1,1,274,149,0.188713,"\int (d+e x)^4 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^4*(a + b*ArcTanh[c*x]),x]","\frac{3 c^4 e^3 x^4 (20 a c d+b e)+20 c^4 d e^2 x^3 (6 a c d+b e)+6 c^2 e x^2 \left(20 a c^3 d^3+b e \left(10 c^2 d^2+e^2\right)\right)+60 c^2 d x \left(a c^3 d^3+b e \left(2 c^2 d^2+e^2\right)\right)+12 a c^5 e^4 x^5+12 b c^5 x \tanh ^{-1}(c x) \left(5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right)+6 b \left(5 c^4 d^4+10 c^3 d^3 e+10 c^2 d^2 e^2+5 c d e^3+e^4\right) \log (1-c x)+6 b \left(5 c^4 d^4-10 c^3 d^3 e+10 c^2 d^2 e^2-5 c d e^3+e^4\right) \log (c x+1)}{60 c^5}","\frac{(d+e x)^5 \left(a+b \tanh ^{-1}(c x)\right)}{5 e}-\frac{b (c d-e)^5 \log (c x+1)}{10 c^5 e}+\frac{b (c d+e)^5 \log (1-c x)}{10 c^5 e}+\frac{b e^2 x^2 \left(10 c^2 d^2+e^2\right)}{10 c^3}+\frac{b d e x \left(2 c^2 d^2+e^2\right)}{c^3}+\frac{b d e^3 x^3}{3 c}+\frac{b e^4 x^4}{20 c}",1,"(60*c^2*d*(a*c^3*d^3 + b*e*(2*c^2*d^2 + e^2))*x + 6*c^2*e*(20*a*c^3*d^3 + b*e*(10*c^2*d^2 + e^2))*x^2 + 20*c^4*d*e^2*(6*a*c*d + b*e)*x^3 + 3*c^4*e^3*(20*a*c*d + b*e)*x^4 + 12*a*c^5*e^4*x^5 + 12*b*c^5*x*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4)*ArcTanh[c*x] + 6*b*(5*c^4*d^4 + 10*c^3*d^3*e + 10*c^2*d^2*e^2 + 5*c*d*e^3 + e^4)*Log[1 - c*x] + 6*b*(5*c^4*d^4 - 10*c^3*d^3*e + 10*c^2*d^2*e^2 - 5*c*d*e^3 + e^4)*Log[1 + c*x])/(60*c^5)","A",1
2,1,205,125,0.1350324,"\int (d+e x)^3 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^3*(a + b*ArcTanh[c*x]),x]","\frac{2 c^3 e^2 x^3 (12 a c d+b e)+12 c^3 d e x^2 (3 a c d+b e)+6 c x \left(4 a c^3 d^3+b e \left(6 c^2 d^2+e^2\right)\right)+6 a c^4 e^3 x^4+6 b c^4 x \tanh ^{-1}(c x) \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+3 b \left(4 c^3 d^3+6 c^2 d^2 e+4 c d e^2+e^3\right) \log (1-c x)+3 b \left(4 c^3 d^3-6 c^2 d^2 e+4 c d e^2-e^3\right) \log (c x+1)}{24 c^4}","\frac{(d+e x)^4 \left(a+b \tanh ^{-1}(c x)\right)}{4 e}-\frac{b (c d-e)^4 \log (c x+1)}{8 c^4 e}+\frac{b (c d+e)^4 \log (1-c x)}{8 c^4 e}+\frac{b e x \left(6 c^2 d^2+e^2\right)}{4 c^3}+\frac{b d e^2 x^2}{2 c}+\frac{b e^3 x^3}{12 c}",1,"(6*c*(4*a*c^3*d^3 + b*e*(6*c^2*d^2 + e^2))*x + 12*c^3*d*e*(3*a*c*d + b*e)*x^2 + 2*c^3*e^2*(12*a*c*d + b*e)*x^3 + 6*a*c^4*e^3*x^4 + 6*b*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcTanh[c*x] + 3*b*(4*c^3*d^3 + 6*c^2*d^2*e + 4*c*d*e^2 + e^3)*Log[1 - c*x] + 3*b*(4*c^3*d^3 - 6*c^2*d^2*e + 4*c*d*e^2 - e^3)*Log[1 + c*x])/(24*c^4)","A",1
3,1,129,96,0.0987128,"\int (d+e x)^2 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^2*(a + b*ArcTanh[c*x]),x]","\frac{1}{6} \left(\frac{e x^2 (6 a c d+b e)}{c}+\frac{6 d x (a c d+b e)}{c}+2 a e^2 x^3+\frac{b \left(3 c^2 d^2+3 c d e+e^2\right) \log (1-c x)}{c^3}+\frac{b \left(3 c^2 d^2-3 c d e+e^2\right) \log (c x+1)}{c^3}+2 b x \tanh ^{-1}(c x) \left(3 d^2+3 d e x+e^2 x^2\right)\right)","\frac{(d+e x)^3 \left(a+b \tanh ^{-1}(c x)\right)}{3 e}-\frac{b (c d-e)^3 \log (c x+1)}{6 c^3 e}+\frac{b (c d+e)^3 \log (1-c x)}{6 c^3 e}+\frac{b d e x}{c}+\frac{b e^2 x^2}{6 c}",1,"((6*d*(a*c*d + b*e)*x)/c + (e*(6*a*c*d + b*e)*x^2)/c + 2*a*e^2*x^3 + 2*b*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTanh[c*x] + (b*(3*c^2*d^2 + 3*c*d*e + e^2)*Log[1 - c*x])/c^3 + (b*(3*c^2*d^2 - 3*c*d*e + e^2)*Log[1 + c*x])/c^3)/6","A",1
4,1,96,84,0.0110863,"\int (d+e x) \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)*(a + b*ArcTanh[c*x]),x]","a d x+\frac{1}{2} a e x^2+\frac{b d \log \left(1-c^2 x^2\right)}{2 c}+\frac{b e \log (1-c x)}{4 c^2}-\frac{b e \log (c x+1)}{4 c^2}+b d x \tanh ^{-1}(c x)+\frac{1}{2} b e x^2 \tanh ^{-1}(c x)+\frac{b e x}{2 c}","\frac{(d+e x)^2 \left(a+b \tanh ^{-1}(c x)\right)}{2 e}-\frac{b (c d-e)^2 \log (c x+1)}{4 c^2 e}+\frac{b (c d+e)^2 \log (1-c x)}{4 c^2 e}+\frac{b e x}{2 c}",1,"a*d*x + (b*e*x)/(2*c) + (a*e*x^2)/2 + b*d*x*ArcTanh[c*x] + (b*e*x^2*ArcTanh[c*x])/2 + (b*e*Log[1 - c*x])/(4*c^2) - (b*e*Log[1 + c*x])/(4*c^2) + (b*d*Log[1 - c^2*x^2])/(2*c)","A",1
5,1,257,114,0.2584594,"\int \frac{a+b \tanh ^{-1}(c x)}{d+e x} \, dx","Integrate[(a + b*ArcTanh[c*x])/(d + e*x),x]","\frac{a \log (d+e x)-\frac{1}{2} i b \left(-\log \left(\frac{2}{\sqrt{1-c^2 x^2}}\right) \left(\pi -2 i \tanh ^{-1}(c x)\right)-i \text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)+i \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)^2+2 i \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-2 i \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right) \log \left(2 i \sinh \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)\right)-i \text{Li}_2\left(-e^{2 \tanh ^{-1}(c x)}\right)-\frac{1}{4} i \left(\pi -2 i \tanh ^{-1}(c x)\right)^2+\left(\pi -2 i \tanh ^{-1}(c x)\right) \log \left(e^{2 \tanh ^{-1}(c x)}+1\right)\right)+b \tanh ^{-1}(c x) \left(\frac{1}{2} \log \left(1-c^2 x^2\right)+\log \left(i \sinh \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)\right)\right)}{e}","\frac{\left(a+b \tanh ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right)}{e}-\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{e}-\frac{b \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 e}+\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 e}",1,"(a*Log[d + e*x] + b*ArcTanh[c*x]*(Log[1 - c^2*x^2]/2 + Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]]) - (I/2)*b*((-1/4*I)*(Pi - (2*I)*ArcTanh[c*x])^2 + I*(ArcTanh[(c*d)/e] + ArcTanh[c*x])^2 + (Pi - (2*I)*ArcTanh[c*x])*Log[1 + E^(2*ArcTanh[c*x])] + (2*I)*(ArcTanh[(c*d)/e] + ArcTanh[c*x])*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - (Pi - (2*I)*ArcTanh[c*x])*Log[2/Sqrt[1 - c^2*x^2]] - (2*I)*(ArcTanh[(c*d)/e] + ArcTanh[c*x])*Log[(2*I)*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - I*PolyLog[2, -E^(2*ArcTanh[c*x])] - I*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/e","C",0
6,1,102,93,0.12334,"\int \frac{a+b \tanh ^{-1}(c x)}{(d+e x)^2} \, dx","Integrate[(a + b*ArcTanh[c*x])/(d + e*x)^2,x]","-\frac{a}{e (d+e x)}-\frac{b c \log (d+e x)}{c^2 d^2-e^2}-\frac{b c \log (1-c x)}{2 e (c d+e)}-\frac{b c \log (c x+1)}{2 e (e-c d)}-\frac{b \tanh ^{-1}(c x)}{e (d+e x)}","-\frac{a+b \tanh ^{-1}(c x)}{e (d+e x)}-\frac{b c \log (d+e x)}{c^2 d^2-e^2}-\frac{b c \log (1-c x)}{2 e (c d+e)}+\frac{b c \log (c x+1)}{2 e (c d-e)}",1,"-(a/(e*(d + e*x))) - (b*ArcTanh[c*x])/(e*(d + e*x)) - (b*c*Log[1 - c*x])/(2*e*(c*d + e)) - (b*c*Log[1 + c*x])/(2*e*(-(c*d) + e)) - (b*c*Log[d + e*x])/(c^2*d^2 - e^2)","A",1
7,1,133,130,0.1555538,"\int \frac{a+b \tanh ^{-1}(c x)}{(d+e x)^3} \, dx","Integrate[(a + b*ArcTanh[c*x])/(d + e*x)^3,x]","\frac{1}{4} \left(-\frac{2 a}{e (d+e x)^2}+\frac{2 b c}{\left(c^2 d^2-e^2\right) (d+e x)}-\frac{b c^2 \log (1-c x)}{e (c d+e)^2}+\frac{b c^2 \log (c x+1)}{e (e-c d)^2}-\frac{4 b c^3 d \log (d+e x)}{\left(e^2-c^2 d^2\right)^2}-\frac{2 b \tanh ^{-1}(c x)}{e (d+e x)^2}\right)","-\frac{a+b \tanh ^{-1}(c x)}{2 e (d+e x)^2}+\frac{b c}{2 \left(c^2 d^2-e^2\right) (d+e x)}-\frac{b c^2 \log (1-c x)}{4 e (c d+e)^2}+\frac{b c^2 \log (c x+1)}{4 e (c d-e)^2}-\frac{b c^3 d \log (d+e x)}{\left(c^2 d^2-e^2\right)^2}",1,"((-2*a)/(e*(d + e*x)^2) + (2*b*c)/((c^2*d^2 - e^2)*(d + e*x)) - (2*b*ArcTanh[c*x])/(e*(d + e*x)^2) - (b*c^2*Log[1 - c*x])/(e*(c*d + e)^2) + (b*c^2*Log[1 + c*x])/(e*(-(c*d) + e)^2) - (4*b*c^3*d*Log[d + e*x])/(-(c^2*d^2) + e^2)^2)/4","A",1
8,1,173,175,0.271435,"\int \frac{a+b \tanh ^{-1}(c x)}{(d+e x)^4} \, dx","Integrate[(a + b*ArcTanh[c*x])/(d + e*x)^4,x]","\frac{1}{6} \left(-\frac{2 a}{e (d+e x)^3}-\frac{b c^3 \log (1-c x)}{e (c d+e)^3}+\frac{b c^3 \log (c x+1)}{e (c d-e)^3}+\frac{b c}{\left(c^2 d^2-e^2\right) (d+e x)^2}+\frac{4 b c^3 d}{\left(e^2-c^2 d^2\right)^2 (d+e x)}-\frac{2 b c^3 \left(3 c^2 d^2+e^2\right) \log (d+e x)}{\left(c^2 d^2-e^2\right)^3}-\frac{2 b \tanh ^{-1}(c x)}{e (d+e x)^3}\right)","-\frac{a+b \tanh ^{-1}(c x)}{3 e (d+e x)^3}-\frac{b c^3 \log (1-c x)}{6 e (c d+e)^3}+\frac{b c^3 \log (c x+1)}{6 e (c d-e)^3}+\frac{b c}{6 \left(c^2 d^2-e^2\right) (d+e x)^2}+\frac{2 b c^3 d}{3 \left(c^2 d^2-e^2\right)^2 (d+e x)}-\frac{b c^3 \left(3 c^2 d^2+e^2\right) \log (d+e x)}{3 (c d-e)^3 (c d+e)^3}",1,"((-2*a)/(e*(d + e*x)^3) + (b*c)/((c^2*d^2 - e^2)*(d + e*x)^2) + (4*b*c^3*d)/((-(c^2*d^2) + e^2)^2*(d + e*x)) - (2*b*ArcTanh[c*x])/(e*(d + e*x)^3) - (b*c^3*Log[1 - c*x])/(e*(c*d + e)^3) + (b*c^3*Log[1 + c*x])/((c*d - e)^3*e) - (2*b*c^3*(3*c^2*d^2 + e^2)*Log[d + e*x])/(c^2*d^2 - e^2)^3)/6","A",1
9,1,506,359,0.930688,"\int (d+e x)^3 \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x)^3*(a + b*ArcTanh[c*x])^2,x]","\frac{12 a^2 c^4 d^3 x+18 a^2 c^4 d^2 e x^2+12 a^2 c^4 d e^2 x^3+3 a^2 c^4 e^3 x^4+36 a b c^3 d^2 e x+12 a b c^3 d e^2 x^2+2 a b c^3 e^3 x^3+18 a b c^2 d^2 e \log (1-c x)-18 a b c^2 d^2 e \log (c x+1)+12 a b c d e^2 \log \left(c^2 x^2-1\right)+12 a b c^3 d^3 \log \left(1-c^2 x^2\right)+2 b c \tanh ^{-1}(c x) \left(3 a c^3 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+b e \left(18 c^2 d^2 x+6 d e \left(c^2 x^2-1\right)+e^2 x \left(c^2 x^2+3\right)\right)-12 b d \left(c^2 d^2+e^2\right) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+6 a b c e^3 x+3 a b e^3 \log (1-c x)-3 a b e^3 \log (c x+1)+12 b^2 c d \left(c^2 d^2+e^2\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+18 b^2 c^2 d^2 e \log \left(1-c^2 x^2\right)+12 b^2 c^2 d e^2 x+b^2 c^2 e^3 x^2+4 b^2 e^3 \log \left(1-c^2 x^2\right)+3 b^2 \tanh ^{-1}(c x)^2 \left(c^4 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)-4 c^3 d^3-6 c^2 d^2 e-4 c d e^2-e^3\right)-b^2 e^3}{12 c^4}","-\frac{\left(c^4 d^4+6 c^2 d^2 e^2+e^4\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{4 c^4 e}+\frac{a b e x \left(6 c^2 d^2+e^2\right)}{2 c^3}+\frac{d \left(c^2 d^2+e^2\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^3}-\frac{2 b d \left(c^2 d^2+e^2\right) \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^3}+\frac{b d e^2 x^2 \left(a+b \tanh ^{-1}(c x)\right)}{c}+\frac{(d+e x)^4 \left(a+b \tanh ^{-1}(c x)\right)^2}{4 e}+\frac{b e^3 x^3 \left(a+b \tanh ^{-1}(c x)\right)}{6 c}-\frac{b^2 d e^2 \tanh ^{-1}(c x)}{c^3}+\frac{b^2 d e^2 x}{c^2}+\frac{b^2 e^3 x^2}{12 c^2}+\frac{b^2 e \left(6 c^2 d^2+e^2\right) \log \left(1-c^2 x^2\right)}{4 c^4}+\frac{b^2 e^3 \log \left(1-c^2 x^2\right)}{12 c^4}-\frac{b^2 d \left(c^2 d^2+e^2\right) \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c^3}+\frac{b^2 e x \left(6 c^2 d^2+e^2\right) \tanh ^{-1}(c x)}{2 c^3}",1,"(-(b^2*e^3) + 12*a^2*c^4*d^3*x + 36*a*b*c^3*d^2*e*x + 12*b^2*c^2*d*e^2*x + 6*a*b*c*e^3*x + 18*a^2*c^4*d^2*e*x^2 + 12*a*b*c^3*d*e^2*x^2 + b^2*c^2*e^3*x^2 + 12*a^2*c^4*d*e^2*x^3 + 2*a*b*c^3*e^3*x^3 + 3*a^2*c^4*e^3*x^4 + 3*b^2*(-4*c^3*d^3 - 6*c^2*d^2*e - 4*c*d*e^2 - e^3 + c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))*ArcTanh[c*x]^2 + 2*b*c*ArcTanh[c*x]*(3*a*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + b*e*(18*c^2*d^2*x + 6*d*e*(-1 + c^2*x^2) + e^2*x*(3 + c^2*x^2)) - 12*b*d*(c^2*d^2 + e^2)*Log[1 + E^(-2*ArcTanh[c*x])]) + 18*a*b*c^2*d^2*e*Log[1 - c*x] + 3*a*b*e^3*Log[1 - c*x] - 18*a*b*c^2*d^2*e*Log[1 + c*x] - 3*a*b*e^3*Log[1 + c*x] + 12*a*b*c^3*d^3*Log[1 - c^2*x^2] + 18*b^2*c^2*d^2*e*Log[1 - c^2*x^2] + 4*b^2*e^3*Log[1 - c^2*x^2] + 12*a*b*c*d*e^2*Log[-1 + c^2*x^2] + 12*b^2*c*d*(c^2*d^2 + e^2)*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(12*c^4)","A",0
10,1,319,257,0.6546251,"\int (d+e x)^2 \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x)^2*(a + b*ArcTanh[c*x])^2,x]","\frac{3 a^2 c^3 d^2 x+3 a^2 c^3 d e x^2+a^2 c^3 e^2 x^3+3 a b c^2 d^2 \log \left(1-c^2 x^2\right)+6 a b c^2 d e x+a b c^2 e^2 x^2+a b e^2 \log \left(c^2 x^2-1\right)+b \tanh ^{-1}(c x) \left(2 a c^3 x \left(3 d^2+3 d e x+e^2 x^2\right)-2 b \left(3 c^2 d^2+e^2\right) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+b e \left(6 c^2 d x+c^2 e x^2-e\right)\right)+3 a b c d e \log (1-c x)-3 a b c d e \log (c x+1)+b^2 \left(3 c^2 d^2+e^2\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+b^2 (c x-1) \tanh ^{-1}(c x)^2 \left(c^2 \left(3 d^2+3 d e x+e^2 x^2\right)+c e (3 d+e x)+e^2\right)+3 b^2 c d e \log \left(1-c^2 x^2\right)+b^2 c e^2 x}{3 c^3}","-\frac{d \left(\frac{3 e^2}{c^2}+d^2\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{3 e}+\frac{\left(3 c^2 d^2+e^2\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{3 c^3}-\frac{2 b \left(3 c^2 d^2+e^2\right) \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{3 c^3}+\frac{2 a b d e x}{c}+\frac{(d+e x)^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{3 e}+\frac{b e^2 x^2 \left(a+b \tanh ^{-1}(c x)\right)}{3 c}-\frac{b^2 e^2 \tanh ^{-1}(c x)}{3 c^3}+\frac{b^2 d e \log \left(1-c^2 x^2\right)}{c^2}+\frac{b^2 e^2 x}{3 c^2}-\frac{b^2 \left(3 c^2 d^2+e^2\right) \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{3 c^3}+\frac{2 b^2 d e x \tanh ^{-1}(c x)}{c}",1,"(3*a^2*c^3*d^2*x + 6*a*b*c^2*d*e*x + b^2*c*e^2*x + 3*a^2*c^3*d*e*x^2 + a*b*c^2*e^2*x^2 + a^2*c^3*e^2*x^3 + b^2*(-1 + c*x)*(e^2 + c*e*(3*d + e*x) + c^2*(3*d^2 + 3*d*e*x + e^2*x^2))*ArcTanh[c*x]^2 + b*ArcTanh[c*x]*(b*e*(-e + 6*c^2*d*x + c^2*e*x^2) + 2*a*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) - 2*b*(3*c^2*d^2 + e^2)*Log[1 + E^(-2*ArcTanh[c*x])]) + 3*a*b*c*d*e*Log[1 - c*x] - 3*a*b*c*d*e*Log[1 + c*x] + 3*a*b*c^2*d^2*Log[1 - c^2*x^2] + 3*b^2*c*d*e*Log[1 - c^2*x^2] + a*b*e^2*Log[-1 + c^2*x^2] + b^2*(3*c^2*d^2 + e^2)*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(3*c^3)","A",0
