1,1,56,73,0.3692278,"\int \sinh ^4(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]^4*(a + b*Tanh[c + d*x]^2),x]","\frac{12 (a+5 b) (c+d x)-8 (a+2 b) \sinh (2 (c+d x))+(a+b) \sinh (4 (c+d x))-32 b \tanh (c+d x)}{32 d}","\frac{(a+b) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}-\frac{(5 a+9 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3}{8} x (a+5 b)-\frac{b \tanh (c+d x)}{d}",1,"(12*(a + 5*b)*(c + d*x) - 8*(a + 2*b)*Sinh[2*(c + d*x)] + (a + b)*Sinh[4*(c + d*x)] - 32*b*Tanh[c + d*x])/(32*d)","A",1
2,1,73,47,0.0538709,"\int \sinh ^3(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^2),x]","-\frac{3 a \cosh (c+d x)}{4 d}+\frac{a \cosh (3 (c+d x))}{12 d}-\frac{7 b \cosh (c+d x)}{4 d}+\frac{b \cosh (3 (c+d x))}{12 d}-\frac{b \text{sech}(c+d x)}{d}","\frac{(a+b) \cosh ^3(c+d x)}{3 d}-\frac{(a+2 b) \cosh (c+d x)}{d}-\frac{b \text{sech}(c+d x)}{d}",1,"(-3*a*Cosh[c + d*x])/(4*d) - (7*b*Cosh[c + d*x])/(4*d) + (a*Cosh[3*(c + d*x)])/(12*d) + (b*Cosh[3*(c + d*x)])/(12*d) - (b*Sech[c + d*x])/d","A",1
3,1,41,44,0.2102766,"\int \sinh ^2(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]^2*(a + b*Tanh[c + d*x]^2),x]","\frac{-2 (a+3 b) (c+d x)+(a+b) \sinh (2 (c+d x))+4 b \tanh (c+d x)}{4 d}","\frac{(a+b) \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{1}{2} x (a+3 b)+\frac{b \tanh (c+d x)}{d}",1,"(-2*(a + 3*b)*(c + d*x) + (a + b)*Sinh[2*(c + d*x)] + 4*b*Tanh[c + d*x])/(4*d)","A",1
4,1,45,25,0.0560831,"\int \sinh (c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]*(a + b*Tanh[c + d*x]^2),x]","\frac{a \sinh (c) \sinh (d x)}{d}+\frac{a \cosh (c) \cosh (d x)}{d}+\frac{b \cosh (c+d x)}{d}+\frac{b \text{sech}(c+d x)}{d}","\frac{(a+b) \cosh (c+d x)}{d}+\frac{b \text{sech}(c+d x)}{d}",1,"(a*Cosh[c]*Cosh[d*x])/d + (b*Cosh[c + d*x])/d + (b*Sech[c + d*x])/d + (a*Sinh[c]*Sinh[d*x])/d","A",1
5,1,52,26,0.0288718,"\int \text{csch}(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Csch[c + d*x]*(a + b*Tanh[c + d*x]^2),x]","\frac{a \log \left(\sinh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{a \log \left(\cosh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{b \text{sech}(c+d x)}{d}","-\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{b \text{sech}(c+d x)}{d}",1,"-((a*Log[Cosh[c/2 + (d*x)/2]])/d) + (a*Log[Sinh[c/2 + (d*x)/2]])/d - (b*Sech[c + d*x])/d","A",1
6,1,24,24,0.0225829,"\int \text{csch}^2(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^2*(a + b*Tanh[c + d*x]^2),x]","\frac{b \tanh (c+d x)}{d}-\frac{a \coth (c+d x)}{d}","\frac{b \tanh (c+d x)}{d}-\frac{a \coth (c+d x)}{d}",1,"-((a*Coth[c + d*x])/d) + (b*Tanh[c + d*x])/d","A",1
7,1,87,51,0.047893,"\int \text{csch}^3(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^2),x]","-\frac{a \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{b \text{sech}(c+d x)}{d}+\frac{b \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{d}","\frac{(a-2 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{b \text{sech}(c+d x)}{d}",1,"-1/8*(a*Csch[(c + d*x)/2]^2)/d - (a*Log[Tanh[(c + d*x)/2]])/(2*d) + (b*Log[Tanh[(c + d*x)/2]])/d - (a*Sech[(c + d*x)/2]^2)/(8*d) + (b*Sech[c + d*x])/d","A",1
8,1,61,44,0.0719477,"\int \text{csch}^4(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^2),x]","\frac{2 a \coth (c+d x)}{3 d}-\frac{a \coth (c+d x) \text{csch}^2(c+d x)}{3 d}-\frac{b \tanh (c+d x)}{d}-\frac{b \coth (c+d x)}{d}","\frac{(a-b) \coth (c+d x)}{d}-\frac{a \coth ^3(c+d x)}{3 d}-\frac{b \tanh (c+d x)}{d}",1,"(2*a*Coth[c + d*x])/(3*d) - (b*Coth[c + d*x])/d - (a*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (b*Tanh[c + d*x])/d","A",1
9,1,94,118,1.4491454,"\int \sinh ^4(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{12 \left(3 a^2+30 a b+35 b^2\right) (c+d x)-24 \left(a^2+4 a b+3 b^2\right) \sinh (2 (c+d x))+3 (a+b)^2 \sinh (4 (c+d x))+32 b \tanh (c+d x) \left(-6 a+b \text{sech}^2(c+d x)-10 b\right)}{96 d}","-\frac{\left(a^2+10 a b+13 b^2\right) \tanh (c+d x)}{4 d}+\frac{1}{8} x \left(3 a^2+30 a b+35 b^2\right)+\frac{(a+b)^2 \sinh ^4(c+d x) \tanh (c+d x)}{4 d}-\frac{(a+b) (a+9 b) \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac{b^2 \tanh ^3(c+d x)}{3 d}",1,"(12*(3*a^2 + 30*a*b + 35*b^2)*(c + d*x) - 24*(a^2 + 4*a*b + 3*b^2)*Sinh[2*(c + d*x)] + 3*(a + b)^2*Sinh[4*(c + d*x)] + 32*b*(-6*a - 10*b + b*Sech[c + d*x]^2)*Tanh[c + d*x])/(96*d)","A",1
10,1,71,77,0.5380698,"\int \sinh ^3(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{-3 \left(3 a^2+14 a b+11 b^2\right) \cosh (c+d x)+(a+b)^2 \cosh (3 (c+d x))+4 b \text{sech}(c+d x) \left(-6 a+b \text{sech}^2(c+d x)-9 b\right)}{12 d}","\frac{(a+b)^2 \cosh ^3(c+d x)}{3 d}-\frac{(a+b) (a+3 b) \cosh (c+d x)}{d}-\frac{b (2 a+3 b) \text{sech}(c+d x)}{d}+\frac{b^2 \text{sech}^3(c+d x)}{3 d}",1,"(-3*(3*a^2 + 14*a*b + 11*b^2)*Cosh[c + d*x] + (a + b)^2*Cosh[3*(c + d*x)] + 4*b*Sech[c + d*x]*(-6*a - 9*b + b*Sech[c + d*x]^2))/(12*d)","A",1
11,1,70,79,0.9016022,"\int \sinh ^2(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{-6 \left(a^2+6 a b+5 b^2\right) (c+d x)+3 (a+b)^2 \sinh (2 (c+d x))+4 b \tanh (c+d x) \left(6 a-b \text{sech}^2(c+d x)+7 b\right)}{12 d}","\frac{(a+b) (a+5 b) \tanh (c+d x)}{2 d}+\frac{(a+b)^2 \sinh ^2(c+d x) \tanh (c+d x)}{2 d}-\frac{1}{2} x (a+b) (a+5 b)+\frac{b^2 \tanh ^3(c+d x)}{3 d}",1,"(-6*(a^2 + 6*a*b + 5*b^2)*(c + d*x) + 3*(a + b)^2*Sinh[2*(c + d*x)] + 4*b*(6*a + 7*b - b*Sech[c + d*x]^2)*Tanh[c + d*x])/(12*d)","A",1
12,1,46,49,0.3406313,"\int \sinh (c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{3 (a+b)^2 \cosh (c+d x)+b \text{sech}(c+d x) \left(6 (a+b)-b \text{sech}^2(c+d x)\right)}{3 d}","\frac{(a+b)^2 \cosh (c+d x)}{d}+\frac{2 b (a+b) \text{sech}(c+d x)}{d}-\frac{b^2 \text{sech}^3(c+d x)}{3 d}",1,"(3*(a + b)^2*Cosh[c + d*x] + b*Sech[c + d*x]*(6*(a + b) - b*Sech[c + d*x]^2))/(3*d)","A",1
13,1,50,51,0.1693913,"\int \text{csch}(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{3 a^2 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-3 b (2 a+b) \text{sech}(c+d x)+b^2 \text{sech}^3(c+d x)}{3 d}","-\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{b (2 a+b) \text{sech}(c+d x)}{d}+\frac{b^2 \text{sech}^3(c+d x)}{3 d}",1,"(3*a^2*Log[Tanh[(c + d*x)/2]] - 3*b*(2*a + b)*Sech[c + d*x] + b^2*Sech[c + d*x]^3)/(3*d)","A",1
14,1,43,46,0.4922519,"\int \text{csch}^2(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{b \tanh (c+d x) \left(6 a-b \text{sech}^2(c+d x)+b\right)-3 a^2 \coth (c+d x)}{3 d}","-\frac{a^2 \coth (c+d x)}{d}+\frac{2 a b \tanh (c+d x)}{d}+\frac{b^2 \tanh ^3(c+d x)}{3 d}",1,"(-3*a^2*Coth[c + d*x] + b*(6*a + b - b*Sech[c + d*x]^2)*Tanh[c + d*x])/(3*d)","A",1
15,1,96,82,1.5912015,"\int \text{csch}^3(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^2,x]","-\frac{3 a^2 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)+3 a^2 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)+12 a^2 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-48 a b \text{sech}(c+d x)-48 a b \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+8 b^2 \text{sech}^3(c+d x)}{24 d}","-\frac{a^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 d}-\frac{a (a-4 b) \text{sech}(c+d x)}{2 d}+\frac{a (a-4 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{b^2 \text{sech}^3(c+d x)}{3 d}",1,"-1/24*(3*a^2*Csch[(c + d*x)/2]^2 + 12*a^2*Log[Tanh[(c + d*x)/2]] - 48*a*b*Log[Tanh[(c + d*x)/2]] + 3*a^2*Sech[(c + d*x)/2]^2 - 48*a*b*Sech[c + d*x] + 8*b^2*Sech[c + d*x]^3)/d","A",1
16,1,59,72,0.5199361,"\int \text{csch}^4(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{b \tanh (c+d x) \left(-6 a+b \text{sech}^2(c+d x)+2 b\right)-a \coth (c+d x) \left(a \text{csch}^2(c+d x)-2 a+6 b\right)}{3 d}","-\frac{a^2 \coth ^3(c+d x)}{3 d}-\frac{b (2 a-b) \tanh (c+d x)}{d}+\frac{a (a-2 b) \coth (c+d x)}{d}-\frac{b^2 \tanh ^3(c+d x)}{3 d}",1,"(-(a*Coth[c + d*x]*(-2*a + 6*b + a*Csch[c + d*x]^2)) + b*(-6*a + 2*b + b*Sech[c + d*x]^2)*Tanh[c + d*x])/(3*d)","A",1
17,1,125,182,3.9245255,"\int \sinh ^4(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{-32 b \tanh (c+d x) \left(15 a^2-b (5 a+7 b) \text{sech}^2(c+d x)+50 a b+b^2 \text{sech}^4(c+d x)+36 b^2\right)+60 \left(a^3+15 a^2 b+35 a b^2+21 b^3\right) (c+d x)+5 (a+b)^3 \sinh (4 (c+d x))-40 (a+4 b) (a+b)^2 \sinh (2 (c+d x))}{160 d}","-\frac{b \left(6 a^2+35 a b+21 b^2\right) \tanh ^3(c+d x)}{8 d}-\frac{3 (a+b) \left(a^2+14 a b+21 b^2\right) \tanh (c+d x)}{8 d}+\frac{3}{8} x (a+b) \left(a^2+14 a b+21 b^2\right)-\frac{3 b^2 (5 a+21 b) \tanh ^5(c+d x)}{40 d}-\frac{3 (a+3 b) \sinh ^2(c+d x) \tanh (c+d x) \left(a+b \tanh ^2(c+d x)\right)^2}{8 d}+\frac{\sinh ^3(c+d x) \cosh (c+d x) \left(a+b \tanh ^2(c+d x)\right)^3}{4 d}",1,"(60*(a^3 + 15*a^2*b + 35*a*b^2 + 21*b^3)*(c + d*x) - 40*(a + b)^2*(a + 4*b)*Sinh[2*(c + d*x)] + 5*(a + b)^3*Sinh[4*(c + d*x)] - 32*b*(15*a^2 + 50*a*b + 36*b^2 - b*(5*a + 7*b)*Sech[c + d*x]^2 + b^2*Sech[c + d*x]^4)*Tanh[c + d*x])/(160*d)","A",1
18,1,91,105,0.3025544,"\int \sinh ^3(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{20 b^2 (3 a+4 b) \text{sech}^3(c+d x)-45 (a+b)^2 (a+5 b) \cosh (c+d x)+5 (a+b)^3 \cosh (3 (c+d x))-180 b (a+b) (a+2 b) \text{sech}(c+d x)-12 b^3 \text{sech}^5(c+d x)}{60 d}","\frac{b^2 (3 a+4 b) \text{sech}^3(c+d x)}{3 d}+\frac{(a+b)^3 \cosh ^3(c+d x)}{3 d}-\frac{(a+b)^2 (a+4 b) \cosh (c+d x)}{d}-\frac{3 b (a+b) (a+2 b) \text{sech}(c+d x)}{d}-\frac{b^3 \text{sech}^5(c+d x)}{5 d}",1,"(-45*(a + b)^2*(a + 5*b)*Cosh[c + d*x] + 5*(a + b)^3*Cosh[3*(c + d*x)] - 180*b*(a + b)*(a + 2*b)*Sech[c + d*x] + 20*b^2*(3*a + 4*b)*Sech[c + d*x]^3 - 12*b^3*Sech[c + d*x]^5)/(60*d)","A",1
19,1,95,122,2.1984494,"\int \sinh ^2(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{4 b \tanh (c+d x) \left(45 a^2-b (15 a+16 b) \text{sech}^2(c+d x)+105 a b+3 b^2 \text{sech}^4(c+d x)+58 b^2\right)-30 (a+7 b) (a+b)^2 (c+d x)+15 (a+b)^3 \sinh (2 (c+d x))}{60 d}","\frac{b^2 (3 a+2 b) \tanh ^3(c+d x)}{3 d}+\frac{3 b (a+b)^2 \tanh (c+d x)}{d}+\frac{(a+b)^3}{4 d (1-\tanh (c+d x))}-\frac{(a+b)^3}{4 d (\tanh (c+d x)+1)}-\frac{1}{2} x (a+b)^2 (a+7 b)+\frac{b^3 \tanh ^5(c+d x)}{5 d}",1,"(-30*(a + b)^2*(a + 7*b)*(c + d*x) + 15*(a + b)^3*Sinh[2*(c + d*x)] + 4*b*(45*a^2 + 105*a*b + 58*b^2 - b*(15*a + 16*b)*Sech[c + d*x]^2 + 3*b^2*Sech[c + d*x]^4)*Tanh[c + d*x])/(60*d)","A",1
20,1,63,70,0.8795438,"\int \sinh (c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{b \text{sech}(c+d x) \left(-5 b (a+b) \text{sech}^2(c+d x)+15 (a+b)^2+b^2 \text{sech}^4(c+d x)\right)+5 (a+b)^3 \cosh (c+d x)}{5 d}","-\frac{b^2 (a+b) \text{sech}^3(c+d x)}{d}+\frac{(a+b)^3 \cosh (c+d x)}{d}+\frac{3 b (a+b)^2 \text{sech}(c+d x)}{d}+\frac{b^3 \text{sech}^5(c+d x)}{5 d}",1,"(5*(a + b)^3*Cosh[c + d*x] + b*Sech[c + d*x]*(15*(a + b)^2 - 5*b*(a + b)*Sech[c + d*x]^2 + b^2*Sech[c + d*x]^4))/(5*d)","A",1
21,1,79,84,0.326307,"\int \text{csch}(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{15 a^3 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-15 b \left(3 a^2+3 a b+b^2\right) \text{sech}(c+d x)+5 b^2 (3 a+2 b) \text{sech}^3(c+d x)-3 b^3 \text{sech}^5(c+d x)}{15 d}","-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{b \left(3 a^2+3 a b+b^2\right) \text{sech}(c+d x)}{d}+\frac{b^2 (3 a+2 b) \text{sech}^3(c+d x)}{3 d}-\frac{b^3 \text{sech}^5(c+d x)}{5 d}",1,"(15*a^3*Log[Tanh[(c + d*x)/2]] - 15*b*(3*a^2 + 3*a*b + b^2)*Sech[c + d*x] + 5*b^2*(3*a + 2*b)*Sech[c + d*x]^3 - 3*b^3*Sech[c + d*x]^5)/(15*d)","A",1
22,1,70,64,0.7143392,"\int \text{csch}^2(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{b \tanh (c+d x) \left(15 a^2-b (5 a+2 b) \text{sech}^2(c+d x)+5 a b+b^2 \text{sech}^4(c+d x)+b^2\right)-5 a^3 \coth (c+d x)}{5 d}","-\frac{a^3 \coth (c+d x)}{d}+\frac{3 a^2 b \tanh (c+d x)}{d}+\frac{a b^2 \tanh ^3(c+d x)}{d}+\frac{b^3 \tanh ^5(c+d x)}{5 d}",1,"(-5*a^3*Coth[c + d*x] + b*(15*a^2 + 5*a*b + b^2 - b*(5*a + 2*b)*Sech[c + d*x]^2 + b^2*Sech[c + d*x]^4)*Tanh[c + d*x])/(5*d)","A",1
23,1,127,152,6.1973777,"\int \text{csch}^3(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3,x]","-\frac{a^3 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a^3 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{3 a^2 b \text{sech}(c+d x)}{d}-\frac{a^2 (a-6 b) \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{b^2 (3 a+b) \text{sech}^3(c+d x)}{3 d}+\frac{b^3 \text{sech}^5(c+d x)}{5 d}","\frac{b \left(81 a^2-28 a b-4 b^2\right) \text{sech}(c+d x)}{30 d}+\frac{a^2 (a-6 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac{7 b \text{sech}(c+d x) \left(a-b \text{sech}^2(c+d x)+b\right)^2}{10 d}+\frac{b (33 a-2 b) \text{sech}(c+d x) \left(a-b \text{sech}^2(c+d x)+b\right)}{30 d}-\frac{\coth (c+d x) \text{csch}(c+d x) \left(a-b \text{sech}^2(c+d x)+b\right)^3}{2 d}",1,"-1/8*(a^3*Csch[(c + d*x)/2]^2)/d - (a^2*(a - 6*b)*Log[Tanh[(c + d*x)/2]])/(2*d) - (a^3*Sech[(c + d*x)/2]^2)/(8*d) + (3*a^2*b*Sech[c + d*x])/d - (b^2*(3*a + b)*Sech[c + d*x]^3)/(3*d) + (b^3*Sech[c + d*x]^5)/(5*d)","A",1
24,1,87,98,1.2554481,"\int \text{csch}^4(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{b \tanh (c+d x) \left(-45 a^2+b (15 a+b) \text{sech}^2(c+d x)+30 a b-3 b^2 \text{sech}^4(c+d x)+2 b^2\right)-5 a^2 \coth (c+d x) \left(a \text{csch}^2(c+d x)-2 a+9 b\right)}{15 d}","-\frac{a^3 \coth ^3(c+d x)}{3 d}+\frac{a^2 (a-3 b) \coth (c+d x)}{d}-\frac{b^2 (3 a-b) \tanh ^3(c+d x)}{3 d}-\frac{3 a b (a-b) \tanh (c+d x)}{d}-\frac{b^3 \tanh ^5(c+d x)}{5 d}",1,"(-5*a^2*Coth[c + d*x]*(-2*a + 9*b + a*Csch[c + d*x]^2) + b*(-45*a^2 + 30*a*b + 2*b^2 + b*(15*a + b)*Sech[c + d*x]^2 - 3*b^2*Sech[c + d*x]^4)*Tanh[c + d*x])/(15*d)","A",1
25,1,93,118,0.2732637,"\int \frac{\sinh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Sinh[c + d*x]^4/(a + b*Tanh[c + d*x]^2),x]","\frac{32 a^{3/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)+4 \left(3 a^2-6 a b-b^2\right) (c+d x)+(a+b)^2 \sinh (4 (c+d x))-8 a (a+b) \sinh (2 (c+d x))}{32 d (a+b)^3}","\frac{a^{3/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{d (a+b)^3}+\frac{x \left(3 a^2-6 a b-b^2\right)}{8 (a+b)^3}+\frac{\sinh (c+d x) \cosh ^3(c+d x)}{4 d (a+b)}-\frac{(5 a+b) \sinh (c+d x) \cosh (c+d x)}{8 d (a+b)^2}",1,"(4*(3*a^2 - 6*a*b - b^2)*(c + d*x) + 32*a^(3/2)*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] - 8*a*(a + b)*Sinh[2*(c + d*x)] + (a + b)^2*Sinh[4*(c + d*x)])/(32*(a + b)^3*d)","A",1
26,1,135,75,0.5773699,"\int \frac{\sinh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Sinh[c + d*x]^3/(a + b*Tanh[c + d*x]^2),x]","\frac{(a+b)^{3/2} \cosh (3 (c+d x))-3 (3 a-b) \sqrt{a+b} \cosh (c+d x)+12 i a \sqrt{b} \left(\tan ^{-1}\left(\frac{-\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)+\tan ^{-1}\left(\frac{\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)\right)}{12 d (a+b)^{5/2}}","\frac{\cosh ^3(c+d x)}{3 d (a+b)}-\frac{a \cosh (c+d x)}{d (a+b)^2}+\frac{a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right)}{d (a+b)^{5/2}}",1,"((12*I)*a*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]]) - 3*(3*a - b)*Sqrt[a + b]*Cosh[c + d*x] + (a + b)^(3/2)*Cosh[3*(c + d*x)])/(12*(a + b)^(5/2)*d)","C",1
27,1,67,78,0.1528749,"\int \frac{\sinh ^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Sinh[c + d*x]^2/(a + b*Tanh[c + d*x]^2),x]","\frac{-2 (a-b) (c+d x)+(a+b) \sinh (2 (c+d x))-4 \sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{4 d (a+b)^2}","-\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{d (a+b)^2}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 d (a+b)}-\frac{x (a-b)}{2 (a+b)^2}",1,"(-2*(a - b)*(c + d*x) - 4*Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] + (a + b)*Sinh[2*(c + d*x)])/(4*(a + b)^2*d)","A",1
28,1,107,53,0.2728961,"\int \frac{\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Sinh[c + d*x]/(a + b*Tanh[c + d*x]^2),x]","\frac{\sqrt{a+b} \cosh (c+d x)-i \sqrt{b} \left(\tan ^{-1}\left(\frac{-\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)+\tan ^{-1}\left(\frac{\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)\right)}{d (a+b)^{3/2}}","\frac{\cosh (c+d x)}{d (a+b)}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right)}{d (a+b)^{3/2}}",1,"((-I)*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]]) + Sqrt[a + b]*Cosh[c + d*x])/((a + b)^(3/2)*d)","C",1
29,1,123,55,0.2114802,"\int \frac{\text{csch}(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Csch[c + d*x]/(a + b*Tanh[c + d*x]^2),x]","\frac{\sqrt{a+b} \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+i \sqrt{b} \tan ^{-1}\left(\frac{-\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)+i \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)}{a d \sqrt{a+b}}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right)}{a d \sqrt{a+b}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a d}",1,"(I*Sqrt[b]*ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + I*Sqrt[b]*ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + Sqrt[a + b]*Log[Tanh[(c + d*x)/2]])/(a*Sqrt[a + b]*d)","C",1
30,1,48,48,0.1283922,"\int \frac{\text{csch}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Csch[c + d*x]^2/(a + b*Tanh[c + d*x]^2),x]","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{3/2} d}-\frac{\coth (c+d x)}{a d}","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{3/2} d}-\frac{\coth (c+d x)}{a d}",1,"-((Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*d)) - Coth[c + d*x]/(a*d)","A",1
31,1,170,85,0.6966474,"\int \frac{\text{csch}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Csch[c + d*x]^3/(a + b*Tanh[c + d*x]^2),x]","-\frac{8 i \sqrt{b} \sqrt{a+b} \tan ^{-1}\left(\frac{-\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)+8 i \sqrt{b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)+a \text{csch}^2\left(\frac{1}{2} (c+d x)\right)+a \text{sech}^2\left(\frac{1}{2} (c+d x)\right)+4 a \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+8 b \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^2 d}","\frac{(a+2 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^2 d}-\frac{\sqrt{b} \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right)}{a^2 d}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d}",1,"-1/8*((8*I)*Sqrt[b]*Sqrt[a + b]*ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + (8*I)*Sqrt[b]*Sqrt[a + b]*ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + a*Csch[(c + d*x)/2]^2 + 4*a*Log[Tanh[(c + d*x)/2]] + 8*b*Log[Tanh[(c + d*x)/2]] + a*Sech[(c + d*x)/2]^2)/(a^2*d)","C",1
32,1,71,70,0.3194206,"\int \frac{\text{csch}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Csch[c + d*x]^4/(a + b*Tanh[c + d*x]^2),x]","\frac{3 \sqrt{b} (a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)+\sqrt{a} \coth (c+d x) \left(-a \text{csch}^2(c+d x)+2 a+3 b\right)}{3 a^{5/2} d}","\frac{\sqrt{b} (a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{(a+b) \coth (c+d x)}{a^2 d}-\frac{\coth ^3(c+d x)}{3 a d}",1,"(3*Sqrt[b]*(a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] + Sqrt[a]*Coth[c + d*x]*(2*a + 3*b - a*Csch[c + d*x]^2))/(3*a^(5/2)*d)","A",1
33,1,132,192,1.038891,"\int \frac{\sinh ^4(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{12 \left(a^2-6 a b+b^2\right) (c+d x)+(a+b)^2 \sinh (4 (c+d x))-8 (a-b) (a+b) \sinh (2 (c+d x))+48 \sqrt{a} \sqrt{b} (a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)+\frac{16 a b (a+b) \sinh (2 (c+d x))}{(a+b) \cosh (2 (c+d x))+a-b}}{32 d (a+b)^4}","\frac{3 x \left(a^2-6 a b+b^2\right)}{8 (a+b)^4}+\frac{3 b (3 a-b) \tanh (c+d x)}{8 d (a+b)^3 \left(a+b \tanh ^2(c+d x)\right)}+\frac{3 \sqrt{a} \sqrt{b} (a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 d (a+b)^4}+\frac{\sinh (c+d x) \cosh ^3(c+d x)}{4 d (a+b) \left(a+b \tanh ^2(c+d x)\right)}-\frac{(5 a-b) \sinh (c+d x) \cosh (c+d x)}{8 d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)}",1,"(12*(a^2 - 6*a*b + b^2)*(c + d*x) + 48*Sqrt[a]*(a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] - 8*(a - b)*(a + b)*Sinh[2*(c + d*x)] + (16*a*b*(a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)]) + (a + b)^2*Sinh[4*(c + d*x)])/(32*(a + b)^4*d)","A",1
34,1,160,124,1.3711684,"\int \frac{\sinh ^3(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\frac{3 \cosh (c+d x) \left(a \left(\frac{4 b}{(a+b) \cosh (2 (c+d x))+a-b}-3\right)+5 b\right)}{(a+b)^3}+\frac{\cosh (3 (c+d x))}{(a+b)^2}+\frac{6 i \sqrt{b} (3 a-2 b) \left(\tan ^{-1}\left(\frac{-\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)+\tan ^{-1}\left(\frac{\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)\right)}{(a+b)^{7/2}}}{12 d}","\frac{\cosh ^3(c+d x)}{3 d (a+b)^2}-\frac{(a-b) \cosh (c+d x)}{d (a+b)^3}+\frac{a b \text{sech}(c+d x)}{2 d (a+b)^3 \left(a-b \text{sech}^2(c+d x)+b\right)}+\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right)}{2 d (a+b)^{7/2}}",1,"(((6*I)*(3*a - 2*b)*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]]))/(a + b)^(7/2) + (3*Cosh[c + d*x]*(5*b + a*(-3 + (4*b)/(a - b + (a + b)*Cosh[2*(c + d*x)]))))/(a + b)^3 + Cosh[3*(c + d*x)]/(a + b)^2)/(12*d)","C",1
35,1,105,132,0.73683,"\int \frac{\sinh ^2(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{-2 (a-3 b) (c+d x)+(a+b) \sinh (2 (c+d x))+\frac{2 \sqrt{b} (b-3 a) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{2 b (a+b) \sinh (2 (c+d x))}{(a+b) \cosh (2 (c+d x))+a-b}}{4 d (a+b)^3}","-\frac{b \tanh (c+d x)}{d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)}-\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 \sqrt{a} d (a+b)^3}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 d (a+b) \left(a+b \tanh ^2(c+d x)\right)}-\frac{x (a-3 b)}{2 (a+b)^3}",1,"(-2*(a - 3*b)*(c + d*x) + (2*Sqrt[b]*(-3*a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/Sqrt[a] + (a + b)*Sinh[2*(c + d*x)] - (2*b*(a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)]))/(4*(a + b)^3*d)","A",1
36,1,133,92,0.7745047,"\int \frac{\sinh (c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\frac{2 \cosh (c+d x) \left(1-\frac{b}{(a+b) \cosh (2 (c+d x))+a-b}\right)}{(a+b)^2}-\frac{3 i \sqrt{b} \left(\tan ^{-1}\left(\frac{-\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)+\tan ^{-1}\left(\frac{\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)\right)}{(a+b)^{5/2}}}{2 d}","\frac{3 \cosh (c+d x)}{2 d (a+b)^2}-\frac{\cosh (c+d x)}{2 d (a+b) \left(a-b \text{sech}^2(c+d x)+b\right)}-\frac{3 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right)}{2 d (a+b)^{5/2}}",1,"(((-3*I)*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]]))/(a + b)^(5/2) + (2*Cosh[c + d*x]*(1 - b/(a - b + (a + b)*Cosh[2*(c + d*x)])))/(a + b)^2)/(2*d)","C",1
37,1,175,103,0.7464764,"\int \frac{\text{csch}(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Csch[c + d*x]/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\frac{2 a b \cosh (c+d x)}{(a+b) ((a+b) \cosh (2 (c+d x))+a-b)}+\frac{i \sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{-\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)}{(a+b)^{3/2}}+\frac{i \sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)}{(a+b)^{3/2}}+2 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^2 d}","\frac{\sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right)}{2 a^2 d (a+b)^{3/2}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a^2 d}+\frac{b \text{sech}(c+d x)}{2 a d (a+b) \left(a-b \text{sech}^2(c+d x)+b\right)}",1,"((I*Sqrt[b]*(3*a + 2*b)*ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/(a + b)^(3/2) + (I*Sqrt[b]*(3*a + 2*b)*ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/(a + b)^(3/2) + (2*a*b*Cosh[c + d*x])/((a + b)*(a - b + (a + b)*Cosh[2*(c + d*x)])) + 2*Log[Tanh[(c + d*x)/2]])/(2*a^2*d)","C",1
38,1,86,82,0.5021001,"\int \frac{\text{csch}^2(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Csch[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{-3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)-\frac{\sqrt{a} b \sinh (2 (c+d x))}{(a+b) \cosh (2 (c+d x))+a-b}-2 \sqrt{a} \coth (c+d x)}{2 a^{5/2} d}","-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{5/2} d}-\frac{3 \coth (c+d x)}{2 a^2 d}+\frac{\coth (c+d x)}{2 a d \left(a+b \tanh ^2(c+d x)\right)}",1,"(-3*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] - 2*Sqrt[a]*Coth[c + d*x] - (Sqrt[a]*b*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)]))/(2*a^(5/2)*d)","A",1
39,1,203,141,4.3461619,"\int \frac{\text{csch}^3(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Csch[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2,x]","-\frac{\frac{8 a b \cosh (c+d x)}{(a+b) \cosh (2 (c+d x))+a-b}+4 (a+4 b) \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 i \sqrt{b} (3 a+4 b) \tan ^{-1}\left(\frac{-\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)}{\sqrt{a+b}}+\frac{4 i \sqrt{b} (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)}{\sqrt{a+b}}+a \text{csch}^2\left(\frac{1}{2} (c+d x)\right)+a \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{8 a^3 d}","\frac{(a+4 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}-\frac{\sqrt{b} (3 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right)}{2 a^3 d \sqrt{a+b}}-\frac{b \text{sech}(c+d x)}{a^2 d \left(a-b \text{sech}^2(c+d x)+b\right)}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left(a-b \text{sech}^2(c+d x)+b\right)}",1,"-1/8*(((4*I)*Sqrt[b]*(3*a + 4*b)*ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/Sqrt[a + b] + ((4*I)*Sqrt[b]*(3*a + 4*b)*ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/Sqrt[a + b] + (8*a*b*Cosh[c + d*x])/(a - b + (a + b)*Cosh[2*(c + d*x)]) + a*Csch[(c + d*x)/2]^2 + 4*(a + 4*b)*Log[Tanh[(c + d*x)/2]] + a*Sech[(c + d*x)/2]^2)/(a^3*d)","C",1
40,1,114,113,0.929361,"\int \frac{\text{csch}^4(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Csch[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{3 \sqrt{b} (3 a+5 b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)+\frac{3 \sqrt{a} b (a+b) \sinh (2 (c+d x))}{(a+b) \cosh (2 (c+d x))+a-b}+2 \sqrt{a} \coth (c+d x) \left(-a \text{csch}^2(c+d x)+2 a+6 b\right)}{6 a^{7/2} d}","\frac{\sqrt{b} (3 a+5 b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{7/2} d}+\frac{b (a+b) \tanh (c+d x)}{2 a^3 d \left(a+b \tanh ^2(c+d x)\right)}+\frac{(a+2 b) \coth (c+d x)}{a^3 d}-\frac{\coth ^3(c+d x)}{3 a^2 d}",1,"(3*Sqrt[b]*(3*a + 5*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] + 2*Sqrt[a]*Coth[c + d*x]*(2*a + 6*b - a*Csch[c + d*x]^2) + (3*Sqrt[a]*b*(a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)]))/(6*a^(7/2)*d)","A",1
41,1,184,240,0.8577946,"\int \frac{\sinh ^4(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{12 \left(a^2-10 a b+5 b^2\right) (c+d x)+\frac{12 \sqrt{b} \left(5 a^2-10 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{16 a b^2 (a+b) \sinh (2 (c+d x))}{((a+b) \cosh (2 (c+d x))+a-b)^2}-8 (a-2 b) (a+b) \sinh (2 (c+d x))+(a+b)^2 \sinh (4 (c+d x))+\frac{4 b (9 a-5 b) (a+b) \sinh (2 (c+d x))}{(a+b) \cosh (2 (c+d x))+a-b}}{32 d (a+b)^5}","\frac{3 \sqrt{b} \left(5 a^2-10 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 \sqrt{a} d (a+b)^5}+\frac{3 x \left(a^2-10 a b+5 b^2\right)}{8 (a+b)^5}+\frac{3 b (a-b) \tanh (c+d x)}{2 d (a+b)^4 \left(a+b \tanh ^2(c+d x)\right)}+\frac{b (7 a-5 b) \tanh (c+d x)}{8 d (a+b)^3 \left(a+b \tanh ^2(c+d x)\right)^2}+\frac{\sinh (c+d x) \cosh ^3(c+d x)}{4 d (a+b) \left(a+b \tanh ^2(c+d x)\right)^2}-\frac{(5 a-3 b) \sinh (c+d x) \cosh (c+d x)}{8 d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)^2}",1,"(12*(a^2 - 10*a*b + 5*b^2)*(c + d*x) + (12*Sqrt[b]*(5*a^2 - 10*a*b + b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/Sqrt[a] - 8*(a - 2*b)*(a + b)*Sinh[2*(c + d*x)] + (16*a*b^2*(a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)])^2 + (4*(9*a - 5*b)*b*(a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)]) + (a + b)^2*Sinh[4*(c + d*x)])/(32*(a + b)^5*d)","A",1
