1,1,97,108,0.0726254,"\int x^3 (d+c d x) \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + c*d*x)*(a + b*ArcTanh[c*x]),x]","\frac{d \left(24 a c^5 x^5+30 a c^4 x^4+6 b c^4 x^4+6 b c^4 x^4 (4 c x+5) \tanh ^{-1}(c x)+10 b c^3 x^3+12 b c^2 x^2+30 b c x+27 b \log (1-c x)-3 b \log (c x+1)\right)}{120 c^4}","\frac{1}{5} c d x^5 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{4} d x^4 \left(a+b \tanh ^{-1}(c x)\right)+\frac{9 b d \log (1-c x)}{40 c^4}-\frac{b d \log (c x+1)}{40 c^4}+\frac{b d x}{4 c^3}+\frac{b d x^2}{10 c^2}+\frac{b d x^3}{12 c}+\frac{1}{20} b d x^4",1,"(d*(30*b*c*x + 12*b*c^2*x^2 + 10*b*c^3*x^3 + 30*a*c^4*x^4 + 6*b*c^4*x^4 + 24*a*c^5*x^5 + 6*b*c^4*x^4*(5 + 4*c*x)*ArcTanh[c*x] + 27*b*Log[1 - c*x] - 3*b*Log[1 + c*x]))/(120*c^4)","A",1
2,1,87,96,0.0641262,"\int x^2 (d+c d x) \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + c*d*x)*(a + b*ArcTanh[c*x]),x]","\frac{d \left(6 a c^4 x^4+8 a c^3 x^3+2 b c^3 x^3+2 b c^3 x^3 (3 c x+4) \tanh ^{-1}(c x)+4 b c^2 x^2+6 b c x+7 b \log (1-c x)+b \log (c x+1)\right)}{24 c^3}","\frac{1}{4} c d x^4 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{3} d x^3 \left(a+b \tanh ^{-1}(c x)\right)+\frac{7 b d \log (1-c x)}{24 c^3}+\frac{b d \log (c x+1)}{24 c^3}+\frac{b d x}{4 c^2}+\frac{b d x^2}{6 c}+\frac{1}{12} b d x^3",1,"(d*(6*b*c*x + 4*b*c^2*x^2 + 8*a*c^3*x^3 + 2*b*c^3*x^3 + 6*a*c^4*x^4 + 2*b*c^3*x^3*(4 + 3*c*x)*ArcTanh[c*x] + 7*b*Log[1 - c*x] + b*Log[1 + c*x]))/(24*c^3)","A",1
3,1,79,84,0.0553228,"\int x (d+c d x) \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x*(d + c*d*x)*(a + b*ArcTanh[c*x]),x]","\frac{d \left(4 a c^3 x^3+6 a c^2 x^2+2 b c^2 x^2+2 b c^2 x^2 (2 c x+3) \tanh ^{-1}(c x)+6 b c x+5 b \log (1-c x)-b \log (c x+1)\right)}{12 c^2}","\frac{1}{3} c d x^3 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{2} d x^2 \left(a+b \tanh ^{-1}(c x)\right)+\frac{5 b d \log (1-c x)}{12 c^2}-\frac{b d \log (c x+1)}{12 c^2}+\frac{b d x}{2 c}+\frac{1}{6} b d x^2",1,"(d*(6*b*c*x + 6*a*c^2*x^2 + 2*b*c^2*x^2 + 4*a*c^3*x^3 + 2*b*c^2*x^2*(3 + 2*c*x)*ArcTanh[c*x] + 5*b*Log[1 - c*x] - b*Log[1 + c*x]))/(12*c^2)","A",1
4,1,95,44,0.0097049,"\int (d+c d x) \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[(d + c*d*x)*(a + b*ArcTanh[c*x]),x]","\frac{1}{2} a c d x^2+a d x+\frac{b d \log \left(1-c^2 x^2\right)}{2 c}+\frac{1}{2} b c d x^2 \tanh ^{-1}(c x)+\frac{b d \log (1-c x)}{4 c}-\frac{b d \log (c x+1)}{4 c}+b d x \tanh ^{-1}(c x)+\frac{b d x}{2}","\frac{d (c x+1)^2 \left(a+b \tanh ^{-1}(c x)\right)}{2 c}+\frac{b d \log (1-c x)}{c}+\frac{b d x}{2}",1,"a*d*x + (b*d*x)/2 + (a*c*d*x^2)/2 + b*d*x*ArcTanh[c*x] + (b*c*d*x^2*ArcTanh[c*x])/2 + (b*d*Log[1 - c*x])/(4*c) - (b*d*Log[1 + c*x])/(4*c) + (b*d*Log[1 - c^2*x^2])/(2*c)","B",1
5,1,54,60,0.0749651,"\int \frac{(d+c d x) \left(a+b \tanh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + c*d*x)*(a + b*ArcTanh[c*x]))/x,x]","\frac{1}{2} d \left(2 a c x+2 a \log (x)+b \log \left(1-c^2 x^2\right)-b \text{Li}_2(-c x)+b \text{Li}_2(c x)+2 b c x \tanh ^{-1}(c x)\right)","a c d x+a d \log (x)+\frac{1}{2} b d \log \left(1-c^2 x^2\right)-\frac{1}{2} b d \text{Li}_2(-c x)+\frac{1}{2} b d \text{Li}_2(c x)+b c d x \tanh ^{-1}(c x)",1,"(d*(2*a*c*x + 2*b*c*x*ArcTanh[c*x] + 2*a*Log[x] + b*Log[1 - c^2*x^2] - b*PolyLog[2, -(c*x)] + b*PolyLog[2, c*x]))/2","A",1
6,1,71,70,0.065747,"\int \frac{(d+c d x) \left(a+b \tanh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + c*d*x)*(a + b*ArcTanh[c*x]))/x^2,x]","a c d \log (x)-\frac{a d}{x}+b c d \left(-\frac{1}{2} \log \left(1-c^2 x^2\right)+\log (c x)-\frac{\tanh ^{-1}(c x)}{c x}\right)+\frac{1}{2} b c d (\text{Li}_2(c x)-\text{Li}_2(-c x))","-\frac{d \left(a+b \tanh ^{-1}(c x)\right)}{x}+a c d \log (x)-\frac{1}{2} b c d \log \left(1-c^2 x^2\right)-\frac{1}{2} b c d \text{Li}_2(-c x)+\frac{1}{2} b c d \text{Li}_2(c x)+b c d \log (x)",1,"-((a*d)/x) + a*c*d*Log[x] + b*c*d*(-(ArcTanh[c*x]/(c*x)) + Log[c*x] - Log[1 - c^2*x^2]/2) + (b*c*d*(-PolyLog[2, -(c*x)] + PolyLog[2, c*x]))/2","A",0
7,1,76,56,0.0612339,"\int \frac{(d+c d x) \left(a+b \tanh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + c*d*x)*(a + b*ArcTanh[c*x]))/x^3,x]","-\frac{d \left(4 a c x+2 a-4 b c^2 x^2 \log (x)+3 b c^2 x^2 \log (1-c x)+b c^2 x^2 \log (c x+1)+2 b c x+2 (2 b c x+b) \tanh ^{-1}(c x)\right)}{4 x^2}","-\frac{d (c x+1)^2 \left(a+b \tanh ^{-1}(c x)\right)}{2 x^2}+b c^2 d \log (x)-b c^2 d \log (1-c x)-\frac{b c d}{2 x}",1,"-1/4*(d*(2*a + 4*a*c*x + 2*b*c*x + 2*(b + 2*b*c*x)*ArcTanh[c*x] - 4*b*c^2*x^2*Log[x] + 3*b*c^2*x^2*Log[1 - c*x] + b*c^2*x^2*Log[1 + c*x]))/x^2","A",1
8,1,86,98,0.0671452,"\int \frac{(d+c d x) \left(a+b \tanh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + c*d*x)*(a + b*ArcTanh[c*x]))/x^4,x]","-\frac{d \left(6 a c x+4 a-4 b c^3 x^3 \log (x)+5 b c^3 x^3 \log (1-c x)-b c^3 x^3 \log (c x+1)+6 b c^2 x^2+2 b c x+2 b (3 c x+2) \tanh ^{-1}(c x)\right)}{12 x^3}","-\frac{d \left(a+b \tanh ^{-1}(c x)\right)}{3 x^3}-\frac{c d \left(a+b \tanh ^{-1}(c x)\right)}{2 x^2}+\frac{1}{3} b c^3 d \log (x)-\frac{5}{12} b c^3 d \log (1-c x)+\frac{1}{12} b c^3 d \log (c x+1)-\frac{b c^2 d}{2 x}-\frac{b c d}{6 x^2}",1,"-1/12*(d*(4*a + 6*a*c*x + 2*b*c*x + 6*b*c^2*x^2 + 2*b*(2 + 3*c*x)*ArcTanh[c*x] - 4*b*c^3*x^3*Log[x] + 5*b*c^3*x^3*Log[1 - c*x] - b*c^3*x^3*Log[1 + c*x]))/x^3","A",1
9,1,94,110,0.0662343,"\int \frac{(d+c d x) \left(a+b \tanh ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((d + c*d*x)*(a + b*ArcTanh[c*x]))/x^5,x]","-\frac{d \left(8 a c x+6 a-8 b c^4 x^4 \log (x)+7 b c^4 x^4 \log (1-c x)+b c^4 x^4 \log (c x+1)+6 b c^3 x^3+4 b c^2 x^2+2 b c x+2 b (4 c x+3) \tanh ^{-1}(c x)\right)}{24 x^4}","-\frac{d \left(a+b \tanh ^{-1}(c x)\right)}{4 x^4}-\frac{c d \left(a+b \tanh ^{-1}(c x)\right)}{3 x^3}+\frac{1}{3} b c^4 d \log (x)-\frac{7}{24} b c^4 d \log (1-c x)-\frac{1}{24} b c^4 d \log (c x+1)-\frac{b c^3 d}{4 x}-\frac{b c^2 d}{6 x^2}-\frac{b c d}{12 x^3}",1,"-1/24*(d*(6*a + 8*a*c*x + 2*b*c*x + 4*b*c^2*x^2 + 6*b*c^3*x^3 + 2*b*(3 + 4*c*x)*ArcTanh[c*x] - 8*b*c^4*x^4*Log[x] + 7*b*c^4*x^4*Log[1 - c*x] + b*c^4*x^4*Log[1 + c*x]))/x^4","A",1
10,1,125,157,0.1081723,"\int x^3 (d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + c*d*x)^2*(a + b*ArcTanh[c*x]),x]","\frac{d^2 \left(60 a c^6 x^6+144 a c^5 x^5+90 a c^4 x^4+12 b c^5 x^5+36 b c^4 x^4+50 b c^3 x^3+72 b c^2 x^2+6 b c^4 x^4 \left(10 c^2 x^2+24 c x+15\right) \tanh ^{-1}(c x)+150 b c x+147 b \log (1-c x)-3 b \log (c x+1)\right)}{360 c^4}","\frac{1}{6} c^2 d^2 x^6 \left(a+b \tanh ^{-1}(c x)\right)+\frac{2}{5} c d^2 x^5 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{4} d^2 x^4 \left(a+b \tanh ^{-1}(c x)\right)+\frac{49 b d^2 \log (1-c x)}{120 c^4}-\frac{b d^2 \log (c x+1)}{120 c^4}+\frac{5 b d^2 x}{12 c^3}+\frac{b d^2 x^2}{5 c^2}+\frac{1}{30} b c d^2 x^5+\frac{5 b d^2 x^3}{36 c}+\frac{1}{10} b d^2 x^4",1,"(d^2*(150*b*c*x + 72*b*c^2*x^2 + 50*b*c^3*x^3 + 90*a*c^4*x^4 + 36*b*c^4*x^4 + 144*a*c^5*x^5 + 12*b*c^5*x^5 + 60*a*c^6*x^6 + 6*b*c^4*x^4*(15 + 24*c*x + 10*c^2*x^2)*ArcTanh[c*x] + 147*b*Log[1 - c*x] - 3*b*Log[1 + c*x]))/(360*c^4)","A",1
11,1,115,143,0.0919733,"\int x^2 (d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + c*d*x)^2*(a + b*ArcTanh[c*x]),x]","\frac{d^2 \left(12 a c^5 x^5+30 a c^4 x^4+20 a c^3 x^3+3 b c^4 x^4+10 b c^3 x^3+16 b c^2 x^2+2 b c^3 x^3 \left(6 c^2 x^2+15 c x+10\right) \tanh ^{-1}(c x)+30 b c x+31 b \log (1-c x)+b \log (c x+1)\right)}{60 c^3}","\frac{1}{5} c^2 d^2 x^5 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{2} c d^2 x^4 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{3} d^2 x^3 \left(a+b \tanh ^{-1}(c x)\right)+\frac{31 b d^2 \log (1-c x)}{60 c^3}+\frac{b d^2 \log (c x+1)}{60 c^3}+\frac{b d^2 x}{2 c^2}+\frac{1}{20} b c d^2 x^4+\frac{4 b d^2 x^2}{15 c}+\frac{1}{6} b d^2 x^3",1,"(d^2*(30*b*c*x + 16*b*c^2*x^2 + 20*a*c^3*x^3 + 10*b*c^3*x^3 + 30*a*c^4*x^4 + 3*b*c^4*x^4 + 12*a*c^5*x^5 + 2*b*c^3*x^3*(10 + 15*c*x + 6*c^2*x^2)*ArcTanh[c*x] + 31*b*Log[1 - c*x] + b*Log[1 + c*x]))/(60*c^3)","A",1
12,1,107,129,0.0861545,"\int x (d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x*(d + c*d*x)^2*(a + b*ArcTanh[c*x]),x]","\frac{d^2 \left(6 a c^4 x^4+16 a c^3 x^3+12 a c^2 x^2+2 b c^3 x^3+8 b c^2 x^2+2 b c^2 x^2 \left(3 c^2 x^2+8 c x+6\right) \tanh ^{-1}(c x)+18 b c x+17 b \log (1-c x)-b \log (c x+1)\right)}{24 c^2}","\frac{1}{4} c^2 d^2 x^4 \left(a+b \tanh ^{-1}(c x)\right)+\frac{2}{3} c d^2 x^3 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{2} d^2 x^2 \left(a+b \tanh ^{-1}(c x)\right)+\frac{17 b d^2 \log (1-c x)}{24 c^2}-\frac{b d^2 \log (c x+1)}{24 c^2}+\frac{1}{12} b c d^2 x^3+\frac{3 b d^2 x}{4 c}+\frac{1}{3} b d^2 x^2",1,"(d^2*(18*b*c*x + 12*a*c^2*x^2 + 8*b*c^2*x^2 + 16*a*c^3*x^3 + 2*b*c^3*x^3 + 6*a*c^4*x^4 + 2*b*c^2*x^2*(6 + 8*c*x + 3*c^2*x^2)*ArcTanh[c*x] + 17*b*Log[1 - c*x] - b*Log[1 + c*x]))/(24*c^2)","A",1
13,1,92,71,0.1060937,"\int (d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[(d + c*d*x)^2*(a + b*ArcTanh[c*x]),x]","\frac{d^2 \left(2 a c^3 x^3+6 a c^2 x^2+6 a c x+b c^2 x^2+b \log \left(1-c^2 x^2\right)+2 b c x \left(c^2 x^2+3 c x+3\right) \tanh ^{-1}(c x)+6 b c x+6 b \log (1-c x)\right)}{6 c}","\frac{d^2 (c x+1)^3 \left(a+b \tanh ^{-1}(c x)\right)}{3 c}+\frac{b d^2 (c x+1)^2}{6 c}+\frac{4 b d^2 \log (1-c x)}{3 c}+\frac{2}{3} b d^2 x",1,"(d^2*(6*a*c*x + 6*b*c*x + 6*a*c^2*x^2 + b*c^2*x^2 + 2*a*c^3*x^3 + 2*b*c*x*(3 + 3*c*x + c^2*x^2)*ArcTanh[c*x] + 6*b*Log[1 - c*x] + b*Log[1 - c^2*x^2]))/(6*c)","A",1
14,1,103,114,0.1033167,"\int \frac{(d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + c*d*x)^2*(a + b*ArcTanh[c*x]))/x,x]","\frac{1}{4} d^2 \left(2 a c^2 x^2+8 a c x+4 a \log (x)+4 b \log \left(1-c^2 x^2\right)+2 b c^2 x^2 \tanh ^{-1}(c x)-2 b \text{Li}_2(-c x)+2 b \text{Li}_2(c x)+2 b c x+b \log (1-c x)-b \log (c x+1)+8 b c x \tanh ^{-1}(c x)\right)","\frac{1}{2} c^2 d^2 x^2 \left(a+b \tanh ^{-1}(c x)\right)+2 a c d^2 x+a d^2 \log (x)+b d^2 \log \left(1-c^2 x^2\right)-\frac{1}{2} b d^2 \text{Li}_2(-c x)+\frac{1}{2} b d^2 \text{Li}_2(c x)+\frac{1}{2} b c d^2 x-\frac{1}{2} b d^2 \tanh ^{-1}(c x)+2 b c d^2 x \tanh ^{-1}(c x)",1,"(d^2*(8*a*c*x + 2*b*c*x + 2*a*c^2*x^2 + 8*b*c*x*ArcTanh[c*x] + 2*b*c^2*x^2*ArcTanh[c*x] + 4*a*Log[x] + b*Log[1 - c*x] - b*Log[1 + c*x] + 4*b*Log[1 - c^2*x^2] - 2*b*PolyLog[2, -(c*x)] + 2*b*PolyLog[2, c*x]))/4","A",0
15,1,73,61,0.1064154,"\int \frac{(d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + c*d*x)^2*(a + b*ArcTanh[c*x]))/x^2,x]","\frac{d^2 \left(a c^2 x^2+2 a c x \log (x)-a+b c^2 x^2 \tanh ^{-1}(c x)-b c x \text{Li}_2(-c x)+b c x \text{Li}_2(c x)+b c x \log (c x)-b \tanh ^{-1}(c x)\right)}{x}","\frac{d^2 \left(c^2 x^2-1\right) \left(a+b \tanh ^{-1}(c x)\right)}{x}+c d^2 (2 a+b) \log (x)-b c d^2 \text{Li}_2(-c x)+b c d^2 \text{Li}_2(c x)",1,"(d^2*(-a + a*c^2*x^2 - b*ArcTanh[c*x] + b*c^2*x^2*ArcTanh[c*x] + 2*a*c*x*Log[x] + b*c*x*Log[c*x] - b*c*x*PolyLog[2, -(c*x)] + b*c*x*PolyLog[2, c*x]))/x","A",0
16,1,143,137,0.0988741,"\int \frac{(d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + c*d*x)^2*(a + b*ArcTanh[c*x]))/x^3,x]","\frac{d^2 \left(4 a c^2 x^2 \log (x)-8 a c x-2 a-2 b c^2 x^2 \text{Li}_2(-c x)+2 b c^2 x^2 \text{Li}_2(c x)+8 b c^2 x^2 \log (c x)-b c^2 x^2 \log (1-c x)+b c^2 x^2 \log (c x+1)-4 b c^2 x^2 \log \left(1-c^2 x^2\right)-2 b c x-8 b c x \tanh ^{-1}(c x)-2 b \tanh ^{-1}(c x)\right)}{4 x^2}","-\frac{d^2 \left(a+b \tanh ^{-1}(c x)\right)}{2 x^2}-\frac{2 c d^2 \left(a+b \tanh ^{-1}(c x)\right)}{x}+a c^2 d^2 \log (x)-\frac{1}{2} b c^2 d^2 \text{Li}_2(-c x)+\frac{1}{2} b c^2 d^2 \text{Li}_2(c x)-b c^2 d^2 \log \left(1-c^2 x^2\right)+2 b c^2 d^2 \log (x)+\frac{1}{2} b c^2 d^2 \tanh ^{-1}(c x)-\frac{b c d^2}{2 x}",1,"(d^2*(-2*a - 8*a*c*x - 2*b*c*x - 2*b*ArcTanh[c*x] - 8*b*c*x*ArcTanh[c*x] + 4*a*c^2*x^2*Log[x] + 8*b*c^2*x^2*Log[c*x] - b*c^2*x^2*Log[1 - c*x] + b*c^2*x^2*Log[1 + c*x] - 4*b*c^2*x^2*Log[1 - c^2*x^2] - 2*b*c^2*x^2*PolyLog[2, -(c*x)] + 2*b*c^2*x^2*PolyLog[2, c*x]))/(4*x^2)","A",0
17,1,103,81,0.0938945,"\int \frac{(d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + c*d*x)^2*(a + b*ArcTanh[c*x]))/x^4,x]","-\frac{d^2 \left(6 a c^2 x^2+6 a c x+2 a-8 b c^3 x^3 \log (x)+7 b c^3 x^3 \log (1-c x)+b c^3 x^3 \log (c x+1)+6 b c^2 x^2+2 b \left(3 c^2 x^2+3 c x+1\right) \tanh ^{-1}(c x)+b c x\right)}{6 x^3}","-\frac{d^2 (c x+1)^3 \left(a+b \tanh ^{-1}(c x)\right)}{3 x^3}+\frac{4}{3} b c^3 d^2 \log (x)-\frac{4}{3} b c^3 d^2 \log (1-c x)-\frac{b c^2 d^2}{x}-\frac{b c d^2}{6 x^2}",1,"-1/6*(d^2*(2*a + 6*a*c*x + b*c*x + 6*a*c^2*x^2 + 6*b*c^2*x^2 + 2*b*(1 + 3*c*x + 3*c^2*x^2)*ArcTanh[c*x] - 8*b*c^3*x^3*Log[x] + 7*b*c^3*x^3*Log[1 - c*x] + b*c^3*x^3*Log[1 + c*x]))/x^3","A",1
18,1,114,147,0.0945531,"\int \frac{(d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((d + c*d*x)^2*(a + b*ArcTanh[c*x]))/x^5,x]","-\frac{d^2 \left(12 a c^2 x^2+16 a c x+6 a-16 b c^4 x^4 \log (x)+17 b c^4 x^4 \log (1-c x)-b c^4 x^4 \log (c x+1)+18 b c^3 x^3+8 b c^2 x^2+2 b \left(6 c^2 x^2+8 c x+3\right) \tanh ^{-1}(c x)+2 b c x\right)}{24 x^4}","-\frac{c^2 d^2 \left(a+b \tanh ^{-1}(c x)\right)}{2 x^2}-\frac{d^2 \left(a+b \tanh ^{-1}(c x)\right)}{4 x^4}-\frac{2 c d^2 \left(a+b \tanh ^{-1}(c x)\right)}{3 x^3}+\frac{2}{3} b c^4 d^2 \log (x)-\frac{17}{24} b c^4 d^2 \log (1-c x)+\frac{1}{24} b c^4 d^2 \log (c x+1)-\frac{3 b c^3 d^2}{4 x}-\frac{b c^2 d^2}{3 x^2}-\frac{b c d^2}{12 x^3}",1,"-1/24*(d^2*(6*a + 16*a*c*x + 2*b*c*x + 12*a*c^2*x^2 + 8*b*c^2*x^2 + 18*b*c^3*x^3 + 2*b*(3 + 8*c*x + 6*c^2*x^2)*ArcTanh[c*x] - 16*b*c^4*x^4*Log[x] + 17*b*c^4*x^4*Log[1 - c*x] - b*c^4*x^4*Log[1 + c*x]))/x^4","A",1
19,1,122,161,0.0968971,"\int \frac{(d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + c*d*x)^2*(a + b*ArcTanh[c*x]))/x^6,x]","-\frac{d^2 \left(20 a c^2 x^2+30 a c x+12 a-32 b c^5 x^5 \log (x)+31 b c^5 x^5 \log (1-c x)+b c^5 x^5 \log (c x+1)+30 b c^4 x^4+16 b c^3 x^3+10 b c^2 x^2+2 b \left(10 c^2 x^2+15 c x+6\right) \tanh ^{-1}(c x)+3 b c x\right)}{60 x^5}","-\frac{c^2 d^2 \left(a+b \tanh ^{-1}(c x)\right)}{3 x^3}-\frac{d^2 \left(a+b \tanh ^{-1}(c x)\right)}{5 x^5}-\frac{c d^2 \left(a+b \tanh ^{-1}(c x)\right)}{2 x^4}+\frac{8}{15} b c^5 d^2 \log (x)-\frac{31}{60} b c^5 d^2 \log (1-c x)-\frac{1}{60} b c^5 d^2 \log (c x+1)-\frac{b c^4 d^2}{2 x}-\frac{4 b c^3 d^2}{15 x^2}-\frac{b c^2 d^2}{6 x^3}-\frac{b c d^2}{20 x^4}",1,"-1/60*(d^2*(12*a + 30*a*c*x + 3*b*c*x + 20*a*c^2*x^2 + 10*b*c^2*x^2 + 16*b*c^3*x^3 + 30*b*c^4*x^4 + 2*b*(6 + 15*c*x + 10*c^2*x^2)*ArcTanh[c*x] - 32*b*c^5*x^5*Log[x] + 31*b*c^5*x^5*Log[1 - c*x] + b*c^5*x^5*Log[1 + c*x]))/x^5","A",1
20,1,151,192,0.1359578,"\int x^3 (d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + c*d*x)^3*(a + b*ArcTanh[c*x]),x]","\frac{d^3 \left(120 a c^7 x^7+420 a c^6 x^6+504 a c^5 x^5+210 a c^4 x^4+20 b c^6 x^6+84 b c^5 x^5+156 b c^4 x^4+210 b c^3 x^3+312 b c^2 x^2+6 b c^4 x^4 \left(20 c^3 x^3+70 c^2 x^2+84 c x+35\right) \tanh ^{-1}(c x)+630 b c x+627 b \log (1-c x)-3 b \log (c x+1)\right)}{840 c^4}","\frac{1}{7} c^3 d^3 x^7 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{2} c^2 d^3 x^6 \left(a+b \tanh ^{-1}(c x)\right)+\frac{3}{5} c d^3 x^5 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{4} d^3 x^4 \left(a+b \tanh ^{-1}(c x)\right)+\frac{209 b d^3 \log (1-c x)}{280 c^4}-\frac{b d^3 \log (c x+1)}{280 c^4}+\frac{3 b d^3 x}{4 c^3}+\frac{1}{42} b c^2 d^3 x^6+\frac{13 b d^3 x^2}{35 c^2}+\frac{1}{10} b c d^3 x^5+\frac{b d^3 x^3}{4 c}+\frac{13}{70} b d^3 x^4",1,"(d^3*(630*b*c*x + 312*b*c^2*x^2 + 210*b*c^3*x^3 + 210*a*c^4*x^4 + 156*b*c^4*x^4 + 504*a*c^5*x^5 + 84*b*c^5*x^5 + 420*a*c^6*x^6 + 20*b*c^6*x^6 + 120*a*c^7*x^7 + 6*b*c^4*x^4*(35 + 84*c*x + 70*c^2*x^2 + 20*c^3*x^3)*ArcTanh[c*x] + 627*b*Log[1 - c*x] - 3*b*Log[1 + c*x]))/(840*c^4)","A",1
21,1,142,178,0.1256748,"\int x^2 (d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + c*d*x)^3*(a + b*ArcTanh[c*x]),x]","\frac{d^3 \left(60 a c^6 x^6+216 a c^5 x^5+270 a c^4 x^4+120 a c^3 x^3+12 b c^5 x^5+54 b c^4 x^4+110 b c^3 x^3+168 b c^2 x^2+6 b c^3 x^3 \left(10 c^3 x^3+36 c^2 x^2+45 c x+20\right) \tanh ^{-1}(c x)+330 b c x+333 b \log (1-c x)+3 b \log (c x+1)\right)}{360 c^3}","\frac{1}{6} c^3 d^3 x^6 \left(a+b \tanh ^{-1}(c x)\right)+\frac{3}{5} c^2 d^3 x^5 \left(a+b \tanh ^{-1}(c x)\right)+\frac{3}{4} c d^3 x^4 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{3} d^3 x^3 \left(a+b \tanh ^{-1}(c x)\right)+\frac{37 b d^3 \log (1-c x)}{40 c^3}+\frac{b d^3 \log (c x+1)}{120 c^3}+\frac{1}{30} b c^2 d^3 x^5+\frac{11 b d^3 x}{12 c^2}+\frac{3}{20} b c d^3 x^4+\frac{7 b d^3 x^2}{15 c}+\frac{11}{36} b d^3 x^3",1,"(d^3*(330*b*c*x + 168*b*c^2*x^2 + 120*a*c^3*x^3 + 110*b*c^3*x^3 + 270*a*c^4*x^4 + 54*b*c^4*x^4 + 216*a*c^5*x^5 + 12*b*c^5*x^5 + 60*a*c^6*x^6 + 6*b*c^3*x^3*(20 + 45*c*x + 36*c^2*x^2 + 10*c^3*x^3)*ArcTanh[c*x] + 333*b*Log[1 - c*x] + 3*b*Log[1 + c*x]))/(360*c^3)","A",1
22,1,133,135,0.1094128,"\int x (d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x*(d + c*d*x)^3*(a + b*ArcTanh[c*x]),x]","\frac{d^3 \left(8 a c^5 x^5+30 a c^4 x^4+40 a c^3 x^3+20 a c^2 x^2+2 b c^4 x^4+10 b c^3 x^3+24 b c^2 x^2+2 b c^2 x^2 \left(4 c^3 x^3+15 c^2 x^2+20 c x+10\right) \tanh ^{-1}(c x)+50 b c x+49 b \log (1-c x)-b \log (c x+1)\right)}{40 c^2}","\frac{d^3 (c x+1)^5 \left(a+b \tanh ^{-1}(c x)\right)}{5 c^2}-\frac{d^3 (c x+1)^4 \left(a+b \tanh ^{-1}(c x)\right)}{4 c^2}+\frac{b d^3 (c x+1)^4}{20 c^2}+\frac{b d^3 (c x+1)^3}{20 c^2}+\frac{3 b d^3 (c x+1)^2}{20 c^2}+\frac{6 b d^3 \log (1-c x)}{5 c^2}+\frac{3 b d^3 x}{5 c}",1,"(d^3*(50*b*c*x + 20*a*c^2*x^2 + 24*b*c^2*x^2 + 40*a*c^3*x^3 + 10*b*c^3*x^3 + 30*a*c^4*x^4 + 2*b*c^4*x^4 + 8*a*c^5*x^5 + 2*b*c^2*x^2*(10 + 20*c*x + 15*c^2*x^2 + 4*c^3*x^3)*ArcTanh[c*x] + 49*b*Log[1 - c*x] - b*Log[1 + c*x]))/(40*c^2)","A",1
23,1,115,84,0.1447119,"\int (d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[(d + c*d*x)^3*(a + b*ArcTanh[c*x]),x]","\frac{d^3 \left(6 a c^4 x^4+24 a c^3 x^3+36 a c^2 x^2+24 a c x+2 b c^3 x^3+12 b c^2 x^2+6 b c x \left(c^3 x^3+4 c^2 x^2+6 c x+4\right) \tanh ^{-1}(c x)+42 b c x+45 b \log (1-c x)+3 b \log (c x+1)\right)}{24 c}","\frac{d^3 (c x+1)^4 \left(a+b \tanh ^{-1}(c x)\right)}{4 c}+\frac{b d^3 (c x+1)^3}{12 c}+\frac{b d^3 (c x+1)^2}{4 c}+\frac{2 b d^3 \log (1-c x)}{c}+b d^3 x",1,"(d^3*(24*a*c*x + 42*b*c*x + 36*a*c^2*x^2 + 12*b*c^2*x^2 + 24*a*c^3*x^3 + 2*b*c^3*x^3 + 6*a*c^4*x^4 + 6*b*c*x*(4 + 6*c*x + 4*c^2*x^2 + c^3*x^3)*ArcTanh[c*x] + 45*b*Log[1 - c*x] + 3*b*Log[1 + c*x]))/(24*c)","A",1
24,1,148,152,0.1333404,"\int \frac{(d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x,x]","\frac{1}{12} d^3 \left(4 a c^3 x^3+18 a c^2 x^2+36 a c x+12 a \log (x)+4 b c^3 x^3 \tanh ^{-1}(c x)+2 b c^2 x^2+18 b \log \left(1-c^2 x^2\right)+2 b \log \left(c^2 x^2-1\right)+18 b c^2 x^2 \tanh ^{-1}(c x)-6 b \text{Li}_2(-c x)+6 b \text{Li}_2(c x)+18 b c x+9 b \log (1-c x)-9 b \log (c x+1)+36 b c x \tanh ^{-1}(c x)\right)","\frac{1}{3} c^3 d^3 x^3 \left(a+b \tanh ^{-1}(c x)\right)+\frac{3}{2} c^2 d^3 x^2 \left(a+b \tanh ^{-1}(c x)\right)+3 a c d^3 x+a d^3 \log (x)+\frac{1}{6} b c^2 d^3 x^2+\frac{5}{3} b d^3 \log \left(1-c^2 x^2\right)-\frac{1}{2} b d^3 \text{Li}_2(-c x)+\frac{1}{2} b d^3 \text{Li}_2(c x)+\frac{3}{2} b c d^3 x-\frac{3}{2} b d^3 \tanh ^{-1}(c x)+3 b c d^3 x \tanh ^{-1}(c x)",1,"(d^3*(36*a*c*x + 18*b*c*x + 18*a*c^2*x^2 + 2*b*c^2*x^2 + 4*a*c^3*x^3 + 36*b*c*x*ArcTanh[c*x] + 18*b*c^2*x^2*ArcTanh[c*x] + 4*b*c^3*x^3*ArcTanh[c*x] + 12*a*Log[x] + 9*b*Log[1 - c*x] - 9*b*Log[1 + c*x] + 18*b*Log[1 - c^2*x^2] + 2*b*Log[-1 + c^2*x^2] - 6*b*PolyLog[2, -(c*x)] + 6*b*PolyLog[2, c*x]))/12","A",0
25,1,149,150,0.14411,"\int \frac{(d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^2,x]","\frac{d^3 \left(2 a c^3 x^3+12 a c^2 x^2+12 a c x \log (x)-4 a+2 b c^3 x^3 \tanh ^{-1}(c x)+2 b c^2 x^2+4 b c x \log \left(1-c^2 x^2\right)+12 b c^2 x^2 \tanh ^{-1}(c x)-6 b c x \text{Li}_2(-c x)+6 b c x \text{Li}_2(c x)+4 b c x \log (c x)+b c x \log (1-c x)-b c x \log (c x+1)-4 b \tanh ^{-1}(c x)\right)}{4 x}","\frac{1}{2} c^3 d^3 x^2 \left(a+b \tanh ^{-1}(c x)\right)-\frac{d^3 \left(a+b \tanh ^{-1}(c x)\right)}{x}+3 a c^2 d^3 x+3 a c d^3 \log (x)+b c d^3 \log \left(1-c^2 x^2\right)+\frac{1}{2} b c^2 d^3 x+3 b c^2 d^3 x \tanh ^{-1}(c x)-\frac{3}{2} b c d^3 \text{Li}_2(-c x)+\frac{3}{2} b c d^3 \text{Li}_2(c x)+b c d^3 \log (x)-\frac{1}{2} b c d^3 \tanh ^{-1}(c x)",1,"(d^3*(-4*a + 12*a*c^2*x^2 + 2*b*c^2*x^2 + 2*a*c^3*x^3 - 4*b*ArcTanh[c*x] + 12*b*c^2*x^2*ArcTanh[c*x] + 2*b*c^3*x^3*ArcTanh[c*x] + 12*a*c*x*Log[x] + 4*b*c*x*Log[c*x] + b*c*x*Log[1 - c*x] - b*c*x*Log[1 + c*x] + 4*b*c*x*Log[1 - c^2*x^2] - 6*b*c*x*PolyLog[2, -(c*x)] + 6*b*c*x*PolyLog[2, c*x]))/(4*x)","A",0
26,1,165,160,0.1448197,"\int \frac{(d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^3,x]","\frac{d^3 \left(4 a c^3 x^3+12 a c^2 x^2 \log (x)-12 a c x-2 a+4 b c^3 x^3 \tanh ^{-1}(c x)-6 b c^2 x^2 \text{Li}_2(-c x)+6 b c^2 x^2 \text{Li}_2(c x)+12 b c^2 x^2 \log (c x)-b c^2 x^2 \log (1-c x)+b c^2 x^2 \log (c x+1)-4 b c^2 x^2 \log \left(1-c^2 x^2\right)-2 b c x-12 b c x \tanh ^{-1}(c x)-2 b \tanh ^{-1}(c x)\right)}{4 x^2}","-\frac{d^3 \left(a+b \tanh ^{-1}(c x)\right)}{2 x^2}-\frac{3 c d^3 \left(a+b \tanh ^{-1}(c x)\right)}{x}+a c^3 d^3 x+3 a c^2 d^3 \log (x)+b c^3 d^3 x \tanh ^{-1}(c x)-\frac{3}{2} b c^2 d^3 \text{Li}_2(-c x)+\frac{3}{2} b c^2 d^3 \text{Li}_2(c x)-b c^2 d^3 \log \left(1-c^2 x^2\right)+3 b c^2 d^3 \log (x)+\frac{1}{2} b c^2 d^3 \tanh ^{-1}(c x)-\frac{b c d^3}{2 x}",1,"(d^3*(-2*a - 12*a*c*x - 2*b*c*x + 4*a*c^3*x^3 - 2*b*ArcTanh[c*x] - 12*b*c*x*ArcTanh[c*x] + 4*b*c^3*x^3*ArcTanh[c*x] + 12*a*c^2*x^2*Log[x] + 12*b*c^2*x^2*Log[c*x] - b*c^2*x^2*Log[1 - c*x] + b*c^2*x^2*Log[1 + c*x] - 4*b*c^2*x^2*Log[1 - c^2*x^2] - 6*b*c^2*x^2*PolyLog[2, -(c*x)] + 6*b*c^2*x^2*PolyLog[2, c*x]))/(4*x^2)","A",0
27,1,175,176,0.1349201,"\int \frac{(d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^4,x]","\frac{d^3 \left(12 a c^3 x^3 \log (x)-36 a c^2 x^2-18 a c x-4 a-6 b c^3 x^3 \text{Li}_2(-c x)+6 b c^3 x^3 \text{Li}_2(c x)+40 b c^3 x^3 \log (c x)-9 b c^3 x^3 \log (1-c x)+9 b c^3 x^3 \log (c x+1)-18 b c^2 x^2-36 b c^2 x^2 \tanh ^{-1}(c x)-20 b c^3 x^3 \log \left(1-c^2 x^2\right)-2 b c x-18 b c x \tanh ^{-1}(c x)-4 b \tanh ^{-1}(c x)\right)}{12 x^3}","-\frac{3 c^2 d^3 \left(a+b \tanh ^{-1}(c x)\right)}{x}-\frac{d^3 \left(a+b \tanh ^{-1}(c x)\right)}{3 x^3}-\frac{3 c d^3 \left(a+b \tanh ^{-1}(c x)\right)}{2 x^2}+a c^3 d^3 \log (x)-\frac{1}{2} b c^3 d^3 \text{Li}_2(-c x)+\frac{1}{2} b c^3 d^3 \text{Li}_2(c x)+\frac{10}{3} b c^3 d^3 \log (x)+\frac{3}{2} b c^3 d^3 \tanh ^{-1}(c x)-\frac{3 b c^2 d^3}{2 x}-\frac{5}{3} b c^3 d^3 \log \left(1-c^2 x^2\right)-\frac{b c d^3}{6 x^2}",1,"(d^3*(-4*a - 18*a*c*x - 2*b*c*x - 36*a*c^2*x^2 - 18*b*c^2*x^2 - 4*b*ArcTanh[c*x] - 18*b*c*x*ArcTanh[c*x] - 36*b*c^2*x^2*ArcTanh[c*x] + 12*a*c^3*x^3*Log[x] + 40*b*c^3*x^3*Log[c*x] - 9*b*c^3*x^3*Log[1 - c*x] + 9*b*c^3*x^3*Log[1 + c*x] - 20*b*c^3*x^3*Log[1 - c^2*x^2] - 6*b*c^3*x^3*PolyLog[2, -(c*x)] + 6*b*c^3*x^3*PolyLog[2, c*x]))/(12*x^3)","A",0
28,1,131,93,0.1176886,"\int \frac{(d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^5,x]","-\frac{d^3 \left(24 a c^3 x^3+36 a c^2 x^2+24 a c x+6 a-48 b c^4 x^4 \log (x)+45 b c^4 x^4 \log (1-c x)+3 b c^4 x^4 \log (c x+1)+42 b c^3 x^3+12 b c^2 x^2+6 b \left(4 c^3 x^3+6 c^2 x^2+4 c x+1\right) \tanh ^{-1}(c x)+2 b c x\right)}{24 x^4}","-\frac{d^3 (c x+1)^4 \left(a+b \tanh ^{-1}(c x)\right)}{4 x^4}+2 b c^4 d^3 \log (x)-2 b c^4 d^3 \log (1-c x)-\frac{7 b c^3 d^3}{4 x}-\frac{b c^2 d^3}{2 x^2}-\frac{b c d^3}{12 x^3}",1,"-1/24*(d^3*(6*a + 24*a*c*x + 2*b*c*x + 36*a*c^2*x^2 + 12*b*c^2*x^2 + 24*a*c^3*x^3 + 42*b*c^3*x^3 + 6*b*(1 + 4*c*x + 6*c^2*x^2 + 4*c^3*x^3)*ArcTanh[c*x] - 48*b*c^4*x^4*Log[x] + 45*b*c^4*x^4*Log[1 - c*x] + 3*b*c^4*x^4*Log[1 + c*x]))/x^4","A",1
29,1,140,137,0.1271975,"\int \frac{(d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^6,x]","-\frac{d^3 \left(20 a c^3 x^3+40 a c^2 x^2+30 a c x+8 a-48 b c^5 x^5 \log (x)+49 b c^5 x^5 \log (1-c x)-b c^5 x^5 \log (c x+1)+50 b c^4 x^4+24 b c^3 x^3+10 b c^2 x^2+2 b \left(10 c^3 x^3+20 c^2 x^2+15 c x+4\right) \tanh ^{-1}(c x)+2 b c x\right)}{40 x^5}","-\frac{d^3 (c x+1)^4 \left(a+b \tanh ^{-1}(c x)\right)}{5 x^5}+\frac{c d^3 (c x+1)^4 \left(a+b \tanh ^{-1}(c x)\right)}{20 x^4}+\frac{6}{5} b c^5 d^3 \log (x)-\frac{6}{5} b c^5 d^3 \log (1-c x)-\frac{5 b c^4 d^3}{4 x}-\frac{3 b c^3 d^3}{5 x^2}-\frac{b c^2 d^3}{4 x^3}-\frac{b c d^3}{20 x^4}",1,"-1/40*(d^3*(8*a + 30*a*c*x + 2*b*c*x + 40*a*c^2*x^2 + 10*b*c^2*x^2 + 20*a*c^3*x^3 + 24*b*c^3*x^3 + 50*b*c^4*x^4 + 2*b*(4 + 15*c*x + 20*c^2*x^2 + 10*c^3*x^3)*ArcTanh[c*x] - 48*b*c^5*x^5*Log[x] + 49*b*c^5*x^5*Log[1 - c*x] - b*c^5*x^5*Log[1 + c*x]))/x^5","A",1
30,1,149,196,0.1299453,"\int \frac{(d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)}{x^7} \, dx","Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^7,x]","-\frac{d^3 \left(120 a c^3 x^3+270 a c^2 x^2+216 a c x+60 a-336 b c^6 x^6 \log (x)+333 b c^6 x^6 \log (1-c x)+3 b c^6 x^6 \log (c x+1)+330 b c^5 x^5+168 b c^4 x^4+110 b c^3 x^3+54 b c^2 x^2+6 b \left(20 c^3 x^3+45 c^2 x^2+36 c x+10\right) \tanh ^{-1}(c x)+12 b c x\right)}{360 x^6}","-\frac{c^3 d^3 \left(a+b \tanh ^{-1}(c x)\right)}{3 x^3}-\frac{3 c^2 d^3 \left(a+b \tanh ^{-1}(c x)\right)}{4 x^4}-\frac{d^3 \left(a+b \tanh ^{-1}(c x)\right)}{6 x^6}-\frac{3 c d^3 \left(a+b \tanh ^{-1}(c x)\right)}{5 x^5}+\frac{14}{15} b c^6 d^3 \log (x)-\frac{37}{40} b c^6 d^3 \log (1-c x)-\frac{1}{120} b c^6 d^3 \log (c x+1)-\frac{11 b c^5 d^3}{12 x}-\frac{7 b c^4 d^3}{15 x^2}-\frac{11 b c^3 d^3}{36 x^3}-\frac{3 b c^2 d^3}{20 x^4}-\frac{b c d^3}{30 x^5}",1,"-1/360*(d^3*(60*a + 216*a*c*x + 12*b*c*x + 270*a*c^2*x^2 + 54*b*c^2*x^2 + 120*a*c^3*x^3 + 110*b*c^3*x^3 + 168*b*c^4*x^4 + 330*b*c^5*x^5 + 6*b*(10 + 36*c*x + 45*c^2*x^2 + 20*c^3*x^3)*ArcTanh[c*x] - 336*b*c^6*x^6*Log[x] + 333*b*c^6*x^6*Log[1 - c*x] + 3*b*c^6*x^6*Log[1 + c*x]))/x^6","A",1
31,1,177,224,0.1587314,"\int x^3 (d+c d x)^4 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + c*d*x)^4*(a + b*ArcTanh[c*x]),x]","\frac{d^4 \left(210 a c^8 x^8+960 a c^7 x^7+1680 a c^6 x^6+1344 a c^5 x^5+420 a c^4 x^4+30 b c^7 x^7+160 b c^6 x^6+378 b c^5 x^5+576 b c^4 x^4+770 b c^3 x^3+1152 b c^2 x^2+6 b c^4 x^4 \left(35 c^4 x^4+160 c^3 x^3+280 c^2 x^2+224 c x+70\right) \tanh ^{-1}(c x)+2310 b c x+2307 b \log (1-c x)-3 b \log (c x+1)\right)}{1680 c^4}","\frac{1}{8} c^4 d^4 x^8 \left(a+b \tanh ^{-1}(c x)\right)+\frac{4}{7} c^3 d^4 x^7 \left(a+b \tanh ^{-1}(c x)\right)+c^2 d^4 x^6 \left(a+b \tanh ^{-1}(c x)\right)+\frac{4}{5} c d^4 x^5 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{4} d^4 x^4 \left(a+b \tanh ^{-1}(c x)\right)+\frac{769 b d^4 \log (1-c x)}{560 c^4}-\frac{b d^4 \log (c x+1)}{560 c^4}+\frac{1}{56} b c^3 d^4 x^7+\frac{11 b d^4 x}{8 c^3}+\frac{2}{21} b c^2 d^4 x^6+\frac{24 b d^4 x^2}{35 c^2}+\frac{9}{40} b c d^4 x^5+\frac{11 b d^4 x^3}{24 c}+\frac{12}{35} b d^4 x^4",1,"(d^4*(2310*b*c*x + 1152*b*c^2*x^2 + 770*b*c^3*x^3 + 420*a*c^4*x^4 + 576*b*c^4*x^4 + 1344*a*c^5*x^5 + 378*b*c^5*x^5 + 1680*a*c^6*x^6 + 160*b*c^6*x^6 + 960*a*c^7*x^7 + 30*b*c^7*x^7 + 210*a*c^8*x^8 + 6*b*c^4*x^4*(70 + 224*c*x + 280*c^2*x^2 + 160*c^3*x^3 + 35*c^4*x^4)*ArcTanh[c*x] + 2307*b*Log[1 - c*x] - 3*b*Log[1 + c*x]))/(1680*c^4)","A",1
32,1,168,171,0.1496315,"\int x^2 (d+c d x)^4 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + c*d*x)^4*(a + b*ArcTanh[c*x]),x]","\frac{d^4 \left(180 a c^7 x^7+840 a c^6 x^6+1512 a c^5 x^5+1260 a c^4 x^4+420 a c^3 x^3+30 b c^6 x^6+168 b c^5 x^5+423 b c^4 x^4+700 b c^3 x^3+1056 b c^2 x^2+12 b c^3 x^3 \left(15 c^4 x^4+70 c^3 x^3+126 c^2 x^2+105 c x+35\right) \tanh ^{-1}(c x)+2100 b c x+2106 b \log (1-c x)+6 b \log (c x+1)\right)}{1260 c^3}","\frac{d^4 (c x+1)^7 \left(a+b \tanh ^{-1}(c x)\right)}{7 c^3}-\frac{d^4 (c x+1)^6 \left(a+b \tanh ^{-1}(c x)\right)}{3 c^3}+\frac{d^4 (c x+1)^5 \left(a+b \tanh ^{-1}(c x)\right)}{5 c^3}+\frac{1}{42} b c^3 d^4 x^6+\frac{176 b d^4 \log (1-c x)}{105 c^3}+\frac{2}{15} b c^2 d^4 x^5+\frac{5 b d^4 x}{3 c^2}+\frac{47}{140} b c d^4 x^4+\frac{88 b d^4 x^2}{105 c}+\frac{5}{9} b d^4 x^3",1,"(d^4*(2100*b*c*x + 1056*b*c^2*x^2 + 420*a*c^3*x^3 + 700*b*c^3*x^3 + 1260*a*c^4*x^4 + 423*b*c^4*x^4 + 1512*a*c^5*x^5 + 168*b*c^5*x^5 + 840*a*c^6*x^6 + 30*b*c^6*x^6 + 180*a*c^7*x^7 + 12*b*c^3*x^3*(35 + 105*c*x + 126*c^2*x^2 + 70*c^3*x^3 + 15*c^4*x^4)*ArcTanh[c*x] + 2106*b*Log[1 - c*x] + 6*b*Log[1 + c*x]))/(1260*c^3)","A",1
33,1,159,153,0.1407613,"\int x (d+c d x)^4 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x*(d + c*d*x)^4*(a + b*ArcTanh[c*x]),x]","\frac{d^4 \left(30 a c^6 x^6+144 a c^5 x^5+270 a c^4 x^4+240 a c^3 x^3+90 a c^2 x^2+6 b c^5 x^5+36 b c^4 x^4+100 b c^3 x^3+192 b c^2 x^2+6 b c^2 x^2 \left(5 c^4 x^4+24 c^3 x^3+45 c^2 x^2+40 c x+15\right) \tanh ^{-1}(c x)+390 b c x+387 b \log (1-c x)-3 b \log (c x+1)\right)}{180 c^2}","\frac{d^4 (c x+1)^6 \left(a+b \tanh ^{-1}(c x)\right)}{6 c^2}-\frac{d^4 (c x+1)^5 \left(a+b \tanh ^{-1}(c x)\right)}{5 c^2}+\frac{b d^4 (c x+1)^5}{30 c^2}+\frac{b d^4 (c x+1)^4}{30 c^2}+\frac{4 b d^4 (c x+1)^3}{45 c^2}+\frac{4 b d^4 (c x+1)^2}{15 c^2}+\frac{32 b d^4 \log (1-c x)}{15 c^2}+\frac{16 b d^4 x}{15 c}",1,"(d^4*(390*b*c*x + 90*a*c^2*x^2 + 192*b*c^2*x^2 + 240*a*c^3*x^3 + 100*b*c^3*x^3 + 270*a*c^4*x^4 + 36*b*c^4*x^4 + 144*a*c^5*x^5 + 6*b*c^5*x^5 + 30*a*c^6*x^6 + 6*b*c^2*x^2*(15 + 40*c*x + 45*c^2*x^2 + 24*c^3*x^3 + 5*c^4*x^4)*ArcTanh[c*x] + 387*b*Log[1 - c*x] - 3*b*Log[1 + c*x]))/(180*c^2)","A",1
34,1,146,107,0.1762713,"\int (d+c d x)^4 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[(d + c*d*x)^4*(a + b*ArcTanh[c*x]),x]","\frac{d^4 \left(12 a c^5 x^5+60 a c^4 x^4+120 a c^3 x^3+120 a c^2 x^2+60 a c x+3 b c^4 x^4+20 b c^3 x^3+66 b c^2 x^2+6 b \log \left(1-c^2 x^2\right)+12 b c x \left(c^4 x^4+5 c^3 x^3+10 c^2 x^2+10 c x+5\right) \tanh ^{-1}(c x)+180 b c x+180 b \log (1-c x)\right)}{60 c}","\frac{d^4 (c x+1)^5 \left(a+b \tanh ^{-1}(c x)\right)}{5 c}+\frac{b d^4 (c x+1)^4}{20 c}+\frac{2 b d^4 (c x+1)^3}{15 c}+\frac{2 b d^4 (c x+1)^2}{5 c}+\frac{16 b d^4 \log (1-c x)}{5 c}+\frac{8}{5} b d^4 x",1,"(d^4*(60*a*c*x + 180*b*c*x + 120*a*c^2*x^2 + 66*b*c^2*x^2 + 120*a*c^3*x^3 + 20*b*c^3*x^3 + 60*a*c^4*x^4 + 3*b*c^4*x^4 + 12*a*c^5*x^5 + 12*b*c*x*(5 + 10*c*x + 10*c^2*x^2 + 5*c^3*x^3 + c^4*x^4)*ArcTanh[c*x] + 180*b*Log[1 - c*x] + 6*b*Log[1 - c^2*x^2]))/(60*c)","A",1
35,1,179,185,0.1708288,"\int \frac{(d+c d x)^4 \left(a+b \tanh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x,x]","\frac{1}{24} d^4 \left(6 a c^4 x^4+32 a c^3 x^3+72 a c^2 x^2+96 a c x+24 a \log (x)+6 b c^4 x^4 \tanh ^{-1}(c x)+2 b c^3 x^3+32 b c^3 x^3 \tanh ^{-1}(c x)+16 b c^2 x^2+48 b \log \left(1-c^2 x^2\right)+16 b \log \left(c^2 x^2-1\right)+72 b c^2 x^2 \tanh ^{-1}(c x)-12 b \text{Li}_2(-c x)+12 b \text{Li}_2(c x)+78 b c x+39 b \log (1-c x)-39 b \log (c x+1)+96 b c x \tanh ^{-1}(c x)\right)","\frac{1}{4} c^4 d^4 x^4 \left(a+b \tanh ^{-1}(c x)\right)+\frac{4}{3} c^3 d^4 x^3 \left(a+b \tanh ^{-1}(c x)\right)+3 c^2 d^4 x^2 \left(a+b \tanh ^{-1}(c x)\right)+4 a c d^4 x+a d^4 \log (x)+\frac{1}{12} b c^3 d^4 x^3+\frac{2}{3} b c^2 d^4 x^2+\frac{8}{3} b d^4 \log \left(1-c^2 x^2\right)-\frac{1}{2} b d^4 \text{Li}_2(-c x)+\frac{1}{2} b d^4 \text{Li}_2(c x)+\frac{13}{4} b c d^4 x-\frac{13}{4} b d^4 \tanh ^{-1}(c x)+4 b c d^4 x \tanh ^{-1}(c x)",1,"(d^4*(96*a*c*x + 78*b*c*x + 72*a*c^2*x^2 + 16*b*c^2*x^2 + 32*a*c^3*x^3 + 2*b*c^3*x^3 + 6*a*c^4*x^4 + 96*b*c*x*ArcTanh[c*x] + 72*b*c^2*x^2*ArcTanh[c*x] + 32*b*c^3*x^3*ArcTanh[c*x] + 6*b*c^4*x^4*ArcTanh[c*x] + 24*a*Log[x] + 39*b*Log[1 - c*x] - 39*b*Log[1 + c*x] + 48*b*Log[1 - c^2*x^2] + 16*b*Log[-1 + c^2*x^2] - 12*b*PolyLog[2, -(c*x)] + 12*b*PolyLog[2, c*x]))/24","A",0
36,1,194,178,0.1776093,"\int \frac{(d+c d x)^4 \left(a+b \tanh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^2,x]","\frac{d^4 \left(2 a c^4 x^4+12 a c^3 x^3+36 a c^2 x^2+24 a c x \log (x)-6 a+2 b c^4 x^4 \tanh ^{-1}(c x)+b c^3 x^3+12 b c^3 x^3 \tanh ^{-1}(c x)+12 b c^2 x^2+15 b c x \log \left(1-c^2 x^2\right)+b c x \log \left(c^2 x^2-1\right)+36 b c^2 x^2 \tanh ^{-1}(c x)-12 b c x \text{Li}_2(-c x)+12 b c x \text{Li}_2(c x)+6 b c x \log (c x)+6 b c x \log (1-c x)-6 b c x \log (c x+1)-6 b \tanh ^{-1}(c x)\right)}{6 x}","\frac{1}{3} c^4 d^4 x^3 \left(a+b \tanh ^{-1}(c x)\right)+2 c^3 d^4 x^2 \left(a+b \tanh ^{-1}(c x)\right)-\frac{d^4 \left(a+b \tanh ^{-1}(c x)\right)}{x}+6 a c^2 d^4 x+4 a c d^4 \log (x)+\frac{1}{6} b c^3 d^4 x^2+\frac{8}{3} b c d^4 \log \left(1-c^2 x^2\right)+2 b c^2 d^4 x+6 b c^2 d^4 x \tanh ^{-1}(c x)-2 b c d^4 \text{Li}_2(-c x)+2 b c d^4 \text{Li}_2(c x)+b c d^4 \log (x)-2 b c d^4 \tanh ^{-1}(c x)",1,"(d^4*(-6*a + 36*a*c^2*x^2 + 12*b*c^2*x^2 + 12*a*c^3*x^3 + b*c^3*x^3 + 2*a*c^4*x^4 - 6*b*ArcTanh[c*x] + 36*b*c^2*x^2*ArcTanh[c*x] + 12*b*c^3*x^3*ArcTanh[c*x] + 2*b*c^4*x^4*ArcTanh[c*x] + 24*a*c*x*Log[x] + 6*b*c*x*Log[c*x] + 6*b*c*x*Log[1 - c*x] - 6*b*c*x*Log[1 + c*x] + 15*b*c*x*Log[1 - c^2*x^2] + b*c*x*Log[-1 + c^2*x^2] - 12*b*c*x*PolyLog[2, -(c*x)] + 12*b*c*x*PolyLog[2, c*x]))/(6*x)","A",0
37,1,143,156,0.1654519,"\int \frac{(d+c d x)^4 \left(a+b \tanh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^3,x]","\frac{d^4 \left(a c^4 x^4+8 a c^3 x^3+12 a c^2 x^2 \log (x)-8 a c x-a+b c^4 x^4 \tanh ^{-1}(c x)+b c^3 x^3+8 b c^3 x^3 \tanh ^{-1}(c x)-6 b c^2 x^2 \text{Li}_2(-c x)+6 b c^2 x^2 \text{Li}_2(c x)+8 b c^2 x^2 \log (c x)-b c x-8 b c x \tanh ^{-1}(c x)-b \tanh ^{-1}(c x)\right)}{2 x^2}","\frac{1}{2} c^4 d^4 x^2 \left(a+b \tanh ^{-1}(c x)\right)-\frac{d^4 \left(a+b \tanh ^{-1}(c x)\right)}{2 x^2}-\frac{4 c d^4 \left(a+b \tanh ^{-1}(c x)\right)}{x}+4 a c^3 d^4 x+6 a c^2 d^4 \log (x)+\frac{1}{2} b c^3 d^4 x+4 b c^3 d^4 x \tanh ^{-1}(c x)-3 b c^2 d^4 \text{Li}_2(-c x)+3 b c^2 d^4 \text{Li}_2(c x)+4 b c^2 d^4 \log (x)-\frac{b c d^4}{2 x}",1,"(d^4*(-a - 8*a*c*x - b*c*x + 8*a*c^3*x^3 + b*c^3*x^3 + a*c^4*x^4 - b*ArcTanh[c*x] - 8*b*c*x*ArcTanh[c*x] + 8*b*c^3*x^3*ArcTanh[c*x] + b*c^4*x^4*ArcTanh[c*x] + 12*a*c^2*x^2*Log[x] + 8*b*c^2*x^2*Log[c*x] - 6*b*c^2*x^2*PolyLog[2, -(c*x)] + 6*b*c^2*x^2*PolyLog[2, c*x]))/(2*x^2)","A",0
38,1,197,189,0.1753809,"\int \frac{(d+c d x)^4 \left(a+b \tanh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^4,x]","\frac{d^4 \left(6 a c^4 x^4+24 a c^3 x^3 \log (x)-36 a c^2 x^2-12 a c x-2 a+6 b c^4 x^4 \tanh ^{-1}(c x)-12 b c^3 x^3 \text{Li}_2(-c x)+12 b c^3 x^3 \text{Li}_2(c x)+38 b c^3 x^3 \log (c x)-6 b c^3 x^3 \log (1-c x)+6 b c^3 x^3 \log (c x+1)-12 b c^2 x^2-36 b c^2 x^2 \tanh ^{-1}(c x)-16 b c^3 x^3 \log \left(1-c^2 x^2\right)-b c x-12 b c x \tanh ^{-1}(c x)-2 b \tanh ^{-1}(c x)\right)}{6 x^3}","-\frac{6 c^2 d^4 \left(a+b \tanh ^{-1}(c x)\right)}{x}-\frac{d^4 \left(a+b \tanh ^{-1}(c x)\right)}{3 x^3}-\frac{2 c d^4 \left(a+b \tanh ^{-1}(c x)\right)}{x^2}+a c^4 d^4 x+4 a c^3 d^4 \log (x)+b c^4 d^4 x \tanh ^{-1}(c x)-2 b c^3 d^4 \text{Li}_2(-c x)+2 b c^3 d^4 \text{Li}_2(c x)+\frac{19}{3} b c^3 d^4 \log (x)+2 b c^3 d^4 \tanh ^{-1}(c x)-\frac{2 b c^2 d^4}{x}-\frac{8}{3} b c^3 d^4 \log \left(1-c^2 x^2\right)-\frac{b c d^4}{6 x^2}",1,"(d^4*(-2*a - 12*a*c*x - b*c*x - 36*a*c^2*x^2 - 12*b*c^2*x^2 + 6*a*c^4*x^4 - 2*b*ArcTanh[c*x] - 12*b*c*x*ArcTanh[c*x] - 36*b*c^2*x^2*ArcTanh[c*x] + 6*b*c^4*x^4*ArcTanh[c*x] + 24*a*c^3*x^3*Log[x] + 38*b*c^3*x^3*Log[c*x] - 6*b*c^3*x^3*Log[1 - c*x] + 6*b*c^3*x^3*Log[1 + c*x] - 16*b*c^3*x^3*Log[1 - c^2*x^2] - 12*b*c^3*x^3*PolyLog[2, -(c*x)] + 12*b*c^3*x^3*PolyLog[2, c*x]))/(6*x^3)","A",0
39,1,206,209,0.160298,"\int \frac{(d+c d x)^4 \left(a+b \tanh ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^5,x]","\frac{d^4 \left(24 a c^4 x^4 \log (x)-96 a c^3 x^3-72 a c^2 x^2-32 a c x-6 a-12 b c^4 x^4 \text{Li}_2(-c x)+12 b c^4 x^4 \text{Li}_2(c x)+128 b c^4 x^4 \log (c x)-39 b c^4 x^4 \log (1-c x)+39 b c^4 x^4 \log (c x+1)-78 b c^3 x^3-96 b c^3 x^3 \tanh ^{-1}(c x)-16 b c^2 x^2-72 b c^2 x^2 \tanh ^{-1}(c x)-64 b c^4 x^4 \log \left(1-c^2 x^2\right)-2 b c x-32 b c x \tanh ^{-1}(c x)-6 b \tanh ^{-1}(c x)\right)}{24 x^4}","-\frac{4 c^3 d^4 \left(a+b \tanh ^{-1}(c x)\right)}{x}-\frac{3 c^2 d^4 \left(a+b \tanh ^{-1}(c x)\right)}{x^2}-\frac{d^4 \left(a+b \tanh ^{-1}(c x)\right)}{4 x^4}-\frac{4 c d^4 \left(a+b \tanh ^{-1}(c x)\right)}{3 x^3}+a c^4 d^4 \log (x)-\frac{1}{2} b c^4 d^4 \text{Li}_2(-c x)+\frac{1}{2} b c^4 d^4 \text{Li}_2(c x)+\frac{16}{3} b c^4 d^4 \log (x)+\frac{13}{4} b c^4 d^4 \tanh ^{-1}(c x)-\frac{13 b c^3 d^4}{4 x}-\frac{2 b c^2 d^4}{3 x^2}-\frac{8}{3} b c^4 d^4 \log \left(1-c^2 x^2\right)-\frac{b c d^4}{12 x^3}",1,"(d^4*(-6*a - 32*a*c*x - 2*b*c*x - 72*a*c^2*x^2 - 16*b*c^2*x^2 - 96*a*c^3*x^3 - 78*b*c^3*x^3 - 6*b*ArcTanh[c*x] - 32*b*c*x*ArcTanh[c*x] - 72*b*c^2*x^2*ArcTanh[c*x] - 96*b*c^3*x^3*ArcTanh[c*x] + 24*a*c^4*x^4*Log[x] + 128*b*c^4*x^4*Log[c*x] - 39*b*c^4*x^4*Log[1 - c*x] + 39*b*c^4*x^4*Log[1 + c*x] - 64*b*c^4*x^4*Log[1 - c^2*x^2] - 12*b*c^4*x^4*PolyLog[2, -(c*x)] + 12*b*c^4*x^4*PolyLog[2, c*x]))/(24*x^4)","A",0
40,1,157,109,0.1458434,"\int \frac{(d+c d x)^4 \left(a+b \tanh ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^6,x]","-\frac{d^4 \left(60 a c^4 x^4+120 a c^3 x^3+120 a c^2 x^2+60 a c x+12 a-192 b c^5 x^5 \log (x)+186 b c^5 x^5 \log (1-c x)+6 b c^5 x^5 \log (c x+1)+180 b c^4 x^4+66 b c^3 x^3+20 b c^2 x^2+12 b \left(5 c^4 x^4+10 c^3 x^3+10 c^2 x^2+5 c x+1\right) \tanh ^{-1}(c x)+3 b c x\right)}{60 x^5}","-\frac{d^4 (c x+1)^5 \left(a+b \tanh ^{-1}(c x)\right)}{5 x^5}+\frac{16}{5} b c^5 d^4 \log (x)-\frac{16}{5} b c^5 d^4 \log (1-c x)-\frac{3 b c^4 d^4}{x}-\frac{11 b c^3 d^4}{10 x^2}-\frac{b c^2 d^4}{3 x^3}-\frac{b c d^4}{20 x^4}",1,"-1/60*(d^4*(12*a + 60*a*c*x + 3*b*c*x + 120*a*c^2*x^2 + 20*b*c^2*x^2 + 120*a*c^3*x^3 + 66*b*c^3*x^3 + 60*a*c^4*x^4 + 180*b*c^4*x^4 + 12*b*(1 + 5*c*x + 10*c^2*x^2 + 10*c^3*x^3 + 5*c^4*x^4)*ArcTanh[c*x] - 192*b*c^5*x^5*Log[x] + 186*b*c^5*x^5*Log[1 - c*x] + 6*b*c^5*x^5*Log[1 + c*x]))/x^5","A",1
41,1,166,151,0.154878,"\int \frac{(d+c d x)^4 \left(a+b \tanh ^{-1}(c x)\right)}{x^7} \, dx","Integrate[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^7,x]","-\frac{d^4 \left(90 a c^4 x^4+240 a c^3 x^3+270 a c^2 x^2+144 a c x+30 a-384 b c^6 x^6 \log (x)+387 b c^6 x^6 \log (1-c x)-3 b c^6 x^6 \log (c x+1)+390 b c^5 x^5+192 b c^4 x^4+100 b c^3 x^3+36 b c^2 x^2+6 b \left(15 c^4 x^4+40 c^3 x^3+45 c^2 x^2+24 c x+5\right) \tanh ^{-1}(c x)+6 b c x\right)}{180 x^6}","-\frac{d^4 (c x+1)^5 \left(a+b \tanh ^{-1}(c x)\right)}{6 x^6}+\frac{c d^4 (c x+1)^5 \left(a+b \tanh ^{-1}(c x)\right)}{30 x^5}+\frac{32}{15} b c^6 d^4 \log (x)-\frac{32}{15} b c^6 d^4 \log (1-c x)-\frac{13 b c^5 d^4}{6 x}-\frac{16 b c^4 d^4}{15 x^2}-\frac{5 b c^3 d^4}{9 x^3}-\frac{b c^2 d^4}{5 x^4}-\frac{b c d^4}{30 x^5}",1,"-1/180*(d^4*(30*a + 144*a*c*x + 6*b*c*x + 270*a*c^2*x^2 + 36*b*c^2*x^2 + 240*a*c^3*x^3 + 100*b*c^3*x^3 + 90*a*c^4*x^4 + 192*b*c^4*x^4 + 390*b*c^5*x^5 + 6*b*(5 + 24*c*x + 45*c^2*x^2 + 40*c^3*x^3 + 15*c^4*x^4)*ArcTanh[c*x] - 384*b*c^6*x^6*Log[x] + 387*b*c^6*x^6*Log[1 - c*x] - 3*b*c^6*x^6*Log[1 + c*x]))/x^6","A",1
42,1,175,229,0.1783623,"\int \frac{(d+c d x)^4 \left(a+b \tanh ^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^8,x]","-\frac{d^4 \left(420 a c^4 x^4+1260 a c^3 x^3+1512 a c^2 x^2+840 a c x+180 a-2112 b c^7 x^7 \log (x)+2106 b c^7 x^7 \log (1-c x)+6 b c^7 x^7 \log (c x+1)+2100 b c^6 x^6+1056 b c^5 x^5+700 b c^4 x^4+423 b c^3 x^3+168 b c^2 x^2+12 b \left(35 c^4 x^4+105 c^3 x^3+126 c^2 x^2+70 c x+15\right) \tanh ^{-1}(c x)+30 b c x\right)}{1260 x^7}","-\frac{c^4 d^4 \left(a+b \tanh ^{-1}(c x)\right)}{3 x^3}-\frac{c^3 d^4 \left(a+b \tanh ^{-1}(c x)\right)}{x^4}-\frac{6 c^2 d^4 \left(a+b \tanh ^{-1}(c x)\right)}{5 x^5}-\frac{d^4 \left(a+b \tanh ^{-1}(c x)\right)}{7 x^7}-\frac{2 c d^4 \left(a+b \tanh ^{-1}(c x)\right)}{3 x^6}+\frac{176}{105} b c^7 d^4 \log (x)-\frac{117}{70} b c^7 d^4 \log (1-c x)-\frac{1}{210} b c^7 d^4 \log (c x+1)-\frac{5 b c^6 d^4}{3 x}-\frac{88 b c^5 d^4}{105 x^2}-\frac{5 b c^4 d^4}{9 x^3}-\frac{47 b c^3 d^4}{140 x^4}-\frac{2 b c^2 d^4}{15 x^5}-\frac{b c d^4}{42 x^6}",1,"-1/1260*(d^4*(180*a + 840*a*c*x + 30*b*c*x + 1512*a*c^2*x^2 + 168*b*c^2*x^2 + 1260*a*c^3*x^3 + 423*b*c^3*x^3 + 420*a*c^4*x^4 + 700*b*c^4*x^4 + 1056*b*c^5*x^5 + 2100*b*c^6*x^6 + 12*b*(15 + 70*c*x + 126*c^2*x^2 + 105*c^3*x^3 + 35*c^4*x^4)*ArcTanh[c*x] - 2112*b*c^7*x^7*Log[x] + 2106*b*c^7*x^7*Log[1 - c*x] + 6*b*c^7*x^7*Log[1 + c*x]))/x^7","A",1
43,1,129,177,0.3910845,"\int \frac{x^3 \left(a+b \tanh ^{-1}(c x)\right)}{d+c d x} \, dx","Integrate[(x^3*(a + b*ArcTanh[c*x]))/(d + c*d*x),x]","\frac{2 a c^3 x^3-3 a c^2 x^2+6 a c x-6 a \log (c x+1)+b c^2 x^2+4 b \log \left(1-c^2 x^2\right)+b \tanh ^{-1}(c x) \left(2 c^3 x^3-3 c^2 x^2+6 c x+6 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+3\right)-3 b \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-3 b c x-b}{6 c^4 d}","\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^4 d}-\frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)}{2 c^2 d}+\frac{x^3 \left(a+b \tanh ^{-1}(c x)\right)}{3 c d}+\frac{a x}{c^3 d}-\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 c^4 d}+\frac{b \tanh ^{-1}(c x)}{2 c^4 d}-\frac{b x}{2 c^3 d}+\frac{b x \tanh ^{-1}(c x)}{c^3 d}+\frac{b x^2}{6 c^2 d}+\frac{2 b \log \left(1-c^2 x^2\right)}{3 c^4 d}",1,"(-b + 6*a*c*x - 3*b*c*x - 3*a*c^2*x^2 + b*c^2*x^2 + 2*a*c^3*x^3 + b*ArcTanh[c*x]*(3 + 6*c*x - 3*c^2*x^2 + 2*c^3*x^3 + 6*Log[1 + E^(-2*ArcTanh[c*x])]) - 6*a*Log[1 + c*x] + 4*b*Log[1 - c^2*x^2] - 3*b*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(6*c^4*d)","A",0
44,1,97,145,0.2469141,"\int \frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)}{d+c d x} \, dx","Integrate[(x^2*(a + b*ArcTanh[c*x]))/(d + c*d*x),x]","\frac{a c^2 x^2-2 a c x+2 a \log (c x+1)-b \log \left(1-c^2 x^2\right)+b \tanh ^{-1}(c x) \left(c^2 x^2-2 c x-2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-1\right)+b \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+b c x}{2 c^3 d}","-\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^3 d}+\frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)}{2 c d}-\frac{a x}{c^2 d}+\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 c^3 d}-\frac{b \tanh ^{-1}(c x)}{2 c^3 d}+\frac{b x}{2 c^2 d}-\frac{b x \tanh ^{-1}(c x)}{c^2 d}-\frac{b \log \left(1-c^2 x^2\right)}{2 c^3 d}",1,"(-2*a*c*x + b*c*x + a*c^2*x^2 + b*ArcTanh[c*x]*(-1 - 2*c*x + c^2*x^2 - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + 2*a*Log[1 + c*x] - b*Log[1 - c^2*x^2] + b*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(2*c^3*d)","A",0
45,1,75,94,0.1559725,"\int \frac{x \left(a+b \tanh ^{-1}(c x)\right)}{d+c d x} \, dx","Integrate[(x*(a + b*ArcTanh[c*x]))/(d + c*d*x),x]","\frac{2 a c x-2 a \log (c x+1)+b \log \left(1-c^2 x^2\right)-b \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+2 b \tanh ^{-1}(c x) \left(c x+\log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)}{2 c^2 d}","\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^2 d}+\frac{a x}{c d}-\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 c^2 d}+\frac{b \log \left(1-c^2 x^2\right)}{2 c^2 d}+\frac{b x \tanh ^{-1}(c x)}{c d}",1,"(2*a*c*x + 2*b*ArcTanh[c*x]*(c*x + Log[1 + E^(-2*ArcTanh[c*x])]) - 2*a*Log[1 + c*x] + b*Log[1 - c^2*x^2] - b*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(2*c^2*d)","A",0
46,1,52,51,0.1005921,"\int \frac{a+b \tanh ^{-1}(c x)}{d+c d x} \, dx","Integrate[(a + b*ArcTanh[c*x])/(d + c*d*x),x]","\frac{2 a \log (c x+1)+b \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-2 b \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)}{2 c d}","\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 c d}-\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c d}",1,"(-2*b*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + 2*a*Log[1 + c*x] + b*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(2*c*d)","A",0
47,1,55,46,0.1126484,"\int \frac{a+b \tanh ^{-1}(c x)}{x (d+c d x)} \, dx","Integrate[(a + b*ArcTanh[c*x])/(x*(d + c*d*x)),x]","\frac{-2 a \log (c x+1)+2 a \log (x)-b \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+2 b \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)}{2 d}","\frac{\log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d}-\frac{b \text{Li}_2\left(\frac{2}{c x+1}-1\right)}{2 d}",1,"(2*b*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] + 2*a*Log[x] - 2*a*Log[1 + c*x] - b*PolyLog[2, E^(-2*ArcTanh[c*x])])/(2*d)","A",0
48,1,93,93,0.1990655,"\int \frac{a+b \tanh ^{-1}(c x)}{x^2 (d+c d x)} \, dx","Integrate[(a + b*ArcTanh[c*x])/(x^2*(d + c*d*x)),x]","\frac{b c x \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-2 \left(a c x \log (x)-a c x \log (c x+1)+a-b c x \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+b \tanh ^{-1}(c x) \left(c x \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+1\right)\right)}{2 d x}","-\frac{a+b \tanh ^{-1}(c x)}{d x}-\frac{c \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d}-\frac{b c \log \left(1-c^2 x^2\right)}{2 d}+\frac{b c \text{Li}_2\left(\frac{2}{c x+1}-1\right)}{2 d}+\frac{b c \log (x)}{d}",1,"(-2*(a + b*ArcTanh[c*x]*(1 + c*x*Log[1 - E^(-2*ArcTanh[c*x])]) + a*c*x*Log[x] - a*c*x*Log[1 + c*x] - b*c*x*Log[(c*x)/Sqrt[1 - c^2*x^2]]) + b*c*x*PolyLog[2, E^(-2*ArcTanh[c*x])])/(2*d*x)","A",0
49,1,133,146,0.3405162,"\int \frac{a+b \tanh ^{-1}(c x)}{x^3 (d+c d x)} \, dx","Integrate[(a + b*ArcTanh[c*x])/(x^3*(d + c*d*x)),x]","-\frac{-2 a c^2 x^2 \log (x)+2 a c^2 x^2 \log (c x+1)-2 a c x+a+b c^2 x^2 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+2 b c^2 x^2 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)-b \tanh ^{-1}(c x) \left(c^2 x^2+2 c^2 x^2 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+2 c x-1\right)+b c x}{2 d x^2}","\frac{c^2 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d}-\frac{a+b \tanh ^{-1}(c x)}{2 d x^2}+\frac{c \left(a+b \tanh ^{-1}(c x)\right)}{d x}-\frac{b c^2 \text{Li}_2\left(\frac{2}{c x+1}-1\right)}{2 d}+\frac{b c^2 \log \left(1-c^2 x^2\right)}{2 d}-\frac{b c^2 \log (x)}{d}+\frac{b c^2 \tanh ^{-1}(c x)}{2 d}-\frac{b c}{2 d x}",1,"-1/2*(a - 2*a*c*x + b*c*x - b*ArcTanh[c*x]*(-1 + 2*c*x + c^2*x^2 + 2*c^2*x^2*Log[1 - E^(-2*ArcTanh[c*x])]) - 2*a*c^2*x^2*Log[x] + 2*a*c^2*x^2*Log[1 + c*x] + 2*b*c^2*x^2*Log[(c*x)/Sqrt[1 - c^2*x^2]] + b*c^2*x^2*PolyLog[2, E^(-2*ArcTanh[c*x])])/(d*x^2)","A",0
50,1,172,185,0.425951,"\int \frac{a+b \tanh ^{-1}(c x)}{x^4 (d+c d x)} \, dx","Integrate[(a + b*ArcTanh[c*x])/(x^4*(d + c*d*x)),x]","\frac{-6 a c^3 x^3 \log (x)+6 a c^3 x^3 \log (c x+1)-6 a c^2 x^2+3 a c x-2 a+3 b c^3 x^3 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+b c^3 x^3+3 b c^2 x^2+8 b c^3 x^3 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)-b \tanh ^{-1}(c x) \left(3 c^3 x^3+6 c^3 x^3 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+6 c^2 x^2-3 c x+2\right)-b c x}{6 d x^3}","-\frac{c^3 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d}-\frac{c^2 \left(a+b \tanh ^{-1}(c x)\right)}{d x}-\frac{a+b \tanh ^{-1}(c x)}{3 d x^3}+\frac{c \left(a+b \tanh ^{-1}(c x)\right)}{2 d x^2}+\frac{b c^3 \text{Li}_2\left(\frac{2}{c x+1}-1\right)}{2 d}+\frac{4 b c^3 \log (x)}{3 d}-\frac{b c^3 \tanh ^{-1}(c x)}{2 d}+\frac{b c^2}{2 d x}-\frac{2 b c^3 \log \left(1-c^2 x^2\right)}{3 d}-\frac{b c}{6 d x^2}",1,"(-2*a + 3*a*c*x - b*c*x - 6*a*c^2*x^2 + 3*b*c^2*x^2 + b*c^3*x^3 - b*ArcTanh[c*x]*(2 - 3*c*x + 6*c^2*x^2 + 3*c^3*x^3 + 6*c^3*x^3*Log[1 - E^(-2*ArcTanh[c*x])]) - 6*a*c^3*x^3*Log[x] + 6*a*c^3*x^3*Log[1 + c*x] + 8*b*c^3*x^3*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 3*b*c^3*x^3*PolyLog[2, E^(-2*ArcTanh[c*x])])/(6*d*x^3)","A",0
51,1,142,181,0.7029678,"\int \frac{x^3 \left(a+b \tanh ^{-1}(c x)\right)}{(d+c d x)^2} \, dx","Integrate[(x^3*(a + b*ArcTanh[c*x]))/(d + c*d*x)^2,x]","\frac{2 a c^2 x^2-8 a c x+\frac{4 a}{c x+1}+12 a \log (c x+1)+b \left(-4 \log \left(1-c^2 x^2\right)+2 \tanh ^{-1}(c x) \left(c^2 x^2-4 c x-6 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-\sinh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)-1\right)+6 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+2 c x-\sinh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)\right)}{4 c^4 d^2}","\frac{a+b \tanh ^{-1}(c x)}{c^4 d^2 (c x+1)}-\frac{3 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^4 d^2}+\frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)}{2 c^2 d^2}-\frac{2 a x}{c^3 d^2}+\frac{3 b \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 c^4 d^2}+\frac{b}{2 c^4 d^2 (c x+1)}-\frac{b \tanh ^{-1}(c x)}{c^4 d^2}+\frac{b x}{2 c^3 d^2}-\frac{2 b x \tanh ^{-1}(c x)}{c^3 d^2}-\frac{b \log \left(1-c^2 x^2\right)}{c^4 d^2}",1,"(-8*a*c*x + 2*a*c^2*x^2 + (4*a)/(1 + c*x) + 12*a*Log[1 + c*x] + b*(2*c*x + Cosh[2*ArcTanh[c*x]] - 4*Log[1 - c^2*x^2] + 6*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 2*ArcTanh[c*x]*(-1 - 4*c*x + c^2*x^2 + Cosh[2*ArcTanh[c*x]] - 6*Log[1 + E^(-2*ArcTanh[c*x])] - Sinh[2*ArcTanh[c*x]]) - Sinh[2*ArcTanh[c*x]]))/(4*c^4*d^2)","A",0
52,1,121,149,0.5791943,"\int \frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)}{(d+c d x)^2} \, dx","Integrate[(x^2*(a + b*ArcTanh[c*x]))/(d + c*d*x)^2,x]","\frac{4 a c x-\frac{4 a}{c x+1}-8 a \log (c x+1)+b \left(2 \log \left(1-c^2 x^2\right)-4 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+\sinh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(2 \tanh ^{-1}(c x)\right)+2 \tanh ^{-1}(c x) \left(2 c x+4 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+\sinh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(2 \tanh ^{-1}(c x)\right)\right)\right)}{4 c^3 d^2}","-\frac{a+b \tanh ^{-1}(c x)}{c^3 d^2 (c x+1)}+\frac{2 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^3 d^2}+\frac{a x}{c^2 d^2}-\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{c^3 d^2}-\frac{b}{2 c^3 d^2 (c x+1)}+\frac{b \tanh ^{-1}(c x)}{2 c^3 d^2}+\frac{b x \tanh ^{-1}(c x)}{c^2 d^2}+\frac{b \log \left(1-c^2 x^2\right)}{2 c^3 d^2}",1,"(4*a*c*x - (4*a)/(1 + c*x) - 8*a*Log[1 + c*x] + b*(-Cosh[2*ArcTanh[c*x]] + 2*Log[1 - c^2*x^2] - 4*PolyLog[2, -E^(-2*ArcTanh[c*x])] + Sinh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]*(2*c*x - Cosh[2*ArcTanh[c*x]] + 4*Log[1 + E^(-2*ArcTanh[c*x])] + Sinh[2*ArcTanh[c*x]])))/(4*c^3*d^2)","A",0
53,1,99,106,0.3727799,"\int \frac{x \left(a+b \tanh ^{-1}(c x)\right)}{(d+c d x)^2} \, dx","Integrate[(x*(a + b*ArcTanh[c*x]))/(d + c*d*x)^2,x]","\frac{\frac{4 a}{c x+1}+4 a \log (c x+1)+b \left(2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-\sinh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)+2 \tanh ^{-1}(c x) \left(-2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-\sinh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)\right)\right)}{4 c^2 d^2}","\frac{a+b \tanh ^{-1}(c x)}{c^2 d^2 (c x+1)}-\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^2 d^2}+\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 c^2 d^2}+\frac{b}{2 c^2 d^2 (c x+1)}-\frac{b \tanh ^{-1}(c x)}{2 c^2 d^2}",1,"((4*a)/(1 + c*x) + 4*a*Log[1 + c*x] + b*(Cosh[2*ArcTanh[c*x]] + 2*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 2*ArcTanh[c*x]*(Cosh[2*ArcTanh[c*x]] - 2*Log[1 + E^(-2*ArcTanh[c*x])] - Sinh[2*ArcTanh[c*x]]) - Sinh[2*ArcTanh[c*x]]))/(4*c^2*d^2)","A",0
54,1,64,57,0.0693021,"\int \frac{a+b \tanh ^{-1}(c x)}{(d+c d x)^2} \, dx","Integrate[(a + b*ArcTanh[c*x])/(d + c*d*x)^2,x]","\frac{-4 a-(b c x+b) \log (1-c x)+b \log (c x+1)+b c x \log (c x+1)-4 b \tanh ^{-1}(c x)-2 b}{4 c d^2 (c x+1)}","-\frac{a+b \tanh ^{-1}(c x)}{c d^2 (c x+1)}-\frac{b}{2 c d^2 (c x+1)}+\frac{b \tanh ^{-1}(c x)}{2 c d^2}",1,"(-4*a - 2*b - 4*b*ArcTanh[c*x] - (b + b*c*x)*Log[1 - c*x] + b*Log[1 + c*x] + b*c*x*Log[1 + c*x])/(4*c*d^2*(1 + c*x))","A",1
55,1,101,124,0.4181747,"\int \frac{a+b \tanh ^{-1}(c x)}{x (d+c d x)^2} \, dx","Integrate[(a + b*ArcTanh[c*x])/(x*(d + c*d*x)^2),x]","\frac{\frac{4 a}{c x+1}-4 a \log (c x+1)+4 a \log (x)+b \left(-2 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-\sinh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)+2 \tanh ^{-1}(c x) \left(2 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-\sinh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)\right)\right)}{4 d^2}","\frac{a+b \tanh ^{-1}(c x)}{d^2 (c x+1)}+\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}+\frac{a \log (x)}{d^2}-\frac{b \text{Li}_2(-c x)}{2 d^2}+\frac{b \text{Li}_2(c x)}{2 d^2}-\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 d^2}+\frac{b}{2 d^2 (c x+1)}-\frac{b \tanh ^{-1}(c x)}{2 d^2}",1,"((4*a)/(1 + c*x) + 4*a*Log[x] - 4*a*Log[1 + c*x] + b*(Cosh[2*ArcTanh[c*x]] - 2*PolyLog[2, E^(-2*ArcTanh[c*x])] + 2*ArcTanh[c*x]*(Cosh[2*ArcTanh[c*x]] + 2*Log[1 - E^(-2*ArcTanh[c*x])] - Sinh[2*ArcTanh[c*x]]) - Sinh[2*ArcTanh[c*x]]))/(4*d^2)","A",0
56,1,140,171,0.8265071,"\int \frac{a+b \tanh ^{-1}(c x)}{x^2 (d+c d x)^2} \, dx","Integrate[(a + b*ArcTanh[c*x])/(x^2*(d + c*d*x)^2),x]","\frac{-\frac{4 a c}{c x+1}-8 a c \log (x)+8 a c \log (c x+1)-\frac{4 a}{x}+b c \left(4 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+4 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+\sinh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(2 \tanh ^{-1}(c x)\right)+\tanh ^{-1}(c x) \left(-\frac{4}{c x}-8 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+2 \sinh \left(2 \tanh ^{-1}(c x)\right)-2 \cosh \left(2 \tanh ^{-1}(c x)\right)\right)\right)}{4 d^2}","-\frac{c \left(a+b \tanh ^{-1}(c x)\right)}{d^2 (c x+1)}-\frac{a+b \tanh ^{-1}(c x)}{d^2 x}-\frac{2 c \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}-\frac{2 a c \log (x)}{d^2}-\frac{b c \log \left(1-c^2 x^2\right)}{2 d^2}+\frac{b c \text{Li}_2(-c x)}{d^2}-\frac{b c \text{Li}_2(c x)}{d^2}+\frac{b c \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{d^2}-\frac{b c}{2 d^2 (c x+1)}+\frac{b c \log (x)}{d^2}+\frac{b c \tanh ^{-1}(c x)}{2 d^2}",1,"((-4*a)/x - (4*a*c)/(1 + c*x) - 8*a*c*Log[x] + 8*a*c*Log[1 + c*x] + b*c*(-Cosh[2*ArcTanh[c*x]] + 4*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 4*PolyLog[2, E^(-2*ArcTanh[c*x])] + Sinh[2*ArcTanh[c*x]] + ArcTanh[c*x]*(-4/(c*x) - 2*Cosh[2*ArcTanh[c*x]] - 8*Log[1 - E^(-2*ArcTanh[c*x])] + 2*Sinh[2*ArcTanh[c*x]])))/(4*d^2)","A",0
57,1,189,212,1.1068558,"\int \frac{a+b \tanh ^{-1}(c x)}{x^3 (d+c d x)^2} \, dx","Integrate[(a + b*ArcTanh[c*x])/(x^3*(d + c*d*x)^2),x]","\frac{\frac{4 a c^2}{c x+1}+12 a c^2 \log (x)-12 a c^2 \log (c x+1)+\frac{8 a c}{x}-\frac{2 a}{x^2}-6 b c^2 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-8 b c^2 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+2 b \tanh ^{-1}(c x) \left(6 c^2 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-c^2 \sinh \left(2 \tanh ^{-1}(c x)\right)+c^2 \cosh \left(2 \tanh ^{-1}(c x)\right)+c^2+\frac{4 c}{x}-\frac{1}{x^2}\right)-b c^2 \sinh \left(2 \tanh ^{-1}(c x)\right)+b c^2 \cosh \left(2 \tanh ^{-1}(c x)\right)-\frac{2 b c}{x}}{4 d^2}","\frac{c^2 \left(a+b \tanh ^{-1}(c x)\right)}{d^2 (c x+1)}+\frac{3 c^2 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}-\frac{a+b \tanh ^{-1}(c x)}{2 d^2 x^2}+\frac{2 c \left(a+b \tanh ^{-1}(c x)\right)}{d^2 x}+\frac{3 a c^2 \log (x)}{d^2}-\frac{3 b c^2 \text{Li}_2(-c x)}{2 d^2}+\frac{3 b c^2 \text{Li}_2(c x)}{2 d^2}-\frac{3 b c^2 \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 d^2}+\frac{b c^2 \log \left(1-c^2 x^2\right)}{d^2}+\frac{b c^2}{2 d^2 (c x+1)}-\frac{2 b c^2 \log (x)}{d^2}-\frac{b c}{2 d^2 x}",1,"((-2*a)/x^2 + (8*a*c)/x - (2*b*c)/x + (4*a*c^2)/(1 + c*x) + b*c^2*Cosh[2*ArcTanh[c*x]] + 12*a*c^2*Log[x] - 12*a*c^2*Log[1 + c*x] - 8*b*c^2*Log[(c*x)/Sqrt[1 - c^2*x^2]] - 6*b*c^2*PolyLog[2, E^(-2*ArcTanh[c*x])] - b*c^2*Sinh[2*ArcTanh[c*x]] + 2*b*ArcTanh[c*x]*(c^2 - x^(-2) + (4*c)/x + c^2*Cosh[2*ArcTanh[c*x]] + 6*c^2*Log[1 - E^(-2*ArcTanh[c*x])] - c^2*Sinh[2*ArcTanh[c*x]]))/(4*d^2)","A",0
58,1,189,227,0.8172093,"\int \frac{x^4 \left(a+b \tanh ^{-1}(c x)\right)}{(d+c d x)^3} \, dx","Integrate[(x^4*(a + b*ArcTanh[c*x]))/(d + c*d*x)^3,x]","\frac{16 a c^2 x^2-96 a c x+\frac{128 a}{c x+1}-\frac{16 a}{(c x+1)^2}+192 a \log (c x+1)+b \left(-48 \log \left(1-c^2 x^2\right)+4 \tanh ^{-1}(c x) \left(4 c^2 x^2-24 c x-48 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-14 \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(4 \tanh ^{-1}(c x)\right)+14 \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(4 \tanh ^{-1}(c x)\right)-4\right)+96 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+16 c x-28 \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(4 \tanh ^{-1}(c x)\right)+28 \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(4 \tanh ^{-1}(c x)\right)\right)}{32 c^5 d^3}","\frac{4 \left(a+b \tanh ^{-1}(c x)\right)}{c^5 d^3 (c x+1)}-\frac{a+b \tanh ^{-1}(c x)}{2 c^5 d^3 (c x+1)^2}-\frac{6 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^5 d^3}+\frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)}{2 c^3 d^3}-\frac{3 a x}{c^4 d^3}+\frac{3 b \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{c^5 d^3}+\frac{15 b}{8 c^5 d^3 (c x+1)}-\frac{b}{8 c^5 d^3 (c x+1)^2}-\frac{19 b \tanh ^{-1}(c x)}{8 c^5 d^3}+\frac{b x}{2 c^4 d^3}-\frac{3 b x \tanh ^{-1}(c x)}{c^4 d^3}-\frac{3 b \log \left(1-c^2 x^2\right)}{2 c^5 d^3}",1,"(-96*a*c*x + 16*a*c^2*x^2 - (16*a)/(1 + c*x)^2 + (128*a)/(1 + c*x) + 192*a*Log[1 + c*x] + b*(16*c*x + 28*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] - 48*Log[1 - c^2*x^2] + 96*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 28*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(-4 - 24*c*x + 4*c^2*x^2 + 14*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] - 48*Log[1 + E^(-2*ArcTanh[c*x])] - 14*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]])))/(32*c^5*d^3)","A",0
59,1,167,194,0.6964317,"\int \frac{x^3 \left(a+b \tanh ^{-1}(c x)\right)}{(d+c d x)^3} \, dx","Integrate[(x^3*(a + b*ArcTanh[c*x]))/(d + c*d*x)^3,x]","\frac{32 a c x-\frac{96 a}{c x+1}+\frac{16 a}{(c x+1)^2}-96 a \log (c x+1)+b \left(16 \log \left(1-c^2 x^2\right)-48 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+20 \sinh \left(2 \tanh ^{-1}(c x)\right)-\sinh \left(4 \tanh ^{-1}(c x)\right)-20 \cosh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(4 \tanh ^{-1}(c x)\right)+4 \tanh ^{-1}(c x) \left(8 c x+24 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+10 \sinh \left(2 \tanh ^{-1}(c x)\right)-\sinh \left(4 \tanh ^{-1}(c x)\right)-10 \cosh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(4 \tanh ^{-1}(c x)\right)\right)\right)}{32 c^4 d^3}","-\frac{3 \left(a+b \tanh ^{-1}(c x)\right)}{c^4 d^3 (c x+1)}+\frac{a+b \tanh ^{-1}(c x)}{2 c^4 d^3 (c x+1)^2}+\frac{3 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^4 d^3}+\frac{a x}{c^3 d^3}-\frac{3 b \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 c^4 d^3}-\frac{11 b}{8 c^4 d^3 (c x+1)}+\frac{b}{8 c^4 d^3 (c x+1)^2}+\frac{11 b \tanh ^{-1}(c x)}{8 c^4 d^3}+\frac{b x \tanh ^{-1}(c x)}{c^3 d^3}+\frac{b \log \left(1-c^2 x^2\right)}{2 c^4 d^3}",1,"(32*a*c*x + (16*a)/(1 + c*x)^2 - (96*a)/(1 + c*x) - 96*a*Log[1 + c*x] + b*(-20*Cosh[2*ArcTanh[c*x]] + Cosh[4*ArcTanh[c*x]] + 16*Log[1 - c^2*x^2] - 48*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 20*Sinh[2*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(8*c*x - 10*Cosh[2*ArcTanh[c*x]] + Cosh[4*ArcTanh[c*x]] + 24*Log[1 + E^(-2*ArcTanh[c*x])] + 10*Sinh[2*ArcTanh[c*x]] - Sinh[4*ArcTanh[c*x]]) - Sinh[4*ArcTanh[c*x]]))/(32*c^4*d^3)","A",0
60,1,145,150,0.4893776,"\int \frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)}{(d+c d x)^3} \, dx","Integrate[(x^2*(a + b*ArcTanh[c*x]))/(d + c*d*x)^3,x]","\frac{\frac{64 a}{c x+1}-\frac{16 a}{(c x+1)^2}+32 a \log (c x+1)+b \left(16 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-12 \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(4 \tanh ^{-1}(c x)\right)+12 \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(4 \tanh ^{-1}(c x)\right)+4 \tanh ^{-1}(c x) \left(-8 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-6 \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(4 \tanh ^{-1}(c x)\right)+6 \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(4 \tanh ^{-1}(c x)\right)\right)\right)}{32 c^3 d^3}","\frac{2 \left(a+b \tanh ^{-1}(c x)\right)}{c^3 d^3 (c x+1)}-\frac{a+b \tanh ^{-1}(c x)}{2 c^3 d^3 (c x+1)^2}-\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^3 d^3}+\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 c^3 d^3}+\frac{7 b}{8 c^3 d^3 (c x+1)}-\frac{b}{8 c^3 d^3 (c x+1)^2}-\frac{7 b \tanh ^{-1}(c x)}{8 c^3 d^3}",1,"((-16*a)/(1 + c*x)^2 + (64*a)/(1 + c*x) + 32*a*Log[1 + c*x] + b*(12*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] + 16*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 12*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(6*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] - 8*Log[1 + E^(-2*ArcTanh[c*x])] - 6*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]])))/(32*c^3*d^3)","A",0
61,1,99,77,0.1111876,"\int \frac{x \left(a+b \tanh ^{-1}(c x)\right)}{(d+c d x)^3} \, dx","Integrate[(x*(a + b*ArcTanh[c*x]))/(d + c*d*x)^3,x]","-\frac{16 a c x+8 a-3 b c^2 x^2 \log (c x+1)+6 b c x-6 b c x \log (c x+1)+3 b (c x+1)^2 \log (1-c x)-3 b \log (c x+1)+8 (2 b c x+b) \tanh ^{-1}(c x)+4 b}{16 c^2 d^3 (c x+1)^2}","\frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)}{2 d^3 (c x+1)^2}-\frac{3 b}{8 c^2 d^3 (c x+1)}+\frac{b}{8 c^2 d^3 (c x+1)^2}-\frac{b \tanh ^{-1}(c x)}{8 c^2 d^3}",1,"-1/16*(8*a + 4*b + 16*a*c*x + 6*b*c*x + 8*(b + 2*b*c*x)*ArcTanh[c*x] + 3*b*(1 + c*x)^2*Log[1 - c*x] - 3*b*Log[1 + c*x] - 6*b*c*x*Log[1 + c*x] - 3*b*c^2*x^2*Log[1 + c*x])/(c^2*d^3*(1 + c*x)^2)","A",1
62,1,86,77,0.0734138,"\int \frac{a+b \tanh ^{-1}(c x)}{(d+c d x)^3} \, dx","Integrate[(a + b*ArcTanh[c*x])/(d + c*d*x)^3,x]","\frac{-8 a+b c^2 x^2 \log (c x+1)-2 b c x+2 b c x \log (c x+1)-b (c x+1)^2 \log (1-c x)+b \log (c x+1)-8 b \tanh ^{-1}(c x)-4 b}{16 c d^3 (c x+1)^2}","-\frac{a+b \tanh ^{-1}(c x)}{2 c d^3 (c x+1)^2}-\frac{b}{8 c d^3 (c x+1)}-\frac{b}{8 c d^3 (c x+1)^2}+\frac{b \tanh ^{-1}(c x)}{8 c d^3}",1,"(-8*a - 4*b - 2*b*c*x - 8*b*ArcTanh[c*x] - b*(1 + c*x)^2*Log[1 - c*x] + b*Log[1 + c*x] + 2*b*c*x*Log[1 + c*x] + b*c^2*x^2*Log[1 + c*x])/(16*c*d^3*(1 + c*x)^2)","A",1
63,1,147,161,0.5306206,"\int \frac{a+b \tanh ^{-1}(c x)}{x (d+c d x)^3} \, dx","Integrate[(a + b*ArcTanh[c*x])/(x*(d + c*d*x)^3),x]","\frac{\frac{32 a}{c x+1}+\frac{16 a}{(c x+1)^2}-32 a \log (c x+1)+32 a \log (x)+b \left(-16 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-12 \sinh \left(2 \tanh ^{-1}(c x)\right)-\sinh \left(4 \tanh ^{-1}(c x)\right)+12 \cosh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(4 \tanh ^{-1}(c x)\right)+4 \tanh ^{-1}(c x) \left(8 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-6 \sinh \left(2 \tanh ^{-1}(c x)\right)-\sinh \left(4 \tanh ^{-1}(c x)\right)+6 \cosh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(4 \tanh ^{-1}(c x)\right)\right)\right)}{32 d^3}","\frac{a+b \tanh ^{-1}(c x)}{d^3 (c x+1)}+\frac{a+b \tanh ^{-1}(c x)}{2 d^3 (c x+1)^2}+\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^3}+\frac{a \log (x)}{d^3}-\frac{b \text{Li}_2(-c x)}{2 d^3}+\frac{b \text{Li}_2(c x)}{2 d^3}-\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 d^3}+\frac{5 b}{8 d^3 (c x+1)}+\frac{b}{8 d^3 (c x+1)^2}-\frac{5 b \tanh ^{-1}(c x)}{8 d^3}",1,"((16*a)/(1 + c*x)^2 + (32*a)/(1 + c*x) + 32*a*Log[x] - 32*a*Log[1 + c*x] + b*(12*Cosh[2*ArcTanh[c*x]] + Cosh[4*ArcTanh[c*x]] - 16*PolyLog[2, E^(-2*ArcTanh[c*x])] - 12*Sinh[2*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(6*Cosh[2*ArcTanh[c*x]] + Cosh[4*ArcTanh[c*x]] + 8*Log[1 - E^(-2*ArcTanh[c*x])] - 6*Sinh[2*ArcTanh[c*x]] - Sinh[4*ArcTanh[c*x]]) - Sinh[4*ArcTanh[c*x]]))/(32*d^3)","A",0
64,1,186,218,1.2382877,"\int \frac{a+b \tanh ^{-1}(c x)}{x^2 (d+c d x)^3} \, dx","Integrate[(a + b*ArcTanh[c*x])/(x^2*(d + c*d*x)^3),x]","\frac{-\frac{64 a c}{c x+1}-\frac{16 a c}{(c x+1)^2}-96 a c \log (x)+96 a c \log (c x+1)-\frac{32 a}{x}+b c \left(32 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+48 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+20 \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(4 \tanh ^{-1}(c x)\right)-20 \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(4 \tanh ^{-1}(c x)\right)+4 \tanh ^{-1}(c x) \left(-\frac{8}{c x}-24 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+10 \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(4 \tanh ^{-1}(c x)\right)-10 \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(4 \tanh ^{-1}(c x)\right)\right)\right)}{32 d^3}","-\frac{2 c \left(a+b \tanh ^{-1}(c x)\right)}{d^3 (c x+1)}-\frac{c \left(a+b \tanh ^{-1}(c x)\right)}{2 d^3 (c x+1)^2}-\frac{a+b \tanh ^{-1}(c x)}{d^3 x}-\frac{3 c \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^3}-\frac{3 a c \log (x)}{d^3}-\frac{b c \log \left(1-c^2 x^2\right)}{2 d^3}+\frac{3 b c \text{Li}_2(-c x)}{2 d^3}-\frac{3 b c \text{Li}_2(c x)}{2 d^3}+\frac{3 b c \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 d^3}-\frac{9 b c}{8 d^3 (c x+1)}-\frac{b c}{8 d^3 (c x+1)^2}+\frac{b c \log (x)}{d^3}+\frac{9 b c \tanh ^{-1}(c x)}{8 d^3}",1,"((-32*a)/x - (16*a*c)/(1 + c*x)^2 - (64*a*c)/(1 + c*x) - 96*a*c*Log[x] + 96*a*c*Log[1 + c*x] + b*c*(-20*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] + 32*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 48*PolyLog[2, E^(-2*ArcTanh[c*x])] + 20*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(-8/(c*x) - 10*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] - 24*Log[1 - E^(-2*ArcTanh[c*x])] + 10*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]])))/(32*d^3)","A",0
65,1,220,268,1.474236,"\int \frac{a+b \tanh ^{-1}(c x)}{x^3 (d+c d x)^3} \, dx","Integrate[(a + b*ArcTanh[c*x])/(x^3*(d + c*d*x)^3),x]","\frac{\frac{96 a c^2}{c x+1}+\frac{16 a c^2}{(c x+1)^2}+192 a c^2 \log (x)-192 a c^2 \log (c x+1)+\frac{96 a c}{x}-\frac{16 a}{x^2}+b c^2 \left(-96 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+4 \tanh ^{-1}(c x) \left(-\frac{4}{c^2 x^2}+\frac{24}{c x}+48 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-14 \sinh \left(2 \tanh ^{-1}(c x)\right)-\sinh \left(4 \tanh ^{-1}(c x)\right)+14 \cosh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(4 \tanh ^{-1}(c x)\right)+4\right)-96 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-\frac{16}{c x}-28 \sinh \left(2 \tanh ^{-1}(c x)\right)-\sinh \left(4 \tanh ^{-1}(c x)\right)+28 \cosh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(4 \tanh ^{-1}(c x)\right)\right)}{32 d^3}","\frac{3 c^2 \left(a+b \tanh ^{-1}(c x)\right)}{d^3 (c x+1)}+\frac{c^2 \left(a+b \tanh ^{-1}(c x)\right)}{2 d^3 (c x+1)^2}+\frac{6 c^2 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^3}-\frac{a+b \tanh ^{-1}(c x)}{2 d^3 x^2}+\frac{3 c \left(a+b \tanh ^{-1}(c x)\right)}{d^3 x}+\frac{6 a c^2 \log (x)}{d^3}-\frac{3 b c^2 \text{Li}_2(-c x)}{d^3}+\frac{3 b c^2 \text{Li}_2(c x)}{d^3}-\frac{3 b c^2 \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{d^3}+\frac{3 b c^2 \log \left(1-c^2 x^2\right)}{2 d^3}+\frac{13 b c^2}{8 d^3 (c x+1)}+\frac{b c^2}{8 d^3 (c x+1)^2}-\frac{3 b c^2 \log (x)}{d^3}-\frac{9 b c^2 \tanh ^{-1}(c x)}{8 d^3}-\frac{b c}{2 d^3 x}",1,"((-16*a)/x^2 + (96*a*c)/x + (16*a*c^2)/(1 + c*x)^2 + (96*a*c^2)/(1 + c*x) + 192*a*c^2*Log[x] - 192*a*c^2*Log[1 + c*x] + b*c^2*(-16/(c*x) + 28*Cosh[2*ArcTanh[c*x]] + Cosh[4*ArcTanh[c*x]] - 96*Log[(c*x)/Sqrt[1 - c^2*x^2]] - 96*PolyLog[2, E^(-2*ArcTanh[c*x])] - 28*Sinh[2*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(4 - 4/(c^2*x^2) + 24/(c*x) + 14*Cosh[2*ArcTanh[c*x]] + Cosh[4*ArcTanh[c*x]] + 48*Log[1 - E^(-2*ArcTanh[c*x])] - 14*Sinh[2*ArcTanh[c*x]] - Sinh[4*ArcTanh[c*x]]) - Sinh[4*ArcTanh[c*x]]))/(32*d^3)","A",0
66,1,75,80,0.1137394,"\int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^4} \, dx","Integrate[(a + b*ArcTanh[c*x])/(1 + c*x)^4,x]","-\frac{48 a+2 b \left(3 c^2 x^2+9 c x+10\right)+3 b (c x+1)^3 \log (1-c x)-3 b (c x+1)^3 \log (c x+1)+48 b \tanh ^{-1}(c x)}{144 c (c x+1)^3}","-\frac{a+b \tanh ^{-1}(c x)}{3 c (c x+1)^3}-\frac{b}{24 c (c x+1)}-\frac{b}{24 c (c x+1)^2}-\frac{b}{18 c (c x+1)^3}+\frac{b \tanh ^{-1}(c x)}{24 c}",1,"-1/144*(48*a + 2*b*(10 + 9*c*x + 3*c^2*x^2) + 48*b*ArcTanh[c*x] + 3*b*(1 + c*x)^3*Log[1 - c*x] - 3*b*(1 + c*x)^3*Log[1 + c*x])/(c*(1 + c*x)^3)","A",1
67,1,39,41,0.0774183,"\int \frac{\tanh ^{-1}(a x)}{c x+a c x^2} \, dx","Integrate[ArcTanh[a*x]/(c*x + a*c*x^2),x]","\frac{\tanh ^{-1}(a x) \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)}{c}-\frac{\text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)}{2 c}","\frac{\log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)}{c}-\frac{\text{Li}_2\left(\frac{2}{a x+1}-1\right)}{2 c}",1,"(ArcTanh[a*x]*Log[1 - E^(-2*ArcTanh[a*x])])/c - PolyLog[2, E^(-2*ArcTanh[a*x])]/(2*c)","A",0
68,1,271,270,0.7597746,"\int x^3 (d+c d x) \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d + c*d*x)*(a + b*ArcTanh[c*x])^2,x]","\frac{d \left(12 a^2 c^5 x^5+15 a^2 c^4 x^4+6 a b c^4 x^4+10 a b c^3 x^3+12 a b c^2 x^2+12 a b \log \left(c^2 x^2-1\right)+2 b \tanh ^{-1}(c x) \left(3 a c^4 x^4 (4 c x+5)+b \left(3 c^4 x^4+5 c^3 x^3+6 c^2 x^2+15 c x-9\right)-12 b \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+30 a b c x+15 a b \log (1-c x)-15 a b \log (c x+1)-18 a b+2 b^2 c^3 x^3+5 b^2 c^2 x^2+20 b^2 \log \left(1-c^2 x^2\right)+3 b^2 \left(4 c^5 x^5+5 c^4 x^4-9\right) \tanh ^{-1}(c x)^2+12 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+18 b^2 c x-5 b^2\right)}{60 c^4}","-\frac{d \left(a+b \tanh ^{-1}(c x)\right)^2}{20 c^4}-\frac{2 b d \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{5 c^4}+\frac{a b d x}{2 c^3}+\frac{b d x^2 \left(a+b \tanh ^{-1}(c x)\right)}{5 c^2}+\frac{1}{5} c d x^5 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{4} d x^4 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{10} b d x^4 \left(a+b \tanh ^{-1}(c x)\right)+\frac{b d x^3 \left(a+b \tanh ^{-1}(c x)\right)}{6 c}-\frac{b^2 d \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{5 c^4}-\frac{3 b^2 d \tanh ^{-1}(c x)}{10 c^4}+\frac{3 b^2 d x}{10 c^3}+\frac{b^2 d x \tanh ^{-1}(c x)}{2 c^3}+\frac{b^2 d x^2}{12 c^2}+\frac{b^2 d \log \left(1-c^2 x^2\right)}{3 c^4}+\frac{b^2 d x^3}{30 c}",1,"(d*(-18*a*b - 5*b^2 + 30*a*b*c*x + 18*b^2*c*x + 12*a*b*c^2*x^2 + 5*b^2*c^2*x^2 + 10*a*b*c^3*x^3 + 2*b^2*c^3*x^3 + 15*a^2*c^4*x^4 + 6*a*b*c^4*x^4 + 12*a^2*c^5*x^5 + 3*b^2*(-9 + 5*c^4*x^4 + 4*c^5*x^5)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(3*a*c^4*x^4*(5 + 4*c*x) + b*(-9 + 15*c*x + 6*c^2*x^2 + 5*c^3*x^3 + 3*c^4*x^4) - 12*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 15*a*b*Log[1 - c*x] - 15*a*b*Log[1 + c*x] + 20*b^2*Log[1 - c^2*x^2] + 12*a*b*Log[-1 + c^2*x^2] + 12*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(60*c^4)","A",0
69,1,234,236,0.5700405,"\int x^2 (d+c d x) \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d + c*d*x)*(a + b*ArcTanh[c*x])^2,x]","\frac{d \left(3 a^2 c^4 x^4+4 a^2 c^3 x^3+2 a b c^3 x^3+4 a b c^2 x^2+4 a b \log \left(c^2 x^2-1\right)+2 b \tanh ^{-1}(c x) \left(a c^3 x^3 (3 c x+4)+b \left(c^3 x^3+2 c^2 x^2+3 c x-2\right)-4 b \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+6 a b c x+3 a b \log (1-c x)-3 a b \log (c x+1)+b^2 c^2 x^2+4 b^2 \log \left(1-c^2 x^2\right)+b^2 \left(3 c^4 x^4+4 c^3 x^3-7\right) \tanh ^{-1}(c x)^2+4 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+4 b^2 c x-b^2\right)}{12 c^3}","\frac{d \left(a+b \tanh ^{-1}(c x)\right)^2}{12 c^3}-\frac{2 b d \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{3 c^3}+\frac{a b d x}{2 c^2}+\frac{1}{4} c d x^4 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{3} d x^3 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{6} b d x^3 \left(a+b \tanh ^{-1}(c x)\right)+\frac{b d x^2 \left(a+b \tanh ^{-1}(c x)\right)}{3 c}-\frac{b^2 d \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{3 c^3}-\frac{b^2 d \tanh ^{-1}(c x)}{3 c^3}+\frac{b^2 d x}{3 c^2}+\frac{b^2 d x \tanh ^{-1}(c x)}{2 c^2}+\frac{b^2 d \log \left(1-c^2 x^2\right)}{3 c^3}+\frac{b^2 d x^2}{12 c}",1,"(d*(-b^2 + 6*a*b*c*x + 4*b^2*c*x + 4*a*b*c^2*x^2 + b^2*c^2*x^2 + 4*a^2*c^3*x^3 + 2*a*b*c^3*x^3 + 3*a^2*c^4*x^4 + b^2*(-7 + 4*c^3*x^3 + 3*c^4*x^4)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(a*c^3*x^3*(4 + 3*c*x) + b*(-2 + 3*c*x + 2*c^2*x^2 + c^3*x^3) - 4*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 3*a*b*Log[1 - c*x] - 3*a*b*Log[1 + c*x] + 4*b^2*Log[1 - c^2*x^2] + 4*a*b*Log[-1 + c^2*x^2] + 4*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(12*c^3)","A",0
70,1,201,196,0.5061178,"\int x (d+c d x) \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d + c*d*x)*(a + b*ArcTanh[c*x])^2,x]","\frac{d \left(2 a^2 c^3 x^3+3 a^2 c^2 x^2+2 a b c^2 x^2+2 a b \log \left(c^2 x^2-1\right)+2 b \tanh ^{-1}(c x) \left(a c^2 x^2 (2 c x+3)+b \left(c^2 x^2+3 c x-1\right)-2 b \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+6 a b c x+3 a b \log (1-c x)-3 a b \log (c x+1)+3 b^2 \log \left(1-c^2 x^2\right)+b^2 \left(2 c^3 x^3+3 c^2 x^2-5\right) \tanh ^{-1}(c x)^2+2 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+2 b^2 c x\right)}{6 c^2}","-\frac{d \left(a+b \tanh ^{-1}(c x)\right)^2}{6 c^2}-\frac{2 b d \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{3 c^2}+\frac{1}{3} c d x^3 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{2} d x^2 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{3} b d x^2 \left(a+b \tanh ^{-1}(c x)\right)+\frac{a b d x}{c}-\frac{b^2 d \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{3 c^2}+\frac{b^2 d \log \left(1-c^2 x^2\right)}{2 c^2}-\frac{b^2 d \tanh ^{-1}(c x)}{3 c^2}+\frac{b^2 d x}{3 c}+\frac{b^2 d x \tanh ^{-1}(c x)}{c}",1,"(d*(6*a*b*c*x + 2*b^2*c*x + 3*a^2*c^2*x^2 + 2*a*b*c^2*x^2 + 2*a^2*c^3*x^3 + b^2*(-5 + 3*c^2*x^2 + 2*c^3*x^3)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(a*c^2*x^2*(3 + 2*c*x) + b*(-1 + 3*c*x + c^2*x^2) - 2*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 3*a*b*Log[1 - c*x] - 3*a*b*Log[1 + c*x] + 3*b^2*Log[1 - c^2*x^2] + 2*a*b*Log[-1 + c^2*x^2] + 2*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(6*c^2)","A",0
71,1,156,112,0.321654,"\int (d+c d x) \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + c*d*x)*(a + b*ArcTanh[c*x])^2,x]","\frac{d \left(a^2 c^2 x^2+2 a^2 c x+2 a b \log \left(1-c^2 x^2\right)+2 a b c x+a b \log (1-c x)-a b \log (c x+1)+2 b \tanh ^{-1}(c x) \left(c x (a c x+2 a+b)-2 b \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+b^2 \log \left(1-c^2 x^2\right)+b^2 \left(c^2 x^2+2 c x-3\right) \tanh ^{-1}(c x)^2+2 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)\right)}{2 c}","\frac{d (c x+1)^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c}-\frac{2 b d \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c}+a b d x+\frac{b^2 d \log \left(1-c^2 x^2\right)}{2 c}-\frac{b^2 d \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c}+b^2 d x \tanh ^{-1}(c x)",1,"(d*(2*a^2*c*x + 2*a*b*c*x + a^2*c^2*x^2 + b^2*(-3 + 2*c*x + c^2*x^2)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(c*x*(2*a + b + a*c*x) - 2*b*Log[1 + E^(-2*ArcTanh[c*x])]) + a*b*Log[1 - c*x] - a*b*Log[1 + c*x] + 2*a*b*Log[1 - c^2*x^2] + b^2*Log[1 - c^2*x^2] + 2*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(2*c)","A",0
72,1,228,191,0.5142063,"\int \frac{(d+c d x) \left(a+b \tanh ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d + c*d*x)*(a + b*ArcTanh[c*x])^2)/x,x]","d \left(a^2 c x+a^2 \log (c x)+a b \left(\log \left(1-c^2 x^2\right)+2 c x \tanh ^{-1}(c x)\right)+a b (\text{Li}_2(c x)-\text{Li}_2(-c x))+b^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)+b^2 \left(\tanh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+\tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)-\frac{2}{3} \tanh ^{-1}(c x)^3-\tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+\tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)+\frac{i \pi ^3}{24}\right)\right)","-b d \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+b d \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)+d \left(a+b \tanh ^{-1}(c x)\right)^2+c d x \left(a+b \tanh ^{-1}(c x)\right)^2+2 d \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2-2 b d \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+b^2 (-d) \text{Li}_2\left(1-\frac{2}{1-c x}\right)+\frac{1}{2} b^2 d \text{Li}_3\left(1-\frac{2}{1-c x}\right)-\frac{1}{2} b^2 d \text{Li}_3\left(\frac{2}{1-c x}-1\right)",1,"d*(a^2*c*x + a^2*Log[c*x] + a*b*(2*c*x*ArcTanh[c*x] + Log[1 - c^2*x^2]) + b^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])]) + a*b*(-PolyLog[2, -(c*x)] + PolyLog[2, c*x]) + b^2*((I/24)*Pi^3 - (2*ArcTanh[c*x]^3)/3 - ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + PolyLog[3, -E^(-2*ArcTanh[c*x])]/2 - PolyLog[3, E^(2*ArcTanh[c*x])]/2))","C",0
73,1,249,201,0.5178358,"\int \frac{(d+c d x) \left(a+b \tanh ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d + c*d*x)*(a + b*ArcTanh[c*x])^2)/x^2,x]","-\frac{d \left(a^2 (-c) x \log (x)+a^2+a b \left(c x \left(\log \left(1-c^2 x^2\right)-2 \log (c x)\right)+2 \tanh ^{-1}(c x)\right)+a b c x (\text{Li}_2(-c x)-\text{Li}_2(c x))+b^2 \left(c x \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+\tanh ^{-1}(c x) \left((1-c x) \tanh ^{-1}(c x)-2 c x \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)\right)\right)-b^2 c x \left(\tanh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+\tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)-\frac{2}{3} \tanh ^{-1}(c x)^3-\tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+\tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)+\frac{i \pi ^3}{24}\right)\right)}{x}","-b c d \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+b c d \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)+c d \left(a+b \tanh ^{-1}(c x)\right)^2-\frac{d \left(a+b \tanh ^{-1}(c x)\right)^2}{x}+2 c d \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2+2 b c d \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)+b^2 (-c) d \text{Li}_2\left(\frac{2}{c x+1}-1\right)+\frac{1}{2} b^2 c d \text{Li}_3\left(1-\frac{2}{1-c x}\right)-\frac{1}{2} b^2 c d \text{Li}_3\left(\frac{2}{1-c x}-1\right)",1,"-((d*(a^2 - a^2*c*x*Log[x] + a*b*(2*ArcTanh[c*x] + c*x*(-2*Log[c*x] + Log[1 - c^2*x^2])) + b^2*(ArcTanh[c*x]*((1 - c*x)*ArcTanh[c*x] - 2*c*x*Log[1 - E^(-2*ArcTanh[c*x])]) + c*x*PolyLog[2, E^(-2*ArcTanh[c*x])]) + a*b*c*x*(PolyLog[2, -(c*x)] - PolyLog[2, c*x]) - b^2*c*x*((I/24)*Pi^3 - (2*ArcTanh[c*x]^3)/3 - ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + PolyLog[3, -E^(-2*ArcTanh[c*x])]/2 - PolyLog[3, E^(2*ArcTanh[c*x])]/2)))/x)","C",0
74,1,206,151,0.2828586,"\int \frac{(d+c d x) \left(a+b \tanh ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d + c*d*x)*(a + b*ArcTanh[c*x])^2)/x^3,x]","-\frac{d \left(2 a^2 c x+a^2-4 a b c^2 x^2 \log (c x)+a b c^2 x^2 \log (1-c x)-a b c^2 x^2 \log (c x+1)+2 a b c^2 x^2 \log \left(1-c^2 x^2\right)+2 b \tanh ^{-1}(c x) \left(2 a c x+a-2 b c^2 x^2 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+b c x\right)+2 a b c x+2 b^2 c^2 x^2 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-2 b^2 c^2 x^2 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+b^2 \left(-3 c^2 x^2+2 c x+1\right) \tanh ^{-1}(c x)^2\right)}{2 x^2}","\frac{3}{2} c^2 d \left(a+b \tanh ^{-1}(c x)\right)^2+2 b c^2 d \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)-\frac{d \left(a+b \tanh ^{-1}(c x)\right)^2}{2 x^2}-\frac{c d \left(a+b \tanh ^{-1}(c x)\right)^2}{x}-\frac{b c d \left(a+b \tanh ^{-1}(c x)\right)}{x}-b^2 c^2 d \text{Li}_2\left(\frac{2}{c x+1}-1\right)-\frac{1}{2} b^2 c^2 d \log \left(1-c^2 x^2\right)+b^2 c^2 d \log (x)",1,"-1/2*(d*(a^2 + 2*a^2*c*x + 2*a*b*c*x + b^2*(1 + 2*c*x - 3*c^2*x^2)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(a + 2*a*c*x + b*c*x - 2*b*c^2*x^2*Log[1 - E^(-2*ArcTanh[c*x])]) - 4*a*b*c^2*x^2*Log[c*x] + a*b*c^2*x^2*Log[1 - c*x] - a*b*c^2*x^2*Log[1 + c*x] - 2*b^2*c^2*x^2*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 2*a*b*c^2*x^2*Log[1 - c^2*x^2] + 2*b^2*c^2*x^2*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x^2","A",0
75,1,246,206,0.4822467,"\int \frac{(d+c d x) \left(a+b \tanh ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d + c*d*x)*(a + b*ArcTanh[c*x])^2)/x^4,x]","-\frac{d \left(3 a^2 c x+2 a^2-4 a b c^3 x^3 \log (c x)+3 a b c^3 x^3 \log (1-c x)-3 a b c^3 x^3 \log (c x+1)+6 a b c^2 x^2+2 a b c^3 x^3 \log \left(1-c^2 x^2\right)+2 b \tanh ^{-1}(c x) \left(a (3 c x+2)-2 b c^3 x^3 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+b c x \left(-c^2 x^2+3 c x+1\right)\right)+2 a b c x+2 b^2 c^3 x^3 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+b^2 \left(-5 c^3 x^3+3 c x+2\right) \tanh ^{-1}(c x)^2+2 b^2 c^2 x^2-6 b^2 c^3 x^3 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)\right)}{6 x^3}","\frac{5}{6} c^3 d \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{2}{3} b c^3 d \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)-\frac{b c^2 d \left(a+b \tanh ^{-1}(c x)\right)}{x}-\frac{d \left(a+b \tanh ^{-1}(c x)\right)^2}{3 x^3}-\frac{c d \left(a+b \tanh ^{-1}(c x)\right)^2}{2 x^2}-\frac{b c d \left(a+b \tanh ^{-1}(c x)\right)}{3 x^2}-\frac{1}{3} b^2 c^3 d \text{Li}_2\left(\frac{2}{c x+1}-1\right)+b^2 c^3 d \log (x)+\frac{1}{3} b^2 c^3 d \tanh ^{-1}(c x)-\frac{b^2 c^2 d}{3 x}-\frac{1}{2} b^2 c^3 d \log \left(1-c^2 x^2\right)",1,"-1/6*(d*(2*a^2 + 3*a^2*c*x + 2*a*b*c*x + 6*a*b*c^2*x^2 + 2*b^2*c^2*x^2 + b^2*(2 + 3*c*x - 5*c^3*x^3)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(a*(2 + 3*c*x) + b*c*x*(1 + 3*c*x - c^2*x^2) - 2*b*c^3*x^3*Log[1 - E^(-2*ArcTanh[c*x])]) - 4*a*b*c^3*x^3*Log[c*x] + 3*a*b*c^3*x^3*Log[1 - c*x] - 3*a*b*c^3*x^3*Log[1 + c*x] - 6*b^2*c^3*x^3*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 2*a*b*c^3*x^3*Log[1 - c^2*x^2] + 2*b^2*c^3*x^3*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x^3","A",0
76,1,329,356,1.0964265,"\int x^3 (d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2,x]","\frac{d^2 \left(30 a^2 c^6 x^6+72 a^2 c^5 x^5+45 a^2 c^4 x^4+12 a b c^5 x^5+36 a b c^4 x^4+50 a b c^3 x^3+72 a b c^2 x^2+72 a b \log \left(c^2 x^2-1\right)+2 b \tanh ^{-1}(c x) \left(3 a c^4 x^4 \left(10 c^2 x^2+24 c x+15\right)+b \left(6 c^5 x^5+18 c^4 x^4+25 c^3 x^3+36 c^2 x^2+75 c x-54\right)-72 b \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+150 a b c x+75 a b \log (1-c x)-75 a b \log (c x+1)-108 a b+3 b^2 c^4 x^4+12 b^2 c^3 x^3+31 b^2 c^2 x^2+106 b^2 \log \left(1-c^2 x^2\right)+3 b^2 \left(10 c^6 x^6+24 c^5 x^5+15 c^4 x^4-49\right) \tanh ^{-1}(c x)^2+72 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+108 b^2 c x-34 b^2\right)}{180 c^4}","-\frac{d^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{60 c^4}-\frac{4 b d^2 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{5 c^4}+\frac{5 a b d^2 x}{6 c^3}+\frac{1}{6} c^2 d^2 x^6 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{2 b d^2 x^2 \left(a+b \tanh ^{-1}(c x)\right)}{5 c^2}+\frac{2}{5} c d^2 x^5 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{15} b c d^2 x^5 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{4} d^2 x^4 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{5} b d^2 x^4 \left(a+b \tanh ^{-1}(c x)\right)+\frac{5 b d^2 x^3 \left(a+b \tanh ^{-1}(c x)\right)}{18 c}-\frac{2 b^2 d^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{5 c^4}-\frac{3 b^2 d^2 \tanh ^{-1}(c x)}{5 c^4}+\frac{3 b^2 d^2 x}{5 c^3}+\frac{5 b^2 d^2 x \tanh ^{-1}(c x)}{6 c^3}+\frac{31 b^2 d^2 x^2}{180 c^2}+\frac{53 b^2 d^2 \log \left(1-c^2 x^2\right)}{90 c^4}+\frac{b^2 d^2 x^3}{15 c}+\frac{1}{60} b^2 d^2 x^4",1,"(d^2*(-108*a*b - 34*b^2 + 150*a*b*c*x + 108*b^2*c*x + 72*a*b*c^2*x^2 + 31*b^2*c^2*x^2 + 50*a*b*c^3*x^3 + 12*b^2*c^3*x^3 + 45*a^2*c^4*x^4 + 36*a*b*c^4*x^4 + 3*b^2*c^4*x^4 + 72*a^2*c^5*x^5 + 12*a*b*c^5*x^5 + 30*a^2*c^6*x^6 + 3*b^2*(-49 + 15*c^4*x^4 + 24*c^5*x^5 + 10*c^6*x^6)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(3*a*c^4*x^4*(15 + 24*c*x + 10*c^2*x^2) + b*(-54 + 75*c*x + 36*c^2*x^2 + 25*c^3*x^3 + 18*c^4*x^4 + 6*c^5*x^5) - 72*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 75*a*b*Log[1 - c*x] - 75*a*b*Log[1 + c*x] + 106*b^2*Log[1 - c^2*x^2] + 72*a*b*Log[-1 + c^2*x^2] + 72*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(180*c^4)","A",0
77,1,297,312,1.0294321,"\int x^2 (d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2,x]","\frac{d^2 \left(6 a^2 c^5 x^5+15 a^2 c^4 x^4+10 a^2 c^3 x^3+3 a b c^4 x^4+10 a b c^3 x^3+16 a b c^2 x^2+16 a b \log \left(c^2 x^2-1\right)+b \tanh ^{-1}(c x) \left(2 a c^3 x^3 \left(6 c^2 x^2+15 c x+10\right)+b \left(3 c^4 x^4+10 c^3 x^3+16 c^2 x^2+30 c x-19\right)-32 b \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+30 a b c x+15 a b \log (1-c x)-15 a b \log (c x+1)-9 a b+b^2 c^3 x^3+5 b^2 c^2 x^2+20 b^2 \log \left(1-c^2 x^2\right)+b^2 \left(6 c^5 x^5+15 c^4 x^4+10 c^3 x^3-31\right) \tanh ^{-1}(c x)^2+16 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+19 b^2 c x-5 b^2\right)}{30 c^3}","\frac{d^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{30 c^3}-\frac{16 b d^2 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{15 c^3}+\frac{1}{5} c^2 d^2 x^5 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{a b d^2 x}{c^2}+\frac{1}{2} c d^2 x^4 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{10} b c d^2 x^4 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{3} d^2 x^3 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{3} b d^2 x^3 \left(a+b \tanh ^{-1}(c x)\right)+\frac{8 b d^2 x^2 \left(a+b \tanh ^{-1}(c x)\right)}{15 c}-\frac{8 b^2 d^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{15 c^3}-\frac{19 b^2 d^2 \tanh ^{-1}(c x)}{30 c^3}+\frac{19 b^2 d^2 x}{30 c^2}+\frac{b^2 d^2 x \tanh ^{-1}(c x)}{c^2}+\frac{2 b^2 d^2 \log \left(1-c^2 x^2\right)}{3 c^3}+\frac{b^2 d^2 x^2}{6 c}+\frac{1}{30} b^2 d^2 x^3",1,"(d^2*(-9*a*b - 5*b^2 + 30*a*b*c*x + 19*b^2*c*x + 16*a*b*c^2*x^2 + 5*b^2*c^2*x^2 + 10*a^2*c^3*x^3 + 10*a*b*c^3*x^3 + b^2*c^3*x^3 + 15*a^2*c^4*x^4 + 3*a*b*c^4*x^4 + 6*a^2*c^5*x^5 + b^2*(-31 + 10*c^3*x^3 + 15*c^4*x^4 + 6*c^5*x^5)*ArcTanh[c*x]^2 + b*ArcTanh[c*x]*(2*a*c^3*x^3*(10 + 15*c*x + 6*c^2*x^2) + b*(-19 + 30*c*x + 16*c^2*x^2 + 10*c^3*x^3 + 3*c^4*x^4) - 32*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 15*a*b*Log[1 - c*x] - 15*a*b*Log[1 + c*x] + 20*b^2*Log[1 - c^2*x^2] + 16*a*b*Log[-1 + c^2*x^2] + 16*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(30*c^3)","A",0