11,1,174,160,0.4397164,"\int (d+e x) \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x)*(a + b*ArcTanh[c*x])^2,x]","\frac{2 a^2 c^2 d x+a^2 c^2 e x^2+2 a b c d \log \left(1-c^2 x^2\right)+2 b c \tanh ^{-1}(c x) \left(a c x (2 d+e x)-2 b d \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+b e x\right)+2 a b c e x+a b e \log (1-c x)-a b e \log (c x+1)+b^2 e \log \left(1-c^2 x^2\right)+b^2 (c x-1) \tanh ^{-1}(c x)^2 (2 c d+c e x+e)+2 b^2 c d \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)}{2 c^2}","-\frac{\left(\frac{e^2}{c^2}+d^2\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{2 e}+\frac{(d+e x)^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 e}+\frac{d \left(a+b \tanh ^{-1}(c x)\right)^2}{c}-\frac{2 b d \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c}+\frac{a b e x}{c}+\frac{b^2 e \log \left(1-c^2 x^2\right)}{2 c^2}-\frac{b^2 d \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c}+\frac{b^2 e x \tanh ^{-1}(c x)}{c}",1,"(2*a^2*c^2*d*x + 2*a*b*c*e*x + a^2*c^2*e*x^2 + b^2*(-1 + c*x)*(2*c*d + e + c*e*x)*ArcTanh[c*x]^2 + 2*b*c*ArcTanh[c*x]*(b*e*x + a*c*x*(2*d + e*x) - 2*b*d*Log[1 + E^(-2*ArcTanh[c*x])]) + a*b*e*Log[1 - c*x] - a*b*e*Log[1 + c*x] + 2*a*b*c*d*Log[1 - c^2*x^2] + b^2*e*Log[1 - c^2*x^2] + 2*b^2*c*d*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(2*c^2)","A",0
12,1,759,188,10.6126489,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{d+e x} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(d + e*x),x]","\frac{6 a^2 \log (d+e x)-6 i a b \left(-\log \left(\frac{2}{\sqrt{1-c^2 x^2}}\right) \left(\pi -2 i \tanh ^{-1}(c x)\right)-i \text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)+i \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)^2+2 i \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-2 i \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right) \log \left(2 i \sinh \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)\right)-i \text{Li}_2\left(-e^{2 \tanh ^{-1}(c x)}\right)-\frac{1}{4} i \left(\pi -2 i \tanh ^{-1}(c x)\right)^2+\left(\pi -2 i \tanh ^{-1}(c x)\right) \log \left(e^{2 \tanh ^{-1}(c x)}+1\right)\right)+6 a b \tanh ^{-1}(c x) \left(\log \left(1-c^2 x^2\right)+2 \log \left(i \sinh \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)\right)\right)+\frac{b^2 \left(-4 e \sqrt{1-\frac{c^2 d^2}{e^2}} \tanh ^{-1}(c x)^3 e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)}-6 c d \tanh ^{-1}(c x)^2 \log \left(\frac{c (d+e x)}{\sqrt{1-c^2 x^2}}\right)-3 i \pi  c d \log \left(1-c^2 x^2\right) \tanh ^{-1}(c x)+12 c d \tanh ^{-1}(c x) \text{Li}_2\left(-e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right)+12 c d \tanh ^{-1}(c x) \text{Li}_2\left(e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right)-12 c d \text{Li}_3\left(-e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right)-12 c d \text{Li}_3\left(e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right)+6 c d \tanh ^{-1}(c x)^2 \log \left(1-e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right)+6 c d \tanh ^{-1}(c x)^2 \log \left(e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}+1\right)+6 c d \tanh ^{-1}(c x)^2 \log \left(\frac{1}{2} e^{-\tanh ^{-1}(c x)} \left(c d \left(e^{2 \tanh ^{-1}(c x)}+1\right)+e \left(e^{2 \tanh ^{-1}(c x)}-1\right)\right)\right)+12 c d \tanh ^{-1}(c x) \tanh ^{-1}\left(\frac{c d}{e}\right) \log \left(\frac{1}{2} i e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)-\tanh ^{-1}(c x)} \left(e^{2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}-1\right)\right)-12 c d \tanh ^{-1}(c x) \tanh ^{-1}\left(\frac{c d}{e}\right) \log \left(i \sinh \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)\right)+6 c d \tanh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+3 c d \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-8 c d \tanh ^{-1}(c x)^3-6 c d \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-6 i \pi  c d \tanh ^{-1}(c x) \log \left(\frac{1}{2} \left(e^{-\tanh ^{-1}(c x)}+e^{\tanh ^{-1}(c x)}\right)\right)+4 e \tanh ^{-1}(c x)^3\right)}{c d}}{6 e}","-\frac{b \left(a+b \tanh ^{-1}(c x)\right) \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{e}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right)}{e}+\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{e}-\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{e}-\frac{b^2 \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 e}+\frac{b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 e}",1,"(6*a^2*Log[d + e*x] + 6*a*b*ArcTanh[c*x]*(Log[1 - c^2*x^2] + 2*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]]) - (6*I)*a*b*((-1/4*I)*(Pi - (2*I)*ArcTanh[c*x])^2 + I*(ArcTanh[(c*d)/e] + ArcTanh[c*x])^2 + (Pi - (2*I)*ArcTanh[c*x])*Log[1 + E^(2*ArcTanh[c*x])] + (2*I)*(ArcTanh[(c*d)/e] + ArcTanh[c*x])*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - (Pi - (2*I)*ArcTanh[c*x])*Log[2/Sqrt[1 - c^2*x^2]] - (2*I)*(ArcTanh[(c*d)/e] + ArcTanh[c*x])*Log[(2*I)*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - I*PolyLog[2, -E^(2*ArcTanh[c*x])] - I*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]) + (b^2*(-8*c*d*ArcTanh[c*x]^3 + 4*e*ArcTanh[c*x]^3 - (4*Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^3)/E^ArcTanh[(c*d)/e] - 6*c*d*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] - (6*I)*c*d*Pi*ArcTanh[c*x]*Log[(E^(-ArcTanh[c*x]) + E^ArcTanh[c*x])/2] + 6*c*d*ArcTanh[c*x]^2*Log[1 - E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 6*c*d*ArcTanh[c*x]^2*Log[1 + E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 12*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x]*Log[(I/2)*E^(-ArcTanh[(c*d)/e] - ArcTanh[c*x])*(-1 + E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c*x])))] + 6*c*d*ArcTanh[c*x]^2*Log[(e*(-1 + E^(2*ArcTanh[c*x])) + c*d*(1 + E^(2*ArcTanh[c*x])))/(2*E^ArcTanh[c*x])] - 6*c*d*ArcTanh[c*x]^2*Log[(c*(d + e*x))/Sqrt[1 - c^2*x^2]] - (3*I)*c*d*Pi*ArcTanh[c*x]*Log[1 - c^2*x^2] - 12*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] + 6*c*d*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 12*c*d*ArcTanh[c*x]*PolyLog[2, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 12*c*d*ArcTanh[c*x]*PolyLog[2, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 3*c*d*PolyLog[3, -E^(-2*ArcTanh[c*x])] - 12*c*d*PolyLog[3, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 12*c*d*PolyLog[3, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])]))/(c*d))/(6*e)","C",0
13,1,317,321,4.5997161,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{(d+e x)^2} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(d + e*x)^2,x]","-\frac{a^2}{e (d+e x)}+\frac{a b c \left(\frac{(e-c d) \log (1-c x)+(c d+e) \log (c x+1)-2 e \log (c (d+e x))}{(c d-e) (c d+e)}-\frac{2 \tanh ^{-1}(c x)}{c d+c e x}\right)}{e}+\frac{b^2 \left(\frac{c d \left(-i \pi  \left(\tanh ^{-1}(c x)-\frac{1}{2} \log \left(1-c^2 x^2\right)\right)+\text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-2 \tanh ^{-1}(c x) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-2 \tanh ^{-1}\left(\frac{c d}{e}\right) \left(\log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-\log \left(i \sinh \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)\right)+\tanh ^{-1}(c x)\right)+i \pi  \log \left(e^{2 \tanh ^{-1}(c x)}+1\right)\right)}{c^2 d^2-e^2}-\frac{\tanh ^{-1}(c x)^2 e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)}}{e \sqrt{1-\frac{c^2 d^2}{e^2}}}+\frac{x \tanh ^{-1}(c x)^2}{d+e x}\right)}{d}","\frac{2 b c \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^2 d^2-e^2}-\frac{2 b c \left(a+b \tanh ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right)}{c^2 d^2-e^2}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{e (d+e x)}+\frac{b c \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{e (c d+e)}-\frac{b c \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{e (c d-e)}-\frac{b^2 c \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{c^2 d^2-e^2}+\frac{b^2 c \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{c^2 d^2-e^2}+\frac{b^2 c \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{2 e (c d+e)}+\frac{b^2 c \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 e (c d-e)}",1,"-(a^2/(e*(d + e*x))) + (a*b*c*((-2*ArcTanh[c*x])/(c*d + c*e*x) + ((-(c*d) + e)*Log[1 - c*x] + (c*d + e)*Log[1 + c*x] - 2*e*Log[c*(d + e*x)])/((c*d - e)*(c*d + e))))/e + (b^2*(-(ArcTanh[c*x]^2/(Sqrt[1 - (c^2*d^2)/e^2]*e*E^ArcTanh[(c*d)/e])) + (x*ArcTanh[c*x]^2)/(d + e*x) + (c*d*(I*Pi*Log[1 + E^(2*ArcTanh[c*x])] - 2*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - I*Pi*(ArcTanh[c*x] - Log[1 - c^2*x^2]/2) - 2*ArcTanh[(c*d)/e]*(ArcTanh[c*x] + Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]]) + PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/(c^2*d^2 - e^2)))/d","C",0
14,1,470,480,7.0637218,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{(d+e x)^3} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(d + e*x)^3,x]","-\frac{a^2}{2 e (d+e x)^2}-\frac{a b c^2 \left(\frac{\frac{2 e \left(c^2 \left(-d^2\right)+2 c^2 d (d+e x) \log (c (d+e x))+e^2\right)}{c (c d+e)^2 (d+e x)}-\log (c x+1)}{(e-c d)^2}+\frac{\log (1-c x)}{(c d+e)^2}+\frac{2 \tanh ^{-1}(c x)}{(c d+c e x)^2}\right)}{2 e}+\frac{b^2 c^2 \left(\frac{2 c d \left(-i \pi  \left(\tanh ^{-1}(c x)-\frac{1}{2} \log \left(1-c^2 x^2\right)\right)+\text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-2 \tanh ^{-1}(c x) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-2 \tanh ^{-1}\left(\frac{c d}{e}\right) \left(\log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-\log \left(i \sinh \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)\right)+\tanh ^{-1}(c x)\right)+i \pi  \log \left(e^{2 \tanh ^{-1}(c x)}+1\right)\right)}{c^2 d^2-e^2}-\frac{2 \tanh ^{-1}(c x)^2 e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)}}{e \sqrt{1-\frac{c^2 d^2}{e^2}}}-\frac{e \left(c^2 x^2-1\right) \tanh ^{-1}(c x)^2}{c^2 (d+e x)^2}+\frac{2 e \left(c d \log \left(\frac{c (d+e x)}{\sqrt{1-c^2 x^2}}\right)-e \tanh ^{-1}(c x)\right)}{c^3 d^3-c d e^2}+\frac{2 x \tanh ^{-1}(c x) \left(c d \tanh ^{-1}(c x)-e\right)}{c d (d+e x)}\right)}{2 (c d-e) (c d+e)}","\frac{2 b c^3 d \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{(c d-e)^2 (c d+e)^2}-\frac{2 b c^3 d \left(a+b \tanh ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right)}{(c d-e)^2 (c d+e)^2}+\frac{b c \left(a+b \tanh ^{-1}(c x)\right)}{\left(c^2 d^2-e^2\right) (d+e x)}+\frac{b c^2 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{2 e (c d+e)^2}-\frac{b c^2 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{2 e (c d-e)^2}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{2 e (d+e x)^2}-\frac{b^2 c^3 d \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{(c d-e)^2 (c d+e)^2}+\frac{b^2 c^3 d \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{(c d-e)^2 (c d+e)^2}+\frac{b^2 c^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{4 e (c d+e)^2}+\frac{b^2 c^2 \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{4 e (c d-e)^2}+\frac{b^2 c^2 \log (1-c x)}{2 (c d-e) (c d+e)^2}-\frac{b^2 c^2 \log (c x+1)}{2 (c d-e)^2 (c d+e)}+\frac{b^2 c^2 e \log (d+e x)}{(c d-e)^2 (c d+e)^2}",1,"-1/2*a^2/(e*(d + e*x)^2) - (a*b*c^2*((2*ArcTanh[c*x])/(c*d + c*e*x)^2 + Log[1 - c*x]/(c*d + e)^2 + (-Log[1 + c*x] + (2*e*(-(c^2*d^2) + e^2 + 2*c^2*d*(d + e*x)*Log[c*(d + e*x)]))/(c*(c*d + e)^2*(d + e*x)))/(-(c*d) + e)^2))/(2*e) + (b^2*c^2*((-2*ArcTanh[c*x]^2)/(Sqrt[1 - (c^2*d^2)/e^2]*e*E^ArcTanh[(c*d)/e]) - (e*(-1 + c^2*x^2)*ArcTanh[c*x]^2)/(c^2*(d + e*x)^2) + (2*x*ArcTanh[c*x]*(-e + c*d*ArcTanh[c*x]))/(c*d*(d + e*x)) + (2*e*(-(e*ArcTanh[c*x]) + c*d*Log[(c*(d + e*x))/Sqrt[1 - c^2*x^2]]))/(c^3*d^3 - c*d*e^2) + (2*c*d*(I*Pi*Log[1 + E^(2*ArcTanh[c*x])] - 2*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - I*Pi*(ArcTanh[c*x] - Log[1 - c^2*x^2]/2) - 2*ArcTanh[(c*d)/e]*(ArcTanh[c*x] + Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]]) + PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/(c^2*d^2 - e^2)))/(2*(c*d - e)*(c*d + e))","C",0