42,1,227,166,2.0035819,"\int \frac{\sinh ^3(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{-\frac{6 \cosh (c+d x) \left(3 a^3-24 a^2 b+\left(6 a^3-27 a^2 b-11 a b^2+22 b^3\right) \cosh (2 (c+d x))+30 a b^2+3 (a-3 b) (a+b)^2 \cosh ^2(2 (c+d x))-13 b^3\right)}{(a+b)^4 ((a+b) \cosh (2 (c+d x))+a-b)^2}+\frac{2 \cosh (3 (c+d x))}{(a+b)^3}+\frac{15 i \sqrt{b} (3 a-4 b) \left(\tan ^{-1}\left(\frac{-\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)+\tan ^{-1}\left(\frac{\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)\right)}{(a+b)^{9/2}}}{24 d}","\frac{\cosh ^3(c+d x)}{3 d (a+b)^3}-\frac{(a-2 b) \cosh (c+d x)}{d (a+b)^4}+\frac{b (7 a-4 b) \text{sech}(c+d x)}{8 d (a+b)^4 \left(a-b \text{sech}^2(c+d x)+b\right)}+\frac{a b \text{sech}(c+d x)}{4 d (a+b)^3 \left(a-b \text{sech}^2(c+d x)+b\right)^2}+\frac{5 \sqrt{b} (3 a-4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right)}{8 d (a+b)^{9/2}}",1,"(((15*I)*(3*a - 4*b)*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]]))/(a + b)^(9/2) - (6*Cosh[c + d*x]*(3*a^3 - 24*a^2*b + 30*a*b^2 - 13*b^3 + (6*a^3 - 27*a^2*b - 11*a*b^2 + 22*b^3)*Cosh[2*(c + d*x)] + 3*(a - 3*b)*(a + b)^2*Cosh[2*(c + d*x)]^2))/((a + b)^4*(a - b + (a + b)*Cosh[2*(c + d*x)])^2) + (2*Cosh[3*(c + d*x)])/(a + b)^3)/(24*d)","C",1
43,1,158,185,1.3399745,"\int \frac{\sinh ^2(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\frac{\sqrt{b} \left(-15 a^2+10 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{3/2}}-\frac{4 b^2 (a+b) \sinh (2 (c+d x))}{((a+b) \cosh (2 (c+d x))+a-b)^2}-4 (a-5 b) (c+d x)+2 (a+b) \sinh (2 (c+d x))-\frac{b (9 a-b) (a+b) \sinh (2 (c+d x))}{a ((a+b) \cosh (2 (c+d x))+a-b)}}{8 d (a+b)^4}","-\frac{\sqrt{b} \left(15 a^2-10 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{3/2} d (a+b)^4}-\frac{b (11 a-b) \tanh (c+d x)}{8 a d (a+b)^3 \left(a+b \tanh ^2(c+d x)\right)}-\frac{3 b \tanh (c+d x)}{4 d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)^2}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 d (a+b) \left(a+b \tanh ^2(c+d x)\right)^2}-\frac{x (a-5 b)}{2 (a+b)^4}",1,"(-4*(a - 5*b)*(c + d*x) + (Sqrt[b]*(-15*a^2 + 10*a*b + b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/a^(3/2) + 2*(a + b)*Sinh[2*(c + d*x)] - (4*b^2*(a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)])^2 - ((9*a - b)*b*(a + b)*Sinh[2*(c + d*x)])/(a*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(8*(a + b)^4*d)","A",1
44,1,157,126,1.8283099,"\int \frac{\sinh (c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\frac{2 \cosh (c+d x) \left(-\frac{4 b^2}{((a+b) \cosh (2 (c+d x))+a-b)^2}-\frac{9 b}{(a+b) \cosh (2 (c+d x))+a-b}+4\right)}{(a+b)^3}-\frac{15 i \sqrt{b} \left(\tan ^{-1}\left(\frac{-\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)+\tan ^{-1}\left(\frac{\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)\right)}{(a+b)^{7/2}}}{8 d}","\frac{15 \cosh (c+d x)}{8 d (a+b)^3}-\frac{5 \cosh (c+d x)}{8 d (a+b)^2 \left(a-b \text{sech}^2(c+d x)+b\right)}-\frac{\cosh (c+d x)}{4 d (a+b) \left(a-b \text{sech}^2(c+d x)+b\right)^2}-\frac{15 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right)}{8 d (a+b)^{7/2}}",1,"(((-15*I)*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]]))/(a + b)^(7/2) + (2*Cosh[c + d*x]*(4 - (4*b^2)/(a - b + (a + b)*Cosh[2*(c + d*x)])^2 - (9*b)/(a - b + (a + b)*Cosh[2*(c + d*x)])))/(a + b)^3)/(8*d)","C",1
45,1,236,156,1.5132872,"\int \frac{\text{csch}(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Csch[c + d*x]/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\frac{8 a^2 b^2 \cosh (c+d x)}{(a+b)^2 ((a+b) \cosh (2 (c+d x))+a-b)^2}+\frac{i \sqrt{b} \left(15 a^2+20 a b+8 b^2\right) \tan ^{-1}\left(\frac{-\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)}{(a+b)^{5/2}}+\frac{i \sqrt{b} \left(15 a^2+20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)}{(a+b)^{5/2}}+\frac{2 a b (9 a+4 b) \cosh (c+d x)}{(a+b)^2 ((a+b) \cosh (2 (c+d x))+a-b)}+8 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^3 d}","-\frac{\tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac{b (7 a+4 b) \text{sech}(c+d x)}{8 a^2 d (a+b)^2 \left(a-b \text{sech}^2(c+d x)+b\right)}+\frac{\sqrt{b} \left(15 a^2+20 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right)}{8 a^3 d (a+b)^{5/2}}+\frac{b \text{sech}(c+d x)}{4 a d (a+b) \left(a-b \text{sech}^2(c+d x)+b\right)^2}",1,"((I*Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/(a + b)^(5/2) + (I*Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/(a + b)^(5/2) + (8*a^2*b^2*Cosh[c + d*x])/((a + b)^2*(a - b + (a + b)*Cosh[2*(c + d*x)])^2) + (2*a*b*(9*a + 4*b)*Cosh[c + d*x])/((a + b)^2*(a - b + (a + b)*Cosh[2*(c + d*x)])) + 8*Log[Tanh[(c + d*x)/2]])/(8*a^3*d)","C",1
46,1,109,112,1.0600746,"\int \frac{\text{csch}^2(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Csch[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{-15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)-\frac{\sqrt{a} b \sinh (2 (c+d x)) ((9 a+7 b) \cosh (2 (c+d x))+9 a-7 b)}{((a+b) \cosh (2 (c+d x))+a-b)^2}-8 \sqrt{a} \coth (c+d x)}{8 a^{7/2} d}","-\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{7/2} d}-\frac{15 \coth (c+d x)}{8 a^3 d}+\frac{5 \coth (c+d x)}{8 a^2 d \left(a+b \tanh ^2(c+d x)\right)}+\frac{\coth (c+d x)}{4 a d \left(a+b \tanh ^2(c+d x)\right)^2}",1,"(-15*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] - 8*Sqrt[a]*Coth[c + d*x] - (Sqrt[a]*b*(9*a - 7*b + (9*a + 7*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)])^2)/(8*a^(7/2)*d)","A",1
47,1,269,196,4.4468019,"\int \frac{\text{csch}^3(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Csch[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3,x]","-\frac{\frac{8 a^2 b^2 \cosh (c+d x)}{(a+b) ((a+b) \cosh (2 (c+d x))+a-b)^2}+\frac{i \sqrt{b} \left(15 a^2+40 a b+24 b^2\right) \tan ^{-1}\left(\frac{-\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)}{(a+b)^{3/2}}+\frac{i \sqrt{b} \left(15 a^2+40 a b+24 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a+b}}{\sqrt{b}}\right)}{(a+b)^{3/2}}+\frac{2 a b (9 a+8 b) \cosh (c+d x)}{(a+b) ((a+b) \cosh (2 (c+d x))+a-b)}+4 (a+6 b) \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+a \text{csch}^2\left(\frac{1}{2} (c+d x)\right)+a \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{8 a^4 d}","\frac{(a+6 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^4 d}-\frac{b (11 a+12 b) \text{sech}(c+d x)}{8 a^3 d (a+b) \left(a-b \text{sech}^2(c+d x)+b\right)}-\frac{3 b \text{sech}(c+d x)}{4 a^2 d \left(a-b \text{sech}^2(c+d x)+b\right)^2}-\frac{\sqrt{b} \left(15 a^2+40 a b+24 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right)}{8 a^4 d (a+b)^{3/2}}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left(a-b \text{sech}^2(c+d x)+b\right)^2}",1,"-1/8*((I*Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)*ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/(a + b)^(3/2) + (I*Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)*ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/(a + b)^(3/2) + (8*a^2*b^2*Cosh[c + d*x])/((a + b)*(a - b + (a + b)*Cosh[2*(c + d*x)])^2) + (2*a*b*(9*a + 8*b)*Cosh[c + d*x])/((a + b)*(a - b + (a + b)*Cosh[2*(c + d*x)])) + a*Csch[(c + d*x)/2]^2 + 4*(a + 6*b)*Log[Tanh[(c + d*x)/2]] + a*Sech[(c + d*x)/2]^2)/(a^4*d)","C",1
48,1,149,151,1.4009823,"\int \frac{\text{csch}^4(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Csch[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\frac{3 \sqrt{a} b \sinh (2 (c+d x)) \left(\left(9 a^2+20 a b+11 b^2\right) \cosh (2 (c+d x))+9 a^2+6 a b-11 b^2\right)}{((a+b) \cosh (2 (c+d x))+a-b)^2}+15 \sqrt{b} (3 a+7 b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)-8 \sqrt{a} \coth (c+d x) \left(a \text{csch}^2(c+d x)-2 a-9 b\right)}{24 a^{9/2} d}","\frac{5 \sqrt{b} (3 a+7 b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{9/2} d}+\frac{b (7 a+11 b) \tanh (c+d x)}{8 a^4 d \left(a+b \tanh ^2(c+d x)\right)}+\frac{(a+3 b) \coth (c+d x)}{a^4 d}+\frac{b (a+b) \tanh (c+d x)}{4 a^3 d \left(a+b \tanh ^2(c+d x)\right)^2}-\frac{\coth ^3(c+d x)}{3 a^3 d}",1,"(15*Sqrt[b]*(3*a + 7*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] - 8*Sqrt[a]*Coth[c + d*x]*(-2*a - 9*b + a*Csch[c + d*x]^2) + (3*Sqrt[a]*b*(9*a^2 + 6*a*b - 11*b^2 + (9*a^2 + 20*a*b + 11*b^2)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)])^2)/(24*a^(9/2)*d)","A",1
49,1,92,132,0.199783,"\int \sinh ^4(c+d x) \left(a+b \tanh ^3(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]^4*(a + b*Tanh[c + d*x]^3),x]","\frac{3 a (c+d x)}{8 d}-\frac{a \sinh (2 (c+d x))}{4 d}+\frac{a \sinh (4 (c+d x))}{32 d}+\frac{b \left(\sinh ^4(c+d x)-4 \sinh ^2(c+d x)+2 \text{sech}^2(c+d x)+12 \log (\cosh (c+d x))\right)}{4 d}","-\frac{3 (a+8 b) \log (1-\tanh (c+d x))}{16 d}+\frac{3 (a-8 b) \log (\tanh (c+d x)+1)}{16 d}+\frac{\sinh ^4(c+d x) (a \tanh (c+d x)+b)}{4 d}-\frac{\sinh ^2(c+d x) \tanh (c+d x) (a+8 b \tanh (c+d x))}{8 d}-\frac{3 a \tanh (c+d x)}{8 d}-\frac{3 b \tanh ^2(c+d x)}{2 d}",1,"(3*a*(c + d*x))/(8*d) + (b*(12*Log[Cosh[c + d*x]] + 2*Sech[c + d*x]^2 - 4*Sinh[c + d*x]^2 + Sinh[c + d*x]^4))/(4*d) - (a*Sinh[2*(c + d*x)])/(4*d) + (a*Sinh[4*(c + d*x)])/(32*d)","A",1
50,1,104,98,0.2764426,"\int \sinh ^3(c+d x) \left(a+b \tanh ^3(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^3),x]","-\frac{3 a \cosh (c+d x)}{4 d}+\frac{a \cosh (3 (c+d x))}{12 d}+\frac{b \sinh ^3(c+d x) \tanh ^2(c+d x)}{3 d}-\frac{5 b \left(2 \sinh (c+d x) \tanh ^2(c+d x)-3 \left(\tan ^{-1}(\sinh (c+d x))-\tanh (c+d x) \text{sech}(c+d x)\right)\right)}{6 d}","\frac{a \cosh ^3(c+d x)}{3 d}-\frac{a \cosh (c+d x)}{d}+\frac{5 b \sinh ^3(c+d x)}{6 d}-\frac{5 b \sinh (c+d x)}{2 d}-\frac{b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}+\frac{5 b \tan ^{-1}(\sinh (c+d x))}{2 d}",1,"(-3*a*Cosh[c + d*x])/(4*d) + (a*Cosh[3*(c + d*x)])/(12*d) + (b*Sinh[c + d*x]^3*Tanh[c + d*x]^2)/(3*d) - (5*b*(2*Sinh[c + d*x]*Tanh[c + d*x]^2 - 3*(ArcTan[Sinh[c + d*x]] - Sech[c + d*x]*Tanh[c + d*x])))/(6*d)","A",1
51,1,69,100,0.1146551,"\int \sinh ^2(c+d x) \left(a+b \tanh ^3(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]^2*(a + b*Tanh[c + d*x]^3),x]","\frac{a (-c-d x)}{2 d}+\frac{a \sinh (2 (c+d x))}{4 d}-\frac{b \left(-\sinh ^2(c+d x)+\text{sech}^2(c+d x)+4 \log (\cosh (c+d x))\right)}{2 d}","\frac{(a+4 b) \log (1-\tanh (c+d x))}{4 d}-\frac{(a-4 b) \log (\tanh (c+d x)+1)}{4 d}+\frac{\sinh ^2(c+d x) (a \tanh (c+d x)+b)}{2 d}+\frac{a \tanh (c+d x)}{2 d}+\frac{b \tanh ^2(c+d x)}{2 d}",1,"(a*(-c - d*x))/(2*d) - (b*(4*Log[Cosh[c + d*x]] + Sech[c + d*x]^2 - Sinh[c + d*x]^2))/(2*d) + (a*Sinh[2*(c + d*x)])/(4*d)","A",1
52,1,72,63,0.1261765,"\int \sinh (c+d x) \left(a+b \tanh ^3(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]*(a + b*Tanh[c + d*x]^3),x]","\frac{a \sinh (c) \sinh (d x)}{d}+\frac{a \cosh (c) \cosh (d x)}{d}+\frac{b \sinh (c+d x) \tanh ^2(c+d x)}{d}-\frac{3 b \left(\tan ^{-1}(\sinh (c+d x))-\tanh (c+d x) \text{sech}(c+d x)\right)}{2 d}","\frac{a \cosh (c+d x)}{d}+\frac{3 b \sinh (c+d x)}{2 d}-\frac{3 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}",1,"(a*Cosh[c]*Cosh[d*x])/d + (a*Sinh[c]*Sinh[d*x])/d + (b*Sinh[c + d*x]*Tanh[c + d*x]^2)/d - (3*b*(ArcTan[Sinh[c + d*x]] - Sech[c + d*x]*Tanh[c + d*x]))/(2*d)","A",1
53,1,75,49,0.0322095,"\int \text{csch}(c+d x) \left(a+b \tanh ^3(c+d x)\right) \, dx","Integrate[Csch[c + d*x]*(a + b*Tanh[c + d*x]^3),x]","\frac{a \log \left(\sinh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{a \log \left(\cosh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d}","-\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d}",1,"(b*ArcTan[Sinh[c + d*x]])/(2*d) - (a*Log[Cosh[c/2 + (d*x)/2]])/d + (a*Log[Sinh[c/2 + (d*x)/2]])/d - (b*Sech[c + d*x]*Tanh[c + d*x])/(2*d)","A",1
54,1,29,29,0.0274397,"\int \text{csch}^2(c+d x) \left(a+b \tanh ^3(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^2*(a + b*Tanh[c + d*x]^3),x]","-\frac{a \coth (c+d x)}{d}-\frac{b \text{sech}^2(c+d x)}{2 d}","\frac{b \tanh ^2(c+d x)}{2 d}-\frac{a \coth (c+d x)}{d}",1,"-((a*Coth[c + d*x])/d) - (b*Sech[c + d*x]^2)/(2*d)","A",1
55,1,95,71,0.0275642,"\int \text{csch}^3(c+d x) \left(a+b \tanh ^3(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^3),x]","-\frac{a \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d}","\frac{a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d}",1,"(b*ArcTan[Sinh[c + d*x]])/(2*d) - (a*Csch[(c + d*x)/2]^2)/(8*d) - (a*Log[Tanh[(c + d*x)/2]])/(2*d) - (a*Sech[(c + d*x)/2]^2)/(8*d) + (b*Sech[c + d*x]*Tanh[c + d*x])/(2*d)","A",1
56,1,74,56,0.2230956,"\int \text{csch}^4(c+d x) \left(a+b \tanh ^3(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^3),x]","\frac{2 a \coth (c+d x)}{3 d}-\frac{a \coth (c+d x) \text{csch}^2(c+d x)}{3 d}-\frac{b \left(-\text{sech}^2(c+d x)-2 \log (\sinh (c+d x))+2 \log (\cosh (c+d x))\right)}{2 d}","-\frac{a \coth ^3(c+d x)}{3 d}+\frac{a \coth (c+d x)}{d}-\frac{b \tanh ^2(c+d x)}{2 d}+\frac{b \log (\tanh (c+d x))}{d}",1,"(2*a*Coth[c + d*x])/(3*d) - (a*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (b*(2*Log[Cosh[c + d*x]] - 2*Log[Sinh[c + d*x]] - Sech[c + d*x]^2))/(2*d)","A",1
57,1,156,170,2.8287143,"\int \sinh ^4(c+d x) \left(a+b \tanh ^3(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]^4*(a + b*Tanh[c + d*x]^3)^2,x]","\frac{60 \left(a^2+21 b^2\right) (c+d x)-40 \left(a^2+4 b^2\right) \sinh (2 (c+d x))+5 \left(a^2+b^2\right) \sinh (4 (c+d x))-200 a b \cosh (2 (c+d x))+10 a b \cosh (4 (c+d x))+160 a b \text{sech}^2(c+d x)+960 a b \log (\cosh (c+d x))-1152 b^2 \tanh (c+d x)-32 b^2 \tanh (c+d x) \text{sech}^4(c+d x)+224 b^2 \tanh (c+d x) \text{sech}^2(c+d x)}{160 d}","\frac{\sinh (c+d x) \cosh ^3(c+d x) \left(a^2+2 a b \tanh (c+d x)+b^2\right)}{4 d}-\frac{\sinh (c+d x) \cosh (c+d x) \left(5 a^2+20 a b \tanh (c+d x)+17 b^2\right)}{8 d}+\frac{3}{8} x \left(a^2+21 b^2\right)-\frac{a b \tanh ^2(c+d x)}{d}+\frac{6 a b \log (\cosh (c+d x))}{d}-\frac{b^2 \tanh ^5(c+d x)}{5 d}-\frac{b^2 \tanh ^3(c+d x)}{d}-\frac{6 b^2 \tanh (c+d x)}{d}",1,"(60*(a^2 + 21*b^2)*(c + d*x) - 200*a*b*Cosh[2*(c + d*x)] + 10*a*b*Cosh[4*(c + d*x)] + 960*a*b*Log[Cosh[c + d*x]] + 160*a*b*Sech[c + d*x]^2 - 40*(a^2 + 4*b^2)*Sinh[2*(c + d*x)] + 5*(a^2 + b^2)*Sinh[4*(c + d*x)] - 1152*b^2*Tanh[c + d*x] + 224*b^2*Sech[c + d*x]^2*Tanh[c + d*x] - 32*b^2*Sech[c + d*x]^4*Tanh[c + d*x])/(160*d)","A",1
58,1,121,182,0.7795728,"\int \sinh ^3(c+d x) \left(a+b \tanh ^3(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^3)^2,x]","\frac{-45 \left(a^2+5 b^2\right) \cosh (c+d x)+5 \left(a^2+b^2\right) \cosh (3 (c+d x))-2 b \left(30 \text{sech}(c+d x) (a \tanh (c+d x)+6 b)-5 a \left(-27 \sinh (c+d x)+\sinh (3 (c+d x))+60 \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)\right)+6 b \text{sech}^5(c+d x)-40 b \text{sech}^3(c+d x)\right)}{60 d}","\frac{a^2 \cosh ^3(c+d x)}{3 d}-\frac{a^2 \cosh (c+d x)}{d}+\frac{5 a b \sinh ^3(c+d x)}{3 d}-\frac{5 a b \sinh (c+d x)}{d}-\frac{a b \sinh ^3(c+d x) \tanh ^2(c+d x)}{d}+\frac{5 a b \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b^2 \cosh ^3(c+d x)}{3 d}-\frac{4 b^2 \cosh (c+d x)}{d}-\frac{b^2 \text{sech}^5(c+d x)}{5 d}+\frac{4 b^2 \text{sech}^3(c+d x)}{3 d}-\frac{6 b^2 \text{sech}(c+d x)}{d}",1,"(-45*(a^2 + 5*b^2)*Cosh[c + d*x] + 5*(a^2 + b^2)*Cosh[3*(c + d*x)] - 2*b*(-40*b*Sech[c + d*x]^3 + 6*b*Sech[c + d*x]^5 - 5*a*(60*ArcTan[Tanh[(c + d*x)/2]] - 27*Sinh[c + d*x] + Sinh[3*(c + d*x)]) + 30*Sech[c + d*x]*(6*b + a*Tanh[c + d*x])))/(60*d)","A",1
59,1,137,129,1.6018944,"\int \sinh ^2(c+d x) \left(a+b \tanh ^3(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]^2*(a + b*Tanh[c + d*x]^3)^2,x]","\frac{15 a^2 \sinh (2 (c+d x))-30 a^2 c-30 a^2 d x+30 a b \cosh (2 (c+d x))-240 a b \log (\cosh (c+d x))-4 b \text{sech}^2(c+d x) (15 a+16 b \tanh (c+d x))+15 b^2 \sinh (2 (c+d x))+232 b^2 \tanh (c+d x)+12 b^2 \tanh (c+d x) \text{sech}^4(c+d x)-210 b^2 c-210 b^2 d x}{60 d}","\frac{\sinh (c+d x) \cosh (c+d x) \left(a^2+2 a b \tanh (c+d x)+b^2\right)}{2 d}-\frac{1}{2} x \left(a^2+7 b^2\right)+\frac{a b \tanh ^2(c+d x)}{d}-\frac{4 a b \log (\cosh (c+d x))}{d}+\frac{b^2 \tanh ^5(c+d x)}{5 d}+\frac{2 b^2 \tanh ^3(c+d x)}{3 d}+\frac{3 b^2 \tanh (c+d x)}{d}",1,"(-30*a^2*c - 210*b^2*c - 30*a^2*d*x - 210*b^2*d*x + 30*a*b*Cosh[2*(c + d*x)] - 240*a*b*Log[Cosh[c + d*x]] + 15*a^2*Sinh[2*(c + d*x)] + 15*b^2*Sinh[2*(c + d*x)] + 232*b^2*Tanh[c + d*x] + 12*b^2*Sech[c + d*x]^4*Tanh[c + d*x] - 4*b*Sech[c + d*x]^2*(15*a + 16*b*Tanh[c + d*x]))/(60*d)","A",1
60,1,90,123,0.437714,"\int \sinh (c+d x) \left(a+b \tanh ^3(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]*(a + b*Tanh[c + d*x]^3)^2,x]","\frac{5 \left(a^2+b^2\right) \cosh (c+d x)+b \left(5 \text{sech}(c+d x) (a \tanh (c+d x)+3 b)+10 a \left(\sinh (c+d x)-3 \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)\right)+b \text{sech}^5(c+d x)-5 b \text{sech}^3(c+d x)\right)}{5 d}","\frac{a^2 \cosh (c+d x)}{d}+\frac{3 a b \sinh (c+d x)}{d}-\frac{3 a b \tan ^{-1}(\sinh (c+d x))}{d}-\frac{a b \sinh (c+d x) \tanh ^2(c+d x)}{d}+\frac{b^2 \cosh (c+d x)}{d}+\frac{b^2 \text{sech}^5(c+d x)}{5 d}-\frac{b^2 \text{sech}^3(c+d x)}{d}+\frac{3 b^2 \text{sech}(c+d x)}{d}",1,"(5*(a^2 + b^2)*Cosh[c + d*x] + b*(-5*b*Sech[c + d*x]^3 + b*Sech[c + d*x]^5 + 10*a*(-3*ArcTan[Tanh[(c + d*x)/2]] + Sinh[c + d*x]) + 5*Sech[c + d*x]*(3*b + a*Tanh[c + d*x])))/(5*d)","A",1
61,1,106,98,0.170585,"\int \text{csch}(c+d x) \left(a+b \tanh ^3(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]*(a + b*Tanh[c + d*x]^3)^2,x]","\frac{a^2 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{2 a b \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{a b \tanh (c+d x) \text{sech}(c+d x)}{d}-\frac{b^2 \text{sech}^5(c+d x)}{5 d}+\frac{2 b^2 \text{sech}^3(c+d x)}{3 d}-\frac{b^2 \text{sech}(c+d x)}{d}","-\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{a b \tan ^{-1}(\sinh (c+d x))}{d}-\frac{a b \tanh (c+d x) \text{sech}(c+d x)}{d}-\frac{b^2 \text{sech}^5(c+d x)}{5 d}+\frac{2 b^2 \text{sech}^3(c+d x)}{3 d}-\frac{b^2 \text{sech}(c+d x)}{d}",1,"(2*a*b*ArcTan[Tanh[(c + d*x)/2]])/d + (a^2*Log[Tanh[(c + d*x)/2]])/d - (b^2*Sech[c + d*x])/d + (2*b^2*Sech[c + d*x]^3)/(3*d) - (b^2*Sech[c + d*x]^5)/(5*d) - (a*b*Sech[c + d*x]*Tanh[c + d*x])/d","A",1
62,1,94,47,0.2521551,"\int \text{csch}^2(c+d x) \left(a+b \tanh ^3(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^2*(a + b*Tanh[c + d*x]^3)^2,x]","-\frac{a^2 \coth (c+d x)}{d}-\frac{a b \text{sech}^2(c+d x)}{d}+\frac{b^2 \tanh (c+d x)}{5 d}+\frac{b^2 \tanh (c+d x) \text{sech}^4(c+d x)}{5 d}-\frac{2 b^2 \tanh (c+d x) \text{sech}^2(c+d x)}{5 d}","-\frac{a^2 \coth (c+d x)}{d}+\frac{a b \tanh ^2(c+d x)}{d}+\frac{b^2 \tanh ^5(c+d x)}{5 d}",1,"-((a^2*Coth[c + d*x])/d) - (a*b*Sech[c + d*x]^2)/d + (b^2*Tanh[c + d*x])/(5*d) - (2*b^2*Sech[c + d*x]^2*Tanh[c + d*x])/(5*d) + (b^2*Sech[c + d*x]^4*Tanh[c + d*x])/(5*d)","A",1
63,1,138,107,0.1637548,"\int \text{csch}^3(c+d x) \left(a+b \tanh ^3(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^3)^2,x]","-\frac{a^2 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a^2 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a^2 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{2 a b \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{a b \tanh (c+d x) \text{sech}(c+d x)}{d}+\frac{b^2 \text{sech}^5(c+d x)}{5 d}-\frac{b^2 \text{sech}^3(c+d x)}{3 d}","\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a^2 \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{a b \tan ^{-1}(\sinh (c+d x))}{d}+\frac{a b \tanh (c+d x) \text{sech}(c+d x)}{d}+\frac{b^2 \text{sech}^5(c+d x)}{5 d}-\frac{b^2 \text{sech}^3(c+d x)}{3 d}",1,"(2*a*b*ArcTan[Tanh[(c + d*x)/2]])/d - (a^2*Csch[(c + d*x)/2]^2)/(8*d) - (a^2*Log[Tanh[(c + d*x)/2]])/(2*d) - (a^2*Sech[(c + d*x)/2]^2)/(8*d) - (b^2*Sech[c + d*x]^3)/(3*d) + (b^2*Sech[c + d*x]^5)/(5*d) + (a*b*Sech[c + d*x]*Tanh[c + d*x])/d","A",1
64,1,147,97,0.218398,"\int \text{csch}^4(c+d x) \left(a+b \tanh ^3(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^3)^2,x]","\frac{2 a^2 \coth (c+d x)}{3 d}-\frac{a^2 \coth (c+d x) \text{csch}^2(c+d x)}{3 d}+\frac{a b \text{sech}^2(c+d x)}{d}+\frac{2 a b \log (\sinh (c+d x))}{d}-\frac{2 a b \log (\cosh (c+d x))}{d}+\frac{2 b^2 \tanh (c+d x)}{15 d}-\frac{b^2 \tanh (c+d x) \text{sech}^4(c+d x)}{5 d}+\frac{b^2 \tanh (c+d x) \text{sech}^2(c+d x)}{15 d}","-\frac{a^2 \coth ^3(c+d x)}{3 d}+\frac{a^2 \coth (c+d x)}{d}-\frac{a b \tanh ^2(c+d x)}{d}+\frac{2 a b \log (\tanh (c+d x))}{d}-\frac{b^2 \tanh ^5(c+d x)}{5 d}+\frac{b^2 \tanh ^3(c+d x)}{3 d}",1,"(2*a^2*Coth[c + d*x])/(3*d) - (a^2*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (2*a*b*Log[Cosh[c + d*x]])/d + (2*a*b*Log[Sinh[c + d*x]])/d + (a*b*Sech[c + d*x]^2)/d + (2*b^2*Tanh[c + d*x])/(15*d) + (b^2*Sech[c + d*x]^2*Tanh[c + d*x])/(15*d) - (b^2*Sech[c + d*x]^4*Tanh[c + d*x])/(5*d)","A",1
65,1,294,275,6.2706128,"\int \sinh ^4(c+d x) \left(a+b \tanh ^3(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]^4*(a + b*Tanh[c + d*x]^3)^3,x]","\frac{3 \left(3 a^2 b+5 b^3\right) \log (\cosh (c+d x))}{d}+\frac{3 a \left(a^2+63 b^2\right) (c+d x)}{8 d}-\frac{a \left(a^2+12 b^2\right) \sinh (2 (c+d x))}{4 d}+\frac{a \left(a^2+3 b^2\right) \sinh (4 (c+d x))}{32 d}-\frac{b \left(15 a^2+11 b^2\right) \cosh (2 (c+d x))}{8 d}+\frac{b \left(3 a^2+b^2\right) \cosh (4 (c+d x))}{32 d}+\frac{b \left(3 a^2+20 b^2\right) \text{sech}^2(c+d x)}{2 d}-\frac{108 a b^2 \tanh (c+d x)}{5 d}-\frac{3 a b^2 \tanh (c+d x) \text{sech}^4(c+d x)}{5 d}+\frac{21 a b^2 \tanh (c+d x) \text{sech}^2(c+d x)}{5 d}-\frac{b^3 \text{sech}^8(c+d x)}{8 d}+\frac{b^3 \text{sech}^6(c+d x)}{d}-\frac{15 b^3 \text{sech}^4(c+d x)}{4 d}","-\frac{b \left(3 a^2+10 b^2\right) \tanh ^2(c+d x)}{2 d}+\frac{3 b \left(3 a^2+5 b^2\right) \log (\cosh (c+d x))}{d}+\frac{\sinh (c+d x) \cosh ^3(c+d x) \left(b \left(3 a^2+b^2\right) \tanh (c+d x)+a \left(a^2+3 b^2\right)\right)}{4 d}-\frac{\sinh (c+d x) \cosh (c+d x) \left(2 b \left(15 a^2+11 b^2\right) \tanh (c+d x)+a \left(5 a^2+51 b^2\right)\right)}{8 d}+\frac{3}{8} a x \left(a^2+63 b^2\right)-\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}-\frac{3 a b^2 \tanh ^3(c+d x)}{d}-\frac{18 a b^2 \tanh (c+d x)}{d}-\frac{b^3 \tanh ^8(c+d x)}{8 d}-\frac{b^3 \tanh ^6(c+d x)}{2 d}-\frac{3 b^3 \tanh ^4(c+d x)}{2 d}",1,"(3*a*(a^2 + 63*b^2)*(c + d*x))/(8*d) - (b*(15*a^2 + 11*b^2)*Cosh[2*(c + d*x)])/(8*d) + (b*(3*a^2 + b^2)*Cosh[4*(c + d*x)])/(32*d) + (3*(3*a^2*b + 5*b^3)*Log[Cosh[c + d*x]])/d + (b*(3*a^2 + 20*b^2)*Sech[c + d*x]^2)/(2*d) - (15*b^3*Sech[c + d*x]^4)/(4*d) + (b^3*Sech[c + d*x]^6)/d - (b^3*Sech[c + d*x]^8)/(8*d) - (a*(a^2 + 12*b^2)*Sinh[2*(c + d*x)])/(4*d) + (a*(a^2 + 3*b^2)*Sinh[4*(c + d*x)])/(32*d) - (108*a*b^2*Tanh[c + d*x])/(5*d) + (21*a*b^2*Sech[c + d*x]^2*Tanh[c + d*x])/(5*d) - (3*a*b^2*Sech[c + d*x]^4*Tanh[c + d*x])/(5*d)","A",1
66,1,291,351,6.614177,"\int \sinh ^3(c+d x) \left(a+b \tanh ^3(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^3)^3,x]","-\frac{3 \text{sech}^2(c+d x) \left(64 a^2 b \sinh (c+d x)+255 b^3 \sinh (c+d x)\right)}{128 d}-\frac{3 b \left(9 a^2+7 b^2\right) \sinh (c+d x)}{4 d}+\frac{b \left(3 a^2+b^2\right) \sinh (3 (c+d x))}{12 d}-\frac{3 a \left(a^2+15 b^2\right) \cosh (c+d x)}{4 d}+\frac{a \left(a^2+3 b^2\right) \cosh (3 (c+d x))}{12 d}+\frac{15 b \left(64 a^2+77 b^2\right) \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}+\frac{4 a b^2 \text{sech}^3(c+d x)}{d}-\frac{18 a b^2 \text{sech}(c+d x)}{d}+\frac{b^3 \tanh (c+d x) \text{sech}^7(c+d x)}{8 d}-\frac{41 b^3 \tanh (c+d x) \text{sech}^5(c+d x)}{48 d}+\frac{515 b^3 \tanh (c+d x) \text{sech}^3(c+d x)}{192 d}","\frac{a^3 \cosh ^3(c+d x)}{3 d}-\frac{a^3 \cosh (c+d x)}{d}+\frac{5 a^2 b \sinh ^3(c+d x)}{2 d}-\frac{15 a^2 b \sinh (c+d x)}{2 d}-\frac{3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}+\frac{15 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{a b^2 \cosh ^3(c+d x)}{d}-\frac{12 a b^2 \cosh (c+d x)}{d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}+\frac{4 a b^2 \text{sech}^3(c+d x)}{d}-\frac{18 a b^2 \text{sech}(c+d x)}{d}+\frac{385 b^3 \sinh ^3(c+d x)}{128 d}-\frac{1155 b^3 \sinh (c+d x)}{128 d}-\frac{b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}-\frac{11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac{33 b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}-\frac{231 b^3 \sinh ^3(c+d x) \tanh ^2(c+d x)}{128 d}+\frac{1155 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}",1,"(15*b*(64*a^2 + 77*b^2)*ArcTan[Tanh[(c + d*x)/2]])/(64*d) - (3*a*(a^2 + 15*b^2)*Cosh[c + d*x])/(4*d) + (a*(a^2 + 3*b^2)*Cosh[3*(c + d*x)])/(12*d) - (18*a*b^2*Sech[c + d*x])/d + (4*a*b^2*Sech[c + d*x]^3)/d - (3*a*b^2*Sech[c + d*x]^5)/(5*d) - (3*b*(9*a^2 + 7*b^2)*Sinh[c + d*x])/(4*d) - (3*Sech[c + d*x]^2*(64*a^2*b*Sinh[c + d*x] + 255*b^3*Sinh[c + d*x]))/(128*d) + (b*(3*a^2 + b^2)*Sinh[3*(c + d*x)])/(12*d) + (515*b^3*Sech[c + d*x]^3*Tanh[c + d*x])/(192*d) - (41*b^3*Sech[c + d*x]^5*Tanh[c + d*x])/(48*d) + (b^3*Sech[c + d*x]^7*Tanh[c + d*x])/(8*d)","A",1
67,1,244,220,6.2529899,"\int \sinh ^2(c+d x) \left(a+b \tanh ^3(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]^2*(a + b*Tanh[c + d*x]^3)^3,x]","\frac{\left(-6 a^2 b-5 b^3\right) \log (\cosh (c+d x))}{d}-\frac{a \left(a^2+21 b^2\right) (c+d x)}{2 d}+\frac{a \left(a^2+3 b^2\right) \sinh (2 (c+d x))}{4 d}+\frac{b \left(3 a^2+b^2\right) \cosh (2 (c+d x))}{4 d}-\frac{b \left(3 a^2+10 b^2\right) \text{sech}^2(c+d x)}{2 d}+\frac{58 a b^2 \tanh (c+d x)}{5 d}+\frac{3 a b^2 \tanh (c+d x) \text{sech}^4(c+d x)}{5 d}-\frac{16 a b^2 \tanh (c+d x) \text{sech}^2(c+d x)}{5 d}+\frac{b^3 \text{sech}^8(c+d x)}{8 d}-\frac{5 b^3 \text{sech}^6(c+d x)}{6 d}+\frac{5 b^3 \text{sech}^4(c+d x)}{2 d}","\frac{b \left(3 a^2+4 b^2\right) \tanh ^2(c+d x)}{2 d}-\frac{b \left(6 a^2+5 b^2\right) \log (\cosh (c+d x))}{d}+\frac{\sinh (c+d x) \cosh (c+d x) \left(b \left(3 a^2+b^2\right) \tanh (c+d x)+a \left(a^2+3 b^2\right)\right)}{2 d}-\frac{1}{2} a x \left(a^2+21 b^2\right)+\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac{2 a b^2 \tanh ^3(c+d x)}{d}+\frac{9 a b^2 \tanh (c+d x)}{d}+\frac{b^3 \tanh ^8(c+d x)}{8 d}+\frac{b^3 \tanh ^6(c+d x)}{3 d}+\frac{3 b^3 \tanh ^4(c+d x)}{4 d}",1,"-1/2*(a*(a^2 + 21*b^2)*(c + d*x))/d + (b*(3*a^2 + b^2)*Cosh[2*(c + d*x)])/(4*d) + ((-6*a^2*b - 5*b^3)*Log[Cosh[c + d*x]])/d - (b*(3*a^2 + 10*b^2)*Sech[c + d*x]^2)/(2*d) + (5*b^3*Sech[c + d*x]^4)/(2*d) - (5*b^3*Sech[c + d*x]^6)/(6*d) + (b^3*Sech[c + d*x]^8)/(8*d) + (a*(a^2 + 3*b^2)*Sinh[2*(c + d*x)])/(4*d) + (58*a*b^2*Tanh[c + d*x])/(5*d) - (16*a*b^2*Sech[c + d*x]^2*Tanh[c + d*x])/(5*d) + (3*a*b^2*Sech[c + d*x]^4*Tanh[c + d*x])/(5*d)","A",1
68,1,233,269,6.3286512,"\int \sinh (c+d x) \left(a+b \tanh ^3(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]*(a + b*Tanh[c + d*x]^3)^3,x]","\frac{\text{sech}^2(c+d x) \left(192 a^2 b \sinh (c+d x)+325 b^3 \sinh (c+d x)\right)}{128 d}+\frac{b \left(3 a^2+b^2\right) \sinh (c+d x)}{d}+\frac{a \left(a^2+3 b^2\right) \cosh (c+d x)}{d}-\frac{9 b \left(64 a^2+35 b^2\right) \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}+\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{3 a b^2 \text{sech}^3(c+d x)}{d}+\frac{9 a b^2 \text{sech}(c+d x)}{d}-\frac{b^3 \tanh (c+d x) \text{sech}^7(c+d x)}{8 d}+\frac{11 b^3 \tanh (c+d x) \text{sech}^5(c+d x)}{16 d}-\frac{105 b^3 \tanh (c+d x) \text{sech}^3(c+d x)}{64 d}","\frac{a^3 \cosh (c+d x)}{d}+\frac{9 a^2 b \sinh (c+d x)}{2 d}-\frac{9 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}+\frac{3 a b^2 \cosh (c+d x)}{d}+\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{3 a b^2 \text{sech}^3(c+d x)}{d}+\frac{9 a b^2 \text{sech}(c+d x)}{d}+\frac{315 b^3 \sinh (c+d x)}{128 d}-\frac{315 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}-\frac{b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}-\frac{3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac{21 b^3 \sinh (c+d x) \tanh ^4(c+d x)}{64 d}-\frac{105 b^3 \sinh (c+d x) \tanh ^2(c+d x)}{128 d}",1,"(-9*b*(64*a^2 + 35*b^2)*ArcTan[Tanh[(c + d*x)/2]])/(64*d) + (a*(a^2 + 3*b^2)*Cosh[c + d*x])/d + (9*a*b^2*Sech[c + d*x])/d - (3*a*b^2*Sech[c + d*x]^3)/d + (3*a*b^2*Sech[c + d*x]^5)/(5*d) + (b*(3*a^2 + b^2)*Sinh[c + d*x])/d + (Sech[c + d*x]^2*(192*a^2*b*Sinh[c + d*x] + 325*b^3*Sinh[c + d*x]))/(128*d) - (105*b^3*Sech[c + d*x]^3*Tanh[c + d*x])/(64*d) + (11*b^3*Sech[c + d*x]^5*Tanh[c + d*x])/(16*d) - (b^3*Sech[c + d*x]^7*Tanh[c + d*x])/(8*d)","A",1
