1,1,68,89,0.1080112,"\int \sinh ^4(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^2),x]","\frac{(45 b-48 a) \sinh (2 (c+d x))+(6 a-9 b) \sinh (4 (c+d x))+72 a c+72 a d x+b \sinh (6 (c+d x))-60 b c-60 b d x}{192 d}","\frac{(6 a-5 b) \sinh ^3(c+d x) \cosh (c+d x)}{24 d}-\frac{(6 a-5 b) \sinh (c+d x) \cosh (c+d x)}{16 d}+\frac{1}{16} x (6 a-5 b)+\frac{b \sinh ^5(c+d x) \cosh (c+d x)}{6 d}",1,"(72*a*c - 60*b*c + 72*a*d*x - 60*b*d*x + (-48*a + 45*b)*Sinh[2*(c + d*x)] + (6*a - 9*b)*Sinh[4*(c + d*x)] + b*Sinh[6*(c + d*x)])/(192*d)","A",1
2,1,77,53,0.0247816,"\int \sinh ^3(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^2),x]","-\frac{3 a \cosh (c+d x)}{4 d}+\frac{a \cosh (3 (c+d x))}{12 d}+\frac{5 b \cosh (c+d x)}{8 d}-\frac{5 b \cosh (3 (c+d x))}{48 d}+\frac{b \cosh (5 (c+d x))}{80 d}","\frac{(a-2 b) \cosh ^3(c+d x)}{3 d}-\frac{(a-b) \cosh (c+d x)}{d}+\frac{b \cosh ^5(c+d x)}{5 d}",1,"(-3*a*Cosh[c + d*x])/(4*d) + (5*b*Cosh[c + d*x])/(8*d) + (a*Cosh[3*(c + d*x)])/(12*d) - (5*b*Cosh[3*(c + d*x)])/(48*d) + (b*Cosh[5*(c + d*x)])/(80*d)","A",1
3,1,47,61,0.086095,"\int \sinh ^2(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^2),x]","\frac{-4 (4 a-3 b) (c+d x)+8 (a-b) \sinh (2 (c+d x))+b \sinh (4 (c+d x))}{32 d}","\frac{(4 a-3 b) \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac{1}{8} x (4 a-3 b)+\frac{b \sinh ^3(c+d x) \cosh (c+d x)}{4 d}",1,"(-4*(4*a - 3*b)*(c + d*x) + 8*(a - b)*Sinh[2*(c + d*x)] + b*Sinh[4*(c + d*x)])/(32*d)","A",1
4,1,53,32,0.0243713,"\int \sinh (c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2),x]","\frac{a \sinh (c) \sinh (d x)}{d}+\frac{a \cosh (c) \cosh (d x)}{d}-\frac{3 b \cosh (c+d x)}{4 d}+\frac{b \cosh (3 (c+d x))}{12 d}","\frac{(a-b) \cosh (c+d x)}{d}+\frac{b \cosh ^3(c+d x)}{3 d}",1,"(a*Cosh[c]*Cosh[d*x])/d - (3*b*Cosh[c + d*x])/(4*d) + (b*Cosh[3*(c + d*x)])/(12*d) + (a*Sinh[c]*Sinh[d*x])/d","A",1
5,1,36,30,0.0294135,"\int \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[a + b*Sinh[c + d*x]^2,x]","a x+\frac{b (-c-d x)}{2 d}+\frac{b \sinh (2 (c+d x))}{4 d}","a x+\frac{b \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b x}{2}",1,"a*x + (b*(-c - d*x))/(2*d) + (b*Sinh[2*(c + d*x)])/(4*d)","A",1
6,1,62,25,0.0305001,"\int \text{csch}(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Csch[c + d*x]*(a + b*Sinh[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 \sinh (c) \sinh (d x)}{d}+\frac{b \cosh (c) \cosh (d x)}{d}","\frac{b \cosh (c+d x)}{d}-\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}",1,"(b*Cosh[c]*Cosh[d*x])/d - (a*Log[Cosh[c/2 + (d*x)/2]])/d + (a*Log[Sinh[c/2 + (d*x)/2]])/d + (b*Sinh[c]*Sinh[d*x])/d","B",1
7,1,16,16,0.0189649,"\int \text{csch}^2(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^2),x]","b x-\frac{a \coth (c+d x)}{d}","b x-\frac{a \coth (c+d x)}{d}",1,"b*x - (a*Coth[c + d*x])/d","A",1
8,1,99,40,0.0320679,"\int \text{csch}^3(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^3*(a + b*Sinh[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 \log \left(\sinh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{b \log \left(\cosh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}","\frac{(a-2 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}",1,"-1/8*(a*Csch[(c + d*x)/2]^2)/d - (b*Log[Cosh[c/2 + (d*x)/2]])/d + (b*Log[Sinh[c/2 + (d*x)/2]])/d - (a*Log[Tanh[(c + d*x)/2]])/(2*d) - (a*Sech[(c + d*x)/2]^2)/(8*d)","B",1
9,1,49,43,0.0289333,"\int \text{csch}^4(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^4*(a + b*Sinh[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 \coth (c+d x)}{d}","\frac{(2 a-3 b) \coth (c+d x)}{3 d}-\frac{a \coth (c+d x) \text{csch}^2(c+d x)}{3 d}",1,"(2*a*Coth[c + d*x])/(3*d) - (b*Coth[c + d*x])/d - (a*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d)","A",1
10,1,133,146,0.2048676,"\int \sinh ^4(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^2,x]","\frac{-96 \left(8 a^2-15 a b+7 b^2\right) \sinh (2 (c+d x))+24 \left(4 a^2-12 a b+7 b^2\right) \sinh (4 (c+d x))+1152 a^2 c+1152 a^2 d x+32 a b \sinh (6 (c+d x))-1920 a b c-1920 a b d x-32 b^2 \sinh (6 (c+d x))+3 b^2 \sinh (8 (c+d x))+840 b^2 c+840 b^2 d x}{3072 d}","\frac{\left(48 a^2-208 a b+139 b^2\right) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}-\frac{\left(80 a^2-176 a b+93 b^2\right) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{1}{128} x \left(48 a^2-80 a b+35 b^2\right)+\frac{b (16 a-13 b) \sinh (c+d x) \cosh ^5(c+d x)}{48 d}+\frac{b^2 \sinh ^5(c+d x) \cosh ^3(c+d x)}{8 d}",1,"(1152*a^2*c - 1920*a*b*c + 840*b^2*c + 1152*a^2*d*x - 1920*a*b*d*x + 840*b^2*d*x - 96*(8*a^2 - 15*a*b + 7*b^2)*Sinh[2*(c + d*x)] + 24*(4*a^2 - 12*a*b + 7*b^2)*Sinh[4*(c + d*x)] + 32*a*b*Sinh[6*(c + d*x)] - 32*b^2*Sinh[6*(c + d*x)] + 3*b^2*Sinh[8*(c + d*x)])/(3072*d)","A",1
11,1,154,85,0.03873,"\int \sinh ^3(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^2,x]","-\frac{3 a^2 \cosh (c+d x)}{4 d}+\frac{a^2 \cosh (3 (c+d x))}{12 d}+\frac{5 a b \cosh (c+d x)}{4 d}-\frac{5 a b \cosh (3 (c+d x))}{24 d}+\frac{a b \cosh (5 (c+d x))}{40 d}-\frac{35 b^2 \cosh (c+d x)}{64 d}+\frac{7 b^2 \cosh (3 (c+d x))}{64 d}-\frac{7 b^2 \cosh (5 (c+d x))}{320 d}+\frac{b^2 \cosh (7 (c+d x))}{448 d}","\frac{b (2 a-3 b) \cosh ^5(c+d x)}{5 d}+\frac{(a-3 b) (a-b) \cosh ^3(c+d x)}{3 d}-\frac{(a-b)^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh ^7(c+d x)}{7 d}",1,"(-3*a^2*Cosh[c + d*x])/(4*d) + (5*a*b*Cosh[c + d*x])/(4*d) - (35*b^2*Cosh[c + d*x])/(64*d) + (a^2*Cosh[3*(c + d*x)])/(12*d) - (5*a*b*Cosh[3*(c + d*x)])/(24*d) + (7*b^2*Cosh[3*(c + d*x)])/(64*d) + (a*b*Cosh[5*(c + d*x)])/(40*d) - (7*b^2*Cosh[5*(c + d*x)])/(320*d) + (b^2*Cosh[7*(c + d*x)])/(448*d)","A",1
12,1,99,110,0.1857958,"\int \sinh ^2(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\left(48 a^2-96 a b+45 b^2\right) \sinh (2 (c+d x))-96 a^2 c-96 a^2 d x+3 b (4 a-3 b) \sinh (4 (c+d x))+144 a b c+144 a b d x+b^2 \sinh (6 (c+d x))-60 b^2 c-60 b^2 d x}{192 d}","\frac{\left(8 a^2-20 a b+11 b^2\right) \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac{1}{16} x \left(8 a^2-12 a b+5 b^2\right)+\frac{b (4 a-3 b) \sinh (c+d x) \cosh ^3(c+d x)}{8 d}+\frac{b^2 \sinh ^3(c+d x) \cosh ^3(c+d x)}{6 d}",1,"(-96*a^2*c + 144*a*b*c - 60*b^2*c - 96*a^2*d*x + 144*a*b*d*x - 60*b^2*d*x + (48*a^2 - 96*a*b + 45*b^2)*Sinh[2*(c + d*x)] + 3*(4*a - 3*b)*b*Sinh[4*(c + d*x)] + b^2*Sinh[6*(c + d*x)])/(192*d)","A",1
13,1,111,57,0.0369881,"\int \sinh (c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2)^2,x]","\frac{a^2 \sinh (c) \sinh (d x)}{d}+\frac{a^2 \cosh (c) \cosh (d x)}{d}-\frac{3 a b \cosh (c+d x)}{2 d}+\frac{a b \cosh (3 (c+d x))}{6 d}+\frac{5 b^2 \cosh (c+d x)}{8 d}-\frac{5 b^2 \cosh (3 (c+d x))}{48 d}+\frac{b^2 \cosh (5 (c+d x))}{80 d}","\frac{2 b (a-b) \cosh ^3(c+d x)}{3 d}+\frac{(a-b)^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh ^5(c+d x)}{5 d}",1,"(a^2*Cosh[c]*Cosh[d*x])/d - (3*a*b*Cosh[c + d*x])/(2*d) + (5*b^2*Cosh[c + d*x])/(8*d) + (a*b*Cosh[3*(c + d*x)])/(6*d) - (5*b^2*Cosh[3*(c + d*x)])/(48*d) + (b^2*Cosh[5*(c + d*x)])/(80*d) + (a^2*Sinh[c]*Sinh[d*x])/d","A",1
14,1,60,72,0.1264778,"\int \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[(a + b*Sinh[c + d*x]^2)^2,x]","\frac{4 \left(8 a^2-8 a b+3 b^2\right) (c+d x)+8 b (2 a-b) \sinh (2 (c+d x))+b^2 \sinh (4 (c+d x))}{32 d}","\frac{1}{8} x \left(8 a^2-8 a b+3 b^2\right)+\frac{b (8 a-3 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{b^2 \sinh ^3(c+d x) \cosh (c+d x)}{4 d}",1,"(4*(8*a^2 - 8*a*b + 3*b^2)*(c + d*x) + 8*(2*a - b)*b*Sinh[2*(c + d*x)] + b^2*Sinh[4*(c + d*x)])/(32*d)","A",1
15,1,104,52,0.0336044,"\int \text{csch}(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]*(a + b*Sinh[c + d*x]^2)^2,x]","\frac{a^2 \log \left(\sinh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{a^2 \log \left(\cosh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}+\frac{2 a b \sinh (c) \sinh (d x)}{d}+\frac{2 a b \cosh (c) \cosh (d x)}{d}-\frac{3 b^2 \cosh (c+d x)}{4 d}+\frac{b^2 \cosh (3 (c+d x))}{12 d}","-\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b (2 a-b) \cosh (c+d x)}{d}+\frac{b^2 \cosh ^3(c+d x)}{3 d}",1,"(2*a*b*Cosh[c]*Cosh[d*x])/d - (3*b^2*Cosh[c + d*x])/(4*d) + (b^2*Cosh[3*(c + d*x)])/(12*d) - (a^2*Log[Cosh[c/2 + (d*x)/2]])/d + (a^2*Log[Sinh[c/2 + (d*x)/2]])/d + (2*a*b*Sinh[c]*Sinh[d*x])/d","A",1
16,1,56,50,0.1637553,"\int \text{csch}^2(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^2,x]","-\frac{a^2 \coth (c+d x)}{d}+2 a b x+\frac{b^2 (-c-d x)}{2 d}+\frac{b^2 \sinh (2 (c+d x))}{4 d}","-\frac{a^2 \coth (c+d x)}{d}+\frac{1}{2} b x (4 a-b)+\frac{b^2 \sinh (c+d x) \cosh (c+d x)}{2 d}",1,"2*a*b*x + (b^2*(-c - d*x))/(2*d) - (a^2*Coth[c + d*x])/d + (b^2*Sinh[2*(c + d*x)])/(4*d)","A",1
17,1,134,56,0.0625669,"\int \text{csch}^3(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^2,x]","-\frac{a^2 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a^2 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a^2 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{2 a b \log \left(\sinh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{2 a b \log \left(\cosh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}+\frac{b^2 \sinh (c) \sinh (d x)}{d}+\frac{b^2 \cosh (c) \cosh (d x)}{d}","-\frac{a^2 \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{a (a-4 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac{b^2 \cosh (c+d x)}{d}",1,"(b^2*Cosh[c]*Cosh[d*x])/d - (a^2*Csch[(c + d*x)/2]^2)/(8*d) - (2*a*b*Log[Cosh[c/2 + (d*x)/2]])/d + (2*a*b*Log[Sinh[c/2 + (d*x)/2]])/d - (a^2*Log[Tanh[(c + d*x)/2]])/(2*d) - (a^2*Sech[(c + d*x)/2]^2)/(8*d) + (b^2*Sinh[c]*Sinh[d*x])/d","B",1
18,1,85,40,0.7208959,"\int \text{csch}^4(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^2,x]","\frac{4 \sinh ^4(c+d x) \left(a \text{csch}^2(c+d x)+b\right)^2 \left(3 b^2 (c+d x)-a \coth (c+d x) \left(a \text{csch}^2(c+d x)-2 a+6 b\right)\right)}{3 d (2 a+b \cosh (2 (c+d x))-b)^2}","-\frac{a^2 \coth ^3(c+d x)}{3 d}+\frac{a (a-2 b) \coth (c+d x)}{d}+b^2 x",1,"(4*(b + a*Csch[c + d*x]^2)^2*(3*b^2*(c + d*x) - a*Coth[c + d*x]*(-2*a + 6*b + a*Csch[c + d*x]^2))*Sinh[c + d*x]^4)/(3*d*(2*a - b + b*Cosh[2*(c + d*x)])^2)","B",1
19,1,162,261,0.4198449,"\int \sinh ^4(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{120 (4 a-3 b) \left(8 a^2-14 a b+7 b^2\right) (c+d x)+10 b \left(16 a^2-32 a b+15 b^2\right) \sinh (6 (c+d x))-20 \left(128 a^3-360 a^2 b+336 a b^2-105 b^3\right) \sinh (2 (c+d x))+40 \left(8 a^3-36 a^2 b+42 a b^2-15 b^3\right) \sinh (4 (c+d x))+5 b^2 (6 a-5 b) \sinh (8 (c+d x))+2 b^3 \sinh (10 (c+d x))}{10240 d}","\frac{3}{256} x (4 a-3 b) \left(8 a^2-14 a b+7 b^2\right)+\frac{\left(48 a^3-272 a^2 b+314 a b^2-105 b^3\right) \sinh (c+d x) \cosh ^3(c+d x)}{640 d}-\frac{\left(576 a^3-1744 a^2 b+1678 a b^2-525 b^3\right) \sinh (c+d x) \cosh (c+d x)}{1280 d}+\frac{\sinh ^3(c+d x) \cosh ^7(c+d x) \left(a-(a-b) \tanh ^2(c+d x)\right)^3}{10 d}+\frac{3 (2 a-3 b) \sinh ^3(c+d x) \cosh ^5(c+d x) \left(a-(a-b) \tanh ^2(c+d x)\right)^2}{80 d}-\frac{b \sinh ^3(c+d x) \cosh ^3(c+d x) \left(a (14 a-9 b)-(22 a-21 b) (a-b) \tanh ^2(c+d x)\right)}{160 d}",1,"(120*(4*a - 3*b)*(8*a^2 - 14*a*b + 7*b^2)*(c + d*x) - 20*(128*a^3 - 360*a^2*b + 336*a*b^2 - 105*b^3)*Sinh[2*(c + d*x)] + 40*(8*a^3 - 36*a^2*b + 42*a*b^2 - 15*b^3)*Sinh[4*(c + d*x)] + 10*b*(16*a^2 - 32*a*b + 15*b^2)*Sinh[6*(c + d*x)] + 5*(6*a - 5*b)*b^2*Sinh[8*(c + d*x)] + 2*b^3*Sinh[10*(c + d*x)])/(10240*d)","A",1
20,1,127,115,0.7977446,"\int \sinh ^3(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{-1890 (4 a-3 b) \left(8 a^2-14 a b+7 b^2\right) \cosh (c+d x)+420 \left(16 a^3-60 a^2 b+63 a b^2-21 b^3\right) \cosh (3 (c+d x))+135 b^2 (4 a-3 b) \cosh (7 (c+d x))+756 b (4 a-3 b) (a-b) \cosh (5 (c+d x))+35 b^3 \cosh (9 (c+d x))}{80640 d}","\frac{b^2 (3 a-4 b) \cosh ^7(c+d x)}{7 d}+\frac{3 b (a-2 b) (a-b) \cosh ^5(c+d x)}{5 d}+\frac{(a-4 b) (a-b)^2 \cosh ^3(c+d x)}{3 d}-\frac{(a-b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^9(c+d x)}{9 d}",1,"(-1890*(4*a - 3*b)*(8*a^2 - 14*a*b + 7*b^2)*Cosh[c + d*x] + 420*(16*a^3 - 60*a^2*b + 63*a*b^2 - 21*b^3)*Cosh[3*(c + d*x)] + 756*(4*a - 3*b)*(a - b)*b*Cosh[5*(c + d*x)] + 135*(4*a - 3*b)*b^2*Cosh[7*(c + d*x)] + 35*b^3*Cosh[9*(c + d*x)])/(80640*d)","A",1
21,1,130,181,0.2905133,"\int \sinh ^2(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{24 b \left(12 a^2-18 a b+7 b^2\right) \sinh (4 (c+d x))-24 \left(64 a^3-144 a^2 b+120 a b^2-35 b^3\right) (c+d x)+48 \left(16 a^3-48 a^2 b+45 a b^2-14 b^3\right) \sinh (2 (c+d x))+16 b^2 (3 a-2 b) \sinh (6 (c+d x))+3 b^3 \sinh (8 (c+d x))}{3072 d}","\frac{b \left(24 a^2-64 a b+35 b^2\right) \sinh ^3(c+d x) \cosh (c+d x)}{192 d}+\frac{\left(96 a^3-376 a^2 b+360 a b^2-105 b^3\right) \sinh (c+d x) \cosh (c+d x)}{384 d}-\frac{1}{128} x \left(64 a^3-144 a^2 b+120 a b^2-35 b^3\right)+\frac{\sinh (c+d x) \cosh (c+d x) \left(a+b \sinh ^2(c+d x)\right)^3}{8 d}+\frac{(6 a-7 b) \sinh (c+d x) \cosh (c+d x) \left(a+b \sinh ^2(c+d x)\right)^2}{48 d}",1,"(-24*(64*a^3 - 144*a^2*b + 120*a*b^2 - 35*b^3)*(c + d*x) + 48*(16*a^3 - 48*a^2*b + 45*a*b^2 - 14*b^3)*Sinh[2*(c + d*x)] + 24*b*(12*a^2 - 18*a*b + 7*b^2)*Sinh[4*(c + d*x)] + 16*(3*a - 2*b)*b^2*Sinh[6*(c + d*x)] + 3*b^3*Sinh[8*(c + d*x)])/(3072*d)","A",1
22,1,94,79,0.2964889,"\int \sinh (c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\cosh (c+d x) \left(1120 a^3+b \left(560 a^2-784 a b+299 b^2\right) \cosh (2 (c+d x))-2800 a^2 b+6 b^2 (14 a-9 b) \cosh (4 (c+d x))+2492 a b^2+5 b^3 \cosh (6 (c+d x))-762 b^3\right)}{1120 d}","\frac{3 b^2 (a-b) \cosh ^5(c+d x)}{5 d}+\frac{b (a-b)^2 \cosh ^3(c+d x)}{d}+\frac{(a-b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^7(c+d x)}{7 d}",1,"(Cosh[c + d*x]*(1120*a^3 - 2800*a^2*b + 2492*a*b^2 - 762*b^3 + b*(560*a^2 - 784*a*b + 299*b^2)*Cosh[2*(c + d*x)] + 6*(14*a - 9*b)*b^2*Cosh[4*(c + d*x)] + 5*b^3*Cosh[6*(c + d*x)]))/(1120*d)","A",1
23,1,95,128,0.2580166,"\int \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[(a + b*Sinh[c + d*x]^2)^3,x]","\frac{12 (2 a-b) \left(8 a^2-8 a b+5 b^2\right) (c+d x)+9 b \left(16 a^2-16 a b+5 b^2\right) \sinh (2 (c+d x))+9 b^2 (2 a-b) \sinh (4 (c+d x))+b^3 \sinh (6 (c+d x))}{192 d}","\frac{b \left(64 a^2-54 a b+15 b^2\right) \sinh (c+d x) \cosh (c+d x)}{48 d}+\frac{1}{16} x (2 a-b) \left(8 a^2-8 a b+5 b^2\right)+\frac{5 b^2 (2 a-b) \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac{b \sinh (c+d x) \cosh (c+d x) \left(a+b \sinh ^2(c+d x)\right)^2}{6 d}",1,"(12*(2*a - b)*(8*a^2 - 8*a*b + 5*b^2)*(c + d*x) + 9*b*(16*a^2 - 16*a*b + 5*b^2)*Sinh[2*(c + d*x)] + 9*(2*a - b)*b^2*Sinh[4*(c + d*x)] + b^3*Sinh[6*(c + d*x)])/(192*d)","A",1
24,1,83,83,0.2151186,"\int \text{csch}(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{3 \left(80 a^3 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+b^3 \cosh (5 (c+d x))\right)+30 b \left(24 a^2-18 a b+5 b^2\right) \cosh (c+d x)+5 b^2 (12 a-5 b) \cosh (3 (c+d x))}{240 d}","-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \left(3 a^2-3 a b+b^2\right) \cosh (c+d x)}{d}+\frac{b^2 (3 a-2 b) \cosh ^3(c+d x)}{3 d}+\frac{b^3 \cosh ^5(c+d x)}{5 d}",1,"(30*b*(24*a^2 - 18*a*b + 5*b^2)*Cosh[c + d*x] + 5*(12*a - 5*b)*b^2*Cosh[3*(c + d*x)] + 3*(b^3*Cosh[5*(c + d*x)] + 80*a^3*Log[Tanh[(c + d*x)/2]]))/(240*d)","A",1
25,1,113,137,1.8817849,"\int \text{csch}^2(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\sinh ^6(c+d x) \left(a \text{csch}^2(c+d x)+b\right)^3 \left(-32 a^3 \coth (c+d x)+12 b \left(8 a^2-4 a b+b^2\right) (c+d x)+8 b^2 (3 a-b) \sinh (2 (c+d x))+b^3 \sinh (4 (c+d x))\right)}{4 d (2 a+b \cosh (2 (c+d x))-b)^3}","\frac{3}{8} b x \left(8 a^2-4 a b+b^2\right)-\frac{a (2 a+b) (4 a+b) \coth (c+d x)}{8 d}+\frac{b \cosh ^4(c+d x) \coth (c+d x) \left(a-(a-b) \tanh ^2(c+d x)\right)^2}{4 d}+\frac{b \cosh ^2(c+d x) \coth (c+d x) \left(a (4 a+b)-(4 a-3 b) (a-b) \tanh ^2(c+d x)\right)}{8 d}",1,"((b + a*Csch[c + d*x]^2)^3*Sinh[c + d*x]^6*(12*b*(8*a^2 - 4*a*b + b^2)*(c + d*x) - 32*a^3*Coth[c + d*x] + 8*(3*a - b)*b^2*Sinh[2*(c + d*x)] + b^3*Sinh[4*(c + d*x)]))/(4*d*(2*a - b + b*Cosh[2*(c + d*x)])^3)","A",1
26,1,210,83,4.5850645,"\int \text{csch}^3(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^3,x]","-\frac{\left(a+b \sinh ^2(c+d x)\right)^3 \left(3 a^3 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)+3 a^3 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)+12 a^3 \log \left(\sinh \left(\frac{1}{2} (c+d x)\right)\right)-12 a^3 \log \left(\cosh \left(\frac{1}{2} (c+d x)\right)\right)-72 a^2 b \log \left(\sinh \left(\frac{1}{2} (c+d x)\right)\right)+72 a^2 b \log \left(\cosh \left(\frac{1}{2} (c+d x)\right)\right)-72 a b^2 \sinh (c) \sinh (d x)-18 b^2 (4 a-b) \cosh (c) \cosh (d x)+18 b^3 \sinh (c) \sinh (d x)-2 b^3 \sinh (3 c) \sinh (3 d x)-2 b^3 \cosh (3 c) \cosh (3 d x)\right)}{3 d (2 a+b \cosh (2 (c+d x))-b)^3}","-\frac{a^3 \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{a^2 (a-6 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac{b^2 (3 a-b) \cosh (c+d x)}{d}+\frac{b^3 \cosh ^3(c+d x)}{3 d}",1,"-1/3*((-18*(4*a - b)*b^2*Cosh[c]*Cosh[d*x] - 2*b^3*Cosh[3*c]*Cosh[3*d*x] + 3*a^3*Csch[(c + d*x)/2]^2 - 12*a^3*Log[Cosh[(c + d*x)/2]] + 72*a^2*b*Log[Cosh[(c + d*x)/2]] + 12*a^3*Log[Sinh[(c + d*x)/2]] - 72*a^2*b*Log[Sinh[(c + d*x)/2]] + 3*a^3*Sech[(c + d*x)/2]^2 - 72*a*b^2*Sinh[c]*Sinh[d*x] + 18*b^3*Sinh[c]*Sinh[d*x] - 2*b^3*Sinh[3*c]*Sinh[3*d*x])*(a + b*Sinh[c + d*x]^2)^3)/(d*(2*a - b + b*Cosh[2*(c + d*x)])^3)","B",1
27,1,107,113,2.4479014,"\int \text{csch}^4(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{2 \sinh ^6(c+d x) \left(a \text{csch}^2(c+d x)+b\right)^3 \left(3 b^2 (2 (6 a-b) (c+d x)+b \sinh (2 (c+d x)))-4 a^2 \coth (c+d x) \left(a \text{csch}^2(c+d x)-2 a+9 b\right)\right)}{3 d (2 a+b \cosh (2 (c+d x))-b)^3}","\frac{a \left(2 a^2-5 a b-2 b^2\right) \coth (c+d x)}{2 d}-\frac{a^2 (2 a+3 b) \coth ^3(c+d x)}{6 d}+\frac{1}{2} b^2 x (6 a-b)+\frac{b \cosh ^2(c+d x) \coth ^3(c+d x) \left(a-(a-b) \tanh ^2(c+d x)\right)^2}{2 d}",1,"(2*(b + a*Csch[c + d*x]^2)^3*Sinh[c + d*x]^6*(-4*a^2*Coth[c + d*x]*(-2*a + 9*b + a*Csch[c + d*x]^2) + 3*b^2*(2*(6*a - b)*(c + d*x) + b*Sinh[2*(c + d*x)])))/(3*d*(2*a - b + b*Cosh[2*(c + d*x)])^3)","A",1
28,1,165,109,0.8850489,"\int \frac{\sinh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Sinh[c + d*x]^7/(a + b*Sinh[c + d*x]^2),x]","\frac{-\frac{240 a^3 \left(\tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)+\tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)\right)}{\sqrt{a-b}}+30 \sqrt{b} \left(8 a^2+6 a b+5 b^2\right) \cosh (c+d x)-5 b^{3/2} (4 a+5 b) \cosh (3 (c+d x))+3 b^{5/2} \cosh (5 (c+d x))}{240 b^{7/2} d}","-\frac{a^3 \tan ^{-1}\left(\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right)}{b^{7/2} d \sqrt{a-b}}+\frac{\left(a^2+a b+b^2\right) \cosh (c+d x)}{b^3 d}-\frac{(a+2 b) \cosh ^3(c+d x)}{3 b^2 d}+\frac{\cosh ^5(c+d x)}{5 b d}",1,"((-240*a^3*(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]))/Sqrt[a - b] + 30*Sqrt[b]*(8*a^2 + 6*a*b + 5*b^2)*Cosh[c + d*x] - 5*b^(3/2)*(4*a + 5*b)*Cosh[3*(c + d*x)] + 3*b^(5/2)*Cosh[5*(c + d*x)])/(240*b^(7/2)*d)","C",1
29,1,97,121,0.4776664,"\int \frac{\sinh ^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Sinh[c + d*x]^6/(a + b*Sinh[c + d*x]^2),x]","\frac{-\frac{32 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a-b}}+4 \left(8 a^2+4 a b+3 b^2\right) (c+d x)-8 b (a+b) \sinh (2 (c+d x))+b^2 \sinh (4 (c+d x))}{32 b^3 d}","-\frac{a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{b^3 d \sqrt{a-b}}+\frac{x \left(8 a^2+4 a b+3 b^2\right)}{8 b^3}-\frac{(4 a+3 b) \sinh (c+d x) \cosh (c+d x)}{8 b^2 d}+\frac{\sinh ^3(c+d x) \cosh (c+d x)}{4 b d}",1,"(4*(8*a^2 + 4*a*b + 3*b^2)*(c + d*x) - (32*a^(5/2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/Sqrt[a - b] - 8*b*(a + b)*Sinh[2*(c + d*x)] + b^2*Sinh[4*(c + d*x)])/(32*b^3*d)","A",1
30,1,134,79,0.4316646,"\int \frac{\sinh ^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Sinh[c + d*x]^5/(a + b*Sinh[c + d*x]^2),x]","\frac{\frac{12 a^2 \left(\tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)+\tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)\right)}{\sqrt{a-b}}-3 \sqrt{b} (4 a+3 b) \cosh (c+d x)+b^{3/2} \cosh (3 (c+d x))}{12 b^{5/2} d}","\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right)}{b^{5/2} d \sqrt{a-b}}-\frac{(a+b) \cosh (c+d x)}{b^2 d}+\frac{\cosh ^3(c+d x)}{3 b d}",1,"((12*a^2*(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]))/Sqrt[a - b] - 3*Sqrt[b]*(4*a + 3*b)*Cosh[c + d*x] + b^(3/2)*Cosh[3*(c + d*x)])/(12*b^(5/2)*d)","C",1
31,1,71,79,0.2466003,"\int \frac{\sinh ^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Sinh[c + d*x]^4/(a + b*Sinh[c + d*x]^2),x]","\frac{\frac{4 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a-b}}-2 (2 a+b) (c+d x)+b \sinh (2 (c+d x))}{4 b^2 d}","\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{b^2 d \sqrt{a-b}}-\frac{x (2 a+b)}{2 b^2}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 b d}",1,"(-2*(2*a + b)*(c + d*x) + (4*a^(3/2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/Sqrt[a - b] + b*Sinh[2*(c + d*x)])/(4*b^2*d)","A",1
32,1,107,56,0.2424157,"\int \frac{\sinh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Sinh[c + d*x]^3/(a + b*Sinh[c + d*x]^2),x]","\frac{\sqrt{b} \cosh (c+d x)-\frac{a \left(\tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)+\tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)\right)}{\sqrt{a-b}}}{b^{3/2} d}","\frac{\cosh (c+d x)}{b d}-\frac{a \tan ^{-1}\left(\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right)}{b^{3/2} d \sqrt{a-b}}",1,"(-((a*(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]))/Sqrt[a - b]) + Sqrt[b]*Cosh[c + d*x])/(b^(3/2)*d)","C",1
33,1,50,50,0.1283997,"\int \frac{\sinh ^2(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Sinh[c + d*x]^2/(a + b*Sinh[c + d*x]^2),x]","\frac{-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a-b}}+c+d x}{b d}","\frac{x}{b}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{b d \sqrt{a-b}}",1,"(c + d*x - (Sqrt[a]*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/Sqrt[a - b])/(b*d)","A",1
34,1,91,40,0.1286296,"\int \frac{\sinh (c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Sinh[c + d*x]/(a + b*Sinh[c + d*x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)+\tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)}{\sqrt{b} d \sqrt{a-b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right)}{\sqrt{b} d \sqrt{a-b}}",1,"(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/(Sqrt[a - b]*Sqrt[b]*d)","C",1
35,1,40,40,0.0713694,"\int \frac{1}{a+b \sinh ^2(c+d x)} \, dx","Integrate[(a + b*Sinh[c + d*x]^2)^(-1),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d \sqrt{a-b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d \sqrt{a-b}}",1,"ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[a - b]*d)","A",1
36,1,124,60,0.2156487,"\int \frac{\text{csch}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Csch[c + d*x]/(a + b*Sinh[c + d*x]^2),x]","\frac{\sqrt{a-b} \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)-\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)}{a d \sqrt{a-b}}","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right)}{a d \sqrt{a-b}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a d}",1,"(-(Sqrt[b]*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]) - Sqrt[b]*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + Sqrt[a - b]*Log[Tanh[(c + d*x)/2]])/(a*Sqrt[a - b]*d)","C",1
37,1,57,57,0.292952,"\int \frac{\text{csch}^2(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Csch[c + d*x]^2/(a + b*Sinh[c + d*x]^2),x]","\frac{-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a-b}}-\sqrt{a} \coth (c+d x)}{a^{3/2} d}","-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{3/2} d \sqrt{a-b}}-\frac{\coth (c+d x)}{a d}",1,"(-((b*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/Sqrt[a - b]) - Sqrt[a]*Coth[c + d*x])/(a^(3/2)*d)","A",1
38,1,201,88,0.6996769,"\int \frac{\text{csch}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Csch[c + d*x]^3/(a + b*Sinh[c + d*x]^2),x]","-\frac{\text{csch}^4(c+d x) (2 a+b \cosh (2 (c+d x))-b) \left(2 a \sqrt{a-b} \cosh (c+d x)-2 \sinh ^2(c+d x) \left(2 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)+2 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)-\sqrt{a-b} (a+2 b) \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{8 a^2 d \sqrt{a-b} \left(a \text{csch}^2(c+d x)+b\right)}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right)}{a^2 d \sqrt{a-b}}+\frac{(a+2 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^2 d}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d}",1,"-1/8*((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^4*(2*a*Sqrt[a - b]*Cosh[c + d*x] - 2*(2*b^(3/2)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + 2*b^(3/2)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] - Sqrt[a - b]*(a + 2*b)*Log[Tanh[(c + d*x)/2]])*Sinh[c + d*x]^2))/(a^2*Sqrt[a - b]*d*(b + a*Csch[c + d*x]^2))","C",1
39,1,126,78,0.690469,"\int \frac{\text{csch}^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Csch[c + d*x]^4/(a + b*Sinh[c + d*x]^2),x]","-\frac{\text{csch}^2(c+d x) (2 a+b \cosh (2 (c+d x))-b) \left(\sqrt{a} \sqrt{a-b} \coth (c+d x) \left(a \text{csch}^2(c+d x)-2 a-3 b\right)-3 b^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)\right)}{6 a^{5/2} d \sqrt{a-b} \left(a \text{csch}^2(c+d x)+b\right)}","\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{5/2} d \sqrt{a-b}}+\frac{(a+b) \coth (c+d x)}{a^2 d}-\frac{\coth ^3(c+d x)}{3 a d}",1,"-1/6*((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2*(-3*b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]] + Sqrt[a]*Sqrt[a - b]*Coth[c + d*x]*(-2*a - 3*b + a*Csch[c + d*x]^2)))/(a^(5/2)*Sqrt[a - b]*d*(b + a*Csch[c + d*x]^2))","A",1
40,1,574,130,6.2844185,"\int \frac{\text{csch}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Csch[c + d*x]^5/(a + b*Sinh[c + d*x]^2),x]","-\frac{b^{5/2} \text{csch}^2(c+d x) (2 a+b \cosh (2 (c+d x))-b) \tan ^{-1}\left(\frac{\text{sech}\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{b} \cosh \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a} \sinh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a-b}}\right)}{2 a^3 d \sqrt{a-b} \left(a \text{csch}^2(c+d x)+b\right)}-\frac{b^{5/2} \text{csch}^2(c+d x) (2 a+b \cosh (2 (c+d x))-b) \tan ^{-1}\left(\frac{\text{sech}\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{b} \cosh \left(\frac{1}{2} (c+d x)\right)+i \sqrt{a} \sinh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a-b}}\right)}{2 a^3 d \sqrt{a-b} \left(a \text{csch}^2(c+d x)+b\right)}+\frac{(3 a+4 b) \text{csch}^2(c+d x) \text{csch}^2\left(\frac{1}{2} (c+d x)\right) (2 a+b \cosh (2 (c+d x))-b)}{64 a^2 d \left(a \text{csch}^2(c+d x)+b\right)}+\frac{(3 a+4 b) \text{csch}^2(c+d x) \text{sech}^2\left(\frac{1}{2} (c+d x)\right) (2 a+b \cosh (2 (c+d x))-b)}{64 a^2 d \left(a \text{csch}^2(c+d x)+b\right)}+\frac{\left(3 a^2+4 a b+8 b^2\right) \text{csch}^2(c+d x) \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right) (2 a+b \cosh (2 (c+d x))-b)}{16 a^3 d \left(a \text{csch}^2(c+d x)+b\right)}-\frac{\text{csch}^2(c+d x) \text{csch}^4\left(\frac{1}{2} (c+d x)\right) (2 a+b \cosh (2 (c+d x))-b)}{128 a d \left(a \text{csch}^2(c+d x)+b\right)}+\frac{\text{csch}^2(c+d x) \text{sech}^4\left(\frac{1}{2} (c+d x)\right) (2 a+b \cosh (2 (c+d x))-b)}{128 a d \left(a \text{csch}^2(c+d x)+b\right)}","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right)}{a^3 d \sqrt{a-b}}+\frac{(3 a+4 b) \coth (c+d x) \text{csch}(c+d x)}{8 a^2 d}-\frac{\left(3 a^2+4 a b+8 b^2\right) \tanh ^{-1}(\cosh (c+d x))}{8 a^3 d}-\frac{\coth (c+d x) \text{csch}^3(c+d x)}{4 a d}",1,"-1/2*(b^(5/2)*ArcTan[(Sech[(c + d*x)/2]*(Sqrt[b]*Cosh[(c + d*x)/2] - I*Sqrt[a]*Sinh[(c + d*x)/2]))/Sqrt[a - b]]*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2)/(a^3*Sqrt[a - b]*d*(b + a*Csch[c + d*x]^2)) - (b^(5/2)*ArcTan[(Sech[(c + d*x)/2]*(Sqrt[b]*Cosh[(c + d*x)/2] + I*Sqrt[a]*Sinh[(c + d*x)/2]))/Sqrt[a - b]]*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2)/(2*a^3*Sqrt[a - b]*d*(b + a*Csch[c + d*x]^2)) + ((3*a + 4*b)*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[(c + d*x)/2]^2*Csch[c + d*x]^2)/(64*a^2*d*(b + a*Csch[c + d*x]^2)) - ((2*a - b + b*Cosh[2*(c + d*x)])*Csch[(c + d*x)/2]^4*Csch[c + d*x]^2)/(128*a*d*(b + a*Csch[c + d*x]^2)) + ((3*a^2 + 4*a*b + 8*b^2)*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2*Log[Tanh[(c + d*x)/2]])/(16*a^3*d*(b + a*Csch[c + d*x]^2)) + ((3*a + 4*b)*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2*Sech[(c + d*x)/2]^2)/(64*a^2*d*(b + a*Csch[c + d*x]^2)) + ((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2*Sech[(c + d*x)/2]^4)/(128*a*d*(b + a*Csch[c + d*x]^2))","C",1
41,1,155,110,1.4796913,"\int \frac{\text{csch}^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Csch[c + d*x]^6/(a + b*Sinh[c + d*x]^2),x]","-\frac{\text{csch}^2(c+d x) (2 a+b \cosh (2 (c+d x))-b) \left(\sqrt{a} \sqrt{a-b} \coth (c+d x) \left(3 a^2 \text{csch}^4(c+d x)+8 a^2-a (4 a+5 b) \text{csch}^2(c+d x)+10 a b+15 b^2\right)+15 b^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)\right)}{30 a^{7/2} d \sqrt{a-b} \left(a \text{csch}^2(c+d x)+b\right)}","-\frac{b^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{7/2} d \sqrt{a-b}}+\frac{(2 a+b) \coth ^3(c+d x)}{3 a^2 d}-\frac{\left(a^2+a b+b^2\right) \coth (c+d x)}{a^3 d}-\frac{\coth ^5(c+d x)}{5 a d}",1,"-1/30*((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2*(15*b^3*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]] + Sqrt[a]*Sqrt[a - b]*Coth[c + d*x]*(8*a^2 + 10*a*b + 15*b^2 - a*(4*a + 5*b)*Csch[c + d*x]^2 + 3*a^2*Csch[c + d*x]^4)))/(a^(7/2)*Sqrt[a - b]*d*(b + a*Csch[c + d*x]^2))","A",1
42,1,99,102,0.8775616,"\int \frac{\sinh ^4(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2,x]","-\frac{\frac{\sqrt{a} (2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{(a-b)^{3/2}}+\frac{a b \sinh (2 (c+d x))}{(a-b) (2 a+b \cosh (2 (c+d x))-b)}-2 (c+d x)}{2 b^2 d}","-\frac{\sqrt{a} (2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 b^2 d (a-b)^{3/2}}-\frac{a \tanh (c+d x)}{2 b d (a-b) \left(a-(a-b) \tanh ^2(c+d x)\right)}+\frac{x}{b^2}",1,"-1/2*(-2*(c + d*x) + (Sqrt[a]*(2*a - 3*b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a - b)^(3/2) + (a*b*Sinh[2*(c + d*x)])/((a - b)*(2*a - b + b*Cosh[2*(c + d*x)])))/(b^2*d)","A",1
43,1,141,90,0.631854,"\int \frac{\sinh ^3(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{-\frac{2 a \sqrt{b} \cosh (c+d x)}{(a-b) (2 a+b \cosh (2 (c+d x))-b)}+\frac{(a-2 b) \left(\tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)+\tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)\right)}{(a-b)^{3/2}}}{2 b^{3/2} d}","\frac{(a-2 b) \tan ^{-1}\left(\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right)}{2 b^{3/2} d (a-b)^{3/2}}-\frac{a \cosh (c+d x)}{2 b d (a-b) \left(a+b \cosh ^2(c+d x)-b\right)}",1,"(((a - 2*b)*(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]))/(a - b)^(3/2) - (2*a*Sqrt[b]*Cosh[c + d*x])/((a - b)*(2*a - b + b*Cosh[2*(c + d*x)])))/(2*b^(3/2)*d)","C",1
44,1,81,84,0.4419989,"\int \frac{\sinh ^2(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\frac{\sinh (2 (c+d x))}{(a-b) (2 a+b \cosh (2 (c+d x))-b)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} (a-b)^{3/2}}}{2 d}","\frac{\sinh (c+d x) \cosh (c+d x)}{2 d (a-b) \left(a+b \sinh ^2(c+d x)\right)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 \sqrt{a} d (a-b)^{3/2}}",1,"(-(ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]/(Sqrt[a]*(a - b)^(3/2))) + Sinh[2*(c + d*x)]/((a - b)*(2*a - b + b*Cosh[2*(c + d*x)])))/(2*d)","A",1
45,1,130,81,0.3395399,"\int \frac{\sinh (c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\frac{2 \cosh (c+d x)}{(a-b) (2 a+b \cosh (2 (c+d x))-b)}+\frac{\tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)+\tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)}{\sqrt{b} (a-b)^{3/2}}}{2 d}","\frac{\cosh (c+d x)}{2 d (a-b) \left(a+b \cosh ^2(c+d x)-b\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right)}{2 \sqrt{b} d (a-b)^{3/2}}",1,"((ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/((a - b)^(3/2)*Sqrt[b]) + (2*Cosh[c + d*x])/((a - b)*(2*a - b + b*Cosh[2*(c + d*x)])))/(2*d)","C",1
46,1,96,95,0.2998408,"\int \frac{1}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[(a + b*Sinh[c + d*x]^2)^(-2),x]","\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^{3/2}}-\frac{b \sinh (2 (c+d x))}{2 a d (a-b) (2 a+b \cosh (2 (c+d x))-b)}","\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^{3/2}}-\frac{b \sinh (c+d x) \cosh (c+d x)}{2 a d (a-b) \left(a+b \sinh ^2(c+d x)\right)}",1,"((2*a - b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^(3/2)*d) - (b*Sinh[2*(c + d*x)])/(2*a*(a - b)*d*(2*a - b + b*Cosh[2*(c + d*x)]))","A",1
47,1,176,110,0.6560434,"\int \frac{\text{csch}(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Csch[c + d*x]/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{-\frac{2 a b \cosh (c+d x)}{(a-b) (2 a+b \cosh (2 (c+d x))-b)}+\frac{\sqrt{b} (2 b-3 a) \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)}{(a-b)^{3/2}}+\frac{\sqrt{b} (2 b-3 a) \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)}{(a-b)^{3/2}}+2 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^2 d}","-\frac{\sqrt{b} (3 a-2 b) \tan ^{-1}\left(\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right)}{2 a^2 d (a-b)^{3/2}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac{b \cosh (c+d x)}{2 a d (a-b) \left(a+b \cosh ^2(c+d x)-b\right)}",1,"((Sqrt[b]*(-3*a + 2*b)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/(a - b)^(3/2) + (Sqrt[b]*(-3*a + 2*b)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/(a - b)^(3/2) - (2*a*b*Cosh[c + d*x])/((a - b)*(2*a - b + b*Cosh[2*(c + d*x)])) + 2*Log[Tanh[(c + d*x)/2]])/(2*a^2*d)","C",1
48,1,170,142,0.8159854,"\int \frac{\text{csch}^2(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Csch[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^2,x]","-\frac{\text{csch}^5(c+d x) (2 a+b \cosh (2 (c+d x))-b) \left(2 \sqrt{a} \sqrt{a-b} \cosh (c+d x) \left(4 a^2+b (2 a-3 b) \cosh (2 (c+d x))-6 a b+3 b^2\right)-2 b (3 b-4 a) \sinh (c+d x) (2 a+b \cosh (2 (c+d x))-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)\right)}{16 a^{5/2} d (a-b)^{3/2} \left(a \text{csch}^2(c+d x)+b\right)^2}","-\frac{b (4 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{5/2} d (a-b)^{3/2}}+\frac{\left(2 a^2-4 a b+3 b^2\right) \tanh (c+d x)}{2 a^2 d (a-b) \left(a-(a-b) \tanh ^2(c+d x)\right)}-\frac{\coth (c+d x)}{a d \left(a-(a-b) \tanh ^2(c+d x)\right)}",1,"-1/16*((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^5*(2*Sqrt[a]*Sqrt[a - b]*Cosh[c + d*x]*(4*a^2 - 6*a*b + 3*b^2 + (2*a - 3*b)*b*Cosh[2*(c + d*x)]) - 2*b*(-4*a + 3*b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]*(2*a - b + b*Cosh[2*(c + d*x)])*Sinh[c + d*x]))/(a^(5/2)*(a - b)^(3/2)*d*(b + a*Csch[c + d*x]^2)^2)","A",1
49,1,350,161,1.3737438,"\int \frac{\text{csch}^3(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Csch[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\text{csch}^3(c+d x) (2 a+b \cosh (2 (c+d x))-b) \left(\frac{4 b^{3/2} (5 a-4 b) \text{csch}(c+d x) (2 a+b \cosh (2 (c+d x))-b) \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)}{(a-b)^{3/2}}+\frac{4 b^{3/2} (5 a-4 b) \text{csch}(c+d x) (2 a+b \cosh (2 (c+d x))-b) \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)}{(a-b)^{3/2}}+\frac{8 a b^2 \coth (c+d x)}{a-b}-a \text{csch}^2\left(\frac{1}{2} (c+d x)\right) \text{csch}(c+d x) (2 a+b \cosh (2 (c+d x))-b)-a \text{csch}(c+d x) \text{sech}^2\left(\frac{1}{2} (c+d x)\right) (2 a+b \cosh (2 (c+d x))-b)-4 (a+4 b) \text{csch}(c+d x) \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right) (2 a+b \cosh (2 (c+d x))-b)\right)}{32 a^3 d \left(a \text{csch}^2(c+d x)+b\right)^2}","\frac{b^{3/2} (5 a-4 b) \tan ^{-1}\left(\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right)}{2 a^3 d (a-b)^{3/2}}+\frac{(a+4 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}-\frac{b (a-2 b) \cosh (c+d x)}{2 a^2 d (a-b) \left(a+b \cosh ^2(c+d x)-b\right)}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left(a+b \cosh ^2(c+d x)-b\right)}",1,"((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^3*((8*a*b^2*Coth[c + d*x])/(a - b) + (4*(5*a - 4*b)*b^(3/2)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x])/(a - b)^(3/2) + (4*(5*a - 4*b)*b^(3/2)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x])/(a - b)^(3/2) - a*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[(c + d*x)/2]^2*Csch[c + d*x] - 4*(a + 4*b)*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]*Log[Tanh[(c + d*x)/2]] - a*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]*Sech[(c + d*x)/2]^2))/(32*a^3*d*(b + a*Csch[c + d*x]^2)^2)","C",1
50,1,210,174,1.2742186,"\int \frac{\text{csch}^4(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Csch[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\text{csch}^4(c+d x) (2 a+b \cosh (2 (c+d x))-b) \left(-2 a^{3/2} \coth (c+d x) \text{csch}^2(c+d x) (2 a+b \cosh (2 (c+d x))-b)-\frac{3 \sqrt{a} b^3 \sinh (2 (c+d x))}{a-b}+\frac{3 b^2 (6 a-5 b) (2 a+b \cosh (2 (c+d x))-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{(a-b)^{3/2}}+4 \sqrt{a} (a+3 b) \coth (c+d x) (2 a+b \cosh (2 (c+d x))-b)\right)}{24 a^{7/2} d \left(a \text{csch}^2(c+d x)+b\right)^2}","\frac{b^2 (6 a-5 b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{7/2} d (a-b)^{3/2}}-\frac{(2 a-5 b) \coth ^3(c+d x)}{6 a^2 d (a-b)}+\frac{\left(2 a^2+a b-5 b^2\right) \coth (c+d x)}{2 a^3 d (a-b)}-\frac{b \text{csch}^3(c+d x) \text{sech}(c+d x)}{2 a d (a-b) \left(a-(a-b) \tanh ^2(c+d x)\right)}",1,"((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^4*((3*(6*a - 5*b)*b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]*(2*a - b + b*Cosh[2*(c + d*x)]))/(a - b)^(3/2) + 4*Sqrt[a]*(a + 3*b)*(2*a - b + b*Cosh[2*(c + d*x)])*Coth[c + d*x] - 2*a^(3/2)*(2*a - b + b*Cosh[2*(c + d*x)])*Coth[c + d*x]*Csch[c + d*x]^2 - (3*Sqrt[a]*b^3*Sinh[2*(c + d*x)])/(a - b)))/(24*a^(7/2)*d*(b + a*Csch[c + d*x]^2)^2)","A",1
51,1,104,124,1.3836633,"\int \frac{\sinh ^4(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} (a-b)^{5/2}}+\frac{\sinh (2 (c+d x)) ((2 a-5 b) \cosh (2 (c+d x))-8 a+5 b)}{(a-b)^2 (2 a+b \cosh (2 (c+d x))-b)^2}}{8 d}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 \sqrt{a} d (a-b)^{5/2}}-\frac{3 \tanh (c+d x)}{8 d (a-b)^2 \left(a-(a-b) \tanh ^2(c+d x)\right)}+\frac{\tanh ^3(c+d x)}{4 d (a-b) \left(a-(a-b) \tanh ^2(c+d x)\right)^2}",1,"((3*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(5/2)) + ((-8*a + 5*b + (2*a - 5*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a - b)^2*(2*a - b + b*Cosh[2*(c + d*x)])^2))/(8*d)","A",1
52,1,170,135,1.33291,"\int \frac{\sinh ^3(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\frac{2 \sqrt{b} \cosh (c+d x) \left(-2 a^2+b (a-4 b) \cosh (2 (c+d x))-5 a b+4 b^2\right)}{(a-b)^2 (2 a+b \cosh (2 (c+d x))-b)^2}+\frac{(a-4 b) \left(\tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)+\tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)\right)}{(a-b)^{5/2}}}{8 b^{3/2} d}","\frac{(a-4 b) \tan ^{-1}\left(\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right)}{8 b^{3/2} d (a-b)^{5/2}}+\frac{(a-4 b) \cosh (c+d x)}{8 b d (a-b)^2 \left(a+b \cosh ^2(c+d x)-b\right)}-\frac{a \cosh (c+d x)}{4 b d (a-b) \left(a+b \cosh ^2(c+d x)-b\right)^2}",1,"(((a - 4*b)*(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]))/(a - b)^(5/2) + (2*Sqrt[b]*Cosh[c + d*x]*(-2*a^2 - 5*a*b + 4*b^2 + (a - 4*b)*b*Cosh[2*(c + d*x)]))/((a - b)^2*(2*a - b + b*Cosh[2*(c + d*x)])^2))/(8*b^(3/2)*d)","C",1
53,1,121,139,1.3576473,"\int \frac{\sinh ^2(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\frac{\sinh (2 (c+d x)) \left(8 a^2+b (2 a+b) \cosh (2 (c+d x))-4 a b-b^2\right)}{a (a-b)^2 (2 a+b \cosh (2 (c+d x))-b)^2}-\frac{(4 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{3/2} (a-b)^{5/2}}}{8 d}","-\frac{(4 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{3/2} d (a-b)^{5/2}}+\frac{(2 a+b) \sinh (c+d x) \cosh (c+d x)}{8 a d (a-b)^2 \left(a+b \sinh ^2(c+d x)\right)}+\frac{\sinh (c+d x) \cosh (c+d x)}{4 d (a-b) \left(a+b \sinh ^2(c+d x)\right)^2}",1,"(-(((4*a - b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*(a - b)^(5/2))) + ((8*a^2 - 4*a*b - b^2 + b*(2*a + b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(a*(a - b)^2*(2*a - b + b*Cosh[2*(c + d*x)])^2))/(8*d)","A",1
54,1,149,118,0.7867831,"\int \frac{\sinh (c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\frac{2 \cosh (c+d x) (10 a+3 b \cosh (2 (c+d x))-7 b)}{(a-b)^2 (2 a+b \cosh (2 (c+d x))-b)^2}+\frac{3 \left(\tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)+\tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)\right)}{\sqrt{b} (a-b)^{5/2}}}{8 d}","\frac{3 \cosh (c+d x)}{8 d (a-b)^2 \left(a+b \cosh ^2(c+d x)-b\right)}+\frac{\cosh (c+d x)}{4 d (a-b) \left(a+b \cosh ^2(c+d x)-b\right)^2}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right)}{8 \sqrt{b} d (a-b)^{5/2}}",1,"((3*(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]))/((a - b)^(5/2)*Sqrt[b]) + (2*Cosh[c + d*x]*(10*a - 7*b + 3*b*Cosh[2*(c + d*x)]))/((a - b)^2*(2*a - b + b*Cosh[2*(c + d*x)])^2))/(8*d)","C",1
55,1,132,154,1.2265912,"\int \frac{1}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[(a + b*Sinh[c + d*x]^2)^(-3),x]","\frac{\frac{\left(8 a^2-8 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{(a-b)^{5/2}}+\frac{\sqrt{a} b \sinh (2 (c+d x)) \left(-16 a^2+3 b (b-2 a) \cosh (2 (c+d x))+16 a b-3 b^2\right)}{(a-b)^2 (2 a+b \cosh (2 (c+d x))-b)^2}}{8 a^{5/2} d}","-\frac{3 b (2 a-b) \sinh (c+d x) \cosh (c+d x)}{8 a^2 d (a-b)^2 \left(a+b \sinh ^2(c+d x)\right)}+\frac{\left(8 a^2-8 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a-b)^{5/2}}-\frac{b \sinh (c+d x) \cosh (c+d x)}{4 a d (a-b) \left(a+b \sinh ^2(c+d x)\right)^2}",1,"(((8*a^2 - 8*a*b + 3*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a - b)^(5/2) + (Sqrt[a]*b*(-16*a^2 + 16*a*b - 3*b^2 + 3*b*(-2*a + b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a - b)^2*(2*a - b + b*Cosh[2*(c + d*x)])^2))/(8*a^(5/2)*d)","A",1
56,1,237,166,3.3700651,"\int \frac{\text{csch}(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Csch[c + d*x]/(a + b*Sinh[c + d*x]^2)^3,x]","-\frac{\frac{\sqrt{b} \left(15 a^2-20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)}{(a-b)^{5/2}}+\frac{\sqrt{b} \left(15 a^2-20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)}{(a-b)^{5/2}}+\frac{8 a^2 b \cosh (c+d x)}{(a-b) (2 a+b \cosh (2 (c+d x))-b)^2}+\frac{2 a b (7 a-4 b) \cosh (c+d x)}{(a-b)^2 (2 a+b \cosh (2 (c+d x))-b)}-8 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^3 d}","-\frac{\tanh ^{-1}(\cosh (c+d x))}{a^3 d}-\frac{b (7 a-4 b) \cosh (c+d x)}{8 a^2 d (a-b)^2 \left(a+b \cosh ^2(c+d x)-b\right)}-\frac{\sqrt{b} \left(15 a^2-20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right)}{8 a^3 d (a-b)^{5/2}}-\frac{b \cosh (c+d x)}{4 a d (a-b) \left(a+b \cosh ^2(c+d x)-b\right)^2}",1,"-1/8*((Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/(a - b)^(5/2) + (Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/(a - b)^(5/2) + (8*a^2*b*Cosh[c + d*x])/((a - b)*(2*a - b + b*Cosh[2*(c + d*x)])^2) + (2*a*(7*a - 4*b)*b*Cosh[c + d*x])/((a - b)^2*(2*a - b + b*Cosh[2*(c + d*x)])) - 8*Log[Tanh[(c + d*x)/2]])/(a^3*d)","C",1
57,1,225,215,1.8137285,"\int \frac{\text{csch}^2(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Csch[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\text{csch}^6(c+d x) (2 a+b \cosh (2 (c+d x))-b) \left(\frac{4 a^{3/2} b^2 \sinh (2 (c+d x))}{a-b}-\frac{3 b \left(8 a^2-12 a b+5 b^2\right) (2 a+b \cosh (2 (c+d x))-b)^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{(a-b)^{5/2}}+\frac{\sqrt{a} b^2 (10 a-7 b) \sinh (2 (c+d x)) (2 a+b \cosh (2 (c+d x))-b)}{(a-b)^2}-8 \sqrt{a} \coth (c+d x) (2 a+b \cosh (2 (c+d x))-b)^2\right)}{64 a^{7/2} d \left(a \text{csch}^2(c+d x)+b\right)^3}","-\frac{(4 a-5 b) (2 a-3 b) \coth (c+d x)}{8 a^3 d (a-b)^2}-\frac{b \coth (c+d x) \left(-\left((4 a-b) \tanh ^2(c+d x)\right)+4 a-5 b\right)}{8 a^2 d (a-b)^2 \left(a-(a-b) \tanh ^2(c+d x)\right)}-\frac{3 b \left(8 a^2-12 a b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{7/2} d (a-b)^{5/2}}-\frac{b \text{csch}(c+d x) \text{sech}^3(c+d x)}{4 a d (a-b) \left(a-(a-b) \tanh ^2(c+d x)\right)^2}",1,"((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^6*((-3*b*(8*a^2 - 12*a*b + 5*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]*(2*a - b + b*Cosh[2*(c + d*x)])^2)/(a - b)^(5/2) - 8*Sqrt[a]*(2*a - b + b*Cosh[2*(c + d*x)])^2*Coth[c + d*x] + (4*a^(3/2)*b^2*Sinh[2*(c + d*x)])/(a - b) + (Sqrt[a]*(10*a - 7*b)*b^2*(2*a - b + b*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(a - b)^2))/(64*a^(7/2)*d*(b + a*Csch[c + d*x]^2)^3)","A",1
58,1,419,224,2.8140579,"\int \frac{\text{csch}^3(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Csch[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\text{csch}^5(c+d x) (2 a+b \cosh (2 (c+d x))-b) \left(\frac{8 a^2 b^2 \coth (c+d x)}{a-b}+\frac{b^{3/2} \left(35 a^2-56 a b+24 b^2\right) \text{csch}(c+d x) (2 a+b \cosh (2 (c+d x))-b)^2 \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)}{(a-b)^{5/2}}+\frac{b^{3/2} \left(35 a^2-56 a b+24 b^2\right) \text{csch}(c+d x) (2 a+b \cosh (2 (c+d x))-b)^2 \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a-b}}\right)}{(a-b)^{5/2}}+\frac{2 a b^2 (11 a-8 b) \coth (c+d x) (2 a+b \cosh (2 (c+d x))-b)}{(a-b)^2}-a \text{csch}^2\left(\frac{1}{2} (c+d x)\right) \text{csch}(c+d x) (2 a+b \cosh (2 (c+d x))-b)^2-a \text{csch}(c+d x) \text{sech}^2\left(\frac{1}{2} (c+d x)\right) (2 a+b \cosh (2 (c+d x))-b)^2-4 (a+6 b) \text{csch}(c+d x) \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right) (2 a+b \cosh (2 (c+d x))-b)^2\right)}{64 a^4 d \left(a \text{csch}^2(c+d x)+b\right)^3}","\frac{(a+6 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^4 d}-\frac{b (a-4 b) (4 a-3 b) \cosh (c+d x)}{8 a^3 d (a-b)^2 \left(a+b \cosh ^2(c+d x)-b\right)}-\frac{b (2 a-3 b) \cosh (c+d x)}{4 a^2 d (a-b) \left(a+b \cosh ^2(c+d x)-b\right)^2}+\frac{b^{3/2} \left(35 a^2-56 a b+24 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right)}{8 a^4 d (a-b)^{5/2}}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left(a+b \cosh ^2(c+d x)-b\right)^2}",1,"((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^5*((8*a^2*b^2*Coth[c + d*x])/(a - b) + (2*a*(11*a - 8*b)*b^2*(2*a - b + b*Cosh[2*(c + d*x)])*Coth[c + d*x])/(a - b)^2 + (b^(3/2)*(35*a^2 - 56*a*b + 24*b^2)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]*(2*a - b + b*Cosh[2*(c + d*x)])^2*Csch[c + d*x])/(a - b)^(5/2) + (b^(3/2)*(35*a^2 - 56*a*b + 24*b^2)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]*(2*a - b + b*Cosh[2*(c + d*x)])^2*Csch[c + d*x])/(a - b)^(5/2) - a*(2*a - b + b*Cosh[2*(c + d*x)])^2*Csch[(c + d*x)/2]^2*Csch[c + d*x] - 4*(a + 6*b)*(2*a - b + b*Cosh[2*(c + d*x)])^2*Csch[c + d*x]*Log[Tanh[(c + d*x)/2]] - a*(2*a - b + b*Cosh[2*(c + d*x)])^2*Csch[c + d*x]*Sech[(c + d*x)/2]^2))/(64*a^4*d*(b + a*Csch[c + d*x]^2)^3)","C",1
59,1,167,259,2.8008098,"\int \frac{\text{csch}^4(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Csch[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\frac{3 b^2 \left(48 a^2-80 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{(a-b)^{5/2}}+\sqrt{a} \left(\frac{3 b^3 \sinh (2 (c+d x)) \left(-32 a^2+b (11 b-14 a) \cosh (2 (c+d x))+40 a b-11 b^2\right)}{(a-b)^2 (2 a+b \cosh (2 (c+d x))-b)^2}-8 \coth (c+d x) \left(a \text{csch}^2(c+d x)-2 a-9 b\right)\right)}{24 a^{9/2} d}","-\frac{b (10 a-7 b) \text{csch}^3(c+d x) \text{sech}(c+d x)}{8 a^2 d (a-b)^2 \left(a-(a-b) \tanh ^2(c+d x)\right)}+\frac{b^2 \left(48 a^2-80 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{9/2} d (a-b)^{5/2}}-\frac{\left(8 a^2-52 a b+35 b^2\right) \coth ^3(c+d x)}{24 a^3 d (a-b)^2}+\frac{\left(8 a^3-4 a^2 b-45 a b^2+35 b^3\right) \coth (c+d x)}{8 a^4 d (a-b)^2}-\frac{b \text{csch}^3(c+d x) \text{sech}^3(c+d x)}{4 a d (a-b) \left(a-(a-b) \tanh ^2(c+d x)\right)^2}",1,"((3*b^2*(48*a^2 - 80*a*b + 35*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a - b)^(5/2) + Sqrt[a]*(-8*Coth[c + d*x]*(-2*a - 9*b + a*Csch[c + d*x]^2) + (3*b^3*(-32*a^2 + 40*a*b - 11*b^2 + b*(-14*a + 11*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a - b)^2*(2*a - b + b*Cosh[2*(c + d*x)])^2)))/(24*a^(9/2)*d)","A",1
60,1,2,2,0.0041245,"\int \frac{1}{1+\sinh ^2(x)} \, dx","Integrate[(1 + Sinh[x]^2)^(-1),x]","\tanh (x)","\tanh (x)",1,"Tanh[x]","A",1
61,1,17,11,0.0026313,"\int \frac{1}{\left(1+\sinh ^2(x)\right)^2} \, dx","Integrate[(1 + Sinh[x]^2)^(-2),x]","\frac{2 \tanh (x)}{3}+\frac{1}{3} \tanh (x) \text{sech}^2(x)","\tanh (x)-\frac{\tanh ^3(x)}{3}",1,"(2*Tanh[x])/3 + (Sech[x]^2*Tanh[x])/3","A",1
62,1,27,19,0.0030436,"\int \frac{1}{\left(1+\sinh ^2(x)\right)^3} \, dx","Integrate[(1 + Sinh[x]^2)^(-3),x]","\frac{8 \tanh (x)}{15}+\frac{1}{5} \tanh (x) \text{sech}^4(x)+\frac{4}{15} \tanh (x) \text{sech}^2(x)","\frac{\tanh ^5(x)}{5}-\frac{2 \tanh ^3(x)}{3}+\tanh (x)",1,"(8*Tanh[x])/15 + (4*Sech[x]^2*Tanh[x])/15 + (Sech[x]^4*Tanh[x])/5","A",1
63,0,0,15,0.0226206,"\int \frac{1}{1-\sinh ^2(x)} \, dx","Integrate[(1 - Sinh[x]^2)^(-1),x]","\int \frac{1}{1-\sinh ^2(x)} \, dx","\frac{\tanh ^{-1}\left(\sqrt{2} \tanh (x)\right)}{\sqrt{2}}",1,"Integrate[(1 - Sinh[x]^2)^(-1), x]","F",-1
64,1,35,37,0.1301032,"\int \frac{1}{\left(1-\sinh ^2(x)\right)^2} \, dx","Integrate[(1 - Sinh[x]^2)^(-2),x]","\frac{3 \tanh ^{-1}\left(\sqrt{2} \tanh (x)\right)}{4 \sqrt{2}}-\frac{\sinh (2 x)}{4 (\cosh (2 x)-3)}","\frac{3 \tanh ^{-1}\left(\sqrt{2} \tanh (x)\right)}{4 \sqrt{2}}+\frac{\sinh (x) \cosh (x)}{4 \left(1-\sinh ^2(x)\right)}",1,"(3*ArcTanh[Sqrt[2]*Tanh[x]])/(4*Sqrt[2]) - Sinh[2*x]/(4*(-3 + Cosh[2*x]))","A",1
65,1,51,55,0.179032,"\int \frac{1}{\left(1-\sinh ^2(x)\right)^3} \, dx","Integrate[(1 - Sinh[x]^2)^(-3),x]","\frac{19 \tanh ^{-1}\left(\sqrt{2} \tanh (x)\right)}{32 \sqrt{2}}-\frac{9 \sinh (2 x)}{32 (\cosh (2 x)-3)}+\frac{\sinh (2 x)}{4 (\cosh (2 x)-3)^2}","\frac{19 \tanh ^{-1}\left(\sqrt{2} \tanh (x)\right)}{32 \sqrt{2}}+\frac{9 \sinh (x) \cosh (x)}{32 \left(1-\sinh ^2(x)\right)}+\frac{\sinh (x) \cosh (x)}{8 \left(1-\sinh ^2(x)\right)^2}",1,"(19*ArcTanh[Sqrt[2]*Tanh[x]])/(32*Sqrt[2]) + Sinh[2*x]/(4*(-3 + Cosh[2*x])^2) - (9*Sinh[2*x])/(32*(-3 + Cosh[2*x]))","A",1
66,1,114,130,0.6679683,"\int \sinh ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Sinh[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\frac{(b-a) (a+3 b) \log \left(\sqrt{2 a+b \cosh (2 (e+f x))-b}+\sqrt{2} \sqrt{b} \cosh (e+f x)\right)}{b^{3/2}}+\frac{\cosh (e+f x) \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b} (a+b \cosh (2 (e+f x))-4 b)}{2 b}}{8 f}","-\frac{(a-b) (a+3 b) \tanh ^{-1}\left(\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{8 b^{3/2} f}+\frac{\cosh (e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^{3/2}}{4 b f}-\frac{(a+3 b) \cosh (e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{8 b f}",1,"((Cosh[e + f*x]*(a - 4*b + b*Cosh[2*(e + f*x)])*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])/(2*b) + ((-a + b)*(a + 3*b)*Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]])/b^(3/2))/(8*f)","A",1
67,1,97,82,0.1513775,"\int \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\cosh (e+f x) \sqrt{2 a+b \cosh (2 (e+f x))-b}}{2 \sqrt{2} f}+\frac{(a-b) \log \left(\sqrt{2 a+b \cosh (2 (e+f x))-b}+\sqrt{2} \sqrt{b} \cosh (e+f x)\right)}{2 \sqrt{b} f}","\frac{\cosh (e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{2 f}+\frac{(a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{2 \sqrt{b} f}",1,"(Cosh[e + f*x]*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])/(2*Sqrt[2]*f) + ((a - b)*Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]])/(2*Sqrt[b]*f)","A",1
68,1,97,84,0.1253849,"\int \text{csch}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Csch[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\sqrt{b} \log \left(\sqrt{2 a+b \cosh (2 (e+f x))-b}+\sqrt{2} \sqrt{b} \cosh (e+f x)\right)-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)}{f}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{f}",1,"(-(Sqrt[a]*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]]) + Sqrt[b]*Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]])/f","A",1
69,1,104,88,0.2906555,"\int \text{csch}^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Csch[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{2 (a-b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)-\sqrt{2} \sqrt{a} \coth (e+f x) \text{csch}(e+f x) \sqrt{2 a+b \cosh (2 (e+f x))-b}}{4 \sqrt{a} f}","\frac{(a-b) \tanh ^{-1}\left(\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{2 \sqrt{a} f}-\frac{\coth (e+f x) \text{csch}(e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{2 f}",1,"(2*(a - b)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] - Sqrt[2]*Sqrt[a]*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]*Coth[e + f*x]*Csch[e + f*x])/(4*Sqrt[a]*f)","A",1
70,1,129,144,0.5392565,"\int \text{csch}^5(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Csch[e + f*x]^5*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\left(-6 a^2+4 a b+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)-\sqrt{2} \sqrt{a} \coth (e+f x) \text{csch}(e+f x) \sqrt{2 a+b \cosh (2 (e+f x))-b} \left(2 a \text{csch}^2(e+f x)-3 a+b\right)}{16 a^{3/2} f}","-\frac{(a-b) (3 a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{8 a^{3/2} f}-\frac{\coth (e+f x) \text{csch}^3(e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^{3/2}}{4 a f}+\frac{(3 a+b) \coth (e+f x) \text{csch}(e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{8 a f}",1,"((-6*a^2 + 4*a*b + 2*b^2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] - Sqrt[2]*Sqrt[a]*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]*Coth[e + f*x]*Csch[e + f*x]*(-3*a + b + 2*a*Csch[e + f*x]^2))/(16*a^(3/2)*f)","A",1
71,1,210,300,1.3781975,"\int \sinh ^4(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Sinh[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\sqrt{2} b \sinh (2 (e+f x)) \left(8 a^2+4 b (4 a-7 b) \cosh (2 (e+f x))-48 a b+3 b^2 \cosh (4 (e+f x))+25 b^2\right)-32 i a \left(a^2+a b-2 b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+16 i a \left(2 a^2+3 a b-8 b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{240 b^2 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{\left(2 a^2+3 a b-8 b^2\right) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 b^2 f}+\frac{\left(2 a^2+3 a b-8 b^2\right) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{15 b^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{(a-4 b) \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 b f}+\frac{\sinh ^3(e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{5 f}-\frac{(a-4 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{15 b f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((16*I)*a*(2*a^2 + 3*a*b - 8*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (32*I)*a*(a^2 + a*b - 2*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(8*a^2 - 48*a*b + 25*b^2 + 4*(4*a - 7*b)*b*Cosh[2*(e + f*x)] + 3*b^2*Cosh[4*(e + f*x)])*Sinh[2*(e + f*x)])/(240*b^2*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
72,1,170,177,0.8437256,"\int \sinh ^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Sinh[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{b \sinh (2 (e+f x)) (2 a+b \cosh (2 (e+f x))-b)+2 i \sqrt{2} a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i \sqrt{2} a (a-2 b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 b f \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","\frac{\sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}+\frac{i a (a-b) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} F\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 b f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{i (a-2 b) \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 b f \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}",1,"((-2*I)*Sqrt[2]*a*(a - 2*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (2*I)*Sqrt[2]*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + b*(2*a - b + b*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(6*b*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","A",1
73,1,69,60,0.0957497,"\int \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{i \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{f \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}",1,"((-I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a])/(f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","A",1
74,1,151,199,0.6084809,"\int \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Csch[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\sqrt{2} \coth (e+f x) (-2 a-b \cosh (2 (e+f x))+b)+2 i (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{2 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{\tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{f}-\frac{\coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{f}+\frac{b \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{a f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"(Sqrt[2]*(-2*a + b - b*Cosh[2*(e + f*x)])*Coth[e + f*x] - (2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (2*I)*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(2*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
75,1,208,276,2.9883287,"\int \text{csch}^4(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Csch[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\frac{\coth (e+f x) \text{csch}^2(e+f x) \left(4 \left(2 a^2-4 a b+b^2\right) \cosh (2 (e+f x))-(2 a-b) (8 a-b \cosh (4 (e+f x))-3 b)\right)}{2 \sqrt{2}}-4 i a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+2 i a (2 a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 a f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{(2 a-b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a f}+\frac{(2 a-b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a f}-\frac{\coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}-\frac{b \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{(2 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"(((4*(2*a^2 - 4*a*b + b^2)*Cosh[2*(e + f*x)] - (2*a - b)*(8*a - 3*b - b*Cosh[4*(e + f*x)]))*Coth[e + f*x]*Csch[e + f*x]^2)/(2*Sqrt[2]) + (2*I)*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (4*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(6*a*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
76,1,151,177,0.5904202,"\int \sinh ^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sinh[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} \sqrt{b} \sqrt{2 a+b \cosh (2 (e+f x))-b} \left(\left(6 a^2-51 a b+37 b^2\right) \cosh (e+f x)+b ((7 a-8 b) \cosh (3 (e+f x))+b \cosh (5 (e+f x)))\right)-12 (a-b)^2 (a+5 b) \log \left(\sqrt{2 a+b \cosh (2 (e+f x))-b}+\sqrt{2} \sqrt{b} \cosh (e+f x)\right)}{192 b^{3/2} f}","-\frac{(a-b)^2 (a+5 b) \tanh ^{-1}\left(\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{16 b^{3/2} f}+\frac{\cosh (e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^{5/2}}{6 b f}-\frac{(a+5 b) \cosh (e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^{3/2}}{24 b f}-\frac{(a-b) (a+5 b) \cosh (e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{16 b f}",1,"(Sqrt[2]*Sqrt[b]*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]*((6*a^2 - 51*a*b + 37*b^2)*Cosh[e + f*x] + b*((7*a - 8*b)*Cosh[3*(e + f*x)] + b*Cosh[5*(e + f*x)])) - 12*(a - b)^2*(a + 5*b)*Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]])/(192*b^(3/2)*f)","A",0
77,1,111,121,0.3238668,"\int \sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\frac{1}{2} \cosh (e+f x) (5 a+b \cosh (2 (e+f x))-4 b) \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}+\frac{3 (a-b)^2 \log \left(\sqrt{2 a+b \cosh (2 (e+f x))-b}+\sqrt{2} \sqrt{b} \cosh (e+f x)\right)}{\sqrt{b}}}{8 f}","\frac{3 (a-b) \cosh (e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{8 f}+\frac{\cosh (e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^{3/2}}{4 f}+\frac{3 (a-b)^2 \tanh ^{-1}\left(\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{8 \sqrt{b} f}",1,"((Cosh[e + f*x]*(5*a - 4*b + b*Cosh[2*(e + f*x)])*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])/2 + (3*(a - b)^2*Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]])/Sqrt[b])/(8*f)","A",1
78,1,136,127,0.5615396,"\int \text{csch}(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csch[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{-4 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)+b \cosh (e+f x) \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}-2 \sqrt{b} (b-3 a) \log \left(\sqrt{2 a+b \cosh (2 (e+f x))-b}+\sqrt{2} \sqrt{b} \cosh (e+f x)\right)}{4 f}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{f}+\frac{b \cosh (e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{2 f}+\frac{\sqrt{b} (3 a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{2 f}",1,"(-4*a^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] + b*Cosh[e + f*x]*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]] - 2*Sqrt[b]*(-3*a + b)*Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]])/(4*f)","A",1
79,1,143,130,0.7119468,"\int \text{csch}^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csch[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{4 b^{3/2} \log \left(\sqrt{2 a+b \cosh (2 (e+f x))-b}+\sqrt{2} \sqrt{b} \cosh (e+f x)\right)+2 \sqrt{a} (a-3 b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)-a \coth (e+f x) \text{csch}(e+f x) \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}{4 f}","\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{f}+\frac{\sqrt{a} (a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{2 f}-\frac{a \coth (e+f x) \text{csch}(e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{2 f}",1,"(2*Sqrt[a]*(a - 3*b)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] - a*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]]*Coth[e + f*x]*Csch[e + f*x] + 4*b^(3/2)*Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]])/(4*f)","A",1