15,1,830,614,2.1105521,"\int (d+e x)^3 \left(a+b \tanh ^{-1}(c x)\right)^3 \, dx","Integrate[(d + e*x)^3*(a + b*ArcTanh[c*x])^3,x]","\frac{2 a^3 e^3 x^4 c^4+6 a^2 b x \left(4 d^3+6 e x d^2+4 e^2 x^2 d+e^3 x^3\right) \tanh ^{-1}(c x) c^4+2 a^2 e^2 (4 a c d+b e) x^3 c^3+12 a^2 d e (a c d+b e) x^2 c^3+24 a b^2 d^3 \left(\tanh ^{-1}(c x) \left((c x-1) \tanh ^{-1}(c x)-2 \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right)\right)+\text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)\right) c^3+8 b^3 d^3 \left(\left((c x-1) \tanh ^{-1}(c x)-3 \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right)\right) \tanh ^{-1}(c x)^2+3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)+\frac{3}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)\right) c^3+36 a b^2 d^2 e \left(\left(c^2 x^2-1\right) \tanh ^{-1}(c x)^2+2 c x \tanh ^{-1}(c x)+\log \left(1-c^2 x^2\right)\right) c^2-12 b^3 d^2 e \left(\tanh ^{-1}(c x) \left(\left(1-c^2 x^2\right) \tanh ^{-1}(c x)^2+(3-3 c x) \tanh ^{-1}(c x)+6 \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right)\right)-3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)\right) c^2+2 a^2 \left(4 a c^3 d^3+3 b e \left(6 c^2 d^2+e^2\right)\right) x c+24 a b^2 d e^2 \left(\left(c^3 x^3-1\right) \tanh ^{-1}(c x)^2+\left(c^2 x^2-2 \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right)-1\right) \tanh ^{-1}(c x)+c x+\text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)\right) c+4 b^3 d e^2 \left(2 c^3 x^3 \tanh ^{-1}(c x)^3-2 \tanh ^{-1}(c x)^3+3 c^2 x^2 \tanh ^{-1}(c x)^2-6 \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)^2-3 \tanh ^{-1}(c x)^2+6 c x \tanh ^{-1}(c x)+6 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)+3 \log \left(1-c^2 x^2\right)+3 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)\right) c+3 a^2 b \left(4 c^3 d^3+6 c^2 e d^2+4 c e^2 d+e^3\right) \log (1-c x)+3 a^2 b \left(4 c^3 d^3-6 c^2 e d^2+4 c e^2 d-e^3\right) \log (c x+1)+2 a b^2 e^3 \left(c^2 x^2+2 c \left(c^2 x^2+3\right) \tanh ^{-1}(c x) x+3 \left(c^4 x^4-1\right) \tanh ^{-1}(c x)^2+4 \log \left(1-c^2 x^2\right)-1\right)+2 b^3 e^3 \left(\left(c^4 x^4-1\right) \tanh ^{-1}(c x)^3+\left(c^3 x^3+3 c x-4\right) \tanh ^{-1}(c x)^2+\left(c^2 x^2-8 \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right)-1\right) \tanh ^{-1}(c x)+c x+4 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)\right)}{8 c^4}","-\frac{b^2 e^3 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{2 c^4}+\frac{3 a b^2 d e^2 x}{c^2}+\frac{b^2 e^3 x^2 \left(a+b \tanh ^{-1}(c x)\right)}{4 c^2}-\frac{3 b^2 e \left(6 c^2 d^2+e^2\right) \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{2 c^4}-\frac{3 b^2 d \left(c^2 d^2+e^2\right) \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^3}+\frac{b e^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{4 c^4}-\frac{3 b d e^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^3}+\frac{3 b e \left(6 c^2 d^2+e^2\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{4 c^4}-\frac{\left(c^4 d^4+6 c^2 d^2 e^2+e^4\right) \left(a+b \tanh ^{-1}(c x)\right)^3}{4 c^4 e}+\frac{d \left(c^2 d^2+e^2\right) \left(a+b \tanh ^{-1}(c x)\right)^3}{c^3}+\frac{3 b e x \left(6 c^2 d^2+e^2\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{4 c^3}-\frac{3 b d \left(c^2 d^2+e^2\right) \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^3}+\frac{3 b d e^2 x^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c}+\frac{(d+e x)^4 \left(a+b \tanh ^{-1}(c x)\right)^3}{4 e}+\frac{b e^3 x^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{4 c}-\frac{b^3 e^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{4 c^4}-\frac{b^3 e^3 \tanh ^{-1}(c x)}{4 c^4}+\frac{b^3 e^3 x}{4 c^3}+\frac{3 b^3 d e^2 x \tanh ^{-1}(c x)}{c^2}-\frac{3 b^3 e \left(6 c^2 d^2+e^2\right) \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{4 c^4}+\frac{3 b^3 d \left(c^2 d^2+e^2\right) \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{2 c^3}+\frac{3 b^3 d e^2 \log \left(1-c^2 x^2\right)}{2 c^3}",1,"(2*a^2*c*(4*a*c^3*d^3 + 3*b*e*(6*c^2*d^2 + e^2))*x + 12*a^2*c^3*d*e*(a*c*d + b*e)*x^2 + 2*a^2*c^3*e^2*(4*a*c*d + b*e)*x^3 + 2*a^3*c^4*e^3*x^4 + 6*a^2*b*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcTanh[c*x] + 3*a^2*b*(4*c^3*d^3 + 6*c^2*d^2*e + 4*c*d*e^2 + e^3)*Log[1 - c*x] + 3*a^2*b*(4*c^3*d^3 - 6*c^2*d^2*e + 4*c*d*e^2 - e^3)*Log[1 + c*x] + 36*a*b^2*c^2*d^2*e*(2*c*x*ArcTanh[c*x] + (-1 + c^2*x^2)*ArcTanh[c*x]^2 + Log[1 - c^2*x^2]) + 2*a*b^2*e^3*(-1 + c^2*x^2 + 2*c*x*(3 + c^2*x^2)*ArcTanh[c*x] + 3*(-1 + c^4*x^4)*ArcTanh[c*x]^2 + 4*Log[1 - c^2*x^2]) - 12*b^3*c^2*d^2*e*(ArcTanh[c*x]*((3 - 3*c*x)*ArcTanh[c*x] + (1 - c^2*x^2)*ArcTanh[c*x]^2 + 6*Log[1 + E^(-2*ArcTanh[c*x])]) - 3*PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 24*a*b^2*c*d*e^2*(c*x + (-1 + c^3*x^3)*ArcTanh[c*x]^2 + ArcTanh[c*x]*(-1 + c^2*x^2 - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 24*a*b^2*c^3*d^3*(ArcTanh[c*x]*((-1 + c*x)*ArcTanh[c*x] - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 2*b^3*e^3*(c*x + (-4 + 3*c*x + c^3*x^3)*ArcTanh[c*x]^2 + (-1 + c^4*x^4)*ArcTanh[c*x]^3 + ArcTanh[c*x]*(-1 + c^2*x^2 - 8*Log[1 + E^(-2*ArcTanh[c*x])]) + 4*PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 8*b^3*c^3*d^3*(ArcTanh[c*x]^2*((-1 + c*x)*ArcTanh[c*x] - 3*Log[1 + E^(-2*ArcTanh[c*x])]) + 3*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + (3*PolyLog[3, -E^(-2*ArcTanh[c*x])])/2) + 4*b^3*c*d*e^2*(6*c*x*ArcTanh[c*x] - 3*ArcTanh[c*x]^2 + 3*c^2*x^2*ArcTanh[c*x]^2 - 2*ArcTanh[c*x]^3 + 2*c^3*x^3*ArcTanh[c*x]^3 - 6*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 3*Log[1 - c^2*x^2] + 6*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 3*PolyLog[3, -E^(-2*ArcTanh[c*x])]))/(8*c^4)","A",0
16,1,591,387,1.3140608,"\int (d+e x)^2 \left(a+b \tanh ^{-1}(c x)\right)^3 \, dx","Integrate[(d + e*x)^2*(a + b*ArcTanh[c*x])^3,x]","\frac{2 a^3 c^3 e^2 x^3+6 a^2 b c^3 x \tanh ^{-1}(c x) \left(3 d^2+3 d e x+e^2 x^2\right)+3 a^2 b \left(3 c^2 d^2+3 c d e+e^2\right) \log (1-c x)+3 a^2 b \left(3 c^2 d^2-3 c d e+e^2\right) \log (c x+1)+3 a^2 c^2 e x^2 (2 a c d+b e)+6 a^2 c^2 d x (a c d+3 b e)+18 a b^2 c^2 d^2 \left(\text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+\tanh ^{-1}(c x) \left((c x-1) \tanh ^{-1}(c x)-2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)\right)+18 a b^2 c d e \left(\log \left(1-c^2 x^2\right)+\left(c^2 x^2-1\right) \tanh ^{-1}(c x)^2+2 c x \tanh ^{-1}(c x)\right)+6 a b^2 e^2 \left(\left(c^3 x^3-1\right) \tanh ^{-1}(c x)^2+\tanh ^{-1}(c x) \left(c^2 x^2-2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-1\right)+\text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+c x\right)+6 b^3 c^2 d^2 \left(3 \tanh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+\frac{3}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)+\tanh ^{-1}(c x)^2 \left((c x-1) \tanh ^{-1}(c x)-3 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)\right)-6 b^3 c d e \left(\tanh ^{-1}(c x) \left(\left(1-c^2 x^2\right) \tanh ^{-1}(c x)^2+(3-3 c x) \tanh ^{-1}(c x)+6 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)-3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)\right)+b^3 e^2 \left(2 c^3 x^3 \tanh ^{-1}(c x)^3+3 \log \left(1-c^2 x^2\right)+3 c^2 x^2 \tanh ^{-1}(c x)^2+6 \tanh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+3 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-2 \tanh ^{-1}(c x)^3-3 \tanh ^{-1}(c x)^2+6 c x \tanh ^{-1}(c x)-6 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)}{6 c^3}","-\frac{6 b^2 d e \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^2}+\frac{a b^2 e^2 x}{c^2}-\frac{b^2 \left(3 c^2 d^2+e^2\right) \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^3}-\frac{b e^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^3}-\frac{d \left(\frac{3 e^2}{c^2}+d^2\right) \left(a+b \tanh ^{-1}(c x)\right)^3}{3 e}+\frac{3 b d e \left(a+b \tanh ^{-1}(c x)\right)^2}{c^2}+\frac{\left(3 c^2 d^2+e^2\right) \left(a+b \tanh ^{-1}(c x)\right)^3}{3 c^3}-\frac{b \left(3 c^2 d^2+e^2\right) \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^3}+\frac{3 b d e x \left(a+b \tanh ^{-1}(c x)\right)^2}{c}+\frac{(d+e x)^3 \left(a+b \tanh ^{-1}(c x)\right)^3}{3 e}+\frac{b e^2 x^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c}-\frac{3 b^3 d e \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c^2}+\frac{b^3 e^2 x \tanh ^{-1}(c x)}{c^2}+\frac{b^3 \left(3 c^2 d^2+e^2\right) \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{2 c^3}+\frac{b^3 e^2 \log \left(1-c^2 x^2\right)}{2 c^3}",1,"(6*a^2*c^2*d*(a*c*d + 3*b*e)*x + 3*a^2*c^2*e*(2*a*c*d + b*e)*x^2 + 2*a^3*c^3*e^2*x^3 + 6*a^2*b*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTanh[c*x] + 3*a^2*b*(3*c^2*d^2 + 3*c*d*e + e^2)*Log[1 - c*x] + 3*a^2*b*(3*c^2*d^2 - 3*c*d*e + e^2)*Log[1 + c*x] + 18*a*b^2*c*d*e*(2*c*x*ArcTanh[c*x] + (-1 + c^2*x^2)*ArcTanh[c*x]^2 + Log[1 - c^2*x^2]) - 6*b^3*c*d*e*(ArcTanh[c*x]*((3 - 3*c*x)*ArcTanh[c*x] + (1 - c^2*x^2)*ArcTanh[c*x]^2 + 6*Log[1 + E^(-2*ArcTanh[c*x])]) - 3*PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 6*a*b^2*e^2*(c*x + (-1 + c^3*x^3)*ArcTanh[c*x]^2 + ArcTanh[c*x]*(-1 + c^2*x^2 - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 18*a*b^2*c^2*d^2*(ArcTanh[c*x]*((-1 + c*x)*ArcTanh[c*x] - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 6*b^3*c^2*d^2*(ArcTanh[c*x]^2*((-1 + c*x)*ArcTanh[c*x] - 3*Log[1 + E^(-2*ArcTanh[c*x])]) + 3*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + (3*PolyLog[3, -E^(-2*ArcTanh[c*x])])/2) + b^3*e^2*(6*c*x*ArcTanh[c*x] - 3*ArcTanh[c*x]^2 + 3*c^2*x^2*ArcTanh[c*x]^2 - 2*ArcTanh[c*x]^3 + 2*c^3*x^3*ArcTanh[c*x]^3 - 6*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 3*Log[1 - c^2*x^2] + 6*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 3*PolyLog[3, -E^(-2*ArcTanh[c*x])]))/(6*c^3)","A",0
17,1,331,244,0.7457554,"\int (d+e x) \left(a+b \tanh ^{-1}(c x)\right)^3 \, dx","Integrate[(d + e*x)*(a + b*ArcTanh[c*x])^3,x]","\frac{2 a^3 c^2 e x^2+6 a^2 b c^2 x \tanh ^{-1}(c x) (2 d+e x)+2 a^2 c x (2 a c d+3 b e)+3 a^2 b (2 c d+e) \log (1-c x)+3 a^2 b (2 c d-e) \log (c x+1)+6 a b^2 e \left(\log \left(1-c^2 x^2\right)+\left(c^2 x^2-1\right) \tanh ^{-1}(c x)^2+2 c x \tanh ^{-1}(c x)\right)+12 a b^2 c d \left(\text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+\tanh ^{-1}(c x) \left((c x-1) \tanh ^{-1}(c x)-2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)\right)-2 b^3 e \left(\tanh ^{-1}(c x) \left(\left(1-c^2 x^2\right) \tanh ^{-1}(c x)^2+(3-3 c x) \tanh ^{-1}(c x)+6 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)-3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)\right)+4 b^3 c d \left(3 \tanh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+\frac{3}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)+\tanh ^{-1}(c x)^2 \left((c x-1) \tanh ^{-1}(c x)-3 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)\right)}{4 c^2}","-\frac{3 b^2 e \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^2}-\frac{3 b^2 d \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c}-\frac{\left(\frac{e^2}{c^2}+d^2\right) \left(a+b \tanh ^{-1}(c x)\right)^3}{2 e}+\frac{3 b e \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^2}+\frac{(d+e x)^2 \left(a+b \tanh ^{-1}(c x)\right)^3}{2 e}+\frac{d \left(a+b \tanh ^{-1}(c x)\right)^3}{c}-\frac{3 b d \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c}+\frac{3 b e x \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c}-\frac{3 b^3 e \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{2 c^2}+\frac{3 b^3 d \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{2 c}",1,"(2*a^2*c*(2*a*c*d + 3*b*e)*x + 2*a^3*c^2*e*x^2 + 6*a^2*b*c^2*x*(2*d + e*x)*ArcTanh[c*x] + 3*a^2*b*(2*c*d + e)*Log[1 - c*x] + 3*a^2*b*(2*c*d - e)*Log[1 + c*x] + 6*a*b^2*e*(2*c*x*ArcTanh[c*x] + (-1 + c^2*x^2)*ArcTanh[c*x]^2 + Log[1 - c^2*x^2]) - 2*b^3*e*(ArcTanh[c*x]*((3 - 3*c*x)*ArcTanh[c*x] + (1 - c^2*x^2)*ArcTanh[c*x]^2 + 6*Log[1 + E^(-2*ArcTanh[c*x])]) - 3*PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 12*a*b^2*c*d*(ArcTanh[c*x]*((-1 + c*x)*ArcTanh[c*x] - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 4*b^3*c*d*(ArcTanh[c*x]^2*((-1 + c*x)*ArcTanh[c*x] - 3*Log[1 + E^(-2*ArcTanh[c*x])]) + 3*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + (3*PolyLog[3, -E^(-2*ArcTanh[c*x])])/2))/(4*c^2)","A",0
18,0,0,272,98.0235631,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{d+e x} \, dx","Integrate[(a + b*ArcTanh[c*x])^3/(d + e*x),x]","\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{d+e x} \, dx","-\frac{3 b^2 \left(a+b \tanh ^{-1}(c x)\right) \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 e}+\frac{3 b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{2 e}-\frac{3 b \left(a+b \tanh ^{-1}(c x)\right)^2 \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 e}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^3 \log \left(\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right)}{e}+\frac{3 b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{2 e}-\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^3}{e}-\frac{3 b^3 \text{Li}_4\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{4 e}+\frac{3 b^3 \text{Li}_4\left(1-\frac{2}{c x+1}\right)}{4 e}",1,"Integrate[(a + b*ArcTanh[c*x])^3/(d + e*x), x]","F",-1
19,1,813,517,13.851236,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{(d+e x)^2} \, dx","Integrate[(a + b*ArcTanh[c*x])^3/(d + e*x)^2,x]","-\frac{a^3}{e (d+e x)}-\frac{3 b \tanh ^{-1}(c x) a^2}{e (d+e x)}-\frac{3 b c \log (1-c x) a^2}{2 e (c d+e)}+\frac{3 b c \log (c x+1) a^2}{2 c d e-2 e^2}-\frac{3 b c \log (d+e x) a^2}{c^2 d^2-e^2}+\frac{3 b^2 \left(-\frac{e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)} \tanh ^{-1}(c x)^2}{\sqrt{1-\frac{c^2 d^2}{e^2}} e}+\frac{x \tanh ^{-1}(c x)^2}{d+e x}+\frac{c d \left(i \pi  \log \left(1+e^{2 \tanh ^{-1}(c x)}\right)-2 \tanh ^{-1}(c x) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-i \pi  \left(\tanh ^{-1}(c x)-\frac{1}{2} \log \left(1-c^2 x^2\right)\right)-2 \tanh ^{-1}\left(\frac{c d}{e}\right) \left(\tanh ^{-1}(c x)+\log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-\log \left(i \sinh \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)\right)\right)+\text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)\right)}{c^2 d^2-e^2}\right) a}{d}+\frac{b^3 \left(\frac{x \tanh ^{-1}(c x)^3}{d+e x}+\frac{3 \left(2 \sqrt{1-\frac{c^2 d^2}{e^2}} e e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)} \tanh ^{-1}(c x)^3+3 c d \tanh ^{-1}(c x)^3-e \tanh ^{-1}(c x)^3-3 c d \log \left(1-e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)^2-3 c d \log \left(1+e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)^2-3 c d \log \left(\frac{1}{2} e^{-\tanh ^{-1}(c x)} \left(e \left(-1+e^{2 \tanh ^{-1}(c x)}\right)+c d \left(1+e^{2 \tanh ^{-1}(c x)}\right)\right)\right) \tanh ^{-1}(c x)^2+3 c d \log \left(\frac{c (d+e x)}{\sqrt{1-c^2 x^2}}\right) \tanh ^{-1}(c x)^2+3 i c d \pi  \log \left(\frac{1}{2} \left(e^{-\tanh ^{-1}(c x)}+e^{\tanh ^{-1}(c x)}\right)\right) \tanh ^{-1}(c x)-6 c d \tanh ^{-1}\left(\frac{c d}{e}\right) \log \left(\frac{1}{2} i e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)-\tanh ^{-1}(c x)} \left(-1+e^{2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)\right) \tanh ^{-1}(c x)+\frac{3}{2} i c d \pi  \log \left(1-c^2 