1,1,81,59,0.0106515,"\int x^5 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x^5*(a + b*ArcTanh[c*x]),x]","\frac{a x^6}{6}+\frac{b \log (1-c x)}{12 c^6}-\frac{b \log (c x+1)}{12 c^6}+\frac{b x}{6 c^5}+\frac{b x^3}{18 c^3}+\frac{1}{6} b x^6 \tanh ^{-1}(c x)+\frac{b x^5}{30 c}","\frac{1}{6} x^6 \left(a+b \tanh ^{-1}(c x)\right)-\frac{b \tanh ^{-1}(c x)}{6 c^6}+\frac{b x}{6 c^5}+\frac{b x^3}{18 c^3}+\frac{b x^5}{30 c}",1,"(b*x)/(6*c^5) + (b*x^3)/(18*c^3) + (b*x^5)/(30*c) + (a*x^6)/6 + (b*x^6*ArcTanh[c*x])/6 + (b*Log[1 - c*x])/(12*c^6) - (b*Log[1 + c*x])/(12*c^6)","A",1
2,1,62,57,0.0091184,"\int x^4 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x^4*(a + b*ArcTanh[c*x]),x]","\frac{a x^5}{5}+\frac{b x^2}{10 c^3}+\frac{b \log \left(1-c^2 x^2\right)}{10 c^5}+\frac{1}{5} b x^5 \tanh ^{-1}(c x)+\frac{b x^4}{20 c}","\frac{1}{5} x^5 \left(a+b \tanh ^{-1}(c x)\right)+\frac{b x^2}{10 c^3}+\frac{b \log \left(1-c^2 x^2\right)}{10 c^5}+\frac{b x^4}{20 c}",1,"(b*x^2)/(10*c^3) + (b*x^4)/(20*c) + (a*x^5)/5 + (b*x^5*ArcTanh[c*x])/5 + (b*Log[1 - c^2*x^2])/(10*c^5)","A",1
3,1,70,48,0.0095441,"\int x^3 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x^3*(a + b*ArcTanh[c*x]),x]","\frac{a x^4}{4}+\frac{b \log (1-c x)}{8 c^4}-\frac{b \log (c x+1)}{8 c^4}+\frac{b x}{4 c^3}+\frac{1}{4} b x^4 \tanh ^{-1}(c x)+\frac{b x^3}{12 c}","\frac{1}{4} x^4 \left(a+b \tanh ^{-1}(c x)\right)-\frac{b \tanh ^{-1}(c x)}{4 c^4}+\frac{b x}{4 c^3}+\frac{b x^3}{12 c}",1,"(b*x)/(4*c^3) + (b*x^3)/(12*c) + (a*x^4)/4 + (b*x^4*ArcTanh[c*x])/4 + (b*Log[1 - c*x])/(8*c^4) - (b*Log[1 + c*x])/(8*c^4)","A",1
4,1,51,46,0.0085972,"\int x^2 \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x^2*(a + b*ArcTanh[c*x]),x]","\frac{a x^3}{3}+\frac{b \log \left(1-c^2 x^2\right)}{6 c^3}+\frac{1}{3} b x^3 \tanh ^{-1}(c x)+\frac{b x^2}{6 c}","\frac{1}{3} x^3 \left(a+b \tanh ^{-1}(c x)\right)+\frac{b \log \left(1-c^2 x^2\right)}{6 c^3}+\frac{b x^2}{6 c}",1,"(b*x^2)/(6*c) + (a*x^3)/3 + (b*x^3*ArcTanh[c*x])/3 + (b*Log[1 - c^2*x^2])/(6*c^3)","A",1
5,1,59,37,0.0079698,"\int x \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[x*(a + b*ArcTanh[c*x]),x]","\frac{a x^2}{2}+\frac{b \log (1-c x)}{4 c^2}-\frac{b \log (c x+1)}{4 c^2}+\frac{1}{2} b x^2 \tanh ^{-1}(c x)+\frac{b x}{2 c}","\frac{1}{2} x^2 \left(a+b \tanh ^{-1}(c x)\right)-\frac{b \tanh ^{-1}(c x)}{2 c^2}+\frac{b x}{2 c}",1,"(b*x)/(2*c) + (a*x^2)/2 + (b*x^2*ArcTanh[c*x])/2 + (b*Log[1 - c*x])/(4*c^2) - (b*Log[1 + c*x])/(4*c^2)","A",1
6,1,30,30,0.0037335,"\int \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[a + b*ArcTanh[c*x],x]","a x+\frac{b \log \left(1-c^2 x^2\right)}{2 c}+b x \tanh ^{-1}(c x)","a x+\frac{b \log \left(1-c^2 x^2\right)}{2 c}+b x \tanh ^{-1}(c x)",1,"a*x + b*x*ArcTanh[c*x] + (b*Log[1 - c^2*x^2])/(2*c)","A",1
7,1,24,26,0.010974,"\int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx","Integrate[(a + b*ArcTanh[c*x])/x,x]","a \log (x)+\frac{1}{2} b (\text{Li}_2(c x)-\text{Li}_2(-c x))","a \log (x)-\frac{1}{2} b \text{Li}_2(-c x)+\frac{1}{2} b \text{Li}_2(c x)",1,"a*Log[x] + (b*(-PolyLog[2, -(c*x)] + PolyLog[2, c*x]))/2","A",1
8,1,39,36,0.0081548,"\int \frac{a+b \tanh ^{-1}(c x)}{x^2} \, dx","Integrate[(a + b*ArcTanh[c*x])/x^2,x]","-\frac{a}{x}-\frac{1}{2} b c \log \left(1-c^2 x^2\right)+b c \log (x)-\frac{b \tanh ^{-1}(c x)}{x}","-\frac{a+b \tanh ^{-1}(c x)}{x}-\frac{1}{2} b c \log \left(1-c^2 x^2\right)+b c \log (x)",1,"-(a/x) - (b*ArcTanh[c*x])/x + b*c*Log[x] - (b*c*Log[1 - c^2*x^2])/2","A",1
9,1,59,37,0.0091213,"\int \frac{a+b \tanh ^{-1}(c x)}{x^3} \, dx","Integrate[(a + b*ArcTanh[c*x])/x^3,x]","-\frac{a}{2 x^2}-\frac{1}{4} b c^2 \log (1-c x)+\frac{1}{4} b c^2 \log (c x+1)-\frac{b \tanh ^{-1}(c x)}{2 x^2}-\frac{b c}{2 x}","-\frac{a+b \tanh ^{-1}(c x)}{2 x^2}+\frac{1}{2} b c^2 \tanh ^{-1}(c x)-\frac{b c}{2 x}",1,"-1/2*a/x^2 - (b*c)/(2*x) - (b*ArcTanh[c*x])/(2*x^2) - (b*c^2*Log[1 - c*x])/4 + (b*c^2*Log[1 + c*x])/4","A",1
10,1,59,54,0.0095901,"\int \frac{a+b \tanh ^{-1}(c x)}{x^4} \, dx","Integrate[(a + b*ArcTanh[c*x])/x^4,x]","-\frac{a}{3 x^3}+\frac{1}{3} b c^3 \log (x)-\frac{1}{6} b c^3 \log \left(1-c^2 x^2\right)-\frac{b \tanh ^{-1}(c x)}{3 x^3}-\frac{b c}{6 x^2}","-\frac{a+b \tanh ^{-1}(c x)}{3 x^3}+\frac{1}{3} b c^3 \log (x)-\frac{1}{6} b c^3 \log \left(1-c^2 x^2\right)-\frac{b c}{6 x^2}",1,"-1/3*a/x^3 - (b*c)/(6*x^2) - (b*ArcTanh[c*x])/(3*x^3) + (b*c^3*Log[x])/3 - (b*c^3*Log[1 - c^2*x^2])/6","A",1
11,1,70,48,0.0090519,"\int \frac{a+b \tanh ^{-1}(c x)}{x^5} \, dx","Integrate[(a + b*ArcTanh[c*x])/x^5,x]","-\frac{a}{4 x^4}-\frac{1}{8} b c^4 \log (1-c x)+\frac{1}{8} b c^4 \log (c x+1)-\frac{b c^3}{4 x}-\frac{b \tanh ^{-1}(c x)}{4 x^4}-\frac{b c}{12 x^3}","-\frac{a+b \tanh ^{-1}(c x)}{4 x^4}+\frac{1}{4} b c^4 \tanh ^{-1}(c x)-\frac{b c^3}{4 x}-\frac{b c}{12 x^3}",1,"-1/4*a/x^4 - (b*c)/(12*x^3) - (b*c^3)/(4*x) - (b*ArcTanh[c*x])/(4*x^4) - (b*c^4*Log[1 - c*x])/8 + (b*c^4*Log[1 + c*x])/8","A",1
12,1,70,65,0.0095952,"\int \frac{a+b \tanh ^{-1}(c x)}{x^6} \, dx","Integrate[(a + b*ArcTanh[c*x])/x^6,x]","-\frac{a}{5 x^5}+\frac{1}{5} b c^5 \log (x)-\frac{b c^3}{10 x^2}-\frac{1}{10} b c^5 \log \left(1-c^2 x^2\right)-\frac{b \tanh ^{-1}(c x)}{5 x^5}-\frac{b c}{20 x^4}","-\frac{a+b \tanh ^{-1}(c x)}{5 x^5}+\frac{1}{5} b c^5 \log (x)-\frac{b c^3}{10 x^2}-\frac{1}{10} b c^5 \log \left(1-c^2 x^2\right)-\frac{b c}{20 x^4}",1,"-1/5*a/x^5 - (b*c)/(20*x^4) - (b*c^3)/(10*x^2) - (b*ArcTanh[c*x])/(5*x^5) + (b*c^5*Log[x])/5 - (b*c^5*Log[1 - c^2*x^2])/10","A",1
13,1,164,145,0.0733789,"\int x^5 \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[x^5*(a + b*ArcTanh[c*x])^2,x]","\frac{30 a^2 c^6 x^6+12 a b c^5 x^5+20 a b c^3 x^3+4 b c x \tanh ^{-1}(c x) \left(15 a c^5 x^5+b \left(3 c^4 x^4+5 c^2 x^2+15\right)\right)+60 a b c x+2 b (15 a+23 b) \log (1-c x)-30 a b \log (c x+1)+30 b^2 \left(c^6 x^6-1\right) \tanh ^{-1}(c x)^2+3 b^2 c^4 x^4+16 b^2 c^2 x^2+46 b^2 \log (c x+1)}{180 c^6}","-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{6 c^6}+\frac{a b x}{3 c^5}+\frac{b x^3 \left(a+b \tanh ^{-1}(c x)\right)}{9 c^3}+\frac{1}{6} x^6 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{b x^5 \left(a+b \tanh ^{-1}(c x)\right)}{15 c}+\frac{b^2 x \tanh ^{-1}(c x)}{3 c^5}+\frac{4 b^2 x^2}{45 c^4}+\frac{b^2 x^4}{60 c^2}+\frac{23 b^2 \log \left(1-c^2 x^2\right)}{90 c^6}",1,"(60*a*b*c*x + 16*b^2*c^2*x^2 + 20*a*b*c^3*x^3 + 3*b^2*c^4*x^4 + 12*a*b*c^5*x^5 + 30*a^2*c^6*x^6 + 4*b*c*x*(15*a*c^5*x^5 + b*(15 + 5*c^2*x^2 + 3*c^4*x^4))*ArcTanh[c*x] + 30*b^2*(-1 + c^6*x^6)*ArcTanh[c*x]^2 + 2*b*(15*a + 23*b)*Log[1 - c*x] - 30*a*b*Log[1 + c*x] + 46*b^2*Log[1 + c*x])/(180*c^6)","A",1
14,1,161,162,0.5002401,"\int x^4 \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[x^4*(a + b*ArcTanh[c*x])^2,x]","\frac{6 a^2 c^5 x^5+3 a b c^4 x^4+6 a b c^2 x^2+6 a b \log \left(c^2 x^2-1\right)+3 b \tanh ^{-1}(c x) \left(4 a c^5 x^5+b \left(c^4 x^4+2 c^2 x^2-3\right)-4 b \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)-9 a b+6 b^2 \left(c^5 x^5-1\right) \tanh ^{-1}(c x)^2+b^2 c^3 x^3+6 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+9 b^2 c x}{30 c^5}","\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{5 c^5}-\frac{2 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{5 c^5}+\frac{b x^2 \left(a+b \tanh ^{-1}(c x)\right)}{5 c^3}+\frac{1}{5} x^5 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{b x^4 \left(a+b \tanh ^{-1}(c x)\right)}{10 c}-\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{5 c^5}-\frac{3 b^2 \tanh ^{-1}(c x)}{10 c^5}+\frac{3 b^2 x}{10 c^4}+\frac{b^2 x^3}{30 c^2}",1,"(-9*a*b + 9*b^2*c*x + 6*a*b*c^2*x^2 + b^2*c^3*x^3 + 3*a*b*c^4*x^4 + 6*a^2*c^5*x^5 + 6*b^2*(-1 + c^5*x^5)*ArcTanh[c*x]^2 + 3*b*ArcTanh[c*x]*(4*a*c^5*x^5 + b*(-3 + 2*c^2*x^2 + c^4*x^4) - 4*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 6*a*b*Log[-1 + c^2*x^2] + 6*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(30*c^5)","A",0
15,1,132,113,0.0644211,"\int x^3 \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(a + b*ArcTanh[c*x])^2,x]","\frac{3 a^2 c^4 x^4+2 a b c^3 x^3+2 b c x \tanh ^{-1}(c x) \left(3 a c^3 x^3+b \left(c^2 x^2+3\right)\right)+6 a b c x+b (3 a+4 b) \log (1-c x)-3 a b \log (c x+1)+3 b^2 \left(c^4 x^4-1\right) \tanh ^{-1}(c x)^2+b^2 c^2 x^2+4 b^2 \log (c x+1)}{12 c^4}","-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{4 c^4}+\frac{a b x}{2 c^3}+\frac{1}{4} x^4 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{b x^3 \left(a+b \tanh ^{-1}(c x)\right)}{6 c}+\frac{b^2 x \tanh ^{-1}(c x)}{2 c^3}+\frac{b^2 x^2}{12 c^2}+\frac{b^2 \log \left(1-c^2 x^2\right)}{3 c^4}",1,"(6*a*b*c*x + b^2*c^2*x^2 + 2*a*b*c^3*x^3 + 3*a^2*c^4*x^4 + 2*b*c*x*(3*a*c^3*x^3 + b*(3 + c^2*x^2))*ArcTanh[c*x] + 3*b^2*(-1 + c^4*x^4)*ArcTanh[c*x]^2 + b*(3*a + 4*b)*Log[1 - c*x] - 3*a*b*Log[1 + c*x] + 4*b^2*Log[1 + c*x])/(12*c^4)","A",1
16,1,122,130,0.2760591,"\int x^2 \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(a + b*ArcTanh[c*x])^2,x]","\frac{a^2 c^3 x^3+a b c^2 x^2+a b \log \left(c^2 x^2-1\right)+b \tanh ^{-1}(c x) \left(2 a c^3 x^3+b c^2 x^2-2 b \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-b\right)+b^2 \left(c^3 x^3-1\right) \tanh ^{-1}(c x)^2+b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+b^2 c x}{3 c^3}","\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{3 c^3}-\frac{2 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{3 c^3}+\frac{1}{3} x^3 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{b x^2 \left(a+b \tanh ^{-1}(c x)\right)}{3 c}-\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{3 c^3}-\frac{b^2 \tanh ^{-1}(c x)}{3 c^3}+\frac{b^2 x}{3 c^2}",1,"(b^2*c*x + a*b*c^2*x^2 + a^2*c^3*x^3 + b^2*(-1 + c^3*x^3)*ArcTanh[c*x]^2 + b*ArcTanh[c*x]*(-b + b*c^2*x^2 + 2*a*c^3*x^3 - 2*b*Log[1 + E^(-2*ArcTanh[c*x])]) + a*b*Log[-1 + c^2*x^2] + b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(3*c^3)","A",0
17,1,90,75,0.067062,"\int x \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[x*(a + b*ArcTanh[c*x])^2,x]","\frac{a^2 c^2 x^2+2 a b c x+b (a+b) \log (1-c x)-a b \log (c x+1)+2 b c x \tanh ^{-1}(c x) (a c x+b)+b^2 \left(c^2 x^2-1\right) \tanh ^{-1}(c x)^2+b^2 \log (c x+1)}{2 c^2}","-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^2}+\frac{1}{2} x^2 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{a b x}{c}+\frac{b^2 \log \left(1-c^2 x^2\right)}{2 c^2}+\frac{b^2 x \tanh ^{-1}(c x)}{c}",1,"(2*a*b*c*x + a^2*c^2*x^2 + 2*b*c*x*(b + a*c*x)*ArcTanh[c*x] + b^2*(-1 + c^2*x^2)*ArcTanh[c*x]^2 + b*(a + b)*Log[1 - c*x] - a*b*Log[1 + c*x] + b^2*Log[1 + c*x])/(2*c^2)","A",1
18,1,82,74,0.1488878,"\int \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[(a + b*ArcTanh[c*x])^2,x]","\frac{a \left(a c x+b \log \left(1-c^2 x^2\right)\right)+2 b \tanh ^{-1}(c x) \left(a c x-b \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+b^2 (c x-1) \tanh ^{-1}(c x)^2}{c}","x \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{c}-\frac{2 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c}-\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c}",1,"(b^2*(-1 + c*x)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(a*c*x - b*Log[1 + E^(-2*ArcTanh[c*x])]) + a*(a*c*x + b*Log[1 - c^2*x^2]) + b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])])/c","A",0
19,1,120,117,0.0750655,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/x,x]","\frac{1}{2} b \left(2 \text{Li}_2\left(\frac{c x+1}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)-2 \text{Li}_2\left(\frac{c x+1}{c x-1}\right) \left(a+b \tanh ^{-1}(c x)\right)+b \left(\text{Li}_3\left(\frac{c x+1}{c x-1}\right)-\text{Li}_3\left(\frac{c x+1}{1-c x}\right)\right)\right)+2 \tanh ^{-1}\left(\frac{c x+1}{c x-1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2","-b \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+b \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)+2 \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{2} b^2 \text{Li}_3\left(1-\frac{2}{1-c x}\right)-\frac{1}{2} b^2 \text{Li}_3\left(\frac{2}{1-c x}-1\right)",1,"2*(a + b*ArcTanh[c*x])^2*ArcTanh[(1 + c*x)/(-1 + c*x)] + (b*(2*(a + b*ArcTanh[c*x])*PolyLog[2, (1 + c*x)/(1 - c*x)] - 2*(a + b*ArcTanh[c*x])*PolyLog[2, (1 + c*x)/(-1 + c*x)] + b*(-PolyLog[3, (1 + c*x)/(1 - c*x)] + PolyLog[3, (1 + c*x)/(-1 + c*x)])))/2","A",1
20,1,94,71,0.1527597,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/x^2,x]","\frac{-a \left(a+b c x \log \left(1-c^2 x^2\right)-2 b c x \log (c x)\right)+2 b \tanh ^{-1}(c x) \left(b c x \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-a\right)-b^2 c x \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+b^2 (c x-1) \tanh ^{-1}(c x)^2}{x}","c \left(a+b \tanh ^{-1}(c x)\right)^2-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x}+2 b c \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)+b^2 (-c) \text{Li}_2\left(\frac{2}{c x+1}-1\right)",1,"(b^2*(-1 + c*x)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(-a + b*c*x*Log[1 - E^(-2*ArcTanh[c*x])]) - a*(a - 2*b*c*x*Log[c*x] + b*c*x*Log[1 - c^2*x^2]) - b^2*c*x*PolyLog[2, E^(-2*ArcTanh[c*x])])/x","A",0
21,1,101,80,0.0724415,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/x^3,x]","-\frac{a^2+b c^2 x^2 (a+b) \log (1-c x)-b c^2 x^2 (a-b) \log (c x+1)+2 a b c x+2 b \tanh ^{-1}(c x) (a+b c x)-2 b^2 c^2 x^2 \log (x)-b^2 \left(c^2 x^2-1\right) \tanh ^{-1}(c x)^2}{2 x^2}","\frac{1}{2} c^2 \left(a+b \tanh ^{-1}(c x)\right)^2-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{2 x^2}-\frac{b c \left(a+b \tanh ^{-1}(c x)\right)}{x}-\frac{1}{2} b^2 c^2 \log \left(1-c^2 x^2\right)+b^2 c^2 \log (x)",1,"-1/2*(a^2 + 2*a*b*c*x + 2*b*(a + b*c*x)*ArcTanh[c*x] - b^2*(-1 + c^2*x^2)*ArcTanh[c*x]^2 - 2*b^2*c^2*x^2*Log[x] + b*(a + b)*c^2*x^2*Log[1 - c*x] - (a - b)*b*c^2*x^2*Log[1 + c*x])/x^2","A",1
22,1,145,130,0.3793244,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/x^4,x]","-\frac{a^2-2 a b c^3 x^3 \log (c x)+b \tanh ^{-1}(c x) \left(2 a-b c^3 x^3-2 b c^3 x^3 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+b c x\right)+a b c^3 x^3 \log \left(1-c^2 x^2\right)+a b c x+b^2 c^3 x^3 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+b^2 \left(1-c^3 x^3\right) \tanh ^{-1}(c x)^2+b^2 c^2 x^2}{3 x^3}","\frac{1}{3} c^3 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{2}{3} b c^3 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{3 x^3}-\frac{b c \left(a+b \tanh ^{-1}(c x)\right)}{3 x^2}-\frac{1}{3} b^2 c^3 \text{Li}_2\left(\frac{2}{c x+1}-1\right)+\frac{1}{3} b^2 c^3 \tanh ^{-1}(c x)-\frac{b^2 c^2}{3 x}",1,"-1/3*(a^2 + a*b*c*x + b^2*c^2*x^2 + b^2*(1 - c^3*x^3)*ArcTanh[c*x]^2 + b*ArcTanh[c*x]*(2*a + b*c*x - b*c^3*x^3 - 2*b*c^3*x^3*Log[1 - E^(-2*ArcTanh[c*x])]) - 2*a*b*c^3*x^3*Log[c*x] + a*b*c^3*x^3*Log[1 - c^2*x^2] + b^2*c^3*x^3*PolyLog[2, E^(-2*ArcTanh[c*x])])/x^3","A",0
23,1,164,117,0.0758434,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{x^5} \, dx","Integrate[(a + b*ArcTanh[c*x])^2/x^5,x]","-\frac{3 a^2+3 a b c^4 x^4 \log (1-c x)-3 a b c^4 x^4 \log (c x+1)+6 a b c^3 x^3+2 b \tanh ^{-1}(c x) \left(3 a+3 b c^3 x^3+b c x\right)+2 a b c x-8 b^2 c^4 x^4 \log (x)+4 b^2 c^4 x^4 \log (1-c x)+4 b^2 c^4 x^4 \log (c x+1)-3 b^2 \left(c^4 x^4-1\right) \tanh ^{-1}(c x)^2+b^2 c^2 x^2}{12 x^4}","\frac{1}{4} c^4 \left(a+b \tanh ^{-1}(c x)\right)^2-\frac{b c^3 \left(a+b \tanh ^{-1}(c x)\right)}{2 x}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^2}{4 x^4}-\frac{b c \left(a+b \tanh ^{-1}(c x)\right)}{6 x^3}+\frac{2}{3} b^2 c^4 \log (x)-\frac{b^2 c^2}{12 x^2}-\frac{1}{3} b^2 c^4 \log \left(1-c^2 x^2\right)",1,"-1/12*(3*a^2 + 2*a*b*c*x + b^2*c^2*x^2 + 6*a*b*c^3*x^3 + 2*b*(3*a + b*c*x + 3*b*c^3*x^3)*ArcTanh[c*x] - 3*b^2*(-1 + c^4*x^4)*ArcTanh[c*x]^2 - 8*b^2*c^4*x^4*Log[x] + 3*a*b*c^4*x^4*Log[1 - c*x] + 4*b^2*c^4*x^4*Log[1 - c*x] - 3*a*b*c^4*x^4*Log[1 + c*x] + 4*b^2*c^4*x^4*Log[1 + c*x])/x^4","A",1
24,1,305,247,0.7822345,"\int x^5 \left(a+b \tanh ^{-1}(c x)\right)^3 \, dx","Integrate[x^5*(a + b*ArcTanh[c*x])^3,x]","\frac{10 a^3 c^6 x^6+b \tanh ^{-1}(c x) \left(30 a^2 c^6 x^6+4 a b c x \left(3 c^4 x^4+5 c^2 x^2+15\right)+b^2 \left(3 c^4 x^4+16 c^2 x^2-19\right)-92 b^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+6 a^2 b c^5 x^5+10 a^2 b c^3 x^3+30 a^2 b c x+15 a^2 b \log (1-c x)-15 a^2 b \log (c x+1)+3 a b^2 c^4 x^4+16 a b^2 c^2 x^2+46 a b^2 \log \left(1-c^2 x^2\right)+2 b^2 \tanh ^{-1}(c x)^2 \left(15 a \left(c^6 x^6-1\right)+b \left(3 c^5 x^5+5 c^3 x^3+15 c x-23\right)\right)-19 a b^2+10 b^3 \left(c^6 x^6-1\right) \tanh ^{-1}(c x)^3+b^3 c^3 x^3+46 b^3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+19 b^3 c x}{60 c^6}","-\frac{23 b^2 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{15 c^6}+\frac{4 b^2 x^2 \left(a+b \tanh ^{-1}(c x)\right)}{15 c^4}+\frac{b^2 x^4 \left(a+b \tanh ^{-1}(c x)\right)}{20 c^2}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{6 c^6}+\frac{23 b \left(a+b \tanh ^{-1}(c x)\right)^2}{30 c^6}+\frac{b x \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^5}+\frac{b x^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{6 c^3}+\frac{1}{6} x^6 \left(a+b \tanh ^{-1}(c x)\right)^3+\frac{b x^5 \left(a+b \tanh ^{-1}(c x)\right)^2}{10 c}-\frac{23 b^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{30 c^6}-\frac{19 b^3 \tanh ^{-1}(c x)}{60 c^6}+\frac{19 b^3 x}{60 c^5}+\frac{b^3 x^3}{60 c^3}",1,"(-19*a*b^2 + 30*a^2*b*c*x + 19*b^3*c*x + 16*a*b^2*c^2*x^2 + 10*a^2*b*c^3*x^3 + b^3*c^3*x^3 + 3*a*b^2*c^4*x^4 + 6*a^2*b*c^5*x^5 + 10*a^3*c^6*x^6 + 2*b^2*(b*(-23 + 15*c*x + 5*c^3*x^3 + 3*c^5*x^5) + 15*a*(-1 + c^6*x^6))*ArcTanh[c*x]^2 + 10*b^3*(-1 + c^6*x^6)*ArcTanh[c*x]^3 + b*ArcTanh[c*x]*(30*a^2*c^6*x^6 + 4*a*b*c*x*(15 + 5*c^2*x^2 + 3*c^4*x^4) + b^2*(-19 + 16*c^2*x^2 + 3*c^4*x^4) - 92*b^2*Log[1 + E^(-2*ArcTanh[c*x])]) + 15*a^2*b*Log[1 - c*x] - 15*a^2*b*Log[1 + c*x] + 46*a*b^2*Log[1 - c^2*x^2] + 46*b^3*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(60*c^6)","A",0
25,1,383,262,0.808902,"\int x^4 \left(a+b \tanh ^{-1}(c x)\right)^3 \, dx","Integrate[x^4*(a + b*ArcTanh[c*x])^3,x]","\frac{4 a^3 c^5 x^5+12 a^2 b c^5 x^5 \tanh ^{-1}(c x)+3 a^2 b c^4 x^4+6 a^2 b c^2 x^2+6 a^2 b \log \left(1-c^2 x^2\right)+12 a b^2 c^5 x^5 \tanh ^{-1}(c x)^2+6 a b^2 c^4 x^4 \tanh ^{-1}(c x)+2 a b^2 c^3 x^3+12 a b^2 c^2 x^2 \tanh ^{-1}(c x)+12 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right) \left(a+b \tanh ^{-1}(c x)\right)+18 a b^2 c x-12 a b^2 \tanh ^{-1}(c x)^2-18 a b^2 \tanh ^{-1}(c x)-24 a b^2 \tanh ^{-1}(c x) \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)+4 b^3 c^5 x^5 \tanh ^{-1}(c x)^3+3 b^3 c^4 x^4 \tanh ^{-1}(c x)^2+2 b^3 c^3 x^3 \tanh ^{-1}(c x)+b^3 c^2 x^2+10 b^3 \log \left(1-c^2 x^2\right)+6 b^3 c^2 x^2 \tanh ^{-1}(c x)^2+6 b^3 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)+18 b^3 c x \tanh ^{-1}(c x)-4 b^3 \tanh ^{-1}(c x)^3-9 b^3 \tanh ^{-1}(c x)^2-12 b^3 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-b^3}{20 c^5}","-\frac{3 b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{5 c^5}+\frac{9 a b^2 x}{10 c^4}+\frac{b^2 x^3 \left(a+b \tanh ^{-1}(c x)\right)}{10 c^2}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{5 c^5}-\frac{9 b \left(a+b \tanh ^{-1}(c x)\right)^2}{20 c^5}-\frac{3 b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{5 c^5}+\frac{3 b x^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{10 c^3}+\frac{1}{5} x^5 \left(a+b \tanh ^{-1}(c x)\right)^3+\frac{3 b x^4 \left(a+b \tanh ^{-1}(c x)\right)^2}{20 c}+\frac{3 b^3 \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{10 c^5}+\frac{9 b^3 x \tanh ^{-1}(c x)}{10 c^4}+\frac{b^3 x^2}{20 c^3}+\frac{b^3 \log \left(1-c^2 x^2\right)}{2 c^5}",1,"(-b^3 + 18*a*b^2*c*x + 6*a^2*b*c^2*x^2 + b^3*c^2*x^2 + 2*a*b^2*c^3*x^3 + 3*a^2*b*c^4*x^4 + 4*a^3*c^5*x^5 - 18*a*b^2*ArcTanh[c*x] + 18*b^3*c*x*ArcTanh[c*x] + 12*a*b^2*c^2*x^2*ArcTanh[c*x] + 2*b^3*c^3*x^3*ArcTanh[c*x] + 6*a*b^2*c^4*x^4*ArcTanh[c*x] + 12*a^2*b*c^5*x^5*ArcTanh[c*x] - 12*a*b^2*ArcTanh[c*x]^2 - 9*b^3*ArcTanh[c*x]^2 + 6*b^3*c^2*x^2*ArcTanh[c*x]^2 + 3*b^3*c^4*x^4*ArcTanh[c*x]^2 + 12*a*b^2*c^5*x^5*ArcTanh[c*x]^2 - 4*b^3*ArcTanh[c*x]^3 + 4*b^3*c^5*x^5*ArcTanh[c*x]^3 - 24*a*b^2*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 12*b^3*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 6*a^2*b*Log[1 - c^2*x^2] + 10*b^3*Log[1 - c^2*x^2] + 12*b^2*(a + b*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 6*b^3*PolyLog[3, -E^(-2*ArcTanh[c*x])])/(20*c^5)","A",0
26,1,245,185,0.6000699,"\int x^3 \left(a+b \tanh ^{-1}(c x)\right)^3 \, dx","Integrate[x^3*(a + b*ArcTanh[c*x])^3,x]","\frac{2 a^3 c^4 x^4+2 b \tanh ^{-1}(c x) \left(3 a^2 c^4 x^4+2 a b c x \left(c^2 x^2+3\right)+b^2 \left(c^2 x^2-1\right)-8 b^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+2 a^2 b c^3 x^3+6 a^2 b c x+3 a^2 b \log (1-c x)-3 a^2 b \log (c x+1)+2 a b^2 c^2 x^2+8 a b^2 \log \left(1-c^2 x^2\right)+2 b^2 \tanh ^{-1}(c x)^2 \left(3 a \left(c^4 x^4-1\right)+b \left(c^3 x^3+3 c x-4\right)\right)-2 a b^2+2 b^3 \left(c^4 x^4-1\right) \tanh ^{-1}(c x)^3+8 b^3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+2 b^3 c x}{8 c^4}","-\frac{2 b^2 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^4}+\frac{b^2 x^2 \left(a+b \tanh ^{-1}(c x)\right)}{4 c^2}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{4 c^4}+\frac{b \left(a+b \tanh ^{-1}(c x)\right)^2}{c^4}+\frac{3 b x \left(a+b \tanh ^{-1}(c x)\right)^2}{4 c^3}+\frac{1}{4} x^4 \left(a+b \tanh ^{-1}(c x)\right)^3+\frac{b x^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{4 c}-\frac{b^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{c^4}-\frac{b^3 \tanh ^{-1}(c x)}{4 c^4}+\frac{b^3 x}{4 c^3}",1,"(-2*a*b^2 + 6*a^2*b*c*x + 2*b^3*c*x + 2*a*b^2*c^2*x^2 + 2*a^2*b*c^3*x^3 + 2*a^3*c^4*x^4 + 2*b^2*(b*(-4 + 3*c*x + c^3*x^3) + 3*a*(-1 + c^4*x^4))*ArcTanh[c*x]^2 + 2*b^3*(-1 + c^4*x^4)*ArcTanh[c*x]^3 + 2*b*ArcTanh[c*x]*(3*a^2*c^4*x^4 + b^2*(-1 + c^2*x^2) + 2*a*b*c*x*(3 + c^2*x^2) - 8*b^2*Log[1 + E^(-2*ArcTanh[c*x])]) + 3*a^2*b*Log[1 - c*x] - 3*a^2*b*Log[1 + c*x] + 8*a*b^2*Log[1 - c^2*x^2] + 8*b^3*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(8*c^4)","A",0
27,1,250,197,0.5609451,"\int x^2 \left(a+b \tanh ^{-1}(c x)\right)^3 \, dx","Integrate[x^2*(a + b*ArcTanh[c*x])^3,x]","\frac{2 a^3 c^3 x^3+6 a^2 b c^3 x^3 \tanh ^{-1}(c x)+3 a^2 b c^2 x^2+3 a^2 b \log \left(1-c^2 x^2\right)+6 a b^2 \left(\left(c^3 x^3-1\right) \tanh ^{-1}(c x)^2+\tanh ^{-1}(c x) \left(c^2 x^2-2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)-1\right)+\text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+c x\right)+b^3 \left(2 c^3 x^3 \tanh ^{-1}(c x)^3+3 \log \left(1-c^2 x^2\right)+3 c^2 x^2 \tanh ^{-1}(c x)^2+6 \tanh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+3 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)-2 \tanh ^{-1}(c x)^3-3 \tanh ^{-1}(c x)^2+6 c x \tanh ^{-1}(c x)-6 \tanh ^{-1}(c x)^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)}{6 c^3}","-\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^3}+\frac{a b^2 x}{c^2}-\frac{b \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^3}+\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{3 c^3}-\frac{b \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2}{c^3}+\frac{1}{3} x^3 \left(a+b \tanh ^{-1}(c x)\right)^3+\frac{b x^2 \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c}+\frac{b^3 \text{Li}_3\left(1-\frac{2}{1-c x}\right)}{2 c^3}+\frac{b^3 x \tanh ^{-1}(c x)}{c^2}+\frac{b^3 \log \left(1-c^2 x^2\right)}{2 c^3}",1,"(3*a^2*b*c^2*x^2 + 2*a^3*c^3*x^3 + 6*a^2*b*c^3*x^3*ArcTanh[c*x] + 3*a^2*b*Log[1 - c^2*x^2] + 6*a*b^2*(c*x + (-1 + c^3*x^3)*ArcTanh[c*x]^2 + ArcTanh[c*x]*(-1 + c^2*x^2 - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + b^3*(6*c*x*ArcTanh[c*x] - 3*ArcTanh[c*x]^2 + 3*c^2*x^2*ArcTanh[c*x]^2 - 2*ArcTanh[c*x]^3 + 2*c^3*x^3*ArcTanh[c*x]^3 - 6*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 3*Log[1 - c^2*x^2] + 6*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 3*PolyLog[3, -E^(-2*ArcTanh[c*x])]))/(6*c^3)","A",0
28,1,161,123,0.3020734,"\int x \left(a+b \tanh ^{-1}(c x)\right)^3 \, dx","Integrate[x*(a + b*ArcTanh[c*x])^3,x]","\frac{a \left(2 a^2 c^2 x^2+6 a b c x+3 a b \log (1-c x)-3 a b \log (c x+1)+6 b^2 \log \left(1-c^2 x^2\right)\right)+6 b^2 (c x-1) \tanh ^{-1}(c x)^2 (a c x+a+b)+6 b \tanh ^{-1}(c x) \left(a c x (a c x+2 b)-2 b^2 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)+2 b^3 \left(c^2 x^2-1\right) \tanh ^{-1}(c x)^3+6 b^3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)}{4 c^2}","-\frac{3 b^2 \log \left(\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)}{c^2}+\frac{3 b \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c^2}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{2 c^2}+\frac{1}{2} x^2 \left(a+b \tanh ^{-1}(c x)\right)^3+\frac{3 b x \left(a+b \tanh ^{-1}(c x)\right)^2}{2 c}-\frac{3 b^3 \text{Li}_2\left(1-\frac{2}{1-c x}\right)}{2 c^2}",1,"(6*b^2*(-1 + c*x)*(a + b + a*c*x)*ArcTanh[c*x]^2 + 2*b^3*(-1 + c^2*x^2)*ArcTanh[c*x]^3 + 6*b*ArcTanh[c*x]*(a*c*x*(2*b + a*c*x) - 2*b^2*Log[1 + E^(-2*ArcTanh[c*x])]) + a*(6*a*b*c*x + 2*a^2*c^2*x^2 + 3*a*b*Log[1 - c*x] - 3*a*b*Log[1 + c*x] + 6*b^2*Log[1 - c^2*x^2]) + 6*b^3*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(4*c^2)","A",0
29,1,161,108,0.2878985,"\int \left(a+b \tanh ^{-1}(c x)\right)^3 \, dx","Integrate[(a + b*ArcTanh[c*x])^3,x]","\frac{2 a^3 c x+3 a^2 b \log \left(1-c^2 x^2\right)+6 a^2 b c x \tanh ^{-1}(c x)+6 a 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^3 \left(6 \tanh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c x)}\right)+3 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c x)}\right)+2 \tanh ^{-1}(c x)^2 \left((c x-1) \tanh ^{-1}(c x)-3 \log \left(e^{-2 \tanh ^{-1}(c x)}+1\right)\right)\right)}{2 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}+x \left(a+b \tanh ^{-1}(c x)\right)^3+\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{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}_3\left(1-\frac{2}{1-c x}\right)}{2 c}",1,"(2*a^3*c*x + 6*a^2*b*c*x*ArcTanh[c*x] + 3*a^2*b*Log[1 - c^2*x^2] + 6*a*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])]) + b^3*(2*ArcTanh[c*x]^2*((-1 + c*x)*ArcTanh[c*x] - 3*Log[1 + E^(-2*ArcTanh[c*x])]) + 6*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 3*PolyLog[3, -E^(-2*ArcTanh[c*x])]))/(2*c)","A",0
30,1,178,184,0.1371308,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{x} \, dx","Integrate[(a + b*ArcTanh[c*x])^3/x,x]","\frac{3}{4} b \left(2 \text{Li}_2\left(\frac{c x+1}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2-2 \text{Li}_2\left(\frac{c x+1}{c x-1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2+b \left(-2 \text{Li}_3\left(\frac{c x+1}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)+2 \text{Li}_3\left(\frac{c x+1}{c x-1}\right) \left(a+b \tanh ^{-1}(c x)\right)+b \left(\text{Li}_4\left(\frac{c x+1}{1-c x}\right)-\text{Li}_4\left(\frac{c x+1}{c x-1}\right)\right)\right)\right)+2 \tanh ^{-1}\left(\frac{c x+1}{c x-1}\right) \left(a+b \tanh ^{-1}(c x)\right)^3","\frac{3}{2} b^2 \text{Li}_3\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)-\frac{3}{2} b^2 \text{Li}_3\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)-\frac{3}{2} b \text{Li}_2\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{3}{2} b \text{Li}_2\left(\frac{2}{1-c x}-1\right) \left(a+b \tanh ^{-1}(c x)\right)^2+2 \tanh ^{-1}\left(1-\frac{2}{1-c x}\right) \left(a+b \tanh ^{-1}(c x)\right)^3-\frac{3}{4} b^3 \text{Li}_4\left(1-\frac{2}{1-c x}\right)+\frac{3}{4} b^3 \text{Li}_4\left(\frac{2}{1-c x}-1\right)",1,"2*(a + b*ArcTanh[c*x])^3*ArcTanh[(1 + c*x)/(-1 + c*x)] + (3*b*(2*(a + b*ArcTanh[c*x])^2*PolyLog[2, (1 + c*x)/(1 - c*x)] - 2*(a + b*ArcTanh[c*x])^2*PolyLog[2, (1 + c*x)/(-1 + c*x)] + b*(-2*(a + b*ArcTanh[c*x])*PolyLog[3, (1 + c*x)/(1 - c*x)] + 2*(a + b*ArcTanh[c*x])*PolyLog[3, (1 + c*x)/(-1 + c*x)] + b*(PolyLog[4, (1 + c*x)/(1 - c*x)] - PolyLog[4, (1 + c*x)/(-1 + c*x)]))))/4","A",1
31,1,196,102,0.3436109,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{x^2} \, dx","Integrate[(a + b*ArcTanh[c*x])^3/x^2,x]","-\frac{a^3}{x}-\frac{3}{2} a^2 b c \log \left(1-c^2 x^2\right)+3 a^2 b c \log (x)-\frac{3 a^2 b \tanh ^{-1}(c x)}{x}+3 a b^2 c \left(\tanh ^{-1}(c x) \left(-\frac{\tanh ^{-1}(c x)}{c x}+\tanh ^{-1}(c x)+2 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)\right)-\text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)\right)+b^3 c \left(3 \tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)-\frac{3}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)-\frac{\tanh ^{-1}(c x)^3}{c x}-\tanh ^{-1}(c x)^3+3 \tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)+\frac{i \pi ^3}{8}\right)","-3 b^2 c \text{Li}_2\left(\frac{2}{c x+1}-1\right) \left(a+b \tanh ^{-1}(c x)\right)+c \left(a+b \tanh ^{-1}(c x)\right)^3-\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{x}+3 b c \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2-\frac{3}{2} b^3 c \text{Li}_3\left(\frac{2}{c x+1}-1\right)",1,"-(a^3/x) - (3*a^2*b*ArcTanh[c*x])/x + 3*a^2*b*c*Log[x] - (3*a^2*b*c*Log[1 - c^2*x^2])/2 + 3*a*b^2*c*(ArcTanh[c*x]*(ArcTanh[c*x] - ArcTanh[c*x]/(c*x) + 2*Log[1 - E^(-2*ArcTanh[c*x])]) - PolyLog[2, E^(-2*ArcTanh[c*x])]) + b^3*c*((I/8)*Pi^3 - ArcTanh[c*x]^3 - ArcTanh[c*x]^3/(c*x) + 3*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 3*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - (3*PolyLog[3, E^(2*ArcTanh[c*x])])/2)","C",0
32,1,192,123,0.2927277,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{x^3} \, dx","Integrate[(a + b*ArcTanh[c*x])^3/x^3,x]","\frac{a \left(-2 a^2-3 a b c^2 x^2 \log (1-c x)+3 a b c^2 x^2 \log (c x+1)-6 a b c x+12 b^2 c^2 x^2 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)\right)-6 b \tanh ^{-1}(c x) \left(a^2+2 a b c x-2 b^2 c^2 x^2 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)\right)+6 b^2 (c x-1) \tanh ^{-1}(c x)^2 (a c x+a+b c x)-6 b^3 c^2 x^2 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+2 b^3 \left(c^2 x^2-1\right) \tanh ^{-1}(c x)^3}{4 x^2}","3 b^2 c^2 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)+\frac{3}{2} b c^2 \left(a+b \tanh ^{-1}(c x)\right)^2+\frac{1}{2} c^2 \left(a+b \tanh ^{-1}(c x)\right)^3-\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{2 x^2}-\frac{3 b c \left(a+b \tanh ^{-1}(c x)\right)^2}{2 x}-\frac{3}{2} b^3 c^2 \text{Li}_2\left(\frac{2}{c x+1}-1\right)",1,"(6*b^2*(-1 + c*x)*(a + a*c*x + b*c*x)*ArcTanh[c*x]^2 + 2*b^3*(-1 + c^2*x^2)*ArcTanh[c*x]^3 - 6*b*ArcTanh[c*x]*(a^2 + 2*a*b*c*x - 2*b^2*c^2*x^2*Log[1 - E^(-2*ArcTanh[c*x])]) + a*(-2*a^2 - 6*a*b*c*x - 3*a*b*c^2*x^2*Log[1 - c*x] + 3*a*b*c^2*x^2*Log[1 + c*x] + 12*b^2*c^2*x^2*Log[(c*x)/Sqrt[1 - c^2*x^2]]) - 6*b^3*c^2*x^2*PolyLog[2, E^(-2*ArcTanh[c*x])])/(4*x^2)","A",0
33,1,323,200,0.9369438,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{x^4} \, dx","Integrate[(a + b*ArcTanh[c*x])^3/x^4,x]","-\frac{8 a^3-24 a^2 b c^3 x^3 \log (x)+12 a^2 b c^3 x^3 \log \left(1-c^2 x^2\right)+12 a^2 b c x+24 a^2 b \tanh ^{-1}(c x)+24 a b^2 \left(c^3 x^3 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)+\left(1-c^3 x^3\right) \tanh ^{-1}(c x)^2+c^2 x^2-c x \tanh ^{-1}(c x) \left(c^2 x^2+2 c^2 x^2 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)-1\right)\right)+b^3 \left(-24 c^3 x^3 \tanh ^{-1}(c x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c x)}\right)+12 c^3 x^3 \text{Li}_3\left(e^{2 \tanh ^{-1}(c x)}\right)-i \pi ^3 c^3 x^3+8 c^3 x^3 \tanh ^{-1}(c x)^3-12 c^3 x^3 \tanh ^{-1}(c x)^2-24 c^3 x^3 \tanh ^{-1}(c x)^2 \log \left(1-e^{2 \tanh ^{-1}(c x)}\right)+24 c^2 x^2 \tanh ^{-1}(c x)-24 c^3 x^3 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+12 c x \tanh ^{-1}(c x)^2+8 \tanh ^{-1}(c x)^3\right)}{24 x^3}","-b^2 c^3 \text{Li}_2\left(\frac{2}{c x+1}-1\right) \left(a+b \tanh ^{-1}(c x)\right)-\frac{b^2 c^2 \left(a+b \tanh ^{-1}(c x)\right)}{x}+\frac{1}{3} c^3 \left(a+b \tanh ^{-1}(c x)\right)^3+\frac{1}{2} b c^3 \left(a+b \tanh ^{-1}(c x)\right)^2+b c^3 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)^2-\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{3 x^3}-\frac{b c \left(a+b \tanh ^{-1}(c x)\right)^2}{2 x^2}-\frac{1}{2} b^3 c^3 \text{Li}_3\left(\frac{2}{c x+1}-1\right)+b^3 c^3 \log (x)-\frac{1}{2} b^3 c^3 \log \left(1-c^2 x^2\right)",1,"-1/24*(8*a^3 + 12*a^2*b*c*x + 24*a^2*b*ArcTanh[c*x] - 24*a^2*b*c^3*x^3*Log[x] + 12*a^2*b*c^3*x^3*Log[1 - c^2*x^2] + 24*a*b^2*(c^2*x^2 + (1 - c^3*x^3)*ArcTanh[c*x]^2 - c*x*ArcTanh[c*x]*(-1 + c^2*x^2 + 2*c^2*x^2*Log[1 - E^(-2*ArcTanh[c*x])]) + c^3*x^3*PolyLog[2, E^(-2*ArcTanh[c*x])]) + b^3*((-I)*c^3*Pi^3*x^3 + 24*c^2*x^2*ArcTanh[c*x] + 12*c*x*ArcTanh[c*x]^2 - 12*c^3*x^3*ArcTanh[c*x]^2 + 8*ArcTanh[c*x]^3 + 8*c^3*x^3*ArcTanh[c*x]^3 - 24*c^3*x^3*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] - 24*c^3*x^3*Log[(c*x)/Sqrt[1 - c^2*x^2]] - 24*c^3*x^3*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + 12*c^3*x^3*PolyLog[3, E^(2*ArcTanh[c*x])]))/x^3","C",0
34,1,295,187,0.6990069,"\int \frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{x^5} \, dx","Integrate[(a + b*ArcTanh[c*x])^3/x^5,x]","-\frac{2 a^3+2 b \tanh ^{-1}(c x) \left(3 a^2+2 a b c x \left(3 c^2 x^2+1\right)-8 b^2 c^4 x^4 \log \left(1-e^{-2 \tanh ^{-1}(c x)}\right)+b^2 c^2 x^2 \left(1-c^2 x^2\right)\right)+3 a^2 b c^4 x^4 \log (1-c x)-3 a^2 b c^4 x^4 \log (c x+1)+6 a^2 b c^3 x^3+2 a^2 b c x-2 a b^2 c^4 x^4+2 a b^2 c^2 x^2-16 a b^2 c^4 x^4 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+2 b^2 \tanh ^{-1}(c x)^2 \left(a \left(3-3 c^4 x^4\right)+b c x \left(-4 c^3 x^3+3 c^2 x^2+1\right)\right)+8 b^3 c^4 x^4 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c x)}\right)-2 b^3 \left(c^4 x^4-1\right) \tanh ^{-1}(c x)^3+2 b^3 c^3 x^3}{8 x^4}","2 b^2 c^4 \log \left(2-\frac{2}{c x+1}\right) \left(a+b \tanh ^{-1}(c x)\right)-\frac{b^2 c^2 \left(a+b \tanh ^{-1}(c x)\right)}{4 x^2}+\frac{1}{4} c^4 \left(a+b \tanh ^{-1}(c x)\right)^3+b c^4 \left(a+b \tanh ^{-1}(c x)\right)^2-\frac{3 b c^3 \left(a+b \tanh ^{-1}(c x)\right)^2}{4 x}-\frac{\left(a+b \tanh ^{-1}(c x)\right)^3}{4 x^4}-\frac{b c \left(a+b \tanh ^{-1}(c x)\right)^2}{4 x^3}-b^3 c^4 \text{Li}_2\left(\frac{2}{c x+1}-1\right)+\frac{1}{4} b^3 c^4 \tanh ^{-1}(c x)-\frac{b^3 c^3}{4 x}",1,"-1/8*(2*a^3 + 2*a^2*b*c*x + 2*a*b^2*c^2*x^2 + 6*a^2*b*c^3*x^3 + 2*b^3*c^3*x^3 - 2*a*b^2*c^4*x^4 + 2*b^2*(b*c*x*(1 + 3*c^2*x^2 - 4*c^3*x^3) + a*(3 - 3*c^4*x^4))*ArcTanh[c*x]^2 - 2*b^3*(-1 + c^4*x^4)*ArcTanh[c*x]^3 + 2*b*ArcTanh[c*x]*(3*a^2 + b^2*c^2*x^2*(1 - c^2*x^2) + 2*a*b*c*x*(1 + 3*c^2*x^2) - 8*b^2*c^4*x^4*Log[1 - E^(-2*ArcTanh[c*x])]) + 3*a^2*b*c^4*x^4*Log[1 - c*x] - 3*a^2*b*c^4*x^4*Log[1 + c*x] - 16*a*b^2*c^4*x^4*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 8*b^3*c^4*x^4*PolyLog[2, E^(-2*ArcTanh[c*x])])/x^4","A",0
35,1,128,124,0.0780857,"\int (d x)^{5/2} \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[(d*x)^(5/2)*(a + b*ArcTanh[c*x]),x]","\frac{(d x)^{5/2} \left(10 a c^{7/2} x^{7/2}+4 b c^{5/2} x^{5/2}+10 b c^{7/2} x^{7/2} \tanh ^{-1}(c x)+20 b \sqrt{c} \sqrt{x}+5 b \log \left(1-\sqrt{c} \sqrt{x}\right)-5 b \log \left(\sqrt{c} \sqrt{x}+1\right)-10 b \tan ^{-1}\left(\sqrt{c} \sqrt{x}\right)\right)}{35 c^{7/2} x^{5/2}}","\frac{2 (d x)^{7/2} \left(a+b \tanh ^{-1}(c x)\right)}{7 d}-\frac{2 b d^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{7 c^{7/2}}-\frac{2 b d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{7 c^{7/2}}+\frac{4 b d^2 \sqrt{d x}}{7 c^3}+\frac{4 b (d x)^{5/2}}{35 c}",1,"((d*x)^(5/2)*(20*b*Sqrt[c]*Sqrt[x] + 4*b*c^(5/2)*x^(5/2) + 10*a*c^(7/2)*x^(7/2) - 10*b*ArcTan[Sqrt[c]*Sqrt[x]] + 10*b*c^(7/2)*x^(7/2)*ArcTanh[c*x] + 5*b*Log[1 - Sqrt[c]*Sqrt[x]] - 5*b*Log[1 + Sqrt[c]*Sqrt[x]]))/(35*c^(7/2)*x^(5/2))","A",1
36,1,115,106,0.0607039,"\int (d x)^{3/2} \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[(d*x)^(3/2)*(a + b*ArcTanh[c*x]),x]","\frac{(d x)^{3/2} \left(6 a c^{5/2} x^{5/2}+4 b c^{3/2} x^{3/2}+6 b c^{5/2} x^{5/2} \tanh ^{-1}(c x)+3 b \log \left(1-\sqrt{c} \sqrt{x}\right)-3 b \log \left(\sqrt{c} \sqrt{x}+1\right)+6 b \tan ^{-1}\left(\sqrt{c} \sqrt{x}\right)\right)}{15 c^{5/2} x^{3/2}}","\frac{2 (d x)^{5/2} \left(a+b \tanh ^{-1}(c x)\right)}{5 d}+\frac{2 b d^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{5 c^{5/2}}-\frac{2 b d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{5 c^{5/2}}+\frac{4 b (d x)^{3/2}}{15 c}",1,"((d*x)^(3/2)*(4*b*c^(3/2)*x^(3/2) + 6*a*c^(5/2)*x^(5/2) + 6*b*ArcTan[Sqrt[c]*Sqrt[x]] + 6*b*c^(5/2)*x^(5/2)*ArcTanh[c*x] + 3*b*Log[1 - Sqrt[c]*Sqrt[x]] - 3*b*Log[1 + Sqrt[c]*Sqrt[x]]))/(15*c^(5/2)*x^(3/2))","A",1
37,1,114,106,0.0526241,"\int \sqrt{d x} \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d*x]*(a + b*ArcTanh[c*x]),x]","\frac{\sqrt{d x} \left(2 a c^{3/2} x^{3/2}+2 b c^{3/2} x^{3/2} \tanh ^{-1}(c x)+4 b \sqrt{c} \sqrt{x}+b \log \left(1-\sqrt{c} \sqrt{x}\right)-b \log \left(\sqrt{c} \sqrt{x}+1\right)-2 b \tan ^{-1}\left(\sqrt{c} \sqrt{x}\right)\right)}{3 c^{3/2} \sqrt{x}}","\frac{2 (d x)^{3/2} \left(a+b \tanh ^{-1}(c x)\right)}{3 d}-\frac{2 b \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{3 c^{3/2}}-\frac{2 b \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{3 c^{3/2}}+\frac{4 b \sqrt{d x}}{3 c}",1,"(Sqrt[d*x]*(4*b*Sqrt[c]*Sqrt[x] + 2*a*c^(3/2)*x^(3/2) - 2*b*ArcTan[Sqrt[c]*Sqrt[x]] + 2*b*c^(3/2)*x^(3/2)*ArcTanh[c*x] + b*Log[1 - Sqrt[c]*Sqrt[x]] - b*Log[1 + Sqrt[c]*Sqrt[x]]))/(3*c^(3/2)*Sqrt[x])","A",1
38,1,98,85,0.0349456,"\int \frac{a+b \tanh ^{-1}(c x)}{\sqrt{d x}} \, dx","Integrate[(a + b*ArcTanh[c*x])/Sqrt[d*x],x]","\frac{\sqrt{x} \left(2 a \sqrt{c} \sqrt{x}+b \log \left(1-\sqrt{c} \sqrt{x}\right)-b \log \left(\sqrt{c} \sqrt{x}+1\right)+2 b \tan ^{-1}\left(\sqrt{c} \sqrt{x}\right)+2 b \sqrt{c} \sqrt{x} \tanh ^{-1}(c x)\right)}{\sqrt{c} \sqrt{d x}}","\frac{2 \sqrt{d x} \left(a+b \tanh ^{-1}(c x)\right)}{d}+\frac{2 b \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{\sqrt{c} \sqrt{d}}-\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{\sqrt{c} \sqrt{d}}",1,"(Sqrt[x]*(2*a*Sqrt[c]*Sqrt[x] + 2*b*ArcTan[Sqrt[c]*Sqrt[x]] + 2*b*Sqrt[c]*Sqrt[x]*ArcTanh[c*x] + b*Log[1 - Sqrt[c]*Sqrt[x]] - b*Log[1 + Sqrt[c]*Sqrt[x]]))/(Sqrt[c]*Sqrt[d*x])","A",1