80,1,123,135,0.6444126,"\int \text{csch}^5(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csch[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} \sqrt{a} \coth (e+f x) \text{csch}(e+f x) \sqrt{2 a+b \cosh (2 (e+f x))-b} \left(-2 a \text{csch}^2(e+f x)+3 a-5 b\right)-6 (a-b)^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)}{16 \sqrt{a} f}","-\frac{3 (a-b)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{8 \sqrt{a} f}-\frac{\coth (e+f x) \text{csch}^3(e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^{3/2}}{4 f}+\frac{3 (a-b) \coth (e+f x) \text{csch}(e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{8 f}",1,"(-6*(a - b)^2*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] + Sqrt[2]*Sqrt[a]*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]*Coth[e + f*x]*Csch[e + f*x]*(3*a - 5*b - 2*a*Csch[e + f*x]^2))/(16*Sqrt[a]*f)","A",1
81,1,174,199,1.0322041,"\int \text{csch}^7(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csch[e + f*x]^7*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\frac{(a-b)^2 (5 a+b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)}{a^{3/2}}-\frac{\coth (e+f x) \text{csch}^5(e+f x) \sqrt{a+\frac{1}{2} b \cosh (2 (e+f x))-\frac{b}{2}} \left(-4 \left(25 a^2-36 a b+3 b^2\right) \cosh (2 (e+f x))+\left(15 a^2-22 a b+3 b^2\right) \cosh (4 (e+f x))+149 a^2-122 a b+9 b^2\right)}{24 a}}{16 f}","\frac{(a-b)^2 (5 a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{16 a^{3/2} f}-\frac{\coth (e+f x) \text{csch}^5(e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^{5/2}}{6 a f}+\frac{(5 a+b) \coth (e+f x) \text{csch}^3(e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^{3/2}}{24 a f}-\frac{(a-b) (5 a+b) \coth (e+f x) \text{csch}(e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{16 a f}",1,"(((a - b)^2*(5*a + b)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]])/a^(3/2) - (Sqrt[a - b/2 + (b*Cosh[2*(e + f*x)])/2]*(149*a^2 - 122*a*b + 9*b^2 - 4*(25*a^2 - 36*a*b + 3*b^2)*Cosh[2*(e + f*x)] + (15*a^2 - 22*a*b + 3*b^2)*Cosh[4*(e + f*x)])*Coth[e + f*x]*Csch[e + f*x]^5)/(24*a))/(16*f)","A",1
82,1,262,367,2.817595,"\int \sinh ^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sinh[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} b \sinh (2 (e+f x)) \left(32 a^3+b \left(144 a^2-480 a b+299 b^2\right) \cosh (2 (e+f x))-496 a^2 b+2 b^2 (26 a-27 b) \cosh (4 (e+f x))+684 a b^2+5 b^3 \cosh (6 (e+f x))-250 b^3\right)-64 i a \left(2 a^3+3 a^2 b-13 a b^2+8 b^3\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+128 i a \left(a^3+2 a^2 b-12 a b^2+8 b^3\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{2240 b^2 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{2 (a-2 b) \left(a^2+4 a b-4 b^2\right) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 b^2 f}+\frac{\left(a^2-11 a b+8 b^2\right) \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 b f}-\frac{\left(a^2-11 a b+8 b^2\right) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{35 b f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{2 (a-2 b) \left(a^2+4 a b-4 b^2\right) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{35 b^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{b \sinh ^5(e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{7 f}+\frac{2 (4 a-3 b) \sinh ^3(e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 f}",1,"((128*I)*a*(a^3 + 2*a^2*b - 12*a*b^2 + 8*b^3)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (64*I)*a*(2*a^3 + 3*a^2*b - 13*a*b^2 + 8*b^3)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(32*a^3 - 496*a^2*b + 684*a*b^2 - 250*b^3 + b*(144*a^2 - 480*a*b + 299*b^2)*Cosh[2*(e + f*x)] + 2*(26*a - 27*b)*b^2*Cosh[4*(e + f*x)] + 5*b^3*Cosh[6*(e + f*x)])*Sinh[2*(e + f*x)])/(2240*b^2*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
83,1,213,236,1.3713324,"\int \sinh ^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sinh[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} b \sinh (2 (e+f x)) \left(48 a^2+4 b (9 a-7 b) \cosh (2 (e+f x))-68 a b+3 b^2 \cosh (4 (e+f x))+25 b^2\right)+16 i a \left(3 a^2-7 a b+4 b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-16 i a \left(3 a^2-13 a b+8 b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{240 b f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{i \left(3 a^2-13 a b+8 b^2\right) \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{15 b f \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}+\frac{\sinh (e+f x) \cosh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{5 f}+\frac{(3 a-4 b) \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 f}+\frac{i a (3 a-4 b) (a-b) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} F\left(i e+i f x\left|\frac{b}{a}\right.\right)}{15 b f \sqrt{a+b \sinh ^2(e+f x)}}",1,"((-16*I)*a*(3*a^2 - 13*a*b + 8*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (16*I)*a*(3*a^2 - 7*a*b + 4*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(48*a^2 - 68*a*b + 25*b^2 + 4*(9*a - 7*b)*b*Cosh[2*(e + f*x)] + 3*b^2*Cosh[4*(e + f*x)])*Sinh[2*(e + f*x)])/(240*b*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","A",1
84,1,169,174,0.7812049,"\int \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{b \sinh (2 (e+f x)) (2 a+b \cosh (2 (e+f x))-b)+2 i \sqrt{2} a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-4 i \sqrt{2} a (2 a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 f \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","\frac{b \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}+\frac{i a (a-b) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} F\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{2 i (2 a-b) \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 f \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}",1,"((-4*I)*Sqrt[2]*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (2*I)*Sqrt[2]*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + b*(2*a - b + b*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(6*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","A",1
85,1,155,204,0.9610751,"\int \text{csch}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csch[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2),x]","-\frac{a \left(\sqrt{2} \coth (e+f x) (2 a+b \cosh (2 (e+f x))-b)-2 i (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+2 i (a+b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)\right)}{2 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{(a+b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{f}-\frac{a \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{f}+\frac{2 b \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{(a+b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"-1/2*(a*(Sqrt[2]*(2*a - b + b*Cosh[2*(e + f*x)])*Coth[e + f*x] + (2*I)*(a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (2*I)*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a]))/(f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
86,1,213,267,4.0436536,"\int \text{csch}^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csch[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{-2 i \left(2 a^2-5 a b+3 b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+\frac{\coth (e+f x) \text{csch}^2(e+f x) \left(2 \left(2 a^2-7 a b+4 b^2\right) \cosh (2 (e+f x))-8 a^2+b (a-2 b) \cosh (4 (e+f x))+13 a b-6 b^2\right)}{\sqrt{2}}+4 i a (a-2 b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{2 (a-2 b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}+\frac{2 (a-2 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}-\frac{a \coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}-\frac{b (a-3 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{2 (a-2 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"(((-8*a^2 + 13*a*b - 6*b^2 + 2*(2*a^2 - 7*a*b + 4*b^2)*Cosh[2*(e + f*x)] + (a - 2*b)*b*Cosh[4*(e + f*x)])*Coth[e + f*x]*Csch[e + f*x]^2)/Sqrt[2] + (4*I)*a*(a - 2*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (2*I)*(2*a^2 - 5*a*b + 3*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(6*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
87,1,208,232,1.4126207,"\int \left(a+b \sinh ^2(c+d x)\right)^{5/2} \, dx","Integrate[(a + b*Sinh[c + d*x]^2)^(5/2),x]","\frac{\sqrt{2} b \sinh (2 (c+d x)) \left(88 a^2+28 b (2 a-b) \cosh (2 (c+d x))-88 a b+3 b^2 \cosh (4 (c+d x))+25 b^2\right)+64 i a \left(2 a^2-3 a b+b^2\right) \sqrt{\frac{2 a+b \cosh (2 (c+d x))-b}{a}} F\left(i (c+d x)\left|\frac{b}{a}\right.\right)-16 i a \left(23 a^2-23 a b+8 b^2\right) \sqrt{\frac{2 a+b \cosh (2 (c+d x))-b}{a}} E\left(i (c+d x)\left|\frac{b}{a}\right.\right)}{240 d \sqrt{2 a+b \cosh (2 (c+d x))-b}}","-\frac{i \left(23 a^2-23 a b+8 b^2\right) \sqrt{a+b \sinh ^2(c+d x)} E\left(i c+i d x\left|\frac{b}{a}\right.\right)}{15 d \sqrt{\frac{b \sinh ^2(c+d x)}{a}+1}}+\frac{b \sinh (c+d x) \cosh (c+d x) \left(a+b \sinh ^2(c+d x)\right)^{3/2}}{5 d}+\frac{4 b (2 a-b) \sinh (c+d x) \cosh (c+d x) \sqrt{a+b \sinh ^2(c+d x)}}{15 d}+\frac{4 i a (a-b) (2 a-b) \sqrt{\frac{b \sinh ^2(c+d x)}{a}+1} F\left(i c+i d x\left|\frac{b}{a}\right.\right)}{15 d \sqrt{a+b \sinh ^2(c+d x)}}",1,"((-16*I)*a*(23*a^2 - 23*a*b + 8*b^2)*Sqrt[(2*a - b + b*Cosh[2*(c + d*x)])/a]*EllipticE[I*(c + d*x), b/a] + (64*I)*a*(2*a^2 - 3*a*b + b^2)*Sqrt[(2*a - b + b*Cosh[2*(c + d*x)])/a]*EllipticF[I*(c + d*x), b/a] + Sqrt[2]*b*(88*a^2 - 88*a*b + 25*b^2 + 28*(2*a - b)*b*Cosh[2*(c + d*x)] + 3*b^2*Cosh[4*(c + d*x)])*Sinh[2*(c + d*x)])/(240*d*Sqrt[2*a - b + b*Cosh[2*(c + d*x)]])","A",1
88,1,11,11,0.0062142,"\int \sqrt{1+\sinh ^2(x)} \, dx","Integrate[Sqrt[1 + Sinh[x]^2],x]","\sqrt{\cosh ^2(x)} \tanh (x)","\sqrt{\cosh ^2(x)} \tanh (x)",1,"Sqrt[Cosh[x]^2]*Tanh[x]","A",1
89,1,13,13,0.0054014,"\int \sqrt{-1-\sinh ^2(x)} \, dx","Integrate[Sqrt[-1 - Sinh[x]^2],x]","\sqrt{-\cosh ^2(x)} \tanh (x)","\sqrt{-\cosh ^2(x)} \tanh (x)",1,"Sqrt[-Cosh[x]^2]*Tanh[x]","A",1
90,1,11,11,0.0247072,"\int \sqrt{1-\sinh ^2(x)} \, dx","Integrate[Sqrt[1 - Sinh[x]^2],x]","-i E(i x|-1)","-i E(i x|-1)",1,"(-I)*EllipticE[I*x, -1]","A",1
91,1,33,33,0.0344293,"\int \sqrt{-1+\sinh ^2(x)} \, dx","Integrate[Sqrt[-1 + Sinh[x]^2],x]","\frac{i \sqrt{3-\cosh (2 x)} E(i x|-1)}{\sqrt{\cosh (2 x)-3}}","-\frac{i \sqrt{\sinh ^2(x)-1} E(i x|-1)}{\sqrt{1-\sinh ^2(x)}}",1,"(I*Sqrt[3 - Cosh[2*x]]*EllipticE[I*x, -1])/Sqrt[-3 + Cosh[2*x]]","A",1
92,1,54,42,0.0440399,"\int \sqrt{a+b \sinh ^2(x)} \, dx","Integrate[Sqrt[a + b*Sinh[x]^2],x]","-\frac{i a \sqrt{\frac{2 a+b \cosh (2 x)-b}{a}} E\left(i x\left|\frac{b}{a}\right.\right)}{\sqrt{2 a+b \cosh (2 x)-b}}","-\frac{i \sqrt{a+b \sinh ^2(x)} E\left(i x\left|\frac{b}{a}\right.\right)}{\sqrt{\frac{b \sinh ^2(x)}{a}+1}}",1,"((-I)*a*Sqrt[(2*a - b + b*Cosh[2*x])/a]*EllipticE[I*x, b/a])/Sqrt[2*a - b + b*Cosh[2*x]]","A",1
93,1,23,29,0.02066,"\int \left(1+\sinh ^2(x)\right)^{3/2} \, dx","Integrate[(1 + Sinh[x]^2)^(3/2),x]","\frac{1}{12} (9 \sinh (x)+\sinh (3 x)) \sqrt{\cosh ^2(x)} \text{sech}(x)","\frac{1}{3} \cosh ^2(x)^{3/2} \tanh (x)+\frac{2}{3} \sqrt{\cosh ^2(x)} \tanh (x)",1,"(Sqrt[Cosh[x]^2]*Sech[x]*(9*Sinh[x] + Sinh[3*x]))/12","A",1
94,1,25,33,0.0065623,"\int \left(-1-\sinh ^2(x)\right)^{3/2} \, dx","Integrate[(-1 - Sinh[x]^2)^(3/2),x]","-\frac{1}{12} (9 \sinh (x)+\sinh (3 x)) \sqrt{-\cosh ^2(x)} \text{sech}(x)","\frac{1}{3} \left(-\cosh ^2(x)\right)^{3/2} \tanh (x)-\frac{2}{3} \sqrt{-\cosh ^2(x)} \tanh (x)",1,"-1/12*(Sqrt[-Cosh[x]^2]*Sech[x]*(9*Sinh[x] + Sinh[3*x]))","A",1
95,1,45,45,0.0719933,"\int \left(1-\sinh ^2(x)\right)^{3/2} \, dx","Integrate[(1 - Sinh[x]^2)^(3/2),x]","\frac{1}{12} \left(8 i F(i x|-1)-24 i E(i x|-1)-\sinh (2 x) \sqrt{6-2 \cosh (2 x)}\right)","\frac{2}{3} i F(i x|-1)-2 i E(i x|-1)-\frac{1}{3} \sinh (x) \sqrt{1-\sinh ^2(x)} \cosh (x)",1,"((-24*I)*EllipticE[I*x, -1] + (8*I)*EllipticF[I*x, -1] - Sqrt[6 - 2*Cosh[2*x]]*Sinh[2*x])/12","A",1
96,1,78,87,0.123485,"\int \left(-1+\sinh ^2(x)\right)^{3/2} \, dx","Integrate[(-1 + Sinh[x]^2)^(3/2),x]","\frac{\frac{\sinh (4 x)-6 \sinh (2 x)}{\sqrt{2}}+8 i \sqrt{3-\cosh (2 x)} F(i x|-1)-24 i \sqrt{3-\cosh (2 x)} E(i x|-1)}{12 \sqrt{\cosh (2 x)-3}}","\frac{1}{3} \sinh (x) \sqrt{\sinh ^2(x)-1} \cosh (x)+\frac{2 i \sqrt{1-\sinh ^2(x)} F(i x|-1)}{3 \sqrt{\sinh ^2(x)-1}}+\frac{2 i \sqrt{\sinh ^2(x)-1} E(i x|-1)}{\sqrt{1-\sinh ^2(x)}}",1,"((-24*I)*Sqrt[3 - Cosh[2*x]]*EllipticE[I*x, -1] + (8*I)*Sqrt[3 - Cosh[2*x]]*EllipticF[I*x, -1] + (-6*Sinh[2*x] + Sinh[4*x])/Sqrt[2])/(12*Sqrt[-3 + Cosh[2*x]])","A",1
97,1,132,123,0.3878259,"\int \left(a+b \sinh ^2(x)\right)^{3/2} \, dx","Integrate[(a + b*Sinh[x]^2)^(3/2),x]","\frac{\sqrt{2} b \sinh (2 x) (2 a+b \cosh (2 x)-b)+4 i a (a-b) \sqrt{\frac{2 a+b \cosh (2 x)-b}{a}} F\left(i x\left|\frac{b}{a}\right.\right)-8 i a (2 a-b) \sqrt{\frac{2 a+b \cosh (2 x)-b}{a}} E\left(i x\left|\frac{b}{a}\right.\right)}{12 \sqrt{2 a+b \cosh (2 x)-b}}","\frac{1}{3} b \sinh (x) \cosh (x) \sqrt{a+b \sinh ^2(x)}+\frac{i a (a-b) \sqrt{\frac{b \sinh ^2(x)}{a}+1} F\left(i x\left|\frac{b}{a}\right.\right)}{3 \sqrt{a+b \sinh ^2(x)}}-\frac{2 i (2 a-b) \sqrt{a+b \sinh ^2(x)} E\left(i x\left|\frac{b}{a}\right.\right)}{3 \sqrt{\frac{b \sinh ^2(x)}{a}+1}}",1,"((-8*I)*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*x])/a]*EllipticE[I*x, b/a] + (4*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*x])/a]*EllipticF[I*x, b/a] + Sqrt[2]*b*(2*a - b + b*Cosh[2*x])*Sinh[2*x])/(12*Sqrt[2*a - b + b*Cosh[2*x]])","A",1
98,1,98,83,0.2796981,"\int \frac{\sinh ^3(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Sinh[e + f*x]^3/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\cosh (e+f x) \sqrt{2 a+b \cosh (2 (e+f x))-b}}{2 \sqrt{2} b f}-\frac{(a+b) \log \left(\sqrt{2 a+b \cosh (2 (e+f x))-b}+\sqrt{2} \sqrt{b} \cosh (e+f x)\right)}{2 b^{3/2} f}","\frac{\cosh (e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{2 b f}-\frac{(a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{2 b^{3/2} f}",1,"(Cosh[e + f*x]*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])/(2*Sqrt[2]*b*f) - ((a + b)*Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]])/(2*b^(3/2)*f)","A",1
99,1,49,41,0.1109841,"\int \frac{\sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Sinh[e + f*x]/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\log \left(\sqrt{2 a+b \cosh (2 (e+f x))-b}+\sqrt{2} \sqrt{b} \cosh (e+f x)\right)}{\sqrt{b} f}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{\sqrt{b} f}",1,"Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]]/(Sqrt[b]*f)","A",1
100,1,49,42,0.1797507,"\int \frac{\text{csch}(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Csch[e + f*x]/Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)}{\sqrt{a} f}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{\sqrt{a} f}",1,"-(ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]]/(Sqrt[a]*f))","A",1
101,1,102,89,0.3409053,"\int \frac{\text{csch}^3(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Csch[e + f*x]^3/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{2 (a+b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)-\sqrt{2} \sqrt{a} \coth (e+f x) \text{csch}(e+f x) \sqrt{2 a+b \cosh (2 (e+f x))-b}}{4 a^{3/2} f}","\frac{(a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{2 a^{3/2} f}-\frac{\coth (e+f x) \text{csch}(e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{2 a f}",1,"(2*(a + b)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] - Sqrt[2]*Sqrt[a]*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]*Coth[e + f*x]*Csch[e + f*x])/(4*a^(3/2)*f)","A",1
102,1,168,229,0.9565136,"\int \frac{\sinh ^4(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Sinh[e + f*x]^4/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{b \sinh (2 (e+f x)) (2 a+b \cosh (2 (e+f x))-b)-2 i \sqrt{2} a (2 a+b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+4 i \sqrt{2} a (a+b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 b^2 f \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","-\frac{2 (a+b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 b^2 f}+\frac{2 (a+b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 b^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{\sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 b f}-\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 b f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((4*I)*Sqrt[2]*a*(a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (2*I)*Sqrt[2]*a*(2*a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + b*(2*a - b + b*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(6*b^2*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","C",1
103,1,89,128,0.2501607,"\int \frac{\sinh ^2(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Sinh[e + f*x]^2/Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{i \sqrt{2 a+b \cosh (2 (e+f x))-b} \left(E\left(i (e+f x)\left|\frac{b}{a}\right.\right)-F\left(i (e+f x)\left|\frac{b}{a}\right.\right)\right)}{b f \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}}}","\frac{i a \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} F\left(i e+i f x\left|\frac{b}{a}\right.\right)}{b f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{i \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{b f \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}",1,"((-I)*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]*(EllipticE[I*(e + f*x), b/a] - EllipticF[I*(e + f*x), b/a]))/(b*f*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a])","A",1
104,1,68,60,0.0833274,"\int \frac{1}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[1/Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{i \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{i \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} F\left(i e+i f x\left|\frac{b}{a}\right.\right)}{f \sqrt{a+b \sinh ^2(e+f x)}}",1,"((-I)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","A",1
105,1,150,134,0.5318285,"\int \frac{\text{csch}^2(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Csch[e + f*x]^2/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\sqrt{2} \coth (e+f x) (-2 a-b \cosh (2 (e+f x))+b)+2 i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{2 a f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{\tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{a f}-\frac{\coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{a f}-\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{a f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"(Sqrt[2]*(-2*a + b - b*Cosh[2*(e + f*x)])*Coth[e + f*x] - (2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(2*a*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
106,1,201,267,3.5400864,"\int \frac{\text{csch}^4(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Csch[e + f*x]^4/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\frac{\coth (e+f x) \text{csch}^2(e+f x) \left(\left(4 a^2-2 a b-4 b^2\right) \cosh (2 (e+f x))-8 a^2+b (a+b) \cosh (4 (e+f x))+a b+3 b^2\right)}{\sqrt{2}}-2 i a (2 a+b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+4 i a (a+b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 a^2 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{2 (a+b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^2 f}+\frac{2 (a+b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^2 f}-\frac{b \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{2 (a+b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{\coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a f}",1,"(((-8*a^2 + a*b + 3*b^2 + (4*a^2 - 2*a*b - 4*b^2)*Cosh[2*(e + f*x)] + b*(a + b)*Cosh[4*(e + f*x)])*Coth[e + f*x]*Csch[e + f*x]^2)/Sqrt[2] + (4*I)*a*(a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (2*I)*a*(2*a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(6*a^2*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
107,1,98,83,0.4112507,"\int \frac{\sinh ^3(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sinh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\log \left(\sqrt{2 a+b \cosh (2 (e+f x))-b}+\sqrt{2} \sqrt{b} \cosh (e+f x)\right)}{b^{3/2} f}-\frac{\sqrt{2} a \cosh (e+f x)}{b f (a-b) \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{b^{3/2} f}-\frac{a \cosh (e+f x)}{b f (a-b) \sqrt{a+b \cosh ^2(e+f x)-b}}",1,"-((Sqrt[2]*a*Cosh[e + f*x])/((a - b)*b*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])) + Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]]/(b^(3/2)*f)","A",1
108,1,43,36,0.149202,"\int \frac{\sinh (e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sinh[e + f*x]/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} \cosh (e+f x)}{f (a-b) \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{\cosh (e+f x)}{f (a-b) \sqrt{a+b \cosh ^2(e+f x)-b}}",1,"(Sqrt[2]*Cosh[e + f*x])/((a - b)*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","A",1
109,1,98,84,0.3767441,"\int \frac{\text{csch}(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csch[e + f*x]/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{-\frac{\sqrt{2} \sqrt{a} b \cosh (e+f x)}{(a-b) \sqrt{2 a+b \cosh (2 (e+f x))-b}}-\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)}{a^{3/2} f}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{a^{3/2} f}-\frac{b \cosh (e+f x)}{a f (a-b) \sqrt{a+b \cosh ^2(e+f x)-b}}",1,"(-ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] - (Sqrt[2]*Sqrt[a]*b*Cosh[e + f*x])/((a - b)*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]))/(a^(3/2)*f)","A",1
110,1,134,139,0.715238,"\int \frac{\text{csch}^3(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csch[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\frac{(a+3 b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)}{a^{5/2}}-\frac{\coth (e+f x) \text{csch}(e+f x) \left(2 a^2+b (a-3 b) \cosh (2 (e+f x))-3 a b+3 b^2\right)}{a^2 (a-b) \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}}{2 f}","\frac{(a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{2 a^{5/2} f}-\frac{b (a-3 b) \cosh (e+f x)}{2 a^2 f (a-b) \sqrt{a+b \cosh ^2(e+f x)-b}}-\frac{\coth (e+f x) \text{csch}(e+f x)}{2 a f \sqrt{a+b \cosh ^2(e+f x)-b}}",1,"(((a + 3*b)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]])/a^(5/2) - ((2*a^2 - 3*a*b + 3*b^2 + (a - 3*b)*b*Cosh[2*(e + f*x)])*Coth[e + f*x]*Csch[e + f*x])/(a^2*(a - b)*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]]))/(2*f)","A",1
111,1,211,341,1.2418599,"\int \frac{\sinh ^6(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sinh[e + f*x]^6/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{-b \sinh (2 (e+f x)) \left(-8 a^2+b (b-a) \cosh (2 (e+f x))+3 a b-b^2\right)-2 i \sqrt{2} a \left(8 a^2-7 a b-b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+2 i \sqrt{2} a \left(8 a^2-3 a b-2 b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 b^3 f (a-b) \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","-\frac{\left(8 a^2-3 a b-2 b^2\right) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 b^3 f (a-b)}+\frac{\left(8 a^2-3 a b-2 b^2\right) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 b^3 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{(4 a-b) \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 b^2 f (a-b)}-\frac{(4 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 b^2 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{a \sinh ^3(e+f x) \cosh (e+f x)}{b f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}",1,"((2*I)*Sqrt[2]*a*(8*a^2 - 3*a*b - 2*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (2*I)*Sqrt[2]*a*(8*a^2 - 7*a*b - b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] - b*(-8*a^2 + 3*a*b - b^2 + b*(-a + b)*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(6*(a - b)*b^3*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","C",1
112,1,156,256,1.0465615,"\int \frac{\sinh ^4(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sinh[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{a \left(4 i (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i (2 a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)-\sqrt{2} b \sinh (2 (e+f x))\right)}{2 b^2 f (a-b) \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{(2 a-b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{b^2 f (a-b)}-\frac{(2 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{b^2 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{a \sinh (e+f x) \cosh (e+f x)}{b f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{b f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"(a*((-2*I)*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (4*I)*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] - Sqrt[2]*b*Sinh[2*(e + f*x)]))/(2*(a - b)*b^2*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
113,1,151,173,0.4858431,"\int \frac{\sinh ^2(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sinh[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{-i \sqrt{2} (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+i \sqrt{2} a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)+b \sinh (2 (e+f x))}{b f (a-b) \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","\frac{\sinh (e+f x) \cosh (e+f x)}{f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}-\frac{i \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} F\left(i e+i f x\left|\frac{b}{a}\right.\right)}{b f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{i \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{b f (a-b) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}",1,"(I*Sqrt[2]*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - I*Sqrt[2]*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + b*Sinh[2*(e + f*x)])/((a - b)*b*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","A",1
114,1,100,115,0.1793096,"\int \frac{1}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^(-3/2),x]","\frac{-\sqrt{2} b \sinh (2 (e+f x))-2 i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{2 a f (a-b) \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{b \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}-\frac{i \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{a f (a-b) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}",1,"((-2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - Sqrt[2]*b*Sinh[2*(e + f*x)])/(2*a*(a - b)*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","A",1
115,1,185,290,1.2941328,"\int \frac{\text{csch}^2(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csch[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{-\left(\coth (e+f x) \left(2 a^2+b (a-2 b) \cosh (2 (e+f x))-3 a b+2 b^2\right)\right)+i \sqrt{2} a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-i \sqrt{2} a (a-2 b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{a^2 f (a-b) \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","\frac{(a-2 b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{a^2 f (a-b)}-\frac{(a-2 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{a^2 f (a-b)}-\frac{b \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{a^2 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{(a-2 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{a^2 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{b \coth (e+f x)}{a f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}",1,"(-((2*a^2 - 3*a*b + 2*b^2 + (a - 2*b)*b*Cosh[2*(e + f*x)])*Coth[e + f*x]) - I*Sqrt[2]*a*(a - 2*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + I*Sqrt[2]*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(a^2*(a - b)*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","C",1
116,1,130,143,0.8882162,"\int \frac{\sinh ^5(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sinh[e + f*x]^5/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{\frac{\log \left(\sqrt{2 a+b \cosh (2 (e+f x))-b}+\sqrt{2} \sqrt{b} \cosh (e+f x)\right)}{b^{5/2}}-\frac{2 \sqrt{2} a \cosh (e+f x) \left(3 a^2+b (2 a-3 b) \cosh (2 (e+f x))-7 a b+3 b^2\right)}{3 b^2 (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}}{f}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{b^{5/2} f}-\frac{a (3 a-5 b) \cosh (e+f x)}{3 b^2 f (a-b)^2 \sqrt{a+b \cosh ^2(e+f x)-b}}-\frac{a \sinh ^2(e+f x) \cosh (e+f x)}{3 b f (a-b) \left(a+b \cosh ^2(e+f x)-b\right)^{3/2}}",1,"((-2*Sqrt[2]*a*Cosh[e + f*x]*(3*a^2 - 7*a*b + 3*b^2 + (2*a - 3*b)*b*Cosh[2*(e + f*x)]))/(3*(a - b)^2*b^2*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2)) + Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]]/b^(5/2))/f","A",0
117,1,67,87,0.3329231,"\int \frac{\sinh ^3(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sinh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{\sqrt{2} \cosh (e+f x) ((a-3 b) \cosh (2 (e+f x))-5 a+3 b)}{3 f (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","\frac{\sinh ^2(e+f x) \cosh (e+f x)}{3 f (a-b) \left(a+b \cosh ^2(e+f x)-b\right)^{3/2}}-\frac{2 \cosh (e+f x)}{3 f (a-b)^2 \sqrt{a+b \cosh ^2(e+f x)-b}}",1,"(Sqrt[2]*Cosh[e + f*x]*(-5*a + 3*b + (a - 3*b)*Cosh[2*(e + f*x)]))/(3*(a - b)^2*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","A",1
118,1,63,79,0.1756071,"\int \frac{\sinh (e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sinh[e + f*x]/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{2 \sqrt{2} \cosh (e+f x) (3 a+b \cosh (2 (e+f x))-2 b)}{3 f (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","\frac{2 \cosh (e+f x)}{3 f (a-b)^2 \sqrt{a+b \cosh ^2(e+f x)-b}}+\frac{\cosh (e+f x)}{3 f (a-b) \left(a+b \cosh ^2(e+f x)-b\right)^{3/2}}",1,"(2*Sqrt[2]*Cosh[e + f*x]*(3*a - 2*b + b*Cosh[2*(e + f*x)]))/(3*(a - b)^2*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","A",1
119,1,130,136,0.793524,"\int \frac{\text{csch}(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csch[e + f*x]/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{\frac{\sqrt{2} b \cosh (e+f x) \left(-12 a^2+b (3 b-5 a) \cosh (2 (e+f x))+13 a b-3 b^2\right)}{3 a^2 (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)}{a^{5/2}}}{f}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right)}{a^{5/2} f}-\frac{b (5 a-3 b) \cosh (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \cosh ^2(e+f x)-b}}-\frac{b \cosh (e+f x)}{3 a f (a-b) \left(a+b \cosh ^2(e+f x)-b\right)^{3/2}}",1,"(-(ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]]/a^(5/2)) + (Sqrt[2]*b*Cosh[e + f*x]*(-12*a^2 + 13*a*b - 3*b^2 + b*(-5*a + 3*b)*Cosh[2*(e + f*x)]))/(3*a^2*(a - b)^2*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2)))/f","A",1
120,1,207,344,2.0892098,"\int \frac{\sinh ^6(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sinh[e + f*x]^6/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{a \left(\sqrt{2} b \sinh (2 (e+f x)) \left(-8 a^2+b (7 b-5 a) \cosh (2 (e+f x))+17 a b-7 b^2\right)+2 i a \left(8 a^2-17 a b+9 b^2\right) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i a \left(8 a^2-13 a b+3 b^2\right) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)\right)}{6 b^3 f (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","\frac{\left(8 a^2-13 a b+3 b^2\right) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 b^3 f (a-b)^2}-\frac{\left(8 a^2-13 a b+3 b^2\right) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 b^3 f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{2 a (2 a-3 b) \sinh (e+f x) \cosh (e+f x)}{3 b^2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{2 (2 a-3 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 b^2 f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{a \sinh ^3(e+f x) \cosh (e+f x)}{3 b f (a-b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"(a*((-2*I)*a*(8*a^2 - 13*a*b + 3*b^2)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] + (2*I)*a*(8*a^2 - 17*a*b + 9*b^2)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(-8*a^2 + 17*a*b - 7*b^2 + b*(-5*a + 7*b)*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)]))/(6*(a - b)^2*b^3*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","C",1
121,1,198,244,1.6698402,"\int \frac{\sinh ^4(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sinh[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{-\sqrt{2} b \sinh (2 (e+f x)) \left(-a^2-b (a-2 b) \cosh (2 (e+f x))+4 a b-2 b^2\right)-i a \left(2 a^2-5 a b+3 b^2\right) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+2 i a^2 (a-2 b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{3 b^2 f (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","\frac{2 \sqrt{a} (a-2 b) \cosh (e+f x) E\left(\tan ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right)|1-\frac{a}{b}\right)}{3 b^{3/2} f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)} \sqrt{\frac{a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}-\frac{a \sinh (e+f x) \cosh (e+f x)}{3 b f (a-b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}-\frac{(a-3 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a b f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((2*I)*a^2*(a - 2*b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] - I*a*(2*a^2 - 5*a*b + 3*b^2)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] - Sqrt[2]*b*(-a^2 + 4*a*b - 2*b^2 - (a - 2*b)*b*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(3*(a - b)^2*b^2*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","C",1
122,1,187,241,1.4400669,"\int \frac{\sinh ^2(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sinh[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{\sqrt{2} b \sinh (2 (e+f x)) \left(4 a^2+b (a+b) \cosh (2 (e+f x))-a b-b^2\right)-2 i a^2 (a-b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+2 i a^2 (a+b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 a b f (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","\frac{(a+b) \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\sinh (e+f x) \cosh (e+f x)}{3 f (a-b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}-\frac{i \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} F\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 b f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}+\frac{i (a+b) \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 a b f (a-b)^2 \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}",1,"((2*I)*a^2*(a + b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] - (2*I)*a^2*(a - b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(4*a^2 - a*b - b^2 + b*(a + b)*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(6*a*(a - b)^2*b*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","A",1
123,1,190,251,1.3750785,"\int \frac{1}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^(-5/2),x]","\frac{\sqrt{2} b \sinh (2 (e+f x)) \left(-5 a^2+b (b-2 a) \cosh (2 (e+f x))+5 a b-b^2\right)+i a^2 (a-b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i a^2 (2 a-b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{3 a^2 f (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","-\frac{2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}-\frac{2 i (2 a-b) \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 a^2 f (a-b)^2 \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}-\frac{b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}+\frac{i \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} F\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 a f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}",1,"((-2*I)*a^2*(2*a - b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] + I*a^2*(a - b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(-5*a^2 + 5*a*b - b^2 + b*(-2*a + b)*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(3*a^2*(a - b)^2*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","A",1
124,1,234,385,2.4000018,"\int \frac{\text{csch}^2(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csch[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{i \left(4 a^2 \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} \left(\left(3 a^2-7 a b+4 b^2\right) F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+\left(-3 a^2+13 a b-8 b^2\right) E\left(i (e+f x)\left|\frac{b}{a}\right.\right)\right)+2 i \sqrt{2} \left(-2 a b^2 (a-b) \sinh (2 (e+f x))-b^2 (7 a-5 b) \sinh (2 (e+f x)) (2 a+b \cosh (2 (e+f x))-b)+3 (a-b)^2 \coth (e+f x) (2 a+b \cosh (2 (e+f x))-b)^2\right)\right)}{12 a^3 f (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","-\frac{2 b (3 a-2 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^3 f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{2 b (3 a-2 b) \coth (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\left(3 a^2-13 a b+8 b^2\right) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)^2}-\frac{\left(3 a^2-13 a b+8 b^2\right) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)^2}-\frac{\left(3 a^2-13 a b+8 b^2\right) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^3 f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{b \coth (e+f x)}{3 a f (a-b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"((I/12)*(4*a^2*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*((-3*a^2 + 13*a*b - 8*b^2)*EllipticE[I*(e + f*x), b/a] + (3*a^2 - 7*a*b + 4*b^2)*EllipticF[I*(e + f*x), b/a]) + (2*I)*Sqrt[2]*(3*(a - b)^2*(2*a - b + b*Cosh[2*(e + f*x)])^2*Coth[e + f*x] - 2*a*(a - b)*b^2*Sinh[2*(e + f*x)] - (7*a - 5*b)*b^2*(2*a - b + b*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])))/(a^3*(a - b)^2*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","C",1
125,1,19,14,0.0110953,"\int \frac{1}{\sqrt{1+\sinh ^2(x)}} \, dx","Integrate[1/Sqrt[1 + Sinh[x]^2],x]","\frac{2 \cosh (x) \tan ^{-1}\left(\tanh \left(\frac{x}{2}\right)\right)}{\sqrt{\cosh ^2(x)}}","\frac{\cosh (x) \tan ^{-1}(\sinh (x))}{\sqrt{\cosh ^2(x)}}",1,"(2*ArcTan[Tanh[x/2]]*Cosh[x])/Sqrt[Cosh[x]^2]","A",1
126,1,11,11,0.0400887,"\int \frac{1}{\sqrt{1-\sinh ^2(x)}} \, dx","Integrate[1/Sqrt[1 - Sinh[x]^2],x]","-i F(i x|-1)","-i F(i x|-1)",1,"(-I)*EllipticF[I*x, -1]","A",1
127,1,33,33,0.0435849,"\int \frac{1}{\sqrt{-1+\sinh ^2(x)}} \, dx","Integrate[1/Sqrt[-1 + Sinh[x]^2],x]","-\frac{i \sqrt{3-\cosh (2 x)} F(i x|-1)}{\sqrt{\cosh (2 x)-3}}","-\frac{i \sqrt{1-\sinh ^2(x)} F(i x|-1)}{\sqrt{\sinh ^2(x)-1}}",1,"((-I)*Sqrt[3 - Cosh[2*x]]*EllipticF[I*x, -1])/Sqrt[-3 + Cosh[2*x]]","A",1
128,1,21,16,0.0087772,"\int \frac{1}{\sqrt{-1-\sinh ^2(x)}} \, dx","Integrate[1/Sqrt[-1 - Sinh[x]^2],x]","\frac{2 \cosh (x) \tan ^{-1}\left(\tanh \left(\frac{x}{2}\right)\right)}{\sqrt{-\cosh ^2(x)}}","\frac{\cosh (x) \tan ^{-1}(\sinh (x))}{\sqrt{-\cosh ^2(x)}}",1,"(2*ArcTan[Tanh[x/2]]*Cosh[x])/Sqrt[-Cosh[x]^2]","A",1
129,1,53,42,0.0532915,"\int \frac{1}{\sqrt{a+b \sinh ^2(x)}} \, dx","Integrate[1/Sqrt[a + b*Sinh[x]^2],x]","-\frac{i \sqrt{\frac{2 a+b \cosh (2 x)-b}{a}} F\left(i x\left|\frac{b}{a}\right.\right)}{\sqrt{2 a+b \cosh (2 x)-b}}","-\frac{i \sqrt{\frac{b \sinh ^2(x)}{a}+1} F\left(i x\left|\frac{b}{a}\right.\right)}{\sqrt{a+b \sinh ^2(x)}}",1,"((-I)*Sqrt[(2*a - b + b*Cosh[2*x])/a]*EllipticF[I*x, b/a])/Sqrt[2*a - b + b*Cosh[2*x]]","A",1
130,0,0,128,9.649139,"\int (d \sinh (e+f x))^m \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[(d*Sinh[e + f*x])^m*(a + b*Sinh[e + f*x]^2)^p,x]","\int (d \sinh (e+f x))^m \left(a+b \sinh ^2(e+f x)\right)^p \, dx","\frac{d \cosh (e+f x) \left(-\sinh ^2(e+f x)\right)^{\frac{1-m}{2}} (d \sinh (e+f x))^{m-1} \left(a+b \cosh ^2(e+f x)-b\right)^p \left(\frac{b \cosh ^2(e+f x)}{a-b}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{1-m}{2},-p;\frac{3}{2};\cosh ^2(e+f x),-\frac{b \cosh ^2(e+f x)}{a-b}\right)}{f}",1,"Integrate[(d*Sinh[e + f*x])^m*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
131,0,0,226,11.8042102,"\int \sinh ^5(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Sinh[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^p,x]","\int \sinh ^5(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","\frac{\left(3 a^2+4 a b (p+1)+4 b^2 \left(p^2+3 p+2\right)\right) \cosh (e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^p \left(\frac{b \cosh ^2(e+f x)}{a-b}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \cosh ^2(e+f x)}{a-b}\right)}{b^2 f (2 p+3) (2 p+5)}-\frac{(3 a+2 b (p+2)) \cosh (e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^{p+1}}{b^2 f (2 p+3) (2 p+5)}+\frac{\sinh ^2(e+f x) \cosh (e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^{p+1}}{b f (2 p+5)}",1,"Integrate[Sinh[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
132,0,0,137,15.9579255,"\int \sinh ^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Sinh[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^p,x]","\int \sinh ^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","\frac{\cosh (e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^{p+1}}{b f (2 p+3)}-\frac{(a+2 b (p+1)) \cosh (e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^p \left(\frac{b \cosh ^2(e+f x)}{a-b}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \cosh ^2(e+f x)}{a-b}\right)}{b f (2 p+3)}",1,"Integrate[Sinh[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
133,1,78,78,0.2168661,"\int \sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^p,x]","\frac{\cosh (e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^p \left(\frac{b \cosh ^2(e+f x)}{a-b}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \cosh ^2(e+f x)}{a-b}\right)}{f}","\frac{\cosh (e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^p \left(\frac{b \cosh ^2(e+f x)}{a-b}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \cosh ^2(e+f x)}{a-b}\right)}{f}",1,"(Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*Cosh[e + f*x]^2)/(a - b))])/(f*(1 + (b*Cosh[e + f*x]^2)/(a - b))^p)","A",1
134,0,0,88,4.4691628,"\int \text{csch}(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Csch[e + f*x]*(a + b*Sinh[e + f*x]^2)^p,x]","\int \text{csch}(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","-\frac{\cosh (e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^p \left(\frac{b \cosh ^2(e+f x)}{a-b}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};\cosh ^2(e+f x),-\frac{b \cosh ^2(e+f x)}{a-b}\right)}{f}",1,"Integrate[Csch[e + f*x]*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
135,0,0,87,131.2896495,"\int \text{csch}^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Csch[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^p,x]","\int \text{csch}^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","\frac{\cosh (e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^p \left(\frac{b \cosh ^2(e+f x)}{a-b}+1\right)^{-p} F_1\left(\frac{1}{2};2,-p;\frac{3}{2};\cosh ^2(e+f x),-\frac{b \cosh ^2(e+f x)}{a-b}\right)}{f}",1,"Integrate[Csch[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
136,-1,0,88,180.0012283,"\int \text{csch}^5(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Csch[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^p,x]","\text{\$Aborted}","-\frac{\cosh (e+f x) \left(a+b \cosh ^2(e+f x)-b\right)^p \left(\frac{b \cosh ^2(e+f x)}{a-b}+1\right)^{-p} F_1\left(\frac{1}{2};3,-p;\frac{3}{2};\cosh ^2(e+f x),-\frac{b \cosh ^2(e+f x)}{a-b}\right)}{f}",1,"$Aborted","F",-1
137,0,0,103,11.0030275,"\int \sinh ^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Sinh[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^p,x]","\int \sinh ^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","\frac{\sinh ^4(e+f x) \sqrt{\cosh ^2(e+f x)} \tanh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{5}{2};\frac{1}{2},-p;\frac{7}{2};-\sinh ^2(e+f x),-\frac{b \sinh ^2(e+f x)}{a}\right)}{5 f}",1,"Integrate[Sinh[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
138,1,250,101,0.7845614,"\int \sinh ^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Sinh[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^p,x]","\frac{2^{-p-2} \text{csch}(2 (e+f x)) \sqrt{-\frac{b \sinh ^2(e+f x)}{a}} \sqrt{\frac{b \cosh ^2(e+f x)}{b-a}} (2 a+b \cosh (2 (e+f x))-b)^{p+1} \left((p+1) (2 a+b \cosh (2 (e+f x))-b) F_1\left(p+2;\frac{1}{2},\frac{1}{2};p+3;\frac{2 a-b+b \cosh (2 (e+f x))}{2 a},\frac{2 a-b+b \cosh (2 (e+f x))}{2 (a-b)}\right)-2 a (p+2) F_1\left(p+1;\frac{1}{2},\frac{1}{2};p+2;\frac{2 a-b+b \cosh (2 (e+f x))}{2 a},\frac{2 a-b+b \cosh (2 (e+f x))}{2 (a-b)}\right)\right)}{b^2 f (p+1) (p+2)}","\frac{\tanh ^3(e+f x) \text{sech}^2(e+f x)^p \left(a+b \sinh ^2(e+f x)\right)^p \left(1-\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{-p} F_1\left(\frac{3}{2};p+2,-p;\frac{5}{2};\tanh ^2(e+f x),\frac{(a-b) \tanh ^2(e+f x)}{a}\right)}{3 f}",1,"(2^(-2 - p)*Sqrt[(b*Cosh[e + f*x]^2)/(-a + b)]*(2*a - b + b*Cosh[2*(e + f*x)])^(1 + p)*(-2*a*(2 + p)*AppellF1[1 + p, 1/2, 1/2, 2 + p, (2*a - b + b*Cosh[2*(e + f*x)])/(2*a), (2*a - b + b*Cosh[2*(e + f*x)])/(2*(a - b))] + (1 + p)*AppellF1[2 + p, 1/2, 1/2, 3 + p, (2*a - b + b*Cosh[2*(e + f*x)])/(2*a), (2*a - b + b*Cosh[2*(e + f*x)])/(2*(a - b))]*(2*a - b + b*Cosh[2*(e + f*x)]))*Csch[2*(e + f*x)]*Sqrt[-((b*Sinh[e + f*x]^2)/a)])/(b^2*f*(1 + p)*(2 + p))","B",0
139,0,0,99,5.8136033,"\int \text{csch}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Csch[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^p,x]","\int \text{csch}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","-\frac{\sqrt{\cosh ^2(e+f x)} \text{csch}(e+f x) \text{sech}(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} F_1\left(-\frac{1}{2};\frac{1}{2},-p;\frac{1}{2};-\sinh ^2(e+f x),-\frac{b \sinh ^2(e+f x)}{a}\right)}{f}",1,"Integrate[Csch[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
140,0,0,103,10.8632604,"\int \text{csch}^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Csch[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^p,x]","\int \text{csch}^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","-\frac{\sqrt{\cosh ^2(e+f x)} \text{csch}^3(e+f x) \text{sech}(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} F_1\left(-\frac{3}{2};\frac{1}{2},-p;-\frac{1}{2};-\sinh ^2(e+f x),-\frac{b \sinh ^2(e+f x)}{a}\right)}{3 f}",1,"Integrate[Csch[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
141,1,81,106,0.1009426,"\int \sinh ^4(c+d x) \left(a+b \sinh ^3(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^3),x]","\frac{-560 a \sinh (2 (c+d x))+70 a \sinh (4 (c+d x))+840 a c+840 a d x-1225 b \cosh (c+d x)+245 b \cosh (3 (c+d x))-49 b \cosh (5 (c+d x))+5 b \cosh (7 (c+d x))}{2240 d}","\frac{a \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{3 a \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3 a x}{8}+\frac{b \cosh ^7(c+d x)}{7 d}-\frac{3 b \cosh ^5(c+d x)}{5 d}+\frac{b \cosh ^3(c+d x)}{d}-\frac{b \cosh (c+d x)}{d}",1,"(840*a*c + 840*a*d*x - 1225*b*Cosh[c + d*x] + 245*b*Cosh[3*(c + d*x)] - 49*b*Cosh[5*(c + d*x)] + 5*b*Cosh[7*(c + d*x)] - 560*a*Sinh[2*(c + d*x)] + 70*a*Sinh[4*(c + d*x)])/(2240*d)","A",1
142,1,66,99,0.1290075,"\int \sinh ^3(c+d x) \left(a+b \sinh ^3(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^3),x]","\frac{-144 a \cosh (c+d x)+16 a \cosh (3 (c+d x))+b (45 \sinh (2 (c+d x))-9 \sinh (4 (c+d x))+\sinh (6 (c+d x))-60 c-60 d x)}{192 d}","\frac{a \cosh ^3(c+d x)}{3 d}-\frac{a \cosh (c+d x)}{d}+\frac{b \sinh ^5(c+d x) \cosh (c+d x)}{6 d}-\frac{5 b \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac{5 b \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac{5 b x}{16}",1,"(-144*a*Cosh[c + d*x] + 16*a*Cosh[3*(c + d*x)] + b*(-60*c - 60*d*x + 45*Sinh[2*(c + d*x)] - 9*Sinh[4*(c + d*x)] + Sinh[6*(c + d*x)]))/(192*d)","A",1
143,1,79,70,0.0962078,"\int \sinh ^2(c+d x) \left(a+b \sinh ^3(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^3),x]","\frac{a (-c-d x)}{2 d}+\frac{a \sinh (2 (c+d x))}{4 d}+\frac{5 b \cosh (c+d x)}{8 d}-\frac{5 b \cosh (3 (c+d x))}{48 d}+\frac{b \cosh (5 (c+d x))}{80 d}","\frac{a \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{a x}{2}+\frac{b \cosh ^5(c+d x)}{5 d}-\frac{2 b \cosh ^3(c+d x)}{3 d}+\frac{b \cosh (c+d x)}{d}",1,"(a*(-c - d*x))/(2*d) + (5*b*Cosh[c + d*x])/(8*d) - (5*b*Cosh[3*(c + d*x)])/(48*d) + (b*Cosh[5*(c + d*x)])/(80*d) + (a*Sinh[2*(c + d*x)])/(4*d)","A",1
144,1,45,60,0.1285542,"\int \sinh (c+d x) \left(a+b \sinh ^3(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^3),x]","\frac{32 a \cosh (c+d x)+b (12 (c+d x)-8 \sinh (2 (c+d x))+\sinh (4 (c+d x)))}{32 d}","\frac{a \cosh (c+d x)}{d}+\frac{b \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{3 b \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3 b x}{8}",1,"(32*a*Cosh[c + d*x] + b*(12*(c + d*x) - 8*Sinh[2*(c + d*x)] + Sinh[4*(c + d*x)]))/(32*d)","A",1
145,1,34,32,0.0117459,"\int \left(a+b \sinh ^3(c+d x)\right) \, dx","Integrate[a + b*Sinh[c + d*x]^3,x]","a x-\frac{3 b \cosh (c+d x)}{4 d}+\frac{b \cosh (3 (c+d x))}{12 d}","a x+\frac{b \cosh ^3(c+d x)}{3 d}-\frac{b \cosh (c+d x)}{d}",1,"a*x - (3*b*Cosh[c + d*x])/(4*d) + (b*Cosh[3*(c + d*x)])/(12*d)","A",1
146,1,72,40,0.0531163,"\int \text{csch}(c+d x) \left(a+b \sinh ^3(c+d x)\right) \, dx","Integrate[Csch[c + d*x]*(a + b*Sinh[c + d*x]^3),x]","\frac{a \log \left(\sinh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{a \log \left(\cosh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}+\frac{b (-c-d x)}{2 d}+\frac{b \sinh (2 (c+d x))}{4 d}","-\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b x}{2}",1,"(b*(-c - d*x))/(2*d) - (a*Log[Cosh[c/2 + (d*x)/2]])/d + (a*Log[Sinh[c/2 + (d*x)/2]])/d + (b*Sinh[2*(c + d*x)])/(4*d)","A",1
147,1,35,24,0.0290388,"\int \text{csch}^2(c+d x) \left(a+b \sinh ^3(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^3),x]","-\frac{a \coth (c+d x)}{d}+\frac{b \sinh (c) \sinh (d x)}{d}+\frac{b \cosh (c) \cosh (d x)}{d}","\frac{b \cosh (c+d x)}{d}-\frac{a \coth (c+d x)}{d}",1,"(b*Cosh[c]*Cosh[d*x])/d - (a*Coth[c + d*x])/d + (b*Sinh[c]*Sinh[d*x])/d","A",1
148,1,63,39,0.0132115,"\int \text{csch}^3(c+d x) \left(a+b \sinh ^3(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^3),x]","-\frac{a \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+b x","\frac{a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}+b x",1,"b*x - (a*Csch[(c + d*x)/2]^2)/(8*d) - (a*Log[Tanh[(c + d*x)/2]])/(2*d) - (a*Sech[(c + d*x)/2]^2)/(8*d)","A",1
149,1,76,41,0.0302509,"\int \text{csch}^4(c+d x) \left(a+b \sinh ^3(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^3),x]","\frac{2 a \coth (c+d x)}{3 d}-\frac{a \coth (c+d x) \text{csch}^2(c+d x)}{3 d}+\frac{b \log \left(\sinh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{b \log \left(\cosh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}","-\frac{a \coth ^3(c+d x)}{3 d}+\frac{a \coth (c+d x)}{d}-\frac{b \tanh ^{-1}(\cosh (c+d x))}{d}",1,"(2*a*Coth[c + d*x])/(3*d) - (a*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (b*Log[Cosh[c/2 + (d*x)/2]])/d + (b*Log[Sinh[c/2 + (d*x)/2]])/d","A",1
150,1,125,192,0.6833735,"\int \sinh ^3(c+d x) \left(a+b \sinh ^3(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^3)^2,x]","\frac{-1890 \left(32 a^2-21 b^2\right) \cosh (c+d x)+420 \left(16 a^2-21 b^2\right) \cosh (3 (c+d x))+b (-840 a (-45 \sinh (2 (c+d x))+9 \sinh (4 (c+d x))-\sinh (6 (c+d x))+60 c+60 d x)+2268 b \cosh (5 (c+d x))-405 b \cosh (7 (c+d x))+35 b \cosh (9 (c+d x)))}{80640 d}","\frac{a^2 \cosh ^3(c+d x)}{3 d}-\frac{a^2 \cosh (c+d x)}{d}+\frac{a b \sinh ^5(c+d x) \cosh (c+d x)}{3 d}-\frac{5 a b \sinh ^3(c+d x) \cosh (c+d x)}{12 d}+\frac{5 a b \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac{5 a b x}{8}+\frac{b^2 \cosh ^9(c+d x)}{9 d}-\frac{4 b^2 \cosh ^7(c+d x)}{7 d}+\frac{6 b^2 \cosh ^5(c+d x)}{5 d}-\frac{4 b^2 \cosh ^3(c+d x)}{3 d}+\frac{b^2 \cosh (c+d x)}{d}",1,"(-1890*(32*a^2 - 21*b^2)*Cosh[c + d*x] + 420*(16*a^2 - 21*b^2)*Cosh[3*(c + d*x)] + b*(2268*b*Cosh[5*(c + d*x)] - 405*b*Cosh[7*(c + d*x)] + 35*b*Cosh[9*(c + d*x)] - 840*a*(60*c + 60*d*x - 45*Sinh[2*(c + d*x)] + 9*Sinh[4*(c + d*x)] - Sinh[6*(c + d*x)])))/(80640*d)","A",1
151,1,133,180,0.1332902,"\int \sinh ^2(c+d x) \left(a+b \sinh ^3(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^3)^2,x]","\frac{3840 a^2 \sinh (2 (c+d x))-7680 a^2 c-7680 a^2 d x+19200 a b \cosh (c+d x)-3200 a b \cosh (3 (c+d x))+384 a b \cosh (5 (c+d x))-3360 b^2 \sinh (2 (c+d x))+840 b^2 \sinh (4 (c+d x))-160 b^2 \sinh (6 (c+d x))+15 b^2 \sinh (8 (c+d x))+4200 b^2 c+4200 b^2 d x}{15360 d}","\frac{a^2 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{a^2 x}{2}+\frac{2 a b \cosh ^5(c+d x)}{5 d}-\frac{4 a b \cosh ^3(c+d x)}{3 d}+\frac{2 a b \cosh (c+d x)}{d}+\frac{b^2 \sinh ^7(c+d x) \cosh (c+d x)}{8 d}-\frac{7 b^2 \sinh ^5(c+d x) \cosh (c+d x)}{48 d}+\frac{35 b^2 \sinh ^3(c+d x) \cosh (c+d x)}{192 d}-\frac{35 b^2 \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{35 b^2 x}{128}",1,"(-7680*a^2*c + 4200*b^2*c - 7680*a^2*d*x + 4200*b^2*d*x + 19200*a*b*Cosh[c + d*x] - 3200*a*b*Cosh[3*(c + d*x)] + 384*a*b*Cosh[5*(c + d*x)] + 3840*a^2*Sinh[2*(c + d*x)] - 3360*b^2*Sinh[2*(c + d*x)] + 840*b^2*Sinh[4*(c + d*x)] - 160*b^2*Sinh[6*(c + d*x)] + 15*b^2*Sinh[8*(c + d*x)])/(15360*d)","A",1
152,1,92,130,0.4150869,"\int \sinh (c+d x) \left(a+b \sinh ^3(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^3)^2,x]","\frac{35 \left(64 a^2-35 b^2\right) \cosh (c+d x)+b (140 a (12 (c+d x)-8 \sinh (2 (c+d x))+\sinh (4 (c+d x)))+245 b \cosh (3 (c+d x))-49 b \cosh (5 (c+d x))+5 b \cosh (7 (c+d x)))}{2240 d}","\frac{a^2 \cosh (c+d x)}{d}+\frac{a b \sinh ^3(c+d x) \cosh (c+d x)}{2 d}-\frac{3 a b \sinh (c+d x) \cosh (c+d x)}{4 d}+\frac{3 a b x}{4}+\frac{b^2 \cosh ^7(c+d x)}{7 d}-\frac{3 b^2 \cosh ^5(c+d x)}{5 d}+\frac{b^2 \cosh ^3(c+d x)}{d}-\frac{b^2 \cosh (c+d x)}{d}",1,"(35*(64*a^2 - 35*b^2)*Cosh[c + d*x] + b*(245*b*Cosh[3*(c + d*x)] - 49*b*Cosh[5*(c + d*x)] + 5*b*Cosh[7*(c + d*x)] + 140*a*(12*(c + d*x) - 8*Sinh[2*(c + d*x)] + Sinh[4*(c + d*x)])))/(2240*d)","A",1
153,1,94,114,0.148716,"\int \left(a+b \sinh ^3(c+d x)\right)^2 \, dx","Integrate[(a + b*Sinh[c + d*x]^3)^2,x]","\frac{192 a^2 c+192 a^2 d x-288 a b \cosh (c+d x)+32 a b \cosh (3 (c+d x))+45 b^2 \sinh (2 (c+d x))-9 b^2 \sinh (4 (c+d x))+b^2 \sinh (6 (c+d x))-60 b^2 c-60 b^2 d x}{192 d}","a^2 x+\frac{2 a b \cosh ^3(c+d x)}{3 d}-\frac{2 a b \cosh (c+d x)}{d}+\frac{b^2 \sinh ^5(c+d x) \cosh (c+d x)}{6 d}-\frac{5 b^2 \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac{5 b^2 \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac{5 b^2 x}{16}",1,"(192*a^2*c - 60*b^2*c + 192*a^2*d*x - 60*b^2*d*x - 288*a*b*Cosh[c + d*x] + 32*a*b*Cosh[3*(c + d*x)] + 45*b^2*Sinh[2*(c + d*x)] - 9*b^2*Sinh[4*(c + d*x)] + b^2*Sinh[6*(c + d*x)])/(192*d)","A",1
154,1,96,88,0.3135254,"\int \text{csch}(c+d x) \left(a+b \sinh ^3(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]*(a + b*Sinh[c + d*x]^3)^2,x]","\frac{120 a \left(b \sinh (2 (c+d x))-2 \left(-a \log \left(\sinh \left(\frac{1}{2} (c+d x)\right)\right)+a \log \left(\cosh \left(\frac{1}{2} (c+d x)\right)\right)+b c+b d x\right)\right)+150 b^2 \cosh (c+d x)-25 b^2 \cosh (3 (c+d x))+3 b^2 \cosh (5 (c+d x))}{240 d}","-\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{a b \sinh (c+d x) \cosh (c+d x)}{d}-a b x+\frac{b^2 \cosh ^5(c+d x)}{5 d}-\frac{2 b^2 \cosh ^3(c+d x)}{3 d}+\frac{b^2 \cosh (c+d x)}{d}",1,"(150*b^2*Cosh[c + d*x] - 25*b^2*Cosh[3*(c + d*x)] + 3*b^2*Cosh[5*(c + d*x)] + 120*a*(-2*(b*c + b*d*x + a*Log[Cosh[(c + d*x)/2]] - a*Log[Sinh[(c + d*x)/2]]) + b*Sinh[2*(c + d*x)]))/(240*d)","A",1
155,1,92,82,0.2753778,"\int \text{csch}^2(c+d x) \left(a+b \sinh ^3(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^3)^2,x]","-\frac{a^2 \coth (c+d x)}{d}+\frac{2 a b \sinh (c) \sinh (d x)}{d}+\frac{2 a b \cosh (c) \cosh (d x)}{d}+\frac{3 b^2 (c+d x)}{8 d}-\frac{b^2 \sinh (2 (c+d x))}{4 d}+\frac{b^2 \sinh (4 (c+d x))}{32 d}","-\frac{a^2 \coth (c+d x)}{d}+\frac{2 a b \cosh (c+d x)}{d}+\frac{b^2 \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{3 b^2 \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3 b^2 x}{8}",1,"(3*b^2*(c + d*x))/(8*d) + (2*a*b*Cosh[c]*Cosh[d*x])/d - (a^2*Coth[c + d*x])/d + (2*a*b*Sinh[c]*Sinh[d*x])/d - (b^2*Sinh[2*(c + d*x)])/(4*d) + (b^2*Sinh[4*(c + d*x)])/(32*d)","A",1
156,1,105,77,0.0310019,"\int \text{csch}^3(c+d x) \left(a+b \sinh ^3(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^3)^2,x]","-\frac{a^2 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a^2 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a^2 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+2 a b x-\frac{3 b^2 \cosh (c+d x)}{4 d}+\frac{b^2 \cosh (3 (c+d x))}{12 d}","\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a^2 \coth (c+d x) \text{csch}(c+d x)}{2 d}+2 a b x+\frac{b^2 \cosh ^3(c+d x)}{3 d}-\frac{b^2 \cosh (c+d x)}{d}",1,"2*a*b*x - (3*b^2*Cosh[c + d*x])/(4*d) + (b^2*Cosh[3*(c + d*x)])/(12*d) - (a^2*Csch[(c + d*x)/2]^2)/(8*d) - (a^2*Log[Tanh[(c + d*x)/2]])/(2*d) - (a^2*Sech[(c + d*x)/2]^2)/(8*d)","A",1
157,1,81,76,0.4066308,"\int \text{csch}^4(c+d x) \left(a+b \sinh ^3(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^3)^2,x]","\frac{3 b \left(b \sinh (2 (c+d x))-2 \left(-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)+b c+b d x\right)\right)-4 a^2 \coth (c+d x) \left(\text{csch}^2(c+d x)-2\right)}{12 d}","-\frac{a^2 \coth ^3(c+d x)}{3 d}+\frac{a^2 \coth (c+d x)}{d}-\frac{2 a b \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b^2 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b^2 x}{2}",1,"(-4*a^2*Coth[c + d*x]*(-2 + Csch[c + d*x]^2) + 3*b*(-2*(b*c + b*d*x + 4*a*Log[Cosh[(c + d*x)/2]] - 4*a*Log[Sinh[(c + d*x)/2]]) + b*Sinh[2*(c + d*x)]))/(12*d)","A",1
158,1,149,90,0.040092,"\int \text{csch}^5(c+d x) \left(a+b \sinh ^3(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^3)^2,x]","-\frac{a^2 \text{csch}^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{3 a^2 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{a^2 \text{sech}^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{3 a^2 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{3 a^2 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{2 a b \coth (c+d x)}{d}+\frac{b^2 \sinh (c) \sinh (d x)}{d}+\frac{b^2 \cosh (c) \cosh (d x)}{d}","-\frac{3 a^2 \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac{a^2 \coth (c+d x) \text{csch}^3(c+d x)}{4 d}+\frac{3 a^2 \coth (c+d x) \text{csch}(c+d x)}{8 d}-\frac{2 a b \coth (c+d x)}{d}+\frac{b^2 \cosh (c+d x)}{d}",1,"(b^2*Cosh[c]*Cosh[d*x])/d - (2*a*b*Coth[c + d*x])/d + (3*a^2*Csch[(c + d*x)/2]^2)/(32*d) - (a^2*Csch[(c + d*x)/2]^4)/(64*d) + (3*a^2*Log[Tanh[(c + d*x)/2]])/(8*d) + (3*a^2*Sech[(c + d*x)/2]^2)/(32*d) + (a^2*Sech[(c + d*x)/2]^4)/(64*d) + (b^2*Sinh[c]*Sinh[d*x])/d","A",1
159,1,197,88,0.9390632,"\int \text{csch}^6(c+d x) \left(a+b \sinh ^3(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^3)^2,x]","\frac{16 \left(-16 a^2 \tanh \left(\frac{1}{2} (c+d x)\right)-12 a^2 \sinh ^6\left(\frac{1}{2} (c+d x)\right) \text{csch}^5(c+d x)-19 a^2 \sinh ^4\left(\frac{1}{2} (c+d x)\right) \text{csch}^3(c+d x)-15 a b \text{sech}^2\left(\frac{1}{2} (c+d x)\right)-60 a b \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+60 b^2 c+60 b^2 d x\right)-256 a^2 \coth \left(\frac{1}{2} (c+d x)\right)-3 a^2 \sinh (c+d x) \text{csch}^6\left(\frac{1}{2} (c+d x)\right)+19 a^2 \sinh (c+d x) \text{csch}^4\left(\frac{1}{2} (c+d x)\right)-240 a b \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{960 d}","-\frac{a^2 \coth ^5(c+d x)}{5 d}+\frac{2 a^2 \coth ^3(c+d x)}{3 d}-\frac{a^2 \coth (c+d x)}{d}+\frac{a b \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{a b \coth (c+d x) \text{csch}(c+d x)}{d}+b^2 x",1,"(-256*a^2*Coth[(c + d*x)/2] - 240*a*b*Csch[(c + d*x)/2]^2 + 19*a^2*Csch[(c + d*x)/2]^4*Sinh[c + d*x] - 3*a^2*Csch[(c + d*x)/2]^6*Sinh[c + d*x] + 16*(60*b^2*c + 60*b^2*d*x - 60*a*b*Log[Tanh[(c + d*x)/2]] - 15*a*b*Sech[(c + d*x)/2]^2 - 19*a^2*Csch[c + d*x]^3*Sinh[(c + d*x)/2]^4 - 12*a^2*Csch[c + d*x]^5*Sinh[(c + d*x)/2]^6 - 16*a^2*Tanh[(c + d*x)/2]))/(960*d)","B",1
160,1,235,133,0.0544452,"\int \text{csch}^7(c+d x) \left(a+b \sinh ^3(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^3)^2,x]","-\frac{a^2 \text{csch}^6\left(\frac{1}{2} (c+d x)\right)}{384 d}+\frac{a^2 \text{csch}^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 a^2 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{a^2 \text{sech}^6\left(\frac{1}{2} (c+d x)\right)}{384 d}-\frac{a^2 \text{sech}^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 a^2 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 a^2 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}+\frac{4 a b \coth (c+d x)}{3 d}-\frac{2 a b \coth (c+d x) \text{csch}^2(c+d x)}{3 d}+\frac{b^2 \log \left(\sinh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{b^2 \log \left(\cosh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}","\frac{5 a^2 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac{a^2 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{5 a^2 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{5 a^2 \coth (c+d x) \text{csch}(c+d x)}{16 d}-\frac{2 a b \coth ^3(c+d x)}{3 d}+\frac{2 a b \coth (c+d x)}{d}-\frac{b^2 \tanh ^{-1}(\cosh (c+d x))}{d}",1,"(4*a*b*Coth[c + d*x])/(3*d) - (5*a^2*Csch[(c + d*x)/2]^2)/(64*d) + (a^2*Csch[(c + d*x)/2]^4)/(64*d) - (a^2*Csch[(c + d*x)/2]^6)/(384*d) - (2*a*b*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (b^2*Log[Cosh[c/2 + (d*x)/2]])/d + (b^2*Log[Sinh[c/2 + (d*x)/2]])/d - (5*a^2*Log[Tanh[(c + d*x)/2]])/(16*d) - (5*a^2*Sech[(c + d*x)/2]^2)/(64*d) - (a^2*Sech[(c + d*x)/2]^4)/(64*d) - (a^2*Sech[(c + d*x)/2]^6)/(384*d)","A",1
161,1,194,291,0.4709424,"\int \sinh ^2(c+d x) \left(a+b \sinh ^3(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^3)^3,x]","\frac{-27720 a \left(64 a^2-105 b^2\right) (c+d x)+110880 a \left(8 a^2-21 b^2\right) \sinh (2 (c+d x))-20790 b \left(77 b^2-320 a^2\right) \cosh (c+d x)+34650 b \left(11 b^2-32 a^2\right) \cosh (3 (c+d x))-2079 b \left(55 b^2-64 a^2\right) \cosh (5 (c+d x))+582120 a b^2 \sinh (4 (c+d x))-110880 a b^2 \sinh (6 (c+d x))+10395 a b^2 \sinh (8 (c+d x))+27225 b^3 \cosh (7 (c+d x))-4235 b^3 \cosh (9 (c+d x))+315 b^3 \cosh (11 (c+d x))}{3548160 d}","\frac{a^3 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{a^3 x}{2}+\frac{3 a^2 b \cosh ^5(c+d x)}{5 d}-\frac{2 a^2 b \cosh ^3(c+d x)}{d}+\frac{3 a^2 b \cosh (c+d x)}{d}+\frac{3 a b^2 \sinh ^7(c+d x) \cosh (c+d x)}{8 d}-\frac{7 a b^2 \sinh ^5(c+d x) \cosh (c+d x)}{16 d}+\frac{35 a b^2 \sinh ^3(c+d x) \cosh (c+d x)}{64 d}-\frac{105 a b^2 \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{105}{128} a b^2 x+\frac{b^3 \cosh ^{11}(c+d x)}{11 d}-\frac{5 b^3 \cosh ^9(c+d x)}{9 d}+\frac{10 b^3 \cosh ^7(c+d x)}{7 d}-\frac{2 b^3 \cosh ^5(c+d x)}{d}+\frac{5 b^3 \cosh ^3(c+d x)}{3 d}-\frac{b^3 \cosh (c+d x)}{d}",1,"(-27720*a*(64*a^2 - 105*b^2)*(c + d*x) - 20790*b*(-320*a^2 + 77*b^2)*Cosh[c + d*x] + 34650*b*(-32*a^2 + 11*b^2)*Cosh[3*(c + d*x)] - 2079*b*(-64*a^2 + 55*b^2)*Cosh[5*(c + d*x)] + 27225*b^3*Cosh[7*(c + d*x)] - 4235*b^3*Cosh[9*(c + d*x)] + 315*b^3*Cosh[11*(c + d*x)] + 110880*a*(8*a^2 - 21*b^2)*Sinh[2*(c + d*x)] + 582120*a*b^2*Sinh[4*(c + d*x)] - 110880*a*b^2*Sinh[6*(c + d*x)] + 10395*a*b^2*Sinh[8*(c + d*x)])/(3548160*d)","A",1