69,1,154,219,3.2059141,"\int \text{csch}(c+d x) \left(a+b \tanh ^3(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]*(a + b*Tanh[c + d*x]^3)^3,x]","\frac{-45 b \text{sech}(c+d x) \left(\left(64 a^2+31 b^2\right) \tanh (c+d x)+128 a b\right)+30 \left(64 a^3 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+b \left(192 a^2+35 b^2\right) \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)\right)-8 b^2 \text{sech}^5(c+d x) (144 a+125 b \tanh (c+d x))+10 b^2 \text{sech}^3(c+d x) (384 a+163 b \tanh (c+d x))+240 b^3 \tanh (c+d x) \text{sech}^7(c+d x)}{1920 d}","-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{3 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{3 a^2 b \tanh (c+d x) \text{sech}(c+d x)}{2 d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}+\frac{2 a b^2 \text{sech}^3(c+d x)}{d}-\frac{3 a b^2 \text{sech}(c+d x)}{d}+\frac{35 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}-\frac{b^3 \tanh ^7(c+d x) \text{sech}(c+d x)}{8 d}-\frac{7 b^3 \tanh ^5(c+d x) \text{sech}(c+d x)}{48 d}-\frac{35 b^3 \tanh ^3(c+d x) \text{sech}(c+d x)}{192 d}-\frac{35 b^3 \tanh (c+d x) \text{sech}(c+d x)}{128 d}",1,"(30*(b*(192*a^2 + 35*b^2)*ArcTan[Tanh[(c + d*x)/2]] + 64*a^3*Log[Tanh[(c + d*x)/2]]) + 240*b^3*Sech[c + d*x]^7*Tanh[c + d*x] - 8*b^2*Sech[c + d*x]^5*(144*a + 125*b*Tanh[c + d*x]) + 10*b^2*Sech[c + d*x]^3*(384*a + 163*b*Tanh[c + d*x]) - 45*b*Sech[c + d*x]*(128*a*b + (64*a^2 + 31*b^2)*Tanh[c + d*x]))/(1920*d)","A",1
70,1,113,71,0.7696329,"\int \text{csch}^2(c+d x) \left(a+b \tanh ^3(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^2*(a + b*Tanh[c + d*x]^3)^3,x]","\frac{b \left(-4 \text{sech}^2(c+d x) \left(15 a^2+12 a b \tanh (c+d x)+5 b^2\right)+24 a b \tanh (c+d x)+6 b \text{sech}^4(c+d x) (4 a \tanh (c+d x)+5 b)+5 b^2 \text{sech}^8(c+d x)-20 b^2 \text{sech}^6(c+d x)\right)-40 a^3 \coth (c+d x)}{40 d}","-\frac{a^3 \coth (c+d x)}{d}+\frac{3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac{b^3 \tanh ^8(c+d x)}{8 d}",1,"(-40*a^3*Coth[c + d*x] + b*(-20*b^2*Sech[c + d*x]^6 + 5*b^2*Sech[c + d*x]^8 + 24*a*b*Tanh[c + d*x] + 6*b*Sech[c + d*x]^4*(5*b + 4*a*Tanh[c + d*x]) - 4*Sech[c + d*x]^2*(15*a^2 + 5*b^2 + 12*a*b*Tanh[c + d*x])))/(40*d)","A",1
71,1,243,232,6.365278,"\int \text{csch}^3(c+d x) \left(a+b \tanh ^3(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^3)^3,x]","-\frac{a^3 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a^3 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a^3 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{\text{sech}^2(c+d x) \left(192 a^2 b \sinh (c+d x)+5 b^3 \sinh (c+d x)\right)}{128 d}+\frac{b \left(192 a^2+5 b^2\right) \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}+\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{a b^2 \text{sech}^3(c+d x)}{d}-\frac{b^3 \tanh (c+d x) \text{sech}^7(c+d x)}{8 d}+\frac{17 b^3 \tanh (c+d x) \text{sech}^5(c+d x)}{48 d}-\frac{59 b^3 \tanh (c+d x) \text{sech}^3(c+d x)}{192 d}","\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a^3 \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{3 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{3 a^2 b \tanh (c+d x) \text{sech}(c+d x)}{2 d}+\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{a b^2 \text{sech}^3(c+d x)}{d}+\frac{5 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}-\frac{b^3 \tanh ^5(c+d x) \text{sech}^3(c+d x)}{8 d}-\frac{5 b^3 \tanh ^3(c+d x) \text{sech}^3(c+d x)}{48 d}-\frac{5 b^3 \tanh (c+d x) \text{sech}^3(c+d x)}{64 d}+\frac{5 b^3 \tanh (c+d x) \text{sech}(c+d x)}{128 d}",1,"(b*(192*a^2 + 5*b^2)*ArcTan[Tanh[(c + d*x)/2]])/(64*d) - (a^3*Csch[(c + d*x)/2]^2)/(8*d) - (a^3*Log[Tanh[(c + d*x)/2]])/(2*d) - (a^3*Sech[(c + d*x)/2]^2)/(8*d) - (a*b^2*Sech[c + d*x]^3)/d + (3*a*b^2*Sech[c + d*x]^5)/(5*d) + (Sech[c + d*x]^2*(192*a^2*b*Sinh[c + d*x] + 5*b^3*Sinh[c + d*x]))/(128*d) - (59*b^3*Sech[c + d*x]^3*Tanh[c + d*x])/(192*d) + (17*b^3*Sech[c + d*x]^5*Tanh[c + d*x])/(48*d) - (b^3*Sech[c + d*x]^7*Tanh[c + d*x])/(8*d)","A",1
72,1,213,138,0.197409,"\int \text{csch}^4(c+d x) \left(a+b \tanh ^3(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^3)^3,x]","\frac{2 a^3 \coth (c+d x)}{3 d}-\frac{a^3 \coth (c+d x) \text{csch}^2(c+d x)}{3 d}+\frac{3 a^2 b \text{sech}^2(c+d x)}{2 d}+\frac{3 a^2 b \log (\sinh (c+d x))}{d}-\frac{3 a^2 b \log (\cosh (c+d x))}{d}+\frac{2 a b^2 \tanh (c+d x)}{5 d}-\frac{3 a b^2 \tanh (c+d x) \text{sech}^4(c+d x)}{5 d}+\frac{a b^2 \tanh (c+d x) \text{sech}^2(c+d x)}{5 d}-\frac{b^3 \text{sech}^8(c+d x)}{8 d}+\frac{b^3 \text{sech}^6(c+d x)}{3 d}-\frac{b^3 \text{sech}^4(c+d x)}{4 d}","-\frac{a^3 \coth ^3(c+d x)}{3 d}+\frac{a^3 \coth (c+d x)}{d}-\frac{3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac{3 a^2 b \log (\tanh (c+d x))}{d}-\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac{a b^2 \tanh ^3(c+d x)}{d}-\frac{b^3 \tanh ^8(c+d x)}{8 d}+\frac{b^3 \tanh ^6(c+d x)}{6 d}",1,"(2*a^3*Coth[c + d*x])/(3*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (3*a^2*b*Log[Cosh[c + d*x]])/d + (3*a^2*b*Log[Sinh[c + d*x]])/d + (3*a^2*b*Sech[c + d*x]^2)/(2*d) - (b^3*Sech[c + d*x]^4)/(4*d) + (b^3*Sech[c + d*x]^6)/(3*d) - (b^3*Sech[c + d*x]^8)/(8*d) + (2*a*b^2*Tanh[c + d*x])/(5*d) + (a*b^2*Sech[c + d*x]^2*Tanh[c + d*x])/(5*d) - (3*a*b^2*Sech[c + d*x]^4*Tanh[c + d*x])/(5*d)","A",1
73,1,645,491,4.8308149,"\int \frac{\sinh ^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx","Integrate[Sinh[c + d*x]^4/(a + b*Tanh[c + d*x]^3),x]","\frac{3 \left(a (a-b) \left(12 \left(a^2-6 a b+5 b^2\right) (c+d x)+(a+b)^2 \sinh (4 (c+d x))\right)-8 a \left(a^3+a^2 b+2 a b^2+2 b^3\right) \sinh (2 (c+d x))+4 b \left(5 a^3+5 a^2 b+a b^2+b^3\right) \cosh (2 (c+d x))-\left(b (a-b) (a+b)^2 \cosh (4 (c+d x))\right)\right)-32 a b \text{RootSum}\left[\text{$\#$1}^3 a+\text{$\#$1}^3 b+3 \text{$\#$1}^2 a-3 \text{$\#$1}^2 b+3 \text{$\#$1} a+3 \text{$\#$1} b+a-b\&,\frac{5 \text{$\#$1}^2 a^3 \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)-10 \text{$\#$1}^2 a^3 c-10 \text{$\#$1}^2 a^3 d x-10 \text{$\#$1}^2 a^2 b \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)+20 \text{$\#$1}^2 a^2 b c+20 \text{$\#$1}^2 a^2 b d x+10 \text{$\#$1}^2 a b^2 \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)-20 \text{$\#$1}^2 a b^2 c-20 \text{$\#$1}^2 a b^2 d x-2 \text{$\#$1}^2 b^3 \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)+4 \text{$\#$1}^2 b^3 c+4 \text{$\#$1}^2 b^3 d x+3 a^3 \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)+4 \text{$\#$1} a^3 \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)-8 \text{$\#$1} a^3 c-8 \text{$\#$1} a^3 d x-2 \text{$\#$1} a^2 b \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)+4 \text{$\#$1} a^2 b c+4 \text{$\#$1} a^2 b d x+6 a b^2 \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)-4 \text{$\#$1} a b^2 \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)+8 \text{$\#$1} a b^2 c+8 \text{$\#$1} a b^2 d x+2 \text{$\#$1} b^3 \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)-4 \text{$\#$1} b^3 c-4 \text{$\#$1} b^3 d x-6 a^3 c-6 a^3 d x-12 a b^2 c-12 a b^2 d x}{\text{$\#$1}^2 a-\text{$\#$1}^2 b+2 \text{$\#$1} a+2 \text{$\#$1} b+a-b}\&\right]}{96 d (a-b)^2 (a+b)^3}","-\frac{a^2 b \left(a^2+2 b^2\right) \log \left(a+b \tanh ^3(c+d x)\right)}{d \left(a^2-b^2\right)^3}-\frac{a^{2/3} \sqrt[3]{b} \left(3 a^{4/3} b^{2/3}+a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tanh (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} d \left(a^{2/3} b^{2/3}+a^{4/3}+b^{4/3}\right)^3}+\frac{a^{2/3} \sqrt[3]{b} \left(a^4+7 a^2 b^2+3 a^{2/3} b^{4/3} \left(2 a^2+b^2\right)+b^4\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right)}{6 d \left(a^2-b^2\right)^3}-\frac{a^{2/3} \sqrt[3]{b} \left(a^4+7 a^2 b^2+3 a^{2/3} b^{4/3} \left(2 a^2+b^2\right)+b^4\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tanh (c+d x)\right)}{3 d \left(a^2-b^2\right)^3}-\frac{5 a-b}{16 d (a+b)^2 (1-\tanh (c+d x))}+\frac{5 a+b}{16 d (a-b)^2 (\tanh (c+d x)+1)}+\frac{1}{16 d (a+b) (1-\tanh (c+d x))^2}-\frac{1}{16 d (a-b) (\tanh (c+d x)+1)^2}-\frac{3 a (a-5 b) \log (1-\tanh (c+d x))}{16 d (a+b)^3}+\frac{3 a (a+5 b) \log (\tanh (c+d x)+1)}{16 d (a-b)^3}",1,"(-32*a*b*RootSum[a - b + 3*a*#1 + 3*b*#1 + 3*a*#1^2 - 3*b*#1^2 + a*#1^3 + b*#1^3 & , (-6*a^3*c - 12*a*b^2*c - 6*a^3*d*x - 12*a*b^2*d*x + 3*a^3*Log[E^(2*(c + d*x)) - #1] + 6*a*b^2*Log[E^(2*(c + d*x)) - #1] - 8*a^3*c*#1 + 4*a^2*b*c*#1 + 8*a*b^2*c*#1 - 4*b^3*c*#1 - 8*a^3*d*x*#1 + 4*a^2*b*d*x*#1 + 8*a*b^2*d*x*#1 - 4*b^3*d*x*#1 + 4*a^3*Log[E^(2*(c + d*x)) - #1]*#1 - 2*a^2*b*Log[E^(2*(c + d*x)) - #1]*#1 - 4*a*b^2*Log[E^(2*(c + d*x)) - #1]*#1 + 2*b^3*Log[E^(2*(c + d*x)) - #1]*#1 - 10*a^3*c*#1^2 + 20*a^2*b*c*#1^2 - 20*a*b^2*c*#1^2 + 4*b^3*c*#1^2 - 10*a^3*d*x*#1^2 + 20*a^2*b*d*x*#1^2 - 20*a*b^2*d*x*#1^2 + 4*b^3*d*x*#1^2 + 5*a^3*Log[E^(2*(c + d*x)) - #1]*#1^2 - 10*a^2*b*Log[E^(2*(c + d*x)) - #1]*#1^2 + 10*a*b^2*Log[E^(2*(c + d*x)) - #1]*#1^2 - 2*b^3*Log[E^(2*(c + d*x)) - #1]*#1^2)/(a - b + 2*a*#1 + 2*b*#1 + a*#1^2 - b*#1^2) & ] + 3*(4*b*(5*a^3 + 5*a^2*b + a*b^2 + b^3)*Cosh[2*(c + d*x)] - (a - b)*b*(a + b)^2*Cosh[4*(c + d*x)] - 8*a*(a^3 + a^2*b + 2*a*b^2 + 2*b^3)*Sinh[2*(c + d*x)] + a*(a - b)*(12*(a^2 - 6*a*b + 5*b^2)*(c + d*x) + (a + b)^2*Sinh[4*(c + d*x)])))/(96*(a - b)^2*(a + b)^3*d)","C",1
74,1,826,33,0.5349422,"\int \frac{\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx","Integrate[Sinh[c + d*x]^3/(a + b*Tanh[c + d*x]^3),x]","\frac{\cosh (3 (c+d x)) a^3+27 b \sinh (c+d x) a^2-b \sinh (3 (c+d x)) a^2-9 \left(a^2+3 b^2\right) \cosh (c+d x) a-b^2 \cosh (3 (c+d x)) a-2 b \text{RootSum}\left[a \text{$\#$1}^6+b \text{$\#$1}^6+3 a \text{$\#$1}^4-3 b \text{$\#$1}^4+3 a \text{$\#$1}^2+3 b \text{$\#$1}^2+a-b\&,\frac{3 a^2 c \text{$\#$1}^4+3 b^2 c \text{$\#$1}^4-3 a b c \text{$\#$1}^4+3 a^2 d x \text{$\#$1}^4+3 b^2 d x \text{$\#$1}^4-3 a b d x \text{$\#$1}^4+6 a^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4+6 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4-6 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4+2 a^2 c \text{$\#$1}^2-2 b^2 c \text{$\#$1}^2+2 a^2 d x \text{$\#$1}^2-2 b^2 d x \text{$\#$1}^2+4 a^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2-4 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2+3 a^2 c+3 b^2 c+3 a b c+3 a^2 d x+3 b^2 d x+3 a b d x+6 a^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)+6 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)+6 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)}{a \text{$\#$1}^5+b \text{$\#$1}^5+2 a \text{$\#$1}^3-2 b \text{$\#$1}^3+a \text{$\#$1}+b \text{$\#$1}}\&\right] a+9 b^3 \sinh (c+d x)+b^3 \sinh (3 (c+d x))}{12 (a-b)^2 (a+b)^2 d}","i \text{Int}\left(-\frac{i \sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)},x\right)",0,"(-9*a*(a^2 + 3*b^2)*Cosh[c + d*x] + a^3*Cosh[3*(c + d*x)] - a*b^2*Cosh[3*(c + d*x)] - 2*a*b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 + b*#1^6 & , (3*a^2*c + 3*a*b*c + 3*b^2*c + 3*a^2*d*x + 3*a*b*d*x + 3*b^2*d*x + 6*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 6*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 6*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 2*a^2*c*#1^2 - 2*b^2*c*#1^2 + 2*a^2*d*x*#1^2 - 2*b^2*d*x*#1^2 + 4*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 4*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 3*a^2*c*#1^4 - 3*a*b*c*#1^4 + 3*b^2*c*#1^4 + 3*a^2*d*x*#1^4 - 3*a*b*d*x*#1^4 + 3*b^2*d*x*#1^4 + 6*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 6*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 6*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4)/(a*#1 + b*#1 + 2*a*#1^3 - 2*b*#1^3 + a*#1^5 + b*#1^5) & ] + 27*a^2*b*Sinh[c + d*x] + 9*b^3*Sinh[c + d*x] - a^2*b*Sinh[3*(c + d*x)] + b^3*Sinh[3*(c + d*x)])/(12*(a - b)^2*(a + b)^2*d)","B",1
75,1,423,384,3.813312,"\int \frac{\sinh ^2(c+d x)}{a+b \tanh ^3(c+d x)} \, dx","Integrate[Sinh[c + d*x]^2/(a + b*Tanh[c + d*x]^3),x]","-\frac{4 b \text{RootSum}\left[\text{$\#$1}^3 a+\text{$\#$1}^3 b+3 \text{$\#$1}^2 a-3 \text{$\#$1}^2 b+3 \text{$\#$1} a+3 \text{$\#$1} b+a-b\&,\frac{-4 \text{$\#$1}^2 a^2 \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)+8 \text{$\#$1}^2 a^2 c+8 \text{$\#$1}^2 a^2 d x+4 \text{$\#$1}^2 a b \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)-8 \text{$\#$1}^2 a b c-8 \text{$\#$1}^2 a b d x-\text{$\#$1}^2 b^2 \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)+2 \text{$\#$1}^2 b^2 c+2 \text{$\#$1}^2 b^2 d x-2 a^2 \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)-2 \text{$\#$1} a^2 \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)+4 \text{$\#$1} a^2 c+4 \text{$\#$1} a^2 d x-b^2 \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)+2 \text{$\#$1} b^2 \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)-4 \text{$\#$1} b^2 c-4 \text{$\#$1} b^2 d x+4 a^2 c+4 a^2 d x+2 b^2 c+2 b^2 d x}{\text{$\#$1}^2 a-\text{$\#$1}^2 b+2 \text{$\#$1} a+2 \text{$\#$1} b+a-b}\&\right]+6 \left(a^2-3 a b+2 b^2\right) (c+d x)-3 a (a+b) \sinh (2 (c+d x))+3 b (a+b) \cosh (2 (c+d x))}{12 d (a-b) (a+b)^2}","\frac{b \left(2 a^2+b^2\right) \log \left(a+b \tanh ^3(c+d x)\right)}{3 d \left(a^2-b^2\right)^2}-\frac{a^{2/3} \sqrt[3]{b} \left(3 a^{2/3} b^{4/3}+a^2+2 b^2\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right)}{6 d \left(a^2-b^2\right)^2}+\frac{a^{2/3} \sqrt[3]{b} \left(3 a^{2/3} b^{4/3}+a^2+2 b^2\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tanh (c+d x)\right)}{3 d \left(a^2-b^2\right)^2}+\frac{a^{2/3} \sqrt[3]{b} \left(-3 a^{2/3} b^{4/3}+a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tanh (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} d \left(a^2-b^2\right)^2}+\frac{1}{4 d (a+b) (1-\tanh (c+d x))}-\frac{1}{4 d (a-b) (\tanh (c+d x)+1)}+\frac{(a-2 b) \log (1-\tanh (c+d x))}{4 d (a+b)^2}-\frac{(a+2 b) \log (\tanh (c+d x)+1)}{4 d (a-b)^2}",1,"-1/12*(6*(a^2 - 3*a*b + 2*b^2)*(c + d*x) + 3*b*(a + b)*Cosh[2*(c + d*x)] + 4*b*RootSum[a - b + 3*a*#1 + 3*b*#1 + 3*a*#1^2 - 3*b*#1^2 + a*#1^3 + b*#1^3 & , (4*a^2*c + 2*b^2*c + 4*a^2*d*x + 2*b^2*d*x - 2*a^2*Log[E^(2*(c + d*x)) - #1] - b^2*Log[E^(2*(c + d*x)) - #1] + 4*a^2*c*#1 - 4*b^2*c*#1 + 4*a^2*d*x*#1 - 4*b^2*d*x*#1 - 2*a^2*Log[E^(2*(c + d*x)) - #1]*#1 + 2*b^2*Log[E^(2*(c + d*x)) - #1]*#1 + 8*a^2*c*#1^2 - 8*a*b*c*#1^2 + 2*b^2*c*#1^2 + 8*a^2*d*x*#1^2 - 8*a*b*d*x*#1^2 + 2*b^2*d*x*#1^2 - 4*a^2*Log[E^(2*(c + d*x)) - #1]*#1^2 + 4*a*b*Log[E^(2*(c + d*x)) - #1]*#1^2 - b^2*Log[E^(2*(c + d*x)) - #1]*#1^2)/(a - b + 2*a*#1 + 2*b*#1 + a*#1^2 - b*#1^2) & ] - 3*a*(a + b)*Sinh[2*(c + d*x)])/((a - b)*(a + b)^2*d)","C",1
76,1,409,31,0.26943,"\int \frac{\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx","Integrate[Sinh[c + d*x]/(a + b*Tanh[c + d*x]^3),x]","\frac{b \text{RootSum}\left[\text{$\#$1}^6 a+\text{$\#$1}^6 b+3 \text{$\#$1}^4 a-3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 a+3 \text{$\#$1}^2 b+a-b\&,\frac{4 \text{$\#$1}^4 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+2 \text{$\#$1}^4 a c+2 \text{$\#$1}^4 a d x-2 \text{$\#$1}^4 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-\text{$\#$1}^4 b c-\text{$\#$1}^4 b d x+4 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+2 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+2 a c+2 a d x+b c+b d x}{\text{$\#$1}^5 a+\text{$\#$1}^5 b+2 \text{$\#$1}^3 a-2 \text{$\#$1}^3 b+\text{$\#$1} a+\text{$\#$1} b}\&\right]+6 a \cosh (c+d x)-6 b \sinh (c+d x)}{6 d (a-b) (a+b)}","-i \text{Int}\left(\frac{i \sinh (c+d x)}{a+b \tanh ^3(c+d x)},x\right)",0,"(6*a*Cosh[c + d*x] + b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 + b*#1^6 & , (2*a*c + b*c + 2*a*d*x + b*d*x + 4*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 2*a*c*#1^4 - b*c*#1^4 + 2*a*d*x*#1^4 - b*d*x*#1^4 + 4*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4)/(a*#1 + b*#1 + 2*a*#1^3 - 2*b*#1^3 + a*#1^5 + b*#1^5) & ] - 6*b*Sinh[c + d*x])/(6*(a - b)*(a + b)*d)","B",1
77,1,319,31,0.1791766,"\int \frac{\text{csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx","Integrate[Csch[c + d*x]/(a + b*Tanh[c + d*x]^3),x]","\frac{6 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-b \text{RootSum}\left[\text{$\#$1}^6 a+\text{$\#$1}^6 b+3 \text{$\#$1}^4 a-3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 a+3 \text{$\#$1}^2 b+a-b\&,\frac{2 \text{$\#$1}^4 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^4 c+\text{$\#$1}^4 d x-4 \text{$\#$1}^2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-2 \text{$\#$1}^2 c-2 \text{$\#$1}^2 d x+2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+c+d x}{\text{$\#$1}^5 a+\text{$\#$1}^5 b+2 \text{$\#$1}^3 a-2 \text{$\#$1}^3 b+\text{$\#$1} a+\text{$\#$1} b}\&\right]}{6 a d}","i \text{Int}\left(-\frac{i \text{csch}(c+d x)}{a+b \tanh ^3(c+d x)},x\right)",0,"(6*Log[Tanh[(c + d*x)/2]] - b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 + b*#1^6 & , (c + d*x + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 2*c*#1^2 - 2*d*x*#1^2 - 4*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + c*#1^4 + d*x*#1^4 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4)/(a*#1 + b*#1 + 2*a*#1^3 - 2*b*#1^3 + a*#1^5 + b*#1^5) & ])/(6*a*d)","B",1
78,1,190,157,0.1476868,"\int \frac{\text{csch}^2(c+d x)}{a+b \tanh ^3(c+d x)} \, dx","Integrate[Csch[c + d*x]^2/(a + b*Tanh[c + d*x]^3),x]","-\frac{2 b \text{RootSum}\left[\text{$\#$1}^3 a+\text{$\#$1}^3 b+3 \text{$\#$1}^2 a-3 \text{$\#$1}^2 b+3 \text{$\#$1} a+3 \text{$\#$1} b+a-b\&,\frac{-\log (-\text{$\#$1} \sinh (c+d x)+\text{$\#$1} \cosh (c+d x)-\sinh (c+d x)-\cosh (c+d x))+\text{$\#$1} \log (-\text{$\#$1} \sinh (c+d x)+\text{$\#$1} \cosh (c+d x)-\sinh (c+d x)-\cosh (c+d x))+\text{$\#$1} c+\text{$\#$1} d x-c-d x}{\text{$\#$1}^2 a+\text{$\#$1}^2 b+2 \text{$\#$1} a-2 \text{$\#$1} b+a+b}\&\right]+3 \coth (c+d x)}{3 a d}","-\frac{\sqrt[3]{b} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right)}{6 a^{4/3} d}+\frac{\sqrt[3]{b} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tanh (c+d x)\right)}{3 a^{4/3} d}+\frac{\sqrt[3]{b} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tanh (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{4/3} d}-\frac{\coth (c+d x)}{a d}",1,"-1/3*(3*Coth[c + d*x] + 2*b*RootSum[a - b + 3*a*#1 + 3*b*#1 + 3*a*#1^2 - 3*b*#1^2 + a*#1^3 + b*#1^3 & , (-c - d*x - Log[-Cosh[c + d*x] - Sinh[c + d*x] + Cosh[c + d*x]*#1 - Sinh[c + d*x]*#1] + c*#1 + d*x*#1 + Log[-Cosh[c + d*x] - Sinh[c + d*x] + Cosh[c + d*x]*#1 - Sinh[c + d*x]*#1]*#1)/(a + b + 2*a*#1 - 2*b*#1 + a*#1^2 + b*#1^2) & ])/(a*d)","C",1
79,1,201,33,0.3862721,"\int \frac{\text{csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx","Integrate[Csch[c + d*x]^3/(a + b*Tanh[c + d*x]^3),x]","-\frac{16 b \text{RootSum}\left[\text{$\#$1}^6 a+\text{$\#$1}^6 b+3 \text{$\#$1}^4 a-3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 a+3 \text{$\#$1}^2 b+a-b\&,\frac{2 \text{$\#$1} \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1} c+\text{$\#$1} d x}{\text{$\#$1}^4 a+\text{$\#$1}^4 b+2 \text{$\#$1}^2 a-2 \text{$\#$1}^2 b+a+b}\&\right]+3 \left(\text{csch}^2\left(\frac{1}{2} (c+d x)\right)+\text{sech}^2\left(\frac{1}{2} (c+d x)\right)+4 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)\right)}{24 a d}","-i \text{Int}\left(\frac{i \text{csch}^3(c+d x)}{a+b \tanh ^3(c+d x)},x\right)",0,"-1/24*(16*b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 + b*#1^6 & , (c*#1 + d*x*#1 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1)/(a + b + 2*a*#1^2 - 2*b*#1^2 + a*#1^4 + b*#1^4) & ] + 3*(Csch[(c + d*x)/2]^2 + 4*Log[Tanh[(c + d*x)/2]] + Sech[(c + d*x)/2]^2))/(a*d)","B",1
80,1,322,215,3.61752,"\int \frac{\text{csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx","Integrate[Csch[c + d*x]^4/(a + b*Tanh[c + d*x]^3),x]","\frac{b \text{RootSum}\left[\text{$\#$1}^3 a+\text{$\#$1}^3 b+3 \text{$\#$1}^2 a-3 \text{$\#$1}^2 b+3 \text{$\#$1} a+3 \text{$\#$1} b+a-b\&,\frac{-\text{$\#$1}^2 a \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)+2 \text{$\#$1}^2 a c+2 \text{$\#$1}^2 a d x-\text{$\#$1}^2 b \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)+2 \text{$\#$1}^2 b c+2 \text{$\#$1}^2 b d x+4 \text{$\#$1} a \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)+a \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)-8 \text{$\#$1} a c-8 \text{$\#$1} a d x+2 \text{$\#$1} b \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)-b \log \left(e^{2 (c+d x)}-\text{$\#$1}\right)-4 \text{$\#$1} b c-4 \text{$\#$1} b d x-2 a c-2 a d x+2 b c+2 b d x}{\text{$\#$1}^2 a-\text{$\#$1}^2 b+2 \text{$\#$1} a+2 \text{$\#$1} b+a-b}\&\right]-a \coth (c+d x) \left(\text{csch}^2(c+d x)-2\right)+3 b (-\log (\sinh (c+d x))+c+d x)}{3 a^2 d}","\frac{\sqrt[3]{b} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right)}{6 a^{4/3} d}-\frac{\sqrt[3]{b} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tanh (c+d x)\right)}{3 a^{4/3} d}-\frac{\sqrt[3]{b} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tanh (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{4/3} d}+\frac{b \log \left(a+b \tanh ^3(c+d x)\right)}{3 a^2 d}-\frac{b \log (\tanh (c+d x))}{a^2 d}-\frac{\coth ^3(c+d x)}{3 a d}+\frac{\coth (c+d x)}{a d}",1,"(-(a*Coth[c + d*x]*(-2 + Csch[c + d*x]^2)) + 3*b*(c + d*x - Log[Sinh[c + d*x]]) + b*RootSum[a - b + 3*a*#1 + 3*b*#1 + 3*a*#1^2 - 3*b*#1^2 + a*#1^3 + b*#1^3 & , (-2*a*c + 2*b*c - 2*a*d*x + 2*b*d*x + a*Log[E^(2*(c + d*x)) - #1] - b*Log[E^(2*(c + d*x)) - #1] - 8*a*c*#1 - 4*b*c*#1 - 8*a*d*x*#1 - 4*b*d*x*#1 + 4*a*Log[E^(2*(c + d*x)) - #1]*#1 + 2*b*Log[E^(2*(c + d*x)) - #1]*#1 + 2*a*c*#1^2 + 2*b*c*#1^2 + 2*a*d*x*#1^2 + 2*b*d*x*#1^2 - a*Log[E^(2*(c + d*x)) - #1]*#1^2 - b*Log[E^(2*(c + d*x)) - #1]*#1^2)/(a - b + 2*a*#1 + 2*b*#1 + a*#1^2 - b*#1^2) & ])/(3*a^2*d)","C",1
81,1,44,63,0.1689076,"\int \cosh ^4(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Cosh[c + d*x]^4*(a + b*Tanh[c + d*x]^2),x]","\frac{(a+b) \sinh (4 (c+d x))+12 a (c+d x)+8 a \sinh (2 (c+d x))-4 b d x}{32 d}","\frac{(a+b) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac{(3 a-b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{1}{8} x (3 a-b)",1,"(-4*b*d*x + 12*a*(c + d*x) + 8*a*Sinh[2*(c + d*x)] + (a + b)*Sinh[4*(c + d*x)])/(32*d)","A",1
82,1,44,30,0.0175674,"\int \cosh ^3(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Cosh[c + d*x]^3*(a + b*Tanh[c + d*x]^2),x]","\frac{a \sinh ^3(c+d x)}{3 d}+\frac{a \sinh (c+d x)}{d}+\frac{b \sinh ^3(c+d x)}{3 d}","\frac{(a+b) \sinh ^3(c+d x)}{3 d}+\frac{a \sinh (c+d x)}{d}",1,"(a*Sinh[c + d*x])/d + (a*Sinh[c + d*x]^3)/(3*d) + (b*Sinh[c + d*x]^3)/(3*d)","A",1
83,1,32,33,0.0595813,"\int \cosh ^2(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Cosh[c + d*x]^2*(a + b*Tanh[c + d*x]^2),x]","\frac{2 (a-b) (c+d x)+(a+b) \sinh (2 (c+d x))}{4 d}","\frac{(a+b) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac{1}{2} x (a-b)",1,"(2*(a - b)*(c + d*x) + (a + b)*Sinh[2*(c + d*x)])/(4*d)","A",1
84,1,47,27,0.0367123,"\int \cosh (c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Cosh[c + d*x]*(a + b*Tanh[c + d*x]^2),x]","\frac{a \sinh (c) \cosh (d x)}{d}+\frac{a \cosh (c) \sinh (d x)}{d}+\frac{b \sinh (c+d x)}{d}-\frac{b \tan ^{-1}(\sinh (c+d x))}{d}","\frac{(a+b) \sinh (c+d x)}{d}-\frac{b \tan ^{-1}(\sinh (c+d x))}{d}",1,"-((b*ArcTan[Sinh[c + d*x]])/d) + (a*Cosh[d*x]*Sinh[c])/d + (a*Cosh[c]*Sinh[d*x])/d + (b*Sinh[c + d*x])/d","A",1
85,1,48,40,0.0268745,"\int \text{sech}(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Sech[c + d*x]*(a + b*Tanh[c + d*x]^2),x]","\frac{a \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d}","\frac{(2 a+b) \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d}",1,"(a*ArcTan[Sinh[c + d*x]])/d + (b*ArcTan[Sinh[c + d*x]])/(2*d) - (b*Sech[c + d*x]*Tanh[c + d*x])/(2*d)","A",1
86,1,28,28,0.0126451,"\int \text{sech}^2(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Sech[c + d*x]^2*(a + b*Tanh[c + d*x]^2),x]","\frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^3(c+d x)}{3 d}","\frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^3(c+d x)}{3 d}",1,"(a*Tanh[c + d*x])/d + (b*Tanh[c + d*x]^3)/(3*d)","A",1
87,1,93,66,0.0343849,"\int \text{sech}^3(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Sech[c + d*x]^3*(a + b*Tanh[c + d*x]^2),x]","\frac{a \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{a \tanh (c+d x) \text{sech}(c+d x)}{2 d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{8 d}-\frac{b \tanh (c+d x) \text{sech}^3(c+d x)}{4 d}+\frac{b \tanh (c+d x) \text{sech}(c+d x)}{8 d}","\frac{(4 a+b) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{(4 a+b) \tanh (c+d x) \text{sech}(c+d x)}{8 d}-\frac{b \tanh (c+d x) \text{sech}^3(c+d x)}{4 d}",1,"(a*ArcTan[Sinh[c + d*x]])/(2*d) + (b*ArcTan[Sinh[c + d*x]])/(8*d) + (a*Sech[c + d*x]*Tanh[c + d*x])/(2*d) + (b*Sech[c + d*x]*Tanh[c + d*x])/(8*d) - (b*Sech[c + d*x]^3*Tanh[c + d*x])/(4*d)","A",1
88,1,86,48,0.052541,"\int \text{sech}^4(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Sech[c + d*x]^4*(a + b*Tanh[c + d*x]^2),x]","-\frac{a \tanh ^3(c+d x)}{3 d}+\frac{a \tanh (c+d x)}{d}+\frac{2 b \tanh (c+d x)}{15 d}-\frac{b \tanh (c+d x) \text{sech}^4(c+d x)}{5 d}+\frac{b \tanh (c+d x) \text{sech}^2(c+d x)}{15 d}","-\frac{(a-b) \tanh ^3(c+d x)}{3 d}+\frac{a \tanh (c+d x)}{d}-\frac{b \tanh ^5(c+d x)}{5 d}",1,"(a*Tanh[c + d*x])/d + (2*b*Tanh[c + d*x])/(15*d) + (b*Sech[c + d*x]^2*Tanh[c + d*x])/(15*d) - (b*Sech[c + d*x]^4*Tanh[c + d*x])/(5*d) - (a*Tanh[c + d*x]^3)/(3*d)","A",1
89,1,63,85,0.3268165,"\int \cosh ^4(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Cosh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{4 \left(3 a^2-2 a b+3 b^2\right) (c+d x)+8 \left(a^2-b^2\right) \sinh (2 (c+d x))+(a+b)^2 \sinh (4 (c+d x))}{32 d}","\frac{3 \left(a^2-b^2\right) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{1}{8} x \left(3 a^2-2 a b+3 b^2\right)+\frac{(a+b) \sinh (c+d x) \cosh ^3(c+d x) \left(a+b \tanh ^2(c+d x)\right)}{4 d}",1,"(4*(3*a^2 - 2*a*b + 3*b^2)*(c + d*x) + 8*(a^2 - b^2)*Sinh[2*(c + d*x)] + (a + b)^2*Sinh[4*(c + d*x)])/(32*d)","A",1
90,1,71,54,0.4575969,"\int \cosh ^3(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Cosh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\sinh (c+d x) \left((a+b) ((a+b) \cosh (2 (c+d x))+5 a-7 b)+\frac{6 b^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{\sqrt{-\sinh ^2(c+d x)}}\right)}{6 d}","\frac{\left(a^2-b^2\right) \sinh (c+d x)}{d}+\frac{(a+b)^2 \sinh ^3(c+d x)}{3 d}+\frac{b^2 \tan ^{-1}(\sinh (c+d x))}{d}",1,"(Sinh[c + d*x]*((a + b)*(5*a - 7*b + (a + b)*Cosh[2*(c + d*x)]) + (6*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/Sqrt[-Sinh[c + d*x]^2]))/(6*d)","A",1
91,1,54,51,0.4105617,"\int \cosh ^2(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Cosh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{(a-3 b) (a+b) (c+d x)}{2 d}+\frac{(a+b)^2 \sinh (2 (c+d x))}{4 d}+\frac{b^2 \tanh (c+d x)}{d}","\frac{(a+b)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac{1}{2} x (a-3 b) (a+b)+\frac{b^2 \tanh (c+d x)}{d}",1,"((a - 3*b)*(a + b)*(c + d*x))/(2*d) + ((a + b)^2*Sinh[2*(c + d*x)])/(4*d) + (b^2*Tanh[c + d*x])/d","A",1
92,1,54,60,0.1776789,"\int \cosh (c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Cosh[c + d*x]*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{2 (a+b)^2 \sinh (c+d x)+b \left(b \tanh (c+d x) \text{sech}(c+d x)-(4 a+3 b) \tan ^{-1}(\sinh (c+d x))\right)}{2 d}","\frac{(a+b)^2 \sinh (c+d x)}{d}-\frac{b (4 a+3 b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b^2 \tanh (c+d x) \text{sech}(c+d x)}{2 d}",1,"(2*(a + b)^2*Sinh[c + d*x] + b*(-((4*a + 3*b)*ArcTan[Sinh[c + d*x]]) + b*Sech[c + d*x]*Tanh[c + d*x]))/(2*d)","A",1
