1,1,56,72,0.0222482,"\int (c e+d e x)^3 \left(a+b \tan ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcTan[c + d*x]),x]","\frac{e^3 \left(\frac{1}{4} (c+d x)^4 \left(a+b \tan ^{-1}(c+d x)\right)-\frac{1}{4} b \left(\frac{1}{3} (c+d x)^3+\tan ^{-1}(c+d x)-d x\right)\right)}{d}","\frac{e^3 (c+d x)^4 \left(a+b \tan ^{-1}(c+d x)\right)}{4 d}-\frac{b e^3 (c+d x)^3}{12 d}-\frac{b e^3 \tan ^{-1}(c+d x)}{4 d}+\frac{1}{4} b e^3 x",1,"(e^3*(-1/4*(b*(-(d*x) + (c + d*x)^3/3 + ArcTan[c + d*x])) + ((c + d*x)^4*(a + b*ArcTan[c + d*x]))/4))/d","A",1
2,1,54,67,0.0229991,"\int (c e+d e x)^2 \left(a+b \tan ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcTan[c + d*x]),x]","\frac{e^2 \left(\frac{1}{3} (c+d x)^3 \left(a+b \tan ^{-1}(c+d x)\right)-\frac{1}{6} b \left((c+d x)^2-\log \left((c+d x)^2+1\right)\right)\right)}{d}","\frac{e^2 (c+d x)^3 \left(a+b \tan ^{-1}(c+d x)\right)}{3 d}-\frac{b e^2 (c+d x)^2}{6 d}+\frac{b e^2 \log \left((c+d x)^2+1\right)}{6 d}",1,"(e^2*(((c + d*x)^3*(a + b*ArcTan[c + d*x]))/3 - (b*((c + d*x)^2 - Log[1 + (c + d*x)^2]))/6))/d","A",1
3,1,40,48,0.0162102,"\int (c e+d e x) \left(a+b \tan ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcTan[c + d*x]),x]","\frac{e \left((c+d x)^2 \left(a+b \tan ^{-1}(c+d x)\right)+b \left(\tan ^{-1}(c+d x)-d x\right)\right)}{2 d}","\frac{e (c+d x)^2 \left(a+b \tan ^{-1}(c+d x)\right)}{2 d}+\frac{b e \tan ^{-1}(c+d x)}{2 d}-\frac{b e x}{2}",1,"(e*(b*(-(d*x) + ArcTan[c + d*x]) + (c + d*x)^2*(a + b*ArcTan[c + d*x])))/(2*d)","A",1
4,1,52,63,0.0179811,"\int \frac{a+b \tan ^{-1}(c+d x)}{c e+d e x} \, dx","Integrate[(a + b*ArcTan[c + d*x])/(c*e + d*e*x),x]","\frac{a \log (c+d x)+\frac{1}{2} i b \text{Li}_2(-i (c+d x))-\frac{1}{2} i b \text{Li}_2(i (c+d x))}{d e}","\frac{a \log (c+d x)}{d e}+\frac{i b \text{Li}_2(-i (c+d x))}{2 d e}-\frac{i b \text{Li}_2(i (c+d x))}{2 d e}",1,"(a*Log[c + d*x] + (I/2)*b*PolyLog[2, (-I)*(c + d*x)] - (I/2)*b*PolyLog[2, I*(c + d*x)])/(d*e)","A",1
5,1,50,61,0.0273931,"\int \frac{a+b \tan ^{-1}(c+d x)}{(c e+d e x)^2} \, dx","Integrate[(a + b*ArcTan[c + d*x])/(c*e + d*e*x)^2,x]","\frac{-\frac{a+b \tan ^{-1}(c+d x)}{c+d x}+b \log (c+d x)-\frac{1}{2} b \log \left((c+d x)^2+1\right)}{d e^2}","-\frac{a+b \tan ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac{b \log (c+d x)}{d e^2}-\frac{b \log \left((c+d x)^2+1\right)}{2 d e^2}",1,"(-((a + b*ArcTan[c + d*x])/(c + d*x)) + b*Log[c + d*x] - (b*Log[1 + (c + d*x)^2])/2)/(d*e^2)","A",1
6,1,51,63,0.0221802,"\int \frac{a+b \tan ^{-1}(c+d x)}{(c e+d e x)^3} \, dx","Integrate[(a + b*ArcTan[c + d*x])/(c*e + d*e*x)^3,x]","-\frac{a+b (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-(c+d x)^2\right)+b \tan ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}","-\frac{a+b \tan ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac{b}{2 d e^3 (c+d x)}-\frac{b \tan ^{-1}(c+d x)}{2 d e^3}",1,"-1/2*(a + b*ArcTan[c + d*x] + b*(c + d*x)*Hypergeometric2F1[-1/2, 1, 1/2, -(c + d*x)^2])/(d*e^3*(c + d*x)^2)","C",1
7,1,216,157,0.1203959,"\int (c e+d e x)^3 \left(a+b \tan ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcTan[c + d*x])^2,x]","\frac{e^3 \left((c+d x) \left(3 a^2 (c+d x)^3-2 a b \left(c^2+2 c d x+d^2 x^2-3\right)+b^2 (c+d x)\right)+2 b \tan ^{-1}(c+d x) \left(3 a \left(c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4-1\right)-b \left(c^3+3 c^2 d x+3 c d^2 x^2-3 c+d^3 x^3-3 d x\right)\right)+3 b^2 \left(c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4-1\right) \tan ^{-1}(c+d x)^2-4 b^2 \log \left((c+d x)^2+1\right)\right)}{12 d}","\frac{e^3 (c+d x)^4 \left(a+b \tan ^{-1}(c+d x)\right)^2}{4 d}-\frac{b e^3 (c+d x)^3 \left(a+b \tan ^{-1}(c+d x)\right)}{6 d}-\frac{e^3 \left(a+b \tan ^{-1}(c+d x)\right)^2}{4 d}+\frac{1}{2} a b e^3 x+\frac{b^2 e^3 (c+d x)^2}{12 d}-\frac{b^2 e^3 \log \left((c+d x)^2+1\right)}{3 d}+\frac{b^2 e^3 (c+d x) \tan ^{-1}(c+d x)}{2 d}",1,"(e^3*((c + d*x)*(b^2*(c + d*x) + 3*a^2*(c + d*x)^3 - 2*a*b*(-3 + c^2 + 2*c*d*x + d^2*x^2)) + 2*b*(-(b*(-3*c + c^3 - 3*d*x + 3*c^2*d*x + 3*c*d^2*x^2 + d^3*x^3)) + 3*a*(-1 + c^4 + 4*c^3*d*x + 6*c^2*d^2*x^2 + 4*c*d^3*x^3 + d^4*x^4))*ArcTan[c + d*x] + 3*b^2*(-1 + c^4 + 4*c^3*d*x + 6*c^2*d^2*x^2 + 4*c*d^3*x^3 + d^4*x^4)*ArcTan[c + d*x]^2 - 4*b^2*Log[1 + (c + d*x)^2]))/(12*d)","A",1
8,1,163,183,0.4309024,"\int (c e+d e x)^2 \left(a+b \tan ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcTan[c + d*x])^2,x]","\frac{e^2 \left(a^2 (c+d x)^3+a b \left(-(c+d x)^2+\log \left((c+d x)^2+1\right)+2 (c+d x)^3 \tan ^{-1}(c+d x)\right)+b^2 \left(i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)+(c+d x)^3 \tan ^{-1}(c+d x)^2-(c+d x)^2 \tan ^{-1}(c+d x)+i \tan ^{-1}(c+d x)^2-\tan ^{-1}(c+d x)-2 \tan ^{-1}(c+d x) \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)+c+d x\right)\right)}{3 d}","\frac{e^2 (c+d x)^3 \left(a+b \tan ^{-1}(c+d x)\right)^2}{3 d}-\frac{b e^2 (c+d x)^2 \left(a+b \tan ^{-1}(c+d x)\right)}{3 d}-\frac{i e^2 \left(a+b \tan ^{-1}(c+d x)\right)^2}{3 d}-\frac{2 b e^2 \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{3 d}-\frac{i b^2 e^2 \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right)}{3 d}-\frac{b^2 e^2 \tan ^{-1}(c+d x)}{3 d}+\frac{1}{3} b^2 e^2 x",1,"(e^2*(a^2*(c + d*x)^3 + a*b*(-(c + d*x)^2 + 2*(c + d*x)^3*ArcTan[c + d*x] + Log[1 + (c + d*x)^2]) + b^2*(c + d*x - ArcTan[c + d*x] - (c + d*x)^2*ArcTan[c + d*x] + I*ArcTan[c + d*x]^2 + (c + d*x)^3*ArcTan[c + d*x]^2 - 2*ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + I*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])])))/(3*d)","A",0
9,1,107,95,0.0759314,"\int (c e+d e x) \left(a+b \tan ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcTan[c + d*x])^2,x]","\frac{e \left(2 b \tan ^{-1}(c+d x) \left(a \left(c^2+2 c d x+d^2 x^2+1\right)-b (c+d x)\right)+a (c+d x) (a c+a d x-2 b)+b^2 \left(c^2+2 c d x+d^2 x^2+1\right) \tan ^{-1}(c+d x)^2+b^2 \log \left((c+d x)^2+1\right)\right)}{2 d}","\frac{e (c+d x)^2 \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d}+\frac{e \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d}-a b e x+\frac{b^2 e \log \left((c+d x)^2+1\right)}{2 d}-\frac{b^2 e (c+d x) \tan ^{-1}(c+d x)}{d}",1,"(e*(a*(c + d*x)*(-2*b + a*c + a*d*x) + 2*b*(-(b*(c + d*x)) + a*(1 + c^2 + 2*c*d*x + d^2*x^2))*ArcTan[c + d*x] + b^2*(1 + c^2 + 2*c*d*x + d^2*x^2)*ArcTan[c + d*x]^2 + b^2*Log[1 + (c + d*x)^2]))/(2*d)","A",1
10,1,170,183,0.0794448,"\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^2}{c e+d e x} \, dx","Integrate[(a + b*ArcTan[c + d*x])^2/(c*e + d*e*x),x]","\frac{2 i b \text{Li}_2\left(-\frac{c+d x+i}{c+d x-i}\right) \left(a+b \tan ^{-1}(c+d x)\right)-2 i b \text{Li}_2\left(\frac{c+d x+i}{c+d x-i}\right) \left(a+b \tan ^{-1}(c+d x)\right)+4 \tanh ^{-1}\left(\frac{c+d x+i}{c+d x-i}\right) \left(a+b \tan ^{-1}(c+d x)\right)^2+b^2 \text{Li}_3\left(-\frac{c+d x+i}{c+d x-i}\right)-b^2 \text{Li}_3\left(\frac{c+d x+i}{c+d x-i}\right)}{2 d e}","-\frac{i b \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{d e}+\frac{i b \text{Li}_2\left(\frac{2}{i (c+d x)+1}-1\right) \left(a+b \tan ^{-1}(c+d x)\right)}{d e}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{1+i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)^2}{d e}-\frac{b^2 \text{Li}_3\left(1-\frac{2}{i (c+d x)+1}\right)}{2 d e}+\frac{b^2 \text{Li}_3\left(\frac{2}{i (c+d x)+1}-1\right)}{2 d e}",1,"(4*(a + b*ArcTan[c + d*x])^2*ArcTanh[(I + c + d*x)/(-I + c + d*x)] + (2*I)*b*(a + b*ArcTan[c + d*x])*PolyLog[2, -((I + c + d*x)/(-I + c + d*x))] - (2*I)*b*(a + b*ArcTan[c + d*x])*PolyLog[2, (I + c + d*x)/(-I + c + d*x)] + b^2*PolyLog[3, -((I + c + d*x)/(-I + c + d*x))] - b^2*PolyLog[3, (I + c + d*x)/(-I + c + d*x)])/(2*d*e)","A",1
11,1,135,119,0.2290758,"\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^2}{(c e+d e x)^2} \, dx","Integrate[(a + b*ArcTan[c + d*x])^2/(c*e + d*e*x)^2,x]","\frac{a \left(2 b (c+d x) \log \left(\frac{c+d x}{\sqrt{(c+d x)^2+1}}\right)-a\right)+2 b \tan ^{-1}(c+d x) \left(-a+b (c+d x) \log \left(1-e^{2 i \tan ^{-1}(c+d x)}\right)\right)-i b^2 (c+d x) \text{Li}_2\left(e^{2 i \tan ^{-1}(c+d x)}\right)-i b^2 (c+d x-i) \tan ^{-1}(c+d x)^2}{d e^2 (c+d x)}","-\frac{\left(a+b \tan ^{-1}(c+d x)\right)^2}{d e^2 (c+d x)}-\frac{i \left(a+b \tan ^{-1}(c+d x)\right)^2}{d e^2}+\frac{2 b \log \left(2-\frac{2}{1-i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{d e^2}-\frac{i b^2 \text{Li}_2\left(\frac{2}{1-i (c+d x)}-1\right)}{d e^2}",1,"((-I)*b^2*(-I + c + d*x)*ArcTan[c + d*x]^2 + 2*b*ArcTan[c + d*x]*(-a + b*(c + d*x)*Log[1 - E^((2*I)*ArcTan[c + d*x])]) + a*(-a + 2*b*(c + d*x)*Log[(c + d*x)/Sqrt[1 + (c + d*x)^2]]) - I*b^2*(c + d*x)*PolyLog[2, E^((2*I)*ArcTan[c + d*x])])/(d*e^2*(c + d*x))","A",0
12,1,194,117,0.1321061,"\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^2}{(c e+d e x)^3} \, dx","Integrate[(a + b*ArcTan[c + d*x])^2/(c*e + d*e*x)^3,x]","-\frac{a^2+2 b \tan ^{-1}(c+d x) \left(a \left(c^2+2 c d x+d^2 x^2+1\right)+b (c+d x)\right)+2 a b c+2 a b d x+b^2 c^2 \log \left(c^2+2 c d x+d^2 x^2+1\right)+b^2 d^2 x^2 \log \left(c^2+2 c d x+d^2 x^2+1\right)+2 b^2 c d x \log \left(c^2+2 c d x+d^2 x^2+1\right)+b^2 \left(c^2+2 c d x+d^2 x^2+1\right) \tan ^{-1}(c+d x)^2-2 b^2 (c+d x)^2 \log (c+d x)}{2 d e^3 (c+d x)^2}","-\frac{b \left(a+b \tan ^{-1}(c+d x)\right)}{d e^3 (c+d x)}-\frac{\left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d e^3 (c+d x)^2}-\frac{\left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d e^3}+\frac{b^2 \log (c+d x)}{d e^3}-\frac{b^2 \log \left((c+d x)^2+1\right)}{2 d e^3}",1,"-1/2*(a^2 + 2*a*b*c + 2*a*b*d*x + 2*b*(b*(c + d*x) + a*(1 + c^2 + 2*c*d*x + d^2*x^2))*ArcTan[c + d*x] + b^2*(1 + c^2 + 2*c*d*x + d^2*x^2)*ArcTan[c + d*x]^2 - 2*b^2*(c + d*x)^2*Log[c + d*x] + b^2*c^2*Log[1 + c^2 + 2*c*d*x + d^2*x^2] + 2*b^2*c*d*x*Log[1 + c^2 + 2*c*d*x + d^2*x^2] + b^2*d^2*x^2*Log[1 + c^2 + 2*c*d*x + d^2*x^2])/(d*e^3*(c + d*x)^2)","A",1
13,1,163,194,0.7496912,"\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^2}{(c e+d e x)^4} \, dx","Integrate[(a + b*ArcTan[c + d*x])^2/(c*e + d*e*x)^4,x]","-\frac{\frac{a^2}{(c+d x)^3}+\frac{a b}{(c+d x)^2}+2 a b \log \left(\frac{c+d x}{\sqrt{(c+d x)^2+1}}\right)+b \tan ^{-1}(c+d x) \left(\frac{2 a}{(c+d x)^3}+\frac{b}{(c+d x)^2}+2 b \log \left(1-e^{2 i \tan ^{-1}(c+d x)}\right)+b\right)+a b-i b^2 \text{Li}_2\left(e^{2 i \tan ^{-1}(c+d x)}\right)+\frac{b^2}{c+d x}+b^2 \left(\frac{1}{(c+d x)^3}-i\right) \tan ^{-1}(c+d x)^2}{3 d e^4}","-\frac{b \left(a+b \tan ^{-1}(c+d x)\right)}{3 d e^4 (c+d x)^2}-\frac{\left(a+b \tan ^{-1}(c+d x)\right)^2}{3 d e^4 (c+d x)^3}+\frac{i \left(a+b \tan ^{-1}(c+d x)\right)^2}{3 d e^4}-\frac{2 b \log \left(2-\frac{2}{1-i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{3 d e^4}+\frac{i b^2 \text{Li}_2\left(\frac{2}{1-i (c+d x)}-1\right)}{3 d e^4}-\frac{b^2}{3 d e^4 (c+d x)}-\frac{b^2 \tan ^{-1}(c+d x)}{3 d e^4}",1,"-1/3*(a*b + a^2/(c + d*x)^3 + (a*b)/(c + d*x)^2 + b^2/(c + d*x) + b^2*(-I + (c + d*x)^(-3))*ArcTan[c + d*x]^2 + b*ArcTan[c + d*x]*(b + (2*a)/(c + d*x)^3 + b/(c + d*x)^2 + 2*b*Log[1 - E^((2*I)*ArcTan[c + d*x])]) + 2*a*b*Log[(c + d*x)/Sqrt[1 + (c + d*x)^2]] - I*b^2*PolyLog[2, E^((2*I)*ArcTan[c + d*x])])/(d*e^4)","A",0
14,1,245,170,0.3508947,"\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^2}{(c e+d e x)^5} \, dx","Integrate[(a + b*ArcTan[c + d*x])^2/(c*e + d*e*x)^5,x]","-\frac{3 a^2-2 b \tan ^{-1}(c+d x) \left(3 a \left(c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4-1\right)+b \left(3 c^3+9 c^2 d x+9 c d^2 x^2-c+3 d^3 x^3-d x\right)\right)-6 a b (c+d x)^3+2 a b (c+d x)-4 b^2 (c+d x)^4 \log \left(c^2+2 c d x+d^2 x^2+1\right)-3 b^2 \left(c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4-1\right) \tan ^{-1}(c+d x)^2+b^2 (c+d x)^2+8 b^2 (c+d x)^4 \log (c+d x)}{12 d e^5 (c+d x)^4}","\frac{b \left(a+b \tan ^{-1}(c+d x)\right)}{2 d e^5 (c+d x)}-\frac{b \left(a+b \tan ^{-1}(c+d x)\right)}{6 d e^5 (c+d x)^3}-\frac{\left(a+b \tan ^{-1}(c+d x)\right)^2}{4 d e^5 (c+d x)^4}+\frac{\left(a+b \tan ^{-1}(c+d x)\right)^2}{4 d e^5}-\frac{b^2}{12 d e^5 (c+d x)^2}-\frac{2 b^2 \log (c+d x)}{3 d e^5}+\frac{b^2 \log \left((c+d x)^2+1\right)}{3 d e^5}",1,"-1/12*(3*a^2 + 2*a*b*(c + d*x) + b^2*(c + d*x)^2 - 6*a*b*(c + d*x)^3 - 2*b*(b*(-c + 3*c^3 - d*x + 9*c^2*d*x + 9*c*d^2*x^2 + 3*d^3*x^3) + 3*a*(-1 + c^4 + 4*c^3*d*x + 6*c^2*d^2*x^2 + 4*c*d^3*x^3 + d^4*x^4))*ArcTan[c + d*x] - 3*b^2*(-1 + c^4 + 4*c^3*d*x + 6*c^2*d^2*x^2 + 4*c*d^3*x^3 + d^4*x^4)*ArcTan[c + d*x]^2 + 8*b^2*(c + d*x)^4*Log[c + d*x] - 4*b^2*(c + d*x)^4*Log[1 + c^2 + 2*c*d*x + d^2*x^2])/(d*e^5*(c + d*x)^4)","A",1
15,1,349,271,0.7657773,"\int (c e+d e x)^2 \left(a+b \tan ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcTan[c + d*x])^3,x]","\frac{e^2 \left(2 a^3 (c+d x)^3-3 a^2 b (c+d x)^2+3 a^2 b \log \left((c+d x)^2+1\right)+6 a^2 b (c+d x)^3 \tan ^{-1}(c+d x)+6 a b^2 \left(i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)+(c+d x)^3 \tan ^{-1}(c+d x)^2-(c+d x)^2 \tan ^{-1}(c+d x)+i \tan ^{-1}(c+d x)^2-\tan ^{-1}(c+d x)-2 \tan ^{-1}(c+d x) \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)+c+d x\right)+b^3 \left(6 i \tan ^{-1}(c+d x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)-3 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c+d x)}\right)+6 \log \left(\frac{1}{\sqrt{(c+d x)^2+1}}\right)-2 (c+d x) \tan ^{-1}(c+d x)^3+2 (c+d x) \left((c+d x)^2+1\right) \tan ^{-1}(c+d x)^3+2 i \tan ^{-1}(c+d x)^3-3 \left((c+d x)^2+1\right) \tan ^{-1}(c+d x)^2+6 (c+d x) \tan ^{-1}(c+d x)-6 \tan ^{-1}(c+d x)^2 \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)\right)\right)}{6 d}","-\frac{i b^2 e^2 \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{d}+a b^2 e^2 x-\frac{b e^2 \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d}-\frac{b e^2 (c+d x)^2 \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d}+\frac{e^2 (c+d x)^3 \left(a+b \tan ^{-1}(c+d x)\right)^3}{3 d}-\frac{i e^2 \left(a+b \tan ^{-1}(c+d x)\right)^3}{3 d}-\frac{b e^2 \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)^2}{d}-\frac{b^3 e^2 \text{Li}_3\left(1-\frac{2}{i (c+d x)+1}\right)}{2 d}-\frac{b^3 e^2 \log \left((c+d x)^2+1\right)}{2 d}+\frac{b^3 e^2 (c+d x) \tan ^{-1}(c+d x)}{d}",1,"(e^2*(-3*a^2*b*(c + d*x)^2 + 2*a^3*(c + d*x)^3 + 6*a^2*b*(c + d*x)^3*ArcTan[c + d*x] + 3*a^2*b*Log[1 + (c + d*x)^2] + 6*a*b^2*(c + d*x - ArcTan[c + d*x] - (c + d*x)^2*ArcTan[c + d*x] + I*ArcTan[c + d*x]^2 + (c + d*x)^3*ArcTan[c + d*x]^2 - 2*ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + I*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]) + b^3*(6*(c + d*x)*ArcTan[c + d*x] - 3*(1 + (c + d*x)^2)*ArcTan[c + d*x]^2 + (2*I)*ArcTan[c + d*x]^3 - 2*(c + d*x)*ArcTan[c + d*x]^3 + 2*(c + d*x)*(1 + (c + d*x)^2)*ArcTan[c + d*x]^3 - 6*ArcTan[c + d*x]^2*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 6*Log[1/Sqrt[1 + (c + d*x)^2]] + (6*I)*ArcTan[c + d*x]*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])] - 3*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])])))/(6*d)","A",0
