1,1,64,59,0.0034089,"\int x^5 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^5*(a + b*ArcTan[c*x]),x]","\frac{a x^6}{6}+\frac{b \tan ^{-1}(c x)}{6 c^6}-\frac{b x}{6 c^5}+\frac{b x^3}{18 c^3}+\frac{1}{6} b x^6 \tan ^{-1}(c x)-\frac{b x^5}{30 c}","\frac{1}{6} x^6 \left(a+b \tan ^{-1}(c x)\right)+\frac{b \tan ^{-1}(c x)}{6 c^6}-\frac{b x}{6 c^5}+\frac{b x^3}{18 c^3}-\frac{b x^5}{30 c}",1,"-1/6*(b*x)/c^5 + (b*x^3)/(18*c^3) - (b*x^5)/(30*c) + (a*x^6)/6 + (b*ArcTan[c*x])/(6*c^6) + (b*x^6*ArcTan[c*x])/6","A",1
2,1,61,56,0.0135263,"\int x^4 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^4*(a + b*ArcTan[c*x]),x]","\frac{a x^5}{5}+\frac{b x^2}{10 c^3}-\frac{b \log \left(c^2 x^2+1\right)}{10 c^5}+\frac{1}{5} b x^5 \tan ^{-1}(c x)-\frac{b x^4}{20 c}","\frac{1}{5} x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{b x^2}{10 c^3}-\frac{b \log \left(c^2 x^2+1\right)}{10 c^5}-\frac{b x^4}{20 c}",1,"(b*x^2)/(10*c^3) - (b*x^4)/(20*c) + (a*x^5)/5 + (b*x^5*ArcTan[c*x])/5 - (b*Log[1 + c^2*x^2])/(10*c^5)","A",1
3,1,53,48,0.002762,"\int x^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^3*(a + b*ArcTan[c*x]),x]","\frac{a x^4}{4}-\frac{b \tan ^{-1}(c x)}{4 c^4}+\frac{b x}{4 c^3}+\frac{1}{4} b x^4 \tan ^{-1}(c x)-\frac{b x^3}{12 c}","\frac{1}{4} x^4 \left(a+b \tan ^{-1}(c x)\right)-\frac{b \tan ^{-1}(c x)}{4 c^4}+\frac{b x}{4 c^3}-\frac{b x^3}{12 c}",1,"(b*x)/(4*c^3) - (b*x^3)/(12*c) + (a*x^4)/4 - (b*ArcTan[c*x])/(4*c^4) + (b*x^4*ArcTan[c*x])/4","A",1
4,1,50,45,0.0090779,"\int x^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^2*(a + b*ArcTan[c*x]),x]","\frac{a x^3}{3}+\frac{b \log \left(c^2 x^2+1\right)}{6 c^3}+\frac{1}{3} b x^3 \tan ^{-1}(c x)-\frac{b x^2}{6 c}","\frac{1}{3} x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{b \log \left(c^2 x^2+1\right)}{6 c^3}-\frac{b x^2}{6 c}",1,"-1/6*(b*x^2)/c + (a*x^3)/3 + (b*x^3*ArcTan[c*x])/3 + (b*Log[1 + c^2*x^2])/(6*c^3)","A",1
5,1,42,37,0.0024158,"\int x \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x*(a + b*ArcTan[c*x]),x]","\frac{a x^2}{2}+\frac{b \tan ^{-1}(c x)}{2 c^2}+\frac{1}{2} b x^2 \tan ^{-1}(c x)-\frac{b x}{2 c}","\frac{1}{2} x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{b \tan ^{-1}(c x)}{2 c^2}-\frac{b x}{2 c}",1,"-1/2*(b*x)/c + (a*x^2)/2 + (b*ArcTan[c*x])/(2*c^2) + (b*x^2*ArcTan[c*x])/2","A",1
6,1,29,29,0.0030305,"\int \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[a + b*ArcTan[c*x],x]","a x-\frac{b \log \left(c^2 x^2+1\right)}{2 c}+b x \tan ^{-1}(c x)","a x-\frac{b \log \left(c^2 x^2+1\right)}{2 c}+b x \tan ^{-1}(c x)",1,"a*x + b*x*ArcTan[c*x] - (b*Log[1 + c^2*x^2])/(2*c)","A",1
7,1,35,35,0.0025689,"\int \frac{a+b \tan ^{-1}(c x)}{x} \, dx","Integrate[(a + b*ArcTan[c*x])/x,x]","a \log (x)+\frac{1}{2} i b \text{Li}_2(-i c x)-\frac{1}{2} i b \text{Li}_2(i c x)","a \log (x)+\frac{1}{2} i b \text{Li}_2(-i c x)-\frac{1}{2} i b \text{Li}_2(i c x)",1,"a*Log[x] + (I/2)*b*PolyLog[2, (-I)*c*x] - (I/2)*b*PolyLog[2, I*c*x]","A",1
8,1,38,35,0.0027705,"\int \frac{a+b \tan ^{-1}(c x)}{x^2} \, dx","Integrate[(a + b*ArcTan[c*x])/x^2,x]","-\frac{a}{x}-\frac{1}{2} b c \log \left(c^2 x^2+1\right)+b c \log (x)-\frac{b \tan ^{-1}(c x)}{x}","-\frac{a+b \tan ^{-1}(c x)}{x}-\frac{1}{2} b c \log \left(c^2 x^2+1\right)+b c \log (x)",1,"-(a/x) - (b*ArcTan[c*x])/x + b*c*Log[x] - (b*c*Log[1 + c^2*x^2])/2","A",1
9,1,46,37,0.0030102,"\int \frac{a+b \tan ^{-1}(c x)}{x^3} \, dx","Integrate[(a + b*ArcTan[c*x])/x^3,x]","-\frac{a}{2 x^2}-\frac{b c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)}{2 x}-\frac{b \tan ^{-1}(c x)}{2 x^2}","-\frac{a+b \tan ^{-1}(c x)}{2 x^2}-\frac{1}{2} b c^2 \tan ^{-1}(c x)-\frac{b c}{2 x}",1,"-1/2*a/x^2 - (b*ArcTan[c*x])/(2*x^2) - (b*c*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/(2*x)","C",1
10,1,54,53,0.0158275,"\int \frac{a+b \tan ^{-1}(c x)}{x^4} \, dx","Integrate[(a + b*ArcTan[c*x])/x^4,x]","-\frac{a}{3 x^3}+\frac{1}{6} b c \left(c^2 \log \left(c^2 x^2+1\right)-2 c^2 \log (x)-\frac{1}{x^2}\right)-\frac{b \tan ^{-1}(c x)}{3 x^3}","-\frac{a+b \tan ^{-1}(c x)}{3 x^3}-\frac{1}{3} b c^3 \log (x)+\frac{1}{6} b c^3 \log \left(c^2 x^2+1\right)-\frac{b c}{6 x^2}",1,"-1/3*a/x^3 - (b*ArcTan[c*x])/(3*x^3) + (b*c*(-x^(-2) - 2*c^2*Log[x] + c^2*Log[1 + c^2*x^2]))/6","A",1
11,1,46,48,0.0031939,"\int \frac{a+b \tan ^{-1}(c x)}{x^5} \, dx","Integrate[(a + b*ArcTan[c*x])/x^5,x]","-\frac{a}{4 x^4}-\frac{b c \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)}{12 x^3}-\frac{b \tan ^{-1}(c x)}{4 x^4}","-\frac{a+b \tan ^{-1}(c x)}{4 x^4}+\frac{1}{4} b c^4 \tan ^{-1}(c x)+\frac{b c^3}{4 x}-\frac{b c}{12 x^3}",1,"-1/4*a/x^4 - (b*ArcTan[c*x])/(4*x^4) - (b*c*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)])/(12*x^3)","C",1
12,1,64,64,0.0176721,"\int \frac{a+b \tan ^{-1}(c x)}{x^6} \, dx","Integrate[(a + b*ArcTan[c*x])/x^6,x]","-\frac{a}{5 x^5}+\frac{1}{10} b c \left(2 c^4 \log (x)+\frac{c^2}{x^2}-c^4 \log \left(c^2 x^2+1\right)-\frac{1}{2 x^4}\right)-\frac{b \tan ^{-1}(c x)}{5 x^5}","-\frac{a+b \tan ^{-1}(c x)}{5 x^5}+\frac{1}{5} b c^5 \log (x)+\frac{b c^3}{10 x^2}-\frac{1}{10} b c^5 \log \left(c^2 x^2+1\right)-\frac{b c}{20 x^4}",1,"-1/5*a/x^5 - (b*ArcTan[c*x])/(5*x^5) + (b*c*(-1/2*1/x^4 + c^2/x^2 + 2*c^4*Log[x] - c^4*Log[1 + c^2*x^2]))/10","A",1
13,1,138,144,0.1394739,"\int x^5 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x^5*(a + b*ArcTan[c*x])^2,x]","\frac{c x \left(30 a^2 c^5 x^5-4 a b \left(3 c^4 x^4-5 c^2 x^2+15\right)+b^2 c x \left(3 c^2 x^2-16\right)\right)+4 b \tan ^{-1}(c x) \left(15 a \left(c^6 x^6+1\right)+b c x \left(-3 c^4 x^4+5 c^2 x^2-15\right)\right)+30 b^2 \left(c^6 x^6+1\right) \tan ^{-1}(c x)^2+46 b^2 \log \left(c^2 x^2+1\right)}{180 c^6}","\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{6 c^6}-\frac{a b x}{3 c^5}+\frac{b x^3 \left(a+b \tan ^{-1}(c x)\right)}{9 c^3}+\frac{1}{6} x^6 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b x^5 \left(a+b \tan ^{-1}(c x)\right)}{15 c}-\frac{b^2 x \tan ^{-1}(c x)}{3 c^5}-\frac{4 b^2 x^2}{45 c^4}+\frac{b^2 x^4}{60 c^2}+\frac{23 b^2 \log \left(c^2 x^2+1\right)}{90 c^6}",1,"(c*x*(30*a^2*c^5*x^5 + b^2*c*x*(-16 + 3*c^2*x^2) - 4*a*b*(15 - 5*c^2*x^2 + 3*c^4*x^4)) + 4*b*(b*c*x*(-15 + 5*c^2*x^2 - 3*c^4*x^4) + 15*a*(1 + c^6*x^6))*ArcTan[c*x] + 30*b^2*(1 + c^6*x^6)*ArcTan[c*x]^2 + 46*b^2*Log[1 + c^2*x^2])/(180*c^6)","A",1
14,1,169,170,0.5138643,"\int x^4 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x^4*(a + b*ArcTan[c*x])^2,x]","\frac{6 a^2 c^5 x^5-3 a b c^4 x^4+6 a b c^2 x^2-6 a b \log \left(c^2 x^2+1\right)-3 b \tan ^{-1}(c x) \left(-4 a c^5 x^5+b \left(c^4 x^4-2 c^2 x^2-3\right)-4 b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+9 a b+6 b^2 \left(c^5 x^5-i\right) \tan ^{-1}(c x)^2+b^2 c^3 x^3-6 i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-9 b^2 c x}{30 c^5}","\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{5 c^5}+\frac{2 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{5 c^5}+\frac{b x^2 \left(a+b \tan ^{-1}(c x)\right)}{5 c^3}+\frac{1}{5} x^5 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b x^4 \left(a+b \tan ^{-1}(c x)\right)}{10 c}+\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{5 c^5}+\frac{3 b^2 \tan ^{-1}(c x)}{10 c^5}-\frac{3 b^2 x}{10 c^4}+\frac{b^2 x^3}{30 c^2}",1,"(9*a*b - 9*b^2*c*x + 6*a*b*c^2*x^2 + b^2*c^3*x^3 - 3*a*b*c^4*x^4 + 6*a^2*c^5*x^5 + 6*b^2*(-I + c^5*x^5)*ArcTan[c*x]^2 - 3*b*ArcTan[c*x]*(-4*a*c^5*x^5 + b*(-3 - 2*c^2*x^2 + c^4*x^4) - 4*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - 6*a*b*Log[1 + c^2*x^2] - (6*I)*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(30*c^5)","A",0
15,1,111,112,0.0847077,"\int x^3 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(a + b*ArcTan[c*x])^2,x]","\frac{c x \left(3 a^2 c^3 x^3-2 a b c^2 x^2+6 a b+b^2 c x\right)-2 b \tan ^{-1}(c x) \left(a \left(3-3 c^4 x^4\right)+b c x \left(c^2 x^2-3\right)\right)+3 b^2 \left(c^4 x^4-1\right) \tan ^{-1}(c x)^2-4 b^2 \log \left(c^2 x^2+1\right)}{12 c^4}","-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4}+\frac{a b x}{2 c^3}+\frac{1}{4} x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b x^3 \left(a+b \tan ^{-1}(c x)\right)}{6 c}+\frac{b^2 x \tan ^{-1}(c x)}{2 c^3}+\frac{b^2 x^2}{12 c^2}-\frac{b^2 \log \left(c^2 x^2+1\right)}{3 c^4}",1,"(c*x*(6*a*b + b^2*c*x - 2*a*b*c^2*x^2 + 3*a^2*c^3*x^3) - 2*b*(b*c*x*(-3 + c^2*x^2) + a*(3 - 3*c^4*x^4))*ArcTan[c*x] + 3*b^2*(-1 + c^4*x^4)*ArcTan[c*x]^2 - 4*b^2*Log[1 + c^2*x^2])/(12*c^4)","A",1
16,1,131,138,0.2947653,"\int x^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(a + b*ArcTan[c*x])^2,x]","\frac{a^2 c^3 x^3-a b c^2 x^2+a b \log \left(c^2 x^2+1\right)-b \tan ^{-1}(c x) \left(-2 a c^3 x^3+b c^2 x^2+2 b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+b\right)+b^2 \left(c^3 x^3+i\right) \tan ^{-1}(c x)^2+i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+b^2 c x}{3 c^3}","-\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^3}-\frac{2 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}+\frac{1}{3} x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c}-\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{3 c^3}-\frac{b^2 \tan ^{-1}(c x)}{3 c^3}+\frac{b^2 x}{3 c^2}",1,"(b^2*c*x - a*b*c^2*x^2 + a^2*c^3*x^3 + b^2*(I + c^3*x^3)*ArcTan[c*x]^2 - b*ArcTan[c*x]*(b + b*c^2*x^2 - 2*a*c^3*x^3 + 2*b*Log[1 + E^((2*I)*ArcTan[c*x])]) + a*b*Log[1 + c^2*x^2] + I*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(3*c^3)","A",0
17,1,75,76,0.0687655,"\int x \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x*(a + b*ArcTan[c*x])^2,x]","\frac{2 b \tan ^{-1}(c x) \left(a c^2 x^2+a-b c x\right)+a c x (a c x-2 b)+b^2 \log \left(c^2 x^2+1\right)+b^2 \left(c^2 x^2+1\right) \tan ^{-1}(c x)^2}{2 c^2}","\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2}+\frac{1}{2} x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{a b x}{c}+\frac{b^2 \log \left(c^2 x^2+1\right)}{2 c^2}-\frac{b^2 x \tan ^{-1}(c x)}{c}",1,"(a*c*x*(-2*b + a*c*x) + 2*b*(a - b*c*x + a*c^2*x^2)*ArcTan[c*x] + b^2*(1 + c^2*x^2)*ArcTan[c*x]^2 + b^2*Log[1 + c^2*x^2])/(2*c^2)","A",1
18,1,90,83,0.0810455,"\int \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[(a + b*ArcTan[c*x])^2,x]","\frac{a \left(a c x-b \log \left(c^2 x^2+1\right)\right)+2 b \tan ^{-1}(c x) \left(a c x+b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)-i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+b^2 (c x-i) \tan ^{-1}(c x)^2}{c}","x \left(a+b \tan ^{-1}(c x)\right)^2+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{c}+\frac{2 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c}",1,"(b^2*(-I + c*x)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(a*c*x + b*Log[1 + E^((2*I)*ArcTan[c*x])]) + a*(a*c*x - b*Log[1 + c^2*x^2]) - I*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/c","A",0
19,1,144,132,0.0780415,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x} \, dx","Integrate[(a + b*ArcTan[c*x])^2/x,x]","\frac{1}{2} b \left(2 i \text{Li}_2\left(\frac{c x+i}{i-c x}\right) \left(a+b \tan ^{-1}(c x)\right)-2 i \text{Li}_2\left(\frac{c x+i}{c x-i}\right) \left(a+b \tan ^{-1}(c x)\right)+b \left(\text{Li}_3\left(\frac{c x+i}{i-c x}\right)-\text{Li}_3\left(\frac{c x+i}{c x-i}\right)\right)\right)+2 \tanh ^{-1}\left(\frac{c x+i}{c x-i}\right) \left(a+b \tan ^{-1}(c x)\right)^2","-i b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)+i b \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)+2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{2} b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)+\frac{1}{2} b^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\right)",1,"2*(a + b*ArcTan[c*x])^2*ArcTanh[(I + c*x)/(-I + c*x)] + (b*((2*I)*(a + b*ArcTan[c*x])*PolyLog[2, (I + c*x)/(I - c*x)] - (2*I)*(a + b*ArcTan[c*x])*PolyLog[2, (I + c*x)/(-I + c*x)] + b*(PolyLog[3, (I + c*x)/(I - c*x)] - PolyLog[3, (I + c*x)/(-I + c*x)])))/2","A",1
20,1,102,82,0.1490599,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[(a + b*ArcTan[c*x])^2/x^2,x]","\frac{-a \left(a+b c x \log \left(c^2 x^2+1\right)-2 b c x \log (c x)\right)+2 b \tan ^{-1}(c x) \left(-a+b c x \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)\right)-i b^2 c x \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+b^2 (-1-i c x) \tan ^{-1}(c x)^2}{x}","-i c \left(a+b \tan ^{-1}(c x)\right)^2-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x}+2 b c \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-i b^2 c \text{Li}_2\left(\frac{2}{1-i c x}-1\right)",1,"(b^2*(-1 - I*c*x)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(-a + b*c*x*Log[1 - E^((2*I)*ArcTan[c*x])]) - a*(a - 2*b*c*x*Log[c*x] + b*c*x*Log[1 + c^2*x^2]) - I*b^2*c*x*PolyLog[2, E^((2*I)*ArcTan[c*x])])/x","A",0
21,1,90,79,0.0663536,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[(a + b*ArcTan[c*x])^2/x^3,x]","-\frac{a^2+2 b \tan ^{-1}(c x) \left(a c^2 x^2+a+b c x\right)+2 a b c x-2 b^2 c^2 x^2 \log (x)+b^2 c^2 x^2 \log \left(c^2 x^2+1\right)+b^2 \left(c^2 x^2+1\right) \tan ^{-1}(c x)^2}{2 x^2}","-\frac{1}{2} c^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{1}{2} b^2 c^2 \log \left(c^2 x^2+1\right)+b^2 c^2 \log (x)",1,"-1/2*(a^2 + 2*a*b*c*x + 2*b*(a + b*c*x + a*c^2*x^2)*ArcTan[c*x] + b^2*(1 + c^2*x^2)*ArcTan[c*x]^2 - 2*b^2*c^2*x^2*Log[x] + b^2*c^2*x^2*Log[1 + c^2*x^2])/x^2","A",1
22,1,153,140,0.3894815,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[(a + b*ArcTan[c*x])^2/x^4,x]","-\frac{a^2+2 a b c^3 x^3 \log (c x)+b \tan ^{-1}(c x) \left(2 a+b c^3 x^3+2 b c^3 x^3 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+b c x\right)-a b c^3 x^3 \log \left(c^2 x^2+1\right)+a b c x-i b^2 c^3 x^3 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+b^2 \left(1-i c^3 x^3\right) \tan ^{-1}(c x)^2+b^2 c^2 x^2}{3 x^3}","\frac{1}{3} i c^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{2}{3} b c^3 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{3 x^3}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{3 x^2}+\frac{1}{3} i b^2 c^3 \text{Li}_2\left(\frac{2}{1-i c x}-1\right)-\frac{1}{3} b^2 c^3 \tan ^{-1}(c x)-\frac{b^2 c^2}{3 x}",1,"-1/3*(a^2 + a*b*c*x + b^2*c^2*x^2 + b^2*(1 - I*c^3*x^3)*ArcTan[c*x]^2 + b*ArcTan[c*x]*(2*a + b*c*x + b*c^3*x^3 + 2*b*c^3*x^3*Log[1 - E^((2*I)*ArcTan[c*x])]) + 2*a*b*c^3*x^3*Log[c*x] - a*b*c^3*x^3*Log[1 + c^2*x^2] - I*b^2*c^3*x^3*PolyLog[2, E^((2*I)*ArcTan[c*x])])/x^3","A",0
23,1,128,116,0.092948,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^5} \, dx","Integrate[(a + b*ArcTan[c*x])^2/x^5,x]","\frac{-3 a^2+6 a b c^3 x^3+2 b \tan ^{-1}(c x) \left(3 a \left(c^4 x^4-1\right)+b c x \left(3 c^2 x^2-1\right)\right)-2 a b c x-8 b^2 c^4 x^4 \log (x)+3 b^2 \left(c^4 x^4-1\right) \tan ^{-1}(c x)^2-b^2 c^2 x^2+4 b^2 c^4 x^4 \log \left(c^2 x^2+1\right)}{12 x^4}","\frac{1}{4} c^4 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{b c^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 x^4}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{6 x^3}-\frac{2}{3} b^2 c^4 \log (x)-\frac{b^2 c^2}{12 x^2}+\frac{1}{3} b^2 c^4 \log \left(c^2 x^2+1\right)",1,"(-3*a^2 - 2*a*b*c*x - b^2*c^2*x^2 + 6*a*b*c^3*x^3 + 2*b*(b*c*x*(-1 + 3*c^2*x^2) + 3*a*(-1 + c^4*x^4))*ArcTan[c*x] + 3*b^2*(-1 + c^4*x^4)*ArcTan[c*x]^2 - 8*b^2*c^4*x^4*Log[x] + 4*b^2*c^4*x^4*Log[1 + c^2*x^2])/(12*x^4)","A",1
24,1,291,255,0.8287662,"\int x^5 \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Integrate[x^5*(a + b*ArcTan[c*x])^3,x]","\frac{10 a^3 c^6 x^6+b \tan ^{-1}(c x) \left(30 a^2 \left(c^6 x^6+1\right)-4 a b c x \left(3 c^4 x^4-5 c^2 x^2+15\right)+b^2 \left(3 c^4 x^4-16 c^2 x^2-19\right)-92 b^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)-6 a^2 b c^5 x^5+10 a^2 b c^3 x^3-30 a^2 b c x+3 a b^2 c^4 x^4-16 a b^2 c^2 x^2+46 a b^2 \log \left(c^2 x^2+1\right)+2 b^2 \tan ^{-1}(c x)^2 \left(15 a \left(c^6 x^6+1\right)+b \left(-3 c^5 x^5+5 c^3 x^3-15 c x+23 i\right)\right)-19 a b^2+10 b^3 \left(c^6 x^6+1\right) \tan ^{-1}(c x)^3-b^3 c^3 x^3+46 i b^3 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+19 b^3 c x}{60 c^6}","-\frac{23 b^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{15 c^6}-\frac{4 b^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{15 c^4}+\frac{b^2 x^4 \left(a+b \tan ^{-1}(c x)\right)}{20 c^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^3}{6 c^6}-\frac{23 i b \left(a+b \tan ^{-1}(c x)\right)^2}{30 c^6}-\frac{b x \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^5}+\frac{b x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{6 c^3}+\frac{1}{6} x^6 \left(a+b \tan ^{-1}(c x)\right)^3-\frac{b x^5 \left(a+b \tan ^{-1}(c x)\right)^2}{10 c}-\frac{23 i b^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{30 c^6}-\frac{19 b^3 \tan ^{-1}(c x)}{60 c^6}+\frac{19 b^3 x}{60 c^5}-\frac{b^3 x^3}{60 c^3}",1,"(-19*a*b^2 - 30*a^2*b*c*x + 19*b^3*c*x - 16*a*b^2*c^2*x^2 + 10*a^2*b*c^3*x^3 - b^3*c^3*x^3 + 3*a*b^2*c^4*x^4 - 6*a^2*b*c^5*x^5 + 10*a^3*c^6*x^6 + 2*b^2*(b*(23*I - 15*c*x + 5*c^3*x^3 - 3*c^5*x^5) + 15*a*(1 + c^6*x^6))*ArcTan[c*x]^2 + 10*b^3*(1 + c^6*x^6)*ArcTan[c*x]^3 + b*ArcTan[c*x]*(b^2*(-19 - 16*c^2*x^2 + 3*c^4*x^4) - 4*a*b*c*x*(15 - 5*c^2*x^2 + 3*c^4*x^4) + 30*a^2*(1 + c^6*x^6) - 92*b^2*Log[1 + E^((2*I)*ArcTan[c*x])]) + 46*a*b^2*Log[1 + c^2*x^2] + (46*I)*b^3*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(60*c^6)","A",0
25,1,396,271,0.899545,"\int x^4 \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Integrate[x^4*(a + b*ArcTan[c*x])^3,x]","\frac{4 a^3 c^5 x^5+12 a^2 b c^5 x^5 \tan ^{-1}(c x)-3 a^2 b c^4 x^4+6 a^2 b c^2 x^2-6 a^2 b \log \left(c^2 x^2+1\right)+12 a b^2 c^5 x^5 \tan ^{-1}(c x)^2-6 a b^2 c^4 x^4 \tan ^{-1}(c x)+2 a b^2 c^3 x^3+12 a b^2 c^2 x^2 \tan ^{-1}(c x)-12 i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right) \left(a+b \tan ^{-1}(c x)\right)-18 a b^2 c x-12 i a b^2 \tan ^{-1}(c x)^2+18 a b^2 \tan ^{-1}(c x)+24 a b^2 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+4 b^3 c^5 x^5 \tan ^{-1}(c x)^3-3 b^3 c^4 x^4 \tan ^{-1}(c x)^2+2 b^3 c^3 x^3 \tan ^{-1}(c x)-b^3 c^2 x^2+10 b^3 \log \left(c^2 x^2+1\right)+6 b^3 c^2 x^2 \tan ^{-1}(c x)^2+6 b^3 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)-18 b^3 c x \tan ^{-1}(c x)-4 i b^3 \tan ^{-1}(c x)^3+9 b^3 \tan ^{-1}(c x)^2+12 b^3 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-b^3}{20 c^5}","\frac{3 i b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{5 c^5}-\frac{9 a b^2 x}{10 c^4}+\frac{b^2 x^3 \left(a+b \tan ^{-1}(c x)\right)}{10 c^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{5 c^5}+\frac{9 b \left(a+b \tan ^{-1}(c x)\right)^2}{20 c^5}+\frac{3 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{5 c^5}+\frac{3 b x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{10 c^3}+\frac{1}{5} x^5 \left(a+b \tan ^{-1}(c x)\right)^3-\frac{3 b x^4 \left(a+b \tan ^{-1}(c x)\right)^2}{20 c}+\frac{3 b^3 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{10 c^5}-\frac{9 b^3 x \tan ^{-1}(c x)}{10 c^4}-\frac{b^3 x^2}{20 c^3}+\frac{b^3 \log \left(c^2 x^2+1\right)}{2 c^5}",1,"(-b^3 - 18*a*b^2*c*x + 6*a^2*b*c^2*x^2 - b^3*c^2*x^2 + 2*a*b^2*c^3*x^3 - 3*a^2*b*c^4*x^4 + 4*a^3*c^5*x^5 + 18*a*b^2*ArcTan[c*x] - 18*b^3*c*x*ArcTan[c*x] + 12*a*b^2*c^2*x^2*ArcTan[c*x] + 2*b^3*c^3*x^3*ArcTan[c*x] - 6*a*b^2*c^4*x^4*ArcTan[c*x] + 12*a^2*b*c^5*x^5*ArcTan[c*x] - (12*I)*a*b^2*ArcTan[c*x]^2 + 9*b^3*ArcTan[c*x]^2 + 6*b^3*c^2*x^2*ArcTan[c*x]^2 - 3*b^3*c^4*x^4*ArcTan[c*x]^2 + 12*a*b^2*c^5*x^5*ArcTan[c*x]^2 - (4*I)*b^3*ArcTan[c*x]^3 + 4*b^3*c^5*x^5*ArcTan[c*x]^3 + 24*a*b^2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 12*b^3*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - 6*a^2*b*Log[1 + c^2*x^2] + 10*b^3*Log[1 + c^2*x^2] - (12*I)*b^2*(a + b*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 6*b^3*PolyLog[3, -E^((2*I)*ArcTan[c*x])])/(20*c^5)","A",0