162,1,184,267,0.4476373,"\int \sinh (c+d x) \left(a+b \sinh ^3(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^3)^3,x]","\frac{1120 a \left(64 a^2-105 b^2\right) \cosh (c+d x)+b \left(-53760 a^2 \sinh (2 (c+d x))+6720 a^2 \sinh (4 (c+d x))+80640 a^2 c+80640 a^2 d x+23520 a b \cosh (3 (c+d x))-4704 a b \cosh (5 (c+d x))+480 a b \cosh (7 (c+d x))+14700 b^2 \sinh (2 (c+d x))-4200 b^2 \sinh (4 (c+d x))+1050 b^2 \sinh (6 (c+d x))-175 b^2 \sinh (8 (c+d x))+14 b^2 \sinh (10 (c+d x))-17640 b^2 c-17640 b^2 d x\right)}{71680 d}","\frac{a^3 \cosh (c+d x)}{d}+\frac{3 a^2 b \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{9 a^2 b \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{9}{8} a^2 b x+\frac{3 a b^2 \cosh ^7(c+d x)}{7 d}-\frac{9 a b^2 \cosh ^5(c+d x)}{5 d}+\frac{3 a b^2 \cosh ^3(c+d x)}{d}-\frac{3 a b^2 \cosh (c+d x)}{d}+\frac{b^3 \sinh ^9(c+d x) \cosh (c+d x)}{10 d}-\frac{9 b^3 \sinh ^7(c+d x) \cosh (c+d x)}{80 d}+\frac{21 b^3 \sinh ^5(c+d x) \cosh (c+d x)}{160 d}-\frac{21 b^3 \sinh ^3(c+d x) \cosh (c+d x)}{128 d}+\frac{63 b^3 \sinh (c+d x) \cosh (c+d x)}{256 d}-\frac{63 b^3 x}{256}",1,"(1120*a*(64*a^2 - 105*b^2)*Cosh[c + d*x] + b*(80640*a^2*c - 17640*b^2*c + 80640*a^2*d*x - 17640*b^2*d*x + 23520*a*b*Cosh[3*(c + d*x)] - 4704*a*b*Cosh[5*(c + d*x)] + 480*a*b*Cosh[7*(c + d*x)] - 53760*a^2*Sinh[2*(c + d*x)] + 14700*b^2*Sinh[2*(c + d*x)] + 6720*a^2*Sinh[4*(c + d*x)] - 4200*b^2*Sinh[4*(c + d*x)] + 1050*b^2*Sinh[6*(c + d*x)] - 175*b^2*Sinh[8*(c + d*x)] + 14*b^2*Sinh[10*(c + d*x)]))/(71680*d)","A",1
163,1,159,204,0.2570638,"\int \left(a+b \sinh ^3(c+d x)\right)^3 \, dx","Integrate[(a + b*Sinh[c + d*x]^3)^3,x]","\frac{80640 a^3 c+80640 a^3 d x+1260 \left(16 a^2 b-7 b^3\right) \cosh (3 (c+d x))+5670 b \left(7 b^2-32 a^2\right) \cosh (c+d x)+56700 a b^2 \sinh (2 (c+d x))-11340 a b^2 \sinh (4 (c+d x))+1260 a b^2 \sinh (6 (c+d x))-75600 a b^2 c-75600 a b^2 d x+2268 b^3 \cosh (5 (c+d x))-405 b^3 \cosh (7 (c+d x))+35 b^3 \cosh (9 (c+d x))}{80640 d}","a^3 x+\frac{a^2 b \cosh ^3(c+d x)}{d}-\frac{3 a^2 b \cosh (c+d x)}{d}+\frac{a b^2 \sinh ^5(c+d x) \cosh (c+d x)}{2 d}-\frac{5 a b^2 \sinh ^3(c+d x) \cosh (c+d x)}{8 d}+\frac{15 a b^2 \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac{15}{16} a b^2 x+\frac{b^3 \cosh ^9(c+d x)}{9 d}-\frac{4 b^3 \cosh ^7(c+d x)}{7 d}+\frac{6 b^3 \cosh ^5(c+d x)}{5 d}-\frac{4 b^3 \cosh ^3(c+d x)}{3 d}+\frac{b^3 \cosh (c+d x)}{d}",1,"(80640*a^3*c - 75600*a*b^2*c + 80640*a^3*d*x - 75600*a*b^2*d*x + 5670*b*(-32*a^2 + 7*b^2)*Cosh[c + d*x] + 1260*(16*a^2*b - 7*b^3)*Cosh[3*(c + d*x)] + 2268*b^3*Cosh[5*(c + d*x)] - 405*b^3*Cosh[7*(c + d*x)] + 35*b^3*Cosh[9*(c + d*x)] + 56700*a*b^2*Sinh[2*(c + d*x)] - 11340*a*b^2*Sinh[4*(c + d*x)] + 1260*a*b^2*Sinh[6*(c + d*x)])/(80640*d)","A",1
164,1,158,201,0.2154428,"\int \text{csch}(c+d x) \left(a+b \sinh ^3(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]*(a + b*Sinh[c + d*x]^3)^3,x]","\frac{15360 a^3 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+11520 a^2 b \sinh (2 (c+d x))-23040 a^2 b c-23040 a^2 b d x+28800 a b^2 \cosh (c+d x)-4800 a b^2 \cosh (3 (c+d x))+576 a b^2 \cosh (5 (c+d x))-3360 b^3 \sinh (2 (c+d x))+840 b^3 \sinh (4 (c+d x))-160 b^3 \sinh (6 (c+d x))+15 b^3 \sinh (8 (c+d x))+4200 b^3 c+4200 b^3 d x}{15360 d}","-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{3 a^2 b \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{3}{2} a^2 b x+\frac{3 a b^2 \cosh ^5(c+d x)}{5 d}-\frac{2 a b^2 \cosh ^3(c+d x)}{d}+\frac{3 a b^2 \cosh (c+d x)}{d}+\frac{b^3 \sinh ^7(c+d x) \cosh (c+d x)}{8 d}-\frac{7 b^3 \sinh ^5(c+d x) \cosh (c+d x)}{48 d}+\frac{35 b^3 \sinh ^3(c+d x) \cosh (c+d x)}{192 d}-\frac{35 b^3 \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{35 b^3 x}{128}",1,"(-23040*a^2*b*c + 4200*b^3*c - 23040*a^2*b*d*x + 4200*b^3*d*x + 28800*a*b^2*Cosh[c + d*x] - 4800*a*b^2*Cosh[3*(c + d*x)] + 576*a*b^2*Cosh[5*(c + d*x)] + 15360*a^3*Log[Tanh[(c + d*x)/2]] + 11520*a^2*b*Sinh[2*(c + d*x)] - 3360*b^3*Sinh[2*(c + d*x)] + 840*b^3*Sinh[4*(c + d*x)] - 160*b^3*Sinh[6*(c + d*x)] + 15*b^3*Sinh[8*(c + d*x)])/(15360*d)","A",1
165,1,140,152,1.1346737,"\int \text{csch}^2(c+d x) \left(a+b \sinh ^3(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^3)^3,x]","\frac{-1120 a^3 \tanh \left(\frac{1}{2} (c+d x)\right)-1120 a^3 \coth \left(\frac{1}{2} (c+d x)\right)+35 b \left(192 a^2-35 b^2\right) \cosh (c+d x)-1680 a b^2 \sinh (2 (c+d x))+210 a b^2 \sinh (4 (c+d x))+2520 a b^2 c+2520 a b^2 d x+245 b^3 \cosh (3 (c+d x))-49 b^3 \cosh (5 (c+d x))+5 b^3 \cosh (7 (c+d x))}{2240 d}","-\frac{a^3 \coth (c+d x)}{d}+\frac{3 a^2 b \cosh (c+d x)}{d}+\frac{3 a b^2 \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{9 a b^2 \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{9}{8} a b^2 x+\frac{b^3 \cosh ^7(c+d x)}{7 d}-\frac{3 b^3 \cosh ^5(c+d x)}{5 d}+\frac{b^3 \cosh ^3(c+d x)}{d}-\frac{b^3 \cosh (c+d x)}{d}",1,"(2520*a*b^2*c + 2520*a*b^2*d*x + 35*b*(192*a^2 - 35*b^2)*Cosh[c + d*x] + 245*b^3*Cosh[3*(c + d*x)] - 49*b^3*Cosh[5*(c + d*x)] + 5*b^3*Cosh[7*(c + d*x)] - 1120*a^3*Coth[(c + d*x)/2] - 1680*a*b^2*Sinh[2*(c + d*x)] + 210*a*b^2*Sinh[4*(c + d*x)] - 1120*a^3*Tanh[(c + d*x)/2])/(2240*d)","A",1
166,1,150,156,3.4410945,"\int \text{csch}^3(c+d x) \left(a+b \sinh ^3(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^3)^3,x]","\frac{-24 a^3 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)-24 a^3 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)-96 a^3 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+576 a^2 b c+576 a^2 b d x-432 a b^2 \cosh (c+d x)+48 a b^2 \cosh (3 (c+d x))+45 b^3 \sinh (2 (c+d x))-9 b^3 \sinh (4 (c+d x))+b^3 \sinh (6 (c+d x))-60 b^3 c-60 b^3 d x}{192 d}","\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a^3 \coth (c+d x) \text{csch}(c+d x)}{2 d}+3 a^2 b x+\frac{a b^2 \cosh ^3(c+d x)}{d}-\frac{3 a b^2 \cosh (c+d x)}{d}+\frac{b^3 \sinh ^5(c+d x) \cosh (c+d x)}{6 d}-\frac{5 b^3 \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac{5 b^3 \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac{5 b^3 x}{16}",1,"(576*a^2*b*c - 60*b^3*c + 576*a^2*b*d*x - 60*b^3*d*x - 432*a*b^2*Cosh[c + d*x] + 48*a*b^2*Cosh[3*(c + d*x)] - 24*a^3*Csch[(c + d*x)/2]^2 - 96*a^3*Log[Tanh[(c + d*x)/2]] - 24*a^3*Sech[(c + d*x)/2]^2 + 45*b^3*Sinh[2*(c + d*x)] - 9*b^3*Sinh[4*(c + d*x)] + b^3*Sinh[6*(c + d*x)])/(192*d)","A",1
167,1,169,129,0.4367262,"\int \text{csch}^4(c+d x) \left(a+b \sinh ^3(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^3)^3,x]","\frac{80 a^3 \tanh \left(\frac{1}{2} (c+d x)\right)+80 a^3 \coth \left(\frac{1}{2} (c+d x)\right)-5 a^3 \sinh (c+d x) \text{csch}^4\left(\frac{1}{2} (c+d x)\right)+80 a^3 \sinh ^4\left(\frac{1}{2} (c+d x)\right) \text{csch}^3(c+d x)+720 a^2 b \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+180 a b^2 \sinh (2 (c+d x))-360 a b^2 c-360 a b^2 d x+150 b^3 \cosh (c+d x)-25 b^3 \cosh (3 (c+d x))+3 b^3 \cosh (5 (c+d x))}{240 d}","-\frac{a^3 \coth ^3(c+d x)}{3 d}+\frac{a^3 \coth (c+d x)}{d}-\frac{3 a^2 b \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{3 a b^2 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{3}{2} a b^2 x+\frac{b^3 \cosh ^5(c+d x)}{5 d}-\frac{2 b^3 \cosh ^3(c+d x)}{3 d}+\frac{b^3 \cosh (c+d x)}{d}",1,"(-360*a*b^2*c - 360*a*b^2*d*x + 150*b^3*Cosh[c + d*x] - 25*b^3*Cosh[3*(c + d*x)] + 3*b^3*Cosh[5*(c + d*x)] + 80*a^3*Coth[(c + d*x)/2] + 720*a^2*b*Log[Tanh[(c + d*x)/2]] + 80*a^3*Csch[c + d*x]^3*Sinh[(c + d*x)/2]^4 - 5*a^3*Csch[(c + d*x)/2]^4*Sinh[c + d*x] + 180*a*b^2*Sinh[2*(c + d*x)] + 80*a^3*Tanh[(c + d*x)/2])/(240*d)","A",1
168,1,218,148,6.1363718,"\int \text{csch}^5(c+d x) \left(a+b \sinh ^3(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^3)^3,x]","-\frac{a^3 \text{csch}^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{3 a^3 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{a^3 \text{sech}^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{3 a^3 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{3 a^3 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{3 a^2 b \tanh \left(\frac{1}{2} (c+d x)\right)}{2 d}-\frac{3 a^2 b \coth \left(\frac{1}{2} (c+d x)\right)}{2 d}+\frac{3 a b^2 \cosh (c+d x)}{d}+\frac{3 b^3 (c+d x)}{8 d}-\frac{b^3 \sinh (2 (c+d x))}{4 d}+\frac{b^3 \sinh (4 (c+d x))}{32 d}","-\frac{3 a^3 \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac{a^3 \coth (c+d x) \text{csch}^3(c+d x)}{4 d}+\frac{3 a^3 \coth (c+d x) \text{csch}(c+d x)}{8 d}-\frac{3 a^2 b \coth (c+d x)}{d}+\frac{3 a b^2 \cosh (c+d x)}{d}+\frac{b^3 \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{3 b^3 \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3 b^3 x}{8}",1,"(3*b^3*(c + d*x))/(8*d) + (3*a*b^2*Cosh[c + d*x])/d - (3*a^2*b*Coth[(c + d*x)/2])/(2*d) + (3*a^3*Csch[(c + d*x)/2]^2)/(32*d) - (a^3*Csch[(c + d*x)/2]^4)/(64*d) + (3*a^3*Log[Tanh[(c + d*x)/2]])/(8*d) + (3*a^3*Sech[(c + d*x)/2]^2)/(32*d) + (a^3*Sech[(c + d*x)/2]^4)/(64*d) - (b^3*Sinh[2*(c + d*x)])/(4*d) + (b^3*Sinh[4*(c + d*x)])/(32*d) - (3*a^2*b*Tanh[(c + d*x)/2])/(2*d)","A",1
169,1,225,131,1.8720084,"\int \text{csch}^6(c+d x) \left(a+b \sinh ^3(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^3)^3,x]","\frac{\frac{1}{2} a \left(8 \left(-32 a^2 \tanh \left(\frac{1}{2} (c+d x)\right)-24 a^2 \sinh ^6\left(\frac{1}{2} (c+d x)\right) \text{csch}^5(c+d x)-38 a^2 \sinh ^4\left(\frac{1}{2} (c+d x)\right) \text{csch}^3(c+d x)-45 a b \text{sech}^2\left(\frac{1}{2} (c+d x)\right)+180 b \left(2 b (c+d x)-a \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)-256 a^2 \coth \left(\frac{1}{2} (c+d x)\right)-3 a^2 \sinh (c+d x) \text{csch}^6\left(\frac{1}{2} (c+d x)\right)+19 a^2 \sinh (c+d x) \text{csch}^4\left(\frac{1}{2} (c+d x)\right)-360 a b \text{csch}^2\left(\frac{1}{2} (c+d x)\right)\right)-360 b^3 \cosh (c+d x)+40 b^3 \cosh (3 (c+d x))}{480 d}","-\frac{a^3 \coth ^5(c+d x)}{5 d}+\frac{2 a^3 \coth ^3(c+d x)}{3 d}-\frac{a^3 \coth (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{3 a^2 b \coth (c+d x) \text{csch}(c+d x)}{2 d}+3 a b^2 x+\frac{b^3 \cosh ^3(c+d x)}{3 d}-\frac{b^3 \cosh (c+d x)}{d}",1,"(-360*b^3*Cosh[c + d*x] + 40*b^3*Cosh[3*(c + d*x)] + (a*(-256*a^2*Coth[(c + d*x)/2] - 360*a*b*Csch[(c + d*x)/2]^2 + 19*a^2*Csch[(c + d*x)/2]^4*Sinh[c + d*x] - 3*a^2*Csch[(c + d*x)/2]^6*Sinh[c + d*x] + 8*(180*b*(2*b*(c + d*x) - a*Log[Tanh[(c + d*x)/2]]) - 45*a*b*Sech[(c + d*x)/2]^2 - 38*a^2*Csch[c + d*x]^3*Sinh[(c + d*x)/2]^4 - 24*a^2*Csch[c + d*x]^5*Sinh[(c + d*x)/2]^6 - 32*a^2*Tanh[(c + d*x)/2])))/2)/(480*d)","A",1
170,1,236,166,1.5456069,"\int \text{csch}^7(c+d x) \left(a+b \sinh ^3(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^3)^3,x]","-\frac{a^3 \text{csch}^6\left(\frac{1}{2} (c+d x)\right)+30 a^3 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)+a^3 \text{sech}^6\left(\frac{1}{2} (c+d x)\right)+6 a^3 \text{sech}^4\left(\frac{1}{2} (c+d x)\right)+30 a^3 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)+120 a^3 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-384 a^2 b \tanh \left(\frac{1}{2} (c+d x)\right)-384 a^2 b \coth \left(\frac{1}{2} (c+d x)\right)-6 a^2 \text{csch}^4\left(\frac{1}{2} (c+d x)\right) (a-4 b \sinh (c+d x))-384 a^2 b \sinh ^4\left(\frac{1}{2} (c+d x)\right) \text{csch}^3(c+d x)-1152 a b^2 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-96 b^3 \sinh (2 (c+d x))+192 b^3 c+192 b^3 d x}{384 d}","\frac{5 a^3 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac{a^3 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{5 a^3 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{5 a^3 \coth (c+d x) \text{csch}(c+d x)}{16 d}-\frac{a^2 b \coth ^3(c+d x)}{d}+\frac{3 a^2 b \coth (c+d x)}{d}-\frac{3 a b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b^3 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b^3 x}{2}",1,"-1/384*(192*b^3*c + 192*b^3*d*x - 384*a^2*b*Coth[(c + d*x)/2] + 30*a^3*Csch[(c + d*x)/2]^2 + a^3*Csch[(c + d*x)/2]^6 + 120*a^3*Log[Tanh[(c + d*x)/2]] - 1152*a*b^2*Log[Tanh[(c + d*x)/2]] + 30*a^3*Sech[(c + d*x)/2]^2 + 6*a^3*Sech[(c + d*x)/2]^4 + a^3*Sech[(c + d*x)/2]^6 - 384*a^2*b*Csch[c + d*x]^3*Sinh[(c + d*x)/2]^4 - 6*a^2*Csch[(c + d*x)/2]^4*(a - 4*b*Sinh[c + d*x]) - 96*b^3*Sinh[2*(c + d*x)] - 384*a^2*b*Tanh[(c + d*x)/2])/d","A",1
171,1,168,328,0.3655533,"\int \frac{\sinh ^6(c+d x)}{a+b \sinh ^3(c+d x)} \, dx","Integrate[Sinh[c + d*x]^6/(a + b*Sinh[c + d*x]^3),x]","\frac{8 a^2 \text{RootSum}\left[\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 \text{$\#$1}^2 b-b\&,\frac{2 \text{$\#$1} \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1} c+\text{$\#$1} d x}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b+4 \text{$\#$1} a+b}\&\right]-12 a c-12 a d x-9 b \cosh (c+d x)+b \cosh (3 (c+d x))}{12 b^2 d}","-\frac{2 a^{4/3} \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}+b^{2/3}}}\right)}{3 b^2 d \sqrt{a^{2/3}+b^{2/3}}}-\frac{2 (-1)^{2/3} a^{4/3} \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left(\sqrt[6]{-1} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b^2 d \sqrt{\sqrt[3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{2 (-1)^{2/3} a^{4/3} \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left((-1)^{5/6} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}\right)}{3 b^2 d \sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}-\frac{a x}{b^2}+\frac{\cosh ^3(c+d x)}{3 b d}-\frac{\cosh (c+d x)}{b d}",1,"(-12*a*c - 12*a*d*x - 9*b*Cosh[c + d*x] + b*Cosh[3*(c + d*x)] + 8*a^2*RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (c*#1 + d*x*#1 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1)/(b + 4*a*#1 - 2*b*#1^2 + b*#1^4) & ])/(12*b^2*d)","C",1
172,1,299,295,0.3321255,"\int \frac{\sinh ^5(c+d x)}{a+b \sinh ^3(c+d x)} \, dx","Integrate[Sinh[c + d*x]^5/(a + b*Sinh[c + d*x]^3),x]","\frac{-2 a \text{RootSum}\left[\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 \text{$\#$1}^2 b-b\&,\frac{2 \text{$\#$1}^4 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^4 c+\text{$\#$1}^4 d x-4 \text{$\#$1}^2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-2 \text{$\#$1}^2 c-2 \text{$\#$1}^2 d x+2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+c+d x}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b+4 \text{$\#$1}^2 a+\text{$\#$1} b}\&\right]-6 (c+d x)+3 \sinh (2 (c+d x))}{12 b d}","\frac{2 a \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}+b^{2/3}}}\right)}{3 b^{5/3} d \sqrt{a^{2/3}+b^{2/3}}}+\frac{2 a \tan ^{-1}\left(\frac{(-1)^{5/6} \left(\sqrt[6]{-1} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right)}{3 b^{5/3} d \sqrt{-(-1)^{2/3} a^{2/3}-b^{2/3}}}+\frac{2 a \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left((-1)^{5/6} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}\right)}{3 b^{5/3} d \sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac{x}{2 b}",1,"(-6*(c + d*x) - 2*a*RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (c + d*x + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 2*c*#1^2 - 2*d*x*#1^2 - 4*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + c*#1^4 + d*x*#1^4 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4)/(b*#1 + 4*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + 3*Sinh[2*(c + d*x)])/(12*b*d)","C",1
173,1,214,303,0.3646848,"\int \frac{\sinh ^4(c+d x)}{a+b \sinh ^3(c+d x)} \, dx","Integrate[Sinh[c + d*x]^4/(a + b*Sinh[c + d*x]^3),x]","\frac{3 \cosh (c+d x)-a \text{RootSum}\left[\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 \text{$\#$1}^2 b-b\&,\frac{2 \text{$\#$1}^2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^2 c+\text{$\#$1}^2 d x-2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-c-d x}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b+4 \text{$\#$1} a+b}\&\right]}{3 b d}","-\frac{2 a^{2/3} \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}+b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}+b^{2/3}}}-\frac{2 a^{2/3} \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left(\sqrt[6]{-1} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{\sqrt[3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 \sqrt[3]{-1} a^{2/3} \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left((-1)^{5/6} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}+\frac{\cosh (c+d x)}{b d}",1,"(3*Cosh[c + d*x] - a*RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (-c - d*x - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + c*#1^2 + d*x*#1^2 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2)/(b + 4*a*#1 - 2*b*#1^2 + b*#1^4) & ])/(3*b*d)","C",1
174,1,145,294,0.2327292,"\int \frac{\sinh ^3(c+d x)}{a+b \sinh ^3(c+d x)} \, dx","Integrate[Sinh[c + d*x]^3/(a + b*Sinh[c + d*x]^3),x]","\frac{-2 a \text{RootSum}\left[\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 \text{$\#$1}^2 b-b\&,\frac{2 \text{$\#$1} \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1} c+\text{$\#$1} d x}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b+4 \text{$\#$1} a+b}\&\right]+3 c+3 d x}{3 b d}","\frac{2 \sqrt[3]{a} \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}+b^{2/3}}}\right)}{3 b d \sqrt{a^{2/3}+b^{2/3}}}+\frac{2 (-1)^{2/3} \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left(\sqrt[6]{-1} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b d \sqrt{\sqrt[3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 (-1)^{2/3} \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left((-1)^{5/6} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}\right)}{3 b d \sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}+\frac{x}{b}",1,"(3*c + 3*d*x - 2*a*RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (c*#1 + d*x*#1 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1)/(b + 4*a*#1 - 2*b*#1^2 + b*#1^4) & ])/(3*b*d)","C",1
175,1,275,262,0.1828333,"\int \frac{\sinh ^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx","Integrate[Sinh[c + d*x]^2/(a + b*Sinh[c + d*x]^3),x]","\frac{\text{RootSum}\left[\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 \text{$\#$1}^2 b-b\&,\frac{2 \text{$\#$1}^4 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^4 c+\text{$\#$1}^4 d x-4 \text{$\#$1}^2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-2 \text{$\#$1}^2 c-2 \text{$\#$1}^2 d x+2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+c+d x}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b+4 \text{$\#$1}^2 a+\text{$\#$1} b}\&\right]}{6 d}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}+b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{a^{2/3}+b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{(-1)^{5/6} \left(\sqrt[6]{-1} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{-(-1)^{2/3} a^{2/3}-b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left((-1)^{5/6} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}",1,"RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (c + d*x + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 2*c*#1^2 - 2*d*x*#1^2 - 4*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + c*#1^4 + d*x*#1^4 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4)/(b*#1 + 4*a*#1^2 - 2*b*#1^3 + b*#1^5) & ]/(6*d)","C",1
176,1,199,290,0.2527423,"\int \frac{\sinh (c+d x)}{a+b \sinh ^3(c+d x)} \, dx","Integrate[Sinh[c + d*x]/(a + b*Sinh[c + d*x]^3),x]","\frac{\text{RootSum}\left[\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 \text{$\#$1}^2 b-b\&,\frac{2 \text{$\#$1}^2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^2 c+\text{$\#$1}^2 d x-2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-c-d x}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b+4 \text{$\#$1} a+b}\&\right]}{3 d}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}+b^{2/3}}}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} d \sqrt{a^{2/3}+b^{2/3}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left(\sqrt[6]{-1} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} d \sqrt{\sqrt[3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{2 \sqrt[3]{-1} \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left((-1)^{5/6} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} d \sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}",1,"RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (-c - d*x - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + c*#1^2 + d*x*#1^2 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2)/(b + 4*a*#1 - 2*b*#1^2 + b*#1^4) & ]/(3*d)","C",1
177,1,131,280,0.1727498,"\int \frac{1}{a+b \sinh ^3(c+d x)} \, dx","Integrate[(a + b*Sinh[c + d*x]^3)^(-1),x]","\frac{2 \text{RootSum}\left[\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 \text{$\#$1}^2 b-b\&,\frac{2 \text{$\#$1} \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1} c+\text{$\#$1} d x}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b+4 \text{$\#$1} a+b}\&\right]}{3 d}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}+b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}+b^{2/3}}}-\frac{2 (-1)^{2/3} \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left(\sqrt[6]{-1} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{\sqrt[3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{2 (-1)^{2/3} \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left((-1)^{5/6} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}",1,"(2*RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (c*#1 + d*x*#1 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1)/(b + 4*a*#1 - 2*b*#1^2 + b*#1^4) & ])/(3*d)","C",1
178,1,295,286,0.263581,"\int \frac{\text{csch}(c+d x)}{a+b \sinh ^3(c+d x)} \, dx","Integrate[Csch[c + d*x]/(a + b*Sinh[c + d*x]^3),x]","\frac{6 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-b \text{RootSum}\left[\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 \text{$\#$1}^2 b-b\&,\frac{2 \text{$\#$1}^4 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^4 c+\text{$\#$1}^4 d x-4 \text{$\#$1}^2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-2 \text{$\#$1}^2 c-2 \text{$\#$1}^2 d x+2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+c+d x}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b+4 \text{$\#$1}^2 a+\text{$\#$1} b}\&\right]}{6 a d}","\frac{2 \sqrt[3]{b} \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}+b^{2/3}}}\right)}{3 a d \sqrt{a^{2/3}+b^{2/3}}}+\frac{2 \sqrt[3]{b} \tan ^{-1}\left(\frac{(-1)^{5/6} \left(\sqrt[6]{-1} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right)}{3 a d \sqrt{-(-1)^{2/3} a^{2/3}-b^{2/3}}}+\frac{2 \sqrt[3]{b} \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left((-1)^{5/6} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}\right)}{3 a d \sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a d}",1,"(6*Log[Tanh[(c + d*x)/2]] - b*RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (c + d*x + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 2*c*#1^2 - 2*d*x*#1^2 - 4*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + c*#1^4 + d*x*#1^4 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4)/(b*#1 + 4*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/(6*a*d)","C",1
179,1,230,304,0.4179412,"\int \frac{\text{csch}^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx","Integrate[Csch[c + d*x]^2/(a + b*Sinh[c + d*x]^3),x]","-\frac{2 b \text{RootSum}\left[\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 \text{$\#$1}^2 b-b\&,\frac{2 \text{$\#$1}^2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^2 c+\text{$\#$1}^2 d x-2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-c-d x}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b+4 \text{$\#$1} a+b}\&\right]+3 \tanh \left(\frac{1}{2} (c+d x)\right)+3 \coth \left(\frac{1}{2} (c+d x)\right)}{6 a d}","-\frac{2 b^{2/3} \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}+b^{2/3}}}\right)}{3 a^{4/3} d \sqrt{a^{2/3}+b^{2/3}}}-\frac{2 b^{2/3} \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left(\sqrt[6]{-1} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{4/3} d \sqrt{\sqrt[3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 \sqrt[3]{-1} b^{2/3} \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left((-1)^{5/6} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}\right)}{3 a^{4/3} d \sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}-\frac{\coth (c+d x)}{a d}",1,"-1/6*(3*Coth[(c + d*x)/2] + 2*b*RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (-c - d*x - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + c*#1^2 + d*x*#1^2 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2)/(b + 4*a*#1 - 2*b*#1^2 + b*#1^4) & ] + 3*Tanh[(c + d*x)/2])/(a*d)","C",1
180,1,178,322,0.5129213,"\int \frac{\text{csch}^3(c+d x)}{a+b \sinh ^3(c+d x)} \, dx","Integrate[Csch[c + d*x]^3/(a + b*Sinh[c + d*x]^3),x]","-\frac{16 b \text{RootSum}\left[\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 \text{$\#$1}^2 b-b\&,\frac{2 \text{$\#$1} \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1} c+\text{$\#$1} d x}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b+4 \text{$\#$1} a+b}\&\right]+3 \left(\text{csch}^2\left(\frac{1}{2} (c+d x)\right)+\text{sech}^2\left(\frac{1}{2} (c+d x)\right)+4 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)\right)}{24 a d}","\frac{2 b \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}+b^{2/3}}}\right)}{3 a^{5/3} d \sqrt{a^{2/3}+b^{2/3}}}+\frac{2 (-1)^{2/3} b \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left(\sqrt[6]{-1} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{5/3} d \sqrt{\sqrt[3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 (-1)^{2/3} b \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left((-1)^{5/6} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}\right)}{3 a^{5/3} d \sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}+\frac{\tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d}",1,"-1/24*(16*b*RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (c*#1 + d*x*#1 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1)/(b + 4*a*#1 - 2*b*#1^2 + b*#1^4) & ] + 3*(Csch[(c + d*x)/2]^2 + 4*Log[Tanh[(c + d*x)/2]] + Sech[(c + d*x)/2]^2))/(a*d)","C",1
181,1,370,317,5.7877554,"\int \frac{\text{csch}^4(c+d x)}{a+b \sinh ^3(c+d x)} \, dx","Integrate[Csch[c + d*x]^4/(a + b*Sinh[c + d*x]^3),x]","\frac{4 b^2 \text{RootSum}\left[\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 \text{$\#$1}^2 b-b\&,\frac{2 \text{$\#$1}^4 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^4 c+\text{$\#$1}^4 d x-4 \text{$\#$1}^2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-2 \text{$\#$1}^2 c-2 \text{$\#$1}^2 d x+2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+c+d x}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b+4 \text{$\#$1}^2 a+\text{$\#$1} b}\&\right]+8 a \tanh \left(\frac{1}{2} (c+d x)\right)+8 a \coth \left(\frac{1}{2} (c+d x)\right)-\frac{1}{2} a \sinh (c+d x) \text{csch}^4\left(\frac{1}{2} (c+d x)\right)+8 a \sinh ^4\left(\frac{1}{2} (c+d x)\right) \text{csch}^3(c+d x)-24 b \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{24 a^2 d}","\frac{b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac{2 b^{4/3} \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}+b^{2/3}}}\right)}{3 a^2 d \sqrt{a^{2/3}+b^{2/3}}}-\frac{2 b^{4/3} \tan ^{-1}\left(\frac{(-1)^{5/6} \left(\sqrt[6]{-1} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right)}{3 a^2 d \sqrt{-(-1)^{2/3} a^{2/3}-b^{2/3}}}-\frac{2 b^{4/3} \tan ^{-1}\left(\frac{\sqrt[6]{-1} \left((-1)^{5/6} \sqrt[3]{b}+i \sqrt[3]{a} \tanh \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}\right)}{3 a^2 d \sqrt{\sqrt[3]{-1} a^{2/3}-b^{2/3}}}-\frac{\coth ^3(c+d x)}{3 a d}+\frac{\coth (c+d x)}{a d}",1,"(8*a*Coth[(c + d*x)/2] - 24*b*Log[Tanh[(c + d*x)/2]] + 4*b^2*RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (c + d*x + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 2*c*#1^2 - 2*d*x*#1^2 - 4*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + c*#1^4 + d*x*#1^4 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4)/(b*#1 + 4*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + 8*a*Csch[c + d*x]^3*Sinh[(c + d*x)/2]^4 - (a*Csch[(c + d*x)/2]^4*Sinh[c + d*x])/2 + 8*a*Tanh[(c + d*x)/2])/(24*a^2*d)","C",1
182,1,156,139,1.5048897,"\int \frac{1}{1+\sinh ^3(x)} \, dx","Integrate[(1 + Sinh[x]^3)^(-1),x]","\frac{2 \tanh ^{-1}\left(\frac{\tanh \left(\frac{x}{2}\right)-1}{\sqrt{2}}\right)+i \sqrt{-1-i \sqrt{3}} \left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{2+\left(1-i \sqrt{3}\right) \tanh \left(\frac{x}{2}\right)}{\sqrt{-2+2 i \sqrt{3}}}\right)+\left(-1-i \sqrt{3}\right) \sqrt{-1+i \sqrt{3}} \tan ^{-1}\left(\frac{2+\left(1+i \sqrt{3}\right) \tanh \left(\frac{x}{2}\right)}{\sqrt{-2-2 i \sqrt{3}}}\right)}{3 \sqrt{2}}","-\frac{1}{3} \sqrt{2} \tanh ^{-1}\left(\frac{1-\tanh \left(\frac{x}{2}\right)}{\sqrt{2}}\right)-\frac{1}{3} \sqrt[6]{-1} \log \left(-\sqrt[6]{-1} \tanh \left(\frac{x}{2}\right)+(-1)^{5/6}+1\right)+\frac{1}{3} \sqrt[6]{-1} \log \left(\sqrt[3]{-1} \tanh \left(\frac{x}{2}\right)+\sqrt[6]{-1}+1\right)-\frac{2 \sqrt[6]{-1} \tan ^{-1}\left(\frac{\sqrt[6]{-1} \tanh \left(\frac{x}{2}\right)+i}{\sqrt{1-\sqrt[3]{-1}}}\right)}{3 \sqrt{1-\sqrt[3]{-1}}}",1,"(I*Sqrt[-1 - I*Sqrt[3]]*(I + Sqrt[3])*ArcTan[(2 + (1 - I*Sqrt[3])*Tanh[x/2])/Sqrt[-2 + (2*I)*Sqrt[3]]] + (-1 - I*Sqrt[3])*Sqrt[-1 + I*Sqrt[3]]*ArcTan[(2 + (1 + I*Sqrt[3])*Tanh[x/2])/Sqrt[-2 - (2*I)*Sqrt[3]]] + 2*ArcTanh[(-1 + Tanh[x/2])/Sqrt[2]])/(3*Sqrt[2])","A",1
183,1,156,133,1.2808655,"\int \frac{1}{1-\sinh ^3(x)} \, dx","Integrate[(1 - Sinh[x]^3)^(-1),x]","\frac{2 \tanh ^{-1}\left(\frac{\tanh \left(\frac{x}{2}\right)+1}{\sqrt{2}}\right)+\sqrt{-1+i \sqrt{3}} \left(1+i \sqrt{3}\right) \tan ^{-1}\left(\frac{2+\left(-1-i \sqrt{3}\right) \tanh \left(\frac{x}{2}\right)}{\sqrt{-2-2 i \sqrt{3}}}\right)+\sqrt{-1-i \sqrt{3}} \left(1-i \sqrt{3}\right) \tan ^{-1}\left(\frac{2+i \left(\sqrt{3}+i\right) \tanh \left(\frac{x}{2}\right)}{\sqrt{-2+2 i \sqrt{3}}}\right)}{3 \sqrt{2}}","\frac{1}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\tanh \left(\frac{x}{2}\right)+1}{\sqrt{2}}\right)-\frac{1}{3} (-1)^{5/6} \log \left((-1)^{2/3} \tanh \left(\frac{x}{2}\right)+(-1)^{5/6}+1\right)+\frac{1}{3} (-1)^{5/6} \log \left((-1)^{5/6} \tanh \left(\frac{x}{2}\right)+\sqrt[6]{-1}+1\right)+\frac{2 (-1)^{5/6} \tan ^{-1}\left(\frac{-(-1)^{5/6} \tanh \left(\frac{x}{2}\right)+i}{\sqrt{1+(-1)^{2/3}}}\right)}{3 \sqrt{1+(-1)^{2/3}}}",1,"(Sqrt[-1 + I*Sqrt[3]]*(1 + I*Sqrt[3])*ArcTan[(2 + (-1 - I*Sqrt[3])*Tanh[x/2])/Sqrt[-2 - (2*I)*Sqrt[3]]] + Sqrt[-1 - I*Sqrt[3]]*(1 - I*Sqrt[3])*ArcTan[(2 + I*(I + Sqrt[3])*Tanh[x/2])/Sqrt[-2 + (2*I)*Sqrt[3]]] + 2*ArcTanh[(1 + Tanh[x/2])/Sqrt[2]])/(3*Sqrt[2])","A",1
184,1,82,111,0.1366782,"\int \sinh ^4(c+d x) \left(a+b \sinh ^4(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^4),x]","\frac{-96 (8 a+7 b) \sinh (2 (c+d x))+24 (4 a+7 b) \sinh (4 (c+d x))+1152 a c+1152 a d x-32 b \sinh (6 (c+d x))+3 b \sinh (8 (c+d x))+840 b c+840 b d x}{3072 d}","\frac{(48 a+163 b) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}-\frac{(80 a+93 b) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{1}{128} x (48 a+35 b)+\frac{b \sinh (c+d x) \cosh ^7(c+d x)}{8 d}-\frac{25 b \sinh (c+d x) \cosh ^5(c+d x)}{48 d}",1,"(1152*a*c + 840*b*c + 1152*a*d*x + 840*b*d*x - 96*(8*a + 7*b)*Sinh[2*(c + d*x)] + 24*(4*a + 7*b)*Sinh[4*(c + d*x)] - 32*b*Sinh[6*(c + d*x)] + 3*b*Sinh[8*(c + d*x)])/(3072*d)","A",1
185,1,93,67,0.0277881,"\int \sinh ^3(c+d x) \left(a+b \sinh ^4(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^4),x]","-\frac{3 a \cosh (c+d x)}{4 d}+\frac{a \cosh (3 (c+d x))}{12 d}-\frac{35 b \cosh (c+d x)}{64 d}+\frac{7 b \cosh (3 (c+d x))}{64 d}-\frac{7 b \cosh (5 (c+d x))}{320 d}+\frac{b \cosh (7 (c+d x))}{448 d}","\frac{(a+3 b) \cosh ^3(c+d x)}{3 d}-\frac{(a+b) \cosh (c+d x)}{d}+\frac{b \cosh ^7(c+d x)}{7 d}-\frac{3 b \cosh ^5(c+d x)}{5 d}",1,"(-3*a*Cosh[c + d*x])/(4*d) - (35*b*Cosh[c + d*x])/(64*d) + (a*Cosh[3*(c + d*x)])/(12*d) + (7*b*Cosh[3*(c + d*x)])/(64*d) - (7*b*Cosh[5*(c + d*x)])/(320*d) + (b*Cosh[7*(c + d*x)])/(448*d)","A",1
186,1,63,83,0.078728,"\int \sinh ^2(c+d x) \left(a+b \sinh ^4(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^4),x]","\frac{(48 a+45 b) \sinh (2 (c+d x))-96 a c-96 a d x-9 b \sinh (4 (c+d x))+b \sinh (6 (c+d x))-60 b c-60 b d x}{192 d}","\frac{(8 a+11 b) \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac{1}{16} x (8 a+5 b)+\frac{b \sinh (c+d x) \cosh ^5(c+d x)}{6 d}-\frac{13 b \sinh (c+d x) \cosh ^3(c+d x)}{24 d}",1,"(-96*a*c - 60*b*c - 96*a*d*x - 60*b*d*x + (48*a + 45*b)*Sinh[2*(c + d*x)] - 9*b*Sinh[4*(c + d*x)] + b*Sinh[6*(c + d*x)])/(192*d)","A",1
187,1,69,46,0.0233471,"\int \sinh (c+d x) \left(a+b \sinh ^4(c+d x)\right) \, dx","Integrate[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^4),x]","\frac{a \sinh (c) \sinh (d x)}{d}+\frac{a \cosh (c) \cosh (d x)}{d}+\frac{5 b \cosh (c+d x)}{8 d}-\frac{5 b \cosh (3 (c+d x))}{48 d}+\frac{b \cosh (5 (c+d x))}{80 d}","\frac{(a+b) \cosh (c+d x)}{d}+\frac{b \cosh ^5(c+d x)}{5 d}-\frac{2 b \cosh ^3(c+d x)}{3 d}",1,"(a*Cosh[c]*Cosh[d*x])/d + (5*b*Cosh[c + d*x])/(8*d) - (5*b*Cosh[3*(c + d*x)])/(48*d) + (b*Cosh[5*(c + d*x)])/(80*d) + (a*Sinh[c]*Sinh[d*x])/d","A",1
188,1,49,52,0.0576415,"\int \left(a+b \sinh ^4(c+d x)\right) \, dx","Integrate[a + b*Sinh[c + d*x]^4,x]","a x+\frac{3 b (c+d x)}{8 d}-\frac{b \sinh (2 (c+d x))}{4 d}+\frac{b \sinh (4 (c+d x))}{32 d}","a x+\frac{b \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{3 b \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3 b x}{8}",1,"a*x + (3*b*(c + d*x))/(8*d) - (b*Sinh[2*(c + d*x)])/(4*d) + (b*Sinh[4*(c + d*x)])/(32*d)","A",1
189,1,70,42,0.024208,"\int \text{csch}(c+d x) \left(a+b \sinh ^4(c+d x)\right) \, dx","Integrate[Csch[c + d*x]*(a + b*Sinh[c + d*x]^4),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{3 b \cosh (c+d x)}{4 d}+\frac{b \cosh (3 (c+d x))}{12 d}","-\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \cosh ^3(c+d x)}{3 d}-\frac{b \cosh (c+d x)}{d}",1,"(-3*b*Cosh[c + d*x])/(4*d) + (b*Cosh[3*(c + d*x)])/(12*d) - (a*Log[Cosh[c/2 + (d*x)/2]])/d + (a*Log[Sinh[c/2 + (d*x)/2]])/d","A",1
190,1,45,39,0.1179563,"\int \text{csch}^2(c+d x) \left(a+b \sinh ^4(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^4),x]","-\frac{a \coth (c+d x)}{d}+\frac{b (-c-d x)}{2 d}+\frac{b \sinh (2 (c+d x))}{4 d}","-\frac{a \coth (c+d x)}{d}+\frac{b \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b x}{2}",1,"(b*(-c - d*x))/(2*d) - (a*Coth[c + d*x])/d + (b*Sinh[2*(c + d*x)])/(4*d)","A",1
191,1,82,47,0.0263845,"\int \text{csch}^3(c+d x) \left(a+b \sinh ^4(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^4),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 \sinh (c) \sinh (d x)}{d}+\frac{b \cosh (c) \cosh (d x)}{d}","\frac{a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{b \cosh (c+d x)}{d}",1,"(b*Cosh[c]*Cosh[d*x])/d - (a*Csch[(c + d*x)/2]^2)/(8*d) - (a*Log[Tanh[(c + d*x)/2]])/(2*d) - (a*Sech[(c + d*x)/2]^2)/(8*d) + (b*Sinh[c]*Sinh[d*x])/d","A",1
192,1,40,31,0.0158558,"\int \text{csch}^4(c+d x) \left(a+b \sinh ^4(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^4),x]","\frac{2 a \coth (c+d x)}{3 d}-\frac{a \coth (c+d x) \text{csch}^2(c+d x)}{3 d}+b x","-\frac{a \coth ^3(c+d x)}{3 d}+\frac{a \coth (c+d x)}{d}+b x",1,"b*x + (2*a*Coth[c + d*x])/(3*d) - (a*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d)","A",1
193,1,139,64,0.034513,"\int \text{csch}^5(c+d x) \left(a+b \sinh ^4(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^4),x]","-\frac{a \text{csch}^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{3 a \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{a \text{sech}^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{3 a \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{3 a \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}+\frac{b \log \left(\sinh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{b \log \left(\cosh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}","-\frac{(3 a+8 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac{a \coth (c+d x) \text{csch}^3(c+d x)}{4 d}+\frac{3 a \coth (c+d x) \text{csch}(c+d x)}{8 d}",1,"(3*a*Csch[(c + d*x)/2]^2)/(32*d) - (a*Csch[(c + d*x)/2]^4)/(64*d) - (b*Log[Cosh[c/2 + (d*x)/2]])/d + (b*Log[Sinh[c/2 + (d*x)/2]])/d + (3*a*Log[Tanh[(c + d*x)/2]])/(8*d) + (3*a*Sech[(c + d*x)/2]^2)/(32*d) + (a*Sech[(c + d*x)/2]^4)/(64*d)","B",1
194,1,71,47,0.034668,"\int \text{csch}^6(c+d x) \left(a+b \sinh ^4(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^4),x]","-\frac{8 a \coth (c+d x)}{15 d}-\frac{a \coth (c+d x) \text{csch}^4(c+d x)}{5 d}+\frac{4 a \coth (c+d x) \text{csch}^2(c+d x)}{15 d}-\frac{b \coth (c+d x)}{d}","-\frac{(a+b) \coth (c+d x)}{d}-\frac{a \coth ^5(c+d x)}{5 d}+\frac{2 a \coth ^3(c+d x)}{3 d}",1,"(-8*a*Coth[c + d*x])/(15*d) - (b*Coth[c + d*x])/d + (4*a*Coth[c + d*x]*Csch[c + d*x]^2)/(15*d) - (a*Coth[c + d*x]*Csch[c + d*x]^4)/(5*d)","A",1
195,1,199,92,0.0355964,"\int \text{csch}^7(c+d x) \left(a+b \sinh ^4(c+d x)\right) \, dx","Integrate[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^4),x]","-\frac{a \text{csch}^6\left(\frac{1}{2} (c+d x)\right)}{384 d}+\frac{a \text{csch}^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 a \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{a \text{sech}^6\left(\frac{1}{2} (c+d x)\right)}{384 d}-\frac{a \text{sech}^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 a \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 a \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{16 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 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}","\frac{(5 a+8 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac{(5 a+8 b) \coth (c+d x) \text{csch}(c+d x)}{16 d}-\frac{a \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{5 a \coth (c+d x) \text{csch}^3(c+d x)}{24 d}",1,"(-5*a*Csch[(c + d*x)/2]^2)/(64*d) - (b*Csch[(c + d*x)/2]^2)/(8*d) + (a*Csch[(c + d*x)/2]^4)/(64*d) - (a*Csch[(c + d*x)/2]^6)/(384*d) - (5*a*Log[Tanh[(c + d*x)/2]])/(16*d) - (b*Log[Tanh[(c + d*x)/2]])/(2*d) - (5*a*Sech[(c + d*x)/2]^2)/(64*d) - (b*Sech[(c + d*x)/2]^2)/(8*d) - (a*Sech[(c + d*x)/2]^4)/(64*d) - (a*Sech[(c + d*x)/2]^6)/(384*d)","B",1
196,1,207,120,0.0598879,"\int \sinh ^3(c+d x) \left(a+b \sinh ^4(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^2,x]","-\frac{3 a^2 \cosh (c+d x)}{4 d}+\frac{a^2 \cosh (3 (c+d x))}{12 d}-\frac{35 a b \cosh (c+d x)}{32 d}+\frac{7 a b \cosh (3 (c+d x))}{32 d}-\frac{7 a b \cosh (5 (c+d x))}{160 d}+\frac{a b \cosh (7 (c+d x))}{224 d}-\frac{231 b^2 \cosh (c+d x)}{512 d}+\frac{55 b^2 \cosh (3 (c+d x))}{512 d}-\frac{33 b^2 \cosh (5 (c+d x))}{1024 d}+\frac{55 b^2 \cosh (7 (c+d x))}{7168 d}-\frac{11 b^2 \cosh (9 (c+d x))}{9216 d}+\frac{b^2 \cosh (11 (c+d x))}{11264 d}","\frac{2 b (a+5 b) \cosh ^7(c+d x)}{7 d}-\frac{2 b (3 a+5 b) \cosh ^5(c+d x)}{5 d}+\frac{(a+b) (a+5 b) \cosh ^3(c+d x)}{3 d}-\frac{(a+b)^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh ^{11}(c+d x)}{11 d}-\frac{5 b^2 \cosh ^9(c+d x)}{9 d}",1,"(-3*a^2*Cosh[c + d*x])/(4*d) - (35*a*b*Cosh[c + d*x])/(32*d) - (231*b^2*Cosh[c + d*x])/(512*d) + (a^2*Cosh[3*(c + d*x)])/(12*d) + (7*a*b*Cosh[3*(c + d*x)])/(32*d) + (55*b^2*Cosh[3*(c + d*x)])/(512*d) - (7*a*b*Cosh[5*(c + d*x)])/(160*d) - (33*b^2*Cosh[5*(c + d*x)])/(1024*d) + (a*b*Cosh[7*(c + d*x)])/(224*d) + (55*b^2*Cosh[7*(c + d*x)])/(7168*d) - (11*b^2*Cosh[9*(c + d*x)])/(9216*d) + (b^2*Cosh[11*(c + d*x)])/(11264*d)","A",1
197,1,139,161,0.3443046,"\int \sinh ^2(c+d x) \left(a+b \sinh ^4(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^4)^2,x]","-\frac{-60 \left(128 a^2+240 a b+105 b^2\right) \sinh (2 (c+d x))+15360 a^2 c+15360 a^2 d x-320 a b \sinh (6 (c+d x))+360 b (8 a+5 b) \sinh (4 (c+d x))+19200 a b c+19200 a b d x-450 b^2 \sinh (6 (c+d x))+75 b^2 \sinh (8 (c+d x))-6 b^2 \sinh (10 (c+d x))+7560 b^2 c+7560 b^2 d x}{30720 d}","\frac{\left(128 a^2+352 a b+193 b^2\right) \sinh (c+d x) \cosh (c+d x)}{256 d}-\frac{1}{256} x \left(128 a^2+160 a b+63 b^2\right)+\frac{b (160 a+513 b) \sinh (c+d x) \cosh ^5(c+d x)}{480 d}-\frac{b (416 a+447 b) \sinh (c+d x) \cosh ^3(c+d x)}{384 d}+\frac{b^2 \sinh (c+d x) \cosh ^9(c+d x)}{10 d}-\frac{41 b^2 \sinh (c+d x) \cosh ^7(c+d x)}{80 d}",1,"-1/30720*(15360*a^2*c + 19200*a*b*c + 7560*b^2*c + 15360*a^2*d*x + 19200*a*b*d*x + 7560*b^2*d*x - 60*(128*a^2 + 240*a*b + 105*b^2)*Sinh[2*(c + d*x)] + 360*b*(8*a + 5*b)*Sinh[4*(c + d*x)] - 320*a*b*Sinh[6*(c + d*x)] - 450*b^2*Sinh[6*(c + d*x)] + 75*b^2*Sinh[8*(c + d*x)] - 6*b^2*Sinh[10*(c + d*x)])/d","A",1
198,1,164,92,0.0390475,"\int \sinh (c+d x) \left(a+b \sinh ^4(c+d x)\right)^2 \, dx","Integrate[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^4)^2,x]","\frac{a^2 \sinh (c) \sinh (d x)}{d}+\frac{a^2 \cosh (c) \cosh (d x)}{d}+\frac{5 a b \cosh (c+d x)}{4 d}-\frac{5 a b \cosh (3 (c+d x))}{24 d}+\frac{a b \cosh (5 (c+d x))}{40 d}+\frac{63 b^2 \cosh (c+d x)}{128 d}-\frac{7 b^2 \cosh (3 (c+d x))}{64 d}+\frac{9 b^2 \cosh (5 (c+d x))}{320 d}-\frac{9 b^2 \cosh (7 (c+d x))}{1792 d}+\frac{b^2 \cosh (9 (c+d x))}{2304 d}","\frac{2 b (a+3 b) \cosh ^5(c+d x)}{5 d}-\frac{4 b (a+b) \cosh ^3(c+d x)}{3 d}+\frac{(a+b)^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh ^9(c+d x)}{9 d}-\frac{4 b^2 \cosh ^7(c+d x)}{7 d}",1,"(a^2*Cosh[c]*Cosh[d*x])/d + (5*a*b*Cosh[c + d*x])/(4*d) + (63*b^2*Cosh[c + d*x])/(128*d) - (5*a*b*Cosh[3*(c + d*x)])/(24*d) - (7*b^2*Cosh[3*(c + d*x)])/(64*d) + (a*b*Cosh[5*(c + d*x)])/(40*d) + (9*b^2*Cosh[5*(c + d*x)])/(320*d) - (9*b^2*Cosh[7*(c + d*x)])/(1792*d) + (b^2*Cosh[9*(c + d*x)])/(2304*d) + (a^2*Sinh[c]*Sinh[d*x])/d","A",1
199,1,92,125,0.1661113,"\int \left(a+b \sinh ^4(c+d x)\right)^2 \, dx","Integrate[(a + b*Sinh[c + d*x]^4)^2,x]","\frac{24 \left(128 a^2+96 a b+35 b^2\right) (c+d x)-96 b (16 a+7 b) \sinh (2 (c+d x))+24 b (8 a+7 b) \sinh (4 (c+d x))-32 b^2 \sinh (6 (c+d x))+3 b^2 \sinh (8 (c+d x))}{3072 d}","\frac{1}{128} x \left(128 a^2+96 a b+35 b^2\right)+\frac{b (96 a+163 b) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}-\frac{b (160 a+93 b) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{b^2 \sinh (c+d x) \cosh ^7(c+d x)}{8 d}-\frac{25 b^2 \sinh (c+d x) \cosh ^5(c+d x)}{48 d}",1,"(24*(128*a^2 + 96*a*b + 35*b^2)*(c + d*x) - 96*b*(16*a + 7*b)*Sinh[2*(c + d*x)] + 24*b*(8*a + 7*b)*Sinh[4*(c + d*x)] - 32*b^2*Sinh[6*(c + d*x)] + 3*b^2*Sinh[8*(c + d*x)])/(3072*d)","A",1
200,1,146,92,0.0415715,"\int \text{csch}(c+d x) \left(a+b \sinh ^4(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]*(a + b*Sinh[c + d*x]^4)^2,x]","\frac{a^2 \log \left(\sinh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{a^2 \log \left(\cosh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{3 a b \cosh (c+d x)}{2 d}+\frac{a b \cosh (3 (c+d x))}{6 d}-\frac{35 b^2 \cosh (c+d x)}{64 d}+\frac{7 b^2 \cosh (3 (c+d x))}{64 d}-\frac{7 b^2 \cosh (5 (c+d x))}{320 d}+\frac{b^2 \cosh (7 (c+d x))}{448 d}","-\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b (2 a+3 b) \cosh ^3(c+d x)}{3 d}-\frac{b (2 a+b) \cosh (c+d x)}{d}+\frac{b^2 \cosh ^7(c+d x)}{7 d}-\frac{3 b^2 \cosh ^5(c+d x)}{5 d}",1,"(-3*a*b*Cosh[c + d*x])/(2*d) - (35*b^2*Cosh[c + d*x])/(64*d) + (a*b*Cosh[3*(c + d*x)])/(6*d) + (7*b^2*Cosh[3*(c + d*x)])/(64*d) - (7*b^2*Cosh[5*(c + d*x)])/(320*d) + (b^2*Cosh[7*(c + d*x)])/(448*d) - (a^2*Log[Cosh[c/2 + (d*x)/2]])/d + (a^2*Log[Sinh[c/2 + (d*x)/2]])/d","A",1
201,1,77,103,0.3051507,"\int \text{csch}^2(c+d x) \left(a+b \sinh ^4(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^4)^2,x]","\frac{b ((96 a+45 b) \sinh (2 (c+d x))-192 a c-192 a d x-9 b \sinh (4 (c+d x))+b \sinh (6 (c+d x))-60 b c-60 b d x)-192 a^2 \coth (c+d x)}{192 d}","-\frac{a^2 \coth (c+d x)}{d}+\frac{b (16 a+11 b) \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac{1}{16} b x (16 a+5 b)+\frac{b^2 \sinh (c+d x) \cosh ^5(c+d x)}{6 d}-\frac{13 b^2 \sinh (c+d x) \cosh ^3(c+d x)}{24 d}",1,"(-192*a^2*Coth[c + d*x] + b*(-192*a*c - 60*b*c - 192*a*d*x - 60*b*d*x + (96*a + 45*b)*Sinh[2*(c + d*x)] - 9*b*Sinh[4*(c + d*x)] + b*Sinh[6*(c + d*x)]))/(192*d)","A",1
202,1,144,92,0.0514909,"\int \text{csch}^3(c+d x) \left(a+b \sinh ^4(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^2,x]","-\frac{a^2 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a^2 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a^2 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{2 a b \sinh (c) \sinh (d x)}{d}+\frac{2 a b \cosh (c) \cosh (d x)}{d}+\frac{5 b^2 \cosh (c+d x)}{8 d}-\frac{5 b^2 \cosh (3 (c+d x))}{48 d}+\frac{b^2 \cosh (5 (c+d x))}{80 d}","\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a^2 \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{b (2 a+b) \cosh (c+d x)}{d}+\frac{b^2 \cosh ^5(c+d x)}{5 d}-\frac{2 b^2 \cosh ^3(c+d x)}{3 d}",1,"(2*a*b*Cosh[c]*Cosh[d*x])/d + (5*b^2*Cosh[c + d*x])/(8*d) - (5*b^2*Cosh[3*(c + d*x)])/(48*d) + (b^2*Cosh[5*(c + d*x)])/(80*d) - (a^2*Csch[(c + d*x)/2]^2)/(8*d) - (a^2*Log[Tanh[(c + d*x)/2]])/(2*d) - (a^2*Sech[(c + d*x)/2]^2)/(8*d) + (2*a*b*Sinh[c]*Sinh[d*x])/d","A",1
203,1,68,91,0.327076,"\int \text{csch}^4(c+d x) \left(a+b \sinh ^4(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^4)^2,x]","\frac{3 b (64 a d x-8 b \sinh (2 (c+d x))+b \sinh (4 (c+d x))+12 b c+12 b d x)-32 a^2 \coth (c+d x) \left(\text{csch}^2(c+d x)-2\right)}{96 d}","-\frac{a^2 \coth ^3(c+d x)}{3 d}+\frac{a^2 \coth (c+d x)}{d}+\frac{1}{8} b x (16 a+3 b)+\frac{b^2 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}-\frac{5 b^2 \sinh (c+d x) \cosh (c+d x)}{8 d}",1,"(-32*a^2*Coth[c + d*x]*(-2 + Csch[c + d*x]^2) + 3*b*(12*b*c + 64*a*d*x + 12*b*d*x - 8*b*Sinh[2*(c + d*x)] + b*Sinh[4*(c + d*x)]))/(96*d)","A",1
204,1,186,101,0.0434419,"\int \text{csch}^5(c+d x) \left(a+b \sinh ^4(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^4)^2,x]","-\frac{a^2 \text{csch}^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{3 a^2 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{a^2 \text{sech}^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{3 a^2 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{3 a^2 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}+\frac{2 a b \log \left(\sinh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{2 a b \log \left(\cosh \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{3 b^2 \cosh (c+d x)}{4 d}+\frac{b^2 \cosh (3 (c+d x))}{12 d}","-\frac{a^2 \coth (c+d x) \text{csch}^3(c+d x)}{4 d}+\frac{3 a^2 \coth (c+d x) \text{csch}(c+d x)}{8 d}-\frac{a (3 a+16 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}+\frac{b^2 \cosh ^3(c+d x)}{3 d}-\frac{b^2 \cosh (c+d x)}{d}",1,"(-3*b^2*Cosh[c + d*x])/(4*d) + (b^2*Cosh[3*(c + d*x)])/(12*d) + (3*a^2*Csch[(c + d*x)/2]^2)/(32*d) - (a^2*Csch[(c + d*x)/2]^4)/(64*d) - (2*a*b*Log[Cosh[c/2 + (d*x)/2]])/d + (2*a*b*Log[Sinh[c/2 + (d*x)/2]])/d + (3*a^2*Log[Tanh[(c + d*x)/2]])/(8*d) + (3*a^2*Sech[(c + d*x)/2]^2)/(32*d) + (a^2*Sech[(c + d*x)/2]^4)/(64*d)","A",1
205,1,67,84,0.8308165,"\int \text{csch}^6(c+d x) \left(a+b \sinh ^4(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^4)^2,x]","\frac{15 b^2 (\sinh (2 (c+d x))-2 (c+d x))-4 a \coth (c+d x) \left(3 a \text{csch}^4(c+d x)-4 a \text{csch}^2(c+d x)+8 a+30 b\right)}{60 d}","-\frac{a^2 \coth ^5(c+d x)}{5 d}+\frac{2 a^2 \coth ^3(c+d x)}{3 d}-\frac{a (a+2 b) \coth (c+d x)}{d}+\frac{b^2 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b^2 x}{2}",1,"(-4*a*Coth[c + d*x]*(8*a + 30*b - 4*a*Csch[c + d*x]^2 + 3*a*Csch[c + d*x]^4) + 15*b^2*(-2*(c + d*x) + Sinh[2*(c + d*x)]))/(60*d)","A",1
206,1,240,111,0.0424023,"\int \text{csch}^7(c+d x) \left(a+b \sinh ^4(c+d x)\right)^2 \, dx","Integrate[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^4)^2,x]","-\frac{a^2 \text{csch}^6\left(\frac{1}{2} (c+d x)\right)}{384 d}+\frac{a^2 \text{csch}^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 a^2 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{a^2 \text{sech}^6\left(\frac{1}{2} (c+d x)\right)}{384 d}-\frac{a^2 \text{sech}^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 a^2 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 a^2 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}-\frac{a b \text{csch}^2\left(\frac{1}{2} (c+d x)\right)}{4 d}-\frac{a b \text{sech}^2\left(\frac{1}{2} (c+d x)\right)}{4 d}-\frac{a b \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{b^2 \sinh (c) \sinh (d x)}{d}+\frac{b^2 \cosh (c) \cosh (d x)}{d}","-\frac{a^2 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{5 a^2 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}+\frac{a (5 a+16 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac{a (5 a+16 b) \coth (c+d x) \text{csch}(c+d x)}{16 d}+\frac{b^2 \cosh (c+d x)}{d}",1,"(b^2*Cosh[c]*Cosh[d*x])/d - (5*a^2*Csch[(c + d*x)/2]^2)/(64*d) - (a*b*Csch[(c + d*x)/2]^2)/(4*d) + (a^2*Csch[(c + d*x)/2]^4)/(64*d) - (a^2*Csch[(c + d*x)/2]^6)/(384*d) - (5*a^2*Log[Tanh[(c + d*x)/2]])/(16*d) - (a*b*Log[Tanh[(c + d*x)/2]])/d - (5*a^2*Sech[(c + d*x)/2]^2)/(64*d) - (a*b*Sech[(c + d*x)/2]^2)/(4*d) - (a^2*Sech[(c + d*x)/2]^4)/(64*d) - (a^2*Sech[(c + d*x)/2]^6)/(384*d) + (b^2*Sinh[c]*Sinh[d*x])/d","B",1
207,1,288,220,2.2783659,"\int \sinh ^5(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]^5*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{627314688 a^3 \cosh (5 (c+d x))+4234374144 a^2 b \cosh (5 (c+d x))-756138240 a^2 b \cosh (7 (c+d x))+65345280 a^2 b \cosh (9 (c+d x))+1531530 \left(20480 a^3+48384 a^2 b+41184 a b^2+12155 b^3\right) \cosh (c+d x)-2042040 \left(2560 a^3+8064 a^2 b+7722 a b^2+2431 b^3\right) \cosh (3 (c+d x))+5256210960 a b^2 \cosh (5 (c+d x))-1501774560 a b^2 \cosh (7 (c+d x))+318558240 a b^2 \cosh (9 (c+d x))-43439760 a b^2 \cosh (11 (c+d x))+2827440 a b^2 \cosh (13 (c+d x))+1895421528 b^3 \cosh (5 (c+d x))-676936260 b^3 \cosh (7 (c+d x))+202502300 b^3 \cosh (9 (c+d x))-47338200 b^3 \cosh (11 (c+d x))+8011080 b^3 \cosh (13 (c+d x))-867867 b^3 \cosh (15 (c+d x))+45045 b^3 \cosh (17 (c+d x))}{50185175040 d}","\frac{b \left(3 a^2+45 a b+70 b^2\right) \cosh ^9(c+d x)}{9 d}-\frac{4 b \left(3 a^2+15 a b+14 b^2\right) \cosh ^7(c+d x)}{7 d}+\frac{(a+b) \left(a^2+17 a b+28 b^2\right) \cosh ^5(c+d x)}{5 d}+\frac{b^2 (3 a+28 b) \cosh ^{13}(c+d x)}{13 d}-\frac{2 b^2 (9 a+28 b) \cosh ^{11}(c+d x)}{11 d}-\frac{2 (a+b)^2 (a+4 b) \cosh ^3(c+d x)}{3 d}+\frac{(a+b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^{17}(c+d x)}{17 d}-\frac{8 b^3 \cosh ^{15}(c+d x)}{15 d}",1,"(1531530*(20480*a^3 + 48384*a^2*b + 41184*a*b^2 + 12155*b^3)*Cosh[c + d*x] - 2042040*(2560*a^3 + 8064*a^2*b + 7722*a*b^2 + 2431*b^3)*Cosh[3*(c + d*x)] + 627314688*a^3*Cosh[5*(c + d*x)] + 4234374144*a^2*b*Cosh[5*(c + d*x)] + 5256210960*a*b^2*Cosh[5*(c + d*x)] + 1895421528*b^3*Cosh[5*(c + d*x)] - 756138240*a^2*b*Cosh[7*(c + d*x)] - 1501774560*a*b^2*Cosh[7*(c + d*x)] - 676936260*b^3*Cosh[7*(c + d*x)] + 65345280*a^2*b*Cosh[9*(c + d*x)] + 318558240*a*b^2*Cosh[9*(c + d*x)] + 202502300*b^3*Cosh[9*(c + d*x)] - 43439760*a*b^2*Cosh[11*(c + d*x)] - 47338200*b^3*Cosh[11*(c + d*x)] + 2827440*a*b^2*Cosh[13*(c + d*x)] + 8011080*b^3*Cosh[13*(c + d*x)] - 867867*b^3*Cosh[15*(c + d*x)] + 45045*b^3*Cosh[17*(c + d*x)])/(50185175040*d)","A",1
208,1,185,183,2.457652,"\int \sinh ^3(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{b \left(-27027 \left(1792 a^2+2640 a b+1001 b^2\right) \cosh (5 (c+d x))+19305 \left(256 a^2+880 a b+455 b^2\right) \cosh (7 (c+d x))-7 b (715 (528 a+455 b) \cosh (9 (c+d x))-1755 (16 a+35 b) \cosh (11 (c+d x))+7425 b \cosh (13 (c+d x))-429 b \cosh (15 (c+d x)))\right)-135135 \left(4096 a^3+8960 a^2 b+7392 a b^2+2145 b^3\right) \cosh (c+d x)+15015 \left(4096 a^3+16128 a^2 b+15840 a b^2+5005 b^3\right) \cosh (3 (c+d x))}{738017280 d}","\frac{b \left(3 a^2+30 a b+35 b^2\right) \cosh ^7(c+d x)}{7 d}+\frac{3 b^2 (a+7 b) \cosh ^{11}(c+d x)}{11 d}-\frac{5 b^2 (3 a+7 b) \cosh ^9(c+d x)}{9 d}-\frac{3 b (a+b) (3 a+7 b) \cosh ^5(c+d x)}{5 d}+\frac{(a+b)^2 (a+7 b) \cosh ^3(c+d x)}{3 d}-\frac{(a+b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^{15}(c+d x)}{15 d}-\frac{7 b^3 \cosh ^{13}(c+d x)}{13 d}",1,"(-135135*(4096*a^3 + 8960*a^2*b + 7392*a*b^2 + 2145*b^3)*Cosh[c + d*x] + 15015*(4096*a^3 + 16128*a^2*b + 15840*a*b^2 + 5005*b^3)*Cosh[3*(c + d*x)] + b*(-27027*(1792*a^2 + 2640*a*b + 1001*b^2)*Cosh[5*(c + d*x)] + 19305*(256*a^2 + 880*a*b + 455*b^2)*Cosh[7*(c + d*x)] - 7*b*(715*(528*a + 455*b)*Cosh[9*(c + d*x)] - 1755*(16*a + 35*b)*Cosh[11*(c + d*x)] + 7425*b*Cosh[13*(c + d*x)] - 429*b*Cosh[15*(c + d*x)])))/(738017280*d)","A",1
209,1,157,143,0.9049179,"\int \sinh (c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{-15015 b \left(1280 a^2+1344 a b+429 b^2\right) \cosh (3 (c+d x))+3003 b \left(768 a^2+1728 a b+715 b^2\right) \cosh (5 (c+d x))+60060 \left(1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right) \cosh (c+d x)-4290 b^2 (216 a+143 b) \cosh (7 (c+d x))+10010 b^2 (8 a+13 b) \cosh (9 (c+d x))-17745 b^3 \cosh (11 (c+d x))+1155 b^3 \cosh (13 (c+d x))}{61501440 d}","\frac{b^2 (a+5 b) \cosh ^9(c+d x)}{3 d}-\frac{4 b^2 (3 a+5 b) \cosh ^7(c+d x)}{7 d}+\frac{3 b (a+b) (a+5 b) \cosh ^5(c+d x)}{5 d}-\frac{2 b (a+b)^2 \cosh ^3(c+d x)}{d}+\frac{(a+b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^{13}(c+d x)}{13 d}-\frac{6 b^3 \cosh ^{11}(c+d x)}{11 d}",1,"(60060*(1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*Cosh[c + d*x] - 15015*b*(1280*a^2 + 1344*a*b + 429*b^2)*Cosh[3*(c + d*x)] + 3003*b*(768*a^2 + 1728*a*b + 715*b^2)*Cosh[5*(c + d*x)] - 4290*b^2*(216*a + 143*b)*Cosh[7*(c + d*x)] + 10010*b^2*(8*a + 13*b)*Cosh[9*(c + d*x)] - 17745*b^3*Cosh[11*(c + d*x)] + 1155*b^3*Cosh[13*(c + d*x)])/(61501440*d)","A",1
210,1,139,158,0.3700972,"\int \text{csch}(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{3548160 a^3 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-20790 b \left(384 a^2+280 a b+77 b^2\right) \cosh (c+d x)-2079 b^2 (112 a+55 b) \cosh (5 (c+d x))+495 b^2 (48 a+55 b) \cosh (7 (c+d x))+6930 b (8 a+5 b) (16 a+11 b) \cosh (3 (c+d x))-4235 b^3 \cosh (9 (c+d x))+315 b^3 \cosh (11 (c+d x))}{3548160 d}","-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \left(3 a^2+9 a b+5 b^2\right) \cosh ^3(c+d x)}{3 d}-\frac{b \left(3 a^2+3 a b+b^2\right) \cosh (c+d x)}{d}+\frac{b^2 (3 a+10 b) \cosh ^7(c+d x)}{7 d}-\frac{b^2 (9 a+10 b) \cosh ^5(c+d x)}{5 d}+\frac{b^3 \cosh ^{11}(c+d x)}{11 d}-\frac{5 b^3 \cosh ^9(c+d x)}{9 d}",1,"(-20790*b*(384*a^2 + 280*a*b + 77*b^2)*Cosh[c + d*x] + 6930*b*(8*a + 5*b)*(16*a + 11*b)*Cosh[3*(c + d*x)] - 2079*b^2*(112*a + 55*b)*Cosh[5*(c + d*x)] + 495*b^2*(48*a + 55*b)*Cosh[7*(c + d*x)] - 4235*b^3*Cosh[9*(c + d*x)] + 315*b^3*Cosh[11*(c + d*x)] + 3548160*a^3*Log[Tanh[(c + d*x)/2]])/(3548160*d)","A",1
211,1,155,148,0.3493688,"\int \text{csch}^3(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{-10080 a^3 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)-10080 a^3 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)-40320 a^3 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+1890 b \left(128 a^2+80 a b+21 b^2\right) \cosh (c+d x)+3024 a b^2 \cosh (5 (c+d x))-1260 b^2 (20 a+7 b) \cosh (3 (c+d x))+2268 b^3 \cosh (5 (c+d x))-405 b^3 \cosh (7 (c+d x))+35 b^3 \cosh (9 (c+d x))}{80640 d}","\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a^3 \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{b \left(3 a^2+3 a b+b^2\right) \cosh (c+d x)}{d}+\frac{3 b^2 (a+2 b) \cosh ^5(c+d x)}{5 d}-\frac{2 b^2 (3 a+2 b) \cosh ^3(c+d x)}{3 d}+\frac{b^3 \cosh ^9(c+d x)}{9 d}-\frac{4 b^3 \cosh ^7(c+d x)}{7 d}",1,"(1890*b*(128*a^2 + 80*a*b + 21*b^2)*Cosh[c + d*x] - 1260*b^2*(20*a + 7*b)*Cosh[3*(c + d*x)] + 3024*a*b^2*Cosh[5*(c + d*x)] + 2268*b^3*Cosh[5*(c + d*x)] - 405*b^3*Cosh[7*(c + d*x)] + 35*b^3*Cosh[9*(c + d*x)] - 10080*a^3*Csch[(c + d*x)/2]^2 - 40320*a^3*Log[Tanh[(c + d*x)/2]] - 10080*a^3*Sech[(c + d*x)/2]^2)/(80640*d)","A",1
212,1,173,142,0.3892328,"\int \text{csch}^5(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{-35 a^3 \text{csch}^4\left(\frac{1}{2} (c+d x)\right)+210 a^3 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)+35 a^3 \text{sech}^4\left(\frac{1}{2} (c+d x)\right)+210 a^3 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)+840 a^3 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+6720 a^2 b \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-35 b^2 (144 a+35 b) \cosh (c+d x)+35 b^2 (16 a+7 b) \cosh (3 (c+d x))-49 b^3 \cosh (5 (c+d x))+5 b^3 \cosh (7 (c+d x))}{2240 d}","-\frac{a^3 \coth (c+d x) \text{csch}^3(c+d x)}{4 d}+\frac{3 a^3 \coth (c+d x) \text{csch}(c+d x)}{8 d}-\frac{3 a^2 (a+8 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}+\frac{b^2 (a+b) \cosh ^3(c+d x)}{d}-\frac{b^2 (3 a+b) \cosh (c+d x)}{d}+\frac{b^3 \cosh ^7(c+d x)}{7 d}-\frac{3 b^3 \cosh ^5(c+d x)}{5 d}",1,"(-35*b^2*(144*a + 35*b)*Cosh[c + d*x] + 35*b^2*(16*a + 7*b)*Cosh[3*(c + d*x)] - 49*b^3*Cosh[5*(c + d*x)] + 5*b^3*Cosh[7*(c + d*x)] + 210*a^3*Csch[(c + d*x)/2]^2 - 35*a^3*Csch[(c + d*x)/2]^4 + 840*a^3*Log[Tanh[(c + d*x)/2]] + 6720*a^2*b*Log[Tanh[(c + d*x)/2]] + 210*a^3*Sech[(c + d*x)/2]^2 + 35*a^3*Sech[(c + d*x)/2]^4)/(2240*d)","A",1
213,1,223,156,0.6967581,"\int \text{csch}^7(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^4)^3,x]","-\frac{5 a^3 \text{csch}^6\left(\frac{1}{2} (c+d x)\right)-30 a^3 \text{csch}^4\left(\frac{1}{2} (c+d x)\right)+150 a^3 \text{csch}^2\left(\frac{1}{2} (c+d x)\right)+5 a^3 \text{sech}^6\left(\frac{1}{2} (c+d x)\right)+30 a^3 \text{sech}^4\left(\frac{1}{2} (c+d x)\right)+150 a^3 \text{sech}^2\left(\frac{1}{2} (c+d x)\right)+600 a^3 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+720 a^2 b \text{csch}^2\left(\frac{1}{2} (c+d x)\right)+720 a^2 b \text{sech}^2\left(\frac{1}{2} (c+d x)\right)+2880 a^2 b \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-240 b^2 (24 a+5 b) \cosh (c+d x)+200 b^3 \cosh (3 (c+d x))-24 b^3 \cosh (5 (c+d x))}{1920 d}","-\frac{a^3 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{5 a^3 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}+\frac{a^2 (5 a+24 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac{a^2 (5 a+24 b) \coth (c+d x) \text{csch}(c+d x)}{16 d}+\frac{b^2 (3 a+b) \cosh (c+d x)}{d}+\frac{b^3 \cosh ^5(c+d x)}{5 d}-\frac{2 b^3 \cosh ^3(c+d x)}{3 d}",1,"-1/1920*(-240*b^2*(24*a + 5*b)*Cosh[c + d*x] + 200*b^3*Cosh[3*(c + d*x)] - 24*b^3*Cosh[5*(c + d*x)] + 150*a^3*Csch[(c + d*x)/2]^2 + 720*a^2*b*Csch[(c + d*x)/2]^2 - 30*a^3*Csch[(c + d*x)/2]^4 + 5*a^3*Csch[(c + d*x)/2]^6 + 600*a^3*Log[Tanh[(c + d*x)/2]] + 2880*a^2*b*Log[Tanh[(c + d*x)/2]] + 150*a^3*Sech[(c + d*x)/2]^2 + 720*a^2*b*Sech[(c + d*x)/2]^2 + 30*a^3*Sech[(c + d*x)/2]^4 + 5*a^3*Sech[(c + d*x)/2]^6)/d","A",1
214,1,219,171,1.6791907,"\int \text{csch}^9(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^9*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{a \left(48 \left(35 a^2+144 a b+384 b^2\right) \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-3 a^2 \text{csch}^8\left(\frac{1}{2} (c+d x)\right)+20 a^2 \text{csch}^6\left(\frac{1}{2} (c+d x)\right)+3 a^2 \text{sech}^8\left(\frac{1}{2} (c+d x)\right)+20 a^2 \text{sech}^6\left(\frac{1}{2} (c+d x)\right)-18 a (5 a+16 b) \text{csch}^4\left(\frac{1}{2} (c+d x)\right)+12 a (35 a+144 b) \text{csch}^2\left(\frac{1}{2} (c+d x)\right)+18 a (5 a+16 b) \text{sech}^4\left(\frac{1}{2} (c+d x)\right)+12 a (35 a+144 b) \text{sech}^2\left(\frac{1}{2} (c+d x)\right)\right)-4608 b^3 \cosh (c+d x)+512 b^3 \cosh (3 (c+d x))}{6144 d}","-\frac{a^3 \coth (c+d x) \text{csch}^7(c+d x)}{8 d}+\frac{7 a^3 \coth (c+d x) \text{csch}^5(c+d x)}{48 d}-\frac{a \left(35 a^2+144 a b+384 b^2\right) \tanh ^{-1}(\cosh (c+d x))}{128 d}-\frac{a^2 (35 a+144 b) \coth (c+d x) \text{csch}^3(c+d x)}{192 d}+\frac{a^2 (35 a+144 b) \coth (c+d x) \text{csch}(c+d x)}{128 d}+\frac{b^3 \cosh ^3(c+d x)}{3 d}-\frac{b^3 \cosh (c+d x)}{d}",1,"(-4608*b^3*Cosh[c + d*x] + 512*b^3*Cosh[3*(c + d*x)] + a*(12*a*(35*a + 144*b)*Csch[(c + d*x)/2]^2 - 18*a*(5*a + 16*b)*Csch[(c + d*x)/2]^4 + 20*a^2*Csch[(c + d*x)/2]^6 - 3*a^2*Csch[(c + d*x)/2]^8 + 48*(35*a^2 + 144*a*b + 384*b^2)*Log[Tanh[(c + d*x)/2]] + 12*a*(35*a + 144*b)*Sech[(c + d*x)/2]^2 + 18*a*(5*a + 16*b)*Sech[(c + d*x)/2]^4 + 20*a^2*Sech[(c + d*x)/2]^6 + 3*a^2*Sech[(c + d*x)/2]^8))/(6144*d)","A",1
215,1,265,189,2.4613778,"\int \text{csch}^{11}(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^11*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{b^3 \cosh (c+d x)}{d}-\frac{a \left(60 \left(21 a^2+80 a b+128 b^2\right) \text{csch}^2\left(\frac{1}{2} (c+d x)\right)+60 \left(21 a^2+80 a b+128 b^2\right) \text{sech}^2\left(\frac{1}{2} (c+d x)\right)+240 \left(21 a^2+80 a b+128 b^2\right) \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+2 a^2 \text{csch}^{10}\left(\frac{1}{2} (c+d x)\right)-15 a^2 \text{csch}^8\left(\frac{1}{2} (c+d x)\right)+2 a^2 \text{sech}^{10}\left(\frac{1}{2} (c+d x)\right)+15 a^2 \text{sech}^8\left(\frac{1}{2} (c+d x)\right)+10 a (7 a+16 b) \text{csch}^6\left(\frac{1}{2} (c+d x)\right)-40 a (7 a+24 b) \text{csch}^4\left(\frac{1}{2} (c+d x)\right)+10 a (7 a+16 b) \text{sech}^6\left(\frac{1}{2} (c+d x)\right)+40 a (7 a+24 b) \text{sech}^4\left(\frac{1}{2} (c+d x)\right)\right)}{20480 d}","-\frac{a^3 \coth (c+d x) \text{csch}^9(c+d x)}{10 d}+\frac{9 a^3 \coth (c+d x) \text{csch}^7(c+d x)}{80 d}+\frac{3 a \left(21 a^2+80 a b+128 b^2\right) \tanh ^{-1}(\cosh (c+d x))}{256 d}-\frac{3 a \left(21 a^2+80 a b+128 b^2\right) \coth (c+d x) \text{csch}(c+d x)}{256 d}-\frac{a^2 (21 a+80 b) \coth (c+d x) \text{csch}^5(c+d x)}{160 d}+\frac{a^2 (21 a+80 b) \coth (c+d x) \text{csch}^3(c+d x)}{128 d}+\frac{b^3 \cosh (c+d x)}{d}",1,"(b^3*Cosh[c + d*x])/d - (a*(60*(21*a^2 + 80*a*b + 128*b^2)*Csch[(c + d*x)/2]^2 - 40*a*(7*a + 24*b)*Csch[(c + d*x)/2]^4 + 10*a*(7*a + 16*b)*Csch[(c + d*x)/2]^6 - 15*a^2*Csch[(c + d*x)/2]^8 + 2*a^2*Csch[(c + d*x)/2]^10 + 240*(21*a^2 + 80*a*b + 128*b^2)*Log[Tanh[(c + d*x)/2]] + 60*(21*a^2 + 80*a*b + 128*b^2)*Sech[(c + d*x)/2]^2 + 40*a*(7*a + 24*b)*Sech[(c + d*x)/2]^4 + 10*a*(7*a + 16*b)*Sech[(c + d*x)/2]^6 + 15*a^2*Sech[(c + d*x)/2]^8 + 2*a^2*Sech[(c + d*x)/2]^10))/(20480*d)","A",1