16,1,196,164,0.5222421,"\int (c e+d e x) \left(a+b \tan ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcTan[c + d*x])^3,x]","\frac{e \left(3 b \tan ^{-1}(c+d x) \left(a \left(a \left(c^2+2 c d x+d^2 x^2+1\right)-2 b (c+d x)\right)-2 b^2 \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)\right)+a \left(a (c+d x) (a c+a d x-3 b)-6 b^2 \log \left(\frac{1}{\sqrt{(c+d x)^2+1}}\right)\right)+3 b^2 (c+d x-i) \tan ^{-1}(c+d x)^2 (-b+a (c+d x+i))+b^3 \left(c^2+2 c d x+d^2 x^2+1\right) \tan ^{-1}(c+d x)^3+3 i b^3 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)\right)}{2 d}","-\frac{3 b^2 e \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{d}-\frac{3 i b e \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d}-\frac{3 b e (c+d x) \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d}+\frac{e (c+d x)^2 \left(a+b \tan ^{-1}(c+d x)\right)^3}{2 d}+\frac{e \left(a+b \tan ^{-1}(c+d x)\right)^3}{2 d}-\frac{3 i b^3 e \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right)}{2 d}",1,"(e*(3*b^2*(-I + c + d*x)*(-b + a*(I + c + d*x))*ArcTan[c + d*x]^2 + b^3*(1 + c^2 + 2*c*d*x + d^2*x^2)*ArcTan[c + d*x]^3 + 3*b*ArcTan[c + d*x]*(a*(-2*b*(c + d*x) + a*(1 + c^2 + 2*c*d*x + d^2*x^2)) - 2*b^2*Log[1 + E^((2*I)*ArcTan[c + d*x])]) + a*(a*(c + d*x)*(-3*b + a*c + a*d*x) - 6*b^2*Log[1/Sqrt[1 + (c + d*x)^2]]) + (3*I)*b^3*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]))/(2*d)","A",0
17,1,252,279,0.1532166,"\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^3}{c e+d e x} \, dx","Integrate[(a + b*ArcTan[c + d*x])^3/(c*e + d*e*x),x]","\frac{6 b^2 \text{Li}_3\left(-\frac{c+d x+i}{c+d x-i}\right) \left(a+b \tan ^{-1}(c+d x)\right)-6 b^2 \text{Li}_3\left(\frac{c+d x+i}{c+d x-i}\right) \left(a+b \tan ^{-1}(c+d x)\right)+6 i b \text{Li}_2\left(-\frac{c+d x+i}{c+d x-i}\right) \left(a+b \tan ^{-1}(c+d x)\right)^2-6 i b \text{Li}_2\left(\frac{c+d x+i}{c+d x-i}\right) \left(a+b \tan ^{-1}(c+d x)\right)^2+8 \tanh ^{-1}\left(\frac{c+d x+i}{c+d x-i}\right) \left(a+b \tan ^{-1}(c+d x)\right)^3-3 i b^3 \text{Li}_4\left(-\frac{c+d x+i}{c+d x-i}\right)+3 i b^3 \text{Li}_4\left(\frac{c+d x+i}{c+d x-i}\right)}{4 d e}","-\frac{3 b^2 \text{Li}_3\left(1-\frac{2}{i (c+d x)+1}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{2 d e}+\frac{3 b^2 \text{Li}_3\left(\frac{2}{i (c+d x)+1}-1\right) \left(a+b \tan ^{-1}(c+d x)\right)}{2 d e}-\frac{3 i b \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right) \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d e}+\frac{3 i b \text{Li}_2\left(\frac{2}{i (c+d x)+1}-1\right) \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d e}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{1+i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)^3}{d e}+\frac{3 i b^3 \text{Li}_4\left(1-\frac{2}{i (c+d x)+1}\right)}{4 d e}-\frac{3 i b^3 \text{Li}_4\left(\frac{2}{i (c+d x)+1}-1\right)}{4 d e}",1,"(8*(a + b*ArcTan[c + d*x])^3*ArcTanh[(I + c + d*x)/(-I + c + d*x)] + (6*I)*b*(a + b*ArcTan[c + d*x])^2*PolyLog[2, -((I + c + d*x)/(-I + c + d*x))] - (6*I)*b*(a + b*ArcTan[c + d*x])^2*PolyLog[2, (I + c + d*x)/(-I + c + d*x)] + 6*b^2*(a + b*ArcTan[c + d*x])*PolyLog[3, -((I + c + d*x)/(-I + c + d*x))] - 6*b^2*(a + b*ArcTan[c + d*x])*PolyLog[3, (I + c + d*x)/(-I + c + d*x)] - (3*I)*b^3*PolyLog[4, -((I + c + d*x)/(-I + c + d*x))] + (3*I)*b^3*PolyLog[4, (I + c + d*x)/(-I + c + d*x)])/(4*d*e)","A",1
18,1,263,163,0.6903286,"\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^3}{(c e+d e x)^2} \, dx","Integrate[(a + b*ArcTan[c + d*x])^3/(c*e + d*e*x)^2,x]","\frac{-\frac{2 a^3}{c+d x}-3 a^2 b \log \left(c^2+2 c d x+d^2 x^2+1\right)+6 a^2 b \log (c+d x)-\frac{6 a^2 b \tan ^{-1}(c+d x)}{c+d x}+6 a b^2 \left(\tan ^{-1}(c+d x) \left(\left(-\frac{1}{c+d x}-i\right) \tan ^{-1}(c+d x)+2 \log \left(1-e^{2 i \tan ^{-1}(c+d x)}\right)\right)-i \text{Li}_2\left(e^{2 i \tan ^{-1}(c+d x)}\right)\right)+2 b^3 \left(3 i \tan ^{-1}(c+d x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c+d x)}\right)+\frac{3}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(c+d x)}\right)-\frac{\tan ^{-1}(c+d x)^3}{c+d x}+i \tan ^{-1}(c+d x)^3+3 \tan ^{-1}(c+d x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c+d x)}\right)-\frac{i \pi ^3}{8}\right)}{2 d e^2}","-\frac{3 i b^2 \text{Li}_2\left(\frac{2}{1-i (c+d x)}-1\right) \left(a+b \tan ^{-1}(c+d x)\right)}{d e^2}-\frac{\left(a+b \tan ^{-1}(c+d x)\right)^3}{d e^2 (c+d x)}-\frac{i \left(a+b \tan ^{-1}(c+d x)\right)^3}{d e^2}+\frac{3 b \log \left(2-\frac{2}{1-i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)^2}{d e^2}+\frac{3 b^3 \text{Li}_3\left(\frac{2}{1-i (c+d x)}-1\right)}{2 d e^2}",1,"((-2*a^3)/(c + d*x) - (6*a^2*b*ArcTan[c + d*x])/(c + d*x) + 6*a^2*b*Log[c + d*x] - 3*a^2*b*Log[1 + c^2 + 2*c*d*x + d^2*x^2] + 6*a*b^2*(ArcTan[c + d*x]*((-I - (c + d*x)^(-1))*ArcTan[c + d*x] + 2*Log[1 - E^((2*I)*ArcTan[c + d*x])]) - I*PolyLog[2, E^((2*I)*ArcTan[c + d*x])]) + 2*b^3*((-1/8*I)*Pi^3 + I*ArcTan[c + d*x]^3 - ArcTan[c + d*x]^3/(c + d*x) + 3*ArcTan[c + d*x]^2*Log[1 - E^((-2*I)*ArcTan[c + d*x])] + (3*I)*ArcTan[c + d*x]*PolyLog[2, E^((-2*I)*ArcTan[c + d*x])] + (3*PolyLog[3, E^((-2*I)*ArcTan[c + d*x])])/2))/(2*d*e^2)","A",0
19,1,225,180,0.3074846,"\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^3}{(c e+d e x)^3} \, dx","Integrate[(a + b*ArcTan[c + d*x])^3/(c*e + d*e*x)^3,x]","-\frac{a^3+3 a^2 b \left(\left((c+d x)^2+1\right) \tan ^{-1}(c+d x)+c+d x\right)+3 a b^2 \left(-2 (c+d x)^2 \log \left(\frac{c+d x}{\sqrt{(c+d x)^2+1}}\right)+2 (c+d x) \tan ^{-1}(c+d x)+\left((c+d x)^2+1\right) \tan ^{-1}(c+d x)^2\right)+b^3 \left(c^2+2 c d x+d^2 x^2+1\right) \tan ^{-1}(c+d x)^3+3 b^3 (c+d x) \left(i (c+d x) \left(\tan ^{-1}(c+d x)^2+\text{Li}_2\left(e^{2 i \tan ^{-1}(c+d x)}\right)\right)+\tan ^{-1}(c+d x)^2-2 (c+d x) \tan ^{-1}(c+d x) \log \left(1-e^{2 i \tan ^{-1}(c+d x)}\right)\right)}{2 d e^3 (c+d x)^2}","\frac{3 b^2 \log \left(2-\frac{2}{1-i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{d e^3}-\frac{3 b \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d e^3 (c+d x)}-\frac{3 i b \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d e^3}-\frac{\left(a+b \tan ^{-1}(c+d x)\right)^3}{2 d e^3 (c+d x)^2}-\frac{\left(a+b \tan ^{-1}(c+d x)\right)^3}{2 d e^3}-\frac{3 i b^3 \text{Li}_2\left(\frac{2}{1-i (c+d x)}-1\right)}{2 d e^3}",1,"-1/2*(a^3 + b^3*(1 + c^2 + 2*c*d*x + d^2*x^2)*ArcTan[c + d*x]^3 + 3*a^2*b*(c + d*x + (1 + (c + d*x)^2)*ArcTan[c + d*x]) + 3*a*b^2*(2*(c + d*x)*ArcTan[c + d*x] + (1 + (c + d*x)^2)*ArcTan[c + d*x]^2 - 2*(c + d*x)^2*Log[(c + d*x)/Sqrt[1 + (c + d*x)^2]]) + 3*b^3*(c + d*x)*(ArcTan[c + d*x]^2 - 2*(c + d*x)*ArcTan[c + d*x]*Log[1 - E^((2*I)*ArcTan[c + d*x])] + I*(c + d*x)*(ArcTan[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcTan[c + d*x])])))/(d*e^3*(c + d*x)^2)","A",0
20,1,360,287,1.2249168,"\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^3}{(c e+d e x)^4} \, dx","Integrate[(a + b*ArcTan[c + d*x])^3/(c*e + d*e*x)^4,x]","\frac{-\frac{8 a^3}{(c+d x)^3}+12 a^2 b \log \left(c^2+2 c d x+d^2 x^2+1\right)-\frac{12 a^2 b}{(c+d x)^2}-24 a^2 b \log (c+d x)-\frac{24 a^2 b \tan ^{-1}(c+d x)}{(c+d x)^3}+24 a b^2 \left(i \text{Li}_2\left(e^{2 i \tan ^{-1}(c+d x)}\right)-\frac{(c+d x)^2+\tan ^{-1}(c+d x)^2}{(c+d x)^3}+\tan ^{-1}(c+d x) \left(-\frac{1}{(c+d x)^2}+i \tan ^{-1}(c+d x)-2 \log \left(1-e^{2 i \tan ^{-1}(c+d x)}\right)-1\right)\right)+b^3 \left(-24 i \tan ^{-1}(c+d x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c+d x)}\right)-12 \text{Li}_3\left(e^{-2 i \tan ^{-1}(c+d x)}\right)+24 \log \left(\frac{c+d x}{\sqrt{(c+d x)^2+1}}\right)-\frac{8 \tan ^{-1}(c+d x)^3}{(c+d x)^3}-8 i \tan ^{-1}(c+d x)^3-\frac{12 \tan ^{-1}(c+d x)^2}{(c+d x)^2}-12 \tan ^{-1}(c+d x)^2-\frac{24 \tan ^{-1}(c+d x)}{c+d x}-24 \tan ^{-1}(c+d x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c+d x)}\right)+i \pi ^3\right)}{24 d e^4}","\frac{i b^2 \text{Li}_2\left(\frac{2}{1-i (c+d x)}-1\right) \left(a+b \tan ^{-1}(c+d x)\right)}{d e^4}-\frac{b^2 \left(a+b \tan ^{-1}(c+d x)\right)}{d e^4 (c+d x)}-\frac{b \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d e^4 (c+d x)^2}-\frac{b \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d e^4}-\frac{\left(a+b \tan ^{-1}(c+d x)\right)^3}{3 d e^4 (c+d x)^3}+\frac{i \left(a+b \tan ^{-1}(c+d x)\right)^3}{3 d e^4}-\frac{b \log \left(2-\frac{2}{1-i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)^2}{d e^4}-\frac{b^3 \text{Li}_3\left(\frac{2}{1-i (c+d x)}-1\right)}{2 d e^4}+\frac{b^3 \log (c+d x)}{d e^4}-\frac{b^3 \log \left((c+d x)^2+1\right)}{2 d e^4}",1,"((-8*a^3)/(c + d*x)^3 - (12*a^2*b)/(c + d*x)^2 - (24*a^2*b*ArcTan[c + d*x])/(c + d*x)^3 - 24*a^2*b*Log[c + d*x] + 12*a^2*b*Log[1 + c^2 + 2*c*d*x + d^2*x^2] + 24*a*b^2*(-(((c + d*x)^2 + ArcTan[c + d*x]^2)/(c + d*x)^3) + ArcTan[c + d*x]*(-1 - (c + d*x)^(-2) + I*ArcTan[c + d*x] - 2*Log[1 - E^((2*I)*ArcTan[c + d*x])]) + I*PolyLog[2, E^((2*I)*ArcTan[c + d*x])]) + b^3*(I*Pi^3 - (24*ArcTan[c + d*x])/(c + d*x) - 12*ArcTan[c + d*x]^2 - (12*ArcTan[c + d*x]^2)/(c + d*x)^2 - (8*I)*ArcTan[c + d*x]^3 - (8*ArcTan[c + d*x]^3)/(c + d*x)^3 - 24*ArcTan[c + d*x]^2*Log[1 - E^((-2*I)*ArcTan[c + d*x])] + 24*Log[(c + d*x)/Sqrt[1 + (c + d*x)^2]] - (24*I)*ArcTan[c + d*x]*PolyLog[2, E^((-2*I)*ArcTan[c + d*x])] - 12*PolyLog[3, E^((-2*I)*ArcTan[c + d*x])]))/(24*d*e^4)","A",0
21,1,31,31,0.0039701,"\int \frac{\tan ^{-1}(1+x)}{2+2 x} \, dx","Integrate[ArcTan[1 + x]/(2 + 2*x),x]","\frac{1}{4} i \text{Li}_2(-i (x+1))-\frac{1}{4} i \text{Li}_2(i (x+1))","\frac{1}{4} i \text{Li}_2(-i (x+1))-\frac{1}{4} i \text{Li}_2(i (x+1))",1,"(I/4)*PolyLog[2, (-I)*(1 + x)] - (I/4)*PolyLog[2, I*(1 + x)]","A",1
22,1,34,41,0.0080262,"\int \frac{\tan ^{-1}(a+b x)}{\frac{a d}{b}+d x} \, dx","Integrate[ArcTan[a + b*x]/((a*d)/b + d*x),x]","\frac{i (\text{Li}_2(-i (a+b x))-\text{Li}_2(i (a+b x)))}{2 d}","\frac{i \text{Li}_2(-i (a+b x))}{2 d}-\frac{i \text{Li}_2(i (a+b x))}{2 d}",1,"((I/2)*(PolyLog[2, (-I)*(a + b*x)] - PolyLog[2, I*(a + b*x)]))/d","A",1
23,0,0,21,6.1097261,"\int (a+b x)^2 \sqrt{\tan ^{-1}(a+b x)} \, dx","Integrate[(a + b*x)^2*Sqrt[ArcTan[a + b*x]],x]","\int (a+b x)^2 \sqrt{\tan ^{-1}(a+b x)} \, dx","\text{Int}\left((a+b x)^2 \sqrt{\tan ^{-1}(a+b x)},x\right)",0,"Integrate[(a + b*x)^2*Sqrt[ArcTan[a + b*x]], x]","A",-1
24,1,157,233,0.2977537,"\int (e+f x)^3 \left(a+b \tan ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)^3*(a + b*ArcTan[c + d*x]),x]","\frac{(e+f x)^4 \left(a+b \tan ^{-1}(c+d x)\right)-\frac{b \left(6 d f^2 x \left(\left(6 c^2-1\right) f^2-12 c d e f+6 d^2 e^2\right)+12 f^3 (c+d x)^2 (d e-c f)-3 i (d e-(c-i) f)^4 \log (-c-d x+i)+3 i (d e-(c+i) f)^4 \log (c+d x+i)+2 f^4 (c+d x)^3\right)}{6 d^4}}{4 f}","\frac{(e+f x)^4 \left(a+b \tan ^{-1}(c+d x)\right)}{4 f}-\frac{b f x \left(-\left(1-6 c^2\right) f^2-12 c d e f+6 d^2 e^2\right)}{4 d^3}-\frac{b \left(-6 \left(1-c^2\right) d^2 e^2 f^2+4 c \left(3-c^2\right) d e f^3+\left(c^4-6 c^2+1\right) f^4-4 c d^3 e^3 f+d^4 e^4\right) \tan ^{-1}(c+d x)}{4 d^4 f}-\frac{b f^2 (c+d x)^2 (d e-c f)}{2 d^4}-\frac{b (d e-c f) (-c f+d e+f) (d e-(c+1) f) \log \left((c+d x)^2+1\right)}{2 d^4}-\frac{b f^3 (c+d x)^3}{12 d^4}",1,"((e + f*x)^4*(a + b*ArcTan[c + d*x]) - (b*(6*d*f^2*(6*d^2*e^2 - 12*c*d*e*f + (-1 + 6*c^2)*f^2)*x + 12*f^3*(d*e - c*f)*(c + d*x)^2 + 2*f^4*(c + d*x)^3 - (3*I)*(d*e - (-I + c)*f)^4*Log[I - c - d*x] + (3*I)*(d*e - (I + c)*f)^4*Log[I + c + d*x]))/(6*d^4))/(4*f)","C",1
25,1,118,155,0.1598823,"\int (e+f x)^2 \left(a+b \tan ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)^2*(a + b*ArcTan[c + d*x]),x]","\frac{(e+f x)^3 \left(a+b \tan ^{-1}(c+d x)\right)-\frac{b \left(6 d f^2 x (d e-c f)-i (d e-(c-i) f)^3 \log (-c-d x+i)+i (d e-(c+i) f)^3 \log (c+d x+i)+f^3 (c+d x)^2\right)}{2 d^3}}{3 f}","\frac{(e+f x)^3 \left(a+b \tan ^{-1}(c+d x)\right)}{3 f}-\frac{b \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \log \left((c+d x)^2+1\right)}{6 d^3}-\frac{b (d e-c f) \left(-\left(3-c^2\right) f^2-2 c d e f+d^2 e^2\right) \tan ^{-1}(c+d x)}{3 d^3 f}-\frac{b f^2 (c+d x)^2}{6 d^3}-\frac{b f x (d e-c f)}{d^2}",1,"((e + f*x)^3*(a + b*ArcTan[c + d*x]) - (b*(6*d*f^2*(d*e - c*f)*x + f^3*(c + d*x)^2 - I*(d*e - (-I + c)*f)^3*Log[I - c - d*x] + I*(d*e - (I + c)*f)^3*Log[I + c + d*x]))/(2*d^3))/(3*f)","C",1
26,1,163,97,0.0654915,"\int (e+f x) \left(a+b \tan ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)*(a + b*ArcTan[c + d*x]),x]","a e x+\frac{1}{2} a f x^2-\frac{b e \left(\log \left(c^2+2 c d x+d^2 x^2+1\right)-2 c \tan ^{-1}(c+d x)\right)}{2 d}+\frac{b f \left(\frac{1}{2} d \left(\frac{c+d x}{d}-\frac{c}{d}\right)^2 \tan ^{-1}(c+d x)-\frac{1}{2} d \left(-\frac{i (-c+i)^2 \log (-c-d x+i)}{2 d^2}+\frac{i (c+i)^2 \log (c+d x+i)}{2 d^2}+\frac{x}{d}\right)\right)}{d}+b e x \tan ^{-1}(c+d x)","\frac{(e+f x)^2 \left(a+b \tan ^{-1}(c+d x)\right)}{2 f}-\frac{b (d e-c f) \log \left((c+d x)^2+1\right)}{2 d^2}-\frac{b (-c f+d e+f) (d e-(c+1) f) \tan ^{-1}(c+d x)}{2 d^2 f}-\frac{b f x}{2 d}",1,"a*e*x + (a*f*x^2)/2 + b*e*x*ArcTan[c + d*x] + (b*f*((d*(-(c/d) + (c + d*x)/d)^2*ArcTan[c + d*x])/2 - (d*(x/d - ((I/2)*(I - c)^2*Log[I - c - d*x])/d^2 + ((I/2)*(I + c)^2*Log[I + c + d*x])/d^2))/2))/d - (b*e*(-2*c*ArcTan[c + d*x] + Log[1 + c^2 + 2*c*d*x + d^2*x^2]))/(2*d)","C",1
27,1,49,38,0.0160892,"\int \left(a+b \tan ^{-1}(c+d x)\right) \, dx","Integrate[a + b*ArcTan[c + d*x],x]","a x-\frac{b \left(\log \left(c^2+2 c d x+d^2 x^2+1\right)-2 c \tan ^{-1}(c+d x)\right)}{2 d}+b x \tan ^{-1}(c+d x)","a x-\frac{b \log \left((c+d x)^2+1\right)}{2 d}+\frac{b (c+d x) \tan ^{-1}(c+d x)}{d}",1,"a*x + b*x*ArcTan[c + d*x] - (b*(-2*c*ArcTan[c + d*x] + Log[1 + c^2 + 2*c*d*x + d^2*x^2]))/(2*d)","A",1
28,1,160,162,0.1219507,"\int \frac{a+b \tan ^{-1}(c+d x)}{e+f x} \, dx","Integrate[(a + b*ArcTan[c + d*x])/(e + f*x),x]","\frac{2 a \log (d (e+f x))-i b \text{Li}_2\left(\frac{f (c+d x-i)}{(c-i) f-d e}\right)+i b \text{Li}_2\left(\frac{f (c+d x+i)}{(c+i) f-d e}\right)+i b \log (1-i (c+d x)) \log \left(\frac{d (e+f x)}{d e-(c+i) f}\right)-i b \log (1+i (c+d x)) \log \left(\frac{d (e+f x)}{-c f+d e+i f}\right)}{2 f}","\frac{\left(a+b \tan ^{-1}(c+d x)\right) \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{f}-\frac{\log \left(\frac{2}{1-i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{f}-\frac{i b \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right)}{2 f}+\frac{i b \text{Li}_2\left(1-\frac{2}{1-i (c+d x)}\right)}{2 f}",1,"(2*a*Log[d*(e + f*x)] + I*b*Log[(d*(e + f*x))/(d*e - (I + c)*f)]*Log[1 - I*(c + d*x)] - I*b*Log[(d*(e + f*x))/(d*e + I*f - c*f)]*Log[1 + I*(c + d*x)] - I*b*PolyLog[2, (f*(-I + c + d*x))/(-(d*e) + (-I + c)*f)] + I*b*PolyLog[2, (f*(I + c + d*x))/(-(d*e) + (I + c)*f)])/(2*f)","A",0
29,1,121,151,0.2059522,"\int \frac{a+b \tan ^{-1}(c+d x)}{(e+f x)^2} \, dx","Integrate[(a + b*ArcTan[c + d*x])/(e + f*x)^2,x]","\frac{-\frac{a+b \tan ^{-1}(c+d x)}{e+f x}+\frac{b d (i (-d e+(c+i) f) \log (-c-d x+i)+i (-c f+d e+i f) \log (c+d x+i)+2 f \log (d (e+f x)))}{2 \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}}{f}","-\frac{a+b \tan ^{-1}(c+d x)}{f (e+f x)}-\frac{b d \log \left(c^2+2 c d x+d^2 x^2+1\right)}{2 \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}+\frac{b d \log (e+f x)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{b d (d e-c f) \tan ^{-1}(c+d x)}{f \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}",1,"(-((a + b*ArcTan[c + d*x])/(e + f*x)) + (b*d*(I*(-(d*e) + (I + c)*f)*Log[I - c - d*x] + I*(d*e + I*f - c*f)*Log[I + c + d*x] + 2*f*Log[d*(e + f*x)]))/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)))/f","C",1