26,1,225,194,0.5593328,"\int x^3 \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Integrate[x^3*(a + b*ArcTan[c*x])^3,x]","\frac{a^3 c^4 x^4+b \tan ^{-1}(c x) \left(3 a^2 \left(c^4 x^4-1\right)-2 a b c x \left(c^2 x^2-3\right)+b^2 \left(c^2 x^2+1\right)+8 b^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)-a^2 b c^3 x^3+3 a^2 b c x+a b^2 c^2 x^2-4 a b^2 \log \left(c^2 x^2+1\right)-b^2 \tan ^{-1}(c x)^2 \left(a \left(3-3 c^4 x^4\right)+b \left(c^3 x^3-3 c x+4 i\right)\right)+a b^2+b^3 \left(c^4 x^4-1\right) \tan ^{-1}(c x)^3-4 i b^3 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-b^3 c x}{4 c^4}","\frac{2 b^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4}+\frac{b^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{4 c^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^3}{4 c^4}+\frac{i b \left(a+b \tan ^{-1}(c x)\right)^2}{c^4}+\frac{3 b x \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^3}+\frac{1}{4} x^4 \left(a+b \tan ^{-1}(c x)\right)^3-\frac{b x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{4 c}+\frac{i b^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c^4}+\frac{b^3 \tan ^{-1}(c x)}{4 c^4}-\frac{b^3 x}{4 c^3}",1,"(a*b^2 + 3*a^2*b*c*x - b^3*c*x + a*b^2*c^2*x^2 - a^2*b*c^3*x^3 + a^3*c^4*x^4 - b^2*(b*(4*I - 3*c*x + c^3*x^3) + a*(3 - 3*c^4*x^4))*ArcTan[c*x]^2 + b^3*(-1 + c^4*x^4)*ArcTan[c*x]^3 + b*ArcTan[c*x]*(-2*a*b*c*x*(-3 + c^2*x^2) + b^2*(1 + c^2*x^2) + 3*a^2*(-1 + c^4*x^4) + 8*b^2*Log[1 + E^((2*I)*ArcTan[c*x])]) - 4*a*b^2*Log[1 + c^2*x^2] - (4*I)*b^3*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(4*c^4)","A",0
27,1,269,206,0.6175232,"\int x^2 \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Integrate[x^2*(a + b*ArcTan[c*x])^3,x]","\frac{2 a^3 c^3 x^3+6 a^2 b c^3 x^3 \tan ^{-1}(c x)-3 a^2 b c^2 x^2+3 a^2 b \log \left(c^2 x^2+1\right)+6 a b^2 \left(\left(c^3 x^3+i\right) \tan ^{-1}(c x)^2-\tan ^{-1}(c x) \left(c^2 x^2+2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+1\right)+i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+c x\right)+b^3 \left(2 c^3 x^3 \tan ^{-1}(c x)^3-3 \log \left(c^2 x^2+1\right)-3 c^2 x^2 \tan ^{-1}(c x)^2+6 i \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-3 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+2 i \tan ^{-1}(c x)^3-3 \tan ^{-1}(c x)^2+6 c x \tan ^{-1}(c x)-6 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)}{6 c^3}","-\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3}+\frac{a b^2 x}{c^2}-\frac{b \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{3 c^3}-\frac{b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^3}+\frac{1}{3} x^3 \left(a+b \tan ^{-1}(c x)\right)^3-\frac{b x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c}-\frac{b^3 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 c^3}+\frac{b^3 x \tan ^{-1}(c x)}{c^2}-\frac{b^3 \log \left(c^2 x^2+1\right)}{2 c^3}",1,"(-3*a^2*b*c^2*x^2 + 2*a^3*c^3*x^3 + 6*a^2*b*c^3*x^3*ArcTan[c*x] + 3*a^2*b*Log[1 + c^2*x^2] + 6*a*b^2*(c*x + (I + c^3*x^3)*ArcTan[c*x]^2 - ArcTan[c*x]*(1 + c^2*x^2 + 2*Log[1 + E^((2*I)*ArcTan[c*x])]) + I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + b^3*(6*c*x*ArcTan[c*x] - 3*ArcTan[c*x]^2 - 3*c^2*x^2*ArcTan[c*x]^2 + (2*I)*ArcTan[c*x]^3 + 2*c^3*x^3*ArcTan[c*x]^3 - 6*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - 3*Log[1 + c^2*x^2] + (6*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/(6*c^3)","A",0
28,1,152,131,0.303113,"\int x \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Integrate[x*(a + b*ArcTan[c*x])^3,x]","\frac{a \left(a c x (a c x-3 b)+3 b^2 \log \left(c^2 x^2+1\right)\right)+3 b^2 \tan ^{-1}(c x)^2 \left(a c^2 x^2+a+b (-c x+i)\right)+3 b \tan ^{-1}(c x) \left(a \left(a c^2 x^2+a-2 b c x\right)-2 b^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+b^3 \left(c^2 x^2+1\right) \tan ^{-1}(c x)^3+3 i b^3 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)}{2 c^2}","-\frac{3 b^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2}-\frac{3 i b \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^3}{2 c^2}+\frac{1}{2} x^2 \left(a+b \tan ^{-1}(c x)\right)^3-\frac{3 b x \left(a+b \tan ^{-1}(c x)\right)^2}{2 c}-\frac{3 i b^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{2 c^2}",1,"(3*b^2*(a + a*c^2*x^2 + b*(I - c*x))*ArcTan[c*x]^2 + b^3*(1 + c^2*x^2)*ArcTan[c*x]^3 + 3*b*ArcTan[c*x]*(a*(a - 2*b*c*x + a*c^2*x^2) - 2*b^2*Log[1 + E^((2*I)*ArcTan[c*x])]) + a*(a*c*x*(-3*b + a*c*x) + 3*b^2*Log[1 + c^2*x^2]) + (3*I)*b^3*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(2*c^2)","A",0
29,1,192,119,0.0983331,"\int \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Integrate[(a + b*ArcTan[c*x])^3,x]","a^3 x-\frac{3 a^2 b \log \left(c^2 x^2+1\right)}{2 c}+3 a^2 b x \tan ^{-1}(c x)+\frac{3 a b^2 \left(-i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+c x \tan ^{-1}(c x)^2-i \tan ^{-1}(c x)^2+2 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)}{c}+\frac{b^3 \left(-3 i \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+\frac{3}{2} \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+c x \tan ^{-1}(c x)^3-i \tan ^{-1}(c x)^3+3 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)}{c}","\frac{3 i b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+x \left(a+b \tan ^{-1}(c x)\right)^3+\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{c}+\frac{3 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c}+\frac{3 b^3 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 c}",1,"a^3*x + 3*a^2*b*x*ArcTan[c*x] - (3*a^2*b*Log[1 + c^2*x^2])/(2*c) + (3*a*b^2*((-I)*ArcTan[c*x]^2 + c*x*ArcTan[c*x]^2 + 2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/c + (b^3*((-I)*ArcTan[c*x]^3 + c*x*ArcTan[c*x]^3 + 3*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - (3*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (3*PolyLog[3, -E^((2*I)*ArcTan[c*x])])/2))/c","A",0
30,1,212,206,0.1509005,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{x} \, dx","Integrate[(a + b*ArcTan[c*x])^3/x,x]","\frac{3}{4} i b \left(2 \text{Li}_2\left(\frac{c x+i}{i-c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-2 \text{Li}_2\left(\frac{c x+i}{c x-i}\right) \left(a+b \tan ^{-1}(c x)\right)^2+b \left(-2 i \text{Li}_3\left(\frac{c x+i}{i-c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 i \text{Li}_3\left(\frac{c x+i}{c x-i}\right) \left(a+b \tan ^{-1}(c x)\right)+b \left(\text{Li}_4\left(\frac{c x+i}{c x-i}\right)-\text{Li}_4\left(\frac{c x+i}{i-c x}\right)\right)\right)\right)+2 \tanh ^{-1}\left(\frac{c x+i}{c x-i}\right) \left(a+b \tan ^{-1}(c x)\right)^3","-\frac{3}{2} b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)+\frac{3}{2} b^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{3}{2} i b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)^2+\frac{3}{2} i b \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)^2+2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3+\frac{3}{4} i b^3 \text{Li}_4\left(1-\frac{2}{i c x+1}\right)-\frac{3}{4} i b^3 \text{Li}_4\left(\frac{2}{i c x+1}-1\right)",1,"2*(a + b*ArcTan[c*x])^3*ArcTanh[(I + c*x)/(-I + c*x)] + ((3*I)/4)*b*(2*(a + b*ArcTan[c*x])^2*PolyLog[2, (I + c*x)/(I - c*x)] - 2*(a + b*ArcTan[c*x])^2*PolyLog[2, (I + c*x)/(-I + c*x)] + b*((-2*I)*(a + b*ArcTan[c*x])*PolyLog[3, (I + c*x)/(I - c*x)] + (2*I)*(a + b*ArcTan[c*x])*PolyLog[3, (I + c*x)/(-I + c*x)] + b*(-PolyLog[4, (I + c*x)/(I - c*x)] + PolyLog[4, (I + c*x)/(-I + c*x)])))","A",1
31,1,214,116,0.3994644,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{x^2} \, dx","Integrate[(a + b*ArcTan[c*x])^3/x^2,x]","-\frac{a^3}{x}-\frac{3}{2} a^2 b c \log \left(c^2 x^2+1\right)+3 a^2 b c \log (x)-\frac{3 a^2 b \tan ^{-1}(c x)}{x}+3 a b^2 c \left(-i \left(\tan ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)\right)-\frac{\tan ^{-1}(c x)^2}{c x}+2 \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)\right)+b^3 c \left(3 i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+\frac{3}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-\frac{\tan ^{-1}(c x)^3}{c x}+i \tan ^{-1}(c x)^3+3 \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-\frac{i \pi ^3}{8}\right)","-3 i b^2 c \text{Li}_2\left(\frac{2}{1-i c x}-1\right) \left(a+b \tan ^{-1}(c x)\right)-i c \left(a+b \tan ^{-1}(c x)\right)^3-\frac{\left(a+b \tan ^{-1}(c x)\right)^3}{x}+3 b c \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2+\frac{3}{2} b^3 c \text{Li}_3\left(\frac{2}{1-i c x}-1\right)",1,"-(a^3/x) - (3*a^2*b*ArcTan[c*x])/x + 3*a^2*b*c*Log[x] - (3*a^2*b*c*Log[1 + c^2*x^2])/2 + 3*a*b^2*c*(-(ArcTan[c*x]^2/(c*x)) + 2*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] - I*(ArcTan[c*x]^2 + PolyLog[2, E^((2*I)*ArcTan[c*x])])) + b^3*c*((-1/8*I)*Pi^3 + I*ArcTan[c*x]^3 - ArcTan[c*x]^3/(c*x) + 3*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + (3*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + (3*PolyLog[3, E^((-2*I)*ArcTan[c*x])])/2)","A",0
32,1,176,133,0.3410506,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{x^3} \, dx","Integrate[(a + b*ArcTan[c*x])^3/x^3,x]","-\frac{a \left(a (a+3 b c x)-6 b^2 c^2 x^2 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)\right)+3 b^2 \tan ^{-1}(c x)^2 \left(a c^2 x^2+a+b c x (1+i c x)\right)+3 b \tan ^{-1}(c x) \left(a \left(a c^2 x^2+a+2 b c x\right)-2 b^2 c^2 x^2 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)\right)+3 i b^3 c^2 x^2 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+b^3 \left(c^2 x^2+1\right) \tan ^{-1}(c x)^3}{2 x^2}","3 b^2 c^2 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{3}{2} i b c^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{2} c^2 \left(a+b \tan ^{-1}(c x)\right)^3-\frac{\left(a+b \tan ^{-1}(c x)\right)^3}{2 x^2}-\frac{3 b c \left(a+b \tan ^{-1}(c x)\right)^2}{2 x}-\frac{3}{2} i b^3 c^2 \text{Li}_2\left(\frac{2}{1-i c x}-1\right)",1,"-1/2*(3*b^2*(a + a*c^2*x^2 + b*c*x*(1 + I*c*x))*ArcTan[c*x]^2 + b^3*(1 + c^2*x^2)*ArcTan[c*x]^3 + 3*b*ArcTan[c*x]*(a*(a + 2*b*c*x + a*c^2*x^2) - 2*b^2*c^2*x^2*Log[1 - E^((2*I)*ArcTan[c*x])]) + a*(a*(a + 3*b*c*x) - 6*b^2*c^2*x^2*Log[(c*x)/Sqrt[1 + c^2*x^2]]) + (3*I)*b^3*c^2*x^2*PolyLog[2, E^((2*I)*ArcTan[c*x])])/x^2","A",0
33,1,321,213,0.8807344,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{x^4} \, dx","Integrate[(a + b*ArcTan[c*x])^3/x^4,x]","-\frac{a^3}{3 x^3}-a^2 b c^3 \log (x)+\frac{1}{2} a^2 b c^3 \log \left(c^2 x^2+1\right)-\frac{a^2 b \tan ^{-1}(c x)}{x^3}-\frac{a^2 b c}{2 x^2}+\frac{i a b^2 \left(c^3 x^3 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+\left(c^3 x^3+i\right) \tan ^{-1}(c x)^2+i c^2 x^2+i c x \tan ^{-1}(c x) \left(c^2 x^2+2 c^2 x^2 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+1\right)\right)}{x^3}+\frac{1}{24} b^3 c^3 \left(-\frac{8 \tan ^{-1}(c x)^3}{c^3 x^3}+24 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)-\frac{12 \tan ^{-1}(c x)^2}{c^2 x^2}-24 i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)-12 \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-8 i \tan ^{-1}(c x)^3-12 \tan ^{-1}(c x)^2-\frac{24 \tan ^{-1}(c x)}{c x}-24 \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)+i \pi ^3\right)","i b^2 c^3 \text{Li}_2\left(\frac{2}{1-i c x}-1\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{b^2 c^2 \left(a+b \tan ^{-1}(c x)\right)}{x}+\frac{1}{3} i c^3 \left(a+b \tan ^{-1}(c x)\right)^3-\frac{1}{2} b c^3 \left(a+b \tan ^{-1}(c x)\right)^2-b c^3 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{\left(a+b \tan ^{-1}(c x)\right)^3}{3 x^3}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{1}{2} b^3 c^3 \text{Li}_3\left(\frac{2}{1-i c x}-1\right)+b^3 c^3 \log (x)-\frac{1}{2} b^3 c^3 \log \left(c^2 x^2+1\right)",1,"-1/3*a^3/x^3 - (a^2*b*c)/(2*x^2) - (a^2*b*ArcTan[c*x])/x^3 - a^2*b*c^3*Log[x] + (a^2*b*c^3*Log[1 + c^2*x^2])/2 + (I*a*b^2*(I*c^2*x^2 + (I + c^3*x^3)*ArcTan[c*x]^2 + I*c*x*ArcTan[c*x]*(1 + c^2*x^2 + 2*c^2*x^2*Log[1 - E^((2*I)*ArcTan[c*x])]) + c^3*x^3*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/x^3 + (b^3*c^3*(I*Pi^3 - (24*ArcTan[c*x])/(c*x) - 12*ArcTan[c*x]^2 - (12*ArcTan[c*x]^2)/(c^2*x^2) - (8*I)*ArcTan[c*x]^3 - (8*ArcTan[c*x]^3)/(c^3*x^3) - 24*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + 24*Log[(c*x)/Sqrt[1 + c^2*x^2]] - (24*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - 12*PolyLog[3, E^((-2*I)*ArcTan[c*x])]))/24","A",0
34,1,265,198,0.7243682,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{x^5} \, dx","Integrate[(a + b*ArcTan[c*x])^3/x^5,x]","-\frac{a^3+b \tan ^{-1}(c x) \left(a^2 \left(3-3 c^4 x^4\right)+a b \left(2 c x-6 c^3 x^3\right)+8 b^2 c^4 x^4 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+b^2 c^2 x^2 \left(c^2 x^2+1\right)\right)-3 a^2 b c^3 x^3+a^2 b c x+a b^2 c^4 x^4+a b^2 c^2 x^2+8 a b^2 c^4 x^4 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+b^2 \tan ^{-1}(c x)^2 \left(a \left(3-3 c^4 x^4\right)+b c x \left(-4 i c^3 x^3-3 c^2 x^2+1\right)\right)-4 i b^3 c^4 x^4 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)-b^3 \left(c^4 x^4-1\right) \tan ^{-1}(c x)^3+b^3 c^3 x^3}{4 x^4}","-2 b^2 c^4 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{b^2 c^2 \left(a+b \tan ^{-1}(c x)\right)}{4 x^2}+\frac{1}{4} c^4 \left(a+b \tan ^{-1}(c x)\right)^3+i b c^4 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{3 b c^3 \left(a+b \tan ^{-1}(c x)\right)^2}{4 x}-\frac{\left(a+b \tan ^{-1}(c x)\right)^3}{4 x^4}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)^2}{4 x^3}+i b^3 c^4 \text{Li}_2\left(\frac{2}{1-i c x}-1\right)-\frac{1}{4} b^3 c^4 \tan ^{-1}(c x)-\frac{b^3 c^3}{4 x}",1,"-1/4*(a^3 + a^2*b*c*x + a*b^2*c^2*x^2 - 3*a^2*b*c^3*x^3 + b^3*c^3*x^3 + a*b^2*c^4*x^4 + b^2*(b*c*x*(1 - 3*c^2*x^2 - (4*I)*c^3*x^3) + a*(3 - 3*c^4*x^4))*ArcTan[c*x]^2 - b^3*(-1 + c^4*x^4)*ArcTan[c*x]^3 + b*ArcTan[c*x]*(b^2*c^2*x^2*(1 + c^2*x^2) + a*b*(2*c*x - 6*c^3*x^3) + a^2*(3 - 3*c^4*x^4) + 8*b^2*c^4*x^4*Log[1 - E^((2*I)*ArcTan[c*x])]) + 8*a*b^2*c^4*x^4*Log[(c*x)/Sqrt[1 + c^2*x^2]] - (4*I)*b^3*c^4*x^4*PolyLog[2, E^((2*I)*ArcTan[c*x])])/x^4","A",0
35,0,0,11,0.6517403,"\int \frac{x}{\tan ^{-1}(a x)} \, dx","Integrate[x/ArcTan[a*x],x]","\int \frac{x}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x}{\tan ^{-1}(a x)},x\right)",0,"Integrate[x/ArcTan[a*x], x]","A",-1
36,0,0,9,0.0118079,"\int \frac{1}{\tan ^{-1}(a x)} \, dx","Integrate[ArcTan[a*x]^(-1),x]","\int \frac{1}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{\tan ^{-1}(a x)},x\right)",0,"Integrate[ArcTan[a*x]^(-1), x]","A",-1
37,0,0,13,0.4477577,"\int \frac{1}{x \tan ^{-1}(a x)} \, dx","Integrate[1/(x*ArcTan[a*x]),x]","\int \frac{1}{x \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x \tan ^{-1}(a x)},x\right)",0,"Integrate[1/(x*ArcTan[a*x]), x]","A",-1
38,0,0,11,0.6190273,"\int \frac{x}{\tan ^{-1}(a x)^2} \, dx","Integrate[x/ArcTan[a*x]^2,x]","\int \frac{x}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x}{\tan ^{-1}(a x)^2},x\right)",0,"Integrate[x/ArcTan[a*x]^2, x]","A",-1
39,0,0,9,0.9139254,"\int \frac{1}{\tan ^{-1}(a x)^2} \, dx","Integrate[ArcTan[a*x]^(-2),x]","\int \frac{1}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{1}{\tan ^{-1}(a x)^2},x\right)",0,"Integrate[ArcTan[a*x]^(-2), x]","A",-1
40,0,0,13,1.0626578,"\int \frac{1}{x \tan ^{-1}(a x)^2} \, dx","Integrate[1/(x*ArcTan[a*x]^2),x]","\int \frac{1}{x \tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{1}{x \tan ^{-1}(a x)^2},x\right)",0,"Integrate[1/(x*ArcTan[a*x]^2), x]","A",-1
41,0,0,13,2.1073565,"\int x \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[x*Sqrt[ArcTan[a*x]],x]","\int x \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(x \sqrt{\tan ^{-1}(a x)},x\right)",0,"Integrate[x*Sqrt[ArcTan[a*x]], x]","A",-1
42,0,0,11,1.9275794,"\int \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[Sqrt[ArcTan[a*x]],x]","\int \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\sqrt{\tan ^{-1}(a x)},x\right)",0,"Integrate[Sqrt[ArcTan[a*x]], x]","A",-1
43,0,0,15,1.3494382,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x} \, dx","Integrate[Sqrt[ArcTan[a*x]]/x,x]","\int \frac{\sqrt{\tan ^{-1}(a x)}}{x} \, dx","\text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x},x\right)",0,"Integrate[Sqrt[ArcTan[a*x]]/x, x]","A",-1
44,0,0,13,1.0215555,"\int x \tan ^{-1}(a x)^{3/2} \, dx","Integrate[x*ArcTan[a*x]^(3/2),x]","\int x \tan ^{-1}(a x)^{3/2} \, dx","\text{Int}\left(x \tan ^{-1}(a x)^{3/2},x\right)",0,"Integrate[x*ArcTan[a*x]^(3/2), x]","A",-1
45,0,0,11,2.370962,"\int \tan ^{-1}(a x)^{3/2} \, dx","Integrate[ArcTan[a*x]^(3/2),x]","\int \tan ^{-1}(a x)^{3/2} \, dx","\text{Int}\left(\tan ^{-1}(a x)^{3/2},x\right)",0,"Integrate[ArcTan[a*x]^(3/2), x]","A",-1
46,0,0,15,1.1723146,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x} \, dx","Integrate[ArcTan[a*x]^(3/2)/x,x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{x} \, dx","\text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x},x\right)",0,"Integrate[ArcTan[a*x]^(3/2)/x, x]","A",-1
47,0,0,13,1.3401749,"\int \frac{x}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[x/Sqrt[ArcTan[a*x]],x]","\int \frac{x}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Integrate[x/Sqrt[ArcTan[a*x]], x]","A",-1
48,0,0,11,0.0194255,"\int \frac{1}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[1/Sqrt[ArcTan[a*x]],x]","\int \frac{1}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Integrate[1/Sqrt[ArcTan[a*x]], x]","A",-1
49,0,0,15,1.2176846,"\int \frac{1}{x \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[1/(x*Sqrt[ArcTan[a*x]]),x]","\int \frac{1}{x \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{x \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Integrate[1/(x*Sqrt[ArcTan[a*x]]), x]","A",-1
50,0,0,13,1.5815347,"\int \frac{x}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[x/ArcTan[a*x]^(3/2),x]","\int \frac{x}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Integrate[x/ArcTan[a*x]^(3/2), x]","A",-1
51,0,0,11,2.7170459,"\int \frac{1}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[ArcTan[a*x]^(-3/2),x]","\int \frac{1}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{1}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Integrate[ArcTan[a*x]^(-3/2), x]","A",-1
52,0,0,15,2.9506597,"\int \frac{1}{x \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[1/(x*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x \tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{1}{x \tan ^{-1}(a x)^{3/2}},x\right)",0,"Integrate[1/(x*ArcTan[a*x]^(3/2)), x]","A",-1