93,1,427,91,7.2769123,"\int \text{sech}(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Sech[c + d*x]*(a + b*Tanh[c + d*x]^2)^2,x]","-\frac{\text{csch}^3(c+d x) \left(128 \sinh ^6(c+d x) \left(a^2 \left(5 \sinh ^4(c+d x)+12 \sinh ^2(c+d x)+7\right)+2 a b \left(5 \sinh ^2(c+d x)+6\right) \sinh ^2(c+d x)+5 b^2 \sinh ^4(c+d x)\right) \, _4F_3\left(\frac{3}{2},2,2,2;1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)+128 \sinh ^6(c+d x) \left(a \sinh ^2(c+d x)+a+b \sinh ^2(c+d x)\right)^2 \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)+35 \left(a^2 \left(485 \sinh ^6(c+d x)+3161 \sinh ^4(c+d x)+5907 \sinh ^2(c+d x)+3375\right)+2 a b \left(485 \sinh ^4(c+d x)+2554 \sinh ^2(c+d x)+2625\right) \sinh ^2(c+d x)+b^2 \left(485 \sinh ^2(c+d x)+1947\right) \sinh ^4(c+d x)\right)-\frac{105 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \left(a^2 \left(9 \sinh ^8(c+d x)+400 \sinh ^6(c+d x)+1674 \sinh ^4(c+d x)+2344 \sinh ^2(c+d x)+1125\right)+2 a b \left(9 \sinh ^6(c+d x)+389 \sinh ^4(c+d x)+1143 \sinh ^2(c+d x)+875\right) \sinh ^2(c+d x)+b^2 \left(9 \sinh ^4(c+d x)+378 \sinh ^2(c+d x)+649\right) \sinh ^4(c+d x)\right)}{\sqrt{-\sinh ^2(c+d x)}}\right)}{6720 d}","\frac{\left(8 a^2+8 a b+3 b^2\right) \tan ^{-1}(\sinh (c+d x))}{8 d}-\frac{3 b (2 a+b) \tanh (c+d x) \text{sech}(c+d x)}{8 d}-\frac{b \tanh (c+d x) \text{sech}^3(c+d x) \left((a+b) \sinh ^2(c+d x)+a\right)}{4 d}",1,"-1/6720*(Csch[c + d*x]^3*(128*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(a + a*Sinh[c + d*x]^2 + b*Sinh[c + d*x]^2)^2 + 128*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(5*b^2*Sinh[c + d*x]^4 + 2*a*b*Sinh[c + d*x]^2*(6 + 5*Sinh[c + d*x]^2) + a^2*(7 + 12*Sinh[c + d*x]^2 + 5*Sinh[c + d*x]^4)) + 35*(b^2*Sinh[c + d*x]^4*(1947 + 485*Sinh[c + d*x]^2) + 2*a*b*Sinh[c + d*x]^2*(2625 + 2554*Sinh[c + d*x]^2 + 485*Sinh[c + d*x]^4) + a^2*(3375 + 5907*Sinh[c + d*x]^2 + 3161*Sinh[c + d*x]^4 + 485*Sinh[c + d*x]^6)) - (105*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(b^2*Sinh[c + d*x]^4*(649 + 378*Sinh[c + d*x]^2 + 9*Sinh[c + d*x]^4) + 2*a*b*Sinh[c + d*x]^2*(875 + 1143*Sinh[c + d*x]^2 + 389*Sinh[c + d*x]^4 + 9*Sinh[c + d*x]^6) + a^2*(1125 + 2344*Sinh[c + d*x]^2 + 1674*Sinh[c + d*x]^4 + 400*Sinh[c + d*x]^6 + 9*Sinh[c + d*x]^8)))/Sqrt[-Sinh[c + d*x]^2]))/d","C",0
94,1,49,49,0.2134062,"\int \text{sech}^2(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Sech[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{a^2 \tanh (c+d x)}{d}+\frac{2 a b \tanh ^3(c+d x)}{3 d}+\frac{b^2 \tanh ^5(c+d x)}{5 d}","\frac{a^2 \tanh (c+d x)}{d}+\frac{2 a b \tanh ^3(c+d x)}{3 d}+\frac{b^2 \tanh ^5(c+d x)}{5 d}",1,"(a^2*Tanh[c + d*x])/d + (2*a*b*Tanh[c + d*x]^3)/(3*d) + (b^2*Tanh[c + d*x]^5)/(5*d)","A",1
95,1,792,125,8.7605912,"\int \text{sech}^3(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Sech[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{a^2 \sinh (c+d x) \left(-\frac{380 (a+b)^2 \sinh ^6(c+d x) \, _4F_3\left(\frac{3}{2},2,2,2;1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)}{a^2}-\frac{128 (a+b)^2 \sinh ^6(c+d x) \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)}{a^2}-\frac{16 (a+b)^2 \sinh ^6(c+d x) \, _6F_5\left(\frac{3}{2},2,2,2,2,2;1,1,1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)}{a^2}-\frac{968 (a+b) \sinh ^4(c+d x) \, _4F_3\left(\frac{3}{2},2,2,2;1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)}{a}-\frac{288 (a+b) \sinh ^4(c+d x) \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)}{a}-\frac{32 (a+b) \sinh ^4(c+d x) \, _6F_5\left(\frac{3}{2},2,2,2,2,2;1,1,1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)}{a}-620 \sinh ^2(c+d x) \, _4F_3\left(\frac{3}{2},2,2,2;1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)-160 \sinh ^2(c+d x) \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)-16 \sinh ^2(c+d x) \, _6F_5\left(\frac{3}{2},2,2,2,2,2;1,1,1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)-\frac{8855 (a+b)^2 \sinh ^2(c+d x)}{a^2}-\frac{19845 (a+b)^2 \sqrt{-\sinh ^2(c+d x)} \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{a^2}+\frac{32970 (a+b)^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{a^2 \sqrt{-\sinh ^2(c+d x)}}+\frac{525 (a+b)^2 \sinh ^4(c+d x) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{a^2 \sqrt{-\sinh ^2(c+d x)}}-\frac{32970 (a+b)^2}{a^2}-\frac{91875 (a+b) \text{csch}^2(c+d x)}{a}-\frac{1365 (a+b) \sqrt{-\sinh ^2(c+d x)} \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{a}+\frac{54180 (a+b) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{a \sqrt{-\sinh ^2(c+d x)}}-\frac{91875 (a+b) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{a \left(-\sinh ^2(c+d x)\right)^{3/2}}-\frac{23555 (a+b)}{a}-65625 \text{csch}^4(c+d x)-14980 \text{csch}^2(c+d x)-\frac{36855 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{\left(-\sinh ^2(c+d x)\right)^{3/2}}+\frac{65625 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{\left(-\sinh ^2(c+d x)\right)^{5/2}}+\frac{1680 \sinh ^4(c+d x) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{\left(-\sinh ^2(c+d x)\right)^{5/2}}\right)}{2520 d}","\frac{\left(8 a^2+4 a b+b^2\right) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac{\left(8 a^2+4 a b+b^2\right) \tanh (c+d x) \text{sech}(c+d x)}{16 d}-\frac{b (8 a+3 b) \tanh (c+d x) \text{sech}^3(c+d x)}{24 d}-\frac{b \tanh (c+d x) \text{sech}^5(c+d x) \left((a+b) \sinh ^2(c+d x)+a\right)}{6 d}",1,"(a^2*Sinh[c + d*x]*((-23555*(a + b))/a - (32970*(a + b)^2)/a^2 - 14980*Csch[c + d*x]^2 - (91875*(a + b)*Csch[c + d*x]^2)/a - 65625*Csch[c + d*x]^4 - (8855*(a + b)^2*Sinh[c + d*x]^2)/a^2 - 620*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^2 - 160*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^2 - 16*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^2 - (968*(a + b)*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^4)/a - (288*(a + b)*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^4)/a - (32*(a + b)*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^4)/a - (380*(a + b)^2*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6)/a^2 - (128*(a + b)^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6)/a^2 - (16*(a + b)^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6)/a^2 + (65625*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/(-Sinh[c + d*x]^2)^(5/2) + (1680*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4)/(-Sinh[c + d*x]^2)^(5/2) - (36855*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/(-Sinh[c + d*x]^2)^(3/2) - (91875*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/(a*(-Sinh[c + d*x]^2)^(3/2)) + (54180*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/(a*Sqrt[-Sinh[c + d*x]^2]) + (32970*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/(a^2*Sqrt[-Sinh[c + d*x]^2]) + (525*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4)/(a^2*Sqrt[-Sinh[c + d*x]^2]) - (1365*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sqrt[-Sinh[c + d*x]^2])/a - (19845*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sqrt[-Sinh[c + d*x]^2])/a^2))/(2520*d)","C",0
96,1,83,76,0.5769488,"\int \text{sech}^4(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Sech[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\tanh (c+d x) \left(\left(35 a^2+14 a b+3 b^2\right) \text{sech}^2(c+d x)+70 a^2-6 b (7 a+4 b) \text{sech}^4(c+d x)+28 a b+15 b^2 \text{sech}^6(c+d x)+6 b^2\right)}{105 d}","\frac{a^2 \tanh (c+d x)}{d}-\frac{b (2 a-b) \tanh ^5(c+d x)}{5 d}-\frac{a (a-2 b) \tanh ^3(c+d x)}{3 d}-\frac{b^2 \tanh ^7(c+d x)}{7 d}",1,"((70*a^2 + 28*a*b + 6*b^2 + (35*a^2 + 14*a*b + 3*b^2)*Sech[c + d*x]^2 - 6*b*(7*a + 4*b)*Sech[c + d*x]^4 + 15*b^2*Sech[c + d*x]^6)*Tanh[c + d*x])/(105*d)","A",1
97,1,81,91,0.7025248,"\int \cosh ^4(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Cosh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{12 \left(a^3-a^2 b+3 a b^2+5 b^3\right) (c+d x)+8 (a-2 b) (a+b)^2 \sinh (2 (c+d x))+(a+b)^3 \sinh (4 (c+d x))-32 b^3 \tanh (c+d x)}{32 d}","\frac{3}{8} x (a+b) \left(a^2-2 a b+5 b^2\right)+\frac{(a+b)^3 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac{3 (a-3 b) (a+b)^2 \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac{b^3 \tanh (c+d x)}{d}",1,"(12*(a^3 - a^2*b + 3*a*b^2 + 5*b^3)*(c + d*x) + 8*(a - 2*b)*(a + b)^2*Sinh[2*(c + d*x)] + (a + b)^3*Sinh[4*(c + d*x)] - 32*b^3*Tanh[c + d*x])/(32*d)","A",1
98,1,494,87,6.8854947,"\int \cosh ^3(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Cosh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\text{csch}^5(c+d x) \left(-256 \sinh ^8(c+d x) \left(a \sinh ^2(c+d x)+a+b \sinh ^2(c+d x)\right)^3 \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right)+21 \left(a^3 \left(753 \sinh ^{10}(c+d x)+19579 \sinh ^8(c+d x)+89514 \sinh ^6(c+d x)+157878 \sinh ^4(c+d x)+124165 \sinh ^2(c+d x)+36015\right)+3 a^2 b \left(753 \sinh ^8(c+d x)+18826 \sinh ^6(c+d x)+69728 \sinh ^4(c+d x)+88150 \sinh ^2(c+d x)+36015\right) \sinh ^2(c+d x)+3 a b^2 \left(753 \sinh ^6(c+d x)+18073 \sinh ^4(c+d x)+50695 \sinh ^2(c+d x)+36015\right) \sinh ^4(c+d x)+b^3 \left(753 \sinh ^4(c+d x)+17320 \sinh ^2(c+d x)+32415\right) \sinh ^6(c+d x)\right)-\frac{315 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \left(a^3 \left(\sinh ^6(c+d x)+243 \sinh ^4(c+d x)+1875 \sinh ^2(c+d x)+2401\right) \cosh ^6(c+d x)+3 a^2 b \left(\sinh ^3(c+d x)+\sinh (c+d x)\right)^2 \left(\sinh ^6(c+d x)+243 \sinh ^4(c+d x)+1875 \sinh ^2(c+d x)+2401\right)+3 a b^2 \sinh ^4(c+d x) \left(\sinh ^8(c+d x)+244 \sinh ^6(c+d x)+2118 \sinh ^4(c+d x)+4180 \sinh ^2(c+d x)+2401\right)+b^3 \sinh ^6(c+d x) \left(\sinh ^6(c+d x)+243 \sinh ^4(c+d x)+1875 \sinh ^2(c+d x)+2161\right)\right)}{\sqrt{-\sinh ^2(c+d x)}}\right)}{30240 d}","\frac{b^2 (6 a+5 b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{(a+b)^3 \sinh ^3(c+d x)}{3 d}+\frac{(a-2 b) (a+b)^2 \sinh (c+d x)}{d}-\frac{b^3 \tanh (c+d x) \text{sech}(c+d x)}{2 d}",1,"(Csch[c + d*x]^5*(-256*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^8*(a + a*Sinh[c + d*x]^2 + b*Sinh[c + d*x]^2)^3 - (315*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(b^3*Sinh[c + d*x]^6*(2161 + 1875*Sinh[c + d*x]^2 + 243*Sinh[c + d*x]^4 + Sinh[c + d*x]^6) + a^3*Cosh[c + d*x]^6*(2401 + 1875*Sinh[c + d*x]^2 + 243*Sinh[c + d*x]^4 + Sinh[c + d*x]^6) + 3*a^2*b*(Sinh[c + d*x] + Sinh[c + d*x]^3)^2*(2401 + 1875*Sinh[c + d*x]^2 + 243*Sinh[c + d*x]^4 + Sinh[c + d*x]^6) + 3*a*b^2*Sinh[c + d*x]^4*(2401 + 4180*Sinh[c + d*x]^2 + 2118*Sinh[c + d*x]^4 + 244*Sinh[c + d*x]^6 + Sinh[c + d*x]^8)))/Sqrt[-Sinh[c + d*x]^2] + 21*(b^3*Sinh[c + d*x]^6*(32415 + 17320*Sinh[c + d*x]^2 + 753*Sinh[c + d*x]^4) + 3*a*b^2*Sinh[c + d*x]^4*(36015 + 50695*Sinh[c + d*x]^2 + 18073*Sinh[c + d*x]^4 + 753*Sinh[c + d*x]^6) + 3*a^2*b*Sinh[c + d*x]^2*(36015 + 88150*Sinh[c + d*x]^2 + 69728*Sinh[c + d*x]^4 + 18826*Sinh[c + d*x]^6 + 753*Sinh[c + d*x]^8) + a^3*(36015 + 124165*Sinh[c + d*x]^2 + 157878*Sinh[c + d*x]^4 + 89514*Sinh[c + d*x]^6 + 19579*Sinh[c + d*x]^8 + 753*Sinh[c + d*x]^10))))/(30240*d)","C",0
99,1,69,78,0.9687602,"\int \cosh ^2(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Cosh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{4 b^2 \tanh (c+d x) \left(9 a-b \text{sech}^2(c+d x)+7 b\right)+6 (a-5 b) (a+b)^2 (c+d x)+3 (a+b)^3 \sinh (2 (c+d x))}{12 d}","\frac{b^2 (3 a+2 b) \tanh (c+d x)}{d}+\frac{(a+b)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac{1}{2} x (a-5 b) (a+b)^2+\frac{b^3 \tanh ^3(c+d x)}{3 d}",1,"(6*(a - 5*b)*(a + b)^2*(c + d*x) + 3*(a + b)^3*Sinh[2*(c + d*x)] + 4*b^2*(9*a + 7*b - b*Sech[c + d*x]^2)*Tanh[c + d*x])/(12*d)","A",1
100,1,89,99,0.4220497,"\int \cosh (c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Cosh[c + d*x]*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{-3 b \left(8 a^2+12 a b+5 b^2\right) \tan ^{-1}(\sinh (c+d x))+3 b^2 (4 a+3 b) \tanh (c+d x) \text{sech}(c+d x)+8 (a+b)^3 \sinh (c+d x)-2 b^3 \tanh (c+d x) \text{sech}^3(c+d x)}{8 d}","\frac{3 b^2 (4 a+3 b) \tanh (c+d x) \text{sech}(c+d x)}{8 d}+\frac{(a+b)^3 \sinh (c+d x)}{d}-\frac{3 b \left(4 (a+b)^2+(2 a+b)^2\right) \tan ^{-1}(\sinh (c+d x))}{8 d}-\frac{b^3 \tanh (c+d x) \text{sech}^3(c+d x)}{4 d}",1,"(-3*b*(8*a^2 + 12*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]] + 8*(a + b)^3*Sinh[c + d*x] + 3*b^2*(4*a + 3*b)*Sech[c + d*x]*Tanh[c + d*x] - 2*b^3*Sech[c + d*x]^3*Tanh[c + d*x])/(8*d)","A",1
101,1,1341,149,16.6757554,"\int \text{sech}(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{a^3 \sinh (c+d x) \left(-\frac{44800 (a+b)^3 \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \sinh ^8(c+d x)}{a^3}-\frac{10752 (a+b)^3 \, _6F_5\left(\frac{3}{2},2,2,2,2,2;1,1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \sinh ^8(c+d x)}{a^3}-\frac{1024 (a+b)^3 \, _7F_6\left(\frac{3}{2},2,2,2,2,2,2;1,1,1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \sinh ^8(c+d x)}{a^3}+\frac{48825 (a+b)^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^8(c+d x)}{a^3 \left(-\sinh ^2(c+d x)\right)^{3/2}}+\frac{56109375 (a+b)^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^8(c+d x)}{a^3 \left(-\sinh ^2(c+d x)\right)^{7/2}}+\frac{33756345 (a+b)^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^8(c+d x)}{a^2 \left(-\sinh ^2(c+d x)\right)^{7/2}}+\frac{142065 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^8(c+d x)}{\left(-\sinh ^2(c+d x)\right)^{9/2}}-\frac{158976 (a+b)^2 \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \sinh ^6(c+d x)}{a^2}-\frac{35328 (a+b)^2 \, _6F_5\left(\frac{3}{2},2,2,2,2,2;1,1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \sinh ^6(c+d x)}{a^2}-\frac{3072 (a+b)^2 \, _7F_6\left(\frac{3}{2},2,2,2,2,2,2;1,1,1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \sinh ^6(c+d x)}{a^2}+\frac{9261945 (a+b)^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^6(c+d x)}{a^3 \left(-\sinh ^2(c+d x)\right)^{3/2}}+\frac{174825 (a+b)^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^6(c+d x)}{a^2 \left(-\sinh ^2(c+d x)\right)^{3/2}}+\frac{2160711 (a+b)^3 \sinh ^4(c+d x)}{a^3}-\frac{189696 (a+b) \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \sinh ^4(c+d x)}{a}-\frac{38400 (a+b) \, _6F_5\left(\frac{3}{2},2,2,2,2,2;1,1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \sinh ^4(c+d x)}{a}-\frac{3072 (a+b) \, _7F_6\left(\frac{3}{2},2,2,2,2,2,2;1,1,1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \sinh ^4(c+d x)}{a}+\frac{17069535 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^4(c+d x)}{\left(-\sinh ^2(c+d x)\right)^{7/2}}+\frac{36772890 (a+b)^3 \sinh ^2(c+d x)}{a^3}+\frac{7808535 (a+b)^2 \sinh ^2(c+d x)}{a^2}-75520 \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \sinh ^2(c+d x)-13824 \, _6F_5\left(\frac{3}{2},2,2,2,2,2;1,1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \sinh ^2(c+d x)-1024 \, _7F_6\left(\frac{3}{2},2,2,2,2,2,2;1,1,1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \sinh ^2(c+d x)+117228825 \text{csch}^6(c+d x)+\frac{274542345 (a+b) \text{csch}^4(c+d x)}{a}+70189350 \text{csch}^4(c+d x)+\frac{58009455 (a+b)^3}{a^3}+\frac{135323370 (a+b)^2}{a^2}+\frac{215549775 (a+b)^2 \text{csch}^2(c+d x)}{a^2}+\frac{168951510 (a+b) \text{csch}^2(c+d x)}{a}+4093425 \text{csch}^2(c+d x)+\frac{9514449 (a+b)}{a}+\frac{210735 (a+b) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sqrt{-\sinh ^2(c+d x)}}{a}-\frac{58009455 (a+b)^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{a^3 \sqrt{-\sinh ^2(c+d x)}}-\frac{207173295 (a+b)^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{a^2 \sqrt{-\sinh ^2(c+d x)}}-\frac{41427855 (a+b) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{a \sqrt{-\sinh ^2(c+d x)}}+\frac{215549775 (a+b)^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{a^2 \left(-\sinh ^2(c+d x)\right)^{3/2}}+\frac{260465625 (a+b) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{a \left(-\sinh ^2(c+d x)\right)^{3/2}}-\frac{274542345 (a+b) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{a \left(-\sinh ^2(c+d x)\right)^{5/2}}-\frac{109265625 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{\left(-\sinh ^2(c+d x)\right)^{5/2}}+\frac{117228825 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{\left(-\sinh ^2(c+d x)\right)^{7/2}}\right)}{725760 d}","\frac{(2 a+b) \left(8 a^2+8 a b+5 b^2\right) \tan ^{-1}(\sinh (c+d x))}{16 d}-\frac{b \left(44 a^2+44 a b+15 b^2\right) \tanh (c+d x) \text{sech}(c+d x)}{48 d}-\frac{b \tanh (c+d x) \text{sech}^5(c+d x) \left((a+b) \sinh ^2(c+d x)+a\right)^2}{6 d}-\frac{5 b (2 a+b) \tanh (c+d x) \text{sech}^3(c+d x) \left((a+b) \sinh ^2(c+d x)+a\right)}{24 d}",1,"(a^3*Sinh[c + d*x]*((9514449*(a + b))/a + (135323370*(a + b)^2)/a^2 + (58009455*(a + b)^3)/a^3 + 4093425*Csch[c + d*x]^2 + (168951510*(a + b)*Csch[c + d*x]^2)/a + (215549775*(a + b)^2*Csch[c + d*x]^2)/a^2 + 70189350*Csch[c + d*x]^4 + (274542345*(a + b)*Csch[c + d*x]^4)/a + 117228825*Csch[c + d*x]^6 + (7808535*(a + b)^2*Sinh[c + d*x]^2)/a^2 + (36772890*(a + b)^3*Sinh[c + d*x]^2)/a^3 - 75520*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^2 - 13824*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^2 - 1024*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^2 + (2160711*(a + b)^3*Sinh[c + d*x]^4)/a^3 - (189696*(a + b)*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^4)/a - (38400*(a + b)*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^4)/a - (3072*(a + b)*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^4)/a - (158976*(a + b)^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6)/a^2 - (35328*(a + b)^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6)/a^2 - (3072*(a + b)^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6)/a^2 - (44800*(a + b)^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^8)/a^3 - (10752*(a + b)^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^8)/a^3 - (1024*(a + b)^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^8)/a^3 + (142065*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8)/(-Sinh[c + d*x]^2)^(9/2) + (117228825*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/(-Sinh[c + d*x]^2)^(7/2) + (17069535*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4)/(-Sinh[c + d*x]^2)^(7/2) + (33756345*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8)/(a^2*(-Sinh[c + d*x]^2)^(7/2)) + (56109375*(a + b)^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8)/(a^3*(-Sinh[c + d*x]^2)^(7/2)) - (109265625*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/(-Sinh[c + d*x]^2)^(5/2) - (274542345*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/(a*(-Sinh[c + d*x]^2)^(5/2)) + (260465625*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/(a*(-Sinh[c + d*x]^2)^(3/2)) + (215549775*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/(a^2*(-Sinh[c + d*x]^2)^(3/2)) + (174825*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6)/(a^2*(-Sinh[c + d*x]^2)^(3/2)) + (9261945*(a + b)^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6)/(a^3*(-Sinh[c + d*x]^2)^(3/2)) + (48825*(a + b)^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8)/(a^3*(-Sinh[c + d*x]^2)^(3/2)) - (41427855*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/(a*Sqrt[-Sinh[c + d*x]^2]) - (207173295*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/(a^2*Sqrt[-Sinh[c + d*x]^2]) - (58009455*(a + b)^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/(a^3*Sqrt[-Sinh[c + d*x]^2]) + (210735*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sqrt[-Sinh[c + d*x]^2])/a))/(725760*d)","C",0
102,1,67,67,0.1794925,"\int \text{sech}^2(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{a^3 \tanh (c+d x)}{d}+\frac{a^2 b \tanh ^3(c+d x)}{d}+\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac{b^3 \tanh ^7(c+d x)}{7 d}","\frac{a^3 \tanh (c+d x)}{d}+\frac{a^2 b \tanh ^3(c+d x)}{d}+\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac{b^3 \tanh ^7(c+d x)}{7 d}",1,"(a^3*Tanh[c + d*x])/d + (a^2*b*Tanh[c + d*x]^3)/d + (3*a*b^2*Tanh[c + d*x]^5)/(5*d) + (b^3*Tanh[c + d*x]^7)/(7*d)","A",1
103,1,158,198,11.8368299,"\int \text{sech}^3(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{-2 b \left(144 a^2+168 a b+59 b^2\right) \tanh (c+d x) \text{sech}^3(c+d x)+6 \left(64 a^3+48 a^2 b+24 a b^2+5 b^3\right) \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+3 \left(64 a^3+48 a^2 b+24 a b^2+5 b^3\right) \tanh (c+d x) \text{sech}(c+d x)+8 b^2 (24 a+17 b) \tanh (c+d x) \text{sech}^5(c+d x)-48 b^3 \tanh (c+d x) \text{sech}^7(c+d x)}{384 d}","-\frac{b \left(72 a^2+52 a b+15 b^2\right) \tanh (c+d x) \text{sech}^3(c+d x)}{192 d}+\frac{\left(64 a^3+48 a^2 b+24 a b^2+5 b^3\right) \tan ^{-1}(\sinh (c+d x))}{128 d}+\frac{\left(64 a^3+48 a^2 b+24 a b^2+5 b^3\right) \tanh (c+d x) \text{sech}(c+d x)}{128 d}-\frac{b \tanh (c+d x) \text{sech}^7(c+d x) \left((a+b) \sinh ^2(c+d x)+a\right)^2}{8 d}-\frac{b (12 a+5 b) \tanh (c+d x) \text{sech}^5(c+d x) \left((a+b) \sinh ^2(c+d x)+a\right)}{48 d}",1,"(6*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*ArcTan[Tanh[(c + d*x)/2]] + 3*(64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*Sech[c + d*x]*Tanh[c + d*x] - 2*b*(144*a^2 + 168*a*b + 59*b^2)*Sech[c + d*x]^3*Tanh[c + d*x] + 8*b^2*(24*a + 17*b)*Sech[c + d*x]^5*Tanh[c + d*x] - 48*b^3*Sech[c + d*x]^7*Tanh[c + d*x])/(384*d)","A",1
104,1,218,102,0.9099804,"\int \text{sech}^4(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\tanh (c+d x) \text{sech}^8(c+d x) \left(1050 a^3 \cosh (6 (c+d x))+105 a^3 \cosh (8 (c+d x))+5775 a^3+630 a^2 b \cosh (6 (c+d x))+63 a^2 b \cosh (8 (c+d x))-1071 a^2 b+10 \left(903 a^3-63 a^2 b-27 a b^2+107 b^3\right) \cosh (2 (c+d x))+8 \left(525 a^3+126 a^2 b-81 a b^2-50 b^3\right) \cosh (4 (c+d x))+270 a b^2 \cosh (6 (c+d x))+27 a b^2 \cosh (8 (c+d x))+621 a b^2+50 b^3 \cosh (6 (c+d x))+5 b^3 \cosh (8 (c+d x))-725 b^3\right)}{20160 d}","\frac{a^3 \tanh (c+d x)}{d}-\frac{a^2 (a-3 b) \tanh ^3(c+d x)}{3 d}-\frac{b^2 (3 a-b) \tanh ^7(c+d x)}{7 d}-\frac{3 a b (a-b) \tanh ^5(c+d x)}{5 d}-\frac{b^3 \tanh ^9(c+d x)}{9 d}",1,"((5775*a^3 - 1071*a^2*b + 621*a*b^2 - 725*b^3 + 10*(903*a^3 - 63*a^2*b - 27*a*b^2 + 107*b^3)*Cosh[2*(c + d*x)] + 8*(525*a^3 + 126*a^2*b - 81*a*b^2 - 50*b^3)*Cosh[4*(c + d*x)] + 1050*a^3*Cosh[6*(c + d*x)] + 630*a^2*b*Cosh[6*(c + d*x)] + 270*a*b^2*Cosh[6*(c + d*x)] + 50*b^3*Cosh[6*(c + d*x)] + 105*a^3*Cosh[8*(c + d*x)] + 63*a^2*b*Cosh[8*(c + d*x)] + 27*a*b^2*Cosh[8*(c + d*x)] + 5*b^3*Cosh[8*(c + d*x)])*Sech[c + d*x]^8*Tanh[c + d*x])/(20160*d)","B",1
105,1,115,120,0.3025721,"\int \frac{\cosh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Cosh[c + d*x]^4/(a + b*Tanh[c + d*x]^2),x]","\frac{\left(3 a^2+10 a b+15 b^2\right) (c+d x)}{8 d (a+b)^3}+\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a+b)^3}+\frac{(a+2 b) \sinh (2 (c+d x))}{4 d (a+b)^2}+\frac{\sinh (4 (c+d x))}{32 d (a+b)}","\frac{x \left(3 a^2+10 a b+15 b^2\right)}{8 (a+b)^3}+\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a+b)^3}+\frac{\sinh (c+d x) \cosh ^3(c+d x)}{4 d (a+b)}+\frac{(3 a+7 b) \sinh (c+d x) \cosh (c+d x)}{8 d (a+b)^2}",1,"((3*a^2 + 10*a*b + 15*b^2)*(c + d*x))/(8*(a + b)^3*d) + (b^(5/2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)^3*d) + ((a + 2*b)*Sinh[2*(c + d*x)])/(4*(a + b)^2*d) + Sinh[4*(c + d*x)]/(32*(a + b)*d)","A",1
106,1,79,80,0.4297828,"\int \frac{\cosh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Cosh[c + d*x]^3/(a + b*Tanh[c + d*x]^2),x]","\frac{-\frac{12 b^2 \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right)}{\sqrt{a} (a+b)^{5/2}}+\frac{3 (3 a+7 b) \sinh (c+d x)}{(a+b)^2}+\frac{\sinh (3 (c+d x))}{a+b}}{12 d}","\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a+b)^{5/2}}+\frac{\sinh ^3(c+d x)}{3 d (a+b)}+\frac{(a+2 b) \sinh (c+d x)}{d (a+b)^2}",1,"((-12*b^2*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(5/2)) + (3*(3*a + 7*b)*Sinh[c + d*x])/(a + b)^2 + Sinh[3*(c + d*x)]/(a + b))/(12*d)","A",1
107,1,77,77,0.1605732,"\int \frac{\cosh ^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Cosh[c + d*x]^2/(a + b*Tanh[c + d*x]^2),x]","\frac{4 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)+2 \sqrt{a} (a+3 b) (c+d x)+\sqrt{a} (a+b) \sinh (2 (c+d x))}{4 \sqrt{a} d (a+b)^2}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a+b)^2}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 d (a+b)}+\frac{x (a+3 b)}{2 (a+b)^2}",1,"(2*Sqrt[a]*(a + 3*b)*(c + d*x) + 4*b^(3/2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] + Sqrt[a]*(a + b)*Sinh[2*(c + d*x)])/(4*Sqrt[a]*(a + b)^2*d)","A",1
108,1,53,53,0.1067786,"\int \frac{\cosh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Cosh[c + d*x]/(a + b*Tanh[c + d*x]^2),x]","\frac{\sinh (c+d x)}{d (a+b)}+\frac{b \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a+b)^{3/2}}","\frac{\sinh (c+d x)}{d (a+b)}+\frac{b \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a+b)^{3/2}}",1,"(b*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(3/2)*d) + Sinh[c + d*x]/((a + b)*d)","A",1
109,1,36,36,0.0471417,"\int \frac{\text{sech}(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Sech[c + d*x]/(a + b*Tanh[c + d*x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d \sqrt{a+b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d \sqrt{a+b}}",1,"ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[a + b]*d)","A",1
110,1,32,32,0.0579792,"\int \frac{\text{sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Sech[c + d*x]^2/(a + b*Tanh[c + d*x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d}",1,"ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*d)","A",1
111,1,55,55,0.20802,"\int \frac{\text{sech}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Sech[c + d*x]^3/(a + b*Tanh[c + d*x]^2),x]","-\frac{\frac{\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right)}{\sqrt{a}}+2 \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{b d}","\frac{\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b d}-\frac{\tan ^{-1}(\sinh (c+d x))}{b d}",1,"-(((Sqrt[a + b]*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/Sqrt[a] + 2*ArcTan[Tanh[(c + d*x)/2]])/(b*d))","A",1
112,1,50,50,0.146127,"\int \frac{\text{sech}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Sech[c + d*x]^4/(a + b*Tanh[c + d*x]^2),x]","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2} d}-\frac{\tanh (c+d x)}{b d}","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2} d}-\frac{\tanh (c+d x)}{b d}",1,"((a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(3/2)*d) - Tanh[c + d*x]/(b*d)","A",1
113,1,79,86,0.601446,"\int \frac{\text{sech}^5(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Sech[c + d*x]^5/(a + b*Tanh[c + d*x]^2),x]","-\frac{2 (2 a+3 b) \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 (a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right)}{\sqrt{a}}+b \tanh (c+d x) \text{sech}(c+d x)}{2 b^2 d}","\frac{(a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^2 d}-\frac{(2 a+3 b) \tan ^{-1}(\sinh (c+d x))}{2 b^2 d}-\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 b d}",1,"-1/2*((2*(a + b)^(3/2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/Sqrt[a] + 2*(2*a + 3*b)*ArcTan[Tanh[(c + d*x)/2]] + b*Sech[c + d*x]*Tanh[c + d*x])/(b^2*d)","A",1
114,1,71,75,0.4010134,"\int \frac{\text{sech}^6(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Sech[c + d*x]^6/(a + b*Tanh[c + d*x]^2),x]","\frac{(a+b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{5/2} d}-\frac{\tanh (c+d x) \left(3 a+b \text{sech}^2(c+d x)+5 b\right)}{3 b^2 d}","\frac{(a+b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{5/2} d}-\frac{(a+2 b) \tanh (c+d x)}{b^2 d}+\frac{\tanh ^3(c+d x)}{3 b d}",1,"((a + b)^2*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)*d) - ((3*a + 5*b + b*Sech[c + d*x]^2)*Tanh[c + d*x])/(3*b^2*d)","A",1
115,1,111,128,1.1857502,"\int \frac{\cosh ^3(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Cosh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{-\frac{6 b^2 (6 a+b) \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right)}{a^{3/2} (a+b)^{7/2}}+\frac{3 \sinh (c+d x) \left(\frac{4 b^3}{a ((a+b) \cosh (2 (c+d x))+a-b)}+3 a+11 b\right)}{(a+b)^3}+\frac{\sinh (3 (c+d x))}{(a+b)^2}}{12 d}","\frac{b^2 (6 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a+b)^{7/2}}+\frac{b^3 \sinh (c+d x)}{2 a d (a+b)^3 \left((a+b) \sinh ^2(c+d x)+a\right)}+\frac{\sinh ^3(c+d x)}{3 d (a+b)^2}+\frac{(a+3 b) \sinh (c+d x)}{d (a+b)^3}",1,"((-6*b^2*(6*a + b)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/(a^(3/2)*(a + b)^(7/2)) + (3*(3*a + 11*b + (4*b^3)/(a*(a - b + (a + b)*Cosh[2*(c + d*x)])))*Sinh[c + d*x])/(a + b)^3 + Sinh[3*(c + d*x)]/(a + b)^2)/(12*d)","A",1
116,1,110,140,0.8360635,"\int \frac{\cosh ^2(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Cosh[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\frac{2 b^{3/2} (5 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{3/2}}+\frac{2 b^2 (a+b) \sinh (2 (c+d x))}{a ((a+b) \cosh (2 (c+d x))+a-b)}+2 (a+5 b) (c+d x)+(a+b) \sinh (2 (c+d x))}{4 d (a+b)^3}","\frac{b^{3/2} (5 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a+b)^3}-\frac{b (a-b) \tanh (c+d x)}{2 a d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 d (a+b) \left(a+b \tanh ^2(c+d x)\right)}+\frac{x (a+5 b)}{2 (a+b)^3}",1,"(2*(a + 5*b)*(c + d*x) + (2*b^(3/2)*(5*a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/a^(3/2) + (a + b)*Sinh[2*(c + d*x)] + (2*b^2*(a + b)*Sinh[2*(c + d*x)])/(a*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(4*(a + b)^3*d)","A",1
