2}{1+i (c+d x)}\right)}{2 d}+\frac{(c+d x) \left(a+b \cot ^{-1}(c+d x)\right)^3}{d}+\frac{i \left(a+b \cot ^{-1}(c+d x)\right)^3}{d}-\frac{3 b \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d}","\frac{3 i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d}-\frac{3 b^3 \text{PolyLog}\left(3,1-\frac{2}{1+i (c+d x)}\right)}{2 d}+\frac{(c+d x) \left(a+b \cot ^{-1}(c+d x)\right)^3}{d}+\frac{i \left(a+b \cot ^{-1}(c+d x)\right)^3}{d}-\frac{3 b \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d}",1,"(I*(a + b*ArcCot[c + d*x])^3)/d + ((c + d*x)*(a + b*ArcCot[c + d*x])^3)/d - (3*b*(a + b*ArcCot[c + d*x])^2*Log[2/(1 + I*(c + d*x))])/d + ((3*I)*b^2*(a + b*ArcCot[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d - (3*b^3*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*d)","A",6,7,12,0.5833,1,"{5040, 4847, 4921, 4855, 4885, 4995, 6610}"
144,1,372,0,0.2157064,"\int \frac{\left(a+b \cot ^{-1}(c+d x)\right)^3}{e+f x} \, dx","Int[(a + b*ArcCot[c + d*x])^3/(e + f*x),x]","\frac{3 b^2 \left(a+b \cot ^{-1}(c+d x)\right) \text{PolyLog}\left(3,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{2 f}-\frac{3 b^2 \text{PolyLog}\left(3,1-\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{2 f}+\frac{3 i b \left(a+b \cot ^{-1}(c+d x)\right)^2 \text{PolyLog}\left(2,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{2 f}-\frac{3 i b \text{PolyLog}\left(2,1-\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 f}-\frac{3 i b^3 \text{PolyLog}\left(4,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{4 f}+\frac{3 i b^3 \text{PolyLog}\left(4,1-\frac{2}{1-i (c+d x)}\right)}{4 f}+\frac{\left(a+b \cot ^{-1}(c+d x)\right)^3 \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{f}-\frac{\log \left(\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^3}{f}","\frac{3 b^2 \left(a+b \cot ^{-1}(c+d x)\right) \text{PolyLog}\left(3,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{2 f}-\frac{3 b^2 \text{PolyLog}\left(3,1-\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{2 f}+\frac{3 i b \left(a+b \cot ^{-1}(c+d x)\right)^2 \text{PolyLog}\left(2,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{2 f}-\frac{3 i b \text{PolyLog}\left(2,1-\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 f}-\frac{3 i b^3 \text{PolyLog}\left(4,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{4 f}+\frac{3 i b^3 \text{PolyLog}\left(4,1-\frac{2}{1-i (c+d x)}\right)}{4 f}+\frac{\left(a+b \cot ^{-1}(c+d x)\right)^3 \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{f}-\frac{\log \left(\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^3}{f}",1,"-(((a + b*ArcCot[c + d*x])^3*Log[2/(1 - I*(c + d*x))])/f) + ((a + b*ArcCot[c + d*x])^3*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f - (((3*I)/2)*b*(a + b*ArcCot[c + d*x])^2*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/f + (((3*I)/2)*b*(a + b*ArcCot[c + d*x])^2*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f - (3*b^2*(a + b*ArcCot[c + d*x])*PolyLog[3, 1 - 2/(1 - I*(c + d*x))])/(2*f) + (3*b^2*(a + b*ArcCot[c + d*x])*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(2*f) + (((3*I)/4)*b^3*PolyLog[4, 1 - 2/(1 - I*(c + d*x))])/f - (((3*I)/4)*b^3*PolyLog[4, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f","A",2,2,20,0.1000,1,"{5048, 4861}"
145,1,1233,0,2.2536702,"\int \frac{\left(a+b \cot ^{-1}(c+d x)\right)^3}{(e+f x)^2} \, dx","Int[(a + b*ArcCot[c + d*x])^3/(e + f*x)^2,x]","\frac{i d \cot ^{-1}(c+d x)^3 b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{d (d e-c f) \cot ^{-1}(c+d x)^3 b^3}{f \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}+\frac{3 d \cot ^{-1}(c+d x)^2 \log \left(\frac{2}{1-i (c+d x)}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 d \cot ^{-1}(c+d x)^2 \log \left(\frac{2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 d \cot ^{-1}(c+d x)^2 \log \left(\frac{2}{i (c+d x)+1}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i d \cot ^{-1}(c+d x) \text{PolyLog}\left(2,1-\frac{2}{1-i (c+d x)}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 i d \cot ^{-1}(c+d x) \text{PolyLog}\left(2,1-\frac{2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i d \cot ^{-1}(c+d x) \text{PolyLog}\left(2,1-\frac{2}{i (c+d x)+1}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 d \text{PolyLog}\left(3,1-\frac{2}{1-i (c+d x)}\right) b^3}{2 \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}-\frac{3 d \text{PolyLog}\left(3,1-\frac{2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right) b^3}{2 \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}-\frac{3 d \text{PolyLog}\left(3,1-\frac{2}{i (c+d x)+1}\right) b^3}{2 \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}+\frac{3 i a d \cot ^{-1}(c+d x)^2 b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 a d (d e-c f) \cot ^{-1}(c+d x)^2 b^2}{f \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}+\frac{6 a d \cot ^{-1}(c+d x) \log \left(\frac{2}{1-i (c+d x)}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{6 a d \cot ^{-1}(c+d x) \log \left(\frac{2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{6 a d \cot ^{-1}(c+d x) \log \left(\frac{2}{i (c+d x)+1}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i a d \text{PolyLog}\left(2,1-\frac{2}{1-i (c+d x)}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 i a d \text{PolyLog}\left(2,1-\frac{2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i a d \text{PolyLog}\left(2,1-\frac{2}{i (c+d x)+1}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 a^2 d (d e-c f) \tan ^{-1}(c+d x) b}{f \left(f^2+(d e-c f)^2\right)}-\frac{3 a^2 d \log (e+f x) b}{f^2+(d e-c f)^2}+\frac{3 a^2 d \log \left((c+d x)^2+1\right) b}{2 \left(f^2+(d e-c f)^2\right)}-\frac{\left(a+b \cot ^{-1}(c+d x)\right)^3}{f (e+f x)}","\frac{i d \cot ^{-1}(c+d x)^3 b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{d (d e-c f) \cot ^{-1}(c+d x)^3 b^3}{f \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}+\frac{3 d \cot ^{-1}(c+d x)^2 \log \left(\frac{2}{1-i (c+d x)}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 d \cot ^{-1}(c+d x)^2 \log \left(\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 d \cot ^{-1}(c+d x)^2 \log \left(\frac{2}{i (c+d x)+1}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i d \cot ^{-1}(c+d x) \text{PolyLog}\left(2,1-\frac{2}{1-i (c+d x)}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 i d \cot ^{-1}(c+d x) \text{PolyLog}\left(2,1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i d \cot ^{-1}(c+d x) \text{PolyLog}\left(2,1-\frac{2}{i (c+d x)+1}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 d \text{PolyLog}\left(3,1-\frac{2}{1-i (c+d x)}\right) b^3}{2 \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}-\frac{3 d \text{PolyLog}\left(3,1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right) b^3}{2 \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}-\frac{3 d \text{PolyLog}\left(3,1-\frac{2}{i (c+d x)+1}\right) b^3}{2 \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}+\frac{3 i a d \cot ^{-1}(c+d x)^2 b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 a d (d e-c f) \cot ^{-1}(c+d x)^2 b^2}{f \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}+\frac{6 a d \cot ^{-1}(c+d x) \log \left(\frac{2}{1-i (c+d x)}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{6 a d \cot ^{-1}(c+d x) \log \left(\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{6 a d \cot ^{-1}(c+d x) \log \left(\frac{2}{i (c+d x)+1}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i a d \text{PolyLog}\left(2,1-\frac{2}{1-i (c+d x)}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 i a d \text{PolyLog}\left(2,1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i a d \text{PolyLog}\left(2,1-\frac{2}{i (c+d x)+1}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 a^2 d (d e-c f) \tan ^{-1}(c+d x) b}{f \left(f^2+(d e-c f)^2\right)}-\frac{3 a^2 d \log (e+f x) b}{f^2+(d e-c f)^2}+\frac{3 a^2 d \log \left((c+d x)^2+1\right) b}{2 \left(f^2+(d e-c f)^2\right)}-\frac{\left(a+b \cot ^{-1}(c+d x)\right)^3}{f (e+f x)}",1,"((3*I)*a*b^2*d*ArcCot[c + d*x]^2)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*a*b^2*d*(d*e - c*f)*ArcCot[c + d*x]^2)/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) + (I*b^3*d*ArcCot[c + d*x]^3)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (b^3*d*(d*e - c*f)*ArcCot[c + d*x]^3)/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (a + b*ArcCot[c + d*x])^3/(f*(e + f*x)) - (3*a^2*b*d*(d*e - c*f)*ArcTan[c + d*x])/(f*(f^2 + (d*e - c*f)^2)) - (3*a^2*b*d*Log[e + f*x])/(f^2 + (d*e - c*f)^2) + (6*a*b^2*d*ArcCot[c + d*x]*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*b^3*d*ArcCot[c + d*x]^2*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (6*a*b^2*d*ArcCot[c + d*x]*Log[(2*d*(e + f*x))/((d*e + (I - c)*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (3*b^3*d*ArcCot[c + d*x]^2*Log[(2*d*(e + f*x))/((d*e + (I - c)*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (6*a*b^2*d*ArcCot[c + d*x]*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (3*b^3*d*ArcCot[c + d*x]^2*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*a^2*b*d*Log[1 + (c + d*x)^2])/(2*(f^2 + (d*e - c*f)^2)) + ((3*I)*a*b^2*d*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + ((3*I)*b^3*d*ArcCot[c + d*x]*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - ((3*I)*a*b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + (I - c)*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - ((3*I)*b^3*d*ArcCot[c + d*x]*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + (I - c)*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + ((3*I)*a*b^2*d*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + ((3*I)*b^3*d*ArcCot[c + d*x]*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*b^3*d*PolyLog[3, 1 - 2/(1 - I*(c + d*x))])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (3*b^3*d*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + (I - c)*f)*(1 - I*(c + d*x)))])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (3*b^3*d*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2))","A",35,22,20,1.100,1,"{5046, 6741, 5058, 6688, 12, 6725, 706, 31, 635, 203, 260, 4857, 2402, 2315, 2447, 4985, 4885, 4921, 4855, 4859, 4995, 6610}"
146,1,177,0,0.2436357,"\int (e+f x)^m \left(a+b \cot ^{-1}(c+d x)\right) \, dx","Int[(e + f*x)^m*(a + b*ArcCot[c + d*x]),x]","\frac{(e+f x)^{m+1} \left(a+b \cot ^{-1}(c+d x)\right)}{f (m+1)}+\frac{i b d (e+f x)^{m+2} \, _2F_1\left(1,m+2;m+3;\frac{d (e+f x)}{d e-c f+i f}\right)}{2 f (m+1) (m+2) (d e+(-c+i) f)}-\frac{i b d (e+f x)^{m+2} \, _2F_1\left(1,m+2;m+3;\frac{d (e+f x)}{d e-(c+i) f}\right)}{2 f (m+1) (m+2) (d e-(c+i) f)}","\frac{(e+f x)^{m+1} \left(a+b \cot ^{-1}(c+d x)\right)}{f (m+1)}+\frac{i b d (e+f x)^{m+2} \, _2F_1\left(1,m+2;m+3;\frac{d (e+f x)}{d e-c f+i f}\right)}{2 f (m+1) (m+2) (d e+(-c+i) f)}-\frac{i b d (e+f x)^{m+2} \, _2F_1\left(1,m+2;m+3;\frac{d (e+f x)}{d e-(c+i) f}\right)}{2 f (m+1) (m+2) (d e-(c+i) f)}",1,"((e + f*x)^(1 + m)*(a + b*ArcCot[c + d*x]))/(f*(1 + m)) + ((I/2)*b*d*(e + f*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e + I*f - c*f)])/(f*(d*e + (I - c)*f)*(1 + m)*(2 + m)) - ((I/2)*b*d*(e + f*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e - (I + c)*f)])/(f*(d*e - (I + c)*f)*(1 + m)*(2 + m))","A",6,4,18,0.2222,1,"{5048, 4863, 712, 68}"