78,1,263,280,0.777222,"\int x (d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2,x]","\frac{d^2 \left(3 a^2 c^4 x^4+8 a^2 c^3 x^3+6 a^2 c^2 x^2+2 a b c^3 x^3+8 a b c^2 x^2+8 a b \log \left(c^2 x^2-1\right)+2 b \tanh ^{-1}(c x) \left(a c^2 x^2 \left(3 c^2 x^2+8 c x+6\right)+b \left(c^3 x^3+4 c^2 x^2+9 c x-4\right)-8 b \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+18 a b c x+9 a b \log (1-c x)-9 a b \log (c x+1)+b^2 c^2 x^2+10 b^2 \log \left(1-c^2 x^2\right)+b^2 \left(3 c^4 x^4+8 c^3 x^3+6 c^2 x^2-17\right) \tanh ^{-1}(c x)^2+8 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+8 b^2 c x-b^2\right)}{12 c^2}","\frac{1}{4} c^2 d^2 x^4 \left(a+b \tanh ^{-1}(c x)\right)^2-\frac{d^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{12 c^2}-\frac{4 b d^2 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{3 c^2}+\frac{2}{3} c d^2 x^3 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{6} b c d^2 x^3 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{2} d^2 x^2 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{2}{3} b d^2 x^2 \left(a+b \tanh ^{-1}(c x)\right)+\frac{3 a b d^2 x}{2 c}-\frac{2 b^2 d^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{3 c^2}+\frac{5 b^2 d^2 \log \left(1-c^2 x^2\right)}{6 c^2}-\frac{2 b^2 d^2 \tanh ^{-1}(c x)}{3 c^2}+\frac{2 b^2 d^2 x}{3 c}+\frac{3 b^2 d^2 x \tanh ^{-1}(c x)}{2 c}+\frac{1}{12} b^2 d^2 x^2",1,"(d^2*(-b^2 + 18*a*b*c*x + 8*b^2*c*x + 6*a^2*c^2*x^2 + 8*a*b*c^2*x^2 + b^2*c^2*x^2 + 8*a^2*c^3*x^3 + 2*a*b*c^3*x^3 + 3*a^2*c^4*x^4 + b^2*(-17 + 6*c^2*x^2 + 8*c^3*x^3 + 3*c^4*x^4)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(a*c^2*x^2*(6 + 8*c*x + 3*c^2*x^2) + b*(-4 + 9*c*x + 4*c^2*x^2 + c^3*x^3) - 8*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 9*a*b*Log[1 - c*x] - 9*a*b*Log[1 + c*x] + 10*b^2*Log[1 - c^2*x^2] + 8*a*b*Log[-1 + c^2*x^2] + 8*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(12*c^2)","A",0
79,1,227,175,0.6633365,"\int (d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2,x]","\frac{d^2 \left(a^2 c^3 x^3+3 a^2 c^2 x^2+3 a^2 c x+a b c^2 x^2+3 a b \log \left(1-c^2 x^2\right)+a b \log \left(c^2 x^2-1\right)+b \tanh ^{-1}(c x) \left(2 a c x \left(c^2 x^2+3 c x+3\right)+b \left(c^2 x^2+6 c x-1\right)-8 b \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+6 a b c x+3 a b \log (1-c x)-3 a b \log (c x+1)+3 b^2 \log \left(1-c^2 x^2\right)+b^2 \left(c^3 x^3+3 c^2 x^2+3 c x-7\right) \tanh ^{-1}(c x)^2+4 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+b^2 c x\right)}{3 c}","\frac{1}{3} b c d^2 x^2 \left(a+b \tanh ^{-1}(c x)\right)+\frac{d^2 (c x+1)^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{3 c}-\frac{8 b d^2 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{3 c}+2 a b d^2 x+\frac{b^2 d^2 \log \left(1-c^2 x^2\right)}{c}-\frac{4 b^2 d^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{3 c}-\frac{b^2 d^2 \tanh ^{-1}(c x)}{3 c}+2 b^2 d^2 x \tanh ^{-1}(c x)+\frac{1}{3} b^2 d^2 x",1,"(d^2*(3*a^2*c*x + 6*a*b*c*x + b^2*c*x + 3*a^2*c^2*x^2 + a*b*c^2*x^2 + a^2*c^3*x^3 + b^2*(-7 + 3*c*x + 3*c^2*x^2 + c^3*x^3)*ArcTanh[c*x]^2 + b*ArcTanh[c*x]*(2*a*c*x*(3 + 3*c*x + c^2*x^2) + b*(-1 + 6*c*x + c^2*x^2) - 8*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 3*a*b*Log[1 - c*x] - 3*a*b*Log[1 + c*x] + 3*a*b*Log[1 - c^2*x^2] + 3*b^2*Log[1 - c^2*x^2] + a*b*Log[-1 + c^2*x^2] + 4*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(3*c)","A",0
80,1,324,278,0.7074574,"\int \frac{(d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d + c*d*x)^2*(a + b*ArcTanh[c*x])^2)/x,x]","\frac{1}{2} d^2 \left(a^2 c^2 x^2+4 a^2 c x+2 a^2 \log (c x)+a b \left(2 c^2 x^2 \tanh ^{-1}(c x)+2 c x+\log (1-c x)-\log (c x+1)\right)+4 a b \left(\log \left(1-c^2 x^2\right)+2 c x \tanh ^{-1}(c x)\right)+2 a b (\text{Li}_2(c x)-\text{Li}_2(-c x))+b^2 \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)+4 b^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)+2 b^2 \left(\tanh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+\tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)-\frac{2}{3} \tanh ^{-1}(c x)^3-\tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+\tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)+\frac{i \pi ^3}{24}\right)\right)","\frac{1}{2} c^2 d^2 x^2 \left(a+b \tanh ^{-1}(c x)\right)^2-b d^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+b d^2 \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)+a b c d^2 x+2 c d^2 x \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{3}{2} d^2 \left(a+b \tanh ^{-1}(c x)\right)^2+2 d^2 \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2-4 b d^2 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{2} b^2 d^2 \log \left(1-c^2 x^2\right)-2 b^2 d^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)+\frac{1}{2} b^2 d^2 \text{Li}_3\left(1-\frac{2}{1-c x}\right)-\frac{1}{2} b^2 d^2 \text{Li}_3\left(\frac{2}{1-c x}-1\right)+b^2 c d^2 x \tanh ^{-1}(c x)",1,"(d^2*(4*a^2*c*x + a^2*c^2*x^2 + 2*a^2*Log[c*x] + a*b*(2*c*x + 2*c^2*x^2*ArcTanh[c*x] + Log[1 - c*x] - Log[1 + c*x]) + 4*a*b*(2*c*x*ArcTanh[c*x] + Log[1 - c^2*x^2]) + b^2*(2*c*x*ArcTanh[c*x] + (-1 + c^2*x^2)*ArcTanh[c*x]^2 + Log[1 - c^2*x^2]) + 4*b^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])]) + 2*a*b*(-PolyLog[2, -(c*x)] + PolyLog[2, c*x]) + 2*b^2*((I/24)*Pi^3 - (2*ArcTanh[c*x]^3)/3 - ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + PolyLog[3, -E^(-2*ArcTanh[c*x])]/2 - PolyLog[3, E^(2*ArcTanh[c*x])]/2)))/2","C",0
81,1,341,283,0.5248596,"\int \frac{(d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d + c*d*x)^2*(a + b*ArcTanh[c*x])^2)/x^2,x]","\frac{d^2 \left(12 a^2 c^2 x^2+24 a^2 c x \log (x)-12 a^2+24 a b c^2 x^2 \tanh ^{-1}(c x)-24 a b c x \text{Li}_2(-c x)+24 a b c x \text{Li}_2(c x)+24 a b c x \log (c x)-24 a b \tanh ^{-1}(c x)+12 b^2 c^2 x^2 \tanh ^{-1}(c x)^2+24 b^2 c x \tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)+12 b^2 c x \left(2 \tanh ^{-1}(c x)+1\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-12 b^2 c x \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+12 b^2 c x \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-12 b^2 c x \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)+i \pi ^3 b^2 c x-16 b^2 c x \tanh ^{-1}(c x)^3-12 b^2 \tanh ^{-1}(c x)^2-24 b^2 c x \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+24 b^2 c x \tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)+24 b^2 c x \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-24 b^2 c x \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)}{12 x}","c^2 d^2 x \left(a+b \tanh ^{-1}(c x)\right)^2-2 b c d^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+2 b c d^2 \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)+2 c d^2 \left(a+b \tanh ^{-1}(c x)\right)^2-\frac{d^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{x}+4 c d^2 \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2-2 b c d^2 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+2 b c d^2 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)-b^2 c d^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)-b^2 c d^2 \text{Li}_2\left(\frac{2}{c x+1}-1\right)+b^2 c d^2 \text{Li}_3\left(1-\frac{2}{1-c x}\right)-b^2 c d^2 \text{Li}_3\left(\frac{2}{1-c x}-1\right)",1,"(d^2*(-12*a^2 + I*b^2*c*Pi^3*x + 12*a^2*c^2*x^2 - 24*a*b*ArcTanh[c*x] + 24*a*b*c^2*x^2*ArcTanh[c*x] - 12*b^2*ArcTanh[c*x]^2 + 12*b^2*c^2*x^2*ArcTanh[c*x]^2 - 16*b^2*c*x*ArcTanh[c*x]^3 + 24*b^2*c*x*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - 24*b^2*c*x*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 24*b^2*c*x*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 24*b^2*c*x*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 24*a^2*c*x*Log[x] + 24*a*b*c*x*Log[c*x] + 12*b^2*c*x*(1 + 2*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 12*b^2*c*x*PolyLog[2, E^(-2*ArcTanh[c*x])] + 24*b^2*c*x*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 24*a*b*c*x*PolyLog[2, -(c*x)] + 24*a*b*c*x*PolyLog[2, c*x] + 12*b^2*c*x*PolyLog[3, -E^(-2*ArcTanh[c*x])] - 12*b^2*c*x*PolyLog[3, E^(2*ArcTanh[c*x])]))/(12*x)","C",0
82,1,370,313,0.7540069,"\int \frac{(d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d + c*d*x)^2*(a + b*ArcTanh[c*x])^2)/x^3,x]","-\frac{d^2 \left(-2 a^2 c^2 x^2 \log (x)+4 a^2 c x+a^2+2 a b c^2 x^2 (\text{Li}_2(-c x)-\text{Li}_2(c x))+4 a b c x \left(c x \left(\log \left(1-c^2 x^2\right)-2 \log (c x)\right)+2 \tanh ^{-1}(c x)\right)+a b \left(c x (c x \log (1-c x)-c x \log (c x+1)+2)+2 \tanh ^{-1}(c x)\right)-2 b^2 c^2 x^2 \left(\tanh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+\tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)-\frac{2}{3} \tanh ^{-1}(c x)^3-\tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+\tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)+\frac{i \pi ^3}{24}\right)+b^2 \left(-2 c^2 x^2 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+\left(1-c^2 x^2\right) \tanh ^{-1}(c x)^2+2 c x \tanh ^{-1}(c x)\right)+4 b^2 c x \left(c x \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+\tanh ^{-1}(c x) \left((1-c x) \tanh ^{-1}(c x)-2 c x \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)\right)\right)\right)}{2 x^2}","-b c^2 d^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+b c^2 d^2 \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)+\frac{5}{2} c^2 d^2 \left(a+b \tanh ^{-1}(c x)\right)^2+2 c^2 d^2 \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2+4 b c^2 d^2 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)-\frac{d^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 x^2}-\frac{2 c d^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{x}-\frac{b c d^2 \left(a+b \tanh ^{-1}(c x)\right)}{x}-2 b^2 c^2 d^2 \text{Li}_2\left(\frac{2}{c x+1}-1\right)+\frac{1}{2} b^2 c^2 d^2 \text{Li}_3\left(1-\frac{2}{1-c x}\right)-\frac{1}{2} b^2 c^2 d^2 \text{Li}_3\left(\frac{2}{1-c x}-1\right)-\frac{1}{2} b^2 c^2 d^2 \log \left(1-c^2 x^2\right)+b^2 c^2 d^2 \log (x)",1,"-1/2*(d^2*(a^2 + 4*a^2*c*x - 2*a^2*c^2*x^2*Log[x] + a*b*(2*ArcTanh[c*x] + c*x*(2 + c*x*Log[1 - c*x] - c*x*Log[1 + c*x])) + b^2*(2*c*x*ArcTanh[c*x] + (1 - c^2*x^2)*ArcTanh[c*x]^2 - 2*c^2*x^2*Log[(c*x)/Sqrt[1 - c^2*x^2]]) + 4*a*b*c*x*(2*ArcTanh[c*x] + c*x*(-2*Log[c*x] + Log[1 - c^2*x^2])) + 4*b^2*c*x*(ArcTanh[c*x]*((1 - c*x)*ArcTanh[c*x] - 2*c*x*Log[1 - E^(-2*ArcTanh[c*x])]) + c*x*PolyLog[2, E^(-2*ArcTanh[c*x])]) + 2*a*b*c^2*x^2*(PolyLog[2, -(c*x)] - PolyLog[2, c*x]) - 2*b^2*c^2*x^2*((I/24)*Pi^3 - (2*ArcTanh[c*x]^3)/3 - ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + PolyLog[3, -E^(-2*ArcTanh[c*x])]/2 - PolyLog[3, E^(2*ArcTanh[c*x])]/2)))/x^2","C",0
83,1,270,244,0.6347974,"\int \frac{(d+c d x)^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d + c*d*x)^2*(a + b*ArcTanh[c*x])^2)/x^4,x]","-\frac{d^2 \left(3 a^2 c^2 x^2+3 a^2 c x+a^2-8 a b c^3 x^3 \log (c x)+3 a b c^3 x^3 \log (1-c x)-3 a b c^3 x^3 \log (c x+1)+6 a b c^2 x^2+4 a b c^3 x^3 \log \left(1-c^2 x^2\right)+b \tanh ^{-1}(c x) \left(a \left(6 c^2 x^2+6 c x+2\right)-8 b c^3 x^3 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+b c x \left(-c^2 x^2+6 c x+1\right)\right)+a b c x+4 b^2 c^3 x^3 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+b^2 c^2 x^2-6 b^2 c^3 x^3 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+b^2 \left(-7 c^3 x^3+3 c^2 x^2+3 c x+1\right) \tanh ^{-1}(c x)^2\right)}{3 x^3}","\frac{8}{3} a b c^3 d^2 \log (x)+\frac{8}{3} b c^3 d^2 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)-\frac{2 b c^2 d^2 \left(a+b \tanh ^{-1}(c x)\right)}{x}-\frac{d^2 (c x+1)^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{3 x^3}-\frac{b c d^2 \left(a+b \tanh ^{-1}(c x)\right)}{3 x^2}-\frac{4}{3} b^2 c^3 d^2 \text{Li}_2(-c x)+\frac{4}{3} b^2 c^3 d^2 \text{Li}_2(c x)+\frac{4}{3} b^2 c^3 d^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)+2 b^2 c^3 d^2 \log (x)+\frac{1}{3} b^2 c^3 d^2 \tanh ^{-1}(c x)-\frac{b^2 c^2 d^2}{3 x}-b^2 c^3 d^2 \log \left(1-c^2 x^2\right)",1,"-1/3*(d^2*(a^2 + 3*a^2*c*x + a*b*c*x + 3*a^2*c^2*x^2 + 6*a*b*c^2*x^2 + b^2*c^2*x^2 + b^2*(1 + 3*c*x + 3*c^2*x^2 - 7*c^3*x^3)*ArcTanh[c*x]^2 + b*ArcTanh[c*x]*(b*c*x*(1 + 6*c*x - c^2*x^2) + a*(2 + 6*c*x + 6*c^2*x^2) - 8*b*c^3*x^3*Log[1 - E^(-2*ArcTanh[c*x])]) - 8*a*b*c^3*x^3*Log[c*x] + 3*a*b*c^3*x^3*Log[1 - c*x] - 3*a*b*c^3*x^3*Log[1 + c*x] - 6*b^2*c^3*x^3*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 4*a*b*c^3*x^3*Log[1 - c^2*x^2] + 4*b^2*c^3*x^3*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x^3","A",0
84,1,385,415,1.6642917,"\int x^3 (d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2,x]","\frac{d^3 \left(180 a^2 c^7 x^7+630 a^2 c^6 x^6+756 a^2 c^5 x^5+315 a^2 c^4 x^4+60 a b c^6 x^6+252 a b c^5 x^5+468 a b c^4 x^4+630 a b c^3 x^3+936 a b c^2 x^2+936 a b \log \left(c^2 x^2-1\right)+6 b \tanh ^{-1}(c x) \left(3 a c^4 x^4 \left(20 c^3 x^3+70 c^2 x^2+84 c x+35\right)+b \left(10 c^6 x^6+42 c^5 x^5+78 c^4 x^4+105 c^3 x^3+156 c^2 x^2+315 c x-244\right)-312 b \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+1890 a b c x+945 a b \log (1-c x)-945 a b \log (c x+1)-1464 a b+12 b^2 c^5 x^5+63 b^2 c^4 x^4+176 b^2 c^3 x^3+441 b^2 c^2 x^2+1386 b^2 \log \left(1-c^2 x^2\right)+9 b^2 \left(20 c^7 x^7+70 c^6 x^6+84 c^5 x^5+35 c^4 x^4-209\right) \tanh ^{-1}(c x)^2+936 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+1464 b^2 c x-504 b^2\right)}{1260 c^4}","-\frac{d^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{140 c^4}-\frac{52 b d^3 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{35 c^4}+\frac{1}{7} c^3 d^3 x^7 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{3 a b d^3 x}{2 c^3}+\frac{1}{2} c^2 d^3 x^6 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{21} b c^2 d^3 x^6 \left(a+b \tanh ^{-1}(c x)\right)+\frac{26 b d^3 x^2 \left(a+b \tanh ^{-1}(c x)\right)}{35 c^2}+\frac{3}{5} c d^3 x^5 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{5} b c d^3 x^5 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{4} d^3 x^4 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{13}{35} b d^3 x^4 \left(a+b \tanh ^{-1}(c x)\right)+\frac{b d^3 x^3 \left(a+b \tanh ^{-1}(c x)\right)}{2 c}-\frac{26 b^2 d^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{35 c^4}-\frac{122 b^2 d^3 \tanh ^{-1}(c x)}{105 c^4}+\frac{122 b^2 d^3 x}{105 c^3}+\frac{3 b^2 d^3 x \tanh ^{-1}(c x)}{2 c^3}+\frac{7 b^2 d^3 x^2}{20 c^2}+\frac{11 b^2 d^3 \log \left(1-c^2 x^2\right)}{10 c^4}+\frac{1}{105} b^2 c d^3 x^5+\frac{44 b^2 d^3 x^3}{315 c}+\frac{1}{20} b^2 d^3 x^4",1,"(d^3*(-1464*a*b - 504*b^2 + 1890*a*b*c*x + 1464*b^2*c*x + 936*a*b*c^2*x^2 + 441*b^2*c^2*x^2 + 630*a*b*c^3*x^3 + 176*b^2*c^3*x^3 + 315*a^2*c^4*x^4 + 468*a*b*c^4*x^4 + 63*b^2*c^4*x^4 + 756*a^2*c^5*x^5 + 252*a*b*c^5*x^5 + 12*b^2*c^5*x^5 + 630*a^2*c^6*x^6 + 60*a*b*c^6*x^6 + 180*a^2*c^7*x^7 + 9*b^2*(-209 + 35*c^4*x^4 + 84*c^5*x^5 + 70*c^6*x^6 + 20*c^7*x^7)*ArcTanh[c*x]^2 + 6*b*ArcTanh[c*x]*(3*a*c^4*x^4*(35 + 84*c*x + 70*c^2*x^2 + 20*c^3*x^3) + b*(-244 + 315*c*x + 156*c^2*x^2 + 105*c^3*x^3 + 78*c^4*x^4 + 42*c^5*x^5 + 10*c^6*x^6) - 312*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 945*a*b*Log[1 - c*x] - 945*a*b*Log[1 + c*x] + 1386*b^2*Log[1 - c^2*x^2] + 936*a*b*Log[-1 + c^2*x^2] + 936*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(1260*c^4)","A",0
85,1,356,377,1.3295332,"\int x^2 (d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2,x]","\frac{d^3 \left(30 a^2 c^6 x^6+108 a^2 c^5 x^5+135 a^2 c^4 x^4+60 a^2 c^3 x^3+12 a b c^5 x^5+54 a b c^4 x^4+110 a b c^3 x^3+168 a b c^2 x^2+168 a b \log \left(c^2 x^2-1\right)+2 b \tanh ^{-1}(c x) \left(3 a c^3 x^3 \left(10 c^3 x^3+36 c^2 x^2+45 c x+20\right)+b \left(6 c^5 x^5+27 c^4 x^4+55 c^3 x^3+84 c^2 x^2+165 c x-111\right)-168 b \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+330 a b c x+165 a b \log (1-c x)-165 a b \log (c x+1)-162 a b+3 b^2 c^4 x^4+18 b^2 c^3 x^3+61 b^2 c^2 x^2+226 b^2 \log \left(1-c^2 x^2\right)+3 b^2 \left(10 c^6 x^6+36 c^5 x^5+45 c^4 x^4+20 c^3 x^3-111\right) \tanh ^{-1}(c x)^2+168 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+222 b^2 c x-64 b^2\right)}{180 c^3}","\frac{1}{6} c^3 d^3 x^6 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{d^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{60 c^3}-\frac{28 b d^3 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{15 c^3}+\frac{3}{5} c^2 d^3 x^5 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{15} b c^2 d^3 x^5 \left(a+b \tanh ^{-1}(c x)\right)+\frac{11 a b d^3 x}{6 c^2}+\frac{3}{4} c d^3 x^4 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{3}{10} b c d^3 x^4 \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{3} d^3 x^3 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{11}{18} b d^3 x^3 \left(a+b \tanh ^{-1}(c x)\right)+\frac{14 b d^3 x^2 \left(a+b \tanh ^{-1}(c x)\right)}{15 c}-\frac{14 b^2 d^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{15 c^3}-\frac{37 b^2 d^3 \tanh ^{-1}(c x)}{30 c^3}+\frac{37 b^2 d^3 x}{30 c^2}+\frac{11 b^2 d^3 x \tanh ^{-1}(c x)}{6 c^2}+\frac{113 b^2 d^3 \log \left(1-c^2 x^2\right)}{90 c^3}+\frac{1}{60} b^2 c d^3 x^4+\frac{61 b^2 d^3 x^2}{180 c}+\frac{1}{10} b^2 d^3 x^3",1,"(d^3*(-162*a*b - 64*b^2 + 330*a*b*c*x + 222*b^2*c*x + 168*a*b*c^2*x^2 + 61*b^2*c^2*x^2 + 60*a^2*c^3*x^3 + 110*a*b*c^3*x^3 + 18*b^2*c^3*x^3 + 135*a^2*c^4*x^4 + 54*a*b*c^4*x^4 + 3*b^2*c^4*x^4 + 108*a^2*c^5*x^5 + 12*a*b*c^5*x^5 + 30*a^2*c^6*x^6 + 3*b^2*(-111 + 20*c^3*x^3 + 45*c^4*x^4 + 36*c^5*x^5 + 10*c^6*x^6)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(3*a*c^3*x^3*(20 + 45*c*x + 36*c^2*x^2 + 10*c^3*x^3) + b*(-111 + 165*c*x + 84*c^2*x^2 + 55*c^3*x^3 + 27*c^4*x^4 + 6*c^5*x^5) - 168*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 165*a*b*Log[1 - c*x] - 165*a*b*Log[1 + c*x] + 226*b^2*Log[1 - c^2*x^2] + 168*a*b*Log[-1 + c^2*x^2] + 168*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(180*c^3)","A",0
86,1,325,286,1.2068359,"\int x (d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2,x]","\frac{d^3 \left(12 a^2 c^5 x^5+45 a^2 c^4 x^4+60 a^2 c^3 x^3+30 a^2 c^2 x^2+6 a b c^4 x^4+30 a b c^3 x^3+72 a b c^2 x^2+72 a b \log \left(c^2 x^2-1\right)+6 b \tanh ^{-1}(c x) \left(a c^2 x^2 \left(4 c^3 x^3+15 c^2 x^2+20 c x+10\right)+b \left(c^4 x^4+5 c^3 x^3+12 c^2 x^2+25 c x-13\right)-24 b \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+150 a b c x+75 a b \log (1-c x)-75 a b \log (c x+1)-18 a b+2 b^2 c^3 x^3+15 b^2 c^2 x^2+90 b^2 \log \left(1-c^2 x^2\right)+3 b^2 \left(4 c^5 x^5+15 c^4 x^4+20 c^3 x^3+10 c^2 x^2-49\right) \tanh ^{-1}(c x)^2+72 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+78 b^2 c x-15 b^2\right)}{60 c^2}","\frac{1}{10} b c^2 d^3 x^4 \left(a+b \tanh ^{-1}(c x)\right)+\frac{d^3 (c x+1)^5 \left(a+b \tanh ^{-1}(c x)\right)^2}{5 c^2}-\frac{d^3 (c x+1)^4 \left(a+b \tanh ^{-1}(c x)\right)^2}{4 c^2}-\frac{12 b d^3 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{5 c^2}+\frac{1}{2} b c d^3 x^3 \left(a+b \tanh ^{-1}(c x)\right)+\frac{6}{5} b d^3 x^2 \left(a+b \tanh ^{-1}(c x)\right)+\frac{5 a b d^3 x}{2 c}-\frac{6 b^2 d^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{5 c^2}+\frac{3 b^2 d^3 \log \left(1-c^2 x^2\right)}{2 c^2}-\frac{13 b^2 d^3 \tanh ^{-1}(c x)}{10 c^2}+\frac{1}{30} b^2 c d^3 x^3+\frac{13 b^2 d^3 x}{10 c}+\frac{5 b^2 d^3 x \tanh ^{-1}(c x)}{2 c}+\frac{1}{4} b^2 d^3 x^2",1,"(d^3*(-18*a*b - 15*b^2 + 150*a*b*c*x + 78*b^2*c*x + 30*a^2*c^2*x^2 + 72*a*b*c^2*x^2 + 15*b^2*c^2*x^2 + 60*a^2*c^3*x^3 + 30*a*b*c^3*x^3 + 2*b^2*c^3*x^3 + 45*a^2*c^4*x^4 + 6*a*b*c^4*x^4 + 12*a^2*c^5*x^5 + 3*b^2*(-49 + 10*c^2*x^2 + 20*c^3*x^3 + 15*c^4*x^4 + 4*c^5*x^5)*ArcTanh[c*x]^2 + 6*b*ArcTanh[c*x]*(a*c^2*x^2*(10 + 20*c*x + 15*c^2*x^2 + 4*c^3*x^3) + b*(-13 + 25*c*x + 12*c^2*x^2 + 5*c^3*x^3 + c^4*x^4) - 24*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 75*a*b*Log[1 - c*x] - 75*a*b*Log[1 + c*x] + 90*b^2*Log[1 - c^2*x^2] + 72*a*b*Log[-1 + c^2*x^2] + 72*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(60*c^2)","A",0
87,1,293,206,0.8753438,"\int (d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2,x]","\frac{d^3 \left(3 a^2 c^4 x^4+12 a^2 c^3 x^3+18 a^2 c^2 x^2+12 a^2 c x+2 a b c^3 x^3+12 a b c^2 x^2+12 a b \log \left(1-c^2 x^2\right)+12 a b \log \left(c^2 x^2-1\right)+2 b \tanh ^{-1}(c x) \left(3 a c x \left(c^3 x^3+4 c^2 x^2+6 c x+4\right)+b \left(c^3 x^3+6 c^2 x^2+21 c x-6\right)-24 b \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+42 a b c x+21 a b \log (1-c x)-21 a b \log (c x+1)+b^2 c^2 x^2+22 b^2 \log \left(1-c^2 x^2\right)+3 b^2 \left(c^4 x^4+4 c^3 x^3+6 c^2 x^2+4 c x-15\right) \tanh ^{-1}(c x)^2+24 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+12 b^2 c x-b^2\right)}{12 c}","\frac{1}{6} b c^2 d^3 x^3 \left(a+b \tanh ^{-1}(c x)\right)+b c d^3 x^2 \left(a+b \tanh ^{-1}(c x)\right)+\frac{d^3 (c x+1)^4 \left(a+b \tanh ^{-1}(c x)\right)^2}{4 c}-\frac{4 b d^3 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c}+\frac{7}{2} a b d^3 x+\frac{11 b^2 d^3 \log \left(1-c^2 x^2\right)}{6 c}-\frac{2 b^2 d^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c}+\frac{1}{12} b^2 c d^3 x^2-\frac{b^2 d^3 \tanh ^{-1}(c x)}{c}+\frac{7}{2} b^2 d^3 x \tanh ^{-1}(c x)+b^2 d^3 x",1,"(d^3*(-b^2 + 12*a^2*c*x + 42*a*b*c*x + 12*b^2*c*x + 18*a^2*c^2*x^2 + 12*a*b*c^2*x^2 + b^2*c^2*x^2 + 12*a^2*c^3*x^3 + 2*a*b*c^3*x^3 + 3*a^2*c^4*x^4 + 3*b^2*(-15 + 4*c*x + 6*c^2*x^2 + 4*c^3*x^3 + c^4*x^4)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(3*a*c*x*(4 + 6*c*x + 4*c^2*x^2 + c^3*x^3) + b*(-6 + 21*c*x + 6*c^2*x^2 + c^3*x^3) - 24*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 21*a*b*Log[1 - c*x] - 21*a*b*Log[1 + c*x] + 12*a*b*Log[1 - c^2*x^2] + 22*b^2*Log[1 - c^2*x^2] + 12*a*b*Log[-1 + c^2*x^2] + 24*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(12*c)","A",0
88,1,448,355,0.833219,"\int \frac{(d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x,x]","\frac{1}{24} d^3 \left(8 a^2 c^3 x^3+36 a^2 c^2 x^2+72 a^2 c x+24 a^2 \log (c x)+16 a b c^3 x^3 \tanh ^{-1}(c x)+8 a b c^2 x^2+72 a b \log \left(1-c^2 x^2\right)+8 a b \log \left(c^2 x^2-1\right)+72 a b c^2 x^2 \tanh ^{-1}(c x)-24 a b \text{Li}_2(-c x)+24 a b \text{Li}_2(c x)+72 a b c x+36 a b \log (1-c x)-36 a b \log (c x+1)+144 a b c x \tanh ^{-1}(c x)+8 b^2 c^3 x^3 \tanh ^{-1}(c x)^2+36 b^2 \log \left(1-c^2 x^2\right)+36 b^2 c^2 x^2 \tanh ^{-1}(c x)^2+8 b^2 c^2 x^2 \tanh ^{-1}(c x)+8 b^2 \left(3 \tanh ^{-1}(c x)+10\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+24 b^2 \tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)+12 b^2 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-12 b^2 \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)+8 b^2 c x+72 b^2 c x \tanh ^{-1}(c x)^2+72 b^2 c x \tanh ^{-1}(c x)-16 b^2 \tanh ^{-1}(c x)^3-116 b^2 \tanh ^{-1}(c x)^2-8 b^2 \tanh ^{-1}(c x)-24 b^2 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-160 b^2 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+24 b^2 \tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)+i \pi ^3 b^2\right)","\frac{1}{3} c^3 d^3 x^3 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{3}{2} c^2 d^3 x^2 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{3} b c^2 d^3 x^2 \left(a+b \tanh ^{-1}(c x)\right)-b d^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+b d^3 \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)+3 a b c d^3 x+3 c d^3 x \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{11}{6} d^3 \left(a+b \tanh ^{-1}(c x)\right)^2+2 d^3 \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2-\frac{20}{3} b d^3 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+\frac{3}{2} b^2 d^3 \log \left(1-c^2 x^2\right)-\frac{10}{3} b^2 d^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)+\frac{1}{2} b^2 d^3 \text{Li}_3\left(1-\frac{2}{1-c x}\right)-\frac{1}{2} b^2 d^3 \text{Li}_3\left(\frac{2}{1-c x}-1\right)+\frac{1}{3} b^2 c d^3 x-\frac{1}{3} b^2 d^3 \tanh ^{-1}(c x)+3 b^2 c d^3 x \tanh ^{-1}(c x)",1,"(d^3*(I*b^2*Pi^3 + 72*a^2*c*x + 72*a*b*c*x + 8*b^2*c*x + 36*a^2*c^2*x^2 + 8*a*b*c^2*x^2 + 8*a^2*c^3*x^3 - 8*b^2*ArcTanh[c*x] + 144*a*b*c*x*ArcTanh[c*x] + 72*b^2*c*x*ArcTanh[c*x] + 72*a*b*c^2*x^2*ArcTanh[c*x] + 8*b^2*c^2*x^2*ArcTanh[c*x] + 16*a*b*c^3*x^3*ArcTanh[c*x] - 116*b^2*ArcTanh[c*x]^2 + 72*b^2*c*x*ArcTanh[c*x]^2 + 36*b^2*c^2*x^2*ArcTanh[c*x]^2 + 8*b^2*c^3*x^3*ArcTanh[c*x]^2 - 16*b^2*ArcTanh[c*x]^3 - 160*b^2*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 24*b^2*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 24*b^2*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 24*a^2*Log[c*x] + 36*a*b*Log[1 - c*x] - 36*a*b*Log[1 + c*x] + 72*a*b*Log[1 - c^2*x^2] + 36*b^2*Log[1 - c^2*x^2] + 8*a*b*Log[-1 + c^2*x^2] + 8*b^2*(10 + 3*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 24*b^2*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 24*a*b*PolyLog[2, -(c*x)] + 24*a*b*PolyLog[2, c*x] + 12*b^2*PolyLog[3, -E^(-2*ArcTanh[c*x])] - 12*b^2*PolyLog[3, E^(2*ArcTanh[c*x])]))/24","C",0
89,1,479,361,0.6412282,"\int \frac{(d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^2,x]","\frac{d^3 \left(4 a^2 c^3 x^3+24 a^2 c^2 x^2+24 a^2 c x \log (x)-8 a^2+8 a b c^3 x^3 \tanh ^{-1}(c x)+8 a b c^2 x^2+16 a b c x \log \left(1-c^2 x^2\right)+48 a b c^2 x^2 \tanh ^{-1}(c x)-24 a b c x \text{Li}_2(-c x)+24 a b c x \text{Li}_2(c x)+16 a b c x \log (c x)+4 a b c x \log (1-c x)-4 a b c x \log (c x+1)-16 a b \tanh ^{-1}(c x)+4 b^2 c^3 x^3 \tanh ^{-1}(c x)^2+4 b^2 c x \log \left(1-c^2 x^2\right)+24 b^2 c^2 x^2 \tanh ^{-1}(c x)^2+8 b^2 c^2 x^2 \tanh ^{-1}(c x)+24 b^2 c x \left(\tanh ^{-1}(c x)+1\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-8 b^2 c x \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+24 b^2 c x \tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)+12 b^2 c x \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-12 b^2 c x \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)+i \pi ^3 b^2 c x-16 b^2 c x \tanh ^{-1}(c x)^3-20 b^2 c x \tanh ^{-1}(c x)^2-8 b^2 \tanh ^{-1}(c x)^2+16 b^2 c x \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-24 b^2 c x \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-48 b^2 c x \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+24 b^2 c x \tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)\right)}{8 x}","\frac{1}{2} c^3 d^3 x^2 \left(a+b \tanh ^{-1}(c x)\right)^2+a b c^2 d^3 x+3 c^2 d^3 x \left(a+b \tanh ^{-1}(c x)\right)^2-3 b c d^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+3 b c d^3 \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)+\frac{7}{2} c d^3 \left(a+b \tanh ^{-1}(c x)\right)^2-\frac{d^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{x}+6 c d^3 \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2-6 b c d^3 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+2 b c d^3 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{2} b^2 c d^3 \log \left(1-c^2 x^2\right)+b^2 c^2 d^3 x \tanh ^{-1}(c x)-3 b^2 c d^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)-b^2 c d^3 \text{Li}_2\left(\frac{2}{c x+1}-1\right)+\frac{3}{2} b^2 c d^3 \text{Li}_3\left(1-\frac{2}{1-c x}\right)-\frac{3}{2} b^2 c d^3 \text{Li}_3\left(\frac{2}{1-c x}-1\right)",1,"(d^3*(-8*a^2 + I*b^2*c*Pi^3*x + 24*a^2*c^2*x^2 + 8*a*b*c^2*x^2 + 4*a^2*c^3*x^3 - 16*a*b*ArcTanh[c*x] + 48*a*b*c^2*x^2*ArcTanh[c*x] + 8*b^2*c^2*x^2*ArcTanh[c*x] + 8*a*b*c^3*x^3*ArcTanh[c*x] - 8*b^2*ArcTanh[c*x]^2 - 20*b^2*c*x*ArcTanh[c*x]^2 + 24*b^2*c^2*x^2*ArcTanh[c*x]^2 + 4*b^2*c^3*x^3*ArcTanh[c*x]^2 - 16*b^2*c*x*ArcTanh[c*x]^3 + 16*b^2*c*x*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - 48*b^2*c*x*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 24*b^2*c*x*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 24*b^2*c*x*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 24*a^2*c*x*Log[x] + 16*a*b*c*x*Log[c*x] + 4*a*b*c*x*Log[1 - c*x] - 4*a*b*c*x*Log[1 + c*x] + 16*a*b*c*x*Log[1 - c^2*x^2] + 4*b^2*c*x*Log[1 - c^2*x^2] + 24*b^2*c*x*(1 + ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 8*b^2*c*x*PolyLog[2, E^(-2*ArcTanh[c*x])] + 24*b^2*c*x*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 24*a*b*c*x*PolyLog[2, -(c*x)] + 24*a*b*c*x*PolyLog[2, c*x] + 12*b^2*c*x*PolyLog[3, -E^(-2*ArcTanh[c*x])] - 12*b^2*c*x*PolyLog[3, E^(2*ArcTanh[c*x])]))/(8*x)","C",0
90,1,461,385,1.023704,"\int \frac{(d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^3,x]","\frac{1}{2} d^3 \left(2 a^2 c^3 x+6 a^2 c^2 \log (x)-\frac{6 a^2 c}{x}-\frac{a^2}{x^2}-6 a b c^2 (\text{Li}_2(-c x)-\text{Li}_2(c x))+2 a b c^2 \left(\log \left(1-c^2 x^2\right)+2 c x \tanh ^{-1}(c x)\right)-\frac{6 a b c \left(c x \left(\log \left(1-c^2 x^2\right)-2 \log (c x)\right)+2 \tanh ^{-1}(c x)\right)}{x}-\frac{a b \left(c x (c x \log (1-c x)-c x \log (c x+1)+2)+2 \tanh ^{-1}(c x)\right)}{x^2}+2 b^2 c^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)+6 b^2 c^2 \left(\tanh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+\tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)-\frac{2}{3} \tanh ^{-1}(c x)^3-\tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+\tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)+\frac{i \pi ^3}{24}\right)+\frac{b^2 \left(2 c^2 x^2 \log \left(\frac{c x}{\sqrt{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)}{x^2}+\frac{6 b^2 c \left(\tanh ^{-1}(c x) \left((c x-1) \tanh ^{-1}(c x)+2 c x \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)\right)-c x \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)\right)}{x}\right)","c^3 d^3 x \left(a+b \tanh ^{-1}(c x)\right)^2-3 b c^2 d^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+3 b c^2 d^3 \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)+\frac{9}{2} c^2 d^3 \left(a+b \tanh ^{-1}(c x)\right)^2+6 c^2 d^3 \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2-2 b c^2 d^3 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+6 b c^2 d^3 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)-\frac{d^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 x^2}-\frac{3 c d^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{x}-\frac{b c d^3 \left(a+b \tanh ^{-1}(c x)\right)}{x}-b^2 c^2 d^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)-3 b^2 c^2 d^3 \text{Li}_2\left(\frac{2}{c x+1}-1\right)+\frac{3}{2} b^2 c^2 d^3 \text{Li}_3\left(1-\frac{2}{1-c x}\right)-\frac{3}{2} b^2 c^2 d^3 \text{Li}_3\left(\frac{2}{1-c x}-1\right)-\frac{1}{2} b^2 c^2 d^3 \log \left(1-c^2 x^2\right)+b^2 c^2 d^3 \log (x)",1,"(d^3*(-(a^2/x^2) - (6*a^2*c)/x + 2*a^2*c^3*x + 6*a^2*c^2*Log[x] - (a*b*(2*ArcTanh[c*x] + c*x*(2 + c*x*Log[1 - c*x] - c*x*Log[1 + c*x])))/x^2 + (b^2*(-2*c*x*ArcTanh[c*x] + (-1 + c^2*x^2)*ArcTanh[c*x]^2 + 2*c^2*x^2*Log[(c*x)/Sqrt[1 - c^2*x^2]]))/x^2 + 2*a*b*c^2*(2*c*x*ArcTanh[c*x] + Log[1 - c^2*x^2]) - (6*a*b*c*(2*ArcTanh[c*x] + c*x*(-2*Log[c*x] + Log[1 - c^2*x^2])))/x + 2*b^2*c^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^2*c*(ArcTanh[c*x]*((-1 + c*x)*ArcTanh[c*x] + 2*c*x*Log[1 - E^(-2*ArcTanh[c*x])]) - c*x*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x - 6*a*b*c^2*(PolyLog[2, -(c*x)] - PolyLog[2, c*x]) + 6*b^2*c^2*((I/24)*Pi^3 - (2*ArcTanh[c*x]^3)/3 - ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + PolyLog[3, -E^(-2*ArcTanh[c*x])]/2 - PolyLog[3, E^(2*ArcTanh[c*x])]/2)))/2","C",0
91,1,569,396,0.7331421,"\int \frac{(d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^4,x]","\frac{d^3 \left(24 a^2 c^3 x^3 \log (x)-72 a^2 c^2 x^2-36 a^2 c x-8 a^2-24 a b c^3 x^3 \text{Li}_2(-c x)+24 a b c^3 x^3 \text{Li}_2(c x)+160 a b c^3 x^3 \log (c x)-36 a b c^3 x^3 \log (1-c x)+36 a b c^3 x^3 \log (c x+1)-72 a b c^2 x^2-144 a b c^2 x^2 \tanh ^{-1}(c x)-80 a b c^3 x^3 \log \left(1-c^2 x^2\right)-8 a b c x-72 a b c x \tanh ^{-1}(c x)-16 a b \tanh ^{-1}(c x)+24 b^2 c^3 x^3 \tanh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-80 b^2 c^3 x^3 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+24 b^2 c^3 x^3 \tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)+12 b^2 c^3 x^3 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-12 b^2 c^3 x^3 \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)+i \pi ^3 b^2 c^3 x^3-16 b^2 c^3 x^3 \tanh ^{-1}(c x)^3+116 b^2 c^3 x^3 \tanh ^{-1}(c x)^2+8 b^2 c^3 x^3 \tanh ^{-1}(c x)+160 b^2 c^3 x^3 \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-24 b^2 c^3 x^3 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+24 b^2 c^3 x^3 \tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)-8 b^2 c^2 x^2-72 b^2 c^2 x^2 \tanh ^{-1}(c x)^2-72 b^2 c^2 x^2 \tanh ^{-1}(c x)+72 b^2 c^3 x^3 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)-36 b^2 c x \tanh ^{-1}(c x)^2-8 b^2 c x \tanh ^{-1}(c x)-8 b^2 \tanh ^{-1}(c x)^2\right)}{24 x^3}","-b c^3 d^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+b c^3 d^3 \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)+\frac{29}{6} c^3 d^3 \left(a+b \tanh ^{-1}(c x)\right)^2+2 c^3 d^3 \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{20}{3} b c^3 d^3 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)-\frac{3 c^2 d^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{x}-\frac{3 b c^2 d^3 \left(a+b \tanh ^{-1}(c x)\right)}{x}-\frac{d^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{3 x^3}-\frac{3 c d^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 x^2}-\frac{b c d^3 \left(a+b \tanh ^{-1}(c x)\right)}{3 x^2}-\frac{10}{3} b^2 c^3 d^3 \text{Li}_2\left(\frac{2}{c x+1}-1\right)+\frac{1}{2} b^2 c^3 d^3 \text{Li}_3\left(1-\frac{2}{1-c x}\right)-\frac{1}{2} b^2 c^3 d^3 \text{Li}_3\left(\frac{2}{1-c x}-1\right)+3 b^2 c^3 d^3 \log (x)+\frac{1}{3} b^2 c^3 d^3 \tanh ^{-1}(c x)-\frac{b^2 c^2 d^3}{3 x}-\frac{3}{2} b^2 c^3 d^3 \log \left(1-c^2 x^2\right)",1,"(d^3*(-8*a^2 - 36*a^2*c*x - 8*a*b*c*x - 72*a^2*c^2*x^2 - 72*a*b*c^2*x^2 - 8*b^2*c^2*x^2 + I*b^2*c^3*Pi^3*x^3 - 16*a*b*ArcTanh[c*x] - 72*a*b*c*x*ArcTanh[c*x] - 8*b^2*c*x*ArcTanh[c*x] - 144*a*b*c^2*x^2*ArcTanh[c*x] - 72*b^2*c^2*x^2*ArcTanh[c*x] + 8*b^2*c^3*x^3*ArcTanh[c*x] - 8*b^2*ArcTanh[c*x]^2 - 36*b^2*c*x*ArcTanh[c*x]^2 - 72*b^2*c^2*x^2*ArcTanh[c*x]^2 + 116*b^2*c^3*x^3*ArcTanh[c*x]^2 - 16*b^2*c^3*x^3*ArcTanh[c*x]^3 + 160*b^2*c^3*x^3*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - 24*b^2*c^3*x^3*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 24*b^2*c^3*x^3*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 24*a^2*c^3*x^3*Log[x] + 160*a*b*c^3*x^3*Log[c*x] - 36*a*b*c^3*x^3*Log[1 - c*x] + 36*a*b*c^3*x^3*Log[1 + c*x] + 72*b^2*c^3*x^3*Log[(c*x)/Sqrt[1 - c^2*x^2]] - 80*a*b*c^3*x^3*Log[1 - c^2*x^2] + 24*b^2*c^3*x^3*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 80*b^2*c^3*x^3*PolyLog[2, E^(-2*ArcTanh[c*x])] + 24*b^2*c^3*x^3*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 24*a*b*c^3*x^3*PolyLog[2, -(c*x)] + 24*a*b*c^3*x^3*PolyLog[2, c*x] + 12*b^2*c^3*x^3*PolyLog[3, -E^(-2*ArcTanh[c*x])] - 12*b^2*c^3*x^3*PolyLog[3, E^(2*ArcTanh[c*x])]))/(24*x^3)","C",0
92,1,343,271,0.8049327,"\int \frac{(d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{x^5} \, dx","Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^5,x]","-\frac{d^3 \left(12 a^2 c^3 x^3+18 a^2 c^2 x^2+12 a^2 c x+3 a^2-48 a b c^4 x^4 \log (c x)+21 a b c^4 x^4 \log (1-c x)-21 a b c^4 x^4 \log (c x+1)+42 a b c^3 x^3+12 a b c^2 x^2+24 a b c^4 x^4 \log \left(1-c^2 x^2\right)+2 b \tanh ^{-1}(c x) \left(3 a \left(4 c^3 x^3+6 c^2 x^2+4 c x+1\right)-24 b c^4 x^4 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+b c x \left(-6 c^3 x^3+21 c^2 x^2+6 c x+1\right)\right)+2 a b c x+24 b^2 c^4 x^4 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-b^2 c^4 x^4+12 b^2 c^3 x^3+b^2 c^2 x^2-44 b^2 c^4 x^4 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+3 b^2 \left(-15 c^4 x^4+4 c^3 x^3+6 c^2 x^2+4 c x+1\right) \tanh ^{-1}(c x)^2\right)}{12 x^4}","4 a b c^4 d^3 \log (x)+4 b c^4 d^3 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)-\frac{7 b c^3 d^3 \left(a+b \tanh ^{-1}(c x)\right)}{2 x}-\frac{b c^2 d^3 \left(a+b \tanh ^{-1}(c x)\right)}{x^2}-\frac{d^3 (c x+1)^4 \left(a+b \tanh ^{-1}(c x)\right)^2}{4 x^4}-\frac{b c d^3 \left(a+b \tanh ^{-1}(c x)\right)}{6 x^3}-2 b^2 c^4 d^3 \text{Li}_2(-c x)+2 b^2 c^4 d^3 \text{Li}_2(c x)+2 b^2 c^4 d^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)+\frac{11}{3} b^2 c^4 d^3 \log (x)+b^2 c^4 d^3 \tanh ^{-1}(c x)-\frac{b^2 c^3 d^3}{x}-\frac{b^2 c^2 d^3}{12 x^2}-\frac{11}{6} b^2 c^4 d^3 \log \left(1-c^2 x^2\right)",1,"-1/12*(d^3*(3*a^2 + 12*a^2*c*x + 2*a*b*c*x + 18*a^2*c^2*x^2 + 12*a*b*c^2*x^2 + b^2*c^2*x^2 + 12*a^2*c^3*x^3 + 42*a*b*c^3*x^3 + 12*b^2*c^3*x^3 - b^2*c^4*x^4 + 3*b^2*(1 + 4*c*x + 6*c^2*x^2 + 4*c^3*x^3 - 15*c^4*x^4)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(b*c*x*(1 + 6*c*x + 21*c^2*x^2 - 6*c^3*x^3) + 3*a*(1 + 4*c*x + 6*c^2*x^2 + 4*c^3*x^3) - 24*b*c^4*x^4*Log[1 - E^(-2*ArcTanh[c*x])]) - 48*a*b*c^4*x^4*Log[c*x] + 21*a*b*c^4*x^4*Log[1 - c*x] - 21*a*b*c^4*x^4*Log[1 + c*x] - 44*b^2*c^4*x^4*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 24*a*b*c^4*x^4*Log[1 - c^2*x^2] + 24*b^2*c^4*x^4*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x^4","A",0
93,1,372,352,1.2041223,"\int \frac{(d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{x^6} \, dx","Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^6,x]","-\frac{d^3 \left(30 a^2 c^3 x^3+60 a^2 c^2 x^2+45 a^2 c x+12 a^2-144 a b c^5 x^5 \log (c x)+75 a b c^5 x^5 \log (1-c x)-75 a b c^5 x^5 \log (c x+1)+150 a b c^4 x^4+72 a b c^3 x^3+30 a b c^2 x^2+72 a b c^5 x^5 \log \left(1-c^2 x^2\right)+6 b \tanh ^{-1}(c x) \left(a \left(10 c^3 x^3+20 c^2 x^2+15 c x+4\right)-24 b c^5 x^5 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+b c x \left(-13 c^4 x^4+25 c^3 x^3+12 c^2 x^2+5 c x+1\right)\right)+6 a b c x+72 b^2 c^5 x^5 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-15 b^2 c^5 x^5+78 b^2 c^4 x^4+15 b^2 c^3 x^3+2 b^2 c^2 x^2-180 b^2 c^5 x^5 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+3 b^2 \left(-49 c^5 x^5+10 c^3 x^3+20 c^2 x^2+15 c x+4\right) \tanh ^{-1}(c x)^2\right)}{60 x^5}","\frac{12}{5} a b c^5 d^3 \log (x)+\frac{12}{5} b c^5 d^3 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)-\frac{5 b c^4 d^3 \left(a+b \tanh ^{-1}(c x)\right)}{2 x}-\frac{6 b c^3 d^3 \left(a+b \tanh ^{-1}(c x)\right)}{5 x^2}-\frac{b c^2 d^3 \left(a+b \tanh ^{-1}(c x)\right)}{2 x^3}-\frac{d^3 (c x+1)^4 \left(a+b \tanh ^{-1}(c x)\right)^2}{5 x^5}+\frac{c d^3 (c x+1)^4 \left(a+b \tanh ^{-1}(c x)\right)^2}{20 x^4}-\frac{b c d^3 \left(a+b \tanh ^{-1}(c x)\right)}{10 x^4}-\frac{6}{5} b^2 c^5 d^3 \text{Li}_2(-c x)+\frac{6}{5} b^2 c^5 d^3 \text{Li}_2(c x)+\frac{6}{5} b^2 c^5 d^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)+3 b^2 c^5 d^3 \log (x)+\frac{13}{10} b^2 c^5 d^3 \tanh ^{-1}(c x)-\frac{13 b^2 c^4 d^3}{10 x}-\frac{b^2 c^3 d^3}{4 x^2}-\frac{b^2 c^2 d^3}{30 x^3}-\frac{3}{2} b^2 c^5 d^3 \log \left(1-c^2 x^2\right)",1,"-1/60*(d^3*(12*a^2 + 45*a^2*c*x + 6*a*b*c*x + 60*a^2*c^2*x^2 + 30*a*b*c^2*x^2 + 2*b^2*c^2*x^2 + 30*a^2*c^3*x^3 + 72*a*b*c^3*x^3 + 15*b^2*c^3*x^3 + 150*a*b*c^4*x^4 + 78*b^2*c^4*x^4 - 15*b^2*c^5*x^5 + 3*b^2*(4 + 15*c*x + 20*c^2*x^2 + 10*c^3*x^3 - 49*c^5*x^5)*ArcTanh[c*x]^2 + 6*b*ArcTanh[c*x]*(a*(4 + 15*c*x + 20*c^2*x^2 + 10*c^3*x^3) + b*c*x*(1 + 5*c*x + 12*c^2*x^2 + 25*c^3*x^3 - 13*c^4*x^4) - 24*b*c^5*x^5*Log[1 - E^(-2*ArcTanh[c*x])]) - 144*a*b*c^5*x^5*Log[c*x] + 75*a*b*c^5*x^5*Log[1 - c*x] - 75*a*b*c^5*x^5*Log[1 + c*x] - 180*b^2*c^5*x^5*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 72*a*b*c^5*x^5*Log[1 - c^2*x^2] + 72*b^2*c^5*x^5*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x^5","A",0
94,1,402,479,1.3942421,"\int \frac{(d+c d x)^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{x^7} \, dx","Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^7,x]","-\frac{d^3 \left(60 a^2 c^3 x^3+135 a^2 c^2 x^2+108 a^2 c x+30 a^2-336 a b c^6 x^6 \log (c x)+165 a b c^6 x^6 \log (1-c x)-165 a b c^6 x^6 \log (c x+1)+330 a b c^5 x^5+168 a b c^4 x^4+110 a b c^3 x^3+54 a b c^2 x^2+168 a b c^6 x^6 \log \left(1-c^2 x^2\right)+2 b \tanh ^{-1}(c x) \left(3 a \left(20 c^3 x^3+45 c^2 x^2+36 c x+10\right)-168 b c^6 x^6 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+b c x \left(-111 c^5 x^5+165 c^4 x^4+84 c^3 x^3+55 c^2 x^2+27 c x+6\right)\right)+12 a b c x+168 b^2 c^6 x^6 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-64 b^2 c^6 x^6+222 b^2 c^5 x^5+61 b^2 c^4 x^4+18 b^2 c^3 x^3+3 b^2 c^2 x^2-452 b^2 c^6 x^6 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+3 b^2 \left(-111 c^6 x^6+20 c^3 x^3+45 c^2 x^2+36 c x+10\right) \tanh ^{-1}(c x)^2\right)}{180 x^6}","\frac{28}{15} a b c^6 d^3 \log (x)+\frac{37}{20} b c^6 d^3 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+\frac{1}{60} b c^6 d^3 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)-\frac{11 b c^5 d^3 \left(a+b \tanh ^{-1}(c x)\right)}{6 x}-\frac{14 b c^4 d^3 \left(a+b \tanh ^{-1}(c x)\right)}{15 x^2}-\frac{c^3 d^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{3 x^3}-\frac{11 b c^3 d^3 \left(a+b \tanh ^{-1}(c x)\right)}{18 x^3}-\frac{3 c^2 d^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{4 x^4}-\frac{3 b c^2 d^3 \left(a+b \tanh ^{-1}(c x)\right)}{10 x^4}-\frac{d^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{6 x^6}-\frac{3 c d^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{5 x^5}-\frac{b c d^3 \left(a+b \tanh ^{-1}(c x)\right)}{15 x^5}-\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(-c x)+\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(c x)+\frac{37}{40} b^2 c^6 d^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)-\frac{1}{120} b^2 c^6 d^3 \text{Li}_2\left(1-\frac{2}{c x+1}\right)+\frac{113}{45} b^2 c^6 d^3 \log (x)+\frac{37}{30} b^2 c^6 d^3 \tanh ^{-1}(c x)-\frac{37 b^2 c^5 d^3}{30 x}-\frac{61 b^2 c^4 d^3}{180 x^2}-\frac{b^2 c^3 d^3}{10 x^3}-\frac{b^2 c^2 d^3}{60 x^4}-\frac{113}{90} b^2 c^6 d^3 \log \left(1-c^2 x^2\right)",1,"-1/180*(d^3*(30*a^2 + 108*a^2*c*x + 12*a*b*c*x + 135*a^2*c^2*x^2 + 54*a*b*c^2*x^2 + 3*b^2*c^2*x^2 + 60*a^2*c^3*x^3 + 110*a*b*c^3*x^3 + 18*b^2*c^3*x^3 + 168*a*b*c^4*x^4 + 61*b^2*c^4*x^4 + 330*a*b*c^5*x^5 + 222*b^2*c^5*x^5 - 64*b^2*c^6*x^6 + 3*b^2*(10 + 36*c*x + 45*c^2*x^2 + 20*c^3*x^3 - 111*c^6*x^6)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(3*a*(10 + 36*c*x + 45*c^2*x^2 + 20*c^3*x^3) + b*c*x*(6 + 27*c*x + 55*c^2*x^2 + 84*c^3*x^3 + 165*c^4*x^4 - 111*c^5*x^5) - 168*b*c^6*x^6*Log[1 - E^(-2*ArcTanh[c*x])]) - 336*a*b*c^6*x^6*Log[c*x] + 165*a*b*c^6*x^6*Log[1 - c*x] - 165*a*b*c^6*x^6*Log[1 + c*x] - 452*b^2*c^6*x^6*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 168*a*b*c^6*x^6*Log[1 - c^2*x^2] + 168*b^2*c^6*x^6*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x^6","A",0
95,1,347,329,0.8308553,"\int \frac{x^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{d+c d x} \, dx","Integrate[(x^3*(a + b*ArcTanh[c*x])^2)/(d + c*d*x),x]","-\frac{a^2 \log (c x+1)}{c^4 d}+\frac{a^2 x}{c^3 d}-\frac{a^2 x^2}{2 c^2 d}+\frac{a^2 x^3}{3 c d}+\frac{a b \left(-8 \log \left(\frac{1}{\sqrt{1-c^2 x^2}}\right)+\left(1-c^2 x^2\right) \left(-2 c x \tanh ^{-1}(c x)+3 \tanh ^{-1}(c x)-1\right)-3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-3 c x+8 c x \tanh ^{-1}(c x)+6 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)}{3 c^4 d}+\frac{b^2 \left(6 \log \left(\frac{1}{\sqrt{1-c^2 x^2}}\right)-2 c x \left(1-c^2 x^2\right) \tanh ^{-1}(c x)^2+3 \left(1-c^2 x^2\right) \tanh ^{-1}(c x)^2-2 \left(1-c^2 x^2\right) \tanh ^{-1}(c x)+\left(8-6 \tanh ^{-1}(c x)\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-3 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)+2 c x+8 c x \tanh ^{-1}(c x)^2-8 \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)-16 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)}{6 c^4 d}","-\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^4 d}+\frac{11 \left(a+b \tanh ^{-1}(c x)\right)^2}{6 c^4 d}-\frac{8 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{3 c^4 d}+\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^4 d}-\frac{a b x}{c^3 d}+\frac{x \left(a+b \tanh ^{-1}(c x)\right)^2}{c^3 d}-\frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^2 d}+\frac{b x^2 \left(a+b \tanh ^{-1}(c x)\right)}{3 c^2 d}+\frac{x^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{3 c d}-\frac{4 b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{3 c^4 d}-\frac{b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 c^4 d}-\frac{b^2 \tanh ^{-1}(c x)}{3 c^4 d}+\frac{b^2 x}{3 c^3 d}-\frac{b^2 x \tanh ^{-1}(c x)}{c^3 d}-\frac{b^2 \log \left(1-c^2 x^2\right)}{2 c^4 d}",1,"(a^2*x)/(c^3*d) - (a^2*x^2)/(2*c^2*d) + (a^2*x^3)/(3*c*d) - (a^2*Log[1 + c*x])/(c^4*d) + (a*b*(-3*c*x + 8*c*x*ArcTanh[c*x] + (1 - c^2*x^2)*(-1 + 3*ArcTanh[c*x] - 2*c*x*ArcTanh[c*x]) + 6*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 8*Log[1/Sqrt[1 - c^2*x^2]] - 3*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(3*c^4*d) + (b^2*(2*c*x - 6*c*x*ArcTanh[c*x] - 2*(1 - c^2*x^2)*ArcTanh[c*x] - 8*ArcTanh[c*x]^2 + 8*c*x*ArcTanh[c*x]^2 + 3*(1 - c^2*x^2)*ArcTanh[c*x]^2 - 2*c*x*(1 - c^2*x^2)*ArcTanh[c*x]^2 - 16*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + 6*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 6*Log[1/Sqrt[1 - c^2*x^2]] + (8 - 6*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 3*PolyLog[3, -E^(-2*ArcTanh[c*x])]))/(6*c^4*d)","A",0
96,1,260,247,0.5168775,"\int \frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{d+c d x} \, dx","Integrate[(x^2*(a + b*ArcTanh[c*x])^2)/(d + c*d*x),x]","\frac{a^2 c^2 x^2-2 a^2 c x+2 a^2 \log (c x+1)-2 a b \log \left(1-c^2 x^2\right)+2 a b c^2 x^2 \tanh ^{-1}(c x)+2 b \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right) \left(a+b \tanh ^{-1}(c x)-b\right)+2 a b c x-2 a b \tanh ^{-1}(c x)-4 a b c x \tanh ^{-1}(c x)-4 a b \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+b^2 \log \left(1-c^2 x^2\right)+b^2 c^2 x^2 \tanh ^{-1}(c x)^2+b^2 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)+b^2 \tanh ^{-1}(c x)^2-2 b^2 c x \tanh ^{-1}(c x)^2+2 b^2 c x \tanh ^{-1}(c x)-2 b^2 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+4 b^2 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)}{2 c^3 d}","\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^3 d}-\frac{3 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^3 d}+\frac{2 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^3 d}-\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^3 d}+\frac{a b x}{c^2 d}-\frac{x \left(a+b \tanh ^{-1}(c x)\right)^2}{c^2 d}+\frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c d}+\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c^3 d}+\frac{b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 c^3 d}+\frac{b^2 x \tanh ^{-1}(c x)}{c^2 d}+\frac{b^2 \log \left(1-c^2 x^2\right)}{2 c^3 d}",1,"(-2*a^2*c*x + 2*a*b*c*x + a^2*c^2*x^2 - 2*a*b*ArcTanh[c*x] - 4*a*b*c*x*ArcTanh[c*x] + 2*b^2*c*x*ArcTanh[c*x] + 2*a*b*c^2*x^2*ArcTanh[c*x] + b^2*ArcTanh[c*x]^2 - 2*b^2*c*x*ArcTanh[c*x]^2 + b^2*c^2*x^2*ArcTanh[c*x]^2 - 4*a*b*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + 4*b^2*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 2*b^2*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 2*a^2*Log[1 + c*x] - 2*a*b*Log[1 - c^2*x^2] + b^2*Log[1 - c^2*x^2] + 2*b*(a - b + b*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + b^2*PolyLog[3, -E^(-2*ArcTanh[c*x])])/(2*c^3*d)","A",0
97,1,140,172,0.4863558,"\int \frac{x \left(a+b \tanh ^{-1}(c x)\right)^2}{d+c d x} \, dx","Integrate[(x*(a + b*ArcTanh[c*x])^2)/(d + c*d*x),x]","\frac{2 a \left(a c x-a \log (c x+1)+b \log \left(1-c^2 x^2\right)\right)-2 b \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right) \left(a+b \tanh ^{-1}(c x)-b\right)+4 b \tanh ^{-1}(c x) \left((a-b) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+a c x\right)-b^2 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)+2 b^2 \tanh ^{-1}(c x)^2 \left(c x+\log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-1\right)}{2 c^2 d}","-\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^2 d}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{c^2 d}-\frac{2 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^2 d}+\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^2 d}+\frac{x \left(a+b \tanh ^{-1}(c x)\right)^2}{c d}-\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c^2 d}-\frac{b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 c^2 d}",1,"(2*b^2*ArcTanh[c*x]^2*(-1 + c*x + Log[1 + E^(-2*ArcTanh[c*x])]) + 4*b*ArcTanh[c*x]*(a*c*x + (a - b)*Log[1 + E^(-2*ArcTanh[c*x])]) + 2*a*(a*c*x - a*Log[1 + c*x] + b*Log[1 - c^2*x^2]) - 2*b*(a - b + b*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] - b^2*PolyLog[3, -E^(-2*ArcTanh[c*x])])/(2*c^2*d)","A",0
98,1,102,84,0.2090161,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{d+c d x} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(d + c*d*x),x]","\frac{2 a^2 \log (c x+1)+2 b \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right) \left(a+b \tanh ^{-1}(c x)\right)-4 a b \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+b^2 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-2 b^2 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)}{2 c d}","\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c d}-\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c d}+\frac{b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 c d}",1,"(-4*a*b*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 2*b^2*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 2*a^2*Log[1 + c*x] + 2*b*(a + b*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + b^2*PolyLog[3, -E^(-2*ArcTanh[c*x])])/(2*c*d)","A",0
99,1,132,77,0.3887167,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x (d+c d x)} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(x*(d + c*d*x)),x]","\frac{a^2 \log (c x)-a^2 \log (c x+1)+a b \left(2 \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)\right)+b^2 \left(\tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)-\frac{2}{3} \tanh ^{-1}(c x)^3+\tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)+\frac{i \pi ^3}{24}\right)}{d}","-\frac{b \text{Li}_2\left(\frac{2}{c x+1}-1\right) \left(a+b \tanh ^{-1}(c x)\right)}{d}+\frac{\log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d}-\frac{b^2 \text{Li}_3\left(\frac{2}{c x+1}-1\right)}{2 d}",1,"(a^2*Log[c*x] - a^2*Log[1 + c*x] + a*b*(2*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - PolyLog[2, E^(-2*ArcTanh[c*x])]) + b^2*((I/24)*Pi^3 - (2*ArcTanh[c*x]^3)/3 + ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - PolyLog[3, E^(2*ArcTanh[c*x])]/2))/d","C",0
100,1,225,162,0.6168795,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x^2 (d+c d x)} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(x^2*(d + c*d*x)),x]","\frac{a^2 (-c) \log (x)+a^2 c \log (c x+1)-\frac{a^2}{x}+\frac{a b \left(2 c x \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+c x \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-2 \tanh ^{-1}(c x) \left(c x \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+1\right)\right)}{x}+b^2 c \left(-\tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)+\frac{2}{3} \tanh ^{-1}(c x)^3-\frac{\tanh ^{-1}(c x)^2}{c x}+\tanh ^{-1}(c x)^2-\tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)+2 \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-\frac{i \pi ^3}{24}\right)}{d}","\frac{b c \text{Li}_2\left(\frac{2}{c x+1}-1\right) \left(a+b \tanh ^{-1}(c x)\right)}{d}+\frac{c \left(a+b \tanh ^{-1}(c x)\right)^2}{d}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{d x}+\frac{2 b c \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d}-\frac{c \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d}-\frac{b^2 c \text{Li}_2\left(\frac{2}{c x+1}-1\right)}{d}+\frac{b^2 c \text{Li}_3\left(\frac{2}{c x+1}-1\right)}{2 d}",1,"(-(a^2/x) - a^2*c*Log[x] + a^2*c*Log[1 + c*x] + (a*b*(-2*ArcTanh[c*x]*(1 + c*x*Log[1 - E^(-2*ArcTanh[c*x])]) + 2*c*x*Log[(c*x)/Sqrt[1 - c^2*x^2]] + c*x*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x + b^2*c*((-1/24*I)*Pi^3 + ArcTanh[c*x]^2 - ArcTanh[c*x]^2/(c*x) + (2*ArcTanh[c*x]^3)/3 + 2*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] - PolyLog[2, E^(-2*ArcTanh[c*x])] - ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + PolyLog[3, E^(2*ArcTanh[c*x])]/2))/d","C",0
101,1,317,250,1.0362033,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x^3 (d+c d x)} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(x^3*(d + c*d*x)),x]","\frac{2 a^2 c^2 \log (x)-2 a^2 c^2 \log (c x+1)+\frac{2 a^2 c}{x}-\frac{a^2}{x^2}+\frac{2 a b \left(-c^2 x^2 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-c x \left(2 c x \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+1\right)+\tanh ^{-1}(c x) \left(c^2 x^2+2 c^2 x^2 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+2 c x-1\right)\right)}{x^2}+2 b^2 c^2 \left(\log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)-\frac{\tanh ^{-1}(c x)^2}{2 c^2 x^2}+\tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)+\text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)-\frac{2}{3} \tanh ^{-1}(c x)^3+\frac{\tanh ^{-1}(c x)^2}{c x}-\frac{1}{2} \tanh ^{-1}(c x)^2-\frac{\tanh ^{-1}(c x)}{c x}+\tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)-2 \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+\frac{i \pi ^3}{24}\right)}{2 d}","-\frac{b c^2 \text{Li}_2\left(\frac{2}{c x+1}-1\right) \left(a+b \tanh ^{-1}(c x)\right)}{d}-\frac{c^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 d}+\frac{c^2 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d}-\frac{2 b c^2 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{2 d x^2}+\frac{c \left(a+b \tanh ^{-1}(c x)\right)^2}{d x}-\frac{b c \left(a+b \tanh ^{-1}(c x)\right)}{d x}+\frac{b^2 c^2 \text{Li}_2\left(\frac{2}{c x+1}-1\right)}{d}-\frac{b^2 c^2 \text{Li}_3\left(\frac{2}{c x+1}-1\right)}{2 d}-\frac{b^2 c^2 \log \left(1-c^2 x^2\right)}{2 d}+\frac{b^2 c^2 \log (x)}{d}",1,"(-(a^2/x^2) + (2*a^2*c)/x + 2*a^2*c^2*Log[x] - 2*a^2*c^2*Log[1 + c*x] + (2*a*b*(ArcTanh[c*x]*(-1 + 2*c*x + c^2*x^2 + 2*c^2*x^2*Log[1 - E^(-2*ArcTanh[c*x])]) - c*x*(1 + 2*c*x*Log[(c*x)/Sqrt[1 - c^2*x^2]]) - c^2*x^2*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x^2 + 2*b^2*c^2*((I/24)*Pi^3 - ArcTanh[c*x]/(c*x) - ArcTanh[c*x]^2/2 - ArcTanh[c*x]^2/(2*c^2*x^2) + ArcTanh[c*x]^2/(c*x) - (2*ArcTanh[c*x]^3)/3 - 2*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + Log[(c*x)/Sqrt[1 - c^2*x^2]] + PolyLog[2, E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - PolyLog[3, E^(2*ArcTanh[c*x])]/2))/(2*d)","C",0
102,1,388,334,1.3778933,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x^4 (d+c d x)} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(x^4*(d + c*d*x)),x]","\frac{-24 a^2 c^3 \log (x)+24 a^2 c^3 \log (c x+1)-\frac{24 a^2 c^2}{x}+\frac{12 a^2 c}{x^2}-\frac{8 a^2}{x^3}-\frac{8 a b \left(-3 c^3 x^3 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-c x \left(c^2 x^2+8 c^2 x^2 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+3 c x-1\right)+\tanh ^{-1}(c x) \left(3 c^3 x^3+6 c^3 x^3 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+6 c^2 x^2-3 c x+2\right)\right)}{x^3}+b^2 c^3 \left(-\frac{8 \tanh ^{-1}(c x)^2}{c^3 x^3}-24 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+\frac{12 \tanh ^{-1}(c x)^2}{c^2 x^2}-\frac{8 \tanh ^{-1}(c x)}{c^2 x^2}-24 \tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)-32 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+12 \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)-\frac{8}{c x}+16 \tanh ^{-1}(c x)^3-\frac{24 \tanh ^{-1}(c x)^2}{c x}+20 \tanh ^{-1}(c x)^2+\frac{24 \tanh ^{-1}(c x)}{c x}+8 \tanh ^{-1}(c x)-24 \tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)+64 \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-i \pi ^3\right)}{24 d}","\frac{b c^3 \text{Li}_2\left(\frac{2}{c x+1}-1\right) \left(a+b \tanh ^{-1}(c x)\right)}{d}+\frac{5 c^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{6 d}-\frac{c^3 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d}+\frac{8 b c^3 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{3 d}-\frac{c^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{d x}+\frac{b c^2 \left(a+b \tanh ^{-1}(c x)\right)}{d x}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{3 d x^3}+\frac{c \left(a+b \tanh ^{-1}(c x)\right)^2}{2 d x^2}-\frac{b c \left(a+b \tanh ^{-1}(c x)\right)}{3 d x^2}-\frac{4 b^2 c^3 \text{Li}_2\left(\frac{2}{c x+1}-1\right)}{3 d}+\frac{b^2 c^3 \text{Li}_3\left(\frac{2}{c x+1}-1\right)}{2 d}-\frac{b^2 c^3 \log (x)}{d}+\frac{b^2 c^3 \tanh ^{-1}(c x)}{3 d}-\frac{b^2 c^2}{3 d x}+\frac{b^2 c^3 \log \left(1-c^2 x^2\right)}{2 d}",1,"((-8*a^2)/x^3 + (12*a^2*c)/x^2 - (24*a^2*c^2)/x - 24*a^2*c^3*Log[x] + 24*a^2*c^3*Log[1 + c*x] - (8*a*b*(ArcTanh[c*x]*(2 - 3*c*x + 6*c^2*x^2 + 3*c^3*x^3 + 6*c^3*x^3*Log[1 - E^(-2*ArcTanh[c*x])]) - c*x*(-1 + 3*c*x + c^2*x^2 + 8*c^2*x^2*Log[(c*x)/Sqrt[1 - c^2*x^2]]) - 3*c^3*x^3*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x^3 + b^2*c^3*((-I)*Pi^3 - 8/(c*x) + 8*ArcTanh[c*x] - (8*ArcTanh[c*x])/(c^2*x^2) + (24*ArcTanh[c*x])/(c*x) + 20*ArcTanh[c*x]^2 - (8*ArcTanh[c*x]^2)/(c^3*x^3) + (12*ArcTanh[c*x]^2)/(c^2*x^2) - (24*ArcTanh[c*x]^2)/(c*x) + 16*ArcTanh[c*x]^3 + 64*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - 24*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] - 24*Log[(c*x)/Sqrt[1 - c^2*x^2]] - 32*PolyLog[2, E^(-2*ArcTanh[c*x])] - 24*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + 12*PolyLog[3, E^(2*ArcTanh[c*x])]))/(24*d)","C",0
103,1,425,394,1.7057173,"\int \frac{x^4 \left(a+b \tanh ^{-1}(c x)\right)^2}{(d+c d x)^2} \, dx","Integrate[(x^4*(a + b*ArcTanh[c*x])^2)/(d + c*d*x)^2,x]","\frac{4 a^2 c^3 x^3-12 a^2 c^2 x^2+36 a^2 c x-\frac{12 a^2}{c x+1}-48 a^2 \log (c x+1)+2 a b \left(2 c^2 x^2+20 \log \left(1-c^2 x^2\right)+2 \tanh ^{-1}(c x) \left(2 c^3 x^3-6 c^2 x^2+18 c x+24 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+3 \sinh \left(2 \tanh ^{-1}(c x)\right)-3 \cosh \left(2 \tanh ^{-1}(c x)\right)+6\right)-24 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-12 c x+3 \sinh \left(2 \tanh ^{-1}(c x)\right)-3 \cosh \left(2 \tanh ^{-1}(c x)\right)-2\right)+b^2 \left(4 c^3 x^3 \tanh ^{-1}(c x)^2-12 \log \left(1-c^2 x^2\right)-12 c^2 x^2 \tanh ^{-1}(c x)^2+4 c^2 x^2 \tanh ^{-1}(c x)-8 \left(6 \tanh ^{-1}(c x)-5\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-24 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)+4 c x+36 c x \tanh ^{-1}(c x)^2-24 c x \tanh ^{-1}(c x)-28 \tanh ^{-1}(c x)^2-4 \tanh ^{-1}(c x)+48 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-80 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+6 \tanh ^{-1}(c x)^2 \sinh \left(2 \tanh ^{-1}(c x)\right)+6 \tanh ^{-1}(c x) \sinh \left(2 \tanh ^{-1}(c x)\right)+3 \sinh \left(2 \tanh ^{-1}(c x)\right)-6 \tanh ^{-1}(c x)^2 \cosh \left(2 \tanh ^{-1}(c x)\right)-6 \tanh ^{-1}(c x) \cosh \left(2 \tanh ^{-1}(c x)\right)-3 \cosh \left(2 \tanh ^{-1}(c x)\right)\right)}{12 c^5 d^2}","-\frac{4 b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^5 d^2}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{c^5 d^2 (c x+1)}+\frac{29 \left(a+b \tanh ^{-1}(c x)\right)^2}{6 c^5 d^2}-\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{c^5 d^2 (c x+1)}-\frac{20 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{3 c^5 d^2}+\frac{4 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^5 d^2}-\frac{2 a b x}{c^4 d^2}+\frac{3 x \left(a+b \tanh ^{-1}(c x)\right)^2}{c^4 d^2}-\frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{c^3 d^2}+\frac{b x^2 \left(a+b \tanh ^{-1}(c x)\right)}{3 c^3 d^2}+\frac{x^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{3 c^2 d^2}-\frac{10 b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{3 c^5 d^2}-\frac{2 b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{c^5 d^2}-\frac{b^2}{2 c^5 d^2 (c x+1)}+\frac{b^2 \tanh ^{-1}(c x)}{6 c^5 d^2}+\frac{b^2 x}{3 c^4 d^2}-\frac{2 b^2 x \tanh ^{-1}(c x)}{c^4 d^2}-\frac{b^2 \log \left(1-c^2 x^2\right)}{c^5 d^2}",1,"(36*a^2*c*x - 12*a^2*c^2*x^2 + 4*a^2*c^3*x^3 - (12*a^2)/(1 + c*x) - 48*a^2*Log[1 + c*x] + b^2*(4*c*x - 4*ArcTanh[c*x] - 24*c*x*ArcTanh[c*x] + 4*c^2*x^2*ArcTanh[c*x] - 28*ArcTanh[c*x]^2 + 36*c*x*ArcTanh[c*x]^2 - 12*c^2*x^2*ArcTanh[c*x]^2 + 4*c^3*x^3*ArcTanh[c*x]^2 - 3*Cosh[2*ArcTanh[c*x]] - 6*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] - 6*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] - 80*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + 48*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] - 12*Log[1 - c^2*x^2] - 8*(-5 + 6*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 24*PolyLog[3, -E^(-2*ArcTanh[c*x])] + 3*Sinh[2*ArcTanh[c*x]] + 6*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] + 6*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]]) + 2*a*b*(-2 - 12*c*x + 2*c^2*x^2 - 3*Cosh[2*ArcTanh[c*x]] + 20*Log[1 - c^2*x^2] - 24*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 3*Sinh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]*(6 + 18*c*x - 6*c^2*x^2 + 2*c^3*x^3 - 3*Cosh[2*ArcTanh[c*x]] + 24*Log[1 + E^(-2*ArcTanh[c*x])] + 3*Sinh[2*ArcTanh[c*x]])))/(12*c^5*d^2)","A",0
104,1,354,331,1.2747604,"\int \frac{x^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{(d+c d x)^2} \, dx","Integrate[(x^3*(a + b*ArcTanh[c*x])^2)/(d + c*d*x)^2,x]","\frac{2 a^2 c^2 x^2-8 a^2 c x+\frac{4 a^2}{c x+1}+12 a^2 \log (c x+1)+2 a b \left(-4 \log \left(1-c^2 x^2\right)+2 \tanh ^{-1}(c x) \left(c^2 x^2-4 c x-6 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-\sinh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)-1\right)+6 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+2 c x-\sinh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)\right)+b^2 \left(2 \log \left(1-c^2 x^2\right)+2 c^2 x^2 \tanh ^{-1}(c x)^2+4 \left(3 \tanh ^{-1}(c x)-2\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+6 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-8 c x \tanh ^{-1}(c x)^2+6 \tanh ^{-1}(c x)^2+4 c x \tanh ^{-1}(c x)-12 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+16 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-2 \tanh ^{-1}(c x)^2 \sinh \left(2 \tanh ^{-1}(c x)\right)-2 \tanh ^{-1}(c x) \sinh \left(2 \tanh ^{-1}(c x)\right)-\sinh \left(2 \tanh ^{-1}(c x)\right)+2 \tanh ^{-1}(c x)^2 \cosh \left(2 \tanh ^{-1}(c x)\right)+2 \tanh ^{-1}(c x) \cosh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)\right)}{4 c^4 d^2}","\frac{3 b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^4 d^2}+\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{c^4 d^2 (c x+1)}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{c^4 d^2 (c x+1)}-\frac{3 \left(a+b \tanh ^{-1}(c x)\right)^2}{c^4 d^2}+\frac{4 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^4 d^2}-\frac{3 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^4 d^2}+\frac{a b x}{c^3 d^2}-\frac{2 x \left(a+b \tanh ^{-1}(c x)\right)^2}{c^3 d^2}+\frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^2 d^2}+\frac{2 b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c^4 d^2}+\frac{3 b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 c^4 d^2}+\frac{b^2}{2 c^4 d^2 (c x+1)}-\frac{b^2 \tanh ^{-1}(c x)}{2 c^4 d^2}+\frac{b^2 x \tanh ^{-1}(c x)}{c^3 d^2}+\frac{b^2 \log \left(1-c^2 x^2\right)}{2 c^4 d^2}",1,"(-8*a^2*c*x + 2*a^2*c^2*x^2 + (4*a^2)/(1 + c*x) + 12*a^2*Log[1 + c*x] + 2*a*b*(2*c*x + Cosh[2*ArcTanh[c*x]] - 4*Log[1 - c^2*x^2] + 6*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 2*ArcTanh[c*x]*(-1 - 4*c*x + c^2*x^2 + Cosh[2*ArcTanh[c*x]] - 6*Log[1 + E^(-2*ArcTanh[c*x])] - Sinh[2*ArcTanh[c*x]]) - Sinh[2*ArcTanh[c*x]]) + b^2*(4*c*x*ArcTanh[c*x] + 6*ArcTanh[c*x]^2 - 8*c*x*ArcTanh[c*x]^2 + 2*c^2*x^2*ArcTanh[c*x]^2 + Cosh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] + 16*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 12*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 2*Log[1 - c^2*x^2] + 4*(-2 + 3*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 6*PolyLog[3, -E^(-2*ArcTanh[c*x])] - Sinh[2*ArcTanh[c*x]] - 2*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] - 2*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]]))/(4*c^4*d^2)","A",0
105,1,295,260,0.9179661,"\int \frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{(d+c d x)^2} \, dx","Integrate[(x^2*(a + b*ArcTanh[c*x])^2)/(d + c*d*x)^2,x]","\frac{4 a^2 c x-\frac{4 a^2}{c x+1}-8 a^2 \log (c x+1)+2 a b \left(2 \log \left(1-c^2 x^2\right)-4 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+\sinh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(2 \tanh ^{-1}(c x)\right)+2 \tanh ^{-1}(c x) \left(2 c x+4 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+\sinh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(2 \tanh ^{-1}(c x)\right)\right)\right)+b^2 \left(\left(4-8 \tanh ^{-1}(c x)\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-4 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)+4 c x \tanh ^{-1}(c x)^2-4 \tanh ^{-1}(c x)^2+8 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-8 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+2 \tanh ^{-1}(c x)^2 \sinh \left(2 \tanh ^{-1}(c x)\right)+2 \tanh ^{-1}(c x) \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(2 \tanh ^{-1}(c x)\right)-2 \tanh ^{-1}(c x)^2 \cosh \left(2 \tanh ^{-1}(c x)\right)-2 \tanh ^{-1}(c x) \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(2 \tanh ^{-1}(c x)\right)\right)}{4 c^3 d^2}","-\frac{2 b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^3 d^2}-\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{c^3 d^2 (c x+1)}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{c^3 d^2 (c x+1)}+\frac{3 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^3 d^2}-\frac{2 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^3 d^2}+\frac{2 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^3 d^2}+\frac{x \left(a+b \tanh ^{-1}(c x)\right)^2}{c^2 d^2}-\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c^3 d^2}-\frac{b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{c^3 d^2}-\frac{b^2}{2 c^3 d^2 (c x+1)}+\frac{b^2 \tanh ^{-1}(c x)}{2 c^3 d^2}",1,"(4*a^2*c*x - (4*a^2)/(1 + c*x) - 8*a^2*Log[1 + c*x] + b^2*(-4*ArcTanh[c*x]^2 + 4*c*x*ArcTanh[c*x]^2 - Cosh[2*ArcTanh[c*x]] - 2*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] - 2*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] - 8*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + 8*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + (4 - 8*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 4*PolyLog[3, -E^(-2*ArcTanh[c*x])] + Sinh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]]) + 2*a*b*(-Cosh[2*ArcTanh[c*x]] + 2*Log[1 - c^2*x^2] - 4*PolyLog[2, -E^(-2*ArcTanh[c*x])] + Sinh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]*(2*c*x - Cosh[2*ArcTanh[c*x]] + 4*Log[1 + E^(-2*ArcTanh[c*x])] + Sinh[2*ArcTanh[c*x]])))/(4*c^3*d^2)","A",0
106,1,233,188,0.6199787,"\int \frac{x \left(a+b \tanh ^{-1}(c x)\right)^2}{(d+c d x)^2} \, dx","Integrate[(x*(a + b*ArcTanh[c*x])^2)/(d + c*d*x)^2,x]","\frac{\frac{4 a^2}{c x+1}+4 a^2 \log (c x+1)+2 a b \left(2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-\sinh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)+2 \tanh ^{-1}(c x) \left(-2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-\sinh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)\right)\right)+b^2 \left(4 \tanh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+2 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-4 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-2 \tanh ^{-1}(c x)^2 \sinh \left(2 \tanh ^{-1}(c x)\right)-2 \tanh ^{-1}(c x) \sinh \left(2 \tanh ^{-1}(c x)\right)-\sinh \left(2 \tanh ^{-1}(c x)\right)+2 \tanh ^{-1}(c x)^2 \cosh \left(2 \tanh ^{-1}(c x)\right)+2 \tanh ^{-1}(c x) \cosh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)\right)}{4 c^2 d^2}","\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^2 d^2}+\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{c^2 d^2 (c x+1)}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{c^2 d^2 (c x+1)}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^2 d^2}-\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^2 d^2}+\frac{b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 c^2 d^2}+\frac{b^2}{2 c^2 d^2 (c x+1)}-\frac{b^2 \tanh ^{-1}(c x)}{2 c^2 d^2}",1,"((4*a^2)/(1 + c*x) + 4*a^2*Log[1 + c*x] + 2*a*b*(Cosh[2*ArcTanh[c*x]] + 2*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 2*ArcTanh[c*x]*(Cosh[2*ArcTanh[c*x]] - 2*Log[1 + E^(-2*ArcTanh[c*x])] - Sinh[2*ArcTanh[c*x]]) - Sinh[2*ArcTanh[c*x]]) + b^2*(Cosh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] - 4*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 4*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 2*PolyLog[3, -E^(-2*ArcTanh[c*x])] - Sinh[2*ArcTanh[c*x]] - 2*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] - 2*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]]))/(4*c^2*d^2)","A",0
107,1,124,107,0.139063,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{(d+c d x)^2} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(d + c*d*x)^2,x]","\frac{-4 a^2+2 a b \log (c x+1)+2 a b c x \log (c x+1)-b (2 a+b) (c x+1) \log (1-c x)-4 b (2 a+b) \tanh ^{-1}(c x)-4 a b+b^2 \log (c x+1)+b^2 c x \log (c x+1)+2 b^2 (c x-1) \tanh ^{-1}(c x)^2-2 b^2}{4 c d^2 (c x+1)}","-\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{c d^2 (c x+1)}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{c d^2 (c x+1)}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{2 c d^2}-\frac{b^2}{2 c d^2 (c x+1)}+\frac{b^2 \tanh ^{-1}(c x)}{2 c d^2}",1,"(-4*a^2 - 4*a*b - 2*b^2 - 4*b*(2*a + b)*ArcTanh[c*x] + 2*b^2*(-1 + c*x)*ArcTanh[c*x]^2 - b*(2*a + b)*(1 + c*x)*Log[1 - c*x] + 2*a*b*Log[1 + c*x] + b^2*Log[1 + c*x] + 2*a*b*c*x*Log[1 + c*x] + b^2*c*x*Log[1 + c*x])/(4*c*d^2*(1 + c*x))","A",1
108,1,254,295,0.8598071,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x (d+c d x)^2} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(x*(d + c*d*x)^2),x]","\frac{\frac{24 a^2}{c x+1}+24 a^2 \log (c x)-24 a^2 \log (c x+1)+12 a b \left(-2 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-\sinh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)+2 \tanh ^{-1}(c x) \left(2 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-\sinh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)\right)\right)+b^2 \left(24 \tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)-12 \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)-16 \tanh ^{-1}(c x)^3+24 \tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)-12 \tanh ^{-1}(c x)^2 \sinh \left(2 \tanh ^{-1}(c x)\right)-12 \tanh ^{-1}(c x) \sinh \left(2 \tanh ^{-1}(c x)\right)-6 \sinh \left(2 \tanh ^{-1}(c x)\right)+12 \tanh ^{-1}(c x)^2 \cosh \left(2 \tanh ^{-1}(c x)\right)+12 \tanh ^{-1}(c x) \cosh \left(2 \tanh ^{-1}(c x)\right)+6 \cosh \left(2 \tanh ^{-1}(c x)\right)+i \pi ^3\right)}{24 d^2}","-\frac{b \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}+\frac{b \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}-\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}+\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{d^2 (c x+1)}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{d^2 (c x+1)}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{2 d^2}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d^2}+\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d^2}+\frac{b^2 \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{2 d^2}-\frac{b^2 \text{Li}_3\left(\frac{2}{1-c x}-1\right)}{2 d^2}-\frac{b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 d^2}+\frac{b^2}{2 d^2 (c x+1)}-\frac{b^2 \tanh ^{-1}(c x)}{2 d^2}",1,"((24*a^2)/(1 + c*x) + 24*a^2*Log[c*x] - 24*a^2*Log[1 + c*x] + 12*a*b*(Cosh[2*ArcTanh[c*x]] - 2*PolyLog[2, E^(-2*ArcTanh[c*x])] + 2*ArcTanh[c*x]*(Cosh[2*ArcTanh[c*x]] + 2*Log[1 - E^(-2*ArcTanh[c*x])] - Sinh[2*ArcTanh[c*x]]) - Sinh[2*ArcTanh[c*x]]) + b^2*(I*Pi^3 - 16*ArcTanh[c*x]^3 + 6*Cosh[2*ArcTanh[c*x]] + 12*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] + 12*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] + 24*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 24*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 12*PolyLog[3, E^(2*ArcTanh[c*x])] - 6*Sinh[2*ArcTanh[c*x]] - 12*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] - 12*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]]))/(24*d^2)","C",0
109,1,347,371,1.7215855,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x^2 (d+c d x)^2} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(x^2*(d + c*d*x)^2),x]","\frac{-\frac{12 a^2 c}{c x+1}-24 a^2 c \log (x)+24 a^2 c \log (c x+1)-\frac{12 a^2}{x}+6 a b c \left(4 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+4 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+\sinh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(2 \tanh ^{-1}(c x)\right)+\tanh ^{-1}(c x) \left(-\frac{4}{c x}-8 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+2 \sinh \left(2 \tanh ^{-1}(c x)\right)-2 \cosh \left(2 \tanh ^{-1}(c x)\right)\right)\right)+b^2 c \left(-24 \tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)-12 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+12 \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)+16 \tanh ^{-1}(c x)^3-\frac{12 \tanh ^{-1}(c x)^2}{c x}+12 \tanh ^{-1}(c x)^2-24 \tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)+24 \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+6 \tanh ^{-1}(c x)^2 \sinh \left(2 \tanh ^{-1}(c x)\right)+6 \tanh ^{-1}(c x) \sinh \left(2 \tanh ^{-1}(c x)\right)+3 \sinh \left(2 \tanh ^{-1}(c x)\right)-6 \tanh ^{-1}(c x)^2 \cosh \left(2 \tanh ^{-1}(c x)\right)-6 \tanh ^{-1}(c x) \cosh \left(2 \tanh ^{-1}(c x)\right)-3 \cosh \left(2 \tanh ^{-1}(c x)\right)-i \pi ^3\right)}{12 d^2}","\frac{2 b c \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}-\frac{2 b c \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}+\frac{2 b c \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}-\frac{b c \left(a+b \tanh ^{-1}(c x)\right)}{d^2 (c x+1)}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{d^2 x}-\frac{c \left(a+b \tanh ^{-1}(c x)\right)^2}{d^2 (c x+1)}+\frac{3 c \left(a+b \tanh ^{-1}(c x)\right)^2}{2 d^2}-\frac{4 c \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d^2}+\frac{2 b c \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}-\frac{2 c \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d^2}-\frac{b^2 c \text{Li}_2\left(\frac{2}{c x+1}-1\right)}{d^2}-\frac{b^2 c \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{d^2}+\frac{b^2 c \text{Li}_3\left(\frac{2}{1-c x}-1\right)}{d^2}+\frac{b^2 c \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{d^2}-\frac{b^2 c}{2 d^2 (c x+1)}+\frac{b^2 c \tanh ^{-1}(c x)}{2 d^2}",1,"((-12*a^2)/x - (12*a^2*c)/(1 + c*x) - 24*a^2*c*Log[x] + 24*a^2*c*Log[1 + c*x] + b^2*c*((-I)*Pi^3 + 12*ArcTanh[c*x]^2 - (12*ArcTanh[c*x]^2)/(c*x) + 16*ArcTanh[c*x]^3 - 3*Cosh[2*ArcTanh[c*x]] - 6*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] - 6*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] + 24*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - 24*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] - 12*PolyLog[2, E^(-2*ArcTanh[c*x])] - 24*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + 12*PolyLog[3, E^(2*ArcTanh[c*x])] + 3*Sinh[2*ArcTanh[c*x]] + 6*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] + 6*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]]) + 6*a*b*c*(-Cosh[2*ArcTanh[c*x]] + 4*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 4*PolyLog[2, E^(-2*ArcTanh[c*x])] + Sinh[2*ArcTanh[c*x]] + ArcTanh[c*x]*(-4/(c*x) - 2*Cosh[2*ArcTanh[c*x]] - 8*Log[1 - E^(-2*ArcTanh[c*x])] + 2*Sinh[2*ArcTanh[c*x]])))/(12*d^2)","C",0
110,1,452,480,2.1696605,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x^3 (d+c d x)^2} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(x^3*(d + c*d*x)^2),x]","\frac{\frac{8 a^2 c^2}{c x+1}+24 a^2 c^2 \log (x)-24 a^2 c^2 \log (c x+1)+\frac{16 a^2 c}{x}-\frac{4 a^2}{x^2}+\frac{4 a b \left(-6 c^2 x^2 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+c x \left(-8 c x \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)-c x \sinh \left(2 \tanh ^{-1}(c x)\right)+c x \cosh \left(2 \tanh ^{-1}(c x)\right)-2\right)+2 \tanh ^{-1}(c x) \left(c^2 x^2+6 c^2 x^2 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-c^2 x^2 \sinh \left(2 \tanh ^{-1}(c x)\right)+c^2 x^2 \cosh \left(2 \tanh ^{-1}(c x)\right)+4 c x-1\right)\right)}{x^2}+b^2 c^2 \left(8 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)-\frac{4 \tanh ^{-1}(c x)^2}{c^2 x^2}+24 \tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)+16 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-12 \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)-16 \tanh ^{-1}(c x)^3+\frac{16 \tanh ^{-1}(c x)^2}{c x}-12 \tanh ^{-1}(c x)^2-\frac{8 \tanh ^{-1}(c x)}{c x}+24 \tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)-32 \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-4 \tanh ^{-1}(c x)^2 \sinh \left(2 \tanh ^{-1}(c x)\right)-4 \tanh ^{-1}(c x) \sinh \left(2 \tanh ^{-1}(c x)\right)-2 \sinh \left(2 \tanh ^{-1}(c x)\right)+4 \tanh ^{-1}(c x)^2 \cosh \left(2 \tanh ^{-1}(c x)\right)+4 \tanh ^{-1}(c x) \cosh \left(2 \tanh ^{-1}(c x)\right)+2 \cosh \left(2 \tanh ^{-1}(c x)\right)+i \pi ^3\right)}{8 d^2}","-\frac{3 b c^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}+\frac{3 b c^2 \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}-\frac{3 b c^2 \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}+\frac{c^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{d^2 (c x+1)}-\frac{2 c^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{d^2}+\frac{b c^2 \left(a+b \tanh ^{-1}(c x)\right)}{d^2 (c x+1)}+\frac{6 c^2 \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d^2}+\frac{3 c^2 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d^2}-\frac{4 b c^2 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{2 d^2 x^2}+\frac{2 c \left(a+b \tanh ^{-1}(c x)\right)^2}{d^2 x}-\frac{b c \left(a+b \tanh ^{-1}(c x)\right)}{d^2 x}+\frac{2 b^2 c^2 \text{Li}_2\left(\frac{2}{c x+1}-1\right)}{d^2}+\frac{3 b^2 c^2 \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{2 d^2}-\frac{3 b^2 c^2 \text{Li}_3\left(\frac{2}{1-c x}-1\right)}{2 d^2}-\frac{3 b^2 c^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 d^2}-\frac{b^2 c^2 \log \left(1-c^2 x^2\right)}{2 d^2}+\frac{b^2 c^2}{2 d^2 (c x+1)}+\frac{b^2 c^2 \log (x)}{d^2}-\frac{b^2 c^2 \tanh ^{-1}(c x)}{2 d^2}",1,"((-4*a^2)/x^2 + (16*a^2*c)/x + (8*a^2*c^2)/(1 + c*x) + 24*a^2*c^2*Log[x] - 24*a^2*c^2*Log[1 + c*x] + b^2*c^2*(I*Pi^3 - (8*ArcTanh[c*x])/(c*x) - 12*ArcTanh[c*x]^2 - (4*ArcTanh[c*x]^2)/(c^2*x^2) + (16*ArcTanh[c*x]^2)/(c*x) - 16*ArcTanh[c*x]^3 + 2*Cosh[2*ArcTanh[c*x]] + 4*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] + 4*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] - 32*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] + 24*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 8*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 16*PolyLog[2, E^(-2*ArcTanh[c*x])] + 24*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 12*PolyLog[3, E^(2*ArcTanh[c*x])] - 2*Sinh[2*ArcTanh[c*x]] - 4*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] - 4*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]]) + (4*a*b*(-6*c^2*x^2*PolyLog[2, E^(-2*ArcTanh[c*x])] + c*x*(-2 + c*x*Cosh[2*ArcTanh[c*x]] - 8*c*x*Log[(c*x)/Sqrt[1 - c^2*x^2]] - c*x*Sinh[2*ArcTanh[c*x]]) + 2*ArcTanh[c*x]*(-1 + 4*c*x + c^2*x^2 + c^2*x^2*Cosh[2*ArcTanh[c*x]] + 6*c^2*x^2*Log[1 - E^(-2*ArcTanh[c*x])] - c^2*x^2*Sinh[2*ArcTanh[c*x]])))/x^2)/(8*d^2)","C",0
111,1,420,408,2.0032715,"\int \frac{x^4 \left(a+b \tanh ^{-1}(c x)\right)^2}{(d+c d x)^3} \, dx","Integrate[(x^4*(a + b*ArcTanh[c*x])^2)/(d + c*d*x)^3,x]","\frac{8 a^2 c^2 x^2-48 a^2 c x+\frac{64 a^2}{c x+1}-\frac{8 a^2}{(c x+1)^2}+96 a^2 \log (c x+1)+a b \left(-48 \log \left(1-c^2 x^2\right)+4 \tanh ^{-1}(c x) \left(4 c^2 x^2-24 c x-48 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-14 \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(4 \tanh ^{-1}(c x)\right)+14 \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(4 \tanh ^{-1}(c x)\right)-4\right)+96 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+16 c x-28 \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(4 \tanh ^{-1}(c x)\right)+28 \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(4 \tanh ^{-1}(c x)\right)\right)+16 b^2 \left(\frac{1}{64} \left(32 \log \left(1-c^2 x^2\right)+8 \tanh ^{-1}(c x)^2 \left(4 c^2 x^2-24 c x-48 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-14 \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(4 \tanh ^{-1}(c x)\right)+14 \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(4 \tanh ^{-1}(c x)\right)+20\right)+192 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-56 \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(4 \tanh ^{-1}(c x)\right)+56 \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(4 \tanh ^{-1}(c x)\right)+4 \tanh ^{-1}(c x) \left(16 c x+96 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-28 \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(4 \tanh ^{-1}(c x)\right)+28 \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(4 \tanh ^{-1}(c x)\right)\right)\right)+\left(6 \tanh ^{-1}(c x)-3\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)\right)}{16 c^5 d^3}","\frac{6 b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^5 d^3}+\frac{15 b \left(a+b \tanh ^{-1}(c x)\right)}{4 c^5 d^3 (c x+1)}-\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{4 c^5 d^3 (c x+1)^2}+\frac{4 \left(a+b \tanh ^{-1}(c x)\right)^2}{c^5 d^3 (c x+1)}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^5 d^3 (c x+1)^2}-\frac{43 \left(a+b \tanh ^{-1}(c x)\right)^2}{8 c^5 d^3}+\frac{6 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^5 d^3}-\frac{6 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^5 d^3}+\frac{a b x}{c^4 d^3}-\frac{3 x \left(a+b \tanh ^{-1}(c x)\right)^2}{c^4 d^3}+\frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^3 d^3}+\frac{3 b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c^5 d^3}+\frac{3 b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{c^5 d^3}+\frac{29 b^2}{16 c^5 d^3 (c x+1)}-\frac{b^2}{16 c^5 d^3 (c x+1)^2}-\frac{29 b^2 \tanh ^{-1}(c x)}{16 c^5 d^3}+\frac{b^2 x \tanh ^{-1}(c x)}{c^4 d^3}+\frac{b^2 \log \left(1-c^2 x^2\right)}{2 c^5 d^3}",1,"(-48*a^2*c*x + 8*a^2*c^2*x^2 - (8*a^2)/(1 + c*x)^2 + (64*a^2)/(1 + c*x) + 96*a^2*Log[1 + c*x] + a*b*(16*c*x + 28*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] - 48*Log[1 - c^2*x^2] + 96*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 28*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(-4 - 24*c*x + 4*c^2*x^2 + 14*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] - 48*Log[1 + E^(-2*ArcTanh[c*x])] - 14*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]])) + 16*b^2*((-3 + 6*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + (56*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] + 32*Log[1 - c^2*x^2] + 192*PolyLog[3, -E^(-2*ArcTanh[c*x])] - 56*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(16*c*x + 28*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] + 96*Log[1 + E^(-2*ArcTanh[c*x])] - 28*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]]) + 8*ArcTanh[c*x]^2*(20 - 24*c*x + 4*c^2*x^2 + 14*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] - 48*Log[1 + E^(-2*ArcTanh[c*x])] - 14*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]]))/64))/(16*c^5*d^3)","A",0
