1,1,231,160,2.8981763,"\int \left(a+b \coth ^2(c+d x)\right)^5 \, dx","Integrate[(a + b*Coth[c + d*x]^2)^5,x]","-\frac{b^5 \coth ^9(c+d x) \left(\frac{1575 a^4 \tanh ^8(c+d x)}{b^4}+\frac{1050 a^3 \left(3 \tanh ^2(c+d x)+1\right) \tanh ^6(c+d x)}{b^3}+\frac{210 a^2 \left(15 \tanh ^4(c+d x)+5 \tanh ^2(c+d x)+3\right) \tanh ^4(c+d x)}{b^2}-\frac{315 (a+b)^5 \tanh ^{-1}\left(\sqrt{\tanh ^2(c+d x)}\right) \tanh ^{10}(c+d x)}{b^5 \sqrt{\tanh ^2(c+d x)}}+\frac{15 a \left(105 \tanh ^6(c+d x)+35 \tanh ^4(c+d x)+21 \tanh ^2(c+d x)+15\right) \tanh ^2(c+d x)}{b}+315 \tanh ^8(c+d x)+105 \tanh ^6(c+d x)+63 \tanh ^4(c+d x)+45 \tanh ^2(c+d x)+35\right)}{315 d}","-\frac{b^3 \left(10 a^2+5 a b+b^2\right) \coth ^5(c+d x)}{5 d}-\frac{b^2 \left(10 a^3+10 a^2 b+5 a b^2+b^3\right) \coth ^3(c+d x)}{3 d}-\frac{b \left(5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right) \coth (c+d x)}{d}-\frac{b^4 (5 a+b) \coth ^7(c+d x)}{7 d}+x (a+b)^5-\frac{b^5 \coth ^9(c+d x)}{9 d}",1,"-1/315*(b^5*Coth[c + d*x]^9*(35 + 45*Tanh[c + d*x]^2 + 63*Tanh[c + d*x]^4 + 105*Tanh[c + d*x]^6 + 315*Tanh[c + d*x]^8 + (1575*a^4*Tanh[c + d*x]^8)/b^4 - (315*(a + b)^5*ArcTanh[Sqrt[Tanh[c + d*x]^2]]*Tanh[c + d*x]^10)/(b^5*Sqrt[Tanh[c + d*x]^2]) + (1050*a^3*Tanh[c + d*x]^6*(1 + 3*Tanh[c + d*x]^2))/b^3 + (210*a^2*Tanh[c + d*x]^4*(3 + 5*Tanh[c + d*x]^2 + 15*Tanh[c + d*x]^4))/b^2 + (15*a*Tanh[c + d*x]^2*(15 + 21*Tanh[c + d*x]^2 + 35*Tanh[c + d*x]^4 + 105*Tanh[c + d*x]^6))/b))/d","A",1
2,1,127,110,2.0826958,"\int \left(a+b \coth ^2(c+d x)\right)^4 \, dx","Integrate[(a + b*Coth[c + d*x]^2)^4,x]","-\frac{\coth (c+d x) \left(b \left(35 b \left(6 a^2+4 a b+b^2\right) \coth ^2(c+d x)+105 \left(4 a^3+6 a^2 b+4 a b^2+b^3\right)+21 b^2 (4 a+b) \coth ^4(c+d x)+15 b^3 \coth ^6(c+d x)\right)-105 (a+b)^4 \tanh ^{-1}\left(\sqrt{\tanh ^2(c+d x)}\right) \sqrt{\tanh ^2(c+d x)}\right)}{105 d}","-\frac{b^2 \left(6 a^2+4 a b+b^2\right) \coth ^3(c+d x)}{3 d}-\frac{b (2 a+b) \left(2 a^2+2 a b+b^2\right) \coth (c+d x)}{d}-\frac{b^3 (4 a+b) \coth ^5(c+d x)}{5 d}+x (a+b)^4-\frac{b^4 \coth ^7(c+d x)}{7 d}",1,"-1/105*(Coth[c + d*x]*(b*(105*(4*a^3 + 6*a^2*b + 4*a*b^2 + b^3) + 35*b*(6*a^2 + 4*a*b + b^2)*Coth[c + d*x]^2 + 21*b^2*(4*a + b)*Coth[c + d*x]^4 + 15*b^3*Coth[c + d*x]^6) - 105*(a + b)^4*ArcTanh[Sqrt[Tanh[c + d*x]^2]]*Sqrt[Tanh[c + d*x]^2]))/d","A",1
3,1,100,74,1.4493574,"\int \left(a+b \coth ^2(c+d x)\right)^3 \, dx","Integrate[(a + b*Coth[c + d*x]^2)^3,x]","\frac{(a+b)^3 \tanh ^{-1}\left(\sqrt{\tanh ^2(c+d x)}\right) \tanh (c+d x)}{d \sqrt{\tanh ^2(c+d x)}}-\frac{b \coth (c+d x) \left(15 \left(3 a^2+3 a b+b^2\right)+5 b (3 a+b) \coth ^2(c+d x)+3 b^2 \coth ^4(c+d x)\right)}{15 d}","-\frac{b \left(3 a^2+3 a b+b^2\right) \coth (c+d x)}{d}-\frac{b^2 (3 a+b) \coth ^3(c+d x)}{3 d}+x (a+b)^3-\frac{b^3 \coth ^5(c+d x)}{5 d}",1,"-1/15*(b*Coth[c + d*x]*(15*(3*a^2 + 3*a*b + b^2) + 5*b*(3*a + b)*Coth[c + d*x]^2 + 3*b^2*Coth[c + d*x]^4))/d + ((a + b)^3*ArcTanh[Sqrt[Tanh[c + d*x]^2]]*Tanh[c + d*x])/(d*Sqrt[Tanh[c + d*x]^2])","A",1
4,1,65,43,0.5065246,"\int \left(a+b \coth ^2(c+d x)\right)^2 \, dx","Integrate[(a + b*Coth[c + d*x]^2)^2,x]","-\frac{\coth (c+d x) \left(b \left(6 a+b \coth ^2(c+d x)+3 b\right)-3 (a+b)^2 \tanh ^{-1}\left(\sqrt{\tanh ^2(c+d x)}\right) \sqrt{\tanh ^2(c+d x)}\right)}{3 d}","-\frac{b (2 a+b) \coth (c+d x)}{d}+x (a+b)^2-\frac{b^2 \coth ^3(c+d x)}{3 d}",1,"-1/3*(Coth[c + d*x]*(b*(6*a + 3*b + b*Coth[c + d*x]^2) - 3*(a + b)^2*ArcTanh[Sqrt[Tanh[c + d*x]^2]]*Sqrt[Tanh[c + d*x]^2]))/d","A",1
5,1,47,46,0.0626442,"\int \frac{1}{a+b \coth ^2(c+d x)} \, dx","Integrate[(a + b*Coth[c + d*x]^2)^(-1),x]","\frac{\tanh ^{-1}(\tanh (c+d x))-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{b}}\right)}{\sqrt{a}}}{d (a+b)}","\frac{x}{a+b}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{b}}\right)}{\sqrt{a} d (a+b)}",1,"(-((Sqrt[b]*ArcTan[(Sqrt[a]*Tanh[c + d*x])/Sqrt[b]])/Sqrt[a]) + ArcTanh[Tanh[c + d*x]])/((a + b)*d)","A",1
6,1,97,89,0.6094463,"\int \frac{1}{\left(a+b \coth ^2(c+d x)\right)^2} \, dx","Integrate[(a + b*Coth[c + d*x]^2)^(-2),x]","\frac{\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \coth (c+d x)}{\sqrt{a}}\right)}{a^{3/2}}+\frac{b (a+b) \coth (c+d x)}{a \left(a+b \coth ^2(c+d x)\right)}-\log (1-\coth (c+d x))+\log (\coth (c+d x)+1)}{2 d (a+b)^2}","-\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{b}}\right)}{2 a^{3/2} d (a+b)^2}+\frac{b \coth (c+d x)}{2 a d (a+b) \left(a+b \coth ^2(c+d x)\right)}+\frac{x}{(a+b)^2}",1,"((Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[b]*Coth[c + d*x])/Sqrt[a]])/a^(3/2) + (b*(a + b)*Coth[c + d*x])/(a*(a + b*Coth[c + d*x]^2)) - Log[1 - Coth[c + d*x]] + Log[1 + Coth[c + d*x]])/(2*(a + b)^2*d)","A",1