39,1,99,85,0.0456076,"\int \frac{a+b \tanh ^{-1}(c x)}{(d x)^{3/2}} \, dx","Integrate[(a + b*ArcTanh[c*x])/(d*x)^(3/2),x]","\frac{x \left(-2 a-b \sqrt{c} \sqrt{x} \log \left(1-\sqrt{c} \sqrt{x}\right)+b \sqrt{c} \sqrt{x} \log \left(\sqrt{c} \sqrt{x}+1\right)+2 b \sqrt{c} \sqrt{x} \tan ^{-1}\left(\sqrt{c} \sqrt{x}\right)-2 b \tanh ^{-1}(c x)\right)}{(d x)^{3/2}}","-\frac{2 \left(a+b \tanh ^{-1}(c x)\right)}{d \sqrt{d x}}+\frac{2 b \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{d^{3/2}}+\frac{2 b \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{d^{3/2}}",1,"(x*(-2*a + 2*b*Sqrt[c]*Sqrt[x]*ArcTan[Sqrt[c]*Sqrt[x]] - 2*b*ArcTanh[c*x] - b*Sqrt[c]*Sqrt[x]*Log[1 - Sqrt[c]*Sqrt[x]] + b*Sqrt[c]*Sqrt[x]*Log[1 + Sqrt[c]*Sqrt[x]]))/(d*x)^(3/2)","A",1
40,1,107,107,0.0660797,"\int \frac{a+b \tanh ^{-1}(c x)}{(d x)^{5/2}} \, dx","Integrate[(a + b*ArcTanh[c*x])/(d*x)^(5/2),x]","-\frac{x \left(2 a+b c^{3/2} x^{3/2} \log \left(1-\sqrt{c} \sqrt{x}\right)-b c^{3/2} x^{3/2} \log \left(\sqrt{c} \sqrt{x}+1\right)+2 b c^{3/2} x^{3/2} \tan ^{-1}\left(\sqrt{c} \sqrt{x}\right)+4 b c x+2 b \tanh ^{-1}(c x)\right)}{3 (d x)^{5/2}}","-\frac{2 \left(a+b \tanh ^{-1}(c x)\right)}{3 d (d x)^{3/2}}-\frac{2 b c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{3 d^{5/2}}+\frac{2 b c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{3 d^{5/2}}-\frac{4 b c}{3 d^2 \sqrt{d x}}",1,"-1/3*(x*(2*a + 4*b*c*x + 2*b*c^(3/2)*x^(3/2)*ArcTan[Sqrt[c]*Sqrt[x]] + 2*b*ArcTanh[c*x] + b*c^(3/2)*x^(3/2)*Log[1 - Sqrt[c]*Sqrt[x]] - b*c^(3/2)*x^(3/2)*Log[1 + Sqrt[c]*Sqrt[x]]))/(d*x)^(5/2)","A",1
41,1,108,107,0.0531312,"\int \frac{a+b \tanh ^{-1}(c x)}{(d x)^{7/2}} \, dx","Integrate[(a + b*ArcTanh[c*x])/(d*x)^(7/2),x]","\frac{x \left(-6 a-3 b c^{5/2} x^{5/2} \log \left(1-\sqrt{c} \sqrt{x}\right)+3 b c^{5/2} x^{5/2} \log \left(\sqrt{c} \sqrt{x}+1\right)+6 b c^{5/2} x^{5/2} \tan ^{-1}\left(\sqrt{c} \sqrt{x}\right)-4 b c x-6 b \tanh ^{-1}(c x)\right)}{15 (d x)^{7/2}}","-\frac{2 \left(a+b \tanh ^{-1}(c x)\right)}{5 d (d x)^{5/2}}+\frac{2 b c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{5 d^{7/2}}+\frac{2 b c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{5 d^{7/2}}-\frac{4 b c}{15 d^2 (d x)^{3/2}}",1,"(x*(-6*a - 4*b*c*x + 6*b*c^(5/2)*x^(5/2)*ArcTan[Sqrt[c]*Sqrt[x]] - 6*b*ArcTanh[c*x] - 3*b*c^(5/2)*x^(5/2)*Log[1 - Sqrt[c]*Sqrt[x]] + 3*b*c^(5/2)*x^(5/2)*Log[1 + Sqrt[c]*Sqrt[x]]))/(15*(d*x)^(7/2))","A",1
42,1,122,125,0.0731713,"\int \frac{a+b \tanh ^{-1}(c x)}{(d x)^{9/2}} \, dx","Integrate[(a + b*ArcTanh[c*x])/(d*x)^(9/2),x]","-\frac{\sqrt{d x} \left(10 a+5 b c^{7/2} x^{7/2} \log \left(1-\sqrt{c} \sqrt{x}\right)-5 b c^{7/2} x^{7/2} \log \left(\sqrt{c} \sqrt{x}+1\right)+10 b c^{7/2} x^{7/2} \tan ^{-1}\left(\sqrt{c} \sqrt{x}\right)+20 b c^3 x^3+4 b c x+10 b \tanh ^{-1}(c x)\right)}{35 d^5 x^4}","-\frac{2 \left(a+b \tanh ^{-1}(c x)\right)}{7 d (d x)^{7/2}}-\frac{2 b c^{7/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{7 d^{9/2}}+\frac{2 b c^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)}{7 d^{9/2}}-\frac{4 b c^3}{7 d^4 \sqrt{d x}}-\frac{4 b c}{35 d^2 (d x)^{5/2}}",1,"-1/35*(Sqrt[d*x]*(10*a + 4*b*c*x + 20*b*c^3*x^3 + 10*b*c^(7/2)*x^(7/2)*ArcTan[Sqrt[c]*Sqrt[x]] + 10*b*ArcTanh[c*x] + 5*b*c^(7/2)*x^(7/2)*Log[1 - Sqrt[c]*Sqrt[x]] - 5*b*c^(7/2)*x^(7/2)*Log[1 + Sqrt[c]*Sqrt[x]]))/(d^5*x^4)","A",1
43,0,0,19,3.8312325,"\int (d x)^m \left(a+b \tanh ^{-1}(c x)\right)^3 \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c*x])^3,x]","\int (d x)^m \left(a+b \tanh ^{-1}(c x)\right)^3 \, dx","\text{Int}\left((d x)^m \left(a+b \tanh ^{-1}(c x)\right)^3,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTanh[c*x])^3, x]","A",-1
44,0,0,19,2.5185084,"\int (d x)^m \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c*x])^2,x]","\int (d x)^m \left(a+b \tanh ^{-1}(c x)\right)^2 \, dx","\text{Int}\left((d x)^m \left(a+b \tanh ^{-1}(c x)\right)^2,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTanh[c*x])^2, x]","A",-1
45,1,59,72,0.0709844,"\int (d x)^m \left(a+b \tanh ^{-1}(c x)\right) \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c*x]),x]","-\frac{x (d x)^m \left(b c x \, _2F_1\left(1,\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)-(m+2) \left(a+b \tanh ^{-1}(c x)\right)\right)}{(m+1) (m+2)}","\frac{(d x)^{m+1} \left(a+b \tanh ^{-1}(c x)\right)}{d (m+1)}-\frac{b c (d x)^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{d^2 (m+1) (m+2)}",1,"-((x*(d*x)^m*(-((2 + m)*(a + b*ArcTanh[c*x])) + b*c*x*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, c^2*x^2]))/((1 + m)*(2 + m)))","A",1
46,0,0,19,0.2728296,"\int \frac{(d x)^m}{a+b \tanh ^{-1}(c x)} \, dx","Integrate[(d*x)^m/(a + b*ArcTanh[c*x]),x]","\int \frac{(d x)^m}{a+b \tanh ^{-1}(c x)} \, dx","\text{Int}\left(\frac{(d x)^m}{a+b \tanh ^{-1}(c x)},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcTanh[c*x]), x]","A",-1
47,0,0,19,0.5465326,"\int \frac{(d x)^m}{\left(a+b \tanh ^{-1}(c x)\right)^2} \, dx","Integrate[(d*x)^m/(a + b*ArcTanh[c*x])^2,x]","\int \frac{(d x)^m}{\left(a+b \tanh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{(d x)^m}{\left(a+b \tanh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcTanh[c*x])^2, x]","A",-1
48,0,0,13,0.4768614,"\int \left(a+b \tanh ^{-1}(c x)\right)^p \, dx","Integrate[(a + b*ArcTanh[c*x])^p,x]","\int \left(a+b \tanh ^{-1}(c x)\right)^p \, dx","\text{Int}\left(\left(a+b \tanh ^{-1}(c x)\right)^p,x\right)",0,"Integrate[(a + b*ArcTanh[c*x])^p, x]","A",-1
49,0,0,19,0.3551535,"\int (d x)^m \left(a+b \tanh ^{-1}(c x)\right)^p \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c*x])^p,x]","\int (d x)^m \left(a+b \tanh ^{-1}(c x)\right)^p \, dx","\text{Int}\left((d x)^m \left(a+b \tanh ^{-1}(c x)\right)^p,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTanh[c*x])^p, x]","A",-1
50,1,78,54,0.0179923,"\int x^7 \left(a+b \tanh ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^7*(a + b*ArcTanh[c*x^2]),x]","\frac{a x^8}{8}+\frac{b \log \left(1-c x^2\right)}{16 c^4}-\frac{b \log \left(c x^2+1\right)}{16 c^4}+\frac{b x^2}{8 c^3}+\frac{b x^6}{24 c}+\frac{1}{8} b x^8 \tanh ^{-1}\left(c x^2\right)","\frac{1}{8} x^8 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)-\frac{b \tanh ^{-1}\left(c x^2\right)}{8 c^4}+\frac{b x^2}{8 c^3}+\frac{b x^6}{24 c}",1,"(b*x^2)/(8*c^3) + (b*x^6)/(24*c) + (a*x^8)/8 + (b*x^8*ArcTanh[c*x^2])/8 + (b*Log[1 - c*x^2])/(16*c^4) - (b*Log[1 + c*x^2])/(16*c^4)","A",1
51,1,53,48,0.0161377,"\int x^5 \left(a+b \tanh ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^5*(a + b*ArcTanh[c*x^2]),x]","\frac{a x^6}{6}+\frac{b \log \left(1-c^2 x^4\right)}{12 c^3}+\frac{b x^4}{12 c}+\frac{1}{6} b x^6 \tanh ^{-1}\left(c x^2\right)","\frac{1}{6} x^6 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)+\frac{b \log \left(1-c^2 x^4\right)}{12 c^3}+\frac{b x^4}{12 c}",1,"(b*x^4)/(12*c) + (a*x^6)/6 + (b*x^6*ArcTanh[c*x^2])/6 + (b*Log[1 - c^2*x^4])/(12*c^3)","A",1
52,1,67,43,0.0138359,"\int x^3 \left(a+b \tanh ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^3*(a + b*ArcTanh[c*x^2]),x]","\frac{a x^4}{4}+\frac{b \log \left(1-c x^2\right)}{8 c^2}-\frac{b \log \left(c x^2+1\right)}{8 c^2}+\frac{b x^2}{4 c}+\frac{1}{4} b x^4 \tanh ^{-1}\left(c x^2\right)","\frac{1}{4} x^4 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)-\frac{b \tanh ^{-1}\left(c x^2\right)}{4 c^2}+\frac{b x^2}{4 c}",1,"(b*x^2)/(4*c) + (a*x^4)/4 + (b*x^4*ArcTanh[c*x^2])/4 + (b*Log[1 - c*x^2])/(8*c^2) - (b*Log[1 + c*x^2])/(8*c^2)","A",1
53,1,42,37,0.0080205,"\int x \left(a+b \tanh ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x*(a + b*ArcTanh[c*x^2]),x]","\frac{a x^2}{2}+\frac{b \log \left(1-c^2 x^4\right)}{4 c}+\frac{1}{2} b x^2 \tanh ^{-1}\left(c x^2\right)","\frac{1}{2} x^2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)+\frac{b \log \left(1-c^2 x^4\right)}{4 c}",1,"(a*x^2)/2 + (b*x^2*ArcTanh[c*x^2])/2 + (b*Log[1 - c^2*x^4])/(4*c)","A",1
54,1,28,30,0.014448,"\int \frac{a+b \tanh ^{-1}\left(c x^2\right)}{x} \, dx","Integrate[(a + b*ArcTanh[c*x^2])/x,x]","a \log (x)+\frac{1}{4} b \left(\text{Li}_2\left(c x^2\right)-\text{Li}_2\left(-c x^2\right)\right)","a \log (x)-\frac{1}{4} b \text{Li}_2\left(-c x^2\right)+\frac{1}{4} b \text{Li}_2\left(c x^2\right)",1,"a*Log[x] + (b*(-PolyLog[2, -(c*x^2)] + PolyLog[2, c*x^2]))/4","A",1
55,1,45,40,0.0121725,"\int \frac{a+b \tanh ^{-1}\left(c x^2\right)}{x^3} \, dx","Integrate[(a + b*ArcTanh[c*x^2])/x^3,x]","-\frac{a}{2 x^2}-\frac{1}{4} b c \log \left(1-c^2 x^4\right)-\frac{b \tanh ^{-1}\left(c x^2\right)}{2 x^2}+b c \log (x)","-\frac{a+b \tanh ^{-1}\left(c x^2\right)}{2 x^2}-\frac{1}{4} b c \log \left(1-c^2 x^4\right)+b c \log (x)",1,"-1/2*a/x^2 - (b*ArcTanh[c*x^2])/(2*x^2) + b*c*Log[x] - (b*c*Log[1 - c^2*x^4])/4","A",1
56,1,65,41,0.0125763,"\int \frac{a+b \tanh ^{-1}\left(c x^2\right)}{x^5} \, dx","Integrate[(a + b*ArcTanh[c*x^2])/x^5,x]","-\frac{a}{4 x^4}-\frac{1}{8} b c^2 \log \left(1-c x^2\right)+\frac{1}{8} b c^2 \log \left(c x^2+1\right)-\frac{b c}{4 x^2}-\frac{b \tanh ^{-1}\left(c x^2\right)}{4 x^4}","-\frac{a+b \tanh ^{-1}\left(c x^2\right)}{4 x^4}+\frac{1}{4} b c^2 \tanh ^{-1}\left(c x^2\right)-\frac{b c}{4 x^2}",1,"-1/4*a/x^4 - (b*c)/(4*x^2) - (b*ArcTanh[c*x^2])/(4*x^4) - (b*c^2*Log[1 - c*x^2])/8 + (b*c^2*Log[1 + c*x^2])/8","A",1
57,1,61,56,0.0125045,"\int \frac{a+b \tanh ^{-1}\left(c x^2\right)}{x^7} \, dx","Integrate[(a + b*ArcTanh[c*x^2])/x^7,x]","-\frac{a}{6 x^6}+\frac{1}{3} b c^3 \log (x)-\frac{1}{12} b c^3 \log \left(1-c^2 x^4\right)-\frac{b c}{12 x^4}-\frac{b \tanh ^{-1}\left(c x^2\right)}{6 x^6}","-\frac{a+b \tanh ^{-1}\left(c x^2\right)}{6 x^6}+\frac{1}{3} b c^3 \log (x)-\frac{1}{12} b c^3 \log \left(1-c^2 x^4\right)-\frac{b c}{12 x^4}",1,"-1/6*a/x^6 - (b*c)/(12*x^4) - (b*ArcTanh[c*x^2])/(6*x^6) + (b*c^3*Log[x])/3 - (b*c^3*Log[1 - c^2*x^4])/12","A",1
58,1,93,65,0.0256996,"\int x^4 \left(a+b \tanh ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^4*(a + b*ArcTanh[c*x^2]),x]","\frac{a x^5}{5}+\frac{b \log \left(1-\sqrt{c} x\right)}{10 c^{5/2}}-\frac{b \log \left(\sqrt{c} x+1\right)}{10 c^{5/2}}+\frac{b \tan ^{-1}\left(\sqrt{c} x\right)}{5 c^{5/2}}+\frac{2 b x^3}{15 c}+\frac{1}{5} b x^5 \tanh ^{-1}\left(c x^2\right)","\frac{1}{5} x^5 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)+\frac{b \tan ^{-1}\left(\sqrt{c} x\right)}{5 c^{5/2}}-\frac{b \tanh ^{-1}\left(\sqrt{c} x\right)}{5 c^{5/2}}+\frac{2 b x^3}{15 c}",1,"(2*b*x^3)/(15*c) + (a*x^5)/5 + (b*ArcTan[Sqrt[c]*x])/(5*c^(5/2)) + (b*x^5*ArcTanh[c*x^2])/5 + (b*Log[1 - Sqrt[c]*x])/(10*c^(5/2)) - (b*Log[1 + Sqrt[c]*x])/(10*c^(5/2))","A",1
59,1,91,63,0.0203767,"\int x^2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^2*(a + b*ArcTanh[c*x^2]),x]","\frac{a x^3}{3}+\frac{b \log \left(1-\sqrt{c} x\right)}{6 c^{3/2}}-\frac{b \log \left(\sqrt{c} x+1\right)}{6 c^{3/2}}-\frac{b \tan ^{-1}\left(\sqrt{c} x\right)}{3 c^{3/2}}+\frac{1}{3} b x^3 \tanh ^{-1}\left(c x^2\right)+\frac{2 b x}{3 c}","\frac{1}{3} x^3 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)-\frac{b \tan ^{-1}\left(\sqrt{c} x\right)}{3 c^{3/2}}-\frac{b \tanh ^{-1}\left(\sqrt{c} x\right)}{3 c^{3/2}}+\frac{2 b x}{3 c}",1,"(2*b*x)/(3*c) + (a*x^3)/3 - (b*ArcTan[Sqrt[c]*x])/(3*c^(3/2)) + (b*x^3*ArcTanh[c*x^2])/3 + (b*Log[1 - Sqrt[c]*x])/(6*c^(3/2)) - (b*Log[1 + Sqrt[c]*x])/(6*c^(3/2))","A",1
60,1,57,44,0.0219259,"\int \left(a+b \tanh ^{-1}\left(c x^2\right)\right) \, dx","Integrate[a + b*ArcTanh[c*x^2],x]","a x+b x \tanh ^{-1}\left(c x^2\right)+\frac{b \left(\log \left(1-\sqrt{c} x\right)-\log \left(\sqrt{c} x+1\right)+2 \tan ^{-1}\left(\sqrt{c} x\right)\right)}{2 \sqrt{c}}","a x+b x \tanh ^{-1}\left(c x^2\right)+\frac{b \tan ^{-1}\left(\sqrt{c} x\right)}{\sqrt{c}}-\frac{b \tanh ^{-1}\left(\sqrt{c} x\right)}{\sqrt{c}}",1,"a*x + b*x*ArcTanh[c*x^2] + (b*(2*ArcTan[Sqrt[c]*x] + Log[1 - Sqrt[c]*x] - Log[1 + Sqrt[c]*x]))/(2*Sqrt[c])","A",1
61,1,75,46,0.0190694,"\int \frac{a+b \tanh ^{-1}\left(c x^2\right)}{x^2} \, dx","Integrate[(a + b*ArcTanh[c*x^2])/x^2,x]","-\frac{a}{x}-\frac{b \tanh ^{-1}\left(c x^2\right)}{x}-\frac{1}{2} b \sqrt{c} \log \left(1-\sqrt{c} x\right)+\frac{1}{2} b \sqrt{c} \log \left(\sqrt{c} x+1\right)+b \sqrt{c} \tan ^{-1}\left(\sqrt{c} x\right)","-\frac{a+b \tanh ^{-1}\left(c x^2\right)}{x}+b \sqrt{c} \tan ^{-1}\left(\sqrt{c} x\right)+b \sqrt{c} \tanh ^{-1}\left(\sqrt{c} x\right)",1,"-(a/x) + b*Sqrt[c]*ArcTan[Sqrt[c]*x] - (b*ArcTanh[c*x^2])/x - (b*Sqrt[c]*Log[1 - Sqrt[c]*x])/2 + (b*Sqrt[c]*Log[1 + Sqrt[c]*x])/2","A",1
62,1,91,63,0.0288184,"\int \frac{a+b \tanh ^{-1}\left(c x^2\right)}{x^4} \, dx","Integrate[(a + b*ArcTanh[c*x^2])/x^4,x]","-\frac{a}{3 x^3}-\frac{1}{6} b c^{3/2} \log \left(1-\sqrt{c} x\right)+\frac{1}{6} b c^{3/2} \log \left(\sqrt{c} x+1\right)-\frac{1}{3} b c^{3/2} \tan ^{-1}\left(\sqrt{c} x\right)-\frac{b \tanh ^{-1}\left(c x^2\right)}{3 x^3}-\frac{2 b c}{3 x}","-\frac{a+b \tanh ^{-1}\left(c x^2\right)}{3 x^3}-\frac{1}{3} b c^{3/2} \tan ^{-1}\left(\sqrt{c} x\right)+\frac{1}{3} b c^{3/2} \tanh ^{-1}\left(\sqrt{c} x\right)-\frac{2 b c}{3 x}",1,"-1/3*a/x^3 - (2*b*c)/(3*x) - (b*c^(3/2)*ArcTan[Sqrt[c]*x])/3 - (b*ArcTanh[c*x^2])/(3*x^3) - (b*c^(3/2)*Log[1 - Sqrt[c]*x])/6 + (b*c^(3/2)*Log[1 + Sqrt[c]*x])/6","A",1
63,1,91,63,0.029949,"\int \frac{a+b \tanh ^{-1}\left(c x^2\right)}{x^6} \, dx","Integrate[(a + b*ArcTanh[c*x^2])/x^6,x]","-\frac{a}{5 x^5}-\frac{1}{10} b c^{5/2} \log \left(1-\sqrt{c} x\right)+\frac{1}{10} b c^{5/2} \log \left(\sqrt{c} x+1\right)+\frac{1}{5} b c^{5/2} \tan ^{-1}\left(\sqrt{c} x\right)-\frac{2 b c}{15 x^3}-\frac{b \tanh ^{-1}\left(c x^2\right)}{5 x^5}","-\frac{a+b \tanh ^{-1}\left(c x^2\right)}{5 x^5}+\frac{1}{5} b c^{5/2} \tan ^{-1}\left(\sqrt{c} x\right)+\frac{1}{5} b c^{5/2} \tanh ^{-1}\left(\sqrt{c} x\right)-\frac{2 b c}{15 x^3}",1,"-1/5*a/x^5 - (2*b*c)/(15*x^3) + (b*c^(5/2)*ArcTan[Sqrt[c]*x])/5 - (b*ArcTanh[c*x^2])/(5*x^5) - (b*c^(5/2)*Log[1 - Sqrt[c]*x])/10 + (b*c^(5/2)*Log[1 + Sqrt[c]*x])/10","A",1
64,1,146,125,0.0801153,"\int x^7 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[x^7*(a + b*ArcTanh[c*x^2])^2,x]","\frac{3 a^2 c^4 x^8+2 a b c^3 x^6+2 b c x^2 \tanh ^{-1}\left(c x^2\right) \left(3 a c^3 x^6+b \left(c^2 x^4+3\right)\right)+6 a b c x^2+b (3 a+4 b) \log \left(1-c x^2\right)-3 a b \log \left(c x^2+1\right)+3 b^2 \left(c^4 x^8-1\right) \tanh ^{-1}\left(c x^2\right)^2+b^2 c^2 x^4+4 b^2 \log \left(c x^2+1\right)}{24 c^4}","-\frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{8 c^4}+\frac{a b x^2}{4 c^3}+\frac{1}{8} x^8 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2+\frac{b x^6 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{12 c}+\frac{b^2 x^2 \tanh ^{-1}\left(c x^2\right)}{4 c^3}+\frac{b^2 x^4}{24 c^2}+\frac{b^2 \log \left(1-c^2 x^4\right)}{6 c^4}",1,"(6*a*b*c*x^2 + b^2*c^2*x^4 + 2*a*b*c^3*x^6 + 3*a^2*c^4*x^8 + 2*b*c*x^2*(3*a*c^3*x^6 + b*(3 + c^2*x^4))*ArcTanh[c*x^2] + 3*b^2*(-1 + c^4*x^8)*ArcTanh[c*x^2]^2 + b*(3*a + 4*b)*Log[1 - c*x^2] - 3*a*b*Log[1 + c*x^2] + 4*b^2*Log[1 + c*x^2])/(24*c^4)","A",1
65,1,132,146,0.3026858,"\int x^5 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[x^5*(a + b*ArcTanh[c*x^2])^2,x]","\frac{a^2 c^3 x^6+a b c^2 x^4+a b \log \left(c^2 x^4-1\right)+b \tanh ^{-1}\left(c x^2\right) \left(2 a c^3 x^6+b c^2 x^4-2 b \log \left(e^{-2 \tanh ^{-1}\left(c x^2\right)}+1\right)-b\right)+b^2 \left(c^3 x^6-1\right) \tanh ^{-1}\left(c x^2\right)^2+b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c x^2\right)}\right)+b^2 c x^2}{6 c^3}","\frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{6 c^3}-\frac{b \log \left(\frac{2}{1-c x^2}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{3 c^3}+\frac{1}{6} x^6 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2+\frac{b x^4 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{6 c}-\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-c x^2}\right)}{6 c^3}-\frac{b^2 \tanh ^{-1}\left(c x^2\right)}{6 c^3}+\frac{b^2 x^2}{6 c^2}",1,"(b^2*c*x^2 + a*b*c^2*x^4 + a^2*c^3*x^6 + b^2*(-1 + c^3*x^6)*ArcTanh[c*x^2]^2 + b*ArcTanh[c*x^2]*(-b + b*c^2*x^4 + 2*a*c^3*x^6 - 2*b*Log[1 + E^(-2*ArcTanh[c*x^2])]) + a*b*Log[-1 + c^2*x^4] + b^2*PolyLog[2, -E^(-2*ArcTanh[c*x^2])])/(6*c^3)","A",0
66,1,106,91,0.0619509,"\int x^3 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[x^3*(a + b*ArcTanh[c*x^2])^2,x]","\frac{a^2 c^2 x^4+2 a b c x^2+b (a+b) \log \left(1-c x^2\right)-a b \log \left(c x^2+1\right)+2 b c x^2 \tanh ^{-1}\left(c x^2\right) \left(a c x^2+b\right)+b^2 \left(c^2 x^4-1\right) \tanh ^{-1}\left(c x^2\right)^2+b^2 \log \left(c x^2+1\right)}{4 c^2}","-\frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{4 c^2}+\frac{a b x^2}{2 c}+\frac{1}{4} x^4 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2+\frac{b^2 \log \left(1-c^2 x^4\right)}{4 c^2}+\frac{b^2 x^2 \tanh ^{-1}\left(c x^2\right)}{2 c}",1,"(2*a*b*c*x^2 + a^2*c^2*x^4 + 2*b*c*x^2*(b + a*c*x^2)*ArcTanh[c*x^2] + b^2*(-1 + c^2*x^4)*ArcTanh[c*x^2]^2 + b*(a + b)*Log[1 - c*x^2] - a*b*Log[1 + c*x^2] + b^2*Log[1 + c*x^2])/(4*c^2)","A",1
67,1,99,94,0.1625082,"\int x \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[x*(a + b*ArcTanh[c*x^2])^2,x]","\frac{a \left(a c x^2+b \log \left(1-c^2 x^4\right)\right)+2 b \tanh ^{-1}\left(c x^2\right) \left(a c x^2-b \log \left(e^{-2 \tanh ^{-1}\left(c x^2\right)}+1\right)\right)+b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c x^2\right)}\right)+b^2 \left(c x^2-1\right) \tanh ^{-1}\left(c x^2\right)^2}{2 c}","\frac{1}{2} x^2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2+\frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{2 c}-\frac{b \log \left(\frac{2}{1-c x^2}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{c}-\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-c x^2}\right)}{2 c}",1,"(b^2*(-1 + c*x^2)*ArcTanh[c*x^2]^2 + 2*b*ArcTanh[c*x^2]*(a*c*x^2 - b*Log[1 + E^(-2*ArcTanh[c*x^2])]) + a*(a*c*x^2 + b*Log[1 - c^2*x^4]) + b^2*PolyLog[2, -E^(-2*ArcTanh[c*x^2])])/(2*c)","A",0
68,1,183,137,0.0763311,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{x} \, dx","Integrate[(a + b*ArcTanh[c*x^2])^2/x,x]","\frac{1}{2} \left(2 \tanh ^{-1}\left(1-\frac{2}{1-c x^2}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2-4 b c \left(\frac{1}{2} \left(\frac{\text{Li}_2\left(\frac{-c x^2-1}{c x^2-1}\right) \left(-a-b \tanh ^{-1}\left(c x^2\right)\right)}{2 c}+\frac{b \text{Li}_3\left(\frac{-c x^2-1}{c x^2-1}\right)}{4 c}\right)+\frac{1}{2} \left(-\frac{\text{Li}_2\left(\frac{c x^2+1}{c x^2-1}\right) \left(-a-b \tanh ^{-1}\left(c x^2\right)\right)}{2 c}-\frac{b \text{Li}_3\left(\frac{c x^2+1}{c x^2-1}\right)}{4 c}\right)\right)\right)","-\frac{1}{2} b \text{Li}_2\left(1-\frac{2}{1-c x^2}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)+\frac{1}{2} b \text{Li}_2\left(\frac{2}{1-c x^2}-1\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)+\tanh ^{-1}\left(1-\frac{2}{1-c x^2}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2+\frac{1}{4} b^2 \text{Li}_3\left(1-\frac{2}{1-c x^2}\right)-\frac{1}{4} b^2 \text{Li}_3\left(\frac{2}{1-c x^2}-1\right)",1,"(2*(a + b*ArcTanh[c*x^2])^2*ArcTanh[1 - 2/(1 - c*x^2)] - 4*b*c*((((-a - b*ArcTanh[c*x^2])*PolyLog[2, (-1 - c*x^2)/(-1 + c*x^2)])/(2*c) + (b*PolyLog[3, (-1 - c*x^2)/(-1 + c*x^2)])/(4*c))/2 + (-1/2*((-a - b*ArcTanh[c*x^2])*PolyLog[2, (1 + c*x^2)/(-1 + c*x^2)])/c - (b*PolyLog[3, (1 + c*x^2)/(-1 + c*x^2)])/(4*c))/2))/2","A",1
69,1,119,87,0.1637002,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{x^3} \, dx","Integrate[(a + b*ArcTanh[c*x^2])^2/x^3,x]","-\frac{a^2}{2 x^2}+a b c \left(-\frac{1}{2} \log \left(1-c^2 x^4\right)+\log \left(c x^2\right)-\frac{\tanh ^{-1}\left(c x^2\right)}{c x^2}\right)+\frac{1}{2} b^2 c \left(\tanh ^{-1}\left(c x^2\right) \left(-\frac{\tanh ^{-1}\left(c x^2\right)}{c x^2}+\tanh ^{-1}\left(c x^2\right)+2 \log \left(1-e^{-2 \tanh ^{-1}\left(c x^2\right)}\right)\right)-\text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c x^2\right)}\right)\right)","\frac{1}{2} c \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2-\frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{2 x^2}+b c \log \left(2-\frac{2}{c x^2+1}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)-\frac{1}{2} b^2 c \text{Li}_2\left(\frac{2}{c x^2+1}-1\right)",1,"-1/2*a^2/x^2 + a*b*c*(-(ArcTanh[c*x^2]/(c*x^2)) + Log[c*x^2] - Log[1 - c^2*x^4]/2) + (b^2*c*(ArcTanh[c*x^2]*(ArcTanh[c*x^2] - ArcTanh[c*x^2]/(c*x^2) + 2*Log[1 - E^(-2*ArcTanh[c*x^2])]) - PolyLog[2, E^(-2*ArcTanh[c*x^2])]))/2","A",0
70,1,111,88,0.0969837,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{x^5} \, dx","Integrate[(a + b*ArcTanh[c*x^2])^2/x^5,x]","\frac{1}{4} \left(-\frac{a^2}{x^4}-b c^2 (a+b) \log \left(1-c x^2\right)+b c^2 (a-b) \log \left(c x^2+1\right)-\frac{2 a b c}{x^2}-\frac{2 b \tanh ^{-1}\left(c x^2\right) \left(a+b c x^2\right)}{x^4}+\frac{b^2 \left(c^2 x^4-1\right) \tanh ^{-1}\left(c x^2\right)^2}{x^4}+4 b^2 c^2 \log (x)\right)","\frac{1}{4} c^2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2-\frac{b c \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{2 x^2}-\frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{4 x^4}-\frac{1}{4} b^2 c^2 \log \left(1-c^2 x^4\right)+b^2 c^2 \log (x)",1,"(-(a^2/x^4) - (2*a*b*c)/x^2 - (2*b*(a + b*c*x^2)*ArcTanh[c*x^2])/x^4 + (b^2*(-1 + c^2*x^4)*ArcTanh[c*x^2]^2)/x^4 + 4*b^2*c^2*Log[x] - b*(a + b)*c^2*Log[1 - c*x^2] + (a - b)*b*c^2*Log[1 + c*x^2])/4","A",1
71,0,0,1173,5.6195361,"\int x^4 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[x^4*(a + b*ArcTanh[c*x^2])^2,x]","\int x^4 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2 \, dx","\frac{1}{20} \left(2 a-b \log \left(1-c x^2\right)\right)^2 x^5+\frac{1}{20} b^2 \log ^2\left(c x^2+1\right) x^5-\frac{2}{25} a b x^5+\frac{1}{25} b^2 \log \left(1-c x^2\right) x^5+\frac{1}{25} b \left(2 a-b \log \left(1-c x^2\right)\right) x^5+\frac{1}{5} a b \log \left(c x^2+1\right) x^5-\frac{1}{10} b^2 \log \left(1-c x^2\right) \log \left(c x^2+1\right) x^5-\frac{b^2 \log \left(1-c x^2\right) x^3}{15 c}+\frac{b \left(2 a-b \log \left(1-c x^2\right)\right) x^3}{15 c}+\frac{2 b^2 \log \left(c x^2+1\right) x^3}{15 c}+\frac{2 a b x^3}{15 c}+\frac{8 b^2 x}{15 c^2}+\frac{i b^2 \tan ^{-1}\left(\sqrt{c} x\right)^2}{5 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left(\sqrt{c} x\right)^2}{5 c^{5/2}}-\frac{4 b^2 \tan ^{-1}\left(\sqrt{c} x\right)}{15 c^{5/2}}+\frac{2 a b \tan ^{-1}\left(\sqrt{c} x\right)}{5 c^{5/2}}-\frac{4 b^2 \tanh ^{-1}\left(\sqrt{c} x\right)}{15 c^{5/2}}+\frac{2 b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{1-\sqrt{c} x}\right)}{5 c^{5/2}}-\frac{2 b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{1-i \sqrt{c} x}\right)}{5 c^{5/2}}+\frac{b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{(1+i) \left(1-\sqrt{c} x\right)}{1-i \sqrt{c} x}\right)}{5 c^{5/2}}+\frac{2 b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{i \sqrt{c} x+1}\right)}{5 c^{5/2}}-\frac{2 b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{\sqrt{c} x+1}\right)}{5 c^{5/2}}+\frac{b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(-\frac{2 \sqrt{c} \left(1-\sqrt{-c} x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right)}{5 c^{5/2}}+\frac{b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2 \sqrt{c} \left(\sqrt{-c} x+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right)}{5 c^{5/2}}+\frac{b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{(1-i) \left(\sqrt{c} x+1\right)}{1-i \sqrt{c} x}\right)}{5 c^{5/2}}-\frac{b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(1-c x^2\right)}{5 c^{5/2}}-\frac{b \tanh ^{-1}\left(\sqrt{c} x\right) \left(2 a-b \log \left(1-c x^2\right)\right)}{5 c^{5/2}}+\frac{b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(c x^2+1\right)}{5 c^{5/2}}-\frac{b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(c x^2+1\right)}{5 c^{5/2}}+\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-\sqrt{c} x}\right)}{5 c^{5/2}}+\frac{i b^2 \text{Li}_2\left(1-\frac{2}{1-i \sqrt{c} x}\right)}{5 c^{5/2}}-\frac{i b^2 \text{Li}_2\left(1-\frac{(1+i) \left(1-\sqrt{c} x\right)}{1-i \sqrt{c} x}\right)}{10 c^{5/2}}+\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i \sqrt{c} x+1}\right)}{5 c^{5/2}}+\frac{b^2 \text{Li}_2\left(1-\frac{2}{\sqrt{c} x+1}\right)}{5 c^{5/2}}-\frac{b^2 \text{Li}_2\left(\frac{2 \sqrt{c} \left(1-\sqrt{-c} x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\sqrt{c} x+1\right)}+1\right)}{10 c^{5/2}}-\frac{b^2 \text{Li}_2\left(1-\frac{2 \sqrt{c} \left(\sqrt{-c} x+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right)}{10 c^{5/2}}-\frac{i b^2 \text{Li}_2\left(1-\frac{(1-i) \left(\sqrt{c} x+1\right)}{1-i \sqrt{c} x}\right)}{10 c^{5/2}}",1,"Integrate[x^4*(a + b*ArcTanh[c*x^2])^2, x]","F",-1
72,0,0,1129,3.3052457,"\int x^2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[x^2*(a + b*ArcTanh[c*x^2])^2,x]","\int x^2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2 \, dx","\frac{1}{12} \left(2 a-b \log \left(1-c x^2\right)\right)^2 x^3+\frac{1}{12} b^2 \log ^2\left(c x^2+1\right) x^3-\frac{2}{9} a b x^3+\frac{1}{9} b^2 \log \left(1-c x^2\right) x^3+\frac{1}{9} b \left(2 a-b \log \left(1-c x^2\right)\right) x^3+\frac{1}{3} a b \log \left(c x^2+1\right) x^3-\frac{1}{6} b^2 \log \left(1-c x^2\right) \log \left(c x^2+1\right) x^3-\frac{2 b^2 \log \left(1-c x^2\right) x}{3 c}+\frac{2 b^2 \log \left(c x^2+1\right) x}{3 c}+\frac{4 a b x}{3 c}-\frac{i b^2 \tan ^{-1}\left(\sqrt{c} x\right)^2}{3 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left(\sqrt{c} x\right)^2}{3 c^{3/2}}+\frac{4 b^2 \tan ^{-1}\left(\sqrt{c} x\right)}{3 c^{3/2}}-\frac{2 a b \tan ^{-1}\left(\sqrt{c} x\right)}{3 c^{3/2}}-\frac{4 b^2 \tanh ^{-1}\left(\sqrt{c} x\right)}{3 c^{3/2}}+\frac{2 b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{1-\sqrt{c} x}\right)}{3 c^{3/2}}+\frac{2 b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{1-i \sqrt{c} x}\right)}{3 c^{3/2}}-\frac{b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{(1+i) \left(1-\sqrt{c} x\right)}{1-i \sqrt{c} x}\right)}{3 c^{3/2}}-\frac{2 b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{i \sqrt{c} x+1}\right)}{3 c^{3/2}}-\frac{2 b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{\sqrt{c} x+1}\right)}{3 c^{3/2}}+\frac{b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(-\frac{2 \sqrt{c} \left(1-\sqrt{-c} x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right)}{3 c^{3/2}}+\frac{b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2 \sqrt{c} \left(\sqrt{-c} x+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right)}{3 c^{3/2}}-\frac{b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{(1-i) \left(\sqrt{c} x+1\right)}{1-i \sqrt{c} x}\right)}{3 c^{3/2}}+\frac{b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(1-c x^2\right)}{3 c^{3/2}}-\frac{b \tanh ^{-1}\left(\sqrt{c} x\right) \left(2 a-b \log \left(1-c x^2\right)\right)}{3 c^{3/2}}-\frac{b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(c x^2+1\right)}{3 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(c x^2+1\right)}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-\sqrt{c} x}\right)}{3 c^{3/2}}-\frac{i b^2 \text{Li}_2\left(1-\frac{2}{1-i \sqrt{c} x}\right)}{3 c^{3/2}}+\frac{i b^2 \text{Li}_2\left(1-\frac{(1+i) \left(1-\sqrt{c} x\right)}{1-i \sqrt{c} x}\right)}{6 c^{3/2}}-\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i \sqrt{c} x+1}\right)}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left(1-\frac{2}{\sqrt{c} x+1}\right)}{3 c^{3/2}}-\frac{b^2 \text{Li}_2\left(\frac{2 \sqrt{c} \left(1-\sqrt{-c} x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\sqrt{c} x+1\right)}+1\right)}{6 c^{3/2}}-\frac{b^2 \text{Li}_2\left(1-\frac{2 \sqrt{c} \left(\sqrt{-c} x+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right)}{6 c^{3/2}}+\frac{i b^2 \text{Li}_2\left(1-\frac{(1-i) \left(\sqrt{c} x+1\right)}{1-i \sqrt{c} x}\right)}{6 c^{3/2}}",1,"Integrate[x^2*(a + b*ArcTanh[c*x^2])^2, x]","F",-1
73,1,566,958,2.7411781,"\int \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[(a + b*ArcTanh[c*x^2])^2,x]","\frac{1}{2} x \left(2 a^2+4 a b \tanh ^{-1}\left(c x^2\right)+\frac{4 a b \left(\tan ^{-1}\left(\sqrt{c x^2}\right)-\tanh ^{-1}\left(\sqrt{c x^2}\right)\right)}{\sqrt{c x^2}}+\frac{b^2 \left(\text{Li}_2\left(\frac{1}{2} \left(1-\sqrt{c x^2}\right)\right)-\text{Li}_2\left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\sqrt{c x^2}-1\right)\right)-\text{Li}_2\left(\left(-\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{c x^2}-1\right)\right)-\text{Li}_2\left(\frac{1}{2} \left(\sqrt{c x^2}+1\right)\right)+\text{Li}_2\left(\left(\frac{1}{2}-\frac{i}{2}\right) \left(\sqrt{c x^2}+1\right)\right)+\text{Li}_2\left(\left(\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{c x^2}+1\right)\right)-\frac{1}{2} i \text{Li}_2\left(-e^{4 i \tan ^{-1}\left(\sqrt{c x^2}\right)}\right)+\frac{1}{2} \log ^2\left(1-\sqrt{c x^2}\right)-\frac{1}{2} \log ^2\left(\sqrt{c x^2}+1\right)-\log (2) \log \left(1-\sqrt{c x^2}\right)-\log \left(1-\sqrt{c x^2}\right) \log \left(\left(\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{c x^2}-i\right)\right)+\log \left(\frac{1}{2} \left((1+i)-(1-i) \sqrt{c x^2}\right)\right) \log \left(\sqrt{c x^2}+1\right)+\log \left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\sqrt{c x^2}+i\right)\right) \log \left(\sqrt{c x^2}+1\right)+\log (2) \log \left(\sqrt{c x^2}+1\right)-\log \left(1-\sqrt{c x^2}\right) \log \left(\frac{1}{2} \left((1-i) \sqrt{c x^2}+(1+i)\right)\right)-2 i \tan ^{-1}\left(\sqrt{c x^2}\right)^2+2 \sqrt{c x^2} \tanh ^{-1}\left(c x^2\right)^2+2 \tan ^{-1}\left(\sqrt{c x^2}\right) \log \left(1+e^{4 i \tan ^{-1}\left(\sqrt{c x^2}\right)}\right)+2 \log \left(1-\sqrt{c x^2}\right) \tanh ^{-1}\left(c x^2\right)-2 \log \left(\sqrt{c x^2}+1\right) \tanh ^{-1}\left(c x^2\right)+4 \tan ^{-1}\left(\sqrt{c x^2}\right) \tanh ^{-1}\left(c x^2\right)\right)}{\sqrt{c x^2}}\right)","x a^2+\frac{2 b \tan ^{-1}\left(\sqrt{c} x\right) a}{\sqrt{c}}-\frac{2 b \tanh ^{-1}\left(\sqrt{c} x\right) a}{\sqrt{c}}-b x \log \left(1-c x^2\right) a+b x \log \left(c x^2+1\right) a+\frac{i b^2 \tan ^{-1}\left(\sqrt{c} x\right)^2}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left(\sqrt{c} x\right)^2}{\sqrt{c}}+\frac{1}{4} b^2 x \log ^2\left(1-c x^2\right)+\frac{1}{4} b^2 x \log ^2\left(c x^2+1\right)+\frac{2 b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{1-\sqrt{c} x}\right)}{\sqrt{c}}-\frac{2 b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{1-i \sqrt{c} x}\right)}{\sqrt{c}}+\frac{b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{(1+i) \left(1-\sqrt{c} x\right)}{1-i \sqrt{c} x}\right)}{\sqrt{c}}+\frac{2 b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{i \sqrt{c} x+1}\right)}{\sqrt{c}}-\frac{2 b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{\sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(-\frac{2 \sqrt{c} \left(1-\sqrt{-c} x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right)}{\sqrt{c}}+\frac{b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2 \sqrt{c} \left(\sqrt{-c} x+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right)}{\sqrt{c}}+\frac{b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{(1-i) \left(\sqrt{c} x+1\right)}{1-i \sqrt{c} x}\right)}{\sqrt{c}}-\frac{b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(1-c x^2\right)}{\sqrt{c}}+\frac{b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(1-c x^2\right)}{\sqrt{c}}+\frac{b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(c x^2+1\right)}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(c x^2+1\right)}{\sqrt{c}}-\frac{1}{2} b^2 x \log \left(1-c x^2\right) \log \left(c x^2+1\right)+\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-\sqrt{c} x}\right)}{\sqrt{c}}+\frac{i b^2 \text{Li}_2\left(1-\frac{2}{1-i \sqrt{c} x}\right)}{\sqrt{c}}-\frac{i b^2 \text{Li}_2\left(1-\frac{(1+i) \left(1-\sqrt{c} x\right)}{1-i \sqrt{c} x}\right)}{2 \sqrt{c}}+\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i \sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{b^2 \text{Li}_2\left(1-\frac{2}{\sqrt{c} x+1}\right)}{\sqrt{c}}-\frac{b^2 \text{Li}_2\left(\frac{2 \sqrt{c} \left(1-\sqrt{-c} x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\sqrt{c} x+1\right)}+1\right)}{2 \sqrt{c}}-\frac{b^2 \text{Li}_2\left(1-\frac{2 \sqrt{c} \left(\sqrt{-c} x+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right)}{2 \sqrt{c}}-\frac{i b^2 \text{Li}_2\left(1-\frac{(1-i) \left(\sqrt{c} x+1\right)}{1-i \sqrt{c} x}\right)}{2 \sqrt{c}}",1,"(x*(2*a^2 + 4*a*b*ArcTanh[c*x^2] + (4*a*b*(ArcTan[Sqrt[c*x^2]] - ArcTanh[Sqrt[c*x^2]]))/Sqrt[c*x^2] + (b^2*((-2*I)*ArcTan[Sqrt[c*x^2]]^2 + 4*ArcTan[Sqrt[c*x^2]]*ArcTanh[c*x^2] + 2*Sqrt[c*x^2]*ArcTanh[c*x^2]^2 + 2*ArcTan[Sqrt[c*x^2]]*Log[1 + E^((4*I)*ArcTan[Sqrt[c*x^2]])] + 2*ArcTanh[c*x^2]*Log[1 - Sqrt[c*x^2]] - Log[2]*Log[1 - Sqrt[c*x^2]] + Log[1 - Sqrt[c*x^2]]^2/2 - Log[1 - Sqrt[c*x^2]]*Log[(1/2 + I/2)*(-I + Sqrt[c*x^2])] - 2*ArcTanh[c*x^2]*Log[1 + Sqrt[c*x^2]] + Log[2]*Log[1 + Sqrt[c*x^2]] + Log[((1 + I) - (1 - I)*Sqrt[c*x^2])/2]*Log[1 + Sqrt[c*x^2]] + Log[(-1/2 - I/2)*(I + Sqrt[c*x^2])]*Log[1 + Sqrt[c*x^2]] - Log[1 + Sqrt[c*x^2]]^2/2 - Log[1 - Sqrt[c*x^2]]*Log[((1 + I) + (1 - I)*Sqrt[c*x^2])/2] - (I/2)*PolyLog[2, -E^((4*I)*ArcTan[Sqrt[c*x^2]])] + PolyLog[2, (1 - Sqrt[c*x^2])/2] - PolyLog[2, (-1/2 - I/2)*(-1 + Sqrt[c*x^2])] - PolyLog[2, (-1/2 + I/2)*(-1 + Sqrt[c*x^2])] - PolyLog[2, (1 + Sqrt[c*x^2])/2] + PolyLog[2, (1/2 - I/2)*(1 + Sqrt[c*x^2])] + PolyLog[2, (1/2 + I/2)*(1 + Sqrt[c*x^2])]))/Sqrt[c*x^2]))/2","A",0
74,1,566,942,3.7225873,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{x^2} \, dx","Integrate[(a + b*ArcTanh[c*x^2])^2/x^2,x]","\frac{-2 a^2-4 a b \tanh ^{-1}\left(c x^2\right)+4 a b \sqrt{c x^2} \left(\tan ^{-1}\left(\sqrt{c x^2}\right)+\tanh ^{-1}\left(\sqrt{c x^2}\right)\right)+b^2 \sqrt{c x^2} \left(-\text{Li}_2\left(\frac{1}{2} \left(1-\sqrt{c x^2}\right)\right)+\text{Li}_2\left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\sqrt{c x^2}-1\right)\right)+\text{Li}_2\left(\left(-\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{c x^2}-1\right)\right)+\text{Li}_2\left(\frac{1}{2} \left(\sqrt{c x^2}+1\right)\right)-\text{Li}_2\left(\left(\frac{1}{2}-\frac{i}{2}\right) \left(\sqrt{c x^2}+1\right)\right)-\text{Li}_2\left(\left(\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{c x^2}+1\right)\right)-\frac{1}{2} i \text{Li}_2\left(-e^{4 i \tan ^{-1}\left(\sqrt{c x^2}\right)}\right)-\frac{1}{2} \log ^2\left(1-\sqrt{c x^2}\right)+\frac{1}{2} \log ^2\left(\sqrt{c x^2}+1\right)+\log (2) \log \left(1-\sqrt{c x^2}\right)+\log \left(1-\sqrt{c x^2}\right) \log \left(\left(\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{c x^2}-i\right)\right)-\log \left(\frac{1}{2} \left((1+i)-(1-i) \sqrt{c x^2}\right)\right) \log \left(\sqrt{c x^2}+1\right)-\log \left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\sqrt{c x^2}+i\right)\right) \log \left(\sqrt{c x^2}+1\right)-\log (2) \log \left(\sqrt{c x^2}+1\right)+\log \left(1-\sqrt{c x^2}\right) \log \left(\frac{1}{2} \left((1-i) \sqrt{c x^2}+(1+i)\right)\right)-2 i \tan ^{-1}\left(\sqrt{c x^2}\right)^2-\frac{2 \tanh ^{-1}\left(c x^2\right)^2}{\sqrt{c x^2}}+2 \tan ^{-1}\left(\sqrt{c x^2}\right) \log \left(1+e^{4 i \tan ^{-1}\left(\sqrt{c x^2}\right)}\right)-2 \log \left(1-\sqrt{c x^2}\right) \tanh ^{-1}\left(c x^2\right)+2 \log \left(\sqrt{c x^2}+1\right) \tanh ^{-1}\left(c x^2\right)+4 \tan ^{-1}\left(\sqrt{c x^2}\right) \tanh ^{-1}\left(c x^2\right)\right)}{2 x}","i \sqrt{c} \tan ^{-1}\left(\sqrt{c} x\right)^2 b^2+\sqrt{c} \tanh ^{-1}\left(\sqrt{c} x\right)^2 b^2-\frac{\log ^2\left(c x^2+1\right) b^2}{4 x}-2 \sqrt{c} \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{1-\sqrt{c} x}\right) b^2-2 \sqrt{c} \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{1-i \sqrt{c} x}\right) b^2+\sqrt{c} \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{(1+i) \left(1-\sqrt{c} x\right)}{1-i \sqrt{c} x}\right) b^2+2 \sqrt{c} \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{i \sqrt{c} x+1}\right) b^2+2 \sqrt{c} \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{\sqrt{c} x+1}\right) b^2-\sqrt{c} \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(-\frac{2 \sqrt{c} \left(1-\sqrt{-c} x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right) b^2-\sqrt{c} \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2 \sqrt{c} \left(\sqrt{-c} x+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right) b^2+\sqrt{c} \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{(1-i) \left(\sqrt{c} x+1\right)}{1-i \sqrt{c} x}\right) b^2-\sqrt{c} \tan ^{-1}\left(\sqrt{c} x\right) \log \left(1-c x^2\right) b^2+\sqrt{c} \tan ^{-1}\left(\sqrt{c} x\right) \log \left(c x^2+1\right) b^2+\sqrt{c} \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(c x^2+1\right) b^2+\frac{\log \left(1-c x^2\right) \log \left(c x^2+1\right) b^2}{2 x}-\sqrt{c} \text{Li}_2\left(1-\frac{2}{1-\sqrt{c} x}\right) b^2+i \sqrt{c} \text{Li}_2\left(1-\frac{2}{1-i \sqrt{c} x}\right) b^2-\frac{1}{2} i \sqrt{c} \text{Li}_2\left(1-\frac{(1+i) \left(1-\sqrt{c} x\right)}{1-i \sqrt{c} x}\right) b^2+i \sqrt{c} \text{Li}_2\left(1-\frac{2}{i \sqrt{c} x+1}\right) b^2-\sqrt{c} \text{Li}_2\left(1-\frac{2}{\sqrt{c} x+1}\right) b^2+\frac{1}{2} \sqrt{c} \text{Li}_2\left(\frac{2 \sqrt{c} \left(1-\sqrt{-c} x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\sqrt{c} x+1\right)}+1\right) b^2+\frac{1}{2} \sqrt{c} \text{Li}_2\left(1-\frac{2 \sqrt{c} \left(\sqrt{-c} x+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right) b^2-\frac{1}{2} i \sqrt{c} \text{Li}_2\left(1-\frac{(1-i) \left(\sqrt{c} x+1\right)}{1-i \sqrt{c} x}\right) b^2+2 a \sqrt{c} \tan ^{-1}\left(\sqrt{c} x\right) b+\sqrt{c} \tanh ^{-1}\left(\sqrt{c} x\right) \left(2 a-b \log \left(1-c x^2\right)\right) b-\frac{a \log \left(c x^2+1\right) b}{x}-\frac{\left(2 a-b \log \left(1-c x^2\right)\right)^2}{4 x}",1,"(-2*a^2 - 4*a*b*ArcTanh[c*x^2] + 4*a*b*Sqrt[c*x^2]*(ArcTan[Sqrt[c*x^2]] + ArcTanh[Sqrt[c*x^2]]) + b^2*Sqrt[c*x^2]*((-2*I)*ArcTan[Sqrt[c*x^2]]^2 + 4*ArcTan[Sqrt[c*x^2]]*ArcTanh[c*x^2] - (2*ArcTanh[c*x^2]^2)/Sqrt[c*x^2] + 2*ArcTan[Sqrt[c*x^2]]*Log[1 + E^((4*I)*ArcTan[Sqrt[c*x^2]])] - 2*ArcTanh[c*x^2]*Log[1 - Sqrt[c*x^2]] + Log[2]*Log[1 - Sqrt[c*x^2]] - Log[1 - Sqrt[c*x^2]]^2/2 + Log[1 - Sqrt[c*x^2]]*Log[(1/2 + I/2)*(-I + Sqrt[c*x^2])] + 2*ArcTanh[c*x^2]*Log[1 + Sqrt[c*x^2]] - Log[2]*Log[1 + Sqrt[c*x^2]] - Log[((1 + I) - (1 - I)*Sqrt[c*x^2])/2]*Log[1 + Sqrt[c*x^2]] - Log[(-1/2 - I/2)*(I + Sqrt[c*x^2])]*Log[1 + Sqrt[c*x^2]] + Log[1 + Sqrt[c*x^2]]^2/2 + Log[1 - Sqrt[c*x^2]]*Log[((1 + I) + (1 - I)*Sqrt[c*x^2])/2] - (I/2)*PolyLog[2, -E^((4*I)*ArcTan[Sqrt[c*x^2]])] - PolyLog[2, (1 - Sqrt[c*x^2])/2] + PolyLog[2, (-1/2 - I/2)*(-1 + Sqrt[c*x^2])] + PolyLog[2, (-1/2 + I/2)*(-1 + Sqrt[c*x^2])] + PolyLog[2, (1 + Sqrt[c*x^2])/2] - PolyLog[2, (1/2 - I/2)*(1 + Sqrt[c*x^2])] - PolyLog[2, (1/2 + I/2)*(1 + Sqrt[c*x^2])]))/(2*x)","A",0
