1,1,54,70,0.3325016,"\int \left(a+b \text{sech}^2(c+d x)\right) \sinh ^4(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)*Sinh[c + d*x]^4,x]","\frac{12 (a-4 b) (c+d x)-8 (a-b) \sinh (2 (c+d x))+a \sinh (4 (c+d x))+32 b \tanh (c+d x)}{32 d}","-\frac{(5 a-4 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3}{8} x (a-4 b)+\frac{a \sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac{b \tanh (c+d x)}{d}",1,"(12*(a - 4*b)*(c + d*x) - 8*(a - b)*Sinh[2*(c + d*x)] + a*Sinh[4*(c + d*x)] + 32*b*Tanh[c + d*x])/(32*d)","A",1
2,1,53,44,0.0516431,"\int \left(a+b \text{sech}^2(c+d x)\right) \sinh ^3(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)*Sinh[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 \cosh (c+d x)}{d}+\frac{b \text{sech}(c+d x)}{d}","-\frac{(a-b) \cosh (c+d x)}{d}+\frac{a \cosh ^3(c+d x)}{3 d}+\frac{b \text{sech}(c+d x)}{d}",1,"(-3*a*Cosh[c + d*x])/(4*d) + (b*Cosh[c + d*x])/d + (a*Cosh[3*(c + d*x)])/(12*d) + (b*Sech[c + d*x])/d","A",1
3,1,57,43,0.1941045,"\int \left(a+b \text{sech}^2(c+d x)\right) \sinh ^2(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)*Sinh[c + d*x]^2,x]","\frac{a (-c-d x)}{2 d}+\frac{a \sinh (2 (c+d x))}{4 d}+\frac{b \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{b \tanh (c+d x)}{d}","-\frac{1}{2} x (a-2 b)+\frac{a \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b \tanh (c+d x)}{d}",1,"(a*(-c - d*x))/(2*d) + (b*ArcTanh[Tanh[c + d*x]])/d + (a*Sinh[2*(c + d*x)])/(4*d) - (b*Tanh[c + d*x])/d","A",1
4,1,35,24,0.0219893,"\int \left(a+b \text{sech}^2(c+d x)\right) \sinh (c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)*Sinh[c + d*x],x]","\frac{a \sinh (c) \sinh (d x)}{d}+\frac{a \cosh (c) \cosh (d x)}{d}-\frac{b \text{sech}(c+d x)}{d}","\frac{a \cosh (c+d x)}{d}-\frac{b \text{sech}(c+d x)}{d}",1,"(a*Cosh[c]*Cosh[d*x])/d - (b*Sech[c + d*x])/d + (a*Sinh[c]*Sinh[d*x])/d","A",1
5,1,67,27,0.047743,"\int \text{csch}(c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Csch[c + d*x]*(a + b*Sech[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{b \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{d}","\frac{b \text{sech}(c+d x)}{d}-\frac{(a+b) \tanh ^{-1}(\cosh (c+d x))}{d}",1,"-((a*Log[Cosh[c/2 + (d*x)/2]])/d) + (a*Log[Sinh[c/2 + (d*x)/2]])/d + (b*Log[Tanh[(c + d*x)/2]])/d + (b*Sech[c + d*x])/d","B",1
6,1,37,27,0.0730566,"\int \text{csch}^2(c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^2*(a + b*Sech[c + d*x]^2),x]","-\frac{a \coth (c+d x)}{d}-\frac{b \tanh (c+d x)}{d}-\frac{b \coth (c+d x)}{d}","-\frac{(a+b) \coth (c+d x)}{d}-\frac{b \tanh (c+d x)}{d}",1,"-((a*Coth[c + d*x])/d) - (b*Coth[c + d*x])/d - (b*Tanh[c + d*x])/d","A",1
7,1,131,54,0.0490436,"\int \text{csch}^3(c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^3*(a + b*Sech[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{csch}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{b \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{b \text{sech}(c+d x)}{d}-\frac{3 b \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}","\frac{(a+3 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{(a+b) \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 - (b*Csch[(c + d*x)/2]^2)/(8*d) - (a*Log[Tanh[(c + d*x)/2]])/(2*d) - (3*b*Log[Tanh[(c + d*x)/2]])/(2*d) - (a*Sech[(c + d*x)/2]^2)/(8*d) - (b*Sech[(c + d*x)/2]^2)/(8*d) - (b*Sech[c + d*x])/d","B",1
8,1,84,45,0.0541938,"\int \text{csch}^4(c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^4*(a + b*Sech[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{5 b \coth (c+d x)}{3 d}-\frac{b \coth (c+d x) \text{csch}^2(c+d x)}{3 d}","-\frac{(a+b) \coth ^3(c+d x)}{3 d}+\frac{(a+2 b) \coth (c+d x)}{d}+\frac{b \tanh (c+d x)}{d}",1,"(2*a*Coth[c + d*x])/(3*d) + (5*b*Coth[c + d*x])/(3*d) - (a*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (b*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) + (b*Tanh[c + d*x])/d","A",1
9,1,153,114,1.6628694,"\int \left(a+b \text{sech}^2(c+d x)\right)^2 \sinh ^4(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^2*Sinh[c + d*x]^4,x]","\frac{\text{sech}^3(c+d x) \left(a \cosh ^2(c+d x)+b\right)^2 \left(3 \cosh ^3(c+d x) \left(4 d x \left(3 a^2-24 a b+8 b^2\right)+a^2 \sinh (4 (c+d x))-8 a (a-2 b) \sinh (2 (c+d x))\right)+64 b (3 a-2 b) \text{sech}(c) \sinh (d x) \cosh ^2(c+d x)+32 b^2 \tanh (c) \cosh (c+d x)+32 b^2 \text{sech}(c) \sinh (d x)\right)}{24 d (a \cosh (2 (c+d x))+a+2 b)^2}","-\frac{\left(a^2-8 a b+4 b^2\right) \tanh (c+d x)}{4 d}+\frac{1}{8} x \left(3 a^2-24 a b+8 b^2\right)+\frac{a^2 \sinh ^4(c+d x) \tanh (c+d x)}{4 d}-\frac{a (a-8 b) \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac{b^2 \tanh ^3(c+d x)}{3 d}",1,"((b + a*Cosh[c + d*x]^2)^2*Sech[c + d*x]^3*(32*b^2*Sech[c]*Sinh[d*x] + 64*(3*a - 2*b)*b*Cosh[c + d*x]^2*Sech[c]*Sinh[d*x] + 3*Cosh[c + d*x]^3*(4*(3*a^2 - 24*a*b + 8*b^2)*d*x - 8*a*(a - 2*b)*Sinh[2*(c + d*x)] + a^2*Sinh[4*(c + d*x)]) + 32*b^2*Cosh[c + d*x]*Tanh[c]))/(24*d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)","A",1
10,1,83,72,0.5258157,"\int \left(a+b \text{sech}^2(c+d x)\right)^2 \sinh ^3(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^2*Sinh[c + d*x]^3,x]","\frac{\text{sech}^3(c+d x) \left(-3 \left(11 a^2-64 a b+16 b^2\right) \cosh (2 (c+d x))+a^2 \cosh (6 (c+d x))-26 a^2-6 a (a-4 b) \cosh (4 (c+d x))+168 a b-16 b^2\right)}{96 d}","\frac{a^2 \cosh ^3(c+d x)}{3 d}-\frac{a (a-2 b) \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,"((-26*a^2 + 168*a*b - 16*b^2 - 3*(11*a^2 - 64*a*b + 16*b^2)*Cosh[2*(c + d*x)] - 6*a*(a - 4*b)*Cosh[4*(c + d*x)] + a^2*Cosh[6*(c + d*x)])*Sech[c + d*x]^3)/(96*d)","A",1
11,1,126,73,0.9518201,"\int \left(a+b \text{sech}^2(c+d x)\right)^2 \sinh ^2(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^2*Sinh[c + d*x]^2,x]","\frac{\text{sech}^3(c+d x) \left(a \cosh ^2(c+d x)+b\right)^2 \left(3 a \cosh ^3(c+d x) (a \sinh (2 (c+d x))-2 d x (a-4 b))-4 b (6 a-b) \text{sech}(c) \sinh (d x) \cosh ^2(c+d x)-4 b^2 \tanh (c) \cosh (c+d x)-4 b^2 \text{sech}(c) \sinh (d x)\right)}{3 d (a \cosh (2 (c+d x))+a+2 b)^2}","\frac{a^2 \sinh ^2(c+d x) \tanh (c+d x)}{2 d}+\frac{a (a-4 b) \tanh (c+d x)}{2 d}-\frac{1}{2} a x (a-4 b)+\frac{b^2 \tanh ^3(c+d x)}{3 d}",1,"((b + a*Cosh[c + d*x]^2)^2*Sech[c + d*x]^3*(-4*b^2*Sech[c]*Sinh[d*x] - 4*(6*a - b)*b*Cosh[c + d*x]^2*Sech[c]*Sinh[d*x] + 3*a*Cosh[c + d*x]^3*(-2*(a - 4*b)*d*x + a*Sinh[2*(c + d*x)]) - 4*b^2*Cosh[c + d*x]*Tanh[c]))/(3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)","A",1
12,1,59,45,0.1732985,"\int \left(a+b \text{sech}^2(c+d x)\right)^2 \sinh (c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^2*Sinh[c + d*x],x]","\frac{\text{sech}^3(c+d x) \left(3 a^2 \cosh (4 (c+d x))+9 a^2+12 a (a-2 b) \cosh (2 (c+d x))-24 a b-8 b^2\right)}{24 d}","\frac{a^2 \cosh (c+d x)}{d}-\frac{2 a b \text{sech}(c+d x)}{d}-\frac{b^2 \text{sech}^3(c+d x)}{3 d}",1,"((9*a^2 - 24*a*b - 8*b^2 + 12*a*(a - 2*b)*Cosh[2*(c + d*x)] + 3*a^2*Cosh[4*(c + d*x)])*Sech[c + d*x]^3)/(24*d)","A",1
13,1,108,52,0.5765642,"\int \text{csch}(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]*(a + b*Sech[c + d*x]^2)^2,x]","-\frac{4 \text{sech}^3(c+d x) \left(a \cosh ^2(c+d x)+b\right)^2 \left(-3 b (2 a+b) \cosh ^2(c+d x)+3 (a+b)^2 \cosh ^3(c+d x) \left(\log \left(\cosh \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sinh \left(\frac{1}{2} (c+d x)\right)\right)\right)-b^2\right)}{3 d (a \cosh (2 (c+d x))+a+2 b)^2}","\frac{b (2 a+b) \text{sech}(c+d x)}{d}-\frac{(a+b)^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b^2 \text{sech}^3(c+d x)}{3 d}",1,"(-4*(b + a*Cosh[c + d*x]^2)^2*(-b^2 - 3*b*(2*a + b)*Cosh[c + d*x]^2 + 3*(a + b)^2*Cosh[c + d*x]^3*(Log[Cosh[(c + d*x)/2]] - Log[Sinh[(c + d*x)/2]]))*Sech[c + d*x]^3)/(3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)","B",1
14,1,109,50,1.766876,"\int \text{csch}^2(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^2*(a + b*Sech[c + d*x]^2)^2,x]","-\frac{4 \text{sech}^3(c+d x) \left(a \cosh ^2(c+d x)+b\right)^2 \left(\sinh (d x) \cosh ^2(c+d x) \left(b (6 a+5 b) \text{sech}(c)-3 (a+b)^2 \text{csch}(c) \coth (c+d x)\right)+b^2 \tanh (c) \cosh (c+d x)+b^2 \text{sech}(c) \sinh (d x)\right)}{3 d (a \cosh (2 (c+d x))+a+2 b)^2}","-\frac{2 b (a+b) \tanh (c+d x)}{d}-\frac{(a+b)^2 \coth (c+d x)}{d}+\frac{b^2 \tanh ^3(c+d x)}{3 d}",1,"(-4*(b + a*Cosh[c + d*x]^2)^2*Sech[c + d*x]^3*(b^2*Sech[c]*Sinh[d*x] + Cosh[c + d*x]^2*(-3*(a + b)^2*Coth[c + d*x]*Csch[c] + b*(6*a + 5*b)*Sech[c])*Sinh[d*x] + b^2*Cosh[c + d*x]*Tanh[c]))/(3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)","B",1
15,1,144,104,1.6445696,"\int \text{csch}^3(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^3*(a + b*Sech[c + d*x]^2)^2,x]","-\frac{\text{sech}^3(c+d x) \left(a \cosh ^2(c+d x)+b\right)^2 \left(48 b (a+b) \cosh ^2(c+d x)+3 (a+b) \cosh ^3(c+d x) \left((a+b) \text{csch}^2\left(\frac{1}{2} (c+d x)\right)+(a+b) \text{sech}^2\left(\frac{1}{2} (c+d x)\right)-4 (a+5 b) \left(\log \left(\cosh \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sinh \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)+8 b^2\right)}{6 d (a \cosh (2 (c+d x))+a+2 b)^2}","-\frac{\left(3 a^2+6 a b+5 b^2\right) \coth (c+d x) \text{csch}(c+d x)}{6 d}-\frac{b (6 a+5 b) \text{sech}(c+d x)}{3 d}+\frac{(a+b) (a+5 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac{b^2 \text{csch}^2(c+d x) \text{sech}^3(c+d x)}{3 d}",1,"-1/6*((b + a*Cosh[c + d*x]^2)^2*(8*b^2 + 48*b*(a + b)*Cosh[c + d*x]^2 + 3*(a + b)*Cosh[c + d*x]^3*((a + b)*Csch[(c + d*x)/2]^2 - 4*(a + 5*b)*(Log[Cosh[(c + d*x)/2]] - Log[Sinh[(c + d*x)/2]]) + (a + b)*Sech[(c + d*x)/2]^2))*Sech[c + d*x]^3)/(d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)","A",1
16,1,151,75,1.3195017,"\int \text{csch}^4(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^4*(a + b*Sech[c + d*x]^2)^2,x]","-\frac{\text{csch}(2 c) \text{csch}^3(2 (c+d x)) \left(-3 a^2 \sinh (2 (c+d x))+a^2 \sinh (6 (c+d x))+3 a^2 \sinh (4 c+2 d x)+a^2 \sinh (4 c+6 d x)-6 a b \sinh (2 (c+d x))+2 a b \sinh (6 (c+d x))+8 a b \sinh (4 c+6 d x)+8 a (a+2 b) \sinh (2 c)-6 (a+2 b)^2 \sinh (2 d x)+8 b^2 \sinh (4 c+6 d x)\right)}{6 d}","\frac{b (2 a+3 b) \tanh (c+d x)}{d}-\frac{(a+b)^2 \coth ^3(c+d x)}{3 d}+\frac{(a+b) (a+3 b) \coth (c+d x)}{d}-\frac{b^2 \tanh ^3(c+d x)}{3 d}",1,"-1/6*(Csch[2*c]*Csch[2*(c + d*x)]^3*(8*a*(a + 2*b)*Sinh[2*c] - 6*(a + 2*b)^2*Sinh[2*d*x] - 3*a^2*Sinh[2*(c + d*x)] - 6*a*b*Sinh[2*(c + d*x)] + a^2*Sinh[6*(c + d*x)] + 2*a*b*Sinh[6*(c + d*x)] + 3*a^2*Sinh[4*c + 2*d*x] + a^2*Sinh[4*c + 6*d*x] + 8*a*b*Sinh[4*c + 6*d*x] + 8*b^2*Sinh[4*c + 6*d*x]))/d","B",1
17,1,651,182,2.5149169,"\int \left(a+b \text{sech}^2(c+d x)\right)^3 \sinh ^4(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^3*Sinh[c + d*x]^4,x]","\frac{\text{sech}(c) \text{sech}^5(c+d x) \left(a \cosh ^2(c+d x)+b\right)^3 \left(-180 a^3 \sinh (2 c+d x)-310 a^3 \sinh (2 c+3 d x)-310 a^3 \sinh (4 c+3 d x)-150 a^3 \sinh (4 c+5 d x)-150 a^3 \sinh (6 c+5 d x)-15 a^3 \sinh (6 c+7 d x)-15 a^3 \sinh (8 c+7 d x)+5 a^3 \sinh (8 c+9 d x)+5 a^3 \sinh (10 c+9 d x)+600 a^3 d x \cosh (2 c+3 d x)+600 a^3 d x \cosh (4 c+3 d x)+120 a^3 d x \cosh (4 c+5 d x)+120 a^3 d x \cosh (6 c+5 d x)-180 a^3 \sinh (d x)+1200 a d x \left(a^2-12 a b+8 b^2\right) \cosh (2 c+d x)+1200 a d x \left(a^2-12 a b+8 b^2\right) \cosh (d x)-7080 a^2 b \sinh (2 c+d x)+8760 a^2 b \sinh (2 c+3 d x)-840 a^2 b \sinh (4 c+3 d x)+2520 a^2 b \sinh (4 c+5 d x)+600 a^2 b \sinh (6 c+5 d x)+120 a^2 b \sinh (6 c+7 d x)+120 a^2 b \sinh (8 c+7 d x)-7200 a^2 b d x \cosh (2 c+3 d x)-7200 a^2 b d x \cosh (4 c+3 d x)-1440 a^2 b d x \cosh (4 c+5 d x)-1440 a^2 b d x \cosh (6 c+5 d x)+12120 a^2 b \sinh (d x)+11520 a b^2 \sinh (2 c+d x)-8960 a b^2 \sinh (2 c+3 d x)+3840 a b^2 \sinh (4 c+3 d x)-2560 a b^2 \sinh (4 c+5 d x)+4800 a b^2 d x \cosh (2 c+3 d x)+4800 a b^2 d x \cosh (4 c+3 d x)+960 a b^2 d x \cosh (4 c+5 d x)+960 a b^2 d x \cosh (6 c+5 d x)-14080 a b^2 \sinh (d x)-640 b^3 \sinh (4 c+3 d x)+128 b^3 \sinh (4 c+5 d x)+1280 b^3 \sinh (d x)\right)}{1280 d (a \cosh (2 (c+d x))+a+2 b)^3}","\frac{b \left(6 a^2-23 a b-8 b^2\right) \tanh ^3(c+d x)}{8 d}-\frac{3 a \left(a^2-12 a b+8 b^2\right) \tanh (c+d x)}{8 d}+\frac{3}{8} a x \left(a^2-12 a b+8 b^2\right)-\frac{3 b^2 (5 a-16 b) \tanh ^5(c+d x)}{40 d}-\frac{3 (a-2 b) \sinh ^2(c+d x) \tanh (c+d x) \left(a-b \tanh ^2(c+d x)+b\right)^2}{8 d}+\frac{\sinh ^3(c+d x) \cosh (c+d x) \left(a-b \tanh ^2(c+d x)+b\right)^3}{4 d}",1,"((b + a*Cosh[c + d*x]^2)^3*Sech[c]*Sech[c + d*x]^5*(1200*a*(a^2 - 12*a*b + 8*b^2)*d*x*Cosh[d*x] + 1200*a*(a^2 - 12*a*b + 8*b^2)*d*x*Cosh[2*c + d*x] + 600*a^3*d*x*Cosh[2*c + 3*d*x] - 7200*a^2*b*d*x*Cosh[2*c + 3*d*x] + 4800*a*b^2*d*x*Cosh[2*c + 3*d*x] + 600*a^3*d*x*Cosh[4*c + 3*d*x] - 7200*a^2*b*d*x*Cosh[4*c + 3*d*x] + 4800*a*b^2*d*x*Cosh[4*c + 3*d*x] + 120*a^3*d*x*Cosh[4*c + 5*d*x] - 1440*a^2*b*d*x*Cosh[4*c + 5*d*x] + 960*a*b^2*d*x*Cosh[4*c + 5*d*x] + 120*a^3*d*x*Cosh[6*c + 5*d*x] - 1440*a^2*b*d*x*Cosh[6*c + 5*d*x] + 960*a*b^2*d*x*Cosh[6*c + 5*d*x] - 180*a^3*Sinh[d*x] + 12120*a^2*b*Sinh[d*x] - 14080*a*b^2*Sinh[d*x] + 1280*b^3*Sinh[d*x] - 180*a^3*Sinh[2*c + d*x] - 7080*a^2*b*Sinh[2*c + d*x] + 11520*a*b^2*Sinh[2*c + d*x] - 310*a^3*Sinh[2*c + 3*d*x] + 8760*a^2*b*Sinh[2*c + 3*d*x] - 8960*a*b^2*Sinh[2*c + 3*d*x] - 310*a^3*Sinh[4*c + 3*d*x] - 840*a^2*b*Sinh[4*c + 3*d*x] + 3840*a*b^2*Sinh[4*c + 3*d*x] - 640*b^3*Sinh[4*c + 3*d*x] - 150*a^3*Sinh[4*c + 5*d*x] + 2520*a^2*b*Sinh[4*c + 5*d*x] - 2560*a*b^2*Sinh[4*c + 5*d*x] + 128*b^3*Sinh[4*c + 5*d*x] - 150*a^3*Sinh[6*c + 5*d*x] + 600*a^2*b*Sinh[6*c + 5*d*x] - 15*a^3*Sinh[6*c + 7*d*x] + 120*a^2*b*Sinh[6*c + 7*d*x] - 15*a^3*Sinh[8*c + 7*d*x] + 120*a^2*b*Sinh[8*c + 7*d*x] + 5*a^3*Sinh[8*c + 9*d*x] + 5*a^3*Sinh[10*c + 9*d*x]))/(1280*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)","B",1
18,1,119,99,1.2532504,"\int \left(a+b \text{sech}^2(c+d x)\right)^3 \sinh ^3(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^3*Sinh[c + d*x]^3,x]","\frac{4 \text{sech}^5(c+d x) \left(a \cosh ^2(c+d x)+b\right)^3 \left(5 a^2 \cosh ^6(c+d x) (a \cosh (2 (c+d x))-5 a+18 b)+10 b^2 (3 a-b) \cosh ^2(c+d x)+90 a b (a-b) \cosh ^4(c+d x)+6 b^3\right)}{15 d (a \cosh (2 (c+d x))+a+2 b)^3}","\frac{a^3 \cosh ^3(c+d x)}{3 d}-\frac{a^2 (a-3 b) \cosh (c+d x)}{d}+\frac{b^2 (3 a-b) \text{sech}^3(c+d x)}{3 d}+\frac{3 a b (a-b) \text{sech}(c+d x)}{d}+\frac{b^3 \text{sech}^5(c+d x)}{5 d}",1,"(4*(b + a*Cosh[c + d*x]^2)^3*(6*b^3 + 10*(3*a - b)*b^2*Cosh[c + d*x]^2 + 90*a*(a - b)*b*Cosh[c + d*x]^4 + 5*a^2*Cosh[c + d*x]^6*(-5*a + 18*b + a*Cosh[2*(c + d*x)]))*Sech[c + d*x]^5)/(15*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)","A",1
19,1,480,112,1.9156853,"\int \left(a+b \text{sech}^2(c+d x)\right)^3 \sinh ^2(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^3*Sinh[c + d*x]^2,x]","\frac{\text{sech}(c) \text{sech}^5(c+d x) \left(75 a^3 \sinh (2 c+d x)+135 a^3 \sinh (2 c+3 d x)+135 a^3 \sinh (4 c+3 d x)+75 a^3 \sinh (4 c+5 d x)+75 a^3 \sinh (6 c+5 d x)+15 a^3 \sinh (6 c+7 d x)+15 a^3 \sinh (8 c+7 d x)-300 a^3 d x \cosh (2 c+3 d x)-300 a^3 d x \cosh (4 c+3 d x)-60 a^3 d x \cosh (4 c+5 d x)-60 a^3 d x \cosh (6 c+5 d x)+75 a^3 \sinh (d x)+2880 a^2 b \sinh (2 c+d x)-2880 a^2 b \sinh (2 c+3 d x)+720 a^2 b \sinh (4 c+3 d x)-720 a^2 b \sinh (4 c+5 d x)-600 a^2 d x (a-6 b) \cosh (2 c+d x)+1800 a^2 b d x \cosh (2 c+3 d x)+1800 a^2 b d x \cosh (4 c+3 d x)+360 a^2 b d x \cosh (4 c+5 d x)+360 a^2 b d x \cosh (6 c+5 d x)-4320 a^2 b \sinh (d x)-600 a^2 d x (a-6 b) \cosh (d x)-1440 a b^2 \sinh (2 c+d x)+480 a b^2 \sinh (2 c+3 d x)-720 a b^2 \sinh (4 c+3 d x)+240 a b^2 \sinh (4 c+5 d x)+960 a b^2 \sinh (d x)-480 b^3 \sinh (2 c+d x)+160 b^3 \sinh (2 c+3 d x)+32 b^3 \sinh (4 c+5 d x)-160 b^3 \sinh (d x)\right)}{3840 d}","\frac{a^3}{4 d (1-\tanh (c+d x))}-\frac{a^3}{4 d (\tanh (c+d x)+1)}-\frac{3 a^2 b \tanh (c+d x)}{d}-\frac{1}{2} a^2 x (a-6 b)+\frac{b^2 (3 a+b) \tanh ^3(c+d x)}{3 d}-\frac{b^3 \tanh ^5(c+d x)}{5 d}",1,"(Sech[c]*Sech[c + d*x]^5*(-600*a^2*(a - 6*b)*d*x*Cosh[d*x] - 600*a^2*(a - 6*b)*d*x*Cosh[2*c + d*x] - 300*a^3*d*x*Cosh[2*c + 3*d*x] + 1800*a^2*b*d*x*Cosh[2*c + 3*d*x] - 300*a^3*d*x*Cosh[4*c + 3*d*x] + 1800*a^2*b*d*x*Cosh[4*c + 3*d*x] - 60*a^3*d*x*Cosh[4*c + 5*d*x] + 360*a^2*b*d*x*Cosh[4*c + 5*d*x] - 60*a^3*d*x*Cosh[6*c + 5*d*x] + 360*a^2*b*d*x*Cosh[6*c + 5*d*x] + 75*a^3*Sinh[d*x] - 4320*a^2*b*Sinh[d*x] + 960*a*b^2*Sinh[d*x] - 160*b^3*Sinh[d*x] + 75*a^3*Sinh[2*c + d*x] + 2880*a^2*b*Sinh[2*c + d*x] - 1440*a*b^2*Sinh[2*c + d*x] - 480*b^3*Sinh[2*c + d*x] + 135*a^3*Sinh[2*c + 3*d*x] - 2880*a^2*b*Sinh[2*c + 3*d*x] + 480*a*b^2*Sinh[2*c + 3*d*x] + 160*b^3*Sinh[2*c + 3*d*x] + 135*a^3*Sinh[4*c + 3*d*x] + 720*a^2*b*Sinh[4*c + 3*d*x] - 720*a*b^2*Sinh[4*c + 3*d*x] + 75*a^3*Sinh[4*c + 5*d*x] - 720*a^2*b*Sinh[4*c + 5*d*x] + 240*a*b^2*Sinh[4*c + 5*d*x] + 32*b^3*Sinh[4*c + 5*d*x] + 75*a^3*Sinh[6*c + 5*d*x] + 15*a^3*Sinh[6*c + 7*d*x] + 15*a^3*Sinh[8*c + 7*d*x]))/(3840*d)","B",1
20,1,93,64,0.2574099,"\int \left(a+b \text{sech}^2(c+d x)\right)^3 \sinh (c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^3*Sinh[c + d*x],x]","\frac{8 \text{sech}^5(c+d x) \left(a \cosh ^2(c+d x)+b\right)^3 \left(5 a^3 \cosh ^6(c+d x)-15 a^2 b \cosh ^4(c+d x)-5 a b^2 \cosh ^2(c+d x)-b^3\right)}{5 d (a \cosh (2 (c+d x))+a+2 b)^3}","\frac{a^3 \cosh (c+d x)}{d}-\frac{3 a^2 b \text{sech}(c+d x)}{d}-\frac{a b^2 \text{sech}^3(c+d x)}{d}-\frac{b^3 \text{sech}^5(c+d x)}{5 d}",1,"(8*(b + a*Cosh[c + d*x]^2)^3*(-b^3 - 5*a*b^2*Cosh[c + d*x]^2 - 15*a^2*b*Cosh[c + d*x]^4 + 5*a^3*Cosh[c + d*x]^6)*Sech[c + d*x]^5)/(5*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)","A",1
21,1,134,83,1.2809171,"\int \text{csch}(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]*(a + b*Sech[c + d*x]^2)^3,x]","-\frac{8 \text{sech}^5(c+d x) \left(a \cosh ^2(c+d x)+b\right)^3 \left(-15 b \left(3 a^2+3 a b+b^2\right) \cosh ^4(c+d x)-5 b^2 (3 a+b) \cosh ^2(c+d x)+15 (a+b)^3 \cosh ^5(c+d x) \left(\log \left(\cosh \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sinh \left(\frac{1}{2} (c+d x)\right)\right)\right)-3 b^3\right)}{15 d (a \cosh (2 (c+d x))+a+2 b)^3}","\frac{b \left(3 a^2+3 a b+b^2\right) \text{sech}(c+d x)}{d}+\frac{b^2 (3 a+b) \text{sech}^3(c+d x)}{3 d}-\frac{(a+b)^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b^3 \text{sech}^5(c+d x)}{5 d}",1,"(-8*(b + a*Cosh[c + d*x]^2)^3*(-3*b^3 - 5*b^2*(3*a + b)*Cosh[c + d*x]^2 - 15*b*(3*a^2 + 3*a*b + b^2)*Cosh[c + d*x]^4 + 15*(a + b)^3*Cosh[c + d*x]^5*(Log[Cosh[(c + d*x)/2]] - Log[Sinh[(c + d*x)/2]]))*Sech[c + d*x]^5)/(15*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)","A",1
22,1,380,70,2.7926788,"\int \text{csch}^2(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^2*(a + b*Sech[c + d*x]^2)^3,x]","-\frac{\text{csch}(c) \text{sech}(c) \coth (c+d x) \left(-25 a^3 \sinh (2 (c+d x))-20 a^3 \sinh (4 (c+d x))-5 a^3 \sinh (6 (c+d x))-25 a^3 \sinh (2 (c+2 d x))+25 a^3 \sinh (4 c+2 d x)+5 a^3 \sinh (6 c+4 d x)-5 a^3 \sinh (4 c+6 d x)+10 a \left(5 a^2+12 a b+8 b^2\right) \sinh (2 c)-120 a^2 b \sinh (2 (c+2 d x))+30 a^2 b \sinh (4 c+2 d x)-30 a^2 b \sinh (4 c+6 d x)-10 \left(5 a^3+18 a^2 b+20 a b^2+8 b^3\right) \sinh (2 d x)+50 a b^2 \sinh (2 (c+d x))+40 a b^2 \sinh (4 (c+d x))+10 a b^2 \sinh (6 (c+d x))-160 a b^2 \sinh (2 (c+2 d x))-40 a b^2 \sinh (4 c+6 d x)+30 b^3 \sinh (2 (c+d x))+24 b^3 \sinh (4 (c+d x))+6 b^3 \sinh (6 (c+d x))-64 b^3 \sinh (2 (c+2 d x))-16 b^3 \sinh (4 c+6 d x)\right) \left(a+b \text{sech}^2(c+d x)\right)^3}{40 d (a \cosh (2 (c+d x))+a+2 b)^3}","\frac{b^2 (a+b) \tanh ^3(c+d x)}{d}-\frac{3 b (a+b)^2 \tanh (c+d x)}{d}-\frac{(a+b)^3 \coth (c+d x)}{d}-\frac{b^3 \tanh ^5(c+d x)}{5 d}",1,"-1/40*(Coth[c + d*x]*Csch[c]*Sech[c]*(a + b*Sech[c + d*x]^2)^3*(10*a*(5*a^2 + 12*a*b + 8*b^2)*Sinh[2*c] - 10*(5*a^3 + 18*a^2*b + 20*a*b^2 + 8*b^3)*Sinh[2*d*x] - 25*a^3*Sinh[2*(c + d*x)] + 50*a*b^2*Sinh[2*(c + d*x)] + 30*b^3*Sinh[2*(c + d*x)] - 20*a^3*Sinh[4*(c + d*x)] + 40*a*b^2*Sinh[4*(c + d*x)] + 24*b^3*Sinh[4*(c + d*x)] - 5*a^3*Sinh[6*(c + d*x)] + 10*a*b^2*Sinh[6*(c + d*x)] + 6*b^3*Sinh[6*(c + d*x)] - 25*a^3*Sinh[2*(c + 2*d*x)] - 120*a^2*b*Sinh[2*(c + 2*d*x)] - 160*a*b^2*Sinh[2*(c + 2*d*x)] - 64*b^3*Sinh[2*(c + 2*d*x)] + 25*a^3*Sinh[4*c + 2*d*x] + 30*a^2*b*Sinh[4*c + 2*d*x] + 5*a^3*Sinh[6*c + 4*d*x] - 5*a^3*Sinh[4*c + 6*d*x] - 30*a^2*b*Sinh[4*c + 6*d*x] - 40*a*b^2*Sinh[4*c + 6*d*x] - 16*b^3*Sinh[4*c + 6*d*x]))/(d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)","B",1
23,1,224,144,3.8872411,"\int \text{csch}^3(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^3*(a + b*Sech[c + d*x]^2)^3,x]","-\frac{\text{sech}^6(c+d x) \left(a \cosh ^2(c+d x)+b\right)^3 \left(\frac{3}{4} \left(75 a^3+195 a^2 b+165 a b^2+77 b^3\right) \sinh (4 (c+d x)) \text{csch}^3(c+d x)+\coth (c+d x) \text{csch}(c+d x) \left(150 a^3+270 a^2 b+10 \left(9 a^3+45 a^2 b+75 a b^2+35 b^3\right) \cosh (4 (c+d x))-30 a b^2+15 (a+b)^2 (a+7 b) \cosh (6 (c+d x))-206 b^3\right)-480 (a+b)^2 (a+7 b) \cosh ^6(c+d x) \left(\log \left(\cosh \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sinh \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{120 d (a \cosh (2 (c+d x))+a+2 b)^3}","-\frac{b \left(6 a^2+15 a b+7 b^2\right) \text{sech}^3(c+d x)}{6 d}-\frac{b^2 (5 a+7 b) \text{sech}^5(c+d x)}{10 d}-\frac{(a+b)^2 (a+7 b) \text{sech}(c+d x)}{2 d}+\frac{(a+b)^2 (a+7 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{(a+b) \text{csch}^2(c+d x) \text{sech}^5(c+d x) \left(a \cosh ^2(c+d x)+b\right)^2}{2 d}",1,"-1/120*((b + a*Cosh[c + d*x]^2)^3*Sech[c + d*x]^6*((150*a^3 + 270*a^2*b - 30*a*b^2 - 206*b^3 + 10*(9*a^3 + 45*a^2*b + 75*a*b^2 + 35*b^3)*Cosh[4*(c + d*x)] + 15*(a + b)^2*(a + 7*b)*Cosh[6*(c + d*x)])*Coth[c + d*x]*Csch[c + d*x] - 480*(a + b)^2*(a + 7*b)*Cosh[c + d*x]^6*(Log[Cosh[(c + d*x)/2]] - Log[Sinh[(c + d*x)/2]]) + (3*(75*a^3 + 195*a^2*b + 165*a*b^2 + 77*b^3)*Csch[c + d*x]^3*Sinh[4*(c + d*x)])/4))/(d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)","A",1
24,1,213,104,2.4656569,"\int \text{csch}^4(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^4*(a + b*Sech[c + d*x]^2)^3,x]","-\frac{8 \tanh (c) \text{sech}^5(c+d x) \left(a \cosh ^2(c+d x)+b\right)^3 \left(\text{csch}(c) \sinh (d x) \cosh ^4(c+d x) \left(5 (a+b)^2 (2 a+11 b) \coth (c) \coth (c+d x)-b \left(45 a^2+120 a b+73 b^2\right)\right)+\cosh ^3(c+d x) \left(5 (a+b)^3 \coth ^2(c) \coth ^2(c+d x)-b^2 (15 a+14 b)\right)-\text{csch}(c) \sinh (d x) \cosh ^2(c+d x) \left(b^2 (15 a+14 b)+5 (a+b)^3 \coth (c) \coth ^3(c+d x)\right)-3 b^3 \cosh (c+d x)-3 b^3 \text{csch}(c) \sinh (d x)\right)}{15 d (a \cosh (2 (c+d x))+a+2 b)^3}","-\frac{b^2 (3 a+4 b) \tanh ^3(c+d x)}{3 d}+\frac{3 b (a+b) (a+2 b) \tanh (c+d x)}{d}-\frac{(a+b)^3 \coth ^3(c+d x)}{3 d}+\frac{(a+b)^2 (a+4 b) \coth (c+d x)}{d}+\frac{b^3 \tanh ^5(c+d x)}{5 d}",1,"(-8*(b + a*Cosh[c + d*x]^2)^3*Sech[c + d*x]^5*(-3*b^3*Cosh[c + d*x] + Cosh[c + d*x]^3*(-(b^2*(15*a + 14*b)) + 5*(a + b)^3*Coth[c]^2*Coth[c + d*x]^2) - 3*b^3*Csch[c]*Sinh[d*x] + Cosh[c + d*x]^4*(-(b*(45*a^2 + 120*a*b + 73*b^2)) + 5*(a + b)^2*(2*a + 11*b)*Coth[c]*Coth[c + d*x])*Csch[c]*Sinh[d*x] - Cosh[c + d*x]^2*(b^2*(15*a + 14*b) + 5*(a + b)^3*Coth[c]*Coth[c + d*x]^3)*Csch[c]*Sinh[d*x])*Tanh[c])/(15*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)","B",1
25,1,294,117,2.4825248,"\int \frac{\sinh ^4(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Sinh[c + d*x]^4/(a + b*Sech[c + d*x]^2),x]","-\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\sqrt{b} \left(3 a^3+34 a^2 b+64 a b^2+32 b^3\right) (\cosh (2 c)-\sinh (2 c)) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)-\sqrt{b (\cosh (c)-\sinh (c))^4} \left(\sqrt{b} \sqrt{a+b} \left(a^2 \sinh (4 (c+d x))-2 a^2 c+12 a^2 d x-8 a (a+b) \sinh (2 (c+d x))+48 a b d x+32 b^2 d x\right)+a^2 (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)\right)\right)}{64 a^3 \sqrt{b} d \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4} \left(a+b \text{sech}^2(c+d x)\right)}","-\frac{\sqrt{b} (a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{a^3 d}-\frac{(5 a+4 b) \sinh (c+d x) \cosh (c+d x)}{8 a^2 d}+\frac{x \left(3 a^2+12 a b+8 b^2\right)}{8 a^3}+\frac{\sinh (c+d x) \cosh ^3(c+d x)}{4 a d}",1,"-1/64*((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(Sqrt[b]*(3*a^3 + 34*a^2*b + 64*a*b^2 + 32*b^3)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]) - Sqrt[b*(Cosh[c] - Sinh[c])^4]*(a^2*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]] + Sqrt[b]*Sqrt[a + b]*(-2*a^2*c + 12*a^2*d*x + 48*a*b*d*x + 32*b^2*d*x - 8*a*(a + b)*Sinh[2*(c + d*x)] + a^2*Sinh[4*(c + d*x)]))))/(a^3*Sqrt[b]*Sqrt[a + b]*d*(a + b*Sech[c + d*x]^2)*Sqrt[b*(Cosh[c] - Sinh[c])^4])","B",1
26,1,372,71,2.2257121,"\int \frac{\sinh ^3(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Sinh[c + d*x]^3/(a + b*Sech[c + d*x]^2),x]","\frac{(a \cosh (2 (c+d x))+a+2 b) \left(2 a^{3/2} \sqrt{b} \cosh (3 (c+d x))+3 \left(a^2+8 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)+3 \left(a^2+8 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)-3 a^2 \left(\tan ^{-1}\left(\frac{\sqrt{a}-i \sqrt{a+b} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b}}\right)+\tan ^{-1}\left(\frac{\sqrt{a}+i \sqrt{a+b} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b}}\right)\right)-6 \sqrt{a} \sqrt{b} (3 a+4 b) \cosh (c+d x)\right)}{48 a^{5/2} \sqrt{b} d \left(a \cosh ^2(c+d x)+b\right)}","\frac{\sqrt{b} (a+b) \tan ^{-1}\left(\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right)}{a^{5/2} d}-\frac{(a+b) \cosh (c+d x)}{a^2 d}+\frac{\cosh ^3(c+d x)}{3 a d}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*(3*(a^2 + 8*a*b + 8*b^2)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + 3*(a^2 + 8*a*b + 8*b^2)*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] - 3*a^2*(ArcTan[(Sqrt[a] - I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[(Sqrt[a] + I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]]) - 6*Sqrt[a]*Sqrt[b]*(3*a + 4*b)*Cosh[c + d*x] + 2*a^(3/2)*Sqrt[b]*Cosh[3*(c + d*x)]))/(48*a^(5/2)*Sqrt[b]*d*(b + a*Cosh[c + d*x]^2))","C",1
27,1,236,75,0.9969975,"\int \frac{\sinh ^2(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Sinh[c + d*x]^2/(a + b*Sech[c + d*x]^2),x]","\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{\frac{\left(a^2+8 a b+8 b^2\right) (\cosh (2 c)-\sinh (2 c)) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{d \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}-4 x (a+2 b)+\frac{2 a \sinh (2 c) \cosh (2 d x)}{d}+\frac{2 a \cosh (2 c) \sinh (2 d x)}{d}}{a^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{\sqrt{b} d \sqrt{a+b}}\right)}{16 \left(a+b \text{sech}^2(c+d x)\right)}","\frac{\sqrt{b} \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{a^2 d}-\frac{x (a+2 b)}{2 a^2}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 a d}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(-(ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b]*d)) + (-4*(a + 2*b)*x + ((a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (2*a*Cosh[2*d*x]*Sinh[2*c])/d + (2*a*Cosh[2*c]*Sinh[2*d*x])/d)/a^2))/(16*(a + b*Sech[c + d*x]^2))","B",1
28,1,328,47,1.015406,"\int \frac{\sinh (c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Sinh[c + d*x]/(a + b*Sech[c + d*x]^2),x]","\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{a \left(\tan ^{-1}\left(\frac{\sqrt{a}-i \sqrt{a+b} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b}}\right)+\tan ^{-1}\left(\frac{\sqrt{a}+i \sqrt{a+b} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b}}\right)\right)}{\sqrt{b}}-\frac{(a+4 b) \left(\tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)+\tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)\right)}{\sqrt{b}}+4 \sqrt{a} \cosh (c+d x)\right)}{8 a^{3/2} d \left(a+b \text{sech}^2(c+d x)\right)}","\frac{\cosh (c+d x)}{a d}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right)}{a^{3/2} d}",1,"((-(((a + 4*b)*(ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]))/Sqrt[b]) + (a*(ArcTan[(Sqrt[a] - I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[(Sqrt[a] + I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]]))/Sqrt[b] + 4*Sqrt[a]*Cosh[c + d*x])*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2)/(8*a^(3/2)*d*(a + b*Sech[c + d*x]^2))","C",1
29,1,232,55,0.9284401,"\int \frac{\text{csch}(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Csch[c + d*x]/(a + b*Sech[c + d*x]^2),x]","\frac{\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)}{\sqrt{a}}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)}{\sqrt{a}}+\log \left(\sinh \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cosh \left(\frac{1}{2} (c+d x)\right)\right)}{d (a+b)}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right)}{\sqrt{a} d (a+b)}-\frac{\tanh ^{-1}(\cosh (c+d x))}{d (a+b)}",1,"((Sqrt[b]*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]])/Sqrt[a] + (Sqrt[b]*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]])/Sqrt[a] - Log[Cosh[(c + d*x)/2]] + Log[Sinh[(c + d*x)/2]])/((a + b)*d)","C",1
30,1,179,53,0.7521667,"\int \frac{\text{csch}^2(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Csch[c + d*x]^2/(a + b*Sech[c + d*x]^2),x]","\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\sqrt{a+b} \text{csch}(c) \sinh (d x) \sqrt{b (\cosh (c)-\sinh (c))^4} \text{csch}(c+d x)+b (\cosh (2 c)-\sinh (2 c)) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)\right)}{2 d (a+b)^{3/2} \sqrt{b (\cosh (c)-\sinh (c))^4} \left(a+b \text{sech}^2(c+d x)\right)}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{d (a+b)^{3/2}}-\frac{\coth (c+d x)}{d (a+b)}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(b*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]) + Sqrt[a + b]*Csch[c]*Csch[c + d*x]*Sqrt[b*(Cosh[c] - Sinh[c])^4]*Sinh[d*x]))/(2*(a + b)^(3/2)*d*(a + b*Sech[c + d*x]^2)*Sqrt[b*(Cosh[c] - Sinh[c])^4])","B",1
31,1,338,87,2.0501895,"\int \frac{\text{csch}^3(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Csch[c + d*x]^3/(a + b*Sech[c + d*x]^2),x]","-\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left((a+b) \text{csch}^2\left(\frac{1}{2} (c+d x)\right)+(a+b) \text{sech}^2\left(\frac{1}{2} (c+d x)\right)+8 \sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)+8 \sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)+4 a \log \left(\sinh \left(\frac{1}{2} (c+d x)\right)\right)-4 a \log \left(\cosh \left(\frac{1}{2} (c+d x)\right)\right)-4 b \log \left(\sinh \left(\frac{1}{2} (c+d x)\right)\right)+4 b \log \left(\cosh \left(\frac{1}{2} (c+d x)\right)\right)\right)}{16 d (a+b)^2 \left(a+b \text{sech}^2(c+d x)\right)}","-\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right)}{d (a+b)^2}+\frac{(a-b) \tanh ^{-1}(\cosh (c+d x))}{2 d (a+b)^2}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 d (a+b)}",1,"-1/16*((a + 2*b + a*Cosh[2*(c + d*x)])*(8*Sqrt[a]*Sqrt[b]*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + 8*Sqrt[a]*Sqrt[b]*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + (a + b)*Csch[(c + d*x)/2]^2 - 4*a*Log[Cosh[(c + d*x)/2]] + 4*b*Log[Cosh[(c + d*x)/2]] + 4*a*Log[Sinh[(c + d*x)/2]] - 4*b*Log[Sinh[(c + d*x)/2]] + (a + b)*Sech[(c + d*x)/2]^2)*Sech[c + d*x]^2)/((a + b)^2*d*(a + b*Sech[c + d*x]^2))","C",1
32,1,216,75,2.081985,"\int \frac{\text{csch}^4(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Csch[c + d*x]^4/(a + b*Sech[c + d*x]^2),x]","\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{1}{4} \sqrt{a+b} \text{csch}(c) \sqrt{b (\cosh (c)-\sinh (c))^4} \text{csch}^3(c+d x) ((b-2 a) \sinh (2 c+3 d x)+6 a \sinh (d x)-3 b \sinh (2 c+d x))+3 a b (\sinh (2 c)-\cosh (2 c)) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)\right)}{6 d (a+b)^{5/2} \sqrt{b (\cosh (c)-\sinh (c))^4} \left(a+b \text{sech}^2(c+d x)\right)}","-\frac{a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{d (a+b)^{5/2}}-\frac{\coth ^3(c+d x)}{3 d (a+b)}+\frac{a \coth (c+d x)}{d (a+b)^2}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(3*a*b*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(-Cosh[2*c] + Sinh[2*c]) + (Sqrt[a + b]*Csch[c]*Csch[c + d*x]^3*Sqrt[b*(Cosh[c] - Sinh[c])^4]*(6*a*Sinh[d*x] - 3*b*Sinh[2*c + d*x] + (-2*a + b)*Sinh[2*c + 3*d*x]))/4))/(6*(a + b)^(5/2)*d*(a + b*Sech[c + d*x]^2)*Sqrt[b*(Cosh[c] - Sinh[c])^4])","B",1