30,1,175,227,0.7950741,"\int \frac{a+b \tan ^{-1}(c+d x)}{(e+f x)^3} \, dx","Integrate[(a + b*ArcTan[c + d*x])/(e + f*x)^3,x]","\frac{-\frac{a+b \tan ^{-1}(c+d x)}{(e+f x)^2}+\frac{1}{2} b d^2 \left(-\frac{2 f}{d (e+f x) \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}-\frac{4 f (c f-d e) \log (d (e+f x))}{\left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)^2}-\frac{i \log (-c-d x+i)}{(d e-(c-i) f)^2}+\frac{i \log (c+d x+i)}{(d e-(c+i) f)^2}\right)}{2 f}","-\frac{a+b \tan ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac{b d^2 (d e-c f) \log \left(c^2+2 c d x+d^2 x^2+1\right)}{2 \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)^2}-\frac{b d}{2 (e+f x) \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}+\frac{b d^2 (d e-c f) \log (e+f x)}{\left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)^2}+\frac{b d^2 (-c f+d e+f) (d e-(c+1) f) \tan ^{-1}(c+d x)}{2 f \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)^2}",1,"(-((a + b*ArcTan[c + d*x])/(e + f*x)^2) + (b*d^2*((-2*f)/(d*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*(e + f*x)) - (I*Log[I - c - d*x])/(d*e - (-I + c)*f)^2 + (I*Log[I + c + d*x])/(d*e - (I + c)*f)^2 - (4*f*(-(d*e) + c*f)*Log[d*(e + f*x)])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^2))/2)/(2*f)","C",1
31,1,801,382,4.2296466,"\int (e+f x)^2 \left(a+b \tan ^{-1}(c+d x)\right)^2 \, dx","Integrate[(e + f*x)^2*(a + b*ArcTan[c + d*x])^2,x]","\frac{1}{3} a^2 f^2 x^3+a^2 e f x^2+a^2 e^2 x+\frac{a b \left(-d f x (6 d e-4 c f+d f x)+2 \left(f^2 c^3-3 d e f c^2+3 \left(d^2 e^2-f^2\right) c+3 d e f+d^3 x \left(3 e^2+3 f x e+f^2 x^2\right)\right) \tan ^{-1}(c+d x)+\left(-3 d^2 e^2+6 c d f e+\left(1-3 c^2\right) f^2\right) \log \left((c+d x)^2+1\right)\right)}{3 d^3}+\frac{b^2 e^2 \left(\tan ^{-1}(c+d x) \left((c+d x-i) \tan ^{-1}(c+d x)+2 \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)\right)-i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)\right)}{d}+\frac{b^2 e f \left(\left(-c^2+2 i c+d^2 x^2+1\right) \tan ^{-1}(c+d x)^2-2 \left(2 \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right) c+c+d x\right) \tan ^{-1}(c+d x)+\log \left((c+d x)^2+1\right)+2 i c \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)\right)}{d^2}+\frac{b^2 f^2 \left((c+d x)^2+1\right)^{3/2} \left(\frac{3 (c+d x) \tan ^{-1}(c+d x)^2 c^2}{\sqrt{(c+d x)^2+1}}-3 i \tan ^{-1}(c+d x)^2 \cos \left(3 \tan ^{-1}(c+d x)\right) c^2+6 \tan ^{-1}(c+d x) \cos \left(3 \tan ^{-1}(c+d x)\right) \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right) c^2+3 \tan ^{-1}(c+d x)^2 \sin \left(3 \tan ^{-1}(c+d x)\right) c^2+\frac{6 (c+d x) \tan ^{-1}(c+d x) c}{\sqrt{(c+d x)^2+1}}+6 \cos \left(3 \tan ^{-1}(c+d x)\right) \log \left(\frac{1}{\sqrt{(c+d x)^2+1}}\right) c+6 \tan ^{-1}(c+d x) \sin \left(3 \tan ^{-1}(c+d x)\right) c+\frac{3 (c+d x) \tan ^{-1}(c+d x)^2}{\sqrt{(c+d x)^2+1}}+i \tan ^{-1}(c+d x)^2 \cos \left(3 \tan ^{-1}(c+d x)\right)-2 \tan ^{-1}(c+d x) \cos \left(3 \tan ^{-1}(c+d x)\right) \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)+\frac{\left(-9 i c^2-12 c+3 i\right) \tan ^{-1}(c+d x)^2+2 \left(\left(9 c^2-3\right) \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)-2\right) \tan ^{-1}(c+d x)+18 c \log \left(\frac{1}{\sqrt{(c+d x)^2+1}}\right)}{\sqrt{(c+d x)^2+1}}-\frac{4 i \left(3 c^2-1\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)}{\left((c+d x)^2+1\right)^{3/2}}-\tan ^{-1}(c+d x)^2 \sin \left(3 \tan ^{-1}(c+d x)\right)+\sin \left(3 \tan ^{-1}(c+d x)\right)+\frac{c+d x}{\sqrt{(c+d x)^2+1}}\right)}{12 d^3}","\frac{i \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \left(a+b \tan ^{-1}(c+d x)\right)^2}{3 d^3}-\frac{(d e-c f) \left(-\left(3-c^2\right) f^2-2 c d e f+d^2 e^2\right) \left(a+b \tan ^{-1}(c+d x)\right)^2}{3 d^3 f}+\frac{2 b \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{3 d^3}-\frac{b f^2 (c+d x)^2 \left(a+b \tan ^{-1}(c+d x)\right)}{3 d^3}-\frac{2 a b f x (d e-c f)}{d^2}+\frac{(e+f x)^3 \left(a+b \tan ^{-1}(c+d x)\right)^2}{3 f}+\frac{i b^2 \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right)}{3 d^3}+\frac{b^2 f (d e-c f) \log \left((c+d x)^2+1\right)}{d^3}-\frac{2 b^2 f (c+d x) (d e-c f) \tan ^{-1}(c+d x)}{d^3}-\frac{b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}+\frac{b^2 f^2 x}{3 d^2}",1,"a^2*e^2*x + a^2*e*f*x^2 + (a^2*f^2*x^3)/3 + (a*b*(-(d*f*x*(6*d*e - 4*c*f + d*f*x)) + 2*(3*d*e*f - 3*c^2*d*e*f + c^3*f^2 + 3*c*(d^2*e^2 - f^2) + d^3*x*(3*e^2 + 3*e*f*x + f^2*x^2))*ArcTan[c + d*x] + (-3*d^2*e^2 + 6*c*d*e*f + (1 - 3*c^2)*f^2)*Log[1 + (c + d*x)^2]))/(3*d^3) + (b^2*e^2*(ArcTan[c + d*x]*((-I + c + d*x)*ArcTan[c + d*x] + 2*Log[1 + E^((2*I)*ArcTan[c + d*x])]) - I*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]))/d + (b^2*e*f*((1 + (2*I)*c - c^2 + d^2*x^2)*ArcTan[c + d*x]^2 - 2*ArcTan[c + d*x]*(c + d*x + 2*c*Log[1 + E^((2*I)*ArcTan[c + d*x])]) + Log[1 + (c + d*x)^2] + (2*I)*c*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]))/d^2 + (b^2*f^2*(1 + (c + d*x)^2)^(3/2)*((c + d*x)/Sqrt[1 + (c + d*x)^2] + (6*c*(c + d*x)*ArcTan[c + d*x])/Sqrt[1 + (c + d*x)^2] + (3*(c + d*x)*ArcTan[c + d*x]^2)/Sqrt[1 + (c + d*x)^2] + (3*c^2*(c + d*x)*ArcTan[c + d*x]^2)/Sqrt[1 + (c + d*x)^2] + I*ArcTan[c + d*x]^2*Cos[3*ArcTan[c + d*x]] - (3*I)*c^2*ArcTan[c + d*x]^2*Cos[3*ArcTan[c + d*x]] - 2*ArcTan[c + d*x]*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 6*c^2*ArcTan[c + d*x]*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 6*c*Cos[3*ArcTan[c + d*x]]*Log[1/Sqrt[1 + (c + d*x)^2]] + ((3*I - 12*c - (9*I)*c^2)*ArcTan[c + d*x]^2 + 2*ArcTan[c + d*x]*(-2 + (-3 + 9*c^2)*Log[1 + E^((2*I)*ArcTan[c + d*x])]) + 18*c*Log[1/Sqrt[1 + (c + d*x)^2]])/Sqrt[1 + (c + d*x)^2] - ((4*I)*(-1 + 3*c^2)*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])])/(1 + (c + d*x)^2)^(3/2) + Sin[3*ArcTan[c + d*x]] + 6*c*ArcTan[c + d*x]*Sin[3*ArcTan[c + d*x]] - ArcTan[c + d*x]^2*Sin[3*ArcTan[c + d*x]] + 3*c^2*ArcTan[c + d*x]^2*Sin[3*ArcTan[c + d*x]]))/(12*d^3)","B",0
32,1,264,222,0.4548586,"\int (e+f x) \left(a+b \tan ^{-1}(c+d x)\right)^2 \, dx","Integrate[(e + f*x)*(a + b*ArcTan[c + d*x])^2,x]","\frac{-a^2 c^2 f+2 a^2 c d e+2 a^2 d^2 e x+a^2 d^2 f x^2-2 b \tan ^{-1}(c+d x) \left(a \left(c^2 f-2 c d e-2 d^2 e x-f \left(d^2 x^2+1\right)\right)-2 b (d e-c f) \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)+b f (c+d x)\right)+4 a b d e \log \left(\frac{1}{\sqrt{(c+d x)^2+1}}\right)-4 a b c f \log \left(\frac{1}{\sqrt{(c+d x)^2+1}}\right)-2 a b c f-2 a b d f x-2 i b^2 (d e-c f) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)+b^2 (c+d x-i) \tan ^{-1}(c+d x)^2 (-c f+2 d e+d f x+i f)-2 b^2 f \log \left(\frac{1}{\sqrt{(c+d x)^2+1}}\right)}{2 d^2}","\frac{i (d e-c f) \left(a+b \tan ^{-1}(c+d x)\right)^2}{d^2}-\frac{(-c f+d e+f) (d e-(c+1) f) \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d^2 f}+\frac{2 b (d e-c f) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{d^2}+\frac{(e+f x)^2 \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 f}-\frac{a b f x}{d}+\frac{i b^2 (d e-c f) \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right)}{d^2}+\frac{b^2 f \log \left((c+d x)^2+1\right)}{2 d^2}-\frac{b^2 f (c+d x) \tan ^{-1}(c+d x)}{d^2}",1,"(2*a^2*c*d*e - 2*a*b*c*f - a^2*c^2*f + 2*a^2*d^2*e*x - 2*a*b*d*f*x + a^2*d^2*f*x^2 + b^2*(-I + c + d*x)*(2*d*e + I*f - c*f + d*f*x)*ArcTan[c + d*x]^2 - 2*b*ArcTan[c + d*x]*(b*f*(c + d*x) + a*(-2*c*d*e + c^2*f - 2*d^2*e*x - f*(1 + d^2*x^2)) - 2*b*(d*e - c*f)*Log[1 + E^((2*I)*ArcTan[c + d*x])]) + 4*a*b*d*e*Log[1/Sqrt[1 + (c + d*x)^2]] - 2*b^2*f*Log[1/Sqrt[1 + (c + d*x)^2]] - 4*a*b*c*f*Log[1/Sqrt[1 + (c + d*x)^2]] - (2*I)*b^2*(d*e - c*f)*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])])/(2*d^2)","A",0
33,1,109,102,0.0945901,"\int \left(a+b \tan ^{-1}(c+d x)\right)^2 \, dx","Integrate[(a + b*ArcTan[c + d*x])^2,x]","\frac{a \left(a c+a d x+2 b \log \left(\frac{1}{\sqrt{(c+d x)^2+1}}\right)\right)+2 b \tan ^{-1}(c+d x) \left(a c+a d x+b \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)\right)-i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)+b^2 (c+d x-i) \tan ^{-1}(c+d x)^2}{d}","\frac{(c+d x) \left(a+b \tan ^{-1}(c+d x)\right)^2}{d}+\frac{i \left(a+b \tan ^{-1}(c+d x)\right)^2}{d}+\frac{2 b \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{d}+\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right)}{d}",1,"(b^2*(-I + c + d*x)*ArcTan[c + d*x]^2 + 2*b*ArcTan[c + d*x]*(a*c + a*d*x + b*Log[1 + E^((2*I)*ArcTan[c + d*x])]) + a*(a*c + a*d*x + 2*b*Log[1/Sqrt[1 + (c + d*x)^2]]) - I*b^2*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])])/d","A",0
34,0,0,261,6.5099766,"\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^2}{e+f x} \, dx","Integrate[(a + b*ArcTan[c + d*x])^2/(e + f*x),x]","\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^2}{e+f x} \, dx","-\frac{i b \left(a+b \tan ^{-1}(c+d x)\right) \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right)}{f}+\frac{\left(a+b \tan ^{-1}(c+d x)\right)^2 \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{f}+\frac{i b \text{Li}_2\left(1-\frac{2}{1-i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{f}-\frac{\log \left(\frac{2}{1-i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)^2}{f}+\frac{b^2 \text{Li}_3\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right)}{2 f}-\frac{b^2 \text{Li}_3\left(1-\frac{2}{1-i (c+d x)}\right)}{2 f}",1,"Integrate[(a + b*ArcTan[c + d*x])^2/(e + f*x), x]","F",-1
35,1,419,568,7.4924311,"\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^2}{(e+f x)^2} \, dx","Integrate[(a + b*ArcTan[c + d*x])^2/(e + f*x)^2,x]","\frac{-\frac{a^2}{f}+\frac{2 a b \left(d (e+f x) \log \left(\frac{d (e+f x)}{\sqrt{(c+d x)^2+1}}\right)-\tan ^{-1}(c+d x) \left(c^2 f-c d e+c d f x-d^2 e x+f\right)\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{b^2 d (e+f x) \left(-\frac{(d e-c f) \left(i \text{Li}_2\left(\exp \left(2 i \left(\tan ^{-1}\left(\frac{d e-c f}{f}\right)+\tan ^{-1}(c+d x)\right)\right)\right)-2 \left(\tan ^{-1}\left(\frac{d e-c f}{f}\right)+\tan ^{-1}(c+d x)\right) \log \left(1-\exp \left(2 i \left(\tan ^{-1}\left(\frac{d e-c f}{f}\right)+\tan ^{-1}(c+d x)\right)\right)\right)-i \tan ^{-1}(c+d x) \left(\pi -2 \tan ^{-1}\left(\frac{d e-c f}{f}\right)\right)+2 \tan ^{-1}\left(\frac{d e-c f}{f}\right) \log \left(\sin \left(\tan ^{-1}\left(\frac{d e-c f}{f}\right)+\tan ^{-1}(c+d x)\right)\right)+\pi  \log \left(\frac{1}{\sqrt{(c+d x)^2+1}}\right)-\pi  \log \left(1+e^{-2 i \tan ^{-1}(c+d x)}\right)\right)}{f^2 \left(\frac{(d e-c f)^2}{f^2}+1\right)}-\frac{\tan ^{-1}(c+d x)^2 e^{i \tan ^{-1}\left(\frac{d e-c f}{f}\right)}}{f \sqrt{\frac{(d e-c f)^2}{f^2}+1}}+\frac{(c+d x) \tan ^{-1}(c+d x)^2}{d (e+f x)}\right)}{d e-c f}}{e+f x}","\frac{2 a b d \log (e+f x)}{(d e-c f)^2+f^2}-\frac{a b d \log \left((c+d x)^2+1\right)}{(d e-c f)^2+f^2}+\frac{2 a b d (d e-c f) \tan ^{-1}(c+d x)}{f \left((d e-c f)^2+f^2\right)}-\frac{\left(a+b \tan ^{-1}(c+d x)\right)^2}{f (e+f x)}+\frac{i b^2 d \text{Li}_2\left(1-\frac{2}{1-i (c+d x)}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}-\frac{i b^2 d \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{i b^2 d \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{i b^2 d \tan ^{-1}(c+d x)^2}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{b^2 d (d e-c f) \tan ^{-1}(c+d x)^2}{f \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}-\frac{2 b^2 d \log \left(\frac{2}{1-i (c+d x)}\right) \tan ^{-1}(c+d x)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{2 b^2 d \tan ^{-1}(c+d x) \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{2 b^2 d \log \left(\frac{2}{1+i (c+d x)}\right) \tan ^{-1}(c+d x)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}",1,"(-(a^2/f) + (2*a*b*(-((-(c*d*e) + f + c^2*f - d^2*e*x + c*d*f*x)*ArcTan[c + d*x]) + d*(e + f*x)*Log[(d*(e + f*x))/Sqrt[1 + (c + d*x)^2]]))/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (b^2*d*(e + f*x)*(-((E^(I*ArcTan[(d*e - c*f)/f])*ArcTan[c + d*x]^2)/(f*Sqrt[1 + (d*e - c*f)^2/f^2])) + ((c + d*x)*ArcTan[c + d*x]^2)/(d*(e + f*x)) - ((d*e - c*f)*((-I)*(Pi - 2*ArcTan[(d*e - c*f)/f])*ArcTan[c + d*x] - Pi*Log[1 + E^((-2*I)*ArcTan[c + d*x])] - 2*(ArcTan[(d*e - c*f)/f] + ArcTan[c + d*x])*Log[1 - E^((2*I)*(ArcTan[(d*e - c*f)/f] + ArcTan[c + d*x]))] + Pi*Log[1/Sqrt[1 + (c + d*x)^2]] + 2*ArcTan[(d*e - c*f)/f]*Log[Sin[ArcTan[(d*e - c*f)/f] + ArcTan[c + d*x]]] + I*PolyLog[2, E^((2*I)*(ArcTan[(d*e - c*f)/f] + ArcTan[c + d*x]))]))/(f^2*(1 + (d*e - c*f)^2/f^2))))/(d*e - c*f))/(e + f*x)","A",0
36,1,1844,564,9.9781889,"\int (e+f x)^2 \left(a+b \tan ^{-1}(c+d x)\right)^3 \, dx","Integrate[(e + f*x)^2*(a + b*ArcTan[c + d*x])^3,x]","\frac{e^2 \left((c+d x) \tan ^{-1}(c+d x)^3-i \tan ^{-1}(c+d x)^3+3 \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right) \tan ^{-1}(c+d x)^2-3 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right) \tan ^{-1}(c+d x)+\frac{3}{2} \text{Li}_3\left(-e^{2 i \tan ^{-1}(c+d x)}\right)\right) b^3}{d}+\frac{e f \left(\tan ^{-1}(c+d x) \left(2 i c \tan ^{-1}(c+d x)^2+\left((c+d x)^2+1\right) \tan ^{-1}(c+d x)^2-(c+d x) \left(2 c \tan ^{-1}(c+d x)+3\right) \tan ^{-1}(c+d x)-6 c \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right) \tan ^{-1}(c+d x)+3 i \tan ^{-1}(c+d x)-6 \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)\right)+3 i \left(2 c \tan ^{-1}(c+d x)+1\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)-3 c \text{Li}_3\left(-e^{2 i \tan ^{-1}(c+d x)}\right)\right) b^3}{d^2}+\frac{f^2 \left(\frac{1}{12} \left((c+d x)^2+1\right)^{3/2} \left(-3 i c^2 \cos \left(3 \tan ^{-1}(c+d x)\right) \tan ^{-1}(c+d x)^3+i \cos \left(3 \tan ^{-1}(c+d x)\right) \tan ^{-1}(c+d x)^3+3 c^2 \sin \left(3 \tan ^{-1}(c+d x)\right) \tan ^{-1}(c+d x)^3-\sin \left(3 \tan ^{-1}(c+d x)\right) \tan ^{-1}(c+d x)^3+\frac{3 c^2 (c+d x) \tan ^{-1}(c+d x)^3}{\sqrt{(c+d x)^2+1}}+\frac{3 (c+d x) \tan ^{-1}(c+d x)^3}{\sqrt{(c+d x)^2+1}}-9 i c \cos \left(3 \tan ^{-1}(c+d x)\right) \tan ^{-1}(c+d x)^2+9 c^2 \cos \left(3 \tan ^{-1}(c+d x)\right) \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right) \tan ^{-1}(c+d x)^2-3 \cos \left(3 \tan ^{-1}(c+d x)\right) \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right) \tan ^{-1}(c+d x)^2+9 c \sin \left(3 \tan ^{-1}(c+d x)\right) \tan ^{-1}(c+d x)^2+\frac{9 c (c+d x) \tan ^{-1}(c+d x)^2}{\sqrt{(c+d x)^2+1}}+18 c \cos \left(3 \tan ^{-1}(c+d x)\right) \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right) \tan ^{-1}(c+d x)+3 \sin \left(3 \tan ^{-1}(c+d x)\right) \tan ^{-1}(c+d x)+\frac{3 (c+d x) \tan ^{-1}(c+d x)}{\sqrt{(c+d x)^2+1}}+3 \cos \left(3 \tan ^{-1}(c+d x)\right) \log \left(\frac{1}{\sqrt{(c+d x)^2+1}}\right)+\frac{3 \left(\left(-3 i \tan ^{-1}(c+d x) c^2-4 \tan ^{-1}(c+d x) c-9 i c+i \tan ^{-1}(c+d x)-2\right) \tan ^{-1}(c+d x)^2+3 \left(3 \tan ^{-1}(c+d x) c^2+6 c-\tan ^{-1}(c+d x)\right) \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right) \tan ^{-1}(c+d x)+3 \log \left(\frac{1}{\sqrt{(c+d x)^2+1}}\right)\right)}{\sqrt{(c+d x)^2+1}}+\frac{6 \left(3 c^2-1\right) \text{Li}_3\left(-e^{2 i \tan ^{-1}(c+d x)}\right)}{\left((c+d x)^2+1\right)^{3/2}}\right)-i \left(3 \tan ^{-1}(c+d x) c^2+3 c-\tan ^{-1}(c+d x)\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)\right) b^3}{d^3}+\frac{3 a e^2 \left((c+d