117,1,84,101,0.7621361,"\int \frac{\cosh (c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Cosh[c + d*x]/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\frac{b (4 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{a^{3/2} (a+b)^{5/2}}+\frac{\sinh (c+d x) \left(\frac{b^2}{a \left((a+b) \sinh ^2(c+d x)+a\right)}+2\right)}{(a+b)^2}}{2 d}","\frac{b (4 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a+b)^{5/2}}+\frac{b^2 \sinh (c+d x)}{2 a d (a+b)^2 \left((a+b) \sinh ^2(c+d x)+a\right)}+\frac{\sinh (c+d x)}{d (a+b)^2}",1,"((b*(4*a + b)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(a^(3/2)*(a + b)^(5/2)) + (Sinh[c + d*x]*(2 + b^2/(a*(a + (a + b)*Sinh[c + d*x]^2))))/(a + b)^2)/(2*d)","A",1
118,1,78,83,0.3062307,"\int \frac{\text{sech}(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\frac{(2 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{a^{3/2} \sqrt{a+b}}+\frac{b \sinh (c+d x)}{a \left((a+b) \sinh ^2(c+d x)+a\right)}}{2 d (a+b)}","\frac{(2 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a+b)^{3/2}}+\frac{b \sinh (c+d x)}{2 a d (a+b) \left((a+b) \sinh ^2(c+d x)+a\right)}",1,"(((2*a + b)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(a^(3/2)*Sqrt[a + b]) + (b*Sinh[c + d*x])/(a*(a + (a + b)*Sinh[c + d*x]^2)))/(2*(a + b)*d)","A",1
119,1,63,66,0.2678484,"\int \frac{\text{sech}^2(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\frac{\sqrt{a} \tanh (c+d x)}{a+b \tanh ^2(c+d x)}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{b}}}{2 a^{3/2} d}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b} d}+\frac{\tanh (c+d x)}{2 a d \left(a+b \tanh ^2(c+d x)\right)}",1,"(ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]]/Sqrt[b] + (Sqrt[a]*Tanh[c + d*x])/(a + b*Tanh[c + d*x]^2))/(2*a^(3/2)*d)","A",1
120,1,69,72,0.1404628,"\int \frac{\text{sech}^3(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\frac{\sqrt{a} \sinh (c+d x)}{(a+b) \sinh ^2(c+d x)+a}+\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a+b}}}{2 a^{3/2} d}","\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d \sqrt{a+b}}+\frac{\sinh (c+d x)}{2 a d \left((a+b) \sinh ^2(c+d x)+a\right)}",1,"(ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]]/Sqrt[a + b] + (Sqrt[a]*Sinh[c + d*x])/(a + (a + b)*Sinh[c + d*x]^2))/(2*a^(3/2)*d)","A",1
121,1,83,77,0.3270043,"\int \frac{\text{sech}^4(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{(b-a) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)+\frac{\sqrt{a} \sqrt{b} (a+b) \sinh (2 (c+d x))}{(a+b) \cosh (2 (c+d x))+a-b}}{2 a^{3/2} b^{3/2} d}","\frac{(a+b) \tanh (c+d x)}{2 a b d \left(a+b \tanh ^2(c+d x)\right)}-\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2} d}",1,"((-a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] + (Sqrt[a]*Sqrt[b]*(a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)]))/(2*a^(3/2)*b^(3/2)*d)","A",1
122,1,203,102,0.5554662,"\int \frac{\text{sech}^5(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]^5/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{(a-b) \left(4 a^{3/2} \sqrt{a+b} \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+\left(2 a^2+a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right)\right)+(a+b) \cosh (2 (c+d x)) \left(4 a^{3/2} \sqrt{a+b} \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+\left(2 a^2+a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right)\right)+2 \sqrt{a} b (a+b)^{3/2} \sinh (c+d x)}{2 a^{3/2} b^2 d \sqrt{a+b} ((a+b) \cosh (2 (c+d x))+a-b)}","-\frac{(2 a-b) \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^2 d}+\frac{(a+b) \sinh (c+d x)}{2 a b d \left((a+b) \sinh ^2(c+d x)+a\right)}+\frac{\tan ^{-1}(\sinh (c+d x))}{b^2 d}",1,"((a - b)*((2*a^2 + a*b - b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]] + 4*a^(3/2)*Sqrt[a + b]*ArcTan[Tanh[(c + d*x)/2]]) + (a + b)*((2*a^2 + a*b - b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]] + 4*a^(3/2)*Sqrt[a + b]*ArcTan[Tanh[(c + d*x)/2]])*Cosh[2*(c + d*x)] + 2*Sqrt[a]*b*(a + b)^(3/2)*Sinh[c + d*x])/(2*a^(3/2)*b^2*Sqrt[a + b]*d*(a - b + (a + b)*Cosh[2*(c + d*x)]))","A",1
123,1,102,97,0.6542918,"\int \frac{\text{sech}^6(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]^6/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{-\frac{(3 a-b) (a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{3/2}}+\frac{\sqrt{b} (a+b)^2 \sinh (2 (c+d x))}{a ((a+b) \cosh (2 (c+d x))+a-b)}+2 \sqrt{b} \tanh (c+d x)}{2 b^{5/2} d}","-\frac{(3 a-b) (a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{5/2} d}+\frac{(a+b)^2 \tanh (c+d x)}{2 a b^2 d \left(a+b \tanh ^2(c+d x)\right)}+\frac{\tanh (c+d x)}{b^2 d}",1,"(-(((3*a - b)*(a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/a^(3/2)) + (Sqrt[b]*(a + b)^2*Sinh[2*(c + d*x)])/(a*(a - b + (a + b)*Cosh[2*(c + d*x)])) + 2*Sqrt[b]*Tanh[c + d*x])/(2*b^(5/2)*d)","A",1
124,1,265,155,1.4136177,"\int \frac{\text{sech}^7(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]^7/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{(a-b) \left(2 a^{3/2} (4 a+5 b) \sqrt{a+b} \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+a^{3/2} b \sqrt{a+b} \tanh (c+d x) \text{sech}(c+d x)+(4 a-b) (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right)\right)+(a+b) \cosh (2 (c+d x)) \left(2 a^{3/2} (4 a+5 b) \sqrt{a+b} \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+a^{3/2} b \sqrt{a+b} \tanh (c+d x) \text{sech}(c+d x)+(4 a-b) (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right)\right)+2 \sqrt{a} b (a+b)^{5/2} \sinh (c+d x)}{2 a^{3/2} b^3 d \sqrt{a+b} ((a+b) \cosh (2 (c+d x))+a-b)}","-\frac{(4 a-b) (a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^3 d}+\frac{(4 a+5 b) \tan ^{-1}(\sinh (c+d x))}{2 b^3 d}+\frac{(2 a+b) (a+b) \sinh (c+d x)}{2 a b^2 d \left((a+b) \sinh ^2(c+d x)+a\right)}-\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 b d \left((a+b) \sinh ^2(c+d x)+a\right)}",1,"(2*Sqrt[a]*b*(a + b)^(5/2)*Sinh[c + d*x] + (a - b)*((4*a - b)*(a + b)^2*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]] + 2*a^(3/2)*Sqrt[a + b]*(4*a + 5*b)*ArcTan[Tanh[(c + d*x)/2]] + a^(3/2)*b*Sqrt[a + b]*Sech[c + d*x]*Tanh[c + d*x]) + (a + b)*Cosh[2*(c + d*x)]*((4*a - b)*(a + b)^2*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]] + 2*a^(3/2)*Sqrt[a + b]*(4*a + 5*b)*ArcTan[Tanh[(c + d*x)/2]] + a^(3/2)*b*Sqrt[a + b]*Sech[c + d*x]*Tanh[c + d*x]))/(2*a^(3/2)*b^3*Sqrt[a + b]*d*(a - b + (a + b)*Cosh[2*(c + d*x)]))","A",1
125,1,164,198,1.4239457,"\int \frac{\cosh ^2(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Cosh[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\frac{b^2 (a+b) (13 a+3 b) \sinh (2 (c+d x))}{a^2 ((a+b) \cosh (2 (c+d x))+a-b)}+\frac{b^{3/2} \left(35 a^2+14 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{5/2}}+\frac{4 b^3 (a+b) \sinh (2 (c+d x))}{a ((a+b) \cosh (2 (c+d x))+a-b)^2}+4 (a+7 b) (c+d x)+2 (a+b) \sinh (2 (c+d x))}{8 d (a+b)^4}","-\frac{b (a-3 b) (4 a+b) \tanh (c+d x)}{8 a^2 d (a+b)^3 \left(a+b \tanh ^2(c+d x)\right)}+\frac{b^{3/2} \left(35 a^2+14 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a+b)^4}-\frac{b (2 a-b) \tanh (c+d x)}{4 a d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)^2}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 d (a+b) \left(a+b \tanh ^2(c+d x)\right)^2}+\frac{x (a+7 b)}{2 (a+b)^4}",1,"(4*(a + 7*b)*(c + d*x) + (b^(3/2)*(35*a^2 + 14*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/a^(5/2) + 2*(a + b)*Sinh[2*(c + d*x)] + (4*b^3*(a + b)*Sinh[2*(c + d*x)])/(a*(a - b + (a + b)*Cosh[2*(c + d*x)])^2) + (b^2*(a + b)*(13*a + 3*b)*Sinh[2*(c + d*x)])/(a^2*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(8*(a + b)^4*d)","A",1
126,1,136,154,2.1369404,"\int \frac{\cosh (c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Cosh[c + d*x]/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\frac{\sinh (c+d x) \left(\frac{3 b^3}{a^2 \left((a+b) \sinh ^2(c+d x)+a\right)}+\frac{2 b^2 \left(6 (a+b) \sinh ^2(c+d x)+6 a+b\right)}{a \left((a+b) \sinh ^2(c+d x)+a\right)^2}+8\right)}{(a+b)^3}+\frac{3 b \left(8 a^2+4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{a^{5/2} (a+b)^{7/2}}}{8 d}","\frac{3 b^2 (4 a+b) \sinh (c+d x)}{8 a^2 d (a+b)^3 \left((a+b) \sinh ^2(c+d x)+a\right)}+\frac{3 b \left(8 a^2+4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a+b)^{7/2}}+\frac{b^3 \sinh (c+d x)}{4 a d (a+b)^3 \left((a+b) \sinh ^2(c+d x)+a\right)^2}+\frac{\sinh (c+d x)}{d (a+b)^3}",1,"((3*b*(8*a^2 + 4*a*b + b^2)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(a^(5/2)*(a + b)^(7/2)) + (Sinh[c + d*x]*(8 + (3*b^3)/(a^2*(a + (a + b)*Sinh[c + d*x]^2)) + (2*b^2*(6*a + b + 6*(a + b)*Sinh[c + d*x]^2))/(a*(a + (a + b)*Sinh[c + d*x]^2)^2)))/(a + b)^3)/(8*d)","A",1
127,1,134,144,1.1837348,"\int \frac{\text{sech}(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\frac{2 \sqrt{a} b \sinh (c+d x) \left(\left(8 a^2+11 a b+3 b^2\right) \cosh (2 (c+d x))+8 a^2-a b-3 b^2\right)}{(a+b)^2 ((a+b) \cosh (2 (c+d x))+a-b)^2}-\frac{\left(8 a^2+8 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}}{8 a^{5/2} d}","\frac{3 b (2 a+b) \sinh (c+d x)}{8 a^2 d (a+b)^2 \left((a+b) \sinh ^2(c+d x)+a\right)}+\frac{\left(8 a^2+8 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a+b)^{5/2}}+\frac{b \sinh (c+d x) \cosh ^2(c+d x)}{4 a d (a+b) \left((a+b) \sinh ^2(c+d x)+a\right)^2}",1,"(-(((8*a^2 + 8*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2)) + (2*Sqrt[a]*b*(8*a^2 - a*b - 3*b^2 + (8*a^2 + 11*a*b + 3*b^2)*Cosh[2*(c + d*x)])*Sinh[c + d*x])/((a + b)^2*(a - b + (a + b)*Cosh[2*(c + d*x)])^2))/(8*a^(5/2)*d)","A",1
128,1,77,96,0.8284836,"\int \frac{\text{sech}^2(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{5/2} \sqrt{b}}+\frac{\tanh (c+d x) \left(5 a+3 b \tanh ^2(c+d x)\right)}{a^2 \left(a+b \tanh ^2(c+d x)\right)^2}}{8 d}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} \sqrt{b} d}+\frac{3 \tanh (c+d x)}{8 a^2 d \left(a+b \tanh ^2(c+d x)\right)}+\frac{\tanh (c+d x)}{4 a d \left(a+b \tanh ^2(c+d x)\right)^2}",1,"((3*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(5/2)*Sqrt[b]) + (Tanh[c + d*x]*(5*a + 3*b*Tanh[c + d*x]^2))/(a^2*(a + b*Tanh[c + d*x]^2)^2))/(8*d)","A",1
129,1,123,129,0.7238196,"\int \frac{\text{sech}^3(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{(4 a+3 b) \left(\frac{3 \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{a^{5/2} \sqrt{a+b}}+\frac{3 (a+b) \sinh ^3(c+d x)+5 a \sinh (c+d x)}{a^2 \left((a+b) \sinh ^2(c+d x)+a\right)^2}\right)-\frac{8 \sinh (c+d x)}{\left((a+b) \sinh ^2(c+d x)+a\right)^2}}{24 d (a+b)}","\frac{(4 a+3 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a+b)^{3/2}}+\frac{(4 a+3 b) \sinh (c+d x)}{8 a^2 d (a+b) \left((a+b) \sinh ^2(c+d x)+a\right)}+\frac{b \sinh (c+d x)}{4 a d (a+b) \left((a+b) \sinh ^2(c+d x)+a\right)^2}",1,"((-8*Sinh[c + d*x])/(a + (a + b)*Sinh[c + d*x]^2)^2 + (4*a + 3*b)*((3*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(a^(5/2)*Sqrt[a + b]) + (5*a*Sinh[c + d*x] + 3*(a + b)*Sinh[c + d*x]^3)/(a^2*(a + (a + b)*Sinh[c + d*x]^2)^2)))/(24*(a + b)*d)","A",1
130,1,115,115,1.0505123,"\int \frac{\text{sech}^4(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\frac{\sqrt{a} \sinh (2 (c+d x)) \left(\left(a^2+4 a b+3 b^2\right) \cosh (2 (c+d x))+a^2+6 a b-3 b^2\right)}{b ((a+b) \cosh (2 (c+d x))+a-b)^2}+\frac{(3 b-a) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{b^{3/2}}}{8 a^{5/2} d}","-\frac{(a-3 b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} b^{3/2} d}-\frac{(a-3 b) \tanh (c+d x)}{8 a^2 b d \left(a+b \tanh ^2(c+d x)\right)}+\frac{(a+b) \tanh (c+d x)}{4 a b d \left(a+b \tanh ^2(c+d x)\right)^2}",1,"(((-a + 3*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/b^(3/2) + (Sqrt[a]*(a^2 + 6*a*b - 3*b^2 + (a^2 + 4*a*b + 3*b^2)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(b*(a - b + (a + b)*Cosh[2*(c + d*x)])^2))/(8*a^(5/2)*d)","A",1
131,1,88,104,0.3111355,"\int \frac{\text{sech}^5(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]^5/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\frac{3 \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{a^{3/2} \sqrt{a+b}}+\frac{3 (a+b) \sinh ^3(c+d x)+5 a \sinh (c+d x)}{a \left((a+b) \sinh ^2(c+d x)+a\right)^2}}{8 a d}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d \sqrt{a+b}}+\frac{3 \sinh (c+d x)}{8 a^2 d \left((a+b) \sinh ^2(c+d x)+a\right)}+\frac{\sinh (c+d x)}{4 a d \left((a+b) \sinh ^2(c+d x)+a\right)^2}",1,"((3*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(a^(3/2)*Sqrt[a + b]) + (5*a*Sinh[c + d*x] + 3*(a + b)*Sinh[c + d*x]^3)/(a*(a + (a + b)*Sinh[c + d*x]^2)^2))/(8*a*d)","A",1
132,1,128,131,0.9950711,"\int \frac{\text{sech}^6(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]^6/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\left(3 a^2-2 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)-\frac{\sqrt{a} \sqrt{b} (a+b) \sinh (2 (c+d x)) \left(3 \left(a^2-b^2\right) \cosh (2 (c+d x))+3 a^2-10 a b+3 b^2\right)}{((a+b) \cosh (2 (c+d x))+a-b)^2}}{8 a^{5/2} b^{5/2} d}","\frac{3 \left(\frac{1}{a^2}-\frac{1}{b^2}\right) \tanh (c+d x)}{8 d \left(a+b \tanh ^2(c+d x)\right)}+\frac{\left(3 a^2-2 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} b^{5/2} d}+\frac{(a+b) \tanh (c+d x) \text{sech}^2(c+d x)}{4 a b d \left(a+b \tanh ^2(c+d x)\right)^2}",1,"((3*a^2 - 2*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] - (Sqrt[a]*Sqrt[b]*(a + b)*(3*a^2 - 10*a*b + 3*b^2 + 3*(a^2 - b^2)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)])^2)/(8*a^(5/2)*b^(5/2)*d)","A",1
133,1,317,156,3.3816487,"\int \frac{\text{sech}^7(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]^7/(a + b*Tanh[c + d*x]^2)^3,x]","-\frac{\frac{8 b \left(4 a^2+a b-3 b^2\right) \sinh (c+d x)}{a^2 ((a+b) \cosh (2 (c+d x))+a-b)}+\frac{i \sqrt{a+b} \left(8 a^2-4 a b+3 b^2\right) \log ((a+b) \cosh (2 (c+d x))+a-b)}{a^{5/2}}+\frac{2 \sqrt{a+b} \left(8 a^2-4 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right)}{a^{5/2}}-\frac{i \left(8 a^3+4 a^2 b-a b^2+3 b^3\right) \log ((a+b) \cosh (2 (c+d x))+a-b)}{a^{5/2} \sqrt{a+b}}+\frac{2 \left(8 a^3+4 a^2 b-a b^2+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right)}{a^{5/2} \sqrt{a+b}}-\frac{32 b^2 (a+b) \sinh (c+d x)}{a ((a+b) \cosh (2 (c+d x))+a-b)^2}+64 \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{32 b^3 d}","-\frac{(4 a-3 b) (a+b) \sinh (c+d x)}{8 a^2 b^2 d \left((a+b) \sinh ^2(c+d x)+a\right)}+\frac{\sqrt{a+b} \left(8 a^2-4 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} b^3 d}+\frac{(a+b) \sinh (c+d x)}{4 a b d \left((a+b) \sinh ^2(c+d x)+a\right)^2}-\frac{\tan ^{-1}(\sinh (c+d x))}{b^3 d}",1,"-1/32*((2*Sqrt[a + b]*(8*a^2 - 4*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/a^(5/2) + (2*(8*a^3 + 4*a^2*b - a*b^2 + 3*b^3)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/(a^(5/2)*Sqrt[a + b]) + 64*ArcTan[Tanh[(c + d*x)/2]] + (I*Sqrt[a + b]*(8*a^2 - 4*a*b + 3*b^2)*Log[a - b + (a + b)*Cosh[2*(c + d*x)]])/a^(5/2) - (I*(8*a^3 + 4*a^2*b - a*b^2 + 3*b^3)*Log[a - b + (a + b)*Cosh[2*(c + d*x)]])/(a^(5/2)*Sqrt[a + b]) - (32*b^2*(a + b)*Sinh[c + d*x])/(a*(a - b + (a + b)*Cosh[2*(c + d*x)])^2) + (8*b*(4*a^2 + a*b - 3*b^2)*Sinh[c + d*x])/(a^2*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(b^3*d)","C",1
134,1,97,54,0.0385427,"\int \tanh ^4(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Tanh[c + d*x]^4*(a + b*Tanh[c + d*x]^2),x]","\frac{a \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{a \tanh ^3(c+d x)}{3 d}-\frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{b \tanh ^5(c+d x)}{5 d}-\frac{b \tanh ^3(c+d x)}{3 d}-\frac{b \tanh (c+d x)}{d}","-\frac{(a+b) \tanh ^3(c+d x)}{3 d}-\frac{(a+b) \tanh (c+d x)}{d}+x (a+b)-\frac{b \tanh ^5(c+d x)}{5 d}",1,"(a*ArcTanh[Tanh[c + d*x]])/d + (b*ArcTanh[Tanh[c + d*x]])/d - (a*Tanh[c + d*x])/d - (b*Tanh[c + d*x])/d - (a*Tanh[c + d*x]^3)/(3*d) - (b*Tanh[c + d*x]^3)/(3*d) - (b*Tanh[c + d*x]^5)/(5*d)","A",1
135,1,43,49,0.2493791,"\int \tanh ^3(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Tanh[c + d*x]^3*(a + b*Tanh[c + d*x]^2),x]","-\frac{2 (a+b) \tanh ^2(c+d x)-4 (a+b) \log (\cosh (c+d x))+b \tanh ^4(c+d x)}{4 d}","-\frac{(a+b) \tanh ^2(c+d x)}{2 d}+\frac{(a+b) \log (\cosh (c+d x))}{d}-\frac{b \tanh ^4(c+d x)}{4 d}",1,"-1/4*(-4*(a + b)*Log[Cosh[c + d*x]] + 2*(a + b)*Tanh[c + d*x]^2 + b*Tanh[c + d*x]^4)/d","A",1
136,1,65,36,0.0275548,"\int \tanh ^2(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Tanh[c + d*x]^2*(a + b*Tanh[c + d*x]^2),x]","\frac{a \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{b \tanh ^3(c+d x)}{3 d}-\frac{b \tanh (c+d x)}{d}","-\frac{(a+b) \tanh (c+d x)}{d}+x (a+b)-\frac{b \tanh ^3(c+d x)}{3 d}",1,"(a*ArcTanh[Tanh[c + d*x]])/d + (b*ArcTanh[Tanh[c + d*x]])/d - (a*Tanh[c + d*x])/d - (b*Tanh[c + d*x])/d - (b*Tanh[c + d*x]^3)/(3*d)","A",1
137,1,41,31,0.0242919,"\int \tanh (c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Tanh[c + d*x]*(a + b*Tanh[c + d*x]^2),x]","\frac{a \log (\cosh (c+d x))}{d}-\frac{b \tanh ^2(c+d x)}{2 d}+\frac{b \log (\cosh (c+d x))}{d}","\frac{(a+b) \log (\cosh (c+d x))}{d}-\frac{b \tanh ^2(c+d x)}{2 d}",1,"(a*Log[Cosh[c + d*x]])/d + (b*Log[Cosh[c + d*x]])/d - (b*Tanh[c + d*x]^2)/(2*d)","A",1
138,1,28,19,0.0073387,"\int \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[a + b*Tanh[c + d*x]^2,x]","a x+\frac{b \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{b \tanh (c+d x)}{d}","a x-\frac{b \tanh (c+d x)}{d}+b x",1,"a*x + (b*ArcTanh[Tanh[c + d*x]])/d - (b*Tanh[c + d*x])/d","A",1
139,1,33,25,0.0422324,"\int \coth (c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Coth[c + d*x]*(a + b*Tanh[c + d*x]^2),x]","\frac{a (\log (\tanh (c+d x))+\log (\cosh (c+d x)))}{d}+\frac{b \log (\cosh (c+d x))}{d}","\frac{a \log (\sinh (c+d x))}{d}+\frac{b \log (\cosh (c+d x))}{d}",1,"(b*Log[Cosh[c + d*x]])/d + (a*(Log[Cosh[c + d*x]] + Log[Tanh[c + d*x]]))/d","A",1
140,1,32,18,0.0269371,"\int \coth ^2(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Coth[c + d*x]^2*(a + b*Tanh[c + d*x]^2),x]","b x-\frac{a \coth (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\tanh ^2(c+d x)\right)}{d}","x (a+b)-\frac{a \coth (c+d x)}{d}",1,"b*x - (a*Coth[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, Tanh[c + d*x]^2])/d","C",1
141,1,39,31,0.1215496,"\int \coth ^3(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Coth[c + d*x]^3*(a + b*Tanh[c + d*x]^2),x]","\frac{2 (a+b) (\log (\tanh (c+d x))+\log (\cosh (c+d x)))-a \coth ^2(c+d x)}{2 d}","\frac{(a+b) \log (\sinh (c+d x))}{d}-\frac{a \coth ^2(c+d x)}{2 d}",1,"(-(a*Coth[c + d*x]^2) + 2*(a + b)*(Log[Cosh[c + d*x]] + Log[Tanh[c + d*x]]))/(2*d)","A",1
142,1,61,36,0.0399866,"\int \coth ^4(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Coth[c + d*x]^4*(a + b*Tanh[c + d*x]^2),x]","-\frac{a \coth ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\tanh ^2(c+d x)\right)}{3 d}-\frac{b \coth (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\tanh ^2(c+d x)\right)}{d}","-\frac{(a+b) \coth (c+d x)}{d}+x (a+b)-\frac{a \coth ^3(c+d x)}{3 d}",1,"-1/3*(a*Coth[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, Tanh[c + d*x]^2])/d - (b*Coth[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, Tanh[c + d*x]^2])/d","C",1
143,1,51,49,0.3350515,"\int \coth ^5(c+d x) \left(a+b \tanh ^2(c+d x)\right) \, dx","Integrate[Coth[c + d*x]^5*(a + b*Tanh[c + d*x]^2),x]","-\frac{2 (a+b) \coth ^2(c+d x)-4 (a+b) (\log (\tanh (c+d x))+\log (\cosh (c+d x)))+a \coth ^4(c+d x)}{4 d}","-\frac{(a+b) \coth ^2(c+d x)}{2 d}+\frac{(a+b) \log (\sinh (c+d x))}{d}-\frac{a \coth ^4(c+d x)}{4 d}",1,"-1/4*(2*(a + b)*Coth[c + d*x]^2 + a*Coth[c + d*x]^4 - 4*(a + b)*(Log[Cosh[c + d*x]] + Log[Tanh[c + d*x]]))/d","A",1
144,1,190,83,0.081627,"\int \tanh ^4(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Tanh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{a^2 \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{a^2 \tanh ^3(c+d x)}{3 d}-\frac{a^2 \tanh (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{2 a b \tanh ^5(c+d x)}{5 d}-\frac{2 a b \tanh ^3(c+d x)}{3 d}-\frac{2 a b \tanh (c+d x)}{d}+\frac{b^2 \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{b^2 \tanh ^7(c+d x)}{7 d}-\frac{b^2 \tanh ^5(c+d x)}{5 d}-\frac{b^2 \tanh ^3(c+d x)}{3 d}-\frac{b^2 \tanh (c+d x)}{d}","-\frac{b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac{(a+b)^2 \tanh ^3(c+d x)}{3 d}-\frac{(a+b)^2 \tanh (c+d x)}{d}+x (a+b)^2-\frac{b^2 \tanh ^7(c+d x)}{7 d}",1,"(a^2*ArcTanh[Tanh[c + d*x]])/d + (2*a*b*ArcTanh[Tanh[c + d*x]])/d + (b^2*ArcTanh[Tanh[c + d*x]])/d - (a^2*Tanh[c + d*x])/d - (2*a*b*Tanh[c + d*x])/d - (b^2*Tanh[c + d*x])/d - (a^2*Tanh[c + d*x]^3)/(3*d) - (2*a*b*Tanh[c + d*x]^3)/(3*d) - (b^2*Tanh[c + d*x]^3)/(3*d) - (2*a*b*Tanh[c + d*x]^5)/(5*d) - (b^2*Tanh[c + d*x]^5)/(5*d) - (b^2*Tanh[c + d*x]^7)/(7*d)","B",1
145,1,66,76,0.4260741,"\int \tanh ^3(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Tanh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^2,x]","-\frac{3 b (2 a+b) \tanh ^4(c+d x)+6 (a+b)^2 \tanh ^2(c+d x)-12 (a+b)^2 \log (\cosh (c+d x))+2 b^2 \tanh ^6(c+d x)}{12 d}","-\frac{b (2 a+b) \tanh ^4(c+d x)}{4 d}-\frac{(a+b)^2 \tanh ^2(c+d x)}{2 d}+\frac{(a+b)^2 \log (\cosh (c+d x))}{d}-\frac{b^2 \tanh ^6(c+d x)}{6 d}",1,"-1/12*(-12*(a + b)^2*Log[Cosh[c + d*x]] + 6*(a + b)^2*Tanh[c + d*x]^2 + 3*b*(2*a + b)*Tanh[c + d*x]^4 + 2*b^2*Tanh[c + d*x]^6)/d","A",1
146,1,137,63,0.0529944,"\int \tanh ^2(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Tanh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{a^2 \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{a^2 \tanh (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{2 a b \tanh ^3(c+d x)}{3 d}-\frac{2 a b \tanh (c+d x)}{d}+\frac{b^2 \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{b^2 \tanh ^5(c+d x)}{5 d}-\frac{b^2 \tanh ^3(c+d x)}{3 d}-\frac{b^2 \tanh (c+d x)}{d}","-\frac{b (2 a+b) \tanh ^3(c+d x)}{3 d}-\frac{(a+b)^2 \tanh (c+d x)}{d}+x (a+b)^2-\frac{b^2 \tanh ^5(c+d x)}{5 d}",1,"(a^2*ArcTanh[Tanh[c + d*x]])/d + (2*a*b*ArcTanh[Tanh[c + d*x]])/d + (b^2*ArcTanh[Tanh[c + d*x]])/d - (a^2*Tanh[c + d*x])/d - (2*a*b*Tanh[c + d*x])/d - (b^2*Tanh[c + d*x])/d - (2*a*b*Tanh[c + d*x]^3)/(3*d) - (b^2*Tanh[c + d*x]^3)/(3*d) - (b^2*Tanh[c + d*x]^5)/(5*d)","B",1
147,1,50,57,0.3229347,"\int \tanh (c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Tanh[c + d*x]*(a + b*Tanh[c + d*x]^2)^2,x]","-\frac{2 b (2 a+b) \tanh ^2(c+d x)-4 (a+b)^2 \log (\cosh (c+d x))+b^2 \tanh ^4(c+d x)}{4 d}","-\frac{b (a+b) \tanh ^2(c+d x)}{2 d}-\frac{\left(a+b \tanh ^2(c+d x)\right)^2}{4 d}+\frac{(a+b)^2 \log (\cosh (c+d x))}{d}",1,"-1/4*(-4*(a + b)^2*Log[Cosh[c + d*x]] + 2*b*(2*a + b)*Tanh[c + d*x]^2 + b^2*Tanh[c + d*x]^4)/d","A",1
148,1,65,43,0.7497962,"\int \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\tanh (c+d x) \left(\frac{3 (a+b)^2 \tanh ^{-1}\left(\sqrt{\tanh ^2(c+d x)}\right)}{\sqrt{\tanh ^2(c+d x)}}-b \left(6 a+b \left(\tanh ^2(c+d x)+3\right)\right)\right)}{3 d}","-\frac{b (2 a+b) \tanh (c+d x)}{d}+x (a+b)^2-\frac{b^2 \tanh ^3(c+d x)}{3 d}",1,"(Tanh[c + d*x]*((3*(a + b)^2*ArcTanh[Sqrt[Tanh[c + d*x]^2]])/Sqrt[Tanh[c + d*x]^2] - b*(6*a + b*(3 + Tanh[c + d*x]^2))))/(3*d)","A",1
149,1,48,49,0.1354299,"\int \coth (c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Coth[c + d*x]*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{2 \left(a^2 \log (\tanh (c+d x))+(a+b)^2 \log (\cosh (c+d x))\right)-b^2 \tanh ^2(c+d x)}{2 d}","\frac{a^2 \log (\tanh (c+d x))}{d}+\frac{(a+b)^2 \log (\cosh (c+d x))}{d}-\frac{b^2 \tanh ^2(c+d x)}{2 d}",1,"(2*((a + b)^2*Log[Cosh[c + d*x]] + a^2*Log[Tanh[c + d*x]]) - b^2*Tanh[c + d*x]^2)/(2*d)","A",1
150,1,64,36,0.1068987,"\int \coth ^2(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Coth[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^2,x]","-\frac{a^2 \coth (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\tanh ^2(c+d x)\right)}{d}+2 a b x+\frac{b^2 \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{b^2 \tanh (c+d x)}{d}","-\frac{a^2 \coth (c+d x)}{d}+x (a+b)^2-\frac{b^2 \tanh (c+d x)}{d}",1,"2*a*b*x + (b^2*ArcTanh[Tanh[c + d*x]])/d - (a^2*Coth[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, Tanh[c + d*x]^2])/d - (b^2*Tanh[c + d*x])/d","C",1
151,1,50,52,0.1708904,"\int \coth ^3(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Coth[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^2,x]","\frac{-a^2 \coth ^2(c+d x)+2 a (a+2 b) \log (\tanh (c+d x))+2 (a+b)^2 \log (\cosh (c+d x))}{2 d}","-\frac{a^2 \coth ^2(c+d x)}{2 d}+\frac{a (a+2 b) \log (\tanh (c+d x))}{d}+\frac{(a+b)^2 \log (\cosh (c+d x))}{d}",1,"(-(a^2*Coth[c + d*x]^2) + 2*(a + b)^2*Log[Cosh[c + d*x]] + 2*a*(a + 2*b)*Log[Tanh[c + d*x]])/(2*d)","A",1
152,1,65,43,0.619184,"\int \coth ^4(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Coth[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^2,x]","-\frac{\coth (c+d x) \left(a \left(a \coth ^2(c+d x)+3 a+6 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{a^2 \coth ^3(c+d x)}{3 d}-\frac{a (a+2 b) \coth (c+d x)}{d}+x (a+b)^2",1,"-1/3*(Coth[c + d*x]*(a*(3*a + 6*b + a*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
153,1,58,72,0.4291797,"\int \coth ^5(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Coth[c + d*x]^5*(a + b*Tanh[c + d*x]^2)^2,x]","-\frac{a^2 \coth ^4(c+d x)+2 a (a+2 b) \coth ^2(c+d x)-4 (a+b)^2 (\log (\tanh (c+d x))+\log (\cosh (c+d x)))}{4 d}","-\frac{a^2 \coth ^4(c+d x)}{4 d}-\frac{a (a+2 b) \coth ^2(c+d x)}{2 d}+\frac{(a+b)^2 \log (\tanh (c+d x))}{d}+\frac{(a+b)^2 \log (\cosh (c+d x))}{d}",1,"-1/4*(2*a*(a + 2*b)*Coth[c + d*x]^2 + a^2*Coth[c + d*x]^4 - 4*(a + b)^2*(Log[Cosh[c + d*x]] + Log[Tanh[c + d*x]]))/d","A",1
154,1,98,63,0.0942328,"\int \coth ^6(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Coth[c + d*x]^6*(a + b*Tanh[c + d*x]^2)^2,x]","-\frac{a^2 \coth ^5(c+d x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\tanh ^2(c+d x)\right)}{5 d}-\frac{2 a b \coth ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\tanh ^2(c+d x)\right)}{3 d}-\frac{b^2 \coth (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\tanh ^2(c+d x)\right)}{d}","-\frac{a^2 \coth ^5(c+d x)}{5 d}-\frac{a (a+2 b) \coth ^3(c+d x)}{3 d}-\frac{(a+b)^2 \coth (c+d x)}{d}+x (a+b)^2",1,"-1/5*(a^2*Coth[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, Tanh[c + d*x]^2])/d - (2*a*b*Coth[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, Tanh[c + d*x]^2])/(3*d) - (b^2*Coth[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, Tanh[c + d*x]^2])/d","C",1
155,1,74,92,0.5172681,"\int \coth ^7(c+d x) \left(a+b \tanh ^2(c+d x)\right)^2 \, dx","Integrate[Coth[c + d*x]^7*(a + b*Tanh[c + d*x]^2)^2,x]","-\frac{2 a^2 \coth ^6(c+d x)+3 a (a+2 b) \coth ^4(c+d x)+6 (a+b)^2 \coth ^2(c+d x)-12 (a+b)^2 (\log (\tanh (c+d x))+\log (\cosh (c+d x)))}{12 d}","-\frac{a^2 \coth ^6(c+d x)}{6 d}-\frac{a (a+2 b) \coth ^4(c+d x)}{4 d}-\frac{(a+b)^2 \coth ^2(c+d x)}{2 d}+\frac{(a+b)^2 \log (\tanh (c+d x))}{d}+\frac{(a+b)^2 \log (\cosh (c+d x))}{d}",1,"-1/12*(6*(a + b)^2*Coth[c + d*x]^2 + 3*a*(a + 2*b)*Coth[c + d*x]^4 + 2*a^2*Coth[c + d*x]^6 - 12*(a + b)^2*(Log[Cosh[c + d*x]] + Log[Tanh[c + d*x]]))/d","A",1
156,1,123,114,1.5189876,"\int \tanh ^4(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Tanh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\tanh (c+d x) \left(-63 b \left(3 a^2+3 a b+b^2\right) \tanh ^4(c+d x)-45 b^2 (3 a+b) \tanh ^6(c+d x)-105 (a+b)^3 \tanh ^2(c+d x)+\frac{315 (a+b)^3 \tanh ^{-1}\left(\sqrt{\tanh ^2(c+d x)}\right)}{\sqrt{\tanh ^2(c+d x)}}-315 (a+b)^3-35 b^3 \tanh ^8(c+d x)\right)}{315 d}","-\frac{b \left(3 a^2+3 a b+b^2\right) \tanh ^5(c+d x)}{5 d}-\frac{b^2 (3 a+b) \tanh ^7(c+d x)}{7 d}-\frac{(a+b)^3 \tanh ^3(c+d x)}{3 d}-\frac{(a+b)^3 \tanh (c+d x)}{d}+x (a+b)^3-\frac{b^3 \tanh ^9(c+d x)}{9 d}",1,"(Tanh[c + d*x]*(-315*(a + b)^3 - 105*(a + b)^3*Tanh[c + d*x]^2 - 63*b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^4 - 45*b^2*(3*a + b)*Tanh[c + d*x]^6 - 35*b^3*Tanh[c + d*x]^8 + (315*(a + b)^3*ArcTanh[Sqrt[Tanh[c + d*x]^2]])/Sqrt[Tanh[c + d*x]^2]))/(315*d)","A",1