147,0,0,0,0.0605398,"\int (e+f x)^m \left(a+b \cot ^{-1}(c+d x)\right)^2 \, dx","Int[(e + f*x)^m*(a + b*ArcCot[c + d*x])^2,x]","\int (e+f x)^m \left(a+b \cot ^{-1}(c+d x)\right)^2 \, dx","\text{Int}\left((e+f x)^m \left(a+b \cot ^{-1}(c+d x)\right)^2,x\right)",0,"Defer[Subst][Defer[Int][((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCot[x])^2, x], x, c + d*x]/d","A",0,0,0,0,-1,"{}"
148,0,0,0,0.0598165,"\int (e+f x)^m \left(a+b \cot ^{-1}(c+d x)\right)^3 \, dx","Int[(e + f*x)^m*(a + b*ArcCot[c + d*x])^3,x]","\int (e+f x)^m \left(a+b \cot ^{-1}(c+d x)\right)^3 \, dx","\text{Int}\left((e+f x)^m \left(a+b \cot ^{-1}(c+d x)\right)^3,x\right)",0,"Defer[Subst][Defer[Int][((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCot[x])^3, x], x, c + d*x]/d","A",0,0,0,0,-1,"{}"
149,1,42,0,0.0435711,"\int x^3 \cot ^{-1}\left(a+b x^4\right) \, dx","Int[x^3*ArcCot[a + b*x^4],x]","\frac{\log \left(\left(a+b x^4\right)^2+1\right)}{8 b}+\frac{\left(a+b x^4\right) \cot ^{-1}\left(a+b x^4\right)}{4 b}","\frac{\log \left(\left(a+b x^4\right)^2+1\right)}{8 b}+\frac{\left(a+b x^4\right) \cot ^{-1}\left(a+b x^4\right)}{4 b}",1,"((a + b*x^4)*ArcCot[a + b*x^4])/(4*b) + Log[1 + (a + b*x^4)^2]/(8*b)","A",4,4,12,0.3333,1,"{6715, 5040, 4847, 260}"
150,1,45,0,0.0444314,"\int x^{-1+n} \cot ^{-1}\left(a+b x^n\right) \, dx","Int[x^(-1 + n)*ArcCot[a + b*x^n],x]","\frac{\log \left(\left(a+b x^n\right)^2+1\right)}{2 b n}+\frac{\left(a+b x^n\right) \cot ^{-1}\left(a+b x^n\right)}{b n}","\frac{\log \left(\left(a+b x^n\right)^2+1\right)}{2 b n}+\frac{\left(a+b x^n\right) \cot ^{-1}\left(a+b x^n\right)}{b n}",1,"((a + b*x^n)*ArcCot[a + b*x^n])/(b*n) + Log[1 + (a + b*x^n)^2]/(2*b*n)","A",4,4,14,0.2857,1,"{6715, 5040, 4847, 260}"
151,0,0,0,0.0458797,"\int \frac{\left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^n}{1-c^2 x^2} \, dx","Int[(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2),x]","\int \frac{\left(a+b \cot ^{-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 \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^n}{1-c^2 x^2},x\right)",0,"Defer[Int][(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x]","A",0,0,0,0,-1,"{}"
152,1,488,0,0.5227369,"\int \frac{\left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^3}{1-c^2 x^2} \, dx","Int[(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2),x]","\frac{3 b^2 \text{PolyLog}\left(3,1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{2 c}-\frac{3 b^2 \text{PolyLog}\left(3,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right)}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{2 c}+\frac{3 i b \text{PolyLog}\left(2,1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{2 c}-\frac{3 i b \text{PolyLog}\left(2,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right)}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{2 c}-\frac{3 i b^3 \text{PolyLog}\left(4,1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right)}{4 c}+\frac{3 i b^3 \text{PolyLog}\left(4,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right)}\right)}{4 c}-\frac{2 \coth ^{-1}\left(1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^3}{c}","\frac{3 b^2 \text{PolyLog}\left(3,1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{2 c}-\frac{3 b^2 \text{PolyLog}\left(3,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right)}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{2 c}+\frac{3 i b \text{PolyLog}\left(2,1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{2 c}-\frac{3 i b \text{PolyLog}\left(2,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right)}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{2 c}-\frac{3 i b^3 \text{PolyLog}\left(4,1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right)}{4 c}+\frac{3 i b^3 \text{PolyLog}\left(4,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right)}\right)}{4 c}-\frac{2 \coth ^{-1}\left(1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^3}{c}",1,"(-2*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*ArcCoth[1 - 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/c + (((3*I)/2)*b*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, 1 - (2*I)/(I + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/c - (((3*I)/2)*b*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(I + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/c + (3*b^2*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, 1 - (2*I)/(I + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) - (3*b^2*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(I + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/(2*c) - (((3*I)/4)*b^3*PolyLog[4, 1 - (2*I)/(I + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/c + (((3*I)/4)*b^3*PolyLog[4, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(I + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/c","A",9,7,40,0.1750,1,"{6681, 4851, 4989, 4885, 4993, 4997, 6610}"
153,1,321,0,0.3198209,"\int \frac{\left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2}{1-c^2 x^2} \, dx","Int[(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2),x]","\frac{i b \text{PolyLog}\left(2,1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}-\frac{i b \text{PolyLog}\left(2,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right)}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right)}{2 c}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right)}\right)}{2 c}-\frac{2 \coth ^{-1}\left(1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{c}","\frac{i b \text{PolyLog}\left(2,1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}-\frac{i b \text{PolyLog}\left(2,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right)}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right)}{2 c}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right)}\right)}{2 c}-\frac{2 \coth ^{-1}\left(1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{c}",1,"(-2*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*ArcCoth[1 - 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/c + (I*b*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, 1 - (2*I)/(I + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/c - (I*b*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(I + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/c + (b^2*PolyLog[3, 1 - (2*I)/(I + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) - (b^2*PolyLog[3, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(I + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/(2*c)","A",7,6,40,0.1500,1,"{6681, 4851, 4989, 4885, 4993, 6610}"
154,1,98,0,0.0690427,"\int \frac{a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)}{1-c^2 x^2} \, dx","Int[(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(1 - c^2*x^2),x]","\frac{i b \text{PolyLog}\left(2,-\frac{i \sqrt{c x+1}}{\sqrt{1-c x}}\right)}{2 c}-\frac{i b \text{PolyLog}\left(2,\frac{i \sqrt{c x+1}}{\sqrt{1-c x}}\right)}{2 c}-\frac{a \log \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}{c}","\frac{i b \text{PolyLog}\left(2,-\frac{i \sqrt{c x+1}}{\sqrt{1-c x}}\right)}{2 c}-\frac{i b \text{PolyLog}\left(2,\frac{i \sqrt{c x+1}}{\sqrt{1-c x}}\right)}{2 c}-\frac{a \log \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}{c}",1,"-((a*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/c) + ((I/2)*b*PolyLog[2, ((-I)*Sqrt[1 + c*x])/Sqrt[1 - c*x]])/c - ((I/2)*b*PolyLog[2, (I*Sqrt[1 + c*x])/Sqrt[1 - c*x]])/c","A",4,4,38,0.1053,1,"{206, 6681, 4849, 2391}"
155,0,0,0,0.0457182,"\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)} \, dx","Int[1/((1 - c^2*x^2)*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])),x]","\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \cot ^{-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 \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)},x\right)",0,"Defer[Int][1/((1 - c^2*x^2)*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), x]","A",0,0,0,0,-1,"{}"
156,0,0,0,0.0415443,"\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2} \, dx","Int[1/((1 - c^2*x^2)*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2),x]","\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \cot ^{-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 \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2},x\right)",0,"Defer[Int][1/((1 - c^2*x^2)*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]","A",0,0,0,0,-1,"{}"
157,1,16,0,0.0082204,"\int \cot ^{-1}(\tan (a+b x)) \, dx","Int[ArcCot[Tan[a + b*x]],x]","-\frac{\cot ^{-1}(\tan (a+b x))^2}{2 b}","-\frac{\cot ^{-1}(\tan (a+b x))^2}{2 b}",1,"-ArcCot[Tan[a + b*x]]^2/(2*b)","A",2,2,7,0.2857,1,"{2157, 30}"
158,1,403,0,0.5116016,"\int x^2 \cot ^{-1}(c+d \tan (a+b x)) \, dx","Int[x^2*ArcCot[c + d*Tan[a + b*x]],x]","-\frac{i x \text{PolyLog}\left(3,-\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{PolyLog}\left(3,-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{4 b^2}+\frac{\text{PolyLog}\left(4,-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{8 b^3}-\frac{\text{PolyLog}\left(4,-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{8 b^3}-\frac{x^2 \text{PolyLog}\left(2,-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{4 b}+\frac{x^2 \text{PolyLog}\left(2,-\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 \cot ^{-1}(d \tan (a+b x)+c)","-\frac{i x \text{PolyLog}\left(3,-\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{PolyLog}\left(3,-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{4 b^2}+\frac{\text{PolyLog}\left(4,-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{8 b^3}-\frac{\text{PolyLog}\left(4,-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{8 b^3}-\frac{x^2 \text{PolyLog}\left(2,-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{4 b}+\frac{x^2 \text{PolyLog}\left(2,-\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 \cot ^{-1}(d \tan (a+b x)+c)",1,"(x^3*ArcCot[c + d*Tan[a + b*x]])/3 - (I/6)*x^3*Log[1 + ((1 + I*c + d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c - d)] + (I/6)*x^3*Log[1 + ((c + I*(1 - d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 + d))] - (x^2*PolyLog[2, -(((1 + I*c + d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c - d))])/(4*b) + (x^2*PolyLog[2, -(((c + I*(1 - d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 + d)))])/(4*b) - ((I/4)*x*PolyLog[3, -(((1 + I*c + d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c - d))])/b^2 + ((I/4)*x*PolyLog[3, -(((c + I*(1 - d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 + d)))])/b^2 + PolyLog[4, -(((1 + I*c + d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c - d))]/(8*b^3) - PolyLog[4, -(((c + I*(1 - d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 + d)))]/(8*b^3)","A",11,6,15,0.4000,1,"{5176, 2190, 2531, 6609, 2282, 6589}"