x) \tan ^{-1}(c+d x)^2-i \tan ^{-1}(c+d x)^2+2 \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right) \tan ^{-1}(c+d x)-i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)\right) b^2}{d}+6 a e f \left(-\frac{c (c+d x) \tan ^{-1}(c+d x)^2}{d^2}+\frac{\left((c+d x)^2+1\right) \tan ^{-1}(c+d x)^2}{2 d^2}+\frac{i c \tan ^{-1}(c+d x)^2}{d^2}-\frac{(c+d x) \tan ^{-1}(c+d x)}{d^2}-\frac{2 c \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right) \tan ^{-1}(c+d x)}{d^2}-\frac{\log \left(\frac{1}{\sqrt{(c+d x)^2+1}}\right)}{d^2}+\frac{i c \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)}{d^2}\right) b^2+\frac{a f^2 \left((c+d x)^2+1\right)^{3/2} \left(\frac{3 (c+d x) \tan ^{-1}(c+d x)^2 c^2}{\sqrt{(c+d x)^2+1}}-3 i \tan ^{-1}(c+d x)^2 \cos \left(3 \tan ^{-1}(c+d x)\right) c^2+6 \tan ^{-1}(c+d x) \cos \left(3 \tan ^{-1}(c+d x)\right) \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right) c^2+3 \tan ^{-1}(c+d x)^2 \sin \left(3 \tan ^{-1}(c+d x)\right) c^2+\frac{6 (c+d x) \tan ^{-1}(c+d x) c}{\sqrt{(c+d x)^2+1}}+6 \cos \left(3 \tan ^{-1}(c+d x)\right) \log \left(\frac{1}{\sqrt{(c+d x)^2+1}}\right) c+6 \tan ^{-1}(c+d x) \sin \left(3 \tan ^{-1}(c+d x)\right) c+\frac{3 (c+d x) \tan ^{-1}(c+d x)^2}{\sqrt{(c+d x)^2+1}}+i \tan ^{-1}(c+d x)^2 \cos \left(3 \tan ^{-1}(c+d x)\right)-2 \tan ^{-1}(c+d x) \cos \left(3 \tan ^{-1}(c+d x)\right) \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)+\frac{\tan ^{-1}(c+d x) \left(\left(-9 i c^2-12 c+3 i\right) \tan ^{-1}(c+d x)-4\right)+6 \left(3 c^2-1\right) \tan ^{-1}(c+d x) \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)+18 c \log \left(\frac{1}{\sqrt{(c+d x)^2+1}}\right)}{\sqrt{(c+d x)^2+1}}-\frac{4 i \left(3 c^2-1\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)}{\left((c+d x)^2+1\right)^{3/2}}-\tan ^{-1}(c+d x)^2 \sin \left(3 \tan ^{-1}(c+d x)\right)+\sin \left(3 \tan ^{-1}(c+d x)\right)+\frac{c+d x}{\sqrt{(c+d x)^2+1}}\right) b^2}{4 d^3}+a^2 x \left(3 e^2+3 f x e+f^2 x^2\right) \tan ^{-1}(c+d x) b+\frac{1}{3} a^3 f^2 x^3-\frac{a^2 f (b f-2 a d e) x^2}{2 d}+\frac{a^2 \left(a d^2 e^2-3 b d f e+2 b c f^2\right) x}{d^2}+\frac{\left(a^2 b f^2 c^3-3 a^2 b d e f c^2+3 a^2 b d^2 e^2 c-3 a^2 b f^2 c+3 a^2 b d e f\right) \tan ^{-1}(c+d x)}{d^3}+\frac{\left(-3 b d^2 e^2 a^2-3 b c^2 f^2 a^2+b f^2 a^2+6 b c d e f a^2\right) \log \left(c^2+2 d x c+d^2 x^2+1\right)}{2 d^3}","\frac{i b^2 \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{d^3}-\frac{6 b^2 f (d e-c f) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{d^3}+\frac{a b^2 f^2 x}{d^2}+\frac{i \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \left(a+b \tan ^{-1}(c+d x)\right)^3}{3 d^3}-\frac{(d e-c f) \left(-\left(3-c^2\right) f^2-2 c d e f+d^2 e^2\right) \left(a+b \tan ^{-1}(c+d x)\right)^3}{3 d^3 f}+\frac{b \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)^2}{d^3}-\frac{3 i b f (d e-c f) \left(a+b \tan ^{-1}(c+d x)\right)^2}{d^3}-\frac{3 b f (c+d x) (d e-c f) \left(a+b \tan ^{-1}(c+d x)\right)^2}{d^3}-\frac{b f^2 \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d^3}-\frac{b f^2 (c+d x)^2 \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d^3}+\frac{(e+f x)^3 \left(a+b \tan ^{-1}(c+d x)\right)^3}{3 f}+\frac{b^3 \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \text{Li}_3\left(1-\frac{2}{i (c+d x)+1}\right)}{2 d^3}-\frac{3 i b^3 f (d e-c f) \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right)}{d^3}-\frac{b^3 f^2 \log \left((c+d x)^2+1\right)}{2 d^3}+\frac{b^3 f^2 (c+d x) \tan ^{-1}(c+d x)}{d^3}",1,"(a^2*(a*d^2*e^2 - 3*b*d*e*f + 2*b*c*f^2)*x)/d^2 - (a^2*f*(-2*a*d*e + b*f)*x^2)/(2*d) + (a^3*f^2*x^3)/3 + ((3*a^2*b*c*d^2*e^2 + 3*a^2*b*d*e*f - 3*a^2*b*c^2*d*e*f - 3*a^2*b*c*f^2 + a^2*b*c^3*f^2)*ArcTan[c + d*x])/d^3 + a^2*b*x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcTan[c + d*x] + ((-3*a^2*b*d^2*e^2 + 6*a^2*b*c*d*e*f + a^2*b*f^2 - 3*a^2*b*c^2*f^2)*Log[1 + c^2 + 2*c*d*x + d^2*x^2])/(2*d^3) + (3*a*b^2*e^2*((-I)*ArcTan[c + d*x]^2 + (c + d*x)*ArcTan[c + d*x]^2 + 2*ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])] - I*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]))/d + 6*a*b^2*e*f*(-(((c + d*x)*ArcTan[c + d*x])/d^2) + (I*c*ArcTan[c + d*x]^2)/d^2 - (c*(c + d*x)*ArcTan[c + d*x]^2)/d^2 + ((1 + (c + d*x)^2)*ArcTan[c + d*x]^2)/(2*d^2) - (2*c*ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])])/d^2 - Log[1/Sqrt[1 + (c + d*x)^2]]/d^2 + (I*c*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])])/d^2) + (b^3*e^2*((-I)*ArcTan[c + d*x]^3 + (c + d*x)*ArcTan[c + d*x]^3 + 3*ArcTan[c + d*x]^2*Log[1 + E^((2*I)*ArcTan[c + d*x])] - (3*I)*ArcTan[c + d*x]*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])] + (3*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])])/2))/d + (b^3*e*f*(ArcTan[c + d*x]*((3*I)*ArcTan[c + d*x] + (2*I)*c*ArcTan[c + d*x]^2 + (1 + (c + d*x)^2)*ArcTan[c + d*x]^2 - (c + d*x)*ArcTan[c + d*x]*(3 + 2*c*ArcTan[c + d*x]) - 6*Log[1 + E^((2*I)*ArcTan[c + d*x])] - 6*c*ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])]) + (3*I)*(1 + 2*c*ArcTan[c + d*x])*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])] - 3*c*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])]))/d^2 + (a*b^2*f^2*(1 + (c + d*x)^2)^(3/2)*((c + d*x)/Sqrt[1 + (c + d*x)^2] + (6*c*(c + d*x)*ArcTan[c + d*x])/Sqrt[1 + (c + d*x)^2] + (3*(c + d*x)*ArcTan[c + d*x]^2)/Sqrt[1 + (c + d*x)^2] + (3*c^2*(c + d*x)*ArcTan[c + d*x]^2)/Sqrt[1 + (c + d*x)^2] + I*ArcTan[c + d*x]^2*Cos[3*ArcTan[c + d*x]] - (3*I)*c^2*ArcTan[c + d*x]^2*Cos[3*ArcTan[c + d*x]] - 2*ArcTan[c + d*x]*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 6*c^2*ArcTan[c + d*x]*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 6*c*Cos[3*ArcTan[c + d*x]]*Log[1/Sqrt[1 + (c + d*x)^2]] + (ArcTan[c + d*x]*(-4 + (3*I - 12*c - (9*I)*c^2)*ArcTan[c + d*x]) + 6*(-1 + 3*c^2)*ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 18*c*Log[1/Sqrt[1 + (c + d*x)^2]])/Sqrt[1 + (c + d*x)^2] - ((4*I)*(-1 + 3*c^2)*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])])/(1 + (c + d*x)^2)^(3/2) + Sin[3*ArcTan[c + d*x]] + 6*c*ArcTan[c + d*x]*Sin[3*ArcTan[c + d*x]] - ArcTan[c + d*x]^2*Sin[3*ArcTan[c + d*x]] + 3*c^2*ArcTan[c + d*x]^2*Sin[3*ArcTan[c + d*x]]))/(4*d^3) + (b^3*f^2*((-I)*(3*c - ArcTan[c + d*x] + 3*c^2*ArcTan[c + d*x])*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])] + ((1 + (c + d*x)^2)^(3/2)*((3*(c + d*x)*ArcTan[c + d*x])/Sqrt[1 + (c + d*x)^2] + (9*c*(c + d*x)*ArcTan[c + d*x]^2)/Sqrt[1 + (c + d*x)^2] + (3*(c + d*x)*ArcTan[c + d*x]^3)/Sqrt[1 + (c + d*x)^2] + (3*c^2*(c + d*x)*ArcTan[c + d*x]^3)/Sqrt[1 + (c + d*x)^2] - (9*I)*c*ArcTan[c + d*x]^2*Cos[3*ArcTan[c + d*x]] + I*ArcTan[c + d*x]^3*Cos[3*ArcTan[c + d*x]] - (3*I)*c^2*ArcTan[c + d*x]^3*Cos[3*ArcTan[c + d*x]] + 18*c*ArcTan[c + d*x]*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*ArcTan[c + d*x])] - 3*ArcTan[c + d*x]^2*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 9*c^2*ArcTan[c + d*x]^2*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 3*Cos[3*ArcTan[c + d*x]]*Log[1/Sqrt[1 + (c + d*x)^2]] + (3*(ArcTan[c + d*x]^2*(-2 - (9*I)*c + I*ArcTan[c + d*x] - 4*c*ArcTan[c + d*x] - (3*I)*c^2*ArcTan[c + d*x]) + 3*ArcTan[c + d*x]*(6*c - ArcTan[c + d*x] + 3*c^2*ArcTan[c + d*x])*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 3*Log[1/Sqrt[1 + (c + d*x)^2]]))/Sqrt[1 + (c + d*x)^2] + (6*(-1 + 3*c^2)*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])])/(1 + (c + d*x)^2)^(3/2) + 3*ArcTan[c + d*x]*Sin[3*ArcTan[c + d*x]] + 9*c*ArcTan[c + d*x]^2*Sin[3*ArcTan[c + d*x]] - ArcTan[c + d*x]^3*Sin[3*ArcTan[c + d*x]] + 3*c^2*ArcTan[c + d*x]^3*Sin[3*ArcTan[c + d*x]]))/12))/d^3","B",0
37,1,592,337,0.8194021,"\int (e+f x) \left(a+b \tan ^{-1}(c+d x)\right)^3 \, dx","Integrate[(e + f*x)*(a + b*ArcTan[c + d*x])^3,x]","\frac{a^3 f (c+d x)^2+a^2 (c+d x) (-2 a c f+2 a d e-3 b f)-3 a^2 b (d e-c f) \log \left((c+d x)^2+1\right)-3 a^2 b (c+d x) \tan ^{-1}(c+d x) (c f-d (2 e+f x))+3 a^2 b f \tan ^{-1}(c+d x)+6 a b^2 d e \left(\tan ^{-1}(c+d x) \left((c+d x-i) \tan ^{-1}(c+d x)+2 \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)\right)-i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)\right)-6 a b^2 c f \left(\tan ^{-1}(c+d x) \left((c+d x-i) \tan ^{-1}(c+d x)+2 \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)\right)-i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)\right)+6 a b^2 f \left(-\log \left(\frac{1}{\sqrt{(c+d x)^2+1}}\right)+\frac{1}{2} \left((c+d x)^2+1\right) \tan ^{-1}(c+d x)^2-(c+d x) \tan ^{-1}(c+d x)\right)+2 b^3 d e \left(-3 i \tan ^{-1}(c+d x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)+\frac{3}{2} \text{Li}_3\left(-e^{2 i \tan ^{-1}(c+d x)}\right)+\tan ^{-1}(c+d x)^2 \left((c+d x-i) \tan ^{-1}(c+d x)+3 \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)\right)\right)+b^3 f \left(3 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)+\tan ^{-1}(c+d x) \left(\left((c+d x)^2+1\right) \tan ^{-1}(c+d x)^2-3 (c+d x) \tan ^{-1}(c+d x)+3 i \tan ^{-1}(c+d x)-6 \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)\right)\right)-2 b^3 c f \left(-3 i \tan ^{-1}(c+d x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right)+\frac{3}{2} \text{Li}_3\left(-e^{2 i \tan ^{-1}(c+d x)}\right)+\tan ^{-1}(c+d x)^2 \left((c+d x-i) \tan ^{-1}(c+d x)+3 \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)\right)\right)}{2 d^2}","\frac{3 i b^2 (d e-c f) \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{d^2}-\frac{3 b^2 f \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{d^2}+\frac{i (d e-c f) \left(a+b \tan ^{-1}(c+d x)\right)^3}{d^2}-\frac{(-c f+d e+f) (d e-(c+1) f) \left(a+b \tan ^{-1}(c+d x)\right)^3}{2 d^2 f}+\frac{3 b (d e-c f) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)^2}{d^2}-\frac{3 i b f \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d^2}-\frac{3 b f (c+d x) \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 d^2}+\frac{(e+f x)^2 \left(a+b \tan ^{-1}(c+d x)\right)^3}{2 f}+\frac{3 b^3 (d e-c f) \text{Li}_3\left(1-\frac{2}{i (c+d x)+1}\right)}{2 d^2}-\frac{3 i b^3 f \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right)}{2 d^2}",1,"(a^2*(2*a*d*e - 3*b*f - 2*a*c*f)*(c + d*x) + a^3*f*(c + d*x)^2 + 3*a^2*b*f*ArcTan[c + d*x] - 3*a^2*b*(c + d*x)*(c*f - d*(2*e + f*x))*ArcTan[c + d*x] + 6*a*b^2*f*(-((c + d*x)*ArcTan[c + d*x]) + ((1 + (c + d*x)^2)*ArcTan[c + d*x]^2)/2 - Log[1/Sqrt[1 + (c + d*x)^2]]) - 3*a^2*b*(d*e - c*f)*Log[1 + (c + d*x)^2] + 6*a*b^2*d*e*(ArcTan[c + d*x]*((-I + c + d*x)*ArcTan[c + d*x] + 2*Log[1 + E^((2*I)*ArcTan[c + d*x])]) - I*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]) - 6*a*b^2*c*f*(ArcTan[c + d*x]*((-I + c + d*x)*ArcTan[c + d*x] + 2*Log[1 + E^((2*I)*ArcTan[c + d*x])]) - I*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]) + b^3*f*(ArcTan[c + d*x]*((3*I)*ArcTan[c + d*x] - 3*(c + d*x)*ArcTan[c + d*x] + (1 + (c + d*x)^2)*ArcTan[c + d*x]^2 - 6*Log[1 + E^((2*I)*ArcTan[c + d*x])]) + (3*I)*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]) + 2*b^3*d*e*(ArcTan[c + d*x]^2*((-I + c + d*x)*ArcTan[c + d*x] + 3*Log[1 + E^((2*I)*ArcTan[c + d*x])]) - (3*I)*ArcTan[c + d*x]*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])] + (3*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])])/2) - 2*b^3*c*f*(ArcTan[c + d*x]^2*((-I + c + d*x)*ArcTan[c + d*x] + 3*Log[1 + E^((2*I)*ArcTan[c + d*x])]) - (3*I)*ArcTan[c + d*x]*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])] + (3*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])])/2))/(2*d^2)","A",0
38,1,266,143,0.1542906,"\int \left(a+b \tan ^{-1}(c+d x)\right)^3 \, dx","Integrate[(a + b*ArcTan[c + d*x])^3,x]","\frac{2 a^3 d x-3 a^2 b \log \left(c^2+2 c d x+d^2 x^2+1\right)+6 a^2 b c \tan ^{-1}(c+d x)+6 a^2 b d x \tan ^{-1}(c+d x)-6 i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)-6 i a b^2 \tan ^{-1}(c+d x)^2+6 a b^2 c \tan ^{-1}(c+d x)^2+6 a b^2 d x \tan ^{-1}(c+d x)^2+12 a b^2 \tan ^{-1}(c+d x) \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)+3 b^3 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c+d x)}\right)-2 i b^3 \tan ^{-1}(c+d x)^3+2 b^3 c \tan ^{-1}(c+d x)^3+2 b^3 d x \tan ^{-1}(c+d x)^3+6 b^3 \tan ^{-1}(c+d x)^2 \log \left(1+e^{2 i \tan ^{-1}(c+d x)}\right)}{2 d}","\frac{3 i b^2 \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{d}+\frac{(c+d x) \left(a+b \tan ^{-1}(c+d x)\right)^3}{d}+\frac{i \left(a+b \tan ^{-1}(c+d x)\right)^3}{d}+\frac{3 b \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)^2}{d}+\frac{3 b^3 \text{Li}_3\left(1-\frac{2}{i (c+d x)+1}\right)}{2 d}",1,"(2*a^3*d*x + 6*a^2*b*c*ArcTan[c + d*x] + 6*a^2*b*d*x*ArcTan[c + d*x] - (6*I)*a*b^2*ArcTan[c + d*x]^2 + 6*a*b^2*c*ArcTan[c + d*x]^2 + 6*a*b^2*d*x*ArcTan[c + d*x]^2 - (2*I)*b^3*ArcTan[c + d*x]^3 + 2*b^3*c*ArcTan[c + d*x]^3 + 2*b^3*d*x*ArcTan[c + d*x]^3 + 12*a*b^2*ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 6*b^3*ArcTan[c + d*x]^2*Log[1 + E^((2*I)*ArcTan[c + d*x])] - 3*a^2*b*Log[1 + c^2 + 2*c*d*x + d^2*x^2] - (6*I)*b^2*(a + b*ArcTan[c + d*x])*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])] + 3*b^3*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])])/(2*d)","A",0
39,0,0,372,8.9467586,"\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^3}{e+f x} \, dx","Integrate[(a + b*ArcTan[c + d*x])^3/(e + f*x),x]","\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^3}{e+f x} \, dx","\frac{3 b^2 \left(a+b \tan ^{-1}(c+d x)\right) \text{Li}_3\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right)}{2 f}-\frac{3 b^2 \text{Li}_3\left(1-\frac{2}{1-i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)}{2 f}-\frac{3 i b \left(a+b \tan ^{-1}(c+d x)\right)^2 \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right)}{2 f}+\frac{\left(a+b \tan ^{-1}(c+d x)\right)^3 \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{f}+\frac{3 i b \text{Li}_2\left(1-\frac{2}{1-i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)^2}{2 f}-\frac{\log \left(\frac{2}{1-i (c+d x)}\right) \left(a+b \tan ^{-1}(c+d x)\right)^3}{f}+\frac{3 i b^3 \text{Li}_4\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right)}{4 f}-\frac{3 i b^3 \text{Li}_4\left(1-\frac{2}{1-i (c+d x)}\right)}{4 f}",1,"Integrate[(a + b*ArcTan[c + d*x])^3/(e + f*x), x]","F",-1