157,1,98,107,0.3126746,"\int \tanh ^3(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Tanh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{-\frac{1}{2} b \left(3 a^2+3 a b+b^2\right) \tanh ^4(c+d x)-\frac{1}{3} b^2 (3 a+b) \tanh ^6(c+d x)-(a+b)^3 \tanh ^2(c+d x)+2 (a+b)^3 \log (\cosh (c+d x))-\frac{1}{4} b^3 \tanh ^8(c+d x)}{2 d}","-\frac{b \left(3 a^2+3 a b+b^2\right) \tanh ^4(c+d x)}{4 d}-\frac{b^2 (3 a+b) \tanh ^6(c+d x)}{6 d}-\frac{(a+b)^3 \tanh ^2(c+d x)}{2 d}+\frac{(a+b)^3 \log (\cosh (c+d x))}{d}-\frac{b^3 \tanh ^8(c+d x)}{8 d}",1,"(2*(a + b)^3*Log[Cosh[c + d*x]] - (a + b)^3*Tanh[c + d*x]^2 - (b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^4)/2 - (b^2*(3*a + b)*Tanh[c + d*x]^6)/3 - (b^3*Tanh[c + d*x]^8)/4)/(2*d)","A",1
158,1,108,94,1.7241244,"\int \tanh ^2(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Tanh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\tanh (c+d x) \left(-35 b \left(3 a^2+3 a b+b^2\right) \tanh ^2(c+d x)-21 b^2 (3 a+b) \tanh ^4(c+d x)+\frac{105 (a+b)^3 \tanh ^{-1}\left(\sqrt{\tanh ^2(c+d x)}\right)}{\sqrt{\tanh ^2(c+d x)}}-105 (a+b)^3-15 b^3 \tanh ^6(c+d x)\right)}{105 d}","-\frac{b \left(3 a^2+3 a b+b^2\right) \tanh ^3(c+d x)}{3 d}-\frac{b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac{(a+b)^3 \tanh (c+d x)}{d}+x (a+b)^3-\frac{b^3 \tanh ^7(c+d x)}{7 d}",1,"(Tanh[c + d*x]*(-105*(a + b)^3 - 35*b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^2 - 21*b^2*(3*a + b)*Tanh[c + d*x]^4 - 15*b^3*Tanh[c + d*x]^6 + (105*(a + b)^3*ArcTanh[Sqrt[Tanh[c + d*x]^2]])/Sqrt[Tanh[c + d*x]^2]))/(105*d)","A",1
159,1,76,83,0.2490651,"\int \tanh (c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Tanh[c + d*x]*(a + b*Tanh[c + d*x]^2)^3,x]","-\frac{b (a+b)^2 \tanh ^2(c+d x)+\frac{1}{2} (a+b) \left(a+b \tanh ^2(c+d x)\right)^2+\frac{1}{3} \left(a+b \tanh ^2(c+d x)\right)^3-2 (a+b)^3 \log (\cosh (c+d x))}{2 d}","-\frac{b (a+b)^2 \tanh ^2(c+d x)}{2 d}-\frac{(a+b) \left(a+b \tanh ^2(c+d x)\right)^2}{4 d}-\frac{\left(a+b \tanh ^2(c+d x)\right)^3}{6 d}+\frac{(a+b)^3 \log (\cosh (c+d x))}{d}",1,"-1/2*(-2*(a + b)^3*Log[Cosh[c + d*x]] + b*(a + b)^2*Tanh[c + d*x]^2 + ((a + b)*(a + b*Tanh[c + d*x]^2)^2)/2 + (a + b*Tanh[c + d*x]^2)^3/3)/d","A",1
160,1,95,74,0.6146435,"\int \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\tanh (c+d x) \left(\frac{15 (a+b)^3 \tanh ^{-1}\left(\sqrt{\tanh ^2(c+d x)}\right)}{\sqrt{\tanh ^2(c+d x)}}-b \left(45 a^2+15 a b \left(\tanh ^2(c+d x)+3\right)+b^2 \left(3 \tanh ^4(c+d x)+5 \tanh ^2(c+d x)+15\right)\right)\right)}{15 d}","-\frac{b \left(3 a^2+3 a b+b^2\right) \tanh (c+d x)}{d}-\frac{b^2 (3 a+b) \tanh ^3(c+d x)}{3 d}+x (a+b)^3-\frac{b^3 \tanh ^5(c+d x)}{5 d}",1,"(Tanh[c + d*x]*((15*(a + b)^3*ArcTanh[Sqrt[Tanh[c + d*x]^2]])/Sqrt[Tanh[c + d*x]^2] - b*(45*a^2 + 15*a*b*(3 + Tanh[c + d*x]^2) + b^2*(15 + 5*Tanh[c + d*x]^2 + 3*Tanh[c + d*x]^4))))/(15*d)","A",1
161,1,67,72,0.546585,"\int \coth (c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Coth[c + d*x]*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{2 a^3 \log (\tanh (c+d x))-b^2 (3 a+b) \tanh ^2(c+d x)+2 (a+b)^3 \log (\cosh (c+d x))-\frac{1}{2} b^3 \tanh ^4(c+d x)}{2 d}","\frac{a^3 \log (\tanh (c+d x))}{d}-\frac{b^2 (3 a+b) \tanh ^2(c+d x)}{2 d}+\frac{(a+b)^3 \log (\cosh (c+d x))}{d}-\frac{b^3 \tanh ^4(c+d x)}{4 d}",1,"(2*(a + b)^3*Log[Cosh[c + d*x]] + 2*a^3*Log[Tanh[c + d*x]] - b^2*(3*a + b)*Tanh[c + d*x]^2 - (b^3*Tanh[c + d*x]^4)/2)/(2*d)","A",1
162,1,81,59,2.3119498,"\int \coth ^2(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Coth[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\tanh (c+d x) \left(-3 a^3 \coth ^2(c+d x)-b^2 \left(9 a+b \tanh ^2(c+d x)+3 b\right)+3 (a+b)^3 \sqrt{\coth ^2(c+d x)} \tanh ^{-1}\left(\sqrt{\coth ^2(c+d x)}\right)\right)}{3 d}","-\frac{a^3 \coth (c+d x)}{d}-\frac{b^2 (3 a+b) \tanh (c+d x)}{d}+x (a+b)^3-\frac{b^3 \tanh ^3(c+d x)}{3 d}",1,"(Tanh[c + d*x]*(-3*a^3*Coth[c + d*x]^2 + 3*(a + b)^3*ArcTanh[Sqrt[Coth[c + d*x]^2]]*Sqrt[Coth[c + d*x]^2] - b^2*(9*a + 3*b + b*Tanh[c + d*x]^2)))/(3*d)","A",1
163,1,63,72,0.4619947,"\int \coth ^3(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Coth[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3,x]","-\frac{a^3 \coth ^2(c+d x)-2 a^2 (a+3 b) \log (\tanh (c+d x))-2 (a+b)^3 \log (\cosh (c+d x))+b^3 \tanh ^2(c+d x)}{2 d}","-\frac{a^3 \coth ^2(c+d x)}{2 d}+\frac{a^2 (a+3 b) \log (\tanh (c+d x))}{d}+\frac{(a+b)^3 \log (\cosh (c+d x))}{d}-\frac{b^3 \tanh ^2(c+d x)}{2 d}",1,"-1/2*(a^3*Coth[c + d*x]^2 - 2*(a + b)^3*Log[Cosh[c + d*x]] - 2*a^2*(a + 3*b)*Log[Tanh[c + d*x]] + b^3*Tanh[c + d*x]^2)/d","A",1
164,1,82,59,1.283771,"\int \coth ^4(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Coth[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\tanh (c+d x) \left(-a^3 \coth ^4(c+d x)-3 a^2 (a+3 b) \coth ^2(c+d x)+3 (a+b)^3 \sqrt{\coth ^2(c+d x)} \tanh ^{-1}\left(\sqrt{\coth ^2(c+d x)}\right)-3 b^3\right)}{3 d}","-\frac{a^3 \coth ^3(c+d x)}{3 d}-\frac{a^2 (a+3 b) \coth (c+d x)}{d}+x (a+b)^3-\frac{b^3 \tanh (c+d x)}{d}",1,"((-3*b^3 - 3*a^2*(a + 3*b)*Coth[c + d*x]^2 - a^3*Coth[c + d*x]^4 + 3*(a + b)^3*ArcTanh[Sqrt[Coth[c + d*x]^2]]*Sqrt[Coth[c + d*x]^2])*Tanh[c + d*x])/(3*d)","A",1
165,1,67,83,0.5832629,"\int \coth ^5(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Coth[c + d*x]^5*(a + b*Tanh[c + d*x]^2)^3,x]","\frac{-\frac{1}{2} a^3 \coth ^4(c+d x)-a^2 (a+3 b) \coth ^2(c+d x)+2 (a+b)^3 \log (\sinh (c+d x))-2 b^3 \log (\tanh (c+d x))}{2 d}","-\frac{a^3 \coth ^4(c+d x)}{4 d}+\frac{a \left(a^2+3 a b+3 b^2\right) \log (\tanh (c+d x))}{d}-\frac{a^2 (a+3 b) \coth ^2(c+d x)}{2 d}+\frac{(a+b)^3 \log (\cosh (c+d x))}{d}",1,"(-(a^2*(a + 3*b)*Coth[c + d*x]^2) - (a^3*Coth[c + d*x]^4)/2 + 2*(a + b)^3*Log[Sinh[c + d*x]] - 2*b^3*Log[Tanh[c + d*x]])/(2*d)","A",1
166,1,100,74,1.7805872,"\int \coth ^6(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Coth[c + d*x]^6*(a + b*Tanh[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{a \coth (c+d x) \left(15 \left(a^2+3 a b+3 b^2\right)+3 a^2 \coth ^4(c+d x)+5 a (a+3 b) \coth ^2(c+d x)\right)}{15 d}","-\frac{a^3 \coth ^5(c+d x)}{5 d}-\frac{a \left(a^2+3 a b+3 b^2\right) \coth (c+d x)}{d}-\frac{a^2 (a+3 b) \coth ^3(c+d x)}{3 d}+x (a+b)^3",1,"-1/15*(a*Coth[c + d*x]*(15*(a^2 + 3*a*b + 3*b^2) + 5*a*(a + 3*b)*Coth[c + d*x]^2 + 3*a^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
167,1,76,103,0.254008,"\int \coth ^7(c+d x) \left(a+b \tanh ^2(c+d x)\right)^3 \, dx","Integrate[Coth[c + d*x]^7*(a + b*Tanh[c + d*x]^2)^3,x]","-\frac{a (a+b)^2 \coth ^2(c+d x)+\frac{1}{2} (a+b) \left(a \coth ^2(c+d x)+b\right)^2+\frac{1}{3} \left(a \coth ^2(c+d x)+b\right)^3-2 (a+b)^3 \log (\sinh (c+d x))}{2 d}","-\frac{a^3 \coth ^6(c+d x)}{6 d}-\frac{a \left(a^2+3 a b+3 b^2\right) \coth ^2(c+d x)}{2 d}-\frac{a^2 (a+3 b) \coth ^4(c+d x)}{4 d}+\frac{(a+b)^3 \log (\tanh (c+d x))}{d}+\frac{(a+b)^3 \log (\cosh (c+d x))}{d}",1,"-1/2*(a*(a + b)^2*Coth[c + d*x]^2 + ((a + b)*(b + a*Coth[c + d*x]^2)^2)/2 + (b + a*Coth[c + d*x]^2)^3/3 - 2*(a + b)^3*Log[Sinh[c + d*x]])/d","A",1
168,1,128,110,1.7929752,"\int \left(a+b \tanh ^2(c+d x)\right)^4 \, dx","Integrate[(a + b*Tanh[c + d*x]^2)^4,x]","\frac{\tanh (c+d x) \left(\frac{105 (a+b)^4 \tanh ^{-1}\left(\sqrt{\tanh ^2(c+d x)}\right)}{\sqrt{\tanh ^2(c+d x)}}-b \left(35 b \left(6 a^2+4 a b+b^2\right) \tanh ^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) \tanh ^4(c+d x)+15 b^3 \tanh ^6(c+d x)\right)\right)}{105 d}","-\frac{b^2 \left(6 a^2+4 a b+b^2\right) \tanh ^3(c+d x)}{3 d}-\frac{b (2 a+b) \left(2 a^2+2 a b+b^2\right) \tanh (c+d x)}{d}-\frac{b^3 (4 a+b) \tanh ^5(c+d x)}{5 d}+x (a+b)^4-\frac{b^4 \tanh ^7(c+d x)}{7 d}",1,"(Tanh[c + d*x]*((105*(a + b)^4*ArcTanh[Sqrt[Tanh[c + d*x]^2]])/Sqrt[Tanh[c + d*x]^2] - 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)*Tanh[c + d*x]^2 + 21*b^2*(4*a + b)*Tanh[c + d*x]^4 + 15*b^3*Tanh[c + d*x]^6)))/(105*d)","A",1
169,1,170,160,2.3381713,"\int \left(a+b \tanh ^2(c+d x)\right)^5 \, dx","Integrate[(a + b*Tanh[c + d*x]^2)^5,x]","\frac{\tanh (c+d x) \left(\frac{315 (a+b)^5 \tanh ^{-1}\left(\sqrt{\tanh ^2(c+d x)}\right)}{\sqrt{\tanh ^2(c+d x)}}-b \left(63 b^2 \left(10 a^2+5 a b+b^2\right) \tanh ^4(c+d x)+105 b \left(10 a^3+10 a^2 b+5 a b^2+b^3\right) \tanh ^2(c+d x)+315 \left(5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right)+45 b^3 (5 a+b) \tanh ^6(c+d x)+35 b^4 \tanh ^8(c+d x)\right)\right)}{315 d}","-\frac{b^3 \left(10 a^2+5 a b+b^2\right) \tanh ^5(c+d x)}{5 d}-\frac{b^2 \left(10 a^3+10 a^2 b+5 a b^2+b^3\right) \tanh ^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) \tanh (c+d x)}{d}-\frac{b^4 (5 a+b) \tanh ^7(c+d x)}{7 d}+x (a+b)^5-\frac{b^5 \tanh ^9(c+d x)}{9 d}",1,"(Tanh[c + d*x]*((315*(a + b)^5*ArcTanh[Sqrt[Tanh[c + d*x]^2]])/Sqrt[Tanh[c + d*x]^2] - b*(315*(5*a^4 + 10*a^3*b + 10*a^2*b^2 + 5*a*b^3 + b^4) + 105*b*(10*a^3 + 10*a^2*b + 5*a*b^2 + b^3)*Tanh[c + d*x]^2 + 63*b^2*(10*a^2 + 5*a*b + b^2)*Tanh[c + d*x]^4 + 45*b^3*(5*a + b)*Tanh[c + d*x]^6 + 35*b^4*Tanh[c + d*x]^8)))/(315*d)","A",1
170,1,60,66,0.1797644,"\int \frac{\tanh ^5(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Tanh[c + d*x]^5/(a + b*Tanh[c + d*x]^2),x]","-\frac{-\frac{a^2 \log \left(a+b \tanh ^2(c+d x)\right)}{b^2 (a+b)}-\frac{2 \log (\cosh (c+d x))}{a+b}+\frac{\tanh ^2(c+d x)}{b}}{2 d}","\frac{a^2 \log \left(a+b \tanh ^2(c+d x)\right)}{2 b^2 d (a+b)}+\frac{\log (\cosh (c+d x))}{d (a+b)}-\frac{\tanh ^2(c+d x)}{2 b d}",1,"-1/2*((-2*Log[Cosh[c + d*x]])/(a + b) - (a^2*Log[a + b*Tanh[c + d*x]^2])/(b^2*(a + b)) + Tanh[c + d*x]^2/b)/d","A",1
171,1,66,59,0.1985754,"\int \frac{\tanh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Tanh[c + d*x]^4/(a + b*Tanh[c + d*x]^2),x]","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{b^{3/2} d (a+b)}+\frac{c+d x}{d (a+b)}-\frac{\tanh (c+d x)}{b d}","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{b^{3/2} d (a+b)}+\frac{x}{a+b}-\frac{\tanh (c+d x)}{b d}",1,"(c + d*x)/((a + b)*d) + (a^(3/2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(b^(3/2)*(a + b)*d) - Tanh[c + d*x]/(b*d)","A",1
172,1,42,46,0.0386914,"\int \frac{\tanh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Tanh[c + d*x]^3/(a + b*Tanh[c + d*x]^2),x]","\frac{2 b \log (\cosh (c+d x))-a \log \left(a+b \tanh ^2(c+d x)\right)}{2 a b d+2 b^2 d}","\frac{\log (\cosh (c+d x))}{d (a+b)}-\frac{a \log \left(a+b \tanh ^2(c+d x)\right)}{2 b d (a+b)}",1,"(2*b*Log[Cosh[c + d*x]] - a*Log[a + b*Tanh[c + d*x]^2])/(2*a*b*d + 2*b^2*d)","A",1
173,1,47,46,0.030563,"\int \frac{\tanh ^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Tanh[c + d*x]^2/(a + b*Tanh[c + d*x]^2),x]","\frac{\tanh ^{-1}(\tanh (c+d x))-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{b}}}{d (a+b)}","\frac{x}{a+b}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{b} d (a+b)}",1,"(-((Sqrt[a]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/Sqrt[b]) + ArcTanh[Tanh[c + d*x]])/((a + b)*d)","A",1
174,1,35,42,0.0366427,"\int \frac{\tanh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Tanh[c + d*x]/(a + b*Tanh[c + d*x]^2),x]","\frac{\log \left(a+b \tanh ^2(c+d x)\right)+2 \log (\cosh (c+d x))}{2 a d+2 b d}","\frac{\log \left(a+b \tanh ^2(c+d x)\right)}{2 d (a+b)}+\frac{\log (\cosh (c+d x))}{d (a+b)}",1,"(2*Log[Cosh[c + d*x]] + Log[a + b*Tanh[c + d*x]^2])/(2*a*d + 2*b*d)","A",1
175,1,65,45,0.0844242,"\int \frac{1}{a+b \tanh ^2(c+d x)} \, dx","Integrate[(a + b*Tanh[c + d*x]^2)^(-1),x]","\frac{\frac{2 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a}}-\log (1-\tanh (c+d x))+\log (\tanh (c+d x)+1)}{2 a d+2 b d}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a+b)}+\frac{x}{a+b}",1,"((2*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/Sqrt[a] - Log[1 - Tanh[c + d*x]] + Log[1 + Tanh[c + d*x]])/(2*a*d + 2*b*d)","A",1
176,1,54,60,0.0757002,"\int \frac{\coth (c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Coth[c + d*x]/(a + b*Tanh[c + d*x]^2),x]","\frac{-b \log \left(a+b \tanh ^2(c+d x)\right)+2 (a+b) \log (\tanh (c+d x))+2 a \log (\cosh (c+d x))}{2 a d (a+b)}","-\frac{b \log \left(a+b \tanh ^2(c+d x)\right)}{2 a d (a+b)}+\frac{\log (\cosh (c+d x))}{d (a+b)}+\frac{\log (\tanh (c+d x))}{a d}",1,"(2*a*Log[Cosh[c + d*x]] + 2*(a + b)*Log[Tanh[c + d*x]] - b*Log[a + b*Tanh[c + d*x]^2])/(2*a*(a + b)*d)","A",1
177,1,67,60,0.1911414,"\int \frac{\coth ^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Coth[c + d*x]^2/(a + b*Tanh[c + d*x]^2),x]","-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{3/2} d (a+b)}+\frac{c+d x}{d (a+b)}-\frac{\coth (c+d x)}{a d}","-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{3/2} d (a+b)}+\frac{x}{a+b}-\frac{\coth (c+d x)}{a d}",1,"(c + d*x)/((a + b)*d) - (b^(3/2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*(a + b)*d) - Coth[c + d*x]/(a*d)","A",1
178,1,60,85,0.1821896,"\int \frac{\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Coth[c + d*x]^3/(a + b*Tanh[c + d*x]^2),x]","-\frac{-\frac{b^2 \log \left(a \coth ^2(c+d x)+b\right)}{a^2 (a+b)}-\frac{2 \log (\sinh (c+d x))}{a+b}+\frac{\coth ^2(c+d x)}{a}}{2 d}","\frac{b^2 \log \left(a+b \tanh ^2(c+d x)\right)}{2 a^2 d (a+b)}+\frac{(a-b) \log (\tanh (c+d x))}{a^2 d}+\frac{\log (\cosh (c+d x))}{d (a+b)}-\frac{\coth ^2(c+d x)}{2 a d}",1,"-1/2*(Coth[c + d*x]^2/a - (b^2*Log[b + a*Coth[c + d*x]^2])/(a^2*(a + b)) - (2*Log[Sinh[c + d*x]])/(a + b))/d","A",1
179,1,91,82,0.7098064,"\int \frac{\coth ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx","Integrate[Coth[c + d*x]^4/(a + b*Tanh[c + d*x]^2),x]","\frac{\frac{6 \left(\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{5/2}}+c+d x\right)}{a+b}-\frac{\coth (c+d x) \text{csch}^2(c+d x) ((4 a-3 b) \cosh (2 (c+d x))-2 a+3 b)}{a^2}}{6 d}","\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{5/2} d (a+b)}-\frac{(a-b) \coth (c+d x)}{a^2 d}+\frac{x}{a+b}-\frac{\coth ^3(c+d x)}{3 a d}",1,"((6*(c + d*x + (b^(5/2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/a^(5/2)))/(a + b) - ((-2*a + 3*b + (4*a - 3*b)*Cosh[2*(c + d*x)])*Coth[c + d*x]*Csch[c + d*x]^2)/a^2)/(6*d)","A",1
180,1,69,83,0.5603449,"\int \frac{\tanh ^5(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Tanh[c + d*x]^5/(a + b*Tanh[c + d*x]^2)^2,x]","-\frac{\frac{a^2 (a+b)}{b^2 \left(a+b \tanh ^2(c+d x)\right)}+\frac{a (a+2 b) \log \left(a+b \tanh ^2(c+d x)\right)}{b^2}-2 \log (\cosh (c+d x))}{2 d (a+b)^2}","-\frac{a^2}{2 b^2 d (a+b) \left(a+b \tanh ^2(c+d x)\right)}-\frac{a (a+2 b) \log \left(a+b \tanh ^2(c+d x)\right)}{2 b^2 d (a+b)^2}+\frac{\log (\cosh (c+d x))}{d (a+b)^2}",1,"-1/2*(-2*Log[Cosh[c + d*x]] + (a*(a + 2*b)*Log[a + b*Tanh[c + d*x]^2])/b^2 + (a^2*(a + b))/(b^2*(a + b*Tanh[c + d*x]^2)))/((a + b)^2*d)","A",1
181,1,90,89,0.5993727,"\int \frac{\tanh ^4(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Tanh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{-\frac{\sqrt{a} (a+3 b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{b^{3/2}}+\frac{a (a+b) \sinh (2 (c+d x))}{b ((a+b) \cosh (2 (c+d x))+a-b)}+2 (c+d x)}{2 d (a+b)^2}","-\frac{\sqrt{a} (a+3 b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 b^{3/2} d (a+b)^2}+\frac{a \tanh (c+d x)}{2 b d (a+b) \left(a+b \tanh ^2(c+d x)\right)}+\frac{x}{(a+b)^2}",1,"(2*(c + d*x) - (Sqrt[a]*(a + 3*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/b^(3/2) + (a*(a + b)*Sinh[2*(c + d*x)])/(b*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(2*(a + b)^2*d)","A",1
182,1,57,72,0.5304618,"\int \frac{\tanh ^3(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Tanh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\frac{a (a+b)}{b \left(a+b \tanh ^2(c+d x)\right)}+\log \left(a+b \tanh ^2(c+d x)\right)+2 \log (\cosh (c+d x))}{2 d (a+b)^2}","\frac{a}{2 b d (a+b) \left(a+b \tanh ^2(c+d x)\right)}+\frac{\log \left(a+b \tanh ^2(c+d x)\right)}{2 d (a+b)^2}+\frac{\log (\cosh (c+d x))}{d (a+b)^2}",1,"(2*Log[Cosh[c + d*x]] + Log[a + b*Tanh[c + d*x]^2] + (a*(a + b))/(b*(a + b*Tanh[c + d*x]^2)))/(2*(a + b)^2*d)","A",1
183,1,86,85,0.4723889,"\int \frac{\tanh ^2(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Tanh[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\frac{(b-a) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}-\frac{(a+b) \sinh (2 (c+d x))}{(a+b) \cosh (2 (c+d x))+a-b}+2 (c+d x)}{2 d (a+b)^2}","-\frac{\tanh (c+d x)}{2 d (a+b) \left(a+b \tanh ^2(c+d x)\right)}-\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b} d (a+b)^2}+\frac{x}{(a+b)^2}",1,"(2*(c + d*x) + ((-a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) - ((a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)]))/(2*(a + b)^2*d)","A",1
184,1,55,68,0.4656743,"\int \frac{\tanh (c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Tanh[c + d*x]/(a + b*Tanh[c + d*x]^2)^2,x]","-\frac{\frac{a+b}{a+b \tanh ^2(c+d x)}-\log \left(a+b \tanh ^2(c+d x)\right)-2 \log (\cosh (c+d x))}{2 d (a+b)^2}","-\frac{1}{2 d (a+b) \left(a+b \tanh ^2(c+d x)\right)}+\frac{\log \left(a+b \tanh ^2(c+d x)\right)}{2 d (a+b)^2}+\frac{\log (\cosh (c+d x))}{d (a+b)^2}",1,"-1/2*(-2*Log[Cosh[c + d*x]] - Log[a + b*Tanh[c + d*x]^2] + (a + b)/(a + b*Tanh[c + d*x]^2))/((a + b)^2*d)","A",1
185,1,97,89,0.5784092,"\int \frac{1}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[(a + b*Tanh[c + d*x]^2)^(-2),x]","\frac{\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{3/2}}+\frac{b (a+b) \tanh (c+d x)}{a \left(a+b \tanh ^2(c+d x)\right)}-\log (1-\tanh (c+d x))+\log (\tanh (c+d x)+1)}{2 d (a+b)^2}","\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a+b)^2}+\frac{b \tanh (c+d x)}{2 a d (a+b) \left(a+b \tanh ^2(c+d x)\right)}+\frac{x}{(a+b)^2}",1,"((Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/a^(3/2) - Log[1 - Tanh[c + d*x]] + Log[1 + Tanh[c + d*x]] + (b*(a + b)*Tanh[c + d*x])/(a*(a + b*Tanh[c + d*x]^2)))/(2*(a + b)^2*d)","A",1
186,1,83,95,2.16421,"\int \frac{\coth (c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Coth[c + d*x]/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\frac{\frac{b \left(\frac{a (a+b)}{a+b \tanh ^2(c+d x)}-(2 a+b) \log \left(a+b \tanh ^2(c+d x)\right)\right)}{(a+b)^2}+2 \log (\tanh (c+d x))}{a^2}+\frac{2 \log (\cosh (c+d x))}{(a+b)^2}}{2 d}","-\frac{b (2 a+b) \log \left(a+b \tanh ^2(c+d x)\right)}{2 a^2 d (a+b)^2}+\frac{\log (\tanh (c+d x))}{a^2 d}+\frac{b}{2 a d (a+b) \left(a+b \tanh ^2(c+d x)\right)}+\frac{\log (\cosh (c+d x))}{d (a+b)^2}",1,"((2*Log[Cosh[c + d*x]])/(a + b)^2 + (2*Log[Tanh[c + d*x]] + (b*(-((2*a + b)*Log[a + b*Tanh[c + d*x]^2]) + (a*(a + b))/(a + b*Tanh[c + d*x]^2)))/(a + b)^2)/a^2)/(2*d)","A",1
187,1,111,119,1.8633509,"\int \frac{\coth ^2(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Coth[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^2,x]","-\frac{\frac{b^{3/2} (5 a+3 b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{5/2} (a+b)^2}+\frac{b^2 \sinh (2 (c+d x))}{a^2 (a+b) ((a+b) \cosh (2 (c+d x))+a-b)}+\frac{2 \coth (c+d x)}{a^2}-\frac{2 (c+d x)}{(a+b)^2}}{2 d}","-\frac{b^{3/2} (5 a+3 b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{5/2} d (a+b)^2}-\frac{(2 a+3 b) \coth (c+d x)}{2 a^2 d (a+b)}+\frac{b \coth (c+d x)}{2 a d (a+b) \left(a+b \tanh ^2(c+d x)\right)}+\frac{x}{(a+b)^2}",1,"-1/2*((-2*(c + d*x))/(a + b)^2 + (b^(3/2)*(5*a + 3*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(5/2)*(a + b)^2) + (2*Coth[c + d*x])/a^2 + (b^2*Sinh[2*(c + d*x)])/(a^2*(a + b)*(a - b + (a + b)*Cosh[2*(c + d*x)])))/d","A",1
188,1,93,124,0.9199947,"\int \frac{\coth ^3(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Coth[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\frac{b^3}{a^3 (a+b) \left(a \coth ^2(c+d x)+b\right)}+\frac{b^2 (3 a+2 b) \log \left(a \coth ^2(c+d x)+b\right)}{a^3 (a+b)^2}-\frac{\coth ^2(c+d x)}{a^2}+\frac{2 \log (\sinh (c+d x))}{(a+b)^2}}{2 d}","\frac{b^2 (3 a+2 b) \log \left(a+b \tanh ^2(c+d x)\right)}{2 a^3 d (a+b)^2}+\frac{(a-2 b) \log (\tanh (c+d x))}{a^3 d}-\frac{b^2}{2 a^2 d (a+b) \left(a+b \tanh ^2(c+d x)\right)}-\frac{\coth ^2(c+d x)}{2 a^2 d}+\frac{\log (\cosh (c+d x))}{d (a+b)^2}",1,"(-(Coth[c + d*x]^2/a^2) + b^3/(a^3*(a + b)*(b + a*Coth[c + d*x]^2)) + (b^2*(3*a + 2*b)*Log[b + a*Coth[c + d*x]^2])/(a^3*(a + b)^2) + (2*Log[Sinh[c + d*x]])/(a + b)^2)/(2*d)","A",1
189,1,139,159,1.5645628,"\int \frac{\coth ^4(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^2} \, dx","Integrate[Coth[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2,x]","\frac{\frac{3 b^{5/2} (7 a+5 b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{7/2} (a+b)^2}+\frac{3 b^3 \sinh (2 (c+d x))}{a^3 (a+b) ((a+b) \cosh (2 (c+d x))+a-b)}+\frac{4 (3 b-2 a) \coth (c+d x)}{a^3}-\frac{2 \coth (c+d x) \text{csch}^2(c+d x)}{a^2}+\frac{6 (c+d x)}{(a+b)^2}}{6 d}","\frac{b^{5/2} (7 a+5 b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{7/2} d (a+b)^2}-\frac{(2 a+5 b) \coth ^3(c+d x)}{6 a^2 d (a+b)}-\frac{\left(2 a^2-2 a b-5 b^2\right) \coth (c+d x)}{2 a^3 d (a+b)}+\frac{b \coth ^3(c+d x)}{2 a d (a+b) \left(a+b \tanh ^2(c+d x)\right)}+\frac{x}{(a+b)^2}",1,"((6*(c + d*x))/(a + b)^2 + (3*b^(5/2)*(7*a + 5*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(7/2)*(a + b)^2) + (4*(-2*a + 3*b)*Coth[c + d*x])/a^3 - (2*Coth[c + d*x]*Csch[c + d*x]^2)/a^2 + (3*b^3*Sinh[2*(c + d*x)])/(a^3*(a + b)*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(6*d)","A",1
190,1,144,144,1.292677,"\int \frac{\tanh ^6(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Tanh[c + d*x]^6/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{-\frac{\sqrt{a} \left(3 a^2+10 a b+15 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{b^{5/2}}-\frac{4 a^2 (a+b) \sinh (2 (c+d x))}{b ((a+b) \cosh (2 (c+d x))+a-b)^2}+\frac{3 a (a+b) (a+3 b) \sinh (2 (c+d x))}{b^2 ((a+b) \cosh (2 (c+d x))+a-b)}+8 (c+d x)}{8 d (a+b)^3}","-\frac{\sqrt{a} \left(3 a^2+10 a b+15 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 b^{5/2} d (a+b)^3}+\frac{a (3 a+7 b) \tanh (c+d x)}{8 b^2 d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)}+\frac{a \tanh ^3(c+d x)}{4 b d (a+b) \left(a+b \tanh ^2(c+d x)\right)^2}+\frac{x}{(a+b)^3}",1,"(8*(c + d*x) - (Sqrt[a]*(3*a^2 + 10*a*b + 15*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/b^(5/2) - (4*a^2*(a + b)*Sinh[2*(c + d*x)])/(b*(a - b + (a + b)*Cosh[2*(c + d*x)])^2) + (3*a*(a + b)*(a + 3*b)*Sinh[2*(c + d*x)])/(b^2*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(8*(a + b)^3*d)","A",1
191,1,91,109,1.0979475,"\int \frac{\tanh ^5(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Tanh[c + d*x]^5/(a + b*Tanh[c + d*x]^2)^3,x]","-\frac{\frac{a^2 (a+b)^2}{b^2 \left(a+b \tanh ^2(c+d x)\right)^2}-\frac{2 a (a+2 b) (a+b)}{b^2 \left(a+b \tanh ^2(c+d x)\right)}-2 \log \left(a+b \tanh ^2(c+d x)\right)-4 \log (\cosh (c+d x))}{4 d (a+b)^3}","-\frac{a^2}{4 b^2 d (a+b) \left(a+b \tanh ^2(c+d x)\right)^2}+\frac{a (a+2 b)}{2 b^2 d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)}+\frac{\log \left(a+b \tanh ^2(c+d x)\right)}{2 d (a+b)^3}+\frac{\log (\cosh (c+d x))}{d (a+b)^3}",1,"-1/4*(-4*Log[Cosh[c + d*x]] - 2*Log[a + b*Tanh[c + d*x]^2] + (a^2*(a + b)^2)/(b^2*(a + b*Tanh[c + d*x]^2)^2) - (2*a*(a + b)*(a + 2*b))/(b^2*(a + b*Tanh[c + d*x]^2)))/((a + b)^3*d)","A",1
192,1,135,137,1.1501513,"\int \frac{\tanh ^4(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Tanh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{-\frac{\left(a^2+6 a b-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}+\frac{(a-5 b) (a+b) \sinh (2 (c+d x))}{b ((a+b) \cosh (2 (c+d x))+a-b)}+\frac{4 a (a+b) \sinh (2 (c+d x))}{((a+b) \cosh (2 (c+d x))+a-b)^2}+8 (c+d x)}{8 d (a+b)^3}","-\frac{\left(a^2+6 a b-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 \sqrt{a} b^{3/2} d (a+b)^3}-\frac{(a+5 b) \tanh (c+d x)}{8 b d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)}+\frac{a \tanh (c+d x)}{4 b d (a+b) \left(a+b \tanh ^2(c+d x)\right)^2}+\frac{x}{(a+b)^3}",1,"(8*(c + d*x) - ((a^2 + 6*a*b - 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(3/2)) + (4*a*(a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)])^2 + ((a - 5*b)*(a + b)*Sinh[2*(c + d*x)])/(b*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(8*(a + b)^3*d)","A",1
193,1,80,98,0.5045525,"\int \frac{\tanh ^3(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Tanh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\frac{a (a+b)^2}{b \left(a+b \tanh ^2(c+d x)\right)^2}-\frac{2 (a+b)}{a+b \tanh ^2(c+d x)}+2 \log \left(a+b \tanh ^2(c+d x)\right)+4 \log (\cosh (c+d x))}{4 d (a+b)^3}","\frac{a}{4 b d (a+b) \left(a+b \tanh ^2(c+d x)\right)^2}-\frac{1}{2 d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)}+\frac{\log \left(a+b \tanh ^2(c+d x)\right)}{2 d (a+b)^3}+\frac{\log (\cosh (c+d x))}{d (a+b)^3}",1,"(4*Log[Cosh[c + d*x]] + 2*Log[a + b*Tanh[c + d*x]^2] + (a*(a + b)^2)/(b*(a + b*Tanh[c + d*x]^2)^2) - (2*(a + b))/(a + b*Tanh[c + d*x]^2))/(4*(a + b)^3*d)","A",1
194,1,137,137,1.1658769,"\int \frac{\tanh ^2(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Tanh[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\frac{\left(-3 a^2+6 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{3/2} \sqrt{b}}-\frac{(5 a-b) (a+b) \sinh (2 (c+d x))}{a ((a+b) \cosh (2 (c+d x))+a-b)}-\frac{4 b (a+b) \sinh (2 (c+d x))}{((a+b) \cosh (2 (c+d x))+a-b)^2}+8 (c+d x)}{8 d (a+b)^3}","-\frac{\left(3 a^2-6 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{3/2} \sqrt{b} d (a+b)^3}-\frac{(3 a-b) \tanh (c+d x)}{8 a d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)}-\frac{\tanh (c+d x)}{4 d (a+b) \left(a+b \tanh ^2(c+d x)\right)^2}+\frac{x}{(a+b)^3}",1,"(8*(c + d*x) + ((-3*a^2 + 6*a*b + b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*Sqrt[b]) - (4*b*(a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)])^2 - ((5*a - b)*(a + b)*Sinh[2*(c + d*x)])/(a*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(8*(a + b)^3*d)","A",1
195,1,77,94,0.5916864,"\int \frac{\tanh (c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Tanh[c + d*x]/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{-\frac{(a+b)^2}{\left(a+b \tanh ^2(c+d x)\right)^2}-\frac{2 (a+b)}{a+b \tanh ^2(c+d x)}+2 \log \left(a+b \tanh ^2(c+d x)\right)+4 \log (\cosh (c+d x))}{4 d (a+b)^3}","-\frac{1}{2 d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)}-\frac{1}{4 d (a+b) \left(a+b \tanh ^2(c+d x)\right)^2}+\frac{\log \left(a+b \tanh ^2(c+d x)\right)}{2 d (a+b)^3}+\frac{\log (\cosh (c+d x))}{d (a+b)^3}",1,"(4*Log[Cosh[c + d*x]] + 2*Log[a + b*Tanh[c + d*x]^2] - (a + b)^2/(a + b*Tanh[c + d*x]^2)^2 - (2*(a + b))/(a + b*Tanh[c + d*x]^2))/(4*(a + b)^3*d)","A",1
196,1,147,142,0.3209402,"\int \frac{1}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[(a + b*Tanh[c + d*x]^2)^(-3),x]","\frac{\frac{b (7 a+3 b) (a+b) \tanh (c+d x)}{a^2 \left(a+b \tanh ^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} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{5/2}}+\frac{2 b (a+b)^2 \tanh (c+d x)}{a \left(a+b \tanh ^2(c+d x)\right)^2}-4 \log (1-\tanh (c+d x))+4 \log (\tanh (c+d x)+1)}{8 d (a+b)^3}","\frac{b (7 a+3 b) \tanh (c+d x)}{8 a^2 d (a+b)^2 \left(a+b \tanh ^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} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a+b)^3}+\frac{b \tanh (c+d x)}{4 a d (a+b) \left(a+b \tanh ^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]*Tanh[c + d*x])/Sqrt[a]])/a^(5/2) - 4*Log[1 - Tanh[c + d*x]] + 4*Log[1 + Tanh[c + d*x]] + (2*b*(a + b)^2*Tanh[c + d*x])/(a*(a + b*Tanh[c + d*x]^2)^2) + (b*(a + b)*(7*a + 3*b)*Tanh[c + d*x])/(a^2*(a + b*Tanh[c + d*x]^2)))/(8*(a + b)^3*d)","A",1
197,1,117,138,1.7562052,"\int \frac{\coth (c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Coth[c + d*x]/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\frac{\frac{b \left(\frac{a (a+b) \left(2 b (2 a+b) \tanh ^2(c+d x)+a (5 a+3 b)\right)}{\left(a+b \tanh ^2(c+d x)\right)^2}-2 \left(3 a^2+3 a b+b^2\right) \log \left(a+b \tanh ^2(c+d x)\right)\right)}{(a+b)^3}+4 \log (\tanh (c+d x))}{a^3}+\frac{4 \log (\cosh (c+d x))}{(a+b)^3}}{4 d}","\frac{\log (\tanh (c+d x))}{a^3 d}+\frac{b (2 a+b)}{2 a^2 d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)}-\frac{b \left(3 a^2+3 a b+b^2\right) \log \left(a+b \tanh ^2(c+d x)\right)}{2 a^3 d (a+b)^3}+\frac{b}{4 a d (a+b) \left(a+b \tanh ^2(c+d x)\right)^2}+\frac{\log (\cosh (c+d x))}{d (a+b)^3}",1,"((4*Log[Cosh[c + d*x]])/(a + b)^3 + (4*Log[Tanh[c + d*x]] + (b*(-2*(3*a^2 + 3*a*b + b^2)*Log[a + b*Tanh[c + d*x]^2] + (a*(a + b)*(a*(5*a + 3*b) + 2*b*(2*a + b)*Tanh[c + d*x]^2))/(a + b*Tanh[c + d*x]^2)^2))/(a + b)^3)/a^3)/(4*d)","A",1