53,1,108,117,0.0278103,"\int \sqrt{x} \tan ^{-1}(x) \, dx","Integrate[Sqrt[x]*ArcTan[x],x]","\frac{1}{6} \left(4 x^{3/2} \tan ^{-1}(x)-8 \sqrt{x}-\sqrt{2} \log \left(x-\sqrt{2} \sqrt{x}+1\right)+\sqrt{2} \log \left(x+\sqrt{2} \sqrt{x}+1\right)-2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)+2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)\right)","\frac{2}{3} x^{3/2} \tan ^{-1}(x)-\frac{4 \sqrt{x}}{3}-\frac{\log \left(x-\sqrt{2} \sqrt{x}+1\right)}{3 \sqrt{2}}+\frac{\log \left(x+\sqrt{2} \sqrt{x}+1\right)}{3 \sqrt{2}}-\frac{1}{3} \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)+\frac{1}{3} \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)",1,"(-8*Sqrt[x] - 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]] + 4*x^(3/2)*ArcTan[x] - Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] + Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[x] + x])/6","A",1
54,0,0,19,4.1150221,"\int (d x)^m \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Integrate[(d*x)^m*(a + b*ArcTan[c*x])^3,x]","\int (d x)^m \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","\text{Int}\left((d x)^m \left(a+b \tan ^{-1}(c x)\right)^3,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTan[c*x])^3, x]","A",-1
55,0,0,19,2.6549798,"\int (d x)^m \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[(d*x)^m*(a + b*ArcTan[c*x])^2,x]","\int (d x)^m \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","\text{Int}\left((d x)^m \left(a+b \tan ^{-1}(c x)\right)^2,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTan[c*x])^2, x]","A",-1
56,1,60,73,0.0325681,"\int (d x)^m \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[(d*x)^m*(a + b*ArcTan[c*x]),x]","-\frac{x (d x)^m \left(b c x \, _2F_1\left(1,\frac{m}{2}+1;\frac{m}{2}+2;-c^2 x^2\right)-(m+2) \left(a+b \tan ^{-1}(c x)\right)\right)}{(m+1) (m+2)}","\frac{(d x)^{m+1} \left(a+b \tan ^{-1}(c x)\right)}{d (m+1)}-\frac{b c (d x)^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{d^2 (m+1) (m+2)}",1,"-((x*(d*x)^m*(-((2 + m)*(a + b*ArcTan[c*x])) + b*c*x*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, -(c^2*x^2)]))/((1 + m)*(2 + m)))","A",1
57,0,0,19,0.2799689,"\int \frac{(d x)^m}{a+b \tan ^{-1}(c x)} \, dx","Integrate[(d*x)^m/(a + b*ArcTan[c*x]),x]","\int \frac{(d x)^m}{a+b \tan ^{-1}(c x)} \, dx","\text{Int}\left(\frac{(d x)^m}{a+b \tan ^{-1}(c x)},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcTan[c*x]), x]","A",-1
58,0,0,13,0.4606057,"\int \left(a+b \tan ^{-1}(c x)\right)^p \, dx","Integrate[(a + b*ArcTan[c*x])^p,x]","\int \left(a+b \tan ^{-1}(c x)\right)^p \, dx","\text{Int}\left(\left(a+b \tan ^{-1}(c x)\right)^p,x\right)",0,"Integrate[(a + b*ArcTan[c*x])^p, x]","A",-1
59,0,0,19,0.3600661,"\int (d x)^m \left(a+b \tan ^{-1}(c x)\right)^p \, dx","Integrate[(d*x)^m*(a + b*ArcTan[c*x])^p,x]","\int (d x)^m \left(a+b \tan ^{-1}(c x)\right)^p \, dx","\text{Int}\left((d x)^m \left(a+b \tan ^{-1}(c x)\right)^p,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTan[c*x])^p, x]","A",-1
60,1,59,54,0.0083778,"\int x^7 \left(a+b \tan ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^7*(a + b*ArcTan[c*x^2]),x]","\frac{a x^8}{8}-\frac{b \tan ^{-1}\left(c x^2\right)}{8 c^4}+\frac{b x^2}{8 c^3}-\frac{b x^6}{24 c}+\frac{1}{8} b x^8 \tan ^{-1}\left(c x^2\right)","\frac{1}{8} x^8 \left(a+b \tan ^{-1}\left(c x^2\right)\right)-\frac{b \tan ^{-1}\left(c x^2\right)}{8 c^4}+\frac{b x^2}{8 c^3}-\frac{b x^6}{24 c}",1,"(b*x^2)/(8*c^3) - (b*x^6)/(24*c) + (a*x^8)/8 - (b*ArcTan[c*x^2])/(8*c^4) + (b*x^8*ArcTan[c*x^2])/8","A",1
61,1,52,47,0.0132848,"\int x^5 \left(a+b \tan ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^5*(a + b*ArcTan[c*x^2]),x]","\frac{a x^6}{6}+\frac{b \log \left(c^2 x^4+1\right)}{12 c^3}-\frac{b x^4}{12 c}+\frac{1}{6} b x^6 \tan ^{-1}\left(c x^2\right)","\frac{1}{6} x^6 \left(a+b \tan ^{-1}\left(c x^2\right)\right)+\frac{b \log \left(c^2 x^4+1\right)}{12 c^3}-\frac{b x^4}{12 c}",1,"-1/12*(b*x^4)/c + (a*x^6)/6 + (b*x^6*ArcTan[c*x^2])/6 + (b*Log[1 + c^2*x^4])/(12*c^3)","A",1
62,1,48,43,0.0056019,"\int x^3 \left(a+b \tan ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^3*(a + b*ArcTan[c*x^2]),x]","\frac{a x^4}{4}+\frac{b \tan ^{-1}\left(c x^2\right)}{4 c^2}-\frac{b x^2}{4 c}+\frac{1}{4} b x^4 \tan ^{-1}\left(c x^2\right)","\frac{1}{4} x^4 \left(a+b \tan ^{-1}\left(c x^2\right)\right)+\frac{b \tan ^{-1}\left(c x^2\right)}{4 c^2}-\frac{b x^2}{4 c}",1,"-1/4*(b*x^2)/c + (a*x^4)/4 + (b*ArcTan[c*x^2])/(4*c^2) + (b*x^4*ArcTan[c*x^2])/4","A",1
63,1,41,36,0.0067958,"\int x \left(a+b \tan ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x*(a + b*ArcTan[c*x^2]),x]","\frac{a x^2}{2}-\frac{b \log \left(c^2 x^4+1\right)}{4 c}+\frac{1}{2} b x^2 \tan ^{-1}\left(c x^2\right)","\frac{1}{2} x^2 \left(a+b \tan ^{-1}\left(c x^2\right)\right)-\frac{b \log \left(c^2 x^4+1\right)}{4 c}",1,"(a*x^2)/2 + (b*x^2*ArcTan[c*x^2])/2 - (b*Log[1 + c^2*x^4])/(4*c)","A",1
64,1,39,39,0.0054546,"\int \frac{a+b \tan ^{-1}\left(c x^2\right)}{x} \, dx","Integrate[(a + b*ArcTan[c*x^2])/x,x]","a \log (x)+\frac{1}{4} i b \text{Li}_2\left(-i c x^2\right)-\frac{1}{4} i b \text{Li}_2\left(i c x^2\right)","a \log (x)+\frac{1}{4} i b \text{Li}_2\left(-i c x^2\right)-\frac{1}{4} i b \text{Li}_2\left(i c x^2\right)",1,"a*Log[x] + (I/4)*b*PolyLog[2, (-I)*c*x^2] - (I/4)*b*PolyLog[2, I*c*x^2]","A",1
65,1,44,39,0.0068323,"\int \frac{a+b \tan ^{-1}\left(c x^2\right)}{x^3} \, dx","Integrate[(a + b*ArcTan[c*x^2])/x^3,x]","-\frac{a}{2 x^2}-\frac{1}{4} b c \log \left(c^2 x^4+1\right)-\frac{b \tan ^{-1}\left(c x^2\right)}{2 x^2}+b c \log (x)","-\frac{a+b \tan ^{-1}\left(c x^2\right)}{2 x^2}-\frac{1}{4} b c \log \left(c^2 x^4+1\right)+b c \log (x)",1,"-1/2*a/x^2 - (b*ArcTan[c*x^2])/(2*x^2) + b*c*Log[x] - (b*c*Log[1 + c^2*x^4])/4","A",1
66,1,48,41,0.0068661,"\int \frac{a+b \tan ^{-1}\left(c x^2\right)}{x^5} \, dx","Integrate[(a + b*ArcTan[c*x^2])/x^5,x]","-\frac{a}{4 x^4}-\frac{b c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^4\right)}{4 x^2}-\frac{b \tan ^{-1}\left(c x^2\right)}{4 x^4}","-\frac{a+b \tan ^{-1}\left(c x^2\right)}{4 x^4}-\frac{1}{4} b c^2 \tan ^{-1}\left(c x^2\right)-\frac{b c}{4 x^2}",1,"-1/4*a/x^4 - (b*ArcTan[c*x^2])/(4*x^4) - (b*c*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^4)])/(4*x^2)","C",1
67,1,60,55,0.0132086,"\int \frac{a+b \tan ^{-1}\left(c x^2\right)}{x^7} \, dx","Integrate[(a + b*ArcTan[c*x^2])/x^7,x]","-\frac{a}{6 x^6}-\frac{1}{3} b c^3 \log (x)+\frac{1}{12} b c^3 \log \left(c^2 x^4+1\right)-\frac{b c}{12 x^4}-\frac{b \tan ^{-1}\left(c x^2\right)}{6 x^6}","-\frac{a+b \tan ^{-1}\left(c x^2\right)}{6 x^6}-\frac{1}{3} b c^3 \log (x)+\frac{1}{12} b c^3 \log \left(c^2 x^4+1\right)-\frac{b c}{12 x^4}",1,"-1/6*a/x^6 - (b*c)/(12*x^4) - (b*ArcTan[c*x^2])/(6*x^6) - (b*c^3*Log[x])/3 + (b*c^3*Log[1 + c^2*x^4])/12","A",1
68,1,179,161,0.0637157,"\int x^4 \left(a+b \tan ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^4*(a + b*ArcTan[c*x^2]),x]","\frac{a x^5}{5}+\frac{b \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)}{10 \sqrt{2} c^{5/2}}-\frac{b \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)}{10 \sqrt{2} c^{5/2}}+\frac{b \tan ^{-1}\left(\frac{2 \sqrt{c} x-\sqrt{2}}{\sqrt{2}}\right)}{5 \sqrt{2} c^{5/2}}+\frac{b \tan ^{-1}\left(\frac{2 \sqrt{c} x+\sqrt{2}}{\sqrt{2}}\right)}{5 \sqrt{2} c^{5/2}}-\frac{2 b x^3}{15 c}+\frac{1}{5} b x^5 \tan ^{-1}\left(c x^2\right)","\frac{1}{5} x^5 \left(a+b \tan ^{-1}\left(c x^2\right)\right)+\frac{b \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)}{10 \sqrt{2} c^{5/2}}-\frac{b \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)}{10 \sqrt{2} c^{5/2}}-\frac{b \tan ^{-1}\left(1-\sqrt{2} \sqrt{c} x\right)}{5 \sqrt{2} c^{5/2}}+\frac{b \tan ^{-1}\left(\sqrt{2} \sqrt{c} x+1\right)}{5 \sqrt{2} c^{5/2}}-\frac{2 b x^3}{15 c}",1,"(-2*b*x^3)/(15*c) + (a*x^5)/5 + (b*x^5*ArcTan[c*x^2])/5 + (b*ArcTan[(-Sqrt[2] + 2*Sqrt[c]*x)/Sqrt[2]])/(5*Sqrt[2]*c^(5/2)) + (b*ArcTan[(Sqrt[2] + 2*Sqrt[c]*x)/Sqrt[2]])/(5*Sqrt[2]*c^(5/2)) + (b*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(10*Sqrt[2]*c^(5/2)) - (b*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(10*Sqrt[2]*c^(5/2))","A",1
69,1,177,159,0.0371843,"\int x^2 \left(a+b \tan ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^2*(a + b*ArcTan[c*x^2]),x]","\frac{a x^3}{3}-\frac{b \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)}{6 \sqrt{2} c^{3/2}}+\frac{b \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)}{6 \sqrt{2} c^{3/2}}+\frac{b \tan ^{-1}\left(\frac{2 \sqrt{c} x-\sqrt{2}}{\sqrt{2}}\right)}{3 \sqrt{2} c^{3/2}}+\frac{b \tan ^{-1}\left(\frac{2 \sqrt{c} x+\sqrt{2}}{\sqrt{2}}\right)}{3 \sqrt{2} c^{3/2}}+\frac{1}{3} b x^3 \tan ^{-1}\left(c x^2\right)-\frac{2 b x}{3 c}","\frac{1}{3} x^3 \left(a+b \tan ^{-1}\left(c x^2\right)\right)-\frac{b \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)}{6 \sqrt{2} c^{3/2}}+\frac{b \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)}{6 \sqrt{2} c^{3/2}}-\frac{b \tan ^{-1}\left(1-\sqrt{2} \sqrt{c} x\right)}{3 \sqrt{2} c^{3/2}}+\frac{b \tan ^{-1}\left(\sqrt{2} \sqrt{c} x+1\right)}{3 \sqrt{2} c^{3/2}}-\frac{2 b x}{3 c}",1,"(-2*b*x)/(3*c) + (a*x^3)/3 + (b*x^3*ArcTan[c*x^2])/3 + (b*ArcTan[(-Sqrt[2] + 2*Sqrt[c]*x)/Sqrt[2]])/(3*Sqrt[2]*c^(3/2)) + (b*ArcTan[(Sqrt[2] + 2*Sqrt[c]*x)/Sqrt[2]])/(3*Sqrt[2]*c^(3/2)) - (b*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(6*Sqrt[2]*c^(3/2)) + (b*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(6*Sqrt[2]*c^(3/2))","A",1
70,1,107,140,0.0457324,"\int \left(a+b \tan ^{-1}\left(c x^2\right)\right) \, dx","Integrate[a + b*ArcTan[c*x^2],x]","a x+b x \tan ^{-1}\left(c x^2\right)-\frac{b \left(\log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)-\log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)-2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{c} x\right)+2 \tan ^{-1}\left(\sqrt{2} \sqrt{c} x+1\right)\right)}{2 \sqrt{2} \sqrt{c}}","a x-\frac{b \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)}{2 \sqrt{2} \sqrt{c}}+\frac{b \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)}{2 \sqrt{2} \sqrt{c}}+b x \tan ^{-1}\left(c x^2\right)+\frac{b \tan ^{-1}\left(1-\sqrt{2} \sqrt{c} x\right)}{\sqrt{2} \sqrt{c}}-\frac{b \tan ^{-1}\left(\sqrt{2} \sqrt{c} x+1\right)}{\sqrt{2} \sqrt{c}}",1,"a*x + b*x*ArcTan[c*x^2] - (b*(-2*ArcTan[1 - Sqrt[2]*Sqrt[c]*x] + 2*ArcTan[1 + Sqrt[2]*Sqrt[c]*x] + Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2] - Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2]))/(2*Sqrt[2]*Sqrt[c])","A",1
71,1,158,143,0.0457368,"\int \frac{a+b \tan ^{-1}\left(c x^2\right)}{x^2} \, dx","Integrate[(a + b*ArcTan[c*x^2])/x^2,x]","-\frac{a}{x}-\frac{b \sqrt{c} \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)}{2 \sqrt{2}}+\frac{b \sqrt{c} \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)}{2 \sqrt{2}}-\frac{b \tan ^{-1}\left(c x^2\right)}{x}+\frac{b \sqrt{c} \tan ^{-1}\left(\frac{2 \sqrt{c} x-\sqrt{2}}{\sqrt{2}}\right)}{\sqrt{2}}+\frac{b \sqrt{c} \tan ^{-1}\left(\frac{2 \sqrt{c} x+\sqrt{2}}{\sqrt{2}}\right)}{\sqrt{2}}","-\frac{a+b \tan ^{-1}\left(c x^2\right)}{x}-\frac{b \sqrt{c} \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)}{2 \sqrt{2}}+\frac{b \sqrt{c} \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)}{2 \sqrt{2}}-\frac{b \sqrt{c} \tan ^{-1}\left(1-\sqrt{2} \sqrt{c} x\right)}{\sqrt{2}}+\frac{b \sqrt{c} \tan ^{-1}\left(\sqrt{2} \sqrt{c} x+1\right)}{\sqrt{2}}",1,"-(a/x) - (b*ArcTan[c*x^2])/x + (b*Sqrt[c]*ArcTan[(-Sqrt[2] + 2*Sqrt[c]*x)/Sqrt[2]])/Sqrt[2] + (b*Sqrt[c]*ArcTan[(Sqrt[2] + 2*Sqrt[c]*x)/Sqrt[2]])/Sqrt[2] - (b*Sqrt[c]*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2]) + (b*Sqrt[c]*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2])","A",1
72,1,177,159,0.0570035,"\int \frac{a+b \tan ^{-1}\left(c x^2\right)}{x^4} \, dx","Integrate[(a + b*ArcTan[c*x^2])/x^4,x]","-\frac{a}{3 x^3}-\frac{b c^{3/2} \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)}{6 \sqrt{2}}+\frac{b c^{3/2} \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)}{6 \sqrt{2}}-\frac{b c^{3/2} \tan ^{-1}\left(\frac{2 \sqrt{c} x-\sqrt{2}}{\sqrt{2}}\right)}{3 \sqrt{2}}-\frac{b c^{3/2} \tan ^{-1}\left(\frac{2 \sqrt{c} x+\sqrt{2}}{\sqrt{2}}\right)}{3 \sqrt{2}}-\frac{b \tan ^{-1}\left(c x^2\right)}{3 x^3}-\frac{2 b c}{3 x}","-\frac{a+b \tan ^{-1}\left(c x^2\right)}{3 x^3}-\frac{b c^{3/2} \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)}{6 \sqrt{2}}+\frac{b c^{3/2} \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)}{6 \sqrt{2}}+\frac{b c^{3/2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{c} x\right)}{3 \sqrt{2}}-\frac{b c^{3/2} \tan ^{-1}\left(\sqrt{2} \sqrt{c} x+1\right)}{3 \sqrt{2}}-\frac{2 b c}{3 x}",1,"-1/3*a/x^3 - (2*b*c)/(3*x) - (b*ArcTan[c*x^2])/(3*x^3) - (b*c^(3/2)*ArcTan[(-Sqrt[2] + 2*Sqrt[c]*x)/Sqrt[2]])/(3*Sqrt[2]) - (b*c^(3/2)*ArcTan[(Sqrt[2] + 2*Sqrt[c]*x)/Sqrt[2]])/(3*Sqrt[2]) - (b*c^(3/2)*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(6*Sqrt[2]) + (b*c^(3/2)*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(6*Sqrt[2])","A",1
73,1,177,159,0.0538314,"\int \frac{a+b \tan ^{-1}\left(c x^2\right)}{x^6} \, dx","Integrate[(a + b*ArcTan[c*x^2])/x^6,x]","-\frac{a}{5 x^5}+\frac{b c^{5/2} \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)}{10 \sqrt{2}}-\frac{b c^{5/2} \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)}{10 \sqrt{2}}-\frac{b c^{5/2} \tan ^{-1}\left(\frac{2 \sqrt{c} x-\sqrt{2}}{\sqrt{2}}\right)}{5 \sqrt{2}}-\frac{b c^{5/2} \tan ^{-1}\left(\frac{2 \sqrt{c} x+\sqrt{2}}{\sqrt{2}}\right)}{5 \sqrt{2}}-\frac{2 b c}{15 x^3}-\frac{b \tan ^{-1}\left(c x^2\right)}{5 x^5}","-\frac{a+b \tan ^{-1}\left(c x^2\right)}{5 x^5}+\frac{b c^{5/2} \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)}{10 \sqrt{2}}-\frac{b c^{5/2} \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)}{10 \sqrt{2}}+\frac{b c^{5/2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{c} x\right)}{5 \sqrt{2}}-\frac{b c^{5/2} \tan ^{-1}\left(\sqrt{2} \sqrt{c} x+1\right)}{5 \sqrt{2}}-\frac{2 b c}{15 x^3}",1,"-1/5*a/x^5 - (2*b*c)/(15*x^3) - (b*ArcTan[c*x^2])/(5*x^5) - (b*c^(5/2)*ArcTan[(-Sqrt[2] + 2*Sqrt[c]*x)/Sqrt[2]])/(5*Sqrt[2]) - (b*c^(5/2)*ArcTan[(Sqrt[2] + 2*Sqrt[c]*x)/Sqrt[2]])/(5*Sqrt[2]) + (b*c^(5/2)*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(10*Sqrt[2]) - (b*c^(5/2)*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(10*Sqrt[2])","A",1
74,1,121,124,0.1250041,"\int x^7 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[x^7*(a + b*ArcTan[c*x^2])^2,x]","\frac{c x^2 \left(3 a^2 c^3 x^6-2 a b c^2 x^4+6 a b+b^2 c x^2\right)-2 b \tan ^{-1}\left(c x^2\right) \left(a \left(3-3 c^4 x^8\right)+b c x^2 \left(c^2 x^4-3\right)\right)+3 b^2 \left(c^4 x^8-1\right) \tan ^{-1}\left(c x^2\right)^2-4 b^2 \log \left(c^2 x^4+1\right)}{24 c^4}","-\frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{8 c^4}+\frac{a b x^2}{4 c^3}+\frac{1}{8} x^8 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2-\frac{b x^6 \left(a+b \tan ^{-1}\left(c x^2\right)\right)}{12 c}+\frac{b^2 x^2 \tan ^{-1}\left(c x^2\right)}{4 c^3}+\frac{b^2 x^4}{24 c^2}-\frac{b^2 \log \left(c^2 x^4+1\right)}{6 c^4}",1,"(c*x^2*(6*a*b + b^2*c*x^2 - 2*a*b*c^2*x^4 + 3*a^2*c^3*x^6) - 2*b*(b*c*x^2*(-3 + c^2*x^4) + a*(3 - 3*c^4*x^8))*ArcTan[c*x^2] + 3*b^2*(-1 + c^4*x^8)*ArcTan[c*x^2]^2 - 4*b^2*Log[1 + c^2*x^4])/(24*c^4)","A",1
75,1,141,154,0.3349085,"\int x^5 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[x^5*(a + b*ArcTan[c*x^2])^2,x]","\frac{a^2 c^3 x^6-a b c^2 x^4+a b \log \left(c^2 x^4+1\right)-b \tan ^{-1}\left(c x^2\right) \left(-2 a c^3 x^6+b c^2 x^4+2 b \log \left(1+e^{2 i \tan ^{-1}\left(c x^2\right)}\right)+b\right)+b^2 \left(c^3 x^6+i\right) \tan ^{-1}\left(c x^2\right)^2+i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}\left(c x^2\right)}\right)+b^2 c x^2}{6 c^3}","-\frac{i \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{6 c^3}-\frac{b \log \left(\frac{2}{1+i c x^2}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)}{3 c^3}+\frac{1}{6} x^6 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2-\frac{b x^4 \left(a+b \tan ^{-1}\left(c x^2\right)\right)}{6 c}-\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i c x^2+1}\right)}{6 c^3}-\frac{b^2 \tan ^{-1}\left(c x^2\right)}{6 c^3}+\frac{b^2 x^2}{6 c^2}",1,"(b^2*c*x^2 - a*b*c^2*x^4 + a^2*c^3*x^6 + b^2*(I + c^3*x^6)*ArcTan[c*x^2]^2 - b*ArcTan[c*x^2]*(b + b*c^2*x^4 - 2*a*c^3*x^6 + 2*b*Log[1 + E^((2*I)*ArcTan[c*x^2])]) + a*b*Log[1 + c^2*x^4] + I*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x^2])])/(6*c^3)","A",0
76,1,85,90,0.0700778,"\int x^3 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[x^3*(a + b*ArcTan[c*x^2])^2,x]","\frac{2 b \tan ^{-1}\left(c x^2\right) \left(a c^2 x^4+a-b c x^2\right)+a c x^2 \left(a c x^2-2 b\right)+b^2 \log \left(c^2 x^4+1\right)+b^2 \left(c^2 x^4+1\right) \tan ^{-1}\left(c x^2\right)^2}{4 c^2}","\frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{4 c^2}-\frac{a b x^2}{2 c}+\frac{1}{4} x^4 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2+\frac{b^2 \log \left(c^2 x^4+1\right)}{4 c^2}-\frac{b^2 x^2 \tan ^{-1}\left(c x^2\right)}{2 c}",1,"(a*c*x^2*(-2*b + a*c*x^2) + 2*b*(a - b*c*x^2 + a*c^2*x^4)*ArcTan[c*x^2] + b^2*(1 + c^2*x^4)*ArcTan[c*x^2]^2 + b^2*Log[1 + c^2*x^4])/(4*c^2)","A",1
77,1,107,101,0.0741235,"\int x \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[x*(a + b*ArcTan[c*x^2])^2,x]","\frac{a \left(a c x^2-b \log \left(c^2 x^4+1\right)\right)+2 b \tan ^{-1}\left(c x^2\right) \left(a c x^2+b \log \left(1+e^{2 i \tan ^{-1}\left(c x^2\right)}\right)\right)-i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}\left(c x^2\right)}\right)+b^2 \left(c x^2-i\right) \tan ^{-1}\left(c x^2\right)^2}{2 c}","\frac{1}{2} x^2 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2+\frac{i \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{2 c}+\frac{b \log \left(\frac{2}{1+i c x^2}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)}{c}+\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i c x^2+1}\right)}{2 c}",1,"(b^2*(-I + c*x^2)*ArcTan[c*x^2]^2 + 2*b*ArcTan[c*x^2]*(a*c*x^2 + b*Log[1 + E^((2*I)*ArcTan[c*x^2])]) + a*(a*c*x^2 - b*Log[1 + c^2*x^4]) - I*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x^2])])/(2*c)","A",0
78,1,165,151,0.1115764,"\int \frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{x} \, dx","Integrate[(a + b*ArcTan[c*x^2])^2/x,x]","\frac{1}{4} b \left(2 i \text{Li}_2\left(\frac{c x^2+i}{i-c x^2}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)-2 i \text{Li}_2\left(\frac{c x^2+i}{c x^2-i}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)+b \left(\text{Li}_3\left(\frac{c x^2+i}{i-c x^2}\right)-\text{Li}_3\left(\frac{c x^2+i}{c x^2-i}\right)\right)\right)+\tanh ^{-1}\left(1+\frac{2 i}{c x^2-i}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2","-\frac{1}{2} i b \text{Li}_2\left(1-\frac{2}{i c x^2+1}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)+\frac{1}{2} i b \text{Li}_2\left(\frac{2}{i c x^2+1}-1\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)+\tanh ^{-1}\left(1-\frac{2}{1+i c x^2}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2-\frac{1}{4} b^2 \text{Li}_3\left(1-\frac{2}{i c x^2+1}\right)+\frac{1}{4} b^2 \text{Li}_3\left(\frac{2}{i c x^2+1}-1\right)",1,"(a + b*ArcTan[c*x^2])^2*ArcTanh[1 + (2*I)/(-I + c*x^2)] + (b*((2*I)*(a + b*ArcTan[c*x^2])*PolyLog[2, (I + c*x^2)/(I - c*x^2)] - (2*I)*(a + b*ArcTan[c*x^2])*PolyLog[2, (I + c*x^2)/(-I + c*x^2)] + b*(PolyLog[3, (I + c*x^2)/(I - c*x^2)] - PolyLog[3, (I + c*x^2)/(-I + c*x^2)])))/4","A",1