112,1,418,337,1.2986684,"\int \frac{x^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{(d+c d x)^3} \, dx","Integrate[(x^3*(a + b*ArcTanh[c*x])^2)/(d + c*d*x)^3,x]","\frac{64 a^2 c x-\frac{192 a^2}{c x+1}+\frac{32 a^2}{(c x+1)^2}-192 a^2 \log (c x+1)+4 a b \left(16 \log \left(1-c^2 x^2\right)-48 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+20 \sinh \left(2 \tanh ^{-1}(c x)\right)-\sinh \left(4 \tanh ^{-1}(c x)\right)-20 \cosh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(4 \tanh ^{-1}(c x)\right)+4 \tanh ^{-1}(c x) \left(8 c x+24 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+10 \sinh \left(2 \tanh ^{-1}(c x)\right)-\sinh \left(4 \tanh ^{-1}(c x)\right)-10 \cosh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(4 \tanh ^{-1}(c x)\right)\right)\right)+b^2 \left(-64 \left(3 \tanh ^{-1}(c x)-1\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-96 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)+64 c x \tanh ^{-1}(c x)^2-64 \tanh ^{-1}(c x)^2+192 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-128 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+80 \tanh ^{-1}(c x)^2 \sinh \left(2 \tanh ^{-1}(c x)\right)-8 \tanh ^{-1}(c x)^2 \sinh \left(4 \tanh ^{-1}(c x)\right)+80 \tanh ^{-1}(c x) \sinh \left(2 \tanh ^{-1}(c x)\right)-4 \tanh ^{-1}(c x) \sinh \left(4 \tanh ^{-1}(c x)\right)+40 \sinh \left(2 \tanh ^{-1}(c x)\right)-\sinh \left(4 \tanh ^{-1}(c x)\right)-80 \tanh ^{-1}(c x)^2 \cosh \left(2 \tanh ^{-1}(c x)\right)+8 \tanh ^{-1}(c x)^2 \cosh \left(4 \tanh ^{-1}(c x)\right)-80 \tanh ^{-1}(c x) \cosh \left(2 \tanh ^{-1}(c x)\right)+4 \tanh ^{-1}(c x) \cosh \left(4 \tanh ^{-1}(c x)\right)-40 \cosh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(4 \tanh ^{-1}(c x)\right)\right)}{64 c^4 d^3}","-\frac{3 b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^4 d^3}-\frac{11 b \left(a+b \tanh ^{-1}(c x)\right)}{4 c^4 d^3 (c x+1)}+\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{4 c^4 d^3 (c x+1)^2}-\frac{3 \left(a+b \tanh ^{-1}(c x)\right)^2}{c^4 d^3 (c x+1)}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^4 d^3 (c x+1)^2}+\frac{19 \left(a+b \tanh ^{-1}(c x)\right)^2}{8 c^4 d^3}-\frac{2 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^4 d^3}+\frac{3 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^4 d^3}+\frac{x \left(a+b \tanh ^{-1}(c x)\right)^2}{c^3 d^3}-\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c^4 d^3}-\frac{3 b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 c^4 d^3}-\frac{21 b^2}{16 c^4 d^3 (c x+1)}+\frac{b^2}{16 c^4 d^3 (c x+1)^2}+\frac{21 b^2 \tanh ^{-1}(c x)}{16 c^4 d^3}",1,"(64*a^2*c*x + (32*a^2)/(1 + c*x)^2 - (192*a^2)/(1 + c*x) - 192*a^2*Log[1 + c*x] + 4*a*b*(-20*Cosh[2*ArcTanh[c*x]] + Cosh[4*ArcTanh[c*x]] + 16*Log[1 - c^2*x^2] - 48*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 20*Sinh[2*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(8*c*x - 10*Cosh[2*ArcTanh[c*x]] + Cosh[4*ArcTanh[c*x]] + 24*Log[1 + E^(-2*ArcTanh[c*x])] + 10*Sinh[2*ArcTanh[c*x]] - Sinh[4*ArcTanh[c*x]]) - Sinh[4*ArcTanh[c*x]]) + b^2*(-64*ArcTanh[c*x]^2 + 64*c*x*ArcTanh[c*x]^2 - 40*Cosh[2*ArcTanh[c*x]] - 80*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] - 80*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] + Cosh[4*ArcTanh[c*x]] + 4*ArcTanh[c*x]*Cosh[4*ArcTanh[c*x]] + 8*ArcTanh[c*x]^2*Cosh[4*ArcTanh[c*x]] - 128*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + 192*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] - 64*(-1 + 3*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 96*PolyLog[3, -E^(-2*ArcTanh[c*x])] + 40*Sinh[2*ArcTanh[c*x]] + 80*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] + 80*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]] - Sinh[4*ArcTanh[c*x]] - 4*ArcTanh[c*x]*Sinh[4*ArcTanh[c*x]] - 8*ArcTanh[c*x]^2*Sinh[4*ArcTanh[c*x]]))/(64*c^4*d^3)","A",0
113,1,310,265,1.3974213,"\int \frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{(d+c d x)^3} \, dx","Integrate[(x^2*(a + b*ArcTanh[c*x])^2)/(d + c*d*x)^3,x]","\frac{\frac{32 a^2}{c x+1}-\frac{8 a^2}{(c x+1)^2}+16 a^2 \log (c x+1)+a b \left(16 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-12 \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(4 \tanh ^{-1}(c x)\right)+12 \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(4 \tanh ^{-1}(c x)\right)+4 \tanh ^{-1}(c x) \left(-8 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-6 \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(4 \tanh ^{-1}(c x)\right)+6 \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(4 \tanh ^{-1}(c x)\right)\right)\right)+16 b^2 \left(\tanh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)+\frac{1}{64} \left(\sinh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(2 \tanh ^{-1}(c x)\right)\right) \left(-\sinh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)+4 \tanh ^{-1}(c x) \left(-\sinh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(2 \tanh ^{-1}(c x)\right)-12\right)+8 \tanh ^{-1}(c x)^2 \left(\left(8 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-1\right) \sinh \left(2 \tanh ^{-1}(c x)\right)+\left(8 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+1\right) \cosh \left(2 \tanh ^{-1}(c x)\right)-6\right)-24\right)\right)}{16 c^3 d^3}","\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^3 d^3}+\frac{7 b \left(a+b \tanh ^{-1}(c x)\right)}{4 c^3 d^3 (c x+1)}-\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{4 c^3 d^3 (c x+1)^2}+\frac{2 \left(a+b \tanh ^{-1}(c x)\right)^2}{c^3 d^3 (c x+1)}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^3 d^3 (c x+1)^2}-\frac{7 \left(a+b \tanh ^{-1}(c x)\right)^2}{8 c^3 d^3}-\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^3 d^3}+\frac{b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 c^3 d^3}+\frac{13 b^2}{16 c^3 d^3 (c x+1)}-\frac{b^2}{16 c^3 d^3 (c x+1)^2}-\frac{13 b^2 \tanh ^{-1}(c x)}{16 c^3 d^3}",1,"((-8*a^2)/(1 + c*x)^2 + (32*a^2)/(1 + c*x) + 16*a^2*Log[1 + c*x] + 16*b^2*(ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + PolyLog[3, -E^(-2*ArcTanh[c*x])]/2 + ((-Cosh[2*ArcTanh[c*x]] + Sinh[2*ArcTanh[c*x]])*(-24 + Cosh[2*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(-12 + Cosh[2*ArcTanh[c*x]] - Sinh[2*ArcTanh[c*x]]) - Sinh[2*ArcTanh[c*x]] + 8*ArcTanh[c*x]^2*(-6 + Cosh[2*ArcTanh[c*x]]*(1 + 8*Log[1 + E^(-2*ArcTanh[c*x])]) + (-1 + 8*Log[1 + E^(-2*ArcTanh[c*x])])*Sinh[2*ArcTanh[c*x]])))/64) + a*b*(12*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] + 16*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 12*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(6*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] - 8*Log[1 + E^(-2*ArcTanh[c*x])] - 6*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]])))/(16*c^3*d^3)","A",0
114,1,150,157,0.3085365,"\int \frac{x \left(a+b \tanh ^{-1}(c x)\right)^2}{(d+c d x)^3} \, dx","Integrate[(x*(a + b*ArcTanh[c*x])^2)/(d + c*d*x)^3,x]","\frac{-2 \left(16 a^2+12 a b+5 b^2\right) (c x+1)+2 \left(8 a^2+4 a b+b^2\right)-b (12 a+5 b) (c x+1)^2 \log (1-c x)+b (12 a+5 b) (c x+1)^2 \log (c x+1)-8 b \tanh ^{-1}(c x) (a (8 c x+4)+b (3 c x+2))+4 b^2 \left(3 c^2 x^2-2 c x-1\right) \tanh ^{-1}(c x)^2}{32 c^2 d^3 (c x+1)^2}","-\frac{3 b \left(a+b \tanh ^{-1}(c x)\right)}{4 c^2 d^3 (c x+1)}+\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{4 c^2 d^3 (c x+1)^2}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{8 c^2 d^3}+\frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 d^3 (c x+1)^2}-\frac{5 b^2}{16 c^2 d^3 (c x+1)}+\frac{b^2}{16 c^2 d^3 (c x+1)^2}+\frac{5 b^2 \tanh ^{-1}(c x)}{16 c^2 d^3}",1,"(2*(8*a^2 + 4*a*b + b^2) - 2*(16*a^2 + 12*a*b + 5*b^2)*(1 + c*x) - 8*b*(b*(2 + 3*c*x) + a*(4 + 8*c*x))*ArcTanh[c*x] + 4*b^2*(-1 - 2*c*x + 3*c^2*x^2)*ArcTanh[c*x]^2 - b*(12*a + 5*b)*(1 + c*x)^2*Log[1 - c*x] + b*(12*a + 5*b)*(1 + c*x)^2*Log[1 + c*x])/(32*c^2*d^3*(1 + c*x)^2)","A",1
115,1,183,157,0.1364284,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{(d+c d x)^3} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(d + c*d*x)^3,x]","\frac{-8 a^2-4 a b-b^2}{16 c d^3 (c x+1)^2}+\frac{\left(-4 a b-3 b^2\right) \log (1-c x)}{32 c d^3}+\frac{\left(4 a b+3 b^2\right) \log (c x+1)}{32 c d^3}-\frac{b (4 a+3 b)}{16 c d^3 (c x+1)}-\frac{b \tanh ^{-1}(c x) (4 a+b c x+2 b)}{4 c d^3 (c x+1)^2}+\frac{b^2 \left(c^2 x^2+2 c x-3\right) \tanh ^{-1}(c x)^2}{8 c d^3 (c x+1)^2}","-\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{4 c d^3 (c x+1)}-\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{4 c d^3 (c x+1)^2}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{2 c d^3 (c x+1)^2}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{8 c d^3}-\frac{3 b^2}{16 c d^3 (c x+1)}-\frac{b^2}{16 c d^3 (c x+1)^2}+\frac{3 b^2 \tanh ^{-1}(c x)}{16 c d^3}",1,"(-8*a^2 - 4*a*b - b^2)/(16*c*d^3*(1 + c*x)^2) - (b*(4*a + 3*b))/(16*c*d^3*(1 + c*x)) - (b*(4*a + 2*b + b*c*x)*ArcTanh[c*x])/(4*c*d^3*(1 + c*x)^2) + (b^2*(-3 + 2*c*x + c^2*x^2)*ArcTanh[c*x]^2)/(8*c*d^3*(1 + c*x)^2) + ((-4*a*b - 3*b^2)*Log[1 - c*x])/(32*c*d^3) + ((4*a*b + 3*b^2)*Log[1 + c*x])/(32*c*d^3)","A",1
116,1,376,362,1.4619114,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x (d+c d x)^3} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(x*(d + c*d*x)^3),x]","\frac{\frac{192 a^2}{c x+1}+\frac{96 a^2}{(c x+1)^2}+192 a^2 \log (c x)-192 a^2 \log (c x+1)+12 a b \left(-16 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-12 \sinh \left(2 \tanh ^{-1}(c x)\right)-\sinh \left(4 \tanh ^{-1}(c x)\right)+12 \cosh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(4 \tanh ^{-1}(c x)\right)+4 \tanh ^{-1}(c x) \left(8 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-6 \sinh \left(2 \tanh ^{-1}(c x)\right)-\sinh \left(4 \tanh ^{-1}(c x)\right)+6 \cosh \left(2 \tanh ^{-1}(c x)\right)+\cosh \left(4 \tanh ^{-1}(c x)\right)\right)\right)+b^2 \left(192 \tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)-96 \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)-128 \tanh ^{-1}(c x)^3+192 \tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)-144 \tanh ^{-1}(c x)^2 \sinh \left(2 \tanh ^{-1}(c x)\right)-24 \tanh ^{-1}(c x)^2 \sinh \left(4 \tanh ^{-1}(c x)\right)-144 \tanh ^{-1}(c x) \sinh \left(2 \tanh ^{-1}(c x)\right)-12 \tanh ^{-1}(c x) \sinh \left(4 \tanh ^{-1}(c x)\right)-72 \sinh \left(2 \tanh ^{-1}(c x)\right)-3 \sinh \left(4 \tanh ^{-1}(c x)\right)+144 \tanh ^{-1}(c x)^2 \cosh \left(2 \tanh ^{-1}(c x)\right)+24 \tanh ^{-1}(c x)^2 \cosh \left(4 \tanh ^{-1}(c x)\right)+144 \tanh ^{-1}(c x) \cosh \left(2 \tanh ^{-1}(c x)\right)+12 \tanh ^{-1}(c x) \cosh \left(4 \tanh ^{-1}(c x)\right)+72 \cosh \left(2 \tanh ^{-1}(c x)\right)+3 \cosh \left(4 \tanh ^{-1}(c x)\right)+8 i \pi ^3\right)}{192 d^3}","-\frac{b \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^3}+\frac{b \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^3}-\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^3}+\frac{5 b \left(a+b \tanh ^{-1}(c x)\right)}{4 d^3 (c x+1)}+\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{4 d^3 (c x+1)^2}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{d^3 (c x+1)}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{2 d^3 (c x+1)^2}-\frac{5 \left(a+b \tanh ^{-1}(c x)\right)^2}{8 d^3}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d^3}+\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d^3}+\frac{b^2 \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{2 d^3}-\frac{b^2 \text{Li}_3\left(\frac{2}{1-c x}-1\right)}{2 d^3}-\frac{b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 d^3}+\frac{11 b^2}{16 d^3 (c x+1)}+\frac{b^2}{16 d^3 (c x+1)^2}-\frac{11 b^2 \tanh ^{-1}(c x)}{16 d^3}",1,"((96*a^2)/(1 + c*x)^2 + (192*a^2)/(1 + c*x) + 192*a^2*Log[c*x] - 192*a^2*Log[1 + c*x] + 12*a*b*(12*Cosh[2*ArcTanh[c*x]] + Cosh[4*ArcTanh[c*x]] - 16*PolyLog[2, E^(-2*ArcTanh[c*x])] - 12*Sinh[2*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(6*Cosh[2*ArcTanh[c*x]] + Cosh[4*ArcTanh[c*x]] + 8*Log[1 - E^(-2*ArcTanh[c*x])] - 6*Sinh[2*ArcTanh[c*x]] - Sinh[4*ArcTanh[c*x]]) - Sinh[4*ArcTanh[c*x]]) + b^2*((8*I)*Pi^3 - 128*ArcTanh[c*x]^3 + 72*Cosh[2*ArcTanh[c*x]] + 144*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] + 144*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] + 3*Cosh[4*ArcTanh[c*x]] + 12*ArcTanh[c*x]*Cosh[4*ArcTanh[c*x]] + 24*ArcTanh[c*x]^2*Cosh[4*ArcTanh[c*x]] + 192*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 192*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 96*PolyLog[3, E^(2*ArcTanh[c*x])] - 72*Sinh[2*ArcTanh[c*x]] - 144*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] - 144*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]] - 3*Sinh[4*ArcTanh[c*x]] - 12*ArcTanh[c*x]*Sinh[4*ArcTanh[c*x]] - 24*ArcTanh[c*x]^2*Sinh[4*ArcTanh[c*x]]))/(192*d^3)","C",0
117,1,479,448,2.3517815,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x^2 (d+c d x)^3} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(x^2*(d + c*d*x)^3),x]","\frac{-\frac{128 a^2 c}{c x+1}-\frac{32 a^2 c}{(c x+1)^2}-192 a^2 c \log (x)+192 a^2 c \log (c x+1)-\frac{64 a^2}{x}+\frac{4 a b \left(c x \left(32 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+20 \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(4 \tanh ^{-1}(c x)\right)-20 \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(4 \tanh ^{-1}(c x)\right)\right)+48 c x \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-4 \tanh ^{-1}(c x) \left(24 c x \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-10 c x \sinh \left(2 \tanh ^{-1}(c x)\right)-c x \sinh \left(4 \tanh ^{-1}(c x)\right)+10 c x \cosh \left(2 \tanh ^{-1}(c x)\right)+c x \cosh \left(4 \tanh ^{-1}(c x)\right)+8\right)\right)}{x}+b^2 c \left(-192 \tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)-64 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+96 \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)+128 \tanh ^{-1}(c x)^3-\frac{64 \tanh ^{-1}(c x)^2}{c x}+64 \tanh ^{-1}(c x)^2-192 \tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)+128 \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+80 \tanh ^{-1}(c x)^2 \sinh \left(2 \tanh ^{-1}(c x)\right)+8 \tanh ^{-1}(c x)^2 \sinh \left(4 \tanh ^{-1}(c x)\right)+80 \tanh ^{-1}(c x) \sinh \left(2 \tanh ^{-1}(c x)\right)+4 \tanh ^{-1}(c x) \sinh \left(4 \tanh ^{-1}(c x)\right)+40 \sinh \left(2 \tanh ^{-1}(c x)\right)+\sinh \left(4 \tanh ^{-1}(c x)\right)-80 \tanh ^{-1}(c x)^2 \cosh \left(2 \tanh ^{-1}(c x)\right)-8 \tanh ^{-1}(c x)^2 \cosh \left(4 \tanh ^{-1}(c x)\right)-80 \tanh ^{-1}(c x) \cosh \left(2 \tanh ^{-1}(c x)\right)-4 \tanh ^{-1}(c x) \cosh \left(4 \tanh ^{-1}(c x)\right)-40 \cosh \left(2 \tanh ^{-1}(c x)\right)-\cosh \left(4 \tanh ^{-1}(c x)\right)-8 i \pi ^3\right)}{64 d^3}","\frac{3 b c \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^3}-\frac{3 b c \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^3}+\frac{3 b c \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^3}-\frac{9 b c \left(a+b \tanh ^{-1}(c x)\right)}{4 d^3 (c x+1)}-\frac{b c \left(a+b \tanh ^{-1}(c x)\right)}{4 d^3 (c x+1)^2}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{d^3 x}-\frac{2 c \left(a+b \tanh ^{-1}(c x)\right)^2}{d^3 (c x+1)}-\frac{c \left(a+b \tanh ^{-1}(c x)\right)^2}{2 d^3 (c x+1)^2}+\frac{17 c \left(a+b \tanh ^{-1}(c x)\right)^2}{8 d^3}-\frac{6 c \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d^3}+\frac{2 b c \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^3}-\frac{3 c \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d^3}-\frac{b^2 c \text{Li}_2\left(\frac{2}{c x+1}-1\right)}{d^3}-\frac{3 b^2 c \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{2 d^3}+\frac{3 b^2 c \text{Li}_3\left(\frac{2}{1-c x}-1\right)}{2 d^3}+\frac{3 b^2 c \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 d^3}-\frac{19 b^2 c}{16 d^3 (c x+1)}-\frac{b^2 c}{16 d^3 (c x+1)^2}+\frac{19 b^2 c \tanh ^{-1}(c x)}{16 d^3}",1,"((-64*a^2)/x - (32*a^2*c)/(1 + c*x)^2 - (128*a^2*c)/(1 + c*x) - 192*a^2*c*Log[x] + 192*a^2*c*Log[1 + c*x] + b^2*c*((-8*I)*Pi^3 + 64*ArcTanh[c*x]^2 - (64*ArcTanh[c*x]^2)/(c*x) + 128*ArcTanh[c*x]^3 - 40*Cosh[2*ArcTanh[c*x]] - 80*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] - 80*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] - 4*ArcTanh[c*x]*Cosh[4*ArcTanh[c*x]] - 8*ArcTanh[c*x]^2*Cosh[4*ArcTanh[c*x]] + 128*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - 192*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] - 64*PolyLog[2, E^(-2*ArcTanh[c*x])] - 192*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + 96*PolyLog[3, E^(2*ArcTanh[c*x])] + 40*Sinh[2*ArcTanh[c*x]] + 80*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] + 80*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]] + 4*ArcTanh[c*x]*Sinh[4*ArcTanh[c*x]] + 8*ArcTanh[c*x]^2*Sinh[4*ArcTanh[c*x]]) + (4*a*b*(48*c*x*PolyLog[2, E^(-2*ArcTanh[c*x])] + c*x*(-20*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] + 32*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 20*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]]) - 4*ArcTanh[c*x]*(8 + 10*c*x*Cosh[2*ArcTanh[c*x]] + c*x*Cosh[4*ArcTanh[c*x]] + 24*c*x*Log[1 - E^(-2*ArcTanh[c*x])] - 10*c*x*Sinh[2*ArcTanh[c*x]] - c*x*Sinh[4*ArcTanh[c*x]])))/x)/(64*d^3)","C",0
118,1,168,176,0.1781829,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{(1+c x)^4} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(1 + c*x)^4,x]","-\frac{16 \left(18 a^2+6 a b+b^2\right)+24 b \tanh ^{-1}(c x) \left(24 a+b \left(3 c^2 x^2+9 c x+10\right)\right)+6 b (12 a+11 b) (c x+1)^2+6 b (12 a+5 b) (c x+1)+3 b (12 a+11 b) (c x+1)^3 \log (1-c x)-3 b (12 a+11 b) (c x+1)^3 \log (c x+1)-36 b^2 \left(c^3 x^3+3 c^2 x^2+3 c x-7\right) \tanh ^{-1}(c x)^2}{864 c (c x+1)^3}","-\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{12 c (c x+1)}-\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{12 c (c x+1)^2}-\frac{b \left(a+b \tanh ^{-1}(c x)\right)}{9 c (c x+1)^3}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{24 c}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{3 c (c x+1)^3}-\frac{11 b^2}{144 c (c x+1)}-\frac{5 b^2}{144 c (c x+1)^2}-\frac{b^2}{54 c (c x+1)^3}+\frac{11 b^2 \tanh ^{-1}(c x)}{144 c}",1,"-1/864*(16*(18*a^2 + 6*a*b + b^2) + 6*b*(12*a + 5*b)*(1 + c*x) + 6*b*(12*a + 11*b)*(1 + c*x)^2 + 24*b*(24*a + b*(10 + 9*c*x + 3*c^2*x^2))*ArcTanh[c*x] - 36*b^2*(-7 + 3*c*x + 3*c^2*x^2 + c^3*x^3)*ArcTanh[c*x]^2 + 3*b*(12*a + 11*b)*(1 + c*x)^3*Log[1 - c*x] - 3*b*(12*a + 11*b)*(1 + c*x)^3*Log[1 + c*x])/(c*(1 + c*x)^3)","A",1
119,1,59,67,0.1539424,"\int \frac{\tanh ^{-1}(a x)^2}{c x-a c x^2} \, dx","Integrate[ArcTanh[a*x]^2/(c*x - a*c*x^2),x]","\frac{\tanh ^{-1}(a x) \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)}{c}-\frac{\text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)}{2 c}+\frac{\tanh ^{-1}(a x)^2 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)}{c}","-\frac{\text{Li}_3\left(\frac{2}{1-a x}-1\right)}{2 c}+\frac{\text{Li}_2\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)}{c}+\frac{\log \left(2-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{c}",1,"(ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])])/c + (ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])])/c - PolyLog[3, E^(2*ArcTanh[a*x])]/(2*c)","A",0
120,1,644,306,1.471957,"\int (1+c x)^3 \left(a+b \tanh ^{-1}(c x)\right)^3 \, dx","Integrate[(1 + c*x)^3*(a + b*ArcTanh[c*x])^3,x]","\frac{2 a^3 c^4 x^4+8 a^3 c^3 x^3+12 a^3 c^2 x^2+8 a^3 c x+6 a^2 b c^4 x^4 \tanh ^{-1}(c x)+2 a^2 b c^3 x^3+24 a^2 b c^3 x^3 \tanh ^{-1}(c x)+12 a^2 b c^2 x^2+36 a^2 b c^2 x^2 \tanh ^{-1}(c x)+42 a^2 b c x+45 a^2 b \log (1-c x)+3 a^2 b \log (c x+1)+24 a^2 b c x \tanh ^{-1}(c x)+6 a b^2 c^4 x^4 \tanh ^{-1}(c x)^2+24 a b^2 c^3 x^3 \tanh ^{-1}(c x)^2+4 a b^2 c^3 x^3 \tanh ^{-1}(c x)+2 a b^2 c^2 x^2+44 a b^2 \log \left(1-c^2 x^2\right)+36 a b^2 c^2 x^2 \tanh ^{-1}(c x)^2+24 a b^2 c^2 x^2 \tanh ^{-1}(c x)+4 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right) \left(12 a+12 b \tanh ^{-1}(c x)+11 b\right)+24 a b^2 c x+24 a b^2 c x \tanh ^{-1}(c x)^2+84 a b^2 c x \tanh ^{-1}(c x)-90 a b^2 \tanh ^{-1}(c x)^2-24 a b^2 \tanh ^{-1}(c x)-96 a b^2 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-2 a b^2+2 b^3 c^4 x^4 \tanh ^{-1}(c x)^3+8 b^3 c^3 x^3 \tanh ^{-1}(c x)^3+2 b^3 c^3 x^3 \tanh ^{-1}(c x)^2+12 b^3 \log \left(1-c^2 x^2\right)+12 b^3 c^2 x^2 \tanh ^{-1}(c x)^3+12 b^3 c^2 x^2 \tanh ^{-1}(c x)^2+2 b^3 c^2 x^2 \tanh ^{-1}(c x)+24 b^3 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)+2 b^3 c x+8 b^3 c x \tanh ^{-1}(c x)^3+42 b^3 c x \tanh ^{-1}(c x)^2+24 b^3 c x \tanh ^{-1}(c x)-30 b^3 \tanh ^{-1}(c x)^3-56 b^3 \tanh ^{-1}(c x)^2-2 b^3 \tanh ^{-1}(c x)-48 b^3 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-88 b^3 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)}{8 c}","-\frac{6 b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c}+\frac{1}{4} b^2 c x^2 \left(a+b \tanh ^{-1}(c x)\right)-\frac{11 b^2 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c}+3 a b^2 x+\frac{1}{4} b c^2 x^3 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{3}{2} b c x^2 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{(c x+1)^4 \left(a+b \tanh ^{-1}(c x)\right)^3}{4 c}+\frac{21}{4} b x \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{4 b \left(a+b \tanh ^{-1}(c x)\right)^2}{c}-\frac{6 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c}+\frac{3 b^3 \log \left(1-c^2 x^2\right)}{2 c}-\frac{11 b^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{2 c}+\frac{3 b^3 \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{c}+3 b^3 x \tanh ^{-1}(c x)-\frac{b^3 \tanh ^{-1}(c x)}{4 c}+\frac{b^3 x}{4}",1,"(-2*a*b^2 + 8*a^3*c*x + 42*a^2*b*c*x + 24*a*b^2*c*x + 2*b^3*c*x + 12*a^3*c^2*x^2 + 12*a^2*b*c^2*x^2 + 2*a*b^2*c^2*x^2 + 8*a^3*c^3*x^3 + 2*a^2*b*c^3*x^3 + 2*a^3*c^4*x^4 - 24*a*b^2*ArcTanh[c*x] - 2*b^3*ArcTanh[c*x] + 24*a^2*b*c*x*ArcTanh[c*x] + 84*a*b^2*c*x*ArcTanh[c*x] + 24*b^3*c*x*ArcTanh[c*x] + 36*a^2*b*c^2*x^2*ArcTanh[c*x] + 24*a*b^2*c^2*x^2*ArcTanh[c*x] + 2*b^3*c^2*x^2*ArcTanh[c*x] + 24*a^2*b*c^3*x^3*ArcTanh[c*x] + 4*a*b^2*c^3*x^3*ArcTanh[c*x] + 6*a^2*b*c^4*x^4*ArcTanh[c*x] - 90*a*b^2*ArcTanh[c*x]^2 - 56*b^3*ArcTanh[c*x]^2 + 24*a*b^2*c*x*ArcTanh[c*x]^2 + 42*b^3*c*x*ArcTanh[c*x]^2 + 36*a*b^2*c^2*x^2*ArcTanh[c*x]^2 + 12*b^3*c^2*x^2*ArcTanh[c*x]^2 + 24*a*b^2*c^3*x^3*ArcTanh[c*x]^2 + 2*b^3*c^3*x^3*ArcTanh[c*x]^2 + 6*a*b^2*c^4*x^4*ArcTanh[c*x]^2 - 30*b^3*ArcTanh[c*x]^3 + 8*b^3*c*x*ArcTanh[c*x]^3 + 12*b^3*c^2*x^2*ArcTanh[c*x]^3 + 8*b^3*c^3*x^3*ArcTanh[c*x]^3 + 2*b^3*c^4*x^4*ArcTanh[c*x]^3 - 96*a*b^2*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 88*b^3*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 48*b^3*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 45*a^2*b*Log[1 - c*x] + 3*a^2*b*Log[1 + c*x] + 44*a*b^2*Log[1 - c^2*x^2] + 12*b^3*Log[1 - c^2*x^2] + 4*b^2*(12*a + 11*b + 12*b*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 24*b^3*PolyLog[3, -E^(-2*ArcTanh[c*x])])/(8*c)","B",0
121,1,488,240,0.9933747,"\int (1+c x)^2 \left(a+b \tanh ^{-1}(c x)\right)^3 \, dx","Integrate[(1 + c*x)^2*(a + b*ArcTanh[c*x])^3,x]","\frac{2 a^3 c^3 x^3+6 a^3 c^2 x^2+6 a^3 c x+6 a^2 b c^3 x^3 \tanh ^{-1}(c x)+3 a^2 b c^2 x^2+18 a^2 b c^2 x^2 \tanh ^{-1}(c x)+18 a^2 b c x+21 a^2 b \log (1-c x)+3 a^2 b \log (c x+1)+18 a^2 b c x \tanh ^{-1}(c x)+6 a b^2 c^3 x^3 \tanh ^{-1}(c x)^2+18 a b^2 \log \left(1-c^2 x^2\right)+18 a b^2 c^2 x^2 \tanh ^{-1}(c x)^2+6 a b^2 c^2 x^2 \tanh ^{-1}(c x)+6 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right) \left(4 a+4 b \tanh ^{-1}(c x)+3 b\right)+6 a b^2 c x-42 a b^2 \tanh ^{-1}(c x)^2+18 a b^2 c x \tanh ^{-1}(c x)^2-6 a b^2 \tanh ^{-1}(c x)+36 a b^2 c x \tanh ^{-1}(c x)-48 a b^2 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+2 b^3 c^3 x^3 \tanh ^{-1}(c x)^3+3 b^3 \log \left(1-c^2 x^2\right)+6 b^3 c^2 x^2 \tanh ^{-1}(c x)^3+3 b^3 c^2 x^2 \tanh ^{-1}(c x)^2+12 b^3 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-14 b^3 \tanh ^{-1}(c x)^3+6 b^3 c x \tanh ^{-1}(c x)^3-21 b^3 \tanh ^{-1}(c x)^2+18 b^3 c x \tanh ^{-1}(c x)^2+6 b^3 c x \tanh ^{-1}(c x)-24 b^3 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-36 b^3 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)}{6 c}","-\frac{4 b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c}-\frac{6 b^2 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c}+a b^2 x+\frac{1}{2} b c x^2 \left(a+b \tanh ^{-1}(c x)\right)^2+3 b x \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{5 b \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c}+\frac{(c x+1)^3 \left(a+b \tanh ^{-1}(c x)\right)^3}{3 c}-\frac{4 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c}+\frac{b^3 \log \left(1-c^2 x^2\right)}{2 c}-\frac{3 b^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c}+\frac{2 b^3 \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{c}+b^3 x \tanh ^{-1}(c x)",1,"(6*a^3*c*x + 18*a^2*b*c*x + 6*a*b^2*c*x + 6*a^3*c^2*x^2 + 3*a^2*b*c^2*x^2 + 2*a^3*c^3*x^3 - 6*a*b^2*ArcTanh[c*x] + 18*a^2*b*c*x*ArcTanh[c*x] + 36*a*b^2*c*x*ArcTanh[c*x] + 6*b^3*c*x*ArcTanh[c*x] + 18*a^2*b*c^2*x^2*ArcTanh[c*x] + 6*a*b^2*c^2*x^2*ArcTanh[c*x] + 6*a^2*b*c^3*x^3*ArcTanh[c*x] - 42*a*b^2*ArcTanh[c*x]^2 - 21*b^3*ArcTanh[c*x]^2 + 18*a*b^2*c*x*ArcTanh[c*x]^2 + 18*b^3*c*x*ArcTanh[c*x]^2 + 18*a*b^2*c^2*x^2*ArcTanh[c*x]^2 + 3*b^3*c^2*x^2*ArcTanh[c*x]^2 + 6*a*b^2*c^3*x^3*ArcTanh[c*x]^2 - 14*b^3*ArcTanh[c*x]^3 + 6*b^3*c*x*ArcTanh[c*x]^3 + 6*b^3*c^2*x^2*ArcTanh[c*x]^3 + 2*b^3*c^3*x^3*ArcTanh[c*x]^3 - 48*a*b^2*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 36*b^3*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 24*b^3*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 21*a^2*b*Log[1 - c*x] + 3*a^2*b*Log[1 + c*x] + 18*a*b^2*Log[1 - c^2*x^2] + 3*b^3*Log[1 - c^2*x^2] + 6*b^2*(4*a + 3*b + 4*b*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 12*b^3*PolyLog[3, -E^(-2*ArcTanh[c*x])])/(6*c)","B",0
122,1,334,191,0.5009523,"\int (1+c x) \left(a+b \tanh ^{-1}(c x)\right)^3 \, dx","Integrate[(1 + c*x)*(a + b*ArcTanh[c*x])^3,x]","\frac{2 a^3 c^2 x^2+4 a^3 c x+6 a^2 b c^2 x^2 \tanh ^{-1}(c x)+6 a^2 b c x+9 a^2 b \log (1-c x)+3 a^2 b \log (c x+1)+12 a^2 b c x \tanh ^{-1}(c x)+6 a b^2 \log \left(1-c^2 x^2\right)+6 a b^2 c^2 x^2 \tanh ^{-1}(c x)^2+6 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right) \left(2 a+2 b \tanh ^{-1}(c x)+b\right)-18 a b^2 \tanh ^{-1}(c x)^2+12 a b^2 c x \tanh ^{-1}(c x)^2+12 a b^2 c x \tanh ^{-1}(c x)-24 a b^2 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+2 b^3 c^2 x^2 \tanh ^{-1}(c x)^3+6 b^3 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-6 b^3 \tanh ^{-1}(c x)^3+4 b^3 c x \tanh ^{-1}(c x)^3-6 b^3 \tanh ^{-1}(c x)^2+6 b^3 c x \tanh ^{-1}(c x)^2-12 b^3 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-12 b^3 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)}{4 c}","-\frac{3 b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c}-\frac{3 b^2 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c}+\frac{3}{2} b x \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{3 b \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c}+\frac{(c x+1)^2 \left(a+b \tanh ^{-1}(c x)\right)^3}{2 c}-\frac{3 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c}-\frac{3 b^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{2 c}+\frac{3 b^3 \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{2 c}",1,"(4*a^3*c*x + 6*a^2*b*c*x + 2*a^3*c^2*x^2 + 12*a^2*b*c*x*ArcTanh[c*x] + 12*a*b^2*c*x*ArcTanh[c*x] + 6*a^2*b*c^2*x^2*ArcTanh[c*x] - 18*a*b^2*ArcTanh[c*x]^2 - 6*b^3*ArcTanh[c*x]^2 + 12*a*b^2*c*x*ArcTanh[c*x]^2 + 6*b^3*c*x*ArcTanh[c*x]^2 + 6*a*b^2*c^2*x^2*ArcTanh[c*x]^2 - 6*b^3*ArcTanh[c*x]^3 + 4*b^3*c*x*ArcTanh[c*x]^3 + 2*b^3*c^2*x^2*ArcTanh[c*x]^3 - 24*a*b^2*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 12*b^3*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 12*b^3*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 9*a^2*b*Log[1 - c*x] + 3*a^2*b*Log[1 + c*x] + 6*a*b^2*Log[1 - c^2*x^2] + 6*b^2*(2*a + b + 2*b*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 6*b^3*PolyLog[3, -E^(-2*ArcTanh[c*x])])/(4*c)","A",0
123,1,152,111,0.2913321,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{1+c x} \, dx","Integrate[(a + b*ArcTanh[c*x])^3/(1 + c*x),x]","\frac{4 a^3 \log (c x+1)-12 a^2 b \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+6 b^2 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right) \left(a+b \tanh ^{-1}(c x)\right)-12 a b^2 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+6 b \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right) \left(a+b \tanh ^{-1}(c x)\right)^2+3 b^3 \text{Li}_4\left(-e^{-2 \tanh ^{-1}(c x)}\right)-4 b^3 \tanh ^{-1}(c x)^3 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)}{4 c}","\frac{3 b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{2 c}+\frac{3 b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c}-\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^3}{c}+\frac{3 b^3 \text{Li}_4\left(1-\frac{2}{c x+1}\right)}{4 c}",1,"(-12*a^2*b*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 12*a*b^2*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] - 4*b^3*ArcTanh[c*x]^3*Log[1 + E^(-2*ArcTanh[c*x])] + 4*a^3*Log[1 + c*x] + 6*b*(a + b*ArcTanh[c*x])^2*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 6*b^2*(a + b*ArcTanh[c*x])*PolyLog[3, -E^(-2*ArcTanh[c*x])] + 3*b^3*PolyLog[4, -E^(-2*ArcTanh[c*x])])/(4*c)","A",0
124,1,198,139,0.1532681,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{(1+c x)^2} \, dx","Integrate[(a + b*ArcTanh[c*x])^3/(1 + c*x)^2,x]","\frac{-8 a^3-3 b \left(2 a^2+2 a b+b^2\right) (c x+1) \log (1-c x)-12 b \left(2 a^2+2 a b+b^2\right) \tanh ^{-1}(c x)+6 a^2 b \log (c x+1)+6 a^2 b c x \log (c x+1)-12 a^2 b+6 a b^2 \log (c x+1)+6 a b^2 c x \log (c x+1)+6 b^2 (2 a+b) (c x-1) \tanh ^{-1}(c x)^2-12 a b^2+3 b^3 \log (c x+1)+3 b^3 c x \log (c x+1)+4 b^3 (c x-1) \tanh ^{-1}(c x)^3-6 b^3}{8 c (c x+1)}","-\frac{3 b^2 \left(a+b \tanh ^{-1}(c x)\right)}{2 c (c x+1)}+\frac{3 b \left(a+b \tanh ^{-1}(c x)\right)^2}{4 c}-\frac{3 b \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c (c x+1)}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{2 c}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{c (c x+1)}-\frac{3 b^3}{4 c (c x+1)}+\frac{3 b^3 \tanh ^{-1}(c x)}{4 c}",1,"(-8*a^3 - 12*a^2*b - 12*a*b^2 - 6*b^3 - 12*b*(2*a^2 + 2*a*b + b^2)*ArcTanh[c*x] + 6*b^2*(2*a + b)*(-1 + c*x)*ArcTanh[c*x]^2 + 4*b^3*(-1 + c*x)*ArcTanh[c*x]^3 - 3*b*(2*a^2 + 2*a*b + b^2)*(1 + c*x)*Log[1 - c*x] + 6*a^2*b*Log[1 + c*x] + 6*a*b^2*Log[1 + c*x] + 3*b^3*Log[1 + c*x] + 6*a^2*b*c*x*Log[1 + c*x] + 6*a*b^2*c*x*Log[1 + c*x] + 3*b^3*c*x*Log[1 + c*x])/(8*c*(1 + c*x))","A",1
125,1,215,208,0.2185517,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{(1+c x)^3} \, dx","Integrate[(a + b*ArcTanh[c*x])^3/(1 + c*x)^3,x]","\frac{-6 b \left(8 a^2+12 a b+7 b^2\right) (c x+1)-3 b \left(8 a^2+12 a b+7 b^2\right) (c x+1)^2 \log (1-c x)+3 b \left(8 a^2+12 a b+7 b^2\right) (c x+1)^2 \log (c x+1)-24 b \tanh ^{-1}(c x) \left(8 a^2+4 a b (c x+2)+b^2 (3 c x+4)\right)-2 \left(32 a^3+24 a^2 b+12 a b^2+3 b^3\right)+12 b^2 (c x-1) \tanh ^{-1}(c x)^2 (4 a (c x+3)+b (3 c x+5))+16 b^3 \left(c^2 x^2+2 c x-3\right) \tanh ^{-1}(c x)^3}{128 c (c x+1)^2}","-\frac{9 b^2 \left(a+b \tanh ^{-1}(c x)\right)}{16 c (c x+1)}-\frac{3 b^2 \left(a+b \tanh ^{-1}(c x)\right)}{16 c (c x+1)^2}+\frac{9 b \left(a+b \tanh ^{-1}(c x)\right)^2}{32 c}-\frac{3 b \left(a+b \tanh ^{-1}(c x)\right)^2}{8 c (c x+1)}-\frac{3 b \left(a+b \tanh ^{-1}(c x)\right)^2}{8 c (c x+1)^2}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{8 c}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{2 c (c x+1)^2}-\frac{21 b^3}{64 c (c x+1)}-\frac{3 b^3}{64 c (c x+1)^2}+\frac{21 b^3 \tanh ^{-1}(c x)}{64 c}",1,"(-2*(32*a^3 + 24*a^2*b + 12*a*b^2 + 3*b^3) - 6*b*(8*a^2 + 12*a*b + 7*b^2)*(1 + c*x) - 24*b*(8*a^2 + 4*a*b*(2 + c*x) + b^2*(4 + 3*c*x))*ArcTanh[c*x] + 12*b^2*(-1 + c*x)*(4*a*(3 + c*x) + b*(5 + 3*c*x))*ArcTanh[c*x]^2 + 16*b^3*(-3 + 2*c*x + c^2*x^2)*ArcTanh[c*x]^3 - 3*b*(8*a^2 + 12*a*b + 7*b^2)*(1 + c*x)^2*Log[1 - c*x] + 3*b*(8*a^2 + 12*a*b + 7*b^2)*(1 + c*x)^2*Log[1 + c*x])/(128*c*(1 + c*x)^2)","A",1
126,1,279,275,0.2227758,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{(1+c x)^4} \, dx","Integrate[(a + b*ArcTanh[c*x])^3/(1 + c*x)^4,x]","-\frac{24 b \tanh ^{-1}(c x) \left(144 a^2+12 a b \left(3 c^2 x^2+9 c x+10\right)+b^2 \left(33 c^2 x^2+81 c x+56\right)\right)+6 b \left(72 a^2+132 a b+85 b^2\right) (c x+1)^2+6 b \left(72 a^2+60 a b+19 b^2\right) (c x+1)+3 b \left(72 a^2+132 a b+85 b^2\right) (c x+1)^3 \log (1-c x)-3 b \left(72 a^2+132 a b+85 b^2\right) (c x+1)^3 \log (c x+1)+32 \left(36 a^3+18 a^2 b+6 a b^2+b^3\right)-36 b^2 (c x-1) \tanh ^{-1}(c x)^2 \left(12 a \left(c^2 x^2+4 c x+7\right)+b \left(11 c^2 x^2+32 c x+29\right)\right)-144 b^3 \left(c^3 x^3+3 c^2 x^2+3 c x-7\right) \tanh ^{-1}(c x)^3}{3456 c (c x+1)^3}","-\frac{11 b^2 \left(a+b \tanh ^{-1}(c x)\right)}{48 c (c x+1)}-\frac{5 b^2 \left(a+b \tanh ^{-1}(c x)\right)}{48 c (c x+1)^2}-\frac{b^2 \left(a+b \tanh ^{-1}(c x)\right)}{18 c (c x+1)^3}+\frac{11 b \left(a+b \tanh ^{-1}(c x)\right)^2}{96 c}-\frac{b \left(a+b \tanh ^{-1}(c x)\right)^2}{8 c (c x+1)}-\frac{b \left(a+b \tanh ^{-1}(c x)\right)^2}{8 c (c x+1)^2}-\frac{b \left(a+b \tanh ^{-1}(c x)\right)^2}{6 c (c x+1)^3}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{24 c}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{3 c (c x+1)^3}-\frac{85 b^3}{576 c (c x+1)}-\frac{19 b^3}{576 c (c x+1)^2}-\frac{b^3}{108 c (c x+1)^3}+\frac{85 b^3 \tanh ^{-1}(c x)}{576 c}",1,"-1/3456*(32*(36*a^3 + 18*a^2*b + 6*a*b^2 + b^3) + 6*b*(72*a^2 + 60*a*b + 19*b^2)*(1 + c*x) + 6*b*(72*a^2 + 132*a*b + 85*b^2)*(1 + c*x)^2 + 24*b*(144*a^2 + 12*a*b*(10 + 9*c*x + 3*c^2*x^2) + b^2*(56 + 81*c*x + 33*c^2*x^2))*ArcTanh[c*x] - 36*b^2*(-1 + c*x)*(12*a*(7 + 4*c*x + c^2*x^2) + b*(29 + 32*c*x + 11*c^2*x^2))*ArcTanh[c*x]^2 - 144*b^3*(-7 + 3*c*x + 3*c^2*x^2 + c^3*x^3)*ArcTanh[c*x]^3 + 3*b*(72*a^2 + 132*a*b + 85*b^2)*(1 + c*x)^3*Log[1 - c*x] - 3*b*(72*a^2 + 132*a*b + 85*b^2)*(1 + c*x)^3*Log[1 + c*x])/(c*(1 + c*x)^3)","A",1
127,1,172,309,0.3501708,"\int \frac{x^2 \tanh ^{-1}(a x)^3}{c+a c x} \, dx","Integrate[(x^2*ArcTanh[a*x]^3)/(c + a*c*x),x]","\frac{2 a^2 x^2 \tanh ^{-1}(a x)^3+6 \left(\tanh ^{-1}(a x)-1\right)^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+6 \left(\tanh ^{-1}(a x)-1\right) \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)+3 \text{Li}_4\left(-e^{-2 \tanh ^{-1}(a x)}\right)-4 a x \tanh ^{-1}(a x)^3+2 \tanh ^{-1}(a x)^3+6 a x \tanh ^{-1}(a x)^2-6 \tanh ^{-1}(a x)^2-4 \tanh ^{-1}(a x)^3 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)+12 \tanh ^{-1}(a x)^2 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)-12 \tanh ^{-1}(a x) \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)}{4 a^3 c}","-\frac{3 \text{Li}_2\left(1-\frac{2}{1-a x}\right)}{2 a^3 c}-\frac{3 \text{Li}_3\left(1-\frac{2}{1-a x}\right)}{2 a^3 c}+\frac{3 \text{Li}_4\left(1-\frac{2}{a x+1}\right)}{4 a^3 c}+\frac{3 \text{Li}_2\left(1-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^2}{2 a^3 c}+\frac{3 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^3 c}+\frac{3 \text{Li}_3\left(1-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)}{2 a^3 c}-\frac{3 \tanh ^{-1}(a x)^3}{2 a^3 c}+\frac{3 \tanh ^{-1}(a x)^2}{2 a^3 c}-\frac{\log \left(\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^3}{a^3 c}+\frac{3 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{a^3 c}-\frac{3 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^3 c}-\frac{x \tanh ^{-1}(a x)^3}{a^2 c}+\frac{3 x \tanh ^{-1}(a x)^2}{2 a^2 c}+\frac{x^2 \tanh ^{-1}(a x)^3}{2 a c}",1,"(-6*ArcTanh[a*x]^2 + 6*a*x*ArcTanh[a*x]^2 + 2*ArcTanh[a*x]^3 - 4*a*x*ArcTanh[a*x]^3 + 2*a^2*x^2*ArcTanh[a*x]^3 - 12*ArcTanh[a*x]*Log[1 + E^(-2*ArcTanh[a*x])] + 12*ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] - 4*ArcTanh[a*x]^3*Log[1 + E^(-2*ArcTanh[a*x])] + 6*(-1 + ArcTanh[a*x])^2*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 6*(-1 + ArcTanh[a*x])*PolyLog[3, -E^(-2*ArcTanh[a*x])] + 3*PolyLog[4, -E^(-2*ArcTanh[a*x])])/(4*a^3*c)","A",0
128,1,126,205,0.2712116,"\int \frac{x \tanh ^{-1}(a x)^3}{c+a c x} \, dx","Integrate[(x*ArcTanh[a*x]^3)/(c + a*c*x),x]","\frac{-\frac{3}{2} \left(\tanh ^{-1}(a x)-2\right) \tanh ^{-1}(a x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)-\frac{3}{2} \left(\tanh ^{-1}(a x)-1\right) \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)-\frac{3}{4} \text{Li}_4\left(-e^{-2 \tanh ^{-1}(a x)}\right)+a x \tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^3+\tanh ^{-1}(a x)^3 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)-3 \tanh ^{-1}(a x)^2 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)}{a^2 c}","\frac{3 \text{Li}_3\left(1-\frac{2}{1-a x}\right)}{2 a^2 c}-\frac{3 \text{Li}_4\left(1-\frac{2}{a x+1}\right)}{4 a^2 c}-\frac{3 \text{Li}_2\left(1-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^2}{2 a^2 c}-\frac{3 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^2 c}-\frac{3 \text{Li}_3\left(1-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)}{2 a^2 c}+\frac{\tanh ^{-1}(a x)^3}{a^2 c}+\frac{\log \left(\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^3}{a^2 c}-\frac{3 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{a^2 c}+\frac{x \tanh ^{-1}(a x)^3}{a c}",1,"(-ArcTanh[a*x]^3 + a*x*ArcTanh[a*x]^3 - 3*ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] + ArcTanh[a*x]^3*Log[1 + E^(-2*ArcTanh[a*x])] - (3*(-2 + ArcTanh[a*x])*ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])])/2 - (3*(-1 + ArcTanh[a*x])*PolyLog[3, -E^(-2*ArcTanh[a*x])])/2 - (3*PolyLog[4, -E^(-2*ArcTanh[a*x])])/4)/(a^2*c)","A",0
129,1,82,104,0.0890044,"\int \frac{\tanh ^{-1}(a x)^3}{c+a c x} \, dx","Integrate[ArcTanh[a*x]^3/(c + a*c*x),x]","\frac{6 \tanh ^{-1}(a x)^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+6 \tanh ^{-1}(a x) \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)+3 \text{Li}_4\left(-e^{-2 \tanh ^{-1}(a x)}\right)-4 \tanh ^{-1}(a x)^3 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)}{4 a c}","\frac{3 \text{Li}_4\left(1-\frac{2}{a x+1}\right)}{4 a c}+\frac{3 \text{Li}_2\left(1-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^2}{2 a c}+\frac{3 \text{Li}_3\left(1-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)}{2 a c}-\frac{\log \left(\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^3}{a c}",1,"(-4*ArcTanh[a*x]^3*Log[1 + E^(-2*ArcTanh[a*x])] + 6*ArcTanh[a*x]^2*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 6*ArcTanh[a*x]*PolyLog[3, -E^(-2*ArcTanh[a*x])] + 3*PolyLog[4, -E^(-2*ArcTanh[a*x])])/(4*a*c)","A",0
130,1,86,93,0.151209,"\int \frac{\tanh ^{-1}(a x)^3}{x (c+a c x)} \, dx","Integrate[ArcTanh[a*x]^3/(x*(c + a*c*x)),x]","\frac{96 \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)-96 \tanh ^{-1}(a x) \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)+48 \text{Li}_4\left(e^{2 \tanh ^{-1}(a x)}\right)-32 \tanh ^{-1}(a x)^4+64 \tanh ^{-1}(a x)^3 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)+\pi ^4}{64 c}","-\frac{3 \text{Li}_4\left(\frac{2}{a x+1}-1\right)}{4 c}-\frac{3 \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)^2}{2 c}-\frac{3 \text{Li}_3\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)}{2 c}+\frac{\log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^3}{c}",1,"(Pi^4 - 32*ArcTanh[a*x]^4 + 64*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] + 96*ArcTanh[a*x]^2*PolyLog[2, E^(2*ArcTanh[a*x])] - 96*ArcTanh[a*x]*PolyLog[3, E^(2*ArcTanh[a*x])] + 48*PolyLog[4, E^(2*ArcTanh[a*x])])/(64*c)","A",0
131,1,86,93,0.0704624,"\int \frac{\tanh ^{-1}(a x)^3}{c x+a c x^2} \, dx","Integrate[ArcTanh[a*x]^3/(c*x + a*c*x^2),x]","\frac{96 \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)-96 \tanh ^{-1}(a x) \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)+48 \text{Li}_4\left(e^{2 \tanh ^{-1}(a x)}\right)-32 \tanh ^{-1}(a x)^4+64 \tanh ^{-1}(a x)^3 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)+\pi ^4}{64 c}","-\frac{3 \text{Li}_4\left(\frac{2}{a x+1}-1\right)}{4 c}-\frac{3 \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)^2}{2 c}-\frac{3 \text{Li}_3\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)}{2 c}+\frac{\log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^3}{c}",1,"(Pi^4 - 32*ArcTanh[a*x]^4 + 64*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] + 96*ArcTanh[a*x]^2*PolyLog[2, E^(2*ArcTanh[a*x])] - 96*ArcTanh[a*x]*PolyLog[3, E^(2*ArcTanh[a*x])] + 48*PolyLog[4, E^(2*ArcTanh[a*x])])/(64*c)","A",0
132,1,154,191,0.2922232,"\int \frac{\tanh ^{-1}(a x)^3}{x^2 (c+a c x)} \, dx","Integrate[ArcTanh[a*x]^3/(x^2*(c + a*c*x)),x]","\frac{a \left(-\frac{3}{2} \left(\tanh ^{-1}(a x)-2\right) \tanh ^{-1}(a x) \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)+\frac{3}{2} \left(\tanh ^{-1}(a x)-1\right) \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)-\frac{3}{4} \text{Li}_4\left(e^{2 \tanh ^{-1}(a x)}\right)+\frac{1}{2} \tanh ^{-1}(a x)^4-\frac{\tanh ^{-1}(a x)^3}{a x}-\tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^3 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)+3 \tanh ^{-1}(a x)^2 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)-\frac{\pi ^4}{64}+\frac{i \pi ^3}{8}\right)}{c}","-\frac{3 a \text{Li}_3\left(\frac{2}{a x+1}-1\right)}{2 c}+\frac{3 a \text{Li}_4\left(\frac{2}{a x+1}-1\right)}{4 c}+\frac{3 a \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)^2}{2 c}-\frac{3 a \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)}{c}+\frac{3 a \text{Li}_3\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)}{2 c}+\frac{a \tanh ^{-1}(a x)^3}{c}-\frac{\tanh ^{-1}(a x)^3}{c x}-\frac{a \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^3}{c}+\frac{3 a \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^2}{c}",1,"(a*((I/8)*Pi^3 - Pi^4/64 - ArcTanh[a*x]^3 - ArcTanh[a*x]^3/(a*x) + ArcTanh[a*x]^4/2 + 3*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] - ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] - (3*(-2 + ArcTanh[a*x])*ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])])/2 + (3*(-1 + ArcTanh[a*x])*PolyLog[3, E^(2*ArcTanh[a*x])])/2 - (3*PolyLog[4, E^(2*ArcTanh[a*x])])/4))/c","C",0
133,1,222,305,0.6407677,"\int \frac{\tanh ^{-1}(a x)^3}{x^3 (c+a c x)} \, dx","Integrate[ArcTanh[a*x]^3/(x^3*(c + a*c*x)),x]","\frac{a^2 \left(-\frac{32 \tanh ^{-1}(a x)^3}{a^2 x^2}+96 \left(\tanh ^{-1}(a x)-2\right) \tanh ^{-1}(a x) \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)-96 \tanh ^{-1}(a x) \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)-96 \text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)+96 \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)+48 \text{Li}_4\left(e^{2 \tanh ^{-1}(a x)}\right)-32 \tanh ^{-1}(a x)^4+\frac{64 \tanh ^{-1}(a x)^3}{a x}+96 \tanh ^{-1}(a x)^3-\frac{96 \tanh ^{-1}(a x)^2}{a x}+96 \tanh ^{-1}(a x)^2+64 \tanh ^{-1}(a x)^3 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)-192 \tanh ^{-1}(a x)^2 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)+192 \tanh ^{-1}(a x) \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)+\pi ^4-8 i \pi ^3\right)}{64 c}","-\frac{3 a^2 \text{Li}_2\left(\frac{2}{a x+1}-1\right)}{2 c}+\frac{3 a^2 \text{Li}_3\left(\frac{2}{a x+1}-1\right)}{2 c}-\frac{3 a^2 \text{Li}_4\left(\frac{2}{a x+1}-1\right)}{4 c}-\frac{3 a^2 \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)^2}{2 c}+\frac{3 a^2 \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)}{c}-\frac{3 a^2 \text{Li}_3\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)}{2 c}-\frac{a^2 \tanh ^{-1}(a x)^3}{2 c}+\frac{3 a^2 \tanh ^{-1}(a x)^2}{2 c}+\frac{a^2 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^3}{c}-\frac{3 a^2 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^2}{c}+\frac{3 a^2 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)}{c}-\frac{\tanh ^{-1}(a x)^3}{2 c x^2}+\frac{a \tanh ^{-1}(a x)^3}{c x}-\frac{3 a \tanh ^{-1}(a x)^2}{2 c x}",1,"(a^2*((-8*I)*Pi^3 + Pi^4 + 96*ArcTanh[a*x]^2 - (96*ArcTanh[a*x]^2)/(a*x) + 96*ArcTanh[a*x]^3 - (32*ArcTanh[a*x]^3)/(a^2*x^2) + (64*ArcTanh[a*x]^3)/(a*x) - 32*ArcTanh[a*x]^4 + 192*ArcTanh[a*x]*Log[1 - E^(-2*ArcTanh[a*x])] - 192*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] + 64*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] - 96*PolyLog[2, E^(-2*ArcTanh[a*x])] + 96*(-2 + ArcTanh[a*x])*ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] + 96*PolyLog[3, E^(2*ArcTanh[a*x])] - 96*ArcTanh[a*x]*PolyLog[3, E^(2*ArcTanh[a*x])] + 48*PolyLog[4, E^(2*ArcTanh[a*x])]))/(64*c)","C",0
134,1,233,384,0.4125133,"\int \frac{x^2 \tanh ^{-1}(a x)^4}{c-a c x} \, dx","Integrate[(x^2*ArcTanh[a*x]^4)/(c - a*c*x),x]","-\frac{-\frac{1}{2} \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^4+2 \left(\tanh ^{-1}(a x)^2+3 \tanh ^{-1}(a x)+3\right) \tanh ^{-1}(a x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+3 \tanh ^{-1}(a x) \text{Li}_4\left(-e^{-2 \tanh ^{-1}(a x)}\right)+3 \left(\tanh ^{-1}(a x)+1\right)^2 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)+3 \text{Li}_4\left(-e^{-2 \tanh ^{-1}(a x)}\right)+\frac{3}{2} \text{Li}_5\left(-e^{-2 \tanh ^{-1}(a x)}\right)-\frac{2}{5} \tanh ^{-1}(a x)^5+a x \tanh ^{-1}(a x)^4-\tanh ^{-1}(a x)^4+2 a x \tanh ^{-1}(a x)^3-2 \tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^4 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)-4 \tanh ^{-1}(a x)^3 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)-6 \tanh ^{-1}(a x)^2 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)}{a^3 c}","-\frac{3 \text{Li}_3\left(1-\frac{2}{1-a x}\right)}{a^3 c}+\frac{3 \text{Li}_4\left(1-\frac{2}{1-a x}\right)}{a^3 c}-\frac{3 \text{Li}_5\left(1-\frac{2}{1-a x}\right)}{2 a^3 c}+\frac{2 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^3}{a^3 c}+\frac{6 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{a^3 c}-\frac{3 \text{Li}_3\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{a^3 c}+\frac{6 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^3 c}-\frac{6 \text{Li}_3\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^3 c}+\frac{3 \text{Li}_4\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^3 c}-\frac{\tanh ^{-1}(a x)^4}{2 a^3 c}-\frac{2 \tanh ^{-1}(a x)^3}{a^3 c}+\frac{\log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^4}{a^3 c}+\frac{4 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^3}{a^3 c}+\frac{6 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{a^3 c}-\frac{x \tanh ^{-1}(a x)^4}{a^2 c}-\frac{2 x \tanh ^{-1}(a x)^3}{a^2 c}-\frac{x^2 \tanh ^{-1}(a x)^4}{2 a c}",1,"-((-2*ArcTanh[a*x]^3 + 2*a*x*ArcTanh[a*x]^3 - ArcTanh[a*x]^4 + a*x*ArcTanh[a*x]^4 - ((1 - a^2*x^2)*ArcTanh[a*x]^4)/2 - (2*ArcTanh[a*x]^5)/5 - 6*ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] - 4*ArcTanh[a*x]^3*Log[1 + E^(-2*ArcTanh[a*x])] - ArcTanh[a*x]^4*Log[1 + E^(-2*ArcTanh[a*x])] + 2*ArcTanh[a*x]*(3 + 3*ArcTanh[a*x] + ArcTanh[a*x]^2)*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 3*(1 + ArcTanh[a*x])^2*PolyLog[3, -E^(-2*ArcTanh[a*x])] + 3*PolyLog[4, -E^(-2*ArcTanh[a*x])] + 3*ArcTanh[a*x]*PolyLog[4, -E^(-2*ArcTanh[a*x])] + (3*PolyLog[5, -E^(-2*ArcTanh[a*x])])/2)/(a^3*c))","A",0
135,1,172,261,0.2666476,"\int \frac{x \tanh ^{-1}(a x)^4}{c-a c x} \, dx","Integrate[(x*ArcTanh[a*x]^4)/(c - a*c*x),x]","-\frac{2 \left(\tanh ^{-1}(a x)+3\right) \tanh ^{-1}(a x)^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+3 \left(\tanh ^{-1}(a x)+2\right) \tanh ^{-1}(a x) \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)+3 \tanh ^{-1}(a x) \text{Li}_4\left(-e^{-2 \tanh ^{-1}(a x)}\right)+3 \text{Li}_4\left(-e^{-2 \tanh ^{-1}(a x)}\right)+\frac{3}{2} \text{Li}_5\left(-e^{-2 \tanh ^{-1}(a x)}\right)-\frac{2}{5} \tanh ^{-1}(a x)^5+a x \tanh ^{-1}(a x)^4-\tanh ^{-1}(a x)^4-\tanh ^{-1}(a x)^4 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)-4 \tanh ^{-1}(a x)^3 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)}{a^2 c}","\frac{3 \text{Li}_4\left(1-\frac{2}{1-a x}\right)}{a^2 c}-\frac{3 \text{Li}_5\left(1-\frac{2}{1-a x}\right)}{2 a^2 c}+\frac{2 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^3}{a^2 c}+\frac{6 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{a^2 c}-\frac{3 \text{Li}_3\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{a^2 c}-\frac{6 \text{Li}_3\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^2 c}+\frac{3 \text{Li}_4\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^2 c}-\frac{\tanh ^{-1}(a x)^4}{a^2 c}+\frac{\log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^4}{a^2 c}+\frac{4 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^3}{a^2 c}-\frac{x \tanh ^{-1}(a x)^4}{a c}",1,"-((-ArcTanh[a*x]^4 + a*x*ArcTanh[a*x]^4 - (2*ArcTanh[a*x]^5)/5 - 4*ArcTanh[a*x]^3*Log[1 + E^(-2*ArcTanh[a*x])] - ArcTanh[a*x]^4*Log[1 + E^(-2*ArcTanh[a*x])] + 2*ArcTanh[a*x]^2*(3 + ArcTanh[a*x])*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 3*ArcTanh[a*x]*(2 + ArcTanh[a*x])*PolyLog[3, -E^(-2*ArcTanh[a*x])] + 3*PolyLog[4, -E^(-2*ArcTanh[a*x])] + 3*ArcTanh[a*x]*PolyLog[4, -E^(-2*ArcTanh[a*x])] + (3*PolyLog[5, -E^(-2*ArcTanh[a*x])])/2)/(a^2*c))","A",0
136,1,112,131,0.114704,"\int \frac{\tanh ^{-1}(a x)^4}{c-a c x} \, dx","Integrate[ArcTanh[a*x]^4/(c - a*c*x),x]","-\frac{2 \tanh ^{-1}(a x)^3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+3 \tanh ^{-1}(a x)^2 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)+3 \tanh ^{-1}(a x) \text{Li}_4\left(-e^{-2 \tanh ^{-1}(a x)}\right)+\frac{3}{2} \text{Li}_5\left(-e^{-2 \tanh ^{-1}(a x)}\right)-\frac{2}{5} \tanh ^{-1}(a x)^5-\tanh ^{-1}(a x)^4 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)}{a c}","-\frac{3 \text{Li}_5\left(1-\frac{2}{1-a x}\right)}{2 a c}+\frac{2 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^3}{a c}-\frac{3 \text{Li}_3\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{a c}+\frac{3 \text{Li}_4\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a c}+\frac{\log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^4}{a c}",1,"-(((-2*ArcTanh[a*x]^5)/5 - ArcTanh[a*x]^4*Log[1 + E^(-2*ArcTanh[a*x])] + 2*ArcTanh[a*x]^3*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 3*ArcTanh[a*x]^2*PolyLog[3, -E^(-2*ArcTanh[a*x])] + 3*ArcTanh[a*x]*PolyLog[4, -E^(-2*ArcTanh[a*x])] + (3*PolyLog[5, -E^(-2*ArcTanh[a*x])])/2)/(a*c))","A",0
137,1,102,118,0.1732442,"\int \frac{\tanh ^{-1}(a x)^4}{x (c-a c x)} \, dx","Integrate[ArcTanh[a*x]^4/(x*(c - a*c*x)),x]","\frac{2 \tanh ^{-1}(a x)^3 \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)}{c}-\frac{3 \tanh ^{-1}(a x)^2 \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)}{c}+\frac{3 \tanh ^{-1}(a x) \text{Li}_4\left(e^{2 \tanh ^{-1}(a x)}\right)}{c}-\frac{3 \text{Li}_5\left(e^{2 \tanh ^{-1}(a x)}\right)}{2 c}+\frac{\tanh ^{-1}(a x)^4 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)}{c}","-\frac{3 \text{Li}_5\left(\frac{2}{1-a x}-1\right)}{2 c}+\frac{2 \text{Li}_2\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)^3}{c}-\frac{3 \text{Li}_3\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)^2}{c}+\frac{3 \text{Li}_4\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)}{c}+\frac{\log \left(2-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^4}{c}",1,"(ArcTanh[a*x]^4*Log[1 - E^(2*ArcTanh[a*x])])/c + (2*ArcTanh[a*x]^3*PolyLog[2, E^(2*ArcTanh[a*x])])/c - (3*ArcTanh[a*x]^2*PolyLog[3, E^(2*ArcTanh[a*x])])/c + (3*ArcTanh[a*x]*PolyLog[4, E^(2*ArcTanh[a*x])])/c - (3*PolyLog[5, E^(2*ArcTanh[a*x])])/(2*c)","A",0
138,1,102,118,0.0765898,"\int \frac{\tanh ^{-1}(a x)^4}{c x-a c x^2} \, dx","Integrate[ArcTanh[a*x]^4/(c*x - a*c*x^2),x]","\frac{2 \tanh ^{-1}(a x)^3 \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)}{c}-\frac{3 \tanh ^{-1}(a x)^2 \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)}{c}+\frac{3 \tanh ^{-1}(a x) \text{Li}_4\left(e^{2 \tanh ^{-1}(a x)}\right)}{c}-\frac{3 \text{Li}_5\left(e^{2 \tanh ^{-1}(a x)}\right)}{2 c}+\frac{\tanh ^{-1}(a x)^4 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)}{c}","-\frac{3 \text{Li}_5\left(\frac{2}{1-a x}-1\right)}{2 c}+\frac{2 \text{Li}_2\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)^3}{c}-\frac{3 \text{Li}_3\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)^2}{c}+\frac{3 \text{Li}_4\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)}{c}+\frac{\log \left(2-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^4}{c}",1,"(ArcTanh[a*x]^4*Log[1 - E^(2*ArcTanh[a*x])])/c + (2*ArcTanh[a*x]^3*PolyLog[2, E^(2*ArcTanh[a*x])])/c - (3*ArcTanh[a*x]^2*PolyLog[3, E^(2*ArcTanh[a*x])])/c + (3*ArcTanh[a*x]*PolyLog[4, E^(2*ArcTanh[a*x])])/c - (3*PolyLog[5, E^(2*ArcTanh[a*x])])/(2*c)","A",0
139,1,172,239,0.4944786,"\int \frac{\tanh ^{-1}(a x)^4}{x^2 (c-a c x)} \, dx","Integrate[ArcTanh[a*x]^4/(x^2*(c - a*c*x)),x]","-\frac{a \left(-2 \left(\tanh ^{-1}(a x)+3\right) \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)+3 \left(\tanh ^{-1}(a x)+2\right) \tanh ^{-1}(a x) \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)-3 \tanh ^{-1}(a x) \text{Li}_4\left(e^{2 \tanh ^{-1}(a x)}\right)-3 \text{Li}_4\left(e^{2 \tanh ^{-1}(a x)}\right)+\frac{3}{2} \text{Li}_5\left(e^{2 \tanh ^{-1}(a x)}\right)+\frac{\tanh ^{-1}(a x)^4}{a x}+\tanh ^{-1}(a x)^4+\tanh ^{-1}(a x)^4 \left(-\log \left(1-e^{2 \tanh ^{-1}(a x)}\right)\right)-4 \tanh ^{-1}(a x)^3 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)+\frac{i \pi ^5}{160}-\frac{\pi ^4}{16}\right)}{c}","-\frac{3 a \text{Li}_4\left(\frac{2}{a x+1}-1\right)}{c}-\frac{3 a \text{Li}_5\left(\frac{2}{1-a x}-1\right)}{2 c}+\frac{2 a \text{Li}_2\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)^3}{c}-\frac{6 a \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)^2}{c}-\frac{3 a \text{Li}_3\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)^2}{c}-\frac{6 a \text{Li}_3\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)}{c}+\frac{3 a \text{Li}_4\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)}{c}+\frac{a \tanh ^{-1}(a x)^4}{c}-\frac{\tanh ^{-1}(a x)^4}{c x}+\frac{a \log \left(2-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^4}{c}+\frac{4 a \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^3}{c}",1,"-((a*(-1/16*Pi^4 + (I/160)*Pi^5 + ArcTanh[a*x]^4 + ArcTanh[a*x]^4/(a*x) - 4*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] - ArcTanh[a*x]^4*Log[1 - E^(2*ArcTanh[a*x])] - 2*ArcTanh[a*x]^2*(3 + ArcTanh[a*x])*PolyLog[2, E^(2*ArcTanh[a*x])] + 3*ArcTanh[a*x]*(2 + ArcTanh[a*x])*PolyLog[3, E^(2*ArcTanh[a*x])] - 3*PolyLog[4, E^(2*ArcTanh[a*x])] - 3*ArcTanh[a*x]*PolyLog[4, E^(2*ArcTanh[a*x])] + (3*PolyLog[5, E^(2*ArcTanh[a*x])])/2))/c)","C",0
140,1,250,380,0.8799723,"\int \frac{\tanh ^{-1}(a x)^4}{x^3 (c-a c x)} \, dx","Integrate[ArcTanh[a*x]^4/(x^3*(c - a*c*x)),x]","-\frac{a^2 \left(\frac{\tanh ^{-1}(a x)^4}{2 a^2 x^2}-2 \left(\tanh ^{-1}(a x)^2+3 \tanh ^{-1}(a x)+3\right) \tanh ^{-1}(a x) \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)-3 \tanh ^{-1}(a x) \text{Li}_4\left(e^{2 \tanh ^{-1}(a x)}\right)+3 \left(\tanh ^{-1}(a x)+1\right)^2 \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)-3 \text{Li}_4\left(e^{2 \tanh ^{-1}(a x)}\right)+\frac{3}{2} \text{Li}_5\left(e^{2 \tanh ^{-1}(a x)}\right)+\frac{\tanh ^{-1}(a x)^4}{a x}+\frac{1}{2} \tanh ^{-1}(a x)^4+\frac{2 \tanh ^{-1}(a x)^3}{a x}+2 \tanh ^{-1}(a x)^3+\tanh ^{-1}(a x)^4 \left(-\log \left(1-e^{2 \tanh ^{-1}(a x)}\right)\right)-4 \tanh ^{-1}(a x)^3 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)-6 \tanh ^{-1}(a x)^2 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)+\frac{i \pi ^5}{160}-\frac{\pi ^4}{16}-\frac{i \pi ^3}{4}\right)}{c}","-\frac{3 a^2 \text{Li}_3\left(\frac{2}{a x+1}-1\right)}{c}-\frac{3 a^2 \text{Li}_4\left(\frac{2}{a x+1}-1\right)}{c}-\frac{3 a^2 \text{Li}_5\left(\frac{2}{1-a x}-1\right)}{2 c}+\frac{2 a^2 \text{Li}_2\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)^3}{c}-\frac{6 a^2 \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)^2}{c}-\frac{3 a^2 \text{Li}_3\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)^2}{c}-\frac{6 a^2 \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)}{c}-\frac{6 a^2 \text{Li}_3\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)}{c}+\frac{3 a^2 \text{Li}_4\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)}{c}+\frac{3 a^2 \tanh ^{-1}(a x)^4}{2 c}+\frac{2 a^2 \tanh ^{-1}(a x)^3}{c}+\frac{a^2 \log \left(2-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^4}{c}+\frac{4 a^2 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^3}{c}+\frac{6 a^2 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^2}{c}-\frac{\tanh ^{-1}(a x)^4}{2 c x^2}-\frac{a \tanh ^{-1}(a x)^4}{c x}-\frac{2 a \tanh ^{-1}(a x)^3}{c x}",1,"-((a^2*((-1/4*I)*Pi^3 - Pi^4/16 + (I/160)*Pi^5 + 2*ArcTanh[a*x]^3 + (2*ArcTanh[a*x]^3)/(a*x) + ArcTanh[a*x]^4/2 + ArcTanh[a*x]^4/(2*a^2*x^2) + ArcTanh[a*x]^4/(a*x) - 6*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] - 4*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] - ArcTanh[a*x]^4*Log[1 - E^(2*ArcTanh[a*x])] - 2*ArcTanh[a*x]*(3 + 3*ArcTanh[a*x] + ArcTanh[a*x]^2)*PolyLog[2, E^(2*ArcTanh[a*x])] + 3*(1 + ArcTanh[a*x])^2*PolyLog[3, E^(2*ArcTanh[a*x])] - 3*PolyLog[4, E^(2*ArcTanh[a*x])] - 3*ArcTanh[a*x]*PolyLog[4, E^(2*ArcTanh[a*x])] + (3*PolyLog[5, E^(2*ArcTanh[a*x])])/2))/c)","C",0
141,0,0,19,2.3965939,"\int \frac{x}{(c+a c x) \tanh ^{-1}(a x)} \, dx","Integrate[x/((c + a*c*x)*ArcTanh[a*x]),x]","\int \frac{x}{(c+a c x) \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x}{(a c x+c) \tanh ^{-1}(a x)},x\right)",0,"Integrate[x/((c + a*c*x)*ArcTanh[a*x]), x]","A",-1
142,0,0,18,0.1153024,"\int \frac{1}{(c+a c x) \tanh ^{-1}(a x)} \, dx","Integrate[1/((c + a*c*x)*ArcTanh[a*x]),x]","\int \frac{1}{(c+a c x) \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{(a c x+c) \tanh ^{-1}(a x)},x\right)",0,"Integrate[1/((c + a*c*x)*ArcTanh[a*x]), x]","A",-1
143,0,0,21,0.143414,"\int \frac{1}{x (c+a c x) \tanh ^{-1}(a x)} \, dx","Integrate[1/(x*(c + a*c*x)*ArcTanh[a*x]),x]","\int \frac{1}{x (c+a c x) \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x (a c x+c) \tanh ^{-1}(a x)},x\right)",0,"Integrate[1/(x*(c + a*c*x)*ArcTanh[a*x]), x]","A",-1
144,0,0,19,1.9690611,"\int \frac{x}{(c+a c x) \tanh ^{-1}(a x)^2} \, dx","Integrate[x/((c + a*c*x)*ArcTanh[a*x]^2),x]","\int \frac{x}{(c+a c x) \tanh ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x}{(a c x+c) \tanh ^{-1}(a x)^2},x\right)",0,"Integrate[x/((c + a*c*x)*ArcTanh[a*x]^2), x]","A",-1
145,0,0,18,1.0440199,"\int \frac{1}{(c+a c x) \tanh ^{-1}(a x)^2} \, dx","Integrate[1/((c + a*c*x)*ArcTanh[a*x]^2),x]","\int \frac{1}{(c+a c x) \tanh ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{1}{(a c x+c) \tanh ^{-1}(a x)^2},x\right)",0,"Integrate[1/((c + a*c*x)*ArcTanh[a*x]^2), x]","A",-1
146,0,0,21,1.2822483,"\int \frac{1}{x (c+a c x) \tanh ^{-1}(a x)^2} \, dx","Integrate[1/(x*(c + a*c*x)*ArcTanh[a*x]^2),x]","\int \frac{1}{x (c+a c x) \tanh ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{1}{x (a c x+c) \tanh ^{-1}(a x)^2},x\right)",0,"Integrate[1/(x*(c + a*c*x)*ArcTanh[a*x]^2), x]","A",-1
147,1,474,275,6.5923494,"\int \frac{x^3 \left(a+b \tanh ^{-1}(c x)\right)}{d+e x} \, dx","Integrate[(x^3*(a + b*ArcTanh[c*x]))/(d + e*x),x]","\frac{-6 a d^3 \log (d+e x)+6 a d^2 e x-3 a d e^2 x^2+2 a e^3 x^3-\frac{b e^3}{c^3}+\frac{3}{2} i \pi  b d^3 \log \left(1-c^2 x^2\right)+\frac{3 b d^2 e \sqrt{1-\frac{c^2 d^2}{e^2}} \tanh ^{-1}(c x)^2 e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)}}{c}+\frac{3 b d^2 e \log \left(1-c^2 x^2\right)}{c}+\frac{3 b d e^2 \tanh ^{-1}(c x)}{c^2}+\frac{b e^3 \log \left(1-c^2 x^2\right)}{c^3}+3 b d^3 \text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-6 b d^3 \tanh ^{-1}(c x) \tanh ^{-1}\left(\frac{c d}{e}\right)-6 b d^3 \tanh ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-6 b d^3 \tanh ^{-1}(c x) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)+6 b d^3 \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)-3 b d^3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+3 b d^3 \tanh ^{-1}(c x)^2-3 i \pi  b d^3 \tanh ^{-1}(c x)+6 b d^3 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+3 i \pi  b d^3 \log \left(e^{2 \tanh ^{-1}(c x)}+1\right)-\frac{3 b d^2 e \tanh ^{-1}(c x)^2}{c}+6 b d^2 e x \tanh ^{-1}(c x)-3 b d e^2 x^2 \tanh ^{-1}(c x)-\frac{3 b d e^2 x}{c}+2 b e^3 x^3 \tanh ^{-1}(c x)+\frac{b e^3 x^2}{c}}{6 e^4}","\frac{d^3 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{e^4}-\frac{d^3 \left(a+b \tanh ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right)}{e^4}-\frac{d x^2 \left(a+b \tanh ^{-1}(c x)\right)}{2 e^2}+\frac{x^3 \left(a+b \tanh ^{-1}(c x)\right)}{3 e}+\frac{a d^2 x}{e^3}+\frac{b d^2 \log \left(1-c^2 x^2\right)}{2 c e^3}+\frac{b d \tanh ^{-1}(c x)}{2 c^2 e^2}+\frac{b \log \left(1-c^2 x^2\right)}{6 c^3 e}-\frac{b d^3 \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 e^4}+\frac{b d^3 \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 e^4}+\frac{b d^2 x \tanh ^{-1}(c x)}{e^3}-\frac{b d x}{2 c e^2}+\frac{b x^2}{6 c e}",1,"(-((b*e^3)/c^3) + 6*a*d^2*e*x - (3*b*d*e^2*x)/c - 3*a*d*e^2*x^2 + (b*e^3*x^2)/c + 2*a*e^3*x^3 + (3*b*d*e^2*ArcTanh[c*x])/c^2 - (3*I)*b*d^3*Pi*ArcTanh[c*x] + 6*b*d^2*e*x*ArcTanh[c*x] - 3*b*d*e^2*x^2*ArcTanh[c*x] + 2*b*e^3*x^3*ArcTanh[c*x] - 6*b*d^3*ArcTanh[(c*d)/e]*ArcTanh[c*x] + 3*b*d^3*ArcTanh[c*x]^2 - (3*b*d^2*e*ArcTanh[c*x]^2)/c + (3*b*d^2*Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^2)/(c*E^ArcTanh[(c*d)/e]) + 6*b*d^3*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + (3*I)*b*d^3*Pi*Log[1 + E^(2*ArcTanh[c*x])] - 6*b*d^3*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - 6*b*d^3*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - 6*a*d^3*Log[d + e*x] + (3*b*d^2*e*Log[1 - c^2*x^2])/c + (b*e^3*Log[1 - c^2*x^2])/c^3 + ((3*I)/2)*b*d^3*Pi*Log[1 - c^2*x^2] + 6*b*d^3*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - 3*b*d^3*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 3*b*d^3*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))])/(6*e^4)","C",0
148,1,394,214,3.0525984,"\int \frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)}{d+e x} \, dx","Integrate[(x^2*(a + b*ArcTanh[c*x]))/(d + e*x),x]","\frac{2 a d^2 \log (d+e x)-2 a d e x+a e^2 x^2-\frac{b d e \sqrt{1-\frac{c^2 d^2}{e^2}} \tanh ^{-1}(c x)^2 e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)}}{c}-\frac{1}{2} i \pi  b d^2 \log \left(1-c^2 x^2\right)-\frac{b d e \log \left(1-c^2 x^2\right)}{c}-\frac{b e^2 \tanh ^{-1}(c x)}{c^2}-b d^2 \text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)+2 b d^2 \tanh ^{-1}(c x) \tanh ^{-1}\left(\frac{c d}{e}\right)+2 b d^2 \tanh ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)+2 b d^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 b d^2 \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)+b d^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-b d^2 \tanh ^{-1}(c x)^2+i \pi  b d^2 \tanh ^{-1}(c x)-2 b d^2 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-i \pi  b d^2 \log \left(e^{2 \tanh ^{-1}(c x)}+1\right)+\frac{b d e \tanh ^{-1}(c x)^2}{c}-2 b d e x \tanh ^{-1}(c x)+b e^2 x^2 \tanh ^{-1}(c x)+\frac{b e^2 x}{c}}{2 e^3}","-\frac{d^2 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{e^3}+\frac{d^2 \left(a+b \tanh ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right)}{e^3}+\frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)}{2 e}-\frac{a d x}{e^2}-\frac{b d \log \left(1-c^2 x^2\right)}{2 c e^2}-\frac{b \tanh ^{-1}(c x)}{2 c^2 e}+\frac{b d^2 \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 e^3}-\frac{b d^2 \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 e^3}-\frac{b d x \tanh ^{-1}(c x)}{e^2}+\frac{b x}{2 c e}",1,"(-2*a*d*e*x + (b*e^2*x)/c + a*e^2*x^2 - (b*e^2*ArcTanh[c*x])/c^2 + I*b*d^2*Pi*ArcTanh[c*x] - 2*b*d*e*x*ArcTanh[c*x] + b*e^2*x^2*ArcTanh[c*x] + 2*b*d^2*ArcTanh[(c*d)/e]*ArcTanh[c*x] - b*d^2*ArcTanh[c*x]^2 + (b*d*e*ArcTanh[c*x]^2)/c - (b*d*Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^2)/(c*E^ArcTanh[(c*d)/e]) - 2*b*d^2*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - I*b*d^2*Pi*Log[1 + E^(2*ArcTanh[c*x])] + 2*b*d^2*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 2*b*d^2*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 2*a*d^2*Log[d + e*x] - (b*d*e*Log[1 - c^2*x^2])/c - (I/2)*b*d^2*Pi*Log[1 - c^2*x^2] - 2*b*d^2*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] + b*d^2*PolyLog[2, -E^(-2*ArcTanh[c*x])] - b*d^2*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))])/(2*e^3)","C",0
149,1,315,156,2.4711876,"\int \frac{x \left(a+b \tanh ^{-1}(c x)\right)}{d+e x} \, dx","Integrate[(x*(a + b*ArcTanh[c*x]))/(d + e*x),x]","\frac{-2 a d \log (d+e x)+2 a e x+\frac{b \left(e \sqrt{1-\frac{c^2 d^2}{e^2}} \tanh ^{-1}(c x)^2 e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)}+\frac{1}{2} i \pi  c d \log \left(1-c^2 x^2\right)+e \log \left(1-c^2 x^2\right)+c d \text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-2 c d \tanh ^{-1}(c x) \tanh ^{-1}\left(\frac{c d}{e}\right)-2 c d \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 c d \tanh ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)+2 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)-c d \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+c d \tanh ^{-1}(c x)^2-i \pi  c d \tanh ^{-1}(c x)+2 c d \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+i \pi  c d \log \left(e^{2 \tanh ^{-1}(c x)}+1\right)-e \tanh ^{-1}(c x)^2+2 c e x \tanh ^{-1}(c x)\right)}{c}}{2 e^2}","\frac{d \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{e^2}-\frac{d \left(a+b \tanh ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right)}{e^2}+\frac{a x}{e}+\frac{b \log \left(1-c^2 x^2\right)}{2 c e}-\frac{b d \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 e^2}+\frac{b d \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 e^2}+\frac{b x \tanh ^{-1}(c x)}{e}",1,"(2*a*e*x - 2*a*d*Log[d + e*x] + (b*((-I)*c*d*Pi*ArcTanh[c*x] + 2*c*e*x*ArcTanh[c*x] - 2*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x] + c*d*ArcTanh[c*x]^2 - e*ArcTanh[c*x]^2 + (Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^2)/E^ArcTanh[(c*d)/e] + 2*c*d*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + I*c*d*Pi*Log[1 + E^(2*ArcTanh[c*x])] - 2*c*d*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - 2*c*d*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + e*Log[1 - c^2*x^2] + (I/2)*c*d*Pi*Log[1 - c^2*x^2] + 2*c*d*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - c*d*PolyLog[2, -E^(-2*ArcTanh[c*x])] + c*d*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/c)/(2*e^2)","C",0
150,1,257,114,0.2387055,"\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
151,1,294,148,1.632291,"\int \frac{a+b \tanh ^{-1}(c x)}{x (d+e x)} \, dx","Integrate[(a + b*ArcTanh[c*x])/(x*(d + e*x)),x]","\frac{-2 a d \log (d+e x)+2 a d \log (x)+\frac{b \left(e \sqrt{1-\frac{c^2 d^2}{e^2}} \tanh ^{-1}(c x)^2 e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)}+\frac{1}{2} i \pi  c d \log \left(1-c^2 x^2\right)+c d \text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-2 c d \tanh ^{-1}(c x) \tanh ^{-1}\left(\frac{c d}{e}\right)-2 c d \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 c d \tanh ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)+2 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)-c d \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+c d \tanh ^{-1}(c x)^2-i \pi  c d \tanh ^{-1}(c x)+2 c d \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+i \pi  c d \log \left(e^{2 \tanh ^{-1}(c x)}+1\right)-e \tanh ^{-1}(c x)^2\right)}{c}}{2 d^2}","-\frac{\left(a+b \tanh ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right)}{d}+\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d}+\frac{a \log (x)}{d}+\frac{b \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 d}-\frac{b \text{Li}_2(-c x)}{2 d}+\frac{b \text{Li}_2(c x)}{2 d}-\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 d}",1,"(2*a*d*Log[x] - 2*a*d*Log[d + e*x] + (b*((-I)*c*d*Pi*ArcTanh[c*x] - 2*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x] + c*d*ArcTanh[c*x]^2 - e*ArcTanh[c*x]^2 + (Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^2)/E^ArcTanh[(c*d)/e] + 2*c*d*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] + I*c*d*Pi*Log[1 + E^(2*ArcTanh[c*x])] - 2*c*d*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - 2*c*d*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + (I/2)*c*d*Pi*Log[1 - c^2*x^2] + 2*c*d*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - c*d*PolyLog[2, E^(-2*ArcTanh[c*x])] + c*d*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/c)/(2*d^2)","C",0
152,1,360,200,3.2156054,"\int \frac{a+b \tanh ^{-1}(c x)}{x^2 (d+e x)} \, dx","Integrate[(a + b*ArcTanh[c*x])/(x^2*(d + e*x)),x]","-\frac{\frac{2 a d^2}{x}+2 a d e \log (x)-2 a d e \log (d+e x)+\frac{b e^2 \sqrt{1-\frac{c^2 d^2}{e^2}} \tanh ^{-1}(c x)^2 e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)}}{c}-2 b c d^2 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+\frac{1}{2} i \pi  b d e \log \left(1-c^2 x^2\right)+\frac{2 b d^2 \tanh ^{-1}(c x)}{x}-b d e \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+b d e \text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)+b d e \tanh ^{-1}(c x)^2-2 b d e \tanh ^{-1}(c x) \tanh ^{-1}\left(\frac{c d}{e}\right)-i \pi  b d e \tanh ^{-1}(c x)+2 b d e \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+i \pi  b d e \log \left(e^{2 \tanh ^{-1}(c x)}+1\right)-2 b d e \tanh ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-2 b d e \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 b d e \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)-\frac{b e^2 \tanh ^{-1}(c x)^2}{c}}{2 d^3}","-\frac{e \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}+\frac{e \left(a+b \tanh ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right)}{d^2}-\frac{a+b \tanh ^{-1}(c x)}{d x}-\frac{a e \log (x)}{d^2}-\frac{b c \log \left(1-c^2 x^2\right)}{2 d}+\frac{b e \text{Li}_2(-c x)}{2 d^2}-\frac{b e \text{Li}_2(c x)}{2 d^2}+\frac{b e \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 d^2}-\frac{b e \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 d^2}+\frac{b c \log (x)}{d}",1,"-1/2*((2*a*d^2)/x - I*b*d*e*Pi*ArcTanh[c*x] + (2*b*d^2*ArcTanh[c*x])/x - 2*b*d*e*ArcTanh[(c*d)/e]*ArcTanh[c*x] + b*d*e*ArcTanh[c*x]^2 - (b*e^2*ArcTanh[c*x]^2)/c + (b*Sqrt[1 - (c^2*d^2)/e^2]*e^2*ArcTanh[c*x]^2)/(c*E^ArcTanh[(c*d)/e]) + 2*b*d*e*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] + I*b*d*e*Pi*Log[1 + E^(2*ArcTanh[c*x])] - 2*b*d*e*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - 2*b*d*e*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 2*a*d*e*Log[x] - 2*a*d*e*Log[d + e*x] - 2*b*c*d^2*Log[(c*x)/Sqrt[1 - c^2*x^2]] + (I/2)*b*d*e*Pi*Log[1 - c^2*x^2] + 2*b*d*e*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - b*d*e*PolyLog[2, E^(-2*ArcTanh[c*x])] + b*d*e*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))])/d^3","C",0
153,1,435,261,6.0872893,"\int \frac{a+b \tanh ^{-1}(c x)}{x^3 (d+e x)} \, dx","Integrate[(a + b*ArcTanh[c*x])/(x^3*(d + e*x)),x]","\frac{a e^2 \log (x)}{d^3}-\frac{a e^2 \log (d+e x)}{d^3}+\frac{a e}{d^2 x}-\frac{a}{2 d x^2}-\frac{b \left(\frac{c d^3 \left(1-c^2 x^2\right) \tanh ^{-1}(c x)}{x^2}+\frac{c^2 d^3}{x}-e^3 \sqrt{1-\frac{c^2 d^2}{e^2}} \tanh ^{-1}(c x)^2 e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)}+2 c^2 d^2 e \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+i \pi  c d e^2 \log \left(\frac{1}{\sqrt{1-c^2 x^2}}\right)-\frac{2 c d^2 e \tanh ^{-1}(c x)}{x}+c d e^2 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-c d e^2 \text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-c d e^2 \tanh ^{-1}(c x)^2+2 c d e^2 \tanh ^{-1}(c x) \tanh ^{-1}\left(\frac{c d}{e}\right)+i \pi  c d e^2 \tanh ^{-1}(c x)-2 c d e^2 \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-i \pi  c d e^2 \log \left(e^{2 \tanh ^{-1}(c x)}+1\right)+2 c d e^2 \tanh ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)+2 c d e^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 c d e^2 \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)+e^3 \tanh ^{-1}(c x)^2\right)}{2 c d^4}","\frac{e^2 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^3}-\frac{e^2 \left(a+b \tanh ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right)}{d^3}+\frac{e \left(a+b \tanh ^{-1}(c x)\right)}{d^2 x}-\frac{a+b \tanh ^{-1}(c x)}{2 d x^2}+\frac{a e^2 \log (x)}{d^3}+\frac{b c e \log \left(1-c^2 x^2\right)}{2 d^2}+\frac{b c^2 \tanh ^{-1}(c x)}{2 d}-\frac{b e^2 \text{Li}_2(-c x)}{2 d^3}+\frac{b e^2 \text{Li}_2(c x)}{2 d^3}-\frac{b e^2 \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 d^3}+\frac{b e^2 \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 d^3}-\frac{b c e \log (x)}{d^2}-\frac{b c}{2 d x}",1,"-1/2*a/(d*x^2) + (a*e)/(d^2*x) + (a*e^2*Log[x])/d^3 - (a*e^2*Log[d + e*x])/d^3 - (b*((c^2*d^3)/x + I*c*d*e^2*Pi*ArcTanh[c*x] - (2*c*d^2*e*ArcTanh[c*x])/x + (c*d^3*(1 - c^2*x^2)*ArcTanh[c*x])/x^2 + 2*c*d*e^2*ArcTanh[(c*d)/e]*ArcTanh[c*x] - c*d*e^2*ArcTanh[c*x]^2 + e^3*ArcTanh[c*x]^2 - (Sqrt[1 - (c^2*d^2)/e^2]*e^3*ArcTanh[c*x]^2)/E^ArcTanh[(c*d)/e] - 2*c*d*e^2*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - I*c*d*e^2*Pi*Log[1 + E^(2*ArcTanh[c*x])] + 2*c*d*e^2*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 2*c*d*e^2*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + I*c*d*e^2*Pi*Log[1/Sqrt[1 - c^2*x^2]] + 2*c^2*d^2*e*Log[(c*x)/Sqrt[1 - c^2*x^2]] - 2*c*d*e^2*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] + c*d*e^2*PolyLog[2, E^(-2*ArcTanh[c*x])] - c*d*e^2*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/(2*c*d^4)","C",0
154,1,1072,385,15.7581866,"\int \frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{d+e x} \, dx","Integrate[(x^2*(a + b*ArcTanh[c*x])^2)/(d + e*x),x]","\frac{3 e^2 x^2 a^2-6 d e x a^2+6 d^2 \log (d+e x) a^2+\frac{6 b \left(-d^2 \tanh ^{-1}(c x)^2 c^2-2 d e x \tanh ^{-1}(c x) c^2+2 d^2 \tanh ^{-1}\left(\frac{c d}{e}\right) \tanh ^{-1}(c x) c^2+i d^2 \pi  \tanh ^{-1}(c x) c^2-2 d^2 \tanh ^{-1}(c x) \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right) c^2-i d^2 \pi  \log \left(1+e^{2 \tanh ^{-1}(c x)}\right) c^2+2 d^2 \tanh ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right) c^2+2 d^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) c^2-\frac{1}{2} i d^2 \pi  \log \left(1-c^2 x^2\right) c^2-2 d^2 \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) c^2+d^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right) c^2-d^2 \text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right) c^2-d \sqrt{1-\frac{c^2 d^2}{e^2}} e e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)} \tanh ^{-1}(c x)^2 c+d e \tanh ^{-1}(c x)^2 c+e^2 x c-d e \log \left(1-c^2 x^2\right) c+e^2 \left(c^2 x^2-1\right) \tanh ^{-1}(c x)\right) a}{c^2}+\frac{b^2 \left(-2 c^2 d^2 \tanh ^{-1}(c x)^3+2 c d e \tanh ^{-1}(c x)^3+6 c d e \tanh ^{-1}(c x)^2-6 c^2 d e x \tanh ^{-1}(c x)^2+3 e^2 \left(c^2 x^2-1\right) \tanh ^{-1}(c x)^2-6 c^2 d^2 \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)^2+6 c e^2 x \tanh ^{-1}(c x)+12 c d e \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)+3 e^2 \log \left(1-c^2 x^2\right)+6 c d \left(c d \tanh ^{-1}(c x)-e\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+3 c^2 d^2 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-\frac{6 c d (e-c d) (c d+e) \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)}{c^2}}{6 e^3}","-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^2 e}+\frac{b d^2 \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{e^3}-\frac{b d^2 \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^3}-\frac{d^2 \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{e^3}+\frac{d^2 \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^3}-\frac{d x \left(a+b \tanh ^{-1}(c x)\right)^2}{e^2}-\frac{d \left(a+b \tanh ^{-1}(c x)\right)^2}{c e^2}+\frac{2 b d \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c e^2}+\frac{x^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 e}+\frac{a b x}{c e}+\frac{b^2 \log \left(1-c^2 x^2\right)}{2 c^2 e}+\frac{b^2 d^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 e^3}-\frac{b^2 d^2 \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 e^3}+\frac{b^2 d \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c e^2}+\frac{b^2 x \tanh ^{-1}(c x)}{c e}",1,"(-6*a^2*d*e*x + 3*a^2*e^2*x^2 + 6*a^2*d^2*Log[d + e*x] + (6*a*b*(c*e^2*x + I*c^2*d^2*Pi*ArcTanh[c*x] - 2*c^2*d*e*x*ArcTanh[c*x] + e^2*(-1 + c^2*x^2)*ArcTanh[c*x] + 2*c^2*d^2*ArcTanh[(c*d)/e]*ArcTanh[c*x] - c^2*d^2*ArcTanh[c*x]^2 + c*d*e*ArcTanh[c*x]^2 - (c*d*Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^2)/E^ArcTanh[(c*d)/e] - 2*c^2*d^2*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - I*c^2*d^2*Pi*Log[1 + E^(2*ArcTanh[c*x])] + 2*c^2*d^2*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 2*c^2*d^2*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - c*d*e*Log[1 - c^2*x^2] - (I/2)*c^2*d^2*Pi*Log[1 - c^2*x^2] - 2*c^2*d^2*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] + c^2*d^2*PolyLog[2, -E^(-2*ArcTanh[c*x])] - c^2*d^2*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/c^2 + (b^2*(6*c*e^2*x*ArcTanh[c*x] + 6*c*d*e*ArcTanh[c*x]^2 - 6*c^2*d*e*x*ArcTanh[c*x]^2 + 3*e^2*(-1 + c^2*x^2)*ArcTanh[c*x]^2 - 2*c^2*d^2*ArcTanh[c*x]^3 + 2*c*d*e*ArcTanh[c*x]^3 + 12*c*d*e*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 6*c^2*d^2*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 3*e^2*Log[1 - c^2*x^2] + 6*c*d*(-e + c*d*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 3*c^2*d^2*PolyLog[3, -E^(-2*ArcTanh[c*x])] - (6*c*d*(-(c*d) + e)*(c*d + e)*(-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)))/c^2)/(6*e^3)","C",0
155,1,861,279,13.5497807,"\int \frac{x \left(a+b \tanh ^{-1}(c x)\right)^2}{d+e x} \, dx","Integrate[(x*(a + b*ArcTanh[c*x])^2)/(d + e*x),x]","\frac{3 e x a^2-3 d \log (d+e x) a^2+\frac{3 b \left(\sqrt{1-\frac{c^2 d^2}{e^2}} e e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)} \tanh ^{-1}(c x)^2+c d \tanh ^{-1}(c x)^2-e \tanh ^{-1}(c x)^2+2 c e x \tanh ^{-1}(c x)-2 c d \tanh ^{-1}\left(\frac{c d}{e}\right) \tanh ^{-1}(c x)+2 c d \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)-2 c d \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right) \tanh ^{-1}(c x)-i c d \pi  \tanh ^{-1}(c x)+i c d \pi  \log \left(1+e^{2 \tanh ^{-1}(c x)}\right)-2 c d \tanh ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)+e \log \left(1-c^2 x^2\right)+\frac{1}{2} i c d \pi  \log \left(1-c^2 x^2\right)+2 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)-c d \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+c d \text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)\right) a}{c}+\frac{b^2 \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+4 c d \tanh ^{-1}(c x)^3-2 e \tanh ^{-1}(c x)^3-3 e \tanh ^{-1}(c x)^2+3 c e x \tanh ^{-1}(c x)^2+3 c d \log \left(1+e^{-2 \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(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)^2-6 e \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)+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)+3 \left(e-c d \tanh ^{-1}(c x)\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-\frac{3}{2} c d \text{Li}_3\left(-e^{-2 \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)+6 c d \text{Li}_3\left(e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right)\right)}{c}}{3 e^2}","-\frac{b d \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{e^2}+\frac{b 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)}{e^2}+\frac{d \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{e^2}-\frac{d \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^2}+\frac{x \left(a+b \tanh ^{-1}(c x)\right)^2}{e}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{c e}-\frac{2 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c e}-\frac{b^2 d \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 e^2}+\frac{b^2 d \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 e^2}-\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c e}",1,"(3*a^2*e*x - 3*a^2*d*Log[d + e*x] + (3*a*b*((-I)*c*d*Pi*ArcTanh[c*x] + 2*c*e*x*ArcTanh[c*x] - 2*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x] + c*d*ArcTanh[c*x]^2 - e*ArcTanh[c*x]^2 + (Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^2)/E^ArcTanh[(c*d)/e] + 2*c*d*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + I*c*d*Pi*Log[1 + E^(2*ArcTanh[c*x])] - 2*c*d*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - 2*c*d*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + e*Log[1 - c^2*x^2] + (I/2)*c*d*Pi*Log[1 - c^2*x^2] + 2*c*d*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - c*d*PolyLog[2, -E^(-2*ArcTanh[c*x])] + c*d*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/c + (b^2*(-3*e*ArcTanh[c*x]^2 + 3*c*e*x*ArcTanh[c*x]^2 + 4*c*d*ArcTanh[c*x]^3 - 2*e*ArcTanh[c*x]^3 + (2*Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^3)/E^ArcTanh[(c*d)/e] - 6*e*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + 3*c*d*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + (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]]] + 3*(e - c*d*ArcTanh[c*x])*PolyLog[2, -E^(-2*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])] - (3*c*d*PolyLog[3, -E^(-2*ArcTanh[c*x])])/2 + 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])]))/c)/(3*e^2)","C",0
156,1,759,188,10.1920562,"\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
157,1,818,319,12.6101829,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x (d+e x)} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(x*(d + e*x)),x]","\frac{24 c d \log (x) a^2-24 c d \log (d+e x) a^2-24 b \left(-\sqrt{1-\frac{c^2 d^2}{e^2}} e e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)} \tanh ^{-1}(c x)^2-c d \tanh ^{-1}(c x)^2+e \tanh ^{-1}(c x)^2+2 c d \tanh ^{-1}\left(\frac{c d}{e}\right) \tanh ^{-1}(c x)-2 c d \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)+2 c d \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right) \tanh ^{-1}(c x)+i c d \pi  \tanh ^{-1}(c x)-i c d \pi  \log \left(1+e^{2 \tanh ^{-1}(c x)}\right)+2 c d \tanh ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)-\frac{1}{2} i c d \pi  \log \left(1-c^2 x^2\right)-2 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)+c d \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-c d \text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right)\right) a+b^2 \left(16 \sqrt{1-\frac{c^2 d^2}{e^2}} e e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)} \tanh ^{-1}(c x)^3+16 c d \tanh ^{-1}(c x)^3-16 e \tanh ^{-1}(c x)^3+24 c d \log \left(1-e^{2 \tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)^2-24 c d \log \left(1-e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)^2-24 c d \log \left(1+e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)^2-24 c d \log \left(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+24 c d \log \left(\frac{c (d+e x)}{\sqrt{1-c^2 x^2}}\right) \tanh ^{-1}(c x)^2+c d \log (16777216) \tanh ^{-1}(c x)^2+24 i c d \pi  \log \left(e^{-\tanh ^{-1}(c x)}+e^{\tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)-48 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)+12 i c d \pi  \log \left(1-c^2 x^2\right) \tanh ^{-1}(c x)+48 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)+24 c d \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)-48 c d \text{Li}_2\left(-e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)-48 c d \text{Li}_2\left(e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)-24 i c d \pi  \log (2) \tanh ^{-1}(c x)-12 c d \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)+48 c d \text{Li}_3\left(-e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right)+48 c d \text{Li}_3\left(e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right)+i c d \pi ^3\right)}{24 c d^2}","\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)}{d}-\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)}{d}-\frac{b \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d}+\frac{b \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)}{d}-\frac{b \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d}+\frac{\log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d}+\frac{b^2 \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 d}+\frac{b^2 \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{2 d}-\frac{b^2 \text{Li}_3\left(\frac{2}{1-c x}-1\right)}{2 d}-\frac{b^2 \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 d}",1,"(24*a^2*c*d*Log[x] - 24*a^2*c*d*Log[d + e*x] - 24*a*b*(I*c*d*Pi*ArcTanh[c*x] + 2*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x] - c*d*ArcTanh[c*x]^2 + e*ArcTanh[c*x]^2 - (Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^2)/E^ArcTanh[(c*d)/e] - 2*c*d*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - I*c*d*Pi*Log[1 + E^(2*ArcTanh[c*x])] + 2*c*d*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 2*c*d*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - (I/2)*c*d*Pi*Log[1 - c^2*x^2] - 2*c*d*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] + c*d*PolyLog[2, E^(-2*ArcTanh[c*x])] - c*d*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]) + b^2*(I*c*d*Pi^3 + 16*c*d*ArcTanh[c*x]^3 - 16*e*ArcTanh[c*x]^3 + (16*Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^3)/E^ArcTanh[(c*d)/e] - (24*I)*c*d*Pi*ArcTanh[c*x]*Log[2] + c*d*ArcTanh[c*x]^2*Log[16777216] + (24*I)*c*d*Pi*ArcTanh[c*x]*Log[E^(-ArcTanh[c*x]) + E^ArcTanh[c*x]] + 24*c*d*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] - 24*c*d*ArcTanh[c*x]^2*Log[1 - E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 24*c*d*ArcTanh[c*x]^2*Log[1 + E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 48*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])))] - 24*c*d*ArcTanh[c*x]^2*Log[(e*(-1 + E^(2*ArcTanh[c*x])) + c*d*(1 + E^(2*ArcTanh[c*x])))/E^ArcTanh[c*x]] + 24*c*d*ArcTanh[c*x]^2*Log[(c*(d + e*x))/Sqrt[1 - c^2*x^2]] + (12*I)*c*d*Pi*ArcTanh[c*x]*Log[1 - c^2*x^2] + 48*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] + 24*c*d*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 48*c*d*ArcTanh[c*x]*PolyLog[2, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 48*c*d*ArcTanh[c*x]*PolyLog[2, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 12*c*d*PolyLog[3, E^(2*ArcTanh[c*x])] + 48*c*d*PolyLog[3, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 48*c*d*PolyLog[3, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])]))/(24*c*d^2)","C",0