40,0,0,1233,17.4314319,"\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^3}{(e+f x)^2} \, dx","Integrate[(a + b*ArcTan[c + d*x])^3/(e + f*x)^2,x]","\int \frac{\left(a+b \tan ^{-1}(c+d x)\right)^3}{(e+f x)^2} \, dx","\frac{i d \tan ^{-1}(c+d x)^3 b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{d (d e-c f) \tan ^{-1}(c+d x)^3 b^3}{f \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}-\frac{3 d \tan ^{-1}(c+d x)^2 \log \left(\frac{2}{1-i (c+d x)}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 d \tan ^{-1}(c+d x)^2 \log \left(\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 d \tan ^{-1}(c+d x)^2 \log \left(\frac{2}{i (c+d x)+1}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i d \tan ^{-1}(c+d x) \text{Li}_2\left(1-\frac{2}{1-i (c+d x)}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 i d \tan ^{-1}(c+d x) \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i d \tan ^{-1}(c+d x) \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 d \text{Li}_3\left(1-\frac{2}{1-i (c+d x)}\right) b^3}{2 \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}+\frac{3 d \text{Li}_3\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right) b^3}{2 \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}+\frac{3 d \text{Li}_3\left(1-\frac{2}{i (c+d x)+1}\right) b^3}{2 \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}+\frac{3 i a d \tan ^{-1}(c+d x)^2 b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 a d (d e-c f) \tan ^{-1}(c+d x)^2 b^2}{f \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}-\frac{6 a d \tan ^{-1}(c+d x) \log \left(\frac{2}{1-i (c+d x)}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{6 a d \tan ^{-1}(c+d x) \log \left(\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{6 a d \tan ^{-1}(c+d x) \log \left(\frac{2}{i (c+d x)+1}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i a d \text{Li}_2\left(1-\frac{2}{1-i (c+d x)}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 i a d \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i a d \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 a^2 d (d e-c f) \tan ^{-1}(c+d x) b}{f \left(f^2+(d e-c f)^2\right)}+\frac{3 a^2 d \log (e+f x) b}{f^2+(d e-c f)^2}-\frac{3 a^2 d \log \left((c+d x)^2+1\right) b}{2 \left(f^2+(d e-c f)^2\right)}-\frac{\left(a+b \tan ^{-1}(c+d x)\right)^3}{f (e+f x)}",1,"Integrate[(a + b*ArcTan[c + d*x])^3/(e + f*x)^2, x]","F",-1
41,1,162,177,0.3929051,"\int (e+f x)^m \left(a+b \tan ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)^m*(a + b*ArcTan[c + d*x]),x]","\frac{(e+f x)^{m+1} \left(2 \left(a+b \tan ^{-1}(c+d x)\right)+\frac{b d (e+f x) \left((d e-(c+i) f) \, _2F_1\left(1,m+2;m+3;\frac{d (e+f x)}{d e-(c-i) f}\right)+(-d e+(c-i) f) \, _2F_1\left(1,m+2;m+3;\frac{d (e+f x)}{d e-(c+i) f}\right)\right)}{(m+2) (-i c f+i d e+f) (d e-(c-i) f)}\right)}{2 f (m+1)}","\frac{(e+f x)^{m+1} \left(a+b \tan ^{-1}(c+d x)\right)}{f (m+1)}-\frac{i b d (e+f x)^{m+2} \, _2F_1\left(1,m+2;m+3;\frac{d (e+f x)}{d e-c f+i f}\right)}{2 f (m+1) (m+2) (d e+(-c+i) f)}+\frac{i b d (e+f x)^{m+2} \, _2F_1\left(1,m+2;m+3;\frac{d (e+f x)}{d e-(c+i) f}\right)}{2 f (m+1) (m+2) (d e-(c+i) f)}",1,"((e + f*x)^(1 + m)*(2*(a + b*ArcTan[c + d*x]) + (b*d*(e + f*x)*((d*e - (I + c)*f)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e - (-I + c)*f)] + (-(d*e) + (-I + c)*f)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e - (I + c)*f)]))/((I*d*e + f - I*c*f)*(d*e - (-I + c)*f)*(2 + m))))/(2*f*(1 + m))","A",1
42,0,0,23,4.9954431,"\int (e+f x)^m \left(a+b \tan ^{-1}(c+d x)\right)^2 \, dx","Integrate[(e + f*x)^m*(a + b*ArcTan[c + d*x])^2,x]","\int (e+f x)^m \left(a+b \tan ^{-1}(c+d x)\right)^2 \, dx","\text{Int}\left((e+f x)^m \left(a+b \tan ^{-1}(c+d x)\right)^2,x\right)",0,"Integrate[(e + f*x)^m*(a + b*ArcTan[c + d*x])^2, x]","A",-1
43,0,0,23,0.5068543,"\int (e+f x)^m \left(a+b \tan ^{-1}(c+d x)\right)^3 \, dx","Integrate[(e + f*x)^m*(a + b*ArcTan[c + d*x])^3,x]","\int (e+f x)^m \left(a+b \tan ^{-1}(c+d x)\right)^3 \, dx","\text{Int}\left((e+f x)^m \left(a+b \tan ^{-1}(c+d x)\right)^3,x\right)",0,"Integrate[(e + f*x)^m*(a + b*ArcTan[c + d*x])^3, x]","A",-1
44,1,95,106,0.0770792,"\int x^3 \tan ^{-1}(a+b x) \, dx","Integrate[x^3*ArcTan[a + b*x],x]","\frac{6 \left(1-6 a^2\right) b x+6 b^4 x^4 \tan ^{-1}(a+b x)-2 (a+b x)^3+12 a (a+b x)^2+3 i (a-i)^4 \log (-a-b x+i)-3 i (a+i)^4 \log (a+b x+i)}{24 b^4}","-\frac{a \left(1-a^2\right) \log \left((a+b x)^2+1\right)}{2 b^4}+\frac{\left(1-6 a^2\right) x}{4 b^3}-\frac{\left(a^4-6 a^2+1\right) \tan ^{-1}(a+b x)}{4 b^4}-\frac{(a+b x)^3}{12 b^4}+\frac{a (a+b x)^2}{2 b^4}+\frac{1}{4} x^4 \tan ^{-1}(a+b x)",1,"(6*(1 - 6*a^2)*b*x + 12*a*(a + b*x)^2 - 2*(a + b*x)^3 + 6*b^4*x^4*ArcTan[a + b*x] + (3*I)*(-I + a)^4*Log[I - a - b*x] - (3*I)*(I + a)^4*Log[I + a + b*x])/(24*b^4)","C",1
45,1,114,79,0.0570233,"\int x^2 \tan ^{-1}(a+b x) \, dx","Integrate[x^2*ArcTan[a + b*x],x]","\frac{\frac{1}{3} b \left(\frac{a+b x}{b}-\frac{a}{b}\right)^3 \tan ^{-1}(a+b x)-\frac{1}{3} b \left(\frac{(a+b x)^2}{2 b^3}-\frac{(1-i a)^3 \log (a+b x+i)}{2 b^3}-\frac{(1+i a)^3 \log (-a-b x+i)}{2 b^3}-\frac{3 a x}{b^2}\right)}{b}","\frac{\left(1-3 a^2\right) \log \left((a+b x)^2+1\right)}{6 b^3}-\frac{a \left(3-a^2\right) \tan ^{-1}(a+b x)}{3 b^3}-\frac{(a+b x)^2}{6 b^3}+\frac{a x}{b^2}+\frac{1}{3} x^3 \tan ^{-1}(a+b x)",1,"((b*(-(a/b) + (a + b*x)/b)^3*ArcTan[a + b*x])/3 - (b*((-3*a*x)/b^2 + (a + b*x)^2/(2*b^3) - ((1 + I*a)^3*Log[I - a - b*x])/(2*b^3) - ((1 - I*a)^3*Log[I + a + b*x])/(2*b^3)))/3)/b","C",1
46,1,90,60,0.0362423,"\int x \tan ^{-1}(a+b x) \, dx","Integrate[x*ArcTan[a + b*x],x]","\frac{-i a^2 \log (a+b x+i)+2 b^2 x^2 \tan ^{-1}(a+b x)+2 a \log (a+b x+i)+i (a-i)^2 \log (-a-b x+i)+i \log (a+b x+i)-2 b x}{4 b^2}","\frac{\left(1-a^2\right) \tan ^{-1}(a+b x)}{2 b^2}+\frac{a \log \left((a+b x)^2+1\right)}{2 b^2}+\frac{1}{2} x^2 \tan ^{-1}(a+b x)-\frac{x}{2 b}",1,"(-2*b*x + 2*b^2*x^2*ArcTan[a + b*x] + I*(-I + a)^2*Log[I - a - b*x] + I*Log[I + a + b*x] + 2*a*Log[I + a + b*x] - I*a^2*Log[I + a + b*x])/(4*b^2)","C",1
47,1,39,33,0.0159558,"\int \tan ^{-1}(a+b x) \, dx","Integrate[ArcTan[a + b*x],x]","-\frac{\log \left(a^2+2 a b x+b^2 x^2+1\right)-2 (a+b x) \tan ^{-1}(a+b x)}{2 b}","\frac{(a+b x) \tan ^{-1}(a+b x)}{b}-\frac{\log \left((a+b x)^2+1\right)}{2 b}",1,"-1/2*(-2*(a + b*x)*ArcTan[a + b*x] + Log[1 + a^2 + 2*a*b*x + b^2*x^2])/b","A",1
48,1,171,120,0.0089704,"\int \frac{\tan ^{-1}(a+b x)}{x} \, dx","Integrate[ArcTan[a + b*x]/x,x]","\frac{1}{2} i \text{Li}_2\left(\frac{i (1-i (a+b x))}{a+i}\right)-\frac{1}{2} i \text{Li}_2\left(-\frac{i (i (a+b x)+1)}{a-i}\right)-\frac{1}{2} i \log (1+i (a+b x)) \log \left(\frac{i \left(\frac{a+b x}{b}-\frac{a}{b}\right)}{-\frac{1}{b}-\frac{i a}{b}}\right)+\frac{1}{2} i \log (1-i (a+b x)) \log \left(-\frac{i \left(\frac{a+b x}{b}-\frac{a}{b}\right)}{-\frac{1}{b}+\frac{i a}{b}}\right)","\frac{1}{2} i \text{Li}_2\left(1-\frac{2}{1-i (a+b x)}\right)-\frac{1}{2} i \text{Li}_2\left(1-\frac{2 b x}{(i-a) (1-i (a+b x))}\right)+\log \left(\frac{2}{1-i (a+b x)}\right) \left(-\tan ^{-1}(a+b x)\right)+\log \left(\frac{2 b x}{(-a+i) (1-i (a+b x))}\right) \tan ^{-1}(a+b x)",1,"(-1/2*I)*Log[1 + I*(a + b*x)]*Log[(I*(-(a/b) + (a + b*x)/b))/(-b^(-1) - (I*a)/b)] + (I/2)*Log[1 - I*(a + b*x)]*Log[((-I)*(-(a/b) + (a + b*x)/b))/(-b^(-1) + (I*a)/b)] + (I/2)*PolyLog[2, (I*(1 - I*(a + b*x)))/(I + a)] - (I/2)*PolyLog[2, ((-I)*(1 + I*(a + b*x)))/(-I + a)]","A",0
49,1,67,62,0.0624389,"\int \frac{\tan ^{-1}(a+b x)}{x^2} \, dx","Integrate[ArcTan[a + b*x]/x^2,x]","-\frac{\tan ^{-1}(a+b x)}{x}+\frac{b (i (a+i) \log (-a-b x+i)+(-1-i a) \log (a+b x+i)+2 \log (x))}{2 \left(a^2+1\right)}","\frac{b \log (x)}{a^2+1}-\frac{b \log \left((a+b x)^2+1\right)}{2 \left(a^2+1\right)}-\frac{a b \tan ^{-1}(a+b x)}{a^2+1}-\frac{\tan ^{-1}(a+b x)}{x}",1,"-(ArcTan[a + b*x]/x) + (b*(2*Log[x] + I*(I + a)*Log[I - a - b*x] + (-1 - I*a)*Log[I + a + b*x]))/(2*(1 + a^2))","C",1
50,1,92,96,0.1050284,"\int \frac{\tan ^{-1}(a+b x)}{x^3} \, dx","Integrate[ArcTan[a + b*x]/x^3,x]","\frac{-2 \tan ^{-1}(a+b x)+\frac{b x \left(-i (a+i)^2 b x \log (-a-b x+i)-4 a b x \log (x)+(a-i) ((1+i a) b x \log (a+b x+i)-2 (a+i))\right)}{\left(a^2+1\right)^2}}{4 x^2}","-\frac{a b^2 \log (x)}{\left(a^2+1\right)^2}+\frac{a b^2 \log \left((a+b x)^2+1\right)}{2 \left(a^2+1\right)^2}-\frac{\left(1-a^2\right) b^2 \tan ^{-1}(a+b x)}{2 \left(a^2+1\right)^2}-\frac{b}{2 \left(a^2+1\right) x}-\frac{\tan ^{-1}(a+b x)}{2 x^2}",1,"(-2*ArcTan[a + b*x] + (b*x*(-4*a*b*x*Log[x] - I*(I + a)^2*b*x*Log[I - a - b*x] + (-I + a)*(-2*(I + a) + (1 + I*a)*b*x*Log[I + a + b*x])))/(1 + a^2)^2)/(4*x^2)","C",1
51,1,128,129,0.1486971,"\int \frac{\tan ^{-1}(a+b x)}{x^4} \, dx","Integrate[ArcTan[a + b*x]/x^4,x]","\frac{2 \left(3 a^2-1\right) b^3 x^3 \log (x)-(a-i) b x \left((a+i) \left(a^2-4 a b x+1\right)+i (a-i)^2 b^2 x^2 \log (a+b x+i)\right)-2 \left(a^2+1\right)^3 \tan ^{-1}(a+b x)+i (a+i)^3 b^3 x^3 \log (-a-b x+i)}{6 \left(a^2+1\right)^3 x^3}","-\frac{\left(1-3 a^2\right) b^3 \log (x)}{3 \left(a^2+1\right)^3}+\frac{\left(1-3 a^2\right) b^3 \log \left((a+b x)^2+1\right)}{6 \left(a^2+1\right)^3}+\frac{a \left(3-a^2\right) b^3 \tan ^{-1}(a+b x)}{3 \left(a^2+1\right)^3}+\frac{2 a b^2}{3 \left(a^2+1\right)^2 x}-\frac{b}{6 \left(a^2+1\right) x^2}-\frac{\tan ^{-1}(a+b x)}{3 x^3}",1,"(-2*(1 + a^2)^3*ArcTan[a + b*x] + 2*(-1 + 3*a^2)*b^3*x^3*Log[x] + I*(I + a)^3*b^3*x^3*Log[I - a - b*x] - (-I + a)*b*x*((I + a)*(1 + a^2 - 4*a*b*x) + I*(-I + a)^2*b^2*x^2*Log[I + a + b*x]))/(6*(1 + a^2)^3*x^3)","C",1
52,1,701,863,0.9128513,"\int \frac{\tan ^{-1}(a+b x)}{c+d x^3} \, dx","Integrate[ArcTan[a + b*x]/(c + d*x^3),x]","\frac{-i \text{Li}_2\left(\frac{\sqrt[3]{d} (a+b x-i)}{(a-i) \sqrt[3]{d}-b \sqrt[3]{c}}\right)+(-1)^{5/6} \text{Li}_2\left(\frac{\sqrt[6]{-1} \sqrt[3]{d} (a+b x-i)}{\sqrt[6]{-1} \sqrt[3]{d} (a-i)+i b \sqrt[3]{c}}\right)+\sqrt[6]{-1} \text{Li}_2\left(\frac{\sqrt[3]{-1} \sqrt[3]{d} (a+b x-i)}{\sqrt[3]{-1} \sqrt[3]{d} (a-i)+b \sqrt[3]{c}}\right)+i \text{Li}_2\left(\frac{\sqrt[3]{d} (a+b x+i)}{(a+i) \sqrt[3]{d}-b \sqrt[3]{c}}\right)-\sqrt[6]{-1} \text{Li}_2\left(\frac{\sqrt[3]{-1} \sqrt[3]{d} (a+b x+i)}{\sqrt[3]{-1} \sqrt[3]{d} (a+i)+b \sqrt[3]{c}}\right)-(-1)^{5/6} \text{Li}_2\left(\frac{(-1)^{2/3} \sqrt[3]{d} (a+b x+i)}{(-1)^{2/3} (a+i) \sqrt[3]{d}-b \sqrt[3]{c}}\right)-i \log (i a+i b x+1) \log \left(\frac{b \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{b \sqrt[3]{c}-(a-i) \sqrt[3]{d}}\right)+i \log (-i (a+b x+i)) \log \left(\frac{b \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{b \sqrt[3]{c}-(a+i) \sqrt[3]{d}}\right)+\sqrt[6]{-1} \log (i a+i b x+1) \log \left(\frac{b \left(\sqrt[3]{c}-\sqrt[3]{-1} \sqrt[3]{d} x\right)}{b \sqrt[3]{c}+\sqrt[3]{-1} (a-i) \sqrt[3]{d}}\right)-\sqrt[6]{-1} \log (-i (a+b x+i)) \log \left(\frac{b \left(\sqrt[3]{c}-\sqrt[3]{-1} \sqrt[3]{d} x\right)}{b \sqrt[3]{c}+\sqrt[3]{-1} (a+i) \sqrt[3]{d}}\right)-(-1)^{5/6} \log (-i (a+b x+i)) \log \left(\frac{b \left(\sqrt[3]{c}+(-1)^{2/3} \sqrt[3]{d} x\right)}{b \sqrt[3]{c}+\sqrt[6]{-1} (1-i a) \sqrt[3]{d}}\right)+(-1)^{5/6} \log (i a+i b x+1) \log \left(\frac{b \left(\sqrt[3]{c}+(-1)^{2/3} \sqrt[3]{d} x\right)}{b \sqrt[3]{c}-(-1)^{2/3} (a-i) \sqrt[3]{d}}\right)}{6 c^{2/3} \sqrt[3]{d}}","-\frac{i \log (i a+i b x+1) \log \left(\frac{b \left(\sqrt[3]{d} x+\sqrt[3]{c}\right)}{\sqrt[3]{d} (i-a)+b \sqrt[3]{c}}\right)}{6 c^{2/3} \sqrt[3]{d}}+\frac{i \log (-i a-i b x+1) \log \left(\frac{b \left(\sqrt[3]{d} x+\sqrt[3]{c}\right)}{b \sqrt[3]{c}-(a+i) \sqrt[3]{d}}\right)}{6 c^{2/3} \sqrt[3]{d}}+\frac{\sqrt[6]{-1} \log (i a+i b x+1) \log \left(\frac{b \left(\sqrt[3]{c}-\sqrt[3]{-1} \sqrt[3]{d} x\right)}{b \sqrt[3]{c}-\sqrt[3]{-1} (i-a) \sqrt[3]{d}}\right)}{6 c^{2/3} \sqrt[3]{d}}-\frac{\sqrt[6]{-1} \log (-i a-i b x+1) \log \left(\frac{b \left(\sqrt[3]{c}-\sqrt[3]{-1} \sqrt[3]{d} x\right)}{\sqrt[3]{-1} \sqrt[3]{d} (a+i)+b \sqrt[3]{c}}\right)}{6 c^{2/3} \sqrt[3]{d}}+\frac{(-1)^{5/6} \log (i a+i b x+1) \log \left(\frac{b \left((-1)^{2/3} \sqrt[3]{d} x+\sqrt[3]{c}\right)}{(-1)^{2/3} \sqrt[3]{d} (i-a)+b \sqrt[3]{c}}\right)}{6 c^{2/3} \sqrt[3]{d}}-\frac{(-1)^{5/6} \log (-i a-i b x+1) \log \left(\frac{b \left((-1)^{2/3} \sqrt[3]{d} x+\sqrt[3]{c}\right)}{\sqrt[6]{-1} \sqrt[3]{d} (1-i a)+b \sqrt[3]{c}}\right)}{6 c^{2/3} \sqrt[3]{d}}-\frac{i \text{Li}_2\left(\frac{\sqrt[3]{d} (-a-b x+i)}{\sqrt[3]{d} (i-a)+b \sqrt[3]{c}}\right)}{6 c^{2/3} \sqrt[3]{d}}+\frac{(-1)^{5/6} \text{Li}_2\left(-\frac{\sqrt[6]{-1} \sqrt[3]{d} (-a-b x+i)}{i b \sqrt[3]{c}-\sqrt[6]{-1} (i-a) \sqrt[3]{d}}\right)}{6 c^{2/3} \sqrt[3]{d}}+\frac{\sqrt[6]{-1} \text{Li}_2\left(-\frac{\sqrt[3]{-1} \sqrt[3]{d} (-a-b x+i)}{b \sqrt[3]{c}-\sqrt[3]{-1} (i-a) \sqrt[3]{d}}\right)}{6 c^{2/3} \sqrt[3]{d}}+\frac{i \text{Li}_2\left(-\frac{\sqrt[3]{d} (a+b x+i)}{b \sqrt[3]{c}-(a+i) \sqrt[3]{d}}\right)}{6 c^{2/3} \sqrt[3]{d}}-\frac{\sqrt[6]{-1} \text{Li}_2\left(\frac{\sqrt[3]{-1} \sqrt[3]{d} (a+b x+i)}{\sqrt[3]{-1} \sqrt[3]{d} (a+i)+b \sqrt[3]{c}}\right)}{6 c^{2/3} \sqrt[3]{d}}-\frac{(-1)^{5/6} \text{Li}_2\left(-\frac{(-1)^{2/3} \sqrt[3]{d} (a+b x+i)}{b \sqrt[3]{c}-(-1)^{2/3} (a+i) \sqrt[3]{d}}\right)}{6 c^{2/3} \sqrt[3]{d}}",1,"((-I)*Log[1 + I*a + I*b*x]*Log[(b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) - (-I + a)*d^(1/3))] + I*Log[(-I)*(I + a + b*x)]*Log[(b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) - (I + a)*d^(1/3))] + (-1)^(1/6)*Log[1 + I*a + I*b*x]*Log[(b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/3)*(-I + a)*d^(1/3))] - (-1)^(1/6)*Log[(-I)*(I + a + b*x)]*Log[(b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/3)*(I + a)*d^(1/3))] - (-1)^(5/6)*Log[(-I)*(I + a + b*x)]*Log[(b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/6)*(1 - I*a)*d^(1/3))] + (-1)^(5/6)*Log[1 + I*a + I*b*x]*Log[(b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) - (-1)^(2/3)*(-I + a)*d^(1/3))] - I*PolyLog[2, (d^(1/3)*(-I + a + b*x))/(-(b*c^(1/3)) + (-I + a)*d^(1/3))] + (-1)^(5/6)*PolyLog[2, ((-1)^(1/6)*d^(1/3)*(-I + a + b*x))/(I*b*c^(1/3) + (-1)^(1/6)*(-I + a)*d^(1/3))] + (-1)^(1/6)*PolyLog[2, ((-1)^(1/3)*d^(1/3)*(-I + a + b*x))/(b*c^(1/3) + (-1)^(1/3)*(-I + a)*d^(1/3))] + I*PolyLog[2, (d^(1/3)*(I + a + b*x))/(-(b*c^(1/3)) + (I + a)*d^(1/3))] - (-1)^(1/6)*PolyLog[2, ((-1)^(1/3)*d^(1/3)*(I + a + b*x))/(b*c^(1/3) + (-1)^(1/3)*(I + a)*d^(1/3))] - (-1)^(5/6)*PolyLog[2, ((-1)^(2/3)*d^(1/3)*(I + a + b*x))/(-(b*c^(1/3)) + (-1)^(2/3)*(I + a)*d^(1/3))])/(6*c^(2/3)*d^(1/3))","A",1
53,1,409,543,0.3618213,"\int \frac{\tan ^{-1}(a+b x)}{c+d x^2} \, dx","Integrate[ArcTan[a + b*x]/(c + d*x^2),x]","-\frac{i \left(-\text{Li}_2\left(\frac{\sqrt{d} (a+b x-i)}{(a-i) \sqrt{d}-b \sqrt{-c}}\right)+\text{Li}_2\left(\frac{\sqrt{d} (a+b x-i)}{\sqrt{d} (a-i)+b \sqrt{-c}}\right)+\text{Li}_2\left(\frac{\sqrt{d} (a+b x+i)}{(a+i) \sqrt{d}-b \sqrt{-c}}\right)-\text{Li}_2\left(\frac{\sqrt{d} (a+b x+i)}{\sqrt{d} (a+i)+b \sqrt{-c}}\right)+\log (i a+i b x+1) \log \left(\frac{b \left(\sqrt{-c}-\sqrt{d} x\right)}{b \sqrt{-c}+(a-i) \sqrt{d}}\right)-\log (-i (a+b x+i)) \log \left(\frac{b \left(\sqrt{-c}-\sqrt{d} x\right)}{b \sqrt{-c}+(a+i) \sqrt{d}}\right)-\log (i a+i b x+1) \log \left(\frac{b \left(\sqrt{-c}+\sqrt{d} x\right)}{b \sqrt{-c}-(a-i) \sqrt{d}}\right)+\log (-i (a+b x+i)) \log \left(\frac{b \left(\sqrt{-c}+\sqrt{d} x\right)}{b \sqrt{-c}-(a+i) \sqrt{d}}\right)\right)}{4 \sqrt{-c} \sqrt{d}}","-\frac{i \text{Li}_2\left(-\frac{\sqrt{d} (-a-b x+i)}{b \sqrt{-c}-(i-a) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{i \text{Li}_2\left(\frac{\sqrt{d} (-a-b x+i)}{\sqrt{d} (i-a)+b \sqrt{-c}}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{i \text{Li}_2\left(-\frac{\sqrt{d} (a+b x+i)}{b \sqrt{-c}-(a+i) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{i \text{Li}_2\left(\frac{\sqrt{d} (a+b x+i)}{\sqrt{d} (a+i)+b \sqrt{-c}}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{i \log (i a+i b x+1) \log \left(\frac{b \left(\sqrt{-c}-\sqrt{d} x\right)}{b \sqrt{-c}-(-a+i) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{i \log (-i a-i b x+1) \log \left(\frac{b \left(\sqrt{-c}-\sqrt{d} x\right)}{b \sqrt{-c}+(a+i) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{i \log (i a+i b x+1) \log \left(\frac{b \left(\sqrt{-c}+\sqrt{d} x\right)}{b \sqrt{-c}+(-a+i) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{i \log (-i a-i b x+1) \log \left(\frac{b \left(\sqrt{-c}+\sqrt{d} x\right)}{b \sqrt{-c}-(a+i) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}",1,"((-1/4*I)*(Log[1 + I*a + I*b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (-I + a)*Sqrt[d])] - Log[(-I)*(I + a + b*x)]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (I + a)*Sqrt[d])] - Log[1 + I*a + I*b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (-I + a)*Sqrt[d])] + Log[(-I)*(I + a + b*x)]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (I + a)*Sqrt[d])] - PolyLog[2, (Sqrt[d]*(-I + a + b*x))/(-(b*Sqrt[-c]) + (-I + a)*Sqrt[d])] + PolyLog[2, (Sqrt[d]*(-I + a + b*x))/(b*Sqrt[-c] + (-I + a)*Sqrt[d])] + PolyLog[2, (Sqrt[d]*(I + a + b*x))/(-(b*Sqrt[-c]) + (I + a)*Sqrt[d])] - PolyLog[2, (Sqrt[d]*(I + a + b*x))/(b*Sqrt[-c] + (I + a)*Sqrt[d])]))/(Sqrt[-c]*Sqrt[d])","A",1