x^2\right) \tanh ^{-1}(c x)+6 c d \tanh ^{-1}\left(\frac{c d}{e}\right) \log \left(i \sinh \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)\right) \tanh ^{-1}(c x)-6 c d \text{Li}_2\left(-e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)-6 c d \text{Li}_2\left(e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)+6 c d \text{Li}_3\left(-e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right)+6 c d \text{Li}_3\left(e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right)\right)}{3 c^2 d^2-3 e^2}\right)}{d}","-\frac{3 b^2 c \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^2 d^2-e^2}+\frac{3 b^2 c \left(a+b \tanh ^{-1}(c x)\right) \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{c^2 d^2-e^2}+\frac{3 b^2 c \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{2 e (c d+e)}+\frac{3 b^2 c \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{2 e (c d-e)}+\frac{3 b c \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^2 d^2-e^2}-\frac{3 b c \left(a+b \tanh ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right)}{c^2 d^2-e^2}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{e (d+e x)}+\frac{3 b c \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{2 e (c d+e)}-\frac{3 b c \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{2 e (c d-e)}-\frac{3 b^3 c \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 \left(c^2 d^2-e^2\right)}+\frac{3 b^3 c \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 \left(c^2 d^2-e^2\right)}-\frac{3 b^3 c \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{4 e (c d+e)}+\frac{3 b^3 c \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{4 e (c d-e)}",1,"-(a^3/(e*(d + e*x))) - (3*a^2*b*ArcTanh[c*x])/(e*(d + e*x)) - (3*a^2*b*c*Log[1 - c*x])/(2*e*(c*d + e)) + (3*a^2*b*c*Log[1 + c*x])/(2*c*d*e - 2*e^2) - (3*a^2*b*c*Log[d + e*x])/(c^2*d^2 - e^2) + (3*a*b^2*(-(ArcTanh[c*x]^2/(Sqrt[1 - (c^2*d^2)/e^2]*e*E^ArcTanh[(c*d)/e])) + (x*ArcTanh[c*x]^2)/(d + e*x) + (c*d*(I*Pi*Log[1 + E^(2*ArcTanh[c*x])] - 2*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - I*Pi*(ArcTanh[c*x] - Log[1 - c^2*x^2]/2) - 2*ArcTanh[(c*d)/e]*(ArcTanh[c*x] + Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]]) + PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/(c^2*d^2 - e^2)))/d + (b^3*((x*ArcTanh[c*x]^3)/(d + e*x) + (3*(3*c*d*ArcTanh[c*x]^3 - e*ArcTanh[c*x]^3 + (2*Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^3)/E^ArcTanh[(c*d)/e] + (3*I)*c*d*Pi*ArcTanh[c*x]*Log[(E^(-ArcTanh[c*x]) + E^ArcTanh[c*x])/2] - 3*c*d*ArcTanh[c*x]^2*Log[1 - E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 3*c*d*ArcTanh[c*x]^2*Log[1 + E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 6*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x]*Log[(I/2)*E^(-ArcTanh[(c*d)/e] - ArcTanh[c*x])*(-1 + E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c*x])))] - 3*c*d*ArcTanh[c*x]^2*Log[(e*(-1 + E^(2*ArcTanh[c*x])) + c*d*(1 + E^(2*ArcTanh[c*x])))/(2*E^ArcTanh[c*x])] + 3*c*d*ArcTanh[c*x]^2*Log[(c*(d + e*x))/Sqrt[1 - c^2*x^2]] + ((3*I)/2)*c*d*Pi*ArcTanh[c*x]*Log[1 - c^2*x^2] + 6*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - 6*c*d*ArcTanh[c*x]*PolyLog[2, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 6*c*d*ArcTanh[c*x]*PolyLog[2, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 6*c*d*PolyLog[3, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 6*c*d*PolyLog[3, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])]))/(3*c^2*d^2 - 3*e^2)))/d","C",0
20,0,0,953,85.8937199,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{(d+e x)^3} \, dx","Integrate[(a + b*ArcTanh[c*x])^3/(d + e*x)^3,x]","\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{(d+e x)^3} \, dx","-\frac{3 c^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right) b^3}{4 (c d-e) (c d+e)^2}-\frac{3 c^2 \text{Li}_2\left(1-\frac{2}{c x+1}\right) b^3}{4 (c d-e)^2 (c d+e)}+\frac{3 c^2 e \text{Li}_2\left(1-\frac{2}{c x+1}\right) b^3}{2 (c d-e)^2 (c d+e)^2}-\frac{3 c^2 e \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right) b^3}{2 (c d-e)^2 (c d+e)^2}-\frac{3 c^2 \text{Li}_3\left(1-\frac{2}{1-c x}\right) b^3}{8 e (c d+e)^2}+\frac{3 c^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right) b^3}{8 (c d-e)^2 e}-\frac{3 c^3 d \text{Li}_3\left(1-\frac{2}{c x+1}\right) b^3}{2 (c d-e)^2 (c d+e)^2}+\frac{3 c^3 d \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right) b^3}{2 (c d-e)^2 (c d+e)^2}-\frac{3 c^2 \left(a+b \tanh ^{-1}(c x)\right) \log \left(\frac{2}{1-c x}\right) b^2}{2 (c d-e) (c d+e)^2}+\frac{3 c^2 \left(a+b \tanh ^{-1}(c x)\right) \log \left(\frac{2}{c x+1}\right) b^2}{2 (c d-e)^2 (c d+e)}-\frac{3 c^2 e \left(a+b \tanh ^{-1}(c x)\right) \log \left(\frac{2}{c x+1}\right) b^2}{(c d-e)^2 (c d+e)^2}+\frac{3 c^2 e \left(a+b \tanh ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right) b^2}{(c d-e)^2 (c d+e)^2}+\frac{3 c^2 \left(a+b \tanh ^{-1}(c x)\right) \text{Li}_2\left(1-\frac{2}{1-c x}\right) b^2}{4 e (c d+e)^2}+\frac{3 c^2 \left(a+b \tanh ^{-1}(c x)\right) \text{Li}_2\left(1-\frac{2}{c x+1}\right) b^2}{4 (c d-e)^2 e}-\frac{3 c^3 d \left(a+b \tanh ^{-1}(c x)\right) \text{Li}_2\left(1-\frac{2}{c x+1}\right) b^2}{(c d-e)^2 (c d+e)^2}+\frac{3 c^3 d \left(a+b \tanh ^{-1}(c x)\right) \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right) b^2}{(c d-e)^2 (c d+e)^2}+\frac{3 c \left(a+b \tanh ^{-1}(c x)\right)^2 b}{2 \left(c^2 d^2-e^2\right) (d+e x)}+\frac{3 c^2 \left(a+b \tanh ^{-1}(c x)\right)^2 \log \left(\frac{2}{1-c x}\right) b}{4 e (c d+e)^2}-\frac{3 c^2 \left(a+b \tanh ^{-1}(c x)\right)^2 \log \left(\frac{2}{c x+1}\right) b}{4 (c d-e)^2 e}+\frac{3 c^3 d \left(a+b \tanh ^{-1}(c x)\right)^2 \log \left(\frac{2}{c x+1}\right) b}{(c d-e)^2 (c d+e)^2}-\frac{3 c^3 d \left(a+b \tanh ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right) b}{(c d-e)^2 (c d+e)^2}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{2 e (d+e x)^2}",1,"Integrate[(a + b*ArcTanh[c*x])^3/(d + e*x)^3, x]","F",-1
21,1,240,67,0.2776503,"\int \frac{a+b \tanh ^{-1}(c x)}{1+2 c x} \, dx","Integrate[(a + b*ArcTanh[c*x])/(1 + 2*c*x),x]","\frac{a \log (2 c x+1)-\frac{1}{2} i b \left(-\log \left(\frac{2}{\sqrt{1-c^2 x^2}}\right) \left(\pi -2 i \tanh ^{-1}(c x)\right)-i \text{Li}_2\left(-e^{2 \tanh ^{-1}(c x)}\right)-i \text{Li}_2\left(e^{-2 \left(\tanh ^{-1}(c x)+\tanh ^{-1}\left(\frac{1}{2}\right)\right)}\right)-\frac{1}{4} i \left(\pi -2 i \tanh ^{-1}(c x)\right)^2+i \left(\tanh ^{-1}(c x)+\tanh ^{-1}\left(\frac{1}{2}\right)\right)^2+\left(\pi -2 i \tanh ^{-1}(c x)\right) \log \left(e^{2 \tanh ^{-1}(c x)}+1\right)+2 i \left(\tanh ^{-1}(c x)+\tanh ^{-1}\left(\frac{1}{2}\right)\right) \log \left(1-e^{-2 \left(\tanh ^{-1}(c x)+\tanh ^{-1}\left(\frac{1}{2}\right)\right)}\right)-2 i \left(\tanh ^{-1}(c x)+\tanh ^{-1}\left(\frac{1}{2}\right)\right) \log \left(2 i \sinh \left(\tanh ^{-1}(c x)+\tanh ^{-1}\left(\frac{1}{2}\right)\right)\right)\right)+b \tanh ^{-1}(c x) \left(\frac{1}{2} \log \left(1-c^2 x^2\right)+\log \left(i \sinh \left(\tanh ^{-1}(c x)+\tanh ^{-1}\left(\frac{1}{2}\right)\right)\right)\right)}{2 c}","\frac{\left(a-b \tanh ^{-1}\left(\frac{1}{2}\right)\right) \log \left(-\frac{2 c x+1}{2 d}\right)}{2 c}-\frac{b \text{Li}_2(-2 c x-1)}{4 c}+\frac{b \text{Li}_2\left(\frac{1}{3} (2 c x+1)\right)}{4 c}",1,"(a*Log[1 + 2*c*x] + b*ArcTanh[c*x]*(Log[1 - c^2*x^2]/2 + Log[I*Sinh[ArcTanh[1/2] + ArcTanh[c*x]]]) - (I/2)*b*((-1/4*I)*(Pi - (2*I)*ArcTanh[c*x])^2 + I*(ArcTanh[1/2] + ArcTanh[c*x])^2 + (Pi - (2*I)*ArcTanh[c*x])*Log[1 + E^(2*ArcTanh[c*x])] + (2*I)*(ArcTanh[1/2] + ArcTanh[c*x])*Log[1 - E^(-2*(ArcTanh[1/2] + ArcTanh[c*x]))] - (Pi - (2*I)*ArcTanh[c*x])*Log[2/Sqrt[1 - c^2*x^2]] - (2*I)*(ArcTanh[1/2] + ArcTanh[c*x])*Log[(2*I)*Sinh[ArcTanh[1/2] + ArcTanh[c*x]]] - I*PolyLog[2, -E^(2*ArcTanh[c*x])] - I*PolyLog[2, E^(-2*(ArcTanh[1/2] + ArcTanh[c*x]))]))/(2*c)","C",0
22,1,272,88,0.0999542,"\int \frac{\tanh ^{-1}(x)}{1-\sqrt{2} x} \, dx","Integrate[ArcTanh[x]/(1 - Sqrt[2]*x),x]","\frac{4 \text{Li}_2\left(e^{2 \tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right)-2 \tanh ^{-1}(x)}\right)+4 \text{Li}_2\left(-e^{2 \tanh ^{-1}(x)}\right)-4 i \pi  \log \left(\frac{2}{\sqrt{1-x^2}}\right)-8 \log \left(\frac{2}{\sqrt{1-x^2}}\right) \tanh ^{-1}(x)-4 \log \left(1-x^2\right) \tanh ^{-1}(x)-8 \tanh ^{-1}(x)^2+8 \tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right) \tanh ^{-1}(x)-4 i \pi  \tanh ^{-1}(x)-8 \tanh ^{-1}(x) \log \left(1-e^{2 \tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right)-2 \tanh ^{-1}(x)}\right)+8 \tanh ^{-1}(x) \log \left(e^{2 \tanh ^{-1}(x)}+1\right)+8 \tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right) \log \left(1-e^{2 \tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right)-2 \tanh ^{-1}(x)}\right)+4 i \pi  \log \left(e^{2 \tanh ^{-1}(x)}+1\right)-8 \tanh ^{-1}(x) \log \left(-i \sinh \left(\tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right)-\tanh ^{-1}(x)\right)\right)+8 \tanh ^{-1}(x) \log \left(-2 i \sinh \left(\tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right)-\tanh ^{-1}(x)\right)\right)-8 \tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right) \log \left(-2 i \sinh \left(\tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right)-\tanh ^{-1}(x)\right)\right)+\pi ^2-4 \tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right)^2}{8 \sqrt{2}}","-\frac{\text{Li}_2\left(-\frac{\sqrt{2}-2 x}{2-\sqrt{2}}\right)}{2 \sqrt{2}}+\frac{\text{Li}_2\left(\frac{\sqrt{2}-2 x}{2+\sqrt{2}}\right)}{2 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right) \log \left(1-\sqrt{2} x\right)}{\sqrt{2}}",1,"(Pi^2 - 4*ArcTanh[1/Sqrt[2]]^2 - (4*I)*Pi*ArcTanh[x] + 8*ArcTanh[1/Sqrt[2]]*ArcTanh[x] - 8*ArcTanh[x]^2 + 8*ArcTanh[1/Sqrt[2]]*Log[1 - E^(2*ArcTanh[1/Sqrt[2]] - 2*ArcTanh[x])] - 8*ArcTanh[x]*Log[1 - E^(2*ArcTanh[1/Sqrt[2]] - 2*ArcTanh[x])] + (4*I)*Pi*Log[1 + E^(2*ArcTanh[x])] + 8*ArcTanh[x]*Log[1 + E^(2*ArcTanh[x])] - (4*I)*Pi*Log[2/Sqrt[1 - x^2]] - 8*ArcTanh[x]*Log[2/Sqrt[1 - x^2]] - 4*ArcTanh[x]*Log[1 - x^2] - 8*ArcTanh[x]*Log[(-I)*Sinh[ArcTanh[1/Sqrt[2]] - ArcTanh[x]]] - 8*ArcTanh[1/Sqrt[2]]*Log[(-2*I)*Sinh[ArcTanh[1/Sqrt[2]] - ArcTanh[x]]] + 8*ArcTanh[x]*Log[(-2*I)*Sinh[ArcTanh[1/Sqrt[2]] - ArcTanh[x]]] + 4*PolyLog[2, E^(2*ArcTanh[1/Sqrt[2]] - 2*ArcTanh[x])] + 4*PolyLog[2, -E^(2*ArcTanh[x])])/(8*Sqrt[2])","C",0
23,1,254,182,0.2768016,"\int (d+e x)^3 \left(a+b \tanh ^{-1}\left(c x^2\right)\right) \, dx","Integrate[(d + e*x)^3*(a + b*ArcTanh[c*x^2]),x]","\frac{1}{8} \left(\frac{2 e x^2 \left(6 a c d^2+b e^2\right)}{c}+\frac{8 d x \left(a c d^2+2 b e^2\right)}{c}+8 a d e^2 x^3+2 a e^3 x^4+\frac{8 b d \left(c d^2-e^2\right) \tan ^{-1}\left(\sqrt{c} x\right)}{c^{3/2}}+\frac{6 b d^2 e \log \left(1-c^2 x^4\right)}{c}-\frac{b e^3 \log \left(c x^2+1\right)}{c^2}+\frac{b \left(4 c^{3/2} d^3+4 \sqrt{c} d e^2+e^3\right) \log \left(1-\sqrt{c} x\right)}{c^2}+\frac{b \left(-4 c^2 d^3-4 c d e^2+\sqrt{c} e^3\right) \log \left(\sqrt{c} x+1\right)}{c^{5/2}}+2 b x \tanh ^{-1}\left(c x^2\right) \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)\right)","\frac{(d+e x)^4 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{4 e}+\frac{b d \left(c d^2-e^2\right) \tan ^{-1}\left(\sqrt{c} x\right)}{c^{3/2}}-\frac{b d \left(c d^2+e^2\right) \tanh ^{-1}\left(\sqrt{c} x\right)}{c^{3/2}}+\frac{b \left(c^2 d^4+6 c d^2 e^2+e^4\right) \log \left(1-c x^2\right)}{8 c^2 e}-\frac{b \left(c^2 d^4-6 c d^2 e^2+e^4\right) \log \left(c x^2+1\right)}{8 c^2 e}+\frac{2 b d e^2 x}{c}+\frac{b e^3 x^2}{4 c}",1,"((8*d*(a*c*d^2 + 2*b*e^2)*x)/c + (2*e*(6*a*c*d^2 + b*e^2)*x^2)/c + 8*a*d*e^2*x^3 + 2*a*e^3*x^4 + (8*b*d*(c*d^2 - e^2)*ArcTan[Sqrt[c]*x])/c^(3/2) + 2*b*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcTanh[c*x^2] + (b*(4*c^(3/2)*d^3 + 4*Sqrt[c]*d*e^2 + e^3)*Log[1 - Sqrt[c]*x])/c^2 + (b*(-4*c^2*d^3 - 4*c*d*e^2 + Sqrt[c]*e^3)*Log[1 + Sqrt[c]*x])/c^(5/2) - (b*e^3*Log[1 + c*x^2])/c^2 + (6*b*d^2*e*Log[1 - c^2*x^4])/c)/8","A",1
24,1,170,158,0.1709484,"\int (d+e x)^2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right) \, dx","Integrate[(d + e*x)^2*(a + b*ArcTanh[c*x^2]),x]","\frac{1}{6} \left(6 a d^2 x+6 a d e x^2+2 a e^2 x^3+\frac{b \left(3 c d^2+e^2\right) \log \left(1-\sqrt{c} x\right)}{c^{3/2}}-\frac{b \left(3 c d^2+e^2\right) \log \left(\sqrt{c} x+1\right)}{c^{3/2}}+\frac{2 b \left(3 c d^2-e^2\right) \tan ^{-1}\left(\sqrt{c} x\right)}{c^{3/2}}+\frac{3 b d e \log \left(1-c^2 x^4\right)}{c}+2 b x \tanh ^{-1}\left(c x^2\right) \left(3 d^2+3 d e x+e^2 x^2\right)+\frac{4 b e^2 x}{c}\right)","\frac{(d+e x)^3 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{3 e}+\frac{b \left(3 c d^2-e^2\right) \tan ^{-1}\left(\sqrt{c} x\right)}{3 c^{3/2}}-\frac{b \left(3 c d^2+e^2\right) \tanh ^{-1}\left(\sqrt{c} x\right)}{3 c^{3/2}}+\frac{b d \left(c d^2+3 e^2\right) \log \left(1-c x^2\right)}{6 c e}-\frac{b d \left(c d^2-3 e^2\right) \log \left(c x^2+1\right)}{6 c e}+\frac{2 b e^2 x}{3 c}",1,"(6*a*d^2*x + (4*b*e^2*x)/c + 6*a*d*e*x^2 + 2*a*e^2*x^3 + (2*b*(3*c*d^2 - e^2)*ArcTan[Sqrt[c]*x])/c^(3/2) + 2*b*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTanh[c*x^2] + (b*(3*c*d^2 + e^2)*Log[1 - Sqrt[c]*x])/c^(3/2) - (b*(3*c*d^2 + e^2)*Log[1 + Sqrt[c]*x])/c^(3/2) + (3*b*d*e*Log[1 - c^2*x^4])/c)/6","A",1