7,1,147,142,0.3605208,"\int \frac{1}{\left(a+b \coth ^2(c+d x)\right)^3} \, dx","Integrate[(a + b*Coth[c + d*x]^2)^(-3),x]","\frac{\frac{b (7 a+3 b) (a+b) \coth (c+d x)}{a^2 \left(a+b \coth ^2(c+d x)\right)}+\frac{\sqrt{b} \left(15 a^2+10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \coth (c+d x)}{\sqrt{a}}\right)}{a^{5/2}}+\frac{2 b (a+b)^2 \coth (c+d x)}{a \left(a+b \coth ^2(c+d x)\right)^2}-4 \log (1-\coth (c+d x))+4 \log (\coth (c+d x)+1)}{8 d (a+b)^3}","\frac{b (7 a+3 b) \coth (c+d x)}{8 a^2 d (a+b)^2 \left(a+b \coth ^2(c+d x)\right)}-\frac{\sqrt{b} \left(15 a^2+10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{b}}\right)}{8 a^{5/2} d (a+b)^3}+\frac{b \coth (c+d x)}{4 a d (a+b) \left(a+b \coth ^2(c+d x)\right)^2}+\frac{x}{(a+b)^3}",1,"((Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Coth[c + d*x])/Sqrt[a]])/a^(5/2) + (2*b*(a + b)^2*Coth[c + d*x])/(a*(a + b*Coth[c + d*x]^2)^2) + (b*(a + b)*(7*a + 3*b)*Coth[c + d*x])/(a^2*(a + b*Coth[c + d*x]^2)) - 4*Log[1 - Coth[c + d*x]] + 4*Log[1 + Coth[c + d*x]])/(8*(a + b)^3*d)","A",1
8,1,203,201,0.6330176,"\int \frac{1}{\left(a+b \coth ^2(c+d x)\right)^4} \, dx","Integrate[(a + b*Coth[c + d*x]^2)^(-4),x]","\frac{\frac{2 b (11 a+5 b) (a+b)^2 \coth (c+d x)}{a^2 \left(a+b \coth ^2(c+d x)\right)^2}+\frac{3 b \left(19 a^2+16 a b+5 b^2\right) (a+b) \coth (c+d x)}{a^3 \left(a+b \coth ^2(c+d x)\right)}+\frac{3 \sqrt{b} \left(35 a^3+35 a^2 b+21 a b^2+5 b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \coth (c+d x)}{\sqrt{a}}\right)}{a^{7/2}}+\frac{8 b (a+b)^3 \coth (c+d x)}{a \left(a+b \coth ^2(c+d x)\right)^3}-24 \log (1-\coth (c+d x))+24 \log (\coth (c+d x)+1)}{48 d (a+b)^4}","\frac{b (11 a+5 b) \coth (c+d x)}{24 a^2 d (a+b)^2 \left(a+b \coth ^2(c+d x)\right)^2}+\frac{b \left(19 a^2+16 a b+5 b^2\right) \coth (c+d x)}{16 a^3 d (a+b)^3 \left(a+b \coth ^2(c+d x)\right)}-\frac{\sqrt{b} \left(35 a^3+35 a^2 b+21 a b^2+5 b^3\right) \tan ^{-1}\left(\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{b}}\right)}{16 a^{7/2} d (a+b)^4}+\frac{b \coth (c+d x)}{6 a d (a+b) \left(a+b \coth ^2(c+d x)\right)^3}+\frac{x}{(a+b)^4}",1,"((3*Sqrt[b]*(35*a^3 + 35*a^2*b + 21*a*b^2 + 5*b^3)*ArcTan[(Sqrt[b]*Coth[c + d*x])/Sqrt[a]])/a^(7/2) + (8*b*(a + b)^3*Coth[c + d*x])/(a*(a + b*Coth[c + d*x]^2)^3) + (2*b*(a + b)^2*(11*a + 5*b)*Coth[c + d*x])/(a^2*(a + b*Coth[c + d*x]^2)^2) + (3*b*(a + b)*(19*a^2 + 16*a*b + 5*b^2)*Coth[c + d*x])/(a^3*(a + b*Coth[c + d*x]^2)) - 24*Log[1 - Coth[c + d*x]] + 24*Log[1 + Coth[c + d*x]])/(48*(a + b)^4*d)","A",1
9,1,19,19,0.0988197,"\int \frac{1}{1-2 \coth ^2(x)} \, dx","Integrate[(1 - 2*Coth[x]^2)^(-1),x]","\sqrt{2} \tanh ^{-1}\left(\frac{\tanh (x)}{\sqrt{2}}\right)-x","\sqrt{2} \tanh ^{-1}\left(\frac{\tanh (x)}{\sqrt{2}}\right)-x",1,"-x + Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]","A",1
10,1,20,3,0.0068259,"\int \sqrt{1-\coth ^2(x)} \, dx","Integrate[Sqrt[1 - Coth[x]^2],x]","\sinh (x) \sqrt{-\text{csch}^2(x)} \log \left(\tanh \left(\frac{x}{2}\right)\right)","\sin ^{-1}(\coth (x))",1,"Sqrt[-Csch[x]^2]*Log[Tanh[x/2]]*Sinh[x]","B",1
11,1,18,14,0.0076232,"\int \sqrt{-1+\coth ^2(x)} \, dx","Integrate[Sqrt[-1 + Coth[x]^2],x]","\sinh (x) \sqrt{\text{csch}^2(x)} \log \left(\tanh \left(\frac{x}{2}\right)\right)","-\tanh ^{-1}\left(\frac{\coth (x)}{\sqrt{\text{csch}^2(x)}}\right)",1,"Sqrt[Csch[x]^2]*Log[Tanh[x/2]]*Sinh[x]","A",1
12,1,41,24,0.0608682,"\int \left(1-\coth ^2(x)\right)^{3/2} \, dx","Integrate[(1 - Coth[x]^2)^(3/2),x]","\frac{1}{4} \text{csch}\left(\frac{x}{2}\right) \sqrt{-\text{csch}^2(x)} \text{sech}\left(\frac{x}{2}\right) \left(\cosh (x)+\sinh ^2(x) \log \left(\tanh \left(\frac{x}{2}\right)\right)\right)","\frac{1}{2} \sin ^{-1}(\coth (x))+\frac{1}{2} \coth (x) \sqrt{-\text{csch}^2(x)}",1,"(Csch[x/2]*Sqrt[-Csch[x]^2]*Sech[x/2]*(Cosh[x] + Log[Tanh[x/2]]*Sinh[x]^2))/4","A",1
13,1,40,31,0.0401757,"\int \left(-1+\coth ^2(x)\right)^{3/2} \, dx","Integrate[(-1 + Coth[x]^2)^(3/2),x]","-\frac{1}{8} \sinh (x) \sqrt{\text{csch}^2(x)} \left(\text{csch}^2\left(\frac{x}{2}\right)+\text{sech}^2\left(\frac{x}{2}\right)+4 \log \left(\tanh \left(\frac{x}{2}\right)\right)\right)","\frac{1}{2} \tanh ^{-1}\left(\frac{\coth (x)}{\sqrt{\text{csch}^2(x)}}\right)-\frac{1}{2} \coth (x) \sqrt{\text{csch}^2(x)}",1,"-1/8*(Sqrt[Csch[x]^2]*(Csch[x/2]^2 + 4*Log[Tanh[x/2]] + Sech[x/2]^2)*Sinh[x])","A",1