216,1,246,220,2.0391368,"\int \text{csch}^{13}(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^13*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{2 a \left(750629 a^2+2074200 a b+1422720 b^2\right) \cosh (3 (c+d x)) \text{csch}^{12}(c+d x)-9 a \left(77099 a^2+280360 a b+246400 b^2\right) \cosh (5 (c+d x)) \text{csch}^{12}(c+d x)+63 a \left(3421 a^2+12440 a b+14720 b^2\right) \cosh (7 (c+d x)) \text{csch}^{12}(c+d x)-525 a \left(77 a^2+280 a b+384 b^2\right) \cosh (9 (c+d x)) \text{csch}^{12}(c+d x)+45 a \left(77 a^2+280 a b+384 b^2\right) \cosh (11 (c+d x)) \text{csch}^{12}(c+d x)-30 a \left(76555 a^2+75816 a b+45696 b^2\right) \coth (c+d x) \text{csch}^{11}(c+d x)+15360 \left(231 a^3+840 a^2 b+1152 a b^2+1024 b^3\right) \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{15728640 d}","-\frac{a^3 \coth (c+d x) \text{csch}^{11}(c+d x)}{12 d}+\frac{11 a^3 \coth (c+d x) \text{csch}^9(c+d x)}{120 d}-\frac{a \left(77 a^2+280 a b+384 b^2\right) \coth (c+d x) \text{csch}^3(c+d x)}{512 d}+\frac{3 a \left(77 a^2+280 a b+384 b^2\right) \coth (c+d x) \text{csch}(c+d x)}{1024 d}-\frac{3 a^2 (11 a+40 b) \coth (c+d x) \text{csch}^7(c+d x)}{320 d}+\frac{7 a^2 (11 a+40 b) \coth (c+d x) \text{csch}^5(c+d x)}{640 d}-\frac{\left(231 a^3+840 a^2 b+1152 a b^2+1024 b^3\right) \tanh ^{-1}(\cosh (c+d x))}{1024 d}",1,"(-30*a*(76555*a^2 + 75816*a*b + 45696*b^2)*Coth[c + d*x]*Csch[c + d*x]^11 + 2*a*(750629*a^2 + 2074200*a*b + 1422720*b^2)*Cosh[3*(c + d*x)]*Csch[c + d*x]^12 - 9*a*(77099*a^2 + 280360*a*b + 246400*b^2)*Cosh[5*(c + d*x)]*Csch[c + d*x]^12 + 63*a*(3421*a^2 + 12440*a*b + 14720*b^2)*Cosh[7*(c + d*x)]*Csch[c + d*x]^12 - 525*a*(77*a^2 + 280*a*b + 384*b^2)*Cosh[9*(c + d*x)]*Csch[c + d*x]^12 + 45*a*(77*a^2 + 280*a*b + 384*b^2)*Cosh[11*(c + d*x)]*Csch[c + d*x]^12 + 15360*(231*a^3 + 840*a^2*b + 1152*a*b^2 + 1024*b^3)*Log[Tanh[(c + d*x)/2]])/(15728640*d)","A",1
217,1,189,255,0.6589409,"\int \sinh ^2(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{-105 b \left(2304 a^2+2880 a b+1001 b^2\right) \sinh (4 (c+d x))+35 b \left(768 a^2+2160 a b+1001 b^2\right) \sinh (6 (c+d x))-840 \left(1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right) (c+d x)+105 \left(4096 a^3+11520 a^2 b+10080 a b^2+3003 b^3\right) \sinh (2 (c+d x))-105 b^2 (120 a+91 b) \sinh (8 (c+d x))+21 b^2 (48 a+91 b) \sinh (10 (c+d x))-245 b^3 \sinh (12 (c+d x))+15 b^3 \sinh (14 (c+d x))}{1720320 d}","\frac{b \left(1920 a^2+12312 a b+10579 b^2\right) \sinh (c+d x) \cosh ^5(c+d x)}{3840 d}-\frac{b \left(4992 a^2+10728 a b+5549 b^2\right) \sinh (c+d x) \cosh ^3(c+d x)}{3072 d}+\frac{\left(1024 a^3+4224 a^2 b+4632 a b^2+1619 b^3\right) \sinh (c+d x) \cosh (c+d x)}{2048 d}-\frac{x \left(1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right)}{2048}+\frac{b^2 (504 a+2593 b) \sinh (c+d x) \cosh ^9(c+d x)}{1680 d}-\frac{b^2 (6888 a+11821 b) \sinh (c+d x) \cosh ^7(c+d x)}{4480 d}+\frac{b^3 \sinh (c+d x) \cosh ^{13}(c+d x)}{14 d}-\frac{85 b^3 \sinh (c+d x) \cosh ^{11}(c+d x)}{168 d}",1,"(-840*(1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*(c + d*x) + 105*(4096*a^3 + 11520*a^2*b + 10080*a*b^2 + 3003*b^3)*Sinh[2*(c + d*x)] - 105*b*(2304*a^2 + 2880*a*b + 1001*b^2)*Sinh[4*(c + d*x)] + 35*b*(768*a^2 + 2160*a*b + 1001*b^2)*Sinh[6*(c + d*x)] - 105*b^2*(120*a + 91*b)*Sinh[8*(c + d*x)] + 21*b^2*(48*a + 91*b)*Sinh[10*(c + d*x)] - 245*b^3*Sinh[12*(c + d*x)] + 15*b^3*Sinh[14*(c + d*x)])/(1720320*d)","A",1
218,1,156,211,0.3896573,"\int \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[(a + b*Sinh[c + d*x]^4)^3,x]","\frac{-720 b \left(128 a^2+112 a b+33 b^2\right) \sinh (2 (c+d x))+45 b \left(256 a^2+448 a b+165 b^2\right) \sinh (4 (c+d x))+120 \left(1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right) (c+d x)-40 b^2 (96 a+55 b) \sinh (6 (c+d x))+45 b^2 (8 a+11 b) \sinh (8 (c+d x))-72 b^3 \sinh (10 (c+d x))+5 b^3 \sinh (12 (c+d x))}{122880 d}","\frac{b \left(1152 a^2+3912 a b+2279 b^2\right) \sinh (c+d x) \cosh ^3(c+d x)}{1536 d}-\frac{b \left(1920 a^2+2232 a b+793 b^2\right) \sinh (c+d x) \cosh (c+d x)}{1024 d}+\frac{x \left(1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right)}{1024}+\frac{3 b^2 (40 a+139 b) \sinh (c+d x) \cosh ^7(c+d x)}{320 d}-\frac{b^2 (3000 a+3481 b) \sinh (c+d x) \cosh ^5(c+d x)}{1920 d}+\frac{b^3 \sinh (c+d x) \cosh ^{11}(c+d x)}{12 d}-\frac{61 b^3 \sinh (c+d x) \cosh ^9(c+d x)}{120 d}",1,"(120*(1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*(c + d*x) - 720*b*(128*a^2 + 112*a*b + 33*b^2)*Sinh[2*(c + d*x)] + 45*b*(256*a^2 + 448*a*b + 165*b^2)*Sinh[4*(c + d*x)] - 40*b^2*(96*a + 55*b)*Sinh[6*(c + d*x)] + 45*b^2*(8*a + 11*b)*Sinh[8*(c + d*x)] - 72*b^3*Sinh[10*(c + d*x)] + 5*b^3*Sinh[12*(c + d*x)])/(122880*d)","A",1
219,1,134,181,0.7837008,"\int \text{csch}^2(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{-10240 a^3 \coth (c+d x)-120 b \left(128 a^2+80 a b+21 b^2\right) (c+d x)+60 b \left(128 a^2+120 a b+35 b^2\right) \sinh (2 (c+d x))-120 b^2 (12 a+5 b) \sinh (4 (c+d x))+10 b^2 (16 a+15 b) \sinh (6 (c+d x))-25 b^3 \sinh (8 (c+d x))+2 b^3 \sinh (10 (c+d x))}{10240 d}","-\frac{a^3 \coth (c+d x)}{d}+\frac{b \left(384 a^2+528 a b+193 b^2\right) \sinh (c+d x) \cosh (c+d x)}{256 d}-\frac{3}{256} b x \left(128 a^2+80 a b+21 b^2\right)+\frac{b^2 (80 a+171 b) \sinh (c+d x) \cosh ^5(c+d x)}{160 d}-\frac{b^2 (208 a+149 b) \sinh (c+d x) \cosh ^3(c+d x)}{128 d}+\frac{b^3 \sinh (c+d x) \cosh ^9(c+d x)}{10 d}-\frac{41 b^3 \sinh (c+d x) \cosh ^7(c+d x)}{80 d}",1,"(-120*b*(128*a^2 + 80*a*b + 21*b^2)*(c + d*x) - 10240*a^3*Coth[c + d*x] + 60*b*(128*a^2 + 120*a*b + 35*b^2)*Sinh[2*(c + d*x)] - 120*b^2*(12*a + 5*b)*Sinh[4*(c + d*x)] + 10*b^2*(16*a + 15*b)*Sinh[6*(c + d*x)] - 25*b^3*Sinh[8*(c + d*x)] + 2*b^3*Sinh[10*(c + d*x)])/(10240*d)","A",1
220,1,131,161,0.7791996,"\int \text{csch}^4(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{b \left(9216 a^2 c+9216 a^2 d x-96 b (24 a+7 b) \sinh (2 (c+d x))+24 b (12 a+7 b) \sinh (4 (c+d x))+3456 a b c+3456 a b d x-32 b^2 \sinh (6 (c+d x))+3 b^2 \sinh (8 (c+d x))+840 b^2 c+840 b^2 d x\right)-1024 a^3 \coth (c+d x) \left(\text{csch}^2(c+d x)-2\right)}{3072 d}","-\frac{a^3 \coth ^3(c+d x)}{3 d}+\frac{a^3 \coth (c+d x)}{d}+\frac{1}{128} b x \left(384 a^2+144 a b+35 b^2\right)+\frac{b^2 (144 a+163 b) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}-\frac{3 b^2 (80 a+31 b) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{b^3 \sinh (c+d x) \cosh ^7(c+d x)}{8 d}-\frac{25 b^3 \sinh (c+d x) \cosh ^5(c+d x)}{48 d}",1,"(-1024*a^3*Coth[c + d*x]*(-2 + Csch[c + d*x]^2) + b*(9216*a^2*c + 3456*a*b*c + 840*b^2*c + 9216*a^2*d*x + 3456*a*b*d*x + 840*b^2*d*x - 96*b*(24*a + 7*b)*Sinh[2*(c + d*x)] + 24*b*(12*a + 7*b)*Sinh[4*(c + d*x)] - 32*b^2*Sinh[6*(c + d*x)] + 3*b^2*Sinh[8*(c + d*x)]))/(3072*d)","A",1
221,1,110,148,1.0790825,"\int \text{csch}^6(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{5 b^2 (9 (16 a+5 b) \sinh (2 (c+d x))-288 a c-288 a d x-9 b \sinh (4 (c+d x))+b \sinh (6 (c+d x))-60 b c-60 b d x)-64 a^2 \coth (c+d x) \left(3 a \text{csch}^4(c+d x)-4 a \text{csch}^2(c+d x)+8 a+45 b\right)}{960 d}","-\frac{a^3 \coth ^5(c+d x)}{5 d}+\frac{2 a^3 \coth ^3(c+d x)}{3 d}-\frac{a^2 (a+3 b) \coth (c+d x)}{d}+\frac{b^2 (24 a+11 b) \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac{1}{16} b^2 x (24 a+5 b)+\frac{b^3 \sinh (c+d x) \cosh ^5(c+d x)}{6 d}-\frac{13 b^3 \sinh (c+d x) \cosh ^3(c+d x)}{24 d}",1,"(-64*a^2*Coth[c + d*x]*(8*a + 45*b - 4*a*Csch[c + d*x]^2 + 3*a*Csch[c + d*x]^4) + 5*b^2*(-288*a*c - 60*b*c - 288*a*d*x - 60*b*d*x + 9*(16*a + 5*b)*Sinh[2*(c + d*x)] - 9*b*Sinh[4*(c + d*x)] + b*Sinh[6*(c + d*x)]))/(960*d)","A",1
222,1,106,133,0.7220989,"\int \text{csch}^8(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^8*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{35 b^2 (12 (8 a+b) (c+d x)-8 b \sinh (2 (c+d x))+b \sinh (4 (c+d x)))-32 a^2 \coth (c+d x) \left((8 a+35 b) \text{csch}^2(c+d x)-2 (8 a+35 b)+5 a \text{csch}^6(c+d x)-6 a \text{csch}^4(c+d x)\right)}{1120 d}","-\frac{a^3 \coth ^7(c+d x)}{7 d}+\frac{3 a^3 \coth ^5(c+d x)}{5 d}-\frac{a^2 (a+b) \coth ^3(c+d x)}{d}+\frac{a^2 (a+3 b) \coth (c+d x)}{d}+\frac{3}{8} b^2 x (8 a+b)+\frac{b^3 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}-\frac{5 b^3 \sinh (c+d x) \cosh (c+d x)}{8 d}",1,"(-32*a^2*Coth[c + d*x]*(-2*(8*a + 35*b) + (8*a + 35*b)*Csch[c + d*x]^2 - 6*a*Csch[c + d*x]^4 + 5*a*Csch[c + d*x]^6) + 35*b^2*(12*(8*a + b)*(c + d*x) - 8*b*Sinh[2*(c + d*x)] + b*Sinh[4*(c + d*x)]))/(1120*d)","A",1
223,1,115,140,0.6127634,"\int \text{csch}^{10}(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^10*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{315 b^3 (\sinh (2 (c+d x))-2 (c+d x))-4 a \coth (c+d x) \left(35 a^2 \text{csch}^8(c+d x)-40 a^2 \text{csch}^6(c+d x)+128 a^2+3 a (16 a+63 b) \text{csch}^4(c+d x)-4 a (16 a+63 b) \text{csch}^2(c+d x)+504 a b+945 b^2\right)}{1260 d}","-\frac{a^3 \coth ^9(c+d x)}{9 d}+\frac{4 a^3 \coth ^7(c+d x)}{7 d}-\frac{a \left(a^2+3 a b+3 b^2\right) \coth (c+d x)}{d}-\frac{3 a^2 (2 a+b) \coth ^5(c+d x)}{5 d}+\frac{2 a^2 (2 a+3 b) \coth ^3(c+d x)}{3 d}+\frac{b^3 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b^3 x}{2}",1,"(-4*a*Coth[c + d*x]*(128*a^2 + 504*a*b + 945*b^2 - 4*a*(16*a + 63*b)*Csch[c + d*x]^2 + 3*a*(16*a + 63*b)*Csch[c + d*x]^4 - 40*a^2*Csch[c + d*x]^6 + 35*a^2*Csch[c + d*x]^8) + 315*b^3*(-2*(c + d*x) + Sinh[2*(c + d*x)]))/(1260*d)","A",1
224,1,239,147,6.1049802,"\int \text{csch}^{12}(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^12*(a + b*Sinh[c + d*x]^4)^3,x]","-\frac{a^3 \coth (c+d x) \text{csch}^{10}(c+d x)}{11 d}+\frac{10 a^3 \coth (c+d x) \text{csch}^8(c+d x)}{99 d}+\frac{\text{csch}^3(c+d x) \left(-640 a^3 \cosh (c+d x)-2376 a^2 b \cosh (c+d x)-3465 a b^2 \cosh (c+d x)\right)}{3465 d}+\frac{2 \text{csch}(c+d x) \left(640 a^3 \cosh (c+d x)+2376 a^2 b \cosh (c+d x)+3465 a b^2 \cosh (c+d x)\right)}{3465 d}+\frac{\text{csch}^7(c+d x) \left(-80 a^3 \cosh (c+d x)-297 a^2 b \cosh (c+d x)\right)}{693 d}+\frac{2 \text{csch}^5(c+d x) \left(80 a^3 \cosh (c+d x)+297 a^2 b \cosh (c+d x)\right)}{1155 d}+\frac{b^3 (c+d x)}{d}","-\frac{a^3 \coth ^{11}(c+d x)}{11 d}+\frac{5 a^3 \coth ^9(c+d x)}{9 d}-\frac{a \left(5 a^2+9 a b+3 b^2\right) \coth ^3(c+d x)}{3 d}+\frac{a \left(a^2+3 a b+3 b^2\right) \coth (c+d x)}{d}-\frac{a^2 (10 a+3 b) \coth ^7(c+d x)}{7 d}+\frac{a^2 (10 a+9 b) \coth ^5(c+d x)}{5 d}+b^3 x",1,"(b^3*(c + d*x))/d + (2*(640*a^3*Cosh[c + d*x] + 2376*a^2*b*Cosh[c + d*x] + 3465*a*b^2*Cosh[c + d*x])*Csch[c + d*x])/(3465*d) + ((-640*a^3*Cosh[c + d*x] - 2376*a^2*b*Cosh[c + d*x] - 3465*a*b^2*Cosh[c + d*x])*Csch[c + d*x]^3)/(3465*d) + (2*(80*a^3*Cosh[c + d*x] + 297*a^2*b*Cosh[c + d*x])*Csch[c + d*x]^5)/(1155*d) + ((-80*a^3*Cosh[c + d*x] - 297*a^2*b*Cosh[c + d*x])*Csch[c + d*x]^7)/(693*d) + (10*a^3*Coth[c + d*x]*Csch[c + d*x]^8)/(99*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^10)/(11*d)","A",1
225,1,350,144,3.2440672,"\int \text{csch}^{14}(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^14*(a + b*Sinh[c + d*x]^4)^3,x]","-\frac{\text{csch}^{13}(c+d x) \left(3660800 a^3 \cosh (5 (c+d x))-1464320 a^3 \cosh (7 (c+d x))+399360 a^3 \cosh (9 (c+d x))-66560 a^3 \cosh (11 (c+d x))+5120 a^3 \cosh (13 (c+d x))+13087360 a^2 b \cosh (5 (c+d x))-5234944 a^2 b \cosh (7 (c+d x))+1427712 a^2 b \cosh (9 (c+d x))-237952 a^2 b \cosh (11 (c+d x))+18304 a^2 b \cosh (13 (c+d x))+8580 \left(1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right) \cosh (c+d x)-6435 \left(1024 a^3+2944 a^2 b+2408 a b^2+693 b^3\right) \cosh (3 (c+d x))+13093080 a b^2 \cosh (5 (c+d x))-6390384 a b^2 \cosh (7 (c+d x))+1873872 a b^2 \cosh (9 (c+d x))-312312 a b^2 \cosh (11 (c+d x))+24024 a b^2 \cosh (13 (c+d x))+4129125 b^3 \cosh (5 (c+d x))-2312310 b^3 \cosh (7 (c+d x))+810810 b^3 \cosh (9 (c+d x))-165165 b^3 \cosh (11 (c+d x))+15015 b^3 \cosh (13 (c+d x))\right)}{61501440 d}","-\frac{a^3 \coth ^{13}(c+d x)}{13 d}+\frac{6 a^3 \coth ^{11}(c+d x)}{11 d}-\frac{a^2 (5 a+b) \coth ^9(c+d x)}{3 d}+\frac{4 a^2 (5 a+3 b) \coth ^7(c+d x)}{7 d}-\frac{3 a (a+b) (5 a+b) \coth ^5(c+d x)}{5 d}+\frac{2 a (a+b)^2 \coth ^3(c+d x)}{d}-\frac{(a+b)^3 \coth (c+d x)}{d}",1,"-1/61501440*((8580*(1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*Cosh[c + d*x] - 6435*(1024*a^3 + 2944*a^2*b + 2408*a*b^2 + 693*b^3)*Cosh[3*(c + d*x)] + 3660800*a^3*Cosh[5*(c + d*x)] + 13087360*a^2*b*Cosh[5*(c + d*x)] + 13093080*a*b^2*Cosh[5*(c + d*x)] + 4129125*b^3*Cosh[5*(c + d*x)] - 1464320*a^3*Cosh[7*(c + d*x)] - 5234944*a^2*b*Cosh[7*(c + d*x)] - 6390384*a*b^2*Cosh[7*(c + d*x)] - 2312310*b^3*Cosh[7*(c + d*x)] + 399360*a^3*Cosh[9*(c + d*x)] + 1427712*a^2*b*Cosh[9*(c + d*x)] + 1873872*a*b^2*Cosh[9*(c + d*x)] + 810810*b^3*Cosh[9*(c + d*x)] - 66560*a^3*Cosh[11*(c + d*x)] - 237952*a^2*b*Cosh[11*(c + d*x)] - 312312*a*b^2*Cosh[11*(c + d*x)] - 165165*b^3*Cosh[11*(c + d*x)] + 5120*a^3*Cosh[13*(c + d*x)] + 18304*a^2*b*Cosh[13*(c + d*x)] + 24024*a*b^2*Cosh[13*(c + d*x)] + 15015*b^3*Cosh[13*(c + d*x)])*Csch[c + d*x]^13)/d","B",1
226,1,404,182,4.6635251,"\int \text{csch}^{16}(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^16*(a + b*Sinh[c + d*x]^4)^3,x]","-\frac{\text{csch}^{15}(c+d x) \left(21525504 a^3 \cosh (5 (c+d x))-9784320 a^3 \cosh (7 (c+d x))+3261440 a^3 \cosh (9 (c+d x))-752640 a^3 \cosh (11 (c+d x))+107520 a^3 \cosh (13 (c+d x))-7168 a^3 \cosh (15 (c+d x))+74954880 a^2 b \cosh (5 (c+d x))-34070400 a^2 b \cosh (7 (c+d x))+11356800 a^2 b \cosh (9 (c+d x))-2620800 a^2 b \cosh (11 (c+d x))+374400 a^2 b \cosh (13 (c+d x))-24960 a^2 b \cosh (15 (c+d x))+45045 \left(1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right) \cosh (c+d x)-5005 \left(7168 a^3+20352 a^2 b+16632 a b^2+4785 b^3\right) \cosh (3 (c+d x))+74162088 a b^2 \cosh (5 (c+d x))-39999960 a b^2 \cosh (7 (c+d x))+14054040 a b^2 \cosh (9 (c+d x))-3243240 a b^2 \cosh (11 (c+d x))+463320 a b^2 \cosh (13 (c+d x))-30888 a b^2 \cosh (15 (c+d x))+23288265 b^3 \cosh (5 (c+d x))-14189175 b^3 \cosh (7 (c+d x))+5720715 b^3 \cosh (9 (c+d x))-1486485 b^3 \cosh (11 (c+d x))+225225 b^3 \cosh (13 (c+d x))-15015 b^3 \cosh (15 (c+d x))\right)}{369008640 d}","-\frac{a^3 \coth ^{15}(c+d x)}{15 d}+\frac{7 a^3 \coth ^{13}(c+d x)}{13 d}-\frac{a \left(35 a^2+30 a b+3 b^2\right) \coth ^7(c+d x)}{7 d}-\frac{3 a^2 (7 a+b) \coth ^{11}(c+d x)}{11 d}+\frac{5 a^2 (7 a+3 b) \coth ^9(c+d x)}{9 d}+\frac{3 a (a+b) (7 a+3 b) \coth ^5(c+d x)}{5 d}-\frac{(a+b)^2 (7 a+b) \coth ^3(c+d x)}{3 d}+\frac{(a+b)^3 \coth (c+d x)}{d}",1,"-1/369008640*((45045*(1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*Cosh[c + d*x] - 5005*(7168*a^3 + 20352*a^2*b + 16632*a*b^2 + 4785*b^3)*Cosh[3*(c + d*x)] + 21525504*a^3*Cosh[5*(c + d*x)] + 74954880*a^2*b*Cosh[5*(c + d*x)] + 74162088*a*b^2*Cosh[5*(c + d*x)] + 23288265*b^3*Cosh[5*(c + d*x)] - 9784320*a^3*Cosh[7*(c + d*x)] - 34070400*a^2*b*Cosh[7*(c + d*x)] - 39999960*a*b^2*Cosh[7*(c + d*x)] - 14189175*b^3*Cosh[7*(c + d*x)] + 3261440*a^3*Cosh[9*(c + d*x)] + 11356800*a^2*b*Cosh[9*(c + d*x)] + 14054040*a*b^2*Cosh[9*(c + d*x)] + 5720715*b^3*Cosh[9*(c + d*x)] - 752640*a^3*Cosh[11*(c + d*x)] - 2620800*a^2*b*Cosh[11*(c + d*x)] - 3243240*a*b^2*Cosh[11*(c + d*x)] - 1486485*b^3*Cosh[11*(c + d*x)] + 107520*a^3*Cosh[13*(c + d*x)] + 374400*a^2*b*Cosh[13*(c + d*x)] + 463320*a*b^2*Cosh[13*(c + d*x)] + 225225*b^3*Cosh[13*(c + d*x)] - 7168*a^3*Cosh[15*(c + d*x)] - 24960*a^2*b*Cosh[15*(c + d*x)] - 30888*a*b^2*Cosh[15*(c + d*x)] - 15015*b^3*Cosh[15*(c + d*x)])*Csch[c + d*x]^15)/d","B",1
227,1,494,221,6.1854279,"\int \text{csch}^{18}(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^18*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{\text{csch}^{17}(c+d x) \left(-697016320 a^3 \cosh (c+d x)+557613056 a^3 \cosh (3 (c+d x))-354844672 a^3 \cosh (5 (c+d x))+177422336 a^3 \cosh (7 (c+d x))-68239360 a^3 \cosh (9 (c+d x))+19496960 a^3 \cosh (11 (c+d x))-3899392 a^3 \cosh (13 (c+d x))+487424 a^3 \cosh (15 (c+d x))-28672 a^3 \cosh (17 (c+d x))-784143360 a^2 b \cosh (c+d x)+1568286720 a^2 b \cosh (3 (c+d x))-1211857920 a^2 b \cosh (5 (c+d x))+605928960 a^2 b \cosh (7 (c+d x))-233049600 a^2 b \cosh (9 (c+d x))+66585600 a^2 b \cosh (11 (c+d x))-13317120 a^2 b \cosh (13 (c+d x))+1664640 a^2 b \cosh (15 (c+d x))-97920 a^2 b \cosh (17 (c+d x))-571771200 a b^2 \cosh (c+d x)+1280767488 a b^2 \cosh (3 (c+d x))-1189284096 a b^2 \cosh (5 (c+d x))+692659968 a b^2 \cosh (7 (c+d x))-277717440 a b^2 \cosh (9 (c+d x))+79347840 a b^2 \cosh (11 (c+d x))-15869568 a b^2 \cosh (13 (c+d x))+1983696 a b^2 \cosh (15 (c+d x))-116688 a b^2 \cosh (17 (c+d x))-157237080 b^3 \cosh (c+d x)+368384016 b^3 \cosh (3 (c+d x))-372263892 b^3 \cosh (5 (c+d x))+242288046 b^3 \cosh (7 (c+d x))-108738630 b^3 \cosh (9 (c+d x))+33693660 b^3 \cosh (11 (c+d x))-6942936 b^3 \cosh (13 (c+d x))+867867 b^3 \cosh (15 (c+d x))-51051 b^3 \cosh (17 (c+d x))\right)}{6273146880 d}","-\frac{a^3 \coth ^{17}(c+d x)}{17 d}+\frac{8 a^3 \coth ^{15}(c+d x)}{15 d}-\frac{a \left(70 a^2+45 a b+3 b^2\right) \coth ^9(c+d x)}{9 d}+\frac{4 a \left(14 a^2+15 a b+3 b^2\right) \coth ^7(c+d x)}{7 d}-\frac{(a+b) \left(28 a^2+17 a b+b^2\right) \coth ^5(c+d x)}{5 d}-\frac{a^2 (28 a+3 b) \coth ^{13}(c+d x)}{13 d}+\frac{2 a^2 (28 a+9 b) \coth ^{11}(c+d x)}{11 d}+\frac{2 (a+b)^2 (4 a+b) \coth ^3(c+d x)}{3 d}-\frac{(a+b)^3 \coth (c+d x)}{d}",1,"((-697016320*a^3*Cosh[c + d*x] - 784143360*a^2*b*Cosh[c + d*x] - 571771200*a*b^2*Cosh[c + d*x] - 157237080*b^3*Cosh[c + d*x] + 557613056*a^3*Cosh[3*(c + d*x)] + 1568286720*a^2*b*Cosh[3*(c + d*x)] + 1280767488*a*b^2*Cosh[3*(c + d*x)] + 368384016*b^3*Cosh[3*(c + d*x)] - 354844672*a^3*Cosh[5*(c + d*x)] - 1211857920*a^2*b*Cosh[5*(c + d*x)] - 1189284096*a*b^2*Cosh[5*(c + d*x)] - 372263892*b^3*Cosh[5*(c + d*x)] + 177422336*a^3*Cosh[7*(c + d*x)] + 605928960*a^2*b*Cosh[7*(c + d*x)] + 692659968*a*b^2*Cosh[7*(c + d*x)] + 242288046*b^3*Cosh[7*(c + d*x)] - 68239360*a^3*Cosh[9*(c + d*x)] - 233049600*a^2*b*Cosh[9*(c + d*x)] - 277717440*a*b^2*Cosh[9*(c + d*x)] - 108738630*b^3*Cosh[9*(c + d*x)] + 19496960*a^3*Cosh[11*(c + d*x)] + 66585600*a^2*b*Cosh[11*(c + d*x)] + 79347840*a*b^2*Cosh[11*(c + d*x)] + 33693660*b^3*Cosh[11*(c + d*x)] - 3899392*a^3*Cosh[13*(c + d*x)] - 13317120*a^2*b*Cosh[13*(c + d*x)] - 15869568*a*b^2*Cosh[13*(c + d*x)] - 6942936*b^3*Cosh[13*(c + d*x)] + 487424*a^3*Cosh[15*(c + d*x)] + 1664640*a^2*b*Cosh[15*(c + d*x)] + 1983696*a*b^2*Cosh[15*(c + d*x)] + 867867*b^3*Cosh[15*(c + d*x)] - 28672*a^3*Cosh[17*(c + d*x)] - 97920*a^2*b*Cosh[17*(c + d*x)] - 116688*a*b^2*Cosh[17*(c + d*x)] - 51051*b^3*Cosh[17*(c + d*x)])*Csch[c + d*x]^17)/(6273146880*d)","B",1
228,1,548,248,6.1817879,"\int \text{csch}^{20}(c+d x) \left(a+b \sinh ^4(c+d x)\right)^3 \, dx","Integrate[Csch[c + d*x]^20*(a + b*Sinh[c + d*x]^4)^3,x]","\frac{\text{csch}^{19}(c+d x) \left(-7945986048 a^3 \cosh (c+d x)+6501261312 a^3 \cosh (3 (c+d x))-4334174208 a^3 \cosh (5 (c+d x))+2333786112 a^3 \cosh (7 (c+d x))-1000194048 a^3 \cosh (9 (c+d x))+333398016 a^3 \cosh (11 (c+d x))-83349504 a^3 \cosh (13 (c+d x))+14708736 a^3 \cosh (15 (c+d x))-1634304 a^3 \cosh (17 (c+d x))+86016 a^3 \cosh (19 (c+d x))-8939234304 a^2 b \cosh (c+d x)+18149354496 a^2 b \cosh (3 (c+d x))-14582690304 a^2 b \cosh (5 (c+d x))+7852217856 a^2 b \cosh (7 (c+d x))-3365236224 a^2 b \cosh (9 (c+d x))+1121745408 a^2 b \cosh (11 (c+d x))-280436352 a^2 b \cosh (13 (c+d x))+49488768 a^2 b \cosh (15 (c+d x))-5498752 a^2 b \cosh (17 (c+d x))+289408 a^2 b \cosh (19 (c+d x))-6518191680 a b^2 \cosh (c+d x)+14814072000 a b^2 \cosh (3 (c+d x))-14221509120 a b^2 \cosh (5 (c+d x))+8803791360 a b^2 \cosh (7 (c+d x))-3906077760 a b^2 \cosh (9 (c+d x))+1302025920 a b^2 \cosh (11 (c+d x))-325506480 a b^2 \cosh (13 (c+d x))+57442320 a b^2 \cosh (15 (c+d x))-6382480 a b^2 \cosh (17 (c+d x))+335920 a b^2 \cosh (19 (c+d x))-1792502712 b^3 \cosh (c+d x)+4260103848 b^3 \cosh (3 (c+d x))-4440518082 b^3 \cosh (5 (c+d x))+3047642598 b^3 \cosh (7 (c+d x))-1489040982 b^3 \cosh (9 (c+d x))+527386002 b^3 \cosh (11 (c+d x))-134271423 b^3 \cosh (13 (c+d x))+23694957 b^3 \cosh (15 (c+d x))-2632773 b^3 \cosh (17 (c+d x))+138567 b^3 \cosh (19 (c+d x))\right)}{79459860480 d}","-\frac{a^3 \coth ^{19}(c+d x)}{19 d}+\frac{9 a^3 \coth ^{17}(c+d x)}{17 d}-\frac{3 a \left(42 a^2+21 a b+b^2\right) \coth ^{11}(c+d x)}{11 d}+\frac{a \left(42 a^2+35 a b+5 b^2\right) \coth ^9(c+d x)}{3 d}+\frac{3 (a+b) \left(12 a^2+9 a b+b^2\right) \coth ^5(c+d x)}{5 d}-\frac{a^2 (12 a+b) \coth ^{15}(c+d x)}{5 d}+\frac{21 a^2 (4 a+b) \coth ^{13}(c+d x)}{13 d}-\frac{\left(84 a^3+105 a^2 b+30 a b^2+b^3\right) \coth ^7(c+d x)}{7 d}-\frac{(a+b)^2 (3 a+b) \coth ^3(c+d x)}{d}+\frac{(a+b)^3 \coth (c+d x)}{d}",1,"((-7945986048*a^3*Cosh[c + d*x] - 8939234304*a^2*b*Cosh[c + d*x] - 6518191680*a*b^2*Cosh[c + d*x] - 1792502712*b^3*Cosh[c + d*x] + 6501261312*a^3*Cosh[3*(c + d*x)] + 18149354496*a^2*b*Cosh[3*(c + d*x)] + 14814072000*a*b^2*Cosh[3*(c + d*x)] + 4260103848*b^3*Cosh[3*(c + d*x)] - 4334174208*a^3*Cosh[5*(c + d*x)] - 14582690304*a^2*b*Cosh[5*(c + d*x)] - 14221509120*a*b^2*Cosh[5*(c + d*x)] - 4440518082*b^3*Cosh[5*(c + d*x)] + 2333786112*a^3*Cosh[7*(c + d*x)] + 7852217856*a^2*b*Cosh[7*(c + d*x)] + 8803791360*a*b^2*Cosh[7*(c + d*x)] + 3047642598*b^3*Cosh[7*(c + d*x)] - 1000194048*a^3*Cosh[9*(c + d*x)] - 3365236224*a^2*b*Cosh[9*(c + d*x)] - 3906077760*a*b^2*Cosh[9*(c + d*x)] - 1489040982*b^3*Cosh[9*(c + d*x)] + 333398016*a^3*Cosh[11*(c + d*x)] + 1121745408*a^2*b*Cosh[11*(c + d*x)] + 1302025920*a*b^2*Cosh[11*(c + d*x)] + 527386002*b^3*Cosh[11*(c + d*x)] - 83349504*a^3*Cosh[13*(c + d*x)] - 280436352*a^2*b*Cosh[13*(c + d*x)] - 325506480*a*b^2*Cosh[13*(c + d*x)] - 134271423*b^3*Cosh[13*(c + d*x)] + 14708736*a^3*Cosh[15*(c + d*x)] + 49488768*a^2*b*Cosh[15*(c + d*x)] + 57442320*a*b^2*Cosh[15*(c + d*x)] + 23694957*b^3*Cosh[15*(c + d*x)] - 1634304*a^3*Cosh[17*(c + d*x)] - 5498752*a^2*b*Cosh[17*(c + d*x)] - 6382480*a*b^2*Cosh[17*(c + d*x)] - 2632773*b^3*Cosh[17*(c + d*x)] + 86016*a^3*Cosh[19*(c + d*x)] + 289408*a^2*b*Cosh[19*(c + d*x)] + 335920*a*b^2*Cosh[19*(c + d*x)] + 138567*b^3*Cosh[19*(c + d*x)])*Csch[c + d*x]^19)/(79459860480*d)","B",1
229,1,390,148,0.347377,"\int \frac{\sinh ^7(c+d x)}{a-b \sinh ^4(c+d x)} \, dx","Integrate[Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4),x]","\frac{-3 a \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^6 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^6 c+\text{$\#$1}^6 d x-6 \text{$\#$1}^4 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-3 \text{$\#$1}^4 c-3 \text{$\#$1}^4 d x+6 \text{$\#$1}^2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+3 \text{$\#$1}^2 c+3 \text{$\#$1}^2 d x-2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-c-d x}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]+18 \cosh (c+d x)-2 \cosh (3 (c+d x))}{24 b d}","-\frac{a \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 b^{7/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 b^{7/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\cosh ^3(c+d x)}{3 b d}+\frac{\cosh (c+d x)}{b d}",1,"(18*Cosh[c + d*x] - 2*Cosh[3*(c + d*x)] - 3*a*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-c - d*x - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 3*c*#1^2 + 3*d*x*#1^2 + 6*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 3*c*#1^4 - 3*d*x*#1^4 - 6*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + c*#1^6 + d*x*#1^6 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(24*b*d)","C",1
230,1,235,139,0.2709526,"\int \frac{\sinh ^5(c+d x)}{a-b \sinh ^4(c+d x)} \, dx","Integrate[Sinh[c + d*x]^5/(a - b*Sinh[c + d*x]^4),x]","-\frac{a \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^3 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^3 c+\text{$\#$1}^3 d x-2 \text{$\#$1} \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-\text{$\#$1} c-\text{$\#$1} d x}{\text{$\#$1}^6 b-3 \text{$\#$1}^4 b-8 \text{$\#$1}^2 a+3 \text{$\#$1}^2 b-b}\&\right]+2 \cosh (c+d x)}{2 b d}","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 b^{5/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 b^{5/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\cosh (c+d x)}{b d}",1,"-1/2*(2*Cosh[c + d*x] + a*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-(c*#1) - d*x*#1 - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1 + c*#1^3 + d*x*#1^3 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^3)/(-b - 8*a*#1^2 + 3*b*#1^2 - 3*b*#1^4 + b*#1^6) & ])/(b*d)","C",1
231,1,365,115,0.1754333,"\int \frac{\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx","Integrate[Sinh[c + d*x]^3/(a - b*Sinh[c + d*x]^4),x]","-\frac{\text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^6 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^6 c+\text{$\#$1}^6 d x-6 \text{$\#$1}^4 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-3 \text{$\#$1}^4 c-3 \text{$\#$1}^4 d x+6 \text{$\#$1}^2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+3 \text{$\#$1}^2 c+3 \text{$\#$1}^2 d x-2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-c-d x}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]}{8 d}","\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 b^{3/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 b^{3/4} d \sqrt{\sqrt{a}-\sqrt{b}}}",1,"-1/8*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-c - d*x - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 3*c*#1^2 + 3*d*x*#1^2 + 6*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 3*c*#1^4 - 3*d*x*#1^4 - 6*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + c*#1^6 + d*x*#1^6 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ]/d","C",1
232,1,221,125,0.1698161,"\int \frac{\sinh (c+d x)}{a-b \sinh ^4(c+d x)} \, dx","Integrate[Sinh[c + d*x]/(a - b*Sinh[c + d*x]^4),x]","-\frac{\text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^3 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^3 c+\text{$\#$1}^3 d x-2 \text{$\#$1} \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-\text{$\#$1} c-\text{$\#$1} d x}{\text{$\#$1}^6 b-3 \text{$\#$1}^4 b-8 \text{$\#$1}^2 a+3 \text{$\#$1}^2 b-b}\&\right]}{2 d}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 \sqrt{a} \sqrt[4]{b} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 \sqrt{a} \sqrt[4]{b} d \sqrt{\sqrt{a}+\sqrt{b}}}",1,"-1/2*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-(c*#1) - d*x*#1 - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1 + c*#1^3 + d*x*#1^3 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^3)/(-b - 8*a*#1^2 + 3*b*#1^2 - 3*b*#1^4 + b*#1^6) & ]/d","C",1
233,1,385,136,0.2540551,"\int \frac{\text{csch}(c+d x)}{a-b \sinh ^4(c+d x)} \, dx","Integrate[Csch[c + d*x]/(a - b*Sinh[c + d*x]^4),x]","\frac{8 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-b \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^6 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^6 c+\text{$\#$1}^6 d x-6 \text{$\#$1}^4 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-3 \text{$\#$1}^4 c-3 \text{$\#$1}^4 d x+6 \text{$\#$1}^2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+3 \text{$\#$1}^2 c+3 \text{$\#$1}^2 d x-2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-c-d x}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]}{8 a d}","-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a d}",1,"(8*Log[Tanh[(c + d*x)/2]] - b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-c - d*x - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 3*c*#1^2 + 3*d*x*#1^2 + 6*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 3*c*#1^4 - 3*d*x*#1^4 - 6*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + c*#1^6 + d*x*#1^6 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(8*a*d)","C",1
234,1,265,184,0.3892326,"\int \frac{\text{csch}^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx","Integrate[Csch[c + d*x]^3/(a - b*Sinh[c + d*x]^4),x]","-\frac{4 b \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^3 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^3 c+\text{$\#$1}^3 d x-2 \text{$\#$1} \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-\text{$\#$1} c-\text{$\#$1} d x}{\text{$\#$1}^6 b-3 \text{$\#$1}^4 b-8 \text{$\#$1}^2 a+3 \text{$\#$1}^2 b-b}\&\right]+\text{csch}^2\left(\frac{1}{2} (c+d x)\right)+\text{sech}^2\left(\frac{1}{2} (c+d x)\right)+4 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{8 a d}","\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a^{3/2} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a^{3/2} d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{1}{4 a d (1-\cosh (c+d x))}-\frac{1}{4 a d (\cosh (c+d x)+1)}+\frac{\tanh ^{-1}(\cosh (c+d x))}{2 a d}",1,"-1/8*(Csch[(c + d*x)/2]^2 + 4*Log[Tanh[(c + d*x)/2]] + 4*b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-(c*#1) - d*x*#1 - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1 + c*#1^3 + d*x*#1^3 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^3)/(-b - 8*a*#1^2 + 3*b*#1^2 - 3*b*#1^4 + b*#1^6) & ] + Sech[(c + d*x)/2]^2)/(a*d)","C",1
235,1,158,175,0.8875242,"\int \frac{\sinh ^6(c+d x)}{a-b \sinh ^4(c+d x)} \, dx","Integrate[Sinh[c + d*x]^6/(a - b*Sinh[c + d*x]^4),x]","\frac{\frac{2 a \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{2 a \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}+2 \sqrt{b} (c+d x)-\sqrt{b} \sinh (2 (c+d x))}{4 b^{3/2} d}","-\frac{a^{3/4} \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{2 b^{3/2} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{a^{3/4} \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{2 b^{3/2} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{1}{4 b d (1-\tanh (c+d x))}+\frac{1}{4 b d (\tanh (c+d x)+1)}+\frac{x}{2 b}",1,"(2*Sqrt[b]*(c + d*x) + (2*a*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + (2*a*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - Sqrt[b]*Sinh[2*(c + d*x)])/(4*b^(3/2)*d)","A",1
236,1,143,127,0.473466,"\int \frac{\sinh ^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx","Integrate[Sinh[c + d*x]^4/(a - b*Sinh[c + d*x]^4),x]","\frac{\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}-2 (c+d x)}{2 b d}","\frac{\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{2 b d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{2 b d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{x}{b}",1,"(-2*(c + d*x) - (Sqrt[a]*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + (Sqrt[a]*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]])/(2*b*d)","A",1
237,1,127,125,0.356242,"\int \frac{\sinh ^2(c+d x)}{a-b \sinh ^4(c+d x)} \, dx","Integrate[Sinh[c + d*x]^2/(a - b*Sinh[c + d*x]^4),x]","\frac{\frac{\tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}}{2 \sqrt{b} d}","\frac{\tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{2 \sqrt[4]{a} \sqrt{b} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{2 \sqrt[4]{a} \sqrt{b} d \sqrt{\sqrt{a}-\sqrt{b}}}",1,"(ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]]/Sqrt[-a + Sqrt[a]*Sqrt[b]] + ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]]/Sqrt[a + Sqrt[a]*Sqrt[b]])/(2*Sqrt[b]*d)","A",1
238,1,128,115,0.2494906,"\int \frac{1}{a-b \sinh ^4(c+d x)} \, dx","Integrate[(a - b*Sinh[c + d*x]^4)^(-1),x]","\frac{\frac{\tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{\tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}}{2 \sqrt{a} d}","\frac{\tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \sqrt{\sqrt{a}+\sqrt{b}}}",1,"(-(ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]]/Sqrt[-a + Sqrt[a]*Sqrt[b]]) + ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]]/Sqrt[a + Sqrt[a]*Sqrt[b]])/(2*Sqrt[a]*d)","A",1
239,1,143,139,0.8097404,"\int \frac{\text{csch}^2(c+d x)}{a-b \sinh ^4(c+d x)} \, dx","Integrate[Csch[c + d*x]^2/(a - b*Sinh[c + d*x]^4),x]","\frac{\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}-2 \coth (c+d x)}{2 a d}","-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{5/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{5/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\coth (c+d x)}{a d}",1,"((Sqrt[b]*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + (Sqrt[b]*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - 2*Coth[c + d*x])/(2*a*d)","A",1
240,1,165,148,2.2783205,"\int \frac{\text{csch}^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx","Integrate[Csch[c + d*x]^4/(a - b*Sinh[c + d*x]^4),x]","\frac{\frac{3 b \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{3 b \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}+4 \sqrt{a} \coth (c+d x)-2 \sqrt{a} \coth (c+d x) \text{csch}^2(c+d x)}{6 a^{3/2} d}","\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{7/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{7/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\coth ^3(c+d x)}{3 a d}+\frac{\coth (c+d x)}{a d}",1,"((-3*b*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + (3*b*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] + 4*Sqrt[a]*Coth[c + d*x] - 2*Sqrt[a]*Coth[c + d*x]*Csch[c + d*x]^2)/(6*a^(3/2)*d)","A",1
241,1,615,235,0.9727825,"\int \frac{\sinh ^9(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^9/(a - b*Sinh[c + d*x]^4)^2,x]","\frac{\frac{a \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^6 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^6 b c+\text{$\#$1}^6 b d x+40 \text{$\#$1}^4 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+20 \text{$\#$1}^4 a c+20 \text{$\#$1}^4 a d x-54 \text{$\#$1}^4 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-27 \text{$\#$1}^4 b c-27 \text{$\#$1}^4 b d x-40 \text{$\#$1}^2 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-20 \text{$\#$1}^2 a c-20 \text{$\#$1}^2 a d x+54 \text{$\#$1}^2 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+27 \text{$\#$1}^2 b c+27 \text{$\#$1}^2 b d x-2 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-b c-b d x}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]}{a-b}+\frac{32 a \cosh (c+d x) (2 a-b \cosh (2 (c+d x))+b)}{(a-b) (8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b)}+32 \cosh (c+d x)}{32 b^2 d}","-\frac{\sqrt{a} \left(5 \sqrt{a}-6 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 b^{9/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\sqrt{a} \left(5 \sqrt{a}+6 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 b^{9/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}+\frac{a \cosh (c+d x) \left(a-b \cosh ^2(c+d x)+b\right)}{4 b^2 d (a-b) \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)}+\frac{\cosh (c+d x)}{b^2 d}",1,"(32*Cosh[c + d*x] + (32*a*Cosh[c + d*x]*(2*a + b - b*Cosh[2*(c + d*x)]))/((a - b)*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) + (a*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-(b*c) - b*d*x - 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 20*a*c*#1^2 + 27*b*c*#1^2 - 20*a*d*x*#1^2 + 27*b*d*x*#1^2 - 40*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 54*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 20*a*c*#1^4 - 27*b*c*#1^4 + 20*a*d*x*#1^4 - 27*b*d*x*#1^4 + 40*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 54*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + b*c*#1^6 + b*d*x*#1^6 + 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a - b))/(32*b^2*d)","C",1
242,1,737,210,0.6975428,"\int \frac{\sinh ^7(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4)^2,x]","-\frac{\text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{-6 \text{$\#$1}^6 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-3 \text{$\#$1}^6 a c-3 \text{$\#$1}^6 a d x+8 \text{$\#$1}^6 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+4 \text{$\#$1}^6 b c+4 \text{$\#$1}^6 b d x+10 \text{$\#$1}^4 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+5 \text{$\#$1}^4 a c+5 \text{$\#$1}^4 a d x-24 \text{$\#$1}^4 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-12 \text{$\#$1}^4 b c-12 \text{$\#$1}^4 b d x-10 \text{$\#$1}^2 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-5 \text{$\#$1}^2 a c-5 \text{$\#$1}^2 a d x+24 \text{$\#$1}^2 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+12 \text{$\#$1}^2 b c+12 \text{$\#$1}^2 b d x+6 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-8 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+3 a c+3 a d x-4 b c-4 b d x}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]-\frac{16 a (\cosh (3 (c+d x))-5 \cosh (c+d x))}{8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b}}{32 b d (a-b)}","\frac{\left(3 \sqrt{a}-4 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 b^{7/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\left(3 \sqrt{a}+4 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 b^{7/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{a \cosh (c+d x) \left(2-\cosh ^2(c+d x)\right)}{4 b d (a-b) \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)}",1,"-1/32*((-16*a*(-5*Cosh[c + d*x] + Cosh[3*(c + d*x)]))/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]) + RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (3*a*c - 4*b*c + 3*a*d*x - 4*b*d*x + 6*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 8*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 5*a*c*#1^2 + 12*b*c*#1^2 - 5*a*d*x*#1^2 + 12*b*d*x*#1^2 - 10*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 24*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 5*a*c*#1^4 - 12*b*c*#1^4 + 5*a*d*x*#1^4 - 12*b*d*x*#1^4 + 10*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 24*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 3*a*c*#1^6 + 4*b*c*#1^6 - 3*a*d*x*#1^6 + 4*b*d*x*#1^6 - 6*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6 + 8*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/((a - b)*b*d)","C",1
243,1,597,217,0.579182,"\int \frac{\sinh ^5(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^5/(a - b*Sinh[c + d*x]^4)^2,x]","\frac{\text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^6 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^6 b c+\text{$\#$1}^6 b d x+8 \text{$\#$1}^4 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+4 \text{$\#$1}^4 a c+4 \text{$\#$1}^4 a d x-22 \text{$\#$1}^4 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-11 \text{$\#$1}^4 b c-11 \text{$\#$1}^4 b d x-8 \text{$\#$1}^2 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-4 \text{$\#$1}^2 a c-4 \text{$\#$1}^2 a d x+22 \text{$\#$1}^2 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+11 \text{$\#$1}^2 b c+11 \text{$\#$1}^2 b d x-2 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-b c-b d x}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]+\frac{32 \cosh (c+d x) (2 a-b \cosh (2 (c+d x))+b)}{8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b}}{32 b d (a-b)}","-\frac{\left(\sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 \sqrt{a} b^{5/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\left(\sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 \sqrt{a} b^{5/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}+\frac{\cosh (c+d x) \left(a-b \cosh ^2(c+d x)+b\right)}{4 b d (a-b) \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)}",1,"((32*Cosh[c + d*x]*(2*a + b - b*Cosh[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]) + RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-(b*c) - b*d*x - 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 4*a*c*#1^2 + 11*b*c*#1^2 - 4*a*d*x*#1^2 + 11*b*d*x*#1^2 - 8*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 22*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 4*a*c*#1^4 - 11*b*c*#1^4 + 4*a*d*x*#1^4 - 11*b*d*x*#1^4 + 8*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 22*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + b*c*#1^6 + b*d*x*#1^6 + 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(32*(a - b)*b*d)","C",1
244,1,422,186,0.5115238,"\int \frac{\sinh ^3(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^3/(a - b*Sinh[c + d*x]^4)^2,x]","-\frac{\text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^6 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^6 c+\text{$\#$1}^6 d x-14 \text{$\#$1}^4 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-7 \text{$\#$1}^4 c-7 \text{$\#$1}^4 d x+14 \text{$\#$1}^2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+7 \text{$\#$1}^2 c+7 \text{$\#$1}^2 d x-2 \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-c-d x}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]+\frac{16 (\cosh (3 (c+d x))-5 \cosh (c+d x))}{-8 a-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x))+3 b}}{32 d (a-b)}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 \sqrt{a} b^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 \sqrt{a} b^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\cosh (c+d x) \left(2-\cosh ^2(c+d x)\right)}{4 d (a-b) \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)}",1,"-1/32*((16*(-5*Cosh[c + d*x] + Cosh[3*(c + d*x)]))/(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)]) + RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-c - d*x - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 7*c*#1^2 + 7*d*x*#1^2 + 14*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 7*c*#1^4 - 7*d*x*#1^4 - 14*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + c*#1^6 + d*x*#1^6 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/((a - b)*d)","C",1
245,1,597,221,0.3836284,"\int \frac{\sinh (c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]/(a - b*Sinh[c + d*x]^4)^2,x]","\frac{\text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^6 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+\text{$\#$1}^6 b c+\text{$\#$1}^6 b d x-24 \text{$\#$1}^4 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-12 \text{$\#$1}^4 a c-12 \text{$\#$1}^4 a d x+10 \text{$\#$1}^4 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+5 \text{$\#$1}^4 b c+5 \text{$\#$1}^4 b d x+24 \text{$\#$1}^2 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+12 \text{$\#$1}^2 a c+12 \text{$\#$1}^2 a d x-10 \text{$\#$1}^2 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-5 \text{$\#$1}^2 b c-5 \text{$\#$1}^2 b d x-2 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-b c-b d x}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]+\frac{32 \cosh (c+d x) (2 a-b \cosh (2 (c+d x))+b)}{8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b}}{32 a d (a-b)}","\frac{\left(3 \sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 a^{3/2} \sqrt[4]{b} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\left(3 \sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 a^{3/2} \sqrt[4]{b} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}+\frac{\cosh (c+d x) \left(a-b \cosh ^2(c+d x)+b\right)}{4 a d (a-b) \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)}",1,"((32*Cosh[c + d*x]*(2*a + b - b*Cosh[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]) + RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-(b*c) - b*d*x - 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 12*a*c*#1^2 - 5*b*c*#1^2 + 12*a*d*x*#1^2 - 5*b*d*x*#1^2 + 24*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 10*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 12*a*c*#1^4 + 5*b*c*#1^4 - 12*a*d*x*#1^4 + 5*b*d*x*#1^4 - 24*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 10*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + b*c*#1^6 + b*d*x*#1^6 + 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(32*a*(a - b)*d)","C",1
246,1,761,325,0.8806241,"\int \frac{\text{csch}(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^2} \, dx","Integrate[Csch[c + d*x]/(a - b*Sinh[c + d*x]^4)^2,x]","\frac{-\frac{b \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{10 \text{$\#$1}^6 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+5 \text{$\#$1}^6 a c+5 \text{$\#$1}^6 a d x-8 \text{$\#$1}^6 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-4 \text{$\#$1}^6 b c-4 \text{$\#$1}^6 b d x-38 \text{$\#$1}^4 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-19 \text{$\#$1}^4 a c-19 \text{$\#$1}^4 a d x+24 \text{$\#$1}^4 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+12 \text{$\#$1}^4 b c+12 \text{$\#$1}^4 b d x+38 \text{$\#$1}^2 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+19 \text{$\#$1}^2 a c+19 \text{$\#$1}^2 a d x-24 \text{$\#$1}^2 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-12 \text{$\#$1}^2 b c-12 \text{$\#$1}^2 b d x-10 a \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)+8 b \log \left(-\text{$\#$1} \sinh \left(\frac{1}{2} (c+d x)\right)+\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)\right)-5 a c-5 a d x+4 b c+4 b d x}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]}{a-b}+\frac{16 a b (\cosh (3 (c+d x))-5 \cosh (c+d x))}{(a-b) (8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b)}+32 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{32 a^2 d}","-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 a^{3/2} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 a^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a^2 d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a^2 d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac{b \cosh (c+d x) \left(2-\cosh ^2(c+d x)\right)}{4 a d (a-b) \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)}",1,"((16*a*b*(-5*Cosh[c + d*x] + Cosh[3*(c + d*x)]))/((a - b)*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) + 32*Log[Tanh[(c + d*x)/2]] - (b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-5*a*c + 4*b*c - 5*a*d*x + 4*b*d*x - 10*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 8*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 19*a*c*#1^2 - 12*b*c*#1^2 + 19*a*d*x*#1^2 - 12*b*d*x*#1^2 + 38*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 24*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 19*a*c*#1^4 + 12*b*c*#1^4 - 19*a*d*x*#1^4 + 12*b*d*x*#1^4 - 38*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 24*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 5*a*c*#1^6 - 4*b*c*#1^6 + 5*a*d*x*#1^6 - 4*b*d*x*#1^6 + 10*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6 - 8*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a - b))/(32*a^2*d)","C",1
247,1,262,320,4.7750969,"\int \frac{\sinh ^8(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^8/(a - b*Sinh[c + d*x]^4)^2,x]","\frac{-\frac{\sqrt{a} \left(4 \sqrt{a}+5 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\left(\sqrt{a}+\sqrt{b}\right) \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\sqrt{a} \left(4 \sqrt{a}-5 \sqrt{b}\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\left(\sqrt{a}-\sqrt{b}\right) \sqrt{\sqrt{a} \sqrt{b}-a}}+\frac{2 a b (\sinh (4 (c+d x))-6 \sinh (2 (c+d x)))}{(a-b) (8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b)}+8 (c+d x)}{8 b^2 d}","\frac{\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{8 b^{3/2} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{8 b^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{2 b^2 d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{2 b^2 d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\tanh ^5(c+d x)}{4 b d \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)}-\frac{\tanh (c+d x)}{4 b d (a-b)}+\frac{x}{b^2}",1,"(8*(c + d*x) + (Sqrt[a]*(4*Sqrt[a] - 5*Sqrt[b])*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] - Sqrt[b])*Sqrt[-a + Sqrt[a]*Sqrt[b]]) - (Sqrt[a]*(4*Sqrt[a] + 5*Sqrt[b])*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] + Sqrt[b])*Sqrt[a + Sqrt[a]*Sqrt[b]]) + (2*a*b*(-6*Sinh[2*(c + d*x)] + Sinh[4*(c + d*x)]))/((a - b)*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])))/(8*b^2*d)","A",0
248,1,238,233,2.6115154,"\int \frac{\sinh ^6(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^6/(a - b*Sinh[c + d*x]^4)^2,x]","\frac{\frac{\sqrt{b} \left(-\sqrt{a} \sqrt{b}-2 a+3 b\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\sqrt{b} \left(\sqrt{a} \sqrt{b}-2 a+3 b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}-\frac{4 b \sinh (2 (c+d x)) (-2 a+b \cosh (2 (c+d x))-b)}{8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b}}{8 b^2 d (a-b)}","\frac{\left(2 \sqrt{a}-3 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{8 \sqrt[4]{a} b^{3/2} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\left(2 \sqrt{a}+3 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{8 \sqrt[4]{a} b^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}+\frac{\tanh (c+d x)}{4 b d (a-b)}+\frac{\tanh ^3(c+d x) \text{sech}^2(c+d x)}{4 b d \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)}",1,"((Sqrt[b]*(-2*a + Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + (Sqrt[b]*(-2*a - Sqrt[a]*Sqrt[b] + 3*b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - (4*b*(-2*a - b + b*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]))/(8*(a - b)*b^2*d)","A",1
249,1,225,195,4.2056182,"\int \frac{\sinh ^4(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^4/(a - b*Sinh[c + d*x]^4)^2,x]","-\frac{\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{a} \sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{a} \sqrt{b} \sqrt{\sqrt{a} \sqrt{b}-a}}-\frac{2 (\sinh (4 (c+d x))-6 \sinh (2 (c+d x)))}{8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b}}{8 d (a-b)}","\frac{\tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{3/4} \sqrt{b} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{3/4} \sqrt{b} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}+\frac{\tanh ^5(c+d x)}{4 a d \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)}-\frac{\tanh (c+d x)}{4 a d (a-b)}",1,"-1/8*(((Sqrt[a] + Sqrt[b])*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) + ((Sqrt[a] - Sqrt[b])*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) - (2*(-6*Sinh[2*(c + d*x)] + Sinh[4*(c + d*x)]))/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]))/((a - b)*d)","A",1
250,1,253,220,2.0864626,"\int \frac{\sinh ^2(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^2} \, dx","Integrate[Sinh[c + d*x]^2/(a - b*Sinh[c + d*x]^4)^2,x]","\frac{\frac{\sqrt{a} \left(-\sqrt{a} \sqrt{b}+2 a-b\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\sqrt{a} \left(\sqrt{a} \sqrt{b}+2 a-b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{b} \sqrt{\sqrt{a} \sqrt{b}-a}}+\frac{4 \sqrt{a} \sinh (2 (c+d x)) (2 a-b \cosh (2 (c+d x))+b)}{8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b}}{8 a^{3/2} d (a-b)}","-\frac{\left(2 \sqrt{a}-\sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{5/4} \sqrt{b} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\left(2 \sqrt{a}+\sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{5/4} \sqrt{b} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}+\frac{\tanh (c+d x) \left(a-(a+b) \tanh ^2(c+d x)\right)}{4 a d (a-b) \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)}",1,"((Sqrt[a]*(2*a + Sqrt[a]*Sqrt[b] - b)*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) + (Sqrt[a]*(2*a - Sqrt[a]*Sqrt[b] - b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) + (4*Sqrt[a]*(2*a + b - b*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]))/(8*a^(3/2)*(a - b)*d)","A",1
251,1,230,210,2.8709551,"\int \frac{1}{\left(a-b \sinh ^4(c+d x)\right)^2} \, dx","Integrate[(a - b*Sinh[c + d*x]^4)^(-2),x]","\frac{\frac{\left(-\sqrt{a} \sqrt{b}+4 a-3 b\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{\left(\sqrt{a} \sqrt{b}+4 a-3 b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}+\frac{2 \sqrt{a} b (\sinh (4 (c+d x))-6 \sinh (2 (c+d x)))}{8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b}}{8 a^{3/2} d (a-b)}","\frac{\left(4 \sqrt{a}-3 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{7/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\left(4 \sqrt{a}+3 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{7/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{b \tanh (c+d x) \left(1-2 \tanh ^2(c+d x)\right)}{4 a d (a-b) \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)}",1,"(-(((4*a + Sqrt[a]*Sqrt[b] - 3*b)*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]]) + ((4*a - Sqrt[a]*Sqrt[b] - 3*b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] + (2*Sqrt[a]*b*(-6*Sinh[2*(c + d*x)] + Sinh[4*(c + d*x)]))/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]))/(8*a^(3/2)*(a - b)*d)","A",1
252,1,272,237,1.9151628,"\int \frac{\text{csch}^2(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^2} \, dx","Integrate[Csch[c + d*x]^2/(a - b*Sinh[c + d*x]^4)^2,x]","\frac{\frac{\left(6 a \sqrt{b}+5 \sqrt{a} b\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\left(\sqrt{a}+\sqrt{b}\right) \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\left(6 a \sqrt{b}-5 \sqrt{a} b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\left(\sqrt{a}-\sqrt{b}\right) \sqrt{\sqrt{a} \sqrt{b}-a}}+\frac{4 \sqrt{a} b \sinh (2 (c+d x)) (2 a-b \cosh (2 (c+d x))+b)}{(a-b) (8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b)}-8 \sqrt{a} \coth (c+d x)}{8 a^{5/2} d}","-\frac{\sqrt{b} \left(6 \sqrt{a}-5 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{9/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\sqrt{b} \left(6 \sqrt{a}+5 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{9/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}+\frac{b \tanh (c+d x) \left(a-(a+b) \tanh ^2(c+d x)\right)}{4 a^2 d (a-b) \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)}-\frac{\coth (c+d x)}{a^2 d}",1,"(((6*a*Sqrt[b] - 5*Sqrt[a]*b)*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] - Sqrt[b])*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + ((6*a*Sqrt[b] + 5*Sqrt[a]*b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] + Sqrt[b])*Sqrt[a + Sqrt[a]*Sqrt[b]]) - 8*Sqrt[a]*Coth[c + d*x] + (4*Sqrt[a]*b*(2*a + b - b*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a - b)*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])))/(8*a^(5/2)*d)","A",1
253,1,1021,315,1.7449791,"\int \frac{\sinh ^9(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^9/(a - b*Sinh[c + d*x]^4)^3,x]","\frac{\frac{512 a (a-b) \cosh (c+d x) (2 a+b-b \cosh (2 (c+d x)))}{(-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}-\text{RootSum}\left[b \text{$\#$1}^8-4 b \text{$\#$1}^6-16 a \text{$\#$1}^4+6 b \text{$\#$1}^4-4 b \text{$\#$1}^2+b\&,\frac{-5 b^2 c \text{$\#$1}^6+2 a b c \text{$\#$1}^6-5 b^2 d x \text{$\#$1}^6+2 a b d x \text{$\#$1}^6-10 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^6+4 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^6+10 a^2 c \text{$\#$1}^4+39 b^2 c \text{$\#$1}^4-28 a b c \text{$\#$1}^4+10 a^2 d x \text{$\#$1}^4+39 b^2 d x \text{$\#$1}^4-28 a b d x \text{$\#$1}^4+20 a^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4+78 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4-56 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4-10 a^2 c \text{$\#$1}^2-39 b^2 c \text{$\#$1}^2+28 a b c \text{$\#$1}^2-10 a^2 d x \text{$\#$1}^2-39 b^2 d x \text{$\#$1}^2+28 a b d x \text{$\#$1}^2-20 a^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2-78 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2+56 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2+5 b^2 c-2 a b c+5 b^2 d x-2 a b d x+10 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)-4 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)}{b \text{$\#$1}^7-3 b \text{$\#$1}^5-8 a \text{$\#$1}^3+3 b \text{$\#$1}^3-b \text{$\#$1}}\&\right]+\frac{32 \cosh (c+d x) \left(-9 a^2+13 b a+5 b^2+(2 a-5 b) b \cosh (2 (c+d x))\right)}{8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))}}{128 (a-b)^2 b^2 d}","-\frac{\cosh (c+d x) \left(9 a^2-2 b (2 a-5 b) \cosh ^2(c+d x)-11 a b-10 b^2\right)}{32 b^2 d (a-b)^2 \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)}+\frac{\left(-14 \sqrt{a} \sqrt{b}+5 a+12 b\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 \sqrt{a} b^{9/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\left(14 \sqrt{a} \sqrt{b}+5 a+12 b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 \sqrt{a} b^{9/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}+\frac{a \cosh (c+d x) \left(a-b \cosh ^2(c+d x)+b\right)}{8 b^2 d (a-b) \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)^2}",1,"((32*Cosh[c + d*x]*(-9*a^2 + 13*a*b + 5*b^2 + (2*a - 5*b)*b*Cosh[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]) + (512*a*(a - b)*Cosh[c + d*x]*(2*a + b - b*Cosh[2*(c + d*x)]))/(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2 - RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*a*b*c + 5*b^2*c - 2*a*b*d*x + 5*b^2*d*x - 4*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 10*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 10*a^2*c*#1^2 + 28*a*b*c*#1^2 - 39*b^2*c*#1^2 - 10*a^2*d*x*#1^2 + 28*a*b*d*x*#1^2 - 39*b^2*d*x*#1^2 - 20*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 56*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 78*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 10*a^2*c*#1^4 - 28*a*b*c*#1^4 + 39*b^2*c*#1^4 + 10*a^2*d*x*#1^4 - 28*a*b*d*x*#1^4 + 39*b^2*d*x*#1^4 + 20*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 56*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 78*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 2*a*b*c*#1^6 - 5*b^2*c*#1^6 + 2*a*b*d*x*#1^6 - 5*b^2*d*x*#1^6 + 4*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6 - 10*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(128*(a - b)^2*b^2*d)","C",1
254,1,802,290,1.3302052,"\int \frac{\sinh ^7(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4)^3,x]","\frac{-\frac{32 \cosh (c+d x) (-7 a+25 b+3 (a-3 b) \cosh (2 (c+d x)))}{8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))}-3 \text{RootSum}\left[b \text{$\#$1}^8-4 b \text{$\#$1}^6-16 a \text{$\#$1}^4+6 b \text{$\#$1}^4-4 b \text{$\#$1}^2+b\&,\frac{-a c \text{$\#$1}^6+3 b c \text{$\#$1}^6-a d x \text{$\#$1}^6+3 b d x \text{$\#$1}^6-2 a \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^6+6 b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^6+3 a c \text{$\#$1}^4-17 b c \text{$\#$1}^4+3 a d x \text{$\#$1}^4-17 b d x \text{$\#$1}^4+6 a \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4-34 b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4-3 a c \text{$\#$1}^2+17 b c \text{$\#$1}^2-3 a d x \text{$\#$1}^2+17 b d x \text{$\#$1}^2-6 a \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2+34 b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2+a c-3 b c+a d x-3 b d x+2 a \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)-6 b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)}{b \text{$\#$1}^7-3 b \text{$\#$1}^5-8 a \text{$\#$1}^3+3 b \text{$\#$1}^3-b \text{$\#$1}}\&\right]+\frac{512 a (a-b) (\cosh (3 (c+d x))-5 \cosh (c+d x))}{(-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}}{256 (a-b)^2 b d}","\frac{3 \left(\sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 \sqrt{a} b^{7/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{3 \left(\sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 \sqrt{a} b^{7/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}+\frac{\cosh (c+d x) \left(-3 (a-3 b) \cosh ^2(c+d x)+5 a-17 b\right)}{32 b d (a-b)^2 \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)}-\frac{a \cosh (c+d x) \left(2-\cosh ^2(c+d x)\right)}{8 b d (a-b) \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)^2}",1,"((-32*Cosh[c + d*x]*(-7*a + 25*b + 3*(a - 3*b)*Cosh[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]) + (512*a*(a - b)*(-5*Cosh[c + d*x] + Cosh[3*(c + d*x)]))/(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2 - 3*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (a*c - 3*b*c + a*d*x - 3*b*d*x + 2*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 6*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 3*a*c*#1^2 + 17*b*c*#1^2 - 3*a*d*x*#1^2 + 17*b*d*x*#1^2 - 6*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 34*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 3*a*c*#1^4 - 17*b*c*#1^4 + 3*a*d*x*#1^4 - 17*b*d*x*#1^4 + 6*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 34*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - a*c*#1^6 + 3*b*c*#1^6 - a*d*x*#1^6 + 3*b*d*x*#1^6 - 2*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6 + 6*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(256*(a - b)^2*b*d)","C",1
255,1,1019,313,1.9613038,"\int \frac{\sinh ^5(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^5/(a - b*Sinh[c + d*x]^4)^3,x]","-\frac{-\frac{512 (a-b) \cosh (c+d x) (2 a+b-b \cosh (2 (c+d x)))}{(-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}+\frac{\text{RootSum}\left[b \text{$\#$1}^8-4 b \text{$\#$1}^6-16 a \text{$\#$1}^4+6 b \text{$\#$1}^4-4 b \text{$\#$1}^2+b\&,\frac{-b^2 c \text{$\#$1}^6-2 a b c \text{$\#$1}^6-b^2 d x \text{$\#$1}^6-2 a b d x \text{$\#$1}^6-2 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^6-4 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^6-6 a^2 c \text{$\#$1}^4-5 b^2 c \text{$\#$1}^4+32 a b c \text{$\#$1}^4-6 a^2 d x \text{$\#$1}^4-5 b^2 d x \text{$\#$1}^4+32 a b d x \text{$\#$1}^4-12 a^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4-10 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4+64 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4+6 a^2 c \text{$\#$1}^2+5 b^2 c \text{$\#$1}^2-32 a b c \text{$\#$1}^2+6 a^2 d x \text{$\#$1}^2+5 b^2 d x \text{$\#$1}^2-32 a b d x \text{$\#$1}^2+12 a^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2+10 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2-64 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2+b^2 c+2 a b c+b^2 d x+2 a b d x+2 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)+4 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)}{b \text{$\#$1}^7-3 b \text{$\#$1}^5-8 a \text{$\#$1}^3+3 b \text{$\#$1}^3-b \text{$\#$1}}\&\right]}{a}+\frac{32 \cosh (c+d x) \left(a^2-9 b a-b^2+b (2 a+b) \cosh (2 (c+d x))\right)}{a (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}}{128 (a-b)^2 b d}","-\frac{\left(-10 \sqrt{a} \sqrt{b}+3 a+4 b\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 a^{3/2} b^{5/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{\left(10 \sqrt{a} \sqrt{b}+3 a+4 b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 a^{3/2} b^{5/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{\cosh (c+d x) \left(a^2+2 b (2 a+b) \cosh ^2(c+d x)-11 a b-2 b^2\right)}{32 a b d (a-b)^2 \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)}+\frac{\cosh (c+d x) \left(a-b \cosh ^2(c+d x)+b\right)}{8 b d (a-b) \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)^2}",1,"-1/128*((32*Cosh[c + d*x]*(a^2 - 9*a*b - b^2 + b*(2*a + b)*Cosh[2*(c + d*x)]))/(a*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) - (512*(a - b)*Cosh[c + d*x]*(2*a + b - b*Cosh[2*(c + d*x)]))/(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2 + RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (2*a*b*c + b^2*c + 2*a*b*d*x + b^2*d*x + 4*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 2*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 6*a^2*c*#1^2 - 32*a*b*c*#1^2 + 5*b^2*c*#1^2 + 6*a^2*d*x*#1^2 - 32*a*b*d*x*#1^2 + 5*b^2*d*x*#1^2 + 12*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 64*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 10*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 6*a^2*c*#1^4 + 32*a*b*c*#1^4 - 5*b^2*c*#1^4 - 6*a^2*d*x*#1^4 + 32*a*b*d*x*#1^4 - 5*b^2*d*x*#1^4 - 12*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 64*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 10*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 2*a*b*c*#1^6 - b^2*c*#1^6 - 2*a*b*d*x*#1^6 - b^2*d*x*#1^6 - 4*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6 - 2*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ]/a)/((a - b)^2*b*d)","C",1