33,1,1330,194,14.2515604,"\int \frac{\sinh ^4(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2,x]","-\frac{(\cosh (2 c+2 d x) a+a+2 b)^2 \left(16 x+\frac{\left(a^3-6 b a^2-24 b^2 a-16 b^3\right) \tanh ^{-1}\left(\frac{\text{sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right) (\cosh (2 c)-\sinh (2 c))}{b (a+b)^{3/2} d \sqrt{b (\cosh (c)-\sinh (c))^4}}+\frac{\left(a^2+8 b a+8 b^2\right) \text{sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{b (a+b) d (\cosh (2 (c+d x)) a+a+2 b)}\right) \text{sech}^4(c+d x)}{256 a^2 \left(b \text{sech}^2(c+d x)+a\right)^2}+\frac{3 (\cosh (2 c+2 d x) a+a+2 b)^2 \left(\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 b^{3/2} (a+b)^{3/2} d}-\frac{a \sinh (2 (c+d x))}{8 b (a+b) d (\cosh (2 (c+d x)) a+a+2 b)}\right) \text{sech}^4(c+d x)}{128 \left(b \text{sech}^2(c+d x)+a\right)^2}+\frac{(\cosh (2 c+2 d x) a+a+2 b)^2 \left(\frac{\left(a^5-30 b a^4-480 b^2 a^3-1600 b^3 a^2-1920 b^4 a-768 b^5\right) \left(\frac{i \tan ^{-1}\left(\text{sech}(d x) \left(\frac{i \sinh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}-\frac{i \cosh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}\right) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right) \sinh (2 c)}{8 a^4 b \sqrt{a+b} d \sqrt{b \cosh (4 c)-b \sinh (4 c)}}-\frac{i \tan ^{-1}\left(\text{sech}(d x) \left(\frac{i \sinh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}-\frac{i \cosh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}\right) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right) \cosh (2 c)}{8 a^4 b \sqrt{a+b} d \sqrt{b \cosh (4 c)-b \sinh (4 c)}}\right)}{a+b}+\frac{\text{sech}(2 c) \left(\sinh (2 c) a^5-\sinh (2 d x) a^5+160 b d x \cosh (2 c) a^4+80 b d x \cosh (2 d x) a^4+80 b d x \cosh (4 c+2 d x) a^4+34 b \sinh (2 c) a^4-62 b \sinh (2 d x) a^4-30 b \sinh (4 c+2 d x) a^4-12 b \sinh (2 c+4 d x) a^4-12 b \sinh (6 c+4 d x) a^4+2 b \sinh (4 c+6 d x) a^4+2 b \sinh (8 c+6 d x) a^4+1248 b^2 d x \cosh (2 c) a^3+464 b^2 d x \cosh (2 d x) a^3+464 b^2 d x \cosh (4 c+2 d x) a^3+224 b^2 \sinh (2 c) a^3-318 b^2 \sinh (2 d x) a^3-158 b^2 \sinh (4 c+2 d x) a^3-36 b^2 \sinh (2 c+4 d x) a^3-36 b^2 \sinh (6 c+4 d x) a^3+2 b^2 \sinh (4 c+6 d x) a^3+2 b^2 \sinh (8 c+6 d x) a^3+3392 b^3 d x \cosh (2 c) a^2+768 b^3 d x \cosh (2 d x) a^2+768 b^3 d x \cosh (4 c+2 d x) a^2+576 b^3 \sinh (2 c) a^2-512 b^3 \sinh (2 d x) a^2-256 b^3 \sinh (4 c+2 d x) a^2-24 b^3 \sinh (2 c+4 d x) a^2-24 b^3 \sinh (6 c+4 d x) a^2+3840 b^4 d x \cosh (2 c) a+384 b^4 d x \cosh (2 d x) a+384 b^4 d x \cosh (4 c+2 d x) a+640 b^4 \sinh (2 c) a-256 b^4 \sinh (2 d x) a-128 b^4 \sinh (4 c+2 d x) a+1536 b^5 d x \cosh (2 c)+256 b^5 \sinh (2 c)\right)}{8 a^4 b (a+b) d (\cosh (2 c+2 d x) a+a+2 b)}\right) \text{sech}^4(c+d x)}{128 \left(b \text{sech}^2(c+d x)+a\right)^2}","-\frac{3 \sqrt{b} \sqrt{a+b} (a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{2 a^4 d}-\frac{3 b (3 a+4 b) \tanh (c+d x)}{8 a^3 d \left(a-b \tanh ^2(c+d x)+b\right)}-\frac{(5 a+6 b) \sinh (c+d x) \cosh (c+d x)}{8 a^2 d \left(a-b \tanh ^2(c+d x)+b\right)}+\frac{3 x \left(a^2+8 a b+8 b^2\right)}{8 a^4}+\frac{\sinh (c+d x) \cosh ^3(c+d x)}{4 a d \left(a-b \tanh ^2(c+d x)+b\right)}",1,"-1/256*((a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(16*x + ((a^3 - 6*a^2*b - 24*a*b^2 - 16*b^3)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(b*(a + b)^(3/2)*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + ((a^2 + 8*a*b + 8*b^2)*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/(b*(a + b)*d*(a + 2*b + a*Cosh[2*(c + d*x)]))))/(a^2*(a + b*Sech[c + d*x]^2)^2) + (3*(a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(((a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*b^(3/2)*(a + b)^(3/2)*d) - (a*Sinh[2*(c + d*x)])/(8*b*(a + b)*d*(a + 2*b + a*Cosh[2*(c + d*x)]))))/(128*(a + b*Sech[c + d*x]^2)^2) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(((a^5 - 30*a^4*b - 480*a^3*b^2 - 1600*a^2*b^3 - 1920*a*b^4 - 768*b^5)*(((-1/8*I)*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Cosh[2*c])/(a^4*b*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/8)*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Sinh[2*c])/(a^4*b*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/(a + b) + (Sech[2*c]*(160*a^4*b*d*x*Cosh[2*c] + 1248*a^3*b^2*d*x*Cosh[2*c] + 3392*a^2*b^3*d*x*Cosh[2*c] + 3840*a*b^4*d*x*Cosh[2*c] + 1536*b^5*d*x*Cosh[2*c] + 80*a^4*b*d*x*Cosh[2*d*x] + 464*a^3*b^2*d*x*Cosh[2*d*x] + 768*a^2*b^3*d*x*Cosh[2*d*x] + 384*a*b^4*d*x*Cosh[2*d*x] + 80*a^4*b*d*x*Cosh[4*c + 2*d*x] + 464*a^3*b^2*d*x*Cosh[4*c + 2*d*x] + 768*a^2*b^3*d*x*Cosh[4*c + 2*d*x] + 384*a*b^4*d*x*Cosh[4*c + 2*d*x] + a^5*Sinh[2*c] + 34*a^4*b*Sinh[2*c] + 224*a^3*b^2*Sinh[2*c] + 576*a^2*b^3*Sinh[2*c] + 640*a*b^4*Sinh[2*c] + 256*b^5*Sinh[2*c] - a^5*Sinh[2*d*x] - 62*a^4*b*Sinh[2*d*x] - 318*a^3*b^2*Sinh[2*d*x] - 512*a^2*b^3*Sinh[2*d*x] - 256*a*b^4*Sinh[2*d*x] - 30*a^4*b*Sinh[4*c + 2*d*x] - 158*a^3*b^2*Sinh[4*c + 2*d*x] - 256*a^2*b^3*Sinh[4*c + 2*d*x] - 128*a*b^4*Sinh[4*c + 2*d*x] - 12*a^4*b*Sinh[2*c + 4*d*x] - 36*a^3*b^2*Sinh[2*c + 4*d*x] - 24*a^2*b^3*Sinh[2*c + 4*d*x] - 12*a^4*b*Sinh[6*c + 4*d*x] - 36*a^3*b^2*Sinh[6*c + 4*d*x] - 24*a^2*b^3*Sinh[6*c + 4*d*x] + 2*a^4*b*Sinh[4*c + 6*d*x] + 2*a^3*b^2*Sinh[4*c + 6*d*x] + 2*a^4*b*Sinh[8*c + 6*d*x] + 2*a^3*b^2*Sinh[8*c + 6*d*x]))/(8*a^4*b*(a + b)*d*(a + 2*b + a*Cosh[2*c + 2*d*x]))))/(128*(a + b*Sech[c + d*x]^2)^2)","C",0
34,1,861,114,4.8191492,"\int \frac{\sinh ^3(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2,x]","\frac{(\cosh (2 (c+d x)) a+a+2 b)^2 \text{sech}^4(c+d x) \left(\frac{9 \tan ^{-1}\left(\frac{\left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right) \sinh (c) \tanh \left(\frac{d x}{2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right) a^3}{b^{3/2}}+\frac{9 \tan ^{-1}\left(\frac{\left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right) \sinh (c) \tanh \left(\frac{d x}{2}\right)+\cosh (c) \left(i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)+\sqrt{a}\right)}{\sqrt{b}}\right) a^3}{b^{3/2}}-\frac{9 \tan ^{-1}\left(\frac{\sqrt{a}-i \sqrt{a+b} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b}}\right) a^3}{b^{3/2}}-\frac{9 \tan ^{-1}\left(\frac{i \sqrt{a+b} \tanh \left(\frac{1}{2} (c+d x)\right)+\sqrt{a}}{\sqrt{b}}\right) a^3}{b^{3/2}}+32 \cosh (3 c) \cosh (3 d x) a^{3/2}-288 \sinh (c) \sinh (d x) a^{3/2}+32 \sinh (3 c) \sinh (3 d x) a^{3/2}-\frac{384 b \cosh (c+d x) a^{3/2}}{\cosh (2 (c+d x)) a+a+2 b}+576 \sqrt{b} \tan ^{-1}\left(\frac{\left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right) \sinh (c) \tanh \left(\frac{d x}{2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right) a+576 \sqrt{b} \tan ^{-1}\left(\frac{\left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right) \sinh (c) \tanh \left(\frac{d x}{2}\right)+\cosh (c) \left(i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)+\sqrt{a}\right)}{\sqrt{b}}\right) a-96 (3 a+8 b) \cosh (c) \cosh (d x) \sqrt{a}-768 b \sinh (c) \sinh (d x) \sqrt{a}-\frac{384 b^2 \cosh (c+d x) \sqrt{a}}{\cosh (2 (c+d x)) a+a+2 b}+960 b^{3/2} \tan ^{-1}\left(\frac{\left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right) \sinh (c) \tanh \left(\frac{d x}{2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)+960 b^{3/2} \tan ^{-1}\left(\frac{\left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right) \sinh (c) \tanh \left(\frac{d x}{2}\right)+\cosh (c) \left(i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)+\sqrt{a}\right)}{\sqrt{b}}\right)\right)}{1536 a^{7/2} d \left(b \text{sech}^2(c+d x)+a\right)^2}","\frac{\sqrt{b} (3 a+5 b) \tan ^{-1}\left(\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right)}{2 a^{7/2} d}-\frac{b (a+b) \cosh (c+d x)}{2 a^3 d \left(a \cosh ^2(c+d x)+b\right)}-\frac{(a+2 b) \cosh (c+d x)}{a^3 d}+\frac{\cosh ^3(c+d x)}{3 a^2 d}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]^4*((9*a^3*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]])/b^(3/2) + 576*a*Sqrt[b]*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + 960*b^(3/2)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + (9*a^3*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]])/b^(3/2) + 576*a*Sqrt[b]*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + 960*b^(3/2)*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] - (9*a^3*ArcTan[(Sqrt[a] - I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]])/b^(3/2) - (9*a^3*ArcTan[(Sqrt[a] + I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]])/b^(3/2) - 96*Sqrt[a]*(3*a + 8*b)*Cosh[c]*Cosh[d*x] + 32*a^(3/2)*Cosh[3*c]*Cosh[3*d*x] - (384*a^(3/2)*b*Cosh[c + d*x])/(a + 2*b + a*Cosh[2*(c + d*x)]) - (384*Sqrt[a]*b^2*Cosh[c + d*x])/(a + 2*b + a*Cosh[2*(c + d*x)]) - 288*a^(3/2)*Sinh[c]*Sinh[d*x] - 768*Sqrt[a]*b*Sinh[c]*Sinh[d*x] + 32*a^(3/2)*Sinh[3*c]*Sinh[3*d*x]))/(1536*a^(7/2)*d*(a + b*Sech[c + d*x]^2)^2)","C",0
35,1,791,131,11.3321145,"\int \frac{\sinh ^2(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^2/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^4(c+d x) (a \cosh (2 c+2 d x)+a+2 b)^2 \left(\frac{\left(a^2+8 a b+8 b^2\right) \text{sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{b d (a+b) (a \cosh (2 (c+d x))+a+2 b)}+\frac{\left(a^3-6 a^2 b-24 a b^2-16 b^3\right) (\cosh (2 c)-\sinh (2 c)) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{b d (a+b)^{3/2} \sqrt{b (\cosh (c)-\sinh (c))^4}}+16 x\right)}{128 a^2 \left(a+b \text{sech}^2(c+d x)\right)^2}+\frac{\text{sech}^4(c+d x) (a \cosh (2 c+2 d x)+a+2 b)^2 \left(-\frac{\left(a^3+18 a^2 b+48 a b^2+32 b^3\right) \text{sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{b d (a+b) (a \cosh (2 (c+d x))+a+2 b)}+\frac{\left(-a^4+16 a^3 b+144 a^2 b^2+256 a b^3+128 b^4\right) (\cosh (2 c)-\sinh (2 c)) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{b d (a+b)^{3/2} \sqrt{b (\cosh (c)-\sinh (c))^4}}-64 x (a+2 b)+\frac{16 a \sinh (2 c) \cosh (2 d x)}{d}+\frac{16 a \cosh (2 c) \sinh (2 d x)}{d}\right)}{256 a^3 \left(a+b \text{sech}^2(c+d x)\right)^2}-\frac{\text{sech}^4(c+d x) (a \cosh (2 c+2 d x)+a+2 b)^2 \left(\frac{\sqrt{b} (a+2 b) \sinh (2 (c+d x))}{(a+b) (a \cosh (2 (c+d x))+a+2 b)}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}\right)}{256 b^{3/2} d \left(a+b \text{sech}^2(c+d x)\right)^2}+\frac{\text{sech}^4(c+d x) (a \cosh (2 c+2 d x)+a+2 b)^2 \left(\frac{a \sinh (2 (c+d x))}{8 b d (a+b) (a \cosh (2 (c+d x))+a+2 b)}-\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 b^{3/2} d (a+b)^{3/2}}\right)}{16 \left(a+b \text{sech}^2(c+d x)\right)^2}","\frac{\sqrt{b} (3 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{2 a^3 d \sqrt{a+b}}-\frac{x (a+4 b)}{2 a^3}+\frac{b \tanh (c+d x)}{a^2 d \left(a-b \tanh ^2(c+d x)+b\right)}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 a d \left(a-b \tanh ^2(c+d x)+b\right)}",1,"((a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(16*x + ((a^3 - 6*a^2*b - 24*a*b^2 - 16*b^3)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(b*(a + b)^(3/2)*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + ((a^2 + 8*a*b + 8*b^2)*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/(b*(a + b)*d*(a + 2*b + a*Cosh[2*(c + d*x)]))))/(128*a^2*(a + b*Sech[c + d*x]^2)^2) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(-64*(a + 2*b)*x + ((-a^4 + 16*a^3*b + 144*a^2*b^2 + 256*a*b^3 + 128*b^4)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(b*(a + b)^(3/2)*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (16*a*Cosh[2*d*x]*Sinh[2*c])/d + (16*a*Cosh[2*c]*Sinh[2*d*x])/d - ((a^3 + 18*a^2*b + 48*a*b^2 + 32*b^3)*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/(b*(a + b)*d*(a + 2*b + a*Cosh[2*(c + d*x)]))))/(256*a^3*(a + b*Sech[c + d*x]^2)^2) - ((a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(-((a*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(3/2)) + (Sqrt[b]*(a + 2*b)*Sinh[2*(c + d*x)])/((a + b)*(a + 2*b + a*Cosh[2*(c + d*x)]))))/(256*b^(3/2)*d*(a + b*Sech[c + d*x]^2)^2) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(-1/8*((a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(b^(3/2)*(a + b)^(3/2)*d) + (a*Sinh[2*(c + d*x)])/(8*b*(a + b)*d*(a + 2*b + a*Cosh[2*(c + d*x)]))))/(16*(a + b*Sech[c + d*x]^2)^2)","B",0
36,1,479,84,2.7634432,"\int \frac{\sinh (c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^4(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \left(\frac{32 b \cosh (c+d x)}{a^2 (a \cosh (2 (c+d x))+a+2 b)}+\frac{32 \cosh (c) \cosh (d x)}{a^2}+\frac{2 \left(-\left(a^2+24 b^2\right) \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)+a^2 \tan ^{-1}\left(\frac{\sqrt{a}-i \sqrt{a+b} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b}}\right)+a^2 \tan ^{-1}\left(\frac{\sqrt{a}+i \sqrt{a+b} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b}}\right)+a^2 \left(-\tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)\right)+16 \sqrt{a} b^{3/2} \sinh (c) \sinh (d x)-24 b^2 \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)\right)}{a^{5/2} b^{3/2}}\right)}{128 d \left(a+b \text{sech}^2(c+d x)\right)^2}","-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right)}{2 a^{5/2} d}+\frac{3 \cosh (c+d x)}{2 a^2 d}-\frac{\cosh ^3(c+d x)}{2 a d \left(a \cosh ^2(c+d x)+b\right)}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]^4*((32*Cosh[c]*Cosh[d*x])/a^2 + (32*b*Cosh[c + d*x])/(a^2*(a + 2*b + a*Cosh[2*(c + d*x)])) + (2*(-((a^2 + 24*b^2)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]) - a^2*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] - 24*b^2*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + a^2*ArcTan[(Sqrt[a] - I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]] + a^2*ArcTan[(Sqrt[a] + I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]] + 16*Sqrt[a]*b^(3/2)*Sinh[c]*Sinh[d*x]))/(a^(5/2)*b^(3/2))))/(128*d*(a + b*Sech[c + d*x]^2)^2)","C",1
37,1,377,99,1.2509536,"\int \frac{\text{csch}(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Csch[c + d*x]/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{\sqrt{b} (3 a+b) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)}{a^{3/2}}+\frac{\sqrt{b} (3 a+b) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)}{a^{3/2}}-2 \text{sech}(c+d x) \log \left(\cosh \left(\frac{1}{2} (c+d x)\right)\right) (a \cosh (2 (c+d x))+a+2 b)+2 \text{sech}(c+d x) \log \left(\sinh \left(\frac{1}{2} (c+d x)\right)\right) (a \cosh (2 (c+d x))+a+2 b)-\frac{2 b (a+b)}{a}\right)}{8 d (a+b)^2 \left(a+b \text{sech}^2(c+d x)\right)^2}","\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right)}{2 a^{3/2} d (a+b)^2}-\frac{b \cosh (c+d x)}{2 a d (a+b) \left(a \cosh ^2(c+d x)+b\right)}-\frac{\tanh ^{-1}(\cosh (c+d x))}{d (a+b)^2}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^3*((-2*b*(a + b))/a + (Sqrt[b]*(3*a + b)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x])/a^(3/2) + (Sqrt[b]*(3*a + b)*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x])/a^(3/2) - 2*(a + 2*b + a*Cosh[2*(c + d*x)])*Log[Cosh[(c + d*x)/2]]*Sech[c + d*x] + 2*(a + 2*b + a*Cosh[2*(c + d*x)])*Log[Sinh[(c + d*x)/2]]*Sech[c + d*x]))/(8*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^2)","C",1
38,1,220,92,2.6248255,"\int \frac{\text{csch}^2(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Csch[c + d*x]^2/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^4(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(2 \text{csch}(c) \sinh (d x) \text{csch}(c+d x) (a \cosh (2 (c+d x))+a+2 b)+\frac{3 b (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}-\frac{b (a+2 b) \tanh (2 c)}{a}+b \text{sech}(2 c) \sinh (2 d x)\right)}{8 d (a+b)^2 \left(a+b \text{sech}^2(c+d x)\right)^2}","\frac{3 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{2 d (a+b)^{5/2}}-\frac{3 \coth (c+d x)}{2 d (a+b)^2}+\frac{\coth (c+d x)}{2 d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*((3*b*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + 2*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]*Sinh[d*x] + b*Sech[2*c]*Sinh[2*d*x] - (b*(a + 2*b)*Tanh[2*c])/a))/(8*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^2)","B",1
39,1,462,147,2.2270583,"\int \frac{\text{csch}^3(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Csch[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(-(a+b) \text{sech}^2\left(\frac{1}{2} (c+d x)\right) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)+4 (a-3 b) \text{sech}(c+d x) \log \left(\cosh \left(\frac{1}{2} (c+d x)\right)\right) (a \cosh (2 (c+d x))+a+2 b)-\left((a+b) \text{csch}^2\left(\frac{1}{2} (c+d x)\right) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)\right)-4 (a-3 b) \text{sech}(c+d x) \log \left(\sinh \left(\frac{1}{2} (c+d x)\right)\right) (a \cosh (2 (c+d x))+a+2 b)+\frac{4 \sqrt{b} (b-3 a) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)}{\sqrt{a}}+\frac{4 \sqrt{b} (b-3 a) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)}{\sqrt{a}}+8 b (a+b)\right)}{32 d (a+b)^3 \left(a+b \text{sech}^2(c+d x)\right)^2}","-\frac{(a-b) \cosh (c+d x)}{2 d (a+b)^2 \left(a \cosh ^2(c+d x)+b\right)}-\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right)}{2 \sqrt{a} d (a+b)^3}+\frac{(a-3 b) \tanh ^{-1}(\cosh (c+d x))}{2 d (a+b)^3}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 d (a+b) \left(a \cosh ^2(c+d x)+b\right)}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^3*(8*b*(a + b) + (4*Sqrt[b]*(-3*a + b)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x])/Sqrt[a] + (4*Sqrt[b]*(-3*a + b)*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x])/Sqrt[a] - (a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[(c + d*x)/2]^2*Sech[c + d*x] + 4*(a - 3*b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Log[Cosh[(c + d*x)/2]]*Sech[c + d*x] - 4*(a - 3*b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Log[Sinh[(c + d*x)/2]]*Sech[c + d*x] - (a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[(c + d*x)/2]^2*Sech[c + d*x]))/(32*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^2)","C",1
40,1,295,123,6.0777124,"\int \frac{\text{csch}^4(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Csch[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^4(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(-3 a b \text{sech}(2 c) \sinh (2 d x)-2 (a+b) \coth (c) \text{csch}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b)+2 (a+b) \text{csch}(c) \sinh (d x) \text{csch}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b)-4 (a-2 b) \text{csch}(c) \sinh (d x) \text{csch}(c+d x) (a \cosh (2 (c+d x))+a+2 b)-\frac{3 b (3 a-2 b) (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}+3 b (a+2 b) \tanh (2 c)\right)}{24 d (a+b)^3 \left(a+b \text{sech}^2(c+d x)\right)^2}","-\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{2 d (a+b)^{7/2}}-\frac{a b \tanh (c+d x)}{2 d (a+b)^3 \left(a-b \tanh ^2(c+d x)+b\right)}-\frac{\coth ^3(c+d x)}{3 d (a+b)^2}+\frac{(a-b) \coth (c+d x)}{d (a+b)^3}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*(-2*(a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Coth[c]*Csch[c + d*x]^2 - (3*(3*a - 2*b)*b*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) - 4*(a - 2*b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]*Sinh[d*x] + 2*(a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]^3*Sinh[d*x] - 3*a*b*Sech[2*c]*Sinh[2*d*x] + 3*b*(a + 2*b)*Tanh[2*c]))/(24*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^2)","B",1
41,1,3080,242,27.3157922,"\int \frac{\sinh ^4(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]","\text{Result too large to show}","-\frac{3 b (a+2 b) \tanh (c+d x)}{2 a^4 d \left(a-b \tanh ^2(c+d x)+b\right)}-\frac{b (7 a+12 b) \tanh (c+d x)}{8 a^3 d \left(a-b \tanh ^2(c+d x)+b\right)^2}-\frac{(5 a+8 b) \sinh (c+d x) \cosh (c+d x)}{8 a^2 d \left(a-b \tanh ^2(c+d x)+b\right)^2}-\frac{3 \sqrt{b} \left(5 a^2+20 a b+16 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 a^5 d \sqrt{a+b}}+\frac{3 x \left(a^2+12 a b+16 b^2\right)}{8 a^5}+\frac{\sinh (c+d x) \cosh ^3(c+d x)}{4 a d \left(a-b \tanh ^2(c+d x)+b\right)^2}",1,"(3*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*(((3*a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(16384*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-3*a*(a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) + (Sqrt[b]*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a*b + 4*b^2)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(16384*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3) - (3*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-2*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 + 256*b^5)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (Sech[2*c]*(256*b^2*(a + b)^2*(3*a^2 + 8*a*b + 8*b^2)*d*x*Cosh[2*c] + 512*a*b^2*(a + b)^2*(a + 2*b)*d*x*Cosh[2*d*x] + 128*a^4*b^2*d*x*Cosh[2*(c + 2*d*x)] + 256*a^3*b^3*d*x*Cosh[2*(c + 2*d*x)] + 128*a^2*b^4*d*x*Cosh[2*(c + 2*d*x)] + 512*a^4*b^2*d*x*Cosh[4*c + 2*d*x] + 2048*a^3*b^3*d*x*Cosh[4*c + 2*d*x] + 2560*a^2*b^4*d*x*Cosh[4*c + 2*d*x] + 1024*a*b^5*d*x*Cosh[4*c + 2*d*x] + 128*a^4*b^2*d*x*Cosh[6*c + 4*d*x] + 256*a^3*b^3*d*x*Cosh[6*c + 4*d*x] + 128*a^2*b^4*d*x*Cosh[6*c + 4*d*x] - 9*a^6*Sinh[2*c] + 12*a^5*b*Sinh[2*c] + 684*a^4*b^2*Sinh[2*c] + 2880*a^3*b^3*Sinh[2*c] + 5280*a^2*b^4*Sinh[2*c] + 4608*a*b^5*Sinh[2*c] + 1536*b^6*Sinh[2*c] + 9*a^6*Sinh[2*d*x] - 14*a^5*b*Sinh[2*d*x] - 608*a^4*b^2*Sinh[2*d*x] - 2112*a^3*b^3*Sinh[2*d*x] - 2560*a^2*b^4*Sinh[2*d*x] - 1024*a*b^5*Sinh[2*d*x] + 3*a^6*Sinh[2*(c + 2*d*x)] - 12*a^5*b*Sinh[2*(c + 2*d*x)] - 204*a^4*b^2*Sinh[2*(c + 2*d*x)] - 384*a^3*b^3*Sinh[2*(c + 2*d*x)] - 192*a^2*b^4*Sinh[2*(c + 2*d*x)] - 3*a^6*Sinh[4*c + 2*d*x] + 10*a^5*b*Sinh[4*c + 2*d*x] + 304*a^4*b^2*Sinh[4*c + 2*d*x] + 1056*a^3*b^3*Sinh[4*c + 2*d*x] + 1280*a^2*b^4*Sinh[4*c + 2*d*x] + 512*a*b^5*Sinh[4*c + 2*d*x]))/(a + 2*b + a*Cosh[2*(c + d*x)])^2))/(65536*a^3*b^2*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((6*(a^6 - 8*a^5*b + 120*a^4*b^2 + 1280*a^3*b^3 + 3200*a^2*b^4 + 3072*a*b^5 + 1024*b^6)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (Sech[2*c]*(-1536*b^2*(a + b)^2*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3)*d*x*Cosh[2*c] - 3072*a*b^2*(a^2 + 3*a*b + 2*b^2)^2*d*x*Cosh[2*d*x] - 768*a^5*b^2*d*x*Cosh[2*(c + 2*d*x)] - 3072*a^4*b^3*d*x*Cosh[2*(c + 2*d*x)] - 3840*a^3*b^4*d*x*Cosh[2*(c + 2*d*x)] - 1536*a^2*b^5*d*x*Cosh[2*(c + 2*d*x)] - 3072*a^5*b^2*d*x*Cosh[4*c + 2*d*x] - 18432*a^4*b^3*d*x*Cosh[4*c + 2*d*x] - 39936*a^3*b^4*d*x*Cosh[4*c + 2*d*x] - 36864*a^2*b^5*d*x*Cosh[4*c + 2*d*x] - 12288*a*b^6*d*x*Cosh[4*c + 2*d*x] - 768*a^5*b^2*d*x*Cosh[6*c + 4*d*x] - 3072*a^4*b^3*d*x*Cosh[6*c + 4*d*x] - 3840*a^3*b^4*d*x*Cosh[6*c + 4*d*x] - 1536*a^2*b^5*d*x*Cosh[6*c + 4*d*x] + 9*a^7*Sinh[2*c] - 54*a^6*b*Sinh[2*c] - 2392*a^5*b^2*Sinh[2*c] - 13968*a^4*b^3*Sinh[2*c] - 36480*a^3*b^4*Sinh[2*c] - 50432*a^2*b^5*Sinh[2*c] - 35840*a*b^6*Sinh[2*c] - 10240*b^7*Sinh[2*c] - 9*a^7*Sinh[2*d*x] + 56*a^6*b*Sinh[2*d*x] + 2552*a^5*b^2*Sinh[2*d*x] + 13184*a^4*b^3*Sinh[2*d*x] + 27072*a^3*b^4*Sinh[2*d*x] + 24576*a^2*b^5*Sinh[2*d*x] + 8192*a*b^6*Sinh[2*d*x] - 3*a^7*Sinh[2*(c + 2*d*x)] + 26*a^6*b*Sinh[2*(c + 2*d*x)] + 992*a^5*b^2*Sinh[2*(c + 2*d*x)] + 3648*a^4*b^3*Sinh[2*(c + 2*d*x)] + 4480*a^3*b^4*Sinh[2*(c + 2*d*x)] + 1792*a^2*b^5*Sinh[2*(c + 2*d*x)] + 3*a^7*Sinh[4*c + 2*d*x] - 24*a^6*b*Sinh[4*c + 2*d*x] - 600*a^5*b^2*Sinh[4*c + 2*d*x] - 3200*a^4*b^3*Sinh[4*c + 2*d*x] - 6720*a^3*b^4*Sinh[4*c + 2*d*x] - 6144*a^2*b^5*Sinh[4*c + 2*d*x] - 2048*a*b^6*Sinh[4*c + 2*d*x] + 256*a^5*b^2*Sinh[6*c + 4*d*x] + 1024*a^4*b^3*Sinh[6*c + 4*d*x] + 1280*a^3*b^4*Sinh[6*c + 4*d*x] + 512*a^2*b^5*Sinh[6*c + 4*d*x] + 64*a^5*b^2*Sinh[4*c + 6*d*x] + 128*a^4*b^3*Sinh[4*c + 6*d*x] + 64*a^3*b^4*Sinh[4*c + 6*d*x] + 64*a^5*b^2*Sinh[8*c + 6*d*x] + 128*a^4*b^3*Sinh[8*c + 6*d*x] + 64*a^3*b^4*Sinh[8*c + 6*d*x]))/(a + 2*b + a*Cosh[2*(c + d*x)])^2))/(32768*a^4*b^2*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3) - ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((6*a^2*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (a*Sech[2*c]*((-9*a^4 - 16*a^3*b + 48*a^2*b^2 + 128*a*b^3 + 64*b^4)*Sinh[2*d*x] + a*(-3*a^3 + 2*a^2*b + 24*a*b^2 + 16*b^3)*Sinh[2*(c + 2*d*x)] + (3*a^4 - 64*a^2*b^2 - 128*a*b^3 - 64*b^4)*Sinh[4*c + 2*d*x]) + (9*a^5 + 18*a^4*b - 64*a^3*b^2 - 256*a^2*b^3 - 320*a*b^4 - 128*b^5)*Tanh[2*c])/(a^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(8192*b^2*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*(768*(7*a^2 + 32*a*b + 32*b^2)*x - (3*(a^7 - 14*a^6*b + 336*a^5*b^2 + 5600*a^4*b^3 + 22400*a^3*b^4 + 37632*a^2*b^5 + 28672*a*b^6 + 8192*b^7)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(b^2*(a + b)^(5/2)*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) - (4*(a^5 + 50*a^4*b + 400*a^3*b^2 + 1120*a^2*b^3 + 1280*a*b^4 + 512*b^5)*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/(b*(a + b)*d*(a + 2*b + a*Cosh[2*(c + d*x)])^2) + (768*a*(a + 2*b)*(Cosh[2*(c + d*x)] - Sinh[2*(c + d*x)]))/d - (768*a*(a + 2*b)*(Cosh[2*(c + d*x)] + Sinh[2*(c + d*x)]))/d + (128*a^2*Sinh[4*(c + d*x)])/d + (a*(3*a^6 - 44*a^5*b - 1900*a^4*b^2 - 10880*a^3*b^3 - 23360*a^2*b^4 - 21504*a*b^5 - 7168*b^6)*Sech[2*c]*Sinh[2*d*x] + (-3*a^7 + 42*a^6*b + 2192*a^5*b^2 + 16480*a^4*b^3 + 51200*a^3*b^4 + 77824*a^2*b^5 + 57344*a*b^6 + 16384*b^7)*Tanh[2*c])/(b^2*(a + b)^2*d*(a + 2*b + a*Cosh[2*(c + d*x)]))))/(32768*a^5*(a + b*Sech[c + d*x]^2)^3)","B",0
42,1,1217,154,10.3822097,"\int \frac{\sinh ^3(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3,x]","\frac{(\cosh (2 (c+d x)) a+a+2 b)^3 \text{sech}^6(c+d x) \left(\frac{24 (3 a-4 b) \left(\tan ^{-1}\left(\frac{\left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right) \sinh (c) \tanh \left(\frac{d x}{2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)+\tan ^{-1}\left(\frac{\left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right) \sinh (c) \tanh \left(\frac{d x}{2}\right)+\cosh (c) \left(i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)+\sqrt{a}\right)}{\sqrt{b}}\right)\right)}{a^{3/2} b^{5/2}}-\frac{54 \left(\tan ^{-1}\left(\frac{\sqrt{a}-i \sqrt{a+b} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b}}\right)+\tan ^{-1}\left(\frac{i \sqrt{a+b} \tanh \left(\frac{1}{2} (c+d x)\right)+\sqrt{a}}{\sqrt{b}}\right)\right)}{\sqrt{a} b^{5/2}}-\frac{36 \cosh (c+d x) (3 \cosh (2 (c+d x)) a+3 a+10 b)}{b^2 (\cosh (2 (c+d x)) a+a+2 b)^2}+\frac{48 \cosh (c+d x) \left(3 a^2+6 b a+(3 a-4 b) \cosh (2 (c+d x)) a+8 b^2\right)}{a b^2 (\cosh (2 (c+d x)) a+a+2 b)^2}+\frac{3 \left(3 a^4-40 b a^3+720 b^2 a^2+6720 b^3 a+8960 b^4\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right) \sinh (c) \tanh \left(\frac{d x}{2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)+3 \left(3 a^4-40 b a^3+720 b^2 a^2+6720 b^3 a+8960 b^4\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right) \sinh (c) \tanh \left(\frac{d x}{2}\right)+\cosh (c) \left(i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)+\sqrt{a}\right)}{\sqrt{b}}\right)+\frac{2 \sqrt{a} \sqrt{b} \cosh (c+d x) \left(9 a^5-90 b a^4-10144 b^2 a^3+128 b^2 \cosh (6 (c+d x)) a^3-48672 b^3 a^2-128 b^2 (15 a+28 b) \cosh (4 (c+d x)) a^2-85120 b^4 a+\left(9 a^4-120 b a^3-12432 b^2 a^2-47936 b^3 a-44800 b^4\right) \cosh (2 (c+d x)) a-53760 b^5\right)}{(\cosh (2 (c+d x)) a+a+2 b)^2}}{a^{9/2} b^{5/2}}+\frac{9 \left(-\frac{3 \left(a^3-8 b a^2+80 b^2 a+320 b^3\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right) \sinh (c) \tanh \left(\frac{d x}{2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)}{b^{5/2}}-\frac{3 \left(a^3-8 b a^2+80 b^2 a+320 b^3\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right) \sinh (c) \tanh \left(\frac{d x}{2}\right)+\cosh (c) \left(i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)+\sqrt{a}\right)}{\sqrt{b}}\right)}{b^{5/2}}+512 \sqrt{a} \cosh (c) \cosh (d x)+512 \sqrt{a} \sinh (c) \sinh (d x)-\frac{2 \sqrt{a} \left(3 a^3-24 b a^2-400 b^2 a-576 b^3\right) \cosh (c+d x)}{b^2 (\cosh (2 (c+d x)) a+a+2 b)}-\frac{8 \sqrt{a} \left(a^3+24 b a^2+80 b^2 a+64 b^3\right) \cosh (c+d x)}{b (\cosh (2 (c+d x)) a+a+2 b)^2}\right)}{a^{7/2}}\right)}{49152 d \left(b \text{sech}^2(c+d x)+a\right)^3}","\frac{5 \sqrt{b} (3 a+7 b) \tan ^{-1}\left(\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right)}{8 a^{9/2} d}+\frac{b^2 (a+b) \cosh (c+d x)}{4 a^4 d \left(a \cosh ^2(c+d x)+b\right)^2}-\frac{b (9 a+13 b) \cosh (c+d x)}{8 a^4 d \left(a \cosh ^2(c+d x)+b\right)}-\frac{(a+3 b) \cosh (c+d x)}{a^4 d}+\frac{\cosh ^3(c+d x)}{3 a^3 d}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*((24*(3*a - 4*b)*(ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]))/(a^(3/2)*b^(5/2)) - (54*(ArcTan[(Sqrt[a] - I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[(Sqrt[a] + I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]]))/(Sqrt[a]*b^(5/2)) - (36*Cosh[c + d*x]*(3*a + 10*b + 3*a*Cosh[2*(c + d*x)]))/(b^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2) + (48*Cosh[c + d*x]*(3*a^2 + 6*a*b + 8*b^2 + a*(3*a - 4*b)*Cosh[2*(c + d*x)]))/(a*b^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2) + (3*(3*a^4 - 40*a^3*b + 720*a^2*b^2 + 6720*a*b^3 + 8960*b^4)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + 3*(3*a^4 - 40*a^3*b + 720*a^2*b^2 + 6720*a*b^3 + 8960*b^4)*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + (2*Sqrt[a]*Sqrt[b]*Cosh[c + d*x]*(9*a^5 - 90*a^4*b - 10144*a^3*b^2 - 48672*a^2*b^3 - 85120*a*b^4 - 53760*b^5 + a*(9*a^4 - 120*a^3*b - 12432*a^2*b^2 - 47936*a*b^3 - 44800*b^4)*Cosh[2*(c + d*x)] - 128*a^2*b^2*(15*a + 28*b)*Cosh[4*(c + d*x)] + 128*a^3*b^2*Cosh[6*(c + d*x)]))/(a + 2*b + a*Cosh[2*(c + d*x)])^2)/(a^(9/2)*b^(5/2)) + (9*((-3*(a^3 - 8*a^2*b + 80*a*b^2 + 320*b^3)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]])/b^(5/2) - (3*(a^3 - 8*a^2*b + 80*a*b^2 + 320*b^3)*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]])/b^(5/2) + 512*Sqrt[a]*Cosh[c]*Cosh[d*x] - (8*Sqrt[a]*(a^3 + 24*a^2*b + 80*a*b^2 + 64*b^3)*Cosh[c + d*x])/(b*(a + 2*b + a*Cosh[2*(c + d*x)])^2) - (2*Sqrt[a]*(3*a^3 - 24*a^2*b - 400*a*b^2 - 576*b^3)*Cosh[c + d*x])/(b^2*(a + 2*b + a*Cosh[2*(c + d*x)])) + 512*Sqrt[a]*Sinh[c]*Sinh[d*x]))/a^(7/2)))/(49152*d*(a + b*Sech[c + d*x]^2)^3)","C",0
43,1,2544,187,17.8445709,"\int \frac{\sinh ^2(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]","\text{Result too large to show}","-\frac{x (a+6 b)}{2 a^4}+\frac{b (11 a+12 b) \tanh (c+d x)}{8 a^3 d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)}+\frac{3 b \tanh (c+d x)}{4 a^2 d \left(a-b \tanh ^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} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 a^4 d (a+b)^{3/2}}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 a d \left(a-b \tanh ^2(c+d x)+b\right)^2}",1,"(-5*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*(((3*a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(8192*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3) - ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-3*a*(a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) + (Sqrt[b]*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a*b + 4*b^2)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(2048*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-2*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 + 256*b^5)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (Sech[2*c]*(256*b^2*(a + b)^2*(3*a^2 + 8*a*b + 8*b^2)*d*x*Cosh[2*c] + 512*a*b^2*(a + b)^2*(a + 2*b)*d*x*Cosh[2*d*x] + 128*a^4*b^2*d*x*Cosh[2*(c + 2*d*x)] + 256*a^3*b^3*d*x*Cosh[2*(c + 2*d*x)] + 128*a^2*b^4*d*x*Cosh[2*(c + 2*d*x)] + 512*a^4*b^2*d*x*Cosh[4*c + 2*d*x] + 2048*a^3*b^3*d*x*Cosh[4*c + 2*d*x] + 2560*a^2*b^4*d*x*Cosh[4*c + 2*d*x] + 1024*a*b^5*d*x*Cosh[4*c + 2*d*x] + 128*a^4*b^2*d*x*Cosh[6*c + 4*d*x] + 256*a^3*b^3*d*x*Cosh[6*c + 4*d*x] + 128*a^2*b^4*d*x*Cosh[6*c + 4*d*x] - 9*a^6*Sinh[2*c] + 12*a^5*b*Sinh[2*c] + 684*a^4*b^2*Sinh[2*c] + 2880*a^3*b^3*Sinh[2*c] + 5280*a^2*b^4*Sinh[2*c] + 4608*a*b^5*Sinh[2*c] + 1536*b^6*Sinh[2*c] + 9*a^6*Sinh[2*d*x] - 14*a^5*b*Sinh[2*d*x] - 608*a^4*b^2*Sinh[2*d*x] - 2112*a^3*b^3*Sinh[2*d*x] - 2560*a^2*b^4*Sinh[2*d*x] - 1024*a*b^5*Sinh[2*d*x] + 3*a^6*Sinh[2*(c + 2*d*x)] - 12*a^5*b*Sinh[2*(c + 2*d*x)] - 204*a^4*b^2*Sinh[2*(c + 2*d*x)] - 384*a^3*b^3*Sinh[2*(c + 2*d*x)] - 192*a^2*b^4*Sinh[2*(c + 2*d*x)] - 3*a^6*Sinh[4*c + 2*d*x] + 10*a^5*b*Sinh[4*c + 2*d*x] + 304*a^4*b^2*Sinh[4*c + 2*d*x] + 1056*a^3*b^3*Sinh[4*c + 2*d*x] + 1280*a^2*b^4*Sinh[4*c + 2*d*x] + 512*a*b^5*Sinh[4*c + 2*d*x]))/(a + 2*b + a*Cosh[2*(c + d*x)])^2))/(4096*a^3*b^2*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((6*(a^6 - 8*a^5*b + 120*a^4*b^2 + 1280*a^3*b^3 + 3200*a^2*b^4 + 3072*a*b^5 + 1024*b^6)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (Sech[2*c]*(-1536*b^2*(a + b)^2*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3)*d*x*Cosh[2*c] - 3072*a*b^2*(a^2 + 3*a*b + 2*b^2)^2*d*x*Cosh[2*d*x] - 768*a^5*b^2*d*x*Cosh[2*(c + 2*d*x)] - 3072*a^4*b^3*d*x*Cosh[2*(c + 2*d*x)] - 3840*a^3*b^4*d*x*Cosh[2*(c + 2*d*x)] - 1536*a^2*b^5*d*x*Cosh[2*(c + 2*d*x)] - 3072*a^5*b^2*d*x*Cosh[4*c + 2*d*x] - 18432*a^4*b^3*d*x*Cosh[4*c + 2*d*x] - 39936*a^3*b^4*d*x*Cosh[4*c + 2*d*x] - 36864*a^2*b^5*d*x*Cosh[4*c + 2*d*x] - 12288*a*b^6*d*x*Cosh[4*c + 2*d*x] - 768*a^5*b^2*d*x*Cosh[6*c + 4*d*x] - 3072*a^4*b^3*d*x*Cosh[6*c + 4*d*x] - 3840*a^3*b^4*d*x*Cosh[6*c + 4*d*x] - 1536*a^2*b^5*d*x*Cosh[6*c + 4*d*x] + 9*a^7*Sinh[2*c] - 54*a^6*b*Sinh[2*c] - 2392*a^5*b^2*Sinh[2*c] - 13968*a^4*b^3*Sinh[2*c] - 36480*a^3*b^4*Sinh[2*c] - 50432*a^2*b^5*Sinh[2*c] - 35840*a*b^6*Sinh[2*c] - 10240*b^7*Sinh[2*c] - 9*a^7*Sinh[2*d*x] + 56*a^6*b*Sinh[2*d*x] + 2552*a^5*b^2*Sinh[2*d*x] + 13184*a^4*b^3*Sinh[2*d*x] + 27072*a^3*b^4*Sinh[2*d*x] + 24576*a^2*b^5*Sinh[2*d*x] + 8192*a*b^6*Sinh[2*d*x] - 3*a^7*Sinh[2*(c + 2*d*x)] + 26*a^6*b*Sinh[2*(c + 2*d*x)] + 992*a^5*b^2*Sinh[2*(c + 2*d*x)] + 3648*a^4*b^3*Sinh[2*(c + 2*d*x)] + 4480*a^3*b^4*Sinh[2*(c + 2*d*x)] + 1792*a^2*b^5*Sinh[2*(c + 2*d*x)] + 3*a^7*Sinh[4*c + 2*d*x] - 24*a^6*b*Sinh[4*c + 2*d*x] - 600*a^5*b^2*Sinh[4*c + 2*d*x] - 3200*a^4*b^3*Sinh[4*c + 2*d*x] - 6720*a^3*b^4*Sinh[4*c + 2*d*x] - 6144*a^2*b^5*Sinh[4*c + 2*d*x] - 2048*a*b^6*Sinh[4*c + 2*d*x] + 256*a^5*b^2*Sinh[6*c + 4*d*x] + 1024*a^4*b^3*Sinh[6*c + 4*d*x] + 1280*a^3*b^4*Sinh[6*c + 4*d*x] + 512*a^2*b^5*Sinh[6*c + 4*d*x] + 64*a^5*b^2*Sinh[4*c + 6*d*x] + 128*a^4*b^3*Sinh[4*c + 6*d*x] + 64*a^3*b^4*Sinh[4*c + 6*d*x] + 64*a^5*b^2*Sinh[8*c + 6*d*x] + 128*a^4*b^3*Sinh[8*c + 6*d*x] + 64*a^3*b^4*Sinh[8*c + 6*d*x]))/(a + 2*b + a*Cosh[2*(c + d*x)])^2))/(16384*a^4*b^2*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((6*a^2*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (a*Sech[2*c]*((-9*a^4 - 16*a^3*b + 48*a^2*b^2 + 128*a*b^3 + 64*b^4)*Sinh[2*d*x] + a*(-3*a^3 + 2*a^2*b + 24*a*b^2 + 16*b^3)*Sinh[2*(c + 2*d*x)] + (3*a^4 - 64*a^2*b^2 - 128*a*b^3 - 64*b^4)*Sinh[4*c + 2*d*x]) + (9*a^5 + 18*a^4*b - 64*a^3*b^2 - 256*a^2*b^3 - 320*a*b^4 - 128*b^5)*Tanh[2*c])/(a^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(4096*b^2*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3)","B",0