79,1,127,97,0.15704,"\int \frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{x^3} \, dx","Integrate[(a + b*ArcTan[c*x^2])^2/x^3,x]","-\frac{a^2}{2 x^2}+a b c \left(-\frac{1}{2} \log \left(c^2 x^4+1\right)+\log \left(c x^2\right)-\frac{\tan ^{-1}\left(c x^2\right)}{c x^2}\right)+\frac{1}{2} b^2 c \left(-i \left(\tan ^{-1}\left(c x^2\right)^2+\text{Li}_2\left(e^{2 i \tan ^{-1}\left(c x^2\right)}\right)\right)-\frac{\tan ^{-1}\left(c x^2\right)^2}{c x^2}+2 \tan ^{-1}\left(c x^2\right) \log \left(1-e^{2 i \tan ^{-1}\left(c x^2\right)}\right)\right)","-\frac{1}{2} i c \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2-\frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{2 x^2}+b c \log \left(2-\frac{2}{1-i c x^2}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)-\frac{1}{2} i b^2 c \text{Li}_2\left(\frac{2}{1-i c x^2}-1\right)",1,"-1/2*a^2/x^2 + a*b*c*(-(ArcTan[c*x^2]/(c*x^2)) + Log[c*x^2] - Log[1 + c^2*x^4]/2) + (b^2*c*(-(ArcTan[c*x^2]^2/(c*x^2)) + 2*ArcTan[c*x^2]*Log[1 - E^((2*I)*ArcTan[c*x^2])] - I*(ArcTan[c*x^2]^2 + PolyLog[2, E^((2*I)*ArcTan[c*x^2])])))/2","A",0
80,1,98,87,0.0828315,"\int \frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{x^5} \, dx","Integrate[(a + b*ArcTan[c*x^2])^2/x^5,x]","-\frac{a^2+2 b \tan ^{-1}\left(c x^2\right) \left(a c^2 x^4+a+b c x^2\right)+2 a b c x^2-4 b^2 c^2 x^4 \log (x)+b^2 c^2 x^4 \log \left(c^2 x^4+1\right)+b^2 \left(c^2 x^4+1\right) \tan ^{-1}\left(c x^2\right)^2}{4 x^4}","-\frac{1}{4} c^2 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2-\frac{b c \left(a+b \tan ^{-1}\left(c x^2\right)\right)}{2 x^2}-\frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{4 x^4}-\frac{1}{4} b^2 c^2 \log \left(c^2 x^4+1\right)+b^2 c^2 \log (x)",1,"-1/4*(a^2 + 2*a*b*c*x^2 + 2*b*(a + b*c*x^2 + a*c^2*x^4)*ArcTan[c*x^2] + b^2*(1 + c^2*x^4)*ArcTan[c*x^2]^2 - 4*b^2*c^2*x^4*Log[x] + b^2*c^2*x^4*Log[1 + c^2*x^4])/x^4","A",1
81,0,0,1393,5.1753927,"\int x^2 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[x^2*(a + b*ArcTan[c*x^2])^2,x]","\int x^2 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2 \, dx","\frac{1}{12} \left(2 a+i b \log \left(1-i c x^2\right)\right)^2 x^3-\frac{1}{12} b^2 \log ^2\left(i c x^2+1\right) x^3+\frac{2}{9} i a b x^3-\frac{1}{9} b^2 \log \left(1-i c x^2\right) x^3-\frac{1}{9} i b \left(2 a+i b \log \left(1-i c x^2\right)\right) x^3-\frac{1}{3} i a b \log \left(i c x^2+1\right) x^3+\frac{1}{6} b^2 \log \left(1-i c x^2\right) \log \left(i c x^2+1\right) x^3-\frac{2 i b^2 \log \left(1-i c x^2\right) x}{3 c}+\frac{2 i b^2 \log \left(i c x^2+1\right) x}{3 c}-\frac{4 a b x}{3 c}+\frac{\sqrt[4]{-1} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right)^2}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right)^2}{3 c^{3/2}}+\frac{4 (-1)^{3/4} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right)}{3 c^{3/2}}-\frac{4 (-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right)}{3 c^{3/2}}-\frac{2 \sqrt[4]{-1} a b \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right)}{3 c^{3/2}}-\frac{2 (-1)^{3/4} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{1-\sqrt[4]{-1} \sqrt{c} x}\right)}{3 c^{3/2}}+\frac{2 (-1)^{3/4} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{\sqrt{2} \left(\sqrt{c} x+\sqrt[4]{-1}\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{3 c^{3/2}}+\frac{2 (-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right)}{3 c^{3/2}}-\frac{2 (-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{(-1)^{3/4} \sqrt{c} x+1}\right)}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(-\frac{\sqrt{2} \left(\sqrt{c} x+(-1)^{3/4}\right)}{(-1)^{3/4} \sqrt{c} x+1}\right)}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{(1+i) \left(\sqrt[4]{-1} \sqrt{c} x+1\right)}{(-1)^{3/4} \sqrt{c} x+1}\right)}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{(1-i) \left((-1)^{3/4} \sqrt{c} x+1\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(1-i c x^2\right)}{3 c^{3/2}}-\frac{\sqrt[4]{-1} b \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \left(2 a+i b \log \left(1-i c x^2\right)\right)}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(i c x^2+1\right)}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(i c x^2+1\right)}{3 c^{3/2}}+\frac{\sqrt[4]{-1} b^2 \text{Li}_2\left(1-\frac{2}{1-\sqrt[4]{-1} \sqrt{c} x}\right)}{3 c^{3/2}}+\frac{\sqrt[4]{-1} b^2 \text{Li}_2\left(1-\frac{2}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{3 c^{3/2}}-\frac{\sqrt[4]{-1} b^2 \text{Li}_2\left(1-\frac{\sqrt{2} \left(\sqrt{c} x+\sqrt[4]{-1}\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{6 c^{3/2}}+\frac{(-1)^{3/4} b^2 \text{Li}_2\left(1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right)}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \text{Li}_2\left(1-\frac{2}{(-1)^{3/4} \sqrt{c} x+1}\right)}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \text{Li}_2\left(\frac{\sqrt{2} \left(\sqrt{c} x+(-1)^{3/4}\right)}{(-1)^{3/4} \sqrt{c} x+1}+1\right)}{6 c^{3/2}}-\frac{(-1)^{3/4} b^2 \text{Li}_2\left(1-\frac{(1+i) \left(\sqrt[4]{-1} \sqrt{c} x+1\right)}{(-1)^{3/4} \sqrt{c} x+1}\right)}{6 c^{3/2}}-\frac{\sqrt[4]{-1} b^2 \text{Li}_2\left(1-\frac{(1-i) \left((-1)^{3/4} \sqrt{c} x+1\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{6 c^{3/2}}",1,"Integrate[x^2*(a + b*ArcTan[c*x^2])^2, x]","F",-1
82,1,5504,1191,31.5395664,"\int \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[(a + b*ArcTan[c*x^2])^2,x]","\text{Result too large to show}","x a^2-\frac{2 (-1)^{3/4} b \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) a}{\sqrt{c}}+\frac{2 (-1)^{3/4} b \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) a}{\sqrt{c}}+i b x \log \left(1-i c x^2\right) a-i b x \log \left(i c x^2+1\right) a+\frac{(-1)^{3/4} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right)^2}{\sqrt{c}}-\frac{\sqrt[4]{-1} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right)^2}{\sqrt{c}}-\frac{1}{4} b^2 x \log ^2\left(1-i c x^2\right)-\frac{1}{4} b^2 x \log ^2\left(i c x^2+1\right)+\frac{2 \sqrt[4]{-1} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{1-\sqrt[4]{-1} \sqrt{c} x}\right)}{\sqrt{c}}-\frac{2 \sqrt[4]{-1} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{\sqrt[4]{-1} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{\sqrt{2} \left(\sqrt{c} x+\sqrt[4]{-1}\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{2 \sqrt[4]{-1} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right)}{\sqrt{c}}-\frac{2 \sqrt[4]{-1} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{(-1)^{3/4} \sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{\sqrt[4]{-1} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(-\frac{\sqrt{2} \left(\sqrt{c} x+(-1)^{3/4}\right)}{(-1)^{3/4} \sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{\sqrt[4]{-1} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{(1+i) \left(\sqrt[4]{-1} \sqrt{c} x+1\right)}{(-1)^{3/4} \sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{\sqrt[4]{-1} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{(1-i) \left((-1)^{3/4} \sqrt{c} x+1\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{\sqrt[4]{-1} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(1-i c x^2\right)}{\sqrt{c}}-\frac{\sqrt[4]{-1} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(1-i c x^2\right)}{\sqrt{c}}-\frac{\sqrt[4]{-1} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(i c x^2+1\right)}{\sqrt{c}}+\frac{\sqrt[4]{-1} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(i c x^2+1\right)}{\sqrt{c}}+\frac{1}{2} b^2 x \log \left(1-i c x^2\right) \log \left(i c x^2+1\right)+\frac{(-1)^{3/4} b^2 \text{Li}_2\left(1-\frac{2}{1-\sqrt[4]{-1} \sqrt{c} x}\right)}{\sqrt{c}}+\frac{(-1)^{3/4} b^2 \text{Li}_2\left(1-\frac{2}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{\sqrt{c}}-\frac{(-1)^{3/4} b^2 \text{Li}_2\left(1-\frac{\sqrt{2} \left(\sqrt{c} x+\sqrt[4]{-1}\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{2 \sqrt{c}}+\frac{\sqrt[4]{-1} b^2 \text{Li}_2\left(1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right)}{\sqrt{c}}+\frac{\sqrt[4]{-1} b^2 \text{Li}_2\left(1-\frac{2}{(-1)^{3/4} \sqrt{c} x+1}\right)}{\sqrt{c}}-\frac{\sqrt[4]{-1} b^2 \text{Li}_2\left(\frac{\sqrt{2} \left(\sqrt{c} x+(-1)^{3/4}\right)}{(-1)^{3/4} \sqrt{c} x+1}+1\right)}{2 \sqrt{c}}-\frac{\sqrt[4]{-1} b^2 \text{Li}_2\left(1-\frac{(1+i) \left(\sqrt[4]{-1} \sqrt{c} x+1\right)}{(-1)^{3/4} \sqrt{c} x+1}\right)}{2 \sqrt{c}}-\frac{(-1)^{3/4} b^2 \text{Li}_2\left(1-\frac{(1-i) \left((-1)^{3/4} \sqrt{c} x+1\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{2 \sqrt{c}}",1,"Result too large to show","C",0
83,1,5535,1164,30.7546413,"\int \frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{x^2} \, dx","Integrate[(a + b*ArcTan[c*x^2])^2/x^2,x]","\text{Result too large to show}","\sqrt[4]{-1} \sqrt{c} \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right)^2 b^2-(-1)^{3/4} \sqrt{c} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right)^2 b^2+\frac{\log ^2\left(i c x^2+1\right) b^2}{4 x}-2 (-1)^{3/4} \sqrt{c} \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{1-\sqrt[4]{-1} \sqrt{c} x}\right) b^2+2 (-1)^{3/4} \sqrt{c} \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{\sqrt[4]{-1} \sqrt{c} x+1}\right) b^2-(-1)^{3/4} \sqrt{c} \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{\sqrt{2} \left(\sqrt{c} x+\sqrt[4]{-1}\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right) b^2+2 (-1)^{3/4} \sqrt{c} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right) b^2-2 (-1)^{3/4} \sqrt{c} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{(-1)^{3/4} \sqrt{c} x+1}\right) b^2+(-1)^{3/4} \sqrt{c} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(-\frac{\sqrt{2} \left(\sqrt{c} x+(-1)^{3/4}\right)}{(-1)^{3/4} \sqrt{c} x+1}\right) b^2+(-1)^{3/4} \sqrt{c} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{(1+i) \left(\sqrt[4]{-1} \sqrt{c} x+1\right)}{(-1)^{3/4} \sqrt{c} x+1}\right) b^2-(-1)^{3/4} \sqrt{c} \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{(1-i) \left((-1)^{3/4} \sqrt{c} x+1\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right) b^2-(-1)^{3/4} \sqrt{c} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(1-i c x^2\right) b^2+(-1)^{3/4} \sqrt{c} \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(i c x^2+1\right) b^2+(-1)^{3/4} \sqrt{c} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(i c x^2+1\right) b^2-\frac{\log \left(1-i c x^2\right) \log \left(i c x^2+1\right) b^2}{2 x}+\sqrt[4]{-1} \sqrt{c} \text{Li}_2\left(1-\frac{2}{1-\sqrt[4]{-1} \sqrt{c} x}\right) b^2+\sqrt[4]{-1} \sqrt{c} \text{Li}_2\left(1-\frac{2}{\sqrt[4]{-1} \sqrt{c} x+1}\right) b^2-\frac{1}{2} \sqrt[4]{-1} \sqrt{c} \text{Li}_2\left(1-\frac{\sqrt{2} \left(\sqrt{c} x+\sqrt[4]{-1}\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right) b^2+(-1)^{3/4} \sqrt{c} \text{Li}_2\left(1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right) b^2+(-1)^{3/4} \sqrt{c} \text{Li}_2\left(1-\frac{2}{(-1)^{3/4} \sqrt{c} x+1}\right) b^2-\frac{1}{2} (-1)^{3/4} \sqrt{c} \text{Li}_2\left(\frac{\sqrt{2} \left(\sqrt{c} x+(-1)^{3/4}\right)}{(-1)^{3/4} \sqrt{c} x+1}+1\right) b^2-\frac{1}{2} (-1)^{3/4} \sqrt{c} \text{Li}_2\left(1-\frac{(1+i) \left(\sqrt[4]{-1} \sqrt{c} x+1\right)}{(-1)^{3/4} \sqrt{c} x+1}\right) b^2-\frac{1}{2} \sqrt[4]{-1} \sqrt{c} \text{Li}_2\left(1-\frac{(1-i) \left((-1)^{3/4} \sqrt{c} x+1\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right) b^2-2 \sqrt[4]{-1} a \sqrt{c} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) b-\sqrt[4]{-1} \sqrt{c} \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \left(2 a+i b \log \left(1-i c x^2\right)\right) b+\frac{i a \log \left(i c x^2+1\right) b}{x}-\frac{\left(2 a+i b \log \left(1-i c x^2\right)\right)^2}{4 x}",1,"Result too large to show","B",0
84,0,0,1360,2.9420309,"\int \frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{x^4} \, dx","Integrate[(a + b*ArcTan[c*x^2])^2/x^4,x]","\int \frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{x^4} \, dx","\frac{1}{3} (-1)^{3/4} c^{3/2} \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right)^2 b^2-\frac{1}{3} \sqrt[4]{-1} c^{3/2} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right)^2 b^2+\frac{\log ^2\left(i c x^2+1\right) b^2}{12 x^3}-\frac{4}{3} \sqrt[4]{-1} c^{3/2} \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) b^2-\frac{4}{3} \sqrt[4]{-1} c^{3/2} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) b^2+\frac{2}{3} \sqrt[4]{-1} c^{3/2} \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{1-\sqrt[4]{-1} \sqrt{c} x}\right) b^2-\frac{2}{3} \sqrt[4]{-1} c^{3/2} \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{\sqrt[4]{-1} \sqrt{c} x+1}\right) b^2+\frac{1}{3} \sqrt[4]{-1} c^{3/2} \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{\sqrt{2} \left(\sqrt{c} x+\sqrt[4]{-1}\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right) b^2+\frac{2}{3} \sqrt[4]{-1} c^{3/2} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right) b^2-\frac{2}{3} \sqrt[4]{-1} c^{3/2} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{(-1)^{3/4} \sqrt{c} x+1}\right) b^2+\frac{1}{3} \sqrt[4]{-1} c^{3/2} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(-\frac{\sqrt{2} \left(\sqrt{c} x+(-1)^{3/4}\right)}{(-1)^{3/4} \sqrt{c} x+1}\right) b^2+\frac{1}{3} \sqrt[4]{-1} c^{3/2} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{(1+i) \left(\sqrt[4]{-1} \sqrt{c} x+1\right)}{(-1)^{3/4} \sqrt{c} x+1}\right) b^2+\frac{1}{3} \sqrt[4]{-1} c^{3/2} \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{(1-i) \left((-1)^{3/4} \sqrt{c} x+1\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right) b^2-\frac{1}{3} \sqrt[4]{-1} c^{3/2} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(1-i c x^2\right) b^2-\frac{i c \log \left(1-i c x^2\right) b^2}{3 x}-\frac{1}{3} \sqrt[4]{-1} c^{3/2} \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(i c x^2+1\right) b^2+\frac{1}{3} \sqrt[4]{-1} c^{3/2} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(i c x^2+1\right) b^2-\frac{\log \left(1-i c x^2\right) \log \left(i c x^2+1\right) b^2}{6 x^3}+\frac{2 i c \log \left(i c x^2+1\right) b^2}{3 x}+\frac{1}{3} (-1)^{3/4} c^{3/2} \text{Li}_2\left(1-\frac{2}{1-\sqrt[4]{-1} \sqrt{c} x}\right) b^2+\frac{1}{3} (-1)^{3/4} c^{3/2} \text{Li}_2\left(1-\frac{2}{\sqrt[4]{-1} \sqrt{c} x+1}\right) b^2-\frac{1}{6} (-1)^{3/4} c^{3/2} \text{Li}_2\left(1-\frac{\sqrt{2} \left(\sqrt{c} x+\sqrt[4]{-1}\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right) b^2+\frac{1}{3} \sqrt[4]{-1} c^{3/2} \text{Li}_2\left(1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right) b^2+\frac{1}{3} \sqrt[4]{-1} c^{3/2} \text{Li}_2\left(1-\frac{2}{(-1)^{3/4} \sqrt{c} x+1}\right) b^2-\frac{1}{6} \sqrt[4]{-1} c^{3/2} \text{Li}_2\left(\frac{\sqrt{2} \left(\sqrt{c} x+(-1)^{3/4}\right)}{(-1)^{3/4} \sqrt{c} x+1}+1\right) b^2-\frac{1}{6} \sqrt[4]{-1} c^{3/2} \text{Li}_2\left(1-\frac{(1+i) \left(\sqrt[4]{-1} \sqrt{c} x+1\right)}{(-1)^{3/4} \sqrt{c} x+1}\right) b^2-\frac{1}{6} (-1)^{3/4} c^{3/2} \text{Li}_2\left(1-\frac{(1-i) \left((-1)^{3/4} \sqrt{c} x+1\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right) b^2+\frac{2}{3} (-1)^{3/4} a c^{3/2} \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) b-\frac{1}{3} (-1)^{3/4} c^{3/2} \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \left(2 a+i b \log \left(1-i c x^2\right)\right) b-\frac{c \left(2 a+i b \log \left(1-i c x^2\right)\right) b}{3 x}+\frac{i a \log \left(i c x^2+1\right) b}{3 x^3}-\frac{2 a c b}{3 x}-\frac{\left(2 a+i b \log \left(1-i c x^2\right)\right)^2}{12 x^3}",1,"Integrate[(a + b*ArcTan[c*x^2])^2/x^4, x]","F",-1
85,0,0,1444,2.8913549,"\int \frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{x^6} \, dx","Integrate[(a + b*ArcTan[c*x^2])^2/x^6,x]","\int \frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{x^6} \, dx","-\frac{1}{5} \sqrt[4]{-1} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right)^2 c^{5/2}+\frac{1}{5} (-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right)^2 c^{5/2}-\frac{4}{15} (-1)^{3/4} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) c^{5/2}+\frac{4}{15} (-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) c^{5/2}+\frac{2}{5} \sqrt[4]{-1} a b \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) c^{5/2}+\frac{2}{5} (-1)^{3/4} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{1-\sqrt[4]{-1} \sqrt{c} x}\right) c^{5/2}-\frac{2}{5} (-1)^{3/4} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{\sqrt[4]{-1} \sqrt{c} x+1}\right) c^{5/2}+\frac{1}{5} (-1)^{3/4} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{\sqrt{2} \left(\sqrt{c} x+\sqrt[4]{-1}\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right) c^{5/2}-\frac{2}{5} (-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right) c^{5/2}+\frac{2}{5} (-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{(-1)^{3/4} \sqrt{c} x+1}\right) c^{5/2}-\frac{1}{5} (-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(-\frac{\sqrt{2} \left(\sqrt{c} x+(-1)^{3/4}\right)}{(-1)^{3/4} \sqrt{c} x+1}\right) c^{5/2}-\frac{1}{5} (-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{(1+i) \left(\sqrt[4]{-1} \sqrt{c} x+1\right)}{(-1)^{3/4} \sqrt{c} x+1}\right) c^{5/2}+\frac{1}{5} (-1)^{3/4} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{(1-i) \left((-1)^{3/4} \sqrt{c} x+1\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right) c^{5/2}+\frac{1}{5} (-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(1-i c x^2\right) c^{5/2}+\frac{1}{5} \sqrt[4]{-1} b \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \left(2 a+i b \log \left(1-i c x^2\right)\right) c^{5/2}-\frac{1}{5} (-1)^{3/4} b^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(i c x^2+1\right) c^{5/2}-\frac{1}{5} (-1)^{3/4} b^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(i c x^2+1\right) c^{5/2}-\frac{1}{5} \sqrt[4]{-1} b^2 \text{Li}_2\left(1-\frac{2}{1-\sqrt[4]{-1} \sqrt{c} x}\right) c^{5/2}-\frac{1}{5} \sqrt[4]{-1} b^2 \text{Li}_2\left(1-\frac{2}{\sqrt[4]{-1} \sqrt{c} x+1}\right) c^{5/2}+\frac{1}{10} \sqrt[4]{-1} b^2 \text{Li}_2\left(1-\frac{\sqrt{2} \left(\sqrt{c} x+\sqrt[4]{-1}\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right) c^{5/2}-\frac{1}{5} (-1)^{3/4} b^2 \text{Li}_2\left(1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right) c^{5/2}-\frac{1}{5} (-1)^{3/4} b^2 \text{Li}_2\left(1-\frac{2}{(-1)^{3/4} \sqrt{c} x+1}\right) c^{5/2}+\frac{1}{10} (-1)^{3/4} b^2 \text{Li}_2\left(\frac{\sqrt{2} \left(\sqrt{c} x+(-1)^{3/4}\right)}{(-1)^{3/4} \sqrt{c} x+1}+1\right) c^{5/2}+\frac{1}{10} (-1)^{3/4} b^2 \text{Li}_2\left(1-\frac{(1+i) \left(\sqrt[4]{-1} \sqrt{c} x+1\right)}{(-1)^{3/4} \sqrt{c} x+1}\right) c^{5/2}+\frac{1}{10} \sqrt[4]{-1} b^2 \text{Li}_2\left(1-\frac{(1-i) \left((-1)^{3/4} \sqrt{c} x+1\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right) c^{5/2}-\frac{b^2 \log \left(1-i c x^2\right) c^2}{5 x}-\frac{i b \left(2 a+i b \log \left(1-i c x^2\right)\right) c^2}{5 x}-\frac{8 b^2 c^2}{15 x}+\frac{2 i a b c^2}{5 x}-\frac{i b^2 \log \left(1-i c x^2\right) c}{15 x^3}-\frac{b \left(2 a+i b \log \left(1-i c x^2\right)\right) c}{15 x^3}+\frac{2 i b^2 \log \left(i c x^2+1\right) c}{15 x^3}-\frac{2 a b c}{15 x^3}-\frac{\left(2 a+i b \log \left(1-i c x^2\right)\right)^2}{20 x^5}+\frac{b^2 \log ^2\left(i c x^2+1\right)}{20 x^5}-\frac{b^2 \log \left(1-i c x^2\right) \log \left(i c x^2+1\right)}{10 x^5}+\frac{i a b \log \left(i c x^2+1\right)}{5 x^5}",1,"Integrate[(a + b*ArcTan[c*x^2])^2/x^6, x]","F",-1