75,0,0,1102,2.769385,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{x^4} \, dx","Integrate[(a + b*ArcTanh[c*x^2])^2/x^4,x]","\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{x^4} \, dx","-\frac{1}{3} i c^{3/2} \tan ^{-1}\left(\sqrt{c} x\right)^2 b^2+\frac{1}{3} c^{3/2} \tanh ^{-1}\left(\sqrt{c} x\right)^2 b^2-\frac{\log ^2\left(c x^2+1\right) b^2}{12 x^3}+\frac{4}{3} c^{3/2} \tan ^{-1}\left(\sqrt{c} x\right) b^2+\frac{4}{3} c^{3/2} \tanh ^{-1}\left(\sqrt{c} x\right) b^2-\frac{2}{3} c^{3/2} \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{1-\sqrt{c} x}\right) b^2+\frac{2}{3} c^{3/2} \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{1-i \sqrt{c} x}\right) b^2-\frac{1}{3} c^{3/2} \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{(1+i) \left(1-\sqrt{c} x\right)}{1-i \sqrt{c} x}\right) b^2-\frac{2}{3} c^{3/2} \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{i \sqrt{c} x+1}\right) b^2+\frac{2}{3} c^{3/2} \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{\sqrt{c} x+1}\right) b^2-\frac{1}{3} c^{3/2} \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(-\frac{2 \sqrt{c} \left(1-\sqrt{-c} x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right) b^2-\frac{1}{3} c^{3/2} \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2 \sqrt{c} \left(\sqrt{-c} x+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right) b^2-\frac{1}{3} c^{3/2} \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{(1-i) \left(\sqrt{c} x+1\right)}{1-i \sqrt{c} x}\right) b^2+\frac{1}{3} c^{3/2} \tan ^{-1}\left(\sqrt{c} x\right) \log \left(1-c x^2\right) b^2+\frac{c \log \left(1-c x^2\right) b^2}{3 x}-\frac{1}{3} c^{3/2} \tan ^{-1}\left(\sqrt{c} x\right) \log \left(c x^2+1\right) b^2+\frac{1}{3} c^{3/2} \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(c x^2+1\right) b^2+\frac{\log \left(1-c x^2\right) \log \left(c x^2+1\right) b^2}{6 x^3}-\frac{2 c \log \left(c x^2+1\right) b^2}{3 x}-\frac{1}{3} c^{3/2} \text{Li}_2\left(1-\frac{2}{1-\sqrt{c} x}\right) b^2-\frac{1}{3} i c^{3/2} \text{Li}_2\left(1-\frac{2}{1-i \sqrt{c} x}\right) b^2+\frac{1}{6} i c^{3/2} \text{Li}_2\left(1-\frac{(1+i) \left(1-\sqrt{c} x\right)}{1-i \sqrt{c} x}\right) b^2-\frac{1}{3} i c^{3/2} \text{Li}_2\left(1-\frac{2}{i \sqrt{c} x+1}\right) b^2-\frac{1}{3} c^{3/2} \text{Li}_2\left(1-\frac{2}{\sqrt{c} x+1}\right) b^2+\frac{1}{6} c^{3/2} \text{Li}_2\left(\frac{2 \sqrt{c} \left(1-\sqrt{-c} x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\sqrt{c} x+1\right)}+1\right) b^2+\frac{1}{6} c^{3/2} \text{Li}_2\left(1-\frac{2 \sqrt{c} \left(\sqrt{-c} x+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right) b^2+\frac{1}{6} i c^{3/2} \text{Li}_2\left(1-\frac{(1-i) \left(\sqrt{c} x+1\right)}{1-i \sqrt{c} x}\right) b^2-\frac{2}{3} a c^{3/2} \tan ^{-1}\left(\sqrt{c} x\right) b+\frac{1}{3} c^{3/2} \tanh ^{-1}\left(\sqrt{c} x\right) \left(2 a-b \log \left(1-c x^2\right)\right) b-\frac{c \left(2 a-b \log \left(1-c x^2\right)\right) b}{3 x}-\frac{a \log \left(c x^2+1\right) b}{3 x^3}-\frac{2 a c b}{3 x}-\frac{\left(2 a-b \log \left(1-c x^2\right)\right)^2}{12 x^3}",1,"Integrate[(a + b*ArcTanh[c*x^2])^2/x^4, x]","F",-1
76,0,0,1176,3.2694649,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{x^6} \, dx","Integrate[(a + b*ArcTanh[c*x^2])^2/x^6,x]","\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{x^6} \, dx","\frac{1}{5} i b^2 \tan ^{-1}\left(\sqrt{c} x\right)^2 c^{5/2}+\frac{1}{5} b^2 \tanh ^{-1}\left(\sqrt{c} x\right)^2 c^{5/2}-\frac{4}{15} b^2 \tan ^{-1}\left(\sqrt{c} x\right) c^{5/2}+\frac{2}{5} a b \tan ^{-1}\left(\sqrt{c} x\right) c^{5/2}+\frac{4}{15} b^2 \tanh ^{-1}\left(\sqrt{c} x\right) c^{5/2}-\frac{2}{5} b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{1-\sqrt{c} x}\right) c^{5/2}-\frac{2}{5} b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{1-i \sqrt{c} x}\right) c^{5/2}+\frac{1}{5} b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{(1+i) \left(1-\sqrt{c} x\right)}{1-i \sqrt{c} x}\right) c^{5/2}+\frac{2}{5} b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{i \sqrt{c} x+1}\right) c^{5/2}+\frac{2}{5} b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2}{\sqrt{c} x+1}\right) c^{5/2}-\frac{1}{5} b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(-\frac{2 \sqrt{c} \left(1-\sqrt{-c} x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right) c^{5/2}-\frac{1}{5} b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{2 \sqrt{c} \left(\sqrt{-c} x+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right) c^{5/2}+\frac{1}{5} b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(\frac{(1-i) \left(\sqrt{c} x+1\right)}{1-i \sqrt{c} x}\right) c^{5/2}-\frac{1}{5} b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(1-c x^2\right) c^{5/2}+\frac{1}{5} b \tanh ^{-1}\left(\sqrt{c} x\right) \left(2 a-b \log \left(1-c x^2\right)\right) c^{5/2}+\frac{1}{5} b^2 \tan ^{-1}\left(\sqrt{c} x\right) \log \left(c x^2+1\right) c^{5/2}+\frac{1}{5} b^2 \tanh ^{-1}\left(\sqrt{c} x\right) \log \left(c x^2+1\right) c^{5/2}-\frac{1}{5} b^2 \text{Li}_2\left(1-\frac{2}{1-\sqrt{c} x}\right) c^{5/2}+\frac{1}{5} i b^2 \text{Li}_2\left(1-\frac{2}{1-i \sqrt{c} x}\right) c^{5/2}-\frac{1}{10} i b^2 \text{Li}_2\left(1-\frac{(1+i) \left(1-\sqrt{c} x\right)}{1-i \sqrt{c} x}\right) c^{5/2}+\frac{1}{5} i b^2 \text{Li}_2\left(1-\frac{2}{i \sqrt{c} x+1}\right) c^{5/2}-\frac{1}{5} b^2 \text{Li}_2\left(1-\frac{2}{\sqrt{c} x+1}\right) c^{5/2}+\frac{1}{10} b^2 \text{Li}_2\left(\frac{2 \sqrt{c} \left(1-\sqrt{-c} x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\sqrt{c} x+1\right)}+1\right) c^{5/2}+\frac{1}{10} b^2 \text{Li}_2\left(1-\frac{2 \sqrt{c} \left(\sqrt{-c} x+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\sqrt{c} x+1\right)}\right) c^{5/2}-\frac{1}{10} i b^2 \text{Li}_2\left(1-\frac{(1-i) \left(\sqrt{c} x+1\right)}{1-i \sqrt{c} x}\right) c^{5/2}-\frac{b^2 \log \left(1-c x^2\right) c^2}{5 x}-\frac{b \left(2 a-b \log \left(1-c x^2\right)\right) c^2}{5 x}-\frac{8 b^2 c^2}{15 x}+\frac{2 a b c^2}{5 x}+\frac{b^2 \log \left(1-c x^2\right) c}{15 x^3}-\frac{b \left(2 a-b \log \left(1-c x^2\right)\right) c}{15 x^3}-\frac{2 b^2 \log \left(c x^2+1\right) c}{15 x^3}-\frac{2 a b c}{15 x^3}-\frac{\left(2 a-b \log \left(1-c x^2\right)\right)^2}{20 x^5}-\frac{b^2 \log ^2\left(c x^2+1\right)}{20 x^5}+\frac{b^2 \log \left(1-c x^2\right) \log \left(c x^2+1\right)}{10 x^5}-\frac{a b \log \left(c x^2+1\right)}{5 x^5}",1,"Integrate[(a + b*ArcTanh[c*x^2])^2/x^6, x]","F",-1
77,1,185,141,0.4985432,"\int x^3 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3 \, dx","Integrate[x^3*(a + b*ArcTanh[c*x^2])^3,x]","\frac{a \left(2 a^2 c^2 x^4+6 a b c x^2+3 a b \log \left(1-c x^2\right)-3 a b \log \left(c x^2+1\right)+6 b^2 \log \left(1-c^2 x^4\right)\right)+6 b^2 \left(c x^2-1\right) \tanh ^{-1}\left(c x^2\right)^2 \left(a c x^2+a+b\right)+6 b \tanh ^{-1}\left(c x^2\right) \left(a c x^2 \left(a c x^2+2 b\right)-2 b^2 \log \left(e^{-2 \tanh ^{-1}\left(c x^2\right)}+1\right)\right)+2 b^3 \left(c^2 x^4-1\right) \tanh ^{-1}\left(c x^2\right)^3+6 b^3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c x^2\right)}\right)}{8 c^2}","-\frac{3 b^2 \log \left(\frac{2}{1-c x^2}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{2 c^2}-\frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3}{4 c^2}+\frac{3 b \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{4 c^2}+\frac{3 b x^2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{4 c}+\frac{1}{4} x^4 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3-\frac{3 b^3 \text{Li}_2\left(1-\frac{2}{1-c x^2}\right)}{4 c^2}",1,"(6*b^2*(-1 + c*x^2)*(a + b + a*c*x^2)*ArcTanh[c*x^2]^2 + 2*b^3*(-1 + c^2*x^4)*ArcTanh[c*x^2]^3 + 6*b*ArcTanh[c*x^2]*(a*c*x^2*(2*b + a*c*x^2) - 2*b^2*Log[1 + E^(-2*ArcTanh[c*x^2])]) + a*(6*a*b*c*x^2 + 2*a^2*c^2*x^4 + 3*a*b*Log[1 - c*x^2] - 3*a*b*Log[1 + c*x^2] + 6*b^2*Log[1 - c^2*x^4]) + 6*b^3*PolyLog[2, -E^(-2*ArcTanh[c*x^2])])/(8*c^2)","A",0
78,1,213,134,0.0857204,"\int x \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3 \, dx","Integrate[x*(a + b*ArcTanh[c*x^2])^3,x]","\frac{a^3 x^2}{2}+\frac{3 a^2 b \log \left(1-c^2 x^4\right)}{4 c}+\frac{3}{2} a^2 b x^2 \tanh ^{-1}\left(c x^2\right)+\frac{3 a b^2 \left(\text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c x^2\right)}\right)+\tanh ^{-1}\left(c x^2\right) \left(c x^2 \tanh ^{-1}\left(c x^2\right)-\tanh ^{-1}\left(c x^2\right)-2 \log \left(e^{-2 \tanh ^{-1}\left(c x^2\right)}+1\right)\right)\right)}{2 c}+\frac{b^3 \left(3 \tanh ^{-1}\left(c x^2\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c x^2\right)}\right)+\frac{3}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}\left(c x^2\right)}\right)+\tanh ^{-1}\left(c x^2\right)^2 \left(c x^2 \tanh ^{-1}\left(c x^2\right)-\tanh ^{-1}\left(c x^2\right)-3 \log \left(e^{-2 \tanh ^{-1}\left(c x^2\right)}+1\right)\right)\right)}{2 c}","-\frac{3 b^2 \text{Li}_2\left(1-\frac{2}{1-c x^2}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{2 c}+\frac{1}{2} x^2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3+\frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3}{2 c}-\frac{3 b \log \left(\frac{2}{1-c x^2}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{2 c}+\frac{3 b^3 \text{Li}_3\left(1-\frac{2}{1-c x^2}\right)}{4 c}",1,"(a^3*x^2)/2 + (3*a^2*b*x^2*ArcTanh[c*x^2])/2 + (3*a^2*b*Log[1 - c^2*x^4])/(4*c) + (3*a*b^2*(ArcTanh[c*x^2]*(-ArcTanh[c*x^2] + c*x^2*ArcTanh[c*x^2] - 2*Log[1 + E^(-2*ArcTanh[c*x^2])]) + PolyLog[2, -E^(-2*ArcTanh[c*x^2])]))/(2*c) + (b^3*(ArcTanh[c*x^2]^2*(-ArcTanh[c*x^2] + c*x^2*ArcTanh[c*x^2] - 3*Log[1 + E^(-2*ArcTanh[c*x^2])]) + 3*ArcTanh[c*x^2]*PolyLog[2, -E^(-2*ArcTanh[c*x^2])] + (3*PolyLog[3, -E^(-2*ArcTanh[c*x^2])])/2))/(2*c)","A",0
79,1,211,207,0.2096408,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3}{x} \, dx","Integrate[(a + b*ArcTanh[c*x^2])^3/x,x]","\frac{3}{8} b \left(2 \text{Li}_2\left(\frac{c x^2+1}{1-c x^2}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2-2 \text{Li}_2\left(\frac{c x^2+1}{c x^2-1}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2+b \left(-2 \text{Li}_3\left(\frac{c x^2+1}{1-c x^2}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)+2 \text{Li}_3\left(\frac{c x^2+1}{c x^2-1}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)+b \left(\text{Li}_4\left(\frac{c x^2+1}{1-c x^2}\right)-\text{Li}_4\left(\frac{c x^2+1}{c x^2-1}\right)\right)\right)\right)+\tanh ^{-1}\left(\frac{2}{c x^2-1}+1\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3","\frac{3}{4} b^2 \text{Li}_3\left(1-\frac{2}{1-c x^2}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)-\frac{3}{4} b^2 \text{Li}_3\left(\frac{2}{1-c x^2}-1\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)-\frac{3}{4} b \text{Li}_2\left(1-\frac{2}{1-c x^2}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2+\frac{3}{4} b \text{Li}_2\left(\frac{2}{1-c x^2}-1\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2+\tanh ^{-1}\left(1-\frac{2}{1-c x^2}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3-\frac{3}{8} b^3 \text{Li}_4\left(1-\frac{2}{1-c x^2}\right)+\frac{3}{8} b^3 \text{Li}_4\left(\frac{2}{1-c x^2}-1\right)",1,"(a + b*ArcTanh[c*x^2])^3*ArcTanh[1 + 2/(-1 + c*x^2)] + (3*b*(2*(a + b*ArcTanh[c*x^2])^2*PolyLog[2, (1 + c*x^2)/(1 - c*x^2)] - 2*(a + b*ArcTanh[c*x^2])^2*PolyLog[2, (1 + c*x^2)/(-1 + c*x^2)] + b*(-2*(a + b*ArcTanh[c*x^2])*PolyLog[3, (1 + c*x^2)/(1 - c*x^2)] + 2*(a + b*ArcTanh[c*x^2])*PolyLog[3, (1 + c*x^2)/(-1 + c*x^2)] + b*(PolyLog[4, (1 + c*x^2)/(1 - c*x^2)] - PolyLog[4, (1 + c*x^2)/(-1 + c*x^2)]))))/8","A",1
80,1,222,125,0.4332075,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3}{x^3} \, dx","Integrate[(a + b*ArcTanh[c*x^2])^3/x^3,x]","\frac{1}{4} \left(-\frac{2 a^3}{x^2}-3 a^2 b c \log \left(1-c^2 x^4\right)-\frac{6 a^2 b \tanh ^{-1}\left(c x^2\right)}{x^2}+12 a^2 b c \log (x)+6 a b^2 c \left(\tanh ^{-1}\left(c x^2\right) \left(\left(1-\frac{1}{c x^2}\right) \tanh ^{-1}\left(c x^2\right)+2 \log \left(1-e^{-2 \tanh ^{-1}\left(c x^2\right)}\right)\right)-\text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c x^2\right)}\right)\right)+2 b^3 c \left(3 \tanh ^{-1}\left(c x^2\right) \text{Li}_2\left(e^{2 \tanh ^{-1}\left(c x^2\right)}\right)-\frac{3}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}\left(c x^2\right)}\right)-\frac{\tanh ^{-1}\left(c x^2\right)^3}{c x^2}-\tanh ^{-1}\left(c x^2\right)^3+3 \tanh ^{-1}\left(c x^2\right)^2 \log \left(1-e^{2 \tanh ^{-1}\left(c x^2\right)}\right)+\frac{i \pi ^3}{8}\right)\right)","-\frac{3}{2} b^2 c \text{Li}_2\left(\frac{2}{c x^2+1}-1\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)+\frac{1}{2} c \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3-\frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3}{2 x^2}+\frac{3}{2} b c \log \left(2-\frac{2}{c x^2+1}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2-\frac{3}{4} b^3 c \text{Li}_3\left(\frac{2}{c x^2+1}-1\right)",1,"((-2*a^3)/x^2 - (6*a^2*b*ArcTanh[c*x^2])/x^2 + 12*a^2*b*c*Log[x] - 3*a^2*b*c*Log[1 - c^2*x^4] + 6*a*b^2*c*(ArcTanh[c*x^2]*((1 - 1/(c*x^2))*ArcTanh[c*x^2] + 2*Log[1 - E^(-2*ArcTanh[c*x^2])]) - PolyLog[2, E^(-2*ArcTanh[c*x^2])]) + 2*b^3*c*((I/8)*Pi^3 - ArcTanh[c*x^2]^3 - ArcTanh[c*x^2]^3/(c*x^2) + 3*ArcTanh[c*x^2]^2*Log[1 - E^(2*ArcTanh[c*x^2])] + 3*ArcTanh[c*x^2]*PolyLog[2, E^(2*ArcTanh[c*x^2])] - (3*PolyLog[3, E^(2*ArcTanh[c*x^2])])/2))/4","C",0
81,1,218,139,0.2945732,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3}{x^5} \, dx","Integrate[(a + b*ArcTanh[c*x^2])^3/x^5,x]","\frac{a \left(-2 a^2-3 a b c^2 x^4 \log \left(1-c x^2\right)+3 a b c^2 x^4 \log \left(c x^2+1\right)-6 a b c x^2+12 b^2 c^2 x^4 \log \left(\frac{c x^2}{\sqrt{1-c^2 x^4}}\right)\right)-6 b \tanh ^{-1}\left(c x^2\right) \left(a^2+2 a b c x^2-2 b^2 c^2 x^4 \log \left(1-e^{-2 \tanh ^{-1}\left(c x^2\right)}\right)\right)+6 b^2 \left(c x^2-1\right) \tanh ^{-1}\left(c x^2\right)^2 \left(a c x^2+a+b c x^2\right)-6 b^3 c^2 x^4 \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c x^2\right)}\right)+2 b^3 \left(c^2 x^4-1\right) \tanh ^{-1}\left(c x^2\right)^3}{8 x^4}","\frac{3}{2} b^2 c^2 \log \left(2-\frac{2}{c x^2+1}\right) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)+\frac{3}{4} b c^2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2+\frac{1}{4} c^2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3-\frac{3 b c \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{4 x^2}-\frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3}{4 x^4}-\frac{3}{4} b^3 c^2 \text{Li}_2\left(\frac{2}{c x^2+1}-1\right)",1,"(6*b^2*(-1 + c*x^2)*(a + a*c*x^2 + b*c*x^2)*ArcTanh[c*x^2]^2 + 2*b^3*(-1 + c^2*x^4)*ArcTanh[c*x^2]^3 - 6*b*ArcTanh[c*x^2]*(a^2 + 2*a*b*c*x^2 - 2*b^2*c^2*x^4*Log[1 - E^(-2*ArcTanh[c*x^2])]) + a*(-2*a^2 - 6*a*b*c*x^2 - 3*a*b*c^2*x^4*Log[1 - c*x^2] + 3*a*b*c^2*x^4*Log[1 + c*x^2] + 12*b^2*c^2*x^4*Log[(c*x^2)/Sqrt[1 - c^2*x^4]]) - 6*b^3*c^2*x^4*PolyLog[2, E^(-2*ArcTanh[c*x^2])])/(8*x^4)","A",0
82,1,241,317,0.1312816,"\int (d x)^{5/2} \left(a+b \tanh ^{-1}\left(c x^2\right)\right) \, dx","Integrate[(d*x)^(5/2)*(a + b*ArcTanh[c*x^2]),x]","\frac{(d x)^{5/2} \left(12 a c^{7/4} x^{7/2}+16 b c^{3/4} x^{3/2}+12 b c^{7/4} x^{7/2} \tanh ^{-1}\left(c x^2\right)+6 b \log \left(1-\sqrt[4]{c} \sqrt{x}\right)-6 b \log \left(\sqrt[4]{c} \sqrt{x}+1\right)-3 \sqrt{2} b \log \left(\sqrt{c} x-\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)+3 \sqrt{2} b \log \left(\sqrt{c} x+\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)+6 \sqrt{2} b \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{c} \sqrt{x}\right)-6 \sqrt{2} b \tan ^{-1}\left(\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)+12 b \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)\right)}{42 c^{7/4} x^{5/2}}","\frac{2 (d x)^{7/2} \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{7 d}-\frac{b d^{5/2} \log \left(\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{7 \sqrt{2} c^{7/4}}+\frac{b d^{5/2} \log \left(\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{7 \sqrt{2} c^{7/4}}+\frac{2 b d^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{7 c^{7/4}}+\frac{\sqrt{2} b d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{7 c^{7/4}}-\frac{\sqrt{2} b d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}+1\right)}{7 c^{7/4}}-\frac{2 b d^{5/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{7 c^{7/4}}+\frac{8 b d (d x)^{3/2}}{21 c}",1,"((d*x)^(5/2)*(16*b*c^(3/4)*x^(3/2) + 12*a*c^(7/4)*x^(7/2) + 6*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]] - 6*Sqrt[2]*b*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]] + 12*b*ArcTan[c^(1/4)*Sqrt[x]] + 12*b*c^(7/4)*x^(7/2)*ArcTanh[c*x^2] + 6*b*Log[1 - c^(1/4)*Sqrt[x]] - 6*b*Log[1 + c^(1/4)*Sqrt[x]] - 3*Sqrt[2]*b*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 3*Sqrt[2]*b*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/(42*c^(7/4)*x^(5/2))","A",1
83,1,240,317,0.0997959,"\int (d x)^{3/2} \left(a+b \tanh ^{-1}\left(c x^2\right)\right) \, dx","Integrate[(d*x)^(3/2)*(a + b*ArcTanh[c*x^2]),x]","\frac{(d x)^{3/2} \left(4 a c^{5/4} x^{5/2}+4 b c^{5/4} x^{5/2} \tanh ^{-1}\left(c x^2\right)+16 b \sqrt[4]{c} \sqrt{x}+2 b \log \left(1-\sqrt[4]{c} \sqrt{x}\right)-2 b \log \left(\sqrt[4]{c} \sqrt{x}+1\right)+\sqrt{2} b \log \left(\sqrt{c} x-\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)-\sqrt{2} b \log \left(\sqrt{c} x+\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)+2 \sqrt{2} b \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{c} \sqrt{x}\right)-2 \sqrt{2} b \tan ^{-1}\left(\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)-4 b \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)\right)}{10 c^{5/4} x^{3/2}}","\frac{2 (d x)^{5/2} \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{5 d}+\frac{b d^{3/2} \log \left(\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{5 \sqrt{2} c^{5/4}}-\frac{b d^{3/2} \log \left(\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{5 \sqrt{2} c^{5/4}}-\frac{2 b d^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{5 c^{5/4}}+\frac{\sqrt{2} b d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{5 c^{5/4}}-\frac{\sqrt{2} b d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}+1\right)}{5 c^{5/4}}-\frac{2 b d^{3/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{5 c^{5/4}}+\frac{8 b d \sqrt{d x}}{5 c}",1,"((d*x)^(3/2)*(16*b*c^(1/4)*Sqrt[x] + 4*a*c^(5/4)*x^(5/2) + 2*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]] - 2*Sqrt[2]*b*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]] - 4*b*ArcTan[c^(1/4)*Sqrt[x]] + 4*b*c^(5/4)*x^(5/2)*ArcTanh[c*x^2] + 2*b*Log[1 - c^(1/4)*Sqrt[x]] - 2*b*Log[1 + c^(1/4)*Sqrt[x]] + Sqrt[2]*b*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - Sqrt[2]*b*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/(10*c^(5/4)*x^(3/2))","A",1
84,1,227,301,0.0796503,"\int \sqrt{d x} \left(a+b \tanh ^{-1}\left(c x^2\right)\right) \, dx","Integrate[Sqrt[d*x]*(a + b*ArcTanh[c*x^2]),x]","\frac{\sqrt{d x} \left(4 a c^{3/4} x^{3/2}+4 b c^{3/4} x^{3/2} \tanh ^{-1}\left(c x^2\right)+2 b \log \left(1-\sqrt[4]{c} \sqrt{x}\right)-2 b \log \left(\sqrt[4]{c} \sqrt{x}+1\right)+\sqrt{2} b \log \left(\sqrt{c} x-\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)-\sqrt{2} b \log \left(\sqrt{c} x+\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)-2 \sqrt{2} b \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{c} \sqrt{x}\right)+2 \sqrt{2} b \tan ^{-1}\left(\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)+4 b \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)\right)}{6 c^{3/4} \sqrt{x}}","\frac{2 (d x)^{3/2} \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{3 d}+\frac{b \sqrt{d} \log \left(\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{3 \sqrt{2} c^{3/4}}-\frac{b \sqrt{d} \log \left(\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{3 \sqrt{2} c^{3/4}}+\frac{2 b \sqrt{d} \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{3 c^{3/4}}-\frac{\sqrt{2} b \sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{3 c^{3/4}}+\frac{\sqrt{2} b \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}+1\right)}{3 c^{3/4}}-\frac{2 b \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{3 c^{3/4}}",1,"(Sqrt[d*x]*(4*a*c^(3/4)*x^(3/2) - 2*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]] + 2*Sqrt[2]*b*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]] + 4*b*ArcTan[c^(1/4)*Sqrt[x]] + 4*b*c^(3/4)*x^(3/2)*ArcTanh[c*x^2] + 2*b*Log[1 - c^(1/4)*Sqrt[x]] - 2*b*Log[1 + c^(1/4)*Sqrt[x]] + Sqrt[2]*b*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - Sqrt[2]*b*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/(6*c^(3/4)*Sqrt[x])","A",1
85,1,227,285,0.0649211,"\int \frac{a+b \tanh ^{-1}\left(c x^2\right)}{\sqrt{d x}} \, dx","Integrate[(a + b*ArcTanh[c*x^2])/Sqrt[d*x],x]","\frac{\sqrt{x} \left(4 a \sqrt[4]{c} \sqrt{x}+4 b \sqrt[4]{c} \sqrt{x} \tanh ^{-1}\left(c x^2\right)+2 b \log \left(1-\sqrt[4]{c} \sqrt{x}\right)-2 b \log \left(\sqrt[4]{c} \sqrt{x}+1\right)-\sqrt{2} b \log \left(\sqrt{c} x-\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)+\sqrt{2} b \log \left(\sqrt{c} x+\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)-2 \sqrt{2} b \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{c} \sqrt{x}\right)+2 \sqrt{2} b \tan ^{-1}\left(\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)-4 b \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)\right)}{2 \sqrt[4]{c} \sqrt{d x}}","\frac{2 \sqrt{d x} \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{d}-\frac{b \log \left(\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{\sqrt{2} \sqrt[4]{c} \sqrt{d}}+\frac{b \log \left(\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{\sqrt{2} \sqrt[4]{c} \sqrt{d}}-\frac{2 b \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{\sqrt[4]{c} \sqrt{d}}-\frac{\sqrt{2} b \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{\sqrt[4]{c} \sqrt{d}}+\frac{\sqrt{2} b \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}+1\right)}{\sqrt[4]{c} \sqrt{d}}-\frac{2 b \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{\sqrt[4]{c} \sqrt{d}}",1,"(Sqrt[x]*(4*a*c^(1/4)*Sqrt[x] - 2*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]] + 2*Sqrt[2]*b*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]] - 4*b*ArcTan[c^(1/4)*Sqrt[x]] + 4*b*c^(1/4)*Sqrt[x]*ArcTanh[c*x^2] + 2*b*Log[1 - c^(1/4)*Sqrt[x]] - 2*b*Log[1 + c^(1/4)*Sqrt[x]] - Sqrt[2]*b*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + Sqrt[2]*b*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/(2*c^(1/4)*Sqrt[d*x])","A",1
86,1,268,285,0.1109116,"\int \frac{a+b \tanh ^{-1}\left(c x^2\right)}{(d x)^{3/2}} \, dx","Integrate[(a + b*ArcTanh[c*x^2])/(d*x)^(3/2),x]","-\frac{x \left(4 a+4 b \tanh ^{-1}\left(c x^2\right)+2 b \sqrt[4]{c} \sqrt{x} \log \left(1-\sqrt[4]{c} \sqrt{x}\right)-2 b \sqrt[4]{c} \sqrt{x} \log \left(\sqrt[4]{c} \sqrt{x}+1\right)-\sqrt{2} b \sqrt[4]{c} \sqrt{x} \log \left(\sqrt{c} x-\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)+\sqrt{2} b \sqrt[4]{c} \sqrt{x} \log \left(\sqrt{c} x+\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)+2 \sqrt{2} b \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{c} \sqrt{x}\right)-2 \sqrt{2} b \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)+4 b \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)\right)}{2 (d x)^{3/2}}","-\frac{2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{d \sqrt{d x}}+\frac{b \sqrt[4]{c} \log \left(\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{\sqrt{2} d^{3/2}}-\frac{b \sqrt[4]{c} \log \left(\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{\sqrt{2} d^{3/2}}-\frac{2 b \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{d^{3/2}}-\frac{\sqrt{2} b \sqrt[4]{c} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{d^{3/2}}+\frac{\sqrt{2} b \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}+1\right)}{d^{3/2}}+\frac{2 b \sqrt[4]{c} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{d^{3/2}}",1,"-1/2*(x*(4*a + 2*Sqrt[2]*b*c^(1/4)*Sqrt[x]*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]] - 2*Sqrt[2]*b*c^(1/4)*Sqrt[x]*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]] + 4*b*c^(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]] + 4*b*ArcTanh[c*x^2] + 2*b*c^(1/4)*Sqrt[x]*Log[1 - c^(1/4)*Sqrt[x]] - 2*b*c^(1/4)*Sqrt[x]*Log[1 + c^(1/4)*Sqrt[x]] - Sqrt[2]*b*c^(1/4)*Sqrt[x]*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + Sqrt[2]*b*c^(1/4)*Sqrt[x]*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/(d*x)^(3/2)","A",1
87,1,268,301,0.112499,"\int \frac{a+b \tanh ^{-1}\left(c x^2\right)}{(d x)^{5/2}} \, dx","Integrate[(a + b*ArcTanh[c*x^2])/(d*x)^(5/2),x]","-\frac{x \left(4 a+2 b c^{3/4} x^{3/2} \log \left(1-\sqrt[4]{c} \sqrt{x}\right)-2 b c^{3/4} x^{3/2} \log \left(\sqrt[4]{c} \sqrt{x}+1\right)+\sqrt{2} b c^{3/4} x^{3/2} \log \left(\sqrt{c} x-\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)-\sqrt{2} b c^{3/4} x^{3/2} \log \left(\sqrt{c} x+\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)+2 \sqrt{2} b c^{3/4} x^{3/2} \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{c} \sqrt{x}\right)-2 \sqrt{2} b c^{3/4} x^{3/2} \tan ^{-1}\left(\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)-4 b c^{3/4} x^{3/2} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)+4 b \tanh ^{-1}\left(c x^2\right)\right)}{6 (d x)^{5/2}}","-\frac{2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{3 d (d x)^{3/2}}-\frac{b c^{3/4} \log \left(\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{3 \sqrt{2} d^{5/2}}+\frac{b c^{3/4} \log \left(\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{3 \sqrt{2} d^{5/2}}+\frac{2 b c^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{3 d^{5/2}}-\frac{\sqrt{2} b c^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{3 d^{5/2}}+\frac{\sqrt{2} b c^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}+1\right)}{3 d^{5/2}}+\frac{2 b c^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{3 d^{5/2}}",1,"-1/6*(x*(4*a + 2*Sqrt[2]*b*c^(3/4)*x^(3/2)*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]] - 2*Sqrt[2]*b*c^(3/4)*x^(3/2)*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]] - 4*b*c^(3/4)*x^(3/2)*ArcTan[c^(1/4)*Sqrt[x]] + 4*b*ArcTanh[c*x^2] + 2*b*c^(3/4)*x^(3/2)*Log[1 - c^(1/4)*Sqrt[x]] - 2*b*c^(3/4)*x^(3/2)*Log[1 + c^(1/4)*Sqrt[x]] + Sqrt[2]*b*c^(3/4)*x^(3/2)*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - Sqrt[2]*b*c^(3/4)*x^(3/2)*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/(d*x)^(5/2)","A",1
88,1,275,317,0.0937609,"\int \frac{a+b \tanh ^{-1}\left(c x^2\right)}{(d x)^{7/2}} \, dx","Integrate[(a + b*ArcTanh[c*x^2])/(d*x)^(7/2),x]","\frac{x \left(-4 a-2 b c^{5/4} x^{5/2} \log \left(1-\sqrt[4]{c} \sqrt{x}\right)+2 b c^{5/4} x^{5/2} \log \left(\sqrt[4]{c} \sqrt{x}+1\right)-\sqrt{2} b c^{5/4} x^{5/2} \log \left(\sqrt{c} x-\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)+\sqrt{2} b c^{5/4} x^{5/2} \log \left(\sqrt{c} x+\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)+2 \sqrt{2} b c^{5/4} x^{5/2} \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{c} \sqrt{x}\right)-2 \sqrt{2} b c^{5/4} x^{5/2} \tan ^{-1}\left(\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)-4 b c^{5/4} x^{5/2} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)-16 b c x^2-4 b \tanh ^{-1}\left(c x^2\right)\right)}{10 (d x)^{7/2}}","-\frac{2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{5 d (d x)^{5/2}}-\frac{b c^{5/4} \log \left(\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{5 \sqrt{2} d^{7/2}}+\frac{b c^{5/4} \log \left(\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{5 \sqrt{2} d^{7/2}}-\frac{2 b c^{5/4} \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{5 d^{7/2}}+\frac{\sqrt{2} b c^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{5 d^{7/2}}-\frac{\sqrt{2} b c^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}+1\right)}{5 d^{7/2}}+\frac{2 b c^{5/4} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{5 d^{7/2}}-\frac{8 b c}{5 d^3 \sqrt{d x}}",1,"(x*(-4*a - 16*b*c*x^2 + 2*Sqrt[2]*b*c^(5/4)*x^(5/2)*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]] - 2*Sqrt[2]*b*c^(5/4)*x^(5/2)*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]] - 4*b*c^(5/4)*x^(5/2)*ArcTan[c^(1/4)*Sqrt[x]] - 4*b*ArcTanh[c*x^2] - 2*b*c^(5/4)*x^(5/2)*Log[1 - c^(1/4)*Sqrt[x]] + 2*b*c^(5/4)*x^(5/2)*Log[1 + c^(1/4)*Sqrt[x]] - Sqrt[2]*b*c^(5/4)*x^(5/2)*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + Sqrt[2]*b*c^(5/4)*x^(5/2)*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/(10*(d*x)^(7/2))","A",1
89,1,281,317,0.1089715,"\int \frac{a+b \tanh ^{-1}\left(c x^2\right)}{(d x)^{9/2}} \, dx","Integrate[(a + b*ArcTanh[c*x^2])/(d*x)^(9/2),x]","\frac{\sqrt{d x} \left(-12 a-6 b c^{7/4} x^{7/2} \log \left(1-\sqrt[4]{c} \sqrt{x}\right)+6 b c^{7/4} x^{7/2} \log \left(\sqrt[4]{c} \sqrt{x}+1\right)+3 \sqrt{2} b c^{7/4} x^{7/2} \log \left(\sqrt{c} x-\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)-3 \sqrt{2} b c^{7/4} x^{7/2} \log \left(\sqrt{c} x+\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)+6 \sqrt{2} b c^{7/4} x^{7/2} \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{c} \sqrt{x}\right)-6 \sqrt{2} b c^{7/4} x^{7/2} \tan ^{-1}\left(\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right)+12 b c^{7/4} x^{7/2} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)-16 b c x^2-12 b \tanh ^{-1}\left(c x^2\right)\right)}{42 d^5 x^4}","-\frac{2 \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{7 d (d x)^{7/2}}+\frac{b c^{7/4} \log \left(\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{7 \sqrt{2} d^{9/2}}-\frac{b c^{7/4} \log \left(\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt[4]{c} \sqrt{d x}+\sqrt{d}\right)}{7 \sqrt{2} d^{9/2}}+\frac{2 b c^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{7 d^{9/2}}+\frac{\sqrt{2} b c^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{7 d^{9/2}}-\frac{\sqrt{2} b c^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}+1\right)}{7 d^{9/2}}+\frac{2 b c^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d x}}{\sqrt{d}}\right)}{7 d^{9/2}}-\frac{8 b c}{21 d^3 (d x)^{3/2}}",1,"(Sqrt[d*x]*(-12*a - 16*b*c*x^2 + 6*Sqrt[2]*b*c^(7/4)*x^(7/2)*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]] - 6*Sqrt[2]*b*c^(7/4)*x^(7/2)*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]] + 12*b*c^(7/4)*x^(7/2)*ArcTan[c^(1/4)*Sqrt[x]] - 12*b*ArcTanh[c*x^2] - 6*b*c^(7/4)*x^(7/2)*Log[1 - c^(1/4)*Sqrt[x]] + 6*b*c^(7/4)*x^(7/2)*Log[1 + c^(1/4)*Sqrt[x]] + 3*Sqrt[2]*b*c^(7/4)*x^(7/2)*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 3*Sqrt[2]*b*c^(7/4)*x^(7/2)*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/(42*d^5*x^4)","A",1
90,0,0,6327,63.7151029,"\int \sqrt{d x} \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[Sqrt[d*x]*(a + b*ArcTanh[c*x^2])^2,x]","\int \sqrt{d x} \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2 \, dx","-\frac{2 i \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right)^2 b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{2 i \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)^2 b^2}{3 c^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right)^2 b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)^2 b^2}{3 c^{3/4} \sqrt{x}}+\frac{1}{6} x \sqrt{d x} \log ^2\left(c x^2+1\right) b^2+\frac{4 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{1-\sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{4 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{(1+i) \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{4 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{i \sqrt[4]{-c} \sqrt{x}+1}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{4 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{\sqrt[4]{-c} \sqrt{x}+1}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{(1-i) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{4 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{1-\sqrt[4]{c} \sqrt{x}}\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{4 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(i \sqrt[4]{-c}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(i \sqrt[4]{-c}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{(1+i) \left(1-\sqrt[4]{c} \sqrt{x}\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{4 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{i \sqrt[4]{c} \sqrt{x}+1}\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{4 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{\sqrt[4]{c} \sqrt{x}+1}\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{(1-i) \left(\sqrt[4]{c} \sqrt{x}+1\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(1-c x^2\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(1-c x^2\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{4}{9} x \sqrt{d x} \log \left(1-c x^2\right) b^2-\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(c x^2+1\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(c x^2+1\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(c x^2+1\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(c x^2+1\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{1}{3} x \sqrt{d x} \log \left(1-c x^2\right) \log \left(c x^2+1\right) b^2+\frac{2 \sqrt{d x} \text{Li}_2\left(1-\frac{2}{1-\sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{2 i \sqrt{d x} \text{Li}_2\left(1-\frac{2}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{i \sqrt{d x} \text{Li}_2\left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}+1\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{i \sqrt{d x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{i \sqrt{d x} \text{Li}_2\left(1-\frac{(1+i) \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{2 i \sqrt{d x} \text{Li}_2\left(1-\frac{2}{i \sqrt[4]{-c} \sqrt{x}+1}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \text{Li}_2\left(1-\frac{2}{\sqrt[4]{-c} \sqrt{x}+1}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{\sqrt{d x} \text{Li}_2\left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}+1\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{\sqrt{d x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{\sqrt{d x} \text{Li}_2\left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}+1\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{\sqrt{d x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{i \sqrt{d x} \text{Li}_2\left(1-\frac{(1-i) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \text{Li}_2\left(1-\frac{2}{1-\sqrt[4]{c} \sqrt{x}}\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{i \sqrt{d x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{\sqrt{d x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{2 i \sqrt{d x} \text{Li}_2\left(1-\frac{2}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{i \sqrt{d x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}+1\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{i \sqrt{d x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{i \sqrt{d x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(i \sqrt[4]{-c}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}+1\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{i \sqrt{d x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(i \sqrt[4]{-c}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{i \sqrt{d x} \text{Li}_2\left(1-\frac{(1+i) \left(1-\sqrt[4]{c} \sqrt{x}\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{2 i \sqrt{d x} \text{Li}_2\left(1-\frac{2}{i \sqrt[4]{c} \sqrt{x}+1}\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \text{Li}_2\left(1-\frac{2}{\sqrt[4]{c} \sqrt{x}+1}\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{\sqrt{d x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}+1\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{\sqrt{d x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{\sqrt{d x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}+1\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{\sqrt{d x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{\sqrt{d x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}+1\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{\sqrt{d x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 c^{3/4} \sqrt{x}}+\frac{i \sqrt{d x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{\sqrt{d x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 (-c)^{3/4} \sqrt{x}}-\frac{i \sqrt{d x} \text{Li}_2\left(1-\frac{(1-i) \left(\sqrt[4]{c} \sqrt{x}+1\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{3 c^{3/4} \sqrt{x}}-\frac{2 \sqrt{2} a \sqrt{d x} \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{c} \sqrt{x}\right) b}{3 c^{3/4} \sqrt{x}}+\frac{2 \sqrt{2} a \sqrt{d x} \tan ^{-1}\left(\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right) b}{3 c^{3/4} \sqrt{x}}+\frac{\sqrt{2} a \sqrt{d x} \log \left(\sqrt{c} x-\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right) b}{3 c^{3/4} \sqrt{x}}-\frac{\sqrt{2} a \sqrt{d x} \log \left(\sqrt{c} x+\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right) b}{3 c^{3/4} \sqrt{x}}+\frac{2 \sqrt{d x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \left(2 a-b \log \left(1-c x^2\right)\right) b}{3 c^{3/4} \sqrt{x}}-\frac{2 \sqrt{d x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \left(2 a-b \log \left(1-c x^2\right)\right) b}{3 c^{3/4} \sqrt{x}}+\frac{4}{9} x \sqrt{d x} \left(2 a-b \log \left(1-c x^2\right)\right) b+\frac{2}{3} a x \sqrt{d x} \log \left(c x^2+1\right) b-\frac{8}{9} a x \sqrt{d x} b+\frac{1}{6} x \sqrt{d x} \left(2 a-b \log \left(1-c x^2\right)\right)^2",1,"Integrate[Sqrt[d*x]*(a + b*ArcTanh[c*x^2])^2, x]","F",-1