44,1,453,116,9.6038178,"\int \frac{\sinh (c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]/(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b)^3 \left(\frac{512 \sinh (c) \sinh (d x)}{a^3}+\frac{512 \cosh (c) \cosh (d x)}{a^3}-\frac{15 \left(\left(a^3+64 b^3\right) \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)+\left(a^3+64 b^3\right) \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)-\left(a^3 \left(\tan ^{-1}\left(\frac{\sqrt{a}-i \sqrt{a+b} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b}}\right)+\tan ^{-1}\left(\frac{\sqrt{a}+i \sqrt{a+b} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b}}\right)\right)\right)\right)}{a^{7/2} b^{5/2}}+\frac{8 \cosh (c+d x) \left(3 \left(a^4+48 a b^3\right) \cosh (2 (c+d x))+16 b^3 (9 a+14 b)\right)}{a^3 b^2 (a \cosh (2 (c+d x))+a+2 b)^2}-\frac{6 a \sinh (4 (c+d x)) \text{csch}(c+d x)}{b^2 (a \cosh (2 (c+d x))+a+2 b)^2}\right)}{4096 d \left(a+b \text{sech}^2(c+d x)\right)^3}","-\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right)}{8 a^{7/2} d}+\frac{15 \cosh (c+d x)}{8 a^3 d}-\frac{5 \cosh ^3(c+d x)}{8 a^2 d \left(a \cosh ^2(c+d x)+b\right)}-\frac{\cosh ^5(c+d x)}{4 a d \left(a \cosh ^2(c+d x)+b\right)^2}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*((-15*((a^3 + 64*b^3)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + (a^3 + 64*b^3)*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] - a^3*(ArcTan[(Sqrt[a] - I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[(Sqrt[a] + I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]])))/(a^(7/2)*b^(5/2)) + (512*Cosh[c]*Cosh[d*x])/a^3 + (8*Cosh[c + d*x]*(16*b^3*(9*a + 14*b) + 3*(a^4 + 48*a*b^3)*Cosh[2*(c + d*x)]))/(a^3*b^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2) + (512*Sinh[c]*Sinh[d*x])/a^3 - (6*a*Csch[c + d*x]*Sinh[4*(c + d*x)])/(b^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(4096*d*(a + b*Sech[c + d*x]^2)^3)","C",0
45,1,440,154,2.4975037,"\int \frac{\text{csch}(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Csch[c + d*x]/(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{sech}^5(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{8 b^2 (a+b)^2}{a^2}-\frac{2 b (9 a+5 b) (a+b) (a \cosh (2 (c+d x))+a+2 b)}{a^2}+\frac{\sqrt{b} \left(15 a^2+10 a b+3 b^2\right) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)}{a^{5/2}}+\frac{\sqrt{b} \left(15 a^2+10 a b+3 b^2\right) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)}{a^{5/2}}-8 \text{sech}(c+d x) \log \left(\cosh \left(\frac{1}{2} (c+d x)\right)\right) (a \cosh (2 (c+d x))+a+2 b)^2+8 \text{sech}(c+d x) \log \left(\sinh \left(\frac{1}{2} (c+d x)\right)\right) (a \cosh (2 (c+d x))+a+2 b)^2\right)}{64 d (a+b)^3 \left(a+b \text{sech}^2(c+d x)\right)^3}","-\frac{b (7 a+3 b) \cosh (c+d x)}{8 a^2 d (a+b)^2 \left(a \cosh ^2(c+d x)+b\right)}+\frac{\sqrt{b} \left(15 a^2+10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right)}{8 a^{5/2} d (a+b)^3}-\frac{b \cosh ^3(c+d x)}{4 a d (a+b) \left(a \cosh ^2(c+d x)+b\right)^2}-\frac{\tanh ^{-1}(\cosh (c+d x))}{d (a+b)^3}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^5*((8*b^2*(a + b)^2)/a^2 - (2*b*(a + b)*(9*a + 5*b)*(a + 2*b + a*Cosh[2*(c + d*x)]))/a^2 + (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x])/a^(5/2) + (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x])/a^(5/2) - 8*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Log[Cosh[(c + d*x)/2]]*Sech[c + d*x] + 8*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Log[Sinh[(c + d*x)/2]]*Sech[c + d*x]))/(64*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^3)","C",1
46,1,981,126,6.9202442,"\int \frac{\text{csch}^2(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Csch[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]","\frac{(\cosh (2 c+2 d x) a+a+2 b)^3 \left(\frac{15 i b \tan ^{-1}\left(\text{sech}(d x) \left(\frac{i \sinh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}-\frac{i \cosh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}\right) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right) \sinh (2 c)}{64 \sqrt{a+b} d \sqrt{b \cosh (4 c)-b \sinh (4 c)}}-\frac{15 i b \tan ^{-1}\left(\text{sech}(d x) \left(\frac{i \sinh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}-\frac{i \cosh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}\right) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right) \cosh (2 c)}{64 \sqrt{a+b} d \sqrt{b \cosh (4 c)-b \sinh (4 c)}}\right) \text{sech}^6(c+d x)}{(a+b)^3 \left(b \text{sech}^2(c+d x)+a\right)^3}+\frac{(\cosh (2 c+2 d x) a+a+2 b) \text{csch}(c) \text{csch}(c+d x) \text{sech}(2 c) \left(-32 \sinh (d x) a^4+32 \sinh (3 d x) a^4-48 \sinh (2 c-d x) a^4+48 \sinh (2 c+d x) a^4-32 \sinh (4 c+d x) a^4-8 \sinh (2 c+3 d x) a^4+32 \sinh (4 c+3 d x) a^4-8 \sinh (6 c+3 d x) a^4+8 \sinh (2 c+5 d x) a^4+8 \sinh (6 c+5 d x) a^4-64 b \sinh (d x) a^3+46 b \sinh (3 d x) a^3-128 b \sinh (2 c-d x) a^3+146 b \sinh (2 c+d x) a^3-82 b \sinh (4 c+d x) a^3+18 b \sinh (2 c+3 d x) a^3+73 b \sinh (4 c+3 d x) a^3-9 b \sinh (6 c+3 d x) a^3-9 b \sinh (2 c+5 d x) a^3+9 b \sinh (4 c+5 d x) a^3+22 b^2 \sinh (d x) a^2-54 b^2 \sinh (3 d x) a^2-106 b^2 \sinh (2 c-d x) a^2+182 b^2 \sinh (2 c+d x) a^2-54 b^2 \sinh (4 c+d x) a^2+54 b^2 \sinh (2 c+3 d x) a^2+24 b^2 \sinh (4 c+3 d x) a^2-24 b^2 \sinh (6 c+3 d x) a^2-2 b^2 \sinh (2 c+5 d x) a^2+2 b^2 \sinh (4 c+5 d x) a^2+80 b^3 \sinh (d x) a-8 b^3 \sinh (3 d x) a+80 b^3 \sinh (2 c-d x) a+80 b^3 \sinh (2 c+d x) a-80 b^3 \sinh (4 c+d x) a+8 b^3 \sinh (2 c+3 d x) a+8 b^3 \sinh (4 c+3 d x) a-8 b^3 \sinh (6 c+3 d x) a+16 b^4 \sinh (d x)+16 b^4 \sinh (2 c-d x)+16 b^4 \sinh (2 c+d x)-16 b^4 \sinh (4 c+d x)\right) \text{sech}^6(c+d x)}{512 a^2 (a+b)^3 d \left(b \text{sech}^2(c+d x)+a\right)^3}","\frac{15 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 d (a+b)^{7/2}}-\frac{15 \coth (c+d x)}{8 d (a+b)^3}+\frac{5 \coth (c+d x)}{8 d (a+b)^2 \left(a-b \tanh ^2(c+d x)+b\right)}+\frac{\coth (c+d x)}{4 d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)^2}",1,"((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((((-15*I)/64)*b*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Cosh[2*c])/(Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + (((15*I)/64)*b*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Sinh[2*c])/(Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/((a + b)^3*(a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Csch[c]*Csch[c + d*x]*Sech[2*c]*Sech[c + d*x]^6*(-32*a^4*Sinh[d*x] - 64*a^3*b*Sinh[d*x] + 22*a^2*b^2*Sinh[d*x] + 80*a*b^3*Sinh[d*x] + 16*b^4*Sinh[d*x] + 32*a^4*Sinh[3*d*x] + 46*a^3*b*Sinh[3*d*x] - 54*a^2*b^2*Sinh[3*d*x] - 8*a*b^3*Sinh[3*d*x] - 48*a^4*Sinh[2*c - d*x] - 128*a^3*b*Sinh[2*c - d*x] - 106*a^2*b^2*Sinh[2*c - d*x] + 80*a*b^3*Sinh[2*c - d*x] + 16*b^4*Sinh[2*c - d*x] + 48*a^4*Sinh[2*c + d*x] + 146*a^3*b*Sinh[2*c + d*x] + 182*a^2*b^2*Sinh[2*c + d*x] + 80*a*b^3*Sinh[2*c + d*x] + 16*b^4*Sinh[2*c + d*x] - 32*a^4*Sinh[4*c + d*x] - 82*a^3*b*Sinh[4*c + d*x] - 54*a^2*b^2*Sinh[4*c + d*x] - 80*a*b^3*Sinh[4*c + d*x] - 16*b^4*Sinh[4*c + d*x] - 8*a^4*Sinh[2*c + 3*d*x] + 18*a^3*b*Sinh[2*c + 3*d*x] + 54*a^2*b^2*Sinh[2*c + 3*d*x] + 8*a*b^3*Sinh[2*c + 3*d*x] + 32*a^4*Sinh[4*c + 3*d*x] + 73*a^3*b*Sinh[4*c + 3*d*x] + 24*a^2*b^2*Sinh[4*c + 3*d*x] + 8*a*b^3*Sinh[4*c + 3*d*x] - 8*a^4*Sinh[6*c + 3*d*x] - 9*a^3*b*Sinh[6*c + 3*d*x] - 24*a^2*b^2*Sinh[6*c + 3*d*x] - 8*a*b^3*Sinh[6*c + 3*d*x] + 8*a^4*Sinh[2*c + 5*d*x] - 9*a^3*b*Sinh[2*c + 5*d*x] - 2*a^2*b^2*Sinh[2*c + 5*d*x] + 9*a^3*b*Sinh[4*c + 5*d*x] + 2*a^2*b^2*Sinh[4*c + 5*d*x] + 8*a^4*Sinh[6*c + 5*d*x]))/(512*a^2*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^3)","C",0
47,1,524,213,4.1168504,"\int \frac{\text{csch}^3(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Csch[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{sech}^5(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{\sqrt{b} \left(-15 a^2+10 a b+b^2\right) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)}{a^{3/2}}+\frac{\sqrt{b} \left(-15 a^2+10 a b+b^2\right) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left(\frac{\sinh (c) \tanh \left(\frac{d x}{2}\right) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right)+\cosh (c) \left(\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left(\frac{d x}{2}\right)\right)}{\sqrt{b}}\right)}{a^{3/2}}-\frac{8 b^2 (a+b)^2}{a}+\frac{2 b (9 a+b) (a+b) (a \cosh (2 (c+d x))+a+2 b)}{a}-(a+b) \text{sech}^2\left(\frac{1}{2} (c+d x)\right) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2+4 (a-5 b) \text{sech}(c+d x) \log \left(\cosh \left(\frac{1}{2} (c+d x)\right)\right) (a \cosh (2 (c+d x))+a+2 b)^2-(a+b) \text{csch}^2\left(\frac{1}{2} (c+d x)\right) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2-4 (a-5 b) \text{sech}(c+d x) \log \left(\sinh \left(\frac{1}{2} (c+d x)\right)\right) (a \cosh (2 (c+d x))+a+2 b)^2\right)}{64 d (a+b)^4 \left(a+b \text{sech}^2(c+d x)\right)^3}","-\frac{\left(4 a^2-9 a b-b^2\right) \cosh (c+d x)}{8 a d (a+b)^3 \left(a \cosh ^2(c+d x)+b\right)}-\frac{\sqrt{b} \left(15 a^2-10 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right)}{8 a^{3/2} d (a+b)^4}+\frac{b (2 a-b) \cosh (c+d x)}{4 a d (a+b)^2 \left(a \cosh ^2(c+d x)+b\right)^2}-\frac{\cosh (c+d x) \coth ^2(c+d x)}{2 d (a+b) \left(a \cosh ^2(c+d x)+b\right)^2}+\frac{(a-5 b) \tanh ^{-1}(\cosh (c+d x))}{2 d (a+b)^4}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^5*((-8*b^2*(a + b)^2)/a + (2*b*(a + b)*(9*a + b)*(a + 2*b + a*Cosh[2*(c + d*x)]))/a + (Sqrt[b]*(-15*a^2 + 10*a*b + b^2)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x])/a^(3/2) + (Sqrt[b]*(-15*a^2 + 10*a*b + b^2)*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x])/a^(3/2) - (a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Csch[(c + d*x)/2]^2*Sech[c + d*x] + 4*(a - 5*b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Log[Cosh[(c + d*x)/2]]*Sech[c + d*x] - 4*(a - 5*b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Log[Sinh[(c + d*x)/2]]*Sech[c + d*x] - (a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[(c + d*x)/2]^2*Sech[c + d*x]))/(64*(a + b)^4*d*(a + b*Sech[c + d*x]^2)^3)","C",1
48,1,985,165,5.5295359,"\int \frac{\text{csch}^4(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Csch[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]","-\frac{(\cosh (2 (c+d x)) a+a+2 b) \text{sech}^6(c+d x) \left(\frac{\text{csch}(c) \text{sech}(2 c) \left(224 \sinh (2 c-d x) a^4-224 \sinh (2 c+d x) a^4+176 \sinh (4 c+d x) a^4+48 \sinh (2 c+3 d x) a^4-96 \sinh (4 c+3 d x) a^4+48 \sinh (6 c+3 d x) a^4+16 \sinh (2 c+5 d x) a^4+16 \sinh (6 c+5 d x) a^4+16 \sinh (4 c+7 d x) a^4+16 \sinh (8 c+7 d x) a^4+576 b \sinh (2 c-d x) a^3-657 b \sinh (2 c+d x) a^3+569 b \sinh (4 c+d x) a^3+111 b \sinh (2 c+3 d x) a^3-152 b \sinh (4 c+3 d x) a^3+192 b \sinh (6 c+3 d x) a^3+72 b \sinh (4 c+5 d x) a^3+27 b \sinh (6 c+5 d x) a^3+45 b \sinh (8 c+5 d x) a^3-83 b \sinh (4 c+7 d x) a^3+27 b \sinh (6 c+7 d x) a^3-56 b \sinh (8 c+7 d x) a^3+124 b^2 \sinh (2 c-d x) a^2-538 b^2 \sinh (2 c+d x) a^2+666 b^2 \sinh (4 c+d x) a^2+360 b^2 \sinh (2 c+3 d x) a^2+146 b^2 \sinh (4 c+3 d x) a^2+558 b^2 \sinh (6 c+3 d x) a^2-598 b^2 \sinh (2 c+5 d x) a^2+150 b^2 \sinh (4 c+5 d x) a^2-388 b^2 \sinh (6 c+5 d x) a^2-60 b^2 \sinh (8 c+5 d x) a^2+6 b^2 \sinh (4 c+7 d x) a^2-6 b^2 \sinh (6 c+7 d x) a^2-2184 b^3 \sinh (2 c-d x) a+984 b^3 \sinh (2 c+d x) a+1704 b^3 \sinh (4 c+d x) a+312 b^3 \sinh (2 c+3 d x) a-728 b^3 \sinh (4 c+3 d x) a-168 b^3 \sinh (6 c+3 d x) a+48 b^3 \sinh (2 c+5 d x) a-48 b^3 \sinh (4 c+5 d x) a+4 \left(44 a^4+122 b a^3+63 b^2 a^2+126 b^3 a+36 b^4\right) \sinh (d x)+\left(-96 a^4-71 b a^3+344 b^2 a^2-1208 b^3 a+48 b^4\right) \sinh (3 d x)+144 b^4 \sinh (2 c-d x)+144 b^4 \sinh (2 c+d x)-144 b^4 \sinh (4 c+d x)-48 b^4 \sinh (2 c+3 d x)-48 b^4 \sinh (4 c+3 d x)+48 b^4 \sinh (6 c+3 d x)\right) \text{csch}^3(c+d x)}{a}+\frac{480 (3 a-4 b) b \tanh ^{-1}\left(\frac{\text{sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right) (\cosh (2 (c+d x)) a+a+2 b)^2 (\cosh (2 c)-\sinh (2 c))}{\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{6144 (a+b)^4 d \left(b \text{sech}^2(c+d x)+a\right)^3}","-\frac{5 \sqrt{b} (3 a-4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 d (a+b)^{9/2}}-\frac{b (7 a-4 b) \tanh (c+d x)}{8 d (a+b)^4 \left(a-b \tanh ^2(c+d x)+b\right)}-\frac{a b \tanh (c+d x)}{4 d (a+b)^3 \left(a-b \tanh ^2(c+d x)+b\right)^2}-\frac{\coth ^3(c+d x)}{3 d (a+b)^3}+\frac{(a-2 b) \coth (c+d x)}{d (a+b)^4}",1,"-1/6144*((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*((480*(3*a - 4*b)*b*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (Csch[c]*Csch[c + d*x]^3*Sech[2*c]*(4*(44*a^4 + 122*a^3*b + 63*a^2*b^2 + 126*a*b^3 + 36*b^4)*Sinh[d*x] + (-96*a^4 - 71*a^3*b + 344*a^2*b^2 - 1208*a*b^3 + 48*b^4)*Sinh[3*d*x] + 224*a^4*Sinh[2*c - d*x] + 576*a^3*b*Sinh[2*c - d*x] + 124*a^2*b^2*Sinh[2*c - d*x] - 2184*a*b^3*Sinh[2*c - d*x] + 144*b^4*Sinh[2*c - d*x] - 224*a^4*Sinh[2*c + d*x] - 657*a^3*b*Sinh[2*c + d*x] - 538*a^2*b^2*Sinh[2*c + d*x] + 984*a*b^3*Sinh[2*c + d*x] + 144*b^4*Sinh[2*c + d*x] + 176*a^4*Sinh[4*c + d*x] + 569*a^3*b*Sinh[4*c + d*x] + 666*a^2*b^2*Sinh[4*c + d*x] + 1704*a*b^3*Sinh[4*c + d*x] - 144*b^4*Sinh[4*c + d*x] + 48*a^4*Sinh[2*c + 3*d*x] + 111*a^3*b*Sinh[2*c + 3*d*x] + 360*a^2*b^2*Sinh[2*c + 3*d*x] + 312*a*b^3*Sinh[2*c + 3*d*x] - 48*b^4*Sinh[2*c + 3*d*x] - 96*a^4*Sinh[4*c + 3*d*x] - 152*a^3*b*Sinh[4*c + 3*d*x] + 146*a^2*b^2*Sinh[4*c + 3*d*x] - 728*a*b^3*Sinh[4*c + 3*d*x] - 48*b^4*Sinh[4*c + 3*d*x] + 48*a^4*Sinh[6*c + 3*d*x] + 192*a^3*b*Sinh[6*c + 3*d*x] + 558*a^2*b^2*Sinh[6*c + 3*d*x] - 168*a*b^3*Sinh[6*c + 3*d*x] + 48*b^4*Sinh[6*c + 3*d*x] + 16*a^4*Sinh[2*c + 5*d*x] - 598*a^2*b^2*Sinh[2*c + 5*d*x] + 48*a*b^3*Sinh[2*c + 5*d*x] + 72*a^3*b*Sinh[4*c + 5*d*x] + 150*a^2*b^2*Sinh[4*c + 5*d*x] - 48*a*b^3*Sinh[4*c + 5*d*x] + 16*a^4*Sinh[6*c + 5*d*x] + 27*a^3*b*Sinh[6*c + 5*d*x] - 388*a^2*b^2*Sinh[6*c + 5*d*x] + 45*a^3*b*Sinh[8*c + 5*d*x] - 60*a^2*b^2*Sinh[8*c + 5*d*x] + 16*a^4*Sinh[4*c + 7*d*x] - 83*a^3*b*Sinh[4*c + 7*d*x] + 6*a^2*b^2*Sinh[4*c + 7*d*x] + 27*a^3*b*Sinh[6*c + 7*d*x] - 6*a^2*b^2*Sinh[6*c + 7*d*x] + 16*a^4*Sinh[8*c + 7*d*x] - 56*a^3*b*Sinh[8*c + 7*d*x]))/a))/((a + b)^4*d*(a + b*Sech[c + d*x]^2)^3)","B",0
49,1,45,61,0.0927122,"\int \cosh ^4(c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Cosh[c + d*x]^4*(a + b*Sech[c + d*x]^2),x]","\frac{4 (3 a+4 b) (c+d x)+8 (a+b) \sinh (2 (c+d x))+a \sinh (4 (c+d x))}{32 d}","\frac{(3 a+4 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{1}{8} x (3 a+4 b)+\frac{a \sinh (c+d x) \cosh ^3(c+d x)}{4 d}",1,"(4*(3*a + 4*b)*(c + d*x) + 8*(a + b)*Sinh[2*(c + d*x)] + a*Sinh[4*(c + d*x)])/(32*d)","A",1
50,1,50,30,0.0177598,"\int \cosh ^3(c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Cosh[c + d*x]^3*(a + b*Sech[c + d*x]^2),x]","\frac{a \sinh ^3(c+d x)}{3 d}+\frac{a \sinh (c+d x)}{d}+\frac{b \sinh (c) \cosh (d x)}{d}+\frac{b \cosh (c) \sinh (d x)}{d}","\frac{(a+b) \sinh (c+d x)}{d}+\frac{a \sinh ^3(c+d x)}{3 d}",1,"(b*Cosh[d*x]*Sinh[c])/d + (b*Cosh[c]*Sinh[d*x])/d + (a*Sinh[c + d*x])/d + (a*Sinh[c + d*x]^3)/(3*d)","A",1
51,1,33,31,0.0301433,"\int \cosh ^2(c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Cosh[c + d*x]^2*(a + b*Sech[c + d*x]^2),x]","\frac{a (c+d x)}{2 d}+\frac{a \sinh (2 (c+d x))}{4 d}+b x","\frac{1}{2} x (a+2 b)+\frac{a \sinh (c+d x) \cosh (c+d x)}{2 d}",1,"b*x + (a*(c + d*x))/(2*d) + (a*Sinh[2*(c + d*x)])/(4*d)","A",1
52,1,35,24,0.0176128,"\int \cosh (c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Cosh[c + d*x]*(a + b*Sech[c + d*x]^2),x]","\frac{a \sinh (c) \cosh (d x)}{d}+\frac{a \cosh (c) \sinh (d x)}{d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{d}","\frac{a \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","A",1
53,1,48,40,0.0236981,"\int \text{sech}(c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Sech[c + d*x]*(a + b*Sech[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
54,1,39,30,0.0120577,"\int \text{sech}^2(c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Sech[c + d*x]^2*(a + b*Sech[c + d*x]^2),x]","\frac{a \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}-\frac{b \tanh ^3(c+d x)}{3 d}",1,"(a*Tanh[c + d*x])/d + (b*Tanh[c + d*x])/d - (b*Tanh[c + d*x]^3)/(3*d)","A",1
55,1,60,70,0.1050023,"\int \text{sech}^3(c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Sech[c + d*x]^3*(a + b*Sech[c + d*x]^2),x]","\frac{(4 a+3 b) \tan ^{-1}(\sinh (c+d x))+(4 a+3 b) \tanh (c+d x) \text{sech}(c+d x)+2 b \tanh (c+d x) \text{sech}^3(c+d x)}{8 d}","\frac{(4 a+3 b) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{(4 a+3 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,"((4*a + 3*b)*ArcTan[Sinh[c + d*x]] + (4*a + 3*b)*Sech[c + d*x]*Tanh[c + d*x] + 2*b*Sech[c + d*x]^3*Tanh[c + d*x])/(8*d)","A",1
56,1,71,50,0.0140917,"\int \text{sech}^4(c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Sech[c + d*x]^4*(a + b*Sech[c + d*x]^2),x]","-\frac{a \tanh ^3(c+d x)}{3 d}+\frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^5(c+d x)}{5 d}-\frac{2 b \tanh ^3(c+d x)}{3 d}+\frac{b \tanh (c+d x)}{d}","-\frac{(a+2 b) \tanh ^3(c+d x)}{3 d}+\frac{(a+b) \tanh (c+d x)}{d}+\frac{b \tanh ^5(c+d x)}{5 d}",1,"(a*Tanh[c + d*x])/d + (b*Tanh[c + d*x])/d - (a*Tanh[c + d*x]^3)/(3*d) - (2*b*Tanh[c + d*x]^3)/(3*d) + (b*Tanh[c + d*x]^5)/(5*d)","A",1
57,1,58,82,0.1313526,"\int \cosh ^4(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Cosh[c + d*x]^4*(a + b*Sech[c + d*x]^2)^2,x]","\frac{4 \left(3 a^2+8 a b+8 b^2\right) (c+d x)+a^2 \sinh (4 (c+d x))+8 a (a+2 b) \sinh (2 (c+d x))}{32 d}","\frac{1}{8} x \left(3 a^2+8 a b+8 b^2\right)+\frac{3 a (a+2 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{a \sinh (c+d x) \cosh ^3(c+d x) \left(a-b \tanh ^2(c+d x)+b\right)}{4 d}",1,"(4*(3*a^2 + 8*a*b + 8*b^2)*(c + d*x) + 8*a*(a + 2*b)*Sinh[2*(c + d*x)] + a^2*Sinh[4*(c + d*x)])/(32*d)","A",1
58,1,72,49,0.0244882,"\int \cosh ^3(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Cosh[c + d*x]^3*(a + b*Sech[c + d*x]^2)^2,x]","\frac{a^2 \sinh ^3(c+d x)}{3 d}+\frac{a^2 \sinh (c+d x)}{d}+\frac{2 a b \sinh (c) \cosh (d x)}{d}+\frac{2 a b \cosh (c) \sinh (d x)}{d}+\frac{b^2 \tan ^{-1}(\sinh (c+d x))}{d}","\frac{a^2 \sinh ^3(c+d x)}{3 d}+\frac{a (a+2 b) \sinh (c+d x)}{d}+\frac{b^2 \tan ^{-1}(\sinh (c+d x))}{d}",1,"(b^2*ArcTan[Sinh[c + d*x]])/d + (2*a*b*Cosh[d*x]*Sinh[c])/d + (2*a*b*Cosh[c]*Sinh[d*x])/d + (a^2*Sinh[c + d*x])/d + (a^2*Sinh[c + d*x]^3)/(3*d)","A",1
59,1,52,47,0.1494018,"\int \cosh ^2(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Cosh[c + d*x]^2*(a + b*Sech[c + d*x]^2)^2,x]","\frac{a^2 (c+d x)}{2 d}+\frac{a^2 \sinh (2 (c+d x))}{4 d}+2 a b x+\frac{b^2 \tanh (c+d x)}{d}","\frac{a^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac{1}{2} a x (a+4 b)+\frac{b^2 \tanh (c+d x)}{d}",1,"2*a*b*x + (a^2*(c + d*x))/(2*d) + (a^2*Sinh[2*(c + d*x)])/(4*d) + (b^2*Tanh[c + d*x])/d","A",1
60,1,80,56,0.0409111,"\int \cosh (c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Cosh[c + d*x]*(a + b*Sech[c + d*x]^2)^2,x]","\frac{a^2 \sinh (c) \cosh (d x)}{d}+\frac{a^2 \cosh (c) \sinh (d x)}{d}+\frac{2 a b \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b^2 \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b^2 \tanh (c+d x) \text{sech}(c+d x)}{2 d}","\frac{a^2 \sinh (c+d x)}{d}+\frac{b (4 a+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*ArcTan[Sinh[c + d*x]])/d + (b^2*ArcTan[Sinh[c + d*x]])/(2*d) + (a^2*Cosh[d*x]*Sinh[c])/d + (a^2*Cosh[c]*Sinh[d*x])/d + (b^2*Sech[c + d*x]*Tanh[c + d*x])/(2*d)","A",1
61,1,71,90,0.1332563,"\int \text{sech}(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Sech[c + d*x]*(a + b*Sech[c + d*x]^2)^2,x]","\frac{\left(8 a^2+8 a b+3 b^2\right) \tan ^{-1}(\sinh (c+d x))+b (8 a+3 b) \tanh (c+d x) \text{sech}(c+d x)+2 b^2 \tanh (c+d x) \text{sech}^3(c+d x)}{8 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 \sinh ^2(c+d x)+a+b\right)}{4 d}",1,"((8*a^2 + 8*a*b + 3*b^2)*ArcTan[Sinh[c + d*x]] + b*(8*a + 3*b)*Sech[c + d*x]*Tanh[c + d*x] + 2*b^2*Sech[c + d*x]^3*Tanh[c + d*x])/(8*d)","A",1
62,1,93,53,0.0309265,"\int \text{sech}^2(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Sech[c + d*x]^2*(a + b*Sech[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{2 a b \tanh (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{b^2 \tanh (c+d x)}{d}","-\frac{2 b (a+b) \tanh ^3(c+d x)}{3 d}+\frac{(a+b)^2 \tanh (c+d x)}{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])/d + (b^2*Tanh[c + d*x])/d - (2*a*b*Tanh[c + d*x]^3)/(3*d) - (2*b^2*Tanh[c + d*x]^3)/(3*d) + (b^2*Tanh[c + d*x]^5)/(5*d)","A",1
63,1,104,128,0.2438536,"\int \text{sech}^3(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Sech[c + d*x]^3*(a + b*Sech[c + d*x]^2)^2,x]","\frac{3 \left(8 a^2+12 a b+5 b^2\right) \tan ^{-1}(\sinh (c+d x))+3 \left(8 a^2+12 a b+5 b^2\right) \tanh (c+d x) \text{sech}(c+d x)+2 b (12 a+5 b) \tanh (c+d x) \text{sech}^3(c+d x)+8 b^2 \tanh (c+d x) \text{sech}^5(c+d x)}{48 d}","\frac{\left(8 a^2+12 a b+5 b^2\right) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac{\left(8 a^2+12 a b+5 b^2\right) \tanh (c+d x) \text{sech}(c+d x)}{16 d}+\frac{b (8 a+5 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 \sinh ^2(c+d x)+a+b\right)}{6 d}",1,"(3*(8*a^2 + 12*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]] + 3*(8*a^2 + 12*a*b + 5*b^2)*Sech[c + d*x]*Tanh[c + d*x] + 2*b*(12*a + 5*b)*Sech[c + d*x]^3*Tanh[c + d*x] + 8*b^2*Sech[c + d*x]^5*Tanh[c + d*x])/(48*d)","A",1
64,1,144,80,0.0235933,"\int \text{sech}^4(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Sech[c + d*x]^4*(a + b*Sech[c + d*x]^2)^2,x]","-\frac{a^2 \tanh ^3(c+d x)}{3 d}+\frac{a^2 \tanh (c+d x)}{d}+\frac{2 a b \tanh ^5(c+d x)}{5 d}-\frac{4 a b \tanh ^3(c+d x)}{3 d}+\frac{2 a b \tanh (c+d x)}{d}-\frac{b^2 \tanh ^7(c+d x)}{7 d}+\frac{3 b^2 \tanh ^5(c+d x)}{5 d}-\frac{b^2 \tanh ^3(c+d x)}{d}+\frac{b^2 \tanh (c+d x)}{d}","\frac{b (2 a+3 b) \tanh ^5(c+d x)}{5 d}-\frac{(a+b) (a+3 b) \tanh ^3(c+d x)}{3 d}+\frac{(a+b)^2 \tanh (c+d x)}{d}-\frac{b^2 \tanh ^7(c+d x)}{7 d}",1,"(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) - (4*a*b*Tanh[c + d*x]^3)/(3*d) - (b^2*Tanh[c + d*x]^3)/d + (2*a*b*Tanh[c + d*x]^5)/(5*d) + (3*b^2*Tanh[c + d*x]^5)/(5*d) - (b^2*Tanh[c + d*x]^7)/(7*d)","A",1
65,1,70,84,0.4267136,"\int \cosh ^4(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Cosh[c + d*x]^4*(a + b*Sech[c + d*x]^2)^3,x]","\frac{a^3 \sinh (4 (c+d x))+12 a \left(a^2+4 a b+8 b^2\right) (c+d x)+8 a^2 (a+3 b) \sinh (2 (c+d x))+32 b^3 \tanh (c+d x)}{32 d}","\frac{a^3 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac{3}{8} a x \left(a^2+4 a b+8 b^2\right)+\frac{3 a^2 (a+4 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{b^3 \tanh (c+d x)}{d}",1,"(12*a*(a^2 + 4*a*b + 8*b^2)*(c + d*x) + 8*a^2*(a + 3*b)*Sinh[2*(c + d*x)] + a^3*Sinh[4*(c + d*x)] + 32*b^3*Tanh[c + d*x])/(32*d)","A",1
66,1,483,81,6.8818081,"\int \cosh ^3(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Cosh[c + d*x]^3*(a + b*Sech[c + d*x]^2)^3,x]","\frac{\coth ^3(c+d x) \text{csch}^2(c+d x) (a \cosh (c+d x)+b \text{sech}(c+d x))^3 \left(-256 \sinh ^8(c+d x) \left(a \sinh ^2(c+d x)+a+b\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)+19786 \sinh ^6(c+d x)+69728 \sinh ^4(c+d x)+88150 \sinh ^2(c+d x)+36015\right)+3 a b^2 \left(753 \sinh ^6(c+d x)+17593 \sinh ^4(c+d x)+52135 \sinh ^2(c+d x)+36015\right)+b^3 \left(1473 \sinh ^4(c+d x)+16120 \sinh ^2(c+d x)+36015\right)\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 ^6(c+d x)+243 \sinh ^4(c+d x)+1875 \sinh ^2(c+d x)+2401\right) \cosh ^4(c+d x)+3 a b^2 \left(\sinh ^8(c+d x)+148 \sinh ^6(c+d x)+2118 \sinh ^4(c+d x)+4276 \sinh ^2(c+d x)+2401\right)+b^3 \left(-47 \sinh ^6(c+d x)+243 \sinh ^4(c+d x)+1875 \sinh ^2(c+d x)+2401\right)\right)}{\sqrt{-\sinh ^2(c+d x)}}\right)}{3780 d (a \cosh (2 c+2 d x)+a+2 b)^3}","\frac{a^3 \sinh ^3(c+d x)}{3 d}+\frac{a^2 (a+3 b) \sinh (c+d x)}{d}+\frac{b^2 (6 a+b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b^3 \tanh (c+d x) \text{sech}(c+d x)}{2 d}",1,"(Coth[c + d*x]^3*Csch[c + d*x]^2*(a*Cosh[c + d*x] + b*Sech[c + d*x])^3*(-256*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^8*(a + b + a*Sinh[c + d*x]^2)^3 - (315*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(b^3*(2401 + 1875*Sinh[c + d*x]^2 + 243*Sinh[c + d*x]^4 - 47*Sinh[c + d*x]^6) + 3*a^2*b*Cosh[c + d*x]^4*(2401 + 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*b^2*(2401 + 4276*Sinh[c + d*x]^2 + 2118*Sinh[c + d*x]^4 + 148*Sinh[c + d*x]^6 + Sinh[c + d*x]^8)))/Sqrt[-Sinh[c + d*x]^2] + 21*(b^3*(36015 + 16120*Sinh[c + d*x]^2 + 1473*Sinh[c + d*x]^4) + 3*a*b^2*(36015 + 52135*Sinh[c + d*x]^2 + 17593*Sinh[c + d*x]^4 + 753*Sinh[c + d*x]^6) + 3*a^2*b*(36015 + 88150*Sinh[c + d*x]^2 + 69728*Sinh[c + d*x]^4 + 19786*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))))/(3780*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^3)","C",0
67,1,64,72,0.5084119,"\int \cosh ^2(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Cosh[c + d*x]^2*(a + b*Sech[c + d*x]^2)^3,x]","\frac{3 a^3 \sinh (2 (c+d x))+6 a^2 (a+6 b) (c+d x)+4 b^2 \tanh (c+d x) \left(9 a+b \text{sech}^2(c+d x)+2 b\right)}{12 d}","\frac{a^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac{1}{2} a^2 x (a+6 b)+\frac{b^2 (3 a+b) \tanh (c+d x)}{d}-\frac{b^3 \tanh ^3(c+d x)}{3 d}",1,"(6*a^2*(a + 6*b)*(c + d*x) + 3*a^3*Sinh[2*(c + d*x)] + 4*b^2*(9*a + 2*b + b*Sech[c + d*x]^2)*Tanh[c + d*x])/(12*d)","A",1
68,1,575,93,7.8750655,"\int \cosh (c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Cosh[c + d*x]*(a + b*Sech[c + d*x]^2)^3,x]","-\frac{\cosh (c+d x) \coth ^5(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \left(256 \sinh ^8(c+d x) \left(a \sinh ^2(c+d x)+a+b\right)^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)+384 \sinh ^8(c+d x) \left(a \sinh ^2(c+d x)+a+b\right)^2 \left(a \left(5 \sinh ^2(c+d x)+7\right)+7 b\right) \, _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(4887 \sinh ^{10}(c+d x)+107725 \sinh ^8(c+d x)+491574 \sinh ^6(c+d x)+922986 \sinh ^4(c+d x)+789235 \sinh ^2(c+d x)+252105\right)+3 a^2 b \left(5640 \sinh ^8(c+d x)+120431 \sinh ^6(c+d x)+437991 \sinh ^4(c+d x)+573145 \sinh ^2(c+d x)+252105\right)+3 a b^2 \left(6393 \sinh ^6(c+d x)+133071 \sinh ^4(c+d x)+357055 \sinh ^2(c+d x)+252105\right)+b^3 \left(8226 \sinh ^4(c+d x)+140965 \sinh ^2(c+d x)+252105\right)\right)+\frac{315 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \left(a^3 \left(7 \sinh ^8(c+d x)+1468 \sinh ^6(c+d x)+11562 \sinh ^4(c+d x)+24604 \sinh ^2(c+d x)+16807\right) \cosh ^4(c+d x)+3 a^2 b \left(8 \sinh ^{10}(c+d x)+1719 \sinh ^8(c+d x)+14956 \sinh ^6(c+d x)+40442 \sinh ^4(c+d x)+43812 \sinh ^2(c+d x)+16807\right)+3 a b^2 \left(9 \sinh ^8(c+d x)+1858 \sinh ^6(c+d x)+15312 \sinh ^4(c+d x)+29406 \sinh ^2(c+d x)+16807\right)+b^3 \left(-62 \sinh ^6(c+d x)+2187 \sinh ^4(c+d x)+15000 \sinh ^2(c+d x)+16807\right)\right)}{\sqrt{-\sinh ^2(c+d x)}}\right)}{7560 d (a \cosh (2 c+2 d x)+a+2 b)^3}","\frac{a^3 \sinh (c+d x)}{d}+\frac{3 b \left(8 a^2+4 a b+b^2\right) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{3 b^2 (4 a+b) \tanh (c+d x) \text{sech}(c+d x)}{8 d}+\frac{b^3 \tanh (c+d x) \text{sech}^3(c+d x)}{4 d}",1,"-1/7560*(Cosh[c + d*x]*Coth[c + d*x]^5*(a + b*Sech[c + d*x]^2)^3*(256*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 + b + a*Sinh[c + d*x]^2)^3 + 384*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^8*(a + b + a*Sinh[c + d*x]^2)^2*(7*b + a*(7 + 5*Sinh[c + d*x]^2)) + (315*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(b^3*(16807 + 15000*Sinh[c + d*x]^2 + 2187*Sinh[c + d*x]^4 - 62*Sinh[c + d*x]^6) + a^3*Cosh[c + d*x]^4*(16807 + 24604*Sinh[c + d*x]^2 + 11562*Sinh[c + d*x]^4 + 1468*Sinh[c + d*x]^6 + 7*Sinh[c + d*x]^8) + 3*a*b^2*(16807 + 29406*Sinh[c + d*x]^2 + 15312*Sinh[c + d*x]^4 + 1858*Sinh[c + d*x]^6 + 9*Sinh[c + d*x]^8) + 3*a^2*b*(16807 + 43812*Sinh[c + d*x]^2 + 40442*Sinh[c + d*x]^4 + 14956*Sinh[c + d*x]^6 + 1719*Sinh[c + d*x]^8 + 8*Sinh[c + d*x]^10)))/Sqrt[-Sinh[c + d*x]^2] - 21*(b^3*(252105 + 140965*Sinh[c + d*x]^2 + 8226*Sinh[c + d*x]^4) + 3*a*b^2*(252105 + 357055*Sinh[c + d*x]^2 + 133071*Sinh[c + d*x]^4 + 6393*Sinh[c + d*x]^6) + 3*a^2*b*(252105 + 573145*Sinh[c + d*x]^2 + 437991*Sinh[c + d*x]^4 + 120431*Sinh[c + d*x]^6 + 5640*Sinh[c + d*x]^8) + a^3*(252105 + 789235*Sinh[c + d*x]^2 + 922986*Sinh[c + d*x]^4 + 491574*Sinh[c + d*x]^6 + 107725*Sinh[c + d*x]^8 + 4887*Sinh[c + d*x]^10))))/(d*(a + 2*b + a*Cosh[2*c + 2*d*x])^3)","C",0
69,1,1430,147,9.8361564,"\int \text{sech}(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]*(a + b*Sech[c + d*x]^2)^3,x]","\frac{\coth ^6(c+d x) \text{csch}(c+d x) \left(b \text{sech}^2(c+d x)+a\right)^3 \left(-44800 a^3 \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \sinh ^{16}(c+d x)-10752 a^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 ^{16}(c+d x)-1024 a^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 ^{16}(c+d x)-158976 a^2 (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 ^{14}(c+d x)-35328 a^2 (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 ^{14}(c+d x)-3072 a^2 (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 ^{14}(c+d x)+2160711 a^3 \sinh ^{12}(c+d x)-189696 a (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 ^{12}(c+d x)-38400 a (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 ^{12}(c+d x)-3072 a (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 ^{12}(c+d x)+48825 a^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sqrt{-\sinh ^2(c+d x)} \sinh ^{12}(c+d x)+36772890 a^3 \sinh ^{10}(c+d x)+7808535 a^2 (a+b) \sinh ^{10}(c+d x)-75520 (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 ^{10}(c+d x)-13824 (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 ^{10}(c+d x)-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 ^{10}(c+d x)+9261945 a^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sqrt{-\sinh ^2(c+d x)} \sinh ^{10}(c+d x)+174825 a^2 (a+b) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sqrt{-\sinh ^2(c+d x)} \sinh ^{10}(c+d x)+58009455 a^3 \sinh ^8(c+d x)+9514449 a (a+b)^2 \sinh ^8(c+d x)+135323370 a^2 (a+b) \sinh ^8(c+d x)+56109375 a^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sqrt{-\sinh ^2(c+d x)} \sinh ^8(c+d x)+210735 a (a+b)^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sqrt{-\sinh ^2(c+d x)} \sinh ^8(c+d x)+33756345 a^2 (a+b) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sqrt{-\sinh ^2(c+d x)} \sinh ^8(c+d x)+4093425 (a+b)^3 \sinh ^6(c+d x)+168951510 a (a+b)^2 \sinh ^6(c+d x)+215549775 a^2 (a+b) \sinh ^6(c+d x)+58009455 a^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sqrt{-\sinh ^2(c+d x)} \sinh ^6(c+d x)+41427855 a (a+b)^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sqrt{-\sinh ^2(c+d x)} \sinh ^6(c+d x)+207173295 a^2 (a+b) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sqrt{-\sinh ^2(c+d x)} \sinh ^6(c+d x)+70189350 (a+b)^3 \sinh ^4(c+d x)+274542345 a (a+b)^2 \sinh ^4(c+d x)+142065 (a+b)^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \left(-\sinh ^2(c+d x)\right)^{3/2} \sinh ^4(c+d x)+17069535 (a+b)^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sqrt{-\sinh ^2(c+d x)} \sinh ^4(c+d x)+260465625 a (a+b)^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sqrt{-\sinh ^2(c+d x)} \sinh ^4(c+d x)+215549775 a^2 (a+b) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sqrt{-\sinh ^2(c+d x)} \sinh ^4(c+d x)+117228825 (a+b)^3 \sinh ^2(c+d x)-109265625 (a+b)^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \left(-\sinh ^2(c+d x)\right)^{3/2}-274542345 a (a+b)^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \left(-\sinh ^2(c+d x)\right)^{3/2}+117228825 (a+b)^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sqrt{-\sinh ^2(c+d x)}\right)}{90720 d (\cosh (2 c+2 d x) a+a+2 b)^3}","\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 \sinh ^2(c+d x)+a+b\right)^2}{6 d}+\frac{5 b (2 a+b) \tanh (c+d x) \text{sech}^3(c+d x) \left(a \sinh ^2(c+d x)+a+b\right)}{24 d}",1,"(Coth[c + d*x]^6*Csch[c + d*x]*(a + b*Sech[c + d*x]^2)^3*(117228825*(a + b)^3*Sinh[c + d*x]^2 + 274542345*a*(a + b)^2*Sinh[c + d*x]^4 + 70189350*(a + b)^3*Sinh[c + d*x]^4 + 215549775*a^2*(a + b)*Sinh[c + d*x]^6 + 168951510*a*(a + b)^2*Sinh[c + d*x]^6 + 4093425*(a + b)^3*Sinh[c + d*x]^6 + 58009455*a^3*Sinh[c + d*x]^8 + 135323370*a^2*(a + b)*Sinh[c + d*x]^8 + 9514449*a*(a + b)^2*Sinh[c + d*x]^8 + 36772890*a^3*Sinh[c + d*x]^10 + 7808535*a^2*(a + b)*Sinh[c + d*x]^10 - 75520*(a + b)^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^10 - 13824*(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]^10 - 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]^10 + 2160711*a^3*Sinh[c + d*x]^12 - 189696*a*(a + b)^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^12 - 38400*a*(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]^12 - 3072*a*(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]^12 - 158976*a^2*(a + b)*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^14 - 35328*a^2*(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]^14 - 3072*a^2*(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]^14 - 44800*a^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^16 - 10752*a^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^16 - 1024*a^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]^16 + 117228825*(a + b)^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sqrt[-Sinh[c + d*x]^2] + 215549775*a^2*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 260465625*a*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 17069535*(a + b)^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 58009455*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 207173295*a^2*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 41427855*a*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 56109375*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8*Sqrt[-Sinh[c + d*x]^2] + 33756345*a^2*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8*Sqrt[-Sinh[c + d*x]^2] + 210735*a*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8*Sqrt[-Sinh[c + d*x]^2] + 9261945*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^10*Sqrt[-Sinh[c + d*x]^2] + 174825*a^2*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^10*Sqrt[-Sinh[c + d*x]^2] + 48825*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^12*Sqrt[-Sinh[c + d*x]^2] - 274542345*a*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(-Sinh[c + d*x]^2)^(3/2) - 109265625*(a + b)^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(-Sinh[c + d*x]^2)^(3/2) + 142065*(a + b)^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4*(-Sinh[c + d*x]^2)^(3/2)))/(90720*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^3)","C",0