14,1,13,13,0.0075812,"\int \frac{1}{\sqrt{1-\coth ^2(x)}} \, dx","Integrate[1/Sqrt[1 - Coth[x]^2],x]","\frac{\coth (x)}{\sqrt{-\text{csch}^2(x)}}","\frac{\coth (x)}{\sqrt{-\text{csch}^2(x)}}",1,"Coth[x]/Sqrt[-Csch[x]^2]","A",1
15,1,11,11,0.0079812,"\int \frac{1}{\sqrt{-1+\coth ^2(x)}} \, dx","Integrate[1/Sqrt[-1 + Coth[x]^2],x]","\frac{\coth (x)}{\sqrt{\text{csch}^2(x)}}","\frac{\coth (x)}{\sqrt{\text{csch}^2(x)}}",1,"Coth[x]/Sqrt[Csch[x]^2]","A",1
16,1,60,63,0.1860337,"\int \coth ^3(x) \sqrt{a+b \coth ^2(x)} \, dx","Integrate[Coth[x]^3*Sqrt[a + b*Coth[x]^2],x]","\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)-\frac{\sqrt{a+b \coth ^2(x)} \left(a+b \coth ^2(x)+3 b\right)}{3 b}","-\frac{\left(a+b \coth ^2(x)\right)^{3/2}}{3 b}-\sqrt{a+b \coth ^2(x)}+\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)",1,"Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - (Sqrt[a + b*Coth[x]^2]*(a + 3*b + b*Coth[x]^2))/(3*b)","A",1
17,1,191,85,0.8464166,"\int \coth ^2(x) \sqrt{a+b \coth ^2(x)} \, dx","Integrate[Coth[x]^2*Sqrt[a + b*Coth[x]^2],x]","-\frac{\sinh (x) \sqrt{\text{csch}^2(x) ((a+b) \cosh (2 x)-a+b)} \left(\sqrt{2} \sqrt{a+b} (a+2 b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \cosh (x)}{\sqrt{(a+b) \cosh (2 x)-a+b}}\right)+\sqrt{b} \left(\sqrt{a+b} \coth (x) \text{csch}(x) \sqrt{(a+b) \cosh (2 x)-a+b}-2 \sqrt{2} (a+b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{(a+b) \cosh (2 x)-a+b}}\right)\right)\right)}{2 \sqrt{2} \sqrt{b} \sqrt{a+b} \sqrt{(a+b) \cosh (2 x)-a+b}}","-\frac{1}{2} \coth (x) \sqrt{a+b \coth ^2(x)}-\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)}{2 \sqrt{b}}+\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)",1,"-1/2*(Sqrt[(-a + b + (a + b)*Cosh[2*x])*Csch[x]^2]*(Sqrt[2]*Sqrt[a + b]*(a + 2*b)*ArcTanh[(Sqrt[2]*Sqrt[b]*Cosh[x])/Sqrt[-a + b + (a + b)*Cosh[2*x]]] + Sqrt[b]*(-2*Sqrt[2]*(a + b)*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[-a + b + (a + b)*Cosh[2*x]]] + Sqrt[a + b]*Sqrt[-a + b + (a + b)*Cosh[2*x]]*Coth[x]*Csch[x]))*Sinh[x])/(Sqrt[2]*Sqrt[b]*Sqrt[a + b]*Sqrt[-a + b + (a + b)*Cosh[2*x]])","B",1
18,1,44,44,0.0318287,"\int \coth (x) \sqrt{a+b \coth ^2(x)} \, dx","Integrate[Coth[x]*Sqrt[a + b*Coth[x]^2],x]","\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)-\sqrt{a+b \coth ^2(x)}","\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)-\sqrt{a+b \coth ^2(x)}",1,"Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - Sqrt[a + b*Coth[x]^2]","A",1
19,1,137,60,0.2208124,"\int \sqrt{a+b \coth ^2(x)} \, dx","Integrate[Sqrt[a + b*Coth[x]^2],x]","\frac{1}{2} \left(-2 \sqrt{b} \log \left(\sqrt{b} \sqrt{a+b \coth ^2(x)}+b \coth (x)\right)-\sqrt{a+b} \log \left(\sqrt{a+b} \sqrt{a+b \coth ^2(x)}+a-b \coth (x)\right)+\sqrt{a+b} \log \left(\sqrt{a+b} \sqrt{a+b \coth ^2(x)}+a+b \coth (x)\right)-\sqrt{a+b} \log (1-\coth (x))+\sqrt{a+b} \log (\coth (x)+1)\right)","\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)-\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)",1,"(-(Sqrt[a + b]*Log[1 - Coth[x]]) + Sqrt[a + b]*Log[1 + Coth[x]] - 2*Sqrt[b]*Log[b*Coth[x] + Sqrt[b]*Sqrt[a + b*Coth[x]^2]] - Sqrt[a + b]*Log[a - b*Coth[x] + Sqrt[a + b]*Sqrt[a + b*Coth[x]^2]] + Sqrt[a + b]*Log[a + b*Coth[x] + Sqrt[a + b]*Sqrt[a + b*Coth[x]^2]])/2","B",1
20,1,56,56,0.032688,"\int \sqrt{a+b \coth ^2(x)} \tanh (x) \, dx","Integrate[Sqrt[a + b*Coth[x]^2]*Tanh[x],x]","\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a}}\right)","\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a}}\right)",1,"-(Sqrt[a]*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a]]) + Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]","A",1
21,1,42,48,0.0921013,"\int \sqrt{a+b \coth ^2(x)} \tanh ^2(x) \, dx","Integrate[Sqrt[a + b*Coth[x]^2]*Tanh[x]^2,x]","\tanh (x) \left(-\sqrt{a+b \coth ^2(x)}\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{(a+b) \coth ^2(x)}{b \coth ^2(x)+a}\right)","\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)-\tanh (x) \sqrt{a+b \coth ^2(x)}",1,"-(Sqrt[a + b*Coth[x]^2]*Hypergeometric2F1[-1/2, 1, 1/2, ((a + b)*Coth[x]^2)/(a + b*Coth[x]^2)]*Tanh[x])","C",1
22,1,86,82,0.4311889,"\int \coth ^3(x) \left(a+b \coth ^2(x)\right)^{3/2} \, dx","Integrate[Coth[x]^3*(a + b*Coth[x]^2)^(3/2),x]","(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)-\frac{\sqrt{a+b \coth ^2(x)} \left(3 a^2+b (6 a+5 b) \coth ^2(x)+20 a b+3 b^2 \coth ^4(x)+15 b^2\right)}{15 b}","-\frac{\left(a+b \coth ^2(x)\right)^{5/2}}{5 b}-\frac{1}{3} \left(a+b \coth ^2(x)\right)^{3/2}-(a+b) \sqrt{a+b \coth ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)",1,"(a + b)^(3/2)*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - (Sqrt[a + b*Coth[x]^2]*(3*a^2 + 20*a*b + 15*b^2 + b*(6*a + 5*b)*Coth[x]^2 + 3*b^2*Coth[x]^4))/(15*b)","A",1