91,0,0,6177,62.6661511,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{\sqrt{d x}} \, dx","Integrate[(a + b*ArcTanh[c*x^2])^2/Sqrt[d*x],x]","\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{\sqrt{d x}} \, dx","\frac{2 x a^2}{\sqrt{d x}}-\frac{2 \sqrt{2} b \sqrt{x} \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{c} \sqrt{x}\right) a}{\sqrt[4]{c} \sqrt{d x}}+\frac{2 \sqrt{2} b \sqrt{x} \tan ^{-1}\left(\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right) a}{\sqrt[4]{c} \sqrt{d x}}-\frac{4 b \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) a}{\sqrt[4]{c} \sqrt{d x}}-\frac{4 b \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) a}{\sqrt[4]{c} \sqrt{d x}}-\frac{\sqrt{2} b \sqrt{x} \log \left(\sqrt{c} x-\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right) a}{\sqrt[4]{c} \sqrt{d x}}+\frac{\sqrt{2} b \sqrt{x} \log \left(\sqrt{c} x+\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right) a}{\sqrt[4]{c} \sqrt{d x}}-\frac{2 b x \log \left(1-c x^2\right) a}{\sqrt{d x}}+\frac{2 b x \log \left(c x^2+1\right) a}{\sqrt{d x}}+\frac{2 i b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right)^2}{\sqrt[4]{-c} \sqrt{d x}}+\frac{2 i b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)^2}{\sqrt[4]{c} \sqrt{d x}}-\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right)^2}{\sqrt[4]{-c} \sqrt{d x}}-\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)^2}{\sqrt[4]{c} \sqrt{d x}}+\frac{b^2 x \log ^2\left(1-c x^2\right)}{2 \sqrt{d x}}+\frac{b^2 x \log ^2\left(c x^2+1\right)}{2 \sqrt{d x}}+\frac{4 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{1-\sqrt[4]{-c} \sqrt{x}}\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{4 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{1-i \sqrt[4]{-c} \sqrt{x}}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{(1+i) \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{4 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{i \sqrt[4]{-c} \sqrt{x}+1}\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{4 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{\sqrt[4]{-c} \sqrt{x}+1}\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{(1-i) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{4 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{1-\sqrt[4]{c} \sqrt{x}}\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{4 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{1-i \sqrt[4]{c} \sqrt{x}}\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(i \sqrt[4]{-c}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(i \sqrt[4]{-c}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{(1+i) \left(1-\sqrt[4]{c} \sqrt{x}\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{4 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{i \sqrt[4]{c} \sqrt{x}+1}\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{4 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{\sqrt[4]{c} \sqrt{x}+1}\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{(1-i) \left(\sqrt[4]{c} \sqrt{x}+1\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(1-c x^2\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(1-c x^2\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(1-c x^2\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(1-c x^2\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(c x^2+1\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{2 b^2 \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(c x^2+1\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(c x^2+1\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(c x^2+1\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{b^2 x \log \left(1-c x^2\right) \log \left(c x^2+1\right)}{\sqrt{d x}}+\frac{2 b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2}{1-\sqrt[4]{-c} \sqrt{x}}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{2 i b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2}{1-i \sqrt[4]{-c} \sqrt{x}}\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{i b^2 \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}+1\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{i b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{i b^2 \sqrt{x} \text{Li}_2\left(1-\frac{(1+i) \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{2 i b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2}{i \sqrt[4]{-c} \sqrt{x}+1}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2}{\sqrt[4]{-c} \sqrt{x}+1}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{b^2 \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}+1\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{b^2 \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}+1\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{i b^2 \sqrt{x} \text{Li}_2\left(1-\frac{(1-i) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2}{1-\sqrt[4]{c} \sqrt{x}}\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{i b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{2 i b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2}{1-i \sqrt[4]{c} \sqrt{x}}\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{i b^2 \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}+1\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{i b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{i b^2 \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(i \sqrt[4]{-c}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}+1\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{i b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(i \sqrt[4]{-c}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{i b^2 \sqrt{x} \text{Li}_2\left(1-\frac{(1+i) \left(1-\sqrt[4]{c} \sqrt{x}\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{2 i b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2}{i \sqrt[4]{c} \sqrt{x}+1}\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{2 b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2}{\sqrt[4]{c} \sqrt{x}+1}\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{b^2 \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}+1\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{b^2 \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}+1\right)}{\sqrt[4]{c} \sqrt{d x}}+\frac{b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{b^2 \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}+1\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{c} \sqrt{d x}}-\frac{i b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}-\frac{b^2 \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right)}{\sqrt[4]{-c} \sqrt{d x}}+\frac{i b^2 \sqrt{x} \text{Li}_2\left(1-\frac{(1-i) \left(\sqrt[4]{c} \sqrt{x}+1\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right)}{\sqrt[4]{c} \sqrt{d x}}",1,"Integrate[(a + b*ArcTanh[c*x^2])^2/Sqrt[d*x], x]","F",-1
92,0,0,6334,99.1052845,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{(d x)^{3/2}} \, dx","Integrate[(a + b*ArcTanh[c*x^2])^2/(d*x)^(3/2),x]","\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{(d x)^{3/2}} \, dx","\frac{2 i \sqrt[4]{-c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right)^2 b^2}{d \sqrt{d x}}+\frac{2 i \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)^2 b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{-c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right)^2 b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)^2 b^2}{d \sqrt{d x}}-\frac{\log ^2\left(c x^2+1\right) b^2}{2 d \sqrt{d x}}-\frac{4 \sqrt[4]{-c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{1-\sqrt[4]{-c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}-\frac{4 \sqrt[4]{-c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{-c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{-c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{-c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{(1+i) \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}+\frac{4 \sqrt[4]{-c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{i \sqrt[4]{-c} \sqrt{x}+1}\right) b^2}{d \sqrt{d x}}+\frac{4 \sqrt[4]{-c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{\sqrt[4]{-c} \sqrt{x}+1}\right) b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{-c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{-c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{-c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{-c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{-c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{(1-i) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}-\frac{4 \sqrt[4]{c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{1-\sqrt[4]{c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{-c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{-c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}-\frac{4 \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(i \sqrt[4]{-c}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(i \sqrt[4]{-c}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{(1+i) \left(1-\sqrt[4]{c} \sqrt{x}\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}+\frac{4 \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{i \sqrt[4]{c} \sqrt{x}+1}\right) b^2}{d \sqrt{d x}}+\frac{4 \sqrt[4]{c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{\sqrt[4]{c} \sqrt{x}+1}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{-c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{-c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{(1-i) \left(\sqrt[4]{c} \sqrt{x}+1\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{-c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(1-c x^2\right) b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{-c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(1-c x^2\right) b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{-c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(c x^2+1\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(c x^2+1\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{-c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(c x^2+1\right) b^2}{d \sqrt{d x}}+\frac{2 \sqrt[4]{c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(c x^2+1\right) b^2}{d \sqrt{d x}}+\frac{\log \left(1-c x^2\right) \log \left(c x^2+1\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(1-\frac{2}{1-\sqrt[4]{-c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}+\frac{2 i \sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(1-\frac{2}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}-\frac{i \sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}+1\right) b^2}{d \sqrt{d x}}-\frac{i \sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{d \sqrt{d x}}+\frac{i \sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(1-\frac{(1+i) \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}+\frac{2 i \sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(1-\frac{2}{i \sqrt[4]{-c} \sqrt{x}+1}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(1-\frac{2}{\sqrt[4]{-c} \sqrt{x}+1}\right) b^2}{d \sqrt{d x}}-\frac{\sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}+1\right) b^2}{d \sqrt{d x}}-\frac{\sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}+\frac{\sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}+1\right) b^2}{d \sqrt{d x}}+\frac{\sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}+\frac{i \sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(1-\frac{(1-i) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{c} \sqrt{x} \text{Li}_2\left(1-\frac{2}{1-\sqrt[4]{c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}-\frac{i \sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{d \sqrt{d x}}+\frac{\sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}+\frac{2 i \sqrt[4]{c} \sqrt{x} \text{Li}_2\left(1-\frac{2}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}-\frac{i \sqrt[4]{c} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}+1\right) b^2}{d \sqrt{d x}}-\frac{i \sqrt[4]{c} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{d \sqrt{d x}}-\frac{i \sqrt[4]{c} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(i \sqrt[4]{-c}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}+1\right) b^2}{d \sqrt{d x}}-\frac{i \sqrt[4]{c} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(i \sqrt[4]{-c}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{d \sqrt{d x}}+\frac{i \sqrt[4]{c} \sqrt{x} \text{Li}_2\left(1-\frac{(1+i) \left(1-\sqrt[4]{c} \sqrt{x}\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}+\frac{2 i \sqrt[4]{c} \sqrt{x} \text{Li}_2\left(1-\frac{2}{i \sqrt[4]{c} \sqrt{x}+1}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt[4]{c} \sqrt{x} \text{Li}_2\left(1-\frac{2}{\sqrt[4]{c} \sqrt{x}+1}\right) b^2}{d \sqrt{d x}}+\frac{\sqrt[4]{c} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}+1\right) b^2}{d \sqrt{d x}}+\frac{\sqrt[4]{c} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}-\frac{\sqrt[4]{c} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}+1\right) b^2}{d \sqrt{d x}}-\frac{\sqrt[4]{c} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}+\frac{\sqrt[4]{c} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}+1\right) b^2}{d \sqrt{d x}}+\frac{\sqrt[4]{c} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}-\frac{i \sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{d \sqrt{d x}}+\frac{\sqrt[4]{-c} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{d \sqrt{d x}}+\frac{i \sqrt[4]{c} \sqrt{x} \text{Li}_2\left(1-\frac{(1-i) \left(\sqrt[4]{c} \sqrt{x}+1\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{d \sqrt{d x}}-\frac{2 \sqrt{2} a \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{c} \sqrt{x}\right) b}{d \sqrt{d x}}+\frac{2 \sqrt{2} a \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right) b}{d \sqrt{d x}}+\frac{\sqrt{2} a \sqrt[4]{c} \sqrt{x} \log \left(\sqrt{c} x-\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right) b}{d \sqrt{d x}}-\frac{\sqrt{2} a \sqrt[4]{c} \sqrt{x} \log \left(\sqrt{c} x+\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right) b}{d \sqrt{d x}}-\frac{2 \sqrt[4]{c} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \left(2 a-b \log \left(1-c x^2\right)\right) b}{d \sqrt{d x}}+\frac{2 \sqrt[4]{c} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \left(2 a-b \log \left(1-c x^2\right)\right) b}{d \sqrt{d x}}-\frac{2 a \log \left(c x^2+1\right) b}{d \sqrt{d x}}-\frac{\left(2 a-b \log \left(1-c x^2\right)\right)^2}{2 d \sqrt{d x}}",1,"Integrate[(a + b*ArcTanh[c*x^2])^2/(d*x)^(3/2), x]","F",-1
93,0,0,6520,52.2245715,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{(d x)^{5/2}} \, dx","Integrate[(a + b*ArcTanh[c*x^2])^2/(d*x)^(5/2),x]","\int \frac{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2}{(d x)^{5/2}} \, dx","-\frac{2 i (-c)^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right)^2 b^2}{3 d^2 \sqrt{d x}}-\frac{2 i c^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)^2 b^2}{3 d^2 \sqrt{d x}}+\frac{2 (-c)^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right)^2 b^2}{3 d^2 \sqrt{d x}}+\frac{2 c^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right)^2 b^2}{3 d^2 \sqrt{d x}}-\frac{\log ^2\left(c x^2+1\right) b^2}{6 d^2 x \sqrt{d x}}-\frac{4 (-c)^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{1-\sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{4 (-c)^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 (-c)^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 (-c)^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{2 (-c)^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{(1+i) \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{4 (-c)^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{i \sqrt[4]{-c} \sqrt{x}+1}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{4 (-c)^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2}{\sqrt[4]{-c} \sqrt{x}+1}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{2 (-c)^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{2 (-c)^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 (-c)^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 (-c)^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{2 (-c)^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{(1-i) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{4 c^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{1-\sqrt[4]{c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 (-c)^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 (-c)^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{4 c^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 c^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 c^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 c^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(i \sqrt[4]{-c}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 c^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(i \sqrt[4]{-c}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{2 c^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{(1+i) \left(1-\sqrt[4]{c} \sqrt{x}\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{4 c^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{i \sqrt[4]{c} \sqrt{x}+1}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{4 c^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2}{\sqrt[4]{c} \sqrt{x}+1}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 c^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 c^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{2 c^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{2 c^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 c^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(-\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 c^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 (-c)^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 (-c)^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{2 c^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(\frac{(1-i) \left(\sqrt[4]{c} \sqrt{x}+1\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{2 (-c)^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(1-c x^2\right) b^2}{3 d^2 \sqrt{d x}}+\frac{2 (-c)^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(1-c x^2\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 (-c)^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(c x^2+1\right) b^2}{3 d^2 \sqrt{d x}}+\frac{2 c^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(c x^2+1\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 (-c)^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{-c} \sqrt{x}\right) \log \left(c x^2+1\right) b^2}{3 d^2 \sqrt{d x}}+\frac{2 c^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \log \left(c x^2+1\right) b^2}{3 d^2 \sqrt{d x}}+\frac{\log \left(1-c x^2\right) \log \left(c x^2+1\right) b^2}{3 d^2 x \sqrt{d x}}-\frac{2 (-c)^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2}{1-\sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 i (-c)^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{i (-c)^{3/4} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}+1\right) b^2}{3 d^2 \sqrt{d x}}+\frac{i (-c)^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{i (-c)^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{(1+i) \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 i (-c)^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2}{i \sqrt[4]{-c} \sqrt{x}+1}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 (-c)^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2}{\sqrt[4]{-c} \sqrt{x}+1}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{(-c)^{3/4} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}+1\right) b^2}{3 d^2 \sqrt{d x}}-\frac{(-c)^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{(-c)^{3/4} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{-c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}+1\right) b^2}{3 d^2 \sqrt{d x}}+\frac{(-c)^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{-c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{i (-c)^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{(1-i) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{1-i \sqrt[4]{-c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 c^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2}{1-\sqrt[4]{c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{i (-c)^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{(-c)^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(1-\sqrt[4]{c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 i c^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{i c^{3/4} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(i \sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}+1\right) b^2}{3 d^2 \sqrt{d x}}+\frac{i c^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(i \sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{i c^{3/4} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(i \sqrt[4]{-c}-\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}+1\right) b^2}{3 d^2 \sqrt{d x}}+\frac{i c^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(i \sqrt[4]{-c}+\sqrt[4]{c}\right) \left(1-i \sqrt[4]{c} \sqrt{x}\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{i c^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{(1+i) \left(1-\sqrt[4]{c} \sqrt{x}\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 i c^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2}{i \sqrt[4]{c} \sqrt{x}+1}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 c^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2}{\sqrt[4]{c} \sqrt{x}+1}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{c^{3/4} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{-c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{-c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}+1\right) b^2}{3 d^2 \sqrt{d x}}+\frac{c^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{-c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{-c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{c^{3/4} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt{-\sqrt{c}} \sqrt{x}\right)}{\left(\sqrt{-\sqrt{c}}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}+1\right) b^2}{3 d^2 \sqrt{d x}}-\frac{c^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt{-\sqrt{c}} \sqrt{x}+1\right)}{\left(\sqrt{-\sqrt{c}}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{c^{3/4} \sqrt{x} \text{Li}_2\left(\frac{2 \sqrt[4]{c} \left(1-\sqrt[4]{-c} \sqrt{x}\right)}{\left(\sqrt[4]{-c}-\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}+1\right) b^2}{3 d^2 \sqrt{d x}}+\frac{c^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{c} \left(\sqrt[4]{-c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{i (-c)^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+i \sqrt[4]{c}\right) \left(1-i \sqrt[4]{-c} \sqrt{x}\right)}\right) b^2}{3 d^2 \sqrt{d x}}+\frac{(-c)^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{2 \sqrt[4]{-c} \left(\sqrt[4]{c} \sqrt{x}+1\right)}{\left(\sqrt[4]{-c}+\sqrt[4]{c}\right) \left(\sqrt[4]{-c} \sqrt{x}+1\right)}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{i c^{3/4} \sqrt{x} \text{Li}_2\left(1-\frac{(1-i) \left(\sqrt[4]{c} \sqrt{x}+1\right)}{1-i \sqrt[4]{c} \sqrt{x}}\right) b^2}{3 d^2 \sqrt{d x}}-\frac{2 \sqrt{2} a c^{3/4} \sqrt{x} \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{c} \sqrt{x}\right) b}{3 d^2 \sqrt{d x}}+\frac{2 \sqrt{2} a c^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right) b}{3 d^2 \sqrt{d x}}-\frac{\sqrt{2} a c^{3/4} \sqrt{x} \log \left(\sqrt{c} x-\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right) b}{3 d^2 \sqrt{d x}}+\frac{\sqrt{2} a c^{3/4} \sqrt{x} \log \left(\sqrt{c} x+\sqrt{2} \sqrt[4]{c} \sqrt{x}+1\right) b}{3 d^2 \sqrt{d x}}+\frac{2 c^{3/4} \sqrt{x} \tan ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \left(2 a-b \log \left(1-c x^2\right)\right) b}{3 d^2 \sqrt{d x}}+\frac{2 c^{3/4} \sqrt{x} \tanh ^{-1}\left(\sqrt[4]{c} \sqrt{x}\right) \left(2 a-b \log \left(1-c x^2\right)\right) b}{3 d^2 \sqrt{d x}}-\frac{2 a \log \left(c x^2+1\right) b}{3 d^2 x \sqrt{d x}}-\frac{\left(2 a-b \log \left(1-c x^2\right)\right)^2}{6 d^2 x \sqrt{d x}}",1,"Integrate[(a + b*ArcTanh[c*x^2])^2/(d*x)^(5/2), x]","F",-1
94,0,0,21,1.8538942,"\int (d x)^m \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3 \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c*x^2])^3,x]","\int (d x)^m \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3 \, dx","\text{Int}\left((d x)^m \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^3,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTanh[c*x^2])^3, x]","A",-1
95,0,0,21,1.2500422,"\int (d x)^m \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c*x^2])^2,x]","\int (d x)^m \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2 \, dx","\text{Int}\left((d x)^m \left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTanh[c*x^2])^2, x]","A",-1
96,1,64,74,0.0585892,"\int (d x)^m \left(a+b \tanh ^{-1}\left(c x^2\right)\right) \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c*x^2]),x]","-\frac{x (d x)^m \left(2 b c x^2 \, _2F_1\left(1,\frac{m+3}{4};\frac{m+7}{4};c^2 x^4\right)-(m+3) \left(a+b \tanh ^{-1}\left(c x^2\right)\right)\right)}{(m+1) (m+3)}","\frac{(d x)^{m+1} \left(a+b \tanh ^{-1}\left(c x^2\right)\right)}{d (m+1)}-\frac{2 b c (d x)^{m+3} \, _2F_1\left(1,\frac{m+3}{4};\frac{m+7}{4};c^2 x^4\right)}{d^3 (m+1) (m+3)}",1,"-((x*(d*x)^m*(-((3 + m)*(a + b*ArcTanh[c*x^2])) + 2*b*c*x^2*Hypergeometric2F1[1, (3 + m)/4, (7 + m)/4, c^2*x^4]))/((1 + m)*(3 + m)))","A",1
97,0,0,21,0.3560675,"\int \frac{(d x)^m}{a+b \tanh ^{-1}\left(c x^2\right)} \, dx","Integrate[(d*x)^m/(a + b*ArcTanh[c*x^2]),x]","\int \frac{(d x)^m}{a+b \tanh ^{-1}\left(c x^2\right)} \, dx","\text{Int}\left(\frac{(d x)^m}{a+b \tanh ^{-1}\left(c x^2\right)},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcTanh[c*x^2]), x]","A",-1
98,0,0,21,0.3719112,"\int \frac{(d x)^m}{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2} \, dx","Integrate[(d*x)^m/(a + b*ArcTanh[c*x^2])^2,x]","\int \frac{(d x)^m}{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2} \, dx","\text{Int}\left(\frac{(d x)^m}{\left(a+b \tanh ^{-1}\left(c x^2\right)\right)^2},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcTanh[c*x^2])^2, x]","A",-1
99,1,78,54,0.0182889,"\int x^{11} \left(a+b \tanh ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x^11*(a + b*ArcTanh[c*x^3]),x]","\frac{a x^{12}}{12}+\frac{b \log \left(1-c x^3\right)}{24 c^4}-\frac{b \log \left(c x^3+1\right)}{24 c^4}+\frac{b x^3}{12 c^3}+\frac{b x^9}{36 c}+\frac{1}{12} b x^{12} \tanh ^{-1}\left(c x^3\right)","\frac{1}{12} x^{12} \left(a+b \tanh ^{-1}\left(c x^3\right)\right)-\frac{b \tanh ^{-1}\left(c x^3\right)}{12 c^4}+\frac{b x^3}{12 c^3}+\frac{b x^9}{36 c}",1,"(b*x^3)/(12*c^3) + (b*x^9)/(36*c) + (a*x^12)/12 + (b*x^12*ArcTanh[c*x^3])/12 + (b*Log[1 - c*x^3])/(24*c^4) - (b*Log[1 + c*x^3])/(24*c^4)","A",1
100,1,53,48,0.0152109,"\int x^8 \left(a+b \tanh ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x^8*(a + b*ArcTanh[c*x^3]),x]","\frac{a x^9}{9}+\frac{b \log \left(1-c^2 x^6\right)}{18 c^3}+\frac{b x^6}{18 c}+\frac{1}{9} b x^9 \tanh ^{-1}\left(c x^3\right)","\frac{1}{9} x^9 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)+\frac{b \log \left(1-c^2 x^6\right)}{18 c^3}+\frac{b x^6}{18 c}",1,"(b*x^6)/(18*c) + (a*x^9)/9 + (b*x^9*ArcTanh[c*x^3])/9 + (b*Log[1 - c^2*x^6])/(18*c^3)","A",1
101,1,67,43,0.0136441,"\int x^5 \left(a+b \tanh ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x^5*(a + b*ArcTanh[c*x^3]),x]","\frac{a x^6}{6}+\frac{b \log \left(1-c x^3\right)}{12 c^2}-\frac{b \log \left(c x^3+1\right)}{12 c^2}+\frac{b x^3}{6 c}+\frac{1}{6} b x^6 \tanh ^{-1}\left(c x^3\right)","\frac{1}{6} x^6 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)-\frac{b \tanh ^{-1}\left(c x^3\right)}{6 c^2}+\frac{b x^3}{6 c}",1,"(b*x^3)/(6*c) + (a*x^6)/6 + (b*x^6*ArcTanh[c*x^3])/6 + (b*Log[1 - c*x^3])/(12*c^2) - (b*Log[1 + c*x^3])/(12*c^2)","A",1
102,1,42,37,0.0082943,"\int x^2 \left(a+b \tanh ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x^2*(a + b*ArcTanh[c*x^3]),x]","\frac{a x^3}{3}+\frac{b \log \left(1-c^2 x^6\right)}{6 c}+\frac{1}{3} b x^3 \tanh ^{-1}\left(c x^3\right)","\frac{1}{3} x^3 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)+\frac{b \log \left(1-c^2 x^6\right)}{6 c}",1,"(a*x^3)/3 + (b*x^3*ArcTanh[c*x^3])/3 + (b*Log[1 - c^2*x^6])/(6*c)","A",1
103,1,28,30,0.0139518,"\int \frac{a+b \tanh ^{-1}\left(c x^3\right)}{x} \, dx","Integrate[(a + b*ArcTanh[c*x^3])/x,x]","a \log (x)+\frac{1}{6} b \left(\text{Li}_2\left(c x^3\right)-\text{Li}_2\left(-c x^3\right)\right)","a \log (x)-\frac{1}{6} b \text{Li}_2\left(-c x^3\right)+\frac{1}{6} b \text{Li}_2\left(c x^3\right)",1,"a*Log[x] + (b*(-PolyLog[2, -(c*x^3)] + PolyLog[2, c*x^3]))/6","A",1
104,1,45,40,0.0122016,"\int \frac{a+b \tanh ^{-1}\left(c x^3\right)}{x^4} \, dx","Integrate[(a + b*ArcTanh[c*x^3])/x^4,x]","-\frac{a}{3 x^3}-\frac{1}{6} b c \log \left(1-c^2 x^6\right)-\frac{b \tanh ^{-1}\left(c x^3\right)}{3 x^3}+b c \log (x)","-\frac{a+b \tanh ^{-1}\left(c x^3\right)}{3 x^3}-\frac{1}{6} b c \log \left(1-c^2 x^6\right)+b c \log (x)",1,"-1/3*a/x^3 - (b*ArcTanh[c*x^3])/(3*x^3) + b*c*Log[x] - (b*c*Log[1 - c^2*x^6])/6","A",1
105,1,65,41,0.0124843,"\int \frac{a+b \tanh ^{-1}\left(c x^3\right)}{x^7} \, dx","Integrate[(a + b*ArcTanh[c*x^3])/x^7,x]","-\frac{a}{6 x^6}-\frac{1}{12} b c^2 \log \left(1-c x^3\right)+\frac{1}{12} b c^2 \log \left(c x^3+1\right)-\frac{b c}{6 x^3}-\frac{b \tanh ^{-1}\left(c x^3\right)}{6 x^6}","-\frac{a+b \tanh ^{-1}\left(c x^3\right)}{6 x^6}+\frac{1}{6} b c^2 \tanh ^{-1}\left(c x^3\right)-\frac{b c}{6 x^3}",1,"-1/6*a/x^6 - (b*c)/(6*x^3) - (b*ArcTanh[c*x^3])/(6*x^6) - (b*c^2*Log[1 - c*x^3])/12 + (b*c^2*Log[1 + c*x^3])/12","A",1
106,1,61,56,0.0120373,"\int \frac{a+b \tanh ^{-1}\left(c x^3\right)}{x^{10}} \, dx","Integrate[(a + b*ArcTanh[c*x^3])/x^10,x]","-\frac{a}{9 x^9}+\frac{1}{3} b c^3 \log (x)-\frac{1}{18} b c^3 \log \left(1-c^2 x^6\right)-\frac{b c}{18 x^6}-\frac{b \tanh ^{-1}\left(c x^3\right)}{9 x^9}","-\frac{a+b \tanh ^{-1}\left(c x^3\right)}{9 x^9}+\frac{1}{3} b c^3 \log (x)-\frac{1}{18} b c^3 \log \left(1-c^2 x^6\right)-\frac{b c}{18 x^6}",1,"-1/9*a/x^9 - (b*c)/(18*x^6) - (b*ArcTanh[c*x^3])/(9*x^9) + (b*c^3*Log[x])/3 - (b*c^3*Log[1 - c^2*x^6])/18","A",1
107,1,196,174,0.0435595,"\int x^3 \left(a+b \tanh ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x^3*(a + b*ArcTanh[c*x^3]),x]","\frac{a x^4}{4}+\frac{b \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)}{16 c^{4/3}}-\frac{b \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)}{16 c^{4/3}}+\frac{b \log \left(1-\sqrt[3]{c} x\right)}{8 c^{4/3}}-\frac{b \log \left(\sqrt[3]{c} x+1\right)}{8 c^{4/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x-1}{\sqrt{3}}\right)}{8 c^{4/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x+1}{\sqrt{3}}\right)}{8 c^{4/3}}+\frac{1}{4} b x^4 \tanh ^{-1}\left(c x^3\right)+\frac{3 b x}{4 c}","\frac{1}{4} x^4 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)+\frac{b \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)}{16 c^{4/3}}-\frac{b \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)}{16 c^{4/3}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{c} x}{\sqrt{3}}\right)}{8 c^{4/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{8 c^{4/3}}-\frac{b \tanh ^{-1}\left(\sqrt[3]{c} x\right)}{4 c^{4/3}}+\frac{3 b x}{4 c}",1,"(3*b*x)/(4*c) + (a*x^4)/4 - (Sqrt[3]*b*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]])/(8*c^(4/3)) - (Sqrt[3]*b*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/(8*c^(4/3)) + (b*x^4*ArcTanh[c*x^3])/4 + (b*Log[1 - c^(1/3)*x])/(8*c^(4/3)) - (b*Log[1 + c^(1/3)*x])/(8*c^(4/3)) + (b*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(16*c^(4/3)) - (b*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(16*c^(4/3))","A",1
108,1,136,101,0.0434474,"\int \left(a+b \tanh ^{-1}\left(c x^3\right)\right) \, dx","Integrate[a + b*ArcTanh[c*x^3],x]","a x-\frac{b \left(\log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)+\log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)-2 \log \left(1-\sqrt[3]{c} x\right)-2 \log \left(\sqrt[3]{c} x+1\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x-1}{\sqrt{3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x+1}{\sqrt{3}}\right)\right)}{4 \sqrt[3]{c}}+b x \tanh ^{-1}\left(c x^3\right)","a x+\frac{b \log \left(1-c^{2/3} x^2\right)}{2 \sqrt[3]{c}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 c^{2/3} x^2+1}{\sqrt{3}}\right)}{2 \sqrt[3]{c}}-\frac{b \log \left(c^{4/3} x^4+c^{2/3} x^2+1\right)}{4 \sqrt[3]{c}}+b x \tanh ^{-1}\left(c x^3\right)",1,"a*x + b*x*ArcTanh[c*x^3] - (b*(-2*Sqrt[3]*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]] + 2*Sqrt[3]*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]] - 2*Log[1 - c^(1/3)*x] - 2*Log[1 + c^(1/3)*x] + Log[1 - c^(1/3)*x + c^(2/3)*x^2] + Log[1 + c^(1/3)*x + c^(2/3)*x^2]))/(4*c^(1/3))","A",1
109,1,187,165,0.0559688,"\int \frac{a+b \tanh ^{-1}\left(c x^3\right)}{x^3} \, dx","Integrate[(a + b*ArcTanh[c*x^3])/x^3,x]","-\frac{a}{2 x^2}-\frac{1}{8} b c^{2/3} \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)+\frac{1}{8} b c^{2/3} \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)-\frac{1}{4} b c^{2/3} \log \left(1-\sqrt[3]{c} x\right)+\frac{1}{4} b c^{2/3} \log \left(\sqrt[3]{c} x+1\right)+\frac{1}{4} \sqrt{3} b c^{2/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x-1}{\sqrt{3}}\right)+\frac{1}{4} \sqrt{3} b c^{2/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x+1}{\sqrt{3}}\right)-\frac{b \tanh ^{-1}\left(c x^3\right)}{2 x^2}","-\frac{a+b \tanh ^{-1}\left(c x^3\right)}{2 x^2}-\frac{1}{8} b c^{2/3} \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)+\frac{1}{8} b c^{2/3} \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)-\frac{1}{4} \sqrt{3} b c^{2/3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{c} x}{\sqrt{3}}\right)+\frac{1}{4} \sqrt{3} b c^{2/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)+\frac{1}{2} b c^{2/3} \tanh ^{-1}\left(\sqrt[3]{c} x\right)",1,"-1/2*a/x^2 + (Sqrt[3]*b*c^(2/3)*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]])/4 + (Sqrt[3]*b*c^(2/3)*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/4 - (b*ArcTanh[c*x^3])/(2*x^2) - (b*c^(2/3)*Log[1 - c^(1/3)*x])/4 + (b*c^(2/3)*Log[1 + c^(1/3)*x])/4 - (b*c^(2/3)*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/8 + (b*c^(2/3)*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/8","A",1
110,1,196,115,0.0509822,"\int \frac{a+b \tanh ^{-1}\left(c x^3\right)}{x^6} \, dx","Integrate[(a + b*ArcTanh[c*x^3])/x^6,x]","-\frac{a}{5 x^5}+\frac{1}{20} b c^{5/3} \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)+\frac{1}{20} b c^{5/3} \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)-\frac{1}{10} b c^{5/3} \log \left(1-\sqrt[3]{c} x\right)-\frac{1}{10} b c^{5/3} \log \left(\sqrt[3]{c} x+1\right)-\frac{1}{10} \sqrt{3} b c^{5/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x-1}{\sqrt{3}}\right)+\frac{1}{10} \sqrt{3} b c^{5/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x+1}{\sqrt{3}}\right)-\frac{3 b c}{10 x^2}-\frac{b \tanh ^{-1}\left(c x^3\right)}{5 x^5}","-\frac{a+b \tanh ^{-1}\left(c x^3\right)}{5 x^5}-\frac{1}{10} b c^{5/3} \log \left(1-c^{2/3} x^2\right)-\frac{1}{10} \sqrt{3} b c^{5/3} \tan ^{-1}\left(\frac{2 c^{2/3} x^2+1}{\sqrt{3}}\right)+\frac{1}{20} b c^{5/3} \log \left(c^{4/3} x^4+c^{2/3} x^2+1\right)-\frac{3 b c}{10 x^2}",1,"-1/5*a/x^5 - (3*b*c)/(10*x^2) - (Sqrt[3]*b*c^(5/3)*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]])/10 + (Sqrt[3]*b*c^(5/3)*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/10 - (b*ArcTanh[c*x^3])/(5*x^5) - (b*c^(5/3)*Log[1 - c^(1/3)*x])/10 - (b*c^(5/3)*Log[1 + c^(1/3)*x])/10 + (b*c^(5/3)*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/20 + (b*c^(5/3)*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/20","A",1
111,1,198,176,0.0307706,"\int x^7 \left(a+b \tanh ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x^7*(a + b*ArcTanh[c*x^3]),x]","\frac{a x^8}{8}+\frac{b \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)}{32 c^{8/3}}-\frac{b \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)}{32 c^{8/3}}+\frac{b \log \left(1-\sqrt[3]{c} x\right)}{16 c^{8/3}}-\frac{b \log \left(\sqrt[3]{c} x+1\right)}{16 c^{8/3}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x-1}{\sqrt{3}}\right)}{16 c^{8/3}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x+1}{\sqrt{3}}\right)}{16 c^{8/3}}+\frac{3 b x^5}{40 c}+\frac{1}{8} b x^8 \tanh ^{-1}\left(c x^3\right)","\frac{1}{8} x^8 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)+\frac{b \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)}{32 c^{8/3}}-\frac{b \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)}{32 c^{8/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{c} x}{\sqrt{3}}\right)}{16 c^{8/3}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{16 c^{8/3}}-\frac{b \tanh ^{-1}\left(\sqrt[3]{c} x\right)}{8 c^{8/3}}+\frac{3 b x^5}{40 c}",1,"(3*b*x^5)/(40*c) + (a*x^8)/8 + (Sqrt[3]*b*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]])/(16*c^(8/3)) + (Sqrt[3]*b*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/(16*c^(8/3)) + (b*x^8*ArcTanh[c*x^3])/8 + (b*Log[1 - c^(1/3)*x])/(16*c^(8/3)) - (b*Log[1 + c^(1/3)*x])/(16*c^(8/3)) + (b*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(32*c^(8/3)) - (b*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(32*c^(8/3))","A",1