70,1,319,74,1.4856365,"\int \text{sech}^2(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]^2*(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{sech}(c) \text{sech}(c+d x) \left(525 a^3 \sinh (2 c+3 d x)-210 a^3 \sinh (4 c+3 d x)+210 a^3 \sinh (4 c+5 d x)-35 a^3 \sinh (6 c+5 d x)+35 a^3 \sinh (6 c+7 d x)-35 a \left(15 a^2+26 a b+16 b^2\right) \sinh (2 c+d x)+1260 a^2 b \sinh (2 c+3 d x)-210 a^2 b \sinh (4 c+3 d x)+490 a^2 b \sinh (4 c+5 d x)+70 a^2 b \sinh (6 c+7 d x)+140 \left(5 a^3+11 a^2 b+10 a b^2+4 b^3\right) \sinh (d x)+1176 a b^2 \sinh (2 c+3 d x)+392 a b^2 \sinh (4 c+5 d x)+56 a b^2 \sinh (6 c+7 d x)+336 b^3 \sinh (2 c+3 d x)+112 b^3 \sinh (4 c+5 d x)+16 b^3 \sinh (6 c+7 d x)\right) \left(a+b \text{sech}^2(c+d x)\right)^3}{280 d (a \cosh (2 (c+d x))+a+2 b)^3}","\frac{3 b^2 (a+b) \tanh ^5(c+d x)}{5 d}-\frac{b (a+b)^2 \tanh ^3(c+d x)}{d}+\frac{(a+b)^3 \tanh (c+d x)}{d}-\frac{b^3 \tanh ^7(c+d x)}{7 d}",1,"(Sech[c]*Sech[c + d*x]*(a + b*Sech[c + d*x]^2)^3*(140*(5*a^3 + 11*a^2*b + 10*a*b^2 + 4*b^3)*Sinh[d*x] - 35*a*(15*a^2 + 26*a*b + 16*b^2)*Sinh[2*c + d*x] + 525*a^3*Sinh[2*c + 3*d*x] + 1260*a^2*b*Sinh[2*c + 3*d*x] + 1176*a*b^2*Sinh[2*c + 3*d*x] + 336*b^3*Sinh[2*c + 3*d*x] - 210*a^3*Sinh[4*c + 3*d*x] - 210*a^2*b*Sinh[4*c + 3*d*x] + 210*a^3*Sinh[4*c + 5*d*x] + 490*a^2*b*Sinh[4*c + 5*d*x] + 392*a*b^2*Sinh[4*c + 5*d*x] + 112*b^3*Sinh[4*c + 5*d*x] - 35*a^3*Sinh[6*c + 5*d*x] + 35*a^3*Sinh[6*c + 7*d*x] + 70*a^2*b*Sinh[6*c + 7*d*x] + 56*a*b^2*Sinh[6*c + 7*d*x] + 16*b^3*Sinh[6*c + 7*d*x]))/(280*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)","B",1
71,1,297,196,9.3958428,"\int \text{sech}^3(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]^3*(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{sech}^8(c+d x) \left(a \cosh ^2(c+d x)+b\right)^3 \left(2 b \left(144 a^2+120 a b+35 b^2\right) \tanh (c) \cosh ^5(c+d x)+2 b \left(144 a^2+120 a b+35 b^2\right) \text{sech}(c) \sinh (d x) \cosh ^4(c+d x)+3 \left(64 a^3+144 a^2 b+120 a b^2+35 b^3\right) \tanh (c) \cosh ^7(c+d x)+6 \left(64 a^3+144 a^2 b+120 a b^2+35 b^3\right) \cosh ^8(c+d x) \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+3 \left(64 a^3+144 a^2 b+120 a b^2+35 b^3\right) \text{sech}(c) \sinh (d x) \cosh ^6(c+d x)+8 b^2 (24 a+7 b) \tanh (c) \cosh ^3(c+d x)+8 b^2 (24 a+7 b) \text{sech}(c) \sinh (d x) \cosh ^2(c+d x)+48 b^3 \tanh (c) \cosh (c+d x)+48 b^3 \text{sech}(c) \sinh (d x)\right)}{48 d (a \cosh (2 (c+d x))+a+2 b)^3}","\frac{b \left(72 a^2+92 a b+35 b^2\right) \tanh (c+d x) \text{sech}^3(c+d x)}{192 d}+\frac{\left(64 a^3+144 a^2 b+120 a b^2+35 b^3\right) \tan ^{-1}(\sinh (c+d x))}{128 d}+\frac{\left(64 a^3+144 a^2 b+120 a b^2+35 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 \sinh ^2(c+d x)+a+b\right)^2}{8 d}+\frac{b (12 a+7 b) \tanh (c+d x) \text{sech}^5(c+d x) \left(a \sinh ^2(c+d x)+a+b\right)}{48 d}",1,"((b + a*Cosh[c + d*x]^2)^3*Sech[c + d*x]^8*(6*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*ArcTan[Tanh[(c + d*x)/2]]*Cosh[c + d*x]^8 + 48*b^3*Sech[c]*Sinh[d*x] + 8*b^2*(24*a + 7*b)*Cosh[c + d*x]^2*Sech[c]*Sinh[d*x] + 2*b*(144*a^2 + 120*a*b + 35*b^2)*Cosh[c + d*x]^4*Sech[c]*Sinh[d*x] + 3*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*Cosh[c + d*x]^6*Sech[c]*Sinh[d*x] + 48*b^3*Cosh[c + d*x]*Tanh[c] + 8*b^2*(24*a + 7*b)*Cosh[c + d*x]^3*Tanh[c] + 2*b*(144*a^2 + 120*a*b + 35*b^2)*Cosh[c + d*x]^5*Tanh[c] + 3*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*Cosh[c + d*x]^7*Tanh[c]))/(48*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)","A",1
72,1,348,108,1.7559171,"\int \text{sech}^4(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]^4*(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{sech}(c) \text{sech}^9(c+d x) \left(6825 a^3 \sinh (2 c+3 d x)-1995 a^3 \sinh (4 c+3 d x)+3465 a^3 \sinh (4 c+5 d x)-315 a^3 \sinh (6 c+5 d x)+945 a^3 \sinh (6 c+7 d x)+105 a^3 \sinh (8 c+9 d x)-315 a \left(17 a^2+36 a b+24 b^2\right) \sinh (2 c+d x)+18648 a^2 b \sinh (2 c+3 d x)-2520 a^2 b \sinh (4 c+3 d x)+9072 a^2 b \sinh (4 c+5 d x)+2268 a^2 b \sinh (6 c+7 d x)+252 a^2 b \sinh (8 c+9 d x)+63 \left(125 a^3+324 a^2 b+312 a b^2+128 b^3\right) \sinh (d x)+18144 a b^2 \sinh (2 c+3 d x)+7776 a b^2 \sinh (4 c+5 d x)+1944 a b^2 \sinh (6 c+7 d x)+216 a b^2 \sinh (8 c+9 d x)+5376 b^3 \sinh (2 c+3 d x)+2304 b^3 \sinh (4 c+5 d x)+576 b^3 \sinh (6 c+7 d x)+64 b^3 \sinh (8 c+9 d x)\right)}{40320 d}","-\frac{b^2 (3 a+4 b) \tanh ^7(c+d x)}{7 d}+\frac{3 b (a+b) (a+2 b) \tanh ^5(c+d x)}{5 d}-\frac{(a+b)^2 (a+4 b) \tanh ^3(c+d x)}{3 d}+\frac{(a+b)^3 \tanh (c+d x)}{d}+\frac{b^3 \tanh ^9(c+d x)}{9 d}",1,"(Sech[c]*Sech[c + d*x]^9*(63*(125*a^3 + 324*a^2*b + 312*a*b^2 + 128*b^3)*Sinh[d*x] - 315*a*(17*a^2 + 36*a*b + 24*b^2)*Sinh[2*c + d*x] + 6825*a^3*Sinh[2*c + 3*d*x] + 18648*a^2*b*Sinh[2*c + 3*d*x] + 18144*a*b^2*Sinh[2*c + 3*d*x] + 5376*b^3*Sinh[2*c + 3*d*x] - 1995*a^3*Sinh[4*c + 3*d*x] - 2520*a^2*b*Sinh[4*c + 3*d*x] + 3465*a^3*Sinh[4*c + 5*d*x] + 9072*a^2*b*Sinh[4*c + 5*d*x] + 7776*a*b^2*Sinh[4*c + 5*d*x] + 2304*b^3*Sinh[4*c + 5*d*x] - 315*a^3*Sinh[6*c + 5*d*x] + 945*a^3*Sinh[6*c + 7*d*x] + 2268*a^2*b*Sinh[6*c + 7*d*x] + 1944*a*b^2*Sinh[6*c + 7*d*x] + 576*b^3*Sinh[6*c + 7*d*x] + 105*a^3*Sinh[8*c + 9*d*x] + 252*a^2*b*Sinh[8*c + 9*d*x] + 216*a*b^2*Sinh[8*c + 9*d*x] + 64*b^3*Sinh[8*c + 9*d*x]))/(40320*d)","B",1
73,1,95,117,0.4951156,"\int \frac{\cosh ^4(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Cosh[c + d*x]^4/(a + b*Sech[c + d*x]^2),x]","\frac{4 \left(3 a^2-4 a b+8 b^2\right) (c+d x)+a^2 \sinh (4 (c+d x))-\frac{32 b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{\sqrt{a+b}}+8 a (a-b) \sinh (2 (c+d x))}{32 a^3 d}","-\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{a^3 d \sqrt{a+b}}+\frac{(3 a-4 b) \sinh (c+d x) \cosh (c+d x)}{8 a^2 d}+\frac{x \left(3 a^2-4 a b+8 b^2\right)}{8 a^3}+\frac{\sinh (c+d x) \cosh ^3(c+d x)}{4 a d}",1,"(4*(3*a^2 - 4*a*b + 8*b^2)*(c + d*x) - (32*b^(5/2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/Sqrt[a + b] + 8*a*(a - b)*Sinh[2*(c + d*x)] + a^2*Sinh[4*(c + d*x)])/(32*a^3*d)","A",1
74,1,79,76,0.2735852,"\int \frac{\cosh ^3(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Cosh[c + d*x]^3/(a + b*Sech[c + d*x]^2),x]","\frac{a^{3/2} \sinh (3 (c+d x))-\frac{12 b^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \text{csch}(c+d x)}{\sqrt{a}}\right)}{\sqrt{a+b}}+3 \sqrt{a} (3 a-4 b) \sinh (c+d x)}{12 a^{5/2} d}","\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{a^{5/2} d \sqrt{a+b}}+\frac{(a-b) \sinh (c+d x)}{a^2 d}+\frac{\sinh ^3(c+d x)}{3 a d}",1,"((-12*b^2*ArcTan[(Sqrt[a + b]*Csch[c + d*x])/Sqrt[a]])/Sqrt[a + b] + 3*Sqrt[a]*(3*a - 4*b)*Sinh[c + d*x] + a^(3/2)*Sinh[3*(c + d*x)])/(12*a^(5/2)*d)","A",1
75,1,67,75,0.2350714,"\int \frac{\cosh ^2(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Cosh[c + d*x]^2/(a + b*Sech[c + d*x]^2),x]","\frac{\frac{4 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{\sqrt{a+b}}+2 (a-2 b) (c+d x)+a \sinh (2 (c+d x))}{4 a^2 d}","\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a+b}}+\frac{x (a-2 b)}{2 a^2}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 a d}",1,"(2*(a - 2*b)*(c + d*x) + (4*b^(3/2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/Sqrt[a + b] + a*Sinh[2*(c + d*x)])/(4*a^2*d)","A",1
76,1,52,52,0.1241656,"\int \frac{\cosh (c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Cosh[c + d*x]/(a + b*Sech[c + d*x]^2),x]","\frac{\sqrt{a} \sinh (c+d x)-\frac{b \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{\sqrt{a+b}}}{a^{3/2} d}","\frac{\sinh (c+d x)}{a d}-\frac{b \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{a^{3/2} d \sqrt{a+b}}",1,"(-((b*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/Sqrt[a + b]) + Sqrt[a]*Sinh[c + d*x])/(a^(3/2)*d)","A",1
77,1,36,36,0.0656581,"\int \frac{\text{sech}(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Sech[c + d*x]/(a + b*Sech[c + d*x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{\sqrt{a} d \sqrt{a+b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{\sqrt{a} d \sqrt{a+b}}",1,"ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]]/(Sqrt[a]*Sqrt[a + b]*d)","A",1
78,1,36,36,0.0799649,"\int \frac{\text{sech}^2(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Sech[c + d*x]^2/(a + b*Sech[c + d*x]^2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{\sqrt{b} d \sqrt{a+b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{\sqrt{b} d \sqrt{a+b}}",1,"ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b]*d)","A",1
79,1,194,55,0.7120179,"\int \frac{\text{sech}^3(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Sech[c + d*x]^3/(a + b*Sech[c + d*x]^2),x]","\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(2 \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+\sqrt{a} \cosh (c) \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right)-\sqrt{a} \sinh (c) \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right)\right)}{2 b d \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \left(a+b \text{sech}^2(c+d x)\right)}","\frac{\tan ^{-1}(\sinh (c+d x))}{b d}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{b d \sqrt{a+b}}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(Sqrt[a]*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*Cosh[c] + 2*Sqrt[a + b]*ArcTan[Tanh[(c + d*x)/2]]*Sqrt[(Cosh[c] - Sinh[c])^2] - Sqrt[a]*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*Sinh[c]))/(2*b*Sqrt[a + b]*d*(a + b*Sech[c + d*x]^2)*Sqrt[(Cosh[c] - Sinh[c])^2])","B",1
80,1,182,52,0.6320235,"\int \frac{\text{sech}^4(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Sech[c + d*x]^4/(a + b*Sech[c + d*x]^2),x]","\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\sqrt{a+b} \text{sech}(c) \sinh (d x) \sqrt{b (\cosh (c)-\sinh (c))^4} \text{sech}(c+d x)+a (\sinh (2 c)-\cosh (2 c)) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)\right)}{2 b d \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4} \left(a+b \text{sech}^2(c+d x)\right)}","\frac{\tanh (c+d x)}{b d}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{b^{3/2} d \sqrt{a+b}}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(a*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(-Cosh[2*c] + Sinh[2*c]) + Sqrt[a + b]*Sech[c]*Sech[c + d*x]*Sqrt[b*(Cosh[c] - Sinh[c])^4]*Sinh[d*x]))/(2*b*Sqrt[a + b]*d*(a + b*Sech[c + d*x]^2)*Sqrt[b*(Cosh[c] - Sinh[c])^4])","B",1
81,1,213,86,1.8360326,"\int \frac{\text{sech}^5(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Sech[c + d*x]^5/(a + b*Sech[c + d*x]^2),x]","\frac{\cosh (c) \text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(2 a^{3/2} (\tanh (c)-1) \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right)+b \sqrt{a+b} \text{sech}^2(c) \sqrt{(\cosh (c)-\sinh (c))^2} \sinh (d x) \text{sech}^2(c+d x)-\sqrt{a+b} \text{sech}(c) \sqrt{(\cosh (c)-\sinh (c))^2} \left(2 (2 a-b) \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-b \tanh (c) \text{sech}(c+d x)\right)\right)}{4 b^2 d \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \left(a+b \text{sech}^2(c+d x)\right)}","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{b^2 d \sqrt{a+b}}-\frac{(2 a-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,"(Cosh[c]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(b*Sqrt[a + b]*Sech[c]^2*Sech[c + d*x]^2*Sqrt[(Cosh[c] - Sinh[c])^2]*Sinh[d*x] + 2*a^(3/2)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(-1 + Tanh[c]) - Sqrt[a + b]*Sech[c]*Sqrt[(Cosh[c] - Sinh[c])^2]*(2*(2*a - b)*ArcTan[Tanh[(c + d*x)/2]] - b*Sech[c + d*x]*Tanh[c])))/(4*b^2*Sqrt[a + b]*d*(a + b*Sech[c + d*x]^2)*Sqrt[(Cosh[c] - Sinh[c])^2])","B",1
82,1,214,77,2.1781515,"\int \frac{\text{sech}^6(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Sech[c + d*x]^6/(a + b*Sech[c + d*x]^2),x]","\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(3 a^2 (\cosh (2 c)-\sinh (2 c)) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)+\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4} \text{sech}(c+d x) \left(\text{sech}(c) \sinh (d x) \left(-3 a+b \text{sech}^2(c+d x)+2 b\right)+b \tanh (c) \text{sech}(c+d x)\right)\right)}{6 b^2 d \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4} \left(a+b \text{sech}^2(c+d x)\right)}","\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{b^{5/2} d \sqrt{a+b}}-\frac{(a-b) \tanh (c+d x)}{b^2 d}-\frac{\tanh ^3(c+d x)}{3 b d}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(3*a^2*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]) + Sqrt[a + b]*Sech[c + d*x]*Sqrt[b*(Cosh[c] - Sinh[c])^4]*(Sech[c]*(-3*a + 2*b + b*Sech[c + d*x]^2)*Sinh[d*x] + b*Sech[c + d*x]*Tanh[c])))/(6*b^2*Sqrt[a + b]*d*(a + b*Sech[c + d*x]^2)*Sqrt[b*(Cosh[c] - Sinh[c])^4])","B",1
83,1,113,125,0.9190362,"\int \frac{\cosh ^3(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Cosh[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2,x]","\frac{a^{3/2} \sinh (3 (c+d x))+3 \sqrt{a} \sinh (c+d x) \left(-\frac{4 b^3}{(a+b) (a \cosh (2 (c+d x))+a+2 b)}+3 a-8 b\right)-\frac{6 b^2 (6 a+5 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \text{csch}(c+d x)}{\sqrt{a}}\right)}{(a+b)^{3/2}}}{12 a^{7/2} d}","\frac{b^2 (6 a+5 b) \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{2 a^{7/2} d (a+b)^{3/2}}-\frac{b^3 \sinh (c+d x)}{2 a^3 d (a+b) \left(a \sinh ^2(c+d x)+a+b\right)}+\frac{(a-2 b) \sinh (c+d x)}{a^3 d}+\frac{\sinh ^3(c+d x)}{3 a^2 d}",1,"((-6*b^2*(6*a + 5*b)*ArcTan[(Sqrt[a + b]*Csch[c + d*x])/Sqrt[a]])/(a + b)^(3/2) + 3*Sqrt[a]*(3*a - 8*b - (4*b^3)/((a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])))*Sinh[c + d*x] + a^(3/2)*Sinh[3*(c + d*x)])/(12*a^(7/2)*d)","A",1
84,1,103,144,1.3793902,"\int \frac{\cosh ^2(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Cosh[c + d*x]^2/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\frac{2 b^{3/2} (5 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}+\sinh (2 (c+d x)) \left(\frac{2 a b^2}{(a+b) (a \cosh (2 (c+d x))+a+2 b)}+a\right)+2 (a-4 b) (c+d x)}{4 a^3 d}","\frac{b^{3/2} (5 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{2 a^3 d (a+b)^{3/2}}+\frac{x (a-4 b)}{2 a^3}+\frac{b (a+2 b) \tanh (c+d x)}{2 a^2 d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 a d \left(a-b \tanh ^2(c+d x)+b\right)}",1,"(2*(a - 4*b)*(c + d*x) + (2*b^(3/2)*(5*a + 4*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(3/2) + (a + (2*a*b^2)/((a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])))*Sinh[2*(c + d*x)])/(4*a^3*d)","A",1
85,1,234,100,1.8115603,"\int \frac{\cosh (c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Cosh[c + d*x]/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{2 \sqrt{a} b^2 \tanh (c+d x)}{a+b}+2 \sqrt{a} \sinh (c) \cosh (d x) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)+2 \sqrt{a} \cosh (c) \sinh (d x) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)+\frac{b (4 a+3 b) (\cosh (c)-\sinh (c)) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right)}{(a+b)^{3/2} \sqrt{(\cosh (c)-\sinh (c))^2}}\right)}{8 a^{5/2} d \left(a+b \text{sech}^2(c+d x)\right)^2}","-\frac{b (4 a+3 b) \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{2 a^{5/2} d (a+b)^{3/2}}+\frac{b^2 \sinh (c+d x)}{2 a^2 d (a+b) \left(a \sinh ^2(c+d x)+a+b\right)}+\frac{\sinh (c+d x)}{a^2 d}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^3*((b*(4*a + 3*b)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]*(Cosh[c] - Sinh[c]))/((a + b)^(3/2)*Sqrt[(Cosh[c] - Sinh[c])^2]) + 2*Sqrt[a]*Cosh[d*x]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]*Sinh[c] + 2*Sqrt[a]*Cosh[c]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]*Sinh[d*x] + (2*Sqrt[a]*b^2*Tanh[c + d*x])/(a + b)))/(8*a^(5/2)*d*(a + b*Sech[c + d*x]^2)^2)","B",1
86,1,124,82,0.3143085,"\int \frac{\text{sech}(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\left(2 a^2+3 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)-\sqrt{a} b \sqrt{a+b} \sinh (c+d x)+a (2 a+b) \sinh ^2(c+d x) \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{a^{3/2} d (a+b)^{3/2} (a \cosh (2 (c+d x))+a+2 b)}","\frac{(2 a+b) \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{2 a^{3/2} d (a+b)^{3/2}}-\frac{b \sinh (c+d x)}{2 a d (a+b) \left(a \sinh ^2(c+d x)+a+b\right)}",1,"((2*a^2 + 3*a*b + b^2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]] - Sqrt[a]*b*Sqrt[a + b]*Sinh[c + d*x] + a*(2*a + b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]]*Sinh[c + d*x]^2)/(a^(3/2)*(a + b)^(3/2)*d*(a + 2*b + a*Cosh[2*(c + d*x)]))","A",1
87,1,187,74,1.0145564,"\int \frac{\text{sech}^2(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]^2/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^4(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{(\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}-\frac{(a+2 b) \tanh (2 c)}{a}+\text{sech}(2 c) \sinh (2 d x)\right)}{8 d (a+b) \left(a+b \text{sech}^2(c+d x)\right)^2}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{2 \sqrt{b} d (a+b)^{3/2}}+\frac{\tanh (c+d x)}{2 d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*((ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + Sech[2*c]*Sinh[2*d*x] - ((a + 2*b)*Tanh[2*c])/a))/(8*(a + b)*d*(a + b*Sech[c + d*x]^2)^2)","B",1
88,1,108,73,0.2278594,"\int \frac{\text{sech}^3(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^4(c+d x) (a \cosh (2 c+2 d x)+a+2 b)^2 \left(\frac{\sinh (c+d x)}{(a+b) \left(a \sinh ^2(c+d x)+a+b\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{\sqrt{a} (a+b)^{3/2}}\right)}{8 d \left(a+b \text{sech}^2(c+d x)\right)^2}","\frac{\sinh (c+d x)}{2 d (a+b) \left(a \sinh ^2(c+d x)+a+b\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{2 \sqrt{a} d (a+b)^{3/2}}",1,"((a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]]/(Sqrt[a]*(a + b)^(3/2)) + Sinh[c + d*x]/((a + b)*(a + b + a*Sinh[c + d*x]^2))))/(8*d*(a + b*Sech[c + d*x]^2)^2)","A",1
89,1,88,83,0.2161381,"\int \frac{\text{sech}^4(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2,x]","4 \left(\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 b^{3/2} d (a+b)^{3/2}}-\frac{a \sinh (2 (c+d x))}{8 b d (a+b) (a \cosh (2 (c+d x))+a+2 b)}\right)","\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{2 b^{3/2} d (a+b)^{3/2}}-\frac{a \tanh (c+d x)}{2 b d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)}",1,"4*(((a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*b^(3/2)*(a + b)^(3/2)*d) - (a*Sinh[2*(c + d*x)])/(8*b*(a + b)*d*(a + 2*b + a*Cosh[2*(c + d*x)])))","A",1
90,1,282,101,2.5289789,"\int \frac{\text{sech}^5(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]^5/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(-2 a b \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh (c+d x)+\sqrt{a} (2 a+3 b) \cosh (c) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right)-\text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\sqrt{a} (2 a+3 b) \sinh (c) \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right)-4 (a+b)^{3/2} \sqrt{(\cosh (c)-\sinh (c))^2} \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{8 b^2 d (a+b)^{3/2} \sqrt{(\cosh (c)-\sinh (c))^2} \left(a+b \text{sech}^2(c+d x)\right)^2}","-\frac{\sqrt{a} (2 a+3 b) \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{2 b^2 d (a+b)^{3/2}}-\frac{a \sinh (c+d x)}{2 b d (a+b) \left(a \sinh ^2(c+d x)+a+b\right)}+\frac{\tan ^{-1}(\sinh (c+d x))}{b^2 d}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^3*(Sqrt[a]*(2*a + 3*b)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*Cosh[c]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x] - (a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]*(-4*(a + b)^(3/2)*ArcTan[Tanh[(c + d*x)/2]]*Sqrt[(Cosh[c] - Sinh[c])^2] + Sqrt[a]*(2*a + 3*b)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*Sinh[c]) - 2*a*b*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[c + d*x]))/(8*b^2*(a + b)^(3/2)*d*(a + b*Sech[c + d*x]^2)^2*Sqrt[(Cosh[c] - Sinh[c])^2])","B",1
91,1,229,101,3.8126624,"\int \frac{\text{sech}^6(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]^6/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^4(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(2 \text{sech}(c) \sinh (d x) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)+\frac{a (a \text{sech}(2 c) \sinh (2 d x)-(a+2 b) \tanh (2 c))}{a+b}-\frac{a (3 a+4 b) (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{(a+b)^{3/2} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{8 b^2 d \left(a+b \text{sech}^2(c+d x)\right)^2}","\frac{a^2 \tanh (c+d x)}{2 b^2 d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)}-\frac{a (3 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{2 b^{5/2} d (a+b)^{3/2}}+\frac{\tanh (c+d x)}{b^2 d}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*(-((a*(3*a + 4*b)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])*(Cosh[2*c] - Sinh[2*c]))/((a + b)^(3/2)*Sqrt[b*(Cosh[c] - Sinh[c])^4])) + 2*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c]*Sech[c + d*x]*Sinh[d*x] + (a*(a*Sech[2*c]*Sinh[2*d*x] - (a + 2*b)*Tanh[2*c]))/(a + b)))/(8*b^2*d*(a + b*Sech[c + d*x]^2)^2)","B",1
92,1,489,153,5.1933671,"\int \frac{\text{sech}^7(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]^7/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}(c) \text{sech}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(-a^{3/2} (4 a+5 b) \cosh ^2(c) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right)+2 a^2 b \sqrt{a+b} \cosh (c) \sqrt{(\cosh (c)-\sinh (c))^2} \tanh (c+d x)+a^{3/2} \sinh (2 c) \text{sech}(c+d x) \left(2 a^2 \cosh (2 (c+d x))+2 a^2+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right)-\cosh (c) \text{sech}(c+d x) \left(2 \sqrt{a+b} \left(4 a^2+3 a b-b^2\right) \sqrt{(\cosh (c)-\sinh (c))^2} \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right) (a \cosh (2 (c+d x))+a+2 b)-a^{5/2} b \sinh (c) (5 \cosh (2 (c+d x))+13) \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right)\right)+b (a+b)^{3/2} \sqrt{(\cosh (c)-\sinh (c))^2} \sinh (d x) \text{sech}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b)+b (a+b)^{3/2} \sinh (c) \sqrt{(\cosh (c)-\sinh (c))^2} \text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b)\right)}{8 b^3 d (a+b)^{3/2} \sqrt{(\cosh (c)-\sinh (c))^2} \left(a+b \text{sech}^2(c+d x)\right)^2}","\frac{a^{3/2} (4 a+5 b) \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{2 b^3 d (a+b)^{3/2}}-\frac{(4 a-b) \tan ^{-1}(\sinh (c+d x))}{2 b^3 d}+\frac{a (2 a+b) \sinh (c+d x)}{2 b^2 d (a+b) \left(a \sinh ^2(c+d x)+a+b\right)}+\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 b d \left(a \sinh ^2(c+d x)+a+b\right)}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c]*Sech[c + d*x]^3*(-(a^(3/2)*(4*a + 5*b)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*Cosh[c]^2*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]) + b*(a + b)^(3/2)*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*Sqrt[(Cosh[c] - Sinh[c])^2]*Sinh[c] - Cosh[c]*Sech[c + d*x]*(2*Sqrt[a + b]*(4*a^2 + 3*a*b - b^2)*ArcTan[Tanh[(c + d*x)/2]]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sqrt[(Cosh[c] - Sinh[c])^2] - a^(5/2)*b*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(13 + 5*Cosh[2*(c + d*x)])*Sinh[c]) + a^(3/2)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(2*a^2 + 5*b^2 + 2*a^2*Cosh[2*(c + d*x)])*Sech[c + d*x]*Sinh[2*c] + b*(a + b)^(3/2)*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^3*Sqrt[(Cosh[c] - Sinh[c])^2]*Sinh[d*x] + 2*a^2*b*Sqrt[a + b]*Cosh[c]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[c + d*x]))/(8*b^3*(a + b)^(3/2)*d*(a + b*Sech[c + d*x]^2)^2*Sqrt[(Cosh[c] - Sinh[c])^2])","B",1
93,1,156,204,4.2294614,"\int \frac{\cosh ^2(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Cosh[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]","\frac{\frac{b^{3/2} \left(35 a^2+56 a b+24 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}+a \sinh (2 (c+d x)) \left(\frac{2 b^3 (5 a \cosh (2 (c+d x))+3 a+8 b)}{(a+b)^2 (a \cosh (2 (c+d x))+a+2 b)^2}+\frac{13 a b^2}{(a+b)^2 (a \cosh (2 (c+d x))+a+2 b)}+2\right)+4 (a-6 b) (c+d x)}{8 a^4 d}","\frac{x (a-6 b)}{2 a^4}+\frac{b (4 a+3 b) (a+4 b) \tanh (c+d x)}{8 a^3 d (a+b)^2 \left(a-b \tanh ^2(c+d x)+b\right)}+\frac{b (2 a+3 b) \tanh (c+d x)}{4 a^2 d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)^2}+\frac{b^{3/2} \left(35 a^2+56 a b+24 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 a^4 d (a+b)^{5/2}}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 a d \left(a-b \tanh ^2(c+d x)+b\right)^2}",1,"(4*(a - 6*b)*(c + d*x) + (b^(3/2)*(35*a^2 + 56*a*b + 24*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) + a*(2 + (13*a*b^2)/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])) + (2*b^3*(3*a + 8*b + 5*a*Cosh[2*(c + d*x)]))/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2))*Sinh[2*(c + d*x)])/(8*a^4*d)","A",1
94,1,292,154,3.653038,"\int \frac{\cosh (c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Cosh[c + d*x]/(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{sech}^5(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{3 b \left(8 a^2+12 a b+5 b^2\right) (\cosh (c)-\sinh (c)) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right)}{(a+b)^{5/2} \sqrt{(\cosh (c)-\sinh (c))^2}}-\frac{8 \sqrt{a} b^3 \tanh (c+d x)}{a+b}+\frac{6 \sqrt{a} b^2 (4 a+3 b) \tanh (c+d x) (a \cosh (2 (c+d x))+a+2 b)}{(a+b)^2}+8 \sqrt{a} \sinh (c) \cosh (d x) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2+8 \sqrt{a} \cosh (c) \sinh (d x) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2\right)}{64 a^{7/2} d \left(a+b \text{sech}^2(c+d x)\right)^3}","-\frac{3 b \left(4 (a+b)^2+(2 a+b)^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{8 a^{7/2} d (a+b)^{5/2}}-\frac{b^3 \sinh (c+d x)}{4 a^3 d (a+b) \left(a \sinh ^2(c+d x)+a+b\right)^2}+\frac{3 b^2 (4 a+3 b) \sinh (c+d x)}{8 a^3 d (a+b)^2 \left(a \sinh ^2(c+d x)+a+b\right)}+\frac{\sinh (c+d x)}{a^3 d}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^5*((3*b*(8*a^2 + 12*a*b + 5*b^2)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]*(Cosh[c] - Sinh[c]))/((a + b)^(5/2)*Sqrt[(Cosh[c] - Sinh[c])^2]) + 8*Sqrt[a]*Cosh[d*x]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]*Sinh[c] + 8*Sqrt[a]*Cosh[c]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]*Sinh[d*x] - (8*Sqrt[a]*b^3*Tanh[c + d*x])/(a + b) + (6*Sqrt[a]*b^2*(4*a + 3*b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Tanh[c + d*x])/(a + b)^2))/(64*a^(7/2)*d*(a + b*Sech[c + d*x]^2)^3)","A",1
95,1,214,142,2.9360189,"\int \frac{\text{sech}(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]/(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{sech}^5(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{\left(8 a^2+8 a b+3 b^2\right) (\sinh (c)-\cosh (c)) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right)}{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}}+8 \sqrt{a} b^2 (a+b) \tanh (c+d x)-2 \sqrt{a} b (8 a+5 b) \tanh (c+d x) (a \cosh (2 (c+d x))+a+2 b)\right)}{64 a^{5/2} d (a+b)^2 \left(a+b \text{sech}^2(c+d x)\right)^3}","-\frac{3 b (2 a+b) \sinh (c+d x)}{8 a^2 d (a+b)^2 \left(a \sinh ^2(c+d x)+a+b\right)}+\frac{\left(8 a^2+8 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\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 \sinh ^2(c+d x)+a+b\right)^2}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^5*(((8*a^2 + 8*a*b + 3*b^2)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]*(-Cosh[c] + Sinh[c]))/(Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]) + 8*Sqrt[a]*b^2*(a + b)*Tanh[c + d*x] - 2*Sqrt[a]*b*(8*a + 5*b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Tanh[c + d*x]))/(64*a^(5/2)*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3)","A",1
96,1,258,108,2.1690246,"\int \frac{\text{sech}^2(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(-\frac{\text{sech}(2 c) \left(\left(5 a^2+16 a b+8 b^2\right) \sinh (2 c)-a (5 a+2 b) \sinh (2 d x)\right) (a \cosh (2 (c+d x))+a+2 b)}{a^2}+\frac{4 b (a+b) \text{sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{a^2}+\frac{3 (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b)^2 \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{64 d (a+b)^2 \left(a+b \text{sech}^2(c+d x)\right)^3}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 \sqrt{b} d (a+b)^{5/2}}+\frac{3 \tanh (c+d x)}{8 d (a+b)^2 \left(a-b \tanh ^2(c+d x)+b\right)}+\frac{\tanh (c+d x)}{4 d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)^2}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*((3*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (4*b*(a + b)*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/a^2 - ((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[2*c]*((5*a^2 + 16*a*b + 8*b^2)*Sinh[2*c] - a*(5*a + 2*b)*Sinh[2*d*x]))/a^2))/(64*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3)","B",1
97,1,159,123,0.7892033,"\int \frac{\text{sech}^3(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3,x]","-\frac{\text{sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b)^3 \left(\frac{8 \sinh (c+d x)}{\left(a \sinh ^2(c+d x)+a+b\right)^2}-(4 a+b) \left(\frac{5 (a+b) \sinh (c+d x)+3 a \sinh ^3(c+d x)}{(a+b)^2 \left(a \sinh ^2(c+d x)+a+b\right)^2}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{\sqrt{a} (a+b)^{5/2}}\right)\right)}{192 a d \left(a+b \text{sech}^2(c+d x)\right)^3}","\frac{(4 a+b) \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{8 a^{3/2} d (a+b)^{5/2}}+\frac{(4 a+b) \sinh (c+d x)}{8 a d (a+b)^2 \left(a \sinh ^2(c+d x)+a+b\right)}-\frac{b \sinh (c+d x)}{4 a d (a+b) \left(a \sinh ^2(c+d x)+a+b\right)^2}",1,"-1/192*((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*((8*Sinh[c + d*x])/(a + b + a*Sinh[c + d*x]^2)^2 - (4*a + b)*((3*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(5/2)) + (5*(a + b)*Sinh[c + d*x] + 3*a*Sinh[c + d*x]^3)/((a + b)^2*(a + b + a*Sinh[c + d*x]^2)^2))))/(a*d*(a + b*Sech[c + d*x]^2)^3)","A",1
98,1,250,125,3.3858485,"\int \frac{\text{sech}^4(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(-\frac{4 (a+b) \text{sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{a}+\frac{\text{sech}(2 c) ((a+4 b) \sinh (2 c)-(a-2 b) \sinh (2 d x)) (a \cosh (2 (c+d x))+a+2 b)}{b}+\frac{(a+4 b) (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b)^2 \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{b \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{64 d (a+b)^2 \left(a+b \text{sech}^2(c+d x)\right)^3}","\frac{(a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 b^{3/2} d (a+b)^{5/2}}+\frac{(a+4 b) \tanh (c+d x)}{8 b d (a+b)^2 \left(a-b \tanh ^2(c+d x)+b\right)}-\frac{a \tanh (c+d x)}{4 b d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)^2}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*(((a + 4*b)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*(Cosh[2*c] - Sinh[2*c]))/(b*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) - (4*(a + b)*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/a + ((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[2*c]*((a + 4*b)*Sinh[2*c] - (a - 2*b)*Sinh[2*d*x]))/b))/(64*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3)","A",1
99,1,125,106,0.2866229,"\int \frac{\text{sech}^5(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]^5/(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b)^3 \left(\frac{5 (a+b) \sinh (c+d x)+3 a \sinh ^3(c+d x)}{(a+b)^2 \left(a \sinh ^2(c+d x)+a+b\right)^2}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{\sqrt{a} (a+b)^{5/2}}\right)}{64 d \left(a+b \text{sech}^2(c+d x)\right)^3}","\frac{3 \sinh (c+d x)}{8 d (a+b)^2 \left(a \sinh ^2(c+d x)+a+b\right)}+\frac{\sinh (c+d x)}{4 d (a+b) \left(a \sinh ^2(c+d x)+a+b\right)^2}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{8 \sqrt{a} d (a+b)^{5/2}}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*((3*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(5/2)) + (5*(a + b)*Sinh[c + d*x] + 3*a*Sinh[c + d*x]^3)/((a + b)^2*(a + b + a*Sinh[c + d*x]^2)^2)))/(64*d*(a + b*Sech[c + d*x]^2)^3)","A",1
100,1,125,144,1.0117681,"\int \frac{\text{sech}^6(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]^6/(a + b*Sech[c + d*x]^2)^3,x]","\frac{\frac{\left(3 a^2+8 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}-\frac{a \sqrt{b} \sinh (2 (c+d x)) \left(3 a^2+3 a (a+2 b) \cosh (2 (c+d x))+16 a b+16 b^2\right)}{(a+b)^2 (a \cosh (2 (c+d x))+a+2 b)^2}}{8 b^{5/2} d}","\frac{\left(3 a^2+8 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 b^{5/2} d (a+b)^{5/2}}-\frac{3 a (a+2 b) \tanh (c+d x)}{8 b^2 d (a+b)^2 \left(a-b \tanh ^2(c+d x)+b\right)}-\frac{a \tanh (c+d x) \text{sech}^2(c+d x)}{4 b d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)^2}",1,"(((3*a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2))/(8*b^(5/2)*d)","A",1
101,1,247,153,4.7213458,"\int \frac{\text{sech}^7(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]^7/(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{sech}^5(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{\sqrt{a} \left(8 a^2+20 a b+15 b^2\right) (\cosh (c)-\sinh (c)) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right)}{(a+b)^{5/2} \sqrt{(\cosh (c)-\sinh (c))^2}}-\frac{8 a b^2 \tanh (c+d x)}{a+b}-\frac{2 a b (4 a+7 b) \tanh (c+d x) (a \cosh (2 (c+d x))+a+2 b)}{(a+b)^2}+16 \text{sech}(c+d x) \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right) (a \cosh (2 (c+d x))+a+2 b)^2\right)}{64 b^3 d \left(a+b \text{sech}^2(c+d x)\right)^3}","-\frac{\sqrt{a} \left(8 a^2+20 a b+15 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right)}{8 b^3 d (a+b)^{5/2}}-\frac{a (4 a+7 b) \sinh (c+d x)}{8 b^2 d (a+b)^2 \left(a \sinh ^2(c+d x)+a+b\right)}-\frac{a \sinh (c+d x)}{4 b d (a+b) \left(a \sinh ^2(c+d x)+a+b\right)^2}+\frac{\tan ^{-1}(\sinh (c+d x))}{b^3 d}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^5*(16*ArcTan[Tanh[(c + d*x)/2]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x] + (Sqrt[a]*(8*a^2 + 20*a*b + 15*b^2)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]*(Cosh[c] - Sinh[c]))/((a + b)^(5/2)*Sqrt[(Cosh[c] - Sinh[c])^2]) - (8*a*b^2*Tanh[c + d*x])/(a + b) - (2*a*b*(4*a + 7*b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Tanh[c + d*x])/(a + b)^2))/(64*b^3*d*(a + b*Sech[c + d*x]^2)^3)","A",1