159,1,305,0,0.4055476,"\int x \cot ^{-1}(c+d \tan (a+b x)) \, dx","Int[x*ArcCot[c + d*Tan[a + b*x]],x]","-\frac{i \text{PolyLog}\left(3,-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{8 b^2}+\frac{i \text{PolyLog}\left(3,-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{8 b^2}-\frac{x \text{PolyLog}\left(2,-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{4 b}+\frac{x \text{PolyLog}\left(2,-\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 \cot ^{-1}(d \tan (a+b x)+c)","-\frac{i \text{PolyLog}\left(3,-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{8 b^2}+\frac{i \text{PolyLog}\left(3,-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{8 b^2}-\frac{x \text{PolyLog}\left(2,-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{4 b}+\frac{x \text{PolyLog}\left(2,-\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 \cot ^{-1}(d \tan (a+b x)+c)",1,"(x^2*ArcCot[c + d*Tan[a + b*x]])/2 - (I/4)*x^2*Log[1 + ((1 + I*c + d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c - d)] + (I/4)*x^2*Log[1 + ((c + I*(1 - d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 + d))] - (x*PolyLog[2, -(((1 + I*c + d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c - d))])/(4*b) + (x*PolyLog[2, -(((c + I*(1 - d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 + d)))])/(4*b) - ((I/8)*PolyLog[3, -(((1 + I*c + d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c - d))])/b^2 + ((I/8)*PolyLog[3, -(((c + I*(1 - d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 + d)))])/b^2","A",9,5,13,0.3846,1,"{5176, 2190, 2531, 2282, 6589}"
160,1,198,0,0.2432274,"\int \cot ^{-1}(c+d \tan (a+b x)) \, dx","Int[ArcCot[c + d*Tan[a + b*x]],x]","-\frac{\text{PolyLog}\left(2,-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{4 b}+\frac{\text{PolyLog}\left(2,-\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 \cot ^{-1}(d \tan (a+b x)+c)","-\frac{\text{PolyLog}\left(2,-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{4 b}+\frac{\text{PolyLog}\left(2,-\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 \cot ^{-1}(d \tan (a+b x)+c)",1,"x*ArcCot[c + d*Tan[a + b*x]] - (I/2)*x*Log[1 + ((1 + I*c + d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c - d)] + (I/2)*x*Log[1 + ((c + I*(1 - d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 + d))] - PolyLog[2, -(((1 + I*c + d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c - d))]/(4*b) + PolyLog[2, -(((c + I*(1 - d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 + d)))]/(4*b)","A",7,4,11,0.3636,1,"{5168, 2190, 2279, 2391}"
161,0,0,0,0.0748455,"\int \frac{\cot ^{-1}(c+d \tan (a+b x))}{x} \, dx","Int[ArcCot[c + d*Tan[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+d \tan (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(d \tan (a+b x)+c)}{x},x\right)",0,"Defer[Int][ArcCot[c + d*Tan[a + b*x]]/x, x]","A",0,0,0,0,-1,"{}"
162,1,154,0,0.2527637,"\int x^2 \cot ^{-1}(c+(1+i c) \tan (a+b x)) \, dx","Int[x^2*ArcCot[c + (1 + I*c)*Tan[a + b*x]],x]","\frac{i x \text{PolyLog}\left(3,i c e^{2 i a+2 i b x}\right)}{4 b^2}-\frac{\text{PolyLog}\left(4,i c e^{2 i a+2 i b x}\right)}{8 b^3}+\frac{x^2 \text{PolyLog}\left(2,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 \cot ^{-1}(c+(1+i c) \tan (a+b x))+\frac{b x^4}{12}","\frac{i x \text{PolyLog}\left(3,i c e^{2 i a+2 i b x}\right)}{4 b^2}-\frac{\text{PolyLog}\left(4,i c e^{2 i a+2 i b x}\right)}{8 b^3}+\frac{x^2 \text{PolyLog}\left(2,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 \cot ^{-1}(c+(1+i c) \tan (a+b x))+\frac{b x^4}{12}",1,"(b*x^4)/12 + (x^3*ArcCot[c + (1 + I*c)*Tan[a + b*x]])/3 + (I/6)*x^3*Log[1 - I*c*E^((2*I)*a + (2*I)*b*x)] + (x^2*PolyLog[2, I*c*E^((2*I)*a + (2*I)*b*x)])/(4*b) + ((I/4)*x*PolyLog[3, I*c*E^((2*I)*a + (2*I)*b*x)])/b^2 - PolyLog[4, I*c*E^((2*I)*a + (2*I)*b*x)]/(8*b^3)","A",7,7,21,0.3333,1,"{5172, 2184, 2190, 2531, 6609, 2282, 6589}"
163,1,123,0,0.2203514,"\int x \cot ^{-1}(c+(1+i c) \tan (a+b x)) \, dx","Int[x*ArcCot[c + (1 + I*c)*Tan[a + b*x]],x]","\frac{i \text{PolyLog}\left(3,i c e^{2 i a+2 i b x}\right)}{8 b^2}+\frac{x \text{PolyLog}\left(2,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 \cot ^{-1}(c+(1+i c) \tan (a+b x))+\frac{b x^3}{6}","\frac{i \text{PolyLog}\left(3,i c e^{2 i a+2 i b x}\right)}{8 b^2}+\frac{x \text{PolyLog}\left(2,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 \cot ^{-1}(c+(1+i c) \tan (a+b x))+\frac{b x^3}{6}",1,"(b*x^3)/6 + (x^2*ArcCot[c + (1 + I*c)*Tan[a + b*x]])/2 + (I/4)*x^2*Log[1 - I*c*E^((2*I)*a + (2*I)*b*x)] + (x*PolyLog[2, I*c*E^((2*I)*a + (2*I)*b*x)])/(4*b) + ((I/8)*PolyLog[3, I*c*E^((2*I)*a + (2*I)*b*x)])/b^2","A",6,6,19,0.3158,1,"{5172, 2184, 2190, 2531, 2282, 6589}"
164,1,85,0,0.1315994,"\int \cot ^{-1}(c+(1+i c) \tan (a+b x)) \, dx","Int[ArcCot[c + (1 + I*c)*Tan[a + b*x]],x]","\frac{\text{PolyLog}\left(2,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 \cot ^{-1}(c+(1+i c) \tan (a+b x))+\frac{b x^2}{2}","\frac{\text{PolyLog}\left(2,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 \cot ^{-1}(c+(1+i c) \tan (a+b x))+\frac{b x^2}{2}",1,"(b*x^2)/2 + x*ArcCot[c + (1 + I*c)*Tan[a + b*x]] + (I/2)*x*Log[1 - I*c*E^((2*I)*a + (2*I)*b*x)] + PolyLog[2, I*c*E^((2*I)*a + (2*I)*b*x)]/(4*b)","A",5,5,17,0.2941,1,"{5164, 2184, 2190, 2279, 2391}"
165,0,0,0,0.1043702,"\int \frac{\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx","Int[ArcCot[c + (1 + I*c)*Tan[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x},x\right)",0,"Defer[Int][ArcCot[c + (1 + I*c)*Tan[a + b*x]]/x, x]","A",0,0,0,0,-1,"{}"
166,1,155,0,0.2609404,"\int x^2 \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx","Int[x^2*ArcCot[c - (1 - I*c)*Tan[a + b*x]],x]","-\frac{i x \text{PolyLog}\left(3,-i c e^{2 i a+2 i b x}\right)}{4 b^2}+\frac{\text{PolyLog}\left(4,-i c e^{2 i a+2 i b x}\right)}{8 b^3}-\frac{x^2 \text{PolyLog}\left(2,-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 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac{b x^4}{12}","-\frac{i x \text{PolyLog}\left(3,-i c e^{2 i a+2 i b x}\right)}{4 b^2}+\frac{\text{PolyLog}\left(4,-i c e^{2 i a+2 i b x}\right)}{8 b^3}-\frac{x^2 \text{PolyLog}\left(2,-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 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac{b x^4}{12}",1,"-(b*x^4)/12 + (x^3*ArcCot[c - (1 - I*c)*Tan[a + b*x]])/3 - (I/6)*x^3*Log[1 + I*c*E^((2*I)*a + (2*I)*b*x)] - (x^2*PolyLog[2, (-I)*c*E^((2*I)*a + (2*I)*b*x)])/(4*b) - ((I/4)*x*PolyLog[3, (-I)*c*E^((2*I)*a + (2*I)*b*x)])/b^2 + PolyLog[4, (-I)*c*E^((2*I)*a + (2*I)*b*x)]/(8*b^3)","A",7,7,22,0.3182,1,"{5172, 2184, 2190, 2531, 6609, 2282, 6589}"
167,1,124,0,0.2392327,"\int x \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx","Int[x*ArcCot[c - (1 - I*c)*Tan[a + b*x]],x]","-\frac{i \text{PolyLog}\left(3,-i c e^{2 i a+2 i b x}\right)}{8 b^2}-\frac{x \text{PolyLog}\left(2,-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 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac{b x^3}{6}","-\frac{i \text{PolyLog}\left(3,-i c e^{2 i a+2 i b x}\right)}{8 b^2}-\frac{x \text{PolyLog}\left(2,-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 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac{b x^3}{6}",1,"-(b*x^3)/6 + (x^2*ArcCot[c - (1 - I*c)*Tan[a + b*x]])/2 - (I/4)*x^2*Log[1 + I*c*E^((2*I)*a + (2*I)*b*x)] - (x*PolyLog[2, (-I)*c*E^((2*I)*a + (2*I)*b*x)])/(4*b) - ((I/8)*PolyLog[3, (-I)*c*E^((2*I)*a + (2*I)*b*x)])/b^2","A",6,6,20,0.3000,1,"{5172, 2184, 2190, 2531, 2282, 6589}"
168,1,86,0,0.1415771,"\int \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx","Int[ArcCot[c - (1 - I*c)*Tan[a + b*x]],x]","-\frac{\text{PolyLog}\left(2,-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 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac{b x^2}{2}","-\frac{\text{PolyLog}\left(2,-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 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac{b x^2}{2}",1,"-(b*x^2)/2 + x*ArcCot[c - (1 - I*c)*Tan[a + b*x]] - (I/2)*x*Log[1 + I*c*E^((2*I)*a + (2*I)*b*x)] - PolyLog[2, (-I)*c*E^((2*I)*a + (2*I)*b*x)]/(4*b)","A",5,5,18,0.2778,1,"{5164, 2184, 2190, 2279, 2391}"
169,0,0,0,0.1639224,"\int \frac{\cot ^{-1}(c-(1-i c) \tan (a+b x))}{x} \, dx","Int[ArcCot[c - (1 - I*c)*Tan[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c-(1-i c) \tan (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c-(1-i c) \tan (a+b x))}{x},x\right)",0,"Defer[Int][ArcCot[c - (1 - I*c)*Tan[a + b*x]]/x, x]","A",0,0,0,0,-1,"{}"
170,1,16,0,0.0128469,"\int \cot ^{-1}(\cot (a+b x)) \, dx","Int[ArcCot[Cot[a + b*x]],x]","\frac{\cot ^{-1}(\cot (a+b x))^2}{2 b}","\frac{\cot ^{-1}(\cot (a+b x))^2}{2 b}",1,"ArcCot[Cot[a + b*x]]^2/(2*b)","A",2,2,7,0.2857,1,"{2157, 30}"
171,1,399,0,0.5079462,"\int x^2 \cot ^{-1}(c+d \cot (a+b x)) \, dx","Int[x^2*ArcCot[c + d*Cot[a + b*x]],x]","-\frac{i x \text{PolyLog}\left(3,\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{PolyLog}\left(3,\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{4 b^2}+\frac{\text{PolyLog}\left(4,\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{8 b^3}-\frac{\text{PolyLog}\left(4,\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{8 b^3}-\frac{x^2 \text{PolyLog}\left(2,\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{4 b}+\frac{x^2 \text{PolyLog}\left(2,\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 \cot ^{-1}(d \cot (a+b x)+c)","-\frac{i x \text{PolyLog}\left(3,\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{PolyLog}\left(3,\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{4 b^2}+\frac{\text{PolyLog}\left(4,\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{8 b^3}-\frac{\text{PolyLog}\left(4,\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{8 b^3}-\frac{x^2 \text{PolyLog}\left(2,\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{4 b}+\frac{x^2 \text{PolyLog}\left(2,\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 \cot ^{-1}(d \cot (a+b x)+c)",1,"(x^3*ArcCot[c + d*Cot[a + b*x]])/3 - (I/6)*x^3*Log[1 - ((1 + I*c - d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c + d)] + (I/6)*x^3*Log[1 - ((c + I*(1 + d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 - d))] - (x^2*PolyLog[2, ((1 + I*c - d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c + d)])/(4*b) + (x^2*PolyLog[2, ((c + I*(1 + d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 - d))])/(4*b) - ((I/4)*x*PolyLog[3, ((1 + I*c - d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c + d)])/b^2 + ((I/4)*x*PolyLog[3, ((c + I*(1 + d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 - d))])/b^2 + PolyLog[4, ((1 + I*c - d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c + d)]/(8*b^3) - PolyLog[4, ((c + I*(1 + d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 - d))]/(8*b^3)","A",11,6,15,0.4000,1,"{5178, 2190, 2531, 6609, 2282, 6589}"