158,1,1010,412,12.7535872,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x^2 (d+e x)} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/(x^2*(d + e*x)),x]","-\frac{e \log (x) a^2}{d^2}+\frac{e \log (d+e x) a^2}{d^2}-\frac{a^2}{d x}+\frac{b \left(-\frac{2 c \tanh ^{-1}(c x) d^2}{x}+2 c^2 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right) d^2-c e \tanh ^{-1}(c x)^2 d+2 c e \tanh ^{-1}\left(\frac{c d}{e}\right) \tanh ^{-1}(c x) d+i c e \pi  \tanh ^{-1}(c x) d-2 c e \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right) d-i c e \pi  \log \left(1+e^{2 \tanh ^{-1}(c x)}\right) d+2 c e \tanh ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right) d+2 c e \tanh ^{-1}(c x) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right) d-\frac{1}{2} i c e \pi  \log \left(1-c^2 x^2\right) d-2 c e \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) d+c e \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right) d-c e \text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)\right)}\right) d-\sqrt{1-\frac{c^2 d^2}{e^2}} e^2 e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)} \tanh ^{-1}(c x)^2+e^2 \tanh ^{-1}(c x)^2\right) a}{c d^3}+\frac{b^2 \left(8 e^2 \tanh ^{-1}(c x)^3+8 c d e \tanh ^{-1}(c x)^3+24 c^2 d^2 \tanh ^{-1}(c x)^2-24 c d e \log \left(1-e^{2 \tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)^2-\frac{24 c d^2 \tanh ^{-1}(c x)^2}{x}+48 c^2 d^2 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)-24 c d e \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right) \tanh ^{-1}(c x)-24 c^2 d^2 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+12 c d e \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)-i c d e \pi ^3\right)}{24 c d^3}+\frac{b^2 (c d-e) e (c d+e) \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)}{d^3 \left(3 c^3 d^2-3 c e^2\right)}","\frac{b e \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}-\frac{b e \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}+\frac{b e \text{Li}_2\left(1-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d^2}-\frac{b e \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)}{d^2}-\frac{2 e \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d^2}-\frac{e \log \left(\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{d^2}+\frac{e \left(a+b \tanh ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right)}{d^2}+\frac{c \left(a+b \tanh ^{-1}(c x)\right)^2}{d}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{d x}+\frac{2 b c \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)}{d}-\frac{b^2 e \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{2 d^2}+\frac{b^2 e \text{Li}_3\left(\frac{2}{1-c x}-1\right)}{2 d^2}+\frac{b^2 e \text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 d^2}-\frac{b^2 e \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 d^2}-\frac{b^2 c \text{Li}_2\left(\frac{2}{c x+1}-1\right)}{d}",1,"-(a^2/(d*x)) - (a^2*e*Log[x])/d^2 + (a^2*e*Log[d + e*x])/d^2 + (a*b*(I*c*d*e*Pi*ArcTanh[c*x] - (2*c*d^2*ArcTanh[c*x])/x + 2*c*d*e*ArcTanh[(c*d)/e]*ArcTanh[c*x] - c*d*e*ArcTanh[c*x]^2 + e^2*ArcTanh[c*x]^2 - (Sqrt[1 - (c^2*d^2)/e^2]*e^2*ArcTanh[c*x]^2)/E^ArcTanh[(c*d)/e] - 2*c*d*e*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - I*c*d*e*Pi*Log[1 + E^(2*ArcTanh[c*x])] + 2*c*d*e*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 2*c*d*e*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 2*c^2*d^2*Log[(c*x)/Sqrt[1 - c^2*x^2]] - (I/2)*c*d*e*Pi*Log[1 - c^2*x^2] - 2*c*d*e*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] + c*d*e*PolyLog[2, E^(-2*ArcTanh[c*x])] - c*d*e*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/(c*d^3) + (b^2*((-I)*c*d*e*Pi^3 + 24*c^2*d^2*ArcTanh[c*x]^2 - (24*c*d^2*ArcTanh[c*x]^2)/x + 8*c*d*e*ArcTanh[c*x]^3 + 8*e^2*ArcTanh[c*x]^3 + 48*c^2*d^2*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - 24*c*d*e*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] - 24*c^2*d^2*PolyLog[2, E^(-2*ArcTanh[c*x])] - 24*c*d*e*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + 12*c*d*e*PolyLog[3, E^(2*ArcTanh[c*x])]))/(24*c*d^3) + (b^2*(c*d - e)*e*(c*d + e)*(-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])]))/(d^3*(3*c^3*d^2 - 3*c*e^2))","C",0
159,1,521,275,9.3698835,"\int \frac{\tanh ^{-1}(c x)^2}{x (d+e x)} \, dx","Integrate[ArcTanh[c*x]^2/(x*(d + e*x)),x]","\frac{16 e \sqrt{1-\frac{c^2 d^2}{e^2}} \tanh ^{-1}(c x)^3 e^{-\tanh ^{-1}\left(\frac{c d}{e}\right)}+24 c d \tanh ^{-1}(c x)^2 \log \left(\frac{c (d+e x)}{\sqrt{1-c^2 x^2}}\right)+12 i \pi  c d \log \left(1-c^2 x^2\right) \tanh ^{-1}(c x)-48 c d \tanh ^{-1}(c x) \text{Li}_2\left(-e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right)-48 c d \tanh ^{-1}(c x) \text{Li}_2\left(e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right)+48 c d \text{Li}_3\left(-e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right)+48 c d \text{Li}_3\left(e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right)-24 c d \tanh ^{-1}(c x)^2 \log \left(1-e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}\right)-24 c d \tanh ^{-1}(c x)^2 \log \left(e^{\tanh ^{-1}\left(\frac{c d}{e}\right)+\tanh ^{-1}(c x)}+1\right)-24 c d \tanh ^{-1}(c x)^2 \log \left(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)-48 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)+48 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)+24 c d \tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)-12 c d \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)+16 c d \tanh ^{-1}(c x)^3+24 c d \tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)+c d \log (16777216) \tanh ^{-1}(c x)^2+24 i \pi  c d \tanh ^{-1}(c x) \log \left(e^{-\tanh ^{-1}(c x)}+e^{\tanh ^{-1}(c x)}\right)-24 i \pi  c d \log (2) \tanh ^{-1}(c x)+i \pi ^3 c d-16 e \tanh ^{-1}(c x)^3}{24 c d^2}","\frac{\text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 d}+\frac{\tanh ^{-1}(c x) \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{d}-\frac{\tanh ^{-1}(c x)^2 \log \left(\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right)}{d}+\frac{\text{Li}_3\left(1-\frac{2}{1-c x}\right)}{2 d}-\frac{\text{Li}_3\left(\frac{2}{1-c x}-1\right)}{2 d}-\frac{\text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 d}-\frac{\text{Li}_2\left(1-\frac{2}{1-c x}\right) \tanh ^{-1}(c x)}{d}+\frac{\text{Li}_2\left(\frac{2}{1-c x}-1\right) \tanh ^{-1}(c x)}{d}-\frac{\text{Li}_2\left(1-\frac{2}{c x+1}\right) \tanh ^{-1}(c x)}{d}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \tanh ^{-1}(c x)^2}{d}+\frac{\log \left(\frac{2}{c x+1}\right) \tanh ^{-1}(c x)^2}{d}",1,"(I*c*d*Pi^3 + 16*c*d*ArcTanh[c*x]^3 - 16*e*ArcTanh[c*x]^3 + (16*Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^3)/E^ArcTanh[(c*d)/e] - (24*I)*c*d*Pi*ArcTanh[c*x]*Log[2] + c*d*ArcTanh[c*x]^2*Log[16777216] + (24*I)*c*d*Pi*ArcTanh[c*x]*Log[E^(-ArcTanh[c*x]) + E^ArcTanh[c*x]] + 24*c*d*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] - 24*c*d*ArcTanh[c*x]^2*Log[1 - E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 24*c*d*ArcTanh[c*x]^2*Log[1 + E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 48*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])))] - 24*c*d*ArcTanh[c*x]^2*Log[(e*(-1 + E^(2*ArcTanh[c*x])) + c*d*(1 + E^(2*ArcTanh[c*x])))/E^ArcTanh[c*x]] + 24*c*d*ArcTanh[c*x]^2*Log[(c*(d + e*x))/Sqrt[1 - c^2*x^2]] + (12*I)*c*d*Pi*ArcTanh[c*x]*Log[1 - c^2*x^2] + 48*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] + 24*c*d*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 48*c*d*ArcTanh[c*x]*PolyLog[2, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 48*c*d*ArcTanh[c*x]*PolyLog[2, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 12*c*d*PolyLog[3, E^(2*ArcTanh[c*x])] + 48*c*d*PolyLog[3, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 48*c*d*PolyLog[3, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])])/(24*c*d^2)","C",0
160,0,0,21,0.5391217,"\int \frac{1}{(d+e x) \left(a+b \tan ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x)*(a + b*ArcTan[c*x])),x]","\int \frac{1}{(d+e x) \left(a+b \tan ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{(d+e x) \left(a+b \tan ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d + e*x)*(a + b*ArcTan[c*x])), x]","A",-1
161,1,72,72,0.0192146,"\int x^4 \left(1-a^2 x^2\right) \tanh ^{-1}(a x) \, dx","Integrate[x^4*(1 - a^2*x^2)*ArcTanh[a*x],x]","\frac{x^2}{35 a^3}-\frac{1}{7} a^2 x^7 \tanh ^{-1}(a x)+\frac{\log \left(1-a^2 x^2\right)}{35 a^5}-\frac{a x^6}{42}+\frac{1}{5} x^5 \tanh ^{-1}(a x)+\frac{x^4}{70 a}","\frac{x^2}{35 a^3}-\frac{1}{7} a^2 x^7 \tanh ^{-1}(a x)+\frac{\log \left(1-a^2 x^2\right)}{35 a^5}-\frac{a x^6}{42}+\frac{1}{5} x^5 \tanh ^{-1}(a x)+\frac{x^4}{70 a}",1,"x^2/(35*a^3) + x^4/(70*a) - (a*x^6)/42 + (x^5*ArcTanh[a*x])/5 - (a^2*x^7*ArcTanh[a*x])/7 + Log[1 - a^2*x^2]/(35*a^5)","A",1
162,1,79,63,0.0183534,"\int x^3 \left(1-a^2 x^2\right) \tanh ^{-1}(a x) \, dx","Integrate[x^3*(1 - a^2*x^2)*ArcTanh[a*x],x]","\frac{\log (1-a x)}{24 a^4}-\frac{\log (a x+1)}{24 a^4}+\frac{x}{12 a^3}-\frac{1}{6} a^2 x^6 \tanh ^{-1}(a x)-\frac{a x^5}{30}+\frac{1}{4} x^4 \tanh ^{-1}(a x)+\frac{x^3}{36 a}","-\frac{\tanh ^{-1}(a x)}{12 a^4}+\frac{x}{12 a^3}-\frac{1}{6} a^2 x^6 \tanh ^{-1}(a x)-\frac{a x^5}{30}+\frac{1}{4} x^4 \tanh ^{-1}(a x)+\frac{x^3}{36 a}",1,"x/(12*a^3) + x^3/(36*a) - (a*x^5)/30 + (x^4*ArcTanh[a*x])/4 - (a^2*x^6*ArcTanh[a*x])/6 + Log[1 - a*x]/(24*a^4) - Log[1 + a*x]/(24*a^4)","A",1
163,1,62,62,0.0157296,"\int x^2 \left(1-a^2 x^2\right) \tanh ^{-1}(a x) \, dx","Integrate[x^2*(1 - a^2*x^2)*ArcTanh[a*x],x]","-\frac{1}{5} a^2 x^5 \tanh ^{-1}(a x)+\frac{\log \left(1-a^2 x^2\right)}{15 a^3}-\frac{a x^4}{20}+\frac{1}{3} x^3 \tanh ^{-1}(a x)+\frac{x^2}{15 a}","-\frac{1}{5} a^2 x^5 \tanh ^{-1}(a x)+\frac{\log \left(1-a^2 x^2\right)}{15 a^3}-\frac{a x^4}{20}+\frac{1}{3} x^3 \tanh ^{-1}(a x)+\frac{x^2}{15 a}",1,"x^2/(15*a) - (a*x^4)/20 + (x^3*ArcTanh[a*x])/3 - (a^2*x^5*ArcTanh[a*x])/5 + Log[1 - a^2*x^2]/(15*a^3)","A",1
164,1,69,40,0.0150706,"\int x \left(1-a^2 x^2\right) \tanh ^{-1}(a x) \, dx","Integrate[x*(1 - a^2*x^2)*ArcTanh[a*x],x]","-\frac{1}{4} a^2 x^4 \tanh ^{-1}(a x)+\frac{\log (1-a x)}{8 a^2}-\frac{\log (a x+1)}{8 a^2}-\frac{a x^3}{12}+\frac{1}{2} x^2 \tanh ^{-1}(a x)+\frac{x}{4 a}","-\frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}{4 a^2}-\frac{a x^3}{12}+\frac{x}{4 a}",1,"x/(4*a) - (a*x^3)/12 + (x^2*ArcTanh[a*x])/2 - (a^2*x^4*ArcTanh[a*x])/4 + Log[1 - a*x]/(8*a^2) - Log[1 + a*x]/(8*a^2)","A",1
165,1,47,64,0.0099486,"\int \left(1-a^2 x^2\right) \tanh ^{-1}(a x) \, dx","Integrate[(1 - a^2*x^2)*ArcTanh[a*x],x]","-\frac{1}{3} a^2 x^3 \tanh ^{-1}(a x)+\frac{\log \left(1-a^2 x^2\right)}{3 a}-\frac{a x^2}{6}+x \tanh ^{-1}(a x)","\frac{1-a^2 x^2}{6 a}+\frac{\log \left(1-a^2 x^2\right)}{3 a}+\frac{1}{3} x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)+\frac{2}{3} x \tanh ^{-1}(a x)",1,"-1/6*(a*x^2) + x*ArcTanh[a*x] - (a^2*x^3*ArcTanh[a*x])/3 + Log[1 - a^2*x^2]/(3*a)","A",1
166,1,60,48,0.0180626,"\int \frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)}{x} \, dx","Integrate[((1 - a^2*x^2)*ArcTanh[a*x])/x,x]","-\frac{1}{2} a^2 x^2 \tanh ^{-1}(a x)+\frac{1}{2} (\text{Li}_2(a x)-\text{Li}_2(-a x))-\frac{a x}{2}-\frac{1}{4} \log (1-a x)+\frac{1}{4} \log (a x+1)","-\frac{1}{2} a^2 x^2 \tanh ^{-1}(a x)-\frac{\text{Li}_2(-a x)}{2}+\frac{\text{Li}_2(a x)}{2}-\frac{a x}{2}+\frac{1}{2} \tanh ^{-1}(a x)",1,"-1/2*(a*x) - (a^2*x^2*ArcTanh[a*x])/2 - Log[1 - a*x]/4 + Log[1 + a*x]/4 + (-PolyLog[2, -(a*x)] + PolyLog[2, a*x])/2","A",0
167,1,38,38,0.0102621,"\int \frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)}{x^2} \, dx","Integrate[((1 - a^2*x^2)*ArcTanh[a*x])/x^2,x]","-a \log \left(1-a^2 x^2\right)+a^2 (-x) \tanh ^{-1}(a x)+a \log (x)-\frac{\tanh ^{-1}(a x)}{x}","-a \log \left(1-a^2 x^2\right)+a^2 (-x) \tanh ^{-1}(a x)+a \log (x)-\frac{\tanh ^{-1}(a x)}{x}",1,"-(ArcTanh[a*x]/x) - a^2*x*ArcTanh[a*x] + a*Log[x] - a*Log[1 - a^2*x^2]","A",1
168,1,68,56,0.0333598,"\int \frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)}{x^3} \, dx","Integrate[((1 - a^2*x^2)*ArcTanh[a*x])/x^3,x]","-\frac{1}{2} a^2 (\text{Li}_2(a x)-\text{Li}_2(-a x))-\frac{1}{4} a^2 \log (1-a x)+\frac{1}{4} a^2 \log (a x+1)-\frac{\tanh ^{-1}(a x)}{2 x^2}-\frac{a}{2 x}","\frac{1}{2} a^2 \text{Li}_2(-a x)-\frac{1}{2} a^2 \text{Li}_2(a x)+\frac{1}{2} a^2 \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)}{2 x^2}-\frac{a}{2 x}",1,"-1/2*a/x - ArcTanh[a*x]/(2*x^2) - (a^2*Log[1 - a*x])/4 + (a^2*Log[1 + a*x])/4 - (a^2*(-PolyLog[2, -(a*x)] + PolyLog[2, a*x]))/2","A",0
169,1,58,58,0.0163443,"\int \frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)}{x^4} \, dx","Integrate[((1 - a^2*x^2)*ArcTanh[a*x])/x^4,x]","-\frac{2}{3} a^3 \log (x)+\frac{a^2 \tanh ^{-1}(a x)}{x}+\frac{1}{3} a^3 \log \left(1-a^2 x^2\right)-\frac{\tanh ^{-1}(a x)}{3 x^3}-\frac{a}{6 x^2}","-\frac{2}{3} a^3 \log (x)+\frac{a^2 \tanh ^{-1}(a x)}{x}+\frac{1}{3} a^3 \log \left(1-a^2 x^2\right)-\frac{\tanh ^{-1}(a x)}{3 x^3}-\frac{a}{6 x^2}",1,"-1/6*a/x^2 - ArcTanh[a*x]/(3*x^3) + (a^2*ArcTanh[a*x])/x - (2*a^3*Log[x])/3 + (a^3*Log[1 - a^2*x^2])/3","A",1
170,1,71,42,0.0154587,"\int \frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)}{x^5} \, dx","Integrate[((1 - a^2*x^2)*ArcTanh[a*x])/x^5,x]","\frac{1}{8} a^4 \log (1-a x)-\frac{1}{8} a^4 \log (a x+1)+\frac{a^3}{4 x}+\frac{a^2 \tanh ^{-1}(a x)}{2 x^2}-\frac{\tanh ^{-1}(a x)}{4 x^4}-\frac{a}{12 x^3}","\frac{a^3}{4 x}-\frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}{4 x^4}-\frac{a}{12 x^3}",1,"-1/12*a/x^3 + a^3/(4*x) - ArcTanh[a*x]/(4*x^4) + (a^2*ArcTanh[a*x])/(2*x^2) + (a^4*Log[1 - a*x])/8 - (a^4*Log[1 + a*x])/8","A",1
171,1,71,71,0.0166155,"\int \frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)}{x^6} \, dx","Integrate[((1 - a^2*x^2)*ArcTanh[a*x])/x^6,x]","-\frac{2}{15} a^5 \log (x)+\frac{a^3}{15 x^2}+\frac{a^2 \tanh ^{-1}(a x)}{3 x^3}+\frac{1}{15} a^5 \log \left(1-a^2 x^2\right)-\frac{\tanh ^{-1}(a x)}{5 x^5}-\frac{a}{20 x^4}","-\frac{2}{15} a^5 \log (x)+\frac{a^3}{15 x^2}+\frac{a^2 \tanh ^{-1}(a x)}{3 x^3}+\frac{1}{15} a^5 \log \left(1-a^2 x^2\right)-\frac{\tanh ^{-1}(a x)}{5 x^5}-\frac{a}{20 x^4}",1,"-1/20*a/x^4 + a^3/(15*x^2) - ArcTanh[a*x]/(5*x^5) + (a^2*ArcTanh[a*x])/(3*x^3) - (2*a^5*Log[x])/15 + (a^5*Log[1 - a^2*x^2])/15","A",1
172,1,113,162,0.9581687,"\int x^4 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2 \, dx","Integrate[x^4*(1 - a^2*x^2)*ArcTanh[a*x]^2,x]","-\frac{3 a^5 x^5+2 a^3 x^3+9 \left(5 a^7 x^7-7 a^5 x^5+2\right) \tanh ^{-1}(a x)^2+3 \tanh ^{-1}(a x) \left(5 a^6 x^6-3 a^4 x^4-6 a^2 x^2+12 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)+4\right)-18 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)-12 a x}{315 a^5}","-\frac{2 \text{Li}_2\left(1-\frac{2}{1-a x}\right)}{35 a^5}+\frac{2 \tanh ^{-1}(a x)^2}{35 a^5}-\frac{4 \tanh ^{-1}(a x)}{105 a^5}-\frac{4 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{35 a^5}+\frac{4 x}{105 a^4}+\frac{2 x^2 \tanh ^{-1}(a x)}{35 a^3}-\frac{1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac{2 x^3}{315 a^2}-\frac{1}{21} a x^6 \tanh ^{-1}(a x)+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2+\frac{x^4 \tanh ^{-1}(a x)}{35 a}-\frac{x^5}{105}",1,"-1/315*(-12*a*x + 2*a^3*x^3 + 3*a^5*x^5 + 9*(2 - 7*a^5*x^5 + 5*a^7*x^7)*ArcTanh[a*x]^2 + 3*ArcTanh[a*x]*(4 - 6*a^2*x^2 - 3*a^4*x^4 + 5*a^6*x^6 + 12*Log[1 + E^(-2*ArcTanh[a*x])]) - 18*PolyLog[2, -E^(-2*ArcTanh[a*x])])/a^5","A",0
173,1,88,116,0.0441454,"\int x^3 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2 \, dx","Integrate[x^3*(1 - a^2*x^2)*ArcTanh[a*x]^2,x]","-\frac{3 a^4 x^4+a^2 x^2-14 \log \left(1-a^2 x^2\right)+15 \left(2 a^6 x^6-3 a^4 x^4+1\right) \tanh ^{-1}(a x)^2+2 a x \left(6 a^4 x^4-5 a^2 x^2-15\right) \tanh ^{-1}(a x)}{180 a^4}","-\frac{\tanh ^{-1}(a x)^2}{12 a^4}+\frac{x \tanh ^{-1}(a x)}{6 a^3}-\frac{1}{6} a^2 x^6 \tanh ^{-1}(a x)^2-\frac{x^2}{180 a^2}+\frac{7 \log \left(1-a^2 x^2\right)}{90 a^4}-\frac{1}{15} a x^5 \tanh ^{-1}(a x)+\frac{1}{4} x^4 \tanh ^{-1}(a x)^2+\frac{x^3 \tanh ^{-1}(a x)}{18 a}-\frac{x^4}{60}",1,"-1/180*(a^2*x^2 + 3*a^4*x^4 + 2*a*x*(-15 - 5*a^2*x^2 + 6*a^4*x^4)*ArcTanh[a*x] + 15*(1 - 3*a^4*x^4 + 2*a^6*x^6)*ArcTanh[a*x]^2 - 14*Log[1 - a^2*x^2])/a^4","A",1
174,1,95,138,0.2542589,"\int x^2 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2 \, dx","Integrate[x^2*(1 - a^2*x^2)*ArcTanh[a*x]^2,x]","-\frac{a^3 x^3+2 \left(3 a^5 x^5-5 a^3 x^3+2\right) \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \left(3 a^4 x^4-4 a^2 x^2+8 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)+1\right)-4 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)-a x}{30 a^3}","-\frac{2 \text{Li}_2\left(1-\frac{2}{1-a x}\right)}{15 a^3}+\frac{2 \tanh ^{-1}(a x)^2}{15 a^3}-\frac{\tanh ^{-1}(a x)}{30 a^3}-\frac{4 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{15 a^3}-\frac{1}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac{x}{30 a^2}-\frac{1}{10} a x^4 \tanh ^{-1}(a x)+\frac{1}{3} x^3 \tanh ^{-1}(a x)^2+\frac{2 x^2 \tanh ^{-1}(a x)}{15 a}-\frac{x^3}{30}",1,"-1/30*(-(a*x) + a^3*x^3 + 2*(2 - 5*a^3*x^3 + 3*a^5*x^5)*ArcTanh[a*x]^2 + ArcTanh[a*x]*(1 - 4*a^2*x^2 + 3*a^4*x^4 + 8*Log[1 + E^(-2*ArcTanh[a*x])]) - 4*PolyLog[2, -E^(-2*ArcTanh[a*x])])/a^3","A",0
175,1,66,95,0.0294388,"\int x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2 \, dx","Integrate[x*(1 - a^2*x^2)*ArcTanh[a*x]^2,x]","\frac{\left(6 a x-2 a^3 x^3\right) \tanh ^{-1}(a x)-a^2 x^2+2 \log \left(1-a^2 x^2\right)-3 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2}{12 a^2}","\frac{1-a^2 x^2}{12 a^2}+\frac{\log \left(1-a^2 x^2\right)}{6 a^2}-\frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}{4 a^2}+\frac{x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}{6 a}+\frac{x \tanh ^{-1}(a x)}{3 a}",1,"(-(a^2*x^2) + (6*a*x - 2*a^3*x^3)*ArcTanh[a*x] - 3*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2 + 2*Log[1 - a^2*x^2])/(12*a^2)","A",1
176,1,71,115,0.1176126,"\int \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2 \, dx","Integrate[(1 - a^2*x^2)*ArcTanh[a*x]^2,x]","-\frac{\tanh ^{-1}(a x) \left(a^2 x^2+4 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)-1\right)-2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+a x+(a x-1)^2 (a x+2) \tanh ^{-1}(a x)^2}{3 a}","\frac{1}{3} x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2+\frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)}{3 a}-\frac{2 \text{Li}_2\left(1-\frac{2}{1-a x}\right)}{3 a}+\frac{2}{3} x \tanh ^{-1}(a x)^2+\frac{2 \tanh ^{-1}(a x)^2}{3 a}-\frac{4 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{3 a}-\frac{x}{3}",1,"-1/3*(a*x + (-1 + a*x)^2*(2 + a*x)*ArcTanh[a*x]^2 + ArcTanh[a*x]*(-1 + a^2*x^2 + 4*Log[1 + E^(-2*ArcTanh[a*x])]) - 2*PolyLog[2, -E^(-2*ArcTanh[a*x])])/a","A",0
177,1,145,146,0.0444039,"\int \frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}{x} \, dx","Integrate[((1 - a^2*x^2)*ArcTanh[a*x]^2)/x,x]","-\frac{1}{2} \log \left(1-a^2 x^2\right)-\frac{1}{2} \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^2-\frac{1}{2} \text{Li}_3\left(\frac{-a x-1}{a x-1}\right)+\frac{1}{2} \text{Li}_3\left(\frac{a x+1}{a x-1}\right)+\text{Li}_2\left(\frac{-a x-1}{a x-1}\right) \tanh ^{-1}(a x)-\text{Li}_2\left(\frac{a x+1}{a x-1}\right) \tanh ^{-1}(a x)+2 \tanh ^{-1}\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2-a x \tanh ^{-1}(a x)","-\frac{1}{2} \log \left(1-a^2 x^2\right)-\frac{1}{2} a^2 x^2 \tanh ^{-1}(a x)^2+\frac{1}{2} \text{Li}_3\left(1-\frac{2}{1-a x}\right)-\frac{1}{2} \text{Li}_3\left(\frac{2}{1-a x}-1\right)-\text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)+\text{Li}_2\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)+2 \tanh ^{-1}\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2+\frac{1}{2} \tanh ^{-1}(a x)^2-a x \tanh ^{-1}(a x)",1,"-(a*x*ArcTanh[a*x]) - ((-1 + a^2*x^2)*ArcTanh[a*x]^2)/2 + 2*ArcTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)] - Log[1 - a^2*x^2]/2 + ArcTanh[a*x]*PolyLog[2, (-1 - a*x)/(-1 + a*x)] - ArcTanh[a*x]*PolyLog[2, (1 + a*x)/(-1 + a*x)] - PolyLog[3, (-1 - a*x)/(-1 + a*x)]/2 + PolyLog[3, (1 + a*x)/(-1 + a*x)]/2","A",1
178,1,102,93,0.1373363,"\int \frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}{x^2} \, dx","Integrate[((1 - a^2*x^2)*ArcTanh[a*x]^2)/x^2,x]","-a \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+a \left(\tanh ^{-1}(a x) \left(-\frac{\tanh ^{-1}(a x)}{a x}+\tanh ^{-1}(a x)+2 \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)\right)-\text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)\right)-a \tanh ^{-1}(a x) \left(a x \tanh ^{-1}(a x)-\tanh ^{-1}(a x)-2 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)\right)","-a^2 x \tanh ^{-1}(a x)^2+a \text{Li}_2\left(1-\frac{2}{1-a x}\right)-a \text{Li}_2\left(\frac{2}{a x+1}-1\right)-\frac{\tanh ^{-1}(a x)^2}{x}+2 a \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)+2 a \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)",1,"-(a*ArcTanh[a*x]*(-ArcTanh[a*x] + a*x*ArcTanh[a*x] - 2*Log[1 + E^(-2*ArcTanh[a*x])])) - a*PolyLog[2, -E^(-2*ArcTanh[a*x])] + a*(ArcTanh[a*x]*(ArcTanh[a*x] - ArcTanh[a*x]/(a*x) + 2*Log[1 - E^(-2*ArcTanh[a*x])]) - PolyLog[2, E^(-2*ArcTanh[a*x])])","A",0
179,1,174,172,0.081536,"\int \frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}{x^3} \, dx","Integrate[((1 - a^2*x^2)*ArcTanh[a*x]^2)/x^3,x]","\frac{1}{2} a^2 \text{Li}_3\left(\frac{-a x-1}{a x-1}\right)-\frac{1}{2} a^2 \text{Li}_3\left(\frac{a x+1}{a x-1}\right)-a^2 \text{Li}_2\left(\frac{-a x-1}{a x-1}\right) \tanh ^{-1}(a x)+a^2 \text{Li}_2\left(\frac{a x+1}{a x-1}\right) \tanh ^{-1}(a x)-\frac{1}{2} a^2 \log \left(1-a^2 x^2\right)+\frac{\left(a^2 x^2-1\right) \tanh ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)-2 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1-a x}\right)-\frac{a \tanh ^{-1}(a x)}{x}","-\frac{1}{2} a^2 \text{Li}_3\left(1-\frac{2}{1-a x}\right)+\frac{1}{2} a^2 \text{Li}_3\left(\frac{2}{1-a x}-1\right)+a^2 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)-a^2 \text{Li}_2\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)-\frac{1}{2} a^2 \log \left(1-a^2 x^2\right)+a^2 \log (x)+\frac{1}{2} a^2 \tanh ^{-1}(a x)^2-2 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1-a x}\right)-\frac{\tanh ^{-1}(a x)^2}{2 x^2}-\frac{a \tanh ^{-1}(a x)}{x}",1,"-((a*ArcTanh[a*x])/x) + ((-1 + a^2*x^2)*ArcTanh[a*x]^2)/(2*x^2) - 2*a^2*ArcTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)] + a^2*Log[x] - (a^2*Log[1 - a^2*x^2])/2 - a^2*ArcTanh[a*x]*PolyLog[2, (-1 - a*x)/(-1 + a*x)] + a^2*ArcTanh[a*x]*PolyLog[2, (1 + a*x)/(-1 + a*x)] + (a^2*PolyLog[3, (-1 - a*x)/(-1 + a*x)])/2 - (a^2*PolyLog[3, (1 + a*x)/(-1 + a*x)])/2","A",1
180,1,93,116,0.3055437,"\int \frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}{x^4} \, dx","Integrate[((1 - a^2*x^2)*ArcTanh[a*x]^2)/x^4,x]","\frac{2 a^3 x^3 \text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x) \left(a^3 x^3-4 a^3 x^3 \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)-a x\right)-a^2 x^2-(a x-1)^2 (2 a x+1) \tanh ^{-1}(a x)^2}{3 x^3}","\frac{2}{3} a^3 \text{Li}_2\left(\frac{2}{a x+1}-1\right)-\frac{2}{3} a^3 \tanh ^{-1}(a x)^2+\frac{1}{3} a^3 \tanh ^{-1}(a x)-\frac{4}{3} a^3 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)-\frac{a^2}{3 x}+\frac{a^2 \tanh ^{-1}(a x)^2}{x}-\frac{\tanh ^{-1}(a x)^2}{3 x^3}-\frac{a \tanh ^{-1}(a x)}{3 x^2}",1,"(-(a^2*x^2) - (-1 + a*x)^2*(1 + 2*a*x)*ArcTanh[a*x]^2 + ArcTanh[a*x]*(-(a*x) + a^3*x^3 - 4*a^3*x^3*Log[1 - E^(-2*ArcTanh[a*x])]) + 2*a^3*x^3*PolyLog[2, E^(-2*ArcTanh[a*x])])/(3*x^3)","A",0
181,1,82,89,0.035811,"\int \frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}{x^5} \, dx","Integrate[((1 - a^2*x^2)*ArcTanh[a*x]^2)/x^5,x]","\frac{-4 a^4 x^4 \log (x)+\left(6 a^3 x^3-2 a x\right) \tanh ^{-1}(a x)-a^2 x^2-3 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2+2 a^4 x^4 \log \left(1-a^2 x^2\right)}{12 x^4}","-\frac{1}{3} a^4 \log (x)+\frac{a^3 \tanh ^{-1}(a x)}{2 x}-\frac{a^2}{12 x^2}-\frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}{4 x^4}+\frac{1}{6} a^4 \log \left(1-a^2 x^2\right)-\frac{a \tanh ^{-1}(a x)}{6 x^3}",1,"(-(a^2*x^2) + (-2*a*x + 6*a^3*x^3)*ArcTanh[a*x] - 3*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2 - 4*a^4*x^4*Log[x] + 2*a^4*x^4*Log[1 - a^2*x^2])/(12*x^4)","A",1
182,1,114,143,0.6223398,"\int \frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}{x^6} \, dx","Integrate[((1 - a^2*x^2)*ArcTanh[a*x]^2)/x^6,x]","\frac{4 a^5 x^5 \text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)+a^2 x^2 \left(a^2 x^2-1\right)-2 \left(2 a^5 x^5-5 a^2 x^2+3\right) \tanh ^{-1}(a x)^2-a x \tanh ^{-1}(a x) \left(a^4 x^4+8 a^4 x^4 \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)-4 a^2 x^2+3\right)}{30 x^5}","\frac{2}{15} a^5 \text{Li}_2\left(\frac{2}{a x+1}-1\right)-\frac{2}{15} a^5 \tanh ^{-1}(a x)^2-\frac{1}{30} a^5 \tanh ^{-1}(a x)-\frac{4}{15} a^5 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)+\frac{a^4}{30 x}+\frac{2 a^3 \tanh ^{-1}(a x)}{15 x^2}-\frac{a^2}{30 x^3}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^3}-\frac{\tanh ^{-1}(a x)^2}{5 x^5}-\frac{a \tanh ^{-1}(a x)}{10 x^4}",1,"(a^2*x^2*(-1 + a^2*x^2) - 2*(3 - 5*a^2*x^2 + 2*a^5*x^5)*ArcTanh[a*x]^2 - a*x*ArcTanh[a*x]*(3 - 4*a^2*x^2 + a^4*x^4 + 8*a^4*x^4*Log[1 - E^(-2*ArcTanh[a*x])]) + 4*a^5*x^5*PolyLog[2, E^(-2*ArcTanh[a*x])])/(30*x^5)","A",0
183,1,134,157,0.2970888,"\int \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3 \, dx","Integrate[(1 - a^2*x^2)*ArcTanh[a*x]^3,x]","-\frac{2 a^3 x^3 \tanh ^{-1}(a x)^3+3 \log \left(1-a^2 x^2\right)+3 a^2 x^2 \tanh ^{-1}(a x)^2-12 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)-6 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)-6 a x \tanh ^{-1}(a x)^3+4 \tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2+6 a x \tanh ^{-1}(a x)+12 \tanh ^{-1}(a x)^2 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)}{6 a}","-\frac{\log \left(1-a^2 x^2\right)}{2 a}+\frac{1}{3} x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3+\frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}{2 a}+\frac{\text{Li}_3\left(1-\frac{2}{1-a x}\right)}{a}-\frac{2 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a}+\frac{2}{3} x \tanh ^{-1}(a x)^3+\frac{2 \tanh ^{-1}(a x)^3}{3 a}-x \tanh ^{-1}(a x)-\frac{2 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{a}",1,"-1/6*(6*a*x*ArcTanh[a*x] - 3*ArcTanh[a*x]^2 + 3*a^2*x^2*ArcTanh[a*x]^2 + 4*ArcTanh[a*x]^3 - 6*a*x*ArcTanh[a*x]^3 + 2*a^3*x^3*ArcTanh[a*x]^3 + 12*ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] + 3*Log[1 - a^2*x^2] - 12*ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] - 6*PolyLog[3, -E^(-2*ArcTanh[a*x])])/a","A",0
184,1,232,193,0.2804683,"\int \frac{x \tanh ^{-1}\left(\frac{x}{\sqrt{2}}\right)}{1-x^2} \, dx","Integrate[(x*ArcTanh[x/Sqrt[2]])/(1 - x^2),x]","\frac{1}{4} \left(-2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(\frac{x}{\sqrt{2}}\right)}\right)+\text{Li}_2\left(\left(3-2 \sqrt{2}\right) e^{-2 \tanh ^{-1}\left(\frac{x}{\sqrt{2}}\right)}\right)+\text{Li}_2\left(\left(3+2 \sqrt{2}\right) e^{-2 \tanh ^{-1}\left(\frac{x}{\sqrt{2}}\right)}\right)+4 \tanh ^{-1}\left(\frac{x}{\sqrt{2}}\right) \log \left(e^{-2 \tanh ^{-1}\left(\frac{x}{\sqrt{2}}\right)}+1\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt{2}}\right) \log \left(\left(2 \sqrt{2}-3\right) e^{-2 \tanh ^{-1}\left(\frac{x}{\sqrt{2}}\right)}+1\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt{2}}\right) \log \left(1-\left(3+2 \sqrt{2}\right) e^{-2 \tanh ^{-1}\left(\frac{x}{\sqrt{2}}\right)}\right)-4 \sinh ^{-1}(1) \tanh ^{-1}(x)+2 \sinh ^{-1}(1) \log \left(\left(2 \sqrt{2}-3\right) e^{-2 \tanh ^{-1}\left(\frac{x}{\sqrt{2}}\right)}+1\right)-2 \sinh ^{-1}(1) \log \left(1-\left(3+2 \sqrt{2}\right) e^{-2 \tanh ^{-1}\left(\frac{x}{\sqrt{2}}\right)}\right)\right)","-\frac{1}{2} \text{Li}_2\left(1-\frac{2 \sqrt{2}}{x+\sqrt{2}}\right)+\frac{1}{4} \text{Li}_2\left(\frac{4 (1-x)}{\left(2-\sqrt{2}\right) \left(x+\sqrt{2}\right)}+1\right)+\frac{1}{4} \text{Li}_2\left(1-\frac{4 (x+1)}{\left(2+\sqrt{2}\right) \left(x+\sqrt{2}\right)}\right)+\log \left(\frac{2 \sqrt{2}}{x+\sqrt{2}}\right) \tanh ^{-1}\left(\frac{x}{\sqrt{2}}\right)-\frac{1}{2} \log \left(-\frac{4 (1-x)}{\left(2-\sqrt{2}\right) \left(x+\sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{x}{\sqrt{2}}\right)-\frac{1}{2} \log \left(\frac{4 (x+1)}{\left(2+\sqrt{2}\right) \left(x+\sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{x}{\sqrt{2}}\right)",1,"(-4*ArcSinh[1]*ArcTanh[x] + 4*ArcTanh[x/Sqrt[2]]*Log[1 + E^(-2*ArcTanh[x/Sqrt[2]])] + 2*ArcSinh[1]*Log[1 + (-3 + 2*Sqrt[2])/E^(2*ArcTanh[x/Sqrt[2]])] - 2*ArcTanh[x/Sqrt[2]]*Log[1 + (-3 + 2*Sqrt[2])/E^(2*ArcTanh[x/Sqrt[2]])] - 2*ArcSinh[1]*Log[1 - (3 + 2*Sqrt[2])/E^(2*ArcTanh[x/Sqrt[2]])] - 2*ArcTanh[x/Sqrt[2]]*Log[1 - (3 + 2*Sqrt[2])/E^(2*ArcTanh[x/Sqrt[2]])] - 2*PolyLog[2, -E^(-2*ArcTanh[x/Sqrt[2]])] + PolyLog[2, (3 - 2*Sqrt[2])/E^(2*ArcTanh[x/Sqrt[2]])] + PolyLog[2, (3 + 2*Sqrt[2])/E^(2*ArcTanh[x/Sqrt[2]])])/4","A",0
185,0,0,21,0.7838889,"\int \frac{x \left(1-a^2 x^2\right)}{\tanh ^{-1}(a x)} \, dx","Integrate[(x*(1 - a^2*x^2))/ArcTanh[a*x],x]","\int \frac{x \left(1-a^2 x^2\right)}{\tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x \left(1-a^2 x^2\right)}{\tanh ^{-1}(a x)},x\right)",0,"Integrate[(x*(1 - a^2*x^2))/ArcTanh[a*x], x]","A",-1
186,0,0,20,0.3378346,"\int \frac{1-a^2 x^2}{\tanh ^{-1}(a x)} \, dx","Integrate[(1 - a^2*x^2)/ArcTanh[a*x],x]","\int \frac{1-a^2 x^2}{\tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1-a^2 x^2}{\tanh ^{-1}(a x)},x\right)",0,"Integrate[(1 - a^2*x^2)/ArcTanh[a*x], x]","A",-1
187,0,0,23,1.145163,"\int \frac{1-a^2 x^2}{x \tanh ^{-1}(a x)} \, dx","Integrate[(1 - a^2*x^2)/(x*ArcTanh[a*x]),x]","\int \frac{1-a^2 x^2}{x \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1-a^2 x^2}{x \tanh ^{-1}(a x)},x\right)",0,"Integrate[(1 - a^2*x^2)/(x*ArcTanh[a*x]), x]","A",-1
188,0,0,21,1.0159703,"\int \frac{x \left(1-a^2 x^2\right)}{\tanh ^{-1}(a x)^2} \, dx","Integrate[(x*(1 - a^2*x^2))/ArcTanh[a*x]^2,x]","\int \frac{x \left(1-a^2 x^2\right)}{\tanh ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x \left(1-a^2 x^2\right)}{\tanh ^{-1}(a x)^2},x\right)",0,"Integrate[(x*(1 - a^2*x^2))/ArcTanh[a*x]^2, x]","A",-1
189,0,0,20,1.3965493,"\int \frac{1-a^2 x^2}{\tanh ^{-1}(a x)^2} \, dx","Integrate[(1 - a^2*x^2)/ArcTanh[a*x]^2,x]","\int \frac{1-a^2 x^2}{\tanh ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{1-a^2 x^2}{\tanh ^{-1}(a x)^2},x\right)",0,"Integrate[(1 - a^2*x^2)/ArcTanh[a*x]^2, x]","A",-1
190,0,0,23,1.18161,"\int \frac{1-a^2 x^2}{x \tanh ^{-1}(a x)^2} \, dx","Integrate[(1 - a^2*x^2)/(x*ArcTanh[a*x]^2),x]","\int \frac{1-a^2 x^2}{x \tanh ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{1-a^2 x^2}{x \tanh ^{-1}(a x)^2},x\right)",0,"Integrate[(1 - a^2*x^2)/(x*ArcTanh[a*x]^2), x]","A",-1
191,0,0,20,1.1638937,"\int \frac{1-a^2 x^2}{\tanh ^{-1}(a x)^3} \, dx","Integrate[(1 - a^2*x^2)/ArcTanh[a*x]^3,x]","\int \frac{1-a^2 x^2}{\tanh ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{1-a^2 x^2}{\tanh ^{-1}(a x)^3},x\right)",0,"Integrate[(1 - a^2*x^2)/ArcTanh[a*x]^3, x]","A",-1
192,1,96,96,0.0275356,"\int x^4 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x) \, dx","Integrate[x^4*(1 - a^2*x^2)^2*ArcTanh[a*x],x]","\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)+\frac{a^3 x^8}{72}+\frac{4 x^2}{315 a^3}-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)+\frac{4 \log \left(1-a^2 x^2\right)}{315 a^5}-\frac{11 a x^6}{378}+\frac{1}{5} x^5 \tanh ^{-1}(a x)+\frac{2 x^4}{315 a}","\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)+\frac{a^3 x^8}{72}+\frac{4 x^2}{315 a^3}-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)+\frac{4 \log \left(1-a^2 x^2\right)}{315 a^5}-\frac{11 a x^6}{378}+\frac{1}{5} x^5 \tanh ^{-1}(a x)+\frac{2 x^4}{315 a}",1,"(4*x^2)/(315*a^3) + (2*x^4)/(315*a) - (11*a*x^6)/378 + (a^3*x^8)/72 + (x^5*ArcTanh[a*x])/5 - (2*a^2*x^7*ArcTanh[a*x])/7 + (a^4*x^9*ArcTanh[a*x])/9 + (4*Log[1 - a^2*x^2])/(315*a^5)","A",1
193,1,103,87,0.0266272,"\int x^3 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x) \, dx","Integrate[x^3*(1 - a^2*x^2)^2*ArcTanh[a*x],x]","\frac{1}{8} a^4 x^8 \tanh ^{-1}(a x)+\frac{\log (1-a x)}{48 a^4}-\frac{\log (a x+1)}{48 a^4}+\frac{a^3 x^7}{56}+\frac{x}{24 a^3}-\frac{1}{3} a^2 x^6 \tanh ^{-1}(a x)-\frac{a x^5}{24}+\frac{1}{4} x^4 \tanh ^{-1}(a x)+\frac{x^3}{72 a}","\frac{1}{8} a^4 x^8 \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)}{24 a^4}+\frac{a^3 x^7}{56}+\frac{x}{24 a^3}-\frac{1}{3} a^2 x^6 \tanh ^{-1}(a x)-\frac{a x^5}{24}+\frac{1}{4} x^4 \tanh ^{-1}(a x)+\frac{x^3}{72 a}",1,"x/(24*a^3) + x^3/(72*a) - (a*x^5)/24 + (a^3*x^7)/56 + (x^4*ArcTanh[a*x])/4 - (a^2*x^6*ArcTanh[a*x])/3 + (a^4*x^8*ArcTanh[a*x])/8 + Log[1 - a*x]/(48*a^4) - Log[1 + a*x]/(48*a^4)","A",1
194,1,86,86,0.0217448,"\int x^2 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x) \, dx","Integrate[x^2*(1 - a^2*x^2)^2*ArcTanh[a*x],x]","\frac{1}{7} a^4 x^7 \tanh ^{-1}(a x)+\frac{a^3 x^6}{42}-\frac{2}{5} a^2 x^5 \tanh ^{-1}(a x)+\frac{4 \log \left(1-a^2 x^2\right)}{105 a^3}-\frac{9 a x^4}{140}+\frac{1}{3} x^3 \tanh ^{-1}(a x)+\frac{4 x^2}{105 a}","\frac{1}{7} a^4 x^7 \tanh ^{-1}(a x)+\frac{a^3 x^6}{42}-\frac{2}{5} a^2 x^5 \tanh ^{-1}(a x)+\frac{4 \log \left(1-a^2 x^2\right)}{105 a^3}-\frac{9 a x^4}{140}+\frac{1}{3} x^3 \tanh ^{-1}(a x)+\frac{4 x^2}{105 a}",1,"(4*x^2)/(105*a) - (9*a*x^4)/140 + (a^3*x^6)/42 + (x^3*ArcTanh[a*x])/3 - (2*a^2*x^5*ArcTanh[a*x])/5 + (a^4*x^7*ArcTanh[a*x])/7 + (4*Log[1 - a^2*x^2])/(105*a^3)","A",1
195,1,93,50,0.0226551,"\int x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x) \, dx","Integrate[x*(1 - a^2*x^2)^2*ArcTanh[a*x],x]","\frac{1}{6} a^4 x^6 \tanh ^{-1}(a x)+\frac{a^3 x^5}{30}-\frac{1}{2} a^2 x^4 \tanh ^{-1}(a x)+\frac{\log (1-a x)}{12 a^2}-\frac{\log (a x+1)}{12 a^2}-\frac{a x^3}{9}+\frac{1}{2} x^2 \tanh ^{-1}(a x)+\frac{x}{6 a}","\frac{a^3 x^5}{30}-\frac{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)}{6 a^2}-\frac{a x^3}{9}+\frac{x}{6 a}",1,"x/(6*a) - (a*x^3)/9 + (a^3*x^5)/30 + (x^2*ArcTanh[a*x])/2 - (a^2*x^4*ArcTanh[a*x])/2 + (a^4*x^6*ArcTanh[a*x])/6 + Log[1 - a*x]/(12*a^2) - Log[1 + a*x]/(12*a^2)","A",1
196,1,71,104,0.0170531,"\int \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x) \, dx","Integrate[(1 - a^2*x^2)^2*ArcTanh[a*x],x]","\frac{1}{5} a^4 x^5 \tanh ^{-1}(a x)+\frac{a^3 x^4}{20}-\frac{2}{3} a^2 x^3 \tanh ^{-1}(a x)+\frac{4 \log \left(1-a^2 x^2\right)}{15 a}-\frac{7 a x^2}{30}+x \tanh ^{-1}(a x)","\frac{\left(1-a^2 x^2\right)^2}{20 a}+\frac{2 \left(1-a^2 x^2\right)}{15 a}+\frac{4 \log \left(1-a^2 x^2\right)}{15 a}+\frac{1}{5} x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)+\frac{4}{15} x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)+\frac{8}{15} x \tanh ^{-1}(a x)",1,"(-7*a*x^2)/30 + (a^3*x^4)/20 + x*ArcTanh[a*x] - (2*a^2*x^3*ArcTanh[a*x])/3 + (a^4*x^5*ArcTanh[a*x])/5 + (4*Log[1 - a^2*x^2])/(15*a)","A",1
197,1,73,70,0.0717743,"\int \frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}{x} \, dx","Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x])/x,x]","\frac{1}{24} \left(6 a^4 x^4 \tanh ^{-1}(a x)+2 a^3 x^3-24 a^2 x^2 \tanh ^{-1}(a x)-12 \text{Li}_2(-a x)+12 \text{Li}_2(a x)-18 a x-9 \log (1-a x)+9 \log (a x+1)\right)","\frac{1}{4} a^4 x^4 \tanh ^{-1}(a x)+\frac{a^3 x^3}{12}-a^2 x^2 \tanh ^{-1}(a x)-\frac{\text{Li}_2(-a x)}{2}+\frac{\text{Li}_2(a x)}{2}-\frac{3 a x}{4}+\frac{3}{4} \tanh ^{-1}(a x)",1,"(-18*a*x + 2*a^3*x^3 - 24*a^2*x^2*ArcTanh[a*x] + 6*a^4*x^4*ArcTanh[a*x] - 9*Log[1 - a*x] + 9*Log[1 + a*x] - 12*PolyLog[2, -(a*x)] + 12*PolyLog[2, a*x])/24","A",0
198,1,64,64,0.0166955,"\int \frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}{x^2} \, dx","Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x])/x^2,x]","\frac{1}{3} a^4 x^3 \tanh ^{-1}(a x)+\frac{a^3 x^2}{6}-\frac{4}{3} a \log \left(1-a^2 x^2\right)-2 a^2 x \tanh ^{-1}(a x)+a \log (x)-\frac{\tanh ^{-1}(a x)}{x}","\frac{1}{3} a^4 x^3 \tanh ^{-1}(a x)+\frac{a^3 x^2}{6}-\frac{4}{3} a \log \left(1-a^2 x^2\right)-2 a^2 x \tanh ^{-1}(a x)+a \log (x)-\frac{\tanh ^{-1}(a x)}{x}",1,"(a^3*x^2)/6 - ArcTanh[a*x]/x - 2*a^2*x*ArcTanh[a*x] + (a^4*x^3*ArcTanh[a*x])/3 + a*Log[x] - (4*a*Log[1 - a^2*x^2])/3","A",1
199,1,62,62,0.0622381,"\int \frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}{x^3} \, dx","Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x])/x^3,x]","-\frac{-a^4 x^4 \tanh ^{-1}(a x)-a^3 x^3-2 a^2 x^2 \text{Li}_2(-a x)+2 a^2 x^2 \text{Li}_2(a x)+a x+\tanh ^{-1}(a x)}{2 x^2}","\frac{1}{2} a^4 x^2 \tanh ^{-1}(a x)+\frac{a^3 x}{2}+a^2 \text{Li}_2(-a x)-a^2 \text{Li}_2(a x)-\frac{\tanh ^{-1}(a x)}{2 x^2}-\frac{a}{2 x}",1,"-1/2*(a*x - a^3*x^3 + ArcTanh[a*x] - a^4*x^4*ArcTanh[a*x] - 2*a^2*x^2*PolyLog[2, -(a*x)] + 2*a^2*x^2*PolyLog[2, a*x])/x^2","A",1
200,1,68,68,0.0178244,"\int \frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}{x^4} \, dx","Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x])/x^4,x]","a^4 x \tanh ^{-1}(a x)-\frac{5}{3} a^3 \log (x)+\frac{2 a^2 \tanh ^{-1}(a x)}{x}+\frac{4}{3} a^3 \log \left(1-a^2 x^2\right)-\frac{\tanh ^{-1}(a x)}{3 x^3}-\frac{a}{6 x^2}","a^4 x \tanh ^{-1}(a x)-\frac{5}{3} a^3 \log (x)+\frac{2 a^2 \tanh ^{-1}(a x)}{x}+\frac{4}{3} a^3 \log \left(1-a^2 x^2\right)-\frac{\tanh ^{-1}(a x)}{3 x^3}-\frac{a}{6 x^2}",1,"-1/6*a/x^2 - ArcTanh[a*x]/(3*x^3) + (2*a^2*ArcTanh[a*x])/x + a^4*x*ArcTanh[a*x] - (5*a^3*Log[x])/3 + (4*a^3*Log[1 - a^2*x^2])/3","A",1
201,1,84,77,0.0808042,"\int \frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}{x^5} \, dx","Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x])/x^5,x]","\frac{1}{24} \left(-12 a^4 \text{Li}_2(-a x)+12 a^4 \text{Li}_2(a x)+9 a^4 \log (1-a x)-9 a^4 \log (a x+1)+\frac{18 a^3}{x}+\frac{24 a^2 \tanh ^{-1}(a x)}{x^2}-\frac{6 \tanh ^{-1}(a x)}{x^4}-\frac{2 a}{x^3}\right)","-\frac{1}{2} a^4 \text{Li}_2(-a x)+\frac{1}{2} a^4 \text{Li}_2(a x)-\frac{3}{4} a^4 \tanh ^{-1}(a x)+\frac{3 a^3}{4 x}+\frac{a^2 \tanh ^{-1}(a x)}{x^2}-\frac{\tanh ^{-1}(a x)}{4 x^4}-\frac{a}{12 x^3}",1,"((-2*a)/x^3 + (18*a^3)/x - (6*ArcTanh[a*x])/x^4 + (24*a^2*ArcTanh[a*x])/x^2 + 9*a^4*Log[1 - a*x] - 9*a^4*Log[1 + a*x] - 12*a^4*PolyLog[2, -(a*x)] + 12*a^4*PolyLog[2, a*x])/24","A",0
202,1,83,83,0.024565,"\int \frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}{x^6} \, dx","Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x])/x^6,x]","\frac{8}{15} a^5 \log (x)-\frac{a^4 \tanh ^{-1}(a x)}{x}+\frac{7 a^3}{30 x^2}+\frac{2 a^2 \tanh ^{-1}(a x)}{3 x^3}-\frac{4}{15} a^5 \log \left(1-a^2 x^2\right)-\frac{\tanh ^{-1}(a x)}{5 x^5}-\frac{a}{20 x^4}","\frac{8}{15} a^5 \log (x)-\frac{a^4 \tanh ^{-1}(a x)}{x}+\frac{7 a^3}{30 x^2}+\frac{2 a^2 \tanh ^{-1}(a x)}{3 x^3}-\frac{4}{15} a^5 \log \left(1-a^2 x^2\right)-\frac{\tanh ^{-1}(a x)}{5 x^5}-\frac{a}{20 x^4}",1,"-1/20*a/x^4 + (7*a^3)/(30*x^2) - ArcTanh[a*x]/(5*x^5) + (2*a^2*ArcTanh[a*x])/(3*x^3) - (a^4*ArcTanh[a*x])/x + (8*a^5*Log[x])/15 - (4*a^5*Log[1 - a^2*x^2])/15","A",1
203,1,138,202,1.9892289,"\int x^4 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2 \, dx","Integrate[x^4*(1 - a^2*x^2)^2*ArcTanh[a*x]^2,x]","\frac{36 \left(35 a^9 x^9-90 a^7 x^7+63 a^5 x^5-8\right) \tanh ^{-1}(a x)^2+a x \left(45 a^6 x^6-69 a^4 x^4-67 a^2 x^2+87\right)+3 \tanh ^{-1}(a x) \left(105 a^8 x^8-220 a^6 x^6+48 a^4 x^4+96 a^2 x^2-192 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)-29\right)+288 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)}{11340 a^5}","-\frac{8 \text{Li}_2\left(1-\frac{2}{1-a x}\right)}{315 a^5}+\frac{8 \tanh ^{-1}(a x)^2}{315 a^5}-\frac{29 \tanh ^{-1}(a x)}{3780 a^5}-\frac{16 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{315 a^5}+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2+\frac{29 x}{3780 a^4}+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac{8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac{a^2 x^7}{252}-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac{67 x^3}{11340 a^2}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2+\frac{4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac{23 x^5}{3780}",1,"(a*x*(87 - 67*a^2*x^2 - 69*a^4*x^4 + 45*a^6*x^6) + 36*(-8 + 63*a^5*x^5 - 90*a^7*x^7 + 35*a^9*x^9)*ArcTanh[a*x]^2 + 3*ArcTanh[a*x]*(-29 + 96*a^2*x^2 + 48*a^4*x^4 - 220*a^6*x^6 + 105*a^8*x^8 - 192*Log[1 + E^(-2*ArcTanh[a*x])]) + 288*PolyLog[2, -E^(-2*ArcTanh[a*x])])/(11340*a^5)","A",0
204,1,108,156,0.066516,"\int x^3 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2 \, dx","Integrate[x^3*(1 - a^2*x^2)^2*ArcTanh[a*x]^2,x]","\frac{3 a^6 x^6-6 a^4 x^4-5 a^2 x^2+16 \log \left(1-a^2 x^2\right)+21 \left(a^2 x^2-1\right)^3 \left(3 a^2 x^2+1\right) \tanh ^{-1}(a x)^2+2 a x \left(9 a^6 x^6-21 a^4 x^4+7 a^2 x^2+21\right) \tanh ^{-1}(a x)}{504 a^4}","\frac{1}{8} a^4 x^8 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{24 a^4}+\frac{1}{28} a^3 x^7 \tanh ^{-1}(a x)+\frac{x \tanh ^{-1}(a x)}{12 a^3}+\frac{a^2 x^6}{168}-\frac{1}{3} a^2 x^6 \tanh ^{-1}(a x)^2-\frac{5 x^2}{504 a^2}+\frac{2 \log \left(1-a^2 x^2\right)}{63 a^4}-\frac{1}{12} a x^5 \tanh ^{-1}(a x)+\frac{1}{4} x^4 \tanh ^{-1}(a x)^2+\frac{x^3 \tanh ^{-1}(a x)}{36 a}-\frac{x^4}{84}",1,"(-5*a^2*x^2 - 6*a^4*x^4 + 3*a^6*x^6 + 2*a*x*(21 + 7*a^2*x^2 - 21*a^4*x^4 + 9*a^6*x^6)*ArcTanh[a*x] + 21*(-1 + a^2*x^2)^3*(1 + 3*a^2*x^2)*ArcTanh[a*x]^2 + 16*Log[1 - a^2*x^2])/(504*a^4)","A",1
205,1,121,178,1.2556701,"\int x^2 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2 \, dx","Integrate[x^2*(1 - a^2*x^2)^2*ArcTanh[a*x]^2,x]","\frac{a x \left(6 a^4 x^4-17 a^2 x^2-3\right)+6 \left(15 a^7 x^7-42 a^5 x^5+35 a^3 x^3-8\right) \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \left(30 a^6 x^6-81 a^4 x^4+48 a^2 x^2-96 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)+3\right)+48 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)}{630 a^3}","\frac{1}{7} a^4 x^7 \tanh ^{-1}(a x)^2-\frac{8 \text{Li}_2\left(1-\frac{2}{1-a x}\right)}{105 a^3}+\frac{1}{21} a^3 x^6 \tanh ^{-1}(a x)+\frac{8 \tanh ^{-1}(a x)^2}{105 a^3}+\frac{\tanh ^{-1}(a x)}{210 a^3}-\frac{16 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{105 a^3}+\frac{a^2 x^5}{105}-\frac{2}{5} a^2 x^5 \tanh ^{-1}(a x)^2-\frac{x}{210 a^2}-\frac{9}{70} a x^4 \tanh ^{-1}(a x)+\frac{1}{3} x^3 \tanh ^{-1}(a x)^2+\frac{8 x^2 \tanh ^{-1}(a x)}{105 a}-\frac{17 x^3}{630}",1,"(a*x*(-3 - 17*a^2*x^2 + 6*a^4*x^4) + 6*(-8 + 35*a^3*x^3 - 42*a^5*x^5 + 15*a^7*x^7)*ArcTanh[a*x]^2 + ArcTanh[a*x]*(3 + 48*a^2*x^2 - 81*a^4*x^4 + 30*a^6*x^6 - 96*Log[1 + E^(-2*ArcTanh[a*x])]) + 48*PolyLog[2, -E^(-2*ArcTanh[a*x])])/(630*a^3)","A",0
206,1,82,138,0.0552318,"\int x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2 \, dx","Integrate[x*(1 - a^2*x^2)^2*ArcTanh[a*x]^2,x]","\frac{3 a^4 x^4-14 a^2 x^2+16 \log \left(1-a^2 x^2\right)+30 \left(a^2 x^2-1\right)^3 \tanh ^{-1}(a x)^2+4 a x \left(3 a^4 x^4-10 a^2 x^2+15\right) \tanh ^{-1}(a x)}{180 a^2}","\frac{\left(1-a^2 x^2\right)^2}{60 a^2}+\frac{2 \left(1-a^2 x^2\right)}{45 a^2}+\frac{4 \log \left(1-a^2 x^2\right)}{45 a^2}-\frac{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2}{6 a^2}+\frac{x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}{15 a}+\frac{4 x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}{45 a}+\frac{8 x \tanh ^{-1}(a x)}{45 a}",1,"(-14*a^2*x^2 + 3*a^4*x^4 + 4*a*x*(15 - 10*a^2*x^2 + 3*a^4*x^4)*ArcTanh[a*x] + 30*(-1 + a^2*x^2)^3*ArcTanh[a*x]^2 + 16*Log[1 - a^2*x^2])/(180*a^2)","A",1
207,1,99,171,0.6907831,"\int \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2 \, dx","Integrate[(1 - a^2*x^2)^2*ArcTanh[a*x]^2,x]","\frac{a x \left(a^2 x^2-11\right)+2 \left(3 a^2 x^2+9 a x+8\right) (a x-1)^3 \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \left(3 a^4 x^4-14 a^2 x^2-32 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)+11\right)+16 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)}{30 a}","\frac{a^2 x^3}{30}+\frac{1}{5} x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2+\frac{4}{15} x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2+\frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}{10 a}+\frac{4 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}{15 a}-\frac{8 \text{Li}_2\left(1-\frac{2}{1-a x}\right)}{15 a}+\frac{8}{15} x \tanh ^{-1}(a x)^2+\frac{8 \tanh ^{-1}(a x)^2}{15 a}-\frac{16 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{15 a}-\frac{11 x}{30}",1,"(a*x*(-11 + a^2*x^2) + 2*(-1 + a*x)^3*(8 + 9*a*x + 3*a^2*x^2)*ArcTanh[a*x]^2 + ArcTanh[a*x]*(11 - 14*a^2*x^2 + 3*a^4*x^4 - 32*Log[1 + E^(-2*ArcTanh[a*x])]) + 16*PolyLog[2, -E^(-2*ArcTanh[a*x])])/(30*a)","A",0
208,1,191,186,0.0645627,"\int \frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}{x} \, dx","Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x,x]","\frac{1}{4} \left(a^4 x^4-1\right) \tanh ^{-1}(a x)^2+\frac{a^2 x^2}{12}-\frac{2}{3} \log \left(1-a^2 x^2\right)+\frac{1}{6} a x \left(a^2 x^2+3\right) \tanh ^{-1}(a x)-\left(a^2 x^2-1\right) \tanh ^{-1}(a x)^2-\frac{1}{2} \text{Li}_3\left(\frac{-a x-1}{a x-1}\right)+\frac{1}{2} \text{Li}_3\left(\frac{a x+1}{a x-1}\right)+\text{Li}_2\left(\frac{-a x-1}{a x-1}\right) \tanh ^{-1}(a x)-\text{Li}_2\left(\frac{a x+1}{a x-1}\right) \tanh ^{-1}(a x)-2 a x \tanh ^{-1}(a x)+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1-a x}\right)","\frac{1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+\frac{1}{6} a^3 x^3 \tanh ^{-1}(a x)+\frac{a^2 x^2}{12}-\frac{2}{3} \log \left(1-a^2 x^2\right)-a^2 x^2 \tanh ^{-1}(a x)^2+\frac{1}{2} \text{Li}_3\left(1-\frac{2}{1-a x}\right)-\frac{1}{2} \text{Li}_3\left(\frac{2}{1-a x}-1\right)-\text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)+\text{Li}_2\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)-\frac{3}{2} a x \tanh ^{-1}(a x)+\frac{3}{4} \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1-a x}\right)",1,"(a^2*x^2)/12 - 2*a*x*ArcTanh[a*x] + (a*x*(3 + a^2*x^2)*ArcTanh[a*x])/6 - (-1 + a^2*x^2)*ArcTanh[a*x]^2 + ((-1 + a^4*x^4)*ArcTanh[a*x]^2)/4 + 2*ArcTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)] - (2*Log[1 - a^2*x^2])/3 + ArcTanh[a*x]*PolyLog[2, (-1 - a*x)/(-1 + a*x)] - ArcTanh[a*x]*PolyLog[2, (1 + a*x)/(-1 + a*x)] - PolyLog[3, (-1 - a*x)/(-1 + a*x)]/2 + PolyLog[3, (1 + a*x)/(-1 + a*x)]/2","A",1
209,1,182,156,0.057192,"\int \frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}{x^2} \, dx","Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^2,x]","\frac{1}{3} a \left(-\left(1-a^2 x^2\right) \left(a x \tanh ^{-1}(a x)+1\right) \tanh ^{-1}(a x)+\text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+a x+a x \tanh ^{-1}(a x)^2-\tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x) \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)\right)-2 a \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+a \left(\tanh ^{-1}(a x) \left(-\frac{\tanh ^{-1}(a x)}{a x}+\tanh ^{-1}(a x)+2 \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)\right)-\text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)\right)-2 a \tanh ^{-1}(a x) \left(a x \tanh ^{-1}(a x)-\tanh ^{-1}(a x)-2 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)\right)","\frac{1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+\frac{1}{3} a^3 x^2 \tanh ^{-1}(a x)+\frac{a^2 x}{3}-2 a^2 x \tanh ^{-1}(a x)^2+\frac{5}{3} a \text{Li}_2\left(1-\frac{2}{1-a x}\right)-a \text{Li}_2\left(\frac{2}{a x+1}-1\right)-\frac{2}{3} a \tanh ^{-1}(a x)^2-\frac{1}{3} a \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{10}{3} a \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)+2 a \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)",1,"-2*a*ArcTanh[a*x]*(-ArcTanh[a*x] + a*x*ArcTanh[a*x] - 2*Log[1 + E^(-2*ArcTanh[a*x])]) - 2*a*PolyLog[2, -E^(-2*ArcTanh[a*x])] + (a*(a*x - ArcTanh[a*x]^2 + a*x*ArcTanh[a*x]^2 - (1 - a^2*x^2)*ArcTanh[a*x]*(1 + a*x*ArcTanh[a*x]) - 2*ArcTanh[a*x]*Log[1 + E^(-2*ArcTanh[a*x])] + PolyLog[2, -E^(-2*ArcTanh[a*x])]))/3 + a*(ArcTanh[a*x]*(ArcTanh[a*x] - ArcTanh[a*x]/(a*x) + 2*Log[1 - E^(-2*ArcTanh[a*x])]) - PolyLog[2, E^(-2*ArcTanh[a*x])])","A",0
210,1,183,162,0.069684,"\int \frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}{x^3} \, dx","Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^3,x]","a^3 x \tanh ^{-1}(a x)+a^2 \text{Li}_3\left(\frac{-a x-1}{a x-1}\right)-a^2 \text{Li}_3\left(\frac{a x+1}{a x-1}\right)-2 a^2 \text{Li}_2\left(\frac{-a x-1}{a x-1}\right) \tanh ^{-1}(a x)+2 a^2 \text{Li}_2\left(\frac{a x+1}{a x-1}\right) \tanh ^{-1}(a x)+\frac{1}{2} a^2 \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^2+\frac{\left(a^2 x^2-1\right) \tanh ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)-4 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1-a x}\right)-\frac{a \tanh ^{-1}(a x)}{x}","\frac{1}{2} a^4 x^2 \tanh ^{-1}(a x)^2+a^3 x \tanh ^{-1}(a x)-a^2 \text{Li}_3\left(1-\frac{2}{1-a x}\right)+a^2 \text{Li}_3\left(\frac{2}{1-a x}-1\right)+2 a^2 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)-2 a^2 \text{Li}_2\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)+a^2 \log (x)-4 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1-a x}\right)-\frac{\tanh ^{-1}(a x)^2}{2 x^2}-\frac{a \tanh ^{-1}(a x)}{x}",1,"-((a*ArcTanh[a*x])/x) + a^3*x*ArcTanh[a*x] + (a^2*(-1 + a^2*x^2)*ArcTanh[a*x]^2)/2 + ((-1 + a^2*x^2)*ArcTanh[a*x]^2)/(2*x^2) - 4*a^2*ArcTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)] + a^2*Log[x] - 2*a^2*ArcTanh[a*x]*PolyLog[2, (-1 - a*x)/(-1 + a*x)] + 2*a^2*ArcTanh[a*x]*PolyLog[2, (1 + a*x)/(-1 + a*x)] + a^2*PolyLog[3, (-1 - a*x)/(-1 + a*x)] - a^2*PolyLog[3, (1 + a*x)/(-1 + a*x)]","A",1
211,1,153,167,0.0738534,"\int \frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}{x^4} \, dx","Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^4,x]","\frac{1}{3} \left(3 a^4 x \tanh ^{-1}(a x)^2+3 a^3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+5 a^3 \text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)-8 a^3 \tanh ^{-1}(a x)^2+a^3 \tanh ^{-1}(a x)-10 a^3 \tanh ^{-1}(a x) \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)-6 a^3 \tanh ^{-1}(a x) \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)-\frac{a^2}{x}+\frac{6 a^2 \tanh ^{-1}(a x)^2}{x}-\frac{\tanh ^{-1}(a x)^2}{x^3}-\frac{a \tanh ^{-1}(a x)}{x^2}\right)","a^4 x \tanh ^{-1}(a x)^2-a^3 \text{Li}_2\left(1-\frac{2}{1-a x}\right)+\frac{5}{3} a^3 \text{Li}_2\left(\frac{2}{a x+1}-1\right)-\frac{2}{3} a^3 \tanh ^{-1}(a x)^2+\frac{1}{3} a^3 \tanh ^{-1}(a x)-2 a^3 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)-\frac{10}{3} a^3 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)-\frac{a^2}{3 x}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{x}-\frac{\tanh ^{-1}(a x)^2}{3 x^3}-\frac{a \tanh ^{-1}(a x)}{3 x^2}",1,"(-(a^2/x) + a^3*ArcTanh[a*x] - (a*ArcTanh[a*x])/x^2 - 8*a^3*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/x^3 + (6*a^2*ArcTanh[a*x]^2)/x + 3*a^4*x*ArcTanh[a*x]^2 - 10*a^3*ArcTanh[a*x]*Log[1 - E^(-2*ArcTanh[a*x])] - 6*a^3*ArcTanh[a*x]*Log[1 + E^(-2*ArcTanh[a*x])] + 3*a^3*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 5*a^3*PolyLog[2, E^(-2*ArcTanh[a*x])])/3","A",0
212,1,238,214,0.3674925,"\int \frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}{x^5} \, dx","Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^5,x]","\frac{1}{24} \left(24 a^4 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+24 a^4 \tanh ^{-1}(a x) \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)+12 a^4 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)-12 a^4 \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)-16 a^4 \tanh ^{-1}(a x)^3-18 a^4 \tanh ^{-1}(a x)^2-24 a^4 \tanh ^{-1}(a x)^2 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)+24 a^4 \tanh ^{-1}(a x)^2 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)+i \pi ^3 a^4+2 a^4+\frac{36 a^3 \tanh ^{-1}(a x)}{x}-\frac{2 a^2}{x^2}+\frac{24 a^2 \tanh ^{-1}(a x)^2}{x^2}-32 a^4 \log \left(\frac{a x}{\sqrt{1-a^2 x^2}}\right)-\frac{6 \tanh ^{-1}(a x)^2}{x^4}-\frac{4 a \tanh ^{-1}(a x)}{x^3}\right)","\frac{1}{2} a^4 \text{Li}_3\left(1-\frac{2}{1-a x}\right)-\frac{1}{2} a^4 \text{Li}_3\left(\frac{2}{1-a x}-1\right)-a^4 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)+a^4 \text{Li}_2\left(\frac{2}{1-a x}-1\right) \tanh ^{-1}(a x)-\frac{4}{3} a^4 \log (x)-\frac{3}{4} a^4 \tanh ^{-1}(a x)^2+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1-a x}\right)+\frac{3 a^3 \tanh ^{-1}(a x)}{2 x}-\frac{a^2}{12 x^2}+\frac{a^2 \tanh ^{-1}(a x)^2}{x^2}+\frac{2}{3} a^4 \log \left(1-a^2 x^2\right)-\frac{\tanh ^{-1}(a x)^2}{4 x^4}-\frac{a \tanh ^{-1}(a x)}{6 x^3}",1,"(2*a^4 + I*a^4*Pi^3 - (2*a^2)/x^2 - (4*a*ArcTanh[a*x])/x^3 + (36*a^3*ArcTanh[a*x])/x - 18*a^4*ArcTanh[a*x]^2 - (6*ArcTanh[a*x]^2)/x^4 + (24*a^2*ArcTanh[a*x]^2)/x^2 - 16*a^4*ArcTanh[a*x]^3 - 24*a^4*ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] + 24*a^4*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] - 32*a^4*Log[(a*x)/Sqrt[1 - a^2*x^2]] + 24*a^4*ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 24*a^4*ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] + 12*a^4*PolyLog[3, -E^(-2*ArcTanh[a*x])] - 12*a^4*PolyLog[3, E^(2*ArcTanh[a*x])])/24","C",0
213,1,118,157,0.8166661,"\int \frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}{x^6} \, dx","Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^6,x]","\frac{-16 a^5 x^5 \text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)+a^2 x^2 \left(11 a^2 x^2-1\right)+2 (a x-1)^3 \left(8 a^2 x^2+9 a x+3\right) \tanh ^{-1}(a x)^2+a x \tanh ^{-1}(a x) \left(-11 a^4 x^4+32 a^4 x^4 \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)+14 a^2 x^2-3\right)}{30 x^5}","-\frac{8}{15} a^5 \text{Li}_2\left(\frac{2}{a x+1}-1\right)+\frac{8}{15} a^5 \tanh ^{-1}(a x)^2-\frac{11}{30} a^5 \tanh ^{-1}(a x)+\frac{16}{15} a^5 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)+\frac{11 a^4}{30 x}-\frac{a^4 \tanh ^{-1}(a x)^2}{x}+\frac{7 a^3 \tanh ^{-1}(a x)}{15 x^2}-\frac{a^2}{30 x^3}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{3 x^3}-\frac{\tanh ^{-1}(a x)^2}{5 x^5}-\frac{a \tanh ^{-1}(a x)}{10 x^4}",1,"(a^2*x^2*(-1 + 11*a^2*x^2) + 2*(-1 + a*x)^3*(3 + 9*a*x + 8*a^2*x^2)*ArcTanh[a*x]^2 + a*x*ArcTanh[a*x]*(-3 + 14*a^2*x^2 - 11*a^4*x^4 + 32*a^4*x^4*Log[1 - E^(-2*ArcTanh[a*x])]) - 16*a^5*x^5*PolyLog[2, E^(-2*ArcTanh[a*x])])/(30*x^5)","A",0
214,1,99,113,0.0611235,"\int \frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}{x^7} \, dx","Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^7,x]","\frac{30 \left(a^2 x^2-1\right)^3 \tanh ^{-1}(a x)^2+a^2 x^2 \left(32 a^4 x^4 \log (x)+14 a^2 x^2-16 a^4 x^4 \log \left(1-a^2 x^2\right)-3\right)-4 a x \left(15 a^4 x^4-10 a^2 x^2+3\right) \tanh ^{-1}(a x)}{180 x^6}","\frac{8}{45} a^6 \log (x)-\frac{a^5 \tanh ^{-1}(a x)}{3 x}+\frac{7 a^4}{90 x^2}+\frac{2 a^3 \tanh ^{-1}(a x)}{9 x^3}-\frac{a^2}{60 x^4}-\frac{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2}{6 x^6}-\frac{4}{45} a^6 \log \left(1-a^2 x^2\right)-\frac{a \tanh ^{-1}(a x)}{15 x^5}",1,"(-4*a*x*(3 - 10*a^2*x^2 + 15*a^4*x^4)*ArcTanh[a*x] + 30*(-1 + a^2*x^2)^3*ArcTanh[a*x]^2 + a^2*x^2*(-3 + 14*a^2*x^2 + 32*a^4*x^4*Log[x] - 16*a^4*x^4*Log[1 - a^2*x^2]))/(180*x^6)","A",1
215,1,140,183,1.4571092,"\int \frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}{x^8} \, dx","Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^8,x]","\frac{-48 a^7 x^7 \text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)+a^2 x^2 \left(3 a^4 x^4+17 a^2 x^2-6\right)+6 \left(8 a^7 x^7-35 a^4 x^4+42 a^2 x^2-15\right) \tanh ^{-1}(a x)^2+3 a x \tanh ^{-1}(a x) \left(-a^6 x^6+32 a^6 x^6 \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)-16 a^4 x^4+27 a^2 x^2-10\right)}{630 x^7}","-\frac{8}{105} a^7 \text{Li}_2\left(\frac{2}{a x+1}-1\right)+\frac{8}{105} a^7 \tanh ^{-1}(a x)^2-\frac{1}{210} a^7 \tanh ^{-1}(a x)+\frac{16}{105} a^7 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)+\frac{a^6}{210 x}-\frac{8 a^5 \tanh ^{-1}(a x)}{105 x^2}+\frac{17 a^4}{630 x^3}-\frac{a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac{9 a^3 \tanh ^{-1}(a x)}{70 x^4}-\frac{a^2}{105 x^5}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac{\tanh ^{-1}(a x)^2}{7 x^7}-\frac{a \tanh ^{-1}(a x)}{21 x^6}",1,"(a^2*x^2*(-6 + 17*a^2*x^2 + 3*a^4*x^4) + 6*(-15 + 42*a^2*x^2 - 35*a^4*x^4 + 8*a^7*x^7)*ArcTanh[a*x]^2 + 3*a*x*ArcTanh[a*x]*(-10 + 27*a^2*x^2 - 16*a^4*x^4 - a^6*x^6 + 32*a^6*x^6*Log[1 - E^(-2*ArcTanh[a*x])]) - 48*a^7*x^7*PolyLog[2, E^(-2*ArcTanh[a*x])])/(630*x^7)","A",0
216,1,124,170,0.0668028,"\int \frac{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}{x^9} \, dx","Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^9,x]","\frac{21 \left(a^2 x^2+3\right) \left(a^2 x^2-1\right)^3 \tanh ^{-1}(a x)^2+a^2 x^2 \left(32 a^6 x^6 \log (x)+5 a^4 x^4+6 a^2 x^2-16 a^6 x^6 \log \left(1-a^2 x^2\right)-3\right)-2 a x \left(21 a^6 x^6+7 a^4 x^4-21 a^2 x^2+9\right) \tanh ^{-1}(a x)}{504 x^8}","\frac{4}{63} a^8 \log (x)+\frac{1}{24} a^8 \tanh ^{-1}(a x)^2-\frac{a^7 \tanh ^{-1}(a x)}{12 x}+\frac{5 a^6}{504 x^2}-\frac{a^5 \tanh ^{-1}(a x)}{36 x^3}+\frac{a^4}{84 x^4}-\frac{a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac{a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac{a^2}{168 x^6}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac{2}{63} a^8 \log \left(1-a^2 x^2\right)-\frac{\tanh ^{-1}(a x)^2}{8 x^8}-\frac{a \tanh ^{-1}(a x)}{28 x^7}",1,"(-2*a*x*(9 - 21*a^2*x^2 + 7*a^4*x^4 + 21*a^6*x^6)*ArcTanh[a*x] + 21*(-1 + a^2*x^2)^3*(3 + a^2*x^2)*ArcTanh[a*x]^2 + a^2*x^2*(-3 + 6*a^2*x^2 + 5*a^4*x^4 + 32*a^6*x^6*Log[x] - 16*a^6*x^6*Log[1 - a^2*x^2]))/(504*x^8)","A",1
217,1,183,248,0.5993754,"\int \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^3 \, dx","Integrate[(1 - a^2*x^2)^2*ArcTanh[a*x]^3,x]","\frac{12 a^5 x^5 \tanh ^{-1}(a x)^3+9 a^4 x^4 \tanh ^{-1}(a x)^2-40 a^3 x^3 \tanh ^{-1}(a x)^3+6 a^3 x^3 \tanh ^{-1}(a x)+3 a^2 x^2-30 \log \left(1-a^2 x^2\right)-42 a^2 x^2 \tanh ^{-1}(a x)^2+96 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+48 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)+60 a x \tanh ^{-1}(a x)^3-66 a x \tanh ^{-1}(a x)-32 \tanh ^{-1}(a x)^3+33 \tanh ^{-1}(a x)^2-96 \tanh ^{-1}(a x)^2 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)-3}{60 a}","-\frac{1-a^2 x^2}{20 a}-\frac{\log \left(1-a^2 x^2\right)}{2 a}+\frac{1}{5} x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^3+\frac{4}{15} x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3+\frac{3 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}{20 a}+\frac{2 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}{5 a}-\frac{1}{10} x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)+\frac{4 \text{Li}_3\left(1-\frac{2}{1-a x}\right)}{5 a}-\frac{8 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{5 a}+\frac{8}{15} x \tanh ^{-1}(a x)^3+\frac{8 \tanh ^{-1}(a x)^3}{15 a}-x \tanh ^{-1}(a x)-\frac{8 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{5 a}",1,"(-3 + 3*a^2*x^2 - 66*a*x*ArcTanh[a*x] + 6*a^3*x^3*ArcTanh[a*x] + 33*ArcTanh[a*x]^2 - 42*a^2*x^2*ArcTanh[a*x]^2 + 9*a^4*x^4*ArcTanh[a*x]^2 - 32*ArcTanh[a*x]^3 + 60*a*x*ArcTanh[a*x]^3 - 40*a^3*x^3*ArcTanh[a*x]^3 + 12*a^5*x^5*ArcTanh[a*x]^3 - 96*ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] - 30*Log[1 - a^2*x^2] + 96*ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 48*PolyLog[3, -E^(-2*ArcTanh[a*x])])/(60*a)","A",0
218,0,0,23,0.951754,"\int \frac{x \left(1-a^2 x^2\right)^2}{\tanh ^{-1}(a x)} \, dx","Integrate[(x*(1 - a^2*x^2)^2)/ArcTanh[a*x],x]","\int \frac{x \left(1-a^2 x^2\right)^2}{\tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x \left(1-a^2 x^2\right)^2}{\tanh ^{-1}(a x)},x\right)",0,"Integrate[(x*(1 - a^2*x^2)^2)/ArcTanh[a*x], x]","A",-1
219,0,0,22,0.492581,"\int \frac{\left(1-a^2 x^2\right)^2}{\tanh ^{-1}(a x)} \, dx","Integrate[(1 - a^2*x^2)^2/ArcTanh[a*x],x]","\int \frac{\left(1-a^2 x^2\right)^2}{\tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{\left(1-a^2 x^2\right)^2}{\tanh ^{-1}(a x)},x\right)",0,"Integrate[(1 - a^2*x^2)^2/ArcTanh[a*x], x]","A",-1
220,0,0,25,1.4624319,"\int \frac{\left(1-a^2 x^2\right)^2}{x \tanh ^{-1}(a x)} \, dx","Integrate[(1 - a^2*x^2)^2/(x*ArcTanh[a*x]),x]","\int \frac{\left(1-a^2 x^2\right)^2}{x \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{\left(1-a^2 x^2\right)^2}{x \tanh ^{-1}(a x)},x\right)",0,"Integrate[(1 - a^2*x^2)^2/(x*ArcTanh[a*x]), x]","A",-1
221,0,0,23,0.7781462,"\int \frac{x \left(1-a^2 x^2\right)^2}{\tanh ^{-1}(a x)^2} \, dx","Integrate[(x*(1 - a^2*x^2)^2)/ArcTanh[a*x]^2,x]","\int \frac{x \left(1-a^2 x^2\right)^2}{\tanh ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x \left(1-a^2 x^2\right)^2}{\tanh ^{-1}(a x)^2},x\right)",0,"Integrate[(x*(1 - a^2*x^2)^2)/ArcTanh[a*x]^2, x]","A",-1
222,0,0,22,1.1494591,"\int \frac{\left(1-a^2 x^2\right)^2}{\tanh ^{-1}(a x)^2} \, dx","Integrate[(1 - a^2*x^2)^2/ArcTanh[a*x]^2,x]","\int \frac{\left(1-a^2 x^2\right)^2}{\tanh ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{\left(1-a^2 x^2\right)^2}{\tanh ^{-1}(a x)^2},x\right)",0,"Integrate[(1 - a^2*x^2)^2/ArcTanh[a*x]^2, x]","A",-1
223,0,0,25,1.0119014,"\int \frac{\left(1-a^2 x^2\right)^2}{x \tanh ^{-1}(a x)^2} \, dx","Integrate[(1 - a^2*x^2)^2/(x*ArcTanh[a*x]^2),x]","\int \frac{\left(1-a^2 x^2\right)^2}{x \tanh ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{\left(1-a^2 x^2\right)^2}{x \tanh ^{-1}(a x)^2},x\right)",0,"Integrate[(1 - a^2*x^2)^2/(x*ArcTanh[a*x]^2), x]","A",-1
224,1,79,144,0.0445162,"\int \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x) \, dx","Integrate[(1 - a^2*x^2)^3*ArcTanh[a*x],x]","\frac{-5 a^6 x^6+24 a^4 x^4-57 a^2 x^2+48 \log \left(1-a^2 x^2\right)-6 a x \left(5 a^6 x^6-21 a^4 x^4+35 a^2 x^2-35\right) \tanh ^{-1}(a x)}{210 a}","\frac{\left(1-a^2 x^2\right)^3}{42 a}+\frac{3 \left(1-a^2 x^2\right)^2}{70 a}+\frac{4 \left(1-a^2 x^2\right)}{35 a}+\frac{8 \log \left(1-a^2 x^2\right)}{35 a}+\frac{1}{7} x \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)+\frac{6}{35} x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)+\frac{8}{35} x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)+\frac{16}{35} x \tanh ^{-1}(a x)",1,"(-57*a^2*x^2 + 24*a^4*x^4 - 5*a^6*x^6 - 6*a*x*(-35 + 35*a^2*x^2 - 21*a^4*x^4 + 5*a^6*x^6)*ArcTanh[a*x] + 48*Log[1 - a^2*x^2])/(210*a)","A",1
225,1,124,227,1.2231779,"\int \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2 \, dx","Integrate[(1 - a^2*x^2)^3*ArcTanh[a*x]^2,x]","-\frac{3 a^5 x^5-19 a^3 x^3+9 (a x-1)^4 \left(5 a^3 x^3+20 a^2 x^2+29 a x+16\right) \tanh ^{-1}(a x)^2+3 \tanh ^{-1}(a x) \left(5 a^6 x^6-24 a^4 x^4+57 a^2 x^2+96 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)-38\right)-144 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+114 a x}{315 a}","-\frac{1}{105} a^4 x^5+\frac{19 a^2 x^3}{315}+\frac{1}{7} x \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2+\frac{6}{35} x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2+\frac{8}{35} x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2+\frac{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)}{21 a}+\frac{3 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}{35 a}+\frac{8 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}{35 a}-\frac{16 \text{Li}_2\left(1-\frac{2}{1-a x}\right)}{35 a}+\frac{16}{35} x \tanh ^{-1}(a x)^2+\frac{16 \tanh ^{-1}(a x)^2}{35 a}-\frac{32 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{35 a}-\frac{38 x}{105}",1,"-1/315*(114*a*x - 19*a^3*x^3 + 3*a^5*x^5 + 9*(-1 + a*x)^4*(16 + 29*a*x + 20*a^2*x^2 + 5*a^3*x^3)*ArcTanh[a*x]^2 + 3*ArcTanh[a*x]*(-38 + 57*a^2*x^2 - 24*a^4*x^4 + 5*a^6*x^6 + 96*Log[1 + E^(-2*ArcTanh[a*x])]) - 144*PolyLog[2, -E^(-2*ArcTanh[a*x])])/a","A",0
226,1,231,338,1.0997654,"\int \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^3 \, dx","Integrate[(1 - a^2*x^2)^3*ArcTanh[a*x]^3,x]","-\frac{60 a^7 x^7 \tanh ^{-1}(a x)^3+30 a^6 x^6 \tanh ^{-1}(a x)^2-252 a^5 x^5 \tanh ^{-1}(a x)^3+12 a^5 x^5 \tanh ^{-1}(a x)+3 a^4 x^4-144 a^4 x^4 \tanh ^{-1}(a x)^2+420 a^3 x^3 \tanh ^{-1}(a x)^3-76 a^3 x^3 \tanh ^{-1}(a x)-32 a^2 x^2+196 \log \left(1-a^2 x^2\right)+342 a^2 x^2 \tanh ^{-1}(a x)^2-576 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)-288 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)-420 a x \tanh ^{-1}(a x)^3+456 a x \tanh ^{-1}(a x)+192 \tanh ^{-1}(a x)^3-228 \tanh ^{-1}(a x)^2+576 \tanh ^{-1}(a x)^2 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)+29}{420 a}","-\frac{\left(1-a^2 x^2\right)^2}{140 a}-\frac{13 \left(1-a^2 x^2\right)}{210 a}-\frac{7 \log \left(1-a^2 x^2\right)}{15 a}+\frac{1}{7} x \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^3+\frac{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2}{14 a}+\frac{6}{35} x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^3+\frac{9 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}{70 a}-\frac{1}{35} x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)+\frac{8}{35} x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3+\frac{12 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}{35 a}-\frac{13}{105} x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)+\frac{24 \text{Li}_3\left(1-\frac{2}{1-a x}\right)}{35 a}-\frac{48 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{35 a}+\frac{16}{35} x \tanh ^{-1}(a x)^3+\frac{16 \tanh ^{-1}(a x)^3}{35 a}-\frac{14}{15} x \tanh ^{-1}(a x)-\frac{48 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{35 a}",1,"-1/420*(29 - 32*a^2*x^2 + 3*a^4*x^4 + 456*a*x*ArcTanh[a*x] - 76*a^3*x^3*ArcTanh[a*x] + 12*a^5*x^5*ArcTanh[a*x] - 228*ArcTanh[a*x]^2 + 342*a^2*x^2*ArcTanh[a*x]^2 - 144*a^4*x^4*ArcTanh[a*x]^2 + 30*a^6*x^6*ArcTanh[a*x]^2 + 192*ArcTanh[a*x]^3 - 420*a*x*ArcTanh[a*x]^3 + 420*a^3*x^3*ArcTanh[a*x]^3 - 252*a^5*x^5*ArcTanh[a*x]^3 + 60*a^7*x^7*ArcTanh[a*x]^3 + 576*ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] + 196*Log[1 - a^2*x^2] - 576*ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] - 288*PolyLog[3, -E^(-2*ArcTanh[a*x])])/a","A",0
227,1,60,87,0.1395029,"\int \frac{x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx","Integrate[(x^3*ArcTanh[a*x])/(1 - a^2*x^2),x]","\frac{\tanh ^{-1}(a x) \left(-a^2 x^2+2 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)+1\right)-\text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)-a x+\tanh ^{-1}(a x)^2}{2 a^4}","\frac{\text{Li}_2\left(1-\frac{2}{1-a x}\right)}{2 a^4}-\frac{\tanh ^{-1}(a x)^2}{2 a^4}+\frac{\tanh ^{-1}(a x)}{2 a^4}+\frac{\log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^4}-\frac{x}{2 a^3}-\frac{x^2 \tanh ^{-1}(a x)}{2 a^2}",1,"(-(a*x) + ArcTanh[a*x]^2 + ArcTanh[a*x]*(1 - a^2*x^2 + 2*Log[1 + E^(-2*ArcTanh[a*x])]) - PolyLog[2, -E^(-2*ArcTanh[a*x])])/(2*a^4)","A",0
228,1,42,42,0.0379009,"\int \frac{x^2 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx","Integrate[(x^2*ArcTanh[a*x])/(1 - a^2*x^2),x]","\frac{\tanh ^{-1}(a x)^2}{2 a^3}-\frac{x \tanh ^{-1}(a x)}{a^2}-\frac{\log \left(1-a^2 x^2\right)}{2 a^3}","\frac{\tanh ^{-1}(a x)^2}{2 a^3}-\frac{x \tanh ^{-1}(a x)}{a^2}-\frac{\log \left(1-a^2 x^2\right)}{2 a^3}",1,"-((x*ArcTanh[a*x])/a^2) + ArcTanh[a*x]^2/(2*a^3) - Log[1 - a^2*x^2]/(2*a^3)","A",1
229,1,44,54,0.0514552,"\int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx","Integrate[(x*ArcTanh[a*x])/(1 - a^2*x^2),x]","-\frac{\text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)-\tanh ^{-1}(a x) \left(\tanh ^{-1}(a x)+2 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)\right)}{2 a^2}","\frac{\text{Li}_2\left(1-\frac{2}{1-a x}\right)}{2 a^2}-\frac{\tanh ^{-1}(a x)^2}{2 a^2}+\frac{\log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^2}",1,"-1/2*(-(ArcTanh[a*x]*(ArcTanh[a*x] + 2*Log[1 + E^(-2*ArcTanh[a*x])])) + PolyLog[2, -E^(-2*ArcTanh[a*x])])/a^2","A",0
230,1,13,13,0.0042636,"\int \frac{\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx","Integrate[ArcTanh[a*x]/(1 - a^2*x^2),x]","\frac{\tanh ^{-1}(a x)^2}{2 a}","\frac{\tanh ^{-1}(a x)^2}{2 a}",1,"ArcTanh[a*x]^2/(2*a)","A",1
231,1,42,45,0.0669562,"\int \frac{\tanh ^{-1}(a x)}{x \left(1-a^2 x^2\right)} \, dx","Integrate[ArcTanh[a*x]/(x*(1 - a^2*x^2)),x]","\frac{1}{2} \left(\tanh ^{-1}(a x) \left(\tanh ^{-1}(a x)+2 \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)\right)-\text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)\right)","-\frac{1}{2} \text{Li}_2\left(\frac{2}{a x+1}-1\right)+\frac{1}{2} \tanh ^{-1}(a x)^2+\log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)",1,"(ArcTanh[a*x]*(ArcTanh[a*x] + 2*Log[1 - E^(-2*ArcTanh[a*x])]) - PolyLog[2, E^(-2*ArcTanh[a*x])])/2","A",0
232,1,41,41,0.0401311,"\int \frac{\tanh ^{-1}(a x)}{x^2 \left(1-a^2 x^2\right)} \, dx","Integrate[ArcTanh[a*x]/(x^2*(1 - a^2*x^2)),x]","-\frac{1}{2} a \log \left(1-a^2 x^2\right)+a \log (x)+\frac{1}{2} a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)}{x}","-\frac{1}{2} a \log \left(1-a^2 x^2\right)+a \log (x)+\frac{1}{2} a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)}{x}",1,"-(ArcTanh[a*x]/x) + (a*ArcTanh[a*x]^2)/2 + a*Log[x] - (a*Log[1 - a^2*x^2])/2","A",1
233,1,60,84,0.2689603,"\int \frac{\tanh ^{-1}(a x)}{x^3 \left(1-a^2 x^2\right)} \, dx","Integrate[ArcTanh[a*x]/(x^3*(1 - a^2*x^2)),x]","-\frac{1}{2} a^2 \left(-\tanh ^{-1}(a x) \left(-\frac{1}{a^2 x^2}+\tanh ^{-1}(a x)+2 \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)+1\right)+\text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)+\frac{1}{a x}\right)","-\frac{1}{2} a^2 \text{Li}_2\left(\frac{2}{a x+1}-1\right)+\frac{1}{2} a^2 \tanh ^{-1}(a x)^2+\frac{1}{2} a^2 \tanh ^{-1}(a x)+a^2 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)}{2 x^2}-\frac{a}{2 x}",1,"-1/2*(a^2*(1/(a*x) - ArcTanh[a*x]*(1 - 1/(a^2*x^2) + ArcTanh[a*x] + 2*Log[1 - E^(-2*ArcTanh[a*x])]) + PolyLog[2, E^(-2*ArcTanh[a*x])]))","A",0
234,1,112,135,0.1193621,"\int \frac{x^3 \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx","Integrate[(x^3*ArcTanh[a*x]^2)/(1 - a^2*x^2),x]","-\frac{-\log \left(\frac{1}{\sqrt{1-a^2 x^2}}\right)-\frac{1}{2} \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)-\frac{1}{3} \tanh ^{-1}(a x)^3+a x \tanh ^{-1}(a x)-\tanh ^{-1}(a x)^2 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)}{a^4}","-\frac{\text{Li}_3\left(1-\frac{2}{1-a x}\right)}{2 a^4}+\frac{\text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^4}-\frac{\tanh ^{-1}(a x)^3}{3 a^4}+\frac{\tanh ^{-1}(a x)^2}{2 a^4}+\frac{\log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{a^4}-\frac{x \tanh ^{-1}(a x)}{a^3}-\frac{x^2 \tanh ^{-1}(a x)^2}{2 a^2}-\frac{\log \left(1-a^2 x^2\right)}{2 a^4}",1,"-((a*x*ArcTanh[a*x] - ((1 - a^2*x^2)*ArcTanh[a*x]^2)/2 - ArcTanh[a*x]^3/3 - ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] - Log[1/Sqrt[1 - a^2*x^2]] + ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] + PolyLog[3, -E^(-2*ArcTanh[a*x])]/2)/a^4)","A",0
235,1,59,75,0.1849203,"\int \frac{x^2 \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx","Integrate[(x^2*ArcTanh[a*x]^2)/(1 - a^2*x^2),x]","-\frac{\text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)-\frac{1}{3} \tanh ^{-1}(a x) \left(-3 a x \tanh ^{-1}(a x)+\left(\tanh ^{-1}(a x)+3\right) \tanh ^{-1}(a x)+6 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)\right)}{a^3}","\frac{\text{Li}_2\left(1-\frac{2}{1-a x}\right)}{a^3}+\frac{\tanh ^{-1}(a x)^3}{3 a^3}-\frac{\tanh ^{-1}(a x)^2}{a^3}+\frac{2 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^3}-\frac{x \tanh ^{-1}(a x)^2}{a^2}",1,"-((-1/3*(ArcTanh[a*x]*(-3*a*x*ArcTanh[a*x] + ArcTanh[a*x]*(3 + ArcTanh[a*x]) + 6*Log[1 + E^(-2*ArcTanh[a*x])])) + PolyLog[2, -E^(-2*ArcTanh[a*x])])/a^3)","A",0
236,1,68,78,0.0705559,"\int \frac{x \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx","Integrate[(x*ArcTanh[a*x]^2)/(1 - a^2*x^2),x]","-\frac{\tanh ^{-1}(a x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)-\frac{1}{3} \tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^2 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)}{a^2}","-\frac{\text{Li}_3\left(1-\frac{2}{1-a x}\right)}{2 a^2}+\frac{\text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^2}-\frac{\tanh ^{-1}(a x)^3}{3 a^2}+\frac{\log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{a^2}",1,"-((-1/3*ArcTanh[a*x]^3 - ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] + ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] + PolyLog[3, -E^(-2*ArcTanh[a*x])]/2)/a^2)","A",0
237,1,13,13,0.0054362,"\int \frac{\tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx","Integrate[ArcTanh[a*x]^2/(1 - a^2*x^2),x]","\frac{\tanh ^{-1}(a x)^3}{3 a}","\frac{\tanh ^{-1}(a x)^3}{3 a}",1,"ArcTanh[a*x]^3/(3*a)","A",1
238,1,60,66,0.0673196,"\int \frac{\tanh ^{-1}(a x)^2}{x \left(1-a^2 x^2\right)} \, dx","Integrate[ArcTanh[a*x]^2/(x*(1 - a^2*x^2)),x]","\tanh ^{-1}(a x) \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)-\frac{1}{3} \tanh ^{-1}(a x)^3+\tanh ^{-1}(a x)^2 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)","-\frac{1}{2} \text{Li}_3\left(\frac{2}{a x+1}-1\right)-\text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)+\frac{1}{3} \tanh ^{-1}(a x)^3+\log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^2",1,"-1/3*ArcTanh[a*x]^3 + ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] + ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] - PolyLog[3, E^(2*ArcTanh[a*x])]/2","A",0
239,1,61,66,0.2276939,"\int \frac{\tanh ^{-1}(a x)^2}{x^2 \left(1-a^2 x^2\right)} \, dx","Integrate[ArcTanh[a*x]^2/(x^2*(1 - a^2*x^2)),x]","-a \left(\text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)-\frac{1}{3} \tanh ^{-1}(a x) \left(\left(\tanh ^{-1}(a x)+3\right) \tanh ^{-1}(a x)-\frac{3 \tanh ^{-1}(a x)}{a x}+6 \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)\right)\right)","-a \text{Li}_2\left(\frac{2}{a x+1}-1\right)+\frac{1}{3} a \tanh ^{-1}(a x)^3+a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}+2 a \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)",1,"-(a*(-1/3*(ArcTanh[a*x]*((-3*ArcTanh[a*x])/(a*x) + ArcTanh[a*x]*(3 + ArcTanh[a*x]) + 6*Log[1 - E^(-2*ArcTanh[a*x])])) + PolyLog[2, E^(-2*ArcTanh[a*x])]))","A",0
240,1,133,138,0.3655877,"\int \frac{\tanh ^{-1}(a x)^2}{x^3 \left(1-a^2 x^2\right)} \, dx","Integrate[ArcTanh[a*x]^2/(x^3*(1 - a^2*x^2)),x]","-a^2 \left(-\log \left(\frac{a x}{\sqrt{1-a^2 x^2}}\right)+\frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}{2 a^2 x^2}-\tanh ^{-1}(a x) \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)+\frac{1}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)+\frac{1}{3} \tanh ^{-1}(a x)^3+\frac{\tanh ^{-1}(a x)}{a x}-\tanh ^{-1}(a x)^2 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)-\frac{i \pi ^3}{24}\right)","-\frac{1}{2} a^2 \text{Li}_3\left(\frac{2}{a x+1}-1\right)-a^2 \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)-\frac{1}{2} a^2 \log \left(1-a^2 x^2\right)+a^2 \log (x)+\frac{1}{3} a^2 \tanh ^{-1}(a x)^3+\frac{1}{2} a^2 \tanh ^{-1}(a x)^2+a^2 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{2 x^2}-\frac{a \tanh ^{-1}(a x)}{x}",1,"-(a^2*((-1/24*I)*Pi^3 + ArcTanh[a*x]/(a*x) + ((1 - a^2*x^2)*ArcTanh[a*x]^2)/(2*a^2*x^2) + ArcTanh[a*x]^3/3 - ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] - Log[(a*x)/Sqrt[1 - a^2*x^2]] - ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] + PolyLog[3, E^(2*ArcTanh[a*x])]/2))","C",0
241,1,142,205,0.2805779,"\int \frac{x^3 \tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx","Integrate[(x^3*ArcTanh[a*x]^3)/(1 - a^2*x^2),x]","-\frac{-2 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3+6 \tanh ^{-1}(a x) \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)+6 \left(\tanh ^{-1}(a x)^2+1\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+3 \text{Li}_4\left(-e^{-2 \tanh ^{-1}(a x)}\right)-\tanh ^{-1}(a x)^4+6 a x \tanh ^{-1}(a x)^2-6 \tanh ^{-1}(a x)^2-4 \tanh ^{-1}(a x)^3 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)-12 \tanh ^{-1}(a x) \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)}{4 a^4}","\frac{3 \text{Li}_2\left(1-\frac{2}{1-a x}\right)}{2 a^4}+\frac{3 \text{Li}_4\left(1-\frac{2}{1-a x}\right)}{4 a^4}+\frac{3 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{2 a^4}-\frac{3 \text{Li}_3\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{2 a^4}-\frac{\tanh ^{-1}(a x)^4}{4 a^4}+\frac{\tanh ^{-1}(a x)^3}{2 a^4}-\frac{3 \tanh ^{-1}(a x)^2}{2 a^4}+\frac{\log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^3}{a^4}+\frac{3 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^4}-\frac{3 x \tanh ^{-1}(a x)^2}{2 a^3}-\frac{x^2 \tanh ^{-1}(a x)^3}{2 a^2}",1,"-1/4*(-6*ArcTanh[a*x]^2 + 6*a*x*ArcTanh[a*x]^2 - 2*(1 - a^2*x^2)*ArcTanh[a*x]^3 - ArcTanh[a*x]^4 - 12*ArcTanh[a*x]*Log[1 + E^(-2*ArcTanh[a*x])] - 4*ArcTanh[a*x]^3*Log[1 + E^(-2*ArcTanh[a*x])] + 6*(1 + ArcTanh[a*x]^2)*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 6*ArcTanh[a*x]*PolyLog[3, -E^(-2*ArcTanh[a*x])] + 3*PolyLog[4, -E^(-2*ArcTanh[a*x])])/a^4","A",0
242,1,78,103,0.2564292,"\int \frac{x^2 \tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx","Integrate[(x^2*ArcTanh[a*x]^3)/(1 - a^2*x^2),x]","\frac{-12 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)-6 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x)^2 \left(\tanh ^{-1}(a x)^2+(4-4 a x) \tanh ^{-1}(a x)+12 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)\right)}{4 a^3}","-\frac{3 \text{Li}_3\left(1-\frac{2}{1-a x}\right)}{2 a^3}+\frac{3 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^3}+\frac{\tanh ^{-1}(a x)^4}{4 a^3}-\frac{\tanh ^{-1}(a x)^3}{a^3}+\frac{3 \log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{a^3}-\frac{x \tanh ^{-1}(a x)^3}{a^2}",1,"(ArcTanh[a*x]^2*((4 - 4*a*x)*ArcTanh[a*x] + ArcTanh[a*x]^2 + 12*Log[1 + E^(-2*ArcTanh[a*x])]) - 12*ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] - 6*PolyLog[3, -E^(-2*ArcTanh[a*x])])/(4*a^3)","A",0
243,1,87,108,0.0688693,"\int \frac{x \tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx","Integrate[(x*ArcTanh[a*x]^3)/(1 - a^2*x^2),x]","-\frac{6 \tanh ^{-1}(a x)^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+6 \tanh ^{-1}(a x) \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)+3 \text{Li}_4\left(-e^{-2 \tanh ^{-1}(a x)}\right)-\tanh ^{-1}(a x)^4-4 \tanh ^{-1}(a x)^3 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)}{4 a^2}","\frac{3 \text{Li}_4\left(1-\frac{2}{1-a x}\right)}{4 a^2}+\frac{3 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{2 a^2}-\frac{3 \text{Li}_3\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{2 a^2}-\frac{\tanh ^{-1}(a x)^4}{4 a^2}+\frac{\log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^3}{a^2}",1,"-1/4*(-ArcTanh[a*x]^4 - 4*ArcTanh[a*x]^3*Log[1 + E^(-2*ArcTanh[a*x])] + 6*ArcTanh[a*x]^2*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 6*ArcTanh[a*x]*PolyLog[3, -E^(-2*ArcTanh[a*x])] + 3*PolyLog[4, -E^(-2*ArcTanh[a*x])])/a^2","A",0
244,1,13,13,0.0053042,"\int \frac{\tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx","Integrate[ArcTanh[a*x]^3/(1 - a^2*x^2),x]","\frac{\tanh ^{-1}(a x)^4}{4 a}","\frac{\tanh ^{-1}(a x)^4}{4 a}",1,"ArcTanh[a*x]^4/(4*a)","A",1
245,1,83,91,0.072835,"\int \frac{\tanh ^{-1}(a x)^3}{x \left(1-a^2 x^2\right)} \, dx","Integrate[ArcTanh[a*x]^3/(x*(1 - a^2*x^2)),x]","\frac{3}{2} \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)-\frac{3}{2} \tanh ^{-1}(a x) \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)+\frac{3}{4} \text{Li}_4\left(e^{2 \tanh ^{-1}(a x)}\right)-\frac{1}{4} \tanh ^{-1}(a x)^4+\tanh ^{-1}(a x)^3 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)","-\frac{3}{4} \text{Li}_4\left(\frac{2}{a x+1}-1\right)-\frac{3}{2} \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)^2-\frac{3}{2} \text{Li}_3\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)+\frac{1}{4} \tanh ^{-1}(a x)^4+\log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^3",1,"-1/4*ArcTanh[a*x]^4 + ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] + (3*ArcTanh[a*x]^2*PolyLog[2, E^(2*ArcTanh[a*x])])/2 - (3*ArcTanh[a*x]*PolyLog[3, E^(2*ArcTanh[a*x])])/2 + (3*PolyLog[4, E^(2*ArcTanh[a*x])])/4","A",0
246,1,93,90,0.211586,"\int \frac{\tanh ^{-1}(a x)^3}{x^2 \left(1-a^2 x^2\right)} \, dx","Integrate[ArcTanh[a*x]^3/(x^2*(1 - a^2*x^2)),x]","-a \left(-3 \tanh ^{-1}(a x) \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)+\frac{3}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)-\frac{1}{4} \tanh ^{-1}(a x)^4+\frac{\tanh ^{-1}(a x)^3}{a x}+\tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)-\frac{i \pi ^3}{8}\right)","-\frac{3}{2} a \text{Li}_3\left(\frac{2}{a x+1}-1\right)-3 a \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)+\frac{1}{4} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+3 a \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^2",1,"-(a*((-1/8*I)*Pi^3 + ArcTanh[a*x]^3 + ArcTanh[a*x]^3/(a*x) - ArcTanh[a*x]^4/4 - 3*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] - 3*ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] + (3*PolyLog[3, E^(2*ArcTanh[a*x])])/2))","C",0
247,1,165,200,0.4602658,"\int \frac{\tanh ^{-1}(a x)^3}{x^3 \left(1-a^2 x^2\right)} \, dx","Integrate[ArcTanh[a*x]^3/(x^3*(1 - a^2*x^2)),x]","-\frac{1}{64} a^2 \left(\frac{32 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3}{a^2 x^2}-96 \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)+96 \tanh ^{-1}(a x) \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)+96 \text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)-48 \text{Li}_4\left(e^{2 \tanh ^{-1}(a x)}\right)+16 \tanh ^{-1}(a x)^4+\frac{96 \tanh ^{-1}(a x)^2}{a x}-96 \tanh ^{-1}(a x)^2-64 \tanh ^{-1}(a x)^3 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)-192 \tanh ^{-1}(a x) \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)-\pi ^4\right)","-\frac{3}{2} a^2 \text{Li}_2\left(\frac{2}{a x+1}-1\right)-\frac{3}{4} a^2 \text{Li}_4\left(\frac{2}{a x+1}-1\right)-\frac{3}{2} a^2 \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)^2-\frac{3}{2} a^2 \text{Li}_3\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)+\frac{1}{4} a^2 \tanh ^{-1}(a x)^4+\frac{1}{2} a^2 \tanh ^{-1}(a x)^3+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2+a^2 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^3+3 a^2 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)^3}{2 x^2}-\frac{3 a \tanh ^{-1}(a x)^2}{2 x}",1,"-1/64*(a^2*(-Pi^4 - 96*ArcTanh[a*x]^2 + (96*ArcTanh[a*x]^2)/(a*x) + (32*(1 - a^2*x^2)*ArcTanh[a*x]^3)/(a^2*x^2) + 16*ArcTanh[a*x]^4 - 192*ArcTanh[a*x]*Log[1 - E^(-2*ArcTanh[a*x])] - 64*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] + 96*PolyLog[2, E^(-2*ArcTanh[a*x])] - 96*ArcTanh[a*x]^2*PolyLog[2, E^(2*ArcTanh[a*x])] + 96*ArcTanh[a*x]*PolyLog[3, E^(2*ArcTanh[a*x])] - 48*PolyLog[4, E^(2*ArcTanh[a*x])]))","A",0
248,1,15,15,0.008554,"\int \frac{\sqrt{\tanh ^{-1}(a x)}}{1-a^2 x^2} \, dx","Integrate[Sqrt[ArcTanh[a*x]]/(1 - a^2*x^2),x]","\frac{2 \tanh ^{-1}(a x)^{3/2}}{3 a}","\frac{2 \tanh ^{-1}(a x)^{3/2}}{3 a}",1,"(2*ArcTanh[a*x]^(3/2))/(3*a)","A",1
249,0,0,23,1.5012841,"\int \frac{x}{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)} \, dx","Integrate[x/((1 - a^2*x^2)*ArcTanh[a*x]),x]","\int \frac{x}{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x}{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)},x\right)",0,"Integrate[x/((1 - a^2*x^2)*ArcTanh[a*x]), x]","A",-1
250,1,9,9,0.0311038,"\int \frac{1}{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)} \, dx","Integrate[1/((1 - a^2*x^2)*ArcTanh[a*x]),x]","\frac{\log \left(\tanh ^{-1}(a x)\right)}{a}","\frac{\log \left(\tanh ^{-1}(a x)\right)}{a}",1,"Log[ArcTanh[a*x]]/a","A",1
251,0,0,25,0.2306281,"\int \frac{1}{x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)} \, dx","Integrate[1/(x*(1 - a^2*x^2)*ArcTanh[a*x]),x]","\int \frac{1}{x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)},x\right)",0,"Integrate[1/(x*(1 - a^2*x^2)*ArcTanh[a*x]), x]","A",-1
252,0,0,26,0.1356379,"\int \frac{x}{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2} \, dx","Integrate[x/((1 - a^2*x^2)*ArcTanh[a*x]^2),x]","\int \frac{x}{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2} \, dx","\frac{\text{Int}\left(\frac{1}{\tanh ^{-1}(a x)},x\right)}{a}-\frac{x}{a \tanh ^{-1}(a x)}",0,"Integrate[x/((1 - a^2*x^2)*ArcTanh[a*x]^2), x]","A",-1
253,1,11,11,0.0050231,"\int \frac{1}{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2} \, dx","Integrate[1/((1 - a^2*x^2)*ArcTanh[a*x]^2),x]","-\frac{1}{a \tanh ^{-1}(a x)}","-\frac{1}{a \tanh ^{-1}(a x)}",1,"-(1/(a*ArcTanh[a*x]))","A",1
254,0,0,33,0.1494007,"\int \frac{1}{x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2} \, dx","Integrate[1/(x*(1 - a^2*x^2)*ArcTanh[a*x]^2),x]","\int \frac{1}{x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \tanh ^{-1}(a x)},x\right)}{a}-\frac{1}{a x \tanh ^{-1}(a x)}",0,"Integrate[1/(x*(1 - a^2*x^2)*ArcTanh[a*x]^2), x]","A",-1
255,0,0,31,0.6111875,"\int \frac{x}{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3} \, dx","Integrate[x/((1 - a^2*x^2)*ArcTanh[a*x]^3),x]","\int \frac{x}{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3} \, dx","\frac{\text{Int}\left(\frac{1}{\tanh ^{-1}(a x)^2},x\right)}{2 a}-\frac{x}{2 a \tanh ^{-1}(a x)^2}",0,"Integrate[x/((1 - a^2*x^2)*ArcTanh[a*x]^3), x]","A",-1
256,1,13,13,0.0065289,"\int \frac{1}{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3} \, dx","Integrate[1/((1 - a^2*x^2)*ArcTanh[a*x]^3),x]","-\frac{1}{2 a \tanh ^{-1}(a x)^2}","-\frac{1}{2 a \tanh ^{-1}(a x)^2}",1,"-1/2*1/(a*ArcTanh[a*x]^2)","A",1
257,0,0,37,0.7369007,"\int \frac{1}{x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3} \, dx","Integrate[1/(x*(1 - a^2*x^2)*ArcTanh[a*x]^3),x]","\int \frac{1}{x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \tanh ^{-1}(a x)^2},x\right)}{2 a}-\frac{1}{2 a x \tanh ^{-1}(a x)^2}",0,"Integrate[1/(x*(1 - a^2*x^2)*ArcTanh[a*x]^3), x]","A",-1
258,1,17,17,0.0115198,"\int \frac{\tanh ^{-1}(a x)^p}{1-a^2 x^2} \, dx","Integrate[ArcTanh[a*x]^p/(1 - a^2*x^2),x]","\frac{\tanh ^{-1}(a x)^{p+1}}{a (p+1)}","\frac{\tanh ^{-1}(a x)^{p+1}}{a (p+1)}",1,"ArcTanh[a*x]^(1 + p)/(a*(1 + p))","A",1
259,1,64,109,0.1694494,"\int \frac{x^3 \tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^2} \, dx","Integrate[(x^3*ArcTanh[a*x])/(1 - a^2*x^2)^2,x]","-\frac{-4 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+4 \tanh ^{-1}(a x)^2+\sinh \left(2 \tanh ^{-1}(a x)\right)-2 \tanh ^{-1}(a x) \left(\cosh \left(2 \tanh ^{-1}(a x)\right)-4 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)\right)}{8 a^4}","-\frac{\text{Li}_2\left(1-\frac{2}{1-a x}\right)}{2 a^4}+\frac{\tanh ^{-1}(a x)^2}{2 a^4}-\frac{\tanh ^{-1}(a x)}{4 a^4}-\frac{\log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^4}+\frac{\tanh ^{-1}(a x)}{2 a^4 \left(1-a^2 x^2\right)}-\frac{x}{4 a^3 \left(1-a^2 x^2\right)}",1,"-1/8*(4*ArcTanh[a*x]^2 - 2*ArcTanh[a*x]*(Cosh[2*ArcTanh[a*x]] - 4*Log[1 + E^(-2*ArcTanh[a*x])]) - 4*PolyLog[2, -E^(-2*ArcTanh[a*x])] + Sinh[2*ArcTanh[a*x]])/a^4","A",0