86,1,170,149,0.1719584,"\int x^3 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^3 \, dx","Integrate[x^3*(a + b*ArcTan[c*x^2])^3,x]","\frac{a \left(a c x^2 \left(a c x^2-3 b\right)+3 b^2 \log \left(c^2 x^4+1\right)\right)+3 b^2 \tan ^{-1}\left(c x^2\right)^2 \left(a c^2 x^4+a+b \left(-c x^2+i\right)\right)+3 b \tan ^{-1}\left(c x^2\right) \left(a \left(a c^2 x^4+a-2 b c x^2\right)-2 b^2 \log \left(1+e^{2 i \tan ^{-1}\left(c x^2\right)}\right)\right)+b^3 \left(c^2 x^4+1\right) \tan ^{-1}\left(c x^2\right)^3+3 i b^3 \text{Li}_2\left(-e^{2 i \tan ^{-1}\left(c x^2\right)}\right)}{4 c^2}","-\frac{3 b^2 \log \left(\frac{2}{1+i c x^2}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)}{2 c^2}+\frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^3}{4 c^2}-\frac{3 i b \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{4 c^2}-\frac{3 b x^2 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{4 c}+\frac{1}{4} x^4 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^3-\frac{3 i b^3 \text{Li}_2\left(1-\frac{2}{i c x^2+1}\right)}{4 c^2}",1,"(3*b^2*(a + a*c^2*x^4 + b*(I - c*x^2))*ArcTan[c*x^2]^2 + b^3*(1 + c^2*x^4)*ArcTan[c*x^2]^3 + 3*b*ArcTan[c*x^2]*(a*(a - 2*b*c*x^2 + a*c^2*x^4) - 2*b^2*Log[1 + E^((2*I)*ArcTan[c*x^2])]) + a*(a*c*x^2*(-3*b + a*c*x^2) + 3*b^2*Log[1 + c^2*x^4]) + (3*I)*b^3*PolyLog[2, -E^((2*I)*ArcTan[c*x^2])])/(4*c^2)","A",0
87,1,224,144,0.1259448,"\int x \left(a+b \tan ^{-1}\left(c x^2\right)\right)^3 \, dx","Integrate[x*(a + b*ArcTan[c*x^2])^3,x]","\frac{2 a^3 c x^2-3 a^2 b \log \left(c^2 x^4+1\right)+6 a^2 b c x^2 \tan ^{-1}\left(c x^2\right)-6 i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}\left(c x^2\right)}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)-6 i a b^2 \tan ^{-1}\left(c x^2\right)^2+6 a b^2 c x^2 \tan ^{-1}\left(c x^2\right)^2+12 a b^2 \tan ^{-1}\left(c x^2\right) \log \left(1+e^{2 i \tan ^{-1}\left(c x^2\right)}\right)+3 b^3 \text{Li}_3\left(-e^{2 i \tan ^{-1}\left(c x^2\right)}\right)-2 i b^3 \tan ^{-1}\left(c x^2\right)^3+2 b^3 c x^2 \tan ^{-1}\left(c x^2\right)^3+6 b^3 \tan ^{-1}\left(c x^2\right)^2 \log \left(1+e^{2 i \tan ^{-1}\left(c x^2\right)}\right)}{4 c}","\frac{3 i b^2 \text{Li}_2\left(1-\frac{2}{i c x^2+1}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)}{2 c}+\frac{1}{2} x^2 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^3+\frac{i \left(a+b \tan ^{-1}\left(c x^2\right)\right)^3}{2 c}+\frac{3 b \log \left(\frac{2}{1+i c x^2}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{2 c}+\frac{3 b^3 \text{Li}_3\left(1-\frac{2}{i c x^2+1}\right)}{4 c}",1,"(2*a^3*c*x^2 + 6*a^2*b*c*x^2*ArcTan[c*x^2] - (6*I)*a*b^2*ArcTan[c*x^2]^2 + 6*a*b^2*c*x^2*ArcTan[c*x^2]^2 - (2*I)*b^3*ArcTan[c*x^2]^3 + 2*b^3*c*x^2*ArcTan[c*x^2]^3 + 12*a*b^2*ArcTan[c*x^2]*Log[1 + E^((2*I)*ArcTan[c*x^2])] + 6*b^3*ArcTan[c*x^2]^2*Log[1 + E^((2*I)*ArcTan[c*x^2])] - 3*a^2*b*Log[1 + c^2*x^4] - (6*I)*b^2*(a + b*ArcTan[c*x^2])*PolyLog[2, -E^((2*I)*ArcTan[c*x^2])] + 3*b^3*PolyLog[3, -E^((2*I)*ArcTan[c*x^2])])/(4*c)","A",0
88,1,245,229,0.2073245,"\int \frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^3}{x} \, dx","Integrate[(a + b*ArcTan[c*x^2])^3/x,x]","\frac{3}{8} i b \left(2 \text{Li}_2\left(\frac{c x^2+i}{i-c x^2}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2-2 \text{Li}_2\left(\frac{c x^2+i}{c x^2-i}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2+b \left(-2 i \text{Li}_3\left(\frac{c x^2+i}{i-c x^2}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)+2 i \text{Li}_3\left(\frac{c x^2+i}{c x^2-i}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)+b \left(\text{Li}_4\left(\frac{c x^2+i}{c x^2-i}\right)-\text{Li}_4\left(\frac{c x^2+i}{i-c x^2}\right)\right)\right)\right)+\tanh ^{-1}\left(1+\frac{2 i}{c x^2-i}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)^3","-\frac{3}{4} b^2 \text{Li}_3\left(1-\frac{2}{i c x^2+1}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)+\frac{3}{4} b^2 \text{Li}_3\left(\frac{2}{i c x^2+1}-1\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)-\frac{3}{4} i b \text{Li}_2\left(1-\frac{2}{i c x^2+1}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2+\frac{3}{4} i b \text{Li}_2\left(\frac{2}{i c x^2+1}-1\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2+\tanh ^{-1}\left(1-\frac{2}{1+i c x^2}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)^3+\frac{3}{8} i b^3 \text{Li}_4\left(1-\frac{2}{i c x^2+1}\right)-\frac{3}{8} i b^3 \text{Li}_4\left(\frac{2}{i c x^2+1}-1\right)",1,"(a + b*ArcTan[c*x^2])^3*ArcTanh[1 + (2*I)/(-I + c*x^2)] + ((3*I)/8)*b*(2*(a + b*ArcTan[c*x^2])^2*PolyLog[2, (I + c*x^2)/(I - c*x^2)] - 2*(a + b*ArcTan[c*x^2])^2*PolyLog[2, (I + c*x^2)/(-I + c*x^2)] + b*((-2*I)*(a + b*ArcTan[c*x^2])*PolyLog[3, (I + c*x^2)/(I - c*x^2)] + (2*I)*(a + b*ArcTan[c*x^2])*PolyLog[3, (I + c*x^2)/(-I + c*x^2)] + b*(-PolyLog[4, (I + c*x^2)/(I - c*x^2)] + PolyLog[4, (I + c*x^2)/(-I + c*x^2)])))","A",1
89,1,239,138,0.4606575,"\int \frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^3}{x^3} \, dx","Integrate[(a + b*ArcTan[c*x^2])^3/x^3,x]","\frac{1}{4} \left(-\frac{2 a^3}{x^2}-3 a^2 b c \log \left(c^2 x^4+1\right)-\frac{6 a^2 b \tan ^{-1}\left(c x^2\right)}{x^2}+12 a^2 b c \log (x)+6 a b^2 c \left(\tan ^{-1}\left(c x^2\right) \left(\left(-\frac{1}{c x^2}-i\right) \tan ^{-1}\left(c x^2\right)+2 \log \left(1-e^{2 i \tan ^{-1}\left(c x^2\right)}\right)\right)-i \text{Li}_2\left(e^{2 i \tan ^{-1}\left(c x^2\right)}\right)\right)+2 b^3 c \left(3 i \tan ^{-1}\left(c x^2\right) \text{Li}_2\left(e^{-2 i \tan ^{-1}\left(c x^2\right)}\right)+\frac{3}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}\left(c x^2\right)}\right)-\frac{\tan ^{-1}\left(c x^2\right)^3}{c x^2}+i \tan ^{-1}\left(c x^2\right)^3+3 \tan ^{-1}\left(c x^2\right)^2 \log \left(1-e^{-2 i \tan ^{-1}\left(c x^2\right)}\right)-\frac{i \pi ^3}{8}\right)\right)","-\frac{3}{2} i b^2 c \text{Li}_2\left(\frac{2}{1-i c x^2}-1\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)-\frac{1}{2} i c \left(a+b \tan ^{-1}\left(c x^2\right)\right)^3-\frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^3}{2 x^2}+\frac{3}{2} b c \log \left(2-\frac{2}{1-i c x^2}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2+\frac{3}{4} b^3 c \text{Li}_3\left(\frac{2}{1-i c x^2}-1\right)",1,"((-2*a^3)/x^2 - (6*a^2*b*ArcTan[c*x^2])/x^2 + 12*a^2*b*c*Log[x] - 3*a^2*b*c*Log[1 + c^2*x^4] + 6*a*b^2*c*(ArcTan[c*x^2]*((-I - 1/(c*x^2))*ArcTan[c*x^2] + 2*Log[1 - E^((2*I)*ArcTan[c*x^2])]) - I*PolyLog[2, E^((2*I)*ArcTan[c*x^2])]) + 2*b^3*c*((-1/8*I)*Pi^3 + I*ArcTan[c*x^2]^3 - ArcTan[c*x^2]^3/(c*x^2) + 3*ArcTan[c*x^2]^2*Log[1 - E^((-2*I)*ArcTan[c*x^2])] + (3*I)*ArcTan[c*x^2]*PolyLog[2, E^((-2*I)*ArcTan[c*x^2])] + (3*PolyLog[3, E^((-2*I)*ArcTan[c*x^2])])/2))/4","A",0
90,1,196,149,0.3485255,"\int \frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^3}{x^5} \, dx","Integrate[(a + b*ArcTan[c*x^2])^3/x^5,x]","-\frac{a \left(a \left(a+3 b c x^2\right)-6 b^2 c^2 x^4 \log \left(\frac{c x^2}{\sqrt{c^2 x^4+1}}\right)\right)+3 b^2 \tan ^{-1}\left(c x^2\right)^2 \left(a c^2 x^4+a+b c x^2 \left(1+i c x^2\right)\right)+3 b \tan ^{-1}\left(c x^2\right) \left(a \left(a c^2 x^4+a+2 b c x^2\right)-2 b^2 c^2 x^4 \log \left(1-e^{2 i \tan ^{-1}\left(c x^2\right)}\right)\right)+3 i b^3 c^2 x^4 \text{Li}_2\left(e^{2 i \tan ^{-1}\left(c x^2\right)}\right)+b^3 \left(c^2 x^4+1\right) \tan ^{-1}\left(c x^2\right)^3}{4 x^4}","\frac{3}{2} b^2 c^2 \log \left(2-\frac{2}{1-i c x^2}\right) \left(a+b \tan ^{-1}\left(c x^2\right)\right)-\frac{3}{4} i b c^2 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2-\frac{1}{4} c^2 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^3-\frac{3 b c \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{4 x^2}-\frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^3}{4 x^4}-\frac{3}{4} i b^3 c^2 \text{Li}_2\left(\frac{2}{1-i c x^2}-1\right)",1,"-1/4*(3*b^2*(a + a*c^2*x^4 + b*c*x^2*(1 + I*c*x^2))*ArcTan[c*x^2]^2 + b^3*(1 + c^2*x^4)*ArcTan[c*x^2]^3 + 3*b*ArcTan[c*x^2]*(a*(a + 2*b*c*x^2 + a*c^2*x^4) - 2*b^2*c^2*x^4*Log[1 - E^((2*I)*ArcTan[c*x^2])]) + a*(a*(a + 3*b*c*x^2) - 6*b^2*c^2*x^4*Log[(c*x^2)/Sqrt[1 + c^2*x^4]]) + (3*I)*b^3*c^2*x^4*PolyLog[2, E^((2*I)*ArcTan[c*x^2])])/x^4","A",0
91,0,0,21,1.9624166,"\int (d x)^m \left(a+b \tan ^{-1}\left(c x^2\right)\right)^3 \, dx","Integrate[(d*x)^m*(a + b*ArcTan[c*x^2])^3,x]","\int (d x)^m \left(a+b \tan ^{-1}\left(c x^2\right)\right)^3 \, dx","\text{Int}\left((d x)^m \left(a+b \tan ^{-1}\left(c x^2\right)\right)^3,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTan[c*x^2])^3, x]","A",-1
92,0,0,21,1.2840937,"\int (d x)^m \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[(d*x)^m*(a + b*ArcTan[c*x^2])^2,x]","\int (d x)^m \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2 \, dx","\text{Int}\left((d x)^m \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTan[c*x^2])^2, x]","A",-1
93,1,65,75,0.0656414,"\int (d x)^m \left(a+b \tan ^{-1}\left(c x^2\right)\right) \, dx","Integrate[(d*x)^m*(a + b*ArcTan[c*x^2]),x]","-\frac{x (d x)^m \left(2 b c x^2 \, _2F_1\left(1,\frac{m+3}{4};\frac{m+7}{4};-c^2 x^4\right)-(m+3) \left(a+b \tan ^{-1}\left(c x^2\right)\right)\right)}{(m+1) (m+3)}","\frac{(d x)^{m+1} \left(a+b \tan ^{-1}\left(c x^2\right)\right)}{d (m+1)}-\frac{2 b c (d x)^{m+3} \, _2F_1\left(1,\frac{m+3}{4};\frac{m+7}{4};-c^2 x^4\right)}{d^3 (m+1) (m+3)}",1,"-((x*(d*x)^m*(-((3 + m)*(a + b*ArcTan[c*x^2])) + 2*b*c*x^2*Hypergeometric2F1[1, (3 + m)/4, (7 + m)/4, -(c^2*x^4)]))/((1 + m)*(3 + m)))","A",1
94,0,0,21,0.3348391,"\int \frac{(d x)^m}{a+b \tan ^{-1}\left(c x^2\right)} \, dx","Integrate[(d*x)^m/(a + b*ArcTan[c*x^2]),x]","\int \frac{(d x)^m}{a+b \tan ^{-1}\left(c x^2\right)} \, dx","\text{Int}\left(\frac{(d x)^m}{a+b \tan ^{-1}\left(c x^2\right)},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcTan[c*x^2]), x]","A",-1
95,0,0,21,0.3565736,"\int \frac{(d x)^m}{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2} \, dx","Integrate[(d*x)^m/(a + b*ArcTan[c*x^2])^2,x]","\int \frac{(d x)^m}{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2} \, dx","\text{Int}\left(\frac{(d x)^m}{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcTan[c*x^2])^2, x]","A",-1
96,1,59,54,0.0090201,"\int x^{11} \left(a+b \tan ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x^11*(a + b*ArcTan[c*x^3]),x]","\frac{a x^{12}}{12}-\frac{b \tan ^{-1}\left(c x^3\right)}{12 c^4}+\frac{b x^3}{12 c^3}-\frac{b x^9}{36 c}+\frac{1}{12} b x^{12} \tan ^{-1}\left(c x^3\right)","\frac{1}{12} x^{12} \left(a+b \tan ^{-1}\left(c x^3\right)\right)-\frac{b \tan ^{-1}\left(c x^3\right)}{12 c^4}+\frac{b x^3}{12 c^3}-\frac{b x^9}{36 c}",1,"(b*x^3)/(12*c^3) - (b*x^9)/(36*c) + (a*x^12)/12 - (b*ArcTan[c*x^3])/(12*c^4) + (b*x^12*ArcTan[c*x^3])/12","A",1
97,1,52,47,0.0136814,"\int x^8 \left(a+b \tan ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x^8*(a + b*ArcTan[c*x^3]),x]","\frac{a x^9}{9}+\frac{b \log \left(c^2 x^6+1\right)}{18 c^3}-\frac{b x^6}{18 c}+\frac{1}{9} b x^9 \tan ^{-1}\left(c x^3\right)","\frac{1}{9} x^9 \left(a+b \tan ^{-1}\left(c x^3\right)\right)+\frac{b \log \left(c^2 x^6+1\right)}{18 c^3}-\frac{b x^6}{18 c}",1,"-1/18*(b*x^6)/c + (a*x^9)/9 + (b*x^9*ArcTan[c*x^3])/9 + (b*Log[1 + c^2*x^6])/(18*c^3)","A",1
98,1,48,43,0.0057312,"\int x^5 \left(a+b \tan ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x^5*(a + b*ArcTan[c*x^3]),x]","\frac{a x^6}{6}+\frac{b \tan ^{-1}\left(c x^3\right)}{6 c^2}-\frac{b x^3}{6 c}+\frac{1}{6} b x^6 \tan ^{-1}\left(c x^3\right)","\frac{1}{6} x^6 \left(a+b \tan ^{-1}\left(c x^3\right)\right)+\frac{b \tan ^{-1}\left(c x^3\right)}{6 c^2}-\frac{b x^3}{6 c}",1,"-1/6*(b*x^3)/c + (a*x^6)/6 + (b*ArcTan[c*x^3])/(6*c^2) + (b*x^6*ArcTan[c*x^3])/6","A",1
99,1,41,36,0.0071062,"\int x^2 \left(a+b \tan ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x^2*(a + b*ArcTan[c*x^3]),x]","\frac{a x^3}{3}-\frac{b \log \left(c^2 x^6+1\right)}{6 c}+\frac{1}{3} b x^3 \tan ^{-1}\left(c x^3\right)","\frac{1}{3} x^3 \left(a+b \tan ^{-1}\left(c x^3\right)\right)-\frac{b \log \left(c^2 x^6+1\right)}{6 c}",1,"(a*x^3)/3 + (b*x^3*ArcTan[c*x^3])/3 - (b*Log[1 + c^2*x^6])/(6*c)","A",1
100,1,39,39,0.0050503,"\int \frac{a+b \tan ^{-1}\left(c x^3\right)}{x} \, dx","Integrate[(a + b*ArcTan[c*x^3])/x,x]","a \log (x)+\frac{1}{6} i b \text{Li}_2\left(-i c x^3\right)-\frac{1}{6} i b \text{Li}_2\left(i c x^3\right)","a \log (x)+\frac{1}{6} i b \text{Li}_2\left(-i c x^3\right)-\frac{1}{6} i b \text{Li}_2\left(i c x^3\right)",1,"a*Log[x] + (I/6)*b*PolyLog[2, (-I)*c*x^3] - (I/6)*b*PolyLog[2, I*c*x^3]","A",1
101,1,44,39,0.0070682,"\int \frac{a+b \tan ^{-1}\left(c x^3\right)}{x^4} \, dx","Integrate[(a + b*ArcTan[c*x^3])/x^4,x]","-\frac{a}{3 x^3}-\frac{1}{6} b c \log \left(c^2 x^6+1\right)-\frac{b \tan ^{-1}\left(c x^3\right)}{3 x^3}+b c \log (x)","-\frac{a+b \tan ^{-1}\left(c x^3\right)}{3 x^3}-\frac{1}{6} b c \log \left(c^2 x^6+1\right)+b c \log (x)",1,"-1/3*a/x^3 - (b*ArcTan[c*x^3])/(3*x^3) + b*c*Log[x] - (b*c*Log[1 + c^2*x^6])/6","A",1
102,1,48,41,0.0073348,"\int \frac{a+b \tan ^{-1}\left(c x^3\right)}{x^7} \, dx","Integrate[(a + b*ArcTan[c*x^3])/x^7,x]","-\frac{a}{6 x^6}-\frac{b c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^6\right)}{6 x^3}-\frac{b \tan ^{-1}\left(c x^3\right)}{6 x^6}","-\frac{a+b \tan ^{-1}\left(c x^3\right)}{6 x^6}-\frac{1}{6} b c^2 \tan ^{-1}\left(c x^3\right)-\frac{b c}{6 x^3}",1,"-1/6*a/x^6 - (b*ArcTan[c*x^3])/(6*x^6) - (b*c*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^6)])/(6*x^3)","C",1
103,1,60,55,0.0140548,"\int \frac{a+b \tan ^{-1}\left(c x^3\right)}{x^{10}} \, dx","Integrate[(a + b*ArcTan[c*x^3])/x^10,x]","-\frac{a}{9 x^9}-\frac{1}{3} b c^3 \log (x)+\frac{1}{18} b c^3 \log \left(c^2 x^6+1\right)-\frac{b c}{18 x^6}-\frac{b \tan ^{-1}\left(c x^3\right)}{9 x^9}","-\frac{a+b \tan ^{-1}\left(c x^3\right)}{9 x^9}-\frac{1}{3} b c^3 \log (x)+\frac{1}{18} b c^3 \log \left(c^2 x^6+1\right)-\frac{b c}{18 x^6}",1,"-1/9*a/x^9 - (b*c)/(18*x^6) - (b*ArcTan[c*x^3])/(9*x^9) - (b*c^3*Log[x])/3 + (b*c^3*Log[1 + c^2*x^6])/18","A",1
104,1,179,174,0.0567538,"\int x^3 \left(a+b \tan ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x^3*(a + b*ArcTan[c*x^3]),x]","\frac{a x^4}{4}-\frac{\sqrt{3} b \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)}{16 c^{4/3}}+\frac{\sqrt{3} b \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)}{16 c^{4/3}}+\frac{b \tan ^{-1}\left(\sqrt[3]{c} x\right)}{4 c^{4/3}}-\frac{b \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)}{8 c^{4/3}}+\frac{b \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)}{8 c^{4/3}}+\frac{1}{4} b x^4 \tan ^{-1}\left(c x^3\right)-\frac{3 b x}{4 c}","\frac{1}{4} x^4 \left(a+b \tan ^{-1}\left(c x^3\right)\right)-\frac{\sqrt{3} b \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)}{16 c^{4/3}}+\frac{\sqrt{3} b \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)}{16 c^{4/3}}+\frac{b \tan ^{-1}\left(\sqrt[3]{c} x\right)}{4 c^{4/3}}-\frac{b \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)}{8 c^{4/3}}+\frac{b \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)}{8 c^{4/3}}-\frac{3 b x}{4 c}",1,"(-3*b*x)/(4*c) + (a*x^4)/4 + (b*ArcTan[c^(1/3)*x])/(4*c^(4/3)) + (b*x^4*ArcTan[c*x^3])/4 - (b*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(8*c^(4/3)) + (b*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(8*c^(4/3)) - (Sqrt[3]*b*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(16*c^(4/3)) + (Sqrt[3]*b*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(16*c^(4/3))","A",1
105,1,131,101,0.0454237,"\int \left(a+b \tan ^{-1}\left(c x^3\right)\right) \, dx","Integrate[a + b*ArcTan[c*x^3],x]","a x-\frac{b \left(-2 \log \left(c^{2/3} x^2+1\right)+\log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)+\log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)-2 \sqrt{3} \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)-2 \sqrt{3} \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)\right)}{4 \sqrt[3]{c}}+b x \tan ^{-1}\left(c x^3\right)","a x+\frac{b \log \left(c^{2/3} x^2+1\right)}{2 \sqrt[3]{c}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{1-2 c^{2/3} x^2}{\sqrt{3}}\right)}{2 \sqrt[3]{c}}-\frac{b \log \left(c^{4/3} x^4-c^{2/3} x^2+1\right)}{4 \sqrt[3]{c}}+b x \tan ^{-1}\left(c x^3\right)",1,"a*x + b*x*ArcTan[c*x^3] - (b*(-2*Sqrt[3]*ArcTan[Sqrt[3] - 2*c^(1/3)*x] - 2*Sqrt[3]*ArcTan[Sqrt[3] + 2*c^(1/3)*x] - 2*Log[1 + c^(2/3)*x^2] + Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2] + Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2]))/(4*c^(1/3))","A",1
106,1,170,165,0.0516806,"\int \frac{a+b \tan ^{-1}\left(c x^3\right)}{x^3} \, dx","Integrate[(a + b*ArcTan[c*x^3])/x^3,x]","-\frac{a}{2 x^2}-\frac{1}{8} \sqrt{3} b c^{2/3} \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)+\frac{1}{8} \sqrt{3} b c^{2/3} \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)+\frac{1}{2} b c^{2/3} \tan ^{-1}\left(\sqrt[3]{c} x\right)-\frac{1}{4} b c^{2/3} \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)+\frac{1}{4} b c^{2/3} \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)-\frac{b \tan ^{-1}\left(c x^3\right)}{2 x^2}","-\frac{a+b \tan ^{-1}\left(c x^3\right)}{2 x^2}-\frac{1}{8} \sqrt{3} b c^{2/3} \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)+\frac{1}{8} \sqrt{3} b c^{2/3} \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)+\frac{1}{2} b c^{2/3} \tan ^{-1}\left(\sqrt[3]{c} x\right)-\frac{1}{4} b c^{2/3} \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)+\frac{1}{4} b c^{2/3} \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)",1,"-1/2*a/x^2 + (b*c^(2/3)*ArcTan[c^(1/3)*x])/2 - (b*ArcTan[c*x^3])/(2*x^2) - (b*c^(2/3)*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/4 + (b*c^(2/3)*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/4 - (Sqrt[3]*b*c^(2/3)*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/8 + (Sqrt[3]*b*c^(2/3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/8","A",1