112,1,198,117,0.0291135,"\int x^4 \left(a+b \tanh ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x^4*(a + b*ArcTanh[c*x^3]),x]","\frac{a x^5}{5}-\frac{b \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)}{20 c^{5/3}}-\frac{b \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)}{20 c^{5/3}}+\frac{b \log \left(1-\sqrt[3]{c} x\right)}{10 c^{5/3}}+\frac{b \log \left(\sqrt[3]{c} x+1\right)}{10 c^{5/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x-1}{\sqrt{3}}\right)}{10 c^{5/3}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x+1}{\sqrt{3}}\right)}{10 c^{5/3}}+\frac{3 b x^2}{10 c}+\frac{1}{5} b x^5 \tanh ^{-1}\left(c x^3\right)","\frac{1}{5} x^5 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)+\frac{b \log \left(1-c^{2/3} x^2\right)}{10 c^{5/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 c^{2/3} x^2+1}{\sqrt{3}}\right)}{10 c^{5/3}}-\frac{b \log \left(c^{4/3} x^4+c^{2/3} x^2+1\right)}{20 c^{5/3}}+\frac{3 b x^2}{10 c}",1,"(3*b*x^2)/(10*c) + (a*x^5)/5 - (Sqrt[3]*b*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]])/(10*c^(5/3)) + (Sqrt[3]*b*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/(10*c^(5/3)) + (b*x^5*ArcTanh[c*x^3])/5 + (b*Log[1 - c^(1/3)*x])/(10*c^(5/3)) + (b*Log[1 + c^(1/3)*x])/(10*c^(5/3)) - (b*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(20*c^(5/3)) - (b*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(20*c^(5/3))","A",1
113,1,187,165,0.026139,"\int x \left(a+b \tanh ^{-1}\left(c x^3\right)\right) \, dx","Integrate[x*(a + b*ArcTanh[c*x^3]),x]","\frac{a x^2}{2}+\frac{b \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)}{8 c^{2/3}}-\frac{b \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)}{8 c^{2/3}}+\frac{b \log \left(1-\sqrt[3]{c} x\right)}{4 c^{2/3}}-\frac{b \log \left(\sqrt[3]{c} x+1\right)}{4 c^{2/3}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x-1}{\sqrt{3}}\right)}{4 c^{2/3}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x+1}{\sqrt{3}}\right)}{4 c^{2/3}}+\frac{1}{2} b x^2 \tanh ^{-1}\left(c x^3\right)","\frac{1}{2} x^2 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)+\frac{b \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)}{8 c^{2/3}}-\frac{b \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)}{8 c^{2/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{c} x}{\sqrt{3}}\right)}{4 c^{2/3}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)}{4 c^{2/3}}-\frac{b \tanh ^{-1}\left(\sqrt[3]{c} x\right)}{2 c^{2/3}}",1,"(a*x^2)/2 + (Sqrt[3]*b*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]])/(4*c^(2/3)) + (Sqrt[3]*b*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/(4*c^(2/3)) + (b*x^2*ArcTanh[c*x^3])/2 + (b*Log[1 - c^(1/3)*x])/(4*c^(2/3)) - (b*Log[1 + c^(1/3)*x])/(4*c^(2/3)) + (b*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) - (b*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3))","A",1
114,1,183,104,0.0272761,"\int \frac{a+b \tanh ^{-1}\left(c x^3\right)}{x^2} \, dx","Integrate[(a + b*ArcTanh[c*x^3])/x^2,x]","-\frac{a}{x}+\frac{1}{4} b \sqrt[3]{c} \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)+\frac{1}{4} b \sqrt[3]{c} \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)-\frac{b \tanh ^{-1}\left(c x^3\right)}{x}-\frac{1}{2} b \sqrt[3]{c} \log \left(1-\sqrt[3]{c} x\right)-\frac{1}{2} b \sqrt[3]{c} \log \left(\sqrt[3]{c} x+1\right)+\frac{1}{2} \sqrt{3} b \sqrt[3]{c} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x-1}{\sqrt{3}}\right)-\frac{1}{2} \sqrt{3} b \sqrt[3]{c} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x+1}{\sqrt{3}}\right)","-\frac{a+b \tanh ^{-1}\left(c x^3\right)}{x}-\frac{1}{2} b \sqrt[3]{c} \log \left(1-c^{2/3} x^2\right)+\frac{1}{2} \sqrt{3} b \sqrt[3]{c} \tan ^{-1}\left(\frac{2 c^{2/3} x^2+1}{\sqrt{3}}\right)+\frac{1}{4} b \sqrt[3]{c} \log \left(c^{4/3} x^4+c^{2/3} x^2+1\right)",1,"-(a/x) + (Sqrt[3]*b*c^(1/3)*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]])/2 - (Sqrt[3]*b*c^(1/3)*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/2 - (b*ArcTanh[c*x^3])/x - (b*c^(1/3)*Log[1 - c^(1/3)*x])/2 - (b*c^(1/3)*Log[1 + c^(1/3)*x])/2 + (b*c^(1/3)*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/4 + (b*c^(1/3)*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/4","A",1
115,1,196,174,0.0528433,"\int \frac{a+b \tanh ^{-1}\left(c x^3\right)}{x^5} \, dx","Integrate[(a + b*ArcTanh[c*x^3])/x^5,x]","-\frac{a}{4 x^4}-\frac{1}{16} b c^{4/3} \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)+\frac{1}{16} b c^{4/3} \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)-\frac{1}{8} b c^{4/3} \log \left(1-\sqrt[3]{c} x\right)+\frac{1}{8} b c^{4/3} \log \left(\sqrt[3]{c} x+1\right)-\frac{1}{8} \sqrt{3} b c^{4/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x-1}{\sqrt{3}}\right)-\frac{1}{8} \sqrt{3} b c^{4/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x+1}{\sqrt{3}}\right)-\frac{b \tanh ^{-1}\left(c x^3\right)}{4 x^4}-\frac{3 b c}{4 x}","-\frac{a+b \tanh ^{-1}\left(c x^3\right)}{4 x^4}-\frac{1}{16} b c^{4/3} \log \left(c^{2/3} x^2-\sqrt[3]{c} x+1\right)+\frac{1}{16} b c^{4/3} \log \left(c^{2/3} x^2+\sqrt[3]{c} x+1\right)+\frac{1}{8} \sqrt{3} b c^{4/3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{c} x}{\sqrt{3}}\right)-\frac{1}{8} \sqrt{3} b c^{4/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} x}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right)+\frac{1}{4} b c^{4/3} \tanh ^{-1}\left(\sqrt[3]{c} x\right)-\frac{3 b c}{4 x}",1,"-1/4*a/x^4 - (3*b*c)/(4*x) - (Sqrt[3]*b*c^(4/3)*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]])/8 - (Sqrt[3]*b*c^(4/3)*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/8 - (b*ArcTanh[c*x^3])/(4*x^4) - (b*c^(4/3)*Log[1 - c^(1/3)*x])/8 + (b*c^(4/3)*Log[1 + c^(1/3)*x])/8 - (b*c^(4/3)*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/16 + (b*c^(4/3)*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/16","A",1
116,1,146,125,0.0798264,"\int x^{11} \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2 \, dx","Integrate[x^11*(a + b*ArcTanh[c*x^3])^2,x]","\frac{3 a^2 c^4 x^{12}+2 a b c^3 x^9+2 b c x^3 \tanh ^{-1}\left(c x^3\right) \left(3 a c^3 x^9+b \left(c^2 x^6+3\right)\right)+6 a b c x^3+b (3 a+4 b) \log \left(1-c x^3\right)-3 a b \log \left(c x^3+1\right)+3 b^2 \left(c^4 x^{12}-1\right) \tanh ^{-1}\left(c x^3\right)^2+b^2 c^2 x^6+4 b^2 \log \left(c x^3+1\right)}{36 c^4}","-\frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{12 c^4}+\frac{a b x^3}{6 c^3}+\frac{1}{12} x^{12} \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2+\frac{b x^9 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)}{18 c}+\frac{b^2 x^3 \tanh ^{-1}\left(c x^3\right)}{6 c^3}+\frac{b^2 x^6}{36 c^2}+\frac{b^2 \log \left(1-c^2 x^6\right)}{9 c^4}",1,"(6*a*b*c*x^3 + b^2*c^2*x^6 + 2*a*b*c^3*x^9 + 3*a^2*c^4*x^12 + 2*b*c*x^3*(3*a*c^3*x^9 + b*(3 + c^2*x^6))*ArcTanh[c*x^3] + 3*b^2*(-1 + c^4*x^12)*ArcTanh[c*x^3]^2 + b*(3*a + 4*b)*Log[1 - c*x^3] - 3*a*b*Log[1 + c*x^3] + 4*b^2*Log[1 + c*x^3])/(36*c^4)","A",1
117,1,132,146,0.2869746,"\int x^8 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2 \, dx","Integrate[x^8*(a + b*ArcTanh[c*x^3])^2,x]","\frac{a^2 c^3 x^9+a b c^2 x^6+a b \log \left(c^2 x^6-1\right)+b \tanh ^{-1}\left(c x^3\right) \left(2 a c^3 x^9+b c^2 x^6-2 b \log \left(e^{-2 \tanh ^{-1}\left(c x^3\right)}+1\right)-b\right)+b^2 \left(c^3 x^9-1\right) \tanh ^{-1}\left(c x^3\right)^2+b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c x^3\right)}\right)+b^2 c x^3}{9 c^3}","\frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{9 c^3}-\frac{2 b \log \left(\frac{2}{1-c x^3}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)}{9 c^3}+\frac{1}{9} x^9 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2+\frac{b x^6 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)}{9 c}-\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-c x^3}\right)}{9 c^3}-\frac{b^2 \tanh ^{-1}\left(c x^3\right)}{9 c^3}+\frac{b^2 x^3}{9 c^2}",1,"(b^2*c*x^3 + a*b*c^2*x^6 + a^2*c^3*x^9 + b^2*(-1 + c^3*x^9)*ArcTanh[c*x^3]^2 + b*ArcTanh[c*x^3]*(-b + b*c^2*x^6 + 2*a*c^3*x^9 - 2*b*Log[1 + E^(-2*ArcTanh[c*x^3])]) + a*b*Log[-1 + c^2*x^6] + b^2*PolyLog[2, -E^(-2*ArcTanh[c*x^3])])/(9*c^3)","A",0
118,1,106,91,0.0586339,"\int x^5 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2 \, dx","Integrate[x^5*(a + b*ArcTanh[c*x^3])^2,x]","\frac{a^2 c^2 x^6+2 a b c x^3+b (a+b) \log \left(1-c x^3\right)-a b \log \left(c x^3+1\right)+2 b c x^3 \tanh ^{-1}\left(c x^3\right) \left(a c x^3+b\right)+b^2 \left(c^2 x^6-1\right) \tanh ^{-1}\left(c x^3\right)^2+b^2 \log \left(c x^3+1\right)}{6 c^2}","-\frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{6 c^2}+\frac{a b x^3}{3 c}+\frac{1}{6} x^6 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2+\frac{b^2 \log \left(1-c^2 x^6\right)}{6 c^2}+\frac{b^2 x^3 \tanh ^{-1}\left(c x^3\right)}{3 c}",1,"(2*a*b*c*x^3 + a^2*c^2*x^6 + 2*b*c*x^3*(b + a*c*x^3)*ArcTanh[c*x^3] + b^2*(-1 + c^2*x^6)*ArcTanh[c*x^3]^2 + b*(a + b)*Log[1 - c*x^3] - a*b*Log[1 + c*x^3] + b^2*Log[1 + c*x^3])/(6*c^2)","A",1
119,1,99,96,0.159285,"\int x^2 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2 \, dx","Integrate[x^2*(a + b*ArcTanh[c*x^3])^2,x]","\frac{a \left(a c x^3+b \log \left(1-c^2 x^6\right)\right)+2 b \tanh ^{-1}\left(c x^3\right) \left(a c x^3-b \log \left(e^{-2 \tanh ^{-1}\left(c x^3\right)}+1\right)\right)+b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c x^3\right)}\right)+b^2 \left(c x^3-1\right) \tanh ^{-1}\left(c x^3\right)^2}{3 c}","\frac{1}{3} x^3 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2+\frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{3 c}-\frac{2 b \log \left(\frac{2}{1-c x^3}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)}{3 c}-\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-c x^3}\right)}{3 c}",1,"(b^2*(-1 + c*x^3)*ArcTanh[c*x^3]^2 + 2*b*ArcTanh[c*x^3]*(a*c*x^3 - b*Log[1 + E^(-2*ArcTanh[c*x^3])]) + a*(a*c*x^3 + b*Log[1 - c^2*x^6]) + b^2*PolyLog[2, -E^(-2*ArcTanh[c*x^3])])/(3*c)","A",0
120,1,183,140,0.0722941,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{x} \, dx","Integrate[(a + b*ArcTanh[c*x^3])^2/x,x]","\frac{1}{3} \left(2 \tanh ^{-1}\left(1-\frac{2}{1-c x^3}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2-4 b c \left(\frac{1}{2} \left(\frac{\text{Li}_2\left(\frac{-c x^3-1}{c x^3-1}\right) \left(-a-b \tanh ^{-1}\left(c x^3\right)\right)}{2 c}+\frac{b \text{Li}_3\left(\frac{-c x^3-1}{c x^3-1}\right)}{4 c}\right)+\frac{1}{2} \left(-\frac{\text{Li}_2\left(\frac{c x^3+1}{c x^3-1}\right) \left(-a-b \tanh ^{-1}\left(c x^3\right)\right)}{2 c}-\frac{b \text{Li}_3\left(\frac{c x^3+1}{c x^3-1}\right)}{4 c}\right)\right)\right)","-\frac{1}{3} b \text{Li}_2\left(1-\frac{2}{1-c x^3}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)+\frac{1}{3} b \text{Li}_2\left(\frac{2}{1-c x^3}-1\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)+\frac{2}{3} \tanh ^{-1}\left(1-\frac{2}{1-c x^3}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2+\frac{1}{6} b^2 \text{Li}_3\left(1-\frac{2}{1-c x^3}\right)-\frac{1}{6} b^2 \text{Li}_3\left(\frac{2}{1-c x^3}-1\right)",1,"(2*(a + b*ArcTanh[c*x^3])^2*ArcTanh[1 - 2/(1 - c*x^3)] - 4*b*c*((((-a - b*ArcTanh[c*x^3])*PolyLog[2, (-1 - c*x^3)/(-1 + c*x^3)])/(2*c) + (b*PolyLog[3, (-1 - c*x^3)/(-1 + c*x^3)])/(4*c))/2 + (-1/2*((-a - b*ArcTanh[c*x^3])*PolyLog[2, (1 + c*x^3)/(-1 + c*x^3)])/c - (b*PolyLog[3, (1 + c*x^3)/(-1 + c*x^3)])/(4*c))/2))/3","A",1
121,1,117,90,0.1633948,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{x^4} \, dx","Integrate[(a + b*ArcTanh[c*x^3])^2/x^4,x]","\frac{-a \left(a+b c x^3 \log \left(1-c^2 x^6\right)-2 b c x^3 \log \left(c x^3\right)\right)+2 b \tanh ^{-1}\left(c x^3\right) \left(b c x^3 \log \left(1-e^{-2 \tanh ^{-1}\left(c x^3\right)}\right)-a\right)-b^2 c x^3 \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c x^3\right)}\right)+b^2 \left(c x^3-1\right) \tanh ^{-1}\left(c x^3\right)^2}{3 x^3}","\frac{1}{3} c \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2-\frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{3 x^3}+\frac{2}{3} b c \log \left(2-\frac{2}{c x^3+1}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)-\frac{1}{3} b^2 c \text{Li}_2\left(\frac{2}{c x^3+1}-1\right)",1,"(b^2*(-1 + c*x^3)*ArcTanh[c*x^3]^2 + 2*b*ArcTanh[c*x^3]*(-a + b*c*x^3*Log[1 - E^(-2*ArcTanh[c*x^3])]) - a*(a - 2*b*c*x^3*Log[c*x^3] + b*c*x^3*Log[1 - c^2*x^6]) - b^2*c*x^3*PolyLog[2, E^(-2*ArcTanh[c*x^3])])/(3*x^3)","A",0
122,1,111,88,0.0875716,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{x^7} \, dx","Integrate[(a + b*ArcTanh[c*x^3])^2/x^7,x]","\frac{1}{6} \left(-\frac{a^2}{x^6}-b c^2 (a+b) \log \left(1-c x^3\right)+b c^2 (a-b) \log \left(c x^3+1\right)-\frac{2 a b c}{x^3}-\frac{2 b \tanh ^{-1}\left(c x^3\right) \left(a+b c x^3\right)}{x^6}+\frac{b^2 \left(c^2 x^6-1\right) \tanh ^{-1}\left(c x^3\right)^2}{x^6}+6 b^2 c^2 \log (x)\right)","\frac{1}{6} c^2 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2-\frac{b c \left(a+b \tanh ^{-1}\left(c x^3\right)\right)}{3 x^3}-\frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{6 x^6}-\frac{1}{6} b^2 c^2 \log \left(1-c^2 x^6\right)+b^2 c^2 \log (x)",1,"(-(a^2/x^6) - (2*a*b*c)/x^3 - (2*b*(a + b*c*x^3)*ArcTanh[c*x^3])/x^6 + (b^2*(-1 + c^2*x^6)*ArcTanh[c*x^3]^2)/x^6 + 6*b^2*c^2*Log[x] - b*(a + b)*c^2*Log[1 - c*x^3] + (a - b)*b*c^2*Log[1 + c*x^3])/6","A",1
123,1,159,144,0.3816004,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{x^{10}} \, dx","Integrate[(a + b*ArcTanh[c*x^3])^2/x^10,x]","-\frac{a^2-2 a b c^3 x^9 \log \left(c x^3\right)+b \tanh ^{-1}\left(c x^3\right) \left(2 a-b c^3 x^9-2 b c^3 x^9 \log \left(1-e^{-2 \tanh ^{-1}\left(c x^3\right)}\right)+b c x^3\right)+a b c^3 x^9 \log \left(1-c^2 x^6\right)+a b c x^3+b^2 c^3 x^9 \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c x^3\right)}\right)+b^2 \left(1-c^3 x^9\right) \tanh ^{-1}\left(c x^3\right)^2+b^2 c^2 x^6}{9 x^9}","\frac{1}{9} c^3 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2+\frac{2}{9} b c^3 \log \left(2-\frac{2}{c x^3+1}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)-\frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{9 x^9}-\frac{b c \left(a+b \tanh ^{-1}\left(c x^3\right)\right)}{9 x^6}-\frac{1}{9} b^2 c^3 \text{Li}_2\left(\frac{2}{c x^3+1}-1\right)+\frac{1}{9} b^2 c^3 \tanh ^{-1}\left(c x^3\right)-\frac{b^2 c^2}{9 x^3}",1,"-1/9*(a^2 + a*b*c*x^3 + b^2*c^2*x^6 + b^2*(1 - c^3*x^9)*ArcTanh[c*x^3]^2 + b*ArcTanh[c*x^3]*(2*a + b*c*x^3 - b*c^3*x^9 - 2*b*c^3*x^9*Log[1 - E^(-2*ArcTanh[c*x^3])]) - 2*a*b*c^3*x^9*Log[c*x^3] + a*b*c^3*x^9*Log[1 - c^2*x^6] + b^2*c^3*x^9*PolyLog[2, E^(-2*ArcTanh[c*x^3])])/x^9","A",0
124,1,334,231,0.4960157,"\int x^8 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3 \, dx","Integrate[x^8*(a + b*ArcTanh[c*x^3])^3,x]","\frac{2 a^3 c^3 x^9+6 a^2 b c^3 x^9 \tanh ^{-1}\left(c x^3\right)+3 a^2 b c^2 x^6+3 a^2 b \log \left(1-c^2 x^6\right)+6 a b^2 c^3 x^9 \tanh ^{-1}\left(c x^3\right)^2+6 a b^2 c^2 x^6 \tanh ^{-1}\left(c x^3\right)+6 b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c x^3\right)}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)+6 a b^2 c x^3-6 a b^2 \tanh ^{-1}\left(c x^3\right)^2-6 a b^2 \tanh ^{-1}\left(c x^3\right)-12 a b^2 \tanh ^{-1}\left(c x^3\right) \log \left(e^{-2 \tanh ^{-1}\left(c x^3\right)}+1\right)+2 b^3 c^3 x^9 \tanh ^{-1}\left(c x^3\right)^3+3 b^3 \log \left(1-c^2 x^6\right)+3 b^3 c^2 x^6 \tanh ^{-1}\left(c x^3\right)^2+3 b^3 \text{Li}_3\left(-e^{-2 \tanh ^{-1}\left(c x^3\right)}\right)+6 b^3 c x^3 \tanh ^{-1}\left(c x^3\right)-2 b^3 \tanh ^{-1}\left(c x^3\right)^3-3 b^3 \tanh ^{-1}\left(c x^3\right)^2-6 b^3 \tanh ^{-1}\left(c x^3\right)^2 \log \left(e^{-2 \tanh ^{-1}\left(c x^3\right)}+1\right)}{18 c^3}","-\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-c x^3}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)}{3 c^3}+\frac{a b^2 x^3}{3 c^2}+\frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3}{9 c^3}-\frac{b \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{6 c^3}-\frac{b \log \left(\frac{2}{1-c x^3}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{3 c^3}+\frac{1}{9} x^9 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3+\frac{b x^6 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{6 c}+\frac{b^3 \text{Li}_3\left(1-\frac{2}{1-c x^3}\right)}{6 c^3}+\frac{b^3 x^3 \tanh ^{-1}\left(c x^3\right)}{3 c^2}+\frac{b^3 \log \left(1-c^2 x^6\right)}{6 c^3}",1,"(6*a*b^2*c*x^3 + 3*a^2*b*c^2*x^6 + 2*a^3*c^3*x^9 - 6*a*b^2*ArcTanh[c*x^3] + 6*b^3*c*x^3*ArcTanh[c*x^3] + 6*a*b^2*c^2*x^6*ArcTanh[c*x^3] + 6*a^2*b*c^3*x^9*ArcTanh[c*x^3] - 6*a*b^2*ArcTanh[c*x^3]^2 - 3*b^3*ArcTanh[c*x^3]^2 + 3*b^3*c^2*x^6*ArcTanh[c*x^3]^2 + 6*a*b^2*c^3*x^9*ArcTanh[c*x^3]^2 - 2*b^3*ArcTanh[c*x^3]^3 + 2*b^3*c^3*x^9*ArcTanh[c*x^3]^3 - 12*a*b^2*ArcTanh[c*x^3]*Log[1 + E^(-2*ArcTanh[c*x^3])] - 6*b^3*ArcTanh[c*x^3]^2*Log[1 + E^(-2*ArcTanh[c*x^3])] + 3*a^2*b*Log[1 - c^2*x^6] + 3*b^3*Log[1 - c^2*x^6] + 6*b^2*(a + b*ArcTanh[c*x^3])*PolyLog[2, -E^(-2*ArcTanh[c*x^3])] + 3*b^3*PolyLog[3, -E^(-2*ArcTanh[c*x^3])])/(18*c^3)","A",0
125,1,185,139,0.3156381,"\int x^5 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3 \, dx","Integrate[x^5*(a + b*ArcTanh[c*x^3])^3,x]","\frac{a \left(2 a^2 c^2 x^6+6 a b c x^3+3 a b \log \left(1-c x^3\right)-3 a b \log \left(c x^3+1\right)+6 b^2 \log \left(1-c^2 x^6\right)\right)+6 b^2 \left(c x^3-1\right) \tanh ^{-1}\left(c x^3\right)^2 \left(a c x^3+a+b\right)+6 b \tanh ^{-1}\left(c x^3\right) \left(a c x^3 \left(a c x^3+2 b\right)-2 b^2 \log \left(e^{-2 \tanh ^{-1}\left(c x^3\right)}+1\right)\right)+2 b^3 \left(c^2 x^6-1\right) \tanh ^{-1}\left(c x^3\right)^3+6 b^3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c x^3\right)}\right)}{12 c^2}","-\frac{b^2 \log \left(\frac{2}{1-c x^3}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)}{c^2}-\frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3}{6 c^2}+\frac{b \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{2 c^2}+\frac{b x^3 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{2 c}+\frac{1}{6} x^6 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3-\frac{b^3 \text{Li}_2\left(1-\frac{2}{1-c x^3}\right)}{2 c^2}",1,"(6*b^2*(-1 + c*x^3)*(a + b + a*c*x^3)*ArcTanh[c*x^3]^2 + 2*b^3*(-1 + c^2*x^6)*ArcTanh[c*x^3]^3 + 6*b*ArcTanh[c*x^3]*(a*c*x^3*(2*b + a*c*x^3) - 2*b^2*Log[1 + E^(-2*ArcTanh[c*x^3])]) + a*(6*a*b*c*x^3 + 2*a^2*c^2*x^6 + 3*a*b*Log[1 - c*x^3] - 3*a*b*Log[1 + c*x^3] + 6*b^2*Log[1 - c^2*x^6]) + 6*b^3*PolyLog[2, -E^(-2*ArcTanh[c*x^3])])/(12*c^2)","A",0
126,1,208,130,0.2571243,"\int x^2 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3 \, dx","Integrate[x^2*(a + b*ArcTanh[c*x^3])^3,x]","\frac{a^3 x^3}{3}+\frac{a^2 b \left(c x^3 \tanh ^{-1}\left(c x^3\right)-\log \left(\frac{1}{\sqrt{1-c^2 x^6}}\right)\right)}{c}+\frac{a b^2 \left(\text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c x^3\right)}\right)+\tanh ^{-1}\left(c x^3\right) \left(c x^3 \tanh ^{-1}\left(c x^3\right)-\tanh ^{-1}\left(c x^3\right)-2 \log \left(e^{-2 \tanh ^{-1}\left(c x^3\right)}+1\right)\right)\right)}{c}+\frac{b^3 \left(3 \tanh ^{-1}\left(c x^3\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c x^3\right)}\right)+\frac{3}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}\left(c x^3\right)}\right)+\tanh ^{-1}\left(c x^3\right)^2 \left(c x^3 \tanh ^{-1}\left(c x^3\right)-\tanh ^{-1}\left(c x^3\right)-3 \log \left(e^{-2 \tanh ^{-1}\left(c x^3\right)}+1\right)\right)\right)}{3 c}","-\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-c x^3}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)}{c}+\frac{1}{3} x^3 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3+\frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3}{3 c}-\frac{b \log \left(\frac{2}{1-c x^3}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{c}+\frac{b^3 \text{Li}_3\left(1-\frac{2}{1-c x^3}\right)}{2 c}",1,"(a^3*x^3)/3 + (a^2*b*(c*x^3*ArcTanh[c*x^3] - Log[1/Sqrt[1 - c^2*x^6]]))/c + (a*b^2*(ArcTanh[c*x^3]*(-ArcTanh[c*x^3] + c*x^3*ArcTanh[c*x^3] - 2*Log[1 + E^(-2*ArcTanh[c*x^3])]) + PolyLog[2, -E^(-2*ArcTanh[c*x^3])]))/c + (b^3*(ArcTanh[c*x^3]^2*(-ArcTanh[c*x^3] + c*x^3*ArcTanh[c*x^3] - 3*Log[1 + E^(-2*ArcTanh[c*x^3])]) + 3*ArcTanh[c*x^3]*PolyLog[2, -E^(-2*ArcTanh[c*x^3])] + (3*PolyLog[3, -E^(-2*ArcTanh[c*x^3])])/2))/(3*c)","A",0
127,1,214,210,0.1900805,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3}{x} \, dx","Integrate[(a + b*ArcTanh[c*x^3])^3/x,x]","\frac{1}{4} b \left(2 \text{Li}_2\left(\frac{c x^3+1}{1-c x^3}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2-2 \text{Li}_2\left(\frac{c x^3+1}{c x^3-1}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2+b \left(-2 \text{Li}_3\left(\frac{c x^3+1}{1-c x^3}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)+2 \text{Li}_3\left(\frac{c x^3+1}{c x^3-1}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)+b \left(\text{Li}_4\left(\frac{c x^3+1}{1-c x^3}\right)-\text{Li}_4\left(\frac{c x^3+1}{c x^3-1}\right)\right)\right)\right)+\frac{2}{3} \tanh ^{-1}\left(\frac{2}{c x^3-1}+1\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3","\frac{1}{2} b^2 \text{Li}_3\left(1-\frac{2}{1-c x^3}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)-\frac{1}{2} b^2 \text{Li}_3\left(\frac{2}{1-c x^3}-1\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)-\frac{1}{2} b \text{Li}_2\left(1-\frac{2}{1-c x^3}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2+\frac{1}{2} b \text{Li}_2\left(\frac{2}{1-c x^3}-1\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2+\frac{2}{3} \tanh ^{-1}\left(1-\frac{2}{1-c x^3}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3-\frac{1}{4} b^3 \text{Li}_4\left(1-\frac{2}{1-c x^3}\right)+\frac{1}{4} b^3 \text{Li}_4\left(\frac{2}{1-c x^3}-1\right)",1,"(2*(a + b*ArcTanh[c*x^3])^3*ArcTanh[1 + 2/(-1 + c*x^3)])/3 + (b*(2*(a + b*ArcTanh[c*x^3])^2*PolyLog[2, (1 + c*x^3)/(1 - c*x^3)] - 2*(a + b*ArcTanh[c*x^3])^2*PolyLog[2, (1 + c*x^3)/(-1 + c*x^3)] + b*(-2*(a + b*ArcTanh[c*x^3])*PolyLog[3, (1 + c*x^3)/(1 - c*x^3)] + 2*(a + b*ArcTanh[c*x^3])*PolyLog[3, (1 + c*x^3)/(-1 + c*x^3)] + b*(PolyLog[4, (1 + c*x^3)/(1 - c*x^3)] - PolyLog[4, (1 + c*x^3)/(-1 + c*x^3)]))))/4","A",1
128,1,223,120,0.409204,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3}{x^4} \, dx","Integrate[(a + b*ArcTanh[c*x^3])^3/x^4,x]","-\frac{a^3}{3 x^3}-\frac{1}{2} a^2 b c \log \left(1-c^2 x^6\right)-\frac{a^2 b \tanh ^{-1}\left(c x^3\right)}{x^3}+3 a^2 b c \log (x)+a b^2 c \left(\tanh ^{-1}\left(c x^3\right) \left(\left(1-\frac{1}{c x^3}\right) \tanh ^{-1}\left(c x^3\right)+2 \log \left(1-e^{-2 \tanh ^{-1}\left(c x^3\right)}\right)\right)-\text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c x^3\right)}\right)\right)+\frac{1}{3} b^3 c \left(3 \tanh ^{-1}\left(c x^3\right) \text{Li}_2\left(e^{2 \tanh ^{-1}\left(c x^3\right)}\right)-\frac{3}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}\left(c x^3\right)}\right)-\frac{\tanh ^{-1}\left(c x^3\right)^3}{c x^3}-\tanh ^{-1}\left(c x^3\right)^3+3 \tanh ^{-1}\left(c x^3\right)^2 \log \left(1-e^{2 \tanh ^{-1}\left(c x^3\right)}\right)+\frac{i \pi ^3}{8}\right)","-b^2 c \text{Li}_2\left(\frac{2}{c x^3+1}-1\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)+\frac{1}{3} c \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3-\frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3}{3 x^3}+b c \log \left(2-\frac{2}{c x^3+1}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2-\frac{1}{2} b^3 c \text{Li}_3\left(\frac{2}{c x^3+1}-1\right)",1,"-1/3*a^3/x^3 - (a^2*b*ArcTanh[c*x^3])/x^3 + 3*a^2*b*c*Log[x] - (a^2*b*c*Log[1 - c^2*x^6])/2 + a*b^2*c*(ArcTanh[c*x^3]*((1 - 1/(c*x^3))*ArcTanh[c*x^3] + 2*Log[1 - E^(-2*ArcTanh[c*x^3])]) - PolyLog[2, E^(-2*ArcTanh[c*x^3])]) + (b^3*c*((I/8)*Pi^3 - ArcTanh[c*x^3]^3 - ArcTanh[c*x^3]^3/(c*x^3) + 3*ArcTanh[c*x^3]^2*Log[1 - E^(2*ArcTanh[c*x^3])] + 3*ArcTanh[c*x^3]*PolyLog[2, E^(2*ArcTanh[c*x^3])] - (3*PolyLog[3, E^(2*ArcTanh[c*x^3])])/2))/3","C",0
129,1,218,136,0.2976157,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3}{x^7} \, dx","Integrate[(a + b*ArcTanh[c*x^3])^3/x^7,x]","\frac{a \left(-2 a^2-3 a b c^2 x^6 \log \left(1-c x^3\right)+3 a b c^2 x^6 \log \left(c x^3+1\right)-6 a b c x^3+12 b^2 c^2 x^6 \log \left(\frac{c x^3}{\sqrt{1-c^2 x^6}}\right)\right)-6 b \tanh ^{-1}\left(c x^3\right) \left(a^2+2 a b c x^3-2 b^2 c^2 x^6 \log \left(1-e^{-2 \tanh ^{-1}\left(c x^3\right)}\right)\right)+6 b^2 \left(c x^3-1\right) \tanh ^{-1}\left(c x^3\right)^2 \left(a c x^3+a+b c x^3\right)-6 b^3 c^2 x^6 \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c x^3\right)}\right)+2 b^3 \left(c^2 x^6-1\right) \tanh ^{-1}\left(c x^3\right)^3}{12 x^6}","b^2 c^2 \log \left(2-\frac{2}{c x^3+1}\right) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)+\frac{1}{2} b c^2 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2+\frac{1}{6} c^2 \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3-\frac{b c \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2}{2 x^3}-\frac{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3}{6 x^6}-\frac{1}{2} b^3 c^2 \text{Li}_2\left(\frac{2}{c x^3+1}-1\right)",1,"(6*b^2*(-1 + c*x^3)*(a + a*c*x^3 + b*c*x^3)*ArcTanh[c*x^3]^2 + 2*b^3*(-1 + c^2*x^6)*ArcTanh[c*x^3]^3 - 6*b*ArcTanh[c*x^3]*(a^2 + 2*a*b*c*x^3 - 2*b^2*c^2*x^6*Log[1 - E^(-2*ArcTanh[c*x^3])]) + a*(-2*a^2 - 6*a*b*c*x^3 - 3*a*b*c^2*x^6*Log[1 - c*x^3] + 3*a*b*c^2*x^6*Log[1 + c*x^3] + 12*b^2*c^2*x^6*Log[(c*x^3)/Sqrt[1 - c^2*x^6]]) - 6*b^3*c^2*x^6*PolyLog[2, E^(-2*ArcTanh[c*x^3])])/(12*x^6)","A",0
130,0,0,21,1.8759536,"\int (d x)^m \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3 \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c*x^3])^3,x]","\int (d x)^m \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3 \, dx","\text{Int}\left((d x)^m \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^3,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTanh[c*x^3])^3, x]","A",-1
131,0,0,21,1.2453278,"\int (d x)^m \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2 \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c*x^3])^2,x]","\int (d x)^m \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2 \, dx","\text{Int}\left((d x)^m \left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTanh[c*x^3])^2, x]","A",-1
132,1,64,74,0.0609493,"\int (d x)^m \left(a+b \tanh ^{-1}\left(c x^3\right)\right) \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c*x^3]),x]","-\frac{x (d x)^m \left(3 b c x^3 \, _2F_1\left(1,\frac{m+4}{6};\frac{m+10}{6};c^2 x^6\right)-(m+4) \left(a+b \tanh ^{-1}\left(c x^3\right)\right)\right)}{(m+1) (m+4)}","\frac{(d x)^{m+1} \left(a+b \tanh ^{-1}\left(c x^3\right)\right)}{d (m+1)}-\frac{3 b c (d x)^{m+4} \, _2F_1\left(1,\frac{m+4}{6};\frac{m+10}{6};c^2 x^6\right)}{d^4 (m+1) (m+4)}",1,"-((x*(d*x)^m*(-((4 + m)*(a + b*ArcTanh[c*x^3])) + 3*b*c*x^3*Hypergeometric2F1[1, (4 + m)/6, (10 + m)/6, c^2*x^6]))/((1 + m)*(4 + m)))","A",1
133,0,0,21,0.3634303,"\int \frac{(d x)^m}{a+b \tanh ^{-1}\left(c x^3\right)} \, dx","Integrate[(d*x)^m/(a + b*ArcTanh[c*x^3]),x]","\int \frac{(d x)^m}{a+b \tanh ^{-1}\left(c x^3\right)} \, dx","\text{Int}\left(\frac{(d x)^m}{a+b \tanh ^{-1}\left(c x^3\right)},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcTanh[c*x^3]), x]","A",-1
134,0,0,21,0.3759322,"\int \frac{(d x)^m}{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2} \, dx","Integrate[(d*x)^m/(a + b*ArcTanh[c*x^3])^2,x]","\int \frac{(d x)^m}{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2} \, dx","\text{Int}\left(\frac{(d x)^m}{\left(a+b \tanh ^{-1}\left(c x^3\right)\right)^2},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcTanh[c*x^3])^2, x]","A",-1
135,1,67,50,0.0108386,"\int x^3 \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right) \, dx","Integrate[x^3*(a + b*ArcTanh[c/x]),x]","\frac{a x^4}{4}+\frac{1}{8} b c^4 \log (x-c)-\frac{1}{8} b c^4 \log (c+x)+\frac{1}{4} b c^3 x+\frac{1}{4} b x^4 \tanh ^{-1}\left(\frac{c}{x}\right)+\frac{1}{12} b c x^3","\frac{1}{4} x^4 \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)-\frac{1}{4} b c^4 \tanh ^{-1}\left(\frac{x}{c}\right)+\frac{1}{4} b c^3 x+\frac{1}{12} b c x^3",1,"(b*c^3*x)/4 + (b*c*x^3)/12 + (a*x^4)/4 + (b*x^4*ArcTanh[c/x])/4 + (b*c^4*Log[-c + x])/8 - (b*c^4*Log[c + x])/8","A",1
136,1,50,45,0.0083548,"\int x^2 \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right) \, dx","Integrate[x^2*(a + b*ArcTanh[c/x]),x]","\frac{a x^3}{3}+\frac{1}{6} b c^3 \log \left(x^2-c^2\right)+\frac{1}{3} b x^3 \tanh ^{-1}\left(\frac{c}{x}\right)+\frac{1}{6} b c x^2","\frac{1}{3} x^3 \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)+\frac{1}{6} b c^3 \log \left(c^2-x^2\right)+\frac{1}{6} b c x^2",1,"(b*c*x^2)/6 + (a*x^3)/3 + (b*x^3*ArcTanh[c/x])/3 + (b*c^3*Log[-c^2 + x^2])/6","A",1
137,1,56,39,0.0082375,"\int x \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right) \, dx","Integrate[x*(a + b*ArcTanh[c/x]),x]","\frac{a x^2}{2}+\frac{1}{4} b c^2 \log (x-c)-\frac{1}{4} b c^2 \log (c+x)+\frac{1}{2} b x^2 \tanh ^{-1}\left(\frac{c}{x}\right)+\frac{b c x}{2}","\frac{1}{2} x^2 \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)-\frac{1}{2} b c^2 \tanh ^{-1}\left(\frac{x}{c}\right)+\frac{b c x}{2}",1,"(b*c*x)/2 + (a*x^2)/2 + (b*x^2*ArcTanh[c/x])/2 + (b*c^2*Log[-c + x])/4 - (b*c^2*Log[c + x])/4","A",1
138,1,29,29,0.0031507,"\int \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right) \, dx","Integrate[a + b*ArcTanh[c/x],x]","a x+\frac{1}{2} b c \log \left(c^2-x^2\right)+b x \tanh ^{-1}\left(\frac{c}{x}\right)","a x+\frac{1}{2} b c \log \left(c^2-x^2\right)+b x \tanh ^{-1}\left(\frac{c}{x}\right)",1,"a*x + b*x*ArcTanh[c/x] + (b*c*Log[c^2 - x^2])/2","A",1
139,1,28,30,0.013144,"\int \frac{a+b \tanh ^{-1}\left(\frac{c}{x}\right)}{x} \, dx","Integrate[(a + b*ArcTanh[c/x])/x,x]","a \log (x)+\frac{1}{2} b \left(\text{Li}_2\left(-\frac{c}{x}\right)-\text{Li}_2\left(\frac{c}{x}\right)\right)","a \log (x)+\frac{1}{2} b \text{Li}_2\left(-\frac{c}{x}\right)-\frac{1}{2} b \text{Li}_2\left(\frac{c}{x}\right)",1,"a*Log[x] + (b*(PolyLog[2, -(c/x)] - PolyLog[2, c/x]))/2","A",1
140,1,38,35,0.0088527,"\int \frac{a+b \tanh ^{-1}\left(\frac{c}{x}\right)}{x^2} \, dx","Integrate[(a + b*ArcTanh[c/x])/x^2,x]","-\frac{a}{x}-\frac{b \log \left(1-\frac{c^2}{x^2}\right)}{2 c}-\frac{b \tanh ^{-1}\left(\frac{c}{x}\right)}{x}","-\frac{a+b \tanh ^{-1}\left(\frac{c}{x}\right)}{x}-\frac{b \log \left(1-\frac{c^2}{x^2}\right)}{2 c}",1,"-(a/x) - (b*ArcTanh[c/x])/x - (b*Log[1 - c^2/x^2])/(2*c)","A",1
141,1,60,43,0.0094073,"\int \frac{a+b \tanh ^{-1}\left(\frac{c}{x}\right)}{x^3} \, dx","Integrate[(a + b*ArcTanh[c/x])/x^3,x]","-\frac{a}{2 x^2}-\frac{b \log (x-c)}{4 c^2}+\frac{b \log (c+x)}{4 c^2}-\frac{b \tanh ^{-1}\left(\frac{c}{x}\right)}{2 x^2}-\frac{b}{2 c x}","-\frac{a+b \tanh ^{-1}\left(\frac{c}{x}\right)}{2 x^2}+\frac{b \tanh ^{-1}\left(\frac{x}{c}\right)}{2 c^2}-\frac{b}{2 c x}",1,"-1/2*a/x^2 - b/(2*c*x) - (b*ArcTanh[c/x])/(2*x^2) - (b*Log[-c + x])/(4*c^2) + (b*Log[c + x])/(4*c^2)","A",1
142,1,62,57,0.0103199,"\int \frac{a+b \tanh ^{-1}\left(\frac{c}{x}\right)}{x^4} \, dx","Integrate[(a + b*ArcTanh[c/x])/x^4,x]","-\frac{a}{3 x^3}+\frac{b \log (x)}{3 c^3}-\frac{b \log \left(x^2-c^2\right)}{6 c^3}-\frac{b \tanh ^{-1}\left(\frac{c}{x}\right)}{3 x^3}-\frac{b}{6 c x^2}","-\frac{a+b \tanh ^{-1}\left(\frac{c}{x}\right)}{3 x^3}+\frac{b \log (x)}{3 c^3}-\frac{b \log \left(c^2-x^2\right)}{6 c^3}-\frac{b}{6 c x^2}",1,"-1/3*a/x^3 - b/(6*c*x^2) - (b*ArcTanh[c/x])/(3*x^3) + (b*Log[x])/(3*c^3) - (b*Log[-c^2 + x^2])/(6*c^3)","A",1
143,1,131,123,0.0640753,"\int x^3 \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^2 \, dx","Integrate[x^3*(a + b*ArcTanh[c/x])^2,x]","\frac{1}{12} \left(3 a^2 x^4+b c^4 (3 a+4 b) \log (x-c)-3 a b c^4 \log (c+x)+6 a b c^3 x+2 b x \tanh ^{-1}\left(\frac{c}{x}\right) \left(3 a x^3+b c \left(3 c^2+x^2\right)\right)+2 a b c x^3+3 b^2 \left(x^4-c^4\right) \tanh ^{-1}\left(\frac{c}{x}\right)^2+4 b^2 c^4 \log (c+x)+b^2 c^2 x^2\right)","-\frac{1}{4} c^4 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{1}{2} b c^3 x \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)+\frac{1}{4} x^4 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{1}{6} b c x^3 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)+\frac{2}{3} b^2 c^4 \log (x)+\frac{1}{12} b^2 c^2 x^2+\frac{1}{3} b^2 c^4 \log \left(1-\frac{c^2}{x^2}\right)",1,"(6*a*b*c^3*x + b^2*c^2*x^2 + 2*a*b*c*x^3 + 3*a^2*x^4 + 2*b*x*(3*a*x^3 + b*c*(3*c^2 + x^2))*ArcTanh[c/x] + 3*b^2*(-c^4 + x^4)*ArcTanh[c/x]^2 + b*(3*a + 4*b)*c^4*Log[-c + x] - 3*a*b*c^4*Log[c + x] + 4*b^2*c^4*Log[c + x])/12","A",1
144,1,145,142,0.3265322,"\int x^2 \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^2 \, dx","Integrate[x^2*(a + b*ArcTanh[c/x])^2,x]","\frac{1}{3} \left(a^2 x^3+b \tanh ^{-1}\left(\frac{c}{x}\right) \left(2 a x^3-2 b c^3 \log \left(1-e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)-b c^3+b c x^2\right)-2 a b c^3 \log \left(\frac{c}{x}\right)+a b c^3 \log \left(1-\frac{c^2}{x^2}\right)+a b c x^2+b^2 c^3 \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)+b^2 \left(x^3-c^3\right) \tanh ^{-1}\left(\frac{c}{x}\right)^2+b^2 c^2 x\right)","-\frac{1}{3} c^3 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2-\frac{2}{3} b c^3 \log \left(2-\frac{2}{\frac{c}{x}+1}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)+\frac{1}{3} x^3 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{1}{3} b c x^2 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)+\frac{1}{3} b^2 c^3 \text{Li}_2\left(\frac{2}{\frac{c}{x}+1}-1\right)-\frac{1}{3} b^2 c^3 \coth ^{-1}\left(\frac{x}{c}\right)+\frac{1}{3} b^2 c^2 x",1,"(b^2*c^2*x + a*b*c*x^2 + a^2*x^3 + b^2*(-c^3 + x^3)*ArcTanh[c/x]^2 + b*ArcTanh[c/x]*(-(b*c^3) + b*c*x^2 + 2*a*x^3 - 2*b*c^3*Log[1 - E^(-2*ArcTanh[c/x])]) + a*b*c^3*Log[1 - c^2/x^2] - 2*a*b*c^3*Log[c/x] + b^2*c^3*PolyLog[2, E^(-2*ArcTanh[c/x])])/3","A",0
145,1,92,83,0.0496917,"\int x \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^2 \, dx","Integrate[x*(a + b*ArcTanh[c/x])^2,x]","\frac{1}{2} \left(a^2 x^2+b c^2 (a+b) \log (x-c)-a b c^2 \log (c+x)+2 a b c x+2 b x \tanh ^{-1}\left(\frac{c}{x}\right) (a x+b c)+b^2 \left(x^2-c^2\right) \tanh ^{-1}\left(\frac{c}{x}\right)^2+b^2 c^2 \log (c+x)\right)","-\frac{1}{2} c^2 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{1}{2} x^2 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2+b c x \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)+\frac{1}{2} b^2 c^2 \log \left(1-\frac{c^2}{x^2}\right)+b^2 c^2 \log (x)",1,"(2*a*b*c*x + a^2*x^2 + 2*b*x*(b*c + a*x)*ArcTanh[c/x] + b^2*(-c^2 + x^2)*ArcTanh[c/x]^2 + b*(a + b)*c^2*Log[-c + x] - a*b*c^2*Log[c + x] + b^2*c^2*Log[c + x])/2","A",1
146,1,97,74,0.1256343,"\int \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^2 \, dx","Integrate[(a + b*ArcTanh[c/x])^2,x]","a \left(a x+b c \log \left(1-\frac{c^2}{x^2}\right)-2 b c \log \left(\frac{c}{x}\right)\right)+2 b \tanh ^{-1}\left(\frac{c}{x}\right) \left(a x-b c \log \left(1-e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)\right)+b^2 c \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)+b^2 (x-c) \tanh ^{-1}\left(\frac{c}{x}\right)^2","c \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2+x \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2-2 b c \log \left(\frac{2 c}{c-x}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)+b^2 (-c) \text{Li}_2\left(-\frac{c+x}{c-x}\right)",1,"b^2*(-c + x)*ArcTanh[c/x]^2 + 2*b*ArcTanh[c/x]*(a*x - b*c*Log[1 - E^(-2*ArcTanh[c/x])]) + a*(a*x + b*c*Log[1 - c^2/x^2] - 2*b*c*Log[c/x]) + b^2*c*PolyLog[2, E^(-2*ArcTanh[c/x])]","A",0
147,1,114,133,0.0946299,"\int \frac{\left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^2}{x} \, dx","Integrate[(a + b*ArcTanh[c/x])^2/x,x]","\frac{1}{2} b \left(2 \text{Li}_2\left(\frac{c+x}{c-x}\right) \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)-2 \text{Li}_2\left(\frac{c+x}{x-c}\right) \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)+b \left(\text{Li}_3\left(\frac{c+x}{x-c}\right)-\text{Li}_3\left(\frac{c+x}{c-x}\right)\right)\right)-2 \tanh ^{-1}\left(\frac{c+x}{c-x}\right) \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^2","b \text{Li}_2\left(1-\frac{2}{1-\frac{c}{x}}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)-b \text{Li}_2\left(\frac{2}{1-\frac{c}{x}}-1\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)-2 \tanh ^{-1}\left(1-\frac{2}{1-\frac{c}{x}}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2-\frac{1}{2} b^2 \text{Li}_3\left(1-\frac{2}{1-\frac{c}{x}}\right)+\frac{1}{2} b^2 \text{Li}_3\left(\frac{2}{1-\frac{c}{x}}-1\right)",1,"-2*(a + b*ArcTanh[c/x])^2*ArcTanh[(c + x)/(c - x)] + (b*(2*(a + b*ArcTanh[c/x])*PolyLog[2, (c + x)/(c - x)] - 2*(a + b*ArcTanh[c/x])*PolyLog[2, (c + x)/(-c + x)] + b*(-PolyLog[3, (c + x)/(c - x)] + PolyLog[3, (c + x)/(-c + x)])))/2","A",0