256,1,802,288,1.3225514,"\int \frac{\sinh ^3(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^3/(a - b*Sinh[c + d*x]^4)^3,x]","\frac{\frac{32 \cosh (c+d x) (-17 a-b+(5 a+b) \cosh (2 (c+d x)))}{a (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}+\frac{\text{RootSum}\left[b \text{$\#$1}^8-4 b \text{$\#$1}^6-16 a \text{$\#$1}^4+6 b \text{$\#$1}^4-4 b \text{$\#$1}^2+b\&,\frac{-5 a c \text{$\#$1}^6-b c \text{$\#$1}^6-5 a d x \text{$\#$1}^6-b d x \text{$\#$1}^6-10 a \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^6-2 b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^6+47 a c \text{$\#$1}^4-5 b c \text{$\#$1}^4+47 a d x \text{$\#$1}^4-5 b d x \text{$\#$1}^4+94 a \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4-10 b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4-47 a c \text{$\#$1}^2+5 b c \text{$\#$1}^2-47 a d x \text{$\#$1}^2+5 b d x \text{$\#$1}^2-94 a \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2+10 b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2+5 a c+b c+5 a d x+b d x+10 a \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)+2 b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)}{b \text{$\#$1}^7-3 b \text{$\#$1}^5-8 a \text{$\#$1}^3+3 b \text{$\#$1}^3-b \text{$\#$1}}\&\right]}{a}+\frac{512 (a-b) (\cosh (3 (c+d x))-5 \cosh (c+d x))}{(-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}}{256 (a-b)^2 d}","-\frac{\left(5 \sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 a^{3/2} b^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\left(5 \sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 a^{3/2} b^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{\cosh (c+d x) \left(-\left((5 a+b) \cosh ^2(c+d x)\right)+11 a+b\right)}{32 a d (a-b)^2 \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)}-\frac{\cosh (c+d x) \left(2-\cosh ^2(c+d x)\right)}{8 d (a-b) \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)^2}",1,"((32*Cosh[c + d*x]*(-17*a - b + (5*a + b)*Cosh[2*(c + d*x)]))/(a*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) + (512*(a - b)*(-5*Cosh[c + d*x] + Cosh[3*(c + d*x)]))/(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2 + RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (5*a*c + b*c + 5*a*d*x + b*d*x + 10*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 47*a*c*#1^2 + 5*b*c*#1^2 - 47*a*d*x*#1^2 + 5*b*d*x*#1^2 - 94*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 10*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 47*a*c*#1^4 - 5*b*c*#1^4 + 47*a*d*x*#1^4 - 5*b*d*x*#1^4 + 94*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 10*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 5*a*c*#1^6 - b*c*#1^6 - 5*a*d*x*#1^6 - b*d*x*#1^6 - 10*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6 - 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ]/a)/(256*(a - b)^2*d)","C",1
257,1,1018,313,1.4202901,"\int \frac{\sinh (c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]/(a - b*Sinh[c + d*x]^4)^3,x]","\frac{\frac{512 a (a-b) \cosh (c+d x) (2 a+b-b \cosh (2 (c+d x)))}{(-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}+3 \text{RootSum}\left[b \text{$\#$1}^8-4 b \text{$\#$1}^6-16 a \text{$\#$1}^4+6 b \text{$\#$1}^4-4 b \text{$\#$1}^2+b\&,\frac{-b^2 c \text{$\#$1}^6+2 a b c \text{$\#$1}^6-b^2 d x \text{$\#$1}^6+2 a b d x \text{$\#$1}^6-2 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^6+4 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^6-14 a^2 c \text{$\#$1}^4-5 b^2 c \text{$\#$1}^4+12 a b c \text{$\#$1}^4-14 a^2 d x \text{$\#$1}^4-5 b^2 d x \text{$\#$1}^4+12 a b d x \text{$\#$1}^4-28 a^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4-10 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4+24 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4+14 a^2 c \text{$\#$1}^2+5 b^2 c \text{$\#$1}^2-12 a b c \text{$\#$1}^2+14 a^2 d x \text{$\#$1}^2+5 b^2 d x \text{$\#$1}^2-12 a b d x \text{$\#$1}^2+28 a^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2+10 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2-24 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2+b^2 c-2 a b c+b^2 d x-2 a b d x+2 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)-4 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)}{b \text{$\#$1}^7-3 b \text{$\#$1}^5-8 a \text{$\#$1}^3+3 b \text{$\#$1}^3-b \text{$\#$1}}\&\right]+\frac{32 \cosh (c+d x) \left(7 a^2+5 b a-3 b^2+3 b (b-2 a) \cosh (2 (c+d x))\right)}{8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))}}{128 a^2 (a-b)^2 d}","\frac{3 \left(-10 \sqrt{a} \sqrt{b}+7 a+4 b\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 a^{5/2} \sqrt[4]{b} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{3 \left(10 \sqrt{a} \sqrt{b}+7 a+4 b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 a^{5/2} \sqrt[4]{b} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}+\frac{\cosh (c+d x) \left((7 a-3 b) (a+2 b)-6 b (2 a-b) \cosh ^2(c+d x)\right)}{32 a^2 d (a-b)^2 \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)}+\frac{\cosh (c+d x) \left(a-b \cosh ^2(c+d x)+b\right)}{8 a d (a-b) \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)^2}",1,"((32*Cosh[c + d*x]*(7*a^2 + 5*a*b - 3*b^2 + 3*b*(-2*a + b)*Cosh[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]) + (512*a*(a - b)*Cosh[c + d*x]*(2*a + b - b*Cosh[2*(c + d*x)]))/(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2 + 3*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*a*b*c + b^2*c - 2*a*b*d*x + b^2*d*x - 4*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 2*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 14*a^2*c*#1^2 - 12*a*b*c*#1^2 + 5*b^2*c*#1^2 + 14*a^2*d*x*#1^2 - 12*a*b*d*x*#1^2 + 5*b^2*d*x*#1^2 + 28*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 24*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 10*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 14*a^2*c*#1^4 + 12*a*b*c*#1^4 - 5*b^2*c*#1^4 - 14*a^2*d*x*#1^4 + 12*a*b*d*x*#1^4 - 5*b^2*d*x*#1^4 - 28*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 24*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 10*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 2*a*b*c*#1^6 - b^2*c*#1^6 + 2*a*b*d*x*#1^6 - b^2*d*x*#1^6 + 4*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6 - 2*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(128*a^2*(a - b)^2*d)","C",1
258,1,1189,617,5.6180859,"\int \frac{\text{csch}(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^3} \, dx","Integrate[Csch[c + d*x]/(a - b*Sinh[c + d*x]^4)^3,x]","\frac{\frac{512 b (\cosh (3 (c+d x))-5 \cosh (c+d x)) a^2}{(a-b) (-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}+\frac{32 b \cosh (c+d x) (-41 a+23 b+(13 a-7 b) \cosh (2 (c+d x))) a}{(a-b)^2 (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}+256 \log \left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-\frac{b \text{RootSum}\left[b \text{$\#$1}^8-4 b \text{$\#$1}^6-16 a \text{$\#$1}^4+6 b \text{$\#$1}^4-4 b \text{$\#$1}^2+b\&,\frac{45 a^2 c \text{$\#$1}^6+32 b^2 c \text{$\#$1}^6-71 a b c \text{$\#$1}^6+45 a^2 d x \text{$\#$1}^6+32 b^2 d x \text{$\#$1}^6-71 a b d x \text{$\#$1}^6+90 a^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^6+64 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^6-142 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^6-199 a^2 c \text{$\#$1}^4-96 b^2 c \text{$\#$1}^4+253 a b c \text{$\#$1}^4-199 a^2 d x \text{$\#$1}^4-96 b^2 d x \text{$\#$1}^4+253 a b d x \text{$\#$1}^4-398 a^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4-192 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4+506 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^4+199 a^2 c \text{$\#$1}^2+96 b^2 c \text{$\#$1}^2-253 a b c \text{$\#$1}^2+199 a^2 d x \text{$\#$1}^2+96 b^2 d x \text{$\#$1}^2-253 a b d x \text{$\#$1}^2+398 a^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2+192 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2-506 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right) \text{$\#$1}^2-45 a^2 c-32 b^2 c+71 a b c-45 a^2 d x-32 b^2 d x+71 a b d x-90 a^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)-64 b^2 \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)+142 a b \log \left(\text{$\#$1} \cosh \left(\frac{1}{2} (c+d x)\right)-\cosh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right)-\sinh \left(\frac{1}{2} (c+d x)\right) \text{$\#$1}\right)}{b \text{$\#$1}^7-3 b \text{$\#$1}^5-8 a \text{$\#$1}^3+3 b \text{$\#$1}^3-b \text{$\#$1}}\&\right]}{(a-b)^2}}{256 a^3 d}","-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 a^{5/2} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\sqrt[4]{b} \left(5 \sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 a^{5/2} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\sqrt[4]{b} \left(5 \sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 a^{5/2} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 a^{5/2} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a^3 d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a^3 d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a^3 d}-\frac{b \cosh (c+d x) \left(2-\cosh ^2(c+d x)\right)}{4 a^2 d (a-b) \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)}-\frac{b \cosh (c+d x) \left(-\left((5 a+b) \cosh ^2(c+d x)\right)+11 a+b\right)}{32 a^2 d (a-b)^2 \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)}-\frac{b \cosh (c+d x) \left(2-\cosh ^2(c+d x)\right)}{8 a d (a-b) \left(a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right)^2}",1,"((32*a*b*Cosh[c + d*x]*(-41*a + 23*b + (13*a - 7*b)*Cosh[2*(c + d*x)]))/((a - b)^2*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) + (512*a^2*b*(-5*Cosh[c + d*x] + Cosh[3*(c + d*x)]))/((a - b)*(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2) + 256*Log[Tanh[(c + d*x)/2]] - (b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-45*a^2*c + 71*a*b*c - 32*b^2*c - 45*a^2*d*x + 71*a*b*d*x - 32*b^2*d*x - 90*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 142*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 64*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 199*a^2*c*#1^2 - 253*a*b*c*#1^2 + 96*b^2*c*#1^2 + 199*a^2*d*x*#1^2 - 253*a*b*d*x*#1^2 + 96*b^2*d*x*#1^2 + 398*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 506*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 192*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 199*a^2*c*#1^4 + 253*a*b*c*#1^4 - 96*b^2*c*#1^4 - 199*a^2*d*x*#1^4 + 253*a*b*d*x*#1^4 - 96*b^2*d*x*#1^4 - 398*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 506*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 192*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 45*a^2*c*#1^6 - 71*a*b*c*#1^6 + 32*b^2*c*#1^6 + 45*a^2*d*x*#1^6 - 71*a*b*d*x*#1^6 + 32*b^2*d*x*#1^6 + 90*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6 - 142*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6 + 64*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a - b)^2)/(256*a^3*d)","C",1
259,1,331,319,4.0882072,"\int \frac{\sinh ^8(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^8/(a - b*Sinh[c + d*x]^4)^3,x]","\frac{\frac{\left(2 a^{3/2} \sqrt{b}-8 \sqrt{a} b^{3/2}+a b+5 b^2\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\sqrt{b} \left(2 \sqrt{a}-5 \sqrt{b}\right) \left(\sqrt{a}+\sqrt{b}\right)^2 \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}}+\frac{8 b \sinh (2 (c+d x)) ((5 b-2 a) \cosh (2 (c+d x))+5 a-14 b)}{8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b}+\frac{64 a b (a-b) (\sinh (4 (c+d x))-6 \sinh (2 (c+d x)))}{(-8 a-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x))+3 b)^2}}{64 b^2 d (a-b)^2}","-\frac{\left(2 \sqrt{a}-5 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{3/4} b^{3/2} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\left(2 \sqrt{a}+5 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{3/4} b^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{\tanh ^3(c+d x)}{32 a b d (a-b)}+\frac{\tanh ^9(c+d x)}{8 a d \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)^2}-\frac{(a+5 b) \tanh (c+d x)}{32 a b d (a-b)^2}-\frac{\tanh ^5(c+d x) \text{sech}^2(c+d x)}{32 a b d \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)}",1,"(((2*Sqrt[a] - 5*Sqrt[b])*(Sqrt[a] + Sqrt[b])^2*Sqrt[b]*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + ((2*a^(3/2)*Sqrt[b] + a*b - 8*Sqrt[a]*b^(3/2) + 5*b^2)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) + (8*b*(5*a - 14*b + (-2*a + 5*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]) + (64*a*(a - b)*b*(-6*Sinh[2*(c + d*x)] + Sinh[4*(c + d*x)]))/(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2)/(64*(a - b)^2*b^2*d)","A",1
260,1,351,345,3.3729069,"\int \frac{\sinh ^6(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^6/(a - b*Sinh[c + d*x]^4)^3,x]","-\frac{\frac{4 \sinh (2 (c+d x)) \left(4 a^2+3 b (a+b) \cosh (2 (c+d x))-19 a b-3 b^2\right)}{a b (8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b)}+\frac{\left(10 \sqrt{a} \sqrt{b}+4 a+3 b\right) \left(\sqrt{a}-\sqrt{b}\right)^2 \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{a b^{3/2} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\left(\sqrt{a}+\sqrt{b}\right)^2 \left(-10 \sqrt{a} \sqrt{b}+4 a+3 b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{a b^{3/2} \sqrt{\sqrt{a} \sqrt{b}-a}}-\frac{128 (a-b) \sinh (2 (c+d x)) (2 a-b \cosh (2 (c+d x))+b)}{b (-8 a-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x))+3 b)^2}}{64 d (a-b)^2}","\frac{\left(-10 \sqrt{a} \sqrt{b}+4 a+3 b\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{5/4} b^{3/2} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{\left(10 \sqrt{a} \sqrt{b}+4 a+3 b\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{5/4} b^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}+\frac{\tanh (c+d x) \left(\frac{2 a \left(a^2-a b-8 b^2\right)}{(a-b)^3}-\frac{\left(2 a^2+15 a b+3 b^2\right) \tanh ^2(c+d x)}{(a-b)^2}\right)}{32 a b d \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)}+\frac{\tanh (c+d x) \left(a (a+3 b)-\left(a^2+6 a b+b^2\right) \tanh ^2(c+d x)\right)}{8 d (a-b)^3 \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)^2}",1,"-1/64*(((Sqrt[a] + Sqrt[b])^2*(4*a - 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(a*Sqrt[-a + Sqrt[a]*Sqrt[b]]*b^(3/2)) + ((Sqrt[a] - Sqrt[b])^2*(4*a + 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(a*Sqrt[a + Sqrt[a]*Sqrt[b]]*b^(3/2)) + (4*(4*a^2 - 19*a*b - 3*b^2 + 3*b*(a + b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(a*b*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) - (128*(a - b)*(2*a + b - b*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(b*(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2))/((a - b)^2*d)","A",1
261,1,316,314,4.8822626,"\int \frac{\sinh ^4(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^4/(a - b*Sinh[c + d*x]^4)^3,x]","\frac{-\frac{3 \left(2 a^{3/2}-3 a \sqrt{b}+b^{3/2}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{a^{3/2} \sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{3 \left(2 a^{3/2}+3 a \sqrt{b}-b^{3/2}\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{a^{3/2} \sqrt{b} \sqrt{\sqrt{a} \sqrt{b}-a}}+\frac{8 \sinh (2 (c+d x)) ((2 a+b) \cosh (2 (c+d x))-7 a-2 b)}{a (8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b)}+\frac{64 (a-b) (\sinh (4 (c+d x))-6 \sinh (2 (c+d x)))}{(-8 a-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x))+3 b)^2}}{64 d (a-b)^2}","\frac{3 \left(2 \sqrt{a}-\sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{7/4} \sqrt{b} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{3 \left(2 \sqrt{a}+\sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{7/4} \sqrt{b} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{\tanh (c+d x) \left(\frac{9 a^2-24 a b-b^2}{(a-b)^3}-\frac{(17 a+3 b) \tanh ^2(c+d x)}{(a-b)^2}\right)}{32 a d \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)}-\frac{b \tanh (c+d x) \left(-4 (a+b) \tanh ^2(c+d x)+3 a+b\right)}{8 d (a-b)^3 \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)^2}",1,"((-3*(2*a^(3/2) + 3*a*Sqrt[b] - b^(3/2))*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(a^(3/2)*Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) - (3*(2*a^(3/2) - 3*a*Sqrt[b] + b^(3/2))*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(a^(3/2)*Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) + (8*(-7*a - 2*b + (2*a + b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(a*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) + (64*(a - b)*(-6*Sinh[2*(c + d*x)] + Sinh[4*(c + d*x)]))/(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2)/(64*(a - b)^2*d)","A",1
262,1,343,348,5.0028904,"\int \frac{\sinh ^2(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^3} \, dx","Integrate[Sinh[c + d*x]^2/(a - b*Sinh[c + d*x]^4)^3,x]","\frac{\frac{4 \sinh (2 (c+d x)) \left(12 a^2+b (5 b-11 a) \cosh (2 (c+d x))+11 a b-5 b^2\right)}{8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b}+\frac{\left(14 \sqrt{a} \sqrt{b}+12 a+5 b\right) \left(\sqrt{a}-\sqrt{b}\right)^2 \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\left(\sqrt{a}+\sqrt{b}\right)^2 \left(-14 \sqrt{a} \sqrt{b}+12 a+5 b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{b} \sqrt{\sqrt{a} \sqrt{b}-a}}+\frac{128 a (a-b) \sinh (2 (c+d x)) (2 a-b \cosh (2 (c+d x))+b)}{(-8 a-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x))+3 b)^2}}{64 a^2 d (a-b)^2}","-\frac{\left(-14 \sqrt{a} \sqrt{b}+12 a+5 b\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{9/4} \sqrt{b} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\left(14 \sqrt{a} \sqrt{b}+12 a+5 b\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{9/4} \sqrt{b} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}+\frac{\tanh (c+d x) \left(\frac{2 a \left(5 a^2-9 a b-4 b^2\right)}{(a-b)^3}-\frac{5 \left(2 a^2+3 a b-b^2\right) \tanh ^2(c+d x)}{(a-b)^2}\right)}{32 a^2 d \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)}+\frac{b \tanh (c+d x) \left(a (a+3 b)-\left(a^2+6 a b+b^2\right) \tanh ^2(c+d x)\right)}{8 a d (a-b)^3 \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)^2}",1,"(((Sqrt[a] + Sqrt[b])^2*(12*a - 14*Sqrt[a]*Sqrt[b] + 5*b)*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) + ((Sqrt[a] - Sqrt[b])^2*(12*a + 14*Sqrt[a]*Sqrt[b] + 5*b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) + (4*(12*a^2 + 11*a*b - 5*b^2 + b*(-11*a + 5*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]) + (128*a*(a - b)*(2*a + b - b*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2)/(64*a^2*(a - b)^2*d)","A",1
263,1,333,320,3.0407367,"\int \frac{1}{\left(a-b \sinh ^4(c+d x)\right)^3} \, dx","Integrate[(a - b*Sinh[c + d*x]^4)^(-3),x]","\frac{\frac{64 a^{3/2} b (a-b) (\sinh (4 (c+d x))-6 \sinh (2 (c+d x)))}{(-8 a-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x))+3 b)^2}+\frac{\left(50 \sqrt{a} \sqrt{b}+32 a+21 b\right) \left(\sqrt{a}-\sqrt{b}\right)^2 \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{\left(\sqrt{a}+\sqrt{b}\right)^2 \left(-50 \sqrt{a} \sqrt{b}+32 a+21 b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}+\frac{8 \sqrt{a} b \sinh (2 (c+d x)) ((6 a-3 b) \cosh (2 (c+d x))-19 a+10 b)}{8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b}}{64 a^{5/2} d (a-b)^2}","\frac{\left(-50 \sqrt{a} \sqrt{b}+32 a+21 b\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{11/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\left(50 \sqrt{a} \sqrt{b}+32 a+21 b\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{11/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{b \tanh (c+d x) \left(\frac{17 a^2-40 a b+7 b^2}{(a-b)^3}-\frac{(33 a-13 b) \tanh ^2(c+d x)}{(a-b)^2}\right)}{32 a^2 d \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)}-\frac{b^2 \tanh (c+d x) \left(-4 (a+b) \tanh ^2(c+d x)+3 a+b\right)}{8 a d (a-b)^3 \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)^2}",1,"(-(((Sqrt[a] + Sqrt[b])^2*(32*a - 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]]) + ((Sqrt[a] - Sqrt[b])^2*(32*a + 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] + (8*Sqrt[a]*b*(-19*a + 10*b + (6*a - 3*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]) + (64*a^(3/2)*(a - b)*b*(-6*Sinh[2*(c + d*x)] + Sinh[4*(c + d*x)]))/(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2)/(64*a^(5/2)*(a - b)^2*d)","A",1
264,1,357,359,3.5066928,"\int \frac{\text{csch}^2(c+d x)}{\left(a-b \sinh ^4(c+d x)\right)^3} \, dx","Integrate[Csch[c + d*x]^2/(a - b*Sinh[c + d*x]^4)^3,x]","\frac{\frac{4 b \sinh (2 (c+d x)) \left(28 a^2+b (13 b-19 a) \cosh (2 (c+d x))+3 a b-13 b^2\right)}{(a-b)^2 (8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b)}+\frac{3 \sqrt{b} \left(34 \sqrt{a} \sqrt{b}+20 a+15 b\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\left(\sqrt{a}+\sqrt{b}\right)^2 \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{3 \sqrt{b} \left(-34 \sqrt{a} \sqrt{b}+20 a+15 b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\left(\sqrt{a}-\sqrt{b}\right)^2 \sqrt{\sqrt{a} \sqrt{b}-a}}+\frac{128 a b \sinh (2 (c+d x)) (2 a-b \cosh (2 (c+d x))+b)}{(a-b) (-8 a-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x))+3 b)^2}-64 \coth (c+d x)}{64 a^3 d}","-\frac{3 \sqrt{b} \left(-34 \sqrt{a} \sqrt{b}+20 a+15 b\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{13/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{3 \sqrt{b} \left(34 \sqrt{a} \sqrt{b}+20 a+15 b\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{13/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{\coth (c+d x)}{a^3 d}+\frac{b^2 \tanh (c+d x) \left(a (a+3 b)-\left(a^2+6 a b+b^2\right) \tanh ^2(c+d x)\right)}{8 a^2 d (a-b)^3 \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)^2}+\frac{b \tanh (c+d x) \left(\frac{2 a^2 (9 a-17 b)}{(a-b)^3}-\frac{\left(18 a^2+15 a b-13 b^2\right) \tanh ^2(c+d x)}{(a-b)^2}\right)}{32 a^3 d \left((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right)}",1,"((3*Sqrt[b]*(20*a - 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] - Sqrt[b])^2*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + (3*Sqrt[b]*(20*a + 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] + Sqrt[b])^2*Sqrt[a + Sqrt[a]*Sqrt[b]]) - 64*Coth[c + d*x] + (4*b*(28*a^2 + 3*a*b - 13*b^2 + b*(-19*a + 13*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a - b)^2*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) + (128*a*b*(2*a + b - b*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a - b)*(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2))/(64*a^3*d)","A",1
265,1,24,25,0.104396,"\int \frac{1}{1-\sinh ^4(x)} \, dx","Integrate[(1 - Sinh[x]^4)^(-1),x]","\frac{1}{4} \left(\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \tanh (x)\right)+2 \tanh (x)\right)","\frac{\tanh ^{-1}\left(\sqrt{2} \tanh (x)\right)}{2 \sqrt{2}}+\frac{\tanh (x)}{2}",1,"(Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]] + 2*Tanh[x])/4","A",1
266,1,45,176,0.0768802,"\int \frac{1}{1+\sinh ^4(x)} \, dx","Integrate[(1 + Sinh[x]^4)^(-1),x]","\frac{\tanh ^{-1}\left(\sqrt{1-i} \tanh (x)\right)}{2 \sqrt{1-i}}+\frac{\tanh ^{-1}\left(\sqrt{1+i} \tanh (x)\right)}{2 \sqrt{1+i}}","-\frac{1}{8} \sqrt{1+\sqrt{2}} \log \left(2 \tanh ^2(x)-2 \sqrt{1+\sqrt{2}} \tanh (x)+\sqrt{2}\right)+\frac{1}{8} \sqrt{1+\sqrt{2}} \log \left(\sqrt{2} \tanh ^2(x)+\sqrt{2 \left(1+\sqrt{2}\right)} \tanh (x)+1\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{1+\sqrt{2}}-2 \tanh (x)}{\sqrt{\sqrt{2}-1}}\right)}{4 \sqrt{1+\sqrt{2}}}+\frac{\tan ^{-1}\left(\frac{2 \tanh (x)+\sqrt{1+\sqrt{2}}}{\sqrt{\sqrt{2}-1}}\right)}{4 \sqrt{1+\sqrt{2}}}",1,"ArcTanh[Sqrt[1 - I]*Tanh[x]]/(2*Sqrt[1 - I]) + ArcTanh[Sqrt[1 + I]*Tanh[x]]/(2*Sqrt[1 + I])","C",1
267,1,141,435,0.3347611,"\int \frac{1}{a+b \sinh ^5(x)} \, dx","Integrate[(a + b*Sinh[x]^5)^(-1),x]","\frac{8}{5} \text{RootSum}\left[\text{$\#$1}^{10} b-5 \text{$\#$1}^8 b+10 \text{$\#$1}^6 b+32 \text{$\#$1}^5 a-10 \text{$\#$1}^4 b+5 \text{$\#$1}^2 b-b\&,\frac{\text{$\#$1}^3 x+2 \text{$\#$1}^3 \log \left(-\text{$\#$1} \sinh \left(\frac{x}{2}\right)+\text{$\#$1} \cosh \left(\frac{x}{2}\right)-\sinh \left(\frac{x}{2}\right)-\cosh \left(\frac{x}{2}\right)\right)}{\text{$\#$1}^8 b-4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b+16 \text{$\#$1}^3 a-4 \text{$\#$1}^2 b+b}\&\right]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[5]{b}-\sqrt[5]{a} \tanh \left(\frac{x}{2}\right)}{\sqrt{a^{2/5}+b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}+b^{2/5}}}+\frac{2 (-1)^{9/10} \tanh ^{-1}\left(\frac{(-1)^{9/10} \left(\sqrt[5]{a} \tanh \left(\frac{x}{2}\right)+\sqrt[5]{-1} \sqrt[5]{b}\right)}{\sqrt{\sqrt[5]{-1} b^{2/5}-(-1)^{4/5} a^{2/5}}}\right)}{5 a^{4/5} \sqrt{\sqrt[5]{-1} b^{2/5}-(-1)^{4/5} a^{2/5}}}+\frac{2 \sqrt[5]{-1} \tanh ^{-1}\left(\frac{\sqrt[5]{-1} \sqrt[5]{a} \tanh \left(\frac{x}{2}\right)+\sqrt[5]{b}}{\sqrt{(-1)^{2/5} a^{2/5}+b^{2/5}}}\right)}{5 a^{4/5} \sqrt{(-1)^{2/5} a^{2/5}+b^{2/5}}}+\frac{2 (-1)^{9/10} \tanh ^{-1}\left(\frac{(-1)^{3/10} \left((-1)^{3/5} \sqrt[5]{a} \tanh \left(\frac{x}{2}\right)+\sqrt[5]{b}\right)}{\sqrt{(-1)^{3/5} b^{2/5}-(-1)^{4/5} a^{2/5}}}\right)}{5 a^{4/5} \sqrt{(-1)^{3/5} b^{2/5}-(-1)^{4/5} a^{2/5}}}-\frac{2 (-1)^{9/10} \tanh ^{-1}\left(\frac{-(-1)^{9/10} \sqrt[5]{a} \tanh \left(\frac{x}{2}\right)+i \sqrt[5]{b}}{\sqrt{-(-1)^{4/5} a^{2/5}-b^{2/5}}}\right)}{5 a^{4/5} \sqrt{-(-1)^{4/5} a^{2/5}-b^{2/5}}}",1,"(8*RootSum[-b + 5*b*#1^2 - 10*b*#1^4 + 32*a*#1^5 + 10*b*#1^6 - 5*b*#1^8 + b*#1^10 & , (x*#1^3 + 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^3)/(b - 4*b*#1^2 + 16*a*#1^3 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8) & ])/5","C",1
268,1,134,175,0.1760462,"\int \frac{1}{a+b \sinh ^6(x)} \, dx","Integrate[(a + b*Sinh[x]^6)^(-1),x]","\frac{16}{3} \text{RootSum}\left[\text{$\#$1}^6 b-6 \text{$\#$1}^5 b+15 \text{$\#$1}^4 b+64 \text{$\#$1}^3 a-20 \text{$\#$1}^3 b+15 \text{$\#$1}^2 b-6 \text{$\#$1} b+b\&,\frac{\text{$\#$1}^2 x+\text{$\#$1}^2 \log (-\text{$\#$1} \sinh (x)+\text{$\#$1} \cosh (x)-\sinh (x)-\cosh (x))}{\text{$\#$1}^5 b-5 \text{$\#$1}^4 b+10 \text{$\#$1}^3 b+32 \text{$\#$1}^2 a-10 \text{$\#$1}^2 b+5 \text{$\#$1} b-b}\&\right]","\frac{\tanh ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}-\sqrt[3]{b}} \tanh (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}-\sqrt[3]{b}}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b}} \tanh (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b}}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b}} \tanh (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b}}}",1,"(16*RootSum[b - 6*b*#1 + 15*b*#1^2 + 64*a*#1^3 - 20*b*#1^3 + 15*b*#1^4 - 6*b*#1^5 + b*#1^6 & , (x*#1^2 + Log[-Cosh[x] - Sinh[x] + Cosh[x]*#1 - Sinh[x]*#1]*#1^2)/(-b + 5*b*#1 + 32*a*#1^2 - 10*b*#1^2 + 10*b*#1^3 - 5*b*#1^4 + b*#1^5) & ])/3","C",1
269,1,160,245,0.2575518,"\int \frac{1}{a+b \sinh ^8(x)} \, dx","Integrate[(a + b*Sinh[x]^8)^(-1),x]","16 \text{RootSum}\left[\text{$\#$1}^8 b-8 \text{$\#$1}^7 b+28 \text{$\#$1}^6 b-56 \text{$\#$1}^5 b+256 \text{$\#$1}^4 a+70 \text{$\#$1}^4 b-56 \text{$\#$1}^3 b+28 \text{$\#$1}^2 b-8 \text{$\#$1} b+b\&,\frac{\text{$\#$1}^3 x+\text{$\#$1}^3 \log (-\text{$\#$1} \sinh (x)+\text{$\#$1} \cosh (x)-\sinh (x)-\cosh (x))}{\text{$\#$1}^7 b-7 \text{$\#$1}^6 b+21 \text{$\#$1}^5 b-35 \text{$\#$1}^4 b+128 \text{$\#$1}^3 a+35 \text{$\#$1}^3 b-21 \text{$\#$1}^2 b+7 \text{$\#$1} b-b}\&\right]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}-\sqrt[4]{b}} \tanh (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}-\sqrt[4]{b}}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}-i \sqrt[4]{b}} \tanh (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}-i \sqrt[4]{b}}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}+i \sqrt[4]{b}} \tanh (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}+i \sqrt[4]{b}}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}+\sqrt[4]{b}} \tanh (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}+\sqrt[4]{b}}}",1,"16*RootSum[b - 8*b*#1 + 28*b*#1^2 - 56*b*#1^3 + 256*a*#1^4 + 70*b*#1^4 - 56*b*#1^5 + 28*b*#1^6 - 8*b*#1^7 + b*#1^8 & , (x*#1^3 + Log[-Cosh[x] - Sinh[x] + Cosh[x]*#1 - Sinh[x]*#1]*#1^3)/(-b + 7*b*#1 - 21*b*#1^2 + 128*a*#1^3 + 35*b*#1^3 - 35*b*#1^4 + 21*b*#1^5 - 7*b*#1^6 + b*#1^7) & ]","C",1
270,1,439,242,1.0267488,"\int \frac{1}{1+\sinh ^5(x)} \, dx","Integrate[(1 + Sinh[x]^5)^(-1),x]","\frac{1}{10} \left(2 \sqrt{2} \tanh ^{-1}\left(\frac{\tanh \left(\frac{x}{2}\right)-1}{\sqrt{2}}\right)-\text{RootSum}\left[\text{$\#$1}^8-2 \text{$\#$1}^7-2 \text{$\#$1}^5+14 \text{$\#$1}^4+2 \text{$\#$1}^3+2 \text{$\#$1}+1\&,\frac{\text{$\#$1}^6 x+2 \text{$\#$1}^6 \log \left(-\text{$\#$1} \sinh \left(\frac{x}{2}\right)+\text{$\#$1} \cosh \left(\frac{x}{2}\right)-\sinh \left(\frac{x}{2}\right)-\cosh \left(\frac{x}{2}\right)\right)-4 \text{$\#$1}^5 x-8 \text{$\#$1}^5 \log \left(-\text{$\#$1} \sinh \left(\frac{x}{2}\right)+\text{$\#$1} \cosh \left(\frac{x}{2}\right)-\sinh \left(\frac{x}{2}\right)-\cosh \left(\frac{x}{2}\right)\right)+9 \text{$\#$1}^4 x+18 \text{$\#$1}^4 \log \left(-\text{$\#$1} \sinh \left(\frac{x}{2}\right)+\text{$\#$1} \cosh \left(\frac{x}{2}\right)-\sinh \left(\frac{x}{2}\right)-\cosh \left(\frac{x}{2}\right)\right)-24 \text{$\#$1}^3 x-48 \text{$\#$1}^3 \log \left(-\text{$\#$1} \sinh \left(\frac{x}{2}\right)+\text{$\#$1} \cosh \left(\frac{x}{2}\right)-\sinh \left(\frac{x}{2}\right)-\cosh \left(\frac{x}{2}\right)\right)-9 \text{$\#$1}^2 x-18 \text{$\#$1}^2 \log \left(-\text{$\#$1} \sinh \left(\frac{x}{2}\right)+\text{$\#$1} \cosh \left(\frac{x}{2}\right)-\sinh \left(\frac{x}{2}\right)-\cosh \left(\frac{x}{2}\right)\right)-4 \text{$\#$1} x-8 \text{$\#$1} \log \left(-\text{$\#$1} \sinh \left(\frac{x}{2}\right)+\text{$\#$1} \cosh \left(\frac{x}{2}\right)-\sinh \left(\frac{x}{2}\right)-\cosh \left(\frac{x}{2}\right)\right)-2 \log \left(-\text{$\#$1} \sinh \left(\frac{x}{2}\right)+\text{$\#$1} \cosh \left(\frac{x}{2}\right)-\sinh \left(\frac{x}{2}\right)-\cosh \left(\frac{x}{2}\right)\right)-x}{4 \text{$\#$1}^7-7 \text{$\#$1}^6-5 \text{$\#$1}^4+28 \text{$\#$1}^3+3 \text{$\#$1}^2+1}\&\right]\right)","-\frac{1}{5} \sqrt{2} \tanh ^{-1}\left(\frac{1-\tanh \left(\frac{x}{2}\right)}{\sqrt{2}}\right)+\frac{2 (-1)^{9/10} \tanh ^{-1}\left(\frac{(-1)^{7/10} \left(\sqrt[5]{-1} \tanh \left(\frac{x}{2}\right)+1\right)}{\sqrt{-(-1)^{2/5} \left(1+(-1)^{2/5}\right)}}\right)}{5 \sqrt{-(-1)^{2/5} \left(1+(-1)^{2/5}\right)}}-\frac{2 (-1)^{4/5} \tanh ^{-1}\left(\frac{1-(-1)^{4/5} \tanh \left(\frac{x}{2}\right)}{\sqrt{1-(-1)^{3/5}}}\right)}{5 \sqrt{1-(-1)^{3/5}}}-\frac{2 (-1)^{3/5} \tan ^{-1}\left(\frac{(-1)^{3/5} \tanh \left(\frac{x}{2}\right)+1}{\sqrt{\sqrt[5]{-1}-1}}\right)}{5 \sqrt{\sqrt[5]{-1}-1}}+\frac{2 (-1)^{9/10} \tan ^{-1}\left(\frac{-(-1)^{9/10} \tanh \left(\frac{x}{2}\right)+i}{\sqrt{1+(-1)^{4/5}}}\right)}{5 \sqrt{1+(-1)^{4/5}}}",1,"(2*Sqrt[2]*ArcTanh[(-1 + Tanh[x/2])/Sqrt[2]] - RootSum[1 + 2*#1 + 2*#1^3 + 14*#1^4 - 2*#1^5 - 2*#1^7 + #1^8 & , (-x - 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1] - 4*x*#1 - 8*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1 - 9*x*#1^2 - 18*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^2 - 24*x*#1^3 - 48*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^3 + 9*x*#1^4 + 18*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^4 - 4*x*#1^5 - 8*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^5 + x*#1^6 + 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^6)/(1 + 3*#1^2 + 28*#1^3 - 5*#1^4 - 7*#1^6 + 4*#1^7) & ])/10","C",1
271,1,87,71,0.2165871,"\int \frac{1}{1+\sinh ^6(x)} \, dx","Integrate[(1 + Sinh[x]^6)^(-1),x]","\frac{1}{18} \left(6 \tanh (x)+\sqrt[4]{-3} \left(\left(-3-i \sqrt{3}\right) \tan ^{-1}\left(\frac{1}{2} \sqrt[4]{-3} \left(1+i \sqrt{3}\right) \tanh (x)\right)-\left(\sqrt{3}+3 i\right) \tan ^{-1}\left(\frac{1}{2} \sqrt[4]{-\frac{1}{3}} \left(3+i \sqrt{3}\right) \tanh (x)\right)\right)\right)","\frac{\tanh ^{-1}\left(\sqrt{1+\sqrt[3]{-1}} \tanh (x)\right)}{3 \sqrt{1+\sqrt[3]{-1}}}+\frac{\tanh ^{-1}\left(\sqrt{1-(-1)^{2/3}} \tanh (x)\right)}{3 \sqrt{1-(-1)^{2/3}}}+\frac{\tanh (x)}{3}",1,"((-3)^(1/4)*((-3 - I*Sqrt[3])*ArcTan[((-3)^(1/4)*(1 + I*Sqrt[3])*Tanh[x])/2] - (3*I + Sqrt[3])*ArcTan[((-1/3)^(1/4)*(3 + I*Sqrt[3])*Tanh[x])/2]) + 6*Tanh[x])/18","C",1
272,1,127,129,0.1247073,"\int \frac{1}{1+\sinh ^8(x)} \, dx","Integrate[(1 + Sinh[x]^8)^(-1),x]","16 \text{RootSum}\left[\text{$\#$1}^8-8 \text{$\#$1}^7+28 \text{$\#$1}^6-56 \text{$\#$1}^5+326 \text{$\#$1}^4-56 \text{$\#$1}^3+28 \text{$\#$1}^2-8 \text{$\#$1}+1\&,\frac{\text{$\#$1}^3 x+\text{$\#$1}^3 \log (-\text{$\#$1} \sinh (x)+\text{$\#$1} \cosh (x)-\sinh (x)-\cosh (x))}{\text{$\#$1}^7-7 \text{$\#$1}^6+21 \text{$\#$1}^5-35 \text{$\#$1}^4+163 \text{$\#$1}^3-21 \text{$\#$1}^2+7 \text{$\#$1}-1}\&\right]","\frac{\tanh ^{-1}\left(\sqrt{1-\sqrt[4]{-1}} \tanh (x)\right)}{4 \sqrt{1-\sqrt[4]{-1}}}+\frac{\tanh ^{-1}\left(\sqrt{1+\sqrt[4]{-1}} \tanh (x)\right)}{4 \sqrt{1+\sqrt[4]{-1}}}+\frac{\tanh ^{-1}\left(\sqrt{1-(-1)^{3/4}} \tanh (x)\right)}{4 \sqrt{1-(-1)^{3/4}}}+\frac{\tanh ^{-1}\left(\sqrt{1+(-1)^{3/4}} \tanh (x)\right)}{4 \sqrt{1+(-1)^{3/4}}}",1,"16*RootSum[1 - 8*#1 + 28*#1^2 - 56*#1^3 + 326*#1^4 - 56*#1^5 + 28*#1^6 - 8*#1^7 + #1^8 & , (x*#1^3 + Log[-Cosh[x] - Sinh[x] + Cosh[x]*#1 - Sinh[x]*#1]*#1^3)/(-1 + 7*#1 - 21*#1^2 + 163*#1^3 - 35*#1^4 + 21*#1^5 - 7*#1^6 + #1^7) & ]","C",1
273,1,437,228,0.9496761,"\int \frac{1}{1-\sinh ^5(x)} \, dx","Integrate[(1 - Sinh[x]^5)^(-1),x]","\frac{1}{10} \left(\text{RootSum}\left[\text{$\#$1}^8+2 \text{$\#$1}^7+2 \text{$\#$1}^5+14 \text{$\#$1}^4-2 \text{$\#$1}^3-2 \text{$\#$1}+1\&,\frac{\text{$\#$1}^6 x+2 \text{$\#$1}^6 \log \left(-\text{$\#$1} \sinh \left(\frac{x}{2}\right)+\text{$\#$1} \cosh \left(\frac{x}{2}\right)-\sinh \left(\frac{x}{2}\right)-\cosh \left(\frac{x}{2}\right)\right)+4 \text{$\#$1}^5 x+8 \text{$\#$1}^5 \log \left(-\text{$\#$1} \sinh \left(\frac{x}{2}\right)+\text{$\#$1} \cosh \left(\frac{x}{2}\right)-\sinh \left(\frac{x}{2}\right)-\cosh \left(\frac{x}{2}\right)\right)+9 \text{$\#$1}^4 x+18 \text{$\#$1}^4 \log \left(-\text{$\#$1} \sinh \left(\frac{x}{2}\right)+\text{$\#$1} \cosh \left(\frac{x}{2}\right)-\sinh \left(\frac{x}{2}\right)-\cosh \left(\frac{x}{2}\right)\right)+24 \text{$\#$1}^3 x+48 \text{$\#$1}^3 \log \left(-\text{$\#$1} \sinh \left(\frac{x}{2}\right)+\text{$\#$1} \cosh \left(\frac{x}{2}\right)-\sinh \left(\frac{x}{2}\right)-\cosh \left(\frac{x}{2}\right)\right)-9 \text{$\#$1}^2 x-18 \text{$\#$1}^2 \log \left(-\text{$\#$1} \sinh \left(\frac{x}{2}\right)+\text{$\#$1} \cosh \left(\frac{x}{2}\right)-\sinh \left(\frac{x}{2}\right)-\cosh \left(\frac{x}{2}\right)\right)+4 \text{$\#$1} x+8 \text{$\#$1} \log \left(-\text{$\#$1} \sinh \left(\frac{x}{2}\right)+\text{$\#$1} \cosh \left(\frac{x}{2}\right)-\sinh \left(\frac{x}{2}\right)-\cosh \left(\frac{x}{2}\right)\right)-2 \log \left(-\text{$\#$1} \sinh \left(\frac{x}{2}\right)+\text{$\#$1} \cosh \left(\frac{x}{2}\right)-\sinh \left(\frac{x}{2}\right)-\cosh \left(\frac{x}{2}\right)\right)-x}{4 \text{$\#$1}^7+7 \text{$\#$1}^6+5 \text{$\#$1}^4+28 \text{$\#$1}^3-3 \text{$\#$1}^2-1}\&\right]+2 \sqrt{2} \tanh ^{-1}\left(\frac{\tanh \left(\frac{x}{2}\right)+1}{\sqrt{2}}\right)\right)","-\frac{2 \tanh ^{-1}\left(\frac{(-1)^{3/5}-\tanh \left(\frac{x}{2}\right)}{\sqrt{1-\sqrt[5]{-1}}}\right)}{5 \sqrt{1-\sqrt[5]{-1}}}+\frac{1}{5} \sqrt{2} \tanh ^{-1}\left(\frac{\tanh \left(\frac{x}{2}\right)+1}{\sqrt{2}}\right)+\frac{2 \tanh ^{-1}\left(\frac{\tanh \left(\frac{x}{2}\right)+(-1)^{4/5}}{\sqrt{1-(-1)^{3/5}}}\right)}{5 \sqrt{1-(-1)^{3/5}}}-\frac{2 \sqrt[10]{-1} \tanh ^{-1}\left(\frac{(-1)^{3/10} \left((-1)^{4/5} \tanh \left(\frac{x}{2}\right)+1\right)}{\sqrt{\sqrt[5]{-1}+(-1)^{3/5}}}\right)}{5 \sqrt{\sqrt[5]{-1}+(-1)^{3/5}}}-\frac{2 \sqrt[10]{-1} \tan ^{-1}\left(\frac{\sqrt[10]{-1} \tanh \left(\frac{x}{2}\right)+i}{\sqrt{1-\sqrt[5]{-1}}}\right)}{5 \sqrt{1-\sqrt[5]{-1}}}",1,"(2*Sqrt[2]*ArcTanh[(1 + Tanh[x/2])/Sqrt[2]] + RootSum[1 - 2*#1 - 2*#1^3 + 14*#1^4 + 2*#1^5 + 2*#1^7 + #1^8 & , (-x - 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1] + 4*x*#1 + 8*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1 - 9*x*#1^2 - 18*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^2 + 24*x*#1^3 + 48*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^3 + 9*x*#1^4 + 18*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^4 + 4*x*#1^5 + 8*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^5 + x*#1^6 + 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^6)/(-1 - 3*#1^2 + 28*#1^3 + 5*#1^4 + 7*#1^6 + 4*#1^7) & ])/10","C",1
274,1,70,83,0.4498074,"\int \frac{1}{1-\sinh ^6(x)} \, dx","Integrate[(1 - Sinh[x]^6)^(-1),x]","\frac{1}{6} \left(\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \tanh (x)\right)+i \sqrt{3} \left(\tan ^{-1}\left(\frac{1-2 i \tanh (x)}{\sqrt{3}}\right)-\tan ^{-1}\left(\frac{1+2 i \tanh (x)}{\sqrt{3}}\right)\right)-\tan ^{-1}(\text{csch}(x) \text{sech}(x))\right)","\frac{\tanh ^{-1}\left(\sqrt{2} \tanh (x)\right)}{3 \sqrt{2}}+\frac{\tanh ^{-1}\left(\sqrt{1-\sqrt[3]{-1}} \tanh (x)\right)}{3 \sqrt{1-\sqrt[3]{-1}}}+\frac{\tanh ^{-1}\left(\sqrt{1+(-1)^{2/3}} \tanh (x)\right)}{3 \sqrt{1+(-1)^{2/3}}}",1,"(-ArcTan[Csch[x]*Sech[x]] + I*Sqrt[3]*(ArcTan[(1 - (2*I)*Tanh[x])/Sqrt[3]] - ArcTan[(1 + (2*I)*Tanh[x])/Sqrt[3]]) + Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]])/6","C",1
275,1,64,69,0.503578,"\int \frac{1}{1-\sinh ^8(x)} \, dx","Integrate[(1 - Sinh[x]^8)^(-1),x]","\frac{1}{8} \left(\frac{2 \tanh ^{-1}\left(\sqrt{1-i} \tanh (x)\right)}{\sqrt{1-i}}+\frac{2 \tanh ^{-1}\left(\sqrt{1+i} \tanh (x)\right)}{\sqrt{1+i}}+\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \tanh (x)\right)+2 \tanh (x)\right)","\frac{\tanh ^{-1}\left(\sqrt{1-i} \tanh (x)\right)}{4 \sqrt{1-i}}+\frac{\tanh ^{-1}\left(\sqrt{1+i} \tanh (x)\right)}{4 \sqrt{1+i}}+\frac{\tanh ^{-1}\left(\sqrt{2} \tanh (x)\right)}{4 \sqrt{2}}+\frac{\tanh (x)}{4}",1,"((2*ArcTanh[Sqrt[1 - I]*Tanh[x]])/Sqrt[1 - I] + (2*ArcTanh[Sqrt[1 + I]*Tanh[x]])/Sqrt[1 + I] + Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]] + 2*Tanh[x])/8","A",1
276,1,19,18,0.0038851,"\int \frac{\cosh ^5(x)}{a+a \sinh ^2(x)} \, dx","Integrate[Cosh[x]^5/(a + a*Sinh[x]^2),x]","\frac{\frac{3 \sinh (x)}{4}+\frac{1}{12} \sinh (3 x)}{a}","\frac{\sinh ^3(x)}{3 a}+\frac{\sinh (x)}{a}",1,"((3*Sinh[x])/4 + Sinh[3*x]/12)/a","A",1
277,1,18,20,0.0026375,"\int \frac{\cosh ^4(x)}{a+a \sinh ^2(x)} \, dx","Integrate[Cosh[x]^4/(a + a*Sinh[x]^2),x]","\frac{\frac{x}{2}+\frac{1}{4} \sinh (2 x)}{a}","\frac{x}{2 a}+\frac{\sinh (x) \cosh (x)}{2 a}",1,"(x/2 + Sinh[2*x]/4)/a","A",1
278,1,6,6,0.0021112,"\int \frac{\cosh ^3(x)}{a+a \sinh ^2(x)} \, dx","Integrate[Cosh[x]^3/(a + a*Sinh[x]^2),x]","\frac{\sinh (x)}{a}","\frac{\sinh (x)}{a}",1,"Sinh[x]/a","A",1
279,1,5,5,0.0004074,"\int \frac{\cosh ^2(x)}{a+a \sinh ^2(x)} \, dx","Integrate[Cosh[x]^2/(a + a*Sinh[x]^2),x]","\frac{x}{a}","\frac{x}{a}",1,"x/a","A",1
280,1,12,7,0.0031244,"\int \frac{\cosh (x)}{a+a \sinh ^2(x)} \, dx","Integrate[Cosh[x]/(a + a*Sinh[x]^2),x]","\frac{2 \tan ^{-1}\left(\tanh \left(\frac{x}{2}\right)\right)}{a}","\frac{\tan ^{-1}(\sinh (x))}{a}",1,"(2*ArcTan[Tanh[x/2]])/a","A",1
281,1,20,22,0.0040708,"\int \frac{\text{sech}(x)}{a+a \sinh ^2(x)} \, dx","Integrate[Sech[x]/(a + a*Sinh[x]^2),x]","\frac{\tan ^{-1}\left(\tanh \left(\frac{x}{2}\right)\right)+\frac{1}{2} \tanh (x) \text{sech}(x)}{a}","\frac{\tan ^{-1}(\sinh (x))}{2 a}+\frac{\tanh (x) \text{sech}(x)}{2 a}",1,"(ArcTan[Tanh[x/2]] + (Sech[x]*Tanh[x])/2)/a","A",1
282,1,34,35,0.0048511,"\int \frac{\text{sech}^3(x)}{a+a \sinh ^2(x)} \, dx","Integrate[Sech[x]^3/(a + a*Sinh[x]^2),x]","\frac{\frac{3}{4} \tan ^{-1}\left(\tanh \left(\frac{x}{2}\right)\right)+\frac{1}{4} \tanh (x) \text{sech}^3(x)+\frac{3}{8} \tanh (x) \text{sech}(x)}{a}","\frac{3 \tan ^{-1}(\sinh (x))}{8 a}+\frac{\tanh (x) \text{sech}^3(x)}{4 a}+\frac{3 \tanh (x) \text{sech}(x)}{8 a}",1,"((3*ArcTan[Tanh[x/2]])/4 + (3*Sech[x]*Tanh[x])/8 + (Sech[x]^3*Tanh[x])/4)/a","A",1
283,1,63,89,0.1756171,"\int \cosh ^4(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Cosh[c + d*x]^4*(a + b*Sinh[c + d*x]^2),x]","\frac{(48 a-3 b) \sinh (2 (c+d x))+3 (2 a+b) \sinh (4 (c+d x))+72 a c+72 a d x+b \sinh (6 (c+d x))-12 b d x}{192 d}","\frac{(6 a-b) \sinh (c+d x) \cosh ^3(c+d x)}{24 d}+\frac{(6 a-b) \sinh (c+d x) \cosh (c+d x)}{16 d}+\frac{1}{16} x (6 a-b)+\frac{b \sinh (c+d x) \cosh ^5(c+d x)}{6 d}",1,"(72*a*c + 72*a*d*x - 12*b*d*x + (48*a - 3*b)*Sinh[2*(c + d*x)] + 3*(2*a + b)*Sinh[4*(c + d*x)] + b*Sinh[6*(c + d*x)])/(192*d)","A",1
284,1,48,46,0.1398084,"\int \cosh ^3(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Cosh[c + d*x]^3*(a + b*Sinh[c + d*x]^2),x]","\frac{\sinh (c+d x) (4 (5 a+2 b) \cosh (2 (c+d x))+100 a+3 b \cosh (4 (c+d x))-11 b)}{120 d}","\frac{(a+b) \sinh ^3(c+d x)}{3 d}+\frac{a \sinh (c+d x)}{d}+\frac{b \sinh ^5(c+d x)}{5 d}",1,"((100*a - 11*b + 4*(5*a + 2*b)*Cosh[2*(c + d*x)] + 3*b*Cosh[4*(c + d*x)])*Sinh[c + d*x])/(120*d)","A",1
285,1,43,61,0.0719812,"\int \cosh ^2(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Cosh[c + d*x]^2*(a + b*Sinh[c + d*x]^2),x]","\frac{8 a \sinh (2 (c+d x))+16 a c+16 a d x+b \sinh (4 (c+d x))-4 b d x}{32 d}","\frac{(4 a-b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{1}{8} x (4 a-b)+\frac{b \sinh (c+d x) \cosh ^3(c+d x)}{4 d}",1,"(16*a*c + 16*a*d*x - 4*b*d*x + 8*a*Sinh[2*(c + d*x)] + b*Sinh[4*(c + d*x)])/(32*d)","A",1
286,1,39,28,0.0121191,"\int \cosh (c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Cosh[c + d*x]*(a + b*Sinh[c + d*x]^2),x]","\frac{a \sinh (c) \cosh (d x)}{d}+\frac{a \cosh (c) \sinh (d x)}{d}+\frac{b \sinh ^3(c+d x)}{3 d}","\frac{a \sinh (c+d x)}{d}+\frac{b \sinh ^3(c+d x)}{3 d}",1,"(a*Cosh[d*x]*Sinh[c])/d + (a*Cosh[c]*Sinh[d*x])/d + (b*Sinh[c + d*x]^3)/(3*d)","A",1
287,1,37,28,0.0141242,"\int \text{sech}(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Sech[c + d*x]*(a + b*Sinh[c + d*x]^2),x]","\frac{a \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b \sinh (c+d x)}{d}-\frac{b \tan ^{-1}(\sinh (c+d x))}{d}","\frac{(a-b) \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b \sinh (c+d x)}{d}",1,"(a*ArcTan[Sinh[c + d*x]])/d - (b*ArcTan[Sinh[c + d*x]])/d + (b*Sinh[c + d*x])/d","A",1
288,1,36,19,0.0131331,"\int \text{sech}^2(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Sech[c + d*x]^2*(a + b*Sinh[c + d*x]^2),x]","\frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{b \tanh (c+d x)}{d}","\frac{(a-b) \tanh (c+d x)}{d}+b x",1,"(b*ArcTanh[Tanh[c + d*x]])/d + (a*Tanh[c + d*x])/d - (b*Tanh[c + d*x])/d","A",1
289,1,71,42,0.0241818,"\int \text{sech}^3(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Sech[c + d*x]^3*(a + b*Sinh[c + d*x]^2),x]","\frac{a \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{a \tanh (c+d x) \text{sech}(c+d x)}{2 d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d}","\frac{(a+b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{(a-b) \tanh (c+d x) \text{sech}(c+d x)}{2 d}",1,"(a*ArcTan[Sinh[c + d*x]])/(2*d) + (b*ArcTan[Sinh[c + d*x]])/(2*d) + (a*Sech[c + d*x]*Tanh[c + d*x])/(2*d) - (b*Sech[c + d*x]*Tanh[c + d*x])/(2*d)","A",1
290,1,44,32,0.0079337,"\int \text{sech}^4(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Sech[c + d*x]^4*(a + b*Sinh[c + d*x]^2),x]","-\frac{a \tanh ^3(c+d x)}{3 d}+\frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^3(c+d x)}{3 d}","\frac{a \tanh (c+d x)}{d}-\frac{(a-b) \tanh ^3(c+d x)}{3 d}",1,"(a*Tanh[c + d*x])/d - (a*Tanh[c + d*x]^3)/(3*d) + (b*Tanh[c + d*x]^3)/(3*d)","A",1
291,1,60,70,0.1565541,"\int \text{sech}^5(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2),x]","\frac{(3 a+b) \tan ^{-1}(\sinh (c+d x))+2 (a-b) \tanh (c+d x) \text{sech}^3(c+d x)+(3 a+b) \tanh (c+d x) \text{sech}(c+d x)}{8 d}","\frac{(3 a+b) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{(a-b) \tanh (c+d x) \text{sech}^3(c+d x)}{4 d}+\frac{(3 a+b) \tanh (c+d x) \text{sech}(c+d x)}{8 d}",1,"((3*a + b)*ArcTan[Sinh[c + d*x]] + (3*a + b)*Sech[c + d*x]*Tanh[c + d*x] + 2*(a - b)*Sech[c + d*x]^3*Tanh[c + d*x])/(8*d)","A",1
292,1,102,54,0.0554846,"\int \text{sech}^6(c+d x) \left(a+b \sinh ^2(c+d x)\right) \, dx","Integrate[Sech[c + d*x]^6*(a + b*Sinh[c + d*x]^2),x]","\frac{a \tanh ^5(c+d x)}{5 d}-\frac{2 a \tanh ^3(c+d x)}{3 d}+\frac{a \tanh (c+d x)}{d}+\frac{2 b \tanh (c+d x)}{15 d}-\frac{b \tanh (c+d x) \text{sech}^4(c+d x)}{5 d}+\frac{b \tanh (c+d x) \text{sech}^2(c+d x)}{15 d}","\frac{(a-b) \tanh ^5(c+d x)}{5 d}-\frac{(2 a-b) \tanh ^3(c+d x)}{3 d}+\frac{a \tanh (c+d x)}{d}",1,"(a*Tanh[c + d*x])/d + (2*b*Tanh[c + d*x])/(15*d) + (b*Sech[c + d*x]^2*Tanh[c + d*x])/(15*d) - (b*Sech[c + d*x]^4*Tanh[c + d*x])/(5*d) - (2*a*Tanh[c + d*x]^3)/(3*d) + (a*Tanh[c + d*x]^5)/(5*d)","A",1
293,1,98,159,0.3209125,"\int \cosh ^4(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Cosh[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^2,x]","\frac{24 \left(48 a^2-16 a b+3 b^2\right) (c+d x)+24 \left(4 a^2+4 a b-b^2\right) \sinh (4 (c+d x))+32 a b \sinh (6 (c+d x))+96 a (8 a-b) \sinh (2 (c+d x))+3 b^2 \sinh (8 (c+d x))}{3072 d}","\frac{\left(48 a^2-16 a b+3 b^2\right) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}+\frac{\left(48 a^2-16 a b+3 b^2\right) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{1}{128} x \left(48 a^2-16 a b+3 b^2\right)+\frac{b (10 a-3 b) \sinh (c+d x) \cosh ^5(c+d x)}{48 d}+\frac{b \sinh (c+d x) \cosh ^7(c+d x) \left(a-(a-b) \tanh ^2(c+d x)\right)}{8 d}",1,"(24*(48*a^2 - 16*a*b + 3*b^2)*(c + d*x) + 96*a*(8*a - b)*Sinh[2*(c + d*x)] + 24*(4*a^2 + 4*a*b - b^2)*Sinh[4*(c + d*x)] + 32*a*b*Sinh[6*(c + d*x)] + 3*b^2*Sinh[8*(c + d*x)])/(3072*d)","A",1
294,1,64,74,0.114479,"\int \cosh ^3(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Cosh[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^2,x]","\frac{105 a^2 \sinh (c+d x)+21 b (2 a+b) \sinh ^5(c+d x)+35 a (a+2 b) \sinh ^3(c+d x)+15 b^2 \sinh ^7(c+d x)}{105 d}","\frac{a^2 \sinh (c+d x)}{d}+\frac{b (2 a+b) \sinh ^5(c+d x)}{5 d}+\frac{a (a+2 b) \sinh ^3(c+d x)}{3 d}+\frac{b^2 \sinh ^7(c+d x)}{7 d}",1,"(105*a^2*Sinh[c + d*x] + 35*a*(a + 2*b)*Sinh[c + d*x]^3 + 21*b*(2*a + b)*Sinh[c + d*x]^5 + 15*b^2*Sinh[c + d*x]^7)/(105*d)","A",1
295,1,79,119,0.2923226,"\int \cosh ^2(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Cosh[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^2,x]","\frac{12 \left(8 a^2-4 a b+b^2\right) (c+d x)+3 \left(16 a^2-b^2\right) \sinh (2 (c+d x))+3 b (4 a-b) \sinh (4 (c+d x))+b^2 \sinh (6 (c+d x))}{192 d}","\frac{\left(8 a^2-4 a b+b^2\right) \sinh (c+d x) \cosh (c+d x)}{16 d}+\frac{1}{16} x \left(8 a^2-4 a b+b^2\right)+\frac{b (8 a-3 b) \sinh (c+d x) \cosh ^3(c+d x)}{24 d}+\frac{b \sinh (c+d x) \cosh ^5(c+d x) \left(a-(a-b) \tanh ^2(c+d x)\right)}{6 d}",1,"(12*(8*a^2 - 4*a*b + b^2)*(c + d*x) + 3*(16*a^2 - b^2)*Sinh[2*(c + d*x)] + 3*(4*a - b)*b*Sinh[4*(c + d*x)] + b^2*Sinh[6*(c + d*x)])/(192*d)","A",1
296,1,44,49,0.0523208,"\int \cosh (c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Cosh[c + d*x]*(a + b*Sinh[c + d*x]^2)^2,x]","\frac{a^2 \sinh (c+d x)+\frac{2}{3} a b \sinh ^3(c+d x)+\frac{1}{5} b^2 \sinh ^5(c+d x)}{d}","\frac{a^2 \sinh (c+d x)}{d}+\frac{2 a b \sinh ^3(c+d x)}{3 d}+\frac{b^2 \sinh ^5(c+d x)}{5 d}",1,"(a^2*Sinh[c + d*x] + (2*a*b*Sinh[c + d*x]^3)/3 + (b^2*Sinh[c + d*x]^5)/5)/d","A",1
297,1,70,55,0.2934789,"\int \text{sech}(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Sech[c + d*x]*(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\sinh (c+d x) \left(b \left(6 a+b \left(\sinh ^2(c+d x)-3\right)\right)+\frac{3 (a-b)^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{\sqrt{-\sinh ^2(c+d x)}}\right)}{3 d}","\frac{b (2 a-b) \sinh (c+d x)}{d}+\frac{(a-b)^2 \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b^2 \sinh ^3(c+d x)}{3 d}",1,"(Sinh[c + d*x]*((3*(a - b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/Sqrt[-Sinh[c + d*x]^2] + b*(6*a + b*(-3 + Sinh[c + d*x]^2))))/(3*d)","A",1
298,1,50,53,0.3051402,"\int \text{sech}^2(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Sech[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^2,x]","\frac{2 b (4 a-3 b) (c+d x)+4 (a-b)^2 \tanh (c+d x)+b^2 \sinh (2 (c+d x))}{4 d}","\frac{(a-b)^2 \tanh (c+d x)}{d}+\frac{1}{2} b x (4 a-3 b)+\frac{b^2 \sinh (c+d x) \cosh (c+d x)}{2 d}",1,"(2*(4*a - 3*b)*b*(c + d*x) + b^2*Sinh[2*(c + d*x)] + 4*(a - b)^2*Tanh[c + d*x])/(4*d)","A",1
299,1,233,64,6.8407485,"\int \text{sech}^3(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Sech[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\text{csch}^3(c+d x) \left(-64 \sinh ^6(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, _4F_3\left(\frac{3}{2},2,2,2;1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)+\frac{105 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \left(\left(9 a^2+124 a b+101 b^2\right) \sinh ^4(c+d x)+125 a^2+2 b (a+27 b) \sinh ^6(c+d x)+2 a (27 a+125 b) \sinh ^2(c+d x)+b^2 \sinh ^8(c+d x)\right)}{\sqrt{-\sinh ^2(c+d x)}}-35 \left(375 a^2+a \sinh ^2(c+d x) (37 a+61 b \cosh (2 (c+d x))+689 b)+61 b^2 \sinh ^6(c+d x)+303 b^2 \sinh ^4(c+d x)\right)\right)}{1680 d}","\frac{(a+3 b) (a-b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{(a-b)^2 \tanh (c+d x) \text{sech}(c+d x)}{2 d}+\frac{b^2 \sinh (c+d x)}{d}",1,"(Csch[c + d*x]^3*(-64*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(a + b*Sinh[c + d*x]^2)^2 - 35*(375*a^2 + a*(37*a + 689*b + 61*b*Cosh[2*(c + d*x)])*Sinh[c + d*x]^2 + 303*b^2*Sinh[c + d*x]^4 + 61*b^2*Sinh[c + d*x]^6) + (105*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(125*a^2 + 2*a*(27*a + 125*b)*Sinh[c + d*x]^2 + (9*a^2 + 124*a*b + 101*b^2)*Sinh[c + d*x]^4 + 2*b*(a + 27*b)*Sinh[c + d*x]^6 + b^2*Sinh[c + d*x]^8))/Sqrt[-Sinh[c + d*x]^2]))/(1680*d)","C",0
300,1,57,47,0.346334,"\int \text{sech}^4(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Sech[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^2,x]","\frac{(a-b) \tanh (c+d x) \text{sech}^2(c+d x) ((a+2 b) \cosh (2 (c+d x))+2 a+b)+3 b^2 (c+d x)}{3 d}","\frac{\left(a^2-b^2\right) \tanh (c+d x)}{d}-\frac{(a-b)^2 \tanh ^3(c+d x)}{3 d}+b^2 x",1,"(3*b^2*(c + d*x) + (a - b)*(2*a + b + (a + 2*b)*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*Tanh[c + d*x])/(3*d)","A",1
301,1,303,96,5.8727771,"\int \text{sech}^5(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2)^2,x]","-\frac{\text{csch}^3(c+d x) \left(128 \sinh ^6(c+d x) \left(7 a^2+12 a b \sinh ^2(c+d x)+5 b^2 \sinh ^4(c+d x)\right) \, _4F_3\left(\frac{3}{2},2,2,2;1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)+128 \sinh ^6(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)-\frac{105 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \left(\left(37 a^2+988 a b+649 b^2\right) \sinh ^4(c+d x)+1125 a^2+2 b (11 a+189 b) \sinh ^6(c+d x)+2 a (297 a+875 b) \sinh ^2(c+d x)+9 b^2 \sinh ^8(c+d x)\right)}{\sqrt{-\sinh ^2(c+d x)}}+35 \left(3375 a^2+a \sinh ^2(c+d x) (657 a+607 b \cosh (2 (c+d x))+4643 b)+485 b^2 \sinh ^6(c+d x)+1947 b^2 \sinh ^4(c+d x)\right)\right)}{6720 d}","\frac{\left(3 a^2+2 a b+3 b^2\right) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{3 \left(a^2-b^2\right) \tanh (c+d x) \text{sech}(c+d x)}{8 d}+\frac{(a-b) \tanh (c+d x) \text{sech}^3(c+d x) \left(a+b \sinh ^2(c+d x)\right)}{4 d}",1,"-1/6720*(Csch[c + d*x]^3*(128*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(a + b*Sinh[c + d*x]^2)^2 + 128*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(7*a^2 + 12*a*b*Sinh[c + d*x]^2 + 5*b^2*Sinh[c + d*x]^4) + 35*(3375*a^2 + a*(657*a + 4643*b + 607*b*Cosh[2*(c + d*x)])*Sinh[c + d*x]^2 + 1947*b^2*Sinh[c + d*x]^4 + 485*b^2*Sinh[c + d*x]^6) - (105*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(1125*a^2 + 2*a*(297*a + 875*b)*Sinh[c + d*x]^2 + (37*a^2 + 988*a*b + 649*b^2)*Sinh[c + d*x]^4 + 2*b*(11*a + 189*b)*Sinh[c + d*x]^6 + 9*b^2*Sinh[c + d*x]^8))/Sqrt[-Sinh[c + d*x]^2]))/d","C",0
302,1,69,57,0.3924495,"\int \text{sech}^6(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Sech[c + d*x]^6*(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\tanh (c+d x) \left(2 \left(2 a^2+a b-3 b^2\right) \text{sech}^2(c+d x)+8 a^2+3 (a-b)^2 \text{sech}^4(c+d x)+4 a b+3 b^2\right)}{15 d}","\frac{a^2 \tanh (c+d x)}{d}+\frac{(a-b)^2 \tanh ^5(c+d x)}{5 d}-\frac{2 a (a-b) \tanh ^3(c+d x)}{3 d}",1,"((8*a^2 + 4*a*b + 3*b^2 + 2*(2*a^2 + a*b - 3*b^2)*Sech[c + d*x]^2 + 3*(a - b)^2*Sech[c + d*x]^4)*Tanh[c + d*x])/(15*d)","A",1
303,1,715,131,10.2260193,"\int \text{sech}^7(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2 \, dx","Integrate[Sech[c + d*x]^7*(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\text{csch}^3(c+d x) \left(32 \left(-\sinh ^2(c+d x)\right)^{3/2} \sinh ^4(c+d x) \left(5 a^2+9 a b \sinh ^2(c+d x)+4 b^2 \sinh ^4(c+d x)\right) \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)+4 \left(-\sinh ^2(c+d x)\right)^{3/2} \sinh ^4(c+d x) \left(155 a^2+242 a b \sinh ^2(c+d x)+95 b^2 \sinh ^4(c+d x)\right) \, _4F_3\left(\frac{3}{2},2,2,2;1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)+16 a^2 \left(-\sinh ^2(c+d x)\right)^{3/2} \sinh ^4(c+d x) \, _6F_5\left(\frac{3}{2},2,2,2,2,2;1,1,1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)+32 a b \left(-\sinh ^2(c+d x)\right)^{3/2} \sinh ^6(c+d x) \, _6F_5\left(\frac{3}{2},2,2,2,2,2;1,1,1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)+16 b^2 \left(-\sinh ^2(c+d x)\right)^{3/2} \sinh ^8(c+d x) \, _6F_5\left(\frac{3}{2},2,2,2,2,2;1,1,1,1,\frac{9}{2};-\sinh ^2(c+d x)\right)+14980 a^2 \left(-\sinh ^2(c+d x)\right)^{3/2}-65625 a^2 \sqrt{-\sinh ^2(c+d x)}+36855 a^2 \sinh ^2(c+d x) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)+65625 a^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)+1680 a^2 \sinh ^4(c+d x) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)-23555 a b \left(-\sinh ^2(c+d x)\right)^{5/2}+91875 a b \left(-\sinh ^2(c+d x)\right)^{3/2}+91875 a b \sinh ^2(c+d x) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)+1365 a b \sinh ^6(c+d x) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)+54180 a b \sinh ^4(c+d x) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)-32970 b^2 \left(-\sinh ^2(c+d x)\right)^{5/2}+8855 b^2 \left(-\sinh ^2(c+d x)\right)^{3/2} \sinh ^4(c+d x)+525 b^2 \sinh ^8(c+d x) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)+19845 b^2 \sinh ^6(c+d x) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)+32970 b^2 \sinh ^4(c+d x) \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)\right)}{2520 d \sqrt{-\sinh ^2(c+d x)}}","\frac{\left(5 a^2+2 a b+b^2\right) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac{\left(5 a^2+2 a b+b^2\right) \tanh (c+d x) \text{sech}(c+d x)}{16 d}+\frac{(a-b) (5 a+3 b) \tanh (c+d x) \text{sech}^3(c+d x)}{24 d}+\frac{(a-b) \tanh (c+d x) \text{sech}^5(c+d x) \left(a+b \sinh ^2(c+d x)\right)}{6 d}",1,"(Csch[c + d*x]^3*(65625*a^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]] + 36855*a^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^2 + 91875*a*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^2 + 1680*a^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 + 54180*a*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 + 32970*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 + 1365*a*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 + 19845*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 + 525*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8 - 65625*a^2*Sqrt[-Sinh[c + d*x]^2] + 14980*a^2*(-Sinh[c + d*x]^2)^(3/2) + 91875*a*b*(-Sinh[c + d*x]^2)^(3/2) + 8855*b^2*Sinh[c + d*x]^4*(-Sinh[c + d*x]^2)^(3/2) + 16*a^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^4*(-Sinh[c + d*x]^2)^(3/2) + 32*a*b*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(-Sinh[c + d*x]^2)^(3/2) + 16*b^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^8*(-Sinh[c + d*x]^2)^(3/2) - 23555*a*b*(-Sinh[c + d*x]^2)^(5/2) - 32970*b^2*(-Sinh[c + d*x]^2)^(5/2) + 32*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^4*(-Sinh[c + d*x]^2)^(3/2)*(5*a^2 + 9*a*b*Sinh[c + d*x]^2 + 4*b^2*Sinh[c + d*x]^4) + 4*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^4*(-Sinh[c + d*x]^2)^(3/2)*(155*a^2 + 242*a*b*Sinh[c + d*x]^2 + 95*b^2*Sinh[c + d*x]^4)))/(2520*d*Sqrt[-Sinh[c + d*x]^2])","C",0
304,1,144,238,0.5313166,"\int \cosh ^4(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Cosh[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{120 (4 a-b) \left(8 a^2-2 a b+b^2\right) (c+d x)-10 b \left(b^2-16 a^2\right) \sinh (6 (c+d x))+20 \left(128 a^3-24 a^2 b+b^3\right) \sinh (2 (c+d x))+40 \left(8 a^3+12 a^2 b-6 a b^2+b^3\right) \sinh (4 (c+d x))+5 b^2 (6 a-b) \sinh (8 (c+d x))+2 b^3 \sinh (10 (c+d x))}{10240 d}","\frac{b \left(44 a^2-28 a b+5 b^2\right) \sinh (c+d x) \cosh ^5(c+d x)}{160 d}+\frac{(4 a-b) \left(8 a^2-2 a b+b^2\right) \sinh (c+d x) \cosh ^3(c+d x)}{128 d}+\frac{3 (4 a-b) \left(8 a^2-2 a b+b^2\right) \sinh (c+d x) \cosh (c+d x)}{256 d}+\frac{3}{256} x (4 a-b) \left(8 a^2-2 a b+b^2\right)+\frac{b \sinh (c+d x) \cosh ^9(c+d x) \left(a-(a-b) \tanh ^2(c+d x)\right)^2}{10 d}+\frac{b \sinh (c+d x) \cosh ^7(c+d x) \left(a (10 a-b)-5 (a-b) (2 a-b) \tanh ^2(c+d x)\right)}{80 d}",1,"(120*(4*a - b)*(8*a^2 - 2*a*b + b^2)*(c + d*x) + 20*(128*a^3 - 24*a^2*b + b^3)*Sinh[2*(c + d*x)] + 40*(8*a^3 + 12*a^2*b - 6*a*b^2 + b^3)*Sinh[4*(c + d*x)] - 10*b*(-16*a^2 + b^2)*Sinh[6*(c + d*x)] + 5*(6*a - b)*b^2*Sinh[8*(c + d*x)] + 2*b^3*Sinh[10*(c + d*x)])/(10240*d)","A",1
305,1,83,98,0.2404179,"\int \cosh ^3(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Cosh[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{315 a^3 \sinh (c+d x)+105 a^2 (a+3 b) \sinh ^3(c+d x)+45 b^2 (3 a+b) \sinh ^7(c+d x)+189 a b (a+b) \sinh ^5(c+d x)+35 b^3 \sinh ^9(c+d x)}{315 d}","\frac{a^3 \sinh (c+d x)}{d}+\frac{a^2 (a+3 b) \sinh ^3(c+d x)}{3 d}+\frac{b^2 (3 a+b) \sinh ^7(c+d x)}{7 d}+\frac{3 a b (a+b) \sinh ^5(c+d x)}{5 d}+\frac{b^3 \sinh ^9(c+d x)}{9 d}",1,"(315*a^3*Sinh[c + d*x] + 105*a^2*(a + 3*b)*Sinh[c + d*x]^3 + 189*a*b*(a + b)*Sinh[c + d*x]^5 + 45*b^2*(3*a + b)*Sinh[c + d*x]^7 + 35*b^3*Sinh[c + d*x]^9)/(315*d)","A",1
306,1,120,203,0.3384109,"\int \cosh ^2(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Cosh[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{48 \left(16 a^3-3 a b^2+b^3\right) \sinh (2 (c+d x))+24 b \left(12 a^2-6 a b+b^2\right) \sinh (4 (c+d x))+24 \left(64 a^3-48 a^2 b+24 a b^2-5 b^3\right) (c+d x)+16 b^2 (3 a-b) \sinh (6 (c+d x))+3 b^3 \sinh (8 (c+d x))}{3072 d}","\frac{b \left(88 a^2-68 a b+15 b^2\right) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}+\frac{\left(64 a^3-48 a^2 b+24 a b^2-5 b^3\right) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{1}{128} x \left(64 a^3-48 a^2 b+24 a b^2-5 b^3\right)+\frac{b \sinh (c+d x) \cosh ^7(c+d x) \left(a-(a-b) \tanh ^2(c+d x)\right)^2}{8 d}+\frac{b \sinh (c+d x) \cosh ^5(c+d x) \left(a (8 a-b)-(8 a-5 b) (a-b) \tanh ^2(c+d x)\right)}{48 d}",1,"(24*(64*a^3 - 48*a^2*b + 24*a*b^2 - 5*b^3)*(c + d*x) + 48*(16*a^3 - 3*a*b^2 + b^3)*Sinh[2*(c + d*x)] + 24*b*(12*a^2 - 6*a*b + b^2)*Sinh[4*(c + d*x)] + 16*(3*a - b)*b^2*Sinh[6*(c + d*x)] + 3*b^3*Sinh[8*(c + d*x)])/(3072*d)","A",1
307,1,59,67,0.1138468,"\int \cosh (c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Cosh[c + d*x]*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{a^3 \sinh (c+d x)+a^2 b \sinh ^3(c+d x)+\frac{3}{5} a b^2 \sinh ^5(c+d x)+\frac{1}{7} b^3 \sinh ^7(c+d x)}{d}","\frac{a^3 \sinh (c+d x)}{d}+\frac{a^2 b \sinh ^3(c+d x)}{d}+\frac{3 a b^2 \sinh ^5(c+d x)}{5 d}+\frac{b^3 \sinh ^7(c+d x)}{7 d}",1,"(a^3*Sinh[c + d*x] + a^2*b*Sinh[c + d*x]^3 + (3*a*b^2*Sinh[c + d*x]^5)/5 + (b^3*Sinh[c + d*x]^7)/7)/d","A",1
308,1,100,86,0.5489064,"\int \text{sech}(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\sinh (c+d x) \left(b \left(45 a^2+15 a b \left(\sinh ^2(c+d x)-3\right)+b^2 \left(3 \sinh ^4(c+d x)-5 \sinh ^2(c+d x)+15\right)\right)+\frac{15 (a-b)^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)}{\sqrt{-\sinh ^2(c+d x)}}\right)}{15 d}","\frac{b \left(3 a^2-3 a b+b^2\right) \sinh (c+d x)}{d}+\frac{b^2 (3 a-b) \sinh ^3(c+d x)}{3 d}+\frac{(a-b)^3 \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b^3 \sinh ^5(c+d x)}{5 d}",1,"(Sinh[c + d*x]*((15*(a - b)^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/Sqrt[-Sinh[c + d*x]^2] + b*(45*a^2 + 15*a*b*(-3 + Sinh[c + d*x]^2) + b^2*(15 - 5*Sinh[c + d*x]^2 + 3*Sinh[c + d*x]^4))))/(15*d)","A",1
309,1,78,92,0.5299995,"\int \text{sech}^2(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{12 b \left(8 a^2-12 a b+5 b^2\right) (c+d x)+8 b^2 (3 a-2 b) \sinh (2 (c+d x))+32 (a-b)^3 \tanh (c+d x)+b^3 \sinh (4 (c+d x))}{32 d}","\frac{3}{8} b x \left(8 a^2-12 a b+5 b^2\right)+\frac{3 b^2 (4 a-3 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{(a-b)^3 \tanh (c+d x)}{d}+\frac{b^3 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}",1,"(12*b*(8*a^2 - 12*a*b + 5*b^2)*(c + d*x) + 8*(3*a - 2*b)*b^2*Sinh[2*(c + d*x)] + b^3*Sinh[4*(c + d*x)] + 32*(a - b)^3*Tanh[c + d*x])/(32*d)","A",1