107,1,183,115,0.0517291,"\int \frac{a+b \tan ^{-1}\left(c x^3\right)}{x^6} \, dx","Integrate[(a + b*ArcTan[c*x^3])/x^6,x]","-\frac{a}{5 x^5}+\frac{1}{10} b c^{5/3} \log \left(c^{2/3} x^2+1\right)-\frac{1}{20} b c^{5/3} \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)-\frac{1}{20} b c^{5/3} \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)+\frac{1}{10} \sqrt{3} b c^{5/3} \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)+\frac{1}{10} \sqrt{3} b c^{5/3} \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)-\frac{3 b c}{10 x^2}-\frac{b \tan ^{-1}\left(c x^3\right)}{5 x^5}","-\frac{a+b \tan ^{-1}\left(c x^3\right)}{5 x^5}+\frac{1}{10} b c^{5/3} \log \left(c^{2/3} x^2+1\right)+\frac{1}{10} \sqrt{3} b c^{5/3} \tan ^{-1}\left(\frac{1-2 c^{2/3} x^2}{\sqrt{3}}\right)-\frac{1}{20} b c^{5/3} \log \left(c^{4/3} x^4-c^{2/3} x^2+1\right)-\frac{3 b c}{10 x^2}",1,"-1/5*a/x^5 - (3*b*c)/(10*x^2) - (b*ArcTan[c*x^3])/(5*x^5) + (Sqrt[3]*b*c^(5/3)*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/10 + (Sqrt[3]*b*c^(5/3)*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/10 + (b*c^(5/3)*Log[1 + c^(2/3)*x^2])/10 - (b*c^(5/3)*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/20 - (b*c^(5/3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/20","A",1
108,1,181,176,0.0857772,"\int x^7 \left(a+b \tan ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x^7*(a + b*ArcTan[c*x^3]),x]","\frac{a x^8}{8}+\frac{\sqrt{3} b \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)}{32 c^{8/3}}-\frac{\sqrt{3} b \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)}{32 c^{8/3}}+\frac{b \tan ^{-1}\left(\sqrt[3]{c} x\right)}{8 c^{8/3}}-\frac{b \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)}{16 c^{8/3}}+\frac{b \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)}{16 c^{8/3}}-\frac{3 b x^5}{40 c}+\frac{1}{8} b x^8 \tan ^{-1}\left(c x^3\right)","\frac{1}{8} x^8 \left(a+b \tan ^{-1}\left(c x^3\right)\right)+\frac{\sqrt{3} b \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)}{32 c^{8/3}}-\frac{\sqrt{3} b \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)}{32 c^{8/3}}+\frac{b \tan ^{-1}\left(\sqrt[3]{c} x\right)}{8 c^{8/3}}-\frac{b \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)}{16 c^{8/3}}+\frac{b \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)}{16 c^{8/3}}-\frac{3 b x^5}{40 c}",1,"(-3*b*x^5)/(40*c) + (a*x^8)/8 + (b*ArcTan[c^(1/3)*x])/(8*c^(8/3)) + (b*x^8*ArcTan[c*x^3])/8 - (b*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(16*c^(8/3)) + (b*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(16*c^(8/3)) + (Sqrt[3]*b*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(32*c^(8/3)) - (Sqrt[3]*b*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(32*c^(8/3))","A",1
109,1,185,117,0.0291827,"\int x^4 \left(a+b \tan ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x^4*(a + b*ArcTan[c*x^3]),x]","\frac{a x^5}{5}+\frac{b \log \left(c^{2/3} x^2+1\right)}{10 c^{5/3}}-\frac{b \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)}{20 c^{5/3}}-\frac{b \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)}{20 c^{5/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)}{10 c^{5/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)}{10 c^{5/3}}-\frac{3 b x^2}{10 c}+\frac{1}{5} b x^5 \tan ^{-1}\left(c x^3\right)","\frac{1}{5} x^5 \left(a+b \tan ^{-1}\left(c x^3\right)\right)+\frac{b \log \left(c^{2/3} x^2+1\right)}{10 c^{5/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{1-2 c^{2/3} x^2}{\sqrt{3}}\right)}{10 c^{5/3}}-\frac{b \log \left(c^{4/3} x^4-c^{2/3} x^2+1\right)}{20 c^{5/3}}-\frac{3 b x^2}{10 c}",1,"(-3*b*x^2)/(10*c) + (a*x^5)/5 + (b*x^5*ArcTan[c*x^3])/5 - (Sqrt[3]*b*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(10*c^(5/3)) - (Sqrt[3]*b*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(10*c^(5/3)) + (b*Log[1 + c^(2/3)*x^2])/(10*c^(5/3)) - (b*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(20*c^(5/3)) - (b*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(20*c^(5/3))","A",1
110,1,170,165,0.027887,"\int x \left(a+b \tan ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x*(a + b*ArcTan[c*x^3]),x]","\frac{a x^2}{2}-\frac{\sqrt{3} b \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)}{8 c^{2/3}}+\frac{\sqrt{3} b \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)}{8 c^{2/3}}-\frac{b \tan ^{-1}\left(\sqrt[3]{c} x\right)}{2 c^{2/3}}+\frac{b \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)}{4 c^{2/3}}-\frac{b \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)}{4 c^{2/3}}+\frac{1}{2} b x^2 \tan ^{-1}\left(c x^3\right)","\frac{1}{2} x^2 \left(a+b \tan ^{-1}\left(c x^3\right)\right)-\frac{\sqrt{3} b \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)}{8 c^{2/3}}+\frac{\sqrt{3} b \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)}{8 c^{2/3}}-\frac{b \tan ^{-1}\left(\sqrt[3]{c} x\right)}{2 c^{2/3}}+\frac{b \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)}{4 c^{2/3}}-\frac{b \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)}{4 c^{2/3}}",1,"(a*x^2)/2 - (b*ArcTan[c^(1/3)*x])/(2*c^(2/3)) + (b*x^2*ArcTan[c*x^3])/2 + (b*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(4*c^(2/3)) - (b*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(4*c^(2/3)) - (Sqrt[3]*b*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) + (Sqrt[3]*b*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3))","A",1
111,1,170,104,0.0404958,"\int \frac{a+b \tan ^{-1}\left(c x^3\right)}{x^2} \, dx","Integrate[(a + b*ArcTan[c*x^3])/x^2,x]","-\frac{a}{x}+\frac{1}{2} b \sqrt[3]{c} \log \left(c^{2/3} x^2+1\right)-\frac{1}{4} b \sqrt[3]{c} \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)-\frac{1}{4} b \sqrt[3]{c} \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)-\frac{b \tan ^{-1}\left(c x^3\right)}{x}-\frac{1}{2} \sqrt{3} b \sqrt[3]{c} \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)-\frac{1}{2} \sqrt{3} b \sqrt[3]{c} \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)","-\frac{a+b \tan ^{-1}\left(c x^3\right)}{x}+\frac{1}{2} b \sqrt[3]{c} \log \left(c^{2/3} x^2+1\right)-\frac{1}{2} \sqrt{3} b \sqrt[3]{c} \tan ^{-1}\left(\frac{1-2 c^{2/3} x^2}{\sqrt{3}}\right)-\frac{1}{4} b \sqrt[3]{c} \log \left(c^{4/3} x^4-c^{2/3} x^2+1\right)",1,"-(a/x) - (b*ArcTan[c*x^3])/x - (Sqrt[3]*b*c^(1/3)*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/2 - (Sqrt[3]*b*c^(1/3)*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/2 + (b*c^(1/3)*Log[1 + c^(2/3)*x^2])/2 - (b*c^(1/3)*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/4 - (b*c^(1/3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/4","A",1
112,1,179,174,0.0544483,"\int \frac{a+b \tan ^{-1}\left(c x^3\right)}{x^5} \, dx","Integrate[(a + b*ArcTan[c*x^3])/x^5,x]","-\frac{a}{4 x^4}-\frac{1}{16} \sqrt{3} b c^{4/3} \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)+\frac{1}{16} \sqrt{3} b c^{4/3} \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)-\frac{1}{4} b c^{4/3} \tan ^{-1}\left(\sqrt[3]{c} x\right)+\frac{1}{8} b c^{4/3} \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)-\frac{1}{8} b c^{4/3} \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)-\frac{b \tan ^{-1}\left(c x^3\right)}{4 x^4}-\frac{3 b c}{4 x}","-\frac{a+b \tan ^{-1}\left(c x^3\right)}{4 x^4}-\frac{1}{16} \sqrt{3} b c^{4/3} \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)+\frac{1}{16} \sqrt{3} b c^{4/3} \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)-\frac{1}{4} b c^{4/3} \tan ^{-1}\left(\sqrt[3]{c} x\right)+\frac{1}{8} b c^{4/3} \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)-\frac{1}{8} b c^{4/3} \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)-\frac{3 b c}{4 x}",1,"-1/4*a/x^4 - (3*b*c)/(4*x) - (b*c^(4/3)*ArcTan[c^(1/3)*x])/4 - (b*ArcTan[c*x^3])/(4*x^4) + (b*c^(4/3)*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/8 - (b*c^(4/3)*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/8 - (Sqrt[3]*b*c^(4/3)*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/16 + (Sqrt[3]*b*c^(4/3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/16","A",1
113,1,121,124,0.1016975,"\int x^{11} \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2 \, dx","Integrate[x^11*(a + b*ArcTan[c*x^3])^2,x]","\frac{c x^3 \left(3 a^2 c^3 x^9-2 a b c^2 x^6+6 a b+b^2 c x^3\right)-2 b \tan ^{-1}\left(c x^3\right) \left(a \left(3-3 c^4 x^{12}\right)+b c x^3 \left(c^2 x^6-3\right)\right)+3 b^2 \left(c^4 x^{12}-1\right) \tan ^{-1}\left(c x^3\right)^2-4 b^2 \log \left(c^2 x^6+1\right)}{36 c^4}","-\frac{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{12 c^4}+\frac{a b x^3}{6 c^3}+\frac{1}{12} x^{12} \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2-\frac{b x^9 \left(a+b \tan ^{-1}\left(c x^3\right)\right)}{18 c}+\frac{b^2 x^3 \tan ^{-1}\left(c x^3\right)}{6 c^3}+\frac{b^2 x^6}{36 c^2}-\frac{b^2 \log \left(c^2 x^6+1\right)}{9 c^4}",1,"(c*x^3*(6*a*b + b^2*c*x^3 - 2*a*b*c^2*x^6 + 3*a^2*c^3*x^9) - 2*b*(b*c*x^3*(-3 + c^2*x^6) + a*(3 - 3*c^4*x^12))*ArcTan[c*x^3] + 3*b^2*(-1 + c^4*x^12)*ArcTan[c*x^3]^2 - 4*b^2*Log[1 + c^2*x^6])/(36*c^4)","A",1
114,1,141,154,0.3142883,"\int x^8 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2 \, dx","Integrate[x^8*(a + b*ArcTan[c*x^3])^2,x]","\frac{a^2 c^3 x^9-a b c^2 x^6+a b \log \left(c^2 x^6+1\right)-b \tan ^{-1}\left(c x^3\right) \left(-2 a c^3 x^9+b c^2 x^6+2 b \log \left(1+e^{2 i \tan ^{-1}\left(c x^3\right)}\right)+b\right)+b^2 \left(c^3 x^9+i\right) \tan ^{-1}\left(c x^3\right)^2+i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}\left(c x^3\right)}\right)+b^2 c x^3}{9 c^3}","-\frac{i \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{9 c^3}-\frac{2 b \log \left(\frac{2}{1+i c x^3}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)}{9 c^3}+\frac{1}{9} x^9 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2-\frac{b x^6 \left(a+b \tan ^{-1}\left(c x^3\right)\right)}{9 c}-\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i c x^3+1}\right)}{9 c^3}-\frac{b^2 \tan ^{-1}\left(c x^3\right)}{9 c^3}+\frac{b^2 x^3}{9 c^2}",1,"(b^2*c*x^3 - a*b*c^2*x^6 + a^2*c^3*x^9 + b^2*(I + c^3*x^9)*ArcTan[c*x^3]^2 - b*ArcTan[c*x^3]*(b + b*c^2*x^6 - 2*a*c^3*x^9 + 2*b*Log[1 + E^((2*I)*ArcTan[c*x^3])]) + a*b*Log[1 + c^2*x^6] + I*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x^3])])/(9*c^3)","A",0
115,1,85,90,0.0684904,"\int x^5 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2 \, dx","Integrate[x^5*(a + b*ArcTan[c*x^3])^2,x]","\frac{2 b \tan ^{-1}\left(c x^3\right) \left(a c^2 x^6+a-b c x^3\right)+a c x^3 \left(a c x^3-2 b\right)+b^2 \log \left(c^2 x^6+1\right)+b^2 \left(c^2 x^6+1\right) \tan ^{-1}\left(c x^3\right)^2}{6 c^2}","\frac{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{6 c^2}-\frac{a b x^3}{3 c}+\frac{1}{6} x^6 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2+\frac{b^2 \log \left(c^2 x^6+1\right)}{6 c^2}-\frac{b^2 x^3 \tan ^{-1}\left(c x^3\right)}{3 c}",1,"(a*c*x^3*(-2*b + a*c*x^3) + 2*b*(a - b*c*x^3 + a*c^2*x^6)*ArcTan[c*x^3] + b^2*(1 + c^2*x^6)*ArcTan[c*x^3]^2 + b^2*Log[1 + c^2*x^6])/(6*c^2)","A",1
116,1,107,104,0.0738071,"\int x^2 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2 \, dx","Integrate[x^2*(a + b*ArcTan[c*x^3])^2,x]","\frac{a \left(a c x^3-b \log \left(c^2 x^6+1\right)\right)+2 b \tan ^{-1}\left(c x^3\right) \left(a c x^3+b \log \left(1+e^{2 i \tan ^{-1}\left(c x^3\right)}\right)\right)-i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}\left(c x^3\right)}\right)+b^2 \left(c x^3-i\right) \tan ^{-1}\left(c x^3\right)^2}{3 c}","\frac{1}{3} x^3 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2+\frac{i \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{3 c}+\frac{2 b \log \left(\frac{2}{1+i c x^3}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)}{3 c}+\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i c x^3+1}\right)}{3 c}",1,"(b^2*(-I + c*x^3)*ArcTan[c*x^3]^2 + 2*b*ArcTan[c*x^3]*(a*c*x^3 + b*Log[1 + E^((2*I)*ArcTan[c*x^3])]) + a*(a*c*x^3 - b*Log[1 + c^2*x^6]) - I*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x^3])])/(3*c)","A",0
117,1,167,154,0.1241425,"\int \frac{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{x} \, dx","Integrate[(a + b*ArcTan[c*x^3])^2/x,x]","\frac{1}{6} \left(b \left(2 i \text{Li}_2\left(\frac{c x^3+i}{i-c x^3}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)-2 i \text{Li}_2\left(\frac{c x^3+i}{c x^3-i}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)+b \left(\text{Li}_3\left(\frac{c x^3+i}{i-c x^3}\right)-\text{Li}_3\left(\frac{c x^3+i}{c x^3-i}\right)\right)\right)+4 \tanh ^{-1}\left(1+\frac{2 i}{c x^3-i}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2\right)","-\frac{1}{3} i b \text{Li}_2\left(1-\frac{2}{i c x^3+1}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)+\frac{1}{3} i b \text{Li}_2\left(\frac{2}{i c x^3+1}-1\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)+\frac{2}{3} \tanh ^{-1}\left(1-\frac{2}{1+i c x^3}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2-\frac{1}{6} b^2 \text{Li}_3\left(1-\frac{2}{i c x^3+1}\right)+\frac{1}{6} b^2 \text{Li}_3\left(\frac{2}{i c x^3+1}-1\right)",1,"(4*(a + b*ArcTan[c*x^3])^2*ArcTanh[1 + (2*I)/(-I + c*x^3)] + b*((2*I)*(a + b*ArcTan[c*x^3])*PolyLog[2, (I + c*x^3)/(I - c*x^3)] - (2*I)*(a + b*ArcTan[c*x^3])*PolyLog[2, (I + c*x^3)/(-I + c*x^3)] + b*(PolyLog[3, (I + c*x^3)/(I - c*x^3)] - PolyLog[3, (I + c*x^3)/(-I + c*x^3)])))/6","A",1
118,1,125,100,0.1683805,"\int \frac{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{x^4} \, dx","Integrate[(a + b*ArcTan[c*x^3])^2/x^4,x]","\frac{-a \left(a+b c x^3 \log \left(c^2 x^6+1\right)-2 b c x^3 \log \left(c x^3\right)\right)+2 b \tan ^{-1}\left(c x^3\right) \left(-a+b c x^3 \log \left(1-e^{2 i \tan ^{-1}\left(c x^3\right)}\right)\right)-i b^2 c x^3 \text{Li}_2\left(e^{2 i \tan ^{-1}\left(c x^3\right)}\right)+b^2 \left(-1-i c x^3\right) \tan ^{-1}\left(c x^3\right)^2}{3 x^3}","-\frac{1}{3} i c \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2-\frac{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{3 x^3}+\frac{2}{3} b c \log \left(2-\frac{2}{1-i c x^3}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)-\frac{1}{3} i b^2 c \text{Li}_2\left(\frac{2}{1-i c x^3}-1\right)",1,"(b^2*(-1 - I*c*x^3)*ArcTan[c*x^3]^2 + 2*b*ArcTan[c*x^3]*(-a + b*c*x^3*Log[1 - E^((2*I)*ArcTan[c*x^3])]) - a*(a - 2*b*c*x^3*Log[c*x^3] + b*c*x^3*Log[1 + c^2*x^6]) - I*b^2*c*x^3*PolyLog[2, E^((2*I)*ArcTan[c*x^3])])/(3*x^3)","A",0
119,1,98,87,0.083263,"\int \frac{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{x^7} \, dx","Integrate[(a + b*ArcTan[c*x^3])^2/x^7,x]","-\frac{a^2+2 b \tan ^{-1}\left(c x^3\right) \left(a c^2 x^6+a+b c x^3\right)+2 a b c x^3-6 b^2 c^2 x^6 \log (x)+b^2 c^2 x^6 \log \left(c^2 x^6+1\right)+b^2 \left(c^2 x^6+1\right) \tan ^{-1}\left(c x^3\right)^2}{6 x^6}","-\frac{1}{6} c^2 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2-\frac{b c \left(a+b \tan ^{-1}\left(c x^3\right)\right)}{3 x^3}-\frac{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{6 x^6}-\frac{1}{6} b^2 c^2 \log \left(c^2 x^6+1\right)+b^2 c^2 \log (x)",1,"-1/6*(a^2 + 2*a*b*c*x^3 + 2*b*(a + b*c*x^3 + a*c^2*x^6)*ArcTan[c*x^3] + b^2*(1 + c^2*x^6)*ArcTan[c*x^3]^2 - 6*b^2*c^2*x^6*Log[x] + b^2*c^2*x^6*Log[1 + c^2*x^6])/x^6","A",1
120,1,167,154,0.4120053,"\int \frac{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{x^{10}} \, dx","Integrate[(a + b*ArcTan[c*x^3])^2/x^10,x]","-\frac{a^2+2 a b c^3 x^9 \log \left(c x^3\right)+b \tan ^{-1}\left(c x^3\right) \left(2 a+b c^3 x^9+2 b c^3 x^9 \log \left(1-e^{2 i \tan ^{-1}\left(c x^3\right)}\right)+b c x^3\right)-a b c^3 x^9 \log \left(c^2 x^6+1\right)+a b c x^3-i b^2 c^3 x^9 \text{Li}_2\left(e^{2 i \tan ^{-1}\left(c x^3\right)}\right)+b^2 \left(1-i c^3 x^9\right) \tan ^{-1}\left(c x^3\right)^2+b^2 c^2 x^6}{9 x^9}","\frac{1}{9} i c^3 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2-\frac{2}{9} b c^3 \log \left(2-\frac{2}{1-i c x^3}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)-\frac{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{9 x^9}-\frac{b c \left(a+b \tan ^{-1}\left(c x^3\right)\right)}{9 x^6}+\frac{1}{9} i b^2 c^3 \text{Li}_2\left(\frac{2}{1-i c x^3}-1\right)-\frac{1}{9} b^2 c^3 \tan ^{-1}\left(c x^3\right)-\frac{b^2 c^2}{9 x^3}",1,"-1/9*(a^2 + a*b*c*x^3 + b^2*c^2*x^6 + b^2*(1 - I*c^3*x^9)*ArcTan[c*x^3]^2 + b*ArcTan[c*x^3]*(2*a + b*c*x^3 + b*c^3*x^9 + 2*b*c^3*x^9*Log[1 - E^((2*I)*ArcTan[c*x^3])]) + 2*a*b*c^3*x^9*Log[c*x^3] - a*b*c^3*x^9*Log[1 + c^2*x^6] - I*b^2*c^3*x^9*PolyLog[2, E^((2*I)*ArcTan[c*x^3])])/x^9","A",0
121,1,346,240,0.5622205,"\int x^8 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^3 \, dx","Integrate[x^8*(a + b*ArcTan[c*x^3])^3,x]","\frac{2 a^3 c^3 x^9+6 a^2 b c^3 x^9 \tan ^{-1}\left(c x^3\right)-3 a^2 b c^2 x^6+3 a^2 b \log \left(c^2 x^6+1\right)+6 a b^2 c^3 x^9 \tan ^{-1}\left(c x^3\right)^2-6 a b^2 c^2 x^6 \tan ^{-1}\left(c x^3\right)+6 i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}\left(c x^3\right)}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)+6 a b^2 c x^3+6 i a b^2 \tan ^{-1}\left(c x^3\right)^2-6 a b^2 \tan ^{-1}\left(c x^3\right)-12 a b^2 \tan ^{-1}\left(c x^3\right) \log \left(1+e^{2 i \tan ^{-1}\left(c x^3\right)}\right)+2 b^3 c^3 x^9 \tan ^{-1}\left(c x^3\right)^3-3 b^3 \log \left(c^2 x^6+1\right)-3 b^3 c^2 x^6 \tan ^{-1}\left(c x^3\right)^2-3 b^3 \text{Li}_3\left(-e^{2 i \tan ^{-1}\left(c x^3\right)}\right)+6 b^3 c x^3 \tan ^{-1}\left(c x^3\right)+2 i b^3 \tan ^{-1}\left(c x^3\right)^3-3 b^3 \tan ^{-1}\left(c x^3\right)^2-6 b^3 \tan ^{-1}\left(c x^3\right)^2 \log \left(1+e^{2 i \tan ^{-1}\left(c x^3\right)}\right)}{18 c^3}","-\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i c x^3+1}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)}{3 c^3}+\frac{a b^2 x^3}{3 c^2}-\frac{i \left(a+b \tan ^{-1}\left(c x^3\right)\right)^3}{9 c^3}-\frac{b \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{6 c^3}-\frac{b \log \left(\frac{2}{1+i c x^3}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{3 c^3}+\frac{1}{9} x^9 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^3-\frac{b x^6 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{6 c}-\frac{b^3 \text{Li}_3\left(1-\frac{2}{i c x^3+1}\right)}{6 c^3}+\frac{b^3 x^3 \tan ^{-1}\left(c x^3\right)}{3 c^2}-\frac{b^3 \log \left(c^2 x^6+1\right)}{6 c^3}",1,"(6*a*b^2*c*x^3 - 3*a^2*b*c^2*x^6 + 2*a^3*c^3*x^9 - 6*a*b^2*ArcTan[c*x^3] + 6*b^3*c*x^3*ArcTan[c*x^3] - 6*a*b^2*c^2*x^6*ArcTan[c*x^3] + 6*a^2*b*c^3*x^9*ArcTan[c*x^3] + (6*I)*a*b^2*ArcTan[c*x^3]^2 - 3*b^3*ArcTan[c*x^3]^2 - 3*b^3*c^2*x^6*ArcTan[c*x^3]^2 + 6*a*b^2*c^3*x^9*ArcTan[c*x^3]^2 + (2*I)*b^3*ArcTan[c*x^3]^3 + 2*b^3*c^3*x^9*ArcTan[c*x^3]^3 - 12*a*b^2*ArcTan[c*x^3]*Log[1 + E^((2*I)*ArcTan[c*x^3])] - 6*b^3*ArcTan[c*x^3]^2*Log[1 + E^((2*I)*ArcTan[c*x^3])] + 3*a^2*b*Log[1 + c^2*x^6] - 3*b^3*Log[1 + c^2*x^6] + (6*I)*b^2*(a + b*ArcTan[c*x^3])*PolyLog[2, -E^((2*I)*ArcTan[c*x^3])] - 3*b^3*PolyLog[3, -E^((2*I)*ArcTan[c*x^3])])/(18*c^3)","A",0