260,1,45,57,0.0934373,"\int \frac{x^2 \tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^2} \, dx","Integrate[(x^2*ArcTanh[a*x])/(1 - a^2*x^2)^2,x]","\frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2-2 a x \tanh ^{-1}(a x)+1}{4 a^3 \left(a^2 x^2-1\right)}","-\frac{\tanh ^{-1}(a x)^2}{4 a^3}+\frac{x \tanh ^{-1}(a x)}{2 a^2 \left(1-a^2 x^2\right)}-\frac{1}{4 a^3 \left(1-a^2 x^2\right)}",1,"(1 - 2*a*x*ArcTanh[a*x] + (1 - a^2*x^2)*ArcTanh[a*x]^2)/(4*a^3*(-1 + a^2*x^2))","A",1
261,1,66,55,0.0546627,"\int \frac{x \tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^2} \, dx","Integrate[(x*ArcTanh[a*x])/(1 - a^2*x^2)^2,x]","\frac{-a^2 x^2 \log (a x+1)+\left(a^2 x^2-1\right) \log (1-a x)+2 a x+\log (a x+1)-4 \tanh ^{-1}(a x)}{8 a^2 \left(a^2 x^2-1\right)}","-\frac{x}{4 a \left(1-a^2 x^2\right)}+\frac{\tanh ^{-1}(a x)}{2 a^2 \left(1-a^2 x^2\right)}-\frac{\tanh ^{-1}(a x)}{4 a^2}",1,"(2*a*x - 4*ArcTanh[a*x] + (-1 + a^2*x^2)*Log[1 - a*x] + Log[1 + a*x] - a^2*x^2*Log[1 + a*x])/(8*a^2*(-1 + a^2*x^2))","A",1
262,1,44,54,0.0701678,"\int \frac{\tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^2} \, dx","Integrate[ArcTanh[a*x]/(1 - a^2*x^2)^2,x]","\frac{\left(a^2 x^2-1\right) \tanh ^{-1}(a x)^2-2 a x \tanh ^{-1}(a x)+1}{4 a \left(a^2 x^2-1\right)}","-\frac{1}{4 a \left(1-a^2 x^2\right)}+\frac{x \tanh ^{-1}(a x)}{2 \left(1-a^2 x^2\right)}+\frac{\tanh ^{-1}(a x)^2}{4 a}",1,"(1 - 2*a*x*ArcTanh[a*x] + (-1 + a^2*x^2)*ArcTanh[a*x]^2)/(4*a*(-1 + a^2*x^2))","A",1
263,1,63,91,0.1775124,"\int \frac{\tanh ^{-1}(a x)}{x \left(1-a^2 x^2\right)^2} \, dx","Integrate[ArcTanh[a*x]/(x*(1 - a^2*x^2)^2),x]","\frac{1}{8} \left(-4 \text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)+4 \tanh ^{-1}(a x)^2-\sinh \left(2 \tanh ^{-1}(a x)\right)+2 \tanh ^{-1}(a x) \left(4 \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)+\cosh \left(2 \tanh ^{-1}(a x)\right)\right)\right)","-\frac{a x}{4 \left(1-a^2 x^2\right)}+\frac{\tanh ^{-1}(a x)}{2 \left(1-a^2 x^2\right)}-\frac{1}{2} \text{Li}_2\left(\frac{2}{a x+1}-1\right)+\frac{1}{2} \tanh ^{-1}(a x)^2-\frac{1}{4} \tanh ^{-1}(a x)+\log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)",1,"(4*ArcTanh[a*x]^2 + 2*ArcTanh[a*x]*(Cosh[2*ArcTanh[a*x]] + 4*Log[1 - E^(-2*ArcTanh[a*x])]) - 4*PolyLog[2, E^(-2*ArcTanh[a*x])] - Sinh[2*ArcTanh[a*x]])/8","A",0
264,1,77,82,0.1420738,"\int \frac{\tanh ^{-1}(a x)}{x^2 \left(1-a^2 x^2\right)^2} \, dx","Integrate[ArcTanh[a*x]/(x^2*(1 - a^2*x^2)^2),x]","\frac{1}{4} \left(a \left(\frac{1}{a^2 x^2-1}-2 \log \left(1-a^2 x^2\right)+4 \log (a x)\right)-\frac{2 \left(3 a^2 x^2-2\right) \tanh ^{-1}(a x)}{x \left(a^2 x^2-1\right)}+3 a \tanh ^{-1}(a x)^2\right)","-\frac{a}{4 \left(1-a^2 x^2\right)}-\frac{1}{2} a \log \left(1-a^2 x^2\right)+\frac{a^2 x \tanh ^{-1}(a x)}{2 \left(1-a^2 x^2\right)}+a \log (x)+\frac{3}{4} a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)}{x}",1,"((-2*(-2 + 3*a^2*x^2)*ArcTanh[a*x])/(x*(-1 + a^2*x^2)) + 3*a*ArcTanh[a*x]^2 + a*((-1 + a^2*x^2)^(-1) + 4*Log[a*x] - 2*Log[1 - a^2*x^2]))/4","A",1
265,1,83,123,0.3965007,"\int \frac{\tanh ^{-1}(a x)}{x^3 \left(1-a^2 x^2\right)^2} \, dx","Integrate[ArcTanh[a*x]/(x^3*(1 - a^2*x^2)^2),x]","\frac{1}{8} a^2 \left(2 \tanh ^{-1}(a x) \left(-\frac{2}{a^2 x^2}+8 \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)+\cosh \left(2 \tanh ^{-1}(a x)\right)+2\right)-8 \text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)-\frac{4}{a x}+8 \tanh ^{-1}(a x)^2-\sinh \left(2 \tanh ^{-1}(a x)\right)\right)","-a^2 \text{Li}_2\left(\frac{2}{a x+1}-1\right)+\frac{a^2 \tanh ^{-1}(a x)}{2 \left(1-a^2 x^2\right)}+a^2 \tanh ^{-1}(a x)^2+\frac{1}{4} a^2 \tanh ^{-1}(a x)+2 a^2 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)-\frac{a^3 x}{4 \left(1-a^2 x^2\right)}-\frac{\tanh ^{-1}(a x)}{2 x^2}-\frac{a}{2 x}",1,"(a^2*(-4/(a*x) + 8*ArcTanh[a*x]^2 + 2*ArcTanh[a*x]*(2 - 2/(a^2*x^2) + Cosh[2*ArcTanh[a*x]] + 8*Log[1 - E^(-2*ArcTanh[a*x])]) - 8*PolyLog[2, E^(-2*ArcTanh[a*x])] - Sinh[2*ArcTanh[a*x]]))/8","A",0
266,1,103,161,0.1820012,"\int \frac{x^3 \tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^2} \, dx","Integrate[(x^3*ArcTanh[a*x]^2)/(1 - a^2*x^2)^2,x]","\frac{\tanh ^{-1}(a x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)-\frac{1}{3} \tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^2 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)-\frac{1}{4} \tanh ^{-1}(a x) \sinh \left(2 \tanh ^{-1}(a x)\right)+\frac{1}{8} \left(2 \tanh ^{-1}(a x)^2+1\right) \cosh \left(2 \tanh ^{-1}(a x)\right)}{a^4}","\frac{\text{Li}_3\left(1-\frac{2}{1-a x}\right)}{2 a^4}-\frac{\text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{a^4}+\frac{\tanh ^{-1}(a x)^3}{3 a^4}-\frac{\tanh ^{-1}(a x)^2}{4 a^4}-\frac{\log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{a^4}+\frac{1}{4 a^4 \left(1-a^2 x^2\right)}+\frac{\tanh ^{-1}(a x)^2}{2 a^4 \left(1-a^2 x^2\right)}-\frac{x \tanh ^{-1}(a x)}{2 a^3 \left(1-a^2 x^2\right)}",1,"(-1/3*ArcTanh[a*x]^3 + ((1 + 2*ArcTanh[a*x]^2)*Cosh[2*ArcTanh[a*x]])/8 - ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] + ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] + PolyLog[3, -E^(-2*ArcTanh[a*x])]/2 - (ArcTanh[a*x]*Sinh[2*ArcTanh[a*x]])/4)/a^4","A",0
267,1,93,94,0.1348761,"\int \frac{x^2 \tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^2} \, dx","Integrate[(x^2*ArcTanh[a*x]^2)/(1 - a^2*x^2)^2,x]","\frac{-3 \left(\left(a^2 x^2-1\right) \log (1-a x)+\left(1-a^2 x^2\right) \log (a x+1)+2 a x\right)+\left(4-4 a^2 x^2\right) \tanh ^{-1}(a x)^3-12 a x \tanh ^{-1}(a x)^2+12 \tanh ^{-1}(a x)}{24 a^3 \left(a^2 x^2-1\right)}","-\frac{\tanh ^{-1}(a x)^3}{6 a^3}+\frac{\tanh ^{-1}(a x)}{4 a^3}+\frac{x}{4 a^2 \left(1-a^2 x^2\right)}+\frac{x \tanh ^{-1}(a x)^2}{2 a^2 \left(1-a^2 x^2\right)}-\frac{\tanh ^{-1}(a x)}{2 a^3 \left(1-a^2 x^2\right)}",1,"(12*ArcTanh[a*x] - 12*a*x*ArcTanh[a*x]^2 + (4 - 4*a^2*x^2)*ArcTanh[a*x]^3 - 3*(2*a*x + (-1 + a^2*x^2)*Log[1 - a*x] + (1 - a^2*x^2)*Log[1 + a*x]))/(24*a^3*(-1 + a^2*x^2))","A",1
268,1,43,82,0.0549852,"\int \frac{x \tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^2} \, dx","Integrate[(x*ArcTanh[a*x]^2)/(1 - a^2*x^2)^2,x]","\frac{\left(a^2 x^2+1\right) \tanh ^{-1}(a x)^2-2 a x \tanh ^{-1}(a x)+1}{4 a^2-4 a^4 x^2}","\frac{1}{4 a^2 \left(1-a^2 x^2\right)}+\frac{\tanh ^{-1}(a x)^2}{2 a^2 \left(1-a^2 x^2\right)}-\frac{x \tanh ^{-1}(a x)}{2 a \left(1-a^2 x^2\right)}-\frac{\tanh ^{-1}(a x)^2}{4 a^2}",1,"(1 - 2*a*x*ArcTanh[a*x] + (1 + a^2*x^2)*ArcTanh[a*x]^2)/(4*a^2 - 4*a^4*x^2)","A",1
269,1,93,88,0.0948993,"\int \frac{\tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^2} \, dx","Integrate[ArcTanh[a*x]^2/(1 - a^2*x^2)^2,x]","\frac{-3 \left(\left(a^2 x^2-1\right) \log (1-a x)+\left(1-a^2 x^2\right) \log (a x+1)+2 a x\right)+4 \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^3-12 a x \tanh ^{-1}(a x)^2+12 \tanh ^{-1}(a x)}{24 a \left(a^2 x^2-1\right)}","\frac{x}{4 \left(1-a^2 x^2\right)}+\frac{x \tanh ^{-1}(a x)^2}{2 \left(1-a^2 x^2\right)}-\frac{\tanh ^{-1}(a x)}{2 a \left(1-a^2 x^2\right)}+\frac{\tanh ^{-1}(a x)^3}{6 a}+\frac{\tanh ^{-1}(a x)}{4 a}",1,"(12*ArcTanh[a*x] - 12*a*x*ArcTanh[a*x]^2 + 4*(-1 + a^2*x^2)*ArcTanh[a*x]^3 - 3*(2*a*x + (-1 + a^2*x^2)*Log[1 - a*x] + (1 - a^2*x^2)*Log[1 + a*x]))/(24*a*(-1 + a^2*x^2))","A",1
270,1,106,136,0.1825986,"\int \frac{\tanh ^{-1}(a x)^2}{x \left(1-a^2 x^2\right)^2} \, dx","Integrate[ArcTanh[a*x]^2/(x*(1 - a^2*x^2)^2),x]","\frac{1}{24} \left(24 \tanh ^{-1}(a x) \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)-12 \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)-8 \tanh ^{-1}(a x)^3+24 \tanh ^{-1}(a x)^2 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)-6 \tanh ^{-1}(a x) \sinh \left(2 \tanh ^{-1}(a x)\right)+6 \tanh ^{-1}(a x)^2 \cosh \left(2 \tanh ^{-1}(a x)\right)+3 \cosh \left(2 \tanh ^{-1}(a x)\right)+i \pi ^3\right)","\frac{1}{4 \left(1-a^2 x^2\right)}+\frac{\tanh ^{-1}(a x)^2}{2 \left(1-a^2 x^2\right)}-\frac{a x \tanh ^{-1}(a x)}{2 \left(1-a^2 x^2\right)}-\frac{1}{2} \text{Li}_3\left(\frac{2}{a x+1}-1\right)-\text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)+\frac{1}{3} \tanh ^{-1}(a x)^3-\frac{1}{4} \tanh ^{-1}(a x)^2+\log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^2",1,"(I*Pi^3 - 8*ArcTanh[a*x]^3 + 3*Cosh[2*ArcTanh[a*x]] + 6*ArcTanh[a*x]^2*Cosh[2*ArcTanh[a*x]] + 24*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] + 24*ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] - 12*PolyLog[3, E^(2*ArcTanh[a*x])] - 6*ArcTanh[a*x]*Sinh[2*ArcTanh[a*x]])/24","C",0
271,1,97,142,0.3236418,"\int \frac{\tanh ^{-1}(a x)^2}{x^2 \left(1-a^2 x^2\right)^2} \, dx","Integrate[ArcTanh[a*x]^2/(x^2*(1 - a^2*x^2)^2),x]","\frac{-8 a x \text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)+4 a x \tanh ^{-1}(a x)^3+2 \tanh ^{-1}(a x)^2 \left(4 a x+a x \sinh \left(2 \tanh ^{-1}(a x)\right)-4\right)+a x \sinh \left(2 \tanh ^{-1}(a x)\right)-2 a x \tanh ^{-1}(a x) \left(\cosh \left(2 \tanh ^{-1}(a x)\right)-8 \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)\right)}{8 x}","\frac{a^2 x}{4 \left(1-a^2 x^2\right)}+\frac{a^2 x \tanh ^{-1}(a x)^2}{2 \left(1-a^2 x^2\right)}-\frac{a \tanh ^{-1}(a x)}{2 \left(1-a^2 x^2\right)}-a \text{Li}_2\left(\frac{2}{a x+1}-1\right)+\frac{1}{2} a \tanh ^{-1}(a x)^3+a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{1}{4} a \tanh ^{-1}(a x)+2 a \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)",1,"(4*a*x*ArcTanh[a*x]^3 - 2*a*x*ArcTanh[a*x]*(Cosh[2*ArcTanh[a*x]] - 8*Log[1 - E^(-2*ArcTanh[a*x])]) - 8*a*x*PolyLog[2, E^(-2*ArcTanh[a*x])] + a*x*Sinh[2*ArcTanh[a*x]] + 2*ArcTanh[a*x]^2*(-4 + 4*a*x + a*x*Sinh[2*ArcTanh[a*x]]))/(8*x)","A",0
272,1,146,205,1.0157316,"\int \frac{\tanh ^{-1}(a x)^2}{x^3 \left(1-a^2 x^2\right)^2} \, dx","Integrate[ArcTanh[a*x]^2/(x^3*(1 - a^2*x^2)^2),x]","a^2 \left(2 \tanh ^{-1}(a x) \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)+\frac{1}{24} \left(24 \log \left(\frac{a x}{\sqrt{1-a^2 x^2}}\right)+6 \tanh ^{-1}(a x)^2 \left(-\frac{2}{a^2 x^2}+8 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)+\cosh \left(2 \tanh ^{-1}(a x)\right)+2\right)-24 \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)-16 \tanh ^{-1}(a x)^3-\frac{6 \tanh ^{-1}(a x) \left(a x \sinh \left(2 \tanh ^{-1}(a x)\right)+4\right)}{a x}+3 \cosh \left(2 \tanh ^{-1}(a x)\right)+2 i \pi ^3\right)\right)","-a^2 \text{Li}_3\left(\frac{2}{a x+1}-1\right)-2 a^2 \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)+\frac{a^2}{4 \left(1-a^2 x^2\right)}-\frac{1}{2} a^2 \log \left(1-a^2 x^2\right)+\frac{a^2 \tanh ^{-1}(a x)^2}{2 \left(1-a^2 x^2\right)}+a^2 \log (x)+\frac{2}{3} a^2 \tanh ^{-1}(a x)^3+\frac{1}{4} a^2 \tanh ^{-1}(a x)^2+2 a^2 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^2-\frac{a^3 x \tanh ^{-1}(a x)}{2 \left(1-a^2 x^2\right)}-\frac{\tanh ^{-1}(a x)^2}{2 x^2}-\frac{a \tanh ^{-1}(a x)}{x}",1,"a^2*(2*ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] + ((2*I)*Pi^3 - 16*ArcTanh[a*x]^3 + 3*Cosh[2*ArcTanh[a*x]] + 6*ArcTanh[a*x]^2*(2 - 2/(a^2*x^2) + Cosh[2*ArcTanh[a*x]] + 8*Log[1 - E^(2*ArcTanh[a*x])]) + 24*Log[(a*x)/Sqrt[1 - a^2*x^2]] - 24*PolyLog[3, E^(2*ArcTanh[a*x])] - (6*ArcTanh[a*x]*(4 + a*x*Sinh[2*ArcTanh[a*x]]))/(a*x))/24)","C",0
273,1,139,227,0.1882723,"\int \frac{x^3 \tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^2} \, dx","Integrate[(x^3*ArcTanh[a*x]^3)/(1 - a^2*x^2)^2,x]","\frac{24 \tanh ^{-1}(a x)^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a x)}\right)+24 \tanh ^{-1}(a x) \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a x)}\right)+12 \text{Li}_4\left(-e^{-2 \tanh ^{-1}(a x)}\right)-4 \tanh ^{-1}(a x)^4-16 \tanh ^{-1}(a x)^3 \log \left(e^{-2 \tanh ^{-1}(a x)}+1\right)-6 \tanh ^{-1}(a x)^2 \sinh \left(2 \tanh ^{-1}(a x)\right)-3 \sinh \left(2 \tanh ^{-1}(a x)\right)+4 \tanh ^{-1}(a x)^3 \cosh \left(2 \tanh ^{-1}(a x)\right)+6 \tanh ^{-1}(a x) \cosh \left(2 \tanh ^{-1}(a x)\right)}{16 a^4}","-\frac{3 \text{Li}_4\left(1-\frac{2}{1-a x}\right)}{4 a^4}-\frac{3 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^2}{2 a^4}+\frac{3 \text{Li}_3\left(1-\frac{2}{1-a x}\right) \tanh ^{-1}(a x)}{2 a^4}+\frac{\tanh ^{-1}(a x)^4}{4 a^4}-\frac{\tanh ^{-1}(a x)^3}{4 a^4}-\frac{3 \tanh ^{-1}(a x)}{8 a^4}-\frac{\log \left(\frac{2}{1-a x}\right) \tanh ^{-1}(a x)^3}{a^4}+\frac{\tanh ^{-1}(a x)^3}{2 a^4 \left(1-a^2 x^2\right)}+\frac{3 \tanh ^{-1}(a x)}{4 a^4 \left(1-a^2 x^2\right)}-\frac{3 x}{8 a^3 \left(1-a^2 x^2\right)}-\frac{3 x \tanh ^{-1}(a x)^2}{4 a^3 \left(1-a^2 x^2\right)}",1,"(-4*ArcTanh[a*x]^4 + 6*ArcTanh[a*x]*Cosh[2*ArcTanh[a*x]] + 4*ArcTanh[a*x]^3*Cosh[2*ArcTanh[a*x]] - 16*ArcTanh[a*x]^3*Log[1 + E^(-2*ArcTanh[a*x])] + 24*ArcTanh[a*x]^2*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 24*ArcTanh[a*x]*PolyLog[3, -E^(-2*ArcTanh[a*x])] + 12*PolyLog[4, -E^(-2*ArcTanh[a*x])] - 3*Sinh[2*ArcTanh[a*x]] - 6*ArcTanh[a*x]^2*Sinh[2*ArcTanh[a*x]])/(16*a^4)","A",0
274,1,72,121,0.0915272,"\int \frac{x^2 \tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^2} \, dx","Integrate[(x^2*ArcTanh[a*x]^3)/(1 - a^2*x^2)^2,x]","\frac{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^4+3 \left(a^2 x^2+1\right) \tanh ^{-1}(a x)^2-4 a x \tanh ^{-1}(a x)^3-6 a x \tanh ^{-1}(a x)+3}{8 a^3 \left(a^2 x^2-1\right)}","-\frac{\tanh ^{-1}(a x)^4}{8 a^3}+\frac{3 \tanh ^{-1}(a x)^2}{8 a^3}+\frac{x \tanh ^{-1}(a x)^3}{2 a^2 \left(1-a^2 x^2\right)}+\frac{3 x \tanh ^{-1}(a x)}{4 a^2 \left(1-a^2 x^2\right)}-\frac{3}{8 a^3 \left(1-a^2 x^2\right)}-\frac{3 \tanh ^{-1}(a x)^2}{4 a^3 \left(1-a^2 x^2\right)}",1,"(3 - 6*a*x*ArcTanh[a*x] + 3*(1 + a^2*x^2)*ArcTanh[a*x]^2 - 4*a*x*ArcTanh[a*x]^3 + (1 - a^2*x^2)*ArcTanh[a*x]^4)/(8*a^3*(-1 + a^2*x^2))","A",1
275,1,91,119,0.0598963,"\int \frac{x \tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^2} \, dx","Integrate[(x*ArcTanh[a*x]^3)/(1 - a^2*x^2)^2,x]","\frac{3 \left(a^2 x^2-1\right) \log (1-a x)-3 \left(a^2 x^2-1\right) \log (a x+1)-4 \left(a^2 x^2+1\right) \tanh ^{-1}(a x)^3+6 a x+12 a x \tanh ^{-1}(a x)^2-12 \tanh ^{-1}(a x)}{16 a^2 \left(a^2 x^2-1\right)}","-\frac{3 x}{8 a \left(1-a^2 x^2\right)}+\frac{\tanh ^{-1}(a x)^3}{2 a^2 \left(1-a^2 x^2\right)}-\frac{3 x \tanh ^{-1}(a x)^2}{4 a \left(1-a^2 x^2\right)}+\frac{3 \tanh ^{-1}(a x)}{4 a^2 \left(1-a^2 x^2\right)}-\frac{\tanh ^{-1}(a x)^3}{4 a^2}-\frac{3 \tanh ^{-1}(a x)}{8 a^2}",1,"(6*a*x - 12*ArcTanh[a*x] + 12*a*x*ArcTanh[a*x]^2 - 4*(1 + a^2*x^2)*ArcTanh[a*x]^3 + 3*(-1 + a^2*x^2)*Log[1 - a*x] - 3*(-1 + a^2*x^2)*Log[1 + a*x])/(16*a^2*(-1 + a^2*x^2))","A",1
276,1,71,115,0.0602863,"\int \frac{\tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^2} \, dx","Integrate[ArcTanh[a*x]^3/(1 - a^2*x^2)^2,x]","\frac{\left(a^2 x^2-1\right) \tanh ^{-1}(a x)^4+3 \left(a^2 x^2+1\right) \tanh ^{-1}(a x)^2-4 a x \tanh ^{-1}(a x)^3-6 a x \tanh ^{-1}(a x)+3}{8 a \left(a^2 x^2-1\right)}","-\frac{3}{8 a \left(1-a^2 x^2\right)}+\frac{x \tanh ^{-1}(a x)^3}{2 \left(1-a^2 x^2\right)}-\frac{3 \tanh ^{-1}(a x)^2}{4 a \left(1-a^2 x^2\right)}+\frac{3 x \tanh ^{-1}(a x)}{4 \left(1-a^2 x^2\right)}+\frac{\tanh ^{-1}(a x)^4}{8 a}+\frac{3 \tanh ^{-1}(a x)^2}{8 a}",1,"(3 - 6*a*x*ArcTanh[a*x] + 3*(1 + a^2*x^2)*ArcTanh[a*x]^2 - 4*a*x*ArcTanh[a*x]^3 + (-1 + a^2*x^2)*ArcTanh[a*x]^4)/(8*a*(-1 + a^2*x^2))","A",1
277,1,135,193,0.1984071,"\int \frac{\tanh ^{-1}(a x)^3}{x \left(1-a^2 x^2\right)^2} \, dx","Integrate[ArcTanh[a*x]^3/(x*(1 - a^2*x^2)^2),x]","\frac{1}{64} \left(96 \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)-96 \tanh ^{-1}(a x) \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)+48 \text{Li}_4\left(e^{2 \tanh ^{-1}(a x)}\right)-16 \tanh ^{-1}(a x)^4+64 \tanh ^{-1}(a x)^3 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)-24 \tanh ^{-1}(a x)^2 \sinh \left(2 \tanh ^{-1}(a x)\right)-12 \sinh \left(2 \tanh ^{-1}(a x)\right)+16 \tanh ^{-1}(a x)^3 \cosh \left(2 \tanh ^{-1}(a x)\right)+24 \tanh ^{-1}(a x) \cosh \left(2 \tanh ^{-1}(a x)\right)+\pi ^4\right)","-\frac{3 a x}{8 \left(1-a^2 x^2\right)}+\frac{\tanh ^{-1}(a x)^3}{2 \left(1-a^2 x^2\right)}-\frac{3 a x \tanh ^{-1}(a x)^2}{4 \left(1-a^2 x^2\right)}+\frac{3 \tanh ^{-1}(a x)}{4 \left(1-a^2 x^2\right)}-\frac{3}{4} \text{Li}_4\left(\frac{2}{a x+1}-1\right)-\frac{3}{2} \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)^2-\frac{3}{2} \text{Li}_3\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)+\frac{1}{4} \tanh ^{-1}(a x)^4-\frac{1}{4} \tanh ^{-1}(a x)^3-\frac{3}{8} \tanh ^{-1}(a x)+\log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^3",1,"(Pi^4 - 16*ArcTanh[a*x]^4 + 24*ArcTanh[a*x]*Cosh[2*ArcTanh[a*x]] + 16*ArcTanh[a*x]^3*Cosh[2*ArcTanh[a*x]] + 64*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] + 96*ArcTanh[a*x]^2*PolyLog[2, E^(2*ArcTanh[a*x])] - 96*ArcTanh[a*x]*PolyLog[3, E^(2*ArcTanh[a*x])] + 48*PolyLog[4, E^(2*ArcTanh[a*x])] - 12*Sinh[2*ArcTanh[a*x]] - 24*ArcTanh[a*x]^2*Sinh[2*ArcTanh[a*x]])/64","A",0
278,1,144,191,0.3542714,"\int \frac{\tanh ^{-1}(a x)^3}{x^2 \left(1-a^2 x^2\right)^2} \, dx","Integrate[ArcTanh[a*x]^3/(x^2*(1 - a^2*x^2)^2),x]","\frac{1}{16} a \left(48 \tanh ^{-1}(a x) \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)-24 \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)+6 \tanh ^{-1}(a x)^4-\frac{16 \tanh ^{-1}(a x)^3}{a x}-16 \tanh ^{-1}(a x)^3+48 \tanh ^{-1}(a x)^2 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)+4 \tanh ^{-1}(a x)^3 \sinh \left(2 \tanh ^{-1}(a x)\right)+6 \tanh ^{-1}(a x) \sinh \left(2 \tanh ^{-1}(a x)\right)-6 \tanh ^{-1}(a x)^2 \cosh \left(2 \tanh ^{-1}(a x)\right)-3 \cosh \left(2 \tanh ^{-1}(a x)\right)+2 i \pi ^3\right)","-\frac{3 a}{8 \left(1-a^2 x^2\right)}+\frac{a^2 x \tanh ^{-1}(a x)^3}{2 \left(1-a^2 x^2\right)}-\frac{3 a \tanh ^{-1}(a x)^2}{4 \left(1-a^2 x^2\right)}+\frac{3 a^2 x \tanh ^{-1}(a x)}{4 \left(1-a^2 x^2\right)}-\frac{3}{2} a \text{Li}_3\left(\frac{2}{a x+1}-1\right)-3 a \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)+\frac{3}{8} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{3}{8} a \tanh ^{-1}(a x)^2+3 a \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^2",1,"(a*((2*I)*Pi^3 - 16*ArcTanh[a*x]^3 - (16*ArcTanh[a*x]^3)/(a*x) + 6*ArcTanh[a*x]^4 - 3*Cosh[2*ArcTanh[a*x]] - 6*ArcTanh[a*x]^2*Cosh[2*ArcTanh[a*x]] + 48*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] + 48*ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] - 24*PolyLog[3, E^(2*ArcTanh[a*x])] + 6*ArcTanh[a*x]*Sinh[2*ArcTanh[a*x]] + 4*ArcTanh[a*x]^3*Sinh[2*ArcTanh[a*x]]))/16","C",0
279,1,215,302,0.8106125,"\int \frac{\tanh ^{-1}(a x)^3}{x^3 \left(1-a^2 x^2\right)^2} \, dx","Integrate[ArcTanh[a*x]^3/(x^3*(1 - a^2*x^2)^2),x]","\frac{1}{32} a^2 \left(-\frac{16 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3}{a^2 x^2}+96 \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)-96 \tanh ^{-1}(a x) \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)-48 \text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)+48 \text{Li}_4\left(e^{2 \tanh ^{-1}(a x)}\right)-16 \tanh ^{-1}(a x)^4-\frac{48 \tanh ^{-1}(a x)^2}{a x}+48 \tanh ^{-1}(a x)^2+64 \tanh ^{-1}(a x)^3 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)+96 \tanh ^{-1}(a x) \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)-12 \tanh ^{-1}(a x)^2 \sinh \left(2 \tanh ^{-1}(a x)\right)-6 \sinh \left(2 \tanh ^{-1}(a x)\right)+8 \tanh ^{-1}(a x)^3 \cosh \left(2 \tanh ^{-1}(a x)\right)+12 \tanh ^{-1}(a x) \cosh \left(2 \tanh ^{-1}(a x)\right)+\pi ^4\right)","-\frac{3}{2} a^2 \text{Li}_2\left(\frac{2}{a x+1}-1\right)-\frac{3}{2} a^2 \text{Li}_4\left(\frac{2}{a x+1}-1\right)-3 a^2 \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)^2-3 a^2 \text{Li}_3\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)+\frac{a^2 \tanh ^{-1}(a x)^3}{2 \left(1-a^2 x^2\right)}+\frac{3 a^2 \tanh ^{-1}(a x)}{4 \left(1-a^2 x^2\right)}+\frac{1}{2} a^2 \tanh ^{-1}(a x)^4+\frac{1}{4} a^2 \tanh ^{-1}(a x)^3+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2-\frac{3}{8} a^2 \tanh ^{-1}(a x)+2 a^2 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^3+3 a^2 \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)-\frac{3 a^3 x}{8 \left(1-a^2 x^2\right)}-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{4 \left(1-a^2 x^2\right)}-\frac{\tanh ^{-1}(a x)^3}{2 x^2}-\frac{3 a \tanh ^{-1}(a x)^2}{2 x}",1,"(a^2*(Pi^4 + 48*ArcTanh[a*x]^2 - (48*ArcTanh[a*x]^2)/(a*x) - (16*(1 - a^2*x^2)*ArcTanh[a*x]^3)/(a^2*x^2) - 16*ArcTanh[a*x]^4 + 12*ArcTanh[a*x]*Cosh[2*ArcTanh[a*x]] + 8*ArcTanh[a*x]^3*Cosh[2*ArcTanh[a*x]] + 96*ArcTanh[a*x]*Log[1 - E^(-2*ArcTanh[a*x])] + 64*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] - 48*PolyLog[2, E^(-2*ArcTanh[a*x])] + 96*ArcTanh[a*x]^2*PolyLog[2, E^(2*ArcTanh[a*x])] - 96*ArcTanh[a*x]*PolyLog[3, E^(2*ArcTanh[a*x])] + 48*PolyLog[4, E^(2*ArcTanh[a*x])] - 6*Sinh[2*ArcTanh[a*x]] - 12*ArcTanh[a*x]^2*Sinh[2*ArcTanh[a*x]]))/32","A",0
280,1,87,103,0.2355985,"\int \frac{\sqrt{\tanh ^{-1}(a x)}}{\left(1-a^2 x^2\right)^2} \, dx","Integrate[Sqrt[ArcTanh[a*x]]/(1 - a^2*x^2)^2,x]","\sqrt{\tanh ^{-1}(a x)} \left(\frac{\tanh ^{-1}(a x)}{3 a}-\frac{x}{2 \left(a^2 x^2-1\right)}\right)-\frac{\sqrt{\frac{\pi }{2}} \left(\text{erfi}\left(\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right)-\text{erf}\left(\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right)\right)}{16 a}","\frac{x \sqrt{\tanh ^{-1}(a x)}}{2 \left(1-a^2 x^2\right)}+\frac{\sqrt{\frac{\pi }{2}} \text{erf}\left(\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right)}{16 a}-\frac{\sqrt{\frac{\pi }{2}} \text{erfi}\left(\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right)}{16 a}+\frac{\tanh ^{-1}(a x)^{3/2}}{3 a}",1,"Sqrt[ArcTanh[a*x]]*(-1/2*x/(-1 + a^2*x^2) + ArcTanh[a*x]/(3*a)) - (Sqrt[Pi/2]*(-Erf[Sqrt[2]*Sqrt[ArcTanh[a*x]]] + Erfi[Sqrt[2]*Sqrt[ArcTanh[a*x]]]))/(16*a)","A",1
281,0,0,40,4.9334524,"\int \frac{x^4}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)} \, dx","Integrate[x^4/((1 - a^2*x^2)^2*ArcTanh[a*x]),x]","\int \frac{x^4}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)} \, dx","\frac{\text{Int}\left(\frac{1}{\tanh ^{-1}(a x)},x\right)}{a^4}+\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{2 a^5}-\frac{3 \log \left(\tanh ^{-1}(a x)\right)}{2 a^5}",0,"Integrate[x^4/((1 - a^2*x^2)^2*ArcTanh[a*x]), x]","A",-1
282,0,0,43,1.7066905,"\int \frac{x^3}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)} \, dx","Integrate[x^3/((1 - a^2*x^2)^2*ArcTanh[a*x]),x]","\int \frac{x^3}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)} \, dx","\frac{\text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{2 a^4}-\frac{\text{Int}\left(\frac{x}{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)},x\right)}{a^2}",0,"Integrate[x^3/((1 - a^2*x^2)^2*ArcTanh[a*x]), x]","A",-1
283,1,27,27,0.1125532,"\int \frac{x^2}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)} \, dx","Integrate[x^2/((1 - a^2*x^2)^2*ArcTanh[a*x]),x]","\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{2 a^3}-\frac{\log \left(\tanh ^{-1}(a x)\right)}{2 a^3}","\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{2 a^3}-\frac{\log \left(\tanh ^{-1}(a x)\right)}{2 a^3}",1,"CoshIntegral[2*ArcTanh[a*x]]/(2*a^3) - Log[ArcTanh[a*x]]/(2*a^3)","A",1
284,1,14,14,0.0671456,"\int \frac{x}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)} \, dx","Integrate[x/((1 - a^2*x^2)^2*ArcTanh[a*x]),x]","\frac{\text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{2 a^2}","\frac{\text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{2 a^2}",1,"SinhIntegral[2*ArcTanh[a*x]]/(2*a^2)","A",1
285,1,27,27,0.0753634,"\int \frac{1}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)} \, dx","Integrate[1/((1 - a^2*x^2)^2*ArcTanh[a*x]),x]","\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{2 a}+\frac{\log \left(\tanh ^{-1}(a x)\right)}{2 a}","\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{2 a}+\frac{\log \left(\tanh ^{-1}(a x)\right)}{2 a}",1,"CoshIntegral[2*ArcTanh[a*x]]/(2*a) + Log[ArcTanh[a*x]]/(2*a)","A",1
286,0,0,37,1.0587549,"\int \frac{1}{x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)} \, dx","Integrate[1/(x*(1 - a^2*x^2)^2*ArcTanh[a*x]),x]","\int \frac{1}{x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)},x\right)+\frac{1}{2} \text{Shi}\left(2 \tanh ^{-1}(a x)\right)",0,"Integrate[1/(x*(1 - a^2*x^2)^2*ArcTanh[a*x]), x]","A",-1
287,0,0,61,3.4020086,"\int \frac{x^3}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2} \, dx","Integrate[x^3/((1 - a^2*x^2)^2*ArcTanh[a*x]^2),x]","\int \frac{x^3}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{\tanh ^{-1}(a x)},x\right)}{a^3}+\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{a^4}+\frac{x}{a^3 \tanh ^{-1}(a x)}-\frac{x}{a^3 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}",0,"Integrate[x^3/((1 - a^2*x^2)^2*ArcTanh[a*x]^2), x]","A",-1
288,1,36,38,0.1483377,"\int \frac{x^2}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2} \, dx","Integrate[x^2/((1 - a^2*x^2)^2*ArcTanh[a*x]^2),x]","\frac{\text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{a^3}+\frac{x^2}{a \left(a^2 x^2-1\right) \tanh ^{-1}(a x)}","\frac{\text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{a^3}-\frac{x^2}{a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}",1,"x^2/(a*(-1 + a^2*x^2)*ArcTanh[a*x]) + SinhIntegral[2*ArcTanh[a*x]]/a^3","A",1
289,1,32,36,0.0792132,"\int \frac{x}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2} \, dx","Integrate[x/((1 - a^2*x^2)^2*ArcTanh[a*x]^2),x]","\frac{\frac{a x}{\left(a^2 x^2-1\right) \tanh ^{-1}(a x)}+\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{a^2}","\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{a^2}-\frac{x}{a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}",1,"((a*x)/((-1 + a^2*x^2)*ArcTanh[a*x]) + CoshIntegral[2*ArcTanh[a*x]])/a^2","A",1
290,1,30,35,0.0927235,"\int \frac{1}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2} \, dx","Integrate[1/((1 - a^2*x^2)^2*ArcTanh[a*x]^2),x]","\frac{\frac{1}{\left(a^2 x^2-1\right) \tanh ^{-1}(a x)}+\text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{a}","\frac{\text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{a}-\frac{1}{a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}",1,"(1/((-1 + a^2*x^2)*ArcTanh[a*x]) + SinhIntegral[2*ArcTanh[a*x]])/a","A",1
291,0,0,62,4.2738531,"\int \frac{1}{x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2} \, dx","Integrate[1/(x*(1 - a^2*x^2)^2*ArcTanh[a*x]^2),x]","\int \frac{1}{x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \tanh ^{-1}(a x)},x\right)}{a}-\frac{a x}{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)}+\text{Chi}\left(2 \tanh ^{-1}(a x)\right)-\frac{1}{a x \tanh ^{-1}(a x)}",0,"Integrate[1/(x*(1 - a^2*x^2)^2*ArcTanh[a*x]^2), x]","A",-1
292,0,0,102,9.4287134,"\int \frac{x^3}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^3} \, dx","Integrate[x^3/((1 - a^2*x^2)^2*ArcTanh[a*x]^3),x]","\int \frac{x^3}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^3} \, dx","-\frac{\text{Int}\left(\frac{1}{\tanh ^{-1}(a x)^2},x\right)}{2 a^3}+\frac{\text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{a^4}+\frac{x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac{a^2 x^2+1}{2 a^4 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{x}{2 a^3 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}",0,"Integrate[x^3/((1 - a^2*x^2)^2*ArcTanh[a*x]^3), x]","A",-1
293,1,47,64,0.1223068,"\int \frac{x^2}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^3} \, dx","Integrate[x^2/((1 - a^2*x^2)^2*ArcTanh[a*x]^3),x]","\frac{\frac{a x \left(a x+2 \tanh ^{-1}(a x)\right)}{\left(a^2 x^2-1\right) \tanh ^{-1}(a x)^2}+2 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{2 a^3}","\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{a^3}-\frac{x^2}{2 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}-\frac{x}{a^2 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}",1,"((a*x*(a*x + 2*ArcTanh[a*x]))/((-1 + a^2*x^2)*ArcTanh[a*x]^2) + 2*CoshIntegral[2*ArcTanh[a*x]])/(2*a^3)","A",1
294,1,66,72,0.064916,"\int \frac{x}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^3} \, dx","Integrate[x/((1 - a^2*x^2)^2*ArcTanh[a*x]^3),x]","\frac{2 \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^2 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)+\left(a^2 x^2+1\right) \tanh ^{-1}(a x)+a x}{2 a^2 \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^2}","\frac{\text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{a^2}-\frac{x}{2 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}-\frac{a^2 x^2+1}{2 a^2 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}",1,"(a*x + (1 + a^2*x^2)*ArcTanh[a*x] + 2*(-1 + a^2*x^2)*ArcTanh[a*x]^2*SinhIntegral[2*ArcTanh[a*x]])/(2*a^2*(-1 + a^2*x^2)*ArcTanh[a*x]^2)","A",1
295,1,58,58,0.0826859,"\int \frac{1}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^3} \, dx","Integrate[1/((1 - a^2*x^2)^2*ArcTanh[a*x]^3),x]","\frac{2 \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^2 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)+2 a x \tanh ^{-1}(a x)+1}{2 a \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^2}","-\frac{x}{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{1}{2 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}+\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{a}",1,"(1 + 2*a*x*ArcTanh[a*x] + 2*(-1 + a^2*x^2)*ArcTanh[a*x]^2*CoshIntegral[2*ArcTanh[a*x]])/(2*a*(-1 + a^2*x^2)*ArcTanh[a*x]^2)","A",1
296,0,0,99,3.833612,"\int \frac{1}{x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^3} \, dx","Integrate[1/(x*(1 - a^2*x^2)^2*ArcTanh[a*x]^3),x]","\int \frac{1}{x \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^3} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \tanh ^{-1}(a x)^2},x\right)}{2 a}-\frac{a x}{2 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}-\frac{a^2 x^2+1}{2 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}+\text{Shi}\left(2 \tanh ^{-1}(a x)\right)-\frac{1}{2 a x \tanh ^{-1}(a x)^2}",0,"Integrate[1/(x*(1 - a^2*x^2)^2*ArcTanh[a*x]^3), x]","A",-1
297,1,73,97,0.1406226,"\int \frac{1}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^4} \, dx","Integrate[1/((1 - a^2*x^2)^2*ArcTanh[a*x]^4),x]","\frac{2 \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^3 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)+\left(a^2 x^2+1\right) \tanh ^{-1}(a x)^2+a x \tanh ^{-1}(a x)+1}{3 a \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^3}","-\frac{x}{3 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}-\frac{a^2 x^2+1}{3 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{1}{3 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3}+\frac{2 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{3 a}",1,"(1 + a*x*ArcTanh[a*x] + (1 + a^2*x^2)*ArcTanh[a*x]^2 + 2*(-1 + a^2*x^2)*ArcTanh[a*x]^3*SinhIntegral[2*ArcTanh[a*x]])/(3*a*(-1 + a^2*x^2)*ArcTanh[a*x]^3)","A",1
298,1,84,120,0.1009336,"\int \frac{1}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^5} \, dx","Integrate[1/((1 - a^2*x^2)^2*ArcTanh[a*x]^5),x]","\frac{4 \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^4 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)+\left(a^2 x^2+1\right) \tanh ^{-1}(a x)^2+4 a x \tanh ^{-1}(a x)^3+2 a x \tanh ^{-1}(a x)+3}{12 a \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^4}","-\frac{x}{3 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{x}{6 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3}-\frac{a^2 x^2+1}{12 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}-\frac{1}{4 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^4}+\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{3 a}",1,"(3 + 2*a*x*ArcTanh[a*x] + (1 + a^2*x^2)*ArcTanh[a*x]^2 + 4*a*x*ArcTanh[a*x]^3 + 4*(-1 + a^2*x^2)*ArcTanh[a*x]^4*CoshIntegral[2*ArcTanh[a*x]])/(12*a*(-1 + a^2*x^2)*ArcTanh[a*x]^4)","A",1
299,1,101,154,0.1306128,"\int \frac{1}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^6} \, dx","Integrate[1/((1 - a^2*x^2)^2*ArcTanh[a*x]^6),x]","\frac{4 \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^5 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)+2 \left(a^2 x^2+1\right) \tanh ^{-1}(a x)^4+\left(a^2 x^2+1\right) \tanh ^{-1}(a x)^2+2 a x \tanh ^{-1}(a x)^3+3 a x \tanh ^{-1}(a x)+6}{30 a \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^5}","-\frac{x}{15 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}-\frac{x}{10 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^4}-\frac{a^2 x^2+1}{15 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{a^2 x^2+1}{30 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3}-\frac{1}{5 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^5}+\frac{2 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{15 a}",1,"(6 + 3*a*x*ArcTanh[a*x] + (1 + a^2*x^2)*ArcTanh[a*x]^2 + 2*a*x*ArcTanh[a*x]^3 + 2*(1 + a^2*x^2)*ArcTanh[a*x]^4 + 4*(-1 + a^2*x^2)*ArcTanh[a*x]^5*SinhIntegral[2*ArcTanh[a*x]])/(30*a*(-1 + a^2*x^2)*ArcTanh[a*x]^5)","A",1
300,1,112,177,0.1017032,"\int \frac{1}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^7} \, dx","Integrate[1/((1 - a^2*x^2)^2*ArcTanh[a*x]^7),x]","\frac{8 \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^6 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)+2 \left(a^2 x^2+1\right) \tanh ^{-1}(a x)^4+3 \left(a^2 x^2+1\right) \tanh ^{-1}(a x)^2+8 a x \tanh ^{-1}(a x)^5+4 a x \tanh ^{-1}(a x)^3+12 a x \tanh ^{-1}(a x)+30}{180 a \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^6}","-\frac{2 x}{45 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{x}{45 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3}-\frac{x}{15 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^5}-\frac{a^2 x^2+1}{90 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}-\frac{a^2 x^2+1}{60 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^4}-\frac{1}{6 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^6}+\frac{2 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{45 a}",1,"(30 + 12*a*x*ArcTanh[a*x] + 3*(1 + a^2*x^2)*ArcTanh[a*x]^2 + 4*a*x*ArcTanh[a*x]^3 + 2*(1 + a^2*x^2)*ArcTanh[a*x]^4 + 8*a*x*ArcTanh[a*x]^5 + 8*(-1 + a^2*x^2)*ArcTanh[a*x]^6*CoshIntegral[2*ArcTanh[a*x]])/(180*a*(-1 + a^2*x^2)*ArcTanh[a*x]^6)","A",1
301,1,128,211,0.136387,"\int \frac{1}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^8} \, dx","Integrate[1/((1 - a^2*x^2)^2*ArcTanh[a*x]^8),x]","\frac{4 \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^7 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)+2 \left(a^2 x^2+1\right) \tanh ^{-1}(a x)^6+\left(a^2 x^2+1\right) \tanh ^{-1}(a x)^4+3 \left(a^2 x^2+1\right) \tanh ^{-1}(a x)^2+2 a x \tanh ^{-1}(a x)^5+3 a x \tanh ^{-1}(a x)^3+15 a x \tanh ^{-1}(a x)+45}{315 a \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^7}","-\frac{2 x}{315 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}-\frac{x}{105 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^4}-\frac{x}{21 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^6}-\frac{2 \left(a^2 x^2+1\right)}{315 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{a^2 x^2+1}{315 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3}-\frac{a^2 x^2+1}{105 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^5}-\frac{1}{7 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^7}+\frac{4 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{315 a}",1,"(45 + 15*a*x*ArcTanh[a*x] + 3*(1 + a^2*x^2)*ArcTanh[a*x]^2 + 3*a*x*ArcTanh[a*x]^3 + (1 + a^2*x^2)*ArcTanh[a*x]^4 + 2*a*x*ArcTanh[a*x]^5 + 2*(1 + a^2*x^2)*ArcTanh[a*x]^6 + 4*(-1 + a^2*x^2)*ArcTanh[a*x]^7*SinhIntegral[2*ArcTanh[a*x]])/(315*a*(-1 + a^2*x^2)*ArcTanh[a*x]^7)","A",1
302,1,98,77,0.0853759,"\int \frac{x^3 \tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^3} \, dx","Integrate[(x^3*ArcTanh[a*x])/(1 - a^2*x^2)^3,x]","-\frac{5 \log (1-a x)}{64 a^4}+\frac{5 \log (a x+1)}{64 a^4}+\frac{\left(2 a^2 x^2-1\right) \tanh ^{-1}(a x)}{4 a^4 \left(a^2 x^2-1\right)^2}-\frac{5 x}{32 a^3 \left(a^2 x^2-1\right)}-\frac{x}{16 a^3 \left(a^2 x^2-1\right)^2}","-\frac{3 \tanh ^{-1}(a x)}{32 a^4}+\frac{x^4 \tanh ^{-1}(a x)}{4 \left(1-a^2 x^2\right)^2}-\frac{x^3}{16 a \left(1-a^2 x^2\right)^2}+\frac{3 x}{32 a^3 \left(1-a^2 x^2\right)}",1,"-1/16*x/(a^3*(-1 + a^2*x^2)^2) - (5*x)/(32*a^3*(-1 + a^2*x^2)) + ((-1 + 2*a^2*x^2)*ArcTanh[a*x])/(4*a^4*(-1 + a^2*x^2)^2) - (5*Log[1 - a*x])/(64*a^4) + (5*Log[1 + a*x])/(64*a^4)","A",1
303,1,61,100,0.089023,"\int \frac{x^2 \tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^3} \, dx","Integrate[(x^2*ArcTanh[a*x])/(1 - a^2*x^2)^3,x]","-\frac{-2 \left(a^3 x^3+a x\right) \tanh ^{-1}(a x)+a^2 x^2+\left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2}{16 a^3 \left(a^2 x^2-1\right)^2}","-\frac{\tanh ^{-1}(a x)^2}{16 a^3}-\frac{x \tanh ^{-1}(a x)}{8 a^2 \left(1-a^2 x^2\right)}+\frac{x \tanh ^{-1}(a x)}{4 a^2 \left(1-a^2 x^2\right)^2}+\frac{1}{16 a^3 \left(1-a^2 x^2\right)}-\frac{1}{16 a^3 \left(1-a^2 x^2\right)^2}",1,"-1/16*(a^2*x^2 - 2*(a*x + a^3*x^3)*ArcTanh[a*x] + (-1 + a^2*x^2)^2*ArcTanh[a*x]^2)/(a^3*(-1 + a^2*x^2)^2)","A",1
304,1,88,75,0.0600086,"\int \frac{x \tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^3} \, dx","Integrate[(x*ArcTanh[a*x])/(1 - a^2*x^2)^3,x]","\frac{3 x}{32 a \left(a^2 x^2-1\right)}-\frac{x}{16 a \left(a^2 x^2-1\right)^2}+\frac{\tanh ^{-1}(a x)}{4 a^2 \left(a^2 x^2-1\right)^2}+\frac{3 \log (1-a x)}{64 a^2}-\frac{3 \log (a x+1)}{64 a^2}","-\frac{3 x}{32 a \left(1-a^2 x^2\right)}-\frac{x}{16 a \left(1-a^2 x^2\right)^2}+\frac{\tanh ^{-1}(a x)}{4 a^2 \left(1-a^2 x^2\right)^2}-\frac{3 \tanh ^{-1}(a x)}{32 a^2}",1,"-1/16*x/(a*(-1 + a^2*x^2)^2) + (3*x)/(32*a*(-1 + a^2*x^2)) + ArcTanh[a*x]/(4*a^2*(-1 + a^2*x^2)^2) + (3*Log[1 - a*x])/(64*a^2) - (3*Log[1 + a*x])/(64*a^2)","A",1
305,1,65,94,0.0694024,"\int \frac{\tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^3} \, dx","Integrate[ArcTanh[a*x]/(1 - a^2*x^2)^3,x]","\frac{\left(10 a x-6 a^3 x^3\right) \tanh ^{-1}(a x)+3 a^2 x^2+3 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2-4}{16 a \left(a^2 x^2-1\right)^2}","-\frac{3}{16 a \left(1-a^2 x^2\right)}-\frac{1}{16 a \left(1-a^2 x^2\right)^2}+\frac{3 x \tanh ^{-1}(a x)}{8 \left(1-a^2 x^2\right)}+\frac{x \tanh ^{-1}(a x)}{4 \left(1-a^2 x^2\right)^2}+\frac{3 \tanh ^{-1}(a x)^2}{16 a}",1,"(-4 + 3*a^2*x^2 + (10*a*x - 6*a^3*x^3)*ArcTanh[a*x] + 3*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2)/(16*a*(-1 + a^2*x^2)^2)","A",1
306,1,81,129,0.2084683,"\int \frac{\tanh ^{-1}(a x)}{x \left(1-a^2 x^2\right)^3} \, dx","Integrate[ArcTanh[a*x]/(x*(1 - a^2*x^2)^3),x]","\frac{1}{128} \left(-64 \text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)+64 \tanh ^{-1}(a x)^2-24 \sinh \left(2 \tanh ^{-1}(a x)\right)-\sinh \left(4 \tanh ^{-1}(a x)\right)+4 \tanh ^{-1}(a x) \left(32 \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)+12 \cosh \left(2 \tanh ^{-1}(a x)\right)+\cosh \left(4 \tanh ^{-1}(a x)\right)\right)\right)","-\frac{11 a x}{32 \left(1-a^2 x^2\right)}-\frac{a x}{16 \left(1-a^2 x^2\right)^2}+\frac{\tanh ^{-1}(a x)}{2 \left(1-a^2 x^2\right)}+\frac{\tanh ^{-1}(a x)}{4 \left(1-a^2 x^2\right)^2}-\frac{1}{2} \text{Li}_2\left(\frac{2}{a x+1}-1\right)+\frac{1}{2} \tanh ^{-1}(a x)^2-\frac{11}{32} \tanh ^{-1}(a x)+\log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)",1,"(64*ArcTanh[a*x]^2 + 4*ArcTanh[a*x]*(12*Cosh[2*ArcTanh[a*x]] + Cosh[4*ArcTanh[a*x]] + 32*Log[1 - E^(-2*ArcTanh[a*x])]) - 64*PolyLog[2, E^(-2*ArcTanh[a*x])] - 24*Sinh[2*ArcTanh[a*x]] - Sinh[4*ArcTanh[a*x]])/128","A",0
307,1,94,123,0.1708918,"\int \frac{\tanh ^{-1}(a x)}{x^2 \left(1-a^2 x^2\right)^3} \, dx","Integrate[ArcTanh[a*x]/(x^2*(1 - a^2*x^2)^3),x]","\frac{1}{16} \left(a \left(\frac{7 a^2 x^2-8}{\left(a^2 x^2-1\right)^2}-8 \log \left(1-a^2 x^2\right)+16 \log (x)\right)-\frac{2 \left(15 a^4 x^4-25 a^2 x^2+8\right) \tanh ^{-1}(a x)}{x \left(a^2 x^2-1\right)^2}+15 a \tanh ^{-1}(a x)^2\right)","-\frac{7 a}{16 \left(1-a^2 x^2\right)}-\frac{a}{16 \left(1-a^2 x^2\right)^2}-\frac{1}{2} a \log \left(1-a^2 x^2\right)+\frac{7 a^2 x \tanh ^{-1}(a x)}{8 \left(1-a^2 x^2\right)}+\frac{a^2 x \tanh ^{-1}(a x)}{4 \left(1-a^2 x^2\right)^2}+a \log (x)+\frac{15}{16} a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)}{x}",1,"((-2*(8 - 25*a^2*x^2 + 15*a^4*x^4)*ArcTanh[a*x])/(x*(-1 + a^2*x^2)^2) + 15*a*ArcTanh[a*x]^2 + a*((-8 + 7*a^2*x^2)/(-1 + a^2*x^2)^2 + 16*Log[x] - 8*Log[1 - a^2*x^2]))/16","A",1
308,1,71,127,0.0976018,"\int \frac{x^3 \tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^3} \, dx","Integrate[(x^3*ArcTanh[a*x]^2)/(1 - a^2*x^2)^3,x]","\frac{\left(6 a x-10 a^3 x^3\right) \tanh ^{-1}(a x)+5 a^2 x^2+\left(5 a^4 x^4+6 a^2 x^2-3\right) \tanh ^{-1}(a x)^2-4}{32 a^4 \left(a^2 x^2-1\right)^2}","-\frac{3 \tanh ^{-1}(a x)^2}{32 a^4}+\frac{x^4}{32 \left(1-a^2 x^2\right)^2}+\frac{x^4 \tanh ^{-1}(a x)^2}{4 \left(1-a^2 x^2\right)^2}-\frac{x^3 \tanh ^{-1}(a x)}{8 a \left(1-a^2 x^2\right)^2}-\frac{3}{32 a^4 \left(1-a^2 x^2\right)}+\frac{3 x \tanh ^{-1}(a x)}{16 a^3 \left(1-a^2 x^2\right)}",1,"(-4 + 5*a^2*x^2 + (6*a*x - 10*a^3*x^3)*ArcTanh[a*x] + (-3 + 6*a^2*x^2 + 5*a^4*x^4)*ArcTanh[a*x]^2)/(32*a^4*(-1 + a^2*x^2)^2)","A",1
309,1,121,163,0.1213961,"\int \frac{x^2 \tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^3} \, dx","Integrate[(x^2*ArcTanh[a*x]^2)/(1 - a^2*x^2)^3,x]","\frac{48 \left(a^3 x^3+a x\right) \tanh ^{-1}(a x)^2+6 a x \left(a^2 x^2+1\right)+3 \left(a^2 x^2-1\right)^2 \log (1-a x)-3 \left(a^2 x^2-1\right)^2 \log (a x+1)-16 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^3-48 a^2 x^2 \tanh ^{-1}(a x)}{384 a^3 \left(a^2 x^2-1\right)^2}","-\frac{\tanh ^{-1}(a x)^3}{24 a^3}-\frac{\tanh ^{-1}(a x)}{64 a^3}-\frac{x}{64 a^2 \left(1-a^2 x^2\right)}+\frac{x}{32 a^2 \left(1-a^2 x^2\right)^2}-\frac{x \tanh ^{-1}(a x)^2}{8 a^2 \left(1-a^2 x^2\right)}+\frac{x \tanh ^{-1}(a x)^2}{4 a^2 \left(1-a^2 x^2\right)^2}+\frac{\tanh ^{-1}(a x)}{8 a^3 \left(1-a^2 x^2\right)}-\frac{\tanh ^{-1}(a x)}{8 a^3 \left(1-a^2 x^2\right)^2}",1,"(6*a*x*(1 + a^2*x^2) - 48*a^2*x^2*ArcTanh[a*x] + 48*(a*x + a^3*x^3)*ArcTanh[a*x]^2 - 16*(-1 + a^2*x^2)^2*ArcTanh[a*x]^3 + 3*(-1 + a^2*x^2)^2*Log[1 - a*x] - 3*(-1 + a^2*x^2)^2*Log[1 + a*x])/(384*a^3*(-1 + a^2*x^2)^2)","A",1
310,1,71,125,0.0738339,"\int \frac{x \tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^3} \, dx","Integrate[(x*ArcTanh[a*x]^2)/(1 - a^2*x^2)^3,x]","\frac{-3 a^2 x^2+2 a x \left(3 a^2 x^2-5\right) \tanh ^{-1}(a x)+\left(-3 a^4 x^4+6 a^2 x^2+5\right) \tanh ^{-1}(a x)^2+4}{32 a^2 \left(a^2 x^2-1\right)^2}","\frac{3}{32 a^2 \left(1-a^2 x^2\right)}+\frac{1}{32 a^2 \left(1-a^2 x^2\right)^2}+\frac{\tanh ^{-1}(a x)^2}{4 a^2 \left(1-a^2 x^2\right)^2}-\frac{3 x \tanh ^{-1}(a x)}{16 a \left(1-a^2 x^2\right)}-\frac{x \tanh ^{-1}(a x)}{8 a \left(1-a^2 x^2\right)^2}-\frac{3 \tanh ^{-1}(a x)^2}{32 a^2}",1,"(4 - 3*a^2*x^2 + 2*a*x*(-5 + 3*a^2*x^2)*ArcTanh[a*x] + (5 + 6*a^2*x^2 - 3*a^4*x^4)*ArcTanh[a*x]^2)/(32*a^2*(-1 + a^2*x^2)^2)","A",1
311,1,127,151,0.1096238,"\int \frac{\tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^3} \, dx","Integrate[ArcTanh[a*x]^2/(1 - a^2*x^2)^3,x]","\frac{1}{128} \left(-\frac{30 x}{a^2 x^2-1}+\frac{4 x}{\left(a^2 x^2-1\right)^2}-\frac{16 x \left(3 a^2 x^2-5\right) \tanh ^{-1}(a x)^2}{\left(a^2 x^2-1\right)^2}+\frac{16 \left(3 a^2 x^2-4\right) \tanh ^{-1}(a x)}{a \left(a^2 x^2-1\right)^2}-\frac{15 \log (1-a x)}{a}+\frac{15 \log (a x+1)}{a}+\frac{16 \tanh ^{-1}(a x)^3}{a}\right)","\frac{15 x}{64 \left(1-a^2 x^2\right)}+\frac{x}{32 \left(1-a^2 x^2\right)^2}+\frac{3 x \tanh ^{-1}(a x)^2}{8 \left(1-a^2 x^2\right)}+\frac{x \tanh ^{-1}(a x)^2}{4 \left(1-a^2 x^2\right)^2}-\frac{3 \tanh ^{-1}(a x)}{8 a \left(1-a^2 x^2\right)}-\frac{\tanh ^{-1}(a x)}{8 a \left(1-a^2 x^2\right)^2}+\frac{\tanh ^{-1}(a x)^3}{8 a}+\frac{15 \tanh ^{-1}(a x)}{64 a}",1,"((4*x)/(-1 + a^2*x^2)^2 - (30*x)/(-1 + a^2*x^2) + (16*(-4 + 3*a^2*x^2)*ArcTanh[a*x])/(a*(-1 + a^2*x^2)^2) - (16*x*(-5 + 3*a^2*x^2)*ArcTanh[a*x]^2)/(-1 + a^2*x^2)^2 + (16*ArcTanh[a*x]^3)/a - (15*Log[1 - a*x])/a + (15*Log[1 + a*x])/a)/128","A",1
312,1,129,196,0.4082513,"\int \frac{\tanh ^{-1}(a x)^2}{x \left(1-a^2 x^2\right)^3} \, dx","Integrate[ArcTanh[a*x]^2/(x*(1 - a^2*x^2)^3),x]","\tanh ^{-1}(a x) \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)+\frac{1}{768} \left(-384 \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)-256 \tanh ^{-1}(a x)^3-12 \tanh ^{-1}(a x) \left(24 \sinh \left(2 \tanh ^{-1}(a x)\right)+\sinh \left(4 \tanh ^{-1}(a x)\right)\right)+144 \cosh \left(2 \tanh ^{-1}(a x)\right)+3 \cosh \left(4 \tanh ^{-1}(a x)\right)+24 \tanh ^{-1}(a x)^2 \left(32 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)+12 \cosh \left(2 \tanh ^{-1}(a x)\right)+\cosh \left(4 \tanh ^{-1}(a x)\right)\right)+32 i \pi ^3\right)","\frac{11}{32 \left(1-a^2 x^2\right)}+\frac{1}{32 \left(1-a^2 x^2\right)^2}+\frac{\tanh ^{-1}(a x)^2}{2 \left(1-a^2 x^2\right)}+\frac{\tanh ^{-1}(a x)^2}{4 \left(1-a^2 x^2\right)^2}-\frac{11 a x \tanh ^{-1}(a x)}{16 \left(1-a^2 x^2\right)}-\frac{a x \tanh ^{-1}(a x)}{8 \left(1-a^2 x^2\right)^2}-\frac{1}{2} \text{Li}_3\left(\frac{2}{a x+1}-1\right)-\text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)+\frac{1}{3} \tanh ^{-1}(a x)^3-\frac{11}{32} \tanh ^{-1}(a x)^2+\log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^2",1,"ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] + ((32*I)*Pi^3 - 256*ArcTanh[a*x]^3 + 144*Cosh[2*ArcTanh[a*x]] + 3*Cosh[4*ArcTanh[a*x]] + 24*ArcTanh[a*x]^2*(12*Cosh[2*ArcTanh[a*x]] + Cosh[4*ArcTanh[a*x]] + 32*Log[1 - E^(2*ArcTanh[a*x])]) - 384*PolyLog[3, E^(2*ArcTanh[a*x])] - 12*ArcTanh[a*x]*(24*Sinh[2*ArcTanh[a*x]] + Sinh[4*ArcTanh[a*x]]))/768","C",0
313,1,127,209,0.5925479,"\int \frac{\tanh ^{-1}(a x)^2}{x^2 \left(1-a^2 x^2\right)^3} \, dx","Integrate[ArcTanh[a*x]^2/(x^2*(1 - a^2*x^2)^3),x]","-a \left(\tanh ^{-1}(a x)^2 \left(\frac{a x}{a^2 x^2-1}+\frac{1}{a x}-\frac{1}{32} \sinh \left(4 \tanh ^{-1}(a x)\right)-1\right)+\text{Li}_2\left(e^{-2 \tanh ^{-1}(a x)}\right)-\frac{5}{8} \tanh ^{-1}(a x)^3-\frac{1}{4} \sinh \left(2 \tanh ^{-1}(a x)\right)-\frac{1}{256} \sinh \left(4 \tanh ^{-1}(a x)\right)+\frac{1}{64} \tanh ^{-1}(a x) \left(-128 \log \left(1-e^{-2 \tanh ^{-1}(a x)}\right)+32 \cosh \left(2 \tanh ^{-1}(a x)\right)+\cosh \left(4 \tanh ^{-1}(a x)\right)\right)\right)","\frac{31 a^2 x}{64 \left(1-a^2 x^2\right)}+\frac{a^2 x}{32 \left(1-a^2 x^2\right)^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^2}{8 \left(1-a^2 x^2\right)}+\frac{a^2 x \tanh ^{-1}(a x)^2}{4 \left(1-a^2 x^2\right)^2}-\frac{7 a \tanh ^{-1}(a x)}{8 \left(1-a^2 x^2\right)}-\frac{a \tanh ^{-1}(a x)}{8 \left(1-a^2 x^2\right)^2}-a \text{Li}_2\left(\frac{2}{a x+1}-1\right)+\frac{5}{8} a \tanh ^{-1}(a x)^3+a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{31}{64} a \tanh ^{-1}(a x)+2 a \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)",1,"-(a*((-5*ArcTanh[a*x]^3)/8 + (ArcTanh[a*x]*(32*Cosh[2*ArcTanh[a*x]] + Cosh[4*ArcTanh[a*x]] - 128*Log[1 - E^(-2*ArcTanh[a*x])]))/64 + PolyLog[2, E^(-2*ArcTanh[a*x])] - Sinh[2*ArcTanh[a*x]]/4 + ArcTanh[a*x]^2*(-1 + 1/(a*x) + (a*x)/(-1 + a^2*x^2) - Sinh[4*ArcTanh[a*x]]/32) - Sinh[4*ArcTanh[a*x]]/256))","A",1
314,1,135,192,0.1282038,"\int \frac{x^3 \tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^3} \, dx","Integrate[(x^3*ArcTanh[a*x]^3)/(1 - a^2*x^2)^3,x]","\frac{-48 a x \left(5 a^2 x^2-3\right) \tanh ^{-1}(a x)^2+48 \left(5 a^2 x^2-4\right) \tanh ^{-1}(a x)+16 \left(5 a^4 x^4+6 a^2 x^2-3\right) \tanh ^{-1}(a x)^3+3 \left(-34 a^3 x^3-17 \left(a^2 x^2-1\right)^2 \log (1-a x)+17 \left(a^2 x^2-1\right)^2 \log (a x+1)+30 a x\right)}{512 a^4 \left(a^2 x^2-1\right)^2}","-\frac{3 \tanh ^{-1}(a x)^3}{32 a^4}+\frac{27 \tanh ^{-1}(a x)}{256 a^4}+\frac{x^4 \tanh ^{-1}(a x)^3}{4 \left(1-a^2 x^2\right)^2}+\frac{3 x^4 \tanh ^{-1}(a x)}{32 \left(1-a^2 x^2\right)^2}-\frac{3 x^3}{128 a \left(1-a^2 x^2\right)^2}-\frac{3 x^3 \tanh ^{-1}(a x)^2}{16 a \left(1-a^2 x^2\right)^2}-\frac{9 \tanh ^{-1}(a x)}{32 a^4 \left(1-a^2 x^2\right)}+\frac{45 x}{256 a^3 \left(1-a^2 x^2\right)}+\frac{9 x \tanh ^{-1}(a x)^2}{32 a^3 \left(1-a^2 x^2\right)}",1,"(48*(-4 + 5*a^2*x^2)*ArcTanh[a*x] - 48*a*x*(-3 + 5*a^2*x^2)*ArcTanh[a*x]^2 + 16*(-3 + 6*a^2*x^2 + 5*a^4*x^4)*ArcTanh[a*x]^3 + 3*(30*a*x - 34*a^3*x^3 - 17*(-1 + a^2*x^2)^2*Log[1 - a*x] + 17*(-1 + a^2*x^2)^2*Log[1 + a*x]))/(512*a^4*(-1 + a^2*x^2)^2)","A",1
315,1,107,215,0.1193193,"\int \frac{x^2 \tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^3} \, dx","Integrate[(x^2*ArcTanh[a*x]^3)/(1 - a^2*x^2)^3,x]","\frac{16 \left(a^3 x^3+a x\right) \tanh ^{-1}(a x)^3+6 \left(a^3 x^3+a x\right) \tanh ^{-1}(a x)-3 a^2 x^2-4 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^4-3 \left(a^4 x^4+6 a^2 x^2+1\right) \tanh ^{-1}(a x)^2}{128 a^3 \left(a^2 x^2-1\right)^2}","-\frac{\tanh ^{-1}(a x)^4}{32 a^3}-\frac{3 \tanh ^{-1}(a x)^2}{128 a^3}-\frac{x \tanh ^{-1}(a x)^3}{8 a^2 \left(1-a^2 x^2\right)}+\frac{x \tanh ^{-1}(a x)^3}{4 a^2 \left(1-a^2 x^2\right)^2}-\frac{3 x \tanh ^{-1}(a x)}{64 a^2 \left(1-a^2 x^2\right)}+\frac{3 x \tanh ^{-1}(a x)}{32 a^2 \left(1-a^2 x^2\right)^2}+\frac{3}{128 a^3 \left(1-a^2 x^2\right)}-\frac{3}{128 a^3 \left(1-a^2 x^2\right)^2}+\frac{3 \tanh ^{-1}(a x)^2}{16 a^3 \left(1-a^2 x^2\right)}-\frac{3 \tanh ^{-1}(a x)^2}{16 a^3 \left(1-a^2 x^2\right)^2}",1,"(-3*a^2*x^2 + 6*(a*x + a^3*x^3)*ArcTanh[a*x] - 3*(1 + 6*a^2*x^2 + a^4*x^4)*ArcTanh[a*x]^2 + 16*(a*x + a^3*x^3)*ArcTanh[a*x]^3 - 4*(-1 + a^2*x^2)^2*ArcTanh[a*x]^4)/(128*a^3*(-1 + a^2*x^2)^2)","A",1
316,1,148,188,0.0905555,"\int \frac{x \tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^3} \, dx","Integrate[(x*ArcTanh[a*x]^3)/(1 - a^2*x^2)^3,x]","\frac{-45 a^4 x^4 \log (a x+1)+90 a^3 x^3+90 a^2 x^2 \log (a x+1)+45 \left(a^2 x^2-1\right)^2 \log (1-a x)+48 a x \left(3 a^2 x^2-5\right) \tanh ^{-1}(a x)^2-48 \left(3 a^2 x^2-4\right) \tanh ^{-1}(a x)+\left(-48 a^4 x^4+96 a^2 x^2+80\right) \tanh ^{-1}(a x)^3-102 a x-45 \log (a x+1)}{512 a^2 \left(a^2 x^2-1\right)^2}","-\frac{45 x}{256 a \left(1-a^2 x^2\right)}-\frac{3 x}{128 a \left(1-a^2 x^2\right)^2}+\frac{\tanh ^{-1}(a x)^3}{4 a^2 \left(1-a^2 x^2\right)^2}-\frac{9 x \tanh ^{-1}(a x)^2}{32 a \left(1-a^2 x^2\right)}-\frac{3 x \tanh ^{-1}(a x)^2}{16 a \left(1-a^2 x^2\right)^2}+\frac{9 \tanh ^{-1}(a x)}{32 a^2 \left(1-a^2 x^2\right)}+\frac{3 \tanh ^{-1}(a x)}{32 a^2 \left(1-a^2 x^2\right)^2}-\frac{3 \tanh ^{-1}(a x)^3}{32 a^2}-\frac{45 \tanh ^{-1}(a x)}{256 a^2}",1,"(-102*a*x + 90*a^3*x^3 - 48*(-4 + 3*a^2*x^2)*ArcTanh[a*x] + 48*a*x*(-5 + 3*a^2*x^2)*ArcTanh[a*x]^2 + (80 + 96*a^2*x^2 - 48*a^4*x^4)*ArcTanh[a*x]^3 + 45*(-1 + a^2*x^2)^2*Log[1 - a*x] - 45*Log[1 + a*x] + 90*a^2*x^2*Log[1 + a*x] - 45*a^4*x^4*Log[1 + a*x])/(512*a^2*(-1 + a^2*x^2)^2)","A",1
317,1,111,203,0.1649133,"\int \frac{\tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^3} \, dx","Integrate[ArcTanh[a*x]^3/(1 - a^2*x^2)^3,x]","\frac{\left(80 a x-48 a^3 x^3\right) \tanh ^{-1}(a x)^3+\left(102 a x-90 a^3 x^3\right) \tanh ^{-1}(a x)+45 a^2 x^2+12 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^4+3 \left(15 a^4 x^4-6 a^2 x^2-17\right) \tanh ^{-1}(a x)^2-48}{128 a \left(a^2 x^2-1\right)^2}","-\frac{45}{128 a \left(1-a^2 x^2\right)}-\frac{3}{128 a \left(1-a^2 x^2\right)^2}+\frac{3 x \tanh ^{-1}(a x)^3}{8 \left(1-a^2 x^2\right)}+\frac{x \tanh ^{-1}(a x)^3}{4 \left(1-a^2 x^2\right)^2}-\frac{9 \tanh ^{-1}(a x)^2}{16 a \left(1-a^2 x^2\right)}-\frac{3 \tanh ^{-1}(a x)^2}{16 a \left(1-a^2 x^2\right)^2}+\frac{45 x \tanh ^{-1}(a x)}{64 \left(1-a^2 x^2\right)}+\frac{3 x \tanh ^{-1}(a x)}{32 \left(1-a^2 x^2\right)^2}+\frac{3 \tanh ^{-1}(a x)^4}{32 a}+\frac{45 \tanh ^{-1}(a x)^2}{128 a}",1,"(-48 + 45*a^2*x^2 + (102*a*x - 90*a^3*x^3)*ArcTanh[a*x] + 3*(-17 - 6*a^2*x^2 + 15*a^4*x^4)*ArcTanh[a*x]^2 + (80*a*x - 48*a^3*x^3)*ArcTanh[a*x]^3 + 12*(-1 + a^2*x^2)^2*ArcTanh[a*x]^4)/(128*a*(-1 + a^2*x^2)^2)","A",1
318,1,189,277,0.2954722,"\int \frac{\tanh ^{-1}(a x)^3}{x \left(1-a^2 x^2\right)^3} \, dx","Integrate[ArcTanh[a*x]^3/(x*(1 - a^2*x^2)^3),x]","\frac{1536 \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)-1536 \tanh ^{-1}(a x) \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)+768 \text{Li}_4\left(e^{2 \tanh ^{-1}(a x)}\right)-256 \tanh ^{-1}(a x)^4+1024 \tanh ^{-1}(a x)^3 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)-576 \tanh ^{-1}(a x)^2 \sinh \left(2 \tanh ^{-1}(a x)\right)-24 \tanh ^{-1}(a x)^2 \sinh \left(4 \tanh ^{-1}(a x)\right)-288 \sinh \left(2 \tanh ^{-1}(a x)\right)-3 \sinh \left(4 \tanh ^{-1}(a x)\right)+384 \tanh ^{-1}(a x)^3 \cosh \left(2 \tanh ^{-1}(a x)\right)+32 \tanh ^{-1}(a x)^3 \cosh \left(4 \tanh ^{-1}(a x)\right)+576 \tanh ^{-1}(a x) \cosh \left(2 \tanh ^{-1}(a x)\right)+12 \tanh ^{-1}(a x) \cosh \left(4 \tanh ^{-1}(a x)\right)+16 \pi ^4}{1024}","-\frac{141 a x}{256 \left(1-a^2 x^2\right)}-\frac{3 a x}{128 \left(1-a^2 x^2\right)^2}+\frac{\tanh ^{-1}(a x)^3}{2 \left(1-a^2 x^2\right)}+\frac{\tanh ^{-1}(a x)^3}{4 \left(1-a^2 x^2\right)^2}-\frac{33 a x \tanh ^{-1}(a x)^2}{32 \left(1-a^2 x^2\right)}-\frac{3 a x \tanh ^{-1}(a x)^2}{16 \left(1-a^2 x^2\right)^2}+\frac{33 \tanh ^{-1}(a x)}{32 \left(1-a^2 x^2\right)}+\frac{3 \tanh ^{-1}(a x)}{32 \left(1-a^2 x^2\right)^2}-\frac{3}{4} \text{Li}_4\left(\frac{2}{a x+1}-1\right)-\frac{3}{2} \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)^2-\frac{3}{2} \text{Li}_3\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)+\frac{1}{4} \tanh ^{-1}(a x)^4-\frac{11}{32} \tanh ^{-1}(a x)^3-\frac{141}{256} \tanh ^{-1}(a x)+\log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^3",1,"(16*Pi^4 - 256*ArcTanh[a*x]^4 + 576*ArcTanh[a*x]*Cosh[2*ArcTanh[a*x]] + 384*ArcTanh[a*x]^3*Cosh[2*ArcTanh[a*x]] + 12*ArcTanh[a*x]*Cosh[4*ArcTanh[a*x]] + 32*ArcTanh[a*x]^3*Cosh[4*ArcTanh[a*x]] + 1024*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] + 1536*ArcTanh[a*x]^2*PolyLog[2, E^(2*ArcTanh[a*x])] - 1536*ArcTanh[a*x]*PolyLog[3, E^(2*ArcTanh[a*x])] + 768*PolyLog[4, E^(2*ArcTanh[a*x])] - 288*Sinh[2*ArcTanh[a*x]] - 576*ArcTanh[a*x]^2*Sinh[2*ArcTanh[a*x]] - 3*Sinh[4*ArcTanh[a*x]] - 24*ArcTanh[a*x]^2*Sinh[4*ArcTanh[a*x]])/1024","A",0
319,1,218,281,0.6959242,"\int \frac{\tanh ^{-1}(a x)^3}{x^2 \left(1-a^2 x^2\right)^3} \, dx","Integrate[ArcTanh[a*x]^3/(x^2*(1 - a^2*x^2)^3),x]","-a \left(-\frac{a x \tanh ^{-1}(a x)^3}{1-a^2 x^2}-3 \tanh ^{-1}(a x) \text{Li}_2\left(e^{2 \tanh ^{-1}(a x)}\right)+\frac{3}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(a x)}\right)-\frac{15}{32} \tanh ^{-1}(a x)^4+\frac{\tanh ^{-1}(a x)^3}{a x}+\tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2 \log \left(1-e^{2 \tanh ^{-1}(a x)}\right)-\frac{1}{32} \tanh ^{-1}(a x)^3 \sinh \left(4 \tanh ^{-1}(a x)\right)-\frac{3}{4} \tanh ^{-1}(a x) \sinh \left(2 \tanh ^{-1}(a x)\right)-\frac{3}{256} \tanh ^{-1}(a x) \sinh \left(4 \tanh ^{-1}(a x)\right)+\frac{3}{4} \tanh ^{-1}(a x)^2 \cosh \left(2 \tanh ^{-1}(a x)\right)+\frac{3}{128} \tanh ^{-1}(a x)^2 \cosh \left(4 \tanh ^{-1}(a x)\right)+\frac{3}{8} \cosh \left(2 \tanh ^{-1}(a x)\right)+\frac{3 \cosh \left(4 \tanh ^{-1}(a x)\right)}{1024}-\frac{i \pi ^3}{8}\right)","-\frac{93 a}{128 \left(1-a^2 x^2\right)}-\frac{3 a}{128 \left(1-a^2 x^2\right)^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^3}{8 \left(1-a^2 x^2\right)}+\frac{a^2 x \tanh ^{-1}(a x)^3}{4 \left(1-a^2 x^2\right)^2}-\frac{21 a \tanh ^{-1}(a x)^2}{16 \left(1-a^2 x^2\right)}-\frac{3 a \tanh ^{-1}(a x)^2}{16 \left(1-a^2 x^2\right)^2}+\frac{93 a^2 x \tanh ^{-1}(a x)}{64 \left(1-a^2 x^2\right)}+\frac{3 a^2 x \tanh ^{-1}(a x)}{32 \left(1-a^2 x^2\right)^2}-\frac{3}{2} a \text{Li}_3\left(\frac{2}{a x+1}-1\right)-3 a \text{Li}_2\left(\frac{2}{a x+1}-1\right) \tanh ^{-1}(a x)+\frac{15}{32} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{93}{128} a \tanh ^{-1}(a x)^2+3 a \log \left(2-\frac{2}{a x+1}\right) \tanh ^{-1}(a x)^2",1,"-(a*((-1/8*I)*Pi^3 + ArcTanh[a*x]^3 + ArcTanh[a*x]^3/(a*x) - (a*x*ArcTanh[a*x]^3)/(1 - a^2*x^2) - (15*ArcTanh[a*x]^4)/32 + (3*Cosh[2*ArcTanh[a*x]])/8 + (3*ArcTanh[a*x]^2*Cosh[2*ArcTanh[a*x]])/4 + (3*Cosh[4*ArcTanh[a*x]])/1024 + (3*ArcTanh[a*x]^2*Cosh[4*ArcTanh[a*x]])/128 - 3*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] - 3*ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] + (3*PolyLog[3, E^(2*ArcTanh[a*x])])/2 - (3*ArcTanh[a*x]*Sinh[2*ArcTanh[a*x]])/4 - (3*ArcTanh[a*x]*Sinh[4*ArcTanh[a*x]])/256 - (ArcTanh[a*x]^3*Sinh[4*ArcTanh[a*x]])/32))","C",0
320,1,152,168,0.5037775,"\int \frac{\sqrt{\tanh ^{-1}(a x)}}{\left(1-a^2 x^2\right)^3} \, dx","Integrate[Sqrt[ArcTanh[a*x]]/(1 - a^2*x^2)^3,x]","\frac{\frac{32 \sqrt{\tanh ^{-1}(a x)} \left(-3 a^3 x^3+2 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)+5 a x\right)}{\left(a^2 x^2-1\right)^2}+\frac{\sqrt{\tanh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 \tanh ^{-1}(a x)\right)}{\sqrt{-\tanh ^{-1}(a x)}}+\frac{8 \sqrt{2} \sqrt{\tanh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 \tanh ^{-1}(a x)\right)}{\sqrt{-\tanh ^{-1}(a x)}}-8 \sqrt{2} \Gamma \left(\frac{1}{2},2 \tanh ^{-1}(a x)\right)-\Gamma \left(\frac{1}{2},4 \tanh ^{-1}(a x)\right)}{256 a}","\frac{\sqrt{\pi } \text{erf}\left(2 \sqrt{\tanh ^{-1}(a x)}\right)}{256 a}+\frac{\sqrt{\frac{\pi }{2}} \text{erf}\left(\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right)}{16 a}-\frac{\sqrt{\pi } \text{erfi}\left(2 \sqrt{\tanh ^{-1}(a x)}\right)}{256 a}-\frac{\sqrt{\frac{\pi }{2}} \text{erfi}\left(\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right)}{16 a}+\frac{\tanh ^{-1}(a x)^{3/2}}{4 a}+\frac{\sqrt{\tanh ^{-1}(a x)} \sinh \left(2 \tanh ^{-1}(a x)\right)}{4 a}+\frac{\sqrt{\tanh ^{-1}(a x)} \sinh \left(4 \tanh ^{-1}(a x)\right)}{32 a}",1,"((32*Sqrt[ArcTanh[a*x]]*(5*a*x - 3*a^3*x^3 + 2*(-1 + a^2*x^2)^2*ArcTanh[a*x]))/(-1 + a^2*x^2)^2 + (Sqrt[ArcTanh[a*x]]*Gamma[1/2, -4*ArcTanh[a*x]])/Sqrt[-ArcTanh[a*x]] + (8*Sqrt[2]*Sqrt[ArcTanh[a*x]]*Gamma[1/2, -2*ArcTanh[a*x]])/Sqrt[-ArcTanh[a*x]] - 8*Sqrt[2]*Gamma[1/2, 2*ArcTanh[a*x]] - Gamma[1/2, 4*ArcTanh[a*x]])/(256*a)","A",0
321,0,0,25,7.8601119,"\int \frac{x^6}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)} \, dx","Integrate[x^6/((1 - a^2*x^2)^3*ArcTanh[a*x]),x]","\int \frac{x^6}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^6}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)},x\right)",0,"Integrate[x^6/((1 - a^2*x^2)^3*ArcTanh[a*x]), x]","A",-1
322,0,0,25,15.2466051,"\int \frac{x^5}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)} \, dx","Integrate[x^5/((1 - a^2*x^2)^3*ArcTanh[a*x]),x]","\int \frac{x^5}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^5}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)},x\right)",0,"Integrate[x^5/((1 - a^2*x^2)^3*ArcTanh[a*x]), x]","A",-1
323,1,31,41,0.1653381,"\int \frac{x^4}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)} \, dx","Integrate[x^4/((1 - a^2*x^2)^3*ArcTanh[a*x]),x]","\frac{-4 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)+\text{Chi}\left(4 \tanh ^{-1}(a x)\right)+3 \log \left(\tanh ^{-1}(a x)\right)}{8 a^5}","-\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{2 a^5}+\frac{\text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{8 a^5}+\frac{3 \log \left(\tanh ^{-1}(a x)\right)}{8 a^5}",1,"(-4*CoshIntegral[2*ArcTanh[a*x]] + CoshIntegral[4*ArcTanh[a*x]] + 3*Log[ArcTanh[a*x]])/(8*a^5)","A",1
324,1,24,29,0.1397915,"\int \frac{x^3}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)} \, dx","Integrate[x^3/((1 - a^2*x^2)^3*ArcTanh[a*x]),x]","\frac{\text{Shi}\left(4 \tanh ^{-1}(a x)\right)-2 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{8 a^4}","\frac{\text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{8 a^4}-\frac{\text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{4 a^4}",1,"(-2*SinhIntegral[2*ArcTanh[a*x]] + SinhIntegral[4*ArcTanh[a*x]])/(8*a^4)","A",1
325,1,22,27,0.122015,"\int \frac{x^2}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)} \, dx","Integrate[x^2/((1 - a^2*x^2)^3*ArcTanh[a*x]),x]","\frac{\text{Chi}\left(4 \tanh ^{-1}(a x)\right)-\log \left(\tanh ^{-1}(a x)\right)}{8 a^3}","\frac{\text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{8 a^3}-\frac{\log \left(\tanh ^{-1}(a x)\right)}{8 a^3}",1,"(CoshIntegral[4*ArcTanh[a*x]] - Log[ArcTanh[a*x]])/(8*a^3)","A",1
326,1,24,29,0.1357408,"\int \frac{x}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)} \, dx","Integrate[x/((1 - a^2*x^2)^3*ArcTanh[a*x]),x]","\frac{2 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)+\text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{8 a^2}","\frac{\text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{4 a^2}+\frac{\text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{8 a^2}",1,"(2*SinhIntegral[2*ArcTanh[a*x]] + SinhIntegral[4*ArcTanh[a*x]])/(8*a^2)","A",1
327,1,33,41,0.0866466,"\int \frac{1}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)} \, dx","Integrate[1/((1 - a^2*x^2)^3*ArcTanh[a*x]),x]","-\frac{-4 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)-\text{Chi}\left(4 \tanh ^{-1}(a x)\right)-3 \log \left(\tanh ^{-1}(a x)\right)}{8 a}","\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{2 a}+\frac{\text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{8 a}+\frac{3 \log \left(\tanh ^{-1}(a x)\right)}{8 a}",1,"-1/8*(-4*CoshIntegral[2*ArcTanh[a*x]] - CoshIntegral[4*ArcTanh[a*x]] - 3*Log[ArcTanh[a*x]])/a","A",1
328,0,0,49,1.3424803,"\int \frac{1}{x \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)} \, dx","Integrate[1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]),x]","\int \frac{1}{x \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)} \, dx","-\text{Int}\left(\frac{1}{x \left(a^2 x^2-1\right) \tanh ^{-1}(a x)},x\right)+\frac{3}{4} \text{Shi}\left(2 \tanh ^{-1}(a x)\right)+\frac{1}{8} \text{Shi}\left(4 \tanh ^{-1}(a x)\right)",0,"Integrate[1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]), x]","A",-1
329,0,0,102,12.1321142,"\int \frac{x^5}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2} \, dx","Integrate[x^5/((1 - a^2*x^2)^3*ArcTanh[a*x]^2),x]","\int \frac{x^5}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2} \, dx","\frac{\text{Int}\left(\frac{1}{\tanh ^{-1}(a x)},x\right)}{a^5}-\frac{3 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{2 a^6}+\frac{\text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{2 a^6}-\frac{x}{a^5 \tanh ^{-1}(a x)}+\frac{2 x}{a^5 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{x}{a^5 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}",0,"Integrate[x^5/((1 - a^2*x^2)^3*ArcTanh[a*x]^2), x]","A",-1
330,1,49,53,0.2001884,"\int \frac{x^4}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2} \, dx","Integrate[x^4/((1 - a^2*x^2)^3*ArcTanh[a*x]^2),x]","\frac{-\frac{2 a^4 x^4}{\left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)}-2 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)+\text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{2 a^5}","-\frac{\text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{a^5}+\frac{\text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{2 a^5}-\frac{x^4}{a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}",1,"((-2*a^4*x^4)/((-1 + a^2*x^2)^2*ArcTanh[a*x]) - 2*SinhIntegral[2*ArcTanh[a*x]] + SinhIntegral[4*ArcTanh[a*x]])/(2*a^5)","A",1
331,1,80,55,0.1219048,"\int \frac{x^3}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2} \, dx","Integrate[x^3/((1 - a^2*x^2)^3*ArcTanh[a*x]^2),x]","\frac{-2 a^3 x^3-\left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x) \text{Chi}\left(2 \tanh ^{-1}(a x)\right)+\left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x) \text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{2 a^4 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)}","-\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{2 a^4}+\frac{\text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{2 a^4}-\frac{x^3}{a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}",1,"(-2*a^3*x^3 - (-1 + a^2*x^2)^2*ArcTanh[a*x]*CoshIntegral[2*ArcTanh[a*x]] + (-1 + a^2*x^2)^2*ArcTanh[a*x]*CoshIntegral[4*ArcTanh[a*x]])/(2*a^4*(-1 + a^2*x^2)^2*ArcTanh[a*x])","A",1
332,1,56,41,0.2127346,"\int \frac{x^2}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2} \, dx","Integrate[x^2/((1 - a^2*x^2)^3*ArcTanh[a*x]^2),x]","\frac{\left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x) \text{Shi}\left(4 \tanh ^{-1}(a x)\right)-2 a^2 x^2}{2 a^3 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)}","\frac{\text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{2 a^3}-\frac{x^2}{a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}",1,"(-2*a^2*x^2 + (-1 + a^2*x^2)^2*ArcTanh[a*x]*SinhIntegral[4*ArcTanh[a*x]])/(2*a^3*(-1 + a^2*x^2)^2*ArcTanh[a*x])","A",1
333,1,75,53,0.1115996,"\int \frac{x}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2} \, dx","Integrate[x/((1 - a^2*x^2)^3*ArcTanh[a*x]^2),x]","\frac{\left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x) \text{Chi}\left(2 \tanh ^{-1}(a x)\right)+\left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x) \text{Chi}\left(4 \tanh ^{-1}(a x)\right)-2 a x}{2 a^2 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)}","\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{2 a^2}+\frac{\text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{2 a^2}-\frac{x}{a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}",1,"(-2*a*x + (-1 + a^2*x^2)^2*ArcTanh[a*x]*CoshIntegral[2*ArcTanh[a*x]] + (-1 + a^2*x^2)^2*ArcTanh[a*x]*CoshIntegral[4*ArcTanh[a*x]])/(2*a^2*(-1 + a^2*x^2)^2*ArcTanh[a*x])","A",1
334,1,43,49,0.1513593,"\int \frac{1}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2} \, dx","Integrate[1/((1 - a^2*x^2)^3*ArcTanh[a*x]^2),x]","\frac{-\frac{2}{\left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)}+2 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)+\text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{2 a}","-\frac{1}{a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}+\frac{\text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{a}+\frac{\text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{2 a}",1,"(-2/((-1 + a^2*x^2)^2*ArcTanh[a*x]) + 2*SinhIntegral[2*ArcTanh[a*x]] + SinhIntegral[4*ArcTanh[a*x]])/(2*a)","A",1
335,0,0,99,4.6858211,"\int \frac{1}{x \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2} \, dx","Integrate[1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]^2),x]","\int \frac{1}{x \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \tanh ^{-1}(a x)},x\right)}{a}-\frac{a x}{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{a x}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}+\frac{3}{2} \text{Chi}\left(2 \tanh ^{-1}(a x)\right)+\frac{1}{2} \text{Chi}\left(4 \tanh ^{-1}(a x)\right)-\frac{1}{a x \tanh ^{-1}(a x)}",0,"Integrate[1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]^2), x]","A",-1
336,1,60,100,0.2192872,"\int \frac{x^4}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^3} \, dx","Integrate[x^4/((1 - a^2*x^2)^3*ArcTanh[a*x]^3),x]","-\frac{\frac{a^3 x^3 \left(a x+4 \tanh ^{-1}(a x)\right)}{\left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2}+2 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)-2 \text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{2 a^5}","-\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{a^5}+\frac{\text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{a^5}-\frac{x^4}{2 a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}+\frac{2 x}{a^4 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{2 x}{a^4 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}",1,"-1/2*((a^3*x^3*(a*x + 4*ArcTanh[a*x]))/((-1 + a^2*x^2)^2*ArcTanh[a*x]^2) + 2*CoshIntegral[2*ArcTanh[a*x]] - 2*CoshIntegral[4*ArcTanh[a*x]])/a^5","A",1
337,1,66,107,0.2634338,"\int \frac{x^3}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^3} \, dx","Integrate[x^3/((1 - a^2*x^2)^3*ArcTanh[a*x]^3),x]","-\frac{\frac{a^2 x^2 \left(\left(a^2 x^2+3\right) \tanh ^{-1}(a x)+a x\right)}{\left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2}+\text{Shi}\left(2 \tanh ^{-1}(a x)\right)-2 \text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{2 a^4}","-\frac{\text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{2 a^4}+\frac{\text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{a^4}-\frac{3 x^2}{2 a^2 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}-\frac{x^4}{2 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}-\frac{x^3}{2 a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}",1,"-1/2*((a^2*x^2*(a*x + (3 + a^2*x^2)*ArcTanh[a*x]))/((-1 + a^2*x^2)^2*ArcTanh[a*x]^2) + SinhIntegral[2*ArcTanh[a*x]] - 2*SinhIntegral[4*ArcTanh[a*x]])/a^4","A",1
338,1,56,86,0.1728598,"\int \frac{x^2}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^3} \, dx","Integrate[x^2/((1 - a^2*x^2)^3*ArcTanh[a*x]^3),x]","\frac{\text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{a^3}-\frac{x \left(2 \left(a^2 x^2+1\right) \tanh ^{-1}(a x)+a x\right)}{2 a^2 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2}","\frac{\text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{a^3}-\frac{x^2}{2 a \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2}+\frac{x}{a^2 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{2 x}{a^2 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}",1,"-1/2*(x*(a*x + 2*(1 + a^2*x^2)*ArcTanh[a*x]))/(a^2*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2) + CoshIntegral[4*ArcTanh[a*x]]/a^3","A",1
339,1,96,100,0.2047805,"\int \frac{x}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^3} \, dx","Integrate[x/((1 - a^2*x^2)^3*ArcTanh[a*x]^3),x]","-\frac{-\left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)-2 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2 \text{Shi}\left(4 \tanh ^{-1}(a x)\right)+3 a^2 x^2 \tanh ^{-1}(a x)+a x+\tanh ^{-1}(a x)}{2 a^2 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2}","\frac{\text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{2 a^2}+\frac{\text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{a^2}-\frac{x}{2 a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}+\frac{3}{2 a^2 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{2}{a^2 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}",1,"-1/2*(a*x + ArcTanh[a*x] + 3*a^2*x^2*ArcTanh[a*x] - (-1 + a^2*x^2)^2*ArcTanh[a*x]^2*SinhIntegral[2*ArcTanh[a*x]] - 2*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2*SinhIntegral[4*ArcTanh[a*x]])/(a^2*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2)","A",1
340,1,86,69,0.1987674,"\int \frac{1}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^3} \, dx","Integrate[1/((1 - a^2*x^2)^3*ArcTanh[a*x]^3),x]","\frac{2 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)+2 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2 \text{Chi}\left(4 \tanh ^{-1}(a x)\right)-4 a x \tanh ^{-1}(a x)-1}{2 a \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2}","-\frac{2 x}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}-\frac{1}{2 a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}+\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{a}+\frac{\text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{a}",1,"(-1 - 4*a*x*ArcTanh[a*x] + 2*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2*CoshIntegral[2*ArcTanh[a*x]] + 2*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2*CoshIntegral[4*ArcTanh[a*x]])/(2*a*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2)","A",1
341,0,0,176,5.694519,"\int \frac{1}{x \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^3} \, dx","Integrate[1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]^3),x]","\int \frac{1}{x \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^3} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \tanh ^{-1}(a x)^2},x\right)}{2 a}-\frac{a x}{2 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}-\frac{a x}{2 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}-\frac{a^2 x^2+1}{2 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}+\frac{3}{2 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{2}{\left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}+\frac{3}{2} \text{Shi}\left(2 \tanh ^{-1}(a x)\right)+\text{Shi}\left(4 \tanh ^{-1}(a x)\right)-\frac{1}{2 a x \tanh ^{-1}(a x)^2}",0,"Integrate[1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]^3), x]","A",-1
342,1,108,125,0.2567258,"\int \frac{1}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^4} \, dx","Integrate[1/((1 - a^2*x^2)^3*ArcTanh[a*x]^4),x]","-\frac{-2 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^3 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)-4 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^3 \text{Shi}\left(4 \tanh ^{-1}(a x)\right)+6 a^2 x^2 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2+2 a x \tanh ^{-1}(a x)+1}{3 a \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^3}","-\frac{2 x}{3 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}+\frac{2}{a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{8}{3 a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}-\frac{1}{3 a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^3}+\frac{2 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{3 a}+\frac{4 \text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{3 a}",1,"-1/3*(1 + 2*a*x*ArcTanh[a*x] + 2*ArcTanh[a*x]^2 + 6*a^2*x^2*ArcTanh[a*x]^2 - 2*(-1 + a^2*x^2)^2*ArcTanh[a*x]^3*SinhIntegral[2*ArcTanh[a*x]] - 4*(-1 + a^2*x^2)^2*ArcTanh[a*x]^3*SinhIntegral[4*ArcTanh[a*x]])/(a*(-1 + a^2*x^2)^2*ArcTanh[a*x]^3)","A",1
343,1,132,170,0.1750888,"\int \frac{1}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^5} \, dx","Integrate[1/((1 - a^2*x^2)^3*ArcTanh[a*x]^5),x]","-\frac{12 a^3 x^3 \tanh ^{-1}(a x)^3-4 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^4 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)-16 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^4 \text{Chi}\left(4 \tanh ^{-1}(a x)\right)+6 a^2 x^2 \tanh ^{-1}(a x)^2+20 a x \tanh ^{-1}(a x)^3+2 \tanh ^{-1}(a x)^2+4 a x \tanh ^{-1}(a x)+3}{12 a \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^4}","\frac{x}{\left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{8 x}{3 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}-\frac{x}{3 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^3}+\frac{1}{2 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}-\frac{2}{3 a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}-\frac{1}{4 a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^4}+\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{3 a}+\frac{4 \text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{3 a}",1,"-1/12*(3 + 4*a*x*ArcTanh[a*x] + 2*ArcTanh[a*x]^2 + 6*a^2*x^2*ArcTanh[a*x]^2 + 20*a*x*ArcTanh[a*x]^3 + 12*a^3*x^3*ArcTanh[a*x]^3 - 4*(-1 + a^2*x^2)^2*ArcTanh[a*x]^4*CoshIntegral[2*ArcTanh[a*x]] - 16*(-1 + a^2*x^2)^2*ArcTanh[a*x]^4*CoshIntegral[4*ArcTanh[a*x]])/(a*(-1 + a^2*x^2)^2*ArcTanh[a*x]^4)","A",1
344,1,166,257,0.3271505,"\int \frac{1}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^6} \, dx","Integrate[1/((1 - a^2*x^2)^3*ArcTanh[a*x]^6),x]","-\frac{3 a^4 x^4 \tanh ^{-1}(a x)^4+3 a^3 x^3 \tanh ^{-1}(a x)^3-2 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^5 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)-16 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^5 \text{Shi}\left(4 \tanh ^{-1}(a x)\right)+24 a^2 x^2 \tanh ^{-1}(a x)^4+3 a^2 x^2 \tanh ^{-1}(a x)^2+5 \tanh ^{-1}(a x)^4+5 a x \tanh ^{-1}(a x)^3+\tanh ^{-1}(a x)^2+3 a x \tanh ^{-1}(a x)+3}{15 a \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^5}","\frac{x}{5 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2}-\frac{8 x}{15 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^2}-\frac{x}{5 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^4}+\frac{a^2 x^2+1}{5 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}+\frac{8}{5 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)}-\frac{32}{15 a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}+\frac{1}{5 a \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3}-\frac{4}{15 a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^3}-\frac{1}{5 a \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)^5}+\frac{2 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{15 a}+\frac{16 \text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{15 a}",1,"-1/15*(3 + 3*a*x*ArcTanh[a*x] + ArcTanh[a*x]^2 + 3*a^2*x^2*ArcTanh[a*x]^2 + 5*a*x*ArcTanh[a*x]^3 + 3*a^3*x^3*ArcTanh[a*x]^3 + 5*ArcTanh[a*x]^4 + 24*a^2*x^2*ArcTanh[a*x]^4 + 3*a^4*x^4*ArcTanh[a*x]^4 - 2*(-1 + a^2*x^2)^2*ArcTanh[a*x]^5*SinhIntegral[2*ArcTanh[a*x]] - 16*(-1 + a^2*x^2)^2*ArcTanh[a*x]^5*SinhIntegral[4*ArcTanh[a*x]])/(a*(-1 + a^2*x^2)^2*ArcTanh[a*x]^5)","A",1
345,1,81,134,0.1841093,"\int \frac{\tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^4} \, dx","Integrate[ArcTanh[a*x]/(1 - a^2*x^2)^4,x]","\frac{45 a^4 x^4-105 a^2 x^2+45 \left(a^2 x^2-1\right)^3 \tanh ^{-1}(a x)^2-6 a x \left(15 a^4 x^4-40 a^2 x^2+33\right) \tanh ^{-1}(a x)+68}{288 a \left(a^2 x^2-1\right)^3}","-\frac{5}{32 a \left(1-a^2 x^2\right)}-\frac{5}{96 a \left(1-a^2 x^2\right)^2}-\frac{1}{36 a \left(1-a^2 x^2\right)^3}+\frac{5 x \tanh ^{-1}(a x)}{16 \left(1-a^2 x^2\right)}+\frac{5 x \tanh ^{-1}(a x)}{24 \left(1-a^2 x^2\right)^2}+\frac{x \tanh ^{-1}(a x)}{6 \left(1-a^2 x^2\right)^3}+\frac{5 \tanh ^{-1}(a x)^2}{32 a}",1,"(68 - 105*a^2*x^2 + 45*a^4*x^4 - 6*a*x*(33 - 40*a^2*x^2 + 15*a^4*x^4)*ArcTanh[a*x] + 45*(-1 + a^2*x^2)^3*ArcTanh[a*x]^2)/(288*a*(-1 + a^2*x^2)^3)","A",1
346,1,157,214,0.2579315,"\int \frac{\tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^4} \, dx","Integrate[ArcTanh[a*x]^2/(1 - a^2*x^2)^4,x]","\frac{-\frac{1470 x}{a^2 x^2-1}+\frac{260 x}{\left(a^2 x^2-1\right)^2}-\frac{64 x}{\left(a^2 x^2-1\right)^3}-\frac{144 x \left(15 a^4 x^4-40 a^2 x^2+33\right) \tanh ^{-1}(a x)^2}{\left(a^2 x^2-1\right)^3}+\frac{48 \left(45 a^4 x^4-105 a^2 x^2+68\right) \tanh ^{-1}(a x)}{a \left(a^2 x^2-1\right)^3}-\frac{735 \log (1-a x)}{a}+\frac{735 \log (a x+1)}{a}+\frac{720 \tanh ^{-1}(a x)^3}{a}}{6912}","\frac{245 x}{1152 \left(1-a^2 x^2\right)}+\frac{65 x}{1728 \left(1-a^2 x^2\right)^2}+\frac{x}{108 \left(1-a^2 x^2\right)^3}+\frac{5 x \tanh ^{-1}(a x)^2}{16 \left(1-a^2 x^2\right)}+\frac{5 x \tanh ^{-1}(a x)^2}{24 \left(1-a^2 x^2\right)^2}+\frac{x \tanh ^{-1}(a x)^2}{6 \left(1-a^2 x^2\right)^3}-\frac{5 \tanh ^{-1}(a x)}{16 a \left(1-a^2 x^2\right)}-\frac{5 \tanh ^{-1}(a x)}{48 a \left(1-a^2 x^2\right)^2}-\frac{\tanh ^{-1}(a x)}{18 a \left(1-a^2 x^2\right)^3}+\frac{5 \tanh ^{-1}(a x)^3}{48 a}+\frac{245 \tanh ^{-1}(a x)}{1152 a}",1,"((-64*x)/(-1 + a^2*x^2)^3 + (260*x)/(-1 + a^2*x^2)^2 - (1470*x)/(-1 + a^2*x^2) + (48*(68 - 105*a^2*x^2 + 45*a^4*x^4)*ArcTanh[a*x])/(a*(-1 + a^2*x^2)^3) - (144*x*(33 - 40*a^2*x^2 + 15*a^4*x^4)*ArcTanh[a*x]^2)/(-1 + a^2*x^2)^3 + (720*ArcTanh[a*x]^3)/a - (735*Log[1 - a*x])/a + (735*Log[1 + a*x])/a)/6912","A",1
347,1,143,291,0.1528198,"\int \frac{\tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^4} \, dx","Integrate[ArcTanh[a*x]^3/(1 - a^2*x^2)^4,x]","\frac{2205 a^4 x^4-4605 a^2 x^2+540 \left(a^2 x^2-1\right)^3 \tanh ^{-1}(a x)^4-144 a x \left(15 a^4 x^4-40 a^2 x^2+33\right) \tanh ^{-1}(a x)^3-6 a x \left(735 a^4 x^4-1600 a^2 x^2+897\right) \tanh ^{-1}(a x)+9 \left(245 a^6 x^6-375 a^4 x^4-105 a^2 x^2+299\right) \tanh ^{-1}(a x)^2+2432}{6912 a \left(a^2 x^2-1\right)^3}","-\frac{245}{768 a \left(1-a^2 x^2\right)}-\frac{65}{2304 a \left(1-a^2 x^2\right)^2}-\frac{1}{216 a \left(1-a^2 x^2\right)^3}+\frac{5 x \tanh ^{-1}(a x)^3}{16 \left(1-a^2 x^2\right)}+\frac{5 x \tanh ^{-1}(a x)^3}{24 \left(1-a^2 x^2\right)^2}+\frac{x \tanh ^{-1}(a x)^3}{6 \left(1-a^2 x^2\right)^3}-\frac{15 \tanh ^{-1}(a x)^2}{32 a \left(1-a^2 x^2\right)}-\frac{5 \tanh ^{-1}(a x)^2}{32 a \left(1-a^2 x^2\right)^2}-\frac{\tanh ^{-1}(a x)^2}{12 a \left(1-a^2 x^2\right)^3}+\frac{245 x \tanh ^{-1}(a x)}{384 \left(1-a^2 x^2\right)}+\frac{65 x \tanh ^{-1}(a x)}{576 \left(1-a^2 x^2\right)^2}+\frac{x \tanh ^{-1}(a x)}{36 \left(1-a^2 x^2\right)^3}+\frac{5 \tanh ^{-1}(a x)^4}{64 a}+\frac{245 \tanh ^{-1}(a x)^2}{768 a}",1,"(2432 - 4605*a^2*x^2 + 2205*a^4*x^4 - 6*a*x*(897 - 1600*a^2*x^2 + 735*a^4*x^4)*ArcTanh[a*x] + 9*(299 - 105*a^2*x^2 - 375*a^4*x^4 + 245*a^6*x^6)*ArcTanh[a*x]^2 - 144*a*x*(33 - 40*a^2*x^2 + 15*a^4*x^4)*ArcTanh[a*x]^3 + 540*(-1 + a^2*x^2)^3*ArcTanh[a*x]^4)/(6912*a*(-1 + a^2*x^2)^3)","A",1
348,1,257,252,0.7475473,"\int \frac{\sqrt{\tanh ^{-1}(a x)}}{\left(1-a^2 x^2\right)^4} \, dx","Integrate[Sqrt[ArcTanh[a*x]]/(1 - a^2*x^2)^4,x]","\frac{-\frac{3168 x \sqrt{\tanh ^{-1}(a x)}}{\left(a^2 x^2-1\right)^3}+\frac{3840 a^2 x^3 \sqrt{\tanh ^{-1}(a x)}}{\left(a^2 x^2-1\right)^3}-\frac{1440 a^4 x^5 \sqrt{\tanh ^{-1}(a x)}}{\left(a^2 x^2-1\right)^3}+\frac{960 \tanh ^{-1}(a x)^{3/2}}{a}+\frac{\sqrt{6} \sqrt{\tanh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-6 \tanh ^{-1}(a x)\right)}{a \sqrt{-\tanh ^{-1}(a x)}}+\frac{27 \sqrt{\tanh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 \tanh ^{-1}(a x)\right)}{a \sqrt{-\tanh ^{-1}(a x)}}+\frac{135 \sqrt{2} \sqrt{\tanh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 \tanh ^{-1}(a x)\right)}{a \sqrt{-\tanh ^{-1}(a x)}}-\frac{135 \sqrt{2} \Gamma \left(\frac{1}{2},2 \tanh ^{-1}(a x)\right)}{a}-\frac{27 \Gamma \left(\frac{1}{2},4 \tanh ^{-1}(a x)\right)}{a}-\frac{\sqrt{6} \Gamma \left(\frac{1}{2},6 \tanh ^{-1}(a x)\right)}{a}}{4608}","\frac{3 \sqrt{\pi } \text{erf}\left(2 \sqrt{\tanh ^{-1}(a x)}\right)}{512 a}+\frac{15 \sqrt{\frac{\pi }{2}} \text{erf}\left(\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right)}{256 a}+\frac{\sqrt{\frac{\pi }{6}} \text{erf}\left(\sqrt{6} \sqrt{\tanh ^{-1}(a x)}\right)}{768 a}-\frac{3 \sqrt{\pi } \text{erfi}\left(2 \sqrt{\tanh ^{-1}(a x)}\right)}{512 a}-\frac{15 \sqrt{\frac{\pi }{2}} \text{erfi}\left(\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right)}{256 a}-\frac{\sqrt{\frac{\pi }{6}} \text{erfi}\left(\sqrt{6} \sqrt{\tanh ^{-1}(a x)}\right)}{768 a}+\frac{5 \tanh ^{-1}(a x)^{3/2}}{24 a}+\frac{15 \sqrt{\tanh ^{-1}(a x)} \sinh \left(2 \tanh ^{-1}(a x)\right)}{64 a}+\frac{3 \sqrt{\tanh ^{-1}(a x)} \sinh \left(4 \tanh ^{-1}(a x)\right)}{64 a}+\frac{\sqrt{\tanh ^{-1}(a x)} \sinh \left(6 \tanh ^{-1}(a x)\right)}{192 a}",1,"((-3168*x*Sqrt[ArcTanh[a*x]])/(-1 + a^2*x^2)^3 + (3840*a^2*x^3*Sqrt[ArcTanh[a*x]])/(-1 + a^2*x^2)^3 - (1440*a^4*x^5*Sqrt[ArcTanh[a*x]])/(-1 + a^2*x^2)^3 + (960*ArcTanh[a*x]^(3/2))/a + (Sqrt[6]*Sqrt[ArcTanh[a*x]]*Gamma[1/2, -6*ArcTanh[a*x]])/(a*Sqrt[-ArcTanh[a*x]]) + (27*Sqrt[ArcTanh[a*x]]*Gamma[1/2, -4*ArcTanh[a*x]])/(a*Sqrt[-ArcTanh[a*x]]) + (135*Sqrt[2]*Sqrt[ArcTanh[a*x]]*Gamma[1/2, -2*ArcTanh[a*x]])/(a*Sqrt[-ArcTanh[a*x]]) - (135*Sqrt[2]*Gamma[1/2, 2*ArcTanh[a*x]])/a - (27*Gamma[1/2, 4*ArcTanh[a*x]])/a - (Sqrt[6]*Gamma[1/2, 6*ArcTanh[a*x]])/a)/4608","A",0
349,0,0,25,8.5171408,"\int \frac{x^8}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)} \, dx","Integrate[x^8/((1 - a^2*x^2)^4*ArcTanh[a*x]),x]","\int \frac{x^8}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^8}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)},x\right)",0,"Integrate[x^8/((1 - a^2*x^2)^4*ArcTanh[a*x]), x]","A",-1
350,0,0,25,33.2352614,"\int \frac{x^7}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)} \, dx","Integrate[x^7/((1 - a^2*x^2)^4*ArcTanh[a*x]),x]","\int \frac{x^7}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^7}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)},x\right)",0,"Integrate[x^7/((1 - a^2*x^2)^4*ArcTanh[a*x]), x]","A",-1
351,1,40,55,0.1679447,"\int \frac{x^6}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)} \, dx","Integrate[x^6/((1 - a^2*x^2)^4*ArcTanh[a*x]),x]","\frac{15 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)-6 \text{Chi}\left(4 \tanh ^{-1}(a x)\right)+\text{Chi}\left(6 \tanh ^{-1}(a x)\right)-10 \log \left(\tanh ^{-1}(a x)\right)}{32 a^7}","\frac{15 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{32 a^7}-\frac{3 \text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{16 a^7}+\frac{\text{Chi}\left(6 \tanh ^{-1}(a x)\right)}{32 a^7}-\frac{5 \log \left(\tanh ^{-1}(a x)\right)}{16 a^7}",1,"(15*CoshIntegral[2*ArcTanh[a*x]] - 6*CoshIntegral[4*ArcTanh[a*x]] + CoshIntegral[6*ArcTanh[a*x]] - 10*Log[ArcTanh[a*x]])/(32*a^7)","A",1
352,1,33,43,0.1719432,"\int \frac{x^5}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)} \, dx","Integrate[x^5/((1 - a^2*x^2)^4*ArcTanh[a*x]),x]","\frac{5 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)-4 \text{Shi}\left(4 \tanh ^{-1}(a x)\right)+\text{Shi}\left(6 \tanh ^{-1}(a x)\right)}{32 a^6}","\frac{5 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{32 a^6}-\frac{\text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{8 a^6}+\frac{\text{Shi}\left(6 \tanh ^{-1}(a x)\right)}{32 a^6}",1,"(5*SinhIntegral[2*ArcTanh[a*x]] - 4*SinhIntegral[4*ArcTanh[a*x]] + SinhIntegral[6*ArcTanh[a*x]])/(32*a^6)","A",1
353,1,40,55,0.1450039,"\int \frac{x^4}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)} \, dx","Integrate[x^4/((1 - a^2*x^2)^4*ArcTanh[a*x]),x]","\frac{-\text{Chi}\left(2 \tanh ^{-1}(a x)\right)-2 \text{Chi}\left(4 \tanh ^{-1}(a x)\right)+\text{Chi}\left(6 \tanh ^{-1}(a x)\right)+2 \log \left(\tanh ^{-1}(a x)\right)}{32 a^5}","-\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{32 a^5}-\frac{\text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{16 a^5}+\frac{\text{Chi}\left(6 \tanh ^{-1}(a x)\right)}{32 a^5}+\frac{\log \left(\tanh ^{-1}(a x)\right)}{16 a^5}",1,"(-CoshIntegral[2*ArcTanh[a*x]] - 2*CoshIntegral[4*ArcTanh[a*x]] + CoshIntegral[6*ArcTanh[a*x]] + 2*Log[ArcTanh[a*x]])/(32*a^5)","A",1