310,1,399,91,7.994743,"\int \text{sech}^3(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\text{csch}^5(c+d x) \left(-256 \sinh ^8(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right)+21 \left(a^3 \left(1473 \sinh ^4(c+d x)+16120 \sinh ^2(c+d x)+36015\right)+3 a^2 b \left(753 \sinh ^4(c+d x)+16120 \sinh ^2(c+d x)+36015\right) \sinh ^2(c+d x)+3 a b^2 \left(753 \sinh ^4(c+d x)+18280 \sinh ^2(c+d x)+36015\right) \sinh ^4(c+d x)+b^3 \left(753 \sinh ^4(c+d x)+17320 \sinh ^2(c+d x)+32415\right) \sinh ^6(c+d x)\right)-\frac{315 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \left(a^3 \left(-47 \sinh ^6(c+d x)+243 \sinh ^4(c+d x)+1875 \sinh ^2(c+d x)+2401\right)+3 a^2 b \left(\sinh ^6(c+d x)+195 \sinh ^4(c+d x)+1875 \sinh ^2(c+d x)+2401\right) \sinh ^2(c+d x)+3 a b^2 \left(\sinh ^6(c+d x)+243 \sinh ^4(c+d x)+2019 \sinh ^2(c+d x)+2401\right) \sinh ^4(c+d x)+b^3 \left(\sinh ^6(c+d x)+243 \sinh ^4(c+d x)+1875 \sinh ^2(c+d x)+2161\right) \sinh ^6(c+d x)\right)}{\sqrt{-\sinh ^2(c+d x)}}\right)}{30240 d}","\frac{b^2 (3 a-2 b) \sinh (c+d x)}{d}+\frac{(a+5 b) (a-b)^2 \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{(a-b)^3 \tanh (c+d x) \text{sech}(c+d x)}{2 d}+\frac{b^3 \sinh ^3(c+d x)}{3 d}",1,"(Csch[c + d*x]^5*(-256*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^8*(a + b*Sinh[c + d*x]^2)^3 + 21*(3*a^2*b*Sinh[c + d*x]^2*(36015 + 16120*Sinh[c + d*x]^2 + 753*Sinh[c + d*x]^4) + b^3*Sinh[c + d*x]^6*(32415 + 17320*Sinh[c + d*x]^2 + 753*Sinh[c + d*x]^4) + 3*a*b^2*Sinh[c + d*x]^4*(36015 + 18280*Sinh[c + d*x]^2 + 753*Sinh[c + d*x]^4) + a^3*(36015 + 16120*Sinh[c + d*x]^2 + 1473*Sinh[c + d*x]^4)) - (315*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(a^3*(2401 + 1875*Sinh[c + d*x]^2 + 243*Sinh[c + d*x]^4 - 47*Sinh[c + d*x]^6) + 3*a^2*b*Sinh[c + d*x]^2*(2401 + 1875*Sinh[c + d*x]^2 + 195*Sinh[c + d*x]^4 + Sinh[c + d*x]^6) + b^3*Sinh[c + d*x]^6*(2161 + 1875*Sinh[c + d*x]^2 + 243*Sinh[c + d*x]^4 + Sinh[c + d*x]^6) + 3*a*b^2*Sinh[c + d*x]^4*(2401 + 2019*Sinh[c + d*x]^2 + 243*Sinh[c + d*x]^4 + Sinh[c + d*x]^6)))/Sqrt[-Sinh[c + d*x]^2]))/(30240*d)","C",0
311,1,84,82,0.7637199,"\int \text{sech}^4(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{6 b^2 (6 a-5 b) (c+d x)+2 (a-b)^2 \tanh (c+d x) \text{sech}^2(c+d x) ((2 a+7 b) \cosh (2 (c+d x))+4 a+5 b)+3 b^3 \sinh (2 (c+d x))}{12 d}","\frac{1}{2} b^2 x (6 a-5 b)-\frac{(a-b)^3 \tanh ^3(c+d x)}{3 d}+\frac{(a-b)^2 (a+2 b) \tanh (c+d x)}{d}+\frac{b^3 \sinh (c+d x) \cosh (c+d x)}{2 d}",1,"(6*(6*a - 5*b)*b^2*(c + d*x) + 3*b^3*Sinh[2*(c + d*x)] + 2*(a - b)^2*(4*a + 5*b + (2*a + 7*b)*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*Tanh[c + d*x])/(12*d)","A",1
312,1,472,103,10.0674239,"\int \text{sech}^5(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2)^3,x]","-\frac{\text{csch}^5(c+d x) \left(256 \sinh ^8(c+d x) \left(a+b \sinh ^2(c+d x)\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+b \sinh ^2(c+d x)\right)^2 \left(7 a+5 b \sinh ^2(c+d x)\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(8226 \sinh ^4(c+d x)+140965 \sinh ^2(c+d x)+252105\right)+9 a^2 b \left(2131 \sinh ^4(c+d x)+41615 \sinh ^2(c+d x)+72030\right) \sinh ^2(c+d x)+15 a b^2 \left(1128 \sinh ^4(c+d x)+21529 \sinh ^2(c+d x)+36015\right) \sinh ^4(c+d x)+b^3 \left(4887 \sinh ^4(c+d x)+90805 \sinh ^2(c+d x)+149460\right) \sinh ^6(c+d x)\right)+\frac{315 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \left(a^3 \left(-62 \sinh ^6(c+d x)+2187 \sinh ^4(c+d x)+15000 \sinh ^2(c+d x)+16807\right)+9 a^2 b \left(3 \sinh ^6(c+d x)+640 \sinh ^4(c+d x)+4375 \sinh ^2(c+d x)+4802\right) \sinh ^2(c+d x)+3 a b^2 \left(8 \sinh ^6(c+d x)+1701 \sinh ^4(c+d x)+11178 \sinh ^2(c+d x)+12005\right) \sinh ^4(c+d x)+b^3 \left(7 \sinh ^6(c+d x)+1458 \sinh ^4(c+d x)+9375 \sinh ^2(c+d x)+9964\right) \sinh ^6(c+d x)\right)}{\sqrt{-\sinh ^2(c+d x)}}\right)}{60480 d}","\frac{3 (a-b) \left((a+b)^2+4 b^2\right) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{(a-b)^3 \tanh (c+d x) \text{sech}^3(c+d x)}{4 d}+\frac{3 (a-b)^2 (a+3 b) \tanh (c+d x) \text{sech}(c+d x)}{8 d}+\frac{b^3 \sinh (c+d x)}{d}",1,"-1/60480*(Csch[c + d*x]^5*(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*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*Sinh[c + d*x]^2)^2*(7*a + 5*b*Sinh[c + d*x]^2) - 21*(15*a*b^2*Sinh[c + d*x]^4*(36015 + 21529*Sinh[c + d*x]^2 + 1128*Sinh[c + d*x]^4) + 9*a^2*b*Sinh[c + d*x]^2*(72030 + 41615*Sinh[c + d*x]^2 + 2131*Sinh[c + d*x]^4) + b^3*Sinh[c + d*x]^6*(149460 + 90805*Sinh[c + d*x]^2 + 4887*Sinh[c + d*x]^4) + a^3*(252105 + 140965*Sinh[c + d*x]^2 + 8226*Sinh[c + d*x]^4)) + (315*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(a^3*(16807 + 15000*Sinh[c + d*x]^2 + 2187*Sinh[c + d*x]^4 - 62*Sinh[c + d*x]^6) + 9*a^2*b*Sinh[c + d*x]^2*(4802 + 4375*Sinh[c + d*x]^2 + 640*Sinh[c + d*x]^4 + 3*Sinh[c + d*x]^6) + b^3*Sinh[c + d*x]^6*(9964 + 9375*Sinh[c + d*x]^2 + 1458*Sinh[c + d*x]^4 + 7*Sinh[c + d*x]^6) + 3*a*b^2*Sinh[c + d*x]^4*(12005 + 11178*Sinh[c + d*x]^2 + 1701*Sinh[c + d*x]^4 + 8*Sinh[c + d*x]^6)))/Sqrt[-Sinh[c + d*x]^2]))/d","C",0
313,1,86,74,0.7712305,"\int \text{sech}^6(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]^6*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{(a-b) \tanh (c+d x) \left(\left(4 a^2+7 a b-11 b^2\right) \text{sech}^2(c+d x)+8 a^2+3 (a-b)^2 \text{sech}^4(c+d x)+14 a b+23 b^2\right)+15 b^3 (c+d x)}{15 d}","\frac{\left(a^3-b^3\right) \tanh (c+d x)}{d}+\frac{(a-b)^3 \tanh ^5(c+d x)}{5 d}-\frac{(a-b)^2 (2 a+b) \tanh ^3(c+d x)}{3 d}+b^3 x",1,"(15*b^3*(c + d*x) + (a - b)*(8*a^2 + 14*a*b + 23*b^2 + (4*a^2 + 7*a*b - 11*b^2)*Sech[c + d*x]^2 + 3*(a - b)^2*Sech[c + d*x]^4)*Tanh[c + d*x])/(15*d)","A",1
314,1,1192,154,14.0098876,"\int \text{sech}^7(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]^7*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\text{csch}^5(c+d x) \left(1024 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) \left(-\sinh ^2(c+d x)\right)^{3/2} \sinh ^{12}(c+d x)-48825 b^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^{12}(c+d x)+3072 a b^2 \, _7F_6\left(\frac{3}{2},2,2,2,2,2,2;1,1,1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \left(-\sinh ^2(c+d x)\right)^{3/2} \sinh ^{10}(c+d x)-9261945 b^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^{10}(c+d x)-174825 a b^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^{10}(c+d x)+2160711 b^3 \sqrt{-\sinh ^2(c+d x)} \sinh ^{10}(c+d x)+3072 a^2 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) \left(-\sinh ^2(c+d x)\right)^{3/2} \sinh ^8(c+d x)-56109375 b^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^8(c+d x)-33756345 a b^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^8(c+d x)-210735 a^2 b \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^8(c+d x)+36772890 b^3 \sqrt{-\sinh ^2(c+d x)} \sinh ^8(c+d x)+7808535 a b^2 \sqrt{-\sinh ^2(c+d x)} \sinh ^8(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) \left(-\sinh ^2(c+d x)\right)^{3/2} \sinh ^6(c+d x)+142065 a^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^6(c+d x)-58009455 b^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^6(c+d x)-207173295 a b^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^6(c+d x)-41427855 a^2 b \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^6(c+d x)+1536 \, _6F_5\left(\frac{3}{2},2,2,2,2,2;1,1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \left(-\sinh ^2(c+d x)\right)^{3/2} \left(b \sinh ^2(c+d x)+a\right)^2 \left(7 b \sinh ^2(c+d x)+9 a\right) \sinh ^6(c+d x)+256 \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{11}{2};-\sinh ^2(c+d x)\right) \left(-\sinh ^2(c+d x)\right)^{3/2} \left(175 b^3 \sinh ^6(c+d x)+621 a b^2 \sinh ^4(c+d x)+741 a^2 b \sinh ^2(c+d x)+295 a^3\right) \sinh ^6(c+d x)+58009455 b^3 \sqrt{-\sinh ^2(c+d x)} \sinh ^6(c+d x)+135323370 a b^2 \sqrt{-\sinh ^2(c+d x)} \sinh ^6(c+d x)+9514449 a^2 b \sqrt{-\sinh ^2(c+d x)} \sinh ^6(c+d x)-17069535 a^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^4(c+d x)-215549775 a b^2 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^4(c+d x)-260465625 a^2 b \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^4(c+d x)+4093425 a^3 \sqrt{-\sinh ^2(c+d x)} \sinh ^4(c+d x)+215549775 a b^2 \sqrt{-\sinh ^2(c+d x)} \sinh ^4(c+d x)+168951510 a^2 b \sqrt{-\sinh ^2(c+d x)} \sinh ^4(c+d x)-109265625 a^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^2(c+d x)-274542345 a^2 b \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right) \sinh ^2(c+d x)-70189350 a^3 \left(-\sinh ^2(c+d x)\right)^{3/2}-274542345 a^2 b \left(-\sinh ^2(c+d x)\right)^{3/2}-117228825 a^3 \tanh ^{-1}\left(\sqrt{-\sinh ^2(c+d x)}\right)+117228825 a^3 \sqrt{-\sinh ^2(c+d x)}\right)}{725760 d \sqrt{-\sinh ^2(c+d x)}}","\frac{(a+b) \left(5 a^2-2 a b+5 b^2\right) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac{(a-b) \left(15 a^2+14 a b+15 b^2\right) \tanh (c+d x) \text{sech}(c+d x)}{48 d}+\frac{5 \left(a^2-b^2\right) \tanh (c+d x) \text{sech}^3(c+d x) \left(a+b \sinh ^2(c+d x)\right)}{24 d}+\frac{(a-b) \tanh (c+d x) \text{sech}^5(c+d x) \left(a+b \sinh ^2(c+d x)\right)^2}{6 d}",1,"(Csch[c + d*x]^5*(-117228825*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]] - 109265625*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^2 - 274542345*a^2*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^2 - 17069535*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 - 260465625*a^2*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 - 215549775*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 + 142065*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 - 41427855*a^2*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 - 207173295*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 - 58009455*b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 - 210735*a^2*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8 - 33756345*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8 - 56109375*b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8 - 174825*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^10 - 9261945*b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^10 - 48825*b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^12 + 117228825*a^3*Sqrt[-Sinh[c + d*x]^2] + 4093425*a^3*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 168951510*a^2*b*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 215549775*a*b^2*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 9514449*a^2*b*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 135323370*a*b^2*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 58009455*b^3*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 7808535*a*b^2*Sinh[c + d*x]^8*Sqrt[-Sinh[c + d*x]^2] + 36772890*b^3*Sinh[c + d*x]^8*Sqrt[-Sinh[c + d*x]^2] + 2160711*b^3*Sinh[c + d*x]^10*Sqrt[-Sinh[c + d*x]^2] - 70189350*a^3*(-Sinh[c + d*x]^2)^(3/2) - 274542345*a^2*b*(-Sinh[c + d*x]^2)^(3/2) + 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]^6*(-Sinh[c + d*x]^2)^(3/2) + 3072*a^2*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]^8*(-Sinh[c + d*x]^2)^(3/2) + 3072*a*b^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^10*(-Sinh[c + d*x]^2)^(3/2) + 1024*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]^12*(-Sinh[c + d*x]^2)^(3/2) + 1536*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(-Sinh[c + d*x]^2)^(3/2)*(a + b*Sinh[c + d*x]^2)^2*(9*a + 7*b*Sinh[c + d*x]^2) + 256*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(-Sinh[c + d*x]^2)^(3/2)*(295*a^3 + 741*a^2*b*Sinh[c + d*x]^2 + 621*a*b^2*Sinh[c + d*x]^4 + 175*b^3*Sinh[c + d*x]^6)))/(725760*d*Sqrt[-Sinh[c + d*x]^2])","C",0
315,1,163,80,0.5973018,"\int \text{sech}^8(c+d x) \left(a+b \sinh ^2(c+d x)\right)^3 \, dx","Integrate[Sech[c + d*x]^8*(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\tanh (c+d x) \text{sech}^6(c+d x) \left(16 a^3 \cosh (6 (c+d x))+512 a^3+8 a^2 b \cosh (6 (c+d x))-304 a^2 b+\left(464 a^3+232 a^2 b-246 a b^2+75 b^3\right) \cosh (2 (c+d x))+2 \left(64 a^3+32 a^2 b+24 a b^2-15 b^3\right) \cosh (4 (c+d x))+6 a b^2 \cosh (6 (c+d x))+192 a b^2+5 b^3 \cosh (6 (c+d x))-50 b^3\right)}{1120 d}","\frac{a^3 \tanh (c+d x)}{d}-\frac{a^2 (a-b) \tanh ^3(c+d x)}{d}-\frac{(a-b)^3 \tanh ^7(c+d x)}{7 d}+\frac{3 a (a-b)^2 \tanh ^5(c+d x)}{5 d}",1,"((512*a^3 - 304*a^2*b + 192*a*b^2 - 50*b^3 + (464*a^3 + 232*a^2*b - 246*a*b^2 + 75*b^3)*Cosh[2*(c + d*x)] + 2*(64*a^3 + 32*a^2*b + 24*a*b^2 - 15*b^3)*Cosh[4*(c + d*x)] + 16*a^3*Cosh[6*(c + d*x)] + 8*a^2*b*Cosh[6*(c + d*x)] + 6*a*b^2*Cosh[6*(c + d*x)] + 5*b^3*Cosh[6*(c + d*x)])*Sech[c + d*x]^6*Tanh[c + d*x])/(1120*d)","B",1
316,1,117,108,0.4878224,"\int \frac{\cosh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Cosh[c + d*x]^7/(a + b*Sinh[c + d*x]^2),x]","\frac{30 \sqrt{b} \left(8 a^2-22 a b+19 b^2\right) \sinh (c+d x)+5 b^{3/2} (9 b-4 a) \sinh (3 (c+d x))+\frac{3 \left(\sqrt{a} b^{5/2} \sinh (5 (c+d x))+80 (a-b)^3 \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)\right)}{\sqrt{a}}}{240 b^{7/2} d}","\frac{\left(a^2-3 a b+3 b^2\right) \sinh (c+d x)}{b^3 d}-\frac{(a-b)^3 \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2} d}-\frac{(a-3 b) \sinh ^3(c+d x)}{3 b^2 d}+\frac{\sinh ^5(c+d x)}{5 b d}",1,"(30*Sqrt[b]*(8*a^2 - 22*a*b + 19*b^2)*Sinh[c + d*x] + 5*b^(3/2)*(-4*a + 9*b)*Sinh[3*(c + d*x)] + (3*(80*(a - b)^3*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + Sqrt[a]*b^(5/2)*Sinh[5*(c + d*x)]))/Sqrt[a])/(240*b^(7/2)*d)","A",1
317,1,106,121,0.3352861,"\int \frac{\cosh ^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Cosh[c + d*x]^6/(a + b*Sinh[c + d*x]^2),x]","\frac{\sqrt{a} \left(4 \left(8 a^2-20 a b+15 b^2\right) (c+d x)-8 b (a-2 b) \sinh (2 (c+d x))+b^2 \sinh (4 (c+d x))\right)-32 (a-b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{32 \sqrt{a} b^3 d}","\frac{x \left(8 a^2-20 a b+15 b^2\right)}{8 b^3}-\frac{(a-b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^3 d}-\frac{(4 a-7 b) \sinh (c+d x) \cosh (c+d x)}{8 b^2 d}+\frac{\sinh (c+d x) \cosh ^3(c+d x)}{4 b d}",1,"(-32*(a - b)^(5/2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]] + Sqrt[a]*(4*(8*a^2 - 20*a*b + 15*b^2)*(c + d*x) - 8*(a - 2*b)*b*Sinh[2*(c + d*x)] + b^2*Sinh[4*(c + d*x)]))/(32*Sqrt[a]*b^3*d)","A",1
318,1,79,77,0.2519024,"\int \frac{\cosh ^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2),x]","\frac{3 \sqrt{b} (7 b-4 a) \sinh (c+d x)-\frac{12 (a-b)^2 \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)}{\sqrt{a}}+b^{3/2} \sinh (3 (c+d x))}{12 b^{5/2} d}","\frac{(a-b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{5/2} d}-\frac{(a-2 b) \sinh (c+d x)}{b^2 d}+\frac{\sinh ^3(c+d x)}{3 b d}",1,"((-12*(a - b)^2*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]])/Sqrt[a] + 3*Sqrt[b]*(-4*a + 7*b)*Sinh[c + d*x] + b^(3/2)*Sinh[3*(c + d*x)])/(12*b^(5/2)*d)","A",1
319,1,80,81,0.1589088,"\int \frac{\cosh ^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2),x]","\frac{\sqrt{a} (b \sinh (2 (c+d x))-2 (2 a-3 b) (c+d x))+4 (a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{4 \sqrt{a} b^2 d}","\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^2 d}-\frac{x (2 a-3 b)}{2 b^2}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 b d}",1,"(4*(a - b)^(3/2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]] + Sqrt[a]*(-2*(2*a - 3*b)*(c + d*x) + b*Sinh[2*(c + d*x)]))/(4*Sqrt[a]*b^2*d)","A",1
320,1,50,52,0.0458941,"\int \frac{\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^2),x]","\frac{\frac{\sinh (c+d x)}{b}-\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}}{d}","\frac{\sinh (c+d x)}{b d}-\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2} d}",1,"(-(((a - b)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(3/2))) + Sinh[c + d*x]/b)/d","A",1
321,1,50,50,0.086957,"\int \frac{\cosh ^2(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Cosh[c + d*x]^2/(a + b*Sinh[c + d*x]^2),x]","\frac{-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a}}+c+d x}{b d}","\frac{x}{b}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b d}",1,"(c + d*x - (Sqrt[a - b]*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/Sqrt[a])/(b*d)","A",1
322,1,32,32,0.0101619,"\int \frac{\cosh (c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Cosh[c + d*x]/(a + b*Sinh[c + d*x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d}",1,"ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*d)","A",1
323,1,54,59,0.1328576,"\int \frac{\text{sech}(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Sech[c + d*x]/(a + b*Sinh[c + d*x]^2),x]","\frac{\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)}{\sqrt{a}}+2 \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{a d-b d}","\frac{\tan ^{-1}(\sinh (c+d x))}{d (a-b)}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)}",1,"((Sqrt[b]*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]])/Sqrt[a] + 2*ArcTan[Tanh[(c + d*x)/2]])/(a*d - b*d)","A",1
324,1,60,60,0.1621454,"\int \frac{\text{sech}^2(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Sech[c + d*x]^2/(a + b*Sinh[c + d*x]^2),x]","\frac{\tanh (c+d x)}{d (a-b)}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^{3/2}}","\frac{\tanh (c+d x)}{d (a-b)}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^{3/2}}",1,"-((b*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(3/2)*d)) + Tanh[c + d*x]/((a - b)*d)","A",1
325,1,91,92,0.2390708,"\int \frac{\text{sech}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Sech[c + d*x]^3/(a + b*Sinh[c + d*x]^2),x]","\frac{-2 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)+2 \sqrt{a} (a-3 b) \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+\sqrt{a} (a-b) \tanh (c+d x) \text{sech}(c+d x)}{2 \sqrt{a} d (a-b)^2}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^2}+\frac{(a-3 b) \tan ^{-1}(\sinh (c+d x))}{2 d (a-b)^2}+\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 d (a-b)}",1,"(-2*b^(3/2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + 2*Sqrt[a]*(a - 3*b)*ArcTan[Tanh[(c + d*x)/2]] + Sqrt[a]*(a - b)*Sech[c + d*x]*Tanh[c + d*x])/(2*Sqrt[a]*(a - b)^2*d)","A",1
326,1,84,88,0.5351498,"\int \frac{\text{sech}^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Sech[c + d*x]^4/(a + b*Sinh[c + d*x]^2),x]","\frac{\frac{3 b^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} (a-b)^{5/2}}+\frac{\tanh (c+d x) \left((a-b) \text{sech}^2(c+d x)+2 a-5 b\right)}{(a-b)^2}}{3 d}","\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^{5/2}}-\frac{\tanh ^3(c+d x)}{3 d (a-b)}+\frac{(a-2 b) \tanh (c+d x)}{d (a-b)^2}",1,"((3*b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(5/2)) + ((2*a - 5*b + (a - b)*Sech[c + d*x]^2)*Tanh[c + d*x])/(a - b)^2)/(3*d)","A",1
327,1,139,138,0.5462508,"\int \frac{\text{sech}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Sech[c + d*x]^5/(a + b*Sinh[c + d*x]^2),x]","\frac{2 \sqrt{a} \left(3 a^2-10 a b+15 b^2\right) \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+\sqrt{a} \left(3 a^2-10 a b+7 b^2\right) \tanh (c+d x) \text{sech}(c+d x)+8 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)+2 \sqrt{a} (a-b)^2 \tanh (c+d x) \text{sech}^3(c+d x)}{8 \sqrt{a} d (a-b)^3}","\frac{\left(3 a^2-10 a b+15 b^2\right) \tan ^{-1}(\sinh (c+d x))}{8 d (a-b)^3}-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^3}+\frac{\tanh (c+d x) \text{sech}^3(c+d x)}{4 d (a-b)}+\frac{(3 a-7 b) \tanh (c+d x) \text{sech}(c+d x)}{8 d (a-b)^2}",1,"(8*b^(5/2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + 2*Sqrt[a]*(3*a^2 - 10*a*b + 15*b^2)*ArcTan[Tanh[(c + d*x)/2]] + Sqrt[a]*(3*a^2 - 10*a*b + 7*b^2)*Sech[c + d*x]*Tanh[c + d*x] + 2*Sqrt[a]*(a - b)^2*Sech[c + d*x]^3*Tanh[c + d*x])/(8*Sqrt[a]*(a - b)^3*d)","A",1
328,1,119,126,0.931113,"\int \frac{\text{sech}^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx","Integrate[Sech[c + d*x]^6/(a + b*Sinh[c + d*x]^2),x]","\frac{\frac{\tanh (c+d x) \left(\left(4 a^2-13 a b+9 b^2\right) \text{sech}^2(c+d x)+8 a^2+3 (a-b)^2 \text{sech}^4(c+d x)-26 a b+33 b^2\right)}{(a-b)^3}-\frac{15 b^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} (a-b)^{7/2}}}{15 d}","\frac{\left(a^2-3 a b+3 b^2\right) \tanh (c+d x)}{d (a-b)^3}-\frac{b^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^{7/2}}+\frac{\tanh ^5(c+d x)}{5 d (a-b)}-\frac{(2 a-3 b) \tanh ^3(c+d x)}{3 d (a-b)^2}",1,"((-15*b^3*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(7/2)) + ((8*a^2 - 26*a*b + 33*b^2 + (4*a^2 - 13*a*b + 9*b^2)*Sech[c + d*x]^2 + 3*(a - b)^2*Sech[c + d*x]^4)*Tanh[c + d*x])/(a - b)^3)/(15*d)","A",1
329,1,118,158,0.5955161,"\int \frac{\cosh ^6(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Cosh[c + d*x]^6/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\frac{2 (4 a+b) (a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{3/2}}+2 (5 b-4 a) (c+d x)+\frac{2 b (a-b)^2 \sinh (2 (c+d x))}{a (2 a+b \cosh (2 (c+d x))-b)}+b \sinh (2 (c+d x))}{4 b^3 d}","\frac{(4 a+b) (a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^3 d}-\frac{x (4 a-5 b)}{2 b^3}+\frac{(2 a-b) (a-b) \tanh (c+d x)}{2 a b^2 d \left(a-(a-b) \tanh ^2(c+d x)\right)}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 b d \left(a-(a-b) \tanh ^2(c+d x)\right)}",1,"(2*(-4*a + 5*b)*(c + d*x) + (2*(a - b)^(3/2)*(4*a + b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/a^(3/2) + b*Sinh[2*(c + d*x)] + (2*(a - b)^2*b*Sinh[2*(c + d*x)])/(a*(2*a - b + b*Cosh[2*(c + d*x)])))/(4*b^3*d)","A",1
330,1,106,104,0.2998675,"\int \frac{\cosh ^5(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{-\frac{\left(-3 a^2+2 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)}{a^{3/2}}+\frac{2 \sqrt{b} (a-b)^2 \sinh (c+d x)}{a (2 a+b \cosh (2 (c+d x))-b)}+2 \sqrt{b} \sinh (c+d x)}{2 b^{5/2} d}","-\frac{\left(3 a^2-2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{5/2} d}+\frac{(a-b)^2 \sinh (c+d x)}{2 a b^2 d \left(a+b \sinh ^2(c+d x)\right)}+\frac{\sinh (c+d x)}{b^2 d}",1,"(-(((-3*a^2 + 2*a*b + b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]])/a^(3/2)) + 2*Sqrt[b]*Sinh[c + d*x] + (2*(a - b)^2*Sqrt[b]*Sinh[c + d*x])/(a*(2*a - b + b*Cosh[2*(c + d*x)])))/(2*b^(5/2)*d)","A",1
331,1,108,100,0.6172217,"\int \frac{\cosh ^4(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{-\frac{\left(2 a^2-a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{3/2} \sqrt{a-b}}+\frac{b (b-a) \sinh (2 (c+d x))}{a (2 a+b \cosh (2 (c+d x))-b)}+2 (c+d x)}{2 b^2 d}","-\frac{\sqrt{a-b} (2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^2 d}-\frac{(a-b) \tanh (c+d x)}{2 a b d \left(a-(a-b) \tanh ^2(c+d x)\right)}+\frac{x}{b^2}",1,"(2*(c + d*x) - ((2*a^2 - a*b - b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*Sqrt[a - b]) + (b*(-a + b)*Sinh[2*(c + d*x)])/(a*(2*a - b + b*Cosh[2*(c + d*x)])))/(2*b^2*d)","A",1
332,1,75,77,0.3502741,"\int \frac{\cosh ^3(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2}}-\frac{(a-b) \sinh (c+d x)}{2 a b \left(a+b \sinh ^2(c+d x)\right)}}{d}","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2} d}-\frac{(a-b) \sinh (c+d x)}{2 a b d \left(a+b \sinh ^2(c+d x)\right)}",1,"(((a + b)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^(3/2)) - ((a - b)*Sinh[c + d*x])/(2*a*b*(a + b*Sinh[c + d*x]^2)))/d","A",1
333,1,78,79,0.2084802,"\int \frac{\cosh ^2(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Cosh[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a-b}}+\frac{\sqrt{a} \sinh (2 (c+d x))}{2 a+b \cosh (2 (c+d x))-b}}{2 a^{3/2} d}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d \sqrt{a-b}}+\frac{\tanh (c+d x)}{2 a d \left(a-(a-b) \tanh ^2(c+d x)\right)}",1,"(ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]/Sqrt[a - b] + (Sqrt[a]*Sinh[2*(c + d*x)])/(2*a - b + b*Cosh[2*(c + d*x)]))/(2*a^(3/2)*d)","A",1
334,1,64,66,0.0446287,"\int \frac{\cosh (c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Cosh[c + d*x]/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b}}+\frac{\sinh (c+d x)}{2 a \left(a+b \sinh ^2(c+d x)\right)}}{d}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b} d}+\frac{\sinh (c+d x)}{2 a d \left(a+b \sinh ^2(c+d x)\right)}",1,"(ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]) + Sinh[c + d*x]/(2*a*(a + b*Sinh[c + d*x]^2)))/d","A",1
335,1,174,106,0.409023,"\int \frac{\text{sech}(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\cosh (2 (c+d x)) \left(4 a^{3/2} b \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-b^{3/2} (b-3 a) \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)\right)+(2 a-b) \left(4 a^{3/2} \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-\sqrt{b} (b-3 a) \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)\right)-2 \sqrt{a} b (a-b) \sinh (c+d x)}{2 a^{3/2} d (a-b)^2 (2 a+b \cosh (2 (c+d x))-b)}","-\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^2}-\frac{b \sinh (c+d x)}{2 a d (a-b) \left(a+b \sinh ^2(c+d x)\right)}+\frac{\tan ^{-1}(\sinh (c+d x))}{d (a-b)^2}",1,"((2*a - b)*(-(Sqrt[b]*(-3*a + b)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]]) + 4*a^(3/2)*ArcTan[Tanh[(c + d*x)/2]]) + (-(b^(3/2)*(-3*a + b)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]]) + 4*a^(3/2)*b*ArcTan[Tanh[(c + d*x)/2]])*Cosh[2*(c + d*x)] - 2*Sqrt[a]*(a - b)*b*Sinh[c + d*x])/(2*a^(3/2)*(a - b)^2*d*(2*a - b + b*Cosh[2*(c + d*x)]))","A",1
336,1,105,114,0.9220234,"\int \frac{\text{sech}^2(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\frac{\frac{b^2 \sinh (2 (c+d x))}{a (2 a+b \cosh (2 (c+d x))-b)}+2 \tanh (c+d x)}{(a-b)^2}-\frac{b (4 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{3/2} (a-b)^{5/2}}}{2 d}","-\frac{b (4 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^{5/2}}+\frac{b^2 \tanh (c+d x)}{2 a d (a-b)^2 \left(a-(a-b) \tanh ^2(c+d x)\right)}+\frac{\tanh (c+d x)}{d (a-b)^2}",1,"(-(((4*a - b)*b*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*(a - b)^(5/2))) + ((b^2*Sinh[2*(c + d*x)])/(a*(2*a - b + b*Cosh[2*(c + d*x)])) + 2*Tanh[c + d*x])/(a - b)^2)/(2*d)","A",1
337,1,230,157,0.9303772,"\int \frac{\text{sech}^3(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{(2 a-b) \left(2 a^{3/2} (a-5 b) \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+a^{3/2} (a-b) \tanh (c+d x) \text{sech}(c+d x)+b^{3/2} (b-5 a) \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)\right)+b \cosh (2 (c+d x)) \left(2 a^{3/2} (a-5 b) \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+a^{3/2} (a-b) \tanh (c+d x) \text{sech}(c+d x)+b^{3/2} (b-5 a) \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)\right)+2 \sqrt{a} b^2 (a-b) \sinh (c+d x)}{2 a^{3/2} d (a-b)^3 (2 a+b \cosh (2 (c+d x))-b)}","\frac{b^{3/2} (5 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^3}+\frac{b (a+b) \sinh (c+d x)}{2 a d (a-b)^2 \left(a+b \sinh ^2(c+d x)\right)}+\frac{(a-5 b) \tan ^{-1}(\sinh (c+d x))}{2 d (a-b)^3}+\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 d (a-b) \left(a+b \sinh ^2(c+d x)\right)}",1,"(2*Sqrt[a]*(a - b)*b^2*Sinh[c + d*x] + (2*a - b)*(b^(3/2)*(-5*a + b)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + 2*a^(3/2)*(a - 5*b)*ArcTan[Tanh[(c + d*x)/2]] + a^(3/2)*(a - b)*Sech[c + d*x]*Tanh[c + d*x]) + b*Cosh[2*(c + d*x)]*(b^(3/2)*(-5*a + b)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + 2*a^(3/2)*(a - 5*b)*ArcTan[Tanh[(c + d*x)/2]] + a^(3/2)*(a - b)*Sech[c + d*x]*Tanh[c + d*x]))/(2*a^(3/2)*(a - b)^3*d*(2*a - b + b*Cosh[2*(c + d*x)]))","A",1
338,1,130,143,1.9806748,"\int \frac{\text{sech}^4(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2} \, dx","Integrate[Sech[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2,x]","\frac{\frac{3 b^2 (6 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{3/2} (a-b)^{7/2}}+\frac{2 \tanh (c+d x) \left((a-b) \text{sech}^2(c+d x)+2 (a-4 b)\right)-\frac{3 b^3 \sinh (2 (c+d x))}{a (2 a+b \cosh (2 (c+d x))-b)}}{(a-b)^3}}{6 d}","\frac{b^2 (6 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^{7/2}}-\frac{b^3 \tanh (c+d x)}{2 a d (a-b)^3 \left(a-(a-b) \tanh ^2(c+d x)\right)}-\frac{\tanh ^3(c+d x)}{3 d (a-b)^2}+\frac{(a-3 b) \tanh (c+d x)}{d (a-b)^3}",1,"((3*(6*a - b)*b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*(a - b)^(7/2)) + ((-3*b^3*Sinh[2*(c + d*x)])/(a*(2*a - b + b*Cosh[2*(c + d*x)])) + 2*(2*(a - 4*b) + (a - b)*Sech[c + d*x]^2)*Tanh[c + d*x])/(a - b)^3)/(6*d)","A",1
339,1,164,160,1.4226451,"\int \frac{\cosh ^6(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Cosh[c + d*x]^6/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\frac{3 b \left(-2 a^2+a b+b^2\right) \sinh (2 (c+d x))}{a^2 (2 a+b \cosh (2 (c+d x))-b)}-\frac{\left(8 a^3-4 a^2 b-a b^2-3 b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{5/2} \sqrt{a-b}}+\frac{4 b (a-b)^2 \sinh (2 (c+d x))}{a (2 a+b \cosh (2 (c+d x))-b)^2}+8 (c+d x)}{8 b^3 d}","-\frac{(a-b) (4 a+3 b) \tanh (c+d x)}{8 a^2 b^2 d \left(a-(a-b) \tanh ^2(c+d x)\right)}-\frac{\sqrt{a-b} \left(8 a^2+4 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} b^3 d}-\frac{(a-b) \tanh (c+d x)}{4 a b d \left(a-(a-b) \tanh ^2(c+d x)\right)^2}+\frac{x}{b^3}",1,"(8*(c + d*x) - ((8*a^3 - 4*a^2*b - a*b^2 - 3*b^3)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(5/2)*Sqrt[a - b]) + (4*(a - b)^2*b*Sinh[2*(c + d*x)])/(a*(2*a - b + b*Cosh[2*(c + d*x)])^2) + (3*b*(-2*a^2 + a*b + b^2)*Sinh[2*(c + d*x)])/(a^2*(2*a - b + b*Cosh[2*(c + d*x)])))/(8*b^3*d)","A",1
340,1,149,133,0.3708034,"\int \frac{\cosh ^5(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\frac{8 a^{3/2} \sqrt{b} (a-b)^2 \sinh (c+d x)}{(2 a+b \cosh (2 (c+d x))-b)^2}-\frac{2 \sqrt{a} \sqrt{b} \left(5 a^2-2 a b-3 b^2\right) \sinh (c+d x)}{2 a+b \cosh (2 (c+d x))-b}-\left(3 a^2+2 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)}{8 a^{5/2} b^{5/2} d}","\frac{3 \left(\frac{1}{a^2}-\frac{1}{b^2}\right) \sinh (c+d x)}{8 d \left(a+b \sinh ^2(c+d x)\right)}+\frac{\left(3 a^2+2 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} b^{5/2} d}-\frac{(a-b) \sinh (c+d x) \cosh ^2(c+d x)}{4 a b d \left(a+b \sinh ^2(c+d x)\right)^2}",1,"(-((3*a^2 + 2*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]]) + (8*a^(3/2)*(a - b)^2*Sqrt[b]*Sinh[c + d*x])/(2*a - b + b*Cosh[2*(c + d*x)])^2 - (2*Sqrt[a]*Sqrt[b]*(5*a^2 - 2*a*b - 3*b^2)*Sinh[c + d*x])/(2*a - b + b*Cosh[2*(c + d*x)]))/(8*a^(5/2)*b^(5/2)*d)","A",1
341,1,102,114,0.6916091,"\int \frac{\cosh ^4(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{\sqrt{a-b}}+\frac{\sqrt{a} \sinh (2 (c+d x)) ((2 a+3 b) \cosh (2 (c+d x))+8 a-3 b)}{(2 a+b \cosh (2 (c+d x))-b)^2}}{8 a^{5/2} d}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d \sqrt{a-b}}+\frac{3 \tanh (c+d x)}{8 a^2 d \left(a-(a-b) \tanh ^2(c+d x)\right)}+\frac{\tanh (c+d x)}{4 a d \left(a-(a-b) \tanh ^2(c+d x)\right)^2}",1,"((3*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/Sqrt[a - b] + (Sqrt[a]*(8*a - 3*b + (2*a + 3*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(2*a - b + b*Cosh[2*(c + d*x)])^2)/(8*a^(5/2)*d)","A",1
342,1,114,117,0.6843223,"\int \frac{\cosh ^3(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{(a+3 b) \left(\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} \sqrt{b}}+\frac{\sinh (c+d x) \left(5 a+3 b \sinh ^2(c+d x)\right)}{8 a^2 \left(a+b \sinh ^2(c+d x)\right)^2}\right)-\frac{\sinh (c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^2}}{3 b d}","\frac{(a+3 b) \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} b^{3/2} d}+\frac{(a+3 b) \sinh (c+d x)}{8 a^2 b d \left(a+b \sinh ^2(c+d x)\right)}-\frac{(a-b) \sinh (c+d x)}{4 a b d \left(a+b \sinh ^2(c+d x)\right)^2}",1,"(-(Sinh[c + d*x]/(a + b*Sinh[c + d*x]^2)^2) + (a + 3*b)*((3*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]) + (Sinh[c + d*x]*(5*a + 3*b*Sinh[c + d*x]^2))/(8*a^2*(a + b*Sinh[c + d*x]^2)^2)))/(3*b*d)","A",1
343,1,124,143,0.9221655,"\int \frac{\cosh ^2(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Cosh[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\frac{\sqrt{a} \sinh (2 (c+d x)) \left(8 a^2+b (2 a-3 b) \cosh (2 (c+d x))-12 a b+3 b^2\right)}{(a-b) (2 a+b \cosh (2 (c+d x))-b)^2}+\frac{(4 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{(a-b)^{3/2}}}{8 a^{5/2} d}","\frac{(4 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a-b)^{3/2}}+\frac{(4 a-3 b) \tanh (c+d x)}{8 a^2 d (a-b) \left(a-(a-b) \tanh ^2(c+d x)\right)}-\frac{b \tanh (c+d x)}{4 a d (a-b) \left(a-(a-b) \tanh ^2(c+d x)\right)^2}",1,"(((4*a - 3*b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a - b)^(3/2) + (Sqrt[a]*(8*a^2 - 12*a*b + 3*b^2 + (2*a - 3*b)*b*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a - b)*(2*a - b + b*Cosh[2*(c + d*x)])^2))/(8*a^(5/2)*d)","A",1
344,1,79,96,0.1796374,"\int \frac{\cosh (c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Cosh[c + d*x]/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\frac{\sqrt{a} \sinh (c+d x) \left(5 a+3 b \sinh ^2(c+d x)\right)}{\left(a+b \sinh ^2(c+d x)\right)^2}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{\sqrt{b}}}{8 a^{5/2} d}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} \sqrt{b} d}+\frac{3 \sinh (c+d x)}{8 a^2 d \left(a+b \sinh ^2(c+d x)\right)}+\frac{\sinh (c+d x)}{4 a d \left(a+b \sinh ^2(c+d x)\right)^2}",1,"((3*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/Sqrt[b] + (Sqrt[a]*Sinh[c + d*x]*(5*a + 3*b*Sinh[c + d*x]^2))/(a + b*Sinh[c + d*x]^2)^2)/(8*a^(5/2)*d)","A",1
345,1,321,159,0.7809925,"\int \frac{\text{sech}(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{(b-2 a)^2 \left(16 a^{5/2} \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)\right)-2 b \cosh (2 (c+d x)) \left(\sqrt{a} b \left(7 a^2-10 a b+3 b^2\right) \sinh (c+d x)-(2 a-b) \left(16 a^{5/2} \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)\right)\right)+\cosh ^2(2 (c+d x)) \left(16 a^{5/2} b^2 \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)+b^{5/2} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)\right)-2 \sqrt{a} b \left(18 a^3-35 a^2 b+20 a b^2-3 b^3\right) \sinh (c+d x)}{8 a^{5/2} d (a-b)^3 (2 a+b \cosh (2 (c+d x))-b)^2}","-\frac{b (7 a-3 b) \sinh (c+d x)}{8 a^2 d (a-b)^2 \left(a+b \sinh ^2(c+d x)\right)}-\frac{\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a-b)^3}-\frac{b \sinh (c+d x)}{4 a d (a-b) \left(a+b \sinh ^2(c+d x)\right)^2}+\frac{\tan ^{-1}(\sinh (c+d x))}{d (a-b)^3}",1,"((-2*a + b)^2*(Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + 16*a^(5/2)*ArcTan[Tanh[(c + d*x)/2]]) + (b^(5/2)*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + 16*a^(5/2)*b^2*ArcTan[Tanh[(c + d*x)/2]])*Cosh[2*(c + d*x)]^2 - 2*Sqrt[a]*b*(18*a^3 - 35*a^2*b + 20*a*b^2 - 3*b^3)*Sinh[c + d*x] - 2*b*Cosh[2*(c + d*x)]*(-((2*a - b)*(Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + 16*a^(5/2)*ArcTan[Tanh[(c + d*x)/2]])) + Sqrt[a]*b*(7*a^2 - 10*a*b + 3*b^2)*Sinh[c + d*x]))/(8*a^(5/2)*(a - b)^3*d*(2*a - b + b*Cosh[2*(c + d*x)])^2)","B",1
346,1,165,172,1.1412461,"\int \frac{\text{sech}^2(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\frac{b^2 (10 a-3 b) \sinh (2 (c+d x))}{a^2 (a-b)^3 (2 a+b \cosh (2 (c+d x))-b)}-\frac{3 b \left(8 a^2-4 a b+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{5/2} (a-b)^{7/2}}+\frac{4 b^2 \sinh (2 (c+d x))}{a (a-b)^2 (2 a+b \cosh (2 (c+d x))-b)^2}+\frac{8 \tanh (c+d x)}{(a-b)^3}}{8 d}","\frac{3 b^2 (4 a-b) \tanh (c+d x)}{8 a^2 d (a-b)^3 \left(a-(a-b) \tanh ^2(c+d x)\right)}-\frac{3 b \left(8 a^2-4 a b+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a-b)^{7/2}}-\frac{b^3 \tanh (c+d x)}{4 a d (a-b)^3 \left(a-(a-b) \tanh ^2(c+d x)\right)^2}+\frac{\tanh (c+d x)}{d (a-b)^3}",1,"((-3*b*(8*a^2 - 4*a*b + b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(5/2)*(a - b)^(7/2)) + (4*b^2*Sinh[2*(c + d*x)])/(a*(a - b)^2*(2*a - b + b*Cosh[2*(c + d*x)])^2) + ((10*a - 3*b)*b^2*Sinh[2*(c + d*x)])/(a^2*(a - b)^3*(2*a - b + b*Cosh[2*(c + d*x)])) + (8*Tanh[c + d*x])/(a - b)^3)/(8*d)","A",1
347,1,222,217,1.8809126,"\int \frac{\text{sech}^3(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{-\frac{3 b^{7/2} \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)}{a^{5/2}}+\frac{14 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)}{a^{3/2}}+\frac{2 b^2 (a-b) \sinh (c+d x) \left(26 a^2+b (11 a-3 b) \cosh (2 (c+d x))-21 a b+3 b^2\right)}{a^2 (2 a+b \cosh (2 (c+d x))-b)^2}-\frac{35 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right)}{\sqrt{a}}+4 (a-b) \tanh (c+d x) \text{sech}(c+d x)+8 a \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)-56 b \tan ^{-1}\left(\tanh \left(\frac{1}{2} (c+d x)\right)\right)}{8 d (a-b)^4}","\frac{b (4 a-b) (a+3 b) \sinh (c+d x)}{8 a^2 d (a-b)^3 \left(a+b \sinh ^2(c+d x)\right)}+\frac{b^{3/2} \left(35 a^2-14 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a-b)^4}+\frac{b (2 a+b) \sinh (c+d x)}{4 a d (a-b)^2 \left(a+b \sinh ^2(c+d x)\right)^2}+\frac{(a-7 b) \tan ^{-1}(\sinh (c+d x))}{2 d (a-b)^4}+\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 d (a-b) \left(a+b \sinh ^2(c+d x)\right)^2}",1,"((-35*b^(3/2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]])/Sqrt[a] + (14*b^(5/2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]])/a^(3/2) - (3*b^(7/2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]])/a^(5/2) + 8*a*ArcTan[Tanh[(c + d*x)/2]] - 56*b*ArcTan[Tanh[(c + d*x)/2]] + (2*(a - b)*b^2*(26*a^2 - 21*a*b + 3*b^2 + (11*a - 3*b)*b*Cosh[2*(c + d*x)])*Sinh[c + d*x])/(a^2*(2*a - b + b*Cosh[2*(c + d*x)])^2) + 4*(a - b)*Sech[c + d*x]*Tanh[c + d*x])/(8*(a - b)^4*d)","A",1
348,1,169,203,2.3112422,"\int \frac{\text{sech}^4(c+d x)}{\left(a+b \sinh ^2(c+d x)\right)^3} \, dx","Integrate[Sech[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^3,x]","\frac{\frac{\frac{3 b^3 \sinh (2 (c+d x)) \left(-32 a^2+b (3 b-14 a) \cosh (2 (c+d x))+24 a b-3 b^2\right)}{a^2 (2 a+b \cosh (2 (c+d x))-b)^2}+8 \tanh (c+d x) \left((a-b) \text{sech}^2(c+d x)+2 a-11 b\right)}{(a-b)^4}+\frac{3 b^2 \left(48 a^2-16 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{a^{5/2} (a-b)^{9/2}}}{24 d}","-\frac{b^3 (16 a-3 b) \tanh (c+d x)}{8 a^2 d (a-b)^4 \left(a-(a-b) \tanh ^2(c+d x)\right)}+\frac{b^2 \left(48 a^2-16 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a-b)^{9/2}}+\frac{b^4 \tanh (c+d x)}{4 a d (a-b)^4 \left(a-(a-b) \tanh ^2(c+d x)\right)^2}-\frac{\tanh ^3(c+d x)}{3 d (a-b)^3}+\frac{(a-4 b) \tanh (c+d x)}{d (a-b)^4}",1,"((3*b^2*(48*a^2 - 16*a*b + 3*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(5/2)*(a - b)^(9/2)) + ((3*b^3*(-32*a^2 + 24*a*b - 3*b^2 + b*(-14*a + 3*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(a^2*(2*a - b + b*Cosh[2*(c + d*x)])^2) + 8*(2*a - 11*b + (a - b)*Sech[c + d*x]^2)*Tanh[c + d*x])/(a - b)^4)/(24*d)","A",1
349,1,24,19,0.0782834,"\int \frac{\cosh ^2(x)}{1-\sinh ^2(x)} \, dx","Integrate[Cosh[x]^2/(1 - Sinh[x]^2),x]","-2 \left(\frac{x}{2}-\frac{\tanh ^{-1}\left(\sqrt{2} \tanh (x)\right)}{\sqrt{2}}\right)","\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \tanh (x)\right)-x",1,"-2*(x/2 - ArcTanh[Sqrt[2]*Tanh[x]]/Sqrt[2])","A",1
350,1,14,10,0.0100749,"\int \frac{\cosh ^3(x)}{1-\sinh ^2(x)} \, dx","Integrate[Cosh[x]^3/(1 - Sinh[x]^2),x]","-2 \left(\frac{\sinh (x)}{2}-\tanh ^{-1}(\sinh (x))\right)","2 \tanh ^{-1}(\sinh (x))-\sinh (x)",1,"-2*(-ArcTanh[Sinh[x]] + Sinh[x]/2)","A",1
351,1,32,30,0.0724745,"\int \frac{\cosh ^4(x)}{1-\sinh ^2(x)} \, dx","Integrate[Cosh[x]^4/(1 - Sinh[x]^2),x]","-2 \left(\frac{5 x}{4}+\frac{1}{8} \sinh (2 x)-\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \tanh (x)\right)\right)","-\frac{5 x}{2}+2 \sqrt{2} \tanh ^{-1}\left(\sqrt{2} \tanh (x)\right)-\frac{1}{2} \sinh (x) \cosh (x)",1,"-2*((5*x)/4 - Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]] + Sinh[2*x]/8)","A",1
352,1,124,117,0.6850704,"\int \cosh ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Cosh[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\sqrt{a+b \sinh ^2(e+f x)} \left(\sqrt{b} \sinh (e+f x) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} (a+b \cosh (2 (e+f x))+3 b)-\sqrt{a} (a-4 b) \sinh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right)\right)}{8 b^{3/2} f \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}","-\frac{a (a-4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{8 b^{3/2} f}+\frac{\sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{4 b f}-\frac{(a-4 b) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{8 b f}",1,"(Sqrt[a + b*Sinh[e + f*x]^2]*(-(Sqrt[a]*(a - 4*b)*ArcSinh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]]) + Sqrt[b]*(a + 3*b + b*Cosh[2*(e + f*x)])*Sinh[e + f*x]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]))/(8*b^(3/2)*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])","A",1
353,1,96,72,0.2592552,"\int \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Cosh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}+\sqrt{b} \sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)}{2 \sqrt{b} f \sqrt{a+b \sinh ^2(e+f x)}}","\frac{\sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{2 f}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{2 \sqrt{b} f}",1,"(Sqrt[b]*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2) + a^(3/2)*ArcSinh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(2*Sqrt[b]*f*Sqrt[a + b*Sinh[e + f*x]^2])","A",1
354,1,130,85,0.7480007,"\int \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\frac{\sqrt{a} \sqrt{b} \sinh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}}}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}+\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{2 a-2 b} \sinh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)}{f}","\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{f}",1,"(Sqrt[a - b]*ArcTan[(Sqrt[2*a - 2*b]*Sinh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] + (Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]]*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])/f","A",1
355,1,175,86,1.5568668,"\int \text{sech}^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Sech[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\sinh (e+f x) \left(\sqrt{2} a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} \tanh ^{-1}\left(\frac{\sqrt{-\frac{(a-b) \sinh ^2(e+f x)}{a}}}{\sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}\right)+\text{sech}^2(e+f x) \sqrt{-\frac{(a-b) \sinh ^2(e+f x)}{a}} (2 a+b \cosh (2 (e+f x))-b)\right)}{4 f \sqrt{-\frac{(a-b) \sinh ^2(e+f x)}{a}} \sqrt{a+b \sinh ^2(e+f x)}}","\frac{a \tan ^{-1}\left(\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{2 f \sqrt{a-b}}+\frac{\tanh (e+f x) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{2 f}",1,"(Sinh[e + f*x]*(Sqrt[2]*a*ArcTanh[Sqrt[-(((a - b)*Sinh[e + f*x]^2)/a)]/Sqrt[1 + (b*Sinh[e + f*x]^2)/a]]*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a] + (2*a - b + b*Cosh[2*(e + f*x)])*Sech[e + f*x]^2*Sqrt[-(((a - b)*Sinh[e + f*x]^2)/a)]))/(4*f*Sqrt[-(((a - b)*Sinh[e + f*x]^2)/a)]*Sqrt[a + b*Sinh[e + f*x]^2])","B",1
356,1,684,151,11.6837558,"\int \text{sech}^5(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Sech[e + f*x]^5*Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{\tanh (e+f x) \text{sech}^3(e+f x) \left(\frac{b \sinh ^2(e+f x)}{a}+1\right) \left(10 b \sinh ^2(e+f x) \sqrt{\frac{(a-b) \tanh ^2(e+f x) \text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a^2}}+15 a \sqrt{\frac{(a-b) \tanh ^2(e+f x) \text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a^2}}+32 b \sinh ^2(e+f x) \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{7/2} \, _2F_1\left(2,4;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}+32 a \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{7/2} \, _2F_1\left(2,4;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}-32 b \sinh ^2(e+f x) \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{5/2} \, _2F_1\left(2,4;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}-32 a \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{5/2} \, _2F_1\left(2,4;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}-15 a \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right)-10 b \sinh ^2(e+f x) \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right)-20 b \sinh ^2(e+f x) \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{3/2} \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}-30 a \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{3/2} \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}\right)}{40 f \sqrt{a+b \sinh ^2(e+f x)} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{3/2} \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}","\frac{a (3 a-4 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{8 f (a-b)^{3/2}}+\frac{\tanh (e+f x) \text{sech}^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{4 f (a-b)}+\frac{(3 a-4 b) \tanh (e+f x) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{8 f (a-b)}",1,"-1/40*(Sech[e + f*x]^3*(1 + (b*Sinh[e + f*x]^2)/a)*Tanh[e + f*x]*(-15*a*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]] - 10*b*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^2 - 30*a*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2) - 20*b*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2) - 32*a*Hypergeometric2F1[2, 4, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(5/2) - 32*b*Hypergeometric2F1[2, 4, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(5/2) + 32*a*Hypergeometric2F1[2, 4, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2) + 32*b*Hypergeometric2F1[2, 4, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2) + 15*a*Sqrt[((a - b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)/a^2] + 10*b*Sinh[e + f*x]^2*Sqrt[((a - b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)/a^2]))/(f*Sqrt[a + b*Sinh[e + f*x]^2]*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2))","C",0
357,1,211,301,1.4116073,"\int \cosh ^4(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Cosh[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\sqrt{2} b \sinh (2 (e+f x)) \left(8 a^2+4 b (4 a+3 b) \cosh (2 (e+f x))+32 a b+3 b^2 \cosh (4 (e+f x))-15 b^2\right)-32 i a \left(a^2-4 a b+3 b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+16 i a \left(2 a^2-7 a b-3 b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{240 b^2 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{\left(2 a^2-7 a b-3 b^2\right) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 b^2 f}+\frac{\left(2 a^2-7 a b-3 b^2\right) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{15 b^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{\sinh (e+f x) \cosh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{5 b f}-\frac{2 (a-3 b) \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 b f}-\frac{(a-9 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{15 b f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((16*I)*a*(2*a^2 - 7*a*b - 3*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (32*I)*a*(a^2 - 4*a*b + 3*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(8*a^2 + 32*a*b - 15*b^2 + 4*b*(4*a + 3*b)*Cosh[2*(e + f*x)] + 3*b^2*Cosh[4*(e + f*x)])*Sinh[2*(e + f*x)])/(240*b^2*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
358,1,168,223,0.8157334,"\int \cosh ^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Cosh[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{b \sinh (2 (e+f x)) (2 a+b \cosh (2 (e+f x))-b)+2 i \sqrt{2} a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i \sqrt{2} a (a+b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 b f \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","\frac{(a+b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 b f}+\frac{\sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}+\frac{2 \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{(a+b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 b f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((-2*I)*Sqrt[2]*a*(a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (2*I)*Sqrt[2]*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + b*(2*a - b + b*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(6*b*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","C",1
359,1,69,60,0.0866434,"\int \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{i \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{f \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}",1,"((-I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a])/(f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","A",1
360,1,148,70,0.4888665,"\int \text{sech}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Sech[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\sqrt{2} \tanh (e+f x) (2 a+b \cosh (2 (e+f x))-b)-2 i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+2 i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{2 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*(2*a - b + b*Cosh[2*(e + f*x)])*Tanh[e + f*x])/(2*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
361,1,204,206,3.0075404,"\int \text{sech}^4(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Sech[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\sqrt{2} \tanh (e+f x) \text{sech}^2(e+f x) \left(\left(8 a^2-4 b^2\right) \cosh (2 (e+f x))+(2 a-b) (8 a+b \cosh (4 (e+f x))-5 b)\right)-16 i a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+8 i a (2 a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{24 f (a-b) \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{\tanh (e+f x) \text{sech}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}-\frac{b \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{(2 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((8*I)*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (16*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*((8*a^2 - 4*b^2)*Cosh[2*(e + f*x)] + (2*a - b)*(8*a - 5*b + b*Cosh[4*(e + f*x)]))*Sech[e + f*x]^2*Tanh[e + f*x])/(24*(a - b)*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
362,1,149,157,1.2490595,"\int \cosh ^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cosh[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\sqrt{a+b \sinh ^2(e+f x)} \left(\sqrt{b} \sinh (e+f x) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} \left(2 b (7 a+6 b) \sinh ^2(e+f x)+3 a (a+10 b)+8 b^2 \sinh ^4(e+f x)\right)-3 a^{3/2} (a-6 b) \sinh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right)\right)}{48 b^{3/2} f \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}","-\frac{a^2 (a-6 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{16 b^{3/2} f}+\frac{\sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^{5/2}}{6 b f}-\frac{(a-6 b) \sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{24 b f}-\frac{a (a-6 b) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{16 b f}",1,"(Sqrt[a + b*Sinh[e + f*x]^2]*(-3*a^(3/2)*(a - 6*b)*ArcSinh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]] + Sqrt[b]*Sinh[e + f*x]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]*(3*a*(a + 10*b) + 2*b*(7*a + 6*b)*Sinh[e + f*x]^2 + 8*b^2*Sinh[e + f*x]^4)))/(48*b^(3/2)*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])","A",1
363,1,93,104,0.4893323,"\int \cosh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cosh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\sqrt{a+b \sinh ^2(e+f x)} \left(\frac{3 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}+5 a \sinh (e+f x)+2 b \sinh ^3(e+f x)\right)}{8 f}","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{8 \sqrt{b} f}+\frac{3 a \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{8 f}+\frac{\sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{4 f}",1,"(Sqrt[a + b*Sinh[e + f*x]^2]*(5*a*Sinh[e + f*x] + 2*b*Sinh[e + f*x]^3 + (3*a^(3/2)*ArcSinh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]])/(Sqrt[b]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])))/(8*f)","A",1
364,1,142,125,0.4741127,"\int \text{sech}(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sech[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{b \sinh (e+f x) \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}+4 (a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{2 a-2 b} \sinh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)+2 \sqrt{b} (3 a-2 b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sinh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)}{4 f}","\frac{b \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{2 f}+\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{f}+\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{2 f}",1,"(4*(a - b)^(3/2)*ArcTan[(Sqrt[2*a - 2*b]*Sinh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] + 2*(3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[2]*Sqrt[b]*Sinh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] + b*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]]*Sinh[e + f*x])/(4*f)","A",1
365,1,150,133,0.7865019,"\int \text{sech}^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sech[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{4 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sinh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)+2 \sqrt{a-b} (a+2 b) \tan ^{-1}\left(\frac{\sqrt{2 a-2 b} \sinh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)+(a-b) \tanh (e+f x) \text{sech}(e+f x) \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}{4 f}","\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{f}+\frac{\sqrt{a-b} (a+2 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{2 f}+\frac{(a-b) \tanh (e+f x) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{2 f}",1,"(2*Sqrt[a - b]*(a + 2*b)*ArcTan[(Sqrt[2*a - 2*b]*Sinh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] + 4*b^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[b]*Sinh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] + (a - b)*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]]*Sech[e + f*x]*Tanh[e + f*x])/(4*f)","A",1
366,1,66,126,0.1251646,"\int \text{sech}^5(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sech[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{a^2 \sinh (e+f x) \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};-\frac{(a-b) \sinh ^2(e+f x)}{b \sinh ^2(e+f x)+a}\right)}{f \sqrt{a+b \sinh ^2(e+f x)}}","\frac{3 a^2 \tan ^{-1}\left(\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{8 f \sqrt{a-b}}+\frac{\tanh (e+f x) \text{sech}^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{4 f}+\frac{3 a \tanh (e+f x) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{8 f}",1,"(a^2*Hypergeometric2F1[1/2, 3, 3/2, -(((a - b)*Sinh[e + f*x]^2)/(a + b*Sinh[e + f*x]^2))]*Sinh[e + f*x])/(f*Sqrt[a + b*Sinh[e + f*x]^2])","C",1
367,1,959,205,15.1608478,"\int \text{sech}^7(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sech[e + f*x]^7*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{a^2 \text{sech}^3(e+f x) \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^2 \tanh (e+f x) \left(256 b \, _2F_1\left(2,5;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sinh ^2(e+f x) \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{9/2}+256 a \, _2F_1\left(2,5;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{9/2}-512 b \, _2F_1\left(2,5;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sinh ^2(e+f x) \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{7/2}-512 a \, _2F_1\left(2,5;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{7/2}-80 b \sinh ^2(e+f x) \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{5/2}+256 b \, _2F_1\left(2,5;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sinh ^2(e+f x) \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{5/2}-120 a \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{5/2}+256 a \, _2F_1\left(2,5;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{5/2}+140 b \sinh ^2(e+f x) \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{3/2}+210 a \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{3/2}+30 b \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right) \sinh ^2(e+f x)+45 a \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right)-30 b \sinh ^2(e+f x) \sqrt{\frac{(a-b) \text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right) \tanh ^2(e+f x)}{a^2}}-45 a \sqrt{\frac{(a-b) \text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right) \tanh ^2(e+f x)}{a^2}}\right)}{240 f \left(b \sinh ^2(e+f x)+a\right)^{3/2} \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{3/2}}","\frac{a^2 (5 a-6 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{16 f (a-b)^{3/2}}+\frac{\tanh (e+f x) \text{sech}^5(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{5/2}}{6 f (a-b)}+\frac{(5 a-6 b) \tanh (e+f x) \text{sech}^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{24 f (a-b)}+\frac{a (5 a-6 b) \tanh (e+f x) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{16 f (a-b)}",1,"(a^2*Sech[e + f*x]^3*(1 + (b*Sinh[e + f*x]^2)/a)^2*Tanh[e + f*x]*(45*a*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]] + 30*b*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^2 + 210*a*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2) + 140*b*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2) - 120*a*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(5/2) + 256*a*Hypergeometric2F1[2, 5, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(5/2) - 80*b*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(5/2) + 256*b*Hypergeometric2F1[2, 5, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(5/2) - 512*a*Hypergeometric2F1[2, 5, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2) - 512*b*Hypergeometric2F1[2, 5, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2) + 256*a*Hypergeometric2F1[2, 5, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(9/2) + 256*b*Hypergeometric2F1[2, 5, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(9/2) - 45*a*Sqrt[((a - b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)/a^2] - 30*b*Sinh[e + f*x]^2*Sqrt[((a - b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)/a^2]))/(240*f*(a + b*Sinh[e + f*x]^2)^(3/2)*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2))","C",0
368,1,256,357,2.6062306,"\int \cosh ^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cosh[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} b \sinh (2 (e+f x)) \left(32 a^3+b \left(144 a^2+192 a b-37 b^2\right) \cosh (2 (e+f x))+400 a^2 b+2 b^2 (26 a+b) \cosh (4 (e+f x))-212 a b^2+5 b^3 \cosh (6 (e+f x))+30 b^3\right)-64 i a \left(2 a^3-11 a^2 b+8 a b^2+b^3\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+128 i a \left(a^3-5 a^2 b-5 a b^2+b^3\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{2240 b^2 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{2 (a+b) \left(a^2-6 a b+b^2\right) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 b^2 f}+\frac{\left(a^2+9 a b-2 b^2\right) \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 b f}-\frac{\left(a^2-18 a b+b^2\right) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{35 b f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{2 (a+b) \left(a^2-6 a b+b^2\right) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{35 b^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{b \sinh (e+f x) \cosh ^5(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{7 f}+\frac{2 (4 a-b) \sinh (e+f x) \cosh ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 f}",1,"((128*I)*a*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (64*I)*a*(2*a^3 - 11*a^2*b + 8*a*b^2 + b^3)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(32*a^3 + 400*a^2*b - 212*a*b^2 + 30*b^3 + b*(144*a^2 + 192*a*b - 37*b^2)*Cosh[2*(e + f*x)] + 2*b^2*(26*a + b)*Cosh[4*(e + f*x)] + 5*b^3*Cosh[6*(e + f*x)])*Sinh[2*(e + f*x)])/(2240*b^2*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
369,1,213,299,1.3812822,"\int \cosh ^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cosh[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} b \sinh (2 (e+f x)) \left(48 a^2+4 b (9 a-2 b) \cosh (2 (e+f x))-28 a b+3 b^2 \cosh (4 (e+f x))+5 b^2\right)+16 i a \left(3 a^2-2 a b-b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-16 i a \left(3 a^2+7 a b-2 b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{240 b f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{\left(3 a^2+7 a b-2 b^2\right) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 b f}-\frac{\left(3 a^2+7 a b-2 b^2\right) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{15 b f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{b \sinh (e+f x) \cosh ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{5 f}+\frac{2 (3 a-b) \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 f}+\frac{(9 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{15 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((-16*I)*a*(3*a^2 + 7*a*b - 2*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (16*I)*a*(3*a^2 - 2*a*b - b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(48*a^2 - 28*a*b + 5*b^2 + 4*(9*a - 2*b)*b*Cosh[2*(e + f*x)] + 3*b^2*Cosh[4*(e + f*x)])*Sinh[2*(e + f*x)])/(240*b*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
370,1,169,174,0.7631115,"\int \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{b \sinh (2 (e+f x)) (2 a+b \cosh (2 (e+f x))-b)+2 i \sqrt{2} a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-4 i \sqrt{2} a (2 a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 f \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","\frac{b \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}+\frac{i a (a-b) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} F\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{2 i (2 a-b) \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 f \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}",1,"((-4*I)*Sqrt[2]*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (2*I)*Sqrt[2]*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + b*(2*a - b + b*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(6*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","A",1
371,1,160,210,0.9699547,"\int \text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{2 i a (a-2 b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)+(a-b) \left(\sqrt{2} \tanh (e+f x) (2 a+b \cosh (2 (e+f x))-b)-2 i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)\right)}{2 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{(a-2 b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{f}+\frac{(a-b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{f}+\frac{b \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{(a-2 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((2*I)*a*(a - 2*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (a - b)*((-2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*(2*a - b + b*Cosh[2*(e + f*x)])*Tanh[e + f*x]))/(2*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
372,1,197,193,2.0206892,"\int \text{sech}^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sech[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\frac{\tanh (e+f x) \text{sech}^2(e+f x) \left(\left(4 a^2+6 a b-2 b^2\right) \cosh (2 (e+f x))+8 a^2+b (a+b) \cosh (4 (e+f x))-3 a b+b^2\right)}{\sqrt{2}}-2 i a (2 a+b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+4 i a (a+b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{(a-b) \tanh (e+f x) \text{sech}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}-\frac{b \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{2 (a+b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((4*I)*a*(a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (2*I)*a*(2*a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + ((8*a^2 - 3*a*b + b^2 + (4*a^2 + 6*a*b - 2*b^2)*Cosh[2*(e + f*x)] + b*(a + b)*Cosh[4*(e + f*x)])*Sech[e + f*x]^2*Tanh[e + f*x])/Sqrt[2])/(6*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