122,1,170,147,0.1763282,"\int x^5 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^3 \, dx","Integrate[x^5*(a + b*ArcTan[c*x^3])^3,x]","\frac{a \left(a c x^3 \left(a c x^3-3 b\right)+3 b^2 \log \left(c^2 x^6+1\right)\right)+3 b^2 \tan ^{-1}\left(c x^3\right)^2 \left(a c^2 x^6+a+b \left(-c x^3+i\right)\right)+3 b \tan ^{-1}\left(c x^3\right) \left(a \left(a c^2 x^6+a-2 b c x^3\right)-2 b^2 \log \left(1+e^{2 i \tan ^{-1}\left(c x^3\right)}\right)\right)+b^3 \left(c^2 x^6+1\right) \tan ^{-1}\left(c x^3\right)^3+3 i b^3 \text{Li}_2\left(-e^{2 i \tan ^{-1}\left(c x^3\right)}\right)}{6 c^2}","-\frac{b^2 \log \left(\frac{2}{1+i c x^3}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)}{c^2}+\frac{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^3}{6 c^2}-\frac{i b \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{2 c^2}-\frac{b x^3 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{2 c}+\frac{1}{6} x^6 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^3-\frac{i b^3 \text{Li}_2\left(1-\frac{2}{i c x^3+1}\right)}{2 c^2}",1,"(3*b^2*(a + a*c^2*x^6 + b*(I - c*x^3))*ArcTan[c*x^3]^2 + b^3*(1 + c^2*x^6)*ArcTan[c*x^3]^3 + 3*b*ArcTan[c*x^3]*(a*(a - 2*b*c*x^3 + a*c^2*x^6) - 2*b^2*Log[1 + E^((2*I)*ArcTan[c*x^3])]) + a*(a*c*x^3*(-3*b + a*c*x^3) + 3*b^2*Log[1 + c^2*x^6]) + (3*I)*b^3*PolyLog[2, -E^((2*I)*ArcTan[c*x^3])])/(6*c^2)","A",0
123,1,224,139,0.1000378,"\int x^2 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^3 \, dx","Integrate[x^2*(a + b*ArcTan[c*x^3])^3,x]","\frac{2 a^3 c x^3-3 a^2 b \log \left(c^2 x^6+1\right)+6 a^2 b c x^3 \tan ^{-1}\left(c x^3\right)-6 i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}\left(c x^3\right)}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)+6 a b^2 c x^3 \tan ^{-1}\left(c x^3\right)^2-6 i a b^2 \tan ^{-1}\left(c x^3\right)^2+12 a b^2 \tan ^{-1}\left(c x^3\right) \log \left(1+e^{2 i \tan ^{-1}\left(c x^3\right)}\right)+3 b^3 \text{Li}_3\left(-e^{2 i \tan ^{-1}\left(c x^3\right)}\right)+2 b^3 c x^3 \tan ^{-1}\left(c x^3\right)^3-2 i b^3 \tan ^{-1}\left(c x^3\right)^3+6 b^3 \tan ^{-1}\left(c x^3\right)^2 \log \left(1+e^{2 i \tan ^{-1}\left(c x^3\right)}\right)}{6 c}","\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i c x^3+1}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)}{c}+\frac{1}{3} x^3 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^3+\frac{i \left(a+b \tan ^{-1}\left(c x^3\right)\right)^3}{3 c}+\frac{b \log \left(\frac{2}{1+i c x^3}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{c}+\frac{b^3 \text{Li}_3\left(1-\frac{2}{i c x^3+1}\right)}{2 c}",1,"(2*a^3*c*x^3 + 6*a^2*b*c*x^3*ArcTan[c*x^3] - (6*I)*a*b^2*ArcTan[c*x^3]^2 + 6*a*b^2*c*x^3*ArcTan[c*x^3]^2 - (2*I)*b^3*ArcTan[c*x^3]^3 + 2*b^3*c*x^3*ArcTan[c*x^3]^3 + 12*a*b^2*ArcTan[c*x^3]*Log[1 + E^((2*I)*ArcTan[c*x^3])] + 6*b^3*ArcTan[c*x^3]^2*Log[1 + E^((2*I)*ArcTan[c*x^3])] - 3*a^2*b*Log[1 + c^2*x^6] - (6*I)*b^2*(a + b*ArcTan[c*x^3])*PolyLog[2, -E^((2*I)*ArcTan[c*x^3])] + 3*b^3*PolyLog[3, -E^((2*I)*ArcTan[c*x^3])])/(6*c)","A",0
124,1,248,232,0.216761,"\int \frac{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^3}{x} \, dx","Integrate[(a + b*ArcTan[c*x^3])^3/x,x]","\frac{1}{4} i b \left(2 \text{Li}_2\left(\frac{c x^3+i}{i-c x^3}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2-2 \text{Li}_2\left(\frac{c x^3+i}{c x^3-i}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2+b \left(-2 i \text{Li}_3\left(\frac{c x^3+i}{i-c x^3}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)+2 i \text{Li}_3\left(\frac{c x^3+i}{c x^3-i}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)+b \left(\text{Li}_4\left(\frac{c x^3+i}{c x^3-i}\right)-\text{Li}_4\left(\frac{c x^3+i}{i-c x^3}\right)\right)\right)\right)+\frac{2}{3} \tanh ^{-1}\left(1+\frac{2 i}{c x^3-i}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)^3","-\frac{1}{2} b^2 \text{Li}_3\left(1-\frac{2}{i c x^3+1}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)+\frac{1}{2} b^2 \text{Li}_3\left(\frac{2}{i c x^3+1}-1\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)-\frac{1}{2} i b \text{Li}_2\left(1-\frac{2}{i c x^3+1}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2+\frac{1}{2} i b \text{Li}_2\left(\frac{2}{i c x^3+1}-1\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2+\frac{2}{3} \tanh ^{-1}\left(1-\frac{2}{1+i c x^3}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)^3+\frac{1}{4} i b^3 \text{Li}_4\left(1-\frac{2}{i c x^3+1}\right)-\frac{1}{4} i b^3 \text{Li}_4\left(\frac{2}{i c x^3+1}-1\right)",1,"(2*(a + b*ArcTan[c*x^3])^3*ArcTanh[1 + (2*I)/(-I + c*x^3)])/3 + (I/4)*b*(2*(a + b*ArcTan[c*x^3])^2*PolyLog[2, (I + c*x^3)/(I - c*x^3)] - 2*(a + b*ArcTan[c*x^3])^2*PolyLog[2, (I + c*x^3)/(-I + c*x^3)] + b*((-2*I)*(a + b*ArcTan[c*x^3])*PolyLog[3, (I + c*x^3)/(I - c*x^3)] + (2*I)*(a + b*ArcTan[c*x^3])*PolyLog[3, (I + c*x^3)/(-I + c*x^3)] + b*(-PolyLog[4, (I + c*x^3)/(I - c*x^3)] + PolyLog[4, (I + c*x^3)/(-I + c*x^3)])))","A",1
125,1,240,133,0.424076,"\int \frac{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^3}{x^4} \, dx","Integrate[(a + b*ArcTan[c*x^3])^3/x^4,x]","-\frac{a^3}{3 x^3}-\frac{1}{2} a^2 b c \log \left(c^2 x^6+1\right)-\frac{a^2 b \tan ^{-1}\left(c x^3\right)}{x^3}+3 a^2 b c \log (x)+a b^2 c \left(\tan ^{-1}\left(c x^3\right) \left(\left(-\frac{1}{c x^3}-i\right) \tan ^{-1}\left(c x^3\right)+2 \log \left(1-e^{2 i \tan ^{-1}\left(c x^3\right)}\right)\right)-i \text{Li}_2\left(e^{2 i \tan ^{-1}\left(c x^3\right)}\right)\right)+\frac{1}{3} b^3 c \left(3 i \tan ^{-1}\left(c x^3\right) \text{Li}_2\left(e^{-2 i \tan ^{-1}\left(c x^3\right)}\right)+\frac{3}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}\left(c x^3\right)}\right)-\frac{\tan ^{-1}\left(c x^3\right)^3}{c x^3}+i \tan ^{-1}\left(c x^3\right)^3+3 \tan ^{-1}\left(c x^3\right)^2 \log \left(1-e^{-2 i \tan ^{-1}\left(c x^3\right)}\right)-\frac{i \pi ^3}{8}\right)","-i b^2 c \text{Li}_2\left(\frac{2}{1-i c x^3}-1\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)-\frac{1}{3} i c \left(a+b \tan ^{-1}\left(c x^3\right)\right)^3-\frac{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^3}{3 x^3}+b c \log \left(2-\frac{2}{1-i c x^3}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2+\frac{1}{2} b^3 c \text{Li}_3\left(\frac{2}{1-i c x^3}-1\right)",1,"-1/3*a^3/x^3 - (a^2*b*ArcTan[c*x^3])/x^3 + 3*a^2*b*c*Log[x] - (a^2*b*c*Log[1 + c^2*x^6])/2 + a*b^2*c*(ArcTan[c*x^3]*((-I - 1/(c*x^3))*ArcTan[c*x^3] + 2*Log[1 - E^((2*I)*ArcTan[c*x^3])]) - I*PolyLog[2, E^((2*I)*ArcTan[c*x^3])]) + (b^3*c*((-1/8*I)*Pi^3 + I*ArcTan[c*x^3]^3 - ArcTan[c*x^3]^3/(c*x^3) + 3*ArcTan[c*x^3]^2*Log[1 - E^((-2*I)*ArcTan[c*x^3])] + (3*I)*ArcTan[c*x^3]*PolyLog[2, E^((-2*I)*ArcTan[c*x^3])] + (3*PolyLog[3, E^((-2*I)*ArcTan[c*x^3])])/2))/3","A",0
126,1,196,146,0.3522087,"\int \frac{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^3}{x^7} \, dx","Integrate[(a + b*ArcTan[c*x^3])^3/x^7,x]","-\frac{a \left(a \left(a+3 b c x^3\right)-6 b^2 c^2 x^6 \log \left(\frac{c x^3}{\sqrt{c^2 x^6+1}}\right)\right)+3 b^2 \tan ^{-1}\left(c x^3\right)^2 \left(a c^2 x^6+a+b c x^3 \left(1+i c x^3\right)\right)+3 b \tan ^{-1}\left(c x^3\right) \left(a \left(a c^2 x^6+a+2 b c x^3\right)-2 b^2 c^2 x^6 \log \left(1-e^{2 i \tan ^{-1}\left(c x^3\right)}\right)\right)+3 i b^3 c^2 x^6 \text{Li}_2\left(e^{2 i \tan ^{-1}\left(c x^3\right)}\right)+b^3 \left(c^2 x^6+1\right) \tan ^{-1}\left(c x^3\right)^3}{6 x^6}","b^2 c^2 \log \left(2-\frac{2}{1-i c x^3}\right) \left(a+b \tan ^{-1}\left(c x^3\right)\right)-\frac{1}{2} i b c^2 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2-\frac{1}{6} c^2 \left(a+b \tan ^{-1}\left(c x^3\right)\right)^3-\frac{b c \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2}{2 x^3}-\frac{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^3}{6 x^6}-\frac{1}{2} i b^3 c^2 \text{Li}_2\left(\frac{2}{1-i c x^3}-1\right)",1,"-1/6*(3*b^2*(a + a*c^2*x^6 + b*c*x^3*(1 + I*c*x^3))*ArcTan[c*x^3]^2 + b^3*(1 + c^2*x^6)*ArcTan[c*x^3]^3 + 3*b*ArcTan[c*x^3]*(a*(a + 2*b*c*x^3 + a*c^2*x^6) - 2*b^2*c^2*x^6*Log[1 - E^((2*I)*ArcTan[c*x^3])]) + a*(a*(a + 3*b*c*x^3) - 6*b^2*c^2*x^6*Log[(c*x^3)/Sqrt[1 + c^2*x^6]]) + (3*I)*b^3*c^2*x^6*PolyLog[2, E^((2*I)*ArcTan[c*x^3])])/x^6","A",0
127,0,0,21,1.9821238,"\int (d x)^m \left(a+b \tan ^{-1}\left(c x^3\right)\right)^3 \, dx","Integrate[(d*x)^m*(a + b*ArcTan[c*x^3])^3,x]","\int (d x)^m \left(a+b \tan ^{-1}\left(c x^3\right)\right)^3 \, dx","\text{Int}\left((d x)^m \left(a+b \tan ^{-1}\left(c x^3\right)\right)^3,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTan[c*x^3])^3, x]","A",-1
128,0,0,21,1.2982217,"\int (d x)^m \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2 \, dx","Integrate[(d*x)^m*(a + b*ArcTan[c*x^3])^2,x]","\int (d x)^m \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2 \, dx","\text{Int}\left((d x)^m \left(a+b \tan ^{-1}\left(c x^3\right)\right)^2,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTan[c*x^3])^2, x]","A",-1
129,1,65,75,0.0668667,"\int (d x)^m \left(a+b \tan ^{-1}\left(c x^3\right)\right) \, dx","Integrate[(d*x)^m*(a + b*ArcTan[c*x^3]),x]","-\frac{x (d x)^m \left(3 b c x^3 \, _2F_1\left(1,\frac{m+4}{6};\frac{m+10}{6};-c^2 x^6\right)-(m+4) \left(a+b \tan ^{-1}\left(c x^3\right)\right)\right)}{(m+1) (m+4)}","\frac{(d x)^{m+1} \left(a+b \tan ^{-1}\left(c x^3\right)\right)}{d (m+1)}-\frac{3 b c (d x)^{m+4} \, _2F_1\left(1,\frac{m+4}{6};\frac{m+10}{6};-c^2 x^6\right)}{d^4 (m+1) (m+4)}",1,"-((x*(d*x)^m*(-((4 + m)*(a + b*ArcTan[c*x^3])) + 3*b*c*x^3*Hypergeometric2F1[1, (4 + m)/6, (10 + m)/6, -(c^2*x^6)]))/((1 + m)*(4 + m)))","A",1
130,0,0,21,0.3405004,"\int \frac{(d x)^m}{a+b \tan ^{-1}\left(c x^3\right)} \, dx","Integrate[(d*x)^m/(a + b*ArcTan[c*x^3]),x]","\int \frac{(d x)^m}{a+b \tan ^{-1}\left(c x^3\right)} \, dx","\text{Int}\left(\frac{(d x)^m}{a+b \tan ^{-1}\left(c x^3\right)},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcTan[c*x^3]), x]","A",-1
131,0,0,21,0.362868,"\int \frac{(d x)^m}{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^2} \, dx","Integrate[(d*x)^m/(a + b*ArcTan[c*x^3])^2,x]","\int \frac{(d x)^m}{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^2} \, dx","\text{Int}\left(\frac{(d x)^m}{\left(a+b \tan ^{-1}\left(c x^3\right)\right)^2},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcTan[c*x^3])^2, x]","A",-1
132,1,55,50,0.0111027,"\int x^3 \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right) \, dx","Integrate[x^3*(a + b*ArcTan[c/x]),x]","\frac{a x^4}{4}-\frac{1}{4} b c^4 \tan ^{-1}\left(\frac{c}{x}\right)-\frac{1}{4} b c^3 x+\frac{1}{4} b x^4 \tan ^{-1}\left(\frac{c}{x}\right)+\frac{1}{12} b c x^3","\frac{1}{4} x^4 \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)+\frac{1}{4} b c^4 \tan ^{-1}\left(\frac{x}{c}\right)-\frac{1}{4} b c^3 x+\frac{1}{12} b c x^3",1,"-1/4*(b*c^3*x) + (b*c*x^3)/12 + (a*x^4)/4 - (b*c^4*ArcTan[c/x])/4 + (b*x^4*ArcTan[c/x])/4","A",1
133,1,48,43,0.0081553,"\int x^2 \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right) \, dx","Integrate[x^2*(a + b*ArcTan[c/x]),x]","\frac{a x^3}{3}-\frac{1}{6} b c^3 \log \left(c^2+x^2\right)+\frac{1}{3} b x^3 \tan ^{-1}\left(\frac{c}{x}\right)+\frac{1}{6} b c x^2","\frac{1}{3} x^3 \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)-\frac{1}{6} b c^3 \log \left(c^2+x^2\right)+\frac{1}{6} b c x^2",1,"(b*c*x^2)/6 + (a*x^3)/3 + (b*x^3*ArcTan[c/x])/3 - (b*c^3*Log[c^2 + x^2])/6","A",1
134,1,44,39,0.0080436,"\int x \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right) \, dx","Integrate[x*(a + b*ArcTan[c/x]),x]","\frac{a x^2}{2}+\frac{1}{2} b c^2 \tan ^{-1}\left(\frac{c}{x}\right)+\frac{1}{2} b x^2 \tan ^{-1}\left(\frac{c}{x}\right)+\frac{b c x}{2}","\frac{1}{2} x^2 \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)-\frac{1}{2} b c^2 \tan ^{-1}\left(\frac{x}{c}\right)+\frac{b c x}{2}",1,"(b*c*x)/2 + (a*x^2)/2 + (b*c^2*ArcTan[c/x])/2 + (b*x^2*ArcTan[c/x])/2","A",1
135,1,27,27,0.0026806,"\int \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right) \, dx","Integrate[a + b*ArcTan[c/x],x]","a x+\frac{1}{2} b c \log \left(c^2+x^2\right)+b x \tan ^{-1}\left(\frac{c}{x}\right)","a x+\frac{1}{2} b c \log \left(c^2+x^2\right)+b x \tan ^{-1}\left(\frac{c}{x}\right)",1,"a*x + b*x*ArcTan[c/x] + (b*c*Log[c^2 + x^2])/2","A",1
136,1,39,39,0.0062375,"\int \frac{a+b \tan ^{-1}\left(\frac{c}{x}\right)}{x} \, dx","Integrate[(a + b*ArcTan[c/x])/x,x]","a \log (x)-\frac{1}{2} i b \text{Li}_2\left(-\frac{i c}{x}\right)+\frac{1}{2} i b \text{Li}_2\left(\frac{i c}{x}\right)","a \log (x)-\frac{1}{2} i b \text{Li}_2\left(-\frac{i c}{x}\right)+\frac{1}{2} i b \text{Li}_2\left(\frac{i c}{x}\right)",1,"a*Log[x] - (I/2)*b*PolyLog[2, ((-I)*c)/x] + (I/2)*b*PolyLog[2, (I*c)/x]","A",1
137,1,37,34,0.0115315,"\int \frac{a+b \tan ^{-1}\left(\frac{c}{x}\right)}{x^2} \, dx","Integrate[(a + b*ArcTan[c/x])/x^2,x]","-\frac{a}{x}+\frac{b \log \left(\frac{c^2}{x^2}+1\right)}{2 c}-\frac{b \tan ^{-1}\left(\frac{c}{x}\right)}{x}","\frac{b \log \left(\frac{c^2}{x^2}+1\right)}{2 c}-\frac{a+b \tan ^{-1}\left(\frac{c}{x}\right)}{x}",1,"-(a/x) - (b*ArcTan[c/x])/x + (b*Log[1 + c^2/x^2])/(2*c)","A",1
138,1,48,43,0.011247,"\int \frac{a+b \tan ^{-1}\left(\frac{c}{x}\right)}{x^3} \, dx","Integrate[(a + b*ArcTan[c/x])/x^3,x]","-\frac{a}{2 x^2}+\frac{b \tan ^{-1}\left(\frac{x}{c}\right)}{2 c^2}-\frac{b \tan ^{-1}\left(\frac{c}{x}\right)}{2 x^2}+\frac{b}{2 c x}","-\frac{a+b \tan ^{-1}\left(\frac{c}{x}\right)}{2 x^2}+\frac{b \tan ^{-1}\left(\frac{x}{c}\right)}{2 c^2}+\frac{b}{2 c x}",1,"-1/2*a/x^2 + b/(2*c*x) - (b*ArcTan[c/x])/(2*x^2) + (b*ArcTan[x/c])/(2*c^2)","A",1
139,1,60,55,0.0096552,"\int \frac{a+b \tan ^{-1}\left(\frac{c}{x}\right)}{x^4} \, dx","Integrate[(a + b*ArcTan[c/x])/x^4,x]","-\frac{a}{3 x^3}+\frac{b \log (x)}{3 c^3}-\frac{b \log \left(c^2+x^2\right)}{6 c^3}-\frac{b \tan ^{-1}\left(\frac{c}{x}\right)}{3 x^3}+\frac{b}{6 c x^2}","-\frac{a+b \tan ^{-1}\left(\frac{c}{x}\right)}{3 x^3}+\frac{b \log (x)}{3 c^3}-\frac{b \log \left(c^2+x^2\right)}{6 c^3}+\frac{b}{6 c x^2}",1,"-1/3*a/x^3 + b/(6*c*x^2) - (b*ArcTan[c/x])/(3*x^3) + (b*Log[x])/(3*c^3) - (b*Log[c^2 + x^2])/(6*c^3)","A",1
140,1,111,122,0.1006632,"\int x^3 \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^2 \, dx","Integrate[x^3*(a + b*ArcTan[c/x])^2,x]","\frac{1}{12} \left(x \left(3 a^2 x^3+2 a b c \left(x^2-3 c^2\right)+b^2 c^2 x\right)+2 b \tan ^{-1}\left(\frac{c}{x}\right) \left(3 a \left(x^4-c^4\right)+b c x \left(x^2-3 c^2\right)\right)+3 b^2 \left(x^4-c^4\right) \tan ^{-1}\left(\frac{c}{x}\right)^2-4 b^2 c^4 \log \left(c^2+x^2\right)\right)","-\frac{1}{4} c^4 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2-\frac{1}{2} b c^3 x \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)+\frac{1}{4} x^4 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{1}{6} b c x^3 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)-\frac{2}{3} b^2 c^4 \log (x)+\frac{1}{12} b^2 c^2 x^2-\frac{1}{3} b^2 c^4 \log \left(\frac{c^2}{x^2}+1\right)",1,"(x*(b^2*c^2*x + 3*a^2*x^3 + 2*a*b*c*(-3*c^2 + x^2)) + 2*b*(b*c*x*(-3*c^2 + x^2) + 3*a*(-c^4 + x^4))*ArcTan[c/x] + 3*b^2*(-c^4 + x^4)*ArcTan[c/x]^2 - 4*b^2*c^4*Log[c^2 + x^2])/12","A",1
141,1,152,152,0.3552256,"\int x^2 \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^2 \, dx","Integrate[x^2*(a + b*ArcTan[c/x])^2,x]","\frac{1}{3} \left(a^2 x^3+2 a b c^3 \log \left(\frac{c}{x}\right)-a b c^3 \log \left(\frac{c^2}{x^2}+1\right)+b \tan ^{-1}\left(\frac{c}{x}\right) \left(2 a x^3+2 b c^3 \log \left(1-e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)+b c \left(c^2+x^2\right)\right)+a b c x^2-i b^2 c^3 \text{Li}_2\left(e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)+b^2 \left(x^3-i c^3\right) \tan ^{-1}\left(\frac{c}{x}\right)^2+b^2 c^2 x\right)","-\frac{1}{3} i c^3 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{2}{3} b c^3 \log \left(2-\frac{2}{1-\frac{i c}{x}}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)+\frac{1}{3} x^3 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{1}{3} b c x^2 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)-\frac{1}{3} i b^2 c^3 \text{Li}_2\left(\frac{2}{1-\frac{i c}{x}}-1\right)+\frac{1}{3} b^2 c^3 \cot ^{-1}\left(\frac{x}{c}\right)+\frac{1}{3} b^2 c^2 x",1,"(b^2*c^2*x + a*b*c*x^2 + a^2*x^3 + b^2*((-I)*c^3 + x^3)*ArcTan[c/x]^2 + b*ArcTan[c/x]*(2*a*x^3 + b*c*(c^2 + x^2) + 2*b*c^3*Log[1 - E^((2*I)*ArcTan[c/x])]) - a*b*c^3*Log[1 + c^2/x^2] + 2*a*b*c^3*Log[c/x] - I*b^2*c^3*PolyLog[2, E^((2*I)*ArcTan[c/x])])/3","A",0
142,1,73,82,0.0579705,"\int x \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^2 \, dx","Integrate[x*(a + b*ArcTan[c/x])^2,x]","\frac{1}{2} \left(2 b \tan ^{-1}\left(\frac{c}{x}\right) \left(a \left(c^2+x^2\right)+b c x\right)+a x (a x+2 b c)+b^2 c^2 \log \left(c^2+x^2\right)+b^2 \left(c^2+x^2\right) \tan ^{-1}\left(\frac{c}{x}\right)^2\right)","\frac{1}{2} c^2 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{1}{2} x^2 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2+b c x \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)+\frac{1}{2} b^2 c^2 \log \left(\frac{c^2}{x^2}+1\right)+b^2 c^2 \log (x)",1,"(a*x*(2*b*c + a*x) + 2*b*(b*c*x + a*(c^2 + x^2))*ArcTan[c/x] + b^2*(c^2 + x^2)*ArcTan[c/x]^2 + b^2*c^2*Log[c^2 + x^2])/2","A",1
143,1,105,83,0.1149437,"\int \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^2 \, dx","Integrate[(a + b*ArcTan[c/x])^2,x]","a \left(a x+b c \log \left(\frac{c^2}{x^2}+1\right)-2 b c \log \left(\frac{c}{x}\right)\right)+2 b \tan ^{-1}\left(\frac{c}{x}\right) \left(a x-b c \log \left(1-e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)\right)+i b^2 c \text{Li}_2\left(e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)+b^2 (x+i c) \tan ^{-1}\left(\frac{c}{x}\right)^2","i c \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2+x \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2-2 b c \log \left(\frac{2 c}{c+i x}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)+i b^2 c \text{Li}_2\left(1-\frac{2 c}{c+i x}\right)",1,"b^2*(I*c + x)*ArcTan[c/x]^2 + 2*b*ArcTan[c/x]*(a*x - b*c*Log[1 - E^((2*I)*ArcTan[c/x])]) + a*(a*x + b*c*Log[1 + c^2/x^2] - 2*b*c*Log[c/x]) + I*b^2*c*PolyLog[2, E^((2*I)*ArcTan[c/x])]","A",0
144,1,148,148,0.0924063,"\int \frac{\left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^2}{x} \, dx","Integrate[(a + b*ArcTan[c/x])^2/x,x]","\frac{1}{2} b \left(2 i \text{Li}_2\left(\frac{c+i x}{c-i x}\right) \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)-2 i \text{Li}_2\left(\frac{x-i c}{i c+x}\right) \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)+b \left(\text{Li}_3\left(\frac{c+i x}{c-i x}\right)-\text{Li}_3\left(\frac{x-i c}{i c+x}\right)\right)\right)-2 \tanh ^{-1}\left(\frac{c+i x}{c-i x}\right) \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^2","i b \text{Li}_2\left(1-\frac{2}{\frac{i c}{x}+1}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)-i b \text{Li}_2\left(\frac{2}{\frac{i c}{x}+1}-1\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)-2 \tanh ^{-1}\left(1-\frac{2}{1+\frac{i c}{x}}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{1}{2} b^2 \text{Li}_3\left(1-\frac{2}{\frac{i c}{x}+1}\right)-\frac{1}{2} b^2 \text{Li}_3\left(\frac{2}{\frac{i c}{x}+1}-1\right)",1,"-2*(a + b*ArcTan[c/x])^2*ArcTanh[(c + I*x)/(c - I*x)] + (b*((2*I)*(a + b*ArcTan[c/x])*PolyLog[2, (c + I*x)/(c - I*x)] - (2*I)*(a + b*ArcTan[c/x])*PolyLog[2, ((-I)*c + x)/(I*c + x)] + b*(PolyLog[3, (c + I*x)/(c - I*x)] - PolyLog[3, ((-I)*c + x)/(I*c + x)])))/2","A",0