148,1,101,87,0.1877548,"\int \frac{\left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^2}{x^2} \, dx","Integrate[(a + b*ArcTanh[c/x])^2/x^2,x]","\frac{a \left(2 b x \log \left(\frac{1}{\sqrt{1-\frac{c^2}{x^2}}}\right)-a c\right)+2 b \tanh ^{-1}\left(\frac{c}{x}\right) \left(b x \log \left(e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}+1\right)-a c\right)-b^2 x \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)+b^2 (x-c) \tanh ^{-1}\left(\frac{c}{x}\right)^2}{c x}","-\frac{\left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2}{c}-\frac{\left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2}{x}+\frac{2 b \log \left(\frac{2}{1-\frac{c}{x}}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)}{c}+\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-\frac{c}{x}}\right)}{c}",1,"(b^2*(-c + x)*ArcTanh[c/x]^2 + 2*b*ArcTanh[c/x]*(-(a*c) + b*x*Log[1 + E^(-2*ArcTanh[c/x])]) + a*(-(a*c) + 2*b*x*Log[1/Sqrt[1 - c^2/x^2]]) - b^2*x*PolyLog[2, -E^(-2*ArcTanh[c/x])])/(c*x)","A",0
149,1,119,87,0.0716969,"\int \frac{\left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^2}{x^3} \, dx","Integrate[(a + b*ArcTanh[c/x])^2/x^3,x]","-\frac{a^2 c^2+a b x^2 \log (x-c)-a b x^2 \log (c+x)+2 a b c x+2 b c \tanh ^{-1}\left(\frac{c}{x}\right) (a c+b x)+b^2 \left(c^2-x^2\right) \tanh ^{-1}\left(\frac{c}{x}\right)^2+b^2 x^2 \log (x-c)+b^2 x^2 \log (c+x)-2 b^2 x^2 \log (x)}{2 c^2 x^2}","\frac{\left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2}{2 c^2}-\frac{\left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2}{2 x^2}-\frac{a b}{c x}-\frac{b^2 \log \left(1-\frac{c^2}{x^2}\right)}{2 c^2}-\frac{b^2 \coth ^{-1}\left(\frac{x}{c}\right)}{c x}",1,"-1/2*(a^2*c^2 + 2*a*b*c*x + 2*b*c*(a*c + b*x)*ArcTanh[c/x] + b^2*(c^2 - x^2)*ArcTanh[c/x]^2 - 2*b^2*x^2*Log[x] + a*b*x^2*Log[-c + x] + b^2*x^2*Log[-c + x] - a*b*x^2*Log[c + x] + b^2*x^2*Log[c + x])/(c^2*x^2)","A",1
150,1,286,203,0.6753475,"\int x^3 \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^3 \, dx","Integrate[x^3*(a + b*ArcTanh[c/x])^3,x]","\frac{1}{8} \left(2 a^3 x^4+2 b \tanh ^{-1}\left(\frac{c}{x}\right) \left(3 a^2 x^4+2 a b c x \left(3 c^2+x^2\right)-8 b^2 c^4 \log \left(1-e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)+b^2 \left(c^2 x^2-c^4\right)\right)+3 a^2 b c^4 \log \left(1-\frac{c}{x}\right)-3 a^2 b c^4 \log \left(\frac{c+x}{x}\right)+6 a^2 b c^3 x+2 a^2 b c x^3-2 a b^2 c^4+2 a b^2 c^2 x^2-16 a b^2 c^4 \log \left(\frac{c}{x \sqrt{1-\frac{c^2}{x^2}}}\right)+2 b^2 \tanh ^{-1}\left(\frac{c}{x}\right)^2 \left(3 a \left(x^4-c^4\right)+b c \left(-4 c^3+3 c^2 x+x^3\right)\right)+8 b^3 c^4 \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)+2 b^3 \left(x^4-c^4\right) \tanh ^{-1}\left(\frac{c}{x}\right)^3+2 b^3 c^3 x\right)","-2 b^2 c^4 \log \left(2-\frac{2}{\frac{c}{x}+1}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)+\frac{1}{4} b^2 c^2 x^2 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)-\frac{1}{4} c^4 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^3-b c^4 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{3}{4} b c^3 x \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{1}{4} x^4 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^3+\frac{1}{4} b c x^3 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2+b^3 c^4 \text{Li}_2\left(\frac{2}{\frac{c}{x}+1}-1\right)-\frac{1}{4} b^3 c^4 \coth ^{-1}\left(\frac{x}{c}\right)+\frac{1}{4} b^3 c^3 x",1,"(-2*a*b^2*c^4 + 6*a^2*b*c^3*x + 2*b^3*c^3*x + 2*a*b^2*c^2*x^2 + 2*a^2*b*c*x^3 + 2*a^3*x^4 + 2*b^2*(b*c*(-4*c^3 + 3*c^2*x + x^3) + 3*a*(-c^4 + x^4))*ArcTanh[c/x]^2 + 2*b^3*(-c^4 + x^4)*ArcTanh[c/x]^3 + 2*b*ArcTanh[c/x]*(3*a^2*x^4 + 2*a*b*c*x*(3*c^2 + x^2) + b^2*(-c^4 + c^2*x^2) - 8*b^2*c^4*Log[1 - E^(-2*ArcTanh[c/x])]) + 3*a^2*b*c^4*Log[1 - c/x] - 16*a*b^2*c^4*Log[c/(Sqrt[1 - c^2/x^2]*x)] - 3*a^2*b*c^4*Log[(c + x)/x] + 8*b^3*c^4*PolyLog[2, E^(-2*ArcTanh[c/x])])/8","A",0
151,1,316,217,0.7921483,"\int x^2 \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^3 \, dx","Integrate[x^2*(a + b*ArcTanh[c/x])^3,x]","\frac{1}{6} \left(2 a^3 x^3+3 a^2 b c^3 \log \left(x^2-c^2\right)+6 a^2 b x^3 \tanh ^{-1}\left(\frac{c}{x}\right)+3 a^2 b c x^2+6 a b^2 \left(c^3 \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)+\left(x^3-c^3\right) \tanh ^{-1}\left(\frac{c}{x}\right)^2+c \tanh ^{-1}\left(\frac{c}{x}\right) \left(-2 c^2 \log \left(1-e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)-c^2+x^2\right)+c^2 x\right)+\frac{1}{4} b^3 \left(-24 c^3 \tanh ^{-1}\left(\frac{c}{x}\right) \text{Li}_2\left(e^{2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)+12 c^3 \text{Li}_3\left(e^{2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)+8 c^3 \tanh ^{-1}\left(\frac{c}{x}\right)^3-12 c^3 \tanh ^{-1}\left(\frac{c}{x}\right)^2-24 c^3 \tanh ^{-1}\left(\frac{c}{x}\right)^2 \log \left(1-e^{2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)-i \pi ^3 c^3+24 c^2 x \tanh ^{-1}\left(\frac{c}{x}\right)-24 c^3 \log \left(\frac{c}{x \sqrt{1-\frac{c^2}{x^2}}}\right)+8 x^3 \tanh ^{-1}\left(\frac{c}{x}\right)^3+12 c x^2 \tanh ^{-1}\left(\frac{c}{x}\right)^2\right)\right)","b^2 c^3 \text{Li}_2\left(\frac{2}{\frac{c}{x}+1}-1\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)+b^2 c^2 x \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)-\frac{1}{3} c^3 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^3-\frac{1}{2} b c^3 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2-b c^3 \log \left(2-\frac{2}{\frac{c}{x}+1}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{1}{3} x^3 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^3+\frac{1}{2} b c x^2 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{1}{2} b^3 c^3 \text{Li}_3\left(\frac{2}{\frac{c}{x}+1}-1\right)+b^3 c^3 \log (x)+\frac{1}{2} b^3 c^3 \log \left(1-\frac{c^2}{x^2}\right)",1,"(3*a^2*b*c*x^2 + 2*a^3*x^3 + 6*a^2*b*x^3*ArcTanh[c/x] + 3*a^2*b*c^3*Log[-c^2 + x^2] + 6*a*b^2*(c^2*x + (-c^3 + x^3)*ArcTanh[c/x]^2 + c*ArcTanh[c/x]*(-c^2 + x^2 - 2*c^2*Log[1 - E^(-2*ArcTanh[c/x])]) + c^3*PolyLog[2, E^(-2*ArcTanh[c/x])]) + (b^3*((-I)*c^3*Pi^3 + 24*c^2*x*ArcTanh[c/x] - 12*c^3*ArcTanh[c/x]^2 + 12*c*x^2*ArcTanh[c/x]^2 + 8*c^3*ArcTanh[c/x]^3 + 8*x^3*ArcTanh[c/x]^3 - 24*c^3*ArcTanh[c/x]^2*Log[1 - E^(2*ArcTanh[c/x])] - 24*c^3*Log[c/(Sqrt[1 - c^2/x^2]*x)] - 24*c^3*ArcTanh[c/x]*PolyLog[2, E^(2*ArcTanh[c/x])] + 12*c^3*PolyLog[3, E^(2*ArcTanh[c/x])]))/4)/6","C",0
152,1,193,135,0.3363126,"\int x \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^3 \, dx","Integrate[x*(a + b*ArcTanh[c/x])^3,x]","\frac{1}{4} \left(a \left(a \left(2 a x^2-3 b c^2 \log \left(\frac{c+x}{x}\right)+6 b c x\right)+3 a b c^2 \log \left(1-\frac{c}{x}\right)-12 b^2 c^2 \log \left(\frac{c}{x \sqrt{1-\frac{c^2}{x^2}}}\right)\right)+6 b \tanh ^{-1}\left(\frac{c}{x}\right) \left(a x (a x+2 b c)-2 b^2 c^2 \log \left(1-e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)\right)+6 b^2 (x-c) \tanh ^{-1}\left(\frac{c}{x}\right)^2 (a (c+x)+b c)+6 b^3 c^2 \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)+2 b^3 \left(x^2-c^2\right) \tanh ^{-1}\left(\frac{c}{x}\right)^3\right)","-3 b^2 c^2 \log \left(2-\frac{2}{\frac{c}{x}+1}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)-\frac{3}{2} b c^2 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2-\frac{1}{2} c^2 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^3+\frac{1}{2} x^2 \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^3+\frac{3}{2} b c x \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{3}{2} b^3 c^2 \text{Li}_2\left(\frac{2}{\frac{c}{x}+1}-1\right)",1,"(6*b^2*(-c + x)*(b*c + a*(c + x))*ArcTanh[c/x]^2 + 2*b^3*(-c^2 + x^2)*ArcTanh[c/x]^3 + 6*b*ArcTanh[c/x]*(a*x*(2*b*c + a*x) - 2*b^2*c^2*Log[1 - E^(-2*ArcTanh[c/x])]) + a*(3*a*b*c^2*Log[1 - c/x] - 12*b^2*c^2*Log[c/(Sqrt[1 - c^2/x^2]*x)] + a*(6*b*c*x + 2*a*x^2 - 3*b*c^2*Log[(c + x)/x])) + 6*b^3*c^2*PolyLog[2, E^(-2*ArcTanh[c/x])])/4","A",0
153,1,198,108,0.2785032,"\int \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^3 \, dx","Integrate[(a + b*ArcTanh[c/x])^3,x]","a^3 x+\frac{3}{2} a^2 b c \log \left(x^2-c^2\right)+3 a^2 b x \tanh ^{-1}\left(\frac{c}{x}\right)-3 a b^2 \left(\tanh ^{-1}\left(\frac{c}{x}\right) \left((c-x) \tanh ^{-1}\left(\frac{c}{x}\right)+2 c \log \left(1-e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)\right)-c \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)\right)+\frac{1}{8} b^3 \left(-24 c \tanh ^{-1}\left(\frac{c}{x}\right) \text{Li}_2\left(e^{2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)+12 c \text{Li}_3\left(e^{2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)+8 c \tanh ^{-1}\left(\frac{c}{x}\right)^3+8 x \tanh ^{-1}\left(\frac{c}{x}\right)^3-24 c \tanh ^{-1}\left(\frac{c}{x}\right)^2 \log \left(1-e^{2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)-i \pi ^3 c\right)","-3 b^2 c \text{Li}_2\left(1-\frac{2 c}{c-x}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)+c \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^3+x \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^3-3 b c \log \left(\frac{2 c}{c-x}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2+\frac{3}{2} b^3 c \text{Li}_3\left(1-\frac{2 c}{c-x}\right)",1,"a^3*x + 3*a^2*b*x*ArcTanh[c/x] + (3*a^2*b*c*Log[-c^2 + x^2])/2 - 3*a*b^2*(ArcTanh[c/x]*((c - x)*ArcTanh[c/x] + 2*c*Log[1 - E^(-2*ArcTanh[c/x])]) - c*PolyLog[2, E^(-2*ArcTanh[c/x])]) + (b^3*((-I)*c*Pi^3 + 8*c*ArcTanh[c/x]^3 + 8*x*ArcTanh[c/x]^3 - 24*c*ArcTanh[c/x]^2*Log[1 - E^(2*ArcTanh[c/x])] - 24*c*ArcTanh[c/x]*PolyLog[2, E^(2*ArcTanh[c/x])] + 12*c*PolyLog[3, E^(2*ArcTanh[c/x])]))/8","C",0
154,1,171,208,0.1992267,"\int \frac{\left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^3}{x} \, dx","Integrate[(a + b*ArcTanh[c/x])^3/x,x]","\frac{3}{4} b \left(2 \text{Li}_2\left(\frac{c+x}{c-x}\right) \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^2-2 \text{Li}_2\left(\frac{c+x}{x-c}\right) \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^2+b \left(-2 \text{Li}_3\left(\frac{c+x}{c-x}\right) \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)+2 \text{Li}_3\left(\frac{c+x}{x-c}\right) \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)+b \left(\text{Li}_4\left(\frac{c+x}{c-x}\right)-\text{Li}_4\left(\frac{c+x}{x-c}\right)\right)\right)\right)-2 \tanh ^{-1}\left(\frac{c+x}{c-x}\right) \left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^3","-\frac{3}{2} b^2 \text{Li}_3\left(1-\frac{2}{1-\frac{c}{x}}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)+\frac{3}{2} b^2 \text{Li}_3\left(\frac{2}{1-\frac{c}{x}}-1\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)+\frac{3}{2} b \text{Li}_2\left(1-\frac{2}{1-\frac{c}{x}}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2-\frac{3}{2} b \text{Li}_2\left(\frac{2}{1-\frac{c}{x}}-1\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2-2 \tanh ^{-1}\left(1-\frac{2}{1-\frac{c}{x}}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^3+\frac{3}{4} b^3 \text{Li}_4\left(1-\frac{2}{1-\frac{c}{x}}\right)-\frac{3}{4} b^3 \text{Li}_4\left(\frac{2}{1-\frac{c}{x}}-1\right)",1,"-2*(a + b*ArcTanh[c/x])^3*ArcTanh[(c + x)/(c - x)] + (3*b*(2*(a + b*ArcTanh[c/x])^2*PolyLog[2, (c + x)/(c - x)] - 2*(a + b*ArcTanh[c/x])^2*PolyLog[2, (c + x)/(-c + x)] + b*(-2*(a + b*ArcTanh[c/x])*PolyLog[3, (c + x)/(c - x)] + 2*(a + b*ArcTanh[c/x])*PolyLog[3, (c + x)/(-c + x)] + b*(PolyLog[4, (c + x)/(c - x)] - PolyLog[4, (c + x)/(-c + x)]))))/4","A",0
155,1,206,126,0.1168335,"\int \frac{\left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^3}{x^2} \, dx","Integrate[(a + b*ArcTanh[c/x])^3/x^2,x]","-\frac{a^3}{x}-\frac{3 a^2 b \left(\frac{c \tanh ^{-1}\left(\frac{c}{x}\right)}{x}-\log \left(\frac{1}{\sqrt{1-\frac{c^2}{x^2}}}\right)\right)}{c}-\frac{3 a b^2 \left(\text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)+\tanh ^{-1}\left(\frac{c}{x}\right) \left(\frac{c \tanh ^{-1}\left(\frac{c}{x}\right)}{x}-\tanh ^{-1}\left(\frac{c}{x}\right)-2 \log \left(e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}+1\right)\right)\right)}{c}-\frac{b^3 \left(3 \tanh ^{-1}\left(\frac{c}{x}\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)+\frac{3}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)+\tanh ^{-1}\left(\frac{c}{x}\right)^2 \left(\frac{c \tanh ^{-1}\left(\frac{c}{x}\right)}{x}-\tanh ^{-1}\left(\frac{c}{x}\right)-3 \log \left(e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}+1\right)\right)\right)}{c}","\frac{3 b^2 \text{Li}_2\left(1-\frac{2}{1-\frac{c}{x}}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)}{c}-\frac{\left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^3}{c}-\frac{\left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^3}{x}+\frac{3 b \log \left(\frac{2}{1-\frac{c}{x}}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2}{c}-\frac{3 b^3 \text{Li}_3\left(1-\frac{2}{1-\frac{c}{x}}\right)}{2 c}",1,"-(a^3/x) - (3*a^2*b*((c*ArcTanh[c/x])/x - Log[1/Sqrt[1 - c^2/x^2]]))/c - (3*a*b^2*(ArcTanh[c/x]*(-ArcTanh[c/x] + (c*ArcTanh[c/x])/x - 2*Log[1 + E^(-2*ArcTanh[c/x])]) + PolyLog[2, -E^(-2*ArcTanh[c/x])]))/c - (b^3*(ArcTanh[c/x]^2*(-ArcTanh[c/x] + (c*ArcTanh[c/x])/x - 3*Log[1 + E^(-2*ArcTanh[c/x])]) + 3*ArcTanh[c/x]*PolyLog[2, -E^(-2*ArcTanh[c/x])] + (3*PolyLog[3, -E^(-2*ArcTanh[c/x])])/2))/c","A",0
156,1,195,139,0.3519913,"\int \frac{\left(a+b \tanh ^{-1}\left(\frac{c}{x}\right)\right)^3}{x^3} \, dx","Integrate[(a + b*ArcTanh[c/x])^3/x^3,x]","\frac{a \left(12 b^2 x^2 \log \left(\frac{1}{\sqrt{1-\frac{c^2}{x^2}}}\right)-a \left(2 a c^2+3 b x^2 \log \left(1-\frac{c}{x}\right)-3 b x^2 \log \left(\frac{c+x}{x}\right)+6 b c x\right)\right)+6 b \tanh ^{-1}\left(\frac{c}{x}\right) \left(2 b^2 x^2 \log \left(e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}+1\right)-a c (a c+2 b x)\right)+6 b^2 (x-c) \tanh ^{-1}\left(\frac{c}{x}\right)^2 (a (c+x)+b x)+2 b^3 \left(x^2-c^2\right) \tanh ^{-1}\left(\frac{c}{x}\right)^3-6 b^3 x^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(\frac{c}{x}\right)}\right)}{4 c^2 x^2}","\frac{3 b^2 \log \left(\frac{2}{1-\frac{c}{x}}\right) \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)}{c^2}-\frac{3 b \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2}{2 c^2}+\frac{\left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^3}{2 c^2}-\frac{\left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^3}{2 x^2}-\frac{3 b \left(a+b \coth ^{-1}\left(\frac{x}{c}\right)\right)^2}{2 c x}+\frac{3 b^3 \text{Li}_2\left(1-\frac{2}{1-\frac{c}{x}}\right)}{2 c^2}",1,"(6*b^2*(-c + x)*(b*x + a*(c + x))*ArcTanh[c/x]^2 + 2*b^3*(-c^2 + x^2)*ArcTanh[c/x]^3 + 6*b*ArcTanh[c/x]*(-(a*c*(a*c + 2*b*x)) + 2*b^2*x^2*Log[1 + E^(-2*ArcTanh[c/x])]) + a*(12*b^2*x^2*Log[1/Sqrt[1 - c^2/x^2]] - a*(2*a*c^2 + 6*b*c*x + 3*b*x^2*Log[1 - c/x] - 3*b*x^2*Log[(c + x)/x])) - 6*b^3*x^2*PolyLog[2, -E^(-2*ArcTanh[c/x])])/(4*c^2*x^2)","A",0
157,1,73,54,0.0147009,"\int x^7 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right) \, dx","Integrate[x^7*(a + b*ArcTanh[c/x^2]),x]","\frac{a x^8}{8}+\frac{1}{16} b c^4 \log \left(x^2-c\right)-\frac{1}{16} b c^4 \log \left(c+x^2\right)+\frac{1}{8} b c^3 x^2+\frac{1}{24} b c x^6+\frac{1}{8} b x^8 \tanh ^{-1}\left(\frac{c}{x^2}\right)","\frac{1}{8} x^8 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)-\frac{1}{8} b c^4 \tanh ^{-1}\left(\frac{x^2}{c}\right)+\frac{1}{8} b c^3 x^2+\frac{1}{24} b c x^6",1,"(b*c^3*x^2)/8 + (b*c*x^6)/24 + (a*x^8)/8 + (b*x^8*ArcTanh[c/x^2])/8 + (b*c^4*Log[-c + x^2])/16 - (b*c^4*Log[c + x^2])/16","A",1
158,1,50,45,0.0113599,"\int x^5 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right) \, dx","Integrate[x^5*(a + b*ArcTanh[c/x^2]),x]","\frac{a x^6}{6}+\frac{1}{12} b c^3 \log \left(x^4-c^2\right)+\frac{1}{12} b c x^4+\frac{1}{6} b x^6 \tanh ^{-1}\left(\frac{c}{x^2}\right)","\frac{1}{6} x^6 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)+\frac{1}{12} b c^3 \log \left(c^2-x^4\right)+\frac{1}{12} b c x^4",1,"(b*c*x^4)/12 + (a*x^6)/6 + (b*x^6*ArcTanh[c/x^2])/6 + (b*c^3*Log[-c^2 + x^4])/12","A",1
159,1,62,43,0.0111641,"\int x^3 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right) \, dx","Integrate[x^3*(a + b*ArcTanh[c/x^2]),x]","\frac{a x^4}{4}+\frac{1}{8} b c^2 \log \left(x^2-c\right)-\frac{1}{8} b c^2 \log \left(c+x^2\right)+\frac{1}{4} b c x^2+\frac{1}{4} b x^4 \tanh ^{-1}\left(\frac{c}{x^2}\right)","\frac{1}{4} x^4 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)-\frac{1}{4} b c^2 \tanh ^{-1}\left(\frac{x^2}{c}\right)+\frac{1}{4} b c x^2",1,"(b*c*x^2)/4 + (a*x^4)/4 + (b*x^4*ArcTanh[c/x^2])/4 + (b*c^2*Log[-c + x^2])/8 - (b*c^2*Log[c + x^2])/8","A",1
160,1,39,34,0.0070863,"\int x \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right) \, dx","Integrate[x*(a + b*ArcTanh[c/x^2]),x]","\frac{a x^2}{2}+\frac{1}{4} b c \log \left(x^4-c^2\right)+\frac{1}{2} b x^2 \tanh ^{-1}\left(\frac{c}{x^2}\right)","\frac{1}{2} x^2 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)+\frac{1}{4} b c \log \left(c^2-x^4\right)",1,"(a*x^2)/2 + (b*x^2*ArcTanh[c/x^2])/2 + (b*c*Log[-c^2 + x^4])/4","A",1
161,1,28,30,0.0137079,"\int \frac{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{x} \, dx","Integrate[(a + b*ArcTanh[c/x^2])/x,x]","a \log (x)+\frac{1}{4} b \left(\text{Li}_2\left(-\frac{c}{x^2}\right)-\text{Li}_2\left(\frac{c}{x^2}\right)\right)","a \log (x)+\frac{1}{4} b \text{Li}_2\left(-\frac{c}{x^2}\right)-\frac{1}{4} b \text{Li}_2\left(\frac{c}{x^2}\right)",1,"a*Log[x] + (b*(PolyLog[2, -(c/x^2)] - PolyLog[2, c/x^2]))/4","A",1
162,1,42,37,0.009583,"\int \frac{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{x^3} \, dx","Integrate[(a + b*ArcTanh[c/x^2])/x^3,x]","-\frac{a}{2 x^2}-\frac{b \log \left(1-\frac{c^2}{x^4}\right)}{4 c}-\frac{b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{2 x^2}","-\frac{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{2 x^2}-\frac{b \log \left(1-\frac{c^2}{x^4}\right)}{4 c}",1,"-1/2*a/x^2 - (b*ArcTanh[c/x^2])/(2*x^2) - (b*Log[1 - c^2/x^4])/(4*c)","A",1
163,1,64,45,0.0120729,"\int \frac{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{x^5} \, dx","Integrate[(a + b*ArcTanh[c/x^2])/x^5,x]","-\frac{a}{4 x^4}-\frac{b \log \left(x^2-c\right)}{8 c^2}+\frac{b \log \left(c+x^2\right)}{8 c^2}-\frac{b}{4 c x^2}-\frac{b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{4 x^4}","-\frac{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{4 x^4}+\frac{b \tanh ^{-1}\left(\frac{x^2}{c}\right)}{4 c^2}-\frac{b}{4 c x^2}",1,"-1/4*a/x^4 - b/(4*c*x^2) - (b*ArcTanh[c/x^2])/(4*x^4) - (b*Log[-c + x^2])/(8*c^2) + (b*Log[c + x^2])/(8*c^2)","A",1
164,1,62,57,0.0133419,"\int \frac{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{x^7} \, dx","Integrate[(a + b*ArcTanh[c/x^2])/x^7,x]","-\frac{a}{6 x^6}+\frac{b \log (x)}{3 c^3}-\frac{b \log \left(x^4-c^2\right)}{12 c^3}-\frac{b}{12 c x^4}-\frac{b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{6 x^6}","-\frac{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{6 x^6}+\frac{b \log (x)}{3 c^3}-\frac{b \log \left(c^2-x^4\right)}{12 c^3}-\frac{b}{12 c x^4}",1,"-1/6*a/x^6 - b/(12*c*x^4) - (b*ArcTanh[c/x^2])/(6*x^6) + (b*Log[x])/(3*c^3) - (b*Log[-c^2 + x^4])/(12*c^3)","A",1
165,1,88,63,0.0219846,"\int x^4 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right) \, dx","Integrate[x^4*(a + b*ArcTanh[c/x^2]),x]","\frac{a x^5}{5}+\frac{1}{10} b c^{5/2} \log \left(\sqrt{c}-x\right)-\frac{1}{10} b c^{5/2} \log \left(\sqrt{c}+x\right)+\frac{1}{5} b c^{5/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)+\frac{2}{15} b c x^3+\frac{1}{5} b x^5 \tanh ^{-1}\left(\frac{c}{x^2}\right)","\frac{1}{5} x^5 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)+\frac{1}{5} b c^{5/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)-\frac{1}{5} b c^{5/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right)+\frac{2}{15} b c x^3",1,"(2*b*c*x^3)/15 + (a*x^5)/5 + (b*c^(5/2)*ArcTan[x/Sqrt[c]])/5 + (b*x^5*ArcTanh[c/x^2])/5 + (b*c^(5/2)*Log[Sqrt[c] - x])/10 - (b*c^(5/2)*Log[Sqrt[c] + x])/10","A",1
166,1,86,61,0.0191718,"\int x^2 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right) \, dx","Integrate[x^2*(a + b*ArcTanh[c/x^2]),x]","\frac{a x^3}{3}+\frac{1}{6} b c^{3/2} \log \left(\sqrt{c}-x\right)-\frac{1}{6} b c^{3/2} \log \left(\sqrt{c}+x\right)-\frac{1}{3} b c^{3/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)+\frac{1}{3} b x^3 \tanh ^{-1}\left(\frac{c}{x^2}\right)+\frac{2 b c x}{3}","\frac{1}{3} x^3 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)-\frac{1}{3} b c^{3/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)-\frac{1}{3} b c^{3/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right)+\frac{2 b c x}{3}",1,"(2*b*c*x)/3 + (a*x^3)/3 - (b*c^(3/2)*ArcTan[x/Sqrt[c]])/3 + (b*x^3*ArcTanh[c/x^2])/3 + (b*c^(3/2)*Log[Sqrt[c] - x])/6 - (b*c^(3/2)*Log[Sqrt[c] + x])/6","A",1
167,1,54,44,0.0170113,"\int \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right) \, dx","Integrate[a + b*ArcTanh[c/x^2],x]","a x+b x \tanh ^{-1}\left(\frac{c}{x^2}\right)+\frac{1}{2} b \sqrt{c} \left(\log \left(\sqrt{c}-x\right)-\log \left(\sqrt{c}+x\right)+2 \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)\right)","a x+b x \tanh ^{-1}\left(\frac{c}{x^2}\right)+b \sqrt{c} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)-b \sqrt{c} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right)",1,"a*x + b*x*ArcTanh[c/x^2] + (b*Sqrt[c]*(2*ArcTan[x/Sqrt[c]] + Log[Sqrt[c] - x] - Log[Sqrt[c] + x]))/2","A",1
168,1,72,46,0.0183966,"\int \frac{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{x^2} \, dx","Integrate[(a + b*ArcTanh[c/x^2])/x^2,x]","-\frac{a}{x}-\frac{b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{x}-\frac{b \log \left(\sqrt{c}-x\right)}{2 \sqrt{c}}+\frac{b \log \left(\sqrt{c}+x\right)}{2 \sqrt{c}}+\frac{b \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)}{\sqrt{c}}","-\frac{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{x}+\frac{b \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)}{\sqrt{c}}+\frac{b \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right)}{\sqrt{c}}",1,"-(a/x) + (b*ArcTan[x/Sqrt[c]])/Sqrt[c] - (b*ArcTanh[c/x^2])/x - (b*Log[Sqrt[c] - x])/(2*Sqrt[c]) + (b*Log[Sqrt[c] + x])/(2*Sqrt[c])","A",1
169,1,90,65,0.031999,"\int \frac{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{x^4} \, dx","Integrate[(a + b*ArcTanh[c/x^2])/x^4,x]","-\frac{a}{3 x^3}-\frac{b \log \left(\sqrt{c}-x\right)}{6 c^{3/2}}+\frac{b \log \left(\sqrt{c}+x\right)}{6 c^{3/2}}-\frac{b \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)}{3 c^{3/2}}-\frac{b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{3 x^3}-\frac{2 b}{3 c x}","-\frac{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{3 x^3}-\frac{b \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)}{3 c^{3/2}}+\frac{b \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right)}{3 c^{3/2}}-\frac{2 b}{3 c x}",1,"-1/3*a/x^3 - (2*b)/(3*c*x) - (b*ArcTan[x/Sqrt[c]])/(3*c^(3/2)) - (b*ArcTanh[c/x^2])/(3*x^3) - (b*Log[Sqrt[c] - x])/(6*c^(3/2)) + (b*Log[Sqrt[c] + x])/(6*c^(3/2))","A",1
170,1,90,65,0.0249444,"\int \frac{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{x^6} \, dx","Integrate[(a + b*ArcTanh[c/x^2])/x^6,x]","-\frac{a}{5 x^5}-\frac{b \log \left(\sqrt{c}-x\right)}{10 c^{5/2}}+\frac{b \log \left(\sqrt{c}+x\right)}{10 c^{5/2}}+\frac{b \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)}{5 c^{5/2}}-\frac{2 b}{15 c x^3}-\frac{b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{5 x^5}","-\frac{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)}{5 x^5}+\frac{b \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)}{5 c^{5/2}}+\frac{b \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right)}{5 c^{5/2}}-\frac{2 b}{15 c x^3}",1,"-1/5*a/x^5 - (2*b)/(15*c*x^3) + (b*ArcTan[x/Sqrt[c]])/(5*c^(5/2)) - (b*ArcTanh[c/x^2])/(5*x^5) - (b*Log[Sqrt[c] - x])/(10*c^(5/2)) + (b*Log[Sqrt[c] + x])/(10*c^(5/2))","A",1
171,1,104,94,0.0642629,"\int x^3 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2 \, dx","Integrate[x^3*(a + b*ArcTanh[c/x^2])^2,x]","\frac{1}{4} \left(a^2 x^4+b c^2 (a+b) \log \left(x^2-c\right)-a b c^2 \log \left(c+x^2\right)+2 a b c x^2+2 b x^2 \tanh ^{-1}\left(\frac{c}{x^2}\right) \left(a x^2+b c\right)+b^2 c^2 \log \left(c+x^2\right)+b^2 \left(x^4-c^2\right) \tanh ^{-1}\left(\frac{c}{x^2}\right)^2\right)","-\frac{1}{4} c^2 \left(a+b \coth ^{-1}\left(\frac{x^2}{c}\right)\right)^2+\frac{1}{2} b c x^2 \left(a+b \coth ^{-1}\left(\frac{x^2}{c}\right)\right)+\frac{1}{4} x^4 \left(a+b \coth ^{-1}\left(\frac{x^2}{c}\right)\right)^2+\frac{1}{4} b^2 c^2 \log \left(1-\frac{c^2}{x^4}\right)+b^2 c^2 \log (x)",1,"(2*a*b*c*x^2 + a^2*x^4 + 2*b*x^2*(b*c + a*x^2)*ArcTanh[c/x^2] + b^2*(-c^2 + x^4)*ArcTanh[c/x^2]^2 + b*(a + b)*c^2*Log[-c + x^2] - a*b*c^2*Log[c + x^2] + b^2*c^2*Log[c + x^2])/4","A",1
172,1,107,94,0.1409709,"\int x \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2 \, dx","Integrate[x*(a + b*ArcTanh[c/x^2])^2,x]","\frac{1}{2} \left(a \left(a x^2+b c \log \left(1-\frac{c^2}{x^4}\right)-2 b c \log \left(\frac{c}{x^2}\right)\right)+2 b \tanh ^{-1}\left(\frac{c}{x^2}\right) \left(a x^2-b c \log \left(1-e^{-2 \tanh ^{-1}\left(\frac{c}{x^2}\right)}\right)\right)+b^2 c \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(\frac{c}{x^2}\right)}\right)+b^2 \left(x^2-c\right) \tanh ^{-1}\left(\frac{c}{x^2}\right)^2\right)","\frac{1}{2} x^2 \left(a+b \coth ^{-1}\left(\frac{x^2}{c}\right)\right)^2-\frac{1}{2} c \left(a+b \coth ^{-1}\left(\frac{x^2}{c}\right)\right)^2-b c \log \left(2-\frac{2}{\frac{c}{x^2}+1}\right) \left(a+b \coth ^{-1}\left(\frac{x^2}{c}\right)\right)+\frac{1}{2} b^2 c \text{Li}_2\left(\frac{2}{\frac{c}{x^2}+1}-1\right)",1,"(b^2*(-c + x^2)*ArcTanh[c/x^2]^2 + 2*b*ArcTanh[c/x^2]*(a*x^2 - b*c*Log[1 - E^(-2*ArcTanh[c/x^2])]) + a*(a*x^2 + b*c*Log[1 - c^2/x^4] - 2*b*c*Log[c/x^2]) + b^2*c*PolyLog[2, E^(-2*ArcTanh[c/x^2])])/2","A",0
173,1,183,144,0.0696364,"\int \frac{\left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2}{x} \, dx","Integrate[(a + b*ArcTanh[c/x^2])^2/x,x]","\frac{1}{2} \left(4 b c \left(\frac{1}{2} \left(\frac{\text{Li}_2\left(\frac{-\frac{c}{x^2}-1}{\frac{c}{x^2}-1}\right) \left(-a-b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)}{2 c}+\frac{b \text{Li}_3\left(\frac{-\frac{c}{x^2}-1}{\frac{c}{x^2}-1}\right)}{4 c}\right)+\frac{1}{2} \left(-\frac{\text{Li}_2\left(\frac{\frac{c}{x^2}+1}{\frac{c}{x^2}-1}\right) \left(-a-b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)}{2 c}-\frac{b \text{Li}_3\left(\frac{\frac{c}{x^2}+1}{\frac{c}{x^2}-1}\right)}{4 c}\right)\right)-2 \tanh ^{-1}\left(1-\frac{2}{1-\frac{c}{x^2}}\right) \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2\right)","\frac{1}{2} b \text{Li}_2\left(1-\frac{2}{1-\frac{c}{x^2}}\right) \left(a+b \coth ^{-1}\left(\frac{x^2}{c}\right)\right)-\frac{1}{2} b \text{Li}_2\left(\frac{2}{1-\frac{c}{x^2}}-1\right) \left(a+b \coth ^{-1}\left(\frac{x^2}{c}\right)\right)-\tanh ^{-1}\left(1-\frac{2}{1-\frac{c}{x^2}}\right) \left(a+b \coth ^{-1}\left(\frac{x^2}{c}\right)\right)^2-\frac{1}{4} b^2 \text{Li}_3\left(1-\frac{2}{1-\frac{c}{x^2}}\right)+\frac{1}{4} b^2 \text{Li}_3\left(\frac{2}{1-\frac{c}{x^2}}-1\right)",1,"(-2*ArcTanh[1 - 2/(1 - c/x^2)]*(a + b*ArcTanh[c/x^2])^2 + 4*b*c*((((-a - b*ArcTanh[c/x^2])*PolyLog[2, (-1 - c/x^2)/(-1 + c/x^2)])/(2*c) + (b*PolyLog[3, (-1 - c/x^2)/(-1 + c/x^2)])/(4*c))/2 + (-1/2*((-a - b*ArcTanh[c/x^2])*PolyLog[2, (1 + c/x^2)/(-1 + c/x^2)])/c - (b*PolyLog[3, (1 + c/x^2)/(-1 + c/x^2)])/(4*c))/2))/2","A",0
174,1,114,99,0.1842038,"\int \frac{\left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2}{x^3} \, dx","Integrate[(a + b*ArcTanh[c/x^2])^2/x^3,x]","-\frac{a^2}{2 x^2}-\frac{a b \left(\frac{c \tanh ^{-1}\left(\frac{c}{x^2}\right)}{x^2}-\log \left(\frac{1}{\sqrt{1-\frac{c^2}{x^4}}}\right)\right)}{c}-\frac{b^2 \left(\text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(\frac{c}{x^2}\right)}\right)+\tanh ^{-1}\left(\frac{c}{x^2}\right) \left(\frac{c \tanh ^{-1}\left(\frac{c}{x^2}\right)}{x^2}-\tanh ^{-1}\left(\frac{c}{x^2}\right)-2 \log \left(e^{-2 \tanh ^{-1}\left(\frac{c}{x^2}\right)}+1\right)\right)\right)}{2 c}","-\frac{\left(a+b \coth ^{-1}\left(\frac{x^2}{c}\right)\right)^2}{2 c}-\frac{\left(a+b \coth ^{-1}\left(\frac{x^2}{c}\right)\right)^2}{2 x^2}+\frac{b \log \left(\frac{2}{1-\frac{c}{x^2}}\right) \left(a+b \coth ^{-1}\left(\frac{x^2}{c}\right)\right)}{c}+\frac{b^2 \text{Li}_2\left(1-\frac{2}{1-\frac{c}{x^2}}\right)}{2 c}",1,"-1/2*a^2/x^2 - (a*b*((c*ArcTanh[c/x^2])/x^2 - Log[1/Sqrt[1 - c^2/x^4]]))/c - (b^2*(ArcTanh[c/x^2]*(-ArcTanh[c/x^2] + (c*ArcTanh[c/x^2])/x^2 - 2*Log[1 + E^(-2*ArcTanh[c/x^2])]) + PolyLog[2, -E^(-2*ArcTanh[c/x^2])]))/(2*c)","A",0
175,1,131,97,0.0848435,"\int \frac{\left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2}{x^5} \, dx","Integrate[(a + b*ArcTanh[c/x^2])^2/x^5,x]","-\frac{a^2 c^2+2 a b c x^2+2 b c \tanh ^{-1}\left(\frac{c}{x^2}\right) \left(a c+b x^2\right)+a b x^4 \log \left(x^2-c\right)-a b x^4 \log \left(c+x^2\right)+b^2 \left(c^2-x^4\right) \tanh ^{-1}\left(\frac{c}{x^2}\right)^2+b^2 x^4 \log \left(x^2-c\right)+b^2 x^4 \log \left(c+x^2\right)-4 b^2 x^4 \log (x)}{4 c^2 x^4}","\frac{\left(a+b \coth ^{-1}\left(\frac{x^2}{c}\right)\right)^2}{4 c^2}-\frac{a b}{2 c x^2}-\frac{\left(a+b \coth ^{-1}\left(\frac{x^2}{c}\right)\right)^2}{4 x^4}-\frac{b^2 \log \left(1-\frac{c^2}{x^4}\right)}{4 c^2}-\frac{b^2 \coth ^{-1}\left(\frac{x^2}{c}\right)}{2 c x^2}",1,"-1/4*(a^2*c^2 + 2*a*b*c*x^2 + 2*b*c*(a*c + b*x^2)*ArcTanh[c/x^2] + b^2*(c^2 - x^4)*ArcTanh[c/x^2]^2 - 4*b^2*x^4*Log[x] + a*b*x^4*Log[-c + x^2] + b^2*x^4*Log[-c + x^2] - a*b*x^4*Log[c + x^2] + b^2*x^4*Log[c + x^2])/(c^2*x^4)","A",1
176,0,0,1214,5.8954518,"\int x^4 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2 \, dx","Integrate[x^4*(a + b*ArcTanh[c/x^2])^2,x]","\int x^4 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2 \, dx","\frac{1}{20} \left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right)^2 x^5+\frac{1}{20} b^2 \log ^2\left(\frac{c}{x^2}+1\right) x^5+\frac{1}{5} a b \log \left(\frac{c}{x^2}+1\right) x^5-\frac{1}{10} b^2 \log \left(1-\frac{c}{x^2}\right) \log \left(\frac{c}{x^2}+1\right) x^5+\frac{2}{15} a b c x^3-\frac{1}{15} b^2 c \log \left(1-\frac{c}{x^2}\right) x^3+\frac{1}{15} b c \left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right) x^3+\frac{2}{15} b^2 c \log \left(\frac{c}{x^2}+1\right) x^3+\frac{8}{15} b^2 c^2 x-\frac{1}{5} i b^2 c^{5/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)^2+\frac{1}{5} b^2 c^{5/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right)^2-\frac{4}{15} b^2 c^{5/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)+\frac{2}{5} a b c^{5/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)-\frac{4}{15} b^2 c^{5/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right)+\frac{2}{5} b^2 c^{5/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right)-\frac{1}{5} b^2 c^{5/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(1-\frac{c}{x^2}\right)-\frac{1}{5} b c^{5/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right)+\frac{1}{5} b^2 c^{5/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{c}{x^2}+1\right)-\frac{1}{5} b^2 c^{5/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{c}{x^2}+1\right)-\frac{2}{5} b^2 c^{5/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right)+\frac{1}{5} b^2 c^{5/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{(1+i) \left(\sqrt{c}-x\right)}{\sqrt{c}-i x}\right)-\frac{2}{5} b^2 c^{5/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c}}{x+\sqrt{c}}\right)+\frac{1}{5} b^2 c^{5/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c} \left(\sqrt{-c}-x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right)+\frac{1}{5} b^2 c^{5/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c} \left(x+\sqrt{-c}\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right)+\frac{1}{5} b^2 c^{5/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{(1-i) \left(x+\sqrt{c}\right)}{\sqrt{c}-i x}\right)+\frac{2}{5} b^2 c^{5/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(2-\frac{2 \sqrt{c}}{x+\sqrt{c}}\right)+\frac{1}{5} i b^2 c^{5/2} \text{Li}_2\left(1-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right)-\frac{1}{5} i b^2 c^{5/2} \text{Li}_2\left(\frac{2 \sqrt{c}}{\sqrt{c}-i x}-1\right)-\frac{1}{10} i b^2 c^{5/2} \text{Li}_2\left(1-\frac{(1+i) \left(\sqrt{c}-x\right)}{\sqrt{c}-i x}\right)+\frac{1}{5} b^2 c^{5/2} \text{Li}_2\left(-\frac{x}{\sqrt{c}}\right)-\frac{1}{5} i b^2 c^{5/2} \text{Li}_2\left(-\frac{i x}{\sqrt{c}}\right)+\frac{1}{5} i b^2 c^{5/2} \text{Li}_2\left(\frac{i x}{\sqrt{c}}\right)-\frac{1}{5} b^2 c^{5/2} \text{Li}_2\left(\frac{x}{\sqrt{c}}\right)+\frac{1}{5} b^2 c^{5/2} \text{Li}_2\left(1-\frac{2 \sqrt{c}}{x+\sqrt{c}}\right)-\frac{1}{5} b^2 c^{5/2} \text{Li}_2\left(\frac{2 \sqrt{c}}{x+\sqrt{c}}-1\right)-\frac{1}{10} b^2 c^{5/2} \text{Li}_2\left(1-\frac{2 \sqrt{c} \left(\sqrt{-c}-x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right)-\frac{1}{10} b^2 c^{5/2} \text{Li}_2\left(1-\frac{2 \sqrt{c} \left(x+\sqrt{-c}\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right)-\frac{1}{10} i b^2 c^{5/2} \text{Li}_2\left(1-\frac{(1-i) \left(x+\sqrt{c}\right)}{\sqrt{c}-i x}\right)",1,"Integrate[x^4*(a + b*ArcTanh[c/x^2])^2, x]","F",-1
177,0,0,1172,3.1715213,"\int x^2 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2 \, dx","Integrate[x^2*(a + b*ArcTanh[c/x^2])^2,x]","\int x^2 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2 \, dx","\frac{1}{12} \left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right)^2 x^3+\frac{1}{12} b^2 \log ^2\left(\frac{c}{x^2}+1\right) x^3+\frac{1}{3} a b \log \left(\frac{c}{x^2}+1\right) x^3-\frac{1}{6} b^2 \log \left(1-\frac{c}{x^2}\right) \log \left(\frac{c}{x^2}+1\right) x^3+\frac{4}{3} a b c x-\frac{2}{3} b^2 c \log \left(1-\frac{c}{x^2}\right) x+\frac{2}{3} b^2 c \log \left(\frac{c}{x^2}+1\right) x+\frac{1}{3} i b^2 c^{3/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)^2+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right)^2+\frac{4}{3} b^2 c^{3/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)-\frac{2}{3} a b c^{3/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)-\frac{4}{3} b^2 c^{3/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right)-\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right)+\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(1-\frac{c}{x^2}\right)-\frac{1}{3} b c^{3/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right)-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{c}{x^2}+1\right)-\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{c}{x^2}+1\right)+\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right)-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{(1+i) \left(\sqrt{c}-x\right)}{\sqrt{c}-i x}\right)-\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c}}{x+\sqrt{c}}\right)+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c} \left(\sqrt{-c}-x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right)+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c} \left(x+\sqrt{-c}\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right)-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{(1-i) \left(x+\sqrt{c}\right)}{\sqrt{c}-i x}\right)+\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(2-\frac{2 \sqrt{c}}{x+\sqrt{c}}\right)-\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left(1-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right)+\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left(\frac{2 \sqrt{c}}{\sqrt{c}-i x}-1\right)+\frac{1}{6} i b^2 c^{3/2} \text{Li}_2\left(1-\frac{(1+i) \left(\sqrt{c}-x\right)}{\sqrt{c}-i x}\right)+\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left(-\frac{x}{\sqrt{c}}\right)+\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left(-\frac{i x}{\sqrt{c}}\right)-\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left(\frac{i x}{\sqrt{c}}\right)-\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left(\frac{x}{\sqrt{c}}\right)+\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left(1-\frac{2 \sqrt{c}}{x+\sqrt{c}}\right)-\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left(\frac{2 \sqrt{c}}{x+\sqrt{c}}-1\right)-\frac{1}{6} b^2 c^{3/2} \text{Li}_2\left(1-\frac{2 \sqrt{c} \left(\sqrt{-c}-x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right)-\frac{1}{6} b^2 c^{3/2} \text{Li}_2\left(1-\frac{2 \sqrt{c} \left(x+\sqrt{-c}\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right)+\frac{1}{6} i b^2 c^{3/2} \text{Li}_2\left(1-\frac{(1-i) \left(x+\sqrt{c}\right)}{\sqrt{c}-i x}\right)",1,"Integrate[x^2*(a + b*ArcTanh[c/x^2])^2, x]","F",-1