373,1,77,79,0.1083054,"\int \frac{\cosh ^3(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Cosh[e + f*x]^3/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\frac{\sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{2 b}-\frac{(a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{2 b^{3/2}}}{f}","\frac{\sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{2 b f}-\frac{(a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{2 b^{3/2} f}",1,"(-1/2*((a - 2*b)*ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/b^(3/2) + (Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(2*b))/f","A",1
374,1,38,38,0.0165357,"\int \frac{\cosh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Cosh[e + f*x]/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{\sqrt{b} f}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{\sqrt{b} f}",1,"ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]]/(Sqrt[b]*f)","A",1
375,1,46,46,0.0319718,"\int \frac{\text{sech}(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Sech[e + f*x]/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{f \sqrt{a-b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{f \sqrt{a-b}}",1,"ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]]/(Sqrt[a - b]*f)","A",1
376,1,443,97,9.3675566,"\int \frac{\text{sech}^3(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Sech[e + f*x]^3/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\tanh (e+f x) \text{sech}^3(e+f x) \left(\frac{b \sinh ^2(e+f x)}{a}+1\right) \left(-30 b \sinh ^2(e+f x) \sqrt{\frac{\tanh ^2(e+f x) \text{sech}^2(e+f x) \left(a^2+a b \left(\sinh ^2(e+f x)-1\right)-b^2 \sinh ^2(e+f x)\right)}{a^2}}-45 a \sqrt{\frac{\tanh ^2(e+f x) \text{sech}^2(e+f x) \left(a^2+a b \left(\sinh ^2(e+f x)-1\right)-b^2 \sinh ^2(e+f x)\right)}{a^2}}+16 b \sinh ^2(e+f x) \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{5/2} \, _2F_1\left(2,3;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}+16 a \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{5/2} \, _2F_1\left(2,3;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}+45 a \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right)+30 b \sinh ^2(e+f x) \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right)\right)}{30 a f \sqrt{a+b \sinh ^2(e+f x)} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{3/2} \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}","\frac{(a-2 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{2 f (a-b)^{3/2}}+\frac{\tanh (e+f x) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{2 f (a-b)}",1,"(Sech[e + f*x]^3*(1 + (b*Sinh[e + f*x]^2)/a)*Tanh[e + f*x]*(45*a*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]] + 30*b*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^2 + 16*a*Hypergeometric2F1[2, 3, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(5/2) + 16*b*Hypergeometric2F1[2, 3, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(5/2) - 45*a*Sqrt[(Sech[e + f*x]^2*(a^2 - b^2*Sinh[e + f*x]^2 + a*b*(-1 + Sinh[e + f*x]^2))*Tanh[e + f*x]^2)/a^2] - 30*b*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a^2 - b^2*Sinh[e + f*x]^2 + a*b*(-1 + Sinh[e + f*x]^2))*Tanh[e + f*x]^2)/a^2]))/(30*a*f*Sqrt[a + b*Sinh[e + f*x]^2]*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2))","C",0
377,1,179,241,0.8550075,"\int \frac{\cosh ^4(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Cosh[e + f*x]^4/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{-2 i \sqrt{2} \left(2 a^2-5 a b+3 b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+b \sinh (2 (e+f x)) (2 a+b \cosh (2 (e+f x))-b)+4 i \sqrt{2} a (a-2 b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 b^2 f \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","-\frac{2 (a-2 b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 b^2 f}+\frac{2 (a-2 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 b^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{\sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 b f}-\frac{(a-3 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a b f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((4*I)*Sqrt[2]*a*(a - 2*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (2*I)*Sqrt[2]*(2*a^2 - 5*a*b + 3*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + b*(2*a - b + b*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(6*b^2*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","C",1
378,1,95,177,0.2377993,"\int \frac{\cosh ^2(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Cosh[e + f*x]^2/Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{i \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} \left((b-a) F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+a E\left(i (e+f x)\left|\frac{b}{a}\right.\right)\right)}{b f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{\tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{b f}+\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{a f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{b f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((-I)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*(a*EllipticE[I*(e + f*x), b/a] + (-a + b)*EllipticF[I*(e + f*x), b/a]))/(b*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
379,1,68,60,0.0737961,"\int \frac{1}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[1/Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{i \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{i \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} F\left(i e+i f x\left|\frac{b}{a}\right.\right)}{f \sqrt{a+b \sinh ^2(e+f x)}}",1,"((-I)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","A",1
380,1,159,160,0.6456044,"\int \frac{\text{sech}^2(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Sech[e + f*x]^2/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\sqrt{2} \tanh (e+f x) (2 a+b \cosh (2 (e+f x))-b)-2 i (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+2 i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{2 f (a-b) \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{b \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{a f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (2*I)*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*(2*a - b + b*Cosh[2*(e + f*x)])*Tanh[e + f*x])/(2*(a - b)*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
381,1,219,219,2.227076,"\int \frac{\text{sech}^4(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Sech[e + f*x]^4/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{-2 i \left(2 a^2-5 a b+3 b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+\frac{\tanh (e+f x) \text{sech}^2(e+f x) \left(\left(4 a^2-6 a b-2 b^2\right) \cosh (2 (e+f x))+8 a^2+b (a-2 b) \cosh (4 (e+f x))-15 a b+4 b^2\right)}{\sqrt{2}}+4 i a (a-2 b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 f (a-b)^2 \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{\tanh (e+f x) \text{sech}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f (a-b)}-\frac{b (a-3 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{2 (a-2 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((4*I)*a*(a - 2*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (2*I)*(2*a^2 - 5*a*b + 3*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + ((8*a^2 - 15*a*b + 4*b^2 + (4*a^2 - 6*a*b - 2*b^2)*Cosh[2*(e + f*x)] + (a - 2*b)*b*Cosh[4*(e + f*x)])*Sech[e + f*x]^2*Tanh[e + f*x])/Sqrt[2])/(6*(a - b)^2*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
382,1,89,77,0.1715138,"\int \frac{\cosh ^3(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cosh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}+\sqrt{b} (b-a) \sinh (e+f x)}{a b^{3/2} f \sqrt{a+b \sinh ^2(e+f x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{b^{3/2} f}-\frac{(a-b) \sinh (e+f x)}{a b f \sqrt{a+b \sinh ^2(e+f x)}}",1,"(Sqrt[b]*(-a + b)*Sinh[e + f*x] + a^(3/2)*ArcSinh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(a*b^(3/2)*f*Sqrt[a + b*Sinh[e + f*x]^2])","A",1
383,1,29,29,0.0294167,"\int \frac{\cosh (e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cosh[e + f*x]/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\sinh (e+f x)}{a f \sqrt{a+b \sinh ^2(e+f x)}}","\frac{\sinh (e+f x)}{a f \sqrt{a+b \sinh ^2(e+f x)}}",1,"Sinh[e + f*x]/(a*f*Sqrt[a + b*Sinh[e + f*x]^2])","A",1
384,1,315,85,7.5770655,"\int \frac{\text{sech}(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sech[e + f*x]/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\tanh (e+f x) \text{sech}^7(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \left(4 (a-b)^2 \sinh ^4(e+f x) \left(a+b \sinh ^2(e+f x)\right) \, _2F_1\left(2,2;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\tanh ^2(e+f x) \text{sech}^2(e+f x) \left(a^2+a b \left(\sinh ^2(e+f x)-1\right)-b^2 \sinh ^2(e+f x)\right)}{a^2}}+15 a \cosh ^2(e+f x) \left(3 a+2 b \sinh ^2(e+f x)\right) \left(a \cosh ^2(e+f x) \sqrt{\frac{\tanh ^2(e+f x) \text{sech}^2(e+f x) \left(a^2+a b \left(\sinh ^2(e+f x)-1\right)-b^2 \sinh ^2(e+f x)\right)}{a^2}}-\left(a+b \sinh ^2(e+f x)\right) \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right)\right)\right)}{15 a^5 f \left(\frac{(a-b) \tanh ^2(e+f x) \text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a^2}\right)^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \sinh (e+f x)}{a f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}",1,"(Sech[e + f*x]^7*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]*(4*(a - b)^2*Hypergeometric2F1[2, 2, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^4*(a + b*Sinh[e + f*x]^2)*Sqrt[(Sech[e + f*x]^2*(a^2 - b^2*Sinh[e + f*x]^2 + a*b*(-1 + Sinh[e + f*x]^2))*Tanh[e + f*x]^2)/a^2] + 15*a*Cosh[e + f*x]^2*(3*a + 2*b*Sinh[e + f*x]^2)*(-(ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*(a + b*Sinh[e + f*x]^2)) + a*Cosh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a^2 - b^2*Sinh[e + f*x]^2 + a*b*(-1 + Sinh[e + f*x]^2))*Tanh[e + f*x]^2)/a^2])))/(15*a^5*f*(((a - b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)/a^2)^(3/2))","C",0
385,1,231,142,5.2091023,"\int \frac{\text{sech}^3(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sech[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\tanh (e+f x) \text{sech}^5(e+f x) \left(16 (a-b) \sinh ^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^2 \, _3F_2\left(2,2,3;1,\frac{9}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right)+16 (a-b) \sinh ^2(e+f x) \left(4 a^2+7 a b \sinh ^2(e+f x)+3 b^2 \sinh ^4(e+f x)\right) \, _2F_1\left(2,3;\frac{9}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right)+7 a \cosh ^2(e+f x) \left(15 a^2+20 a b \sinh ^2(e+f x)+8 b^2 \sinh ^4(e+f x)\right) \, _2F_1\left(1,2;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right)\right)}{105 a^4 f \sqrt{a+b \sinh ^2(e+f x)}}","\frac{b (a+2 b) \sinh (e+f x)}{2 a f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{(a-4 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{2 f (a-b)^{5/2}}+\frac{\tanh (e+f x) \text{sech}(e+f x)}{2 f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}",1,"(Sech[e + f*x]^5*(16*(a - b)*HypergeometricPFQ[{2, 2, 3}, {1, 9/2}, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^2 + 16*(a - b)*Hypergeometric2F1[2, 3, 9/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*(4*a^2 + 7*a*b*Sinh[e + f*x]^2 + 3*b^2*Sinh[e + f*x]^4) + 7*a*Cosh[e + f*x]^2*Hypergeometric2F1[1, 2, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*(15*a^2 + 20*a*b*Sinh[e + f*x]^2 + 8*b^2*Sinh[e + f*x]^4))*Tanh[e + f*x])/(105*a^4*f*Sqrt[a + b*Sinh[e + f*x]^2])","C",0
386,1,196,325,1.1122296,"\int \frac{\cosh ^6(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cosh[e + f*x]^6/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} b \sinh (2 (e+f x)) \left(8 a^2+a b \cosh (2 (e+f x))-13 a b+6 b^2\right)-4 i a \left(8 a^2-17 a b+9 b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+4 i a \left(8 a^2-13 a b+3 b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{12 a b^3 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{\left(8 a^2-13 a b+3 b^2\right) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a b^3 f}+\frac{\left(8 a^2-13 a b+3 b^2\right) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a b^3 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{(4 a-3 b) \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a b^2 f}-\frac{2 (2 a-3 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a b^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{(a-b) \sinh (e+f x) \cosh ^3(e+f x)}{a b f \sqrt{a+b \sinh ^2(e+f x)}}",1,"((4*I)*a*(8*a^2 - 13*a*b + 3*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (4*I)*a*(8*a^2 - 17*a*b + 9*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(8*a^2 - 13*a*b + 6*b^2 + a*b*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(12*a*b^3*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
387,1,155,244,0.621217,"\int \frac{\cosh ^4(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cosh[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{(a-b) \left(-\sqrt{2} b \sinh (2 (e+f x))+4 i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)\right)-2 i a (2 a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{2 a b^2 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{(2 a-b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{a b^2 f}-\frac{(2 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{a b^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{(a-b) \sinh (e+f x) \cosh (e+f x)}{a b f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{a b f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((-2*I)*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (a - b)*((4*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] - Sqrt[2]*b*Sinh[2*(e + f*x)]))/(2*a*b^2*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
388,1,143,91,0.3201088,"\int \frac{\cosh ^2(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cosh[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{-i \sqrt{2} a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+i \sqrt{2} a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)+b \sinh (2 (e+f x))}{a b f \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","\frac{\cosh (e+f x) E\left(\tan ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right)|1-\frac{a}{b}\right)}{\sqrt{a} \sqrt{b} f \sqrt{a+b \sinh ^2(e+f x)} \sqrt{\frac{a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}",1,"(I*Sqrt[2]*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - I*Sqrt[2]*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + b*Sinh[2*(e + f*x)])/(a*b*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","C",1
389,1,100,115,0.1557591,"\int \frac{1}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^(-3/2),x]","\frac{-\sqrt{2} b \sinh (2 (e+f x))-2 i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{2 a f (a-b) \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{b \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}-\frac{i \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{a f (a-b) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}",1,"((-2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - Sqrt[2]*b*Sinh[2*(e + f*x)])/(2*a*(a - b)*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","A",1
390,1,178,217,1.2186147,"\int \frac{\text{sech}^2(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sech[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\tanh (e+f x) \left(2 a^2+b (a+b) \cosh (2 (e+f x))-a b+b^2\right)-i \sqrt{2} a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+i \sqrt{2} a (a+b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{a f (a-b)^2 \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","\frac{\tanh (e+f x)}{f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\sqrt{b} (a+b) \cosh (e+f x) E\left(\tan ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right)|1-\frac{a}{b}\right)}{\sqrt{a} f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)} \sqrt{\frac{a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}-\frac{2 b \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{a f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"(I*Sqrt[2]*a*(a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - I*Sqrt[2]*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + (2*a^2 - a*b + b^2 + b*(a + b)*Cosh[2*(e + f*x)])*Tanh[e + f*x])/(a*(a - b)^2*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","C",1
391,1,126,134,0.8879838,"\int \frac{\cosh ^5(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cosh[e + f*x]^5/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{\frac{2 \sqrt{2} (b-a) \sinh (e+f x) \left(3 a^2+b (2 a+b) \cosh (2 (e+f x))+a b-b^2\right)}{3 a^2 b^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sinh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right)}{b^{5/2}}}{f}","-\frac{(a-b) (3 a+2 b) \sinh (e+f x)}{3 a^2 b^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{b^{5/2} f}-\frac{(a-b) \sinh (e+f x) \cosh ^2(e+f x)}{3 a b f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"(ArcTanh[(Sqrt[2]*Sqrt[b]*Sinh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]]/b^(5/2) + (2*Sqrt[2]*(-a + b)*(3*a^2 + a*b - b^2 + b*(2*a + b)*Cosh[2*(e + f*x)])*Sinh[e + f*x])/(3*a^2*b^2*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2)))/f","A",1
392,1,50,73,0.10083,"\int \frac{\cosh ^3(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cosh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{(a+2 b) \sinh ^3(e+f x)+3 a \sinh (e+f x)}{3 a^2 f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}","\frac{2 \sinh (e+f x)}{3 a^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\sinh (e+f x) \cosh ^2(e+f x)}{3 a f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"(3*a*Sinh[e + f*x] + (a + 2*b)*Sinh[e + f*x]^3)/(3*a^2*f*(a + b*Sinh[e + f*x]^2)^(3/2))","A",1
393,1,47,65,0.0456863,"\int \frac{\cosh (e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cosh[e + f*x]/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{\sinh (e+f x) \left(3 a+2 b \sinh ^2(e+f x)\right)}{3 a^2 f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}","\frac{2 \sinh (e+f x)}{3 a^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\sinh (e+f x)}{3 a f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"(Sinh[e + f*x]*(3*a + 2*b*Sinh[e + f*x]^2))/(3*a^2*f*(a + b*Sinh[e + f*x]^2)^(3/2))","A",1
394,1,1331,134,9.3660773,"\int \frac{\text{sech}(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sech[e + f*x]/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{\text{sech}(e+f x) \tanh (e+f x) \left(\frac{840 (a-b)^2 b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right) \tanh ^4(e+f x) \sinh ^4(e+f x)}{a^4}+\frac{72 b^2 \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{7/2} \sinh ^4(e+f x)}{a^2}+\frac{24 b^2 \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{7/2} \sinh ^4(e+f x)}{a^2}-\frac{1680 (a-b) b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right) \tanh ^2(e+f x) \sinh ^4(e+f x)}{a^3}+\frac{1120 b^2 \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{3/2} \sinh ^4(e+f x)}{a^2}+\frac{840 b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right) \sinh ^4(e+f x)}{a^2}-\frac{840 b^2 \sqrt{\frac{(a-b) \text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right) \tanh ^2(e+f x)}{a^2}} \sinh ^4(e+f x)}{a^2}+\frac{2100 (a-b)^2 b \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right) \tanh ^4(e+f x) \sinh ^2(e+f x)}{a^3}+\frac{168 b \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{7/2} \sinh ^2(e+f x)}{a}+\frac{48 b \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{7/2} \sinh ^2(e+f x)}{a}-\frac{4200 (a-b) b \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right) \tanh ^2(e+f x) \sinh ^2(e+f x)}{a^2}+\frac{2800 b \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{3/2} \sinh ^2(e+f x)}{a}+\frac{2100 b \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right) \sinh ^2(e+f x)}{a}-\frac{2100 b \sqrt{\frac{(a-b) \text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right) \tanh ^2(e+f x)}{a^2}} \sinh ^2(e+f x)}{a}+\frac{1575 (a-b)^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right) \tanh ^4(e+f x)}{a^2}+96 \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{7/2}+24 \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right) \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{7/2}-\frac{3150 (a-b) \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right) \tanh ^2(e+f x)}{a}+2100 \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{3/2}+1575 \sin ^{-1}\left(\sqrt{\frac{(a-b) \tanh ^2(e+f x)}{a}}\right)-1575 \sqrt{\frac{(a-b) \text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right) \tanh ^2(e+f x)}{a^2}}\right)}{315 a^2 f \sqrt{b \sinh ^2(e+f x)+a} \sqrt{\frac{\text{sech}^2(e+f x) \left(b \sinh ^2(e+f x)+a\right)}{a}} \left(\frac{b \sinh ^2(e+f x)}{a}+1\right) \left(\frac{(a-b) \tanh ^2(e+f x)}{a}\right)^{5/2}}","-\frac{b (5 a-2 b) \sinh (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}-\frac{b \sinh (e+f x)}{3 a f (a-b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right)}{f (a-b)^{5/2}}",1,"(Sech[e + f*x]*Tanh[e + f*x]*(1575*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]] + (2100*b*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^2)/a + (840*b^2*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^4)/a^2 - (3150*(a - b)*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Tanh[e + f*x]^2)/a - (4200*(a - b)*b*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^2*Tanh[e + f*x]^2)/a^2 - (1680*(a - b)*b^2*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^4*Tanh[e + f*x]^2)/a^3 + (1575*(a - b)^2*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Tanh[e + f*x]^4)/a^2 + (2100*(a - b)^2*b*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^2*Tanh[e + f*x]^4)/a^3 + (840*(a - b)^2*b^2*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^4*Tanh[e + f*x]^4)/a^4 + 2100*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2) + (2800*b*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2))/a + (1120*b^2*Sinh[e + f*x]^4*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2))/a^2 + 96*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2) + 24*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Tanh[e + f*x]^2)/a]*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2) + (168*b*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2))/a + (48*b*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2))/a + (72*b^2*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^4*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2))/a^2 + (24*b^2*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^4*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2))/a^2 - 1575*Sqrt[((a - b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)/a^2] - (2100*b*Sinh[e + f*x]^2*Sqrt[((a - b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)/a^2])/a - (840*b^2*Sinh[e + f*x]^4*Sqrt[((a - b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)/a^2])/a^2))/(315*a^2*f*Sqrt[a + b*Sinh[e + f*x]^2]*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(1 + (b*Sinh[e + f*x]^2)/a)*(((a - b)*Tanh[e + f*x]^2)/a)^(5/2))","C",0
395,1,206,330,2.2448799,"\int \frac{\cosh ^6(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cosh[e + f*x]^6/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{\frac{1}{2} (a-b) \left(-2 \sqrt{2} b \sinh (2 (e+f x)) \left(8 a^2+b (5 a+2 b) \cosh (2 (e+f x))+a b-2 b^2\right)+4 i a^2 (8 a+b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)\right)-2 i a^2 \left(8 a^2-3 a b-2 b^2\right) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 a^2 b^3 f (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","-\frac{2 (a-b) (2 a+b) \sinh (e+f x) \cosh (e+f x)}{3 a^2 b^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{(4 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^2 b^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{\left(8 a^2-3 a b-2 b^2\right) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^2 b^3 f}-\frac{\left(8 a^2-3 a b-2 b^2\right) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^2 b^3 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{(a-b) \sinh (e+f x) \cosh ^3(e+f x)}{3 a b f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"((-2*I)*a^2*(8*a^2 - 3*a*b - 2*b^2)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] + ((a - b)*((4*I)*a^2*(8*a + b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] - 2*Sqrt[2]*b*(8*a^2 + a*b - 2*b^2 + b*(5*a + 2*b)*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)]))/2)/(6*a^2*b^3*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","C",1
396,1,178,223,1.4353537,"\int \frac{\cosh ^4(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cosh[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{\sqrt{2} b \sinh (2 (e+f x)) \left(a^2+b (a+b) \cosh (2 (e+f x))+2 a b-b^2\right)-i a^2 (2 a+b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+2 i a^2 (a+b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{3 a^2 b^2 f (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","\frac{2 (a+b) \cosh (e+f x) E\left(\tan ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right)|1-\frac{a}{b}\right)}{3 a^{3/2} b^{3/2} f \sqrt{a+b \sinh ^2(e+f x)} \sqrt{\frac{a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}-\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^2 b f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{(a-b) \sinh (e+f x) \cosh (e+f x)}{3 a b f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"((2*I)*a^2*(a + b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] - I*a^2*(2*a + b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(a^2 + 2*a*b - b^2 + b*(a + b)*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(3*a^2*b^2*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","C",1
397,1,193,228,1.4385219,"\int \frac{\cosh ^2(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cosh[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{-\sqrt{2} b \sinh (2 (e+f x)) \left(-4 a^2-b (a-2 b) \cosh (2 (e+f x))+7 a b-2 b^2\right)-2 i a^2 (a-b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+2 i a^2 (a-2 b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 a^2 b f (a-b) (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","\frac{(a-2 b) \cosh (e+f x) E\left(\tan ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right)|1-\frac{a}{b}\right)}{3 a^{3/2} \sqrt{b} f (a-b) \sqrt{a+b \sinh ^2(e+f x)} \sqrt{\frac{a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}+\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^2 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{\sinh (e+f x) \cosh (e+f x)}{3 a f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"((2*I)*a^2*(a - 2*b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] - (2*I)*a^2*(a - b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] - Sqrt[2]*b*(-4*a^2 + 7*a*b - 2*b^2 - (a - 2*b)*b*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(6*a^2*(a - b)*b*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","C",1
398,1,190,251,1.3008208,"\int \frac{1}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^(-5/2),x]","\frac{\sqrt{2} b \sinh (2 (e+f x)) \left(-5 a^2+b (b-2 a) \cosh (2 (e+f x))+5 a b-b^2\right)+i a^2 (a-b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i a^2 (2 a-b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{3 a^2 f (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","-\frac{2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}-\frac{2 i (2 a-b) \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 a^2 f (a-b)^2 \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}-\frac{b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}+\frac{i \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} F\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 a f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}",1,"((-2*I)*a^2*(2*a - b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] + I*a^2*(a - b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(-5*a^2 + 5*a*b - b^2 + b*(-2*a + b)*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(3*a^2*(a - b)^2*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","A",1
399,1,260,292,3.4569402,"\int \frac{\text{sech}^2(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sech[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{-2 i a^2 \left(3 a^2-2 a b-b^2\right) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+2 i a^2 \left(3 a^2+7 a b-2 b^2\right) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)+\frac{\tanh (e+f x) \left(24 a^4-24 a^3 b+4 a b \left(6 a^2+5 a b-3 b^2\right) \cosh (2 (e+f x))+b^2 \left(3 a^2+7 a b-2 b^2\right) \cosh (4 (e+f x))+41 a^2 b^2-19 a b^3+2 b^4\right)}{\sqrt{2}}}{6 a^2 f (a-b)^3 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","-\frac{b (9 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^2 f (a-b)^3 \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{\sqrt{b} \left(3 a^2+7 a b-2 b^2\right) \cosh (e+f x) E\left(\tan ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right)|1-\frac{a}{b}\right)}{3 a^{3/2} f (a-b)^3 \sqrt{a+b \sinh ^2(e+f x)} \sqrt{\frac{a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}+\frac{\tanh (e+f x)}{f (a-b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}+\frac{b (3 a+b) \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b)^2 \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"((2*I)*a^2*(3*a^2 + 7*a*b - 2*b^2)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] - (2*I)*a^2*(3*a^2 - 2*a*b - b^2)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] + ((24*a^4 - 24*a^3*b + 41*a^2*b^2 - 19*a*b^3 + 2*b^4 + 4*a*b*(6*a^2 + 5*a*b - 3*b^2)*Cosh[2*(e + f*x)] + b^2*(3*a^2 + 7*a*b - 2*b^2)*Cosh[4*(e + f*x)])*Tanh[e + f*x])/Sqrt[2])/(6*a^2*(a - b)^3*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","C",1
400,0,0,117,9.1976139,"\int (d \cosh (e+f x))^m \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[(d*Cosh[e + f*x])^m*(a + b*Sinh[e + f*x]^2)^p,x]","\int (d \cosh (e+f x))^m \left(a+b \sinh ^2(e+f x)\right)^p \, dx","\frac{d \sinh (e+f x) \cosh ^2(e+f x)^{\frac{1-m}{2}} (d \cosh (e+f x))^{m-1} \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{1-m}{2},-p;\frac{3}{2};-\sinh ^2(e+f x),-\frac{b \sinh ^2(e+f x)}{a}\right)}{f}",1,"Integrate[(d*Cosh[e + f*x])^m*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
401,0,0,214,10.7623989,"\int \cosh ^5(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Cosh[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^p,x]","\int \cosh ^5(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","\frac{\left(3 a^2-2 a b (2 p+5)+b^2 \left(4 p^2+16 p+15\right)\right) \sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \sinh ^2(e+f x)}{a}\right)}{b^2 f (2 p+3) (2 p+5)}-\frac{(3 a-b (2 p+7)) \sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^{p+1}}{b^2 f (2 p+3) (2 p+5)}+\frac{\sinh (e+f x) \cosh ^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{p+1}}{b f (2 p+5)}",1,"Integrate[Cosh[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
402,1,120,125,0.2019175,"\int \cosh ^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Cosh[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^p,x]","\frac{\sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} \left((b (2 p+3)-a) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \sinh ^2(e+f x)}{a}\right)+\left(a+b \sinh ^2(e+f x)\right) \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^p\right)}{b f (2 p+3)}","\frac{\sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^{p+1}}{b f (2 p+3)}-\frac{(a-b (2 p+3)) \sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \sinh ^2(e+f x)}{a}\right)}{b f (2 p+3)}",1,"(Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^p*((-a + b*(3 + 2*p))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Sinh[e + f*x]^2)/a)] + (a + b*Sinh[e + f*x]^2)*(1 + (b*Sinh[e + f*x]^2)/a)^p))/(b*f*(3 + 2*p)*(1 + (b*Sinh[e + f*x]^2)/a)^p)","A",1
403,1,67,67,0.0256966,"\int \cosh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Cosh[e + f*x]*(a + b*Sinh[e + f*x]^2)^p,x]","\frac{\sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \sinh ^2(e+f x)}{a}\right)}{f}","\frac{\sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \sinh ^2(e+f x)}{a}\right)}{f}",1,"(Hypergeometric2F1[1/2, -p, 3/2, -((b*Sinh[e + f*x]^2)/a)]*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^p)/(f*(1 + (b*Sinh[e + f*x]^2)/a)^p)","A",1
404,0,0,78,3.5085537,"\int \text{sech}(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Sech[e + f*x]*(a + b*Sinh[e + f*x]^2)^p,x]","\int \text{sech}(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","\frac{\sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};-\sinh ^2(e+f x),-\frac{b \sinh ^2(e+f x)}{a}\right)}{f}",1,"Integrate[Sech[e + f*x]*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
405,0,0,78,5.7186168,"\int \text{sech}^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Sech[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^p,x]","\int \text{sech}^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","\frac{\sinh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};2,-p;\frac{3}{2};-\sinh ^2(e+f x),-\frac{b \sinh ^2(e+f x)}{a}\right)}{f}",1,"Integrate[Sech[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
406,0,0,92,10.0165178,"\int \cosh ^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Cosh[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^p,x]","\int \cosh ^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","\frac{\sqrt{\cosh ^2(e+f x)} \tanh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-\frac{3}{2},-p;\frac{3}{2};-\sinh ^2(e+f x),-\frac{b \sinh ^2(e+f x)}{a}\right)}{f}",1,"Integrate[Cosh[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
407,0,0,92,10.595646,"\int \cosh ^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Cosh[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^p,x]","\int \cosh ^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","\frac{\sqrt{\cosh ^2(e+f x)} \tanh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-\frac{1}{2},-p;\frac{3}{2};-\sinh ^2(e+f x),-\frac{b \sinh ^2(e+f x)}{a}\right)}{f}",1,"Integrate[Cosh[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
408,0,0,92,0.8931301,"\int \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^p,x]","\int \left(a+b \sinh ^2(e+f x)\right)^p \, dx","\frac{\sqrt{\cosh ^2(e+f x)} \tanh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};-\sinh ^2(e+f x),-\frac{b \sinh ^2(e+f x)}{a}\right)}{f}",1,"Integrate[(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
409,0,0,92,4.4336051,"\int \text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^p,x]","\int \text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","\frac{\sqrt{\cosh ^2(e+f x)} \tanh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{3}{2},-p;\frac{3}{2};-\sinh ^2(e+f x),-\frac{b \sinh ^2(e+f x)}{a}\right)}{f}",1,"Integrate[Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
410,0,0,92,8.4000854,"\int \text{sech}^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","Integrate[Sech[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^p,x]","\int \text{sech}^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \, dx","\frac{\sqrt{\cosh ^2(e+f x)} \tanh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{5}{2},-p;\frac{3}{2};-\sinh ^2(e+f x),-\frac{b \sinh ^2(e+f x)}{a}\right)}{f}",1,"Integrate[Sech[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^p, x]","F",-1
411,1,220,259,0.4565614,"\int \frac{\cosh ^5(c+d x)}{a+b \sqrt{\sinh (c+d x)}} \, dx","Integrate[Cosh[c + d*x]^5/(a + b*Sqrt[Sinh[c + d*x]]),x]","\frac{-630 a b^4 \left(a^4+2 b^4\right) \sinh ^2(c+d x)+2520 b \left(a^4+b^4\right)^2 \sqrt{\sinh (c+d x)}-2520 a \left(a^4+b^4\right)^2 \log \left(a+b \sqrt{\sinh (c+d x)}\right)+504 b^5 \left(a^4+2 b^4\right) \sinh ^{\frac{5}{2}}(c+d x)-420 a^3 b^6 \sinh ^3(c+d x)+360 a^2 b^7 \sinh ^{\frac{7}{2}}(c+d x)-1260 a^3 b^2 \left(a^4+2 b^4\right) \sinh (c+d x)+840 a^2 b^3 \left(a^4+2 b^4\right) \sinh ^{\frac{3}{2}}(c+d x)-315 a b^8 \sinh ^4(c+d x)+280 b^9 \sinh ^{\frac{9}{2}}(c+d x)}{1260 b^{10} d}","-\frac{2 a \left(a^4+b^4\right)^2 \log \left(a+b \sqrt{\sinh (c+d x)}\right)}{b^{10} d}+\frac{2 \left(a^4+b^4\right)^2 \sqrt{\sinh (c+d x)}}{b^9 d}-\frac{a \left(a^4+2 b^4\right) \sinh ^2(c+d x)}{2 b^6 d}+\frac{2 \left(a^4+2 b^4\right) \sinh ^{\frac{5}{2}}(c+d x)}{5 b^5 d}-\frac{a^3 \sinh ^3(c+d x)}{3 b^4 d}+\frac{2 a^2 \sinh ^{\frac{7}{2}}(c+d x)}{7 b^3 d}-\frac{a^3 \left(a^4+2 b^4\right) \sinh (c+d x)}{b^8 d}+\frac{2 a^2 \left(a^4+2 b^4\right) \sinh ^{\frac{3}{2}}(c+d x)}{3 b^7 d}-\frac{a \sinh ^4(c+d x)}{4 b^2 d}+\frac{2 \sinh ^{\frac{9}{2}}(c+d x)}{9 b d}",1,"(-2520*a*(a^4 + b^4)^2*Log[a + b*Sqrt[Sinh[c + d*x]]] + 2520*b*(a^4 + b^4)^2*Sqrt[Sinh[c + d*x]] - 1260*a^3*b^2*(a^4 + 2*b^4)*Sinh[c + d*x] + 840*a^2*b^3*(a^4 + 2*b^4)*Sinh[c + d*x]^(3/2) - 630*a*b^4*(a^4 + 2*b^4)*Sinh[c + d*x]^2 + 504*b^5*(a^4 + 2*b^4)*Sinh[c + d*x]^(5/2) - 420*a^3*b^6*Sinh[c + d*x]^3 + 360*a^2*b^7*Sinh[c + d*x]^(7/2) - 315*a*b^8*Sinh[c + d*x]^4 + 280*b^9*Sinh[c + d*x]^(9/2))/(1260*b^10*d)","A",1
412,1,117,136,0.149469,"\int \frac{\cosh ^3(c+d x)}{a+b \sqrt{\sinh (c+d x)}} \, dx","Integrate[Cosh[c + d*x]^3/(a + b*Sqrt[Sinh[c + d*x]]),x]","\frac{60 b \left(a^4+b^4\right) \sqrt{\sinh (c+d x)}-60 a \left(a^4+b^4\right) \log \left(a+b \sqrt{\sinh (c+d x)}\right)-30 a^3 b^2 \sinh (c+d x)+20 a^2 b^3 \sinh ^{\frac{3}{2}}(c+d x)-15 a b^4 \sinh ^2(c+d x)+12 b^5 \sinh ^{\frac{5}{2}}(c+d x)}{30 b^6 d}","-\frac{2 a \left(a^4+b^4\right) \log \left(a+b \sqrt{\sinh (c+d x)}\right)}{b^6 d}+\frac{2 \left(a^4+b^4\right) \sqrt{\sinh (c+d x)}}{b^5 d}-\frac{a^3 \sinh (c+d x)}{b^4 d}+\frac{2 a^2 \sinh ^{\frac{3}{2}}(c+d x)}{3 b^3 d}-\frac{a \sinh ^2(c+d x)}{2 b^2 d}+\frac{2 \sinh ^{\frac{5}{2}}(c+d x)}{5 b d}",1,"(-60*a*(a^4 + b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]] + 60*b*(a^4 + b^4)*Sqrt[Sinh[c + d*x]] - 30*a^3*b^2*Sinh[c + d*x] + 20*a^2*b^3*Sinh[c + d*x]^(3/2) - 15*a*b^4*Sinh[c + d*x]^2 + 12*b^5*Sinh[c + d*x]^(5/2))/(30*b^6*d)","A",1
413,1,41,43,0.0279857,"\int \frac{\cosh (c+d x)}{a+b \sqrt{\sinh (c+d x)}} \, dx","Integrate[Cosh[c + d*x]/(a + b*Sqrt[Sinh[c + d*x]]),x]","\frac{2 \left(\frac{\sqrt{\sinh (c+d x)}}{b}-\frac{a \log \left(a+b \sqrt{\sinh (c+d x)}\right)}{b^2}\right)}{d}","\frac{2 \sqrt{\sinh (c+d x)}}{b d}-\frac{2 a \log \left(a+b \sqrt{\sinh (c+d x)}\right)}{b^2 d}",1,"(2*(-((a*Log[a + b*Sqrt[Sinh[c + d*x]]])/b^2) + Sqrt[Sinh[c + d*x]]/b))/d","A",1
414,1,229,286,0.2514866,"\int \frac{\text{sech}(c+d x)}{a+b \sqrt{\sinh (c+d x)}} \, dx","Integrate[Sech[c + d*x]/(a + b*Sqrt[Sinh[c + d*x]]),x]","\frac{3 \left(4 a^3 \tan ^{-1}(\sinh (c+d x))-8 a b^2 \log \left(a+b \sqrt{\sinh (c+d x)}\right)+4 a b^2 \log (\cosh (c+d x))-\sqrt{2} b^3 \log \left(\sinh (c+d x)-\sqrt{2} \sqrt{\sinh (c+d x)}+1\right)+\sqrt{2} b^3 \log \left(\sinh (c+d x)+\sqrt{2} \sqrt{\sinh (c+d x)}+1\right)-2 \sqrt{2} b^3 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\sinh (c+d x)}\right)+2 \sqrt{2} b^3 \tan ^{-1}\left(\sqrt{2} \sqrt{\sinh (c+d x)}+1\right)\right)-8 a^2 b \sinh ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\sinh ^2(c+d x)\right)}{12 d \left(a^4+b^4\right)}","-\frac{2 a b^2 \log \left(a+b \sqrt{\sinh (c+d x)}\right)}{d \left(a^4+b^4\right)}+\frac{a b^2 \log (\cosh (c+d x))}{d \left(a^4+b^4\right)}+\frac{a^3 \tan ^{-1}(\sinh (c+d x))}{d \left(a^4+b^4\right)}-\frac{b \left(a^2+b^2\right) \log \left(\sinh (c+d x)-\sqrt{2} \sqrt{\sinh (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^4+b^4\right)}+\frac{b \left(a^2+b^2\right) \log \left(\sinh (c+d x)+\sqrt{2} \sqrt{\sinh (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^4+b^4\right)}+\frac{b \left(a^2-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\sinh (c+d x)}\right)}{\sqrt{2} d \left(a^4+b^4\right)}-\frac{b \left(a^2-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\sinh (c+d x)}+1\right)}{\sqrt{2} d \left(a^4+b^4\right)}",1,"(3*(-2*Sqrt[2]*b^3*ArcTan[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]]] + 2*Sqrt[2]*b^3*ArcTan[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]]] + 4*a^3*ArcTan[Sinh[c + d*x]] + 4*a*b^2*Log[Cosh[c + d*x]] - 8*a*b^2*Log[a + b*Sqrt[Sinh[c + d*x]]] - Sqrt[2]*b^3*Log[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]] + Sinh[c + d*x]] + Sqrt[2]*b^3*Log[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]] + Sinh[c + d*x]]) - 8*a^2*b*Hypergeometric2F1[3/4, 1, 7/4, -Sinh[c + d*x]^2]*Sinh[c + d*x]^(3/2))/(12*(a^4 + b^4)*d)","C",1
415,1,288,270,0.6098958,"\int \frac{\cosh ^5(c+d x)}{\left(a+b \sqrt{\sinh (c+d x)}\right)^2} \, dx","Integrate[Cosh[c + d*x]^5/(a + b*Sqrt[Sinh[c + d*x]])^2,x]","\frac{-70 a b^4 \left(9 a^4+10 b^4\right) \sinh ^2(c+d x)+840 b \left(a^4+b^4\right) \sqrt{\sinh (c+d x)} \left(\left(9 a^4+b^4\right) \log \left(a+b \sqrt{\sinh (c+d x)}\right)-8 a^4\right)+840 a \left(a^4+b^4\right) \left(\left(9 a^4+b^4\right) \log \left(a+b \sqrt{\sinh (c+d x)}\right)+a^4+b^4\right)+42 b^5 \left(9 a^4+10 b^4\right) \sinh ^{\frac{5}{2}}(c+d x)-252 a^3 b^6 \sinh ^3(c+d x)+180 a^2 b^7 \sinh ^{\frac{7}{2}}(c+d x)-420 a^3 b^2 \left(9 a^4+10 b^4\right) \sinh (c+d x)+140 a^2 b^3 \left(9 a^4+10 b^4\right) \sinh ^{\frac{3}{2}}(c+d x)-135 a b^8 \sinh ^4(c+d x)+105 b^9 \sinh ^{\frac{9}{2}}(c+d x)}{420 b^{10} d \left(a+b \sqrt{\sinh (c+d x)}\right)}","\frac{2 a \left(a^4+b^4\right)^2}{b^{10} d \left(a+b \sqrt{\sinh (c+d x)}\right)}+\frac{2 \left(a^4+b^4\right) \left(9 a^4+b^4\right) \log \left(a+b \sqrt{\sinh (c+d x)}\right)}{b^{10} d}-\frac{4 a \left(3 a^4+2 b^4\right) \sinh ^{\frac{3}{2}}(c+d x)}{3 b^7 d}+\frac{\left(5 a^4+2 b^4\right) \sinh ^2(c+d x)}{2 b^6 d}-\frac{8 a^3 \sinh ^{\frac{5}{2}}(c+d x)}{5 b^5 d}+\frac{a^2 \sinh ^3(c+d x)}{b^4 d}-\frac{16 a^3 \left(a^4+b^4\right) \sqrt{\sinh (c+d x)}}{b^9 d}+\frac{a^2 \left(7 a^4+6 b^4\right) \sinh (c+d x)}{b^8 d}-\frac{4 a \sinh ^{\frac{7}{2}}(c+d x)}{7 b^3 d}+\frac{\sinh ^4(c+d x)}{4 b^2 d}",1,"(840*a*(a^4 + b^4)*(a^4 + b^4 + (9*a^4 + b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]]) + 840*b*(a^4 + b^4)*(-8*a^4 + (9*a^4 + b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]])*Sqrt[Sinh[c + d*x]] - 420*a^3*b^2*(9*a^4 + 10*b^4)*Sinh[c + d*x] + 140*a^2*b^3*(9*a^4 + 10*b^4)*Sinh[c + d*x]^(3/2) - 70*a*b^4*(9*a^4 + 10*b^4)*Sinh[c + d*x]^2 + 42*b^5*(9*a^4 + 10*b^4)*Sinh[c + d*x]^(5/2) - 252*a^3*b^6*Sinh[c + d*x]^3 + 180*a^2*b^7*Sinh[c + d*x]^(7/2) - 135*a*b^8*Sinh[c + d*x]^4 + 105*b^9*Sinh[c + d*x]^(9/2))/(420*b^10*d*(a + b*Sqrt[Sinh[c + d*x]]))","A",1
416,1,123,142,0.4720213,"\int \frac{\cosh ^3(c+d x)}{\left(a+b \sqrt{\sinh (c+d x)}\right)^2} \, dx","Integrate[Cosh[c + d*x]^3/(a + b*Sqrt[Sinh[c + d*x]])^2,x]","\frac{12 \left(\frac{a \left(a^4+b^4\right)}{a+b \sqrt{\sinh (c+d x)}}+\left(5 a^4+b^4\right) \log \left(a+b \sqrt{\sinh (c+d x)}\right)\right)-48 a^3 b \sqrt{\sinh (c+d x)}+18 a^2 b^2 \sinh (c+d x)-8 a b^3 \sinh ^{\frac{3}{2}}(c+d x)+3 b^4 \sinh ^2(c+d x)}{6 b^6 d}","\frac{2 a \left(a^4+b^4\right)}{b^6 d \left(a+b \sqrt{\sinh (c+d x)}\right)}+\frac{2 \left(5 a^4+b^4\right) \log \left(a+b \sqrt{\sinh (c+d x)}\right)}{b^6 d}-\frac{8 a^3 \sqrt{\sinh (c+d x)}}{b^5 d}+\frac{3 a^2 \sinh (c+d x)}{b^4 d}-\frac{4 a \sinh ^{\frac{3}{2}}(c+d x)}{3 b^3 d}+\frac{\sinh ^2(c+d x)}{2 b^2 d}",1,"(12*((5*a^4 + b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]] + (a*(a^4 + b^4))/(a + b*Sqrt[Sinh[c + d*x]])) - 48*a^3*b*Sqrt[Sinh[c + d*x]] + 18*a^2*b^2*Sinh[c + d*x] - 8*a*b^3*Sinh[c + d*x]^(3/2) + 3*b^4*Sinh[c + d*x]^2)/(6*b^6*d)","A",1
417,1,42,49,0.0614623,"\int \frac{\cosh (c+d x)}{\left(a+b \sqrt{\sinh (c+d x)}\right)^2} \, dx","Integrate[Cosh[c + d*x]/(a + b*Sqrt[Sinh[c + d*x]])^2,x]","\frac{2 \left(\frac{a}{a+b \sqrt{\sinh (c+d x)}}+\log \left(a+b \sqrt{\sinh (c+d x)}\right)\right)}{b^2 d}","\frac{2 a}{b^2 d \left(a+b \sqrt{\sinh (c+d x)}\right)}+\frac{2 \log \left(a+b \sqrt{\sinh (c+d x)}\right)}{b^2 d}",1,"(2*(Log[a + b*Sqrt[Sinh[c + d*x]]] + a/(a + b*Sqrt[Sinh[c + d*x]])))/(b^2*d)","A",1
418,1,280,384,0.7185777,"\int \frac{\text{sech}(c+d x)}{\left(a+b \sqrt{\sinh (c+d x)}\right)^2} \, dx","Integrate[Sech[c + d*x]/(a + b*Sqrt[Sinh[c + d*x]])^2,x]","\frac{-4 a b \left(a^4-b^4\right) \sinh ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\sinh ^2(c+d x)\right)+\frac{6 a b^2 \left(a^4+b^4\right)}{a+b \sqrt{\sinh (c+d x)}}+6 b^2 \left(b^4-3 a^4\right) \log \left(a+b \sqrt{\sinh (c+d x)}\right)-3 b^2 \left(b^4-3 a^4\right) \log (\cosh (c+d x))-3 \sqrt{2} a^3 b^3 \left(\log \left(\sinh (c+d x)-\sqrt{2} \sqrt{\sinh (c+d x)}+1\right)-\log \left(\sinh (c+d x)+\sqrt{2} \sqrt{\sinh (c+d x)}+1\right)\right)-6 \sqrt{2} a^3 b^3 \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\sinh (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\sinh (c+d x)}+1\right)\right)+3 a^2 \left(a^4-3 b^4\right) \tan ^{-1}(\sinh (c+d x))}{3 d \left(a^4+b^4\right)^2}","\frac{2 a b^2}{d \left(a^4+b^4\right) \left(a+b \sqrt{\sinh (c+d x)}\right)}-\frac{2 b^2 \left(3 a^4-b^4\right) \log \left(a+b \sqrt{\sinh (c+d x)}\right)}{d \left(a^4+b^4\right)^2}+\frac{b^2 \left(3 a^4-b^4\right) \log (\cosh (c+d x))}{d \left(a^4+b^4\right)^2}+\frac{a^2 \left(a^4-3 b^4\right) \tan ^{-1}(\sinh (c+d x))}{d \left(a^4+b^4\right)^2}-\frac{a b \left(a^4+2 a^2 b^2-b^4\right) \log \left(\sinh (c+d x)-\sqrt{2} \sqrt{\sinh (c+d x)}+1\right)}{\sqrt{2} d \left(a^4+b^4\right)^2}+\frac{a b \left(a^4+2 a^2 b^2-b^4\right) \log \left(\sinh (c+d x)+\sqrt{2} \sqrt{\sinh (c+d x)}+1\right)}{\sqrt{2} d \left(a^4+b^4\right)^2}+\frac{\sqrt{2} a b \left(a^4-2 a^2 b^2-b^4\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\sinh (c+d x)}\right)}{d \left(a^4+b^4\right)^2}-\frac{\sqrt{2} a b \left(a^4-2 a^2 b^2-b^4\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\sinh (c+d x)}+1\right)}{d \left(a^4+b^4\right)^2}",1,"(-6*Sqrt[2]*a^3*b^3*(ArcTan[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]]]) + 3*a^2*(a^4 - 3*b^4)*ArcTan[Sinh[c + d*x]] - 3*b^2*(-3*a^4 + b^4)*Log[Cosh[c + d*x]] + 6*b^2*(-3*a^4 + b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]] - 3*Sqrt[2]*a^3*b^3*(Log[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]] + Sinh[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]] + Sinh[c + d*x]]) + (6*a*b^2*(a^4 + b^4))/(a + b*Sqrt[Sinh[c + d*x]]) - 4*a*b*(a^4 - b^4)*Hypergeometric2F1[3/4, 1, 7/4, -Sinh[c + d*x]^2]*Sinh[c + d*x]^(3/2))/(3*(a^4 + b^4)^2*d)","C",1
419,1,119,130,0.1441235,"\int \frac{\cosh ^5(c+d x)}{a+b \sinh ^n(c+d x)} \, dx","Integrate[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^n),x]","\frac{15 \sinh (c+d x) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)+3 \sinh ^5(c+d x) \, _2F_1\left(1,\frac{5}{n};\frac{n+5}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)+10 \sinh ^3(c+d x) \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{15 a d}","\frac{\sinh (c+d x) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{a d}+\frac{\sinh ^5(c+d x) \, _2F_1\left(1,\frac{5}{n};\frac{n+5}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{5 a d}+\frac{2 \sinh ^3(c+d x) \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{3 a d}",1,"(15*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x] + 10*Hypergeometric2F1[1, 3/n, (3 + n)/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x]^3 + 3*Hypergeometric2F1[1, 5/n, (5 + n)/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x]^5)/(15*a*d)","A",1
420,1,82,84,0.0555155,"\int \frac{\cosh ^3(c+d x)}{a+b \sinh ^n(c+d x)} \, dx","Integrate[Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^n),x]","\frac{\frac{\sinh (c+d x) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{a}+\frac{\sinh ^3(c+d x) \, _2F_1\left(1,\frac{3}{n};1+\frac{3}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{3 a}}{d}","\frac{\sinh (c+d x) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{a d}+\frac{\sinh ^3(c+d x) \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{3 a d}",1,"((Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x])/a + (Hypergeometric2F1[1, 3/n, 1 + 3/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x]^3)/(3*a))/d","A",1
421,1,37,37,0.0091735,"\int \frac{\cosh (c+d x)}{a+b \sinh ^n(c+d x)} \, dx","Integrate[Cosh[c + d*x]/(a + b*Sinh[c + d*x]^n),x]","\frac{\sinh (c+d x) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{a d}","\frac{\sinh (c+d x) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{a d}",1,"(Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x])/(a*d)","A",1
422,1,119,130,0.1171299,"\int \frac{\cosh ^5(c+d x)}{\left(a+b \sinh ^n(c+d x)\right)^2} \, dx","Integrate[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^n)^2,x]","\frac{15 \sinh (c+d x) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)+3 \sinh ^5(c+d x) \, _2F_1\left(2,\frac{5}{n};\frac{n+5}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)+10 \sinh ^3(c+d x) \, _2F_1\left(2,\frac{3}{n};\frac{n+3}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{15 a^2 d}","\frac{\sinh (c+d x) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{a^2 d}+\frac{\sinh ^5(c+d x) \, _2F_1\left(2,\frac{5}{n};\frac{n+5}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{5 a^2 d}+\frac{2 \sinh ^3(c+d x) \, _2F_1\left(2,\frac{3}{n};\frac{n+3}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{3 a^2 d}",1,"(15*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x] + 10*Hypergeometric2F1[2, 3/n, (3 + n)/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x]^3 + 3*Hypergeometric2F1[2, 5/n, (5 + n)/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x]^5)/(15*a^2*d)","A",1
423,1,82,84,0.0542712,"\int \frac{\cosh ^3(c+d x)}{\left(a+b \sinh ^n(c+d x)\right)^2} \, dx","Integrate[Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^n)^2,x]","\frac{\frac{\sinh (c+d x) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{a^2}+\frac{\sinh ^3(c+d x) \, _2F_1\left(2,\frac{3}{n};1+\frac{3}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{3 a^2}}{d}","\frac{\sinh (c+d x) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{a^2 d}+\frac{\sinh ^3(c+d x) \, _2F_1\left(2,\frac{3}{n};\frac{n+3}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{3 a^2 d}",1,"((Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x])/a^2 + (Hypergeometric2F1[2, 3/n, 1 + 3/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x]^3)/(3*a^2))/d","A",1
424,1,37,37,0.0086691,"\int \frac{\cosh (c+d x)}{\left(a+b \sinh ^n(c+d x)\right)^2} \, dx","Integrate[Cosh[c + d*x]/(a + b*Sinh[c + d*x]^n)^2,x]","\frac{\sinh (c+d x) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{a^2 d}","\frac{\sinh (c+d x) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{b \sinh ^n(c+d x)}{a}\right)}{a^2 d}",1,"(Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x])/(a^2*d)","A",1
425,1,23,17,0.0140356,"\int \frac{\coth (x)}{1-\sinh ^2(x)} \, dx","Integrate[Coth[x]/(1 - Sinh[x]^2),x]","-2 \left(\frac{1}{4} \log \left(1-\sinh ^2(x)\right)-\frac{1}{2} \log (\sinh (x))\right)","\log (\sinh (x))-\frac{1}{2} \log \left(1-\sinh ^2(x)\right)",1,"-2*(-1/2*Log[Sinh[x]] + Log[1 - Sinh[x]^2]/4)","A",1
426,1,51,63,0.1002932,"\int \sqrt{a+a \sinh ^2(e+f x)} \tanh ^5(e+f x) \, dx","Integrate[Sqrt[a + a*Sinh[e + f*x]^2]*Tanh[e + f*x]^5,x]","\frac{\left(3 \cosh ^4(e+f x)+6 \cosh ^2(e+f x)-1\right) \text{sech}^4(e+f x) \sqrt{a \cosh ^2(e+f x)}}{3 f}","-\frac{a^2}{3 f \left(a \cosh ^2(e+f x)\right)^{3/2}}+\frac{2 a}{f \sqrt{a \cosh ^2(e+f x)}}+\frac{\sqrt{a \cosh ^2(e+f x)}}{f}",1,"(Sqrt[a*Cosh[e + f*x]^2]*(-1 + 6*Cosh[e + f*x]^2 + 3*Cosh[e + f*x]^4)*Sech[e + f*x]^4)/(3*f)","A",1
427,1,29,38,0.0858908,"\int \sqrt{a+a \sinh ^2(e+f x)} \tanh ^3(e+f x) \, dx","Integrate[Sqrt[a + a*Sinh[e + f*x]^2]*Tanh[e + f*x]^3,x]","\frac{a \left(\cosh ^2(e+f x)+1\right)}{f \sqrt{a \cosh ^2(e+f x)}}","\frac{a}{f \sqrt{a \cosh ^2(e+f x)}}+\frac{\sqrt{a \cosh ^2(e+f x)}}{f}",1,"(a*(1 + Cosh[e + f*x]^2))/(f*Sqrt[a*Cosh[e + f*x]^2])","A",1
428,1,18,18,0.039934,"\int \sqrt{a+a \sinh ^2(e+f x)} \tanh (e+f x) \, dx","Integrate[Sqrt[a + a*Sinh[e + f*x]^2]*Tanh[e + f*x],x]","\frac{\sqrt{a \cosh ^2(e+f x)}}{f}","\frac{\sqrt{a \cosh ^2(e+f x)}}{f}",1,"Sqrt[a*Cosh[e + f*x]^2]/f","A",1
429,1,42,50,0.0638498,"\int \coth (e+f x) \sqrt{a+a \sinh ^2(e+f x)} \, dx","Integrate[Coth[e + f*x]*Sqrt[a + a*Sinh[e + f*x]^2],x]","\frac{\text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)} \left(\cosh (e+f x)+\log \left(\tanh \left(\frac{1}{2} (e+f x)\right)\right)\right)}{f}","\frac{\sqrt{a \cosh ^2(e+f x)}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right)}{f}",1,"(Sqrt[a*Cosh[e + f*x]^2]*(Cosh[e + f*x] + Log[Tanh[(e + f*x)/2]])*Sech[e + f*x])/f","A",1
430,1,77,87,0.2620278,"\int \coth ^3(e+f x) \sqrt{a+a \sinh ^2(e+f x)} \, dx","Integrate[Coth[e + f*x]^3*Sqrt[a + a*Sinh[e + f*x]^2],x]","\frac{\text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)} \left(8 \cosh (e+f x)-\text{csch}^2\left(\frac{1}{2} (e+f x)\right)-\text{sech}^2\left(\frac{1}{2} (e+f x)\right)+12 \log \left(\tanh \left(\frac{1}{2} (e+f x)\right)\right)\right)}{8 f}","\frac{3 \sqrt{a \cosh ^2(e+f x)}}{2 f}-\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right)}{2 f}-\frac{\text{csch}^2(e+f x) \left(a \cosh ^2(e+f x)\right)^{3/2}}{2 a f}",1,"(Sqrt[a*Cosh[e + f*x]^2]*(8*Cosh[e + f*x] - Csch[(e + f*x)/2]^2 + 12*Log[Tanh[(e + f*x)/2]] - Sech[(e + f*x)/2]^2)*Sech[e + f*x])/(8*f)","A",1
431,1,75,120,0.3603335,"\int \sqrt{a+a \sinh ^2(e+f x)} \tanh ^6(e+f x) \, dx","Integrate[Sqrt[a + a*Sinh[e + f*x]^2]*Tanh[e + f*x]^6,x]","-\frac{\text{sech}^5(e+f x) \sqrt{a \cosh ^2(e+f x)} \left(-5 \sinh (e+f x)-15 \sinh (3 (e+f x))-2 \sinh (5 (e+f x))+60 \cosh ^4(e+f x) \tan ^{-1}(\sinh (e+f x))\right)}{32 f}","-\frac{\tanh ^5(e+f x) \sqrt{a \cosh ^2(e+f x)}}{4 f}-\frac{5 \tanh ^3(e+f x) \sqrt{a \cosh ^2(e+f x)}}{8 f}+\frac{15 \tanh (e+f x) \sqrt{a \cosh ^2(e+f x)}}{8 f}-\frac{15 \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)} \tan ^{-1}(\sinh (e+f x))}{8 f}",1,"-1/32*(Sqrt[a*Cosh[e + f*x]^2]*Sech[e + f*x]^5*(60*ArcTan[Sinh[e + f*x]]*Cosh[e + f*x]^4 - 5*Sinh[e + f*x] - 15*Sinh[3*(e + f*x)] - 2*Sinh[5*(e + f*x)]))/f","A",1
432,1,55,91,0.2004752,"\int \sqrt{a+a \sinh ^2(e+f x)} \tanh ^4(e+f x) \, dx","Integrate[Sqrt[a + a*Sinh[e + f*x]^2]*Tanh[e + f*x]^4,x]","\frac{a \left((\cosh (2 (e+f x))+2) \tanh (e+f x)-3 \cosh (e+f x) \tan ^{-1}(\sinh (e+f x))\right)}{2 f \sqrt{a \cosh ^2(e+f x)}}","-\frac{\tanh ^3(e+f x) \sqrt{a \cosh ^2(e+f x)}}{2 f}+\frac{3 \tanh (e+f x) \sqrt{a \cosh ^2(e+f x)}}{2 f}-\frac{3 \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)} \tan ^{-1}(\sinh (e+f x))}{2 f}",1,"(a*(-3*ArcTan[Sinh[e + f*x]]*Cosh[e + f*x] + (2 + Cosh[2*(e + f*x)])*Tanh[e + f*x]))/(2*f*Sqrt[a*Cosh[e + f*x]^2])","A",1
433,1,40,57,0.0537655,"\int \sqrt{a+a \sinh ^2(e+f x)} \tanh ^2(e+f x) \, dx","Integrate[Sqrt[a + a*Sinh[e + f*x]^2]*Tanh[e + f*x]^2,x]","\frac{\text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)} \left(\sinh (e+f x)-\tan ^{-1}(\sinh (e+f x))\right)}{f}","\frac{\tanh (e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}-\frac{\text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)} \tan ^{-1}(\sinh (e+f x))}{f}",1,"(Sqrt[a*Cosh[e + f*x]^2]*Sech[e + f*x]*(-ArcTan[Sinh[e + f*x]] + Sinh[e + f*x]))/f","A",1
434,1,35,56,0.075232,"\int \coth ^2(e+f x) \sqrt{a+a \sinh ^2(e+f x)} \, dx","Integrate[Coth[e + f*x]^2*Sqrt[a + a*Sinh[e + f*x]^2],x]","-\frac{\tanh (e+f x) \left(\text{csch}^2(e+f x)-1\right) \sqrt{a \cosh ^2(e+f x)}}{f}","\frac{\tanh (e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}-\frac{\text{csch}(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}",1,"-((Sqrt[a*Cosh[e + f*x]^2]*(-1 + Csch[e + f*x]^2)*Tanh[e + f*x])/f)","A",1
435,1,47,91,0.0784563,"\int \coth ^4(e+f x) \sqrt{a+a \sinh ^2(e+f x)} \, dx","Integrate[Coth[e + f*x]^4*Sqrt[a + a*Sinh[e + f*x]^2],x]","-\frac{\tanh (e+f x) \left(\text{csch}^4(e+f x)+6 \text{csch}^2(e+f x)-3\right) \sqrt{a \cosh ^2(e+f x)}}{3 f}","\frac{\tanh (e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}-\frac{\text{csch}^3(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{3 f}-\frac{2 \text{csch}(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}",1,"-1/3*(Sqrt[a*Cosh[e + f*x]^2]*(-3 + 6*Csch[e + f*x]^2 + Csch[e + f*x]^4)*Tanh[e + f*x])/f","A",1
436,1,67,124,0.2010462,"\int \coth ^6(e+f x) \sqrt{a+a \sinh ^2(e+f x)} \, dx","Integrate[Coth[e + f*x]^6*Sqrt[a + a*Sinh[e + f*x]^2],x]","\frac{(235 \cosh (2 (e+f x))-90 \cosh (4 (e+f x))+5 \cosh (6 (e+f x))-182) \text{csch}^5(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{160 f}","\frac{\tanh (e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}-\frac{\text{csch}^5(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{5 f}-\frac{\text{csch}^3(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}-\frac{3 \text{csch}(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}",1,"(Sqrt[a*Cosh[e + f*x]^2]*(-182 + 235*Cosh[2*(e + f*x)] - 90*Cosh[4*(e + f*x)] + 5*Cosh[6*(e + f*x)])*Csch[e + f*x]^5*Sech[e + f*x])/(160*f)","A",1
437,1,43,66,0.0947867,"\int \frac{\tanh ^5(e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx","Integrate[Tanh[e + f*x]^5/Sqrt[a + a*Sinh[e + f*x]^2],x]","\frac{-3 \text{sech}^4(e+f x)+10 \text{sech}^2(e+f x)-15}{15 f \sqrt{a \cosh ^2(e+f x)}}","-\frac{a^2}{5 f \left(a \cosh ^2(e+f x)\right)^{5/2}}+\frac{2 a}{3 f \left(a \cosh ^2(e+f x)\right)^{3/2}}-\frac{1}{f \sqrt{a \cosh ^2(e+f x)}}",1,"(-15 + 10*Sech[e + f*x]^2 - 3*Sech[e + f*x]^4)/(15*f*Sqrt[a*Cosh[e + f*x]^2])","A",1
438,1,31,42,0.0757981,"\int \frac{\tanh ^3(e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx","Integrate[Tanh[e + f*x]^3/Sqrt[a + a*Sinh[e + f*x]^2],x]","\frac{\text{sech}^2(e+f x)-3}{3 f \sqrt{a \cosh ^2(e+f x)}}","\frac{a}{3 f \left(a \cosh ^2(e+f x)\right)^{3/2}}-\frac{1}{f \sqrt{a \cosh ^2(e+f x)}}",1,"(-3 + Sech[e + f*x]^2)/(3*f*Sqrt[a*Cosh[e + f*x]^2])","A",1
439,1,19,19,0.0318911,"\int \frac{\tanh (e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx","Integrate[Tanh[e + f*x]/Sqrt[a + a*Sinh[e + f*x]^2],x]","-\frac{1}{f \sqrt{a \cosh ^2(e+f x)}}","-\frac{1}{f \sqrt{a \cosh ^2(e+f x)}}",1,"-(1/(f*Sqrt[a*Cosh[e + f*x]^2]))","A",1
440,1,49,31,0.0541871,"\int \frac{\coth (e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx","Integrate[Coth[e + f*x]/Sqrt[a + a*Sinh[e + f*x]^2],x]","\frac{\cosh (e+f x) \left(\log \left(\sinh \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\cosh \left(\frac{1}{2} (e+f x)\right)\right)\right)}{f \sqrt{a \cosh ^2(e+f x)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}",1,"(Cosh[e + f*x]*(-Log[Cosh[(e + f*x)/2]] + Log[Sinh[(e + f*x)/2]]))/(f*Sqrt[a*Cosh[e + f*x]^2])","A",1
441,1,65,66,0.1227225,"\int \frac{\coth ^3(e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx","Integrate[Coth[e + f*x]^3/Sqrt[a + a*Sinh[e + f*x]^2],x]","-\frac{\cosh (e+f x) \left(\text{csch}^2\left(\frac{1}{2} (e+f x)\right)+\text{sech}^2\left(\frac{1}{2} (e+f x)\right)-4 \log \left(\tanh \left(\frac{1}{2} (e+f x)\right)\right)\right)}{8 f \sqrt{a \cosh ^2(e+f x)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right)}{2 \sqrt{a} f}-\frac{\text{csch}^2(e+f x) \sqrt{a \cosh ^2(e+f x)}}{2 a f}",1,"-1/8*(Cosh[e + f*x]*(Csch[(e + f*x)/2]^2 - 4*Log[Tanh[(e + f*x)/2]] + Sech[(e + f*x)/2]^2))/(f*Sqrt[a*Cosh[e + f*x]^2])","A",1
442,1,66,91,0.1392051,"\int \frac{\tanh ^4(e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx","Integrate[Tanh[e + f*x]^4/Sqrt[a + a*Sinh[e + f*x]^2],x]","\frac{\tanh (e+f x) \left(-8 \tanh ^2(e+f x)-6 \text{sech}^2(e+f x)+3\right)+3 \cosh (e+f x) \tan ^{-1}(\sinh (e+f x))}{8 f \sqrt{a \cosh ^2(e+f x)}}","-\frac{\tanh ^3(e+f x)}{4 f \sqrt{a \cosh ^2(e+f x)}}-\frac{3 \tanh (e+f x)}{8 f \sqrt{a \cosh ^2(e+f x)}}+\frac{3 \cosh (e+f x) \tan ^{-1}(\sinh (e+f x))}{8 f \sqrt{a \cosh ^2(e+f x)}}",1,"(3*ArcTan[Sinh[e + f*x]]*Cosh[e + f*x] + Tanh[e + f*x]*(3 - 6*Sech[e + f*x]^2 - 8*Tanh[e + f*x]^2))/(8*f*Sqrt[a*Cosh[e + f*x]^2])","A",1
443,1,44,62,0.0583087,"\int \frac{\tanh ^2(e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx","Integrate[Tanh[e + f*x]^2/Sqrt[a + a*Sinh[e + f*x]^2],x]","\frac{\cosh (e+f x) \tan ^{-1}(\sinh (e+f x))-\tanh (e+f x)}{2 f \sqrt{a \cosh ^2(e+f x)}}","\frac{\cosh (e+f x) \tan ^{-1}(\sinh (e+f x))}{2 f \sqrt{a \cosh ^2(e+f x)}}-\frac{\tanh (e+f x)}{2 f \sqrt{a \cosh ^2(e+f x)}}",1,"(ArcTan[Sinh[e + f*x]]*Cosh[e + f*x] - Tanh[e + f*x])/(2*f*Sqrt[a*Cosh[e + f*x]^2])","A",1
444,1,25,25,0.0404976,"\int \frac{\coth ^2(e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx","Integrate[Coth[e + f*x]^2/Sqrt[a + a*Sinh[e + f*x]^2],x]","-\frac{\coth (e+f x)}{f \sqrt{a \cosh ^2(e+f x)}}","-\frac{\coth (e+f x)}{f \sqrt{a \cosh ^2(e+f x)}}",1,"-(Coth[e + f*x]/(f*Sqrt[a*Cosh[e + f*x]^2]))","A",1
445,1,37,61,0.0653945,"\int \frac{\coth ^4(e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx","Integrate[Coth[e + f*x]^4/Sqrt[a + a*Sinh[e + f*x]^2],x]","-\frac{\coth (e+f x) \left(\text{csch}^2(e+f x)+3\right)}{3 f \sqrt{a \cosh ^2(e+f x)}}","-\frac{\coth (e+f x)}{f \sqrt{a \cosh ^2(e+f x)}}-\frac{\coth (e+f x) \text{csch}^2(e+f x)}{3 f \sqrt{a \cosh ^2(e+f x)}}",1,"-1/3*(Coth[e + f*x]*(3 + Csch[e + f*x]^2))/(f*Sqrt[a*Cosh[e + f*x]^2])","A",1
446,1,49,96,0.0922733,"\int \frac{\coth ^6(e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx","Integrate[Coth[e + f*x]^6/Sqrt[a + a*Sinh[e + f*x]^2],x]","-\frac{\coth (e+f x) \left(3 \text{csch}^4(e+f x)+10 \text{csch}^2(e+f x)+15\right)}{15 f \sqrt{a \cosh ^2(e+f x)}}","-\frac{\coth (e+f x)}{f \sqrt{a \cosh ^2(e+f x)}}-\frac{\coth (e+f x) \text{csch}^4(e+f x)}{5 f \sqrt{a \cosh ^2(e+f x)}}-\frac{2 \coth (e+f x) \text{csch}^2(e+f x)}{3 f \sqrt{a \cosh ^2(e+f x)}}",1,"-1/15*(Coth[e + f*x]*(15 + 10*Csch[e + f*x]^2 + 3*Csch[e + f*x]^4))/(f*Sqrt[a*Cosh[e + f*x]^2])","A",1
447,1,51,68,0.1183661,"\int \frac{\tanh ^5(e+f x)}{\left(a+a \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tanh[e + f*x]^5/(a + a*Sinh[e + f*x]^2)^(3/2),x]","\frac{\left(-35 \cosh ^4(e+f x)+42 \cosh ^2(e+f x)-15\right) \text{sech}^4(e+f x)}{105 f \left(a \cosh ^2(e+f x)\right)^{3/2}}","-\frac{a^2}{7 f \left(a \cosh ^2(e+f x)\right)^{7/2}}+\frac{2 a}{5 f \left(a \cosh ^2(e+f x)\right)^{5/2}}-\frac{1}{3 f \left(a \cosh ^2(e+f x)\right)^{3/2}}",1,"((-15 + 42*Cosh[e + f*x]^2 - 35*Cosh[e + f*x]^4)*Sech[e + f*x]^4)/(105*f*(a*Cosh[e + f*x]^2)^(3/2))","A",1