354,1,24,29,0.1500597,"\int \frac{x^3}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)} \, dx","Integrate[x^3/((1 - a^2*x^2)^4*ArcTanh[a*x]),x]","\frac{\text{Shi}\left(6 \tanh ^{-1}(a x)\right)-3 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{32 a^4}","\frac{\text{Shi}\left(6 \tanh ^{-1}(a x)\right)}{32 a^4}-\frac{3 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{32 a^4}",1,"(-3*SinhIntegral[2*ArcTanh[a*x]] + SinhIntegral[6*ArcTanh[a*x]])/(32*a^4)","A",1
355,1,55,55,0.1742189,"\int \frac{x^2}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)} \, dx","Integrate[x^2/((1 - a^2*x^2)^4*ArcTanh[a*x]),x]","-\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{32 a^3}+\frac{\text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{16 a^3}+\frac{\text{Chi}\left(6 \tanh ^{-1}(a x)\right)}{32 a^3}-\frac{\log \left(\tanh ^{-1}(a x)\right)}{16 a^3}","-\frac{\text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{32 a^3}+\frac{\text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{16 a^3}+\frac{\text{Chi}\left(6 \tanh ^{-1}(a x)\right)}{32 a^3}-\frac{\log \left(\tanh ^{-1}(a x)\right)}{16 a^3}",1,"-1/32*CoshIntegral[2*ArcTanh[a*x]]/a^3 + CoshIntegral[4*ArcTanh[a*x]]/(16*a^3) + CoshIntegral[6*ArcTanh[a*x]]/(32*a^3) - Log[ArcTanh[a*x]]/(16*a^3)","A",1
356,1,43,43,0.2163125,"\int \frac{x}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)} \, dx","Integrate[x/((1 - a^2*x^2)^4*ArcTanh[a*x]),x]","\frac{5 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{32 a^2}+\frac{\text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{8 a^2}+\frac{\text{Shi}\left(6 \tanh ^{-1}(a x)\right)}{32 a^2}","\frac{5 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{32 a^2}+\frac{\text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{8 a^2}+\frac{\text{Shi}\left(6 \tanh ^{-1}(a x)\right)}{32 a^2}",1,"(5*SinhIntegral[2*ArcTanh[a*x]])/(32*a^2) + SinhIntegral[4*ArcTanh[a*x]]/(8*a^2) + SinhIntegral[6*ArcTanh[a*x]]/(32*a^2)","A",1
357,1,40,55,0.1844888,"\int \frac{1}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)} \, dx","Integrate[1/((1 - a^2*x^2)^4*ArcTanh[a*x]),x]","\frac{15 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)+6 \text{Chi}\left(4 \tanh ^{-1}(a x)\right)+\text{Chi}\left(6 \tanh ^{-1}(a x)\right)+10 \log \left(\tanh ^{-1}(a x)\right)}{32 a}","\frac{15 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{32 a}+\frac{3 \text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{16 a}+\frac{\text{Chi}\left(6 \tanh ^{-1}(a x)\right)}{32 a}+\frac{5 \log \left(\tanh ^{-1}(a x)\right)}{16 a}",1,"(15*CoshIntegral[2*ArcTanh[a*x]] + 6*CoshIntegral[4*ArcTanh[a*x]] + CoshIntegral[6*ArcTanh[a*x]] + 10*Log[ArcTanh[a*x]])/(32*a)","A",1
358,0,0,25,1.5607287,"\int \frac{1}{x \left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)} \, dx","Integrate[1/(x*(1 - a^2*x^2)^4*ArcTanh[a*x]),x]","\int \frac{1}{x \left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x \left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)},x\right)",0,"Integrate[1/(x*(1 - a^2*x^2)^4*ArcTanh[a*x]), x]","A",-1
359,0,0,25,3.1433795,"\int \frac{1}{x^2 \left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)} \, dx","Integrate[1/(x^2*(1 - a^2*x^2)^4*ArcTanh[a*x]),x]","\int \frac{1}{x^2 \left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x^2 \left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)},x\right)",0,"Integrate[1/(x^2*(1 - a^2*x^2)^4*ArcTanh[a*x]), x]","A",-1
360,1,56,67,0.2000575,"\int \frac{x}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)^2} \, dx","Integrate[x/((1 - a^2*x^2)^4*ArcTanh[a*x]^2),x]","\frac{\frac{16 a x}{\left(a^2 x^2-1\right)^3 \tanh ^{-1}(a x)}+5 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)+8 \text{Chi}\left(4 \tanh ^{-1}(a x)\right)+3 \text{Chi}\left(6 \tanh ^{-1}(a x)\right)}{16 a^2}","\frac{5 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{16 a^2}+\frac{\text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{2 a^2}+\frac{3 \text{Chi}\left(6 \tanh ^{-1}(a x)\right)}{16 a^2}-\frac{x}{a \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)}",1,"((16*a*x)/((-1 + a^2*x^2)^3*ArcTanh[a*x]) + 5*CoshIntegral[2*ArcTanh[a*x]] + 8*CoshIntegral[4*ArcTanh[a*x]] + 3*CoshIntegral[6*ArcTanh[a*x]])/(16*a^2)","A",1
361,1,56,66,0.3015248,"\int \frac{1}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)^2} \, dx","Integrate[1/((1 - a^2*x^2)^4*ArcTanh[a*x]^2),x]","\frac{\frac{1}{\left(a^2 x^2-1\right)^3 \tanh ^{-1}(a x)}+\frac{15}{16} \text{Shi}\left(2 \tanh ^{-1}(a x)\right)+\frac{3}{4} \text{Shi}\left(4 \tanh ^{-1}(a x)\right)+\frac{3}{16} \text{Shi}\left(6 \tanh ^{-1}(a x)\right)}{a}","-\frac{1}{a \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)}+\frac{15 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{16 a}+\frac{3 \text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{4 a}+\frac{3 \text{Shi}\left(6 \tanh ^{-1}(a x)\right)}{16 a}",1,"(1/((-1 + a^2*x^2)^3*ArcTanh[a*x]) + (15*SinhIntegral[2*ArcTanh[a*x]])/16 + (3*SinhIntegral[4*ArcTanh[a*x]])/4 + (3*SinhIntegral[6*ArcTanh[a*x]])/16)/a","A",1
362,1,73,114,0.3765964,"\int \frac{x}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)^3} \, dx","Integrate[x/((1 - a^2*x^2)^4*ArcTanh[a*x]^3),x]","\frac{\frac{8 \left(\left(5 a^2 x^2+1\right) \tanh ^{-1}(a x)+a x\right)}{\left(a^2 x^2-1\right)^3 \tanh ^{-1}(a x)^2}+5 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)+16 \text{Shi}\left(4 \tanh ^{-1}(a x)\right)+9 \text{Shi}\left(6 \tanh ^{-1}(a x)\right)}{16 a^2}","\frac{5 \text{Shi}\left(2 \tanh ^{-1}(a x)\right)}{16 a^2}+\frac{\text{Shi}\left(4 \tanh ^{-1}(a x)\right)}{a^2}+\frac{9 \text{Shi}\left(6 \tanh ^{-1}(a x)\right)}{16 a^2}-\frac{x}{2 a \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2}+\frac{5}{2 a^2 \left(1-a^2 x^2\right)^2 \tanh ^{-1}(a x)}-\frac{3}{a^2 \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)}",1,"((8*(a*x + (1 + 5*a^2*x^2)*ArcTanh[a*x]))/((-1 + a^2*x^2)^3*ArcTanh[a*x]^2) + 5*SinhIntegral[2*ArcTanh[a*x]] + 16*SinhIntegral[4*ArcTanh[a*x]] + 9*SinhIntegral[6*ArcTanh[a*x]])/(16*a^2)","A",1
363,1,83,89,0.203317,"\int \frac{1}{\left(1-a^2 x^2\right)^4 \tanh ^{-1}(a x)^3} \, dx","Integrate[1/((1 - a^2*x^2)^4*ArcTanh[a*x]^3),x]","\frac{1}{16} \left(\frac{48 x}{\left(a^2 x^2-1\right)^3 \tanh ^{-1}(a x)}+\frac{8}{a \left(a^2 x^2-1\right)^3 \tanh ^{-1}(a x)^2}+\frac{15 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{a}+\frac{24 \text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{a}+\frac{9 \text{Chi}\left(6 \tanh ^{-1}(a x)\right)}{a}\right)","-\frac{3 x}{\left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)}-\frac{1}{2 a \left(1-a^2 x^2\right)^3 \tanh ^{-1}(a x)^2}+\frac{15 \text{Chi}\left(2 \tanh ^{-1}(a x)\right)}{16 a}+\frac{3 \text{Chi}\left(4 \tanh ^{-1}(a x)\right)}{2 a}+\frac{9 \text{Chi}\left(6 \tanh ^{-1}(a x)\right)}{16 a}",1,"(8/(a*(-1 + a^2*x^2)^3*ArcTanh[a*x]^2) + (48*x)/((-1 + a^2*x^2)^3*ArcTanh[a*x]) + (15*CoshIntegral[2*ArcTanh[a*x]])/a + (24*CoshIntegral[4*ArcTanh[a*x]])/a + (9*CoshIntegral[6*ArcTanh[a*x]])/a)/16","A",1
364,1,79,139,0.1099327,"\int \frac{x^5 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^5*ArcTanh[a*x])/Sqrt[1 - a^2*x^2],x]","-\frac{a x \sqrt{1-a^2 x^2} \left(6 a^2 x^2+25\right)+8 \sqrt{1-a^2 x^2} \left(3 a^4 x^4+4 a^2 x^2+8\right) \tanh ^{-1}(a x)-89 \sin ^{-1}(a x)}{120 a^6}","\frac{89 \sin ^{-1}(a x)}{120 a^6}-\frac{x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{5 a^2}-\frac{8 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^6}-\frac{5 x \sqrt{1-a^2 x^2}}{24 a^5}-\frac{4 x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac{x^3 \sqrt{1-a^2 x^2}}{20 a^3}",1,"-1/120*(a*x*Sqrt[1 - a^2*x^2]*(25 + 6*a^2*x^2) - 89*ArcSin[a*x] + 8*Sqrt[1 - a^2*x^2]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*ArcTanh[a*x])/a^6","A",1
365,1,160,197,0.5130494,"\int \frac{x^4 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^4*ArcTanh[a*x])/Sqrt[1 - a^2*x^2],x]","\frac{\sqrt{1-a^2 x^2} \left(-\frac{9 i \left(\text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)\right)}{\sqrt{1-a^2 x^2}}-2 a^2 x^2-6 a x \left(a^2 x^2-1\right) \tanh ^{-1}(a x)-\frac{9 i \tanh ^{-1}(a x) \left(\log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)}{\sqrt{1-a^2 x^2}}-15 a x \tanh ^{-1}(a x)-13\right)}{24 a^5}","-\frac{3 i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{8 a^5}+\frac{3 i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{8 a^5}-\frac{3 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{4 a^5}-\frac{x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{4 a^2}+\frac{\left(1-a^2 x^2\right)^{3/2}}{12 a^5}-\frac{5 \sqrt{1-a^2 x^2}}{8 a^5}-\frac{3 x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^4}",1,"(Sqrt[1 - a^2*x^2]*(-13 - 2*a^2*x^2 - 15*a*x*ArcTanh[a*x] - 6*a*x*(-1 + a^2*x^2)*ArcTanh[a*x] - ((9*I)*ArcTanh[a*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2] - ((9*I)*(PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(24*a^5)","A",0
366,1,60,87,0.0734875,"\int \frac{x^3 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^3*ArcTanh[a*x])/Sqrt[1 - a^2*x^2],x]","-\frac{a x \sqrt{1-a^2 x^2}+2 \sqrt{1-a^2 x^2} \left(a^2 x^2+2\right) \tanh ^{-1}(a x)-5 \sin ^{-1}(a x)}{6 a^4}","\frac{5 \sin ^{-1}(a x)}{6 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a^2}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a^4}-\frac{x \sqrt{1-a^2 x^2}}{6 a^3}",1,"-1/6*(a*x*Sqrt[1 - a^2*x^2] - 5*ArcSin[a*x] + 2*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcTanh[a*x])/a^4","A",1
367,1,125,146,0.232697,"\int \frac{x^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^2*ArcTanh[a*x])/Sqrt[1 - a^2*x^2],x]","-\frac{\sqrt{1-a^2 x^2}+a x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+i \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-i \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+i \tanh ^{-1}(a x) \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-i \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)}{2 a^3}","-\frac{i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{2 a^3}+\frac{i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{2 a^3}-\frac{\tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{a^3}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{2 a^2}-\frac{\sqrt{1-a^2 x^2}}{2 a^3}",1,"-1/2*(Sqrt[1 - a^2*x^2] + a*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + I*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] - I*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] + I*PolyLog[2, (-I)/E^ArcTanh[a*x]] - I*PolyLog[2, I/E^ArcTanh[a*x]])/a^3","A",0
368,1,29,32,0.0338,"\int \frac{x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2],x]","\frac{\sin ^{-1}(a x)-\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a^2}","\frac{\sin ^{-1}(a x)}{a^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a^2}",1,"(ArcSin[a*x] - Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/a^2","A",1
369,1,76,95,0.0806714,"\int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[ArcTanh[a*x]/Sqrt[1 - a^2*x^2],x]","-\frac{i \left(\text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x) \left(\log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)\right)}{a}","-\frac{i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a}+\frac{i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{a}",1,"((-I)*(ArcTanh[a*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]]) + PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/a","A",0
370,1,57,75,0.1003245,"\int \frac{\tanh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcTanh[a*x]/(x*Sqrt[1 - a^2*x^2]),x]","\text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x) \left(\log \left(1-e^{-\tanh ^{-1}(a x)}\right)-\log \left(e^{-\tanh ^{-1}(a x)}+1\right)\right)","\text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-2 \tanh ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)",1,"ArcTanh[a*x]*(Log[1 - E^(-ArcTanh[a*x])] - Log[1 + E^(-ArcTanh[a*x])]) + PolyLog[2, -E^(-ArcTanh[a*x])] - PolyLog[2, E^(-ArcTanh[a*x])]","A",0
371,1,48,42,0.0523439,"\int \frac{\tanh ^{-1}(a x)}{x^2 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcTanh[a*x]/(x^2*Sqrt[1 - a^2*x^2]),x]","-a \log \left(\sqrt{1-a^2 x^2}+1\right)-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}+a \log (x)","-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-a \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)",1,"-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x) + a*Log[x] - a*Log[1 + Sqrt[1 - a^2*x^2]]","A",1
372,1,126,137,0.7358077,"\int \frac{\tanh ^{-1}(a x)}{x^3 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcTanh[a*x]/(x^3*Sqrt[1 - a^2*x^2]),x]","\frac{1}{8} a^2 \left(4 \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-4 \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+2 \tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)+4 \tanh ^{-1}(a x) \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-4 \tanh ^{-1}(a x) \log \left(e^{-\tanh ^{-1}(a x)}+1\right)-2 \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)-\tanh ^{-1}(a x) \text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-\tanh ^{-1}(a x) \text{sech}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)","\frac{1}{2} a^2 \text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{1}{2} a^2 \text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{a \sqrt{1-a^2 x^2}}{2 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{2 x^2}+a^2 \left(-\tanh ^{-1}(a x)\right) \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)",1,"(a^2*(-2*Coth[ArcTanh[a*x]/2] - ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^2 + 4*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] - 4*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] + 4*PolyLog[2, -E^(-ArcTanh[a*x])] - 4*PolyLog[2, E^(-ArcTanh[a*x])] - ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^2 + 2*Tanh[ArcTanh[a*x]/2]))/8","A",0
373,1,160,205,0.4308568,"\int \frac{x^3 \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^3*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2],x]","\frac{\sqrt{1-a^2 x^2} \left(-\frac{5 i \left(\text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)\right)}{\sqrt{1-a^2 x^2}}+\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2-\frac{5 i \tanh ^{-1}(a x) \left(\log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)}{\sqrt{1-a^2 x^2}}-3 \tanh ^{-1}(a x)^2-a x \tanh ^{-1}(a x)-1\right)}{3 a^4}","-\frac{5 i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{3 a^4}+\frac{5 i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{3 a^4}-\frac{10 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^2}-\frac{\sqrt{1-a^2 x^2}}{3 a^4}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^4}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a^3}",1,"(Sqrt[1 - a^2*x^2]*(-1 - a*x*ArcTanh[a*x] - 3*ArcTanh[a*x]^2 + (1 - a^2*x^2)*ArcTanh[a*x]^2 - ((5*I)*ArcTanh[a*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2] - ((5*I)*(PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(3*a^4)","A",0
374,1,188,161,0.8212999,"\int \frac{x^2 \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^2*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2],x]","\frac{\sqrt{1-a^2 x^2} \left(-\frac{i \left(2 \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-2 \tanh ^{-1}(a x) \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+2 \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-2 \text{Li}_3\left(i e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x)^2 \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\tanh ^{-1}(a x)^2 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)\right)}{\sqrt{1-a^2 x^2}}-a x \tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x)\right)}{2 a^3}","-\frac{i \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)}{a^3}+\frac{i \tanh ^{-1}(a x) \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)}{a^3}+\frac{i \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)}{a^3}-\frac{i \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)}{a^3}+\frac{\sin ^{-1}(a x)}{a^3}+\frac{\tanh ^{-1}(a x)^2 \tan ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)}{a^3}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a^3}",1,"(Sqrt[1 - a^2*x^2]*(-2*ArcTanh[a*x] - a*x*ArcTanh[a*x]^2 - (I*((4*I)*ArcTan[Tanh[ArcTanh[a*x]/2]] + ArcTanh[a*x]^2*Log[1 - I/E^ArcTanh[a*x]] - ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] + 2*ArcTanh[a*x]*PolyLog[2, (-I)/E^ArcTanh[a*x]] - 2*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] + 2*PolyLog[3, (-I)/E^ArcTanh[a*x]] - 2*PolyLog[3, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(2*a^3)","A",0
375,1,104,120,0.238451,"\int \frac{x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2],x]","-\frac{\tanh ^{-1}(a x) \left(\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+2 i \left(\log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)\right)+2 i \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-2 i \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)}{a^2}","-\frac{2 i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a^2}+\frac{2 i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{a^2}-\frac{4 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{a^2}",1,"-((ArcTanh[a*x]*(Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + (2*I)*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]])) + (2*I)*PolyLog[2, (-I)/E^ArcTanh[a*x]] - (2*I)*PolyLog[2, I/E^ArcTanh[a*x]])/a^2)","A",0
376,1,119,103,0.103652,"\int \frac{\tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","Integrate[ArcTanh[a*x]^2/Sqrt[1 - a^2*x^2],x]","\frac{i \left(-2 \tanh ^{-1}(a x) \left(\text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)\right)-2 \left(\text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_3\left(i e^{-\tanh ^{-1}(a x)}\right)\right)-\left(\tanh ^{-1}(a x)^2 \left(\log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)\right)\right)}{a}","-\frac{2 i \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)}{a}+\frac{2 i \tanh ^{-1}(a x) \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)}{a}+\frac{2 i \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)}{a}-\frac{2 i \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)}{a}+\frac{2 \tanh ^{-1}(a x)^2 \tan ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)}{a}",1,"(I*(-(ArcTanh[a*x]^2*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]])) - 2*ArcTanh[a*x]*(PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]) - 2*(PolyLog[3, (-I)/E^ArcTanh[a*x]] - PolyLog[3, I/E^ArcTanh[a*x]])))/a","A",0
377,1,100,68,0.0910331,"\int \frac{\tanh ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcTanh[a*x]^2/(x*Sqrt[1 - a^2*x^2]),x]","2 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-2 \tanh ^{-1}(a x) \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+2 \text{Li}_3\left(-e^{-\tanh ^{-1}(a x)}\right)-2 \text{Li}_3\left(e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x)^2 \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-\tanh ^{-1}(a x)^2 \log \left(e^{-\tanh ^{-1}(a x)}+1\right)","-2 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{\tanh ^{-1}(a x)}\right)+2 \tanh ^{-1}(a x) \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right)+2 \text{Li}_3\left(-e^{\tanh ^{-1}(a x)}\right)-2 \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right)-2 \tanh ^{-1}\left(e^{\tanh ^{-1}(a x)}\right) \tanh ^{-1}(a x)^2",1,"ArcTanh[a*x]^2*Log[1 - E^(-ArcTanh[a*x])] - ArcTanh[a*x]^2*Log[1 + E^(-ArcTanh[a*x])] + 2*ArcTanh[a*x]*PolyLog[2, -E^(-ArcTanh[a*x])] - 2*ArcTanh[a*x]*PolyLog[2, E^(-ArcTanh[a*x])] + 2*PolyLog[3, -E^(-ArcTanh[a*x])] - 2*PolyLog[3, E^(-ArcTanh[a*x])]","A",0
378,1,89,105,0.5235268,"\int \frac{\tanh ^{-1}(a x)^2}{x^2 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcTanh[a*x]^2/(x^2*Sqrt[1 - a^2*x^2]),x]","-\frac{\tanh ^{-1}(a x) \left(\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+2 a x \left(\log \left(e^{-\tanh ^{-1}(a x)}+1\right)-\log \left(1-e^{-\tanh ^{-1}(a x)}\right)\right)\right)}{x}+2 a \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-2 a \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)","-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}+2 a \text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-2 a \text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-4 a \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)",1,"-((ArcTanh[a*x]*(Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + 2*a*x*(-Log[1 - E^(-ArcTanh[a*x])] + Log[1 + E^(-ArcTanh[a*x])])))/x) + 2*a*PolyLog[2, -E^(-ArcTanh[a*x])] - 2*a*PolyLog[2, E^(-ArcTanh[a*x])]","A",0
379,1,188,152,1.269092,"\int \frac{\tanh ^{-1}(a x)^2}{x^3 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcTanh[a*x]^2/(x^3*Sqrt[1 - a^2*x^2]),x]","\frac{1}{8} a^2 \left(8 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-8 \tanh ^{-1}(a x) \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+8 \text{Li}_3\left(-e^{-\tanh ^{-1}(a x)}\right)-8 \text{Li}_3\left(e^{-\tanh ^{-1}(a x)}\right)+4 \tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right) \tanh ^{-1}(a x)+4 \tanh ^{-1}(a x)^2 \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-4 \tanh ^{-1}(a x)^2 \log \left(e^{-\tanh ^{-1}(a x)}+1\right)+8 \log \left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)-4 \tanh ^{-1}(a x) \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)+\tanh ^{-1}(a x)^2 \left(-\text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)-\tanh ^{-1}(a x)^2 \text{sech}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)","-a^2 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{\tanh ^{-1}(a x)}\right)+a^2 \tanh ^{-1}(a x) \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right)+a^2 \text{Li}_3\left(-e^{\tanh ^{-1}(a x)}\right)-a^2 \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right)-a^2 \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}+a^2 \left(-\tanh ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)\right) \tanh ^{-1}(a x)^2",1,"(a^2*(-4*ArcTanh[a*x]*Coth[ArcTanh[a*x]/2] - ArcTanh[a*x]^2*Csch[ArcTanh[a*x]/2]^2 + 4*ArcTanh[a*x]^2*Log[1 - E^(-ArcTanh[a*x])] - 4*ArcTanh[a*x]^2*Log[1 + E^(-ArcTanh[a*x])] + 8*Log[Tanh[ArcTanh[a*x]/2]] + 8*ArcTanh[a*x]*PolyLog[2, -E^(-ArcTanh[a*x])] - 8*ArcTanh[a*x]*PolyLog[2, E^(-ArcTanh[a*x])] + 8*PolyLog[3, -E^(-ArcTanh[a*x])] - 8*PolyLog[3, E^(-ArcTanh[a*x])] - ArcTanh[a*x]^2*Sech[ArcTanh[a*x]/2]^2 + 4*ArcTanh[a*x]*Tanh[ArcTanh[a*x]/2]))/8","A",0
380,1,215,219,1.0059334,"\int \frac{x^3 \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^3*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2],x]","\frac{\sqrt{1-a^2 x^2} \left(-\frac{3 i \left(10 \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-10 \tanh ^{-1}(a x) \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+10 \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-10 \text{Li}_3\left(i e^{-\tanh ^{-1}(a x)}\right)+5 \tanh ^{-1}(a x)^2 \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-5 \tanh ^{-1}(a x)^2 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)\right)}{\sqrt{1-a^2 x^2}}+2 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^3-3 a x \tanh ^{-1}(a x)^2-6 \left(\tanh ^{-1}(a x)^2+1\right) \tanh ^{-1}(a x)\right)}{6 a^4}","-\frac{5 i \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)}{a^4}+\frac{5 i \tanh ^{-1}(a x) \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)}{a^4}+\frac{5 i \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)}{a^4}-\frac{5 i \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)}{a^4}+\frac{\sin ^{-1}(a x)}{a^4}+\frac{5 \tanh ^{-1}(a x)^2 \tan ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)}{a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{3 a^2}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{3 a^4}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a^4}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a^3}",1,"(Sqrt[1 - a^2*x^2]*(-3*a*x*ArcTanh[a*x]^2 + 2*(1 - a^2*x^2)*ArcTanh[a*x]^3 - 6*ArcTanh[a*x]*(1 + ArcTanh[a*x]^2) - ((3*I)*((4*I)*ArcTan[Tanh[ArcTanh[a*x]/2]] + 5*ArcTanh[a*x]^2*Log[1 - I/E^ArcTanh[a*x]] - 5*ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] + 10*ArcTanh[a*x]*PolyLog[2, (-I)/E^ArcTanh[a*x]] - 10*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] + 10*PolyLog[3, (-I)/E^ArcTanh[a*x]] - 10*PolyLog[3, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(6*a^4)","A",0
381,1,570,305,5.234515,"\int \frac{x^2 \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^2*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2],x]","-\frac{i \left(-64 i a x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3-192 i \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+192 \tanh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)+192 i \pi  \tanh ^{-1}(a x) \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)+384 \tanh ^{-1}(a x) \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-384 \tanh ^{-1}(a x) \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)-48 \left(-4 i \pi  \tanh ^{-1}(a x)-4 \left(\tanh ^{-1}(a x)^2+2\right)+\pi ^2\right) \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-384 \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)-48 \pi ^2 \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)+192 i \pi  \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-192 i \pi  \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)+384 \text{Li}_4\left(-i e^{-\tanh ^{-1}(a x)}\right)+384 \text{Li}_4\left(-i e^{\tanh ^{-1}(a x)}\right)-16 \tanh ^{-1}(a x)^4-32 i \pi  \tanh ^{-1}(a x)^3+24 \pi ^2 \tanh ^{-1}(a x)^2+8 i \pi ^3 \tanh ^{-1}(a x)-64 \tanh ^{-1}(a x)^3 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+64 \tanh ^{-1}(a x)^3 \log \left(1+i e^{\tanh ^{-1}(a x)}\right)-96 i \pi  \tanh ^{-1}(a x)^2 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+96 i \pi  \tanh ^{-1}(a x)^2 \log \left(1-i e^{\tanh ^{-1}(a x)}\right)+384 \tanh ^{-1}(a x) \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)+48 \pi ^2 \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)-384 \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)-48 \pi ^2 \tanh ^{-1}(a x) \log \left(1-i e^{\tanh ^{-1}(a x)}\right)+8 i \pi ^3 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)-8 i \pi ^3 \log \left(1+i e^{\tanh ^{-1}(a x)}\right)+8 i \pi ^3 \log \left(\tan \left(\frac{1}{4} \left(\pi +2 i \tanh ^{-1}(a x)\right)\right)\right)+7 \pi ^4\right)}{128 a^3}","-\frac{3 i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a^3}+\frac{3 i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a^3}-\frac{3 i \tanh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)}{2 a^3}+\frac{3 i \tanh ^{-1}(a x)^2 \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)}{2 a^3}+\frac{3 i \tanh ^{-1}(a x) \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)}{a^3}-\frac{3 i \tanh ^{-1}(a x) \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)}{a^3}-\frac{3 i \text{Li}_4\left(-i e^{\tanh ^{-1}(a x)}\right)}{a^3}+\frac{3 i \text{Li}_4\left(i e^{\tanh ^{-1}(a x)}\right)}{a^3}+\frac{\tanh ^{-1}(a x)^3 \tan ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)}{a^3}-\frac{6 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{a^3}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 a^2}-\frac{3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a^3}",1,"((-1/128*I)*(7*Pi^4 + (8*I)*Pi^3*ArcTanh[a*x] + 24*Pi^2*ArcTanh[a*x]^2 - (192*I)*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2 - (32*I)*Pi*ArcTanh[a*x]^3 - (64*I)*a*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3 - 16*ArcTanh[a*x]^4 + 384*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] + (8*I)*Pi^3*Log[1 + I/E^ArcTanh[a*x]] - 384*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] + 48*Pi^2*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] - (96*I)*Pi*ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] - 64*ArcTanh[a*x]^3*Log[1 + I/E^ArcTanh[a*x]] - 48*Pi^2*ArcTanh[a*x]*Log[1 - I*E^ArcTanh[a*x]] + (96*I)*Pi*ArcTanh[a*x]^2*Log[1 - I*E^ArcTanh[a*x]] - (8*I)*Pi^3*Log[1 + I*E^ArcTanh[a*x]] + 64*ArcTanh[a*x]^3*Log[1 + I*E^ArcTanh[a*x]] + (8*I)*Pi^3*Log[Tan[(Pi + (2*I)*ArcTanh[a*x])/4]] - 48*(Pi^2 - (4*I)*Pi*ArcTanh[a*x] - 4*(2 + ArcTanh[a*x]^2))*PolyLog[2, (-I)/E^ArcTanh[a*x]] - 384*PolyLog[2, I/E^ArcTanh[a*x]] + 192*ArcTanh[a*x]^2*PolyLog[2, (-I)*E^ArcTanh[a*x]] - 48*Pi^2*PolyLog[2, I*E^ArcTanh[a*x]] + (192*I)*Pi*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]] + (192*I)*Pi*PolyLog[3, (-I)/E^ArcTanh[a*x]] + 384*ArcTanh[a*x]*PolyLog[3, (-I)/E^ArcTanh[a*x]] - 384*ArcTanh[a*x]*PolyLog[3, (-I)*E^ArcTanh[a*x]] - (192*I)*Pi*PolyLog[3, I*E^ArcTanh[a*x]] + 384*PolyLog[4, (-I)/E^ArcTanh[a*x]] + 384*PolyLog[4, (-I)*E^ArcTanh[a*x]]))/a^3","A",0
382,1,157,128,0.1929748,"\int \frac{x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2],x]","-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3+6 i \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-6 i \tanh ^{-1}(a x) \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+6 i \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-6 i \text{Li}_3\left(i e^{-\tanh ^{-1}(a x)}\right)+3 i \tanh ^{-1}(a x)^2 \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-3 i \tanh ^{-1}(a x)^2 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)}{a^2}","-\frac{6 i \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)}{a^2}+\frac{6 i \tanh ^{-1}(a x) \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)}{a^2}+\frac{6 i \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)}{a^2}-\frac{6 i \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)}{a^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{a^2}+\frac{6 \tanh ^{-1}(a x)^2 \tan ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)}{a^2}",1,"-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3 + (3*I)*ArcTanh[a*x]^2*Log[1 - I/E^ArcTanh[a*x]] - (3*I)*ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] + (6*I)*ArcTanh[a*x]*PolyLog[2, (-I)/E^ArcTanh[a*x]] - (6*I)*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] + (6*I)*PolyLog[3, (-I)/E^ArcTanh[a*x]] - (6*I)*PolyLog[3, I/E^ArcTanh[a*x]])/a^2)","A",0
383,1,451,153,0.4391984,"\int \frac{\tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","Integrate[ArcTanh[a*x]^3/Sqrt[1 - a^2*x^2],x]","-\frac{i \left(192 \tanh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)+192 i \pi  \tanh ^{-1}(a x) \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)+384 \tanh ^{-1}(a x) \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-384 \tanh ^{-1}(a x) \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)-48 \left(\pi -2 i \tanh ^{-1}(a x)\right)^2 \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-48 \pi ^2 \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)+192 i \pi  \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-192 i \pi  \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)+384 \text{Li}_4\left(-i e^{-\tanh ^{-1}(a x)}\right)+384 \text{Li}_4\left(-i e^{\tanh ^{-1}(a x)}\right)-16 \tanh ^{-1}(a x)^4-32 i \pi  \tanh ^{-1}(a x)^3+24 \pi ^2 \tanh ^{-1}(a x)^2+8 i \pi ^3 \tanh ^{-1}(a x)-64 \tanh ^{-1}(a x)^3 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+64 \tanh ^{-1}(a x)^3 \log \left(1+i e^{\tanh ^{-1}(a x)}\right)-96 i \pi  \tanh ^{-1}(a x)^2 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+96 i \pi  \tanh ^{-1}(a x)^2 \log \left(1-i e^{\tanh ^{-1}(a x)}\right)+48 \pi ^2 \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)-48 \pi ^2 \tanh ^{-1}(a x) \log \left(1-i e^{\tanh ^{-1}(a x)}\right)+8 i \pi ^3 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)-8 i \pi ^3 \log \left(1+i e^{\tanh ^{-1}(a x)}\right)+8 i \pi ^3 \log \left(\tan \left(\frac{1}{4} \left(\pi +2 i \tanh ^{-1}(a x)\right)\right)\right)+7 \pi ^4\right)}{64 a}","-\frac{3 i \tanh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)}{a}+\frac{3 i \tanh ^{-1}(a x)^2 \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)}{a}+\frac{6 i \tanh ^{-1}(a x) \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)}{a}-\frac{6 i \tanh ^{-1}(a x) \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)}{a}-\frac{6 i \text{Li}_4\left(-i e^{\tanh ^{-1}(a x)}\right)}{a}+\frac{6 i \text{Li}_4\left(i e^{\tanh ^{-1}(a x)}\right)}{a}+\frac{2 \tanh ^{-1}(a x)^3 \tan ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)}{a}",1,"((-1/64*I)*(7*Pi^4 + (8*I)*Pi^3*ArcTanh[a*x] + 24*Pi^2*ArcTanh[a*x]^2 - (32*I)*Pi*ArcTanh[a*x]^3 - 16*ArcTanh[a*x]^4 + (8*I)*Pi^3*Log[1 + I/E^ArcTanh[a*x]] + 48*Pi^2*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] - (96*I)*Pi*ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] - 64*ArcTanh[a*x]^3*Log[1 + I/E^ArcTanh[a*x]] - 48*Pi^2*ArcTanh[a*x]*Log[1 - I*E^ArcTanh[a*x]] + (96*I)*Pi*ArcTanh[a*x]^2*Log[1 - I*E^ArcTanh[a*x]] - (8*I)*Pi^3*Log[1 + I*E^ArcTanh[a*x]] + 64*ArcTanh[a*x]^3*Log[1 + I*E^ArcTanh[a*x]] + (8*I)*Pi^3*Log[Tan[(Pi + (2*I)*ArcTanh[a*x])/4]] - 48*(Pi - (2*I)*ArcTanh[a*x])^2*PolyLog[2, (-I)/E^ArcTanh[a*x]] + 192*ArcTanh[a*x]^2*PolyLog[2, (-I)*E^ArcTanh[a*x]] - 48*Pi^2*PolyLog[2, I*E^ArcTanh[a*x]] + (192*I)*Pi*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]] + (192*I)*Pi*PolyLog[3, (-I)/E^ArcTanh[a*x]] + 384*ArcTanh[a*x]*PolyLog[3, (-I)/E^ArcTanh[a*x]] - 384*ArcTanh[a*x]*PolyLog[3, (-I)*E^ArcTanh[a*x]] - (192*I)*Pi*PolyLog[3, I*E^ArcTanh[a*x]] + 384*PolyLog[4, (-I)/E^ArcTanh[a*x]] + 384*PolyLog[4, (-I)*E^ArcTanh[a*x]]))/a","B",0
384,1,146,102,0.1418038,"\int \frac{\tanh ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcTanh[a*x]^3/(x*Sqrt[1 - a^2*x^2]),x]","\frac{1}{8} \left(24 \tanh ^{-1}(a x)^2 \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)+24 \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right)+48 \tanh ^{-1}(a x) \text{Li}_3\left(-e^{-\tanh ^{-1}(a x)}\right)-48 \tanh ^{-1}(a x) \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right)+48 \text{Li}_4\left(-e^{-\tanh ^{-1}(a x)}\right)+48 \text{Li}_4\left(e^{\tanh ^{-1}(a x)}\right)-2 \tanh ^{-1}(a x)^4-8 \tanh ^{-1}(a x)^3 \log \left(e^{-\tanh ^{-1}(a x)}+1\right)+8 \tanh ^{-1}(a x)^3 \log \left(1-e^{\tanh ^{-1}(a x)}\right)+\pi ^4\right)","-3 \tanh ^{-1}(a x)^2 \text{Li}_2\left(-e^{\tanh ^{-1}(a x)}\right)+3 \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right)+6 \tanh ^{-1}(a x) \text{Li}_3\left(-e^{\tanh ^{-1}(a x)}\right)-6 \tanh ^{-1}(a x) \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right)-6 \text{Li}_4\left(-e^{\tanh ^{-1}(a x)}\right)+6 \text{Li}_4\left(e^{\tanh ^{-1}(a x)}\right)-2 \tanh ^{-1}\left(e^{\tanh ^{-1}(a x)}\right) \tanh ^{-1}(a x)^3",1,"(Pi^4 - 2*ArcTanh[a*x]^4 - 8*ArcTanh[a*x]^3*Log[1 + E^(-ArcTanh[a*x])] + 8*ArcTanh[a*x]^3*Log[1 - E^ArcTanh[a*x]] + 24*ArcTanh[a*x]^2*PolyLog[2, -E^(-ArcTanh[a*x])] + 24*ArcTanh[a*x]^2*PolyLog[2, E^ArcTanh[a*x]] + 48*ArcTanh[a*x]*PolyLog[3, -E^(-ArcTanh[a*x])] - 48*ArcTanh[a*x]*PolyLog[3, E^ArcTanh[a*x]] + 48*PolyLog[4, -E^(-ArcTanh[a*x])] + 48*PolyLog[4, E^ArcTanh[a*x]])/8","A",0
385,1,131,98,0.3901569,"\int \frac{\tanh ^{-1}(a x)^3}{x^2 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcTanh[a*x]^3/(x^2*Sqrt[1 - a^2*x^2]),x]","a \left(-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{a x}+6 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-6 \tanh ^{-1}(a x) \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+6 \text{Li}_3\left(-e^{-\tanh ^{-1}(a x)}\right)-6 \text{Li}_3\left(e^{-\tanh ^{-1}(a x)}\right)+3 \tanh ^{-1}(a x)^2 \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-3 \tanh ^{-1}(a x)^2 \log \left(e^{-\tanh ^{-1}(a x)}+1\right)\right)","-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{x}-6 a \tanh ^{-1}(a x) \text{Li}_2\left(-e^{\tanh ^{-1}(a x)}\right)+6 a \tanh ^{-1}(a x) \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right)+6 a \text{Li}_3\left(-e^{\tanh ^{-1}(a x)}\right)-6 a \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right)-6 a \tanh ^{-1}\left(e^{\tanh ^{-1}(a x)}\right) \tanh ^{-1}(a x)^2",1,"a*(-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/(a*x)) + 3*ArcTanh[a*x]^2*Log[1 - E^(-ArcTanh[a*x])] - 3*ArcTanh[a*x]^2*Log[1 + E^(-ArcTanh[a*x])] + 6*ArcTanh[a*x]*PolyLog[2, -E^(-ArcTanh[a*x])] - 6*ArcTanh[a*x]*PolyLog[2, E^(-ArcTanh[a*x])] + 6*PolyLog[3, -E^(-ArcTanh[a*x])] - 6*PolyLog[3, E^(-ArcTanh[a*x])])","A",0
386,1,416,267,8.9718179,"\int \frac{\tanh ^{-1}(a x)^3}{x^3 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcTanh[a*x]^3/(x^3*Sqrt[1 - a^2*x^2]),x]","\frac{1}{16} a^2 \tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right) \left(-\frac{4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{a x}-\frac{a x \tanh ^{-1}(a x)^3 \text{csch}^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{\sqrt{1-a^2 x^2}}+24 \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right) \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)+48 \tanh ^{-1}(a x) \text{Li}_3\left(-e^{-\tanh ^{-1}(a x)}\right) \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)-48 \tanh ^{-1}(a x) \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right) \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)+24 \left(\tanh ^{-1}(a x)^2+2\right) \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right) \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)-48 \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right) \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)+48 \text{Li}_4\left(-e^{-\tanh ^{-1}(a x)}\right) \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)+48 \text{Li}_4\left(e^{\tanh ^{-1}(a x)}\right) \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)+12 \tanh ^{-1}(a x)^2-12 \tanh ^{-1}(a x)^2 \coth ^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-2 \tanh ^{-1}(a x)^4 \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)+\pi ^4 \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)-8 \tanh ^{-1}(a x)^3 \log \left(e^{-\tanh ^{-1}(a x)}+1\right) \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)+8 \tanh ^{-1}(a x)^3 \log \left(1-e^{\tanh ^{-1}(a x)}\right) \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)+48 \tanh ^{-1}(a x) \log \left(1-e^{-\tanh ^{-1}(a x)}\right) \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)-48 \tanh ^{-1}(a x) \log \left(e^{-\tanh ^{-1}(a x)}+1\right) \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)","3 a^2 \text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-3 a^2 \text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left(-e^{\tanh ^{-1}(a x)}\right)+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right)+3 a^2 \tanh ^{-1}(a x) \text{Li}_3\left(-e^{\tanh ^{-1}(a x)}\right)-3 a^2 \tanh ^{-1}(a x) \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right)-3 a^2 \text{Li}_4\left(-e^{\tanh ^{-1}(a x)}\right)+3 a^2 \text{Li}_4\left(e^{\tanh ^{-1}(a x)}\right)-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}-a^2 \tanh ^{-1}\left(e^{\tanh ^{-1}(a x)}\right) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)",1,"(a^2*(12*ArcTanh[a*x]^2 - (4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/(a*x) + Pi^4*Coth[ArcTanh[a*x]/2] - 2*ArcTanh[a*x]^4*Coth[ArcTanh[a*x]/2] - 12*ArcTanh[a*x]^2*Coth[ArcTanh[a*x]/2]^2 - (a*x*ArcTanh[a*x]^3*Csch[ArcTanh[a*x]/2]^4)/Sqrt[1 - a^2*x^2] + 48*ArcTanh[a*x]*Coth[ArcTanh[a*x]/2]*Log[1 - E^(-ArcTanh[a*x])] - 48*ArcTanh[a*x]*Coth[ArcTanh[a*x]/2]*Log[1 + E^(-ArcTanh[a*x])] - 8*ArcTanh[a*x]^3*Coth[ArcTanh[a*x]/2]*Log[1 + E^(-ArcTanh[a*x])] + 8*ArcTanh[a*x]^3*Coth[ArcTanh[a*x]/2]*Log[1 - E^ArcTanh[a*x]] + 24*(2 + ArcTanh[a*x]^2)*Coth[ArcTanh[a*x]/2]*PolyLog[2, -E^(-ArcTanh[a*x])] - 48*Coth[ArcTanh[a*x]/2]*PolyLog[2, E^(-ArcTanh[a*x])] + 24*ArcTanh[a*x]^2*Coth[ArcTanh[a*x]/2]*PolyLog[2, E^ArcTanh[a*x]] + 48*ArcTanh[a*x]*Coth[ArcTanh[a*x]/2]*PolyLog[3, -E^(-ArcTanh[a*x])] - 48*ArcTanh[a*x]*Coth[ArcTanh[a*x]/2]*PolyLog[3, E^ArcTanh[a*x]] + 48*Coth[ArcTanh[a*x]/2]*PolyLog[4, -E^(-ArcTanh[a*x])] + 48*Coth[ArcTanh[a*x]/2]*PolyLog[4, E^ArcTanh[a*x]])*Tanh[ArcTanh[a*x]/2])/16","A",0
387,0,0,25,0.4720718,"\int \frac{x^m \tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[(x^m*ArcTanh[a*x])/(1 - a^2*x^2)^(3/2),x]","\int \frac{x^m \tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^{3/2}},x\right)",0,"Integrate[(x^m*ArcTanh[a*x])/(1 - a^2*x^2)^(3/2), x]","A",-1
388,1,76,74,0.0689088,"\int \frac{x^3 \tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[(x^3*ArcTanh[a*x])/(1 - a^2*x^2)^(3/2),x]","\frac{a x \sqrt{1-a^2 x^2}+\left(1-a^2 x^2\right) \sin ^{-1}(a x)+\sqrt{1-a^2 x^2} \left(a^2 x^2-2\right) \tanh ^{-1}(a x)}{a^4 \left(a^2 x^2-1\right)}","-\frac{\sin ^{-1}(a x)}{a^4}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a^4}+\frac{\tanh ^{-1}(a x)}{a^4 \sqrt{1-a^2 x^2}}-\frac{x}{a^3 \sqrt{1-a^2 x^2}}",1,"(a*x*Sqrt[1 - a^2*x^2] + (1 - a^2*x^2)*ArcSin[a*x] + Sqrt[1 - a^2*x^2]*(-2 + a^2*x^2)*ArcTanh[a*x])/(a^4*(-1 + a^2*x^2))","A",1
389,1,121,137,0.2518571,"\int \frac{x^2 \tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[(x^2*ArcTanh[a*x])/(1 - a^2*x^2)^(3/2),x]","\frac{i \left(\frac{i}{\sqrt{1-a^2 x^2}}-\frac{i a x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+\text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x) \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)}{a^3}","\frac{i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a^3}-\frac{i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a^3}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{a^3}+\frac{x \tanh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}-\frac{1}{a^3 \sqrt{1-a^2 x^2}}",1,"(I*(I/Sqrt[1 - a^2*x^2] - (I*a*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] + ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] - ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] + PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/a^3","A",0
390,1,27,43,0.038905,"\int \frac{x \tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[(x*ArcTanh[a*x])/(1 - a^2*x^2)^(3/2),x]","\frac{\tanh ^{-1}(a x)-a x}{a^2 \sqrt{1-a^2 x^2}}","\frac{\tanh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}-\frac{x}{a \sqrt{1-a^2 x^2}}",1,"(-(a*x) + ArcTanh[a*x])/(a^2*Sqrt[1 - a^2*x^2])","A",1
391,1,27,40,0.0357896,"\int \frac{\tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[ArcTanh[a*x]/(1 - a^2*x^2)^(3/2),x]","\frac{a x \tanh ^{-1}(a x)-1}{a \sqrt{1-a^2 x^2}}","\frac{x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{1}{a \sqrt{1-a^2 x^2}}",1,"(-1 + a*x*ArcTanh[a*x])/(a*Sqrt[1 - a^2*x^2])","A",1
392,1,97,112,0.1931993,"\int \frac{\tanh ^{-1}(a x)}{x \left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[ArcTanh[a*x]/(x*(1 - a^2*x^2)^(3/2)),x]","-\frac{a x}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+\text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x) \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-\tanh ^{-1}(a x) \log \left(e^{-\tanh ^{-1}(a x)}+1\right)","-\frac{a x}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+\text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-2 \tanh ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)",1,"-((a*x)/Sqrt[1 - a^2*x^2]) + ArcTanh[a*x]/Sqrt[1 - a^2*x^2] + ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] - ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] + PolyLog[2, -E^(-ArcTanh[a*x])] - PolyLog[2, E^(-ArcTanh[a*x])]","A",0
393,1,89,82,0.1168752,"\int \frac{\tanh ^{-1}(a x)}{x^2 \left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[ArcTanh[a*x]/(x^2*(1 - a^2*x^2)^(3/2)),x]","\frac{a x \left(\sqrt{1-a^2 x^2} \log (x)-\sqrt{1-a^2 x^2} \log \left(\sqrt{1-a^2 x^2}+1\right)-1\right)+\left(2 a^2 x^2-1\right) \tanh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}}","-\frac{a}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-a \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}",1,"((-1 + 2*a^2*x^2)*ArcTanh[a*x] + a*x*(-1 + Sqrt[1 - a^2*x^2]*Log[x] - Sqrt[1 - a^2*x^2]*Log[1 + Sqrt[1 - a^2*x^2]]))/(x*Sqrt[1 - a^2*x^2])","A",1
394,1,182,179,1.6337695,"\int \frac{\tanh ^{-1}(a x)}{x^3 \left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[ArcTanh[a*x]/(x^3*(1 - a^2*x^2)^(3/2)),x]","\frac{1}{8} a^2 \left(-\frac{8 a x}{\sqrt{1-a^2 x^2}}+\frac{8 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{a x \text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{\sqrt{1-a^2 x^2}}+12 \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-12 \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+2 \tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)+12 \tanh ^{-1}(a x) \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-12 \tanh ^{-1}(a x) \log \left(e^{-\tanh ^{-1}(a x)}+1\right)-\tanh ^{-1}(a x) \text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-\tanh ^{-1}(a x) \text{sech}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)","\frac{3}{2} a^2 \text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{3}{2} a^2 \text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{a \sqrt{1-a^2 x^2}}{2 x}+\frac{a^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{2 x^2}-3 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{a^3 x}{\sqrt{1-a^2 x^2}}",1,"(a^2*((-8*a*x)/Sqrt[1 - a^2*x^2] + (8*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (a*x*Csch[ArcTanh[a*x]/2]^2)/Sqrt[1 - a^2*x^2] - ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^2 + 12*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] - 12*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] + 12*PolyLog[2, -E^(-ArcTanh[a*x])] - 12*PolyLog[2, E^(-ArcTanh[a*x])] - ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^2 + 2*Tanh[ArcTanh[a*x]/2]))/8","A",0
395,0,0,27,0.5252768,"\int \frac{x^m \tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[(x^m*ArcTanh[a*x]^2)/(1 - a^2*x^2)^(3/2),x]","\int \frac{x^m \tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^{3/2}},x\right)",0,"Integrate[(x^m*ArcTanh[a*x]^2)/(1 - a^2*x^2)^(3/2), x]","A",-1
396,1,165,186,0.3974991,"\int \frac{x^3 \tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[(x^3*ArcTanh[a*x]^2)/(1 - a^2*x^2)^(3/2),x]","\frac{\frac{-2 i \sqrt{1-a^2 x^2} \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+\left(2-a^2 x^2\right) \tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x) \left(-i \sqrt{1-a^2 x^2} \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)+i \sqrt{1-a^2 x^2} \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+a x\right)+2}{\sqrt{1-a^2 x^2}}+2 i \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)}{a^4}","\frac{2 i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a^4}-\frac{2 i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a^4}+\frac{4 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{a^4}+\frac{2}{a^4 \sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{a^4}+\frac{\tanh ^{-1}(a x)^2}{a^4 \sqrt{1-a^2 x^2}}-\frac{2 x \tanh ^{-1}(a x)}{a^3 \sqrt{1-a^2 x^2}}",1,"((2*I)*PolyLog[2, (-I)/E^ArcTanh[a*x]] + (2 + (2 - a^2*x^2)*ArcTanh[a*x]^2 - 2*ArcTanh[a*x]*(a*x - I*Sqrt[1 - a^2*x^2]*Log[1 - I/E^ArcTanh[a*x]] + I*Sqrt[1 - a^2*x^2]*Log[1 + I/E^ArcTanh[a*x]]) - (2*I)*Sqrt[1 - a^2*x^2]*PolyLog[2, I/E^ArcTanh[a*x]])/Sqrt[1 - a^2*x^2])/a^4","A",0
397,1,193,171,0.3465228,"\int \frac{x^2 \tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[(x^2*ArcTanh[a*x]^2)/(1 - a^2*x^2)^(3/2),x]","\frac{i \left(-\frac{2 i a x}{\sqrt{1-a^2 x^2}}-\frac{i a x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{2 i \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+2 \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-2 \tanh ^{-1}(a x) \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+2 \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-2 \text{Li}_3\left(i e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x)^2 \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\tanh ^{-1}(a x)^2 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)}{a^3}","\frac{2 i \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)}{a^3}-\frac{2 i \tanh ^{-1}(a x) \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)}{a^3}-\frac{2 i \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)}{a^3}+\frac{2 i \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)}{a^3}-\frac{2 \tanh ^{-1}(a x)^2 \tan ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)}{a^3}+\frac{2 x}{a^2 \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{a^2 \sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{a^3 \sqrt{1-a^2 x^2}}",1,"(I*(((-2*I)*a*x)/Sqrt[1 - a^2*x^2] + ((2*I)*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (I*a*x*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2] + ArcTanh[a*x]^2*Log[1 - I/E^ArcTanh[a*x]] - ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] + 2*ArcTanh[a*x]*PolyLog[2, (-I)/E^ArcTanh[a*x]] - 2*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] + 2*PolyLog[3, (-I)/E^ArcTanh[a*x]] - 2*PolyLog[3, I/E^ArcTanh[a*x]]))/a^3","A",0
398,1,34,68,0.0568032,"\int \frac{x \tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[(x*ArcTanh[a*x]^2)/(1 - a^2*x^2)^(3/2),x]","\frac{\tanh ^{-1}(a x)^2-2 a x \tanh ^{-1}(a x)+2}{a^2 \sqrt{1-a^2 x^2}}","\frac{2}{a^2 \sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)^2}{a^2 \sqrt{1-a^2 x^2}}-\frac{2 x \tanh ^{-1}(a x)}{a \sqrt{1-a^2 x^2}}",1,"(2 - 2*a*x*ArcTanh[a*x] + ArcTanh[a*x]^2)/(a^2*Sqrt[1 - a^2*x^2])","A",1
399,1,38,63,0.0412367,"\int \frac{\tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2),x]","\frac{2 a x+a x \tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x)}{a \sqrt{1-a^2 x^2}}","\frac{2 x}{\sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{a \sqrt{1-a^2 x^2}}",1,"(2*a*x - 2*ArcTanh[a*x] + a*x*ArcTanh[a*x]^2)/(a*Sqrt[1 - a^2*x^2])","A",1
400,1,159,127,0.2558379,"\int \frac{\tanh ^{-1}(a x)^2}{x \left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[ArcTanh[a*x]^2/(x*(1 - a^2*x^2)^(3/2)),x]","\frac{2}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{2 a x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+2 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-2 \tanh ^{-1}(a x) \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+2 \text{Li}_3\left(-e^{-\tanh ^{-1}(a x)}\right)-2 \text{Li}_3\left(e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x)^2 \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-\tanh ^{-1}(a x)^2 \log \left(e^{-\tanh ^{-1}(a x)}+1\right)","\frac{2}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{2 a x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-2 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{\tanh ^{-1}(a x)}\right)+2 \tanh ^{-1}(a x) \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right)+2 \text{Li}_3\left(-e^{\tanh ^{-1}(a x)}\right)-2 \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right)-2 \tanh ^{-1}\left(e^{\tanh ^{-1}(a x)}\right) \tanh ^{-1}(a x)^2",1,"2/Sqrt[1 - a^2*x^2] - (2*a*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] + ArcTanh[a*x]^2/Sqrt[1 - a^2*x^2] + ArcTanh[a*x]^2*Log[1 - E^(-ArcTanh[a*x])] - ArcTanh[a*x]^2*Log[1 + E^(-ArcTanh[a*x])] + 2*ArcTanh[a*x]*PolyLog[2, -E^(-ArcTanh[a*x])] - 2*ArcTanh[a*x]*PolyLog[2, E^(-ArcTanh[a*x])] + 2*PolyLog[3, -E^(-ArcTanh[a*x])] - 2*PolyLog[3, E^(-ArcTanh[a*x])]","A",0
401,1,215,171,1.1505001,"\int \frac{\tanh ^{-1}(a x)^2}{x^2 \left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[ArcTanh[a*x]^2/(x^2*(1 - a^2*x^2)^(3/2)),x]","\frac{a \left(4 \sqrt{1-a^2 x^2} \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-4 \sqrt{1-a^2 x^2} \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \log \left(e^{-\tanh ^{-1}(a x)}+1\right)-\frac{2 \left(a^2 x^2-1\right) \tanh ^{-1}(a x)^2 \sinh ^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{a x}+4 a x+2 a x \tanh ^{-1}(a x)^2-4 \tanh ^{-1}(a x)-\frac{1}{2} a x \tanh ^{-1}(a x)^2 \text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)}{2 \sqrt{1-a^2 x^2}}","\frac{2 a^2 x}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{2 a \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}+2 a \text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-2 a \text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-4 a \tanh ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)",1,"(a*(4*a*x - 4*ArcTanh[a*x] + 2*a*x*ArcTanh[a*x]^2 - (a*x*ArcTanh[a*x]^2*Csch[ArcTanh[a*x]/2]^2)/2 + 4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] - 4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] + 4*Sqrt[1 - a^2*x^2]*PolyLog[2, -E^(-ArcTanh[a*x])] - 4*Sqrt[1 - a^2*x^2]*PolyLog[2, E^(-ArcTanh[a*x])] - (2*(-1 + a^2*x^2)*ArcTanh[a*x]^2*Sinh[ArcTanh[a*x]/2]^2)/(a*x)))/(2*Sqrt[1 - a^2*x^2])","A",0
402,1,266,221,2.846862,"\int \frac{\tanh ^{-1}(a x)^2}{x^3 \left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[ArcTanh[a*x]^2/(x^3*(1 - a^2*x^2)^(3/2)),x]","\frac{1}{8} a^2 \left(\frac{16}{\sqrt{1-a^2 x^2}}+\frac{8 \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{16 a x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{2 a x \tanh ^{-1}(a x) \text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{\sqrt{1-a^2 x^2}}+24 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-24 \tanh ^{-1}(a x) \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+24 \text{Li}_3\left(-e^{-\tanh ^{-1}(a x)}\right)-24 \text{Li}_3\left(e^{-\tanh ^{-1}(a x)}\right)+4 \tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right) \tanh ^{-1}(a x)+12 \tanh ^{-1}(a x)^2 \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-12 \tanh ^{-1}(a x)^2 \log \left(e^{-\tanh ^{-1}(a x)}+1\right)+8 \log \left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)+\tanh ^{-1}(a x)^2 \left(-\text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)-\tanh ^{-1}(a x)^2 \text{sech}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)","-3 a^2 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{\tanh ^{-1}(a x)}\right)+3 a^2 \tanh ^{-1}(a x) \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right)+3 a^2 \text{Li}_3\left(-e^{\tanh ^{-1}(a x)}\right)-3 a^2 \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right)+\frac{2 a^2}{\sqrt{1-a^2 x^2}}+\frac{a^2 \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-a^2 \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-3 a^2 \tanh ^{-1}\left(e^{\tanh ^{-1}(a x)}\right) \tanh ^{-1}(a x)^2-\frac{2 a^3 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}",1,"(a^2*(16/Sqrt[1 - a^2*x^2] - (16*a*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] + (8*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2] - (2*a*x*ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^2)/Sqrt[1 - a^2*x^2] - ArcTanh[a*x]^2*Csch[ArcTanh[a*x]/2]^2 + 12*ArcTanh[a*x]^2*Log[1 - E^(-ArcTanh[a*x])] - 12*ArcTanh[a*x]^2*Log[1 + E^(-ArcTanh[a*x])] + 8*Log[Tanh[ArcTanh[a*x]/2]] + 24*ArcTanh[a*x]*PolyLog[2, -E^(-ArcTanh[a*x])] - 24*ArcTanh[a*x]*PolyLog[2, E^(-ArcTanh[a*x])] + 24*PolyLog[3, -E^(-ArcTanh[a*x])] - 24*PolyLog[3, E^(-ArcTanh[a*x])] - ArcTanh[a*x]^2*Sech[ArcTanh[a*x]/2]^2 + 4*ArcTanh[a*x]*Tanh[ArcTanh[a*x]/2]))/8","A",0
403,0,0,27,0.5276453,"\int \frac{x^m \tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[(x^m*ArcTanh[a*x]^3)/(1 - a^2*x^2)^(3/2),x]","\int \frac{x^m \tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^{3/2}},x\right)",0,"Integrate[(x^m*ArcTanh[a*x]^3)/(1 - a^2*x^2)^(3/2), x]","A",-1
404,1,249,220,0.4666069,"\int \frac{x^3 \tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[(x^3*ArcTanh[a*x]^3)/(1 - a^2*x^2)^(3/2),x]","\frac{\frac{6 i \sqrt{1-a^2 x^2} \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-6 i \sqrt{1-a^2 x^2} \text{Li}_3\left(i e^{-\tanh ^{-1}(a x)}\right)-a^2 x^2 \tanh ^{-1}(a x)^3+3 i \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2 \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-3 i \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)-6 a x+2 \tanh ^{-1}(a x)^3-3 a x \tanh ^{-1}(a x)^2+6 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+6 i \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-6 i \tanh ^{-1}(a x) \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)}{a^4}","\frac{6 i \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)}{a^4}-\frac{6 i \tanh ^{-1}(a x) \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)}{a^4}-\frac{6 i \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)}{a^4}+\frac{6 i \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)}{a^4}-\frac{6 \tanh ^{-1}(a x)^2 \tan ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)}{a^4}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{a^4}+\frac{\tanh ^{-1}(a x)^3}{a^4 \sqrt{1-a^2 x^2}}+\frac{6 \tanh ^{-1}(a x)}{a^4 \sqrt{1-a^2 x^2}}-\frac{6 x}{a^3 \sqrt{1-a^2 x^2}}-\frac{3 x \tanh ^{-1}(a x)^2}{a^3 \sqrt{1-a^2 x^2}}",1,"((6*I)*ArcTanh[a*x]*PolyLog[2, (-I)/E^ArcTanh[a*x]] - (6*I)*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] + (-6*a*x + 6*ArcTanh[a*x] - 3*a*x*ArcTanh[a*x]^2 + 2*ArcTanh[a*x]^3 - a^2*x^2*ArcTanh[a*x]^3 + (3*I)*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2*Log[1 - I/E^ArcTanh[a*x]] - (3*I)*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] + (6*I)*Sqrt[1 - a^2*x^2]*PolyLog[3, (-I)/E^ArcTanh[a*x]] - (6*I)*Sqrt[1 - a^2*x^2]*PolyLog[3, I/E^ArcTanh[a*x]])/Sqrt[1 - a^2*x^2])/a^4","A",0
405,1,541,246,1.0680879,"\int \frac{x^2 \tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[(x^2*ArcTanh[a*x]^3)/(1 - a^2*x^2)^(3/2),x]","\frac{-\frac{384}{\sqrt{1-a^2 x^2}}+\frac{64 a x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{192 \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{384 a x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+192 i \tanh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)-192 \pi  \tanh ^{-1}(a x) \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)+384 i \tanh ^{-1}(a x) \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-384 i \tanh ^{-1}(a x) \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)-48 i \left(\pi -2 i \tanh ^{-1}(a x)\right)^2 \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-48 i \pi ^2 \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)-192 \pi  \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)+192 \pi  \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)+384 i \text{Li}_4\left(-i e^{-\tanh ^{-1}(a x)}\right)+384 i \text{Li}_4\left(-i e^{\tanh ^{-1}(a x)}\right)-16 i \tanh ^{-1}(a x)^4+32 \pi  \tanh ^{-1}(a x)^3+24 i \pi ^2 \tanh ^{-1}(a x)^2-8 \pi ^3 \tanh ^{-1}(a x)-64 i \tanh ^{-1}(a x)^3 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+64 i \tanh ^{-1}(a x)^3 \log \left(1+i e^{\tanh ^{-1}(a x)}\right)+96 \pi  \tanh ^{-1}(a x)^2 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)-96 \pi  \tanh ^{-1}(a x)^2 \log \left(1-i e^{\tanh ^{-1}(a x)}\right)+48 i \pi ^2 \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)-48 i \pi ^2 \tanh ^{-1}(a x) \log \left(1-i e^{\tanh ^{-1}(a x)}\right)-8 \pi ^3 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+8 \pi ^3 \log \left(1+i e^{\tanh ^{-1}(a x)}\right)-8 \pi ^3 \log \left(\tan \left(\frac{1}{4} \left(\pi +2 i \tanh ^{-1}(a x)\right)\right)\right)+7 i \pi ^4}{64 a^3}","\frac{3 i \tanh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)}{a^3}-\frac{3 i \tanh ^{-1}(a x)^2 \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)}{a^3}-\frac{6 i \tanh ^{-1}(a x) \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)}{a^3}+\frac{6 i \tanh ^{-1}(a x) \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)}{a^3}+\frac{6 i \text{Li}_4\left(-i e^{\tanh ^{-1}(a x)}\right)}{a^3}-\frac{6 i \text{Li}_4\left(i e^{\tanh ^{-1}(a x)}\right)}{a^3}-\frac{2 \tanh ^{-1}(a x)^3 \tan ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)}{a^3}+\frac{x \tanh ^{-1}(a x)^3}{a^2 \sqrt{1-a^2 x^2}}+\frac{6 x \tanh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}-\frac{6}{a^3 \sqrt{1-a^2 x^2}}-\frac{3 \tanh ^{-1}(a x)^2}{a^3 \sqrt{1-a^2 x^2}}",1,"((7*I)*Pi^4 - 384/Sqrt[1 - a^2*x^2] - 8*Pi^3*ArcTanh[a*x] + (384*a*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] + (24*I)*Pi^2*ArcTanh[a*x]^2 - (192*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2] + 32*Pi*ArcTanh[a*x]^3 + (64*a*x*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2] - (16*I)*ArcTanh[a*x]^4 - 8*Pi^3*Log[1 + I/E^ArcTanh[a*x]] + (48*I)*Pi^2*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] + 96*Pi*ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] - (64*I)*ArcTanh[a*x]^3*Log[1 + I/E^ArcTanh[a*x]] - (48*I)*Pi^2*ArcTanh[a*x]*Log[1 - I*E^ArcTanh[a*x]] - 96*Pi*ArcTanh[a*x]^2*Log[1 - I*E^ArcTanh[a*x]] + 8*Pi^3*Log[1 + I*E^ArcTanh[a*x]] + (64*I)*ArcTanh[a*x]^3*Log[1 + I*E^ArcTanh[a*x]] - 8*Pi^3*Log[Tan[(Pi + (2*I)*ArcTanh[a*x])/4]] - (48*I)*(Pi - (2*I)*ArcTanh[a*x])^2*PolyLog[2, (-I)/E^ArcTanh[a*x]] + (192*I)*ArcTanh[a*x]^2*PolyLog[2, (-I)*E^ArcTanh[a*x]] - (48*I)*Pi^2*PolyLog[2, I*E^ArcTanh[a*x]] - 192*Pi*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]] - 192*Pi*PolyLog[3, (-I)/E^ArcTanh[a*x]] + (384*I)*ArcTanh[a*x]*PolyLog[3, (-I)/E^ArcTanh[a*x]] - (384*I)*ArcTanh[a*x]*PolyLog[3, (-I)*E^ArcTanh[a*x]] + 192*Pi*PolyLog[3, I*E^ArcTanh[a*x]] + (384*I)*PolyLog[4, (-I)/E^ArcTanh[a*x]] + (384*I)*PolyLog[4, (-I)*E^ArcTanh[a*x]])/(64*a^3)","B",0
406,1,45,94,0.0677328,"\int \frac{x \tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[(x*ArcTanh[a*x]^3)/(1 - a^2*x^2)^(3/2),x]","\frac{-6 a x+\tanh ^{-1}(a x)^3-3 a x \tanh ^{-1}(a x)^2+6 \tanh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}","-\frac{6 x}{a \sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)^3}{a^2 \sqrt{1-a^2 x^2}}-\frac{3 x \tanh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}+\frac{6 \tanh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}",1,"(-6*a*x + 6*ArcTanh[a*x] - 3*a*x*ArcTanh[a*x]^2 + ArcTanh[a*x]^3)/(a^2*Sqrt[1 - a^2*x^2])","A",1
407,1,45,88,0.0545033,"\int \frac{\tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[ArcTanh[a*x]^3/(1 - a^2*x^2)^(3/2),x]","\frac{a x \tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2+6 a x \tanh ^{-1}(a x)-6}{a \sqrt{1-a^2 x^2}}","-\frac{6}{a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{3 \tanh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}+\frac{6 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}",1,"(-6 + 6*a*x*ArcTanh[a*x] - 3*ArcTanh[a*x]^2 + a*x*ArcTanh[a*x]^3)/(a*Sqrt[1 - a^2*x^2])","A",1
408,1,230,185,0.3690026,"\int \frac{\tanh ^{-1}(a x)^3}{x \left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[ArcTanh[a*x]^3/(x*(1 - a^2*x^2)^(3/2)),x]","\frac{1}{8} \left(-\frac{48 a x}{\sqrt{1-a^2 x^2}}+\frac{8 \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{24 a x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{48 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+24 \tanh ^{-1}(a x)^2 \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)+24 \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right)+48 \tanh ^{-1}(a x) \text{Li}_3\left(-e^{-\tanh ^{-1}(a x)}\right)-48 \tanh ^{-1}(a x) \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right)+48 \text{Li}_4\left(-e^{-\tanh ^{-1}(a x)}\right)+48 \text{Li}_4\left(e^{\tanh ^{-1}(a x)}\right)-2 \tanh ^{-1}(a x)^4-8 \tanh ^{-1}(a x)^3 \log \left(e^{-\tanh ^{-1}(a x)}+1\right)+8 \tanh ^{-1}(a x)^3 \log \left(1-e^{\tanh ^{-1}(a x)}\right)+\pi ^4\right)","-\frac{6 a x}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{3 a x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{6 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-3 \tanh ^{-1}(a x)^2 \text{Li}_2\left(-e^{\tanh ^{-1}(a x)}\right)+3 \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right)+6 \tanh ^{-1}(a x) \text{Li}_3\left(-e^{\tanh ^{-1}(a x)}\right)-6 \tanh ^{-1}(a x) \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right)-6 \text{Li}_4\left(-e^{\tanh ^{-1}(a x)}\right)+6 \text{Li}_4\left(e^{\tanh ^{-1}(a x)}\right)-2 \tanh ^{-1}\left(e^{\tanh ^{-1}(a x)}\right) \tanh ^{-1}(a x)^3",1,"(Pi^4 - (48*a*x)/Sqrt[1 - a^2*x^2] + (48*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (24*a*x*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2] + (8*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2] - 2*ArcTanh[a*x]^4 - 8*ArcTanh[a*x]^3*Log[1 + E^(-ArcTanh[a*x])] + 8*ArcTanh[a*x]^3*Log[1 - E^ArcTanh[a*x]] + 24*ArcTanh[a*x]^2*PolyLog[2, -E^(-ArcTanh[a*x])] + 24*ArcTanh[a*x]^2*PolyLog[2, E^ArcTanh[a*x]] + 48*ArcTanh[a*x]*PolyLog[3, -E^(-ArcTanh[a*x])] - 48*ArcTanh[a*x]*PolyLog[3, E^ArcTanh[a*x]] + 48*PolyLog[4, -E^(-ArcTanh[a*x])] + 48*PolyLog[4, E^ArcTanh[a*x]])/8","A",0
409,1,270,187,2.0041947,"\int \frac{\tanh ^{-1}(a x)^3}{x^2 \left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[ArcTanh[a*x]^3/(x^2*(1 - a^2*x^2)^(3/2)),x]","-\frac{6 a}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{3 a \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3 \sinh ^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{x}-\frac{a^2 x \tanh ^{-1}(a x)^3 \text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{4 \sqrt{1-a^2 x^2}}+6 a \tanh ^{-1}(a x) \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-6 a \tanh ^{-1}(a x) \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+6 a \text{Li}_3\left(-e^{-\tanh ^{-1}(a x)}\right)-6 a \text{Li}_3\left(e^{-\tanh ^{-1}(a x)}\right)+3 a \tanh ^{-1}(a x)^2 \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-3 a \tanh ^{-1}(a x)^2 \log \left(e^{-\tanh ^{-1}(a x)}+1\right)","-\frac{6 a}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{x}+\frac{a^2 x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{3 a \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-6 a \tanh ^{-1}(a x) \text{Li}_2\left(-e^{\tanh ^{-1}(a x)}\right)+6 a \tanh ^{-1}(a x) \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right)+6 a \text{Li}_3\left(-e^{\tanh ^{-1}(a x)}\right)-6 a \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right)-6 a \tanh ^{-1}\left(e^{\tanh ^{-1}(a x)}\right) \tanh ^{-1}(a x)^2",1,"(-6*a)/Sqrt[1 - a^2*x^2] + (6*a^2*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (3*a*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2] + (a^2*x*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2] - (a^2*x*ArcTanh[a*x]^3*Csch[ArcTanh[a*x]/2]^2)/(4*Sqrt[1 - a^2*x^2]) + 3*a*ArcTanh[a*x]^2*Log[1 - E^(-ArcTanh[a*x])] - 3*a*ArcTanh[a*x]^2*Log[1 + E^(-ArcTanh[a*x])] + 6*a*ArcTanh[a*x]*PolyLog[2, -E^(-ArcTanh[a*x])] - 6*a*ArcTanh[a*x]*PolyLog[2, E^(-ArcTanh[a*x])] + 6*a*PolyLog[3, -E^(-ArcTanh[a*x])] - 6*a*PolyLog[3, E^(-ArcTanh[a*x])] + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3*Sinh[ArcTanh[a*x]/2]^2)/x","A",0
410,1,555,360,7.8666481,"\int \frac{\tanh ^{-1}(a x)^3}{x^3 \left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[ArcTanh[a*x]^3/(x^3*(1 - a^2*x^2)^(3/2)),x]","\frac{a^2 \left(72 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right)+144 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \text{Li}_3\left(-e^{-\tanh ^{-1}(a x)}\right)-144 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right)+24 \sqrt{1-a^2 x^2} \left(3 \tanh ^{-1}(a x)^2+2\right) \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-48 \sqrt{1-a^2 x^2} \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+144 \sqrt{1-a^2 x^2} \text{Li}_4\left(-e^{-\tanh ^{-1}(a x)}\right)+144 \sqrt{1-a^2 x^2} \text{Li}_4\left(e^{\tanh ^{-1}(a x)}\right)+3 \pi ^4 \sqrt{1-a^2 x^2}-6 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^4+12 \sqrt{1-a^2 x^2} \tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right) \tanh ^{-1}(a x)^2-24 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3 \log \left(e^{-\tanh ^{-1}(a x)}+1\right)+24 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3 \log \left(1-e^{\tanh ^{-1}(a x)}\right)+48 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-48 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \log \left(e^{-\tanh ^{-1}(a x)}+1\right)-12 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2 \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)-2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3 \text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3 \text{sech}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-96 a x+16 \tanh ^{-1}(a x)^3-48 a x \tanh ^{-1}(a x)^2+96 \tanh ^{-1}(a x)\right)}{16 \sqrt{1-a^2 x^2}}","3 a^2 \text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-3 a^2 \text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{9}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left(-e^{\tanh ^{-1}(a x)}\right)+\frac{9}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right)+9 a^2 \tanh ^{-1}(a x) \text{Li}_3\left(-e^{\tanh ^{-1}(a x)}\right)-9 a^2 \tanh ^{-1}(a x) \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right)-9 a^2 \text{Li}_4\left(-e^{\tanh ^{-1}(a x)}\right)+9 a^2 \text{Li}_4\left(e^{\tanh ^{-1}(a x)}\right)+\frac{a^2 \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-3 a^2 \tanh ^{-1}\left(e^{\tanh ^{-1}(a x)}\right) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{6 a^3 x}{\sqrt{1-a^2 x^2}}-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}",1,"(a^2*(-96*a*x + 3*Pi^4*Sqrt[1 - a^2*x^2] + 96*ArcTanh[a*x] - 48*a*x*ArcTanh[a*x]^2 + 16*ArcTanh[a*x]^3 - 6*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^4 - 12*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2*Coth[ArcTanh[a*x]/2] - 2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3*Csch[ArcTanh[a*x]/2]^2 + 48*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] - 48*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] - 24*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3*Log[1 + E^(-ArcTanh[a*x])] + 24*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3*Log[1 - E^ArcTanh[a*x]] + 24*Sqrt[1 - a^2*x^2]*(2 + 3*ArcTanh[a*x]^2)*PolyLog[2, -E^(-ArcTanh[a*x])] - 48*Sqrt[1 - a^2*x^2]*PolyLog[2, E^(-ArcTanh[a*x])] + 72*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2*PolyLog[2, E^ArcTanh[a*x]] + 144*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*PolyLog[3, -E^(-ArcTanh[a*x])] - 144*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*PolyLog[3, E^ArcTanh[a*x]] + 144*Sqrt[1 - a^2*x^2]*PolyLog[4, -E^(-ArcTanh[a*x])] + 144*Sqrt[1 - a^2*x^2]*PolyLog[4, E^ArcTanh[a*x]] - 2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3*Sech[ArcTanh[a*x]/2]^2 + 12*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2*Tanh[ArcTanh[a*x]/2]))/(16*Sqrt[1 - a^2*x^2])","A",0
411,0,0,27,0.5011509,"\int \frac{x^m}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)} \, dx","Integrate[x^m/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]),x]","\int \frac{x^m}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^m}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)},x\right)",0,"Integrate[x^m/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]), x]","A",-1
412,0,0,27,3.1990684,"\int \frac{x^2}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)} \, dx","Integrate[x^2/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]),x]","\int \frac{x^2}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^2}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)},x\right)",0,"Integrate[x^2/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]), x]","A",-1
413,1,9,9,0.074501,"\int \frac{x}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)} \, dx","Integrate[x/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]),x]","\frac{\text{Shi}\left(\tanh ^{-1}(a x)\right)}{a^2}","\frac{\text{Shi}\left(\tanh ^{-1}(a x)\right)}{a^2}",1,"SinhIntegral[ArcTanh[a*x]]/a^2","A",1
414,1,9,9,0.0226486,"\int \frac{1}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)} \, dx","Integrate[1/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]),x]","\frac{\text{Chi}\left(\tanh ^{-1}(a x)\right)}{a}","\frac{\text{Chi}\left(\tanh ^{-1}(a x)\right)}{a}",1,"CoshIntegral[ArcTanh[a*x]]/a","A",1
415,0,0,27,1.1340939,"\int \frac{1}{x \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)} \, dx","Integrate[1/(x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]),x]","\int \frac{1}{x \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)},x\right)",0,"Integrate[1/(x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]), x]","A",-1
416,0,0,27,0.6569082,"\int \frac{x^m}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^2} \, dx","Integrate[x^m/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2),x]","\int \frac{x^m}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x^m}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^2},x\right)",0,"Integrate[x^m/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2), x]","A",-1
417,0,0,64,3.263292,"\int \frac{x^2}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^2} \, dx","Integrate[x^2/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2),x]","\int \frac{x^2}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2},x\right)}{a^2}+\frac{\text{Shi}\left(\tanh ^{-1}(a x)\right)}{a^3}-\frac{1}{a^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}",0,"Integrate[x^2/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2), x]","A",-1
418,1,34,36,0.0879702,"\int \frac{x}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^2} \, dx","Integrate[x/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2),x]","\frac{\text{Chi}\left(\tanh ^{-1}(a x)\right)-\frac{a x}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}}{a^2}","\frac{\text{Chi}\left(\tanh ^{-1}(a x)\right)}{a^2}-\frac{x}{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}",1,"(-((a*x)/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])) + CoshIntegral[ArcTanh[a*x]])/a^2","A",1
419,1,32,35,0.0915201,"\int \frac{1}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^2} \, dx","Integrate[1/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2),x]","\frac{\text{Shi}\left(\tanh ^{-1}(a x)\right)-\frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}}{a}","\frac{\text{Shi}\left(\tanh ^{-1}(a x)\right)}{a}-\frac{1}{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}",1,"(-(1/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])) + SinhIntegral[ArcTanh[a*x]])/a","A",1
420,0,0,90,6.3956953,"\int \frac{1}{x \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^2} \, dx","Integrate[1/(x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2),x]","\int \frac{1}{x \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)},x\right)}{a}-\frac{a x}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}-\frac{\sqrt{1-a^2 x^2}}{a x \tanh ^{-1}(a x)}+\text{Chi}\left(\tanh ^{-1}(a x)\right)",0,"Integrate[1/(x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2), x]","A",-1
421,0,0,27,0.9981109,"\int \frac{x^m}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^3} \, dx","Integrate[x^m/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^3),x]","\int \frac{x^m}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x^m}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^3},x\right)",0,"Integrate[x^m/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^3), x]","A",-1
422,0,0,97,6.3873383,"\int \frac{x^2}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^3} \, dx","Integrate[x^2/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^3),x]","\int \frac{x^2}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^3} \, dx","-\frac{\text{Int}\left(\frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3},x\right)}{a^2}+\frac{\text{Chi}\left(\tanh ^{-1}(a x)\right)}{2 a^3}-\frac{x}{2 a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}-\frac{1}{2 a^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}",0,"Integrate[x^2/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^3), x]","A",-1
423,1,43,68,0.1289194,"\int \frac{x}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^3} \, dx","Integrate[x/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^3),x]","\frac{\text{Shi}\left(\tanh ^{-1}(a x)\right)-\frac{a x+\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}}{2 a^2}","\frac{\text{Shi}\left(\tanh ^{-1}(a x)\right)}{2 a^2}-\frac{x}{2 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac{1}{2 a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}",1,"(-((a*x + ArcTanh[a*x])/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)) + SinhIntegral[ArcTanh[a*x]])/(2*a^2)","A",1
424,1,44,65,0.0955149,"\int \frac{1}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^3} \, dx","Integrate[1/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^3),x]","\frac{\text{Chi}\left(\tanh ^{-1}(a x)\right)-\frac{a x \tanh ^{-1}(a x)+1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}}{2 a}","-\frac{x}{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}-\frac{1}{2 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}+\frac{\text{Chi}\left(\tanh ^{-1}(a x)\right)}{2 a}",1,"(-((1 + a*x*ArcTanh[a*x])/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)) + CoshIntegral[ArcTanh[a*x]])/(2*a)","A",1
425,0,0,124,19.6243869,"\int \frac{1}{x \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^3} \, dx","Integrate[1/(x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^3),x]","\int \frac{1}{x \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^3} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2},x\right)}{2 a}-\frac{a x}{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac{1}{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}-\frac{\sqrt{1-a^2 x^2}}{2 a x \tanh ^{-1}(a x)^2}+\frac{1}{2} \text{Shi}\left(\tanh ^{-1}(a x)\right)",0,"Integrate[1/(x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^3), x]","A",-1
426,1,178,243,0.7217971,"\int x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \, dx","Integrate[x^4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x],x]","\frac{\sqrt{1-a^2 x^2} \left(-\frac{45 i \left(\text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x) \left(\log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)\right)}{\sqrt{1-a^2 x^2}}+24 \left(a^2 x^2-1\right)^2+70 \left(a^2 x^2-1\right)+120 a x \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)+210 a x \left(a^2 x^2-1\right) \tanh ^{-1}(a x)+45 a x \tanh ^{-1}(a x)+45\right)}{720 a^5}","-\frac{i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{16 a^5}+\frac{i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{16 a^5}-\frac{\tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{8 a^5}+\frac{1}{6} x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{24 a^2}+\frac{\left(1-a^2 x^2\right)^{5/2}}{30 a^5}-\frac{7 \left(1-a^2 x^2\right)^{3/2}}{72 a^5}+\frac{\sqrt{1-a^2 x^2}}{16 a^5}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 a^4}",1,"(Sqrt[1 - a^2*x^2]*(45 + 70*(-1 + a^2*x^2) + 24*(-1 + a^2*x^2)^2 + 45*a*x*ArcTanh[a*x] + 210*a*x*(-1 + a^2*x^2)*ArcTanh[a*x] + 120*a*x*(-1 + a^2*x^2)^2*ArcTanh[a*x] - ((45*I)*(ArcTanh[a*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]]) + PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(720*a^5)","A",0
427,1,79,136,0.0711799,"\int x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \, dx","Integrate[x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x],x]","\frac{a x \sqrt{1-a^2 x^2} \left(6 a^2 x^2+5\right)+8 \sqrt{1-a^2 x^2} \left(3 a^4 x^4-a^2 x^2-2\right) \tanh ^{-1}(a x)+11 \sin ^{-1}(a x)}{120 a^4}","\frac{11 \sin ^{-1}(a x)}{120 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^2}+\frac{1}{5} x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{x^3 \sqrt{1-a^2 x^2}}{20 a}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}+\frac{x \sqrt{1-a^2 x^2}}{24 a^3}",1,"(a*x*Sqrt[1 - a^2*x^2]*(5 + 6*a^2*x^2) + 11*ArcSin[a*x] + 8*Sqrt[1 - a^2*x^2]*(-2 - a^2*x^2 + 3*a^4*x^4)*ArcTanh[a*x])/(120*a^4)","A",1
428,1,160,194,0.4758048,"\int x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \, dx","Integrate[x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x],x]","\frac{\sqrt{1-a^2 x^2} \left(-\frac{3 i \left(\text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)\right)}{\sqrt{1-a^2 x^2}}+2 a^2 x^2+6 a x \left(a^2 x^2-1\right) \tanh ^{-1}(a x)-\frac{3 i \tanh ^{-1}(a x) \left(\log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)}{\sqrt{1-a^2 x^2}}+3 a x \tanh ^{-1}(a x)+1\right)}{24 a^3}","-\frac{i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{8 a^3}+\frac{i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{8 a^3}-\frac{\tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{4 a^3}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^2}+\frac{1}{4} x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{\left(1-a^2 x^2\right)^{3/2}}{12 a^3}+\frac{\sqrt{1-a^2 x^2}}{8 a^3}",1,"(Sqrt[1 - a^2*x^2]*(1 + 2*a^2*x^2 + 3*a*x*ArcTanh[a*x] + 6*a*x*(-1 + a^2*x^2)*ArcTanh[a*x] - ((3*I)*ArcTanh[a*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2] - ((3*I)*(PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(24*a^3)","A",0
429,1,49,59,0.0455563,"\int x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \, dx","Integrate[x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x],x]","\frac{a x \sqrt{1-a^2 x^2}-2 \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)+\sin ^{-1}(a x)}{6 a^2}","\frac{x \sqrt{1-a^2 x^2}}{6 a}-\frac{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)}{3 a^2}+\frac{\sin ^{-1}(a x)}{6 a^2}",1,"(a*x*Sqrt[1 - a^2*x^2] + ArcSin[a*x] - 2*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/(6*a^2)","A",1
430,1,117,143,0.2641108,"\int \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \, dx","Integrate[Sqrt[1 - a^2*x^2]*ArcTanh[a*x],x]","\frac{\sqrt{1-a^2 x^2} \left(-\frac{i \left(\text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x) \left(\log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)\right)}{\sqrt{1-a^2 x^2}}+a x \tanh ^{-1}(a x)+1\right)}{2 a}","\frac{\sqrt{1-a^2 x^2}}{2 a}+\frac{1}{2} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{2 a}+\frac{i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{2 a}-\frac{\tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{a}",1,"(Sqrt[1 - a^2*x^2]*(1 + a*x*ArcTanh[a*x] - (I*(ArcTanh[a*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]]) + PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(2*a)","A",0
431,1,91,100,0.1275899,"\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x} \, dx","Integrate[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x,x]","\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x) \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-\tanh ^{-1}(a x) \log \left(e^{-\tanh ^{-1}(a x)}+1\right)-2 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)","\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\sin ^{-1}(a x)-2 \tanh ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)",1,"-2*ArcTan[Tanh[ArcTanh[a*x]/2]] + Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] - ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] + PolyLog[2, -E^(-ArcTanh[a*x])] - PolyLog[2, E^(-ArcTanh[a*x])]","A",0
432,1,121,130,0.3812015,"\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x^2} \, dx","Integrate[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x^2,x]","a \left(-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a x}+i \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-i \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+i \tanh ^{-1}(a x) \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-i \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+\log \left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)\right)","-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-a \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)+i a \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)-i a \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)+2 a \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)",1,"a*(-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(a*x)) + I*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] - I*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] + Log[Tanh[ArcTanh[a*x]/2]] + I*PolyLog[2, (-I)/E^ArcTanh[a*x]] - I*PolyLog[2, I/E^ArcTanh[a*x]])","A",0
433,1,126,136,0.716323,"\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x^3} \, dx","Integrate[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x^3,x]","\frac{1}{8} a^2 \left(-4 \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)+4 \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+2 \tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)-4 \tanh ^{-1}(a x) \log \left(1-e^{-\tanh ^{-1}(a x)}\right)+4 \tanh ^{-1}(a x) \log \left(e^{-\tanh ^{-1}(a x)}+1\right)-2 \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)-\tanh ^{-1}(a x) \text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-\tanh ^{-1}(a x) \text{sech}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)","-\frac{1}{2} a^2 \text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)+\frac{1}{2} a^2 \text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{a \sqrt{1-a^2 x^2}}{2 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{2 x^2}+a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)",1,"(a^2*(-2*Coth[ArcTanh[a*x]/2] - ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^2 - 4*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] + 4*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] - 4*PolyLog[2, -E^(-ArcTanh[a*x])] + 4*PolyLog[2, E^(-ArcTanh[a*x])] - ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^2 + 2*Tanh[ArcTanh[a*x]/2]))/8","A",0
434,1,79,70,0.0782509,"\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x^4} \, dx","Integrate[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x^4,x]","-\frac{a^3 x^3 \log (x)+a x \sqrt{1-a^2 x^2}+2 \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)-a^3 x^3 \log \left(\sqrt{1-a^2 x^2}+1\right)}{6 x^3}","-\frac{a \sqrt{1-a^2 x^2}}{6 x^2}-\frac{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)}{3 x^3}+\frac{1}{6} a^3 \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)",1,"-1/6*(a*x*Sqrt[1 - a^2*x^2] + 2*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x] + a^3*x^3*Log[x] - a^3*x^3*Log[1 + Sqrt[1 - a^2*x^2]])/x^3","A",1