54,1,231,152,0.0228894,"\int \frac{\tan ^{-1}(a+b x)}{c+d x} \, dx","Integrate[ArcTan[a + b*x]/(c + d*x),x]","\frac{i \text{Li}_2\left(-\frac{i d (1-i (a+b x))}{b c-a d-i d}\right)}{2 d}-\frac{i \text{Li}_2\left(\frac{i d (i (a+b x)+1)}{b c-a d+i d}\right)}{2 d}+\frac{i \log (1-i (a+b x)) \log \left(-\frac{i \left(\frac{b c-a d}{b}+\frac{d (a+b x)}{b}\right)}{-\frac{d}{b}-\frac{i (b c-a d)}{b}}\right)}{2 d}-\frac{i \log (1+i (a+b x)) \log \left(\frac{i \left(\frac{b c-a d}{b}+\frac{d (a+b x)}{b}\right)}{-\frac{d}{b}+\frac{i (b c-a d)}{b}}\right)}{2 d}","-\frac{i \text{Li}_2\left(1-\frac{2 b (c+d x)}{(b c-a d+i d) (1-i (a+b x))}\right)}{2 d}+\frac{\tan ^{-1}(a+b x) \log \left(\frac{2 b (c+d x)}{(1-i (a+b x)) (-a d+b c+i d)}\right)}{d}+\frac{i \text{Li}_2\left(1-\frac{2}{1-i (a+b x)}\right)}{2 d}-\frac{\log \left(\frac{2}{1-i (a+b x)}\right) \tan ^{-1}(a+b x)}{d}",1,"((I/2)*Log[1 - I*(a + b*x)]*Log[((-I)*((b*c - a*d)/b + (d*(a + b*x))/b))/(-(d/b) - (I*(b*c - a*d))/b)])/d - ((I/2)*Log[1 + I*(a + b*x)]*Log[(I*((b*c - a*d)/b + (d*(a + b*x))/b))/(-(d/b) + (I*(b*c - a*d))/b)])/d + ((I/2)*PolyLog[2, ((-I)*d*(1 - I*(a + b*x)))/(b*c - I*d - a*d)])/d - ((I/2)*PolyLog[2, (I*d*(1 + I*(a + b*x)))/(b*c + I*d - a*d)])/d","A",0
55,1,771,244,11.8047813,"\int \frac{\tan ^{-1}(a+b x)}{c+\frac{d}{x}} \, dx","Integrate[ArcTan[a + b*x]/(c + d/x),x]","\frac{b c d \sqrt{a^2-\frac{2 a b d}{c}+\frac{b^2 d^2}{c^2}+1} \tan ^{-1}(a+b x)^2 e^{-i \tan ^{-1}\left(a-\frac{b d}{c}\right)}-2 a^2 c^2 \tan ^{-1}(a+b x)+2 b^2 d^2 \tan ^{-1}\left(a-\frac{b d}{c}\right) \log \left(1-\exp \left(2 i \left(\tan ^{-1}(a+b x)-\tan ^{-1}\left(a-\frac{b d}{c}\right)\right)\right)\right)-2 b^2 d^2 \tan ^{-1}(a+b x) \log \left(1-\exp \left(2 i \left(\tan ^{-1}(a+b x)-\tan ^{-1}\left(a-\frac{b d}{c}\right)\right)\right)\right)-2 i b^2 d^2 \tan ^{-1}(a+b x) \tan ^{-1}\left(a-\frac{b d}{c}\right)-2 b^2 d^2 \tan ^{-1}\left(a-\frac{b d}{c}\right) \log \left(-\sin \left(\tan ^{-1}\left(a-\frac{b d}{c}\right)-\tan ^{-1}(a+b x)\right)\right)+2 b^2 c d x \tan ^{-1}(a+b x)+\pi  b^2 d^2 \log \left(\frac{1}{\sqrt{(a+b x)^2+1}}\right)-i b^2 d^2 \tan ^{-1}(a+b x)^2-i \pi  b^2 d^2 \tan ^{-1}(a+b x)-\pi  b^2 d^2 \log \left(1+e^{-2 i \tan ^{-1}(a+b x)}\right)+2 b^2 d^2 \tan ^{-1}(a+b x) \log \left(1+e^{2 i \tan ^{-1}(a+b x)}\right)-2 a c^2 \log \left(\frac{1}{\sqrt{(a+b x)^2+1}}\right)-2 a b c^2 x \tan ^{-1}(a+b x)+i b d (b d-a c) \text{Li}_2\left(\exp \left(2 i \left(\tan ^{-1}(a+b x)-\tan ^{-1}\left(a-\frac{b d}{c}\right)\right)\right)\right)-2 a b c d \tan ^{-1}\left(a-\frac{b d}{c}\right) \log \left(1-\exp \left(2 i \left(\tan ^{-1}(a+b x)-\tan ^{-1}\left(a-\frac{b d}{c}\right)\right)\right)\right)+2 a b c d \tan ^{-1}(a+b x) \log \left(1-\exp \left(2 i \left(\tan ^{-1}(a+b x)-\tan ^{-1}\left(a-\frac{b d}{c}\right)\right)\right)\right)+i b d (a c-b d) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a+b x)}\right)+2 b c d \log \left(\frac{1}{\sqrt{(a+b x)^2+1}}\right)-\pi  a b c d \log \left(\frac{1}{\sqrt{(a+b x)^2+1}}\right)+i a b c d \tan ^{-1}(a+b x)^2-b c d \tan ^{-1}(a+b x)^2+2 a b c d \tan ^{-1}(a+b x)+2 i a b c d \tan ^{-1}(a+b x) \tan ^{-1}\left(a-\frac{b d}{c}\right)+i \pi  a b c d \tan ^{-1}(a+b x)+\pi  a b c d \log \left(1+e^{-2 i \tan ^{-1}(a+b x)}\right)-2 a b c d \tan ^{-1}(a+b x) \log \left(1+e^{2 i \tan ^{-1}(a+b x)}\right)+2 a b c d \tan ^{-1}\left(a-\frac{b d}{c}\right) \log \left(-\sin \left(\tan ^{-1}\left(a-\frac{b d}{c}\right)-\tan ^{-1}(a+b x)\right)\right)}{b c^2 (2 b d-2 a c)}","\frac{i d \text{Li}_2\left(\frac{c (-a-b x+i)}{-a c+i c+b d}\right)}{2 c^2}-\frac{i d \text{Li}_2\left(\frac{c (a+b x+i)}{(a+i) c-b d}\right)}{2 c^2}+\frac{i d \log (i a+i b x+1) \log \left(\frac{b (c x+d)}{b d+(-a+i) c}\right)}{2 c^2}-\frac{i d \log (-i a-i b x+1) \log \left(-\frac{b (c x+d)}{-b d+(a+i) c}\right)}{2 c^2}-\frac{(i a+i b x+1) \log (i a+i b x+1)}{2 b c}-\frac{(-i a-i b x+1) \log (-i (a+b x+i))}{2 b c}",1,"(-2*a^2*c^2*ArcTan[a + b*x] + 2*a*b*c*d*ArcTan[a + b*x] + I*a*b*c*d*Pi*ArcTan[a + b*x] - I*b^2*d^2*Pi*ArcTan[a + b*x] - 2*a*b*c^2*x*ArcTan[a + b*x] + 2*b^2*c*d*x*ArcTan[a + b*x] + (2*I)*a*b*c*d*ArcTan[a - (b*d)/c]*ArcTan[a + b*x] - (2*I)*b^2*d^2*ArcTan[a - (b*d)/c]*ArcTan[a + b*x] - b*c*d*ArcTan[a + b*x]^2 + I*a*b*c*d*ArcTan[a + b*x]^2 - I*b^2*d^2*ArcTan[a + b*x]^2 + (b*c*d*Sqrt[1 + a^2 - (2*a*b*d)/c + (b^2*d^2)/c^2]*ArcTan[a + b*x]^2)/E^(I*ArcTan[a - (b*d)/c]) + a*b*c*d*Pi*Log[1 + E^((-2*I)*ArcTan[a + b*x])] - b^2*d^2*Pi*Log[1 + E^((-2*I)*ArcTan[a + b*x])] - 2*a*b*c*d*ArcTan[a + b*x]*Log[1 + E^((2*I)*ArcTan[a + b*x])] + 2*b^2*d^2*ArcTan[a + b*x]*Log[1 + E^((2*I)*ArcTan[a + b*x])] - 2*a*b*c*d*ArcTan[a - (b*d)/c]*Log[1 - E^((2*I)*(-ArcTan[a - (b*d)/c] + ArcTan[a + b*x]))] + 2*b^2*d^2*ArcTan[a - (b*d)/c]*Log[1 - E^((2*I)*(-ArcTan[a - (b*d)/c] + ArcTan[a + b*x]))] + 2*a*b*c*d*ArcTan[a + b*x]*Log[1 - E^((2*I)*(-ArcTan[a - (b*d)/c] + ArcTan[a + b*x]))] - 2*b^2*d^2*ArcTan[a + b*x]*Log[1 - E^((2*I)*(-ArcTan[a - (b*d)/c] + ArcTan[a + b*x]))] - 2*a*c^2*Log[1/Sqrt[1 + (a + b*x)^2]] + 2*b*c*d*Log[1/Sqrt[1 + (a + b*x)^2]] - a*b*c*d*Pi*Log[1/Sqrt[1 + (a + b*x)^2]] + b^2*d^2*Pi*Log[1/Sqrt[1 + (a + b*x)^2]] + 2*a*b*c*d*ArcTan[a - (b*d)/c]*Log[-Sin[ArcTan[a - (b*d)/c] - ArcTan[a + b*x]]] - 2*b^2*d^2*ArcTan[a - (b*d)/c]*Log[-Sin[ArcTan[a - (b*d)/c] - ArcTan[a + b*x]]] + I*b*d*(a*c - b*d)*PolyLog[2, -E^((2*I)*ArcTan[a + b*x])] + I*b*d*(-(a*c) + b*d)*PolyLog[2, E^((2*I)*(-ArcTan[a - (b*d)/c] + ArcTan[a + b*x]))])/(b*c^2*(-2*a*c + 2*b*d))","B",0
56,1,1536,668,25.8326075,"\int \frac{\tan ^{-1}(a+b x)}{c+\frac{d}{x^2}} \, dx","Integrate[ArcTan[a + b*x]/(c + d/x^2),x]","\frac{(a+b x) \tan ^{-1}(a+b x)+\log \left(\frac{1}{\sqrt{(a+b x)^2+1}}\right)}{b c}-\frac{\sqrt{d} \left(-2 \sqrt{c} \tan ^{-1}\left(\frac{(a-i) \sqrt{c}}{b \sqrt{d}}\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) a^2+2 \sqrt{c} \tan ^{-1}\left(\frac{(a+i) \sqrt{c}}{b \sqrt{d}}\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) a^2+2 i \sqrt{c} \tan ^{-1}\left(\frac{(a-i) \sqrt{c}}{b \sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a-i) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right) a^2+2 i \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a-i) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right) a^2-2 i \sqrt{c} \tan ^{-1}\left(\frac{(a+i) \sqrt{c}}{b \sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a+i) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right) a^2-2 i \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a+i) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right) a^2-2 i \sqrt{c} \tan ^{-1}\left(\frac{(a-i) \sqrt{c}}{b \sqrt{d}}\right) \log \left(-\sin \left(\tan ^{-1}\left(\frac{(a-i) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)\right) a^2+2 i \sqrt{c} \tan ^{-1}\left(\frac{(a+i) \sqrt{c}}{b \sqrt{d}}\right) \log \left(-\sin \left(\tan ^{-1}\left(\frac{(a+i) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)\right) a^2-i b \sqrt{d} \sqrt{\frac{c (a-i)^2+b^2 d}{b^2 d}} e^{-i \tan ^{-1}\left(\frac{(a-i) \sqrt{c}}{b \sqrt{d}}\right)} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)^2 a+i b \sqrt{d} \sqrt{\frac{c (a+i)^2+b^2 d}{b^2 d}} e^{-i \tan ^{-1}\left(\frac{(a+i) \sqrt{c}}{b \sqrt{d}}\right)} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)^2 a+b \sqrt{d} \sqrt{\frac{c (a-i)^2+b^2 d}{b^2 d}} e^{-i \tan ^{-1}\left(\frac{(a-i) \sqrt{c}}{b \sqrt{d}}\right)} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)^2+b \sqrt{d} \sqrt{\frac{c (a+i)^2+b^2 d}{b^2 d}} e^{-i \tan ^{-1}\left(\frac{(a+i) \sqrt{c}}{b \sqrt{d}}\right)} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)^2-2 b \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)^2-2 \sqrt{c} \tan ^{-1}\left(\frac{(a-i) \sqrt{c}}{b \sqrt{d}}\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)+2 \sqrt{c} \tan ^{-1}\left(\frac{(a+i) \sqrt{c}}{b \sqrt{d}}\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)+4 \left(a^2+1\right) \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) \tan ^{-1}(a+b x)+2 i \sqrt{c} \tan ^{-1}\left(\frac{(a-i) \sqrt{c}}{b \sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a-i) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right)+2 i \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a-i) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right)-2 i \sqrt{c} \tan ^{-1}\left(\frac{(a+i) \sqrt{c}}{b \sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a+i) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right)-2 i \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a+i) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right)-2 i \sqrt{c} \tan ^{-1}\left(\frac{(a-i) \sqrt{c}}{b \sqrt{d}}\right) \log \left(-\sin \left(\tan ^{-1}\left(\frac{(a-i) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)\right)+2 i \sqrt{c} \tan ^{-1}\left(\frac{(a+i) \sqrt{c}}{b \sqrt{d}}\right) \log \left(-\sin \left(\tan ^{-1}\left(\frac{(a+i) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)\right)-\left(a^2+1\right) \sqrt{c} \text{Li}_2\left(e^{-2 i \left(\tan ^{-1}\left(\frac{(a-i) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right)+\left(a^2+1\right) \sqrt{c} \text{Li}_2\left(e^{-2 i \left(\tan ^{-1}\left(\frac{(a+i) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right)\right)}{4 \left(a^2+1\right) c^2}","\frac{i \sqrt{d} \text{Li}_2\left(\frac{\sqrt{-c} (-a-b x+i)}{-\sqrt{-c} a+i \sqrt{-c}-b \sqrt{d}}\right)}{4 (-c)^{3/2}}-\frac{i \sqrt{d} \text{Li}_2\left(\frac{\sqrt{-c} (i a+i b x+1)}{(i a+1) \sqrt{-c}-i b \sqrt{d}}\right)}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \text{Li}_2\left(\frac{\sqrt{-c} (a+b x+i)}{\sqrt{-c} a+i \sqrt{-c}-b \sqrt{d}}\right)}{4 (-c)^{3/2}}-\frac{i \sqrt{d} \text{Li}_2\left(\frac{\sqrt{-c} (a+b x+i)}{\sqrt{-c} a+i \sqrt{-c}+b \sqrt{d}}\right)}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \log (i a+i b x+1) \log \left(-\frac{b \left(\sqrt{d}-\sqrt{-c} x\right)}{a \left(-\sqrt{-c}\right)-b \sqrt{d}+i \sqrt{-c}}\right)}{4 (-c)^{3/2}}-\frac{i \sqrt{d} \log (i a+i b x+1) \log \left(\frac{b \left(\sqrt{-c} x+\sqrt{d}\right)}{a \left(-\sqrt{-c}\right)+b \sqrt{d}+i \sqrt{-c}}\right)}{4 (-c)^{3/2}}-\frac{i \sqrt{d} \log (-i a-i b x+1) \log \left(\frac{b \left(\sqrt{d}-\sqrt{-c} x\right)}{a \sqrt{-c}+b \sqrt{d}+i \sqrt{-c}}\right)}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \log (-i a-i b x+1) \log \left(-\frac{b \left(\sqrt{-c} x+\sqrt{d}\right)}{-b \sqrt{d}+(a+i) \sqrt{-c}}\right)}{4 (-c)^{3/2}}-\frac{(i a+i b x+1) \log (i a+i b x+1)}{2 b c}-\frac{(-i a-i b x+1) \log (-i (a+b x+i))}{2 b c}",1,"((a + b*x)*ArcTan[a + b*x] + Log[1/Sqrt[1 + (a + b*x)^2]])/(b*c) - (Sqrt[d]*(-2*Sqrt[c]*ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])]*ArcTan[(Sqrt[c]*x)/Sqrt[d]] - 2*a^2*Sqrt[c]*ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])]*ArcTan[(Sqrt[c]*x)/Sqrt[d]] + 2*Sqrt[c]*ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])]*ArcTan[(Sqrt[c]*x)/Sqrt[d]] + 2*a^2*Sqrt[c]*ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])]*ArcTan[(Sqrt[c]*x)/Sqrt[d]] - 2*b*Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]^2 + (b*Sqrt[d]*Sqrt[((-I + a)^2*c + b^2*d)/(b^2*d)]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]^2)/E^(I*ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])]) - (I*a*b*Sqrt[d]*Sqrt[((-I + a)^2*c + b^2*d)/(b^2*d)]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]^2)/E^(I*ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])]) + (b*Sqrt[d]*Sqrt[((I + a)^2*c + b^2*d)/(b^2*d)]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]^2)/E^(I*ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])]) + (I*a*b*Sqrt[d]*Sqrt[((I + a)^2*c + b^2*d)/(b^2*d)]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]^2)/E^(I*ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])]) + 4*(1 + a^2)*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*ArcTan[a + b*x] + (2*I)*Sqrt[c]*ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])]*Log[1 - E^((-2*I)*(ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] + (2*I)*a^2*Sqrt[c]*ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])]*Log[1 - E^((-2*I)*(ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] + (2*I)*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] + (2*I)*a^2*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - (2*I)*Sqrt[c]*ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])]*Log[1 - E^((-2*I)*(ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - (2*I)*a^2*Sqrt[c]*ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])]*Log[1 - E^((-2*I)*(ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - (2*I)*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - (2*I)*a^2*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - (2*I)*Sqrt[c]*ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])]*Log[-Sin[ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]]] - (2*I)*a^2*Sqrt[c]*ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])]*Log[-Sin[ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]]] + (2*I)*Sqrt[c]*ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])]*Log[-Sin[ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]]] + (2*I)*a^2*Sqrt[c]*ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])]*Log[-Sin[ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]]] - (1 + a^2)*Sqrt[c]*PolyLog[2, E^((-2*I)*(ArcTan[((-I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] + (1 + a^2)*Sqrt[c]*PolyLog[2, E^((-2*I)*(ArcTan[((I + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))]))/(4*(1 + a^2)*c^2)","B",0