448,1,34,44,0.1268913,"\int \frac{\tanh ^3(e+f x)}{\left(a+a \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tanh[e + f*x]^3/(a + a*Sinh[e + f*x]^2)^(3/2),x]","\frac{a \left(3-5 \cosh ^2(e+f x)\right)}{15 f \left(a \cosh ^2(e+f x)\right)^{5/2}}","\frac{a}{5 f \left(a \cosh ^2(e+f x)\right)^{5/2}}-\frac{1}{3 f \left(a \cosh ^2(e+f x)\right)^{3/2}}",1,"(a*(3 - 5*Cosh[e + f*x]^2))/(15*f*(a*Cosh[e + f*x]^2)^(5/2))","A",1
449,1,21,21,0.0398827,"\int \frac{\tanh (e+f x)}{\left(a+a \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tanh[e + f*x]/(a + a*Sinh[e + f*x]^2)^(3/2),x]","-\frac{1}{3 f \left(a \cosh ^2(e+f x)\right)^{3/2}}","-\frac{1}{3 f \left(a \cosh ^2(e+f x)\right)^{3/2}}",1,"-1/3*1/(f*(a*Cosh[e + f*x]^2)^(3/2))","A",1
450,1,41,53,0.0643192,"\int \frac{\coth (e+f x)}{\left(a+a \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Coth[e + f*x]/(a + a*Sinh[e + f*x]^2)^(3/2),x]","\frac{\cosh (e+f x) \log \left(\tanh \left(\frac{1}{2} (e+f x)\right)\right)+1}{a f \sqrt{a \cosh ^2(e+f x)}}","\frac{1}{a f \sqrt{a \cosh ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}",1,"(1 + Cosh[e + f*x]*Log[Tanh[(e + f*x)/2]])/(a*f*Sqrt[a*Cosh[e + f*x]^2])","A",1
451,1,67,66,0.1341689,"\int \frac{\coth ^3(e+f x)}{\left(a+a \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Coth[e + f*x]^3/(a + a*Sinh[e + f*x]^2)^(3/2),x]","-\frac{\cosh ^3(e+f x) \left(\text{csch}^2\left(\frac{1}{2} (e+f x)\right)+\text{sech}^2\left(\frac{1}{2} (e+f x)\right)+4 \log \left(\tanh \left(\frac{1}{2} (e+f x)\right)\right)\right)}{8 f \left(a \cosh ^2(e+f x)\right)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{3/2} f}-\frac{\text{csch}^2(e+f x) \sqrt{a \cosh ^2(e+f x)}}{2 a^2 f}",1,"-1/8*(Cosh[e + f*x]^3*(Csch[(e + f*x)/2]^2 + 4*Log[Tanh[(e + f*x)/2]] + Sech[(e + f*x)/2]^2))/(f*(a*Cosh[e + f*x]^2)^(3/2))","A",1
452,1,58,106,0.1026201,"\int \frac{\tanh ^2(e+f x)}{\left(a+a \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tanh[e + f*x]^2/(a + a*Sinh[e + f*x]^2)^(3/2),x]","\frac{\tanh (e+f x) \left(1-2 \text{sech}^2(e+f x)\right)+\cosh (e+f x) \tan ^{-1}(\sinh (e+f x))}{8 a f \sqrt{a \cosh ^2(e+f x)}}","\frac{\tanh (e+f x)}{8 a f \sqrt{a \cosh ^2(e+f x)}}+\frac{\cosh (e+f x) \tan ^{-1}(\sinh (e+f x))}{8 a f \sqrt{a \cosh ^2(e+f x)}}-\frac{\tanh (e+f x) \text{sech}^2(e+f x)}{4 a f \sqrt{a \cosh ^2(e+f x)}}",1,"(ArcTan[Sinh[e + f*x]]*Cosh[e + f*x] + (1 - 2*Sech[e + f*x]^2)*Tanh[e + f*x])/(8*a*f*Sqrt[a*Cosh[e + f*x]^2])","A",1
453,1,46,64,0.0838309,"\int \frac{\coth ^2(e+f x)}{\left(a+a \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Coth[e + f*x]^2/(a + a*Sinh[e + f*x]^2)^(3/2),x]","-\frac{\coth (e+f x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\sinh ^2(e+f x)\right)}{a f \sqrt{a \cosh ^2(e+f x)}}","-\frac{\coth (e+f x)}{a f \sqrt{a \cosh ^2(e+f x)}}-\frac{\cosh (e+f x) \tan ^{-1}(\sinh (e+f x))}{a f \sqrt{a \cosh ^2(e+f x)}}",1,"-((Coth[e + f*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Sinh[e + f*x]^2])/(a*f*Sqrt[a*Cosh[e + f*x]^2]))","C",1
454,1,29,38,0.0511495,"\int \frac{\coth ^4(e+f x)}{\left(a+a \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Coth[e + f*x]^4/(a + a*Sinh[e + f*x]^2)^(3/2),x]","-\frac{\coth ^3(e+f x)}{3 f \left(a \cosh ^2(e+f x)\right)^{3/2}}","-\frac{\coth (e+f x) \text{csch}^2(e+f x)}{3 a f \sqrt{a \cosh ^2(e+f x)}}",1,"-1/3*Coth[e + f*x]^3/(f*(a*Cosh[e + f*x]^2)^(3/2))","A",1
455,1,41,77,0.1088998,"\int \frac{\coth ^6(e+f x)}{\left(a+a \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Coth[e + f*x]^6/(a + a*Sinh[e + f*x]^2)^(3/2),x]","-\frac{\coth ^3(e+f x) \left(3 \text{csch}^2(e+f x)+5\right)}{15 f \left(a \cosh ^2(e+f x)\right)^{3/2}}","-\frac{\coth (e+f x) \text{csch}^4(e+f x)}{5 a f \sqrt{a \cosh ^2(e+f x)}}-\frac{\coth (e+f x) \text{csch}^2(e+f x)}{3 a f \sqrt{a \cosh ^2(e+f x)}}",1,"-1/15*(Coth[e + f*x]^3*(5 + 3*Csch[e + f*x]^2))/(f*(a*Cosh[e + f*x]^2)^(3/2))","A",1
456,1,51,115,0.1421001,"\int \frac{\coth ^8(e+f x)}{\left(a+a \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Coth[e + f*x]^8/(a + a*Sinh[e + f*x]^2)^(3/2),x]","-\frac{\coth ^3(e+f x) \left(15 \text{csch}^4(e+f x)+42 \text{csch}^2(e+f x)+35\right)}{105 f \left(a \cosh ^2(e+f x)\right)^{3/2}}","-\frac{\coth (e+f x) \text{csch}^6(e+f x)}{7 a f \sqrt{a \cosh ^2(e+f x)}}-\frac{2 \coth (e+f x) \text{csch}^4(e+f x)}{5 a f \sqrt{a \cosh ^2(e+f x)}}-\frac{\coth (e+f x) \text{csch}^2(e+f x)}{3 a f \sqrt{a \cosh ^2(e+f x)}}",1,"-1/105*(Coth[e + f*x]^3*(35 + 42*Csch[e + f*x]^2 + 15*Csch[e + f*x]^4))/(f*(a*Cosh[e + f*x]^2)^(3/2))","A",1
457,1,151,187,0.5638982,"\int \sqrt{a+b \sinh ^2(e+f x)} \tanh ^5(e+f x) \, dx","Integrate[Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]^5,x]","-\frac{\left(8 a^2-24 a b+15 b^2\right) \left(\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)-\sqrt{a+b \sinh ^2(e+f x)}\right)+2 (a-b) \text{sech}^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}-(8 a-7 b) \text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{8 f (a-b)^2}","\frac{\left(8 a^2-24 a b+15 b^2\right) \sqrt{a+b \sinh ^2(e+f x)}}{8 f (a-b)^2}-\frac{\left(8 a^2-24 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{8 f (a-b)^{3/2}}-\frac{\text{sech}^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{4 f (a-b)}+\frac{(8 a-7 b) \text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{8 f (a-b)^2}",1,"-1/8*(-((8*a - 7*b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2)) + 2*(a - b)*Sech[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2) + (8*a^2 - 24*a*b + 15*b^2)*(Sqrt[a - b]*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]] - Sqrt[a + b*Sinh[e + f*x]^2]))/((a - b)^2*f)","A",1
458,1,88,126,0.4223882,"\int \sqrt{a+b \sinh ^2(e+f x)} \tanh ^3(e+f x) \, dx","Integrate[Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]^3,x]","\frac{(\cosh (2 (e+f x))+2) \text{sech}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}-\frac{(2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}}{2 f}","\frac{(2 a-3 b) \sqrt{a+b \sinh ^2(e+f x)}}{2 f (a-b)}-\frac{(2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{2 f \sqrt{a-b}}+\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{2 f (a-b)}",1,"(-(((2*a - 3*b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/Sqrt[a - b]) + (2 + Cosh[2*(e + f*x)])*Sech[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])/(2*f)","A",1
459,1,65,62,0.0542701,"\int \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x) \, dx","Integrate[Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x],x]","\frac{\sqrt{a+b \cosh ^2(e+f x)-b}-\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \cosh ^2(e+f x)-b}}{\sqrt{a-b}}\right)}{f}","\frac{\sqrt{a+b \sinh ^2(e+f x)}}{f}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"(-(Sqrt[a - b]*ArcTanh[Sqrt[a - b + b*Cosh[e + f*x]^2]/Sqrt[a - b]]) + Sqrt[a - b + b*Cosh[e + f*x]^2])/f","A",1
460,1,53,54,0.050151,"\int \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)-\sqrt{a+b \sinh ^2(e+f x)}}{f}","\frac{\sqrt{a+b \sinh ^2(e+f x)}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{f}",1,"-((Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]] - Sqrt[a + b*Sinh[e + f*x]^2])/f)","A",1
461,1,69,106,0.3789412,"\int \coth ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Coth[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a}}+\left(\text{csch}^2(e+f x)-2\right) \sqrt{a+b \sinh ^2(e+f x)}}{2 f}","\frac{(2 a+b) \sqrt{a+b \sinh ^2(e+f x)}}{2 a f}-\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{2 \sqrt{a} f}-\frac{\text{csch}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{2 a f}",1,"-1/2*(((2*a + b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/Sqrt[a] + (-2 + Csch[e + f*x]^2)*Sqrt[a + b*Sinh[e + f*x]^2])/f","A",1
462,1,102,167,0.5800315,"\int \coth ^5(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Coth[e + f*x]^5*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\left(-8 a^2-8 a b+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)-\sqrt{a} \sqrt{a+b \sinh ^2(e+f x)} \left((8 a+b) \text{csch}^2(e+f x)+2 a \text{csch}^4(e+f x)-8 a\right)}{8 a^{3/2} f}","\frac{\left(8 a^2+8 a b-b^2\right) \sqrt{a+b \sinh ^2(e+f x)}}{8 a^2 f}-\frac{(8 a-b) \text{csch}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{8 a^2 f}-\frac{\left(8 a^2+8 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{3/2} f}-\frac{\text{csch}^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{4 a f}",1,"((-8*a^2 - 8*a*b + b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]] - Sqrt[a]*(-8*a + (8*a + b)*Csch[e + f*x]^2 + 2*a*Csch[e + f*x]^4)*Sqrt[a + b*Sinh[e + f*x]^2])/(8*a^(3/2)*f)","A",1
463,1,214,292,2.0290137,"\int \sqrt{a+b \sinh ^2(e+f x)} \tanh ^4(e+f x) \, dx","Integrate[Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]^4,x]","\frac{-\frac{\tanh (e+f x) \text{sech}^2(e+f x) \left(4 \left(4 a^2-6 a b+b^2\right) \cosh (2 (e+f x))+8 a^2+b (4 a-5 b) \cosh (4 (e+f x))-12 a b+b^2\right)}{2 \sqrt{2}}+8 i a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i a (7 a-8 b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 f (a-b) \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{\tanh ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}+\frac{(7 a-8 b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f (a-b)}-\frac{(3 a-4 b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f (a-b)}+\frac{(3 a-4 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{(7 a-8 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((-2*I)*a*(7*a - 8*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (8*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] - ((8*a^2 - 12*a*b + b^2 + 4*(4*a^2 - 6*a*b + b^2)*Cosh[2*(e + f*x)] + (4*a - 5*b)*b*Cosh[4*(e + f*x)])*Sech[e + f*x]^2*Tanh[e + f*x])/(2*Sqrt[2]))/(6*(a - b)*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
464,1,150,168,0.5159348,"\int \sqrt{a+b \sinh ^2(e+f x)} \tanh ^2(e+f x) \, dx","Integrate[Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]^2,x]","\frac{\tanh (e+f x) (-2 a-b \cosh (2 (e+f x))+b)+i \sqrt{2} a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i \sqrt{2} a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{f \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","\frac{\tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{f}+\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{2 \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((-2*I)*Sqrt[2]*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + I*Sqrt[2]*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + (-2*a + b - b*Cosh[2*(e + f*x)])*Tanh[e + f*x])/(f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","C",1
465,1,69,60,0.0892627,"\int \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{i \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{f \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}",1,"((-I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a])/(f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","A",1
466,1,154,202,0.5838989,"\int \coth ^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Coth[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\coth (e+f x) (-2 a-b \cosh (2 (e+f x))+b)+i \sqrt{2} (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i \sqrt{2} a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{f \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","\frac{2 \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{f}-\frac{\coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{f}+\frac{(a+b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{a f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{2 \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((-2*a + b - b*Cosh[2*(e + f*x)])*Coth[e + f*x] - (2*I)*Sqrt[2]*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + I*Sqrt[2]*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","C",1
467,1,210,270,3.0993401,"\int \coth ^4(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx","Integrate[Coth[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{-\frac{\coth (e+f x) \text{csch}^2(e+f x) \left(4 \left(4 a^2-2 a b-b^2\right) \cosh (2 (e+f x))-8 a^2+b (4 a+b) \cosh (4 (e+f x))+4 a b+3 b^2\right)}{2 \sqrt{2}}+8 i a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i a (7 a+b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 a f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{(7 a+b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a f}-\frac{\coth ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}-\frac{(3 a+b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a f}+\frac{(3 a+5 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{(7 a+b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"(-1/2*((-8*a^2 + 4*a*b + 3*b^2 + 4*(4*a^2 - 2*a*b - b^2)*Cosh[2*(e + f*x)] + b*(4*a + b)*Cosh[4*(e + f*x)])*Coth[e + f*x]*Csch[e + f*x]^2)/Sqrt[2] - (2*I)*a*(7*a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (8*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(6*a*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
468,1,169,232,1.6435283,"\int \left(a+b \sinh ^2(e+f x)\right)^{3/2} \tanh ^5(e+f x) \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x]^5,x]","-\frac{-\left(8 a^2-40 a b+35 b^2\right) \left(\sqrt{a+b \sinh ^2(e+f x)} \left(4 a+b \sinh ^2(e+f x)-3 b\right)-3 (a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)\right)+6 (a-b) \text{sech}^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{5/2}-3 (8 a-9 b) \text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{5/2}}{24 f (a-b)^2}","\frac{\left(8 a^2-40 a b+35 b^2\right) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{24 f (a-b)^2}+\frac{\left(8 a^2-40 a b+35 b^2\right) \sqrt{a+b \sinh ^2(e+f x)}}{8 f (a-b)}-\frac{\left(8 a^2-40 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{8 f \sqrt{a-b}}-\frac{\text{sech}^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{5/2}}{4 f (a-b)}+\frac{(8 a-9 b) \text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{5/2}}{8 f (a-b)^2}",1,"-1/24*(-3*(8*a - 9*b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(5/2) + 6*(a - b)*Sech[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(5/2) - (8*a^2 - 40*a*b + 35*b^2)*(-3*(a - b)^(3/2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]] + Sqrt[a + b*Sinh[e + f*x]^2]*(4*a - 3*b + b*Sinh[e + f*x]^2)))/((a - b)^2*f)","A",1
469,1,122,156,0.5609907,"\int \left(a+b \sinh ^2(e+f x)\right)^{3/2} \tanh ^3(e+f x) \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x]^3,x]","\frac{(2 a-5 b) \left(\sqrt{a+b \sinh ^2(e+f x)} \left(4 a+b \sinh ^2(e+f x)-3 b\right)-3 (a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)\right)+3 \text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{5/2}}{6 f (a-b)}","\frac{(2 a-5 b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{6 f (a-b)}+\frac{(2 a-5 b) \sqrt{a+b \sinh ^2(e+f x)}}{2 f}-\frac{(2 a-5 b) \sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{2 f}+\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{5/2}}{2 f (a-b)}",1,"(3*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(5/2) + (2*a - 5*b)*(-3*(a - b)^(3/2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]] + Sqrt[a + b*Sinh[e + f*x]^2]*(4*a - 3*b + b*Sinh[e + f*x]^2)))/(6*(a - b)*f)","A",1
470,1,86,90,0.147544,"\int \left(a+b \sinh ^2(e+f x)\right)^{3/2} \tanh (e+f x) \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x],x]","\frac{\left(4 a+b \cosh ^2(e+f x)-4 b\right) \sqrt{a+b \cosh ^2(e+f x)-b}-3 (a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \cosh ^2(e+f x)-b}}{\sqrt{a-b}}\right)}{3 f}","\frac{(a-b) \sqrt{a+b \sinh ^2(e+f x)}}{f}+\frac{\left(a+b \sinh ^2(e+f x)\right)^{3/2}}{3 f}-\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"(-3*(a - b)^(3/2)*ArcTanh[Sqrt[a - b + b*Cosh[e + f*x]^2]/Sqrt[a - b]] + (4*a - 4*b + b*Cosh[e + f*x]^2)*Sqrt[a - b + b*Cosh[e + f*x]^2])/(3*f)","A",1
471,1,69,78,0.1369469,"\int \coth (e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Coth[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\sqrt{a+b \sinh ^2(e+f x)} \left(4 a+b \sinh ^2(e+f x)\right)-3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{3 f}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{a \sqrt{a+b \sinh ^2(e+f x)}}{f}+\frac{\left(a+b \sinh ^2(e+f x)\right)^{3/2}}{3 f}",1,"(-3*a^(3/2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]] + Sqrt[a + b*Sinh[e + f*x]^2]*(4*a + b*Sinh[e + f*x]^2))/(3*f)","A",1
472,1,90,140,0.4539502,"\int \coth ^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Coth[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\sqrt{a+b \sinh ^2(e+f x)} \left(-3 a \text{csch}^2(e+f x)+8 a+b \cosh (2 (e+f x))+5 b\right)-3 \sqrt{a} (2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{6 f}","\frac{(2 a+3 b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{6 a f}+\frac{(2 a+3 b) \sqrt{a+b \sinh ^2(e+f x)}}{2 f}-\frac{\sqrt{a} (2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{2 f}-\frac{\text{csch}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{5/2}}{2 a f}",1,"(-3*Sqrt[a]*(2*a + 3*b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]] + (8*a + 5*b + b*Cosh[2*(e + f*x)] - 3*a*Csch[e + f*x]^2)*Sqrt[a + b*Sinh[e + f*x]^2])/(6*f)","A",1
473,1,123,203,0.8084257,"\int \coth ^5(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Coth[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\sqrt{a} \sqrt{a+b \sinh ^2(e+f x)} \left(8 \left(4 a+b \sinh ^2(e+f x)+6 b\right)-3 (8 a+5 b) \text{csch}^2(e+f x)-6 a \text{csch}^4(e+f x)\right)-3 \left(8 a^2+24 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{24 \sqrt{a} f}","\frac{\left(8 a^2+3 b (8 a+b)\right) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{24 a^2 f}+\frac{\left(8 a^2+3 b (8 a+b)\right) \sqrt{a+b \sinh ^2(e+f x)}}{8 a f}-\frac{\left(8 a^2+3 b (8 a+b)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{8 \sqrt{a} f}-\frac{(8 a+b) \text{csch}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{5/2}}{8 a^2 f}-\frac{\text{csch}^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{5/2}}{4 a f}",1,"(-3*(8*a^2 + 24*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]] + Sqrt[a]*Sqrt[a + b*Sinh[e + f*x]^2]*(-3*(8*a + 5*b)*Csch[e + f*x]^2 - 6*a*Csch[e + f*x]^4 + 8*(4*a + 6*b + b*Sinh[e + f*x]^2)))/(24*Sqrt[a]*f)","A",1
474,1,224,305,2.8085307,"\int \left(a+b \sinh ^2(e+f x)\right)^{3/2} \tanh ^4(e+f x) \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x]^4,x]","\frac{-\frac{\tanh (e+f x) \text{sech}^2(e+f x) \left(\left(64 a^2-160 a b+17 b^2\right) \cosh (2 (e+f x))+32 a^2+2 b (6 a-17 b) \cosh (4 (e+f x))-108 a b-b^2 \cosh (6 (e+f x))+18 b^2\right)}{4 \sqrt{2}}+4 i a (5 a-8 b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-32 i a (a-2 b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{12 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{\tanh ^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{3 f}+\frac{(a-2 b) \sinh ^2(e+f x) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{f}+\frac{8 (a-2 b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}-\frac{(3 a-8 b) \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}+\frac{(3 a-8 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{8 (a-2 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((-32*I)*a*(a - 2*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (4*I)*a*(5*a - 8*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] - ((32*a^2 - 108*a*b + 18*b^2 + (64*a^2 - 160*a*b + 17*b^2)*Cosh[2*(e + f*x)] + 2*(6*a - 17*b)*b*Cosh[4*(e + f*x)] - b^2*Cosh[6*(e + f*x)])*Sech[e + f*x]^2*Tanh[e + f*x])/(4*Sqrt[2]))/(12*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
475,1,188,260,2.9075756,"\int \left(a+b \sinh ^2(e+f x)\right)^{3/2} \tanh ^2(e+f x) \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x]^2,x]","\frac{\sqrt{2} \tanh (e+f x) \left(-24 a^2-4 b (2 a-3 b) \cosh (2 (e+f x))+40 a b+b^2 \cosh (4 (e+f x))-13 b^2\right)+32 i a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-8 i a (7 a-8 b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{24 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{\tanh (e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{f}+\frac{(7 a-8 b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}+\frac{4 b \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}+\frac{(3 a-4 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{(7 a-8 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((-8*I)*a*(7*a - 8*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (32*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*(-24*a^2 + 40*a*b - 13*b^2 - 4*(2*a - 3*b)*b*Cosh[2*(e + f*x)] + b^2*Cosh[4*(e + f*x)])*Tanh[e + f*x])/(24*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
476,1,169,174,0.7623829,"\int \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{b \sinh (2 (e+f x)) (2 a+b \cosh (2 (e+f x))-b)+2 i \sqrt{2} a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-4 i \sqrt{2} a (2 a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 f \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","\frac{b \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}+\frac{i a (a-b) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} F\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{2 i (2 a-b) \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 f \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}",1,"((-4*I)*Sqrt[2]*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (2*I)*Sqrt[2]*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + b*(2*a - b + b*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(6*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","A",1
477,1,184,256,2.1541244,"\int \coth ^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Coth[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} \coth (e+f x) \left(-24 a^2-4 b (2 a+b) \cosh (2 (e+f x))+8 a b+b^2 \cosh (4 (e+f x))+3 b^2\right)+32 i a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-8 i a (7 a+b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{24 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{(7 a+b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}+\frac{4 b \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}-\frac{\coth (e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{f}+\frac{(3 a+5 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{(7 a+b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"(Sqrt[2]*(-24*a^2 + 8*a*b + 3*b^2 - 4*b*(2*a + b)*Cosh[2*(e + f*x)] + b^2*Cosh[4*(e + f*x)])*Coth[e + f*x] - (8*I)*a*(7*a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (32*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(24*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
478,1,229,306,4.7145176,"\int \coth ^4(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2} \, dx","Integrate[Coth[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{4 i \left(5 a^2-2 a b-3 b^2\right) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-\frac{\coth (e+f x) \text{csch}^2(e+f x) \left(\left(64 a^2+32 a b-79 b^2\right) \cosh (2 (e+f x))-32 a^2+2 b (6 a+11 b) \cosh (4 (e+f x))-44 a b-b^2 \cosh (6 (e+f x))+58 b^2\right)}{4 \sqrt{2}}-32 i a (a+b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{12 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{8 (a+b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}+\frac{(3 a+5 b) \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}-\frac{\coth ^3(e+f x) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}{3 f}-\frac{(a+b) \cosh ^2(e+f x) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{f}+\frac{(3 a+b) (a+3 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{8 (a+b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"(-1/4*((-32*a^2 - 44*a*b + 58*b^2 + (64*a^2 + 32*a*b - 79*b^2)*Cosh[2*(e + f*x)] + 2*b*(6*a + 11*b)*Cosh[4*(e + f*x)] - b^2*Cosh[6*(e + f*x)])*Coth[e + f*x]*Csch[e + f*x]^2)/Sqrt[2] - (32*I)*a*(a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (4*I)*(5*a^2 - 2*a*b - 3*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(12*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
479,1,116,142,0.4739837,"\int \frac{\tanh ^5(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Tanh[e + f*x]^5/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\left(-8 a^2+8 a b-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)+\sqrt{a-b} \text{sech}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \left(-2 (a-b) \text{sech}^2(e+f x)+8 a-5 b\right)}{8 f (a-b)^{5/2}}","-\frac{\left(8 a^2-8 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{8 f (a-b)^{5/2}}-\frac{\text{sech}^4(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{4 f (a-b)}+\frac{(8 a-5 b) \text{sech}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{8 f (a-b)^2}",1,"((-8*a^2 + 8*a*b - 3*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]] + Sqrt[a - b]*Sech[e + f*x]^2*(8*a - 5*b - 2*(a - b)*Sech[e + f*x]^2)*Sqrt[a + b*Sinh[e + f*x]^2])/(8*(a - b)^(5/2)*f)","A",1
480,1,85,89,0.1185867,"\int \frac{\tanh ^3(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Tanh[e + f*x]^3/Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{(a-b)^{3/2}}-\frac{\text{sech}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{a-b}}{2 f}","\frac{\text{sech}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{2 f (a-b)}-\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{2 f (a-b)^{3/2}}",1,"-1/2*(((2*a - b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/(a - b)^(3/2) - (Sech[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])/(a - b))/f","A",1
481,1,44,41,0.0412145,"\int \frac{\tanh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Tanh[e + f*x]/Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cosh ^2(e+f x)-b}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}",1,"-(ArcTanh[Sqrt[a - b + b*Cosh[e + f*x]^2]/Sqrt[a - b]]/(Sqrt[a - b]*f))","A",1
482,1,33,33,0.0380151,"\int \frac{\coth (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Coth[e + f*x]/Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}",1,"-(ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f))","A",1
483,1,72,77,0.2039764,"\int \frac{\coth ^3(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Coth[e + f*x]^3/Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{\text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{a}}{2 f}","-\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{3/2} f}-\frac{\text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{2 a f}",1,"-1/2*(((2*a - b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/a^(3/2) + (Csch[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])/a)/f","A",1
484,1,100,126,0.3716041,"\int \frac{\coth ^5(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Coth[e + f*x]^5/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\left(-8 a^2+8 a b-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)+\sqrt{a} \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \left(-2 a \text{csch}^2(e+f x)-8 a+3 b\right)}{8 a^{5/2} f}","-\frac{(8 a-3 b) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{8 a^2 f}-\frac{\left(8 a^2-8 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{5/2} f}-\frac{\text{csch}^4(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{4 a f}",1,"((-8*a^2 + 8*a*b - 3*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]] + Sqrt[a]*Csch[e + f*x]^2*(-8*a + 3*b - 2*a*Csch[e + f*x]^2)*Sqrt[a + b*Sinh[e + f*x]^2])/(8*a^(5/2)*f)","A",1
485,1,206,219,2.1339782,"\int \frac{\tanh ^4(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Tanh[e + f*x]^4/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{-\frac{\tanh (e+f x) \text{sech}^2(e+f x) \left(2 \left(4 a^2-3 a b+b^2\right) \cosh (2 (e+f x))+(2 a-b) (2 a+b \cosh (4 (e+f x))+b)\right)}{\sqrt{2}}+2 i a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-4 i a (2 a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 f (a-b)^2 \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{\tanh (e+f x) \text{sech}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f (a-b)}+\frac{(3 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{2 (2 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((-4*I)*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (2*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] - ((2*(4*a^2 - 3*a*b + b^2)*Cosh[2*(e + f*x)] + (2*a - b)*(2*a + b + b*Cosh[4*(e + f*x)]))*Sech[e + f*x]^2*Tanh[e + f*x])/Sqrt[2])/(6*(a - b)^2*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
486,1,109,156,0.4146355,"\int \frac{\tanh ^2(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Tanh[e + f*x]^2/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\sqrt{2} \tanh (e+f x) (-2 a-b \cosh (2 (e+f x))+b)-2 i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{2 f (a-b) \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((-2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + Sqrt[2]*(-2*a + b - b*Cosh[2*(e + f*x)])*Tanh[e + f*x])/(2*(a - b)*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
487,1,68,60,0.0705244,"\int \frac{1}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[1/Sqrt[a + b*Sinh[e + f*x]^2],x]","-\frac{i \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{i \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} F\left(i e+i f x\left|\frac{b}{a}\right.\right)}{f \sqrt{a+b \sinh ^2(e+f x)}}",1,"((-I)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","A",1
488,1,105,207,0.3994092,"\int \frac{\coth ^2(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Coth[e + f*x]^2/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{\sqrt{2} \coth (e+f x) (-2 a-b \cosh (2 (e+f x))+b)-2 i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{2 a f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{\tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{a f}-\frac{\coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{a f}+\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{a f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{a f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"(Sqrt[2]*(-2*a + b - b*Cosh[2*(e + f*x)])*Coth[e + f*x] - (2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a])/(2*a*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
489,1,208,285,3.5430286,"\int \frac{\coth ^4(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx","Integrate[Coth[e + f*x]^4/Sqrt[a + b*Sinh[e + f*x]^2],x]","\frac{-\frac{\coth (e+f x) \text{csch}^2(e+f x) \left(2 \left(4 a^2-5 a b+2 b^2\right) \cosh (2 (e+f x))-(2 a-b) (2 a-b \cosh (4 (e+f x))-3 b)\right)}{\sqrt{2}}+2 i a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-4 i a (2 a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 a^2 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{2 (2 a-b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^2 f}-\frac{2 (2 a-b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^2 f}+\frac{(3 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{2 (2 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{\coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a f}",1,"(-(((2*(4*a^2 - 5*a*b + 2*b^2)*Cosh[2*(e + f*x)] - (2*a - b)*(2*a - 3*b - b*Cosh[4*(e + f*x)]))*Coth[e + f*x]*Csch[e + f*x]^2)/Sqrt[2]) - (4*I)*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (2*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(6*a^2*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
490,1,113,187,0.4729254,"\int \frac{\tanh ^5(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tanh[e + f*x]^5/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\left(8 a^2+8 a b-b^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \sinh ^2(e+f x)+a}{a-b}\right)+\frac{1}{2} (a-b) \text{sech}^4(e+f x) ((8 a-3 b) \cosh (2 (e+f x))+4 a+b)}{8 f (a-b)^3 \sqrt{a+b \sinh ^2(e+f x)}}","\frac{8 a^2+8 a b-b^2}{8 f (a-b)^3 \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\left(8 a^2+8 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{8 f (a-b)^{7/2}}-\frac{\text{sech}^4(e+f x)}{4 f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}+\frac{(8 a-3 b) \text{sech}^2(e+f x)}{8 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}",1,"((8*a^2 + 8*a*b - b^2)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sinh[e + f*x]^2)/(a - b)] + ((a - b)*(4*a + b + (8*a - 3*b)*Cosh[2*(e + f*x)])*Sech[e + f*x]^4)/2)/(8*(a - b)^3*f*Sqrt[a + b*Sinh[e + f*x]^2])","C",1
491,1,79,122,0.1190787,"\int \frac{\tanh ^3(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tanh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{(2 a+b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \sinh ^2(e+f x)+a}{a-b}\right)+(a-b) \text{sech}^2(e+f x)}{2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}","\frac{2 a+b}{2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}-\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{2 f (a-b)^{5/2}}+\frac{\text{sech}^2(e+f x)}{2 f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}",1,"((2*a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sinh[e + f*x]^2)/(a - b)] + (a - b)*Sech[e + f*x]^2)/(2*(a - b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2])","C",1
492,1,58,69,0.0772226,"\int \frac{\tanh (e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tanh[e + f*x]/(a + b*Sinh[e + f*x]^2)^(3/2),x]","-\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \cosh ^2(e+f x)}{a-b}+1\right)}{f (b-a) \sqrt{a+b \cosh ^2(e+f x)-b}}","\frac{1}{f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}",1,"-(Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Cosh[e + f*x]^2)/(a - b)]/((-a + b)*f*Sqrt[a - b + b*Cosh[e + f*x]^2]))","C",1
493,1,46,57,0.0576448,"\int \frac{\coth (e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Coth[e + f*x]/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \sinh ^2(e+f x)}{a}+1\right)}{a f \sqrt{a+b \sinh ^2(e+f x)}}","\frac{1}{a f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}",1,"Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Sinh[e + f*x]^2)/a]/(a*f*Sqrt[a + b*Sinh[e + f*x]^2])","C",1
494,1,69,110,0.1068572,"\int \frac{\coth ^3(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Coth[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{(2 a-3 b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \sinh ^2(e+f x)}{a}+1\right)-a \text{csch}^2(e+f x)}{2 a^2 f \sqrt{a+b \sinh ^2(e+f x)}}","-\frac{(2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{5/2} f}+\frac{2 a-3 b}{2 a^2 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\text{csch}^2(e+f x)}{2 a f \sqrt{a+b \sinh ^2(e+f x)}}",1,"(-(a*Csch[e + f*x]^2) + (2*a - 3*b)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Sinh[e + f*x]^2)/a])/(2*a^2*f*Sqrt[a + b*Sinh[e + f*x]^2])","C",1
495,1,94,167,0.3306884,"\int \frac{\coth ^5(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Coth[e + f*x]^5/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{\left(8 a^2-24 a b+15 b^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \sinh ^2(e+f x)}{a}+1\right)+a \text{csch}^2(e+f x) \left(-2 a \text{csch}^2(e+f x)-8 a+5 b\right)}{8 a^3 f \sqrt{a+b \sinh ^2(e+f x)}}","-\frac{(8 a-5 b) \text{csch}^2(e+f x)}{8 a^2 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\left(8 a^2-24 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{7/2} f}+\frac{8 a^2-24 a b+15 b^2}{8 a^3 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\text{csch}^4(e+f x)}{4 a f \sqrt{a+b \sinh ^2(e+f x)}}",1,"(a*Csch[e + f*x]^2*(-8*a + 5*b - 2*a*Csch[e + f*x]^2) + (8*a^2 - 24*a*b + 15*b^2)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Sinh[e + f*x]^2)/a])/(8*a^3*f*Sqrt[a + b*Sinh[e + f*x]^2])","C",1
496,1,212,275,2.2540382,"\int \frac{\tanh ^4(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tanh[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{-\frac{\tanh (e+f x) \text{sech}^2(e+f x) \left(4 \left(4 a^2+3 a b+b^2\right) \cosh (2 (e+f x))+8 a^2+b (7 a+b) \cosh (4 (e+f x))+21 a b-5 b^2\right)}{2 \sqrt{2}}+8 i a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i a (7 a+b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 f (a-b)^3 \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{4 a \tanh (e+f x)}{3 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\tanh (e+f x) \text{sech}^2(e+f x)}{3 f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\sqrt{a} \sqrt{b} (7 a+b) \cosh (e+f x) E\left(\tan ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right)|1-\frac{a}{b}\right)}{3 f (a-b)^3 \sqrt{a+b \sinh ^2(e+f x)} \sqrt{\frac{a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}+\frac{(3 a+5 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 f (a-b)^3 \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((-2*I)*a*(7*a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (8*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] - ((8*a^2 + 21*a*b - 5*b^2 + 4*(4*a^2 + 3*a*b + b^2)*Cosh[2*(e + f*x)] + b*(7*a + b)*Cosh[4*(e + f*x)])*Sech[e + f*x]^2*Tanh[e + f*x])/(2*Sqrt[2]))/(6*(a - b)^3*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
497,1,158,217,1.3286088,"\int \frac{\tanh ^2(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tanh[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{-2 \tanh (e+f x) (a+b \cosh (2 (e+f x)))+i \sqrt{2} (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i \sqrt{2} a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{f (a-b)^2 \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","-\frac{\tanh (e+f x)}{f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}-\frac{2 \sqrt{a} \sqrt{b} \cosh (e+f x) E\left(\tan ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right)|1-\frac{a}{b}\right)}{f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)} \sqrt{\frac{a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}+\frac{(a+b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{a f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((-2*I)*Sqrt[2]*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + I*Sqrt[2]*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] - 2*(a + b*Cosh[2*(e + f*x)])*Tanh[e + f*x])/((a - b)^2*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","C",1
498,1,100,115,0.1619321,"\int \frac{1}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^(-3/2),x]","\frac{-\sqrt{2} b \sinh (2 (e+f x))-2 i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{2 a f (a-b) \sqrt{2 a+b \cosh (2 (e+f x))-b}}","-\frac{b \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}-\frac{i \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{a f (a-b) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}",1,"((-2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - Sqrt[2]*b*Sinh[2*(e + f*x)])/(2*a*(a - b)*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","A",1
499,1,153,237,0.8022448,"\int \frac{\coth ^2(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Coth[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{-2 \coth (e+f x) (a+b \cosh (2 (e+f x))-b)+i \sqrt{2} a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i \sqrt{2} a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{a^2 f \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}}","\frac{2 \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{a^2 f}-\frac{2 \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{a^2 f}+\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{a^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{2 \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{a^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{\coth (e+f x)}{a f \sqrt{a+b \sinh ^2(e+f x)}}",1,"(-2*(a - b + b*Cosh[2*(e + f*x)])*Coth[e + f*x] - (2*I)*Sqrt[2]*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + I*Sqrt[2]*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(a^2*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])","C",1
500,1,214,341,3.2895303,"\int \frac{\coth ^4(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Coth[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(3/2),x]","\frac{-\frac{\coth (e+f x) \text{csch}^2(e+f x) \left(4 \left(4 a^2-11 a b+8 b^2\right) \cosh (2 (e+f x))-8 a^2+b (7 a-8 b) \cosh (4 (e+f x))+37 a b-24 b^2\right)}{2 \sqrt{2}}+8 i a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i a (7 a-8 b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{6 a^3 f \sqrt{2 a+b \cosh (2 (e+f x))-b}}","\frac{(7 a-8 b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f}-\frac{(7 a-8 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f}+\frac{(3 a-4 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^3 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{(7 a-8 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^3 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{(3 a-4 b) \coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^2 b f}-\frac{(a-b) \coth (e+f x) \text{csch}^2(e+f x)}{a b f \sqrt{a+b \sinh ^2(e+f x)}}",1,"(-1/2*((-8*a^2 + 37*a*b - 24*b^2 + 4*(4*a^2 - 11*a*b + 8*b^2)*Cosh[2*(e + f*x)] + (7*a - 8*b)*b*Cosh[4*(e + f*x)])*Coth[e + f*x]*Csch[e + f*x]^2)/Sqrt[2] - (2*I)*a*(7*a - 8*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (8*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(6*a^3*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])","C",1
501,1,114,232,0.5981169,"\int \frac{\tanh ^5(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tanh[e + f*x]^5/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{2 \left(8 a^2+24 a b+3 b^2\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \sinh ^2(e+f x)+a}{a-b}\right)+3 (a-b) \text{sech}^4(e+f x) ((8 a-b) \cosh (2 (e+f x))+4 a+3 b)}{48 f (a-b)^3 \left(a+b \sinh ^2(e+f x)\right)^{3/2}}","\frac{8 a^2+24 a b+3 b^2}{8 f (a-b)^4 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{8 a^2+24 a b+3 b^2}{24 f (a-b)^3 \left(a+b \sinh ^2(e+f x)\right)^{3/2}}-\frac{\left(8 a^2+24 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{8 f (a-b)^{9/2}}-\frac{\text{sech}^4(e+f x)}{4 f (a-b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}+\frac{(8 a-b) \text{sech}^2(e+f x)}{8 f (a-b)^2 \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"(2*(8*a^2 + 24*a*b + 3*b^2)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Sinh[e + f*x]^2)/(a - b)] + 3*(a - b)*(4*a + 3*b + (8*a - b)*Cosh[2*(e + f*x)])*Sech[e + f*x]^4)/(48*(a - b)^3*f*(a + b*Sinh[e + f*x]^2)^(3/2))","C",1
502,1,82,163,0.1271273,"\int \frac{\tanh ^3(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tanh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{(2 a+3 b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \sinh ^2(e+f x)+a}{a-b}\right)+3 (a-b) \text{sech}^2(e+f x)}{6 f (a-b)^2 \left(a+b \sinh ^2(e+f x)\right)^{3/2}}","\frac{2 a+3 b}{2 f (a-b)^3 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{2 a+3 b}{6 f (a-b)^2 \left(a+b \sinh ^2(e+f x)\right)^{3/2}}-\frac{(2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{2 f (a-b)^{7/2}}+\frac{\text{sech}^2(e+f x)}{2 f (a-b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"((2*a + 3*b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Sinh[e + f*x]^2)/(a - b)] + 3*(a - b)*Sech[e + f*x]^2)/(6*(a - b)^2*f*(a + b*Sinh[e + f*x]^2)^(3/2))","C",1
503,1,60,99,0.0871009,"\int \frac{\tanh (e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tanh[e + f*x]/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \cosh ^2(e+f x)}{a-b}+1\right)}{3 f (a-b) \left(a+b \cosh ^2(e+f x)-b\right)^{3/2}}","\frac{1}{f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{1}{3 f (a-b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}",1,"Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Cosh[e + f*x]^2)/(a - b)]/(3*(a - b)*f*(a - b + b*Cosh[e + f*x]^2)^(3/2))","C",1
504,1,49,83,0.0651138,"\int \frac{\coth (e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Coth[e + f*x]/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \sinh ^2(e+f x)}{a}+1\right)}{3 a f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}+\frac{1}{a^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{1}{3 a f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Sinh[e + f*x]^2)/a]/(3*a*f*(a + b*Sinh[e + f*x]^2)^(3/2))","C",1
505,1,69,143,0.3061242,"\int \frac{\coth ^3(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Coth[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(5/2),x]","-\frac{(5 b-2 a) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \sinh ^2(e+f x)}{a}+1\right)+3 a \text{csch}^2(e+f x)}{6 a^2 f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}","-\frac{(2 a-5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{7/2} f}+\frac{2 a-5 b}{2 a^3 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{2 a-5 b}{6 a^2 f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}-\frac{\text{csch}^2(e+f x)}{2 a f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"-1/6*(3*a*Csch[e + f*x]^2 + (-2*a + 5*b)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Sinh[e + f*x]^2)/a])/(a^2*f*(a + b*Sinh[e + f*x]^2)^(3/2))","C",1
506,1,117,208,0.428159,"\int \frac{\coth ^5(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Coth[e + f*x]^5/(a + b*Sinh[e + f*x]^2)^(5/2),x]","-\frac{\text{csch}^2(e+f x) \left(\left(-8 a^2+40 a b-35 b^2\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \sinh ^2(e+f x)}{a}+1\right)+3 a \text{csch}^2(e+f x) \left(2 a \text{csch}^2(e+f x)+8 a-7 b\right)\right)}{24 a^3 f \sqrt{a+b \sinh ^2(e+f x)} \left(a \text{csch}^2(e+f x)+b\right)}","-\frac{(8 a-7 b) \text{csch}^2(e+f x)}{8 a^2 f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}-\frac{\left(8 a^2-40 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{9/2} f}+\frac{8 a^2-40 a b+35 b^2}{8 a^4 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{8 a^2-40 a b+35 b^2}{24 a^3 f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}-\frac{\text{csch}^4(e+f x)}{4 a f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"-1/24*(Csch[e + f*x]^2*(3*a*Csch[e + f*x]^2*(8*a - 7*b + 2*a*Csch[e + f*x]^2) + (-8*a^2 + 40*a*b - 35*b^2)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Sinh[e + f*x]^2)/a]))/(a^3*f*(b + a*Csch[e + f*x]^2)*Sqrt[a + b*Sinh[e + f*x]^2])","C",1
507,1,252,333,3.4608653,"\int \frac{\tanh ^4(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tanh[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(5/2),x]","-\frac{i \left(2 a b \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} \left(\left(-5 a^2+2 a b+3 b^2\right) F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+8 a (a+b) E\left(i (e+f x)\left|\frac{b}{a}\right.\right)\right)-i \sqrt{2} b \left(2 a b (a-b) \sinh (2 (e+f x))+4 b (a+b) \sinh (2 (e+f x)) (2 a+b \cosh (2 (e+f x))-b)+4 (a+b) \tanh (e+f x) (2 a+b \cosh (2 (e+f x))-b)^2-\left((a-b) \tanh (e+f x) \text{sech}^2(e+f x) (2 a+b \cosh (2 (e+f x))-b)^2\right)\right)\right)}{6 b f (a-b)^4 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","-\frac{2 (2 a+b) \tanh (e+f x)}{3 f (a-b)^2 \left(a+b \sinh ^2(e+f x)\right)^{3/2}}-\frac{b (5 a+3 b) \sinh (e+f x) \cosh (e+f x)}{3 f (a-b)^3 \left(a+b \sinh ^2(e+f x)\right)^{3/2}}+\frac{\tanh (e+f x) \text{sech}^2(e+f x)}{3 f (a-b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}-\frac{8 \sqrt{a} \sqrt{b} (a+b) \cosh (e+f x) E\left(\tan ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right)|1-\frac{a}{b}\right)}{3 f (a-b)^4 \sqrt{a+b \sinh ^2(e+f x)} \sqrt{\frac{a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}+\frac{(3 a+b) (a+3 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a f (a-b)^4 \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((-1/6*I)*(2*a*b*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*(8*a*(a + b)*EllipticE[I*(e + f*x), b/a] + (-5*a^2 + 2*a*b + 3*b^2)*EllipticF[I*(e + f*x), b/a]) - I*Sqrt[2]*b*(2*a*(a - b)*b*Sinh[2*(e + f*x)] + 4*b*(a + b)*(2*a - b + b*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)] + 4*(a + b)*(2*a - b + b*Cosh[2*(e + f*x)])^2*Tanh[e + f*x] - (a - b)*(2*a - b + b*Cosh[2*(e + f*x)])^2*Sech[e + f*x]^2*Tanh[e + f*x])))/((a - b)^4*b*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","C",1
508,1,215,274,2.7074708,"\int \frac{\tanh ^2(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tanh[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{8 i a^2 (a-b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i a^2 (7 a+b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)-\frac{\tanh (e+f x) \left(24 a^3-4 a^2 b+b^2 (7 a+b) \cosh (4 (e+f x))+5 a b^2+4 a b (11 a-3 b) \cosh (2 (e+f x))-b^3\right)}{\sqrt{2}}}{6 a f (a-b)^3 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","-\frac{\tanh (e+f x)}{f (a-b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}-\frac{4 b \sinh (e+f x) \cosh (e+f x)}{3 f (a-b)^2 \left(a+b \sinh ^2(e+f x)\right)^{3/2}}-\frac{\sqrt{b} (7 a+b) \cosh (e+f x) E\left(\tan ^{-1}\left(\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right)|1-\frac{a}{b}\right)}{3 \sqrt{a} f (a-b)^3 \sqrt{a+b \sinh ^2(e+f x)} \sqrt{\frac{a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}+\frac{(3 a+5 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a f (a-b)^3 \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}",1,"((-2*I)*a^2*(7*a + b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] + (8*I)*a^2*(a - b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] - ((24*a^3 - 4*a^2*b + 5*a*b^2 - b^3 + 4*a*(11*a - 3*b)*b*Cosh[2*(e + f*x)] + b^2*(7*a + b)*Cosh[4*(e + f*x)])*Tanh[e + f*x])/Sqrt[2])/(6*a*(a - b)^3*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","C",1
509,1,190,251,0.9620651,"\int \frac{1}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^(-5/2),x]","\frac{\sqrt{2} b \sinh (2 (e+f x)) \left(-5 a^2+b (b-2 a) \cosh (2 (e+f x))+5 a b-b^2\right)+i a^2 (a-b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i a^2 (2 a-b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)}{3 a^2 f (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","-\frac{2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}-\frac{2 i (2 a-b) \sqrt{a+b \sinh ^2(e+f x)} E\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 a^2 f (a-b)^2 \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}-\frac{b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left(a+b \sinh ^2(e+f x)\right)^{3/2}}+\frac{i \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} F\left(i e+i f x\left|\frac{b}{a}\right.\right)}{3 a f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}",1,"((-2*I)*a^2*(2*a - b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] + I*a^2*(a - b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(-5*a^2 + 5*a*b - b^2 + b*(-2*a + b)*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(3*a^2*(a - b)^2*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","A",1
510,1,226,351,2.703689,"\int \frac{\coth ^2(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Coth[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2),x]","\frac{8 i a^2 (a-b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} F\left(i (e+f x)\left|\frac{b}{a}\right.\right)-2 i a^2 (7 a-8 b) \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} E\left(i (e+f x)\left|\frac{b}{a}\right.\right)-\frac{\coth (e+f x) \left(24 a^3+4 b \left(11 a^2-19 a b+8 b^2\right) \cosh (2 (e+f x))-68 a^2 b+b^2 (7 a-8 b) \cosh (4 (e+f x))+69 a b^2-24 b^3\right)}{\sqrt{2}}}{6 a^3 f (a-b) (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","\frac{(7 a-8 b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)}-\frac{(7 a-8 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)}+\frac{(3 a-4 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^3 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{(7 a-8 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^3 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{(3 a-4 b) \coth (e+f x)}{3 a^2 f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\coth (e+f x)}{3 a f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"(-(((24*a^3 - 68*a^2*b + 69*a*b^2 - 24*b^3 + 4*b*(11*a^2 - 19*a*b + 8*b^2)*Cosh[2*(e + f*x)] + (7*a - 8*b)*b^2*Cosh[4*(e + f*x)])*Coth[e + f*x])/Sqrt[2]) - (2*I)*a^2*(7*a - 8*b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] + (8*I)*a^2*(a - b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a])/(6*a^3*(a - b)*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","C",1
511,1,247,385,2.8880373,"\int \frac{\coth ^4(e+f x)}{\left(a+b \sinh ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Coth[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(5/2),x]","-\frac{i \left(2 a^2 b \left(\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right)^{3/2} \left((8 b-5 a) F\left(i (e+f x)\left|\frac{b}{a}\right.\right)+8 (a-2 b) E\left(i (e+f x)\left|\frac{b}{a}\right.\right)\right)+\frac{i b \coth (e+f x) \text{csch}^2(e+f x) \left(8 a^3-b \left(13 a^2-36 a b+24 b^2\right) \cosh (4 (e+f x))-63 a^2 b-2 \left(8 a^3-38 a^2 b+63 a b^2-30 b^3\right) \cosh (2 (e+f x))-2 a b^2 \cosh (6 (e+f x))+92 a b^2+4 b^3 \cosh (6 (e+f x))-40 b^3\right)}{\sqrt{2}}\right)}{6 a^4 b f (2 a+b \cosh (2 (e+f x))-b)^{3/2}}","\frac{8 (a-2 b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^4 f}-\frac{8 (a-2 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^4 f}+\frac{(3 a-8 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^4 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}-\frac{8 (a-2 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left(\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right)}{3 a^4 f \sqrt{\frac{\text{sech}^2(e+f x) \left(a+b \sinh ^2(e+f x)\right)}{a}}}+\frac{(3 a-8 b) \coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 b f}-\frac{2 (a-3 b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a^2 b f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{(a-b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a b f \left(a+b \sinh ^2(e+f x)\right)^{3/2}}",1,"((-1/6*I)*((I*b*(8*a^3 - 63*a^2*b + 92*a*b^2 - 40*b^3 - 2*(8*a^3 - 38*a^2*b + 63*a*b^2 - 30*b^3)*Cosh[2*(e + f*x)] - b*(13*a^2 - 36*a*b + 24*b^2)*Cosh[4*(e + f*x)] - 2*a*b^2*Cosh[6*(e + f*x)] + 4*b^3*Cosh[6*(e + f*x)])*Coth[e + f*x]*Csch[e + f*x]^2)/Sqrt[2] + 2*a^2*b*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*(8*(a - 2*b)*EllipticE[I*(e + f*x), b/a] + (-5*a + 8*b)*EllipticF[I*(e + f*x), b/a])))/(a^4*b*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))","C",1
512,0,0,122,10.5918261,"\int \left(a+b \sinh ^2(e+f x)\right)^p (d \tanh (e+f x))^m \, dx","Integrate[(a + b*Sinh[e + f*x]^2)^p*(d*Tanh[e + f*x])^m,x]","\int \left(a+b \sinh ^2(e+f x)\right)^p (d \tanh (e+f x))^m \, dx","\frac{\cosh ^2(e+f x)^{\frac{m+1}{2}} (d \tanh (e+f x))^{m+1} \left(a+b \sinh ^2(e+f x)\right)^p \left(\frac{b \sinh ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2};\frac{m+1}{2},-p;\frac{m+3}{2};-\sinh ^2(e+f x),-\frac{b \sinh ^2(e+f x)}{a}\right)}{d f (m+1)}",1,"Integrate[(a + b*Sinh[e + f*x]^2)^p*(d*Tanh[e + f*x])^m, x]","F",-1
513,1,90,110,0.2630607,"\int \left(a+b \sinh ^2(c+d x)\right)^p \tanh ^3(c+d x) \, dx","Integrate[(a + b*Sinh[c + d*x]^2)^p*Tanh[c + d*x]^3,x]","\frac{\left(a+b \sinh ^2(c+d x)\right)^{p+1} \left((-a+b p+b) \, _2F_1\left(1,p+1;p+2;\frac{b \sinh ^2(c+d x)+a}{a-b}\right)+(p+1) (a-b) \text{sech}^2(c+d x)\right)}{2 d (p+1) (a-b)^2}","\frac{\text{sech}^2(c+d x) \left(a+b \sinh ^2(c+d x)\right)^{p+1}}{2 d (a-b)}-\frac{(a-b (p+1)) \left(a+b \sinh ^2(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sinh ^2(c+d x)+a}{a-b}\right)}{2 d (p+1) (a-b)^2}",1,"(((-a + b + b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sinh[c + d*x]^2)/(a - b)] + (a - b)*(1 + p)*Sech[c + d*x]^2)*(a + b*Sinh[c + d*x]^2)^(1 + p))/(2*(a - b)^2*d*(1 + p))","A",1
514,1,65,63,0.0720786,"\int \left(a+b \sinh ^2(c+d x)\right)^p \tanh (c+d x) \, dx","Integrate[(a + b*Sinh[c + d*x]^2)^p*Tanh[c + d*x],x]","-\frac{\left(a+b \cosh ^2(c+d x)-b\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \cosh ^2(c+d x)}{a-b}+1\right)}{2 d (p+1) (a-b)}","-\frac{\left(a+b \sinh ^2(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sinh ^2(c+d x)+a}{a-b}\right)}{2 d (p+1) (a-b)}",1,"-1/2*((a - b + b*Cosh[c + d*x]^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Cosh[c + d*x]^2)/(a - b)])/((a - b)*d*(1 + p))","A",1
515,1,54,54,0.0613526,"\int \coth (c+d x) \left(a+b \sinh ^2(c+d x)\right)^p \, dx","Integrate[Coth[c + d*x]*(a + b*Sinh[c + d*x]^2)^p,x]","-\frac{\left(a+b \sinh ^2(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sinh ^2(c+d x)}{a}+1\right)}{2 a d (p+1)}","-\frac{\left(a+b \sinh ^2(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sinh ^2(c+d x)}{a}+1\right)}{2 a d (p+1)}",1,"-1/2*(Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sinh[c + d*x]^2)/a]*(a + b*Sinh[c + d*x]^2)^(1 + p))/(a*d*(1 + p))","A",1
516,1,71,94,0.3849936,"\int \coth ^3(c+d x) \left(a+b \sinh ^2(c+d x)\right)^p \, dx","Integrate[Coth[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^p,x]","-\frac{\left(a+b \sinh ^2(c+d x)\right)^{p+1} \left(\frac{(a+b p) \, _2F_1\left(1,p+1;p+2;\frac{b \sinh ^2(c+d x)}{a}+1\right)}{p+1}+a \text{csch}^2(c+d x)\right)}{2 a^2 d}","-\frac{(a+b p) \left(a+b \sinh ^2(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sinh ^2(c+d x)}{a}+1\right)}{2 a^2 d (p+1)}-\frac{\text{csch}^2(c+d x) \left(a+b \sinh ^2(c+d x)\right)^{p+1}}{2 a d}",1,"-1/2*((a*Csch[c + d*x]^2 + ((a + b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sinh[c + d*x]^2)/a])/(1 + p))*(a + b*Sinh[c + d*x]^2)^(1 + p))/(a^2*d)","A",1
517,0,0,103,36.3007333,"\int \left(a+b \sinh ^2(c+d x)\right)^p \tanh ^4(c+d x) \, dx","Integrate[(a + b*Sinh[c + d*x]^2)^p*Tanh[c + d*x]^4,x]","\int \left(a+b \sinh ^2(c+d x)\right)^p \tanh ^4(c+d x) \, dx","\frac{\sinh ^4(c+d x) \sqrt{\cosh ^2(c+d x)} \tanh (c+d x) \left(a+b \sinh ^2(c+d x)\right)^p \left(\frac{b \sinh ^2(c+d x)}{a}+1\right)^{-p} F_1\left(\frac{5}{2};\frac{5}{2},-p;\frac{7}{2};-\sinh ^2(c+d x),-\frac{b \sinh ^2(c+d x)}{a}\right)}{5 d}",1,"Integrate[(a + b*Sinh[c + d*x]^2)^p*Tanh[c + d*x]^4, x]","F",-1
518,0,0,103,5.6494787,"\int \left(a+b \sinh ^2(c+d x)\right)^p \tanh ^2(c+d x) \, dx","Integrate[(a + b*Sinh[c + d*x]^2)^p*Tanh[c + d*x]^2,x]","\int \left(a+b \sinh ^2(c+d x)\right)^p \tanh ^2(c+d x) \, dx","\frac{\sinh ^2(c+d x) \sqrt{\cosh ^2(c+d x)} \tanh (c+d x) \left(a+b \sinh ^2(c+d x)\right)^p \left(\frac{b \sinh ^2(c+d x)}{a}+1\right)^{-p} F_1\left(\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};-\sinh ^2(c+d x),-\frac{b \sinh ^2(c+d x)}{a}\right)}{3 d}",1,"Integrate[(a + b*Sinh[c + d*x]^2)^p*Tanh[c + d*x]^2, x]","F",-1
519,0,0,99,6.7074419,"\int \coth ^2(c+d x) \left(a+b \sinh ^2(c+d x)\right)^p \, dx","Integrate[Coth[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^p,x]","\int \coth ^2(c+d x) \left(a+b \sinh ^2(c+d x)\right)^p \, dx","-\frac{\sqrt{\cosh ^2(c+d x)} \text{csch}(c+d x) \text{sech}(c+d x) \left(a+b \sinh ^2(c+d x)\right)^p \left(\frac{b \sinh ^2(c+d x)}{a}+1\right)^{-p} F_1\left(-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2};-\sinh ^2(c+d x),-\frac{b \sinh ^2(c+d x)}{a}\right)}{d}",1,"Integrate[Coth[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^p, x]","F",-1
520,0,0,103,38.3993535,"\int \coth ^4(c+d x) \left(a+b \sinh ^2(c+d x)\right)^p \, dx","Integrate[Coth[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^p,x]","\int \coth ^4(c+d x) \left(a+b \sinh ^2(c+d x)\right)^p \, dx","-\frac{\sqrt{\cosh ^2(c+d x)} \text{csch}^3(c+d x) \text{sech}(c+d x) \left(a+b \sinh ^2(c+d x)\right)^p \left(\frac{b \sinh ^2(c+d x)}{a}+1\right)^{-p} F_1\left(-\frac{3}{2};-\frac{3}{2},-p;-\frac{1}{2};-\sinh ^2(c+d x),-\frac{b \sinh ^2(c+d x)}{a}\right)}{3 d}",1,"Integrate[Coth[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^p, x]","F",-1
521,1,136,152,0.3547041,"\int \frac{\coth ^3(x)}{a+b \sinh ^3(x)} \, dx","Integrate[Coth[x]^3/(a + b*Sinh[x]^3),x]","-\frac{\left(a^{2/3}+(-1)^{2/3} b^{2/3}\right) \log \left(-(-1)^{2/3} \sqrt[3]{a}-\sqrt[3]{b} \sinh (x)\right)+\left(a^{2/3}+b^{2/3}\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sinh (x)\right)+\left(a^{2/3}-\sqrt[3]{-1} b^{2/3}\right) \log \left(\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} \sinh (x)\right)}{3 a^{5/3}}-\frac{\text{csch}^2(x)}{2 a}+\frac{\log (\sinh (x))}{a}","\frac{b^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sinh (x)+b^{2/3} \sinh ^2(x)\right)}{6 a^{5/3}}-\frac{b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sinh (x)\right)}{3 a^{5/3}}+\frac{b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sinh (x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{5/3}}-\frac{\log \left(a+b \sinh ^3(x)\right)}{3 a}-\frac{\text{csch}^2(x)}{2 a}+\frac{\log (\sinh (x))}{a}",1,"-1/2*Csch[x]^2/a + Log[Sinh[x]]/a - ((a^(2/3) + (-1)^(2/3)*b^(2/3))*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sinh[x]] + (a^(2/3) + b^(2/3))*Log[a^(1/3) + b^(1/3)*Sinh[x]] + (a^(2/3) - (-1)^(1/3)*b^(2/3))*Log[a^(1/3) + (-1)^(2/3)*b^(1/3)*Sinh[x]])/(3*a^(5/3))","A",1
522,1,28,28,0.0251001,"\int \frac{\coth (x)}{\sqrt{a+b \sinh ^3(x)}} \, dx","Integrate[Coth[x]/Sqrt[a + b*Sinh[x]^3],x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^3(x)}}{\sqrt{a}}\right)}{3 \sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^3(x)}}{\sqrt{a}}\right)}{3 \sqrt{a}}",1,"(-2*ArcTanh[Sqrt[a + b*Sinh[x]^3]/Sqrt[a]])/(3*Sqrt[a])","A",1
523,1,45,45,0.0234123,"\int \coth (x) \sqrt{a+b \sinh ^3(x)} \, dx","Integrate[Coth[x]*Sqrt[a + b*Sinh[x]^3],x]","\frac{2}{3} \sqrt{a+b \sinh ^3(x)}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^3(x)}}{\sqrt{a}}\right)","\frac{2}{3} \sqrt{a+b \sinh ^3(x)}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^3(x)}}{\sqrt{a}}\right)",1,"(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[x]^3]/Sqrt[a]])/3 + (2*Sqrt[a + b*Sinh[x]^3])/3","A",1
524,1,29,29,0.0273477,"\int \frac{\coth (x)}{\sqrt{a+b \sinh ^n(x)}} \, dx","Integrate[Coth[x]/Sqrt[a + b*Sinh[x]^n],x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^n(x)}}{\sqrt{a}}\right)}{\sqrt{a} n}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^n(x)}}{\sqrt{a}}\right)}{\sqrt{a} n}",1,"(-2*ArcTanh[Sqrt[a + b*Sinh[x]^n]/Sqrt[a]])/(Sqrt[a]*n)","A",1
525,1,45,47,0.0232396,"\int \coth (x) \sqrt{a+b \sinh ^n(x)} \, dx","Integrate[Coth[x]*Sqrt[a + b*Sinh[x]^n],x]","\frac{2 \sqrt{a+b \sinh ^n(x)}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^n(x)}}{\sqrt{a}}\right)}{n}","\frac{2 \sqrt{a+b \sinh ^n(x)}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sinh ^n(x)}}{\sqrt{a}}\right)}{n}",1,"(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[x]^n]/Sqrt[a]] + 2*Sqrt[a + b*Sinh[x]^n])/n","A",1