435,1,222,191,1.7420114,"\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x^5} \, dx","Integrate[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x^5,x]","\frac{1}{192} a^4 \left(-\frac{a x \text{csch}^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{\sqrt{1-a^2 x^2}}-\frac{16 \left(1-a^2 x^2\right)^{3/2} \sinh ^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{a^3 x^3}-24 \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)+24 \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+8 \tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)-24 \tanh ^{-1}(a x) \log \left(1-e^{-\tanh ^{-1}(a x)}\right)+24 \tanh ^{-1}(a x) \log \left(e^{-\tanh ^{-1}(a x)}+1\right)-8 \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)-3 \tanh ^{-1}(a x) \text{csch}^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-6 \tanh ^{-1}(a x) \text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)+3 \tanh ^{-1}(a x) \text{sech}^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-6 \tanh ^{-1}(a x) \text{sech}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)","-\frac{1}{8} a^4 \text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)+\frac{1}{8} a^4 \text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)+\frac{1}{4} a^4 \tanh ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{8 x^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{4 x^4}-\frac{a \sqrt{1-a^2 x^2}}{12 x^3}-\frac{a^3 \sqrt{1-a^2 x^2}}{24 x}",1,"(a^4*(-8*Coth[ArcTanh[a*x]/2] - 6*ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^2 - (a*x*Csch[ArcTanh[a*x]/2]^4)/Sqrt[1 - a^2*x^2] - 3*ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^4 - 24*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] + 24*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] - 24*PolyLog[2, -E^(-ArcTanh[a*x])] + 24*PolyLog[2, E^(-ArcTanh[a*x])] - 6*ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^2 + 3*ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^4 - (16*(1 - a^2*x^2)^(3/2)*Sinh[ArcTanh[a*x]/2]^4)/(a^3*x^3) + 8*Tanh[ArcTanh[a*x]/2]))/192","A",0
436,1,104,150,0.1365781,"\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x^6} \, dx","Integrate[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x^6,x]","\frac{1}{120} \left(-11 a^5 \log (x)-\frac{a \sqrt{1-a^2 x^2} \left(5 a^2 x^2+6\right)}{x^4}+11 a^5 \log \left(\sqrt{1-a^2 x^2}+1\right)+\frac{8 \sqrt{1-a^2 x^2} \left(2 a^4 x^4+a^2 x^2-3\right) \tanh ^{-1}(a x)}{x^5}\right)","-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{5 x^5}-\frac{a \sqrt{1-a^2 x^2}}{20 x^4}+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 x^3}+\frac{11}{120} a^5 \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)+\frac{2 a^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 x}-\frac{a^3 \sqrt{1-a^2 x^2}}{24 x^2}",1,"(-((a*Sqrt[1 - a^2*x^2]*(6 + 5*a^2*x^2))/x^4) + (8*Sqrt[1 - a^2*x^2]*(-3 + a^2*x^2 + 2*a^4*x^4)*ArcTanh[a*x])/x^5 - 11*a^5*Log[x] + 11*a^5*Log[1 + Sqrt[1 - a^2*x^2]])/120","A",1
437,1,307,243,3.7027932,"\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x^7} \, dx","Integrate[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x^7,x]","\frac{a^6 \left(-\frac{3 a x \text{csch}^6\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{\sqrt{1-a^2 x^2}}-\frac{26 a x \text{csch}^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{\sqrt{1-a^2 x^2}}-\frac{416 \left(1-a^2 x^2\right)^{3/2} \sinh ^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{a^3 x^3}-360 \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)+360 \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+76 \tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)-360 \tanh ^{-1}(a x) \log \left(1-e^{-\tanh ^{-1}(a x)}\right)+360 \tanh ^{-1}(a x) \log \left(e^{-\tanh ^{-1}(a x)}+1\right)-76 \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)-15 \tanh ^{-1}(a x) \text{csch}^6\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-90 \tanh ^{-1}(a x) \text{csch}^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-90 \tanh ^{-1}(a x) \text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-15 \tanh ^{-1}(a x) \text{sech}^6\left(\frac{1}{2} \tanh ^{-1}(a x)\right)+90 \tanh ^{-1}(a x) \text{sech}^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)+6 \tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right) \text{sech}^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-90 \tanh ^{-1}(a x) \text{sech}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)}{5760}","-\frac{1}{16} a^6 \text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)+\frac{1}{16} a^6 \text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)+\frac{1}{8} a^6 \tanh ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}-\frac{a \sqrt{1-a^2 x^2}}{30 x^5}+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}+\frac{a^5 \sqrt{1-a^2 x^2}}{720 x}+\frac{a^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 x^2}-\frac{11 a^3 \sqrt{1-a^2 x^2}}{360 x^3}",1,"(a^6*(-76*Coth[ArcTanh[a*x]/2] - 90*ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^2 - (26*a*x*Csch[ArcTanh[a*x]/2]^4)/Sqrt[1 - a^2*x^2] - 90*ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^4 - (3*a*x*Csch[ArcTanh[a*x]/2]^6)/Sqrt[1 - a^2*x^2] - 15*ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^6 - 360*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] + 360*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] - 360*PolyLog[2, -E^(-ArcTanh[a*x])] + 360*PolyLog[2, E^(-ArcTanh[a*x])] - 90*ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^2 + 90*ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^4 - 15*ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^6 - (416*(1 - a^2*x^2)^(3/2)*Sinh[ArcTanh[a*x]/2]^4)/(a^3*x^3) + 76*Tanh[ArcTanh[a*x]/2] + 6*Sech[ArcTanh[a*x]/2]^4*Tanh[ArcTanh[a*x]/2]))/5760","A",0
438,1,268,336,1.6893006,"\int x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2 \, dx","Integrate[x^4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2,x]","\frac{\sqrt{1-a^2 x^2} \left(-\frac{i \left(90 \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-90 \tanh ^{-1}(a x) \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+90 \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-90 \text{Li}_3\left(i e^{-\tanh ^{-1}(a x)}\right)+45 \tanh ^{-1}(a x)^2 \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-45 \tanh ^{-1}(a x)^2 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)-76 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)\right)}{\sqrt{1-a^2 x^2}}+120 a x \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2+48 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)+140 \left(a^2 x^2-1\right) \tanh ^{-1}(a x)+6 a x \left(a^2 x^2-1\right) \left(35 \tanh ^{-1}(a x)^2+2\right)+90 \tanh ^{-1}(a x)+a x \left(45 \tanh ^{-1}(a x)^2+52\right)\right)}{720 a^5}","-\frac{i \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)}{8 a^5}+\frac{i \tanh ^{-1}(a x) \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)}{8 a^5}+\frac{i \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)}{8 a^5}-\frac{i \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)}{8 a^5}-\frac{19 \sin ^{-1}(a x)}{360 a^5}+\frac{\tanh ^{-1}(a x)^2 \tan ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)}{8 a^5}+\frac{1}{6} x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac{x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{15 a}+\frac{x^3 \sqrt{1-a^2 x^2}}{60 a^2}-\frac{x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{24 a^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{360 a^5}+\frac{x \sqrt{1-a^2 x^2}}{18 a^4}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{16 a^4}+\frac{11 x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{180 a^3}",1,"(Sqrt[1 - a^2*x^2]*(90*ArcTanh[a*x] + 140*(-1 + a^2*x^2)*ArcTanh[a*x] + 48*(-1 + a^2*x^2)^2*ArcTanh[a*x] + 120*a*x*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2 + 6*a*x*(-1 + a^2*x^2)*(2 + 35*ArcTanh[a*x]^2) + a*x*(52 + 45*ArcTanh[a*x]^2) - (I*((-76*I)*ArcTan[Tanh[ArcTanh[a*x]/2]] + 45*ArcTanh[a*x]^2*Log[1 - I/E^ArcTanh[a*x]] - 45*ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] + 90*ArcTanh[a*x]*PolyLog[2, (-I)/E^ArcTanh[a*x]] - 90*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] + 90*PolyLog[3, (-I)/E^ArcTanh[a*x]] - 90*PolyLog[3, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(720*a^5)","A",0
439,1,175,281,0.7037432,"\int x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2 \, dx","Integrate[x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2,x]","\frac{\sqrt{1-a^2 x^2} \left(-\frac{11 i \left(\text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x) \left(\log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)\right)}{\sqrt{1-a^2 x^2}}+12 \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)^2+6 a x \left(a^2 x^2-1\right) \tanh ^{-1}(a x)+2 \left(a^2 x^2-1\right) \left(10 \tanh ^{-1}(a x)^2+1\right)+11 a x \tanh ^{-1}(a x)+11\right)}{60 a^4}","-\frac{11 i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{60 a^4}+\frac{11 i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{60 a^4}-\frac{11 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{30 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{15 a^2}+\frac{1}{5} x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac{x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{10 a}-\frac{\left(1-a^2 x^2\right)^{3/2}}{30 a^4}+\frac{11 \sqrt{1-a^2 x^2}}{60 a^4}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{15 a^4}+\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{12 a^3}",1,"(Sqrt[1 - a^2*x^2]*(11 + 11*a*x*ArcTanh[a*x] + 6*a*x*(-1 + a^2*x^2)*ArcTanh[a*x] + 12*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2 + 2*(-1 + a^2*x^2)*(1 + 10*ArcTanh[a*x]^2) - ((11*I)*(ArcTanh[a*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]]) + PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(60*a^4)","A",0
440,1,228,254,1.2281286,"\int x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2 \, dx","Integrate[x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2,x]","\frac{\sqrt{1-a^2 x^2} \left(-\frac{i \left(6 \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-6 \tanh ^{-1}(a x) \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+6 \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-6 \text{Li}_3\left(i e^{-\tanh ^{-1}(a x)}\right)+3 \tanh ^{-1}(a x)^2 \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-3 \tanh ^{-1}(a x)^2 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)-8 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)\right)}{\sqrt{1-a^2 x^2}}-6 a x \left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2-4 \left(1-a^2 x^2\right) \tanh ^{-1}(a x)+6 \tanh ^{-1}(a x)+a x \left(3 \tanh ^{-1}(a x)^2+2\right)\right)}{24 a^3}","-\frac{i \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)}{4 a^3}+\frac{i \tanh ^{-1}(a x) \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)}{4 a^3}+\frac{i \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)}{4 a^3}-\frac{i \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)}{4 a^3}-\frac{\sin ^{-1}(a x)}{6 a^3}+\frac{\tanh ^{-1}(a x)^2 \tan ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)}{4 a^3}+\frac{x \sqrt{1-a^2 x^2}}{12 a^2}+\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{6 a}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{12 a^3}",1,"(Sqrt[1 - a^2*x^2]*(6*ArcTanh[a*x] - 4*(1 - a^2*x^2)*ArcTanh[a*x] - 6*a*x*(1 - a^2*x^2)*ArcTanh[a*x]^2 + a*x*(2 + 3*ArcTanh[a*x]^2) - (I*((-8*I)*ArcTan[Tanh[ArcTanh[a*x]/2]] + 3*ArcTanh[a*x]^2*Log[1 - I/E^ArcTanh[a*x]] - 3*ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] + 6*ArcTanh[a*x]*PolyLog[2, (-I)/E^ArcTanh[a*x]] - 6*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] + 6*PolyLog[3, (-I)/E^ArcTanh[a*x]] - 6*PolyLog[3, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(24*a^3)","A",0
441,1,135,175,0.426183,"\int x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2 \, dx","Integrate[x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2,x]","\frac{\sqrt{1-a^2 x^2} \left(-\frac{i \left(\text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x) \left(\log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)\right)}{\sqrt{1-a^2 x^2}}-\left(\left(1-a^2 x^2\right) \tanh ^{-1}(a x)^2\right)+a x \tanh ^{-1}(a x)+1\right)}{3 a^2}","-\frac{i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{3 a^2}+\frac{i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{3 a^2}+\frac{\sqrt{1-a^2 x^2}}{3 a^2}-\frac{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^2}{3 a^2}+\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{3 a^2}",1,"(Sqrt[1 - a^2*x^2]*(1 + a*x*ArcTanh[a*x] - (1 - a^2*x^2)*ArcTanh[a*x]^2 - (I*(ArcTanh[a*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]]) + PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(3*a^2)","A",0
442,1,187,158,0.7721779,"\int \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2 \, dx","Integrate[Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2,x]","\frac{\sqrt{1-a^2 x^2} \left(-\frac{i \left(2 \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-2 \tanh ^{-1}(a x) \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+2 \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-2 \text{Li}_3\left(i e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x)^2 \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\tanh ^{-1}(a x)^2 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)-4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)\right)}{\sqrt{1-a^2 x^2}}+a x \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)\right)}{2 a}","\frac{1}{2} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a}-\frac{i \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)}{a}+\frac{i \tanh ^{-1}(a x) \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)}{a}+\frac{i \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)}{a}-\frac{i \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)}{a}-\frac{\sin ^{-1}(a x)}{a}+\frac{\tanh ^{-1}(a x)^2 \tan ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)}{a}",1,"(Sqrt[1 - a^2*x^2]*(2*ArcTanh[a*x] + a*x*ArcTanh[a*x]^2 - (I*((-4*I)*ArcTan[Tanh[ArcTanh[a*x]/2]] + ArcTanh[a*x]^2*Log[1 - I/E^ArcTanh[a*x]] - ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] + 2*ArcTanh[a*x]*PolyLog[2, (-I)/E^ArcTanh[a*x]] - 2*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] + 2*PolyLog[3, (-I)/E^ArcTanh[a*x]] - 2*PolyLog[3, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(2*a)","A",0
443,1,203,174,0.2993893,"\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{x} \, dx","Integrate[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/x,x]","\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-2 \tanh ^{-1}(a x) \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+2 i \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-2 i \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+2 \text{Li}_3\left(-e^{-\tanh ^{-1}(a x)}\right)-2 \text{Li}_3\left(e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x)^2 \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-\tanh ^{-1}(a x)^2 \log \left(e^{-\tanh ^{-1}(a x)}+1\right)+2 i \tanh ^{-1}(a x) \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-2 i \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)","\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+2 i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)-2 i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)-2 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{\tanh ^{-1}(a x)}\right)+2 \tanh ^{-1}(a x) \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right)+2 \text{Li}_3\left(-e^{\tanh ^{-1}(a x)}\right)-2 \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right)-2 \tanh ^{-1}\left(e^{\tanh ^{-1}(a x)}\right) \tanh ^{-1}(a x)^2+4 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)",1,"Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2 + ArcTanh[a*x]^2*Log[1 - E^(-ArcTanh[a*x])] + (2*I)*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] - (2*I)*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] - ArcTanh[a*x]^2*Log[1 + E^(-ArcTanh[a*x])] + 2*ArcTanh[a*x]*PolyLog[2, -E^(-ArcTanh[a*x])] + (2*I)*PolyLog[2, (-I)/E^ArcTanh[a*x]] - (2*I)*PolyLog[2, I/E^ArcTanh[a*x]] - 2*ArcTanh[a*x]*PolyLog[2, E^(-ArcTanh[a*x])] + 2*PolyLog[3, -E^(-ArcTanh[a*x])] - 2*PolyLog[3, E^(-ArcTanh[a*x])]","A",0
444,1,223,197,0.778078,"\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{x^2} \, dx","Integrate[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/x^2,x]","a \left(-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{a x}+2 i \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-2 i \tanh ^{-1}(a x) \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+2 \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-2 \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+2 i \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-2 i \text{Li}_3\left(i e^{-\tanh ^{-1}(a x)}\right)+i \tanh ^{-1}(a x)^2 \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-i \tanh ^{-1}(a x)^2 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+2 \tanh ^{-1}(a x) \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-2 \tanh ^{-1}(a x) \log \left(e^{-\tanh ^{-1}(a x)}+1\right)\right)","-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}+2 a \text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-2 a \text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)+2 i a \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)-2 i a \tanh ^{-1}(a x) \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)-2 i a \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)+2 i a \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)-4 a \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)-2 a \tanh ^{-1}(a x)^2 \tan ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)",1,"a*(-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(a*x)) + 2*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] + I*ArcTanh[a*x]^2*Log[1 - I/E^ArcTanh[a*x]] - I*ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] - 2*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] + 2*PolyLog[2, -E^(-ArcTanh[a*x])] + (2*I)*ArcTanh[a*x]*PolyLog[2, (-I)/E^ArcTanh[a*x]] - (2*I)*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] - 2*PolyLog[2, E^(-ArcTanh[a*x])] + (2*I)*PolyLog[3, (-I)/E^ArcTanh[a*x]] - (2*I)*PolyLog[3, I/E^ArcTanh[a*x]])","A",0
445,1,188,151,1.2622696,"\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{x^3} \, dx","Integrate[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/x^3,x]","\frac{1}{8} a^2 \left(-8 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)+8 \tanh ^{-1}(a x) \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)-8 \text{Li}_3\left(-e^{-\tanh ^{-1}(a x)}\right)+8 \text{Li}_3\left(e^{-\tanh ^{-1}(a x)}\right)+4 \tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right) \tanh ^{-1}(a x)-4 \tanh ^{-1}(a x)^2 \log \left(1-e^{-\tanh ^{-1}(a x)}\right)+4 \tanh ^{-1}(a x)^2 \log \left(e^{-\tanh ^{-1}(a x)}+1\right)+8 \log \left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)-4 \tanh ^{-1}(a x) \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)+\tanh ^{-1}(a x)^2 \left(-\text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)-\tanh ^{-1}(a x)^2 \text{sech}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)","a^2 \tanh ^{-1}(a x) \text{Li}_2\left(-e^{\tanh ^{-1}(a x)}\right)-a^2 \tanh ^{-1}(a x) \text{Li}_2\left(e^{\tanh ^{-1}(a x)}\right)-a^2 \text{Li}_3\left(-e^{\tanh ^{-1}(a x)}\right)+a^2 \text{Li}_3\left(e^{\tanh ^{-1}(a x)}\right)-a^2 \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}+a^2 \tanh ^{-1}\left(e^{\tanh ^{-1}(a x)}\right) \tanh ^{-1}(a x)^2",1,"(a^2*(-4*ArcTanh[a*x]*Coth[ArcTanh[a*x]/2] - ArcTanh[a*x]^2*Csch[ArcTanh[a*x]/2]^2 - 4*ArcTanh[a*x]^2*Log[1 - E^(-ArcTanh[a*x])] + 4*ArcTanh[a*x]^2*Log[1 + E^(-ArcTanh[a*x])] + 8*Log[Tanh[ArcTanh[a*x]/2]] - 8*ArcTanh[a*x]*PolyLog[2, -E^(-ArcTanh[a*x])] + 8*ArcTanh[a*x]*PolyLog[2, E^(-ArcTanh[a*x])] - 8*PolyLog[3, -E^(-ArcTanh[a*x])] + 8*PolyLog[3, E^(-ArcTanh[a*x])] - ArcTanh[a*x]^2*Sech[ArcTanh[a*x]/2]^2 + 4*ArcTanh[a*x]*Tanh[ArcTanh[a*x]/2]))/8","A",0
446,1,177,169,2.0625987,"\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{x^4} \, dx","Integrate[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/x^4,x]","-\frac{1}{3} a^3 \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-\frac{\left(1-a^2 x^2\right)^{3/2} \left(\tanh ^{-1}(a x) \left(\frac{\left(\log \left(1-e^{-\tanh ^{-1}(a x)}\right)-\log \left(e^{-\tanh ^{-1}(a x)}+1\right)\right) \left(\sqrt{1-a^2 x^2} \sinh \left(3 \tanh ^{-1}(a x)\right)-3 a x\right)}{\sqrt{1-a^2 x^2}}+2 \sinh \left(2 \tanh ^{-1}(a x)\right)\right)-\frac{4 a^3 x^3 \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)}{\left(1-a^2 x^2\right)^{3/2}}+4 \tanh ^{-1}(a x)^2+2 \left(\cosh \left(2 \tanh ^{-1}(a x)\right)-1\right)\right)}{12 x^3}","-\frac{1}{3} a^3 \text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)+\frac{1}{3} a^3 \text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)+\frac{2}{3} a^3 \tanh ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{a^2 \sqrt{1-a^2 x^2}}{3 x}-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 x^2}-\frac{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^2}{3 x^3}",1,"-1/3*(a^3*PolyLog[2, -E^(-ArcTanh[a*x])]) - ((1 - a^2*x^2)^(3/2)*(4*ArcTanh[a*x]^2 + 2*(-1 + Cosh[2*ArcTanh[a*x]]) - (4*a^3*x^3*PolyLog[2, E^(-ArcTanh[a*x])])/(1 - a^2*x^2)^(3/2) + ArcTanh[a*x]*(2*Sinh[2*ArcTanh[a*x]] + ((Log[1 - E^(-ArcTanh[a*x])] - Log[1 + E^(-ArcTanh[a*x])])*(-3*a*x + Sqrt[1 - a^2*x^2]*Sinh[3*ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2])))/(12*x^3)","A",0
447,1,272,292,1.4904114,"\int x^4 \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x) \, dx","Integrate[x^4*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x],x]","\frac{218 a^2 x^2 \sqrt{1-a^2 x^2}+121 \sqrt{1-a^2 x^2}-315 a x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-1680 a^7 x^7 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-240 a^6 x^6 \sqrt{1-a^2 x^2}+2520 a^5 x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+216 a^4 x^4 \sqrt{1-a^2 x^2}-210 a^3 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-315 i \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)+315 i \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)-315 i \tanh ^{-1}(a x) \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)+315 i \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)}{13440 a^5}","-\frac{3 i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{128 a^5}+\frac{3 i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{128 a^5}-\frac{3 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{64 a^5}-\frac{1}{8} a^2 x^7 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{3}{16} x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{64 a^2}+\frac{\left(1-a^2 x^2\right)^{7/2}}{56 a^5}-\frac{3 \left(1-a^2 x^2\right)^{5/2}}{80 a^5}+\frac{\left(1-a^2 x^2\right)^{3/2}}{192 a^5}+\frac{3 \sqrt{1-a^2 x^2}}{128 a^5}-\frac{3 x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{128 a^4}",1,"(121*Sqrt[1 - a^2*x^2] + 218*a^2*x^2*Sqrt[1 - a^2*x^2] + 216*a^4*x^4*Sqrt[1 - a^2*x^2] - 240*a^6*x^6*Sqrt[1 - a^2*x^2] - 315*a*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - 210*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + 2520*a^5*x^5*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - 1680*a^7*x^7*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (315*I)*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] + (315*I)*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] - (315*I)*PolyLog[2, (-I)/E^ArcTanh[a*x]] + (315*I)*PolyLog[2, I/E^ArcTanh[a*x]])/(13440*a^5)","A",0
448,1,79,186,0.0894163,"\int x^3 \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x) \, dx","Integrate[x^3*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x],x]","\frac{-48 \left(5 a^2 x^2+2\right) \left(1-a^2 x^2\right)^{5/2} \tanh ^{-1}(a x)+a x \left(-40 a^4 x^4+46 a^2 x^2+45\right) \sqrt{1-a^2 x^2}+51 \sin ^{-1}(a x)}{1680 a^4}","\frac{17 \sin ^{-1}(a x)}{560 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{35 a^2}-\frac{1}{7} a^2 x^6 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{1}{42} a x^5 \sqrt{1-a^2 x^2}+\frac{8}{35} x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{23 x^3 \sqrt{1-a^2 x^2}}{840 a}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{35 a^4}+\frac{3 x \sqrt{1-a^2 x^2}}{112 a^3}",1,"(a*x*Sqrt[1 - a^2*x^2]*(45 + 46*a^2*x^2 - 40*a^4*x^4) + 51*ArcSin[a*x] - 48*(1 - a^2*x^2)^(5/2)*(2 + 5*a^2*x^2)*ArcTanh[a*x])/(1680*a^4)","A",1
449,1,224,243,0.9730477,"\int x^2 \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x) \, dx","Integrate[x^2*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x],x]","\frac{38 a^2 x^2 \sqrt{1-a^2 x^2}+31 \sqrt{1-a^2 x^2}-45 a x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-120 a^5 x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-24 a^4 x^4 \sqrt{1-a^2 x^2}+210 a^3 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-45 i \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)+45 i \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)-45 i \tanh ^{-1}(a x) \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)+45 i \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)}{720 a^3}","-\frac{i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{16 a^3}+\frac{i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{16 a^3}-\frac{\tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{8 a^3}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 a^2}-\frac{1}{6} a^2 x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{7}{24} x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{\left(1-a^2 x^2\right)^{5/2}}{30 a^3}+\frac{\left(1-a^2 x^2\right)^{3/2}}{72 a^3}+\frac{\sqrt{1-a^2 x^2}}{16 a^3}",1,"(31*Sqrt[1 - a^2*x^2] + 38*a^2*x^2*Sqrt[1 - a^2*x^2] - 24*a^4*x^4*Sqrt[1 - a^2*x^2] - 45*a*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + 210*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - 120*a^5*x^5*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (45*I)*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] + (45*I)*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] - (45*I)*PolyLog[2, (-I)/E^ArcTanh[a*x]] + (45*I)*PolyLog[2, I/E^ArcTanh[a*x]])/(720*a^3)","A",0
450,1,61,81,0.0673706,"\int x \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x) \, dx","Integrate[x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x],x]","\frac{a x \left(5-2 a^2 x^2\right) \sqrt{1-a^2 x^2}-8 \left(1-a^2 x^2\right)^{5/2} \tanh ^{-1}(a x)+3 \sin ^{-1}(a x)}{40 a^2}","\frac{x \left(1-a^2 x^2\right)^{3/2}}{20 a}+\frac{3 x \sqrt{1-a^2 x^2}}{40 a}-\frac{\left(1-a^2 x^2\right)^{5/2} \tanh ^{-1}(a x)}{5 a^2}+\frac{3 \sin ^{-1}(a x)}{40 a^2}",1,"(a*x*(5 - 2*a^2*x^2)*Sqrt[1 - a^2*x^2] + 3*ArcSin[a*x] - 8*(1 - a^2*x^2)^(5/2)*ArcTanh[a*x])/(40*a^2)","A",1
451,1,176,189,0.6095525,"\int \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x) \, dx","Integrate[(1 - a^2*x^2)^(3/2)*ArcTanh[a*x],x]","\frac{-2 a^2 x^2 \sqrt{1-a^2 x^2}+11 \sqrt{1-a^2 x^2}+15 a x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-6 a^3 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-9 i \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)+9 i \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)-9 i \tanh ^{-1}(a x) \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)+9 i \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)}{24 a}","\frac{\left(1-a^2 x^2\right)^{3/2}}{12 a}+\frac{3 \sqrt{1-a^2 x^2}}{8 a}+\frac{1}{4} x \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)+\frac{3}{8} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{3 i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{8 a}+\frac{3 i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{8 a}-\frac{3 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{4 a}",1,"(11*Sqrt[1 - a^2*x^2] - 2*a^2*x^2*Sqrt[1 - a^2*x^2] + 15*a*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - 6*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (9*I)*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] + (9*I)*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] - (9*I)*PolyLog[2, (-I)/E^ArcTanh[a*x]] + (9*I)*PolyLog[2, I/E^ArcTanh[a*x]])/(24*a)","A",0
452,1,143,144,0.2303481,"\int \frac{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)}{x} \, dx","Integrate[((1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/x,x]","\frac{1}{6} \left(-a x \sqrt{1-a^2 x^2}-2 a^2 x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+8 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+6 \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-6 \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+6 \tanh ^{-1}(a x) \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-6 \tanh ^{-1}(a x) \log \left(e^{-\tanh ^{-1}(a x)}+1\right)-14 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)\right)","-\frac{1}{6} a x \sqrt{1-a^2 x^2}+\frac{1}{3} \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)+\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{7}{6} \sin ^{-1}(a x)-2 \tanh ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)",1,"(-(a*x*Sqrt[1 - a^2*x^2]) - 14*ArcTan[Tanh[ArcTanh[a*x]/2]] + 8*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - 2*a^2*x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + 6*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] - 6*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] + 6*PolyLog[2, -E^(-ArcTanh[a*x])] - 6*PolyLog[2, E^(-ArcTanh[a*x])])/6","A",0
453,1,168,179,0.625168,"\int \frac{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)}{x^2} \, dx","Integrate[((1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/x^2,x]","\frac{1}{2} \left(-a \sqrt{1-a^2 x^2}+a^2 (-x) \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}+3 i a \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-3 i a \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+3 i a \tanh ^{-1}(a x) \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-3 i a \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+2 a \log \left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)\right)","-\frac{1}{2} a \sqrt{1-a^2 x^2}-\frac{1}{2} a^2 x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-a \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}+\frac{3}{2} i a \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{3}{2} i a \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)+3 a \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)",1,"(-(a*Sqrt[1 - a^2*x^2]) - (2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x - a^2*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + (3*I)*a*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] - (3*I)*a*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] + 2*a*Log[Tanh[ArcTanh[a*x]/2]] + (3*I)*a*PolyLog[2, (-I)/E^ArcTanh[a*x]] - (3*I)*a*PolyLog[2, I/E^ArcTanh[a*x]])/2","A",0
454,1,158,168,0.9994583,"\int \frac{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)}{x^3} \, dx","Integrate[((1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/x^3,x]","\frac{1}{8} a^2 \left(-8 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-12 \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)+12 \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+2 \tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)-12 \tanh ^{-1}(a x) \log \left(1-e^{-\tanh ^{-1}(a x)}\right)+12 \tanh ^{-1}(a x) \log \left(e^{-\tanh ^{-1}(a x)}+1\right)+16 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)-2 \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)-\tanh ^{-1}(a x) \text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-\tanh ^{-1}(a x) \text{sech}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)","-\frac{3}{2} a^2 \text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)+\frac{3}{2} a^2 \text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{a \sqrt{1-a^2 x^2}}{2 x}-a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{2 x^2}+a^2 \sin ^{-1}(a x)+3 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)",1,"(a^2*(16*ArcTan[Tanh[ArcTanh[a*x]/2]] - 8*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - 2*Coth[ArcTanh[a*x]/2] - ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^2 - 12*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] + 12*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] - 12*PolyLog[2, -E^(-ArcTanh[a*x])] + 12*PolyLog[2, E^(-ArcTanh[a*x])] - ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^2 + 2*Tanh[ArcTanh[a*x]/2]))/8","A",0
455,1,199,189,1.2677481,"\int \frac{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)}{x^4} \, dx","Integrate[((1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/x^4,x]","-\frac{\left(1-a^2 x^2\right)^{3/2} \left(\log \left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right) \left(\sinh \left(3 \tanh ^{-1}(a x)\right)-\frac{3 a x}{\sqrt{1-a^2 x^2}}\right)+8 \tanh ^{-1}(a x)+2 \sinh \left(2 \tanh ^{-1}(a x)\right)\right)}{24 x^3}-\left(a^3 \left(-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a x}+i \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-i \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+i \tanh ^{-1}(a x) \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-i \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+\log \left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)\right)\right)","-i a^3 \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)+i a^3 \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)-2 a^3 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)-\frac{a \sqrt{1-a^2 x^2}}{6 x^2}+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)}{3 x^3}+\frac{7}{6} a^3 \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)",1,"-(a^3*(-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(a*x)) + I*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] - I*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] + Log[Tanh[ArcTanh[a*x]/2]] + I*PolyLog[2, (-I)/E^ArcTanh[a*x]] - I*PolyLog[2, I/E^ArcTanh[a*x]])) - ((1 - a^2*x^2)^(3/2)*(8*ArcTanh[a*x] + 2*Sinh[2*ArcTanh[a*x]] + Log[Tanh[ArcTanh[a*x]/2]]*((-3*a*x)/Sqrt[1 - a^2*x^2] + Sinh[3*ArcTanh[a*x]])))/(24*x^3)","A",0
456,1,282,191,3.8328744,"\int \frac{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)}{x^5} \, dx","Integrate[((1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/x^5,x]","\frac{1}{192} a \left(72 a^3 \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-72 a^3 \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)-40 a^3 \tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)+72 a^3 \tanh ^{-1}(a x) \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-72 a^3 \tanh ^{-1}(a x) \log \left(e^{-\tanh ^{-1}(a x)}+1\right)+40 a^3 \coth \left(\frac{1}{2} \tanh ^{-1}(a x)\right)-3 a^3 \tanh ^{-1}(a x) \text{csch}^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)+18 a^3 \tanh ^{-1}(a x) \text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)+3 a^3 \tanh ^{-1}(a x) \text{sech}^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)+18 a^3 \tanh ^{-1}(a x) \text{sech}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)+\frac{16 a^2 \sqrt{1-a^2 x^2} \sinh ^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{x}-\frac{16 \sqrt{1-a^2 x^2} \sinh ^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{x^3}-\frac{a^4 x \text{csch}^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{\sqrt{1-a^2 x^2}}\right)","\frac{3}{8} a^4 \text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{3}{8} a^4 \text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{3}{4} a^4 \tanh ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)+\frac{5 a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{8 x^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{4 x^4}-\frac{a \sqrt{1-a^2 x^2}}{12 x^3}+\frac{11 a^3 \sqrt{1-a^2 x^2}}{24 x}",1,"(a*(40*a^3*Coth[ArcTanh[a*x]/2] + 18*a^3*ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^2 - (a^4*x*Csch[ArcTanh[a*x]/2]^4)/Sqrt[1 - a^2*x^2] - 3*a^3*ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^4 + 72*a^3*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] - 72*a^3*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] + 72*a^3*PolyLog[2, -E^(-ArcTanh[a*x])] - 72*a^3*PolyLog[2, E^(-ArcTanh[a*x])] + 18*a^3*ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^2 + 3*a^3*ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^4 - (16*Sqrt[1 - a^2*x^2]*Sinh[ArcTanh[a*x]/2]^4)/x^3 + (16*a^2*Sqrt[1 - a^2*x^2]*Sinh[ArcTanh[a*x]/2]^4)/x - 40*a^3*Tanh[ArcTanh[a*x]/2]))/192","A",0
457,1,104,94,0.0904324,"\int \frac{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)}{x^6} \, dx","Integrate[((1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/x^6,x]","\frac{3}{40} a^5 \log (x)-\frac{\sqrt{1-a^2 x^2} \left(a^2 x^2-1\right)^2 \tanh ^{-1}(a x)}{5 x^5}-\frac{3}{40} a^5 \log \left(\sqrt{1-a^2 x^2}+1\right)+\left(\frac{a^3}{8 x^2}-\frac{a}{20 x^4}\right) \sqrt{1-a^2 x^2}","-\frac{\left(1-a^2 x^2\right)^{5/2} \tanh ^{-1}(a x)}{5 x^5}-\frac{a \left(1-a^2 x^2\right)^{3/2}}{20 x^4}-\frac{3}{40} a^5 \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)+\frac{3 a^3 \sqrt{1-a^2 x^2}}{40 x^2}",1,"(-1/20*a/x^4 + a^3/(8*x^2))*Sqrt[1 - a^2*x^2] - (Sqrt[1 - a^2*x^2]*(-1 + a^2*x^2)^2*ArcTanh[a*x])/(5*x^5) + (3*a^5*Log[x])/40 - (3*a^5*Log[1 + Sqrt[1 - a^2*x^2]])/40","A",1
458,1,474,243,7.4366373,"\int \frac{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)}{x^7} \, dx","Integrate[((1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/x^7,x]","\frac{64 a^7 x^4 \sinh ^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-3 a^7 x^4 \text{csch}^6\left(\frac{1}{2} \tanh ^{-1}(a x)\right)+4 a^7 x^4 \text{csch}^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)+82 a^7 x^4 \text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-128 a^5 x^2 \sinh ^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)+64 a^3 \sinh ^4\left(\frac{1}{2} \tanh ^{-1}(a x)\right)-\frac{192 a \left(a^2 x^2-1\right)^3 \sinh ^6\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{x^2}-\frac{960 \left(1-a^2 x^2\right)^{7/2} \tanh ^{-1}(a x) \sinh ^6\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{x^3}+360 a^6 x^3 \sqrt{1-a^2 x^2} \text{Li}_2\left(-e^{-\tanh ^{-1}(a x)}\right)-360 a^6 x^3 \sqrt{1-a^2 x^2} \text{Li}_2\left(e^{-\tanh ^{-1}(a x)}\right)+360 a^6 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \log \left(1-e^{-\tanh ^{-1}(a x)}\right)-360 a^6 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \log \left(e^{-\tanh ^{-1}(a x)}+1\right)-15 a^6 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \text{csch}^6\left(\frac{1}{2} \tanh ^{-1}(a x)\right)+90 a^6 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \text{csch}^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)+328 a^5 x^2 \left(a^2 x^2-1\right) \sinh ^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)+360 a^4 x \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x) \sinh ^2\left(\frac{1}{2} \tanh ^{-1}(a x)\right)}{5760 x^3 \sqrt{1-a^2 x^2}}","\frac{1}{16} a^6 \text{Li}_2\left(-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{1}{16} a^6 \text{Li}_2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{1}{8} a^6 \tanh ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}-\frac{a \sqrt{1-a^2 x^2}}{30 x^5}+\frac{7 a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}+\frac{31 a^5 \sqrt{1-a^2 x^2}}{720 x}-\frac{a^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 x^2}+\frac{19 a^3 \sqrt{1-a^2 x^2}}{360 x^3}",1,"(82*a^7*x^4*Csch[ArcTanh[a*x]/2]^2 + 90*a^6*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^2 + 4*a^7*x^4*Csch[ArcTanh[a*x]/2]^4 - 3*a^7*x^4*Csch[ArcTanh[a*x]/2]^6 - 15*a^6*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^6 + 360*a^6*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] - 360*a^6*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] + 360*a^6*x^3*Sqrt[1 - a^2*x^2]*PolyLog[2, -E^(-ArcTanh[a*x])] - 360*a^6*x^3*Sqrt[1 - a^2*x^2]*PolyLog[2, E^(-ArcTanh[a*x])] + 328*a^5*x^2*(-1 + a^2*x^2)*Sinh[ArcTanh[a*x]/2]^2 + 360*a^4*x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]*Sinh[ArcTanh[a*x]/2]^2 + 64*a^3*Sinh[ArcTanh[a*x]/2]^4 - 128*a^5*x^2*Sinh[ArcTanh[a*x]/2]^4 + 64*a^7*x^4*Sinh[ArcTanh[a*x]/2]^4 - (192*a*(-1 + a^2*x^2)^3*Sinh[ArcTanh[a*x]/2]^6)/x^2 - (960*(1 - a^2*x^2)^(7/2)*ArcTanh[a*x]*Sinh[ArcTanh[a*x]/2]^6)/x^3)/(5760*x^3*Sqrt[1 - a^2*x^2])","A",0
459,1,224,233,1.1835412,"\int \left(1-a^2 x^2\right)^{5/2} \tanh ^{-1}(a x) \, dx","Integrate[(1 - a^2*x^2)^(5/2)*ArcTanh[a*x],x]","\frac{-98 a^2 x^2 \sqrt{1-a^2 x^2}+299 \sqrt{1-a^2 x^2}+495 a x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+120 a^5 x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+24 a^4 x^4 \sqrt{1-a^2 x^2}-390 a^3 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-225 i \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)+225 i \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)-225 i \tanh ^{-1}(a x) \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)+225 i \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)}{720 a}","\frac{\left(1-a^2 x^2\right)^{5/2}}{30 a}+\frac{5 \left(1-a^2 x^2\right)^{3/2}}{72 a}+\frac{5 \sqrt{1-a^2 x^2}}{16 a}+\frac{1}{6} x \left(1-a^2 x^2\right)^{5/2} \tanh ^{-1}(a x)+\frac{5}{24} x \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)+\frac{5}{16} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{5 i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{16 a}+\frac{5 i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{16 a}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{8 a}",1,"(299*Sqrt[1 - a^2*x^2] - 98*a^2*x^2*Sqrt[1 - a^2*x^2] + 24*a^4*x^4*Sqrt[1 - a^2*x^2] + 495*a*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - 390*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + 120*a^5*x^5*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (225*I)*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] + (225*I)*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] - (225*I)*PolyLog[2, (-I)/E^ArcTanh[a*x]] + (225*I)*PolyLog[2, I/E^ArcTanh[a*x]])/(720*a)","A",0
460,1,176,189,0.0615218,"\int \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x) \, dx","Integrate[(1 - a^2*x^2)^(3/2)*ArcTanh[a*x],x]","\frac{-2 a^2 x^2 \sqrt{1-a^2 x^2}+11 \sqrt{1-a^2 x^2}+15 a x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-6 a^3 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-9 i \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)+9 i \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)-9 i \tanh ^{-1}(a x) \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)+9 i \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)}{24 a}","\frac{\left(1-a^2 x^2\right)^{3/2}}{12 a}+\frac{3 \sqrt{1-a^2 x^2}}{8 a}+\frac{1}{4} x \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)+\frac{3}{8} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{3 i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{8 a}+\frac{3 i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{8 a}-\frac{3 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{4 a}",1,"(11*Sqrt[1 - a^2*x^2] - 2*a^2*x^2*Sqrt[1 - a^2*x^2] + 15*a*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - 6*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (9*I)*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] + (9*I)*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] - (9*I)*PolyLog[2, (-I)/E^ArcTanh[a*x]] + (9*I)*PolyLog[2, I/E^ArcTanh[a*x]])/(24*a)","A",0
461,1,117,143,0.0679622,"\int \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \, dx","Integrate[Sqrt[1 - a^2*x^2]*ArcTanh[a*x],x]","\frac{\sqrt{1-a^2 x^2} \left(-\frac{i \left(\text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x) \left(\log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)\right)}{\sqrt{1-a^2 x^2}}+a x \tanh ^{-1}(a x)+1\right)}{2 a}","\frac{\sqrt{1-a^2 x^2}}{2 a}+\frac{1}{2} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{2 a}+\frac{i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{2 a}-\frac{\tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{a}",1,"(Sqrt[1 - a^2*x^2]*(1 + a*x*ArcTanh[a*x] - (I*(ArcTanh[a*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]]) + PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(2*a)","A",0
462,1,49,89,0.0523954,"\int \frac{\tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^{5/2}} \, dx","Integrate[ArcTanh[a*x]/(1 - a^2*x^2)^(5/2),x]","-\frac{\left(6 a^3 x^3-9 a x\right) \tanh ^{-1}(a x)-6 a^2 x^2+7}{9 a \left(1-a^2 x^2\right)^{3/2}}","-\frac{2}{3 a \sqrt{1-a^2 x^2}}-\frac{1}{9 a \left(1-a^2 x^2\right)^{3/2}}+\frac{2 x \tanh ^{-1}(a x)}{3 \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)}{3 \left(1-a^2 x^2\right)^{3/2}}",1,"-1/9*(7 - 6*a^2*x^2 + (-9*a*x + 6*a^3*x^3)*ArcTanh[a*x])/(a*(1 - a^2*x^2)^(3/2))","A",1
463,1,65,133,0.0686304,"\int \frac{\tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^{7/2}} \, dx","Integrate[ArcTanh[a*x]/(1 - a^2*x^2)^(7/2),x]","\frac{-120 a^4 x^4+260 a^2 x^2+15 a x \left(8 a^4 x^4-20 a^2 x^2+15\right) \tanh ^{-1}(a x)-149}{225 a \left(1-a^2 x^2\right)^{5/2}}","-\frac{8}{15 a \sqrt{1-a^2 x^2}}-\frac{4}{45 a \left(1-a^2 x^2\right)^{3/2}}-\frac{1}{25 a \left(1-a^2 x^2\right)^{5/2}}+\frac{8 x \tanh ^{-1}(a x)}{15 \sqrt{1-a^2 x^2}}+\frac{4 x \tanh ^{-1}(a x)}{15 \left(1-a^2 x^2\right)^{3/2}}+\frac{x \tanh ^{-1}(a x)}{5 \left(1-a^2 x^2\right)^{5/2}}",1,"(-149 + 260*a^2*x^2 - 120*a^4*x^4 + 15*a*x*(15 - 20*a^2*x^2 + 8*a^4*x^4)*ArcTanh[a*x])/(225*a*(1 - a^2*x^2)^(5/2))","A",1
464,1,81,177,0.0787333,"\int \frac{\tanh ^{-1}(a x)}{\left(1-a^2 x^2\right)^{9/2}} \, dx","Integrate[ArcTanh[a*x]/(1 - a^2*x^2)^(9/2),x]","\frac{1680 a^6 x^6-5320 a^4 x^4+5726 a^2 x^2-105 a x \left(16 a^6 x^6-56 a^4 x^4+70 a^2 x^2-35\right) \tanh ^{-1}(a x)-2161}{3675 a \left(1-a^2 x^2\right)^{7/2}}","-\frac{16}{35 a \sqrt{1-a^2 x^2}}-\frac{8}{105 a \left(1-a^2 x^2\right)^{3/2}}-\frac{6}{175 a \left(1-a^2 x^2\right)^{5/2}}-\frac{1}{49 a \left(1-a^2 x^2\right)^{7/2}}+\frac{16 x \tanh ^{-1}(a x)}{35 \sqrt{1-a^2 x^2}}+\frac{8 x \tanh ^{-1}(a x)}{35 \left(1-a^2 x^2\right)^{3/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 \left(1-a^2 x^2\right)^{5/2}}+\frac{x \tanh ^{-1}(a x)}{7 \left(1-a^2 x^2\right)^{7/2}}",1,"(-2161 + 5726*a^2*x^2 - 5320*a^4*x^4 + 1680*a^6*x^6 - 105*a*x*(-35 + 70*a^2*x^2 - 56*a^4*x^4 + 16*a^6*x^6)*ArcTanh[a*x])/(3675*a*(1 - a^2*x^2)^(7/2))","A",1
465,1,206,291,0.6756331,"\int \left(c-a^2 c x^2\right)^{3/2} \tanh ^{-1}(a x) \, dx","Integrate[(c - a^2*c*x^2)^(3/2)*ArcTanh[a*x],x]","-\frac{c \sqrt{c-a^2 c x^2} \left(2 a^2 x^2 \sqrt{1-a^2 x^2}-11 \sqrt{1-a^2 x^2}-15 a x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+6 a^3 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+9 i \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-9 i \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+9 i \tanh ^{-1}(a x) \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-9 i \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)}{24 a \sqrt{1-a^2 x^2}}","-\frac{3 i c^2 \sqrt{1-a^2 x^2} \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{8 a \sqrt{c-a^2 c x^2}}+\frac{3 i c^2 \sqrt{1-a^2 x^2} \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{8 a \sqrt{c-a^2 c x^2}}-\frac{3 c^2 \sqrt{1-a^2 x^2} \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{4 a \sqrt{c-a^2 c x^2}}+\frac{3 c \sqrt{c-a^2 c x^2}}{8 a}+\frac{\left(c-a^2 c x^2\right)^{3/2}}{12 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \tanh ^{-1}(a x)+\frac{1}{4} x \left(c-a^2 c x^2\right)^{3/2} \tanh ^{-1}(a x)",1,"-1/24*(c*Sqrt[c - a^2*c*x^2]*(-11*Sqrt[1 - a^2*x^2] + 2*a^2*x^2*Sqrt[1 - a^2*x^2] - 15*a*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + 6*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + (9*I)*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] - (9*I)*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] + (9*I)*PolyLog[2, (-I)/E^ArcTanh[a*x]] - (9*I)*PolyLog[2, I/E^ArcTanh[a*x]]))/(a*Sqrt[1 - a^2*x^2])","A",0
466,1,119,235,0.2886297,"\int \sqrt{c-a^2 c x^2} \tanh ^{-1}(a x) \, dx","Integrate[Sqrt[c - a^2*c*x^2]*ArcTanh[a*x],x]","\frac{\sqrt{c \left(1-a^2 x^2\right)} \left(-\frac{i \left(\text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x) \left(\log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)\right)}{\sqrt{1-a^2 x^2}}+a x \tanh ^{-1}(a x)+1\right)}{2 a}","-\frac{i c \sqrt{1-a^2 x^2} \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{2 a \sqrt{c-a^2 c x^2}}+\frac{i c \sqrt{1-a^2 x^2} \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{2 a \sqrt{c-a^2 c x^2}}+\frac{\sqrt{c-a^2 c x^2}}{2 a}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \tanh ^{-1}(a x)-\frac{c \sqrt{1-a^2 x^2} \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{a \sqrt{c-a^2 c x^2}}",1,"(Sqrt[c*(1 - a^2*x^2)]*(1 + a*x*ArcTanh[a*x] - (I*(ArcTanh[a*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]]) + PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(2*a)","A",0
467,1,109,182,0.129572,"\int \frac{\tanh ^{-1}(a x)}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[ArcTanh[a*x]/Sqrt[c - a^2*c*x^2],x]","-\frac{i \sqrt{c \left(1-a^2 x^2\right)} \left(\text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x) \left(\log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\log \left(1+i e^{-\tanh ^{-1}(a x)}\right)\right)\right)}{a c \sqrt{1-a^2 x^2}}","-\frac{i \sqrt{1-a^2 x^2} \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a \sqrt{c-a^2 c x^2}}+\frac{i \sqrt{1-a^2 x^2} \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{1-a^2 x^2} \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{a \sqrt{c-a^2 c x^2}}",1,"((-I)*Sqrt[c*(1 - a^2*x^2)]*(ArcTanh[a*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]]) + PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/(a*c*Sqrt[1 - a^2*x^2])","A",0
468,1,43,48,0.0599009,"\int \frac{\tanh ^{-1}(a x)}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTanh[a*x]/(c - a^2*c*x^2)^(3/2),x]","\frac{\sqrt{c-a^2 c x^2} \left(1-a x \tanh ^{-1}(a x)\right)}{a c^2 \left(a^2 x^2-1\right)}","\frac{x \tanh ^{-1}(a x)}{c \sqrt{c-a^2 c x^2}}-\frac{1}{a c \sqrt{c-a^2 c x^2}}",1,"(Sqrt[c - a^2*c*x^2]*(1 - a*x*ArcTanh[a*x]))/(a*c^2*(-1 + a^2*x^2))","A",1
469,1,64,105,0.0740638,"\int \frac{\tanh ^{-1}(a x)}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcTanh[a*x]/(c - a^2*c*x^2)^(5/2),x]","-\frac{\sqrt{c-a^2 c x^2} \left(\left(6 a^3 x^3-9 a x\right) \tanh ^{-1}(a x)-6 a^2 x^2+7\right)}{9 a c^3 \left(a^2 x^2-1\right)^2}","-\frac{2}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \tanh ^{-1}(a x)}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{1}{9 a c \left(c-a^2 c x^2\right)^{3/2}}+\frac{x \tanh ^{-1}(a x)}{3 c \left(c-a^2 c x^2\right)^{3/2}}",1,"-1/9*(Sqrt[c - a^2*c*x^2]*(7 - 6*a^2*x^2 + (-9*a*x + 6*a^3*x^3)*ArcTanh[a*x]))/(a*c^3*(-1 + a^2*x^2)^2)","A",1
470,1,80,157,0.0910465,"\int \frac{\tanh ^{-1}(a x)}{\left(c-a^2 c x^2\right)^{7/2}} \, dx","Integrate[ArcTanh[a*x]/(c - a^2*c*x^2)^(7/2),x]","\frac{\sqrt{c-a^2 c x^2} \left(120 a^4 x^4-260 a^2 x^2-15 a x \left(8 a^4 x^4-20 a^2 x^2+15\right) \tanh ^{-1}(a x)+149\right)}{225 a c^4 \left(a^2 x^2-1\right)^3}","-\frac{8}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{8 x \tanh ^{-1}(a x)}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{4}{45 a c^2 \left(c-a^2 c x^2\right)^{3/2}}+\frac{4 x \tanh ^{-1}(a x)}{15 c^2 \left(c-a^2 c x^2\right)^{3/2}}-\frac{1}{25 a c \left(c-a^2 c x^2\right)^{5/2}}+\frac{x \tanh ^{-1}(a x)}{5 c \left(c-a^2 c x^2\right)^{5/2}}",1,"(Sqrt[c - a^2*c*x^2]*(149 - 260*a^2*x^2 + 120*a^4*x^4 - 15*a*x*(15 - 20*a^2*x^2 + 8*a^4*x^4)*ArcTanh[a*x]))/(225*a*c^4*(-1 + a^2*x^2)^3)","A",1
471,1,187,158,0.1040138,"\int \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2 \, dx","Integrate[Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2,x]","\frac{\sqrt{1-a^2 x^2} \left(-\frac{i \left(2 \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)-2 \tanh ^{-1}(a x) \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)+2 \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-2 \text{Li}_3\left(i e^{-\tanh ^{-1}(a x)}\right)+\tanh ^{-1}(a x)^2 \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)-\tanh ^{-1}(a x)^2 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)-4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \tanh ^{-1}(a x)\right)\right)\right)}{\sqrt{1-a^2 x^2}}+a x \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)\right)}{2 a}","\frac{1}{2} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a}-\frac{i \tanh ^{-1}(a x) \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)}{a}+\frac{i \tanh ^{-1}(a x) \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)}{a}+\frac{i \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)}{a}-\frac{i \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)}{a}-\frac{\sin ^{-1}(a x)}{a}+\frac{\tanh ^{-1}(a x)^2 \tan ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)}{a}",1,"(Sqrt[1 - a^2*x^2]*(2*ArcTanh[a*x] + a*x*ArcTanh[a*x]^2 - (I*((-4*I)*ArcTan[Tanh[ArcTanh[a*x]/2]] + ArcTanh[a*x]^2*Log[1 - I/E^ArcTanh[a*x]] - ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] + 2*ArcTanh[a*x]*PolyLog[2, (-I)/E^ArcTanh[a*x]] - 2*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] + 2*PolyLog[3, (-I)/E^ArcTanh[a*x]] - 2*PolyLog[3, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(2*a)","A",0
472,1,70,139,0.0844098,"\int \frac{\tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^{5/2}} \, dx","Integrate[ArcTanh[a*x]^2/(1 - a^2*x^2)^(5/2),x]","\frac{-40 a^3 x^3-9 a x \left(2 a^2 x^2-3\right) \tanh ^{-1}(a x)^2+6 \left(6 a^2 x^2-7\right) \tanh ^{-1}(a x)+42 a x}{27 a \left(1-a^2 x^2\right)^{3/2}}","\frac{40 x}{27 \sqrt{1-a^2 x^2}}+\frac{2 x}{27 \left(1-a^2 x^2\right)^{3/2}}+\frac{2 x \tanh ^{-1}(a x)^2}{3 \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{3 \left(1-a^2 x^2\right)^{3/2}}-\frac{4 \tanh ^{-1}(a x)}{3 a \sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{9 a \left(1-a^2 x^2\right)^{3/2}}",1,"(42*a*x - 40*a^3*x^3 + 6*(-7 + 6*a^2*x^2)*ArcTanh[a*x] - 9*a*x*(-3 + 2*a^2*x^2)*ArcTanh[a*x]^2)/(27*a*(1 - a^2*x^2)^(3/2))","A",1
473,1,94,208,0.0972762,"\int \frac{\tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^{7/2}} \, dx","Integrate[ArcTanh[a*x]^2/(1 - a^2*x^2)^(7/2),x]","\frac{4144 a^5 x^5-8560 a^3 x^3+225 a x \left(8 a^4 x^4-20 a^2 x^2+15\right) \tanh ^{-1}(a x)^2-30 \left(120 a^4 x^4-260 a^2 x^2+149\right) \tanh ^{-1}(a x)+4470 a x}{3375 a \left(1-a^2 x^2\right)^{5/2}}","\frac{4144 x}{3375 \sqrt{1-a^2 x^2}}+\frac{272 x}{3375 \left(1-a^2 x^2\right)^{3/2}}+\frac{2 x}{125 \left(1-a^2 x^2\right)^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^2}{15 \sqrt{1-a^2 x^2}}+\frac{4 x \tanh ^{-1}(a x)^2}{15 \left(1-a^2 x^2\right)^{3/2}}+\frac{x \tanh ^{-1}(a x)^2}{5 \left(1-a^2 x^2\right)^{5/2}}-\frac{16 \tanh ^{-1}(a x)}{15 a \sqrt{1-a^2 x^2}}-\frac{8 \tanh ^{-1}(a x)}{45 a \left(1-a^2 x^2\right)^{3/2}}-\frac{2 \tanh ^{-1}(a x)}{25 a \left(1-a^2 x^2\right)^{5/2}}",1,"(4470*a*x - 8560*a^3*x^3 + 4144*a^5*x^5 - 30*(149 - 260*a^2*x^2 + 120*a^4*x^4)*ArcTanh[a*x] + 225*a*x*(15 - 20*a^2*x^2 + 8*a^4*x^4)*ArcTanh[a*x]^2)/(3375*a*(1 - a^2*x^2)^(5/2))","A",1
474,1,120,277,0.122978,"\int \frac{\tanh ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^{9/2}} \, dx","Integrate[ArcTanh[a*x]^2/(1 - a^2*x^2)^(9/2),x]","\frac{2 a x \left(-206656 a^6 x^6+635096 a^4 x^4-654220 a^2 x^2+226905\right)-11025 a x \left(16 a^6 x^6-56 a^4 x^4+70 a^2 x^2-35\right) \tanh ^{-1}(a x)^2+210 \left(1680 a^6 x^6-5320 a^4 x^4+5726 a^2 x^2-2161\right) \tanh ^{-1}(a x)}{385875 a \left(1-a^2 x^2\right)^{7/2}}","\frac{413312 x}{385875 \sqrt{1-a^2 x^2}}+\frac{30256 x}{385875 \left(1-a^2 x^2\right)^{3/2}}+\frac{888 x}{42875 \left(1-a^2 x^2\right)^{5/2}}+\frac{2 x}{343 \left(1-a^2 x^2\right)^{7/2}}+\frac{16 x \tanh ^{-1}(a x)^2}{35 \sqrt{1-a^2 x^2}}+\frac{8 x \tanh ^{-1}(a x)^2}{35 \left(1-a^2 x^2\right)^{3/2}}+\frac{6 x \tanh ^{-1}(a x)^2}{35 \left(1-a^2 x^2\right)^{5/2}}+\frac{x \tanh ^{-1}(a x)^2}{7 \left(1-a^2 x^2\right)^{7/2}}-\frac{32 \tanh ^{-1}(a x)}{35 a \sqrt{1-a^2 x^2}}-\frac{16 \tanh ^{-1}(a x)}{105 a \left(1-a^2 x^2\right)^{3/2}}-\frac{12 \tanh ^{-1}(a x)}{175 a \left(1-a^2 x^2\right)^{5/2}}-\frac{2 \tanh ^{-1}(a x)}{49 a \left(1-a^2 x^2\right)^{7/2}}",1,"(2*a*x*(226905 - 654220*a^2*x^2 + 635096*a^4*x^4 - 206656*a^6*x^6) + 210*(-2161 + 5726*a^2*x^2 - 5320*a^4*x^4 + 1680*a^6*x^6)*ArcTanh[a*x] - 11025*a*x*(-35 + 70*a^2*x^2 - 56*a^4*x^4 + 16*a^6*x^6)*ArcTanh[a*x]^2)/(385875*a*(1 - a^2*x^2)^(7/2))","A",1
475,1,569,302,4.3048137,"\int \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3 \, dx","Integrate[Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3,x]","-\frac{i \left(64 i a x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3+192 i \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2+192 \tanh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)+192 i \pi  \tanh ^{-1}(a x) \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)+384 \tanh ^{-1}(a x) \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-384 \tanh ^{-1}(a x) \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)-48 \left(-4 \tanh ^{-1}(a x)^2-4 i \pi  \tanh ^{-1}(a x)+\pi ^2+8\right) \text{Li}_2\left(-i e^{-\tanh ^{-1}(a x)}\right)+384 \text{Li}_2\left(i e^{-\tanh ^{-1}(a x)}\right)-48 \pi ^2 \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)+192 i \pi  \text{Li}_3\left(-i e^{-\tanh ^{-1}(a x)}\right)-192 i \pi  \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)+384 \text{Li}_4\left(-i e^{-\tanh ^{-1}(a x)}\right)+384 \text{Li}_4\left(-i e^{\tanh ^{-1}(a x)}\right)-16 \tanh ^{-1}(a x)^4-32 i \pi  \tanh ^{-1}(a x)^3+24 \pi ^2 \tanh ^{-1}(a x)^2+8 i \pi ^3 \tanh ^{-1}(a x)-64 \tanh ^{-1}(a x)^3 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+64 \tanh ^{-1}(a x)^3 \log \left(1+i e^{\tanh ^{-1}(a x)}\right)-96 i \pi  \tanh ^{-1}(a x)^2 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+96 i \pi  \tanh ^{-1}(a x)^2 \log \left(1-i e^{\tanh ^{-1}(a x)}\right)-384 \tanh ^{-1}(a x) \log \left(1-i e^{-\tanh ^{-1}(a x)}\right)+48 \pi ^2 \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)+384 \tanh ^{-1}(a x) \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)-48 \pi ^2 \tanh ^{-1}(a x) \log \left(1-i e^{\tanh ^{-1}(a x)}\right)+8 i \pi ^3 \log \left(1+i e^{-\tanh ^{-1}(a x)}\right)-8 i \pi ^3 \log \left(1+i e^{\tanh ^{-1}(a x)}\right)+8 i \pi ^3 \log \left(\tan \left(\frac{1}{4} \left(\pi +2 i \tanh ^{-1}(a x)\right)\right)\right)+7 \pi ^4\right)}{128 a}","\frac{1}{2} x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3+\frac{3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a}+\frac{3 i \text{Li}_2\left(-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a}-\frac{3 i \text{Li}_2\left(\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a}-\frac{3 i \tanh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\tanh ^{-1}(a x)}\right)}{2 a}+\frac{3 i \tanh ^{-1}(a x)^2 \text{Li}_2\left(i e^{\tanh ^{-1}(a x)}\right)}{2 a}+\frac{3 i \tanh ^{-1}(a x) \text{Li}_3\left(-i e^{\tanh ^{-1}(a x)}\right)}{a}-\frac{3 i \tanh ^{-1}(a x) \text{Li}_3\left(i e^{\tanh ^{-1}(a x)}\right)}{a}-\frac{3 i \text{Li}_4\left(-i e^{\tanh ^{-1}(a x)}\right)}{a}+\frac{3 i \text{Li}_4\left(i e^{\tanh ^{-1}(a x)}\right)}{a}+\frac{\tanh ^{-1}(a x)^3 \tan ^{-1}\left(e^{\tanh ^{-1}(a x)}\right)}{a}+\frac{6 \tan ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right) \tanh ^{-1}(a x)}{a}",1,"((-1/128*I)*(7*Pi^4 + (8*I)*Pi^3*ArcTanh[a*x] + 24*Pi^2*ArcTanh[a*x]^2 + (192*I)*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2 - (32*I)*Pi*ArcTanh[a*x]^3 + (64*I)*a*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3 - 16*ArcTanh[a*x]^4 - 384*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] + (8*I)*Pi^3*Log[1 + I/E^ArcTanh[a*x]] + 384*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] + 48*Pi^2*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] - (96*I)*Pi*ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] - 64*ArcTanh[a*x]^3*Log[1 + I/E^ArcTanh[a*x]] - 48*Pi^2*ArcTanh[a*x]*Log[1 - I*E^ArcTanh[a*x]] + (96*I)*Pi*ArcTanh[a*x]^2*Log[1 - I*E^ArcTanh[a*x]] - (8*I)*Pi^3*Log[1 + I*E^ArcTanh[a*x]] + 64*ArcTanh[a*x]^3*Log[1 + I*E^ArcTanh[a*x]] + (8*I)*Pi^3*Log[Tan[(Pi + (2*I)*ArcTanh[a*x])/4]] - 48*(8 + Pi^2 - (4*I)*Pi*ArcTanh[a*x] - 4*ArcTanh[a*x]^2)*PolyLog[2, (-I)/E^ArcTanh[a*x]] + 384*PolyLog[2, I/E^ArcTanh[a*x]] + 192*ArcTanh[a*x]^2*PolyLog[2, (-I)*E^ArcTanh[a*x]] - 48*Pi^2*PolyLog[2, I*E^ArcTanh[a*x]] + (192*I)*Pi*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]] + (192*I)*Pi*PolyLog[3, (-I)/E^ArcTanh[a*x]] + 384*ArcTanh[a*x]*PolyLog[3, (-I)/E^ArcTanh[a*x]] - 384*ArcTanh[a*x]*PolyLog[3, (-I)*E^ArcTanh[a*x]] - (192*I)*Pi*PolyLog[3, I*E^ArcTanh[a*x]] + 384*PolyLog[4, (-I)/E^ArcTanh[a*x]] + 384*PolyLog[4, (-I)*E^ArcTanh[a*x]]))/a","A",0
476,1,87,191,0.0973815,"\int \frac{\tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^{5/2}} \, dx","Integrate[ArcTanh[a*x]^3/(1 - a^2*x^2)^(5/2),x]","\frac{120 a^2 x^2-9 a x \left(2 a^2 x^2-3\right) \tanh ^{-1}(a x)^3+9 \left(6 a^2 x^2-7\right) \tanh ^{-1}(a x)^2-6 a x \left(20 a^2 x^2-21\right) \tanh ^{-1}(a x)-122}{27 a \left(1-a^2 x^2\right)^{3/2}}","-\frac{40}{9 a \sqrt{1-a^2 x^2}}-\frac{2}{27 a \left(1-a^2 x^2\right)^{3/2}}+\frac{2 x \tanh ^{-1}(a x)^3}{3 \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{3 \left(1-a^2 x^2\right)^{3/2}}-\frac{2 \tanh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}-\frac{\tanh ^{-1}(a x)^2}{3 a \left(1-a^2 x^2\right)^{3/2}}+\frac{40 x \tanh ^{-1}(a x)}{9 \sqrt{1-a^2 x^2}}+\frac{2 x \tanh ^{-1}(a x)}{9 \left(1-a^2 x^2\right)^{3/2}}",1,"(-122 + 120*a^2*x^2 - 6*a*x*(-21 + 20*a^2*x^2)*ArcTanh[a*x] + 9*(-7 + 6*a^2*x^2)*ArcTanh[a*x]^2 - 9*a*x*(-3 + 2*a^2*x^2)*ArcTanh[a*x]^3)/(27*a*(1 - a^2*x^2)^(3/2))","A",1
477,1,119,289,0.1148315,"\int \frac{\tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^{7/2}} \, dx","Integrate[ArcTanh[a*x]^3/(1 - a^2*x^2)^(7/2),x]","\frac{-62160 a^4 x^4+125680 a^2 x^2+1125 a x \left(8 a^4 x^4-20 a^2 x^2+15\right) \tanh ^{-1}(a x)^3+30 a x \left(2072 a^4 x^4-4280 a^2 x^2+2235\right) \tanh ^{-1}(a x)-225 \left(120 a^4 x^4-260 a^2 x^2+149\right) \tanh ^{-1}(a x)^2-63682}{16875 a \left(1-a^2 x^2\right)^{5/2}}","-\frac{4144}{1125 a \sqrt{1-a^2 x^2}}-\frac{272}{3375 a \left(1-a^2 x^2\right)^{3/2}}-\frac{6}{625 a \left(1-a^2 x^2\right)^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^3}{15 \sqrt{1-a^2 x^2}}+\frac{4 x \tanh ^{-1}(a x)^3}{15 \left(1-a^2 x^2\right)^{3/2}}+\frac{x \tanh ^{-1}(a x)^3}{5 \left(1-a^2 x^2\right)^{5/2}}-\frac{8 \tanh ^{-1}(a x)^2}{5 a \sqrt{1-a^2 x^2}}-\frac{4 \tanh ^{-1}(a x)^2}{15 a \left(1-a^2 x^2\right)^{3/2}}-\frac{3 \tanh ^{-1}(a x)^2}{25 a \left(1-a^2 x^2\right)^{5/2}}+\frac{4144 x \tanh ^{-1}(a x)}{1125 \sqrt{1-a^2 x^2}}+\frac{272 x \tanh ^{-1}(a x)}{1125 \left(1-a^2 x^2\right)^{3/2}}+\frac{6 x \tanh ^{-1}(a x)}{125 \left(1-a^2 x^2\right)^{5/2}}",1,"(-63682 + 125680*a^2*x^2 - 62160*a^4*x^4 + 30*a*x*(2235 - 4280*a^2*x^2 + 2072*a^4*x^4)*ArcTanh[a*x] - 225*(149 - 260*a^2*x^2 + 120*a^4*x^4)*ArcTanh[a*x]^2 + 1125*a*x*(15 - 20*a^2*x^2 + 8*a^4*x^4)*ArcTanh[a*x]^3)/(16875*a*(1 - a^2*x^2)^(5/2))","A",1
478,1,151,385,0.1408542,"\int \frac{\tanh ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^{9/2}} \, dx","Integrate[ArcTanh[a*x]^3/(1 - a^2*x^2)^(9/2),x]","\frac{43397760 a^6 x^6-131252240 a^4 x^4+132479032 a^2 x^2-385875 a x \left(16 a^6 x^6-56 a^4 x^4+70 a^2 x^2-35\right) \tanh ^{-1}(a x)^3-210 a x \left(206656 a^6 x^6-635096 a^4 x^4+654220 a^2 x^2-226905\right) \tanh ^{-1}(a x)+11025 \left(1680 a^6 x^6-5320 a^4 x^4+5726 a^2 x^2-2161\right) \tanh ^{-1}(a x)^2-44658302}{13505625 a \left(1-a^2 x^2\right)^{7/2}}","-\frac{413312}{128625 a \sqrt{1-a^2 x^2}}-\frac{30256}{385875 a \left(1-a^2 x^2\right)^{3/2}}-\frac{2664}{214375 a \left(1-a^2 x^2\right)^{5/2}}-\frac{6}{2401 a \left(1-a^2 x^2\right)^{7/2}}+\frac{16 x \tanh ^{-1}(a x)^3}{35 \sqrt{1-a^2 x^2}}+\frac{8 x \tanh ^{-1}(a x)^3}{35 \left(1-a^2 x^2\right)^{3/2}}+\frac{6 x \tanh ^{-1}(a x)^3}{35 \left(1-a^2 x^2\right)^{5/2}}+\frac{x \tanh ^{-1}(a x)^3}{7 \left(1-a^2 x^2\right)^{7/2}}-\frac{48 \tanh ^{-1}(a x)^2}{35 a \sqrt{1-a^2 x^2}}-\frac{8 \tanh ^{-1}(a x)^2}{35 a \left(1-a^2 x^2\right)^{3/2}}-\frac{18 \tanh ^{-1}(a x)^2}{175 a \left(1-a^2 x^2\right)^{5/2}}-\frac{3 \tanh ^{-1}(a x)^2}{49 a \left(1-a^2 x^2\right)^{7/2}}+\frac{413312 x \tanh ^{-1}(a x)}{128625 \sqrt{1-a^2 x^2}}+\frac{30256 x \tanh ^{-1}(a x)}{128625 \left(1-a^2 x^2\right)^{3/2}}+\frac{2664 x \tanh ^{-1}(a x)}{42875 \left(1-a^2 x^2\right)^{5/2}}+\frac{6 x \tanh ^{-1}(a x)}{343 \left(1-a^2 x^2\right)^{7/2}}",1,"(-44658302 + 132479032*a^2*x^2 - 131252240*a^4*x^4 + 43397760*a^6*x^6 - 210*a*x*(-226905 + 654220*a^2*x^2 - 635096*a^4*x^4 + 206656*a^6*x^6)*ArcTanh[a*x] + 11025*(-2161 + 5726*a^2*x^2 - 5320*a^4*x^4 + 1680*a^6*x^6)*ArcTanh[a*x]^2 - 385875*a*x*(-35 + 70*a^2*x^2 - 56*a^4*x^4 + 16*a^6*x^6)*ArcTanh[a*x]^3)/(13505625*a*(1 - a^2*x^2)^(7/2))","A",1
479,0,0,24,1.2574578,"\int \frac{\sqrt{1-a^2 x^2}}{\tanh ^{-1}(a x)} \, dx","Integrate[Sqrt[1 - a^2*x^2]/ArcTanh[a*x],x]","\int \frac{\sqrt{1-a^2 x^2}}{\tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{\sqrt{1-a^2 x^2}}{\tanh ^{-1}(a x)},x\right)",0,"Integrate[Sqrt[1 - a^2*x^2]/ArcTanh[a*x], x]","A",-1
480,0,0,24,0.1794503,"\int \frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)} \, dx","Integrate[1/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]),x]","\int \frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)},x\right)",0,"Integrate[1/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]), x]","A",-1
481,1,9,9,0.0273661,"\int \frac{1}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)} \, dx","Integrate[1/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]),x]","\frac{\text{Chi}\left(\tanh ^{-1}(a x)\right)}{a}","\frac{\text{Chi}\left(\tanh ^{-1}(a x)\right)}{a}",1,"CoshIntegral[ArcTanh[a*x]]/a","A",1
482,1,22,27,0.0623027,"\int \frac{1}{\left(1-a^2 x^2\right)^{5/2} \tanh ^{-1}(a x)} \, dx","Integrate[1/((1 - a^2*x^2)^(5/2)*ArcTanh[a*x]),x]","\frac{3 \text{Chi}\left(\tanh ^{-1}(a x)\right)+\text{Chi}\left(3 \tanh ^{-1}(a x)\right)}{4 a}","\frac{3 \text{Chi}\left(\tanh ^{-1}(a x)\right)}{4 a}+\frac{\text{Chi}\left(3 \tanh ^{-1}(a x)\right)}{4 a}",1,"(3*CoshIntegral[ArcTanh[a*x]] + CoshIntegral[3*ArcTanh[a*x]])/(4*a)","A",1
483,1,31,41,0.0652591,"\int \frac{1}{\left(1-a^2 x^2\right)^{7/2} \tanh ^{-1}(a x)} \, dx","Integrate[1/((1 - a^2*x^2)^(7/2)*ArcTanh[a*x]),x]","\frac{10 \text{Chi}\left(\tanh ^{-1}(a x)\right)+5 \text{Chi}\left(3 \tanh ^{-1}(a x)\right)+\text{Chi}\left(5 \tanh ^{-1}(a x)\right)}{16 a}","\frac{5 \text{Chi}\left(\tanh ^{-1}(a x)\right)}{8 a}+\frac{5 \text{Chi}\left(3 \tanh ^{-1}(a x)\right)}{16 a}+\frac{\text{Chi}\left(5 \tanh ^{-1}(a x)\right)}{16 a}",1,"(10*CoshIntegral[ArcTanh[a*x]] + 5*CoshIntegral[3*ArcTanh[a*x]] + CoshIntegral[5*ArcTanh[a*x]])/(16*a)","A",1
484,1,40,55,0.0735561,"\int \frac{1}{\left(1-a^2 x^2\right)^{9/2} \tanh ^{-1}(a x)} \, dx","Integrate[1/((1 - a^2*x^2)^(9/2)*ArcTanh[a*x]),x]","\frac{35 \text{Chi}\left(\tanh ^{-1}(a x)\right)+21 \text{Chi}\left(3 \tanh ^{-1}(a x)\right)+7 \text{Chi}\left(5 \tanh ^{-1}(a x)\right)+\text{Chi}\left(7 \tanh ^{-1}(a x)\right)}{64 a}","\frac{35 \text{Chi}\left(\tanh ^{-1}(a x)\right)}{64 a}+\frac{21 \text{Chi}\left(3 \tanh ^{-1}(a x)\right)}{64 a}+\frac{7 \text{Chi}\left(5 \tanh ^{-1}(a x)\right)}{64 a}+\frac{\text{Chi}\left(7 \tanh ^{-1}(a x)\right)}{64 a}",1,"(35*CoshIntegral[ArcTanh[a*x]] + 21*CoshIntegral[3*ArcTanh[a*x]] + 7*CoshIntegral[5*ArcTanh[a*x]] + CoshIntegral[7*ArcTanh[a*x]])/(64*a)","A",1
485,0,0,24,1.5930329,"\int \frac{\sqrt{1-a^2 x^2}}{\tanh ^{-1}(a x)^2} \, dx","Integrate[Sqrt[1 - a^2*x^2]/ArcTanh[a*x]^2,x]","\int \frac{\sqrt{1-a^2 x^2}}{\tanh ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{\sqrt{1-a^2 x^2}}{\tanh ^{-1}(a x)^2},x\right)",0,"Integrate[Sqrt[1 - a^2*x^2]/ArcTanh[a*x]^2, x]","A",-1
486,0,0,24,0.95896,"\int \frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2} \, dx","Integrate[1/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2),x]","\int \frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2},x\right)",0,"Integrate[1/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2), x]","A",-1
487,1,32,35,0.063819,"\int \frac{1}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^2} \, dx","Integrate[1/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2),x]","\frac{\text{Shi}\left(\tanh ^{-1}(a x)\right)-\frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}}{a}","\frac{\text{Shi}\left(\tanh ^{-1}(a x)\right)}{a}-\frac{1}{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}",1,"(-(1/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])) + SinhIntegral[ArcTanh[a*x]])/a","A",1
488,1,45,52,0.1416161,"\int \frac{1}{\left(1-a^2 x^2\right)^{5/2} \tanh ^{-1}(a x)^2} \, dx","Integrate[1/((1 - a^2*x^2)^(5/2)*ArcTanh[a*x]^2),x]","\frac{3 \left(\text{Shi}\left(\tanh ^{-1}(a x)\right)+\text{Shi}\left(3 \tanh ^{-1}(a x)\right)\right)-\frac{4}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)}}{4 a}","-\frac{1}{a \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)}+\frac{3 \text{Shi}\left(\tanh ^{-1}(a x)\right)}{4 a}+\frac{3 \text{Shi}\left(3 \tanh ^{-1}(a x)\right)}{4 a}",1,"(-4/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]) + 3*(SinhIntegral[ArcTanh[a*x]] + SinhIntegral[3*ArcTanh[a*x]]))/(4*a)","A",1
489,1,56,66,0.1808312,"\int \frac{1}{\left(1-a^2 x^2\right)^{7/2} \tanh ^{-1}(a x)^2} \, dx","Integrate[1/((1 - a^2*x^2)^(7/2)*ArcTanh[a*x]^2),x]","\frac{5 \left(2 \text{Shi}\left(\tanh ^{-1}(a x)\right)+3 \text{Shi}\left(3 \tanh ^{-1}(a x)\right)+\text{Shi}\left(5 \tanh ^{-1}(a x)\right)\right)-\frac{16}{\left(1-a^2 x^2\right)^{5/2} \tanh ^{-1}(a x)}}{16 a}","-\frac{1}{a \left(1-a^2 x^2\right)^{5/2} \tanh ^{-1}(a x)}+\frac{5 \text{Shi}\left(\tanh ^{-1}(a x)\right)}{8 a}+\frac{15 \text{Shi}\left(3 \tanh ^{-1}(a x)\right)}{16 a}+\frac{5 \text{Shi}\left(5 \tanh ^{-1}(a x)\right)}{16 a}",1,"(-16/((1 - a^2*x^2)^(5/2)*ArcTanh[a*x]) + 5*(2*SinhIntegral[ArcTanh[a*x]] + 3*SinhIntegral[3*ArcTanh[a*x]] + SinhIntegral[5*ArcTanh[a*x]]))/(16*a)","A",1
490,1,65,80,0.2191139,"\int \frac{1}{\left(1-a^2 x^2\right)^{9/2} \tanh ^{-1}(a x)^2} \, dx","Integrate[1/((1 - a^2*x^2)^(9/2)*ArcTanh[a*x]^2),x]","\frac{7 \left(5 \text{Shi}\left(\tanh ^{-1}(a x)\right)+9 \text{Shi}\left(3 \tanh ^{-1}(a x)\right)+5 \text{Shi}\left(5 \tanh ^{-1}(a x)\right)+\text{Shi}\left(7 \tanh ^{-1}(a x)\right)\right)-\frac{64}{\left(1-a^2 x^2\right)^{7/2} \tanh ^{-1}(a x)}}{64 a}","-\frac{1}{a \left(1-a^2 x^2\right)^{7/2} \tanh ^{-1}(a x)}+\frac{35 \text{Shi}\left(\tanh ^{-1}(a x)\right)}{64 a}+\frac{63 \text{Shi}\left(3 \tanh ^{-1}(a x)\right)}{64 a}+\frac{35 \text{Shi}\left(5 \tanh ^{-1}(a x)\right)}{64 a}+\frac{7 \text{Shi}\left(7 \tanh ^{-1}(a x)\right)}{64 a}",1,"(-64/((1 - a^2*x^2)^(7/2)*ArcTanh[a*x]) + 7*(5*SinhIntegral[ArcTanh[a*x]] + 9*SinhIntegral[3*ArcTanh[a*x]] + 5*SinhIntegral[5*ArcTanh[a*x]] + SinhIntegral[7*ArcTanh[a*x]]))/(64*a)","A",1
491,0,0,24,1.8246493,"\int \frac{\sqrt{1-a^2 x^2}}{\tanh ^{-1}(a x)^3} \, dx","Integrate[Sqrt[1 - a^2*x^2]/ArcTanh[a*x]^3,x]","\int \frac{\sqrt{1-a^2 x^2}}{\tanh ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{\sqrt{1-a^2 x^2}}{\tanh ^{-1}(a x)^3},x\right)",0,"Integrate[Sqrt[1 - a^2*x^2]/ArcTanh[a*x]^3, x]","A",-1
492,0,0,24,1.1420973,"\int \frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3} \, dx","Integrate[1/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3),x]","\int \frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3},x\right)",0,"Integrate[1/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3), x]","A",-1
493,1,44,65,0.1192557,"\int \frac{1}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^3} \, dx","Integrate[1/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^3),x]","\frac{\text{Chi}\left(\tanh ^{-1}(a x)\right)-\frac{a x \tanh ^{-1}(a x)+1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}}{2 a}","-\frac{x}{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}-\frac{1}{2 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}+\frac{\text{Chi}\left(\tanh ^{-1}(a x)\right)}{2 a}",1,"(-((1 + a*x*ArcTanh[a*x])/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)) + CoshIntegral[ArcTanh[a*x]])/(2*a)","A",1
494,1,56,79,0.2243238,"\int \frac{1}{\left(1-a^2 x^2\right)^{5/2} \tanh ^{-1}(a x)^3} \, dx","Integrate[1/((1 - a^2*x^2)^(5/2)*ArcTanh[a*x]^3),x]","\frac{-\frac{4 \left(3 a x \tanh ^{-1}(a x)+1\right)}{\left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^2}+3 \text{Chi}\left(\tanh ^{-1}(a x)\right)+9 \text{Chi}\left(3 \tanh ^{-1}(a x)\right)}{8 a}","-\frac{3 x}{2 \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)}-\frac{1}{2 a \left(1-a^2 x^2\right)^{3/2} \tanh ^{-1}(a x)^2}+\frac{3 \text{Chi}\left(\tanh ^{-1}(a x)\right)}{8 a}+\frac{9 \text{Chi}\left(3 \tanh ^{-1}(a x)\right)}{8 a}",1,"((-4*(1 + 3*a*x*ArcTanh[a*x]))/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2) + 3*CoshIntegral[ArcTanh[a*x]] + 9*CoshIntegral[3*ArcTanh[a*x]])/(8*a)","A",1
495,1,79,93,0.2493124,"\int \frac{1}{\left(1-a^2 x^2\right)^{7/2} \tanh ^{-1}(a x)^3} \, dx","Integrate[1/((1 - a^2*x^2)^(7/2)*ArcTanh[a*x]^3),x]","\frac{-\frac{80 a x}{\left(1-a^2 x^2\right)^{5/2} \tanh ^{-1}(a x)}-\frac{16}{\left(1-a^2 x^2\right)^{5/2} \tanh ^{-1}(a x)^2}+10 \text{Chi}\left(\tanh ^{-1}(a x)\right)+45 \text{Chi}\left(3 \tanh ^{-1}(a x)\right)+25 \text{Chi}\left(5 \tanh ^{-1}(a x)\right)}{32 a}","-\frac{5 x}{2 \left(1-a^2 x^2\right)^{5/2} \tanh ^{-1}(a x)}-\frac{1}{2 a \left(1-a^2 x^2\right)^{5/2} \tanh ^{-1}(a x)^2}+\frac{5 \text{Chi}\left(\tanh ^{-1}(a x)\right)}{16 a}+\frac{45 \text{Chi}\left(3 \tanh ^{-1}(a x)\right)}{32 a}+\frac{25 \text{Chi}\left(5 \tanh ^{-1}(a x)\right)}{32 a}",1,"(-16/((1 - a^2*x^2)^(5/2)*ArcTanh[a*x]^2) - (80*a*x)/((1 - a^2*x^2)^(5/2)*ArcTanh[a*x]) + 10*CoshIntegral[ArcTanh[a*x]] + 45*CoshIntegral[3*ArcTanh[a*x]] + 25*CoshIntegral[5*ArcTanh[a*x]])/(32*a)","A",1
496,1,99,107,0.2272395,"\int \frac{1}{\left(1-a^2 x^2\right)^{9/2} \tanh ^{-1}(a x)^3} \, dx","Integrate[1/((1 - a^2*x^2)^(9/2)*ArcTanh[a*x]^3),x]","\frac{1}{128} \left(-\frac{448 x}{\left(1-a^2 x^2\right)^{7/2} \tanh ^{-1}(a x)}-\frac{64}{a \left(1-a^2 x^2\right)^{7/2} \tanh ^{-1}(a x)^2}+\frac{35 \text{Chi}\left(\tanh ^{-1}(a x)\right)}{a}+\frac{189 \text{Chi}\left(3 \tanh ^{-1}(a x)\right)}{a}+\frac{175 \text{Chi}\left(5 \tanh ^{-1}(a x)\right)}{a}+\frac{49 \text{Chi}\left(7 \tanh ^{-1}(a x)\right)}{a}\right)","-\frac{7 x}{2 \left(1-a^2 x^2\right)^{7/2} \tanh ^{-1}(a x)}-\frac{1}{2 a \left(1-a^2 x^2\right)^{7/2} \tanh ^{-1}(a x)^2}+\frac{35 \text{Chi}\left(\tanh ^{-1}(a x)\right)}{128 a}+\frac{189 \text{Chi}\left(3 \tanh ^{-1}(a x)\right)}{128 a}+\frac{175 \text{Chi}\left(5 \tanh ^{-1}(a x)\right)}{128 a}+\frac{49 \text{Chi}\left(7 \tanh ^{-1}(a x)\right)}{128 a}",1,"(-64/(a*(1 - a^2*x^2)^(7/2)*ArcTanh[a*x]^2) - (448*x)/((1 - a^2*x^2)^(7/2)*ArcTanh[a*x]) + (35*CoshIntegral[ArcTanh[a*x]])/a + (189*CoshIntegral[3*ArcTanh[a*x]])/a + (175*CoshIntegral[5*ArcTanh[a*x]])/a + (49*CoshIntegral[7*ArcTanh[a*x]])/a)/128","A",1
497,1,193,122,0.4103947,"\int \frac{(d+e x) \left(a+b \tanh ^{-1}(c x)\right)^2}{1-c^2 x^2} \, dx","Integrate[((d + e*x)*(a + b*ArcTanh[c*x])^2)/(1 - c^2*x^2),x]","\frac{-3 a^2 c d \log (1-c x)+3 a^2 c d \log (c x+1)-3 a^2 e \log (1-c x)-3 a^2 e \log (c x+1)+6 a b c d \tanh ^{-1}(c x)^2-6 b e \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right) \left(a+b \tanh ^{-1}(c x)\right)+6 a b e \tanh ^{-1}(c x)^2+12 a b e \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+2 b^2 c d \tanh ^{-1}(c x)^3-3 b^2 e \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)+2 b^2 e \tanh ^{-1}(c x)^3+6 b^2 e \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)}{6 c^2}","\frac{b e \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^2}-\frac{e \left(a+b \tanh ^{-1}(c x)\right)^3}{3 b c^2}+\frac{e \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^2}+\frac{d \left(a+b \tanh ^{-1}(c x)\right)^3}{3 b c}-\frac{b^2 e \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{2 c^2}",1,"(6*a*b*c*d*ArcTanh[c*x]^2 + 6*a*b*e*ArcTanh[c*x]^2 + 2*b^2*c*d*ArcTanh[c*x]^3 + 2*b^2*e*ArcTanh[c*x]^3 + 12*a*b*e*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + 6*b^2*e*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] - 3*a^2*c*d*Log[1 - c*x] - 3*a^2*e*Log[1 - c*x] + 3*a^2*c*d*Log[1 + c*x] - 3*a^2*e*Log[1 + c*x] - 6*b*e*(a + b*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 3*b^2*e*PolyLog[3, -E^(-2*ArcTanh[c*x])])/(6*c^2)","A",0
498,1,213,245,0.1163071,"\int \left(c+d x^2\right)^4 \tanh ^{-1}(a x) \, dx","Integrate[(c + d*x^2)^4*ArcTanh[a*x],x]","\frac{24 a^9 x \tanh ^{-1}(a x) \left(315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right)+a^2 d x^2 \left(3 a^6 \left(1680 c^3+756 c^2 d x^2+240 c d^2 x^4+35 d^3 x^6\right)+4 a^4 d \left(1134 c^2+270 c d x^2+35 d^2 x^4\right)+30 a^2 d^2 \left(72 c+7 d x^2\right)+420 d^3\right)+12 \left(315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right) \log \left(1-a^2 x^2\right)}{7560 a^9}","\frac{d^3 x^6 \left(36 a^2 c+7 d\right)}{378 a^3}+\frac{d^2 x^4 \left(378 a^4 c^2+180 a^2 c d+35 d^2\right)}{1260 a^5}+\frac{d x^2 \left(420 a^6 c^3+378 a^4 c^2 d+180 a^2 c d^2+35 d^3\right)}{630 a^7}+\frac{\left(315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right) \log \left(1-a^2 x^2\right)}{630 a^9}+c^4 x \tanh ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \tanh ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \tanh ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \tanh ^{-1}(a x)+\frac{1}{9} d^4 x^9 \tanh ^{-1}(a x)+\frac{d^4 x^8}{72 a}",1,"(a^2*d*x^2*(420*d^3 + 30*a^2*d^2*(72*c + 7*d*x^2) + 4*a^4*d*(1134*c^2 + 270*c*d*x^2 + 35*d^2*x^4) + 3*a^6*(1680*c^3 + 756*c^2*d*x^2 + 240*c*d^2*x^4 + 35*d^3*x^6)) + 24*a^9*x*(315*c^4 + 420*c^3*d*x^2 + 378*c^2*d^2*x^4 + 180*c*d^3*x^6 + 35*d^4*x^8)*ArcTanh[a*x] + 12*(315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*Log[1 - a^2*x^2])/(7560*a^9)","A",1
499,1,150,169,0.0842346,"\int \left(c+d x^2\right)^3 \tanh ^{-1}(a x) \, dx","Integrate[(c + d*x^2)^3*ArcTanh[a*x],x]","\frac{12 a^7 x \tanh ^{-1}(a x) \left(35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right)+a^2 d x^2 \left(a^4 \left(210 c^2+63 c d x^2+10 d^2 x^4\right)+3 a^2 d \left(42 c+5 d x^2\right)+30 d^2\right)+6 \left(35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right) \log \left(1-a^2 x^2\right)}{420 a^7}","\frac{d^2 x^4 \left(21 a^2 c+5 d\right)}{140 a^3}+\frac{d x^2 \left(35 a^4 c^2+21 a^2 c d+5 d^2\right)}{70 a^5}+\frac{\left(35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right) \log \left(1-a^2 x^2\right)}{70 a^7}+c^3 x \tanh ^{-1}(a x)+c^2 d x^3 \tanh ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \tanh ^{-1}(a x)+\frac{1}{7} d^3 x^7 \tanh ^{-1}(a x)+\frac{d^3 x^6}{42 a}",1,"(a^2*d*x^2*(30*d^2 + 3*a^2*d*(42*c + 5*d*x^2) + a^4*(210*c^2 + 63*c*d*x^2 + 10*d^2*x^4)) + 12*a^7*x*(35*c^3 + 35*c^2*d*x^2 + 21*c*d^2*x^4 + 5*d^3*x^6)*ArcTanh[a*x] + 6*(35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*Log[1 - a^2*x^2])/(420*a^7)","A",1
500,1,98,110,0.0529962,"\int \left(c+d x^2\right)^2 \tanh ^{-1}(a x) \, dx","Integrate[(c + d*x^2)^2*ArcTanh[a*x],x]","\frac{4 a^5 x \tanh ^{-1}(a x) \left(15 c^2+10 c d x^2+3 d^2 x^4\right)+a^2 d x^2 \left(a^2 \left(20 c+3 d x^2\right)+6 d\right)+\left(30 a^4 c^2+20 a^2 c d+6 d^2\right) \log \left(1-a^2 x^2\right)}{60 a^5}","\frac{d x^2 \left(10 a^2 c+3 d\right)}{30 a^3}+\frac{\left(15 a^4 c^2+10 a^2 c d+3 d^2\right) \log \left(1-a^2 x^2\right)}{30 a^5}+c^2 x \tanh ^{-1}(a x)+\frac{2}{3} c d x^3 \tanh ^{-1}(a x)+\frac{1}{5} d^2 x^5 \tanh ^{-1}(a x)+\frac{d^2 x^4}{20 a}",1,"(a^2*d*x^2*(6*d + a^2*(20*c + 3*d*x^2)) + 4*a^5*x*(15*c^2 + 10*c*d*x^2 + 3*d^2*x^4)*ArcTanh[a*x] + (30*a^4*c^2 + 20*a^2*c*d + 6*d^2)*Log[1 - a^2*x^2])/(60*a^5)","A",1
501,1,69,57,0.0105356,"\int \left(c+d x^2\right) \tanh ^{-1}(a x) \, dx","Integrate[(c + d*x^2)*ArcTanh[a*x],x]","\frac{c \log \left(1-a^2 x^2\right)}{2 a}+\frac{d \log \left(1-a^2 x^2\right)}{6 a^3}+c x \tanh ^{-1}(a x)+\frac{1}{3} d x^3 \tanh ^{-1}(a x)+\frac{d x^2}{6 a}","\frac{\left(3 a^2 c+d\right) \log \left(1-a^2 x^2\right)}{6 a^3}+c x \tanh ^{-1}(a x)+\frac{1}{3} d x^3 \tanh ^{-1}(a x)+\frac{d x^2}{6 a}",1,"(d*x^2)/(6*a) + c*x*ArcTanh[a*x] + (d*x^3*ArcTanh[a*x])/3 + (c*Log[1 - a^2*x^2])/(2*a) + (d*Log[1 - a^2*x^2])/(6*a^3)","A",1
502,1,662,429,1.4593037,"\int \frac{\tanh ^{-1}(a x)}{c+d x^2} \, dx","Integrate[ArcTanh[a*x]/(c + d*x^2),x]","-\frac{a \left(i \left(\text{Li}_2\left(\frac{\left(-c a^2+d+2 i \sqrt{a^2 c d}\right) \left(i a c+\sqrt{a^2 c d} x\right)}{\left(c a^2+d\right) \left(\sqrt{a^2 c d} x-i a c\right)}\right)-\text{Li}_2\left(\frac{\left(-c a^2+d-2 i \sqrt{a^2 c d}\right) \left(i a c+\sqrt{a^2 c d} x\right)}{\left(c a^2+d\right) \left(\sqrt{a^2 c d} x-i a c\right)}\right)\right)-2 i \cos ^{-1}\left(\frac{d-a^2 c}{a^2 c+d}\right) \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)+4 \tanh ^{-1}(a x) \tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)-\log \left(\frac{2 i a c (a x-1) \left(\sqrt{a^2 c d}+i d\right)}{\left(a^2 c+d\right) \left(a c+i x \sqrt{a^2 c d}\right)}\right) \left(2 \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)+\cos ^{-1}\left(\frac{d-a^2 c}{a^2 c+d}\right)\right)-\log \left(\frac{2 a c (a x+1) \left(d+i \sqrt{a^2 c d}\right)}{\left(a^2 c+d\right) \left(a c+i x \sqrt{a^2 c d}\right)}\right) \left(\cos ^{-1}\left(\frac{d-a^2 c}{a^2 c+d}\right)-2 \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)+\left(2 \left(\tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)+\tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)+\cos ^{-1}\left(\frac{d-a^2 c}{a^2 c+d}\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{-\tanh ^{-1}(a x)}}{\sqrt{a^2 c+d} \sqrt{\left(a^2 c+d\right) \cosh \left(2 \tanh ^{-1}(a x)\right)+a^2 c-d}}\right)+\left(\cos ^{-1}\left(\frac{d-a^2 c}{a^2 c+d}\right)-2 \left(\tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)+\tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{\tanh ^{-1}(a x)}}{\sqrt{a^2 c+d} \sqrt{\left(a^2 c+d\right) \cosh \left(2 \tanh ^{-1}(a x)\right)+a^2 c-d}}\right)\right)}{4 \sqrt{a^2 c d}}","-\frac{\text{Li}_2\left(-\frac{\sqrt{d} (1-a x)}{a \sqrt{-c}-\sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{Li}_2\left(\frac{\sqrt{d} (1-a x)}{\sqrt{-c} a+\sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{\text{Li}_2\left(-\frac{\sqrt{d} (a x+1)}{a \sqrt{-c}-\sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{Li}_2\left(\frac{\sqrt{d} (a x+1)}{\sqrt{-c} a+\sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{\log (1-a x) \log \left(\frac{a \left(\sqrt{-c}-\sqrt{d} x\right)}{a \sqrt{-c}-\sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{\log (a x+1) \log \left(\frac{a \left(\sqrt{-c}-\sqrt{d} x\right)}{a \sqrt{-c}+\sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{\log (a x+1) \log \left(\frac{a \left(\sqrt{-c}+\sqrt{d} x\right)}{a \sqrt{-c}-\sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{\log (1-a x) \log \left(\frac{a \left(\sqrt{-c}+\sqrt{d} x\right)}{a \sqrt{-c}+\sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}",1,"-1/4*(a*((-2*I)*ArcCos[(-(a^2*c) + d)/(a^2*c + d)]*ArcTan[(a*d*x)/Sqrt[a^2*c*d]] + 4*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)]*ArcTanh[a*x] - (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] + 2*ArcTan[(a*d*x)/Sqrt[a^2*c*d]])*Log[((2*I)*a*c*(I*d + Sqrt[a^2*c*d])*(-1 + a*x))/((a^2*c + d)*(a*c + I*Sqrt[a^2*c*d]*x))] - (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] - 2*ArcTan[(a*d*x)/Sqrt[a^2*c*d]])*Log[(2*a*c*(d + I*Sqrt[a^2*c*d])*(1 + a*x))/((a^2*c + d)*(a*c + I*Sqrt[a^2*c*d]*x))] + (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] + 2*(ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d])/(Sqrt[a^2*c + d]*E^ArcTanh[a*x]*Sqrt[a^2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a*x]]])] + (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] - 2*(ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d]*E^ArcTanh[a*x])/(Sqrt[a^2*c + d]*Sqrt[a^2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a*x]]])] + I*(-PolyLog[2, ((-(a^2*c) + d - (2*I)*Sqrt[a^2*c*d])*(I*a*c + Sqrt[a^2*c*d]*x))/((a^2*c + d)*((-I)*a*c + Sqrt[a^2*c*d]*x))] + PolyLog[2, ((-(a^2*c) + d + (2*I)*Sqrt[a^2*c*d])*(I*a*c + Sqrt[a^2*c*d]*x))/((a^2*c + d)*((-I)*a*c + Sqrt[a^2*c*d]*x))])))/Sqrt[a^2*c*d]","C",1