57,1,933,933,7.3841229,"\int \frac{\tan ^{-1}(a+b x)}{c+\frac{d}{x^3}} \, dx","Integrate[ArcTan[a + b*x]/(c + d/x^3),x]","\frac{6 \left((a+b x) \tan ^{-1}(a+b x)+\log \left(\frac{1}{\sqrt{(a+b x)^2+1}}\right)\right)-b^3 d \text{RootSum}\left[c \text{$\#$1}^3 a^3+3 c \text{$\#$1}^2 a^3+c a^3+3 c \text{$\#$1} a^3+3 i c \text{$\#$1}^3 a^2+3 i c \text{$\#$1}^2 a^2-3 i c a^2-3 i c \text{$\#$1} a^2-3 c \text{$\#$1}^3 a+3 c \text{$\#$1}^2 a-3 c a+3 c \text{$\#$1} a-i c \text{$\#$1}^3-b^3 d \text{$\#$1}^3+3 i c \text{$\#$1}^2-3 b^3 d \text{$\#$1}^2+i c-b^3 d-3 i c \text{$\#$1}-3 b^3 d \text{$\#$1}\&,\frac{-2 \text{$\#$1} \tan ^{-1}(a+b x)^2+2 e^{\tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right)} \sqrt{\frac{\text{$\#$1}}{(\text{$\#$1}+1)^2}} \tan ^{-1}(a+b x)^2+2 e^{\tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right)} \text{$\#$1}^2 \sqrt{\frac{\text{$\#$1}}{(\text{$\#$1}+1)^2}} \tan ^{-1}(a+b x)^2+4 e^{\tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right)} \text{$\#$1} \sqrt{\frac{\text{$\#$1}}{(\text{$\#$1}+1)^2}} \tan ^{-1}(a+b x)^2-2 \tan ^{-1}(a+b x)^2-2 i \tanh ^{-1}\left(\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right) \text{$\#$1}^2 \tan ^{-1}(a+b x)-2 i \log \left(1-e^{2 i \tan ^{-1}(a+b x)-2 \tanh ^{-1}\left(\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right)}\right) \text{$\#$1}^2 \tan ^{-1}(a+b x)+\pi  \text{$\#$1}^2 \tan ^{-1}(a+b x)+2 i \tanh ^{-1}\left(\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right) \tan ^{-1}(a+b x)+2 i \log \left(1-e^{2 i \tan ^{-1}(a+b x)-2 \tanh ^{-1}\left(\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right)}\right) \tan ^{-1}(a+b x)-\pi  \tan ^{-1}(a+b x)-i \pi  \log \left(1+e^{-2 i \tan ^{-1}(a+b x)}\right) \text{$\#$1}^2+2 \tanh ^{-1}\left(\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right) \log \left(1-e^{2 i \tan ^{-1}(a+b x)-2 \tanh ^{-1}\left(\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right)}\right) \text{$\#$1}^2+i \pi  \log \left(\frac{1}{\sqrt{(a+b x)^2+1}}\right) \text{$\#$1}^2-2 \tanh ^{-1}\left(\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right) \log \left(\sin \left(\tan ^{-1}(a+b x)+i \tanh ^{-1}\left(\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right)\right)\right) \text{$\#$1}^2-\text{Li}_2\left(e^{2 i \tan ^{-1}(a+b x)-2 \tanh ^{-1}\left(\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right)}\right) \text{$\#$1}^2+i \pi  \log \left(1+e^{-2 i \tan ^{-1}(a+b x)}\right)-2 \tanh ^{-1}\left(\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right) \log \left(1-e^{2 i \tan ^{-1}(a+b x)-2 \tanh ^{-1}\left(\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right)}\right)-i \pi  \log \left(\frac{1}{\sqrt{(a+b x)^2+1}}\right)+2 \tanh ^{-1}\left(\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right) \log \left(\sin \left(\tan ^{-1}(a+b x)+i \tanh ^{-1}\left(\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right)\right)\right)+\text{Li}_2\left(e^{2 i \tan ^{-1}(a+b x)-2 \tanh ^{-1}\left(\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right)}\right)}{c \text{$\#$1}^2 a^3+c a^3+2 c \text{$\#$1} a^3+2 i c \text{$\#$1}^2 a^2-2 i c a^2-c \text{$\#$1}^2 a-c a+2 c \text{$\#$1} a-b^3 d \text{$\#$1}^2-b^3 d-2 b^3 d \text{$\#$1}}\&\right]}{6 b c}","-\frac{(i a+i b x+1) \log (i a+i b x+1)}{2 b c}+\frac{i \sqrt[3]{d} \log \left(\frac{b \left(\sqrt[3]{c} x+\sqrt[3]{d}\right)}{\sqrt[3]{c} (i-a)+b \sqrt[3]{d}}\right) \log (i a+i b x+1)}{6 c^{4/3}}-\frac{\sqrt[6]{-1} \sqrt[3]{d} \log \left(-\frac{b \left(\sqrt[3]{d}-\sqrt[3]{-1} \sqrt[3]{c} x\right)}{\sqrt[3]{-1} (i-a) \sqrt[3]{c}-b \sqrt[3]{d}}\right) \log (i a+i b x+1)}{6 c^{4/3}}-\frac{(-1)^{5/6} \sqrt[3]{d} \log \left(\frac{b \left((-1)^{2/3} \sqrt[3]{c} x+\sqrt[3]{d}\right)}{(-1)^{2/3} \sqrt[3]{c} (i-a)+b \sqrt[3]{d}}\right) \log (i a+i b x+1)}{6 c^{4/3}}-\frac{(-i a-i b x+1) \log (-i (a+b x+i))}{2 b c}-\frac{i \sqrt[3]{d} \log (-i a-i b x+1) \log \left(-\frac{b \left(\sqrt[3]{c} x+\sqrt[3]{d}\right)}{(a+i) \sqrt[3]{c}-b \sqrt[3]{d}}\right)}{6 c^{4/3}}+\frac{\sqrt[6]{-1} \sqrt[3]{d} \log (-i a-i b x+1) \log \left(\frac{b \left(\sqrt[3]{d}-\sqrt[3]{-1} \sqrt[3]{c} x\right)}{\sqrt[3]{-1} \sqrt[3]{c} (a+i)+b \sqrt[3]{d}}\right)}{6 c^{4/3}}+\frac{(-1)^{5/6} \sqrt[3]{d} \log (-i a-i b x+1) \log \left(\frac{b \left((-1)^{2/3} \sqrt[3]{c} x+\sqrt[3]{d}\right)}{\sqrt[6]{-1} \sqrt[3]{c} (1-i a)+b \sqrt[3]{d}}\right)}{6 c^{4/3}}-\frac{\sqrt[6]{-1} \sqrt[3]{d} \text{Li}_2\left(\frac{\sqrt[3]{-1} \sqrt[3]{c} (-a-b x+i)}{\sqrt[3]{-1} (i-a) \sqrt[3]{c}-b \sqrt[3]{d}}\right)}{6 c^{4/3}}-\frac{(-1)^{5/6} \sqrt[3]{d} \text{Li}_2\left(\frac{\sqrt[6]{-1} \sqrt[3]{c} (-a-b x+i)}{\sqrt[6]{-1} (i-a) \sqrt[3]{c}-i b \sqrt[3]{d}}\right)}{6 c^{4/3}}+\frac{i \sqrt[3]{d} \text{Li}_2\left(\frac{\sqrt[3]{c} (-a-b x+i)}{\sqrt[3]{c} (i-a)+b \sqrt[3]{d}}\right)}{6 c^{4/3}}-\frac{i \sqrt[3]{d} \text{Li}_2\left(\frac{\sqrt[3]{c} (a+b x+i)}{(a+i) \sqrt[3]{c}-b \sqrt[3]{d}}\right)}{6 c^{4/3}}+\frac{(-1)^{5/6} \sqrt[3]{d} \text{Li}_2\left(\frac{(-1)^{2/3} \sqrt[3]{c} (a+b x+i)}{(-1)^{2/3} (a+i) \sqrt[3]{c}-b \sqrt[3]{d}}\right)}{6 c^{4/3}}+\frac{\sqrt[6]{-1} \sqrt[3]{d} \text{Li}_2\left(\frac{\sqrt[3]{-1} \sqrt[3]{c} (a+b x+i)}{\sqrt[3]{-1} \sqrt[3]{c} (a+i)+b \sqrt[3]{d}}\right)}{6 c^{4/3}}",1,"(6*((a + b*x)*ArcTan[a + b*x] + Log[1/Sqrt[1 + (a + b*x)^2]]) - b^3*d*RootSum[I*c - 3*a*c - (3*I)*a^2*c + a^3*c - b^3*d - (3*I)*c*#1 + 3*a*c*#1 - (3*I)*a^2*c*#1 + 3*a^3*c*#1 - 3*b^3*d*#1 + (3*I)*c*#1^2 + 3*a*c*#1^2 + (3*I)*a^2*c*#1^2 + 3*a^3*c*#1^2 - 3*b^3*d*#1^2 - I*c*#1^3 - 3*a*c*#1^3 + (3*I)*a^2*c*#1^3 + a^3*c*#1^3 - b^3*d*#1^3 & , (-(Pi*ArcTan[a + b*x]) - 2*ArcTan[a + b*x]^2 + (2*I)*ArcTan[a + b*x]*ArcTanh[(-1 + #1)/(1 + #1)] + I*Pi*Log[1 + E^((-2*I)*ArcTan[a + b*x])] + (2*I)*ArcTan[a + b*x]*Log[1 - E^((2*I)*ArcTan[a + b*x] - 2*ArcTanh[(-1 + #1)/(1 + #1)])] - 2*ArcTanh[(-1 + #1)/(1 + #1)]*Log[1 - E^((2*I)*ArcTan[a + b*x] - 2*ArcTanh[(-1 + #1)/(1 + #1)])] - I*Pi*Log[1/Sqrt[1 + (a + b*x)^2]] + 2*ArcTanh[(-1 + #1)/(1 + #1)]*Log[Sin[ArcTan[a + b*x] + I*ArcTanh[(-1 + #1)/(1 + #1)]]] + PolyLog[2, E^((2*I)*ArcTan[a + b*x] - 2*ArcTanh[(-1 + #1)/(1 + #1)])] - 2*ArcTan[a + b*x]^2*#1 + Pi*ArcTan[a + b*x]*#1^2 - (2*I)*ArcTan[a + b*x]*ArcTanh[(-1 + #1)/(1 + #1)]*#1^2 - I*Pi*Log[1 + E^((-2*I)*ArcTan[a + b*x])]*#1^2 - (2*I)*ArcTan[a + b*x]*Log[1 - E^((2*I)*ArcTan[a + b*x] - 2*ArcTanh[(-1 + #1)/(1 + #1)])]*#1^2 + 2*ArcTanh[(-1 + #1)/(1 + #1)]*Log[1 - E^((2*I)*ArcTan[a + b*x] - 2*ArcTanh[(-1 + #1)/(1 + #1)])]*#1^2 + I*Pi*Log[1/Sqrt[1 + (a + b*x)^2]]*#1^2 - 2*ArcTanh[(-1 + #1)/(1 + #1)]*Log[Sin[ArcTan[a + b*x] + I*ArcTanh[(-1 + #1)/(1 + #1)]]]*#1^2 - PolyLog[2, E^((2*I)*ArcTan[a + b*x] - 2*ArcTanh[(-1 + #1)/(1 + #1)])]*#1^2 + 2*E^ArcTanh[(1 - #1)/(1 + #1)]*ArcTan[a + b*x]^2*Sqrt[#1/(1 + #1)^2] + 4*E^ArcTanh[(1 - #1)/(1 + #1)]*ArcTan[a + b*x]^2*#1*Sqrt[#1/(1 + #1)^2] + 2*E^ArcTanh[(1 - #1)/(1 + #1)]*ArcTan[a + b*x]^2*#1^2*Sqrt[#1/(1 + #1)^2])/(-(a*c) - (2*I)*a^2*c + a^3*c - b^3*d + 2*a*c*#1 + 2*a^3*c*#1 - 2*b^3*d*#1 - a*c*#1^2 + (2*I)*a^2*c*#1^2 + a^3*c*#1^2 - b^3*d*#1^2) & ])/(6*b*c)","C",0
58,1,604,673,0.5958872,"\int \frac{\tan ^{-1}(a+b x)}{c+d \sqrt{x}} \, dx","Integrate[ArcTan[a + b*x]/(c + d*Sqrt[x]),x]","\frac{i \left(c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{-a-i} d}\right)+c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{-a-i} d}\right)-c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{i-a} d}\right)-c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{i-a} d}\right)+c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{\sqrt{b} c+\sqrt{-a-i} d}\right)-c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{\sqrt{b} c+\sqrt{-a+i} d}\right)+c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{-\sqrt{b} c+\sqrt{-a-i} d}\right)-c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{-\sqrt{b} c+\sqrt{-a+i} d}\right)+c \log (i a+i b x+1) \log \left(c+d \sqrt{x}\right)-c \log (-i (a+b x+i)) \log \left(c+d \sqrt{x}\right)-d \sqrt{x} \log (i a+i b x+1)+d \sqrt{x} \log (-i (a+b x+i))+\frac{2 \sqrt{a+i} d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right)}{\sqrt{b}}-\frac{2 \sqrt{-a+i} d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right)}{\sqrt{b}}\right)}{d^2}","\frac{i c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{-a-i} d}\right)}{d^2}+\frac{i c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{-a-i} d}\right)}{d^2}-\frac{i c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{i-a} d}\right)}{d^2}-\frac{i c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{i-a} d}\right)}{d^2}+\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{\sqrt{b} c+\sqrt{-a-i} d}\right)}{d^2}-\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{\sqrt{b} c+\sqrt{-a+i} d}\right)}{d^2}+\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(-\frac{d \left(\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{\sqrt{b} c-\sqrt{-a-i} d}\right)}{d^2}-\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(-\frac{d \left(\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{\sqrt{b} c-\sqrt{-a+i} d}\right)}{d^2}-\frac{i c \log (-i a-i b x+1) \log \left(c+d \sqrt{x}\right)}{d^2}+\frac{i c \log (i a+i b x+1) \log \left(c+d \sqrt{x}\right)}{d^2}+\frac{i \sqrt{x} \log (-i a-i b x+1)}{d}-\frac{i \sqrt{x} \log (i a+i b x+1)}{d}+\frac{2 i \sqrt{a+i} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right)}{\sqrt{b} d}-\frac{2 i \sqrt{-a+i} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right)}{\sqrt{b} d}",1,"(I*((2*Sqrt[I + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]])/Sqrt[b] - (2*Sqrt[I - a]*d*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[I - a]])/Sqrt[b] + c*Log[(d*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-I - a]*d)]*Log[c + d*Sqrt[x]] - c*Log[(d*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[I - a]*d)]*Log[c + d*Sqrt[x]] + c*Log[(d*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(-(Sqrt[b]*c) + Sqrt[-I - a]*d)]*Log[c + d*Sqrt[x]] - c*Log[(d*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(-(Sqrt[b]*c) + Sqrt[I - a]*d)]*Log[c + d*Sqrt[x]] - d*Sqrt[x]*Log[1 + I*a + I*b*x] + c*Log[c + d*Sqrt[x]]*Log[1 + I*a + I*b*x] + d*Sqrt[x]*Log[(-I)*(I + a + b*x)] - c*Log[c + d*Sqrt[x]]*Log[(-I)*(I + a + b*x)] + c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-I - a]*d)] + c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-I - a]*d)] - c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[I - a]*d)] - c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[I - a]*d)]))/d^2","A",1
59,1,666,770,0.8269687,"\int \frac{\tan ^{-1}(a+b x)}{c+\frac{d}{\sqrt{x}}} \, dx","Integrate[ArcTan[a + b*x]/(c + d/Sqrt[x]),x]","\frac{i \left(-\frac{c^2 (a+b x-i) \log (i a+i b x+1)}{b}+\frac{c^2 (a+b x+i) \log (-i (a+b x+i))}{b}-2 d^2 \left(\text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{b} d-\sqrt{-a-i} c}\right)+\text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{-a-i} c+\sqrt{b} d}\right)+\log \left(c \sqrt{x}+d\right) \left(\log \left(\frac{c \left(-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{\sqrt{b} d+\sqrt{-a-i} c}\right)+\log \left(\frac{c \left(\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{-\sqrt{b} d+\sqrt{-a-i} c}\right)\right)\right)+2 d^2 \left(\text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{b} d-\sqrt{i-a} c}\right)+\text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{i-a} c+\sqrt{b} d}\right)+\log \left(c \sqrt{x}+d\right) \left(\log \left(\frac{c \left(-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{\sqrt{b} d+\sqrt{-a+i} c}\right)+\log \left(\frac{c \left(\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{-\sqrt{b} d+\sqrt{-a+i} c}\right)\right)\right)-2 d^2 \log (i a+i b x+1) \log \left(c \sqrt{x}+d\right)+2 d^2 \log (-i (a+b x+i)) \log \left(c \sqrt{x}+d\right)+2 c d \sqrt{x} \log (i a+i b x+1)-2 c d \sqrt{x} \log (-i (a+b x+i))+4 c d \left(\sqrt{x}-\frac{\sqrt{a+i} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right)}{\sqrt{b}}\right)-4 c d \left(\sqrt{x}-\frac{\sqrt{-a+i} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right)}{\sqrt{b}}\right)\right)}{2 c^3}","-\frac{i d^2 \text{Li}_2\left(-\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{-a-i} c-\sqrt{b} d}\right)}{c^3}+\frac{i d^2 \text{Li}_2\left(-\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{i-a} c-\sqrt{b} d}\right)}{c^3}-\frac{i d^2 \text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{-a-i} c+\sqrt{b} d}\right)}{c^3}+\frac{i d^2 \text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{i-a} c+\sqrt{b} d}\right)}{c^3}-\frac{i d^2 \log \left(c \sqrt{x}+d\right) \log \left(\frac{c \left(-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{\sqrt{b} d+\sqrt{-a-i} c}\right)}{c^3}+\frac{i d^2 \log \left(c \sqrt{x}+d\right) \log \left(\frac{c \left(-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{\sqrt{b} d+\sqrt{-a+i} c}\right)}{c^3}-\frac{i d^2 \log \left(c \sqrt{x}+d\right) \log \left(\frac{c \left(\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{-\sqrt{b} d+\sqrt{-a-i} c}\right)}{c^3}+\frac{i d^2 \log \left(c \sqrt{x}+d\right) \log \left(\frac{c \left(\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{-\sqrt{b} d+\sqrt{-a+i} c}\right)}{c^3}+\frac{i d^2 \log (-i a-i b x+1) \log \left(c \sqrt{x}+d\right)}{c^3}-\frac{i d^2 \log (i a+i b x+1) \log \left(c \sqrt{x}+d\right)}{c^3}-\frac{i d \sqrt{x} \log (-i a-i b x+1)}{c^2}+\frac{i d \sqrt{x} \log (i a+i b x+1)}{c^2}-\frac{2 i \sqrt{a+i} d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right)}{\sqrt{b} c^2}+\frac{2 i \sqrt{-a+i} d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right)}{\sqrt{b} c^2}-\frac{(i a+i b x+1) \log (i a+i b x+1)}{2 b c}-\frac{(-i a-i b x+1) \log (-i (a+b x+i))}{2 b c}",1,"((I/2)*(4*c*d*(Sqrt[x] - (Sqrt[I + a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]])/Sqrt[b]) - 4*c*d*(Sqrt[x] - (Sqrt[I - a]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[I - a]])/Sqrt[b]) + 2*c*d*Sqrt[x]*Log[1 + I*a + I*b*x] - (c^2*(-I + a + b*x)*Log[1 + I*a + I*b*x])/b - 2*d^2*Log[d + c*Sqrt[x]]*Log[1 + I*a + I*b*x] - 2*c*d*Sqrt[x]*Log[(-I)*(I + a + b*x)] + (c^2*(I + a + b*x)*Log[(-I)*(I + a + b*x)])/b + 2*d^2*Log[d + c*Sqrt[x]]*Log[(-I)*(I + a + b*x)] - 2*d^2*((Log[(c*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d)] + Log[(c*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d)])*Log[d + c*Sqrt[x]] + PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(-(Sqrt[-I - a]*c) + Sqrt[b]*d)] + PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d)]) + 2*d^2*((Log[(c*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)] + Log[(c*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c - Sqrt[b]*d)])*Log[d + c*Sqrt[x]] + PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(-(Sqrt[I - a]*c) + Sqrt[b]*d)] + PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)])))/c^3","A",1
60,1,283,274,0.0597144,"\int \frac{\tan ^{-1}(a+b x)}{1+x^2} \, dx","Integrate[ArcTan[a + b*x]/(1 + x^2),x]","-\frac{1}{4} \text{Li}_2\left(\frac{-i a-i b x+1}{-i a-b+1}\right)+\frac{1}{4} \text{Li}_2\left(\frac{-i a-i b x+1}{-i a+b+1}\right)-\frac{1}{4} \text{Li}_2\left(\frac{i a+i b x+1}{i a-b+1}\right)+\frac{1}{4} \text{Li}_2\left(\frac{i a+i b x+1}{i a+b+1}\right)+\frac{1}{4} \log \left(\frac{b (-x+i)}{a+i (b+1)}\right) \log (-i a-i b x+1)-\frac{1}{4} \log \left(-\frac{b (x+i)}{a+i (1-b)}\right) \log (-i a-i b x+1)-\frac{1}{4} \log \left(\frac{b (-x+i)}{a-i (1-b)}\right) \log (i a+i b x+1)+\frac{1}{4} \log \left(-\frac{b (x+i)}{a-i (b+1)}\right) \log (i a+i b x+1)","-\frac{1}{4} \text{Li}_2\left(-\frac{-a-b x+i}{a-i (1-b)}\right)+\frac{1}{4} \text{Li}_2\left(-\frac{-a-b x+i}{a-i (b+1)}\right)-\frac{1}{4} \text{Li}_2\left(\frac{a+b x+i}{a-i b+i}\right)+\frac{1}{4} \text{Li}_2\left(\frac{a+b x+i}{a+i (b+1)}\right)+\frac{1}{4} \log \left(\frac{b (-x+i)}{a+i (b+1)}\right) \log (-i a-i b x+1)-\frac{1}{4} \log \left(-\frac{b (x+i)}{a+i (1-b)}\right) \log (-i a-i b x+1)-\frac{1}{4} \log \left(\frac{b (-x+i)}{a-i (1-b)}\right) \log (i a+i b x+1)+\frac{1}{4} \log \left(-\frac{b (x+i)}{a-i (b+1)}\right) \log (i a+i b x+1)",1,"(Log[(b*(I - x))/(a + I*(1 + b))]*Log[1 - I*a - I*b*x])/4 - (Log[-((b*(I + x))/(a + I*(1 - b)))]*Log[1 - I*a - I*b*x])/4 - (Log[(b*(I - x))/(a - I*(1 - b))]*Log[1 + I*a + I*b*x])/4 + (Log[-((b*(I + x))/(a - I*(1 + b)))]*Log[1 + I*a + I*b*x])/4 - PolyLog[2, (1 - I*a - I*b*x)/(1 - I*a - b)]/4 + PolyLog[2, (1 - I*a - I*b*x)/(1 - I*a + b)]/4 - PolyLog[2, (1 + I*a + I*b*x)/(1 + I*a - b)]/4 + PolyLog[2, (1 + I*a + I*b*x)/(1 + I*a + b)]/4","A",1