102,1,57,48,0.0300978,"\int \left(a+b \text{sech}^2(c+d x)\right) \tanh ^4(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)*Tanh[c + d*x]^4,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 ^5(c+d x)}{5 d}","-\frac{a \tanh ^3(c+d x)}{3 d}-\frac{a \tanh (c+d x)}{d}+a x+\frac{b \tanh ^5(c+d x)}{5 d}",1,"(a*ArcTanh[Tanh[c + d*x]])/d - (a*Tanh[c + d*x])/d - (a*Tanh[c + d*x]^3)/(3*d) + (b*Tanh[c + d*x]^5)/(5*d)","A",1
103,1,45,49,0.0222177,"\int \left(a+b \text{sech}^2(c+d x)\right) \tanh ^3(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)*Tanh[c + d*x]^3,x]","-\frac{a \tanh ^2(c+d x)}{2 d}+\frac{a \log (\cosh (c+d x))}{d}+\frac{b \tanh ^4(c+d x)}{4 d}","\frac{(a-b) \text{sech}^2(c+d x)}{2 d}+\frac{a \log (\cosh (c+d x))}{d}+\frac{b \text{sech}^4(c+d x)}{4 d}",1,"(a*Log[Cosh[c + d*x]])/d - (a*Tanh[c + d*x]^2)/(2*d) + (b*Tanh[c + d*x]^4)/(4*d)","A",1
104,1,41,32,0.0165789,"\int \left(a+b \text{sech}^2(c+d x)\right) \tanh ^2(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)*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 ^3(c+d x)}{3 d}","-\frac{a \tanh (c+d x)}{d}+a x+\frac{b \tanh ^3(c+d x)}{3 d}",1,"(a*ArcTanh[Tanh[c + d*x]])/d - (a*Tanh[c + d*x])/d + (b*Tanh[c + d*x]^3)/(3*d)","A",1
105,1,29,29,0.0214623,"\int \left(a+b \text{sech}^2(c+d x)\right) \tanh (c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)*Tanh[c + d*x],x]","\frac{a \log (\cosh (c+d x))}{d}-\frac{b \text{sech}^2(c+d x)}{2 d}","\frac{a \log (\cosh (c+d x))}{d}-\frac{b \text{sech}^2(c+d x)}{2 d}",1,"(a*Log[Cosh[c + d*x]])/d - (b*Sech[c + d*x]^2)/(2*d)","A",1
106,1,15,15,0.0028999,"\int \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[a + b*Sech[c + d*x]^2,x]","a x+\frac{b \tanh (c+d x)}{d}","a x+\frac{b \tanh (c+d x)}{d}",1,"a*x + (b*Tanh[c + d*x])/d","A",1
107,1,44,28,0.0405782,"\int \coth (c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Coth[c + d*x]*(a + b*Sech[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))-\log (\sinh (c+d x)))}{d}","\frac{(a+b) \log (\sinh (c+d x))}{d}-\frac{b \log (\cosh (c+d x))}{d}",1,"-((b*(Log[Cosh[c + d*x]] - Log[Sinh[c + d*x]]))/d) + (a*(Log[Cosh[c + d*x]] + Log[Tanh[c + d*x]]))/d","A",1
108,1,41,18,0.0307079,"\int \coth ^2(c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Coth[c + d*x]^2*(a + b*Sech[c + d*x]^2),x]","-\frac{a \coth (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\tanh ^2(c+d x)\right)}{d}-\frac{b \coth (c+d x)}{d}","a x-\frac{(a+b) \coth (c+d x)}{d}",1,"-((b*Coth[c + d*x])/d) - (a*Coth[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, Tanh[c + d*x]^2])/d","C",1
109,1,52,31,0.1789548,"\int \coth ^3(c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Coth[c + d*x]^3*(a + b*Sech[c + d*x]^2),x]","-\frac{a \left(\coth ^2(c+d x)-2 \log (\tanh (c+d x))-2 \log (\cosh (c+d x))\right)}{2 d}-\frac{b \text{csch}^2(c+d x)}{2 d}","\frac{a \log (\sinh (c+d x))}{d}-\frac{(a+b) \text{csch}^2(c+d x)}{2 d}",1,"-1/2*(b*Csch[c + d*x]^2)/d - (a*(Coth[c + d*x]^2 - 2*Log[Cosh[c + d*x]] - 2*Log[Tanh[c + d*x]]))/(2*d)","A",1
110,1,49,34,0.0225861,"\int \coth ^4(c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Coth[c + d*x]^4*(a + b*Sech[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 ^3(c+d x)}{3 d}","-\frac{(a+b) \coth ^3(c+d x)}{3 d}-\frac{a \coth (c+d x)}{d}+a x",1,"-1/3*(b*Coth[c + d*x]^3)/d - (a*Coth[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, Tanh[c + d*x]^2])/(3*d)","C",1
111,1,62,51,0.2632869,"\int \coth ^5(c+d x) \left(a+b \text{sech}^2(c+d x)\right) \, dx","Integrate[Coth[c + d*x]^5*(a + b*Sech[c + d*x]^2),x]","-\frac{a \left(\coth ^4(c+d x)+2 \coth ^2(c+d x)-4 \log (\tanh (c+d x))-4 \log (\cosh (c+d x))\right)}{4 d}-\frac{b \coth ^4(c+d x)}{4 d}","-\frac{(a+b) \text{csch}^4(c+d x)}{4 d}-\frac{(2 a+b) \text{csch}^2(c+d x)}{2 d}+\frac{a \log (\sinh (c+d x))}{d}",1,"-1/4*(b*Coth[c + d*x]^4)/d - (a*(2*Coth[c + d*x]^2 + Coth[c + d*x]^4 - 4*Log[Cosh[c + d*x]] - 4*Log[Tanh[c + d*x]]))/(4*d)","A",1
112,1,395,77,1.18474,"\int \left(a+b \text{sech}^2(c+d x)\right)^2 \tanh ^4(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^2*Tanh[c + d*x]^4,x]","\frac{\text{sech}(c) \text{sech}^7(c+d x) \left(4480 a^2 \sinh (2 c+d x)-3780 a^2 \sinh (2 c+3 d x)+2100 a^2 \sinh (4 c+3 d x)-1540 a^2 \sinh (4 c+5 d x)+420 a^2 \sinh (6 c+5 d x)-280 a^2 \sinh (6 c+7 d x)+3675 a^2 d x \cosh (2 c+d x)+2205 a^2 d x \cosh (2 c+3 d x)+2205 a^2 d x \cosh (4 c+3 d x)+735 a^2 d x \cosh (4 c+5 d x)+735 a^2 d x \cosh (6 c+5 d x)+105 a^2 d x \cosh (6 c+7 d x)+105 a^2 d x \cosh (8 c+7 d x)-5320 a^2 \sinh (d x)+3675 a^2 d x \cosh (d x)-1260 a b \sinh (2 c+d x)+924 a b \sinh (2 c+3 d x)-840 a b \sinh (4 c+3 d x)+168 a b \sinh (4 c+5 d x)-420 a b \sinh (6 c+5 d x)+84 a b \sinh (6 c+7 d x)+1680 a b \sinh (d x)+420 b^2 \sinh (2 c+d x)-168 b^2 \sinh (2 c+3 d x)-420 b^2 \sinh (4 c+3 d x)+84 b^2 \sinh (4 c+5 d x)+12 b^2 \sinh (6 c+7 d x)+840 b^2 \sinh (d x)\right)}{13440 d}","-\frac{a^2 \tanh ^3(c+d x)}{3 d}-\frac{a^2 \tanh (c+d x)}{d}+a^2 x+\frac{b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac{b^2 \tanh ^7(c+d x)}{7 d}",1,"(Sech[c]*Sech[c + d*x]^7*(3675*a^2*d*x*Cosh[d*x] + 3675*a^2*d*x*Cosh[2*c + d*x] + 2205*a^2*d*x*Cosh[2*c + 3*d*x] + 2205*a^2*d*x*Cosh[4*c + 3*d*x] + 735*a^2*d*x*Cosh[4*c + 5*d*x] + 735*a^2*d*x*Cosh[6*c + 5*d*x] + 105*a^2*d*x*Cosh[6*c + 7*d*x] + 105*a^2*d*x*Cosh[8*c + 7*d*x] - 5320*a^2*Sinh[d*x] + 1680*a*b*Sinh[d*x] + 840*b^2*Sinh[d*x] + 4480*a^2*Sinh[2*c + d*x] - 1260*a*b*Sinh[2*c + d*x] + 420*b^2*Sinh[2*c + d*x] - 3780*a^2*Sinh[2*c + 3*d*x] + 924*a*b*Sinh[2*c + 3*d*x] - 168*b^2*Sinh[2*c + 3*d*x] + 2100*a^2*Sinh[4*c + 3*d*x] - 840*a*b*Sinh[4*c + 3*d*x] - 420*b^2*Sinh[4*c + 3*d*x] - 1540*a^2*Sinh[4*c + 5*d*x] + 168*a*b*Sinh[4*c + 5*d*x] + 84*b^2*Sinh[4*c + 5*d*x] + 420*a^2*Sinh[6*c + 5*d*x] - 420*a*b*Sinh[6*c + 5*d*x] - 280*a^2*Sinh[6*c + 7*d*x] + 84*a*b*Sinh[6*c + 7*d*x] + 12*b^2*Sinh[6*c + 7*d*x]))/(13440*d)","B",1
113,1,107,77,0.2899787,"\int \left(a+b \text{sech}^2(c+d x)\right)^2 \tanh ^3(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^2*Tanh[c + d*x]^3,x]","\frac{\cosh ^4(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \left(12 a^2 \log (\cosh (c+d x))+3 b (2 a-b) \text{sech}^4(c+d x)+6 a (a-2 b) \text{sech}^2(c+d x)+2 b^2 \text{sech}^6(c+d x)\right)}{3 d (a \cosh (2 c+2 d x)+a+2 b)^2}","\frac{a^2 \log (\cosh (c+d x))}{d}+\frac{b (2 a-b) \text{sech}^4(c+d x)}{4 d}+\frac{a (a-2 b) \text{sech}^2(c+d x)}{2 d}+\frac{b^2 \text{sech}^6(c+d x)}{6 d}",1,"(Cosh[c + d*x]^4*(a + b*Sech[c + d*x]^2)^2*(12*a^2*Log[Cosh[c + d*x]] + 6*a*(a - 2*b)*Sech[c + d*x]^2 + 3*(2*a - b)*b*Sech[c + d*x]^4 + 2*b^2*Sech[c + d*x]^6))/(3*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^2)","A",1
114,1,281,59,0.8639457,"\int \left(a+b \text{sech}^2(c+d x)\right)^2 \tanh ^2(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^2*Tanh[c + d*x]^2,x]","\frac{\text{sech}(c) \text{sech}^5(c+d x) \left(120 a^2 \sinh (2 c+d x)-120 a^2 \sinh (2 c+3 d x)+30 a^2 \sinh (4 c+3 d x)-30 a^2 \sinh (4 c+5 d x)+150 a^2 d x \cosh (2 c+d x)+75 a^2 d x \cosh (2 c+3 d x)+75 a^2 d x \cosh (4 c+3 d x)+15 a^2 d x \cosh (4 c+5 d x)+15 a^2 d x \cosh (6 c+5 d x)-180 a^2 \sinh (d x)+150 a^2 d x \cosh (d x)-120 a b \sinh (2 c+d x)+40 a b \sinh (2 c+3 d x)-60 a b \sinh (4 c+3 d x)+20 a b \sinh (4 c+5 d x)+80 a b \sinh (d x)-60 b^2 \sinh (2 c+d x)+20 b^2 \sinh (2 c+3 d x)+4 b^2 \sinh (4 c+5 d x)-20 b^2 \sinh (d x)\right)}{480 d}","-\frac{a^2 \tanh (c+d x)}{d}+a^2 x+\frac{b (2 a+b) \tanh ^3(c+d x)}{3 d}-\frac{b^2 \tanh ^5(c+d x)}{5 d}",1,"(Sech[c]*Sech[c + d*x]^5*(150*a^2*d*x*Cosh[d*x] + 150*a^2*d*x*Cosh[2*c + d*x] + 75*a^2*d*x*Cosh[2*c + 3*d*x] + 75*a^2*d*x*Cosh[4*c + 3*d*x] + 15*a^2*d*x*Cosh[4*c + 5*d*x] + 15*a^2*d*x*Cosh[6*c + 5*d*x] - 180*a^2*Sinh[d*x] + 80*a*b*Sinh[d*x] - 20*b^2*Sinh[d*x] + 120*a^2*Sinh[2*c + d*x] - 120*a*b*Sinh[2*c + d*x] - 60*b^2*Sinh[2*c + d*x] - 120*a^2*Sinh[2*c + 3*d*x] + 40*a*b*Sinh[2*c + 3*d*x] + 20*b^2*Sinh[2*c + 3*d*x] + 30*a^2*Sinh[4*c + 3*d*x] - 60*a*b*Sinh[4*c + 3*d*x] - 30*a^2*Sinh[4*c + 5*d*x] + 20*a*b*Sinh[4*c + 5*d*x] + 4*b^2*Sinh[4*c + 5*d*x]))/(480*d)","B",1
115,1,81,48,0.144919,"\int \left(a+b \text{sech}^2(c+d x)\right)^2 \tanh (c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^2*Tanh[c + d*x],x]","\frac{\text{sech}^4(c+d x) \left(a \cosh ^2(c+d x)+b\right)^2 \left(4 a^2 \cosh ^4(c+d x) \log (\cosh (c+d x))-4 a b \cosh ^2(c+d x)-b^2\right)}{d (a \cosh (2 (c+d x))+a+2 b)^2}","\frac{a^2 \log (\cosh (c+d x))}{d}-\frac{a b \text{sech}^2(c+d x)}{d}-\frac{b^2 \text{sech}^4(c+d x)}{4 d}",1,"((b + a*Cosh[c + d*x]^2)^2*(-b^2 - 4*a*b*Cosh[c + d*x]^2 + 4*a^2*Cosh[c + d*x]^4*Log[Cosh[c + d*x]])*Sech[c + d*x]^4)/(d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)","A",1
116,1,106,40,0.3768181,"\int \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[(a + b*Sech[c + d*x]^2)^2,x]","\frac{4 \text{sech}^3(c+d x) \left(a \cosh ^2(c+d x)+b\right)^2 \left(3 a^2 d x \cosh ^3(c+d x)+2 b (3 a+b) \text{sech}(c) \sinh (d x) \cosh ^2(c+d x)+b^2 \tanh (c) \cosh (c+d x)+b^2 \text{sech}(c) \sinh (d x)\right)}{3 d (a \cosh (2 (c+d x))+a+2 b)^2}","a^2 x+\frac{b (2 a+b) \tanh (c+d x)}{d}-\frac{b^2 \tanh ^3(c+d x)}{3 d}",1,"(4*(b + a*Cosh[c + d*x]^2)^2*Sech[c + d*x]^3*(3*a^2*d*x*Cosh[c + d*x]^3 + b^2*Sech[c]*Sinh[d*x] + 2*b*(3*a + b)*Cosh[c + d*x]^2*Sech[c]*Sinh[d*x] + b^2*Cosh[c + d*x]*Tanh[c]))/(3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)","B",1
117,1,84,53,0.2460179,"\int \coth (c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Coth[c + d*x]*(a + b*Sech[c + d*x]^2)^2,x]","\frac{2 (a \cosh (c+d x)+b \text{sech}(c+d x))^2 \left(2 \cosh ^2(c+d x) \left((a+b)^2 \log (\sinh (c+d x))-b (2 a+b) \log (\cosh (c+d x))\right)+b^2\right)}{d (a \cosh (2 (c+d x))+a+2 b)^2}","\frac{(a+b)^2 \log (\sinh (c+d x))}{d}-\frac{b (2 a+b) \log (\cosh (c+d x))}{d}+\frac{b^2 \text{sech}^2(c+d x)}{2 d}",1,"(2*(b^2 + 2*Cosh[c + d*x]^2*(-(b*(2*a + b)*Log[Cosh[c + d*x]]) + (a + b)^2*Log[Sinh[c + d*x]]))*(a*Cosh[c + d*x] + b*Sech[c + d*x])^2)/(d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)","A",1
118,1,82,36,0.7410422,"\int \coth ^2(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Coth[c + d*x]^2*(a + b*Sech[c + d*x]^2)^2,x]","\frac{4 \text{sech}(c+d x) \left(a \cosh ^2(c+d x)+b\right)^2 \left(a^2 d x \cosh (c+d x)+\sinh (d x) \left((a+b)^2 \text{csch}(c) \coth (c+d x)-b^2 \text{sech}(c)\right)\right)}{d (a \cosh (2 (c+d x))+a+2 b)^2}","a^2 x-\frac{(a+b)^2 \coth (c+d x)}{d}-\frac{b^2 \tanh (c+d x)}{d}",1,"(4*(b + a*Cosh[c + d*x]^2)^2*Sech[c + d*x]*(a^2*d*x*Cosh[c + d*x] + ((a + b)^2*Coth[c + d*x]*Csch[c] - b^2*Sech[c])*Sinh[d*x]))/(d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)","B",1
119,1,82,55,0.2004717,"\int \coth ^3(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Coth[c + d*x]^3*(a + b*Sech[c + d*x]^2)^2,x]","-\frac{2 \left(a \cosh ^2(c+d x)+b\right)^2 \left((a+b)^2 \text{csch}^2(c+d x)-2 \left(\left(a^2-b^2\right) \log (\sinh (c+d x))+b^2 \log (\cosh (c+d x))\right)\right)}{d (a \cosh (2 (c+d x))+a+2 b)^2}","\frac{\left(a^2-b^2\right) \log (\sinh (c+d x))}{d}-\frac{(a+b)^2 \text{csch}^2(c+d x)}{2 d}+\frac{b^2 \log (\cosh (c+d x))}{d}",1,"(-2*(b + a*Cosh[c + d*x]^2)^2*((a + b)^2*Csch[c + d*x]^2 - 2*(b^2*Log[Cosh[c + d*x]] + (a^2 - b^2)*Log[Sinh[c + d*x]])))/(d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)","A",1
120,1,160,46,0.8095167,"\int \coth ^4(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Coth[c + d*x]^4*(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{csch}(c) \text{csch}^3(c+d x) \left(-12 a^2 \sinh (2 c+d x)+8 a^2 \sinh (2 c+3 d x)-9 a^2 d x \cosh (2 c+d x)-3 a^2 d x \cosh (2 c+3 d x)+3 a^2 d x \cosh (4 c+3 d x)-12 a^2 \sinh (d x)+9 a^2 d x \cosh (d x)-12 a b \sinh (2 c+d x)+4 a b \sinh (2 c+3 d x)-4 b^2 \sinh (2 c+3 d x)+12 b^2 \sinh (d x)\right)}{24 d}","-\frac{\left(a^2-b^2\right) \coth (c+d x)}{d}+a^2 x-\frac{(a+b)^2 \coth ^3(c+d x)}{3 d}",1,"(Csch[c]*Csch[c + d*x]^3*(9*a^2*d*x*Cosh[d*x] - 9*a^2*d*x*Cosh[2*c + d*x] - 3*a^2*d*x*Cosh[2*c + 3*d*x] + 3*a^2*d*x*Cosh[4*c + 3*d*x] - 12*a^2*Sinh[d*x] + 12*b^2*Sinh[d*x] - 12*a^2*Sinh[2*c + d*x] - 12*a*b*Sinh[2*c + d*x] + 8*a^2*Sinh[2*c + 3*d*x] + 4*a*b*Sinh[2*c + 3*d*x] - 4*b^2*Sinh[2*c + 3*d*x]))/(24*d)","B",1
121,1,77,52,0.2324863,"\int \coth ^5(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Coth[c + d*x]^5*(a + b*Sech[c + d*x]^2)^2,x]","-\frac{\left(a \cosh ^2(c+d x)+b\right)^2 \left(-4 a^2 \log (\sinh (c+d x))+(a+b)^2 \text{csch}^4(c+d x)+4 a (a+b) \text{csch}^2(c+d x)\right)}{d (a \cosh (2 (c+d x))+a+2 b)^2}","\frac{a^2 \log (\sinh (c+d x))}{d}-\frac{(a+b)^2 \text{csch}^4(c+d x)}{4 d}-\frac{a (a+b) \text{csch}^2(c+d x)}{d}",1,"-(((b + a*Cosh[c + d*x]^2)^2*(4*a*(a + b)*Csch[c + d*x]^2 + (a + b)^2*Csch[c + d*x]^4 - 4*a^2*Log[Sinh[c + d*x]]))/(d*(a + 2*b + a*Cosh[2*(c + d*x)])^2))","A",1
122,1,256,64,1.1014542,"\int \coth ^6(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Coth[c + d*x]^6*(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{csch}(c) \text{csch}^5(c+d x) \left(180 a^2 \sinh (2 c+d x)-140 a^2 \sinh (2 c+3 d x)-90 a^2 \sinh (4 c+3 d x)+46 a^2 \sinh (4 c+5 d x)+150 a^2 d x \cosh (2 c+d x)+75 a^2 d x \cosh (2 c+3 d x)-75 a^2 d x \cosh (4 c+3 d x)-15 a^2 d x \cosh (4 c+5 d x)+15 a^2 d x \cosh (6 c+5 d x)+280 a^2 \sinh (d x)-150 a^2 d x \cosh (d x)-60 a b \sinh (4 c+3 d x)+12 a b \sinh (4 c+5 d x)+120 a b \sinh (d x)-60 b^2 \sinh (2 c+d x)+20 b^2 \sinh (2 c+3 d x)-4 b^2 \sinh (4 c+5 d x)+20 b^2 \sinh (d x)\right)}{480 d}","-\frac{\left(a^2-b^2\right) \coth ^3(c+d x)}{3 d}-\frac{a^2 \coth (c+d x)}{d}+a^2 x-\frac{(a+b)^2 \coth ^5(c+d x)}{5 d}",1,"(Csch[c]*Csch[c + d*x]^5*(-150*a^2*d*x*Cosh[d*x] + 150*a^2*d*x*Cosh[2*c + d*x] + 75*a^2*d*x*Cosh[2*c + 3*d*x] - 75*a^2*d*x*Cosh[4*c + 3*d*x] - 15*a^2*d*x*Cosh[4*c + 5*d*x] + 15*a^2*d*x*Cosh[6*c + 5*d*x] + 280*a^2*Sinh[d*x] + 120*a*b*Sinh[d*x] + 20*b^2*Sinh[d*x] + 180*a^2*Sinh[2*c + d*x] - 60*b^2*Sinh[2*c + d*x] - 140*a^2*Sinh[2*c + 3*d*x] + 20*b^2*Sinh[2*c + 3*d*x] - 90*a^2*Sinh[4*c + 3*d*x] - 60*a*b*Sinh[4*c + 3*d*x] + 46*a^2*Sinh[4*c + 5*d*x] + 12*a*b*Sinh[4*c + 5*d*x] - 4*b^2*Sinh[4*c + 5*d*x]))/(480*d)","B",1
123,1,107,86,0.5164015,"\int \coth ^7(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^2 \, dx","Integrate[Coth[c + d*x]^7*(a + b*Sech[c + d*x]^2)^2,x]","-\frac{\left(a \cosh ^2(c+d x)+b\right)^2 \left(3 \left(3 a^2+4 a b+b^2\right) \text{csch}^4(c+d x)-12 a^2 \log (\sinh (c+d x))+2 (a+b)^2 \text{csch}^6(c+d x)+6 a (3 a+2 b) \text{csch}^2(c+d x)\right)}{3 d (a \cosh (2 (c+d x))+a+2 b)^2}","\frac{a^2 \log (\sinh (c+d x))}{d}-\frac{(a+b)^2 \text{csch}^4(c+d x)}{4 d}-\frac{a (a+b) \text{csch}^2(c+d x)}{d}-\frac{\text{csch}^6(c+d x) \left(a \cosh ^2(c+d x)+b\right)^3}{6 d (a+b)}",1,"-1/3*((b + a*Cosh[c + d*x]^2)^2*(6*a*(3*a + 2*b)*Csch[c + d*x]^2 + 3*(3*a^2 + 4*a*b + b^2)*Csch[c + d*x]^4 + 2*(a + b)^2*Csch[c + d*x]^6 - 12*a^2*Log[Sinh[c + d*x]]))/(d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)","A",1
124,1,301,110,5.9357793,"\int \left(a+b \text{sech}^2(c+d x)\right)^3 \tanh ^4(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^3*Tanh[c + d*x]^4,x]","\frac{8 \text{sech}^9(c+d x) \left(a \cosh ^2(c+d x)+b\right)^3 \left(315 a^3 d x \cosh ^9(c+d x)+3 b \left(63 a^2-72 a b+b^2\right) \tanh (c) \cosh ^5(c+d x)+3 b \left(63 a^2-72 a b+b^2\right) \text{sech}(c) \sinh (d x) \cosh ^4(c+d x)+\left(105 a^3-378 a^2 b+27 a b^2+4 b^3\right) \tanh (c) \cosh ^7(c+d x)-\left(420 a^3-189 a^2 b-54 a b^2-8 b^3\right) \text{sech}(c) \sinh (d x) \cosh ^8(c+d x)+\left(105 a^3-378 a^2 b+27 a b^2+4 b^3\right) \text{sech}(c) \sinh (d x) \cosh ^6(c+d x)+5 b^2 (27 a-10 b) \tanh (c) \cosh ^3(c+d x)+5 b^2 (27 a-10 b) \text{sech}(c) \sinh (d x) \cosh ^2(c+d x)+35 b^3 \tanh (c) \cosh (c+d x)+35 b^3 \text{sech}(c) \sinh (d x)\right)}{315 d (a \cosh (2 (c+d x))+a+2 b)^3}","-\frac{a^3 \tanh ^3(c+d x)}{3 d}-\frac{a^3 \tanh (c+d x)}{d}+a^3 x+\frac{b \left(3 a^2+3 a b+b^2\right) \tanh ^5(c+d x)}{5 d}-\frac{b^2 (3 a+2 b) \tanh ^7(c+d x)}{7 d}+\frac{b^3 \tanh ^9(c+d x)}{9 d}",1,"(8*(b + a*Cosh[c + d*x]^2)^3*Sech[c + d*x]^9*(315*a^3*d*x*Cosh[c + d*x]^9 + 35*b^3*Sech[c]*Sinh[d*x] + 5*(27*a - 10*b)*b^2*Cosh[c + d*x]^2*Sech[c]*Sinh[d*x] + 3*b*(63*a^2 - 72*a*b + b^2)*Cosh[c + d*x]^4*Sech[c]*Sinh[d*x] + (105*a^3 - 378*a^2*b + 27*a*b^2 + 4*b^3)*Cosh[c + d*x]^6*Sech[c]*Sinh[d*x] - (420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*Cosh[c + d*x]^8*Sech[c]*Sinh[d*x] + 35*b^3*Cosh[c + d*x]*Tanh[c] + 5*(27*a - 10*b)*b^2*Cosh[c + d*x]^3*Tanh[c] + 3*b*(63*a^2 - 72*a*b + b^2)*Cosh[c + d*x]^5*Tanh[c] + (105*a^3 - 378*a^2*b + 27*a*b^2 + 4*b^3)*Cosh[c + d*x]^7*Tanh[c]))/(315*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)","B",1
125,1,128,103,0.7544285,"\int \left(a+b \text{sech}^2(c+d x)\right)^3 \tanh ^3(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^3*Tanh[c + d*x]^3,x]","\frac{\cosh ^6(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \left(24 a^3 \log (\cosh (c+d x))+12 a^2 (a-3 b) \text{sech}^2(c+d x)+4 b^2 (3 a-b) \text{sech}^6(c+d x)+18 a b (a-b) \text{sech}^4(c+d x)+3 b^3 \text{sech}^8(c+d x)\right)}{3 d (a \cosh (2 c+2 d x)+a+2 b)^3}","\frac{a^3 \log (\cosh (c+d x))}{d}-\frac{3 a^2 b \text{sech}^2(c+d x)}{2 d}-\frac{3 a b^2 \text{sech}^4(c+d x)}{4 d}+\frac{\text{sech}^8(c+d x) \left(a \cosh ^2(c+d x)+b\right)^4}{8 b d}-\frac{b^3 \text{sech}^6(c+d x)}{6 d}",1,"(Cosh[c + d*x]^6*(a + b*Sech[c + d*x]^2)^3*(24*a^3*Log[Cosh[c + d*x]] + 12*a^2*(a - 3*b)*Sech[c + d*x]^2 + 18*a*(a - b)*b*Sech[c + d*x]^4 + 4*(3*a - b)*b^2*Sech[c + d*x]^6 + 3*b^3*Sech[c + d*x]^8))/(3*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^3)","A",1
126,1,479,92,1.7900402,"\int \left(a+b \text{sech}^2(c+d x)\right)^3 \tanh ^2(c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^3*Tanh[c + d*x]^2,x]","\frac{\text{sech}(c) \text{sech}^7(c+d x) \left(3150 a^3 \sinh (2 c+d x)-3150 a^3 \sinh (2 c+3 d x)+1260 a^3 \sinh (4 c+3 d x)-1260 a^3 \sinh (4 c+5 d x)+210 a^3 \sinh (6 c+5 d x)-210 a^3 \sinh (6 c+7 d x)+3675 a^3 d x \cosh (2 c+d x)+2205 a^3 d x \cosh (2 c+3 d x)+2205 a^3 d x \cosh (4 c+3 d x)+735 a^3 d x \cosh (4 c+5 d x)+735 a^3 d x \cosh (6 c+5 d x)+105 a^3 d x \cosh (6 c+7 d x)+105 a^3 d x \cosh (8 c+7 d x)-4200 a^3 \sinh (d x)+3675 a^3 d x \cosh (d x)-3990 a^2 b \sinh (2 c+d x)+1890 a^2 b \sinh (2 c+3 d x)-2520 a^2 b \sinh (4 c+3 d x)+840 a^2 b \sinh (4 c+5 d x)-630 a^2 b \sinh (6 c+5 d x)+210 a^2 b \sinh (6 c+7 d x)+3360 a^2 b \sinh (d x)-2100 a b^2 \sinh (2 c+d x)+504 a b^2 \sinh (2 c+3 d x)-1260 a b^2 \sinh (4 c+3 d x)+588 a b^2 \sinh (4 c+5 d x)+84 a b^2 \sinh (6 c+7 d x)+840 a b^2 \sinh (d x)-1120 b^3 \sinh (2 c+d x)+336 b^3 \sinh (2 c+3 d x)+112 b^3 \sinh (4 c+5 d x)+16 b^3 \sinh (6 c+7 d x)-560 b^3 \sinh (d x)\right)}{13440 d}","-\frac{a^3 \tanh (c+d x)}{d}+a^3 x+\frac{b \left(3 a^2+3 a b+b^2\right) \tanh ^3(c+d x)}{3 d}-\frac{b^2 (3 a+2 b) \tanh ^5(c+d x)}{5 d}+\frac{b^3 \tanh ^7(c+d x)}{7 d}",1,"(Sech[c]*Sech[c + d*x]^7*(3675*a^3*d*x*Cosh[d*x] + 3675*a^3*d*x*Cosh[2*c + d*x] + 2205*a^3*d*x*Cosh[2*c + 3*d*x] + 2205*a^3*d*x*Cosh[4*c + 3*d*x] + 735*a^3*d*x*Cosh[4*c + 5*d*x] + 735*a^3*d*x*Cosh[6*c + 5*d*x] + 105*a^3*d*x*Cosh[6*c + 7*d*x] + 105*a^3*d*x*Cosh[8*c + 7*d*x] - 4200*a^3*Sinh[d*x] + 3360*a^2*b*Sinh[d*x] + 840*a*b^2*Sinh[d*x] - 560*b^3*Sinh[d*x] + 3150*a^3*Sinh[2*c + d*x] - 3990*a^2*b*Sinh[2*c + d*x] - 2100*a*b^2*Sinh[2*c + d*x] - 1120*b^3*Sinh[2*c + d*x] - 3150*a^3*Sinh[2*c + 3*d*x] + 1890*a^2*b*Sinh[2*c + 3*d*x] + 504*a*b^2*Sinh[2*c + 3*d*x] + 336*b^3*Sinh[2*c + 3*d*x] + 1260*a^3*Sinh[4*c + 3*d*x] - 2520*a^2*b*Sinh[4*c + 3*d*x] - 1260*a*b^2*Sinh[4*c + 3*d*x] - 1260*a^3*Sinh[4*c + 5*d*x] + 840*a^2*b*Sinh[4*c + 5*d*x] + 588*a*b^2*Sinh[4*c + 5*d*x] + 112*b^3*Sinh[4*c + 5*d*x] + 210*a^3*Sinh[6*c + 5*d*x] - 630*a^2*b*Sinh[6*c + 5*d*x] - 210*a^3*Sinh[6*c + 7*d*x] + 210*a^2*b*Sinh[6*c + 7*d*x] + 84*a*b^2*Sinh[6*c + 7*d*x] + 16*b^3*Sinh[6*c + 7*d*x]))/(13440*d)","B",1
127,1,100,71,0.3005707,"\int \left(a+b \text{sech}^2(c+d x)\right)^3 \tanh (c+d x) \, dx","Integrate[(a + b*Sech[c + d*x]^2)^3*Tanh[c + d*x],x]","\frac{2 \text{sech}^6(c+d x) \left(a \cosh ^2(c+d x)+b\right)^3 \left(12 a^3 \cosh ^6(c+d x) \log (\cosh (c+d x))-18 a^2 b \cosh ^4(c+d x)-9 a b^2 \cosh ^2(c+d x)-2 b^3\right)}{3 d (a \cosh (2 (c+d x))+a+2 b)^3}","\frac{a^3 \log (\cosh (c+d x))}{d}-\frac{3 a^2 b \text{sech}^2(c+d x)}{2 d}-\frac{3 a b^2 \text{sech}^4(c+d x)}{4 d}-\frac{b^3 \text{sech}^6(c+d x)}{6 d}",1,"(2*(b + a*Cosh[c + d*x]^2)^3*(-2*b^3 - 9*a*b^2*Cosh[c + d*x]^2 - 18*a^2*b*Cosh[c + d*x]^4 + 12*a^3*Cosh[c + d*x]^6*Log[Cosh[c + d*x]])*Sech[c + d*x]^6)/(3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)","A",1
128,1,268,73,0.927866,"\int \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{sech}(c) \text{sech}^5(c+d x) \left(150 a^3 d x \cosh (2 c+d x)+75 a^3 d x \cosh (2 c+3 d x)+75 a^3 d x \cosh (4 c+3 d x)+15 a^3 d x \cosh (4 c+5 d x)+15 a^3 d x \cosh (6 c+5 d x)+150 a^3 d x \cosh (d x)-360 a^2 b \sinh (2 c+d x)+360 a^2 b \sinh (2 c+3 d x)-90 a^2 b \sinh (4 c+3 d x)+90 a^2 b \sinh (4 c+5 d x)+540 a^2 b \sinh (d x)-180 a b^2 \sinh (2 c+d x)+300 a b^2 \sinh (2 c+3 d x)+60 a b^2 \sinh (4 c+5 d x)+420 a b^2 \sinh (d x)+80 b^3 \sinh (2 c+3 d x)+16 b^3 \sinh (4 c+5 d x)+160 b^3 \sinh (d x)\right)}{480 d}","a^3 x+\frac{b \left(3 a^2+3 a b+b^2\right) \tanh (c+d x)}{d}-\frac{b^2 (3 a+2 b) \tanh ^3(c+d x)}{3 d}+\frac{b^3 \tanh ^5(c+d x)}{5 d}",1,"(Sech[c]*Sech[c + d*x]^5*(150*a^3*d*x*Cosh[d*x] + 150*a^3*d*x*Cosh[2*c + d*x] + 75*a^3*d*x*Cosh[2*c + 3*d*x] + 75*a^3*d*x*Cosh[4*c + 3*d*x] + 15*a^3*d*x*Cosh[4*c + 5*d*x] + 15*a^3*d*x*Cosh[6*c + 5*d*x] + 540*a^2*b*Sinh[d*x] + 420*a*b^2*Sinh[d*x] + 160*b^3*Sinh[d*x] - 360*a^2*b*Sinh[2*c + d*x] - 180*a*b^2*Sinh[2*c + d*x] + 360*a^2*b*Sinh[2*c + 3*d*x] + 300*a*b^2*Sinh[2*c + 3*d*x] + 80*b^3*Sinh[2*c + 3*d*x] - 90*a^2*b*Sinh[4*c + 3*d*x] + 90*a^2*b*Sinh[4*c + 5*d*x] + 60*a*b^2*Sinh[4*c + 5*d*x] + 16*b^3*Sinh[4*c + 5*d*x]))/(480*d)","B",1
129,1,114,84,0.6633777,"\int \coth (c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Coth[c + d*x]*(a + b*Sech[c + d*x]^2)^3,x]","-\frac{2 \cosh ^6(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \left(4 b \left(3 a^2+3 a b+b^2\right) \log (\cosh (c+d x))-2 b^2 (3 a+b) \text{sech}^2(c+d x)-4 (a+b)^3 \log (\sinh (c+d x))-b^3 \text{sech}^4(c+d x)\right)}{d (a \cosh (2 c+2 d x)+a+2 b)^3}","-\frac{b \left(3 a^2+3 a b+b^2\right) \log (\cosh (c+d x))}{d}+\frac{b^2 (3 a+b) \text{sech}^2(c+d x)}{2 d}+\frac{(a+b)^3 \log (\sinh (c+d x))}{d}+\frac{b^3 \text{sech}^4(c+d x)}{4 d}",1,"(-2*Cosh[c + d*x]^6*(a + b*Sech[c + d*x]^2)^3*(4*b*(3*a^2 + 3*a*b + b^2)*Log[Cosh[c + d*x]] - 4*(a + b)^3*Log[Sinh[c + d*x]] - 2*b^2*(3*a + b)*Sech[c + d*x]^2 - b^3*Sech[c + d*x]^4))/(d*(a + 2*b + a*Cosh[2*c + 2*d*x])^3)","A",1
130,1,126,61,1.7985389,"\int \coth ^2(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Coth[c + d*x]^2*(a + b*Sech[c + d*x]^2)^3,x]","\frac{8 (a \cosh (c+d x)+b \text{sech}(c+d x))^3 \left(3 a^3 d x \cosh ^3(c+d x)+\sinh (d x) \cosh ^2(c+d x) \left(3 (a+b)^3 \text{csch}(c) \coth (c+d x)-b^2 (9 a+5 b) \text{sech}(c)\right)-b^3 \tanh (c) \cosh (c+d x)+b^3 (-\text{sech}(c)) \sinh (d x)\right)}{3 d (a \cosh (2 (c+d x))+a+2 b)^3}","a^3 x-\frac{b^2 (3 a+2 b) \tanh (c+d x)}{d}-\frac{(a+b)^3 \coth (c+d x)}{d}+\frac{b^3 \tanh ^3(c+d x)}{3 d}",1,"(8*(a*Cosh[c + d*x] + b*Sech[c + d*x])^3*(3*a^3*d*x*Cosh[c + d*x]^3 - b^3*Sech[c]*Sinh[d*x] + Cosh[c + d*x]^2*(3*(a + b)^3*Coth[c + d*x]*Csch[c] - b^2*(9*a + 5*b)*Sech[c])*Sinh[d*x] - b^3*Cosh[c + d*x]*Tanh[c]))/(3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)","B",1
131,1,110,81,1.2421105,"\int \coth ^3(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Coth[c + d*x]^3*(a + b*Sech[c + d*x]^2)^3,x]","-\frac{4 \cosh ^6(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \left(-2 b^2 (3 a+2 b) \log (\cosh (c+d x))+(a+b)^3 \text{csch}^2(c+d x)-2 (a-2 b) (a+b)^2 \log (\sinh (c+d x))+b^3 \text{sech}^2(c+d x)\right)}{d (a \cosh (2 c+2 d x)+a+2 b)^3}","\frac{b^2 (3 a+2 b) \log (\cosh (c+d x))}{d}-\frac{(a+b)^3 \text{csch}^2(c+d x)}{2 d}+\frac{(a-2 b) (a+b)^2 \log (\sinh (c+d x))}{d}-\frac{b^3 \text{sech}^2(c+d x)}{2 d}",1,"(-4*Cosh[c + d*x]^6*(a + b*Sech[c + d*x]^2)^3*((a + b)^3*Csch[c + d*x]^2 - 2*b^2*(3*a + 2*b)*Log[Cosh[c + d*x]] - 2*(a - 2*b)*(a + b)^2*Log[Sinh[c + d*x]] + b^3*Sech[c + d*x]^2))/(d*(a + 2*b + a*Cosh[2*c + 2*d*x])^3)","A",1
132,1,343,60,1.726231,"\int \coth ^4(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Coth[c + d*x]^4*(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{csch}(c) \text{sech}(c) \text{csch}^3(c+d x) \text{sech}(c+d x) \left(-16 a^3 \sinh (2 (c+d x))+8 a^3 \sinh (4 (c+d x))+8 a^3 \sinh (2 (c+2 d x))-12 a^3 \sinh (4 c+2 d x)-3 a^3 d x \cosh (2 (c+2 d x))-6 a^3 d x \cosh (4 c+2 d x)+3 a^3 d x \cosh (6 c+4 d x)-4 a^3 \sinh (2 d x)+6 a^3 d x \cosh (2 d x)-12 a^2 b \sinh (2 (c+d x))+6 a^2 b \sinh (4 (c+d x))+6 a^2 b \sinh (2 (c+2 d x))-18 a^2 b \sinh (4 c+2 d x)-18 a^2 b \sinh (2 c)+6 a^2 b \sinh (2 d x)+24 a b^2 \sinh (2 (c+d x))-12 a b^2 \sinh (4 (c+d x))-12 a b^2 \sinh (2 (c+2 d x))-36 a b^2 \sinh (2 c)+24 a b^2 \sinh (2 d x)+8 b^3 \sinh (2 (c+d x))-4 b^3 \sinh (4 (c+d x))-16 b^3 \sinh (2 (c+2 d x))+32 b^3 \sinh (2 d x)\right)}{96 d}","a^3 x-\frac{(a+b)^3 \coth ^3(c+d x)}{3 d}-\frac{(a-2 b) (a+b)^2 \coth (c+d x)}{d}+\frac{b^3 \tanh (c+d x)}{d}",1,"(Csch[c]*Csch[c + d*x]^3*Sech[c]*Sech[c + d*x]*(6*a^3*d*x*Cosh[2*d*x] - 3*a^3*d*x*Cosh[2*(c + 2*d*x)] - 6*a^3*d*x*Cosh[4*c + 2*d*x] + 3*a^3*d*x*Cosh[6*c + 4*d*x] - 18*a^2*b*Sinh[2*c] - 36*a*b^2*Sinh[2*c] - 4*a^3*Sinh[2*d*x] + 6*a^2*b*Sinh[2*d*x] + 24*a*b^2*Sinh[2*d*x] + 32*b^3*Sinh[2*d*x] - 16*a^3*Sinh[2*(c + d*x)] - 12*a^2*b*Sinh[2*(c + d*x)] + 24*a*b^2*Sinh[2*(c + d*x)] + 8*b^3*Sinh[2*(c + d*x)] + 8*a^3*Sinh[4*(c + d*x)] + 6*a^2*b*Sinh[4*(c + d*x)] - 12*a*b^2*Sinh[4*(c + d*x)] - 4*b^3*Sinh[4*(c + d*x)] + 8*a^3*Sinh[2*(c + 2*d*x)] + 6*a^2*b*Sinh[2*(c + 2*d*x)] - 12*a*b^2*Sinh[2*(c + 2*d*x)] - 16*b^3*Sinh[2*(c + 2*d*x)] - 12*a^3*Sinh[4*c + 2*d*x] - 18*a^2*b*Sinh[4*c + 2*d*x]))/(96*d)","B",1
133,1,101,81,0.9072884,"\int \coth ^5(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Coth[c + d*x]^5*(a + b*Sech[c + d*x]^2)^3,x]","-\frac{2 \left(a \cosh ^2(c+d x)+b\right)^3 \left(-4 \left(a^3+b^3\right) \log (\sinh (c+d x))+(a+b)^3 \text{csch}^4(c+d x)+2 (2 a-b) (a+b)^2 \text{csch}^2(c+d x)+4 b^3 \log (\cosh (c+d x))\right)}{d (a \cosh (2 (c+d x))+a+2 b)^3}","\frac{\left(a^3+b^3\right) \log (\sinh (c+d x))}{d}-\frac{(a+b)^3 \text{csch}^4(c+d x)}{4 d}-\frac{(2 a-b) (a+b)^2 \text{csch}^2(c+d x)}{2 d}-\frac{b^3 \log (\cosh (c+d x))}{d}",1,"(-2*(b + a*Cosh[c + d*x]^2)^3*(2*(2*a - b)*(a + b)^2*Csch[c + d*x]^2 + (a + b)^3*Csch[c + d*x]^4 + 4*b^3*Log[Cosh[c + d*x]] - 4*(a^3 + b^3)*Log[Sinh[c + d*x]]))/(d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)","A",1
134,1,303,69,1.0868121,"\int \coth ^6(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Coth[c + d*x]^6*(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{csch}(c) \text{csch}^5(c+d x) \left(180 a^3 \sinh (2 c+d x)-140 a^3 \sinh (2 c+3 d x)-90 a^3 \sinh (4 c+3 d x)+46 a^3 \sinh (4 c+5 d x)+150 a^3 d x \cosh (2 c+d x)+75 a^3 d x \cosh (2 c+3 d x)-75 a^3 d x \cosh (4 c+3 d x)-15 a^3 d x \cosh (4 c+5 d x)+15 a^3 d x \cosh (6 c+5 d x)+280 a^3 \sinh (d x)-150 a^3 d x \cosh (d x)-90 a^2 b \sinh (4 c+3 d x)+18 a^2 b \sinh (4 c+5 d x)+180 a^2 b \sinh (d x)-180 a b^2 \sinh (2 c+d x)+60 a b^2 \sinh (2 c+3 d x)-12 a b^2 \sinh (4 c+5 d x)+60 a b^2 \sinh (d x)-80 b^3 \sinh (2 c+3 d x)+16 b^3 \sinh (4 c+5 d x)+160 b^3 \sinh (d x)\right)}{480 d}","-\frac{\left(a^3+b^3\right) \coth (c+d x)}{d}+a^3 x-\frac{(a+b)^3 \coth ^5(c+d x)}{5 d}-\frac{(a-2 b) (a+b)^2 \coth ^3(c+d x)}{3 d}",1,"(Csch[c]*Csch[c + d*x]^5*(-150*a^3*d*x*Cosh[d*x] + 150*a^3*d*x*Cosh[2*c + d*x] + 75*a^3*d*x*Cosh[2*c + 3*d*x] - 75*a^3*d*x*Cosh[4*c + 3*d*x] - 15*a^3*d*x*Cosh[4*c + 5*d*x] + 15*a^3*d*x*Cosh[6*c + 5*d*x] + 280*a^3*Sinh[d*x] + 180*a^2*b*Sinh[d*x] + 60*a*b^2*Sinh[d*x] + 160*b^3*Sinh[d*x] + 180*a^3*Sinh[2*c + d*x] - 180*a*b^2*Sinh[2*c + d*x] - 140*a^3*Sinh[2*c + 3*d*x] + 60*a*b^2*Sinh[2*c + 3*d*x] - 80*b^3*Sinh[2*c + 3*d*x] - 90*a^3*Sinh[4*c + 3*d*x] - 90*a^2*b*Sinh[4*c + 3*d*x] + 46*a^3*Sinh[4*c + 5*d*x] + 18*a^2*b*Sinh[4*c + 5*d*x] - 12*a*b^2*Sinh[4*c + 5*d*x] + 16*b^3*Sinh[4*c + 5*d*x]))/(480*d)","B",1
135,1,98,77,0.7968304,"\int \coth ^7(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^3 \, dx","Integrate[Coth[c + d*x]^7*(a + b*Sech[c + d*x]^2)^3,x]","-\frac{2 \left(a \cosh ^2(c+d x)+b\right)^3 \left(-12 a^3 \log (\sinh (c+d x))+18 a^2 (a+b) \text{csch}^2(c+d x)+2 (a+b)^3 \text{csch}^6(c+d x)+9 a (a+b)^2 \text{csch}^4(c+d x)\right)}{3 d (a \cosh (2 (c+d x))+a+2 b)^3}","\frac{a^3 \log (\sinh (c+d x))}{d}-\frac{3 a^2 (a+b) \text{csch}^2(c+d x)}{2 d}-\frac{(a+b)^3 \text{csch}^6(c+d x)}{6 d}-\frac{3 a (a+b)^2 \text{csch}^4(c+d x)}{4 d}",1,"(-2*(b + a*Cosh[c + d*x]^2)^3*(18*a^2*(a + b)*Csch[c + d*x]^2 + 9*a*(a + b)^2*Csch[c + d*x]^4 + 2*(a + b)^3*Csch[c + d*x]^6 - 12*a^3*Log[Sinh[c + d*x]]))/(3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)","A",1
136,1,455,111,1.6658523,"\int \left(a+b \text{sech}^2(c+d x)\right)^4 \, dx","Integrate[(a + b*Sech[c + d*x]^2)^4,x]","\frac{\text{sech}(c) \text{sech}^7(c+d x) \left(3675 a^4 d x \cosh (2 c+d x)+2205 a^4 d x \cosh (2 c+3 d x)+2205 a^4 d x \cosh (4 c+3 d x)+735 a^4 d x \cosh (4 c+5 d x)+735 a^4 d x \cosh (6 c+5 d x)+105 a^4 d x \cosh (6 c+7 d x)+105 a^4 d x \cosh (8 c+7 d x)+3675 a^4 d x \cosh (d x)-12600 a^3 b \sinh (2 c+d x)+12600 a^3 b \sinh (2 c+3 d x)-5040 a^3 b \sinh (4 c+3 d x)+5040 a^3 b \sinh (4 c+5 d x)-840 a^3 b \sinh (6 c+5 d x)+840 a^3 b \sinh (6 c+7 d x)+16800 a^3 b \sinh (d x)-10920 a^2 b^2 \sinh (2 c+d x)+15120 a^2 b^2 \sinh (2 c+3 d x)-2520 a^2 b^2 \sinh (4 c+3 d x)+5880 a^2 b^2 \sinh (4 c+5 d x)+840 a^2 b^2 \sinh (6 c+7 d x)+18480 a^2 b^2 \sinh (d x)-4480 a b^3 \sinh (2 c+d x)+9408 a b^3 \sinh (2 c+3 d x)+3136 a b^3 \sinh (4 c+5 d x)+448 a b^3 \sinh (6 c+7 d x)+11200 a b^3 \sinh (d x)+2016 b^4 \sinh (2 c+3 d x)+672 b^4 \sinh (4 c+5 d x)+96 b^4 \sinh (6 c+7 d x)+3360 b^4 \sinh (d x)\right)}{13440 d}","a^4 x-\frac{b^2 \left(6 a^2+8 a b+3 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+3 b) \tanh ^5(c+d x)}{5 d}-\frac{b^4 \tanh ^7(c+d x)}{7 d}",1,"(Sech[c]*Sech[c + d*x]^7*(3675*a^4*d*x*Cosh[d*x] + 3675*a^4*d*x*Cosh[2*c + d*x] + 2205*a^4*d*x*Cosh[2*c + 3*d*x] + 2205*a^4*d*x*Cosh[4*c + 3*d*x] + 735*a^4*d*x*Cosh[4*c + 5*d*x] + 735*a^4*d*x*Cosh[6*c + 5*d*x] + 105*a^4*d*x*Cosh[6*c + 7*d*x] + 105*a^4*d*x*Cosh[8*c + 7*d*x] + 16800*a^3*b*Sinh[d*x] + 18480*a^2*b^2*Sinh[d*x] + 11200*a*b^3*Sinh[d*x] + 3360*b^4*Sinh[d*x] - 12600*a^3*b*Sinh[2*c + d*x] - 10920*a^2*b^2*Sinh[2*c + d*x] - 4480*a*b^3*Sinh[2*c + d*x] + 12600*a^3*b*Sinh[2*c + 3*d*x] + 15120*a^2*b^2*Sinh[2*c + 3*d*x] + 9408*a*b^3*Sinh[2*c + 3*d*x] + 2016*b^4*Sinh[2*c + 3*d*x] - 5040*a^3*b*Sinh[4*c + 3*d*x] - 2520*a^2*b^2*Sinh[4*c + 3*d*x] + 5040*a^3*b*Sinh[4*c + 5*d*x] + 5880*a^2*b^2*Sinh[4*c + 5*d*x] + 3136*a*b^3*Sinh[4*c + 5*d*x] + 672*b^4*Sinh[4*c + 5*d*x] - 840*a^3*b*Sinh[6*c + 5*d*x] + 840*a^3*b*Sinh[6*c + 7*d*x] + 840*a^2*b^2*Sinh[6*c + 7*d*x] + 448*a*b^3*Sinh[6*c + 7*d*x] + 96*b^4*Sinh[6*c + 7*d*x]))/(13440*d)","B",1
137,1,724,163,6.5672026,"\int \left(a+b \text{sech}^2(c+d x)\right)^5 \, dx","Integrate[(a + b*Sech[c + d*x]^2)^5,x]","\frac{32 a^5 x \cosh ^{10}(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^5}{(a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{64 \text{sech}(c) \left(105 a^2 b^3 \sinh (c)+45 a b^4 \sinh (c)+8 b^5 \sinh (c)\right) \cosh ^6(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^5}{105 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{64 \text{sech}(c) \cosh ^5(c+d x) \left(105 a^2 b^3 \sinh (d x)+45 a b^4 \sinh (d x)+8 b^5 \sinh (d x)\right) \left(a+b \text{sech}^2(c+d x)\right)^5}{105 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{64 \text{sech}(c) \left(525 a^3 b^2 \sinh (c)+420 a^2 b^3 \sinh (c)+180 a b^4 \sinh (c)+32 b^5 \sinh (c)\right) \cosh ^8(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^5}{315 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{64 \text{sech}(c) \cosh ^7(c+d x) \left(525 a^3 b^2 \sinh (d x)+420 a^2 b^3 \sinh (d x)+180 a b^4 \sinh (d x)+32 b^5 \sinh (d x)\right) \left(a+b \text{sech}^2(c+d x)\right)^5}{315 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{32 \text{sech}(c) \cosh ^9(c+d x) \left(1575 a^4 b \sinh (d x)+2100 a^3 b^2 \sinh (d x)+1680 a^2 b^3 \sinh (d x)+720 a b^4 \sinh (d x)+128 b^5 \sinh (d x)\right) \left(a+b \text{sech}^2(c+d x)\right)^5}{315 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{32 b^5 \tanh (c) \cosh ^2(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^5}{9 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{32 b^5 \text{sech}(c) \sinh (d x) \cosh (c+d x) \left(a+b \text{sech}^2(c+d x)\right)^5}{9 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{32 \text{sech}(c) \left(45 a b^4 \sinh (c)+8 b^5 \sinh (c)\right) \cosh ^4(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^5}{63 d (a \cosh (2 c+2 d x)+a+2 b)^5}+\frac{32 \text{sech}(c) \cosh ^3(c+d x) \left(45 a b^4 \sinh (d x)+8 b^5 \sinh (d x)\right) \left(a+b \text{sech}^2(c+d x)\right)^5}{63 d (a \cosh (2 c+2 d x)+a+2 b)^5}","a^5 x+\frac{b^3 \left(10 a^2+15 a b+6 b^2\right) \tanh ^5(c+d x)}{5 d}-\frac{b^2 \left(10 a^3+20 a^2 b+15 a b^2+4 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+4 b) \tanh ^7(c+d x)}{7 d}+\frac{b^5 \tanh ^9(c+d x)}{9 d}",1,"(32*a^5*x*Cosh[c + d*x]^10*(a + b*Sech[c + d*x]^2)^5)/(a + 2*b + a*Cosh[2*c + 2*d*x])^5 + (32*Cosh[c + d*x]^4*Sech[c]*(a + b*Sech[c + d*x]^2)^5*(45*a*b^4*Sinh[c] + 8*b^5*Sinh[c]))/(63*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^5) + (64*Cosh[c + d*x]^6*Sech[c]*(a + b*Sech[c + d*x]^2)^5*(105*a^2*b^3*Sinh[c] + 45*a*b^4*Sinh[c] + 8*b^5*Sinh[c]))/(105*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^5) + (64*Cosh[c + d*x]^8*Sech[c]*(a + b*Sech[c + d*x]^2)^5*(525*a^3*b^2*Sinh[c] + 420*a^2*b^3*Sinh[c] + 180*a*b^4*Sinh[c] + 32*b^5*Sinh[c]))/(315*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^5) + (32*b^5*Cosh[c + d*x]*Sech[c]*(a + b*Sech[c + d*x]^2)^5*Sinh[d*x])/(9*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^5) + (32*Cosh[c + d*x]^3*Sech[c]*(a + b*Sech[c + d*x]^2)^5*(45*a*b^4*Sinh[d*x] + 8*b^5*Sinh[d*x]))/(63*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^5) + (64*Cosh[c + d*x]^5*Sech[c]*(a + b*Sech[c + d*x]^2)^5*(105*a^2*b^3*Sinh[d*x] + 45*a*b^4*Sinh[d*x] + 8*b^5*Sinh[d*x]))/(105*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^5) + (64*Cosh[c + d*x]^7*Sech[c]*(a + b*Sech[c + d*x]^2)^5*(525*a^3*b^2*Sinh[d*x] + 420*a^2*b^3*Sinh[d*x] + 180*a*b^4*Sinh[d*x] + 32*b^5*Sinh[d*x]))/(315*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^5) + (32*Cosh[c + d*x]^9*Sech[c]*(a + b*Sech[c + d*x]^2)^5*(1575*a^4*b*Sinh[d*x] + 2100*a^3*b^2*Sinh[d*x] + 1680*a^2*b^3*Sinh[d*x] + 720*a*b^4*Sinh[d*x] + 128*b^5*Sinh[d*x]))/(315*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^5) + (32*b^5*Cosh[c + d*x]^2*(a + b*Sech[c + d*x]^2)^5*Tanh[c])/(9*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^5)","B",1