23,1,219,123,0.9781399,"\int \coth ^2(x) \left(a+b \coth ^2(x)\right)^{3/2} \, dx","Integrate[Coth[x]^2*(a + b*Coth[x]^2)^(3/2),x]","\frac{\sinh (x) \sqrt{\text{csch}^2(x) ((a+b) \cosh (2 x)-a+b)} \left(\sqrt{b} \left(8 \sqrt{2} (a+b)^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{(a+b) \cosh (2 x)-a+b}}\right)-\sqrt{a+b} \coth (x) \text{csch}(x) \sqrt{(a+b) \cosh (2 x)-a+b} \left(5 a+2 b \text{csch}^2(x)+6 b\right)\right)-\sqrt{2} \sqrt{a+b} \left(3 a^2+12 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \cosh (x)}{\sqrt{(a+b) \cosh (2 x)-a+b}}\right)\right)}{8 \sqrt{2} \sqrt{b} \sqrt{a+b} \sqrt{(a+b) \cosh (2 x)-a+b}}","-\frac{\left(3 a^2+12 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)}{8 \sqrt{b}}-\frac{1}{8} (5 a+4 b) \coth (x) \sqrt{a+b \coth ^2(x)}-\frac{1}{4} b \coth ^3(x) \sqrt{a+b \coth ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)",1,"(Sqrt[(-a + b + (a + b)*Cosh[2*x])*Csch[x]^2]*(-(Sqrt[2]*Sqrt[a + b]*(3*a^2 + 12*a*b + 8*b^2)*ArcTanh[(Sqrt[2]*Sqrt[b]*Cosh[x])/Sqrt[-a + b + (a + b)*Cosh[2*x]]]) + Sqrt[b]*(8*Sqrt[2]*(a + b)^2*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[-a + b + (a + b)*Cosh[2*x]]] - Sqrt[a + b]*Sqrt[-a + b + (a + b)*Cosh[2*x]]*Coth[x]*Csch[x]*(5*a + 6*b + 2*b*Csch[x]^2)))*Sinh[x])/(8*Sqrt[2]*Sqrt[b]*Sqrt[a + b]*Sqrt[-a + b + (a + b)*Cosh[2*x]])","A",1
24,1,59,63,0.1635042,"\int \coth (x) \left(a+b \coth ^2(x)\right)^{3/2} \, dx","Integrate[Coth[x]*(a + b*Coth[x]^2)^(3/2),x]","(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)-\frac{1}{3} \sqrt{a+b \coth ^2(x)} \left(4 a+b \coth ^2(x)+3 b\right)","-(a+b) \sqrt{a+b \coth ^2(x)}-\frac{1}{3} \left(a+b \coth ^2(x)\right)^{3/2}+(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)",1,"(a + b)^(3/2)*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - (Sqrt[a + b*Coth[x]^2]*(4*a + 3*b + b*Coth[x]^2))/3","A",1
25,1,161,88,0.2894715,"\int \left(a+b \coth ^2(x)\right)^{3/2} \, dx","Integrate[(a + b*Coth[x]^2)^(3/2),x]","\frac{1}{2} \left(-b \coth (x) \sqrt{a+b \coth ^2(x)}-(a+b)^{3/2} \log \left(\sqrt{a+b} \sqrt{a+b \coth ^2(x)}+a-b \coth (x)\right)+(a+b)^{3/2} \log \left(\sqrt{a+b} \sqrt{a+b \coth ^2(x)}+a+b \coth (x)\right)-\sqrt{b} (3 a+2 b) \log \left(\sqrt{b} \sqrt{a+b \coth ^2(x)}+b \coth (x)\right)+(a+b)^{3/2} (-\log (1-\coth (x)))+(a+b)^{3/2} \log (\coth (x)+1)\right)","-\frac{1}{2} b \coth (x) \sqrt{a+b \coth ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)-\frac{1}{2} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)",1,"(-(b*Coth[x]*Sqrt[a + b*Coth[x]^2]) - (a + b)^(3/2)*Log[1 - Coth[x]] + (a + b)^(3/2)*Log[1 + Coth[x]] - Sqrt[b]*(3*a + 2*b)*Log[b*Coth[x] + Sqrt[b]*Sqrt[a + b*Coth[x]^2]] - (a + b)^(3/2)*Log[a - b*Coth[x] + Sqrt[a + b]*Sqrt[a + b*Coth[x]^2]] + (a + b)^(3/2)*Log[a + b*Coth[x] + Sqrt[a + b]*Sqrt[a + b*Coth[x]^2]])/2","A",1
26,1,71,71,0.062057,"\int \left(a+b \coth ^2(x)\right)^{3/2} \tanh (x) \, dx","Integrate[(a + b*Coth[x]^2)^(3/2)*Tanh[x],x]","a^{3/2} \left(-\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a}}\right)\right)-b \sqrt{a+b \coth ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)","a^{3/2} \left(-\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a}}\right)\right)-b \sqrt{a+b \coth ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)",1,"-(a^(3/2)*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a]]) + (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - b*Sqrt[a + b*Coth[x]^2]","A",1
27,1,180,77,0.4723513,"\int \left(a+b \coth ^2(x)\right)^{3/2} \tanh ^2(x) \, dx","Integrate[(a + b*Coth[x]^2)^(3/2)*Tanh[x]^2,x]","\frac{\tanh (x) \left(-\sqrt{2} b^{3/2} \sqrt{a+b} \cosh (x) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \cosh (x)}{\sqrt{(a+b) \cosh (2 x)-a+b}}\right)-a \sqrt{a+b} \sqrt{(a+b) \cosh (2 x)-a+b}+\sqrt{2} (a+b)^2 \cosh (x) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{(a+b) \cosh (2 x)-a+b}}\right)\right) \sqrt{\text{csch}^2(x) ((a+b) \cosh (2 x)-a+b)}}{\sqrt{2} \sqrt{a+b} \sqrt{(a+b) \cosh (2 x)-a+b}}","b^{3/2} \left(-\tanh ^{-1}\left(\frac{\sqrt{b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)\right)+(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)-a \tanh (x) \sqrt{a+b \coth ^2(x)}",1,"((-(Sqrt[2]*b^(3/2)*Sqrt[a + b]*ArcTanh[(Sqrt[2]*Sqrt[b]*Cosh[x])/Sqrt[-a + b + (a + b)*Cosh[2*x]]]*Cosh[x]) + Sqrt[2]*(a + b)^2*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[-a + b + (a + b)*Cosh[2*x]]]*Cosh[x] - a*Sqrt[a + b]*Sqrt[-a + b + (a + b)*Cosh[2*x]])*Sqrt[(-a + b + (a + b)*Cosh[2*x])*Csch[x]^2]*Tanh[x])/(Sqrt[2]*Sqrt[a + b]*Sqrt[-a + b + (a + b)*Cosh[2*x]])","B",1