172,1,303,0,0.4173584,"\int x \cot ^{-1}(c+d \cot (a+b x)) \, dx","Int[x*ArcCot[c + d*Cot[a + b*x]],x]","-\frac{i \text{PolyLog}\left(3,\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{8 b^2}+\frac{i \text{PolyLog}\left(3,\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{8 b^2}-\frac{x \text{PolyLog}\left(2,\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{4 b}+\frac{x \text{PolyLog}\left(2,\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 \cot ^{-1}(d \cot (a+b x)+c)","-\frac{i \text{PolyLog}\left(3,\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{8 b^2}+\frac{i \text{PolyLog}\left(3,\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{8 b^2}-\frac{x \text{PolyLog}\left(2,\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{4 b}+\frac{x \text{PolyLog}\left(2,\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 \cot ^{-1}(d \cot (a+b x)+c)",1,"(x^2*ArcCot[c + d*Cot[a + b*x]])/2 - (I/4)*x^2*Log[1 - ((1 + I*c - d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c + d)] + (I/4)*x^2*Log[1 - ((c + I*(1 + d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 - d))] - (x*PolyLog[2, ((1 + I*c - d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c + d)])/(4*b) + (x*PolyLog[2, ((c + I*(1 + d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 - d))])/(4*b) - ((I/8)*PolyLog[3, ((1 + I*c - d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c + d)])/b^2 + ((I/8)*PolyLog[3, ((c + I*(1 + d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 - d))])/b^2","A",9,5,13,0.3846,1,"{5178, 2190, 2531, 2282, 6589}"
173,1,198,0,0.2537977,"\int \cot ^{-1}(c+d \cot (a+b x)) \, dx","Int[ArcCot[c + d*Cot[a + b*x]],x]","-\frac{\text{PolyLog}\left(2,\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{4 b}+\frac{\text{PolyLog}\left(2,\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 \cot ^{-1}(d \cot (a+b x)+c)","-\frac{\text{PolyLog}\left(2,\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{4 b}+\frac{\text{PolyLog}\left(2,\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 \cot ^{-1}(d \cot (a+b x)+c)",1,"x*ArcCot[c + d*Cot[a + b*x]] - (I/2)*x*Log[1 - ((1 + I*c - d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c + d)] + (I/2)*x*Log[1 - ((c + I*(1 + d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 - d))] - PolyLog[2, ((1 + I*c - d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c + d)]/(4*b) + PolyLog[2, ((c + I*(1 + d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 - d))]/(4*b)","A",7,4,11,0.3636,1,"{5170, 2190, 2279, 2391}"
174,0,0,0,0.1178406,"\int \frac{\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx","Int[ArcCot[c + d*Cot[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(d \cot (a+b x)+c)}{x},x\right)",0,"Defer[Int][ArcCot[c + d*Cot[a + b*x]]/x, x]","A",0,0,0,0,-1,"{}"
175,1,154,0,0.2659144,"\int x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx","Int[x^2*ArcCot[c + (1 - I*c)*Cot[a + b*x]],x]","-\frac{i x \text{PolyLog}\left(3,i c e^{2 i a+2 i b x}\right)}{4 b^2}+\frac{\text{PolyLog}\left(4,i c e^{2 i a+2 i b x}\right)}{8 b^3}-\frac{x^2 \text{PolyLog}\left(2,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 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac{b x^4}{12}","-\frac{i x \text{PolyLog}\left(3,i c e^{2 i a+2 i b x}\right)}{4 b^2}+\frac{\text{PolyLog}\left(4,i c e^{2 i a+2 i b x}\right)}{8 b^3}-\frac{x^2 \text{PolyLog}\left(2,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 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac{b x^4}{12}",1,"-(b*x^4)/12 + (x^3*ArcCot[c + (1 - I*c)*Cot[a + b*x]])/3 - (I/6)*x^3*Log[1 - I*c*E^((2*I)*a + (2*I)*b*x)] - (x^2*PolyLog[2, I*c*E^((2*I)*a + (2*I)*b*x)])/(4*b) - ((I/4)*x*PolyLog[3, I*c*E^((2*I)*a + (2*I)*b*x)])/b^2 + PolyLog[4, I*c*E^((2*I)*a + (2*I)*b*x)]/(8*b^3)","A",7,7,21,0.3333,1,"{5174, 2184, 2190, 2531, 6609, 2282, 6589}"
176,1,123,0,0.2239255,"\int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx","Int[x*ArcCot[c + (1 - I*c)*Cot[a + b*x]],x]","-\frac{i \text{PolyLog}\left(3,i c e^{2 i a+2 i b x}\right)}{8 b^2}-\frac{x \text{PolyLog}\left(2,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 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac{b x^3}{6}","-\frac{i \text{PolyLog}\left(3,i c e^{2 i a+2 i b x}\right)}{8 b^2}-\frac{x \text{PolyLog}\left(2,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 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac{b x^3}{6}",1,"-(b*x^3)/6 + (x^2*ArcCot[c + (1 - I*c)*Cot[a + b*x]])/2 - (I/4)*x^2*Log[1 - I*c*E^((2*I)*a + (2*I)*b*x)] - (x*PolyLog[2, I*c*E^((2*I)*a + (2*I)*b*x)])/(4*b) - ((I/8)*PolyLog[3, I*c*E^((2*I)*a + (2*I)*b*x)])/b^2","A",6,6,19,0.3158,1,"{5174, 2184, 2190, 2531, 2282, 6589}"
177,1,85,0,0.1359149,"\int \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx","Int[ArcCot[c + (1 - I*c)*Cot[a + b*x]],x]","-\frac{\text{PolyLog}\left(2,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 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac{b x^2}{2}","-\frac{\text{PolyLog}\left(2,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 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac{b x^2}{2}",1,"-(b*x^2)/2 + x*ArcCot[c + (1 - I*c)*Cot[a + b*x]] - (I/2)*x*Log[1 - I*c*E^((2*I)*a + (2*I)*b*x)] - PolyLog[2, I*c*E^((2*I)*a + (2*I)*b*x)]/(4*b)","A",5,5,17,0.2941,1,"{5166, 2184, 2190, 2279, 2391}"
178,0,0,0,0.1139576,"\int \frac{\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx","Int[ArcCot[c + (1 - I*c)*Cot[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x},x\right)",0,"Defer[Int][ArcCot[c + (1 - I*c)*Cot[a + b*x]]/x, x]","A",0,0,0,0,-1,"{}"
179,1,155,0,0.2569275,"\int x^2 \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx","Int[x^2*ArcCot[c - (1 + I*c)*Cot[a + b*x]],x]","\frac{i x \text{PolyLog}\left(3,-i c e^{2 i a+2 i b x}\right)}{4 b^2}-\frac{\text{PolyLog}\left(4,-i c e^{2 i a+2 i b x}\right)}{8 b^3}+\frac{x^2 \text{PolyLog}\left(2,-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 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac{b x^4}{12}","\frac{i x \text{PolyLog}\left(3,-i c e^{2 i a+2 i b x}\right)}{4 b^2}-\frac{\text{PolyLog}\left(4,-i c e^{2 i a+2 i b x}\right)}{8 b^3}+\frac{x^2 \text{PolyLog}\left(2,-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 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac{b x^4}{12}",1,"(b*x^4)/12 + (x^3*ArcCot[c - (1 + I*c)*Cot[a + b*x]])/3 + (I/6)*x^3*Log[1 + I*c*E^((2*I)*a + (2*I)*b*x)] + (x^2*PolyLog[2, (-I)*c*E^((2*I)*a + (2*I)*b*x)])/(4*b) + ((I/4)*x*PolyLog[3, (-I)*c*E^((2*I)*a + (2*I)*b*x)])/b^2 - PolyLog[4, (-I)*c*E^((2*I)*a + (2*I)*b*x)]/(8*b^3)","A",7,7,22,0.3182,1,"{5174, 2184, 2190, 2531, 6609, 2282, 6589}"
180,1,124,0,0.219299,"\int x \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx","Int[x*ArcCot[c - (1 + I*c)*Cot[a + b*x]],x]","\frac{i \text{PolyLog}\left(3,-i c e^{2 i a+2 i b x}\right)}{8 b^2}+\frac{x \text{PolyLog}\left(2,-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 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac{b x^3}{6}","\frac{i \text{PolyLog}\left(3,-i c e^{2 i a+2 i b x}\right)}{8 b^2}+\frac{x \text{PolyLog}\left(2,-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 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac{b x^3}{6}",1,"(b*x^3)/6 + (x^2*ArcCot[c - (1 + I*c)*Cot[a + b*x]])/2 + (I/4)*x^2*Log[1 + I*c*E^((2*I)*a + (2*I)*b*x)] + (x*PolyLog[2, (-I)*c*E^((2*I)*a + (2*I)*b*x)])/(4*b) + ((I/8)*PolyLog[3, (-I)*c*E^((2*I)*a + (2*I)*b*x)])/b^2","A",6,6,20,0.3000,1,"{5174, 2184, 2190, 2531, 2282, 6589}"
181,1,86,0,0.1317322,"\int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx","Int[ArcCot[c - (1 + I*c)*Cot[a + b*x]],x]","\frac{\text{PolyLog}\left(2,-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 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac{b x^2}{2}","\frac{\text{PolyLog}\left(2,-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 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac{b x^2}{2}",1,"(b*x^2)/2 + x*ArcCot[c - (1 + I*c)*Cot[a + b*x]] + (I/2)*x*Log[1 + I*c*E^((2*I)*a + (2*I)*b*x)] + PolyLog[2, (-I)*c*E^((2*I)*a + (2*I)*b*x)]/(4*b)","A",5,5,18,0.2778,1,"{5166, 2184, 2190, 2279, 2391}"
182,0,0,0,0.1165122,"\int \frac{\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx","Int[ArcCot[c - (1 + I*c)*Cot[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x},x\right)",0,"Defer[Int][ArcCot[c - (1 + I*c)*Cot[a + b*x]]/x, x]","A",0,0,0,0,-1,"{}"
183,1,299,0,0.2084367,"\int (e+f x)^3 \cot ^{-1}(\tanh (a+b x)) \, dx","Int[(e + f*x)^3*ArcCot[Tanh[a + b*x]],x]","-\frac{3 i f^2 (e+f x) \text{PolyLog}\left(4,-i e^{2 a+2 b x}\right)}{8 b^3}+\frac{3 i f^2 (e+f x) \text{PolyLog}\left(4,i e^{2 a+2 b x}\right)}{8 b^3}+\frac{3 i f (e+f x)^2 \text{PolyLog}\left(3,-i e^{2 a+2 b x}\right)}{8 b^2}-\frac{3 i f (e+f x)^2 \text{PolyLog}\left(3,i e^{2 a+2 b x}\right)}{8 b^2}+\frac{3 i f^3 \text{PolyLog}\left(5,-i e^{2 a+2 b x}\right)}{16 b^4}-\frac{3 i f^3 \text{PolyLog}\left(5,i e^{2 a+2 b x}\right)}{16 b^4}-\frac{i (e+f x)^3 \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}+\frac{i (e+f x)^3 \text{PolyLog}\left(2,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 \cot ^{-1}(\tanh (a+b x))}{4 f}","-\frac{3 i f^2 (e+f x) \text{PolyLog}\left(4,-i e^{2 a+2 b x}\right)}{8 b^3}+\frac{3 i f^2 (e+f x) \text{PolyLog}\left(4,i e^{2 a+2 b x}\right)}{8 b^3}+\frac{3 i f (e+f x)^2 \text{PolyLog}\left(3,-i e^{2 a+2 b x}\right)}{8 b^2}-\frac{3 i f (e+f x)^2 \text{PolyLog}\left(3,i e^{2 a+2 b x}\right)}{8 b^2}+\frac{3 i f^3 \text{PolyLog}\left(5,-i e^{2 a+2 b x}\right)}{16 b^4}-\frac{3 i f^3 \text{PolyLog}\left(5,i e^{2 a+2 b x}\right)}{16 b^4}-\frac{i (e+f x)^3 \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}+\frac{i (e+f x)^3 \text{PolyLog}\left(2,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 \cot ^{-1}(\tanh (a+b x))}{4 f}",1,"((e + f*x)^4*ArcCot[Tanh[a + b*x]])/(4*f) + ((e + f*x)^4*ArcTan[E^(2*a + 2*b*x)])/(4*f) - ((I/4)*(e + f*x)^3*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/b + ((I/4)*(e + f*x)^3*PolyLog[2, I*E^(2*a + 2*b*x)])/b + (((3*I)/8)*f*(e + f*x)^2*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/b^2 - (((3*I)/8)*f*(e + f*x)^2*PolyLog[3, I*E^(2*a + 2*b*x)])/b^2 - (((3*I)/8)*f^2*(e + f*x)*PolyLog[4, (-I)*E^(2*a + 2*b*x)])/b^3 + (((3*I)/8)*f^2*(e + f*x)*PolyLog[4, I*E^(2*a + 2*b*x)])/b^3 + (((3*I)/16)*f^3*PolyLog[5, (-I)*E^(2*a + 2*b*x)])/b^4 - (((3*I)/16)*f^3*PolyLog[5, I*E^(2*a + 2*b*x)])/b^4","A",12,6,15,0.4000,1,"{5184, 4180, 2531, 6609, 2282, 6589}"