503,1,746,590,7.7855319,"\int \frac{\tanh ^{-1}(a x)}{\left(c+d x^2\right)^2} \, dx","Integrate[ArcTanh[a*x]/(c + d*x^2)^2,x]","\frac{a \left(\frac{i \left(\text{Li}_2\left(\frac{\left(-c a^2+d-2 i \sqrt{a^2 c d}\right) \left(i a c+\sqrt{a^2 c d} x\right)}{\left(c a^2+d\right) \left(\sqrt{a^2 c d} x-i a c\right)}\right)-\text{Li}_2\left(\frac{\left(-c a^2+d+2 i \sqrt{a^2 c d}\right) \left(i a c+\sqrt{a^2 c d} x\right)}{\left(c a^2+d\right) \left(\sqrt{a^2 c d} x-i a c\right)}\right)\right)+2 i \cos ^{-1}\left(\frac{d-a^2 c}{a^2 c+d}\right) \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)-4 \tanh ^{-1}(a x) \tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)+\log \left(\frac{2 i a c (a x-1) \left(\sqrt{a^2 c d}+i d\right)}{\left(a^2 c+d\right) \left(a c+i x \sqrt{a^2 c d}\right)}\right) \left(2 \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)+\cos ^{-1}\left(\frac{d-a^2 c}{a^2 c+d}\right)\right)+\log \left(\frac{2 a c (a x+1) \left(d+i \sqrt{a^2 c d}\right)}{\left(a^2 c+d\right) \left(a c+i x \sqrt{a^2 c d}\right)}\right) \left(\cos ^{-1}\left(\frac{d-a^2 c}{a^2 c+d}\right)-2 \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)-\left(2 \left(\tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)+\tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)+\cos ^{-1}\left(\frac{d-a^2 c}{a^2 c+d}\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{-\tanh ^{-1}(a x)}}{\sqrt{a^2 c+d} \sqrt{\left(a^2 c+d\right) \cosh \left(2 \tanh ^{-1}(a x)\right)+a^2 c-d}}\right)-\left(\cos ^{-1}\left(\frac{d-a^2 c}{a^2 c+d}\right)-2 \left(\tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)+\tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{\tanh ^{-1}(a x)}}{\sqrt{a^2 c+d} \sqrt{\left(a^2 c+d\right) \cosh \left(2 \tanh ^{-1}(a x)\right)+a^2 c-d}}\right)}{\sqrt{a^2 c d}}-\frac{2 \log \left(\frac{\left(a^2 c+d\right) \cosh \left(2 \tanh ^{-1}(a x)\right)}{a^2 c-d}+1\right)}{a^2 c+d}+\frac{4 \tanh ^{-1}(a x) \sinh \left(2 \tanh ^{-1}(a x)\right)}{\left(a^2 c+d\right) \cosh \left(2 \tanh ^{-1}(a x)\right)+a^2 c-d}\right)}{8 c}","\frac{a \log \left(1-a^2 x^2\right)}{4 c \left(a^2 c+d\right)}-\frac{a \log \left(c+d x^2\right)}{4 c \left(a^2 c+d\right)}+\frac{i \text{Li}_2\left(\frac{a \left(\sqrt{c}-i \sqrt{d} x\right)}{a \sqrt{c}-i \sqrt{d}}\right)}{8 c^{3/2} \sqrt{d}}-\frac{i \text{Li}_2\left(\frac{a \left(\sqrt{c}-i \sqrt{d} x\right)}{\sqrt{c} a+i \sqrt{d}}\right)}{8 c^{3/2} \sqrt{d}}+\frac{i \text{Li}_2\left(\frac{a \left(i \sqrt{d} x+\sqrt{c}\right)}{a \sqrt{c}-i \sqrt{d}}\right)}{8 c^{3/2} \sqrt{d}}-\frac{i \text{Li}_2\left(\frac{a \left(i \sqrt{d} x+\sqrt{c}\right)}{\sqrt{c} a+i \sqrt{d}}\right)}{8 c^{3/2} \sqrt{d}}+\frac{i \log \left(1-\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(\frac{\sqrt{d} (1-a x)}{\sqrt{d}+i a \sqrt{c}}\right)}{8 c^{3/2} \sqrt{d}}-\frac{i \log \left(1-\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(-\frac{\sqrt{d} (a x+1)}{-\sqrt{d}+i a \sqrt{c}}\right)}{8 c^{3/2} \sqrt{d}}-\frac{i \log \left(1+\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(-\frac{\sqrt{d} (1-a x)}{-\sqrt{d}+i a \sqrt{c}}\right)}{8 c^{3/2} \sqrt{d}}+\frac{i \log \left(1+\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(\frac{\sqrt{d} (a x+1)}{\sqrt{d}+i a \sqrt{c}}\right)}{8 c^{3/2} \sqrt{d}}+\frac{\tanh ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} \sqrt{d}}+\frac{x \tanh ^{-1}(a x)}{2 c \left(c+d x^2\right)}",1,"(a*((-2*Log[1 + ((a^2*c + d)*Cosh[2*ArcTanh[a*x]])/(a^2*c - d)])/(a^2*c + d) + ((2*I)*ArcCos[(-(a^2*c) + d)/(a^2*c + d)]*ArcTan[(a*d*x)/Sqrt[a^2*c*d]] - 4*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)]*ArcTanh[a*x] + (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] + 2*ArcTan[(a*d*x)/Sqrt[a^2*c*d]])*Log[((2*I)*a*c*(I*d + Sqrt[a^2*c*d])*(-1 + a*x))/((a^2*c + d)*(a*c + I*Sqrt[a^2*c*d]*x))] + (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] - 2*ArcTan[(a*d*x)/Sqrt[a^2*c*d]])*Log[(2*a*c*(d + I*Sqrt[a^2*c*d])*(1 + a*x))/((a^2*c + d)*(a*c + I*Sqrt[a^2*c*d]*x))] - (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] + 2*(ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d])/(Sqrt[a^2*c + d]*E^ArcTanh[a*x]*Sqrt[a^2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a*x]]])] - (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] - 2*(ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d]*E^ArcTanh[a*x])/(Sqrt[a^2*c + d]*Sqrt[a^2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a*x]]])] + I*(PolyLog[2, ((-(a^2*c) + d - (2*I)*Sqrt[a^2*c*d])*(I*a*c + Sqrt[a^2*c*d]*x))/((a^2*c + d)*((-I)*a*c + Sqrt[a^2*c*d]*x))] - PolyLog[2, ((-(a^2*c) + d + (2*I)*Sqrt[a^2*c*d])*(I*a*c + Sqrt[a^2*c*d]*x))/((a^2*c + d)*((-I)*a*c + Sqrt[a^2*c*d]*x))]))/Sqrt[a^2*c*d] + (4*ArcTanh[a*x]*Sinh[2*ArcTanh[a*x]])/(a^2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a*x]])))/(8*c)","A",0
504,1,1828,657,12.9380683,"\int \frac{\tanh ^{-1}(a x)}{\left(c+d x^2\right)^3} \, dx","Integrate[ArcTanh[a*x]/(c + d*x^2)^3,x]","a^5 \left(-\frac{3 d \log \left(1-\frac{\left(-c a^2-d\right) \cosh \left(2 \tanh ^{-1}(a x)\right)}{a^2 c-d}\right)}{16 a^4 c^2 \left(c a^2+d\right)^2}-\frac{5 \log \left(1-\frac{\left(-c a^2-d\right) \cosh \left(2 \tanh ^{-1}(a x)\right)}{a^2 c-d}\right)}{16 a^2 c \left(c a^2+d\right)^2}-\frac{3 d \left(-2 i \cos ^{-1}\left(\frac{d-a^2 c}{c a^2+d}\right) \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)+4 \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right) \tanh ^{-1}(a x)-\left(\cos ^{-1}\left(\frac{d-a^2 c}{c a^2+d}\right)-2 \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right) \log \left(1-\frac{\left(c a^2-d-2 i \sqrt{a^2 c d}\right) \left(2 a^2 c-2 i a \sqrt{a^2 c d} x\right)}{\left(c a^2+d\right) \left(2 c a^2+2 i \sqrt{a^2 c d} x a\right)}\right)+\left(-\cos ^{-1}\left(\frac{d-a^2 c}{c a^2+d}\right)-2 \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right) \log \left(1-\frac{\left(c a^2-d+2 i \sqrt{a^2 c d}\right) \left(2 a^2 c-2 i a \sqrt{a^2 c d} x\right)}{\left(c a^2+d\right) \left(2 c a^2+2 i \sqrt{a^2 c d} x a\right)}\right)+\left(\cos ^{-1}\left(\frac{d-a^2 c}{c a^2+d}\right)+2 i \left(-i \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right)-i \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{-\tanh ^{-1}(a x)}}{\sqrt{c a^2+d} \sqrt{c a^2-d+\left(c a^2+d\right) \cosh \left(2 \tanh ^{-1}(a x)\right)}}\right)+\left(\cos ^{-1}\left(\frac{d-a^2 c}{c a^2+d}\right)-2 i \left(-i \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right)-i \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{\tanh ^{-1}(a x)}}{\sqrt{c a^2+d} \sqrt{c a^2-d+\left(c a^2+d\right) \cosh \left(2 \tanh ^{-1}(a x)\right)}}\right)+i \left(\text{Li}_2\left(\frac{\left(c a^2-d-2 i \sqrt{a^2 c d}\right) \left(2 a^2 c-2 i a \sqrt{a^2 c d} x\right)}{\left(c a^2+d\right) \left(2 c a^2+2 i \sqrt{a^2 c d} x a\right)}\right)-\text{Li}_2\left(\frac{\left(c a^2-d+2 i \sqrt{a^2 c d}\right) \left(2 a^2 c-2 i a \sqrt{a^2 c d} x\right)}{\left(c a^2+d\right) \left(2 c a^2+2 i \sqrt{a^2 c d} x a\right)}\right)\right)\right)}{32 a^4 c^2 \sqrt{a^2 c d} \left(c a^2+d\right)}-\frac{3 \left(-2 i \cos ^{-1}\left(\frac{d-a^2 c}{c a^2+d}\right) \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)+4 \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right) \tanh ^{-1}(a x)-\left(\cos ^{-1}\left(\frac{d-a^2 c}{c a^2+d}\right)-2 \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right) \log \left(1-\frac{\left(c a^2-d-2 i \sqrt{a^2 c d}\right) \left(2 a^2 c-2 i a \sqrt{a^2 c d} x\right)}{\left(c a^2+d\right) \left(2 c a^2+2 i \sqrt{a^2 c d} x a\right)}\right)+\left(-\cos ^{-1}\left(\frac{d-a^2 c}{c a^2+d}\right)-2 \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right) \log \left(1-\frac{\left(c a^2-d+2 i \sqrt{a^2 c d}\right) \left(2 a^2 c-2 i a \sqrt{a^2 c d} x\right)}{\left(c a^2+d\right) \left(2 c a^2+2 i \sqrt{a^2 c d} x a\right)}\right)+\left(\cos ^{-1}\left(\frac{d-a^2 c}{c a^2+d}\right)+2 i \left(-i \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right)-i \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{-\tanh ^{-1}(a x)}}{\sqrt{c a^2+d} \sqrt{c a^2-d+\left(c a^2+d\right) \cosh \left(2 \tanh ^{-1}(a x)\right)}}\right)+\left(\cos ^{-1}\left(\frac{d-a^2 c}{c a^2+d}\right)-2 i \left(-i \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right)-i \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{\tanh ^{-1}(a x)}}{\sqrt{c a^2+d} \sqrt{c a^2-d+\left(c a^2+d\right) \cosh \left(2 \tanh ^{-1}(a x)\right)}}\right)+i \left(\text{Li}_2\left(\frac{\left(c a^2-d-2 i \sqrt{a^2 c d}\right) \left(2 a^2 c-2 i a \sqrt{a^2 c d} x\right)}{\left(c a^2+d\right) \left(2 c a^2+2 i \sqrt{a^2 c d} x a\right)}\right)-\text{Li}_2\left(\frac{\left(c a^2-d+2 i \sqrt{a^2 c d}\right) \left(2 a^2 c-2 i a \sqrt{a^2 c d} x\right)}{\left(c a^2+d\right) \left(2 c a^2+2 i \sqrt{a^2 c d} x a\right)}\right)\right)\right)}{32 a^2 c \sqrt{a^2 c d} \left(c a^2+d\right)}+\frac{d \tanh ^{-1}(a x) \sinh \left(2 \tanh ^{-1}(a x)\right)}{2 a^2 c \left(c a^2+d\right) \left(c a^2+c \cosh \left(2 \tanh ^{-1}(a x)\right) a^2-d+d \cosh \left(2 \tanh ^{-1}(a x)\right)\right)^2}+\frac{5 c^2 \tanh ^{-1}(a x) \sinh \left(2 \tanh ^{-1}(a x)\right) a^4+2 c d a^2+8 c d \tanh ^{-1}(a x) \sinh \left(2 \tanh ^{-1}(a x)\right) a^2+3 d^2 \tanh ^{-1}(a x) \sinh \left(2 \tanh ^{-1}(a x)\right)}{8 a^4 c^2 \left(c a^2+d\right)^2 \left(c a^2+c \cosh \left(2 \tanh ^{-1}(a x)\right) a^2-d+d \cosh \left(2 \tanh ^{-1}(a x)\right)\right)}\right)","\frac{a \left(5 a^2 c+3 d\right) \log \left(1-a^2 x^2\right)}{16 c^2 \left(a^2 c+d\right)^2}-\frac{a \left(5 a^2 c+3 d\right) \log \left(c+d x^2\right)}{16 c^2 \left(a^2 c+d\right)^2}+\frac{a}{8 c \left(a^2 c+d\right) \left(c+d x^2\right)}+\frac{3 i \text{Li}_2\left(\frac{a \left(\sqrt{c}-i \sqrt{d} x\right)}{a \sqrt{c}-i \sqrt{d}}\right)}{32 c^{5/2} \sqrt{d}}-\frac{3 i \text{Li}_2\left(\frac{a \left(\sqrt{c}-i \sqrt{d} x\right)}{\sqrt{c} a+i \sqrt{d}}\right)}{32 c^{5/2} \sqrt{d}}+\frac{3 i \text{Li}_2\left(\frac{a \left(i \sqrt{d} x+\sqrt{c}\right)}{a \sqrt{c}-i \sqrt{d}}\right)}{32 c^{5/2} \sqrt{d}}-\frac{3 i \text{Li}_2\left(\frac{a \left(i \sqrt{d} x+\sqrt{c}\right)}{\sqrt{c} a+i \sqrt{d}}\right)}{32 c^{5/2} \sqrt{d}}+\frac{3 i \log \left(1-\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(\frac{\sqrt{d} (1-a x)}{\sqrt{d}+i a \sqrt{c}}\right)}{32 c^{5/2} \sqrt{d}}-\frac{3 i \log \left(1-\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(-\frac{\sqrt{d} (a x+1)}{-\sqrt{d}+i a \sqrt{c}}\right)}{32 c^{5/2} \sqrt{d}}-\frac{3 i \log \left(1+\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(-\frac{\sqrt{d} (1-a x)}{-\sqrt{d}+i a \sqrt{c}}\right)}{32 c^{5/2} \sqrt{d}}+\frac{3 i \log \left(1+\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(\frac{\sqrt{d} (a x+1)}{\sqrt{d}+i a \sqrt{c}}\right)}{32 c^{5/2} \sqrt{d}}+\frac{3 \tanh ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} \sqrt{d}}+\frac{3 x \tanh ^{-1}(a x)}{8 c^2 \left(c+d x^2\right)}+\frac{x \tanh ^{-1}(a x)}{4 c \left(c+d x^2\right)^2}",1,"a^5*((-5*Log[1 - ((-(a^2*c) - d)*Cosh[2*ArcTanh[a*x]])/(a^2*c - d)])/(16*a^2*c*(a^2*c + d)^2) - (3*d*Log[1 - ((-(a^2*c) - d)*Cosh[2*ArcTanh[a*x]])/(a^2*c - d)])/(16*a^4*c^2*(a^2*c + d)^2) - (3*((-2*I)*ArcCos[(-(a^2*c) + d)/(a^2*c + d)]*ArcTan[(a*d*x)/Sqrt[a^2*c*d]] + 4*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)]*ArcTanh[a*x] - (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] - 2*ArcTan[(a*d*x)/Sqrt[a^2*c*d]])*Log[1 - ((a^2*c - d - (2*I)*Sqrt[a^2*c*d])*(2*a^2*c - (2*I)*a*Sqrt[a^2*c*d]*x))/((a^2*c + d)*(2*a^2*c + (2*I)*a*Sqrt[a^2*c*d]*x))] + (-ArcCos[(-(a^2*c) + d)/(a^2*c + d)] - 2*ArcTan[(a*d*x)/Sqrt[a^2*c*d]])*Log[1 - ((a^2*c - d + (2*I)*Sqrt[a^2*c*d])*(2*a^2*c - (2*I)*a*Sqrt[a^2*c*d]*x))/((a^2*c + d)*(2*a^2*c + (2*I)*a*Sqrt[a^2*c*d]*x))] + (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] + (2*I)*((-I)*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] - I*ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d])/(Sqrt[a^2*c + d]*E^ArcTanh[a*x]*Sqrt[a^2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a*x]]])] + (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] - (2*I)*((-I)*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] - I*ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d]*E^ArcTanh[a*x])/(Sqrt[a^2*c + d]*Sqrt[a^2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a*x]]])] + I*(PolyLog[2, ((a^2*c - d - (2*I)*Sqrt[a^2*c*d])*(2*a^2*c - (2*I)*a*Sqrt[a^2*c*d]*x))/((a^2*c + d)*(2*a^2*c + (2*I)*a*Sqrt[a^2*c*d]*x))] - PolyLog[2, ((a^2*c - d + (2*I)*Sqrt[a^2*c*d])*(2*a^2*c - (2*I)*a*Sqrt[a^2*c*d]*x))/((a^2*c + d)*(2*a^2*c + (2*I)*a*Sqrt[a^2*c*d]*x))])))/(32*a^2*c*Sqrt[a^2*c*d]*(a^2*c + d)) - (3*d*((-2*I)*ArcCos[(-(a^2*c) + d)/(a^2*c + d)]*ArcTan[(a*d*x)/Sqrt[a^2*c*d]] + 4*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)]*ArcTanh[a*x] - (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] - 2*ArcTan[(a*d*x)/Sqrt[a^2*c*d]])*Log[1 - ((a^2*c - d - (2*I)*Sqrt[a^2*c*d])*(2*a^2*c - (2*I)*a*Sqrt[a^2*c*d]*x))/((a^2*c + d)*(2*a^2*c + (2*I)*a*Sqrt[a^2*c*d]*x))] + (-ArcCos[(-(a^2*c) + d)/(a^2*c + d)] - 2*ArcTan[(a*d*x)/Sqrt[a^2*c*d]])*Log[1 - ((a^2*c - d + (2*I)*Sqrt[a^2*c*d])*(2*a^2*c - (2*I)*a*Sqrt[a^2*c*d]*x))/((a^2*c + d)*(2*a^2*c + (2*I)*a*Sqrt[a^2*c*d]*x))] + (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] + (2*I)*((-I)*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] - I*ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d])/(Sqrt[a^2*c + d]*E^ArcTanh[a*x]*Sqrt[a^2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a*x]]])] + (ArcCos[(-(a^2*c) + d)/(a^2*c + d)] - (2*I)*((-I)*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] - I*ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d]*E^ArcTanh[a*x])/(Sqrt[a^2*c + d]*Sqrt[a^2*c - d + (a^2*c + d)*Cosh[2*ArcTanh[a*x]]])] + I*(PolyLog[2, ((a^2*c - d - (2*I)*Sqrt[a^2*c*d])*(2*a^2*c - (2*I)*a*Sqrt[a^2*c*d]*x))/((a^2*c + d)*(2*a^2*c + (2*I)*a*Sqrt[a^2*c*d]*x))] - PolyLog[2, ((a^2*c - d + (2*I)*Sqrt[a^2*c*d])*(2*a^2*c - (2*I)*a*Sqrt[a^2*c*d]*x))/((a^2*c + d)*(2*a^2*c + (2*I)*a*Sqrt[a^2*c*d]*x))])))/(32*a^4*c^2*Sqrt[a^2*c*d]*(a^2*c + d)) + (d*ArcTanh[a*x]*Sinh[2*ArcTanh[a*x]])/(2*a^2*c*(a^2*c + d)*(a^2*c - d + a^2*c*Cosh[2*ArcTanh[a*x]] + d*Cosh[2*ArcTanh[a*x]])^2) + (2*a^2*c*d + 5*a^4*c^2*ArcTanh[a*x]*Sinh[2*ArcTanh[a*x]] + 8*a^2*c*d*ArcTanh[a*x]*Sinh[2*ArcTanh[a*x]] + 3*d^2*ArcTanh[a*x]*Sinh[2*ArcTanh[a*x]])/(8*a^4*c^2*(a^2*c + d)^2*(a^2*c - d + a^2*c*Cosh[2*ArcTanh[a*x]] + d*Cosh[2*ArcTanh[a*x]])))","B",0
505,1,17,17,0.058632,"\int \frac{1}{\left(a-a x^2\right) \left(b-2 b \tanh ^{-1}(x)\right)} \, dx","Integrate[1/((a - a*x^2)*(b - 2*b*ArcTanh[x])),x]","-\frac{\log \left(2 \tanh ^{-1}(x)-1\right)}{2 a b}","-\frac{\log \left(1-2 \tanh ^{-1}(x)\right)}{2 a b}",1,"-1/2*Log[-1 + 2*ArcTanh[x]]/(a*b)","A",1
506,1,576,171,0.9361808,"\int \frac{\tanh ^{-1}(b x)}{1-x^2} \, dx","Integrate[ArcTanh[b*x]/(1 - x^2),x]","-\frac{b \left(i \left(\text{Li}_2\left(\frac{\left(b^2-2 i \sqrt{-b^2}+1\right) \left(b-i \sqrt{-b^2} x\right)}{\left(b^2-1\right) \left(b+i \sqrt{-b^2} x\right)}\right)-\text{Li}_2\left(\frac{\left(b^2+2 i \sqrt{-b^2}+1\right) \left(b-i \sqrt{-b^2} x\right)}{\left(b^2-1\right) \left(b+i \sqrt{-b^2} x\right)}\right)\right)+2 i \cos ^{-1}\left(\frac{b^2+1}{1-b^2}\right) \tan ^{-1}\left(\frac{b x}{\sqrt{-b^2}}\right)-4 \tan ^{-1}\left(\frac{\sqrt{-b^2}}{b x}\right) \tanh ^{-1}(b x)-\log \left(\frac{2 b \left(\sqrt{-b^2}-i\right) (b x-1)}{\left(b^2-1\right) \left(\sqrt{-b^2} x-i b\right)}\right) \left(\cos ^{-1}\left(\frac{b^2+1}{1-b^2}\right)-2 \tan ^{-1}\left(\frac{b x}{\sqrt{-b^2}}\right)\right)-\log \left(\frac{2 b \left(\sqrt{-b^2}+i\right) (b x+1)}{\left(b^2-1\right) \left(\sqrt{-b^2} x-i b\right)}\right) \left(2 \tan ^{-1}\left(\frac{b x}{\sqrt{-b^2}}\right)+\cos ^{-1}\left(\frac{b^2+1}{1-b^2}\right)\right)+\left(\cos ^{-1}\left(\frac{b^2+1}{1-b^2}\right)-2 \left(\tan ^{-1}\left(\frac{\sqrt{-b^2}}{b x}\right)+\tan ^{-1}\left(\frac{b x}{\sqrt{-b^2}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-b^2} e^{-\tanh ^{-1}(b x)}}{\sqrt{b^2-1} \sqrt{\left(b^2-1\right) \cosh \left(2 \tanh ^{-1}(b x)\right)+b^2+1}}\right)+\left(2 \left(\tan ^{-1}\left(\frac{\sqrt{-b^2}}{b x}\right)+\tan ^{-1}\left(\frac{b x}{\sqrt{-b^2}}\right)\right)+\cos ^{-1}\left(\frac{b^2+1}{1-b^2}\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-b^2} e^{\tanh ^{-1}(b x)}}{\sqrt{b^2-1} \sqrt{\left(b^2-1\right) \cosh \left(2 \tanh ^{-1}(b x)\right)+b^2+1}}\right)\right)}{4 \sqrt{-b^2}}","\frac{1}{4} \text{Li}_2\left(\frac{1-b x}{1-b}\right)-\frac{1}{4} \text{Li}_2\left(\frac{1-b x}{b+1}\right)+\frac{1}{4} \text{Li}_2\left(\frac{b x+1}{1-b}\right)-\frac{1}{4} \text{Li}_2\left(\frac{b x+1}{b+1}\right)+\frac{1}{4} \log \left(-\frac{b (1-x)}{1-b}\right) \log (1-b x)-\frac{1}{4} \log \left(\frac{b (x+1)}{b+1}\right) \log (1-b x)-\frac{1}{4} \log \left(\frac{b (1-x)}{b+1}\right) \log (b x+1)+\frac{1}{4} \log \left(-\frac{b (x+1)}{1-b}\right) \log (b x+1)",1,"-1/4*(b*((2*I)*ArcCos[(1 + b^2)/(1 - b^2)]*ArcTan[(b*x)/Sqrt[-b^2]] - 4*ArcTan[Sqrt[-b^2]/(b*x)]*ArcTanh[b*x] - (ArcCos[(1 + b^2)/(1 - b^2)] - 2*ArcTan[(b*x)/Sqrt[-b^2]])*Log[(2*b*(-I + Sqrt[-b^2])*(-1 + b*x))/((-1 + b^2)*((-I)*b + Sqrt[-b^2]*x))] - (ArcCos[(1 + b^2)/(1 - b^2)] + 2*ArcTan[(b*x)/Sqrt[-b^2]])*Log[(2*b*(I + Sqrt[-b^2])*(1 + b*x))/((-1 + b^2)*((-I)*b + Sqrt[-b^2]*x))] + (ArcCos[(1 + b^2)/(1 - b^2)] - 2*(ArcTan[Sqrt[-b^2]/(b*x)] + ArcTan[(b*x)/Sqrt[-b^2]]))*Log[(Sqrt[2]*Sqrt[-b^2])/(Sqrt[-1 + b^2]*E^ArcTanh[b*x]*Sqrt[1 + b^2 + (-1 + b^2)*Cosh[2*ArcTanh[b*x]]])] + (ArcCos[(1 + b^2)/(1 - b^2)] + 2*(ArcTan[Sqrt[-b^2]/(b*x)] + ArcTan[(b*x)/Sqrt[-b^2]]))*Log[(Sqrt[2]*Sqrt[-b^2]*E^ArcTanh[b*x])/(Sqrt[-1 + b^2]*Sqrt[1 + b^2 + (-1 + b^2)*Cosh[2*ArcTanh[b*x]]])] + I*(PolyLog[2, ((1 + b^2 - (2*I)*Sqrt[-b^2])*(b - I*Sqrt[-b^2]*x))/((-1 + b^2)*(b + I*Sqrt[-b^2]*x))] - PolyLog[2, ((1 + b^2 + (2*I)*Sqrt[-b^2])*(b - I*Sqrt[-b^2]*x))/((-1 + b^2)*(b + I*Sqrt[-b^2]*x))])))/Sqrt[-b^2]","C",1
507,1,203,203,0.0418566,"\int \frac{\tanh ^{-1}(a+b x)}{1-x^2} \, dx","Integrate[ArcTanh[a + b*x]/(1 - x^2),x]","\frac{1}{4} \text{Li}_2\left(\frac{-a-b x+1}{-a-b+1}\right)-\frac{1}{4} \text{Li}_2\left(\frac{-a-b x+1}{-a+b+1}\right)+\frac{1}{4} \text{Li}_2\left(\frac{a+b x+1}{a-b+1}\right)-\frac{1}{4} \text{Li}_2\left(\frac{a+b x+1}{a+b+1}\right)+\frac{1}{4} \log \left(-\frac{b (1-x)}{-a-b+1}\right) \log (-a-b x+1)-\frac{1}{4} \log \left(\frac{b (x+1)}{-a+b+1}\right) \log (-a-b x+1)-\frac{1}{4} \log \left(\frac{b (1-x)}{a+b+1}\right) \log (a+b x+1)+\frac{1}{4} \log \left(-\frac{b (x+1)}{a-b+1}\right) \log (a+b x+1)","\frac{1}{4} \text{Li}_2\left(\frac{-a-b x+1}{-a-b+1}\right)-\frac{1}{4} \text{Li}_2\left(\frac{-a-b x+1}{-a+b+1}\right)+\frac{1}{4} \text{Li}_2\left(\frac{a+b x+1}{a-b+1}\right)-\frac{1}{4} \text{Li}_2\left(\frac{a+b x+1}{a+b+1}\right)+\frac{1}{4} \log \left(-\frac{b (1-x)}{-a-b+1}\right) \log (-a-b x+1)-\frac{1}{4} \log \left(\frac{b (x+1)}{-a+b+1}\right) \log (-a-b x+1)-\frac{1}{4} \log \left(\frac{b (1-x)}{a+b+1}\right) \log (a+b x+1)+\frac{1}{4} \log \left(-\frac{b (x+1)}{a-b+1}\right) \log (a+b x+1)",1,"(Log[-((b*(1 - x))/(1 - a - b))]*Log[1 - a - b*x])/4 - (Log[(b*(1 + x))/(1 - a + b)]*Log[1 - a - b*x])/4 - (Log[(b*(1 - x))/(1 + a + b)]*Log[1 + a + b*x])/4 + (Log[-((b*(1 + x))/(1 + a - b))]*Log[1 + a + b*x])/4 + PolyLog[2, (1 - a - b*x)/(1 - a - b)]/4 - PolyLog[2, (1 - a - b*x)/(1 - a + b)]/4 + PolyLog[2, (1 + a + b*x)/(1 + a - b)]/4 - PolyLog[2, (1 + a + b*x)/(1 + a + b)]/4","A",1
508,1,260,86,0.0912739,"\int \frac{\tanh ^{-1}(x)}{a+b x} \, dx","Integrate[ArcTanh[x]/(a + b*x),x]","\frac{-4 \text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{a}{b}\right)+\tanh ^{-1}(x)\right)}\right)+8 \tanh ^{-1}(x) \tanh ^{-1}\left(\frac{a}{b}\right)+8 \tanh ^{-1}\left(\frac{a}{b}\right) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{a}{b}\right)+\tanh ^{-1}(x)\right)}\right)+8 \tanh ^{-1}(x) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{a}{b}\right)+\tanh ^{-1}(x)\right)}\right)-8 \tanh ^{-1}\left(\frac{a}{b}\right) \log \left(2 i \sinh \left(\tanh ^{-1}\left(\frac{a}{b}\right)+\tanh ^{-1}(x)\right)\right)+8 \tanh ^{-1}(x) \log \left(i \sinh \left(\tanh ^{-1}\left(\frac{a}{b}\right)+\tanh ^{-1}(x)\right)\right)-8 \tanh ^{-1}(x) \log \left(2 i \sinh \left(\tanh ^{-1}\left(\frac{a}{b}\right)+\tanh ^{-1}(x)\right)\right)+4 \tanh ^{-1}\left(\frac{a}{b}\right)^2-4 \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+4 i \pi  \tanh ^{-1}(x)-8 \tanh ^{-1}(x) \log \left(e^{2 \tanh ^{-1}(x)}+1\right)-4 i \pi  \log \left(e^{2 \tanh ^{-1}(x)}+1\right)-\pi ^2}{8 b}","-\frac{\text{Li}_2\left(1-\frac{2 (a+b x)}{(a+b) (x+1)}\right)}{2 b}+\frac{\tanh ^{-1}(x) \log \left(\frac{2 (a+b x)}{(x+1) (a+b)}\right)}{b}+\frac{\text{Li}_2\left(1-\frac{2}{x+1}\right)}{2 b}-\frac{\log \left(\frac{2}{x+1}\right) \tanh ^{-1}(x)}{b}",1,"(-Pi^2 + 4*ArcTanh[a/b]^2 + (4*I)*Pi*ArcTanh[x] + 8*ArcTanh[a/b]*ArcTanh[x] + 8*ArcTanh[x]^2 - (4*I)*Pi*Log[1 + E^(2*ArcTanh[x])] - 8*ArcTanh[x]*Log[1 + E^(2*ArcTanh[x])] + 8*ArcTanh[a/b]*Log[1 - E^(-2*(ArcTanh[a/b] + ArcTanh[x]))] + 8*ArcTanh[x]*Log[1 - E^(-2*(ArcTanh[a/b] + 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[a/b] + ArcTanh[x]]] - 8*ArcTanh[a/b]*Log[(2*I)*Sinh[ArcTanh[a/b] + ArcTanh[x]]] - 8*ArcTanh[x]*Log[(2*I)*Sinh[ArcTanh[a/b] + ArcTanh[x]]] - 4*PolyLog[2, -E^(2*ArcTanh[x])] - 4*PolyLog[2, E^(-2*(ArcTanh[a/b] + ArcTanh[x]))])/(8*b)","C",0
509,1,485,397,1.134746,"\int \frac{\tanh ^{-1}(x)}{a+b x^2} \, dx","Integrate[ArcTanh[x]/(a + b*x^2),x]","-\frac{i \left(\text{Li}_2\left(\frac{\left(-a+b+2 i \sqrt{a b}\right) \left(i a+\sqrt{a b} x\right)}{(a+b) \left(\sqrt{a b} x-i a\right)}\right)-\text{Li}_2\left(\frac{\left(-a+b-2 i \sqrt{a b}\right) \left(i a+\sqrt{a b} x\right)}{(a+b) \left(\sqrt{a b} x-i a\right)}\right)\right)-2 i \cos ^{-1}\left(\frac{b-a}{a+b}\right) \tan ^{-1}\left(\frac{b x}{\sqrt{a b}}\right)+4 \tanh ^{-1}(x) \tan ^{-1}\left(\frac{a}{x \sqrt{a b}}\right)-\log \left(\frac{2 i a (x-1) \left(\sqrt{a b}+i b\right)}{(a+b) \left(a+i x \sqrt{a b}\right)}\right) \left(2 \tan ^{-1}\left(\frac{b x}{\sqrt{a b}}\right)+\cos ^{-1}\left(\frac{b-a}{a+b}\right)\right)-\log \left(\frac{2 a (x+1) \left(b+i \sqrt{a b}\right)}{(a+b) \left(a+i x \sqrt{a b}\right)}\right) \left(\cos ^{-1}\left(\frac{b-a}{a+b}\right)-2 \tan ^{-1}\left(\frac{b x}{\sqrt{a b}}\right)\right)+\left(2 \left(\tan ^{-1}\left(\frac{a}{x \sqrt{a b}}\right)+\tan ^{-1}\left(\frac{b x}{\sqrt{a b}}\right)\right)+\cos ^{-1}\left(\frac{b-a}{a+b}\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a b} e^{-\tanh ^{-1}(x)}}{\sqrt{a+b} \sqrt{(a+b) \cosh \left(2 \tanh ^{-1}(x)\right)+a-b}}\right)+\left(\cos ^{-1}\left(\frac{b-a}{a+b}\right)-2 \left(\tan ^{-1}\left(\frac{a}{x \sqrt{a b}}\right)+\tan ^{-1}\left(\frac{b x}{\sqrt{a b}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a b} e^{\tanh ^{-1}(x)}}{\sqrt{a+b} \sqrt{(a+b) \cosh \left(2 \tanh ^{-1}(x)\right)+a-b}}\right)}{4 \sqrt{a b}}","-\frac{\text{Li}_2\left(-\frac{\sqrt{b} (1-x)}{\sqrt{-a}-\sqrt{b}}\right)}{4 \sqrt{-a} \sqrt{b}}+\frac{\text{Li}_2\left(\frac{\sqrt{b} (1-x)}{\sqrt{-a}+\sqrt{b}}\right)}{4 \sqrt{-a} \sqrt{b}}-\frac{\text{Li}_2\left(-\frac{\sqrt{b} (x+1)}{\sqrt{-a}-\sqrt{b}}\right)}{4 \sqrt{-a} \sqrt{b}}+\frac{\text{Li}_2\left(\frac{\sqrt{b} (x+1)}{\sqrt{-a}+\sqrt{b}}\right)}{4 \sqrt{-a} \sqrt{b}}-\frac{\log (1-x) \log \left(\frac{\sqrt{-a}-\sqrt{b} x}{\sqrt{-a}-\sqrt{b}}\right)}{4 \sqrt{-a} \sqrt{b}}+\frac{\log (x+1) \log \left(\frac{\sqrt{-a}-\sqrt{b} x}{\sqrt{-a}+\sqrt{b}}\right)}{4 \sqrt{-a} \sqrt{b}}-\frac{\log (x+1) \log \left(\frac{\sqrt{-a}+\sqrt{b} x}{\sqrt{-a}-\sqrt{b}}\right)}{4 \sqrt{-a} \sqrt{b}}+\frac{\log (1-x) \log \left(\frac{\sqrt{-a}+\sqrt{b} x}{\sqrt{-a}+\sqrt{b}}\right)}{4 \sqrt{-a} \sqrt{b}}",1,"-1/4*((-2*I)*ArcCos[(-a + b)/(a + b)]*ArcTan[(b*x)/Sqrt[a*b]] + 4*ArcTan[a/(Sqrt[a*b]*x)]*ArcTanh[x] - (ArcCos[(-a + b)/(a + b)] + 2*ArcTan[(b*x)/Sqrt[a*b]])*Log[((2*I)*a*(I*b + Sqrt[a*b])*(-1 + x))/((a + b)*(a + I*Sqrt[a*b]*x))] - (ArcCos[(-a + b)/(a + b)] - 2*ArcTan[(b*x)/Sqrt[a*b]])*Log[(2*a*(b + I*Sqrt[a*b])*(1 + x))/((a + b)*(a + I*Sqrt[a*b]*x))] + (ArcCos[(-a + b)/(a + b)] + 2*(ArcTan[a/(Sqrt[a*b]*x)] + ArcTan[(b*x)/Sqrt[a*b]]))*Log[(Sqrt[2]*Sqrt[a*b])/(Sqrt[a + b]*E^ArcTanh[x]*Sqrt[a - b + (a + b)*Cosh[2*ArcTanh[x]]])] + (ArcCos[(-a + b)/(a + b)] - 2*(ArcTan[a/(Sqrt[a*b]*x)] + ArcTan[(b*x)/Sqrt[a*b]]))*Log[(Sqrt[2]*Sqrt[a*b]*E^ArcTanh[x])/(Sqrt[a + b]*Sqrt[a - b + (a + b)*Cosh[2*ArcTanh[x]]])] + I*(-PolyLog[2, ((-a + b - (2*I)*Sqrt[a*b])*(I*a + Sqrt[a*b]*x))/((a + b)*((-I)*a + Sqrt[a*b]*x))] + PolyLog[2, ((-a + b + (2*I)*Sqrt[a*b])*(I*a + Sqrt[a*b]*x))/((a + b)*((-I)*a + Sqrt[a*b]*x))]))/Sqrt[a*b]","C",1
510,1,874,258,19.3211314,"\int \frac{\tanh ^{-1}(x)}{a+b x+c x^2} \, dx","Integrate[ArcTanh[x]/(a + b*x + c*x^2),x]","\frac{\frac{2 \sqrt{4 a c-b^2} \left(b \left(\sqrt{\frac{c (a+b+c)}{4 a c-b^2}} e^{i \tan ^{-1}\left(\frac{-b-2 c}{\sqrt{4 a c-b^2}}\right)}-\sqrt{\frac{c (a-b+c)}{4 a c-b^2}} e^{i \tan ^{-1}\left(\frac{2 c-b}{\sqrt{4 a c-b^2}}\right)}\right)-2 c \left(e^{i \tan ^{-1}\left(\frac{2 c-b}{\sqrt{4 a c-b^2}}\right)} \sqrt{\frac{c (a-b+c)}{4 a c-b^2}}+\sqrt{\frac{c (a+b+c)}{4 a c-b^2}} e^{i \tan ^{-1}\left(\frac{-b-2 c}{\sqrt{4 a c-b^2}}\right)}-1\right)\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)^2}{b^2-4 c^2}+2 \left(-i \tan ^{-1}\left(\frac{-b-2 c}{\sqrt{4 a c-b^2}}\right)+i \tan ^{-1}\left(\frac{2 c-b}{\sqrt{4 a c-b^2}}\right)+2 \tanh ^{-1}(x)+\log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{-b-2 c}{\sqrt{4 a c-b^2}}\right)+\tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)\right)}\right)-\log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{2 c-b}{\sqrt{4 a c-b^2}}\right)+\tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)\right)}\right)\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)+2 \left(\tan ^{-1}\left(\frac{-b-2 c}{\sqrt{4 a c-b^2}}\right) \left(\log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{-b-2 c}{\sqrt{4 a c-b^2}}\right)+\tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)\right)}\right)-\log \left(\sin \left(\tan ^{-1}\left(\frac{-b-2 c}{\sqrt{4 a c-b^2}}\right)+\tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)\right)\right)\right)+\tan ^{-1}\left(\frac{2 c-b}{\sqrt{4 a c-b^2}}\right) \left(\log \left(\sin \left(\tan ^{-1}\left(\frac{2 c-b}{\sqrt{4 a c-b^2}}\right)+\tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)\right)\right)-\log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{2 c-b}{\sqrt{4 a c-b^2}}\right)+\tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)\right)}\right)\right)\right)-i \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{-b-2 c}{\sqrt{4 a c-b^2}}\right)+\tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)\right)}\right)+i \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{2 c-b}{\sqrt{4 a c-b^2}}\right)+\tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)\right)}\right)}{2 \sqrt{4 a c-b^2}}","-\frac{\text{Li}_2\left(1-\frac{2 \left(b+2 c x-\sqrt{b^2-4 a c}\right)}{\left(b+2 c-\sqrt{b^2-4 a c}\right) (x+1)}\right)}{2 \sqrt{b^2-4 a c}}+\frac{\text{Li}_2\left(1-\frac{2 \left(b+2 c x+\sqrt{b^2-4 a c}\right)}{\left(b+2 c+\sqrt{b^2-4 a c}\right) (x+1)}\right)}{2 \sqrt{b^2-4 a c}}+\frac{\tanh ^{-1}(x) \log \left(\frac{2 \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{(x+1) \left(-\sqrt{b^2-4 a c}+b+2 c\right)}\right)}{\sqrt{b^2-4 a c}}-\frac{\tanh ^{-1}(x) \log \left(\frac{2 \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{(x+1) \left(\sqrt{b^2-4 a c}+b+2 c\right)}\right)}{\sqrt{b^2-4 a c}}",1,"((2*Sqrt[-b^2 + 4*a*c]*(b*(Sqrt[(c*(a + b + c))/(-b^2 + 4*a*c)]*E^(I*ArcTan[(-b - 2*c)/Sqrt[-b^2 + 4*a*c]]) - Sqrt[(c*(a - b + c))/(-b^2 + 4*a*c)]*E^(I*ArcTan[(-b + 2*c)/Sqrt[-b^2 + 4*a*c]])) - 2*c*(-1 + Sqrt[(c*(a + b + c))/(-b^2 + 4*a*c)]*E^(I*ArcTan[(-b - 2*c)/Sqrt[-b^2 + 4*a*c]]) + Sqrt[(c*(a - b + c))/(-b^2 + 4*a*c)]*E^(I*ArcTan[(-b + 2*c)/Sqrt[-b^2 + 4*a*c]])))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]]^2)/(b^2 - 4*c^2) + 2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]]*((-I)*ArcTan[(-b - 2*c)/Sqrt[-b^2 + 4*a*c]] + I*ArcTan[(-b + 2*c)/Sqrt[-b^2 + 4*a*c]] + 2*ArcTanh[x] + Log[1 - E^((2*I)*(ArcTan[(-b - 2*c)/Sqrt[-b^2 + 4*a*c]] + ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]]))] - Log[1 - E^((2*I)*(ArcTan[(-b + 2*c)/Sqrt[-b^2 + 4*a*c]] + ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]]))]) + 2*(ArcTan[(-b - 2*c)/Sqrt[-b^2 + 4*a*c]]*(Log[1 - E^((2*I)*(ArcTan[(-b - 2*c)/Sqrt[-b^2 + 4*a*c]] + ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]]))] - Log[Sin[ArcTan[(-b - 2*c)/Sqrt[-b^2 + 4*a*c]] + ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]]]]) + ArcTan[(-b + 2*c)/Sqrt[-b^2 + 4*a*c]]*(-Log[1 - E^((2*I)*(ArcTan[(-b + 2*c)/Sqrt[-b^2 + 4*a*c]] + ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]]))] + Log[Sin[ArcTan[(-b + 2*c)/Sqrt[-b^2 + 4*a*c]] + ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]]]])) - I*PolyLog[2, E^((2*I)*(ArcTan[(-b - 2*c)/Sqrt[-b^2 + 4*a*c]] + ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]]))] + I*PolyLog[2, E^((2*I)*(ArcTan[(-b + 2*c)/Sqrt[-b^2 + 4*a*c]] + ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]]))])/(2*Sqrt[-b^2 + 4*a*c])","C",0
511,0,0,19,4.2547315,"\int \sqrt{c+d x^2} \tanh ^{-1}(a x) \, dx","Integrate[Sqrt[c + d*x^2]*ArcTanh[a*x],x]","\int \sqrt{c+d x^2} \tanh ^{-1}(a x) \, dx","\text{Int}\left(\tanh ^{-1}(a x) \sqrt{c+d x^2},x\right)",0,"Integrate[Sqrt[c + d*x^2]*ArcTanh[a*x], x]","A",-1
512,0,0,19,4.1020701,"\int \frac{\tanh ^{-1}(a x)}{\sqrt{c+d x^2}} \, dx","Integrate[ArcTanh[a*x]/Sqrt[c + d*x^2],x]","\int \frac{\tanh ^{-1}(a x)}{\sqrt{c+d x^2}} \, dx","\text{Int}\left(\frac{\tanh ^{-1}(a x)}{\sqrt{c+d x^2}},x\right)",0,"Integrate[ArcTanh[a*x]/Sqrt[c + d*x^2], x]","A",-1
513,1,119,62,0.1212372,"\int \frac{\tanh ^{-1}(a x)}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[ArcTanh[a*x]/(c + d*x^2)^(3/2),x]","\frac{\frac{-\log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c-d x\right)-\log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c+d x\right)+\log (1-a x)+\log (a x+1)}{\sqrt{a^2 c+d}}+\frac{2 x \tanh ^{-1}(a x)}{\sqrt{c+d x^2}}}{2 c}","\frac{x \tanh ^{-1}(a x)}{c \sqrt{c+d x^2}}-\frac{\tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c+d}}\right)}{c \sqrt{a^2 c+d}}",1,"((2*x*ArcTanh[a*x])/Sqrt[c + d*x^2] + (Log[1 - a*x] + Log[1 + a*x] - Log[a*c - d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]] - Log[a*c + d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]])/Sqrt[a^2*c + d])/(2*c)","A",1
514,1,226,128,0.3111995,"\int \frac{\tanh ^{-1}(a x)}{\left(c+d x^2\right)^{5/2}} \, dx","Integrate[ArcTanh[a*x]/(c + d*x^2)^(5/2),x]","\frac{\frac{2 a c}{\left(a^2 c+d\right) \sqrt{c+d x^2}}-\frac{\left(3 a^2 c+2 d\right) \log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c-d x\right)}{\left(a^2 c+d\right)^{3/2}}-\frac{\left(3 a^2 c+2 d\right) \log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c+d x\right)}{\left(a^2 c+d\right)^{3/2}}+\frac{\left(3 a^2 c+2 d\right) \log (1-a x)}{\left(a^2 c+d\right)^{3/2}}+\frac{\left(3 a^2 c+2 d\right) \log (a x+1)}{\left(a^2 c+d\right)^{3/2}}+\frac{2 x \tanh ^{-1}(a x) \left(3 c+2 d x^2\right)}{\left(c+d x^2\right)^{3/2}}}{6 c^2}","-\frac{\left(3 a^2 c+2 d\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c+d}}\right)}{3 c^2 \left(a^2 c+d\right)^{3/2}}+\frac{a}{3 c \left(a^2 c+d\right) \sqrt{c+d x^2}}+\frac{2 x \tanh ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{x \tanh ^{-1}(a x)}{3 c \left(c+d x^2\right)^{3/2}}",1,"((2*a*c)/((a^2*c + d)*Sqrt[c + d*x^2]) + (2*x*(3*c + 2*d*x^2)*ArcTanh[a*x])/(c + d*x^2)^(3/2) + ((3*a^2*c + 2*d)*Log[1 - a*x])/(a^2*c + d)^(3/2) + ((3*a^2*c + 2*d)*Log[1 + a*x])/(a^2*c + d)^(3/2) - ((3*a^2*c + 2*d)*Log[a*c - d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]])/(a^2*c + d)^(3/2) - ((3*a^2*c + 2*d)*Log[a*c + d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]])/(a^2*c + d)^(3/2))/(6*c^2)","A",1
515,1,329,200,0.5907686,"\int \frac{\tanh ^{-1}(a x)}{\left(c+d x^2\right)^{7/2}} \, dx","Integrate[ArcTanh[a*x]/(c + d*x^2)^(7/2),x]","\frac{2 x \left(a^2 c+d\right)^{5/2} \tanh ^{-1}(a x) \left(15 c^2+20 c d x^2+8 d^2 x^4\right)+2 a c \sqrt{a^2 c+d} \left(c+d x^2\right) \left(a^2 c \left(8 c+7 d x^2\right)+d \left(5 c+4 d x^2\right)\right)+\left(15 a^4 c^2+20 a^2 c d+8 d^2\right) \log (1-a x) \left(c+d x^2\right)^{5/2}+\left(15 a^4 c^2+20 a^2 c d+8 d^2\right) \log (a x+1) \left(c+d x^2\right)^{5/2}-\left(15 a^4 c^2+20 a^2 c d+8 d^2\right) \left(c+d x^2\right)^{5/2} \log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c-d x\right)-\left(15 a^4 c^2+20 a^2 c d+8 d^2\right) \left(c+d x^2\right)^{5/2} \log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c+d x\right)}{30 c^3 \left(a^2 c+d\right)^{5/2} \left(c+d x^2\right)^{5/2}}","\frac{a \left(7 a^2 c+4 d\right)}{15 c^2 \left(a^2 c+d\right)^2 \sqrt{c+d x^2}}+\frac{a}{15 c \left(a^2 c+d\right) \left(c+d x^2\right)^{3/2}}-\frac{\left(15 a^4 c^2+20 a^2 c d+8 d^2\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c+d}}\right)}{15 c^3 \left(a^2 c+d\right)^{5/2}}+\frac{8 x \tanh ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}+\frac{4 x \tanh ^{-1}(a x)}{15 c^2 \left(c+d x^2\right)^{3/2}}+\frac{x \tanh ^{-1}(a x)}{5 c \left(c+d x^2\right)^{5/2}}",1,"(2*a*c*Sqrt[a^2*c + d]*(c + d*x^2)*(d*(5*c + 4*d*x^2) + a^2*c*(8*c + 7*d*x^2)) + 2*(a^2*c + d)^(5/2)*x*(15*c^2 + 20*c*d*x^2 + 8*d^2*x^4)*ArcTanh[a*x] + (15*a^4*c^2 + 20*a^2*c*d + 8*d^2)*(c + d*x^2)^(5/2)*Log[1 - a*x] + (15*a^4*c^2 + 20*a^2*c*d + 8*d^2)*(c + d*x^2)^(5/2)*Log[1 + a*x] - (15*a^4*c^2 + 20*a^2*c*d + 8*d^2)*(c + d*x^2)^(5/2)*Log[a*c - d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]] - (15*a^4*c^2 + 20*a^2*c*d + 8*d^2)*(c + d*x^2)^(5/2)*Log[a*c + d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]])/(30*c^3*(a^2*c + d)^(5/2)*(c + d*x^2)^(5/2))","A",1
516,1,431,283,0.9509065,"\int \frac{\tanh ^{-1}(a x)}{\left(c+d x^2\right)^{9/2}} \, dx","Integrate[ArcTanh[a*x]/(c + d*x^2)^(9/2),x]","\frac{6 x \left(a^2 c+d\right)^{7/2} \tanh ^{-1}(a x) \left(35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right)+2 a c \sqrt{a^2 c+d} \left(c+d x^2\right) \left(3 c^2 \left(a^2 c+d\right)^2+c \left(11 a^2 c+6 d\right) \left(a^2 c+d\right) \left(c+d x^2\right)+3 \left(19 a^4 c^2+22 a^2 c d+8 d^2\right) \left(c+d x^2\right)^2\right)+3 \left(35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right) \log (1-a x) \left(c+d x^2\right)^{7/2}+3 \left(35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right) \log (a x+1) \left(c+d x^2\right)^{7/2}-3 \left(35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right) \left(c+d x^2\right)^{7/2} \log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c-d x\right)-3 \left(35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right) \left(c+d x^2\right)^{7/2} \log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c+d x\right)}{210 c^4 \left(a^2 c+d\right)^{7/2} \left(c+d x^2\right)^{7/2}}","\frac{a \left(11 a^2 c+6 d\right)}{105 c^2 \left(a^2 c+d\right)^2 \left(c+d x^2\right)^{3/2}}+\frac{a}{35 c \left(a^2 c+d\right) \left(c+d x^2\right)^{5/2}}+\frac{a \left(19 a^4 c^2+22 a^2 c d+8 d^2\right)}{35 c^3 \left(a^2 c+d\right)^3 \sqrt{c+d x^2}}-\frac{\left(35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c+d}}\right)}{35 c^4 \left(a^2 c+d\right)^{7/2}}+\frac{16 x \tanh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{8 x \tanh ^{-1}(a x)}{35 c^3 \left(c+d x^2\right)^{3/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 c^2 \left(c+d x^2\right)^{5/2}}+\frac{x \tanh ^{-1}(a x)}{7 c \left(c+d x^2\right)^{7/2}}",1,"(2*a*c*Sqrt[a^2*c + d]*(c + d*x^2)*(3*c^2*(a^2*c + d)^2 + c*(a^2*c + d)*(11*a^2*c + 6*d)*(c + d*x^2) + 3*(19*a^4*c^2 + 22*a^2*c*d + 8*d^2)*(c + d*x^2)^2) + 6*(a^2*c + d)^(7/2)*x*(35*c^3 + 70*c^2*d*x^2 + 56*c*d^2*x^4 + 16*d^3*x^6)*ArcTanh[a*x] + 3*(35*a^6*c^3 + 70*a^4*c^2*d + 56*a^2*c*d^2 + 16*d^3)*(c + d*x^2)^(7/2)*Log[1 - a*x] + 3*(35*a^6*c^3 + 70*a^4*c^2*d + 56*a^2*c*d^2 + 16*d^3)*(c + d*x^2)^(7/2)*Log[1 + a*x] - 3*(35*a^6*c^3 + 70*a^4*c^2*d + 56*a^2*c*d^2 + 16*d^3)*(c + d*x^2)^(7/2)*Log[a*c - d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]] - 3*(35*a^6*c^3 + 70*a^4*c^2*d + 56*a^2*c*d^2 + 16*d^3)*(c + d*x^2)^(7/2)*Log[a*c + d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]])/(210*c^4*(a^2*c + d)^(7/2)*(c + d*x^2)^(7/2))","A",1
517,1,97,186,0.3214106,"\int \sqrt{a-a x^2} \tanh ^{-1}(x) \, dx","Integrate[Sqrt[a - a*x^2]*ArcTanh[x],x]","\frac{1}{2} \sqrt{a \left(1-x^2\right)} \left(-\frac{i \left(\text{Li}_2\left(-i e^{-\tanh ^{-1}(x)}\right)-\text{Li}_2\left(i e^{-\tanh ^{-1}(x)}\right)+\tanh ^{-1}(x) \left(\log \left(1-i e^{-\tanh ^{-1}(x)}\right)-\log \left(1+i e^{-\tanh ^{-1}(x)}\right)\right)\right)}{\sqrt{1-x^2}}+x \tanh ^{-1}(x)+1\right)","-\frac{i a \sqrt{1-x^2} \text{Li}_2\left(-\frac{i \sqrt{1-x}}{\sqrt{x+1}}\right)}{2 \sqrt{a-a x^2}}+\frac{i a \sqrt{1-x^2} \text{Li}_2\left(\frac{i \sqrt{1-x}}{\sqrt{x+1}}\right)}{2 \sqrt{a-a x^2}}+\frac{1}{2} \sqrt{a-a x^2}+\frac{1}{2} x \sqrt{a-a x^2} \tanh ^{-1}(x)-\frac{a \sqrt{1-x^2} \tan ^{-1}\left(\frac{\sqrt{1-x}}{\sqrt{x+1}}\right) \tanh ^{-1}(x)}{\sqrt{a-a x^2}}",1,"(Sqrt[a*(1 - x^2)]*(1 + x*ArcTanh[x] - (I*(ArcTanh[x]*(Log[1 - I/E^ArcTanh[x]] - Log[1 + I/E^ArcTanh[x]]) + PolyLog[2, (-I)/E^ArcTanh[x]] - PolyLog[2, I/E^ArcTanh[x]]))/Sqrt[1 - x^2]))/2","A",0
518,1,90,144,0.1066972,"\int \frac{\tanh ^{-1}(x)}{\sqrt{a-a x^2}} \, dx","Integrate[ArcTanh[x]/Sqrt[a - a*x^2],x]","-\frac{i \sqrt{a \left(1-x^2\right)} \left(\text{Li}_2\left(-i e^{-\tanh ^{-1}(x)}\right)-\text{Li}_2\left(i e^{-\tanh ^{-1}(x)}\right)+\tanh ^{-1}(x) \left(\log \left(1-i e^{-\tanh ^{-1}(x)}\right)-\log \left(1+i e^{-\tanh ^{-1}(x)}\right)\right)\right)}{a \sqrt{1-x^2}}","-\frac{i \sqrt{1-x^2} \text{Li}_2\left(-\frac{i \sqrt{1-x}}{\sqrt{x+1}}\right)}{\sqrt{a-a x^2}}+\frac{i \sqrt{1-x^2} \text{Li}_2\left(\frac{i \sqrt{1-x}}{\sqrt{x+1}}\right)}{\sqrt{a-a x^2}}-\frac{2 \sqrt{1-x^2} \tan ^{-1}\left(\frac{\sqrt{1-x}}{\sqrt{x+1}}\right) \tanh ^{-1}(x)}{\sqrt{a-a x^2}}",1,"((-I)*Sqrt[a*(1 - x^2)]*(ArcTanh[x]*(Log[1 - I/E^ArcTanh[x]] - Log[1 + I/E^ArcTanh[x]]) + PolyLog[2, (-I)/E^ArcTanh[x]] - PolyLog[2, I/E^ArcTanh[x]]))/(a*Sqrt[1 - x^2])","A",0
519,1,30,37,0.0454863,"\int \frac{\tanh ^{-1}(x)}{\left(a-a x^2\right)^{3/2}} \, dx","Integrate[ArcTanh[x]/(a - a*x^2)^(3/2),x]","\frac{\sqrt{a-a x^2} \left(1-x \tanh ^{-1}(x)\right)}{a^2 \left(x^2-1\right)}","\frac{x \tanh ^{-1}(x)}{a \sqrt{a-a x^2}}-\frac{1}{a \sqrt{a-a x^2}}",1,"(Sqrt[a - a*x^2]*(1 - x*ArcTanh[x]))/(a^2*(-1 + x^2))","A",1
520,1,45,83,0.0525887,"\int \frac{\tanh ^{-1}(x)}{\left(a-a x^2\right)^{5/2}} \, dx","Integrate[ArcTanh[x]/(a - a*x^2)^(5/2),x]","-\frac{\sqrt{a-a x^2} \left(\left(6 x^3-9 x\right) \tanh ^{-1}(x)-6 x^2+7\right)}{9 a^3 \left(x^2-1\right)^2}","-\frac{2}{3 a^2 \sqrt{a-a x^2}}+\frac{2 x \tanh ^{-1}(x)}{3 a^2 \sqrt{a-a x^2}}-\frac{1}{9 a \left(a-a x^2\right)^{3/2}}+\frac{x \tanh ^{-1}(x)}{3 a \left(a-a x^2\right)^{3/2}}",1,"-1/9*(Sqrt[a - a*x^2]*(7 - 6*x^2 + (-9*x + 6*x^3)*ArcTanh[x]))/(a^3*(-1 + x^2)^2)","A",1
521,1,55,124,0.0632509,"\int \frac{\tanh ^{-1}(x)}{\left(a-a x^2\right)^{7/2}} \, dx","Integrate[ArcTanh[x]/(a - a*x^2)^(7/2),x]","\frac{\sqrt{a-a x^2} \left(120 x^4-260 x^2-15 \left(8 x^4-20 x^2+15\right) x \tanh ^{-1}(x)+149\right)}{225 a^4 \left(x^2-1\right)^3}","-\frac{8}{15 a^3 \sqrt{a-a x^2}}+\frac{8 x \tanh ^{-1}(x)}{15 a^3 \sqrt{a-a x^2}}-\frac{4}{45 a^2 \left(a-a x^2\right)^{3/2}}+\frac{4 x \tanh ^{-1}(x)}{15 a^2 \left(a-a x^2\right)^{3/2}}-\frac{1}{25 a \left(a-a x^2\right)^{5/2}}+\frac{x \tanh ^{-1}(x)}{5 a \left(a-a x^2\right)^{5/2}}",1,"(Sqrt[a - a*x^2]*(149 - 260*x^2 + 120*x^4 - 15*x*(15 - 20*x^2 + 8*x^4)*ArcTanh[x]))/(225*a^4*(-1 + x^2)^3)","A",1
522,1,236,315,0.167652,"\int x^4 \left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right) \, dx","Integrate[x^4*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]),x]","\frac{30 c^2 e x^2 \log \left(1-c^2 x^2\right) \left(4 a c^3 x^3+4 b c^3 x^3 \tanh ^{-1}(c x)+b \left(c^2 x^2+2\right)\right)+2 \log (1-c x) (-60 a e+30 b d-137 b e)+2 \log (c x+1) (60 a e+30 b d-137 b e)+24 a c^5 x^5 (5 d-2 e)-80 a c^3 e x^3-240 a c e x+3 b c^4 x^4 (10 d-9 e)+2 b c^2 x^2 (30 d-77 e)+30 b e \log ^2\left(1-c^2 x^2\right)-8 b c x \tanh ^{-1}(c x) \left(2 e \left(3 c^4 x^4+5 c^2 x^2+15\right)-15 c^4 d x^4\right)+120 b e \tanh ^{-1}(c x)^2}{600 c^5}","-\frac{e (4 a+3 b) \log (1-c x)}{20 c^5}+\frac{e (4 a-3 b) \log (c x+1)}{20 c^5}+\frac{1}{5} x^5 \left(a+b \tanh ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)-\frac{2 a e x}{5 c^4}-\frac{2 a e x^3}{15 c^2}-\frac{2}{25} a e x^5+\frac{b e \tanh ^{-1}(c x)^2}{5 c^5}-\frac{2 b e x \tanh ^{-1}(c x)}{5 c^4}-\frac{77 b e x^2}{300 c^3}+\frac{b x^4 \left(e \log \left(1-c^2 x^2\right)+d\right)}{20 c}-\frac{2 b e x^3 \tanh ^{-1}(c x)}{15 c^2}+\frac{b \log \left(1-c^2 x^2\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{10 c^5}-\frac{b e \log ^2\left(1-c^2 x^2\right)}{20 c^5}-\frac{23 b e \log \left(1-c^2 x^2\right)}{75 c^5}+\frac{b x^2 \left(e \log \left(1-c^2 x^2\right)+d\right)}{10 c^3}-\frac{2}{25} b e x^5 \tanh ^{-1}(c x)-\frac{9 b e x^4}{200 c}",1,"(-240*a*c*e*x + 2*b*c^2*(30*d - 77*e)*x^2 - 80*a*c^3*e*x^3 + 3*b*c^4*(10*d - 9*e)*x^4 + 24*a*c^5*(5*d - 2*e)*x^5 - 8*b*c*x*(-15*c^4*d*x^4 + 2*e*(15 + 5*c^2*x^2 + 3*c^4*x^4))*ArcTanh[c*x] + 120*b*e*ArcTanh[c*x]^2 + 2*(30*b*d - 60*a*e - 137*b*e)*Log[1 - c*x] + 2*(30*b*d + 60*a*e - 137*b*e)*Log[1 + c*x] + 30*c^2*e*x^2*(4*a*c^3*x^3 + b*(2 + c^2*x^2) + 4*b*c^3*x^3*ArcTanh[c*x])*Log[1 - c^2*x^2] + 30*b*e*Log[1 - c^2*x^2]^2)/(600*c^5)","A",1
523,1,192,225,0.1602359,"\int x^3 \left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right) \, dx","Integrate[x^3*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]),x]","\frac{12 e \log \left(1-c^2 x^2\right) \left(3 a c^4 x^4+3 b \left(c^4 x^4-1\right) \tanh ^{-1}(c x)+b c x \left(c^2 x^2+3\right)\right)+3 \log (1-c x) (-12 a e+6 b d-25 b e)-3 \log (c x+1) (12 a e+6 b d-25 b e)+18 a c^4 x^4 (2 d-e)-36 a c^2 e x^2+2 b c^3 x^3 (6 d-7 e)-18 b c^2 x^2 \tanh ^{-1}(c x) \left(e \left(c^2 x^2+2\right)-2 c^2 d x^2\right)+6 b c x (6 d-25 e)}{144 c^4}","\frac{1}{4} x^4 \left(a+b \tanh ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)-\frac{e x^2 \left(a+b \tanh ^{-1}(c x)\right)}{4 c^2}-\frac{e \log \left(1-c^2 x^2\right) \left(a+b \tanh ^{-1}(c x)\right)}{4 c^4}-\frac{1}{8} e x^4 \left(a+b \tanh ^{-1}(c x)\right)-\frac{b (2 d-3 e) \tanh ^{-1}(c x)}{8 c^4}+\frac{2 b e \tanh ^{-1}(c x)}{3 c^4}+\frac{b x (2 d-3 e)}{8 c^3}-\frac{2 b e x}{3 c^3}+\frac{b e x^3 \log \left(1-c^2 x^2\right)}{12 c}+\frac{b e x \log \left(1-c^2 x^2\right)}{4 c^3}+\frac{b x^3 (2 d-e)}{24 c}-\frac{b e x^3}{18 c}",1,"(6*b*c*(6*d - 25*e)*x - 36*a*c^2*e*x^2 + 2*b*c^3*(6*d - 7*e)*x^3 + 18*a*c^4*(2*d - e)*x^4 - 18*b*c^2*x^2*(-2*c^2*d*x^2 + e*(2 + c^2*x^2))*ArcTanh[c*x] + 3*(6*b*d - 12*a*e - 25*b*e)*Log[1 - c*x] - 3*(6*b*d + 12*a*e - 25*b*e)*Log[1 + c*x] + 12*e*(3*a*c^4*x^4 + b*c*x*(3 + c^2*x^2) + 3*b*(-1 + c^4*x^4)*ArcTanh[c*x])*Log[1 - c^2*x^2])/(144*c^4)","A",1
524,1,183,247,0.1350235,"\int x^2 \left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right) \, dx","Integrate[x^2*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]),x]","\frac{6 c^2 e x^2 \log \left(1-c^2 x^2\right) \left(2 a c x+2 b c x \tanh ^{-1}(c x)+b\right)+2 \log (1-c x) (-6 a e+3 b d-11 b e)+2 \log (c x+1) (6 a e+3 b d-11 b e)+4 a c^3 x^3 (3 d-2 e)-24 a c e x+2 b c^2 x^2 (3 d-5 e)+4 b c x \tanh ^{-1}(c x) \left(3 c^2 d x^2-2 e \left(c^2 x^2+3\right)\right)+3 b e \log ^2\left(1-c^2 x^2\right)+12 b e \tanh ^{-1}(c x)^2}{36 c^3}","-\frac{e (2 a+b) \log (1-c x)}{6 c^3}+\frac{e (2 a-b) \log (c x+1)}{6 c^3}+\frac{1}{3} x^3 \left(a+b \tanh ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)-\frac{2 a e x}{3 c^2}-\frac{2}{9} a e x^3+\frac{b e \tanh ^{-1}(c x)^2}{3 c^3}+\frac{b x^2 \left(e \log \left(1-c^2 x^2\right)+d\right)}{6 c}-\frac{2 b e x \tanh ^{-1}(c x)}{3 c^2}+\frac{b \log \left(1-c^2 x^2\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{6 c^3}-\frac{b e \log ^2\left(1-c^2 x^2\right)}{12 c^3}-\frac{4 b e \log \left(1-c^2 x^2\right)}{9 c^3}-\frac{2}{9} b e x^3 \tanh ^{-1}(c x)-\frac{5 b e x^2}{18 c}",1,"(-24*a*c*e*x + 2*b*c^2*(3*d - 5*e)*x^2 + 4*a*c^3*(3*d - 2*e)*x^3 + 4*b*c*x*(3*c^2*d*x^2 - 2*e*(3 + c^2*x^2))*ArcTanh[c*x] + 12*b*e*ArcTanh[c*x]^2 + 2*(3*b*d - 6*a*e - 11*b*e)*Log[1 - c*x] + 2*(3*b*d + 6*a*e - 11*b*e)*Log[1 + c*x] + 6*c^2*e*x^2*(b + 2*a*c*x + 2*b*c*x*ArcTanh[c*x])*Log[1 - c^2*x^2] + 3*b*e*Log[1 - c^2*x^2]^2)/(36*c^3)","A",1
525,1,129,140,0.1138129,"\int x \left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right) \, dx","Integrate[x*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]),x]","\frac{2 e \log \left(1-c^2 x^2\right) \left(c x (a c x+b)+b \left(c^2 x^2-1\right) \tanh ^{-1}(c x)\right)+\log (1-c x) (b (d-3 e)-2 a e)-\log (c x+1) (2 a e+b (d-3 e))+2 a c^2 x^2 (d-e)+2 b c^2 x^2 (d-e) \tanh ^{-1}(c x)+2 b c x (d-3 e)}{4 c^2}","-\frac{e \left(1-c^2 x^2\right) \log \left(1-c^2 x^2\right) \left(a+b \tanh ^{-1}(c x)\right)}{2 c^2}+\frac{1}{2} d x^2 \left(a+b \tanh ^{-1}(c x)\right)-\frac{1}{2} e x^2 \left(a+b \tanh ^{-1}(c x)\right)-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac{b e x \log \left(1-c^2 x^2\right)}{2 c}+\frac{b e \tanh ^{-1}(c x)}{c^2}+\frac{b x (d-e)}{2 c}-\frac{b e x}{c}",1,"(2*b*c*(d - 3*e)*x + 2*a*c^2*(d - e)*x^2 + 2*b*c^2*(d - e)*x^2*ArcTanh[c*x] + (b*(d - 3*e) - 2*a*e)*Log[1 - c*x] - (b*(d - 3*e) + 2*a*e)*Log[1 + c*x] + 2*e*(c*x*(b + a*c*x) + b*(-1 + c^2*x^2)*ArcTanh[c*x])*Log[1 - c^2*x^2])/(4*c^2)","A",1
526,1,144,104,0.0194308,"\int \left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right) \, dx","Integrate[(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]),x]","a e x \log \left(1-c^2 x^2\right)+\frac{2 a e \tanh ^{-1}(c x)}{c}+a d x-2 a e x+\frac{b d \log \left(1-c^2 x^2\right)}{2 c}+\frac{b e \log ^2\left(1-c^2 x^2\right)}{4 c}-\frac{b e \log \left(1-c^2 x^2\right)}{c}+b e x \log \left(1-c^2 x^2\right) \tanh ^{-1}(c x)+b d x \tanh ^{-1}(c x)+\frac{b e \tanh ^{-1}(c x)^2}{c}-2 b e x \tanh ^{-1}(c x)","x \left(a+b \tanh ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)+\frac{e \left(a+b \tanh ^{-1}(c x)\right)^2}{b c}-2 a e x+\frac{b \left(e \log \left(1-c^2 x^2\right)+d\right)^2}{4 c e}-\frac{b e \log \left(1-c^2 x^2\right)}{c}-2 b e x \tanh ^{-1}(c x)",1,"a*d*x - 2*a*e*x + (2*a*e*ArcTanh[c*x])/c + b*d*x*ArcTanh[c*x] - 2*b*e*x*ArcTanh[c*x] + (b*e*ArcTanh[c*x]^2)/c + (b*d*Log[1 - c^2*x^2])/(2*c) - (b*e*Log[1 - c^2*x^2])/c + a*e*x*Log[1 - c^2*x^2] + b*e*x*ArcTanh[c*x]*Log[1 - c^2*x^2] + (b*e*Log[1 - c^2*x^2]^2)/(4*c)","A",1
527,0,0,216,0.2627672,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x} \, dx","Integrate[((a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/x,x]","\int \frac{\left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x} \, dx","-\frac{1}{2} a e \text{Li}_2\left(c^2 x^2\right)+a d \log (x)+\frac{1}{2} b e \text{Li}_2(-c x) \left(-\log \left(1-c^2 x^2\right)+\log (1-c x)+\log (c x+1)\right)-\frac{1}{2} b e \text{Li}_2(c x) \left(-\log \left(1-c^2 x^2\right)+\log (1-c x)+\log (c x+1)\right)-\frac{1}{2} b d \text{Li}_2(-c x)+\frac{1}{2} b d \text{Li}_2(c x)+b e \text{Li}_3(1-c x)-b e \text{Li}_3(c x+1)-b e \text{Li}_2(1-c x) \log (1-c x)+b e \text{Li}_2(c x+1) \log (c x+1)-\frac{1}{2} b e \log (c x) \log ^2(1-c x)+\frac{1}{2} b e \log (-c x) \log ^2(c x+1)",1,"Integrate[((a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/x, x]","F",-1
528,1,332,105,0.2088601,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x^2} \, dx","Integrate[((a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/x^2,x]","-\frac{4 a e \log \left(1-c^2 x^2\right)+8 a c e x \tanh ^{-1}(c x)+4 a d+2 b c d x \log \left(1-c^2 x^2\right)-4 b c e x \log (x) \log \left(1-c^2 x^2\right)+2 b c e x \log \left(x-\frac{1}{c}\right) \log \left(1-c^2 x^2\right)+2 b c e x \log \left(\frac{1}{c}+x\right) \log \left(1-c^2 x^2\right)+4 b e \log \left(1-c^2 x^2\right) \tanh ^{-1}(c x)-4 b c d x \log (x)+4 b d \tanh ^{-1}(c x)+4 b c e x \text{Li}_2(-c x)+4 b c e x \text{Li}_2(c x)-2 b c e x \text{Li}_2\left(\frac{1}{2}-\frac{c x}{2}\right)-2 b c e x \text{Li}_2\left(\frac{1}{2} (c x+1)\right)-b c e x \log ^2\left(x-\frac{1}{c}\right)-b c e x \log ^2\left(\frac{1}{c}+x\right)-2 b c e x \log \left(\frac{1}{c}+x\right) \log \left(\frac{1}{2} (1-c x)\right)+4 b c e x \log (x) \log (1-c x)-2 b c e x \log \left(x-\frac{1}{c}\right) \log \left(\frac{1}{2} (c x+1)\right)+4 b c e x \log (x) \log (c x+1)+4 b c e x \tanh ^{-1}(c x)^2}{4 x}","-\frac{\left(a+b \tanh ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{x}-\frac{c e \left(a+b \tanh ^{-1}(c x)\right)^2}{b}+\frac{1}{2} b c \log \left(1-\frac{1}{1-c^2 x^2}\right) \left(e \log \left(1-c^2 x^2\right)+d\right)-\frac{1}{2} b c e \text{Li}_2\left(\frac{1}{1-c^2 x^2}\right)",1,"-1/4*(4*a*d + 4*b*d*ArcTanh[c*x] + 8*a*c*e*x*ArcTanh[c*x] + 4*b*c*e*x*ArcTanh[c*x]^2 - 4*b*c*d*x*Log[x] - b*c*e*x*Log[-c^(-1) + x]^2 - b*c*e*x*Log[c^(-1) + x]^2 - 2*b*c*e*x*Log[c^(-1) + x]*Log[(1 - c*x)/2] + 4*b*c*e*x*Log[x]*Log[1 - c*x] - 2*b*c*e*x*Log[-c^(-1) + x]*Log[(1 + c*x)/2] + 4*b*c*e*x*Log[x]*Log[1 + c*x] + 4*a*e*Log[1 - c^2*x^2] + 2*b*c*d*x*Log[1 - c^2*x^2] + 4*b*e*ArcTanh[c*x]*Log[1 - c^2*x^2] - 4*b*c*e*x*Log[x]*Log[1 - c^2*x^2] + 2*b*c*e*x*Log[-c^(-1) + x]*Log[1 - c^2*x^2] + 2*b*c*e*x*Log[c^(-1) + x]*Log[1 - c^2*x^2] + 4*b*c*e*x*PolyLog[2, -(c*x)] + 4*b*c*e*x*PolyLog[2, c*x] - 2*b*c*e*x*PolyLog[2, 1/2 - (c*x)/2] - 2*b*c*e*x*PolyLog[2, (1 + c*x)/2])/x","B",1
529,1,152,157,0.1677282,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x^3} \, dx","Integrate[((a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/x^3,x]","\frac{1}{2} \left(-\frac{e \log \left(1-c^2 x^2\right) \left(a+\left(b-b c^2 x^2\right) \tanh ^{-1}(c x)+b c x\right)}{x^2}+c^2 e (a+b) \log (1-c x)+c^2 e (a-b) \log (c x+1)-2 a c^2 e \log (x)-\frac{a d}{x^2}+b c^2 e (\text{Li}_2(-c x)-\text{Li}_2(c x))-\frac{b d \left(c x (c x \log (1-c x)-c x \log (c x+1)+2)+2 \tanh ^{-1}(c x)\right)}{2 x^2}\right)","-\frac{\left(a+b \tanh ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{2 x^2}+\frac{1}{2} c^2 e (a+b) \log (1-c x)+\frac{1}{2} c^2 e (a-b) \log (c x+1)-a c^2 e \log (x)-\frac{b c \left(e \log \left(1-c^2 x^2\right)+d\right)}{2 x}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left(e \log \left(1-c^2 x^2\right)+d\right)+\frac{1}{2} b c^2 e \text{Li}_2(-c x)-\frac{1}{2} b c^2 e \text{Li}_2(c x)",1,"(-((a*d)/x^2) - 2*a*c^2*e*Log[x] + (a + b)*c^2*e*Log[1 - c*x] + (a - b)*c^2*e*Log[1 + c*x] - (b*d*(2*ArcTanh[c*x] + c*x*(2 + c*x*Log[1 - c*x] - c*x*Log[1 + c*x])))/(2*x^2) - (e*(a + b*c*x + (b - b*c^2*x^2)*ArcTanh[c*x])*Log[1 - c^2*x^2])/x^2 + b*c^2*e*(PolyLog[2, -(c*x)] - PolyLog[2, c*x]))/2","A",0
530,1,460,197,0.3670593,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x^4} \, dx","Integrate[((a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/x^4,x]","\frac{1}{6} \left(-4 a c^3 e \tanh ^{-1}(c x)-\frac{2 a e \log \left(1-c^2 x^2\right)}{x^3}+\frac{4 a c^2 e}{x}-\frac{2 a d}{x^3}+2 b c^3 d \log (x)-2 b c^3 e \text{Li}_2(-c x)-2 b c^3 e \text{Li}_2(c x)+b c^3 e \text{Li}_2\left(\frac{1}{2}-\frac{c x}{2}\right)+b c^3 e \text{Li}_2\left(\frac{1}{2} (c x+1)\right)+\frac{1}{2} b c^3 e \log ^2\left(x-\frac{1}{c}\right)+\frac{1}{2} b c^3 e \log ^2\left(\frac{1}{c}+x\right)-2 b c^3 e \log (x)+b c^3 e \log \left(\frac{1}{c}+x\right) \log \left(\frac{1}{2} (1-c x)\right)-2 b c^3 e \log (x) \log (1-c x)+b c^3 e \log \left(x-\frac{1}{c}\right) \log \left(\frac{1}{2} (c x+1)\right)-2 b c^3 e \log (x) \log (c x+1)-2 b c^3 e \tanh ^{-1}(c x)^2-\frac{b c e \log \left(1-c^2 x^2\right)}{x^2}-\frac{2 b e \log \left(1-c^2 x^2\right) \tanh ^{-1}(c x)}{x^3}+\frac{4 b c^2 e \tanh ^{-1}(c x)}{x}-b c^3 d \log \left(1-c^2 x^2\right)-4 b c^3 e \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+b c^3 e \log \left(1-c^2 x^2\right)+2 b c^3 e \log (x) \log \left(1-c^2 x^2\right)-b c^3 e \log \left(x-\frac{1}{c}\right) \log \left(1-c^2 x^2\right)-b c^3 e \log \left(\frac{1}{c}+x\right) \log \left(1-c^2 x^2\right)-\frac{2 b d \tanh ^{-1}(c x)}{x^3}-\frac{b c d}{x^2}\right)","-\frac{c^3 e \left(a+b \tanh ^{-1}(c x)\right)^2}{3 b}-\frac{\left(a+b \tanh ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{3 x^3}+\frac{2 c^2 e \left(a+b \tanh ^{-1}(c x)\right)}{3 x}-b c^3 e \log (x)-\frac{b c \left(1-c^2 x^2\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{6 x^2}+\frac{1}{6} b c^3 \log \left(1-\frac{1}{1-c^2 x^2}\right) \left(e \log \left(1-c^2 x^2\right)+d\right)-\frac{1}{6} b c^3 e \text{Li}_2\left(\frac{1}{1-c^2 x^2}\right)+\frac{1}{3} b c^3 e \log \left(1-c^2 x^2\right)",1,"((-2*a*d)/x^3 - (b*c*d)/x^2 + (4*a*c^2*e)/x - 4*a*c^3*e*ArcTanh[c*x] - (2*b*d*ArcTanh[c*x])/x^3 + (4*b*c^2*e*ArcTanh[c*x])/x - 2*b*c^3*e*ArcTanh[c*x]^2 + 2*b*c^3*d*Log[x] - 2*b*c^3*e*Log[x] + (b*c^3*e*Log[-c^(-1) + x]^2)/2 + (b*c^3*e*Log[c^(-1) + x]^2)/2 + b*c^3*e*Log[c^(-1) + x]*Log[(1 - c*x)/2] - 2*b*c^3*e*Log[x]*Log[1 - c*x] + b*c^3*e*Log[-c^(-1) + x]*Log[(1 + c*x)/2] - 2*b*c^3*e*Log[x]*Log[1 + c*x] - 4*b*c^3*e*Log[(c*x)/Sqrt[1 - c^2*x^2]] - b*c^3*d*Log[1 - c^2*x^2] + b*c^3*e*Log[1 - c^2*x^2] - (2*a*e*Log[1 - c^2*x^2])/x^3 - (b*c*e*Log[1 - c^2*x^2])/x^2 - (2*b*e*ArcTanh[c*x]*Log[1 - c^2*x^2])/x^3 + 2*b*c^3*e*Log[x]*Log[1 - c^2*x^2] - b*c^3*e*Log[-c^(-1) + x]*Log[1 - c^2*x^2] - b*c^3*e*Log[c^(-1) + x]*Log[1 - c^2*x^2] - 2*b*c^3*e*PolyLog[2, -(c*x)] - 2*b*c^3*e*PolyLog[2, c*x] + b*c^3*e*PolyLog[2, 1/2 - (c*x)/2] + b*c^3*e*PolyLog[2, (1 + c*x)/2])/6","B",1
531,1,299,244,0.1575911,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x^5} \, dx","Integrate[((a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/x^5,x]","\frac{1}{12} \log (1-c x) \left(3 a c^4 e+4 b c^4 e\right)+\frac{1}{12} \log (c x+1) \left(3 a c^4 e-4 b c^4 e\right)+\frac{e \log \left(1-c^2 x^2\right) \left(-3 a+3 b c^4 x^4 \tanh ^{-1}(c x)-3 b c^3 x^3-b c x-3 b \tanh ^{-1}(c x)\right)}{12 x^4}-\frac{1}{2} a c^4 e \log (x)+\frac{a c^2 e}{4 x^2}-\frac{a d}{4 x^4}-\frac{1}{4} b c^4 e (\text{Li}_2(c x)-\text{Li}_2(-c x))+\frac{b c^3 e}{6 x}+b c^4 d \left(\frac{1}{4} \left(-\frac{1}{3 c^3 x^3}-\frac{1}{c x}-\frac{1}{2} \log (1-c x)+\frac{1}{2} \log (c x+1)\right)-\frac{\tanh ^{-1}(c x)}{4 c^4 x^4}\right)-\frac{1}{2} b c^4 e \left(\frac{1}{2} \left(-\frac{1}{c x}-\frac{1}{2} \log (1-c x)+\frac{1}{2} \log (c x+1)\right)-\frac{\tanh ^{-1}(c x)}{2 c^2 x^2}\right)","\frac{1}{12} c^4 e (3 a+4 b) \log (1-c x)+\frac{1}{12} c^4 e (3 a-4 b) \log (c x+1)-\frac{\left(a+b \tanh ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{4 x^4}-\frac{1}{2} a c^4 e \log (x)+\frac{a c^2 e}{4 x^2}+\frac{1}{4} b c^4 e \text{Li}_2(-c x)-\frac{1}{4} b c^4 e \text{Li}_2(c x)-\frac{1}{4} b c^4 e \tanh ^{-1}(c x)+\frac{5 b c^3 e}{12 x}-\frac{b c \left(e \log \left(1-c^2 x^2\right)+d\right)}{12 x^3}+\frac{b c^2 e \tanh ^{-1}(c x)}{4 x^2}+\frac{1}{4} b c^4 \tanh ^{-1}(c x) \left(e \log \left(1-c^2 x^2\right)+d\right)-\frac{b c^3 \left(e \log \left(1-c^2 x^2\right)+d\right)}{4 x}",1,"-1/4*(a*d)/x^4 + (a*c^2*e)/(4*x^2) + (b*c^3*e)/(6*x) - (a*c^4*e*Log[x])/2 + ((3*a*c^4*e + 4*b*c^4*e)*Log[1 - c*x])/12 - (b*c^4*e*(-1/2*ArcTanh[c*x]/(c^2*x^2) + (-(1/(c*x)) - Log[1 - c*x]/2 + Log[1 + c*x]/2)/2))/2 + b*c^4*d*(-1/4*ArcTanh[c*x]/(c^4*x^4) + (-1/3*1/(c^3*x^3) - 1/(c*x) - Log[1 - c*x]/2 + Log[1 + c*x]/2)/4) + ((3*a*c^4*e - 4*b*c^4*e)*Log[1 + c*x])/12 + (e*(-3*a - b*c*x - 3*b*c^3*x^3 - 3*b*ArcTanh[c*x] + 3*b*c^4*x^4*ArcTanh[c*x])*Log[1 - c^2*x^2])/(12*x^4) - (b*c^4*e*(-PolyLog[2, -(c*x)] + PolyLog[2, c*x]))/4","A",0
532,0,0,256,0.3022076,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x^6} \, dx","Integrate[((a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/x^6,x]","\int \frac{\left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x^6} \, dx","-\frac{c^5 e \left(a+b \tanh ^{-1}(c x)\right)^2}{5 b}+\frac{2 c^4 e \left(a+b \tanh ^{-1}(c x)\right)}{5 x}-\frac{\left(a+b \tanh ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{5 x^5}+\frac{2 c^2 e \left(a+b \tanh ^{-1}(c x)\right)}{15 x^3}-\frac{5}{6} b c^5 e \log (x)+\frac{7 b c^3 e}{60 x^2}-\frac{b c \left(e \log \left(1-c^2 x^2\right)+d\right)}{20 x^4}+\frac{1}{10} b c^5 \log \left(1-\frac{1}{1-c^2 x^2}\right) \left(e \log \left(1-c^2 x^2\right)+d\right)-\frac{1}{10} b c^5 e \text{Li}_2\left(\frac{1}{1-c^2 x^2}\right)+\frac{19}{60} b c^5 e \log \left(1-c^2 x^2\right)-\frac{b c^3 \left(1-c^2 x^2\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{10 x^2}",1,"Integrate[((a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/x^6, x]","F",-1
533,1,1376,512,5.8098774,"\int x \left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right) \, dx","Integrate[x*(a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]),x]","\frac{2 a (d-e) g x^2 c^2+2 a e f \log \left(g x^2+f\right) c^2+b d f \left(2 \tanh ^{-1}(c x)^2-2 \left(\tanh ^{-1}(c x)+2 \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right)\right) \tanh ^{-1}(c x)-4 i \sin ^{-1}\left(\sqrt{\frac{c^2 f}{f c^2+g}}\right) \tanh ^{-1}\left(\frac{c g x}{\sqrt{-c^2 f g}}\right)+2 \left(\tanh ^{-1}(c x)-i \sin ^{-1}\left(\sqrt{\frac{c^2 f}{f c^2+g}}\right)\right) \log \left(\frac{e^{-2 \tanh ^{-1}(c x)} \left(\left(1+e^{2 \tanh ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 \tanh ^{-1}(c x)}\right) g-2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)+2 \left(i \sin ^{-1}\left(\sqrt{\frac{c^2 f}{f c^2+g}}\right)+\tanh ^{-1}(c x)\right) \log \left(\frac{e^{-2 \tanh ^{-1}(c x)} \left(\left(1+e^{2 \tanh ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 \tanh ^{-1}(c x)}\right) g+2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)+2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-\text{Li}_2\left(\frac{e^{-2 \tanh ^{-1}(c x)} \left(-f c^2+g-2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)-\text{Li}_2\left(\frac{e^{-2 \tanh ^{-1}(c x)} \left(-f c^2+g+2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)\right) c^2-4 b e g x c+4 b e \sqrt{f} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right) c+2 e g \left(c x (b+a c x)+b \left(c^2 x^2-1\right) \tanh ^{-1}(c x)\right) \log \left(g x^2+f\right)+b d \left(2 \left(f \tanh ^{-1}(c x)^2 c^2-f \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right) c^2+g x c+\tanh ^{-1}(c x) \left(g x^2 c^2+2 f \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right) c^2-g\right)\right)-c^2 f \left(2 \tanh ^{-1}(c x) \left(-\tanh ^{-1}(c x)+\log \left(\frac{e^{2 \tanh ^{-1}(c x)} \left(f c^2+g\right)}{f c^2-2 \sqrt{-f} \sqrt{g} c-g}+1\right)+\log \left(\frac{e^{2 \tanh ^{-1}(c x)} \left(f c^2+g\right)}{f c^2+2 \sqrt{-f} \sqrt{g} c-g}+1\right)\right)+\text{Li}_2\left(-\frac{e^{2 \tanh ^{-1}(c x)} \left(f c^2+g\right)}{f c^2-2 \sqrt{-f} \sqrt{g} c-g}\right)+\text{Li}_2\left(-\frac{e^{2 \tanh ^{-1}(c x)} \left(f c^2+g\right)}{f c^2+2 \sqrt{-f} \sqrt{g} c-g}\right)\right)\right)+b e \left(c^2 f \left(2 \tanh ^{-1}(c x) \left(-\tanh ^{-1}(c x)+\log \left(\frac{e^{2 \tanh ^{-1}(c x)} \left(f c^2+g\right)}{f c^2-2 \sqrt{-f} \sqrt{g} c-g}+1\right)+\log \left(\frac{e^{2 \tanh ^{-1}(c x)} \left(f c^2+g\right)}{f c^2+2 \sqrt{-f} \sqrt{g} c-g}+1\right)\right)+\text{Li}_2\left(-\frac{e^{2 \tanh ^{-1}(c x)} \left(f c^2+g\right)}{f c^2-2 \sqrt{-f} \sqrt{g} c-g}\right)+\text{Li}_2\left(-\frac{e^{2 \tanh ^{-1}(c x)} \left(f c^2+g\right)}{f c^2+2 \sqrt{-f} \sqrt{g} c-g}\right)\right)-2 \left(f \tanh ^{-1}(c x)^2 c^2-f \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right) c^2+g x c+\tanh ^{-1}(c x) \left(g x^2 c^2+2 f \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right) c^2-g\right)\right)\right)+b e g \left(2 \tanh ^{-1}(c x)^2-2 \left(\tanh ^{-1}(c x)+2 \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right)\right) \tanh ^{-1}(c x)-4 i \sin ^{-1}\left(\sqrt{\frac{c^2 f}{f c^2+g}}\right) \tanh ^{-1}\left(\frac{c g x}{\sqrt{-c^2 f g}}\right)+2 \left(\tanh ^{-1}(c x)-i \sin ^{-1}\left(\sqrt{\frac{c^2 f}{f c^2+g}}\right)\right) \log \left(\frac{e^{-2 \tanh ^{-1}(c x)} \left(\left(1+e^{2 \tanh ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 \tanh ^{-1}(c x)}\right) g-2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)+2 \left(i \sin ^{-1}\left(\sqrt{\frac{c^2 f}{f c^2+g}}\right)+\tanh ^{-1}(c x)\right) \log \left(\frac{e^{-2 \tanh ^{-1}(c x)} \left(\left(1+e^{2 \tanh ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 \tanh ^{-1}(c x)}\right) g+2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)+2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)-\text{Li}_2\left(\frac{e^{-2 \tanh ^{-1}(c x)} \left(-f c^2+g-2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)-\text{Li}_2\left(\frac{e^{-2 \tanh ^{-1}(c x)} \left(-f c^2+g+2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)\right)}{4 c^2 g}","\frac{1}{2} d x^2 \left(a+b \tanh ^{-1}(c x)\right)+\frac{e \left(f+g x^2\right) \log \left(f+g x^2\right) \left(a+b \tanh ^{-1}(c x)\right)}{2 g}-\frac{1}{2} e x^2 \left(a+b \tanh ^{-1}(c x)\right)-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac{b e \left(c^2 f+g\right) \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 c^2 g}-\frac{b e \left(c^2 f+g\right) \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{\left(c \sqrt{-f}-\sqrt{g}\right) (c x+1)}\right)}{4 c^2 g}-\frac{b e \left(c^2 f+g\right) \text{Li}_2\left(1-\frac{2 c \left(\sqrt{g} x+\sqrt{-f}\right)}{\left(\sqrt{-f} c+\sqrt{g}\right) (c x+1)}\right)}{4 c^2 g}-\frac{b e \left(c^2 f+g\right) \tanh ^{-1}(c x) \log \left(f+g x^2\right)}{2 c^2 g}-\frac{b e \left(c^2 f+g\right) \log \left(\frac{2}{c x+1}\right) \tanh ^{-1}(c x)}{c^2 g}+\frac{b e \left(c^2 f+g\right) \tanh ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{(c x+1) \left(c \sqrt{-f}-\sqrt{g}\right)}\right)}{2 c^2 g}+\frac{b e \left(c^2 f+g\right) \tanh ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}+\sqrt{g} x\right)}{(c x+1) \left(c \sqrt{-f}+\sqrt{g}\right)}\right)}{2 c^2 g}+\frac{b x (d-e)}{2 c}+\frac{b e x \log \left(f+g x^2\right)}{2 c}+\frac{b e \sqrt{f} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{c \sqrt{g}}-\frac{b e x}{c}",1,"(-4*b*c*e*g*x + 2*a*c^2*(d - e)*g*x^2 + 4*b*c*e*Sqrt[f]*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]] + 2*a*c^2*e*f*Log[f + g*x^2] + 2*e*g*(c*x*(b + a*c*x) + b*(-1 + c^2*x^2)*ArcTanh[c*x])*Log[f + g*x^2] + b*d*(2*(c*g*x + c^2*f*ArcTanh[c*x]^2 + ArcTanh[c*x]*(-g + c^2*g*x^2 + 2*c^2*f*Log[1 + E^(-2*ArcTanh[c*x])]) - c^2*f*PolyLog[2, -E^(-2*ArcTanh[c*x])]) - c^2*f*(2*ArcTanh[c*x]*(-ArcTanh[c*x] + Log[1 + (E^(2*ArcTanh[c*x])*(c^2*f + g))/(c^2*f - 2*c*Sqrt[-f]*Sqrt[g] - g)] + Log[1 + (E^(2*ArcTanh[c*x])*(c^2*f + g))/(c^2*f + 2*c*Sqrt[-f]*Sqrt[g] - g)]) + PolyLog[2, -((E^(2*ArcTanh[c*x])*(c^2*f + g))/(c^2*f - 2*c*Sqrt[-f]*Sqrt[g] - g))] + PolyLog[2, -((E^(2*ArcTanh[c*x])*(c^2*f + g))/(c^2*f + 2*c*Sqrt[-f]*Sqrt[g] - g))])) + b*e*(-2*(c*g*x + c^2*f*ArcTanh[c*x]^2 + ArcTanh[c*x]*(-g + c^2*g*x^2 + 2*c^2*f*Log[1 + E^(-2*ArcTanh[c*x])]) - c^2*f*PolyLog[2, -E^(-2*ArcTanh[c*x])]) + c^2*f*(2*ArcTanh[c*x]*(-ArcTanh[c*x] + Log[1 + (E^(2*ArcTanh[c*x])*(c^2*f + g))/(c^2*f - 2*c*Sqrt[-f]*Sqrt[g] - g)] + Log[1 + (E^(2*ArcTanh[c*x])*(c^2*f + g))/(c^2*f + 2*c*Sqrt[-f]*Sqrt[g] - g)]) + PolyLog[2, -((E^(2*ArcTanh[c*x])*(c^2*f + g))/(c^2*f - 2*c*Sqrt[-f]*Sqrt[g] - g))] + PolyLog[2, -((E^(2*ArcTanh[c*x])*(c^2*f + g))/(c^2*f + 2*c*Sqrt[-f]*Sqrt[g] - g))])) + b*c^2*d*f*(2*ArcTanh[c*x]^2 - (4*I)*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]]*ArcTanh[(c*g*x)/Sqrt[-(c^2*f*g)]] - 2*ArcTanh[c*x]*(ArcTanh[c*x] + 2*Log[1 + E^(-2*ArcTanh[c*x])]) + 2*((-I)*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]] + ArcTanh[c*x])*Log[(c^2*(1 + E^(2*ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))] + 2*(I*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]] + ArcTanh[c*x])*Log[(c^2*(1 + E^(2*ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))] + 2*PolyLog[2, -E^(-2*ArcTanh[c*x])] - PolyLog[2, (-(c^2*f) + g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))] - PolyLog[2, (-(c^2*f) + g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))]) + b*e*g*(2*ArcTanh[c*x]^2 - (4*I)*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]]*ArcTanh[(c*g*x)/Sqrt[-(c^2*f*g)]] - 2*ArcTanh[c*x]*(ArcTanh[c*x] + 2*Log[1 + E^(-2*ArcTanh[c*x])]) + 2*((-I)*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]] + ArcTanh[c*x])*Log[(c^2*(1 + E^(2*ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))] + 2*(I*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]] + ArcTanh[c*x])*Log[(c^2*(1 + E^(2*ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))] + 2*PolyLog[2, -E^(-2*ArcTanh[c*x])] - PolyLog[2, (-(c^2*f) + g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))] - PolyLog[2, (-(c^2*f) + g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))]))/(4*c^2*g)","C",0
534,1,1251,599,3.3618451,"\int \left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right) \, dx","Integrate[(a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]),x]","a d x-2 a e x+b d \tanh ^{-1}(c x) x+a e \log \left(g x^2+f\right) x+\frac{2 a e \sqrt{f} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{g}}+\frac{b d \log \left(1-c^2 x^2\right)}{2 c}+b e \left(x \tanh ^{-1}(c x)+\frac{\log \left(1-c^2 x^2\right)}{2 c}\right) \log \left(g x^2+f\right)-\frac{b e g \left(\frac{\left(-\log \left(x-\frac{1}{c}\right)-\log \left(x+\frac{1}{c}\right)+\log \left(1-c^2 x^2\right)\right) \log \left(g x^2+f\right)}{2 g}+\frac{\log \left(x-\frac{1}{c}\right) \log \left(1-\frac{\sqrt{g} \left(x-\frac{1}{c}\right)}{-i \sqrt{f}-\frac{\sqrt{g}}{c}}\right)+\text{Li}_2\left(\frac{\sqrt{g} \left(x-\frac{1}{c}\right)}{-i \sqrt{f}-\frac{\sqrt{g}}{c}}\right)}{2 g}+\frac{\log \left(x-\frac{1}{c}\right) \log \left(1-\frac{\sqrt{g} \left(x-\frac{1}{c}\right)}{i \sqrt{f}-\frac{\sqrt{g}}{c}}\right)+\text{Li}_2\left(\frac{\sqrt{g} \left(x-\frac{1}{c}\right)}{i \sqrt{f}-\frac{\sqrt{g}}{c}}\right)}{2 g}+\frac{\log \left(x+\frac{1}{c}\right) \log \left(1-\frac{\sqrt{g} \left(x+\frac{1}{c}\right)}{\frac{\sqrt{g}}{c}-i \sqrt{f}}\right)+\text{Li}_2\left(\frac{\sqrt{g} \left(x+\frac{1}{c}\right)}{\frac{\sqrt{g}}{c}-i \sqrt{f}}\right)}{2 g}+\frac{\log \left(x+\frac{1}{c}\right) \log \left(1-\frac{\sqrt{g} \left(x+\frac{1}{c}\right)}{i \sqrt{f}+\frac{\sqrt{g}}{c}}\right)+\text{Li}_2\left(\frac{\sqrt{g} \left(x+\frac{1}{c}\right)}{i \sqrt{f}+\frac{\sqrt{g}}{c}}\right)}{2 g}\right)}{c}-\frac{b e \left(4 c x \tanh ^{-1}(c x)-4 \log \left(\frac{1}{\sqrt{1-c^2 x^2}}\right)+\frac{\sqrt{c^2 f g} \left(-2 i \cos ^{-1}\left(\frac{g-c^2 f}{f c^2+g}\right) \tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)+4 \tan ^{-1}\left(\frac{\sqrt{c^2 f g}}{c g x}\right) \tanh ^{-1}(c x)-\left(\cos ^{-1}\left(\frac{g-c^2 f}{f c^2+g}\right)-2 \tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)\right) \log \left(\frac{2 c^2 f \left(g+i \sqrt{c^2 f g}\right) (c x+1)}{\left(f c^2+g\right) \left(f c^2+i \sqrt{c^2 f g} x c\right)}\right)-\left(\cos ^{-1}\left(\frac{g-c^2 f}{f c^2+g}\right)+2 \tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)\right) \log \left(\frac{2 c^2 f \left(i g+\sqrt{c^2 f g}\right) (c x-1)}{\left(f c^2+g\right) \left(c \sqrt{c^2 f g} x-i c^2 f\right)}\right)+\left(\cos ^{-1}\left(\frac{g-c^2 f}{f c^2+g}\right)+2 \left(\tan ^{-1}\left(\frac{\sqrt{c^2 f g}}{c g x}\right)+\tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)\right)\right) \log \left(\frac{\sqrt{2} e^{-\tanh ^{-1}(c x)} \sqrt{c^2 f g}}{\sqrt{f c^2+g} \sqrt{f c^2-g+\left(f c^2+g\right) \cosh \left(2 \tanh ^{-1}(c x)\right)}}\right)+\left(\cos ^{-1}\left(\frac{g-c^2 f}{f c^2+g}\right)-2 \left(\tan ^{-1}\left(\frac{\sqrt{c^2 f g}}{c g x}\right)+\tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)\right)\right) \log \left(\frac{\sqrt{2} e^{\tanh ^{-1}(c x)} \sqrt{c^2 f g}}{\sqrt{f c^2+g} \sqrt{f c^2-g+\left(f c^2+g\right) \cosh \left(2 \tanh ^{-1}(c x)\right)}}\right)+i \left(\text{Li}_2\left(\frac{\left(-f c^2+g+2 i \sqrt{c^2 f g}\right) \left(i f c^2+\sqrt{c^2 f g} x c\right)}{\left(f c^2+g\right) \left(c \sqrt{c^2 f g} x-i c^2 f\right)}\right)-\text{Li}_2\left(\frac{\left(-f c^2+g-2 i \sqrt{c^2 f g}\right) \left(i f c^2+\sqrt{c^2 f g} x c\right)}{\left(f c^2+g\right) \left(c \sqrt{c^2 f g} x-i c^2 f\right)}\right)\right)\right)}{g}\right)}{2 c}","x \left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)+\frac{2 a e \sqrt{f} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{g}}-2 a e x+\frac{b \log \left(\frac{g \left(1-c^2 x^2\right)}{c^2 f+g}\right) \left(d+e \log \left(f+g x^2\right)\right)}{2 c}+\frac{b e \text{Li}_2\left(\frac{c^2 \left(g x^2+f\right)}{f c^2+g}\right)}{2 c}-\frac{b e \log \left(1-c^2 x^2\right)}{c}+\frac{b e \sqrt{-f} \text{Li}_2\left(-\frac{\sqrt{g} (1-c x)}{c \sqrt{-f}-\sqrt{g}}\right)}{2 \sqrt{g}}-\frac{b e \sqrt{-f} \text{Li}_2\left(\frac{\sqrt{g} (1-c x)}{\sqrt{-f} c+\sqrt{g}}\right)}{2 \sqrt{g}}+\frac{b e \sqrt{-f} \text{Li}_2\left(-\frac{\sqrt{g} (c x+1)}{c \sqrt{-f}-\sqrt{g}}\right)}{2 \sqrt{g}}-\frac{b e \sqrt{-f} \text{Li}_2\left(\frac{\sqrt{g} (c x+1)}{\sqrt{-f} c+\sqrt{g}}\right)}{2 \sqrt{g}}+\frac{b e \sqrt{-f} \log (1-c x) \log \left(\frac{c \left(\sqrt{-f}-\sqrt{g} x\right)}{c \sqrt{-f}-\sqrt{g}}\right)}{2 \sqrt{g}}-\frac{b e \sqrt{-f} \log (c x+1) \log \left(\frac{c \left(\sqrt{-f}-\sqrt{g} x\right)}{c \sqrt{-f}+\sqrt{g}}\right)}{2 \sqrt{g}}+\frac{b e \sqrt{-f} \log (c x+1) \log \left(\frac{c \left(\sqrt{-f}+\sqrt{g} x\right)}{c \sqrt{-f}-\sqrt{g}}\right)}{2 \sqrt{g}}-\frac{b e \sqrt{-f} \log (1-c x) \log \left(\frac{c \left(\sqrt{-f}+\sqrt{g} x\right)}{c \sqrt{-f}+\sqrt{g}}\right)}{2 \sqrt{g}}-2 b e x \tanh ^{-1}(c x)",1,"a*d*x - 2*a*e*x + (2*a*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] + b*d*x*ArcTanh[c*x] + (b*d*Log[1 - c^2*x^2])/(2*c) + a*e*x*Log[f + g*x^2] + b*e*(x*ArcTanh[c*x] + Log[1 - c^2*x^2]/(2*c))*Log[f + g*x^2] - (b*e*g*(((-Log[-c^(-1) + x] - Log[c^(-1) + x] + Log[1 - c^2*x^2])*Log[f + g*x^2])/(2*g) + (Log[-c^(-1) + x]*Log[1 - (Sqrt[g]*(-c^(-1) + x))/((-I)*Sqrt[f] - Sqrt[g]/c)] + PolyLog[2, (Sqrt[g]*(-c^(-1) + x))/((-I)*Sqrt[f] - Sqrt[g]/c)])/(2*g) + (Log[-c^(-1) + x]*Log[1 - (Sqrt[g]*(-c^(-1) + x))/(I*Sqrt[f] - Sqrt[g]/c)] + PolyLog[2, (Sqrt[g]*(-c^(-1) + x))/(I*Sqrt[f] - Sqrt[g]/c)])/(2*g) + (Log[c^(-1) + x]*Log[1 - (Sqrt[g]*(c^(-1) + x))/((-I)*Sqrt[f] + Sqrt[g]/c)] + PolyLog[2, (Sqrt[g]*(c^(-1) + x))/((-I)*Sqrt[f] + Sqrt[g]/c)])/(2*g) + (Log[c^(-1) + x]*Log[1 - (Sqrt[g]*(c^(-1) + x))/(I*Sqrt[f] + Sqrt[g]/c)] + PolyLog[2, (Sqrt[g]*(c^(-1) + x))/(I*Sqrt[f] + Sqrt[g]/c)])/(2*g)))/c - (b*e*(4*c*x*ArcTanh[c*x] - 4*Log[1/Sqrt[1 - c^2*x^2]] + (Sqrt[c^2*f*g]*((-2*I)*ArcCos[(-(c^2*f) + g)/(c^2*f + g)]*ArcTan[(c*g*x)/Sqrt[c^2*f*g]] + 4*ArcTan[Sqrt[c^2*f*g]/(c*g*x)]*ArcTanh[c*x] - (ArcCos[(-(c^2*f) + g)/(c^2*f + g)] - 2*ArcTan[(c*g*x)/Sqrt[c^2*f*g]])*Log[(2*c^2*f*(g + I*Sqrt[c^2*f*g])*(1 + c*x))/((c^2*f + g)*(c^2*f + I*c*Sqrt[c^2*f*g]*x))] - (ArcCos[(-(c^2*f) + g)/(c^2*f + g)] + 2*ArcTan[(c*g*x)/Sqrt[c^2*f*g]])*Log[(2*c^2*f*(I*g + Sqrt[c^2*f*g])*(-1 + c*x))/((c^2*f + g)*((-I)*c^2*f + c*Sqrt[c^2*f*g]*x))] + (ArcCos[(-(c^2*f) + g)/(c^2*f + g)] + 2*(ArcTan[Sqrt[c^2*f*g]/(c*g*x)] + ArcTan[(c*g*x)/Sqrt[c^2*f*g]]))*Log[(Sqrt[2]*Sqrt[c^2*f*g])/(E^ArcTanh[c*x]*Sqrt[c^2*f + g]*Sqrt[c^2*f - g + (c^2*f + g)*Cosh[2*ArcTanh[c*x]]])] + (ArcCos[(-(c^2*f) + g)/(c^2*f + g)] - 2*(ArcTan[Sqrt[c^2*f*g]/(c*g*x)] + ArcTan[(c*g*x)/Sqrt[c^2*f*g]]))*Log[(Sqrt[2]*E^ArcTanh[c*x]*Sqrt[c^2*f*g])/(Sqrt[c^2*f + g]*Sqrt[c^2*f - g + (c^2*f + g)*Cosh[2*ArcTanh[c*x]]])] + I*(-PolyLog[2, ((-(c^2*f) + g - (2*I)*Sqrt[c^2*f*g])*(I*c^2*f + c*Sqrt[c^2*f*g]*x))/((c^2*f + g)*((-I)*c^2*f + c*Sqrt[c^2*f*g]*x))] + PolyLog[2, ((-(c^2*f) + g + (2*I)*Sqrt[c^2*f*g])*(I*c^2*f + c*Sqrt[c^2*f*g]*x))/((c^2*f + g)*((-I)*c^2*f + c*Sqrt[c^2*f*g]*x))])))/g))/(2*c)","C",1
535,0,0,93,0.2337485,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x} \, dx","Integrate[((a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]))/x,x]","\int \frac{\left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x} \, dx","b e \text{Int}\left(\frac{\tanh ^{-1}(c x) \log \left(f+g x^2\right)}{x},x\right)+a d \log (x)+\frac{1}{2} a e \text{Li}_2\left(\frac{g x^2}{f}+1\right)+\frac{1}{2} a e \log \left(-\frac{g x^2}{f}\right) \log \left(f+g x^2\right)-\frac{1}{2} b d \text{Li}_2(-c x)+\frac{1}{2} b d \text{Li}_2(c x)",0,"Integrate[((a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]))/x, x]","A",-1
536,1,1226,613,3.2739791,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x^2} \, dx","Integrate[((a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]))/x^2,x]","-\frac{b \tanh ^{-1}(c x) d}{x}+b c \log (x) d-\frac{1}{2} b c \log \left(1-c^2 x^2\right) d-\frac{a d}{x}+a e \left(\frac{2 \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{f}}-\frac{\log \left(g x^2+f\right)}{x}\right)+\frac{1}{2} b e \left(-\frac{\left(2 \tanh ^{-1}(c x)+c x \left(\log \left(1-c^2 x^2\right)-2 \log (x)\right)\right) \log \left(g x^2+f\right)}{x}-2 c \left(\log (x) \left(\log \left(1-\frac{i \sqrt{g} x}{\sqrt{f}}\right)+\log \left(\frac{i \sqrt{g} x}{\sqrt{f}}+1\right)\right)+\text{Li}_2\left(-\frac{i \sqrt{g} x}{\sqrt{f}}\right)+\text{Li}_2\left(\frac{i \sqrt{g} x}{\sqrt{f}}\right)\right)+c \left(\log \left(x-\frac{1}{c}\right) \log \left(\frac{c \left(\sqrt{f}-i \sqrt{g} x\right)}{c \sqrt{f}-i \sqrt{g}}\right)+\log \left(x+\frac{1}{c}\right) \log \left(\frac{c \left(\sqrt{f}-i \sqrt{g} x\right)}{\sqrt{f} c+i \sqrt{g}}\right)+\log \left(x-\frac{1}{c}\right) \log \left(\frac{c \left(i \sqrt{g} x+\sqrt{f}\right)}{\sqrt{f} c+i \sqrt{g}}\right)-\left(\log \left(x-\frac{1}{c}\right)+\log \left(x+\frac{1}{c}\right)-\log \left(1-c^2 x^2\right)\right) \log \left(g x^2+f\right)+\log \left(x+\frac{1}{c}\right) \log \left(1-\frac{\sqrt{g} (c x+1)}{i \sqrt{f} c+\sqrt{g}}\right)+\text{Li}_2\left(\frac{c \sqrt{g} \left(x+\frac{1}{c}\right)}{i \sqrt{f} c+\sqrt{g}}\right)+\text{Li}_2\left(\frac{i \sqrt{g} (c x-1)}{c \sqrt{f}-i \sqrt{g}}\right)+\text{Li}_2\left(-\frac{i \sqrt{g} (c x-1)}{\sqrt{f} c+i \sqrt{g}}\right)+\text{Li}_2\left(\frac{i \sqrt{g} (c x+1)}{\sqrt{f} c+i \sqrt{g}}\right)\right)+\frac{c g \left(2 i \cos ^{-1}\left(\frac{g-c^2 f}{f c^2+g}\right) \tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)-4 \tan ^{-1}\left(\frac{c f}{\sqrt{c^2 f g} x}\right) \tanh ^{-1}(c x)+\left(\cos ^{-1}\left(\frac{g-c^2 f}{f c^2+g}\right)+2 \tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)\right) \log \left(\frac{2 i c f \left(i g+\sqrt{c^2 f g}\right) (c x-1)}{\left(f c^2+g\right) \left(c f+i \sqrt{c^2 f g} x\right)}\right)+\left(\cos ^{-1}\left(\frac{g-c^2 f}{f c^2+g}\right)-2 \tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)\right) \log \left(\frac{2 c f \left(g+i \sqrt{c^2 f g}\right) (c x+1)}{\left(f c^2+g\right) \left(c f+i \sqrt{c^2 f g} x\right)}\right)-\left(\cos ^{-1}\left(\frac{g-c^2 f}{f c^2+g}\right)+2 \left(\tan ^{-1}\left(\frac{c f}{\sqrt{c^2 f g} x}\right)+\tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)\right)\right) \log \left(\frac{\sqrt{2} e^{-\tanh ^{-1}(c x)} \sqrt{c^2 f g}}{\sqrt{f c^2+g} \sqrt{f c^2-g+\left(f c^2+g\right) \cosh \left(2 \tanh ^{-1}(c x)\right)}}\right)-\left(\cos ^{-1}\left(\frac{g-c^2 f}{f c^2+g}\right)-2 \left(\tan ^{-1}\left(\frac{c f}{\sqrt{c^2 f g} x}\right)+\tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)\right)\right) \log \left(\frac{\sqrt{2} e^{\tanh ^{-1}(c x)} \sqrt{c^2 f g}}{\sqrt{f c^2+g} \sqrt{f c^2-g+\left(f c^2+g\right) \cosh \left(2 \tanh ^{-1}(c x)\right)}}\right)+i \left(\text{Li}_2\left(\frac{\left(-f c^2+g-2 i \sqrt{c^2 f g}\right) \left(i c f+\sqrt{c^2 f g} x\right)}{\left(f c^2+g\right) \left(\sqrt{c^2 f g} x-i c f\right)}\right)-\text{Li}_2\left(\frac{\left(-f c^2+g+2 i \sqrt{c^2 f g}\right) \left(i c f+\sqrt{c^2 f g} x\right)}{\left(f c^2+g\right) \left(\sqrt{c^2 f g} x-i c f\right)}\right)\right)\right)}{\sqrt{c^2 f g}}\right)","-\frac{\left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x}+\frac{2 a e \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{f}}-\frac{1}{2} b c \log \left(\frac{g \left(1-c^2 x^2\right)}{c^2 f+g}\right) \left(d+e \log \left(f+g x^2\right)\right)-\frac{1}{2} b c e \text{Li}_2\left(\frac{c^2 \left(g x^2+f\right)}{f c^2+g}\right)+\frac{1}{2} b c \log \left(-\frac{g x^2}{f}\right) \left(d+e \log \left(f+g x^2\right)\right)+\frac{1}{2} b c e \text{Li}_2\left(\frac{g x^2}{f}+1\right)-\frac{b e \sqrt{g} \text{Li}_2\left(-\frac{\sqrt{g} (1-c x)}{c \sqrt{-f}-\sqrt{g}}\right)}{2 \sqrt{-f}}+\frac{b e \sqrt{g} \text{Li}_2\left(\frac{\sqrt{g} (1-c x)}{\sqrt{-f} c+\sqrt{g}}\right)}{2 \sqrt{-f}}-\frac{b e \sqrt{g} \text{Li}_2\left(-\frac{\sqrt{g} (c x+1)}{c \sqrt{-f}-\sqrt{g}}\right)}{2 \sqrt{-f}}+\frac{b e \sqrt{g} \text{Li}_2\left(\frac{\sqrt{g} (c x+1)}{\sqrt{-f} c+\sqrt{g}}\right)}{2 \sqrt{-f}}-\frac{b e \sqrt{g} \log (1-c x) \log \left(\frac{c \left(\sqrt{-f}-\sqrt{g} x\right)}{c \sqrt{-f}-\sqrt{g}}\right)}{2 \sqrt{-f}}+\frac{b e \sqrt{g} \log (c x+1) \log \left(\frac{c \left(\sqrt{-f}-\sqrt{g} x\right)}{c \sqrt{-f}+\sqrt{g}}\right)}{2 \sqrt{-f}}-\frac{b e \sqrt{g} \log (c x+1) \log \left(\frac{c \left(\sqrt{-f}+\sqrt{g} x\right)}{c \sqrt{-f}-\sqrt{g}}\right)}{2 \sqrt{-f}}+\frac{b e \sqrt{g} \log (1-c x) \log \left(\frac{c \left(\sqrt{-f}+\sqrt{g} x\right)}{c \sqrt{-f}+\sqrt{g}}\right)}{2 \sqrt{-f}}",1,"-((a*d)/x) - (b*d*ArcTanh[c*x])/x + b*c*d*Log[x] - (b*c*d*Log[1 - c^2*x^2])/2 + a*e*((2*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[f] - Log[f + g*x^2]/x) + (b*e*(-(((2*ArcTanh[c*x] + c*x*(-2*Log[x] + Log[1 - c^2*x^2]))*Log[f + g*x^2])/x) - 2*c*(Log[x]*(Log[1 - (I*Sqrt[g]*x)/Sqrt[f]] + Log[1 + (I*Sqrt[g]*x)/Sqrt[f]]) + PolyLog[2, ((-I)*Sqrt[g]*x)/Sqrt[f]] + PolyLog[2, (I*Sqrt[g]*x)/Sqrt[f]]) + c*(Log[-c^(-1) + x]*Log[(c*(Sqrt[f] - I*Sqrt[g]*x))/(c*Sqrt[f] - I*Sqrt[g])] + Log[c^(-1) + x]*Log[(c*(Sqrt[f] - I*Sqrt[g]*x))/(c*Sqrt[f] + I*Sqrt[g])] + Log[-c^(-1) + x]*Log[(c*(Sqrt[f] + I*Sqrt[g]*x))/(c*Sqrt[f] + I*Sqrt[g])] - (Log[-c^(-1) + x] + Log[c^(-1) + x] - Log[1 - c^2*x^2])*Log[f + g*x^2] + Log[c^(-1) + x]*Log[1 - (Sqrt[g]*(1 + c*x))/(I*c*Sqrt[f] + Sqrt[g])] + PolyLog[2, (c*Sqrt[g]*(c^(-1) + x))/(I*c*Sqrt[f] + Sqrt[g])] + PolyLog[2, (I*Sqrt[g]*(-1 + c*x))/(c*Sqrt[f] - I*Sqrt[g])] + PolyLog[2, ((-I)*Sqrt[g]*(-1 + c*x))/(c*Sqrt[f] + I*Sqrt[g])] + PolyLog[2, (I*Sqrt[g]*(1 + c*x))/(c*Sqrt[f] + I*Sqrt[g])]) + (c*g*((2*I)*ArcCos[(-(c^2*f) + g)/(c^2*f + g)]*ArcTan[(c*g*x)/Sqrt[c^2*f*g]] - 4*ArcTan[(c*f)/(Sqrt[c^2*f*g]*x)]*ArcTanh[c*x] + (ArcCos[(-(c^2*f) + g)/(c^2*f + g)] + 2*ArcTan[(c*g*x)/Sqrt[c^2*f*g]])*Log[((2*I)*c*f*(I*g + Sqrt[c^2*f*g])*(-1 + c*x))/((c^2*f + g)*(c*f + I*Sqrt[c^2*f*g]*x))] + (ArcCos[(-(c^2*f) + g)/(c^2*f + g)] - 2*ArcTan[(c*g*x)/Sqrt[c^2*f*g]])*Log[(2*c*f*(g + I*Sqrt[c^2*f*g])*(1 + c*x))/((c^2*f + g)*(c*f + I*Sqrt[c^2*f*g]*x))] - (ArcCos[(-(c^2*f) + g)/(c^2*f + g)] + 2*(ArcTan[(c*f)/(Sqrt[c^2*f*g]*x)] + ArcTan[(c*g*x)/Sqrt[c^2*f*g]]))*Log[(Sqrt[2]*Sqrt[c^2*f*g])/(E^ArcTanh[c*x]*Sqrt[c^2*f + g]*Sqrt[c^2*f - g + (c^2*f + g)*Cosh[2*ArcTanh[c*x]]])] - (ArcCos[(-(c^2*f) + g)/(c^2*f + g)] - 2*(ArcTan[(c*f)/(Sqrt[c^2*f*g]*x)] + ArcTan[(c*g*x)/Sqrt[c^2*f*g]]))*Log[(Sqrt[2]*E^ArcTanh[c*x]*Sqrt[c^2*f*g])/(Sqrt[c^2*f + g]*Sqrt[c^2*f - g + (c^2*f + g)*Cosh[2*ArcTanh[c*x]]])] + I*(PolyLog[2, ((-(c^2*f) + g - (2*I)*Sqrt[c^2*f*g])*(I*c*f + Sqrt[c^2*f*g]*x))/((c^2*f + g)*((-I)*c*f + Sqrt[c^2*f*g]*x))] - PolyLog[2, ((-(c^2*f) + g + (2*I)*Sqrt[c^2*f*g])*(I*c*f + Sqrt[c^2*f*g]*x))/((c^2*f + g)*((-I)*c*f + Sqrt[c^2*f*g]*x))])))/Sqrt[c^2*f*g]))/2","C",0
537,1,982,470,4.9675842,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x^3} \, dx","Integrate[((a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]))/x^3,x]","-\frac{-4 b e g \tanh ^{-1}(c x)^2 x^2-4 b c e \sqrt{f} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right) x^2-2 b c^2 d f \tanh ^{-1}(c x) x^2-4 i b c^2 e f \sin ^{-1}\left(\sqrt{\frac{c^2 f}{f c^2+g}}\right) \tanh ^{-1}\left(\frac{c g x}{\sqrt{-c^2 f g}}\right) x^2-4 b e g \tanh ^{-1}(c x) \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right) x^2-4 b c^2 e f \tanh ^{-1}(c x) \log \left(1+e^{-2 \tanh ^{-1}(c x)}\right) x^2-2 i b c^2 e f \sin ^{-1}\left(\sqrt{\frac{c^2 f}{f c^2+g}}\right) \log \left(\frac{e^{-2 \tanh ^{-1}(c x)} \left(\left(1+e^{2 \tanh ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 \tanh ^{-1}(c x)}\right) g-2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right) x^2+2 b c^2 e f \tanh ^{-1}(c x) \log \left(\frac{e^{-2 \tanh ^{-1}(c x)} \left(\left(1+e^{2 \tanh ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 \tanh ^{-1}(c x)}\right) g-2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right) x^2+2 i b c^2 e f \sin ^{-1}\left(\sqrt{\frac{c^2 f}{f c^2+g}}\right) \log \left(\frac{e^{-2 \tanh ^{-1}(c x)} \left(\left(1+e^{2 \tanh ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 \tanh ^{-1}(c x)}\right) g+2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right) x^2+2 b c^2 e f \tanh ^{-1}(c x) \log \left(\frac{e^{-2 \tanh ^{-1}(c x)} \left(\left(1+e^{2 \tanh ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 \tanh ^{-1}(c x)}\right) g+2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right) x^2+2 b e g \tanh ^{-1}(c x) \log \left(\frac{e^{2 \tanh ^{-1}(c x)} \left(f c^2+g\right)}{f c^2-2 \sqrt{-f} \sqrt{g} c-g}+1\right) x^2+2 b e g \tanh ^{-1}(c x) \log \left(\frac{e^{2 \tanh ^{-1}(c x)} \left(f c^2+g\right)}{f c^2+2 \sqrt{-f} \sqrt{g} c-g}+1\right) x^2-4 a e g \log (x) x^2+2 a e g \log \left(g x^2+f\right) x^2-2 b c^2 e f \tanh ^{-1}(c x) \log \left(g x^2+f\right) x^2+2 b c^2 e f \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right) x^2+2 b e g \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right) x^2+b e g \text{Li}_2\left(-\frac{e^{2 \tanh ^{-1}(c x)} \left(f c^2+g\right)}{f c^2-2 \sqrt{-f} \sqrt{g} c-g}\right) x^2+b e g \text{Li}_2\left(-\frac{e^{2 \tanh ^{-1}(c x)} \left(f c^2+g\right)}{f c^2+2 \sqrt{-f} \sqrt{g} c-g}\right) x^2-b c^2 e f \text{Li}_2\left(\frac{e^{-2 \tanh ^{-1}(c x)} \left(-f c^2+g-2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right) x^2-b c^2 e f \text{Li}_2\left(\frac{e^{-2 \tanh ^{-1}(c x)} \left(-f c^2+g+2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right) x^2+2 b c d f x+2 b c e f \log \left(g x^2+f\right) x+2 a d f+2 b d f \tanh ^{-1}(c x)+2 a e f \log \left(g x^2+f\right)+2 b e f \tanh ^{-1}(c x) \log \left(g x^2+f\right)}{4 f x^2}","-\frac{\left(a+b \tanh ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{2 x^2}-\frac{a e g \log \left(f+g x^2\right)}{2 f}+\frac{a e g \log (x)}{f}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left(d+e \log \left(f+g x^2\right)\right)-\frac{b e \left(c^2 f+g\right) \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 f}+\frac{b e \left(c^2 f+g\right) \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{\left(c \sqrt{-f}-\sqrt{g}\right) (c x+1)}\right)}{4 f}+\frac{b e \left(c^2 f+g\right) \text{Li}_2\left(1-\frac{2 c \left(\sqrt{g} x+\sqrt{-f}\right)}{\left(\sqrt{-f} c+\sqrt{g}\right) (c x+1)}\right)}{4 f}+\frac{b e \left(c^2 f+g\right) \log \left(\frac{2}{c x+1}\right) \tanh ^{-1}(c x)}{f}-\frac{b e \left(c^2 f+g\right) \tanh ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{(c x+1) \left(c \sqrt{-f}-\sqrt{g}\right)}\right)}{2 f}-\frac{b e \left(c^2 f+g\right) \tanh ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}+\sqrt{g} x\right)}{(c x+1) \left(c \sqrt{-f}+\sqrt{g}\right)}\right)}{2 f}-\frac{b c \left(d+e \log \left(f+g x^2\right)\right)}{2 x}-\frac{b e g \text{Li}_2(-c x)}{2 f}+\frac{b e g \text{Li}_2(c x)}{2 f}+\frac{b c e \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{f}}",1,"-1/4*(2*a*d*f + 2*b*c*d*f*x - 4*b*c*e*Sqrt[f]*Sqrt[g]*x^2*ArcTan[(Sqrt[g]*x)/Sqrt[f]] + 2*b*d*f*ArcTanh[c*x] - 2*b*c^2*d*f*x^2*ArcTanh[c*x] - 4*b*e*g*x^2*ArcTanh[c*x]^2 - (4*I)*b*c^2*e*f*x^2*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]]*ArcTanh[(c*g*x)/Sqrt[-(c^2*f*g)]] - 4*b*e*g*x^2*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - 4*b*c^2*e*f*x^2*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - (2*I)*b*c^2*e*f*x^2*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]]*Log[(c^2*(1 + E^(2*ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))] + 2*b*c^2*e*f*x^2*ArcTanh[c*x]*Log[(c^2*(1 + E^(2*ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))] + (2*I)*b*c^2*e*f*x^2*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]]*Log[(c^2*(1 + E^(2*ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))] + 2*b*c^2*e*f*x^2*ArcTanh[c*x]*Log[(c^2*(1 + E^(2*ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))] + 2*b*e*g*x^2*ArcTanh[c*x]*Log[1 + (E^(2*ArcTanh[c*x])*(c^2*f + g))/(c^2*f - 2*c*Sqrt[-f]*Sqrt[g] - g)] + 2*b*e*g*x^2*ArcTanh[c*x]*Log[1 + (E^(2*ArcTanh[c*x])*(c^2*f + g))/(c^2*f + 2*c*Sqrt[-f]*Sqrt[g] - g)] - 4*a*e*g*x^2*Log[x] + 2*a*e*f*Log[f + g*x^2] + 2*b*c*e*f*x*Log[f + g*x^2] + 2*a*e*g*x^2*Log[f + g*x^2] + 2*b*e*f*ArcTanh[c*x]*Log[f + g*x^2] - 2*b*c^2*e*f*x^2*ArcTanh[c*x]*Log[f + g*x^2] + 2*b*c^2*e*f*x^2*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 2*b*e*g*x^2*PolyLog[2, E^(-2*ArcTanh[c*x])] + b*e*g*x^2*PolyLog[2, -((E^(2*ArcTanh[c*x])*(c^2*f + g))/(c^2*f - 2*c*Sqrt[-f]*Sqrt[g] - g))] + b*e*g*x^2*PolyLog[2, -((E^(2*ArcTanh[c*x])*(c^2*f + g))/(c^2*f + 2*c*Sqrt[-f]*Sqrt[g] - g))] - b*c^2*e*f*x^2*PolyLog[2, (-(c^2*f) + g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))] - b*c^2*e*f*x^2*PolyLog[2, (-(c^2*f) + g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))])/(f*x^2)","C",0
538,1,70,78,0.0895319,"\int \frac{\tanh ^{-1}(c x) \left(a+b \tanh ^{-1}(c x)\right)}{(1+c x)^2} \, dx","Integrate[(ArcTanh[c*x]*(a + b*ArcTanh[c*x]))/(1 + c*x)^2,x]","-\frac{(a+b) ((c x+1) \log (1-c x)-(c x+1) \log (c x+1)+2)+4 (a+b) \tanh ^{-1}(c x)-2 b (c x-1) \tanh ^{-1}(c x)^2}{4 c (c x+1)}","-\frac{a+b}{2 c (c x+1)}+\frac{(a+b) \tanh ^{-1}(c x)}{2 c}-\frac{(a+b) \tanh ^{-1}(c x)}{c (c x+1)}-\frac{b (1-c x) \tanh ^{-1}(c x)^2}{2 c (c x+1)}",1,"-1/4*(4*(a + b)*ArcTanh[c*x] - 2*b*(-1 + c*x)*ArcTanh[c*x]^2 + (a + b)*(2 + (1 + c*x)*Log[1 - c*x] - (1 + c*x)*Log[1 + c*x]))/(c*(1 + c*x))","A",1