25,1,104,117,0.0550437,"\int (d+e x) \left(a+b \tanh ^{-1}\left(c x^2\right)\right) \, dx","Integrate[(d + e*x)*(a + b*ArcTanh[c*x^2]),x]","a d x+\frac{1}{2} a e x^2+\frac{b e \log \left(1-c^2 x^4\right)}{4 c}+b d x \tanh ^{-1}\left(c x^2\right)+\frac{b d \left(\log \left(1-\sqrt{c} x\right)-\log \left(\sqrt{c} x+1\right)+2 \tan ^{-1}\left(\sqrt{c} x\right)\right)}{2 \sqrt{c}}+\frac{1}{2} b e x^2 \tanh ^{-1}\left(c x^2\right)","\frac{(d+e x)^2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{2 e}+\frac{b \left(c d^2+e^2\right) \log \left(1-c x^2\right)}{4 c e}-\frac{b \left(c d^2-e^2\right) \log \left(c x^2+1\right)}{4 c e}+\frac{b d \tan ^{-1}\left(\sqrt{c} x\right)}{\sqrt{c}}-\frac{b d \tanh ^{-1}\left(\sqrt{c} x\right)}{\sqrt{c}}",1,"a*d*x + (a*e*x^2)/2 + b*d*x*ArcTanh[c*x^2] + (b*e*x^2*ArcTanh[c*x^2])/2 + (b*d*(2*ArcTan[Sqrt[c]*x] + Log[1 - Sqrt[c]*x] - Log[1 + Sqrt[c]*x]))/(2*Sqrt[c]) + (b*e*Log[1 - c^2*x^4])/(4*c)","A",1
26,1,285,325,17.3651364,"\int \frac{a+b \tanh ^{-1}\left(c x^2\right)}{d+e x} \, dx","Integrate[(a + b*ArcTanh[c*x^2])/(d + e*x),x]","\frac{a \log (d+e x)}{e}+\frac{b \left(\text{Li}_2\left(\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-e}\right)-\text{Li}_2\left(\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-i e}\right)-\text{Li}_2\left(\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+i e}\right)+\text{Li}_2\left(\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+e}\right)+2 \tanh ^{-1}\left(c x^2\right) \log (d+e x)-\log (d+e x) \log \left(\frac{e \left(-\sqrt{c} x+i\right)}{\sqrt{c} d+i e}\right)-\log (d+e x) \log \left(-\frac{e \left(\sqrt{c} x+i\right)}{\sqrt{c} d-i e}\right)+\log (d+e x) \log \left(-\frac{e \left(\sqrt{c} x+1\right)}{\sqrt{c} d-e}\right)+\log (d+e x) \log \left(\frac{e-\sqrt{c} e x}{\sqrt{c} d+e}\right)\right)}{2 e}","\frac{\log (d+e x) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{e}-\frac{b \text{Li}_2\left(\frac{\sqrt{-c} (d+e x)}{\sqrt{-c} d-e}\right)}{2 e}+\frac{b \text{Li}_2\left(\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-e}\right)}{2 e}-\frac{b \text{Li}_2\left(\frac{\sqrt{-c} (d+e x)}{\sqrt{-c} d+e}\right)}{2 e}+\frac{b \text{Li}_2\left(\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+e}\right)}{2 e}-\frac{b \log (d+e x) \log \left(\frac{e \left(1-\sqrt{-c} x\right)}{\sqrt{-c} d+e}\right)}{2 e}-\frac{b \log (d+e x) \log \left(-\frac{e \left(\sqrt{-c} x+1\right)}{\sqrt{-c} d-e}\right)}{2 e}+\frac{b \log (d+e x) \log \left(\frac{e \left(1-\sqrt{c} x\right)}{\sqrt{c} d+e}\right)}{2 e}+\frac{b \log (d+e x) \log \left(-\frac{e \left(\sqrt{c} x+1\right)}{\sqrt{c} d-e}\right)}{2 e}",1,"(a*Log[d + e*x])/e + (b*(2*ArcTanh[c*x^2]*Log[d + e*x] - Log[(e*(I - Sqrt[c]*x))/(Sqrt[c]*d + I*e)]*Log[d + e*x] - Log[-((e*(I + Sqrt[c]*x))/(Sqrt[c]*d - I*e))]*Log[d + e*x] + Log[-((e*(1 + Sqrt[c]*x))/(Sqrt[c]*d - e))]*Log[d + e*x] + Log[d + e*x]*Log[(e - Sqrt[c]*e*x)/(Sqrt[c]*d + e)] + PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - e)] - PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - I*e)] - PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + I*e)] + PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + e)]))/(2*e)","C",1
27,1,261,166,0.3636612,"\int \frac{a+b \tanh ^{-1}\left(c x^2\right)}{(d+e x)^2} \, dx","Integrate[(a + b*ArcTanh[c*x^2])/(d + e*x)^2,x]","\frac{1}{2} \left(-\frac{2 a}{e (d+e x)}-\frac{b c d e \log \left(1-c^2 x^4\right)}{c^2 d^4-e^4}+\frac{4 b c d e \log (d+e x)}{c^2 d^4-e^4}+\frac{b c^2 d^3 \log \left(c x^2+1\right)}{c^2 d^4 e-e^5}+\frac{b \sqrt{c} \left(c^{3/2} d^3-c d^2 e-e^3\right) \log \left(1-\sqrt{c} x\right)}{e^5-c^2 d^4 e}+\frac{b \sqrt{c} \left(c^{3/2} d^3+c d^2 e+e^3\right) \log \left(\sqrt{c} x+1\right)}{e^5-c^2 d^4 e}+\frac{2 b \sqrt{c} \tan ^{-1}\left(\sqrt{c} x\right)}{c d^2+e^2}-\frac{2 b \tanh ^{-1}\left(c x^2\right)}{e (d+e x)}\right)","-\frac{a+b \tanh ^{-1}\left(c x^2\right)}{e (d+e x)}+\frac{2 b c d e \log (d+e x)}{c^2 d^4-e^4}-\frac{b c d \log \left(1-c x^2\right)}{2 e \left(c d^2-e^2\right)}+\frac{b c d \log \left(c x^2+1\right)}{2 e \left(c d^2+e^2\right)}+\frac{b \sqrt{c} \tan ^{-1}\left(\sqrt{c} x\right)}{c d^2+e^2}-\frac{b \sqrt{c} \tanh ^{-1}\left(\sqrt{c} x\right)}{c d^2-e^2}",1,"((-2*a)/(e*(d + e*x)) + (2*b*Sqrt[c]*ArcTan[Sqrt[c]*x])/(c*d^2 + e^2) - (2*b*ArcTanh[c*x^2])/(e*(d + e*x)) + (b*Sqrt[c]*(c^(3/2)*d^3 - c*d^2*e - e^3)*Log[1 - Sqrt[c]*x])/(-(c^2*d^4*e) + e^5) + (b*Sqrt[c]*(c^(3/2)*d^3 + c*d^2*e + e^3)*Log[1 + Sqrt[c]*x])/(-(c^2*d^4*e) + e^5) + (4*b*c*d*e*Log[d + e*x])/(c^2*d^4 - e^4) + (b*c^2*d^3*Log[1 + c*x^2])/(c^2*d^4*e - e^5) - (b*c*d*e*Log[1 - c^2*x^4])/(c^2*d^4 - e^4))/2","A",1
28,1,379,226,0.6813617,"\int \frac{a+b \tanh ^{-1}\left(c x^2\right)}{(d+e x)^3} \, dx","Integrate[(a + b*ArcTanh[c*x^2])/(d + e*x)^3,x]","\frac{1}{4} \left(-\frac{2 a}{e (d+e x)^2}+\frac{4 b c^{3/2} d \tan ^{-1}\left(\sqrt{c} x\right)}{\left(c d^2+e^2\right)^2}-\frac{b c e \left(3 c^2 d^4+e^4\right) \log \left(1-c^2 x^4\right)}{\left(e^4-c^2 d^4\right)^2}-\frac{4 b c d e}{\left(c^2 d^4-e^4\right) (d+e x)}+\frac{4 b c e \left(3 c^2 d^4+e^4\right) \log (d+e x)}{\left(e^4-c^2 d^4\right)^2}+\frac{b c^2 \left(c^2 d^6+3 d^2 e^4\right) \log \left(c x^2+1\right)}{e \left(e^4-c^2 d^4\right)^2}-\frac{b c^{3/2} d \left(c^{5/2} d^5-2 c^2 d^4 e-4 c d^2 e^3+3 \sqrt{c} d e^4-2 e^5\right) \log \left(1-\sqrt{c} x\right)}{e \left(e^4-c^2 d^4\right)^2}-\frac{b c^{3/2} d \left(c^{5/2} d^5+2 c^2 d^4 e+4 c d^2 e^3+3 \sqrt{c} d e^4+2 e^5\right) \log \left(\sqrt{c} x+1\right)}{e \left(e^4-c^2 d^4\right)^2}-\frac{2 b \tanh ^{-1}\left(c x^2\right)}{e (d+e x)^2}\right)","-\frac{a+b \tanh ^{-1}\left(c x^2\right)}{2 e (d+e x)^2}+\frac{b c^{3/2} d \tan ^{-1}\left(\sqrt{c} x\right)}{\left(c d^2+e^2\right)^2}-\frac{b c^{3/2} d \tanh ^{-1}\left(\sqrt{c} x\right)}{\left(c d^2-e^2\right)^2}-\frac{b c d e}{\left(c^2 d^4-e^4\right) (d+e x)}+\frac{b c e \left(3 c^2 d^4+e^4\right) \log (d+e x)}{\left(c^2 d^4-e^4\right)^2}-\frac{b c \left(c d^2+e^2\right) \log \left(1-c x^2\right)}{4 e \left(c d^2-e^2\right)^2}+\frac{b c \left(c d^2-e^2\right) \log \left(c x^2+1\right)}{4 e \left(c d^2+e^2\right)^2}",1,"((-2*a)/(e*(d + e*x)^2) - (4*b*c*d*e)/((c^2*d^4 - e^4)*(d + e*x)) + (4*b*c^(3/2)*d*ArcTan[Sqrt[c]*x])/(c*d^2 + e^2)^2 - (2*b*ArcTanh[c*x^2])/(e*(d + e*x)^2) - (b*c^(3/2)*d*(c^(5/2)*d^5 - 2*c^2*d^4*e - 4*c*d^2*e^3 + 3*Sqrt[c]*d*e^4 - 2*e^5)*Log[1 - Sqrt[c]*x])/(e*(-(c^2*d^4) + e^4)^2) - (b*c^(3/2)*d*(c^(5/2)*d^5 + 2*c^2*d^4*e + 4*c*d^2*e^3 + 3*Sqrt[c]*d*e^4 + 2*e^5)*Log[1 + Sqrt[c]*x])/(e*(-(c^2*d^4) + e^4)^2) + (4*b*c*e*(3*c^2*d^4 + e^4)*Log[d + e*x])/(-(c^2*d^4) + e^4)^2 + (b*c^2*(c^2*d^6 + 3*d^2*e^4)*Log[1 + c*x^2])/(e*(-(c^2*d^4) + e^4)^2) - (b*c*e*(3*c^2*d^4 + e^4)*Log[1 - c^2*x^4])/(-(c^2*d^4) + e^4)^2)/4","A",1
29,1,684,1085,3.0017099,"\int (d+e x) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[(d + e*x)*(a + b*ArcTanh[c*x^2])^2,x]","\frac{2 a^2 c d x^2+a^2 c e x^3+a b e x \left(\log \left(1-c^2 x^4\right)+2 c x^2 \tanh ^{-1}\left(c x^2\right)\right)+4 a b c d x^2 \tanh ^{-1}\left(c x^2\right)+4 a b d \sqrt{c x^2} \left(\tan ^{-1}\left(\sqrt{c x^2}\right)-\tanh ^{-1}\left(\sqrt{c x^2}\right)\right)-b^2 d \sqrt{c x^2} \left(-\text{Li}_2\left(\frac{1}{2} \left(1-\sqrt{c x^2}\right)\right)+\text{Li}_2\left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\sqrt{c x^2}-1\right)\right)+\text{Li}_2\left(\left(-\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{c x^2}-1\right)\right)+\text{Li}_2\left(\frac{1}{2} \left(\sqrt{c x^2}+1\right)\right)-\text{Li}_2\left(\left(\frac{1}{2}-\frac{i}{2}\right) \left(\sqrt{c x^2}+1\right)\right)-\text{Li}_2\left(\left(\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{c x^2}+1\right)\right)+\frac{1}{2} i \text{Li}_2\left(-e^{4 i \tan ^{-1}\left(\sqrt{c x^2}\right)}\right)-\frac{1}{2} \log ^2\left(1-\sqrt{c x^2}\right)+\frac{1}{2} \log ^2\left(\sqrt{c x^2}+1\right)+\log (2) \log \left(1-\sqrt{c x^2}\right)+\log \left(1-\sqrt{c x^2}\right) \log \left(\left(\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{c x^2}-i\right)\right)-\log \left(\frac{1}{2} \left((1+i)-(1-i) \sqrt{c x^2}\right)\right) \log \left(\sqrt{c x^2}+1\right)-\log \left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\sqrt{c x^2}+i\right)\right) \log \left(\sqrt{c x^2}+1\right)-\log (2) \log \left(\sqrt{c x^2}+1\right)+\log \left(1-\sqrt{c x^2}\right) \log \left(\frac{1}{2} \left((1-i) \sqrt{c x^2}+(1+i)\right)\right)+2 i \tan ^{-1}\left(\sqrt{c x^2}\right)^2-2 \sqrt{c x^2} \tanh ^{-1}\left(c x^2\right)^2-2 \tan ^{-1}\left(\sqrt{c x^2}\right) \log \left(1+e^{4 i \tan ^{-1}\left(\sqrt{c x^2}\right)}\right)-2 \log \left(1-\sqrt{c x^2}\right) \tanh ^{-1}\left(c x^2\right)+2 \log \left(\sqrt{c x^2}+1\right) \tanh ^{-1}\left(c x^2\right)-4 \tan ^{-1}\left(\sqrt{c x^2}\right) \tanh ^{-1}\left(c x^2\right)\right)+b^2 e x \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c x^2\right)}\right)+b^2 e x \tanh ^{-1}\left(c x^2\right) \left(\left(c x^2-1\right) \tanh ^{-1}\left(c x^2\right)-2 \log \left(e^{-2 \tanh ^{-1}\left(c x^2\right)}+1\right)\right)}{2 c x}","d x a^2+\frac{2 b d \tan ^{-1}\left(\sqrt{c} x\right) a}{\sqrt{c}}-\frac{2 b d \tanh ^{-1}\left(\sqrt{c} x\right) a}{\sqrt{c}}-b d x \log \left(1-c x^2\right) a+b d x \log \left(c x^2+1\right) a+\frac{i b^2 d \tan ^{-1}\left(\sqrt{c} x\right)^2}{\sqrt{c}}-\frac{b^2 d \tanh ^{-1}\left(\sqrt{c} x\right)^2}{\sqrt{c}}+\frac{1}{2} e x^2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2+\frac{e \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{2 c}+\frac{1}{4} b^2 d x \log ^2\left(1-c x^2\right)+\frac{1}{4} b^2 d x \log ^2\left(c x^2+1\right)+\frac{2 b^2 d \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{1-\sqrt{c} x}\right)}{\sqrt{c}}-\frac{2 b^2 d \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{1-i \sqrt{c} x}\right)}{\sqrt{c}}+\frac{b^2 d \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{(1+i) \left(1-\sqrt{c} x\right)}{1-i \sqrt{c} x}\right)}{\sqrt{c}}+\frac{2 b^2 d \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{i \sqrt{c} x+1}\right)}{\sqrt{c}}-\frac{2 b^2 d \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{\sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{b^2 d \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(-\frac{2 \sqrt{c} \left(1-\sqrt{-c} x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right)}{\sqrt{c}}+\frac{b^2 d \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2 \sqrt{c} \left(\sqrt{-c} x+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right)}{\sqrt{c}}+\frac{b^2 d \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{(1-i) \left(\sqrt{c} x+1\right)}{1-i \sqrt{c} x}\right)}{\sqrt{c}}-\frac{b e \left(a+b \tanh ^{-1}\left(c x^2\right)\right) \log \left(\frac{2}{1-c x^2}\right)}{c}-\frac{b^2 d \tan ^{-1}\left(\sqrt{c} x\right) \log \left(1-c x^2\right)}{\sqrt{c}}+\frac{b^2 d \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(1-c x^2\right)}{\sqrt{c}}+\frac{b^2 d \tan ^{-1}\left(\sqrt{c} x\right) \log \left(c x^2+1\right)}{\sqrt{c}}-\frac{b^2 d \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(c x^2+1\right)}{\sqrt{c}}-\frac{1}{2} b^2 d x \log \left(1-c x^2\right) \log \left(c x^2+1\right)+\frac{b^2 d \text{Li}_2\left(1-\frac{2}{1-\sqrt{c} x}\right)}{\sqrt{c}}+\frac{i b^2 d \text{Li}_2\left(1-\frac{2}{1-i \sqrt{c} x}\right)}{\sqrt{c}}-\frac{i b^2 d \text{Li}_2\left(1-\frac{(1+i) \left(1-\sqrt{c} x\right)}{1-i \sqrt{c} x}\right)}{2 \sqrt{c}}+\frac{i b^2 d \text{Li}_2\left(1-\frac{2}{i \sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{b^2 d \text{Li}_2\left(1-\frac{2}{\sqrt{c} x+1}\right)}{\sqrt{c}}-\frac{b^2 d \text{Li}_2\left(\frac{2 \sqrt{c} \left(1-\sqrt{-c} x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\sqrt{c} x+1\right)}+1\right)}{2 \sqrt{c}}-\frac{b^2 d \text{Li}_2\left(1-\frac{2 \sqrt{c} \left(\sqrt{-c} x+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right)}{2 \sqrt{c}}-\frac{i b^2 d \text{Li}_2\left(1-\frac{(1-i) \left(\sqrt{c} x+1\right)}{1-i \sqrt{c} x}\right)}{2 \sqrt{c}}-\frac{b^2 e \text{Li}_2\left(1-\frac{2}{1-c x^2}\right)}{2 c}",1,"(2*a^2*c*d*x^2 + a^2*c*e*x^3 + 4*a*b*c*d*x^2*ArcTanh[c*x^2] + 4*a*b*d*Sqrt[c*x^2]*(ArcTan[Sqrt[c*x^2]] - ArcTanh[Sqrt[c*x^2]]) + b^2*e*x*ArcTanh[c*x^2]*((-1 + c*x^2)*ArcTanh[c*x^2] - 2*Log[1 + E^(-2*ArcTanh[c*x^2])]) + a*b*e*x*(2*c*x^2*ArcTanh[c*x^2] + Log[1 - c^2*x^4]) + b^2*e*x*PolyLog[2, -E^(-2*ArcTanh[c*x^2])] - b^2*d*Sqrt[c*x^2]*((2*I)*ArcTan[Sqrt[c*x^2]]^2 - 4*ArcTan[Sqrt[c*x^2]]*ArcTanh[c*x^2] - 2*Sqrt[c*x^2]*ArcTanh[c*x^2]^2 - 2*ArcTan[Sqrt[c*x^2]]*Log[1 + E^((4*I)*ArcTan[Sqrt[c*x^2]])] - 2*ArcTanh[c*x^2]*Log[1 - Sqrt[c*x^2]] + Log[2]*Log[1 - Sqrt[c*x^2]] - Log[1 - Sqrt[c*x^2]]^2/2 + Log[1 - Sqrt[c*x^2]]*Log[(1/2 + I/2)*(-I + Sqrt[c*x^2])] + 2*ArcTanh[c*x^2]*Log[1 + Sqrt[c*x^2]] - Log[2]*Log[1 + Sqrt[c*x^2]] - Log[((1 + I) - (1 - I)*Sqrt[c*x^2])/2]*Log[1 + Sqrt[c*x^2]] - Log[(-1/2 - I/2)*(I + Sqrt[c*x^2])]*Log[1 + Sqrt[c*x^2]] + Log[1 + Sqrt[c*x^2]]^2/2 + Log[1 - Sqrt[c*x^2]]*Log[((1 + I) + (1 - I)*Sqrt[c*x^2])/2] + (I/2)*PolyLog[2, -E^((4*I)*ArcTan[Sqrt[c*x^2]])] - PolyLog[2, (1 - Sqrt[c*x^2])/2] + PolyLog[2, (-1/2 - I/2)*(-1 + Sqrt[c*x^2])] + PolyLog[2, (-1/2 + I/2)*(-1 + Sqrt[c*x^2])] + PolyLog[2, (1 + Sqrt[c*x^2])/2] - PolyLog[2, (1/2 - I/2)*(1 + Sqrt[c*x^2])] - PolyLog[2, (1/2 + I/2)*(1 + Sqrt[c*x^2])]))/(2*c*x)","A",0
30,0,0,23,44.3021925,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{d+e x} \, dx","Integrate[(a + b*ArcTanh[c*x^2])^2/(d + e*x),x]","\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{d+e x} \, dx","\text{Int}\left(\frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{d+e x},x\right)",0,"Integrate[(a + b*ArcTanh[c*x^2])^2/(d + e*x), x]","A",-1