138,1,98,70,0.3068418,"\int \frac{\tanh ^5(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Tanh[c + d*x]^5/(a + b*Sech[c + d*x]^2),x]","-\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(a b \text{sech}^2(c+d x)+(a+b)^2 \left(-\log \left(a \sinh ^2(c+d x)+a+b\right)\right)+2 a (a+2 b) \log (\cosh (c+d x))\right)}{4 a b^2 d \left(a+b \text{sech}^2(c+d x)\right)}","\frac{(a+b)^2 \log \left(a \cosh ^2(c+d x)+b\right)}{2 a b^2 d}-\frac{(a+2 b) \log (\cosh (c+d x))}{b^2 d}-\frac{\text{sech}^2(c+d x)}{2 b d}",1,"-1/4*((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(2*a*(a + 2*b)*Log[Cosh[c + d*x]] - (a + b)^2*Log[a + b + a*Sinh[c + d*x]^2] + a*b*Sech[c + d*x]^2))/(a*b^2*d*(a + b*Sech[c + d*x]^2))","A",1
139,1,196,59,1.0858404,"\int \frac{\tanh ^4(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Tanh[c + d*x]^4/(a + b*Sech[c + d*x]^2),x]","\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4} (a \text{sech}(c) \sinh (d x) \text{sech}(c+d x)+b d x)+(a+b)^2 (\sinh (2 c)-\cosh (2 c)) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)\right)}{2 a b d \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4} \left(a+b \text{sech}^2(c+d x)\right)}","-\frac{(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{a b^{3/2} d}+\frac{x}{a}+\frac{\tanh (c+d x)}{b d}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*((a + b)^2*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(-Cosh[2*c] + Sinh[2*c]) + Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]*(b*d*x + a*Sech[c]*Sech[c + d*x]*Sinh[d*x])))/(2*a*b*Sqrt[a + b]*d*(a + b*Sech[c + d*x]^2)*Sqrt[b*(Cosh[c] - Sinh[c])^4])","B",1
140,1,41,45,0.1166266,"\int \frac{\tanh ^3(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Tanh[c + d*x]^3/(a + b*Sech[c + d*x]^2),x]","\frac{(a+b) \log \left(a \cosh ^2(c+d x)+b\right)-2 a \log (\cosh (c+d x))}{2 a b d}","\frac{(a+b) \log \left(a \cosh ^2(c+d x)+b\right)}{2 a b d}-\frac{\log (\cosh (c+d x))}{b d}",1,"(-2*a*Log[Cosh[c + d*x]] + (a + b)*Log[b + a*Cosh[c + d*x]^2])/(2*a*b*d)","A",1
141,1,174,46,0.3038184,"\int \frac{\tanh ^2(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Tanh[c + d*x]^2/(a + b*Sech[c + d*x]^2),x]","\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(d x \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}+(a+b) (\sinh (2 c)-\cosh (2 c)) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)\right)}{2 a d \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4} \left(a+b \text{sech}^2(c+d x)\right)}","\frac{x}{a}-\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{a \sqrt{b} d}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(Sqrt[a + b]*d*x*Sqrt[b*(Cosh[c] - Sinh[c])^4] + (a + b)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(-Cosh[2*c] + Sinh[2*c])))/(2*a*Sqrt[a + b]*d*(a + b*Sech[c + d*x]^2)*Sqrt[b*(Cosh[c] - Sinh[c])^4])","B",1
142,1,26,23,0.1825616,"\int \frac{\tanh (c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Tanh[c + d*x]/(a + b*Sech[c + d*x]^2),x]","\frac{\log (a \cosh (2 (c+d x))+a+2 b)}{2 a d}","\frac{\log \left(a \cosh ^2(c+d x)+b\right)}{2 a d}",1,"Log[a + 2*b + a*Cosh[2*(c + d*x)]]/(2*a*d)","A",1
143,1,172,46,0.2496914,"\int \frac{1}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[(a + b*Sech[c + d*x]^2)^(-1),x]","\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(d x \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}+b (\sinh (2 c)-\cosh (2 c)) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)\right)}{2 a d \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4} \left(a+b \text{sech}^2(c+d x)\right)}","\frac{x}{a}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{a d \sqrt{a+b}}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(Sqrt[a + b]*d*x*Sqrt[b*(Cosh[c] - Sinh[c])^4] + b*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(-Cosh[2*c] + Sinh[2*c])))/(2*a*Sqrt[a + b]*d*(a + b*Sech[c + d*x]^2)*Sqrt[b*(Cosh[c] - Sinh[c])^4])","B",1
144,1,42,46,0.0920758,"\int \frac{\coth (c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Coth[c + d*x]/(a + b*Sech[c + d*x]^2),x]","\frac{b \log \left(a \sinh ^2(c+d x)+a+b\right)+2 a \log (\sinh (c+d x))}{2 a^2 d+2 a b d}","\frac{\log (\sinh (c+d x))}{d (a+b)}+\frac{b \log \left(a \cosh ^2(c+d x)+b\right)}{2 a d (a+b)}",1,"(2*a*Log[Sinh[c + d*x]] + b*Log[a + b + a*Sinh[c + d*x]^2])/(2*a^2*d + 2*a*b*d)","A",1
145,1,193,62,1.1606869,"\int \frac{\coth ^2(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Coth[c + d*x]^2/(a + b*Sech[c + d*x]^2),x]","\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(b^2 (\sinh (2 c)-\cosh (2 c)) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)+\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4} (d x (a+b)+a \text{csch}(c) \sinh (d x) \text{csch}(c+d x))\right)}{2 a d (a+b)^{3/2} \sqrt{b (\cosh (c)-\sinh (c))^4} \left(a+b \text{sech}^2(c+d x)\right)}","-\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{a d (a+b)^{3/2}}-\frac{\coth (c+d x)}{d (a+b)}+\frac{x}{a}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(b^2*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(-Cosh[2*c] + Sinh[2*c]) + Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]*((a + b)*d*x + a*Csch[c]*Csch[c + d*x]*Sinh[d*x])))/(2*a*(a + b)^(3/2)*d*(a + b*Sech[c + d*x]^2)*Sqrt[b*(Cosh[c] - Sinh[c])^4])","B",1
146,1,100,73,0.2358387,"\int \frac{\coth ^3(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Coth[c + d*x]^3/(a + b*Sech[c + d*x]^2),x]","-\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(b^2 \left(-\log \left(a \sinh ^2(c+d x)+a+b\right)\right)+a (a+b) \text{csch}^2(c+d x)-2 a (a+2 b) \log (\sinh (c+d x))\right)}{4 a d (a+b)^2 \left(a+b \text{sech}^2(c+d x)\right)}","\frac{b^2 \log \left(a \cosh ^2(c+d x)+b\right)}{2 a d (a+b)^2}-\frac{\text{csch}^2(c+d x)}{2 d (a+b)}+\frac{(a+2 b) \log (\sinh (c+d x))}{d (a+b)^2}",1,"-1/4*((a + 2*b + a*Cosh[2*(c + d*x)])*(a*(a + b)*Csch[c + d*x]^2 - 2*a*(a + 2*b)*Log[Sinh[c + d*x]] - b^2*Log[a + b + a*Sinh[c + d*x]^2])*Sech[c + d*x]^2)/(a*(a + b)^2*d*(a + b*Sech[c + d*x]^2))","A",1
147,1,380,87,3.3246625,"\int \frac{\coth ^4(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx","Integrate[Coth[c + d*x]^4/(a + b*Sech[c + d*x]^2),x]","\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{1}{8} \sqrt{a+b} \text{csch}(c) \sqrt{b (\cosh (c)-\sinh (c))^4} \text{csch}^3(c+d x) \left(-12 a^2 \sinh (2 c+d x)+8 a^2 \sinh (2 c+3 d x)-3 a^2 d x \cosh (2 c+3 d x)+3 a^2 d x \cosh (4 c+3 d x)-12 a^2 \sinh (d x)-18 a b \sinh (2 c+d x)+14 a b \sinh (2 c+3 d x)-6 a b d x \cosh (2 c+3 d x)+6 a b d x \cosh (4 c+3 d x)-9 d x (a+b)^2 \cosh (2 c+d x)-24 a b \sinh (d x)+9 d x (a+b)^2 \cosh (d x)-3 b^2 d x \cosh (2 c+3 d x)+3 b^2 d x \cosh (4 c+3 d x)\right)+3 b^3 (\sinh (2 c)-\cosh (2 c)) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)\right)}{6 a d (a+b)^{5/2} \sqrt{b (\cosh (c)-\sinh (c))^4} \left(a+b \text{sech}^2(c+d x)\right)}","-\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{a d (a+b)^{5/2}}-\frac{\coth ^3(c+d x)}{3 d (a+b)}-\frac{(a+2 b) \coth (c+d x)}{d (a+b)^2}+\frac{x}{a}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(3*b^3*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(-Cosh[2*c] + Sinh[2*c]) + (Sqrt[a + b]*Csch[c]*Csch[c + d*x]^3*Sqrt[b*(Cosh[c] - Sinh[c])^4]*(9*(a + b)^2*d*x*Cosh[d*x] - 9*(a + b)^2*d*x*Cosh[2*c + d*x] - 3*a^2*d*x*Cosh[2*c + 3*d*x] - 6*a*b*d*x*Cosh[2*c + 3*d*x] - 3*b^2*d*x*Cosh[2*c + 3*d*x] + 3*a^2*d*x*Cosh[4*c + 3*d*x] + 6*a*b*d*x*Cosh[4*c + 3*d*x] + 3*b^2*d*x*Cosh[4*c + 3*d*x] - 12*a^2*Sinh[d*x] - 24*a*b*Sinh[d*x] - 12*a^2*Sinh[2*c + d*x] - 18*a*b*Sinh[2*c + d*x] + 8*a^2*Sinh[2*c + 3*d*x] + 14*a*b*Sinh[2*c + 3*d*x]))/8))/(6*a*(a + b)^(5/2)*d*(a + b*Sech[c + d*x]^2)*Sqrt[b*(Cosh[c] - Sinh[c])^4])","B",1
148,1,109,76,0.4544395,"\int \frac{\tanh ^5(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Tanh[c + d*x]^5/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^4(c+d x) (a \cosh (2 c+2 d x)+a+2 b)^2 \left(\left(\frac{1}{a^2}-\frac{1}{b^2}\right) \log \left(a \cosh ^2(c+d x)+b\right)+\frac{(a+b)^2}{a^2 b \left(a \cosh ^2(c+d x)+b\right)}+\frac{2 \log (\cosh (c+d x))}{b^2}\right)}{8 d \left(a+b \text{sech}^2(c+d x)\right)^2}","\frac{\left(\frac{1}{a^2}-\frac{1}{b^2}\right) \log \left(a \cosh ^2(c+d x)+b\right)}{2 d}+\frac{(a+b)^2}{2 a^2 b d \left(a \cosh ^2(c+d x)+b\right)}+\frac{\log (\cosh (c+d x))}{b^2 d}",1,"((a + 2*b + a*Cosh[2*c + 2*d*x])^2*((a + b)^2/(a^2*b*(b + a*Cosh[c + d*x]^2)) + (2*Log[Cosh[c + d*x]])/b^2 + (a^(-2) - b^(-2))*Log[b + a*Cosh[c + d*x]^2])*Sech[c + d*x]^4)/(8*d*(a + b*Sech[c + d*x]^2)^2)","A",1
149,1,228,91,2.1333309,"\int \frac{\tanh ^4(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Tanh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^4(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{\left(a^2-a b-2 b^2\right) (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{b d \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}+2 x (a \cosh (2 (c+d x))+a+2 b)+\frac{(a+b) \text{sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{b d}\right)}{8 a^2 \left(a+b \text{sech}^2(c+d x)\right)^2}","\frac{(a-2 b) \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{2 a^2 b^{3/2} d}+\frac{x}{a^2}-\frac{(a+b) \tanh (c+d x)}{2 a b d \left(a-b \tanh ^2(c+d x)+b\right)}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*(2*x*(a + 2*b + a*Cosh[2*(c + d*x)]) + ((a^2 - a*b - 2*b^2)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])*(Cosh[2*c] - Sinh[2*c]))/(b*Sqrt[a + b]*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + ((a + b)*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/(b*d)))/(8*a^2*(a + b*Sech[c + d*x]^2)^2)","B",1
150,1,81,51,0.7596461,"\int \frac{\tanh ^3(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Tanh[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2,x]","\frac{(a+2 b) \log (a \cosh (2 (c+d x))+a+2 b)+a \cosh (2 (c+d x)) \log (a \cosh (2 (c+d x))+a+2 b)+2 (a+b)}{2 a^2 d (a \cosh (2 (c+d x))+a+2 b)}","\frac{a+b}{2 a^2 d \left(a \cosh ^2(c+d x)+b\right)}+\frac{\log \left(a \cosh ^2(c+d x)+b\right)}{2 a^2 d}",1,"(2*(a + b) + (a + 2*b)*Log[a + 2*b + a*Cosh[2*(c + d*x)]] + a*Cosh[2*(c + d*x)]*Log[a + 2*b + a*Cosh[2*(c + d*x)]])/(2*a^2*d*(a + 2*b + a*Cosh[2*(c + d*x)]))","A",1
151,1,326,85,4.6290409,"\int \frac{\tanh ^2(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Tanh[c + d*x]^2/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^4(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \left(\frac{\left(a^2+8 a b+8 b^2\right) \text{sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{a^2 b d (a+b) (a \cosh (2 (c+d x))+a+2 b)}+\frac{16 x}{a^2}+\frac{\left(a^3-6 a^2 b-24 a b^2-16 b^3\right) (\cosh (2 c)-\sinh (2 c)) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{a^2 b d (a+b)^{3/2} \sqrt{b (\cosh (c)-\sinh (c))^4}}-\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{b^{3/2} d (a+b)^{3/2}}+\frac{a \sinh (2 (c+d x))}{b d (a+b) (a \cosh (2 (c+d x))+a+2 b)}\right)}{64 \left(a+b \text{sech}^2(c+d x)\right)^2}","-\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{2 a^2 \sqrt{b} d \sqrt{a+b}}+\frac{x}{a^2}-\frac{\tanh (c+d x)}{2 a d \left(a-b \tanh ^2(c+d x)+b\right)}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]^4*((16*x)/a^2 - ((a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(b^(3/2)*(a + b)^(3/2)*d) + ((a^3 - 6*a^2*b - 24*a*b^2 - 16*b^3)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(a^2*b*(a + b)^(3/2)*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + ((a^2 + 8*a*b + 8*b^2)*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/(a^2*b*(a + b)*d*(a + 2*b + a*Cosh[2*(c + d*x)])) + (a*Sinh[2*(c + d*x)])/(b*(a + b)*d*(a + 2*b + a*Cosh[2*(c + d*x)]))))/(64*(a + b*Sech[c + d*x]^2)^2)","B",0
152,1,79,49,0.5930201,"\int \frac{\tanh (c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Tanh[c + d*x]/(a + b*Sech[c + d*x]^2)^2,x]","\frac{(a+2 b) \log (a \cosh (2 (c+d x))+a+2 b)+a \cosh (2 (c+d x)) \log (a \cosh (2 (c+d x))+a+2 b)+2 b}{2 a^2 d (a \cosh (2 (c+d x))+a+2 b)}","\frac{b}{2 a^2 d \left(a \cosh ^2(c+d x)+b\right)}+\frac{\log \left(a \cosh ^2(c+d x)+b\right)}{2 a^2 d}",1,"(2*b + (a + 2*b)*Log[a + 2*b + a*Cosh[2*(c + d*x)]] + a*Cosh[2*(c + d*x)]*Log[a + 2*b + a*Cosh[2*(c + d*x)]])/(2*a^2*d*(a + 2*b + a*Cosh[2*(c + d*x)]))","A",1
153,1,221,93,2.0419656,"\int \frac{1}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[(a + b*Sech[c + d*x]^2)^(-2),x]","\frac{\text{sech}^4(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(2 x (a \cosh (2 (c+d x))+a+2 b)+\frac{b \text{sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{d (a+b)}-\frac{b (3 a+2 b) (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{d (a+b)^{3/2} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{8 a^2 \left(a+b \text{sech}^2(c+d x)\right)^2}","-\frac{\sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{2 a^2 d (a+b)^{3/2}}+\frac{x}{a^2}-\frac{b \tanh (c+d x)}{2 a d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*(2*x*(a + 2*b + a*Cosh[2*(c + d*x)]) - (b*(3*a + 2*b)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])*(Cosh[2*c] - Sinh[2*c]))/((a + b)^(3/2)*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (b*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/((a + b)*d)))/(8*a^2*(a + b*Sech[c + d*x]^2)^2)","B",1
154,1,115,83,0.2951877,"\int \frac{\coth (c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Coth[c + d*x]/(a + b*Sech[c + d*x]^2)^2,x]","\frac{a \sinh ^2(c+d x) \left(2 a^2 \log (\sinh (c+d x))+b (2 a+b) \log \left(a \sinh ^2(c+d x)+a+b\right)\right)+(a+b) \left(2 a^2 \log (\sinh (c+d x))+b \left((2 a+b) \log \left(a \sinh ^2(c+d x)+a+b\right)+b\right)\right)}{a^2 d (a+b)^2 (a \cosh (2 (c+d x))+a+2 b)}","\frac{b^2}{2 a^2 d (a+b) \left(a \cosh ^2(c+d x)+b\right)}+\frac{b (2 a+b) \log \left(a \cosh ^2(c+d x)+b\right)}{2 a^2 d (a+b)^2}+\frac{\log (\sinh (c+d x))}{d (a+b)^2}",1,"((a + b)*(2*a^2*Log[Sinh[c + d*x]] + b*(b + (2*a + b)*Log[a + b + a*Sinh[c + d*x]^2])) + a*(2*a^2*Log[Sinh[c + d*x]] + b*(2*a + b)*Log[a + b + a*Sinh[c + d*x]^2])*Sinh[c + d*x]^2)/(a^2*(a + b)^2*d*(a + 2*b + a*Cosh[2*(c + d*x)]))","A",1
155,1,268,121,2.8242021,"\int \frac{\coth ^2(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Coth[c + d*x]^2/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^4(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{b^2 \text{sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{a^2 d (a+b)^2}-\frac{b^2 (5 a+2 b) (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{a^2 d (a+b)^{5/2} \sqrt{b (\cosh (c)-\sinh (c))^4}}+\frac{2 x (a \cosh (2 (c+d x))+a+2 b)}{a^2}+\frac{2 \text{csch}(c) \sinh (d x) \text{csch}(c+d x) (a \cosh (2 (c+d x))+a+2 b)}{d (a+b)^2}\right)}{8 \left(a+b \text{sech}^2(c+d x)\right)^2}","-\frac{b^{3/2} (5 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{2 a^2 d (a+b)^{5/2}}+\frac{x}{a^2}-\frac{(2 a-b) \coth (c+d x)}{2 a d (a+b)^2}-\frac{b \coth (c+d x)}{2 a d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*((2*x*(a + 2*b + a*Cosh[2*(c + d*x)]))/a^2 - (b^2*(5*a + 2*b)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])*(Cosh[2*c] - Sinh[2*c]))/(a^2*(a + b)^(5/2)*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (2*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]*Sinh[d*x])/((a + b)^2*d) + (b^2*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/(a^2*(a + b)^2*d)))/(8*(a + b*Sech[c + d*x]^2)^2)","B",0
156,1,130,110,1.2764648,"\int \frac{\coth ^3(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Coth[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^4(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \left(\frac{b^3 (a+b)}{a^2 \left(a \sinh ^2(c+d x)+a+b\right)}+\frac{b^2 (3 a+b) \log \left(a \sinh ^2(c+d x)+a+b\right)}{a^2}-(a+b) \text{csch}^2(c+d x)+2 (a+3 b) \log (\sinh (c+d x))\right)}{8 d (a+b)^3 \left(a+b \text{sech}^2(c+d x)\right)^2}","\frac{b^3}{2 a^2 d (a+b)^2 \left(a \cosh ^2(c+d x)+b\right)}+\frac{b^2 (3 a+b) \log \left(a \cosh ^2(c+d x)+b\right)}{2 a^2 d (a+b)^3}-\frac{\text{csch}^2(c+d x)}{2 d (a+b)^2}+\frac{(a+3 b) \log (\sinh (c+d x))}{d (a+b)^3}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]^4*(-((a + b)*Csch[c + d*x]^2) + 2*(a + 3*b)*Log[Sinh[c + d*x]] + (b^2*(3*a + b)*Log[a + b + a*Sinh[c + d*x]^2])/a^2 + (b^3*(a + b))/(a^2*(a + b + a*Sinh[c + d*x]^2))))/(8*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^2)","A",1
157,1,350,161,5.0930873,"\int \frac{\coth ^4(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^2} \, dx","Integrate[Coth[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2,x]","\frac{\text{sech}^4(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{3 b^3 \text{sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{a^2 d (a+b)^3}-\frac{3 b^3 (7 a+2 b) (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b) \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{a^2 d (a+b)^{7/2} \sqrt{b (\cosh (c)-\sinh (c))^4}}+\frac{6 x (a \cosh (2 (c+d x))+a+2 b)}{a^2}-\frac{2 \coth (c) \text{csch}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b)}{d (a+b)^2}+\frac{2 \text{csch}(c) \sinh (d x) \text{csch}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b)}{d (a+b)^2}+\frac{4 (2 a+5 b) \text{csch}(c) \sinh (d x) \text{csch}(c+d x) (a \cosh (2 (c+d x))+a+2 b)}{d (a+b)^3}\right)}{24 \left(a+b \text{sech}^2(c+d x)\right)^2}","-\frac{b^{5/2} (7 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{2 a^2 d (a+b)^{7/2}}-\frac{\left(2 a^2+6 a b-b^2\right) \coth (c+d x)}{2 a d (a+b)^3}+\frac{x}{a^2}-\frac{(2 a-3 b) \coth ^3(c+d x)}{6 a d (a+b)^2}-\frac{b \coth ^3(c+d x)}{2 a d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*((6*x*(a + 2*b + a*Cosh[2*(c + d*x)]))/a^2 - (2*(a + 2*b + a*Cosh[2*(c + d*x)])*Coth[c]*Csch[c + d*x]^2)/((a + b)^2*d) - (3*b^3*(7*a + 2*b)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])*(Cosh[2*c] - Sinh[2*c]))/(a^2*(a + b)^(7/2)*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (4*(2*a + 5*b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]*Sinh[d*x])/((a + b)^3*d) + (2*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]^3*Sinh[d*x])/((a + b)^2*d) + (3*b^3*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/(a^2*(a + b)^3*d)))/(24*(a + b*Sech[c + d*x]^2)^2)","B",0
158,1,515,148,5.7815385,"\int \frac{\tanh ^6(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Tanh[c + d*x]^6/(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\text{sech}(2 c) \left(3 a^4 \sinh (2 (c+2 d x))-3 a^4 \sinh (4 c+2 d x)-9 a^4 \sinh (2 c)+9 a^4 \sinh (2 d x)-3 a^3 b \sinh (2 (c+2 d x))+a^3 b \sinh (4 c+2 d x)-15 a^3 b \sinh (2 c)+13 a^3 b \sinh (2 d x)-6 a^2 b^2 \sinh (2 (c+2 d x))+20 a^2 b^2 \sinh (4 c+2 d x)+4 a^2 b^2 d x \cosh (2 (c+2 d x))+16 a^2 b^2 d x \cosh (4 c+2 d x)+4 a^2 b^2 d x \cosh (6 c+4 d x)+8 b^2 d x \left(3 a^2+8 a b+8 b^2\right) \cosh (2 c)+18 a^2 b^2 \sinh (2 c)-28 a^2 b^2 \sinh (2 d x)+16 a b^3 \sinh (4 c+2 d x)+32 a b^3 d x \cosh (4 c+2 d x)+72 a b^3 \sinh (2 c)-32 a b^3 \sinh (2 d x)+16 a b^2 d x (a+2 b) \cosh (2 d x)+48 b^4 \sinh (2 c)\right)-\frac{2 \left(3 a^3-a^2 b+4 a b^2+8 b^3\right) (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b)^2 \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{128 a^3 b^2 d \left(a+b \text{sech}^2(c+d x)\right)^3}","\frac{x}{a^3}+\frac{(3 a-4 b) (a+b) \tanh (c+d x)}{8 a^2 b^2 d \left(a-b \tanh ^2(c+d x)+b\right)}-\frac{\sqrt{a+b} \left(3 a^2-4 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 a^3 b^{5/2} d}-\frac{(a+b) \tanh ^3(c+d x)}{4 a b d \left(a-b \tanh ^2(c+d x)+b\right)^2}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*((-2*(3*a^3 - a^2*b + 4*a*b^2 + 8*b^3)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + Sech[2*c]*(8*b^2*(3*a^2 + 8*a*b + 8*b^2)*d*x*Cosh[2*c] + 16*a*b^2*(a + 2*b)*d*x*Cosh[2*d*x] + 4*a^2*b^2*d*x*Cosh[2*(c + 2*d*x)] + 16*a^2*b^2*d*x*Cosh[4*c + 2*d*x] + 32*a*b^3*d*x*Cosh[4*c + 2*d*x] + 4*a^2*b^2*d*x*Cosh[6*c + 4*d*x] - 9*a^4*Sinh[2*c] - 15*a^3*b*Sinh[2*c] + 18*a^2*b^2*Sinh[2*c] + 72*a*b^3*Sinh[2*c] + 48*b^4*Sinh[2*c] + 9*a^4*Sinh[2*d*x] + 13*a^3*b*Sinh[2*d*x] - 28*a^2*b^2*Sinh[2*d*x] - 32*a*b^3*Sinh[2*d*x] + 3*a^4*Sinh[2*(c + 2*d*x)] - 3*a^3*b*Sinh[2*(c + 2*d*x)] - 6*a^2*b^2*Sinh[2*(c + 2*d*x)] - 3*a^4*Sinh[4*c + 2*d*x] + a^3*b*Sinh[4*c + 2*d*x] + 20*a^2*b^2*Sinh[4*c + 2*d*x] + 16*a*b^3*Sinh[4*c + 2*d*x])))/(128*a^3*b^2*d*(a + b*Sech[c + d*x]^2)^3)","B",0
159,1,136,77,2.1675907,"\int \frac{\tanh ^5(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Tanh[c + d*x]^5/(a + b*Sech[c + d*x]^2)^3,x]","\frac{2 \left(a^2+4 a b+3 b^2\right)+a^2 \cosh ^2(2 (c+d x)) \log (a \cosh (2 (c+d x))+a+2 b)+(a+2 b)^2 \log (a \cosh (2 (c+d x))+a+2 b)+2 a \cosh (2 (c+d x)) ((a+2 b) \log (a \cosh (2 (c+d x))+a+2 b)+2 (a+b))}{2 a^3 d (a \cosh (2 (c+d x))+a+2 b)^2}","-\frac{(a+b)^2}{4 a^3 d \left(a \cosh ^2(c+d x)+b\right)^2}+\frac{a+b}{a^3 d \left(a \cosh ^2(c+d x)+b\right)}+\frac{\log \left(a \cosh ^2(c+d x)+b\right)}{2 a^3 d}",1,"(2*(a^2 + 4*a*b + 3*b^2) + (a + 2*b)^2*Log[a + 2*b + a*Cosh[2*(c + d*x)]] + a^2*Cosh[2*(c + d*x)]^2*Log[a + 2*b + a*Cosh[2*(c + d*x)]] + 2*a*Cosh[2*(c + d*x)]*(2*(a + b) + (a + 2*b)*Log[a + 2*b + a*Cosh[2*(c + d*x)]]))/(2*a^3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)","A",1
160,1,1457,139,13.0869109,"\int \frac{\tanh ^4(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Tanh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]","\frac{(\cosh (2 c+2 d x) a+a+2 b)^3 \left(\frac{\left(3 a^2+8 b a+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}-\frac{a \sqrt{b} \left(3 a^2+16 b a+3 (a+2 b) \cosh (2 (c+d x)) a+16 b^2\right) \sinh (2 (c+d x))}{(a+b)^2 (\cosh (2 (c+d x)) a+a+2 b)^2}\right) \text{sech}^6(c+d x)}{1024 b^{5/2} d \left(b \text{sech}^2(c+d x)+a\right)^3}-\frac{(\cosh (2 c+2 d x) a+a+2 b)^3 \left(\frac{\sqrt{b} \left(3 a^3+14 b a^2+24 b^2 a+\left(3 a^2+4 b a+4 b^2\right) \cosh (2 (c+d x)) a+16 b^3\right) \sinh (2 (c+d x))}{(a+b)^2 (\cosh (2 (c+d x)) a+a+2 b)^2}-\frac{3 a (a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}\right) \text{sech}^6(c+d x)}{2048 b^{5/2} d \left(b \text{sech}^2(c+d x)+a\right)^3}+\frac{(\cosh (2 c+2 d x) a+a+2 b)^3 \left(\frac{\text{sech}(2 c) \left(-9 \sinh (2 c) a^6+9 \sinh (2 d x) a^6+3 \sinh (2 (c+2 d x)) a^6-3 \sinh (4 c+2 d x) a^6+12 b \sinh (2 c) a^5-14 b \sinh (2 d x) a^5-12 b \sinh (2 (c+2 d x)) a^5+10 b \sinh (4 c+2 d x) a^5+128 b^2 d x \cosh (2 (c+2 d x)) a^4+512 b^2 d x \cosh (4 c+2 d x) a^4+128 b^2 d x \cosh (6 c+4 d x) a^4+684 b^2 \sinh (2 c) a^4-608 b^2 \sinh (2 d x) a^4-204 b^2 \sinh (2 (c+2 d x)) a^4+304 b^2 \sinh (4 c+2 d x) a^4+256 b^3 d x \cosh (2 (c+2 d x)) a^3+2048 b^3 d x \cosh (4 c+2 d x) a^3+256 b^3 d x \cosh (6 c+4 d x) a^3+2880 b^3 \sinh (2 c) a^3-2112 b^3 \sinh (2 d x) a^3-384 b^3 \sinh (2 (c+2 d x)) a^3+1056 b^3 \sinh (4 c+2 d x) a^3+128 b^4 d x \cosh (2 (c+2 d x)) a^2+2560 b^4 d x \cosh (4 c+2 d x) a^2+128 b^4 d x \cosh (6 c+4 d x) a^2+5280 b^4 \sinh (2 c) a^2-2560 b^4 \sinh (2 d x) a^2-192 b^4 \sinh (2 (c+2 d x)) a^2+1280 b^4 \sinh (4 c+2 d x) a^2+512 b^2 (a+b)^2 (a+2 b) d x \cosh (2 d x) a+1024 b^5 d x \cosh (4 c+2 d x) a+4608 b^5 \sinh (2 c) a-1024 b^5 \sinh (2 d x) a+512 b^5 \sinh (4 c+2 d x) a+256 b^2 (a+b)^2 \left(3 a^2+8 b a+8 b^2\right) d x \cosh (2 c)+1536 b^6 \sinh (2 c)\right)}{(\cosh (2 (c+d x)) a+a+2 b)^2}-\frac{2 \left(3 a^5-10 b a^4+80 b^2 a^3+480 b^3 a^2+640 b^4 a+256 b^5\right) \tanh ^{-1}\left(\frac{\text{sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right) (\cosh (2 c)-\sinh (2 c))}{\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right) \text{sech}^6(c+d x)}{4096 a^3 b^2 (a+b)^2 d \left(b \text{sech}^2(c+d x)+a\right)^3}-\frac{(\cosh (2 c+2 d x) a+a+2 b)^3 \left(\frac{6 \tanh ^{-1}\left(\frac{\text{sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right) (\cosh (2 c)-\sinh (2 c)) a^2}{\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}+\frac{a \text{sech}(2 c) \left(\left(-9 a^4-16 b a^3+48 b^2 a^2+128 b^3 a+64 b^4\right) \sinh (2 d x)+a \left(-3 a^3+2 b a^2+24 b^2 a+16 b^3\right) \sinh (2 (c+2 d x))+\left(3 a^4-64 b^2 a^2-128 b^3 a-64 b^4\right) \sinh (4 c+2 d x)\right)+\left(9 a^5+18 b a^4-64 b^2 a^3-256 b^3 a^2-320 b^4 a-128 b^5\right) \tanh (2 c)}{(\cosh (2 (c+d x)) a+a+2 b)^2 a^2}\right) \text{sech}^6(c+d x)}{2048 b^2 (a+b)^2 d \left(b \text{sech}^2(c+d x)+a\right)^3}","\frac{x}{a^3}+\frac{(a-4 b) \tanh (c+d x)}{8 a^2 b d \left(a-b \tanh ^2(c+d x)+b\right)}+\frac{\left(a^2-4 a b-8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 a^3 b^{3/2} d \sqrt{a+b}}-\frac{(a+b) \tanh (c+d x)}{4 a b d \left(a-b \tanh ^2(c+d x)+b\right)^2}",1,"((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*(((3*a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(1024*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3) - ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-3*a*(a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) + (Sqrt[b]*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a*b + 4*b^2)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(2048*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-2*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 + 256*b^5)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (Sech[2*c]*(256*b^2*(a + b)^2*(3*a^2 + 8*a*b + 8*b^2)*d*x*Cosh[2*c] + 512*a*b^2*(a + b)^2*(a + 2*b)*d*x*Cosh[2*d*x] + 128*a^4*b^2*d*x*Cosh[2*(c + 2*d*x)] + 256*a^3*b^3*d*x*Cosh[2*(c + 2*d*x)] + 128*a^2*b^4*d*x*Cosh[2*(c + 2*d*x)] + 512*a^4*b^2*d*x*Cosh[4*c + 2*d*x] + 2048*a^3*b^3*d*x*Cosh[4*c + 2*d*x] + 2560*a^2*b^4*d*x*Cosh[4*c + 2*d*x] + 1024*a*b^5*d*x*Cosh[4*c + 2*d*x] + 128*a^4*b^2*d*x*Cosh[6*c + 4*d*x] + 256*a^3*b^3*d*x*Cosh[6*c + 4*d*x] + 128*a^2*b^4*d*x*Cosh[6*c + 4*d*x] - 9*a^6*Sinh[2*c] + 12*a^5*b*Sinh[2*c] + 684*a^4*b^2*Sinh[2*c] + 2880*a^3*b^3*Sinh[2*c] + 5280*a^2*b^4*Sinh[2*c] + 4608*a*b^5*Sinh[2*c] + 1536*b^6*Sinh[2*c] + 9*a^6*Sinh[2*d*x] - 14*a^5*b*Sinh[2*d*x] - 608*a^4*b^2*Sinh[2*d*x] - 2112*a^3*b^3*Sinh[2*d*x] - 2560*a^2*b^4*Sinh[2*d*x] - 1024*a*b^5*Sinh[2*d*x] + 3*a^6*Sinh[2*(c + 2*d*x)] - 12*a^5*b*Sinh[2*(c + 2*d*x)] - 204*a^4*b^2*Sinh[2*(c + 2*d*x)] - 384*a^3*b^3*Sinh[2*(c + 2*d*x)] - 192*a^2*b^4*Sinh[2*(c + 2*d*x)] - 3*a^6*Sinh[4*c + 2*d*x] + 10*a^5*b*Sinh[4*c + 2*d*x] + 304*a^4*b^2*Sinh[4*c + 2*d*x] + 1056*a^3*b^3*Sinh[4*c + 2*d*x] + 1280*a^2*b^4*Sinh[4*c + 2*d*x] + 512*a*b^5*Sinh[4*c + 2*d*x]))/(a + 2*b + a*Cosh[2*(c + d*x)])^2))/(4096*a^3*b^2*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3) - ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((6*a^2*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (a*Sech[2*c]*((-9*a^4 - 16*a^3*b + 48*a^2*b^2 + 128*a*b^3 + 64*b^4)*Sinh[2*d*x] + a*(-3*a^3 + 2*a^2*b + 24*a*b^2 + 16*b^3)*Sinh[2*(c + 2*d*x)] + (3*a^4 - 64*a^2*b^2 - 128*a*b^3 - 64*b^4)*Sinh[4*c + 2*d*x]) + (9*a^5 + 18*a^4*b - 64*a^3*b^2 - 256*a^2*b^3 - 320*a*b^4 - 128*b^5)*Tanh[2*c])/(a^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(2048*b^2*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3)","B",0
161,1,131,81,1.567772,"\int \frac{\tanh ^3(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Tanh[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3,x]","\frac{2 \left(a^2+3 a b+3 b^2\right)+a^2 \cosh ^2(2 (c+d x)) \log (a \cosh (2 (c+d x))+a+2 b)+(a+2 b)^2 \log (a \cosh (2 (c+d x))+a+2 b)+2 a (a+2 b) \cosh (2 (c+d x)) (\log (a \cosh (2 (c+d x))+a+2 b)+1)}{2 a^3 d (a \cosh (2 (c+d x))+a+2 b)^2}","-\frac{b (a+b)}{4 a^3 d \left(a \cosh ^2(c+d x)+b\right)^2}+\frac{a+2 b}{2 a^3 d \left(a \cosh ^2(c+d x)+b\right)}+\frac{\log \left(a \cosh ^2(c+d x)+b\right)}{2 a^3 d}",1,"(2*(a^2 + 3*a*b + 3*b^2) + (a + 2*b)^2*Log[a + 2*b + a*Cosh[2*(c + d*x)]] + a^2*Cosh[2*(c + d*x)]^2*Log[a + 2*b + a*Cosh[2*(c + d*x)]] + 2*a*(a + 2*b)*Cosh[2*(c + d*x)]*(1 + Log[a + 2*b + a*Cosh[2*(c + d*x)]]))/(2*a^3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)","A",1
162,1,1457,139,12.0556288,"\int \frac{\tanh ^2(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Tanh[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]","-\frac{(\cosh (2 c+2 d x) a+a+2 b)^3 \left(\frac{\left(3 a^2+8 b a+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}-\frac{a \sqrt{b} \left(3 a^2+16 b a+3 (a+2 b) \cosh (2 (c+d x)) a+16 b^2\right) \sinh (2 (c+d x))}{(a+b)^2 (\cosh (2 (c+d x)) a+a+2 b)^2}\right) \text{sech}^6(c+d x)}{1024 b^{5/2} d \left(b \text{sech}^2(c+d x)+a\right)^3}-\frac{(\cosh (2 c+2 d x) a+a+2 b)^3 \left(\frac{\sqrt{b} \left(3 a^3+14 b a^2+24 b^2 a+\left(3 a^2+4 b a+4 b^2\right) \cosh (2 (c+d x)) a+16 b^3\right) \sinh (2 (c+d x))}{(a+b)^2 (\cosh (2 (c+d x)) a+a+2 b)^2}-\frac{3 a (a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}\right) \text{sech}^6(c+d x)}{2048 b^{5/2} d \left(b \text{sech}^2(c+d x)+a\right)^3}+\frac{(\cosh (2 c+2 d x) a+a+2 b)^3 \left(\frac{\text{sech}(2 c) \left(-9 \sinh (2 c) a^6+9 \sinh (2 d x) a^6+3 \sinh (2 (c+2 d x)) a^6-3 \sinh (4 c+2 d x) a^6+12 b \sinh (2 c) a^5-14 b \sinh (2 d x) a^5-12 b \sinh (2 (c+2 d x)) a^5+10 b \sinh (4 c+2 d x) a^5+128 b^2 d x \cosh (2 (c+2 d x)) a^4+512 b^2 d x \cosh (4 c+2 d x) a^4+128 b^2 d x \cosh (6 c+4 d x) a^4+684 b^2 \sinh (2 c) a^4-608 b^2 \sinh (2 d x) a^4-204 b^2 \sinh (2 (c+2 d x)) a^4+304 b^2 \sinh (4 c+2 d x) a^4+256 b^3 d x \cosh (2 (c+2 d x)) a^3+2048 b^3 d x \cosh (4 c+2 d x) a^3+256 b^3 d x \cosh (6 c+4 d x) a^3+2880 b^3 \sinh (2 c) a^3-2112 b^3 \sinh (2 d x) a^3-384 b^3 \sinh (2 (c+2 d x)) a^3+1056 b^3 \sinh (4 c+2 d x) a^3+128 b^4 d x \cosh (2 (c+2 d x)) a^2+2560 b^4 d x \cosh (4 c+2 d x) a^2+128 b^4 d x \cosh (6 c+4 d x) a^2+5280 b^4 \sinh (2 c) a^2-2560 b^4 \sinh (2 d x) a^2-192 b^4 \sinh (2 (c+2 d x)) a^2+1280 b^4 \sinh (4 c+2 d x) a^2+512 b^2 (a+b)^2 (a+2 b) d x \cosh (2 d x) a+1024 b^5 d x \cosh (4 c+2 d x) a+4608 b^5 \sinh (2 c) a-1024 b^5 \sinh (2 d x) a+512 b^5 \sinh (4 c+2 d x) a+256 b^2 (a+b)^2 \left(3 a^2+8 b a+8 b^2\right) d x \cosh (2 c)+1536 b^6 \sinh (2 c)\right)}{(\cosh (2 (c+d x)) a+a+2 b)^2}-\frac{2 \left(3 a^5-10 b a^4+80 b^2 a^3+480 b^3 a^2+640 b^4 a+256 b^5\right) \tanh ^{-1}\left(\frac{\text{sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right) (\cosh (2 c)-\sinh (2 c))}{\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right) \text{sech}^6(c+d x)}{4096 a^3 b^2 (a+b)^2 d \left(b \text{sech}^2(c+d x)+a\right)^3}+\frac{(\cosh (2 c+2 d x) a+a+2 b)^3 \left(\frac{6 \tanh ^{-1}\left(\frac{\text{sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right) (\cosh (2 c)-\sinh (2 c)) a^2}{\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}+\frac{a \text{sech}(2 c) \left(\left(-9 a^4-16 b a^3+48 b^2 a^2+128 b^3 a+64 b^4\right) \sinh (2 d x)+a \left(-3 a^3+2 b a^2+24 b^2 a+16 b^3\right) \sinh (2 (c+2 d x))+\left(3 a^4-64 b^2 a^2-128 b^3 a-64 b^4\right) \sinh (4 c+2 d x)\right)+\left(9 a^5+18 b a^4-64 b^2 a^3-256 b^3 a^2-320 b^4 a-128 b^5\right) \tanh (2 c)}{(\cosh (2 (c+d x)) a+a+2 b)^2 a^2}\right) \text{sech}^6(c+d x)}{2048 b^2 (a+b)^2 d \left(b \text{sech}^2(c+d x)+a\right)^3}","\frac{x}{a^3}-\frac{(3 a+4 b) \tanh (c+d x)}{8 a^2 d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)}-\frac{\left(3 a^2+12 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 a^3 \sqrt{b} d (a+b)^{3/2}}-\frac{\tanh (c+d x)}{4 a d \left(a-b \tanh ^2(c+d x)+b\right)^2}",1,"-1/1024*((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*(((3*a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3) - ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-3*a*(a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) + (Sqrt[b]*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a*b + 4*b^2)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(2048*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-2*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 + 256*b^5)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (Sech[2*c]*(256*b^2*(a + b)^2*(3*a^2 + 8*a*b + 8*b^2)*d*x*Cosh[2*c] + 512*a*b^2*(a + b)^2*(a + 2*b)*d*x*Cosh[2*d*x] + 128*a^4*b^2*d*x*Cosh[2*(c + 2*d*x)] + 256*a^3*b^3*d*x*Cosh[2*(c + 2*d*x)] + 128*a^2*b^4*d*x*Cosh[2*(c + 2*d*x)] + 512*a^4*b^2*d*x*Cosh[4*c + 2*d*x] + 2048*a^3*b^3*d*x*Cosh[4*c + 2*d*x] + 2560*a^2*b^4*d*x*Cosh[4*c + 2*d*x] + 1024*a*b^5*d*x*Cosh[4*c + 2*d*x] + 128*a^4*b^2*d*x*Cosh[6*c + 4*d*x] + 256*a^3*b^3*d*x*Cosh[6*c + 4*d*x] + 128*a^2*b^4*d*x*Cosh[6*c + 4*d*x] - 9*a^6*Sinh[2*c] + 12*a^5*b*Sinh[2*c] + 684*a^4*b^2*Sinh[2*c] + 2880*a^3*b^3*Sinh[2*c] + 5280*a^2*b^4*Sinh[2*c] + 4608*a*b^5*Sinh[2*c] + 1536*b^6*Sinh[2*c] + 9*a^6*Sinh[2*d*x] - 14*a^5*b*Sinh[2*d*x] - 608*a^4*b^2*Sinh[2*d*x] - 2112*a^3*b^3*Sinh[2*d*x] - 2560*a^2*b^4*Sinh[2*d*x] - 1024*a*b^5*Sinh[2*d*x] + 3*a^6*Sinh[2*(c + 2*d*x)] - 12*a^5*b*Sinh[2*(c + 2*d*x)] - 204*a^4*b^2*Sinh[2*(c + 2*d*x)] - 384*a^3*b^3*Sinh[2*(c + 2*d*x)] - 192*a^2*b^4*Sinh[2*(c + 2*d*x)] - 3*a^6*Sinh[4*c + 2*d*x] + 10*a^5*b*Sinh[4*c + 2*d*x] + 304*a^4*b^2*Sinh[4*c + 2*d*x] + 1056*a^3*b^3*Sinh[4*c + 2*d*x] + 1280*a^2*b^4*Sinh[4*c + 2*d*x] + 512*a*b^5*Sinh[4*c + 2*d*x]))/(a + 2*b + a*Cosh[2*(c + d*x)])^2))/(4096*a^3*b^2*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((6*a^2*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (a*Sech[2*c]*((-9*a^4 - 16*a^3*b + 48*a^2*b^2 + 128*a*b^3 + 64*b^4)*Sinh[2*d*x] + a*(-3*a^3 + 2*a^2*b + 24*a*b^2 + 16*b^3)*Sinh[2*(c + 2*d*x)] + (3*a^4 - 64*a^2*b^2 - 128*a*b^3 - 64*b^4)*Sinh[4*c + 2*d*x]) + (9*a^5 + 18*a^4*b - 64*a^3*b^2 - 256*a^2*b^3 - 320*a*b^4 - 128*b^5)*Tanh[2*c])/(a^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(2048*b^2*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3)","B",0