178,1,565,1549,3.4800623,"\int \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2 \, dx","Integrate[(a + b*ArcTanh[c/x^2])^2,x]","a^2 x+2 a b x \tanh ^{-1}\left(\frac{c}{x^2}\right)-2 a b x \sqrt{\frac{c}{x^2}} \left(\tan ^{-1}\left(\sqrt{\frac{c}{x^2}}\right)+\tanh ^{-1}\left(\sqrt{\frac{c}{x^2}}\right)\right)-\frac{1}{2} b^2 x \sqrt{\frac{c}{x^2}} \left(-\text{Li}_2\left(\frac{1}{2} \left(1-\sqrt{\frac{c}{x^2}}\right)\right)+\text{Li}_2\left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\sqrt{\frac{c}{x^2}}-1\right)\right)+\text{Li}_2\left(\left(-\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{\frac{c}{x^2}}-1\right)\right)+\text{Li}_2\left(\frac{1}{2} \left(\sqrt{\frac{c}{x^2}}+1\right)\right)-\text{Li}_2\left(\left(\frac{1}{2}-\frac{i}{2}\right) \left(\sqrt{\frac{c}{x^2}}+1\right)\right)-\text{Li}_2\left(\left(\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{\frac{c}{x^2}}+1\right)\right)-\frac{1}{2} i \text{Li}_2\left(-e^{4 i \tan ^{-1}\left(\sqrt{\frac{c}{x^2}}\right)}\right)-\frac{1}{2} \log ^2\left(1-\sqrt{\frac{c}{x^2}}\right)+\frac{1}{2} \log ^2\left(\sqrt{\frac{c}{x^2}}+1\right)+\log (2) \log \left(1-\sqrt{\frac{c}{x^2}}\right)+\log \left(1-\sqrt{\frac{c}{x^2}}\right) \log \left(\left(\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{\frac{c}{x^2}}-i\right)\right)-\log \left(\frac{1}{2} \left((1+i)-(1-i) \sqrt{\frac{c}{x^2}}\right)\right) \log \left(\sqrt{\frac{c}{x^2}}+1\right)-\log \left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\sqrt{\frac{c}{x^2}}+i\right)\right) \log \left(\sqrt{\frac{c}{x^2}}+1\right)-\log (2) \log \left(\sqrt{\frac{c}{x^2}}+1\right)+\log \left(1-\sqrt{\frac{c}{x^2}}\right) \log \left(\frac{1}{2} \left((1-i) \sqrt{\frac{c}{x^2}}+(1+i)\right)\right)-2 i \tan ^{-1}\left(\sqrt{\frac{c}{x^2}}\right)^2-\frac{2 \tanh ^{-1}\left(\frac{c}{x^2}\right)^2}{\sqrt{\frac{c}{x^2}}}+2 \tan ^{-1}\left(\sqrt{\frac{c}{x^2}}\right) \log \left(1+e^{4 i \tan ^{-1}\left(\sqrt{\frac{c}{x^2}}\right)}\right)-2 \log \left(1-\sqrt{\frac{c}{x^2}}\right) \tanh ^{-1}\left(\frac{c}{x^2}\right)+2 \log \left(\sqrt{\frac{c}{x^2}}+1\right) \tanh ^{-1}\left(\frac{c}{x^2}\right)+4 \tan ^{-1}\left(\sqrt{\frac{c}{x^2}}\right) \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)","x a^2+2 b \sqrt{c} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) a-2 b \sqrt{c} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) a-b x \log \left(1-\frac{c}{x^2}\right) a+b x \log \left(\frac{c}{x^2}+1\right) a+\frac{1}{4} b^2 x \log ^2\left(1-\frac{c}{x^2}\right)+\frac{1}{4} b^2 x \log ^2\left(\frac{c}{x^2}+1\right)+\frac{1}{4} b^2 \sqrt{-c} \log ^2\left(\sqrt{-c}-x\right)+\frac{1}{4} b^2 \sqrt{c} \log ^2\left(\sqrt{c}-x\right)-\frac{1}{4} b^2 \sqrt{-c} \log ^2\left(x+\sqrt{-c}\right)-\frac{1}{4} b^2 \sqrt{c} \log ^2\left(x+\sqrt{c}\right)-b^2 \sqrt{c} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(1-\frac{c}{x^2}\right)-b^2 \sqrt{c} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{c}{x^2}+1\right)-\frac{1}{2} b^2 x \log \left(1-\frac{c}{x^2}\right) \log \left(\frac{c}{x^2}+1\right)-\frac{1}{2} b^2 \sqrt{-c} \log \left(\frac{c}{x^2}+1\right) \log \left(\sqrt{-c}-x\right)-\frac{1}{2} b^2 \sqrt{c} \log \left(1-\frac{c}{x^2}\right) \log \left(\sqrt{c}-x\right)-2 b^2 \sqrt{c} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right)+b^2 \sqrt{c} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{(1+i) \left(\sqrt{c}-x\right)}{\sqrt{c}-i x}\right)-b^2 \sqrt{-c} \log \left(\sqrt{-c}-x\right) \log \left(\frac{x}{\sqrt{-c}}\right)-b^2 \sqrt{c} \log \left(\sqrt{c}-x\right) \log \left(\frac{x}{\sqrt{c}}\right)+\frac{1}{2} b^2 \sqrt{-c} \log \left(\frac{c}{x^2}+1\right) \log \left(x+\sqrt{-c}\right)-\frac{1}{2} b^2 \sqrt{-c} \log \left(\frac{\sqrt{-c}-x}{2 \sqrt{-c}}\right) \log \left(x+\sqrt{-c}\right)+b^2 \sqrt{-c} \log \left(-\frac{x}{\sqrt{-c}}\right) \log \left(x+\sqrt{-c}\right)+\frac{1}{2} b^2 \sqrt{-c} \log \left(\sqrt{-c}-x\right) \log \left(\frac{x+\sqrt{-c}}{2 \sqrt{-c}}\right)-2 b^2 \sqrt{c} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c}}{x+\sqrt{c}}\right)+b^2 \sqrt{c} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c} \left(\sqrt{-c}-x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right)+b^2 \sqrt{c} \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c} \left(x+\sqrt{-c}\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right)+\frac{1}{2} b^2 \sqrt{c} \log \left(1-\frac{c}{x^2}\right) \log \left(x+\sqrt{c}\right)-\frac{1}{2} b^2 \sqrt{c} \log \left(\frac{\sqrt{c}-x}{2 \sqrt{c}}\right) \log \left(x+\sqrt{c}\right)+b^2 \sqrt{c} \log \left(-\frac{x}{\sqrt{c}}\right) \log \left(x+\sqrt{c}\right)+\frac{1}{2} b^2 \sqrt{c} \log \left(\sqrt{c}-x\right) \log \left(\frac{x+\sqrt{c}}{2 \sqrt{c}}\right)+b^2 \sqrt{c} \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{(1-i) \left(x+\sqrt{c}\right)}{\sqrt{c}-i x}\right)+i b^2 \sqrt{c} \text{Li}_2\left(1-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right)-\frac{1}{2} i b^2 \sqrt{c} \text{Li}_2\left(1-\frac{(1+i) \left(\sqrt{c}-x\right)}{\sqrt{c}-i x}\right)+b^2 \sqrt{c} \text{Li}_2\left(-\frac{x}{\sqrt{c}}\right)-i b^2 \sqrt{c} \text{Li}_2\left(-\frac{i x}{\sqrt{c}}\right)+i b^2 \sqrt{c} \text{Li}_2\left(\frac{i x}{\sqrt{c}}\right)-b^2 \sqrt{c} \text{Li}_2\left(\frac{x}{\sqrt{c}}\right)-\frac{1}{2} b^2 \sqrt{c} \text{Li}_2\left(\frac{x+\sqrt{c}}{2 \sqrt{c}}\right)+\frac{1}{2} b^2 \sqrt{-c} \text{Li}_2\left(\frac{1}{2} \left(1-\frac{x}{\sqrt{-c}}\right)\right)-b^2 \sqrt{-c} \text{Li}_2\left(1-\frac{x}{\sqrt{-c}}\right)+b^2 \sqrt{-c} \text{Li}_2\left(\frac{x}{\sqrt{-c}}+1\right)-\frac{1}{2} b^2 \sqrt{-c} \text{Li}_2\left(\frac{c-\sqrt{-c} x}{2 c}\right)-b^2 \sqrt{c} \text{Li}_2\left(1-\frac{x}{\sqrt{c}}\right)+\frac{1}{2} b^2 \sqrt{c} \text{Li}_2\left(\frac{1}{2}-\frac{x}{2 \sqrt{c}}\right)+b^2 \sqrt{c} \text{Li}_2\left(\frac{x}{\sqrt{c}}+1\right)+b^2 \sqrt{c} \text{Li}_2\left(1-\frac{2 \sqrt{c}}{x+\sqrt{c}}\right)-\frac{1}{2} b^2 \sqrt{c} \text{Li}_2\left(1-\frac{2 \sqrt{c} \left(\sqrt{-c}-x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right)-\frac{1}{2} b^2 \sqrt{c} \text{Li}_2\left(1-\frac{2 \sqrt{c} \left(x+\sqrt{-c}\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right)-\frac{1}{2} i b^2 \sqrt{c} \text{Li}_2\left(1-\frac{(1-i) \left(x+\sqrt{c}\right)}{\sqrt{c}-i x}\right)",1,"a^2*x - 2*a*b*Sqrt[c/x^2]*x*(ArcTan[Sqrt[c/x^2]] + ArcTanh[Sqrt[c/x^2]]) + 2*a*b*x*ArcTanh[c/x^2] - (b^2*Sqrt[c/x^2]*x*((-2*I)*ArcTan[Sqrt[c/x^2]]^2 + 4*ArcTan[Sqrt[c/x^2]]*ArcTanh[c/x^2] - (2*ArcTanh[c/x^2]^2)/Sqrt[c/x^2] + 2*ArcTan[Sqrt[c/x^2]]*Log[1 + E^((4*I)*ArcTan[Sqrt[c/x^2]])] - 2*ArcTanh[c/x^2]*Log[1 - Sqrt[c/x^2]] + Log[2]*Log[1 - Sqrt[c/x^2]] - Log[1 - Sqrt[c/x^2]]^2/2 + Log[1 - Sqrt[c/x^2]]*Log[(1/2 + I/2)*(-I + Sqrt[c/x^2])] + 2*ArcTanh[c/x^2]*Log[1 + Sqrt[c/x^2]] - Log[2]*Log[1 + Sqrt[c/x^2]] - Log[((1 + I) - (1 - I)*Sqrt[c/x^2])/2]*Log[1 + Sqrt[c/x^2]] - Log[(-1/2 - I/2)*(I + Sqrt[c/x^2])]*Log[1 + Sqrt[c/x^2]] + Log[1 + Sqrt[c/x^2]]^2/2 + Log[1 - Sqrt[c/x^2]]*Log[((1 + I) + (1 - I)*Sqrt[c/x^2])/2] - (I/2)*PolyLog[2, -E^((4*I)*ArcTan[Sqrt[c/x^2]])] - PolyLog[2, (1 - Sqrt[c/x^2])/2] + PolyLog[2, (-1/2 - I/2)*(-1 + Sqrt[c/x^2])] + PolyLog[2, (-1/2 + I/2)*(-1 + Sqrt[c/x^2])] + PolyLog[2, (1 + Sqrt[c/x^2])/2] - PolyLog[2, (1/2 - I/2)*(1 + Sqrt[c/x^2])] - PolyLog[2, (1/2 + I/2)*(1 + Sqrt[c/x^2])]))/2","A",0
179,1,568,1117,3.0061785,"\int \frac{\left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2}{x^2} \, dx","Integrate[(a + b*ArcTanh[c/x^2])^2/x^2,x]","\frac{-2 a^2-4 a b \tanh ^{-1}\left(\frac{c}{x^2}\right)-\frac{4 a b \left(\tan ^{-1}\left(\sqrt{\frac{c}{x^2}}\right)-\tanh ^{-1}\left(\sqrt{\frac{c}{x^2}}\right)\right)}{\sqrt{\frac{c}{x^2}}}+\frac{b^2 \left(-\text{Li}_2\left(\frac{1}{2} \left(1-\sqrt{\frac{c}{x^2}}\right)\right)+\text{Li}_2\left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\sqrt{\frac{c}{x^2}}-1\right)\right)+\text{Li}_2\left(\left(-\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{\frac{c}{x^2}}-1\right)\right)+\text{Li}_2\left(\frac{1}{2} \left(\sqrt{\frac{c}{x^2}}+1\right)\right)-\text{Li}_2\left(\left(\frac{1}{2}-\frac{i}{2}\right) \left(\sqrt{\frac{c}{x^2}}+1\right)\right)-\text{Li}_2\left(\left(\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{\frac{c}{x^2}}+1\right)\right)+\frac{1}{2} i \text{Li}_2\left(-e^{4 i \tan ^{-1}\left(\sqrt{\frac{c}{x^2}}\right)}\right)-\frac{1}{2} \log ^2\left(1-\sqrt{\frac{c}{x^2}}\right)+\frac{1}{2} \log ^2\left(\sqrt{\frac{c}{x^2}}+1\right)+\log (2) \log \left(1-\sqrt{\frac{c}{x^2}}\right)+\log \left(1-\sqrt{\frac{c}{x^2}}\right) \log \left(\left(\frac{1}{2}+\frac{i}{2}\right) \left(\sqrt{\frac{c}{x^2}}-i\right)\right)-\log \left(\frac{1}{2} \left((1+i)-(1-i) \sqrt{\frac{c}{x^2}}\right)\right) \log \left(\sqrt{\frac{c}{x^2}}+1\right)-\log \left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\sqrt{\frac{c}{x^2}}+i\right)\right) \log \left(\sqrt{\frac{c}{x^2}}+1\right)-\log (2) \log \left(\sqrt{\frac{c}{x^2}}+1\right)+\log \left(1-\sqrt{\frac{c}{x^2}}\right) \log \left(\frac{1}{2} \left((1-i) \sqrt{\frac{c}{x^2}}+(1+i)\right)\right)+2 i \tan ^{-1}\left(\sqrt{\frac{c}{x^2}}\right)^2-2 \sqrt{\frac{c}{x^2}} \tanh ^{-1}\left(\frac{c}{x^2}\right)^2-2 \tan ^{-1}\left(\sqrt{\frac{c}{x^2}}\right) \log \left(1+e^{4 i \tan ^{-1}\left(\sqrt{\frac{c}{x^2}}\right)}\right)-2 \log \left(1-\sqrt{\frac{c}{x^2}}\right) \tanh ^{-1}\left(\frac{c}{x^2}\right)+2 \log \left(\sqrt{\frac{c}{x^2}}+1\right) \tanh ^{-1}\left(\frac{c}{x^2}\right)-4 \tan ^{-1}\left(\sqrt{\frac{c}{x^2}}\right) \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)}{\sqrt{\frac{c}{x^2}}}}{2 x}","-\frac{i \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)^2 b^2}{\sqrt{c}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right)^2 b^2}{\sqrt{c}}-\frac{\log ^2\left(\frac{c}{x^2}+1\right) b^2}{4 x}-\frac{2 \cot ^{-1}\left(\frac{x}{\sqrt{c}}\right) b^2}{\sqrt{c}}-\frac{2 \coth ^{-1}\left(\frac{x}{\sqrt{c}}\right) b^2}{\sqrt{c}}-\frac{2 \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) b^2}{\sqrt{c}}+\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) b^2}{\sqrt{c}}+\frac{2 \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right) b^2}{\sqrt{c}}+\frac{\cot ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(1-\frac{c}{x^2}\right) b^2}{\sqrt{c}}-\frac{\log \left(1-\frac{c}{x^2}\right) b^2}{x}+\frac{\coth ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{c}{x^2}+1\right) b^2}{\sqrt{c}}+\frac{\tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{c}{x^2}+1\right) b^2}{\sqrt{c}}+\frac{\log \left(1-\frac{c}{x^2}\right) \log \left(\frac{c}{x^2}+1\right) b^2}{2 x}+\frac{2 \cot ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2}{1-\frac{i \sqrt{c}}{x}}\right) b^2}{\sqrt{c}}-\frac{\cot ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{(1+i) \left(1-\frac{\sqrt{c}}{x}\right)}{1-\frac{i \sqrt{c}}{x}}\right) b^2}{\sqrt{c}}+\frac{2 \coth ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2}{\frac{\sqrt{c}}{x}+1}\right) b^2}{\sqrt{c}}-\frac{\coth ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(-\frac{2 \sqrt{c} \left(1-\frac{\sqrt{-c}}{x}\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\frac{\sqrt{c}}{x}+1\right)}\right) b^2}{\sqrt{c}}-\frac{\coth ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c} \left(\frac{\sqrt{-c}}{x}+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\frac{\sqrt{c}}{x}+1\right)}\right) b^2}{\sqrt{c}}-\frac{\cot ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{(1-i) \left(\frac{\sqrt{c}}{x}+1\right)}{1-\frac{i \sqrt{c}}{x}}\right) b^2}{\sqrt{c}}-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(2-\frac{2 \sqrt{c}}{x+\sqrt{c}}\right) b^2}{\sqrt{c}}-\frac{i \text{Li}_2\left(1-\frac{2}{1-\frac{i \sqrt{c}}{x}}\right) b^2}{\sqrt{c}}+\frac{i \text{Li}_2\left(1-\frac{(1+i) \left(1-\frac{\sqrt{c}}{x}\right)}{1-\frac{i \sqrt{c}}{x}}\right) b^2}{2 \sqrt{c}}-\frac{\text{Li}_2\left(1-\frac{2}{\frac{\sqrt{c}}{x}+1}\right) b^2}{\sqrt{c}}+\frac{\text{Li}_2\left(\frac{2 \sqrt{c} \left(1-\frac{\sqrt{-c}}{x}\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(\frac{\sqrt{c}}{x}+1\right)}+1\right) b^2}{2 \sqrt{c}}+\frac{\text{Li}_2\left(1-\frac{2 \sqrt{c} \left(\frac{\sqrt{-c}}{x}+1\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(\frac{\sqrt{c}}{x}+1\right)}\right) b^2}{2 \sqrt{c}}+\frac{i \text{Li}_2\left(1-\frac{(1-i) \left(\frac{\sqrt{c}}{x}+1\right)}{1-\frac{i \sqrt{c}}{x}}\right) b^2}{2 \sqrt{c}}-\frac{i \text{Li}_2\left(\frac{2 \sqrt{c}}{\sqrt{c}-i x}-1\right) b^2}{\sqrt{c}}+\frac{\text{Li}_2\left(\frac{2 \sqrt{c}}{x+\sqrt{c}}-1\right) b^2}{\sqrt{c}}-\frac{2 a \cot ^{-1}\left(\frac{x}{\sqrt{c}}\right) b}{\sqrt{c}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right) b}{\sqrt{c}}-\frac{\left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right) b}{x}-\frac{a \log \left(\frac{c}{x^2}+1\right) b}{x}+\frac{2 a b}{x}-\frac{\left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right)^2}{4 x}",1,"(-2*a^2 - (4*a*b*(ArcTan[Sqrt[c/x^2]] - ArcTanh[Sqrt[c/x^2]]))/Sqrt[c/x^2] - 4*a*b*ArcTanh[c/x^2] + (b^2*((2*I)*ArcTan[Sqrt[c/x^2]]^2 - 4*ArcTan[Sqrt[c/x^2]]*ArcTanh[c/x^2] - 2*Sqrt[c/x^2]*ArcTanh[c/x^2]^2 - 2*ArcTan[Sqrt[c/x^2]]*Log[1 + E^((4*I)*ArcTan[Sqrt[c/x^2]])] - 2*ArcTanh[c/x^2]*Log[1 - Sqrt[c/x^2]] + Log[2]*Log[1 - Sqrt[c/x^2]] - Log[1 - Sqrt[c/x^2]]^2/2 + Log[1 - Sqrt[c/x^2]]*Log[(1/2 + I/2)*(-I + Sqrt[c/x^2])] + 2*ArcTanh[c/x^2]*Log[1 + Sqrt[c/x^2]] - Log[2]*Log[1 + Sqrt[c/x^2]] - Log[((1 + I) - (1 - I)*Sqrt[c/x^2])/2]*Log[1 + Sqrt[c/x^2]] - Log[(-1/2 - I/2)*(I + Sqrt[c/x^2])]*Log[1 + Sqrt[c/x^2]] + Log[1 + Sqrt[c/x^2]]^2/2 + Log[1 - Sqrt[c/x^2]]*Log[((1 + I) + (1 - I)*Sqrt[c/x^2])/2] + (I/2)*PolyLog[2, -E^((4*I)*ArcTan[Sqrt[c/x^2]])] - PolyLog[2, (1 - Sqrt[c/x^2])/2] + PolyLog[2, (-1/2 - I/2)*(-1 + Sqrt[c/x^2])] + PolyLog[2, (-1/2 + I/2)*(-1 + Sqrt[c/x^2])] + PolyLog[2, (1 + Sqrt[c/x^2])/2] - PolyLog[2, (1/2 - I/2)*(1 + Sqrt[c/x^2])] - PolyLog[2, (1/2 + I/2)*(1 + Sqrt[c/x^2])]))/Sqrt[c/x^2])/(2*x)","A",0
180,0,0,1263,2.9061355,"\int \frac{\left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2}{x^4} \, dx","Integrate[(a + b*ArcTanh[c/x^2])^2/x^4,x]","\int \frac{\left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2}{x^4} \, dx","\frac{i \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)^2 b^2}{3 c^{3/2}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right)^2 b^2}{3 c^{3/2}}-\frac{\log ^2\left(\frac{c}{x^2}+1\right) b^2}{12 x^3}+\frac{4 \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) b^2}{3 c^{3/2}}+\frac{4 \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) b^2}{3 c^{3/2}}-\frac{2 \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right) b^2}{3 c^{3/2}}+\frac{\tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(1-\frac{c}{x^2}\right) b^2}{3 c^{3/2}}+\frac{\log \left(1-\frac{c}{x^2}\right) b^2}{3 c x}-\frac{\log \left(1-\frac{c}{x^2}\right) b^2}{9 x^3}-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{c}{x^2}+1\right) b^2}{3 c^{3/2}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{c}{x^2}+1\right) b^2}{3 c^{3/2}}+\frac{\log \left(1-\frac{c}{x^2}\right) \log \left(\frac{c}{x^2}+1\right) b^2}{6 x^3}-\frac{2 \log \left(\frac{c}{x^2}+1\right) b^2}{3 c x}+\frac{2 \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right) b^2}{3 c^{3/2}}-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{(1+i) \left(\sqrt{c}-x\right)}{\sqrt{c}-i x}\right) b^2}{3 c^{3/2}}+\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c}}{x+\sqrt{c}}\right) b^2}{3 c^{3/2}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c} \left(\sqrt{-c}-x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right) b^2}{3 c^{3/2}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c} \left(x+\sqrt{-c}\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right) b^2}{3 c^{3/2}}-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{(1-i) \left(x+\sqrt{c}\right)}{\sqrt{c}-i x}\right) b^2}{3 c^{3/2}}-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(2-\frac{2 \sqrt{c}}{x+\sqrt{c}}\right) b^2}{3 c^{3/2}}-\frac{i \text{Li}_2\left(1-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right) b^2}{3 c^{3/2}}+\frac{i \text{Li}_2\left(\frac{2 \sqrt{c}}{\sqrt{c}-i x}-1\right) b^2}{3 c^{3/2}}+\frac{i \text{Li}_2\left(1-\frac{(1+i) \left(\sqrt{c}-x\right)}{\sqrt{c}-i x}\right) b^2}{6 c^{3/2}}-\frac{\text{Li}_2\left(-\frac{x}{\sqrt{c}}\right) b^2}{3 c^{3/2}}+\frac{i \text{Li}_2\left(-\frac{i x}{\sqrt{c}}\right) b^2}{3 c^{3/2}}-\frac{i \text{Li}_2\left(\frac{i x}{\sqrt{c}}\right) b^2}{3 c^{3/2}}+\frac{\text{Li}_2\left(\frac{x}{\sqrt{c}}\right) b^2}{3 c^{3/2}}-\frac{\text{Li}_2\left(1-\frac{2 \sqrt{c}}{x+\sqrt{c}}\right) b^2}{3 c^{3/2}}+\frac{\text{Li}_2\left(\frac{2 \sqrt{c}}{x+\sqrt{c}}-1\right) b^2}{3 c^{3/2}}+\frac{\text{Li}_2\left(1-\frac{2 \sqrt{c} \left(\sqrt{-c}-x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right) b^2}{6 c^{3/2}}+\frac{\text{Li}_2\left(1-\frac{2 \sqrt{c} \left(x+\sqrt{-c}\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right) b^2}{6 c^{3/2}}+\frac{i \text{Li}_2\left(1-\frac{(1-i) \left(x+\sqrt{c}\right)}{\sqrt{c}-i x}\right) b^2}{6 c^{3/2}}-\frac{2 a \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) b}{3 c^{3/2}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right) b}{3 c^{3/2}}-\frac{\left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right) b}{3 c x}-\frac{\left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right) b}{9 x^3}-\frac{a \log \left(\frac{c}{x^2}+1\right) b}{3 x^3}-\frac{2 a b}{3 c x}+\frac{2 a b}{9 x^3}-\frac{\left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right)^2}{12 x^3}",1,"Integrate[(a + b*ArcTanh[c/x^2])^2/x^4, x]","F",-1
181,0,0,1337,2.7729572,"\int \frac{\left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2}{x^6} \, dx","Integrate[(a + b*ArcTanh[c/x^2])^2/x^6,x]","\int \frac{\left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2}{x^6} \, dx","-\frac{i \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right)^2 b^2}{5 c^{5/2}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right)^2 b^2}{5 c^{5/2}}-\frac{\log ^2\left(\frac{c}{x^2}+1\right) b^2}{20 x^5}-\frac{4 \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) b^2}{15 c^{5/2}}+\frac{4 \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) b^2}{15 c^{5/2}}+\frac{2 \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right) b^2}{5 c^{5/2}}-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(1-\frac{c}{x^2}\right) b^2}{5 c^{5/2}}-\frac{\log \left(1-\frac{c}{x^2}\right) b^2}{5 c^2 x}+\frac{\log \left(1-\frac{c}{x^2}\right) b^2}{15 c x^3}-\frac{\log \left(1-\frac{c}{x^2}\right) b^2}{25 x^5}+\frac{\tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{c}{x^2}+1\right) b^2}{5 c^{5/2}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{c}{x^2}+1\right) b^2}{5 c^{5/2}}+\frac{\log \left(1-\frac{c}{x^2}\right) \log \left(\frac{c}{x^2}+1\right) b^2}{10 x^5}-\frac{2 \log \left(\frac{c}{x^2}+1\right) b^2}{15 c x^3}-\frac{2 \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right) b^2}{5 c^{5/2}}+\frac{\tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{(1+i) \left(\sqrt{c}-x\right)}{\sqrt{c}-i x}\right) b^2}{5 c^{5/2}}+\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c}}{x+\sqrt{c}}\right) b^2}{5 c^{5/2}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c} \left(\sqrt{-c}-x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right) b^2}{5 c^{5/2}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c} \left(x+\sqrt{-c}\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right) b^2}{5 c^{5/2}}+\frac{\tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(\frac{(1-i) \left(x+\sqrt{c}\right)}{\sqrt{c}-i x}\right) b^2}{5 c^{5/2}}-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \log \left(2-\frac{2 \sqrt{c}}{x+\sqrt{c}}\right) b^2}{5 c^{5/2}}+\frac{i \text{Li}_2\left(1-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right) b^2}{5 c^{5/2}}-\frac{i \text{Li}_2\left(\frac{2 \sqrt{c}}{\sqrt{c}-i x}-1\right) b^2}{5 c^{5/2}}-\frac{i \text{Li}_2\left(1-\frac{(1+i) \left(\sqrt{c}-x\right)}{\sqrt{c}-i x}\right) b^2}{10 c^{5/2}}-\frac{\text{Li}_2\left(-\frac{x}{\sqrt{c}}\right) b^2}{5 c^{5/2}}-\frac{i \text{Li}_2\left(-\frac{i x}{\sqrt{c}}\right) b^2}{5 c^{5/2}}+\frac{i \text{Li}_2\left(\frac{i x}{\sqrt{c}}\right) b^2}{5 c^{5/2}}+\frac{\text{Li}_2\left(\frac{x}{\sqrt{c}}\right) b^2}{5 c^{5/2}}-\frac{\text{Li}_2\left(1-\frac{2 \sqrt{c}}{x+\sqrt{c}}\right) b^2}{5 c^{5/2}}+\frac{\text{Li}_2\left(\frac{2 \sqrt{c}}{x+\sqrt{c}}-1\right) b^2}{5 c^{5/2}}+\frac{\text{Li}_2\left(1-\frac{2 \sqrt{c} \left(\sqrt{-c}-x\right)}{\left(\sqrt{-c}-\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right) b^2}{10 c^{5/2}}+\frac{\text{Li}_2\left(1-\frac{2 \sqrt{c} \left(x+\sqrt{-c}\right)}{\left(\sqrt{-c}+\sqrt{c}\right) \left(x+\sqrt{c}\right)}\right) b^2}{10 c^{5/2}}-\frac{i \text{Li}_2\left(1-\frac{(1-i) \left(x+\sqrt{c}\right)}{\sqrt{c}-i x}\right) b^2}{10 c^{5/2}}-\frac{8 b^2}{15 c^2 x}+\frac{2 a \tan ^{-1}\left(\frac{x}{\sqrt{c}}\right) b}{5 c^{5/2}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{c}}\right) \left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right) b}{5 c^{5/2}}-\frac{\left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right) b}{5 c^2 x}-\frac{\left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right) b}{15 c x^3}-\frac{\left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right) b}{25 x^5}-\frac{a \log \left(\frac{c}{x^2}+1\right) b}{5 x^5}+\frac{2 a b}{5 c^2 x}-\frac{2 a b}{15 c x^3}+\frac{2 a b}{25 x^5}-\frac{\left(2 a-b \log \left(1-\frac{c}{x^2}\right)\right)^2}{20 x^5}",1,"Integrate[(a + b*ArcTanh[c/x^2])^2/x^6, x]","F",-1
182,0,0,21,2.6682995,"\int (d x)^m \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^3 \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c/x^2])^3,x]","\int (d x)^m \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^3 \, dx","\text{Int}\left((d x)^m \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^3,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTanh[c/x^2])^3, x]","A",-1
183,0,0,21,1.7209901,"\int (d x)^m \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2 \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c/x^2])^2,x]","\int (d x)^m \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2 \, dx","\text{Int}\left((d x)^m \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTanh[c/x^2])^2, x]","A",-1
184,1,68,75,0.0643891,"\int (d x)^m \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right) \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c/x^2]),x]","\frac{(d x)^m \left((m-1) x^2 \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)+2 b c \, _2F_1\left(1,\frac{1}{4}-\frac{m}{4};\frac{5}{4}-\frac{m}{4};\frac{c^2}{x^4}\right)\right)}{(m-1) (m+1) x}","\frac{(d x)^{m+1} \left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)}{d (m+1)}-\frac{2 b c d (d x)^{m-1} \, _2F_1\left(1,\frac{1-m}{4};\frac{5-m}{4};\frac{c^2}{x^4}\right)}{1-m^2}",1,"((d*x)^m*((-1 + m)*x^2*(a + b*ArcTanh[c/x^2]) + 2*b*c*Hypergeometric2F1[1, 1/4 - m/4, 5/4 - m/4, c^2/x^4]))/((-1 + m)*(1 + m)*x)","A",1
185,0,0,21,0.409456,"\int \frac{(d x)^m}{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)} \, dx","Integrate[(d*x)^m/(a + b*ArcTanh[c/x^2]),x]","\int \frac{(d x)^m}{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)} \, dx","\text{Int}\left(\frac{(d x)^m}{a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcTanh[c/x^2]), x]","A",-1
186,0,0,21,0.4110841,"\int \frac{(d x)^m}{\left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2} \, dx","Integrate[(d*x)^m/(a + b*ArcTanh[c/x^2])^2,x]","\int \frac{(d x)^m}{\left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2} \, dx","\text{Int}\left(\frac{(d x)^m}{\left(a+b \tanh ^{-1}\left(\frac{c}{x^2}\right)\right)^2},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcTanh[c/x^2])^2, x]","A",-1
187,1,114,88,0.0336084,"\int x^3 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \, dx","Integrate[x^3*(a + b*ArcTanh[c*Sqrt[x]]),x]","\frac{a x^4}{4}+\frac{b \log \left(1-c \sqrt{x}\right)}{8 c^8}-\frac{b \log \left(c \sqrt{x}+1\right)}{8 c^8}+\frac{b \sqrt{x}}{4 c^7}+\frac{b x^{3/2}}{12 c^5}+\frac{b x^{5/2}}{20 c^3}+\frac{b x^{7/2}}{28 c}+\frac{1}{4} b x^4 \tanh ^{-1}\left(c \sqrt{x}\right)","\frac{1}{4} x^4 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)-\frac{b \tanh ^{-1}\left(c \sqrt{x}\right)}{4 c^8}+\frac{b \sqrt{x}}{4 c^7}+\frac{b x^{3/2}}{12 c^5}+\frac{b x^{5/2}}{20 c^3}+\frac{b x^{7/2}}{28 c}",1,"(b*Sqrt[x])/(4*c^7) + (b*x^(3/2))/(12*c^5) + (b*x^(5/2))/(20*c^3) + (b*x^(7/2))/(28*c) + (a*x^4)/4 + (b*x^4*ArcTanh[c*Sqrt[x]])/4 + (b*Log[1 - c*Sqrt[x]])/(8*c^8) - (b*Log[1 + c*Sqrt[x]])/(8*c^8)","A",1
188,1,101,75,0.0239548,"\int x^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \, dx","Integrate[x^2*(a + b*ArcTanh[c*Sqrt[x]]),x]","\frac{a x^3}{3}+\frac{b \log \left(1-c \sqrt{x}\right)}{6 c^6}-\frac{b \log \left(c \sqrt{x}+1\right)}{6 c^6}+\frac{b \sqrt{x}}{3 c^5}+\frac{b x^{3/2}}{9 c^3}+\frac{b x^{5/2}}{15 c}+\frac{1}{3} b x^3 \tanh ^{-1}\left(c \sqrt{x}\right)","\frac{1}{3} x^3 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)-\frac{b \tanh ^{-1}\left(c \sqrt{x}\right)}{3 c^6}+\frac{b \sqrt{x}}{3 c^5}+\frac{b x^{3/2}}{9 c^3}+\frac{b x^{5/2}}{15 c}",1,"(b*Sqrt[x])/(3*c^5) + (b*x^(3/2))/(9*c^3) + (b*x^(5/2))/(15*c) + (a*x^3)/3 + (b*x^3*ArcTanh[c*Sqrt[x]])/3 + (b*Log[1 - c*Sqrt[x]])/(6*c^6) - (b*Log[1 + c*Sqrt[x]])/(6*c^6)","A",1
189,1,88,62,0.0210009,"\int x \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \, dx","Integrate[x*(a + b*ArcTanh[c*Sqrt[x]]),x]","\frac{a x^2}{2}+\frac{b \log \left(1-c \sqrt{x}\right)}{4 c^4}-\frac{b \log \left(c \sqrt{x}+1\right)}{4 c^4}+\frac{b \sqrt{x}}{2 c^3}+\frac{b x^{3/2}}{6 c}+\frac{1}{2} b x^2 \tanh ^{-1}\left(c \sqrt{x}\right)","\frac{1}{2} x^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)-\frac{b \tanh ^{-1}\left(c \sqrt{x}\right)}{2 c^4}+\frac{b \sqrt{x}}{2 c^3}+\frac{b x^{3/2}}{6 c}",1,"(b*Sqrt[x])/(2*c^3) + (b*x^(3/2))/(6*c) + (a*x^2)/2 + (b*x^2*ArcTanh[c*Sqrt[x]])/2 + (b*Log[1 - c*Sqrt[x]])/(4*c^4) - (b*Log[1 + c*Sqrt[x]])/(4*c^4)","A",1
190,1,42,39,0.0261452,"\int \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right) \, dx","Integrate[a + b*ArcTanh[c*Sqrt[x]],x]","a x-b c \left(\frac{\tanh ^{-1}\left(c \sqrt{x}\right)}{c^3}-\frac{\sqrt{x}}{c^2}\right)+b x \tanh ^{-1}\left(c \sqrt{x}\right)","a x-\frac{b \tanh ^{-1}\left(c \sqrt{x}\right)}{c^2}+\frac{b \sqrt{x}}{c}+b x \tanh ^{-1}\left(c \sqrt{x}\right)",1,"a*x + b*x*ArcTanh[c*Sqrt[x]] - b*c*(-(Sqrt[x]/c^2) + ArcTanh[c*Sqrt[x]]/c^3)","A",1
191,1,29,29,0.0120775,"\int \frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{x} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])/x,x]","a \log (x)-b \text{Li}_2\left(-c \sqrt{x}\right)+b \text{Li}_2\left(c \sqrt{x}\right)","a \log (x)-b \text{Li}_2\left(-c \sqrt{x}\right)+b \text{Li}_2\left(c \sqrt{x}\right)",1,"a*Log[x] - b*PolyLog[2, -(c*Sqrt[x])] + b*PolyLog[2, c*Sqrt[x]]","A",1
192,1,67,40,0.0250132,"\int \frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{x^2} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])/x^2,x]","-\frac{a}{x}-\frac{1}{2} b c^2 \log \left(1-c \sqrt{x}\right)+\frac{1}{2} b c^2 \log \left(c \sqrt{x}+1\right)-\frac{b c}{\sqrt{x}}-\frac{b \tanh ^{-1}\left(c \sqrt{x}\right)}{x}","-\frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{x}+b c^2 \tanh ^{-1}\left(c \sqrt{x}\right)-\frac{b c}{\sqrt{x}}",1,"-(a/x) - (b*c)/Sqrt[x] - (b*ArcTanh[c*Sqrt[x]])/x - (b*c^2*Log[1 - c*Sqrt[x]])/2 + (b*c^2*Log[1 + c*Sqrt[x]])/2","A",1
193,1,86,60,0.0281045,"\int \frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{x^3} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])/x^3,x]","-\frac{a}{2 x^2}-\frac{1}{4} b c^4 \log \left(1-c \sqrt{x}\right)+\frac{1}{4} b c^4 \log \left(c \sqrt{x}+1\right)-\frac{b c^3}{2 \sqrt{x}}-\frac{b c}{6 x^{3/2}}-\frac{b \tanh ^{-1}\left(c \sqrt{x}\right)}{2 x^2}","-\frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{2 x^2}+\frac{1}{2} b c^4 \tanh ^{-1}\left(c \sqrt{x}\right)-\frac{b c^3}{2 \sqrt{x}}-\frac{b c}{6 x^{3/2}}",1,"-1/2*a/x^2 - (b*c)/(6*x^(3/2)) - (b*c^3)/(2*Sqrt[x]) - (b*ArcTanh[c*Sqrt[x]])/(2*x^2) - (b*c^4*Log[1 - c*Sqrt[x]])/4 + (b*c^4*Log[1 + c*Sqrt[x]])/4","A",1
194,1,99,73,0.0310808,"\int \frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{x^4} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])/x^4,x]","-\frac{a}{3 x^3}-\frac{1}{6} b c^6 \log \left(1-c \sqrt{x}\right)+\frac{1}{6} b c^6 \log \left(c \sqrt{x}+1\right)-\frac{b c^5}{3 \sqrt{x}}-\frac{b c^3}{9 x^{3/2}}-\frac{b c}{15 x^{5/2}}-\frac{b \tanh ^{-1}\left(c \sqrt{x}\right)}{3 x^3}","-\frac{a+b \tanh ^{-1}\left(c \sqrt{x}\right)}{3 x^3}+\frac{1}{3} b c^6 \tanh ^{-1}\left(c \sqrt{x}\right)-\frac{b c^5}{3 \sqrt{x}}-\frac{b c^3}{9 x^{3/2}}-\frac{b c}{15 x^{5/2}}",1,"-1/3*a/x^3 - (b*c)/(15*x^(5/2)) - (b*c^3)/(9*x^(3/2)) - (b*c^5)/(3*Sqrt[x]) - (b*ArcTanh[c*Sqrt[x]])/(3*x^3) - (b*c^6*Log[1 - c*Sqrt[x]])/6 + (b*c^6*Log[1 + c*Sqrt[x]])/6","A",1
195,1,224,211,0.117141,"\int x^3 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2 \, dx","Integrate[x^3*(a + b*ArcTanh[c*Sqrt[x]])^2,x]","\frac{105 a^2 c^8 x^4+30 a b c^7 x^{7/2}+42 a b c^5 x^{5/2}+70 a b c^3 x^{3/2}+2 b c \sqrt{x} \tanh ^{-1}\left(c \sqrt{x}\right) \left(105 a c^7 x^{7/2}+b \left(15 c^6 x^3+21 c^4 x^2+35 c^2 x+105\right)\right)+210 a b c \sqrt{x}+b (105 a+176 b) \log \left(1-c \sqrt{x}\right)-105 a b \log \left(c \sqrt{x}+1\right)+105 b^2 \left(c^8 x^4-1\right) \tanh ^{-1}\left(c \sqrt{x}\right)^2+5 b^2 c^6 x^3+18 b^2 c^4 x^2+71 b^2 c^2 x+176 b^2 \log \left(c \sqrt{x}+1\right)}{420 c^8}","-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{4 c^8}+\frac{a b \sqrt{x}}{2 c^7}+\frac{b x^{3/2} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{6 c^5}+\frac{b x^{5/2} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{10 c^3}+\frac{b x^{7/2} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{14 c}+\frac{1}{4} x^4 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2+\frac{b^2 \sqrt{x} \tanh ^{-1}\left(c \sqrt{x}\right)}{2 c^7}+\frac{71 b^2 x}{420 c^6}+\frac{3 b^2 x^2}{70 c^4}+\frac{b^2 x^3}{84 c^2}+\frac{44 b^2 \log \left(1-c^2 x\right)}{105 c^8}",1,"(210*a*b*c*Sqrt[x] + 71*b^2*c^2*x + 70*a*b*c^3*x^(3/2) + 18*b^2*c^4*x^2 + 42*a*b*c^5*x^(5/2) + 5*b^2*c^6*x^3 + 30*a*b*c^7*x^(7/2) + 105*a^2*c^8*x^4 + 2*b*c*Sqrt[x]*(105*a*c^7*x^(7/2) + b*(105 + 35*c^2*x + 21*c^4*x^2 + 15*c^6*x^3))*ArcTanh[c*Sqrt[x]] + 105*b^2*(-1 + c^8*x^4)*ArcTanh[c*Sqrt[x]]^2 + b*(105*a + 176*b)*Log[1 - c*Sqrt[x]] - 105*a*b*Log[1 + c*Sqrt[x]] + 176*b^2*Log[1 + c*Sqrt[x]])/(420*c^8)","A",1
196,1,194,173,0.1076658,"\int x^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2 \, dx","Integrate[x^2*(a + b*ArcTanh[c*Sqrt[x]])^2,x]","\frac{30 a^2 c^6 x^3+12 a b c^5 x^{5/2}+20 a b c^3 x^{3/2}+4 b c \sqrt{x} \tanh ^{-1}\left(c \sqrt{x}\right) \left(15 a c^5 x^{5/2}+b \left(3 c^4 x^2+5 c^2 x+15\right)\right)+60 a b c \sqrt{x}+2 b (15 a+23 b) \log \left(1-c \sqrt{x}\right)-30 a b \log \left(c \sqrt{x}+1\right)+30 b^2 \left(c^6 x^3-1\right) \tanh ^{-1}\left(c \sqrt{x}\right)^2+3 b^2 c^4 x^2+16 b^2 c^2 x+46 b^2 \log \left(c \sqrt{x}+1\right)}{90 c^6}","-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{3 c^6}+\frac{2 a b \sqrt{x}}{3 c^5}+\frac{2 b x^{3/2} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{9 c^3}+\frac{2 b x^{5/2} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{15 c}+\frac{1}{3} x^3 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2+\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(c \sqrt{x}\right)}{3 c^5}+\frac{8 b^2 x}{45 c^4}+\frac{b^2 x^2}{30 c^2}+\frac{23 b^2 \log \left(1-c^2 x\right)}{45 c^6}",1,"(60*a*b*c*Sqrt[x] + 16*b^2*c^2*x + 20*a*b*c^3*x^(3/2) + 3*b^2*c^4*x^2 + 12*a*b*c^5*x^(5/2) + 30*a^2*c^6*x^3 + 4*b*c*Sqrt[x]*(15*a*c^5*x^(5/2) + b*(15 + 5*c^2*x + 3*c^4*x^2))*ArcTanh[c*Sqrt[x]] + 30*b^2*(-1 + c^6*x^3)*ArcTanh[c*Sqrt[x]]^2 + 2*b*(15*a + 23*b)*Log[1 - c*Sqrt[x]] - 30*a*b*Log[1 + c*Sqrt[x]] + 46*b^2*Log[1 + c*Sqrt[x]])/(90*c^6)","A",1
197,1,160,129,0.0918539,"\int x \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2 \, dx","Integrate[x*(a + b*ArcTanh[c*Sqrt[x]])^2,x]","\frac{3 a^2 c^4 x^2+2 a b c^3 x^{3/2}+2 b c \sqrt{x} \tanh ^{-1}\left(c \sqrt{x}\right) \left(3 a c^3 x^{3/2}+b \left(c^2 x+3\right)\right)+6 a b c \sqrt{x}+b (3 a+4 b) \log \left(1-c \sqrt{x}\right)-3 a b \log \left(c \sqrt{x}+1\right)+3 b^2 \left(c^4 x^2-1\right) \tanh ^{-1}\left(c \sqrt{x}\right)^2+b^2 c^2 x+4 b^2 \log \left(c \sqrt{x}+1\right)}{6 c^4}","-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{2 c^4}+\frac{a b \sqrt{x}}{c^3}+\frac{b x^{3/2} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{3 c}+\frac{1}{2} x^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2+\frac{b^2 \sqrt{x} \tanh ^{-1}\left(c \sqrt{x}\right)}{c^3}+\frac{b^2 x}{6 c^2}+\frac{2 b^2 \log \left(1-c^2 x\right)}{3 c^4}",1,"(6*a*b*c*Sqrt[x] + b^2*c^2*x + 2*a*b*c^3*x^(3/2) + 3*a^2*c^4*x^2 + 2*b*c*Sqrt[x]*(3*a*c^3*x^(3/2) + b*(3 + c^2*x))*ArcTanh[c*Sqrt[x]] + 3*b^2*(-1 + c^4*x^2)*ArcTanh[c*Sqrt[x]]^2 + b*(3*a + 4*b)*Log[1 - c*Sqrt[x]] - 3*a*b*Log[1 + c*Sqrt[x]] + 4*b^2*Log[1 + c*Sqrt[x]])/(6*c^4)","A",1
198,1,115,85,0.0647864,"\int \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2 \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])^2,x]","\frac{a^2 c^2 x+2 a b c \sqrt{x}+b (a+b) \log \left(1-c \sqrt{x}\right)-a b \log \left(c \sqrt{x}+1\right)+2 b c \sqrt{x} \tanh ^{-1}\left(c \sqrt{x}\right) \left(a c \sqrt{x}+b\right)+b^2 \left(c^2 x-1\right) \tanh ^{-1}\left(c \sqrt{x}\right)^2+b^2 \log \left(c \sqrt{x}+1\right)}{c^2}","-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{c^2}+\frac{2 a b \sqrt{x}}{c}+x \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2+\frac{b^2 \log \left(1-c^2 x\right)}{c^2}+\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left(c \sqrt{x}\right)}{c}",1,"(2*a*b*c*Sqrt[x] + a^2*c^2*x + 2*b*c*(b + a*c*Sqrt[x])*Sqrt[x]*ArcTanh[c*Sqrt[x]] + b^2*(-1 + c^2*x)*ArcTanh[c*Sqrt[x]]^2 + b*(a + b)*Log[1 - c*Sqrt[x]] - a*b*Log[1 + c*Sqrt[x]] + b^2*Log[1 + c*Sqrt[x]])/c^2","A",1
199,1,164,145,0.0888266,"\int \frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{x} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])^2/x,x]","4 \tanh ^{-1}\left(\frac{2}{c \sqrt{x}-1}+1\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2-b \left(-2 \text{Li}_2\left(\frac{\sqrt{x} c+1}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)+2 \text{Li}_2\left(\frac{\sqrt{x} c+1}{c \sqrt{x}-1}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)+b \left(\text{Li}_3\left(\frac{\sqrt{x} c+1}{1-c \sqrt{x}}\right)-\text{Li}_3\left(\frac{\sqrt{x} c+1}{c \sqrt{x}-1}\right)\right)\right)","-2 b \text{Li}_2\left(1-\frac{2}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)+2 b \text{Li}_2\left(\frac{2}{1-c \sqrt{x}}-1\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)+4 \tanh ^{-1}\left(1-\frac{2}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2+b^2 \text{Li}_3\left(1-\frac{2}{1-c \sqrt{x}}\right)-b^2 \text{Li}_3\left(\frac{2}{1-c \sqrt{x}}-1\right)",1,"4*ArcTanh[1 + 2/(-1 + c*Sqrt[x])]*(a + b*ArcTanh[c*Sqrt[x]])^2 - b*(-2*(a + b*ArcTanh[c*Sqrt[x]])*PolyLog[2, (1 + c*Sqrt[x])/(1 - c*Sqrt[x])] + 2*(a + b*ArcTanh[c*Sqrt[x]])*PolyLog[2, (1 + c*Sqrt[x])/(-1 + c*Sqrt[x])] + b*(PolyLog[3, (1 + c*Sqrt[x])/(1 - c*Sqrt[x])] - PolyLog[3, (1 + c*Sqrt[x])/(-1 + c*Sqrt[x])]))","A",1
200,1,129,85,0.1135929,"\int \frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{x^2} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])^2/x^2,x]","-\frac{a^2-a b c^2 x \log \left(c \sqrt{x}+1\right)+b c^2 x (a+b) \log \left(1-c \sqrt{x}\right)+2 a b c \sqrt{x}+2 b \tanh ^{-1}\left(c \sqrt{x}\right) \left(a+b c \sqrt{x}\right)+b^2 c^2 x \log \left(c \sqrt{x}+1\right)-b^2 c^2 x \log (x)-b^2 \left(c^2 x-1\right) \tanh ^{-1}\left(c \sqrt{x}\right)^2}{x}","c^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2-\frac{2 b c \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{\sqrt{x}}-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{x}+b^2 c^2 \log (x)-b^2 c^2 \log \left(1-c^2 x\right)",1,"-((a^2 + 2*a*b*c*Sqrt[x] + 2*b*(a + b*c*Sqrt[x])*ArcTanh[c*Sqrt[x]] - b^2*(-1 + c^2*x)*ArcTanh[c*Sqrt[x]]^2 + b*(a + b)*c^2*x*Log[1 - c*Sqrt[x]] - a*b*c^2*x*Log[1 + c*Sqrt[x]] + b^2*c^2*x*Log[1 + c*Sqrt[x]] - b^2*c^2*x*Log[x])/x)","A",1
201,1,178,133,0.1318236,"\int \frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{x^3} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])^2/x^3,x]","-\frac{3 a^2+b c^4 x^2 (3 a+4 b) \log \left(1-c \sqrt{x}\right)-3 a b c^4 x^2 \log \left(c \sqrt{x}+1\right)+6 a b c^3 x^{3/2}+2 b \tanh ^{-1}\left(c \sqrt{x}\right) \left(3 a+b c \sqrt{x} \left(3 c^2 x+1\right)\right)+2 a b c \sqrt{x}+4 b^2 c^4 x^2 \log \left(c \sqrt{x}+1\right)-4 b^2 c^4 x^2 \log (x)-3 b^2 \left(c^4 x^2-1\right) \tanh ^{-1}\left(c \sqrt{x}\right)^2+b^2 c^2 x}{6 x^2}","\frac{1}{2} c^4 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2-\frac{b c^3 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{\sqrt{x}}-\frac{b c \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{3 x^{3/2}}-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{2 x^2}+\frac{2}{3} b^2 c^4 \log (x)-\frac{b^2 c^2}{6 x}-\frac{2}{3} b^2 c^4 \log \left(1-c^2 x\right)",1,"-1/6*(3*a^2 + 2*a*b*c*Sqrt[x] + b^2*c^2*x + 6*a*b*c^3*x^(3/2) + 2*b*(3*a + b*c*Sqrt[x]*(1 + 3*c^2*x))*ArcTanh[c*Sqrt[x]] - 3*b^2*(-1 + c^4*x^2)*ArcTanh[c*Sqrt[x]]^2 + b*(3*a + 4*b)*c^4*x^2*Log[1 - c*Sqrt[x]] - 3*a*b*c^4*x^2*Log[1 + c*Sqrt[x]] + 4*b^2*c^4*x^2*Log[1 + c*Sqrt[x]] - 4*b^2*c^4*x^2*Log[x])/x^2","A",1
202,1,418,374,1.2667716,"\int x^3 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3 \, dx","Integrate[x^3*(a + b*ArcTanh[c*Sqrt[x]])^3,x]","\frac{210 a^3 c^8 x^4+6 b \tanh ^{-1}\left(c \sqrt{x}\right) \left(105 a^2 c^8 x^4+2 a b c \sqrt{x} \left(15 c^6 x^3+21 c^4 x^2+35 c^2 x+105\right)+b^2 \left(5 c^6 x^3+18 c^4 x^2+71 c^2 x-94\right)-352 b^2 \log \left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}+1\right)\right)+90 a^2 b c^7 x^{7/2}+126 a^2 b c^5 x^{5/2}+210 a^2 b c^3 x^{3/2}+630 a^2 b c \sqrt{x}+315 a^2 b \log \left(1-c \sqrt{x}\right)-315 a^2 b \log \left(c \sqrt{x}+1\right)+30 a b^2 c^6 x^3+108 a b^2 c^4 x^2+426 a b^2 c^2 x+1056 a b^2 \log \left(1-c^2 x\right)+6 b^2 \tanh ^{-1}\left(c \sqrt{x}\right)^2 \left(105 a \left(c^8 x^4-1\right)+b \left(15 c^7 x^{7/2}+21 c^5 x^{5/2}+35 c^3 x^{3/2}+105 c \sqrt{x}-176\right)\right)-564 a b^2+210 b^3 \left(c^8 x^4-1\right) \tanh ^{-1}\left(c \sqrt{x}\right)^3+6 b^3 c^5 x^{5/2}+46 b^3 c^3 x^{3/2}+1056 b^3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)+564 b^3 c \sqrt{x}}{840 c^8}","-\frac{88 b^2 \log \left(\frac{2}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{35 c^8}+\frac{71 b^2 x \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{140 c^6}+\frac{9 b^2 x^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{70 c^4}+\frac{b^2 x^3 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{28 c^2}-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3}{4 c^8}+\frac{44 b \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{35 c^8}+\frac{3 b \sqrt{x} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{4 c^7}+\frac{b x^{3/2} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{4 c^5}+\frac{3 b x^{5/2} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{20 c^3}+\frac{3 b x^{7/2} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{28 c}+\frac{1}{4} x^4 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3-\frac{44 b^3 \text{Li}_2\left(1-\frac{2}{1-c \sqrt{x}}\right)}{35 c^8}-\frac{47 b^3 \tanh ^{-1}\left(c \sqrt{x}\right)}{70 c^8}+\frac{47 b^3 \sqrt{x}}{70 c^7}+\frac{23 b^3 x^{3/2}}{420 c^5}+\frac{b^3 x^{5/2}}{140 c^3}",1,"(-564*a*b^2 + 630*a^2*b*c*Sqrt[x] + 564*b^3*c*Sqrt[x] + 426*a*b^2*c^2*x + 210*a^2*b*c^3*x^(3/2) + 46*b^3*c^3*x^(3/2) + 108*a*b^2*c^4*x^2 + 126*a^2*b*c^5*x^(5/2) + 6*b^3*c^5*x^(5/2) + 30*a*b^2*c^6*x^3 + 90*a^2*b*c^7*x^(7/2) + 210*a^3*c^8*x^4 + 6*b^2*(b*(-176 + 105*c*Sqrt[x] + 35*c^3*x^(3/2) + 21*c^5*x^(5/2) + 15*c^7*x^(7/2)) + 105*a*(-1 + c^8*x^4))*ArcTanh[c*Sqrt[x]]^2 + 210*b^3*(-1 + c^8*x^4)*ArcTanh[c*Sqrt[x]]^3 + 6*b*ArcTanh[c*Sqrt[x]]*(105*a^2*c^8*x^4 + b^2*(-94 + 71*c^2*x + 18*c^4*x^2 + 5*c^6*x^3) + 2*a*b*c*Sqrt[x]*(105 + 35*c^2*x + 21*c^4*x^2 + 15*c^6*x^3) - 352*b^2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + 315*a^2*b*Log[1 - c*Sqrt[x]] - 315*a^2*b*Log[1 + c*Sqrt[x]] + 1056*a*b^2*Log[1 - c^2*x] + 1056*b^3*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])])/(840*c^8)","A",0