31,0,0,23,42.0113145,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{(d+e x)^2} \, dx","Integrate[(a + b*ArcTanh[c*x^2])^2/(d + e*x)^2,x]","\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{(d+e x)^2} \, dx","\text{Int}\left(\frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{(d+e x)^2},x\right)",0,"Integrate[(a + b*ArcTanh[c*x^2])^2/(d + e*x)^2, x]","A",-1
32,1,299,336,0.2795064,"\int (d+e x)^2 \left(a+b \tanh ^{-1}\left(c x^3\right)\right) \, dx","Integrate[(d + e*x)^2*(a + b*ArcTanh[c*x^3]),x]","\frac{12 a c d^2 x+12 a c d e x^2+4 a c e^2 x^3-3 b \sqrt[3]{c} d \left(\sqrt[3]{c} d-e\right) \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)-3 b \sqrt[3]{c} d \left(\sqrt[3]{c} d+e\right) \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)+2 b e^2 \log \left(1-c^2 x^6\right)+4 b c x \tanh ^{-1}\left(c x^3\right) \left(3 d^2+3 d e x+e^2 x^2\right)+6 b \sqrt[3]{c} d \left(\sqrt[3]{c} d+e\right) \log \left(1-\sqrt[3]{c} x\right)+6 b \sqrt[3]{c} d \left(\sqrt[3]{c} d-e\right) \log \left(\sqrt[3]{c} x+1\right)+6 \sqrt{3} b \sqrt[3]{c} d \left(\sqrt[3]{c} d+e\right) \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x-1}{\sqrt{3}}\right)-6 \sqrt{3} b \sqrt[3]{c} d \left(\sqrt[3]{c} d-e\right) \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x+1}{\sqrt{3}}\right)}{12 c}","\frac{(d+e x)^3 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)}{3 e}+\frac{b d^2 \log \left(1-c^{2/3} x^2\right)}{2 \sqrt[3]{c}}+\frac{\sqrt{3} b d^2 \tan ^{-1}\left(\frac{2 c^{2/3} x^2+1}{\sqrt{3}}\right)}{2 \sqrt[3]{c}}-\frac{b d^2 \log \left(c^{4/3} x^4+c^{2/3} x^2+1\right)}{4 \sqrt[3]{c}}+\frac{b d e \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)}{4 c^{2/3}}-\frac{b d e \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)}{4 c^{2/3}}-\frac{\sqrt{3} b d e \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{c} x}{\sqrt{3}}\right)}{2 c^{2/3}}+\frac{\sqrt{3} b d e \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{2 c^{2/3}}-\frac{b d e \tanh ^{-1}\left(\sqrt[3]{c} x\right)}{c^{2/3}}+\frac{b \left(c d^3+e^3\right) \log \left(1-c x^3\right)}{6 c e}-\frac{b \left(c d^3-e^3\right) \log \left(c x^3+1\right)}{6 c e}",1,"(12*a*c*d^2*x + 12*a*c*d*e*x^2 + 4*a*c*e^2*x^3 + 6*Sqrt[3]*b*c^(1/3)*d*(c^(1/3)*d + e)*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]] - 6*Sqrt[3]*b*c^(1/3)*d*(c^(1/3)*d - e)*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]] + 4*b*c*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTanh[c*x^3] + 6*b*c^(1/3)*d*(c^(1/3)*d + e)*Log[1 - c^(1/3)*x] + 6*b*c^(1/3)*d*(c^(1/3)*d - e)*Log[1 + c^(1/3)*x] - 3*b*c^(1/3)*d*(c^(1/3)*d - e)*Log[1 - c^(1/3)*x + c^(2/3)*x^2] - 3*b*c^(1/3)*d*(c^(1/3)*d + e)*Log[1 + c^(1/3)*x + c^(2/3)*x^2] + 2*b*e^2*Log[1 - c^2*x^6])/(12*c)","A",1
33,1,333,285,0.1044843,"\int (d+e x) \left(a+b \tanh ^{-1}\left(c x^3\right)\right) \, dx","Integrate[(d + e*x)*(a + b*ArcTanh[c*x^3]),x]","a d x+\frac{1}{2} a e x^2-\frac{b d \left(\log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)+\log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)-2 \log \left(1-\sqrt[3]{c} x\right)-2 \log \left(\sqrt[3]{c} x+1\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x-1}{\sqrt{3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x+1}{\sqrt{3}}\right)\right)}{4 \sqrt[3]{c}}+\frac{b e \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)}{8 c^{2/3}}-\frac{b e \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)}{8 c^{2/3}}+\frac{b e \log \left(1-\sqrt[3]{c} x\right)}{4 c^{2/3}}-\frac{b e \log \left(\sqrt[3]{c} x+1\right)}{4 c^{2/3}}+\frac{\sqrt{3} b e \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x-1}{\sqrt{3}}\right)}{4 c^{2/3}}+\frac{\sqrt{3} b e \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x+1}{\sqrt{3}}\right)}{4 c^{2/3}}+b d x \tanh ^{-1}\left(c x^3\right)+\frac{1}{2} b e x^2 \tanh ^{-1}\left(c x^3\right)","\frac{(d+e x)^2 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)}{2 e}+\frac{b d \log \left(1-c^{2/3} x^2\right)}{2 \sqrt[3]{c}}+\frac{\sqrt{3} b d \tan ^{-1}\left(\frac{2 c^{2/3} x^2+1}{\sqrt{3}}\right)}{2 \sqrt[3]{c}}-\frac{b d \log \left(c^{4/3} x^4+c^{2/3} x^2+1\right)}{4 \sqrt[3]{c}}+\frac{b e \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)}{8 c^{2/3}}-\frac{b e \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)}{8 c^{2/3}}-\frac{\sqrt{3} b e \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{c} x}{\sqrt{3}}\right)}{4 c^{2/3}}+\frac{\sqrt{3} b e \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{4 c^{2/3}}-\frac{b e \tanh ^{-1}\left(\sqrt[3]{c} x\right)}{2 c^{2/3}}-\frac{b d^2 \tanh ^{-1}\left(c x^3\right)}{2 e}",1,"a*d*x + (a*e*x^2)/2 + (Sqrt[3]*b*e*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]])/(4*c^(2/3)) + (Sqrt[3]*b*e*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/(4*c^(2/3)) + b*d*x*ArcTanh[c*x^3] + (b*e*x^2*ArcTanh[c*x^3])/2 + (b*e*Log[1 - c^(1/3)*x])/(4*c^(2/3)) - (b*e*Log[1 + c^(1/3)*x])/(4*c^(2/3)) + (b*e*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) - (b*e*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) - (b*d*(-2*Sqrt[3]*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]] + 2*Sqrt[3]*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]] - 2*Log[1 - c^(1/3)*x] - 2*Log[1 + c^(1/3)*x] + Log[1 - c^(1/3)*x + c^(2/3)*x^2] + Log[1 + c^(1/3)*x + c^(2/3)*x^2]))/(4*c^(1/3))","A",1
34,1,515,523,100.8952528,"\int \frac{a+b \tanh ^{-1}\left(c x^3\right)}{d+e x} \, dx","Integrate[(a + b*ArcTanh[c*x^3])/(d + e*x),x]","\frac{a \log (d+e x)}{e}+\frac{b \left(-\text{Li}_2\left(\frac{\sqrt[3]{c} (d+e x)}{\sqrt[3]{c} d-e}\right)+\text{Li}_2\left(\frac{\sqrt[3]{c} (d+e x)}{\sqrt[3]{c} d+e}\right)+\text{Li}_2\left(\frac{2 \sqrt[3]{c} (d+e x)}{2 \sqrt[3]{c} d-i \sqrt{3} e-e}\right)-\text{Li}_2\left(\frac{2 \sqrt[3]{c} (d+e x)}{2 \sqrt[3]{c} d-i \sqrt{3} e+e}\right)+\text{Li}_2\left(\frac{2 \sqrt[3]{c} (d+e x)}{2 \sqrt[3]{c} d+i \sqrt{3} e-e}\right)-\text{Li}_2\left(\frac{2 \sqrt[3]{c} (d+e x)}{2 \sqrt[3]{c} d+i \sqrt{3} e+e}\right)+2 \tanh ^{-1}\left(c x^3\right) \log (d+e x)-\log (d+e x) \log \left(\frac{e \left(-2 \sqrt[3]{c} x-i \sqrt{3}+1\right)}{2 \sqrt[3]{c} d-i \sqrt{3} e+e}\right)+\log (d+e x) \log \left(\frac{e \left(-2 i \sqrt[3]{c} x+\sqrt{3}-i\right)}{2 i \sqrt[3]{c} d+\left(\sqrt{3}-i\right) e}\right)+\log (d+e x) \log \left(\frac{e \left(2 i \sqrt[3]{c} x+\sqrt{3}+i\right)}{\left(\sqrt{3}+i\right) e-2 i \sqrt[3]{c} d}\right)-\log (d+e x) \log \left(-\frac{e \left(\sqrt[3]{c} x+1\right)}{\sqrt[3]{c} d-e}\right)-\log (d+e x) \log \left(-\frac{e \left(2 \sqrt[3]{c} x-i \sqrt{3}-1\right)}{2 \sqrt[3]{c} d+i \sqrt{3} e+e}\right)+\log (d+e x) \log \left(\frac{e-\sqrt[3]{c} e x}{\sqrt[3]{c} d+e}\right)\right)}{2 e}","\frac{\log (d+e x) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)}{e}-\frac{b \text{Li}_2\left(\frac{\sqrt[3]{c} (d+e x)}{\sqrt[3]{c} d-e}\right)}{2 e}+\frac{b \text{Li}_2\left(\frac{\sqrt[3]{c} (d+e x)}{\sqrt[3]{c} d+e}\right)}{2 e}+\frac{b \text{Li}_2\left(\frac{\sqrt[3]{c} (d+e x)}{\sqrt[3]{c} d-\sqrt[3]{-1} e}\right)}{2 e}-\frac{b \text{Li}_2\left(\frac{\sqrt[3]{c} (d+e x)}{\sqrt[3]{c} d+\sqrt[3]{-1} e}\right)}{2 e}-\frac{b \text{Li}_2\left(\frac{\sqrt[3]{c} (d+e x)}{\sqrt[3]{c} d-(-1)^{2/3} e}\right)}{2 e}+\frac{b \text{Li}_2\left(\frac{\sqrt[3]{c} (d+e x)}{\sqrt[3]{c} d+(-1)^{2/3} e}\right)}{2 e}+\frac{b \log (d+e x) \log \left(\frac{e \left(1-\sqrt[3]{c} x\right)}{\sqrt[3]{c} d+e}\right)}{2 e}-\frac{b \log (d+e x) \log \left(-\frac{e \left(\sqrt[3]{c} x+1\right)}{\sqrt[3]{c} d-e}\right)}{2 e}+\frac{b \log (d+e x) \log \left(-\frac{e \left(\sqrt[3]{c} x+\sqrt[3]{-1}\right)}{\sqrt[3]{c} d-\sqrt[3]{-1} e}\right)}{2 e}-\frac{b \log (d+e x) \log \left(-\frac{e \left(\sqrt[3]{c} x+(-1)^{2/3}\right)}{\sqrt[3]{c} d-(-1)^{2/3} e}\right)}{2 e}+\frac{b \log (d+e x) \log \left(\frac{(-1)^{2/3} e \left(\sqrt[3]{-1} \sqrt[3]{c} x+1\right)}{\sqrt[3]{c} d+(-1)^{2/3} e}\right)}{2 e}-\frac{b \log (d+e x) \log \left(\frac{\sqrt[3]{-1} e \left((-1)^{2/3} \sqrt[3]{c} x+1\right)}{\sqrt[3]{c} d+\sqrt[3]{-1} e}\right)}{2 e}",1,"(a*Log[d + e*x])/e + (b*(2*ArcTanh[c*x^3]*Log[d + e*x] - Log[(e*(1 - I*Sqrt[3] - 2*c^(1/3)*x))/(2*c^(1/3)*d + e - I*Sqrt[3]*e)]*Log[d + e*x] + Log[(e*(-I + Sqrt[3] - (2*I)*c^(1/3)*x))/((2*I)*c^(1/3)*d + (-I + Sqrt[3])*e)]*Log[d + e*x] + Log[(e*(I + Sqrt[3] + (2*I)*c^(1/3)*x))/((-2*I)*c^(1/3)*d + (I + Sqrt[3])*e)]*Log[d + e*x] - Log[-((e*(1 + c^(1/3)*x))/(c^(1/3)*d - e))]*Log[d + e*x] - Log[-((e*(-1 - I*Sqrt[3] + 2*c^(1/3)*x))/(2*c^(1/3)*d + e + I*Sqrt[3]*e))]*Log[d + e*x] + Log[d + e*x]*Log[(e - c^(1/3)*e*x)/(c^(1/3)*d + e)] - PolyLog[2, (c^(1/3)*(d + e*x))/(c^(1/3)*d - e)] + PolyLog[2, (c^(1/3)*(d + e*x))/(c^(1/3)*d + e)] + PolyLog[2, (2*c^(1/3)*(d + e*x))/(2*c^(1/3)*d - e - I*Sqrt[3]*e)] - PolyLog[2, (2*c^(1/3)*(d + e*x))/(2*c^(1/3)*d + e - I*Sqrt[3]*e)] + PolyLog[2, (2*c^(1/3)*(d + e*x))/(2*c^(1/3)*d - e + I*Sqrt[3]*e)] - PolyLog[2, (2*c^(1/3)*(d + e*x))/(2*c^(1/3)*d + e + I*Sqrt[3]*e)]))/(2*e)","C",1
35,1,534,414,0.5197501,"\int \frac{a+b \tanh ^{-1}\left(c x^3\right)}{(d+e x)^2} \, dx","Integrate[(a + b*ArcTanh[c*x^3])/(d + e*x)^2,x]","\frac{1}{4} \left(-\frac{4 a}{e (d+e x)}+\frac{2 \sqrt{3} b \sqrt[3]{c} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x-1}{\sqrt{3}}\right)}{c^{2/3} d^2+\sqrt[3]{c} d e+e^2}+\frac{2 b c d^2 e^2 \log \left(1-c^2 x^6\right)}{c^2 d^6-e^6}-\frac{12 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}+\frac{b \sqrt[3]{c} \left(2 c^{5/3} d^5-c^{4/3} d^4 e-c d^3 e^2-\sqrt[3]{c} d e^4-e^5\right) \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)}{c^2 d^6 e-e^7}+\frac{b \sqrt[3]{c} \left(2 c^{5/3} d^5+c^{4/3} d^4 e-c d^3 e^2-\sqrt[3]{c} d e^4+e^5\right) \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)}{e^7-c^2 d^6 e}+\frac{2 b \sqrt[3]{c} \left(c^{5/3} d^5-c^{4/3} d^4 e+c d^3 e^2+\sqrt[3]{c} d e^4-e^5\right) \log \left(1-\sqrt[3]{c} x\right)}{e^7-c^2 d^6 e}-\frac{2 b \sqrt[3]{c} \left(c^{5/3} d^5+c^{4/3} d^4 e+c d^3 e^2+\sqrt[3]{c} d e^4+e^5\right) \log \left(\sqrt[3]{c} x+1\right)}{e^7-c^2 d^6 e}-\frac{2 \sqrt{3} b \sqrt[3]{c} \left(\sqrt[3]{c} d+e\right) \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x+1}{\sqrt{3}}\right)}{c d^3+e^3}-\frac{4 b \tanh ^{-1}\left(c x^3\right)}{e (d+e x)}\right)","-\frac{a+b \tanh ^{-1}\left(c x^3\right)}{e (d+e x)}-\frac{b \sqrt[3]{c} \left(\sqrt[3]{c} d+e\right) \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)}{4 \left(c d^3-e^3\right)}-\frac{b \sqrt[3]{c} \left(\sqrt[3]{c} d-e\right) \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)}{4 \left(c d^3+e^3\right)}-\frac{\sqrt{3} b \sqrt[3]{c} \tan ^{-1}\left(\frac{1-2 \sqrt[3]{c} x}{\sqrt{3}}\right)}{2 \left(c^{2/3} d^2+\sqrt[3]{c} d e+e^2\right)}-\frac{3 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}+\frac{b \sqrt[3]{c} \left(\sqrt[3]{c} d-e\right) \log \left(1-\sqrt[3]{c} x\right)}{2 \left(c d^3+e^3\right)}+\frac{b \sqrt[3]{c} \left(\sqrt[3]{c} d+e\right) \log \left(\sqrt[3]{c} x+1\right)}{2 \left(c d^3-e^3\right)}-\frac{\sqrt{3} b \sqrt[3]{c} \left(\sqrt[3]{c} d+e\right) \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x+1}{\sqrt{3}}\right)}{2 \left(c d^3+e^3\right)}-\frac{b c d^2 \log \left(1-c x^3\right)}{2 e \left(c d^3+e^3\right)}+\frac{b c d^2 \log \left(c x^3+1\right)}{2 e \left(c d^3-e^3\right)}",1,"((-4*a)/(e*(d + e*x)) + (2*Sqrt[3]*b*c^(1/3)*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]])/(c^(2/3)*d^2 + c^(1/3)*d*e + e^2) - (2*Sqrt[3]*b*c^(1/3)*(c^(1/3)*d + e)*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/(c*d^3 + e^3) - (4*b*ArcTanh[c*x^3])/(e*(d + e*x)) + (2*b*c^(1/3)*(c^(5/3)*d^5 - c^(4/3)*d^4*e + c*d^3*e^2 + c^(1/3)*d*e^4 - e^5)*Log[1 - c^(1/3)*x])/(-(c^2*d^6*e) + e^7) - (2*b*c^(1/3)*(c^(5/3)*d^5 + c^(4/3)*d^4*e + c*d^3*e^2 + c^(1/3)*d*e^4 + e^5)*Log[1 + c^(1/3)*x])/(-(c^2*d^6*e) + e^7) - (12*b*c*d^2*e^2*Log[d + e*x])/(c^2*d^6 - e^6) + (b*c^(1/3)*(2*c^(5/3)*d^5 - c^(4/3)*d^4*e - c*d^3*e^2 - c^(1/3)*d*e^4 - e^5)*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(c^2*d^6*e - e^7) + (b*c^(1/3)*(2*c^(5/3)*d^5 + c^(4/3)*d^4*e - c*d^3*e^2 - c^(1/3)*d*e^4 + e^5)*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(-(c^2*d^6*e) + e^7) + (2*b*c*d^2*e^2*Log[1 - c^2*x^6])/(c^2*d^6 - e^6))/4","A",1
36,1,160,195,0.5712044,"\int \frac{x^3 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{1-c^2 x} \, dx","Integrate[(x^3*(a + b*ArcTanh[c*Sqrt[x]]))/(1 - c^2*x),x]","-\frac{30 a c^6 x^3+45 a c^4 x^2+90 a c^2 x+90 a \log \left(1-c^2 x\right)+6 b c^5 x^{5/2}+25 b c^3 x^{3/2}+15 b \tanh ^{-1}\left(c \sqrt{x}\right) \left(2 c^6 x^3+3 c^4 x^2+6 c^2 x-12 \log \left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}+1\right)-11\right)+90 b \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)+165 b c \sqrt{x}-90 b \tanh ^{-1}\left(c \sqrt{x}\right)^2}{90 c^8}","-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{b c^8}+\frac{2 \log \left(\frac{2}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{c^8}-\frac{x \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{c^6}-\frac{x^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{2 c^4}-\frac{x^3 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{3 c^2}+\frac{b \text{Li}_2\left(1-\frac{2}{1-c \sqrt{x}}\right)}{c^8}+\frac{11 b \tanh ^{-1}\left(c \sqrt{x}\right)}{6 c^8}-\frac{11 b \sqrt{x}}{6 c^7}-\frac{5 b x^{3/2}}{18 c^5}-\frac{b x^{5/2}}{15 c^3}",1,"-1/90*(165*b*c*Sqrt[x] + 90*a*c^2*x + 25*b*c^3*x^(3/2) + 45*a*c^4*x^2 + 6*b*c^5*x^(5/2) + 30*a*c^6*x^3 - 90*b*ArcTanh[c*Sqrt[x]]^2 + 15*b*ArcTanh[c*Sqrt[x]]*(-11 + 6*c^2*x + 3*c^4*x^2 + 2*c^6*x^3 - 12*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + 90*a*Log[1 - c^2*x] + 90*b*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])])/c^8","A",0