28,1,60,31,0.0758238,"\int \sqrt{1+\coth ^2(x)} \, dx","Integrate[Sqrt[1 + Coth[x]^2],x]","\frac{\sinh (x) \sqrt{\coth ^2(x)+1} \left(\sqrt{2} \log \left(\sqrt{2} \cosh (x)+\sqrt{\cosh (2 x)}\right)-\tanh ^{-1}\left(\frac{\cosh (x)}{\sqrt{\cosh (2 x)}}\right)\right)}{\sqrt{\cosh (2 x)}}","\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \coth (x)}{\sqrt{\coth ^2(x)+1}}\right)-\sinh ^{-1}(\coth (x))",1,"(Sqrt[1 + Coth[x]^2]*(-ArcTanh[Cosh[x]/Sqrt[Cosh[2*x]]] + Sqrt[2]*Log[Sqrt[2]*Cosh[x] + Sqrt[Cosh[2*x]]])*Sinh[x])/Sqrt[Cosh[2*x]]","A",1
29,1,62,45,0.0511888,"\int \sqrt{-1-\coth ^2(x)} \, dx","Integrate[Sqrt[-1 - Coth[x]^2],x]","\frac{\sinh (x) \sqrt{-\coth ^2(x)-1} \left(\sqrt{2} \log \left(\sqrt{2} \cosh (x)+\sqrt{\cosh (2 x)}\right)-\tanh ^{-1}\left(\frac{\cosh (x)}{\sqrt{\cosh (2 x)}}\right)\right)}{\sqrt{\cosh (2 x)}}","\tan ^{-1}\left(\frac{\coth (x)}{\sqrt{-\coth ^2(x)-1}}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \coth (x)}{\sqrt{-\coth ^2(x)-1}}\right)",1,"(Sqrt[-1 - Coth[x]^2]*(-ArcTanh[Cosh[x]/Sqrt[Cosh[2*x]]] + Sqrt[2]*Log[Sqrt[2]*Cosh[x] + Sqrt[Cosh[2*x]]])*Sinh[x])/Sqrt[Cosh[2*x]]","A",1
30,1,116,50,0.3448468,"\int \left(1+\coth ^2(x)\right)^{3/2} \, dx","Integrate[(1 + Coth[x]^2)^(3/2),x]","-\frac{1}{8} \left(\coth ^2(x)+1\right)^{3/2} \text{sech}^2(2 x) \left(\sinh (4 x)+16 \sinh ^3(x) \sqrt{\cosh (2 x)} \tanh ^{-1}\left(\frac{\cosh (x)}{\sqrt{\cosh (2 x)}}\right)+4 \sinh ^3(x) \left(\sqrt{-\cosh (2 x)} \tan ^{-1}\left(\frac{\cosh (x)}{\sqrt{-\cosh (2 x)}}\right)-4 \sqrt{2} \sqrt{\cosh (2 x)} \log \left(\sqrt{2} \cosh (x)+\sqrt{\cosh (2 x)}\right)\right)\right)","-\frac{1}{2} \coth (x) \sqrt{\coth ^2(x)+1}+2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \coth (x)}{\sqrt{\coth ^2(x)+1}}\right)-\frac{5}{2} \sinh ^{-1}(\coth (x))",1,"-1/8*((1 + Coth[x]^2)^(3/2)*Sech[2*x]^2*(16*ArcTanh[Cosh[x]/Sqrt[Cosh[2*x]]]*Sqrt[Cosh[2*x]]*Sinh[x]^3 + 4*(ArcTan[Cosh[x]/Sqrt[-Cosh[2*x]]]*Sqrt[-Cosh[2*x]] - 4*Sqrt[2]*Sqrt[Cosh[2*x]]*Log[Sqrt[2]*Cosh[x] + Sqrt[Cosh[2*x]]])*Sinh[x]^3 + Sinh[4*x]))","B",1
31,1,118,67,0.1737261,"\int \left(-1-\coth ^2(x)\right)^{3/2} \, dx","Integrate[(-1 - Coth[x]^2)^(3/2),x]","-\frac{1}{8} \left(-\coth ^2(x)-1\right)^{3/2} \text{sech}^2(2 x) \left(\sinh (4 x)+16 \sinh ^3(x) \sqrt{\cosh (2 x)} \tanh ^{-1}\left(\frac{\cosh (x)}{\sqrt{\cosh (2 x)}}\right)+4 \sinh ^3(x) \left(\sqrt{-\cosh (2 x)} \tan ^{-1}\left(\frac{\cosh (x)}{\sqrt{-\cosh (2 x)}}\right)-4 \sqrt{2} \sqrt{\cosh (2 x)} \log \left(\sqrt{2} \cosh (x)+\sqrt{\cosh (2 x)}\right)\right)\right)","\frac{1}{2} \coth (x) \sqrt{-\coth ^2(x)-1}-\frac{5}{2} \tan ^{-1}\left(\frac{\coth (x)}{\sqrt{-\coth ^2(x)-1}}\right)+2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \coth (x)}{\sqrt{-\coth ^2(x)-1}}\right)",1,"-1/8*((-1 - Coth[x]^2)^(3/2)*Sech[2*x]^2*(16*ArcTanh[Cosh[x]/Sqrt[Cosh[2*x]]]*Sqrt[Cosh[2*x]]*Sinh[x]^3 + 4*(ArcTan[Cosh[x]/Sqrt[-Cosh[2*x]]]*Sqrt[-Cosh[2*x]] - 4*Sqrt[2]*Sqrt[Cosh[2*x]]*Log[Sqrt[2]*Cosh[x] + Sqrt[Cosh[2*x]]])*Sinh[x]^3 + Sinh[4*x]))","A",1
32,1,47,47,0.1034463,"\int \frac{\coth ^3(x)}{\sqrt{a+b \coth ^2(x)}} \, dx","Integrate[Coth[x]^3/Sqrt[a + b*Coth[x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}-\frac{\sqrt{a+b \coth ^2(x)}}{b}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}-\frac{\sqrt{a+b \coth ^2(x)}}{b}",1,"ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/Sqrt[a + b] - Sqrt[a + b*Coth[x]^2]/b","A",1
33,1,134,60,0.2485588,"\int \frac{\coth ^2(x)}{\sqrt{a+b \coth ^2(x)}} \, dx","Integrate[Coth[x]^2/Sqrt[a + b*Coth[x]^2],x]","\frac{\sinh (x) \left(\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{(a+b) \cosh (2 x)-a+b}}\right)-\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \cosh (x)}{\sqrt{(a+b) \cosh (2 x)-a+b}}\right)\right) \sqrt{\text{csch}^2(x) ((a+b) \cosh (2 x)-a+b)}}{\sqrt{b} \sqrt{a+b} \sqrt{(a+b) \cosh (2 x)-a+b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)}{\sqrt{a+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)}{\sqrt{b}}",1,"((-(Sqrt[a + b]*ArcTanh[(Sqrt[2]*Sqrt[b]*Cosh[x])/Sqrt[-a + b + (a + b)*Cosh[2*x]]]) + Sqrt[b]*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[-a + b + (a + b)*Cosh[2*x]]])*Sqrt[(-a + b + (a + b)*Cosh[2*x])*Csch[x]^2]*Sinh[x])/(Sqrt[b]*Sqrt[a + b]*Sqrt[-a + b + (a + b)*Cosh[2*x]])","B",1