163,1,129,73,2.0103267,"\int \frac{\tanh (c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Tanh[c + d*x]/(a + b*Sech[c + d*x]^2)^3,x]","\frac{a^2 \cosh ^2(2 (c+d x)) \log (a \cosh (2 (c+d x))+a+2 b)+(a+2 b)^2 \log (a \cosh (2 (c+d x))+a+2 b)+2 a \cosh (2 (c+d x)) ((a+2 b) \log (a \cosh (2 (c+d x))+a+2 b)+2 b)+2 b (2 a+3 b)}{2 a^3 d (a \cosh (2 (c+d x))+a+2 b)^2}","-\frac{b^2}{4 a^3 d \left(a \cosh ^2(c+d x)+b\right)^2}+\frac{b}{a^3 d \left(a \cosh ^2(c+d x)+b\right)}+\frac{\log \left(a \cosh ^2(c+d x)+b\right)}{2 a^3 d}",1,"(2*b*(2*a + 3*b) + (a + 2*b)^2*Log[a + 2*b + a*Cosh[2*(c + d*x)]] + a^2*Cosh[2*(c + d*x)]^2*Log[a + 2*b + a*Cosh[2*(c + d*x)]] + 2*a*Cosh[2*(c + d*x)]*(2*b + (a + 2*b)*Log[a + 2*b + a*Cosh[2*(c + d*x)]]))/(2*a^3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)","A",1
164,1,301,146,6.2086474,"\int \frac{1}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[(a + b*Sech[c + d*x]^2)^(-3),x]","\frac{\text{sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(\frac{b \text{sech}(2 c) \left(\left(9 a^2+28 a b+16 b^2\right) \sinh (2 c)-3 a (3 a+2 b) \sinh (2 d x)\right) (a \cosh (2 (c+d x))+a+2 b)}{d (a+b)^2}-\frac{b \left(15 a^2+20 a b+8 b^2\right) (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b)^2 \tanh ^{-1}\left(\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right)}{d (a+b)^{5/2} \sqrt{b (\cosh (c)-\sinh (c))^4}}-\frac{4 b^2 \text{sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{d (a+b)}+8 x (a \cosh (2 (c+d x))+a+2 b)^2\right)}{64 a^3 \left(a+b \text{sech}^2(c+d x)\right)^3}","\frac{x}{a^3}-\frac{b (7 a+4 b) \tanh (c+d x)}{8 a^2 d (a+b)^2 \left(a-b \tanh ^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} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 a^3 d (a+b)^{5/2}}-\frac{b \tanh (c+d x)}{4 a d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)^2}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*(8*x*(a + 2*b + a*Cosh[2*(c + d*x)])^2 - (b*(15*a^2 + 20*a*b + 8*b^2)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*(Cosh[2*c] - Sinh[2*c]))/((a + b)^(5/2)*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) - (4*b^2*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/((a + b)*d) + (b*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[2*c]*((9*a^2 + 28*a*b + 16*b^2)*Sinh[2*c] - 3*a*(3*a + 2*b)*Sinh[2*d*x]))/((a + b)^2*d)))/(64*a^3*(a + b*Sech[c + d*x]^2)^3)","B",0
165,1,155,130,1.0555106,"\int \frac{\coth (c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Coth[c + d*x]/(a + b*Sech[c + d*x]^2)^3,x]","\frac{\text{sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b)^3 \left(-\frac{b^3 (a+b)^2}{a^3 \left(a \sinh ^2(c+d x)+a+b\right)^2}+\frac{2 b^2 (a+b) (3 a+2 b)}{a^3 \left(a \sinh ^2(c+d x)+a+b\right)}+\frac{2 b \left(3 a^2+3 a b+b^2\right) \log \left(a \sinh ^2(c+d x)+a+b\right)}{a^3}+4 \log (\sinh (c+d x))\right)}{32 d (a+b)^3 \left(a+b \text{sech}^2(c+d x)\right)^3}","-\frac{b^3}{4 a^3 d (a+b) \left(a \cosh ^2(c+d x)+b\right)^2}+\frac{b^2 (3 a+2 b)}{2 a^3 d (a+b)^2 \left(a \cosh ^2(c+d x)+b\right)}+\frac{b \left(3 a^2+3 a b+b^2\right) \log \left(a \cosh ^2(c+d x)+b\right)}{2 a^3 d (a+b)^3}+\frac{\log (\sinh (c+d x))}{d (a+b)^3}",1,"((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*(4*Log[Sinh[c + d*x]] + (2*b*(3*a^2 + 3*a*b + b^2)*Log[a + b + a*Sinh[c + d*x]^2])/a^3 - (b^3*(a + b)^2)/(a^3*(a + b + a*Sinh[c + d*x]^2)^2) + (2*b^2*(a + b)*(3*a + 2*b))/(a^3*(a + b + a*Sinh[c + d*x]^2))))/(32*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^3)","A",1
166,1,2083,182,7.228046,"\int \frac{\coth ^2(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Coth[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]","\text{Result too large to show}","\frac{x}{a^3}-\frac{\left(8 a^2-11 a b-4 b^2\right) \coth (c+d x)}{8 a^2 d (a+b)^3}-\frac{b (9 a+4 b) \coth (c+d x)}{8 a^2 d (a+b)^2 \left(a-b \tanh ^2(c+d x)+b\right)}-\frac{b^{3/2} \left(35 a^2+28 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 a^3 d (a+b)^{7/2}}-\frac{b \coth (c+d x)}{4 a d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)^2}",1,"((35*a^2 + 28*a*b + 8*b^2)*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*(((I/64)*b^2*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Cosh[2*c])/(a^3*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) - ((I/64)*b^2*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Sinh[2*c])/(a^3*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/((a + b)^3*(a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Csch[c]*Csch[c + d*x]*Sech[2*c]*Sech[c + d*x]^6*(8*a^5*d*x*Cosh[d*x] + 56*a^4*b*d*x*Cosh[d*x] + 184*a^3*b^2*d*x*Cosh[d*x] + 296*a^2*b^3*d*x*Cosh[d*x] + 224*a*b^4*d*x*Cosh[d*x] + 64*b^5*d*x*Cosh[d*x] - 12*a^5*d*x*Cosh[3*d*x] - 68*a^4*b*d*x*Cosh[3*d*x] - 132*a^3*b^2*d*x*Cosh[3*d*x] - 108*a^2*b^3*d*x*Cosh[3*d*x] - 32*a*b^4*d*x*Cosh[3*d*x] - 8*a^5*d*x*Cosh[2*c - d*x] - 56*a^4*b*d*x*Cosh[2*c - d*x] - 184*a^3*b^2*d*x*Cosh[2*c - d*x] - 296*a^2*b^3*d*x*Cosh[2*c - d*x] - 224*a*b^4*d*x*Cosh[2*c - d*x] - 64*b^5*d*x*Cosh[2*c - d*x] - 8*a^5*d*x*Cosh[2*c + d*x] - 56*a^4*b*d*x*Cosh[2*c + d*x] - 184*a^3*b^2*d*x*Cosh[2*c + d*x] - 296*a^2*b^3*d*x*Cosh[2*c + d*x] - 224*a*b^4*d*x*Cosh[2*c + d*x] - 64*b^5*d*x*Cosh[2*c + d*x] + 8*a^5*d*x*Cosh[4*c + d*x] + 56*a^4*b*d*x*Cosh[4*c + d*x] + 184*a^3*b^2*d*x*Cosh[4*c + d*x] + 296*a^2*b^3*d*x*Cosh[4*c + d*x] + 224*a*b^4*d*x*Cosh[4*c + d*x] + 64*b^5*d*x*Cosh[4*c + d*x] + 12*a^5*d*x*Cosh[2*c + 3*d*x] + 68*a^4*b*d*x*Cosh[2*c + 3*d*x] + 132*a^3*b^2*d*x*Cosh[2*c + 3*d*x] + 108*a^2*b^3*d*x*Cosh[2*c + 3*d*x] + 32*a*b^4*d*x*Cosh[2*c + 3*d*x] - 12*a^5*d*x*Cosh[4*c + 3*d*x] - 68*a^4*b*d*x*Cosh[4*c + 3*d*x] - 132*a^3*b^2*d*x*Cosh[4*c + 3*d*x] - 108*a^2*b^3*d*x*Cosh[4*c + 3*d*x] - 32*a*b^4*d*x*Cosh[4*c + 3*d*x] + 12*a^5*d*x*Cosh[6*c + 3*d*x] + 68*a^4*b*d*x*Cosh[6*c + 3*d*x] + 132*a^3*b^2*d*x*Cosh[6*c + 3*d*x] + 108*a^2*b^3*d*x*Cosh[6*c + 3*d*x] + 32*a*b^4*d*x*Cosh[6*c + 3*d*x] - 4*a^5*d*x*Cosh[2*c + 5*d*x] - 12*a^4*b*d*x*Cosh[2*c + 5*d*x] - 12*a^3*b^2*d*x*Cosh[2*c + 5*d*x] - 4*a^2*b^3*d*x*Cosh[2*c + 5*d*x] + 4*a^5*d*x*Cosh[4*c + 5*d*x] + 12*a^4*b*d*x*Cosh[4*c + 5*d*x] + 12*a^3*b^2*d*x*Cosh[4*c + 5*d*x] + 4*a^2*b^3*d*x*Cosh[4*c + 5*d*x] - 4*a^5*d*x*Cosh[6*c + 5*d*x] - 12*a^4*b*d*x*Cosh[6*c + 5*d*x] - 12*a^3*b^2*d*x*Cosh[6*c + 5*d*x] - 4*a^2*b^3*d*x*Cosh[6*c + 5*d*x] + 4*a^5*d*x*Cosh[8*c + 5*d*x] + 12*a^4*b*d*x*Cosh[8*c + 5*d*x] + 12*a^3*b^2*d*x*Cosh[8*c + 5*d*x] + 4*a^2*b^3*d*x*Cosh[8*c + 5*d*x] - 32*a^5*Sinh[d*x] - 64*a^4*b*Sinh[d*x] - 30*a^2*b^3*Sinh[d*x] - 120*a*b^4*Sinh[d*x] - 48*b^5*Sinh[d*x] + 32*a^5*Sinh[3*d*x] + 64*a^4*b*Sinh[3*d*x] + 26*a^3*b^2*Sinh[3*d*x] + 86*a^2*b^3*Sinh[3*d*x] + 32*a*b^4*Sinh[3*d*x] - 48*a^5*Sinh[2*c - d*x] - 128*a^4*b*Sinh[2*c - d*x] - 128*a^3*b^2*Sinh[2*c - d*x] - 30*a^2*b^3*Sinh[2*c - d*x] - 120*a*b^4*Sinh[2*c - d*x] - 48*b^5*Sinh[2*c - d*x] + 48*a^5*Sinh[2*c + d*x] + 128*a^4*b*Sinh[2*c + d*x] + 102*a^3*b^2*Sinh[2*c + d*x] - 86*a^2*b^3*Sinh[2*c + d*x] - 136*a*b^4*Sinh[2*c + d*x] - 48*b^5*Sinh[2*c + d*x] - 32*a^5*Sinh[4*c + d*x] - 64*a^4*b*Sinh[4*c + d*x] + 26*a^3*b^2*Sinh[4*c + d*x] + 86*a^2*b^3*Sinh[4*c + d*x] + 136*a*b^4*Sinh[4*c + d*x] + 48*b^5*Sinh[4*c + d*x] - 8*a^5*Sinh[2*c + 3*d*x] - 26*a^3*b^2*Sinh[2*c + 3*d*x] - 86*a^2*b^3*Sinh[2*c + 3*d*x] - 32*a*b^4*Sinh[2*c + 3*d*x] + 32*a^5*Sinh[4*c + 3*d*x] + 64*a^4*b*Sinh[4*c + 3*d*x] - 13*a^3*b^2*Sinh[4*c + 3*d*x] - 36*a^2*b^3*Sinh[4*c + 3*d*x] - 16*a*b^4*Sinh[4*c + 3*d*x] - 8*a^5*Sinh[6*c + 3*d*x] + 13*a^3*b^2*Sinh[6*c + 3*d*x] + 36*a^2*b^3*Sinh[6*c + 3*d*x] + 16*a*b^4*Sinh[6*c + 3*d*x] + 8*a^5*Sinh[2*c + 5*d*x] + 13*a^3*b^2*Sinh[2*c + 5*d*x] + 6*a^2*b^3*Sinh[2*c + 5*d*x] - 13*a^3*b^2*Sinh[4*c + 5*d*x] - 6*a^2*b^3*Sinh[4*c + 5*d*x] + 8*a^5*Sinh[6*c + 5*d*x]))/(512*a^3*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^3)","C",0
167,1,172,152,1.9216298,"\int \frac{\coth ^3(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Coth[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3,x]","-\frac{\text{sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b)^3 \left(\frac{b^4 (a+b)^2}{a^3 \left(a \sinh ^2(c+d x)+a+b\right)^2}-\frac{4 b^3 (a+b) (2 a+b)}{a^3 \left(a \sinh ^2(c+d x)+a+b\right)}-\frac{2 b^2 \left(6 a^2+4 a b+b^2\right) \log \left(a \sinh ^2(c+d x)+a+b\right)}{a^3}+2 (a+b) \text{csch}^2(c+d x)-4 (a+4 b) \log (\sinh (c+d x))\right)}{32 d (a+b)^4 \left(a+b \text{sech}^2(c+d x)\right)^3}","-\frac{b^4}{4 a^3 d (a+b)^2 \left(a \cosh ^2(c+d x)+b\right)^2}+\frac{b^3 (2 a+b)}{a^3 d (a+b)^3 \left(a \cosh ^2(c+d x)+b\right)}+\frac{b^2 \left(6 a^2+4 a b+b^2\right) \log \left(a \cosh ^2(c+d x)+b\right)}{2 a^3 d (a+b)^4}-\frac{\text{csch}^2(c+d x)}{2 d (a+b)^3}+\frac{(a+4 b) \log (\sinh (c+d x))}{d (a+b)^4}",1,"-1/32*((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*(2*(a + b)*Csch[c + d*x]^2 - 4*(a + 4*b)*Log[Sinh[c + d*x]] - (2*b^2*(6*a^2 + 4*a*b + b^2)*Log[a + b + a*Sinh[c + d*x]^2])/a^3 + (b^4*(a + b)^2)/(a^3*(a + b + a*Sinh[c + d*x]^2)^2) - (4*b^3*(a + b)*(2*a + b))/(a^3*(a + b + a*Sinh[c + d*x]^2))))/((a + b)^4*d*(a + b*Sech[c + d*x]^2)^3)","A",1
168,1,3334,232,7.7382709,"\int \frac{\coth ^4(c+d x)}{\left(a+b \text{sech}^2(c+d x)\right)^3} \, dx","Integrate[Coth[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]","\text{Result too large to show}","\frac{x}{a^3}-\frac{\left(8 a^2-39 a b-12 b^2\right) \coth ^3(c+d x)}{24 a^2 d (a+b)^3}-\frac{b (11 a+4 b) \coth ^3(c+d x)}{8 a^2 d (a+b)^2 \left(a-b \tanh ^2(c+d x)+b\right)}-\frac{b^{5/2} \left(63 a^2+36 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{8 a^3 d (a+b)^{9/2}}-\frac{\left(8 a^3+32 a^2 b-15 a b^2-4 b^3\right) \coth (c+d x)}{8 a^2 d (a+b)^4}-\frac{b \coth ^3(c+d x)}{4 a d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)^2}",1,"((63*a^2 + 36*a*b + 8*b^2)*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*(((I/64)*b^3*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Cosh[2*c])/(a^3*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) - ((I/64)*b^3*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Sinh[2*c])/(a^3*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/((a + b)^4*(a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Csch[c]*Csch[c + d*x]^3*Sech[2*c]*Sech[c + d*x]^6*(-36*a^6*d*x*Cosh[d*x] - 336*a^5*b*d*x*Cosh[d*x] - 1560*a^4*b^2*d*x*Cosh[d*x] - 3600*a^3*b^3*d*x*Cosh[d*x] - 4260*a^2*b^4*d*x*Cosh[d*x] - 2496*a*b^5*d*x*Cosh[d*x] - 576*b^6*d*x*Cosh[d*x] + 36*a^6*d*x*Cosh[3*d*x] + 240*a^5*b*d*x*Cosh[3*d*x] + 408*a^4*b^2*d*x*Cosh[3*d*x] - 48*a^3*b^3*d*x*Cosh[3*d*x] - 732*a^2*b^4*d*x*Cosh[3*d*x] - 672*a*b^5*d*x*Cosh[3*d*x] - 192*b^6*d*x*Cosh[3*d*x] + 36*a^6*d*x*Cosh[2*c - d*x] + 336*a^5*b*d*x*Cosh[2*c - d*x] + 1560*a^4*b^2*d*x*Cosh[2*c - d*x] + 3600*a^3*b^3*d*x*Cosh[2*c - d*x] + 4260*a^2*b^4*d*x*Cosh[2*c - d*x] + 2496*a*b^5*d*x*Cosh[2*c - d*x] + 576*b^6*d*x*Cosh[2*c - d*x] + 36*a^6*d*x*Cosh[2*c + d*x] + 336*a^5*b*d*x*Cosh[2*c + d*x] + 1560*a^4*b^2*d*x*Cosh[2*c + d*x] + 3600*a^3*b^3*d*x*Cosh[2*c + d*x] + 4260*a^2*b^4*d*x*Cosh[2*c + d*x] + 2496*a*b^5*d*x*Cosh[2*c + d*x] + 576*b^6*d*x*Cosh[2*c + d*x] - 36*a^6*d*x*Cosh[4*c + d*x] - 336*a^5*b*d*x*Cosh[4*c + d*x] - 1560*a^4*b^2*d*x*Cosh[4*c + d*x] - 3600*a^3*b^3*d*x*Cosh[4*c + d*x] - 4260*a^2*b^4*d*x*Cosh[4*c + d*x] - 2496*a*b^5*d*x*Cosh[4*c + d*x] - 576*b^6*d*x*Cosh[4*c + d*x] - 36*a^6*d*x*Cosh[2*c + 3*d*x] - 240*a^5*b*d*x*Cosh[2*c + 3*d*x] - 408*a^4*b^2*d*x*Cosh[2*c + 3*d*x] + 48*a^3*b^3*d*x*Cosh[2*c + 3*d*x] + 732*a^2*b^4*d*x*Cosh[2*c + 3*d*x] + 672*a*b^5*d*x*Cosh[2*c + 3*d*x] + 192*b^6*d*x*Cosh[2*c + 3*d*x] + 36*a^6*d*x*Cosh[4*c + 3*d*x] + 240*a^5*b*d*x*Cosh[4*c + 3*d*x] + 408*a^4*b^2*d*x*Cosh[4*c + 3*d*x] - 48*a^3*b^3*d*x*Cosh[4*c + 3*d*x] - 732*a^2*b^4*d*x*Cosh[4*c + 3*d*x] - 672*a*b^5*d*x*Cosh[4*c + 3*d*x] - 192*b^6*d*x*Cosh[4*c + 3*d*x] - 36*a^6*d*x*Cosh[6*c + 3*d*x] - 240*a^5*b*d*x*Cosh[6*c + 3*d*x] - 408*a^4*b^2*d*x*Cosh[6*c + 3*d*x] + 48*a^3*b^3*d*x*Cosh[6*c + 3*d*x] + 732*a^2*b^4*d*x*Cosh[6*c + 3*d*x] + 672*a*b^5*d*x*Cosh[6*c + 3*d*x] + 192*b^6*d*x*Cosh[6*c + 3*d*x] - 12*a^6*d*x*Cosh[2*c + 5*d*x] - 144*a^5*b*d*x*Cosh[2*c + 5*d*x] - 456*a^4*b^2*d*x*Cosh[2*c + 5*d*x] - 624*a^3*b^3*d*x*Cosh[2*c + 5*d*x] - 396*a^2*b^4*d*x*Cosh[2*c + 5*d*x] - 96*a*b^5*d*x*Cosh[2*c + 5*d*x] + 12*a^6*d*x*Cosh[4*c + 5*d*x] + 144*a^5*b*d*x*Cosh[4*c + 5*d*x] + 456*a^4*b^2*d*x*Cosh[4*c + 5*d*x] + 624*a^3*b^3*d*x*Cosh[4*c + 5*d*x] + 396*a^2*b^4*d*x*Cosh[4*c + 5*d*x] + 96*a*b^5*d*x*Cosh[4*c + 5*d*x] - 12*a^6*d*x*Cosh[6*c + 5*d*x] - 144*a^5*b*d*x*Cosh[6*c + 5*d*x] - 456*a^4*b^2*d*x*Cosh[6*c + 5*d*x] - 624*a^3*b^3*d*x*Cosh[6*c + 5*d*x] - 396*a^2*b^4*d*x*Cosh[6*c + 5*d*x] - 96*a*b^5*d*x*Cosh[6*c + 5*d*x] + 12*a^6*d*x*Cosh[8*c + 5*d*x] + 144*a^5*b*d*x*Cosh[8*c + 5*d*x] + 456*a^4*b^2*d*x*Cosh[8*c + 5*d*x] + 624*a^3*b^3*d*x*Cosh[8*c + 5*d*x] + 396*a^2*b^4*d*x*Cosh[8*c + 5*d*x] + 96*a*b^5*d*x*Cosh[8*c + 5*d*x] - 12*a^6*d*x*Cosh[4*c + 7*d*x] - 48*a^5*b*d*x*Cosh[4*c + 7*d*x] - 72*a^4*b^2*d*x*Cosh[4*c + 7*d*x] - 48*a^3*b^3*d*x*Cosh[4*c + 7*d*x] - 12*a^2*b^4*d*x*Cosh[4*c + 7*d*x] + 12*a^6*d*x*Cosh[6*c + 7*d*x] + 48*a^5*b*d*x*Cosh[6*c + 7*d*x] + 72*a^4*b^2*d*x*Cosh[6*c + 7*d*x] + 48*a^3*b^3*d*x*Cosh[6*c + 7*d*x] + 12*a^2*b^4*d*x*Cosh[6*c + 7*d*x] - 12*a^6*d*x*Cosh[8*c + 7*d*x] - 48*a^5*b*d*x*Cosh[8*c + 7*d*x] - 72*a^4*b^2*d*x*Cosh[8*c + 7*d*x] - 48*a^3*b^3*d*x*Cosh[8*c + 7*d*x] - 12*a^2*b^4*d*x*Cosh[8*c + 7*d*x] + 12*a^6*d*x*Cosh[10*c + 7*d*x] + 48*a^5*b*d*x*Cosh[10*c + 7*d*x] + 72*a^4*b^2*d*x*Cosh[10*c + 7*d*x] + 48*a^3*b^3*d*x*Cosh[10*c + 7*d*x] + 12*a^2*b^4*d*x*Cosh[10*c + 7*d*x] - 128*a^6*Sinh[d*x] - 440*a^5*b*Sinh[d*x] - 1152*a^4*b^2*Sinh[d*x] - 1920*a^3*b^3*Sinh[d*x] + 228*a^2*b^4*Sinh[d*x] + 1320*a*b^5*Sinh[d*x] + 432*b^6*Sinh[d*x] + 48*a^6*Sinh[3*d*x] + 104*a^5*b*Sinh[3*d*x] + 640*a^4*b^2*Sinh[3*d*x] + 1511*a^3*b^3*Sinh[3*d*x] - 528*a^2*b^4*Sinh[3*d*x] + 264*a*b^5*Sinh[3*d*x] + 144*b^6*Sinh[3*d*x] - 32*a^6*Sinh[2*c - d*x] + 384*a^5*b*Sinh[2*c - d*x] + 2048*a^4*b^2*Sinh[2*c - d*x] + 3072*a^3*b^3*Sinh[2*c - d*x] + 228*a^2*b^4*Sinh[2*c - d*x] + 1320*a*b^5*Sinh[2*c - d*x] + 432*b^6*Sinh[2*c - d*x] + 32*a^6*Sinh[2*c + d*x] - 384*a^5*b*Sinh[2*c + d*x] - 2048*a^4*b^2*Sinh[2*c + d*x] - 2919*a^3*b^3*Sinh[2*c + d*x] + 642*a^2*b^4*Sinh[2*c + d*x] + 1416*a*b^5*Sinh[2*c + d*x] + 432*b^6*Sinh[2*c + d*x] - 128*a^6*Sinh[4*c + d*x] - 440*a^5*b*Sinh[4*c + d*x] - 1152*a^4*b^2*Sinh[4*c + d*x] - 2073*a^3*b^3*Sinh[4*c + d*x] - 642*a^2*b^4*Sinh[4*c + d*x] - 1416*a*b^5*Sinh[4*c + d*x] - 432*b^6*Sinh[4*c + d*x] - 144*a^6*Sinh[2*c + 3*d*x] - 672*a^5*b*Sinh[2*c + 3*d*x] - 960*a^4*b^2*Sinh[2*c + 3*d*x] + 153*a^3*b^3*Sinh[2*c + 3*d*x] + 528*a^2*b^4*Sinh[2*c + 3*d*x] - 264*a*b^5*Sinh[2*c + 3*d*x] - 144*b^6*Sinh[2*c + 3*d*x] + 48*a^6*Sinh[4*c + 3*d*x] + 104*a^5*b*Sinh[4*c + 3*d*x] + 640*a^4*b^2*Sinh[4*c + 3*d*x] + 1664*a^3*b^3*Sinh[4*c + 3*d*x] - 66*a^2*b^4*Sinh[4*c + 3*d*x] - 408*a*b^5*Sinh[4*c + 3*d*x] - 144*b^6*Sinh[4*c + 3*d*x] - 144*a^6*Sinh[6*c + 3*d*x] - 672*a^5*b*Sinh[6*c + 3*d*x] - 960*a^4*b^2*Sinh[6*c + 3*d*x] + 66*a^2*b^4*Sinh[6*c + 3*d*x] + 408*a*b^5*Sinh[6*c + 3*d*x] + 144*b^6*Sinh[6*c + 3*d*x] + 80*a^6*Sinh[2*c + 5*d*x] + 480*a^5*b*Sinh[2*c + 5*d*x] + 832*a^4*b^2*Sinh[2*c + 5*d*x] + 294*a^2*b^4*Sinh[2*c + 5*d*x] + 96*a*b^5*Sinh[2*c + 5*d*x] - 48*a^6*Sinh[4*c + 5*d*x] - 120*a^5*b*Sinh[4*c + 5*d*x] - 294*a^2*b^4*Sinh[4*c + 5*d*x] - 96*a*b^5*Sinh[4*c + 5*d*x] + 80*a^6*Sinh[6*c + 5*d*x] + 480*a^5*b*Sinh[6*c + 5*d*x] + 832*a^4*b^2*Sinh[6*c + 5*d*x] - 51*a^3*b^3*Sinh[6*c + 5*d*x] - 132*a^2*b^4*Sinh[6*c + 5*d*x] - 48*a*b^5*Sinh[6*c + 5*d*x] - 48*a^6*Sinh[8*c + 5*d*x] - 120*a^5*b*Sinh[8*c + 5*d*x] + 51*a^3*b^3*Sinh[8*c + 5*d*x] + 132*a^2*b^4*Sinh[8*c + 5*d*x] + 48*a*b^5*Sinh[8*c + 5*d*x] + 32*a^6*Sinh[4*c + 7*d*x] + 104*a^5*b*Sinh[4*c + 7*d*x] + 51*a^3*b^3*Sinh[4*c + 7*d*x] + 18*a^2*b^4*Sinh[4*c + 7*d*x] - 51*a^3*b^3*Sinh[6*c + 7*d*x] - 18*a^2*b^4*Sinh[6*c + 7*d*x] + 32*a^6*Sinh[8*c + 7*d*x] + 104*a^5*b*Sinh[8*c + 7*d*x]))/(6144*a^3*(a + b)^4*d*(a + b*Sech[c + d*x]^2)^3)","C",0
169,1,1405,207,6.9260164,"\int \frac{1}{\left(a+b \text{sech}^2(c+d x)\right)^4} \, dx","Integrate[(a + b*Sech[c + d*x]^2)^(-4),x]","\frac{\left(35 a^3+70 b a^2+56 b^2 a+16 b^3\right) (\cosh (2 c+2 d x) a+a+2 b)^4 \left(\frac{i b \tan ^{-1}\left(\text{sech}(d x) \left(\frac{i \sinh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}-\frac{i \cosh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}\right) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right) \cosh (2 c)}{256 a^4 \sqrt{a+b} d \sqrt{b \cosh (4 c)-b \sinh (4 c)}}-\frac{i b \tan ^{-1}\left(\text{sech}(d x) \left(\frac{i \sinh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}-\frac{i \cosh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}\right) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right) \sinh (2 c)}{256 a^4 \sqrt{a+b} d \sqrt{b \cosh (4 c)-b \sinh (4 c)}}\right) \text{sech}^8(c+d x)}{(a+b)^3 \left(b \text{sech}^2(c+d x)+a\right)^4}+\frac{(\cosh (2 c+2 d x) a+a+2 b) \text{sech}(2 c) \left(480 d x \cosh (2 c) a^6+360 d x \cosh (2 d x) a^6+360 d x \cosh (4 c+2 d x) a^6+144 d x \cosh (2 c+4 d x) a^6+144 d x \cosh (6 c+4 d x) a^6+24 d x \cosh (4 c+6 d x) a^6+24 d x \cosh (8 c+6 d x) a^6+3168 b d x \cosh (2 c) a^5+2232 b d x \cosh (2 d x) a^5+2232 b d x \cosh (4 c+2 d x) a^5+720 b d x \cosh (2 c+4 d x) a^5+720 b d x \cosh (6 c+4 d x) a^5+72 b d x \cosh (4 c+6 d x) a^5+72 b d x \cosh (8 c+6 d x) a^5+870 b \sinh (2 c) a^5-870 b \sinh (2 d x) a^5+435 b \sinh (4 c+2 d x) a^5-435 b \sinh (2 c+4 d x) a^5+87 b \sinh (6 c+4 d x) a^5-87 b \sinh (4 c+6 d x) a^5+8928 b^2 d x \cosh (2 c) a^4+5688 b^2 d x \cosh (2 d x) a^4+5688 b^2 d x \cosh (4 c+2 d x) a^4+1296 b^2 d x \cosh (2 c+4 d x) a^4+1296 b^2 d x \cosh (6 c+4 d x) a^4+72 b^2 d x \cosh (4 c+6 d x) a^4+72 b^2 d x \cosh (8 c+6 d x) a^4+4292 b^2 \sinh (2 c) a^4-3792 b^2 \sinh (2 d x) a^4+2124 b^2 \sinh (4 c+2 d x) a^4-1374 b^2 \sinh (2 c+4 d x) a^4+366 b^2 \sinh (6 c+4 d x) a^4-116 b^2 \sinh (4 c+6 d x) a^4+14112 b^3 d x \cosh (2 c) a^3+7272 b^3 d x \cosh (2 d x) a^3+7272 b^3 d x \cosh (4 c+2 d x) a^3+1008 b^3 d x \cosh (2 c+4 d x) a^3+1008 b^3 d x \cosh (6 c+4 d x) a^3+24 b^3 d x \cosh (4 c+6 d x) a^3+24 b^3 d x \cosh (8 c+6 d x) a^3+8792 b^3 \sinh (2 c) a^3-6432 b^3 \sinh (2 d x) a^3+3972 b^3 \sinh (4 c+2 d x) a^3-1248 b^3 \sinh (2 c+4 d x) a^3+408 b^3 \sinh (6 c+4 d x) a^3-44 b^3 \sinh (4 c+6 d x) a^3+13248 b^4 d x \cosh (2 c) a^2+4608 b^4 d x \cosh (2 d x) a^2+4608 b^4 d x \cosh (4 c+2 d x) a^2+288 b^4 d x \cosh (2 c+4 d x) a^2+288 b^4 d x \cosh (6 c+4 d x) a^2+9936 b^4 \sinh (2 c) a^2-4608 b^4 \sinh (2 d x) a^2+3072 b^4 \sinh (4 c+2 d x) a^2-384 b^4 \sinh (2 c+4 d x) a^2+144 b^4 \sinh (6 c+4 d x) a^2+6912 b^5 d x \cosh (2 c) a+1152 b^5 d x \cosh (2 d x) a+1152 b^5 d x \cosh (4 c+2 d x) a+5824 b^5 \sinh (2 c) a-1248 b^5 \sinh (2 d x) a+864 b^5 \sinh (4 c+2 d x) a+1536 b^6 d x \cosh (2 c)+1408 b^6 \sinh (2 c)\right) \text{sech}^8(c+d x)}{3072 a^4 (a+b)^3 d \left(b \text{sech}^2(c+d x)+a\right)^4}","\frac{x}{a^4}-\frac{b (11 a+6 b) \tanh (c+d x)}{24 a^2 d (a+b)^2 \left(a-b \tanh ^2(c+d x)+b\right)^2}-\frac{b \left(19 a^2+22 a b+8 b^2\right) \tanh (c+d x)}{16 a^3 d (a+b)^3 \left(a-b \tanh ^2(c+d x)+b\right)}-\frac{\sqrt{b} \left(35 a^3+70 a^2 b+56 a b^2+16 b^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right)}{16 a^4 d (a+b)^{7/2}}-\frac{b \tanh (c+d x)}{6 a d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)^3}",1,"((35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*(a + 2*b + a*Cosh[2*c + 2*d*x])^4*Sech[c + d*x]^8*(((I/256)*b*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Cosh[2*c])/(a^4*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) - ((I/256)*b*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Sinh[2*c])/(a^4*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/((a + b)^3*(a + b*Sech[c + d*x]^2)^4) + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Sech[2*c]*Sech[c + d*x]^8*(480*a^6*d*x*Cosh[2*c] + 3168*a^5*b*d*x*Cosh[2*c] + 8928*a^4*b^2*d*x*Cosh[2*c] + 14112*a^3*b^3*d*x*Cosh[2*c] + 13248*a^2*b^4*d*x*Cosh[2*c] + 6912*a*b^5*d*x*Cosh[2*c] + 1536*b^6*d*x*Cosh[2*c] + 360*a^6*d*x*Cosh[2*d*x] + 2232*a^5*b*d*x*Cosh[2*d*x] + 5688*a^4*b^2*d*x*Cosh[2*d*x] + 7272*a^3*b^3*d*x*Cosh[2*d*x] + 4608*a^2*b^4*d*x*Cosh[2*d*x] + 1152*a*b^5*d*x*Cosh[2*d*x] + 360*a^6*d*x*Cosh[4*c + 2*d*x] + 2232*a^5*b*d*x*Cosh[4*c + 2*d*x] + 5688*a^4*b^2*d*x*Cosh[4*c + 2*d*x] + 7272*a^3*b^3*d*x*Cosh[4*c + 2*d*x] + 4608*a^2*b^4*d*x*Cosh[4*c + 2*d*x] + 1152*a*b^5*d*x*Cosh[4*c + 2*d*x] + 144*a^6*d*x*Cosh[2*c + 4*d*x] + 720*a^5*b*d*x*Cosh[2*c + 4*d*x] + 1296*a^4*b^2*d*x*Cosh[2*c + 4*d*x] + 1008*a^3*b^3*d*x*Cosh[2*c + 4*d*x] + 288*a^2*b^4*d*x*Cosh[2*c + 4*d*x] + 144*a^6*d*x*Cosh[6*c + 4*d*x] + 720*a^5*b*d*x*Cosh[6*c + 4*d*x] + 1296*a^4*b^2*d*x*Cosh[6*c + 4*d*x] + 1008*a^3*b^3*d*x*Cosh[6*c + 4*d*x] + 288*a^2*b^4*d*x*Cosh[6*c + 4*d*x] + 24*a^6*d*x*Cosh[4*c + 6*d*x] + 72*a^5*b*d*x*Cosh[4*c + 6*d*x] + 72*a^4*b^2*d*x*Cosh[4*c + 6*d*x] + 24*a^3*b^3*d*x*Cosh[4*c + 6*d*x] + 24*a^6*d*x*Cosh[8*c + 6*d*x] + 72*a^5*b*d*x*Cosh[8*c + 6*d*x] + 72*a^4*b^2*d*x*Cosh[8*c + 6*d*x] + 24*a^3*b^3*d*x*Cosh[8*c + 6*d*x] + 870*a^5*b*Sinh[2*c] + 4292*a^4*b^2*Sinh[2*c] + 8792*a^3*b^3*Sinh[2*c] + 9936*a^2*b^4*Sinh[2*c] + 5824*a*b^5*Sinh[2*c] + 1408*b^6*Sinh[2*c] - 870*a^5*b*Sinh[2*d*x] - 3792*a^4*b^2*Sinh[2*d*x] - 6432*a^3*b^3*Sinh[2*d*x] - 4608*a^2*b^4*Sinh[2*d*x] - 1248*a*b^5*Sinh[2*d*x] + 435*a^5*b*Sinh[4*c + 2*d*x] + 2124*a^4*b^2*Sinh[4*c + 2*d*x] + 3972*a^3*b^3*Sinh[4*c + 2*d*x] + 3072*a^2*b^4*Sinh[4*c + 2*d*x] + 864*a*b^5*Sinh[4*c + 2*d*x] - 435*a^5*b*Sinh[2*c + 4*d*x] - 1374*a^4*b^2*Sinh[2*c + 4*d*x] - 1248*a^3*b^3*Sinh[2*c + 4*d*x] - 384*a^2*b^4*Sinh[2*c + 4*d*x] + 87*a^5*b*Sinh[6*c + 4*d*x] + 366*a^4*b^2*Sinh[6*c + 4*d*x] + 408*a^3*b^3*Sinh[6*c + 4*d*x] + 144*a^2*b^4*Sinh[6*c + 4*d*x] - 87*a^5*b*Sinh[4*c + 6*d*x] - 116*a^4*b^2*Sinh[4*c + 6*d*x] - 44*a^3*b^3*Sinh[4*c + 6*d*x]))/(3072*a^4*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^4)","C",0
170,1,25,29,0.0231641,"\int \left(1-\text{sech}^2(x)\right)^{3/2} \, dx","Integrate[(1 - Sech[x]^2)^(3/2),x]","\frac{1}{2} \sqrt{\tanh ^2(x)} (\text{csch}(x) \text{sech}(x)+2 \coth (x) \log (\cosh (x)))","\sqrt{\tanh ^2(x)} \coth (x) \log (\cosh (x))-\frac{1}{2} \tanh ^2(x)^{3/2} \coth (x)",1,"((2*Coth[x]*Log[Cosh[x]] + Csch[x]*Sech[x])*Sqrt[Tanh[x]^2])/2","A",1
171,1,14,14,0.0064665,"\int \sqrt{1-\text{sech}^2(x)} \, dx","Integrate[Sqrt[1 - Sech[x]^2],x]","\sqrt{\tanh ^2(x)} \coth (x) \log (\cosh (x))","\sqrt{\tanh ^2(x)} \coth (x) \log (\cosh (x))",1,"Coth[x]*Log[Cosh[x]]*Sqrt[Tanh[x]^2]","A",1
172,1,14,14,0.0099729,"\int \frac{1}{\sqrt{1-\text{sech}^2(x)}} \, dx","Integrate[1/Sqrt[1 - Sech[x]^2],x]","\frac{\tanh (x) \log (\sinh (x))}{\sqrt{\tanh ^2(x)}}","\frac{\tanh (x) \log (\sinh (x))}{\sqrt{\tanh ^2(x)}}",1,"(Log[Sinh[x]]*Tanh[x])/Sqrt[Tanh[x]^2]","A",1
173,1,27,34,0.0140109,"\int \left(-1+\text{sech}^2(x)\right)^{3/2} \, dx","Integrate[(-1 + Sech[x]^2)^(3/2),x]","-\frac{1}{2} \sqrt{-\tanh ^2(x)} (\text{csch}(x) \text{sech}(x)+2 \coth (x) \log (\cosh (x)))","\frac{1}{2} \tanh (x) \sqrt{-\tanh ^2(x)}-\sqrt{-\tanh ^2(x)} \coth (x) \log (\cosh (x))",1,"-1/2*((2*Coth[x]*Log[Cosh[x]] + Csch[x]*Sech[x])*Sqrt[-Tanh[x]^2])","A",1
174,1,16,16,0.0058672,"\int \sqrt{-1+\text{sech}^2(x)} \, dx","Integrate[Sqrt[-1 + Sech[x]^2],x]","\sqrt{-\tanh ^2(x)} \coth (x) \log (\cosh (x))","\sqrt{-\tanh ^2(x)} \coth (x) \log (\cosh (x))",1,"Coth[x]*Log[Cosh[x]]*Sqrt[-Tanh[x]^2]","A",1
175,1,16,16,0.0095357,"\int \frac{1}{\sqrt{-1+\text{sech}^2(x)}} \, dx","Integrate[1/Sqrt[-1 + Sech[x]^2],x]","\frac{\tanh (x) \log (\sinh (x))}{\sqrt{-\tanh ^2(x)}}","\frac{\tanh (x) \log (\sinh (x))}{\sqrt{-\tanh ^2(x)}}",1,"(Log[Sinh[x]]*Tanh[x])/Sqrt[-Tanh[x]^2]","A",1
176,1,114,83,0.7488479,"\int \sqrt{a+b \text{sech}^2(x)} \tanh ^5(x) \, dx","Integrate[Sqrt[a + b*Sech[x]^2]*Tanh[x]^5,x]","\frac{1}{15} \cosh (x) \sqrt{a+b \text{sech}^2(x)} \left(\left(\frac{2 a^2}{b^2}+\frac{10 a}{b}-15\right) \text{sech}(x)+\left(10-\frac{a}{b}\right) \text{sech}^3(x)+\frac{15 \sqrt{2} \sqrt{a} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)}{\sqrt{a \cosh (2 x)+a+2 b}}-3 \text{sech}^5(x)\right)","-\frac{\left(a+b \text{sech}^2(x)\right)^{5/2}}{5 b^2}+\frac{(a+2 b) \left(a+b \text{sech}^2(x)\right)^{3/2}}{3 b^2}-\sqrt{a+b \text{sech}^2(x)}+\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)",1,"(Cosh[x]*Sqrt[a + b*Sech[x]^2]*((15*Sqrt[2]*Sqrt[a]*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])/Sqrt[a + 2*b + a*Cosh[2*x]] + (-15 + (2*a^2)/b^2 + (10*a)/b)*Sech[x] + (10 - a/b)*Sech[x]^3 - 3*Sech[x]^5))/15","A",1
177,1,192,125,0.4703357,"\int \sqrt{a+b \text{sech}^2(x)} \tanh ^4(x) \, dx","Integrate[Sqrt[a + b*Sech[x]^2]*Tanh[x]^4,x]","-\frac{\cosh (x) \sqrt{a+b \text{sech}^2(x)} \left(\sqrt{2} \left(a^2+6 a b-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)-8 \sqrt{2} \sqrt{a} b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)-2 b^{3/2} \tanh (x) \text{sech}^3(x) \sqrt{a \cosh (2 x)+a+2 b}-\sqrt{b} (a-5 b) \tanh (x) \text{sech}(x) \sqrt{a \cosh (2 x)+a+2 b}\right)}{8 b^{3/2} \sqrt{a \cosh (2 x)+a+2 b}}","-\frac{\left(a^2+6 a b-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{8 b^{3/2}}+\frac{(a-3 b) \tanh (x) \sqrt{a-b \tanh ^2(x)+b}}{8 b}+\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)-\frac{1}{4} \tanh ^3(x) \sqrt{a-b \tanh ^2(x)+b}",1,"-1/8*(Cosh[x]*Sqrt[a + b*Sech[x]^2]*(Sqrt[2]*(a^2 + 6*a*b - 3*b^2)*ArcTan[(Sqrt[2]*Sqrt[b]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]] - 8*Sqrt[2]*Sqrt[a]*b^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]] - (a - 5*b)*Sqrt[b]*Sqrt[a + 2*b + a*Cosh[2*x]]*Sech[x]*Tanh[x] - 2*b^(3/2)*Sqrt[a + 2*b + a*Cosh[2*x]]*Sech[x]^3*Tanh[x]))/(b^(3/2)*Sqrt[a + 2*b + a*Cosh[2*x]])","A",1
178,1,90,59,0.3965615,"\int \sqrt{a+b \text{sech}^2(x)} \tanh ^3(x) \, dx","Integrate[Sqrt[a + b*Sech[x]^2]*Tanh[x]^3,x]","\frac{1}{3} \cosh (x) \sqrt{a+b \text{sech}^2(x)} \left(\left(\frac{a}{b}-3\right) \text{sech}(x)+\frac{3 \sqrt{2} \sqrt{a} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)}{\sqrt{a \cosh (2 x)+a+2 b}}+\text{sech}^3(x)\right)","\frac{\left(a+b \text{sech}^2(x)\right)^{3/2}}{3 b}-\sqrt{a+b \text{sech}^2(x)}+\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)",1,"(Cosh[x]*Sqrt[a + b*Sech[x]^2]*((3*Sqrt[2]*Sqrt[a]*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])/Sqrt[a + 2*b + a*Cosh[2*x]] + (-3 + a/b)*Sech[x] + Sech[x]^3))/3","A",1
179,1,150,87,0.2868795,"\int \sqrt{a+b \text{sech}^2(x)} \tanh ^2(x) \, dx","Integrate[Sqrt[a + b*Sech[x]^2]*Tanh[x]^2,x]","-\frac{\cosh (x) \sqrt{a+b \text{sech}^2(x)} \left(\sqrt{2} (a-b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)-2 \sqrt{2} \sqrt{a} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)+\sqrt{b} \tanh (x) \text{sech}(x) \sqrt{a \cosh (2 x)+a+2 b}\right)}{2 \sqrt{b} \sqrt{a \cosh (2 x)+a+2 b}}","-\frac{1}{2} \tanh (x) \sqrt{a-b \tanh ^2(x)+b}+\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)-\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{2 \sqrt{b}}",1,"-1/2*(Cosh[x]*Sqrt[a + b*Sech[x]^2]*(Sqrt[2]*(a - b)*ArcTan[(Sqrt[2]*Sqrt[b]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]] - 2*Sqrt[2]*Sqrt[a]*Sqrt[b]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]] + Sqrt[b]*Sqrt[a + 2*b + a*Cosh[2*x]]*Sech[x]*Tanh[x]))/(Sqrt[b]*Sqrt[a + 2*b + a*Cosh[2*x]])","A",1
180,1,90,40,0.2736572,"\int \sqrt{a+b \text{sech}^2(x)} \tanh (x) \, dx","Integrate[Sqrt[a + b*Sech[x]^2]*Tanh[x],x]","-\frac{\sqrt{a+b \text{sech}^2(x)} \left(-\sqrt{2} \sqrt{a} \sqrt{b} \cosh (x) \sqrt{\frac{a \cosh (2 x)+a+2 b}{b}} \sinh ^{-1}\left(\frac{\sqrt{a} \cosh (x)}{\sqrt{b}}\right)+a \cosh (2 x)+a+2 b\right)}{a \cosh (2 x)+a+2 b}","\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)-\sqrt{a+b \text{sech}^2(x)}",1,"-(((a + 2*b + a*Cosh[2*x] - Sqrt[2]*Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[a]*Cosh[x])/Sqrt[b]]*Cosh[x]*Sqrt[(a + 2*b + a*Cosh[2*x])/b])*Sqrt[a + b*Sech[x]^2])/(a + 2*b + a*Cosh[2*x]))","B",1
181,1,134,59,0.7405144,"\int \sqrt{a+b \text{sech}^2(x)} \, dx","Integrate[Sqrt[a + b*Sech[x]^2],x]","\frac{\sqrt{2} \cosh (x) \sqrt{a+b \text{sech}^2(x)} \left(\sqrt{a} \sqrt{a+b} \sinh ^{-1}\left(\frac{\sqrt{a} \sinh (x)}{\sqrt{a+b}}\right) \sqrt{\frac{a \cosh (2 x)+a+2 b}{a+b}}+\sqrt{b} \sqrt{a \cosh (2 x)+a+2 b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)\right)}{a \cosh (2 x)+a+2 b}","\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)+\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)",1,"(Sqrt[2]*Cosh[x]*(Sqrt[b]*ArcTan[(Sqrt[2]*Sqrt[b]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*Sqrt[a + 2*b + a*Cosh[2*x]] + Sqrt[a]*Sqrt[a + b]*ArcSinh[(Sqrt[a]*Sinh[x])/Sqrt[a + b]]*Sqrt[(a + 2*b + a*Cosh[2*x])/(a + b)])*Sqrt[a + b*Sech[x]^2])/(a + 2*b + a*Cosh[2*x])","B",1
182,1,111,56,0.129553,"\int \coth (x) \sqrt{a+b \text{sech}^2(x)} \, dx","Integrate[Coth[x]*Sqrt[a + b*Sech[x]^2],x]","\frac{\sqrt{2} \cosh (x) \sqrt{a+b \text{sech}^2(x)} \left(\sqrt{a} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)-\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{2 a+2 b} \cosh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)\right)}{\sqrt{a \cosh (2 x)+a+2 b}}","\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)-\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a+b}}\right)",1,"(Sqrt[2]*Cosh[x]*(-(Sqrt[a + b]*ArcTanh[(Sqrt[2*a + 2*b]*Cosh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]) + Sqrt[a]*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])*Sqrt[a + b*Sech[x]^2])/Sqrt[a + 2*b + a*Cosh[2*x]]","A",1
183,1,75,48,0.5214657,"\int \coth ^2(x) \sqrt{a+b \text{sech}^2(x)} \, dx","Integrate[Coth[x]^2*Sqrt[a + b*Sech[x]^2],x]","\sqrt{a+b \text{sech}^2(x)} \left(\frac{\sqrt{2} \sqrt{a} \cosh (x) \sinh ^{-1}\left(\frac{\sqrt{a} \sinh (x)}{\sqrt{a+b}}\right)}{\sqrt{a+b} \sqrt{\frac{a \cosh (2 x)+a+2 b}{a+b}}}-\coth (x)\right)","\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)-\coth (x) \sqrt{a-b \tanh ^2(x)+b}",1,"((Sqrt[2]*Sqrt[a]*ArcSinh[(Sqrt[a]*Sinh[x])/Sqrt[a + b]]*Cosh[x])/(Sqrt[a + b]*Sqrt[(a + 2*b + a*Cosh[2*x])/(a + b)]) - Coth[x])*Sqrt[a + b*Sech[x]^2]","A",1
184,1,156,83,0.5937395,"\int \coth ^3(x) \sqrt{a+b \text{sech}^2(x)} \, dx","Integrate[Coth[x]^3*Sqrt[a + b*Sech[x]^2],x]","-\frac{\sqrt{a+b \text{sech}^2(x)} \left(\sqrt{2} (2 a+b) \cosh (x) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)+\sqrt{a+b} \left(\coth ^2(x) \sqrt{a \cosh (2 x)+a+2 b}-2 \sqrt{2} \sqrt{a} \cosh (x) \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)\right)\right)}{2 \sqrt{a+b} \sqrt{a \cosh (2 x)+a+2 b}}","-\frac{1}{2} \coth ^2(x) \sqrt{a+b \text{sech}^2(x)}+\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)-\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a+b}}\right)}{2 \sqrt{a+b}}",1,"-1/2*((Sqrt[2]*(2*a + b)*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*Cosh[x] + Sqrt[a + b]*(Sqrt[a + 2*b + a*Cosh[2*x]]*Coth[x]^2 - 2*Sqrt[2]*Sqrt[a]*Cosh[x]*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]]))*Sqrt[a + b*Sech[x]^2])/(Sqrt[a + b]*Sqrt[a + 2*b + a*Cosh[2*x]])","A",1
185,1,149,84,0.624699,"\int \coth ^4(x) \sqrt{a+b \text{sech}^2(x)} \, dx","Integrate[Coth[x]^4*Sqrt[a + b*Sech[x]^2],x]","\frac{\sqrt{2} \cosh (x) \sqrt{a \sinh ^2(x)+a+b} \sqrt{a+b \text{sech}^2(x)} \left(3 \sqrt{a} (a+b) \sinh ^{-1}\left(\frac{\sqrt{a} \sinh (x)}{\sqrt{a+b}}\right)-\sqrt{a+b} \text{csch}(x) \sqrt{\frac{a \sinh ^2(x)+a+b}{a+b}} \left((a+b) \text{csch}^2(x)+4 a+3 b\right)\right)}{3 (a+b)^{3/2} \sqrt{\frac{a \sinh ^2(x)+a+b}{a+b}} \sqrt{a \cosh (2 x)+a+2 b}}","\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)-\frac{1}{3} \coth ^3(x) \sqrt{a-b \tanh ^2(x)+b}-\frac{(3 a+2 b) \coth (x) \sqrt{a-b \tanh ^2(x)+b}}{3 (a+b)}",1,"(Sqrt[2]*Cosh[x]*Sqrt[a + b*Sech[x]^2]*Sqrt[a + b + a*Sinh[x]^2]*(3*Sqrt[a]*(a + b)*ArcSinh[(Sqrt[a]*Sinh[x])/Sqrt[a + b]] - Sqrt[a + b]*Csch[x]*(4*a + 3*b + (a + b)*Csch[x]^2)*Sqrt[(a + b + a*Sinh[x]^2)/(a + b)]))/(3*(a + b)^(3/2)*Sqrt[a + 2*b + a*Cosh[2*x]]*Sqrt[(a + b + a*Sinh[x]^2)/(a + b)])","A",1