184,1,229,0,0.1538823,"\int (e+f x)^2 \cot ^{-1}(\tanh (a+b x)) \, dx","Int[(e + f*x)^2*ArcCot[Tanh[a + b*x]],x]","\frac{i f (e+f x) \text{PolyLog}\left(3,-i e^{2 a+2 b x}\right)}{4 b^2}-\frac{i f (e+f x) \text{PolyLog}\left(3,i e^{2 a+2 b x}\right)}{4 b^2}-\frac{i f^2 \text{PolyLog}\left(4,-i e^{2 a+2 b x}\right)}{8 b^3}+\frac{i f^2 \text{PolyLog}\left(4,i e^{2 a+2 b x}\right)}{8 b^3}-\frac{i (e+f x)^2 \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}+\frac{i (e+f x)^2 \text{PolyLog}\left(2,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 \cot ^{-1}(\tanh (a+b x))}{3 f}","\frac{i f (e+f x) \text{PolyLog}\left(3,-i e^{2 a+2 b x}\right)}{4 b^2}-\frac{i f (e+f x) \text{PolyLog}\left(3,i e^{2 a+2 b x}\right)}{4 b^2}-\frac{i f^2 \text{PolyLog}\left(4,-i e^{2 a+2 b x}\right)}{8 b^3}+\frac{i f^2 \text{PolyLog}\left(4,i e^{2 a+2 b x}\right)}{8 b^3}-\frac{i (e+f x)^2 \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}+\frac{i (e+f x)^2 \text{PolyLog}\left(2,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 \cot ^{-1}(\tanh (a+b x))}{3 f}",1,"((e + f*x)^3*ArcCot[Tanh[a + b*x]])/(3*f) + ((e + f*x)^3*ArcTan[E^(2*a + 2*b*x)])/(3*f) - ((I/4)*(e + f*x)^2*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/b + ((I/4)*(e + f*x)^2*PolyLog[2, I*E^(2*a + 2*b*x)])/b + ((I/4)*f*(e + f*x)*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/b^2 - ((I/4)*f*(e + f*x)*PolyLog[3, I*E^(2*a + 2*b*x)])/b^2 - ((I/8)*f^2*PolyLog[4, (-I)*E^(2*a + 2*b*x)])/b^3 + ((I/8)*f^2*PolyLog[4, I*E^(2*a + 2*b*x)])/b^3","A",10,6,15,0.4000,1,"{5184, 4180, 2531, 6609, 2282, 6589}"
185,1,159,0,0.1019382,"\int (e+f x) \cot ^{-1}(\tanh (a+b x)) \, dx","Int[(e + f*x)*ArcCot[Tanh[a + b*x]],x]","\frac{i f \text{PolyLog}\left(3,-i e^{2 a+2 b x}\right)}{8 b^2}-\frac{i f \text{PolyLog}\left(3,i e^{2 a+2 b x}\right)}{8 b^2}-\frac{i (e+f x) \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}+\frac{i (e+f x) \text{PolyLog}\left(2,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 \cot ^{-1}(\tanh (a+b x))}{2 f}","\frac{i f \text{PolyLog}\left(3,-i e^{2 a+2 b x}\right)}{8 b^2}-\frac{i f \text{PolyLog}\left(3,i e^{2 a+2 b x}\right)}{8 b^2}-\frac{i (e+f x) \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}+\frac{i (e+f x) \text{PolyLog}\left(2,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 \cot ^{-1}(\tanh (a+b x))}{2 f}",1,"((e + f*x)^2*ArcCot[Tanh[a + b*x]])/(2*f) + ((e + f*x)^2*ArcTan[E^(2*a + 2*b*x)])/(2*f) - ((I/4)*(e + f*x)*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/b + ((I/4)*(e + f*x)*PolyLog[2, I*E^(2*a + 2*b*x)])/b + ((I/8)*f*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/b^2 - ((I/8)*f*PolyLog[3, I*E^(2*a + 2*b*x)])/b^2","A",8,5,13,0.3846,1,"{5184, 4180, 2531, 2282, 6589}"
186,1,73,0,0.0412203,"\int \cot ^{-1}(\tanh (a+b x)) \, dx","Int[ArcCot[Tanh[a + b*x]],x]","-\frac{i \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}+\frac{i \text{PolyLog}\left(2,i e^{2 a+2 b x}\right)}{4 b}+x \tan ^{-1}\left(e^{2 a+2 b x}\right)+x \cot ^{-1}(\tanh (a+b x))","-\frac{i \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}+\frac{i \text{PolyLog}\left(2,i e^{2 a+2 b x}\right)}{4 b}+x \tan ^{-1}\left(e^{2 a+2 b x}\right)+x \cot ^{-1}(\tanh (a+b x))",1,"x*ArcCot[Tanh[a + b*x]] + x*ArcTan[E^(2*a + 2*b*x)] - ((I/4)*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/b + ((I/4)*PolyLog[2, I*E^(2*a + 2*b*x)])/b","A",6,4,7,0.5714,1,"{5180, 4180, 2279, 2391}"
187,0,0,0,0.0402191,"\int \frac{\cot ^{-1}(\tanh (a+b x))}{e+f x} \, dx","Int[ArcCot[Tanh[a + b*x]]/(e + f*x),x]","\int \frac{\cot ^{-1}(\tanh (a+b x))}{e+f x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(\tanh (a+b x))}{e+f x},x\right)",0,"Defer[Int][ArcCot[Tanh[a + b*x]]/(e + f*x), x]","A",0,0,0,0,-1,"{}"
188,1,355,0,0.4554249,"\int x^2 \cot ^{-1}(c+d \tanh (a+b x)) \, dx","Int[x^2*ArcCot[c + d*Tanh[a + b*x]],x]","\frac{i x \text{PolyLog}\left(3,-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b^2}-\frac{i x \text{PolyLog}\left(3,-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b^2}-\frac{i \text{PolyLog}\left(4,-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^3}+\frac{i \text{PolyLog}\left(4,-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^3}-\frac{i x^2 \text{PolyLog}\left(2,-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i x^2 \text{PolyLog}\left(2,-\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 \cot ^{-1}(d \tanh (a+b x)+c)","\frac{i x \text{PolyLog}\left(3,-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b^2}-\frac{i x \text{PolyLog}\left(3,-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b^2}-\frac{i \text{PolyLog}\left(4,-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^3}+\frac{i \text{PolyLog}\left(4,-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^3}-\frac{i x^2 \text{PolyLog}\left(2,-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i x^2 \text{PolyLog}\left(2,-\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 \cot ^{-1}(d \tanh (a+b x)+c)",1,"(x^3*ArcCot[c + d*Tanh[a + b*x]])/3 - (I/6)*x^3*Log[1 + ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] + (I/6)*x^3*Log[1 + ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] - ((I/4)*x^2*PolyLog[2, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/b + ((I/4)*x^2*PolyLog[2, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/b + ((I/4)*x*PolyLog[3, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/b^2 - ((I/4)*x*PolyLog[3, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/b^2 - ((I/8)*PolyLog[4, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/b^3 + ((I/8)*PolyLog[4, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/b^3","A",11,6,15,0.4000,1,"{5200, 2190, 2531, 6609, 2282, 6589}"
189,1,267,0,0.3759282,"\int x \cot ^{-1}(c+d \tanh (a+b x)) \, dx","Int[x*ArcCot[c + d*Tanh[a + b*x]],x]","\frac{i \text{PolyLog}\left(3,-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^2}-\frac{i \text{PolyLog}\left(3,-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^2}-\frac{i x \text{PolyLog}\left(2,-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i x \text{PolyLog}\left(2,-\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 \cot ^{-1}(d \tanh (a+b x)+c)","\frac{i \text{PolyLog}\left(3,-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^2}-\frac{i \text{PolyLog}\left(3,-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^2}-\frac{i x \text{PolyLog}\left(2,-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i x \text{PolyLog}\left(2,-\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 \cot ^{-1}(d \tanh (a+b x)+c)",1,"(x^2*ArcCot[c + d*Tanh[a + b*x]])/2 - (I/4)*x^2*Log[1 + ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] + (I/4)*x^2*Log[1 + ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] - ((I/4)*x*PolyLog[2, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/b + ((I/4)*x*PolyLog[2, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/b + ((I/8)*PolyLog[3, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/b^2 - ((I/8)*PolyLog[3, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/b^2","A",9,5,13,0.3846,1,"{5200, 2190, 2531, 2282, 6589}"
190,1,174,0,0.2222792,"\int \cot ^{-1}(c+d \tanh (a+b x)) \, dx","Int[ArcCot[c + d*Tanh[a + b*x]],x]","-\frac{i \text{PolyLog}\left(2,-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i \text{PolyLog}\left(2,-\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 \cot ^{-1}(d \tanh (a+b x)+c)","-\frac{i \text{PolyLog}\left(2,-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i \text{PolyLog}\left(2,-\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 \cot ^{-1}(d \tanh (a+b x)+c)",1,"x*ArcCot[c + d*Tanh[a + b*x]] - (I/2)*x*Log[1 + ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] + (I/2)*x*Log[1 + ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] - ((I/4)*PolyLog[2, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/b + ((I/4)*PolyLog[2, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/b","A",7,4,11,0.3636,1,"{5192, 2190, 2279, 2391}"
191,0,0,0,0.0806673,"\int \frac{\cot ^{-1}(c+d \tanh (a+b x))}{x} \, dx","Int[ArcCot[c + d*Tanh[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+d \tanh (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(d \tanh (a+b x)+c)}{x},x\right)",0,"Defer[Int][ArcCot[c + d*Tanh[a + b*x]]/x, x]","A",0,0,0,0,-1,"{}"
192,1,142,0,0.227638,"\int x^2 \cot ^{-1}(c+(i+c) \tanh (a+b x)) \, dx","Int[x^2*ArcCot[c + (I + c)*Tanh[a + b*x]],x]","\frac{i x \text{PolyLog}\left(3,-i c e^{2 a+2 b x}\right)}{4 b^2}-\frac{i \text{PolyLog}\left(4,-i c e^{2 a+2 b x}\right)}{8 b^3}-\frac{i x^2 \text{PolyLog}\left(2,-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 \cot ^{-1}(c+(c+i) \tanh (a+b x))+\frac{1}{12} i b x^4","\frac{i x \text{PolyLog}\left(3,-i c e^{2 a+2 b x}\right)}{4 b^2}-\frac{i \text{PolyLog}\left(4,-i c e^{2 a+2 b x}\right)}{8 b^3}-\frac{i x^2 \text{PolyLog}\left(2,-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 \cot ^{-1}(c+(c+i) \tanh (a+b x))+\frac{1}{12} i b x^4",1,"(I/12)*b*x^4 + (x^3*ArcCot[c + (I + c)*Tanh[a + b*x]])/3 - (I/6)*x^3*Log[1 + I*c*E^(2*a + 2*b*x)] - ((I/4)*x^2*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/b + ((I/4)*x*PolyLog[3, (-I)*c*E^(2*a + 2*b*x)])/b^2 - ((I/8)*PolyLog[4, (-I)*c*E^(2*a + 2*b*x)])/b^3","A",7,7,19,0.3684,1,"{5196, 2184, 2190, 2531, 6609, 2282, 6589}"
193,1,113,0,0.1974447,"\int x \cot ^{-1}(c+(i+c) \tanh (a+b x)) \, dx","Int[x*ArcCot[c + (I + c)*Tanh[a + b*x]],x]","\frac{i \text{PolyLog}\left(3,-i c e^{2 a+2 b x}\right)}{8 b^2}-\frac{i x \text{PolyLog}\left(2,-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 \cot ^{-1}(c+(c+i) \tanh (a+b x))+\frac{1}{6} i b x^3","\frac{i \text{PolyLog}\left(3,-i c e^{2 a+2 b x}\right)}{8 b^2}-\frac{i x \text{PolyLog}\left(2,-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 \cot ^{-1}(c+(c+i) \tanh (a+b x))+\frac{1}{6} i b x^3",1,"(I/6)*b*x^3 + (x^2*ArcCot[c + (I + c)*Tanh[a + b*x]])/2 - (I/4)*x^2*Log[1 + I*c*E^(2*a + 2*b*x)] - ((I/4)*x*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/b + ((I/8)*PolyLog[3, (-I)*c*E^(2*a + 2*b*x)])/b^2","A",6,6,17,0.3529,1,"{5196, 2184, 2190, 2531, 2282, 6589}"
194,1,79,0,0.1195329,"\int \cot ^{-1}(c+(i+c) \tanh (a+b x)) \, dx","Int[ArcCot[c + (I + c)*Tanh[a + b*x]],x]","-\frac{i \text{PolyLog}\left(2,-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 \cot ^{-1}(c+(c+i) \tanh (a+b x))+\frac{1}{2} i b x^2","-\frac{i \text{PolyLog}\left(2,-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 \cot ^{-1}(c+(c+i) \tanh (a+b x))+\frac{1}{2} i b x^2",1,"(I/2)*b*x^2 + x*ArcCot[c + (I + c)*Tanh[a + b*x]] - (I/2)*x*Log[1 + I*c*E^(2*a + 2*b*x)] - ((I/4)*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/b","A",5,5,15,0.3333,1,"{5188, 2184, 2190, 2279, 2391}"
195,0,0,0,0.0930228,"\int \frac{\cot ^{-1}(c+(i+c) \tanh (a+b x))}{x} \, dx","Int[ArcCot[c + (I + c)*Tanh[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+(i+c) \tanh (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c+(c+i) \tanh (a+b x))}{x},x\right)",0,"Defer[Int][ArcCot[c + (I + c)*Tanh[a + b*x]]/x, x]","A",0,0,0,0,-1,"{}"