34,1,29,29,0.0147257,"\int \frac{\coth (x)}{\sqrt{a+b \coth ^2(x)}} \, dx","Integrate[Coth[x]/Sqrt[a + b*Coth[x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}",1,"ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/Sqrt[a + b]","A",1
35,1,77,31,0.1235367,"\int \frac{1}{\sqrt{a+b \coth ^2(x)}} \, dx","Integrate[1/Sqrt[a + b*Coth[x]^2],x]","\frac{\coth (x) \sqrt{\frac{b \coth ^2(x)}{a}+1} \tanh ^{-1}\left(\frac{\sqrt{\frac{(a+b) \coth ^2(x)}{a}}}{\sqrt{\frac{b \coth ^2(x)}{a}+1}}\right)}{\sqrt{\frac{(a+b) \coth ^2(x)}{a}} \sqrt{a+b \coth ^2(x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)}{\sqrt{a+b}}",1,"(ArcTanh[Sqrt[((a + b)*Coth[x]^2)/a]/Sqrt[1 + (b*Coth[x]^2)/a]]*Coth[x]*Sqrt[1 + (b*Coth[x]^2)/a])/(Sqrt[((a + b)*Coth[x]^2)/a]*Sqrt[a + b*Coth[x]^2])","B",1
36,1,56,56,0.0494128,"\int \frac{\tanh (x)}{\sqrt{a+b \coth ^2(x)}} \, dx","Integrate[Tanh[x]/Sqrt[a + b*Coth[x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"-(ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a]]/Sqrt[a]) + ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/Sqrt[a + b]","A",1
37,1,132,51,1.4569905,"\int \frac{\tanh ^2(x)}{\sqrt{a+b \coth ^2(x)}} \, dx","Integrate[Tanh[x]^2/Sqrt[a + b*Coth[x]^2],x]","\frac{\sinh ^2(x) \tanh (x) \left(\frac{b \coth ^2(x)}{a}+1\right) \left(\frac{3 a \left(a+2 b \coth ^2(x)\right) \sin ^{-1}\left(\sqrt{\frac{(a+b) \cosh ^2(x)}{a}}\right)}{\sqrt{-\frac{(a+b) \sinh ^2(x) \cosh ^2(x) \left(a+b \coth ^2(x)\right)}{a^2}}}+2 (a+b) \coth ^2(x) ((a+b) \cosh (2 x)-a+b) \, _2F_1\left(2,2;\frac{5}{2};\frac{(a+b) \cosh ^2(x)}{a}\right)\right)}{3 a^2 \sqrt{a+b \coth ^2(x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)}{\sqrt{a+b}}-\frac{\tanh (x) \sqrt{a+b \coth ^2(x)}}{a}",1,"((1 + (b*Coth[x]^2)/a)*Sinh[x]^2*(2*(a + b)*(-a + b + (a + b)*Cosh[2*x])*Coth[x]^2*Hypergeometric2F1[2, 2, 5/2, ((a + b)*Cosh[x]^2)/a] + (3*a*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*(a + 2*b*Coth[x]^2))/Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)])*Tanh[x])/(3*a^2*Sqrt[a + b*Coth[x]^2])","C",0
38,1,52,52,0.1311125,"\int \frac{\coth ^3(x)}{\left(a+b \coth ^2(x)\right)^{3/2}} \, dx","Integrate[Coth[x]^3/(a + b*Coth[x]^2)^(3/2),x]","\frac{a}{b (a+b) \sqrt{a+b \coth ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}","\frac{a}{b (a+b) \sqrt{a+b \coth ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}",1,"ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) + a/(b*(a + b)*Sqrt[a + b*Coth[x]^2])","A",1
39,1,109,53,0.9959519,"\int \frac{\coth ^2(x)}{\left(a+b \coth ^2(x)\right)^{3/2}} \, dx","Integrate[Coth[x]^2/(a + b*Coth[x]^2)^(3/2),x]","\frac{\frac{\text{csch}(x) \text{sech}(x) ((a+b) \cosh (2 x)-a+b) \sqrt{\frac{(a+b) \coth ^2(x)}{a}} \tanh ^{-1}\left(\frac{\sqrt{\frac{(a+b) \coth ^2(x)}{a}}}{\sqrt{\frac{b \coth ^2(x)}{a}+1}}\right)}{\sqrt{\frac{b \coth ^2(x)}{a}+1}}-2 (a+b) \coth (x)}{2 (a+b)^2 \sqrt{a+b \coth ^2(x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)}{(a+b)^{3/2}}-\frac{\coth (x)}{(a+b) \sqrt{a+b \coth ^2(x)}}",1,"(-2*(a + b)*Coth[x] + (ArcTanh[Sqrt[((a + b)*Coth[x]^2)/a]/Sqrt[1 + (b*Coth[x]^2)/a]]*(-a + b + (a + b)*Cosh[2*x])*Sqrt[((a + b)*Coth[x]^2)/a]*Csch[x]*Sech[x])/Sqrt[1 + (b*Coth[x]^2)/a])/(2*(a + b)^2*Sqrt[a + b*Coth[x]^2])","B",1
40,1,41,49,0.0420868,"\int \frac{\coth (x)}{\left(a+b \coth ^2(x)\right)^{3/2}} \, dx","Integrate[Coth[x]/(a + b*Coth[x]^2)^(3/2),x]","-\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \coth ^2(x)+a}{a+b}\right)}{(a+b) \sqrt{a+b \coth ^2(x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}-\frac{1}{(a+b) \sqrt{a+b \coth ^2(x)}}",1,"-(Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Coth[x]^2)/(a + b)]/((a + b)*Sqrt[a + b*Coth[x]^2]))","C",1
41,1,70,78,0.0576316,"\int \frac{\tanh (x)}{\left(a+b \coth ^2(x)\right)^{3/2}} \, dx","Integrate[Tanh[x]/(a + b*Coth[x]^2)^(3/2),x]","\frac{(a+b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \coth ^2(x)}{a}+1\right)-a \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \coth ^2(x)+a}{a+b}\right)}{a (a+b) \sqrt{a+b \coth ^2(x)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{b}{a (a+b) \sqrt{a+b \coth ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}",1,"(-(a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Coth[x]^2)/(a + b)]) + (a + b)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Coth[x]^2)/a])/(a*(a + b)*Sqrt[a + b*Coth[x]^2])","C",1