198,1,166,178,6.3107612,"\int \frac{\coth ^2(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Coth[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3,x]","-\frac{\frac{b^2 (13 a+7 b) \sinh (2 (c+d x))}{a^3 (a+b)^2 ((a+b) \cosh (2 (c+d x))+a-b)}+\frac{8 \coth (c+d x)}{a^3}+\frac{4 b^3 \sinh (2 (c+d x))}{a^2 (a+b)^2 ((a+b) \cosh (2 (c+d x))+a-b)^2}+\frac{b^{3/2} \left(35 a^2+42 a b+15 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{7/2} (a+b)^3}-\frac{8 (c+d x)}{(a+b)^3}}{8 d}","\frac{b (9 a+5 b) \coth (c+d x)}{8 a^2 d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)}-\frac{b^{3/2} \left(35 a^2+42 a b+15 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{7/2} d (a+b)^3}-\frac{\left(8 a^2+27 a b+15 b^2\right) \coth (c+d x)}{8 a^3 d (a+b)^2}+\frac{b \coth (c+d x)}{4 a d (a+b) \left(a+b \tanh ^2(c+d x)\right)^2}+\frac{x}{(a+b)^3}",1,"-1/8*((-8*(c + d*x))/(a + b)^3 + (b^(3/2)*(35*a^2 + 42*a*b + 15*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(7/2)*(a + b)^3) + (8*Coth[c + d*x])/a^3 + (4*b^3*Sinh[2*(c + d*x)])/(a^2*(a + b)^2*(a - b + (a + b)*Cosh[2*(c + d*x)])^2) + (b^2*(13*a + 7*b)*Sinh[2*(c + d*x)])/(a^3*(a + b)^2*(a - b + (a + b)*Cosh[2*(c + d*x)])))/d","A",1
199,1,138,171,1.9064252,"\int \frac{\coth ^3(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Coth[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3,x]","-\frac{\frac{b^4}{2 a^4 (a+b) \left(a \coth ^2(c+d x)+b\right)^2}-\frac{b^3 (4 a+3 b)}{a^4 (a+b)^2 \left(a \coth ^2(c+d x)+b\right)}+\frac{\coth ^2(c+d x)}{a^3}-\frac{b^2 \left(6 a^2+8 a b+3 b^2\right) \log \left(a \coth ^2(c+d x)+b\right)}{a^4 (a+b)^3}-\frac{2 \log (\sinh (c+d x))}{(a+b)^3}}{2 d}","\frac{(a-3 b) \log (\tanh (c+d x))}{a^4 d}-\frac{b^2 (3 a+2 b)}{2 a^3 d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)}-\frac{\coth ^2(c+d x)}{2 a^3 d}-\frac{b^2}{4 a^2 d (a+b) \left(a+b \tanh ^2(c+d x)\right)^2}+\frac{b^2 \left(6 a^2+8 a b+3 b^2\right) \log \left(a+b \tanh ^2(c+d x)\right)}{2 a^4 d (a+b)^3}+\frac{\log (\cosh (c+d x))}{d (a+b)^3}",1,"-1/2*(Coth[c + d*x]^2/a^3 + b^4/(2*a^4*(a + b)*(b + a*Coth[c + d*x]^2)^2) - (b^3*(4*a + 3*b))/(a^4*(a + b)^2*(b + a*Coth[c + d*x]^2)) - (b^2*(6*a^2 + 8*a*b + 3*b^2)*Log[b + a*Coth[c + d*x]^2])/(a^4*(a + b)^3) - (2*Log[Sinh[c + d*x]])/(a + b)^3)/d","A",1
200,1,194,228,3.8660579,"\int \frac{\coth ^4(c+d x)}{\left(a+b \tanh ^2(c+d x)\right)^3} \, dx","Integrate[Coth[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3,x]","\frac{\frac{3 b^3 (17 a+11 b) \sinh (2 (c+d x))}{a^4 (a+b)^2 ((a+b) \cosh (2 (c+d x))+a-b)}+\frac{8 (9 b-4 a) \coth (c+d x)}{a^4}+\frac{12 b^4 \sinh (2 (c+d x))}{a^3 (a+b)^2 ((a+b) \cosh (2 (c+d x))+a-b)^2}-\frac{8 \coth (c+d x) \text{csch}^2(c+d x)}{a^3}+\frac{3 b^{5/2} \left(63 a^2+90 a b+35 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{9/2} (a+b)^3}+\frac{24 (c+d x)}{(a+b)^3}}{24 d}","\frac{b (11 a+7 b) \coth ^3(c+d x)}{8 a^2 d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)}+\frac{b^{5/2} \left(63 a^2+90 a b+35 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{9/2} d (a+b)^3}-\frac{\left(8 a^2+55 a b+35 b^2\right) \coth ^3(c+d x)}{24 a^3 d (a+b)^2}-\frac{\left(8 a^3-8 a^2 b-55 a b^2-35 b^3\right) \coth (c+d x)}{8 a^4 d (a+b)^2}+\frac{b \coth ^3(c+d x)}{4 a d (a+b) \left(a+b \tanh ^2(c+d x)\right)^2}+\frac{x}{(a+b)^3}",1,"((24*(c + d*x))/(a + b)^3 + (3*b^(5/2)*(63*a^2 + 90*a*b + 35*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(9/2)*(a + b)^3) + (8*(-4*a + 9*b)*Coth[c + d*x])/a^4 - (8*Coth[c + d*x]*Csch[c + d*x]^2)/a^3 + (12*b^4*Sinh[2*(c + d*x)])/(a^3*(a + b)^2*(a - b + (a + b)*Cosh[2*(c + d*x)])^2) + (3*b^3*(17*a + 11*b)*Sinh[2*(c + d*x)])/(a^4*(a + b)^2*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(24*d)","A",1
201,1,203,201,0.6798905,"\int \frac{1}{\left(a+b \tanh ^2(c+d x)\right)^4} \, dx","Integrate[(a + b*Tanh[c + d*x]^2)^(-4),x]","\frac{\frac{2 b (11 a+5 b) (a+b)^2 \tanh (c+d x)}{a^2 \left(a+b \tanh ^2(c+d x)\right)^2}+\frac{3 b \left(19 a^2+16 a b+5 b^2\right) (a+b) \tanh (c+d x)}{a^3 \left(a+b \tanh ^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} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{7/2}}+\frac{8 b (a+b)^3 \tanh (c+d x)}{a \left(a+b \tanh ^2(c+d x)\right)^3}-24 \log (1-\tanh (c+d x))+24 \log (\tanh (c+d x)+1)}{48 d (a+b)^4}","\frac{b (11 a+5 b) \tanh (c+d x)}{24 a^2 d (a+b)^2 \left(a+b \tanh ^2(c+d x)\right)^2}+\frac{b \left(19 a^2+16 a b+5 b^2\right) \tanh (c+d x)}{16 a^3 d (a+b)^3 \left(a+b \tanh ^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{b} \tanh (c+d x)}{\sqrt{a}}\right)}{16 a^{7/2} d (a+b)^4}+\frac{b \tanh (c+d x)}{6 a d (a+b) \left(a+b \tanh ^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]*Tanh[c + d*x])/Sqrt[a]])/a^(7/2) - 24*Log[1 - Tanh[c + d*x]] + 24*Log[1 + Tanh[c + d*x]] + (8*b*(a + b)^3*Tanh[c + d*x])/(a*(a + b*Tanh[c + d*x]^2)^3) + (2*b*(a + b)^2*(11*a + 5*b)*Tanh[c + d*x])/(a^2*(a + b*Tanh[c + d*x]^2)^2) + (3*b*(a + b)*(19*a^2 + 16*a*b + 5*b^2)*Tanh[c + d*x])/(a^3*(a + b*Tanh[c + d*x]^2)))/(48*(a + b)^4*d)","A",1
202,1,19,3,0.0085591,"\int \sqrt{1-\tanh ^2(x)} \, dx","Integrate[Sqrt[1 - Tanh[x]^2],x]","2 \cosh (x) \sqrt{\text{sech}^2(x)} \tan ^{-1}\left(\tanh \left(\frac{x}{2}\right)\right)","\sin ^{-1}(\tanh (x))",1,"2*ArcTan[Tanh[x/2]]*Cosh[x]*Sqrt[Sech[x]^2]","B",1
203,1,21,16,0.0079195,"\int \sqrt{-1+\tanh ^2(x)} \, dx","Integrate[Sqrt[-1 + Tanh[x]^2],x]","2 \cosh (x) \sqrt{-\text{sech}^2(x)} \tan ^{-1}\left(\tanh \left(\frac{x}{2}\right)\right)","-\tanh ^{-1}\left(\frac{\tanh (x)}{\sqrt{-\text{sech}^2(x)}}\right)",1,"2*ArcTan[Tanh[x/2]]*Cosh[x]*Sqrt[-Sech[x]^2]","A",1
204,1,29,22,0.0214335,"\int \left(1-\tanh ^2(x)\right)^{3/2} \, dx","Integrate[(1 - Tanh[x]^2)^(3/2),x]","\frac{\text{sech}(x) \left(2 \tan ^{-1}\left(\tanh \left(\frac{x}{2}\right)\right)+\tanh (x) \text{sech}(x)\right)}{2 \sqrt{\text{sech}^2(x)}}","\frac{1}{2} \sin ^{-1}(\tanh (x))+\frac{1}{2} \tanh (x) \sqrt{\text{sech}^2(x)}",1,"(Sech[x]*(2*ArcTan[Tanh[x/2]] + Sech[x]*Tanh[x]))/(2*Sqrt[Sech[x]^2])","A",1
205,1,28,35,0.0225164,"\int \left(-1+\tanh ^2(x)\right)^{3/2} \, dx","Integrate[(-1 + Tanh[x]^2)^(3/2),x]","-\frac{1}{2} \sqrt{-\text{sech}^2(x)} \left(\tanh (x)+2 \cosh (x) \tan ^{-1}\left(\tanh \left(\frac{x}{2}\right)\right)\right)","\frac{1}{2} \tanh ^{-1}\left(\frac{\tanh (x)}{\sqrt{-\text{sech}^2(x)}}\right)-\frac{1}{2} \tanh (x) \sqrt{-\text{sech}^2(x)}",1,"-1/2*(Sqrt[-Sech[x]^2]*(2*ArcTan[Tanh[x/2]]*Cosh[x] + Tanh[x]))","A",1
206,1,11,11,0.0077692,"\int \frac{1}{\sqrt{1-\tanh ^2(x)}} \, dx","Integrate[1/Sqrt[1 - Tanh[x]^2],x]","\frac{\tanh (x)}{\sqrt{\text{sech}^2(x)}}","\frac{\tanh (x)}{\sqrt{\text{sech}^2(x)}}",1,"Tanh[x]/Sqrt[Sech[x]^2]","A",1
207,1,13,13,0.0074592,"\int \frac{1}{\sqrt{-1+\tanh ^2(x)}} \, dx","Integrate[1/Sqrt[-1 + Tanh[x]^2],x]","\frac{\tanh (x)}{\sqrt{-\text{sech}^2(x)}}","\frac{\tanh (x)}{\sqrt{-\text{sech}^2(x)}}",1,"Tanh[x]/Sqrt[-Sech[x]^2]","A",1
208,1,85,87,0.5205966,"\int \tanh ^5(x) \sqrt{a+b \tanh ^2(x)} \, dx","Integrate[Tanh[x]^5*Sqrt[a + b*Tanh[x]^2],x]","\frac{\sqrt{a+b \tanh ^2(x)} \left(2 a^2-b (a+5 b) \tanh ^2(x)-5 a b-3 b^2 \tanh ^4(x)-15 b^2\right)}{15 b^2}+\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)","-\frac{\left(a+b \tanh ^2(x)\right)^{5/2}}{5 b^2}+\frac{(a-b) \left(a+b \tanh ^2(x)\right)^{3/2}}{3 b^2}-\sqrt{a+b \tanh ^2(x)}+\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)",1,"Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] + (Sqrt[a + b*Tanh[x]^2]*(2*a^2 - 5*a*b - 15*b^2 - b*(a + 5*b)*Tanh[x]^2 - 3*b^2*Tanh[x]^4))/(15*b^2)","A",1
209,1,580,121,6.21676,"\int \tanh ^4(x) \sqrt{a+b \tanh ^2(x)} \, dx","Integrate[Tanh[x]^4*Sqrt[a + b*Tanh[x]^2],x]","\sqrt{\frac{a \cosh (2 x)+a+b \cosh (2 x)-b}{\cosh (2 x)+1}} \left(\frac{\text{sech}(x) (-a \sinh (x)-6 b \sinh (x))}{8 b}+\frac{1}{4} \tanh (x) \text{sech}^2(x)\right)+\frac{-\frac{b \left(a^2-4 b^2\right) \sinh ^4(x) \text{csch}(2 x) \sqrt{\frac{(a+b) \cosh (2 x)+a-b}{\cosh (2 x)+1}} \sqrt{-\frac{a \coth ^2(x)}{b}} \sqrt{-\frac{a (\cosh (2 x)+1) \text{csch}^2(x)}{b}} \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{a ((a+b) \cosh (2 x)+a-b)}-\frac{4 i b \left(4 a b+4 b^2\right) \sqrt{\cosh (2 x)+1} \sqrt{\frac{(a+b) \cosh (2 x)+a-b}{\cosh (2 x)+1}} \left(\frac{i \sinh ^4(x) \text{csch}(2 x) \sqrt{-\frac{a \coth ^2(x)}{b}} \sqrt{-\frac{a (\cosh (2 x)+1) \text{csch}^2(x)}{b}} \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} \Pi \left(\frac{b}{a+b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{2 (a+b) \sqrt{\cosh (2 x)+1} \sqrt{(a+b) \cosh (2 x)+a-b}}-\frac{i \sinh ^4(x) \text{csch}(2 x) \sqrt{-\frac{a \coth ^2(x)}{b}} \sqrt{-\frac{a (\cosh (2 x)+1) \text{csch}^2(x)}{b}} \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{4 a \sqrt{\cosh (2 x)+1} \sqrt{(a+b) \cosh (2 x)+a-b}}\right)}{\sqrt{(a+b) \cosh (2 x)+a-b}}}{4 b}","\frac{\left(a^2-4 a b-8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{8 b^{3/2}}-\frac{(a+4 b) \tanh (x) \sqrt{a+b \tanh ^2(x)}}{8 b}+\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)-\frac{1}{4} \tanh ^3(x) \sqrt{a+b \tanh ^2(x)}",1,"(-((b*(a^2 - 4*b^2)*Sqrt[(a - b + (a + b)*Cosh[2*x])/(1 + Cosh[2*x])]*Sqrt[-((a*Coth[x]^2)/b)]*Sqrt[-((a*(1 + Cosh[2*x])*Csch[x]^2)/b)]*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*Csch[2*x]*EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1]*Sinh[x]^4)/(a*(a - b + (a + b)*Cosh[2*x]))) - ((4*I)*b*(4*a*b + 4*b^2)*Sqrt[1 + Cosh[2*x]]*Sqrt[(a - b + (a + b)*Cosh[2*x])/(1 + Cosh[2*x])]*(((-1/4*I)*Sqrt[-((a*Coth[x]^2)/b)]*Sqrt[-((a*(1 + Cosh[2*x])*Csch[x]^2)/b)]*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*Csch[2*x]*EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1]*Sinh[x]^4)/(a*Sqrt[1 + Cosh[2*x]]*Sqrt[a - b + (a + b)*Cosh[2*x]]) + ((I/2)*Sqrt[-((a*Coth[x]^2)/b)]*Sqrt[-((a*(1 + Cosh[2*x])*Csch[x]^2)/b)]*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*Csch[2*x]*EllipticPi[b/(a + b), ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1]*Sinh[x]^4)/((a + b)*Sqrt[1 + Cosh[2*x]]*Sqrt[a - b + (a + b)*Cosh[2*x]])))/Sqrt[a - b + (a + b)*Cosh[2*x]])/(4*b) + Sqrt[(a - b + a*Cosh[2*x] + b*Cosh[2*x])/(1 + Cosh[2*x])]*((Sech[x]*(-(a*Sinh[x]) - 6*b*Sinh[x]))/(8*b) + (Sech[x]^2*Tanh[x])/4)","C",1
210,1,60,63,0.1763724,"\int \tanh ^3(x) \sqrt{a+b \tanh ^2(x)} \, dx","Integrate[Tanh[x]^3*Sqrt[a + b*Tanh[x]^2],x]","\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)-\frac{\sqrt{a+b \tanh ^2(x)} \left(a+b \tanh ^2(x)+3 b\right)}{3 b}","-\frac{\left(a+b \tanh ^2(x)\right)^{3/2}}{3 b}-\sqrt{a+b \tanh ^2(x)}+\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)",1,"Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - (Sqrt[a + b*Tanh[x]^2]*(a + 3*b + b*Tanh[x]^2))/(3*b)","A",1
211,1,193,85,3.4368705,"\int \tanh ^2(x) \sqrt{a+b \tanh ^2(x)} \, dx","Integrate[Tanh[x]^2*Sqrt[a + b*Tanh[x]^2],x]","\frac{\tanh (x) \left(-\left(\text{sech}^2(x) ((a+b) \cosh (2 x)+a-b)\right)+\sqrt{2} a \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)-2 \sqrt{2} a \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} \Pi \left(\frac{b}{a+b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{2 \sqrt{2} \sqrt{\text{sech}^2(x) ((a+b) \cosh (2 x)+a-b)}}","-\frac{1}{2} \tanh (x) \sqrt{a+b \tanh ^2(x)}-\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{2 \sqrt{b}}+\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)",1,"((Sqrt[2]*a*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1] - 2*Sqrt[2]*a*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*EllipticPi[b/(a + b), ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1] - (a - b + (a + b)*Cosh[2*x])*Sech[x]^2)*Tanh[x])/(2*Sqrt[2]*Sqrt[(a - b + (a + b)*Cosh[2*x])*Sech[x]^2])","C",1
212,1,44,44,0.0278483,"\int \tanh (x) \sqrt{a+b \tanh ^2(x)} \, dx","Integrate[Tanh[x]*Sqrt[a + b*Tanh[x]^2],x]","\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)-\sqrt{a+b \tanh ^2(x)}","\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)-\sqrt{a+b \tanh ^2(x)}",1,"Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - Sqrt[a + b*Tanh[x]^2]","A",1
213,1,137,60,0.2352444,"\int \sqrt{a+b \tanh ^2(x)} \, dx","Integrate[Sqrt[a + b*Tanh[x]^2],x]","\frac{1}{2} \left(-2 \sqrt{b} \log \left(\sqrt{b} \sqrt{a+b \tanh ^2(x)}+b \tanh (x)\right)-\sqrt{a+b} \log \left(\sqrt{a+b} \sqrt{a+b \tanh ^2(x)}+a-b \tanh (x)\right)+\sqrt{a+b} \log \left(\sqrt{a+b} \sqrt{a+b \tanh ^2(x)}+a+b \tanh (x)\right)-\sqrt{a+b} \log (1-\tanh (x))+\sqrt{a+b} \log (\tanh (x)+1)\right)","\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)-\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)",1,"(-(Sqrt[a + b]*Log[1 - Tanh[x]]) + Sqrt[a + b]*Log[1 + Tanh[x]] - 2*Sqrt[b]*Log[b*Tanh[x] + Sqrt[b]*Sqrt[a + b*Tanh[x]^2]] - Sqrt[a + b]*Log[a - b*Tanh[x] + Sqrt[a + b]*Sqrt[a + b*Tanh[x]^2]] + Sqrt[a + b]*Log[a + b*Tanh[x] + Sqrt[a + b]*Sqrt[a + b*Tanh[x]^2]])/2","B",1
214,1,56,56,0.0314084,"\int \coth (x) \sqrt{a+b \tanh ^2(x)} \, dx","Integrate[Coth[x]*Sqrt[a + b*Tanh[x]^2],x]","\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a}}\right)","\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a}}\right)",1,"-(Sqrt[a]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]]) + Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]","A",1
215,1,42,48,0.1349756,"\int \coth ^2(x) \sqrt{a+b \tanh ^2(x)} \, dx","Integrate[Coth[x]^2*Sqrt[a + b*Tanh[x]^2],x]","-\coth (x) \sqrt{a+b \tanh ^2(x)} \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{(a+b) \tanh ^2(x)}{b \tanh ^2(x)+a}\right)","\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)-\coth (x) \sqrt{a+b \tanh ^2(x)}",1,"-(Coth[x]*Hypergeometric2F1[-1/2, 1, 1/2, ((a + b)*Tanh[x]^2)/(a + b*Tanh[x]^2)]*Sqrt[a + b*Tanh[x]^2])","C",1
216,1,83,83,0.2231561,"\int \coth ^3(x) \sqrt{a+b \tanh ^2(x)} \, dx","Integrate[Coth[x]^3*Sqrt[a + b*Tanh[x]^2],x]","-\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a}}\right)}{2 \sqrt{a}}+\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)-\frac{1}{2} \coth ^2(x) \sqrt{a+b \tanh ^2(x)}","-\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a}}\right)}{2 \sqrt{a}}+\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)-\frac{1}{2} \coth ^2(x) \sqrt{a+b \tanh ^2(x)}",1,"-1/2*((2*a + b)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]])/Sqrt[a] + Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - (Coth[x]^2*Sqrt[a + b*Tanh[x]^2])/2","A",1
217,1,161,78,6.2175804,"\int \coth ^4(x) \sqrt{a+b \tanh ^2(x)} \, dx","Integrate[Coth[x]^4*Sqrt[a + b*Tanh[x]^2],x]","\frac{\cosh ^4(x) \coth ^3(x) \left(\frac{b \tanh ^2(x)}{a}+1\right) \left(-\frac{4 (a+b) \left(a \tanh (x)+b \tanh ^3(x)\right)^2 \, _2F_1\left(2,2;\frac{3}{2};-\frac{(a+b) \sinh ^2(x)}{a}\right)}{a^2}-\frac{\text{sech}^4(x) \left(a-2 b \tanh ^2(x)\right) \left(\sqrt{-\frac{(a+b) \sinh ^2(x)}{a}} \sin ^{-1}\left(\sqrt{-\frac{(a+b) \sinh ^2(x)}{a}}\right)+\sqrt{\frac{b \sinh ^2(x)}{a}+\cosh ^2(x)}\right)}{\sqrt{\frac{b \sinh ^2(x)}{a}+\cosh ^2(x)}}\right)}{3 \sqrt{a+b \tanh ^2(x)}}","\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)-\frac{1}{3} \coth ^3(x) \sqrt{a+b \tanh ^2(x)}-\frac{(3 a+b) \coth (x) \sqrt{a+b \tanh ^2(x)}}{3 a}",1,"(Cosh[x]^4*Coth[x]^3*(1 + (b*Tanh[x]^2)/a)*(-((Sech[x]^4*(ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sqrt[-(((a + b)*Sinh[x]^2)/a)] + Sqrt[Cosh[x]^2 + (b*Sinh[x]^2)/a])*(a - 2*b*Tanh[x]^2))/Sqrt[Cosh[x]^2 + (b*Sinh[x]^2)/a]) - (4*(a + b)*Hypergeometric2F1[2, 2, 3/2, -(((a + b)*Sinh[x]^2)/a)]*(a*Tanh[x] + b*Tanh[x]^3)^2)/a^2))/(3*Sqrt[a + b*Tanh[x]^2])","C",0
218,1,111,121,0.6297898,"\int \coth ^5(x) \sqrt{a+b \tanh ^2(x)} \, dx","Integrate[Coth[x]^5*Sqrt[a + b*Tanh[x]^2],x]","\frac{\left(-8 a^2-4 a b+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a}}\right)+\sqrt{a} \left(8 a \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)-\coth ^2(x) \sqrt{a+b \tanh ^2(x)} \left(2 a \coth ^2(x)+4 a+b\right)\right)}{8 a^{3/2}}","-\frac{\left(8 a^2+4 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a}}\right)}{8 a^{3/2}}+\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)-\frac{1}{4} \coth ^4(x) \sqrt{a+b \tanh ^2(x)}-\frac{(4 a+b) \coth ^2(x) \sqrt{a+b \tanh ^2(x)}}{8 a}",1,"((-8*a^2 - 4*a*b + b^2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]] + Sqrt[a]*(8*a*Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - Coth[x]^2*(4*a + b + 2*a*Coth[x]^2)*Sqrt[a + b*Tanh[x]^2]))/(8*a^(3/2))","A",1
219,1,86,82,0.4435403,"\int \tanh ^3(x) \left(a+b \tanh ^2(x)\right)^{3/2} \, dx","Integrate[Tanh[x]^3*(a + b*Tanh[x]^2)^(3/2),x]","(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)-\frac{\sqrt{a+b \tanh ^2(x)} \left(3 a^2+b (6 a+5 b) \tanh ^2(x)+20 a b+3 b^2 \tanh ^4(x)+15 b^2\right)}{15 b}","-\frac{\left(a+b \tanh ^2(x)\right)^{5/2}}{5 b}-\frac{1}{3} \left(a+b \tanh ^2(x)\right)^{3/2}-(a+b) \sqrt{a+b \tanh ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)",1,"(a + b)^(3/2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - (Sqrt[a + b*Tanh[x]^2]*(3*a^2 + 20*a*b + 15*b^2 + b*(6*a + 5*b)*Tanh[x]^2 + 3*b^2*Tanh[x]^4))/(15*b)","A",1
220,1,584,123,6.2704311,"\int \tanh ^2(x) \left(a+b \tanh ^2(x)\right)^{3/2} \, dx","Integrate[Tanh[x]^2*(a + b*Tanh[x]^2)^(3/2),x]","\sqrt{\frac{a \cosh (2 x)+a+b \cosh (2 x)-b}{\cosh (2 x)+1}} \left(\frac{1}{8} \text{sech}(x) (-5 a \sinh (x)-6 b \sinh (x))+\frac{1}{4} b \tanh (x) \text{sech}^2(x)\right)+\frac{1}{4} \left(-\frac{b \left(a^2-4 a b-4 b^2\right) \sinh ^4(x) \text{csch}(2 x) \sqrt{\frac{(a+b) \cosh (2 x)+a-b}{\cosh (2 x)+1}} \sqrt{-\frac{a \coth ^2(x)}{b}} \sqrt{-\frac{a (\cosh (2 x)+1) \text{csch}^2(x)}{b}} \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{a ((a+b) \cosh (2 x)+a-b)}-\frac{4 i b \left(4 a^2+8 a b+4 b^2\right) \sqrt{\cosh (2 x)+1} \sqrt{\frac{(a+b) \cosh (2 x)+a-b}{\cosh (2 x)+1}} \left(\frac{i \sinh ^4(x) \text{csch}(2 x) \sqrt{-\frac{a \coth ^2(x)}{b}} \sqrt{-\frac{a (\cosh (2 x)+1) \text{csch}^2(x)}{b}} \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} \Pi \left(\frac{b}{a+b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{2 (a+b) \sqrt{\cosh (2 x)+1} \sqrt{(a+b) \cosh (2 x)+a-b}}-\frac{i \sinh ^4(x) \text{csch}(2 x) \sqrt{-\frac{a \coth ^2(x)}{b}} \sqrt{-\frac{a (\cosh (2 x)+1) \text{csch}^2(x)}{b}} \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{4 a \sqrt{\cosh (2 x)+1} \sqrt{(a+b) \cosh (2 x)+a-b}}\right)}{\sqrt{(a+b) \cosh (2 x)+a-b}}\right)","-\frac{\left(3 a^2+12 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{8 \sqrt{b}}-\frac{1}{8} (5 a+4 b) \tanh (x) \sqrt{a+b \tanh ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)-\frac{1}{4} b \tanh ^3(x) \sqrt{a+b \tanh ^2(x)}",1,"(-((b*(a^2 - 4*a*b - 4*b^2)*Sqrt[(a - b + (a + b)*Cosh[2*x])/(1 + Cosh[2*x])]*Sqrt[-((a*Coth[x]^2)/b)]*Sqrt[-((a*(1 + Cosh[2*x])*Csch[x]^2)/b)]*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*Csch[2*x]*EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1]*Sinh[x]^4)/(a*(a - b + (a + b)*Cosh[2*x]))) - ((4*I)*b*(4*a^2 + 8*a*b + 4*b^2)*Sqrt[1 + Cosh[2*x]]*Sqrt[(a - b + (a + b)*Cosh[2*x])/(1 + Cosh[2*x])]*(((-1/4*I)*Sqrt[-((a*Coth[x]^2)/b)]*Sqrt[-((a*(1 + Cosh[2*x])*Csch[x]^2)/b)]*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*Csch[2*x]*EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1]*Sinh[x]^4)/(a*Sqrt[1 + Cosh[2*x]]*Sqrt[a - b + (a + b)*Cosh[2*x]]) + ((I/2)*Sqrt[-((a*Coth[x]^2)/b)]*Sqrt[-((a*(1 + Cosh[2*x])*Csch[x]^2)/b)]*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*Csch[2*x]*EllipticPi[b/(a + b), ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1]*Sinh[x]^4)/((a + b)*Sqrt[1 + Cosh[2*x]]*Sqrt[a - b + (a + b)*Cosh[2*x]])))/Sqrt[a - b + (a + b)*Cosh[2*x]])/4 + Sqrt[(a - b + a*Cosh[2*x] + b*Cosh[2*x])/(1 + Cosh[2*x])]*((Sech[x]*(-5*a*Sinh[x] - 6*b*Sinh[x]))/8 + (b*Sech[x]^2*Tanh[x])/4)","C",1
221,1,59,63,0.1752223,"\int \tanh (x) \left(a+b \tanh ^2(x)\right)^{3/2} \, dx","Integrate[Tanh[x]*(a + b*Tanh[x]^2)^(3/2),x]","(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)-\frac{1}{3} \sqrt{a+b \tanh ^2(x)} \left(4 a+b \tanh ^2(x)+3 b\right)","-(a+b) \sqrt{a+b \tanh ^2(x)}-\frac{1}{3} \left(a+b \tanh ^2(x)\right)^{3/2}+(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)",1,"(a + b)^(3/2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - (Sqrt[a + b*Tanh[x]^2]*(4*a + 3*b + b*Tanh[x]^2))/3","A",1
222,1,161,88,0.3669516,"\int \left(a+b \tanh ^2(x)\right)^{3/2} \, dx","Integrate[(a + b*Tanh[x]^2)^(3/2),x]","\frac{1}{2} \left(-b \tanh (x) \sqrt{a+b \tanh ^2(x)}-(a+b)^{3/2} \log \left(\sqrt{a+b} \sqrt{a+b \tanh ^2(x)}+a-b \tanh (x)\right)+(a+b)^{3/2} \log \left(\sqrt{a+b} \sqrt{a+b \tanh ^2(x)}+a+b \tanh (x)\right)-\sqrt{b} (3 a+2 b) \log \left(\sqrt{b} \sqrt{a+b \tanh ^2(x)}+b \tanh (x)\right)+(a+b)^{3/2} (-\log (1-\tanh (x)))+(a+b)^{3/2} \log (\tanh (x)+1)\right)","-\frac{1}{2} b \tanh (x) \sqrt{a+b \tanh ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)-\frac{1}{2} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)",1,"(-((a + b)^(3/2)*Log[1 - Tanh[x]]) + (a + b)^(3/2)*Log[1 + Tanh[x]] - Sqrt[b]*(3*a + 2*b)*Log[b*Tanh[x] + Sqrt[b]*Sqrt[a + b*Tanh[x]^2]] - (a + b)^(3/2)*Log[a - b*Tanh[x] + Sqrt[a + b]*Sqrt[a + b*Tanh[x]^2]] + (a + b)^(3/2)*Log[a + b*Tanh[x] + Sqrt[a + b]*Sqrt[a + b*Tanh[x]^2]] - b*Tanh[x]*Sqrt[a + b*Tanh[x]^2])/2","A",1
223,1,71,71,0.0694246,"\int \coth (x) \left(a+b \tanh ^2(x)\right)^{3/2} \, dx","Integrate[Coth[x]*(a + b*Tanh[x]^2)^(3/2),x]","a^{3/2} \left(-\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a}}\right)\right)-b \sqrt{a+b \tanh ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)","a^{3/2} \left(-\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a}}\right)\right)-b \sqrt{a+b \tanh ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)",1,"-(a^(3/2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]]) + (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - b*Sqrt[a + b*Tanh[x]^2]","A",1
224,1,197,77,3.0049155,"\int \coth ^2(x) \left(a+b \tanh ^2(x)\right)^{3/2} \, dx","Integrate[Coth[x]^2*(a + b*Tanh[x]^2)^(3/2),x]","-\frac{a \tanh (x) \left(\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)-\sqrt{2} (a+2 b) \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)+\sqrt{2} (a+b) \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} \Pi \left(\frac{b}{a+b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{\sqrt{2} \sqrt{\text{sech}^2(x) ((a+b) \cosh (2 x)+a-b)}}","b^{3/2} \left(-\tanh ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)\right)+(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)-a \coth (x) \sqrt{a+b \tanh ^2(x)}",1,"-((a*((a - b + (a + b)*Cosh[2*x])*Csch[x]^2 - Sqrt[2]*(a + 2*b)*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1] + Sqrt[2]*(a + b)*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*EllipticPi[b/(a + b), ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1])*Tanh[x])/(Sqrt[2]*Sqrt[(a - b + (a + b)*Cosh[2*x])*Sech[x]^2]))","C",1
225,1,51,31,0.0626696,"\int \sqrt{1+\tanh ^2(x)} \, dx","Integrate[Sqrt[1 + Tanh[x]^2],x]","\frac{\cosh (x) \sqrt{\tanh ^2(x)+1} \left(\sqrt{2} \sinh ^{-1}\left(\sqrt{2} \sinh (x)\right)-\tanh ^{-1}\left(\frac{\sinh (x)}{\sqrt{\cosh (2 x)}}\right)\right)}{\sqrt{\cosh (2 x)}}","\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \tanh (x)}{\sqrt{\tanh ^2(x)+1}}\right)-\sinh ^{-1}(\tanh (x))",1,"((Sqrt[2]*ArcSinh[Sqrt[2]*Sinh[x]] - ArcTanh[Sinh[x]/Sqrt[Cosh[2*x]]])*Cosh[x]*Sqrt[1 + Tanh[x]^2])/Sqrt[Cosh[2*x]]","A",1
226,1,53,45,0.0372683,"\int \sqrt{-1-\tanh ^2(x)} \, dx","Integrate[Sqrt[-1 - Tanh[x]^2],x]","\frac{\cosh (x) \sqrt{-\tanh ^2(x)-1} \left(\sqrt{2} \sinh ^{-1}\left(\sqrt{2} \sinh (x)\right)-\tanh ^{-1}\left(\frac{\sinh (x)}{\sqrt{\cosh (2 x)}}\right)\right)}{\sqrt{\cosh (2 x)}}","\tan ^{-1}\left(\frac{\tanh (x)}{\sqrt{-\tanh ^2(x)-1}}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \tanh (x)}{\sqrt{-\tanh ^2(x)-1}}\right)",1,"((Sqrt[2]*ArcSinh[Sqrt[2]*Sinh[x]] - ArcTanh[Sinh[x]/Sqrt[Cosh[2*x]]])*Cosh[x]*Sqrt[-1 - Tanh[x]^2])/Sqrt[Cosh[2*x]]","A",1
227,1,74,50,0.1480529,"\int \left(1+\tanh ^2(x)\right)^{3/2} \, dx","Integrate[(1 + Tanh[x]^2)^(3/2),x]","-\frac{\left(\tanh ^2(x)+1\right)^{3/2} \left(-4 \sqrt{2} \sinh ^{-1}\left(\sqrt{2} \sinh (x)\right) \cosh ^3(x)+\sinh (x) \sqrt{\cosh (2 x)} \cosh (x)+5 \cosh ^3(x) \tanh ^{-1}\left(\frac{\sinh (x)}{\sqrt{\cosh (2 x)}}\right)\right)}{2 \cosh ^{\frac{3}{2}}(2 x)}","-\frac{1}{2} \tanh (x) \sqrt{\tanh ^2(x)+1}+2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \tanh (x)}{\sqrt{\tanh ^2(x)+1}}\right)-\frac{5}{2} \sinh ^{-1}(\tanh (x))",1,"-1/2*((-4*Sqrt[2]*ArcSinh[Sqrt[2]*Sinh[x]]*Cosh[x]^3 + 5*ArcTanh[Sinh[x]/Sqrt[Cosh[2*x]]]*Cosh[x]^3 + Cosh[x]*Sqrt[Cosh[2*x]]*Sinh[x])*(1 + Tanh[x]^2)^(3/2))/Cosh[2*x]^(3/2)","A",1
228,1,76,67,0.074529,"\int \left(-1-\tanh ^2(x)\right)^{3/2} \, dx","Integrate[(-1 - Tanh[x]^2)^(3/2),x]","-\frac{\left(-\tanh ^2(x)-1\right)^{3/2} \left(-4 \sqrt{2} \sinh ^{-1}\left(\sqrt{2} \sinh (x)\right) \cosh ^3(x)+\sinh (x) \sqrt{\cosh (2 x)} \cosh (x)+5 \cosh ^3(x) \tanh ^{-1}\left(\frac{\sinh (x)}{\sqrt{\cosh (2 x)}}\right)\right)}{2 \cosh ^{\frac{3}{2}}(2 x)}","\frac{1}{2} \tanh (x) \sqrt{-\tanh ^2(x)-1}-\frac{5}{2} \tan ^{-1}\left(\frac{\tanh (x)}{\sqrt{-\tanh ^2(x)-1}}\right)+2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \tanh (x)}{\sqrt{-\tanh ^2(x)-1}}\right)",1,"-1/2*((-4*Sqrt[2]*ArcSinh[Sqrt[2]*Sinh[x]]*Cosh[x]^3 + 5*ArcTanh[Sinh[x]/Sqrt[Cosh[2*x]]]*Cosh[x]^3 + Cosh[x]*Sqrt[Cosh[2*x]]*Sinh[x])*(-1 - Tanh[x]^2)^(3/2))/Cosh[2*x]^(3/2)","A",1
229,1,68,70,0.5369901,"\int \frac{\tanh ^5(x)}{\sqrt{a+b \tanh ^2(x)}} \, dx","Integrate[Tanh[x]^5/Sqrt[a + b*Tanh[x]^2],x]","\frac{\text{sech}^2(x) ((a-2 b) \cosh (2 x)+a-b) \sqrt{a+b \tanh ^2(x)}}{3 b^2}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}","-\frac{\left(a+b \tanh ^2(x)\right)^{3/2}}{3 b^2}+\frac{(a-b) \sqrt{a+b \tanh ^2(x)}}{b^2}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}",1,"ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/Sqrt[a + b] + ((a - b + (a - 2*b)*Cosh[2*x])*Sech[x]^2*Sqrt[a + b*Tanh[x]^2])/(3*b^2)","A",1
230,1,208,88,4.9000946,"\int \frac{\tanh ^4(x)}{\sqrt{a+b \tanh ^2(x)}} \, dx","Integrate[Tanh[x]^4/Sqrt[a + b*Tanh[x]^2],x]","\frac{\tanh (x) \left(-\left((a+b) \text{sech}^2(x) ((a+b) \cosh (2 x)+a-b)\right)+\sqrt{2} a (a+b) \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)-2 \sqrt{2} a b \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} \Pi \left(\frac{b}{a+b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{2 \sqrt{2} b (a+b) \sqrt{\text{sech}^2(x) ((a+b) \cosh (2 x)+a-b)}}","\frac{(a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{2 b^{3/2}}-\frac{\tanh (x) \sqrt{a+b \tanh ^2(x)}}{2 b}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{\sqrt{a+b}}",1,"((Sqrt[2]*a*(a + b)*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1] - 2*Sqrt[2]*a*b*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*EllipticPi[b/(a + b), ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1] - (a + b)*(a - b + (a + b)*Cosh[2*x])*Sech[x]^2)*Tanh[x])/(2*Sqrt[2]*b*(a + b)*Sqrt[(a - b + (a + b)*Cosh[2*x])*Sech[x]^2])","C",1
231,1,47,47,0.1077228,"\int \frac{\tanh ^3(x)}{\sqrt{a+b \tanh ^2(x)}} \, dx","Integrate[Tanh[x]^3/Sqrt[a + b*Tanh[x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}-\frac{\sqrt{a+b \tanh ^2(x)}}{b}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}-\frac{\sqrt{a+b \tanh ^2(x)}}{b}",1,"ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/Sqrt[a + b] - Sqrt[a + b*Tanh[x]^2]/b","A",1
232,1,101,60,0.5124068,"\int \frac{\tanh ^2(x)}{\sqrt{a+b \tanh ^2(x)}} \, dx","Integrate[Tanh[x]^2/Sqrt[a + b*Tanh[x]^2],x]","-\frac{a \coth (x) \sqrt{\text{sech}^2(x) ((a+b) \cosh (2 x)+a-b)} \Pi \left(\frac{b}{a+b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{b (a+b) \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{\sqrt{a+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{\sqrt{b}}",1,"-((a*Coth[x]*EllipticPi[b/(a + b), ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1]*Sqrt[(a - b + (a + b)*Cosh[2*x])*Sech[x]^2])/(b*(a + b)*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]))","C",1