196,1,145,0,0.2267995,"\int x^2 \cot ^{-1}(c-(i-c) \tanh (a+b x)) \, dx","Int[x^2*ArcCot[c - (I - c)*Tanh[a + b*x]],x]","-\frac{i x \text{PolyLog}\left(3,i c e^{2 a+2 b x}\right)}{4 b^2}+\frac{i \text{PolyLog}\left(4,i c e^{2 a+2 b x}\right)}{8 b^3}+\frac{i x^2 \text{PolyLog}\left(2,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 \cot ^{-1}(c-(-c+i) \tanh (a+b x))-\frac{1}{12} i b x^4","-\frac{i x \text{PolyLog}\left(3,i c e^{2 a+2 b x}\right)}{4 b^2}+\frac{i \text{PolyLog}\left(4,i c e^{2 a+2 b x}\right)}{8 b^3}+\frac{i x^2 \text{PolyLog}\left(2,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 \cot ^{-1}(c-(-c+i) \tanh (a+b x))-\frac{1}{12} i b x^4",1,"(-I/12)*b*x^4 + (x^3*ArcCot[c - (I - c)*Tanh[a + b*x]])/3 + (I/6)*x^3*Log[1 - I*c*E^(2*a + 2*b*x)] + ((I/4)*x^2*PolyLog[2, I*c*E^(2*a + 2*b*x)])/b - ((I/4)*x*PolyLog[3, I*c*E^(2*a + 2*b*x)])/b^2 + ((I/8)*PolyLog[4, I*c*E^(2*a + 2*b*x)])/b^3","A",7,7,22,0.3182,1,"{5196, 2184, 2190, 2531, 6609, 2282, 6589}"
197,1,116,0,0.1943917,"\int x \cot ^{-1}(c-(i-c) \tanh (a+b x)) \, dx","Int[x*ArcCot[c - (I - c)*Tanh[a + b*x]],x]","-\frac{i \text{PolyLog}\left(3,i c e^{2 a+2 b x}\right)}{8 b^2}+\frac{i x \text{PolyLog}\left(2,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 \cot ^{-1}(c-(-c+i) \tanh (a+b x))-\frac{1}{6} i b x^3","-\frac{i \text{PolyLog}\left(3,i c e^{2 a+2 b x}\right)}{8 b^2}+\frac{i x \text{PolyLog}\left(2,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 \cot ^{-1}(c-(-c+i) \tanh (a+b x))-\frac{1}{6} i b x^3",1,"(-I/6)*b*x^3 + (x^2*ArcCot[c - (I - c)*Tanh[a + b*x]])/2 + (I/4)*x^2*Log[1 - I*c*E^(2*a + 2*b*x)] + ((I/4)*x*PolyLog[2, I*c*E^(2*a + 2*b*x)])/b - ((I/8)*PolyLog[3, I*c*E^(2*a + 2*b*x)])/b^2","A",6,6,20,0.3000,1,"{5196, 2184, 2190, 2531, 2282, 6589}"
198,1,82,0,0.118658,"\int \cot ^{-1}(c-(i-c) \tanh (a+b x)) \, dx","Int[ArcCot[c - (I - c)*Tanh[a + b*x]],x]","\frac{i \text{PolyLog}\left(2,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 \cot ^{-1}(c-(-c+i) \tanh (a+b x))-\frac{1}{2} i b x^2","\frac{i \text{PolyLog}\left(2,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 \cot ^{-1}(c-(-c+i) \tanh (a+b x))-\frac{1}{2} i b x^2",1,"(-I/2)*b*x^2 + x*ArcCot[c - (I - c)*Tanh[a + b*x]] + (I/2)*x*Log[1 - I*c*E^(2*a + 2*b*x)] + ((I/4)*PolyLog[2, I*c*E^(2*a + 2*b*x)])/b","A",5,5,18,0.2778,1,"{5188, 2184, 2190, 2279, 2391}"
199,0,0,0,0.0894306,"\int \frac{\cot ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx","Int[ArcCot[c - (I - c)*Tanh[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c-(-c+i) \tanh (a+b x))}{x},x\right)",0,"Defer[Int][ArcCot[c - (I - c)*Tanh[a + b*x]]/x, x]","A",0,0,0,0,-1,"{}"
200,1,299,0,0.2084776,"\int (e+f x)^3 \cot ^{-1}(\coth (a+b x)) \, dx","Int[(e + f*x)^3*ArcCot[Coth[a + b*x]],x]","\frac{3 i f^2 (e+f x) \text{PolyLog}\left(4,-i e^{2 a+2 b x}\right)}{8 b^3}-\frac{3 i f^2 (e+f x) \text{PolyLog}\left(4,i e^{2 a+2 b x}\right)}{8 b^3}-\frac{3 i f (e+f x)^2 \text{PolyLog}\left(3,-i e^{2 a+2 b x}\right)}{8 b^2}+\frac{3 i f (e+f x)^2 \text{PolyLog}\left(3,i e^{2 a+2 b x}\right)}{8 b^2}-\frac{3 i f^3 \text{PolyLog}\left(5,-i e^{2 a+2 b x}\right)}{16 b^4}+\frac{3 i f^3 \text{PolyLog}\left(5,i e^{2 a+2 b x}\right)}{16 b^4}+\frac{i (e+f x)^3 \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}-\frac{i (e+f x)^3 \text{PolyLog}\left(2,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 \cot ^{-1}(\coth (a+b x))}{4 f}","\frac{3 i f^2 (e+f x) \text{PolyLog}\left(4,-i e^{2 a+2 b x}\right)}{8 b^3}-\frac{3 i f^2 (e+f x) \text{PolyLog}\left(4,i e^{2 a+2 b x}\right)}{8 b^3}-\frac{3 i f (e+f x)^2 \text{PolyLog}\left(3,-i e^{2 a+2 b x}\right)}{8 b^2}+\frac{3 i f (e+f x)^2 \text{PolyLog}\left(3,i e^{2 a+2 b x}\right)}{8 b^2}-\frac{3 i f^3 \text{PolyLog}\left(5,-i e^{2 a+2 b x}\right)}{16 b^4}+\frac{3 i f^3 \text{PolyLog}\left(5,i e^{2 a+2 b x}\right)}{16 b^4}+\frac{i (e+f x)^3 \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}-\frac{i (e+f x)^3 \text{PolyLog}\left(2,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 \cot ^{-1}(\coth (a+b x))}{4 f}",1,"((e + f*x)^4*ArcCot[Coth[a + b*x]])/(4*f) - ((e + f*x)^4*ArcTan[E^(2*a + 2*b*x)])/(4*f) + ((I/4)*(e + f*x)^3*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/b - ((I/4)*(e + f*x)^3*PolyLog[2, I*E^(2*a + 2*b*x)])/b - (((3*I)/8)*f*(e + f*x)^2*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/b^2 + (((3*I)/8)*f*(e + f*x)^2*PolyLog[3, I*E^(2*a + 2*b*x)])/b^2 + (((3*I)/8)*f^2*(e + f*x)*PolyLog[4, (-I)*E^(2*a + 2*b*x)])/b^3 - (((3*I)/8)*f^2*(e + f*x)*PolyLog[4, I*E^(2*a + 2*b*x)])/b^3 - (((3*I)/16)*f^3*PolyLog[5, (-I)*E^(2*a + 2*b*x)])/b^4 + (((3*I)/16)*f^3*PolyLog[5, I*E^(2*a + 2*b*x)])/b^4","A",12,6,15,0.4000,1,"{5186, 4180, 2531, 6609, 2282, 6589}"
201,1,229,0,0.1536888,"\int (e+f x)^2 \cot ^{-1}(\coth (a+b x)) \, dx","Int[(e + f*x)^2*ArcCot[Coth[a + b*x]],x]","-\frac{i f (e+f x) \text{PolyLog}\left(3,-i e^{2 a+2 b x}\right)}{4 b^2}+\frac{i f (e+f x) \text{PolyLog}\left(3,i e^{2 a+2 b x}\right)}{4 b^2}+\frac{i f^2 \text{PolyLog}\left(4,-i e^{2 a+2 b x}\right)}{8 b^3}-\frac{i f^2 \text{PolyLog}\left(4,i e^{2 a+2 b x}\right)}{8 b^3}+\frac{i (e+f x)^2 \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}-\frac{i (e+f x)^2 \text{PolyLog}\left(2,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 \cot ^{-1}(\coth (a+b x))}{3 f}","-\frac{i f (e+f x) \text{PolyLog}\left(3,-i e^{2 a+2 b x}\right)}{4 b^2}+\frac{i f (e+f x) \text{PolyLog}\left(3,i e^{2 a+2 b x}\right)}{4 b^2}+\frac{i f^2 \text{PolyLog}\left(4,-i e^{2 a+2 b x}\right)}{8 b^3}-\frac{i f^2 \text{PolyLog}\left(4,i e^{2 a+2 b x}\right)}{8 b^3}+\frac{i (e+f x)^2 \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}-\frac{i (e+f x)^2 \text{PolyLog}\left(2,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 \cot ^{-1}(\coth (a+b x))}{3 f}",1,"((e + f*x)^3*ArcCot[Coth[a + b*x]])/(3*f) - ((e + f*x)^3*ArcTan[E^(2*a + 2*b*x)])/(3*f) + ((I/4)*(e + f*x)^2*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/b - ((I/4)*(e + f*x)^2*PolyLog[2, I*E^(2*a + 2*b*x)])/b - ((I/4)*f*(e + f*x)*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/b^2 + ((I/4)*f*(e + f*x)*PolyLog[3, I*E^(2*a + 2*b*x)])/b^2 + ((I/8)*f^2*PolyLog[4, (-I)*E^(2*a + 2*b*x)])/b^3 - ((I/8)*f^2*PolyLog[4, I*E^(2*a + 2*b*x)])/b^3","A",10,6,15,0.4000,1,"{5186, 4180, 2531, 6609, 2282, 6589}"
202,1,159,0,0.0989876,"\int (e+f x) \cot ^{-1}(\coth (a+b x)) \, dx","Int[(e + f*x)*ArcCot[Coth[a + b*x]],x]","-\frac{i f \text{PolyLog}\left(3,-i e^{2 a+2 b x}\right)}{8 b^2}+\frac{i f \text{PolyLog}\left(3,i e^{2 a+2 b x}\right)}{8 b^2}+\frac{i (e+f x) \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}-\frac{i (e+f x) \text{PolyLog}\left(2,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 \cot ^{-1}(\coth (a+b x))}{2 f}","-\frac{i f \text{PolyLog}\left(3,-i e^{2 a+2 b x}\right)}{8 b^2}+\frac{i f \text{PolyLog}\left(3,i e^{2 a+2 b x}\right)}{8 b^2}+\frac{i (e+f x) \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}-\frac{i (e+f x) \text{PolyLog}\left(2,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 \cot ^{-1}(\coth (a+b x))}{2 f}",1,"((e + f*x)^2*ArcCot[Coth[a + b*x]])/(2*f) - ((e + f*x)^2*ArcTan[E^(2*a + 2*b*x)])/(2*f) + ((I/4)*(e + f*x)*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/b - ((I/4)*(e + f*x)*PolyLog[2, I*E^(2*a + 2*b*x)])/b - ((I/8)*f*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/b^2 + ((I/8)*f*PolyLog[3, I*E^(2*a + 2*b*x)])/b^2","A",8,5,13,0.3846,1,"{5186, 4180, 2531, 2282, 6589}"
203,1,74,0,0.0429558,"\int \cot ^{-1}(\coth (a+b x)) \, dx","Int[ArcCot[Coth[a + b*x]],x]","\frac{i \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}-\frac{i \text{PolyLog}\left(2,i e^{2 a+2 b x}\right)}{4 b}-x \tan ^{-1}\left(e^{2 a+2 b x}\right)+x \cot ^{-1}(\coth (a+b x))","\frac{i \text{PolyLog}\left(2,-i e^{2 a+2 b x}\right)}{4 b}-\frac{i \text{PolyLog}\left(2,i e^{2 a+2 b x}\right)}{4 b}-x \tan ^{-1}\left(e^{2 a+2 b x}\right)+x \cot ^{-1}(\coth (a+b x))",1,"x*ArcCot[Coth[a + b*x]] - x*ArcTan[E^(2*a + 2*b*x)] + ((I/4)*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/b - ((I/4)*PolyLog[2, I*E^(2*a + 2*b*x)])/b","A",6,4,7,0.5714,1,"{5182, 4180, 2279, 2391}"
204,0,0,0,0.0368274,"\int \frac{\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx","Int[ArcCot[Coth[a + b*x]]/(e + f*x),x]","\int \frac{\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(\coth (a+b x))}{e+f x},x\right)",0,"Defer[Int][ArcCot[Coth[a + b*x]]/(e + f*x), x]","A",0,0,0,0,-1,"{}"
205,1,351,0,0.4625131,"\int x^2 \cot ^{-1}(c+d \coth (a+b x)) \, dx","Int[x^2*ArcCot[c + d*Coth[a + b*x]],x]","\frac{i x \text{PolyLog}\left(3,\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b^2}-\frac{i x \text{PolyLog}\left(3,\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b^2}-\frac{i \text{PolyLog}\left(4,\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^3}+\frac{i \text{PolyLog}\left(4,\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^3}-\frac{i x^2 \text{PolyLog}\left(2,\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i x^2 \text{PolyLog}\left(2,\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 \cot ^{-1}(d \coth (a+b x)+c)","\frac{i x \text{PolyLog}\left(3,\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b^2}-\frac{i x \text{PolyLog}\left(3,\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b^2}-\frac{i \text{PolyLog}\left(4,\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^3}+\frac{i \text{PolyLog}\left(4,\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^3}-\frac{i x^2 \text{PolyLog}\left(2,\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i x^2 \text{PolyLog}\left(2,\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 \cot ^{-1}(d \coth (a+b x)+c)",1,"(x^3*ArcCot[c + d*Coth[a + b*x]])/3 - (I/6)*x^3*Log[1 - ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] + (I/6)*x^3*Log[1 - ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] - ((I/4)*x^2*PolyLog[2, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/b + ((I/4)*x^2*PolyLog[2, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/b + ((I/4)*x*PolyLog[3, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/b^2 - ((I/4)*x*PolyLog[3, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/b^2 - ((I/8)*PolyLog[4, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/b^3 + ((I/8)*PolyLog[4, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/b^3","A",11,6,15,0.4000,1,"{5202, 2190, 2531, 6609, 2282, 6589}"
206,1,265,0,0.3764706,"\int x \cot ^{-1}(c+d \coth (a+b x)) \, dx","Int[x*ArcCot[c + d*Coth[a + b*x]],x]","\frac{i \text{PolyLog}\left(3,\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^2}-\frac{i \text{PolyLog}\left(3,\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^2}-\frac{i x \text{PolyLog}\left(2,\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i x \text{PolyLog}\left(2,\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 \cot ^{-1}(d \coth (a+b x)+c)","\frac{i \text{PolyLog}\left(3,\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^2}-\frac{i \text{PolyLog}\left(3,\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^2}-\frac{i x \text{PolyLog}\left(2,\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i x \text{PolyLog}\left(2,\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 \cot ^{-1}(d \coth (a+b x)+c)",1,"(x^2*ArcCot[c + d*Coth[a + b*x]])/2 - (I/4)*x^2*Log[1 - ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] + (I/4)*x^2*Log[1 - ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] - ((I/4)*x*PolyLog[2, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/b + ((I/4)*x*PolyLog[2, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/b + ((I/8)*PolyLog[3, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/b^2 - ((I/8)*PolyLog[3, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/b^2","A",9,5,13,0.3846,1,"{5202, 2190, 2531, 2282, 6589}"