42,1,149,85,6.0066872,"\int \frac{\tanh ^2(x)}{\left(a+b \coth ^2(x)\right)^{3/2}} \, dx","Integrate[Tanh[x]^2/(a + b*Coth[x]^2)^(3/2),x]","\frac{1}{2} \sqrt{\text{csch}^2(x) ((a+b) \cosh (2 x)-a+b)} \left(\frac{2 \sinh (x) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{(a+b) \cosh (2 x)-a+b}}\right)}{(a+b)^{3/2} \sqrt{(a+b) \cosh (2 x)-a+b}}-\frac{\sqrt{2} \tanh (x) \left(\left(a^2+2 a b+2 b^2\right) \cosh (2 x)-a^2+2 b^2\right)}{a^2 (a+b) ((a+b) \cosh (2 x)-a+b)}\right)","-\frac{(a+2 b) \tanh (x) \sqrt{a+b \coth ^2(x)}}{a^2 (a+b)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)}{(a+b)^{3/2}}+\frac{b \tanh (x)}{a (a+b) \sqrt{a+b \coth ^2(x)}}",1,"(Sqrt[(-a + b + (a + b)*Cosh[2*x])*Csch[x]^2]*((2*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[-a + b + (a + b)*Cosh[2*x]]]*Sinh[x])/((a + b)^(3/2)*Sqrt[-a + b + (a + b)*Cosh[2*x]]) - (Sqrt[2]*(-a^2 + 2*b^2 + (a^2 + 2*a*b + 2*b^2)*Cosh[2*x])*Tanh[x])/(a^2*(a + b)*(-a + b + (a + b)*Cosh[2*x]))))/2","A",0
43,1,63,74,0.1008097,"\int \frac{\coth ^3(x)}{\left(a+b \coth ^2(x)\right)^{5/2}} \, dx","Integrate[Coth[x]^3/(a + b*Coth[x]^2)^(5/2),x]","\frac{a (a+b)-3 b \left(a+b \coth ^2(x)\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \coth ^2(x)+a}{a+b}\right)}{3 b (a+b)^2 \left(a+b \coth ^2(x)\right)^{3/2}}","\frac{a}{3 b (a+b) \left(a+b \coth ^2(x)\right)^{3/2}}-\frac{1}{(a+b)^2 \sqrt{a+b \coth ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}",1,"(a*(a + b) - 3*b*(a + b*Coth[x]^2)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Coth[x]^2)/(a + b)])/(3*b*(a + b)^2*(a + b*Coth[x]^2)^(3/2))","C",1
44,1,258,88,7.0921075,"\int \frac{\coth ^2(x)}{\left(a+b \coth ^2(x)\right)^{5/2}} \, dx","Integrate[Coth[x]^2/(a + b*Coth[x]^2)^(5/2),x]","\frac{\sinh ^4(x) \tanh (x) \left(35 a \text{csch}^2(x) \left(-5 a-2 b \coth ^2(x)\right) \left(3 \left(a+b \coth ^2(x)\right)^2 \sin ^{-1}\left(\sqrt{\frac{(a+b) \cosh ^2(x)}{a}}\right)-a \text{csch}^2(x) \left(a \left(-\coth ^2(x)-3\right)-4 b \coth ^2(x)\right) \sqrt{-\frac{\sinh ^2(x) \cosh ^2(x) \left(a^2+a \left(b \coth ^2(x)+b\right)+b^2 \coth ^2(x)\right)}{a^2}}\right)-12 (a+b)^3 \coth ^6(x) \left(a+b \coth ^2(x)\right) \, _2F_1\left(2,2;\frac{9}{2};\frac{(a+b) \cosh ^2(x)}{a}\right) \sqrt{-\frac{\sinh ^2(x) \cosh ^2(x) \left(a^2+a \left(b \coth ^2(x)+b\right)+b^2 \coth ^2(x)\right)}{a^2}}\right)}{315 a^4 (a+b)^2 \sqrt{a+b \coth ^2(x)} \left(\frac{b \coth ^2(x)}{a}+1\right) \sqrt{-\frac{(a+b) \sinh ^2(x) \cosh ^2(x) \left(a+b \coth ^2(x)\right)}{a^2}}}","-\frac{(2 a-b) \coth (x)}{3 a (a+b)^2 \sqrt{a+b \coth ^2(x)}}-\frac{\coth (x)}{3 (a+b) \left(a+b \coth ^2(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)}{(a+b)^{5/2}}",1,"(Sinh[x]^4*(-12*(a + b)^3*Coth[x]^6*(a + b*Coth[x]^2)*Hypergeometric2F1[2, 2, 9/2, ((a + b)*Cosh[x]^2)/a]*Sqrt[-((Cosh[x]^2*(a^2 + b^2*Coth[x]^2 + a*(b + b*Coth[x]^2))*Sinh[x]^2)/a^2)] + 35*a*(-5*a - 2*b*Coth[x]^2)*Csch[x]^2*(3*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*(a + b*Coth[x]^2)^2 - a*(-4*b*Coth[x]^2 + a*(-3 - Coth[x]^2))*Csch[x]^2*Sqrt[-((Cosh[x]^2*(a^2 + b^2*Coth[x]^2 + a*(b + b*Coth[x]^2))*Sinh[x]^2)/a^2)]))*Tanh[x])/(315*a^4*(a + b)^2*Sqrt[a + b*Coth[x]^2]*(1 + (b*Coth[x]^2)/a)*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)])","C",0
45,1,43,70,0.0382802,"\int \frac{\coth (x)}{\left(a+b \coth ^2(x)\right)^{5/2}} \, dx","Integrate[Coth[x]/(a + b*Coth[x]^2)^(5/2),x]","-\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \coth ^2(x)+a}{a+b}\right)}{3 (a+b) \left(a+b \coth ^2(x)\right)^{3/2}}","-\frac{1}{(a+b)^2 \sqrt{a+b \coth ^2(x)}}-\frac{1}{3 (a+b) \left(a+b \coth ^2(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}",1,"-1/3*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Coth[x]^2)/(a + b)]/((a + b)*(a + b*Coth[x]^2)^(3/2))","C",1
46,1,73,108,0.0590749,"\int \frac{\tanh (x)}{\left(a+b \coth ^2(x)\right)^{5/2}} \, dx","Integrate[Tanh[x]/(a + b*Coth[x]^2)^(5/2),x]","\frac{(a+b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \coth ^2(x)}{a}+1\right)-a \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \coth ^2(x)+a}{a+b}\right)}{3 a (a+b) \left(a+b \coth ^2(x)\right)^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a}}\right)}{a^{5/2}}+\frac{b (2 a+b)}{a^2 (a+b)^2 \sqrt{a+b \coth ^2(x)}}+\frac{b}{3 a (a+b) \left(a+b \coth ^2(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}",1,"(-(a*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Coth[x]^2)/(a + b)]) + (a + b)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Coth[x]^2)/a])/(3*a*(a + b)*(a + b*Coth[x]^2)^(3/2))","C",1