145,1,107,96,0.1198984,"\int \frac{\left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^2}{x^2} \, dx","Integrate[(a + b*ArcTan[c/x])^2/x^2,x]","-\frac{a \left(a c+2 b x \log \left(\frac{1}{\sqrt{\frac{c^2}{x^2}+1}}\right)\right)+2 b \tan ^{-1}\left(\frac{c}{x}\right) \left(a c+b x \log \left(1+e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)\right)-i b^2 x \text{Li}_2\left(-e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)+b^2 (c-i x) \tan ^{-1}\left(\frac{c}{x}\right)^2}{c x}","-\frac{i \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2}{c}-\frac{\left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2}{x}-\frac{2 b \log \left(\frac{2}{1+\frac{i c}{x}}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)}{c}-\frac{i b^2 \text{Li}_2\left(1-\frac{2}{\frac{i c}{x}+1}\right)}{c}",1,"-((b^2*(c - I*x)*ArcTan[c/x]^2 + 2*b*ArcTan[c/x]*(a*c + b*x*Log[1 + E^((2*I)*ArcTan[c/x])]) + a*(a*c + 2*b*x*Log[1/Sqrt[1 + c^2/x^2]]) - I*b^2*x*PolyLog[2, -E^((2*I)*ArcTan[c/x])])/(c*x))","A",0
146,1,99,84,0.0757793,"\int \frac{\left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^2}{x^3} \, dx","Integrate[(a + b*ArcTan[c/x])^2/x^3,x]","-\frac{a^2 c^2-2 a b x^2 \tan ^{-1}\left(\frac{x}{c}\right)-2 a b c x+2 b c \tan ^{-1}\left(\frac{c}{x}\right) (a c-b x)+b^2 x^2 \log \left(c^2+x^2\right)+b^2 \left(c^2+x^2\right) \tan ^{-1}\left(\frac{c}{x}\right)^2-2 b^2 x^2 \log (x)}{2 c^2 x^2}","-\frac{\left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2}{2 c^2}-\frac{\left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2}{2 x^2}+\frac{a b}{c x}-\frac{b^2 \log \left(\frac{c^2}{x^2}+1\right)}{2 c^2}+\frac{b^2 \cot ^{-1}\left(\frac{x}{c}\right)}{c x}",1,"-1/2*(a^2*c^2 - 2*a*b*c*x + 2*b*c*(a*c - b*x)*ArcTan[c/x] + b^2*(c^2 + x^2)*ArcTan[c/x]^2 - 2*a*b*x^2*ArcTan[x/c] - 2*b^2*x^2*Log[x] + b^2*x^2*Log[c^2 + x^2])/(c^2*x^2)","A",1
147,1,253,214,0.6526532,"\int x^3 \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^3 \, dx","Integrate[x^3*(a + b*ArcTan[c/x])^3,x]","\frac{1}{4} \left(a^3 x^4+b \tan ^{-1}\left(\frac{c}{x}\right) \left(3 a^2 \left(x^4-c^4\right)+2 a b c x \left(x^2-3 c^2\right)+8 b^2 c^4 \log \left(1-e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)+b^2 c^2 \left(c^2+x^2\right)\right)-3 a^2 b c^3 x+a^2 b c x^3+a b^2 c^4+a b^2 c^2 x^2+8 a b^2 c^4 \log \left(\frac{c}{x \sqrt{\frac{c^2}{x^2}+1}}\right)+b^2 \tan ^{-1}\left(\frac{c}{x}\right)^2 \left(3 a \left(x^4-c^4\right)+b c \left(-4 i c^3-3 c^2 x+x^3\right)\right)-4 i b^3 c^4 \text{Li}_2\left(e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)+b^3 \left(x^4-c^4\right) \tan ^{-1}\left(\frac{c}{x}\right)^3+b^3 c^3 x\right)","2 b^2 c^4 \log \left(2-\frac{2}{1-\frac{i c}{x}}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)+\frac{1}{4} b^2 c^2 x^2 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)-\frac{1}{4} c^4 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^3-i b c^4 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2-\frac{3}{4} b c^3 x \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{1}{4} x^4 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^3+\frac{1}{4} b c x^3 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2-i b^3 c^4 \text{Li}_2\left(\frac{2}{1-\frac{i c}{x}}-1\right)+\frac{1}{4} b^3 c^4 \cot ^{-1}\left(\frac{x}{c}\right)+\frac{1}{4} b^3 c^3 x",1,"(a*b^2*c^4 - 3*a^2*b*c^3*x + b^3*c^3*x + a*b^2*c^2*x^2 + a^2*b*c*x^3 + a^3*x^4 + b^2*(b*c*((-4*I)*c^3 - 3*c^2*x + x^3) + 3*a*(-c^4 + x^4))*ArcTan[c/x]^2 + b^3*(-c^4 + x^4)*ArcTan[c/x]^3 + b*ArcTan[c/x]*(2*a*b*c*x*(-3*c^2 + x^2) + b^2*c^2*(c^2 + x^2) + 3*a^2*(-c^4 + x^4) + 8*b^2*c^4*Log[1 - E^((2*I)*ArcTan[c/x])]) + 8*a*b^2*c^4*Log[c/(Sqrt[1 + c^2/x^2]*x)] - (4*I)*b^3*c^4*PolyLog[2, E^((2*I)*ArcTan[c/x])])/4","A",0
148,1,330,229,0.7312798,"\int x^2 \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^3 \, dx","Integrate[x^2*(a + b*ArcTan[c/x])^3,x]","\frac{a^3 x^3}{3}-\frac{1}{2} a^2 b c^3 \log \left(c^2+x^2\right)+a^2 b x^3 \tan ^{-1}\left(\frac{c}{x}\right)+\frac{1}{2} a^2 b c x^2+a b^2 \left(-i c^3 \text{Li}_2\left(e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)+\left(x^3-i c^3\right) \tan ^{-1}\left(\frac{c}{x}\right)^2+c \tan ^{-1}\left(\frac{c}{x}\right) \left(2 c^2 \log \left(1-e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)+c^2+x^2\right)+c^2 x\right)+\frac{1}{24} b^3 \left(24 i c^3 \tan ^{-1}\left(\frac{c}{x}\right) \text{Li}_2\left(e^{-2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)+12 c^3 \text{Li}_3\left(e^{-2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)+8 i c^3 \tan ^{-1}\left(\frac{c}{x}\right)^3+12 c^3 \tan ^{-1}\left(\frac{c}{x}\right)^2+24 c^3 \tan ^{-1}\left(\frac{c}{x}\right)^2 \log \left(1-e^{-2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)-i \pi ^3 c^3+24 c^2 x \tan ^{-1}\left(\frac{c}{x}\right)-24 c^3 \log \left(\frac{c}{x \sqrt{\frac{c^2}{x^2}+1}}\right)+8 x^3 \tan ^{-1}\left(\frac{c}{x}\right)^3+12 c x^2 \tan ^{-1}\left(\frac{c}{x}\right)^2\right)","-i b^2 c^3 \text{Li}_2\left(\frac{2}{1-\frac{i c}{x}}-1\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)+b^2 c^2 x \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)-\frac{1}{3} i c^3 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^3+\frac{1}{2} b c^3 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2+b c^3 \log \left(2-\frac{2}{1-\frac{i c}{x}}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{1}{3} x^3 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^3+\frac{1}{2} b c x^2 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{1}{2} b^3 c^3 \text{Li}_3\left(\frac{2}{1-\frac{i c}{x}}-1\right)+b^3 c^3 \log (x)+\frac{1}{2} b^3 c^3 \log \left(\frac{c^2}{x^2}+1\right)",1,"(a^2*b*c*x^2)/2 + (a^3*x^3)/3 + a^2*b*x^3*ArcTan[c/x] - (a^2*b*c^3*Log[c^2 + x^2])/2 + a*b^2*(c^2*x + ((-I)*c^3 + x^3)*ArcTan[c/x]^2 + c*ArcTan[c/x]*(c^2 + x^2 + 2*c^2*Log[1 - E^((2*I)*ArcTan[c/x])]) - I*c^3*PolyLog[2, E^((2*I)*ArcTan[c/x])]) + (b^3*((-I)*c^3*Pi^3 + 24*c^2*x*ArcTan[c/x] + 12*c^3*ArcTan[c/x]^2 + 12*c*x^2*ArcTan[c/x]^2 + (8*I)*c^3*ArcTan[c/x]^3 + 8*x^3*ArcTan[c/x]^3 + 24*c^3*ArcTan[c/x]^2*Log[1 - E^((-2*I)*ArcTan[c/x])] - 24*c^3*Log[c/(Sqrt[1 + c^2/x^2]*x)] + (24*I)*c^3*ArcTan[c/x]*PolyLog[2, E^((-2*I)*ArcTan[c/x])] + 12*c^3*PolyLog[3, E^((-2*I)*ArcTan[c/x])]))/24","A",0
149,1,174,145,0.2893006,"\int x \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^3 \, dx","Integrate[x*(a + b*ArcTan[c/x])^3,x]","\frac{1}{2} \left(a \left(a x (a x+3 b c)-6 b^2 c^2 \log \left(\frac{c}{x \sqrt{\frac{c^2}{x^2}+1}}\right)\right)+3 b^2 \tan ^{-1}\left(\frac{c}{x}\right)^2 \left(a \left(c^2+x^2\right)+b c (x+i c)\right)+3 b \tan ^{-1}\left(\frac{c}{x}\right) \left(a \left(a \left(c^2+x^2\right)+2 b c x\right)-2 b^2 c^2 \log \left(1-e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)\right)+3 i b^3 c^2 \text{Li}_2\left(e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)+b^3 \left(c^2+x^2\right) \tan ^{-1}\left(\frac{c}{x}\right)^3\right)","-3 b^2 c^2 \log \left(2-\frac{2}{1-\frac{i c}{x}}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)+\frac{3}{2} i b c^2 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{1}{2} c^2 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^3+\frac{1}{2} x^2 \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^3+\frac{3}{2} b c x \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{3}{2} i b^3 c^2 \text{Li}_2\left(\frac{2}{1-\frac{i c}{x}}-1\right)",1,"(3*b^2*(b*c*(I*c + x) + a*(c^2 + x^2))*ArcTan[c/x]^2 + b^3*(c^2 + x^2)*ArcTan[c/x]^3 + 3*b*ArcTan[c/x]*(a*(2*b*c*x + a*(c^2 + x^2)) - 2*b^2*c^2*Log[1 - E^((2*I)*ArcTan[c/x])]) + a*(a*x*(3*b*c + a*x) - 6*b^2*c^2*Log[c/(Sqrt[1 + c^2/x^2]*x)]) + (3*I)*b^3*c^2*PolyLog[2, E^((2*I)*ArcTan[c/x])])/2","A",0
150,1,215,119,0.2708005,"\int \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^3 \, dx","Integrate[(a + b*ArcTan[c/x])^3,x]","a^3 x+\frac{3}{2} a^2 b c \log \left(c^2+x^2\right)+3 a^2 b x \tan ^{-1}\left(\frac{c}{x}\right)-3 a b^2 \left(-i c \text{Li}_2\left(e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)-\left((x+i c) \tan ^{-1}\left(\frac{c}{x}\right)^2\right)+2 c \tan ^{-1}\left(\frac{c}{x}\right) \log \left(1-e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)\right)-\frac{1}{8} b^3 \left(24 i c \tan ^{-1}\left(\frac{c}{x}\right) \text{Li}_2\left(e^{-2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)+12 c \text{Li}_3\left(e^{-2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)+8 i c \tan ^{-1}\left(\frac{c}{x}\right)^3-8 x \tan ^{-1}\left(\frac{c}{x}\right)^3+24 c \tan ^{-1}\left(\frac{c}{x}\right)^2 \log \left(1-e^{-2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)-i \pi ^3 c\right)","3 i b^2 c \text{Li}_2\left(1-\frac{2 c}{c+i x}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)+i c \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^3+x \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^3-3 b c \log \left(\frac{2 c}{c+i x}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2-\frac{3}{2} b^3 c \text{Li}_3\left(1-\frac{2 c}{c+i x}\right)",1,"a^3*x + 3*a^2*b*x*ArcTan[c/x] + (3*a^2*b*c*Log[c^2 + x^2])/2 - 3*a*b^2*(-((I*c + x)*ArcTan[c/x]^2) + 2*c*ArcTan[c/x]*Log[1 - E^((2*I)*ArcTan[c/x])] - I*c*PolyLog[2, E^((2*I)*ArcTan[c/x])]) - (b^3*((-I)*c*Pi^3 + (8*I)*c*ArcTan[c/x]^3 - 8*x*ArcTan[c/x]^3 + 24*c*ArcTan[c/x]^2*Log[1 - E^((-2*I)*ArcTan[c/x])] + (24*I)*c*ArcTan[c/x]*PolyLog[2, E^((-2*I)*ArcTan[c/x])] + 12*c*PolyLog[3, E^((-2*I)*ArcTan[c/x])]))/8","A",0
151,1,219,230,0.2199191,"\int \frac{\left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^3}{x} \, dx","Integrate[(a + b*ArcTan[c/x])^3/x,x]","\frac{3}{4} i b \left(2 \text{Li}_2\left(\frac{c+i x}{c-i x}\right) \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^2-2 \text{Li}_2\left(\frac{x-i c}{i c+x}\right) \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^2+b \left(-2 i \text{Li}_3\left(\frac{c+i x}{c-i x}\right) \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)+2 i \text{Li}_3\left(\frac{x-i c}{i c+x}\right) \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)+b \left(\text{Li}_4\left(\frac{x-i c}{i c+x}\right)-\text{Li}_4\left(\frac{c+i x}{c-i x}\right)\right)\right)\right)-2 \tanh ^{-1}\left(\frac{c+i x}{c-i x}\right) \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^3","\frac{3}{2} b^2 \text{Li}_3\left(1-\frac{2}{\frac{i c}{x}+1}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)-\frac{3}{2} b^2 \text{Li}_3\left(\frac{2}{\frac{i c}{x}+1}-1\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)+\frac{3}{2} i b \text{Li}_2\left(1-\frac{2}{\frac{i c}{x}+1}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2-\frac{3}{2} i b \text{Li}_2\left(\frac{2}{\frac{i c}{x}+1}-1\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2-2 \tanh ^{-1}\left(1-\frac{2}{1+\frac{i c}{x}}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^3-\frac{3}{4} i b^3 \text{Li}_4\left(1-\frac{2}{\frac{i c}{x}+1}\right)+\frac{3}{4} i b^3 \text{Li}_4\left(\frac{2}{\frac{i c}{x}+1}-1\right)",1,"-2*(a + b*ArcTan[c/x])^3*ArcTanh[(c + I*x)/(c - I*x)] + ((3*I)/4)*b*(2*(a + b*ArcTan[c/x])^2*PolyLog[2, (c + I*x)/(c - I*x)] - 2*(a + b*ArcTan[c/x])^2*PolyLog[2, ((-I)*c + x)/(I*c + x)] + b*((-2*I)*(a + b*ArcTan[c/x])*PolyLog[3, (c + I*x)/(c - I*x)] + (2*I)*(a + b*ArcTan[c/x])*PolyLog[3, ((-I)*c + x)/(I*c + x)] + b*(-PolyLog[4, (c + I*x)/(c - I*x)] + PolyLog[4, ((-I)*c + x)/(I*c + x)])))","A",0
152,1,189,136,0.1894366,"\int \frac{\left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^3}{x^2} \, dx","Integrate[(a + b*ArcTan[c/x])^3/x^2,x]","\frac{-2 \left(a^2 \left(a c+3 b x \log \left(\frac{1}{\sqrt{\frac{c^2}{x^2}+1}}\right)\right)+3 b^2 \tan ^{-1}\left(\frac{c}{x}\right)^2 \left(a c-i a x+b x \log \left(1+e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)\right)+3 a b \tan ^{-1}\left(\frac{c}{x}\right) \left(a c+2 b x \log \left(1+e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)\right)+b^3 (c-i x) \tan ^{-1}\left(\frac{c}{x}\right)^3\right)+6 i b^2 x \text{Li}_2\left(-e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right) \left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)-3 b^3 x \text{Li}_3\left(-e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)}{2 c x}","-\frac{3 i b^2 \text{Li}_2\left(1-\frac{2}{\frac{i c}{x}+1}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)}{c}-\frac{i \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^3}{c}-\frac{\left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^3}{x}-\frac{3 b \log \left(\frac{2}{1+\frac{i c}{x}}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2}{c}-\frac{3 b^3 \text{Li}_3\left(1-\frac{2}{\frac{i c}{x}+1}\right)}{2 c}",1,"(-2*(b^3*(c - I*x)*ArcTan[c/x]^3 + 3*b^2*ArcTan[c/x]^2*(a*c - I*a*x + b*x*Log[1 + E^((2*I)*ArcTan[c/x])]) + 3*a*b*ArcTan[c/x]*(a*c + 2*b*x*Log[1 + E^((2*I)*ArcTan[c/x])]) + a^2*(a*c + 3*b*x*Log[1/Sqrt[1 + c^2/x^2]])) + (6*I)*b^2*x*(a + b*ArcTan[c/x])*PolyLog[2, -E^((2*I)*ArcTan[c/x])] - 3*b^3*x*PolyLog[3, -E^((2*I)*ArcTan[c/x])])/(2*c*x)","A",0
153,1,178,147,0.2927659,"\int \frac{\left(a+b \tan ^{-1}\left(\frac{c}{x}\right)\right)^3}{x^3} \, dx","Integrate[(a + b*ArcTan[c/x])^3/x^3,x]","\frac{a \left(a c (3 b x-a c)+6 b^2 x^2 \log \left(\frac{1}{\sqrt{\frac{c^2}{x^2}+1}}\right)\right)-3 b \tan ^{-1}\left(\frac{c}{x}\right) \left(a \left(a \left(c^2+x^2\right)-2 b c x\right)-2 b^2 x^2 \log \left(1+e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)\right)+3 b^2 (c-i x) \tan ^{-1}\left(\frac{c}{x}\right)^2 (b x-a (c+i x))-\left(b^3 \left(c^2+x^2\right) \tan ^{-1}\left(\frac{c}{x}\right)^3\right)-3 i b^3 x^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}\left(\frac{c}{x}\right)}\right)}{2 c^2 x^2}","\frac{3 b^2 \log \left(\frac{2}{1+\frac{i c}{x}}\right) \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)}{c^2}+\frac{3 i b \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2}{2 c^2}-\frac{\left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^3}{2 c^2}-\frac{\left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^3}{2 x^2}+\frac{3 b \left(a+b \cot ^{-1}\left(\frac{x}{c}\right)\right)^2}{2 c x}+\frac{3 i b^3 \text{Li}_2\left(1-\frac{2}{\frac{i c}{x}+1}\right)}{2 c^2}",1,"(3*b^2*(c - I*x)*(-(a*(c + I*x)) + b*x)*ArcTan[c/x]^2 - b^3*(c^2 + x^2)*ArcTan[c/x]^3 - 3*b*ArcTan[c/x]*(a*(-2*b*c*x + a*(c^2 + x^2)) - 2*b^2*x^2*Log[1 + E^((2*I)*ArcTan[c/x])]) + a*(a*c*(-(a*c) + 3*b*x) + 6*b^2*x^2*Log[1/Sqrt[1 + c^2/x^2]]) - (3*I)*b^3*x^2*PolyLog[2, -E^((2*I)*ArcTan[c/x])])/(2*c^2*x^2)","A",0
154,1,34,51,0.0147498,"\int x^2 \tan ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x^2*ArcTan[Sqrt[x]],x]","\frac{1}{45} \left(15 \left(x^3+1\right) \tan ^{-1}\left(\sqrt{x}\right)+\sqrt{x} \left(-3 x^2+5 x-15\right)\right)","-\frac{x^{5/2}}{15}+\frac{x^{3/2}}{9}+\frac{1}{3} x^3 \tan ^{-1}\left(\sqrt{x}\right)-\frac{\sqrt{x}}{3}+\frac{1}{3} \tan ^{-1}\left(\sqrt{x}\right)",1,"(Sqrt[x]*(-15 + 5*x - 3*x^2) + 15*(1 + x^3)*ArcTan[Sqrt[x]])/45","A",1
155,1,28,42,0.0113448,"\int x \tan ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x*ArcTan[Sqrt[x]],x]","\frac{1}{6} \left(3 \left(x^2-1\right) \tan ^{-1}\left(\sqrt{x}\right)-(x-3) \sqrt{x}\right)","-\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \tan ^{-1}\left(\sqrt{x}\right)+\frac{\sqrt{x}}{2}-\frac{1}{2} \tan ^{-1}\left(\sqrt{x}\right)",1,"(-((-3 + x)*Sqrt[x]) + 3*(-1 + x^2)*ArcTan[Sqrt[x]])/6","A",1
156,1,18,22,0.0067602,"\int \tan ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[ArcTan[Sqrt[x]],x]","(x+1) \tan ^{-1}\left(\sqrt{x}\right)-\sqrt{x}","-\sqrt{x}+x \tan ^{-1}\left(\sqrt{x}\right)+\tan ^{-1}\left(\sqrt{x}\right)",1,"-Sqrt[x] + (1 + x)*ArcTan[Sqrt[x]]","A",1
157,1,31,31,0.0046252,"\int \frac{\tan ^{-1}\left(\sqrt{x}\right)}{x} \, dx","Integrate[ArcTan[Sqrt[x]]/x,x]","i \text{Li}_2\left(-i \sqrt{x}\right)-i \text{Li}_2\left(i \sqrt{x}\right)","i \text{Li}_2\left(-i \sqrt{x}\right)-i \text{Li}_2\left(i \sqrt{x}\right)",1,"I*PolyLog[2, (-I)*Sqrt[x]] - I*PolyLog[2, I*Sqrt[x]]","A",1
158,1,30,27,0.0095127,"\int \frac{\tan ^{-1}\left(\sqrt{x}\right)}{x^2} \, dx","Integrate[ArcTan[Sqrt[x]]/x^2,x]","-\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-x\right)}{\sqrt{x}}-\frac{\tan ^{-1}\left(\sqrt{x}\right)}{x}","-\frac{1}{\sqrt{x}}-\frac{\tan ^{-1}\left(\sqrt{x}\right)}{x}-\tan ^{-1}\left(\sqrt{x}\right)",1,"-(ArcTan[Sqrt[x]]/x) - Hypergeometric2F1[-1/2, 1, 1/2, -x]/Sqrt[x]","C",1
159,1,34,42,0.0123839,"\int \frac{\tan ^{-1}\left(\sqrt{x}\right)}{x^3} \, dx","Integrate[ArcTan[Sqrt[x]]/x^3,x]","-\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-x\right)}{6 x^{3/2}}-\frac{\tan ^{-1}\left(\sqrt{x}\right)}{2 x^2}","-\frac{1}{6 x^{3/2}}-\frac{\tan ^{-1}\left(\sqrt{x}\right)}{2 x^2}+\frac{1}{2 \sqrt{x}}+\frac{1}{2} \tan ^{-1}\left(\sqrt{x}\right)",1,"-1/2*ArcTan[Sqrt[x]]/x^2 - Hypergeometric2F1[-3/2, 1, -1/2, -x]/(6*x^(3/2))","C",1
160,1,30,36,0.0164916,"\int x^{3/2} \tan ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x^(3/2)*ArcTan[Sqrt[x]],x]","\frac{1}{10} \left(4 x^{5/2} \tan ^{-1}\left(\sqrt{x}\right)-(x-2) x-2 \log (x+1)\right)","\frac{2}{5} x^{5/2} \tan ^{-1}\left(\sqrt{x}\right)-\frac{x^2}{10}+\frac{x}{5}-\frac{1}{5} \log (x+1)",1,"(-((-2 + x)*x) + 4*x^(5/2)*ArcTan[Sqrt[x]] - 2*Log[1 + x])/10","A",1
161,1,25,29,0.0107429,"\int \sqrt{x} \tan ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[Sqrt[x]*ArcTan[Sqrt[x]],x]","\frac{1}{3} \left(2 x^{3/2} \tan ^{-1}\left(\sqrt{x}\right)-x+\log (x+1)\right)","\frac{2}{3} x^{3/2} \tan ^{-1}\left(\sqrt{x}\right)-\frac{x}{3}+\frac{1}{3} \log (x+1)",1,"(-x + 2*x^(3/2)*ArcTan[Sqrt[x]] + Log[1 + x])/3","A",1
162,1,20,20,0.0075147,"\int \frac{\tan ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Integrate[ArcTan[Sqrt[x]]/Sqrt[x],x]","2 \sqrt{x} \tan ^{-1}\left(\sqrt{x}\right)-\log (x+1)","2 \sqrt{x} \tan ^{-1}\left(\sqrt{x}\right)-\log (x+1)",1,"2*Sqrt[x]*ArcTan[Sqrt[x]] - Log[1 + x]","A",1
163,1,22,22,0.0119354,"\int \frac{\tan ^{-1}\left(\sqrt{x}\right)}{x^{3/2}} \, dx","Integrate[ArcTan[Sqrt[x]]/x^(3/2),x]","\log (x)-\log (x+1)-\frac{2 \tan ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}}","\log (x)-\log (x+1)-\frac{2 \tan ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}}",1,"(-2*ArcTan[Sqrt[x]])/Sqrt[x] + Log[x] - Log[1 + x]","A",1
164,1,31,37,0.0198724,"\int \frac{\tan ^{-1}\left(\sqrt{x}\right)}{x^{5/2}} \, dx","Integrate[ArcTan[Sqrt[x]]/x^(5/2),x]","\frac{1}{3} \left(-\frac{2 \tan ^{-1}\left(\sqrt{x}\right)}{x^{3/2}}-\frac{1}{x}-\log (x)+\log (x+1)\right)","-\frac{2 \tan ^{-1}\left(\sqrt{x}\right)}{3 x^{3/2}}-\frac{1}{3 x}-\frac{\log (x)}{3}+\frac{1}{3} \log (x+1)",1,"(-x^(-1) - (2*ArcTan[Sqrt[x]])/x^(3/2) - Log[x] + Log[1 + x])/3","A",1
165,1,33,33,0.0065256,"\int \frac{\tan ^{-1}\left(a x^5\right)}{x} \, dx","Integrate[ArcTan[a*x^5]/x,x]","\frac{1}{10} i \text{Li}_2\left(-i a x^5\right)-\frac{1}{10} i \text{Li}_2\left(i a x^5\right)","\frac{1}{10} i \text{Li}_2\left(-i a x^5\right)-\frac{1}{10} i \text{Li}_2\left(i a x^5\right)",1,"(I/10)*PolyLog[2, (-I)*a*x^5] - (I/10)*PolyLog[2, I*a*x^5]","A",1
166,1,32,39,0.0138161,"\int \frac{\tan ^{-1}\left(a x^n\right)}{x} \, dx","Integrate[ArcTan[a*x^n]/x,x]","\frac{i \left(\text{Li}_2\left(-i a x^n\right)-\text{Li}_2\left(i a x^n\right)\right)}{2 n}","\frac{i \text{Li}_2\left(-i a x^n\right)}{2 n}-\frac{i \text{Li}_2\left(i a x^n\right)}{2 n}",1,"((I/2)*(PolyLog[2, (-I)*a*x^n] - PolyLog[2, I*a*x^n]))/n","A",1