61,1,409,543,0.3998353,"\int \frac{\tan ^{-1}(d+e x)}{a+b x^2} \, dx","Integrate[ArcTan[d + e*x]/(a + b*x^2),x]","\frac{i \left(\text{Li}_2\left(\frac{\sqrt{b} (d+e x-i)}{\sqrt{b} (d-i)-\sqrt{-a} e}\right)-\text{Li}_2\left(\frac{\sqrt{b} (d+e x-i)}{\sqrt{b} (d-i)+\sqrt{-a} e}\right)-\text{Li}_2\left(\frac{\sqrt{b} (d+e x+i)}{\sqrt{b} (d+i)-\sqrt{-a} e}\right)+\text{Li}_2\left(\frac{\sqrt{b} (d+e x+i)}{\sqrt{b} (d+i)+\sqrt{-a} e}\right)+\log (i d+i e x+1) \left(-\log \left(\frac{e \left(\sqrt{-a}-\sqrt{b} x\right)}{\sqrt{-a} e+\sqrt{b} (d-i)}\right)\right)+\log (i d+i e x+1) \log \left(\frac{e \left(\sqrt{-a}+\sqrt{b} x\right)}{\sqrt{-a} e-\sqrt{b} (d-i)}\right)+\log (-i (d+e x+i)) \log \left(\frac{e \left(\sqrt{-a}-\sqrt{b} x\right)}{\sqrt{-a} e+\sqrt{b} (d+i)}\right)-\log (-i (d+e x+i)) \log \left(\frac{e \left(\sqrt{-a}+\sqrt{b} x\right)}{\sqrt{-a} e-\sqrt{b} (d+i)}\right)\right)}{4 \sqrt{-a} \sqrt{b}}","-\frac{i \text{Li}_2\left(\frac{\sqrt{b} (-d-e x+i)}{\sqrt{b} (i-d)-\sqrt{-a} e}\right)}{4 \sqrt{-a} \sqrt{b}}+\frac{i \text{Li}_2\left(\frac{\sqrt{b} (-d-e x+i)}{\sqrt{b} (i-d)+\sqrt{-a} e}\right)}{4 \sqrt{-a} \sqrt{b}}-\frac{i \text{Li}_2\left(\frac{\sqrt{b} (d+e x+i)}{\sqrt{b} (d+i)-\sqrt{-a} e}\right)}{4 \sqrt{-a} \sqrt{b}}+\frac{i \text{Li}_2\left(\frac{\sqrt{b} (d+e x+i)}{\sqrt{b} (d+i)+\sqrt{-a} e}\right)}{4 \sqrt{-a} \sqrt{b}}+\frac{i \log (-i d-i e x+1) \log \left(\frac{e \left(\sqrt{-a}-\sqrt{b} x\right)}{\sqrt{-a} e+\sqrt{b} (d+i)}\right)}{4 \sqrt{-a} \sqrt{b}}-\frac{i \log (-i d-i e x+1) \log \left(-\frac{e \left(\sqrt{-a}+\sqrt{b} x\right)}{-\sqrt{-a} e+\sqrt{b} (d+i)}\right)}{4 \sqrt{-a} \sqrt{b}}-\frac{i \log (i d+i e x+1) \log \left(-\frac{e \left(\sqrt{-a}-\sqrt{b} x\right)}{-\sqrt{-a} e+\sqrt{b} (-d+i)}\right)}{4 \sqrt{-a} \sqrt{b}}+\frac{i \log (i d+i e x+1) \log \left(\frac{e \left(\sqrt{-a}+\sqrt{b} x\right)}{\sqrt{-a} e+\sqrt{b} (-d+i)}\right)}{4 \sqrt{-a} \sqrt{b}}",1,"((I/4)*(-(Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*(-I + d) + Sqrt[-a]*e)]*Log[1 + I*d + I*e*x]) + Log[(e*(Sqrt[-a] + Sqrt[b]*x))/(-(Sqrt[b]*(-I + d)) + Sqrt[-a]*e)]*Log[1 + I*d + I*e*x] + Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*(I + d) + Sqrt[-a]*e)]*Log[(-I)*(I + d + e*x)] - Log[(e*(Sqrt[-a] + Sqrt[b]*x))/(-(Sqrt[b]*(I + d)) + Sqrt[-a]*e)]*Log[(-I)*(I + d + e*x)] + PolyLog[2, (Sqrt[b]*(-I + d + e*x))/(Sqrt[b]*(-I + d) - Sqrt[-a]*e)] - PolyLog[2, (Sqrt[b]*(-I + d + e*x))/(Sqrt[b]*(-I + d) + Sqrt[-a]*e)] - PolyLog[2, (Sqrt[b]*(I + d + e*x))/(Sqrt[b]*(I + d) - Sqrt[-a]*e)] + PolyLog[2, (Sqrt[b]*(I + d + e*x))/(Sqrt[b]*(I + d) + Sqrt[-a]*e)]))/(Sqrt[-a]*Sqrt[b])","A",1
62,1,443,367,0.4627328,"\int \frac{\tan ^{-1}(d+e x)}{a+b x+c x^2} \, dx","Integrate[ArcTan[d + e*x]/(a + b*x + c*x^2),x]","\frac{i \left(-\text{Li}_2\left(\frac{2 c (d+e x-i)}{2 c (d-i)+\left(\sqrt{b^2-4 a c}-b\right) e}\right)+\text{Li}_2\left(\frac{2 c (d+e x-i)}{2 c (d-i)-\left(b+\sqrt{b^2-4 a c}\right) e}\right)+\text{Li}_2\left(\frac{2 c (d+e x+i)}{2 c (d+i)+\left(\sqrt{b^2-4 a c}-b\right) e}\right)-\text{Li}_2\left(\frac{2 c (d+e x+i)}{2 c (d+i)-\left(b+\sqrt{b^2-4 a c}\right) e}\right)+\log (1-i (d+e x)) \log \left(\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c (d+i)}\right)-\log (1-i (d+e x)) \log \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c (d+i)}\right)-\log (1+i (d+e x)) \log \left(\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c (d-i)}\right)+\log (1+i (d+e x)) \log \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c (d-i)}\right)\right)}{2 \sqrt{b^2-4 a c}}","-\frac{i \text{Li}_2\left(\frac{2 \left(2 c d-\left(b-\sqrt{b^2-4 a c}\right) e-2 c (d+e x)\right)}{\left(-2 d c+2 i c+b e-\sqrt{b^2-4 a c} e\right) (1-i (d+e x))}+1\right)}{2 \sqrt{b^2-4 a c}}+\frac{i \text{Li}_2\left(\frac{2 \left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e-2 c (d+e x)\right)}{\left(2 c (i-d)+\left(b+\sqrt{b^2-4 a c}\right) e\right) (1-i (d+e x))}+1\right)}{2 \sqrt{b^2-4 a c}}+\frac{\tan ^{-1}(d+e x) \log \left(\frac{2 e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{(1-i (d+e x)) \left(e \left(b-\sqrt{b^2-4 a c}\right)+2 c (-d+i)\right)}\right)}{\sqrt{b^2-4 a c}}-\frac{\tan ^{-1}(d+e x) \log \left(\frac{2 e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{(1-i (d+e x)) \left(e \left(\sqrt{b^2-4 a c}+b\right)+2 c (-d+i)\right)}\right)}{\sqrt{b^2-4 a c}}",1,"((I/2)*(Log[(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*(I + d) + (-b + Sqrt[b^2 - 4*a*c])*e)]*Log[1 - I*(d + e*x)] - Log[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*(I + d) + (b + Sqrt[b^2 - 4*a*c])*e)]*Log[1 - I*(d + e*x)] - Log[(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*(-I + d) + (-b + Sqrt[b^2 - 4*a*c])*e)]*Log[1 + I*(d + e*x)] + Log[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*(-I + d) + (b + Sqrt[b^2 - 4*a*c])*e)]*Log[1 + I*(d + e*x)] - PolyLog[2, (2*c*(-I + d + e*x))/(2*c*(-I + d) + (-b + Sqrt[b^2 - 4*a*c])*e)] + PolyLog[2, (2*c*(-I + d + e*x))/(2*c*(-I + d) - (b + Sqrt[b^2 - 4*a*c])*e)] + PolyLog[2, (2*c*(I + d + e*x))/(2*c*(I + d) + (-b + Sqrt[b^2 - 4*a*c])*e)] - PolyLog[2, (2*c*(I + d + e*x))/(2*c*(I + d) - (b + Sqrt[b^2 - 4*a*c])*e)]))/Sqrt[b^2 - 4*a*c]","A",0
63,1,97,132,0.1169557,"\int \frac{\tan ^{-1}(a+b x)}{\sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx","Integrate[ArcTan[a + b*x]/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2],x]","\frac{i \text{Li}_2\left(-i e^{i \tan ^{-1}(a+b x)}\right)-i \text{Li}_2\left(i e^{i \tan ^{-1}(a+b x)}\right)+\tan ^{-1}(a+b x) \left(\log \left(1-i e^{i \tan ^{-1}(a+b x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a+b x)}\right)\right)}{b}","\frac{i \text{Li}_2\left(-\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{b}-\frac{i \text{Li}_2\left(\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{b}-\frac{2 i \tan ^{-1}(a+b x) \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{b}",1,"(ArcTan[a + b*x]*(Log[1 - I*E^(I*ArcTan[a + b*x])] - Log[1 + I*E^(I*ArcTan[a + b*x])]) + I*PolyLog[2, (-I)*E^(I*ArcTan[a + b*x])] - I*PolyLog[2, I*E^(I*ArcTan[a + b*x])])/b","A",0
64,1,125,216,0.0692725,"\int \frac{\tan ^{-1}(a+b x)}{\sqrt{\left(1+a^2\right) c+2 a b c x+b^2 c x^2}} \, dx","Integrate[ArcTan[a + b*x]/Sqrt[(1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2],x]","\frac{\sqrt{(a+b x)^2+1} \left(i \text{Li}_2\left(-i e^{i \tan ^{-1}(a+b x)}\right)-i \text{Li}_2\left(i e^{i \tan ^{-1}(a+b x)}\right)+\tan ^{-1}(a+b x) \left(\log \left(1-i e^{i \tan ^{-1}(a+b x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a+b x)}\right)\right)\right)}{b \sqrt{c \left((a+b x)^2+1\right)}}","\frac{i \sqrt{(a+b x)^2+1} \text{Li}_2\left(-\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{b \sqrt{c (a+b x)^2+c}}-\frac{i \sqrt{(a+b x)^2+1} \text{Li}_2\left(\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{b \sqrt{c (a+b x)^2+c}}-\frac{2 i \sqrt{(a+b x)^2+1} \tan ^{-1}(a+b x) \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right)}{b \sqrt{c (a+b x)^2+c}}",1,"(Sqrt[1 + (a + b*x)^2]*(ArcTan[a + b*x]*(Log[1 - I*E^(I*ArcTan[a + b*x])] - Log[1 + I*E^(I*ArcTan[a + b*x])]) + I*PolyLog[2, (-I)*E^(I*ArcTan[a + b*x])] - I*PolyLog[2, I*E^(I*ArcTan[a + b*x])]))/(b*Sqrt[c*(1 + (a + b*x)^2)])","A",0
65,1,163,23,0.4217828,"\int \frac{\tan ^{-1}(a+b x)}{\sqrt[3]{1+a^2+2 a b x+b^2 x^2}} \, dx","Integrate[ArcTan[a + b*x]/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3),x]","\frac{\frac{5 \sqrt[3]{2} \sqrt{\pi } \Gamma \left(\frac{5}{3}\right) \, _3F_2\left(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{(a+b x)^2+1}\right)}{(a+b x)^2+1}+6 \Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right) \left(\frac{4 (a+b x) \, _2F_1\left(1,\frac{4}{3};\frac{11}{6};\frac{1}{(a+b x)^2+1}\right) \tan ^{-1}(a+b x)}{(a+b x)^2+1}+10 (a+b x) \tan ^{-1}(a+b x)+15\right)}{20 b \Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right) \sqrt[3]{a^2+2 a b x+b^2 x^2+1}}","\text{Int}\left(\frac{\tan ^{-1}(a+b x)}{\sqrt[3]{(a+b x)^2+1}},x\right)",0,"(6*Gamma[11/6]*Gamma[7/3]*(15 + 10*(a + b*x)*ArcTan[a + b*x] + (4*(a + b*x)*ArcTan[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2)) + (5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2))/(20*b*(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3)*Gamma[11/6]*Gamma[7/3])","B",0
66,1,165,25,0.0997431,"\int \frac{\tan ^{-1}(a+b x)}{\sqrt[3]{\left(1+a^2\right) c+2 a b c x+b^2 c x^2}} \, dx","Integrate[ArcTan[a + b*x]/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]","\frac{\frac{5 \sqrt[3]{2} \sqrt{\pi } \Gamma \left(\frac{5}{3}\right) \, _3F_2\left(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{(a+b x)^2+1}\right)}{(a+b x)^2+1}+6 \Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right) \left(\frac{4 (a+b x) \, _2F_1\left(1,\frac{4}{3};\frac{11}{6};\frac{1}{(a+b x)^2+1}\right) \tan ^{-1}(a+b x)}{(a+b x)^2+1}+10 (a+b x) \tan ^{-1}(a+b x)+15\right)}{20 b \Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right) \sqrt[3]{c \left(a^2+2 a b x+b^2 x^2+1\right)}}","\text{Int}\left(\frac{\tan ^{-1}(a+b x)}{\sqrt[3]{c (a+b x)^2+c}},x\right)",0,"(6*Gamma[11/6]*Gamma[7/3]*(15 + 10*(a + b*x)*ArcTan[a + b*x] + (4*(a + b*x)*ArcTan[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2)) + (5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2))/(20*b*(c*(1 + a^2 + 2*a*b*x + b^2*x^2))^(1/3)*Gamma[11/6]*Gamma[7/3])","B",0
67,1,145,187,0.7236747,"\int \frac{(a+b x)^2 \tan ^{-1}(a+b x)}{\sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx","Integrate[((a + b*x)^2*ArcTan[a + b*x])/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2],x]","\frac{-i \text{Li}_2\left(-i e^{i \tan ^{-1}(a+b x)}\right)+i \text{Li}_2\left(i e^{i \tan ^{-1}(a+b x)}\right)-\sqrt{(a+b x)^2+1}+(a+b x) \sqrt{(a+b x)^2+1} \tan ^{-1}(a+b x)+\tan ^{-1}(a+b x) \left(-\log \left(1-i e^{i \tan ^{-1}(a+b x)}\right)\right)+\tan ^{-1}(a+b x) \log \left(1+i e^{i \tan ^{-1}(a+b x)}\right)}{2 b}","-\frac{i \text{Li}_2\left(-\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{2 b}+\frac{i \text{Li}_2\left(\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{2 b}-\frac{\sqrt{(a+b x)^2+1}}{2 b}+\frac{i \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right) \tan ^{-1}(a+b x)}{b}+\frac{(a+b x) \sqrt{(a+b x)^2+1} \tan ^{-1}(a+b x)}{2 b}",1,"(-Sqrt[1 + (a + b*x)^2] + (a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcTan[a + b*x] - ArcTan[a + b*x]*Log[1 - I*E^(I*ArcTan[a + b*x])] + ArcTan[a + b*x]*Log[1 + I*E^(I*ArcTan[a + b*x])] - I*PolyLog[2, (-I)*E^(I*ArcTan[a + b*x])] + I*PolyLog[2, I*E^(I*ArcTan[a + b*x])])/(2*b)","A",0
68,1,189,281,0.1615348,"\int \frac{(a+b x)^2 \tan ^{-1}(a+b x)}{\sqrt{\left(1+a^2\right) c+2 a b c x+b^2 c x^2}} \, dx","Integrate[((a + b*x)^2*ArcTan[a + b*x])/Sqrt[(1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2],x]","\frac{\sqrt{a^2+2 a b x+b^2 x^2+1} \left(-i \text{Li}_2\left(-i e^{i \tan ^{-1}(a+b x)}\right)+i \text{Li}_2\left(i e^{i \tan ^{-1}(a+b x)}\right)-\sqrt{(a+b x)^2+1}+(a+b x) \sqrt{(a+b x)^2+1} \tan ^{-1}(a+b x)+\tan ^{-1}(a+b x) \left(-\log \left(1-i e^{i \tan ^{-1}(a+b x)}\right)\right)+\tan ^{-1}(a+b x) \log \left(1+i e^{i \tan ^{-1}(a+b x)}\right)\right)}{2 b \sqrt{c \left(a^2+2 a b x+b^2 x^2+1\right)}}","-\frac{i \sqrt{(a+b x)^2+1} \text{Li}_2\left(-\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{2 b \sqrt{c (a+b x)^2+c}}+\frac{i \sqrt{(a+b x)^2+1} \text{Li}_2\left(\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{2 b \sqrt{c (a+b x)^2+c}}-\frac{\sqrt{c (a+b x)^2+c}}{2 b c}+\frac{i \sqrt{(a+b x)^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right) \tan ^{-1}(a+b x)}{b \sqrt{c (a+b x)^2+c}}+\frac{(a+b x) \sqrt{c (a+b x)^2+c} \tan ^{-1}(a+b x)}{2 b c}",1,"(Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(-Sqrt[1 + (a + b*x)^2] + (a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcTan[a + b*x] - ArcTan[a + b*x]*Log[1 - I*E^(I*ArcTan[a + b*x])] + ArcTan[a + b*x]*Log[1 + I*E^(I*ArcTan[a + b*x])] - I*PolyLog[2, (-I)*E^(I*ArcTan[a + b*x])] + I*PolyLog[2, I*E^(I*ArcTan[a + b*x])]))/(2*b*Sqrt[c*(1 + a^2 + 2*a*b*x + b^2*x^2)])","A",0
69,1,181,30,1.5766794,"\int \frac{(a+b x)^2 \tan ^{-1}(a+b x)}{\sqrt[3]{1+a^2+2 a b x+b^2 x^2}} \, dx","Integrate[((a + b*x)^2*ArcTan[a + b*x])/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3),x]","-\frac{3 \left((a+b x)^2+1\right)^{2/3} \left(\frac{5 \sqrt[3]{2} \sqrt{\pi } \Gamma \left(\frac{5}{3}\right) \, _3F_2\left(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{(a+b x)^2+1}\right)}{\left((a+b x)^2+1\right)^2}+\Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right) \left(\frac{24 (a+b x) \, _2F_1\left(1,\frac{4}{3};\frac{11}{6};\frac{1}{(a+b x)^2+1}\right) \tan ^{-1}(a+b x)}{\left((a+b x)^2+1\right)^2}+\frac{90}{(a+b x)^2+1}+5 \tan ^{-1}(a+b x) \left(6 \sin \left(2 \tan ^{-1}(a+b x)\right)-4 (a+b x)\right)+15\right)\right)}{140 b \Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right)}","\text{Int}\left(\frac{(a+b x)^2 \tan ^{-1}(a+b x)}{\sqrt[3]{(a+b x)^2+1}},x\right)",0,"(-3*(1 + (a + b*x)^2)^(2/3)*((5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2)^2 + Gamma[11/6]*Gamma[7/3]*(15 + 90/(1 + (a + b*x)^2) + (24*(a + b*x)*ArcTan[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2)^2 + 5*ArcTan[a + b*x]*(-4*(a + b*x) + 6*Sin[2*ArcTan[a + b*x]]))))/(140*b*Gamma[11/6]*Gamma[7/3])","B",0
70,1,225,32,0.800008,"\int \frac{(a+b x)^2 \tan ^{-1}(a+b x)}{\sqrt[3]{\left(1+a^2\right) c+2 a b c x+b^2 c x^2}} \, dx","Integrate[((a + b*x)^2*ArcTan[a + b*x])/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]","-\frac{3 \sqrt[3]{a^2+2 a b x+b^2 x^2+1} \left((a+b x)^2+1\right)^{2/3} \left(\frac{5 \sqrt[3]{2} \sqrt{\pi } \Gamma \left(\frac{5}{3}\right) \, _3F_2\left(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{(a+b x)^2+1}\right)}{\left((a+b x)^2+1\right)^2}+\Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right) \left(\frac{24 (a+b x) \, _2F_1\left(1,\frac{4}{3};\frac{11}{6};\frac{1}{(a+b x)^2+1}\right) \tan ^{-1}(a+b x)}{\left((a+b x)^2+1\right)^2}+\frac{90}{(a+b x)^2+1}+5 \tan ^{-1}(a+b x) \left(6 \sin \left(2 \tan ^{-1}(a+b x)\right)-4 (a+b x)\right)+15\right)\right)}{140 b \Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right) \sqrt[3]{c \left(a^2+2 a b x+b^2 x^2+1\right)}}","\text{Int}\left(\frac{(a+b x)^2 \tan ^{-1}(a+b x)}{\sqrt[3]{c (a+b x)^2+c}},x\right)",0,"(-3*(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3)*(1 + (a + b*x)^2)^(2/3)*((5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2)^2 + Gamma[11/6]*Gamma[7/3]*(15 + 90/(1 + (a + b*x)^2) + (24*(a + b*x)*ArcTan[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2)^2 + 5*ArcTan[a + b*x]*(-4*(a + b*x) + 6*Sin[2*ArcTan[a + b*x]]))))/(140*b*(c*(1 + a^2 + 2*a*b*x + b^2*x^2))^(1/3)*Gamma[11/6]*Gamma[7/3])","B",0