47,1,171,131,6.7530199,"\int \frac{\tanh ^2(x)}{\left(a+b \coth ^2(x)\right)^{5/2}} \, dx","Integrate[Tanh[x]^2/(a + b*Coth[x]^2)^(5/2),x]","\frac{\sqrt{\text{csch}^2(x) ((a+b) \cosh (2 x)-a+b)} \left(\frac{3 \sqrt{2} \sinh (x) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{(a+b) \cosh (2 x)-a+b}}\right)}{(a+b)^{5/2} \sqrt{(a+b) \cosh (2 x)-a+b}}-\frac{\frac{b^2 \sinh (2 x) \left(\left(9 a^2+14 a b+5 b^2\right) \cosh (2 x)-9 a^2+2 a b+5 b^2\right)}{(a+b)^2 ((a+b) \cosh (2 x)-a+b)^2}+3 \tanh (x)}{a^3}\right)}{3 \sqrt{2}}","-\frac{(3 a+2 b) (a+4 b) \tanh (x) \sqrt{a+b \coth ^2(x)}}{3 a^3 (a+b)^2}+\frac{b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt{a+b \coth ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right)}{(a+b)^{5/2}}+\frac{b \tanh (x)}{3 a (a+b) \left(a+b \coth ^2(x)\right)^{3/2}}",1,"(Sqrt[(-a + b + (a + b)*Cosh[2*x])*Csch[x]^2]*((3*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[-a + b + (a + b)*Cosh[2*x]]]*Sinh[x])/((a + b)^(5/2)*Sqrt[-a + b + (a + b)*Cosh[2*x]]) - ((b^2*(-9*a^2 + 2*a*b + 5*b^2 + (9*a^2 + 14*a*b + 5*b^2)*Cosh[2*x])*Sinh[2*x])/((a + b)^2*(-a + b + (a + b)*Cosh[2*x])^2) + 3*Tanh[x])/a^3))/(3*Sqrt[2])","A",0
48,1,44,25,0.0694515,"\int \frac{1}{\sqrt{1+\coth ^2(x)}} \, dx","Integrate[1/Sqrt[1 + Coth[x]^2],x]","\frac{\sqrt{\cosh (2 x)} \text{csch}(x) \log \left(\sqrt{2} \cosh (x)+\sqrt{\cosh (2 x)}\right)}{\sqrt{2} \sqrt{\coth ^2(x)+1}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \coth (x)}{\sqrt{\coth ^2(x)+1}}\right)}{\sqrt{2}}",1,"(Sqrt[Cosh[2*x]]*Csch[x]*Log[Sqrt[2]*Cosh[x] + Sqrt[Cosh[2*x]]])/(Sqrt[2]*Sqrt[1 + Coth[x]^2])","A",1
49,1,46,27,0.0398969,"\int \frac{1}{\sqrt{-1-\coth ^2(x)}} \, dx","Integrate[1/Sqrt[-1 - Coth[x]^2],x]","\frac{\sqrt{\cosh (2 x)} \text{csch}(x) \log \left(\sqrt{2} \cosh (x)+\sqrt{\cosh (2 x)}\right)}{\sqrt{2} \sqrt{-\coth ^2(x)-1}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \coth (x)}{\sqrt{-\coth ^2(x)-1}}\right)}{\sqrt{2}}",1,"(Sqrt[Cosh[2*x]]*Csch[x]*Log[Sqrt[2]*Cosh[x] + Sqrt[Cosh[2*x]]])/(Sqrt[2]*Sqrt[-1 - Coth[x]^2])","A",1
50,1,40,38,0.0788785,"\int \frac{1}{1+\coth ^3(x)} \, dx","Integrate[(1 + Coth[x]^3)^(-1),x]","\frac{1}{2} \tanh ^{-1}(\tanh (x))+\frac{1}{6 (\tanh (x)+1)}+\frac{2 \tan ^{-1}\left(\frac{1-2 \tanh (x)}{\sqrt{3}}\right)}{3 \sqrt{3}}","\frac{x}{2}-\frac{1}{6 (\coth (x)+1)}-\frac{2 \tan ^{-1}\left(\frac{1-2 \coth (x)}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"(2*ArcTan[(1 - 2*Tanh[x])/Sqrt[3]])/(3*Sqrt[3]) + ArcTanh[Tanh[x]]/2 + 1/(6*(1 + Tanh[x]))","A",1
51,1,86,89,0.1550857,"\int \coth (x) \sqrt{a+b \coth ^4(x)} \, dx","Integrate[Coth[x]*Sqrt[a + b*Coth[x]^4],x]","\frac{1}{2} \left(-\sqrt{a+b \coth ^4(x)}-\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \coth ^2(x)}{\sqrt{a+b \coth ^4(x)}}\right)+\sqrt{a+b} \tanh ^{-1}\left(\frac{a+b \coth ^2(x)}{\sqrt{a+b} \sqrt{a+b \coth ^4(x)}}\right)\right)","-\frac{1}{2} \sqrt{a+b \coth ^4(x)}-\frac{1}{2} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \coth ^2(x)}{\sqrt{a+b \coth ^4(x)}}\right)+\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left(\frac{a+b \coth ^2(x)}{\sqrt{a+b} \sqrt{a+b \coth ^4(x)}}\right)",1,"(-(Sqrt[b]*ArcTanh[(Sqrt[b]*Coth[x]^2)/Sqrt[a + b*Coth[x]^4]]) + Sqrt[a + b]*ArcTanh[(a + b*Coth[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Coth[x]^4])] - Sqrt[a + b*Coth[x]^4])/2","A",1
52,1,40,40,0.0157235,"\int \frac{\coth (x)}{\sqrt{a+b \coth ^4(x)}} \, dx","Integrate[Coth[x]/Sqrt[a + b*Coth[x]^4],x]","\frac{\tanh ^{-1}\left(\frac{a+b \coth ^2(x)}{\sqrt{a+b} \sqrt{a+b \coth ^4(x)}}\right)}{2 \sqrt{a+b}}","\frac{\tanh ^{-1}\left(\frac{a+b \coth ^2(x)}{\sqrt{a+b} \sqrt{a+b \coth ^4(x)}}\right)}{2 \sqrt{a+b}}",1,"ArcTanh[(a + b*Coth[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Coth[x]^4])]/(2*Sqrt[a + b])","A",1
53,1,73,74,0.4971546,"\int \frac{\coth (x)}{\left(a+b \coth ^4(x)\right)^{3/2}} \, dx","Integrate[Coth[x]/(a + b*Coth[x]^4)^(3/2),x]","\frac{1}{2} \left(\frac{\tanh ^{-1}\left(\frac{a+b \coth ^2(x)}{\sqrt{a+b} \sqrt{a+b \coth ^4(x)}}\right)}{(a+b)^{3/2}}-\frac{a-b \coth ^2(x)}{a (a+b) \sqrt{a+b \coth ^4(x)}}\right)","\frac{\tanh ^{-1}\left(\frac{a+b \coth ^2(x)}{\sqrt{a+b} \sqrt{a+b \coth ^4(x)}}\right)}{2 (a+b)^{3/2}}-\frac{a-b \coth ^2(x)}{2 a (a+b) \sqrt{a+b \coth ^4(x)}}",1,"(ArcTanh[(a + b*Coth[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Coth[x]^4])]/(a + b)^(3/2) - (a - b*Coth[x]^2)/(a*(a + b)*Sqrt[a + b*Coth[x]^4]))/2","A",1