203,1,351,304,0.8605117,"\int x^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3 \, dx","Integrate[x^2*(a + b*ArcTanh[c*Sqrt[x]])^3,x]","\frac{10 a^3 c^6 x^3+b \tanh ^{-1}\left(c \sqrt{x}\right) \left(30 a^2 c^6 x^3+4 a b c \sqrt{x} \left(3 c^4 x^2+5 c^2 x+15\right)+b^2 \left(3 c^4 x^2+16 c^2 x-19\right)-92 b^2 \log \left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}+1\right)\right)+6 a^2 b c^5 x^{5/2}+10 a^2 b c^3 x^{3/2}+30 a^2 b c \sqrt{x}+15 a^2 b \log \left(1-c \sqrt{x}\right)-15 a^2 b \log \left(c \sqrt{x}+1\right)+3 a b^2 c^4 x^2+16 a b^2 c^2 x+46 a b^2 \log \left(1-c^2 x\right)+2 b^2 \tanh ^{-1}\left(c \sqrt{x}\right)^2 \left(15 a \left(c^6 x^3-1\right)+b \left(3 c^5 x^{5/2}+5 c^3 x^{3/2}+15 c \sqrt{x}-23\right)\right)-19 a b^2+10 b^3 \left(c^6 x^3-1\right) \tanh ^{-1}\left(c \sqrt{x}\right)^3+b^3 c^3 x^{3/2}+46 b^3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)+19 b^3 c \sqrt{x}}{30 c^6}","-\frac{46 b^2 \log \left(\frac{2}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{15 c^6}+\frac{8 b^2 x \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{15 c^4}+\frac{b^2 x^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{10 c^2}+\frac{23 b \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{15 c^6}-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3}{3 c^6}+\frac{b \sqrt{x} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{c^5}+\frac{b x^{3/2} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{3 c^3}+\frac{b x^{5/2} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{5 c}+\frac{1}{3} x^3 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3-\frac{23 b^3 \text{Li}_2\left(1-\frac{2}{1-c \sqrt{x}}\right)}{15 c^6}-\frac{19 b^3 \tanh ^{-1}\left(c \sqrt{x}\right)}{30 c^6}+\frac{19 b^3 \sqrt{x}}{30 c^5}+\frac{b^3 x^{3/2}}{30 c^3}",1,"(-19*a*b^2 + 30*a^2*b*c*Sqrt[x] + 19*b^3*c*Sqrt[x] + 16*a*b^2*c^2*x + 10*a^2*b*c^3*x^(3/2) + b^3*c^3*x^(3/2) + 3*a*b^2*c^4*x^2 + 6*a^2*b*c^5*x^(5/2) + 10*a^3*c^6*x^3 + 2*b^2*(b*(-23 + 15*c*Sqrt[x] + 5*c^3*x^(3/2) + 3*c^5*x^(5/2)) + 15*a*(-1 + c^6*x^3))*ArcTanh[c*Sqrt[x]]^2 + 10*b^3*(-1 + c^6*x^3)*ArcTanh[c*Sqrt[x]]^3 + b*ArcTanh[c*Sqrt[x]]*(30*a^2*c^6*x^3 + 4*a*b*c*Sqrt[x]*(15 + 5*c^2*x + 3*c^4*x^2) + b^2*(-19 + 16*c^2*x + 3*c^4*x^2) - 92*b^2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + 15*a^2*b*Log[1 - c*Sqrt[x]] - 15*a^2*b*Log[1 + c*Sqrt[x]] + 46*a*b^2*Log[1 - c^2*x] + 46*b^3*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])])/(30*c^6)","A",0
204,1,285,234,0.5585501,"\int x \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3 \, dx","Integrate[x*(a + b*ArcTanh[c*Sqrt[x]])^3,x]","\frac{2 a^3 c^4 x^2+2 b \tanh ^{-1}\left(c \sqrt{x}\right) \left(3 a^2 c^4 x^2+2 a b c \sqrt{x} \left(c^2 x+3\right)+b^2 \left(c^2 x-1\right)-8 b^2 \log \left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}+1\right)\right)+2 a^2 b c^3 x^{3/2}+6 a^2 b c \sqrt{x}+3 a^2 b \log \left(1-c \sqrt{x}\right)-3 a^2 b \log \left(c \sqrt{x}+1\right)+2 a b^2 c^2 x+8 a b^2 \log \left(1-c^2 x\right)+2 b^2 \tanh ^{-1}\left(c \sqrt{x}\right)^2 \left(3 a \left(c^4 x^2-1\right)+b \left(c^3 x^{3/2}+3 c \sqrt{x}-4\right)\right)-2 a b^2+2 b^3 \left(c^4 x^2-1\right) \tanh ^{-1}\left(c \sqrt{x}\right)^3+8 b^3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)+2 b^3 c \sqrt{x}}{4 c^4}","-\frac{4 b^2 \log \left(\frac{2}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{c^4}+\frac{b^2 x \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{2 c^2}+\frac{2 b \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{c^4}-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3}{2 c^4}+\frac{3 b \sqrt{x} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{2 c^3}+\frac{b x^{3/2} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{2 c}+\frac{1}{2} x^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3-\frac{2 b^3 \text{Li}_2\left(1-\frac{2}{1-c \sqrt{x}}\right)}{c^4}-\frac{b^3 \tanh ^{-1}\left(c \sqrt{x}\right)}{2 c^4}+\frac{b^3 \sqrt{x}}{2 c^3}",1,"(-2*a*b^2 + 6*a^2*b*c*Sqrt[x] + 2*b^3*c*Sqrt[x] + 2*a*b^2*c^2*x + 2*a^2*b*c^3*x^(3/2) + 2*a^3*c^4*x^2 + 2*b^2*(b*(-4 + 3*c*Sqrt[x] + c^3*x^(3/2)) + 3*a*(-1 + c^4*x^2))*ArcTanh[c*Sqrt[x]]^2 + 2*b^3*(-1 + c^4*x^2)*ArcTanh[c*Sqrt[x]]^3 + 2*b*ArcTanh[c*Sqrt[x]]*(3*a^2*c^4*x^2 + b^2*(-1 + c^2*x) + 2*a*b*c*Sqrt[x]*(3 + c^2*x) - 8*b^2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + 3*a^2*b*Log[1 - c*Sqrt[x]] - 3*a^2*b*Log[1 + c*Sqrt[x]] + 8*a*b^2*Log[1 - c^2*x] + 8*b^3*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])])/(4*c^4)","A",0
205,1,201,142,0.2865642,"\int \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3 \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])^3,x]","\frac{a \left(2 a^2 c^2 x+6 a b c \sqrt{x}+3 a b \log \left(1-c \sqrt{x}\right)-3 a b \log \left(c \sqrt{x}+1\right)+6 b^2 \log \left(1-c^2 x\right)\right)+6 b \tanh ^{-1}\left(c \sqrt{x}\right) \left(a^2 c^2 x+2 a b c \sqrt{x}-2 b^2 \log \left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}+1\right)\right)+6 b^2 \left(c \sqrt{x}-1\right) \tanh ^{-1}\left(c \sqrt{x}\right)^2 \left(a c \sqrt{x}+a+b\right)+2 b^3 \left(c^2 x-1\right) \tanh ^{-1}\left(c \sqrt{x}\right)^3+6 b^3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)}{2 c^2}","-\frac{6 b^2 \log \left(\frac{2}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{c^2}+\frac{3 b \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{c^2}-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3}{c^2}+\frac{3 b \sqrt{x} \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{c}+x \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3-\frac{3 b^3 \text{Li}_2\left(1-\frac{2}{1-c \sqrt{x}}\right)}{c^2}",1,"(6*b^2*(-1 + c*Sqrt[x])*(a + b + a*c*Sqrt[x])*ArcTanh[c*Sqrt[x]]^2 + 2*b^3*(-1 + c^2*x)*ArcTanh[c*Sqrt[x]]^3 + 6*b*ArcTanh[c*Sqrt[x]]*(2*a*b*c*Sqrt[x] + a^2*c^2*x - 2*b^2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + a*(6*a*b*c*Sqrt[x] + 2*a^2*c^2*x + 3*a*b*Log[1 - c*Sqrt[x]] - 3*a*b*Log[1 + c*Sqrt[x]] + 6*b^2*Log[1 - c^2*x]) + 6*b^3*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])])/(2*c^2)","A",0
206,1,248,224,0.2073514,"\int \frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3}{x} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])^3/x,x]","\frac{3}{2} b \left(2 \text{Li}_2\left(\frac{\sqrt{x} c+1}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2-2 \text{Li}_2\left(\frac{\sqrt{x} c+1}{c \sqrt{x}-1}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2+b \left(-2 \text{Li}_3\left(\frac{\sqrt{x} c+1}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)+2 \text{Li}_3\left(\frac{\sqrt{x} c+1}{c \sqrt{x}-1}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)+b \left(\text{Li}_4\left(\frac{\sqrt{x} c+1}{1-c \sqrt{x}}\right)-\text{Li}_4\left(\frac{\sqrt{x} c+1}{c \sqrt{x}-1}\right)\right)\right)\right)+4 \tanh ^{-1}\left(\frac{2}{c \sqrt{x}-1}+1\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3","3 b^2 \text{Li}_3\left(1-\frac{2}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)-3 b^2 \text{Li}_3\left(\frac{2}{1-c \sqrt{x}}-1\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)-3 b \text{Li}_2\left(1-\frac{2}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2+3 b \text{Li}_2\left(\frac{2}{1-c \sqrt{x}}-1\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2+4 \tanh ^{-1}\left(1-\frac{2}{1-c \sqrt{x}}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3-\frac{3}{2} b^3 \text{Li}_4\left(1-\frac{2}{1-c \sqrt{x}}\right)+\frac{3}{2} b^3 \text{Li}_4\left(\frac{2}{1-c \sqrt{x}}-1\right)",1,"4*ArcTanh[1 + 2/(-1 + c*Sqrt[x])]*(a + b*ArcTanh[c*Sqrt[x]])^3 + (3*b*(2*(a + b*ArcTanh[c*Sqrt[x]])^2*PolyLog[2, (1 + c*Sqrt[x])/(1 - c*Sqrt[x])] - 2*(a + b*ArcTanh[c*Sqrt[x]])^2*PolyLog[2, (1 + c*Sqrt[x])/(-1 + c*Sqrt[x])] + b*(-2*(a + b*ArcTanh[c*Sqrt[x]])*PolyLog[3, (1 + c*Sqrt[x])/(1 - c*Sqrt[x])] + 2*(a + b*ArcTanh[c*Sqrt[x]])*PolyLog[3, (1 + c*Sqrt[x])/(-1 + c*Sqrt[x])] + b*(PolyLog[4, (1 + c*Sqrt[x])/(1 - c*Sqrt[x])] - PolyLog[4, (1 + c*Sqrt[x])/(-1 + c*Sqrt[x])]))))/2","A",1
207,1,230,142,0.3232662,"\int \frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3}{x^2} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])^3/x^2,x]","\frac{a \left(-2 a^2-3 a b c^2 x \log \left(1-c \sqrt{x}\right)+3 a b c^2 x \log \left(c \sqrt{x}+1\right)-6 a b c \sqrt{x}+12 b^2 c^2 x \log \left(\frac{c \sqrt{x}}{\sqrt{1-c^2 x}}\right)\right)-6 b \tanh ^{-1}\left(c \sqrt{x}\right) \left(a^2+2 a b c \sqrt{x}-2 b^2 c^2 x \log \left(1-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)\right)+6 b^2 \left(c \sqrt{x}-1\right) \tanh ^{-1}\left(c \sqrt{x}\right)^2 \left(a c \sqrt{x}+a+b c \sqrt{x}\right)-6 b^3 c^2 x \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)+2 b^3 \left(c^2 x-1\right) \tanh ^{-1}\left(c \sqrt{x}\right)^3}{2 x}","6 b^2 c^2 \log \left(2-\frac{2}{c \sqrt{x}+1}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)+3 b c^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2+c^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3-\frac{3 b c \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{\sqrt{x}}-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3}{x}-3 b^3 c^2 \text{Li}_2\left(\frac{2}{\sqrt{x} c+1}-1\right)",1,"(6*b^2*(-1 + c*Sqrt[x])*(a + a*c*Sqrt[x] + b*c*Sqrt[x])*ArcTanh[c*Sqrt[x]]^2 + 2*b^3*(-1 + c^2*x)*ArcTanh[c*Sqrt[x]]^3 - 6*b*ArcTanh[c*Sqrt[x]]*(a^2 + 2*a*b*c*Sqrt[x] - 2*b^2*c^2*x*Log[1 - E^(-2*ArcTanh[c*Sqrt[x]])]) + a*(-2*a^2 - 6*a*b*c*Sqrt[x] - 3*a*b*c^2*x*Log[1 - c*Sqrt[x]] + 3*a*b*c^2*x*Log[1 + c*Sqrt[x]] + 12*b^2*c^2*x*Log[(c*Sqrt[x])/Sqrt[1 - c^2*x]]) - 6*b^3*c^2*x*PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])])/(2*x)","A",0
208,1,333,234,0.7237922,"\int \frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3}{x^3} \, dx","Integrate[(a + b*ArcTanh[c*Sqrt[x]])^3/x^3,x]","-\frac{2 a^3+2 b \tanh ^{-1}\left(c \sqrt{x}\right) \left(3 a^2+2 a b c \sqrt{x} \left(3 c^2 x+1\right)-8 b^2 c^4 x^2 \log \left(1-e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)+b^2 c^2 x \left(1-c^2 x\right)\right)+3 a^2 b c^4 x^2 \log \left(1-c \sqrt{x}\right)-3 a^2 b c^4 x^2 \log \left(c \sqrt{x}+1\right)+6 a^2 b c^3 x^{3/2}+2 a^2 b c \sqrt{x}-2 a b^2 c^4 x^2+2 a b^2 c^2 x-16 a b^2 c^4 x^2 \log \left(\frac{c \sqrt{x}}{\sqrt{1-c^2 x}}\right)-2 b^2 \tanh ^{-1}\left(c \sqrt{x}\right)^2 \left(3 a \left(c^4 x^2-1\right)+b c \sqrt{x} \left(4 c^3 x^{3/2}-3 c^2 x-1\right)\right)+8 b^3 c^4 x^2 \text{Li}_2\left(e^{-2 \tanh ^{-1}\left(c \sqrt{x}\right)}\right)-2 b^3 \left(c^4 x^2-1\right) \tanh ^{-1}\left(c \sqrt{x}\right)^3+2 b^3 c^3 x^{3/2}}{4 x^2}","4 b^2 c^4 \log \left(2-\frac{2}{c \sqrt{x}+1}\right) \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)-\frac{b^2 c^2 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)}{2 x}+\frac{1}{2} c^4 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3+2 b c^4 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2-\frac{3 b c^3 \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{2 \sqrt{x}}-\frac{b c \left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^2}{2 x^{3/2}}-\frac{\left(a+b \tanh ^{-1}\left(c \sqrt{x}\right)\right)^3}{2 x^2}-2 b^3 c^4 \text{Li}_2\left(\frac{2}{\sqrt{x} c+1}-1\right)+\frac{1}{2} b^3 c^4 \tanh ^{-1}\left(c \sqrt{x}\right)-\frac{b^3 c^3}{2 \sqrt{x}}",1,"-1/4*(2*a^3 + 2*a^2*b*c*Sqrt[x] + 2*a*b^2*c^2*x + 6*a^2*b*c^3*x^(3/2) + 2*b^3*c^3*x^(3/2) - 2*a*b^2*c^4*x^2 - 2*b^2*(b*c*Sqrt[x]*(-1 - 3*c^2*x + 4*c^3*x^(3/2)) + 3*a*(-1 + c^4*x^2))*ArcTanh[c*Sqrt[x]]^2 - 2*b^3*(-1 + c^4*x^2)*ArcTanh[c*Sqrt[x]]^3 + 2*b*ArcTanh[c*Sqrt[x]]*(3*a^2 + b^2*c^2*x*(1 - c^2*x) + 2*a*b*c*Sqrt[x]*(1 + 3*c^2*x) - 8*b^2*c^4*x^2*Log[1 - E^(-2*ArcTanh[c*Sqrt[x]])]) + 3*a^2*b*c^4*x^2*Log[1 - c*Sqrt[x]] - 3*a^2*b*c^4*x^2*Log[1 + c*Sqrt[x]] - 16*a*b^2*c^4*x^2*Log[(c*Sqrt[x])/Sqrt[1 - c^2*x]] + 8*b^3*c^4*x^2*PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])])/x^2","A",0
209,1,31,38,0.0162752,"\int x^{3/2} \tanh ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x^(3/2)*ArcTanh[Sqrt[x]],x]","\frac{1}{10} \left(4 x^{5/2} \tanh ^{-1}\left(\sqrt{x}\right)+(x+2) x+2 \log (1-x)\right)","\frac{2}{5} x^{5/2} \tanh ^{-1}\left(\sqrt{x}\right)+\frac{x^2}{10}+\frac{x}{5}+\frac{1}{5} \log (1-x)",1,"(x*(2 + x) + 4*x^(5/2)*ArcTanh[Sqrt[x]] + 2*Log[1 - x])/10","A",1
210,1,25,31,0.0116232,"\int \sqrt{x} \tanh ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[Sqrt[x]*ArcTanh[Sqrt[x]],x]","\frac{1}{3} \left(2 x^{3/2} \tanh ^{-1}\left(\sqrt{x}\right)+x+\log (1-x)\right)","\frac{2}{3} x^{3/2} \tanh ^{-1}\left(\sqrt{x}\right)+\frac{x}{3}+\frac{1}{3} \log (1-x)",1,"(x + 2*x^(3/2)*ArcTanh[Sqrt[x]] + Log[1 - x])/3","A",1
211,1,20,20,0.0078805,"\int \frac{\tanh ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Integrate[ArcTanh[Sqrt[x]]/Sqrt[x],x]","\log (1-x)+2 \sqrt{x} \tanh ^{-1}\left(\sqrt{x}\right)","\log (1-x)+2 \sqrt{x} \tanh ^{-1}\left(\sqrt{x}\right)",1,"2*Sqrt[x]*ArcTanh[Sqrt[x]] + Log[1 - x]","A",1
212,1,24,24,0.020068,"\int \frac{\tanh ^{-1}\left(\sqrt{x}\right)}{x^{3/2}} \, dx","Integrate[ArcTanh[Sqrt[x]]/x^(3/2),x]","-\log (1-x)+\log (x)-\frac{2 \tanh ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}}","-\log (1-x)+\log (x)-\frac{2 \tanh ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}}",1,"(-2*ArcTanh[Sqrt[x]])/Sqrt[x] - Log[1 - x] + Log[x]","A",1
213,1,222,190,0.0570796,"\int x^3 \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right) \, dx","Integrate[x^3*(a + b*ArcTanh[c*x^(3/2)]),x]","\frac{a x^4}{4}+\frac{b \log \left(1-\sqrt[3]{c} \sqrt{x}\right)}{8 c^{8/3}}-\frac{b \log \left(\sqrt[3]{c} \sqrt{x}+1\right)}{8 c^{8/3}}+\frac{b \log \left(c^{2/3} x-\sqrt[3]{c} \sqrt{x}+1\right)}{16 c^{8/3}}-\frac{b \log \left(c^{2/3} x+\sqrt[3]{c} \sqrt{x}+1\right)}{16 c^{8/3}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} \sqrt{x}-1}{\sqrt{3}}\right)}{8 c^{8/3}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} \sqrt{x}+1}{\sqrt{3}}\right)}{8 c^{8/3}}+\frac{3 b x^{5/2}}{20 c}+\frac{1}{4} b x^4 \tanh ^{-1}\left(c x^{3/2}\right)","\frac{1}{4} x^4 \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)+\frac{b \log \left(c^{2/3} x-\sqrt[3]{c} \sqrt{x}+1\right)}{16 c^{8/3}}-\frac{b \log \left(c^{2/3} x+\sqrt[3]{c} \sqrt{x}+1\right)}{16 c^{8/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{1-2 \sqrt[3]{c} \sqrt{x}}{\sqrt{3}}\right)}{8 c^{8/3}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} \sqrt{x}+1}{\sqrt{3}}\right)}{8 c^{8/3}}-\frac{b \tanh ^{-1}\left(\sqrt[3]{c} \sqrt{x}\right)}{4 c^{8/3}}+\frac{3 b x^{5/2}}{20 c}",1,"(3*b*x^(5/2))/(20*c) + (a*x^4)/4 + (Sqrt[3]*b*ArcTan[(-1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(8*c^(8/3)) + (Sqrt[3]*b*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(8*c^(8/3)) + (b*x^4*ArcTanh[c*x^(3/2)])/4 + (b*Log[1 - c^(1/3)*Sqrt[x]])/(8*c^(8/3)) - (b*Log[1 + c^(1/3)*Sqrt[x]])/(8*c^(8/3)) + (b*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x])/(16*c^(8/3)) - (b*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x])/(16*c^(8/3))","A",1
214,1,75,49,0.0271566,"\int x^2 \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right) \, dx","Integrate[x^2*(a + b*ArcTanh[c*x^(3/2)]),x]","\frac{a x^3}{3}+\frac{b \log \left(1-c x^{3/2}\right)}{6 c^2}-\frac{b \log \left(c x^{3/2}+1\right)}{6 c^2}+\frac{b x^{3/2}}{3 c}+\frac{1}{3} b x^3 \tanh ^{-1}\left(c x^{3/2}\right)","\frac{1}{3} x^3 \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)-\frac{b \tanh ^{-1}\left(c x^{3/2}\right)}{3 c^2}+\frac{b x^{3/2}}{3 c}",1,"(b*x^(3/2))/(3*c) + (a*x^3)/3 + (b*x^3*ArcTanh[c*x^(3/2)])/3 + (b*Log[1 - c*x^(3/2)])/(6*c^2) - (b*Log[1 + c*x^(3/2)])/(6*c^2)","A",1
215,1,222,190,0.0356945,"\int x \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right) \, dx","Integrate[x*(a + b*ArcTanh[c*x^(3/2)]),x]","\frac{a x^2}{2}+\frac{b \log \left(1-\sqrt[3]{c} \sqrt{x}\right)}{4 c^{4/3}}-\frac{b \log \left(\sqrt[3]{c} \sqrt{x}+1\right)}{4 c^{4/3}}+\frac{b \log \left(c^{2/3} x-\sqrt[3]{c} \sqrt{x}+1\right)}{8 c^{4/3}}-\frac{b \log \left(c^{2/3} x+\sqrt[3]{c} \sqrt{x}+1\right)}{8 c^{4/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} \sqrt{x}-1}{\sqrt{3}}\right)}{4 c^{4/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} \sqrt{x}+1}{\sqrt{3}}\right)}{4 c^{4/3}}+\frac{1}{2} b x^2 \tanh ^{-1}\left(c x^{3/2}\right)+\frac{3 b \sqrt{x}}{2 c}","\frac{1}{2} x^2 \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)+\frac{b \log \left(c^{2/3} x-\sqrt[3]{c} \sqrt{x}+1\right)}{8 c^{4/3}}-\frac{b \log \left(c^{2/3} x+\sqrt[3]{c} \sqrt{x}+1\right)}{8 c^{4/3}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{1-2 \sqrt[3]{c} \sqrt{x}}{\sqrt{3}}\right)}{4 c^{4/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} \sqrt{x}+1}{\sqrt{3}}\right)}{4 c^{4/3}}-\frac{b \tanh ^{-1}\left(\sqrt[3]{c} \sqrt{x}\right)}{2 c^{4/3}}+\frac{3 b \sqrt{x}}{2 c}",1,"(3*b*Sqrt[x])/(2*c) + (a*x^2)/2 - (Sqrt[3]*b*ArcTan[(-1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(4*c^(4/3)) - (Sqrt[3]*b*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(4*c^(4/3)) + (b*x^2*ArcTanh[c*x^(3/2)])/2 + (b*Log[1 - c^(1/3)*Sqrt[x]])/(4*c^(4/3)) - (b*Log[1 + c^(1/3)*Sqrt[x]])/(4*c^(4/3)) + (b*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x])/(8*c^(4/3)) - (b*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x])/(8*c^(4/3))","A",1
216,1,141,170,0.1056251,"\int \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right) \, dx","Integrate[a + b*ArcTanh[c*x^(3/2)],x]","a x-\frac{b \left(-\log \left(c^{2/3} x-\sqrt[3]{c} \sqrt{x}+1\right)+\log \left(c^{2/3} x+\sqrt[3]{c} \sqrt{x}+1\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{1-2 \sqrt[3]{c} \sqrt{x}}{\sqrt{3}}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} \sqrt{x}+1}{\sqrt{3}}\right)+4 \tanh ^{-1}\left(\sqrt[3]{c} \sqrt{x}\right)\right)}{4 c^{2/3}}+b x \tanh ^{-1}\left(c x^{3/2}\right)","a x+\frac{b \log \left(c^{2/3} x-\sqrt[3]{c} \sqrt{x}+1\right)}{4 c^{2/3}}-\frac{b \log \left(c^{2/3} x+\sqrt[3]{c} \sqrt{x}+1\right)}{4 c^{2/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{1-2 \sqrt[3]{c} \sqrt{x}}{\sqrt{3}}\right)}{2 c^{2/3}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{2 \sqrt[3]{c} \sqrt{x}+1}{\sqrt{3}}\right)}{2 c^{2/3}}-\frac{b \tanh ^{-1}\left(\sqrt[3]{c} \sqrt{x}\right)}{c^{2/3}}+b x \tanh ^{-1}\left(c x^{3/2}\right)",1,"a*x + b*x*ArcTanh[c*x^(3/2)] - (b*(2*Sqrt[3]*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]] + 4*ArcTanh[c^(1/3)*Sqrt[x]] - Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x] + Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x]))/(4*c^(2/3))","A",1
217,1,32,34,0.0199478,"\int \frac{a+b \tanh ^{-1}\left(c x^{3/2}\right)}{x} \, dx","Integrate[(a + b*ArcTanh[c*x^(3/2)])/x,x]","a \log (x)+\frac{1}{3} b \left(\text{Li}_2\left(c x^{3/2}\right)-\text{Li}_2\left(-c x^{3/2}\right)\right)","a \log (x)-\frac{1}{3} b \text{Li}_2\left(-c x^{3/2}\right)+\frac{1}{3} b \text{Li}_2\left(c x^{3/2}\right)",1,"a*Log[x] + (b*(-PolyLog[2, -(c*x^(3/2))] + PolyLog[2, c*x^(3/2)]))/3","A",1
218,1,205,172,0.0349915,"\int \frac{a+b \tanh ^{-1}\left(c x^{3/2}\right)}{x^2} \, dx","Integrate[(a + b*ArcTanh[c*x^(3/2)])/x^2,x]","-\frac{a}{x}-\frac{1}{2} b c^{2/3} \log \left(1-\sqrt[3]{c} \sqrt{x}\right)+\frac{1}{2} b c^{2/3} \log \left(\sqrt[3]{c} \sqrt{x}+1\right)-\frac{1}{4} b c^{2/3} \log \left(c^{2/3} x-\sqrt[3]{c} \sqrt{x}+1\right)+\frac{1}{4} b c^{2/3} \log \left(c^{2/3} x+\sqrt[3]{c} \sqrt{x}+1\right)+\frac{1}{2} \sqrt{3} b c^{2/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} \sqrt{x}-1}{\sqrt{3}}\right)+\frac{1}{2} \sqrt{3} b c^{2/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} \sqrt{x}+1}{\sqrt{3}}\right)-\frac{b \tanh ^{-1}\left(c x^{3/2}\right)}{x}","-\frac{a+b \tanh ^{-1}\left(c x^{3/2}\right)}{x}-\frac{1}{4} b c^{2/3} \log \left(c^{2/3} x-\sqrt[3]{c} \sqrt{x}+1\right)+\frac{1}{4} b c^{2/3} \log \left(c^{2/3} x+\sqrt[3]{c} \sqrt{x}+1\right)-\frac{1}{2} \sqrt{3} b c^{2/3} \tan ^{-1}\left(\frac{1-2 \sqrt[3]{c} \sqrt{x}}{\sqrt{3}}\right)+\frac{1}{2} \sqrt{3} b c^{2/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} \sqrt{x}+1}{\sqrt{3}}\right)+b c^{2/3} \tanh ^{-1}\left(\sqrt[3]{c} \sqrt{x}\right)",1,"-(a/x) + (Sqrt[3]*b*c^(2/3)*ArcTan[(-1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/2 + (Sqrt[3]*b*c^(2/3)*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/2 - (b*ArcTanh[c*x^(3/2)])/x - (b*c^(2/3)*Log[1 - c^(1/3)*Sqrt[x]])/2 + (b*c^(2/3)*Log[1 + c^(1/3)*Sqrt[x]])/2 - (b*c^(2/3)*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x])/4 + (b*c^(2/3)*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x])/4","A",1
219,1,220,188,0.0603783,"\int \frac{a+b \tanh ^{-1}\left(c x^{3/2}\right)}{x^3} \, dx","Integrate[(a + b*ArcTanh[c*x^(3/2)])/x^3,x]","-\frac{a}{2 x^2}-\frac{1}{4} b c^{4/3} \log \left(1-\sqrt[3]{c} \sqrt{x}\right)+\frac{1}{4} b c^{4/3} \log \left(\sqrt[3]{c} \sqrt{x}+1\right)-\frac{1}{8} b c^{4/3} \log \left(c^{2/3} x-\sqrt[3]{c} \sqrt{x}+1\right)+\frac{1}{8} b c^{4/3} \log \left(c^{2/3} x+\sqrt[3]{c} \sqrt{x}+1\right)-\frac{1}{4} \sqrt{3} b c^{4/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} \sqrt{x}-1}{\sqrt{3}}\right)-\frac{1}{4} \sqrt{3} b c^{4/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} \sqrt{x}+1}{\sqrt{3}}\right)-\frac{b \tanh ^{-1}\left(c x^{3/2}\right)}{2 x^2}-\frac{3 b c}{2 \sqrt{x}}","-\frac{a+b \tanh ^{-1}\left(c x^{3/2}\right)}{2 x^2}-\frac{1}{8} b c^{4/3} \log \left(c^{2/3} x-\sqrt[3]{c} \sqrt{x}+1\right)+\frac{1}{8} b c^{4/3} \log \left(c^{2/3} x+\sqrt[3]{c} \sqrt{x}+1\right)+\frac{1}{4} \sqrt{3} b c^{4/3} \tan ^{-1}\left(\frac{1-2 \sqrt[3]{c} \sqrt{x}}{\sqrt{3}}\right)-\frac{1}{4} \sqrt{3} b c^{4/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c} \sqrt{x}+1}{\sqrt{3}}\right)+\frac{1}{2} b c^{4/3} \tanh ^{-1}\left(\sqrt[3]{c} \sqrt{x}\right)-\frac{3 b c}{2 \sqrt{x}}",1,"-1/2*a/x^2 - (3*b*c)/(2*Sqrt[x]) - (Sqrt[3]*b*c^(4/3)*ArcTan[(-1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/4 - (Sqrt[3]*b*c^(4/3)*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/4 - (b*ArcTanh[c*x^(3/2)])/(2*x^2) - (b*c^(4/3)*Log[1 - c^(1/3)*Sqrt[x]])/4 + (b*c^(4/3)*Log[1 + c^(1/3)*Sqrt[x]])/4 - (b*c^(4/3)*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x])/8 + (b*c^(4/3)*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x])/8","A",1
220,1,73,47,0.0295701,"\int \frac{a+b \tanh ^{-1}\left(c x^{3/2}\right)}{x^4} \, dx","Integrate[(a + b*ArcTanh[c*x^(3/2)])/x^4,x]","-\frac{a}{3 x^3}-\frac{1}{6} b c^2 \log \left(1-c x^{3/2}\right)+\frac{1}{6} b c^2 \log \left(c x^{3/2}+1\right)-\frac{b c}{3 x^{3/2}}-\frac{b \tanh ^{-1}\left(c x^{3/2}\right)}{3 x^3}","-\frac{a+b \tanh ^{-1}\left(c x^{3/2}\right)}{3 x^3}+\frac{1}{3} b c^2 \tanh ^{-1}\left(c x^{3/2}\right)-\frac{b c}{3 x^{3/2}}",1,"-1/3*a/x^3 - (b*c)/(3*x^(3/2)) - (b*ArcTanh[c*x^(3/2)])/(3*x^3) - (b*c^2*Log[1 - c*x^(3/2)])/6 + (b*c^2*Log[1 + c*x^(3/2)])/6","A",1
221,1,122,101,0.1003021,"\int x^2 \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)^2 \, dx","Integrate[x^2*(a + b*ArcTanh[c*x^(3/2)])^2,x]","\frac{a^2 c^2 x^3+2 a b c x^{3/2}+b (a+b) \log \left(1-c x^{3/2}\right)-a b \log \left(c x^{3/2}+1\right)+2 b c x^{3/2} \tanh ^{-1}\left(c x^{3/2}\right) \left(a c x^{3/2}+b\right)+b^2 \left(c^2 x^3-1\right) \tanh ^{-1}\left(c x^{3/2}\right)^2+b^2 \log \left(c x^{3/2}+1\right)}{3 c^2}","-\frac{\left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)^2}{3 c^2}+\frac{2 a b x^{3/2}}{3 c}+\frac{1}{3} x^3 \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)^2+\frac{b^2 \log \left(1-c^2 x^3\right)}{3 c^2}+\frac{2 b^2 x^{3/2} \tanh ^{-1}\left(c x^{3/2}\right)}{3 c}",1,"(2*a*b*c*x^(3/2) + a^2*c^2*x^3 + 2*b*c*x^(3/2)*(b + a*c*x^(3/2))*ArcTanh[c*x^(3/2)] + b^2*(-1 + c^2*x^3)*ArcTanh[c*x^(3/2)]^2 + b*(a + b)*Log[1 - c*x^(3/2)] - a*b*Log[1 + c*x^(3/2)] + b^2*Log[1 + c*x^(3/2)])/(3*c^2)","A",1
222,1,167,156,0.1368457,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)^2}{x} \, dx","Integrate[(a + b*ArcTanh[c*x^(3/2)])^2/x,x]","\frac{1}{3} \left(b \left(2 \text{Li}_2\left(\frac{c x^{3/2}+1}{1-c x^{3/2}}\right) \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)-2 \text{Li}_2\left(\frac{c x^{3/2}+1}{c x^{3/2}-1}\right) \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)+b \left(\text{Li}_3\left(\frac{c x^{3/2}+1}{c x^{3/2}-1}\right)-\text{Li}_3\left(\frac{c x^{3/2}+1}{1-c x^{3/2}}\right)\right)\right)+4 \tanh ^{-1}\left(\frac{2}{c x^{3/2}-1}+1\right) \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)^2\right)","-\frac{2}{3} b \text{Li}_2\left(1-\frac{2}{1-c x^{3/2}}\right) \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)+\frac{2}{3} b \text{Li}_2\left(\frac{2}{1-c x^{3/2}}-1\right) \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)+\frac{4}{3} \tanh ^{-1}\left(1-\frac{2}{1-c x^{3/2}}\right) \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)^2+\frac{1}{3} b^2 \text{Li}_3\left(1-\frac{2}{1-c x^{3/2}}\right)-\frac{1}{3} b^2 \text{Li}_3\left(\frac{2}{1-c x^{3/2}}-1\right)",1,"(4*(a + b*ArcTanh[c*x^(3/2)])^2*ArcTanh[1 + 2/(-1 + c*x^(3/2))] + b*(2*(a + b*ArcTanh[c*x^(3/2)])*PolyLog[2, (1 + c*x^(3/2))/(1 - c*x^(3/2))] - 2*(a + b*ArcTanh[c*x^(3/2)])*PolyLog[2, (1 + c*x^(3/2))/(-1 + c*x^(3/2))] + b*(-PolyLog[3, (1 + c*x^(3/2))/(1 - c*x^(3/2))] + PolyLog[3, (1 + c*x^(3/2))/(-1 + c*x^(3/2))])))/3","A",1
223,1,123,96,0.1421686,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)^2}{x^4} \, dx","Integrate[(a + b*ArcTanh[c*x^(3/2)])^2/x^4,x]","\frac{1}{3} \left(-\frac{a^2}{x^3}-b c^2 (a+b) \log \left(1-c x^{3/2}\right)+b c^2 (a-b) \log \left(c x^{3/2}+1\right)-\frac{2 a b c}{x^{3/2}}-\frac{2 b \tanh ^{-1}\left(c x^{3/2}\right) \left(a+b c x^{3/2}\right)}{x^3}+\frac{b^2 \left(c^2 x^3-1\right) \tanh ^{-1}\left(c x^{3/2}\right)^2}{x^3}+3 b^2 c^2 \log (x)\right)","\frac{1}{3} c^2 \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)^2-\frac{2 b c \left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)}{3 x^{3/2}}-\frac{\left(a+b \tanh ^{-1}\left(c x^{3/2}\right)\right)^2}{3 x^3}-\frac{1}{3} b^2 c^2 \log \left(1-c^2 x^3\right)+b^2 c^2 \log (x)",1,"(-(a^2/x^3) - (2*a*b*c)/x^(3/2) - (2*b*(a + b*c*x^(3/2))*ArcTanh[c*x^(3/2)])/x^3 + (b^2*(-1 + c^2*x^3)*ArcTanh[c*x^(3/2)]^2)/x^3 + 3*b^2*c^2*Log[x] - b*(a + b)*c^2*Log[1 - c*x^(3/2)] + (a - b)*b*c^2*Log[1 + c*x^(3/2)])/3","A",1
224,1,73,66,0.0556707,"\int x^2 \left(a+b \tanh ^{-1}\left(c x^n\right)\right) \, dx","Integrate[x^2*(a + b*ArcTanh[c*x^n]),x]","\frac{a x^3}{3}-\frac{b c n x^{n+3} \, _2F_1\left(1,\frac{n+3}{2 n};\frac{n+3}{2 n}+1;c^2 x^{2 n}\right)}{3 (n+3)}+\frac{1}{3} b x^3 \tanh ^{-1}\left(c x^n\right)","\frac{1}{3} x^3 \left(a+b \tanh ^{-1}\left(c x^n\right)\right)-\frac{b c n x^{n+3} \, _2F_1\left(1,\frac{n+3}{2 n};\frac{3 (n+1)}{2 n};c^2 x^{2 n}\right)}{3 (n+3)}",1,"(a*x^3)/3 + (b*x^3*ArcTanh[c*x^n])/3 - (b*c*n*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/(2*n), 1 + (3 + n)/(2*n), c^2*x^(2*n)])/(3*(3 + n))","A",1
225,1,73,67,0.0473587,"\int x \left(a+b \tanh ^{-1}\left(c x^n\right)\right) \, dx","Integrate[x*(a + b*ArcTanh[c*x^n]),x]","\frac{a x^2}{2}-\frac{b c n x^{n+2} \, _2F_1\left(1,\frac{n+2}{2 n};\frac{n+2}{2 n}+1;c^2 x^{2 n}\right)}{2 (n+2)}+\frac{1}{2} b x^2 \tanh ^{-1}\left(c x^n\right)","\frac{1}{2} x^2 \left(a+b \tanh ^{-1}\left(c x^n\right)\right)-\frac{b c n x^{n+2} \, _2F_1\left(1,\frac{n+2}{2 n};\frac{1}{2} \left(3+\frac{2}{n}\right);c^2 x^{2 n}\right)}{2 (n+2)}",1,"(a*x^2)/2 + (b*x^2*ArcTanh[c*x^n])/2 - (b*c*n*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/(2*n), 1 + (2 + n)/(2*n), c^2*x^(2*n)])/(2*(2 + n))","A",1
226,1,58,58,0.0320683,"\int \left(a+b \tanh ^{-1}\left(c x^n\right)\right) \, dx","Integrate[a + b*ArcTanh[c*x^n],x]","a x-\frac{b c n x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);c^2 x^{2 n}\right)}{n+1}+b x \tanh ^{-1}\left(c x^n\right)","a x-\frac{b c n x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);c^2 x^{2 n}\right)}{n+1}+b x \tanh ^{-1}\left(c x^n\right)",1,"a*x + b*x*ArcTanh[c*x^n] - (b*c*n*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, c^2*x^(2*n)])/(1 + n)","A",1
227,1,39,36,0.07986,"\int \frac{a+b \tanh ^{-1}\left(c x^n\right)}{x} \, dx","Integrate[(a + b*ArcTanh[c*x^n])/x,x]","\frac{b c x^n \, _3F_2\left(\frac{1}{2},\frac{1}{2},1;\frac{3}{2},\frac{3}{2};c^2 x^{2 n}\right)}{n}+a \log (x)","a \log (x)-\frac{b \text{Li}_2\left(-c x^n\right)}{2 n}+\frac{b \text{Li}_2\left(c x^n\right)}{2 n}",1,"(b*c*x^n*HypergeometricPFQ[{1/2, 1/2, 1}, {3/2, 3/2}, c^2*x^(2*n)])/n + a*Log[x]","C",1
228,1,66,67,0.0649631,"\int \frac{a+b \tanh ^{-1}\left(c x^n\right)}{x^2} \, dx","Integrate[(a + b*ArcTanh[c*x^n])/x^2,x]","-\frac{a}{x}+\frac{b c n x^{n-1} \, _2F_1\left(1,\frac{n-1}{2 n};\frac{n-1}{2 n}+1;c^2 x^{2 n}\right)}{n-1}-\frac{b \tanh ^{-1}\left(c x^n\right)}{x}","-\frac{a+b \tanh ^{-1}\left(c x^n\right)}{x}-\frac{b c n x^{n-1} \, _2F_1\left(1,-\frac{1-n}{2 n};\frac{1}{2} \left(3-\frac{1}{n}\right);c^2 x^{2 n}\right)}{1-n}",1,"-(a/x) - (b*ArcTanh[c*x^n])/x + (b*c*n*x^(-1 + n)*Hypergeometric2F1[1, (-1 + n)/(2*n), 1 + (-1 + n)/(2*n), c^2*x^(2*n)])/(-1 + n)","A",1
229,1,73,70,0.044289,"\int \frac{a+b \tanh ^{-1}\left(c x^n\right)}{x^3} \, dx","Integrate[(a + b*ArcTanh[c*x^n])/x^3,x]","-\frac{a}{2 x^2}+\frac{b c n x^{n-2} \, _2F_1\left(1,\frac{n-2}{2 n};\frac{n-2}{2 n}+1;c^2 x^{2 n}\right)}{2 (n-2)}-\frac{b \tanh ^{-1}\left(c x^n\right)}{2 x^2}","-\frac{a+b \tanh ^{-1}\left(c x^n\right)}{2 x^2}-\frac{b c n x^{n-2} \, _2F_1\left(1,\frac{1}{2} \left(1-\frac{2}{n}\right);\frac{1}{2} \left(3-\frac{2}{n}\right);c^2 x^{2 n}\right)}{2 (2-n)}",1,"-1/2*a/x^2 - (b*ArcTanh[c*x^n])/(2*x^2) + (b*c*n*x^(-2 + n)*Hypergeometric2F1[1, (-2 + n)/(2*n), 1 + (-2 + n)/(2*n), c^2*x^(2*n)])/(2*(-2 + n))","A",1
230,1,73,72,0.0439954,"\int \frac{a+b \tanh ^{-1}\left(c x^n\right)}{x^4} \, dx","Integrate[(a + b*ArcTanh[c*x^n])/x^4,x]","-\frac{a}{3 x^3}+\frac{b c n x^{n-3} \, _2F_1\left(1,\frac{n-3}{2 n};\frac{n-3}{2 n}+1;c^2 x^{2 n}\right)}{3 (n-3)}-\frac{b \tanh ^{-1}\left(c x^n\right)}{3 x^3}","-\frac{a+b \tanh ^{-1}\left(c x^n\right)}{3 x^3}-\frac{b c n x^{n-3} \, _2F_1\left(1,-\frac{3-n}{2 n};-\frac{3 (1-n)}{2 n};c^2 x^{2 n}\right)}{3 (3-n)}",1,"-1/3*a/x^3 - (b*ArcTanh[c*x^n])/(3*x^3) + (b*c*n*x^(-3 + n)*Hypergeometric2F1[1, (-3 + n)/(2*n), 1 + (-3 + n)/(2*n), c^2*x^(2*n)])/(3*(-3 + n))","A",1
231,0,0,17,13.8301327,"\int x \left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2 \, dx","Integrate[x*(a + b*ArcTanh[c*x^n])^2,x]","\int x \left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2 \, dx","\text{Int}\left(x \left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2,x\right)",0,"Integrate[x*(a + b*ArcTanh[c*x^n])^2, x]","A",-1
232,0,0,15,1.9732729,"\int \left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2 \, dx","Integrate[(a + b*ArcTanh[c*x^n])^2,x]","\int \left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2 \, dx","\text{Int}\left(\left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2,x\right)",0,"Integrate[(a + b*ArcTanh[c*x^n])^2, x]","A",-1
233,1,183,148,0.0968898,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2}{x} \, dx","Integrate[(a + b*ArcTanh[c*x^n])^2/x,x]","\frac{2 \tanh ^{-1}\left(1-\frac{2}{1-c x^n}\right) \left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2-4 b c \left(\frac{1}{2} \left(\frac{\text{Li}_2\left(\frac{-c x^n-1}{c x^n-1}\right) \left(-a-b \tanh ^{-1}\left(c x^n\right)\right)}{2 c}+\frac{b \text{Li}_3\left(\frac{-c x^n-1}{c x^n-1}\right)}{4 c}\right)+\frac{1}{2} \left(-\frac{\text{Li}_2\left(\frac{c x^n+1}{c x^n-1}\right) \left(-a-b \tanh ^{-1}\left(c x^n\right)\right)}{2 c}-\frac{b \text{Li}_3\left(\frac{c x^n+1}{c x^n-1}\right)}{4 c}\right)\right)}{n}","-\frac{b \text{Li}_2\left(1-\frac{2}{1-c x^n}\right) \left(a+b \tanh ^{-1}\left(c x^n\right)\right)}{n}+\frac{b \text{Li}_2\left(\frac{2}{1-c x^n}-1\right) \left(a+b \tanh ^{-1}\left(c x^n\right)\right)}{n}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{1-c x^n}\right) \left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2}{n}+\frac{b^2 \text{Li}_3\left(1-\frac{2}{1-c x^n}\right)}{2 n}-\frac{b^2 \text{Li}_3\left(\frac{2}{1-c x^n}-1\right)}{2 n}",1,"(2*(a + b*ArcTanh[c*x^n])^2*ArcTanh[1 - 2/(1 - c*x^n)] - 4*b*c*((((-a - b*ArcTanh[c*x^n])*PolyLog[2, (-1 - c*x^n)/(-1 + c*x^n)])/(2*c) + (b*PolyLog[3, (-1 - c*x^n)/(-1 + c*x^n)])/(4*c))/2 + (-1/2*((-a - b*ArcTanh[c*x^n])*PolyLog[2, (1 + c*x^n)/(-1 + c*x^n)])/c - (b*PolyLog[3, (1 + c*x^n)/(-1 + c*x^n)])/(4*c))/2))/n","A",1
234,0,0,19,17.7956883,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2}{x^2} \, dx","Integrate[(a + b*ArcTanh[c*x^n])^2/x^2,x]","\int \frac{\left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2}{x^2} \, dx","\text{Int}\left(\frac{\left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2}{x^2},x\right)",0,"Integrate[(a + b*ArcTanh[c*x^n])^2/x^2, x]","A",-1
235,0,0,19,17.6532005,"\int \frac{\left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2}{x^3} \, dx","Integrate[(a + b*ArcTanh[c*x^n])^2/x^3,x]","\int \frac{\left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2}{x^3} \, dx","\text{Int}\left(\frac{\left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2}{x^3},x\right)",0,"Integrate[(a + b*ArcTanh[c*x^n])^2/x^3, x]","A",-1
236,1,33,30,0.0425082,"\int \frac{\tanh ^{-1}\left(a x^n\right)}{x} \, dx","Integrate[ArcTanh[a*x^n]/x,x]","\frac{a x^n \, _3F_2\left(\frac{1}{2},\frac{1}{2},1;\frac{3}{2},\frac{3}{2};a^2 x^{2 n}\right)}{n}","\frac{\text{Li}_2\left(a x^n\right)}{2 n}-\frac{\text{Li}_2\left(-a x^n\right)}{2 n}",1,"(a*x^n*HypergeometricPFQ[{1/2, 1/2, 1}, {3/2, 3/2}, a^2*x^(2*n)])/n","C",1
237,1,22,24,0.0126594,"\int \frac{\tanh ^{-1}\left(a x^5\right)}{x} \, dx","Integrate[ArcTanh[a*x^5]/x,x]","\frac{1}{10} \left(\text{Li}_2\left(a x^5\right)-\text{Li}_2\left(-a x^5\right)\right)","\frac{\text{Li}_2\left(a x^5\right)}{10}-\frac{1}{10} \text{Li}_2\left(-a x^5\right)",1,"(-PolyLog[2, -(a*x^5)] + PolyLog[2, a*x^5])/10","A",1
238,1,17,19,0.0024427,"\int \tanh ^{-1}\left(\frac{1}{x}\right) \, dx","Integrate[ArcTanh[x^(-1)],x]","\frac{1}{2} \log \left(x^2-1\right)+x \tanh ^{-1}\left(\frac{1}{x}\right)","\frac{1}{2} \log \left(1-x^2\right)+x \tanh ^{-1}\left(\frac{1}{x}\right)",1,"x*ArcTanh[x^(-1)] + Log[-1 + x^2]/2","A",1
239,0,0,21,8.4218004,"\int (d x)^m \left(a+b \tanh ^{-1}\left(c x^n\right)\right)^3 \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c*x^n])^3,x]","\int (d x)^m \left(a+b \tanh ^{-1}\left(c x^n\right)\right)^3 \, dx","\text{Int}\left((d x)^m \left(a+b \tanh ^{-1}\left(c x^n\right)\right)^3,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTanh[c*x^n])^3, x]","A",-1
240,0,0,21,5.9726367,"\int (d x)^m \left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2 \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c*x^n])^2,x]","\int (d x)^m \left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2 \, dx","\text{Int}\left((d x)^m \left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcTanh[c*x^n])^2, x]","A",-1
241,1,77,84,0.0895457,"\int (d x)^m \left(a+b \tanh ^{-1}\left(c x^n\right)\right) \, dx","Integrate[(d*x)^m*(a + b*ArcTanh[c*x^n]),x]","\frac{x (d x)^m \left((m+n+1) \left(a+b \tanh ^{-1}\left(c x^n\right)\right)-b c n x^n \, _2F_1\left(1,\frac{m+n+1}{2 n};\frac{m+3 n+1}{2 n};c^2 x^{2 n}\right)\right)}{(m+1) (m+n+1)}","\frac{x (d x)^m \left(a+b \tanh ^{-1}\left(c x^n\right)\right)}{m+1}-\frac{b c n x^{n+1} (d x)^m \, _2F_1\left(1,\frac{m+n+1}{2 n};\frac{m+3 n+1}{2 n};c^2 x^{2 n}\right)}{(m+1) (m+n+1)}",1,"(x*(d*x)^m*((1 + m + n)*(a + b*ArcTanh[c*x^n]) - b*c*n*x^n*Hypergeometric2F1[1, (1 + m + n)/(2*n), (1 + m + 3*n)/(2*n), c^2*x^(2*n)]))/((1 + m)*(1 + m + n))","A",1
242,0,0,21,0.3822071,"\int \frac{(d x)^m}{a+b \tanh ^{-1}\left(c x^n\right)} \, dx","Integrate[(d*x)^m/(a + b*ArcTanh[c*x^n]),x]","\int \frac{(d x)^m}{a+b \tanh ^{-1}\left(c x^n\right)} \, dx","\text{Int}\left(\frac{(d x)^m}{a+b \tanh ^{-1}\left(c x^n\right)},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcTanh[c*x^n]), x]","A",-1
243,0,0,21,1.830076,"\int \frac{(d x)^m}{\left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2} \, dx","Integrate[(d*x)^m/(a + b*ArcTanh[c*x^n])^2,x]","\int \frac{(d x)^m}{\left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2} \, dx","\text{Int}\left(\frac{(d x)^m}{\left(a+b \tanh ^{-1}\left(c x^n\right)\right)^2},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcTanh[c*x^n])^2, x]","A",-1