207,1,174,0,0.2318695,"\int \cot ^{-1}(c+d \coth (a+b x)) \, dx","Int[ArcCot[c + d*Coth[a + b*x]],x]","-\frac{i \text{PolyLog}\left(2,\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i \text{PolyLog}\left(2,\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 \cot ^{-1}(d \coth (a+b x)+c)","-\frac{i \text{PolyLog}\left(2,\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i \text{PolyLog}\left(2,\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 \cot ^{-1}(d \coth (a+b x)+c)",1,"x*ArcCot[c + d*Coth[a + b*x]] - (I/2)*x*Log[1 - ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] + (I/2)*x*Log[1 - ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] - ((I/4)*PolyLog[2, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/b + ((I/4)*PolyLog[2, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/b","A",7,4,11,0.3636,1,"{5194, 2190, 2279, 2391}"
208,0,0,0,0.0794915,"\int \frac{\cot ^{-1}(c+d \coth (a+b x))}{x} \, dx","Int[ArcCot[c + d*Coth[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+d \coth (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(d \coth (a+b x)+c)}{x},x\right)",0,"Defer[Int][ArcCot[c + d*Coth[a + b*x]]/x, x]","A",0,0,0,0,-1,"{}"
209,1,142,0,0.2313274,"\int x^2 \cot ^{-1}(c+(i+c) \coth (a+b x)) \, dx","Int[x^2*ArcCot[c + (I + c)*Coth[a + b*x]],x]","\frac{i x \text{PolyLog}\left(3,i c e^{2 a+2 b x}\right)}{4 b^2}-\frac{i \text{PolyLog}\left(4,i c e^{2 a+2 b x}\right)}{8 b^3}-\frac{i x^2 \text{PolyLog}\left(2,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 \cot ^{-1}(c+(c+i) \coth (a+b x))+\frac{1}{12} i b x^4","\frac{i x \text{PolyLog}\left(3,i c e^{2 a+2 b x}\right)}{4 b^2}-\frac{i \text{PolyLog}\left(4,i c e^{2 a+2 b x}\right)}{8 b^3}-\frac{i x^2 \text{PolyLog}\left(2,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 \cot ^{-1}(c+(c+i) \coth (a+b x))+\frac{1}{12} i b x^4",1,"(I/12)*b*x^4 + (x^3*ArcCot[c + (I + c)*Coth[a + b*x]])/3 - (I/6)*x^3*Log[1 - I*c*E^(2*a + 2*b*x)] - ((I/4)*x^2*PolyLog[2, I*c*E^(2*a + 2*b*x)])/b + ((I/4)*x*PolyLog[3, I*c*E^(2*a + 2*b*x)])/b^2 - ((I/8)*PolyLog[4, I*c*E^(2*a + 2*b*x)])/b^3","A",7,7,19,0.3684,1,"{5198, 2184, 2190, 2531, 6609, 2282, 6589}"
210,1,113,0,0.2026086,"\int x \cot ^{-1}(c+(i+c) \coth (a+b x)) \, dx","Int[x*ArcCot[c + (I + c)*Coth[a + b*x]],x]","\frac{i \text{PolyLog}\left(3,i c e^{2 a+2 b x}\right)}{8 b^2}-\frac{i x \text{PolyLog}\left(2,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 \cot ^{-1}(c+(c+i) \coth (a+b x))+\frac{1}{6} i b x^3","\frac{i \text{PolyLog}\left(3,i c e^{2 a+2 b x}\right)}{8 b^2}-\frac{i x \text{PolyLog}\left(2,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 \cot ^{-1}(c+(c+i) \coth (a+b x))+\frac{1}{6} i b x^3",1,"(I/6)*b*x^3 + (x^2*ArcCot[c + (I + c)*Coth[a + b*x]])/2 - (I/4)*x^2*Log[1 - I*c*E^(2*a + 2*b*x)] - ((I/4)*x*PolyLog[2, I*c*E^(2*a + 2*b*x)])/b + ((I/8)*PolyLog[3, I*c*E^(2*a + 2*b*x)])/b^2","A",6,6,17,0.3529,1,"{5198, 2184, 2190, 2531, 2282, 6589}"
211,1,79,0,0.1235312,"\int \cot ^{-1}(c+(i+c) \coth (a+b x)) \, dx","Int[ArcCot[c + (I + c)*Coth[a + b*x]],x]","-\frac{i \text{PolyLog}\left(2,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 \cot ^{-1}(c+(c+i) \coth (a+b x))+\frac{1}{2} i b x^2","-\frac{i \text{PolyLog}\left(2,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 \cot ^{-1}(c+(c+i) \coth (a+b x))+\frac{1}{2} i b x^2",1,"(I/2)*b*x^2 + x*ArcCot[c + (I + c)*Coth[a + b*x]] - (I/2)*x*Log[1 - I*c*E^(2*a + 2*b*x)] - ((I/4)*PolyLog[2, I*c*E^(2*a + 2*b*x)])/b","A",5,5,15,0.3333,1,"{5190, 2184, 2190, 2279, 2391}"
212,0,0,0,0.0825572,"\int \frac{\cot ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx","Int[ArcCot[c + (I + c)*Coth[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c+(c+i) \coth (a+b x))}{x},x\right)",0,"Defer[Int][ArcCot[c + (I + c)*Coth[a + b*x]]/x, x]","A",0,0,0,0,-1,"{}"
213,1,145,0,0.2317198,"\int x^2 \cot ^{-1}(c-(i-c) \coth (a+b x)) \, dx","Int[x^2*ArcCot[c - (I - c)*Coth[a + b*x]],x]","-\frac{i x \text{PolyLog}\left(3,-i c e^{2 a+2 b x}\right)}{4 b^2}+\frac{i \text{PolyLog}\left(4,-i c e^{2 a+2 b x}\right)}{8 b^3}+\frac{i x^2 \text{PolyLog}\left(2,-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 \cot ^{-1}(c-(-c+i) \coth (a+b x))-\frac{1}{12} i b x^4","-\frac{i x \text{PolyLog}\left(3,-i c e^{2 a+2 b x}\right)}{4 b^2}+\frac{i \text{PolyLog}\left(4,-i c e^{2 a+2 b x}\right)}{8 b^3}+\frac{i x^2 \text{PolyLog}\left(2,-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 \cot ^{-1}(c-(-c+i) \coth (a+b x))-\frac{1}{12} i b x^4",1,"(-I/12)*b*x^4 + (x^3*ArcCot[c - (I - c)*Coth[a + b*x]])/3 + (I/6)*x^3*Log[1 + I*c*E^(2*a + 2*b*x)] + ((I/4)*x^2*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/b - ((I/4)*x*PolyLog[3, (-I)*c*E^(2*a + 2*b*x)])/b^2 + ((I/8)*PolyLog[4, (-I)*c*E^(2*a + 2*b*x)])/b^3","A",7,7,22,0.3182,1,"{5198, 2184, 2190, 2531, 6609, 2282, 6589}"
214,1,116,0,0.2053177,"\int x \cot ^{-1}(c-(i-c) \coth (a+b x)) \, dx","Int[x*ArcCot[c - (I - c)*Coth[a + b*x]],x]","-\frac{i \text{PolyLog}\left(3,-i c e^{2 a+2 b x}\right)}{8 b^2}+\frac{i x \text{PolyLog}\left(2,-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 \cot ^{-1}(c-(-c+i) \coth (a+b x))-\frac{1}{6} i b x^3","-\frac{i \text{PolyLog}\left(3,-i c e^{2 a+2 b x}\right)}{8 b^2}+\frac{i x \text{PolyLog}\left(2,-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 \cot ^{-1}(c-(-c+i) \coth (a+b x))-\frac{1}{6} i b x^3",1,"(-I/6)*b*x^3 + (x^2*ArcCot[c - (I - c)*Coth[a + b*x]])/2 + (I/4)*x^2*Log[1 + I*c*E^(2*a + 2*b*x)] + ((I/4)*x*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/b - ((I/8)*PolyLog[3, (-I)*c*E^(2*a + 2*b*x)])/b^2","A",6,6,20,0.3000,1,"{5198, 2184, 2190, 2531, 2282, 6589}"
215,1,82,0,0.1242944,"\int \cot ^{-1}(c-(i-c) \coth (a+b x)) \, dx","Int[ArcCot[c - (I - c)*Coth[a + b*x]],x]","\frac{i \text{PolyLog}\left(2,-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 \cot ^{-1}(c-(-c+i) \coth (a+b x))-\frac{1}{2} i b x^2","\frac{i \text{PolyLog}\left(2,-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 \cot ^{-1}(c-(-c+i) \coth (a+b x))-\frac{1}{2} i b x^2",1,"(-I/2)*b*x^2 + x*ArcCot[c - (I - c)*Coth[a + b*x]] + (I/2)*x*Log[1 + I*c*E^(2*a + 2*b*x)] + ((I/4)*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/b","A",5,5,18,0.2778,1,"{5190, 2184, 2190, 2279, 2391}"
216,0,0,0,0.1082972,"\int \frac{\cot ^{-1}(c-(i-c) \coth (a+b x))}{x} \, dx","Int[ArcCot[c - (I - c)*Coth[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c-(i-c) \coth (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c-(-c+i) \coth (a+b x))}{x},x\right)",0,"Defer[Int][ArcCot[c - (I - c)*Coth[a + b*x]]/x, x]","A",0,0,0,0,-1,"{}"
217,1,187,0,0.6079127,"\int \frac{\left(a+b \cot ^{-1}\left(c x^n\right)\right) \left(d+e \log \left(f x^m\right)\right)}{x} \, dx","Int[((a + b*ArcCot[c*x^n])*(d + e*Log[f*x^m]))/x,x]","-\frac{i b d \text{PolyLog}\left(2,-\frac{i x^{-n}}{c}\right)}{2 n}+\frac{i b d \text{PolyLog}\left(2,\frac{i x^{-n}}{c}\right)}{2 n}-\frac{i b e \log \left(f x^m\right) \text{PolyLog}\left(2,-\frac{i x^{-n}}{c}\right)}{2 n}+\frac{i b e \log \left(f x^m\right) \text{PolyLog}\left(2,\frac{i x^{-n}}{c}\right)}{2 n}-\frac{i b e m \text{PolyLog}\left(3,-\frac{i x^{-n}}{c}\right)}{2 n^2}+\frac{i b e m \text{PolyLog}\left(3,\frac{i x^{-n}}{c}\right)}{2 n^2}+a d \log (x)+\frac{a e \log ^2\left(f x^m\right)}{2 m}","-\frac{i b d \text{PolyLog}\left(2,-\frac{i x^{-n}}{c}\right)}{2 n}+\frac{i b d \text{PolyLog}\left(2,\frac{i x^{-n}}{c}\right)}{2 n}-\frac{i b e \log \left(f x^m\right) \text{PolyLog}\left(2,-\frac{i x^{-n}}{c}\right)}{2 n}+\frac{i b e \log \left(f x^m\right) \text{PolyLog}\left(2,\frac{i x^{-n}}{c}\right)}{2 n}-\frac{i b e m \text{PolyLog}\left(3,-\frac{i x^{-n}}{c}\right)}{2 n^2}+\frac{i b e m \text{PolyLog}\left(3,\frac{i x^{-n}}{c}\right)}{2 n^2}+a d \log (x)+\frac{a e \log ^2\left(f x^m\right)}{2 m}",1,"a*d*Log[x] + (a*e*Log[f*x^m]^2)/(2*m) - ((I/2)*b*d*PolyLog[2, (-I)/(c*x^n)])/n - ((I/2)*b*e*Log[f*x^m]*PolyLog[2, (-I)/(c*x^n)])/n + ((I/2)*b*d*PolyLog[2, I/(c*x^n)])/n + ((I/2)*b*e*Log[f*x^m]*PolyLog[2, I/(c*x^n)])/n - ((I/2)*b*e*m*PolyLog[3, (-I)/(c*x^n)])/n^2 + ((I/2)*b*e*m*PolyLog[3, I/(c*x^n)])/n^2","A",13,9,24,0.3750,1,"{2301, 6742, 5032, 4849, 2391, 5008, 5006, 2374, 6589}"
218,1,35,0,0.0284399,"\int \cot ^{-1}\left(e^x\right) \, dx","Int[ArcCot[E^x],x]","\frac{1}{2} i \text{PolyLog}\left(2,i e^{-x}\right)-\frac{1}{2} i \text{PolyLog}\left(2,-i e^{-x}\right)","\frac{1}{2} i \text{PolyLog}\left(2,i e^{-x}\right)-\frac{1}{2} i \text{PolyLog}\left(2,-i e^{-x}\right)",1,"(-I/2)*PolyLog[2, (-I)/E^x] + (I/2)*PolyLog[2, I/E^x]","A",4,3,4,0.7500,1,"{2282, 4849, 2391}"
219,1,71,0,0.0465794,"\int x \cot ^{-1}\left(e^x\right) \, dx","Int[x*ArcCot[E^x],x]","-\frac{1}{2} i x \text{PolyLog}\left(2,-i e^{-x}\right)+\frac{1}{2} i x \text{PolyLog}\left(2,i e^{-x}\right)-\frac{1}{2} i \text{PolyLog}\left(3,-i e^{-x}\right)+\frac{1}{2} i \text{PolyLog}\left(3,i e^{-x}\right)","-\frac{1}{2} i x \text{PolyLog}\left(2,-i e^{-x}\right)+\frac{1}{2} i x \text{PolyLog}\left(2,i e^{-x}\right)-\frac{1}{2} i \text{PolyLog}\left(3,-i e^{-x}\right)+\frac{1}{2} i \text{PolyLog}\left(3,i e^{-x}\right)",1,"(-I/2)*x*PolyLog[2, (-I)/E^x] + (I/2)*x*PolyLog[2, I/E^x] - (I/2)*PolyLog[3, (-I)/E^x] + (I/2)*PolyLog[3, I/E^x]","A",7,4,6,0.6667,1,"{5144, 2531, 2282, 6589}"
220,1,103,0,0.0700694,"\int x^2 \cot ^{-1}\left(e^x\right) \, dx","Int[x^2*ArcCot[E^x],x]","-\frac{1}{2} i x^2 \text{PolyLog}\left(2,-i e^{-x}\right)+\frac{1}{2} i x^2 \text{PolyLog}\left(2,i e^{-x}\right)-i x \text{PolyLog}\left(3,-i e^{-x}\right)+i x \text{PolyLog}\left(3,i e^{-x}\right)-i \text{PolyLog}\left(4,-i e^{-x}\right)+i \text{PolyLog}\left(4,i e^{-x}\right)","-\frac{1}{2} i x^2 \text{PolyLog}\left(2,-i e^{-x}\right)+\frac{1}{2} i x^2 \text{PolyLog}\left(2,i e^{-x}\right)-i x \text{PolyLog}\left(3,-i e^{-x}\right)+i x \text{PolyLog}\left(3,i e^{-x}\right)-i \text{PolyLog}\left(4,-i e^{-x}\right)+i \text{PolyLog}\left(4,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",9,5,8,0.6250,1,"{5144, 2531, 6609, 2282, 6589}"
221,1,51,0,0.0302839,"\int \cot ^{-1}\left(e^{a+b x}\right) \, dx","Int[ArcCot[E^(a + b*x)],x]","\frac{i \text{PolyLog}\left(2,i e^{-a-b x}\right)}{2 b}-\frac{i \text{PolyLog}\left(2,-i e^{-a-b x}\right)}{2 b}","\frac{i \text{PolyLog}\left(2,i e^{-a-b x}\right)}{2 b}-\frac{i \text{PolyLog}\left(2,-i e^{-a-b x}\right)}{2 b}",1,"((-I/2)*PolyLog[2, (-I)*E^(-a - b*x)])/b + ((I/2)*PolyLog[2, I*E^(-a - b*x)])/b","A",4,3,8,0.3750,1,"{2282, 4849, 2391}"