233,1,29,29,0.0173388,"\int \frac{\tanh (x)}{\sqrt{a+b \tanh ^2(x)}} \, dx","Integrate[Tanh[x]/Sqrt[a + b*Tanh[x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}",1,"ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/Sqrt[a + b]","A",1
234,1,31,31,0.0236818,"\int \frac{1}{\sqrt{a+b \tanh ^2(x)}} \, dx","Integrate[1/Sqrt[a + b*Tanh[x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{\sqrt{a+b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{\sqrt{a+b}}",1,"ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/Sqrt[a + b]","A",1
235,1,56,56,0.0503632,"\int \frac{\coth (x)}{\sqrt{a+b \tanh ^2(x)}} \, dx","Integrate[Coth[x]/Sqrt[a + b*Tanh[x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"-(ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]]/Sqrt[a]) + ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/Sqrt[a + b]","A",1
236,1,123,51,6.180469,"\int \frac{\coth ^2(x)}{\sqrt{a+b \tanh ^2(x)}} \, dx","Integrate[Coth[x]^2/Sqrt[a + b*Tanh[x]^2],x]","\frac{\tanh (x) \left(\frac{(a+b)^2 ((a+b) \cosh (2 x)+a-b)^2 \, _2F_1\left(2,2;\frac{5}{2};-\frac{(a+b) \sinh ^2(x)}{a}\right)}{a^3}+3 \text{csch}^2(x) \left(a \coth ^2(x)+2 b\right) \sin ^{-1}\left(\sqrt{-\frac{(a+b) \sinh ^2(x)}{a}}\right) \sqrt{-\frac{(a+b) \sinh ^4(x) \left(a \coth ^2(x)+b\right)}{a^2}}\right)}{3 (a+b) \sqrt{a+b \tanh ^2(x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{\sqrt{a+b}}-\frac{\coth (x) \sqrt{a+b \tanh ^2(x)}}{a}",1,"((((a + b)^2*(a - b + (a + b)*Cosh[2*x])^2*Hypergeometric2F1[2, 2, 5/2, -(((a + b)*Sinh[x]^2)/a)])/a^3 + 3*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*(2*b + a*Coth[x]^2)*Csch[x]^2*Sqrt[-(((a + b)*(b + a*Coth[x]^2)*Sinh[x]^4)/a^2)])*Tanh[x])/(3*(a + b)*Sqrt[a + b*Tanh[x]^2])","C",0
237,1,107,88,0.4686999,"\int \frac{\coth ^3(x)}{\sqrt{a+b \tanh ^2(x)}} \, dx","Integrate[Coth[x]^3/Sqrt[a + b*Tanh[x]^2],x]","\frac{\left(-2 a^2-a b+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a}}\right)+\sqrt{a} \left(2 a \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)-(a+b) \coth ^2(x) \sqrt{a+b \tanh ^2(x)}\right)}{2 a^{3/2} (a+b)}","-\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a}}\right)}{2 a^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}-\frac{\coth ^2(x) \sqrt{a+b \tanh ^2(x)}}{2 a}",1,"((-2*a^2 - a*b + b^2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]] + Sqrt[a]*(2*a*Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - (a + b)*Coth[x]^2*Sqrt[a + b*Tanh[x]^2]))/(2*a^(3/2)*(a + b))","A",1
238,1,67,72,0.1125612,"\int \frac{\tanh ^5(x)}{\left(a+b \tanh ^2(x)\right)^{3/2}} \, dx","Integrate[Tanh[x]^5/(a + b*Tanh[x]^2)^(3/2),x]","\frac{b^2 \left(-\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \tanh ^2(x)+a}{a+b}\right)\right)-(a+b) \left(2 a+b \tanh ^2(x)-b\right)}{b^2 (a+b) \sqrt{a+b \tanh ^2(x)}}","-\frac{a^2}{b^2 (a+b) \sqrt{a+b \tanh ^2(x)}}-\frac{\sqrt{a+b \tanh ^2(x)}}{b^2}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}",1,"(-(b^2*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tanh[x]^2)/(a + b)]) - (a + b)*(2*a - b + b*Tanh[x]^2))/(b^2*(a + b)*Sqrt[a + b*Tanh[x]^2])","C",1
239,1,188,84,2.4521084,"\int \frac{\tanh ^4(x)}{\left(a+b \tanh ^2(x)\right)^{3/2}} \, dx","Integrate[Tanh[x]^4/(a + b*Tanh[x]^2)^(3/2),x]","-\frac{a \tanh (x) \left(\sqrt{2} (a+b) \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)+\sqrt{2} b \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} \Pi \left(\frac{b}{a+b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)-2 a-2 b\right)}{\sqrt{2} b (a+b)^2 \sqrt{\text{sech}^2(x) ((a+b) \cosh (2 x)+a-b)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{b^{3/2}}+\frac{a \tanh (x)}{b (a+b) \sqrt{a+b \tanh ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{(a+b)^{3/2}}",1,"-((a*(-2*a - 2*b + Sqrt[2]*(a + b)*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1] + Sqrt[2]*b*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*EllipticPi[b/(a + b), ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1])*Tanh[x])/(Sqrt[2]*b*(a + b)^2*Sqrt[(a - b + (a + b)*Cosh[2*x])*Sech[x]^2]))","C",1
240,1,52,52,0.1297903,"\int \frac{\tanh ^3(x)}{\left(a+b \tanh ^2(x)\right)^{3/2}} \, dx","Integrate[Tanh[x]^3/(a + b*Tanh[x]^2)^(3/2),x]","\frac{a}{b (a+b) \sqrt{a+b \tanh ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}","\frac{a}{b (a+b) \sqrt{a+b \tanh ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}",1,"ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) + a/(b*(a + b)*Sqrt[a + b*Tanh[x]^2])","A",1
241,1,112,53,1.8327112,"\int \frac{\tanh ^2(x)}{\left(a+b \tanh ^2(x)\right)^{3/2}} \, dx","Integrate[Tanh[x]^2/(a + b*Tanh[x]^2)^(3/2),x]","\frac{\tanh (x) \left(\tanh ^{-1}\left(\frac{\sqrt{\frac{(a+b) \tanh ^2(x)}{a}}}{\sqrt{\frac{b \tanh ^2(x)}{a}+1}}\right) \sqrt{\frac{(a+b) \tanh ^2(x)}{a}} \left(a \coth ^2(x)+b\right)-(a+b) \sqrt{\frac{b \tanh ^2(x)}{a}+1}\right)}{(a+b)^2 \sqrt{a+b \tanh ^2(x)} \sqrt{\frac{b \tanh ^2(x)}{a}+1}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{(a+b)^{3/2}}-\frac{\tanh (x)}{(a+b) \sqrt{a+b \tanh ^2(x)}}",1,"(Tanh[x]*(ArcTanh[Sqrt[((a + b)*Tanh[x]^2)/a]/Sqrt[1 + (b*Tanh[x]^2)/a]]*(b + a*Coth[x]^2)*Sqrt[((a + b)*Tanh[x]^2)/a] - (a + b)*Sqrt[1 + (b*Tanh[x]^2)/a]))/((a + b)^2*Sqrt[a + b*Tanh[x]^2]*Sqrt[1 + (b*Tanh[x]^2)/a])","B",1
242,1,41,49,0.0354585,"\int \frac{\tanh (x)}{\left(a+b \tanh ^2(x)\right)^{3/2}} \, dx","Integrate[Tanh[x]/(a + b*Tanh[x]^2)^(3/2),x]","-\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \tanh ^2(x)+a}{a+b}\right)}{(a+b) \sqrt{a+b \tanh ^2(x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}-\frac{1}{(a+b) \sqrt{a+b \tanh ^2(x)}}",1,"-(Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tanh[x]^2)/(a + b)]/((a + b)*Sqrt[a + b*Tanh[x]^2]))","C",1
243,1,223,56,4.6002213,"\int \frac{1}{\left(a+b \tanh ^2(x)\right)^{3/2}} \, dx","Integrate[(a + b*Tanh[x]^2)^(-3/2),x]","-\frac{\sinh ^2(x) \left(\sqrt{2} a^2 (a+b) \tanh (x) \, _2F_1\left(2,2;\frac{7}{2};-\frac{(a+b) \sinh ^2(x)}{a}\right) \left(-\frac{(a+b) \sinh ^2(x) ((a+b) \cosh (2 x)+a-b)}{a^2}\right)^{3/2}+\frac{15}{4} a \text{csch}(x) \text{sech}(x) ((3 a+2 b) \cosh (2 x)+3 a-2 b) \left(-2 a \sqrt{-\frac{(a+b) \sinh ^4(x) \left(a \coth ^2(x)+b\right)}{a^2}}+(a-b) \sin ^{-1}\left(\sqrt{-\frac{(a+b) \sinh ^2(x)}{a}}\right)+(a+b) \cosh (2 x) \sin ^{-1}\left(\sqrt{-\frac{(a+b) \sinh ^2(x)}{a}}\right)\right)\right)}{15 a^4 \left(-\frac{(a+b) \sinh ^2(x)}{a}\right)^{3/2} \sqrt{a+b \tanh ^2(x)} \sqrt{\frac{b \sinh ^2(x)}{a}+\cosh ^2(x)}}","\frac{b \tanh (x)}{a (a+b) \sqrt{a+b \tanh ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{(a+b)^{3/2}}",1,"-1/15*(Sinh[x]^2*((15*a*(3*a - 2*b + (3*a + 2*b)*Cosh[2*x])*Csch[x]*Sech[x]*((a - b)*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]] + (a + b)*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Cosh[2*x] - 2*a*Sqrt[-(((a + b)*(b + a*Coth[x]^2)*Sinh[x]^4)/a^2)]))/4 + Sqrt[2]*a^2*(a + b)*Hypergeometric2F1[2, 2, 7/2, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*(a - b + (a + b)*Cosh[2*x])*Sinh[x]^2)/a^2))^(3/2)*Tanh[x]))/(a^4*(-(((a + b)*Sinh[x]^2)/a))^(3/2)*Sqrt[Cosh[x]^2 + (b*Sinh[x]^2)/a]*Sqrt[a + b*Tanh[x]^2])","C",0
244,1,70,78,0.0765133,"\int \frac{\coth (x)}{\left(a+b \tanh ^2(x)\right)^{3/2}} \, dx","Integrate[Coth[x]/(a + b*Tanh[x]^2)^(3/2),x]","\frac{(a+b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \tanh ^2(x)}{a}+1\right)-a \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \tanh ^2(x)+a}{a+b}\right)}{a (a+b) \sqrt{a+b \tanh ^2(x)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{b}{a (a+b) \sqrt{a+b \tanh ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}",1,"(-(a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tanh[x]^2)/(a + b)]) + (a + b)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Tanh[x]^2)/a])/(a*(a + b)*Sqrt[a + b*Tanh[x]^2])","C",1
245,1,230,85,7.989928,"\int \frac{\coth ^2(x)}{\left(a+b \tanh ^2(x)\right)^{3/2}} \, dx","Integrate[Coth[x]^2/(a + b*Tanh[x]^2)^(3/2),x]","-\frac{\sinh (2 x) \text{sech}^2(x) \left(\sqrt{2} a^3 \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} \Pi \left(\frac{b}{a+b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)+(a+b) \text{csch}^2(x) \left(\left(a^2+2 a b+2 b^2\right) \cosh (2 x)+a^2-2 b^2\right)-\sqrt{2} a^2 (a+b) \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{2 \sqrt{2} a^2 (a+b)^2 \sqrt{\text{sech}^2(x) ((a+b) \cosh (2 x)+a-b)}}","-\frac{(a+2 b) \coth (x) \sqrt{a+b \tanh ^2(x)}}{a^2 (a+b)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{(a+b)^{3/2}}+\frac{b \coth (x)}{a (a+b) \sqrt{a+b \tanh ^2(x)}}",1,"-1/2*(((a + b)*(a^2 - 2*b^2 + (a^2 + 2*a*b + 2*b^2)*Cosh[2*x])*Csch[x]^2 - Sqrt[2]*a^2*(a + b)*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1] + Sqrt[2]*a^3*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*EllipticPi[b/(a + b), ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1])*Sech[x]^2*Sinh[2*x])/(Sqrt[2]*a^2*(a + b)^2*Sqrt[(a - b + (a + b)*Cosh[2*x])*Sech[x]^2])","C",1
246,1,231,118,2.03279,"\int \frac{\tanh ^6(x)}{\left(a+b \tanh ^2(x)\right)^{5/2}} \, dx","Integrate[Tanh[x]^6/(a + b*Tanh[x]^2)^(5/2),x]","\frac{\sqrt{\text{sech}^2(x) ((a+b) \cosh (2 x)+a-b)} \left(\frac{a (a+b) \sinh (2 x) \left(\left(3 a^2+10 a b+7 b^2\right) \cosh (2 x)+3 a^2+2 a b-7 b^2\right)}{((a+b) \cosh (2 x)+a-b)^2}-\frac{3 \sqrt{2} a \coth (x) \left(\left(a^2+3 a b+2 b^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)+b^2 \Pi \left(\frac{b}{a+b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{b \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}}}\right)}{3 \sqrt{2} b^2 (a+b)^3}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{b^{5/2}}+\frac{a (a+2 b) \tanh (x)}{b^2 (a+b)^2 \sqrt{a+b \tanh ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{(a+b)^{5/2}}+\frac{a \tanh ^3(x)}{3 b (a+b) \left(a+b \tanh ^2(x)\right)^{3/2}}",1,"(Sqrt[(a - b + (a + b)*Cosh[2*x])*Sech[x]^2]*((-3*Sqrt[2]*a*Coth[x]*((a^2 + 3*a*b + 2*b^2)*EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1] + b^2*EllipticPi[b/(a + b), ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1]))/(b*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]) + (a*(a + b)*(3*a^2 + 2*a*b - 7*b^2 + (3*a^2 + 10*a*b + 7*b^2)*Cosh[2*x])*Sinh[2*x])/(a - b + (a + b)*Cosh[2*x])^2))/(3*Sqrt[2]*b^2*(a + b)^3)","C",1
247,1,68,84,0.1158285,"\int \frac{\tanh ^5(x)}{\left(a+b \tanh ^2(x)\right)^{5/2}} \, dx","Integrate[Tanh[x]^5/(a + b*Tanh[x]^2)^(5/2),x]","\frac{(a+b) \left(2 a+3 b \tanh ^2(x)+b\right)-b^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \tanh ^2(x)+a}{a+b}\right)}{3 b^2 (a+b) \left(a+b \tanh ^2(x)\right)^{3/2}}","-\frac{a^2}{3 b^2 (a+b) \left(a+b \tanh ^2(x)\right)^{3/2}}+\frac{a (a+2 b)}{b^2 (a+b)^2 \sqrt{a+b \tanh ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}",1,"(-(b^2*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tanh[x]^2)/(a + b)]) + (a + b)*(2*a + b + 3*b*Tanh[x]^2))/(3*b^2*(a + b)*(a + b*Tanh[x]^2)^(3/2))","C",1
248,1,132,90,3.1261822,"\int \frac{\tanh ^4(x)}{\left(a+b \tanh ^2(x)\right)^{5/2}} \, dx","Integrate[Tanh[x]^4/(a + b*Tanh[x]^2)^(5/2),x]","\frac{\tanh ^3(x) \left(3 \tanh ^{-1}\left(\frac{\sqrt{\frac{(a+b) \tanh ^2(x)}{a}}}{\sqrt{\frac{b \tanh ^2(x)}{a}+1}}\right) \sqrt{\frac{(a+b) \tanh ^2(x)}{a}} \left(a \coth ^2(x)+b\right)^2-(a+b) \sqrt{\frac{b \tanh ^2(x)}{a}+1} \left(3 a \coth ^2(x)+a+4 b\right)\right)}{3 (a+b)^3 \left(a+b \tanh ^2(x)\right)^{3/2} \sqrt{\frac{b \tanh ^2(x)}{a}+1}}","-\frac{(a+4 b) \tanh (x)}{3 b (a+b)^2 \sqrt{a+b \tanh ^2(x)}}+\frac{a \tanh (x)}{3 b (a+b) \left(a+b \tanh ^2(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{(a+b)^{5/2}}",1,"(Tanh[x]^3*(3*ArcTanh[Sqrt[((a + b)*Tanh[x]^2)/a]/Sqrt[1 + (b*Tanh[x]^2)/a]]*(b + a*Coth[x]^2)^2*Sqrt[((a + b)*Tanh[x]^2)/a] - (a + b)*(a + 4*b + 3*a*Coth[x]^2)*Sqrt[1 + (b*Tanh[x]^2)/a]))/(3*(a + b)^3*(a + b*Tanh[x]^2)^(3/2)*Sqrt[1 + (b*Tanh[x]^2)/a])","A",1
249,1,63,74,0.0934742,"\int \frac{\tanh ^3(x)}{\left(a+b \tanh ^2(x)\right)^{5/2}} \, dx","Integrate[Tanh[x]^3/(a + b*Tanh[x]^2)^(5/2),x]","\frac{a (a+b)-3 b \left(a+b \tanh ^2(x)\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \tanh ^2(x)+a}{a+b}\right)}{3 b (a+b)^2 \left(a+b \tanh ^2(x)\right)^{3/2}}","\frac{a}{3 b (a+b) \left(a+b \tanh ^2(x)\right)^{3/2}}-\frac{1}{(a+b)^2 \sqrt{a+b \tanh ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}",1,"(a*(a + b) - 3*b*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tanh[x]^2)/(a + b)]*(a + b*Tanh[x]^2))/(3*b*(a + b)^2*(a + b*Tanh[x]^2)^(3/2))","C",1
250,1,259,88,7.822911,"\int \frac{\tanh ^2(x)}{\left(a+b \tanh ^2(x)\right)^{5/2}} \, dx","Integrate[Tanh[x]^2/(a + b*Tanh[x]^2)^(5/2),x]","\frac{\cosh ^4(x) \coth (x) \left(-12 (a+b)^3 \tanh ^6(x) \left(a+b \tanh ^2(x)\right) \, _2F_1\left(2,2;\frac{9}{2};-\frac{(a+b) \sinh ^2(x)}{a}\right) \sqrt{-\frac{\sinh ^2(x) \cosh ^2(x) \left(a^2+a \left(b \tanh ^2(x)+b\right)+b^2 \tanh ^2(x)\right)}{a^2}}-35 a \text{sech}^2(x) \left(-5 a-2 b \tanh ^2(x)\right) \left(a \text{sech}^2(x) \left(a \left(-\tanh ^2(x)-3\right)-4 b \tanh ^2(x)\right) \sqrt{-\frac{\sinh ^2(x) \cosh ^2(x) \left(a^2+a \left(b \tanh ^2(x)+b\right)+b^2 \tanh ^2(x)\right)}{a^2}}+3 \left(a+b \tanh ^2(x)\right)^2 \sin ^{-1}\left(\sqrt{-\frac{(a+b) \sinh ^2(x)}{a}}\right)\right)\right)}{315 a^4 (a+b)^2 \sqrt{a+b \tanh ^2(x)} \left(\frac{b \tanh ^2(x)}{a}+1\right) \sqrt{-\frac{(a+b) \sinh ^2(x) \cosh ^2(x) \left(a+b \tanh ^2(x)\right)}{a^2}}}","-\frac{(2 a-b) \tanh (x)}{3 a (a+b)^2 \sqrt{a+b \tanh ^2(x)}}-\frac{\tanh (x)}{3 (a+b) \left(a+b \tanh ^2(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{(a+b)^{5/2}}",1,"(Cosh[x]^4*Coth[x]*(-12*(a + b)^3*Hypergeometric2F1[2, 2, 9/2, -(((a + b)*Sinh[x]^2)/a)]*Tanh[x]^6*(a + b*Tanh[x]^2)*Sqrt[-((Cosh[x]^2*Sinh[x]^2*(a^2 + b^2*Tanh[x]^2 + a*(b + b*Tanh[x]^2)))/a^2)] - 35*a*Sech[x]^2*(-5*a - 2*b*Tanh[x]^2)*(3*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*(a + b*Tanh[x]^2)^2 + a*Sech[x]^2*(-4*b*Tanh[x]^2 + a*(-3 - Tanh[x]^2))*Sqrt[-((Cosh[x]^2*Sinh[x]^2*(a^2 + b^2*Tanh[x]^2 + a*(b + b*Tanh[x]^2)))/a^2)])))/(315*a^4*(a + b)^2*Sqrt[a + b*Tanh[x]^2]*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)]*(1 + (b*Tanh[x]^2)/a))","C",0
251,1,43,70,0.0465074,"\int \frac{\tanh (x)}{\left(a+b \tanh ^2(x)\right)^{5/2}} \, dx","Integrate[Tanh[x]/(a + b*Tanh[x]^2)^(5/2),x]","-\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \tanh ^2(x)+a}{a+b}\right)}{3 (a+b) \left(a+b \tanh ^2(x)\right)^{3/2}}","-\frac{1}{(a+b)^2 \sqrt{a+b \tanh ^2(x)}}-\frac{1}{3 (a+b) \left(a+b \tanh ^2(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}",1,"-1/3*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tanh[x]^2)/(a + b)]/((a + b)*(a + b*Tanh[x]^2)^(3/2))","C",1
252,1,976,93,7.7110039,"\int \frac{1}{\left(a+b \tanh ^2(x)\right)^{5/2}} \, dx","Integrate[(a + b*Tanh[x]^2)^(-5/2),x]","\frac{\cosh (x) \sinh (x) \left(\frac{840 b^2 (a+b)^2 \sin ^{-1}\left(\sqrt{-\frac{(a+b) \sinh ^2(x)}{a}}\right) \tanh ^4(x) \sinh ^4(x)}{a^4}+\frac{2100 b (a+b)^2 \sin ^{-1}\left(\sqrt{-\frac{(a+b) \sinh ^2(x)}{a}}\right) \tanh ^2(x) \sinh ^4(x)}{a^3}+\frac{1575 (a+b)^2 \sin ^{-1}\left(\sqrt{-\frac{(a+b) \sinh ^2(x)}{a}}\right) \sinh ^4(x)}{a^2}+\frac{1680 b^2 (a+b) \sin ^{-1}\left(\sqrt{-\frac{(a+b) \sinh ^2(x)}{a}}\right) \tanh ^4(x) \sinh ^2(x)}{a^3}+\frac{4200 b (a+b) \sin ^{-1}\left(\sqrt{-\frac{(a+b) \sinh ^2(x)}{a}}\right) \tanh ^2(x) \sinh ^2(x)}{a^2}+\frac{3150 (a+b) \sin ^{-1}\left(\sqrt{-\frac{(a+b) \sinh ^2(x)}{a}}\right) \sinh ^2(x)}{a}+\frac{840 b^2 \sin ^{-1}\left(\sqrt{-\frac{(a+b) \sinh ^2(x)}{a}}\right) \tanh ^4(x)}{a^2}+\frac{2100 b \sin ^{-1}\left(\sqrt{-\frac{(a+b) \sinh ^2(x)}{a}}\right) \tanh ^2(x)}{a}+1575 \sin ^{-1}\left(\sqrt{-\frac{(a+b) \sinh ^2(x)}{a}}\right)+\frac{72 b^2 \, _2F_1\left(2,2;\frac{9}{2};-\frac{(a+b) \sinh ^2(x)}{a}\right) \left(-\frac{(a+b) \sinh ^2(x)}{a}\right)^{7/2} \tanh ^4(x) \sqrt{\frac{\cosh ^2(x) \left(b \tanh ^2(x)+a\right)}{a}}}{a^2}+\frac{24 b^2 \, _3F_2\left(2,2,2;1,\frac{9}{2};-\frac{(a+b) \sinh ^2(x)}{a}\right) \left(-\frac{(a+b) \sinh ^2(x)}{a}\right)^{7/2} \tanh ^4(x) \sqrt{\frac{\cosh ^2(x) \left(b \tanh ^2(x)+a\right)}{a}}}{a^2}+\frac{1120 b^2 \left(-\frac{(a+b) \sinh ^2(x)}{a}\right)^{3/2} \tanh ^4(x) \sqrt{\frac{\cosh ^2(x) \left(b \tanh ^2(x)+a\right)}{a}}}{a^2}+96 \, _2F_1\left(2,2;\frac{9}{2};-\frac{(a+b) \sinh ^2(x)}{a}\right) \left(-\frac{(a+b) \sinh ^2(x)}{a}\right)^{7/2} \sqrt{\frac{\cosh ^2(x) \left(b \tanh ^2(x)+a\right)}{a}}+24 \, _3F_2\left(2,2,2;1,\frac{9}{2};-\frac{(a+b) \sinh ^2(x)}{a}\right) \left(-\frac{(a+b) \sinh ^2(x)}{a}\right)^{7/2} \sqrt{\frac{\cosh ^2(x) \left(b \tanh ^2(x)+a\right)}{a}}+\frac{168 b \, _2F_1\left(2,2;\frac{9}{2};-\frac{(a+b) \sinh ^2(x)}{a}\right) \left(-\frac{(a+b) \sinh ^2(x)}{a}\right)^{7/2} \tanh ^2(x) \sqrt{\frac{\cosh ^2(x) \left(b \tanh ^2(x)+a\right)}{a}}}{a}+\frac{48 b \, _3F_2\left(2,2,2;1,\frac{9}{2};-\frac{(a+b) \sinh ^2(x)}{a}\right) \left(-\frac{(a+b) \sinh ^2(x)}{a}\right)^{7/2} \tanh ^2(x) \sqrt{\frac{\cosh ^2(x) \left(b \tanh ^2(x)+a\right)}{a}}}{a}+\frac{2800 b \left(-\frac{(a+b) \sinh ^2(x)}{a}\right)^{3/2} \tanh ^2(x) \sqrt{\frac{\cosh ^2(x) \left(b \tanh ^2(x)+a\right)}{a}}}{a}+2100 \left(-\frac{(a+b) \sinh ^2(x)}{a}\right)^{3/2} \sqrt{\frac{\cosh ^2(x) \left(b \tanh ^2(x)+a\right)}{a}}-\frac{840 b^2 \tanh ^4(x) \sqrt{-\frac{(a+b) \cosh ^2(x) \sinh ^2(x) \left(b \tanh ^2(x)+a\right)}{a^2}}}{a^2}-\frac{2100 b \tanh ^2(x) \sqrt{-\frac{(a+b) \cosh ^2(x) \sinh ^2(x) \left(b \tanh ^2(x)+a\right)}{a^2}}}{a}-1575 \sqrt{-\frac{(a+b) \cosh ^2(x) \sinh ^2(x) \left(b \tanh ^2(x)+a\right)}{a^2}}\right)}{315 a^2 \left(-\frac{(a+b) \sinh ^2(x)}{a}\right)^{5/2} \sqrt{b \tanh ^2(x)+a} \sqrt{\frac{\cosh ^2(x) \left(b \tanh ^2(x)+a\right)}{a}} \left(\frac{b \tanh ^2(x)}{a}+1\right)}","\frac{b (5 a+2 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt{a+b \tanh ^2(x)}}+\frac{b \tanh (x)}{3 a (a+b) \left(a+b \tanh ^2(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{(a+b)^{5/2}}",1,"(Cosh[x]*Sinh[x]*(1575*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]] + (3150*(a + b)*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sinh[x]^2)/a + (1575*(a + b)^2*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sinh[x]^4)/a^2 + (2100*b*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Tanh[x]^2)/a + (4200*b*(a + b)*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sinh[x]^2*Tanh[x]^2)/a^2 + (2100*b*(a + b)^2*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sinh[x]^4*Tanh[x]^2)/a^3 + (840*b^2*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Tanh[x]^4)/a^2 + (1680*b^2*(a + b)*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sinh[x]^2*Tanh[x]^4)/a^3 + (840*b^2*(a + b)^2*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sinh[x]^4*Tanh[x]^4)/a^4 + 2100*(-(((a + b)*Sinh[x]^2)/a))^(3/2)*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a] + 96*Hypergeometric2F1[2, 2, 9/2, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a] + 24*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a] + (2800*b*(-(((a + b)*Sinh[x]^2)/a))^(3/2)*Tanh[x]^2*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a])/a + (168*b*Hypergeometric2F1[2, 2, 9/2, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Tanh[x]^2*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a])/a + (48*b*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Tanh[x]^2*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a])/a + (1120*b^2*(-(((a + b)*Sinh[x]^2)/a))^(3/2)*Tanh[x]^4*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a])/a^2 + (72*b^2*Hypergeometric2F1[2, 2, 9/2, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Tanh[x]^4*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a])/a^2 + (24*b^2*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Tanh[x]^4*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a])/a^2 - 1575*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)] - (2100*b*Tanh[x]^2*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)])/a - (840*b^2*Tanh[x]^4*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)])/a^2))/(315*a^2*(-(((a + b)*Sinh[x]^2)/a))^(5/2)*Sqrt[a + b*Tanh[x]^2]*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a]*(1 + (b*Tanh[x]^2)/a))","C",0
253,1,73,108,0.071712,"\int \frac{\coth (x)}{\left(a+b \tanh ^2(x)\right)^{5/2}} \, dx","Integrate[Coth[x]/(a + b*Tanh[x]^2)^(5/2),x]","\frac{(a+b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \tanh ^2(x)}{a}+1\right)-a \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \tanh ^2(x)+a}{a+b}\right)}{3 a (a+b) \left(a+b \tanh ^2(x)\right)^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a}}\right)}{a^{5/2}}+\frac{b (2 a+b)}{a^2 (a+b)^2 \sqrt{a+b \tanh ^2(x)}}+\frac{b}{3 a (a+b) \left(a+b \tanh ^2(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tanh ^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}",1,"(-(a*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tanh[x]^2)/(a + b)]) + (a + b)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Tanh[x]^2)/a])/(3*a*(a + b)*(a + b*Tanh[x]^2)^(3/2))","C",1
254,1,246,131,7.7657732,"\int \frac{\coth ^2(x)}{\left(a+b \tanh ^2(x)\right)^{5/2}} \, dx","Integrate[Coth[x]^2/(a + b*Tanh[x]^2)^(5/2),x]","\frac{\sqrt{\text{sech}^2(x) ((a+b) \cosh (2 x)+a-b)} \left(\frac{3 \sqrt{2} a^3 \coth (x) \left((a+b) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)-a \Pi \left(\frac{b}{a+b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{b \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}}}-\frac{(a+b) \left(2 a b^3 \sinh (2 x)+b^2 (9 a+5 b) \sinh (2 x) ((a+b) \cosh (2 x)+a-b)+3 (a+b)^2 \coth (x) ((a+b) \cosh (2 x)+a-b)^2\right)}{((a+b) \cosh (2 x)+a-b)^2}\right)}{3 \sqrt{2} a^3 (a+b)^3}","-\frac{(3 a+2 b) (a+4 b) \coth (x) \sqrt{a+b \tanh ^2(x)}}{3 a^3 (a+b)^2}+\frac{b (7 a+4 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt{a+b \tanh ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right)}{(a+b)^{5/2}}+\frac{b \coth (x)}{3 a (a+b) \left(a+b \tanh ^2(x)\right)^{3/2}}",1,"(Sqrt[(a - b + (a + b)*Cosh[2*x])*Sech[x]^2]*((3*Sqrt[2]*a^3*Coth[x]*((a + b)*EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1] - a*EllipticPi[b/(a + b), ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1]))/(b*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]) - ((a + b)*(3*(a + b)^2*(a - b + (a + b)*Cosh[2*x])^2*Coth[x] + 2*a*b^3*Sinh[2*x] + b^2*(9*a + 5*b)*(a - b + (a + b)*Cosh[2*x])*Sinh[2*x]))/(a - b + (a + b)*Cosh[2*x])^2))/(3*Sqrt[2]*a^3*(a + b)^3)","C",1
255,1,35,25,0.0226699,"\int \frac{1}{\sqrt{1+\tanh ^2(x)}} \, dx","Integrate[1/Sqrt[1 + Tanh[x]^2],x]","\frac{\sinh ^{-1}\left(\sqrt{2} \sinh (x)\right) \sqrt{\cosh (2 x)} \text{sech}(x)}{\sqrt{2} \sqrt{\tanh ^2(x)+1}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \tanh (x)}{\sqrt{\tanh ^2(x)+1}}\right)}{\sqrt{2}}",1,"(ArcSinh[Sqrt[2]*Sinh[x]]*Sqrt[Cosh[2*x]]*Sech[x])/(Sqrt[2]*Sqrt[1 + Tanh[x]^2])","A",1
256,1,37,27,0.0208985,"\int \frac{1}{\sqrt{-1-\tanh ^2(x)}} \, dx","Integrate[1/Sqrt[-1 - Tanh[x]^2],x]","\frac{\sinh ^{-1}\left(\sqrt{2} \sinh (x)\right) \sqrt{\cosh (2 x)} \text{sech}(x)}{\sqrt{2} \sqrt{-\tanh ^2(x)-1}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \tanh (x)}{\sqrt{-\tanh ^2(x)-1}}\right)}{\sqrt{2}}",1,"(ArcSinh[Sqrt[2]*Sinh[x]]*Sqrt[Cosh[2*x]]*Sech[x])/(Sqrt[2]*Sqrt[-1 - Tanh[x]^2])","A",1
257,1,95,89,0.7607626,"\int \left(a+b \tanh ^3(c+d x)\right)^2 \, dx","Integrate[(a + b*Tanh[c + d*x]^3)^2,x]","-\frac{30 a b \tanh ^2(c+d x)+15 \left((a+b)^2 \log (1-\tanh (c+d x))-(a-b)^2 \log (\tanh (c+d x)+1)\right)+6 b^2 \tanh ^5(c+d x)+10 b^2 \tanh ^3(c+d x)+30 b^2 \tanh (c+d x)}{30 d}","x \left(a^2+b^2\right)-\frac{a b \tanh ^2(c+d x)}{d}+\frac{2 a b \log (\cosh (c+d x))}{d}-\frac{b^2 \tanh ^5(c+d x)}{5 d}-\frac{b^2 \tanh ^3(c+d x)}{3 d}-\frac{b^2 \tanh (c+d x)}{d}",1,"-1/30*(15*((a + b)^2*Log[1 - Tanh[c + d*x]] - (a - b)^2*Log[1 + Tanh[c + d*x]]) + 30*b^2*Tanh[c + d*x] + 30*a*b*Tanh[c + d*x]^2 + 10*b^2*Tanh[c + d*x]^3 + 6*b^2*Tanh[c + d*x]^5)/d","A",1
258,1,40,38,0.072121,"\int \frac{1}{1+\tanh ^3(x)} \, dx","Integrate[(1 + Tanh[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 (\tanh (x)+1)}-\frac{2 \tan ^{-1}\left(\frac{1-2 \tanh (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
259,1,166,124,4.8103232,"\int \tanh (x) \left(a+b \tanh ^4(x)\right)^{3/2} \, dx","Integrate[Tanh[x]*(a + b*Tanh[x]^4)^(3/2),x]","\frac{1}{12} \left(-\sqrt{a+b \tanh ^4(x)} \left(8 a+2 b \tanh ^4(x)+3 b \tanh ^2(x)+6 b\right)+6 (a+b)^{3/2} \tanh ^{-1}\left(\frac{a+b \tanh ^2(x)}{\sqrt{a+b} \sqrt{a+b \tanh ^4(x)}}\right)-6 \sqrt{b} (a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh ^2(x)}{\sqrt{a+b \tanh ^4(x)}}\right)-\frac{3 \sqrt{a} \sqrt{b} \sqrt{a+b \tanh ^4(x)} \sinh ^{-1}\left(\frac{\sqrt{b} \tanh ^2(x)}{\sqrt{a}}\right)}{\sqrt{\frac{b \tanh ^4(x)}{a}+1}}\right)","-\frac{1}{6} \left(a+b \tanh ^4(x)\right)^{3/2}-\frac{1}{4} \left(2 (a+b)+b \tanh ^2(x)\right) \sqrt{a+b \tanh ^4(x)}+\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{a+b \tanh ^2(x)}{\sqrt{a+b} \sqrt{a+b \tanh ^4(x)}}\right)-\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh ^2(x)}{\sqrt{a+b \tanh ^4(x)}}\right)",1,"(-6*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*Tanh[x]^2)/Sqrt[a + b*Tanh[x]^4]] + 6*(a + b)^(3/2)*ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])] - Sqrt[a + b*Tanh[x]^4]*(8*a + 6*b + 3*b*Tanh[x]^2 + 2*b*Tanh[x]^4) - (3*Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Tanh[x]^2)/Sqrt[a]]*Sqrt[a + b*Tanh[x]^4])/Sqrt[1 + (b*Tanh[x]^4)/a])/12","A",1
260,1,86,89,0.0620117,"\int \tanh (x) \sqrt{a+b \tanh ^4(x)} \, dx","Integrate[Tanh[x]*Sqrt[a + b*Tanh[x]^4],x]","\frac{1}{2} \left(-\sqrt{a+b \tanh ^4(x)}-\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh ^2(x)}{\sqrt{a+b \tanh ^4(x)}}\right)+\sqrt{a+b} \tanh ^{-1}\left(\frac{a+b \tanh ^2(x)}{\sqrt{a+b} \sqrt{a+b \tanh ^4(x)}}\right)\right)","-\frac{1}{2} \sqrt{a+b \tanh ^4(x)}-\frac{1}{2} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh ^2(x)}{\sqrt{a+b \tanh ^4(x)}}\right)+\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left(\frac{a+b \tanh ^2(x)}{\sqrt{a+b} \sqrt{a+b \tanh ^4(x)}}\right)",1,"(-(Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[x]^2)/Sqrt[a + b*Tanh[x]^4]]) + Sqrt[a + b]*ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])] - Sqrt[a + b*Tanh[x]^4])/2","A",1
261,1,40,40,0.0177537,"\int \frac{\tanh (x)}{\sqrt{a+b \tanh ^4(x)}} \, dx","Integrate[Tanh[x]/Sqrt[a + b*Tanh[x]^4],x]","\frac{\tanh ^{-1}\left(\frac{a+b \tanh ^2(x)}{\sqrt{a+b} \sqrt{a+b \tanh ^4(x)}}\right)}{2 \sqrt{a+b}}","\frac{\tanh ^{-1}\left(\frac{a+b \tanh ^2(x)}{\sqrt{a+b} \sqrt{a+b \tanh ^4(x)}}\right)}{2 \sqrt{a+b}}",1,"ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])]/(2*Sqrt[a + b])","A",1
262,1,73,74,0.543914,"\int \frac{\tanh (x)}{\left(a+b \tanh ^4(x)\right)^{3/2}} \, dx","Integrate[Tanh[x]/(a + b*Tanh[x]^4)^(3/2),x]","\frac{1}{2} \left(\frac{\tanh ^{-1}\left(\frac{a+b \tanh ^2(x)}{\sqrt{a+b} \sqrt{a+b \tanh ^4(x)}}\right)}{(a+b)^{3/2}}-\frac{a-b \tanh ^2(x)}{a (a+b) \sqrt{a+b \tanh ^4(x)}}\right)","\frac{\tanh ^{-1}\left(\frac{a+b \tanh ^2(x)}{\sqrt{a+b} \sqrt{a+b \tanh ^4(x)}}\right)}{2 (a+b)^{3/2}}-\frac{a-b \tanh ^2(x)}{2 a (a+b) \sqrt{a+b \tanh ^4(x)}}",1,"(ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])]/(a + b)^(3/2) - (a - b*Tanh[x]^2)/(a*(a + b)*Sqrt[a + b*Tanh[x]^4]))/2","A",1
263,1,113,118,0.8530622,"\int \frac{\tanh (x)}{\left(a+b \tanh ^4(x)\right)^{5/2}} \, dx","Integrate[Tanh[x]/(a + b*Tanh[x]^4)^(5/2),x]","\frac{1}{6} \left(\frac{-3 a^2 b \tanh ^4(x)-a^2 (4 a+b)+b^2 (5 a+2 b) \tanh ^6(x)+3 a b (2 a+b) \tanh ^2(x)}{a^2 (a+b)^2 \left(a+b \tanh ^4(x)\right)^{3/2}}+\frac{3 \tanh ^{-1}\left(\frac{a+b \tanh ^2(x)}{\sqrt{a+b} \sqrt{a+b \tanh ^4(x)}}\right)}{(a+b)^{5/2}}\right)","-\frac{3 a^2-b (5 a+2 b) \tanh ^2(x)}{6 a^2 (a+b)^2 \sqrt{a+b \tanh ^4(x)}}-\frac{a-b \tanh ^2(x)}{6 a (a+b) \left(a+b \tanh ^4(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{a+b \tanh ^2(x)}{\sqrt{a+b} \sqrt{a+b \tanh ^4(x)}}\right)}{2 (a+b)^{5/2}}",1,"((3*ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])])/(a + b)^(5/2) + (-(a^2*(4*a + b)) + 3*a*b*(2*a + b)*Tanh[x]^2 - 3*a^2*b*Tanh[x]^4 + b^2*(5*a + 2*b)*Tanh[x]^6)/(a^2*(a + b)^2*(a + b*Tanh[x]^4)^(3/2)))/6","A",1