186,1,191,125,1.0223655,"\int \coth ^5(x) \sqrt{a+b \text{sech}^2(x)} \, dx","Integrate[Coth[x]^5*Sqrt[a + b*Sech[x]^2],x]","-\frac{\cosh (x) \sqrt{a+b \text{sech}^2(x)} \left(\sqrt{2} \left(8 a^2+12 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)+\sqrt{a+b} \left(\frac{1}{2} \coth (x) \text{csch}^3(x) \sqrt{a \cosh (2 x)+a+2 b} ((6 a+5 b) \cosh (2 x)-2 a-b)-8 \sqrt{2} \sqrt{a} (a+b) \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)\right)\right)}{8 (a+b)^{3/2} \sqrt{a \cosh (2 x)+a+2 b}}","-\frac{\left(8 a^2+12 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a+b}}\right)}{8 (a+b)^{3/2}}-\frac{1}{4} \coth ^4(x) \sqrt{a+b \text{sech}^2(x)}-\frac{(4 a+3 b) \coth ^2(x) \sqrt{a+b \text{sech}^2(x)}}{8 (a+b)}+\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)",1,"-1/8*(Cosh[x]*(Sqrt[2]*(8*a^2 + 12*a*b + 3*b^2)*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]] + Sqrt[a + b]*((Sqrt[a + 2*b + a*Cosh[2*x]]*(-2*a - b + (6*a + 5*b)*Cosh[2*x])*Coth[x]*Csch[x]^3)/2 - 8*Sqrt[2]*Sqrt[a]*(a + b)*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]]))*Sqrt[a + b*Sech[x]^2])/((a + b)^(3/2)*Sqrt[a + 2*b + a*Cosh[2*x]])","A",1
187,1,129,76,0.9710313,"\int \left(a+b \text{sech}^2(x)\right)^{3/2} \tanh ^3(x) \, dx","Integrate[(a + b*Sech[x]^2)^(3/2)*Tanh[x]^3,x]","\cosh ^3(x) \left(a+b \text{sech}^2(x)\right)^{3/2} \left(\frac{2 \sqrt{2} a^{3/2} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)}{(a \cosh (2 x)+a+2 b)^{3/2}}+\frac{2 \left(b (6 a-5 b) \text{sech}^3(x)+a (3 a-20 b) \text{sech}(x)+3 b^2 \text{sech}^5(x)\right)}{15 b (a \cosh (2 x)+a+2 b)}\right)","a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)+\frac{\left(a+b \text{sech}^2(x)\right)^{5/2}}{5 b}-\frac{1}{3} \left(a+b \text{sech}^2(x)\right)^{3/2}-a \sqrt{a+b \text{sech}^2(x)}",1,"Cosh[x]^3*(a + b*Sech[x]^2)^(3/2)*((2*Sqrt[2]*a^(3/2)*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])/(a + 2*b + a*Cosh[2*x])^(3/2) + (2*(a*(3*a - 20*b)*Sech[x] + (6*a - 5*b)*b*Sech[x]^3 + 3*b^2*Sech[x]^5))/(15*b*(a + 2*b + a*Cosh[2*x])))","A",1
188,1,197,125,0.8533123,"\int \left(a+b \text{sech}^2(x)\right)^{3/2} \tanh ^2(x) \, dx","Integrate[(a + b*Sech[x]^2)^(3/2)*Tanh[x]^2,x]","-\frac{\cosh ^3(x) \left(a+b \text{sech}^2(x)\right)^{3/2} \left(-8 \sqrt{2} a^{3/2} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)+\sqrt{2} \left(3 a^2-6 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)+2 b^{3/2} \tanh (x) \text{sech}^3(x) \sqrt{a \cosh (2 x)+a+2 b}+\sqrt{b} (5 a-b) \tanh (x) \text{sech}(x) \sqrt{a \cosh (2 x)+a+2 b}\right)}{4 \sqrt{b} (a \cosh (2 x)+a+2 b)^{3/2}}","a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)-\frac{\left(3 a^2-6 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{8 \sqrt{b}}-\frac{1}{8} (5 a+b) \tanh (x) \sqrt{a-b \tanh ^2(x)+b}+\frac{1}{4} b \tanh ^3(x) \sqrt{a-b \tanh ^2(x)+b}",1,"-1/4*(Cosh[x]^3*(a + b*Sech[x]^2)^(3/2)*(Sqrt[2]*(3*a^2 - 6*a*b - b^2)*ArcTan[(Sqrt[2]*Sqrt[b]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]] - 8*Sqrt[2]*a^(3/2)*Sqrt[b]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]] + (5*a - b)*Sqrt[b]*Sqrt[a + 2*b + a*Cosh[2*x]]*Sech[x]*Tanh[x] + 2*b^(3/2)*Sqrt[a + 2*b + a*Cosh[2*x]]*Sech[x]^3*Tanh[x]))/(Sqrt[b]*(a + 2*b + a*Cosh[2*x])^(3/2))","A",1
189,1,65,57,0.1162519,"\int \left(a+b \text{sech}^2(x)\right)^{3/2} \tanh (x) \, dx","Integrate[(a + b*Sech[x]^2)^(3/2)*Tanh[x],x]","-\frac{2 b \left(a+b \text{sech}^2(x)\right)^{3/2} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{a \cosh ^2(x)}{b}\right)}{3 \sqrt{\frac{a \cosh ^2(x)}{b}+1} (a \cosh (2 x)+a+2 b)}","a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)-a \sqrt{a+b \text{sech}^2(x)}-\frac{1}{3} \left(a+b \text{sech}^2(x)\right)^{3/2}",1,"(-2*b*Hypergeometric2F1[-3/2, -3/2, -1/2, -((a*Cosh[x]^2)/b)]*(a + b*Sech[x]^2)^(3/2))/(3*Sqrt[1 + (a*Cosh[x]^2)/b]*(a + 2*b + a*Cosh[2*x]))","C",1
190,1,152,88,0.2745539,"\int \left(a+b \text{sech}^2(x)\right)^{3/2} \, dx","Integrate[(a + b*Sech[x]^2)^(3/2),x]","\frac{\text{sech}(x) \left(a \cosh ^2(x)+b\right) \sqrt{a+b \text{sech}^2(x)} \left(2 \sqrt{2} a^{3/2} \cosh ^2(x) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)+b \sinh (x) \sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{b} (3 a+b) \cosh ^2(x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)\right)}{(a \cosh (2 x)+a+2 b)^{3/2}}","a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)+\frac{1}{2} b \tanh (x) \sqrt{a-b \tanh ^2(x)+b}+\frac{1}{2} \sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)",1,"((b + a*Cosh[x]^2)*Sech[x]*Sqrt[a + b*Sech[x]^2]*(Sqrt[2]*Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[2]*Sqrt[b]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*Cosh[x]^2 + 2*Sqrt[2]*a^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*Cosh[x]^2 + b*Sqrt[a + 2*b + a*Cosh[2*x]]*Sinh[x]))/(a + 2*b + a*Cosh[2*x])^(3/2)","A",1
191,1,159,70,0.5169744,"\int \coth (x) \left(a+b \text{sech}^2(x)\right)^{3/2} \, dx","Integrate[Coth[x]*(a + b*Sech[x]^2)^(3/2),x]","-\frac{2 \left(a \cosh ^2(x)+b\right) \sqrt{a+b \text{sech}^2(x)} \left(\sqrt{2} (a+b)^2 \cosh (x) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)-\sqrt{a+b} \left(\sqrt{2} a^{3/2} \cosh (x) \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)+b \sqrt{a \cosh (2 x)+a+2 b}\right)\right)}{\sqrt{a+b} (a \cosh (2 x)+a+2 b)^{3/2}}","a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)+b \sqrt{a+b \text{sech}^2(x)}-(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a+b}}\right)",1,"(-2*(b + a*Cosh[x]^2)*(Sqrt[2]*(a + b)^2*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*Cosh[x] - Sqrt[a + b]*(b*Sqrt[a + 2*b + a*Cosh[2*x]] + Sqrt[2]*a^(3/2)*Cosh[x]*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]]))*Sqrt[a + b*Sech[x]^2])/(Sqrt[a + b]*(a + 2*b + a*Cosh[2*x])^(3/2))","B",1
192,1,144,81,0.3504783,"\int \coth ^2(x) \left(a+b \text{sech}^2(x)\right)^{3/2} \, dx","Integrate[Coth[x]^2*(a + b*Sech[x]^2)^(3/2),x]","-\frac{2 \left(a \cosh ^2(x)+b\right) \sqrt{a+b \text{sech}^2(x)} \left(-\sqrt{2} a^{3/2} \cosh (x) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)+\sqrt{2} b^{3/2} \cosh (x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)+(a+b) \coth (x) \sqrt{a \cosh (2 x)+a+2 b}\right)}{(a \cosh (2 x)+a+2 b)^{3/2}}","a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)-b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)-(a+b) \coth (x) \sqrt{a-b \tanh ^2(x)+b}",1,"(-2*(b + a*Cosh[x]^2)*(Sqrt[2]*b^(3/2)*ArcTan[(Sqrt[2]*Sqrt[b]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*Cosh[x] - Sqrt[2]*a^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*Cosh[x] + (a + b)*Sqrt[a + 2*b + a*Cosh[2*x]]*Coth[x])*Sqrt[a + b*Sech[x]^2])/(a + 2*b + a*Cosh[2*x])^(3/2)","A",1
193,1,280,170,9.2812776,"\int \left(a+b \text{sech}^2(c+d x)\right)^{5/2} \, dx","Integrate[(a + b*Sech[c + d*x]^2)^(5/2),x]","\frac{\cosh ^5(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^{5/2} \left(8 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a \sinh ^2(c+d x)+a+b}}\right)+\sqrt{b} \left(15 a^2+10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a \sinh ^2(c+d x)+a+b}}\right)\right)}{\sqrt{2} d (a \cosh (2 c+2 d x)+a+2 b)^{5/2}}+\frac{\cosh ^5(c+d x) \left(a+b \text{sech}^2(c+d x)\right)^{5/2} \left(\frac{3 \text{sech}(c) \text{sech}^2(c+d x) \left(3 a b \sinh (d x)+b^2 \sinh (d x)\right)}{2 d}+\frac{3 b (3 a+b) \tanh (c) \text{sech}(c+d x)}{2 d}+\frac{b^2 \text{sech}(c) \sinh (d x) \text{sech}^4(c+d x)}{d}+\frac{b^2 \tanh (c) \text{sech}^3(c+d x)}{d}\right)}{(a \cosh (2 c+2 d x)+a+2 b)^2}","\frac{a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a-b \tanh ^2(c+d x)+b}}\right)}{d}+\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-b \tanh ^2(c+d x)+b}}\right)}{8 d}+\frac{b \tanh (c+d x) \left(a-b \tanh ^2(c+d x)+b\right)^{3/2}}{4 d}+\frac{b (7 a+3 b) \tanh (c+d x) \sqrt{a-b \tanh ^2(c+d x)+b}}{8 d}",1,"((Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a + b + a*Sinh[c + d*x]^2]] + 8*a^(5/2)*ArcTanh[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b + a*Sinh[c + d*x]^2]])*Cosh[c + d*x]^5*(a + b*Sech[c + d*x]^2)^(5/2))/(Sqrt[2]*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^(5/2)) + (Cosh[c + d*x]^5*(a + b*Sech[c + d*x]^2)^(5/2)*((b^2*Sech[c]*Sech[c + d*x]^4*Sinh[d*x])/d + (3*Sech[c]*Sech[c + d*x]^2*(3*a*b*Sinh[d*x] + b^2*Sinh[d*x]))/(2*d) + (3*b*(3*a + b)*Sech[c + d*x]*Tanh[c])/(2*d) + (b^2*Sech[c + d*x]^3*Tanh[c])/d))/(a + 2*b + a*Cosh[2*c + 2*d*x])^2","A",1
194,1,109,66,0.5197142,"\int \frac{\tanh ^5(x)}{\sqrt{a+b \text{sech}^2(x)}} \, dx","Integrate[Tanh[x]^5/Sqrt[a + b*Sech[x]^2],x]","\frac{\text{sech}(x) \left(\frac{\text{sech}(x) (a \cosh (2 x)+a+2 b) \left(2 a-b \text{sech}^2(x)+6 b\right)}{3 b^2}+\frac{\sqrt{2} \sqrt{a \cosh (2 x)+a+2 b} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)}{\sqrt{a}}\right)}{2 \sqrt{a+b \text{sech}^2(x)}}","-\frac{\left(a+b \text{sech}^2(x)\right)^{3/2}}{3 b^2}+\frac{(a+2 b) \sqrt{a+b \text{sech}^2(x)}}{b^2}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"(Sech[x]*((Sqrt[2]*Sqrt[a + 2*b + a*Cosh[2*x]]*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])/Sqrt[a] + ((a + 2*b + a*Cosh[2*x])*Sech[x]*(2*a + 6*b - b*Sech[x]^2))/(3*b^2)))/(2*Sqrt[a + b*Sech[x]^2])","A",1
195,1,169,90,0.529935,"\int \frac{\tanh ^4(x)}{\sqrt{a+b \text{sech}^2(x)}} \, dx","Integrate[Tanh[x]^4/Sqrt[a + b*Sech[x]^2],x]","\frac{\text{sech}(x) \left(2 \sqrt{2} b^{3/2} \sqrt{a \cosh (2 x)+a+2 b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)+\sqrt{a} \left(\sqrt{b} \tanh (x) \text{sech}(x) (a \cosh (2 x)+a+2 b)-\sqrt{2} (a+3 b) \sqrt{a \cosh (2 x)+a+2 b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)\right)\right)}{4 \sqrt{a} b^{3/2} \sqrt{a+b \text{sech}^2(x)}}","-\frac{(a+3 b) \tan ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{2 b^{3/2}}+\frac{\tanh (x) \sqrt{a-b \tanh ^2(x)+b}}{2 b}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{\sqrt{a}}",1,"(Sech[x]*(2*Sqrt[2]*b^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*Sqrt[a + 2*b + a*Cosh[2*x]] + Sqrt[a]*(-(Sqrt[2]*(a + 3*b)*ArcTan[(Sqrt[2]*Sqrt[b]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*Sqrt[a + 2*b + a*Cosh[2*x]]) + Sqrt[b]*(a + 2*b + a*Cosh[2*x])*Sech[x]*Tanh[x])))/(4*Sqrt[a]*b^(3/2)*Sqrt[a + b*Sech[x]^2])","A",1
196,1,105,42,0.2246227,"\int \frac{\tanh ^3(x)}{\sqrt{a+b \text{sech}^2(x)}} \, dx","Integrate[Tanh[x]^3/Sqrt[a + b*Sech[x]^2],x]","\frac{\text{sech}^2(x) (a \cosh (2 x)+a+2 b)}{2 b \sqrt{a+b \text{sech}^2(x)}}+\frac{\text{sech}(x) \sqrt{a \cosh (2 x)+a+2 b} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)}{\sqrt{2} \sqrt{a} \sqrt{a+b \text{sech}^2(x)}}","\frac{\sqrt{a+b \text{sech}^2(x)}}{b}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"(Sqrt[a + 2*b + a*Cosh[2*x]]*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]]*Sech[x])/(Sqrt[2]*Sqrt[a]*Sqrt[a + b*Sech[x]^2]) + ((a + 2*b + a*Cosh[2*x])*Sech[x]^2)/(2*b*Sqrt[a + b*Sech[x]^2])","B",1
197,1,107,60,0.1808731,"\int \frac{\tanh ^2(x)}{\sqrt{a+b \text{sech}^2(x)}} \, dx","Integrate[Tanh[x]^2/Sqrt[a + b*Sech[x]^2],x]","\frac{\text{sech}(x) \sqrt{a \cosh (2 x)+a+2 b} \left(\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)}{\sqrt{a}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)}{\sqrt{b}}\right)}{\sqrt{2} \sqrt{a+b \text{sech}^2(x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{\sqrt{a}}-\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{\sqrt{b}}",1,"((-(ArcTan[(Sqrt[2]*Sqrt[b]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]/Sqrt[b]) + ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]/Sqrt[a])*Sqrt[a + 2*b + a*Cosh[2*x]]*Sech[x])/(Sqrt[2]*Sqrt[a + b*Sech[x]^2])","A",1
198,1,70,25,0.0828158,"\int \frac{\tanh (x)}{\sqrt{a+b \text{sech}^2(x)}} \, dx","Integrate[Tanh[x]/Sqrt[a + b*Sech[x]^2],x]","\frac{\text{sech}(x) \sqrt{a \cosh (2 x)+a+2 b} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)}{\sqrt{2} \sqrt{a} \sqrt{a+b \text{sech}^2(x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"(Sqrt[a + 2*b + a*Cosh[2*x]]*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]]*Sech[x])/(Sqrt[2]*Sqrt[a]*Sqrt[a + b*Sech[x]^2])","B",1
199,1,62,29,0.0412959,"\int \frac{1}{\sqrt{a+b \text{sech}^2(x)}} \, dx","Integrate[1/Sqrt[a + b*Sech[x]^2],x]","\frac{\text{sech}(x) \sqrt{a \cosh (2 x)+a+2 b} \tanh ^{-1}\left(\frac{\sqrt{a} \sinh (x)}{\sqrt{a \sinh ^2(x)+a+b}}\right)}{\sqrt{2} \sqrt{a} \sqrt{a+b \text{sech}^2(x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{\sqrt{a}}",1,"(ArcTanh[(Sqrt[a]*Sinh[x])/Sqrt[a + b + a*Sinh[x]^2]]*Sqrt[a + 2*b + a*Cosh[2*x]]*Sech[x])/(Sqrt[2]*Sqrt[a]*Sqrt[a + b*Sech[x]^2])","B",1
200,1,124,56,0.2610895,"\int \frac{\coth (x)}{\sqrt{a+b \text{sech}^2(x)}} \, dx","Integrate[Coth[x]/Sqrt[a + b*Sech[x]^2],x]","\frac{\text{sech}(x) \sqrt{a \cosh (2 x)+a+2 b} \left(\sqrt{a+b} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)\right)}{\sqrt{2} \sqrt{a} \sqrt{a+b} \sqrt{a+b \text{sech}^2(x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}",1,"(Sqrt[a + 2*b + a*Cosh[2*x]]*(-(Sqrt[a]*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]) + Sqrt[a + b]*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])*Sech[x])/(Sqrt[2]*Sqrt[a]*Sqrt[a + b]*Sqrt[a + b*Sech[x]^2])","B",1
201,1,94,53,0.1115538,"\int \frac{\coth ^2(x)}{\sqrt{a+b \text{sech}^2(x)}} \, dx","Integrate[Coth[x]^2/Sqrt[a + b*Sech[x]^2],x]","\frac{\text{sech}(x) \sqrt{a \cosh (2 x)+a+2 b} \left((a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \sinh (x)}{\sqrt{a \sinh ^2(x)+a+b}}\right)-\sqrt{a} \text{csch}(x) \sqrt{a \sinh ^2(x)+a+b}\right)}{\sqrt{2} \sqrt{a} (a+b) \sqrt{a+b \text{sech}^2(x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{\sqrt{a}}-\frac{\coth (x) \sqrt{a-b \tanh ^2(x)+b}}{a+b}",1,"(Sqrt[a + 2*b + a*Cosh[2*x]]*Sech[x]*((a + b)*ArcTanh[(Sqrt[a]*Sinh[x])/Sqrt[a + b + a*Sinh[x]^2]] - Sqrt[a]*Csch[x]*Sqrt[a + b + a*Sinh[x]^2]))/(Sqrt[2]*Sqrt[a]*(a + b)*Sqrt[a + b*Sech[x]^2])","A",1
202,1,159,90,0.8478,"\int \frac{\coth ^3(x)}{\sqrt{a+b \text{sech}^2(x)}} \, dx","Integrate[Coth[x]^3/Sqrt[a + b*Sech[x]^2],x]","\frac{\frac{\sqrt{2} \text{sech}(x) \sqrt{a \cosh (2 x)+a+2 b} \left(2 (a+b)^{3/2} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)-\sqrt{a} (2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)\right)}{\sqrt{a} \sqrt{a+b}}-\text{csch}^2(x) (a \cosh (2 x)+a+2 b)}{4 (a+b) \sqrt{a+b \text{sech}^2(x)}}","-\frac{\coth ^2(x) \sqrt{a+b \text{sech}^2(x)}}{2 (a+b)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{(2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a+b}}\right)}{2 (a+b)^{3/2}}",1,"(-((a + 2*b + a*Cosh[2*x])*Csch[x]^2) + (Sqrt[2]*Sqrt[a + 2*b + a*Cosh[2*x]]*(-(Sqrt[a]*(2*a + 3*b)*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]) + 2*(a + b)^(3/2)*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])*Sech[x])/(Sqrt[a]*Sqrt[a + b]))/(4*(a + b)*Sqrt[a + b*Sech[x]^2])","A",1
203,1,129,68,0.5312961,"\int \frac{\tanh ^5(x)}{\left(a+b \text{sech}^2(x)\right)^{3/2}} \, dx","Integrate[Tanh[x]^5/(a + b*Sech[x]^2)^(3/2),x]","\frac{\text{sech}^3(x) \left(\frac{\sqrt{2} (a \cosh (2 x)+a+2 b)^{3/2} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)}{a^{3/2}}-\frac{\text{sech}(x) (a \cosh (2 x)+a+2 b) \left(\left(2 a^2+2 a b+b^2\right) \cosh (2 x)+2 a^2+4 a b+b^2\right)}{a b^2}\right)}{4 \left(a+b \text{sech}^2(x)\right)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{(a+b)^2}{a b^2 \sqrt{a+b \text{sech}^2(x)}}-\frac{\sqrt{a+b \text{sech}^2(x)}}{b^2}",1,"(Sech[x]^3*((Sqrt[2]*(a + 2*b + a*Cosh[2*x])^(3/2)*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])/a^(3/2) - ((a + 2*b + a*Cosh[2*x])*(2*a^2 + 4*a*b + b^2 + (2*a^2 + 2*a*b + b^2)*Cosh[2*x])*Sech[x])/(a*b^2)))/(4*(a + b*Sech[x]^2)^(3/2))","A",1
204,1,169,86,0.4088694,"\int \frac{\tanh ^4(x)}{\left(a+b \text{sech}^2(x)\right)^{3/2}} \, dx","Integrate[Tanh[x]^4/(a + b*Sech[x]^2)^(3/2),x]","-\frac{\text{sech}^3(x) (a \cosh (2 x)+a+2 b) \left(\sqrt{a} \left(2 \sqrt{b} (a+b) \sinh (x)-\sqrt{2} a \sqrt{a \cosh (2 x)+a+2 b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)\right)-\sqrt{2} b^{3/2} \sqrt{a \cosh (2 x)+a+2 b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)\right)}{4 a^{3/2} b^{3/2} \left(a+b \text{sech}^2(x)\right)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{a^{3/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{b^{3/2}}-\frac{(a+b) \tanh (x)}{a b \sqrt{a-b \tanh ^2(x)+b}}",1,"-1/4*((a + 2*b + a*Cosh[2*x])*Sech[x]^3*(-(Sqrt[2]*b^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*Sqrt[a + 2*b + a*Cosh[2*x]]) + Sqrt[a]*(-(Sqrt[2]*a*ArcTan[(Sqrt[2]*Sqrt[b]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*Sqrt[a + 2*b + a*Cosh[2*x]]) + 2*Sqrt[b]*(a + b)*Sinh[x])))/(a^(3/2)*b^(3/2)*(a + b*Sech[x]^2)^(3/2))","A",1
205,1,103,49,0.2342268,"\int \frac{\tanh ^3(x)}{\left(a+b \text{sech}^2(x)\right)^{3/2}} \, dx","Integrate[Tanh[x]^3/(a + b*Sech[x]^2)^(3/2),x]","\frac{\text{sech}^3(x) \left(\sqrt{2} (a \cosh (2 x)+a+2 b)^{3/2} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)-\frac{2 \sqrt{a} (a+b) \cosh (x) (a \cosh (2 x)+a+2 b)}{b}\right)}{4 a^{3/2} \left(a+b \text{sech}^2(x)\right)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{a+b}{a b \sqrt{a+b \text{sech}^2(x)}}",1,"(((-2*Sqrt[a]*(a + b)*Cosh[x]*(a + 2*b + a*Cosh[2*x]))/b + Sqrt[2]*(a + 2*b + a*Cosh[2*x])^(3/2)*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])*Sech[x]^3)/(4*a^(3/2)*(a + b*Sech[x]^2)^(3/2))","B",1
206,1,128,51,0.7553199,"\int \frac{\tanh ^2(x)}{\left(a+b \text{sech}^2(x)\right)^{3/2}} \, dx","Integrate[Tanh[x]^2/(a + b*Sech[x]^2)^(3/2),x]","\frac{\text{sech}^2(x) (a \cosh (2 x)+a+2 b) \left(\text{sech}(x) \sinh ^{-1}\left(\frac{\sqrt{a} \sinh (x)}{\sqrt{a+b}}\right) (a \cosh (2 x)+a+2 b)-2 \sqrt{a} \sqrt{a+b} \tanh (x) \sqrt{\frac{a \sinh ^2(x)+a+b}{a+b}}\right)}{4 a^{3/2} \sqrt{a+b} \sqrt{\frac{a \sinh ^2(x)+a+b}{a+b}} \left(a+b \text{sech}^2(x)\right)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{a^{3/2}}-\frac{\tanh (x)}{a \sqrt{a-b \tanh ^2(x)+b}}",1,"((a + 2*b + a*Cosh[2*x])*Sech[x]^2*(ArcSinh[(Sqrt[a]*Sinh[x])/Sqrt[a + b]]*(a + 2*b + a*Cosh[2*x])*Sech[x] - 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(a + b + a*Sinh[x]^2)/(a + b)]*Tanh[x]))/(4*a^(3/2)*Sqrt[a + b]*(a + b*Sech[x]^2)^(3/2)*Sqrt[(a + b + a*Sinh[x]^2)/(a + b)])","B",1
207,1,98,43,0.3959224,"\int \frac{\tanh (x)}{\left(a+b \text{sech}^2(x)\right)^{3/2}} \, dx","Integrate[Tanh[x]/(a + b*Sech[x]^2)^(3/2),x]","-\frac{\text{sech}^3(x) (a \cosh (2 x)+a+2 b) \left(2 \sqrt{a} \cosh (x)-\sqrt{2} \sqrt{a \cosh (2 x)+a+2 b} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)\right)}{4 a^{3/2} \left(a+b \text{sech}^2(x)\right)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{1}{a \sqrt{a+b \text{sech}^2(x)}}",1,"-1/4*((a + 2*b + a*Cosh[2*x])*(2*Sqrt[a]*Cosh[x] - Sqrt[2]*Sqrt[a + 2*b + a*Cosh[2*x]]*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])*Sech[x]^3)/(a^(3/2)*(a + b*Sech[x]^2)^(3/2))","B",1
208,1,107,57,0.8408901,"\int \frac{1}{\left(a+b \text{sech}^2(x)\right)^{3/2}} \, dx","Integrate[(a + b*Sech[x]^2)^(-3/2),x]","\frac{\text{sech}^3(x) (a \cosh (2 x)+a+2 b) \left((a+b)^{3/2} \sinh ^{-1}\left(\frac{\sqrt{a} \sinh (x)}{\sqrt{a+b}}\right) \sqrt{\frac{a \cosh (2 x)+a+2 b}{a+b}}-\sqrt{2} \sqrt{a} b \sinh (x)\right)}{2 \sqrt{2} a^{3/2} (a+b) \left(a+b \text{sech}^2(x)\right)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{a^{3/2}}-\frac{b \tanh (x)}{a (a+b) \sqrt{a-b \tanh ^2(x)+b}}",1,"((a + 2*b + a*Cosh[2*x])*Sech[x]^3*((a + b)^(3/2)*ArcSinh[(Sqrt[a]*Sinh[x])/Sqrt[a + b]]*Sqrt[(a + 2*b + a*Cosh[2*x])/(a + b)] - Sqrt[2]*Sqrt[a]*b*Sinh[x]))/(2*Sqrt[2]*a^(3/2)*(a + b)*(a + b*Sech[x]^2)^(3/2))","A",1
209,1,155,79,0.6856541,"\int \frac{\coth (x)}{\left(a+b \text{sech}^2(x)\right)^{3/2}} \, dx","Integrate[Coth[x]/(a + b*Sech[x]^2)^(3/2),x]","\frac{\text{sech}^2(x) \left(\frac{\sqrt{2} \text{sech}(x) (a \cosh (2 x)+a+2 b)^{3/2} \left((a+b)^{3/2} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)-a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)\right)}{\sqrt{a} \sqrt{a+b}}-2 b (a \cosh (2 x)+a+2 b)\right)}{4 a (a+b) \left(a+b \text{sech}^2(x)\right)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{b}{a (a+b) \sqrt{a+b \text{sech}^2(x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}",1,"(Sech[x]^2*(-2*b*(a + 2*b + a*Cosh[2*x]) + (Sqrt[2]*(a + 2*b + a*Cosh[2*x])^(3/2)*(-(a^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]) + (a + b)^(3/2)*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])*Sech[x])/(Sqrt[a]*Sqrt[a + b])))/(4*a*(a + b)*(a + b*Sech[x]^2)^(3/2))","A",1
210,1,120,88,0.457906,"\int \frac{\coth ^2(x)}{\left(a+b \text{sech}^2(x)\right)^{3/2}} \, dx","Integrate[Coth[x]^2/(a + b*Sech[x]^2)^(3/2),x]","\frac{\text{sech}^3(x) \left(\frac{\sqrt{2} (a \cosh (2 x)+a+2 b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)}{a^{3/2}}-\frac{(a \cosh (2 x)+a+2 b) \left(a \text{csch}(x) (a \cosh (2 x)+a+2 b)+2 b^2 \sinh (x)\right)}{a (a+b)^2}\right)}{4 \left(a+b \text{sech}^2(x)\right)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{a^{3/2}}-\frac{(a-b) \coth (x) \sqrt{a-b \tanh ^2(x)+b}}{a (a+b)^2}-\frac{b \coth (x)}{a (a+b) \sqrt{a-b \tanh ^2(x)+b}}",1,"(Sech[x]^3*((Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*(a + 2*b + a*Cosh[2*x])^(3/2))/a^(3/2) - ((a + 2*b + a*Cosh[2*x])*(a*(a + 2*b + a*Cosh[2*x])*Csch[x] + 2*b^2*Sinh[x]))/(a*(a + b)^2)))/(4*(a + b*Sech[x]^2)^(3/2))","A",1
211,1,178,118,0.747343,"\int \frac{\tanh ^6(x)}{\left(a+b \text{sech}^2(x)\right)^{5/2}} \, dx","Integrate[Tanh[x]^6/(a + b*Sech[x]^2)^(5/2),x]","\frac{\text{sech}^5(x) \left(\frac{\sqrt{2} (a \cosh (2 x)+a+2 b)^{5/2} \left(b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)-a^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)\right)}{a^{5/2} b^{5/2}}+\frac{2 (a+b) \sinh (x) \left(3 a^2+a (3 a-4 b) \cosh (2 x)+4 a b-6 b^2\right) (a \cosh (2 x)+a+2 b)}{3 a^2 b^2}\right)}{8 \left(a+b \text{sech}^2(x)\right)^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{a^{5/2}}-\frac{\left(\frac{1}{a^2}-\frac{1}{b^2}\right) \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}-\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{b^{5/2}}-\frac{(a+b) \tanh ^3(x)}{3 a b \left(a-b \tanh ^2(x)+b\right)^{3/2}}",1,"(Sech[x]^5*((Sqrt[2]*(-(a^(5/2)*ArcTan[(Sqrt[2]*Sqrt[b]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]) + b^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]])*(a + 2*b + a*Cosh[2*x])^(5/2))/(a^(5/2)*b^(5/2)) + (2*(a + b)*(a + 2*b + a*Cosh[2*x])*(3*a^2 + 4*a*b - 6*b^2 + a*(3*a - 4*b)*Cosh[2*x])*Sinh[x])/(3*a^2*b^2)))/(8*(a + b*Sech[x]^2)^(5/2))","A",1
212,1,126,76,0.563591,"\int \frac{\tanh ^5(x)}{\left(a+b \text{sech}^2(x)\right)^{5/2}} \, dx","Integrate[Tanh[x]^5/(a + b*Sech[x]^2)^(5/2),x]","\frac{\text{sech}^5(x) \left(\frac{\sqrt{2} (a \cosh (2 x)+a+2 b)^{5/2} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)}{a^{5/2}}+\frac{4 (a+b) \cosh (x) \left(a^2+a (a-2 b) \cosh (2 x)+a b-3 b^2\right) (a \cosh (2 x)+a+2 b)}{3 a^2 b^2}\right)}{8 \left(a+b \text{sech}^2(x)\right)^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)}{a^{5/2}}-\frac{\frac{1}{a^2}-\frac{1}{b^2}}{\sqrt{a+b \text{sech}^2(x)}}-\frac{(a+b)^2}{3 a b^2 \left(a+b \text{sech}^2(x)\right)^{3/2}}",1,"(((4*(a + b)*Cosh[x]*(a + 2*b + a*Cosh[2*x])*(a^2 + a*b - 3*b^2 + a*(a - 2*b)*Cosh[2*x]))/(3*a^2*b^2) + (Sqrt[2]*(a + 2*b + a*Cosh[2*x])^(5/2)*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])/a^(5/2))*Sech[x]^5)/(8*(a + b*Sech[x]^2)^(5/2))","A",1
213,1,290,90,2.1680589,"\int \frac{\tanh ^4(x)}{\left(a+b \text{sech}^2(x)\right)^{5/2}} \, dx","Integrate[Tanh[x]^4/(a + b*Sech[x]^2)^(5/2),x]","\frac{\text{sech}^4(x) \left(\frac{\sqrt{2} \text{csch}(x) \text{sech}(x) \left(-\frac{16 \left(a \sinh ^2(x)+a+b\right) \left(\frac{a \sinh ^2(x)}{a+b}+1\right) \left(\frac{a^2 (a+b) \sinh ^4(x)}{\left(a \sinh ^2(x)+a+b\right)^2}+\frac{3 a (a+b) \sinh ^2(x)}{a \sinh ^2(x)+a+b}-\frac{3 \sqrt{a} \sqrt{a+b} \sinh (x) \sinh ^{-1}\left(\frac{\sqrt{a} \sinh (x)}{\sqrt{a+b}}\right)}{\sqrt{\frac{a \sinh ^2(x)+a+b}{a+b}}}\right)}{a^3}+\frac{12 \sinh ^4(x)}{a+b}+\frac{2 \sinh ^2(x) \left(a \sinh ^2(x)+a+b\right)}{(a+b)^2}+\frac{\sinh ^2(x)}{a+b}\right) (a \cosh (2 x)+a+2 b)^{5/2}}{\left(a \sinh ^2(x)+a+b\right)^{3/2}}+\frac{8 \tanh (x) (a \cosh (2 x)+2 a+3 b) (a \cosh (2 x)+a+2 b)}{(a+b)^2}-\frac{12 \tanh (x) ((3 a+2 b) \cosh (2 x)+b) (a \cosh (2 x)+a+2 b)}{(a+b)^2}\right)}{384 \left(a+b \text{sech}^2(x)\right)^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{a^{5/2}}+\frac{(a-3 b) \tanh (x)}{3 a^2 b \sqrt{a-b \tanh ^2(x)+b}}-\frac{(a+b) \tanh (x)}{3 a b \left(a-b \tanh ^2(x)+b\right)^{3/2}}",1,"(Sech[x]^4*((Sqrt[2]*(a + 2*b + a*Cosh[2*x])^(5/2)*Csch[x]*Sech[x]*(Sinh[x]^2/(a + b) + (12*Sinh[x]^4)/(a + b) + (2*Sinh[x]^2*(a + b + a*Sinh[x]^2))/(a + b)^2 - (16*(a + b + a*Sinh[x]^2)*(1 + (a*Sinh[x]^2)/(a + b))*((a^2*(a + b)*Sinh[x]^4)/(a + b + a*Sinh[x]^2)^2 + (3*a*(a + b)*Sinh[x]^2)/(a + b + a*Sinh[x]^2) - (3*Sqrt[a]*Sqrt[a + b]*ArcSinh[(Sqrt[a]*Sinh[x])/Sqrt[a + b]]*Sinh[x])/Sqrt[(a + b + a*Sinh[x]^2)/(a + b)]))/a^3))/(a + b + a*Sinh[x]^2)^(3/2) + (8*(a + 2*b + a*Cosh[2*x])*(2*a + 3*b + a*Cosh[2*x])*Tanh[x])/(a + b)^2 - (12*(a + 2*b + a*Cosh[2*x])*(b + (3*a + 2*b)*Cosh[2*x])*Tanh[x])/(a + b)^2))/(384*(a + b*Sech[x]^2)^(5/2))","B",1
214,1,124,68,0.9989145,"\int \frac{\tanh ^3(x)}{\left(a+b \text{sech}^2(x)\right)^{5/2}} \, dx","Integrate[Tanh[x]^3/(a + b*Sech[x]^2)^(5/2),x]","-\frac{\text{sech}^5(x) (a \cosh (2 x)+a+2 b) \left(a^{3/2} (a+4 b) \cosh (3 x)+3 \sqrt{a} (a+2 b)^2 \cosh (x)-3 \sqrt{2} b (a \cosh (2 x)+a+2 b)^{3/2} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)\right)}{24 a^{5/2} b \left(a+b \text{sech}^2(x)\right)^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)}{a^{5/2}}-\frac{1}{a^2 \sqrt{a+b \text{sech}^2(x)}}-\frac{a+b}{3 a b \left(a+b \text{sech}^2(x)\right)^{3/2}}",1,"-1/24*((a + 2*b + a*Cosh[2*x])*(3*Sqrt[a]*(a + 2*b)^2*Cosh[x] + a^(3/2)*(a + 4*b)*Cosh[3*x] - 3*Sqrt[2]*b*(a + 2*b + a*Cosh[2*x])^(3/2)*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])*Sech[x]^5)/(a^(5/2)*b*(a + b*Sech[x]^2)^(5/2))","A",1
215,1,290,88,1.1985954,"\int \frac{\tanh ^2(x)}{\left(a+b \text{sech}^2(x)\right)^{5/2}} \, dx","Integrate[Tanh[x]^2/(a + b*Sech[x]^2)^(5/2),x]","\frac{\text{sech}^4(x) \left(\frac{\sqrt{2} \text{csch}(x) \text{sech}(x) \left(-\frac{16 \left(a \sinh ^2(x)+a+b\right) \left(\frac{a \sinh ^2(x)}{a+b}+1\right) \left(\frac{a^2 (a+b) \sinh ^4(x)}{\left(a \sinh ^2(x)+a+b\right)^2}+\frac{3 a (a+b) \sinh ^2(x)}{a \sinh ^2(x)+a+b}-\frac{3 \sqrt{a} \sqrt{a+b} \sinh (x) \sinh ^{-1}\left(\frac{\sqrt{a} \sinh (x)}{\sqrt{a+b}}\right)}{\sqrt{\frac{a \sinh ^2(x)+a+b}{a+b}}}\right)}{a^3}+\frac{12 \sinh ^4(x)}{a+b}+\frac{2 \sinh ^2(x) \left(a \sinh ^2(x)+a+b\right)}{(a+b)^2}+\frac{\sinh ^2(x)}{a+b}\right) (a \cosh (2 x)+a+2 b)^{5/2}}{\left(a \sinh ^2(x)+a+b\right)^{3/2}}-\frac{8 \tanh (x) (a \cosh (2 x)+2 a+3 b) (a \cosh (2 x)+a+2 b)}{(a+b)^2}+\frac{4 \tanh (x) ((3 a+2 b) \cosh (2 x)+b) (a \cosh (2 x)+a+2 b)}{(a+b)^2}\right)}{384 \left(a+b \text{sech}^2(x)\right)^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{a^{5/2}}-\frac{(2 a+3 b) \tanh (x)}{3 a^2 (a+b) \sqrt{a-b \tanh ^2(x)+b}}-\frac{\tanh (x)}{3 a \left(a-b \tanh ^2(x)+b\right)^{3/2}}",1,"(Sech[x]^4*((Sqrt[2]*(a + 2*b + a*Cosh[2*x])^(5/2)*Csch[x]*Sech[x]*(Sinh[x]^2/(a + b) + (12*Sinh[x]^4)/(a + b) + (2*Sinh[x]^2*(a + b + a*Sinh[x]^2))/(a + b)^2 - (16*(a + b + a*Sinh[x]^2)*(1 + (a*Sinh[x]^2)/(a + b))*((a^2*(a + b)*Sinh[x]^4)/(a + b + a*Sinh[x]^2)^2 + (3*a*(a + b)*Sinh[x]^2)/(a + b + a*Sinh[x]^2) - (3*Sqrt[a]*Sqrt[a + b]*ArcSinh[(Sqrt[a]*Sinh[x])/Sqrt[a + b]]*Sinh[x])/Sqrt[(a + b + a*Sinh[x]^2)/(a + b)]))/a^3))/(a + b + a*Sinh[x]^2)^(3/2) - (8*(a + 2*b + a*Cosh[2*x])*(2*a + 3*b + a*Cosh[2*x])*Tanh[x])/(a + b)^2 + (4*(a + 2*b + a*Cosh[2*x])*(b + (3*a + 2*b)*Cosh[2*x])*Tanh[x])/(a + b)^2))/(384*(a + b*Sech[x]^2)^(5/2))","B",1
216,1,112,62,0.71751,"\int \frac{\tanh (x)}{\left(a+b \text{sech}^2(x)\right)^{5/2}} \, dx","Integrate[Tanh[x]/(a + b*Sech[x]^2)^(5/2),x]","-\frac{\text{sech}^5(x) (a \cosh (2 x)+a+2 b) \left(4 a^{3/2} \cosh (3 x)+12 \sqrt{a} (a+b) \cosh (x)-3 \sqrt{2} (a \cosh (2 x)+a+2 b)^{3/2} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)\right)}{24 a^{5/2} \left(a+b \text{sech}^2(x)\right)^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)}{a^{5/2}}-\frac{1}{a^2 \sqrt{a+b \text{sech}^2(x)}}-\frac{1}{3 a \left(a+b \text{sech}^2(x)\right)^{3/2}}",1,"-1/24*((a + 2*b + a*Cosh[2*x])*(12*Sqrt[a]*(a + b)*Cosh[x] + 4*a^(3/2)*Cosh[3*x] - 3*Sqrt[2]*(a + 2*b + a*Cosh[2*x])^(3/2)*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])*Sech[x]^5)/(a^(5/2)*(a + b*Sech[x]^2)^(5/2))","A",1
217,1,130,95,0.4512521,"\int \frac{1}{\left(a+b \text{sech}^2(x)\right)^{5/2}} \, dx","Integrate[(a + b*Sech[x]^2)^(-5/2),x]","\frac{\text{sech}^5(x) \left(\frac{\sqrt{2} (a \cosh (2 x)+a+2 b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)}{a^{5/2}}-\frac{4 b \sinh (x) (a \cosh (2 x)+a+2 b) \left(3 a^2+a (3 a+2 b) \cosh (2 x)+7 a b+3 b^2\right)}{3 a^2 (a+b)^2}\right)}{8 \left(a+b \text{sech}^2(x)\right)^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{a^{5/2}}-\frac{b (5 a+3 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt{a-b \tanh ^2(x)+b}}-\frac{b \tanh (x)}{3 a (a+b) \left(a-b \tanh ^2(x)+b\right)^{3/2}}",1,"(Sech[x]^5*((Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*(a + 2*b + a*Cosh[2*x])^(5/2))/a^(5/2) - (4*b*(a + 2*b + a*Cosh[2*x])*(3*a^2 + 7*a*b + 3*b^2 + a*(3*a + 2*b)*Cosh[2*x])*Sinh[x])/(3*a^2*(a + b)^2)))/(8*(a + b*Sech[x]^2)^(5/2))","A",1
218,1,242,109,1.1137245,"\int \frac{\coth (x)}{\left(a+b \text{sech}^2(x)\right)^{5/2}} \, dx","Integrate[Coth[x]/(a + b*Sech[x]^2)^(5/2),x]","\frac{\text{sech}^5(x) \left(-\frac{2 b \cosh (x) \left(7 a^2+a (7 a+4 b) \cosh (2 x)+16 a b+6 b^2\right) (a \cosh (2 x)+a+2 b)}{3 a^2 (a+b)^2}-\frac{(a \cosh (2 x)+a+2 b)^{5/2} \left(\sqrt{a} \left(a^2-2 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)+(a+b)^2 \left(\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{2 a+2 b} \cosh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)-2 \sqrt{a+b} \log \left(\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right)\right)\right)}{\sqrt{2} a^{5/2} (a+b)^{5/2}}\right)}{8 \left(a+b \text{sech}^2(x)\right)^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right)}{a^{5/2}}-\frac{b (2 a+b)}{a^2 (a+b)^2 \sqrt{a+b \text{sech}^2(x)}}-\frac{b}{3 a (a+b) \left(a+b \text{sech}^2(x)\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}",1,"(((-2*b*Cosh[x]*(a + 2*b + a*Cosh[2*x])*(7*a^2 + 16*a*b + 6*b^2 + a*(7*a + 4*b)*Cosh[2*x]))/(3*a^2*(a + b)^2) - ((a + 2*b + a*Cosh[2*x])^(5/2)*(Sqrt[a]*(a^2 - 2*a*b - b^2)*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]] + (a + b)^2*(Sqrt[a]*ArcTanh[(Sqrt[2*a + 2*b]*Cosh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]] - 2*Sqrt[a + b]*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])))/(Sqrt[2]*a^(5/2)*(a + b)^(5/2)))*Sech[x]^5)/(8*(a + b*Sech[x]^2)^(5/2))","B",1
219,1,155,133,0.8290424,"\int \frac{\coth ^2(x)}{\left(a+b \text{sech}^2(x)\right)^{5/2}} \, dx","Integrate[Coth[x]^2/(a + b*Sech[x]^2)^(5/2),x]","\frac{\text{sech}^5(x) \left(\frac{\sqrt{2} (a \cosh (2 x)+a+2 b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right)}{a^{5/2}}-\frac{(a \cosh (2 x)+a+2 b) \left(3 a^2 \text{csch}(x) (a \cosh (2 x)+a+2 b)^2-4 b^3 (a+b) \sinh (x)+2 b^2 (9 a+4 b) \sinh (x) (a \cosh (2 x)+a+2 b)\right)}{3 a^2 (a+b)^3}\right)}{8 \left(a+b \text{sech}^2(x)\right)^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right)}{a^{5/2}}-\frac{(a-3 b) (3 a+b) \coth (x) \sqrt{a-b \tanh ^2(x)+b}}{3 a^2 (a+b)^3}-\frac{b (7 a+3 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt{a-b \tanh ^2(x)+b}}-\frac{b \coth (x)}{3 a (a+b) \left(a-b \tanh ^2(x)+b\right)^{3/2}}",1,"(Sech[x]^5*((Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*(a + 2*b + a*Cosh[2*x])^(5/2))/a^(5/2) - ((a + 2*b + a*Cosh[2*x])*(3*a^2*(a + 2*b + a*Cosh[2*x])^2*Csch[x] - 4*b^3*(a + b)*Sinh[x] + 2*b^2*(9*a + 4*b)*(a + 2*b + a*Cosh[2*x])*Sinh[x]))/(3*a^2*(a + b)^3)))/(8*(a + b*Sech[x]^2)^(5/2))","A",1
220,1,330,183,7.9845939,"\int \frac{1}{\left(a+b \text{sech}^2(c+d x)\right)^{7/2}} \, dx","Integrate[(a + b*Sech[c + d*x]^2)^(-7/2),x]","\frac{\text{sech}^7(c+d x) \left(\frac{15}{4} e^{-7 (c+d x)} \left(a \left(e^{2 (c+d x)}+1\right)^2+4 b e^{2 (c+d x)}\right)^{7/2} \left(\tanh ^{-1}\left(\frac{a e^{2 (c+d x)}+a+2 b}{\sqrt{a} \sqrt{a \left(e^{2 (c+d x)}+1\right)^2+4 b e^{2 (c+d x)}}}\right)-\tanh ^{-1}\left(\frac{a e^{2 (c+d x)}+a+2 b e^{2 (c+d x)}}{\sqrt{a} \sqrt{a \left(e^{2 (c+d x)}+1\right)^2+4 b e^{2 (c+d x)}}}\right)\right)-\frac{4 \sqrt{a} b \sinh (c+d x) (a \cosh (2 (c+d x))+a+2 b) \left(135 a^4+480 a^3 b+a^2 \left(45 a^2+60 a b+23 b^2\right) \cosh (4 (c+d x))+709 a^2 b^2+4 a \left(45 a^3+135 a^2 b+117 a b^2+35 b^3\right) \cosh (2 (c+d x))+460 a b^3+120 b^4\right)}{(a+b)^3}\right)}{960 a^{7/2} d \left(a+b \text{sech}^2(c+d x)\right)^{7/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a-b \tanh ^2(c+d x)+b}}\right)}{a^{7/2} d}-\frac{b (9 a+5 b) \tanh (c+d x)}{15 a^2 d (a+b)^2 \left(a-b \tanh ^2(c+d x)+b\right)^{3/2}}-\frac{b \left(33 a^2+40 a b+15 b^2\right) \tanh (c+d x)}{15 a^3 d (a+b)^3 \sqrt{a-b \tanh ^2(c+d x)+b}}-\frac{b \tanh (c+d x)}{5 a d (a+b) \left(a-b \tanh ^2(c+d x)+b\right)^{5/2}}",1,"(Sech[c + d*x]^7*((15*(4*b*E^(2*(c + d*x)) + a*(1 + E^(2*(c + d*x)))^2)^(7/2)*(ArcTanh[(a + 2*b + a*E^(2*(c + d*x)))/(Sqrt[a]*Sqrt[4*b*E^(2*(c + d*x)) + a*(1 + E^(2*(c + d*x)))^2])] - ArcTanh[(a + a*E^(2*(c + d*x)) + 2*b*E^(2*(c + d*x)))/(Sqrt[a]*Sqrt[4*b*E^(2*(c + d*x)) + a*(1 + E^(2*(c + d*x)))^2])]))/(4*E^(7*(c + d*x))) - (4*Sqrt[a]*b*(a + 2*b + a*Cosh[2*(c + d*x)])*(135*a^4 + 480*a^3*b + 709*a^2*b^2 + 460*a*b^3 + 120*b^4 + 4*a*(45*a^3 + 135*a^2*b + 117*a*b^2 + 35*b^3)*Cosh[2*(c + d*x)] + a^2*(45*a^2 + 60*a*b + 23*b^2)*Cosh[4*(c + d*x)])*Sinh[c + d*x])/(a + b)^3))/(960*a^(7/2)*d*(a + b*Sech[c + d*x]^2)^(7/2))","A",1