222,1,103,0,0.0622578,"\int x \cot ^{-1}\left(e^{a+b x}\right) \, dx","Int[x*ArcCot[E^(a + b*x)],x]","-\frac{i \text{PolyLog}\left(3,-i e^{-a-b x}\right)}{2 b^2}+\frac{i \text{PolyLog}\left(3,i e^{-a-b x}\right)}{2 b^2}-\frac{i x \text{PolyLog}\left(2,-i e^{-a-b x}\right)}{2 b}+\frac{i x \text{PolyLog}\left(2,i e^{-a-b x}\right)}{2 b}","-\frac{i \text{PolyLog}\left(3,-i e^{-a-b x}\right)}{2 b^2}+\frac{i \text{PolyLog}\left(3,i e^{-a-b x}\right)}{2 b^2}-\frac{i x \text{PolyLog}\left(2,-i e^{-a-b x}\right)}{2 b}+\frac{i x \text{PolyLog}\left(2,i e^{-a-b x}\right)}{2 b}",1,"((-I/2)*x*PolyLog[2, (-I)*E^(-a - b*x)])/b + ((I/2)*x*PolyLog[2, I*E^(-a - b*x)])/b - ((I/2)*PolyLog[3, (-I)*E^(-a - b*x)])/b^2 + ((I/2)*PolyLog[3, I*E^(-a - b*x)])/b^2","A",7,4,10,0.4000,1,"{5144, 2531, 2282, 6589}"
223,1,151,0,0.0977789,"\int x^2 \cot ^{-1}\left(e^{a+b x}\right) \, dx","Int[x^2*ArcCot[E^(a + b*x)],x]","-\frac{i x \text{PolyLog}\left(3,-i e^{-a-b x}\right)}{b^2}+\frac{i x \text{PolyLog}\left(3,i e^{-a-b x}\right)}{b^2}-\frac{i \text{PolyLog}\left(4,-i e^{-a-b x}\right)}{b^3}+\frac{i \text{PolyLog}\left(4,i e^{-a-b x}\right)}{b^3}-\frac{i x^2 \text{PolyLog}\left(2,-i e^{-a-b x}\right)}{2 b}+\frac{i x^2 \text{PolyLog}\left(2,i e^{-a-b x}\right)}{2 b}","-\frac{i x \text{PolyLog}\left(3,-i e^{-a-b x}\right)}{b^2}+\frac{i x \text{PolyLog}\left(3,i e^{-a-b x}\right)}{b^2}-\frac{i \text{PolyLog}\left(4,-i e^{-a-b x}\right)}{b^3}+\frac{i \text{PolyLog}\left(4,i e^{-a-b x}\right)}{b^3}-\frac{i x^2 \text{PolyLog}\left(2,-i e^{-a-b x}\right)}{2 b}+\frac{i x^2 \text{PolyLog}\left(2,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",9,5,12,0.4167,1,"{5144, 2531, 6609, 2282, 6589}"
224,1,196,0,0.1540109,"\int \cot ^{-1}\left(a+b f^{c+d x}\right) \, dx","Int[ArcCot[a + b*f^(c + d*x)],x]","-\frac{i \text{PolyLog}\left(2,1-\frac{2}{1-i \left(a+b f^{c+d x}\right)}\right)}{2 d \log (f)}+\frac{i \text{PolyLog}\left(2,1-\frac{2 b f^{c+d x}}{(-a+i) \left(1-i \left(a+b f^{c+d x}\right)\right)}\right)}{2 d \log (f)}-\frac{\log \left(\frac{2}{1-i \left(a+b f^{c+d x}\right)}\right) \cot ^{-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) \cot ^{-1}\left(a+b f^{c+d x}\right)}{d \log (f)}","-\frac{i \text{PolyLog}\left(2,1-\frac{2}{1-i \left(a+b f^{c+d x}\right)}\right)}{2 d \log (f)}+\frac{i \text{PolyLog}\left(2,1-\frac{2 b f^{c+d x}}{(-a+i) \left(1-i \left(a+b f^{c+d x}\right)\right)}\right)}{2 d \log (f)}-\frac{\log \left(\frac{2}{1-i \left(a+b f^{c+d x}\right)}\right) \cot ^{-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) \cot ^{-1}\left(a+b f^{c+d x}\right)}{d \log (f)}",1,"-((ArcCot[a + b*f^(c + d*x)]*Log[2/(1 - I*(a + b*f^(c + d*x)))])/(d*Log[f])) + (ArcCot[a + b*f^(c + d*x)]*Log[(2*b*f^(c + d*x))/((I - a)*(1 - I*(a + b*f^(c + d*x))))])/(d*Log[f]) - ((I/2)*PolyLog[2, 1 - 2/(1 - I*(a + b*f^(c + d*x)))])/(d*Log[f]) + ((I/2)*PolyLog[2, 1 - (2*b*f^(c + d*x))/((I - a)*(1 - I*(a + b*f^(c + d*x))))])/(d*Log[f])","A",6,6,12,0.5000,1,"{2282, 5048, 4857, 2402, 2315, 2447}"
225,1,250,0,2.6537525,"\int x \cot ^{-1}\left(a+b f^{c+d x}\right) \, dx","Int[x*ArcCot[a + b*f^(c + d*x)],x]","\frac{i \text{PolyLog}\left(3,\frac{b f^{c+d x}}{-a+i}\right)}{2 d^2 \log ^2(f)}-\frac{i \text{PolyLog}\left(3,-\frac{b f^{c+d x}}{a+i}\right)}{2 d^2 \log ^2(f)}-\frac{i x \text{PolyLog}\left(2,\frac{b f^{c+d x}}{-a+i}\right)}{2 d \log (f)}+\frac{i x \text{PolyLog}\left(2,-\frac{b f^{c+d x}}{a+i}\right)}{2 d \log (f)}-\frac{1}{4} i x^2 \log \left(1-\frac{b f^{c+d x}}{-a+i}\right)+\frac{1}{4} i x^2 \log \left(1+\frac{b f^{c+d x}}{a+i}\right)+\frac{1}{4} i x^2 \log \left(1-\frac{i}{a+b f^{c+d x}}\right)-\frac{1}{4} i x^2 \log \left(1+\frac{i}{a+b f^{c+d x}}\right)","\frac{i \text{PolyLog}\left(3,\frac{b f^{c+d x}}{-a+i}\right)}{2 d^2 \log ^2(f)}-\frac{i \text{PolyLog}\left(3,-\frac{b f^{c+d x}}{a+i}\right)}{2 d^2 \log ^2(f)}-\frac{i x \text{PolyLog}\left(2,\frac{b f^{c+d x}}{-a+i}\right)}{2 d \log (f)}+\frac{i x \text{PolyLog}\left(2,-\frac{b f^{c+d x}}{a+i}\right)}{2 d \log (f)}-\frac{1}{4} i x^2 \log \left(1-\frac{b f^{c+d x}}{-a+i}\right)+\frac{1}{4} i x^2 \log \left(1+\frac{b f^{c+d x}}{a+i}\right)+\frac{1}{4} i x^2 \log \left(1-\frac{i}{a+b f^{c+d x}}\right)-\frac{1}{4} i x^2 \log \left(1+\frac{i}{a+b f^{c+d x}}\right)",1,"(-I/4)*x^2*Log[1 - (b*f^(c + d*x))/(I - a)] + (I/4)*x^2*Log[1 + (b*f^(c + d*x))/(I + a)] + (I/4)*x^2*Log[1 - I/(a + b*f^(c + d*x))] - (I/4)*x^2*Log[1 + I/(a + b*f^(c + d*x))] - ((I/2)*x*PolyLog[2, (b*f^(c + d*x))/(I - a)])/(d*Log[f]) + ((I/2)*x*PolyLog[2, -((b*f^(c + d*x))/(I + a))])/(d*Log[f]) + ((I/2)*PolyLog[3, (b*f^(c + d*x))/(I - a)])/(d^2*Log[f]^2) - ((I/2)*PolyLog[3, -((b*f^(c + d*x))/(I + a))])/(d^2*Log[f]^2)","A",25,8,14,0.5714,1,"{5144, 2551, 12, 6742, 2190, 2531, 2282, 6589}"
226,1,313,0,2.4483279,"\int x^2 \cot ^{-1}\left(a+b f^{c+d x}\right) \, dx","Int[x^2*ArcCot[a + b*f^(c + d*x)],x]","\frac{i x \text{PolyLog}\left(3,\frac{b f^{c+d x}}{-a+i}\right)}{d^2 \log ^2(f)}-\frac{i x \text{PolyLog}\left(3,-\frac{b f^{c+d x}}{a+i}\right)}{d^2 \log ^2(f)}-\frac{i \text{PolyLog}\left(4,\frac{b f^{c+d x}}{-a+i}\right)}{d^3 \log ^3(f)}+\frac{i \text{PolyLog}\left(4,-\frac{b f^{c+d x}}{a+i}\right)}{d^3 \log ^3(f)}-\frac{i x^2 \text{PolyLog}\left(2,\frac{b f^{c+d x}}{-a+i}\right)}{2 d \log (f)}+\frac{i x^2 \text{PolyLog}\left(2,-\frac{b f^{c+d x}}{a+i}\right)}{2 d \log (f)}-\frac{1}{6} i x^3 \log \left(1-\frac{b f^{c+d x}}{-a+i}\right)+\frac{1}{6} i x^3 \log \left(1+\frac{b f^{c+d x}}{a+i}\right)+\frac{1}{6} i x^3 \log \left(1-\frac{i}{a+b f^{c+d x}}\right)-\frac{1}{6} i x^3 \log \left(1+\frac{i}{a+b f^{c+d x}}\right)","\frac{i x \text{PolyLog}\left(3,\frac{b f^{c+d x}}{-a+i}\right)}{d^2 \log ^2(f)}-\frac{i x \text{PolyLog}\left(3,-\frac{b f^{c+d x}}{a+i}\right)}{d^2 \log ^2(f)}-\frac{i \text{PolyLog}\left(4,\frac{b f^{c+d x}}{-a+i}\right)}{d^3 \log ^3(f)}+\frac{i \text{PolyLog}\left(4,-\frac{b f^{c+d x}}{a+i}\right)}{d^3 \log ^3(f)}-\frac{i x^2 \text{PolyLog}\left(2,\frac{b f^{c+d x}}{-a+i}\right)}{2 d \log (f)}+\frac{i x^2 \text{PolyLog}\left(2,-\frac{b f^{c+d x}}{a+i}\right)}{2 d \log (f)}-\frac{1}{6} i x^3 \log \left(1-\frac{b f^{c+d x}}{-a+i}\right)+\frac{1}{6} i x^3 \log \left(1+\frac{b f^{c+d x}}{a+i}\right)+\frac{1}{6} i x^3 \log \left(1-\frac{i}{a+b f^{c+d x}}\right)-\frac{1}{6} i x^3 \log \left(1+\frac{i}{a+b f^{c+d x}}\right)",1,"(-I/6)*x^3*Log[1 - (b*f^(c + d*x))/(I - a)] + (I/6)*x^3*Log[1 + (b*f^(c + d*x))/(I + a)] + (I/6)*x^3*Log[1 - I/(a + b*f^(c + d*x))] - (I/6)*x^3*Log[1 + I/(a + b*f^(c + d*x))] - ((I/2)*x^2*PolyLog[2, (b*f^(c + d*x))/(I - a)])/(d*Log[f]) + ((I/2)*x^2*PolyLog[2, -((b*f^(c + d*x))/(I + a))])/(d*Log[f]) + (I*x*PolyLog[3, (b*f^(c + d*x))/(I - a)])/(d^2*Log[f]^2) - (I*x*PolyLog[3, -((b*f^(c + d*x))/(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",29,9,16,0.5625,1,"{5144, 2551, 12, 6742, 2190, 2531, 6609, 2282, 6589}"
227,1,27,0,0.0203293,"\int e^{-x} \cot ^{-1}\left(e^x\right) \, dx","Int[ArcCot[E^x]/E^x,x]","-x+\frac{1}{2} \log \left(e^{2 x}+1\right)-e^{-x} \cot ^{-1}\left(e^x\right)","-x+\frac{1}{2} \log \left(e^{2 x}+1\right)-e^{-x} \cot ^{-1}\left(e^x\right)",1,"-x - ArcCot[E^x]/E^x + Log[1 + E^(2*x)]/2","A",5,6,10,0.6000,1,"{2194, 5208, 2282, 36, 29, 31}"
228,1,17,0,0.04,"\int \frac{1}{\left(a+a x^2\right) \left(b-2 b \cot ^{-1}(x)\right)} \, dx","Int[1/((a + a*x^2)*(b - 2*b*ArcCot[x])),x]","\frac{\log \left(1-2 \cot ^{-1}(x)\right)}{2 a b}","\frac{\log \left(1-2 \cot ^{-1}(x)\right)}{2 a b}",1,"Log[1 - 2*ArcCot[x]]/(2*a*b)","A",1,1,19,0.05263,1,"{4883}"
229,1,47,0,0.0744283,"\int e^{c (a+b x)} \cot ^{-1}(\sinh (a c+b c x)) \, dx","Int[E^(c*(a + b*x))*ArcCot[Sinh[a*c + b*c*x]],x]","\frac{\log \left(e^{2 c (a+b x)}+1\right)}{b c}+\frac{e^{a c+b c x} \cot ^{-1}(\sinh (c (a+b x)))}{b c}","\frac{\log \left(e^{2 c (a+b x)}+1\right)}{b c}+\frac{e^{a c+b c x} \cot ^{-1}(\sinh (c (a+b x)))}{b c}",1,"(E^(a*c + b*c*x)*ArcCot[Sinh[c*(a + b*x)]])/(b*c) + Log[1 + E^(2*c*(a + b*x))]/(b*c)","A",5,5,20,0.2500,1,"{2194, 5208, 2282, 12, 260}"
230,1,103,0,0.1390052,"\int e^{c (a+b x)} \cot ^{-1}(\cosh (a c+b c x)) \, dx","Int[E^(c*(a + b*x))*ArcCot[Cosh[a*c + b*c*x]],x]","\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} \cot ^{-1}(\cosh (c (a+b x)))}{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} \cot ^{-1}(\cosh (c (a+b x)))}{b c}",1,"(E^(a*c + b*c*x)*ArcCot[Cosh[c*(a + b*x)]])/(b*c) + ((1 - Sqrt[2])*Log[3 - 2*Sqrt[2] + E^(2*c*(a + b*x))])/(2*b*c) + ((1 + Sqrt[2])*Log[3 + 2*Sqrt[2] + E^(2*c*(a + b*x))])/(2*b*c)","A",8,7,20,0.3500,1,"{2194, 5208, 2282, 12, 1247, 632, 31}"
231,1,180,0,0.1775838,"\int e^{c (a+b x)} \cot ^{-1}(\tanh (a c+b c x)) \, dx","Int[E^(c*(a + b*x))*ArcCot[Tanh[a*c + b*c*x]],x]","\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} \cot ^{-1}(\tanh (c (a+b x)))}{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} \cot ^{-1}(\tanh (c (a+b x)))}{b c}",1,"(E^(a*c + b*c*x)*ArcCot[Tanh[c*(a + b*x)]])/(b*c) - ArcTan[1 - Sqrt[2]*E^(a*c + b*c*x)]/(Sqrt[2]*b*c) + ArcTan[1 + Sqrt[2]*E^(a*c + b*c*x)]/(Sqrt[2]*b*c) + Log[1 + E^(2*c*(a + b*x)) - Sqrt[2]*E^(a*c + b*c*x)]/(2*Sqrt[2]*b*c) - Log[1 + E^(2*c*(a + b*x)) + Sqrt[2]*E^(a*c + b*c*x)]/(2*Sqrt[2]*b*c)","A",13,10,20,0.5000,1,"{2194, 5208, 12, 2249, 297, 1162, 617, 204, 1165, 628}"
232,1,180,0,0.1800841,"\int e^{c (a+b x)} \cot ^{-1}(\coth (a c+b c x)) \, dx","Int[E^(c*(a + b*x))*ArcCot[Coth[a*c + b*c*x]],x]","-\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} \cot ^{-1}(\coth (c (a+b x)))}{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} \cot ^{-1}(\coth (c (a+b x)))}{b c}",1,"(E^(a*c + b*c*x)*ArcCot[Coth[c*(a + b*x)]])/(b*c) + ArcTan[1 - Sqrt[2]*E^(a*c + b*c*x)]/(Sqrt[2]*b*c) - ArcTan[1 + Sqrt[2]*E^(a*c + b*c*x)]/(Sqrt[2]*b*c) - Log[1 + E^(2*c*(a + b*x)) - Sqrt[2]*E^(a*c + b*c*x)]/(2*Sqrt[2]*b*c) + Log[1 + E^(2*c*(a + b*x)) + Sqrt[2]*E^(a*c + b*c*x)]/(2*Sqrt[2]*b*c)","A",13,10,20,0.5000,1,"{2194, 5208, 12, 2249, 297, 1162, 617, 204, 1165, 628}"
233,1,103,0,0.1509557,"\int e^{c (a+b x)} \cot ^{-1}(\text{sech}(a c+b c x)) \, dx","Int[E^(c*(a + b*x))*ArcCot[Sech[a*c + b*c*x]],x]","-\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} \cot ^{-1}(\text{sech}(c (a+b x)))}{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} \cot ^{-1}(\text{sech}(c (a+b x)))}{b c}",1,"(E^(a*c + b*c*x)*ArcCot[Sech[c*(a + b*x)]])/(b*c) - ((1 - Sqrt[2])*Log[3 - 2*Sqrt[2] + E^(2*c*(a + b*x))])/(2*b*c) - ((1 + Sqrt[2])*Log[3 + 2*Sqrt[2] + E^(2*c*(a + b*x))])/(2*b*c)","A",8,7,20,0.3500,1,"{2194, 5208, 2282, 12, 1247, 632, 31}"
234,1,48,0,0.0773401,"\int e^{c (a+b x)} \cot ^{-1}(\text{csch}(a c+b c x)) \, dx","Int[E^(c*(a + b*x))*ArcCot[Csch[a*c + b*c*x]],x]","\frac{e^{a c+b c x} \cot ^{-1}(\text{csch}(c (a+b x)))}{b c}-\frac{\log \left(e^{2 c (a+b x)}+1\right)}{b c}","\frac{e^{a c+b c x} \cot ^{-1}(\text{csch}(c (a+b x)))}{b c}-\frac{\log \left(e^{2 c (a+b x)}+1\right)}{b c}",1,"(E^(a*c + b*c*x)*ArcCot[Csch[c*(a + b*x)]])/(b*c) - Log[1 + E^(2*c*(a + b*x))]/(b*c)","A",5,5,20,0.2500,1,"{2194, 5208, 2282, 12, 260}"