37,1,130,160,0.3911241,"\int \frac{x^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{1-c^2 x} \, dx","Integrate[(x^2*(a + b*ArcTanh[c*Sqrt[x]]))/(1 - c^2*x),x]","-\frac{3 a c^4 x^2+6 a c^2 x+6 a \log \left(1-c^2 x\right)+b c^3 x^{3/2}+3 b \tanh ^{-1}\left(c \sqrt{x}\right) \left(c^4 x^2+2 c^2 x-4 \log \left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}+1\right)-3\right)+6 b \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)+9 b c \sqrt{x}-6 b \tanh ^{-1}\left(c \sqrt{x}\right)^2}{6 c^6}","-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{b c^6}+\frac{2 \log \left(\frac{2}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{c^6}-\frac{x \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{c^4}-\frac{x^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{2 c^2}+\frac{b \text{Li}_2\left(1-\frac{2}{1-c \sqrt{x}}\right)}{c^6}+\frac{3 b \tanh ^{-1}\left(c \sqrt{x}\right)}{2 c^6}-\frac{3 b \sqrt{x}}{2 c^5}-\frac{b x^{3/2}}{6 c^3}",1,"-1/6*(9*b*c*Sqrt[x] + 6*a*c^2*x + b*c^3*x^(3/2) + 3*a*c^4*x^2 - 6*b*ArcTanh[c*Sqrt[x]]^2 + 3*b*ArcTanh[c*Sqrt[x]]*(-3 + 2*c^2*x + c^4*x^2 - 4*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + 6*a*Log[1 - c^2*x] + 6*b*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])])/c^6","A",0
38,1,96,120,0.2248664,"\int \frac{x \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{1-c^2 x} \, dx","Integrate[(x*(a + b*ArcTanh[c*Sqrt[x]]))/(1 - c^2*x),x]","-\frac{a c^2 x+a \log \left(1-c^2 x\right)+b \tanh ^{-1}\left(c \sqrt{x}\right) \left(c^2 x-2 \log \left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}+1\right)-1\right)+b \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)+b c \sqrt{x}-b \tanh ^{-1}\left(c \sqrt{x}\right)^2}{c^4}","-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{b c^4}+\frac{2 \log \left(\frac{2}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{c^4}-\frac{x \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{c^2}+\frac{b \text{Li}_2\left(1-\frac{2}{1-c \sqrt{x}}\right)}{c^4}+\frac{b \tanh ^{-1}\left(c \sqrt{x}\right)}{c^4}-\frac{b \sqrt{x}}{c^3}",1,"-((b*c*Sqrt[x] + a*c^2*x - b*ArcTanh[c*Sqrt[x]]^2 + b*ArcTanh[c*Sqrt[x]]*(-1 + c^2*x - 2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + a*Log[1 - c^2*x] + b*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])])/c^4)","A",0
39,1,75,78,0.0982922,"\int \frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{1-c^2 x} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])/(1 - c^2*x),x]","-\frac{a \log \left(1-c^2 x\right)}{c^2}-\frac{b \left(\text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)-\tanh ^{-1}\left(c \sqrt{x}\right) \left(\tanh ^{-1}\left(c \sqrt{x}\right)+2 \log \left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}+1\right)\right)\right)}{c^2}","-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{b c^2}+\frac{2 \log \left(\frac{2}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{c^2}+\frac{b \text{Li}_2\left(1-\frac{2}{1-c \sqrt{x}}\right)}{c^2}",1,"-((a*Log[1 - c^2*x])/c^2) - (b*(-(ArcTanh[c*Sqrt[x]]*(ArcTanh[c*Sqrt[x]] + 2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])])) + PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])]))/c^2","A",0
40,1,72,69,0.1294039,"\int \frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{x \left(1-c^2 x\right)} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])/(x*(1 - c^2*x)),x]","-a \log \left(1-c^2 x\right)+a \log (x)-b \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)+b \tanh ^{-1}\left(c \sqrt{x}\right) \left(\tanh ^{-1}\left(c \sqrt{x}\right)+2 \log \left(1-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)\right)","\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{b}+2 \log \left(2-\frac{2}{c \sqrt{x}+1}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)-b \text{Li}_2\left(\frac{2}{\sqrt{x} c+1}-1\right)",1,"b*ArcTanh[c*Sqrt[x]]*(ArcTanh[c*Sqrt[x]] + 2*Log[1 - E^(-2*ArcTanh[c*Sqrt[x]])]) + a*Log[x] - a*Log[1 - c^2*x] - b*PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])]","A",0
41,1,118,117,0.3358897,"\int \frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{x^2 \left(1-c^2 x\right)} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])/(x^2*(1 - c^2*x)),x]","2 a c^2 \log \left(\sqrt{x}\right)-a c^2 \log \left(1-c^2 x\right)-\frac{a}{x}-b c^2 \left(-\tanh ^{-1}\left(c \sqrt{x}\right) \left(-\frac{1-c^2 x}{c^2 x}+\tanh ^{-1}\left(c \sqrt{x}\right)+2 \log \left(1-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)\right)+\text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)+\frac{1}{c \sqrt{x}}\right)","\frac{c^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{b}+2 c^2 \log \left(2-\frac{2}{c \sqrt{x}+1}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)-\frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{x}-b c^2 \text{Li}_2\left(\frac{2}{\sqrt{x} c+1}-1\right)+b c^2 \tanh ^{-1}\left(c \sqrt{x}\right)-\frac{b c}{\sqrt{x}}",1,"-(a/x) + 2*a*c^2*Log[Sqrt[x]] - a*c^2*Log[1 - c^2*x] - b*c^2*(1/(c*Sqrt[x]) - ArcTanh[c*Sqrt[x]]*(-((1 - c^2*x)/(c^2*x)) + ArcTanh[c*Sqrt[x]] + 2*Log[1 - E^(-2*ArcTanh[c*Sqrt[x]])]) + PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])])","A",0
42,1,158,157,0.5701778,"\int \frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{x^3 \left(1-c^2 x\right)} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])/(x^3*(1 - c^2*x)),x]","-\frac{-6 a c^4 x^2 \log (x)+6 a c^2 x+6 a c^4 x^2 \log \left(1-c^2 x\right)+3 a-6 b c^4 x^2 \tanh ^{-1}\left(c \sqrt{x}\right)^2+9 b c^3 x^{3/2}-3 b \tanh ^{-1}\left(c \sqrt{x}\right) \left(3 c^4 x^2+4 c^4 x^2 \log \left(1-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)-2 c^2 x-1\right)+b c \sqrt{x}}{6 x^2}-b c^4 \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)","\frac{c^4 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{b}+2 c^4 \log \left(2-\frac{2}{c \sqrt{x}+1}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)-\frac{c^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{x}-\frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{2 x^2}-b c^4 \text{Li}_2\left(\frac{2}{\sqrt{x} c+1}-1\right)+\frac{3}{2} b c^4 \tanh ^{-1}\left(c \sqrt{x}\right)-\frac{3 b c^3}{2 \sqrt{x}}-\frac{b c}{6 x^{3/2}}",1,"-1/6*(3*a + b*c*Sqrt[x] + 6*a*c^2*x + 9*b*c^3*x^(3/2) - 6*b*c^4*x^2*ArcTanh[c*Sqrt[x]]^2 - 3*b*ArcTanh[c*Sqrt[x]]*(-1 - 2*c^2*x + 3*c^4*x^2 + 4*c^4*x^2*Log[1 - E^(-2*ArcTanh[c*Sqrt[x]])]) - 6*a*c^4*x^2*Log[x] + 6*a*c^4*x^2*Log[1 - c^2*x])/x^2 - b*c^4*PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])]","A",0
43,1,187,192,0.808766,"\int \frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{x^4 \left(1-c^2 x\right)} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])/(x^4*(1 - c^2*x)),x]","-\frac{-90 a c^6 x^3 \log (x)+90 a c^4 x^2+45 a c^2 x+90 a c^6 x^3 \log \left(1-c^2 x\right)+30 a-90 b c^6 x^3 \tanh ^{-1}\left(c \sqrt{x}\right)^2+165 b c^5 x^{5/2}+25 b c^3 x^{3/2}-15 b \tanh ^{-1}\left(c \sqrt{x}\right) \left(11 c^6 x^3+12 c^6 x^3 \log \left(1-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)-6 c^4 x^2-3 c^2 x-2\right)+6 b c \sqrt{x}}{90 x^3}-b c^6 \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)","\frac{c^6 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{b}+2 c^6 \log \left(2-\frac{2}{c \sqrt{x}+1}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)-\frac{c^4 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{x}-\frac{c^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{2 x^2}-\frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{3 x^3}-b c^6 \text{Li}_2\left(\frac{2}{\sqrt{x} c+1}-1\right)+\frac{11}{6} b c^6 \tanh ^{-1}\left(c \sqrt{x}\right)-\frac{11 b c^5}{6 \sqrt{x}}-\frac{5 b c^3}{18 x^{3/2}}-\frac{b c}{15 x^{5/2}}",1,"-1/90*(30*a + 6*b*c*Sqrt[x] + 45*a*c^2*x + 25*b*c^3*x^(3/2) + 90*a*c^4*x^2 + 165*b*c^5*x^(5/2) - 90*b*c^6*x^3*ArcTanh[c*Sqrt[x]]^2 - 15*b*ArcTanh[c*Sqrt[x]]*(-2 - 3*c^2*x - 6*c^4*x^2 + 11*c^6*x^3 + 12*c^6*x^3*Log[1 - E^(-2*ArcTanh[c*Sqrt[x]])]) - 90*a*c^6*x^3*Log[x] + 90*a*c^6*x^3*Log[1 - c^2*x])/x^3 - b*c^6*PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])]","A",0
44,1,558,460,2.933629,"\int \frac{x^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{d+e x} \, dx","Integrate[(x^2*(a + b*ArcTanh[c*Sqrt[x]]))/(d + e*x),x]","\frac{6 a d^2 \log (d+e x)-6 a d e x+3 a e^2 x^2+\frac{b \left(-6 c^4 d^2 \left(\tanh ^{-1}\left(c \sqrt{x}\right) \left(\tanh ^{-1}\left(c \sqrt{x}\right)+2 \log \left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}+1\right)\right)-\text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)\right)+2 c e \sqrt{x} \left(2 e-3 c^2 d\right)-6 e \left(c^2 x-1\right) \left(c^2 d-e\right) \tanh ^{-1}\left(c \sqrt{x}\right)+c e^2 \sqrt{x} \left(c^2 x-1\right)+3 e^2 \left(c^2 x-1\right)^2 \tanh ^{-1}\left(c \sqrt{x}\right)+3 c^4 d^2 \left(-\text{Li}_2\left(\frac{\left(-d c^2+e-2 \sqrt{-c^2 d e}\right) e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{d c^2+e}\right)-\text{Li}_2\left(\frac{\left(-d c^2+e+2 \sqrt{-c^2 d e}\right) e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{d c^2+e}\right)-4 i \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d+e}}\right) \tanh ^{-1}\left(\frac{c e \sqrt{x}}{\sqrt{-c^2 d e}}\right)+2 \log \left(\frac{e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)} \left(-2 \sqrt{-c^2 d e}+c^2 d \left(e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}+1\right)+e \left(e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}-1\right)\right)}{c^2 d+e}\right) \left(\tanh ^{-1}\left(c \sqrt{x}\right)-i \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d+e}}\right)\right)+2 \log \left(\frac{e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)} \left(2 \sqrt{-c^2 d e}+c^2 d \left(e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}+1\right)+e \left(e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}-1\right)\right)}{c^2 d+e}\right) \left(\tanh ^{-1}\left(c \sqrt{x}\right)+i \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d+e}}\right)\right)+2 \tanh ^{-1}\left(c \sqrt{x}\right)^2\right)\right)}{c^4}}{6 e^3}","-\frac{2 d^2 \log \left(\frac{2}{c \sqrt{x}+1}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{e^3}+\frac{d^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{x}+1\right) \left(c \sqrt{-d}-\sqrt{e}\right)}\right)}{e^3}+\frac{d^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{x}+1\right) \left(c \sqrt{-d}+\sqrt{e}\right)}\right)}{e^3}-\frac{d x \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{e^2}+\frac{x^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{2 e}-\frac{b \tanh ^{-1}\left(c \sqrt{x}\right)}{2 c^4 e}+\frac{b \sqrt{x}}{2 c^3 e}+\frac{b d \tanh ^{-1}\left(c \sqrt{x}\right)}{c^2 e^2}+\frac{b d^2 \text{Li}_2\left(1-\frac{2}{\sqrt{x} c+1}\right)}{e^3}-\frac{b d^2 \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{-d}-\sqrt{e}\right) \left(\sqrt{x} c+1\right)}\right)}{2 e^3}-\frac{b d^2 \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} \sqrt{x}\right)}{\left(\sqrt{-d} c+\sqrt{e}\right) \left(\sqrt{x} c+1\right)}\right)}{2 e^3}-\frac{b d \sqrt{x}}{c e^2}+\frac{b x^{3/2}}{6 c e}",1,"(-6*a*d*e*x + 3*a*e^2*x^2 + 6*a*d^2*Log[d + e*x] + (b*(2*c*e*(-3*c^2*d + 2*e)*Sqrt[x] + c*e^2*Sqrt[x]*(-1 + c^2*x) - 6*(c^2*d - e)*e*(-1 + c^2*x)*ArcTanh[c*Sqrt[x]] + 3*e^2*(-1 + c^2*x)^2*ArcTanh[c*Sqrt[x]] - 6*c^4*d^2*(ArcTanh[c*Sqrt[x]]*(ArcTanh[c*Sqrt[x]] + 2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) - PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])]) + 3*c^4*d^2*(2*ArcTanh[c*Sqrt[x]]^2 - (4*I)*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]]*ArcTanh[(c*e*Sqrt[x])/Sqrt[-(c^2*d*e)]] + 2*((-I)*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]] + ArcTanh[c*Sqrt[x]])*Log[(-2*Sqrt[-(c^2*d*e)] + e*(-1 + E^(2*ArcTanh[c*Sqrt[x]])) + c^2*d*(1 + E^(2*ArcTanh[c*Sqrt[x]])))/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] + 2*(I*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]] + ArcTanh[c*Sqrt[x]])*Log[(2*Sqrt[-(c^2*d*e)] + e*(-1 + E^(2*ArcTanh[c*Sqrt[x]])) + c^2*d*(1 + E^(2*ArcTanh[c*Sqrt[x]])))/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] - PolyLog[2, (-(c^2*d) + e - 2*Sqrt[-(c^2*d*e)])/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] - PolyLog[2, (-(c^2*d) + e + 2*Sqrt[-(c^2*d*e)])/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))])))/c^4)/(6*e^3)","C",0
45,1,337,374,1.4613269,"\int \frac{x \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{d+e x} \, dx","Integrate[(x*(a + b*ArcTanh[c*Sqrt[x]]))/(d + e*x),x]","\frac{-2 a d \log (d+e x)+2 a e x+\frac{2 b \left(\tanh ^{-1}\left(c \sqrt{x}\right) \left(2 c^2 d \log \left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}+1\right)+c^2 e x-e\right)-c^2 d \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)+c^2 d \tanh ^{-1}\left(c \sqrt{x}\right)^2+c e \sqrt{x}\right)}{c^2}-b d \left(\text{Li}_2\left(-\frac{\left(d c^2+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{d c^2-2 \sqrt{-d} \sqrt{e} c-e}\right)+\text{Li}_2\left(-\frac{\left(d c^2+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{d c^2+2 \sqrt{-d} \sqrt{e} c-e}\right)+2 \tanh ^{-1}\left(c \sqrt{x}\right) \left(\log \left(\frac{\left(c^2 d+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{c^2 d-2 c \sqrt{-d} \sqrt{e}-e}+1\right)+\log \left(\frac{\left(c^2 d+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{c^2 d+2 c \sqrt{-d} \sqrt{e}-e}+1\right)-\tanh ^{-1}\left(c \sqrt{x}\right)\right)\right)}{2 e^2}","\frac{2 d \log \left(\frac{2}{c \sqrt{x}+1}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{e^2}-\frac{d \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{x}+1\right) \left(c \sqrt{-d}-\sqrt{e}\right)}\right)}{e^2}-\frac{d \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{x}+1\right) \left(c \sqrt{-d}+\sqrt{e}\right)}\right)}{e^2}+\frac{x \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{e}-\frac{b \tanh ^{-1}\left(c \sqrt{x}\right)}{c^2 e}-\frac{b d \text{Li}_2\left(1-\frac{2}{\sqrt{x} c+1}\right)}{e^2}+\frac{b d \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{-d}-\sqrt{e}\right) \left(\sqrt{x} c+1\right)}\right)}{2 e^2}+\frac{b d \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} \sqrt{x}\right)}{\left(\sqrt{-d} c+\sqrt{e}\right) \left(\sqrt{x} c+1\right)}\right)}{2 e^2}+\frac{b \sqrt{x}}{c e}",1,"(2*a*e*x - 2*a*d*Log[d + e*x] + (2*b*(c*e*Sqrt[x] + c^2*d*ArcTanh[c*Sqrt[x]]^2 + ArcTanh[c*Sqrt[x]]*(-e + c^2*e*x + 2*c^2*d*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) - c^2*d*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])]))/c^2 - b*d*(2*ArcTanh[c*Sqrt[x]]*(-ArcTanh[c*Sqrt[x]] + Log[1 + ((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d - 2*c*Sqrt[-d]*Sqrt[e] - e)] + Log[1 + ((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d + 2*c*Sqrt[-d]*Sqrt[e] - e)]) + PolyLog[2, -(((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d - 2*c*Sqrt[-d]*Sqrt[e] - e))] + PolyLog[2, -(((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d + 2*c*Sqrt[-d]*Sqrt[e] - e))]))/(2*e^2)","A",0
46,1,432,318,1.6173067,"\int \frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{d+e x} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])/(d + e*x),x]","\frac{a \log (d+e x)}{e}-\frac{b \left(\text{Li}_2\left(\frac{\left(-d c^2+e-2 \sqrt{-c^2 d e}\right) e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{d c^2+e}\right)+\text{Li}_2\left(\frac{\left(-d c^2+e+2 \sqrt{-c^2 d e}\right) e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{d c^2+e}\right)+4 i \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d+e}}\right) \tanh ^{-1}\left(\frac{c e \sqrt{x}}{\sqrt{-c^2 d e}}\right)-2 \log \left(\frac{e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)} \left(-2 \sqrt{-c^2 d e}+c^2 d \left(e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}+1\right)+e \left(e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}-1\right)\right)}{c^2 d+e}\right) \left(\tanh ^{-1}\left(c \sqrt{x}\right)-i \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d+e}}\right)\right)-2 \log \left(\frac{e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)} \left(2 \sqrt{-c^2 d e}+c^2 d \left(e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}+1\right)+e \left(e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}-1\right)\right)}{c^2 d+e}\right) \left(\tanh ^{-1}\left(c \sqrt{x}\right)+i \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d+e}}\right)\right)-2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)-2 \tanh ^{-1}\left(c \sqrt{x}\right)^2+2 \tanh ^{-1}\left(c \sqrt{x}\right) \left(\tanh ^{-1}\left(c \sqrt{x}\right)+2 \log \left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}+1\right)\right)\right)}{2 e}","\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{x}+1\right) \left(c \sqrt{-d}-\sqrt{e}\right)}\right)}{e}+\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{x}+1\right) \left(c \sqrt{-d}+\sqrt{e}\right)}\right)}{e}-\frac{2 \log \left(\frac{2}{c \sqrt{x}+1}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{e}-\frac{b \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{-d}-\sqrt{e}\right) \left(\sqrt{x} c+1\right)}\right)}{2 e}-\frac{b \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} \sqrt{x}\right)}{\left(\sqrt{-d} c+\sqrt{e}\right) \left(\sqrt{x} c+1\right)}\right)}{2 e}+\frac{b \text{Li}_2\left(1-\frac{2}{\sqrt{x} c+1}\right)}{e}",1,"(a*Log[d + e*x])/e - (b*(-2*ArcTanh[c*Sqrt[x]]^2 + (4*I)*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]]*ArcTanh[(c*e*Sqrt[x])/Sqrt[-(c^2*d*e)]] + 2*ArcTanh[c*Sqrt[x]]*(ArcTanh[c*Sqrt[x]] + 2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) - 2*((-I)*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]] + ArcTanh[c*Sqrt[x]])*Log[(-2*Sqrt[-(c^2*d*e)] + e*(-1 + E^(2*ArcTanh[c*Sqrt[x]])) + c^2*d*(1 + E^(2*ArcTanh[c*Sqrt[x]])))/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] - 2*(I*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]] + ArcTanh[c*Sqrt[x]])*Log[(2*Sqrt[-(c^2*d*e)] + e*(-1 + E^(2*ArcTanh[c*Sqrt[x]])) + c^2*d*(1 + E^(2*ArcTanh[c*Sqrt[x]])))/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] - 2*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])] + PolyLog[2, (-(c^2*d) + e - 2*Sqrt[-(c^2*d*e)])/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] + PolyLog[2, (-(c^2*d) + e + 2*Sqrt[-(c^2*d*e)])/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))]))/(2*e)","C",0
47,1,302,358,1.1493694,"\int \frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{x (d+e x)} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])/(x*(d + e*x)),x]","-\frac{2 a \log (d+e x)-2 a \log (x)+b \text{Li}_2\left(-\frac{\left(d c^2+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{d c^2-2 \sqrt{-d} \sqrt{e} c-e}\right)+b \text{Li}_2\left(-\frac{\left(d c^2+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{d c^2+2 \sqrt{-d} \sqrt{e} c-e}\right)+2 b \tanh ^{-1}\left(c \sqrt{x}\right) \log \left(\frac{\left(c^2 d+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{c^2 d-2 c \sqrt{-d} \sqrt{e}-e}+1\right)+2 b \tanh ^{-1}\left(c \sqrt{x}\right) \log \left(\frac{\left(c^2 d+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{c^2 d+2 c \sqrt{-d} \sqrt{e}-e}+1\right)+2 b \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)-4 b \tanh ^{-1}\left(c \sqrt{x}\right)^2-4 b \tanh ^{-1}\left(c \sqrt{x}\right) \log \left(1-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)}{2 d}","-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{x}+1\right) \left(c \sqrt{-d}-\sqrt{e}\right)}\right)}{d}-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{x}+1\right) \left(c \sqrt{-d}+\sqrt{e}\right)}\right)}{d}+\frac{2 \log \left(\frac{2}{c \sqrt{x}+1}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{d}+\frac{a \log (x)}{d}+\frac{b \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{-d}-\sqrt{e}\right) \left(\sqrt{x} c+1\right)}\right)}{2 d}+\frac{b \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} \sqrt{x}\right)}{\left(\sqrt{-d} c+\sqrt{e}\right) \left(\sqrt{x} c+1\right)}\right)}{2 d}-\frac{b \text{Li}_2\left(1-\frac{2}{\sqrt{x} c+1}\right)}{d}-\frac{b \text{Li}_2\left(-c \sqrt{x}\right)}{d}+\frac{b \text{Li}_2\left(c \sqrt{x}\right)}{d}",1,"-1/2*(-4*b*ArcTanh[c*Sqrt[x]]^2 - 4*b*ArcTanh[c*Sqrt[x]]*Log[1 - E^(-2*ArcTanh[c*Sqrt[x]])] + 2*b*ArcTanh[c*Sqrt[x]]*Log[1 + ((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d - 2*c*Sqrt[-d]*Sqrt[e] - e)] + 2*b*ArcTanh[c*Sqrt[x]]*Log[1 + ((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d + 2*c*Sqrt[-d]*Sqrt[e] - e)] - 2*a*Log[x] + 2*a*Log[d + e*x] + 2*b*PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])] + b*PolyLog[2, -(((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d - 2*c*Sqrt[-d]*Sqrt[e] - e))] + b*PolyLog[2, -(((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d + 2*c*Sqrt[-d]*Sqrt[e] - e))])/d","A",0
48,1,360,413,1.6436396,"\int \frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{x^2 (d+e x)} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])/(x^2*(d + e*x)),x]","-\frac{2 a e \log \left(\sqrt{x}\right)}{d^2}+\frac{a e \log (d+e x)}{d^2}-\frac{a}{d x}+2 b c^4 \left(\frac{e \left(\text{Li}_2\left(-\frac{\left(d c^2+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{d c^2-2 \sqrt{-d} \sqrt{e} c-e}\right)+\text{Li}_2\left(-\frac{\left(d c^2+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{d c^2+2 \sqrt{-d} \sqrt{e} c-e}\right)+2 \tanh ^{-1}\left(c \sqrt{x}\right) \left(\log \left(\frac{\left(c^2 d+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{c^2 d-2 c \sqrt{-d} \sqrt{e}-e}+1\right)+\log \left(\frac{\left(c^2 d+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{c^2 d+2 c \sqrt{-d} \sqrt{e}-e}+1\right)-\tanh ^{-1}\left(c \sqrt{x}\right)\right)\right)}{4 c^4 d^2}-\frac{\tanh ^{-1}\left(c \sqrt{x}\right) \left(\frac{d \left(1-c^2 x\right)}{x}+e \tanh ^{-1}\left(c \sqrt{x}\right)+2 e \log \left(1-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)\right)+\frac{c d}{\sqrt{x}}-e \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)}{2 c^4 d^2}\right)","-\frac{2 e \log \left(\frac{2}{c \sqrt{x}+1}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{d^2}+\frac{e \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{x}+1\right) \left(c \sqrt{-d}-\sqrt{e}\right)}\right)}{d^2}+\frac{e \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{x}+1\right) \left(c \sqrt{-d}+\sqrt{e}\right)}\right)}{d^2}-\frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{d x}-\frac{a e \log (x)}{d^2}+\frac{b c^2 \tanh ^{-1}\left(c \sqrt{x}\right)}{d}+\frac{b e \text{Li}_2\left(1-\frac{2}{\sqrt{x} c+1}\right)}{d^2}-\frac{b e \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{-d}-\sqrt{e}\right) \left(\sqrt{x} c+1\right)}\right)}{2 d^2}-\frac{b e \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} \sqrt{x}\right)}{\left(\sqrt{-d} c+\sqrt{e}\right) \left(\sqrt{x} c+1\right)}\right)}{2 d^2}+\frac{b e \text{Li}_2\left(-c \sqrt{x}\right)}{d^2}-\frac{b e \text{Li}_2\left(c \sqrt{x}\right)}{d^2}-\frac{b c}{d \sqrt{x}}",1,"-(a/(d*x)) - (2*a*e*Log[Sqrt[x]])/d^2 + (a*e*Log[d + e*x])/d^2 + 2*b*c^4*(-1/2*((c*d)/Sqrt[x] + ArcTanh[c*Sqrt[x]]*((d*(1 - c^2*x))/x + e*ArcTanh[c*Sqrt[x]] + 2*e*Log[1 - E^(-2*ArcTanh[c*Sqrt[x]])]) - e*PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])])/(c^4*d^2) + (e*(2*ArcTanh[c*Sqrt[x]]*(-ArcTanh[c*Sqrt[x]] + Log[1 + ((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d - 2*c*Sqrt[-d]*Sqrt[e] - e)] + Log[1 + ((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d + 2*c*Sqrt[-d]*Sqrt[e] - e)]) + PolyLog[2, -(((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d - 2*c*Sqrt[-d]*Sqrt[e] - e))] + PolyLog[2, -(((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d + 2*c*Sqrt[-d]*Sqrt[e] - e))]))/(4*c^4*d^2))","A",0
49,1,394,506,2.7406252,"\int \frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{x^3 (d+e x)} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])/(x^3*(d + e*x)),x]","-\frac{3 a d^2+6 a e^2 x^2 \log (d+e x)-6 a d e x-6 a e^2 x^2 \log (x)+b \left(3 e^2 x^2 \left(\text{Li}_2\left(-\frac{\left(d c^2+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{d c^2-2 \sqrt{-d} \sqrt{e} c-e}\right)+\text{Li}_2\left(-\frac{\left(d c^2+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{d c^2+2 \sqrt{-d} \sqrt{e} c-e}\right)+2 \tanh ^{-1}\left(c \sqrt{x}\right) \left(\log \left(\frac{\left(c^2 d+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{c^2 d-2 c \sqrt{-d} \sqrt{e}-e}+1\right)+\log \left(\frac{\left(c^2 d+e\right) e^{2 \tanh ^{-1}\left(c \sqrt{x}\right)}}{c^2 d+2 c \sqrt{-d} \sqrt{e}-e}+1\right)-\tanh ^{-1}\left(c \sqrt{x}\right)\right)\right)-3 \tanh ^{-1}\left(c \sqrt{x}\right) \left(d \left(c^2 x-1\right) \left(c^2 d x+d-2 e x\right)+2 e^2 x^2 \tanh ^{-1}\left(c \sqrt{x}\right)+4 e^2 x^2 \log \left(1-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)\right)+c d \sqrt{x} \left(3 c^2 d x+d-6 e x\right)+6 e^2 x^2 \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)\right)}{6 d^3 x^2}","\frac{2 e^2 \log \left(\frac{2}{c \sqrt{x}+1}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{d^3}-\frac{e^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{x}+1\right) \left(c \sqrt{-d}-\sqrt{e}\right)}\right)}{d^3}-\frac{e^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{x}+1\right) \left(c \sqrt{-d}+\sqrt{e}\right)}\right)}{d^3}+\frac{e \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{d^2 x}-\frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{2 d x^2}+\frac{a e^2 \log (x)}{d^3}+\frac{b c^4 \tanh ^{-1}\left(c \sqrt{x}\right)}{2 d}-\frac{b c^3}{2 d \sqrt{x}}-\frac{b c^2 e \tanh ^{-1}\left(c \sqrt{x}\right)}{d^2}-\frac{b e^2 \text{Li}_2\left(1-\frac{2}{\sqrt{x} c+1}\right)}{d^3}+\frac{b e^2 \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} \sqrt{x}\right)}{\left(c \sqrt{-d}-\sqrt{e}\right) \left(\sqrt{x} c+1\right)}\right)}{2 d^3}+\frac{b e^2 \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} \sqrt{x}\right)}{\left(\sqrt{-d} c+\sqrt{e}\right) \left(\sqrt{x} c+1\right)}\right)}{2 d^3}-\frac{b e^2 \text{Li}_2\left(-c \sqrt{x}\right)}{d^3}+\frac{b e^2 \text{Li}_2\left(c \sqrt{x}\right)}{d^3}+\frac{b c e}{d^2 \sqrt{x}}-\frac{b c}{6 d x^{3/2}}",1,"-1/6*(3*a*d^2 - 6*a*d*e*x - 6*a*e^2*x^2*Log[x] + 6*a*e^2*x^2*Log[d + e*x] + b*(c*d*Sqrt[x]*(d + 3*c^2*d*x - 6*e*x) - 3*ArcTanh[c*Sqrt[x]]*(d*(-1 + c^2*x)*(d + c^2*d*x - 2*e*x) + 2*e^2*x^2*ArcTanh[c*Sqrt[x]] + 4*e^2*x^2*Log[1 - E^(-2*ArcTanh[c*Sqrt[x]])]) + 6*e^2*x^2*PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])] + 3*e^2*x^2*(2*ArcTanh[c*Sqrt[x]]*(-ArcTanh[c*Sqrt[x]] + Log[1 + ((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d - 2*c*Sqrt[-d]*Sqrt[e] - e)] + Log[1 + ((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d + 2*c*Sqrt[-d]*Sqrt[e] - e)]) + PolyLog[2, -(((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d - 2*c*Sqrt[-d]*Sqrt[e] - e))] + PolyLog[2, -(((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))/(c^2*d + 2*c*Sqrt[-d]*Sqrt[e] - e))])))/(d^3*x^2)","A",0